Lie Groups, Number Theory, and Vertex Algebras 9781470453510, 1470453517

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768

Lie Groups, Number Theory, and Vertex Algebras Conference on Representation Theory XVI June 24–29, 2019 Inter-University Center Dubrovnik, Croatia

´ Dražen Adamovic Andrej Dujella Antun Milas ´ Pavle Pandžic Editors

Lie Groups, Number Theory, and Vertex Algebras Conference on Representation Theory XVI June 24–29, 2019 Inter-University Center Dubrovnik, Croatia

´ Dražen Adamovic Andrej Dujella Antun Milas ´ Pavle Pandžic Editors

768

Lie Groups, Number Theory, and Vertex Algebras Conference on Representation Theory XVI June 24–29, 2019 Inter-University Center Dubrovnik, Croatia

´ Dražen Adamovic Andrej Dujella Antun Milas ´ Pavle Pandžic Editors

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss

Kailash Misra

Catherine Yan

2020 Mathematics Subject Classification. Primary 17B69, 17B20, 17B67, 22E45, 22E46, 11D09, 11Y35, 11Y50.

For additional information and updates on this book, visit www.ams.org/bookpages/conm-768 Library of Congress Cataloging-in-Publication Data Names: Conference on Representation Theory and Algebraic Geometry in honor of Joseph Bernstein (2019 : Dubrovnik, Croatia), issuing body. | Adamovi´ c, Draˇ zen, 1967- editor. Title: Lie groups, number theory, and vertex algebras : conference on Representation Theory XVI, June 24-29 2019, IUC, Dubrovnik, Croatia / Draˇzen Adamovi´ c, Andrej Dujella, Antun Milas, Pavle Pandˇ zi´ c, editors. Description: Providence, Rhode Island : American Mathematical Society, [2021] | Series: Contemporary mathematics, 0271-4132 ; volume 768 | Includes bibliographical references. Identifiers: LCCN 2020043189 | ISBN 9781470453510 (paperback) | 9781470464240 (ebook) Subjects: LCSH: Representations of groups–Congresses. | Representations of algebras–Congresses. | AMS: Nonassociative rings and algebras – Lie algebras and Lie superalgebras {For Lie groups, see 22Exx} – Vertex operators; vertex operator algebras and related structures. | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Simple, semisimple, reductive (super)algebras. | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. | Topological groups, Lie groups {For transformation groups, see 54H15, 57Sxx, 58-XX. For abstract harmonic analysis, see 43-XX} – Lie groups {For the topology of Lie groups and homogeneous spaces, see | Topological groups, Lie groups {For transformation groups, see 54H15, 57Sxx, 58-XX. For abstract harmonic analysis, see 43-XX} – Lie groups {For the topology of Lie groups and homogeneous spaces, see | Number theory – Diophantine equations [See also 11Gxx, 14Gxx] – Quadratic and bilinear equations. | Number theory – Computational number theory [See also 11-04] – Analytic computations. | Number theory – Computational number theory [See also 11-04] – Computer solution of Diophantine equations. Classification: LCC QA176 .C665 2019 | DDC 512/.22–dc23 LC record available at https://lccn.loc.gov/2020043189 Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2021 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

26 25 24 23 22 21

Contents

Preface

vii

LIE GROUPS

1

The Racah Algebra: An Overview and Recent Results Hendrik De Bie, Plamen Iliev, Wouter van de Vijver, and Luc Vinet

3

Orbit Embedding for Double Flag Varieties and Steinberg Maps Lucas Fresse and Kyo Nishiyama

21

Symplectic Dirac Cohomology and Lifting of Characters to Metaplectic Groups Jing-Song Huang 43 Spectrum of Semisimple Locally Symmetric Spaces and Admissibility of Spherical Representations Salah Mehdi and Martin Olbrich

55

Four Examples of Beilinson-Bernstein Localization Anna Romanov

65

NUMBER THEORY

87

Perfect Powers in Polynomial Power Sums Clemens Fuchs and Sebastian Heintze

89

The Number of Irregular Diophantine Quadruples for a Fixed Diophantine Pair or Triple Yasutsugu Fujita

105

On Ideals Defining Irreducible Representations of Reductive p-adic Groups ´ Goran Muic

119

On Diophantine Properties of Radix Representations in Algebraic Number Fields ˝ Attila Petho

133

VERTEX ALGEBRAS

149

On Certain W -algebras of Type Wk (sl4 , f ) ´, Antun Milas, and Michael Penn Draˇ zen Adamovic

151

On the N = 1 Super Heisenberg-Virasoro Vertex Algebra ´, Berislav Jandric ´, and Gordan Radobolja Draˇ zen Adamovic

167

v

vi

CONTENTS

Character Rings and Fusion Algebras Peter Bantay

179

Continuing Remarks on the Unrolled Quantum Group of sl(2) James F. Clark

187

On the Geometric Interpretation of Certain Vertex Algebras and Their Modules Jesse Corradino

201

Representation Theory of Vertex Operator Algebras and Orbifold Conformal Field Theory Yi-Zhi Huang 221 Further q-series Identities and Conjectures Relating False Theta Functions and Characters Chris Jennings-Shaffer and Antun Milas 253 (1)

Ultra-Discretization of D6 -Geometric Crystal at the Spin Node Kailash C. Misra and Suchada Pongprasert

271

Twisted Exterior Derivative for Universal Enveloping Algebras I ˇ Zoran Skoda

305

The Heisenberg Generalized Vertex Operator Algebra on a Riemann Surface Michael P. Tuite

321

Preface This volume contains the proceedings of the international conference “Representation Theory XVI”, held at the Inter-University Center, Dubrovnik, Croatia, June 24-29, 2019. The volume features nineteen original research articles in representation theory divided into three thematic groups: Lie groups, number theory, and vertex operator algebras. Representations of Lie groups, especially unitary representations, have been and continue to be an important topic in representation theory. Although a lot is known about them, the main problem of classifying unitary representations is still not completely solved, although recently Adams, van Leuven, Trapa and Vogan constructed an algorithm that can be effectively used in low-rank cases. The importance of this subject comes from applications in harmonic analysis, geometry, physics and number theory. The conference featured a number of talks by renowned experts on this subject including Jeffrey Adams, Dan Barbasch, Jing-Song Huang, Dragan Miliˇci´c, Wilfried Schmid and David Vogan. Number theory is one of the oldest topics in mathematics. It can be divided into elementary, algebraic, analytic and computational number theory. It has important connections and applications in algebraic geometry, representation theory and cryptography. The talks in the number theory section reflect the main subjects of interest of the Croatian number theory group, such as Diophantine equations, elliptic curves and modular forms, and contributions of international experts, including our long-time collaborators Yann Bugeaud, Clemens Fuchs, Attila Peth˝o and Robert Tichy. Vertex algebras provide a natural framework for studying representation theory of infinite-dimensional Lie algebras and W-algebras with applications in number theory, combinatorics and theoretical physics. Recently a great impulse for this theory was made through deep connections with the 4-dimensional SCFT in physics. In the last decade, Dubrovnik conferences on Representation Theory have attracted many leading experts on vertex algebras including Tomoyuki Arakawa, Chongying Dong, Yi-Zhi Huang, Ching Hung Lam, Haisheng Li, Victor Kac, James Lepowsky, Masahiko Miyamoto, and others. The Dubrovnik series of conferences is a central place for exchange of ideas between the Croatian vertex algebra group, which now includes many young researchers and students, and the experts in the field. This year’s invited speakers include Tomoyuki Arakawa, Ching Hung Lam, Pierluigi Moseneder Frajria, Yi-Zhi Huang, Cuibo Jiang, Victor Kac, Ivan Mirkovi´c, Masahiko Miyamoto, Anne Moreau, Nils Scheithauer, and Paolo Papi. We thank everyone who participated in the conference, those who helped plan and run the conference, and especially to the contributors and the referees. The vii

viii

PREFACE

conference was made possible through the generous support of the QuantiXLie Center of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). It was also supported by the Ministry of Science and Education of the Republic of Croatia, by the Foundation of the Croatian Academy of Sciences and Arts, and by the InterUniversity Centre in Dubrovnik. Without the hard work of the editorial staff of the American Mathematical Society, this volume would not have been possible. Our thanks go to Christine M. Thivierge for her constant help. We greatly appreciate the staff of the Inter-University Center, Dubrovnik for their efficient and dedicated work on logistics and help during the conference. We are very grateful to Hrvoje Kraljevi´c, Mirko Primc and David Vogan who served with us as directors of the conference. Hrvoje Kraljevi´c also coordinated the technical part of the organization. Draˇzen Adamovi´c, Andrej Dujella Antun Milas, Pavle Pandˇzi´c

Group photo of the participants

ix

Lie groups

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15450

The Racah algebra: An overview and recent results Hendrik De Bie, Plamen Iliev, Wouter van de Vijver, and Luc Vinet Abstract. Recent results on the Racah algebra Rn of rank n−2 are reviewed. Rn is defined in terms of generators and relations and sits in the centralizer of the diagonal action of su(1, 1) in U(su(1, 1))⊗n . Its connections with multivariate Racah polynomials are discussed. It is shown to be the symmetry algebra of the generic superintegrable model on the (n − 1) - sphere and a number of interesting realizations are provided.

1. Introduction It is understood since the seminal work of Zhedanov [50] that the bispectral properties of the families of orthogonal polynomials of the Askey scheme can be encoded into algebras that are generally quadratic. Basically, the generators of these algebras, which bear the names of the different families, are realized by differential or difference operators. These operators have the eponym polynomials as eigenfunctions by acting on the variable or the degree of these polynomials. The operators acting on the variable can be differential or difference depending on the family of polynomials. The difference operators acting on the degree coincide with recurrence operators. When multiplied these operators are taken to be realized in the same representation; that is either acting on the variable or the degree. This is how the Racah algebra (of rank 1) was originally defined. The Racah polynomials are also known to enter in the overlaps between bases associated to the recoupling of three su(1, 1) (or su(2)) irreducible representations; these coefficients are referred to as 6j-symbols. Since these bases are defined by diagonalizing intermediate Casimir elements in U(su(1, 1))⊗3 , this naturally led to the observation that R3 also arises in the centralizer of the diagonal action of su(1, 1) in U(su(1, 1))⊗3 with the generators here represented by these intermediate Casimir operators [25]. This result naturally paves the way for the construction of higher rank generalizations of R3 by considering instead of 3 an arbitrary number of su(1, 1) factors. These are the algebras on which we will focus in this review. The Racah algebra R3 was further seen to occur in a number of interesting situations. For instance, by taking the su(1, 1) copies as realizations of the dynamical algebra of the singular oscillator, one observes that the total Casimir operator is The work of the first and third authors was supported by the Research Foundation Flanders (FWO) under Grant EOS 308894451. The second author was partially supported by the Simons Foundation Grant #635462. The research of the fourth author was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. c 2021 American Mathematical Society

3

4

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

essentially the Hamiltonian of the generic superintegrable system on the 2-sphere [33]. The picture of R3 as a commutant that we described in the preceding paragraph readily leads to the conclusion [20] that the Racah algebra is the symmetry algebra of this superintegrable Hamiltonian. R3 was also found to possess an embedding into U(su(1, 1)) [18]. When using models for su(1, 1) in terms of differential operators, this last embedding gives a one-variable realization of R3 while the embedding in U(su(1, 1))⊗3 provides a three-variable one. The connection between these two models was obtained in [19] through separation of variables. Roughly five years ago, one of us co-authored a paper [21] to introduce the Racah algebra and review many of the facets we mentioned. Since then the subject has much evolved especially regarding the higher rank generalizations. The present paper offers a timely overview of these advances. Interestingly the review [21] contained a picture very much like the one below that we feel appropriate to reproduce here. Generic S.I. model on Sn−1

The Racah algebra

Multivariate Racah polynomials

The Racah problem for su(1, 1)

It depicts the Racah algebra as a central entity connected to orthogonal polynomials, recoupling problems and superintegrable systems. It further indicates that the interrelations between these topics can thus be understood on the basis of the common underlying algebraic structure. This diagram also offers a nice snapshot of the outline of the present article which will unfold as follows. The Racah algebra Rn of rank (n − 2) is defined in Section 2 in the framework of a generalized Racah problem, as the subalgebra of U(su(1, 1))⊗n generated by the intermediate Casimir elements in this tensor product. All commutation relations will be given. Section 3 describes the connections between Rn and multivariate Racah polynomials. These will appear as overlaps between two representation bases diagonalized by different labelling Abelian subalgebras. The case when the polynomials defined by Tratnik occur [45] will be pointed out. With an eye to applications, Section 4 will present four different realizations of Rn . First, upon realizing each su(1, 1) as the conformal algebra in one dimension, it will be explained how Rn arises as the symmetry algebra of the generic superintegrable model on the (n − 1) - sphere. Second a realization of Rn in terms of Dunkl operators [15] will be provided. Third, contrasting the realizations of Rn in terms of n variables in the superintegrable model context or in terms of Dunkl operators, it will be shown how differential operators in (n − 2) variables that satisfy the commutation relations of Rn can be constructed by calling upon

THE RACAH ALGEBRA: AN OVERVIEW AND RECENT RESULTS

5

the Barut-Girardello realization of su(1, 1). Fourth, the loop will be closed if one bears in mind how the Racah algebra was initially identified. Indeed Section 4 will end by extracting directly from properties of the multivariate polynomials a realization of Rn in terms of the difference operators of which the multivariate Racah polynomials are eigenfunctions [23]. Section 5 will offer concluding remarks. 2. The higher rank Racah algebra Let su(1, 1) be the Lie algebra generated by the operators J+ , J− and J0 obeying the following relations [J0 , J± ] = ±J± ,

[J− , J+ ] = 2J0 .

Its Casimir is given by C := J02 − J0 − J+ J− . This operator sits inside the universal enveloping algebra U(su(1, 1)). We consider now the n-fold tensor product U(su(1, 1))⊗n . In this algebra we define the following operators with  = ± or 0: J,k = 1 ⊗ · · · ⊗ 1 ⊗J ⊗ 1 ⊗ · · · ⊗ 1       k−1 times

n−k times

Let K be a subset of [n] := {1, . . . , n}. We define:  J,K = J,k . k∈K

The following lemma is easy to check: Lemma 2.1. Let K ⊂ [n]. The operators J0,K , J+,K and J−,K generate an algebra isomorphic to su(1, 1). We denote this algebra by suK (1, 1). The algebra suK (1, 1) lives in the components of U(su(1, 1))⊗n whose index is in K. Consider its Casimir: 2 CK := J0,K − J0,K − J+,K J−,K .

The Casimirs of all possible suK (1, 1) generate the higher rank Racah algebra. Definition 2.2. The higher rank Racah algebra Rn of rank n − 2 is the subalgebra of U(su(1, 1))⊗n generated by the following set of operators: {CK | K ⊂ [n] and K = ∅}. We exclude the empty set as C∅ = 0. Remark 2.3. Alternatively, one can construct the generators CA from the comultiplication μ∗ of su(1, 1). This is an algebra morphism that embeds su(1, 1) into the tensor product su(1, 1) ⊗ su(1, 1). It is defined as follows on the generators: (2.1)

μ∗ (J0 ) = J0 ⊗ 1 + 1 ⊗ J0 ,

μ∗ (J± ) = J± ⊗ 1 + 1 ⊗ J± .

This map extends to the universal enveloping algebra U(su(1, 1)). This allows us to apply the comultiplication repeatedly on the Casimir C. C1 := C,

Cn := (1 ⊗ . . . ⊗ 1 ⊗μ∗ )(Cn−1 ).   n−2 times

6

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

Each Casimir Ck lives in U(su(1, 1))⊗k . U(su(1, 1))⊗n . Consider the map τk :

m−1 

U(su(1, 1)) →

i=1

We want to lift this Casimir to m 

U(su(1, 1)),

i=1

which acts as follows on homogeneous tensor products: τk (t1 ⊗ . . . ⊗ tm−1 ) := t1 ⊗ . . . ⊗ tk−1 ⊗ 1 ⊗ tk ⊗ . . . ⊗ tm−1 . and extend it by linearity. The map τk adds a 1 at the k-th place. This allows to define the following: ⎛ ⎞ −→



(2.2) τk ⎠ C|A| . CA := ⎝ k∈[n]\A

with A a subset of [n]. This leads to the same generators for the Racah algebra Rn . This alternative pathway has been successfully applied on other (super) Lie algebras and their q-deformations. See for example [7, 10]. For a more general approach see [39]. Lemma 2.4. (See also [39]) The higher rank Racah algebra sits in the centralizer of su[n] (1, 1). Proof. We have [CA , J,[n] ] = [CA , J,A ] + [CA , J,[n]\A ] = 0. The first commutator equals 0 as CA is the Casimir of suA (1, 1) and the second commutator is 0 as both operators act on different components of U(su(1, 1))⊗n .  Example 2.5. We call the simplest non-trivial example R3 the rank one case. It is generated by the set {C1 , C2 , C3 , C12 , C23 , C13 , C123 }. For ease of notation we abbreviate sets of the form {1, 2} by 12 when in the index of a generator. The elements C1 , C2 , C3 and C123 are central elements of Rn . A tedious calculation shows that the generators are not linearly independent (2.3)

C123 = C12 + C13 + C23 − C1 − C2 − C3 .

As C123 is central and by formula (2.3) we have 0 = [C123 , C12 ] = [C13 , C12 ] + [C23 , C12 ], 0 = [C123 , C13 ] = [C12 , C13 ] + [C23 , C13 ], 0 = [C123 , C23 ] = [C12 , C23 ] + [C13 , C23 ]. We conclude that [C12 , C23 ] = [C23 , C13 ] = [C13 , C12 ]. We introduce the following operator 1 1 1 F := [C12 , C23 ] = [C23 , C13 ] = [C13 , C12 ]. 2 2 2 Another tedious computation shows the following relations to be true: [C12 , F ] = C23 C12 − C12 C13 + (C2 − C1 )(C3 − C123 ), (2.4)

[C23 , F ] = C13 C23 − C23 C12 + (C3 − C2 )(C1 − C123 ), [C13 , F ] = C12 C13 − C13 C23 + (C1 − C3 )(C2 − C123 ).

THE RACAH ALGEBRA: AN OVERVIEW AND RECENT RESULTS

7

2.1. Relations for Rn . We wish to find relations for the higher rank Racah algebra Rn for general n. To do so we mention the following lemma: Lemma 2.6. Let {Kp }p=1..k be a set of k disjoint subsets of [n]. Define KB := ∪q∈B Kq with B ⊂ [k]. The following map is an injective morphism: θ : Rk → Rn : CB → CK B . 1 ,...,Kk The image of θ is denoted by RK . k

Proof. See [9, section 4.2] or follow the strategy given in [6, Lemma 2.4] .



Example 2.7. The sets K1 = {2}, K2 = {1, 4} and K3 = {3} are disjoint subsets of the set {1, 2, 3, 4}. By Lemma 2.6 we have an injective morphism of R3 into R4 . Explicitly it is given as follows: θ(C1 ) = CK1 = C2 ,

θ(C12 ) = CK1 K2 = C124 ,

θ(C2 ) = CK2 = C14 ,

θ(C13 ) = CK1 K3 = C23 ,

θ(C3 ) = CK3 = C3 ,

θ(C23 ) = CK2 K3 = C134 ,

θ(C123 ) = CK1 K2 K3 = C1234 . By Lemma 2.6 we can lift the relations of R3 given in Example 2.5 to Rn . Let K, L and M be three disjoint subsets of [n] and consider equality (2.3). By Lemma 2.6 we replace 1 by K, 2 by L and 3 by M : CKLM = CKL + CKM + CLM − CK − CL − CM . As before the notation KL is short for K ∪ L in the index of a generator. We have found a set of linear dependencies between the generators of Rn . By induction one can prove the following: Lemma 2.8. For any set K ⊂ [n], it holds that   Cij − (|K| − 2) Ci . CK = {i,j}⊂K

i∈K

In R3 we also know that C1 is central. In particular we have [C1 , C12 ] = 0 and [C1 , C2 ] = 0. By Lemma 2.6 we find [CK , CKL ] = 0 and [CK , CL ] = 0. The following lemma follows: Lemma 2.9. Let A and B be subsets of [n]. If either A ⊂ B, B ⊂ A or A ∩ B = ∅ then [CA , CB ] = 0. A consequence of this lemma is that the generators Ci , i ∈ [n] and C[n] are central in Rn . This lemma shows the existence of many Abelian subalgebras. Definition 2.10. Consider the following chain A of subsets of [n]: A1 ⊂ A2 ⊂ · · · ⊂ An−2 with |Ak | = k + 1. We define the labeling Abelian algebra YA to be the algebra generated by {CAi | i = 1 . . . n − 2}. We exclude the sets with one element and all the elements because the related generators Ci and C[n] are central. Observe that the number of generators in a labeling Abelian algebra YA equals the rank of the higher rank Racah algebra. In what follows we will give the commutator of any pair of generators. It suffices to do this only for the generators with two indices Cij and one index Ci by Lemma

8

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

2.8. The generators with one index are central so we focus on the generators with two indices. We introduce the following operator: 1 [Cij , Cjk ]. 2 The order of the indices of F is important. Switching any two indices leads to change in sign: Fijk =

Fjik =

1 1 1 1 [Cji , Cik ] = [Ckj , Cji ] = − [Cji , Ckj ] = − [Cij , Cjk ] = −Fijk . 2 2 2 2

We apply Lemma 2.6 on relation (2.4). We are focusing on generators with two indices so we set K = {i}, L = {j} and M = {k}: [Cjk , Fijk ] = Cik Cjk − Cjk Cij + (Ck − Cj )(Ci − Cijk ). The remaining possible commutators are found through straightforward but tedious computations. For notational purposes we introduce Pij = Cij − Ci − Cj . We find: [Pkl , Fijk ] = Pik Pjl − Pil Pjk , [Fijk , Fjkl ] = Fjkl Pij − Fikl (Pjk + 2Cj ) − Fijk Pjl , [Fijk , Fklm ] = Film Pjk − Pik Fjlm . 3. The Racah problem for su(1, 1) and multivariate Racah polynomials Racah problems play an important role in algebras related to quantum systems. Famous solutions to the Racah problem for sl2 or equivalently su(1, 1) are the Clebsch-Gordan coefficients, the 3j-symbols, the 6j-symbols and in general the 3nj-symbols for su(1, 1). We will cast these problems into a single framework. Let us first pose the problem we want to solve. Problem. The Racah problem for su(1, 1) is the following: Let V be an irreducible representation of Rn . Consider two different labeling Abelian algebras Y1 and Y2 . Let the set {ψk } be a basis of V diagonalized by Y1 and {ϕs } be a basis of V diagonalized by Y2 . What are the connection coefficients between these to bases? In other words, find the numbers Rsk such that  Rsk ψk = ϕs . k

We will consider finite dimensional representations. The solution to the Racah problem for the rank one Racah algebra R3 has been known for a long time, see [18, 20, 21, 24]. We will give the result here. In the rank one case we have three labeling Abelian algebras: Y1 = C12 , Y2 = C23 , and Y3 = C13 . The connection coefficients between Y1 and Y2 are Racah polynomials. Definition 3.1. ([38]) Let rn (α, β, γ, δ; x) be the classical univariate Racah polynomials   rn (α, β, γ, δ; x) := (α + 1)n (β + δ + 1)n (γ + 1)n 4 F3 −n,n+α+β+1,−x,x+γ+δ+1 ;1 . α+1,β+δ+1,γ+1

THE RACAH ALGEBRA: AN OVERVIEW AND RECENT RESULTS

9

We will give the exact form for the connection coefficients. To this end we first introduce the following polynomial:    β+1 β−1 κ(x, β) = x + x+ . 2 2 Lemma 3.2. Let V be an irreducible representation of Rn with dim(V ) = N +1. Assume that the central elements on this representation act as the following scalars: C1 = κ(0, β0 ), C2 = κ(0, β1 − β0 − 1), C3 = κ(0, β2 − β1 − 1), C123 = κ(N, β2 ). Assume that {ψk } is a basis of V diagonalized by C12 and {ϕs } is a basis of V diagonalized by C23 with the following eigenvalues: C12 ψk = κ(k, β1 )ψk , C23 ϕs = κ(s, β2 − β0 − 1)ϕs . The connection coefficients are up to a gauge constant equal to Rsk = rs (β1 − β0 − 1, β2 − β1 − 1, −N − 1, β1 + N ; k). The constants β0 , β1 and β2 depend on the representation and can be calculated from the action of the central elements. In what follows we will denote the overlap coefficients depending on these central elements: Rsk (C1 , C2 , C3 , C123 ) Consider now the general case. Let YA1 and YA2 be two labeling Abelian algebras. To find the connection coefficients between bases diagonalized by YA1 and YA2 we will assume that the chains of sets A1 and A2 differ by only one element: A1 : A1 ⊂ · · · ⊂ Ai−1 ⊂ K ⊂ Ai+1 ⊂ · · · ⊂ An−2 , A2 : A1 ⊂ · · · ⊂ Ai−1 ⊂ L ⊂ Ai+1 ⊂ · · · ⊂ An−2 . Let {ψk } be a basis of the representation V diagonalized by YA1 with k = (k1 , . . . , kn−2 ) CAl ψk = λkl ψk , CK ψk = λki ψk . Let {ϕs } be a basis of the representation V diagonalized by YA2 with s = (s1 , . . . , sn−2 ) CAl ϕs = μsl ϕs , CL ϕs = μsi ϕs . We want to describe the connection coefficients Rsk such that  Rsk ψk = ϕs .  k

10

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

Because both bases {ψk } and {ϕs } are diagonalized by the generators CAl with l ∈ [n − 2]\{i} we may assume that μsl = λkl . From this we reduce the connection coefficients as follows

Rsk = Rsi ki δsl k l . l =i

We reduced the problem to finding the coefficients Rsi ki . Consider any eigenspace of the set of operators {CAl | l = i} and denote it by E. The operators CK and CL commute with the set of operators so they will preserve these eigenspaces. Moreover, these eigenspaces are representations for R3 by Lemma 2.6, if we consider the embedding morphism by setting. 1 → K\L, K\L,K∩L,L\K

This algebra R3

CK\L ,

2 → K ∩ L = Ai−1 ,

3 → L\K.

is generated by

CK∩L ,

CAi−1 ,

CL\K ,

CK ,

CL ,

CAi+1

where Ai+1 = K ∪ L and each of the generators commute with {CAl | l = i} preserving the eigenspaces. Consider the intersection of {ψk } with the eigenspace E. This will be a basis of the eigenspace E diagonalized by C12 . Equivalently, the intersection of {ϕs } and the same eigenspace E will be a basis of this eigenspace E diagonalized by C23 . By Lemma 3.2 the connection coefficients between the two bases of the representation E of R3 will therefore be the Racah polynomials Rsi ki (CK\L , CAi−1 , CL\K , CAi+1 ). We conclude that the connection coefficients between two bases diagonalized by two labeling Abelian algebra differing by only one generator is given by

Rsk = Rsi ki (CK\L , CAi−1 , CL\K , CAi+1 ) δsl k l . l =i

Let us now consider two general labeling Abelian algebras. The connection coefficients can be calculated by introducing a sequence of labeling Abelian algebras so that each subsequent pair of algebras only differ by one generator. Each algebra in the sequence will diagonalize a basis of the representation V of Rn . The connection coefficients between two bases diagonalized by two subsequent algebras in the sequence will be Racah polynomials. From this, we can calculate the connection coefficients between any two labeling Abelian algebras. Let us give an example. Example 3.3. Let Y1 = C12 , C123 and Y2 = C34 , C234 be two labeling Abelian algebras of R4 . Consider the following sequence of labeling Abelian algebras: C12 , C123 , C23 , C123 , C23 , C234 , C34 , C234 . Let {ψk1 ,k2 }, {φ11 ,2 }, {φ21 ,2 } and {ϕs1 ,s2 } be the bases diagonalized by the algebras in the sequence respectively. We calculate the connection coefficients:  ϕs1 ,s2 = Rs1 1 (C2 , C3 , C4 , C234 )φ21 ,s2 1

=



Rs1 1 (C2 , C3 , C4 , C234 )Rs2 k2 (C1 , C23 , C4 , C1234 )φ11 ,k2

1 ,k2

=



Rs1 1 (C2 , C3 , C4 , C234 )Rs2 k2 (C1 , C23 , C4 , C1234 )

k1 ,1 ,k2

× R1 k1 (C1 , C2 , C3 , C123 )ψk1 ,k2 .

THE RACAH ALGEBRA: AN OVERVIEW AND RECENT RESULTS

11

The first equality is obtained by considering the eigenspaces of C234 which acts as a representation space for R2,3,4 (3). The second equality is obtained by considering the eigenspaces of C23 which acts as a representation space for R1,23,4 (3) and the final equality is obtained by considering the eigenspaces of C123 which acts as a representation space for R1,2,3 (3). The connection coefficients are Rsk =



Rs1 1 (C2 , C3 , C4 , C234 )Rs2 k2 (C1 , C23 , C4 , C1234 )R1 k1 (C1 , C2 , C3 , C123 ).

1

One can ask the question if for any pair of labeling Abelian algebras we are able to find connection coefficients. To answer this question we introduce the connection graph. The vertices represent labeling Abelian algebras and there is an edge between two vertices if the labeling Abelian algebras differ by one generator. For R4 the connection graph look as follows: (C14 , C124 ) (C24 , C124 )

(C12 , C124 )

(C24 , C234 )

(C12 , C123 )

(C23 , C234 )

(C23 , C123 )

(C34 , C234 )

(C13 , C123 )

(C34 , C134 )

(C13 , C134 ) (C14 , C134 )

This graph is connected so we can find connection coefficients between any two labeling Abelian algebras of R4 by the method shown in Example 3.3. One can show that the connection graph is connected for the Racah algebra of any rank. For a proof see [9]. Other methods to find the connection coefficients have been presented before. See [44] for the tree method as well as [47, 48]. Example 3.4. In this example we show how to obtain the multivariate Racah polynomials as defined by Tratnik [45]. Let Yinitial = C12 , C123 , . . . , C[n−1] and Yfinal = C23 , C234 , . . . , C[2..n] . A sequence of intermediate algebras, each differing by one element with the next, is given as follows: Yi = C23 , . . . , C[2..2+i] , C[2+i] , . . . , C[n−2] . The connection coefficients up to a gauge constant will be given by Rks =

n−2

Rki si (C1 , C[2..i+1] , Ci+2 , C[i+2] ).

i=1

If the action of the central elements on the irreducible representation is given by C1 = κ(0, β0 ), Ci+1 = κ(0, βi − βi−1 − 1) for some number β0 , . . . , βn−1 , then these

12

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

connection coefficients are written explicitly as Rks =

n−2

rkj (2| k|j−1 + βj − β0 − 1, βj+1 − βj − 1, | k|j−1 − sj+1 − 1,

j=1

| k|j−1 + βj + sj+1 , −| k|j−1 + sj ), j where | k|j = i=1 ki . This result was obtained in [13]. These are exactly the multivariate Racah polynomials as defined by Tratnik [13, 23, 45]. In the rank two case one finds bivariate Racah polynomials which coincides with the result in [42]. Remark 3.5. As we shall see in the next section, the Racah algebra has realizations in terms of differential operators. After an appropriate gauge transformation, these operators preserve the space of polynomials [36]. The common eigenfunctions of the labeling Abelian algebras become multivariable Jacobi polynomials which are mutually orthogonal with respect to the Dirichlet distribution. The coefficients Rks connect different bases of Jacobi polynomials, obtained by an appropriate action of the symmetric group, which preserves the Dirichlet distribution. Within this framework, Lemma 3.2 says that the entries of the transition matrix between different bases of two-variable Jacobi polynomials for the Dirichlet distribution of order 3 can be expressed in terms of the Racah polynomials, and this was proved by Dunkl [14]. The extensions to arbitrary dimension and, in particular, techniques to compute Rks , different relations and the explicit formula for the cyclic permutation in Example 3.4 were obtained in [31]. 4. Realizations of the higher rank Racah algebra 4.1. The generic superintegrable system on the sphere. The generic superintegrable system is already well studied. See for example [20, 21, 29, 30, 33–35]. We will introduce this modelhere. Let Sn−1 = {(y1 , . . . , yn ) ∈ Rn | i yi2 = 1} be the sphere in Rn . We have n variables yi and we denote ∂i := ∂yi . The generic superintegrable system on the sphere is the quantum system with Hamiltonian H = ΔLB +

n  bi 2. y i=1 i

The parameters bi are real numbers. The operator ΔLB is the Laplace-Beltrami operator:  ΔLB = (yi ∂j − yj ∂i )2 . 1≤i 3. However, the 2n − 3 operators in the set G = {C1,j : j = 2, . . . , n} ∪ {Ci,n : i = 2, . . . , n − 1} generate the symmetry algebra for the Hamiltonian H, and every operator Cij can be written explicitly as a polynomial of the operators in G, see [29, 30]. Moreover, these constructions can be generalized for a discrete extension of the generic superintegrable

14

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

system on the sphere related to the Hahn polynomials and the hypergeometric distribution [32]. 4.2. The Dunkl model. For a detailed exposition of the Dunkl model see [9]. The Dunkl model is obtained by realizing the algebra su(1, 1) using the Dunkl operators as defined by C.F. Dunkl in [15]. We consider the Dunkl operators related to the reflection group Zn2 . They are defined as follows: Ti := ∂xi + μi

1 − Ri . xi

The operator Ri is the reflection which acts as Ri (f (xi )) = f (−xi ). The number μi > 0 is a deformation parameter. The operators Ti are commutative. With these operators one can realize the algebra su(1, 1):   x2 T2 1 1 J+,i = i , J−,i = i , J0,i = xi ∂i + μi + . 2 2 2 2 This realization of su(1, 1) leads to a new realization of the Racah algebra Rn with 4μ2i − 4μi Ri − 3 , 16  1 Cij = −(xi Tj − xj Ti )2 + (μi Rj + μj Ri )2 − 1 . 4 Observe that the operator xi Tj − xj Ti is the Dunkl angular momentum operator. By Lemma 2.4 this realization of the Racah algebra is in the centralizer of the algebra generated by the following elements:  n n n  1 2 1 2 1 1 J+,[n] = xi ∂i + μi + xi , J−,[n] = Ti , J0,[n] = 2 i=1 2 i=1 2 i=1 2 Ci =

Observe that J−,[n] is the Dunkl-Laplacian ΔDunkl times 1/2. The Racah algebra thus acts as a symmetry  algebra for the Dunkl-Laplacian. Also observe that the Euler operator En = ni=1 xi ∂i appears in J0,[n] . Let Pk be the set of homogeneous polynomials of degree k. These are eigenspaces of J0,[n] . Consider the space of Dunkl-harmonics Hk := Pk ∩ ker(ΔDunkl ). These spaces will act as representations for the higher rank Racah algebra. 4.3. The Barut-Girardello model. For a detailed overview of the BarutGirardello model we refer to the following article [11]. The Barut-Girardello model for the rank one Racah algebra was previously considered in [19]. The previous models realize the Racah algebra Rn in n variables. The Barut-Girardello model has the interesting property that it realizes the Racah algebra in a number of variables equal to the rank n − 2 of said algebra. This is obtained as follows. Consider the following realization of su(1, 1): J+ = x2 ∂x + 2νx,

J− = ∂x ,

J0 = x∂x + ν.

After introducing n variables x1 . . . . , xn , their partial derivatives ∂1 , . . . ∂n and n parameters ν1 , . . . , νn one constructs a realization of the Racah algebra Rn . It is the centralizer of the following su(1, 1) algebra by Lemma 2.4. J+,[n] =

n  i=1

(x2i ∂xi + 2νi xi ),

J−,[n] =

n  i=1

∂i ,

J0,[n] =

n  i=1

(xi ∂i + νi ).

THE RACAH ALGEBRA: AN OVERVIEW AND RECENT RESULTS

15

Let Hk (Rn ) be the kernel of J−,[n] in the space of homogenous polynomials defined on Rn . It has the following basis: j

n−2 ϕj1 ,...,jn−2 = (x1 − x2 )k uj11 uj22 . . . un−2

with

xj+2 − xj+1 , j ∈ {1, . . . , n − 2}. x1 − x2 When one gauges the Racah algebra as follows uj :=

B = (x1 − x2 )−k CB (x1 − x2 )k , C

(4.1)

one obtains an algebra acting on polynomials of at most degree k in the variables u1 , . . . un−2 . Explicit calculation leads to the following realization: Theorem 4.2. The space Πn−2 of all polynomials of degree k in n − 2 variables k carries a realization of the rank n − 2 Racah algebra Rn . This realization is given explicitly by i = νi (νi − 1), C i ∈ [n] and, for i, j ∈ {3, . . . , n},      n−2 n−2 n−2     −k − ∂u1 + C u ∂u u ∂u + 2ν2 k − u ∂u 12 = − k − 1 − =1

 − 2ν1   C 1j = − 1 −

−k − ∂u1 + j−2  =1

1−

+ 2νj

j−2 



n−2 

=1

u ∂u

=1

k−

u



n−2 

=1

+ (ν1 + ν2 )(ν1 + ν2 − 1)

u ∂u

k−1−

u

=1



=1

2 



n−2 

∂uj−2 − ∂uj−1

 u ∂u

 − 2ν1

1−

=1



j−2 



∂uj−2 − ∂uj−1

u

=1

+ (ν1 + νj )(ν1 + νj − 1)  j−2 2  n−2  

  1 − k − ∂u1 + ∂uj−2 − ∂uj−1 u u ∂u C 2j = − =1

+ 2νj

j−2 

=1

 u

k + ∂u 1 −

=1

n−2 

 u ∂u

+ 2ν2

j−2 

=1

 u

∂uj−2 − ∂uj−1

=1

+ (ν2 + νj )(ν2 + νj − 1) ⎛ ⎞2 i−2 

  ij = − ⎝ u ⎠ ∂ui−2 − ∂ui−1 ∂uj−2 − ∂uj−1 C =j−1

⎛

+ 2νj ⎝

i−2 

⎞



⎛

u ⎠ ∂ui−2 − ∂ui−1 − 2νi ⎝

=j−1

i−2 

⎞



u ⎠ ∂uj−2 − ∂uj−1

=j−1

+ (νi + νj )(νi + νj − 1) where we assume i > j and with un−1 = 0 whenever it appears.





16

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

This model has only n − 2 variables which is equal to the rank of the higher rank Racah algebra. It is possible to embed this algebra into a differential operator realization of the universal enveloping algebra of sln−1 , see [12]. 4.4. The discrete model. For a detailed overview see [13, 29]. The discrete model has no known underlying realization of su(1, 1). Instead its action is derived from the action on any irreducible representation of Rn denoted by V . Let Yinitial and Yfinal be two labeling Abelian algebras (Definition 2.10). Consider two bases of V : {ψk } diagonalizing Yinitial and {ϕk } diagonalizing Yfinal as in Example 3.4. The connection coefficients between these two bases are given by the functions Rk defined as:    ϕs ψk =: Rs ( k). By Example 3.4 these functions Rk are multivariate Racah polynomials. We define the action of a generator CA of Rn by      CA Rs ( k) = CA ϕs ψ := ϕs | CA ψ . k

k

We want to describe the operators CA acting of the function Rk . To do so we identify operators whose action on these function Rk coincides with the action of CA . The multivariate Racah polynomials are eigenvectors of the labeling Abelian algebras Yinitial and Yfinal in this realization. The multivariate Racah polynomials are also eigenvectors of the following Racah operators. These were originally introduced in [23]. See also [29, 30, 32, 42]. Definition 4.3. Put Lj =



Gν (Eν − 1).

 ν ∈{−1,0,1}  ν =0

j

Here Eν is a shift operator defined as follows. Let Exνii (f (xj )) = f (xj + δij νi ). Then ν we define Eν = Exν11 Exν22 . . . Exjj . The Gν are rational functions in the variables x0 , x1 , . . . , xj+1 and β0 , . . . , βj+1 and are defined as follows. We introduce the following functions: (βi + 1)(βi+1 − 1) Bi0,0 := xi (xi + βi ) + xi+1 (xi+1 + βi+1 ) + , 2 0,1 Bi := (xi+1 + xi + βi+1 )(xi+1 − xi + βi+1 − βi ), Bi1,0 := (xi+1 − xi )(xi+1 + xi + βi+1 ), Bi1,1 := (xi+1 + xi + βi+1 )(xi+1 + xi + βi+1 + 1). Let Ii f (xi ) := f (−xi − βi ). We extend B s,t by defining: Bi−1,t := Ii (Bi1,t ), Bis,−1 := Ii+1 (Bis,1 ), Bi−1,−1 := Ii (Ii+1 (Bi1,1 )). We also introduce b0i := (2xi + βi + 1)(2xi + βi − 1), b1i := (2xi + βi + 1)(2xi + βi ), := Ii (b1i ). b−1 i

THE RACAH ALGEBRA: AN OVERVIEW AND RECENT RESULTS

17

Let | ν |0 be the number of zeroes appearing in ν . Then Gν is j ν ,ν Bi i i+1 Gν := 2|ν |0 i=0 . j νi i=1 bi The action of these Racah operators coincides up to scalar with the action of Yfinal . In general it can be shown that the action of any C[i..j] coincides with the action of minus a Racah operator up to the addition of a scalar. These operators generate the Racah algebra Rn as Cij = C[i..j] − C[i..j−1] − C[i+1..j] + C[i+1..j−1] + Ci + Cj . To present the discrete model of the higher rank Racah algebra we need to introduce the following map: Definition 4.4. Let σ be the map that adds 1 to any index of an expression: Alg[x0 , . . . , xs ; β0 , . . . βs ; E1 , . . . , Es ] → Alg[x1 , . . . , xs+1 ; β1 , . . . βs+1 ; E2 , . . . , Es+1 ] σ(xi ) = xi+1 , σ(βi ) = βi+1 , σ(Exi ) = Exi+1 , e.g. σ(x1 β22 Ex1 ) = x2 β32 Ex2 . In [13] the following theorem was proven.   x+ Theorem 4.5. Define κ(x, β) = x + β+1 2 above, define the following operators: (4.2) (4.3) (4.4)

β−1 2

 . With Li given as

C[m] = κ(xm−1 , βm−1 ), C[2...m+1] = −Lm−1 + κ(0, βm − β0 − 1), C[p...q] = σ p−2 (C[2...q−p+2] ),

if

p>2

and set x0 = 0. The algebra generated by these operators is a discrete realization of Rn . This result coincides with the realization in rank one which was already known [20, 21] and the result obtained in [42] for the rank 2 case. In [29], the representation of Rn in Theorem 4.5 was constructed by defining in terms of the Racah operators the generators {C1,j : j = 2, . . . , n} ∪ {Ci,n : i = 2, . . . , n − 1} of the symmetry algebra discussed in Remark 4.1. 5. Further results and conclusions The algebraic properties of the higher rank Racah algebra are not well understood yet. The representation theory for the rank one case is being build up in [1–4, 26]. The relationship with other algebras is also being studied. The rank one Racah algebras have the Temperly-Lieb and Brauer algebras as quotients, see [5]. It would be interesting to see this result generalized to higher rank. Howe type dualities have been brought to light involving the higher rank Racah algebra, see [17]. When one replaces su(1, 1) with different algebras in the method given in Section 2, one obtains algebras which are strongly related to the higher rank Racah algebra and for which we can solve the Racah problem. If one takes the Lie super algebra osp(1|2), one obtains the higher rank Bannai-Ito algebra [8, 10]. The q-deformation

18

H. DE BIE, P. ILIEV, W. VAN DE VIJVER, AND L. VINET

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[39] G. I. Lehrer and R. B. Zhang, Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra 306 (2006), no. 1, 138–174, DOI 10.1016/j.jalgebra.2006.03.043. MR2271576 [40] Jean-Marc L´ evy-Leblond and Monique L´ evy-Nahas, Symmetrical coupling of three angular momenta, J. Mathematical Phys. 6 (1965), 1372–1380, DOI 10.1063/1.1704786. MR181340 [41] Willard Miller Jr., Sarah Post, and Pavel Winternitz, Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), no. 42, 423001, 97, DOI 10.1088/17518113/46/42/423001. MR3119484 [42] Sarah Post, Racah polynomials and recoupling schemes of su(1, 1), SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 057, 17, DOI 10.3842/SIGMA.2015.057. MR3372111 [43] Hadewijch De Clercq, Higher rank relations for the Askey-Wilson and q-Bannai-Ito algebra, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 099, 32, DOI 10.3842/SIGMA.2019.099. MR4043929 [44] Fabio Scarabotti, The tree method for multidimensional q-Hahn and q-Racah polynomials, Ramanujan J. 25 (2011), no. 1, 57–91, DOI 10.1007/s11139-010-9245-2. MR2787292 [45] M. V. Tratnik, Some multivariable orthogonal polynomials of the Askey tableau-discrete families, J. Math. Phys. 32 (1991), no. 9, 2337–2342, DOI 10.1063/1.529158. MR1122519 [46] Paul Terwilliger, The equitable presentation for the quantum group Uq (g) associated with a symmetrizable Kac-Moody algebra g, J. Algebra 298 (2006), no. 1, 302–319, DOI 10.1016/j.jalgebra.2005.11.013. MR2215130 [47] J. Van der Jeugt, Coupling coefficients for Lie algebra representations and addition formulas for special functions, J. Math. Phys. 38 (1997), no. 5, 2728–2740, DOI 10.1063/1.531984. MR1447890 [48] Joris Van der Jeugt, 3nj-coefficients and orthogonal polynomials of hypergeometric type, Orthogonal polynomials and special functions (Leuven, 2002), Lecture Notes in Math., vol. 1817, Springer, Berlin, 2003, pp. 25–92, DOI 10.1007/3-540-44945-0 2. MR2022852 [49] Yuan Xu, Intertwining operator and h-harmonics associated with reflection groups, Canad. J. Math. 50 (1998), no. 1, 193–209, DOI 10.4153/CJM-1998-010-9. MR1618819 [50] A. S. Zhedanov, “Hidden symmetry” of Askey-Wilson polynomials (Russian, with English summary), Teoret. Mat. Fiz. 89 (1991), no. 2, 190–204, DOI 10.1007/BF01015906; English transl., Theoret. and Math. Phys. 89 (1991), no. 2, 1146–1157 (1992). MR1151381 Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium Email address: [email protected] School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA 303320160 Email address: [email protected] Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Krijgslaan 281, 9000 Ghent, Belgium Email address: [email protected] ´matiques, Universit´ Centre de Recherches Mathe e de Montr´ eal, P.O. Box 6128, Centre-ville Station, Montr´ eal, QC H3C 3J7, Canada Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15451

Orbit embedding for double flag varieties and Steinberg maps Lucas Fresse and Kyo Nishiyama Abstract. In the first half of this article, we review the Steinberg theory for double flag varieties for symmetric pairs. For a special case of the symmetric + − × GLn /Bn space of type AIII, we will consider X = GL2n /P(n,n) × GLn /Bn on which K = GLn × GLn acts diagonally. We give a classification of K-orbits in X, and explicit combinatorial description of the Steinberg maps. In the latter half, we develop the theory of embedding of a double flag variety into a larger one. This embedding is a powerful tool to study different types of double flag varieties in terms of the known ones. We prove an embedding theorem of orbits in full generality and give an example of type CI which is embedded into type AIII.

Introduction Various combinatorial structures play important roles in representation theory. For example, the set of all equivalence classes of irreducible representations of the symmetric group Sn of order n is classified by the set of partitions P(n) of n. If an irreducible representation σ of Sn is corresponding to a partition λ ∈ P(n), then the set of standard tableaux STabλ gives a basis of the representation space of σ. This is known as the Specht theory. Partitions are also interpreted as highest weights of finite dimensional irreducible representations of GLn and semistandard tableaux give a basis of an irreducible representation. In this respect, the geometry of flag varieties also interacts with combinatorics and representation theory. So let G be a reductive algebraic group over the complex number field, B ⊂ G a Borel subgroup, and consider the full flag variety G/B. Then a symmetric subgroup K of G which is fixed by an involution θ acts on G/B with finitely many orbits and, together with the information of local systems, they actually classify irreducible Harish-Chandra (g, K)-modules with the trivial infinitesimal character in terms of D-modules ([2], see also [18]). The combinatorics of the K-orbits in G/B and their closure relations is deeply related to the category of Harish-Chandra modules (see [14, 17], e.g.). Let us write B = G/B for shorthand. In much earlier time, Springer noticed that the cotangent bundle T ∗ B gives a resolution of singularity of the nilpotent variety Ng consisting of the nilpotent elements in g ([26]). He constructed irreducible 2020 Mathematics Subject Classification. Primary 14M15; Secondary 17B08, 53C35, 05A15. The first author was supported in part by the ANR project GeoLie ANR-15-CE40-0012. The second author was supported by JSPS KAKENHI Grant Number #16K05070. c 2021 American Mathematical Society

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LUCAS FRESSE AND KYO NISHIYAMA

representations of the Weyl group W on the top-degree cohomology space of the so-called Springer fiber Bx associated to a nilpotent element x ∈ Ng , and in that way he created a correspondence between irreducible representations of W and the set of nilpotent orbits Ng /G together with their local systems, which is finite in number ([27], see also [3]). This is a remarkable breakthrough that provides an amount of works on combinatorics related to the geometry of flag varieties as well as the nilpotent varieties (see, e.g., [13, 25] and [21, 30]). Approximately in the same period, Steinberg introduced a variety, now called the Steinberg variety ([28]). He used this variety to study the Springer resolution more deeply. The resolution T ∗ B → Ng actually comes as a moment map arising from the Hamiltonian action of G on the symplectic variety T ∗ B (see [3, 5]). Let us consider the product of the flag variety X := B × B on which G acts diagonally. It is traditional to study a symplectic reduction in the study of Hamiltonian actions on a symplectic variety, and the Steinberg variety is obtained from this recipe. It arises as the null fiber of the moment map μX : T ∗ X → g∗ , and is denoted by ZX = μ−1 X (0). The Steinberg variety can also be interpreted as the fiber product over the resolution maps, which is exhibited below. ZX =T ∗ B×Ng T ∗ B PPP PPP P p1 

T ∗B

p2

ϕ PP

μB

/ T ∗B μB

PPP PP(  / Ng

The variety ZX is highly reducible of equi-dimension, and its irreducible components are parametrized by the Weyl group W , or strictly speaking by the G-orbits {Ow | w ∈ W } in X . The image of an irreducible component by the map ϕ, which is the composite of the first projection to T ∗ B and then by the moment map μB : T ∗ B → Ng , is the closure of a nilpotent orbit. T ∗ B×Ng T ∗ B FF x FF ϕ π xxx FF x FF x FF xx x {x # Ng ⊃ O B×B ⊃ Ow

 Thus we get a combinatorial map Φ : W  (B×B)/G → Ng /G via ϕ π −1 (Ow ) = O (w ∈ W, O ∈ Ng /G). This provides a rich theory involving geometry, combinatorics and representation theory. In fact, it turns out that ZX bears the regular representation of the Weyl group which “doubles” the Springer representations. We started a study which generalizes the above mentioned Steinberg theory to the case of symmetric pairs in [7, 8]. It is related to the triple flag varieties [6, 15, 16, 31] as well as the double flag varieties for symmetric pairs [10, 20] of finite type. In fact, our study is motivated by the paper by Henderson and Trapa [11], although the role of the double flag variety is implicit in their paper. Let P and Q be parabolic subgroups of G and K, respectively. Then we call X := G/P × K/Q a double flag variety of the symmetric pair (G, K). The symmetric subgroup K acts on X diagonally. For this double flag variety, we define

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a conormal variety ZX , which is a direct generalization of the Steinberg variety. If there are only finitely many K-orbits on X, basically we can play the same game as above, and obtain two different combinatorial maps Φ±θ : X/K → N ±θ /K, where N θ = Ng ∩ k is the nilpotent variety of k, and N −θ is the nilpotent variety in the (co)tangent space of G/K (see below for the precise definition). We call the maps Φ±θ defined above Steinberg maps associated to X. If we need to distinguish Φθ and Φ−θ , the map Φθ is called a generalized Steinberg map and Φ−θ an exotic one. In the case of the double flag varieties, it is not obvious if there is a representation theoretic structure on ZX . However, a na¨ıve picture of convolutions (cf. [3, §2.7]) gives us an insight that hopefully we can construct Hecke algebra actions of H(G/B) on the left and those of H(K/BK ) on the right and which make the top Borel-Moore homology space into a Hecke algebra bi-module. See [31] and also [8, Conjecture 7.11]. Let us consider an example of a symmetric pair of type AIII, namely (G, K) = (GL2n , GLn × GLn ). We consider everything over C and basically omit the letter C. Let V = C2n and fix a standard polar decomposition V = V + ⊕ V − , where dim V ± = n and V + = ei | 1 ≤ i ≤ n , V − = ei | n + 1 ≤ i ≤ 2n . Then K is the stabilizer of the polarization. Consider a maximal parabolic subgroup P in G which stabilizes V + so that G/P  Grn (V ), the Grassmannian of n-spaces in V . Also we choose a Borel subgroup of K = GLn × GLn as BK = Bn+ × Bn− , where Bn+ denotes the Borel subgroup of GLn consisting of upper triangular matrices and Bn− its opposite as usual. In this way, our double flag variety is − X = G/P × K/BK  Grn (V ) × F+ n × Fn , ± on which K = GL2n acts. Here F± n is the set of complete flags of subspaces in V . We proved

Theorem 0.1 ([8], see Theorem 2.2 below). There are finitely many K-orbits in X and they are parametrized by pairs of partial permutations of rank n. Namely, we have     τ  X/K  (Tn2 )◦ /Sn , (Tn2 )◦ := ω = τ1  τ1 , τ2 ∈ Tn , rank ω = n , 2

where Tn denotes the set of partial permutation matrices of size n. It is well known that the nilpotent orbits in N θ /K are parametrized by pairs of partitions (λ, μ) ∈ P(n)2 , and those in N −θ /K are parametrized by signed Young diagrams Λ ∈ YD± (n, n) of signature (n, n). See § 2.2 below for details. So we obtain a combinatorial map Φ±θ : (Tn2 )◦ /Sn  X/K −−−→ N ±θ /K 

P(n)2 YD± (n, n)

for N θ , for N −θ .

In [8], we gave partial results which give explicit and efficient algorithms for   τ computing the Steinberg maps Φ±θ for ω = 11 , i.e., one of the partial permun

tation is really a permutation. However, now we have a complete algorithm for all of (Tn2 )◦ /Sn (unpublished, in preparation). In the present paper, we give this complete algorithm without proof for the map Φθ . Proofs and the algorithm for Φ−θ will appear elsewhere soon. Thus we only claim it in an abstract manner here.

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LUCAS FRESSE AND KYO NISHIYAMA

Theorem 0.2 (Theorems 2.3 and 2.6). There exist efficient combinatorial algorithms which describe the maps Φ±θ . In the course of proofs, we obtain a generalization of Robinson-Schensted correspondence for the pairs of partial permutations of full rank (see Theorem 2.6). This correspondence is also interesting in itself, and we noticed1 that there is a strong resemblance of the parameter sets of K-orbits in X and those for the Travkin’s mirabolic triple flag variety ([31]). It seems that they are also related to the parameter sets of Achar-Henderson’s enhanced nilpotent orbits ([1]) and Kato’s exotic nilpotent orbits ([12]). However, up to now, we cannot make out any rigorous geometric interpretations for them. There are many other interesting double flag varieties which admit a finite number of K-orbits (see [10, 20]). However, no unified way to get an explicit and efficient algorithm for the above mentioned Steinberg maps Φ±θ is known up to now. Even for giving parametrizations of K-orbits in X, there is no rigorous theory. In this paper, we propose a technique by which we can embed a double flag variety into a larger double flag variety preserving orbit structures. Thus, if we know the Steinberg theory for a larger double flag variety, we might deduce the theory for a smaller one. In fact, nilpotent orbits of classical Lie algebras are classified in that way. Namely, we first classify the nilpotent orbits in type A, which amounts to establish the theory of Jordan normal form, and we get partitions. For type B, C and D, we embed them as Lie subalgebras of type A. Then nilpotent orbits of these Lie subalgebras can be obtained as non-empty intersection of the ones in type A and these subalgebras themselves. So we can use partitions of special shapes as a parameter set of nilpotent orbits. In Section 3, we discuss the embedding of orbits in X into a larger double flag variety X. The key idea is to use two commuting involutions σ, θ and existence of square roots in the direction of G/K. This idea is originally developed by Takuya Ohta [24] for linear actions, and later for arbitrary actions in [19]. We give a full general theory for embedding in Section 3 without assuming the finiteness of orbits. Then, in Section 4, we give an example of type CI embedded into type AIII discussed above. Let us briefly summarize the main results here. Thus we will consider a larger connected reductive algebraic group G and two commuting involutions σ, θ of G. Define K = Gθ , the fixed point subgroup of θ, and similarly G = Gσ , K = Kσ = Gθ . We assume that all these groups are connected. Take parabolic subgroups P ⊂ G and Q ⊂ K. Then there exist σ-stable parabolic subgroups P ⊂ G and Q ⊂ K which cut out P and Q from G and K respectively. Take a σ-stable subgroup H of G and denote H−σ = {h ∈ H | σ(h) = h−1 }. We say that H admits (−σ)-square roots if for any h ∈ H−σ there exists an f ∈ H−σ such that h = f 2 . We consider the following conditions. (A) P and Q admit (−σ)-square roots. (B) For any σ-stable parabolic subgroups P1 ⊂ G and Q1 ⊂ K which are conjugate to P and Q respectively, the intersection P1 ∩ Q1 admits (−σ)square roots. 1 Actually this was pointed out to the authors by Anthony Henderson (private communication). We thank him for that.

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Theorem 0.3 (Theorem 3.3). In the above setting, let us consider the double flag varieties X := G/P × K/Q and X := G/P × K/Q. If the parabolic subgroups P and Q satisfy the conditions (A) and (B), then there exists a natural embedding X → X which respects the involution σ, and the natural orbit map ι : X/K → X/K defined by ι(O) = K · O for O ∈ X/K is injective, i.e., for any K-orbit O in X, the intersection O ∩ X is either empty or a single K-orbit. Under the same assumption, we prove a theorem on conormal varieties, which essentially says that there exists an embedding of irreducible components of ZX into those of ZX . See Theorem 3.6. In Section 4, we consider (G, K) = (GL2n , GLn × GLn ) of type AIII and take (G, K) = (Sp2n , GLn ). For a Siegel parabolic subgroup P = PS in G and the standard upper triangular Borel subgroup BK = Bn+ in K, we can choose σ-stable parabolic subgroups P and BK which satisfy the assumptions (A) and (B). Thus the embedding theorem holds, and we get an explicit parametrization of K-orbits in X = Sp2n /PS × GLn /Bn+ (see Theorem 4.6). Note that X  LGr(C2n ) × Fn , where LGr(C2n ) is the Lagrangian Grassmannian in the symplectic vector space C2n . 1. Steinberg theory for symmetric pairs: A review In this section, we review the Steinberg theory for symmetric pairs given in [8], although we describe it in a slightly different manner in this article. Let G be a complex connected reductive algebraic group with an involutive automorphism θ : G → G. Let K := Gθ be the fixed-point subgroup of θ. Thus we have a symmetric pair (G, K) and K is called a symmetric subgroup. Assume for simplicity that K is connected. By differentiation, the involution θ induces an involution on the Lie algebra g := Lie (G), which we also denote θ : g → g by abuse of notation. Let k := Lie (K) = gθ the fixed-point subalgebra and put s := g−θ = {x ∈ g | θ(x) = −x}. Let x → xθ and x → x−θ stand for the projections prk : g → k and prs : g → s along the Cartan decomposition g = k ⊕ s. Let N ⊂ g be the cone of nilpotent elements and put N θ := N ∩ k and N −θ := N ∩ s. It is well known that the nilpotent varieties N θ and N −θ consist of finitely many K-orbits. Let us introduce the double flag variety X = G/P × K/Q, where P and Q are parabolic subgroups of G and K respectively [20]. The variety X is a smooth projective variety on which K acts diagonally. Let us denote PG = G/P and QK = K/Q so that X = PG ×QK . As usual, we identifyPG = G/P with the set of parabolic subalgebras p1 which are conjugate to p = Lie (P ). We denote by up1 the nilpotent radical of a parabolic subalgebra p1 . Then the cotangent bundle over PG is isomorphic to T ∗PG = {(p1 , x) | p1 ∈ PG , x ∈ up1 }  G ×P up . We denote by μPG : T ∗PG → N the second projection μPG (p1 , x) = x, which coincides with the moment map2 with respect to a standard symplectic structure 2 The cotangent bundle T ∗P admits a canonical G-invariant symplectic structure. Since the G action of G on the symplectic variety T ∗PG ( G ×P up ) is Hamiltonian, there exists a moment map μ : T ∗PG → g∗ . We fix once and for all a nondegenerate invariant bilinear form on g and identify g  g∗ . With this identification, the conormal direction (g/p1 )⊥ ⊂ g∗ is identified with

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LUCAS FRESSE AND KYO NISHIYAMA

on T ∗PG . Similarly, we have the moment map μQK : T ∗QK = {(q1 , y) | q1 ∈ QK , y ∈ uq1 } → k,

μQK (q1 , y) = y,

with the obvious notation similar to those for PG . Definition 1.1. Let Z := T ∗PG ×N θ T ∗QK be the fiber product over the nilpotent variety N θ : Z = T ∗PG ×N θ T ∗QK LL LL LL LL LL

p1

/ T ∗PG  (p1 , x) 

μPG

Ng  x LL LL LL LL prk LL %  −μQK / N θ  −y = xθ

p2

 (q1 , y) ∈ T ∗QK

ϕθ

We call Z = ZX the conormal variety for the double flag variety X. The definition of the conormal variety Z looks different from that in [8], but they are isomorphic. In fact, we know Fact 1.2. (1) Let μX : T ∗ X → k be the moment map for the canonical Hamiltonian action of K on the cotangent bundle T ∗ X. Then the conormal variety is isomorphic to the null fiber of the moment map: Z  μ−1 X (0). (2) Let O ⊂ X be a K-orbit. We denote by TO∗ X the conormal bundle over O. The ! conormal variety is a disjoint union of the conormal bundles: Z = O∈X/K TO∗ X. (3) The dimension of the conormal variety is equal to dim X if and only if there are only finitely many K-orbits in X. In this case, Z is equidimensional and each irreducible component arises as the closure of a conormal " bundle. Thus Z = O∈X/K TO∗ X gives the decomposition into irreducible components. We are particularly interested in the case where there are only finitely many K-orbits in X. However, for the time being, we do not assume it and develop a general theory. Let us denote the diagonal map in the fiber product by ϕθ : Z → N θ . This map is explicitly described as ϕθ ((p1 , x), (q1 , y)) = xθ = −y ∗

for ((p1 , x), (q1 , y)) ∈ Z.

∗

Note that we consider Z ⊂ T PG ×T QK = T ∗ X here. It is not enough to specify the conormal fiber only by ϕθ and we need another map ϕ−θ ((p1 , x), (q1 , y)) = x−θ = x + y

for ((p1 , x), (q1 , y)) ∈ Z.

We call ϕθ the generalized Steinberg map and ϕ−θ the exotic Steinberg map. Both maps are clearly K-equivariant, but a priori not closed (see [8, Remark 11.3]). up1 which is contained in N . So the image of μ is actually contained in the nilpotent variety N . We do not repeat similar arguments below, but the term “moment map” should be always understood in this way.

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By definition, the image Im ϕθ is contained in the nilpotent variety N θ . There is no guarantee that the image Im ϕ−θ is contained in the nilpotent variety, but in many interesting cases it is so. For this, we refer the readers to [7, §4.1]. Assumption 1.3. We assume Im ϕ−θ ⊂ N −θ throughout in this paper. Note that the sets Gup ⊂ N and Kuq ⊂ N θ are the closures of the Richardson nilpotent orbits associated to P and Q respectively. So the above assumption is equivalent to claiming that prs (Gup ∩ (Kuq + s)) ⊂ N −θ . In other words, x ∈ Gup and xθ ∈ Kuq imply x−θ ∈ N −θ . Let π be the projection from the cotangent bundle T ∗ X to X (the bundle map) and consider the following double fibration maps.

X = QK

Z = T ∗QK ×N θ T ∗PG LLL rr LLL ϕ±θ π rrr LLL r r r LLL r r r L% yrr ×PG N ±θ

Using this diagram, we define orbit maps Φ±θ : X/K −−−→ N ±θ /K,

O → O

by ϕ±θ (π −1 (O)) = O, where O is a K-orbit in X and O is a nilpotent K-orbit in N ±θ . This definition works since there are only finitely many nilpotent K-orbits both in N θ and N −θ . By abuse of the terminology, we also call Φθ the generalized Steinberg map and Φ−θ the exotic Steinberg map. Since π −1 (O) = TO∗ X is the conormal bundle over O, Φ±θ (O) = O if and only if ϕ±θ (TO∗ X) = O. Recall that, if there exist only finitely many K-orbits in X, the closure of TO∗ X can also be interpreted as an irreducible component of the conormal variety Z (Fact 1.2 (3)). 2. Combinatorial Steinberg maps In this section, we will discuss a combinatorial side of the Steinberg theory. Thus we assume that there are finitely many K-orbits in the double flag variety X = PG × QK . In this situation, if we know an explicit classification of the K-orbits in X, both Steinberg maps Φ±θ : X/K → N ±θ might have interesting combinatorial interpretations. 2.1. Classical Steinberg map and the Robinson-Schensted correspondence. Let us first review the results by Steinberg [29]. We consider a special case where G = K (i.e., θ = idG ), and take P = Q = B to be a Borel subgroup of G. Then X = G/B × G/B = BG ×BG , where BG = G/B is the full flag variety. In this case, we see that X/G  B\G/B, and the double coset space on the right hand side is parametrized by the Weyl group W thanks to the Bruhat decomposition. Also, since s = 0, the nilpotent variety N −θ vanishes and ϕ−θ is zero. The nilpotent variety N θ coincides with N = Ng . Thus our map Φθ reduces to Φ : W → N /G. Note that both the Weyl group W and the set of nilpotent orbits N /G provide rich ingredients for combinatorics. Let us examine it in the case of G = GLn = GLn (C). We choose B = Bn+ (the Borel subgroup of upper triangular matrices). The Weyl group W is simply

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LUCAS FRESSE AND KYO NISHIYAMA

the symmetric group Sn of order n and the set of nilpotent GLn -orbits in N is in bijection with the set of partitions of n, which we will denote by P(n). For w ∈ Sn , we take a permutation matrix denoted by the same letter, and write Ow for the G-orbit through (B, wB) ∈ X. Then, as we explained, Zw := π −1 (Ow ) is an irreducible component of the variety Z (called Steinberg variety, in this setting). The image of this irreducible component by ϕ = ϕθ is the closure of a nilpotent orbit in N , which is parametrized by a partition λ ∈ P(n). Thus we get ϕ(Zw ) = Oλ , which establishes the map Φ : Sn → P(n). It is the Robinson-Schensted correspondence that plays another important role of the theory, which establishes a (combinatorial) bijection between Sn and the set of pairs of standard tableaux of same shape (see [9], for example). Thus we have # ∼ RS : Sn −−−→ {(T1 , T2 ) | Ti ∈ STabλ }, λ∈P(n)

where STabλ denotes the set of standard tableaux of the shape λ. Theorem 2.1 (Steinberg [29]). The Steinberg map Φ : Sn  w → λ ∈ P(n) defined by ϕ(Zw ) = Oλ factors through the Robinson-Schensted correspondence. ∼ / ! {(T1 , T2 ) | Ti ∈ STabλ }  (T1 , T2 ) Sn QQQ RS _ QQQ λ∈P(n) QQQ QQQ QQQ QQQ Φ QQQ QQQ   ( P(n)  λ = shape(Ti ) 2.2. A symmetric pair of type AIII. Let us consider the case of the symmetric pair (G, K) = (GL2n , GLn × GLn ). This case is studied in [8]. We take a Siegel parabolic subgroup P = PS ⊂ GL2n of G, which is the stabilizer of the n-dimensional subspace Cn × {0}n ⊂ C2n , and a Borel subgroup Q = Bn+ × Bn− ⊂ GLn × GLn of K, where Bn− denotes the lower triangular Borel subgroup of GLn . Thus the double flag variety X becomes X = GLn /Bn+ × GLn /Bn− × Grn (C2n ), where Grn (C2n ) stands for the Grassmann variety of n-dimensional subspaces in C2n . In [8, Theorem 8.1], the K-orbits on X are completely classified. As a result, there are only finitely many K-orbits in X. Let us review the classification briefly. A partial permutation τ on the set [n] = {1, 2, . . . , n} is an injective map from a subset J ⊂ [n] to [n]. It is convenient to extend τ : J → [n] to τ : [n] → [n] ∪ {0} by putting τ (k) = 0 for k ∈ J. As in the case of permutation, we can associate a matrix in Mn with a partial permutation τ , which we also denote by τ by abuse of notation. Namely the matrix τ is given by (eτ (1) , eτ (2) , . . . , eτ (n) ), where e0 = 0 and e1 , . . . , en denote the elementary basis vectors of Cn . Let us denote by Tn the set of all partial permutation matrices and put     τ  (Tn2 )◦ = ω = τ1  τ1 , τ2 ∈ Tn and rank ω = n ⊂ M2n,n . 2

Then the image [ω] := Im ω generated by the column vectors of ω is an ndimensional vector space, hence it represents a point in Grn (C2n ). Notice that the symmetric group Sn acts on (Tn2 )◦ by the right multiplication (or permutation

ORBIT EMBEDDING FOR STEINBERG MAPS

29

of the column vectors) and this action does not change the corresponding subspace [ω]. Let us denote the parabolic subgroup of G stabilizing [ω] by Pω . Theorem 2.2 ([8, Theorem 8.1]). There exist natural bijections X/K  Grn (C2n )/(Bn+ × Bn− )  (Tn2 )◦ /Sn . The bijections are explicitly given by K · (Bn+ × Bn− , Pω ) ↔ (Bn+ × Bn− ) · [ω] ↔ ωSn . In this setting Assumption 1.3 is satisfied (see [20, Table 3] and [7, Proposition 4.2]). Thus by the machinery which we have already described, we get the Steinberg maps Φθ : (Tn2 )◦ /Sn → N θ /K and Φ−θ : (Tn2 )◦ /Sn → N −θ /K. Since K = GLn × GLn , the nilpotent variety N θ is the product of two copies of the cone of nilpotent matrices of size n. Thus the nilpotent K-orbits in N θ are classified by the pairs of partitions: N θ /K  P(n)2 . The nilpotent orbits in N −θ are classified by the signed Young diagrams of size 2n with signature (n, n) (see [4], or [30]). We denote this set of signed Young diagrams by YD± (n, n). Thus we have the generalized and exotic Steinberg maps Φθ : (Tn2 )◦ /Sn → P(n)2

and

Φ−θ : (Tn2 )◦ /Sn → YD± (n, n),

both of which are combinatorial. Theorem 2.3. There exist efficient combinatorial algorithms which describe the Steinberg maps Φ±θ .   τ Choose ω = τ1 ∈ (Tn2 )◦ . If τ1 or τ2 is a permutation, then up to the action 2

of Sn , we can assume that τ1 or τ2 is equal to 1n (the identity matrix). In these special cases, the above theorem is already proved in Theorems 9.1 and 10.4 in [8]. We will explain the algorithms below but the full proof of the above theorem will appear elsewhere. 2.3. Generalizations of the Robinson-Schensted correspondence to type AIII. To explain the “efficient combinatorial algorithm” in Theorem 2.3, we need some preparation. We choose a partition of [n] = {1, . . . , n} into three disjoint subsets J, M, M  of sizes r = #J, p = #M, q = #M  so that r + p + q = n. Choose another partition into subsets of the same sizes denoted by I, L, L . Let σ : J → I be a bijection and write J = {j1 < · · · < jr } and M = {m1 < · · · < mp }. The same notation applies to the rest of the subsets M  , I, L, L too. Then we put   ··· jr m1 · · · mp m1 · · · mq j1 , τ1 = σ(j1 ) · · · σ(jr ) 0 · · · 0 1 · · · q   j · · · jr m1 · · · mp m1 · · · mq , τ2 = 1 j1 · · · jr m1 · · · mp 0 · · · 0   τ and consider ω = τ1 ∈ (Tn2 )◦ . It is straightforward to check the following lemma. 2

Lemma 2.4. For all the choices of numbers r, p, q, partitions [n] = J M M  = ∼ I  L  L , and bijection σ : J −−→ I as above, {ω} is a complete system of representatives for the quotient space (Tn2 )◦ /Sn .  2    n n n! . , = Corollary 2.5. #X/K = n=r+p+q r! r! p! q! r, p, q r, p, q

30

LUCAS FRESSE AND KYO NISHIYAMA

Let us return to the notation above, and consider ω ∈ (Tn2 )◦ . Since σ : J → I is a bijection, we can apply the classical Robinson-Schensted(-Knuth) algorithm (see [9]) to σ and get a pair of Young tableaux (T1 , T2 ) = (RS1 (σ), RS2 (σ)) of the same shape, where I is the set of entries of T1 and J is that of T2 . There is an algorithm called “rectification” of two Young tableaux T and U denoted as Rect(T ∗ U ) (see [9]). We can rectify more than two tableaux at the same time, so we can write Rect(T ∗ U ∗ V ), for example. For a subset A ⊂ [n], let [A] be the unique vertical Young tableau of shape (1#A ) with entries in A. Theorem 2.6. Let ω ∈ (Tn2 )◦ be a representative specified in Lemma 2.4. Put

  T$1 = Rect([L ] ∗ t RS1 (σ) ∗ [L]) and T$2 = Rect([M  ] ∗ t RS2 (σ) ∗ [M ]), where t T denotes the transpose of the tableau T . Let λ = shape(T$1 ) and μ = shape(T$2 ). Then the image of the generalized Steinberg map is given by Φθ (ω) = (λ, μ).   τ Remark 2.7. In [8], we gave an algorithm for ω = 11 . The formula given n

here is slightly different from it since one of the Borel subgroups is twisted to the lower triangular one. Note that the above theorem clarifies the fiber of Φθ and gives a different parametrization of X/K in terms of standard tableaux. We will describe it shortly. If a pair of partitions λ, ν satisfy νi ≤ λi ≤ νi + 1 for any i ≥ 1, we write ν  λ. This is equivalent to saying that ν ⊂ λ and λ \ ν is a vertical strip. Put Υ(r, p, q) = {(λ, μ; λ , μ ; ν) | λ, μ ∈ P(n), λ , μ ∈ P(r + p), ν ∈ P(r), ν  λ  λ, and ν  μ  μ}. By the construction, it is easy to see that there exists a bijection ∼

gRS : (Tn2 )◦ /Sn −−−→



{(T1 , T2 ; λ , μ ; ν) | (T1 , T2 ) ∈ STabλ ×STabμ }.

n=r+p+q (λ,μ;λ ,μ ;ν)∈Υ(r,p,q)

We call this bijection gRS the generalized Robinson-Schensted correspondence. As before, we get a commutative diagram ∼ / ! {(T$1 , T$2 ; λ , μ ; ν)}  (T$1 , T$2 ; λ , μ ; ν) X/K  (Tn2 )◦ /Sn gRS RRR _ () RRR RRR RRR RRR RRR Φθ RRR RR)   P(n)2  (λ, μ) Here () represents the condition n = r + p + q, (λ, μ; λ , μ ; ν) ∈ Υ(r, p, q), and (T$1 , T$2 ) ∈ STabλ × STabμ . For the formula giving Φ−θ , we refer the readers to [8]. In the quoted paper,   τ we only treated the case where ω = 1 . A complete algorithm will appear n elsewhere. 3. Embedding of the orbits in double flag varieties In this section, we return back to the situation of § 1, and we do not assume the finiteness of the number of K-orbits in X nor Assumption 1.3.

ORBIT EMBEDDING FOR STEINBERG MAPS

31

Thus (G, K) is a symmetric pair and P, Q are parabolic subgroups of G and K respectively. Sometimes a double flag variety X = G/P × K/Q can be embedded into a larger double flag variety. In this respect, we need a more precise setting. Let G be a connected reductive algebraic group over C, and let θ, σ ∈ Aut G be involutions which commute: θσ = σθ. The fixed point subgroup of θ is denoted by K = Gθ so that (G, K) is a symmetric pair. Assume that G is the fixed point subgroup of σ, G = Gσ . Since θ and σ commute, G is stable under θ and K is stable under σ. We put K = Gθ = Kσ = G ∩ K, which is a symmetric subgroup of both G and K. K

⊂

⊃

⊃ K

G

⊂

G

For simplicity, we assume that all the subgroups G, K and K are connected. 3.1. Embedding of double flag varieties. For a parabolic subgroup P of G, we can choose a σ-stable parabolic subgroup P of G which cuts out P from G, i.e., P = Pσ = P ∩ G. Similarly, for a parabolic subgroup Q of K, we can choose a σ-stable parabolic subgroup3 Q of K which satisfies Q = Qσ = Q ∩ K. Denote by X = G/P × K/Q a double flag variety for (G, K). Then X = G/P × K/Q is embedded into X in a natural manner. The embedding X → X sends (gP, kQ) (g ∈ G, k ∈ K) to (gP, kQ), and we do not distinguish X from its embedded image in X. By abuse of notation, let σ also denote the automorphism of X defined by σ(g · P, k · Q) = (σ(g) · P, σ(k) · Q) so that σ(h · x) = σ(h) · σ(x) holds for h ∈ K and x ∈ X. For a σ-stable subgroup H ⊂ G, we will denote H−σ = {h ∈ H | σ(h) = h−1 }. Definition 3.1. For a σ-stable subgroup H of G, we say that H admits (−σ)square roots if for any h ∈ H−σ there exists an f ∈ H−σ which satisfies h = f 2 . Lemma 3.2. If P and Q admit (−σ)-square roots, then X coincides with the fixed point set Xσ := {x ∈ X | σ(x) = x}. Proof. The inclusion X ⊂ Xσ is clear. For the other inclusion, take (gP, kQ) ∈ X . Since (gP, kQ) = σ(gP, kQ) = (σ(g)P, σ(k)Q), we get g −1 σ(g) ∈ P and k−1 σ(k) ∈ Q. This means g −1 σ(g) ∈ P−σ , hence by the assumption there exists an f ∈ P−σ such that g −1 σ(g) = f 2 . From this, we get gf = σ(g)f −1 = σ(gf ). Therefore we conclude that gf ∈ Gσ = G, and gP = (gf )P belongs to G/P ⊂ G/P. Similarly we get kQ = (kh)Q for some h ∈ Q−σ and kh ∈ K. Thus we get (gP, kQ) = ((gf )P, (kh)Q) ∈ X.  σ

3 In most of the literature, the letter Q is reserved to denote the field of rational numbers. However, in this paper, we consider everything over the complex number field C and actually the rational number field does play no rule. For this reason, we choose the letter Q for a parabolic subgroup of K.

32

LUCAS FRESSE AND KYO NISHIYAMA

3.2. Embedding of orbits. Since the embedding X → X is K-equivariant, it induces a natural orbit map defined by 4 ι : X/K 



 O

/ X/K . / O=K·O

There is no reason to expect that ι is an embedding, i.e., to conclude O = O ∩ X. In this setting, there is a criterion which assures that ι is an embedding. We quote Assumption 3.1 from [19] (with terminology re-adapted): “For any x ∈ X, StabK (x) admits (−σ)-square roots.” To translate the condition in the present setting, let x = (gP, kQ) =: (P1 , Q1 ) be a point in X. Note that P1 and Q1 are σ-stable parabolic subgroups of G and K respectively. Then StabK (x) = P1 ∩ Q1 . Thus the above assumption can be rephrased as “P1 ∩ Q1 admits (−σ)-square roots.” Let us summarize the situation: Let G, K = Gθ , G = Gσ , K = Kσ = Gθ be as above and assume they are all connected. Let P and Q be parabolic subgroups of G and K respectively and choose σ-stable parabolic subgroups P ⊂ G and Q ⊂ K such that P = P ∩ G and Q = Q ∩ K. We consider the following two conditions: (A) P and Q admit (−σ)-square roots (see Definition 3.1). (B) For any σ-stable parabolic subgroups P1 ⊂ G and Q1 ⊂ K which are conjugate to P and Q respectively, the intersection P1 ∩ Q1 admits (−σ)square roots. Theorem 3.3. In the above setting, let us consider the double flag varieties X = G/P × K/Q and X = G/P × K/Q. Then the conditions (A) and (B) imply the following. (1) There is a natural embedding X → X defined by (gP, kQ) → (gP, kQ) for any g ∈ G and k ∈ K. The involutive automorphism σ ∈ Aut X given by σ(gP, kQ) = (σ(g)P, σ(k)Q) (g ∈ G, k ∈ K) is well defined and X = Xσ holds. (2) The natural orbit map ι : X/K → X/K defined by ι(O) = K · O for O ∈ X/K is injective, i.e., if we put O = ι(O), then it holds O = O ∩ X = Oσ . (3) For any K-orbit O in X, the intersection O ∩ X is either empty or a single K-orbit. As we have already explained above, this theorem follows from [19, Theorem 3.2]. We make emphasis on the fact that we do not need finiteness of the orbits. If we take Q = K and Q = K, we get the following corollary, which is known to experts on the basis of case-by-case analysis. Corollary 3.4. Let P be a σ-stable parabolic subgroup of G, which admits (−σ)-square roots. Assume that, for any σ-stable parabolic subgroup P1 ⊂ G conjugate to P, the intersection P1 ∩ K admits (−σ)-square roots. Then the natural embedding map G/P → G/P induces an embedding of orbits K\G/P → K\G/P, i.e., for any K-orbit O in G/P, the intersection O ∩ G/P is either empty or a single K-orbit. 4 In the previous sections, we always denote a K-orbit in X by using a black board bold letter like O. However, in this section, we take advantage of systematic notation.

ORBIT EMBEDDING FOR STEINBERG MAPS

33

In this way, the classification of the K-orbits in the partial flag variety G/P reduces to that of K-orbits in G/P together with the determination of the subset of orbits with non-empty intersection with G/P . Note that, in this case, both K\G/P and K\G/P are finite sets. 3.3. Embedding of conormal bundles. We fix a nondegenerate invariant bilinear form on G := Lie (G) which is also invariant under involutions σ and θ. Certainly it always exists and by restrictions it descends to nondegenerate bilinear forms on g, k and K := Lie (K). By this invariant form, we identify G and G∗ , and similar identifications take place for g, k and K. In the following, we will use German capital letters for the Lie algebras of the groups denoted by black board bold letters. Let us consider the cotangent bundles T ∗ X and T ∗ X. Since T ∗ X  (G×P uP )× (K ×Q uQ ) by the identification G  G∗ and K  K∗ , we have a natural extension of σ ∈ Aut X to the whole T ∗ X. We prove Proposition 3.5. Let us assume the condition (A) in § 3.2. Then (T ∗ X)σ = T X holds. ∗

Proof. It is obvious that T ∗ X ⊂ (T ∗ X)σ . Let us prove the reversed inclusion. Take [g, u] ∈ G ×P uP . Then σ([g, u]) = [σ(g), σ(u)] is equal to [g, u] if and only if there exists p ∈ P such that σ(g) = gp and σ(u) = p−1 u. From the first equality, we get p = g −1 σ(g) ∈ P−σ so that there exists a square root f ∈ P−σ of g −1 σ(g). Then [g, u] = [gf, f −1 u] ∈ G ×P up . In fact, σ(gf ) = σ(g)f −1 = gf which implies gf ∈ G. Similarly, σ(f −1 u) = f σ(u) = f p−1 u = f f −2 u = f −1 u, which proves f −1 u ∈ (uP )σ = up . In the same manner, we conclude (K ×Q uQ )σ = K ×Q uq , which proves the proposition.  Let us denote the moment maps by μX : T ∗ X → K∗  K, μX : T ∗ X → k∗  k.  −1 Clearly μX commutes with σ and μX T ∗ X = μX . Let ZX = μ−1 X (0) and ZX = μX (0) 5 be the corresponding conormal varieties . (3.1)

Theorem 3.6. Assume both conditions (A) and (B) in § 3.2. (1) It holds that (ZX )σ = ZX . (2) By Theorem 3.3, the orbit map ι : X/K → X/K is injective. For O ∈ X/K, put O = K · O = ι(O). Then we have (TO∗ X)σ = TO∗ X. −1 −1 ∗ ∗ Proof. (1) (ZX )σ = (T ∗ X)σ ∩ μ−1 X (0) = T X ∩ μX (0) = T X ∩ μX (0) = ZX . σ (2) Since O = O, it suffices to check the equality for the fiber. Take x ∈ O and write x = (P1 , Q1 ) = (P1 , Q1 ) (as a point in G/P ×K/Q embedded into G/P×K/Q).

⊥ Then the fiber of the conormal bundle at x is given by (TO∗ X)x = (P1 , Q1 )+ΔK , where ΔK denotes the diagonal embedding and ⊥ refers to the orthogonal in the dual space. Since all the subspaces which are relevant are σ-stable, we conclude ⊥

⊥ (TO∗ X)σx = (Pσ1 , Qσ1 ) + ΔKσ = (p1 , q1 ) + Δk = (TO∗ X)x .  5 We defined the conormal variety in § 1 in a different way, but they coincide. See Fact 1.2(1) and also [8].

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LUCAS FRESSE AND KYO NISHIYAMA

3.4. Compatibility with Steinberg maps. In this section we assume both conditions (A) and (B), so that Theorem 3.3 provides us with an orbit embedding ι : X/K → X/K. Recall that there are two types of nilpotent cones N ±θ in k = gθ and s = g−θ respectively. The symmetric subgroup K acts on both nilpotent varieties with finitely many orbits. Similarly, we denote the nilpotent variety for G = Lie (G) by N = NG and put Nθ = N ∩ K and N−θ = N ∩ G−θ . The symmetric subgroup K acts on N±θ with finitely many orbits. Clearly N ±θ are fixed point varieties of N±θ by the involution σ. Therefore we have natural orbit maps ι±θ : N ±θ /K → N±θ /K. Namely, for an orbit O ∈ N ±θ /K, the respective map is defined by letting ι±θ (O) = K · O. The maps ι±θ are not injective in general. Assumption 1.3 clearly holds for X whenever it holds for X, and in this case the construction of Section 1 yields the Steinberg maps Φ±θ : X/K → N ±θ /K relative to X and Φ I ±θ : X/K → N±θ /K relative to X. Then it is natural to make the following Conjecture 3.7. The diagram below commutes. In other words, the orbit embedding ι : X/K → X/K is compatible with the Steinberg maps Φ±θ and Φ I ±θ . X/K _ (3.2)

Φ±θ

ι

 X/K

Φ I

±θ

/ N ±θ /K 

ι±θ

/ N±θ /K

The functoriality of the constructions is not sufficient to hold Conjecture 3.7 affirmatively. Let O ⊂ X be a K-orbit and let O := ι(O) be the corresponding K-orbit in X. By definition of the maps Φ I ±θ , there is a dense open subset U of the conormal bundle TO∗ X such that u∈U

=⇒

I θ (O) (uθ ∈ Φ

and u−θ ∈ Φ I −θ (O)).

We may assume that U ⊂ TO∗ X is K-stable and σ-stable. Note that TO∗ X = (TO∗ X)σ by Theorem 3.6. Lemma 3.8. The equality Φ I ±θ (ι(O)) = ι±θ (Φ±θ (O)) holds (and hence the diagram (3.2) commutes) if U intersects TO∗ X. However, in general, it seems difficult to check the condition for U in the above lemma. In the next section, we will discuss an explicit embedding of type CI into type AIII (see (4.1) below). Yet it is still difficult to prove the commutativity of the Steinberg maps in full generality. Direct calculations tell us that up to n = 3 there always exist such U in Lemma 3.8, so that the commutativity holds. 4. Steinberg theory for type CI embedded into type AIII In this section, we apply the theory of embedding to the case of type CI inside type AIII. This section serves as a meaningful example of the embedding theory given in § 3.

ORBIT EMBEDDING FOR STEINBERG MAPS

35

Let us begin with G = GL2n and two commuting involutions σ, θ ∈ Aut G defined by −1 θ(g) = In,n g In,n

In,n = diag(1n , −1n ),   −1n . Jn = 1n

σ(g) = Jn−1 t g −1 Jn Then (4.1)

K = Gθ = GLn × GLn ,

G = Gσ = Sp2n ,

K = Gθ  GLn

are all connected. We use Jn to define a symplectic form on V = C2n by (u, v) = t uJn v. Take Q = B = Bn+ in K = GLn as a Borel subgroup and let P = PS be the Siegel parabolic subgroup of G = Sp2n which stabilizes the Lagrangian subspace V + = e1 , e2 , . . . , en ⊂ V , where ek denotes the k-th elementary basis vector. Similarly we put V − = en+1 , en+2 , . . . , e2n , which is also Lagrangian, and V = V + ⊕ V − is a polarization stable under K. We take BK = Bn+ × Bn− as a σ-stable Borel subgroup of K and put P = StabG (V + ) the stabilizer of V + in G = GL2n . It is easy to see that P is a σ-stable parabolic subgroup. Thus our double flag varieties are (4.2)

X = G/P × K/B = Sp2n /PS × GLn /Bn+  LGr(C2n ) × Fn ,

and (4.3)

X = G/P × K/BK = GL2n /P(n,n) × GLn /Bn+ × GLn /Bn−  Grn (C2n ) × Fn × Fn ,

where LGr(V ) denotes the Lagrangian Grassmannian, the variety of all the Lagrangian subspaces in the symplectic vector space V = C2n with the symplectic form defined by Jn , and Fn is the variety of complete flags in Cn . We use the notation P(n,n) for the parabolic subgroup in GL2n determined by the partition (n, n) of 2n. 4.1. Assumptions (A) and (B). In this setting, Assumptions (A) and (B) in § 3.2 are satisfied as we will see below. So the whole theory of embedding works well. Note that in this case X has finitely many K-orbits (see [8]), hence we conclude that X has also finitely many K-orbits, although the finiteness of orbits is already known by a general classification theory [10]. First, let us check the condition (A). Lemma 4.1. In the setting above, BK ⊂ K and P ⊂ G admit (−σ)-square roots. Proof. We follow the strategy of Ohta [24] (the proof of Theorem 1 there). Take g ∈ P−σ . Then σ(g) = Jn−1 t g −1 Jn = g −1 and thus we get Jn−1 t gJn = g. The expression of the left-hand side can be extended into a mapping on the whole matrix algebra M2n , which we denote by τ temporarily. So we put τ (A) = Jn−1 t AJn for A ∈ M2n , which is an anti-automorphism of M2n . For the above chosen g ∈ P−σ , a standard argument of linear algebra produces a polynomial f (T ) ∈ C[T ] such that f (g)2 = g. Since g is invertible, f (g) is also invertible and f (g) ∈ GL2n = G. Since τ (f (g)) = f (τ (g)) = f (g), this means f (g) ∈ G−σ . Note that g ∈ P stabilizes V + . This forces f (g) to stabilize V + also, hence f (g) ∈ P−σ . This proves that P admits (−σ)-square roots.

36

LUCAS FRESSE AND KYO NISHIYAMA

Using the fact that BK is the stabilizer in K of a complete flag in V ± , we can prove that BK admits (−σ)-square roots in the same way. However, note that K is the stabilizer of the polar decomposition V + ⊕ V − .  Remark 4.2. The same argument proves that any σ-stable parabolic subgroup in G or K admits (−σ)-square roots (cf. Corollary 3.4). To prove the condition (B), we will consider P1 ∩ (BK )1 for an arbitrary σstable parabolic subgroup P1 ⊂ G and a Borel subgroup (BK )1 ⊂ K. An element g ∈ P1 ∩ (BK )1 can be characterized by the property that it stabilizes various subspaces (and also polar decomposition) of V . Thus the literary same arguments in Lemma 4.1 can apply, which proves (B). The conditions (A) and (B) imply the following theorem. Theorem 4.3. Let X and X be the double flag varieties defined in (4.2) and (4.3) respectively. Then the orbit map X/K → X/K is injective, i.e., for any K-orbit O in X, the intersection X ∩ O is either empty or a single K-orbit. 4.2. Explicit embedding of orbits in double flag varieties. Since we get an abstract embedding theorem for orbits, we can use it to classify the orbits in the double flag variety X explicitly in terms of the parametrization of the orbits in X given in Theorem 2.2. First, we clarify the explicit embedding map X → X and the involutive automorphism on X. Recall the various identifications (see § 2.2): X/K  Bn+ × Bn− \GL2n /P(n,n)  Bn+ × Bn− \ Grn (C2n )

(4.4)

 Bn+ × Bn− \M◦2n,n /GLn  (Tn2 )◦ /Sn ,

where M◦2n,n denotes the set of 2n by n matrices of full rank (i.e., rank n). Take   τ ω = τ1 ∈ (Tn2 )◦ as a representative of a K-orbit O, and denote O = Oω . Then 2

Oω is generated by the point (BK , gP), where τ ξ  g = 1 1 ∈ GL2n = G τ2 −ξ2 for some ξ1 , ξ2 ∈ Mn (we put minus sign in front of ξ2 for later convenience). In fact, it is easy to see gP · V + = gV + = [ω], where [ω] denotes the image of the matrix ω, i.e., the n-dimensional subspace generated by the column vectors of ω. We need a lemma. τ   ξ  ξ Lemma 4.4. We can choose g = 1 1 in such a way that ξ1 ∈ (Tn2 )◦ τ2 −ξ2 2 and t ξ1 τ1 − t ξ2 τ2 = 0. Proof. Based on the general description of τ1 and τ2 given in Section 2.3, we can find permutation matrices s1 , s2 , s ∈ Sn such that 1 0 0  1 0 0 r r s1 τ 1 s = 0 0 0 and s2 τ2 s = 0 1p 0 . 0 0 1q 0 0 0

ORBIT EMBEDDING FOR STEINBERG MAPS

Then, choosing ξ1 and ξ2 so  1r s1 ξ1 = 0 0

37

that 0 0 1p 0 0 0



 and

s2 ξ 2 =

 1r 0 0 0 0 0 , 0 0 1q

we get 0 = t (s1 ξ1 )(s1 τ1 s) − t (s2 ξ2 )(s2 τ2 s) = ( t ξ1 τ1 − t ξ2 τ2 )s, whence t ξ1 τ1 − t ξ2 τ2 = 0 as desired.



We define an involution on (Tn2 )◦ /Sn  X/K by σ(Oω ) = Oσ(ω) (ω ∈ (Tn2 )◦ ), which is denoted by the same letter σ by abuse of notation. Note that σ(ω) is determined only modulo the right multiplication by Sn .   τ ξ  τ chosen in Lemma 4.4 and ω = τ1 ∈ Proposition 4.5. For g = 1 1 τ2 −ξ2 2 (Tn2 )◦ , let Oω be the K-orbit in X through the point (BK , gP). Then σ(Oω ) = Oσ(ω) is given by     τ ξ σ(ω) = σ τ1 = ξ2 . 2 1

 Proof. We have σ (BK , gP) = (BK , σ(g)P) and σ(g) = Jn−1 t g −1 Jn . Let us compute t g −1 . We get d 0   tτ t τ2  τ1 ξ1   t τ1 τ1 + t τ2 τ2 t τ1 ξ1 − t τ2 ξ2  1 t = =: , gg = t 1 t 0 d2 ξ1 − t ξ2 τ2 −ξ2 ξ1 τ1 − t ξ2 τ2 t ξ1 ξ1 + t ξ2 ξ2 by the property t ξ1 τ1 − t ξ2 τ2 = 0. An easy calculation tells that d1 = t τ1 τ1 + t τ2 τ2 is a diagonal matrix with diagonal entries 1 or 2, and so is d2 = t ξ1 ξ1 + t ξ2 ξ2 . Thus we get t g −1 = gd−1 with d = diag(d1 , d2 ). From this, we compute   ξ τ −1 σ(g) = Jn−1 t g −1 Jn = Jn−1 gd−1 Jn = 2 2 · diag(−d−1 2 , −d1 ), ξ1 −τ1       ξ τ τ ξ = ξ2 .  and get σ(g)P = 2 2 P. This implies σ τ1 ξ1 −τ1 2 1   τ Theorem 4.6. For ω = τ1 ∈ (Tn2 )◦ /Sn , let Oω be the corresponding K-orbit 2

in X. Then the following (1)–(4) are all equivalent. (1) Oω ∩ X = ∅ (and consequently it is a single K-orbit); (2) σ(Oω ) = Oω , i.e., the K-orbit is σ-stable; (3) t τ1 τ2 ∈ Symn ; (4) t τ2 τ1 ∈ Symn . In particular, the set of K-orbits in the double flag variety X = Sp2n /PS ×GLn /Bn+ of type CI is parametrized by (Cn2 )◦ /Sn , where    % & τ (Cn2 )◦ := ω = τ1 ∈ (Tn2 )◦  t τ1 τ2 = t τ2 τ1 ∈ Symn . 2 As in the case of X/K (see (4.4)), there are natural bijections (4.5)

X/K  (LGr(C2n ) × Fn )/GLn  Bn+ \Sp2n /PS ◦  Bn+ \ LGr(C2n )  Bn+ \S2n,n /GLn ,

38

LUCAS FRESSE AND KYO NISHIYAMA

where (4.6)

◦ S2n,n = {A ∈ M◦2n,n | t AJn A = 0}   A = {A = A1 ∈ M◦2n,n | t A1 A2 = t A2 A1 ∈ Symn }. 2

Note that the actions of b ∈ Bn+ are all defined by the left multiplications by   b 0 . The above theorem tells that all these coset spaces are in bijection 0 t b−1 with (Cn2 )◦ /Sn . We get an interesting corollary as a byproduct.   τ Corollary 4.7. For ω = τ1 ∈ (Tn2 )◦ , the following are all equivalent. 2

(Cn2 )◦ ,

i.e., τ1 τ2 = t τ2 τ1 ∈ Symn holds; (1) ω ∈ t (2) τ1 bτ2 ∈ Symn for some b ∈ Bn− ; (3) t τ1 b τ2 ∈ Symn for some s ∈ Sn and b ∈ sBn− s−1 . (4) t τ1 b τ2 ∈ Symn for any s ∈ Sn and some b ∈ sBn− s−1 .     sτ τ Proof. Obviously τ1 ∈ (Cn2 )◦ if and only if sτ1 ∈ (Cn2 )◦ for some s ∈ Sn t

2

2

if and only if it is so for any s ∈ Sn . Therefore it suffices to prove the equivalence of (1) and (2). Since the implication (1) =⇒ (2) is clear, let us prove that (2)   t  τ bτ1 implies (1). Note that ω = τ1 and = diag( t b, 1) · ω are in the same orbit τ2 2 Oω . Thus we get   t bτ1 is Lagrangian =⇒ Oω ∩ X = ∅ ⇐⇒ (1). (2) ⇐⇒ τ 2

 4.3. Embedding of nilpotent orbits. The embedding of nilpotent orbits in type CI into those of type AIII is well studied. Let us make a quick summary of the known facts. We use the general notation for nilpotent cones introduced in § 3.4. In particular we consider the nilpotent cones N ±θ ⊂ g±θ of the Lie algebra g = Lie (Sp2n ). Note that k := gθ identifies with Lie (GLn ) embedded inside g. The symmetric subgroup K = GLn acts on N ±θ with finitely many orbits. Similarly, the two nilpotent cones for G = Lie (GL2n ) are denoted by Nθ = N∩K and N−θ = N ∩ G−θ . The symmetric subgroup K = GLn × GLn acts on N±θ . Finally recall the natural orbit maps ι±θ : N ±θ /K → N±θ /K, O → K · O. Theorem 4.8 (Ohta [22, 23]). In the above setting, the orbit maps ι±θ are injective and respect the closure ordering. In particular, for any nilpotent K-orbit O ∈ N±θ /K, the intersection O∩N ±θ is either empty or a single nilpotent K-orbit. Moreover, if O is σ-stable, then O ∩ N ±θ is nonempty. Note that the above theorem does not imply that O ∩ N ±θ is irreducible. However, if O ∩ N ±θ = ∅, then O ∩ N ±θ = O ∩ N ±θ is irreducible, and in fact it is the closure of a nilpotent K-orbit. Let us give more explicit information on the embedding of nilpotent orbits together with parametrization.

ORBIT EMBEDDING FOR STEINBERG MAPS

39

Since Nθ is a direct product of the nilpotent varieties of gln , nilpotent K-orbits in Nθ are classified by pairs of partitions of size n. Thus we identify Nθ /K  P(n)2 , and we write O(λ,μ) (λ, μ  n) for the corresponding nilpotent K-orbit. Lemma 4.9. A nilpotent K-orbit O(λ,μ) (λ, μ  n) is σ-stable if and only if λ = μ. In that case, the intersection is O(λ,λ) ∩ N θ = Oλ , which is the nilpotent K-orbit with the Jordan normal form corresponding to λ. Proof. It is easy to see that σ(O(λ,μ) ) = O(μ,λ) by using the explicit form of  the involution σ. If λ = μ, the element diag(x, − t x) ∈ O(λ,λ) belongs to Oλ . Next we consider the nilpotent orbits in N−θ and N −θ . The nilpotent K-orbits in N−θ are parametrized by signed Young diagrams of signature (n, n). We denote by YD± (n, n) the set of all such signed Young diagrams. An element Λ ∈ YD± (n, n) is a Young diagram of size 2n and each box is filled in by either plus or minus sign. In each row, (±)-signs are arranged alternatively. The total number of plus signs is n and that of minus is also n. Moreover, the signed diagram is defined up to permutation of its rows. We denote by OΛ the nilpotent K-orbit corresponding to Λ ∈ YD± (n, n). For example, the set YD± (2, 2) consists of +−+−

−+−+

+−+ −

−+− +

+− +−

+− −+

−+ −+

−+ + −

+− + −

+ + − −

and there are 10 orbits. On the other hand, nilpotent orbits in N −θ for type CI are parametrized by a subset of signed Young diagrams in YD± (n, n) with the property that +−+− · · · −+ () odd rows appear in pairs and their signature must be −+−+ · · · +− We denote the set of such signed Young diagrams by YDCI ± (n, n). For example, YDCI ± (2, 2) consists of +−+−

−+−+

+− +−

+− −+

−+ −+

+− + −

−+ + −

+ + − −

so there are 8 orbits in total. We will denote the nilpotent K-orbit corresponding to Λ ∈ YDCI ± (n, n) by OΛ . Lemma 4.10. A nilpotent K-orbit OΛ for Λ ∈ YD± (n, n) is σ-stable if and only −θ if Λ belongs to YDCI = OΛ . ± (n, n). In that case, OΛ ∩ N Proof. In fact, for any Λ ∈ YD± (n, n), the involution takes OΛ to Oσ(Λ) , where σ(Λ) is obtained from Λ by exchanging plus and minus signs in the rows of odd length. This follows from the consideration below. Let us take a representative  0 z x= ∈ OΛ . Then the shape of Λ is given by the sizes of the Jordan cells in w 0 the Jordan normal form of x. The signatures are controlled by rank(zw)k rank(wz)k

and

rank(zw)k z rank(wz)k w

for k ≥ 1

(see [4] and [8, § 10.1] for details), for instance rank(wz)k − rank(zw)k = #{odd rows of length < 2k starting with “+”} − #{odd rows of length < 2k starting with “−”}.

40

LUCAS FRESSE AND KYO NISHIYAMA

By a direct calculation, we get σ(x) = Jn−1 (− t x)Jn =



0 tz  . w 0

t

Hence the signature of σ(Λ) is controlled by rank( t z t w)k = rank(wz)k rank( t w t z)k = rank(zw)k and rank( t z t w)k t z = rank z(wz)k = rank(zw)k z rank( t w t z)k t w = rank w(zw)k = rank(wz)k w. Since the roles of (wz)k and (zw)k are exchanged, it affects the signature in the odd rows, and in fact, it exchanges the signature of plus and minus. Thus, if OΛ is σ-stable, the odd rows must appear in pair and their signatures are exchanged. Thus we have odd rows of the form appearing in (). The rest of the statements are clear.  Theorem 4.8 gives the orbit embeddings for nilpotent orbits ι±θ : N ±θ /K → N /K in addition to the orbit embedding for orbits in double flag varieties ι : X/K → X/K proved in Theorem 4.3. However, in spite of the explicit description of all these orbit embeddings, the commutativity of the diagram (3.2) is still open. ±θ

Acknowledgments We thank the organizers of the conference held in Dubrovnik, Croatia, June 24–29, 2019. This conference offered the authors a good opportunity to review their previous joint works seriously and it leads to new results reported here. Also we thank Anthony Henderson for the correspondence and Takuya Ohta for discussions on orbit embeddings. References [1] Pramod N. Achar and Anthony Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (2008), no. 1, 27–62, DOI 10.1016/j.aim.2008.04.008. MR2435419 [2] Alexandre Be˘ılinson and Joseph Bernstein, Localisation de g-modules (French, with English summary), C. R. Acad. Sci. Paris S´er. I Math. 292 (1981), no. 1, 15–18. MR610137 [3] Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkh¨ auser Boston, Inc., Boston, MA, 1997. MR1433132 [4] David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR1251060 [5] J. Matthew Douglass and Gerhard R¨ ohrle, The Steinberg variety and representations of reductive groups, J. Algebra 321 (2009), no. 11, 3158–3196, DOI 10.1016/j.jalgebra.2008.10.027. MR2510045 [6] Michael Finkelberg, Victor Ginzburg, and Roman Travkin, Mirabolic affine Grassmannian and character sheaves, Selecta Math. (N.S.) 14 (2009), no. 3-4, 607–628, DOI 10.1007/s00029009-0509-x. MR2511193 [7] Lucas Fresse and Kyo Nishiyama, On the exotic Grassmannian and its nilpotent variety, Represent. Theory 20 (2016), 451–481, DOI 10.1090/ert/489. [Paging previously given as 1–31]. MR3576071 , A Generalization of Steinberg Theory and an Exotic Moment Map, International [8] Mathematics Research Notices (2020), rnaa080 (arXiv preprint arXiv:1904.13156).

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[9] William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR1464693 [10] Xuhua He, Kyo Nishiyama, Hiroyuki Ochiai, and Yoshiki Oshima, On orbits in double flag varieties for symmetric pairs, Transform. Groups 18 (2013), no. 4, 1091–1136, DOI 10.1007/s00031-013-9243-8. MR3127988 [11] Anthony Henderson and Peter E. Trapa, The exotic Robinson-Schensted correspondence, J. Algebra 370 (2012), 32–45, DOI 10.1016/j.jalgebra.2012.06.029. MR2966826 [12] Syu Kato, An exotic Deligne-Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009), no. 2, 305–371, DOI 10.1215/00127094-2009-028. MR2524498 [13] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184, DOI 10.1007/BF01390031. MR560412 [14] George Lusztig and David A. Vogan Jr., Singularities of closures of K-orbits on flag manifolds, Invent. Math. 71 (1983), no. 2, 365–379, DOI 10.1007/BF01389103. MR689649 [15] Peter Magyar, Jerzy Weyman, and Andrei Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 141 (1999), no. 1, 97–118, DOI 10.1006/aima.1998.1776. MR1667147 [16] Peter Magyar, Jerzy Weyman, and Andrei Zelevinsky, Symplectic multiple flag varieties of finite type, J. Algebra 230 (2000), no. 1, 245–265, DOI 10.1006/jabr.2000.8313. MR1774766 [17] William M. McGovern, Cells of Harish-Chandra modules for real classical groups, Amer. J. Math. 120 (1998), no. 1, 211–228. MR1600284 [18] Dragan Miliˇ ci´ c, Localization and Representation Theory of Reductive Lie Groups, available at http://www.math.utah.edu/~milicic/Eprints/book.pdf, 1993. [19] Kyo Nishiyama, Enhanced orbit embedding, Comment. Math. Univ. St. Pauli 63 (2014), no. 1-2, 223–232. MR3328431 [20] Kyo Nishiyama and Hiroyuki Ochiai, Double flag varieties for a symmetric pair and finiteness of orbits, J. Lie Theory 21 (2011), no. 1, 79–99. MR2797821 [21] Kyo Nishiyama, Peter Trapa, and Akihito Wachi, Codimension one connectedness of the graph of associated varieties, Tohoku Math. J. (2) 68 (2016), no. 2, 199–239. MR3514699 [22] Takuya Ohta, The singularities of the closures of nilpotent orbits in certain symmetric pairs, Tohoku Math. J. (2) 38 (1986), no. 3, 441–468, DOI 10.2748/tmj/1178228456. MR854462 [23] Takuya Ohta, The closures of nilpotent orbits in the classical symmetric pairs and their singularities, Tohoku Math. J. (2) 43 (1991), no. 2, 161–211, DOI 10.2748/tmj/1178227492. MR1104427 [24] Takuya Ohta, An inclusion between sets of orbits and surjectivity of the restriction map of rings of invariants, Hokkaido Math. J. 37 (2008), no. 3, 437–454, DOI 10.14492/hokmj/1253539529. MR2441911 [25] R. W. Richardson and T. A. Springer, Combinatorics and geometry of K-orbits on the flag manifold, Linear algebraic groups and their representations (Los Angeles, CA, 1992), Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 109–142, DOI 10.1090/conm/153/01309. MR1247501 [26] T. A. Springer, The unipotent variety of a semi-simple group, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, pp. 373– 391. MR0263830 [27] T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293, DOI 10.1007/BF01403165. MR491988 [28] Robert Steinberg, On the desingularization of the unipotent variety, Invent. Math. 36 (1976), 209–224, DOI 10.1007/BF01390010. MR430094 [29] Robert Steinberg, An occurrence of the Robinson-Schensted correspondence, J. Algebra 113 (1988), no. 2, 523–528, DOI 10.1016/0021-8693(88)90177-9. MR929778 [30] Peter E. Trapa, Richardson orbits for real classical groups, J. Algebra 286 (2005), no. 2, 361–385, DOI 10.1016/j.jalgebra.2003.07.027. MR2128022 [31] Roman Travkin, Mirabolic Robinson-Schensted-Knuth correspondence, Selecta Math. (N.S.) 14 (2009), no. 3-4, 727–758, DOI 10.1007/s00029-009-0508-y. MR2511197

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LUCAS FRESSE AND KYO NISHIYAMA

´ Universit´ e de Lorraine, CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Van`s-Nancy, F-54506, France doeuvre-le Email address: [email protected] Department of Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1, Chuo, Sagamihara 229-8558, Japan Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15452

Symplectic Dirac cohomology and lifting of characters to metaplectic groups Jing-Song Huang Abstract. We formulate the transfer factor of character lifting from orthogonal groups to symplectic groups by Adams in the framework of symplectic Dirac cohomology for the Lie superalgebras and the Rittenberg-Scheunert correspondence of representations of the Lie superalgebra osp(1|2n) and the Lie algebra o(2n + 1). This leads to formulation of a direct lifting of characters from the linear symplectic group Sp(2n, R) to its nonlinear covering metaplectic group M p(2n, R).

1. Introduction A lifting of characters on orthogonal groups and nonlinear metaplectic groups over real numbers was defined and studied by Jeff Adams [Ada98], and it was extended to the p-adic case by Tatiana Howard [How10]. This lifting is closely related to the endoscopic transfer in Langlands functoriality and the theta correspondence in Howe’s reductive dual pairs and also appeared in the work of David Renard [Ren99] and Wen-Wei Li [Li19]. Dirac operators were used for geometric construction of discrete series representations by Parthasarathy [Par72], Atiyah and Schmid [AS], and tempered representations by Wolf [Wol74]. In the late 1990’s, Vogan [V] made a conjecture on the algebraic property of the Dirac operators in the Lie algebra setting. This conjecture was proved by Pandzic and myself in 2002 [HP02]. This led us to study Dirac cohomology of Harish-Chandra modules [HP06] and it became a very useful tool in representation theory. Kostant extended Vogan’s conjecture to the more general setting of the cubic Dirac operator [Kos03]. In the formulation of central problems in Langlands program stable conjugacy plays a pivotal role. The theory of endoscopy investigates the difference between orbital integrals over ordinary and stable conjugacy classes. As Dirac cohomology of a Harish-Chandra module determines its K-character [HPZ13], it corresponds to the dual object of the orbital integrals on elliptic elements. By using Dirac cohomology, the transfer factor is expressed as the difference of characters for the 2020 Mathematics Subject Classification. Primary 22E47; Secondary 22E46. Key words and phrases. Lie superalgebra, symplectic Dirac cohomology, character lifting. The research described in this paper was supported by grants No. 16303218 from Research Grant Council of HKSAR. c 2021 American Mathematical Society

43

44

JING-SONG HUANG

even and odd parts of the spin module, or Dirac index of the trivial representation [Hua15]. The transfer factor of the Adams lifting is the difference of two irreducible components of the oscillator representation. This transfer factor is equal to the symplectic Dirac index of the trivial representation for the Lie superalgebra osp(1|2n). This is analogous to the fact that the endoscopic transfer factor as the difference of characters for the even and odd parts of the spin module is equal to Dirac index of the trivial representation [Hua15]. The corresponding Vogan’s conjecture for symplectic Dirac operators of Lie superalgebras was established by Pandzic and myself in 2005 [HP05]. Let M p(2n, R) denote the nonlinear two-fold covering group of the symplectic group Sp(2n, R). By formulating the Adams lifting in the framework of symplectic Dirac cohomology, we obtain direct lifting of characters from the linear group Sp(2n, R) to the nonlinear group M p(2n, R). This lifting gives a bijection between the discrete series of Sp(2n, R) and the genuine discrete series of M p(2n, R). 2. Preliminaries on symplectic Dirac cohomology for Lie superalgebras Let g = g0 ⊕ g1 be a Lie superalgebra. If g0 is reductive, then the adjoint action of g0 on g1 is completely reducible. In this case, g is called a classical Lie superalgebra. If a classical Lie superalgebra g is also of Riemannian type, i.e., it has nondegenerate supersymmetric invariant bilinear form, then g is called a basic classical Lie superalgebra. Kac classified all simple Lie superalgebras [Kac77]. Besides ordinary Lie algebras the simple basic classical Lie superalgebras list as A(m, n), B(m, n), C(n), D(m, n), D(2, 1, α), F (4) and G(3) in [Kac77]. Now we fix a classical complex Lie superalgebra g = g 0 ⊕ g1 with bracket [· , ·]. We assume that g is of Riemannian type (namely g is a basic classical Lie superalgebra), i.e., there exists a nondegenerate supersymmetric invariant bilinear form B on g. Supersymmetry means that B is symmetric on g0 and skew-symmetric on g1 , and g0 and g1 are orthogonal. Invariance means that B([X, Y ], Z) = B(X, [Y, Z]), for all X, Y, Z ∈ g. We fix an orthonormal basis Wk for g0 with respect to B. Furthermore, we fix a pair of complementary Lagrangean subspaces of g1 , and bases ∂i , xi in them, such that 1 B(∂i , xj ) = δij . 2 This notation is chosen so that ∂i and xi are generators for the Weyl algebra W (g1 ), with only nontrivial commutation relations being [∂i , xi ]W (g1 ) = 1. We put the subscript W (g1 ) to the commutators in W (g1 ) to distinguish them from the (totally different) bracket in g. We see that W (g1 ) gets identified with the algebra of differential operators with polynomial coefficients in the xi ’s, with ∂ ∂i corresponding to ∂x . i Note that if we take ∂1 , . . . , ∂n , x1 , . . . , xn

SYMPLECTIC DIRAC COHOMOLOGY AND LIFTING OF CHARACTERS

45

for a basis of g1 , then the dual basis (with respect to B) is 2x1 , . . . , 2xn , −2∂1 , . . . , −2∂n . Here we say that a basis fi is dual to the basis ei if B(ei , fj ) = δij ; namely, for super spaces it is the identification V ⊗ V ∗ = Hom(V, V ) that involves no signs. In view of this, the Casimir element of g is defined as   Ωg = Wk2 + 2 (xi ∂i − ∂i xi ). i

k

It is easily checked to be an element of the center Z(g) of the enveloping algebra U (g) of g. Using the relation ∂i xi + xi ∂i = [∂i , xi ] in U (g), one can also write it as    Ωg = Wk2 + 4 xi ∂i − 2 [∂i , xi ]. i

k

i

Furthermore, it is independent of the choiceof basis: if ej is any basis of g, with dual basis fj with respect to B, then Ωg = fj ej . The action of g0 on g1 via the bracket defines a map ν : g0 −→ sp(g1 ). On the other hand, sp(g1 ) can be mapped into W (g1 ) as follows. First we note that the symmetrization map σ : S(g1 ) −→ W (g1 ) is a linear isomorphism. Consider the action of σ(S 2 (g1 )) on g1 ⊂ W (g1 ) by commutators in W (g1 ). One readily checks that in our basis (∂i , xj ), the commutators with σ(xi xj ) = xi xj , σ(∂i ∂j ) = ∂i ∂j and σ(∂i xj ) correspond to the following matrices (Ek l is the matrix unit, having the kl entry equal to 1 and other entries equal to 0): σ(xi xj ) ←→ −En+i j − En+j i ; σ(∂i ∂j ) ←→

Ei n+j + Ej n+i ;

σ(∂i xj ) ←→ −Ei j + En+j n+i . For the last one, note that σ(∂i xj ) = 12 (∂i xj + xj ∂i ) = ∂i xj − 12 δij acts in the same way as ∂i xj . Since sp(g1 ) consists of block matrices of the form   A B C −t A where A is an arbitrary n × n matrix and B and C are symmetric n × n matrices, we see that σ(S 2 (g1 )) is a Lie subalgebra of W (g1 ) isomorphic to sp(g1 ) via the isomorphism described above. Combining with the map ν defined above, we get a Lie algebra morphism α : g0 −→ W (g1 ). Compared to [Kos01], our α is his ν∗ followed by the symmetrization map. Now α gives rise to a diagonal embedding g0 −→ U (g) ⊗ W (g1 ), given by X → X ⊗ 1 + 1 ⊗ α(X). We denote the image of this map by g0Δ ; this is a diagonal copy of g0 . We denote by U (g0Δ ), Z(g0Δ ) the corresponding images of U (g0 ) and its center in U (g) ⊗

46

JING-SONG HUANG

W (g1 ).We will be particularly interested in the image of the Casimir element Ωg0 = k Wk2 . It is  Ωg0Δ = (Wk2 ⊗ 1 + 2Wk ⊗ α(Wk ) + 1 ⊗ α(Wk )2 ). k

 Kostant [Kos01] has shown α(Ωg0 ) = k α(Wk )2 is a constant which we denote by C (C is equal to 1/8 of the trace of Ωg0 on g1 ). Kostant actually showed the following: let g0 be a Lie algebra, with a nonsingular invariant symmetric bilinear form B. Let g1 be a vector space with nonsigular alternating bilinear form B. Suppose g1 is a symplectic representation of g0 , i.e., there is a Lie algebra map g0 → sp(g1 ). Then g = g0 ⊕ g1 has a structure of a Lie superalgebra of Riemannian type compatible with the given data if and only if α(Ωg0 ) is a constant. Recall that the symplectic Dirac operator for a basic classical Lie superalgebra corresponding to the decomposition g = g0 ⊕ g1 is defined analogously to the Dirac operator for the Cartan decomposition of a real reductive Lie algebra. It is an element D of U (g) ⊗ W (g1 ) given by  D=2 (∂i ⊗ xi − xi ⊗ ∂i ). i

Then D is easily seen to be independent of the choice of basis, in the sense that if we take any basis ej for g1 with dual basis fj with respect to B, then D = ej ⊗ fj . This also implies that D is g0 -invariant. We also have a formula for D2 , which is analogous to Parthasarathy’s formula for the square of the Dirac operator attached to the Cartan decomposition for a real reductive Lie algebra: D2 = −Ωg ⊗ 1 + Ωg0Δ − C, where C is the constant described above. Recall that a Cartan subalgebra of g reduces to the Cartan subalgebra of the even part g0 . Let h0 be a Cartan subalgebra of g0 . We denote by Δ, Δ0 and Δ1 the sets of all roots, even roots and odd roots respectively. Let b0 be a Borel subalgebra of g0 , containing h0 . We fix a Borel subalgebra b = b0 ⊕ b1 of g. Then there exist subalgebras n+ and n− such that g = n+ ⊕ h0 ⊕ n− , b = h0 ⊕ n+ and +

Δ+ 0

Denote by Δ , respectively. Set

and

ρ0 =

[h0 , n+ ] ⊂ n+ , [h0 , n− ] ⊂ n− . the subsets of positive roots in the sets Δ, Δ0 and Δ1

Δ+ 1

1  1  α, ρ1 = α, and ρ = ρ0 − ρ1 . 2 2 + + α∈Δ0

α∈Δ1

Let Z(g) be the center of the enveloping superalgebra U (g). An element z ∈ Z(g) can be uniquely written in the form:  ± 0 − 0 ± ± z = uz + u+ i ui ui , where uz , ui ∈ U (h0 ) and ui ∈ n U (n ). i

The map z → uz gives a monomorphism β : Z(g) → U (h0 ) = S(h0 )(= C[h∗0 ]).

SYMPLECTIC DIRAC COHOMOLOGY AND LIFTING OF CHARACTERS

47

Let τ be the automorphism of C[h∗0 ] defined by (τ P )(λ) = P (λ − ρ). The composition γ = τ ◦ β : Z(g) → S(h0 ) is called the Harish-Chandra monomorphism. Let W be the Weyl group of g0 . Then γ(Z(g)) is a subalgebra of S(h0 )W . Moreover, the fields of fractions of γ(Z(g)) and S(h0 )W coincide. Let V = V0 ⊕ V1 be a Z2 -graded vector space over C. A linear representation π of a Lie superalgebra g = g0 ⊕ g1 in V is a homomorphism from g into the superalgebra End(V ) π : g → End(V ) = End0 (V ) ⊕ End1 (V ). For brevity we often say that V is a g-module. Let λ ∈ h∗0 be a linear functional on h0 . We say that a g-module V has infinitesimal character λ if Z(g) acts on V via character χλ : Z(g) → C defined by χλ (z) = γ(z)(λ). (To define γ(z)(λ), we identify S(h0 ) with polynomial functions on h∗0 .) Clearly, χλ = χwλ for w ∈ W . If V is a highest weight g-module with highest weight Λ, then the infinitesimal character of V is Λ + ρ, or any element in W · (Λ + ρ). Let M (g1 ) be the Weil representation for the Weyl algebra W (g1 ). It is naturally a sp(g1 )-module (and g0 -module) by the embedding of sp(g1 ) into W (g1 ) (and the homomorphism from g0 to sp(g1 )). Definition 2.1. Let V be a g-module. The Dirac cohomology HD (V ) of V is defined to be the g0 -module Ker D/ Ker D ∩ Im D. Theorem 2.2. [HP05] Let g be a basic classical Lie superalgebra and V a g-module with infinitesimal character χ. If the Dirac cohomology HD (V ) contains a nonzero g0 -module with infinitesimal character λ ∈ h∗0 , then χ is determined by the W -orbit of λ. More precisely, there exists a homomorphism of Z(g) into Z(g0 ) given by the following commuting diagram such that χ(z) = χλ (ζ(z)) for z ∈ Z(g):

H.C.

Z(g) ⏐ ⏐ hom(

ζ

−−−−→ Z(g0 ) ⏐ ⏐ (H.C.

isom

id

S(h0 )W −−−−→ S(h0 )W where the bottom horizontal map is identity and the two vertical maps are the Harish-Chandra monomorphism and isomorphism respectively. If we identify W (g1 ) as the algebra of differential operators with polynomial coefficients in the xi ’s (i = 1, · · · , n), then the Weil representation M (g1 ) can be identified as the polynomial algebra C[x1 , · · · , xn ]. We write M + (g1 ) and M − (g1 ) for the sp(g1 )-submodules of M (g1 ) spanned by homogeneous polynomials of even and odd degrees respectively. Clearly, D : V ⊗ M ± (g1 ) → V ⊗ M ∓ (g1 ). Now we consider D+ : V ⊗ M + → V ⊗ M − and D− : V ⊗ M − → V ⊗ M +

48

JING-SONG HUANG

defined by the restrictions of D. It follows that + − HD (V ) = HD (V ) ⊕ HD (V )

and + − V ⊗ M + − V ⊗ M − = HD (V ) − HD (V ).

In terms of g0 -characters, this reads + − ch V (ch M + − ch M − ) = ch HD (V ) − ch HD (V ).

In particular, set V = 11 (the trivial representation), we have + − ch M + − ch M − = ch HD (11) − ch HD (11).

This gives a g0 -character formula for V analougous to the Weyl character formula. Theorem 2.3. Let g be a basic classical Lie superalgebra and V a finitedimensional g-module. Then we have the following formula for g0 -characters ch V =

+ − + − (V ) − ch HD (V ) (V ) − ch HD (V ) ch HD ch HD . = + − + − ch M − ch M ch HD (11) − ch HD (11)

3. Rittenberg-Scheunert correspondence We now discuss Rittenberg-Scheunert correspondence for representations of osp(1|2n) and o(2n + 1). The roots of the complex basic Lie algebra g = osp(1|2n) fall into the two subsets of even roots Δ0 = {±ei ± ej ; 1 ≤ i < j ≤ n} ∪ {±2ei ; 1 ≤ i ≤ n}, and odd roots Δ1 = {±ei ; 1 ≤ i ≤ n}. We select simple roots e1 − e2 , e2 − e3 , · · · , en−1 − en , en . Then the positive even roots are ei ± ej with 1 ≤ i < j ≤ n, 2ei with 1 ≤ i ≤ n, and positive odd roots are ei with 1 ≤ i ≤ n. The corresponding fundamental weights are 1 (e1 + · · · + en ). 2 Note that the even roots Δ0 are the roots of sp(2n) and ω1 , . . . , ωn−1 , 2ωn are the fundamental weights for sp(2n). Let ρ0 and ρ1 denote the half sum of the positive even roots, respectively odd roots. Then ρ0 = ω1 + · · · + ωn−1 + 2ωn ωi = e1 + · · · + ei (i = 1, . . . , n − 1) and ωn =

and ρ1 = ωn . We set ρ = ρ0 − ρ1 . Then ρ = ω1 + · · · + ωn−1 + ωn .

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Writing in coordinates, 1 1 1 3 1 ρ0 = (n, . . . , 1), ρ1 = ( , . . . , ), ρ = (n − , n − , . . . , ). 2 2 2 2 2 The irreducible finite-dimensional representations of g = osp(1|2n) are uniquely determined by their highest weights of the form Λ = p1 ω1 + · · · + pn ωn , with integers pi ≥ 0 and pn even. These are also the highest weights of finitedimensional representations of g0 = sp(2n). The setting for the complex simple Lie algebra o(2n + 1) is similar. The root system of o(2n + 1) consists of long roots ±ei ± ej with 1 ≤ i < j ≤ n, and short roots ±ei with 1 ≤ i ≤ n. We select simple roots e1 − e2 , e2 − e3 , · · · , en−1 − en , en . Then the positive long roots are ei ± ej with 1 ≤ i < j ≤ n, and positive short roots are ei with 1 ≤ i ≤ n. The corresponding fundamental weights are ωi = e1 + · · · + ei (i = 1, . . . , n − 1) and ωn =

1 (e1 + · · · + en ). 2

Let ρ denote the sum of the positive roots. Then ρ = ω1 + · · · + ωn−1 + ωn The irreducible finite-dimensional representations of g = o(2n + 1) are uniquely determined by their highest weights of the form Λ = p1 ω1 + · · · + pn ωn , with integers pi ≥ 0 (i = 1, . . . , n). For non-spinorial representations, pn must be even, which are exactly the same condition as for the highest weights of finitedimensional representations of g0 = sp(2n) or g = osp(1|2n). Theorem 3.1. [RS82, Sect. 2] The highest weights of both graded irreducible representations of osp(1|2n) and the non-spinorial irreducible representations of o(2n + 1) are the same, taking the form Λ = p1 ω1 + · · · + pn ωn , with integers pi ≥ 0 and pn even. If a graded irreducible representation of osp(1|2n) and a non-spinorial irreducible representation of o(2n + 1) have the same highest weight, then the multiplicity of any weight is the same for both representations.

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Theorem 3.2. [RS82, Note added in proof] If a graded irreducible representation of osp(1|2n) and a non-spinorial irreducible representation of o(2n + 1) have the same highest weight Λ = p1 ω1 + · · · + pn ωn , then they have the same corresponding parameters Λ + ρ as their infinitesimal characters. The Rittenberg-Scheunert correspondence for osp(1|2n) and o(2n + 1) was extended to quantum groups Uq (osp(1|2n)) and U−q (o(2n + 1)) by Ruibin Zhang [Zha92]. 4. Lifting of characters to metaplectic groups Adams defined a lifting of characters between special orthogonal groups SO(2n + 1) over R and the nonlinear metaplectic groups M p(2n, R) [Ada98]. It is closely related to both endoscopy and theta-lifting. If π is an irreducible representation of SO(p, q) with p + q = 2n + 1, then π has a non-zero theta-lift π  to a genuine representation of M p(2n, R). The Adams lifting amounts to addressing the relation between the characters of π and π  . The main idea of [Ada98] is interpreting G = SO(n + 1, n) as an endoscopic group for G = M p(2n, R) and the corresponding character lifting is best illustrated by the examples of the discrete series. Let πλ be a discrete series representation of G = SO(n + 1, n) with Harish-Chandra parameter λ = (a1 , . . . , ak , b1 , . . . bl ) with ai , bj ∈ Z + > · · · > ak > 0, b1 > · · · > bl > 0. Under the theta-lifting, πλ corresponds to a discrete series πλ of M p(2n, R) with Harish-Chandra parameter 1 2 , a1

λ = (a1 , . . . , ak , −bl , . . . , −b1 ). Denote by W the Weyl group of G or G (type Bn or Cn ), WK the Weyl group of K = S(O(n + 1) × O(n)) and WK  the Weyl group of K  = U (n). We consider the two stable sums of discrete series representations  π ¯λ = πwλ w∈WK \W

and π ¯ λ =



πwλ .

w∈WK  \W

Fix an isomorphism of a compact Cartan subgroup T of G = SO(n + 1, n) and a  be the covering group of T  in compact Cartan subgroup T  of Sp(2n, R). Let T  G = M p(2n, R). The character of π ¯λ on T is  sgn(w)ewλ (t) (t ∈ T ), ΘSO (λ)(t) = w∈W DSO (t) and similarly, the character of π ¯λ on  T  is  wλ  (t )   w∈W sgn(w)e   (t ∈ T ). ΘSp (λ )(t ) = DSp (t ) Then the two characters are related by multiplying by the transfer factor ΘSp (λ )(t ) = ΘSO (λ)(t)Φ(t ),

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DSO (t) where the transfer factor Φ(t ) = D  is precisely the difference of two irreducible Sp (t ) components of the oscillator representation (up to a sign)

±Φ(t ) = Ωeven (t ) − Ωodd (t ). Note that the stable character ΘSO (λ)(t) on T coincides with the character of the finite-dimensional representation with infinitesimal character λ + ρ, which equals to the character of the finite-dimensional representation of osp(1|2n) with the same infinitesimal character under the Rittenberg-Scheunert correspondence described in the previous section. The transfer factor is equal to the symplectic Driac index of the trivial repesentation + − Φ = Ωeven − Ωodd = ch HD (11) − ch HD (11),

and the virture character ΘSp (λ )(t ) = ΘSO (λ)(t)Φ(t ) is the symplectic Dirac index of the corresponding finite-dimensional representation of osp(1|2n). In light of the above discussion, we may define a direct lifting of characters of Sp(2n, R) to M p(2n, R). In a sense, we interpret Sp(2n, R) as an endoscopic group for M p(2n, R). We retain the notation of the previous section. Recall that ωn = ρ1 = (1/2, . . . , 1/2). An invariant eigendistribution Θ on Sp(2n, R) can be expressed on a Cartan subgroup by  aw ewλ Θ = w∈W , with aw ∈ C. DSP We may assume λ to be dominant by reorganizing the coefficients aw . Set  aw ew(λ−ωn ) Θ = w∈W . DSP Then Θ is a genuine invariant eigendistribution on M p(2n, R). We define the lifting Γ by Γ : Θ → Θ . If λ = (a1 , . . . , an ) with ai ∈ Z, a1 > · · · > an > 0 is the Harish-Chandra parameter of a holomorphic discrete series representation for Sp(2n, R), then λ = (a1 , . . . , an ) − (1/2, . . . , 1/2) = λ − ωn is the Harish-Chandra parameter of a holomorphic discrete series representation for M p(2n, R). In general, if λ = (a1 , . . . , ak , −bl , . . . , −b1 ) with ai , bj ∈ Z, a1 > · · · > ak > 0, b1 > · · · > bl > 0 is the Harish-Chandra parameter of a discrete series representation for Sp(2n, R), then 1 1 1 1 λ = (a1 − , . . . , ak − , −bl + , . . . , −b1 + ) 2 2 2 2 is the Harish-Chandra parameter for a discrete series representation for M p(2n, R). In other words, if w ∈ W makes wλ dominant, then we set the corresponding λ to be λ = λ − w−1 ωn . Let πλ be a discrete series representation of Sp(2n, R) with the Harish-Chandra parameter λ. Let πλ be the corresponding genuine discrete series representation

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of M p(2n, R) with the Harish-Chandra parameter λ . Then Γ(Θπλ ) = Θπλ . Consequently, we obtain the following theorem. Theorem 4.1. The map Γ is a bijection between stably invariant eigendistributions on Sp(2n, R) and genuine stably invariant eigendistributions on M p(2n, R). It restricts to a bijection between discrete series of Sp(2n, R) and genuine discrete series of M p(2n, R). References [Ada98] Jeffrey Adams, Lifting of characters on orthogonal and metaplectic groups, Duke Math. J. 92 (1998), no. 1, 129–178, DOI 10.1215/S0012-7094-98-09203-1. MR1609329 [AS] Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62, DOI 10.1007/BF01389783. MR463358 [How10] Tatiana K. Howard, Lifting of characters on p-adic orthogonal and metaplectic groups, Compos. Math. 146 (2010), no. 3, 795–810, DOI 10.1112/S0010437X09004618. MR2644935 [Hua15] Jing-Song Huang, Dirac cohomology, elliptic representations and endoscopy, Representations of reductive groups, Progr. Math., vol. 312, Birkh¨ auser/Springer, Cham, 2015, pp. 241–276. MR3495799 [HP02] Jing-Song Huang and Pavle Pandˇzi´ c, Dirac cohomology, unitary representations and a proof of a conjecture of Vogan, J. Amer. Math. Soc. 15 (2002), no. 1, 185–202, DOI 10.1090/S0894-0347-01-00383-6. MR1862801 [HP05] Jing-Song Huang and Pavle Pandˇzi´ c, Dirac cohomology for Lie superalgebras, Transform. Groups 10 (2005), no. 2, 201–209, DOI 10.1007/s00031-005-1006-8. MR2195599 [HP06] Jing-Song Huang and Pavle Pandˇzi´ c, Dirac operators in representation theory, Mathematics: Theory & Applications, Birkh¨ auser Boston, Inc., Boston, MA, 2006. MR2244116 [HPZ13] Jing-Song Huang, Pavle Pandˇzi´ c, and Fuhai Zhu, Dirac cohomology, K-characters and branching laws, Amer. J. Math. 135 (2013), no. 5, 1253–1269, DOI 10.1353/ajm.2013.0041. MR3117306 [Kac77] V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96, DOI 10.1016/0001-8708(77)90017-2. MR486011 [Kac78] V. Kac, Representations of classical Lie superalgebras, Differential geometrical methods in mathematical physics, II (Proc. Conf., Univ. Bonn, Bonn, 1977), Lecture Notes in Math., vol. 676, Springer, Berlin, 1978, pp. 597–626. MR519631 [Kna02] Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkh¨ auser Boston, Inc., Boston, MA, 2002. MR1920389 [Kos01] Bertram Kostant, The Weyl algebra and the structure of all Lie superalgebras of Riemannian type, Transform. Groups 6 (2001), no. 3, 215–226, DOI 10.1007/BF01263090. MR1854711 [Kos03] Bertram Kostant, Dirac cohomology for the cubic Dirac operator, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., vol. 210, Birkh¨ auser Boston, Boston, MA, 2003, pp. 69–93. MR1985723 [Li19] Wen-Wei Li, Spectral transfer for metaplectic groups. I. Local character relations, J. Inst. Math. Jussieu 18 (2019), no. 1, 25–123, DOI 10.1017/S1474748016000384. MR3886154 [Par72] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 1–30, DOI 10.2307/1970892. MR318398 [Ren99] David Renard, Endoscopy for Mp(2n, R), Amer. J. Math. 121 (1999), no. 6, 1215–1243. MR1719818 [RS82] V. Rittenberg and M. Scheunert, A remarkable connection between the representations of the Lie superalgebras osp(1, 2n) and the Lie algebras o(2n + 1), Comm. Math. Phys. 83 (1982), no. 1, 1–9. MR648354 [V] D. A. Vogan, Jr., Dirac operators and unitary representations, 3 talks at MIT Lie groups seminar, Fall 1997. [Wol74] Joseph A. Wolf, Partially harmonic spinors and representations of reductive Lie groups, J. Functional Analysis 15 (1974), 117–154, DOI 10.1016/0022-1236(74)900159. MR0393351

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[Zha92] R. B. Zhang, Finite-dimensional representations of Uq (osp(1/2n)) and its connection with quantum so(2n + 1), Lett. Math. Phys. 25 (1992), no. 4, 317–325, DOI 10.1007/BF00398404. MR1188811 (Huang) Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR, People’s Republic of China Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15453

Spectrum of semisimple locally symmetric spaces and admissibility of spherical representations Salah Mehdi and Martin Olbrich Abstract. We consider compact locally symmetric spaces Γ\G/H where G/H is a non-compact semisimple symmetric space and Γ is a discrete subgroup of G. We discuss some features of the joint spectrum of the (commutative) algebra D(G/H) of invariant differential operators acting, as unbounded operators, on the Hilbert space L2 (Γ\G/H) of square integrable complex functions on Γ\G/H. In the case of the Lorentzian symmetric space SO0 (2, 2n)/SO0 (1, 2n), the representation theoretic spectrum is described explicitly. The strategy is to consider connected reductive Lie groups L acting transitively and cocompactly on G/H, a cocompact lattice Γ ⊂ L and study the spectrum of the algebra D(L/L ∩ H) on L2 (Γ\L/L ∩ H). Though the group G does not act on L2 (Γ\G/H), we explain how (not necessarily unitary) G-representations enter into the spectral decomposition of D(G/H) on L2 (Γ\G/H) and why one should expect a continuous contribution to the spectrum in some cases. As a byproduct, we obtain a result on the L-admissibility of G-representations. These notes contain the statements of the main results, the proofs and the details will appear elsewhere.

1. Introduction Let G be a connected non-compact semisimple real Lie group and H a connected closed subgroup of G with complexified Lie algebras g and h respectively. Suppose that G/H is a semisimple symmetric space of non-compact type with respect to an involution σ and consider the associated decomposition g = h ⊕ q where q := {X ∈ g | σ(X) = −X}. Fix an invariant measure on G/H and consider the Hilbert space L2 (G/H) of square integrable complex functions on G/H. The action by left translations of G on L2 (G/H) defines a unitary representation whose decomposition is known as the Plancherel formula for symmetric spaces [2][3][7][21]: ) ⊕

H  L2 (G/H)  Wρ,−∞ 0 ⊗ Wρ dμ(ρ) H G

  D(G/H)  Z(g)

where Wρ is a unitary irreducible representation of G, Wρ the dual representation, Wρ,−∞ the space of distribution vectors of Wρ, WρH,−∞ the space of H-invariant 2020 Mathematics Subject Classification. Primary 22E46; Secondary 43A85. c 2021 American Mathematical Society

55

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SALAH MEHDI AND MARTIN OLBRICH



distribution vectors, WρH,−∞ 0 is a finite dimensional Hilbert space embedded in $ H is the Hwhose dimension is the multiplicity of Wρ in L2 (G/H), G WH ρ ,−∞

$ | WH spherical dual {ρ ∈ G ρ ,−∞ = {0}} and dμ the Plancherel measure. The (commutative) algebra D(G/H) of G-invariant differential operators on G/H acts

 on WρH,−∞ 0 and the center Z(g) of the enveloping algebra U(g) of g acts via infinitesimal character on Wρ . Except for exceptional cases where G is E6 , E7 and E8 which we will not consider here, there is a surjective map Z(g)  D(G/H) [11]. In other words, harmonic analysis on G/H is closely related to the spectrum of D(G/H) on L2 (G/H): harmonic analysis on G/H ←→ SpecL2 (G/H) (D(G/H)). A compact locally symmetric space is a smooth compact manifold of the form Γ\G/H where Γ is a discrete subgroup of G. As above one may ask about the analysis on Γ\G/H. Let us consider the following group case example. Example 1.1. G = G1 × G1 , H = Δ(G1 × G1 ) and Γ = Γ1 × {e}, where G1 is a connected non-compact semisimple real Lie group, σ is the involution G1 × G1 → G1 × G1 , (a, b) → (b, a) and Γ1 a co-compact torsion free discrete subgroup of G1 . Then L2 (G/H)  L2 (G1 ) is described by the Harish-Chandra Plancherel formula and * 1 $ π L2 (Γ\G/H)  L2 (Γ1 \G1 )  VπΓ,−∞ ⊗V 1 π∈G

< ∞, dim(Vπ ) = ∞ (except for the trivial representation π) $ 1 | V Γ1 and the set {π ∈ G π  ,−∞ = {0}} is discrete. Moreover D(G/H)  Z(g) acts by scalars on π by infinitesimal characters. In particular, the spectrum of D(G/H) on L2 (Γ\G/H) is discrete, consists of eigencharacters only, and (almost) all eigenspaces are infinite dimensional. where

1 dim(VπΓ,−∞ )

Already this example shows that analysis on Γ\G/H is often related to automorphic forms: analysis on Γ\G/H ←→ automorphic forms. Problem 1.2. Describe the spectrum SpecL2 (Γ\G/H) (D(G/H)) of D(G/H) on L (Γ\G/H) in general. 2

Issues: (1) When H is not compact then a discrete subgroup Γ ⊂ G such that Γ\G/H is a smooth compact manifold need not exist as it is illustrated by the Calabi-Markus phenomenon for SO0 (1, n + 1)/SO0 (1, n) [6]. Given such a Γ, the space Γ\G/H will be called a compact Clifford-Klein form of G/H. One can ask for deformations of Γ such that the new quotient is still a Clifford-Klein form (see [17]). (2) The group G does not act on Γ\G/H, and it is not clear how representations of G are involved in SpecL2 (Γ\G/H) (D(G/H)). (3) When H is not compact, the coset space Γ\G cannot be compact if Γ\G/H is compact. In this case, it is not clear if the spectra of L2 (Γ\G) and L2 (Γ\G/H) are related. (4) D(G/H) acts on C ∞ (Γ\G/H) and C ∞ (Γ\G/H) is dense in L2 (Γ\G/H), so D(G/H) acts via unbounded operators on L2 (Γ\G/H). Some care is required with the domains of the operators.

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Problem 1.2 is related with a program developed by Kobayashi to study hidden symmetries (see [14][16] and references therein). In the following we shall describe and state some of the main results we prove in [22]. 2. Existence of compact Clifford-Klein forms and spherical triples When H is compact, the existence of a discrete subgroup Γ of G such that Γ\G/H is a smooth compact manifold is equivalent to the existence of torsion free uniform lattices in G. It turns out that torsion free uniform lattices always exist in G by a result of Borel and Harish-Chandra [5]. For H non-compact, this gives a sufficient condition for the existence of Γ, employed extensively by Kobayashi starting with [19]: ⎫ existence of a closed connected Lie group L ⊂ G such that⎪ ⎬ existence of compact Clifford-Klein (i) L is reductively embedded in G =⇒ forms for G/H (ii) L acts transitively on G/H ⎪ ⎭ (iii) L ∩ H is compact

Indeed, if Γ is a co-compact lattice in L then Γ\G/H is a compact locally symmetric space. Triples (G, H, L) satisfying (i) and (ii) above were classified, when G is simple, by Oniˇsˇcik in the late 1960’s [24] (see also [20]). We will refer to triples (G, H, L) satisfying (i)(ii)(iii) as transitive triples. In fact, the above implication remains true if we weaken transitively in (ii) to cocompactly. However, we are able to show that this does not change anything: cocompactly implies transitively [22]. Next fix P = M AN the Langlands decomposition of a minimal parabolic subν group of G and consider a principal series representation IndG P (τ ⊗ e ⊗ 1) of G, + is a finite dimensional irreducible representation of M and ν is a where τ ∈ M linear form on the complexified Lie algebra of A. Then van den Ban proved in ν [1] that the space of H-invariant distributions vectors in IndG P (τ ⊗ e ⊗ 1) is finite dimensional:

H ν dim IndG P (τ ⊗ e ⊗ 1) −∞ < ∞. Now pick a transitive triple (G, H, L) and let PL = ML AL NL be the Langlands , decomposition of a minimal parabolic of L. Let τL ∈ M L be a finite dimensional irreducible representation of ML and νL a linear form on the complexified Lie algebra of AL . Observe that L is not σ-stable, L/L ∩ H need not be a symmetric

L∩H νL space and the space of L∩H-invariant distributions IndL ⊗1) −∞ could PL (τL ⊗e be infinite dimensional as it is the case for the triple (G1 ×G1 , Δ(G1 ×G1 ), G1 ×{e}). In fact we obtain the following criterion for the space of L∩H-invariant to be finitedimensional. , Proposition 2.1. For any τL ∈ M L a finite dimensional irreducible representation of ML and νL a linear form on the complexified Lie algebra of AL , one has:

L∩H νL dim IndL ⊗ 1) −∞ < ∞ ⇔ PL acts transitively on L/L ∩ H. PL (τL ⊗ e Transitive triples (G, H, L) such that PL acts transitively on L/L ∩ H will be called spherical triples. The use of the qualifier ‘spherical’ is motivated by the following facts: L/L ∩ H is a spherical homogeneous space

⇐=======⇒ if L ∩ H

⇐ ====== ⇒ is compact

PL acts with an open orbit in L/L ∩ H PL acts transitively on L/L ∩ H.

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Example 2.2. (1) (G1 × G1 , Δ(G1 × G1 ), G1 × {e}) is not spherical. (2) (G1 ×G1 , Δ(G1 ×G1 ), G1 ×K1 ) is spherical (where K1 is maximal compact in G1 ). (3) (SO0 (2, 2n), SO0 (1, 2n), U (1, n)) is spherical (n ≥ 2). (4) (SO0 (4, 3), SO0 (4, 1) × SO(2), G2(2) ) is not spherical. 3. L-admissibility 

Suppose G ⊂ G is a connected closed reductive subgroup and π is an irreducible unitary representation of G. As any unitary representation of a reductive Lie group, the restriction of π to G can be decomposed as the direct integral of irreducible unitary representations: ) ⊕ $ τ dμ(τ ) π|G   Mπ (τ )⊗V  G

where Mπ (τ ) is the multiplicity (Hilbert) space. An exposition of the classical decomposition theory of unitary representations can be found in e.g. Chapter 14 of [27] (Thm. 14.10.5 is the key result here, compare also with [15, Fact 2.1]). Definition 3.1. π is G -admissible if π|G  decomposes discretely and dim Mπ (τ ) < ∞. $  π is K-admissible (Harish-Chandra admissibility Example 3.2. (1) G theorem [8]). $H  (2) G = SL(2, R) × SL(2, R) and H = Δ(SL(2, R) × SL(2, R)). Let G π = π1 ⊗ π1∗ where π1 is a holomorphic discrete series representation of SL(2, R) and π1∗ is the dual representation of π1 . Since π1∗ is an antiholomorphic discrete series representation of SL(2, R), we deduce from [25, Section 4] that π|H has a continuous spectrum (but multiplicities are finite). Therefore π is not H-admissible. There is a criterion for H-admissibility in terms of associated varieties due to Kobayashi [18]. We prove the following admissibility result for spherical representations associated with triples (G, H, L). $ H then π is Theorem 3.3. Suppose (G, H, L) is a spherical triple. If π ∈ G L-admissible. Moreover, if π is not trivial then there are infinitely many summands in π|L all having L ∩ H-invariants. 4. Embedding of Casimir operators Fix a transitive triple (G, H, L). Write U(g)H for the H-invariant elements in the enveloping algebra of g and U(g)h the left U(g)-ideal generated by h. The algebra D(G/H) of G-invariant differential operators on G/H is isomorphic to the following quotient algebra: D(G/H)  U(g)H /U(g)h ∩ U(g)H It is known that since G/H is a symmetric space, D(G/H) is commutative (see [10] and references therein). By definition, if (G, H, L) is a transitive triple then one has a diffeomorphism of homogeneous spaces G/H  L/L ∩ H

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and an isomorphism of algebras U(g)  U(h) ⊗U(l∩h) U(l). In particular, we get an embedding of algebras ı : D(G/H) → D(L/L ∩ H) . The algebra D(L/L ∩ H) need not be commutative, though it is for many transitive triples (see [12]). Of special interest in D(G/H) (resp. D(L/L ∩ H), D(L ∩ K/L ∩ H)) is the Casimir element ΩG (resp. ΩL , ΩL∩K ). The algebras D(L/L ∩ H) and D(L ∩ K/L ∩ H) are defined in an analogous way to D(G/H). Observe that when G/H has rank one then D(G/H) is generated by the Casimir ΩG . Proposition 4.1. For each transitive triple (G, H, L) with G/H irreducible, the embedding ı(ΩG ) is computed explicitly in terms of the decomposition of l/l ∩ h into Ad(L ∩ H)-irreducibles. In particular, based on the restriction to l of the Killing form of g, we have: Example 4.2. (1) (G1 × G1 , Δ(G1 × G1 ), G1 × {e}): ı(ΩG ) = 2ΩL . (2) (G1 × G1 , Δ(G1 × G1 ), G1 × K1 ): ı(ΩG ) = 2ΩL − ΩL∩K . (3) (SO0 (2, 2n), SO0 (1, 2n), U (1, n)): ı(ΩG ) = 2ΩL − ΩL∩K (n ≥ 2). (4) (SO0 (4, 3), SO0 (4, 1) × SO(2), G2(2) ): ı(ΩG ) = 3ΩL − 32 ΩL∩K + 2Ωl∩s∩q .  (Here Ωl∩s∩q is the operator j vj2 where {vj } is an orthonormal basis of l ∩ s ∩ q.) In case (3), Schlichtkrull, Trapa and Vogan obtained a similar formula for O(2n)/U (n) [26]. Remark 4.3. By inspection, we observe that ı(ΩG ) involves ‘non-compact terms’ terms from l ∩ q ∩ s only when (G, H, L) is not spherical, but not in the first group case (1). 5. On the spectrum SpecL2 (Γ\G/H) (D(G/H)) Fix a transitive triple (G, H, L). We have the following sequence of isomorphisms:   L2 (Γ\G/H)  L2 (Γ\L/L ∩ H)  L2 (Γ\L)L∩H   ∞

∞

 ∞

C (Γ\G/H)C (Γ\L/L ∩ H)C (Γ\L)  D ∈ D(G/H)

 π∈L

L∩H

  

VπΓ,−∞ ⊗ VπL∩H (Hilbert sum) 

V Γ,−∞  π π∈L

⊗

⎛closure of ⎞ ⎠

L∩H ⎝algebraic Vπ,∞

 ı(D) ∈ D(L/L ∩ H)

direct sum

L∩H νL Suppose now that (G, H, L) is spherical, i.e dim IndL ⊗ 1) −∞ < ∞, PL (τL ⊗ e , for any τL ∈ M L a finite dimensional irreducible representation of ML and νL a linear form on the complexified Lie algebra of AL . Then by Proposition 2.1 and Casselman embedding Theorem, we deduce that Proposition 5.1. For any irreducible unitary representation (π, Vπ ) of L, one has L∩H dim Vπ,∞ < ∞.

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This means that finding the spectrum of D(G/H) on C ∞ (Γ\G/H) may be reduced to the diagonalization of the operators π(ı(D)), D ∈ D(G/H), on the L∩H (infinitely many) finite dimensional blocks Vπ,∞ . Then we obtain the following 2 description of the spectrum of D(G/H) on L (Γ\G/H). Theorem 5.2. Suppose (G, H, L) is a spherical triple. (1) There is direct sum decomposition of C ∞ (Γ\G/H) and L2 (Γ\G/H) into joint eigenspaces of D(G/H). SpecL2 (Γ\G/H) (D(G/H)) consists of the corresponding eigencharacters and their possible accumulation points. (2) In the Lorentzian case (SO0 (2, 2n), SO0 (1, 2n), U (1, n)) for n ≥ 2, $ | V L∩H = {0}} D(G/H) is generated by the Casimir ΩG , the set {π ∈ L π,∞ is computed explicitly and the representations π are identified as well as their contributions to eigenvalues for ı(ΩG ). More precisely, if Δ denotes the Laplace operator on Γ\G/H induced by ΩG , one has: – spec(Δ)∩(−∞, −n2 ] comes from unitary principal series and, at −n2 , also from limits of discrete series of L. – spec(Δ) ∩ (−n2 , 0] consists of contributions from complementary series, ends of complementary series and non-integrable discrete series. (Recall that a discrete series representation of G is integrable if it has a non-zero matrix coefficient belonging to L1 (G).) ∞ {2 −n2 } is the contribution of integrable dis– spec(Δ)∩(0, ∞) = =n+1

crete series. The corresponding eigenspaces are infinite dimensional. The last assertion about the contribution of integrable discrete series with nonzero L ∩ H-invariants generalizes to arbitrary spherical triples (G, H, L). These contributions combine to infinite dimensional eigenspaces of D(G/H) [22]. The shortest proof of this uses Theorem 3.3 instead of the embedding ı : D(G/H) → D(L/L ∩ H) . In view of Theorem 3.3 the result can be rephrased in terms of G-representations as follows. Each integrable discrete series for G/H contributes an infinite dimensional eigenspace of D(G/H) in L2 (Γ\G/H). Compare related results obtained in [13]. 6. Generalized matrix coefficients, unitarity and continuous spectrum The Hilbert spaces L2 (G/H) and L2 (Γ\G) are both unitary G-representations which decompose into direct integrals of irreducible ones. Though the Hilbert space L2 (Γ\G/H) is not a G-module, there is a natural way to involve representations of G. For an admissible representation (ρ, Wρ ) of G of finite length, one may consider generalized matrix coefficients (see [9]): Wρ,−∞ ⊗ Wρ,−∞ → C −∞ (G), w . ⊗ w → cw,w  with = < w, . ρ(f )w > ) f (g) < ρ(g)w, v > dg < ρ(f )w, v > = cw,w  (f )

G

for f ∈ Cc∞ (G), v ∈ Wρ,∞ .

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Then there is a D(G/H)-equivariant map (which is injective if ρ is irreducible): H WρΓ,−∞ ⊗ Wρ,−∞

C −∞ (Γ\G/H) 

(6.1)  D(G/H)

* ,

VΓ,−∞  π π∈L

L∩H ⊗ Vπ,−∞

 D(L/L ∩ H)

A comparison of (6.1) with Theorem 5.2 leads, among other things, to the following funny observation. On one hand, from Theorem 5.2, there are complementary series representations π of U (1, n) contributing to finite dimensional eigenspaces. On the other hand, if ρ were unitary then ρ|L would contain, by Theorem 3.3, infinitely many unitary summands having L ∩ H-invariants. Moreover, a result of Bergeron and Clozel [4], Thm. 6.5.1, tells us that there are indeed lattices Γ ⊂ U (1, n) such that VπΓ,−∞ = {0} for complementary series π with certain integral parameters. Therefore there must exist non-unitary representations ρ of G that are involved in L2 (Γ\G/H) via (6.1). One can also try to use (6.1) to produce families of eigendistributions on Γ\G/H depending continuously on the spectral parameter. Let 0 = v˜ ∈ VπΓ,−∞ and assume we have a continuous nonconstant family t → ρt , t ∈ (a, b) ⊂ R of H-spherical principal series of G with 0 = wt ∈ WρHt ,−∞ and, the crucial ingredient, a continuous family of non-zero intertwining operators Ψt ∈ HomL (Wρt ,∞ , Vπ,∞ ). Then Ψ∗t v ∈ WρΓt ,−∞ and t → cΨ∗t v,wt is the desired continuous family of eigendistributions. Note that the existence of such a family for Γ\G/H excludes the existence of a discrete spectral decomposition as in Theorem 5.2. Therefore such families can only exist for non-spherical triples (G, H, L). One expects that such families are the building blocks for the continuous part of the spectral decomposition of L2 (Γ\G/H). We are able to find such families of intertwiners for some non-spherical triples, and expect their existence for all nonspherical (G, H, L) except for the group case as in Example 2.2 (1). For instance, for the non-spherical triple (SO(8, C), SO0(1, 7), Spin(7, C)) the existence of many families Ψt is ensured by the work of Jan Frahm [23]. Acknowledgments We thank the anonymous referee for careful reading and for suggestions which helped to improve the quality of the manuscript. The results were announced during the conference “Representation Theory XVI” held in Dubrovnik, Croatia, June 24 - 29, 2019. We thank the organizers for their great success in gathering experts and friends to share ideas in such a wonderful place. References [1] Erik P. van den Ban, Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula, Ark. Mat. 25 (1987), no. 2, 175–187, DOI 10.1007/BF02384442. MR923405 [2] E. P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space. I. Spherical functions, Invent. Math. 161 (2005), no. 3, 453–566, DOI 10.1007/s00222-004-0431-y. MR2181715

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[3] E. P. van den Ban and H. Schlichtkrull, The Plancherel decomposition for a reductive symmetric space. II. Representation theory, Invent. Math. 161 (2005), no. 3, 567–628, DOI 10.1007/s00222-004-0432-x. MR2181716 [4] Nicolas Bergeron and Laurent Clozel, Spectre automorphe des vari´ et´ es hyperboliques et applications topologiques (French, with English and French summaries), Ast´ erisque 303 (2005), xx+218. MR2245761 [5] Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535, DOI 10.2307/1970210. MR147566 [6] E. Calabi and L. Markus, Relativistic space forms, Ann. of Math. (2) 75 (1962), 63–76, DOI 10.2307/1970419. MR133789 [7] Patrick Delorme, Formule de Plancherel pour les espaces sym´ etriques r´ eductifs (French), Ann. of Math. (2) 147 (1998), no. 2, 417–452, DOI 10.2307/121014. MR1626757 [8] Harish-Chandra, Representations of semisimple Lie groups. IV, Amer. J. Math. 77 (1955), 743–777, DOI 10.2307/2372596. MR72427 [9] Hongyu He, Generalized matrix coefficients for infinite dimensional unitary representations, J. Ramanujan Math. Soc. 29 (2014), no. 3, 253–272. MR3265060 [10] Sigurdur Helgason, Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000. Integral geometry, invariant differential operators, and spherical functions; Corrected reprint of the 1984 original. MR1790156 [11] Sigurdur Helgason, Some results on invariant differential operators on symmetric spaces, Amer. J. Math. 114 (1992), no. 4, 789–811, DOI 10.2307/2374798. MR1175692 [12] Fanny Kassel and Toshiyuki Kobayashi, Invariant differential operators on spherical homogeneous spaces with overgroups, J. Lie Theory 29 (2019), no. 3, 663–754. MR3973610 [13] Fanny Kassel and Toshiyuki Kobayashi, Poincar´ e series for non-Riemannian locally symmetric spaces, Adv. Math. 287 (2016), 123–236, DOI 10.1016/j.aim.2015.08.029. MR3422677 [14] Toshiyuki Kobayashi, Global analysis by hidden symmetry, Representation theory, number theory, and invariant theory, Progr. Math., vol. 323, Birkh¨ auser/Springer, Cham, 2017, pp. 359–397. MR3753918 [15] Toshiyuki Kobayashi, Branching problems of Zuckerman derived functor modules, Representation theory and mathematical physics, Contemp. Math., vol. 557, Amer. Math. Soc., Providence, RI, 2011, pp. 23–40, DOI 10.1090/conm/557/11024. MR2848919 [16] Toshiyuki Kobayashi, Hidden symmetries and spectrum of the Laplacian on an indefinite Riemannian manifold, Spectral analysis in geometry and number theory, Contemp. Math., vol. 484, Amer. Math. Soc., Providence, RI, 2009, pp. 73–87, DOI 10.1090/conm/484/09466. MR1500139 [17] Toshiyuki Kobayashi, Discontinuous groups for non-Riemannian homogeneous spaces, Mathematics unlimited—2001 and beyond, Springer, Berlin, 2001, pp. 723–747. MR1852186 [18] Toshiyuki Kobayashi, Discrete decomposability of the restriction of Aq (λ) with respect to reductive subgroups. III. Restriction of Harish-Chandra modules and associated varieties, Invent. Math. 131 (1998), no. 2, 229–256, DOI 10.1007/s002220050203. MR1608642 [19] Toshiyuki Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), no. 2, 249–263, DOI 10.1007/BF01443517. MR1016093 [20] Toshiyuki Kobayashi and Taro Yoshino, Compact Clifford-Klein forms of symmetric spaces— revisited, Pure Appl. Math. Q. 1 (2005), no. 3, Special Issue: In memory of Armand Borel., 591–663, DOI 10.4310/PAMQ.2005.v1.n3.a6. MR2201328 ¯ [21] Toshio Oshima and Toshihiko Matsuki, A description of discrete series for semisimple symmetric spaces, Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., vol. 4, North-Holland, Amsterdam, 1984, pp. 331–390, DOI 10.2969/aspm/00410331. MR810636 [22] S. Mehdi and M. Olbrich, Spectrum of pseudo-Riemannian locally symmetric spaces and admissibility of spherical representations, In preparation. [23] J. M¨ ollers, Symmetry breaking operators for strongly spherical reductive pairs and the GrossPrasad conjecture for complex orthogonal groups, arXiv:1705.06109. [24] A. L. Oniˇsˇ cik, Decompositions of reductive Lie groups (Russian), Mat. Sb. (N.S.) 80 (122) (1969), 553–599. MR0277660 [25] Joe Repka, Tensor products of unitary representations of SL2 (R)., Bull. Amer. Math. Soc. 82 (1976), no. 6, 930–932, DOI 10.1090/S0002-9904-1976-14223-1. MR425026

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[26] Henrik Schlichtkrull, Peter E. Trapa, and David A. Vogan Jr., Laplacians on spheres, S˜ ao Paulo J. Math. Sci. 12 (2018), no. 2, 295–358, DOI 10.1007/s40863-018-0100-5. MR3871679 [27] Nolan R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1992. MR1170566 Institut Elie Cartan de Lorraine, CNRS - UMR 7502, Universit´ e de Lorraine Campus de Metz, France Email address: [email protected] Facult´ e des Sciences, de la Technologie et de la Communication , Universit´ e du Luxembourg, Luxembourg Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15454

Four examples of Beilinson–Bernstein localization Anna Romanov Abstract. Let g be a complex semisimple Lie algebra. The Beilinson– Bernstein localization theorem establishes an equivalence of the category of g-modules of a fixed infinitesimal character and a category of modules over a twisted sheaf of differential operators on the flag variety of g. In this expository paper, we give four detailed examples of this theorem when g = sl(2, C). Specifically, we describe the D-modules associated to finite-dimensional irreducible g-modules, Verma modules, Whittaker modules, discrete series representations of SL(2, R), and principal series representations of SL(2, R).

1. Introduction This paper revolves around the following beautiful theorem of Beilinson– Bernstein. Let g be a complex semisimple Lie algebra, U(g) its universal enveloping algebra, and Z(g) ⊂ U(g) the center. Fix a Cartan subalgebra h ⊂ g and let λ ∈ h∗ . The Weyl group orbit θ ⊂ h∗ of λ determines an infinitesimal character χθ : Z(g) → C, and we denote by Uθ the quotient of U(g) by the ideal generated by the kernel of χθ . In [BB81], Beilinson–Bernstein construct a twisted sheaf of differential operators Dλ on the flag variety X of g associated to λ. Denote by M(Uθ ) the category of Uθ -modules and by Mqc (Dλ ) the category of quasi-coherent Dλ -modules. Theorem 1.1. (Beilinson–Bernstein [BB81]) Let λ ∈ h∗ be dominant and regular. There is an equivalence of categories Δλ

M(Uθ )

Mqc (Dλ ) Γ

given by the localization functor Δλ (V ) = Dλ ⊗Uθ V and the global sections functor Γ. Theorem 1.1 lets us transport the study of representations of g to the setting of Dλ -modules, where local techniques of algebraic geometry can be employed. It is difficult to overstate the impact of this theorem on modern representation theory. Starting with its initial use to prove Kazhdan–Lusztig’s conjecture that composition multiplicities of Verma modules are given by Kazhdan–Lusztig polynomials, this 2020 Mathematics Subject Classification. Primary 17B10, Secondary 14F10. Supported by the National Science Foundation Award No. 1803059. c 2021 American Mathematical Society

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theorem fundamentally changed the way that questions in representation theory are approached. The aim of this paper is to provide concrete examples of this powerful theorem for the simplest nontrivial example, g = sl(2, C). We give explicit descriptions of four types of Dλ -modules, then we compute the g-module structure on their global sections to realize them as familiar representations of g. Our first example (Section 5) of finite-dimensional g-modules illustrates the classical Borel-Weil theorem for sl(2, C). In our second example (Section 6), we realize the global sections of standard Harish-Chandra sheaves as Verma modules or dual Verma modules. Our third example (Section 7) describes the Dλ -modules corresponding to discrete series and principal series representations of SL(2, R). This example illustrates two families of representations which arise in the classification of irreducible admissible representations of a real reductive Lie group via Harish-Chandra sheaves. General descriptions of this classification and its applications are discussed in [BB81, HMSW87, LV83, Mil91, Vog82]. Our final example (Section 8) is of a Dλ -module whose global sections have the structure of a Whittaker module. This Dλ -module is an example of a twisted Harish-Chandra sheaf. Whittaker modules were first introduced by Kostant in [Kos78], and a geometric approach to studying them using Dλ -modules was developed in [MS14] and used in [Rom20] to establish the structure of their composition series. 2. Set-up For the remainder of this paper, let g = sl(2, C) with the standard basis       0 0 1 0 0 1 . ,F = ,H = E= 1 0 0 −1 0 0 Then g has a triangular decomposition g = n ⊕ h ⊕ n, where n = CF, h = CH, and n = CE. Let b = h ⊕ n be the upper triangular Borel subalgebra. Denote the corresponding complex Lie groups by G = SL(2, C),  0 /  0 / a b  1 b  × b∈C . B= a ∈ C , b ∈ C , and N = 0 a−1  0 1  Let Σ+ ⊂ Σ be the associated set of positive roots in the root system of g. Denote the single element of Σ+ by α, and by α∨ the corresponding coroot. Let P (Σ) be the weight lattice in h∗ . The Weyl group W of g is isomorphic to Z/2Z. Let X = G/B be the flag variety of g. The variety X is isomorphic to CP1 , which we identify with the set of lines through the origin in C2 . (Indeed, the natural action of G on C2 by matrix multiplication gives a transitive action of G on the set of lines through the origin in C2 , and the stabilizer of the line spanned by (1, 0) is B.) Denote the line through the point (x0 , x1 ) ∈ C2 \{(0, 0)} by [x0 : x1 ], and define a map (2.1)

p : C2 \{(0, 0)} → X, (x0 , x1 ) → [x0 , x1 ].

The group G acts on X  a (2.2) c

via the action  b · [x0 : x1 ] = [ax0 + bx1 : cx0 + dx1 ]. d

We distinguish the points [0 : 1] and [1 : 0] by labeling them ∞ = [0 : 1], and 0 = [1 : 0].

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An open cover of X is given by {U0 , U∞ }, where U0 = CP1 − {∞}, U∞ = CP − {0}. Denote by V = U0 ∩ U∞  C× . We identify U0 with C using the coordinate z : U0 → C given by z([x0 : x1 ]) = x1 /x0 and U∞ with C via the coordinate w([x0 : x1 ]) = x0 /x1 . On V , the coordinates are related by 1 (2.3) w= . z Let OX be the structure sheaf of X, and DX the sheaf of differential operators. For an affine subset U ⊂ X, we denote by R(U ) its ring of regular functions and D(U ) the ring of differential operators on R(U )1 . A C-linear endomorphism T : V → V of an OX -module V is a differential endomorphism of order ≤ n if for any open set U ⊂ X and (n + 1)-tuple of regular functions p1 , . . . , pn in R(U ), we have [. . . [[T, p0 ], p1 ], . . . , pn ] = 0 on U . This generalizes the notion of a differential operator on X. Denote by Diff (V, V) the sheaf of differential endomorphisms of V. 1

3. Serre’s twisting sheaves We start by introducing a family of sheaves of OX -modules on X parameterized by the integers. These sheaves will eventually provide our first examples of Dλ modules in Section 5. ∼ Let L be an invertible OX -module. Then there exist isomorphisms ϕ0 : L|U0 − → ∼ → OU∞ . By restricting ϕ0 and ϕ∞ , we obtain two isomorOU0 and ϕ∞ : L|U∞ − phisms of L|V with OV . Combining these we obtain an OV -module isomorphism ϕ : OV → OV . Taking global sections results in an R(V )-module isomorphism ϕ : R(V ) → R(V ) which we call by the same name. Specifically, ϕ is the R(V )module isomorphism making the following diagram commute. R(V )

ϕ

ϕ−1 ∞

L(V )

R(V ) ϕ0

id

L(V )

Because 1 generates R(V ) as an R(V )-module, this morphism is completely determined by the image of 1; that is, if ϕ(1) = p ∈ R(V ), then ϕ(q) = qp for any q ∈ R(V ). The morphism ϕ is an isomorphism, so ϕ−1 is also given by multiplication by a regular function r = ϕ−1 (1), and rp = 1. Hence r and p have no zeros or poles in V , so in the coordinate z, they must be of the form p(z) = cz n and r(z) = 1c z −n for some c ∈ C× and n ∈ Z. We conclude that the transition function of L is of the form (3.1)

ϕ : R(V ) → R(V ), 1 → cz n .

The integer n determines the sheaf L up to isomorphism [Har77, Cor 6.17], so without loss of generality we can assume c = 1. We denote the invertible sheaf corresponding to n ∈ Z by O(n). These are Serre’s twisting sheaves. Next we’d like to compute the global sections of O(n). Because the transition function ϕ is given by multiplication by z n and O(n)(U∞ )  C[w] and O(n)(U0 )  C[z], a global section of O(n) is a polynomial q(w) ∈ C[w] and a polynomial p(z) ∈ C[z] such that q(1/z)z n = p(z). We can see that a pair of such polynomials only exists when n ≥ 0, and the polynomial q(w) must be of degree less than or equal 1 We align our notation with [Mila], and encourage the reader interested in more details on the theory of algebraic D-modules to consult this excellent reference.

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to n. In this way, the space of global sections of O(n) for n ≥ 0 can be identified with the vector space of polynomials of degree ≤ n. In particular, dim Γ(O(n), X) =

n+1 0

if n ≥ 0; if n < 0.

In the computations that follow, it will be useful to have a more explicit realization of the sheaf O(n) as a sheaf of homogeneous holomorphic functions on C2 \{(0, 0)}. We will describe this realization now. Fix n ∈ Z, and let Fn be the sheaf on X defined by Fn (U ) =

vector space of homogeneous holomorphic functions on p−1 (U ) of degree n

for U ⊂ X open, where p is the map (2.1). This is an invertible sheaf on X. Indeed, for our open cover {U0 , U∞ }, we have isomorphisms (3.2)

∼

→ OX (U0 ), f → p(z) := f (1, z) ϕ0 : Fn (U0 ) −

and (3.3)

∼

ϕ∞ : Fn (U∞ ) − → OX (U∞ ), h → q(w) := h(w, 1)

The invertible sheaf Fn must be isomorphic to one of Serre’s twisting sheaves. A quick computation using (3.2) and (3.3) shows that the transition function ϕ = n ϕ0 ◦ ϕ−1 ∞ maps q → z q, so Fn  O(n). 4. Twisted sheaves of differential operators In [BB81], Beilinson–Bernstein construct a sheaf Dλ of rings on X for each λ ∈ h∗ which is locally isomorphic to the sheaf of differential operators on X. If λ is in the weight lattice, then the sheaf Dλ can be realized very explicitly. For concreteness, we will work in this setting. Full details of the general construction can be found in [Milb, Ch. 2, §1]. Let λ ∈ P (Σ), so t := α∨ (λ) ∈ Z. Define Dt to be the sheaf of differential endomorphisms of the OX -module O(t − 1). That is, in the notation of §2, Dt = Diff (O(t − 1), O(t − 1)). Remark 4.1. The −1 in this definition is a rho shift: if ρ = 12 α, then α∨ (λ − ρ) = t − 1. In general, if X is the flag variety of a semisimple Lie algebra g, ρ is the half sum of positive roots, and λ ∈ P (Σ), then Dλ = Diff (O(λ − ρ), O(λ − ρ)). The sheaf Dt is locally isomorphic to DX , so it is an example of a twisted sheaf of differential operators [Milb, Ch. 1, §1]. Associated to Dt is a ring isomorphism ψ : D(V ) → D(V ), with the property that for T ∈ D(V ), the diagram R(V )

ϕ

ψ(T )

T

R(V )

R(V )

ϕ

R(V )

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commutes. Here ϕ(q) = z t−1 q, as in equation (3.1). This isomorphism completely determines the data of the sheaf Dt , so we would like to compute it explicitly. Since V  C× , the ring D(V ) is generated by multiplication by z and differentiation with respect to z, which we denote by z and ∂z , respectively, and ψ is determined by the image of z and ∂z . We can see that for q ∈ R(V ), ϕ(z · q) = z t−1 zq = zz t−1 q = z · ϕ(q), so ψ(z) = z. Because ψ is an isomorphism of rings of differential operators, it must preserve order, so ψ(∂z ) = a∂z + b for some a, b ∈ R(V ). An easy computation shows that a = 1 and b = −(t − 1)/z. Hence the sheaf Dt is determined up to isomorphism by the ring isomorphism t−1 . (4.1) ψ : D(V ) → D(V ), z → z, ∂z → ∂z − z One can check the equality t−1 = z t−1 ∂z z −(t−1) (4.2) ∂z − z of differential operators. Remark 4.2. If we allow t ∈ C to be arbitrary, equation (4.1) still defines a ring isomorphism. In contrast, the equation (3.1) in Section 3 only defines an R(V )-module isomorphism for n ∈ Z. This reflects the fact that the sheaves O(n) are only defined for integral values of n, whereas we can extend the definition of Dt given above to non-integral values of t by defining a sheaf of rings with gluing given by ψ. However, for t ∈ C\Z, the sheaves Dt can no longer be realized as sheaves of differential endomorphisms of an OX -module. We would like to realize global sections of Dt -modules as g-modules. To do this, we need to give Dt the extra structure of a homogeneous twisted sheaf of differential operators. This extra structure consists of a G-action on Dt and an algebra homomorphism β : U (g) → Γ(X, Dt ) satisfying some compatibility conditions [Milb, Ch. 1 §2]. This additional structure arises naturally if we use the explicit realization of O(t − 1) as a sheaf of homogeneous holomorphic functions given in Section 3. There is a natural action of G on O(t − 1) given by g · f (x0 , x1 ) = f (g −1 · (x0 , x1 )), where g −1 · (x0 , x1 ) is the standard action of G on C2 by matrix multiplication. We can use this to define an action of G on OX which makes the isomorphisms ϕ0 and ϕ∞ (equations (3.2) and (3.3)) G-equivariant. Explicitly, if   a b ∈ SL(2, C) and p(z) ∈ OX (U0 ), q(w) ∈ OX (U∞ ), g= c d this action is given by

 g · p(z) = (d − bz)t−1 p

(4.3) in the U0 chart and (4.4) in the U∞ OX .

−c + az d − bz



 dw − b g · q(w) = (−cw + a) q −cw + a chart. Note that for t = 1, this differs from the standard action of G on 

t−1

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One of the compatibility conditions of a homogeneous twisted sheaf of differential operators is that the g-action obtained by differentiating the G-action on Dt should agree with the g-action obtained from the map β. Hence we can compute the local g-action given by Dt by differentiating the G-actions (4.3) and (4.4). Explicitly, if X ∈ g and p ∈ O(t − 1)(Ui ) for i = 0, ∞,  d  (exp rX) · p, X ·p= dr r=0

(4.5)

where (exp rX) · p is the G-action on OX given by (4.3) and (4.4). Using (4.5), we can associate differential operators to the Lie algebra basis elements E, F, H in each chart. As an example, we will include the calculation for E in the U0 chart:    d  z t−1 (1 − rz) p dr r=0 1 − rz      z z  − z(t + 1)(1 − rz)t−2 p = z 2 (1 − rz)t−3 p 1 − rz 1 − rz r=0

E · p(z) =

= z 2 p (z) − z(t − 1)p(z). We conclude from this calculation that as a differential operator on OX (U0 ), the Lie algebra element E acts in the coordinate z as E0 = z 2 ∂z − z(t − 1). Similar computations result in the following formulas. On the chart U0 , we have (4.6)

E0 = z 2 ∂z − z(t − 1),

(4.7)

F0 = −∂z , and

(4.8)

H0 = 2z∂z − (t − 1).

On the chart U∞ we have (4.9)

E∞ = −∂w ,

(4.10)

F∞ = w2 ∂w − w(t − 1), and

(4.11)

H∞ = −2w∂w + (t − 1),

where ∂w is differentiation with respect to the coordinate w. The ring homomorphism ψ given in equation (4.1) relates these formulas on the intersection V . We include this computation for E as a sanity check, and encourage the suspicious reader to check the other Lie algebra basis elements. First note that by the relationship (2.3) between the coordinates z and w, we have (4.12)

∂w = −z 2 ∂z .

Using (4.12) and (4.1), we can compute the image of the coordinate w and the derivation ∂w under ψ: ψ(w) = w and ψ(∂w ) = w−(t−1) ∂w wt−1 . Finally, we

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71

compute: ψ(E∞ ) = ψ(−∂w ) = ψ(−z 2 ∂z )   t−1 2 = z ∂z − z = z 2 ∂z − z(t − 1) = E0 . The middle equality follows from (4.2). Using the formulas (4.6) - (4.11), we can explicitly describe the g-module structure on the global sections of Dt -modules. The pairs (E0 , E∞ ), (F0 , F∞ ), and (H0 , H∞ ) each define a global section of Dt ; in particular, they are the global sections which are the images of E, F , and H under the map β : U (g) → Γ(X, Dt ) which gives Dt the structure of a homogeneous twisted sheaf of differential operators. In the remaining sections, we describe several families of Dt -modules and use these formulas to realize their global sections as familiar g-modules. 5. Finite dimensional modules With the computations of Section 4, we can realize global sections of Dt modules as g-modules. To warm up, we will do so for the Dt -module O(t − 1). Fix t − 1 ∈ Z≥0 . Recall from our discussion in Section 3 that a global section of O(t − 1) is a pair q(w) ∈ C[w], p(z) ∈ C[z] of polynomials such that (5.1)

p(z) = z t−1 q(1/z).

The polynomial q(w) in any such pair must have degree less than or equal to t − 1, so q ∈ U := span{1, w, w2 , . . . , wt−1 }. A choice of polynomial q ∈ U uniquely determines p(z) ∈ C[z] satisfying equation (5.1), so we can identify Γ(X, O(t − 1)) with U . We can describe the g-module structure of Γ(X, O(t−1)) by computing the action of the differential operators E∞ , F∞ , H∞ on a basis of U using the formulas (4.9) - (4.11). We choose the basis % & (5.2) uk := (−1)k wk k=0,...,t−1 of V for reasons which will soon become apparent. The action of E∞ , F∞ , and H∞ on the basis (5.2) is given by the formulas (5.3)

E∞ · uk = kuk−1 ,

(5.4)

F∞ · uk = ((t − 1) − k)uk+1 ,

(5.5)

H∞ · uk = ((t − 1) − 2k)uk .

We can capture the g-module structure given by the formulas (5.3) - (5.5) with the picture in Figure 1. In this picture, a colored arrow indicates that the corresponding differential operator sends the vector space basis element at the start of the arrow to a scalar multiple of the vector space basis element at the end of the arrow. The scalar is given in the label of the arrow. For example, the red arrow labeled 2 represents the relationship E∞ · u2 = 2u1 . This picture completely describes the g-module structure of Γ(X, O(t − 1)). It is clear that with this g-module structure,

72

ANNA ROMANOV −t + 1

−t + 3 1

ut+1

t−3

2

···

ut t−1

t−5

t−2

u2

t−2

t−3

u1 2

3

t−1

t−1

u0 1

Figure 1. Action of E∞ , F∞ , and H∞ on O(t − 1)(U∞ ) Γ(X, O(t − 1)) is isomorphic to the irreducible t-dimensional g-module of highest weight t − 1. This illustrates the classical Borel-Weil theorem for sl(2, C). Theorem 5.1. (Borel-Weil) Let t ∈ Z≥1 . As g-modules, Γ(X, O(t − 1))  L(t − 1) where L(t − 1) is the irreducible finite-dimensional representation of g of highest weight t − 1. Remark 5.2. In the arguments above, we have done our computations in the chart U∞ . However, we would have arrived at the same conclusion by working in the other chart. Applying ϕ to the basis {uk } results in a basis {wk } of the vector space W of polynomials in z of degree at most t − 1. Explicitly, ϕ(uk ) = (−1)k z t−1−k =: wk . Computing the action of the differential operators E0 , F0 , H0 (equations (4.6) - (4.8)) on the basis {wk } results in an identical picture to the one above. 6. Verma modules A natural source of Dt -modules on the homogeneous space X is a stratification of X by orbits of a group action. In this section, we will describe two Dt -modules which are constructed using the stratification of X by N -orbits, where  0 / 1 b  b ∈ C N= 0 1  is the unipotent subgroup of G from Section 2. These modules are the standard Harish-Chandra sheaves associated to the Harish-Chandra pair (g, N ). We will see that their global sections have the structure of highest weight modules for g. The group N acts on X by the restriction of the action (2.2). There are two orbits: the single point 0 and the open set U∞ .



= 0

U∞

X

For each O ∈ {0, U∞ }, let iO : O → X be inclusion. We can construct a Dt -module associated to each orbit O by using the Dt -module direct image functor to push forward the structure sheaf: (6.1)

IO := iO+ (OO ).

Because each orbit O is a homogeneous space for N , the sheaf OO has a natural N -action. This N -action is compatible with the Dt -action, in the sense that the

FOUR EXAMPLES OF BEILINSON–BERNSTEIN LOCALIZATION

73

differential agrees with the n-action coming from the map β : U (g) → Γ(X, Dt ) when restricted to O, so OO has the structure of an N -homogeneous connection2 on O. The Dt -module IO also carries a compatible N -action, so it is a HarishChandra sheaf (see [Mil93, §3] for a precise definition) for the Harish-Chandra pair (g, N ). Hence its global sections Γ(X, IO ) have the structure of a HarishChandra module3 . The sheaf IO is the standard Harish-Chandra sheaf associated to the orbit O and the connection OO [Milb, Ch. 4 §5]. The goal of this section is to describe the g-module structure on Γ(X, IO ) using the formulas (4.6) - (4.11). We begin with the single point orbit 0. In general, to describe a Dt -module F, we will describe the Dt (Ui )-module structure on the vector spaces F(Ui ) for i = 0, ∞. However, in this first case, it is sufficient just to work only in the chart U0 because the support of the sheaf I0 is contained entirely in the chart U0 . (The support of I0 is 0.) Let j0 : 0 → U0 be inclusion. Then I0 (U0 ) = j0+ (O0 )(U0 ) = j0+ (R(0)). The ring of regular functions R(0) of the single point variety 0 is isomorphic to C, as is the ring D(0) of differential operators. The D(0)-action on R(0) is left multiplication. Because 0 and U0  C are both affine, we can describe the D(U0 )-module structure on the push-forward j0+ (R(0)) directly using the definition of the Dmodule direct image functor for polynomial maps between affine spaces [Mila, Ch. 1 §11]. In the notation of [Mila], we have j0+ (R(0)) = DU0 ←0 ⊗D(0) R(0). Because D(0)  R(0)  C, this is isomorphic to the left D(U0 )-module DU0 ←0 = R(0) ⊗R(U0 ) D(U0 ). Here the D(U0 )-module structure on DU0 ←0 is given by right multiplication on the second tensor factor by the transpose [Mila, Ch. 1 §5] of a differential operator T ∈ D(U0 ). As an R(U0 )(= C[z])-module, R(0) is isomorphic to C[z]/zC[z]. Hence the D(U0 )-module j0+ (R(0)) is isomorphic to the D(U0 )-module * (6.2) D(U0 )/D(U0 )z = ∂zi δ, i≥0

where δ : C → C, 0 → 1, x = 0 → 0 is the Dirac indicator function. From this discussion, we see that to describe the g-module structure on Γ(X, i0+ (O0 )), it suffices to compute the actions of the differential operators E0 , F0 , H0 on a basis for the D(U0 )-module in (6.2). We choose the basis / 0 (−1)k k mk := ∂z δ k! k∈Z≥0 2 Irreducible N -homogeneous connections on O are parameterized by irreducible representations of the component group of stabN x for x ∈ O. In this case, stabN x = 1, so OO is the only N -homogeneous connection on O. In Section 7 we will see an example of a nontrivial homogeneous connection on an orbit. 3 A Harish-Chandra module for the Harish-Chandra pair (g, N ) is a finitely generated U(g)module with an algebraic action of N such that the differential of the N -action agrees with the n-action coming from the g-module structure.

74

ANNA ROMANOV −t − 1 − 2k k+1 k

···

−t − 5

···

mk

−t − 1 − k

−t − k

−t − 3 2

3

m2 −t − 3

−t − 1 1

m1 −t − 2

m0 −t − 1

Figure 2. Action of E0 , F0 , and H0 on I0 (U0 ) of (6.2). We include the computation of the E0 action as an example, then record the remaining formulas. Let k ∈ Z>0 . Using the relationships [∂z , z] = ∂z z−z∂z = 1 and zδ = 0, we compute: (−1)k k ∂z δ k! (−1)k (t − 1) k (−1)k 2 k+1 z ∂z δ − z∂z δ = k! k! k (−1)k+1 (t − 1) k (−1) z(∂zk+1 zδ − (k + 1)∂zk δ) + (∂z zδ − k∂zk−1 δ) = k! k! (−1)k (t − 1) k−1 (−1)k+1 (k + 1) k (∂z zδ − k∂zk−1 δ) + ∂z δ = k! (k − 1)! (−1)k (t − 1) k−1 (−1)k (k + 1) k−1 ∂z δ + ∂z δ = (k − 1)! (k − 1)! = (−t − k)mk−1 .

E0 · mk = (z 2 ∂z − z(t − 1))

Similar computations for F0 and H0 lead to the following formulas. (6.3)

E0 · mk = (−t − k)mk−1 for k = 0, E0 · m0 = 0

(6.4)

F0 · mk = (k + 1)mk+1

(6.5)

H0 · mk = (−t − 1 − 2k)mk

As in Section 5, we can capture this g-module structure with the picture in Figure 2. From the formulas (6.3) - (6.5) and Figure 2, we see that for t ∈ Z≥1 , Γ(X, IO ) is an irreducible Verma module of highest weight −t − 1. Theorem 6.1. Let t ∈ Z≥1 and I0 the standard Harish-Chandra sheaf for Dt attached to the closed N -orbit 0 ∈ X. Then as g-modules, Γ(X, I0 )  M (−t − 1), where M (−t − 1) is the irreducible Verma module of highest weight −t − 1. Remark 6.2. One can see from an inspection of formulas (6.3) - (6.5) and Figure 2 that Theorem 6.1 also holds when t = 0. In this setting, the Verma module M (−1) has singular infinitesimal character, so the statement is not an example of Theorem 1.1. (This is why we don’t include t = 0 in the statement of Theorem 6.1.) Next we examine the standard Harish-Chandra sheaf attached to the open orbit U∞ . Let i∞ : U∞ → X be inclusion. To describe the Dt -module IU∞ := i∞+ (OU∞ ),

FOUR EXAMPLES OF BEILINSON–BERNSTEIN LOCALIZATION t − 1 − 2k t−1−k t−k

···

t−3

···

nk k+1

t−5

t−2

n2

t−1

n1 2

3

k

t−3

75

t−1

n0 1

Figure 3. Action of E∞ , F∞ , and H∞ on IU∞ (U∞ ) we will compute the local g-module structure of the vector spaces IU∞ (U0 ) and IU∞ (U∞ ) using the formulas (4.6) - (4.11). In the chart U∞ , we have (6.6) We choose the basis (6.7)

IU∞ (U∞ ) = R(U∞ ) = C[w]. %

nk := (−1)k wk

& k∈Z≥ 0

of (6.6). The E∞ , F∞ , H∞ actions on the basis (6.7) are given by (6.8)

E∞ · nk = knk−1

(6.9)

F∞ · nk = ((t − 1) − k)nk+1

(6.10)

H∞ · nk = ((t − 1) − 2k)nk

Hence the g-module structure of the vector space IU∞ (U∞ ) is given by Figure 3. It remains to describe the g-module structure in the other chart U0 . Let k0 : V → U0 and k∞ : V → U∞ be inclusion. Then + (R(U∞ )) IU∞ (U0 ) = k0+ k∞

 = k0+ R(V ) ⊗R(U∞ ) R(U∞ )

 = k0+ C[w, w−1 ] ⊗C[w] C[w]

 = k0+ C[w, w−1 ] .

Because the map k0 is an open immersion, the direct image k0+ (C[w, w−1 ]) = C[w, w−1 ] as a vector space, with D(U0 )-module structure given by the restriction of the D(V )-action to the subring D(U0 ) ⊂ D(V ). Hence, as a D(U0 )-module, (6.11)

IU∞ (U0 ) = C[z, z −1 ].

We choose the basis

%

nk := (−1)k z −k

& k∈Z

of (6.11) to align with the basis (6.7) for C[w] given above. The actions of E0 , F0 , H0 on IU∞ (U0 ) are given by the formulas (6.12)

E0 · nk = (t − 1 + k)nk−1 ,

(6.13)

F0 · nk = −knk+1 ,

(6.14)

H0 · nk = (−(t − 1) − 2k)nk .

Figure 4 illustrates this action. With this information, we can describe the g-module structure on Γ(X, IU∞ ). A global section of IU∞ is a pair of functions q(w) ∈ C[w], p(z) ∈ C[z, z −1 ] such

76

ANNA ROMANOV

−2

···

−t − 3

−1

n2 t+2

−t − 1

−t + 1

n1 t+1

−t + 3

n0 t

−t + 5 2

1

n−1 t−1

3

···

n−2 t−2

t−3

Figure 4. Action of E0 , F0 , and H0 on IU∞ (U0 ) that q(1/z) = p(z). A Lie algebra basis element X ∈ {E, F, H} acts on this global section by X · (q, p) = (X∞ · q, X0 · p), where the actions of X∞ and X0 are those given in Figures 3 and 4 and equations (6.8) - (6.14). By construction, these actions are compatible on the intersection; that is, on V , X∞ · q(1/z) = ψ(X∞ ) · p(z) = X0 · p(z), where ψ is the ring isomorphism 4.1 which defines Dt . Because C[w] ⊂ C[z, z −1 ], a choice of a polynomial in C[w] uniquely determines a global section, so the space of global sections can be identified with C[w]. Hence, as a g-module, Γ(X, IU∞ )  C[w] with action as in Figure 3. We can see from formulas (6.8) - (6.10) that this g-module is the dual Verma module of highest weight t − 1. It has an irreducible t-dimensional submodule spanned by {n0 , . . . nt−1 }. Theorem 6.3. Let t ∈ Z≥1 and IU∞ the standard Harish-Chandra sheaf for Dt attached to the open N -orbit U∞ ⊂ X. Then as g-modules, Γ(X, IU∞ )  I(t − 1), where I(t − 1) is the dual Verma module of highest weight t − 1. 7. Admissible representations of SL(2, R) In Section 6, we described the Dt -modules corresponding to Verma modules and dual Verma modules. These Dt -modules were the standard Harish-Chandra sheaves for the Harish-Chandra pair (g, N ). In this section, we will describe four Dt -modules which are constructed in a similar way from K-orbits on X, where  0 / a 0  × a ∈ C . K= 0 a−1  These are the standard Harish-Chandra sheaves for the Harish-Chandra pair (g, K). The group K is the complexification of the maximal compact subgroup of the real Lie group    / 0   1 0 1 0 M∗ = SU(1, 1) = M ∈ SL(2, C) M , 0 −1 0 −1 which is isomorphic to SL(2, R). We will see that the global sections of the standard Harish-Chandra sheaves described in this section are the Harish-Chandra modules attached to discrete series and principal series representations of SL(2, R). The group K acts on X by restriction of the action (2.2). There are three orbits: the two single-point orbits 0 and ∞, and the open orbit V  C× .

FOUR EXAMPLES OF BEILINSON–BERNSTEIN LOCALIZATION



=

77

 ∞

0 X

V

We will begin by describing the standard Harish-Chandra sheaves constructed from the closed orbits. The standard Harish-Chandra sheaf associated to the orbit 0 is I0 := i0+ (O0 ). We described the structure of this Dt -module in Section 6. The g-module structure on its space of global sections is given by formulas (6.3) - (6.5) and Figure 2. The standard Harish-Chandra sheaf I∞ := i∞+ (O∞ ) attached to the closed orbit ∞ has a similar structure. As it is supported entirely in the chart U∞ , it suffices to describe only I∞ (U∞ ). By analogous arguments to those in Section 6, we have * i I∞ (U∞ ) = ∂w δ, i≥0

where δ is the Dirac indicator function. The actions of E∞ , F∞ , and H∞ on the basis / 0 (−1)k k ∂w δ dk := k! k∈Z≥0 are given by the formulas (7.1)

E∞ · dk = (k + 1)dk+1 ,

(7.2)

F∞ · dk = (−t − k)dk−1 for k > 0, F∞ · d0 = 0,

(7.3)

H∞ · dk = (t + 1 + 2k)dk .

Figure 5 illustrates this g-module structure. We can see that for t ∈ Z≥1 , Γ(X, I∞ ) is an irreducible lowest weight module with lowest weight t + 1. Theorem 7.1. Let t ∈ Z≥1 and I0 , I∞ the standard Harish-Chandra sheaves for Dt attached to the closed K-orbits 0 and ∞, respectively. Then as g-modules, Γ(X, I0 )  D+ (−t − 1), and Γ(X, I∞ )  D− (t + 1), where D+ (−t−1) is the Harish-Chandra module of the (holomorphic) discrete series representation of SL(2, R) of highest weight −t − 1 and D− (t + 1) is the HarishChandra module of the (antiholomorphic) discrete series representation of SL(2, R) of lowest weight t + 1. Both of these modules are irreducible. Remark 7.2. When t = 0, Γ(X, I0 ) and Γ(X, I∞ ) are the Harish-Chandra modules of the limits of discrete series representations of SL(2, R). As in Remark 6.2, we exclude this case from the statement of Theorem 7.1 because it is not an example of Theorem 1.1 since the infinitesimal character is singular.

78

ANNA ROMANOV t+1

t+3 −t − 1

d0

t+5

d1 1

t + 1 + 2k −t − k −t − 1 − k

−t − 3

−t − 2

···

d2 2

3

···

dk k

k+1

Figure 5. Action of E∞ , F∞ , and H∞ on I∞ (U∞ ) Now we consider the open orbit V and examine the structure of the standard Harish-Chandra sheaf IV := iV + (OV ). As we did in Section 6, we will describe the local g-module structure of the vector spaces IV (U0 ) and IV (U∞ ). To begin, we will record some basic facts about Dt module functors. Let U0 j0

i0 iV

V j∞

X i∞

U∞ be the natural inclusions of varieties. Because i0 and i∞ are open immersions, the Dt -module push-forward functors i0+ and i∞+ agree with the sheaf-theoretic push-forward functors. Hence for any DV -module F, (7.4)

iV + (F)(Uk ) = jk+ (F)(Uk )

for k = 0, ∞. Moreover, because jk , k = 0, ∞ are affine immersions, (7.5)

jk+ (F)(Uk ) = F(Uk )

as vector spaces, with D(Uk )-module structure coming from the natural inclusions D(Uk ) ⊂ D(V ). Applying (7.4) and (7.5) to the DV -module OV , we see that (7.6)

IV (U0 ) = C[z, z −1 ],

(7.7)

IV (U∞ ) = C[w, w−1 ].

We will compute the local g-module structure on these vector spaces using formulas (4.6) - (4.11) and the basis % & nk := (−1)k wk = (−1)k z −k k∈Z . The actions of E∞ , F∞ , H∞ on IV (U∞ ) are given by (7.8)

E∞ · nk = knk−1 ,

(7.9)

F∞ · nk = ((t − 1) − k)nk+1 ,

(7.10)

H∞ · nk = ((t − 1) − 2k)nk .

The actions of E0 , F0 , H0 on IV (U0 ) are given by (7.11)

E0 · nk = (t − 1 + k)nk−1 ,

(7.12)

F0 · nk = −knk+1 ,

(7.13)

H0 · nk = (−(t − 1) − 2k)nk .

Figures 6 and 7 illustrate the resulting g-modules.

FOUR EXAMPLES OF BEILINSON–BERNSTEIN LOCALIZATION

t−3

···

t−5

t−2

n2

t−3

n1 2

3

t−1

t−1

t+1

n0

t+3 t+1

t

n−1

1

79

t+2

···

n−2 −2

−1

Figure 6. Action of E∞ , F∞ , and H∞ on IV (U∞ ) −t − 3

−2

···

−1

−t + 1

t+1

−t + 3

n0 t

−t + 5 2

1

n1

n2 t+2

−t − 1

n−1 t−1

3

···

n−2 t−2

t−3

Figure 7. Action of E0 , F0 , and H0 on IV (U0 ) One can see from a brief inspection that the two g-modules in Figures 6 and 7 are isomorphic4 . Hence the space of global sections of IV can be identified with C[w, w−1 ], with g-module structure as in Figure 6. Remark 7.3. If t ∈ Z≥1 , we can see from Figure 6 that Γ(X, IV ) has a tdimensional irreducible submodule spanned by {n0 , . . . nt−1 }, and the quotient of Γ(X, IV ) by this submodule is isomorphic to the direct sum of the discrete series representations D+ (−t − 1) and D− (t + 1) from Theorem 7.1. By construction, the g-module Γ(X, IV ) has a compatible action of K which gives it the structure of a Harish-Chandra module for the Harish-Chandra pair (g, K). Explicitly, if we identify Γ(X, IV ) with the g-module in Figure 6, this action can be obtained by exponentiating the H∞ -action given by formula (7.10):   a 0 t−1−2k ∈ K. nk , where k = (7.14) k · nk = a 0 a−1 With this K-action, C[w, w−1 ] = R(V ) obtains the structure of an irreducible K-homogeneous R(V )-module5 . Up to isomorphism, there is exactly one other irreducible K-homogeneous R(V )-module. Theorem 7.4. Up to isomorphism, there are exactly two irreducible Khomogeneous R(V )-modules. Proof. Let M be an irreducible K-homogeneous R(V )-module. As an algebraic representation of K, M has a decomposition * M= Mi i∈I

where I is an indexing set of integers and Mi are K-subrepresentations such that for mi ∈ Mi ,   a 0 ∈ K acts by k · mi = ai mi . k= 0 a−1 explicit isomorphism is given by IV (U∞ ) → IV (U0 ), nk → nk−t+1 . K-homogeneous R(V )-module is an R(V )-module M with an algebraic action of K such that the action map R(V ) ⊗ M → M, p ⊗ m → p · m is K-equivariant. 4 An 5A

80

ANNA ROMANOV

Fix some nonzero mi ∈ Mi . The subspace * Cz n · mi ⊆ M n∈Z

is stable under the actions of R(V ) and K so it forms a K-homogeneous R(V )submodule of M . Since M is assumed to be irreducible, this submodule must be all of M . If i is even, then the assignment mi → z i/2 gives an isomorphism M  R(V ) of K-homogeneous R(V )-modules. If i is odd, then M is not isomorphic to R(V ) because R(V ) contains the trivial representation of K as a subrepresentation and M does not. (Indeed, for any n ∈ Z, k · (z n · mi ) = (k · z n ) · (k · mi ) = a2n+i (z n · mi ), which is not equal to z n · mi for a = 1 because 2n + i is odd.) i−j If i = j are both odd, then the assignment mi → z 2 ·mj gives an isomorphism * * ∼ Cz n · mi − → Cz n · mj n∈Z

n∈Z

of K-homogeneous R(V )-modules. We conclude that up to isomorphism, there are exactly two irreducible K-homogeneous R(V )-modules.  Remark 7.5. As mentioned in Section 6, the irreducible K-homogeneous connections on a K-orbit O are parameterized by representations of the component group of stabK x for x ∈ O. When x ∈ O = V , stabK x = ±1, so there are two irreducible K-homogeneous connections on V . Their global sections are the K-homogeneous R(V )-modules in Theorem 7.4. We will refer to R(V ) as the trivial K-homogeneous R(V )-module and the other module, denoted by P , as the non-trivial K-homogeneous R(V )-module. We can use the non-trivial module P to construct the fourth and final standard HarishChandra sheaf for the pair (g, K). Let * P = P2n+1 n∈Z

be the decomposition of P into irreducible K-representations. Fix p1 ∈ P1 , and let pk := z k · p1 . The set {pk }k∈Z forms a basis for P . The R(V )-module P admits the structure of a D(V )-module6 via the action   1 pk−1 , ∂z · p k = k + 2 z · pk = pk+1 . By construction, P is a K-homogeneous D(V )-module. Since V  C× is affine, there is a corresponding K-homogeneous connection τ . We construct a standard Harish-Chandra sheaf by pushing forward τ to X using the Dt -module pushforward: IV,τ := iV + (τ ). 6 Loosely speaking, we can view P as the ring z 1/2 C[z, z −1 ], which explains the D(V )-module structure below.

FOUR EXAMPLES OF BEILINSON–BERNSTEIN LOCALIZATION

81

t−4 t−2 t+2 t+4 t −t + 1/2 −t + 3/2 −t − 1/2 −t − 3/2 −t − 5/2 −t − 7/2

···

p−2

p−1

−5/2

p0 −1/2

−3/2

p1 1/2

···

p2 3/2

5/2

Figure 8. Action of E∞ , F∞ , and H∞ on IV,τ (U∞ ) 3/2

−t − 2

···

−t

1/2

p−1

p−2

−t − 5/2

−t + 2 −t + 4 −t + 6 −1/2 −7/2 −3/2 −5/2

p0 −t − 1/2

−t − 3/2

p1 −t + 1/2

···

p2 −t + 3/2

−t + 5/2

Figure 9. Action of E0 , F0 , and H0 on IV,τ (U0 ) The local g-module structure on the vector spaces IV,τ (Ui )  P for i = 0, ∞ is given by the formulas 3 + k)pk+1 , 2 1 = (−k − )pk−1 , 2 = (−t + 2 + 2k)pk , 1 = (k + )pk+1 , 2 1 = (−t − − k)pk−1 , 2 = (2k + t)pk .

(7.15)

E0 · pk = (−t +

(7.16)

F0 · p k

(7.17)

H0 · p k

(7.18)

E∞ · pk

(7.19)

F∞ · p k

(7.20)

H∞ · pk

These g-modules are illustrated in Figures 8 and 9. We can see that they are irreducible for any value of t ∈ Z. Theorem 7.6. Let t ∈ Z≥1 and IV , IV,τ the standard Harish-Chandra sheaves for Dt corresponding to the trivial and non-trivial K-homogeneous connections on V , respectively. Then as g-modules, Γ(X, IV )  P+ (t − 1) Γ(X, IV,τ )  P− (t − 1) where P+ (t − 1), P− (t − 1) are the Harish-Chandra modules associated to the reducible and irreducible (resp.) principal series representations of SL(2, R) corresponding to the parameter λ − ρ ∈ h∗ with α∨ (λ) = t. Remark 7.7. Theorems 7.1 and 7.6 describe two families of representations which arise in the classification of irreducible admissible representations of SL(2, R). A complete geometric classification using standard Harish-Chandra sheaves can be obtained by removing the regularity and integrality conditions on λ.

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8. Whittaker modules In Sections 6 and 7, we gave examples of Dt -modules with a Lie group action that was compatible with the Dt -module structure7 . These were examples of Harish-Chandra sheaves. In this section, we will give another example of a Dt module with a Lie group action, but now the two actions will differ by a character of the Lie algebra. This is an example of a twisted Harish-Chandra sheaf. Twisted Harish-Chandra sheaves first arose in [HMSW87, Appendix B] in the study of Harish-Chandra modules for semisimple Lie groups with infinite center, and were later used in [MS14] to provide a geometric description of Whittaker modules. Let N be as in Section 6 and n = Lie N . Fix a Lie algebra morphism η : n → C. Because n is spanned by the matrix E, this morphism is determined by the image of E, which we will also refer to as η: η := η(E) ∈ C. Our starting place is the η-twisted connection OU∞ ,η . As a sheaf of rings on U∞ , OU∞ ,η = OU∞ , but the DU∞ -module structure is twisted by η. It suffices to describe this DU∞ module structure on global sections as U∞ is affine. The D(U∞ )-action on the vector space OU∞ ,η (U∞ ) = C[w] is given by (8.1)

∂w · wk = kwk−1 − ηwk , w · wk = wk+1 .

Remark 8.1. Alternatively, we could have described this D(U∞ )-module in terms of exponential functions. One can see from formula (8.1) that the module W = span{wk e−ηw }k∈Z≥0 with D(U∞ )-action d (wk e−ηw ) dw is isomorphic to the D(U∞ )-module described above. ∂w · (wk e−ηw ) =

We can use the Dt -module direct image functor to push the sheaf OU∞ ,η forward to a Dt -module on X. Let i∞ : U∞ → X be inclusion. Define (8.2)

I∞,η := i∞+ (OU∞ ,η ).

The Dt -module I∞,η is the standard η-twisted Harish-Chandra sheaf [MS14, §3] associated to the Harish-Chandra pair (g, N ) and the open orbit U∞ . Remark 8.2. For η = 0, there is no standard η-twisted Harish-Chandra sheaf associated to the closed orbit 0. Indeed, as O0 and D0 are sheaves on a single point8 , they are simply the data of a vector space, the vector space C. A D0 module structure on O0 is an action of C on itself. As the Lie algebra element E is nilpotent, it must act as multiplication by 0 in any such D0 -module structure on O0 . Moreover, as N fixes 0, in the n-module structure on O0 coming from the action of N on X, the Lie algebra element E also acts as multiplication by 0. Hence for η = 0, it is not possible to give the sheaf O0 the structure of a D0 -module in such a way the n-module structure coming from the D0 -action differs by η from 7 Compatible in the sense that the differential of the group action agrees with the Lie algebra action coming from Dt . 8 Recall that O is the structure sheaf on the single point N -orbit 0 ∈ X and D is the sheaf 0 0 of differential operators on 0 (see set-up in §6).

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Figure 10. Action of E∞ , F∞ , and H∞ on I∞,η (U∞ ) the n-module structure coming from the N -action, as we have done above for the structure sheaf OU∞ on the open orbit U∞ . As we did in Section 6, we will describe the Dt -module structure of I∞,η by computing the local g-module structure on the vector spaces I∞,η (U∞ ) and I∞,η (U0 ) using formulas (4.6) - (4.11). From these local descriptions we can identify the g-action on Γ(X, I∞,η ). In the chart U∞ we have I∞,η (U∞ ) = R(U∞ ) = C[w]

(8.3)

as a vector space with D(U∞ )-action given by (8.1). The action of E∞ , F∞ , and H∞ on the basis % & nk := (−1)k wk k∈Z ≥0

is given by the formulas (8.4)

E∞ · nk = knk−1 + ηnk ,

(8.5)

F∞ · nk = (t − 1 − k)nk+1 − ηnk+2 ,

(8.6)

H∞ · nk = (t − 1 − 2k)nk − 2ηnk+1 .

Remark 8.3. Unlike the previous examples, the differential operators E∞ , F∞ and H∞ do not map basis vectors to scalar multiples of other basis vectors. One might think that this is the result of a poor choice of basis, but this is not the case. In fact, we can not choose a basis for this module such that each operator E∞ , F∞ and H∞ maps basis vectors to scalar multiples of basis vectors. Instead of being disappointed by this more complicated module structure, we should use it as evidence that our previous examples were unusually well-behaved. As we did in Sections 5, 6, and 7, we can capture formulas (8.4) - (8.6) in a picture, see Figure 10. In this figure, colored arrows now represent linear combinations of basis elements. For example, the two blue arrows labeled t + 2 and −η emanating from n1 represent the relationship F∞ · n1 = (t + 2)n2 − ηn3 . Now we turn our attention to the other chart U0 . By the same arguments as in Section 6, as a vector space (8.7)

I∞,η (U0 )  C[z, z −1 ],

with D(U0 )-module structure9 given by (8.8) 9 This

∂z · z k = kz k−1 + ηz k−2 , z · z k = z k+1 action is derived from the relationship (4.12) between ∂z and ∂w on V .

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Figure 11. Action of E0 , F0 , and H0 on I∞,η (U0 ) for k ∈ Z. The actions of E0 , F0 and H0 on the basis % & nk := (−1)k z −k k∈Z are given by the formulas (8.9)

E0 · nk = (t − 1 + k)nk−1 + ηnk ,

(8.10)

F0 · nk = −knk+1 − ηnk+2 ,

(8.11)

H0 · nk = (−(t − 1) − 2k)nk − 2ηnk+1 .

Figure 11 illustrates this action. As we did in Section 6, we can use Figures 10 and 11 and formulas (8.4)-(8.11) to describe the g-module structure on Γ(X, I∞,η ). A global section of I∞,η is a pair (q(w), p(z)) with q(w) ∈ C[w] and p(z) ∈ C[z, z −1 ] such that q(1/z) = p(z), so as a g-module, Γ(X, I∞,η ) is isomorphic to C[w] with E, F , and H actions given by formulas (8.4)-(8.6). Remark 8.4. We draw the reader’s attention to several properties of the gmodule Γ(X, I∞,η ) which can be seen from careful examination of Figures 10 and 11: (1) The module is generated by the vector n0 , which has the property that E acts by a scalar: E · n0 = ηn0 . For a general semisimple Lie algebra g, a vector in a g-module on which n acts by a character is called a Whittaker vector. A g-module which is cyclically generated by a Whittaker vector is a Whittaker module. Hence Γ(X, I∞,η ) is a Whittaker module. (2) The module is irreducible. (3) If η = 0, I∞,η = IU∞ is the standard Harish-Chandra sheaf attached to the open orbit U∞ whose structure we described in Section 6. (4) If η = 0 then the basis {nk } is not a basis of eigenvectors of H. In fact, it is impossible to choose a basis of H-eigenvectors for the module because it is not a weight module. (5) The Casimir operator Ω = H 2 + 2EF + 2F E ∈ Z(g) acts on the module10 by (t − 1)2 + 2(t − 1). 10 In fact, Ω acts on the global sections of any D -module by (t−1)2 +2(t−1) by construction. t This can be checked in each chart with a quick computation using (4.6) - (4.11) and the relationshps [∂z , z] = [∂w , w] = 1. The integer (t − 1)2 + 2(t − 1) is the image of Ω under the infinitesimal character χ : Z(g) → C, z → (λ − ρ)(p0 (z)), where p0 is the Harish-Chandra homomorphism.

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If η = 0, the module Γ(X, I∞,η ) is an irreducible nondegenerate11 Whittaker module. Irreducible nondegenerate Whittaker modules were introduced and classified in [Kos78, §3]. Theorem 8.5. Let t ∈ Z≥1 , η ∈ C and I∞,η the standard η-twisted HarishChandra sheaf for Dt . Then as g-modules, Γ(X, I∞,η )  Y (η, t), where Y (η, t) is the irreducible η-Whittaker module of infinitesimal character χ : Ω → (t − 1)2 + 2(t − 1). References Alexandre Be˘ılinson and Joseph Bernstein, Localisation de g-modules (French, with English summary), C. R. Acad. Sci. Paris S´er. I Math. 292 (1981), no. 1, 15–18. MR610137 [Har77] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR0463157 [HMSW87] Henryk Hecht, Dragan Miliˇ ci´ c, Wilfried Schmid, and Joseph A. Wolf, Localization and standard modules for real semisimple Lie groups. I. The duality theorem, Invent. Math. 90 (1987), no. 2, 297–332, DOI 10.1007/BF01388707. MR910203 [Kos78] Bertram Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184, DOI 10.1007/BF01390249. MR507800 [LV83] George Lusztig and David A. Vogan Jr., Singularities of closures of K-orbits on flag manifolds, Invent. Math. 71 (1983), no. 2, 365–379, DOI 10.1007/BF01389103. MR689649 [Mila] D. Miliˇ ci´ c, Lectures on Algebraic Theory of D-Modules, Unpublished manuscript available at http://math.utah.edu/~milicic. [Milb] D. Miliˇ ci´ c, Localization and representation theory of reductive Lie groups, Unpublished manuscript available at http://math.utah.edu/~milicic. [Mil91] D. Miliˇ ci´ c, Intertwining functors and irreducibility of standard Harish-Chandra sheaves, Harmonic Analysis on Reductive Groups, pages 209–222. Birkh¨ auser, Boston, 1991. [Mil93] John W. Rice, Cousin complexes and resolutions of representations, The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992), Contemp. Math., vol. 154, Amer. Math. Soc., Providence, RI, 1993, pp. 197– 215, DOI 10.1090/conm/154/01364. MR1246385 [MS14] Dragan Miliˇ ci´ c and Wolfgang Soergel, Twisted Harish-Chandra sheaves and Whittaker modules: the nondegenerate case, Developments and retrospectives in Lie theory, Dev. Math., vol. 37, Springer, Cham, 2014, pp. 183–196, DOI 10.1007/978-3-319-09934-7 7. MR3329939 [Rom20] Anna Romanov, A Kazhdan-Lusztig Algorithm for Whittaker Modules, ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–The University of Utah. MR4035117 [Vog82] David A. Vogan Jr., Irreducible characters of semisimple Lie groups. IV. Charactermultiplicity duality, Duke Math. J. 49 (1982), no. 4, 943–1073. MR683010 [BB81]

A14 Quadrangle, Sydney Mathematical Research Institute, The University of Sydney, New South Wales, Australia 2006 Email address: [email protected]

11 For a general semisimple Lie algebra g, a character of n is nondegenerate if it is non-zero on all simple root subspaces. For sl(2, C), all non-zero n-characters are nondegenerate because there is only one simple root.

Number theory

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15456

Perfect powers in polynomial power sums Clemens Fuchs and Sebastian Heintze Abstract. We prove that a non-degenerate simple linear recurrence sequence (Gn (x))∞ n=0 of polynomials satisfying some further conditions cannot contain arbitrary large powers of polynomials if the order of the sequence is at least two. In other words we will show that for m large enough there is no polynomial h(x) of degree ≥ 1 such that (h(x))m is an element of (Gn (x))∞ n=0 . The bound for m depends here only on the sequence (Gn (x))∞ n=0 . In the binary case we prove even more. We show that then there is a bound C on the index n of the sequence (Gn (x))∞ n=0 such that only elements with index n ≤ C can be a proper power.

1. Introduction An interesting question that was studied in several recent papers (e.g. cf. [1, 5–7, 16, 18, 21–23] and also [7–9, 13, 15]) is what one can say about the decomposition of complex polynomials (i.e. elements of the ring C[x] of complex polynomials) regarding the composition operation. The invertible elements w.r.t. decomposition are the linear polynomials. We call f (x) = g ◦ h a non-trivial decomposition if neither g nor h is linear. We call f (x) = g ◦ h an m-decomposition if deg g = m and we say that f is mdecomposable if an m-decomposition exists. We call f indecomposable if f admits only trivial decompositions. A pair (g, h) is called equivalent to (g  , h ) if there are a, b ∈ C, a = 0 such that g(x) = g  (ax + b), h(x) = (h (x) − b)/a. A pair (g, h) is called cyclic if it is equivalent to (g  , xm ) and dihedral if it is equivalent to (g  , Tm (x)) where (Tn )∞ n=0 denotes the sequence of Chebyshev polynomials (defined by Tn (x + 1/x) = xn + 1/xn ) and g  , g  ∈ C[x]. We call f cyclic if it is equivalent to a polynomial g with g(x) = xn for some n > 1, and dihedral if it is equivalent to g = Tn for some n > 2; here equivalent means that there are linear polynomials l1 , l2 such that f = l1 ◦ g ◦ l2 . Mainly, one is interested in non-trivial decompositions (with two factors, an inner and an outer factor) of polynomials with coefficients in C. It is natural to restrict to a subset I of C[x] which is described by a finite amount of data and then to ask whether or not all decompositions in this subset can be described in finite terms depending on the data describing the subset. In [21] and [22] Zannier studied such decompositions with special focus on the number  of terms of the polynomial f . Let us consider for a moment lacunary polynomials, i.e. we set I = {f ∈ C[x]; f has at most  non-constant terms}. Motivated 2020 Mathematics Subject Classification. Primary 11B37, 12Y05, 11R58. c 2021 American Mathematical Society

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by previous work of Erd˝os [3] and Schinzel [17], Zannier [22] finally proved that there are integers p, J depending on  and for every 1 ≤ j ≤ J an algebraic variety Vj defined over Q and a lattice Λj for which equations can be written down explicitly and (Laurent-)polynomials fj , hj ∈ Q[Vj ][z1±1 , . . . , zp±1 ], gj ∈ Q[Vj ][z] with coefficients in the coordinate ring of the variety such that the following holds: a) gj ◦ hj = fj is a (Laurent-)polynomial with  non-constant terms with coefficients in the coordinate ring; b) for every point P ∈ Vj (C) and (u1 , . . . , up ) ∈ Λj one gets a decomposition fj (P, xu1 , . . . , xup ) = gj (P, hj (P, xu1 , . . . , xup )); c) conversely, for every polynomial f ∈ C[x] with  non-constant terms and every non-trivial decomposition f (x) = g ◦ h with h(x) not of the shape axm + b for m ∈ N, a, b ∈ C there is a j, a point P ∈ Vj (C) and (u1 , . . . , up ) ∈ Λj such that f (x) = fj (P, xu1 , . . . , xup ), g(x) = gj (P, x), h(x) = hj (P, xu1 , . . . , xup ). This result is based on an intermediate result [21] that the outer decomposition factor has degree bounded explicitly in terms of  unless the inner decomposition factor is cyclic. Another instance of this approach is given by I = {Gn (x); n ∈ N}, where Gn (x) are elements of a linear recurrence sequence (Gn )∞ n=0 of polynomials in C[x]. To fix terms we shall assume that the recurrence is given by Gn+d (x) = Ad−1 (x)Gn+d−1 (x) + · · · + A0 (x)Gn (x), with A0 , . . . , Ad−1 ∈ C[x] and initial terms G0 , . . . , Gd−1 ∈ C[x]. Denote by α1 , . . . , αt the distinct characteristic roots of the sequence, that is the characteristic polynomial G ∈ C(x)[T ] splits as G(T ) = T d −Ad−1 T d−1 −· · ·−A0 = (T −α1 )k1 (T −α2 )k2 · · · (T −αt )kt , where k1 , . . . , kt ∈ N. Then Gn (x) admits a representation of the form Gn (x) = a1 α1n + a2 α2n + · · · + at αtn . We say that the recurrence is minimal if (Gn )∞ n=0 does not satisfy a recurrence relation with smaller d and coefficients in C[x]. We say that the recurrence is nondegenerate if αi /αj ∈ C∗ for all i = j. We say that the recurrence is simple if k1 = · · · = kt = 1; in this case the ai ’s lie in C(x, α1 , . . . , αt ). We say that the recurrence is a polynomial power sum if a1 , . . . , ad ∈ C and α1 , . . . , αd ∈ C[x]. We say that a polynomial power sum satisfies the dominant root condition if deg(α1 ) > deg(αi ) for i > 1. As an important starting point and motivation we mention that for a given sequence (Gn )∞ n=0 the decompositions of the form Gn (x) = Gm ◦ h for a fixed polynomial h ∈ C[x], deg h ≥ 2 were considered by Peth˝o, Tichy and the first author in a series of papers [4, 10–12]. It was again Zannier [20] who proved in general that this equation has only finitely many solutions (n, m), n = m, unless h is cyclic or dihedral; in this case there are infinitely many solutions coming from a generic equation. Moreover, one has to take the following trivial situations into account: If Gm (x) ∈ C[h(x)] for every m ∈ N, then it is not possible to bound the degree of g independently of n assuming Gn = g ◦ h. If Gn (x) = g(Hn (x)) with g ∈ C[x], deg g = m and (Hn )∞ n=0 is another linear recurrence sequence in C[x], then obviously we again have a sought decomposition for every n ∈ N. Consider as a nice example the Fibonacci polynomials Fn defined by F0 (x) = 0, F1 (x) = 1, Fn+2 (x) = xFn+1 (x) + Fn (x). It is easy to see that for all odd n ≥ 3, Fn is an even polynomial of degree n − 1, and hence if n ≥ 5 is odd, Fn (x) can be written as Fn (x) = g ◦ h, where h(x) = x2 and deg g = (n − 1)/2. Observe that h is cyclic and that the degree of g cannot be bounded independently of n assuming Fn (x) = g(h(x)) and deg h > 1. Also, for Chebyshev polynomials Tn

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it is well-known that Tmn (x) = Tm ◦ Tn for any m, n ∈ N. Observe that h is dihedral and, since deg Tn = n, one cannot bound deg g independently of n assuming Tn (x) = g(h(x)) and deg h > 1. The main result of the first author proved together with Karolus and Kreso in [6] is the following: Let (Gn )∞ n=0 be a minimal non-degenerate simple second order linear recurrence sequence. Assume that Gn is decomposable for some n ∈ N and write Gn (x) = g(h(x)), where h is indecomposable. Since deg h ≥ 2, there exists a root y = x in its splitting field over C(h(x)). Clearly, h(x) = h(y). We have Gn (x) = π1 α1n + π2 α2n . Conjugating (in some fixed algebraic closure of C(x) containing α1 , α2 ) over C(h(x)) via x → y, we get a sequence (Gn (y))∞ n=0 with Gn (y) ∈ C[y], which satisfies the same minimal non-degenerate simple recurrence n relation as (Gn (x))∞ n=0 with x replaced by y. We conclude that Gn (y) = ρ1 β1 + n ρ2 β2 . Since h(x) = h(y), we get Gn (x) = Gn (y), that is ()

π1 α1n + π2 α2n = ρ1 β1n + ρ2 β2n .

Then there is a positive real constant C = C({Ai , Gi ; i = 1, 2}) with the following property: If for some n we have Gn (x) = g(h(x)), where h is indecomposable and neither dihedral nor cyclic, and if () has no proper vanishing subsum, then it holds that deg g ≤ C. We remark that if h is not cyclic, then equation () has a proper vanishing subsum if and only if π1 π2 A0 (x)n ∈ C(h(x)). In particular, the existence of a proper vanishing subsum does not depend on the choice of the conjugate y of x over C(h(x)). However, () clearly depends on n and h for which Gn (x) = g(h(x)) which are not known a priori. Note that if h is not cyclic and A0 (x) = a0 ∈ C, π1 π2 = π ∈ C, then there exists a vanishing subsum of () and one cannot apply the theorem in question; for example, this is the case for Chebyshev polynomials Tn . It is possible to give sufficient conditions in which () has no proper vanishing subsum. We do not give further details here. Furthermore in [5] the first author and Karolus proved the following: Let (Gn )∞ n=0 be a non-degenerate polynomial power sum which satisfies the dominant root condition. Moreover, let m ≥ 2 be an integer. Write m0 for the least integer m /m such that α1 0 ∈ C[x]. Then there is an effectively computable positive constant C such that the following holds: Assume that for some n ∈ N with n > C we have Gn (x) = g ◦ h with deg g = m, deg h > 1. Then there are c1 , . . . , cl ∈ C such that h(x) = c1 γ1 + · · · + cl γl , where m0  = n and l ∈ N is bounded explicitly in terms of m, d and deg(α1 ) + · · · + deg(αd ) and γ1 , . . . , γl ∈ C(x) can be given explicitly in terms of α1 , . . . , αd , both independently of n. Furthermore, it follows that there is an explicitly computable positive constant C, and a subvariety V of Al+m+1 ×Gtm with t, l bounded explicitly in terms of m, d and deg(α1 ) + · · · + deg(αd ) for which a system of polynomialexponential equations in the polynomial variables c1 , . . . , cl , g0 , . . . , gm and the exponential variable  (with coefficients in Q) can be written down explicitly such that the following holds: a) Defining G(x) = g0 xm + g1 xm−1 + · · · + gm ∈ C[V][x] and H = c1 γ1 + c2 γ2 + · · · + cl γl ∈ C[V][x], where γ1 , . . . , γl ∈ C(x) can be given explicitly in terms of α1 , . . . , αd , then Gm0  = G ◦ H holds as an equation in x with coefficients in the coordinate ring of V. In particular, for any point P = (c1 , . . . , cl , g0 , . . . , gm , ) ∈ V(C) we get a decomposition Gn (x) = g ◦ h, g(x) = G(P, x) ∈ C[x] and h(x) = Hl (P, x) ∈ C[x] (with n = m0 ).

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b) Conversely, let Gn (x) = g ◦ h be a decomposition of Gn (x) for some n ∈ N with g, h ∈ C[x], deg g = m, deg h > 1. Then either n ≤ C or there exists a point P = (g0 , . . . , gm , c1 , . . . , cl , ) ∈ V(C) with g(x) = G(P, x) and h(x) = H (P, x) and n = m0 . A number of interesting special cases follow and are discussed, in particular that the results include a description in finite terms of all m-th powers in a linear recurrence sequence of polynomials satisfying the conditions of the theorem. Observe that in [6] only binary recurrences are covered and that in [5] the order is not restricted but instead only polynomial power sums satisfying the dominant root condition are handled. In this paper we revisit the situation when the outer polynomial is fixed to be g(x) = xm . We first quickly review the situation for lacunary polynomials. Here it is natural to consider a non-constant complex polynomial with constant term equal to 1 and with k additional non-constant terms. The results [18] and [22] immediately imply that then m ≤ k. For k ≤ 3 a precise classification of all solutions can be found in [2] (see Lemma 2.1). The analogous result for k = 4 was given in the recent PhD thesis [14]. Now we turn back to polynomial power sums. Assuming the dominant root condition we will prove that the second case which states that h is of a special form cannot occur in this setting. For the binary case we will be able to prove a stronger result than for the general one of order greater than two. We are going to give a counterexample which shows that the stronger result is in general not true for the case of an arbitrary order d of the linear recurrence sequence. 2. Results During the whole paper we are implicitly assuming that the polynomial h(x) has degree deg h ≥ 1. Let us now first state our two results that we are going to prove in the next section: We start with the situation of binary recurrences. Theorem 2.1. Let (Gn (x))∞ n=0 be a non-degenerate simple linear recurrence sequence of order d = 2 with power sum representation Gn (x) = a1 α1n + a2 α2n such that α1 , α2 ∈ C[x] are polynomials and a1 , a2 ∈ C(x) satisfy aa21 ∈ C. Assume furthermore that deg α1 > deg α2 . Then there exists a constant C, which depends only on α1 , α2 , a1 , a2 , such that for all n > C there is no integer m ≥ 2 and no polynomial h(x) ∈ C[x] with the property Gn (x) = (h(x))m . In particular this implies that for m large enough there is no index n and no polynomial h(x) ∈ C[x] such that Gn (x) = (h(x))m . In the case of recurrences of arbitrary large order we prove a slightly weaker result (essentially, the final conclusion in the previous theorem). Theorem 2.2. Let (Gn (x))∞ n=0 be a non-degenerate simple linear recurrence sequence of order d ≥ 3 with power sum representation Gn (x) = a1 α1n + · · · + ad αdn such that α1 , . . . , αd ∈ C[x] are polynomials and a1 , . . . , ad ∈ C are constant. Assume furthermore that deg α1 > deg α2 ≥ deg α3 ≥ · · · ≥ deg αd . Then for m large enough there is no index n and no polynomial h(x) ∈ C[x] such that Gn (x) = (h(x))m . In the case of a non-degenerate simple linear recurrence sequence of order d ≥ 3 we cannot give in general a bound C for the index n such that all elements of the

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sequence (Gn (x))∞ n=0 with index n > C are no proper powers. Consider for instance the third order sequence given by Gn (x) = (xn + 1)2 = (x2 )n + 2xn + 1n which has the property that each element is at least a square. We can easily modify this example to generate counterexamples for any fixed parameter m if we consider Gn (x) = (xn + 1)m . For the proof we mainly follow the proof of [5]. Therefore, we start from Gn (x) = h(x)m . Thus h(x) = ζGn (x)1/m (as formal power series). Then one uses the multinomial series to expand Gn (x)1/m ; in order to justify this multiple expansion, the dominant root condition on the degrees of the characteristic roots is needed. Afterwards a function field variant of the Schmidt subspace theorem (due to Zannier, see Proposition 3.2 below) is used, to find that either n is bounded or h can be expressed in the form c0 t0 + c1 t1 + · · · + cL−1 tL−1 , where ci ∈ C and ti come from a finite set. We have to show that the latter case is impossible. Plugging the expression for h(x) into Gn (x) = h(x)m and comparing degrees and leading coefficients gives the result. The proof of the two theorems are quite similar, the difference involve some subtleties that we try to work out. We remark that the method of proof exactly requires (Gn (x))∞ n=0 to be a polynomial power sum with dominant root condition. 3. Preliminaries In the sequel we will need the following notations: For c ∈ C and f (x) ∈ C(x) where C(x) is the rational function field over C denote by νc (f ) the unique integer such that f (x) = (x−c)νc (f ) p(x)/q(x) with p(x), q(x) ∈ C[x] such that p(c)q(c) = 0. Further denote by ν∞ (f ) = deg q − deg p if f (x) = p(x)/q(x). These functions ν are up to equivalence all valuations on C(x). If νc (f ) > 0, then c is called a zero of f , and if νc (f ) < 0, then c is called a pole of f . For a finite extension F of C(x) each valuation on C(x) can be extended to no more than [F : C(x)] valuations on F . This again gives all valuations on F . Both, in C(x) as well as in F the sum-formula  ν(f ) = 0 

ν

holds, where ν means that the sum is taken over all valuations on the considered function field. Each valuation on a function field corresponds to a place and vice versa. The set of all places of the function field F is denoted by PF . If F  is a finite extension of F , then we say that P  ∈ PF  lies over P ∈ PF if P ⊆ P  and denote this fact by P  | P . In this case there exists an integer e(P  | P ), the so-called ramification index of P  over P , such that for all x ∈ F the equality νP  (x) = e(P  | P ) · νP (x) holds. To prepare the proofs of our two theorems we present subsequently three auxiliary results that are used in [5] as well. The first one also can be found in [19]: Proposition 3.1. Let F/C be a function field in one variable. Suppose that u ∈ F satisfies u = wd for all w ∈ F and d | n, d > 1. Let F  = F (z) with z n = u. Then F  is said to be a Kummer extension of F and we have: a) The polynomial ϕ(T ) = T n − u is the minimal polynomial of z over F (in particular, it is irreducible over F ). The extension F  /F is Galois of degree n; its Galois group is cyclic and all automorphisms of F  /F are given by σ(z) = ζz, where ζ ∈ C is an n-th root of unity.

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C. FUCHS AND S. HEINTZE

b) Let P ∈ PF and P  ∈ PF  be an extension of P . Let rP := gcd (n, νP (u)). Then e(P  |P ) = n/rP . c) Denote by g (resp. g ) the genus of F/C (resp. F  /C). Then g = 1 + n(g − 1) +

1  (n − rP ) deg P. 2 P ∈PF

The following proposition can be seen as a function field analogue of the Schmidt subspace theorem. It will play an important role in our proofs. The reader can find a proof for it in [22]: Proposition 3.2 (Zannier). Let F/C be a function field in one variable, of genus g, let ϕ1 , . . . , ϕn ∈ F be linearly independent over C and let r ∈ {0, 1, . . . , n}. Let S be a finite set of places  of F containing all the poles of ϕ1 , . . . , ϕn and all the zeros of ϕ1 , . . . , ϕr . Put σ = ni=1 ϕi . Then    n   n (|S| + 2g − 2) + ν(σ) − min ν(ϕi ) ≤ deg(ϕi ). 2 i=1,...,n i=r+1

ν∈S

In the next section we will take use of height functions in function fields. Let us therefore define the height of an element f ∈ F ∗ by H(f ) := −



min (0, ν(f )) =

ν



max (0, ν(f ))

ν

where again the sum is taken over all valuations on the function field F/C. Additionally we define H(0) = ∞. These height function satisfies some basic properties that are listed in the following lemma which is proven in [6]: Lemma 3.3. Denote as above by H the height function on F/C. Then for f, g ∈ F ∗ the following properties hold: a) b) c) d) e) f)

H(f ) ≥ 0 and H(f ) = H(1/f ), H(f ) − H(g) ≤ H(f + g) ≤ H(f ) + H(g), H(f ) − H(g) ≤ H(f g) ≤ H(f ) + H(g), H(f n ) = |n| · H(f ), H(f ) = 0 ⇐⇒ f ∈ C∗ , H(A(f )) = deg A · H(f ) for any A ∈ C[T ] \ {0}.

4. Proofs We are now ready to prove our two theorems. At this position we remark that our proofs are very similar to the proof of Theorem 1 in [5] where the same procedure is used. Proof of Theorem 2.1. Assume that there exists an index n, an integer m ≥ 2 and a polynomial h(x) such that Gn (x) = (h(x))m . Thus we have h(x) = ζ(Gn (x))1/m for an m-th root of unity ζ. Using the power sum representation of

PERFECT POWERS IN POLYNOMIAL POWER SUMS

95

Gn (x) as well as the binomial series expansion we get h(x) = ζ(Gn (x))1/m = ζ(a1 α1n + a2 α2n )1/m   n 1/m a2 α2 1/m n/m 1+ = ζa1 α1 a1 α1   h2  nh2 ∞   a2 α2 1/m 1/m n/m = ζa1 α1 h2 a1 α1 h2 =0

(4.1)

=

∞ 

th2 (x)

h2 =0

with the definition    h2  nh2 a2 α2 1/m h2 a1 α1  nh2 α2 1/m n/m = bh2 a1 α1 . α1 1/m n/m α1

th2 (x) : = ζa1

Since we have required in the theorem that  bh2 := ζ

a2 a1

∈ C, it holds that

  h2 a2 1/m ∈ C. h2 a1

Let now F = C(x, α1 (x)1/m ) and m0 be the smallest positive integer such that α1 (x)m0 /m ∈ C(x). Applying Proposition 3.1 we get that F is a Kummer extension of C(x) and that T m0 − α1 (x)m0 /m is the minimal polynomial of α1 (x)1/m over C(x). Moreover, we get that only places in F above ∞ and the roots of α1 as a polynomial in C(x) can ramify. Since our field of constants is C and therefore algebraically closed, we have deg P = 1 for all places P . Combined with gC(x) = 0 the genus formula of Proposition 3.1 yields 2gF − 2 = 2m0 (gC(x) − 1) + ≤ −2m0 +





(m0 − rP ) deg P

P ∈PC(x)

m0

P ∈PC(x) :m0 >rP

≤ −2m0 + m0 (1 + deg α1 ) = m0 (deg α1 − 1). Moreover, let F  = F (a1 ) and m1 be the smallest positive integer such that m1 /m a1 ∈ F . Again the application of Proposition 3.1 yields that F  is a Kummer extension of F . Furthermore, we get that only places in F  above ∞ and the zeros and poles of a1 as an element of C(x) can ramify. Since our field of constants is C and therefore algebraically closed, we have deg P = 1 for all places P . Combined 1/m

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C. FUCHS AND S. HEINTZE

with the bound on gF the genus formula of Proposition 3.1 yields 2gF  − 2 = m1 (2gF − 2) +



(m1 − rP ) deg P

P ∈PF



≤ m1 m0 (deg α1 − 1) +

m1

P ∈PF :m1 >rP

≤ m1 m0 (deg α1 − 1) + m1 m0 (1 + 2H(a1 )) = m1 m0 (deg α1 + 2H(a1 )). The next step is to estimate the valuation of the th2 corresponding to the infinite place of C(x). We get the following lower bound:  ν∞ (th2 (x)) = ν∞

 1/m n/m a1 α1

α2 α1

nh2 

  1 1 ν∞ (a1 ) + n ν∞ (α1 ) + h2 (ν∞ (α2 ) − ν∞ (α1 )) m m   − deg α1 1 + h2 (deg α1 − deg α2 ) = ν∞ (a1 ) + n m m   deg α1 1 . ≥ ν∞ (a1 ) + n h2 − m m =

Let J ∈ N be arbitrary. Therefore for h2 ≥ J + degmα1 we have ν∞ (th2 (x)) ≥ 1 m ν∞ (a1 ) + nJ. This allows us to split the above sum representation (4.1) for h(x) in the following way: h(x) = t0 (x) + t1 (x) + · · · + tL−1 (x) +



th2 (x)

1 ν∞ (th2 (x))≥ m ν∞ (a1 )+nJ

with L − 1 < J + degmα1 . We now distinguish between two cases which will be handled in completely different ways. First we assume that {h(x), t0 (x), . . . , tL−1 (x)} is linearly independent over C. Later we will consider the case that {h(x), t0 (x), . . . , tL−1 (x)} is linearly dependent over C. So let us now assume that {h(x), t0 (x), . . . , tL−1 (x)} is linearly independent over C. We aim to apply Proposition 3.2. To do so let us fix a finite set S of places of F  which contains all zeros and poles of t0 (x), . . . , tL−1 (x) as well as all poles of h(x). Therefore S can be chosen in a way such that it contains at most the places above ∞, the zeros of α1 and α2 and the zeros and poles of a1 . This gives an upper bound on the number of elements in S: |S| ≤ m1 m0 (1 + deg α1 + deg α2 + 2H(a1 )). Further we write ϕ0 = −t0 (x), . . . , ϕL−1 = −tL−1 (x) and ϕL = h(x). We also  L  define σ = 1 i=0 ϕi = ν∞ (th2 (x))≥ m ν∞ (a1 )+nJ th2 (x). Since deg(h(x)) = [F :  C(h(x))] = [F : F ] · [F : C(h(x))] = m1 H(h(x)) = m1 deg h · H(x) = m1 deg h · [F :

PERFECT POWERS IN POLYNOMIAL POWER SUMS

97

C(x)] = m1 m0 deg h Proposition 3.2 implies   ν(σ) − min ν(ϕi ) ≤ ν∈S

i=0,...,L

  L+1 (|S| + 2gF  − 2) + deg(h(x)) ≤ 2 1 ≤ L(L + 1)m1 m0 (1 + 2 deg α1 + deg α2 + 4H(a1 )) + m1 m0 deg h 2 ≤ L(L + 1)m1 m0 (1 + deg α1 + deg α2 + 2H(a1 )) + m1 m0 deg h.

On the other hand we have ν(σ) ≥ mini=0,...,L ν(ϕi ) for every valuation ν and thus the lower bound     ν(σ) − min ν(ϕi ) ≥ νP (σ) − min νP (ϕi ) ν∈S

i=0,...,L

i=0,...,L

P |∞

≥



(νP (σ) − νP (h(x)))

P |∞

=



νP (σ) −

P |∞

=





e(P | ∞) · ν∞ (h(x))

P |∞

νP (σ) − m1 m0 ν∞ (h(x))

P |∞

=



e(P | ∞) · ν∞ (σ) + m1 m0 deg h

P |∞

= m1 m0 ν∞ (σ) + m1 m0 deg h m1 m0 ν∞ (a1 ) + m1 m0 nJ + m1 m0 deg h. ≥ m Let us now compare the upper and lower bounds. Since m1 m0 deg h appears on both sides, we can subtract it and get m1 m0 ν∞ (a1 ) + m1 m0 nJ ≤ L(L + 1)m1 m0 (1 + deg α1 + deg α2 + 2H(a1 )). m Dividing by m1 m0 and isolating the term containing n yields nJ ≤ L(L + 1)(1 + deg α1 + deg α2 + 2H(a1 )) + |ν∞ (a1 )| . Since J ∈ N was arbitrary we can now choose J = 1. Remember that L − 1 < J + degmα1 and therefore L ≤ 1 + J + deg α1 = 2 + deg α1 . Hence (4.2)

n ≤ (2 + deg α1 )(3 + deg α1 )(1 + deg α1 + deg α2 + 2H(a1 )) + |ν∞ (a1 )| .

After this we consider now the case that {h(x), t0 (x), . . . , tL−1 (x)} is linearly dependent over C. We can assume that {t0 (x), . . . , tL−1 (x)} is linearly independent, since otherwise we are able to group them together and the first case is still working if the th2 (x) have constant coefficients. This implies that in a relation of linear dependence h(x) has a nonzero coefficient. Thus there exist complex numbers ci ∈ C such that (4.3)

h(x) =

L−1  i=0

ci ti (x).

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C. FUCHS AND S. HEINTZE

What we are doing subsequently is a reverse induction. We will show cL−1 = 0, then cL−2 = 0 and so on until only c0 remains. During the following calculations we will use the abbreviations β1 := α1n and β2 := α2n . Furthermore let di = bi ci . We start with equation (4.3) and get h(x) = c0 t0 (x) + · · · + cL−1 tL−1 (x) 1/m n/m α1

= d0 a1 =

1/m 1/m d0 a1 β1

 =

1/m 1/m a 1 β1



1/m n/m α1

+ d1 a1 +

1/m 1/m β2 d1 a1 β1 β1

α2 α1

n

 1/m n/m α1

+ · · · + dL−1 a1

 1/m 1/m dL−1 a1 β1

+ ···+

β2 d0 + d1 + · · · + dL−1 β1



β2 β1

L−1 

β2 β1

α2 α1

n(L−1)

L−1

as well as  a1 β1 + a2 β2 = (h(x)) m(L−1)

Multiplying with β1 1+m(L−1)

β1

+

m

= a 1 β1

β2 d0 + d1 + · · · + dL−1 β1



β2 β1

L−1 m .

and dividing by a1 yields

m a2 m(L−1) β1 β2 = β1 d0 β1L−1 + d1 β1L−2 β2 + · · · + dL−1 β2L−1 a1 1+m(L−1)

m(L−1)

= dm + mdm−1 d1 β1 β2 0 β1 0    m m−2 2 m(L−1)−1 2 d0 d1 + mdm−1 + d2 β1 β2 0 2 m(L−1)

+ · · · + dm L−1 β1 β2

.

Now we take a closer look at the coefficients of the monomials β1i β2j in the above equation. We can rewrite the last equation as   a2 1+m(L−1) m(L−1) m−1 β1 (1 − dm )β = md d − β2 1 0 1 0 a1    m m−2 2 m(L−1)−1 2 d0 d1 + mdm−1 (4.4) + d β2 2 β1 0 2 m(L−1)

+ · · · + dm L−1 β1 β2

.

The left hand side of this equation is either zero or a polynomial of degree equal to (1 + m(L − 1)) deg β1 , whereas the polynomial on the right hand side can have at most degree m(L − 1) deg β1 + deg β2 . Since they are equal both sides must be zero. We get 1 − dm 0 = 0 and rewrite the expression on the right side of (4.4) as 

    a2 m m−2 2 m(L−1) m(L−1)−1 2 m−1 d β − mdm−1 d β = d + md d β2 1 2 2 β1 1 1 0 0 0 2 a1 m(L−1)

+ · · · + dm L−1 β1 β2

.

PERFECT POWERS IN POLYNOMIAL POWER SUMS

We apply the same argument as before to get procedure ends up with

a2 a1

99

− mdm−1 d1 = 0. Repeating this 0

1 − dm 0 =0 a2 − mdm−1 d1 = 0 0 a1

  m m−2 2 d0 d1 + mdm−1 d2 = 0 0 2 .. . dm L−1 = 0. It follows immediately that dL−1 = 0. Hence cL−1 = 0. Thus equation (4.3) reduces to L−2  ci ti (x). h(x) = i=0

Doing the same calculations again with this new sum or putting dL−1 = 0 and inspect the above calculations in more detail gives now step by step cL−2 = 0, . . . , c1 = 0. So it holds that h(x) = c0 t0 (x). Taking the m-th power we get n a1 α1n + a2 α2n = dm 0 a1 α1

which is equivalent to a2 n n α = (dm 0 − 1)α1 . a1 2 The left hand side is a polynomial of degree n deg α2 , but the right hand side is either zero or of degree n deg α1 . This is a contradiction. Therefore the linear dependent case cannot occur. Altogether we must be in the linear independent case and thus have the bound (4.2) for the index n. This proves the theorem.  The proof of the other theorem is very similar but has some subtle differences. Hence for the readers convenience we write it down in detail. Proof of Theorem 2.2. Assume that there exists an index n, an integer m > deg α1 and a polynomial h(x) such that Gn (x) = (h(x))m . Thus again we have h(x) = ζ(Gn (x))1/m for an m-th root of unity ζ. Using the power sum representation of Gn (x) as well as the multinomial series expansion we get h(x) = ζ(Gn (x))1/m = ζ(a1 α1n + · · · + ad αdn )1/m   n  n 1/m a2 α2 ad αd 1/m n/m 1+ = ζa1 α1 + ···+ a1 α1 a1 α1    nh2  hd  nhd ∞ h 2  a2 α2 αd ad 1/m n/m = ζa1 α1 gh2 ,...,hd ··· a1 α1 a1 α1 h2 ,...,hd =0

(4.5)

=

∞  h2 ,...,hd =0

th2 ,...,hd (x)

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C. FUCHS AND S. HEINTZE

with the definition



nh2  hd  nhd α2 αd ad ··· th2 ,...,hd (x) : = α1 a1 α1  nh2  nhd α2 αd n/m = bh2 ,...,hd α1 ··· . α1 α1 1/m n/m ζa1 α1 gh2 ,...,hd

a2 a1

h2 

Since we have required in the theorem that a1 , . . . , ad ∈ C, it holds that  h2  hd a2 ad 1/m bh2 ,...,hd := ζa1 gh2 ,...,hd ··· ∈ C. a1 a1 Let now F = C(x, α1 (x)1/m ) and m0 be the smallest positive integer such that α1 (x)m0 /m ∈ C(x). Applying Proposition 3.1 we get that F is a Kummer extension of C(x) and that T m0 − α1 (x)m0 /m is the minimal polynomial of α1 (x)1/m over C(x). Moreover we get that only places in F above ∞ and the roots of α1 as a polynomial in C(x) can ramify. Since our field of constants is C and therefore algebraically closed, we have deg P = 1 for all places P . Combined with gC(x) = 0 the genus formula of Proposition 3.1 yields  2gF − 2 = 2m0 (gC(x) − 1) + (m0 − rP ) deg P ≤ −2m0 +



P ∈PC(x)

m0

P ∈PC(x) :m0 >rP

≤ −2m0 + m0 (1 + deg α1 ) = m0 (deg α1 − 1). The next step is to estimate the valuation of the th2 ,...,hd corresponding to the infinite place of C(x). We get the following lower bound:   nh2  nhd  α2 αd n/m ν∞ (th2 ,...,hd (x)) = ν∞ α1 ··· α1 α1 ⎛ ⎞ d  1 = n ⎝ ν∞ (α1 ) + hj (ν∞ (αj ) − ν∞ (α1 ))⎠ m j=2 ⎞ ⎛ d  − deg α 1 + hj (deg α1 − deg αj )⎠ = n⎝ m j=2 ⎞ ⎛ ⎞ ⎛ d d   deg α 1⎠ hj − hj − 1⎠ . ≥ n⎝ ≥ n⎝ m j=2 j=2 d Let J ∈ N be arbitrary. Therefore for j=2 hj ≥ J + 1 we have the lower bound ν∞ (th2 ,...,hd (x)) ≥ nJ. This allows us to split the above sum representation (4.5) for h(x) in the following way:  h(x) = t1 (x) + t2 (x) + · · · + tL (x) + th2 ,...,hd (x) ν∞ (th2 ,...,hd (x))≥nJ

with L ≤ (J + 1) . We remark at this point that if J = 1 we can get a better bound. Let us therefore use the notation e1 := (0, . . . , 0), e2 := (1, 0, . . . , 0), e3 := (0, 1, 0, . . . , 0), d−1

PERFECT POWERS IN POLYNOMIAL POWER SUMS

101

. . ., ed := (0, . . . , 0, 1) ∈ Nd−1 . Among the t1 (x), . . . , tL (x) occur in this case at 0 most some elements of {te1 (x), . . . , ted (x)} and it holds L ≤ d. Therefore from now on we fix J = 1. We now distinguish between two cases which will be handled in completely different ways. First we assume that {h(x), t1 (x), . . . , tL (x)} is linearly independent over C. Later we will consider the case that {h(x), t1 (x), . . . , tL (x)} is linearly dependent over C. So let us now assume that {h(x), t1 (x), . . . , tL (x)} is linearly independent over C. We aim to apply Proposition 3.2. To do so let us fix a finite set S of places of F which contains all zeros and poles of t1 (x), . . . , tL (x) as well as all poles of h(x). Therefore S can be chosen in a way such that it contains at most the places above ∞ and the zeros of α1 , . . . , αd . This gives an upper bound on the number of elements in S: ⎛ ⎞ d  |S| ≤ m0 ⎝1 + deg αj ⎠ . j=1

Further we write ϕ1 = −t1 (x), . . . , ϕL = −tL (x) and ϕL+1 = h(x). We also  L+1 define σ = i=1 ϕi = ν∞ (th2 ,...,hd (x))≥nJ th2 ,...,hd (x). Since deg(h(x)) = [F : C(h(x))] = H(h(x)) = deg h · H(x) = deg h · [F : C(x)] = m0 deg h Proposition 3.2 implies   ν(σ) − min ν(ϕi ) ≤ ν∈S

i=1,...,L+1

  L+1 (|S| + 2gF − 2) + deg(h(x)) ≤ 2 ⎛ ⎞ d  1 deg αj ⎠ + m0 deg h ≤ L(L + 1)m0 ⎝deg α1 + 2 j=1 ≤ L(L + 1)m0

d 

deg αj + m0 deg h.

j=1

On the other hand we have ν(σ) ≥ mini=1,...,L+1 ν(ϕi ) for every valuation ν and thus the lower bound     ν(σ) − min ν(ϕi ) ≥ νP (σ) − min νP (ϕi ) ν∈S

i=1,...,L+1

i=1,...,L+1

P |∞

≥



(νP (σ) − νP (h(x)))

P |∞

=



P |∞

=



νP (σ) −



e(P | ∞) · ν∞ (h(x))

P |∞

νP (σ) − m0 ν∞ (h(x))

P |∞

=



P |∞

e(P | ∞) · ν∞ (σ) + m0 deg h

= m0 ν∞ (σ) + m0 deg h ≥ m0 nJ + m0 deg h.

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C. FUCHS AND S. HEINTZE

Let us now compare the upper and lower bounds. Since m0 deg h appears on both sides, we can subtract it and get m0 nJ ≤ L(L + 1)m0

d 

deg αj .

j=1

Dividing by m0 and remembering J = 1 as well as L ≤ d yields n ≤ d(d + 1)

(4.6)

d 

deg αj .

j=1

After this we consider now the case that {h(x), t1 (x), . . . , tL (x)} is linearly dependent over C. We can assume that {t1 (x), . . . , tL (x)} is linearly independent, since otherwise we are able to group them together and the first case is still working if the th2 ,...,hd (x) have constant coefficients. This implies that in a relation of linear dependence h(x) has a nonzero coefficient. Thus there exist complex numbers ci ∈ C such that (4.7)

h(x) =

d 

ci tei (x).

i=1

Here we have used the above mentioned restriction on the possible elements of {t1 (x), . . . , tL (x)}. During the following calculation we will use the abbreviation di = bei ci . We start with equation (4.7) and get

(4.8)

h(x) = c1 te1 (x) + · · · + cd ted (x)  n  n α2 αd n/m n/m n/m + · · · + dd α1 . = d1 α1 + d2 α1 α1 α1 n/m

n/m

must be an We see that h(x) is of the form α1 · R with an R ∈ C(x). Thus α1 element of C(x). By the definition of m0 it follows that m0 | n and therefore there n/m n/m exists an integer  such that n = m0 . Moreover α1 ∈ C(x) implies α1 ∈ C[x] since α1 is a polynomial. Let us now rewrite equation (4.8) as follows (4.9)

n/m

h(x) − d1 α1

=

n/m n α2

d2 α1

n/m

+ · · · + dd α1 α1n

αdn

.

The left hand side of equation (4.9) is a polynomial. So the right hand side must be, too. Since the denominator has degree n deg α1 and the numerator degree at n most m deg α1 + n deg α2 < n + n deg α2 ≤ n deg α1 , the only possibility is that both sides are zero. Hence n a1 α1n + · · · + ad αdn = (h(x))m = dm 1 α1

which is equivalent to n a2 α2n + · · · + ad αdn = (dm 1 − a1 )α1 .

The left hand side is a polynomial of degree at most n deg α2 , but the right hand side is either zero or of degree n deg α1 . Thus both sides must be zero. Therefore, by Corollary 2 in [8], the index n is bounded above by an effectively computable constant C  . Altogether we must be either in the linear independent case and thus have the bound (4.6) for the index n or in the linear dependent case and thus have the bound

PERFECT POWERS IN POLYNOMIAL POWER SUMS

103

C  for the index n. Hence we only need to choose m > deg α1 large enough such that for all ⎛ ⎞ d  n ≤ max ⎝d(d + 1) deg αj , C  ⎠ j=1

the polynomial Gn (x) is not an m-th power.



References [1] Arnaud Bodin, Decomposition of polynomials and approximate roots, Proc. Amer. Math. Soc. 138 (2010), no. 6, 1989–1994, DOI 10.1090/S0002-9939-10-10245-7. MR2596034 [2] Pietro Corvaja and Umberto Zannier, Finiteness of odd perfect powers with four nonzero binary digits (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 63 (2013), no. 2, 715–731. MR3112846 [3] P. Erd¨ os, On the number of terms of the square of a polynomial, Nieuw Arch. Wiskunde (2) 23 (1949), 63–65. MR0027779 [4] Clemens Fuchs, On the Diophantine equation Gn (x) = Gm (P (x)) for third order linear recurring sequences, Port. Math. (N.S.) 61 (2004), no. 1, 1–24. MR2040240 [5] C. Fuchs and C. Karolus, Composite values of polynomial power sums, Ann. Math. Blaise Pascal 26 (2019), 1–24. [6] Clemens Fuchs, Christina Karolus, and Dijana Kreso, Decomposable polynomials in second order linear recurrence sequences, Manuscripta Math. 159 (2019), no. 3-4, 321–346, DOI 10.1007/s00229-018-1070-8. MR3959265 [7] Clemens Fuchs, Vincenzo Mantova, and Umberto Zannier, On fewnomials, integral points, and a toric version of Bertini’s theorem, J. Amer. Math. Soc. 31 (2018), no. 1, 107–134, DOI 10.1090/jams/878. MR3718452 [8] Clemens Fuchs and Attila Peth¨ o, Effective bounds for the zeros of linear recurrences in function fields (English, with English and French summaries), J. Th´eor. Nombres Bordeaux 17 (2005), no. 3, 749–766. MR2212123 [9] Clemens Fuchs and Attila Peth˝ o, Composite rational functions having a bounded number of zeros and poles, Proc. Amer. Math. Soc. 139 (2011), no. 1, 31–38, DOI 10.1090/S0002-99392010-10684-6. MR2729068 [10] Clemens Fuchs, Attila Peth˝ o, and Robert F. Tichy, On the Diophantine equation Gn (x) = Gm (P (x)), Monatsh. Math. 137 (2002), no. 3, 173–196, DOI 10.1007/s00605-002-0497-9. MR1942618 [11] Clemens Fuchs, Attila Peth˝ o, and Robert F. Tichy, On the Diophantine equation Gn (x) = Gm (P (x)): higher-order recurrences, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4657–4681, DOI 10.1090/S0002-9947-03-03325-7. MR1990766 [12] Clemens Fuchs, Attila Peth¨ o, and Robert F. Tichy, On the Diophantine equation Gn (x) = Gm (y) with Q(x, y) = 0, Diophantine approximation, Dev. Math., vol. 16, SpringerWienNewYork, Vienna, 2008, pp. 199–209, DOI 10.1007/978-3-211-74280-8 10. MR2487694 [13] Clemens Fuchs and Umberto Zannier, Composite rational functions expressible with few terms, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 1, 175–208, DOI 10.4171/JEMS/299. MR2862037 [14] A. Moscariello, Lacunary polynomials and compositions, PhD thesis, University of Pisa, 2019. [15] Attila Peth˝ o and Szabolcs Tengely, On composite rational functions, Number theory, analysis, and combinatorics, De Gruyter Proc. Math., De Gruyter, Berlin, 2014, pp. 241–259. MR3220123 [16] James Rickards, When is a polynomial a composition of other polynomials?, Amer. Math. Monthly 118 (2011), no. 4, 358–363, DOI 10.4169/amer.math.monthly.118.04.358. MR2800347 [17] A. Schinzel, On the number of terms of a power of a polynomial, Acta Arith. 49 (1987), no. 1, 55–70, DOI 10.4064/aa-49-1-55-70. MR913764 [18] A. Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, Cambridge, 2000. With an appendix by Umberto Zannier. MR1770638

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[19] Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR1251961 [20] Umberto Zannier, On the integer solutions of exponential equations in function fields (English, with English and French summaries), Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 849–874. MR2111014 [21] Umberto Zannier, On the number of terms of a composite polynomial, Acta Arith. 127 (2007), no. 2, 157–167, DOI 10.4064/aa127-2-5. MR2289981 [22] Umberto Zannier, On composite lacunary polynomials and the proof of a conjecture of Schinzel, Invent. Math. 174 (2008), no. 1, 127–138, DOI 10.1007/s00222-008-0136-8. MR2430978 [23] Umberto Zannier, Addendum to the paper: “On the number of terms of a composite polynomial” [MR2289981], Acta Arith. 140 (2009), no. 1, 93–99, DOI 10.4064/aa140-1-6. MR2557855 University of Salzburg, Department of Mathematics, Hellbrunnerstr. 34, A-5020 Salzburg, Austria Email address: [email protected] University of Salzburg, Department of Mathematics, Hellbrunnerstr. 34, A-5020 Salzburg, Austria Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15457

The number of irregular Diophantine quadruples for a fixed Diophantine pair or triple Yasutsugu Fujita Abstract. This paper considers the number of extensions of a fixed Diophantine pair or triple to irregular Diophantine quadruples. The results give absolute upper bounds for the number under certain assumptions on a pair or triple.

1. Introduction A set of m positive integers {a1 , . . . , am } is called a Diophantine m-tuple if ai aj + 1 is a perfect square for all i, j with 1 ≤ i < j ≤ m. The first example {1, 3, 8, 120} of such a set of four numbers, i.e., of a Diophantine quadruple, was found by Fermat. More than three centuries later, Baker and Davenport (see [2]) proved that if {1, 3, 8, d} is a Diophantine quadruple, then d must be 120. On the other hand, Fermat’s quadruple can be regarded as a special case of the Diophantine quadruple {a, b, c, d+ } for a Diophantine triple {a, b, c}, where d+ := d+ (a, b, c) = a + b + c + 2abc + 2rst, √ √ √ r := ab + 1, s := ac + 1, t := bc + 1 (cf. [1], [21]). In other words, d+ (1, 3, 8) = 120. Such a quadruple is called regular, and the following conjecture is believed to be true. Conjecture 1.1. (cf. [1], [21]) Any Diophantine quadruple is regular. This conjecture is still open, while the folklore conjecture asserting that there does not exist a Diophantine quintuple, which follows from Conjecture 1.1, is now the theorem of He, Togb´e and Ziegler (see [26]). Note that d+ (a, b, c) is the smallest among all the possible fourth elements d with max{a, b, c} < d for a fixed Diophantine triple {a, b, c} (cf. [14, Proposition 1]). It is also to be noted that if we define d− := d− (a, b, c) = a + b + c + 2abc − 2rst for a Diophantine triple {a, b, c}, then any of ad− + 1, bd− + 1 and cd− + 1 is a perfect square. In addition, it is easy to check that 0 ≤ d− < c and that d− = 0 if and only if c = a + b + 2r, which is known to be the smallest among all the possible third elements c with max{a, b} < c for a fixed Diophantine pair {a, b}. Hence, we 2020 Mathematics Subject Classification. Primary 11D45, 11D09; Secondary 11B37. The author was supported by JSPS KAKENHI Grant Number 16K05079. c 2021 American Mathematical Society

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see that if c > a + b + 2r, then {a, b, c, d− } is a regular Diophantine quadruple with c = d+ (a, b, d− ). The Baker-Davenport theorem mentioned above has several generalizations. For example, any Diophantine quadruples {a, b, c, d} with a < b < c < d containing the following Diophantine pairs {a, b} or triples {a, b, c} are known to be regular: (i) (ii) (iii) (iv) (v) (vi)

{k − 1, k + 1} for an integer k > 1 (cf. [4], [12], [15], [18]); {k, 4k ± 4} for a positive √ integer k (cf. [17], [23]); {a, b} with a < b < a + 4 a (cf. [17]); {1, b} with b −

1 a prime√power (cf.  [22]); √ {a, b} with a a + 72 − 12 4a + 13 ≤ b ≤ 4a2 + a + 2 a (cf. [9]); {k, A2 k ± 2A, (A + 1)2 k ± 2(A + 1)} for positive integers k, A (cf. [10], [24], [25]).

Note that the results except for (v) do not need the assumption b < c < d, i.e., any Diophantine quadruple {a, b, c, d} containing the pair or triple in (i), (ii), (iii), (iv) or (vi) is regular. In general, even the finiteness of irregular Diophantine quadruples has not been shown yet. However, it is shown in [20] that the number of irregular Diophantine quadruples {a, b, c, d} with max{a, b, c} < d for a fixed {a, b, c} is at most 10. Such an upper bound is updated to 7 in [11]. The aim of this paper is to give upper bounds for the number of irregular Diophantine quadruples {a, b, c, d} for a fixed pair {a, b} or triple {a, b, c} without assuming max{a, b, c} < d. Denote by Q(a, b) (resp. Q(a, b, c)) the number of irregular Diophantine quadruples containing a fixed pair {a, b} (resp. a fixed triple {a, b, c}). As mentioned above, Q(a, b) = 0 or Q(a, b, c) = 0 holds for any pair in (i), (ii), (iii), (iv) or triple in (vi). The first result of this paper asserts that Q(a, b) = 0 holds for any pair in (v). Theorem 1.2. Let {a, b} be a Diophantine pair with   √ 7 1√ (1.1) 4a + 13 ≤ b ≤ 4a2 + a + 2 a. a a+ − 2 2 Then, Q(a, b) = 0. The following theorem gives a condition under which Q(a, b, c) = 0 holds. Theorem 1.3. Let {a, b, c} be a Diophantine triple with a < b and 200b4 ≤ c ≤ min{8.1a3.5 b4.5 , 256b5.5 }. Then, Q(a, b, c) = 0. Let a, b, r with a < r < b be positive integers such that ab + 1 = r 2 . We define the integer c = cλν by (1.2) cλν =

 √ √ √ √ √ 1 √ ( b + λ a)2 (r + ab)2ν + ( b − λ a)2 (r − ab)2ν − 2(a + b) 4ab

with ν a non-negative integer and λ ∈ {±}. Then, {a, b, cλν } is a Diophantine triple for each ν ≥ 1 and λ (see the beginning of Section 3). We next examine the extendibility of a Diophantine triple {a, b, cλν } or a Diophantine pair {a, b} which can be extended only to a Diophantine triple of the form {a, b, cλν }.

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Theorem 1.4. Let {a, b} be a Diophantine pair. Assume that any Diophantine triple {a, b, c} containing {a, b} can be obtained by adding a positive integer c satisfying c = cλν with ν a positive integer and λ ∈ {±}, where cλν is defined by (1.2). Then, the following hold: (1) Q(a, b) ≤ 30. (2) If c = cλ1 with λ ∈ {±}, then Q(a, b, c) ≤ 3. + (3) If c− 2 ≤ c ≤ c3 , then Q(a, b, c) ≤ 6. − (4) If c4 ≤ c ≤ c+ 4 , then Q(a, b, c) ≤ 1. + ≤ c ≤ c (5) If c− 5 6 , then Q(a, b, c) ≤ 2. − (6) If c7 ≤ c ≤ c+ 9 , then Q(a, b, c) ≤ 3. + ≤ c ≤ c (7) If c− 10 11 , then Q(a, b, c) ≤ 4. − (8) If c12 ≤ c ≤ c+ 13 , then Q(a, b, c) ≤ 5. , the Q(a, b, c) ≤ 6. (9) If c ≥ c− 14 Corollary 1.5. Let {a, b} be a Diophantine pair. If a < b ≤ 13a, then the assertions in Theorem 1.4 hold. In order to bound Q(a, b) in a range wider than (1.1), we ameliorate [6, Theorem 1.4] to show the following. Theorem 1.6. Let {a1 , b, c} and {a2 , b, c} be Diophantine triples with a1 < min{a2 , b},

max{a2 , b} < c < 16b3 .

Then, {a1 , a2 , b, c} is a Diophantine quadruple. Theorem 1.6 enables us to limit the possibilities for the third elements c (see Proposition 4.2). Therefore, we obtain an upper bound for Q(a, b) as follows. Theorem 1.7. Let {a, b} be a Diophantine pair with (1.3)

a < b ≤ 4a3 + 16a2 + 22a + 8.

Then, Q(a, b) ≤ 76. The proofs of theorems stated above, including the proofs of results used there, need basic properties of (generalized) Pellian equations, especially Nagell’s argument (see [30, Theorem 108a] and [13, Lemma 1]), and of Diophantine quadruples remarkably developed in [14]. Note that a close investigation of the properties above is crucial for reducing the upper bound for Q(a, b) in Theorem 1.7 (see the proofs of Theorems 1.6, Lemma 4.1 and Propositions 4.2, 4.3). Besides them, key ingredients used in this paper are: • Baker’s method on linear forms in two logarithms (see the proof of Theorem 1.2 in Section 2); • the hypergeometric method explicitly described in [32] and [3] (see the proof of Proposition 4.3 in Section 4); • the Baker-Davenport reduction method (see the proofs of Theorem 1.2 and Proposition 4.3). 2. Proofs of Theorems 1.2 and 1.3 Proof of Theorem 1.2. We may assume that c < d. If b < c, then the assertion follows from [9, Theorem 1.1]. Thus, it remains to consider the case where c < b. Then, by [9, Lemma 3.1] and the proof of [9, Theorem 1.3], either

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2 2 4 3 c = c− 1 or c ∈ {1, T − 1} with {a, b} = {T + 2T, 4T + 8T − 4T } for some positive integer T . Assume first that {a, b} = {T 2 + 2T, 4T 4 + 8T 3 − 4T } for some T . If c = 2 T − 1, then the assertion follows from [17, Corollary 1.6]. If c = 1, then we obtain {a, c, b} = {1, T 2 − 1, 4T 4 + 8T 3 − 4T }. If we redefine a := 1 and b := T 2 − 1, then it holds 4T 4 + 8T 3 − 4T = c− 2 . It follows from [22, Theorem] that {a, b, c, d} is regular. Assume second that c = c− 1 . If   7 7 1√ 4a + 13 + 2a3/2 + a1/2 , b≥a a+ − 2 2 2 √ then from the fact that c = a + b − 2 ab + 1 is an increasing function of b, it is not difficult to see that   7 1√ c>a a+ − 4a + 13 . 2 2

It follows from [9, Theorem 1.1] that {a, b, c, d} is regular. Hence, we may assume that   2 7 1√ (2.1) 4a + 13 < 2a3/2 + a1/2 . 0≤b−a a+ − 2 2 7 In this case, noting that s = s− 1 = r − a, we have 1 7 1√ s>a a+ − 4a + 13 − a 2 2 13 3 83 > a3/2 − a + a1/2 − , 8 56 2 2    √ 7 1 1 s < a2 a + − +1−a 4a + 13 + 2a3/2 a + 2 2 7 13 1 83 < a3/2 − a + a1/2 − 2 8 56

for a ≥ 26,

in other words, inequalities (2.2)

  13 3 83 0 < s − a3/2 − a + a1/2 − 4000 (cf. [7, Lemma 3.4]). Now, following the argument in the proof of [9, Theorem 1.1] with b and c switched, the upper bounds for a and c can be easily obtained. More precisely, let x, y, z be positive integers satisfying ad + 1 = x2 ,

cd + 1 = y 2 ,

bd + 1 = z 2 .

Eliminating d from these equalities, we obtain the following system of Pellian equations: ay 2 − cx2 = a − c, az 2 − bx2 = a − b.

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109

By [9, Lemma 4.1], one can express x as x = Vl = Wm for some even integers l and m, where {Vl } and {Wm } are recurrence sequences defined by V0 = 1, V1 = s + a, Vl+2 = 2sVl+1 − Vl , W0 = 1, W1 = r ± a, Wm+2 = 2rWm+1 − Wm . The standard technique tells us that (2.3)

0 < Λ := l log β − m log α + log χ < α1−2m ,

where √ α = r + ab,

β =s+

√

ac,

√ √ √ b( c + a) √ χ= √ √ . c( b ± a)

Using the fact that b and c = c− 1 = a + b − 2r are close to each other, one can obtain the lower bound for m: 1 c (2.4) log β, m>Δ a where Δ := l − m ≥ 2. We rewrite Λ as 

α Λ = log β Δ χ − m log β and apply Laurent’s theorem ([28, Theorem 2]) with b1 = m, b2 = 1, α1 = α/β, α2 = β Δ χ. Suppose now that b > 1014 , which means a > 9 · 106 . Since 1 a α < < (0.998b)−1/4 = log log α1 β c and

 1 1 h(α1 ) = log α, h(χ) ≤ log cb2 (b − a) , 2 4 with the notation used in [28, Theorem 2] it suffices to have a1 > 4 log α + (ρ − 1)(0.998b)−1/4 ,

 a2 ≥ Δ(ρ + 3) log β + 2 log cb2 (b − a) + (ρ − 1) log χ. Putting ρ = 37 and μ = 0.63, we may take a1 = 4.001 log α, a2 = (40Δ + 16) log β,   m + 9.999Δ + 4 + 11.913, h = 4 log (40Δ + 16) log β which are almost the same as [9, (17)] (the difference is only that “9.998” is replaced by “9.999” in h). If h ≤ 28.74, then inequality (2.4) together with Δ ≥ 2 implies that m/a2 < 67.14. If h > 28.74, then comparing the inequality asserted in [28, Theorem 2] with (2.3) shows that m/a2 < 67.26. Thus, in both cases, it follows from (2.4) that c/a < 1.0424 · 107 . Since c = a + b − 2r > 0.999a2 by (2.1) with a > 9 · 106 , we obtain a < 1.044 · 107 and c < 1.089 · 1014 . Although these ranges are similar to the ones obtained in the proof of [9, Theorem 1.1], we in fact need shorter time to perform the reduction procedure based on [2, Lemma], developed in [15, Lemma 5], because s runs over the range (2.2) only with width a. Starting with the bound m < M = 1021 obtained by using Matveev’s result ([29, Corollary 2.3]), the computation time for the reduction was around 20 days. In any case, a program implemented in PARI/GP ([31]) showed

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3 √ m < 6. On the other hand, inequality (2.4) yields m > 2 c/a log(2 ac) > 83 for a ≥ 58, which is a contradiction. This completes the proof of Theorem 1.2  Proof of Theorem 1.3. In view of [11, Theorem 1.4], it suffices to show that if c ≤ min{8.1a3.5 b4.5 , 256b5.5 }, then there does not exist an integer d with d < c such that {a, b, c, d} is an irregular Diophantine quadruple. Assume that {a, b, c, d} is an irregular Diophantine quadruple with d < c. Then, [20, Lemma 4.2] implies that c > min{8.1a3.5 b4.5 , 256b5.5 }. The contraposition of this assertion gives the upper bound for c, as in the statement of the theorem.  3. Proof of Theorem 1.4 Let a, b, r with a < r < b be positive integers such that ab+1 = r 2 . Let {a, b, c} be a Diophantine triple. Then, ac+1 = s2 and bc+1 = t2 for some positive integers s and t. Eliminating c from these equations, we have the Pellian equation at2 − bs2 = a − b.

(3.1)

Nagell’s argument (see [30, Theorem 108a] and [13, Lemma 1]) tells us that for any positive solution to (3.1) there exist a solution (t0 , s0 ) to (3.1) and a non-negative integer ν such that √ √ √ √ √ t a + s b = (t0 a + s0 b)(r + ab)ν (3.2) with (3.3)

1 0 < |t0 | ≤

(r − 1)(b − a) , 2a

2 0 < s0 ≤

a(b − a) . 2(r − 1)

By (3.2) we may write t = τνλ and s = σνλ with λ ∈ {±}, where (3.4)

τ0 := τ0λ = |t0 |, τ1λ = bs0 + λr|t0 |,

(3.5)

σ0 := σ0λ = s0 , σ1λ = rs0 + λa|t0 |,

λ λ τν+2 = 2rτν+1 − τνλ , λ λ σν+2 = 2rσν+1 − σνλ ,

and t0 = λ|t0 |. In the case where (t0 , s0 ) = (±1, 1), put (sλν )2 − 1 . a Then, it is easy to see that cλν can be expressed as (1.2) and {a, b, cλν } is a Diophantine triple, for any ν ≥ 1 and λ ∈ {±}. Note that c0 := cλ0 = 0 and that if s0 = 1, then γ0 := (s20 − 1)/a > 0, which gives the Diophantine triple {a, b, γ0 }. Although it seems well-known to the researchers in this field, we prove the following fact for the sake of the completeness. sλν = σνλ

(3.6)

and

cλν =

Lemma 3.1. Let {a, b} be a Diophantine pair and {σνλ } the sequence given by (3.5) with some solution (t0 , s0 ) to (3.1) satisfying (3.3). Define the sequence {γνλ }  λ λ 2 λ by γν = (σν ) − 1 /a for ν ≥ 0 and λ ∈ {±}. Then, {a, b, γνλ , γν+1 } is a regular Diophantine quadruple for any ν and λ. Proof. It is clear from (3.3) that γ0 := γ0λ = (s20 − 1)/a < b. We also have (rs0 + a|t0 |)2 − 1 (rs0 − a|t0 |)2 − 1 4000, we see the following: 2 3 c+ 3 < 65a b ,

3 4 c+ 4 < 257a b ,

5 6 c+ 6 < 4102a b ,

5 8 9 c+ 9 < 3 · 10 a b ,

6 10 11 c+ 11 < 5 · 10 a b ,

7 12 13 c+ 13 < 7 · 10 a b .

In addition, we know by [20, Lemma 4.2] that if {A, B, C, D} is an irregular Diophantine quadruple with A < B < C < D, then (3.7)

D > min{81B 3.5 C 4.5 , 256A4.5 C 5.5 }.

Thus, in view of Lemma 3.1 and the fact that d+ (A, B, C) is the smallest among all the possible fourth elements D with max{A, B, C} < D, it is easy to check the following: if c ≤ c+ 3 , then Q− (a, b, c) = 0; − ≤ c ≤ c+ if c− 4 4 , then d ≤ c1 and Q− (a, b, c) ≤ 1; − + if c5 ≤ c ≤ c6 , then d ≤ c+ 1 and Q− (a, b, c) ≤ 2

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2

(estimate the right-hand side of (3.7) by dividing the cases b ≤ a2 and

b > a ); + − if c− 7 ≤ c ≤ c9 , then d ≤ c2 and Q− (a, b, c) ≤ 3 (estimate the right-hand side of (3.7) by dividing the cases b ≤ 2a and b > 2a); + + if c− 10 ≤ c ≤ c11 , then d ≤ c2 and Q− (a, b, c) ≤ 4; − + if c12 ≤ c ≤ c13 , then d ≤ c− 3 and Q− (a, b, c) ≤ 5; + , then d ≤ c and Q− (a, b, c) ≤ 6. if c ≥ c− 14 3 Here, the last case can be found by noting that if C ≥ c− 4 , then Q+ (A, B, C) = 0 by [11, Proposition 1.5], which also implies that Q(a, b, c) = Q− (a, b, c) for c ≥ c− 4. This completes the proof of Theorem 1.4.  Proof of Corollary 1.5. As in the proof of [11, Corollary 1.6], if {a, b, c} is a Diophantine triple with a < b ≤ 13a, then we have c = cλν for some ν and λ. Hence, the assertion is an immediate consequence of Theorem 1.4.  4. Proofs of Theorems 1.6 and 1.7 Proof of Theorem 1.6. On the assumptions in the theorem, we have a1 d1 = a2 d2

and

(b + c − a1 − d1 )2 = (b + c − a2 − d2 )2 ,

as seen in the proof of [6, Theorem 1.4], where di := d− (ai , b, c) = ai + b + c + 2ai bc − 2ri si t and ri :=

3 ai b + 1,

si :=

√ ai c + 1,

t :=

√

bc + 1

for i ∈ {1, 2}. Note that 0 ≤ di < c and ai di + 1 = x2i with a positive integer xi for i ∈ {1, 2}. If b + c − a1 − d1 = b + c − a2 − d2 holds, then a1 + d1 = a2 + d2 , which together with a1 d1 = a2 d2 yields d1 = a2 and d2 = a1 . Since a1 a2 + 1 = a1 d1 + 1 = x21 , it follows that {a1 , a2 , b, c} is a Diophantine quadruple. Suppose now that b + c − a1 − d1 = a2 + d2 − b − c, i.e., (4.1)

a1 + a2 + d1 + d2 = 2b + 2c.

Note that inequalities a1 + d1 < b + c < a2 + d2 hold by assumption. Eliminating d2 from a1 d1 = a2 d2 and (4.1), we have a22 − (2b + 2c − a1 − d1 )a2 + a1 d1 = 0, which implies that  3 1 2b + 2c − a1 − d1 ± (2b + 2c − a1 − d1 )2 − 4a1 d1 . (4.2) a2 = 2 Since a1 < b and d1 < c, we have a1 d1 < bc, which together with a1 + d1 < b + c shows that (2b + 2c − a1 − d1 )2 − 4a1 d1 > (b + c)2 − 4bc = (c − b)2 . Thus, if (4.2) holds with the plus sign, then a2 >

1 (b + c + c − b) = c, 2

THE NUMBER OF IRREGULAR DIOPHANTINE QUADRUPLES

113

which contradicts the assumption. Hence, (4.2) holds with the minus sign, and from a1 d1 = a2 d2 and (4.1) we obtain 1 (4.3) a2 = (a2 + d2 − |a2 − d2 |) . 2 Similarly, noting d2 < c and eliminating a2 from a1 d1 = a2 d2 and (4.1), we have 1 (4.4) d2 = (a2 + d2 − |a2 − d2 |) . 2 It follows from (4.3) and (4.4) that a2 = d2 , which contradicts the fact that a2 d2 + 1 = x22 with an integer x2 . Therefore, b + c − a1 − d1 = a2 + d2 − b − c does not hold.  In order to prove Theorem 1.7 using Theorem 1.6, we need the following lemma and propositions. Lemma 4.1. Let {a, b} be a Diophantine pair with (1.3) and {σνλ } the sequence given by (3.5) with some

solution (t0 , s0 ) to (3.1) satisfying (3.3). Define the sequence {γνλ } by γνλ := (σνλ )2 − 1 /a for ν ≥ 0 and λ ∈ {±}. If (t0 , s0 ) = (±1, 1) (i.e., {γνλ } = {cλν }), then either 1 ≤ γ0 := γ0λ < a or 1 ≤ γ1− < a holds. Proof. Assume first that γ1− ≥ a. Then, by γ1− = ((σ1− )2 − 1)/a we have > a, i.e., σ1− ≥ a + 1. Hence we have rs0 − a − 1 ≥ a|t0 |. Squaring both sides yields s20 − 2(a + 1)rs0 + ab + 2a + 1 ≥ 0. 3 Since s0 < (r + 1)/2 by (3.3), we obtain 3 s0 ≤ (a + 1)r − a (a + 2)b + 1. 3 If s0 ≥ a + 1, then a (a + 2)b + 1 ≤ (a + 1)(r − 1). Squaring both sides twice, we see that b2 − (4a3 + 16a2 + 22a + 8)b − (3a2 + 8a + 4) ≥ 0, which obviously contradicts (1.3). Since we always have γ0 = (s20 − 1)/a > 0, i.e., γ0 ≥ 1, we conclude that if γ1− ≥ a, then 1 ≤ γ0 < a. Assume next that γ0 ≥ a, which means that s0 > a, i.e., s0 ≥ a + 1. Taking the contraposition of the consequence above, we have γ1− < a. Thus, it remains to show that γ1− ≥ 1. Suppose on the contrary that γ1− < 1. Then, γ1− = 0 and hence σ1− = 1, which implies that rs0 − 1 = a|t0 |. Squaring both sides yields σ1−

s20 − 2rs0 + a(b − a) + 1 = 0, 3 which together with s0 < (r + 1)/2 implies that 1 r+1 , s0 = r − a < 2 that is, 2ab + 2a2 + 1 < (4a + 1)r. Again, squaring both sides yields (4.5)

4ab2 − (8a2 + 8a − 3)b + (4a3 − 12a − 8) < 0.

On the other hand, s0 = r − a and s0 ≥ a + 1 together imply b ≥ 4a + 4, which is incompatible with (4.5). Hence, γ1− ≥ 1. Therefore, we have shown that if γ0 ≥ a,  then 1 ≤ γ1− < a. This completes the proof of Lemma 4.1.

114

YASUTSUGU FUJITA

Proposition 4.2. Let {a, b} be a Diophantine pair with (1.3). Then, there exists a solution (t0 , s0 ) to (3.1) satisfying (3.3) such that any Diophantine triple {a, b, c} satisfies either c = cλν or c = ανλ for some ν and λ, where {ανλ } is the sequence defined by ανλ = ((σνλ )2 − 1)/a with {σνλ } a sequence given by (3.5). Proof. We begin by noting that if a = 1, then the assertion holds. Indeed, if a = 1, then (1.3) shows that b ≤ 50, and hence b ≤ 48. It follows from (3.3) that 0 < s0 < 2, i.e., s0 = 1, which implies that c = cλν for some ν and λ. Thus, we assume that a ≥ 2 in the following. We may assume that (t0 , s0 ) = (±1, 1) and {ανλ } = {cλν }. Suppose that we need a solution (t0 , s0 ) ∈ {(±1, 1), (±t0 , s0 )} to (3.1) satisfying (3.3) to express the third element c in a Diophantine triple {a, b, c}. Denote by {βνλ } the sequence defined by βνλ = (((σν )λ )2 − 1)/a with (σ0 )λ = s0 ,

(σ1 )λ = rs0 + λa|t0 |,

  (σν+2 )λ = 2r(σν+1 )λ − (σν )λ .

By [27, Theorem 8] we may assume that a + 2 < b. Then we have 0 < c− 1 = a + b − 2r < b. Moreover, Lemma 4.1 shows that 1≤α , s0

>

THE NUMBER OF IRREGULAR DIOPHANTINE QUADRUPLES

we see from (3.3) that (4.6)

α4−

115



 16a2 (b − a)2 (ab + 1)2 −1 s20 1 ≥ 32(b − a)(ab + 1)2 (r − 1) − a  0  /  1 1 a √ 1− − 7/2 7/2 a5/2 b7/2 . > 32 1 − b a b ab

1 (σ − )2 − 1 > = 4 a a

Noting (1.3) with b > 4000 (cf. [7, Lemma 3.4]), we have a ≥ 9 and a3/2 > 0.406b1/2 . Moreover, we may assume that b > 13a (see the proof of [11, Corollary 1.6]). Thus we have α4− > 29a5/2 b7/2 . If a10/3 ≥ b, then α4− > 29 · 40001/4 b4 > 200b4 , which contradicts [11, Theorem 1.4 (4)]. Hence, b > a10/3 , that is, a/b < b−7/10 . It follows from (4.6) that α4− > 31.735 · 0.406ab4 > 115b4 .

(4.7)

Since it is easy to see that α4+ > 200b4 , it suffices to consider the case where c = α4− . Now [20, Lemmas 3.4 and 4.1] together show that (4.8) 2.778b−3/4 c1/4
a10/3 , b > 4000 and b ≤ 4a3 + 16a2 + 22a + 8 ≤ 6.061a3 the right-hand side of (4.8) is less than 8 log(8.406 · 1013 b53/20 c) log(1.648b−7/20 c) . log(4bc) log(0.148b−11/3 c) Since this is a decreasing function of c = α4− , inequality (4.7) gives 9.097b1/4
1 holds for all conjugates of γ. This fact was proved first by Vince [20], and rediscovered several times, see e.g. Kov´ acs [11] and Peth˝ o [13]. 2. Periodic S-integers For an alphabet (set) A denote A∗ the set of finite words on A including the empty word λ. The set A∗ is equipped with the concatenation operation. For w ∈ A∗ and k ≥ 1 we write wk = w . . . w, the k-times concatenation of w. This definition is extended to k = 0 by setting w0 = λ for all w ∈ A∗ . If a word can be written as w1 wk then it is called periodic, furthermore w1 is called its preperiod and w its period. In this section we are dealing with elements having periodic representation in a number system (γ, D). For any given w, w1 ∈ D ∗ there are infinitely many elements whose representation has preperiod w1 and period w. They are the elements with (β)γ = w1 wk for some k ≥ 0. We prove that some sets have finite intersection with this set. S-unit equations play a vital role in our proofs. Now we define them. For an algebraic number field K denote MK its set of places. Let S ⊂ MK be finite

DIOPHANTINE PROPERTIES OF RADIX REPRESENTATIONS

135

including all archimedean places, and denote ZS the set of S-integers of K, i.e., the set of those elements α ∈ K with |α|v ≤ 1 for all v ∈ MK \ S. The set ZS forms a ring, its group of units is denoted by Z∗S . Consider the weighted S-unit equation (2.1)

α1 X1 + · · · + αs Xs = 1,

where s ≥ 2, α1 , . . . , αs are non-zero elements of K and the solutions x1 , . . . , xs belong to Z∗S . A solution x1 , . . . , xs of (2.1)  is called degenerate if there exists a proper subset I of {1, . . . , s} such that i∈I αi xi = 0. The next theorem was proved by Evertse [3] and independently by van der Poorten and Schlickewei [19], see also [5]. Theorem 2.1. Equation (2.1) has only finitely many non-degenerate solutions in x1 , . . . , xs ∈ Z∗S . For a finite set S ⊂ ZK we will denote by Γ(S), Γ∗ (S) the multiplicative semigroup, the multiplicative group generated by S respectively. Let 0 ∈ / A, B ⊂ ZK be finite. Put S(A, B, s) = {α1 μ1 + · · · + αs μs : αj ∈ A, μj ∈ Γ(B)}. For example, if K = Q, A = {1}, B = {2, 3} then S(A, B, 2) = {2a 3b + 2c 3d : a, b, c, d ≥ 0}. The elements α1 , . . . , αs ∈ K are called multiplicatively dependent if there exist u1 , . . . , us ∈ Z, u1 = 0 such that α1u1 · · · αsus = 1. Otherwise they are called multiplicatively independent. Now we are in the position to formulate our first result. Theorem 2.2. Let (γ, D) be a number system with finiteness property in ZK , and w, w1 ∈ D∗ . Let 0 ∈ / A, B ⊂ ZK be finite such that the elements of {γ} ∪ B are multiplicatively independent. Then there are only finitely many U ∈ S(A, B, s) such that (U )γ = w1 wk . Remark 2.3. This is the finite version of Corollary 2.3 of [15]. More precisely we derived Corollary 2.3 of [15] from Theorem 2.2 without explicitly stating it. We realized this fact only after the publication of [15]. Because, by our opinion, Theorem 2.2 is interesting itself we decided to formulate it here. Proof. Let w1 ∈ D ∗ be given. By unicity of expansions there is at most one U with (U )γ = w1 . Thus our statement is true if w = λ. From here on we assume w = λ. Let w = d0 . . . dh−1 and q = d0 + d1 γ + . . . + dh−1 γ h−1 . Set q0 = 0 if w1 = λ, and q0 = f0 + f1 γ + . . . + fg−1 γ g−1 provided w1 = f0 . . . fg−1 . Finally let U = α1 μ1 + · · · + αs μs , αj ∈ A, μj ∈ Γ(B). It is enough to show that if (U )γ = w1 wk then k is bounded. Indeed, if k is bounded then w1 wk can take finitely many values, but by our first claim (U )γ = v, v ∈ D∗ holds for at most one U ∈ S(A, B, s).

˝ ATTILA PETHO

136

Now assume that (U )γ = w1 wk holds for some k > 0. It means nothing else then α1 μ1 + · · · + αs μs

=

q0 + γ g

k−1 

γ ih

i=0

= = =

q0 + γ g q0 +

k−1 

h−1 

dj γ j

j=0

qγ ih

i=0 γ hk − 1 qγ g h γ −1

qγ g hk qγ g γ + q0 − h . −1 γ −1

γh

Setting αs+1 =

qγ g , γh − 1

αs+2 = q0 −

qγ g γh − 1

we get the equation α1 μ1 + · · · + αs μs = αs+1 γ hk + αs+2 . By the fact mentioned at the end of the Introduction |γ| > 1, hence γ h = 1 and αs+1 , αs+2 are well defined. Plainly αj ∈ K, j = 1, . . . , s+2 and αj = 0, k = 1, . . . , s by assumption. It is easy to see that αs+1 = 0 holds too. In the sequel we have to distinguish the cases αs+2 = 0 and αs+2 = 0. As the argumentation is similar in both cases we detail here only the case αs+2 = 0. Dividing by αs+2 = 0 and letting α ˆ j = αj /αs+2 , j = 1, . . . , s and α ˆ s+1 = −αs+1 /αs+2 , μs+1 = γ hk we get (2.2)

ˆ s+1 μs+1 = 1. α ˆ 1 μ1 + · · · + α

Let S be the set of places of K, which includes the archimedean ones, and those which correspond to prime ideal divisors of γ or some element of B. Plainly S is a finite set. Set Γ1 = Γ∗ (γ, B), Then the elements of Γ1 are S-units and, hence, (2.2) an S-unit equation. If there are infinitely many U ∈ S(A, B, s) such that (U )γ = w1 wk then k has to take arbitrary large values and, hence, (2.2) has infinitely many solutions in (μ1 , . . . , μs+1 ) ∈ Γ∗ (B)s × Γ1 ⊂ Γs+1 1 , which means that γ appears solely in μs+1 and its exponents in the solutions are not bounded. In the sequel we derive from (2.2) new equations in less unknowns such that all but one coordinate of the solution vectors belong to Γ∗ (B), only one - μs+1 - belongs to Γ1 . As (2.2) has infinitely many solutions in Γs+1 1 , i.e. in S-units, Theorem 2.1 implies that there is a proper subset I ⊂ {1, . . . , s + 1} such that the equation  (2.3) α ˆ i μi = 0 i∈I |I|

has infinitely many solutions (μi )i∈I ∈ Γ1 , where |I| denotes the size of I. We show that there is such a subset which contains s + 1. Indeed, assume that s + 1 ∈ / I for all I ⊂ {1, . . . , s + 1} such that the equation |I| (2.3) admits infinitely many solutions (μi )i∈I ∈ Γ1 . As the number of such sets is

DIOPHANTINE PROPERTIES OF RADIX REPRESENTATIONS

137

at most 2s , there is an ∅ = I ⊆ {1, . . . , s} such that the equation  α ˆ i μi = 1 (2.4) i∈{1,...,s+1}\I

is again a S-unit equation of the shape (2.2), but in less summands. Moreover if (μ1 , . . . , μs+1 ) ∈ Γ∗ (B) × Γ1 is a solution of (2.2) such that μi , i ∈ I satisfy (2.3) then μi , i ∈ {1, . . . , s + 1} \ I satisfy (2.4), thus it has infinitely many solutions in s−|I| × Γ1 . Assume that I is maximal in the sense that if (μi )i∈{1,...,s+1}\I ∈ Γ∗ (B)  I ⊂ I ⊆ {1, . . . , s} then i∈{1,...,s+1}\I  α ˆ i μi = 1 has only finitely many solutions in μi ∈ Γ∗ (B), i ∈ {1, . . . , s} \ I  and μs+1 ∈ Γ1 . As (2.4) admits infinitely many S-unit solutions μi ∈ Γ1 , i ∈ {1, . . . , s + 1} \ I, Theorem 2.1 implies the existence of I1 ⊂ {1, . . . , s+1}\I such that i∈I1 α ˆ i μi = 0 / I1 , has infinitely many solutions μi ∈Γ1 , i ∈ I1 . Thus, by our assumption s + 1 ∈ and as μs+1 = γ hk , the equation i∈({1,...,s+1}\I)\I1 α ˆ i μi = 1 has to have infinitely many solutions μi ∈ Γ1 , i ∈ ({1, . . . , s + 1} \ I) \ I1 . Set I  = I ∪ I1 . Then I ⊂ I  ⊆ {1, . . . , s} and ({1, . . . , s+1}\I)\I1 = {1, . . . , s+1}\I  , which contradicts the maximality of I, i.e. I = {1, . . . , s}. Hence αs+1 μs+1 = 1 holds for infinitely many μs+1 = γ hk ∈ Γ1 . This means that αs+1 γ hk = 1 holds for infinitely many k. Thus γ is a root of unity, which is impossible because |γ| > 1. In the sequel we assume s + 1 ∈ I. By possible renumbering we may assume I = {1, . . . , s1 , s + 1} with some 1 ≤ s1 < s. (The case I = {s + 1} is impossible.) Hence s1  α ˆ i μi + α ˆ s+1 μs+1 = 0 i=1

has infinitely many solutions in μi ∈ Γ∗ (B), i = 1, . . . , s1 , μs+1 ∈ Γ1 . As the elements of B ∪ {γ} is multiplicatively independent there is a prime ideal ℘ of ZK , which divides γ, but no elements of B. Dividing this equation by −α ˆ s1 μs1 and setting α ˆ j ← −α ˆ j /α ˆ s1 , μj ← μj /μs1 , j = 1, . . . , s1 −1 and α ˆ s 1 ← −α ˆ s+1 /α ˆ s 1 , μs 1 ← hk μs+1 /μs1 we see that ℘ divides μs1 , further the obtained equation has the shape (2.2), but with s1 < s + 1 summands. Moreover it has infinitely many solutions in (μ1 , . . . , μs1 ) ∈ Γ∗ (B)s1 −1 × Γ1 . After repeated application of this argument we arrive at s1 = 1, i.e., an equation α ˆ 1 μ1 + α ˆ 2 μ2 = 0, which has infinitely many solutions in μ1 ∈ Γ∗ (B) and μ2 ∈ Γ1 such that ℘hk divides μ2 , i.e. the exponent of ℘ in μ2 is not bounded. Dividing by α ˆ 1 μ1 we obtain an equation of shape αμ = 1, which has infinitely many solutions in μ ∈ Γ1 such that the exponent of ℘ in μ is still at least hk, which is not bounded. Fixing one of the solutions, say μ0 , we have that μ/μ0 = 1 holds for infinitely many μ ∈ Γ1 , which is impossible because the elements of {γ} ∪ B, which generate Γ1 are multiplicatively independent.  3. Periodic rational integers The elements of (A, B, s) are sums of powerproduct, thus, by a theorem of van der Poorten and Schlickewei [19] are growing exponentially. In this section we prove that under certain assumptions the set of values of polynomials at rational integers behave similarly, i.e., cannot have arbitrary long periodic expansions, provided the preperiod and the period are given. More precisely we prove

˝ ATTILA PETHO

138

Theorem 3.1. Let K be an algebraic number field of degree k ≥ 2, (γ, D) be a number system with finiteness property in ZK , and w, w1 ∈ D∗ . Let t(X) ∈ ZK [X] be of degree v ≥ 0. Assume that γ has two conjugates whose quotient is not a root of unity. Then there exist only finitely many effectively computable rational integers n such that (t(n))γ = w1 wu . Remark 3.2. You may find a characterization of algebraic integers whose conjugates lie on a circle in Robinson [17]. Assume that γ  = m for some integers  ≥ 1, and m. As (γ, D) is a number system with finiteness property in ZK we have K = Q(γ), i.e., the degree of γ is exactly k. Thus γ can be a zero of an integer polynomial only if its degree is at least k. Hence  ≥ k. Let 0 = d ∈ D. Then  the rational integers ji=0 dγ i admit the periodic representation wj , j ≥ 1 with the word w = d0 . On the other hand, if γ  = m then the quotients of different conjugates of γ are roots of unity. Thus our assumption is necessary. Proof. We showed K = Q(γ) in the remark above. Denote α(j) , j = 1, . . . , k the conjugates of α ∈ K. Using the same notation as in the proof of Theorem 2.2 and following the same line we obtain the equation t(n) = αγ hu + β,

(3.1) with α=

qγ g = 0, γh − 1

β = q0 − α.

Taking conjugates we obtain the system of equations t(1) (n) =

α(1) (γ (1)h )u + β (1) ,

t(2) (n) =

α(2) (γ (2)h )u + β (2)

in the unknown integers n, u. This is a system of polynomial equations for any fixed u, hence it has a common solution if and only if its resultant with respect to (i) n is zero. Let t(X) = tv X v + . . . + t0 . Putting Yi = t0 − α(i) (γ (i)h )u − β (i) , i = 1, 2 our resultant is the determinant of the 2v × 2v matrix ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

(1)

tv 0 ··· 0 (2) tv 0 ··· 0

(1)

tv−1 (1) tv ··· ··· (2) tv−1 (2) tv ··· ···

··· (1) tv−1 ··· 0 ··· (2) tv−1 ··· 0

(1)

t1 ··· ··· (1) tv (2) t1 ··· ··· (2) tv

Y1 (1) t1 ··· (1) tv−1 Y2 (2) t1 ··· (2) tv−1

0 Y1 ··· ··· 0 Y2 ··· ···

··· 0 ··· (1) t1 ··· 0 ··· (2) t1

0 ··· ··· Y1 0 ··· ··· Y2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

DIOPHANTINE PROPERTIES OF RADIX REPRESENTATIONS (1)

139

(2)

Subtracting the tv /tv -times the v + j-th row from the j-t one, j = 1, . . . , v our matrix is equivalent to ⎛ ⎞ 0 tv−1,1 ··· t1,1 Y1,1 0 ··· 0 ⎜ 0 0 tv−1,1 · · · t1,1 Y1,1 0 ··· ⎟ ⎜ ⎟ ⎜ ··· ··· ··· ··· ··· ··· ··· ··· ⎟ ⎜ ⎟ ⎜ 0 ··· 0 0 tv−1,1 · · · t1,1 Y1,1 ⎟ ⎜ ⎟ (2) ⎜ t(2) t(2) ··· t1 Y2 0 ··· 0 ⎟ ⎜ v ⎟ v−1 ⎜ ⎟ (2) (2) (2) ⎜ 0 tv tv−1 · · · t1 Y2 0 ··· ⎟ ⎜ ⎟ ⎝ ··· ··· ··· ··· ··· ··· ··· ··· ⎠ (2) (2) (2) 0 ··· 0 tv tv−1 · · · t1 Y2 (1)

(2)

(1)

(2)

(1)

(2)

where tj,1 = tj −tj ·tv /tv , j = 1, . . . , v −1; Y1,1 = Y1 −Y2 ·tv /tv . Subtracting (2)

the tv−1,1 /tv -times the v + j + 1-th finally we get the matrix ⎛ 0 0 ··· 0 ⎜ 0 0 0 · ·· ⎜ ⎜ ··· ··· · · · · ·· ⎜ ⎜ 0 · · · 0 0 ⎜ (2) ⎜ t(2) t(2) · · · t1 ⎜ v v−1 ⎜ (2) (2) ⎜ 0 tv tv−1 · · · ⎜ ⎝ ··· ··· ··· ··· (2) 0 ··· 0 tv

row from the j-t one, j = 2, . . . , v and so one, Y1,v+1 t2,v+1 ··· tv,v+1 Y2 (2) t1 ··· (2) tv−1

Y1,v+2 Y2,v+2 ··· ··· 0 Y2 ··· ···

··· Y2,v+3 ··· tv,2v−1 ··· 0 ··· (2) t1

Y1,2v ··· ··· Yv,2v 0 ··· ··· Y2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠

where Yi,v+i = Y1,1 + βi,i , i = 1, . . . , v, Yi,v+j = αi,j Y2 + βi,j , i = 1, . . . , v, j > i and αi,j , βi,j , ti,j ∈ K. The determinant of this matrix is  v 

v  (2)v tv Y1,1 + F2 (Y1 , Y2 ) , (Y1,1 + βi,i ) + F1 (Y1 , Y2 ) = t(2)v v i=1

with F1 , F2 ∈ K[Y1 , Y2 ] and such that the total degree of its terms are less than v. (2) As tv = 0 our resultant is zero if and only if v + F2 (Y1 , Y2 ) = 0. Y1,1

Using the definition of Y1 , Y2 we have

  (1) (2) (2) (2)h u (2) −α /t(2) (γ ) + t − β Y1,1 = −α(1) (γ (1)h )u + t0 − β (1) − t(1) v v . 0

This expression simplifies to Y1,1 = α1 (γ (1)h )u + α2 (γ (2)h )u + β

  (1) (2) (1) (1) (2) (2) by setting α1 = −α(1) , α2 = α(2) tv /tv and β = t0 −β (1) −tv t0 − β (2) /tv . Notice that α1 , α2 = 0, and β ∈ K. Substituting the expressions for Y1 , Y2 in F2 our equation becomes v    (3.2) α1 (γ (1)h )u + α2 (γ (2)h )u + F3 (γ (1)h )u , (γ (2)h )u = 0,

˝ ATTILA PETHO

140

where F3 (X, Y ) denotes a polynomial with coefficients from K and such that the total degree of its monomials is at most v − 1. Thus, if |γ (1) | ≥ |γ (2) | then      (3.3) F3 (γ (1)h )u , (γ (2)h )u  ≤ c1 |γ (1) |hu(v−1) with an effective constant depending only on k, v, h, the digits of w and on the coefficients of t and the defining polynomial of γ. Up to now γ (1) and γ (2) denoted two different conjugates of γ. In the sequel we distinguish two cases. Case I. |γ (1) | = |γ (2) |, but γ (1) /γ (2) is not a root of unity. As γ (1) /γ (2) is not a root of unity there exist by Corollary 3.7. of Shorey and Tijdeman [18] effectively computable constants c2 , c3 , c4 such that     α1 (γ (1)h )u + α2 (γ (2)h )u  ≥ c2 |γ (1) |hu exp(−c3 log u), whenever |u| ≥ c4 . Hence v    α1 (γ (1)h )u + α2 (γ (2)h )u  ≥ cv2 |γ (1) |huv exp(−c3 v log u). This lower bound together with the upper bound given in (3.3) implies that (3.2) has only finitely many effectively computable solutions. Case II. |γ (1) | > |γ (2) |. This case is much simpler as the first one. Indeed |γ (1) | > |γ (2) | implies v    α1 (γ (1)h )u + α2 (γ (2)h )u  ≥ c5 |γ (1) |huv , immediately, whenever |u| ≥ c6 . The rest is the same as in the first case.



4. Rational integers with fixed representation word In the last section we studied how many rational integers have periodic representation in a number system of a given field. In this section we changes the roles of the actors. We fix a finite word of integers w and search for number systems (γ, D) and rational integers n such that (n)(γ,D) = w. The underlying idea is simple. If w = w1 . . . w and (α)γ = w then α = w1 + w2 γ + · · · + w γ −1 . Denote k the degree of γ. If k ≥ −1 then 1, γ, . . . , γ −1 are Q-linearly independent numbers, thus α ∈ Z is only possible if w2 , . . . , w = 0 and α = w1 . The problem k−1 is more interesting if k <  − 1. Then γ j = i=0 gij γ i holds for all j ≥ 0 with suitable integers gji . Thus α

=

−1 

wj+1 γ j

j=0

=

−1 

wj+1

j=0

=

k−1 −1  i=0 j=0

k−1 

gij γ i

i=0

wj+1 gij γ i .

DIOPHANTINE PROPERTIES OF RADIX REPRESENTATIONS

141

Now as 1, γ, . . . , γ k−1 are Q-linearly independent numbers α ∈ Z holds if and only −1 if j=0 wj+1 gij = 0 for i = 1, . . . , k − 1. We show that these condition can be interpreted as diophantine equations in the coefficients of the minimal polynomial of γ. It is well known that the sequences (gij )i≥0 are linearly recursive. More precisely we have Lemma 4.1. Let X k + gk−1 X k−1 + . . . + g0 ∈ Z[X] be the minimal polynomial of γ. The sequences (gij )j≥0 have the initial values gij = δij , 0 ≤ j, i ≤ k − 1 and satisfy the recursion gij = −gk−1 gi,j−1 − . . . − g0 gi,j−k

(4.1)

for j ≥ k. As usual δij denotes Kronecker’s δ, which is 1 for j = i and zero otherwise. For convenience of the reader we present here the easy proof. Proof. The statement about the initial values is obviously true. Further, as k−1 k−1 γ k = i=0 −gi γ i , the recursion holds for j = k. Assume that γ j = i=0 gij γ i holds for all 0 ≤ j ≤ h for some h ≥ k. Then γ h+1

=

γγ h = γ

k−1 

gih γ i

i=0

=

k−2 

gih γ i+1 + gk−1,h γ k

i=0

=

−g0 gk−1,h +

k−2 

(gih − gi+1 gk−1,h )γ i+1 ,

i=0

where we used the expression for γ . As 1, γ, . . . , γ k−1 are Q-linearly independent, and setting g−1,h = 0 we obtain k

gi,h+1 = gi−1,h − gi gk−1,h , i = 0, . . . , k − 1.

(4.2)

Summing these equalities for the pairs (i, h) = (k − 1, h + 1), (k − 2, h), . . . , (0, h − k + 2) we obtain the stated relation for i = k − 1. Next we consider i = 0, whence we have g0,h+1 = −g0 gk−1,h , i.e., (g0j )j≥1 = −g0 (gk−1,j−1 )j≥0 , which proves the statement in this case. Assume finally that (4.1) holds for some 0 ≤ i < k − 2. The relation (4.2) with i + 1 on the place of i means that the sequence (gi+1,h )h≥0 is the sum of the sequences (gi,h )h≥0 and (gk−1,h )h≥0 , which both satisfy the recursion (4.1), thus it must do the same.  Lemma 4.1 means that the coefficient of γ i , 0 ≤ i ≤ k − 1 in γ h is a polynomial with integral coefficients in g0 , . . . , gk−1 , whose degree increase with h. For example for k = 2 we have h g0h g1h

0 1 2 1 0 −g0 0 1 −g1

3 g0 g1 g12 − g0

4 −g0 g12 + g02 −g13 + 2g0 g1

5 g0 g13 − 2g02 g1 g14 − 3g0 g12 + g02

142

The h 0 g0h 1 g1h 0 g2h 0

˝ ATTILA PETHO

same data for k = 3. 1 2 3 4 5 6 0 0 −g0 g0 g2 −g0 g22 + g0 g1 g0 g23 − g0 g12 − g0 g1 g2 + g02 1 0 −g1 g1 g2 − g0 −g1 g22 + g0 g2 + g12 g1 g23 − g0 g22 − 2g12 g2 + 2g0 g1 0 1 −g2 g22 − g1 −g23 + 2g1 g2 − g0 g24 − 3g1 g22 + 2g0 g2 + g12

Let w = w1 . . . w ∈ Z∗ , i.e., w be a finite word, whose letters are integers. We shall consider w the word of digits of the representation of an element in a number system1 . Extending w with a string of zeroes the represented element does not change, thus it is natural to assume that either w = 0 or w = 0. Denote by Dw the set of letters of w. Choosing g ∈ Z larger than the size of Dw and such that d1 ≡ d2 (mod g) for all d1 , d2 ∈ Dw , d1 = d2 one can extend Dw on infinitely many ways to a complete residue system modulo g. Now setting  n = i=1 wi g i−1 we see that (n)(g,D) = w for all Dw ⊆ D ⊂ Z such that D is a complete residue system modulo g. Thus in Z any finite word of integers appears as the word of digits of some integer in an appropriate number system. The situation is completely different if we consider the representations of rational integers in number systems in algebraic number fields. This is what we analyse in the sequel. Our argument above shows that if for a given w there exist g, D ⊃ Dw and n ∈ Z such that (n)(g,D) = w then replacing z ∈ D \ Dw by any z  , with z  ≡ z (mod g) we obtain a different number system (g, D  ) in which the representation of n does not change. Thus such number systems are equivalent from the actual point of view. More generally the number systems (γ, D1 ), (γ, D2 ) are called w-equivalent if there exists n ∈ Z such that (n)(γ,D1 ) = (n)(γ,D2 ) = w. After these preparations we can present a method, which establish for a given w ∈ Z∗ all algebraic integers γ for which there exist D ⊃ Dw and n ∈ Z such that (n)(γ,D) = w. Although we are not able to prove, but the construction indicates that there exist for any w ∈ Z∗ only finitely many w-equivalent number systems (γ, D), provided the degree of γ is less than  − 1. Algorithm Input: w = w1 . . . w ∈ Z∗ such that  ≥ 2 and w = 0. Output: The set S of triplets (γ, D, n) such that [Q(γ) : Q] = k ≥ 2, n ∈ Z and (n)(γ,D) = w. 1. S ← ∅; 2. for k ← 2 to  do { −1 3. for i ← 1 to k − 1 do Li ← j=0 wj+1 gij ; (* The Li are polynomials in g0 , . . . , gk−1 .*) 5. S1 ← set of solutions of the system of equations Li = 0, i = 1, . . . , k − 1 in (g0 , . . . , gk−1 ) ∈ Zk ; 6. for g = (g0 , . . . , gk−1 ) ∈ S1 do { 7. S1 ← S1 \ {g}; 8. if 0 ∈ Dw and g0  vi − vh , 1 ≤ i < h ≤ |Dw |, vi , vh ∈ Dw and P (X) = X k + gk−1 X k−1 + . . . + g0 is irreducible in Q[X] then 9. S ← {(γ, D, n)}, where γ is a zero of P (X), D ⊇ Dw is a complete residue −1 system modulo g0 and n = j=0 wj+1 g0j } } (* end of the k cycle*) 1 In the everyday life the string of digits identifies the number whose decimal representation is the given string. For example 2020 means the number 2 · 103 + 2 · 100 .

DIOPHANTINE PROPERTIES OF RADIX REPRESENTATIONS

143

Remark 4.2. We assume  ≥ 2 because otherwise w = w1 is the representation of the rational integer w1 in any number systems (γ, D), provided 0, w1 ∈ D. Algorithm is not an algorithm in strict sense, because in Step 5 the system of equations (4.3)

Li (g0 , . . . , gk−1 ) =

−1 

wj+1 gij = 0, i = 1, . . . , k − 1

j=0

may have for some k infinitely many solutions. This is not an option, but for k =  − 1,  − 2 typical. The reason is that the unknown gk−2 , . . . , g0 appear in a linear term in Lk−1 , . . . , L1 respectively thus it is possible to express gk−2 , . . . , g0 as polynomials in gk−1 , see the details in the proof of Proposition 4.4. If, however, k <  − 2 then there is no linear terms in Li , thus the system of equations (4.3) can be solved by computing the Gr¨obner basis of the polynomial ideal generated by L1 , . . . , Lk−1 or by successive elimination of the unknowns gk−2 , . . . , g0 by computing the resultant of Lk−2 , . . . , L1 with the previously computed resultant (see Peth˝o [14], especially Sections 6.11 and 8.4.3). On this way we get a polynomial equation in two unknowns, which solutions parameterize the solutions of the original problem. By our experience these curves have high genus, thus only finitely many integer points may lie on them. This justifies our expectation that S1 is finite for k <  − 2, i.e., for k <  − 2 there is no infinite loops in the Algorithm. Theorem 4.3. The Algorithm is correct. If w = w1 . . . w ∈ Z∗ with  ≥ 2, w = 0 then the set S includes all γ, n and a Dw ⊆ D ⊂ Z such that [Q(γ) : Q] = k ≥ 2, n ∈ Z and (n)(γ,D) = w. Proof. Let w = w1 . . . w ∈ Z∗ with w = 0. Assume that there exists an algebraic integer γ with minimal polynomial P (X) = X k + gk−1 X k−1 + . . . + g0 ∈ Z[X] such that |g0 | ≥ |Dw |. Assume further that there is a rational integer n such that −1  n= wj+1 γ j . j=0

At this stage g0 , . . . , gk−1 are unknown integers. Using the notations of Lemma 4.1 we obtain ⎛ ⎞ k−1 −1   ⎝ n= wj+1 gij ⎠ γ i . i=0

j=0

By our assumption γ is of degree k ≥ 2, thus 1, γ, . . . , γ k−1 are Q-linear independent, hence the last equation holds if and only if the coefficients of γ i , i = 1, . . . , k−1 are zero, which is the system of equations (4.3). This justifies Steps 4. and 5. Moreover, if  ≤ k − 1 then Li = wi+1 for i = 1, . . . , , i.e., wi = 0 for i = 2, . . . , , hence w = w1 0−1 , which is not an allowed input. Thus non-trivial solutions can appear only if  > k − 1, i.e., k ≤  which justifies the choice of the upper limit of the loop in Step 2. Note that from Step 6. g0 , . . . , gk−1 denote concrete integers. In Step 8 we check the extendability of Dw to a complete residue system modulo g0 . This is possible only if the elements of Dw belong to different residue classes modulo g0 . (This excludes |g0 | < |Dw | too.) In the same step we test the irreducibility of P (X),

˝ ATTILA PETHO

144

which is equivalent to that the degree of γ is k. After P (X) and Dw pass all tests −1 then the rational integer n = j=0 wj+1 g0j satisfies (n)(γ,D) = w.  Now we investigate the cases  = k + 1 and k + 2. Proposition 4.4. Let w2 . . . w ∈ Z∗ with  ≥ 3, w = 0. If w |wi , i = 2, . . . ,  then for all w1 ∈ Z and for k = ,  − 1 there exist g1 , . . . , gk−1 ∈ Z, infinitely many g0 ∈ Z, and {w1 , . . . , w } ⊆ D ⊂ Z such that if γ is a zero if the polynomial −1 P (X) = X k + gk−1 X k−1 + . . . + g0 then n = i=0 wi+1 γ i ∈ Z. In particular, if P is irreducible then (n)(γ,D) = w1 . . . w . Moreover • if  = k + 2 then not only w1 , but also gk−1 ∈ Z can be arbitrary, • if  = k + 1, and there is an 2 ≤ i ≤  such that w  wi then there is no w1 ∈ Z with the above property. Proof. Let the defining polynomial of γ be P (X). From the proof of Theorem 4.3 we know that γ satisfies the requirements of the proposition if and only if the coefficients of P solve the system of equations Li =

−1 

wj+1 gij = 0, i = 1, . . . , k − 1.

j=0

In the sequel we treat the cases  = k + 1 and  = k + 2 separately. Case 1.  = k + 1. By Lemma 4.1 we know that gij = δij , 0 ≤ i, j ≤ k − 1, further gik = −gi , i = 0, . . . , k − 1, hence Li = wi+1 − wk+1 gi , i = 0, . . . , k − 1. We have gi ∈ Z if and only if wk+1 |wi+1 for all i = 1, . . . , k−1. If these relations hold wi+1 then gi = w , i = 1, . . . , k − 1. Now let w1 ∈ Z be arbitrary, and 0, w1 = g0 ∈ Z k+1 such that wi ≡ wj (mod g0 ), 1 ≤ i < j ≤ k + 1, and, finally, D ⊂ Z a complete residue system modulo g0 including Dw then taking n = w1 − wk+1 g0 ∈ Z we have (n)(γ,D) = w. The choice g0 = w1 is excluded because then n = 0. Case 2.  = k + 2. We have by Lemma 4.1 gi,k+1 = −gk−1 gik − gi−1 = gk−1 gi − gi−1 , i = 1, . . . , k − 1 and g0,k+1 = −gk−1 g0k = gk−1 g0 . Hence / wi+1 − wk+1 gi + wk+2 (gk−1 gi − gi−1 ), Li = w1 − wk+1 g0 + wk+2 gk−1 g0 ,

if if

i = 1, . . . , k − 1 i = 0.

As gk−2 , . . . , g0 are integers, the equations Li = 0, i = k − 1, . . . , 1 give conditions for gk−1 . For example Lk−1 = 0 implies wk − wk+1 gk−1 2 gk−2 = + gk−1 , wk+2 hence gk−2 ∈ Z if and only if wk − wk+1 gk−1 ≡ 0 (mod wk+1 ). With increasing k the conditions become more and more ugly, therefor we do not search for the general condition. If wk+2 |wi , i = 1, . . . , k+1, then the situation is simpler, because gi = Gi (gk−1 ), i = 1, . . . , k + 2, with a polynomial Gi ∈ Z[X] of degree k + 3 − i. With the choice Gk+2 = X the claim is true for i = k + 2. Assume that 2 ≤ i0 ≤ k + 2 is such

DIOPHANTINE PROPERTIES OF RADIX REPRESENTATIONS

145

that the claim is true for i0 . The condition Li0 = 0 and the induction hypothesis implies wk+1 wi +1 gi0 −1 = gk−1 gi0 − gi0 + 0 wk+2 wk+2 wk+1 wi +1 = gk−1 Gi0 (gk−1 ) − Gi0 (gk−1 ) + 0 . wk+2 wk+2 w

w 0 +1 Gi0 (X) + wik+2 we see that it has rational Setting Gi0 −1 (X) = XGi0 (X) − wk+1 k+2 integer coefficients, its degree is k + 3 − i0 + 1 = k + 3 − (i0 − 1) and gi0 −1 = Gi0 −1 (gk−1 ) thus our claim is proved. This means that gi = Gi (gk−1 ), i = 1, . . . , k+ 2 are integers for any gk−1 ∈ Z. Choosing w1 ∈ Z arbitrary and finally g0 and D as in the case  = k + 1 finishes the proof of this proposition. 

5. Repunits in number systems Integers with simple decimal expansion fascinate people. Question concerning such numbers lead to interesting and hard diophantine problems. There are numbers whose decimal expansion contains only one repeating digit, in the simplest case this digit is the 1. A rational integer n is called repunit if (n)g = 1 holds with some integers g ≥ 2,  ≥ 1. There are for any fixed g obviously infinitely many repunits. The challenging, and still open, problem is to find all rational integers, which are repunits in two different bases. You can find on this subject recent results and a good overview in the paper of Bugeaud and Shorey [1]. Repunit is a meaningful concept for number systems in algebraic number fields too, provided 1 belongs to the digit set. To be precise; let K be a number field and (γ, D) be a number system in ZK . The element α ∈ ZK is called repunit in (γ, D) if (α)γ = 1 for some  ≥ 0. All repunits in (γ, D) belong to ZK , i.e. infinitely many elements of ZK are repunits in (γ, D). If (γ, D) is fixed then under mild and natural conditions there exit by Theorem 3.1 only finitely many rational integers, which are repunits in (γ, D). Similarly, by Proposition 4.4, if  is fixed then the Algorithm finds up to equivalence all number systems for which there exists a rational integer, which is a repunit of length . We present here that rational integer repunits allow more precise description. To state our first result of this section we have to introduce a polynomial. For i ≥ 0 let Gi (X) =

i 

(X − 1)h =

h=0

(X − 1)i+1 − 1 . X −2

Corollary 5.1. Let k ≥ 2 and K be a number field of degree k. The only rational integer, which is a repunit of length  ≤ k in a number system in ZK is 1. k • If γ is a zero of Qm (X) = i=1 X i + m, 0, ±1 = m ∈ Z then n = 1 − m is a repunit in (γ, D) of length k + 1 provided D is a complete residue system modulo m including 0, 1. k • For 0, 1 = m ∈ Z let Pm (X) = i=0 Gi (m)X k−i , γ be a zero of Pm (X) and D be a complete residue system modulo Gk (m) including 0, 1. Then Gk+1 (m) is a repunit in (γ, D) of length k + 2. Proof. The first assertion is true by the proof of Theorem 4.3. Specifying the notations of Proposition 4.4 we have wi = 1, i = 1, . . . , k + 1. This, together with Li = 1 − gi = 0 implies gi = 1, i = 1, . . . , k. Finally choosing g0 = m, and taking into consideration |g0 | ≥ 2 we obtain the second assertion.

˝ ATTILA PETHO

146

To prove the third assertion we could proceed similarly, but we have chosen a different approach. The polynomials Gi (X) satisfy the recursion Gi+1 (X) = (X − 1)Gi (X) + 1, i ≥ 0. Using this we get (m − 1)Pm (X) =

k 

(m − 1)Gi (m)X k−i

i=0

=

k 

(Gi+1 (m) − 1)X k−i

i=0

=

Gk+1 (m) + XPm (X) −

k+1 

X i,

i=0

hence Gk+1 (m) ≡

k+1 

Xi

(mod Pm (X)),

i=0

which means Gk+1 (m) =

k+1 

γi.

i=0

The cases m = 0, 1 are excluded because Gk (1) = 1 for k ≥ 0, and Gk (0) = 0 if k is odd, and Gk (0) = 1 if k is even, thus the roots of P0 (X) and P1 (X) cannot be bases of number systems.  Remark 5.2. By using the Algorithm, an elementary computation shows that there is no rational integer, which is a repunit of length five in a quadratic number field. As this approach became more and more complicated we returned to the approach of the proof of Theorem 3.1. For k = 2 we have Qm (X) = X 2 +X +m and Pm (X) = X 2 +mX +m2 −m+1. Then 1 − m as well as m3 − 2m2 + 2m is a repunit of length 3 and 4 respectively in some number system generated by the roots of Pm (X), Qm (X). Their discriminants are 1 − 4m and −3m2 + 4m − 4 respectively. Notice that the first is positive for all m < 0, but the second never. Our next result show that the number systems generated by Qm (X), m < 0 are exceptional. Theorem 5.3. Let K be a number field. If  > 0 is odd and K has at least two, or  > 0 is even and K has at least three real conjugates, then there is no rational integer which is a repunit with respect to any number system in K. Proof. Plainly 1 is a repunit of length one in any number systems, whose digit set includes 1. Let γ be an algebraic integer and assume that 1 = n ∈ Z be a repunit in a number system (γ, D). Then there is an 2 ≤  ∈ Z such that n=

−1  i=0

γi =

γ − 1 . γ−1

Let γ  be a conjugate of γ. As the rational numbers are inert with respect to algebraic conjugation we get γ − 1 γ  − 1 =  . γ−1 γ −1

DIOPHANTINE PROPERTIES OF RADIX REPRESENTATIONS

147

−1 If  is odd then the function f (x) = xx−1 is strictly monotonically increasing for  x < −1 and x > 1. We have |γ|, |γ | > 1, hence the last equality is impossible. If  is even, precisely  = 2, then the zeroes of Qm (X) = X 2 + X + m satisfy f (γ) = f )γ  ). Now f (x) is strictly decreasing over (−∞, −1) and strictly increasing over (1, ∞), hence for fixed y ∈ R the equation f (x) = y, |x| > 1 may have at most two real solutions.  

Imre K´ atai and J´ ulia Szab´ o [8] characterized the CNS in the imaginary, and K´atai and Kov´acs [7] in the real quadratic number fields. Summarizing their results is Theorem 5.4. Let γ be a zero of the irreducible polynomial X 2 +aX +b ∈ Z[X], and set K = Q(γ). Then (γ, {0, 1, . . . , |b| − 1}) is a CNS in ZK if and only if 1 ≤ a ≤ b, and b ≥ 2. The roots of the polynomials Qm (X) = X 2 + X + m and Pm (X) = X 2 + mX + m2 − m + 1 generate CNS in which 1 − m as well as m3 − 2m2 + 2m are repunits of length 3 and 4 respectively. We found these examples with the help of our Algorithm. If 1 ≤ a ≤ b, b ≥ 2 be fixed then, by Theorem 3.1, there are only  finitely √ −a+ a2 −4b many rational integer repunits in the CNS , {0, 1, . . . , b − 1} . We did 2 not found any other CNS, in which some rational integer is a repunit. Therefore I propose the following conjectures.  √ −1+ 1−4m Conjecture 5.5. The only rational integer repunits in , 2   √ 2 +4m−4 {0, 1, . . . , m − 1}) and −m+ −3m , {0, 1, . . . , m2 − m} are 1−m and m3 − 2 2m2 + 2m respectively. Probably the following much stronger conjecture is still true. Conjecture 5.6. Apart from the examples of Conjecture 5.5 there are no CNS in quadratic number fields in which there are rational integer repunits. References n

−1 −1 [1] Yann Bugeaud and T. N. Shorey, On the Diophantine equation xx−1 = yy−1 , Pacific J. Math. 207 (2002), no. 1, 61–75, DOI 10.2140/pjm.2002.207.61. MR1974466 [2] Jean Marie Dumont, Peter J. Grabner, and Alain Thomas, Distribution of the digits in the expansions of rational integers in algebraic bases, Acta Sci. Math. (Szeged) 65 (1999), no. 3-4, 469–492. MR1737265 [3] Jan-Hendrik Evertse, On sums of S-units and linear recurrences, Compositio Math. 53 (1984), no. 2, 225–244. MR766298 [4] J.-H. Evertse, K. Gy˝ ory, A. Peth˝ o, and J. M. Thuswaldner, Number systems over general orders, Acta Math. Hungar. 159 (2019), no. 1, 187–205, DOI 10.1007/s10474-019-00958-x. MR4003703 [5] Jan-Hendrik Evertse and K´ alm´ an Gy˝ ory, Discriminant equations in Diophantine number theory, New Mathematical Monographs, vol. 32, Cambridge University Press, Cambridge, 2017. MR3586280 ¨nwald, Intorno all’aritmetica dei sistemi numerici a base negativa con partico[6] V. Gru lare riguardo al sis- tema numerico a base negativo-decimale per lo studio delle sue analogie coll’aritmetica ordinaria (decimale), Giornale di matematiche di Battaglini, 23 (1885), pp. 203–221, 367. [7] I. K´ atai and B. Kov´ acs, Kanonische Zahlensysteme in der Theorie der quadratischen algebraischen Zahlen (German), Acta Sci. Math. (Szeged) 42 (1980), no. 1-2, 99–107. MR576942 m

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[8] I. K´ atai and J. Szab´ o, Canonical number systems for complex integers, Acta Sci. Math. (Szeged) 37 (1975), no. 3-4, 255–260. MR389759 [9] Peter Kirschenhofer and J¨ org M. Thuswaldner, Shift radix systems—a survey, Numeration and substitution 2012, RIMS Kˆ okyˆ uroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, pp. 1–59. MR3330559 [10] Donald E. Knuth, An imaginary number system, Comm. ACM 3 (1960), 245–247, DOI 10.1145/367177.367233. MR0127508 [11] B. Kov´ acs, Canonical number systems in algebraic number fields, Acta Math. Acad. Sci. Hungar. 37 (1981), no. 4, 405–407, DOI 10.1007/BF01895142. MR619892 [12] W. Penney, A “binary” system for complex numbers, J. ACM, 12 (1965), pp. 247–248. [13] A. Peth¨ o, On a polynomial transformation and its application to the construction of a public key cryptosystem, Computational number theory (Debrecen, 1989), de Gruyter, Berlin, 1991, pp. 31–43. MR1151853 [14] Attila Peth¨ o, Algebraische algorithmen (German, with German summary), Friedr. Vieweg & Sohn, Braunschweig, 1999. MR1711312 [15] Attila Peth˝ o, Variations on a theme of K. Mahler, I, Ann. Univ. Sci. Budapest. Sect. Comput. 48 (2018), 137–149. MR3853712 [16] Attila Peth˝ o and J¨ org Thuswaldner, Number systems over orders, Monatsh. Math. 187 (2018), no. 4, 681–704, DOI 10.1007/s00605-018-1191-x. MR3861324 [17] Raphael M. Robinson, Conjugate algebraic integers on a circle, Math. Z. 110 (1969), 41–51, DOI 10.1007/BF01114639. MR246826 [18] T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, vol. 87, Cambridge University Press, Cambridge, 1986. MR891406 [19] A. J. van der Poorten and H. P. Schlickewei, The growth condition for recurrence sequences, Macquarie University Math. Rep., 82-0041 (1982). [20] Andrew Vince, Replicating tessellations, SIAM J. Discrete Math. 6 (1993), no. 3, 501–521, DOI 10.1137/0406040. MR1229702 Department of Computer Science, University of Debrecen, H-4002 Debrecen, P.O. Box 400, Hungary Email address: [email protected]

Vertex algebras

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15461

On certain W -algebras of type Wk (sl4 , f ) Draˇzen Adamovi´c, Antun Milas, and Michael Penn Abstract. We study several examples of simple W -algebras Wk (sl4 , f ) in connection to collapsing levels of affine vertex algebras and N = 3 conformal superalgebras.

1. Introduction In their important work [12, 13], B. Feigin and E. Frenkel showed that a vertex algebra W k (g) (called affine W -algebra) can be associated to a simple Lie algebra g by means of a quantum reduction of the corresponding level k universal affine vertex algebra V k (g). This construction made use of a principal nilpotent element of g. Further properties of W k (g) and its simple quotient Wk (g) were investigated by E. Frenkel, V. Kac, M.Wakimoto [14] and T. Arakawa [4]. This construction can be generalized to any nilpotent element f of g, which has been used by many authors culminating in the treatment of V. Kac, S. Roan, and M. Wakimoto ([15], [16]) who proved important structure theorems. It is generally unknown the structure of the simple vertex algebras Wk (g, f ) for any nilpotent element, e.g. a minimal generating set. In this short paper, our modest goal is to attempt to answer this question for a special class of W -algebras coming from the simple Lie algebra sl4 at certain special levels and nilpotent elements. We should point out that affine W -algebras associated to sl4 have already appeared in the literature [9] but associated to different nilpotent elements. Let us briefly outline the content of the paper. After setting up notation and structural results (e.g. Proposition 2.1), we first consider the level k = − 83 . This level is collapsing and we have an isomorphism with the affine vertex algebra of type sl2 (cf. Theorem 3.1) . We also have a related result for the level k = − 32 (see Remark 3.2). Next we consider k = − 52 . This case is no longer collapsing. Instead, we get an isomorphism with a certain subalgebra of the rank two βγ system, which is of type 13 , 23 . This isomorphism was already stated in [10]; here we only provide further details. 2020 Mathematics Subject Classification. Primary 17B69, 81R10. The first author was partially supported by the Croatian Science Foundation under the project 2634 and by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme. The second author acknowledges the support from the NSF grant DMS 1601070. c 2021 American Mathematical Society

151

´ A. MILAS, AND M. PENN D. ADAMOVIC,

152

In Section 5, we consider the W -algebra coming from W k (spo(2|3), fθ ), also known as the N = 3 superconformal algebra. Here we first outline the proof of Theorem 5.1 giving the structure of the even part of W k (spo(2|3), fθ ), being of type (13 , 2, 33 , 46 , 53 ). We then use this result to construct an isomorphism between W−1/3 (spo(2|3), fθ )Z2 and the simple affine VOA L−2/3 (sl2 ), cf. Corollary 5.1. The N = 3 vertex superalgebra is often tensored with free fermions F. The internal structure of the even part of the corresponding algebra is given in Theorem 5.2.This is then used to prove the main result of Section 5 (cf. Theorem 5.3): (W−1/3 (spo(2|3), fθ ) ⊗ F)Z2 ∼ = W−7/3 (sl4 , f ). 2. Affine W -algebras We begin with a review of the necessary background in order to motivate our construction, which involves the Lie algebra sl4 and a ”short” nilpotent element. 2.1. W -algebra W k (g, f ). Let g be a simple, finite dimensional Lie algebra and h, f ∈ g be two parts of an sl2 -triple, where f is nilpotent, and [h, f ] = −f. Now decompose g=

*

gj

j∈Z

by eigenvalues of ad h, where a ∈ gj if [h, a] = ja. Set gf = {a ∈ g : [f, a] = 0} and observe that all eigenvalues of ad x on gf are non-positive. Let A be the vector 7 super-space of g+ = j>0 gj , where the parity has been reversed, and A be the dual space. Let {bα |α ∈ S+ } and {cα |α ∈ S+ } be bases for A and A respectively and set Ach = A ⊕ A and form the associated vertex algebra of charged free fermions F(Ach ) where the nontrivial OPE is given by bα (z)cβ (z) ∼

δα,β . z−w

Now let V k (g) be the level k = −h∨ universal affine vertex algebra associated to g. We denote its simple quotient by Lk (g) (another standard notation is Lg (kΛ0 )). Now we form C(g, f, x) = Vk (g) ⊗ F(Ach ). and the following field of this vertex algebra d(z) = (2.1)



1  cγα,β ⊗ ◦◦ bγ (z)cα (z)cβ (z)◦◦ 2 α,β,γ∈S+  + (f |uα ) ⊗ cα (z),

uα (z) ⊗ cα (z) −

α∈S+

α∈S+

 are defined by [uα , uβ ] = γ cγα,β uγ where {uα } where the structure constants forms a basis of g. Define D = Resz d(z). It is well known that D is a vertex algebra homomorphism and defines a Z-graded homology complex (C(g, f, x), D). The associated homology is denoted Wk (g, f ) and is called the quantum reduction of Vk (g) with respect to the nilpotent element f ∈ g (see [11–13]. cγα,β

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Consider the following fields, for v ∈ g,  ◦ β ◦ cα J (v) (z) = v(z) ⊗ 1 + β (v) ⊗ ◦ bα (z)c (z)◦ , α,β∈S+

 α where are structure constants defined by [v, uβ ] = α∈S cβ (v)uα where {uα |α ∈ S} is a basis of g. By Theorem 4.1 of [KW] we know that for v ∈ gf the homology classes of the fields J (v) (z) strongly generate Wk (g, f ), where the class of J (v) (z) has conformal weight 1 − j if v ∈ gj . cα β (v)

Remark 1. In order to find the homology class for J (v) (z) we follow the strategy outlined in [6]. This involves splitting d(z) = d1 (z) + d2 (z), where  1  uα (z) ⊗ cα (z) − cγα,β ⊗ ◦◦ bγ (z)cα (z)cβ (z)◦◦ (2.2) d1 (z) = 2 α∈S+

α,β,γ∈S+



and d2 (z) =

(f |uα ) ⊗ cα (z),

α∈S+

where we will set Dj = Resz dj (z). Now, for v ∈ g we define W (v) (z) =

m 

(v)

(−1)j Wj (z),

j=0

where (2.3)

(v) W0 (z)

(v)

= J (v) (z) and for j ≥ 1, Wj (z) is chosen so that (v)

(v)

D2 (Wj−1 (z)) = D1 (Wj (z)), (v)

where this process ends in finitely many steps, where D2 (Wm (z)) = 0. It is easy to check that D1 (J (v) (z)) = 0 for v ∈ gf and thus D(W (v) (z)) = 0. 2.2. A W-algebra associated to sl4 and a short nilpotent element. We now move to the example of a quantum reduction that will be of interest in our setting. Denote the positive roots of sl4 by Δ+ = {α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 } and the root system Δ = Δ+ ∪ Δ− , where Δ− = −Δ+ . We may realize g = sl4 by * xα . g=h⊕ α∈Δ

We use non-degenerate bilinear form to identify h with h∗ . Definition 2.1. We say that a nilpotent element f ∈ g is called short if the adf eigenvalues are −1, 0, 1, namely we have deomposition g = g−1 ⊕ g0 ⊕ g1 . Now consider the short nipotent element f = x−α1 −α2 + x−α2 −α3 and its corresponding sl2 triple completed by e = xα1 +α2 + xα2 +α3 and h = 12 (α1 + 2α2 + α3 ). We decompose g into ad h eigenspaces g−1 = span {x−α2 , x−α1 −α2 , x−α2 −α3 , x−α1 −α2 −α3 } (2.4)

g0 = span {xα1 , xα3 , x−α1 , x−α3 , α1 , α2 , α3 } g1 = span {xα2 , xα1 +α2 , xα2 α3 , xα1 +α2 +α3 }.

Also consider the centralizer of f gf = span {α1 + α3 , xα1 + xα3 , x−α2 , x−α1 + x−α3 , x−α1 +α2 , x−α2 −α3 , x−α1 −α2 −α3 }.

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and its decomposition into ad h eigenspaces gf(−1) = span {x−α2 , x−α1 +α2 , x−α2 −α3 , x−α1 −α2 −α3 } gf(0) = span {α1 + α3 , xα1 + xα3 , x−α1 + x−α3 }. Set S+ = {xα2 , xα1 +α2 , xα2 α3 , xα1 +α2 +α3 } and the corresponding vector spaces A = span {bα2 , bα1 +α2 , bα1 +α2 +α3 } (2.5)

A∗ = span {cα2 , cα1 +α2 , cα1 +α2 +α3 } Ach = A ⊕ A∗ ,

and construct the vertex algebra of free charged fermions F(Ach ) as described above. In this setting we have d1 (z) = cα1 +α2 + cα2 +α3 and d2 (z) = ◦◦ xα2 cα2 ◦◦ + ◦◦ xα1 +α2 cα1 +α2 ◦◦ + ◦◦ xα2 +α3 cα2 +α3 ◦◦ + ◦◦ xα1 +α2 +α3 cα1 +α2 +α3 ◦◦ . An application of Theorem 4.1 of [16] gives (see also [7]): Proposition 2.1. The vertex algebra W k (sl4 , f ) is strongly generated by the homology classes of the fields J (α1 +α3 ) (z) =α1 (z) + α3 (z) + 2◦◦ bα1 +α2 cα1 +α2 ◦◦ + 2◦◦ bα1 +α2 +α3 cα1 +α2 +α3 ◦◦ J (xα1 +xα3 ) (z) =xα1 (z) + xα3 (z) + ◦◦ bα1 +α2 +α3 cα1 +α2 ◦◦ + ◦◦ bα2 +α3 cα2 ◦◦ − ◦◦ bα1 +α2 cα2 ◦◦ − ◦◦ bα1 +α2 +α3 cα2 +α3 ◦◦ J (x−α1 +x−α3 ) (z) =x−α1 (z) + x−α3 (z) + ◦◦ bα2 cα2 +α3 ◦◦ + ◦◦ bα1 +α2 cα1 +α2 +α3 ◦◦ − ◦◦ bα2 cα1 +α2 ◦◦ + ◦◦ bα2 +α3 cα1 +α2 +α3 ◦◦ J (x−α2 ) (z) = x−α2 (z) J (x−α1 −α2 ) (z) = x−α1 −α2 (z) J (x−α2 −α3 ) (z) = x−α2 −α3 (z) J (x−α1 −α2 −α3 ) (z) = x−α1 −α2 −α3 (z) of conformal weights 1, 1, 1, 2, 2, 2, 2 respectively. Let us denote by Wk (sl4 , f ) the simple quotient of W k (sl4 , f ).

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155

Following the iterative process described above (2.2)-2.3 we calculate the following generators: (2.6) (xα +xα ) 1 3 (z)

=J

(xα +xα ) 1 3 (z)

(x−α +x−α ) 1 3 (z)

=J

(x−α +x−α ) 1 3 (z)

W W

W

(α1 +α3 )

(z) =J (α1 +α3 ) (z)

1 ◦ (α1 ) (α1 )◦ 1 ◦ (α2 ) (α1 ) ◦ 1 ◦ (α2 ) (α2 )◦ J J J J J J ◦ − ◦ − ◦ 4◦ 2◦ 4◦ 1 1 ◦ (x−α ) (xα )◦ (α2 ) (α3 ) 3 2 J (k + 2)∂J − ◦J + (k + 2)∂J ◦ − 2 2 1 ◦ (α2 ) (α2 )◦ 1 ◦ (α3 ) (α2 ) ◦ 1 ◦ (α3 ) (α3 )◦ (x−α −α ) (x−α −α ) 2 3 (z) =J 2 3 (z) − W J J J J J J ◦ − ◦ − ◦ 4◦ 2◦ 4◦ 1 1 (x−α ) (xα )◦ 2 3 J −◦ (k + 2)∂J (α2 ) − (k + 2)∂J (α3 ) ◦J ◦ − 2 2 1 ◦ (α2 ) (xα )◦ 1 ◦ (α2 ) (xα )◦ (x ) (x ) (α1 ) (x−α )◦ 1 3 3 W −α1 −α2 −α3 (z) =J −α1 −α2 −α3 (z) − ◦ J J + J J J J ◦ ◦ ◦ − ◦ 2◦ 2◦ 1 ◦ (α3 ) (x−α )◦ 1 ◦ (α3 ) (x−α )◦ (x−α ) 1 3 1 + ◦J J J J ◦ − ◦ + (k + 2)∂J 2 2◦ 1 1 1 (x ) (x ) ◦ (α ) (x )◦ ◦ (α ) (x )◦ ◦ (α ) (x )◦ W −α2 =J −α2 + ◦ J 2 J α2 ◦ − ◦ J 2 J α3 ◦ − ◦ J 3 J α2 ◦ 2 2 2 1 ◦ (α ) (x )◦ (x ) − ◦ J 3 J α3 ◦ + (k + 2)∂J α2 . 2 W

(x−α −α ) 1 2 (z)

=J

(x−α −α ) 1 2 (z)

−

where the Virasoro field is given by (2.7) 1  (x−α −α ) (x−α2 −α3 ) 1 2 (z) + W L(z) = − W (z) − ◦◦ W (xα1 +xα3 ) W (x−α1 +x−α3 ) ◦◦ k+4  1 ◦ (α1 +α3 ) (α1 +α3 ) ◦ k + 1 (α1 +α3 ) ∂W − ◦W W ◦ − 8 2 and has central charge 12k2 + 41k + 32 . k+4 We make the following change of variables on the remaining fields for notational convenience into the following primary fields of weight 1, 1, 1, 2, 2, 2 respectively. cV = −

(2.8)

J + (z) = W (xα1 +xα3 ) (z) J − z) = W (x−α1 +x−α3 ) J 0 (z) = W (α1 +α3 ) (z) 1 H(z) = W (x−α1 −α2 ) (z) − W (x−α2 −α3 ) (z) − (k + 2)∂W (α1 +α3 ) (z) 2 (2.9) 1 ◦ (α1 +α3 ) (xα +xα ) ◦ 1 (x−α2 ) 3 (k + 2)∂W (xα1 +xα3 ) (z) E(z) = W + ◦W W 1 ◦ − 4 2 1 F (z) = W (x−α1 −α2 −α3 ) (z) + ◦◦ W (α1 +α3 ) W (x−α1 +x−α3 ) ◦◦ 4 1 (x−α1 +x−α3 ) − (k + 2)∂W (z) 2 The fields J + , J − , and J 0 generate a sub-VOA that is isomorphic to V 2k+4 (sl2 ), i.e., the level 2k +4 universal VOA associated to sl2 . The remaining nontrivial OPE are given by 1 : 1 For

brevity we write ab instead of a(z)b(w).

´ A. MILAS, AND M. PENN D. ADAMOVIC,

156

−H z−w H − J E∼ z−w 2E J0E ∼ z−w

2E , z−w −2F − , J H ∼ z−w −2F , J0F ∼ z−w +

+

J H ∼

HF ∼

− 12 (24 + 25k + 6k2 )J −

+

− ◦ (4 + k)◦ ◦ J L◦ − 1 4 (2

1 ◦ 0 0 −◦ 4 ◦J J J ◦

0 −◦ − k)◦ ◦ ∂J J ◦ −

+ 16k + 3k2 )∂J −

+ − −◦ −◦ ◦J J J ◦ −

1 2 (4

0 −◦ + k)◦ ◦ J ∂J ◦

+ 5k + k2 )∂ 2 J −

1 2 (6

z−w + 25k + k2 )J + (z −

+

1 2 (16

(z − w)2 z−w

1 2 (24

+

0 −◦ − 14 (8 + 3k)◦ ◦J J ◦ −

+

(z − w)3 +

HE ∼

J F ∼

+

w)3

+ ◦ −(4 + k)◦ ◦ J L◦ +

0 +◦ − 14 (8 + 3k)◦ ◦J J ◦ +

(z

1 ◦ 0 0 +◦ 4 ◦J J J ◦

1 2 (16 − w)2

+ + −◦ +◦ ◦J J J ◦ −

+ 14k + 3k2 )∂J + 1 2 (4

0 +◦ + k)◦ ◦ J ∂J ◦

z−w 0 +◦ − 14 (6 + k)◦ ◦ ∂J J ◦ +

1 2 (8

+ 5k + k2 )∂ 2 J +

z−w

+ −◦ 2(2 + k)(4 + k)L − (8 + 4k)◦ −(48 + 74k + 37k2 + 6k3 ) ◦J J ◦ − HH ∼ + (z − w)4 (z − w)2

+ EF ∼

(z − w)4

+ + EE ∼ =

0 0◦ k◦ 4 ◦ ∂J J ◦

+ (4 + 2k)∂J 0

+ (2 + k)∂ 2 J 0

z−w

− 12 (48 + 74k + 37k2 + 6k3 )

+

FF ∼

+ −◦ ◦ + −◦ (2 + k)(4 + k)∂L − (4 + 2k)◦ ◦ J ∂J ◦ − (4 + 2k)◦ ∂J J ◦ −

k ◦ 0 0◦ 4 ◦J J ◦

(2 + k)(4 + k)L −

1 2 (2

+

− 14 (24 + 25k + 6k2 )J 0 (z − w)3

0 0◦ + k)◦ ◦J J ◦ −

1 4 (8

+ −◦ + 5k)◦ ◦J J ◦ −

1 4 (8

+ 10k + 3k2 )∂J 0

(z − w)2 1 2 (2

+ k)(4 + k)∂L +

1◦ 0 ◦ 2 ◦ J L◦

−

1 ◦ 0 0 0◦ 8 ◦J J J ◦

−

1 ◦ 0 + −◦ 2 ◦J J J ◦

−

1 2 (2

+ −◦ + k)◦ ◦ J ∂J ◦

z−w + −◦ − 14 (4 + 3k)◦ ◦ ∂J J ◦ −

1 4 (3

0 0◦ + 2k)◦ ◦ ∂J J ◦ −

1 4 (2

+ 3k + k2 )∂ 2 J 0

z−w 1 4 (8

1 4 (8

+ +◦ + 3k)◦ ◦J J ◦

(z −

w)2

− −◦ + 3k)◦ ◦J J ◦

(z − w)2

+

+

1 4 (8

1 4 (8

+ +◦ + 3k)◦ ◦ ∂J J ◦

z−w − −◦ + 3k)◦ ◦ ∂J J ◦

z−w

3. The collapsing level k = − 83 We now examine a special case of Wk (sl4 , f ). Here we set k = − 83 and to motivate the upcoming result consider the following elements $ = 1 (−9◦◦ HH ◦◦ − 24◦◦ XY ◦◦ − 4∂H) A 48 $ = 1 (−3◦◦ HH ◦◦ − 8◦◦ XY ◦◦ + 4∂H) B 16 (3.1) 1 (−3◦◦ HY ◦◦ − 4∂Y ) P$ = 12 $ = 1 (◦◦ HX ◦◦ − 4∂X). Q 12 and consider the map(not a vertex algebra homomorphism) from W− 83 (sl4 , f ) to its subalgebra generated by X, Y, and H that sends each field to its “hatted” version,

ON CERTAIN W -ALGEBRAS

157

$ = X, Y$ = Y , and H $ = H The OPE of these fields only differ from the where X OPE for the fields A, B, P, and Q in the following ways −A P ∼ AP −A Q ∼ AQ −B  P ∼ BP

(3.2)

−B Q ∼ BQ  ∼ P Q − PQ

1 (−9◦◦ HHY ◦◦ − 48◦◦ H∂Y ◦◦ − 36◦◦ XY Y ◦◦ + 66◦◦ ∂HY ◦◦ − 4∂ 2 Y ) 72(z − w) 1 (9◦ HHX ◦◦ − 48◦◦ H∂X ◦◦ − 36◦◦ XXY ◦◦ − 6◦◦ ∂HX ◦◦ + 40∂ 2 X) 72(z − w) ◦ 1 (9◦ HHY ◦◦ + 48◦◦ H∂Y ◦◦ + 36◦◦ XY Y ◦◦ − 66◦◦ ∂HY ◦◦ + 4∂ 2 Y ) 72(z − w) ◦ 1 (−9◦◦ HHX ◦◦ + 48◦◦ H∂X ◦◦ − 36◦◦ XXY ◦◦ + 6◦◦ ∂HX ◦◦ − 40∂ 2 X) 72(z − w) 1 (9◦ HHH ◦◦ − 24◦◦ X∂Y ◦◦ + 36◦◦ HXY ◦◦ + 24◦◦ ∂XY ◦◦ + 4∂ 2 H). 72(z − w) ◦

 Fortunately, a routine calculation checks that for the level 2 − 83 + 2 = − 43 universal affine vertex algebra V −4/3 (sl2 ) these fields are in the maximal ideal used to form the corresponding simple VOA, L−4/3 (sl2 ). Therefore we obtained the following result. Theorem 3.1. We have a surjective vertex algebra homomorphism ϕ : W− 83 (sl4 , f ) → L−4/3 (sl2 ) In particular, 8 W − 3 (sl4 , f )/I ∼ = L−4/3 (sl2 )

where

8 9 $ B − B, $ P − P$ , Q − Q $ I = J + A − A,

and J is the maximal ideal of the copy of V −4/3 (sl2 ) in W− 83 (sl4 , f ) such that L−4/3 (sl2 ) = V −4/3 (sl2 )/J. 3.1. A different proof of Proposition 3.1. Remark 3.1. One can give an alternative proof of Proposition 3.1 based on the following facts: (i) Since VOA L−8/3 (sl4 ) is admissible, it is known that the maximal ideal J −8/3 in the universal affine voa V −8/3 (sl4 ) is generated by a singular vector Ω−8/3 of sl4 –weight ω1 + ω3 = θ (=highest root) and conformal ∞

weight 3. By using exact functor Hf2 from [5] (all details are covered in ∞

[5]), we get that Hf2 (J −8/3 ) is a non-trivial ideal in W −8/3 (sl4 , f ). (ii) Recall that W k (sl4 , f ) is strongly generated by the generators of V k (sl2 ), by the Virasoro vector ω and by the three dimensional subspace U2 ⊂ W k (sl4 , f )2 such that V k (sl2 ).U2 ∼ = V k (2ω1 ). ∞

(iii) The lowest graded component of Hf2 (J −8/3 ) has conformal weight 2 (= conformal weight of Ω−8/3 minus one, which is 3 − 1 = 2) and sl2 –weight θ  . Because there are no singular vectors of conformal weight less than three in V−4/3 (sl2 ), we get ∞

U2 ⊂ Hf2 (J −8/3 ).

´ A. MILAS, AND M. PENN D. ADAMOVIC,

158

∞

(iv) Therefore W −8/3 (sl4 , f )/Hf2 (J −8/3 ) is generated by V −8/3 (sl2 ) and possible the Virasoro vector ω. Since x(n)(ω − ωsug ) = L(n)(ω − ωsug ) = 0 (n ≥ 0, x ∈ sl2 ) we conclude ω − ωsug also belongs to this quotient. So ω = ωsug . ∞ (v) Therefore W −8/3 (sl4 , f )/Hf2 (J −8/3 ) collapses to an affine VOA of level ∞

−4/3. But using results from [5] again, we get that Hf2 (L−8/3 (sl4 )) is a simple VOA, and therefore we have that W−8/3 (sl4 , f ) = L−4/3 (sl2 ). 3.2. Collapsing level: k = −3/2. Remark 3.2. Using the analysis in Remark 3.1, we can easily prove that W−3/2 (sl4 , f ) ∼ = L1 (sl4 ). We consider the singular vector Ω−3/2 in the admissible affine vertex algebra V −3/2 (sl4 ) of conformal weight 4 and the sl4 –weight 2ω1 . 4. The level k = −5/2 and a certain Heisenberg coset In this section we prove that, if k = −5/2, our algebra is isomorphic to a coset of a Heisenberg vertex algebra inside of a rank two βγ system. This result was originally state without a proof in Remark 5.3 of [10]. Here we provide additional details. We recall the results of the previous construction of this algebra, calculate the OPE of the generating fields, and describe the relation with our algebra. Consider the rank 2 βγ system, S(2), generated by even, weight 1/2 fields β1 , β2 , γ1 , γ2 subject to the non-trivial OPE δi,j . βi γj ∼ z−w Consider the field h = ◦◦ β1 γ1 ◦◦ + ◦◦ β2 γ2 ◦◦ , which generates a rank 1 Heisenberg subalgebra of S(2), which we denote by H. Finally consider the coset C(2) = Com(H, S(2)). We now recall a theorem of [10]. Theorem 4.1 ([10] Theorem 5.3). C(2) is simple and of type W(1, 1, 1, 2, 2, 2). In fact, it is the simple quotient of an algebra of type W(1, 1, 1, 2, 2, 2, 2) where the Virasoro field in weight 2 coincides with the Sugawara field. Moreover, explicit primary generators are given by (4.1) x1,2 = −◦◦ β1 γ2 ◦◦ x2,1 = −◦◦ β2 γ1 ◦◦ h1 = −◦◦ β1 γ1 ◦◦ + ◦◦ β2 γ2 ◦◦ 1◦ 2 β1 β1 γ1 γ2 ◦◦ + ◦◦ β1 β2 γ2 γ2 ◦◦ 3◦ 3 1◦ 2◦ ◦ ◦ ◦ ◦ ◦ Q = ◦ β2 ∂γ1 ◦ − ◦ (∂β2 )γ1 ◦ + ◦ β1 β2 γ1 γ1 ◦ + ◦ β2 β2 γ1 γ2 ◦◦ 3 3 R = ◦◦ β1 β1 γ1 γ1 ◦◦ − ◦◦ β2 β2 γ2 γ2 ◦◦ +2◦◦ β1 ∂γ1 ◦◦ −2◦◦ β2 ∂γ2 ◦◦ −2◦◦ (∂β1 )γ1 ◦◦ +2◦◦ (∂β2 )γ2 ◦◦ , P = ◦◦ β1 ∂γ2 ◦◦ − ◦◦ (∂β1 )γ2 ◦◦ +

where x1,2 , x2,1 , h1 generate a subalgebra isomorphic to V−1 (sl2 ) and the Virasoro field is given by L = ◦◦ x1,2 x2,1 ◦◦ + 14 ◦◦ h1 h1 ◦◦ − 12 ∂h1 .

ON CERTAIN W -ALGEBRAS

159

The remaining nontrivial OPE of the generators (4.1) are given by: 2 3 x1,2

h1 P ∼

+ (z − w)2 2 x2,1 h1 Q ∼ 3 + (z − w)2 1◦ x1,2 x1,2 ◦◦ x1,2 P ∼ 3 ◦ z−w 1 3 h1 + x1,2 Q ∼ (z − w)2 x1,2 R ∼ x2,1 P ∼ x2,1 Q ∼ x2,1 R ∼ PP ∼ PQ ∼ + + PR ∼ + QQ ∼

−4P −

◦ 2◦ 3 ◦ h1 x1,2 ◦

− 23 L + 12 R + ◦◦ x1,2 x2,1 ◦◦ − 12 ∂h1 z−w

2 3 ∂x1,2

1 3 h1

(z −

w)2

2 3L

+

−

−◦ ◦ x1,2 x2,1 +

1 2R

◦ 1 2 ∂h1 ◦

z−w

◦ − 13 ◦ ◦ x2,1 x1,2 ◦

z−w 4Q +

◦ 2◦ 3 ◦ h1 x2,1 ◦

+

2 3 ∂x2,1

z−w ◦ 2◦ 9 ◦ x1,2 x1,2 ◦ 2 (z − w)

− 89 (z − w)4

+

+

◦ 2◦ 9 ◦ x1,2 ∂x1,2 ◦

z−w

8 9 h1

(z − w)3

◦ − 16 ◦ ◦ h1 R ◦ +

1 6R

+

◦ 2◦ 3 ◦ x2,1 P ◦

+

◦ 13 ◦ 9 ◦ x1,2 x2,1 ◦

+

◦ 1◦ 6 ◦ h1 h1 ◦

5 + − 18 ∂h1

(z − w)2

◦ +◦ ◦ h1 x1,2 x2,1 ◦ +

◦ 5 ◦ 18 ◦ h1 h1 h1 ◦

z−w 1 6 ∂R

+

◦ 17 ◦ 9 ◦ (∂x1,2 )x2,1 ◦

−

◦ 1◦ 3 ◦ x1,2 ∂x2,1 ◦

−

◦ 7 ◦ 18 ◦ h1 ∂h1 ◦

−

2 1 18 ∂ h1

z−w −4x1,2 + (z − w)3 ◦ 28 ◦ 9 ◦ h1 ∂x1,2 ◦

2 3P

−

+

◦ 10 ◦ 28 9 ◦ h1 x1,2 ◦ − 9 ∂x1,2 2 (z − w)

+

2 3 ∂P

◦ − 4◦ ◦ x1,2 L◦ +

◦ 1◦ 3 ◦ x1,2 R◦

−

◦ 17 ◦ 9 ◦ (∂h1 )x1,2 ◦

−

◦ 17 ◦ 9 ◦ (∂h1 )x2,1 ◦

z−w

19 2 9 ∂ x1,2

z−w ◦ 2◦ 9 ◦ x2,1 x2,1 ◦ 2 (z − w)

◦ 28 ◦ 9 ◦ h1 ∂x2,1 ◦

+ 2 3Q

+

◦ 2◦ 9 ◦ x2,1 , ∂x2,1 ◦

(z − w) +

◦ 10 ◦ 28 9 ◦ h1 x2,1 ◦ + 9 ∂x2,1 (z − w)2

+

2 3 ∂Q

◦ + 4◦ ◦ x2,1 L◦ −

◦ 1◦ 3 ◦ x2,1 R◦

z−w

19 2 9 ∂ x2,1

z−w

− 184 −8 9 L+ RR ∼ + (z − w)4 +

−2Q z−w

z−w

4x2,1 QR ∼ + (z − w)3 +

+

2P z−w

− 184 9 ∂L +

◦ 58 ◦ 3 ◦ h1 ∂h1 ◦

◦ 82 ◦ 9 ◦ h1 h1 ◦

+

◦ 259 ◦ 9 ◦ x1,2 x2,1 ◦

−

128 9 ∂h1

(z − w)2 +

◦ 104 ◦ 3 ◦ (∂x1,2 )x2,1 ◦

+

◦ 104 ◦ 3 ◦ x1,2 ∂x2,1 ◦

−

52 2 3 ∂ h1

z−w

We now return to our algebra, W−5/2 (sl4 , f ). Consider the following fields:

(4.2)

1 1 P$ = 2E − ◦◦ J 0 J + ◦◦ + ∂J + 6 6 1 1 $ = 2F − ◦◦ J 0 J − ◦◦ − ∂J − Q 6 6 4 4 $ = −4H + L − ◦◦ J + J − ◦◦ − 1 ◦◦ J 0 J 0 ◦◦ + 2 ∂J 0 , R 3 3 3 3

´ A. MILAS, AND M. PENN D. ADAMOVIC,

160

as well as 1 ◦ 0 0◦ 1 0 J J ◦ + ∂J . 4◦ 2 It is straightforward to check that, if we identify x1,2 with J + , x2,1 with J − , and h1 with J 0 the OPE algebra of the algebra W−5/2 (sl4 , f ) agrees with the OPE algebra of C(2) up to fields that are in the ideal containing T . Explicitly, we have the following [10]:

(4.3)

T = L − ◦◦ J + J − ◦◦ −

Theorem 4.2. We have W−5/2 (sl4 , f ) ∼ = C(2). That is, we have a surjective vertex operator algebra homomorphism Φ : W −5/2 (sl4 , f ) → C(2) whose mapping is described above and kernel contains the field T . Proof. It is clear the Φ(T ) = 0 and thus T ∈ ker Φ. As stated above, it is a straightforward albeit involved calculation, to check that the OPE algebra of the algebra W−5/2 (sl4 , f ) agrees with the OPE algebra of C(2) up to fields that are in the ideal containing T , and thus ker Φ. The differences in the OPE algebra are given by the following $ + 29 f (T ) $ = φ(P$)1 Φ(Q) Φ(P$1 Q) 9 $ = Φ(P$)1 Φ(Q) $ − 31 Φ(∂T ) + 29 Φ(◦◦ J 0 T ◦◦ ) + 1 Φ(T0 H) $ = φ(P$)0 φ(Q) Φ(P$0 Q) 18 9 3 $ = Φ(P$)0 Φ(Q) $ − 76 Φ(◦◦ T J + ◦◦ ) − 4Φ(T0 E) $ = Φ(P$)0 Φ(R) Φ(P$0 R) 9 $ $ = Φ(P )0 f (R) $ + 76 Φ(◦◦ T J − ◦◦ ) − 4Φ(T0 F ) $ 0 R) $ = Φ(Q) $ 0 Φ(R) Φ(Q 9 $ $ 0 Φ(R) = Φ(Q) $ + 92 Φ(T ) $0 R) $ = Φ(R) $ 0 Φ(R) Φ(R 9 $ $ = Φ(R)0 Φ(R).  5. Case k = −7/3 In this section we focus on the connection of W−7/3 (sl4 , f ) with the N = 3 superconformal algebra realized by the quantum Drinfeld-Sokolov reduction W−1/3 (spo(2|3), fθ ) as described in [16], where fθ is a minimal nilpotent, that is a lowest root vector of g. Our main result of this section will be the construction of the identification

 Z2 (5.1) W−7/3 (sl4 , f ) ∼ , = W−1/3 (spo(2|3), fθ ) ⊗ F

ON CERTAIN W -ALGEBRAS

161

where F is a rank one neutral free fermion vertex operator (super) algebra. Building up to this result, we find the Z2 orbifold of both W k (spo(2|3), fθ ) and W k (spo(2|3), fθ ) ⊗ F for generic k. Remark 2. By [1, Theorem 8.2] the minimal affine W –algebra W−1/3 (spo(2, 3), fθ ) of central charge c¯= −3/2 is realized as a subalgebra of SW (1)⊗ VL , the tensor product of the supertriplet vertex algebra SW (1) [2] with a lattice vertex algebra of central charge c = 1. It is also proved in [1] and [3] that * W−1/3 (spo(2, 3), fθ ) = V−2/3 (sl2 ) V−2/3 (2ω1 ). In particular, the algebra W−1/3 (spo(2, 3), fθ ) is a superconformal extension of V−2/3 (sl2 ) and it is generated by odd vectors of conformal weight 3/2: G± , G0 and by generators of V−2/3 (sl2 ). In what follows, we present a different proof of the structure of orbifold of W−1/3 (spo(2, 3), fθ ). 5.1. The orbifold W k (spo(2|3), fθ )Z2 . In [16] the algebra W k (spo(2|3), fθ ) is constructed and the OPEs are given. As such, we will only recall that this algebra is generated by three fields of weight 1, j 0 and j ± which generated a subVOA isomorphic to V −2−4k (sl2 ), a Virasoro field, L, of central charge −7/2 − 6k, and three odd weight 3/2 fields G0 and G± . It is clear that Ψ ∼ = Z2 ⊂ Aut(W k (spo(2|3), fθ )) where Ψ(G0 ) = −G0 and Ψ(G± ) = −G± ,

(5.2)

and the action on all other generators is trivial. By the classical invariant theory of Z2 ∼ = O(1) it is clear that in addition to the generators of the parent algebra that are fixed, j 0 , j ± , and L, the orbifold is generated by the fields W i,j (a, b) = ◦◦ ∂ a Gi ∂ b Gj ◦◦ ,

(5.3)

of weight 3 + a + b, where i, j ∈ {0, +, −} i ≤ j with 0 < + < −, for a, b ≥ 0. Following the reduction strategy in [8], using the ∂ operator, we make an initial reduction of this set to W i,i (0, 2a + 1) and W i,j (0, b),

(5.4)

for i, j ∈ {0, +, −} with i < j and a, b ≥ 0. Further, we know that the classical odd relations will give way to expressions of the form (5.5)

◦ ◦

W i0 ,j0 (a0 , b0 )W i1 ,j1 (a1 , b1 )◦◦ + ◦◦ W i0 ,j1 (a0 , b1 )W i1 ,j0 (a1 , b0 )◦◦

with (i0 , j0 , a0 , b0 ) < (i1 , j1 , a1 , b1 ) ordered lexicographically, with quantum corrections that will provide a meaningful reduction of the generating set (5.4). Using the standard methods of decoupling relations and boot-strapping operator we obtain the following result (full details will appear in a separate publication [17]): √ Theorem 5.1. For k = − 18 (5 ± 3i 7), the orbifold W k (spo(2|3), fθ )Z2 is minimally generated by the fields j 0 , j ± , L, W 0,0 (0, 1), W ±,± (0, 1), W 0,± (0, 0), W 0,± (0, 1), W 0,± (0, 2), W +,− (0, 0), W +,− (0, 1), and W +,− (0, 2) and is of type (13 , 2, 33 , 46 , 53 ).

´ A. MILAS, AND M. PENN D. ADAMOVIC,

162

Using results on conformal embeddings from [3], we obtain that, for k = − 13 , the maximal ideal I, such that (5.6) W−1/3 (spo(2|3), fθ ) ∼ = W −1/3 (spo(2|3), fθ )/I contains the difference T = L − Lsug , where 3 ◦ 0 0◦ 3 ◦ + −◦ 3 0 j j ◦ + ◦ j j ◦ − ∂j , (5.7) Lsug = 16 ◦ 4 8 is the conformal field of the sl2 sub-VOA. Further direct calculations such as W +,+ (0, 1) = − 2T1 W +,+ (0, 1) − (5.8) +

3 ◦ + 0,+ 3 j W (0, 0)◦◦ − ◦◦ ∂j 0 j + j + ◦◦ ◦ 2 32

1 ◦ + +◦ 1 ◦ 2 + +◦ ∂j ∂j ◦ + ◦ ∂ j j ◦ 4◦ 8

and 2 1 1 W 0,+ (0, 0) = T1 W 0,+ (0, 0) − ◦◦ j 0 ∂j + ◦◦ − ◦◦ j + L◦◦ 3 12 12 (5.9) 7 7 − ◦◦ ∂j 0 j + ◦◦ + ∂ 2 j + 48 72 allow us to remove the fields W i,j (a, b) from the generating set described in Theorem 5.1 in this setting leading to the following. As a consequence we obtain a different proof of result obtained earlier in [3, Theorem 6.8]: Corollary 5.1. We have W−1/3 (spo(2|3), fθ )Z2 ∼ = L−2/3 (sl2 ). 5.2. The orbifold (W k (spo(2|3), fθ ) ⊗ F)Z2 . The N = 3 algebra W (spo(2|3), fθ ) is often considered tensored with a rank one free fermion algebra F, generated by one odd field ϕ with OPE given by k

− 12 (2k + 1) . z−w The automorphism Ψ described in (5.2) can be extended to this setting by imposing ϕ(z)ϕ(w) ∼

(5.10)

Ψ(ϕ) = −ϕ.

(5.11)

It is clear the in this case the orbifold inherits the generators described by (5.4) in addition to the fields (5.12)

ωj (a, b) = ◦◦ ∂ a ϕ∂ b Gj ◦◦ and ωϕ (a, b) = ◦◦ ∂ a ϕ∂ b ϕ◦◦ ,

where j ∈ {0, +, −} and a, b ≥ 0. In this setting the expressions (5.5) will have their usefulness supersceded by the expressions of lower weight (5.13)

◦ ◦

ωi (a0 , b0 )ωj (a1 , b1 )◦◦ + ◦◦ ωi (a0 , b1 )ωj (a1 , b0 )◦◦

and (5.14)

◦ ◦

ωi (a0 , b0 )ωϕ (a1 , b1 )◦◦ + ◦◦ ωi (a0 , b1 )ωϕ (a1 , b0 )◦◦ ,

for a0 , a1 , b0 , b1 ≥ 0 and i, j ∈ {0, +, −}. Theorem 5.2. For k = − 12 the orbifold (W k (spo(2|3), fθ ) ⊗ F)Z2 is minimally generated by the fields j 0 , j ± , L, W 0,± (0, 0), W +,− (0, 0), ωϕ (0, 1), ω0 (0, 0), ω0 (0, 1), ω± (0, 0), and ω± (0, 1) and is of type (13 , 24 , 36 ).

ON CERTAIN W -ALGEBRAS

163

∼ L(− 1 , 0) that the Proof. It follows from the well-known fact that F Z2 = 2 generators of the form ωϕ (a, b) can be reduced to the single field ωϕ (0, 1). Next, we focus on the fields of the form W i,j (a, b) where i, j ∈ {0, +, −}, with explicit calculations given for the case when i = j = 0 as the others follow similarly. An argument similar to that found in [9] and [8], using the translation operator, allows us to reduce to the fields W i,j (0, m) for m ≥ 0 with m ∈ / 2Z if i = j. Finally, quantum corrections of (5.13), such as (1 + 2k)W 0,0 (0, m + 1) = −2◦◦ ω0 (0, 0)ω0 (0, m)◦◦ − (1 + 2k)∂W 0,0 (0, m) (1 + 2k)(3 + 4k) ωϕ (0, m + 3) 8m + 12   m  1 m ◦ m− ∂ Aωϕ (0,  + 1)◦◦ , +2 +1 m− ◦ +

(5.15)

=0

1 ◦ 0 0◦ + 2k)L − 16 for m ≥ 0 with A = ◦ j j ◦ , allow us to eliminate all fields of i,j the form W (0, m + 1) from our generating set, leaving us with only the fields W 0,+ (0, 0), W 0,− (0, 0), and W +,− (0, 0) remaining from this family of generators. Now we move on to the generators of the form ωi (a, b) where i ∈ {0, +, −} and a, b ≥ 0. Again, as above, we may only consider the fields of the form ωi (0, m) where m ≥ 0. A quantum correction of (5.14), namely,

− 12 (1

4◦ 2 ◦ (1 + 2k)∂ωi (0, m + 1), ◦ ωi (0, m)ωϕ (0, 1)◦ + 3 3 which holds for all m ≥ 0, eliminates the generators ωi (0, m + 2), leaving only ωi (0, 0) and ωi (0, 1).  (5.16)

(1 + 2k)ωi (0, m + 2) =

Now we move towards our main result, which will follow from Corollary 5.1 and Theorem 5.2. Again, we set k= − 13 and now consider the algebra W−1/3 (spo(2|3), fθ ) ⊗ F, which inherits the singular vector, T , described above (5.7), from the corresponding algebra without the free fermion. In this setting, we begin with the generating set described by Corollary 5.1 with the addition of the fields ωj (0, 0), ωj (0, 1), and ωϕ (0, 1) for j ∈ {0, +, −}. Now, equations such as 3◦ 0 3 j ω+ (0, 0)◦◦ − ◦◦ j + ω0 (0, 0)◦◦ , 4◦ 4 will allow us to remove the fields ωj (0, 1) for j ∈ {0, +, −}. Observe that we may replace the generator ωϕ (0, 1) with the field

(5.17)

(5.18)

ω+ (0, 1) = T0 ω+ (0, 0) +

˜ = Lsug + 3ωϕ (0, 1), L

setting up the following result, which is the main result of this section. Theorem 5.3. The orbifold (W−1/3 (spo(2|3), fθ ) ⊗ F)Z2 has a minimal strong ˜ ω0 (0, 0), and ω± (0, 0) and is of type generating set of the fields j 0 , j ± , L, (1, 1, 1, 2, 2, 2, 2). Moreover, we have (W−1/3 (spo(2|3), fθ ) ⊗ F)Z2 ∼ = W−7/3 (sl4 , f ). Proof. It is straightforward to check that the identification (W−1/3 (spo(2|3), fθ ) ⊗ F)Z2 → W−7/3 (sl4 , f ).

164

´ A. MILAS, AND M. PENN D. ADAMOVIC,

defined by j 0 → J 0 , (5.19)

j ± → J ± ,

i ω0 (0, 0) → − √ H, 2 6 i i ω+ (0, 0) → √ E, ω− (0, 0) → − √ F, 2 6 2 6 ˜ → L, L

preserves OPE and since (W−1/3 (spo(2|3), fθ ) ⊗ F)Z2 is simple this defines a surjective (and thus bijective) vertex algebra homomorphism.  Acknowledgments Results of the paper were presented by the third author at the conference ”Representation Theory XVI”, Dubrovnik, June 2019. We thank A. Linshaw for discussion regarding Theorems 4.1 and 4.2. Many computations in the paper are performed using Thielemans’ OPE Mathematica package [18]. References [1] Draˇ zen Adamovi´ c, Realizations of simple affine vertex algebras and their modules: the cases  and osp(1, 2), Comm. Math. Phys. 366 (2019), no. 3, 1025–1067, DOI 10.1007/s00220sl(2) 019-03328-4. MR3927085 [2] Draˇ zen Adamovi´ c and Antun Milas, The N = 1 triplet vertex operator superalgebras, Comm. Math. Phys. 288 (2009), no. 1, 225–270, DOI 10.1007/s00220-009-0735-2. MR2491623 [3] Draˇ zen Adamovi´ c, Victor G. Kac, Pierluigi M¨ oseneder Frajria, Paolo Papi, and Ozren Perˇse, Conformal embeddings of affine vertex algebras in minimal W -algebras II: decompositions, Jpn. J. Math. 12 (2017), no. 2, 261–315, DOI 10.1007/s11537-017-1621-x. MR3694933 [4] Tomoyuki Arakawa, Rationality of W -algebras: principal nilpotent cases, Ann. of Math. (2) 182 (2015), no. 2, 565–604, DOI 10.4007/annals.2015.182.2.4. MR3418525 [5] Tomoyuki Arakawa and Anne Moreau, Sheets and associated varieties of affine vertex algebras, Adv. Math. 320 (2017), 157–209, DOI 10.1016/j.aim.2017.08.039. MR3709103 [6] Jan de Boer and Tjark Tjin, The relation between quantum W algebras and Lie algebras, Comm. Math. Phys. 160 (1994), no. 2, 317–332. MR1262200 [7] Tomoyuki Arakawa and Alexander Molev, Explicit generators in rectangular affine Walgebras of type A, Lett. Math. Phys. 107 (2017), no. 1, 47–59, DOI 10.1007/s11005-0160890-2. MR3598875 [8] Olivia Chandrasekhar, Michael Penn, and Hanbo Shao, Z/2Z invariants of the free fermion algebra, Comm. Algebra 46 (2018), no. 10, 4201–4222, DOI 10.1080/00927872.2017.1388810. MR3847109 [9] Thomas Creutzig and Andrew R. Linshaw, Orbifolds of symplectic fermion algebras, Trans. Amer. Math. Soc. 369 (2017), no. 1, 467–494, DOI 10.1090/tran6664. MR3557781 [10] T. Creutzig, S. Kanade, A. R. Linshaw, and D. Ridout, Schur-Weyl duality for Heisenberg cosets, Transform. Groups 24 (2019), no. 2, 301–354, DOI 10.1007/s00031-018-9497-2. MR3948937 [11] Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, 2nd ed., Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2004. MR2082709 [12] Boris Feigin and Edward Frenkel, Quantization of the Drinfel d-Sokolov reduction, Phys. Lett. B 246 (1990), no. 1-2, 75–81, DOI 10.1016/0370-2693(90)91310-8. MR1071340 [13] Boris L. Feigin and Edward V. Frenkel, Representations of affine Kac-Moody algebras, bosonization and resolutions, Lett. Math. Phys. 19 (1990), no. 4, 307–317, DOI 10.1007/BF00429950. MR1051813 [14] Edward Frenkel, Victor Kac, and Minoru Wakimoto, Characters and fusion rules for W algebras via quantized Drinfel d-Sokolov reduction, Comm. Math. Phys. 147 (1992), no. 2, 295–328. MR1174415

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[15] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto, Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003), no. 2-3, 307–342, DOI 10.1007/s00220-003-0926-1. MR2013802 [16] Victor G. Kac and Minoru Wakimoto, Quantum reduction and representation theory of superconformal algebras, Adv. Math. 185 (2004), no. 2, 400–458, DOI 10.1016/j.aim.2003.12.005. MR2060475 [17] M. Penn, in preparation. [18] K. Thielemans, A Mathematica package for computing operator product expansions, Internat. J. Modern Phys. C 2 (1991), no. 3, 787–798, DOI 10.1142/S0129183191001001. MR1130315 Department of Mathematics, University of Zagreb, Croatia Email address: [email protected] Department of Mathematics and Statistics, SUNY-Albany, Albany, New York 12222 Email address: [email protected] Mathematics Department, Randolph College, Lynchburg, Virginia 24503 Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15462

On the N = 1 super Heisenberg-Virasoro vertex algebra Draˇzen Adamovi´c, Berislav Jandri´c, and Gordan Radobolja Abstract. We introduce the N = 1 super Heisenberg-Virasoro vertex algebra V SH (cL , cα , cL,α ), which is a super-analog of the Heisenberg-Virasoro vertex algebra. We also define the N = 1 super Heisenberg-Virasoro conformal algebra, so that V SH (cL , cα , cL,α ) is its universal enveloping vertex algebra. We show that when cα = 0, the simple vertex algebra LSH (cL , cα , cL,α ) is isomorphic to the tensor product of Clifford-Heisenberg vertex algebra and simple N = 1 Neveu-Schwarz vertex algebra of suitable central charge. This generalises results from [1]. In the case of level zero, i.e, cα = 0, we also find a free-field realization which generalises the realisation from [3]. In our forthcoming paper, we shall study the representation theory of LSH (cL , cα = 0, cL,α ) at level zero.

1. Introduction The Heisenberg-Virasoro Lie algebra H naturally arises from a free field realization of Virasoro algebra within a rank one Heisenberg algebra. It was introduced by Arbarello et all. in 1988 [1]. Determinant formula and irreducibility criteria for Verma modules were obtained by realising highest weight H–modules as tensor products of highest weight modules over Virasoro and Heisenberg algebras. These results hold when level, i.e. central charge corresponding to the central element of Heisenberg subalgebra is nonzero. Other representations of this algebra have been studied in many papers over the last decades. Representations at level zero have first appeared in construction of vertex operator representations for full toroidal Lie algebras [7]. The structures of Verma and irreducible modules were studied in [6] where the author obtained determinant formula for level zero case by taking limit of formula in non-zero case. Free field realization of some indecomposable modules was constructed in [10]. The Fock space realizations of all highest weight modules and formulas for screening operators were obtained in [3] and [5]. From these constructions, the authors have recovered explicit formulas for singular vectors in Verma modules, calculated fusion rules among irreducible quotients and constructed several families 2020 Mathematics Subject Classification. Primary 17B69; Secondary 17B20, 17B65. The first and third authors were partially supported by the QuantiXLie Centre of Excellence, a project coffinanced by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004). c 2021 American Mathematical Society

167

168

ˇ ´ BERISLAV JANDRIC, ´ AND GORDAN RADOBOLJA DRAZEN ADAMOVIC,

of logarithmic modules. Finally, an interesting application to Galilean Virasoro (or W (2, 2)) algebra was discussed in [3]-[5]. In this paper, we introduce a super-generalisation of the Heisenberg-Virasoro Lie algebra and associated vertex algebra. To the best of our knowledge, these generalizations are novel. There are two natural ways to associate the (universal) vertex algebra. One is to define the N = 1 Heisenberg-Virasoro conformal algebra (cf. [11]) and consider its universal enveloping vertex algebra. Alternatively, one can use the theory of local fields [12] and get the vertex algebra structure on an appropriate quotient of Verma module of highest weight zero. In this paper we first recall the definition and free field realization of HeisenbergVirasoro vertex algebra. In Section 4 we define the N = 1 Heisenberg-Virasoro vertex superalgebra as a universal enveloping algebra associated to the HeisenbergVirasoro conformal algebra and find its structure as a module over the Lie algebra. Free field realization at non-zero level is given in Section 5. We prove that the universal (resp. simple) N = 1 Heisenberg-Virasoro vertex algebra is isomorphic to the tensor product of a rank one Heisenberg algebra, rank one fermionic algebra and universal (resp. simple) N = 1 Neveu-Schwarz vertex algebra. In Section 6 we present a realization at zero level which generalises the constructions from [3] and [5]. Finally, in Section 7 we obtain a determinant formula for Verma modules and announce the structure results for zero level which will be published in [2] 2. Preliminaries Definition 2.1. [11] Lie conformal superalgebra is a Z2 -graded C[D]-module R = R¯0 ⊕ R¯1 , on which a C-bilinear map λ-bracket is defined: [·λ ·] : R × R → C[λ] ⊗ R, which satisfies the following properties • conformal sesquilinearity (derivation property): [(Da)λ b] D [aλ b]

= −λ [aλ b] = [(Da)λ b] + [aλ (Db)]

• skew-symmetry: [aλ b] = −p(a, b) [b−λ−D a] • Jacobi identity: [aλ [bμ c]] − p(a, b) [bμ [aλ c]] = [[aλ b]λ+μ c] Here, we use the notation C[λ] ⊗ R for the Z2 -graded vector space of polynomials in the formal variable λ with coefficients in R. Next we fix notation for algebras appearing below. • Let V V ir (c, 0) (resp. LV ir (c, 0)) denote the universal (resp. simple) Virasoro vertex algebra. • Let V ns (c, 0) (resp. Lns (c, 0)) denote the universal (resp. simple) N = 1 Neveu-Schwarz vertex algebra. • Denote by M (1) the Heisenberg vertex algebra which is freely generated by an even field h satisfying [hλ h] = λ. • Denote by M (1)a the Heisenberg vertex algebra with conformal vector ωa = 12 h(−1)2 + ah(−2) of central charge ca = 1 − 12a2 (cf. [11].

THE N = 1 SUPER HEISENBERG-VIRASORO VERTEX ALGEBRA

169

• Denote by F the fermionic vertex algebra which is freely generated by an odd field φ satisfying [φλ φ] = 1. • Let SM (1) := F ⊗ M (1). The PBW basis of SM (1) is given by φ(−m1 − 1/2) · · · φ(−ms − 1/2)h(−j1 − 1) · · · h(−jt − 1)1 where m1 > · · · > ms ≥ 0, j1 ≥ · · · ≥ jt ≥ 0. [11] The vertex algebra SM (1) contains a subalgebra isomorphic to the universal N = 1 Neveu-Schwarz vertex algebra V ns (cμ , 0) where cμ = 3/2 − 3μ2 . The N = 1 superconformal vector and the corresponding Virasoro vector are given by: (2.1)

ωμ

(2.2)

τμ

 1 1 τ0 τ = h(−1)2 + μh(−2) + φ(−3/2)φ(−1/2) 1, 2 2 = (h(−1)φ(−1/2) + μφ(−3/2))1. =

Denote by SM (1)μ the vertex algebra SM (1) with this super-conformal structure. 3. Heisenberg-Virasoro vertex algebra In this section we recall basic definition and results for the Heisenberg-Virasoro vertex algebra. Definition 3.1. The Heisenberg-Virasoro conformal algebra H is generated by two even generators ω, α and even central elements CL , Cα , CL,α , along with the derivation D: C[D] ⊗ spanC {ω, α, CL , Cα , CL,α } satisfying the following λ-bracket: (3.1)

[αλ α] = λCα ,

(3.2)

[ωλ ω] = (D + 2λ)ω +

(3.3)

1 3 λ CL ; 12 [ωλ α] = Dα + λα − λ2 CL,α

Let us denote by V H (cL , cα , cL,α ) the universal enveloping vertex algebra associated tothe Heisenberg-Virasoro conformal algebra H. It is generated by the fields  α(z) = n∈Z α(n)z −n−1 and L(z) = n∈Z L(n)z −n−2 whose components satisfy the commutation relations for the Heisenberg-Virasoro Lie algebra (cf. [6], [8], [3]) such that central elements CL , Cα , CL,α acts as scalars cL , cα , cL,α respectively. By LH (cL , cα , cL,α ) we will denote the simple quotient of V H (cL , cα , cL,α ). The following theorem is a vertex-algebraic reformulation of the results from [1] (see also [8]): Theorem 3.2. Assume that cα = 0. Then we have the following isomorphism of vertex algebras: (1) V H (cL , cα , cL,α ) ∼ = M (1)a ⊗ V V ir (c, 0), H ∼ M (1)a ⊗ LV ir (c, 0), (2) L (cL , cα , cL,α ) = √ V ir + ωa , α = cα h where such that ω = ω √ cL = c + 1 − 12a2 , cL,α = a cα

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170

Using the embedding of the Virasoro vertex algebra in M (1), we get the free field realization of the Heisenberg-Virasoro vertex algebra when cα = 0. In the case of level zero, i.e., cα = 0, the free field realization of LH (cL , cL,α ) was given in [3]. Here we recall the construction. Consider lattice L = Zc + Zd such that c, c = d, d = 0, c, d = 2. Let h = C ⊗Z L and extend the form ·, · on h. We can consider h as an abelian Lie algebra. Let $ h = C[t, t−1 ] ⊗ h ⊕ CK be the affinization of h. Let M2 (1) be the Heisenberg vertex algebra generated by the fields c(z) and d(z). Theorem 3.3. [3] The vertex subalgebra of M2 (1) generated by −cL,α c(−1), cL − 2 1 1 (3.5) c(−1)d(−1) + c(−2) − d(−2). ω = 2 24 2 is isomorphic to the simple universal Heisenberg-Virasoro vertex algebra LH (cL , cL,α ). (3.4)

α

=

4. Definition of the N = 1 Heisenberg-Virasoro vertex algebra Definition 4.1. The N = 1 Heisenberg-Virasoro algebra SH is an infinite dimensional Lie algebra with even generators L(n), α(n), n ∈ Z, odd generators G(n + 12 ), Ψ(n + 12 ), and three central elements CL , Cα , CL,α , subject to the following super-commutator relations: (4.1)

[α(m), α(n)] = δm+n,0 mCα

(4.2)

[L(m), α(n)] = −nα(m + n) − δm+n,0 (m2 + m)CL,α

(4.3)

[L(m), L(n)] = (m − n)L(m + n) + δm+n,0

(4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11)

m3 − m CL 12

1 1 [Ψ(m + ), Ψ(n + )]+ = δm+n+1,0 Cα 2 2 1 [α(m), Ψ(n + )] = 0 2 m2 + m 1 1 CL [G(m + ), G(n + )]+ = 2L(m + n + 1) + δm+n+1,0 2 2 3 m 1 1 1 [L(m), G(n + )] = ( − n − )G(m + n + ) 2 2 2 2 1 1 [α(m), G(n + )] = mΨ(m + n + ) 2 2 1 2m + n + 1 1 [Ψ(m + ), L(n)] = Ψ(m + n + ); 2 2 2 1 1 [Ψ(m + ), G(n + )]+ = α(m + n + 1) + 2mδm+n+1,0 CL,α 2 2 [SH, Cα ] = [SH, CL ] = [SH, CL,α ] = 0

Definition 4.2. The N = 1 Heisenberg-Virasoro conformal algebra SH is defined by two even generators ω, α, even central elements CL , Cα , CL,α and two odd generators τ , Ψ, along with the derivation D: C[D] ⊗ (spanC {ω, α, CL , Cα , CL,α } ⊕Z2 spanC {τ, Ψ})

THE N = 1 SUPER HEISENBERG-VIRASORO VERTEX ALGEBRA

171

satisfying the following λ-bracket: (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.19)

[ωλ ω] = (D + 2λ)ω +

1 3 λ CL ; 12

3 [ωλ τ ] = (D + λ)τ 2 [ωλ α] = Dα + λα − λ2 CL,α 1 [ωλ Ψ] = (D + λ)Ψ 2 λ2 [τλ τ ] = 2ω + CL 3 [Ψλ τ ] = α + 2λCL,α [αλ τ ] = λΨ [αλ α] = λCα , [Ψλ Ψ] = Cα ,

[αλ Ψ] = 0.

SH

Let us denote by V (cL , cα , cL,α ), the universal enveloping vertex algebra associated to the N = 1 Heisenberg-Virasoro conformal algebra SH. It is generated by the fields  α(z) = α(n)z −n−1 n∈Z

L(z)

=



L(n)z −n−2

n∈Z

Ψ(z)

=



n∈Z

G(z)

=



n∈Z

1 Ψ(n + )z −n−1 2 1 G(n + )z −n−2 2

SH By LSH (cL , cα , cL,α ) we will denote the simple (cL , cα , cL,α ). When

m of V ∞quotient := the commutator formula [a(m), b(n)] j=0 j (a(j) b)(m+n−j) is applied, one directly sees that every V SH (cL , cα , cL,α )–module is a module for the N = 1 Heisenberg-Virasoro Lie algebra SH. Let us describe the vertex algebra V SH (cL , cα , cL,α ) as SH–module. Let SH+ be the Lie subalgebra of SH:

SH+ = spanC {L(n), G(n + 1/2), α(n), Ψ(n + 1/2), CL , Cα , CL,α | n ∈ Z≥0 }. For every (h, hα , cL , cα , cL,α ) ∈ C5 let CvcL ,cα ,cL,α ,h,hI be the 1–dimensional SH+ – module such that for n ∈ Z≥0 : L(n)vcL ,cα ,cL,α ,h,hα = δn,0 hvcL ,cα ,cL,α ,h,hα , α(n)vcL ,cα ,cL,α ,h,hα = δn,0 hα vcL ,cα ,cL,α ,h,hα , G(n + 12 )vcL ,cα ,cL,α ,h,hα = Ψ(n + 12 )vcL ,cα ,cL,α ,h,hα = 0, and CL , Cα , CL,α act by scalars cL , cα , cL,α . The Verma module of highest weight (h, hα ) and central charge (cL , cα , cL,α ) is the following induced SH–module V SH (cL , cα , cL,α , h, hα ) := U (SH) ⊗U(SH+ ) CvcL ,cα ,cL,α ,h,hα . For simplicity of notation, we omit central charges (cL , cα , cL,α ) and write V SH (h, hα ) and vh,hα .

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Consider the case h = hα = 0. Then G(−1/2)v0,0 is a singular vector in V SH (0, 0) and we have the quotient module: V SH (0, 0) =

V SH (0, 0) , G(−1/2)v0,0

where G(−1/2)v0,0 is the SH–submodule generated by G(−1/2)v0,0 . Let 1 be the highest weight vector in V SH (0, 0). Clearly L(−1)1 = G(−1/2)1 = 0. Proposition 4.3. As SH–module we have: V SH (cL , cα , cL,α ) ∼ = V SH (0, 0). As a vector space V SH (cL , cα , cL,α ) is isomorphic to : V N S (c, 0) ⊗ (Ψ(−n − 1/2)| n ∈ Z≥0 ) ⊗ C[α(−n) | n ∈ Z≥1 ]. Proof. By the definition of universal vertex algebra generated by a conformal Lie superalgebra we see that the PBW basis of V SH (cL , cα , cL,α ) consists of monomials Ψ(−m1 − 1/2) · · · Ψ(−ms − 1/2)α(−j1 − 1) · · · α(−jt − 1) · ·G(−n1 − 3/2) · · · G(−nr − 3/2)L(−k1 − 2) · · · L(−ku − 2)1 where n1 > · · · nr ≥ 0, m1 > . . . ms ≥ 0, j1 ≥ · · · ≥ jt ≥ 0, k1 ≥ · · · ≥ ku ≥ 0. But, this is exactly the PBW basis of V SH (0, 0). The proof follows.  5. Realization at non-zero level Proposition 4.3 shows that the universal N = 1 Heisenberg-Virasoro vertex algebra is isomorphic to V N S (c, 0) ⊗ SM (1) as a vector space. In this section we shall lift this isomorphism to the level of vertex algebras. Recall first that SM (1) has the structure of N = 1 Neveu-Schwarz vertex algebra, where the conformal and superconformal vectors are given by (2.1) and (2.2). The next result shows that SM (1) is a simple N = 1 Heisenberg-Virasoro vertex algebra. √ √ Theorem 5.1. For a fixed μ ∈ C, cα = 0, the vectors α = cα h, Ψ = cα φ, ω and τ generate on SM (1)μ the structure of a simple vertex algebra LSH (cμ , cα , cL,α ) where 1 √ cμ = 3/2 − 3μ2 , cL,α = μ cα . 2 Proof. It suffices to show that these four vectors satisfy the λ−bracket from Definition 4.2. Note that τ and ω are the same as in (2.2) and (2.1) and therefore, they satisfy λ–bracket for the universal N = 1 Neveu-Schwarz vertex algebra of central charge cμ = 3/2 − 3μ2 : [τλ τ ] = [τλ τ ] = 2ω +

λ2 cμ 6

3 [ωλ τ ] = (D + λ)τ 2 [ωλ ω] = (D + 2λ)ω +

λ3 cμ 12

THE N = 1 SUPER HEISENBERG-VIRASORO VERTEX ALGEBRA

173

The following λ–bracket can be easily determined: [αλ α] = cα λ 1 [ωλ Ψ] = (D + λ)Ψ 2 √ λ2 cα ω2 h 2 √ λ2 (D + λ)α − cα μ 2 Ψ(1/2)τ + λΨ(3/2)τ √ cα (φ(1/2)τ + λφ(3/2)τ ) √ √ cα h + cα μλ α + 2cL,α λ λΨ

[ωλ α] = (D + λ)α + = [Ψλ τ ] = = = = [αλ τ ] =

Since SM (1) is a simple vertex algebra generated by α and Ψ, it is clear that SM (1) ∼  = LSH (cμ , cα , cL,α ). The proof follows. For c = 0 let LN S (c, 0) (resp. V N S (c, 0)) denote the simple (resp. universal) N = 1 Neveu-Schwarz vertex algebra generated by τ¯ and ω ¯ = 12 τ¯0 τ¯ satisfying the following λ–brackets: [¯ τλ τ¯] = [¯ ωλ ω ¯] =

λ2 c 3

3 [¯ ωλ τ¯] = (D + λ )¯ τ, 2 c (D + 2λ)¯ ω + λ3 . 12 2¯ ω+

Corollary 5.2. For any ordered triple (cL , cα , cL,α ) ∈ C3 , cα = 0, we have the isomorphism of vertex algebras: LSH (cL , cα , cL,α ) ∼ (5.1) = SM (1)μ ⊗ LN S (cns , 0), where (5.2)

2cL,α μ= √ , cα

cns = cL − cμ .

Proof. Define μ and cns by (5.2). Then the following four vectors in SM (1)μ ⊗ LN S (cns , 0) ∼ = LSH (cμ , cα , cL,α ) ⊗ LN S (cns , 0): α ˜ = α ∈ SM (1) ˜ Ψ = Ψ ∈ SM (1) ¯ ω ˜ = ωμ + ω τ˜ = τμ + τ¯ satisfy the same λ−bracket as generators of V SH (cL , cα , cL,α ). This gives a vertex algebra homomorphism f : V SH (cL , cα , cL,α ) → SM (1)μ ⊗ LN S (cns , 0). Next we notice that all generators of SM (1)μ ⊗ LN S (cns , 0) belong to Im(f ) so we conclude that f is surjective. (Indeed α and Ψ belong to Im(f ) by definition, so generators of SM (1) belong to Im(f ). Since ωμ and τμ are expressed in terms of α, Ψ we conclude that they also belong to Im(f ). Therefore ω ¯ and τ¯ are elements of Im(f ).)

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174

On the other hand, the vertex algebra SM (1)μ ⊗ LN S (cns , 0) is the tensor product of two simple vertex algebras, so it must be simple. Therefore (5.1) holds. The proof follows.  Theorem 5.3. We have the following isomorphism of vertex algebras: V SH (cL , cα , cL,α ) ∼ (5.3) = SM (1)μ ⊗ V N S (cns , 0), where cα = 0 and cns is as in Corollary 5.2. Proof. As in Corollary 5.2 we see that SM (1)μ ⊗ V N S (cns , 0) is a cyclic SH– module. This implies that SM (1)μ ⊗ V N S (cns , 0) is a highest weight SH–module with highest weight (0, 0) such that G(−1/2)(1 ⊗ 1) = L(−1)(1 ⊗ 1) = 0. Therefore SM (1)μ ⊗ V N S (cns , 0) must be a certain quotient of V SH (0, 0). But one can easily see from Proposition 4.3 that V SH (cL , cα , cL,α ) is as a vector space isomorphic to SM (1)μ ⊗ V N S (cns , 0), which proves the assertion.  6. Realization at level zero In this section we give a realization at level zero. A detailed analysis of the representation theory at level zero will be given in our forthcoming paper [2]. As in the case of Heisenberg-Virasoro vertex algebra, we consider lattice L = Zc + Zd be a lattice such that c, c = d, d = 0, c, d = 2, let VL = C[L] ⊗ M2 (1) be the corresponding lattice vertex algebra, where M2 (1) is a Heisenberg vertex algebra generated by c(z) and d(z). Let F (2) be the fermionic vertex algebra generated by fields  1 Ψi (n + )z −n−1 Ψi (z) = 2 n∈Z

such that for i = 1, 2, r, s ∈

1 2

+ Z we have the following anti-commutator relation

{Ψi (r), Ψi (s)} = 0, {Ψ1 (r), Ψ2 (s)} = δr+s,0 . The vertex algebra F (2) has the following Virasoro vector of central charge cf er = 1:   1 3 1 3 1 Ψ1 (− )Ψ2 (− ) + Ψ2 (− )Ψ1 (− ) 1. ωf er = 2 2 2 2 2 Define the following four vectors in the vertex algebra M2 (1) ⊗ F (2) : −cL,α c(−1) √ 1 1 cL − 3 1 1 3 3 Ψ2 (− ) − Ψ1 (− )) 2( c(−1)Ψ1 (− ) + d(−1)Ψ2 (− ) + τ = 2 2 2 2 12 2 2 1 cL − 3 1 ω = c(−1)d(−1) + c(−2) − d(−2) + ωf er 2 24 2 √ 1 Ψ = − 2cL,α Ψ2 (− ). 2 Let V(cL , cL,α ) denote the vertex subalgebra of M2 (1)⊗F (2) generated by α, Ψ, τ, ω. In the rest of the paper we use the following shorter notation for level zero algebra V SH (cL , cL,α ) := V SH (cL , 0, cL,α ). α

=

THE N = 1 SUPER HEISENBERG-VIRASORO VERTEX ALGEBRA

175

Theorem 6.1. The vertex algebra V(cL , cL,α ) is isomorphic to (a certain quotient of ) the N = 1 Heisenberg-Virasoro vertex algebra at level zero V SH (cL , cL,α ). Proof. The proof follows from λ–brackets cL cL [ωλ ω] = (D + 2λ)ω + λ3 , [τλ τ ] = 2ω + λ2 (6.1) 12 3 3 (6.2) [ωλ τ ] = (D + )τ 2 [αλ α] = [Ψλ Ψ] = [αλ Ψ] = 0, (6.3) (6.4)

[αλ τ ] = Ψλ,

[αλ ω] = αλ + cL,α λ2 ,

1 [ωλ Ψ] = (D + λ)Ψ 2 The conformal vector ω can be written as a sum of two commuting conformal vectors ω = ωH + ωf er where cL − 3 1 1 c(−2) − d(−2) ωH = c(−1)d(−1) + 2 24 2 is a conformal vector of central charge cL − 1 (cf. [3], [5]) and ωf er is a conformal vector in F (2) of central charge c = 1. Therefore, ω is a conformal vector of central charge cL . Let us now prove that 2 τ0 τ = 2ω, τ1 τ = 0, τ2 τ = cL 1. 3 which give relation (6.1). We have   1 1 3 1 3 1 τ0 τ = Ψ1 (− )Ψ2 (− ) + Ψ2 (− )Ψ1 (− ) 1 2 2 2 2 2 2 cL − 3 1 1 + c(−2) − d(−2) + c(−1)d(−1) = ω 24 2 2   1 1 1 1 Ψ1 (− )Ψ2 (− ) + Ψ2 (− )Ψ1 (− ) 1 = 0 τ1 τ = 2 2 2 2 2 cL − 3 )1 = cL . τ2 τ = 2(1 + 3 3 The proof of λ–brackets relations (6.2)-(6.5) is easy. 

(6.5)

[Ψλ τ ] = α + 2cL,α λ,

Remark 6.2. We shall prove in [2] that V(cL , cL,α ) = V (cL , cL,α ) is a simple vertex superalgebra. 7. Determinant formula The goal of this section is to derive the determinant formulae for the N = 1 super Heisenberg-Virasoro algebra at cα = 0. We follow the approach from [6] for the Heisenberg-Virasoro algebra, where the author used the determinant formula from [1], and specialised it at cα = 0. Here we shall use the determinant formulae for the Neveu-Schwarz algebra from [13], and use same methods. We omit some details since the calculations are completely analogous to the case of the HeisenbergVirasoro algebra of level zero.

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ˇ ´ BERISLAV JANDRIC, ´ AND GORDAN RADOBOLJA DRAZEN ADAMOVIC,

First we consider the Verma module V SH (cL , cα , cL,α , h, hα ) and assume that cα = 0. Then V SH (cL , cα , cL,α , h, hα ) ∼ = SM (1, h1 ) ⊗ V N S (cns , h2 ) for certain h1 , h2 ∈ C. Recall that V SH (cL , cα , cL,α ) ∼ = SM (1) ⊗ V N S (cns , 0) and generator of V N S (cns , 0) is τ

=

τ − τSM (1)

where τSM (1) = a(−1)φ(−1/2) + μφ(−3/2), √ √ a = α/ cα , φ = Φ/ cα . Moreover, ω

=

ωSM (1)

=

ω − ωSM (1)  1 a(−1)2 + μa(−2) + φ(−3/2)φ(−1/2) 1. 2

This implies that h1 h2

hα √ , cα 1 = h − (h21 − μh1 ) 2 √ 1 (h2 − cα μhα ) = h− 2cα α 1 (h2 − 2cL,α hα ) = h− 2cα α    c2L,α hα hα . −2 = h− 2cα cL,α cL,α =

Now we can use the determinant formula for Neveu-Schwarz algebra obtained by A. Meurman and A. Rocha-Caridi in order to obtain the determinant formula for V SH (cL , cα , cL,α , h, hα ). The determinant formula for the Verma module V N S (cns , hns ) is given by

n−kl det[V N S (cns , hns )](·, ·)n/2 = Const Φk,l (cns , hns )p( 2 ) k, l ∈ Z≥0 kl ≤ n k ≡ l mod2

1 where p( n−kl 2 ) is Kostant partition function in 2 Z≥0 , and   5(1 − k2 ) − 4(1 − kl) (k2 − 1)cns + · hns + Φk,l (cns , hns ) = 24 16   2 (k2 − l2 )2 (l − 1)cns 5(1 − l2 ) − 4(1 − kl) + · (7.1) + . 24 16 16

Using the fact that SM (1, h1 ) is irreducible module for SM (1) we conclude that the determinant formula for V SH (cL , cα , cL,α , h, hα ) is obtained from (7.1) by taking    c2L,α hα hα , hns = h − −2 2cα cL,α cL,α

THE N = 1 SUPER HEISENBERG-VIRASORO VERTEX ALGEBRA

cns = c −

177

c2L,α 3 3 + 3μ2 = c − + 12 . 2 2 cα

We get

(7.2)

=

det[V SH (cL , cα , cL,α , h, hα )](·, ·)n/2

n−kl Const ϕk,l (cL , cα , cL,α , h, hα )p2 ( 2 ) k, l ∈ Z≥0 kl ≤ n k ≡ l mod2

where ϕk,l (cL , cL,α , cα , h, hα ) = c2α Φk,l (cns , hns ). Taking constant term in cα we get the determinant formula for cα = 0:      c4L,α hα hα hα hα 1+k− −1 + k + 1+l− −1 + l + 4 cL,α cL,α cL,α cL,α From the determinant formula for level zero case we conclude: hα cL,α SH

Theorem 7.1. Let n =

− 1. Let p = |n|.

• Verma module V (h, hα ) is irreducible if and only if n ∈ / Z or n = 0. • If n is odd integer, then V SH (h, hα ) contains a singular vector at conformal weight p/2. • If n is even integer, then V SH (h, hα ) contains a singular vector at conformal weight p. Remark 7.2. In our forthcoming publication [2] we shall present a detailed structure of highest weight SH–modules with explicit formulae for singular and subsingular vectors . In particular, if n is even, or negative odd, the singular vector from Theorem 7.1 generates the maximal submodule in Verma module V SH (h, (1 + n)cL,α ). If n > 0 is odd, then the maximal submodule is generated by a subsingular vector at conformal weight p. We shall obtain these formulas by means of free field realisation and screening operators. Acknowledgments This paper is based in part on the PhD thesis of the second named author [9]. We would like to thank A. Meurman for discussion related to determinant formula. References [1] E. Arbarello, C. De Concini, V. G. Kac, and C. Procesi, Moduli spaces of curves and representation theory, Comm. Math. Phys. 117 (1988), no. 1, 1–36. MR946992 [2] D. Adamovi´ c, B. Jandri´ c, and G. Radobolja, The N=1 super Heisenberg-Virasoro algebra at level zero, to appear [3] Draˇ zen Adamovi´ c and Gordan Radobolja, Free field realization of the twisted HeisenbergVirasoro algebra at level zero and its applications, J. Pure Appl. Algebra 219 (2015), no. 10, 4322–4342, DOI 10.1016/j.jpaa.2015.02.019. MR3346493 [4] Draˇ zen Adamovi´ c and Gordan Radobolja, On free field realizations of W (2, 2)-modules, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 113, 13, DOI 10.3842/SIGMA.2016.113. MR3579703 [5] Draˇ zen Adamovi´ c and Gordan Radobolja, Self-dual and logarithmic representations of the twisted Heisenberg-Virasoro algebra at level zero, Commun. Contemp. Math. 21 (2019), no. 2, 1850008, 26, DOI 10.1142/S0219199718500086. MR3918045 [6] Yuly Billig, Representations of the twisted Heisenberg-Virasoro algebra at level zero, Canad. Math. Bull. 46 (2003), no. 4, 529–537, DOI 10.4153/CMB-2003-050-8. MR2011391

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[7] Y. Billig, Energy-momentum tensor for the toroidal Lie algebras, arXiv:math/0201313 [8] Hongyan Guo and Qing Wang, Twisted Heisenberg-Virasoro vertex operator algebra, Glas. Mat. Ser. III 54(74) (2019), no. 2, 369–407. MR4043135 [9] B. Jandri´ c, Vertex algebras associated to representations of the N = 1 Super HeisenbergVirasoro and Schr¨ odinger-Virasoro algebra, Ph.D. dissertation (in Croatian), University of Zagreb (2019) [10] Qifen Jiang and Cuipo Jiang, Representations of the twisted Heisenberg-Virasoro algebra and the full toroidal Lie algebras, Algebra Colloq. 14 (2007), no. 1, 117–134, DOI 10.1142/S1005386707000120. MR2278115 [11] Victor Kac, Vertex algebras for beginners, 2nd ed., University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1998. MR1651389 [12] James Lepowsky and Haisheng Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, vol. 227, Birkh¨ auser Boston, Inc., Boston, MA, 2004. MR2023933 [13] A. Meurman and A. Rocha-Caridi, Highest weight representations of the Neveu-Schwarz and Ramond algebras, Comm. Math. Phys. 107 (1986), no. 2, 263–294. MR863643 ˇka Department of Mathematics, Faculty of Science, University of Zagreb, Bijenic 30, 10 000, Zagreb, Croatia Email address: [email protected] Email address: [email protected] ´a 33, 21 000 Split, Croatia Faculty of Science, University of Split, Ru¯dera Boˇ skovic Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15463

Character rings and fusion algebras Peter Bantay Abstract. We present an overview of the close analogies between the character rings of finite groups and the fusion rings of rational conformal models, which follow from general principles related to orbifold deconstruction.

1. Introduction Analogies between group representation theory and 2D conformal field theory have been noticed by several authors over the years. Some of these have a natural interpretation because of the group theoretic origin of the relevant conformal models (e.g. WZNW models based on affine Lie algebras [4, 14], or holomorphic orbifolds based on finite groups [7]), but in other cases the relation is less obvious. A new approach to the subject is provided by recent advances in orbifold deconstruction, and the aim of the present note is to give a sketchy overview of the relevant results. Orbifold deconstruction [1, 3] is a procedure aimed at recognizing whether a given 2D conformal model is an orbifold [8, 11] of another one, and if so, to identify (up to isomorphism) the relevant twist group and the original model. The basic ideas have been described in [1,3], focusing on the conceptual issues, while (part of) the relevant mathematical details have been discussed in [2]. The basic observation is that every orbifold has a distinguished set of primaries, the so-called vacuum block, consisting of the descendants of the vacuum, and that this vacuum block has quite special properties: it is closed under the fusion product, and all its elements have integral conformal weight and quantum dimension. Such sets of primaries, dubbed ‘twisters’ because of their relation to twisted boundary conditions, provide the input for the deconstruction procedure: each twister corresponds to a different deconstruction, with a different twist group and/or deconstructed model. Most of the basic notions related to twisters may be formulated in the much more general setting of sets of primaries closed under the fusion product [2], called FC sets for short. As it turns out, these behave in many ways as character rings of finite groups, especially those – the integral ones – all of whose elements have integer quantum dimension. In particular, one may show that the collection of all FC sets of a conformal model is a modular lattice (with the integral ones forming a sublattice), allowing the generalization to FC sets of such group theoretic notions as nilpotency, solubility, being Abelian, and so on. Moreover, there is a welldefined notion of center and of central extensions, which fit perfectly in the above Key words and phrases. Conformal symmetry, orbifold models, modular tensor categories, character rings. c 2021 American Mathematical Society

179

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PETER BANTAY

mentioned analogy with group theory. Of course, all this is completely natural for twisters, which are nothing but character rings of twist groups according to the general principles of orbifold deconstruction, but their meaning for general FC sets still needs to be clarified. In the sequel, we shall try to sketch the highlights of this circle of questions. 2. FC sets and their classes Let’s consider a rational unitary conformal model [9, 12]: we’ll denote by dp and hp the quantum dimension and conformal weight of a primary p, and by N(p) the associated fusion matrix, whose matrix elements are given by the fusion rules r

r [N(p)]q = Npq

(2.1)

The fusion matrices generate a commutative matrix algebra over C, the Verlinde algebra V [17], whose irreducible representations ρp (all of dimension 1) are in one-to-one correspondence with the primaries of the model. An FC set is a set g of primaries that contains the vacuum and is fusion closed, γ which means that Nαβ > 0 and α, β ∈ g implies γ ∈ g: taking into account that quantum dimensions are positive numbers, this is tantamount to the requirement  γ Nαβ dγ = dα dβ (2.2) γ∈g

for all α, β ∈ g. The fusion matrices N(α) associated to elements α ∈ g generate a subalgebra Vg of the Verlinde algebra, and because V is commutative, the irreducible representations of this subalgebra coincide with the different restrictions of the representations ρp of V. A g-class is defined to be the set C of all those primaries p whose associated representations ρp coincide when restricted to the subalgebra Vg ; we shall denote by ρC this common restriction, and use the notation α(C) = ρC (α) for α ∈ g. Clearly, the collection C(g) of g-classes (whose cardinality equals that of g) provides a partition of the set of all primaries. The first analogy with character rings of finite groups comes from the existence of the orthogonality relations  α(C) β(C) = C

(2.3)

1 if α = β; 0 otherwise

C∈C(g)

for α, β ∈ g, and (2.4)



C1  0

α(C1 ) α(C2 ) =

α∈g

for C1 , C2 ∈ C(g), where (2.5)

 C = 

p

if C1 = C2 ; otherwise

d2p

p∈C

d2p

is the extent of the class C ∈ C(g), which may be shown to be an algebraic integer. The class containing the vacuum primary is of special importance: we shall denote it by g⊥ and call it the trivial class. Note that α(g⊥ ) = dα for α ∈ g, and  d2α (2.6) g⊥  = α∈g

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One can show that the trivial class maximizes both extent and the product of size and extent, i.e. C ≤ g⊥  and |C|C ≤ |g⊥ |g⊥  for every class C ∈ C(g). r A most important property of the trivial class is the product rule: if Npq >0 ⊥ for some p ∈ g , then the primaries q and r belong to the same g-class. It follows at once that the trivial class g⊥ is itself an FC set, the dual of g, hence all previous notions and results go over verbatim to it. In particular, the set of all primaries is partitioned into g⊥ -classes, which we shall call g-blocks (or simply blocks) to avoid confusion with g-classes. It follows from what has been said for classes that the set B(g) of g-blocks provides a partition of the set of all primaries. A simple q argument shows that the primaries p and q belong to the same g-block iff Nαp >0 for some α ∈ g, and in particular, the g-block containing the vacuum is g itself, that is (g⊥)⊥ = g. This illustrates the inherent duality of FC sets: g and its dual g⊥ determine each other, while their extents are, roughly speaking, reciprocal, since the product gg⊥  can be shown to be the same for every FC set g. This duality implies that any result proven about classes gives a corresponding result about blocks, and vice versa, a seemingly trivial observation that turns out to be quite useful in several instances. The inclusion relation makes the collection L of FC sets partially ordered, with maximal element the FC set containing all primaries, and minimal element the trivial FC set consisting of the vacuum primary solely. Because the intersection of two FC sets is obviously an FC set again, L is actually a finite lattice, and one may show that the join g∨h of the FC sets g and h (the smallest FC set that contains both of them) is the dual of the intersection of their duals, i.e. (2.7)

g∨h = (g⊥ ∩ h⊥ )

⊥

If g and h are FC sets such that h ⊆ g, then every h-class is a union of g-classes, in particular g⊥ ⊆ h⊥ , and every g-block is a union of h-blocks; moreover, the number of g-classes contained in h⊥ equals the number of h-blocks contained in g. It follows that the map sending each FC set to its dual is an isomorphism between the lattice L and its dual. An important consequence of the above results, crucial from the viewpoint of orbifold deconstruction, is that L is a modular (even Arguesian) lattice, but usually neither distributive nor complemented [2]. A better understanding of the lattice theoretic properties of L would be highly desirable. For an FC set g, the restriction of the indices of the fusion matrices N(α) to the primaries belonging to a given block b ∈ B(g) results in non-negative integer matrices Nb (α) that form a representation Δb of the subalgebra Vg . This representation decomposes into a direct sum of the irreducible representations ρC , and   the overlap b, C of b with the class C ∈ C(g) is defined  as the multiplicity of ρC in the irreducible decomposition of Δb . The overlap b, C may be shown to equal the rank of the minor of the modular S-matrix obtained by restricting the row indices to b ∈ B(g) and the column indices to C ∈ C(g). Alternatively, one has the expression    (2.8) b, C = |Spq |2 p∈b q∈C

  In particular, this means that b, C = 0 implies Spq = 0 for all p ∈ b and q ∈ C, explaining the appearance of large blocks of zeroes in the modular S-matrix of many conformal models. We note that for twisters (to be introduced in Section 4) the overlap has another, much more profound group theoretic interpretation [1, 3].

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3. Central extensions As mentioned previously, the extent C of a class C ∈ C(g) is bounded from above by the extent of the trivial class: C ≤ g⊥ . Those classes that saturate this bound – the central ones – form the center Z(g) = {z ∈ C(g) | z = g⊥ } of the FC set g ∈ L . Since g⊥ ∈ Z(g), the center is never empty, and a straightforward argument shows that z ∈ Z(g) iff |α(z)| = dα for all α ∈ g. One may show that Z(g) = g, i.e. all classes are central precisely when dα = 1 for all α ∈ g. Such FC sets are called (for obvious reasons) Abelian; in the language of 2D CFT, they correspond to groups of simple currents. The following generalization of the product rule holds: if the primary p belongs to the class C ∈ C(g) and q belongs to the central class z ∈ Z(g), then all primaries r r for which Npq > 0 belong to the same g-class, denoted zC. What is more, if z1 ,z2 ∈ Z(g) are central classes, then z1 z2 is also central, and z1 z2 = z2 z1 . It follows that the center Z(g) of an FC set g ∈ L is an Abelian group, and since (z1 z2 )C = z1 (z2 C) for any class C ∈ C(g), the center permutes the g-classes. For a subgroup Z < Z(g) of the center, the subset {α ∈ g | α(z)= dα for all z ∈ Z} is again an FC set, the central quotient of g by Z, denoted g/Z. The usefulness of this notion rests on the explicit knowledge of its structure, for one has complete control over its classes and blocks in terms of those of g, and in particular, its dual consists of all those primaries that belong to some class z ∈ Z ⊥ (3.1) (g/Z) = z z∈Z

It may be shown that for every h ∈ L such that g/Z ⊆ h ⊆ g there exists some subgroup H < Z for which h = g/H, hence one has an order reversing one-to-one correspondence between central quotients of g ∈ L and subgroups of its center Z(g). Given an FC set g, it is natural to ask whether it is a central quotient of another FC set. For an Abelian group A, an A-extension of g is an FC set h ∈ L such that h/Z = g for some central subgroup Z < Z(h) isomorphic to A. One may show that the different A-extensions of g are in one-to-one correspondence with subgroups of Z(g⊥ ) isomorphic to A, and in particular, every g ∈ L has a maximal central extension, namely (g⊥ /Z(g⊥ ))⊥ , the dual of the maximal central quotient of g⊥ . Note that, while central quotients are related to groups of central classes, central extensions have a similar relation to groups of central blocks, i.e. blocks b ∈ B(g) that satisfy b = g, which form the center Z(g⊥ ) of the dual of g. An interesting observation is that any class (hence any block) that contains a simple current (a primary of dimension 1) is automatically central, but the converse need not be true. 4. Local FC sets It follows from the results discussed in Section 2 that every g-class is a union of g-blocks precisely when g ⊆ g⊥ . It turns out that such FC sets play a basic role in orbifold deconstruction [1, 3], hence they deserve a special name: we’ll call them local FC sets. The point is that the vacuum block of an orbifold model (the set of primaries that originate from the vacuum) is an FC set whose classes correspond to the different twisted sectors, i.e. collections of twisted modules whose twist element belong to the same conjugacy class, while its blocks correspond to orbits of twisted modules. Since the conjugacy class of a twist element is the same for all twisted modules on the same orbit, every block should be included in a well-defined

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class, hence the vacuum block, considered as an FC set, should be local by the above. Note that, while the intersection of local FC sets is clearly local, this is not necessarily the case for their join, hence generally they do not form a lattice. Actually, the vacuum block of an orbifold belongs to a special class of local FC sets – termed twisters because of their relation to twisted boundary conditions – all of whose elements have integral conformal weight. Indeed, since elements of the vacuum block descend from the vacuum primary, they all have integral conformal weight, hence the vacuum block is necessarily a twister. One may show that a local FC set is either a twister or a Z2 -extension of a twister. More precisely, every local FC set that is not a twister has a distinguished central class R, the so-called Ramond class, such that the corresponding central quotient is a twister. The rationale for this appellation is that, in a suitable fermionic generalization of orbifold deconstruction, the blocks contained in the trivial class account for the Neveu-Schwarz (bosonic) sector of the deconstructed model, while those contained in the Ramond class describe the fermionic (Ramond) sector. This is substantiated by the observation that a block is contained in the Ramond class precisely when the conformal weights of its elements differ by integers, and that the number of blocks contained in the trivial and Ramond classes (the number of bosonic and fermionic degrees of freedom) are always equal. Local FC sets have many striking properties. For example, one may show that all of their elements have (rational) integer dimension, and either integer or halfinteger conformal weight1 . Ultimately, all this follows from the observation that, γ if the primaries α, β belong to a local FC set and Nαβ > 0, then the conformal weight hγ differs by an integer from the sum hα + hβ . From a categorical point of view this means that the elements of a local FC set are the simple objects of a Tannakian subcategory of the modular tensor category associated to the model. According to a result of Deligne [6], such a category is (tensor-)equivalent to the representation category of some finite group, hence the associated subalgebra may be identified with the character ring of that group. This has a natural interpretation in terms of orbifold deconstruction: the elements of the vacuum block correspond to irreducible representations of the twist group, hence their fusion rules describe the decomposition of tensor products of the latter, implying that the subalgebra Vg of the Verlinde algebra is nothing but the character ring of the twist group. It follows from the above considerations that results from character theory [13,15,16] should go over to local FC sets, and this observation allows to generalize many group theoretic concepts to arbitrary FC sets, providing a host of non-trivial conjectural statements that seem to hold in greater generality. In this vein one can define for FC sets such concepts as nilpotency, (super)solubility, and so on [2]. For example, an FC set g is nilpotent if it can be obtained from the trivial FC set by a sequence of central extensions, the rationale for this terminology being that if g is local, hence the algebra Vg is isomorphic to the character ring of some finite group G, then g is nilpotent precisely when the corresponding group G is. One may show that if the FC set g is nilpotent, then g⊥  ∈ Z, and this in turn implies that the quantum dimension dα of any element α ∈ g is either an integer or the square root of

1 Note that the converse is not true, for there are many FC sets in which all conformal weights belong to 12 Z, but are nevertheless not local; on the other hand, the integrality of conformal weights implies locality, hence every twister is automatically local.

184

PETER BANTAY

an integer2 . We conjecture that many results about (finite) nilpotent groups would carry over to nilpotent FC sets, e.g. if g is nilpotent and d is an integer dividing g⊥ , then there would exist an FC set h ⊆ g such that h⊥  = d. As explained before, for a local FC set g the algebra Vg is isomorphic to the character ring of some finite group, hence usual properties of character rings should apply to it. This means in particular that, cf. [13, 15, 16] (1) the extent C of any g-class C ∈ C(g) is a rational integer dividing g⊥ ; g⊥  (2) the dimension dα of every α ∈ g is a rational integer dividing |Z(g)| ; (3) if α ∈ g has dimension dα > 1, then α(C) = 0 for some class C ∈ C(g); g⊥  (4) if the class C ∈ C(g) is such that C is coprime to dα for some α ∈ g, then either |α(C)| = dα or α(C) = 0. All these assertions are well-known properties of character rings, e.g. the first one just states that the size of a conjugacy class is an integer dividing the order of the group, while the second one is the content of Ito’s celebrated theorem [13]. What is really amazing is that, as suggested by extensive computational evidence, these properties (and many similar ones) seem to hold for a much larger class of FC sets [2], the so-called integral ones characterized by the property that all of their elements have integer dimension3 . This is truly surprising, as one can show explicit examples of integral FC sets which are not the character ring of any finite group. From this point of view, one may consider integral FC sets as ’character rings’ of some suitable generalization of the group concept. 5. Summary As we have seen, fusion closed sets of primaries of a conformal model (or a modular tensor category) have a fairly deep structure, generalizing many aspects of the character theory of finite groups. Of course, this is no accident, since the vacuum block of an orbifold model, which corresponds on general grounds to the character ring of the twist group, is a special kind of FC set, a so-called twister. But it turns out that the parallel with character theory goes much further, even for FC sets that have no group theoretic interpretation. Many notions from group theory (like nilpotency, solubility, etc.) may be generalized to arbitrary FC sets, and the corresponding properties seem to hold almost verbatim in this more general setting. But it should be stressed that FC sets are more than just some fancy generalization of the group concept, since they possess genuinely new structures and properties, which await careful study. References [1] Peter Bantay, Orbifold deconstruction: a computational approach, Vertex operator algebras, number theory and related topics, Contemp. Math., vol. 753, Amer. Math. Soc., Providence, c RI, [2020] 2020, pp. 1–15, DOI 10.1090/conm/753/15161. MR4139234 [2] Peter Bantay, FC sets and twisters: the basics of orbifold deconstruction, Comm. Math. Phys. 379 (2020), no. 2, 693–721, DOI 10.1007/s00220-020-03855-5. MR4156220 2 That

the latter possibility can occur is exemplified by the maximal FC set of the Ising model (the minimal Virasoro model of central charge 12 ), which is nilpotent while having a primary of √ dimension 2. 3 Actually, they seem to hold (after suitable amendments) in the more general case of FC sets whose elements have integer squared dimension.

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[3] Peter Bantay, A short guide to orbifold deconstruction, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), 027, 10 pages, DOI 10.3842/SIGMA.2019.027. MR3936906 [4] Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071, DOI 10.1073/pnas.83.10.3068. MR843307 [5] S. Carnahan and M. Miyamoto, Regularity of fixed-point vertex operator subalgebras, arXiv:1603.05645. [6] P. Deligne, Cat´ egories tannakiennes (French), The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkh¨ auser Boston, Boston, MA, 1990, pp. 111–195. MR1106898 [7] Robbert Dijkgraaf, Cumrun Vafa, Erik Verlinde, and Herman Verlinde, The operator algebra of orbifold models, Comm. Math. Phys. 123 (1989), no. 3, 485–526. MR1003430 [8] L.J. Dixon, J.A. Harvey, C. Vafa, and E. Witten. Strings on orbifolds, Nucl. Phys, B261:678– 686, 1985. [9] Philippe Di Francesco, Pierre Mathieu, and David S´en´ echal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997. MR1424041 [10] Chongying Dong, Li Ren, and Feng Xu, On orbifold theory, Adv. Math. 321 (2017), 1–30, DOI 10.1016/j.aim.2017.09.032. MR3715704 [11] Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR996026 [12] P. Ginsparg, Curiosities at c = 1, Nuclear Phys. B 295 (1988), no. 2, FS21, 153–170, DOI 10.1016/0550-3213(88)90249-0. MR936581 [13] I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR0460423 [14] Victor Kac, Vertex algebras for beginners, University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1997. MR1417941 [15] Klaus Lux and Herbert Pahlings, Representations of groups, Cambridge Studies in Advanced Mathematics, vol. 124, Cambridge University Press, Cambridge, 2010. A computational approach. MR2680716 [16] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New YorkHeidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR0450380 [17] Erik Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nuclear Phys. B 300 (1988), no. 3, 360–376, DOI 10.1016/0550-3213(88)90603-7. MR954762 ¨ tvo ¨ s Lo ´ ra ´nd University, H-1117 Budapest, Institute for Theoretical Physics, Eo ´zma ´ny P´ Pa eter s. 1/A, Hungary Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15464

Continuing Remarks on the Unrolled Quantum Group of sl(2) James F. Clark Abstract. This note expands on the work of Constantino, Geer, and Patureau-Mirand [9], as well as Creutzig, Milas, and Rupert [11] on exploring H the unrolled quantum group of sl(2) (denoted by Uq (sl2 )) and the category H

of finite dimensional weight Uq (sl2 )-modules (denoted by C ) for any pth root of unity where p > 2. This unrolled quantum group and associated category has connections to topological quantum field theory (TQFT) and to vertex operator algebras (VOAs). New results include showing that the C is a ribbon category, properties about its center, finding all simple and projective modules of C , and finding logarithmic typical modules and their tensor decomposition.

1. Introduction H

In this note, we study the unrolled quantum group of sl(2), denoted U q (sl2 ), when q is a pth root of unity that is defined in [9, 11] and it’s associated category, H C , of fintie dimensional weight U q (sl2 )-modules. This specific quantum group and category are particularly interesting and was first used by Geer et al. to construct invariants of 3-manifolds and knots for TQFTs [7–9, 14, 15]. We are also interested in its category of weight and logarithmic modules, denoted by Clog , and expect it to be in Kazhdan-Lusztig’s correspondence with certain categories for the (1, p)-singlet W -algebra studied extensively in the vertex operator algebra literature [3, 10–12]. As discussed in [11], this inclusion of logarithmic modules into C is motivated by logarithmic conformal field theory. The authors of [11] are interested in finding a 2πi Kazhdan-Lusztig correspondence between Clog when q = e p when p is even and the category of finite length (logarithmic) dimensional modules of the “singlet” 2 VOA, M(r) of central charge 1 − 6(r−1) where r = p2 . More precisely, they have r the following conjecture: 2πi

Conjecture. [11] For q = e p when p is even and r = p2 , there is an equivalence of categories between M(r) − Mod and Clog . Creutzig, Milas, and Rupert also look at the subcategory of logarithmic modules whose generalized weights are in λ ∈ (C \ Z) ∪ rZ, denoted by Clog,typ , and call objects of this subcategory logarithmic typical modules. This conjecture has been investigated in [3, 11] and an important aspect of this correspondence is that it does not require semisimplification on either side. ©2021 American Mathematical Society

187

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JAMES F. CLARK

In the next section we define the unrolled quantum group for any root of unity, its associated category C , and show that C is a ribbon category. In the following H section, we look at some of the properties of the center of U q (sl2 ). In Section 4 we classify all simple and projective modules of C and look at some of their properties. We also find new results such as defining simple modules that are not weight modules in Section 4.1, as well as finding the projective covers, resolutions, and extensions of the atypical modules of C in Section 5. In Section 6, we expand on their construction of logarithmic typical modules for even roots of unity to any root of unity and find the tensor decomposition of logarithmic typical modules in certain cases. 2. The unrolled quantum group and its associated category In this section, we will recall the definition of the unrolled quantum group of sl(2) and look at its associated category of modules. First, we will define some of the notation that will be used throughout the paper. Definition 2.1. We will have that k = C or Q(q). For q ∈ C and n, k ∈ N we shall define the following notation: {n}:= q n − q −n

[n]:=

{n} {1}

{n}!:= {n}{n − 1} · · · {1}

[n]!:= [n][n − 1] · · · [1],

  2πi If q is a pth root of unity q = e p with p > 2, then we shall define the following notation: r :=

2p = 3 + (−1)p



p 2

p

if p is even if p is odd

d :=

p = 2r



1 1 2

if p is even if p is odd

¨ := (C\dZ) ∪ drZ. C

Definition 2.2 (Drinfel’d-Jimbo Quantum Group of sl(2) at Root of Unity). 2πi Let q = e p , p > 2. We shall define Uq (sl2 ) to be the C-algebra generated by E, F , K, and K −1 with the following relations: (2.1)

KK −1 = 1 = K −1 K

KE = q 2 EK

KF = q −2 F K

[E, F ] =

K − K −1 . q − q −1

Uq (sl2 ) is a Hopf algebra with comultiplication, counit, and antipode defined by the following: (2.2)

Δ(E)= 1 ⊗ E + E ⊗ K

ε(E)= 0

S(E)= −EK −1 ,

(2.3)

Δ(F )= K −1 ⊗ F + F ⊗ 1

ε(F )= 0

S(F )= −KF ,

(2.4)

Δ(K)= K ⊗ K

ε(K)= 1

S(K)= K −1 .

Definition 2.3 (Restricted and Small Quantum Group at Root of Unity). We shall define the restricted quantum group U q (sl2 ) as the algebra Uq (sl2 )/ .q (sl2 ) as the algebra Uq (sl2 )/ E r = F r = 0 and the small quantum group U r r p . E = F = 0, K = 1. Both U q (sl2 ) and Uq (sl2 ) are Hopf algebras, since the relations E r = F r = 0 and E r = F r = 0, K p = 1 generate Hopf ideals with comultiplication, counit, and antipode defined in (2.2)–(2.4). Remark. In CFT literature, the restricted quantum group is defined as the small quantum group in the previous definition and is finite dimensional. The restricted quantum group used in this note is defined as in [5, 16, 17], is infinite dimensional, and will allow modules to have non-integral weights.

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Definition 2.4 (Unrolled Quantum Group of sl(2) at Root of Unity [9]). Let 2πi q = e p , p > 2. We shall define the modified version of Uq (sl2 ), denoted by UqH (sl2 ), as the C-algebra generated by E, F , K, K −1 , and H with the relations in Equation (2.1) along with the additional relations: (2.5)

HK = KH

[H, F ] = −2F

[H, E] = 2E

UqH (sl2 )

is a Hopf algebra with comultiplication, counit, and antipode defined in Equations (2.2)–(2.4) along with (2.6)

Δ(H)= H ⊗ 1 + 1 ⊗ H

S(H)= −H

ε(H)= 0

As with Uq (sl2 ), the relation E r = F r = 0 generates a Hopf ideal on UqH (sl2 ). H

We shall define U q (sl2 ) as the Hopf algebra UqH (sl2 )/E r = F r = 0, which we shall call the unrolled quantum group. For the purposes of this paper, we are interested at looking at the category of H finite dimensional weight U q (sl2 )-modules, which we will denote by C . If V is a module, we define a weight of V by H : V → V acts as an eigenvalue operator and the associated eigenspace is called the weight space. We say a vector vi ∈ V is in the λi -eigenspace of H is called a weight vector of weight λi , meaning H.vi = λi vi . A weight module is a module that splits as a direct sum of its weight spaces and K = q H as operators (meaning that if H.vi = λi vi , then K.vi = q λi vi ). Notice that we have no requirements on H or K, which will allow modules to have non-integral weights. We are able to show that C is a ribbon category by defining the structures that are needed for a rigid, braided category and define its ribbon element structure. First we can show that C is a rigid category by defining the left and right evaluation and coevalution maps used to define left and right duals by the following: ! evV : V ∗ ⊗ V → 

! coevV :  → V ⊗ V ∗

vi∗ ⊗ vi → vi∗ .vi

1 →



vi ⊗ vi∗

i

" evV : V ⊗ V ∗ → 

" coevV :  → V ∗ ⊗ V

vi ⊗ vi∗ → vi∗ .(K 1−r vi )

1 →



vi∗ ⊗ (K r−1 vi )

i

In order to prove that C is a braided category, we will look at the following theorem: Theorem 2.5. [17] Let (A, μ, η, Δ, ε) be a bialgebra. If A is braided then the finite dimensional modules of A forms a braided category. The universal R-matrix R can be used to build a braiding cV,W and its inverse by cV,W (v ⊗ w) = τV,W (R(v ⊗ w))

−1 c−1 (τV,W (v ⊗ w)) V,W (v ⊗ w) = R H

We constructed the universal R-matrix for U q (sl2 ) and showed that R=q

H⊗H 2

r−1  n=0

q

n(n − 1)/2

1 {1}n E n ⊗ F n . [n]! H

The universal R-matrix R does not make U q (sl2 ) braided - or even a quasiH

H

cocommutative - Hopf algebra because R ∈ U q (sl2 ) ⊗ U q (sl2 ) since q

H⊗H 2

=

190

JAMES F. CLARK

∞  (H ⊗ H)n 2n · n! n=0

H

is an infinite sum and is therefore in the completion of U q (sl2 ). Even

H

though U q (sl2 ) is not braided, if we define cV,W (v ⊗ w) = τV,W (R(v ⊗ w)) for H

U q (sl2 ), as in Theorem 2.5, we will have that cV,W will satisfy all braiding axH

H⊗H

ioms as q 2 can act on a finite dimensional U q (sl2 )-module as this infinite sum becomes finite when acting on a finite dimensional module. So we can define the H braiding on U q (sl2 )-modules by cV,W (v ⊗ w) = q

H⊗H 2

r−1 

q

n(n−1) 2

n=0

{1}n n F .w ⊗ E n .v [n]!

.

We can also construct the Drinfel’d element, u, and S(u) by the following: (2.7) (2.8)

u= q

−H 2 r−1  2

S(u)= uK

3n(n − 1)

q

n=0 2r−2

2

[n]!

{−1}n F n K −n E n ,

,

The Drinfel’d morphism uV sends vi → u.vi . The term q

−H 2 2

H

∈ U q (sl2 ), but it

H

can act on finite dimensional U q (sl2 )-modules by the same reasoning applied to H⊗H 2

. In a similar manner to how we proved C is a braided category, we can use the following definition and theorem to help to construct θV so that C is a ribbon category. q

Proposition 2.6. [17] For any ribbon algebra (A, μ, η, Δ, ε, S, S −1 , R, θ), the tensor category finite dimensional A-modules is a ribbon category with twist θV defined by θV (v) = θ −1 .v −1 .v, we shall Remark 2.7. Since θV maps v → θ −1 .v and u−1 V maps v → u −1 −1 −1 −1 define δV such that v → δ .v so that θ = u δ . H

We can then construct the ribbon element of U q (sl2 ) to be θ = K r−1 u = q − H

2/2

r−1  n=0

q n(3n + 1)/2 {−1}n F n K r−1−n E n [n]! H

which means θ −1 = u−1 K 1−r . θ is in the completion of U q (sl2 ) and can act on H

finite dimensional U q (sl2 )-modules by the same logic used to define the braiding of C . So if we define (2.9)

θV (v) = θ −1 .v = u−1 K 1−r .v,

then θV will satisfy all of the requirements to be a ribbon element and thus C will be a ribbon category. We can also define the pivotal structure δV (v) = K 1−r .v by using Remark 2.7 and Equation (2.9).

CONTINUING REMARKS ON THE UNROLLED QUANTUM GROUP OF sl(2)

191

H

3. The center of U q (sl2 ) .q (sl2 ) are known and have been wellThe centers of Uq (sl2 ), U q (sl2 ) and U studied. The centers of Uq (sl2 ) and U q (sl2 ) when q is not a root of unity is finitely generated by the Casimir element C. The centers of Uq (sl2 ) and U q (sl2 ) change when we set q to be a pth root of unity, but will still be finitely generated. Their cenH ters are generated by C, E r , F r , K ±r and C, K ±r , respectively. Since U q (sl2 ) has an extra generator, H, compared to U q (sl2 ), we are interested to see how the inclusion of this new generator would potentially change what the center would look like and if the center would still be finitely generated. Some of the following lemmas and propositions are modeled off of similar lemmas and propositions in [17, §6.4-6.5] for finding the center and its properties for U q (sl2 ). First we recall that the quantum Casimir element, denoted by C, and the r th Chebychev polynomial, denoted by Tr , are defined by qK + q −1 K −1 q −1 K + qK −1 = EF + {1}2 {1}2   −1 r −r X +X X +X Tr = 2 2

(3.1)

C = FE +

We are able to prove the following results: H

Proposition 3.1. The center of U q (sl2 ) contains a C-algebra generated by C, K r , and K −r with relation  Tr

{1}2 C 2



⎧ K r + K −r ⎪ ⎨− 2 −r = r ⎪ ⎩K + K 2

if q is even root of unity if q is odd root of unity

Proof. Since [9] has a proof for this proposition when p is even, the proof is the same with a slight modification at the end to cover any root of unity (q r = 1 if p is odd, q r = −1 if p is even).  H

Lemma 3.2. Given any pth root of unity, every element of the center of U q (sl2 ) can be written as

r−1 

E i pi (H, K)F i

where pi (H, K) ∈ C[H, K ±1 ].

i=0

H

H

Lemma 3.3. Let Z be the center of U q (sl2 ), which is a subalgebra of U q (sl2 ) H

and I is the left ideal I = U q (sl2 )F ∩ Z. Then Z = C[H, K ±1 ] ⊕ I. Since

r−1 

E i pi (H, K)F i

can be uniquely written, we can write Z = C[H, K ±1 ]⊕I,

i=0

and since I is a two-sided ideal we can construct an algebra morphism ϕ, called the Harish-Chandra homomorphism, that projects Z onto C[H, K ±1 ]. We can then use this morphism to express the action of the center on a highest weight module. H

Lemma 3.4. If we let V be a highest weight U q (sl2 )-module with highest weight  H  λ, then for any central element z ∈ Z U q (sl2 ) and v ∈ V we have that z.v = ϕ(z)(λ)v0 and that ϕ(z) is injective. Proof of Lemma 3.2−3.4. Kassel has similar results for Uq (sl2 ) in [17], so H  with modifications to those proofs we can show it holds for U q (sl2 ).

192

JAMES F. CLARK

 H  When p = 4, we are able to show that elements of Z U q (sl2 ) are of the form Ep1 (H, K)F +

n+1 

H j (K − K −1 )p j (K) + p 0 (K)

j=1

where the highest power of H in p1 (H, K) is H n , p.j (K) ∈ C[K ±1 ], and p.0 ∈ C[K ±2  ].HUsing Lemmas 3.3 and 3.4, the problem of finding the set of generators of Z U q (sl2 ) can be reduced to finding the set of generators x · k[x, y] which is infinitely generated. Thus, we get the following theorem: H

Theorem 3.5. If p = 4, then the center of U q (sl2 ) is not finitely generated. Remark 3.6. We are also able to show that the center is not finitely generated for any root of unity and the proof will appear in a future paper. 4. Simple and projective modules of C Since Uq (sl2 ) is well known, we know what the simple modules are when q = e , which are listed in [16, 17]. We also know that the simple modules of Uq (sl2 ) must map to simple moduels in U q (sl2 ), because U q (sl2 ) is a quotient of Uq (sl2 ). H In order to find the simple modules in U q (sl2 ), we know that they must also be 2πi p

H

simple modules in U q (sl2 ), since a U q (sl2 )-module can be viewed as a U q (sl2 )module. Once this is done, we can see that there are two classes of simple modules. A highest weight module of highest weight λ is a weight module that has a highest weight vector v0 such that E.v0 = 0 and H.v0 = λv0 . The first class of simple modules are a (n + 1)-dimensional highest weight module called atypical modules with highest weight n + dr, where n ∈ {0, . . . , r − 1} and  ∈ Z. They are denoted by Sn ⊗ CH dr with basis {v0 , . . . , vn } and actions defined by (4.1)

H.vi = (n + dr − 2i)vi ,

E.vi = [i][n + dr + 1 − i]vi−1 ,

F.vi = vi+1 .

= (−1) [i][n + 1 − i]vi−1 

H We define CH dr to be the one-dimensional module Cdr where E and F acts by 0 H and H acts by dr. So if v ∈ Cdr then H.v = drv and  2πi ±dr v = (−1) v, K ±1 .v = q (dr)(±1) v = e p

for any root of unity. We can also define the degree of CH dr as r mod 2. In the .q (sl2 )-modules, Sn (where  = 0) are the only simple modules up to category of U isomorphism [9, 16, 17], but in C , we can have two (n+1)-dimensional simple modules which will not be isomorphic to each other. Also, if

K − K −1 .v = [E, F ].v = 0, {1}

then this implies that K 2 .v = 1 or K.v = ±v, which is why CH dr is isomorphic to .q (sl2 ). These one-dimensional modules are important tools the trivial module in U in finding invariants in TQFT [4, 7]. Next, we consider a larger class of finite dimensional highest weight modules ¨ then we shall define V  to be the r-dimensional called typical modules. If α ∈ C, α highest weight module of weight α + dr − 1 with basis {v0 , . . . , vr−1 } whose actions are given by (4.2)

H.vi = (α + dr − 1 − 2i)vi

E.vi = [i][i − α]vi−1

F.vi = vi+1

CONTINUING REMARKS ON THE UNROLLED QUANTUM GROUP OF sl(2)

193

¨ and is atypical otherwise. When V  is typical, We say that Vα is typical if α ∈ C α then it is simple, since it is generated by any basis vector vi . When p is even we can see that V0 is the only self-dual module, but when p is odd we can see that V r is the only self-dual module. We will redefine this class of 2 simple modules as  Vα := Vα+(1−d)r

(4.3)

which allows us to have V0 be our self dual module for any root of unity, defines the action of H to be H.vi = (α + r − 1 − 2i)vi , and is independent of the root of unity. Thus, we can use Equations (4.2)−(4.3) to define the actions of Vα as (4.4)

H.vi = (α + r − 1 − 2i)vi

E.vi = [i][i − α]vi−1

F.vi = vi+1

Lemma 4.1. If V ∈ obj (C ) is simple, it is either isomorphic to an atypical module or a typical module. Proof. The proof for any root of unity is similar to the proof in [9] for even roots of unity.  Therefore, a simple module that is k-dimensional will look like vk−1

E F

vk−2

E F

···

E F

v1

E F

v0

If we define Cg as the full subcategory of weight modules whose weights are all in class g (mod 2Z), then C = {Cg }g∈C/2Z has a C/2Z-grading. If V ∈ Cg and f ∈ V ∗ , then we can see by the action of H that (Hf ).v = f (S(H).v) = −f (H.v) and so V ∗ ∈ C−g (which is why we chose to define V0 to be the self-dual module for any root of unity as we did in Equation (4.3)). If V1 ∈ Cg1 and V2 ∈ Cg2 where g1 = g2 , then Hom(V1 , V2 ) = 0, since morphisms preserve weights. Proposition 4.2. If α ∈ (C/2Z) \ (dZ/2Z) then Cα is semisimple. Proposition 4.3. All typical modules are also projective modules. Proof of Proposition 4.2−4.3. The proof when p is even is given in [9]. If you modify the proof so that α ∈ C\dZ, then the proof holds for any root of unity.  A vector v is called a dominant weight if (F E)2 .v = 0. While it is well known that a highest weight vector v generates a submodule with basis {F i v}, we are interested in finding what submodule is generated by a dominant weight vector. We were then able to show that our projective, indecomposable modules would be generated by a dominant weight vector and are defined by the following relations: Let wiH := w ⊗ v ∈ Pi ⊗ CH dr be a dominant vector of weight i + dr where i ∈ {0, 1, . . . , r − 2},  ∈ Z. Then the dominant weight vector wiH will generate a 2r-dimensional module with basis R R R S S S H H H L L L {w−2r+i+2 , w−2r+i+4 , . . . , w−i−2 , w−i , w−i+2 , . . . , wi , w−i , w−i+2 , . . . , wi , wi+2 , wi+4 , . . . , w2r−i−2 }

and actions defined by the following: (4.5)

wiH = w ⊗ v ,

R wi+2 = (−1) EwiH ,

R wiS = F wi+2 ,

L H w−i−2 = F w−i ,

194

JAMES F. CLARK H wi−2k = F k wiH ,

(4.6) (4.7)

R wi+2+2k =

(4.8)

H.wkX =

(4.9)

E.wkR =

(4.10) (4.11) (4.12)

L E.w−i−2 =

k

(−1) E

k ∈ {0, . . . , i},

S wi−2k = F k wiS , k

R wi+2

(k +

dr)wkX

(−1)



R wk+2

(−1)



S w−i

,

L w−i−2−2k =

,

,

,

F

k

L w−i−2  k

K.wkX =

(−1) q

F.wkY =

Y wk−2

R F.wi+2 =

R L E.w2r−2−i = F.w−2r+2+i = 0,

wiS

k ∈ {0, . . . , r − 2 − i},

,

wkX

X ∈ {L, H, S, R},

,

Y ∈ {H, S, L},

,

, S E.wiS = F.w−i = 0,

 H S H E.wi−2k = (−1) wi−2(k−1) + [k][i + 1 − k]wi−2(k−1) , S S E.wi−2k = (−1) [k][i + 1 − k]wi−2(k−1)

(4.13) R R L L F.wi+2+2k = −[k][i + 1 + k]wi+2+2(k−1) = −(−1) [k][i + 1 + k]w−i−2−2(k−1) , E.w−i−2−2k .

Remark. While [9] has a similarly defined projective indecomposable module when p is even, the way we define the module uses a different notation that can be found in [11] and also includes CH dr from the start. This way we can define this module generated by a dominant weight for any root of unity upfront. Proposition 4.4. Any projective indecomposable weight module with dominant weight i + dr is isomorphic to Pi ⊗ CH dr . Proof. The proof for when q is an odd root of unity is almost the same as the proof for when q is an even root of unity that is covered in [9]. The only two differences are the values of d in CH dr , and using the odd case in Proposition 3.1 when needed.  4.1. Simple modules beyond category C . It is not difficult to describe all simple modules for the unrolled quantum group (not necessarily in the weight category). Results of this section will not be used in the rest of the paper. H

Proposition 4.5. Any finite dimensional simple U q (sl2 )-module is H and K-diagonalizable. Proof. Let V be such a module and consider the subspace U of V annihilated by E. Since H and K commute, there is a weight vector v ∈ U such that H.v = μv H and K.v = λv. This means that v, F.v, . . . , F r−1 .v is U q (sl2 )-invariant and thus U = V . Therefore, the module is H and K-diagonalizable by construction.  ¨ If V is an Theorem 4.6. Let μ ∈ C, n ∈ {0, 1, . . . , r − 2}, and α ∈ C. H irreducible finite dimensional U q (sl2 )-modules, then V is isomorphic to one of the following (where v0 denotes the highest weight vector): (a) Let Sμ,n be defined as the atypical, (n + 1)-dimensional irreducible module with highest weight (μ, q n ), basis {v0 , . . . , vn }, and actions defined by H.vi = (μ − 2i)vi

K.vi = q n−2i vi

E.vi = [i][n + 1 − i]vi−1

F.vi = vi+1

(b) Let Vμ,α be defined as the typical, r-dimensional irreducible module with highest weight (μ, q α+r−1 ), basis {v0 , . . . , vr−1 }, and actions defined by H.vi = (μ − 2i)vi

K.vi = q α+r−1−2i vi

E.vi = [i][i − α]vi−1

F.vi = vi+1

Proof. This follows directly from the classification of irreducible module Uq (sl2 )-modules subject to E r = F r = 0 and the previous proposition. 

CONTINUING REMARKS ON THE UNROLLED QUANTUM GROUP OF sl(2)

195

Hi,dr L−i−2,dr

R2r−2−i,dr Si,dr

Figure 1. Diagram of Pi ⊗ CH dr 5. Projective covers, resolutions, and extensions of Sn ⊗ CH dr A simplified diagram of Pi ⊗ CH dr is given in Figure 1. We can see that this module has four subquotients: H (“head”), S (“socle”), L (“left”), and R (“right”). The basis can be chosen as H S L R {wi−2k , wi−2k }0≤k≤i ∪ {w−i−2−2k , wi+2+2k }0≤k≤r−2−i

where {wkX } corresponds to the basis of the Xm,n subquotient and the weight vector with the largest weight in this basis has weight m + n. Given an indecomposable, projective module, the “head” subquotient is isomorphic to a unique simple module. We say that the indecomposable, projective module is a projective cover of this H simple module. In the case of Pi ⊗ CH dr , it is the projective cover of Si ⊗ Cdr . ¨ is both simple and projecRemark. Since a typical module, Vα where α ∈ C, tive, it is its own unique subquotient and we say that Vα is a projective cover of itself. We can construct projective and injective resolutions using splicing. Recall that Pn ⊗CH dr is a projective module with diagram given in Equation (1) and the actions on these subquotients are given (4.5)−(4.13). For ease of notation, ⎛ in Equations ⎞ s−1 ⎜* ⎟ s ⎟ we shall define Pn,l = Pn ⊗ ⎜ CH (l+k)r ⎠ where l ∈ dZ and the basis and actions ⎝ k=−s+1 by 2

s Pn,l

given in Equations (4.5)−(4.13) by substituting i = n and d = (l + k) for of s each Pn ⊗ CH (l+k)r ∈ Pn,l . Using the idea of zig-zag modules for Uq (sl2 ) in [13], we shall define the two s indecomposable modules Wn,l and Msn,l for 1 ≤ n ≤ r − 2, l ∈ dZ, and s ∈ N, s as follows. The generators and actions of the module Wn,l will be the same as s s Pn,l minus the generators and actions of H in Pn,l . The module Msn,l will be ⎛ ⎞ * ⎜ l+s−1 ⎟ s /⎜ Sn,jr ⎟ the quotient module Pn,l ⎝ ⎠ with its generators and actions being the j=l−s+1 by 2

s minus the generators and actions of S same as the generators and actions of Pn,l s s s in Pn,l . The diagrams for Wn,l and Mn,l are given in Figures 2−3. We shall now use these modules to find the projective and injective resolutions of our simple modules, Sn ⊗ CH rl .

196

JAMES F. CLARK R2r−2−n,(l−s+1)r +L−n−2,(l−s+3)r

L−n−2,(l−s+1)r

R2r−2−n,(l−s+1)r +L−n−2,(l−s+3)r

R2r−2−n,(l+s−1)r

··· Sn,(l−s+3)r

Sn,(l−s+1)r

Sn,(l+s−3)r

Sn,(l+s−1)r

s Figure 2. Diagram of Wn,l Hn,(l−s+1)r

Hn,(l−s+3)r

Hn,(l+s−3)r

Hn,(l+s−1)r

··· R2r−2−n,(l−s+1)r +L−n−2,(l−s+3)r

L−n−2,(l−s+1)r

R2r−2−n,(l+s−3)r +L−n−2,(l+s−1)r

R2r−2−n,(l+s−1)r

Figure 3. Diagram of Msn,l Lemma 5.1. For each 1 ≤ n ≤ r − 2 and l ∈ dZ, the module Sn ⊗ CH rl has a projective resolution of (5.1)

···

∂3

∂2

3 Pn,l

∂1

2 Pr−2−n,l



1 Pn,l

S n ⊗ CH rl

Lemma 5.2. For each 1 ≤ n ≤ r − 1 and l ∈ dZ, the module Sn ⊗ CH rl has an injective resolution of (5.2)

S n ⊗ CH rl

δ

∂1

1 Pn,l

∂2

2 Pr−2−n,l

3 Pn,l

∂3

···

We can then use the projective resolutions found in Lemma 5.1 to get the following result: Proposition 5.3. For 1 ≤ n ≤ r − 1 and l ∈ dZ, then

 H ∼ ExtiC Sn ⊗ CH rl , Sn ⊗ Crl =   H ∼ ExtiC Sn ⊗ CH rl , Sr−2−n ⊗ Cr(l±1) =

C 0

i is even i is odd

0 C

i is even i is odd

Notice that for i = 1, Sn ⊗ CH rl will have no non-split self extensions and H S n ⊗ CH rl extended by Sr−2−n ⊗ Cr(l±1) will only have one non-split extension that is r-dimensional and can be pictured by the following diagram. vr−1

E F

vr−2

E F

···

E F

va+1

X

va

E F

···

E F

v1

E F

v0

H For Sn ⊗ CH rl extended by Sr−2−n ⊗ Cr(l+1) we would substitute a = r − 2 − n and H X = F , while for Sn ⊗ Crl extended by Sr−2−n ⊗ CH r(l−1) we would substitute a = n and X = E.

CONTINUING REMARKS ON THE UNROLLED QUANTUM GROUP OF sl(2)

197

6. Logarithmic typical modules In this section, we define the category Clog as the category of finite dimensional modules where q H = K as operators, but is not necessarily H-diagonalizable. As discussed in [11], Clog indicates the inclusion of logarithmic modules into C , which is motivated by logarithmic conformal field theory. In this section, we define the ¨ denoted subcategory of logarithmic modules whose generalized weights are in λ ∈ C, by Clog,typ , and call objects of this subcategory logarithmic typical modules, and we expand on a construction of logarithmic typical modules for even roots of unity in [11] to any root of unity. Furthermore, we are able to show an equivalence of categories between Clog,typ and the category of C[x]-modules and are able to use this equivalence in order to find the tensor decomposition of logarithmic typical modules in certain cases. H ¨ λ with λ ∈ C, Proposition 6.1. Consider a 2r-dimensional U q (sl2 )-module V 0 1 basis {vk , vk } where k ∈ {0, . . . , r − 1}, and actions

H.vk0 = (λ − 2k)vk0 + vk1

H.vk1 = (λ − 2k)vk1

2πi λ−2k 1 q vk p 2πiα 2k−λ 1 K −1 .vk0 = q 2k−λ vk0 − vk q p K.vk0 = q λ−2k vk0 +

K.vk1 = q λ−2k vk1 K −1 .vk1 = q 2k−λ vk1

0 1 E.vk0 = [k][λ + 1 − k]vk−1 + βk vk−1

1 E.vk1 = [k][λ + 1 − k]vk−1

0 F.vk0 = vk+1

1 F.vk1 = vk+1

where βk =

k 2πiα  λ−2(j−1) q + q 2(j−1)−λ . p{1} j=1

λ is an indecomposable U H (sl2 )Then V q

module that is a non-split self extension to Vλ . λ is a non-split self extension to Vλ , then it must be 2rLemma 6.2. If V dimensional. Corollary 6.3. There does not exist a non-split self-extension of Sn for any n < r − 1. Lemma 6.4. The only extension of Vμ by Vλ where μ = λ is Vλ ⊕ Vμ . λ are denoted by V λ (1) and Now, if we redefine the notation so that Vλ and V λ V

(s)

, respectively, then we can inductively extend the process above to define a λ (s) . sr-dimensional logarithmic typical module which we will denote by V H λ Theorem 6.5. Consider a sr-dimensional U q (sl2 )-module V s−1 0 1 {vk , vk , . . . , vk } where k ∈ {0, 1, . . . , r − 1} and actions

(s)

with basis

H.vkl = (λ − 2k)vkl + vkl+1 K.vkl = q λ−2k vkl +

s−1−l  j=1

E.vkl =

(2πi)j λ−2k l+j q vk j!pj

{k}{λ + 1 − k} l vk−1 + {1}2

l F.vkl = vk+1

s−1−l  j=1

l+j βkj vk−1 ,

K −1 .vkl = q 2k−λ vkl +

s−1−l  j=1

βkj =

(2πi)j

k 

j!pj {1}

m=1

(−2πi)j 2k−λ l+j q vk j!pj

q λ−2(m−1) − (−1)j q 2(m−1)−λ

where l ∈ {0, . . . , s − 1}

198

JAMES F. CLARK

λ Then V

(s)

H

is an indecomposable weight U q (sl2 )-module. If t, u < s such that

λ t + u = s, then V

(s)

λ is a non-split extension of V

(t)

λ by V

(u)

.

μ (t) where λ = μ is V λ (s) ⊕ λ (s) by V Corollary 6.6. The only extension of V (t) (s) (t) (u) μ . Also, if V λ is a non-split extension of V λ by V λ V where s = t + u, then (s)  Vλ must be sr-dimensional. H

λ Next, if we define Clog,typ to be the subcategory of logarithmic typical U q (sl2 )modules of finite length where each quotient is isomorphic to Vλ , we get the following corollary. 7 λ Corollary 6.7. Clog,typ = λ∈C¨ Clog,typ

Proof. Since Proposition 6.1 and Theorem 6.5 are defined based on their weights and there does not exist a logarithmic module containing two distinct μ (s) ⊕ V λ (t) , the corollary is proven. weights μ and λ except V  Finally, we are able to show that the category Clog,typ is equivalent to another category that is well-studied. λ Lemma 6.8. The subcategory of Clog,typ is equivalent as categories to the category of C[x](x−λ) − Mod, the (x − λ)-primary part of C[x] − Mod.

We use this lemma when we decompose the tensor product of logarithmic typical modules. The idea behind this will be to look at the tensor decomposition of the category of C[x]-modules, which is known, and use the equivalence of categories to find the decomposition for Clog,typ . ¨ such that λ + μ ∈ dZ, then Theorem 6.9. If λ, μ ∈ C λ V

(s)

μ ⊗V

(t)

=

s−1 r−1 * *

(s+t−1−2j) V.λ+μ+r−1−2i

i=0 j=0

Remark 6.10. The results of the tensor decomposition when λ + μ ∈ dZ and its proof can be worked out using similar methods. References [1] Draˇ zen Adamovi´ c and Antun Milas, Logarithmic intertwining operators and W(2, 2p − 1) algebras, J. Math. Phys. 48 (2007), no. 7, 073503, 20, DOI 10.1063/1.2747725. MR2337684 [2] Draˇ zen Adamovi´ c and Antun Milas, Lattice construction of logarithmic modules for certain vertex algebras, Selecta Math. (N.S.) 15 (2009), no. 4, 535–561, DOI 10.1007/s00029-0090009-z. MR2565050 [3] Draˇ zen Adamovi´ c and Antun Milas, Some applications and constructions of intertwining operators in logarithmic conformal field theory, Lie algebras, vertex operator algebras, and related topics, Contemp. Math., vol. 695, Amer. Math. Soc., Providence, RI, 2017, pp. 15–27, DOI 10.1090/conm/695/13992. MR3709702 [4] Christian Blanchet, Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand, Non-semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants, Adv. Math. 301 (2016), 1–78, DOI 10.1016/j.aim.2016.06.003. MR3539369 [5] Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1995. Corrected reprint of the 1994 original. MR1358358 [6] James Clark, The Unrolled Quantum Group of sl(2) with Connections to Topological Quantum Field Theory and Vertex Operator Algebras, ProQuest LLC, Ann Arbor, MI, 2020. Thesis (Ph.D.)–State University of New York at Albany. MR4119904

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[7] Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand, Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories, J. Topol. 7 (2014), no. 4, 1005–1053, DOI 10.1112/jtopol/jtu006. MR3286896 [8] Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand, Relations between Witten-Reshetikhin-Turaev and nonsemisimple sl(2) 3-manifold invariants, Algebr. Geom. Topol. 15 (2015), no. 3, 1363–1386, DOI 10.2140/agt.2015.15.1363. MR3361139 [9] Francesco Costantino, Nathan Geer, and Bertrand Patureau-Mirand, Some remarks on the unrolled quantum group of sl(2), J. Pure Appl. Algebra 219 (2015), no. 8, 3238–3262, DOI 10.1016/j.jpaa.2014.10.012. MR3320217 [10] Thomas Creutzig and Terry Gannon, Logarithmic conformal field theory, log-modular tensor categories and modular forms, J. Phys. A 50 (2017), no. 40, 404004, 37, DOI 10.1088/17518121/aa8538. MR3708086 H [11] Thomas Creutzig, Antun Milas, and Matt Rupert, Logarithmic link invariants of U q (sl2 ) and asymptotic dimensions of singlet vertex algebras, J. Pure Appl. Algebra 222 (2018), no. 10, 3224–3247, DOI 10.1016/j.jpaa.2017.12.004. MR3795642 [12] Thomas Creutzig, Antun Milas, and Simon Wood, On regularised quantum dimensions of the singlet vertex operator algebra and false theta functions, Int. Math. Res. Not. IMRN 5 (2017), 1390–1432, DOI 10.1093/imrn/rnw037. MR3658169 [13] A. M. Ga˘ınutdinov, A. M. Semikhatov, I. Yu. Tipunin, and B. L. Fe˘ıgin, The KazhdanLusztig correspondence for the representation category of the triplet W -algebra in logorithmic conformal field theories (Russian, with Russian summary), Teoret. Mat. Fiz. 148 (2006), no. 3, 398–427, DOI 10.1007/s11232-006-0113-6; English transl., Theoret. and Math. Phys. 148 (2006), no. 3, 1210–1235. MR2283660 [14] Nathan Geer, Jonathan Kujawa, and Bertrand Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories, Selecta Math. (N.S.) 17 (2011), no. 2, 453–504, DOI 10.1007/s00029-010-0046-7. MR2803849 [15] Nathan Geer, Bertrand Patureau-Mirand, and Vladimir Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145 (2009), no. 1, 196–212, DOI 10.1112/S0010437X08003795. MR2480500 [16] Jens Carsten Jantzen, Lectures on quantum groups, Graduate Studies in Mathematics, vol. 6, American Mathematical Society, Providence, RI, 1996. MR1359532 [17] Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR1321145 [18] Matthew Rupert, Categories of weight modules for unrolled restricted quantum groups at roots of unity, arXiv preprint arXiv:1910.05922, 2019. [19] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR1269324 University at Albany (SUNY), 1400 Washington Avenue, Albany, NY 12222

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15465

On the geometric interpretation of certain vertex algebras and their modules Jesse Corradino Abstract. In this note we consider the geometric interpretation of certain vertex algebras and their modules within the framework of the formal geometry of Gel’fand and Kazhdan. Our main result is that we introduce an extension of the torsor of formal coordinates whereby a vertex algebra VE is localized as an OX -module with a flat connection. A further application is to certain representations that are not, in general, conformal. This result extends a result of Frenkel and Ben-Zvi to this particular case.

1. Overview Here we provide a brief overview of the following note. In section 2 we review formal geometry in the sense of Gel’fand and Kahzdan. Therein, we provide non-standard constructions of its main elements with Lie-Rinehart algebroids. In addition, we recall the realization theorem of Guillimen and Sternberg and explain its relevance to formal geometry in this work. Section 3 recalls the definitions and construction we require from the literature on vertex algebras. Section 4 is a description of our results. In particular, we introduce a transitive structure into the geometric formalism as well as a lift of this structure in order to apply the associated vector bundle construction to a vertex algebra we construct from local data on a qcqs κ-scheme X. All results presented here are based on the author’s PhD thesis [15]. 2. Formal geometry Formal geometry is the study of the torsor of formal coordinates CorX as a (g, K)-structure [4] over the Harish-Chandra pair loc.cit (w, Aut(O)) of formal vector fields and power series automorphisms of the formal n-disc D and various constructions related to the same by lifts and extensions thereof. Formal Geometry was introduced by Gel’fand, Kahzdan, and Fuks [22] to study the cohomology of infinite dimensional Lie algebras in the seventies. At the same time, closely related work was studied by Guillemin, Sternberg, and others, on the infinite dimensional Lie groups of Cartan under the guise of pseudo-groups [28, 54]. In particular, much effort was expended by these authors to investigate the structure of the Lie algebra w. Bernstein and Rozenf’eld showed how these works were related through the notion of abstract linearly compact Lie algebras that ”formalized” a sheaf of Lie c 2021 American Mathematical Society

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algebras associated to a pseudogroup [5], thus connecting seemingly disparate paths of inquiry. The subject was taken up later by Nest, Tsygan, and several others, to study deformation quantization [10, 36, 50] and by Beilinson and Drinfeld to study Geometric Langlands [4]. In the latter work, the efficacy of formal geometry as a tool for studying D-modules of arbitrary type and representations of the affine KacMoody Lie algebra, specifically vertex algebras, was revealed, through their role in Geometric Langlands. Thereafter, Frenkel and Ben-Zvi [20] used the framework of formal geometry to realize vertex algebras as geometric objects, specifically as OX modules with flat connection, over an algebraic curve X of genus greater than or equal to two. The involvement of formal geometry in loc.cit depended in a crucial way upon an important result of Huang’s [29, 30] that demonstrated the action of the pro-algebraic group Aut(O) on a conformal vertex algebra. Moreover, Frenkel and Ben-Zvi’s description of the bundle associated to a conformal vertex algebra within the context of formal geometry as a chiral algebra over an algebraic curve was anticipated by a similar theorem [31] of Huang and Lepowsky’s which demonstrated the same result for an open subset of C. The presentation in [20] is the primary source of influence in this note. Formal geometry vis-a-vis vertex algebras then resurfaced more than a decade later in the work of Gorbounov, Gwilliam, and Williams [25] in order to relate chiral differential operators predicated upon the βγ-system to factorization algebras in the works of Costello and Gwilliam [11]. One reason that formal geometry expresses itself across these various applications is because the construction automates the globalization of locally coordinate dependent constructions, thereby fulfilling a common objective in geometry-that is, to globalize local constructions. One adopts the perspective that in a formal neighborhood of a point, smooth spaces are the same up to a non-canonical identification of formal neighborhoods. One expresses various coordinate dependent constructions in a formal neighborhood or disc about a point in some ambient space X and then identifies these expressions within the infinite dimensional space CorX . The application of formal geometry globalizes by ”spreading out” this identification over the base X via CorX , as one regards the torsor of formal coordinates as a ”cover by formal neighborhoods” of X, in the sense of Cech cohomology [36]. Guillemin and Sternberg’s realization theorem [7, 18, 28] augments formal geometry in this note when viewed as an instance of geometry with Harish-Chandra pairs by furnishing it with a useful method whereby the torsor of formal coordinates may be lifted and extended. In particular, we regard pro-algebraic groups as κ-functors represented by objects in the category of linearly compact κ-algebras to make use of their realization morphism. Indeed, according to their realization theorem, the torsor of formal coordinates CorX under the Harish-Chandra pair (w, Aut(O)), possesses a universal mapping property in the category of HarishChandra pairs by asserting the existence of a morphism of Harish-Chandra pairs (g, K) → (w, Aut(O)) whenever Lie(K) = k ⊂ g is transitive. One is therefore able to extend the localization functor to both Harish-Chandra (g, K)-modules and structures given a morphism of g → w. We begin by discussing how a finite dimensional qcqs κ-scheme X may be coordinatized in a manner familiar from differential geometry, which is to say that locally the scheme X “looks like” n-dimensional affine space. Such a coordinatization at a point x ∈ X is obtained by means of etale neighborhoods U of the same.

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Recall that X is locally etale if for every x ∈ X, there exists an open neighborhood U ⊂ X of x and sections f1 , . . . , fn ∈ OX (U ) such that the morphism

f= fi : U → An i

is etale. We can choose constants so that the affine coordinates Xi of x with respect to f are contained in its ideal sheaf, so that f maps x to the origin in An . Accordingly we define the set {xi = fi − f (x)} ∈ OX (U ) to be a coordinate system at x if f maps x to the origin [49]. We shall use our definition of coordinates to define formal coordinates. Let Δ : X → X ×X be the diagonal embedding. A choice of coordinates at x determines a free basis of the conormal sheaf of this embedding. Indeed, define vi = fi ⊗ IΔ + IΔ ⊗ fi mod(IΔ )2 , where IΔ is the ideal sheaf of the same. Then the vi are a free basis of the cotangent sheaf ΩX = IΔ /(IΔ )2 . By duality, the tangent sheaf TX is locally free, too, since the canonical pairing is non-degenerate. Denote by {∂xi } the dual free basis. We regard the tangent sheaf as the canonical Lie-Rinehart algebroid [2]. Let UX be the universal enveloping algebroid of TX . Recall that UX is equipped with an increasing filtration [48]  U≤d it ≤ d} X = Span{∂xi1 · · · ∂xip | ∂xit ∈ TX , t

as TX is projective of finite rank over X. The universal enveloping algebroid UX is the inductive limit over d of this filtration. Moreover, UX is furnished with left and right counits e∨ and an OX -bialgebra structure with respect to these. These constructions are all local [48]. The OX -dual of the universal enveloping algebroid is the jet algebra JX = HomOX (UX , OX ) on X. By the above, the jet algebra is a sheaf of OX -algebras complete with respect to the ideal Jc given by the kernel of the morphism e : JX → OX dual to the counit morphism. The sheafification of this algebra by the standard Spec construction for OX algebras entails that the OX -algebra spectrum of JX is a formal groupoid X-scheme, denoted JetX . Global sections of its structure sheaf are isomorphic to the algebra of functions obtained by the completion of the square X × X along the diagonal Δ(X), rendering this construction of JetX equivalent to the standard one cf. [12]. To compare constructions, observe that the ideal Jc corresponds to the completion of IΔ in the IΔ -adic topology on OX×X . We wish to use this description of the jet scheme in terms of the Lie-Rinehart algebroid (OX , TX ) to define formal coordinates. Lie-Rinehart algebroids are discussed in more detail below. Consider a Lie-Rinehart algebra (R, L). There is a convenient interpretation of its jet algebra, due to Rinehart [51] supplied by his Poincare-Birkhoff-Wit theorem for Lie-Rinehart algebras he proved in loc.cit. One ∨ ˆ has JL ∼ ), where the right hand side indicates the completion of the sym= Sym(L metric algebra as an R-algebra with respect to the augmentation ideal, that is, the

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kernel of the projection of Sym(L) onto R. Furthermore, there is an isomorphism of graded R-algebras gr(JL ) ∼ = Sym(L∨ ) In particular, we apply the PBW theorem of Rinehart to the special case of the universal enveloping algebroid of the Lie-algebroid TX and the corresponding jet algebra JX . Under the auspices of the PBW theorem, one has the jet algebra is isomorphic to the completion of the cotangent sheaf ΩX , so given a choice of coordinate system at x, one has Jc is generated by the vi , because Jc /(Jc )2 ∼ = Ω1X and one lifts the free basis. These facts motivate the first of two definitions of formal coordinates. Trivialize the jet scheme in an etale neighborhood U of a point x in terms of a system of coordinates at x. We say a trivialization of the jet scheme JetX is a system of formal coordinates, viz. φX : JetX (U ) ∼ = Spec(OX (U )[[v1 , . . . , vn ]]) where φX maps Jc onto the maximal ideal m ⊂ OX (U )[[v1 , . . . , vn ]]. We specialize this definition to the fibre of JetX and say we have a system of formal coordinates at x. The space whose rational points are systems of formal coordinates at x is the torsor of formal coordinates, below. We define a formal neighborhood of a point x ∈ X, specifically the formal disc centered at a point x. Let x ∈ U be an etale neighborhood of x and {xi } a system of coordinates. A choice of coordinate system determines a basis {vi } of (mx /m2x )∨ , the fibre of the contangent sheaf at x. Then Spec(Ox ) =: Dx is the formal disc centered 2 ∨ ˆ ∼ at x, where Ox = Sym((m x /mx ) ). Of course, one has Ox = κ(x)[[v1 , . . . , vn ]]. Our description of the disc is somewhat non-standard, as many authors prefer to simply use the completed stalk of the structure sheaf at x cf. [4, 20, 36]. However, we prefer this non-standard description, as it synchronizes it with the constructions presented in the literature on transitive linearly compact Lie algebras cf. [7,28] and our use of Lie-Rinehart algebroids. However, we still emphasize that as a scheme Dx is non-canonically isomorphic to Spec(κ(x)[[x1 , . . . , xn ]]). Let us now consider the Lie algebra of derivations of the formal disc. First, however, a few observations about Ox are in order to make sense of its vector fields. Write elements of Ox as ∞  a(v1 , . . . , vn ) = ad (v1 , . . . , vn ) ∈ Ox d=0

where ad (v1 , . . . , vn ) are homogeneous polynomials of degree d in the vi , abbreviated ad . Expressing elements in this manner suggests how to define the order of an element of Ox . The order of a ∈ Ox to be the minimum d such that ad = 0, denoted ord(a). The order of elements induces a filtration Ox Ox = F0 Ox ⊃ F1 Ox ⊃ . . . ⊃ Fk Ox ⊃ Fk+1 Ox ⊃ . . . where Fk Ox = {a ∈ Ox | ord(a) ≥ k}. Notice that the unique maximal ideal mx of the local ring Ox is given by F1 Ox with respect to this filtration and that {Fk Ox } can be regarded as a system of neighborhoods of the 0, thus rendering Ox as a linearly compact κ-algebra. The Lie algebra of derivations of Dx , denoted w, then has the following description with respect to the order filtration of Ox . Define w(k) = Symk ((mx /m2x )∨ ) ⊗ mx /m2x

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where mx /m2x is the dual vector space with dual basis {∂xi }. The algebra Symk ((mx /m2x )∨ ) should be regarded as the space of k-th degree homogeneous polynomials in the vi coordinates. Take the direct sum ∞ * w(k) gr(w) = k=0

The direct sum gr(w) is a graded Lie algebra via the bracket w(k) × w(l) → w(k + l) so by imparting the discrete topology to w(k) for each k ≥ 0, we can form its completion by ∞

w= w(k) k=0 2 ∨ 2 ˆ = Sym((m x /mx ) ) ⊗ mx /mx

to be the Lie algebra of formal vector fields. We refer to elements of this Lie algebra as formal vector fields. A formal vector field ξ ∈ w is completely determined by where it maps the vi , so suppose ξ(vi ) = ai (v1 , . . . , vn ); then, the formal vector fields have the following coordinate dependent rendering as n  ξ= ai ∂xi i=1

where ai ∈ Ox . Moreover, let ξ  ∈ w with a similar description, then the formula n  {∂xj (fi )gj − ∂xj (gi )fj } ζ = [ξ, ξ  ] = i=1

defines its Lie bracket, where fi , gi ∈ Ox , which is continuous in the linearly compact topology. Hence, w is a linearly compact Lie algebra. The subspace of formal derivations which vanish at the origin, that is, formal vector fields of order greater than 0, form a fundamental subalgebra, which we shall denote by w0 . It induces a canonical filtration of w, denoted {Fi w0 }. Furthermore, w0 is a pro-nilpotent Lie algebra with respect to this filtration, so there exists a corresponding pro-unipotent algebraic group. In particular, we have Exp(w0 ) = GLn  Exp(F1 w0 ), denoted hereafter by Aut(O) and say Aut(O) is the group of formal automorphisms of Dx . Elements of this group, called power series automorphisms, are n-tuples of formal power series (φ1 , . . . , φn ) such that φi (v1 , . . . , vn ) = vj , the φi have zero constant term, and det(Jacφ1 ,...,φn ) is a unit. A power series automorphism is induced by a κ-algebra automorphism of κ(x)[[v1 , . . . , vn ]] that preserves the maximal ideal. In terms of formal geometry, recognize that this description imparts an action of Aut(O) on the set of formal coordinates centered at x. The compatibility of the Lie algebra w and Aut(O) along w0 determine a Harish-Chandra pair (w, Aut(O)). We shall refer to this special Harish-Chandra pair as the Gel’fand-Kazhdan pair hereafter. The torsor of formal coordinates CorX is the (g, K)-structure under the Gel’fand-Kazhdan pair (w, Aut(O)). The torsor of formal coordinates is given by, for U ⊂ X an etale neighborhood of x, to be the set of formal coordinates φX centered at x. In particular, its fibre over a point x ∈ X is the set of systems of

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formal coordinates at x. Immediately one observes these are nothing more than automorphisms of fibres of the jet scheme, that is, automorphisms of Dx . Accordingly, viewed as the union of its fibres, this X-scheme consists of pairs (φx , x) and the structure morphism π : CorX → X is given by (φx , x) → x. The following theorem exhibits CorX as a principal w-space. In the literature, this theorem goes by the name the fundamental theorem of formal geometry. Theorem 1. There exists an isomorphism OCorX ⊗ w → TCorX restricting on fibres to an isomorphism w∼ = T(x,φx ) CorX for (x, φx ) ∈ CorX . Furthermore, the inverse of the this isomorphism furnishes a w-valued 1-form ∇w ∈ Ω1 (CorX ; w) satisfying the Maurer-Cartan equation thereby imparting a flat connection to CorX . The consequence of the fundamental theorem is that CorX is a (g, K)-structure with respect to the Gel’fand-Kazhdan pair (w, Aut(O)). We can define transitive structures of CorX with this theorem together with the results of Guillemin, Sternberg et al. [7, 18, 28]. The result of Guillemin and Sternberg establishes the existence of a morphism of linearly compact Lie algebras g and w given the existence of Lie subalgebra k ⊂ g of codimension n. Indeed, for a pair of such Lie algebras (g, k) over κ such that g is a linearly compact Lie algebra and k ⊂ g is of codimension n, there exists a realization loc.cit of the pair (g, k): a Lie algebra homomorphism & : g → w such that &−1 (w0 ) = k. Furthermore, a realization is transitive if the image of k under & is a transitive Lie subalgebra of w. One defines a Harish-Chandra pair (g, K) to be transitive if k is a transitive subalgebra of g and of codimension n. Then, by the realization theorem, there exists a morphism of transitive Harish-Chandra pairs Φ = (&e , φ) : (g, K) → (w, Aut(O)) unique up to a change of formal coordinate systems at x. Therefore one defines a transitive structure of CorX to be the reduction of the structure group [55] of CorX by Φ . We shall use transitive structures below as intermediary spaces to construct sheaves of vertex algebras. 3. Vertex algebras and their modules Recall the definition of a vertex algebra [20, 39]. Definition 1. A vertex algebra is a quadruple (V, |0 , Y, D), consisting of (I) a vector space V over κ (II) a distinguished vector |0 , referred to as the vacuum state (III) a linear operator, referred to as the state-field correspondence, Y : V → EndV [[z, z −1 ]] assigning to each a ∈ V a field Y(a, z) =



a(n) z −n−1

n∈Z

(IV) an endomorphism D of V , referred to as the translation operator. satisfying the following axioms:

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(Axiom (Axiom (Axiom (Axiom (Axiom

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I) Y(|0 , z) = IdV II) Y(a, z) · |0 ∈ V [[z]] for every a ∈ V , so that Y(a, z)|0 |z=0 = a. III) [D, Y(a, z)] = ∂z Y(a, z) for every a ∈ V IV) D(|0 ) = 0 V) All fields are local, so for any pair a, b ∈ V , there exists N such that (z − w)N [Y(a, z), Y(b, w)] = 0

A remark is that one can ignore D in the definition of a vertex algebra, as Da = a(−2) |0 . A vertex algebra is N-graded if there exists a decomposition of V = ⊕n∈N V n into weight spaces V n of weight n. Typically, such a decomposition is furnished by a Hamiltonian operator on V . Given a subspace V  ⊂ V , we say V  is a vertex subalgebra if V  is an D invariant subspace containing the vacuum vector, such that, for every a, b ∈ V  , Y(a, z) · b ∈ V  ((z)). A subspace V  is said to strongly generate V if V is spanned by vectors 1 r aj(−n · · · aj(−n |0 1) r)

with ajs ∈ V  , r ≥ 0, js ∈ J, ns ≥ 1. We say V is finitely strongly generated if the cardinality of J can be taken to be finite. We shall consider certain vertex algebras finitely strongly generated by the weight 0 and 1 spaces below locally by coordinate dependent sections of OX -modules. The definition of a module M over a vertex algebra V is similar cf. [39] for further direction discussion. In particular, we shall localize certain modules of the vertex algebra we construct that are pullbacks to its Zhu algebra. Given a vertex algebra V , its Zhu algebra, A(V ), is an associative algebra that is a partial invariant of the representation theory of V . Let M be a V -module. Zhu [57] defined an operation  such that the assignment of states a ∈ V to Fourier modes aM (j) ∈ End(M ) of M is an associative algebra map. Moreover, by restriction, one has an assignment of states of V to endomorphisms of the weight 0 space when M is graded, viz. a → aM (0)|M 0 . Assume hereafter that M is N-graded V -module, then this algebra morphism is obtained by the Borcherd’s identity for modules by setting m = |a| and n = −1. In fact, when both sides are degree 0 morphisms, we have  |a|  M M M (a(−1+j) b)M (aM 0 = −j bj + b−j−1 aj+1 )0 j j≥0

j≥0

|a| |a| M Restrict this relation to M 0 and one sees that the sum j=0 j (a(j−1) b)0 is M equal to aM (0) b(0) by Borcherd’s identity. Therefore, to obtain an algebra morphism from V to End(M 0 ), loc.cit defined the operation  |a|   |a| (a(j−1) b) a  b := j j=0 called the Zhu product. The operation  is in general not associative on V , but Zhu shows the existence of a subspace V  ðV ⊂ V , where ð is determined by a Hamiltonian operator, which is a two-sided ideal with respect to  and such that a  (b  c) − (a  b)  c ∈ V  ðV

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Accordingly, the Zhu Algebra of V is the quotient A(V ) := V /V  ðV One has that for an N-graded V -module M , its weight 0 space M 0 is a module over the associative algebra A(V ) [52, 57]. Furthermore, there is a functorial correspondence between representations of A(V )-modules and modules over V whose weight 0 space is an A(V )-module that restricts to a bijective correspondence between the same for simple modules loc.cit. We stress that the correspondence between V -modules and A(V )-modules is far from an equivalence of categories, but nevertheless, one may still utilize the correspondence between A(V )-modules and V -modules with respect to indecomposable weight 0 spaces. It is from this perspective that we provide a partial geometric interpretation of so-called logarithmic (weak) modules discovered in [46, 47] and whose properties are exposed in detail in [32] as a part of the tensor product theory of vertex algebras. Observe that the Borcherds identity canonically attaches a Lie algebra g(V ) ⊂ End(V ) to a vertex algebra V whose bracket is given by the Borcherds identity. If V is a graded vertex algebra, this Lie algebra g(V ) is graded as well. In particular, g(V )0 → A(V ) is a Lie algebra homomorphism. Therefore, if M 0 is a module over A(V ), we have a g(V ) module by pullback, viz. M = U (g(V )) ⊗U(g(V )≤ ) M 0 Dividing by the submodule of Fourier modes of Y(a(−1) b, z)− : Y(a, z)Y(b, z) : one obtains an N-graded V -module M whose weight 0 space is M 0 . 4. The geometric interpretation of the vertex algebra VE and its modules We provide a short reminder of the objects on X that locally strongly finitely generate an N-graded vertex algebra VE in weights 0 and 1. Let R be a finitely generated commutative κ-algebra and L an R-module. The module L is a LieRinehart algebra over R if L is equipped with a κ-linear Lie algebra bracket, an anchor morphism of R-modules ρ : L → Derκ (R) such that the anchor is a morphism of Lie algebras whose Lie bracket is compatible with its R-module structure according to the formula [λ, rλ ] = r[λ, λ ] + ρ(λ )(r)λ for λ, λ ∈ L and r ∈ R. We denote a Lie-Rinehart algebra over R by the pair (R, L) if we wish to emphasize both pieces of data. Lie-Rinehart algebras form categories over both the same algebra and distinct algebras. We give the general definition of morphisms and take the case where the underlying algebras are equal as a special case thereof. We formulate the definition so that it fits into a broader pattern of morphisms of anchored modules, that is, R-modules equipped with a morphism to Derκ (R). Beside Lie-Rinehart algebras, Leibniz algebras are other well-known examples. Definition 2. Let (R, L) and (R , L ) be Lie-Rinehart algebras over R and R , respectively. An Lie-Rinehart morphism (ϕ, ) : (R, L) ⇒ (R , L ) is a pair consisting of (Axiom I) a κ-algebra morphism ϕ : R → R (Axiom II) a κ-Lie algebra morphism  : L → L (Axiom III) an additive map  : L → L such that (rλ) = ϕ(r)(λ) (Axiom IV) ϕ(ρ(λ)(r)) = ρ ((λ))(ϕ(r)) 

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The definition of Lie-Rinehart algebra we have given is somewhat general, so hereafter we shall impose the additional hypothesis that L is a finitely generated projective R-module. We shall consider a sheaf of OX -modules L over a ringed space (X, OX ) such that (OX (U ), L(U )) to be a Lie-Rinehart algebra for U → X etale. We call such a coherent OX -module a Lie-Rinehart algebroid. In particular, for an affine scheme X = Spec(R), the category of Lie-Rinehart algebras and the category Lie-Rinehart algebroids are equivalent. One can show that a representation of a Lie-Rinehart algebra L is equivalent to an R-module M such that the direct sum M ⊕ L is a Lie-Rinehart algebra and contains L as a Lie-Rinehart subalgebra and M as an abelian ideal. In the particular case where R = M , we say the direct sum is the canonical module. A key construction with Lie-Rinehart algebras is that of the universal enveloping algebra of the canonical L-module. This construction determines a functor from the category of Lie-Rinehart algebras to the category of associative κ-algebras, and bears much the same relationship to representations of (R, L) as the classical universal enveloping algebra bears to that of its underlying Lie algebra. One defines for the R-module structure on the canonical L-module a semi-direct product Lie algebra structure with bracket [r + λ, r + λ ] := (ρ(λ)(r  ) − ρ(λ )(r)), [λ, λ ]) Let U (R ⊕ L) denote its universal enveloping algebra in the sense of Lie algebras. We embed R and L under the maps ιR and ιL in U (R ⊕ L) in the standard way, viz. ιR (r)ιL (λ) = ιL (rλ) and ιL (λ)ιR (r) − ιR (r)ιL (λ) = ιR (ρ(λ)(r)) These embeddings are injective by our hypothesis that L is projective. Write UR L− for the subalgebra generated by the image of these embeddings. Let r1 ∈ R and r2 + l ∈ R ⊕ L and consider the two sided ideal IR ⊆ UR L− generated by the differences (r1 (r2 + l))− − r1− (r2 + l)− , where − denotes projection from U (R ⊕ L) onto UR L− . Then the quotient UR L = UR L− /IR is the universal enveloping algebroid of the Lie-Rinehart algbera L. The L-jets JR L are the R-module dual of the universal enveloping algebroid. Last, we observe that the L-de Rham complex (Ω• (L), dL ) complex dualizes the anchor map to a morphism of differentially graded algebras ρ∨ : ΩR → Ω• (L), where ΩR := Homκ (Derκ (R), R). In particular, one may recover the original LieRinehart algebra structure from this co-anchor via dL (r)(λ) = ρ(λ)(r) and, for an R-basis {λi } of L, we have dL (ηk )(λi , λj ) = −ηk ([λi , λj ]) where λi ∈ L and ηk ∈ L∨ . Therefore, the pair (Ω• (L), dL ) recapitulates the original Lie-Rinehart algebra (R, L). This perspective motivates the definition of a Lie bialgebroid [44]. First, observe that if L∨ is itself furnished with a LieRinehart algebra structure (R, L∨ ), then we obtain mutatis mutandi its L∨ -de Rham complex Ω• (L∨ ), with de Rham differential denoted d∨ . We say that (L, L∨ )

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is a Lie-Rinehart bialgebra over R if L ∼ = L∨ and if d∨ is a derivation of the Shouten• Nijeenhus bracket of Ω (L), viz. d∨ [λ1 , λ2 ] = [d∨ (λ1 ), λ2 ] + [λ1 , d∨ (λ2 )] ;p ; L → p+1 L via the duality hypothesis. for λ1 , λ2 ∈ L and d∨ : Given a Lie-Rinehart bialgebra (L, L∨ ) consider the associated double E = L ⊕ L∨ . Unlike the category of Lie algebras, the category of Lie-Rinehart algebras is not closed under the formation of doubles. Indeed, E is no longer a Lie-Rinehart algebra, but rather a Courant-Dorfman algebra, defined as follows. Definition 3. A Courant-Dorfman algebra is a Leibniz algebra together with a symmetric R-bilinear form  | : E ⊗κ E → R satisfying the following axioms for r ∈ R and e1 , e2 ∈ E (Axiom I) ρ(e3 )(e1 |e2 ) = e1 |[e1 , e2 ] + [e2 , e1 ] (Axiom II) ρ(e3 )(e1 |e2 ) = [e3 , e1 ]|e2 + e1 |[e3 , e2 ] (Axiom III) a derivation ∂ : R → E We continue to follow the above heuristic used to describe representations of Lie-Rinehart algebras. Morphisms then are maps of E-modules respecting the additional structure of the Courant-Dorfman algebras. In particular we denote the group of automorphisms of the canonical E-module by Aut(E). There is an infinitesimal structure preserving morphism corresponding to automorphisms. Definition 4. Let (R, E) be a Courant-Dorfman algebra over R. A derivation δ of E consists of an endomorphism δ and a vector field ξ ∈ Derκ (R) such that the following axioms are satisfied. (Axiom I) for e, e ∈ E δ[e, e ] = [δ(e), e ] + [e, δ(e )] (Axiom II) for ξ ∈ Derκ (R) ξe|e = δ(e)|e + e|δ(e ) There are two consequences of this definition we consider. Firstly, the vector field ξ in the definition is determined by δ according to the axioms of a CourantDorfman algebra. As such, the space of derivations forms a Lie algebra. Secondly, given that ξ is determined by δ, one has the identities δ(re) = ξ(r)e + rδ(e) and ρ([δ(e)]) = [ξ, ρ(e)]. Therefore, we have a Lie algebra homomorphism Ξ : DerCD (E) → Derκ (R). Moreover, the assignment of e ∈ E → ade ∈ Der(E) embeds E in Der(E). The image of this embedding is the subalgebra of inner derivations. The geometric rendering of Courant-Dorfman algebras is precisely the same as our approach to that of Lie-Rinehart algebras. We say a sheaf of OX -modules E over a ringed space (X, OX ) such that (OX (U ), E(U )) is a Courant-Dorfman algebra for U → X etale is a Courant-Dorfman algebroid. Again, for an affine scheme X = Spec(R) the category of Courant-Dorfman algebras and the category of Courant-Dorfman algebroids are equivalent. Returning to a Lie-Rinehart bialgebra (L, L∨ ) and its double, an important theorem of [44] asserts that E = L ⊕ L∨ has the structure of a Courant-Dorfman

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algebra. Indeed, the definition given above was their solution to understanding the notion of Lie-Rinehart double analogous to that of Lie algebra double. We say a Courant-Dorfman algebra E is transitive if the anchor map ρ : E → Derκ (R) is a surjection. Less generally, when (R, E) fits into an exact sequence 0 → ΩR → E → Derκ (R) → 0 we say (R, E) is exact. Notice that the anchor map induces a morphism ρ∗ : ΩR → E ∗ by the universal derivation d0 : R → ΩR . We say that ρ∗ : ΩR → E is the coanchor. In the case that E is the Courant-Dorfman algebra corresponding to the double of a Lie-Rinehart bilalgebra (L, L∨ ), E is self-dual. Indeed, E ∼ = E ∨ by the symmetric bilinear pairing. As such, we identify ΩR with its image under the coanchor. One has the quotient E/ΩR is a Lie-Rinehart algebra and E/∂(R) is a Lie algebra. Hereafter, we fix the Courant-Dorfman algebra corresponding to the double of a Lie-Rinehart algebra defined locally on X. We construct of a vertex algebra from this transitive Courant-Dorfman algebra following [42]. Let R ⊕ E be the canonical E-module for E = L ⊕ L∨ and C = R ⊕ E. Define the C[t± ] = (R ⊕ E) ⊗ κ[t± ] as a vector space over κ. The embeddings R ⊗ κ[t± ] := R[t± ] → C[t± ] and E ⊗ κ[t± ] = E[t± ] → C[t± ]. Both R[t± ] and E[t± ] are graded subspaces of C[t± ], with homogenous elements r ⊗ tn and e ⊗ tn of degrees −n − 1 and −n, respectively. With this convention, C[t± ] is a graded κ-vector space * C[t± ]n C[t± ] = n∈Z

where C[t± ]n = R ⊗ κt−n−1 ⊕ E ⊗ κt−n Let d = ∂ ⊗ 1 + IdC ⊗ ∂t : R[t± ] → C[t± ] be a partially defined linear operator of C[t± ]. Notice that d is a linear operator of degree 1 with our grading convention. Define the bilinear product [ , ] : C[t± ]×C[t± ] → C[t± ] on C[t± ] for r, ri ∈ R, e, ei ∈ E, as follows [r1 ⊗ tm , r2 ⊗ tn ] = 0 [r ⊗ tm , e ⊗ tn ] = μr (e)(r) ⊗ tm+n [e ⊗ tm , r ⊗ tn ] = μl (e)(r) ⊗ tm+n [e1 ⊗ tm , e2 ⊗ tn ] = [e1 , e2 ] ⊗ tm+n + me1 |e2 ⊗ tm+n−1 where μ denote the left and right actions of E according to their superscript. One can show that the loop algebra c := C[t± ]/Imd the is a Lie algebra [38]. The projection τ : C[t± ] → c, where τ (c ⊗ tn ) = c ⊗ tn + Im d := c(n) for c ∈ C, allows one to form the generating series  Y(c, z) = c(n) z −n−1 n∈Z ±

in c[[z ]]. Later, we shall abuse notation by using the same notation for a c-module V. Observe, with the grading obtained by restriction of τ , c decomposes as * c = c+ c−

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7 7 where c− = R(−1) n≥1 E(−n) and c+ = n≥0 E(n). One writes R(n) = {r(n) | r ∈ R} and E(n) = {e(n) | e ∈ E} in this notation. Take κ as the trivial one dimensional representation of c+ and the induced c-module over k, viz. U(c)

IndU(c+ ) κ where U denotes the universal enveloping algebra. Applying the PBW-theorem, we have V := U (c− ) which is the vacuum module associated to C. Assign to κ degree zero, then the vacuum module is N-graded as a c-module according to the embedding C → C(−1) ⊂ V where, c → r−1 1 + e−1 1. The vacuum module is a locally nilpotent c-module because c(n) · v = 0 for n sufficiently large by its N-grading. Thus, the power series Y(c, z) associated to arbitrary c ∈ R ⊕ E by this construction acts as a field on V. In particular, V is strongly finitely generated by R ⊕ E. We have the following consequence of the reconstruction theorem [20, 39]. Theorem 2. The assignment Y(·, z) : V → EndV[[z ± ]] is a state-field correspondence, rendering V as an an N-graded vertex algebra, strongly finitely generated by R ⊕ E, with vacuum the unit element of R and translation operator given by D(c) = c(−2) 1, for c ∈ R ⊕ E, extending the derivation ∂ of R ⊕ E. In the reconstruction theorem, we have 1 αm −j1 −1 α1 1 Y(cα c (z) · · · ∂z−ns −1 cαm (z) : (−n1 ) · · · c(−nr ) |0 , z) = (n − 1)! · · · (n − 1)! : ∂z 1 r for c ∈ R ⊕ E. The PBW theorem gives us that the vacuum module is isomorphic as a κ-vector space V∼ = Sym(R(−1)) ⊗ Sym(E(−)) 0 ∼ One has such a map, for V = Sym(R(−1)) via the PBW embedding of R into V. This module is rather large. The following theorem [42] establishes the existence of a quotient of the vacuum module that is a vertex algebra small enough for our construction of a sheaf of vertex algebras. Theorem 3. Let V be the vaccum module and I the c-submodule of V generated by

1 − 1R , r(−1) r  − rr  , r(−1) e − re

then the quotient V/I = VE is an N-graded vertex algebra, strongly finitely generated by its weight 0 and 1 spaces V0E = R V1E = E and whose weight n spaces are VnE = Span{e1 (−n1 ) · · · ek (−nk )1}  where ei ∈ E, n1 ≥ · · · nk ≥ 1 and i ni = n for n ≥ 2.

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To globalize this construction to a sheaf of OX -modules, we shall appeal to the localization technique with respect to a (g, K)-structure such that VE is a (g, K)-module. In order to demonstrate that VE is a Harish-Chandra module, a result of Li and Yamskulna [43], shows the grading preserving automorphisms of the vertex O-algebra are a subgroup of AutV A (V) given by AutCD (E), the group of automorphisms of the Courant-Dorfman algebra E. We construct a bundle of vertex algebras VX (E) by the localization construction applied to VE with respect to an extension of a transitive structure on CorX . Further, we also construct a bundle of indecomposable modules MX of VE by the same construction. In these regards, the author follows the marvelous work [6]. There exists a transitive Harish-Chandra pair which is an extension by the tautological Harish-Chandra pair associated to the ring of gl(E)-valued formal power series, where gl(E) is the Lie algebra of derivations of E as an R-module. First, we include a reminder on of the construction of the JX -module JX (E) of jets of a finitely generated projective OX -module, E. Let pi : X × X → X be the projections onto the i-th factors. The k-th sheaf of E-jets is JkX (E) = k+1 ⊗p−1 OX p−1 (p1 )∗ (OX×X /IΔ 2 E). One has that the k-th sheaf of E-jets is a union of 2 its fibres over points x ∈ X, where the fibre of JkX (E) is given by JkX (E)/mkx = Jkx (E) and mx is the ideal sheaf of the point x. The definition shows that JkX (E) is a JkX -module. According to the embedding of OX in its jet algebra s∗ : OX → JkX → JX , one has by tensoring this arrow with E over OX that one may expand sections of E as kjets of sections. In particular, this furnishes a morphism JkX (E) → Jk−1 X (E). A more geometric perspective, perhaps, of the expansion of sections in k-jets is to recognize that the morphism s∗k corresponds to an evaluatution map X × V(E) → JetkX (E). Next, to define jets of sections, we have that, since powers of the ideal sheaf IΔ of the diagonal form a projective system of OX×X -modules, the k-th jet sheaves JkX (E) form a projective system of OX -modules. Hence we define the sheaf of E-jets to be the projective limit JX (E) = lim JkX (E) k

of this projective system. Observe that one may identify a compatibility condition between derivations of fibres and formal vector fields, in the special case E is a Courant-Dorfman algebroid. We construct this extension of Harish-Chandra pairs as an extension of a transitive pair by a tautological pair. Further, we obtain a transitive structure on CorX under the extension. Our last piece of geometry is therefore to construct a (g, K)-structure adopted to the extension, for it will be this Harish-Chandra pair that supports the vertex O-algebra as a Harish-Chandra module. Ergo, we may localize VE along the same. Let us describe the extension of Harish-Chandra pairs. Let E be the CourantDorfman algebroid corresponding to the Lie-Rinehart bialgebroid (L, L∨ ), as above. Consider the Lie algebra wE of derivations of Jx (E) consisting of derivations χ such that the image of bracketed sections of E interpolates the derivation, χ([e, e ]) = [χ(e), e ] + [e, χ(e )] and a formal vector field ξ ∈ w interpolating the symmetric bilinear form of sections of E, viz. ξ(e|e ) =  χ(e) | e +  e | χ(e )

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This consideration is analogous to considering the derivations of Jx , that is, the Lie algebra w. The formal vector field ξ is uniquely determined by χ by the definition of the derivation of a Courant-Dorfman algebra and so we obtain the following compatibility condition of such formal vector fields and such derivations of Jx (E), viz. χ(f e) = ξ(f )e + f χ(e) for any Ox -linear morphism χ : Jx (E) → Jx (E) interpolating the bracket of sections and ξ ∈ w interpolating the bilinear form. Let wJ ⊂ w be the Lie subalgebra of formal vector fields consisting of elements ξ ∈ w satisfying this compatibility condition. This Lie algebra is specifically obtained as the image of the the morphism of Lie algebras of derivations of the Courant-Dorfman algebroid Ξ : DerCD (E) → TX together with the map of TX to its fibre, for one has Tx → w. The Lie algebra wE consists of pairs of Ox -linear morphisms (ξ, χ) satisfying the above compabitibility condition. This Lie algebra is obtained as a semi-direct product of the Lie algebras wJ and Ox ⊗ gl(E) := O(e), wE = O(e)  wJ . The semi-direct product is formed by the action given by formal vector fields acting on formal gl(E)-valued power series O(e), viz. [ξ1 + χ1 , ξ2 + χ2 ] = [ξ1 , ξ2 ] + [χ1 , χ2 ] + ξ2 (χ1 ) − ξ1 (χ2 )) where χi ∈ O(e) and ξi ∈ w. Notice, such pairs do not necessarily preserve the ideal m ⊗ Ex ⊂ Jx (E), as wJ does not necessarily preserve m. Restricting to w0 ∩ wJ = wJ,0 ⊂ wJ , such pairs preserve the ideal m ⊗ Ex , as wJ,0 preserves m. Denote the Lie subalgebra that preserves m ⊗ Ex by wE,0 . This Lie subalgebra is an ideal since w0 is fundamental. Moreover, it is a pronilpotent Lie subalgebra for it is obtained by restriction of wJ to w0 . Denote the corresponding pro-unipotent group by Aut(E, O). The group Aut(E, O) consists of automorphisms of the fibre Jx (E) preserving the ideal mx ⊗ Ex that are compatible with invertible Ox -linear endomorphisms ψx of Jx (E) and elements of Aut(O), ψx (f e) = φx (f )ψx (e) for f ∈ Ox and e ∈ Jx (E). By construction, we have the following lemma. Lemma 4.1. The pair (wJ , Aut(J)) form a transitive Harish-Chandra pair. Therefore, by the lemma, the space GSX is the reduction of the structure group of CorX to the subgroup Aut(J) ⊂ Aut(O). Then, by the lemma, GSX is a transitive structure on CorX . This gives us the transitive structure on CorX . To obtain an extension of GSX , we require both a Harish-Chandra pair and morphism of the same to (wJ , Aut(J)). This point is the content of the following theorem. Theorem 4. The pair (wE , Aut(O, E)) is a Harish-Chandra pair and the sequence of Harish-Chandra pairs 1 → (O(e), Aut(O(e)) → (wE , Aut(O, E)) → (wJ , Aut(J)) → 1 is exact. One notes that since pairs in wE,0 are Ox -linear and preserve the ideal m ⊗ E ⊂ Jx (E), therefore they act on the quotient Jx (E)/mx ⊗ Ex ∼ = Ox by derivations. Moreover, the Lie subalgebra wE,0 is of finite codimension in wE since wE /wE,0 ∼ = gln ⊕gl(E). Hence, by the realization theorem, there exists morphism of Lie algebras

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& : wE → w such that &−1 (wJ,0 ) = wE,0 , as the realization morphism factors through wJ . Thus, the kernel of & is O(e). Since &−1 (wJ,0 ) = wE,0 , we obtain a corresponding morphism of pro-unipotent groups Φ : Aut(O, E) → Aut(J) with kernel Aut(O(e)). Therefore, the sequence is exact. In particular, we obtain a morphism of Harish-Chandra pairs Φ : (wE , Aut(E, O)) → (wJ , Aut(J)). There exists a (wE , Aut(O, E))-structure corresponding to this Harish-Chandra pair and an analogue of formal coordinates with respect to the the topological module JX (E). Consider U → X an etale neighborhood of x and a system of formal coordinates φX ; in addition, a choice 7 of frame for E say, {e1 , . . . , en }, so n that ψE : JX (E)(U ) ∼ = OX (U )[[v1 , . . . , vn ]] ⊗ i=1 κei and ψE is a trivialization compatible with φX as a morphism of JX -modules, that is, ψE (f m) = φX (f )ψE (e) for f ∈ JX and e ∈ JX (E). We say ψE is a system of formal E-coordinates on X. Define the (g, K)-structure of systems of formal E-coordinates under the HarishChandra extension (wE , Aut(E, O)), as follows. Denote by X the X-scheme of 7Vor n pairs (φx , ψx ) identifying Jx with Ox and Jx (E) with Ox ⊗ j=1 κej such that ψx is compatible with φx as an Ox -module morphism. One can show the this definition determines a functor represented by a formal scheme over X. In addition, one has a theorem. Theorem 5. The X-scheme VorX is a (wE , Aut(E, O))-structure extending the transitive structure GSX on CorX The relevance of this space to the geometric perspective on vertex algebras is that the associated bundle localizes VE in the sense of Beilinson and Bernstein [2]. Indeed, VE is a (wE , Aut(O, E))-module, for both wE → DerV A (VE ) and Aut(E, O) → AutV A (VE ). To see this, one proceeds by the result of [43] cited above. Restrict to the grading preserving automorphisms of VE , given by AutCD (E). Then, it follows from Weierstrass’ preparation theorem [37] that a compatible φx corresponds to a unit of R. Furthermore, an Ox -linear isomorphism induces an automorphism of E as an R-module, by a choice of a free basis. Taken together, one has a pair of an automorphism of R and an R-module automorphism of E and such a pair furnishes an automorphism of the Courant-Dorfman algebra E, that is, an element of AutCD (E). Therefore, one has Aut(E, O) → AutCD (E) ⊂ AutV A (VE ). Similarly, one is able to show wE → DerV A (VE ). The compatibility of these actions is obtained by restriction, which is the condition to be a HarishChandra module. Therefore, VE is a (wE , Aut(O, E))-module. Combining results, we have the following theorem. Theorem 6. Let VE be the vertex O-algebra and VorX the (wE , Aut(O, E))structure defined above. Given the Harish-Chandra module structure on the vaccuum algebra in the above theorem, one has that the associated vector bundle VX (E) := VorX ×Aut(E,O) VE is an OX -module. Moreover, it is flat according to the existence of the connection given by the principal wE -structure on VorX . Finally, we have our main result in the following theorem. Denote the Zhu algebra of VE by A(E) and the projection of VE onto the ideal VE  ðVE by πA . Recall further that for a Lie-Rinehart algebra L, we have its universal enveloping algebroid, UR L. The universal enveloping algebroid UR L enjoys a universal mapping property that we employ to obtain a surjective map of associative κ-algebras

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between UR L and A(E). We obtain a module whose weight 0 space is a module for the Lie-Rinehart algebra, L. In general, there exist indecomposable modules for L, and as such, the corresponding representation of VE is weak. Proposition 1. Let L be the Lie-Rinehart algebra whose double is E as a Courant-Dorfman algebra. Equivalently, the R-submodule of V1E . Given a module N  over A(E), there exists a module M over VE whose weight 0 space M 0 = N , where N is the pull-back to UR L of N  , that is, an L-module. Consider the Lie-Rinehart algebra L and the surjective map of Lie-Rinehart algebras L → E/R∂R ∼ = V1E /V0E ðV0E . Given dL (R) ⊂ L, notice L → L/dL (R) → E/dL (R), so since dL (R) ⊂ R∂R, we E/dL (R) → E/R∂R a surjective map of LieRinehart algebras. The isomorphism E/R∂R ∼ = V1E /V0E  ðV0E is demonstrated 0 0 in [1]. Therefore, since one has VE  ðVE ⊂ VE  ðVE , there exists a morphism preserving brackets L → V1E /V0E  ðV0E → VE /VE  ðVE = A(E) Next, one observes by a calculation that V0E ∩ VE  ðVE = 0 since a  ðb does not have a degree 0 compotent for any a, b ∈ VE . So the projection VE → A(E) is injective on V0E and is a morphism of κ-algebras. Consequently, by the universal mapping property of UR L there exists a map α : UR L → A(E). Since VE generated by R ⊕ E and A(E) is generated by 1 and πA (R ⊕ E) = Span{πA (rj · e1−n1 · · · ek−nk |0 }, where r ∈ R, ei ∈ E and j ≤ −2, the morphism α is a surjection. Let N  be a module over the Zhu algebra A(E), and N be the pullback of N  under α to UR L. Then N is a UR L-module, and so, by definition, an L-module cf. [2]. The quotient map R ⊕ L → c0 = R ⊕ E/∂R furnishes both a map κ-Lie algebras L → c0 ⊂ c and κ-algebras R → c0 ⊂ c. Consequently, again, by the universal mapping property of UR L, there exists a morphism β : UR L → c. Indeed, U(c) let c≤0 act trivially for c |z2 | > 0 and |z2 | > |z1 − z2 | > 0, respectively, to a common rational function in z1 , z2 with the only possible poles at z1 = 0, z2 = 0 and z1 = z2 . Another main axiom is the commutativity (the analogue of the commutativity for a commutative associative algebra in a more subtle sense). Roughly speaking, it says that for u1 , u2 ∈ V , we require that the rational functions obtained by analytically extending u , Y (u1 , z1 )Y (u2 , z2 )u3 and

u , Y (u2 , z2 )Y (u1 , z1 )u3 are the same. In addition, we have the LV (−1)-derivative property d Y (u, z) = Y (LV (−1)u, z) dz for v ∈ V , where LV (−1)v = lim YV (v, z)1 x→0

for v ∈ V . This property means that LV (−1) corresponding to the derivative or infinitesimal translation of the variable z) For a M¨obius vertex algebra, let LV (1), LV (0), LV (−1) be the operators giving the sl2 -module structure. Then LV (−1)v = lim YV (v, z)1 z→0

for v ∈ V , LV (0) is the operator giving the grading of V and [LV (1), YV (v, z)] = YV (LV (1)v, z) + 2zYV (LV (0)v, z) + z 2 YV (LV (−1)v, z). For a vertex operator algebra, we have the axioms for the conformal symmetry. For example, if we let LV (n) = Resz z n+1 YV (ω, z), then c [LV (m), LV (n)] = (m − n)LV (m + n) + (m3 − m)δm+b,0 12 (the Virasoro relation). Also the Virasoro operators LV (1), LV (0) and LV (−1) should be the same as those operators for the underlying M¨obius vertex algebra. A module for a vertex operator algebra V is roughly speaking a C-graded vector space and a vertex operator map YW : V ⊗ W → W [[z, z −1 ]] satisfying all the axioms that still make sense. For three modules W1 , W2 and W3 , an intertwining 3 operator of type WW is a linear map Y : W1 ⊗ W2 → W3 {z}[log z] (here W3 {z} 1 W2 means series in complex powers of z with coefficients in W3 ) satisfying all axioms for modules that still make sense.

3  The intertwining operators of type WW form a vector space. This is in fact 1 W2 the space of conformal blocks on the Riemann sphere with three marked points labeled with the equivalence classes of the modules W1 , W2 and W3 . Its dimension is called a fusion rule. Intertwining operators were called chiral vertex operators in [MS] and were introduced mathematically in [FHL]. If a set of modules for a vertex operator algebra equipped with subspaces of intertwining operators among

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modules in this set satisfying the associativity and commutativity, we call it an intertwining operator algebra (see [H3], [H5] and [H7] and see also [DL] for a special type of intertwining operator algebras called abelian intertwining algebras). Here by associativity of intertwining operators, we mean, roughly, for any intertwining operators Y1 and Y2 , there exist intertwining operators Y3 and Y4 such that for w1 , w2 , w3 and w4 in suitable modules, w4 , Y1 (w1 , z1 )Y2 (w2 , z2 )w3 and w4 , Y3 (Y4 (w1 , z1 − z2 )w2 , z2 )w3 are absolutely convergent in the regions |z1 | > |z2 | > 0 and |z2 | > |z1 − z2 | > 0, respectively. Moreover, these functions can be analytic extended to a common multivalued analytic function with the only possible singular points at z1 = 0, z2 = 0 and z1 = z2 . The associativity of intertwining operators is equivalent to the operator product expansion of chiral vertex operators, one of the two major assumptions or conjectures in the important work [MS] of Moore and Seiberg. Mathematically it was first introduced and proved under suitable conditions in [H1]. By commutativity of intertwining operators, we mean, roughly, for any intertwining operators Y1 and Y2 , there exist intertwining operators Y3 and Y4 such that w4 , Y1 (w1 , z1 )Y2 (w2 , z2 )w3 and w4 , Y3 (w2 , z2 )Y4 (w1 , z1 )w3 are analytic extensions of each other. The commutativity of intertwining operators is an easy consequence of the associativity and skew-symmetry of intertwining operators (see [H7]). Intertwining operator algebras give vertex tensor category structures which in turn give braided tensor category structures. Vertex tensor categories can be viewed as analogues of symmetric tensor categories with their tensor product bifunctors controlled by Riemann surfaces with three punctures and local coordinates vanishing at the punctures. See [HL2], [H14] and [HLZ9] for details. Intertwining operator algebras are equivalent to chiral genus-zero conformal field theories (see [H5]). In the program to construct conformal field theories using the representation theory of vertex operator algebras, after the first step of constructing a vertex operator algebra and studying its modules are finished, the second step of proving the associativity (or operator product expansion) of intertwining operators is equivalent to constructing an intertwining operator algebra. To prove the associativity, the main properties that need to be established first are a convergence and extension property for products of an arbitrary number of intertwining operators (see [H11] in the rational case, [HLZ8] for the adaption of the proof in [H11] in the logarithmic case and [Y] for a generalization to vertex algebras with infinite-dimensional homogeneous subspaces with respect to conformal weights but with finite-dimensional homogeneous subspaces with respect to an additional horizontal grading) and a property stating that suitable lower-bounded generalized modules are in the category of modules that we start with (see [H1] for the rational case, [HLZ8] and [H20] for the generalization to the logarithmic case). The associativity is proved using these properties (see [H1] for the rational case and [HLZ7] for the generalization to the logarithmic case).

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The next step is to construct the chiral genus-one conformal field theories. The main properties that need to be established are the convergence and analytic extensions of q-traces or pseudo-q-traces of products of geometrically-modified intertwining operators (see [H12] for the rational case and [F1] and [F2] for the generalization to the logarithmic case) and the modular invariance of the analytic extensions of these q-traces or pseudo-q-traces (see [H12] for the rational case). The genus-one associativity and commutativity are easy consequences of the convergence and analytic extensions of these q-traces or pseudo-q-traces (see [H12] and [F1] and [F2] for the generalization to the logarithmic case). The main open problem in the construction of chiral higher-genus rational conformal field theories is the convergence of multi q-traces of products of geometricallymodified intertwining operators. The invariance under the action of the mapping class groups is an easy consequence of this convergence, the associativity of intertwining operators and the modular invariance of the q-traces of products of geometrically-modified intertwining operators. The future solution of this problem will depend on the further study of the moduli space of Riemann surfaces with parametrized boundaries, including in particular the study of a conjecture by the author on meromorphic functions on this moduli space. See [RS1]–[RS3] and [RSS1]–[RSS5] for results on this moduli space. Another problem is the construction of locally convex topological completions of modules for the vertex operator algebra such that intertwining operators and higher-genus correlation functions give maps between these completions. Such completions of vertex operator algebras and their modules using only the correlation functions given by the algebras and modules were given in [H6] and [H9]. If we assume the convergence of multi q-traces of products of geometrically-modified intertwining operators discussed above, then the same method as in [H6] and [H9] works when we add those elements coming from genus-zero and genus-one correlation functions obtained using intertwining operators. The author conjectured that these completions obtained using all genus-zero and genus-one correlation functions are the same as the Hilbert space completions if the chiral conformal field theory is unitary (see [H25]). The discussions above are about the construction of chiral conformal field theories. We also need to construct full conformal field theories and open-closed conformal field theories. To construct a genus-zero full rational conformal field theory, the main property that needs to be proved is the nondegeneracy of an invariant bilinear from on the space of intertwining operators. This nondegeneracy in fact needs a formula used by the author to prove the Verlinde formula in [H13] or equivalently the rigidity of the braided tensor category of modules for the vertex operator algebra proved in [H14]. The construction of genus-one and higher-genus full conformal field theories can then be obtained easily from genus-zero full conformal field theories and chrial genus-one and higher-genus conformal field theories (see [HK2] and [HK3]). Finally, to construct open-closed conformal field theories, one needs to construct open-string vertex operator algebras from intertwining operator algebras (see [HK1]). Then one has to prove that with the choices of the open-string vertex algebra (the open part) and the full conformal field theory (the closed part), Cardy’s condition on the compatibility between the closed part and open part is satisfied (see [Ko1] and [Ko2]).

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From the discussion above, we see that intertwining operators are the main objects to study in conformal field theory and also in the representation theory of vertex operator algebras. In fact, conformal field theory is essentially the theory of intertwining operators. Therefore intertwining operators with all their properties can be viewed as a working definition of chiral conformal field theory. Constructing a chiral conformal field theory is the same as proving all the properties of intertwining operators. This will also be our approach in this paper to the construction of orbifold conformal field theories. Before we end this section, the author would like to correct some misunderstandings about conformal field theories and vertex operator algebras because for a long time, opinions formed based on these and other misunderstandings have often been used mistakenly by journals, organizations and the mathematical community to evaluate researches in this area. The first misunderstanding is about the role of vertex operator algebras in conformal field theory. A vertex operator algebra in general is certainly not even a chiral conformal field theory. This can be seen easily from the modular invariance of the space of characters of integrable highest weight modules for affine Lie algebras (see [Ka]) and from Zhu’s modular invariance theorem (see [Z]) on q-traces of vertex operators acting on modules for a vertex operator algebra. Also many poweful methods used to study vertex operator algebras do not work for intertwining operators. For example, since vertex operator algebras involve only rational functions, the method of multiplying a polynomial to cancel the denominator of a rational function works very well. But this method in general does not work for products of at least two intertwining operators. Also for rational functions, one can use the global expressions of rational functions instead of analytic extensions but for multivalued functions obtained from products of at least two intertwining operators, one has to carefully use analytic extensions to obtain the correct results. In fact, fatal mistakes occurred in papers published in major mathematical journals claiming to simplify major results on intertwining operators without using complex analysis exactly because the methods that work only for vertex operator algebras were applied to the study of products of two intertwining operators. One mistake is to assume that intertwining operators involve only integral powers of the variable, which is not even true for non-self-dual lattice vertex operator algebras, the simplest minimal model of central charge 12 and the simplest Wess-Zumino-Witten models for the Lie algebra sl2 . Another mistake is to define maps without using analytic extensions. When working with multivalued analytic functions (not rational functions), one has to use analytic extensions to define a number of maps. Without a careful use of analytic extensions, one cannot even prove that an arbitrarily defined map is linear, not to mention many other properties that these maps should satisfy. Another more subtle but also more important fact is that a vertex operator algebra in general does not even determine a chiral conformal field theory uniquely. Instead, it is in fact the choices of modules and intertwining operators that determine uniquely such a theory. Therefore to construct a chiral conformal field theory, though we need to start with a vertex operator algebra, the more crucial part is to choose a category of modules and spaces of intertwining operators. One simple example is the chiral conformal field theory associated to irrational tori. The vertex operator algebra for such a chiral conformal field theory is a Heisenberg vertex operator algebra, which is also the vertex operator algebra for the conformal

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field theory associated to the corresponding Euclidean space. This vertex operator algebra alone does not lead us to a unique chiral conformal field theory since all different irrational tori and the Euclidean space in the same dimension has the same vertex operator algebra. We have to choose the category of modules for this Heisenberg vertex operator algebra to be the category of finite direct sums of irreducible modules generated by the eigenfunctions of the Laplacian on the given irrational torus and the space of intertwining operators among the irreducible modules to be the spaces of all intertwining operators among these modules. Then we obtain the chiral conformal field theory associated to this particular irrational torus. The second misunderstanding is about modular functors and modular tensor categories. Modular functors are operads formed by holomorphic vector bundles over the moduli space of Riemann surfaces with parametrized boundaries satisfying certain additional conditions and chiral conformal field theories are algebras over such operads satisfying additional conditions (see [S] and also [H3] and [H5] for the genus zero case). But modular functors themselves are not conformal field theories. To construct a chiral conformal field theory, one has to construct a modular functor with a nondegenrate bilinear form. But a modular functor with a nondegenrate bilinear form alone does not give a conformal field theory. Instead, a modular functor with a nondegenrate bilinear form gives a three-dimensional topological field theory. Similarly, modular tensor categories also give only three-dimensional topological field theories and is equivalent to modular functors with nondegenerate bilinear forms. They are far from conformal field theories. Certainly modular functors and modular tensor categories are very useful in the study of conformalfield-theoretic structures. But this happens only when we already proved a lot of results, for example, the convergence, associativity (operator product expansion) and modular invariance, about intertwining operators or some equivalent structures. For example, the modular tensor categories for the Wess-Zumino-Witten models can be constructed using representations of quantum groups. The fusion coefficients of these tensor categories are indeed given by the Verlinde formula. But these modular tensor categories do not give us the convergence, associativity (operator product expansion) and modular invariance for intertwining operators. The third misunderstanding is about the history of intertwining operator algebras. Original papers, books and proposals on intertwining operator algebras have often been rejected based on wrong claims that some much later or nonexistent mathematical works already had this notion or some results before intertwining operator algebras were actually introduced and studied. To correct this misunderstanding, we give a brief history of intertwining operator algebras here. (To be accurate, some of the years below are the years that the papers appeared in the arXiv, not the years that they were published.) In 1984, operator product expansion of chiral conformal fields was studied by Belavin, Polyakov and Zamolodchikov in [BPZ]. In 1988, by assuming that two major conjectures—the operator product expansion and modular invariance (certainly including implicitly the corresponding convergence) of chiral vertex operators (equivalent to intertwining operators in mathematics)—hold, Moore and Seiberg derived a set of polynomial equations which corresponds to a modular tensor category in the sense of Turaev [T1] and obtained the Verlinde formula [V] as a consequence. In 1992, Dong and Lepowsky in [DL] introduced a special type of intertwining operator algebras called abelian intertwining algebras for which the corresponding

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braid group representations are one dimensional and gave examples constructed from lattices. In 1995, the author in [H1] formulated and proved the associativity of intertwining operators assuming that a convergence and extension property for intertwining operators and another algebraic condition hold. In particular, the operator product expansion of intertwining operators was proved assuming these conditions. At the same time in 1995, the author in [H2] proved the convergence and extension property and the other algebraic condition needed in [H1] for minimal models. In the same year, the author introduced in [H3] the mathematical notion of intertwining operator algebra using the associativity of intertwining operators and discussed its role in the construction of conformal field theories in the sense of Kontsevich and Segal [S]. In 1997 the author proved a generalized rationality and a Jacobi identity for intertwining operator algebras. In the same year, Lepowsky and the author in [HL6] proved the convergence and extension property and the other algebraic condition needed in [H1] for the Wess-Zumino-Witten models. Also in the same year, the author constructed genus-zero modular functors from intertwining operator algebras and proved that intertwining operator algebras are algebras over the partial operads of such genus-zero modular functors. In 1999 and 2000, Milas and the author in [HM1] and [HM2] proved the convergence and extension property and the other algebraic condition needed in [H1] for the Neveu-Schwarz sectors of N =1 and N = 2 superconformal minimal models, respectively. In 2001, the author in [H10] introduced a notion of dual of an intertwining operator algebra analogous to the dual of a lattice and the dual of a code such that the dual of a vertex operator algebra satisfying suitable conditions is the intertwining operator algebra obtained from all intertwining operators among all irreducible modules. In 2002, the author proved the convergence and extension property and the other condition needed in [H1] for vertex operator algebras for which irreducible modules are C1 -cofinite (weaker than C2 -cofnite) and N-gradable weak modules are completely reducible. The convergence and extension property proved in this paper in fact also holds for vertex operator algebras for which grading-restricted generalized modules are C1 cofinite but might not be completely reducible (see [HLZ8] for a discussion about this fact). In 2003, the author proved the modular invariance for intertwining operator algebras on the direct sum of all (inequivalent representatives of) irreducible modules for a vertex operator algebra satisfying natural finite reductivity conditions, including in particular the C2 -cofniteness condition and the complete reducibility of N-gradable weak modules. A number of these results on intertwining operator algebras were generalized to the logarithmic and other cases by Lepowsky, Zhang and the author in [HLZ1]–[HLZ9], by the author in [H15] and [H20], by Fiordalisi in [F1] and [F2] and by Yang in [Y]. 3. Basic open problems and conjectures in orbifold conformal field theory In this section, we discuss some basic open problems and conjectures in orbifold conformal field theory. Roughly speaking, given a conformal field theory and a group of automorphisms of this theory, we would like to construct a conformal field theory that can be viewed as the original conformal field theory divided by this group of automorphisms. To formulate this notion precisely, we assume that the given conformal field theory

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is constructed using the representation theory of vertex operator algebras, as is discussed in the preceding section, and the group of automorphisms is a group of automorphisms of the vertex operator algebra. Here is the main open problem on the construction of orbifold conformal field theories: Problem 3.1. Given a vertex operator algebra V and a group G of automorphisms of V , construct and classify all the conformal field theories whose vertex operator algebras contain the fixed point vertex operator algebra V G as subalgebras. It is in fact very difficult to study V G -modules and intertwining operators among V G -modules. On the other hand, twisted V -modules and twisted intertwining operators among twisted V -modules are analogues of V -modules and intertwining operators among V -modules, it is easier to study these than V G -modules and intertwining operators among them. We expect that V G -modules can all be obtained from twisted V -modules (see Theorem 7.1 in [H24] and Theorem 4.8 below for lower-bounded V G -modules in the case that G is the cyclic group generated by an automorphism g of V ). Thus our conjectures and problems below will be mainly on twisted intertwining operators among twisted modules. Now we state the first main conjecture on orbifold conformal field theories (see [H19]): Conjecture 3.2. Assume that V is a simple vertex operator algebra satisfying the following conditions: (1) V(0) = C1, V(n) = 0 for n < 0 and the contragredient V  , as a V -module, is equivalent to V . (2) V is C2 -cofinite, that is, dim V /C2 (V ) < ∞, where C2 (V ) is the subspace of V spanned by the elements of the form Resx x−2 Y (u, x)v for u, v ∈ V and Y : V ⊗ V → V [[x, x−1 ]] is the vertex operator map for V . (3) Every grading-restricted generalized V -module is completely reducible. Let G be a finite group of automorphisms of V . Then the twisted intertwining operators among the g-twisted V -modules for all g ∈ G satisfy the associativity, commutativity and modular invariance property. The following conjecture (see also [H19]) is a consequence of Conjecture 3.2 (cf. Example 5.5 in [Ki]): Conjecture 3.3. Let V be a vertex operator satisfying the three conditions in Conjecture 3.2 and let G be a finite group of automorphisms of V . The the category of g-twisted V -modules for all g ∈ G is a G-crossed modular tensor category (see Turaev [T2] for G-crossed tensor categories). These two conjectures given in [H19] are both under the complete reducibility assumption and are also about finite groups of automorphisms of V . In the case that grading-restricted generalized V -modules are not complete reducible or G is not finite, we have the following conjectures and problems: Conjecture 3.4. Let V be a vertex operator algebra satisfying the first two conditions in Conjecture 3.2 and let G be a finite group of automorphisms of V . Then the twisted intertwining operators among the grading-restricted generalized gtwisted V -modules for all g ∈ G satisfy the associativity, commutativity and modular invariance properties.

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Conjecture 3.5. Let V be a vertex operator algebra satisfying the first two conditions in Conjecture 3.2 and let G be a finite group of automorphisms of V . Then the category of grading-restricted generalized g-twisted V -modules for all g ∈ G has a natural structure of G-crossed tensor category satisfying additional properties. Problem 3.6. Let V be a vertex operator algebra and let G be a group of automorphisms of V . If G is an infinite group, under what conditions do the twisted intertwining operators among the grading-restricted generalized g-twisted V -modules for all g ∈ G satisfy the associativity, commutativity and modular invariance properties? Under what conditions is the category of g-twisted V -modules for all g ∈ G a G-crossed (tensor) category? Remark 3.7. In the case of G = {1}, Conjecture 3.2 is a theorem (see [H11] and [H12]). From this theorem, the author constructed a modular tensor category (see [H14]) and thus Conjecture 3.3 is also a theorem in this case. In the remaining part of this paper, we describe in details the construction and study of twisted modules, the basic and conjectural properties of twisted intertwining operators and some thoughts of the author on further developments of orbifold conformal field theory. 4. Twisted modules, a general construction and existence results To construct orbifold conformal field theories, we first have to understand the structures and properties of twisted modules. Twisted modules associated to automorphisms of finite orders of vertex operator algebras appeared first in the works of Frenkel-Lepowsky-Meurman [FLM3] and Lepowsky [Le]. In [H16], the author introduced twisted modules associated to automorphisms of arbitrary orders (including in particular, infinite orders). In the case of automorphisms of infinite orders, the logarithm of the variable might appear in twisted vertex operators. Here we recall g-twisted modules and their variants from [H16]. Definition 4.1. Let V be a grading-restricted vertex algebra or a vertex operator ! algebra. A generalized g-twisted V -module is a C-graded vector space W = n∈C W[n] (graded by weights) equipped with a linear map g :V ⊗W YW v⊗w

→ →

W {x}[log x], g (v, x)w YW

satisfying the following conditions: (1) The equivariance property: For p ∈ Z, z ∈ C× , v ∈ V and w ∈ W , g p+1 g p ) (gv, z)w = (YW ) (v, z)w, (YW g p g ) (v, z) is the p-th analytic branch of YW (v, x). where for p ∈ Z, (YW g (2) The identity property: For w ∈ W , Y (1, x)w = w. (3) The duality property: For any u, v ∈ V , w ∈ W and w ∈ W  , there exists a multivalued analytic function with preferred branch of the form

f (z1 , z2 ) =

N  i,j,k,l=0

aijkl z1mi z2 j (logz1 )k (logz2 )l (z1 − z2 )−t n

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for N ∈ N, m1 , . . . , mN , n1 , . . . , nN ∈ C and t ∈ Z+ , such that the series g p g p w , (YW ) (u, z1 )(YW ) (v, z2 )w , g p g p w , (YW ) (v, z2 )(YW ) (u, z1 )w , g p w , (YW ) (YV (u, z1 − z2 )v, z2 )w

are absolutely convergent in the regions |z1 | > |z2 | > 0, |z2 | > |z1 | > 0, |z2 | > |z1 − z2 | > 0, respectively, and their sums are equal to the branch f p,p (z1 , z2 ) =

N 

aijkl emi lp (z1 ) enj lp (z2 ) lp (z1 )k lp (z2 )l (z1 − z2 )−t

i,j,k,l=0

of f (z1 , z2 ) in the region |z1 | > |z2 | > 0, the region |z2 | > |z1 | > 0, the region given by |z2 | > |z1 − z2 | > 0 and | arg z1 − arg z2 | < π2 , respectively. (4) The L(0)-grading condition and g-grading condition: Let LgW (0) = [α] g Resx xYW (ω, x). Then for n ∈ C and α ∈ C/Z, w ∈ W[n] , there exists K, Λ ∈ Z+ such that (LgW (0) − n)K w = (g − e2παi )Λ w = 0. (5) The L(−1)-derivative property: For v ∈ V , d g g Y (v, x) = YW (LV (−1)v, x). dx W A lower-bounded generalized g-twisted V -module is a generalized g-twisted V -module satisfying the condition W[n] = 0 for &(n) sufficiently negative. A grading-restricted generalized g-twisted V -module is a lowerbounded generalized g-twisted V -module satisfying in addition the condition dim W[n] < ∞ for n ∈ C. An ordinary g-twisted V -module or simply a g-twisted V -module is a grading-restricted generalized g-twisted V -module such that LW (0) acts on W semisimply. In [Li], Li studied twisted modules associated to automoprhisms of finite order of a vertex operator algebra using the method of weak commutativity and applied this method to such twisted modules for vertex operator (super)algebras obtained from infinite-dimensional Lie (super)algebras. In [Ba], Bakalov gave a Jacobi identity for twisted modules associated to automorphisms of possibly infinite order, which can be used to replace the duality property in the definition above. See [HY] for a proof. In the representation theories of associative algebras and of Lie algebras, free modules and Verma modules, respectively, play fundamental roles. For vertex operator algebras, though there had been constructions of twisted modules for some examples of vertex operator algebras, for more than thirty years, no twisted modules analogous to free modules and Verma modules were constructed. In 2019, the author successfully constructed such analogues. These are lower-bounded generalized g-twisted V -modules satisfying a universal property. For the construction in the case that V is a grading-restricted vertex algebra, we refer the reader to [H23]. A minor modification of the construction in [H23] gives such a universal lower-bounded generalized twisted V -modules in the case that V is a vertex operator algebra (see [H26]). Here we state and discuss the existence results and some properties, including in particular the universal property, of lower-bounded and grading-restricted generalized g-twisted V -modules from [H23] and [H24].

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Theorem 4.2 ([H23]). Let M be a vector space with actions of g and an operator LM (0). Assume that there exist operators Lg , Sg , Ng such that on M , g = e2πiLg and Sg and Ng are the semisimple and nilpotent, respectively, parts of Lg . Also assume that LM (0) can be decomposed as the sum of its semisimple part LM (0)S and nilpotent part LM (0)N and that the real parts of the eigenvalues of LM (0) has a lower bound. Let B ∈ R such that B is less than or equal to the real parts of the eigenvalues of LM (0) on M . Then there exists a lower-bounded +[g] satisfying the following universal property: Let generalized g-twisted V -module M B g (W, YW ) be a lower-bounded generalized g-twisted V -module such that W[n] = 0 when &(n) < B and let M0 be a subspace of W invariant under the actions of g, Sg , Ng , LW (0), LW (0)S and LW (0)N . Assume that there is a linear map f : M → M0 commuting with the actions of g, Sg , Ng , LM (0) and LW (0)|M0 , LM (0)S and LW (0)S |M0 and LM (0)N and LW (0)N )|M0 . Then there exists a unique module map +[g] → W such that fˆ|M = f . If f is surjective and (W, Y g ) is generated by fˆ : M B W g W the coefficients of (Y g )W W V (w0 , x)v for w0 ∈ M0 and v ∈ V , where (Y )W V is the g twist vertex operator map obtained from YW , then fˆ is surjective. +[g] was given in [H23] for An explicit construction, not just the existence, of M B a grading-restricted vertex algebra V . One crucial ingredient in this construction is the twist vertex operators introduced and studied in [H22]. In the case that V is a vertex operator algebra, see Subsection 4.1 of [H26]. The lower-bounded gen+[g] is also unique up to equivalence by the universal eralized g-twisted V -module M B property. One immediate consequence of Theorem 4.2 is the following result: g Corollary 4.3 ([H23]). Let (W, YW ) be a lower-bounded generalized g-twisted g W V -module generated by the coefficients of (Y g )W W V (w, x)v for w ∈ M , where (Y )W V g is the twist vertex operator map obtained from YW and M is a Z2 -graded subspace of W invariant under the actions of g, Sg , Ng , LW (0), LW (0)S and LW (0)N . Let B ∈ R such that W[n] = 0 when &(n) < B. Then there is a generalized g-twisted +[g] such that W is equivalent as a lower-bounded generalized V -submodule J of M B +[g] /J. g-twisted V -module to the quotient module M B

+[g] has been used in [H24] to solve some open problems The construction of M B of more than twenty years. We now discuss these results. These results are proved in [H24] when V is a grading-restricted vertex algebra or a M¨obius vertex algebra. But by using the minor modification in [H26] of the construction of universal lowerbounded generalized g-twisted V -modules in the case that V is a vertex operator algebra, the same proofs also work in this case. The first such open problem is the existence of nonzero lower-bounded generalized g-twisted V -modules. Note that since V itself is a V -module, the existence of nonzero V -modules is obvious. But it is highly nontrivial why nonzero lowerbounded generalized g-twisted V -modules exist. Assuming that the vertex operator algebra is simple and C2 -cofinite and the automorphism is of finite order, Dong, Li and Mason proved the existence of an irreducible twisted module [DLM2]. But no progress has been made in the general case for more than twenty years. On the other hand, some classes of vertex operator algebras that are not C2 -cofinite have a very rich and exciting representation theory. For example, vertex operator algebras associated to affine Lie algebras at admissible levels are not C2 -cofinite.

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But the category of ordinary modules for such a vertex operator algebra has a braided tensor category structure with a twist (see [CHY]), which is also rigid and in many cases even has a modular tensor category structure (see [CHY] for the conjectures and a proof in the case of sl2 and see [C] for a proof of the rigidity in the simply-laced case). It is important to study vertex operator algebras that are not C2 -cofinite. From Theorem 4.2, fora general grading-restricted vertex algebra V (not necessarily C2 -cofinite) we indeed have constructed lower-bounded generalized g-twisted V -modules. But it is not obvious from the construction in [H23] why +[g] is not 0. In [H24], the author solved this problem completely. M B + Theorem 4.4 ([H24]). The lower-bounded generalized g-twisted V -module M B is not 0. In particular, there exists nonzero lower-bounded generalized g-twisted V modules. [g]

In [DLM1], Dong, Li and Mason generalized Zhu’s algebra A(V ) (see [Z]) to a twisted Zhu’s algebra Ag (V ) for a vertex operator algebra V and an automorphism g of V of finite order. In [HY], Yang and the author introduced twisted zero-mode algebra Zg (V ) associated to V and an automorphism g of V not necessarily of finite order and also generalized the twisted Zhu’s algebra Ag (V ) to the case that g is not of finite order. These two associative algebras are in fact isomorphic (see [HY]). In the case that g is of finite order, Ag = 0 is stated explicitly as a conjecture in the beginning of Section 9 of the arXiv version of [DLM1]. In the case that V is C2 -cofinite and g is of finite order, Dong, Li and Mason proved this conjecture in [DLM2]. But in general, this conjecture had been open until it was proved in [H24] as an immediate consequence of Theorem 4.4. Corollary 4.5 ([H24]). The twisted Zhu’s algebra Ag (V ) or the twisted zeromode algebra Zg (V ) is not 0. Another application of Theorem 4.4 is on the following existence of irreducible lower-bounded generalized g-twisted V -module: Theorem 4.6 ([H24]). Let W be a lower-bounded generalized g-twisted V +[g] when M is a one module generated by a nonzero element w (for example, M B dimensional space spanned by an element w and B is less than or equal to the real part of the weight of w). Then there exists a maximal submodule J of W such that J does not contain w and the quotient W/J is irreducible. Though lower-bounded generalized g-twisted V -modules are important in our study of orbifold conformal field theories, we are mainly interested in gradingrestricted generalized g-twisted V -modules and ordinary g-twisted V -modules, because their double contragredients are equivalent to themselves. One important problem is the existence of irreducible grading-restricted generalized g-twisted V modules and irreducible ordinary g-twisted V -modules. In the case that V is simple, C2 -cofinite and g is of finite order, Dong, Li and Mason proved in [DLM2] the existence of an irreducible ordinary g-twisted V -module. Their proof used genus-one 1-point functions. Thus the simplicity and C2 -cofiniteness of V and the finiteness of the order of g are necessary in their approach. Using our construction of the universal lower-bounded generalized g-twisted V -modules, the author proved in [H24] the existence of irreducible grading-restricted generalized g-twisted V -modules and

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irreducible ordinary g-twisted V -modules under some very weak conditions. In particular, the simplicity and C2 -cofininess of V and the finiteness of the order of g are not needed. Theorem 4.7 ([H24]). Let V be a M¨ obius vertex superalgebra and g an automorphism of V . Assume that the set of real parts of the numbers in P (V ) has no accumulation point in R. If the twisted Zhu’s algebra Ag (V ) or the twisted zeromode algebra Zg (V ) is finite dimensional, then there exists an irreducible gradingrestricted generalized g-twisted V -module. Such an irreducible grading-restricted generalized g-twisted V -module is an irreducible ordinary g-twisted V -module if g acts on it semisimply. In particular, if g is of finite order, there exists an irreducible ordinary g-twisted V -module. The author also proved in [H24] that a lower-bounded generalized module for the fixed-point subalgebra V g of V can be extended to a lower-bounded generalized g-twisted V -module. Theorem 4.8 ([H24]). Let V be a grading-restricted vertex algebra and W0 a lower-bounded generalized V g -module (in particular, W0 has a lower-bounded grading by C). Assume that g acts on W0 and there are semisimple and nilpotent operators Sg and Ng , respectively, on W0 such that g = e2πiLg where Lg = Sg + Ng . Then W0 can be extended to a lower-bounded generalized g-twisted V -module, that is, there exists a lower-bounded generalized g-twisted V -module W and an injective module map f : W0 → W of V g -modules. We have the following open problem: Problem 4.9. Finding conditions on the vertex operator algebra such that under these conditions, the universal lower-bounded generalized g-twisted V -modules have irreducible quotients whose homogeneous subspaces are finite dimensional. Remark 4.10. In the case that V is a grading-restricted vertex algebra or vertex operator algebra associated to an affine Lie algebra, the author solved this problem (see [H26]). Since the results in [H26] are only for special types of examples of grading restricted vertex algebras or vertex operator algebras, we shall not discuss the details here. The interested reader is referred to the paper [H26] for details. 5. Twisted intertwining operators To construct orbifold conformal field theories, one approach is to study the representation theory of fixed point vertex operator algebras. If one can prove that intertwining operators among suitable modules for the fixed-point vertex operator algebra of a vertex operator algebra under a group of automorphisms satisfy the convergence and extension property, the associativity (operator product expansion), the modular invariance property and the higher-genus convergence property as is decribed in Section 2, then one can construct and study the corresponding orbifold conformal field theories using the steps described in Section 2. But it is itself a difficult problem to prove these properties using the properties of intertwining operators among modules for the larger vertex operator algebra. On the other hand, as is mentioned in Section 3, since twisted modules for the larger vertex operator algebra are analogues of (untwisted) modules for the vertex

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operator algebra, we expect that the properties of intertwining operators among twisted modules can be studied by generalizing the results and approach for the intertwining operators among modules. Moreover, we expect that every module for the fixed-point vertex operator algebra can be obtained from a twisted module for the larger vertex operator algebra. In particular, intertwining operators among modules for the fixed-point vertex operator algebra can also be obtained from intertwining operators among twisted modules for the larger vertex operator algebra. Therefore instead of studying intertwining operators among modules for the fixed-point vertex operator algebra, we study intertwining operators among twisted modules for the larger vertex operator algebra. For simplicity, as in [H21], we call intertwining operators among twisted modules twisted intertwining operators. We still need a precise definition of twisted intertwining operator. By generalizing the Jacobi identity for intertwining operators, Xu introduced the notion of intertwining operators among twisted modules associated to commuting automorphisms of finite orders (see [X]). But in general, an orbifold conformal field theory might be associated to a nonabelian group of automorphisms. In particular, we have to introduce and study intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms. Also, the group might not be finite. So we also have to introduce and study intertwining operators among twisted modules associated to automorphisms of infinite orders. The formulation used in [FHL] and [X] cannot be generalized directly to give a definition of intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms. For more than twenty years, no such definition was given in the literature. This is the reason why orbifold conformal field theories associated to nonabelian groups had not been studied much mathematically in the past. In [H21], the author found a formulation of such a notion of twisted intertwining operators associated to not-necessarily-commuting automorphisms of possibly infinite orders and proved their basic properties. The general theory and construction of orbifold conformal field theories associated to nonabelian groups (including infinite groups) can now be started from such operators. We first give the precise definition of twisted intertwining operators. Definition 5.1. Let g1 , g2 , g3 be automorphisms of V and let W1 , W2 and W3 be g1 -, g 2 - and g3 -twisted V -modules, respectively. A twisted intertwining operator 3 of type WW is a linear map 1 W2

Y : W 1 ⊗ W2 w 1 ⊗ w2

→

W3 {x}[log x]

→ Y(w1 , x)w2 =

K  

Yn,k (w1 )w2 x−n−1 (log x)k

k=0 n∈C

satisfying the following conditions: (1) The lower truncation property: For w1 ∈ W1 and w2 ∈ W2 , n ∈ C and k = 0, . . . , K, Yn+l,k (w1 )w2 = 0 for l ∈ N sufficiently large.

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(2) The duality property: For u ∈ V , w1 ∈ W1 , w2 ∈ W2 and w3 ∈ W3 , there exists a multivalued analytic function with preferred branch f (z1 , z2 ; u, w1 , w2 , w3 ) N 

=

s

aijklmn z1ri z2j (z1 − z2 )tk (log z1 )l (log z2 )m (log(z1 − z2 ))n

i,j,k,l,m,n=1

for N ∈ N, ri , sj , tk , aijklmn ∈ C, such that for p1 , p2 , p12 ∈ Z, the series g3 p1 w3 , (YW ) (u, z1 )Y p2 (w1 , z2 )w2 , 3 g2 p1 w3 , Y p2 (w1 , z2 )(YW ) (u, z1 )w2 , 2 g1 p12 w3 , Y p2 ((YW ) (u, z1 − z2 )v, z2 )w 1

are absolutely convergent in the regions |z1 | > |z2 | > 0, |z2 | > |z1 | > 0, |z2 | > |z1 − z2 | > 0, respectively. Moreover, their sums are equal to the branches f p1 ,p2 ,p1 (z1 , z2 ; u, w1 , w2 , w3 ) =

N 

aijklmn z1i eri lp1 (z1 ) esj lp2 (z2 ) etk lp1 (z1 −z2 ) (lp1 (z1 ))l (lp2 (z2 ))m (lp1 (z1 − z2 ))n , r

i,j,k,l,m,n=1

f p1 ,p2 ,p2 (z1 , z2 ; u, w1 , w2 , w3 ) =

N 

aijklmn z1i eri lp1 (z1 ) esj lp2 (z2 ) etk lp2 (z1 −z2 ) (lp1 (z1 ))l (lp2 (z2 ))m (lp2 (z1 − z2 ))n , r

i,j,k,l,m,n=1

f p2 ,p2 ,p12 (z1 , z2 ; u, w1 , w2 , w3 ) =

N 

aijklmn z1i eri lp2 (z1 ) esj lp2 (z2 ) etk lp12 (z1 −z2 ) (lp2 (z1 ))l (lp2 (z2 ))m (lp12 (z1 − z2 ))n , r

i,j,k,l,m,n=1

respectively, of f (z1 , z2 ; u, w1 , w2 , w3 ) in the region given by |z1 | > |z2 | > 0 and | arg(z1 − z2 ) − arg z1 | < π2 , the region given by |z2 | > |z1 | > 0 and π − 3π 2 < arg(z1 − z2 ) − arg z2 < − 2 , the region given by |z2 | > |z1 − z2 | > 0 π and | arg z1 − arg z2 | < 2 , respectively. (3) The L(−1)-derivative property: d Y(w1 , x) = Y(L(−1)w1 , x). dx The correct notion of twisted intertwining operator should have some basic properties. These properties were proved in [H21]. In particular, the notion of twisted intertwining operator introduced in [H21] is indeed the correct one. The first property is the following: Theorem 5.2 ([H21]). Let g1 , g2 , g3 be automorphisms of V and let W1 , W2 and W3 be g1 -, g2 - and g3 -twisted V -modules, respectively. Assume that the verg3 tex operator map for W3 given by u → YW (u, x) is injective. If there exists a

W33  twisted intertwining operator Y of type W1 W2 such that the coefficients of the series Y(w1 , x)w2 for w1 ∈ W1 and w2 ∈ W2 span W3 , then g3 = g1 g2 . For the proof of this theorem, we refer the reader to [H21]. Here we give a geometric explanation of the theorem in terms of a picture (Figure 1). From the equivariance property for twisted modules, the monodromy of the twisted vertex operators corresponds to the action of the automorphism on V . Given a twisted

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3  , the monodromy of the twisted vertex operator intertwining operator of type WW 1 W2 for W3 gives g3 . This is described in the left braiding graph in Figure 1. But this braiding graph is topologically the same as the right braiding graph in Figure 1. It is clear that the right braiding graph in Figure 1 is equal to the product of the monodromy of the twisted vertex operator for W1 and the monodromy of the twisted vertex operator for W2 . So the right braiding graph in Figure 1 gives g1 g2 . Thus we see from this geometric picture, g3 should be equal to g1 g2 . g3u w1 w2

u

g1u2u w1 w2

w1 w2

w1 w 2

u

Figure 1. The braiding graphs corresponding to g3 (left) and g1 g2 (right) For the other properties, we first need to recall an action of an automorphism g h of V on a g-twisted V -module. Let (W, YW ) be a g-twisted V -module. Let h be an automorphism of V and let φh (Y g ) : V × W v⊗w

→  →

W {x}[logx] φh (Y g )(v, x)w

be the linear map defined by φh (Y g )(v, x)w = Y g (h−1 v, x)w. Then the pair (W, φh (Y g )) is an hgh−1 -twisted V -module. We shall denote the hgh−1 -twisted V -module in the proposition above by φh (W ). We now discuss the skew-symmetry isomorphism for twisted intertwining operators. Let g1 , g2 be automorphisms of V , W1 , W2 and W3 g1 -, g2- and g1 g2 -twisted 3 V -modules and Y a twisted intertwining operator of type WW . We define linear 1 W2 maps Ω± (Y) : W2 ⊗ W1 w 2 ⊗ w1 by xL(−1)

Ω± (Y)(w2 , x)w1 = e

→  →

W3 {x}[log x] Ω± (Y)(w2 , x)w1

  Y(w1 , y)w2 

y n =e±nπi xn , log y=log x±πi

for w1 ∈ W1 and w2 ∈ W2 . Theorem 5.3 ([H21]). The linearmaps Ω+ (Y) and Ω− (Y) are twisted interW3 , respectively. twining operators of types W2 φ W−13 (W1 ) and φg (W 2 ) W1 g2

1

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From Theorem 5.3, we see that Ω+ and Ω− are indeed ismorphisms between spaces of twisted intertwining operators. W3 W3 Corollary 5.4 ([H21]). The maps Ω+ : VW → VW 1 W2 2φ

and Ω− :

W3 VW → VφWg3 (W2 )W1 are linear isomorphisms. In particular, V 1 W2

, VφWg3 (W2 )W1

W3 and VW 2φ

1

−1 (W1 ) g2

−1 (W1 ) g2 W3 W1 W2

1

are linearly isomorphic.

Finally we discuss the contragredient isomorphism for twisted intertwining opg erators. We first recall contragredient twisted V -modules. Let (W, YW ) be a gtwisted V -module relative to G. Let W  be the graded dual of W . Define a linear map g  (YW ) : V ⊗ W v ⊗ w

→  →

W  {x}[logx], g  (YW ) (v, x)w

by

g  g ) (v, x)w , w = w , YW (exL(1) (−x−2 )L(0) v, x−1 )w (YW g  for v ∈ V , w ∈ W and w ∈ W  . Then the pair (W  , (YW ) ) is a g −1 -twisted g  g  V -module. We call (W , (YW ) ) the contragredient twisted V -module of (W, YW ). Let g1 , g2 be automorphisms of V , W1 , W2 and W3 g 1 -, g2- and g1 g2 -twisted 3 V -modules and Y a twisted intertwining operator of type WW . We define linear 1 W2 maps

A± (Y) : W1 ⊗ W3 w1 ⊗ w3

→  →

W2 {x}[log x] A± (Y)(w1 , x)w3

by A± (Y)(w1 , x)w3 , w2 = w3 , Y(exL(1) e±πiL(0) (x−L(0) )2 w1 , x−1 )w2 for w1 ∈ W1 and w2 ∈ W2 and w3 ∈ W3 . Theorem 5.5 ([H21]). The linear maps A+ (Y) and A− (Y) are twisted inter g (W  )

 W2 2 1 twining operators of types φW and  ) , respectively.  W φ (W W −1 1 1 3

g1

3

From Theorem 5.5, we see that A+ and A− are indeed ismorphisms between spaces of twisted intertwining operators. φ

(W  )

W3 Corollary 5.6 ([H21]). The maps A+ : VW → VWg11W  2 and A− : 1 W2 3

W

W3 → VW12φ VW 1 W2

and

φ

 −1 (W3 ) g1

W VW12φ −1 (W  ) 3 g

(W  )

W3 are linear isomorphisms. In particular, VW , VWg11W  2 1 W2 3

are linearly isomorphic.

1

6. Main conjectural properties of twisted intertwining operators In Section 3, we have recalled some main conjectures on the construction of orbifold conformal field theories using the representation theory of vertex operator algebras. In this section, we formulate precisely and discuss in details the conjectural properties in these conjectures. In several conjectures and problems in Section 3, only associativity, commutativity and modular invariance of twisted intertwining operators are stated. But as in the untwisted case, the statements of these properties make sense only when the

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corresponding convergence properties hold. In fact, the proofs of these convergence properties are one of the main difficult parts of the proofs of these properties in the untwisted case. The twisted case will certainly be the same. So below we shall first give the formulation of these convergence together with the analytic extensions of the convergent series before the formulations of these properties themselves. We first formulate the conjectural convergence and extension property, associativity and commutativity of twisted intertwining operators. We formulate them for lower-bounded generalized twisted V -modules so that they are more flexible. But we warn the reader that these properties in general will not be true for such general twisted modules. Usually the twisted modules should be grading-restricted and satisfy some additional conditions. In the case that the vertex operator algebra is finite reductive and the group of automorphisms is finite, the category of twisted modules for which these properties holds is conjectured to be the category of ordinary twisted modules. In general the correct categories of twisted modules will be given in precise conjectures in the future and are also an important part of the research on the construction of orbifold conformal field theories. Convergence and extension property of products of n twisted intertwining operators. Let g1 , . . . , gn+1 be automorphisms of V . Let W0 ,   W1 , . . . Wn+1 , W1 , . . . , Wn−1 be lower-bounded generalized (g1 · · · gn+1 )-, g1 -, . . . , gn+1 -, (g2 · · · gn+1 )-, . . . , (gn gn+1 )-twisted V -modules, respectively, and

W

W0  i−1  Y1 , . . . , Yi , . . . , Yn twisted intertwining operators of types W W 1 , . . . , Wi W i , 1

W n−1    . . . , Wn Wn+1 , respectively. For w1 ∈ W1 , . . . , wn+1 ∈ Wn+1 and w0 ∈ W0 , the series w0 , Y1 (w1 , z1 ) · · · Yn (wn , zn )wn+1 in complex variables z1 , . . . zn is absolutely convergent in the region |z1 | > · · · > |zn | > 0 and its sum can be analytically continued to a multivalued analytic function F (u1 , Y1 (w1 , z1 ) · · · Yn (wn , zn )un+1 ) on the region {(z1 , . . . , zn ) | zi = 0, zi − zj = 0 for i = j} ⊂ Cn and the only possible singular points zi = 0, ∞ and zi = zj are regular singular points. To formulate the associativity of twisted intertwining operators, we need the following result: Proposition 6.1. Assume that the convergence and extension property of products of 2 twisted intertwining operators holds. Let g1 , . . . , g4 , g be automorphisms of V . Let W1 , W2 , W3 , W4 , W be lower-bounded generalized g1 -, g2 -, g3 -, (g1 g2 g3 ), (g1 g2)-twisted V -modules and Y3 and Y4 twisted intertwining operators of types

W  W4 and , respectively. Then for w1 ∈ W1 , w2 ∈ W2 , w3 ∈ W3 and W1 W2 W W3   w4 ∈ W4 , the series w4 , Y3 (Y4 (w1 , z1 − z2 )w2 , z2 )w3 is absolutely convergent in the region |z2 | > |z1 − z2 | > 0 and its sum can be analytically continued to a multivalued analytic function F (w4 , Y3 (Y4 (w1 , z1 − z2 )w2 , z2 )w3 )

242

YI-ZHI HUANG

on the region {(z1 , z2 ) | z1 , z2 , z1 − z2 = 0} ⊂ C2 and the possible singular points z1 , z2 , z1 − z2 = 0, ∞ are regular singular points. The proof of this proposition is completely the same as the proofs of Proposition 14.1 in [H1] and Proposition 7.3 in [HLZ6]. We are ready to state precisely the conjectural associativity or the operator product expansion of twisted intertwining operators. Associativity of twisted intertwining operators. Let g1 , g2 , g3 be automorphisms of V . Let W1 , W2 , W3 , W4 , W5 be lower-bounded generalized g1 -, g2 -, g3 -, (g  V -modules and Y1 and Y2 intertwining operators

1 g24g3 ), (g2 g3 )-twisted W5 and of types WW W2 W3 , respectively. There exist a lower-bounded (g1 g2 )1 W5 twisted generalized V -module W6 and intertwining operators Y3 and Y4 of the types

W4 

W6  and , respectively, such that for w1 ∈ W1 , w2 ∈ W2 , w3 ∈ W3 and W6 W3 W1 W2   w4 ∈ W4 , F (w4 , Y1 (w1 , z1 )Y2 (w2 , z2 )w3 ) = F (w4 , Y3 (Y4 (w1 , z1 − z2 )w2 , z2 )w3 ). Another important conjectural property following immediately from the associativity of twisted intertwining operators and Theorem 5.3 is the commutativity of twisted intertwining operators: Commutativity of twisted intertwining operators. Let g1 , g2 , g3 be automorphisms of V . Let W1 , W2 , W3 , W4 , W5 be lower-bounded generalized g1 -, g2 -, g3 -, (g1 g2 g 3 ), (g2g3 )-twisted

W5 V-modules and Y1 and Y2 intertwining opera4 and tors of types WW W2 W3 , respectively. There exist a lower-bounded 1 W5 generalized V -module and Y4 of g1 g3 -twisted



W6  W6 and intertwining operators Y3 −1 W4 the types φg (W and , respectively, or a lower-bounded (g 2 g1 g2 g3 )W1 W3 2 ) W6 1 twisted V -module W6 and intertwining operators Y3 and Y4 of the types



W4 generalized W6 and , respectively, such that for w1 ∈ W1 , w2 ∈ W2 , w3 ∈ W3 W2 W6 φ −1 (W1 ) W3 and w4 ∈ W4 ,

g2

F (w4 , Y1 (w1 , z1 )Y2 (w2 , z2 )w3 ) = F (w4 , Y3 (w2 , z2 )Y4 (w1 , z1 )w3 ). The convergence and extension property, the associativity and commutativity of twisted intertwining operators discussed above are genus-zero properties. We now discuss the conjectural genus-one properties. For a conformal field theory, genusone correlation functions should be equal to the analytic extensions of q-traces or more generally pseudo-q-traces of products of geometrically-modified intertwining operators. So in our case, we have to consider q-trace or pseudo-q-traces of products of geometrically-modified twisted intertwining operators. We first have to recall briefly geometrically-modified twisted intertwining

oper 3 ators and pseudo-q-traces. Given a twisted intertwining operator Y of type WW 1 W2 and w1 ∈ W1 , we have an operator (actually a series with linear maps from W2 to W3 as coefficients) Y1 (w1 , z). The corresponding geometrically-modified operator is Y1 (U(qz )w1 , qz ),

THEORY OF VERTEX OPERATOR ALGEBRAS

243

where q z = e2πiz , U(qz ) = (2πiqz )L(0) e−L (A) and Aj ∈ C for j ∈ Z+ are defined by ⎛ ⎛ ⎞⎞  1 ∂ log(1 + 2πiy) = ⎝exp ⎝ Aj y j+1 ⎠⎠ y. 2πi ∂y +

j∈Z+

See [H12] for details. For pseudo-traces, we need to consider grading-restricted twisted V -modules equipped with a projective right module structure for an finite-dimensional associative algebra P over C. We first define pseudo-traces for a finitely generated projective right P -module M . For such a right P -module, there exists a projective basis, that is, a pair of sets {wi }ni=1 ⊆ M , {wi }ni=1 ⊆ HomP (M, P ) such that for all n w ∈ M , w = i=1 wi wi (w). A linear function φ : P → C is said to be symmetric if φ(pq) = φ(qp) for all p, q ∈ P . For a symmetric linear function φ, the pseud-trace TrφM α for α ∈ EndP (M ) associated to φ is the function TrφM defined by  n   φ  wi (α(wi )) . TrM α = φ i=1

For a grading-restricted twisted V -module W equipped with a projective right P module structure, its homogeneous subspaces W[n] for n ∈ C are finitely generated projective right P -modules. Then for a given symmetric linear function φ on P , we have the pseudo-trace TrφM αn of αn ∈ EndP (W[n] ). For α ∈ EndP (W ), we have αn = πn α|W[n] ∈ EndP (W[n] ). We define  φ TrφW α = TrW[n] αn . n∈C

Note that TrφW α a a series of complex numbers, not a complex number. If we want to get a pseudo-trace in C, we have to prove its convergence. Convergence and extension property of pseudo-q-traces of products of n geometrically-modified twisted intertwining operators. Let gi for i = 1, . . . , n + 1 be automorphisms of V . Let Wi for i = 1, . . . , n be grading-restricted ˜ i for i = 1, . . . , n (gi+1 · · · gn+1 )-twisted V generalized gi -twisted V -modules, W 

W ˜ modules, and Yi for i = 1, . . . , n twisted intertwining operators of types W i−1 ˜ , i Wi ˜ n . Let P be a finite-dimensional ˜0 = W respectively, where we use the convention W ˜0 = W ˜n associative algebra and φ a symmetric linear function on P . Assume that W is also a projective right P -module and its twisted vertex operators commute with the action of P . Assume in addition that the product Y1 (w1 , x1 ) · · · Yn (wn , xn ) for w1 ∈ W1 , . . . , wn ∈ Wn commutes with the action of P . For wi ∈ Wi , i = 1, . . . , n, c L(0)− 24

TrφW ˜ Y1 (U(qz1 )w1 , qz1 ) · · · Yn (U(qzn )wn , qzn )qτ n

is absolutely convergent in the region 1 > |qz1 | > . . . > |qzn | > |qτ | > 0 and can be extended to a multivalued analytic function φ

F Y1 ,...,Yn (w1 , . . . , wn ; z1 , . . . , zn ; τ ). in the region '(τ ) > 0, zi = zj + l + mτ for i = j, l, m ∈ Z.

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YI-ZHI HUANG

These multivalued analytic functions are also conjectured to have associativity and commutativity. These properties are consequences of the convergence and extension property of pseudo-q-traces of products of n geometrically-modified twisted intertwining operators and the associativity and commutativity of twisted intertwining operators. Genus-one associativity. In the setting of the convergence and extension property of pseudo-q-traces of products of n geometrically-modified twisted interˆk twining operators, for 1 ≤ k ≤ n − 1, there exist a (gk gk+1 )-twisted V -module W 

W

W ˜ k−1  ˆk and , and twisted intertwining operators Yˆk and Yˆk+1 of types ˆ kW ˜ k+1 W

Wk Wk+1

respectively, such that F

φ

ˆ

ˆk+1 ,Yk+2 ,...,Yn (w1 , . . . , wk−1 , Yk (wk , zk Y1 ,...,Yk−1 ,Y

− zk+1 )wk+1 ,

wk+2 , . . . , wn ; z1 , . . . , zk−1 , zk+1 , . . . , zn ; τ ) is absolutely convergent in the region 1 > |qz1 | > · · · > |qzk−1 | > |qzk+1 | > · · · > |qzn | > |qτ | > 0 and 1 > |q(zk −zk+1 ) − 1| > 0 and is convergent to φ

F Y1 ,...,Yn (w1 , . . . , wn ; z1 , . . . , zn ; τ ) in the region 1 > |qz1 | > · · · > |qzn | > |qτ | > 0 and |q(zk −zk+1 ) | > 1 > |q(zk −zk+1 ) − 1| > 0. Genus-one commutativity. In the setting of the convergence and extension property of pseudo-q-traces of products of n geometrically-modified twisted intertwining operators, for 1 ≤ k ≤ n − 1, there exist a grading-restricted generalized ˆ k and twisted intertwining operators Yˆk and (gk gk+2 · · · gn+1 )-twisted V -module W 



W ˜ k−1 ˆk W and , respectively, or a grading-restricted Yˆk+1 of types ˜ generalized

Wk Wk+1 −1 gk gk+1 gk+2 (gk+1

ing operators Yˆk and Yˆk+1

ˆk φgk (Wk+1 ) W

ˆ k and twisted intertwin· · · gn+1 )-twisted V -module W 

W 

˜ ˆk W and W k−1W of types φ −1 (W ˜ ˆ , respectively, k ) Wk+1 g k+1

k+1

k

such that F

φ

Y1 ,...,Yn (w1 , . . . , wn ; z1 , . . . , zn ; τ ) φ

= F Y1 ,...,Yk−1 ,Yˆk+1 ,Yˆk ,Yk+2 ...,Yn (w1 , . . . , wk−1 , wk+1 , wk , wk+2 , . . . , wn ; z1 , . . . , zk−1 , zk+1 , zk , zk+2 , . . . , zn ; τ ). The most important conjectural property of twisted intertwining operators in the genus-one case is the modular invaraince. Modular invariance of twisted intertwining operators. For automorphisms gi of V , grading-restricted generalized gi -twisted V -modules Wi and wi ∈ Wi for i = 1, . . . , n, let Fw1 ,...,wn be the vector space spanned by functions of the form φ

F Y1 ,...,Yn (w1 , . . . , wn ; z1 , . . . , zn ; τ ) for all finite-dimensional associative algebras P , all symmetric linear functions φ, ˜ i for i = 1, . . . , n + 1, all projective right all (gi+1 · · · gn+1 )-twisted V -modules W ˜ n+1 commuting with its twisted vertex operators, all P -module structures on W

THEORY OF VERTEX OPERATOR ALGEBRAS

twisted intertwining operators Yi of types

245

W ˜ i−1 

for i = 1, . . . , n , respectively, ˜ n+1 . Then for such that their product commutes with the right action of P on W   α β ∈ SL(2, Z), γ δ   L(0) L(0)  zn ατ + β 1 1 z1 φ ,..., ; F Y1 ,...,Yn w1 , . . . , wn ; γτ + δ γτ + δ γτ + δ γτ + δ γτ + δ ˜i Wi W

is in Fw1 ,...,wn . Since the commutativity, genus-one associativity and genus-one commutativity are consequences of the other properties, the main properties that need to be proved are the convergence and extension property of products of twisted intertwining operators, associativity of twisted intertwining operators, convergence and extension property of pseudo-q-traces of products of n geometrically-modified twisted intertwining operators and the modular invariance of twisted intertwining operators. 7. Some thoughts on further developments We discuss in this section briefly some thoughts of the author on further developments based on the conjectural properties in the preceding section. Assuming that the conjectural properties in Section 6 hold for twisted intertwining operators among suitable twisted modules associated to a group of automorphisms of a vertex operator algebra. Then we can generalize the results described in Section 2 to results on what can be called “equivariant chiral and full conformal field theories.” In particular, we should be able to construct equivariant modular functors, equivariant genus-zero and genus-one chrial conformal field theories. In Section 3, we mentioned that to construct higher-genus conformal field theories, the problem of the convergence of multi pseudo-q-traces of products of geometrically-modified intertwining operators is still open. In the case of orbifold conformal field theory, there is also a conjectural convergence of multi q-traces of products of geometrically-modified twisted intertwining operators. If this convergence holds, then we can construct “equivariant (all-genus) chiral and full conformal field theories.” We will also be able to obtain G-crossed tensor category structures. As is mentioned in Section 3, Conjecture 3.3 is a consequence of Conjecture 3.2. In fact, without assuming that Conjecture 3.2 holds, tensor product bifunctors for suitable twisted modules can be defined in the same way as in [HL3], [HL5] and [HLZ3] with intertwining operators replaced by twisted intertwining operators in [H21]. On the other hand, though we do have a Jacobi identity for twisted vertex operators and one twisted intertwining operator obtained as a special case for the Jacobi identity for intertwining operators in [H7], it is not as nice as the one for vertex operators and one intertwining operator in [FHL]. Thus the construction of tensor product modules using the compatibility condition and local gradingrestriction condition has to be modified by using the method of formulating and studying twisted intertwining operators in [H21] instead of the method based on the Jacobi identity in [HL3], [HL4], [HL5] and [HLZ4]. Now assuming that the convergence and extension property of products of twisted intertwining operators and associativity of twisted intertwining operators hold. Then the associativity of the tensor product bifunctors can be proved in the same way as in [H1] and

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[HLZ7]. With this construction, we actually obtain what should be called a “vertex G-crossed tensor category” structure and the G-crossed tensor category structure can be derived from this structure in the same way as how the braided tensor category structure is derived from the vertex tensor category structure in [H14] and [HLZ9]. We also expect that if the convergence and extension property of q-traces of products of n geometrically-modified twisted intertwining operators and the modular invariance of twisted intertwining operators hold, then the rigidity can be proved in the same way as in [H14] in the case that the category of lowerbounded generalized twisted modules are semisimple. The conjectural properties in Section 6 can be proved if we know that the fixed point subalgebra V G of V satisfies conditions in the papers [H11], [H12], [H13] and [H14] or some other conditions. See [Mi] and [CM] for proofs of the conditions for V G in the case that G is finite cyclic and [Mc1] and [Mc2] for some results one can obtain when V G is assumed to satisfy certain conditions, including in particular suitable conditions on suitable categories of V G -modules. But proving these conditions to hold for V G seems to be at least as difficult as proving the conjectural properties in Section 6. Also note that the reason why we want to prove these conditions for V G is exactly that these conditions can be used to prove the the conjectural properties in Section 6. In fact, the proof of the reductivity property of V G in [CM] uses heavily the theory of intertwining operators established in [H11] and [H12]. The author believes that these conditions for V G follow from the conjectures in Section 3. Therefore in the author’s opinion, if possible, we should prove the conjectural properties in Section 6 directly, without using the fixed point subalgebra V G . Then we should be able to derive the algebraic conditions for V G as consequences. In order to derive the conditions on V G as consequences, we need to prove that for a general vertex operator algebra V , the properties of (untwisted) intertwining operators (such as the associativity and modular invariance) imply the algebraic conditions on V . This problem is interesting even without the study of orbifold conformal field theory since it provides a deep understanding of the connection between algebraic properties of the vertex operator algebra V and geometric properties of genus-zero and genus-one correlation functions of the corresponding conformal field theory. In [H10], the author introduced a precise notion of dual of an intertwining operator algebra. Since vertex operator algebra is also an intertwining operator algebra, we also have a dual of a vertex operator algebra. In fact, the dual of a vertex operator algebra is simply the intertwining operator algebra constructed using all intertwining operators among all modules for the vertex operator algebras. In particular, a self-dual vertex operator algebra in this sense means that the only irreducible module is the vertex operator algebra itself and is called in many papers a holomorphic vertex operator algebra. The moonshine module constructed in [FLM3] is a self-dual vertex operator algebra and the uniqueness conjecture of the moonshine module states that a self-dual vertex operator algebra of central charge 24 and without nonzero weight 1 elements must be isomorphic to the moonshine module as a vertex operator algebra. Let V be a self-dual vertex operator algebra of central charge 24 and without nonzero weight 1 elements. It is desirable if we can embed V into a largest possible intertwining operator algebra. Then we can perform all types of operations and constructions in this largest intertwining operator algebra containing V . But taking the dual of V does not work since it is

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self dual. On the other hand, if we let G be the full automorphism group of V , then we can take the dual of the vertex operator subalgebra V G of V . This dual of V G must contain V and is the largest among the duals of all the fixed point subalgebras of V . The construction of the dual of V G is in fact part of Problem 3.1. As is mentioned after Problem 3.1 and in the beginning of Section 5, instead of studying directly V G -modules and intertwining operators among V G -modules, we study twisted V -modules and twisted intertwining operators among twisted V modules. Thus the construction and study of the dual of V G are reduced to the construction and study of the orbifold conformal field theory obtained from the vertex operator algebra V and its automorphsim group G. We hope that if this orbifold conformal field theory is constructed, we can use it to find a strategy to prove the uniqueness of the moonshine module. Note that the construction of such an orbifold conformal field theory from such an arbitrary vertex operator algebra satisfying three conditions must be general and abstract and cannot be worked out for a particular example such as the moonshine module itself. This is in fact one important reason why we have to develop a general orbifold conformal field theory instead of just explicit examples. Acknowledgments The author is grateful to Drazen Adamovic and Antun Milas for their invitation to the conference “Representation Theory XVI” held in Dubrovnik in June 2019. References Bojko Bakalov, Twisted logarithmic modules of vertex algebras, Comm. Math. Phys. 345 (2016), no. 1, 355–383, DOI 10.1007/s00220-015-2503-9. MR3509017 [BPZ] A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), no. 2, 333–380, DOI 10.1016/0550-3213(84)90052-X. MR757857 [Bo] Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071, DOI 10.1073/pnas.83.10.3068. MR843307 [CM] S. Carnahan and M. Miyamoto, Regularity of fixed-point vertex operator subalgebras, to appear; arXiv:1603.05645. [C] Thomas Creutzig, Fusion categories for affine vertex algebras at admissible levels, Selecta Math. (N.S.) 25 (2019), no. 2, Paper No. 27, 21, DOI 10.1007/s00029-0190479-6. MR3932636 [CHY] Thomas Creutzig, Yi-Zhi Huang, and Jinwei Yang, Braided tensor categories of admissible modules for affine Lie algebras, Comm. Math. Phys. 362 (2018), no. 3, 827–854, DOI 10.1007/s00220-018-3217-6. MR3845289 [DHVW1] L. Dixon, J. A. Harvey, C. Vafa, and E. Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985), no. 4, 678–686, DOI 10.1016/0550-3213(85)90593-0. MR818423 [DHVW2] L. Dixon, J. Harvey, C. Vafa, and E. Witten, Strings on orbifolds. II, Nuclear Phys. B 274 (1986), no. 2, 285–314, DOI 10.1016/0550-3213(86)90287-7. MR851703 [DL] Chongying Dong and James Lepowsky, Generalized vertex algebras and relative vertex operators, Progress in Mathematics, vol. 112, Birkh¨ auser Boston, Inc., Boston, MA, 1993. MR1233387 [DLM1] Chongying Dong, Haisheng Li, and Geoffrey Mason, Twisted representations of vertex operator algebras, Math. Ann. 310 (1998), no. 3, 571–600, DOI 10.1007/s002080050161. MR1615132 [DLM2] Chongying Dong, Haisheng Li, and Geoffrey Mason, Modular-invariance of trace functions in orbifold theory and generalized Moonshine, Comm. Math. Phys. 214 (2000), no. 1, 1–56, DOI 10.1007/s002200000242. MR1794264

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Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019 Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15467

Further q-series identities and conjectures relating false theta functions and characters Chris Jennings-Shaffer and Antun Milas Abstract. In this short note, a companion of [21], we discuss several families of q-series identities in connection to false and mock theta functions, characters of modules of vertex algebras, and “sum of tails”.

1. Introduction and previous work In our previous work [21], motivated by character formulas of vertex algebras and superconformal indices in physics, we obtained various identities for false theta functions including the following elegant identity. Theorem 1.1. For k ≥ 1,  (k+1)n2 +kn n∈Z sgn(n)q (1.1) = (q)2k ∞ where as usual (a)n =

n−1 i=0

2k−2

 n1 ,n2 ,...,n2k−1 ≥0

2k−1

q i=1 ni ni+1 + i=1 ni , (q)2n1 (q)2n2 · · · (q)2n2k−1

(1 − aq i ).

We note that these identities have an odd number of summation variables. Interestingly, with an even number of summation variables we obtained a family of modular identities conjectured in [15]. Theorem 1.2. For k ≥ 1, (1.2)

 (q, q 2k+2 , q 2k+3 ; q 2k+3 )∞ = (q)2k+1 ∞ n1 ,n2 ,...,n

2k ≥0

2k−1

2k

q i=1 ni ni+1 + i=1 ni . (q)2n1 (q)2n2 · · · (q)2n2k

In a somewhat different direction, in the same paper, we also examined q-series identities for false theta functions with half-integral characteristics (here k ∈ N and  ∈ {0, 12 }) (1.3)

1 2 1 (−q 2 + )∞  sgn(n)q 2 (2k+1)(n+a) , (q)∞

n∈Z

for some specific rational numbers a. We also considered related identities for certain “shifted” false theta series [21, Section 3]. 2020 Mathematics Subject Classification. Primary 11P84, 17B69. The second author acknowledges the support from the NSF grant DMS 1601070. c 2021 American Mathematical Society

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This paper aims to extend (1.1) and (1.2) in a few directions. Firstly, we would like to study related identities for the false theta functions as in (1.3). Secondly, we relax the condition on the poles in (1.1) and (1.2) and perform a search for identities where the q-hypergeometric side takes the form 

(1.4)

n1 ,n2 ,...,nk ≥0

q n1 +n2 +···+nk +n1 n2 +n2 n3 +···+nk−1 nk , (q)rn11 (q)rn22 · · · (q)rnkk

k

with k ≤ i=1 ri ≤ 2k. Lastly, we consider q-series identities coming from the formal inversion q → q −1 of the q-hypergeometric term in (1.1) and (1.2). This procedure is sometimes used for quantum modular forms to extend a q-series defined in the upper half-plane to the lower half-plane. Our paper is organized as follows. In Sections 2 and 3 we gather several known facts. In Section 4 we prove analogs of Theorems 1.1 and 1.2 for false and classical theta series with half characteristics (Theorem 4.3 and Proposition 4.4). Section 5 is devoted to “inverted identities”, under q → q −1 , associated to the q-hypergeometric series in (1.1) and (1.2). We argue that in both cases we expect modular identities. For the inverted q-series coming from (1.2), this is proven in Proposition 5.1 by (1) reduction to the character formula of a principal subspace of A2k−1 . For (1.1), we expect (see Conjecture 5.2) that the resulting inverted series is modular as it is essentially the level one character of the affine vertex algebra Lsp(2k) (Λ0 ). We show that this is indeed true up to a cubic term (Proposition 5.3). In Section 6, we study more complicated q-hypergeometric series of the form (1.4) with k = 2 and k = 3. Continuing, in Section 7 we consider identities for the series (1.4) with ri = 1 for all i. For 2 ≤ k ≤ 8, except k = 7, we found several interesting “sums of tails” type identities. Then in Section 8 we connect the q-series from Section 7 with characters of modules of principal subspaces and infinite jet schemes. We end with a few remarks for future investigations. 2. Quantum dilogarithm 2.1. Quantum dilogarithm. As in [21], wewill approach several q-series identities using the quantum dilogarithm φ(x) := i≥0 (1 − q i x). Let x and y be non-commutative variables such that xy = qyx, then (2.1)

φ(y)φ(x) = φ(x)φ(−yx)φ(y),

which is Faddeev and Kashaev’s pentagon identity for the quantum dilogarithm. This identity implies that (2.2)

1 1 = , φ(x)φ(y) φ(y)φ(−yx)φ(x)

where φ(x1 )φ(x12 )···φ(xn ) is understood to denote φ(x1 1 ) · φ(x1 2 ) · · · φ(x1n ) . For its relevance in 4d/2d dualities in physics see [14, 15] and references therein. 3. Bailey’s lemma and other known q-series identities As in [21], we require Bailey’s lemma and several standard q-series identities, which we collect in this section. A pair of sequences (αn , βn ) is called a Bailey pair

RELATING FALSE THETA FUNCTIONS AND CHARACTERS

255

relative to a if βn =

n  j=0

αj . (q)n−j (aq)n+j

The k-fold iteration of Bailey’s lemma can be found in its entirety as Theorem 3.4 of [2]. This theorem with k → k − 1, a = q, b1 = b2 = . . . = bk−1 → ∞, c1 = c2 = · · · = ck−2 = q, ck−1 = −w−1 q, N → ∞, and nj → mk−j states that 

m1 (m1 +1)

m1 ,m2 ,...,mk−1 ≥0

m2 (m2 +1)

+ +···+ 2 2 (−w−1 q)m1 (q)mk−1 (−1)m2 +m3 +···+mk−1 q (q)m1 (q)m1 −m2 · · · (q)mk−2 −mk−1

mk−1 (mk−1 +1) 2

wm1 βmk−1

(3.1) (k−1)n(n+1) 2 wn αn (−wq)∞  (−w−1 q)n (−1)kn q = , (q 2 )∞ (−wq)n

n≥0

where (αn , βn ) is any Bailey pair relative to a = q, and w ∈ C. We require a single Bailey pair relative to a = q. Specifically this is the Bailey pair B(3) of Slater [30], which is defined by (3.2)

αnB3

:=

(−1)n q

n(3n+1) 2

(1 − q 2n+1 ) , (1 − q)

βnB3 :=

1 . (q)n

The additional q-series identity we require are as follows. We have two identities of Euler [19, (II.1) and (II.2)],  zn 1 (3.3) = , (q)n (z)∞ n≥0

 (−1)n z n q n(n−1) 2 = (z)∞ . (q)n

(3.4)

n≥0

More generally, the q-binomial theorem [19, (II.3)] states that  (a)n z n (az)∞ = . (q)n (z)∞

(3.5)

n≥0

We also need two forms of Heine’s transformation [19, (III.1) and (III.2)], which are

c   (a, b)n z n (b, az)∞  b , z n bn (3.6) = , (c, q)n (c, z)∞ (az, q)n n≥0 n≥0

c 

 c n  abz , b  (a, b)n z n b , bz ∞ c n b (3.7) = . (c, q)n (c, z)∞ (bz, q)n n≥0

n≥0

Lastly, we use Lemma 1 of [4] written as  n2 n2 (n2 +1) 1 1 n1 +n2 − 21 n1 2  1  (3.8) = (−1) q ζ . 1 (q)2∞ ζq 2 , ζ −1 q 2 n1 ∈Z ∞

n2 ≥|n1 |

We note that the summation bound n2 ≥ |n1 | in (3.8) can be replaced by n2 ≥ n1 .

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4. Identities with half-characteristic In this section we extend Theorems 1.1 and 1.2 from the introduction to halfcharacteristic. Proposition 4.1. Suppose k ≥ 1 and w ∈ C. Then 

q n1 n2 +n2 n3 +···+nk−1 nk +n1 +n2 +···+nk (−w)n1 (q)2n1 (q)2n2 · · · (q)2nk n1 ,n2 ,...,nk ≥0 ⎛  ⎞ 1 k φ −wq 2 ζ1

1 1 ⎠  1 ,  1  ⎝  1   1 = CTζ1 ,ζ2 ,...,ζk −1 φ q 2 ζ1 φ q 2 ζk−1 j=2 φ q 2 ζj φ q 2 ζj−1

(4.1)

where the ζj are non-commuting variables with ζj ζj+1 = qζj+1 ζj for 1 ≤ j ≤ k − 1. Proof. The proof is similar to that of Theorems 1.1 and 1.2. We expand 1 1 φ(−wq 2 ζ1 )/φ(q 2 ζ1 ) with the q-binomial theorem (3.5) and all other products are expanded with Euler’s identity (3.3). By doing so we have ⎞ ⎛  1 k φ −wq 2 ζ1 1 1 ⎜ ⎟ ⎠  1  1 ⎝  1  1 −1 φ q 2 ζ1 φ q 2 ζk−1 j=2 φ q 2 ζj φ q 2 ζj−1 ⎛ ⎞ n1 +m1 +n2 +m2 +···+nk +mk k  2 q (−w)n1 n1  n −m −m = ζ1 ⎝ ζj j ζj−1j−1 ⎠ ζk k (q)n1 (q)m1 (q)n2 (q)m2 · · · (q)nk (q)mk j=2 k n,m∈N0

=



q

n1 +m1 +n2 +m2 +···+nk +mk n −m1 n2 −m2 +n2 m1 +n3 m2 +···+nk mk−1 2 (−w)n1 ζ1 1 ζ2

n −mk

· · · ζk k

.

(q)n1 (q)m1 (q)n2 (q)m2 · · · (q)nk (q)mk

n,m∈Nk 0

The constant term then clearly comes from taking mj = nj and the proposition follows. 

In the lemma below, we give an intermediate identity that is required so that we may apply Bailey’s lemma. Lemma 4.2. Suppose k ≥ 2 and w ∈ C. Then n n +n2 n3 +···+nk−1 nk +n1 +n2 +···+nk q 1 2 (−w)n 1 2 2 (q)2 n1 (q)n2 · · · (q)nk n1 ,n2 ,...,nk ≥0 

1 =



(q)k ∞ m1 ,m2 ,...,mk−1 ≥0

(−1)

m2 +···+mk−1

q

m (mk−1 +1) m1 (m1 +1) m2 (m2 +1) + +···+ k−1 2 2 2 wm1 (−w−1 q)m

(q)m (q)m −m (q)m −m · · · (q)m 1 1 2 2 3 k−2 −mk−1

1 .

Proof. We begin by reevaluating the constant term in (4.1) by applying (2.2) and expanding the products with (3.5), (3.3), and (3.8). For convenience with the

RELATING FALSE THETA FUNCTIONS AND CHARACTERS

257

indices, we instead use ζ0 , ζ1 , . . . , ζk−1 . With this all mind, we find that 

φ

1 −wq 2 ζ0   1 φ q 2 ζ0

 ⎛

⎞

⎜k−1 ⎜  ⎜ ⎜ ⎝ j=1

 φ

1 q 2 ζj

1   φ

1 −1 q2ζ j−1

⎟ ⎟ ⎟ ⎟ ⎠

 φ

1 1 −1 q2ζ k−1



 1 −wq 2 ζ0 k−1  1         =  1 1 1 1 −1 −1 −1 j=1 2 2 φ q ζ0 φ q ζ0 φ −ζj ζ φ q 2 ζj φ q 2 ζ j j−1 

φ



1 =

2k−2 (q)∞

(−1)

2

k−1

k−1

k−1

k−1 mj (mj +1) k−1 nj r +r2 n + mj +  − + 1 j=1 j j=1 j=1 j q j=1 j=1 2 2 2 (−w)r

1

(q)r (q)r (q) (q) · · · (q) 1 2 1 2 k−1

n,m∈Zk−1 k−1 r1 ,r2 ∈N0 ,∈N0 mj ≥nj

k−1 r −r2   −1 j nj × ζ0 1 ζj ζ ζ j−1 j j=1



1 =

(−1)

2

k−1

k−1

k−1

k−1 mj (mj+1) k−1 nj k−1 j (j+1) r1+r2 n + mj+  − + + j=1 j j=1 j=1 j q j=1 j=1 2 j=1 2 2 2 (−w)r

2k−2 (q)∞

1

(q)r (q)r (q) (q) · · · (q) 1 2 1 2 k−1

n,m∈Zk−1 k−1 r1 ,r2 ∈N0 ,∈N 0 mj ≥nj ⎛ ⎞ k−2 n + − r −r2 −1 ⎝  j j j+1 ⎠ nk +k−1 × ζ0 1 ζ ζ . j k−1 j=1

The constant term comes from nj = j+1 − j for 1 ≤ j ≤ k − 2, nk−1 = −k−1 , and r2 = r1 − 1 . For the index bounds, we replace mk−1 ≥ nk−1 with mk−1 ≥ |nk−1 |. Thus by Proposition 4.1,  n1 ,n2 ,...,nk ≥0



1 =

(−1)

2

k−1

k−2

k−1 mj (mj+1) 2

k−1 j (j+1) k−2 (j+1−j )2 k−1 mj+  + 1+ − − +r j=1 j=2 j q j=1 j=2 j=1 2 2 2 2 2 (−w)

2k−2 (q)∞ k−1 m∈Zk−1 ,∈N0 r∈N0 ,r≥1 mk−1≥k−1 mj≥j+1 −j

2k−2 (q)∞

r

(q)r (q)r− (q) (q) · · ·(q) 1 1 2 k−1



1 =

n n +n2 n3 +···+nk−1 nk +n1 +n2 +···+nk q 1 2 (−w)n 1 2 (q)2 · · · (q)2 (q)n n2 nk 1

(−1)

k−1

k−1

k−1 mj (mj +1) k−2 j (j +1) k−1 (k−1 +3) + + mj+  j=1 j=1 j=1 j q j=1 2 2 2 (−w)r+

1

(q)r (q)r+ (q) (q) · · ·(q) 1 1 2 k−1

k−1 r∈N0 ,,m∈N0

(4.2) ×q

k−2 −1 m1 +  (mj−1 −mj )+k−1 (mk−2 +mk−1 )+r j=2 j .

Due to convergence issues in certain calculations below, we view the far righthand side of (4.2) as the x → 1 case of F (x) :=



1 (q)2k−2 ∞

k−1 r∈N0 ,,m∈N0

(4.3)

×q

k−1

k−1

m +  j=1 jq (−1) j=1 j

k−1 mj (mj+1) k−2 j (j+1) k−1 (k−1+3) + + j=1 j=1 2 2 2 (−xw)

(q)1 (q)2 · · ·(q)

k−1

r+1

(q)r (xq)r+1

k−2 −1 m1 +  (mj−1 −mj )+k−1 (mk−2 +mk−1 )+r j=2 j .

We transform the inner sum on r with Heine’s transformation (3.6) with a = 0, b = −xwq 1 , c = xq 1 +1 , and z = q, as  (−xw)r+ q r (−xw)1  (−xwq 1 )r q r (−xw)∞  1 = = (−w−1 q)r (−1)r xr wr q 1 r ,  +1 1 (q) (xq) (xq) (q) (xq ) (xq) (q) r r r ∞ ∞ r+  1 1 r≥0 r≥0 r≥0

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C. JENNINGS-SHAFFER AND A. MILAS

we note that when x = 1, the final series above is not absolutely convergent for all w and 1 . Thus for |xw| < 1,

F (x) =



(−xw)∞

(−1)

k−1 j=1 mj+ j=1 j+r

k−1

1 (xq)∞ (q)2k− ∞

×q

k−1 mj (mj+1) k−2 j (j+1) k−1 (k−1+3) + + 2 2

j=1 2 q j=1 (q)1 (q)2 · · ·(q)k−1

r∈N0 ,,m∈Nk−1 0

k−2 −1 m1 + j=2 j (mj−1 −mj )+k−1 (mk−2 +mk−1 )+1 r r

x wr (−w−1 q)r .

We evaluate the inner sums on each j with (3.4) to find that  (−1)1 q 1 (1+r−m1 ) q (q)1  ≥0

1 (1 −1) 2

= (q 1+r−m1 )∞ =

1

 (−1)j q j (1+mj−1 −mj ) q (q)j  ≥0

j (j −1) 2

(q)∞ , (q)mj−1 −mj

= (q 1+mj−1 −mj ) =

j

 k−1 ≥0

(−1)k−1 q j (2+mk−2 +mk−1 ) q (q)k−1

k−1 (k−1 −1) 2

(q)∞ , (q)r−m1

= (q 2+mk−2 +mk−1 ) =

for 2 ≤ j ≤ k − 2, (q)∞ . (q)mk−2 +mk−1 +1

Thus, for |xw| < 1,

F (x) =

=

(−xw)∞

k−1

m +r (−1) j=1 j q



k−1 mj (mj +1) j=1 2 xr w r (−w−1 q)

r

(xq)∞ (q)k ∞ r,m1 ,m2 ,...,mk−1 ≥0(q)r−m1 (q)m1 −m2 (q)m2 −m3 · · ·(q)mk−3 −mk−2 (q)mk−2 +mk−1 +1 (−xw)∞



k−1 m +r (−1) j=1 j q



k−1 mj (mj +1) j=1 2

xr w r (−w−1 q)r

(xq)∞ (q)k ∞ r,m1 ,m2 ,...,mk−1 ≥0 (q)r−m1 (q)m1 −m2 (q)m2 −m3 · · · (q)mk−2 −mk−1

,

where the second equality follows from Heine’s transformation (3.7) with a → ∞, b = q, c = q 2+mk−2 , z = aq , applied to the inner sum on mk−1 . By (3.5), the sum on r is  (−1)r xr wr (−w−1 q)r  (−1)r xr wr (−w−1 q m1 +1 )r = (−1)m1 xm1 wm1 (−w−1 q)m1 (q) (q)r r−m1 r≥0 r≥0 =

(−1)m1 xm1 wm1 (−w−1 q)m1 (xq)∞ (xq)m1 (−xw)∞

and so 1 F (x) = (q)k∞

 m1 ,m2 ,...,mk−1 ≥0

k−1

k−1

mj (mj +1)

2 (−1) j=2 mj q j=1 xm1 wm1 (−w−1 q)m1 . (xq)m1 (q)m1 −m2 (q)m2 −m3 · · · (q)mk−2 −mk−1

This form of F (x) is well defined for exactly the same values of x as (4.3) and so we find the lemma follows by setting x = 1.  Our extension of Theorems 1.1 and 1.2 to the series in (1.3) is given here.

RELATING FALSE THETA FUNCTIONS AND CHARACTERS

259

Theorem 4.3. Suppose k ≥ 2. Then 

q n1 n2 +n2 n3 +···+nk−1 nk +n1 +n2 +···+nk (−1)n1 (q)2n1 (q)2n2 · · · (q)2nk n1 ,n2 ,...,nk ≥0 ⎛ ⎞  (k+2)n2 +kn (−q)∞ ⎝  k ⎠ (−1)(k+1)n q 2 = +(−1) , (q)k+1 ∞ n −b24 − b34 + b44 + b45 , − b13 − b14 − b15 + b22 > −b24 + b44 , − b13 − b14 − b15 + b22 > −b35 + b44 , − b13 − b14 − b15 + b22 > −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 − b14 − b15 + b22 > −b13 − b14 − b23 + b33 + b34 , − b13 − b14 − b15 + b22 > −b13 − b23 − b25 + b33 + b34 , − b13 − b14 − b15 + b22 > −b13 − b14 + b33 , − b13 − b14 − b15 + b22 > −b13 − b25 + b33 , − b13 − b14 − b15 + b22 > −b24 − b25 + b33 , (E2 ) − b13 − b14 − b23 − b24 + b33 + b34 + b35 > b55 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b13 − b23 + b44 + b45 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b13 − b34 + b44 + b45 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b13 + b44 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b24 − b34 + b44 + b45 ,

274

KAILASH C. MISRA AND SUCHADA PONGPRASERT

− b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b24 + b44 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b35 + b44 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 ≥ −b13 − b14 − b15 + b22 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b13 − b14 − b23 + b33 + b34 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b13 − b23 − b25 + b33 + b34 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b13 − b14 + b33 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b13 − b25 + b33 , − b13 − b14 − b23 − b24 + b33 + b34 + b35 > −b24 − b25 + b33 , (E3 ) − b13 − b14 − b23 + b33 + b34 > b55 , − b13 − b14 − b23 + b33 + b34 > −b13 − b23 + b44 + b45 , − b13 − b14 − b23 + b33 + b34 > −b13 − b34 + b44 + b45 , − b13 − b14 − b23 + b33 + b34 > −b13 + b44 , − b13 − b14 − b23 + b33 + b34 > −b24 − b34 + b44 + b45 , − b13 − b14 − b23 + b33 + b34 > −b24 + b44 , − b13 − b14 − b23 + b33 + b34 > −b35 + b44 , − b13 − b14 − b23 + b33 + b34 ≥ −b13 − b14 − b15 + b22 , − b13 − b14 − b23 + b33 + b34 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 − b14 − b23 + b33 + b34 > −b13 − b23 − b25 + b33 + b34 , − b13 − b14 − b23 + b33 + b34 > −b13 − b14 + b33 , − b13 − b14 − b23 + b33 + b34 > −b13 − b25 + b33 , − b13 − b14 − b23 + b33 + b34 > −b24 − b25 + b33 , (E4 ) − b13 − b23 − b25 + b33 + b34 > b55 , − b13 − b23 − b25 + b33 + b34 > −b13 − b23 + b44 + b45 , − b13 − b23 − b25 + b33 + b34 > −b13 − b34 + b44 + b45 , − b13 − b23 − b25 + b33 + b34 > −b13 + b44 , − b13 − b23 − b25 + b33 + b34 > −b24 − b34 + b44 + b45 , − b13 − b23 − b25 + b33 + b34 > −b24 + b44 , − b13 − b23 − b25 + b33 + b34 > −b35 + b44 , − b13 − b23 − b25 + b33 + b34 ≥ −b13 − b14 − b15 + b22 , − b13 − b23 − b25 + b33 + b34 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 − b23 − b25 + b33 + b34 ≥ −b13 − b14 − b23 + b33 + b34 , − b13 − b23 − b25 + b33 + b34 > −b13 − b14 + b33 , − b13 − b23 − b25 + b33 + b34 > −b13 − b25 + b33 , − b13 − b23 − b25 + b33 + b34 > −b24 − b25 + b33 , (E5 ) − b13 − b14 + b33 > b55 , − b13 − b14 + b33 > −b13 − b23 + b44 + b45 , − b13 − b14 + b33 > −b13 − b34 + b44 + b45 , − b13 − b14 + b33 > −b13 + b44 , − b13 − b14 + b33 > −b24 − b34 + b44 + b45 ,

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

− b13 − b14 + b33 > −b24 + b44 , − b13 − b14 + b33 > −b35 + b44 , − b13 − b14 + b33 ≥ −b13 − b14 − b15 + b22 , − b13 − b14 + b33 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 − b14 + b33 ≥ −b13 − b14 − b23 + b33 + b34 , − b13 − b14 + b33 > −b13 − b23 − b25 + b33 + b34 , − b13 − b14 + b33 > −b13 − b25 + b33 , − b13 − b14 + b33 > −b24 − b25 + b33 , (E6 ) − b13 − b23 + b44 + b45 > b55 , − b13 − b23 + b44 + b45 > −b13 − b34 + b44 + b45 , − b13 − b23 + b44 + b45 > −b13 + b44 , − b13 − b23 + b44 + b45 > −b24 − b34 + b44 + b45 , − b13 − b23 + b44 + b45 > −b24 + b44 , − b13 − b23 + b44 + b45 > −b35 + b44 , − b13 − b23 + b44 + b45 ≥ −b13 − b14 − b15 + b22 , − b13 − b23 + b44 + b45 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 − b23 + b44 + b45 ≥ −b13 − b14 − b23 + b33 + b34 , − b13 − b23 + b44 + b45 ≥ −b13 − b23 − b25 + b33 + b34 , − b13 − b23 + b44 + b45 > −b13 − b14 + b33 , − b13 − b23 + b44 + b45 > −b13 − b25 + b33 , − b13 − b23 + b44 + b45 > −b24 − b25 + b33 , (E7 ) − b13 − b25 + b33 > b55 , − b13 − b25 + b33 > −b13 − b23 + b44 + b45 , − b13 − b25 + b33 > −b13 − b34 + b44 + b45 , − b13 − b25 + b33 > −b13 + b44 , − b13 − b25 + b33 > −b24 − b34 + b44 + b45 , − b13 − b25 + b33 > −b24 + b44 , − b13 − b25 + b33 > −b35 + b44 , − b13 − b25 + b33 ≥ −b13 − b14 − b15 + b22 , − b13 − b25 + b33 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 − b25 + b33 ≥ −b13 − b14 − b23 + b33 + b34 , − b13 − b25 + b33 ≥ −b13 − b23 − b25 + b33 + b34 , − b13 − b25 + b33 ≥ −b13 − b14 + b33 , − b13 − b25 + b33 > −b24 − b25 + b33 , (E8 ) − b24 − b25 + b33 > b55 , − b24 − b25 + b33 > −b13 − b23 + b44 + b45 , − b24 − b25 + b33 > −b13 − b34 + b44 + b45 , − b24 − b25 + b33 > −b13 + b44 , − b24 − b25 + b33 > −b24 − b34 + b44 + b45 ,

275

276

KAILASH C. MISRA AND SUCHADA PONGPRASERT

− b24 − b25 + b33 > −b24 + b44 , − b24 − b25 + b33 > −b35 + b44 , − b24 − b25 + b33 ≥ −b13 − b14 − b15 + b22 , − b24 − b25 + b33 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b24 − b25 + b33 ≥ −b13 − b14 − b23 + b33 + b34 , − b24 − b25 + b33 ≥ −b13 − b23 − b25 + b33 + b34 , − b24 − b25 + b33 ≥ −b13 − b14 + b33 , − b24 − b25 + b33 ≥ −b13 − b25 + b33 , (E9 ) − b13 − b34 + b44 + b45 > b55 , − b13 − b34 + b44 + b45 ≥ −b13 − b23 + b44 + b45 , − b13 − b34 + b44 + b45 > −b13 + b44 , − b13 − b34 + b44 + b45 > −b24 − b34 + b44 + b45 , − b13 − b34 + b44 + b45 > −b24 + b44 , − b13 − b34 + b44 + b45 > −b35 + b44 , − b13 − b34 + b44 + b45 ≥ −b13 − b14 − b15 + b22 , − b13 − b34 + b44 + b45 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 − b34 + b44 + b45 ≥ −b13 − b14 − b23 + b33 + b34 , − b13 − b34 + b44 + b45 ≥ −b13 − b23 − b25 + b33 + b34 , − b13 − b34 + b44 + b45 ≥ −b13 − b14 + b33 , − b13 − b34 + b44 + b45 ≥ −b13 − b25 + b33 , − b13 − b34 + b44 + b45 > −b24 − b25 + b33 , (E10 ) − b13 + b44 > b55 , − b13 + b44 ≥ −b13 − b23 + b44 + b45 , − b13 + b44 ≥ −b13 − b34 + b44 + b45 , − b13 + b44 > −b24 − b34 + b44 + b45 , − b13 + b44 > −b24 + b44 , − b13 + b44 > −b35 + b44 , − b13 + b44 ≥ −b13 − b14 − b15 + b22 , − b13 + b44 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b13 + b44 ≥ −b13 − b14 − b23 + b33 + b34 , − b13 + b44 ≥ −b13 − b23 − b25 + b33 + b34 , − b13 + b44 ≥ −b13 − b14 + b33 , − b13 + b44 ≥ −b13 − b25 + b33 , − b13 + b44 > −b24 − b25 + b33 , (E11 ) − b24 − b34 + b44 + b45 > b55 , − b24 − b34 + b44 + b45 ≥ −b13 − b23 + b44 + b45 , − b24 − b34 + b44 + b45 ≥ −b13 − b34 + b44 + b45 , − b24 − b34 + b44 + b45 > −b13 + b44 , − b24 − b34 + b44 + b45 > −b24 + b44 ,

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ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

− b24 − b34 + b44 + b45 > −b35 + b44 , − b24 − b34 + b44 + b45 ≥ −b13 − b14 − b15 + b22 , − b24 − b34 + b44 + b45 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b24 − b34 + b44 + b45 ≥ −b13 − b14 − b23 + b33 + b34 , − b24 − b34 + b44 + b45 ≥ −b13 − b23 − b25 + b33 + b34 , − b24 − b34 + b44 + b45 ≥ −b13 − b14 + b33 , − b24 − b34 + b44 + b45 ≥ −b13 − b25 + b33 , − b24 − b34 + b44 + b45 ≥ −b24 − b25 + b33 , (E12 ) − b24 + b44 > b55 , − b24 + b44 ≥ −b13 − b23 + b44 + b45 , − b24 + b44 ≥ −b13 − b34 + b44 + b45 , − b24 + b44 ≥ −b13 + b44 , − b24 + b44 ≥ −b24 − b34 + b44 + b45 , − b24 + b44 > −b35 + b44 , − b24 + b44 ≥ −b13 − b14 − b15 + b22 , − b24 + b44 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b24 + b44 ≥ −b13 − b14 − b23 + b33 + b34 , − b24 + b44 ≥ −b13 − b23 − b25 + b33 + b34 , − b24 + b44 ≥ −b13 − b14 + b33 , − b24 + b44 ≥ −b13 − b25 + b33 , − b24 + b44 ≥ −b24 − b25 + b33 , (E13 ) − b35 + b44 > b55 , − b35 + b44 ≥ −b13 − b23 + b44 + b45 , − b35 + b44 ≥ −b13 − b34 + b44 + b45 , − b35 + b44 ≥ −b13 + b44 , − b35 + b44 ≥ −b24 − b34 + b44 + b45 , − b35 + b44 ≥ −b24 + b44 , − b35 + b44 ≥ −b13 − b14 − b15 + b22 , − b35 + b44 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , − b35 + b44 ≥ −b13 − b14 − b23 + b33 + b34 , − b35 + b44 ≥ −b13 − b23 − b25 + b33 + b34 , − b35 + b44 ≥ −b13 − b14 + b33 , − b35 + b44 ≥ −b13 − b25 + b33 , − b35 + b44 ≥ −b24 − b25 + b33 , (E14 ) b55 ≥ −b13 − b23 + b44 + b45 , b55 ≥ −b13 − b34 + b44 + b45 , b55 ≥ −b13 + b44 , b55 ≥ −b24 − b34 + b44 + b45 , b55 ≥ −b24 + b44 ,

277

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KAILASH C. MISRA AND SUCHADA PONGPRASERT

b55 ≥ −b35 + b44 , b55 ≥ −b13 − b14 − b15 + b22 , b55 ≥ −b13 − b14 − b23 − b24 + b33 + b34 + b35 , b55 ≥ −b13 − b14 − b23 + b33 + b34 , b55 ≥ −b13 − b23 − b25 + b33 + b34 , b55 ≥ −b13 − b14 + b33 , b55 ≥ −b13 − b25 + b33 , b55 ≥ −b24 − b25 + b33 . Then we define conditions (Fj ) (1 ≤ j ≤ 14) by replacing > (resp. ≥) with ≥ (resp. >) in (Ej ). Let b = (bij ) ∈ B. Then e˜k (b) = (bij ) where ⎧

b = b11 − 1, b 16 = b16 + 1, b 22 = b22 − 1, b 27 = b27 + 1, b 36 = b36 − 1, ⎪ ⎪ ⎪ 11 ⎪ ⎪ b 38 = b38 + 1, b 47 = b47 − 1, b 49 = b49 + 1, b 59 = b59 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ ⎪

⎪ b69 = b69 − 1, b 6,11 = b6,11 + 1 if (E1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 − 1, b 15 = b15 + 1, b 22 = b22 − 1, b 26 = b26 + 1, b 35 = b35 − 1, ⎪ ⎪ ⎪

⎪ b38 = b38 + 1, b 46 = b46 − 1, b 49 = b49 + 1, b 58 = b58 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b = b69 − 1, b 6,11 = b6,11 + 1 if (E2 ) ⎪ ⎪ ⎪ 69 ⎪ ⎪ ⎪ b 11 = b11 − 1, b 15 = b15 + 1, b 22 = b22 − 1, b 24 = b24 + 1, b 25 = b25 − 1, ⎪ ⎪ ⎪ ⎪ b 26 = b26 + 1, b 34 = b34 − 1, b 38 = b38 + 1, b 46 = b46 − 1, b 47 = b47 + 1, ⎪ ⎪ ⎪ ⎪ ⎪b 48 = b48 − 1, b 49 = b49 + 1b 57 = b57 − 1, b 5,10 = b5,10 + 1, b 69 = b69 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 6,11 = b6,11 + 1 if (E3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 − 1, b 14 = b14 + 1, b 22 = b22 − 1, b 26 = b26 + 1, b 34 = b34 − 1, ⎪ ⎪ ⎪ ⎪b 37 = b37 + 1, b 46 = b46 − 1, b 49 = b49 + 1, b 57 = b57 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 69 = b69 − 1, b 6,11 = b6,11 + 1 if (E4 ) ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 − 1, b 15 = b15 + 1, b 22 = b22 − 1, b 23 = b23 + 1, b 25 = b25 − 1, ⎪ ⎪ ⎪ ⎪b 26 = b26 + 1, b 33 = b33 − 1, b 38 = b38 + 1, b 46 = b46 − 1, b 47 = b47 + 1, ⎪ ⎨ k = 0 : b 48 = b48 − 1, b 49 = b49 + 1b 57 = b57 − 1, b 58 = b58 + 1, b 59 = b59 − 1, ⎪ ⎪ ⎪b 5,10 = b5,10 + 1, b 68 = b68 − 1, b 6,11 = b6,11 + 1 if (E5 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 − 1, b 14 = b14 + 1, b 22 = b22 − 1, b 25 = b25 + 1, b 34 = b34 − 1, ⎪ ⎪ ⎪ ⎪ b 36 = b36 + 1, b 45 = b45 − 1, b 49 = b49 + 1, b 56 = b56 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ ⎪ ⎪b 69 = b69 − 1, b 6,11 = b6,11 + 1 if (E6 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 − 1, b 14 = b14 + 1, b 22 = b22 − 1, b 23 = b23 + 1, b 24 = b24 − 1, ⎪ ⎪ ⎪ ⎪ b 26 = b26 + 1, b 33 = b33 − 1, b 37 = b37 + 1, b 46 = b46 − 1, b 48 = b48 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 57 = b57 − 1, b 58 = b58 + 1, b 59 = b59 − 1, b 5,10 = b5,10 + 1, b 68 = b68 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 6,11 = b6,11 + 1 if (E7 ) ⎪ ⎪ ⎪ ⎪

⎪ b11 = b11 − 1, b 13 = b13 + 1, b 22 = b22 − 1, b 26 = b26 + 1, b 33 = b33 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b = b37 + 1, b 46 = b46 − 1, b 48 = b48 + 1, b 57 = b57 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ 37 ⎪ ⎪ b 68 = b68 − 1, b 6,11 = b6,11 + 1 if (E8 ) ⎪ ⎪ ⎪ ⎪

⎪ ⎪ b = b11 − 1, b 14 = b14 + 1, b 22 = b22 − 1, b 23 = b23 + 1, b 24 = b24 − 1, ⎪ ⎪ 11 ⎪ ⎪ b 25 = b25 + 1, b 33 = b33 − 1, b 36 = b36 + 1, b 45 = b45 − 1, b 49 = b49 + 1, ⎪ ⎪ ⎪ ⎪

⎪ b56 = b56 − 1, b 58 = b58 + 1b 59 = b59 − 1, b 5,10 = b5,10 + 1, b 68 = b68 − 1, ⎪ ⎪ ⎪ ⎩

b6,11 = b6,11 + 1 if (E9 )

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ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

⎧







⎪ ⎪b11 = b11 − 1, b14 = b14 + 1, b22 = b22 − 1, b23 = b23 + 1, b24 = b24 − 1, ⎪ ⎪





⎪ ⎪ b25 = b25 + 1, b33 = b33 − 1, b34 = b34 + 1, b35 = b35 − 1, b 36 = b36 + 1, ⎪ ⎪ ⎪ ⎪ b 44 = b44 − 1, b 49 = b49 + 1b 56 = b56 − 1, b 57 = b57 + 1, b 59 = b59 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 5,10 = b5,10 + 1, b 67 = b67 − 1, b 6,11 = b6,11 + 1 if (E10 ) ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 − 1, b 13 = b13 + 1, b 22 = b22 − 1, b 25 = b25 + 1, b 33 = b33 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 36 = b36 + 1, b 45 = b45 − 1, b 48 = b48 + 1, b 56 = b56 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ ⎪ ⎪b 68 = b68 − 1, b 6,11 = b6,11 + 1 if (E11 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 − 1, b 13 = b13 + 1, b 22 = b22 − 1, b 25 = b25 + 1, b 33 = b33 − 1, ⎨ k = 0 : b 34 = b34 + 1, b 35 = b35 − 1, b 36 = b36 + 1, b 44 = b44 − 1, b 48 = b48 + 1, ⎪ ⎪ ⎪ b 56 = b56 − 1, b 57 = b57 + 1b 58 = b58 − 1, b 5,10 = b5,10 + 1, b 67 = b67 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 6,11 = b6,11 + 1 if (E12 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 − 1, b 13 = b13 + 1, b 22 = b22 − 1, b 24 = b24 + 1, b 33 = b33 − 1, ⎪ ⎪ ⎪ ⎪b 36 = b36 + 1, b 44 = b44 − 1, b 47 = b47 + 1, b 56 = b56 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 67 = b67 − 1, b 6,11 = b6,11 + 1 if (E13 ) ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 − 1, b 13 = b13 + 1, b 22 = b22 − 1, b 24 = b24 + 1, b 33 = b33 − 1, ⎪ ⎪ ⎪ ⎪ b 35 = b35 + 1, b 44 = b44 − 1, b 46 = b46 + 1, b 56 = b56 − 1, b 5,10 = b5,10 + 1, ⎪ ⎪ ⎪ ⎩

b67 = b67 − 1, b 6,11 = b6,11 + 1 if (E14 ) k = 1 : b 11 = b11 + 1, b 12 = b12 − 1, b 6,10 = b6,10 + 1, b 6,11 = b6,11 − 1  b 12 = b12 + 1, b 13 = b13 − 1, b 59 = b59 + 1, b 5,10 = b5,10 − 1 if b12 ≥ b23 k=2: b 22 = b22 + 1, b 23 = b23 − 1, b 69 = b69 + 1, b 6,10 = b6,10 − 1 if b12 < b23 ⎧

b13 = b13 + 1, b 14 = b14 − 1, b 48 = b48 + 1, b 49 = b49 − 1 ⎪ ⎪ ⎪ ⎪ ⎪ if b13 ≥ b24 , b13 + b23 ≥ b24 + b34 ⎪ ⎪ ⎪ ⎨b = b + 1, b = b − 1, b = b + 1, b = b − 1 23 24 58 59 23 24 58 59 k=3: ⎪ < b , b ≥ b if b 13 24 23 34 ⎪ ⎪ ⎪ ⎪ ⎪ b 33 = b33 + 1, b 34 = b34 − 1, b 68 = b68 + 1, b 69 = b69 − 1 ⎪ ⎪ ⎩ if b13 + b23 < b24 + b34 , b23 < b34 ⎧

b14 = b14 + 1, b 15 = b15 − 1, b 37 = b37 + 1, b 38 = b38 − 1 ⎪ ⎪ ⎪ ⎪ ⎪ if b14 ≥ b25 , b14 + b24 ≥ b25 + b35 , b14 + b24 + b34 ≥ b25 + b35 + b45 ⎪ ⎪ ⎪ ⎪





⎪ ⎪ ⎪b24 = b24 + 1, b25 = b25 − 1, b47 = b47 + 1, b48 = b48 − 1 ⎪ ⎨ if b14 < b25 , b24 ≥ b35 , b24 + b34 ≥ b35 + b45 k=4:

⎪ b = b + 1, b 35 = b35 − 1, b 57 = b57 + 1, b 58 = b58 − 1 34 ⎪ 34 ⎪ ⎪ ⎪ ⎪ if b14 + b24 < b25 + b35 , b24 < b35 , b34 ≥ b45 ⎪ ⎪ ⎪

⎪ ⎪b44 = b44 + 1, b 45 = b45 − 1, b 67 = b67 + 1, b 68 = b68 − 1 ⎪ ⎪ ⎩ if b14 + b24 + b34 < b25 + b35 + b45 , b24 + b34 < b35 + b45 , b34 < b45 ⎧

b25 = b25 + 1, b 26 = b26 − 1, b 36 = b36 + 1, b 37 = b37 − 1 ⎪ ⎪ ⎪ ⎨ if b25 + b44 + b45 ≥ b33 + b34 k=5:





⎪ b = b 45 + 1, b46 = b46 − 1, b56 = b56 + 1, b57 = b57 − 1 45 ⎪ ⎪ ⎩ if b25 + b44 + b45 < b33 + b34

279

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KAILASH C. MISRA AND SUCHADA PONGPRASERT

⎧ ⎪ b 15 = b15 + 1, b 16 = b16 − 1, b 26 = b26 + 1, b 27 = b27 − 1 ⎪ ⎪ ⎪ ⎪ ⎪ if b15 + b33 + b34 + b35 ≥ b22 + b23 + b24 , ⎪ ⎪ ⎪ ⎪ ⎪ b15 + b33 + b34 + 2b35 + b55 ≥ b22 + b23 + b24 + b44 ⎨ k = 6 : b 35 = b35 + 1, b 36 = b36 − 1, b 46 = b46 + 1, b 47 = b47 − 1 ⎪ ⎪ ⎪ if b15 + b33 + b34 + b35 < b22 + b23 + b24 , b35 + b55 ≥ b44 ⎪ ⎪ ⎪ ⎪

⎪ b55 = b55 + 1, b 56 = b56 − 1, b 66 = b66 + 1, b 67 = b67 − 1 ⎪ ⎪ ⎪ ⎩ if b15 + b33 + b34 + 2b35 + b55 < b22 + b23 + b24 + b44 , b35 + b55 < b44

and bij = bij otherwise. Also f˜k (b) = (bij ) where ⎧

b = b11 + 1, b 16 = b16 − 1, b 22 = b22 + 1, b 27 = b27 − 1, b 36 = b36 + 1, ⎪ ⎪ ⎪ 11 ⎪ ⎪ b 38 = b38 − 1, b 47 = b47 + 1, b 49 = b49 − 1, b 59 = b59 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ ⎪

⎪ b69 = b69 + 1, b 6,11 = b6,11 − 1 if (F1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 + 1, b 15 = b15 − 1, b 22 = b22 + 1, b 26 = b26 − 1, b 35 = b35 + 1, ⎪ ⎪ ⎪

⎪ b38 = b38 − 1, b 46 = b46 + 1, b 49 = b49 − 1, b 58 = b58 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b = b69 + 1, b 6,11 = b6,11 − 1 if (F2 ) ⎪ ⎪ ⎪ 69 ⎪ ⎪ ⎪ b 11 = b11 + 1, b 15 = b15 − 1, b 22 = b22 + 1, b 24 = b24 − 1, b 25 = b25 + 1, ⎪ ⎪ ⎪ ⎪ b 26 = b26 − 1, b 34 = b34 + 1, b 38 = b38 − 1, b 46 = b46 + 1, b 47 = b47 − 1, ⎪ ⎪ ⎪ ⎪ ⎪b 48 = b48 + 1, b 49 = b49 − 1, b 57 = b57 + 1, b 5,10 = b5,10 − 1, b 69 = b69 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 6,11 = b6,11 − 1 if (F3 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 + 1, b 14 = b14 − 1, b 22 = b22 + 1, b 26 = b26 − 1, b 34 = b34 + 1, ⎪ ⎪ ⎪ ⎪b 37 = b37 − 1, b 46 = b46 + 1, b 49 = b49 − 1, b 57 = b57 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 69 = b69 + 1, b 6,11 = b6,11 − 1 if (F4 ) ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 + 1, b 15 = b15 − 1, b 22 = b22 + 1, b 23 = b23 − 1, b 25 = b25 + 1, ⎪ ⎪ ⎪ ⎪b 26 = b26 − 1, b 33 = b33 + 1, b 38 = b38 − 1, b 46 = b46 + 1, b 47 = b47 − 1, ⎪ ⎨ k = 0 : b 48 = b48 + 1, b 49 = b49 − 1b 57 = b57 + 1, b 58 = b58 − 1, b 59 = b59 + 1, ⎪ ⎪ ⎪b 5,10 = b5,10 − 1, b 68 = b68 + 1, b 6,11 = b6,11 − 1 if (F5 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 + 1, b 14 = b14 − 1, b 22 = b22 + 1, b 25 = b25 − 1, b 34 = b34 + 1, ⎪ ⎪ ⎪ ⎪ b 36 = b36 − 1, b 45 = b45 + 1, b 49 = b49 − 1, b 56 = b56 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ ⎪ ⎪b 69 = b69 + 1, b 6,11 = b6,11 − 1 if (F6 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 + 1, b 14 = b14 − 1, b 22 = b22 + 1, b 23 = b23 − 1, b 24 = b24 + 1, ⎪ ⎪ ⎪ ⎪ b 26 = b26 − 1, b 33 = b33 + 1, b 37 = b37 − 1, b 46 = b46 + 1, b 48 = b48 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 57 = b57 + 1, b 58 = b58 − 1, b 59 = b59 + 1, b 5,10 = b5,10 − 1, b 68 = b68 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 6,11 = b6,11 − 1 if (F7 ) ⎪ ⎪ ⎪ ⎪

⎪ b11 = b11 + 1, b 13 = b13 − 1, b 22 = b22 + 1, b 26 = b26 − 1, b 33 = b33 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b = b37 − 1, b 46 = b46 + 1, b 48 = b48 − 1, b 57 = b57 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ 37 ⎪ ⎪ b 68 = b68 + 1, b 6,11 = b6,11 − 1 if (F8 ) ⎪ ⎪ ⎪ ⎪

⎪ ⎪ b = b11 + 1, b 14 = b14 − 1, b 22 = b22 + 1, b 23 = b23 − 1, b 24 = b24 + 1, ⎪ ⎪ 11 ⎪ ⎪ b 25 = b25 − 1, b 33 = b33 + 1, b 36 = b36 − 1, b 45 = b45 + 1, b 49 = b49 − 1, ⎪ ⎪ ⎪ ⎪

⎪ b56 = b56 + 1, b 58 = b58 − 1b 59 = b59 + 1, b 5,10 = b5,10 − 1, b 68 = b68 + 1, ⎪ ⎪ ⎪ ⎩

b6,11 = b6,11 − 1 if (F9 )

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ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

⎧







⎪ ⎪b11 = b11 + 1, b14 = b14 − 1, b22 = b22 + 1, b23 = b23 − 1, b24 = b24 + 1, ⎪ ⎪





⎪ ⎪ b25 = b25 − 1, b33 = b33 + 1, b34 = b34 − 1, b35 = b35 + 1, b 36 = b36 − 1, ⎪ ⎪ ⎪ ⎪ b 44 = b44 + 1, b 49 = b49 − 1b 56 = b56 + 1, b 57 = b57 − 1, b 59 = b59 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 5,10 = b5,10 − 1, b 67 = b67 + 1, b 6,11 = b6,11 − 1 if (F10 ) ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 + 1, b 13 = b13 − 1, b 22 = b22 + 1, b 25 = b25 − 1, b 33 = b33 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 36 = b36 − 1, b 45 = b45 + 1, b 48 = b48 − 1, b 56 = b56 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ ⎪ ⎪b 68 = b68 + 1, b 6,11 = b6,11 − 1 if (F11 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 11 = b11 + 1, b 13 = b13 − 1, b 22 = b22 + 1, b 25 = b25 − 1, b 33 = b33 + 1, ⎨ k = 0 : b 34 = b34 − 1, b 35 = b35 + 1, b 36 = b36 − 1, b 44 = b44 + 1, b 48 = b48 − 1, ⎪ ⎪ ⎪ b 56 = b56 + 1, b 57 = b57 − 1b 58 = b58 + 1, b 5,10 = b5,10 − 1, b 67 = b67 + 1, ⎪ ⎪ ⎪ ⎪ ⎪ b 6,11 = b6,11 − 1 if (F12 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 + 1, b 13 = b13 − 1, b 22 = b22 + 1, b 24 = b24 − 1, b 33 = b33 + 1, ⎪ ⎪ ⎪ ⎪b 36 = b36 − 1, b 44 = b44 + 1, b 47 = b47 − 1, b 56 = b56 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪b 67 = b67 + 1, b 6,11 = b6,11 − 1 if (F13 ) ⎪ ⎪ ⎪ ⎪ ⎪ b 11 = b11 + 1, b 13 = b13 − 1, b 22 = b22 + 1, b 24 = b24 − 1, b 33 = b33 + 1, ⎪ ⎪ ⎪ ⎪ b 35 = b35 − 1, b 44 = b44 + 1, b 46 = b46 − 1, b 56 = b56 + 1, b 5,10 = b5,10 − 1, ⎪ ⎪ ⎪ ⎩

b67 = b67 + 1, b 6,11 = b6,11 − 1 if (F14 ) k = 1 : b 11 = b11 − 1, b 12 = b12 + 1, b 6,10 = b6,10 − 1, b 6,11 = b6,11 + 1  b 12 = b12 − 1, b 13 = b13 + 1, b 59 = b59 − 1, b 5,10 = b5,10 + 1 if b12 > b23 k=2: b 22 = b22 − 1, b 23 = b23 + 1, b 69 = b69 − 1, b 6,10 = b6,10 + 1 if b12 ≤ b23 ⎧

b13 = b13 − 1, b 14 = b14 + 1, b 48 = b48 − 1, b 49 = b49 + 1 ⎪ ⎪ ⎪ ⎪ ⎪ if b13 > b24 , b13 + b23 > b24 + b34 ⎪ ⎪ ⎪ ⎨b = b − 1, b = b + 1, b = b − 1, b = b + 1 23 24 58 59 23 24 58 59 k=3: ⎪ ≤ b , b > b if b 13 24 23 34 ⎪ ⎪ ⎪ ⎪ ⎪ b 33 = b33 − 1, b 34 = b34 + 1, b 68 = b68 − 1, b 69 = b69 + 1 ⎪ ⎪ ⎩ if b13 + b23 ≤ b24 + b34 , b23 ≤ b34 ⎧

b14 = b14 − 1, b 15 = b15 + 1, b 37 = b37 − 1, b 38 = b38 + 1 ⎪ ⎪ ⎪ ⎪ ⎪ if b14 > b25 , b14 + b24 > b25 + b35 , b14 + b24 + b34 > b25 + b35 + b45 ⎪ ⎪ ⎪ ⎪





⎪ ⎪ ⎪b24 = b24 − 1, b25 = b25 + 1, b47 = b47 − 1, b48 = b48 + 1 ⎪ ⎨ if b14 ≤ b25 , b24 > b35 , b24 + b34 > b35 + b45 k=4:

⎪ b = b − 1, b 35 = b35 + 1, b 57 = b57 − 1, b 58 = b58 + 1 34 ⎪ 34 ⎪ ⎪ ⎪ ⎪ if b14 + b24 ≤ b25 + b35 , b24 ≤ b35 , b34 > b45 ⎪ ⎪ ⎪

⎪ ⎪b44 = b44 − 1, b 45 = b45 + 1, b 67 = b67 − 1, b 68 = b68 + 1 ⎪ ⎪ ⎩ if b14 + b24 + b34 ≤ b25 + b35 + b45 , b24 + b34 ≤ b35 + b45 , b34 ≤ b45 ⎧

b25 = b25 − 1, b 26 = b26 + 1, b 36 = b36 − 1, b 37 = b37 + 1 ⎪ ⎪ ⎪ ⎨ if b25 + b44 + b45 > b33 + b34 k=5:





⎪ b = b 45 − 1, b46 = b46 + 1, b56 = b56 − 1, b57 = b57 + 1 45 ⎪ ⎪ ⎩ if b25 + b44 + b45 ≤ b33 + b34

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KAILASH C. MISRA AND SUCHADA PONGPRASERT

⎧ ⎪ b 15 = b15 − 1, b 16 = b16 + 1, b 26 = b26 − 1, b 27 = b27 + 1 ⎪ ⎪ ⎪ ⎪ ⎪ if b15 + b33 + b34 + b35 > b22 + b23 + b24 , ⎪ ⎪ ⎪ ⎪ ⎪ b15 + b33 + b34 + 2b35 + b55 > b22 + b23 + b24 + b44 ⎨ k = 6 : b 35 = b35 − 1, b 36 = b36 + 1, b 46 = b46 − 1, b 47 = b47 + 1 ⎪ ⎪ ⎪ if b15 + b33 + b34 + b35 ≤ b22 + b23 + b24 , b35 + b55 > b44 ⎪ ⎪ ⎪ ⎪

⎪ b55 = b55 − 1, b 56 = b56 + 1, b 66 = b66 − 1, b 67 = b67 + 1 ⎪ ⎪ ⎪ ⎩ if b15 + b33 + b34 + 2b35 + b55 ≤ b22 + b23 + b24 + b44 , b35 + b55 ≤ b44

and bij = bij otherwise. For b ∈ B 6,l if e˜k (b) or f˜k (b) does not belong to B 6,l , then we assume it to be 0. The maps εk (b), ϕk (b) and wtk (b) for k = 0, 1, 2, 3, 4, 5, 6 are given as follows. We 6 6 observe that wtk (b) = ϕk (b) − εk (b), ϕ(b) = k=0 ϕk (b)Λk , ε(b) = k=0 εk (b)Λk and wt(b) = ϕ(b) − ε(b). ε0 (b) =

l + A1 if b ∈ B 6,l , A1 if b ∈ B 6,∞ ,

where A1 = max{−b56 − b57 − b58 − b59 − b5,10 , −b13 − b23 − b46 − b47 − b48 − b49 , − b13 − b34 − b46 − b47 − b48 − b49 , −b13 − b45 − b46 − b47 − b48 − b49 , − b24 − b34 − b46 − b47 − b48 − b49 , −b24 − b45 − b46 − b47 − b48 − b49 , − b35 − b45 − b46 − b47 − b48 − b49 , −b13 − b14 − b15 − b23 − b24 − b25 − b26 − b27 , −b13 − b14 − b23 − b24 − b36 − b37 − b38 , −b13 − b14 − b23 − b35 − b36 − b37 − b38 , −b13 − b23 − b25 − b35 − b36 − b37 − b38 , −b13 − b14 − b34 − b35 − b36 − b37 − b38 , −b13 − b25 − b34 − b35 − b36 − b37 − b38 , −b24 − b25 − b34 − b35 − b36 − b37 − b38 }, ε1 (b) = b12 , ε2 (b) = max{b13 , −b12 + b13 + b23 }, ε3 (b) = max{b14 , −b13 + b14 + b24 , −b13 + b14 − b23 + b24 + b34 }, ε4 (b) = max{b15 , −b14 + b15 + b25 , −b14 + b15 − b24 + b25 + b35 , − b14 + b15 − b24 + b25 − b34 + b35 + b45 }, ε5 (b) = max{b11 + b12 + b13 + b14 − b22 − b23 − b24 − b25 , b11 + b12 + b13 + b14 − b22 − b23 − b24 − 2b25 + b33 + b34 − b44 − b45 }, ε6 (b) =

l + A2 if b ∈ B 6,l , A2 if b ∈ B 6,∞ ,

where A2 = max{−b11 − b12 − b13 − b14 − b15 , −b11 − b12 − b13 − b14 − 2b15 + b22 + b23 +b24 − b33 −b34 −b35 , −b11 −b12 −b13 −b14 −2b15 +b22 +b23 + b24 − b33 − b34 − 2b35 + b44 − b55 }, ϕ0 (b) =

l + A3 , if b ∈ B 6,l , A3 , if b ∈ B 6,∞ ,

where A3 = max{−b11 − b12 + b23 + b24 + b25 + b26 + b27 − b56 − b57 − b58 − b59 − b5,10 , −b11 − b12 − b13 + b24 + b25 + b26 + b27 − b46 − b47 − b48 − b49 , −b11 − b12 − b13 + b23 + b24 + b25 + b26 + b27 − b34 − b46 − b47

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− b48 − b49 , −b11 − b12 − b13 + b23 + b24 + b25 + b26 + b27 − b45 − b46 − b47 − b48 − b49 , −b11 − b12 + b23 + b25 + b26 + b27 − b34 − b46 − b47 − b48 − b49 , −b11 − b12 + b23 + b25 + b26 + b27 − b45 − b46 − b47 − b48 − b49 , −b11 − b12 + b23 + b24 + b25 + b26 + b27 − b35 − b45 − b46 − b47 − b48 − b49 , −b11 − b12 − b13 − b14 − b15 , −b11 − b12 − b13 − b14 + b25 + b26 + b27 − b36 − b37 − b38 , −b11 − b12 − b13 − b14 + b24 + b25 + b26 + b27 − b35 − b36 − b37 − b38 , −b11 − b12 − b13 + b24 + b26 + b27 − b35 − b36 − b37 − b38 , −b11 − b12 − b13 − b14 + b23 + b24 + b25 + b26 + b27 − b34 − b35 − b36 − b37 − b38 , −b11 − b12 − b13 + b23 + b24 + b26 + b27 − b34 − b35 − b36 − b37 − b38 , −b11 − b12 + b23 + b26 + b27 − b34 − b35 − b36 − b37 − b38 }, ϕ1 (b) = b11 − b22 , ϕ2 (b) = max{b22 − b33 , b12 + b22 − b23 − b33 }, ϕ3 (b) = max{b33 − b44 , b23 + b33 − b34 − b44 , b13 + b23 − b24 + b33 − b34 − b44 }, ϕ4 (b) = max{b44 − b55 , b34 + b44 − b45 − b55 , b24 + b34 − b35 + b44 − b45 − b55 , b14 + b24 − b25 + b34 − b35 + b44 − b45 − b55 }, ϕ5 (b) = max{b45 , b25 −b33 −b34 +b44 +2b45 }, ϕ6 (b) =

l + A4 , if b ∈ B 6,i , A4 , if b ∈ B 6,∞ ,

where A4 = max{−b56 −b57 −b58 −b59 −b5,10 , b35 −b44 +b55 −b56 −b57 −b58 − b59 −b5,10 , b15 −b22 −b23 −b24 +b33 +b34 +2b35 −b44 +b55 −b56 − b57 − b58 − b59 − b5,10 }. wt0 (b) = −b11 − b12 + b23 + b24 + b25 + b26 + b27 , wt1 (b) = b11 − b12 − b22 , wt2 (b) = b12 − b13 + b22 − b23 − b33 , wt3 (b) = b13 − b14 + b23 − b24 + b33 − b34 − b44 , wt4 (b) = b14 − b15 + b24 − b25 + b34 − b35 + b44 − b45 − b55 , wt5 (b) = −b11 − b12 − b13 − b14 +b22 +b23 +b24 +2b25 − b33 − b34 +b44 +2b45 , wt6 (b) = b11 +b12 +b13 +b14 +2b15 − b22 − b23 − b24 +b33 +b34 +2b35 − b44 + b55 − b56 − b57 − b58 − b59 − b5,10 . Choose elements b00 , b01 , b02 , b03 , b04 , b05 , b06 where (b00 )ij = 1

if (i, j) = (1, 6), (2, 7), (3, 8), (4, 9), (5, 10), (6, 11),

(b01 )ij (b02 )ij

=1

if (i, j) = (1, 1), (2, 6), (3, 7), (4, 8), (5, 9), (6, 10),

=1

if (i, j) = (1, 1), (1, 5), (2, 2), (2, 6), (3, 6), (3, 8), (4, 7), (4, 9), (5, 8), (5, 10), (6, 9), (6, 11),

(b03 )ij

=1

if (i, j) = (1, 1), (1, 4), (2, 2), (2, 5), (3, 3), (3, 6), (4, 6), (4, 9), (5, 7), (5, 10), (6, 8), (6, 11),

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KAILASH C. MISRA AND SUCHADA PONGPRASERT

(b04 )ij = 1

if (i, j) = (1, 1), (1, 3), (2, 2), (2, 4), (3, 3), (3, 5), (4, 4), (4, 6), (5, 6), (5, 10), (6, 7), (6, 11),

(b05 )ij (b06 )ij

=1

if (i, j) = (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 11),

=1

if (i, j) = (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6),

= 0 otherwise, for 0 ≤ k ≤ 6. As shown in [6], the crystal B 6,l is a perfect crystal with the set of minimal elements: and

(b0k )ij

(B 6,l )min = {b ∈ B 6,l | c, ε(b) = l} =

6 

C

ak b0k | ak ∈ Z≥0 , a0 + a1 + 2a2 + 2a3 + 2a4 + a5 + a6 = l .

k=0

For λ ∈ Pcl , consider the crystal Tλ = {tλ } with e˜k (tλ ) = f˜k (tλ ) = 0, εk (tλ ) = ϕk (tλ ) = −∞, wt(tλ ) = λ, for k = 0, 1, 2, 3, 4, 5, 6. Then for λ, μ ∈ Pcl , Tλ ⊗ B 6,l ⊗ Tμ is a crystal with the structure given by e˜k (tλ ⊗ b ⊗ tμ ) = tλ ⊗ e˜k b ⊗ tμ , f˜k (tλ ⊗ b ⊗ tμ ) = tλ ⊗ f˜k b ⊗ tμ , ˇ k , λ , εk (tλ ⊗ b ⊗ tμ ) = εk (b) − α

ϕk (tλ ⊗ b ⊗ tμ ) = ϕk (b) + α ˇ k , μ ,

wt(tλ ⊗ b ⊗ tμ ) = λ + μ + wt(b) where tλ ⊗ b ⊗ tμ ∈ Tλ ⊗ B 6,l ⊗ Tμ . The notion of a coherent family of perfect crystals and its limit is defined in (1) [7]. In the following theorem we prove that the family of D6 crystals {B 6,l }l≥1 6,∞ form a coherent family with limit B containing the special vector b∞ = 0 (i.e. ∞ (b )ij = 0 for i ≤ j ≤ i + 5, 1 ≤ i ≤ 6). Theorem 2.1. The family of perfect crystals {B 6,l }l≥1 forms a coherent family and the crystal B 6,∞ is its limit. Proof. Set J = {(l, b)|l ∈ Z>0 , b ∈ (B 6,l )min }. By ([7], Definition 4.1), we need to show that (1) wt(b∞ ) = 0, ε(b∞ ) = ϕ(b∞ ) = 0, (2) for any (l, b) ∈ J, there exists an embedding of crystals f(l,b) : Tε(b) ⊗ B 6,l ⊗ T−ϕ(b) −→ B 6,∞ where f(l,b) (tε(b) ⊗ b ⊗ t−ϕ(b) ) = b∞ , (3) B 6,∞ = ∪(l,b)∈J Im f(l,b) . Since εk (b∞ ) = 0, ϕk (b∞ ) = 0, 0 ≤ k ≤ 6, we have ε(b∞ ) = 0, ϕ(b∞ ) = 0 and hence wt(b∞ ) = 0 which proves (1). Let l ∈ Z>0 and b0 = (b0ij ) be an element of (B 6,l )min . Then there exist ak ∈ Z≥0 , 0 ≤ k ≤ 6 such that a0 + a1 + 2a2 + 2a3 + 2a4 + a5 + a6 = l and b011 = a1 + a2 + a3 + a4 + a6 , b012 = a5 , b013 = a4 , b014 = a3 , b015 = a2 , b016 = a0 , b022 = a2 + a3 + a4 + a6 , b023 = a5 , b024 = a4 , b025 = a3 , b026 = a1 + a2 , b027 = a0 , b033 = a3 + a4 + a6 , b034 = a5 , b035 = a4 , b036 = a2 + a3 , b037 = a1 , b038 = a0 + a2 ,

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b044 = a4 + a6 , b045 = a5 , b046 = a3 + a4 , b047 = a2 , b048 = a1 , b049 = a0 + a2 + a3 , b055 = a6 , b056 = a4 + a5 , b057 = a3 , b058 = a2 , b059 = a1 , b05,10 = a0 +a2 +a3 +a4 , b066 = a6 , b067 = a4 , b068 = a3 , b069 = a2 , b06,10 = a1 , b06,11 = a0 +a2 +a3 +a4 +a5 , ε(b0 ) = a6 Λ0 + a5 Λ1 + a4 Λ2 + a3 Λ3 + a2 Λ4 + a1 Λ5 + a0 Λ6 , ϕ(b0 ) = a0 Λ0 + a1 Λ1 + a2 Λ2 + a3 Λ3 + a4 Λ4 + a5 Λ5 + a6 Λ6 . For any b = (bij ) ∈ B 6,l , we define a map f(l,b0 ) : Tε(b0 ) ⊗ B 6,l ⊗ T−ϕ(b0 ) −→ B 6,∞ by f(l,b0 ) (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) ) = b = (bij ) where bij = bij − b0ij for all i ≤ j ≤ i + 5, 1 ≤ i ≤ 6. Then it is easy to see that εk (b ) = εk (b) − a6−k = εk (b) − α ˇ k , ε(b0 ) for 0 ≤ k ≤ 6, ϕk (b ) = ϕk (b) − ak = ϕk (b) + α ˇ k , −ϕ(b0 ) for 0 ≤ k ≤ 6. Hence we have εk (b ) = εk (b) − α ˇ k , ε(b0 ) = εk (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) ), ϕk (b ) = ϕk (b) + α ˇ k , −ϕ(b0 ) = ϕk (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) ), 6 

wt(b ) =

(ϕk (b ) − εk (b ))Λk = wt(b) +

k=0

6 

α ˇ k , −ϕ(b0 ) Λk +

k=0

6 

α ˇ k , ε(b0 ) Λk

k=0

= wt(b) − ϕ(b0 ) + ε(b0 ) = wt(tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) ). For 0 ≤ k ≤ 6, b ∈ B 6,l , it can be checked easily that the conditions for the action of e˜k on b = b − b0 hold if and only if the conditions for the action of e˜k on b hold. Hence from the defined action of e˜k , we see that e˜k (b ) = e˜k (b)−b0 , 0 ≤ k ≤ 6. This implies that f(l,b0 ) (e˜k (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) )) = f(l,b0 ) (tε(b0 ) ⊗ e˜k (b) ⊗ t−ϕ(b0 ) ) = e˜k (b) − b0 = e˜k (b ) = e˜k (f(l,b0 ) (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) )). Similarly, we have f(l,b0 ) (f˜k (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) )) = f˜k (f(l,b0 ) (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) )). Clearly the map f(l,b0 ) is injective with f(l,b0 ) (tε(b0 ) ⊗ b0 ⊗ t−ϕ(b0 ) ) = b∞ . This proves (2). i+5 i+5 i+5 We observe that j=i bij = j=i bij − j=i b0ij = l − l = 0 for all 1 ≤ i ≤ 6. Also, b 11 = b11 − b011 = b66 + b67 + b68 + b69 + b6,10 − a1 − a2 − a3 − a4 − a6 = b 66 + b 67 + b 68 + b 69 + b 6,10 , b 11

+

b 12

= b11 − b011 + b12 − b012 = b55 + b56 + b57 + b58 + b59 − a1 − a2 − a3 − a4 − a5 − a6 = b 55 + b 56 + b 57 + b 58 + b 59 ,

b 11

+

b 12

+

b 13

= b11 − b011 + b12 − b012 + b13 − b013 = b44 + b45 + b46 + b47 + b48 − a1 − a2 − a3 − 2a4 − a5 − a6 = b 44 + b 45 + b 46 + b 47 + b 48 ,

b 11

+

b 12

+

b 13

+

b 14

= b11 − b011 + b12 − b012 + b13 − b013 + b14 − b014 = b33 + b34 + b35 + b36 + b37 − a1 − a2 − 2a3 − 2a4 − a5 − a6

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KAILASH C. MISRA AND SUCHADA PONGPRASERT

= b 33 + b 34 + b 35 + b 36 + b 37 , b 11 + b 12 + b 13 + b 14 + b 15 = b11 − b011 + b12 − b012 + b13 − b013 + b14 − b014 + b15 − b015 = b22 + b23 + b24 + b25 + b26 − a1 − 2a2 − 2a3 − 2a4 − a5 − a6 = b 22 + b 23 + b 24 + b 25 + b 26 .

 5+t   Similarly, we can show that 6−t j=i bij = j=i+t bi+t,j , for 2 ≤ i ≤ 5, 1 ≤ t ≤ 5. Hence we have B 6,∞ ⊇ ∪(l,b)∈J Im f(l,b) . To prove (3) we also need to show that B 6,∞ ⊆ ∪(l,b)∈J Im f(l,b) . Let b = (bij ) ∈ B 6,∞ . By (2), we can assume that b = b∞ . Set a1 = max{−b11 + b22 , −b11 − b12 + b22 + b23 , −b11 − b12 − b13 + b22 + b23 + b24 , − b11 − b12 − b13 − b14 + b22 + b23 + b24 + b25 , 0}, a2 = max{−b22 + b33 , −b22 − b23 + b33 + b34 , −b22 − b23 − b24 + b33 + b34 + b35 , − b15 , −b26 − a1 , 0}, a3 = max{−b33 + b44 , −b33 − b34 + b44 + b45 , −b14 , −b25 , −b36 − a2 , 0}, a4 = max{−b44 + b55 , −b13 , −b24 , −b35 , −b46 − a3 , 0}, a5 = max{−b12 , −b23 , −b34 , −b45 , −b56 − a4 , 0}, a6 = max{−b11 − a1 − a2 − a3 − a4 , −b22 − a2 − a3 − a4 , −b33 − a3 − a4 , −b44 − a4 , −b55 , 0}, a0 = max{b11 − a2 − a3 − a4 − a5 , b11 + b12 − a2 − a3 − a4 , b11 + b12 + b13 − a2 − a3 , b11 + b12 + b13 + b14 − a2 , b11 + b12 + b13 + b14 + b15 , 0}. Let l = a0 + a1 + 2a2 + 2a3 + 2a4 + a5 + a6 . Let b0 = (b0ij ) where b011 = a1 + a2 + a3 + a4 + a6 , b012 = a5 , b013 = a4 , b014 = a3 , b015 = a2 , b016 = a0 , b022 = a2 + a3 + a4 + a6 , b023 = a5 , b024 = a4 , a025 = a3 , b026 = a1 + a2 , b027 = a0 , b033 = a3 + a4 + a6 , b034 = a5 , b035 = a4 , b036 = a2 + a3 , b037 = a1 , b038 = a0 + a2 , b044 = a4 + a6 , b045 = a5 , b046 = a3 + a4 , b047 = a2 , b048 = a1 , b049 = a0 + a2 + a3 , b055 = a6 , b056 = a4 + a5 , b057 = a3 , b058 = a2 , b05,9 = a1 , b05,10 = a0 + a2 + a3 + a4 , b066 = a6 , b067 = a4 , b068 = a3 , b069 = a2 , b06,10 = a1 , b06,11 == a0 +a2 +a3 +a4 +a5 . Then ε(b0 ) = a6 Λ0 + a5 Λ1 + a4 Λ2 + a3 Λ3 + a2 Λ4 + a1 Λ5 + a0 Λ6 and ϕ(b0 ) = a0 Λ0 +a1 Λ1 +a2 Λ2 +a3 Λ3 +a4 Λ4 +a5 Λ5 +a6 Λ6 . It is easy to see that b0 ∈ (B 6,l )min .  i+5  i+5 0 Set b = (bij ) where bij = bij + b0ij . Then i+5 j=i bij = j=i bij + j=i bij = 0 + l = l, 1 ≤ i ≤ 6 and we observe that b11 = b11 + b011 = b11 + a1 + a2 + a3 + a4 + a6 ≥ 0, since a6 ≥ −b11 − a1 − a2 − a3 − a4 , b12 = b12 + b012 = b12 + a5 ≥ 0, since a5 ≥ −b12 , b13 = b13 + b013 = b13 + a4 ≥ 0, since a4 ≥ −b13 , b14 = b14 + b014 = b14 + a3 ≥ 0, since a3 ≥ −b14 , b15 = b15 + b015 = b14 + a2 ≥ 0, since a2 ≥ −b15 , b16 = b16 + b016 = b15 + a0 = −b11 − b12 − b13 − b14 − b15 + a0 ≥ 0,

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ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

287

since a0 ≥ b11 + b12 + b13 + b14 + b15 . Similarly, we can show that bij ∈ Z≥0 for i ≤ j ≤ i + 5, 2 ≤ i ≤ 6. We also have b44 = b44 + b044 = b66 + b67 + a4 + a6 = b66 + b67 , b44 + b45 = b44 + b044 + b45 + b045 = b55 + b56 + a4 + a5 + a6 = b55 + b56 , b55 = b55 + b055 = b66 + a6 = b66 . Similarly, we see that

6−t

j=i bij

=

5+t

j=i+t bi+t,j ,

1 ≤ i ≤ 3, 1 ≤ t ≤ 5. Also,

b11 = b11 + b011 ≥ b22 + b022 = b22 , since b011 − b022 = a1 ≥ −b11 + b22 , b22 = b22 + b022 ≥ b33 + b033 = b33 , since b022 − b033 = a2 ≥ −b22 + b33 , b33 = b33 + b033 ≥ b44 + b044 = b44 , since b033 − b044 = a3 ≥ −b33 + b44 , b44 = b44 + b044 ≥ b55 + b055 = b55 , since b044 − b055 = a4 ≥ −b44 + b55 . t+1 t Similarly, j=i bij ≥ j=i+1 bi+1,j , 1 ≤ i < t ≤ 5. Hence b ∈ B 6,l . Then f(l,b0 ) (tε(b0 ) ⊗ b ⊗ t−ϕ(b0 ) ) = b , and b ∈ ∪(l,b)∈J Im f(l,b) which proves (3). 

(1)

3. Ultra-discretization of V(D6 ) It is known that the ultra-discretization of a positive geometric crystal is a Kashiwara crystal [1, 19]. In this section we apply the ultra-discretization functor (1) (1) UD to the positive geometric crystal V = V(D6 ) for the affine Lie algebra D6 at the spin node k = 6 in ([18],Theorem 5.1). Then we show that as crystal it is isomorphic to the crystal B 6,∞ given in the last section which proves the conjecture (l) in [12] for this case. As a set X = UD(V) = Z15 . We denote the variables xm in (l) V by the same notation xm in UD(V) = X . (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) (1) Let x=(x6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , x6 ) ∈ X . By applying the ultra-discretization functor UD to the positive geometric crystal V in ([18],Theorem 5.1), we have for 0 ≤ k ≤ 6: ⎧ (2) (1) ⎪ −x2 − x2 , ⎪ ⎪ ⎪ (1) (2) (1) ⎪ ⎪ 2x1 − x2 − x2 , ⎪ ⎪ ⎪ (1) (2) (1) (3) (2) (1) ⎪ ⎪ ⎪−x1 + 2x2 + 2x2 − x3 − x3 − x3 , ⎪ ⎪ (2) (1) (3) (2) (1) (4) (3) ⎪ ⎪ ⎨−x2 − x2 + 2x3 + 2x3 + 2x3 − x4 − x4 (2) (1) UD(γk )(x) = −x4 − x4 , ⎪ ⎪ (3) (2) (1) (4) (3) (2) (1) ⎪ −x3 − x3 − x3 + 2x4 + 2x4 + 2x4 + 2x4 ⎪ ⎪ ⎪ ⎪ (2) (1) (3) (2) (1) ⎪ −x5 − x5 − x6 − x6 − x6 , ⎪ ⎪ ⎪ ⎪ (4) (3) (2) (1) (2) (1) ⎪ −x4 − x4 − x4 − x4 + 2x5 + 2x5 , ⎪ ⎪ ⎪ (3) (2) (1) (3) (2) (1) ⎩ (4) −x4 − x4 − x4 − x4 + 2x6 + 2x6 + 2x6 ,

k = 0, k = 1, k = 2, k = 3, k = 4, k = 5, k = 6.

288

KAILASH C. MISRA AND SUCHADA PONGPRASERT

⎧ (1) (2) (1) (3) (2) (1) (2) (3) ⎪ max{x6 , x2 + x2 − x3 − x3 + x5 , x2 − x3 ⎪ ⎪ ⎪ (1) (2) (1) (2) (3) (1) (3) (2) ⎪ ⎪ +x3 − x4 + x5 , x2 − x3 + x4 , x3 + x3 ⎪ ⎪ ⎪ (3) (2) (1) (2) (3) (1) (2) (1) ⎪ ⎪ −x4 − x4 + x5 , x3 − x4 + x4 , x4 + x4 ⎨ (2) (2) (1) (3) (2) (1) (2) (4) UD(εk )(x) = −x6 , x2 + x2 − x6 , x2 + x2 + x6 − x4 ⎪ ⎪ (3) (2) (1) (2) (4) (2) (2) (1) ⎪ −x4 , x2 + x2 − x3 − x4 + x4 , x2 + x2 ⎪ ⎪ ⎪ ⎪ (3) (2) (3) (2) (2) (2) (1) (4) ⎪ −x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , ⎪ ⎪ ⎪ ⎩ (2) (3) (1) (3) (2) (2) (1) (2) x2 − x3 + x3 + x4 − x5 , x3 + x3 − x5 }, k = 0, ⎧ (1) (2) ⎪ −x1 + x2 , k = 1, ⎪ ⎪ ⎪ (2) (3) (1) (2) (1) (3) (2) ⎪ ⎪ max{−x2 + x3 , x1 − 2x2 − x2 + x3 + x3 }, k = 2, ⎪ ⎪ ⎪ (3) (4) (2) (3) (2) (4) (3) ⎪ ⎪ max{−x3 + x4 , x2 − 2x3 − x3 + x4 + x4 , ⎪ ⎪ ⎪ (2) (1) (3) (2) (1) (4) (3) (2) ⎪ ⎪ x2 + x2 − 2x3 − 2x3 − x3 + x4 + x4 + x4 }, k = 3, ⎪ ⎪ ⎪ (4) (3) (3) (4) (3) (2) (3) ⎪ ⎪ ⎨max{−x4 + x6 , x3 − 2x4 − x4 + x5 + x6 , (3) (2) (4) (3) (2) (2) (3) (2) UD(εk )(x) =: x3 + x3 − 2x4 − 2x4 − x4 + x5 + x6 + x6 , ⎪ ⎪ (3) (2) (1) (4) (3) (2) (1) (2) ⎪ x3 + x3 + x3 − 2x4 − 2x4 − 2x4 − x4 + x5 ⎪ ⎪ ⎪ ⎪ (1) (3) (2) ⎪ +x5 + x6 + x6 }, k = 4, ⎪ ⎪ ⎪ ⎪ (4) (2) (4) (3) (2) (2) (1) ⎪ max{x4 − x5 , x4 + x4 + x4 − 2x5 − x5 }, k = 5, ⎪ ⎪ ⎪ (4) (3) (3) (2) (4) (3) (2) ⎪ ⎪max{−x(3) , x + x − 2x − x , x + x + x ⎪ 6 4 4 6 6 4 4 4 ⎪ ⎪ (1) (3) (2) (1) ⎩ +x4 − 2x6 − 2x6 − x6 }, k = 6. We define (2)

(1)

(2)

(1)

(2)

(1)

(2)

(1)

c˘2 = max{c + x2 + x2 , x3 + x1 } − max{x2 + x2 , x3 + x1 }, (3)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

(3)

(2)

(3)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

(3)

(2)

(3)

(2)

(1)

c˘31 = max{c + x3 + 2x3 + x3 , x2 + x3 + x3 + x4 , x2 + x2 + x4 + x4 } − max{x3 + 2x3 + x3 , x2 + x3 + x3 + x4 , x2 + x2 + x4 + x4 }, (2)

(2)

(1)

(3)

(2)

(1)

(3)

c˘32 = max{c + x3 + 2x3 + x3 , c + x2 + x3 + x3 + x4 , x2 + x2 + x4 (2)

(3)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

+ x4 } − max{c + x3 + 2x3 + x3 , x2 + x3 + x3 + x4 , x2 + x2 (3)

(2)

+ x4 + x4 }, (4)

(3)

(2)

(1)

(3)

(3)

(2)

(1)

(2)

(3)

c˘41 = max{c + x4 + 2x4 + 2x4 + x4 , x3 + x4 + 2x4 + x4 + x5 , x3 (2)

(2)

(1)

(2)

(2)

(3)

(2)

(1)

(2)

(1)

(2)

+ x3 + x4 + x4 + x5 + x6 , x3 + x3 + x3 + x5 + x5 + x6 } (4)

(3)

(2)

(1)

(3)

(3)

(2)

(1)

(2)

(3)

(2)

− max{x4 + 2x4 + 2x4 + x4 , x3 + x4 + 2x4 + x4 + x5 , x3 + x3 (2)

(1)

(2)

(2)

(3)

(2)

(1)

(2)

(1)

(2)

(2)

(1)

+ x4 + x4 + x5 + x6 , x3 + x3 + x3 + x5 + x5 + x6 } (4)

(3)

(2)

(1)

(3)

(3)

(2)

(3)

c˘42 = max{c + x4 + 2x4 + 2x4 + x4 , c + x3 + x4 + 2x4 + x4 + x5 , x3 + (2)

(2)

(1)

(2)

(2)

(3)

(2)

(1)

(2)

(1)

(2)

x3 + x4 + x4 + x5 + x6 , x3 + x3 + x3 + x5 + x5 + x6 }− (4)

(3)

(2)

(1)

(3)

(3)

(2)

(1)

(2)

max{c + x4 + 2x4 + 2x4 + x4 , x3 + x4 + 2x4 + x4 + x5 , (3)

(2)

(2)

(1)

(2)

(2)

(3)

(2)

(1)

(2)

(1)

(2)

x3 + x3 + x4 + x4 + x5 + x6 , x3 + x3 + x3 + x5 + x5 + x6 } (4)

(3)

(2)

(1)

(3)

(3)

(2)

(1)

(2)

c˘43 = max{c + x4 + 2x4 + 2x4 + x4 , c + x3 + x4 + 2x4 + x4 + x5 , (3)

(2)

(2)

(1)

(2)

(2)

(3)

(2)

(1)

(2)

(1)

c + x3 + x3 + x4 + x4 + x5 + x6 , x3 + x3 + x3 + x5 + x5 (2)

(4)

(3)

(2)

(1)

(3)

(3)

(2)

(1)

(2)

(1)

+ x6 } − max{c + x4 + 2x4 + 2x4 + x4 , c + x3 + x4 + 2x4 + x4 (2)

(3)

(2)

(2)

(1)

(2)

(2)

(3)

(2)

(1)

+ x5 , x3 + x3 + x4 + x4 + x5 + x6 , x3 + x3 + x3 + x5 + x5

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

289

(2)

+ x6 } (2)

(1)

(3)

(2)

(2)

(1)

(3)

(2)

c˘5 = max{c + x5 + x5 , x4 + x4 } − max{x5 + x5 , x4 + x4 }, (3)

(2)

(1)

(4)

(3)

(2)

(1)

(4)

(3)

(2)

(1)

(3)

(2)

(1)

(4)

(3)

(2)

(1)

(4)

(3)

(2)

(1)

(3)

(2)

(1)

c˘61 = max{c + x6 + 2x6 + x6 , x4 + x4 + x6 + x6 , x4 + x4 + x4 + x4 } − max{x6 + 2x6 + x6 , x4 + x4 + x6 + x6 , x4 + x4 + x4 + x4 }, (4)

(3)

(2)

(1)

(4)

(3)

(2)

c˘62 = max{c + x6 + 2x6 + x6 , c + x4 + x4 + x6 + x6 , x4 + x4 + x4 (1)

(3)

(2)

(1)

(4)

(3)

(2)

(1)

(4)

(3)

+ x4 } − max{c + x6 + 2x6 + x6 , x4 + x4 + x6 + x6 , x4 + x4 (2)

(1)

+ x4 + x4 }, ˘ = max{x(1) , x(2) + x(1) − x(3) − x(2) + x(1) , x(2) − x(3) + x(1) − x(2) + x(1) , K 6 2 2 3 3 5 2 3 3 4 5 (2)

(3)

(1)

(3)

(2)

(3)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

(2)

(1)

(4)

(2)

(1)

(2)

(3)

(1)

(2)

(4)

(3)

x2 − x3 + x4 , x3 + x3 − x4 − x4 + x5 , x3 − x4 + x4 , (2)

(1)

x4 + x4 − x6 , x2 + x2 − x6 , x2 + x2 + x6 − x4 − x4 , (4)

(2)

(2)

(1)

(3)

(1)

(3)

(2)

(2)

(3)

(2)

(2)

x2 + x2 − x3 − x4 + x4 , x2 + x2 − x3 − x3 + x4 + x4 − x5 , (2)

(3)

(2)

(1)

(2)

x2 + x3 − x4 , x2 − x3 + x3 + x4 − x5 , x3 + x3 − x5 },

Then we have

⎧ (3) (4) (3) (2)  (3) (2) (2) (2) (1) ⎪ (x6 , x4 , x3 , x2 − c, x(2) , ⎪ 5 , x4 , x3 , x6 , x4 , x5 ⎪ ⎪ (1) (1) (1) (1) (1) ⎪ ⎪ x − c, x − c, x , x , x ), ⎪ 3 4 6 ⎪ (3) 1 (4) (3)2 (2) (2) ⎪ (3) (2) (2) (2) (1) (1) ⎪ ⎪ (x6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 + c, ⎪ ⎪ ⎪ (1) (1) (1) (1) ⎪ ⎪ x2 , x3 , x4 , x6 ), ⎪ ⎪ ⎪ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) ⎪ ⎪ , x , x , x2 + c˘2 , x5 , x4 , x3 , x6 , x4 , x5 , (x ⎪ ⎪ 6 (1)4 (1)3 ⎪ (1) (1) (1) ⎪ x1 , x2 + c − c˘2 , x3 , x4 , x6 ), ⎪ ⎪ ⎪ ⎨ (3) (4) (3) (2) (2) (3) (2) (2) (2) (x6 , x4 , x3 + c˘31 , x2 , x5 , x4 , x3 + c˘32 , x6 , x4 , c U D(ek )(x) = (1) (1) (1) (1) (1) (1) ⎪ x5 , x1 , x2 , x3 + c − c˘31 − c˘32 , x4 , x6 ), ⎪ ⎪ ⎪ (3) (4) (3) (2) (2) (3) (2) (2) (2) ⎪ ⎪ (x 41 , x3 , x2 , x5 , x4 + c˘ 4 2 , x3 , x6 , x4 ⎪ 6 , x4 + c˘ ⎪ ⎪ (1) (1) (1) (1) (1) (1) ⎪ ⎪ +c˘43 , x5 , x1 , x2 , x3 , x4 + c − c˘41 − c˘42 − c˘43 , x6 ), ⎪ ⎪ ⎪ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) ⎪ ⎪ (x6 , x4 , x3 , x2 , x5 + c˘5 , x4 , x3 , x6 , x4 , x5 ⎪ ⎪ ⎪ (1) (1) (1) (1) (1) ⎪ ⎪ +c − c˘5 , x1 , x2 , x3 , x4 , x6 ), ⎪ ⎪ ⎪ (3) (4) (3) (2) (2) (3) (2) (2) ⎪ ⎪(x6 + c˘61 , x4 , x3 , x2 , x5 , x4 , x(2) 6 2 , x4 , 3 , x6 + c˘ ⎪ ⎪ ⎩ (1) (1) (1) (1) (1) (1) x5 , x1 , x2 , x3 , x4 , x6 + c − c˘61 − c˘62 ),

k = 0, k = 1, k = 2, k = 3, k = 4, k = 5, k = 6,

where (1)

x3

(1) ˘ − max{c + x(1) , x(2) + x(1) − x(3) − x(2) + x(1) , c + x(2) = x3 + K 6 2 2 3 3 5 2 (3)

(1)

(2)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

− x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 (2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

− x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 (3)

(2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

(2)

− x 6 , x2 + x 2 + x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 , (2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

x2 + x2 − x3 − x3 + x4 + x4 − x5 , c + x2 + x3 − x4 , c (2)

(3)

(1)

(3)

(2)

(2)

(1)

(2)

+ x2 − x3 + x3 + x4 − x5 , c + x3 + x3 − x5 }, (2)

x3

(2)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

= −c + x3 + max{c + x6 , x2 + x2 − x3 − x3 + x5 , c + x2 (3)

(1)

(2)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

− x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 (2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

− x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 (3)

(2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

(2)

− x 6 , x2 + x 2 + x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 ,

290

KAILASH C. MISRA AND SUCHADA PONGPRASERT (2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

x2 + x2 − x3 − x3 + x4 + x4 − x5 , c + x2 + x3 − x4 , (2)

(3)

(1)

(3)

(2)

(2)

(1)

(2)

c + x2 − x3 + x3 + x4 − x5 , c + x3 + x3 − x5 } − max{c (1)

(2)

(1)

(3)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(1)

(2)

+ x6 , c + x2 + x2 − x3 − x3 + x5 , x2 − x3 + x3 − x4 (1)

(2)

(3)

(1)

(3)

(2)

(2)

+ x5 , x2 − x3 + x4 , c + x3 + x3 − x4 − x4 + x5 , c + x3 (3)

(1)

(2)

(1)

(1)

(2)

(4)

(3)

(1)

(3)

(2)

(3)

(1)

(3)

(2)

(2)

(2)

(1)

(3)

(2)

− x4 + x4 , c + x4 + x4 − x6 , c + x2 + x2 − x6 , c + x2 (2)

(1)

(2)

(4)

(2)

(2)

+ x2 + x6 − x4 − x4 , c + x2 + x2 − x3 − x4 + x4 , c + x2 (2)

(2)

(2)

(1)

(4)

(2)

(3)

+ x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 − x3 (2)

(1)

(2)

+ x3 + x4 − x5 , c + x3 + x3 − x5 }, (3)

x3

(3)

(1)

(2)

(1)

(3)

(2)

(1)

(3)

(2)

(2)

= −c + x3 + max{c + x6 , c + x2 + x2 − x3 − x3 + x5 , x2 (3)

(1)

(2)

(1)

(2)

(3)

(1)

(2)

(1)

(2)

(3)

(1)

(1)

(3)

(2)

(1)

(2)

(4)

(3)

(2)

(4)

(2)

(1)

(3)

(2)

(3)

(1)

(3)

(2)

(3)

− x3 + x3 − x4 + x5 , x2 − x3 + x4 , c + x3 + x3 − x4 (2)

(1)

(2)

(2)

− x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , c + x2 (2)

(1)

+ x2 − x6 , c + x2 + x2 + x6 − x4 − x4 , c + x2 + x2 (2)

(3)

(2)

(2)

− x3 − x4 + x4 , c + x2 + x2 − x3 − x3 + x4 + x4 − x5 , (2)

(1)

(4)

(2)

(2)

(1)

x2 + x3 − x4 , x2 − x3 + x3 + x4 − x5 , c + x3 + x3 (2) ˘ − x5 } − K, (1)

x4

(1) ˘ − max{c + x(1) , x(2) + x(1) − x(3) − x(2) + x(1) , x(2) − x(3) = x4 + K 6 2 2 3 3 5 2 3 (1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

(2)

+ x3 − x4 + x5 , c + x2 − x3 + x4 , x3 + x3 − x4 − x4 (1)

(2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(4)

(2)

(3)

+ x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 − x6 , (2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(2)

x2 + x2 + x6 − x4 − x4 , x2 + x2 − x3 − x4 + x4 , x2 (1)

(3)

(2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

(2)

(3)

+ x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 − x3 (2)

(1)

(2)

+ x3 + x4 − x5 , x3 + x3 − x5 }, (2)

x4



(2) ˘ + max{c + x(1) , x(2) + x(1) − x(3) − x(2) + x(1) , x(2) − x(3) = x4 + K 6 2 2 3 3 5 2 3 (1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

(2)

+ x3 − x4 + x5 , c + x2 − x3 + x4 , x3 + x3 − x4 − x4 (1)

(2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(4)

(2)

(3)

+ x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 − x6 , (2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(2)

x2 + x2 + x6 − x4 − x4 , x2 + x2 − x3 − x4 + x4 , x2 (1)

(3)

(2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

(2)

(3)

+ x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 − x3 (2)

(1)

(2)

(1)

+ x3 + x4 − x5 , x3 + x3 − x5 } − max{c + x6 + max{c (1)

(2)

(1)

(3)

(2)

(2)

(3)

(1)

(3)

(1)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

(2)

(1)

(2)

+ x6 , c + x2 + x2 − x3 − x3 + x5 , c + x2 − x3 + x3 (2)

(1)

− x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 − x4 (1)

(2)

(2)

(2)

(1)

(3)

+ x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 − x6 , (2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

(2)

x2 + x2 + x6 − x4 − x4 , c + x2 + x2 − x3 − x4 + x4 , c (2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

+ x2 + x2 − x3 − x3 + x4 + x4 − x5 , c + x2 + x3 − x4 , (2)

(3)

(1)

(3)

(2)

(2)

(1)

(2)

(2)

c + x2 − x3 + x3 + x4 − x5 , c + x3 + x3 − x5 }, c + x2

(1) (3) (2) (1) ˘ c + x(2) − x(3) + x(1) − x(2) + x(1) + x2 − x3 − x3 + x5 + K, 2 3 3 4 5

˘ c + x(2) − x(3) + x(1) + K, ˘ c + x(3) + x(2) − x(3) − x(2) + x(1) + K, 2 3 4 3 3 4 4 5

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL (2)

(3)

(1)

(1)

(2)

(1)

291

(3)

(2)

˘ c + x − x + x + max{x , x + x − x − x + K, 3 4 4 6 2 2 3 3 (1)

(2)

(3)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

+ x5 , x2 − x3 + x3 − x4 + x5 , x2 − x3 + x4 , x3 + x3 (3)

(2)

(1)

− x4 − x4 + x5 , x3 − x4 + x4 , x4 + x4 − x6 , x2 + x2 (3)

(2)

(1)

(2)

(4)

(3)

(2)

(2)

(1)

(3)

(2)

(3)

(2)

(1)

(2)

(4)

− x6 , x2 + x2 + x6 − x4 − x4 , c + x2 + x2 − x3 − x4 (2)

(2)

(2)

(1)

(4)

+ x 4 , x2 + x 2 − x 3 − x 3 + x 4 + x 4 − x 5 , x2 + x 3 − x 4 , (2)

(3)

(1)

(3)

(2)

(2)

(1)

(2)

(2)

(1)

x2 − x3 + x3 + x4 − x5 , x3 + x3 − x5 }, c + x4 + x4 (2)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

− x6 + max{c + x6 , c + x2 + x2 − x3 − x3 + x5 , c + x2 (3)

(1)

(2)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

− x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 (2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

− x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 (3)

(2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

− x6 , x2 + x2 + x6 − x4 − x4 , c + x2 + x2 − x3 − x4 (2)

(2)

(1)

(3)

(2)

(3)

(4)

(2)

(3)

(1)

(3)

(2)

(2)

(2)

(2)

(1)

+ x4 , c + x2 + x2 − x3 − x3 + x4 + x4 − x5 , c + x2 + x3 (2)

(1)

(2)

(2)

− x4 , c + x2 − x3 + x3 + x4 − x5 , c + x3 + x3 − x5 }, x2 (1)

(3)

(3)

(1)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

+ x2 − x6 + max{c + x6 , x2 + x2 − x3 − x3 + x5 , x2 (2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

(2)

− x3 + x3 − x4 + x5 , x2 − x3 + x4 , x3 + x3 − x4 − x4 (1)

(2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(3)

+ x5 , x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 − x6 , (2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

(2)

(2)

x 2 + x 2 + x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 , x2 (1)

(3)

(2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

(2)

(3)

+ x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 − x3 (2)

(1)

(2)

(2)

(1)

(2)

(4)

+ x3 + x4 − x5 , x3 + x3 − x5 }, x2 + x2 + x6 − x4 (3)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

(3)

− x4 + max{c + x6 , x2 + x2 − x3 − x3 + x5 , x2 − x3 (1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

(2)

(1)

+ x 3 − x 4 + x 5 , x2 − x 3 + x 4 , x3 + x 3 − x 4 − x 4 + x 5 , (2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

x3 − x4 + x4 , x4 + x4 − x6 , x2 + x2 − x6 , x2 + x2 (2)

(4)

(3)

(2)

(3)

(2)

(2)

(1)

(2)

(4)

(2)

(1)

(4)

(2)

(2)

(1)

(3)

(3)

(1)

(3)

+ x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 , x2 + x 2 − x 3 (2)

(2)

− x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 − x3 + x3 + x4 (2) (2) (1) (2) (2) (1) (2) (4) (2) ˘ − x5 , x3 + x3 − x5 }, c + x2 + x2 − x3 − x4 + x4 + K, (2) (1) (3) (2) (3) (2) (2) ˘ c + x(2) + x(1) c + x2 + x2 − x3 − x3 + x4 + x4 − x5 + K, 2 3 (4) ˘ c + x(2) − x(3) + x(1) + x(3) − x(2) + K, ˘ c + x(2) + x(1) − x4 + K, 2 3 3 4 5 3 3 (2)

˘ − x5 + K}, (3)

x4

(3)

(1)

(1)

(2)

(1)

(3)

(2)

= −c + x4 + max{c + x6 + max{c + x6 , c + x2 + x2 − x3 − x3 (1)

(2)

(3)

(1)

(2)

(1)

(2)

(3)

(1)

+ x5 , c + x2 − x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c (3)

(2)

(1)

(2)

(2)

(1)

(2)

(2)

(3)

(2)

(1)

(2)

(3)

(1)

(2)

+ x3 + x3 − x4 − x4 + x5 , c + x3 − x4 + x4 , c + x4 (2)

(1)

(3)

(2)

(1)

(2)

(4)

(3)

+ x4 − x6 , x2 + x2 − x6 , x2 + x2 + x6 − x4 − x4 , c (2)

(4)

(2)

(2)

(1)

(3)

(2)

(3)

+ x2 + x2 − x3 − x4 + x4 , c + x2 + x2 − x3 − x3 + x4 (2)

(1)

(1)

(2)

(4)

(2)

(3)

(1)

(3)

+ x4 − x5 , c + x2 + x3 − x4 , c + x2 − x3 + x3 + x4 (2)

(2)

(2)

(1)

(3)

(2)

(1)

− x5 , c + x3 + x3 − x5 }, c + x2 + x2 − x3 − x3 + x5 (2)

(3)

(1)

(2)

(1)

(2)

(3)

(1)

˘ c + x − x + x − x + x + K, ˘ c+x −x +x + K, 2 3 3 4 5 2 3 4

292

KAILASH C. MISRA AND SUCHADA PONGPRASERT (3)

(2)

(3)

(2)

(1)

(2)

(3)

(1)

˘ c + x + x − x − x + x + K, ˘ c+x −x +x + K, 3 3 4 4 5 3 4 4 (1)

(2)

(1)

(3)

(2)

(1)

(2)

(3)

(1)

(2)

+ max{x6 , x2 + x2 − x3 − x3 + x5 , x2 − x3 + x3 − x4 (1)

(2)

(3)

(1)

(3)

(2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

(3)

(2)

(4)

+ x5 , x2 − x3 + x4 , x3 + x3 − x4 − x4 + x5 , x3 − x4 (2)

(1)

+ x4 , x4 + x4 − x6 , x2 + x2 − x6 , x2 + x2 + x6 − x4 (3)

(2)

(1)

(2)

(4)

(2)

(2)

(1)

(3)

(2)

− x4 , c + x2 + x2 − x3 − x4 + x4 , x2 + x2 − x3 − x3 (3)

(2)

(2)

(2)

(1)

(4)

(2)

(3)

(1)

(3)

(2)

+ x 4 + x 4 − x 5 , x2 + x 3 − x 4 , x2 − x 3 + x 3 + x 4 − x 5 , (2)

(1)

(2)

(2)

(1)

(2)

(1)

(2)

x3 + x3 − x5 }, c + x4 + x4 − x6 + max{c + x6 , c + x2 (1)

(3)

(2)

(2)

(3)

(1)

(1)

(2)

(3)

(1)

(2)

(2)

(3)

(2)

(1)

(1)

+ x2 − x3 − x3 + x5 , c + x2 − x3 + x3 − x4 + x5 , c (3)

(2)

(3)

+ x2 − x3 + x4 , c + x3 + x3 − x4 − x4 + x5 , c + x3 − x4 (1)

(2)

(1)

(2)

(2)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

+ x4 , c + x4 + x4 − x6 , x2 + x2 − x6 , x2 + x2 + x6 (4)

(3)

(2)

(3)

(2)

(1)

(3)

(2)

(2)

(4)

(2)

(2)

(1)

(2)

(1)

(4)

(2)

(3)

(1)

(2)

(3)

− x4 − x4 , c + x2 + x2 − x3 − x4 + x4 , c + x2 + x2 − x3 (2)

− x3 + x4 + x4 − x5 , c + x2 + x3 − x4 , c + x2 − x3 (2)

(2)

(1)

(3)

+ x3 + x4 − x5 , c + x3 + x3 − x5 }, x2 + x2 − x6 + max{c (1)

(2)

(1)

(3)

(2)

(1)

(2)

(3)

(1)

(2)

(1)

+ x 6 , x2 + x 2 − x 3 − x 3 + x 5 , x2 − x 3 + x 3 − x 4 + x 5 , (2)

(3)

(1)

(3)

(2)

(3)

(2)

(1)

(2)

(3)

(1)

x2 − x3 + x4 , x3 + x3 − x4 − x4 + x5 , x3 − x4 + x4 , (2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

(4)

c + x4 + x4 − x6 , x2 + x2 − x6 , x2 + x2 + x6 − x4 (3)

(2)

(1)

(2)

(4)

(2)

(1)

(4)

(2)

(2)

(1)

(3)

(2)

(3)

(1)

(3)

(2)

(3)

− x 4 , x2 + x 2 − x 3 − x 4 + x 4 , x2 + x 2 − x 3 − x 3 + x 4 (2)

(2)

(1)

(2)

(1)

(3)

(2)

(2)

+ x4 − x5 , x2 + x3 − x4 , x2 − x3 + x3 + x4 − x5 , x3 (2)

(1)

(2)

(4)

(3)

(1)

(2)

+ x3 − x5 }, x2 + x2 + x6 − x4 − x4 + max{c + x6 , x2 (2)

(1)

(2)

(3)

(2)

(1)

(1)

(2)

(1)

(2)

(3)

+ x2 − x3 − x3 + x5 , x2 − x3 + x3 − x4 + x5 , x2 − x3 (1)

(3)

(2)

(3)

(2)

(2)

(1)

(3)

(2)

(3)

(1)

(2)

(4)

(3)

(2)

(1)

+ x4 , x3 + x3 − x4 − x4 + x5 , x3 − x4 + x4 , x4 + x4 (2)

(1)

(2)

(1)

− x6 , x2 + x2 − x6 , x2 + x2 + x6 − x4 − x4 , x2 + x2 (2)

(4)

(1)

(4)

(2)

(2)

(1)

(3)

(2)

(3)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

− x3 − x4 + x4 , x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 (2)

(2)

(1)

(2)

+ x3 − x4 , x2 − x3 + x3 + x4 − x5 , x3 + x3 − x5 }, c (2) (1) (2) (4) (2) ˘ c + x(2) + x(1) − x(3) − x(2) + x2 + x2 − x3 − x4 + x4 + K, 2 2 3 3 (3) (2) (2) ˘ c + x(2) + x(1) − x(4) + K, ˘ c + x(2) − x(3) + x4 + x4 − x5 + K, 2 3 4 2 3 (1)

(3)

(2)

(2)

(1)

(2)

˘ c + x + x − x + K} ˘ −K ˘ − max{c + x3 + x4 − x5 + K, 3 3 5 (1)

(2)

(1)

(3)

(1)

(2)

(3)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(1)

(2)

+ x6 , c + x2 + x2 − x3 − x3 + x5 , c + x2 − x3 + x3 − x4 (3)

(2)

+ x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 − x4 + x5 , c (2)

(3)

(1)

(2)

(1)

(2)

(4)

(3)

(1)

(3)

(2)

(3)

(3)

(1)

(3)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

+ x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 − x6 , x2 (2)

(1)

(2)

(4)

(2)

(2)

(2)

(1)

(4)

(2)

+ x2 + x6 − x4 − x4 , x2 + x2 − x3 − x4 + x4 , c + x2 (2)

(2)

+ x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , c + x2 (2)

(1)

(2)

− x3 + x3 + x4 − x5 , c + x3 + x3 − x5 } (4)

x4

(4)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

= −c + x4 + max{c + x6 , c + x2 + x2 − x3 − x3 + x5 , c + x2 (3)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

− x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL (3)

(2)

(1)

(3)

(1)

(2)

(3)

(1)

(2)

293

(1)

(2)

(2)

− x4 − x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 (2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

+ x 2 − x 6 , x2 + x 2 + x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 (2)

(2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

+ x4 , c + x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3

(4) (2) (3) (1) (3) (2) (2) (1) (2) ˘ − x4 , c + x2 − x3 + x3 + x4 − x5 , c + x3 + x3 − x5 } − K, (1)

x5



(1)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

˘ − max{c + x , c + x + x − x − x + x , c + x = x5 + K 6 2 2 3 3 5 2 (3)

(1)

(2)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

− x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 (2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

− x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 (3)

(2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

(2)

− x 6 , x2 + x 2 + x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 , (2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

(2)

x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 (3)

(1)

(3)

(2)

(2)

(1)

(2)

− x3 + x3 + x4 − x5 , x3 + x3 − x5 }, (2)

x5



(2)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

= −c + x5 + max{c + x6 , c + x2 + x2 − x3 − x3 + x5 , c + x2 (3)

(1)

(2)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

− x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 (2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

− x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 (3)

(2)

(1)

(2)

(4)

(3)

(2)

(1)

(2)

(4)

(2)

− x 6 , x2 + x 2 + x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 , (2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(4)

(2)

x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 (3) (1) (3) (2) (2) (1) (2) ˘ − x3 + x3 + x4 − x5 , x3 + x3 − x5 } − K, (1)

x6



(1)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

(3)

˘ − max{c + x , x + x − x − x + x , x − x = x6 + K 6 2 2 3 3 5 2 3 (1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

(2)

(1)

+ x 3 − x 4 + x 5 , x2 − x 3 + x 4 , x3 + x 3 − x 4 − x 4 + x 5 , (2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

x3 − x4 + x4 , x4 + x4 − x6 , x2 + x2 − x6 , x2 + x2 (2)

(4)

(3)

(2)

(3)

(2)

(2)

(1)

(2)

(4)

(2)

(1)

(4)

(2)

(2)

(1)

(3)

(3)

(1)

(3)

+ x 6 − x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 , x2 + x 2 − x 3 (2)

(2)

− x3 + x4 + x4 − x5 , x2 + x3 − x4 , x2 − x3 + x3 + x4 (2)

(2)

(1)

(2)

− x5 , x3 + x3 − x5 }, (2)

x6

(2)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

(3)

(1)

= x6 + max{c + x6 , x2 + x2 − x3 − x3 + x5 , x2 − x3 + x3 (2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(3)

(4)

(3)

(2)

(1)

(2)

(4)

(3)

(2)

(2)

(1)

(4)

(2)

(1)

(2)

− x4 + x5 , x2 − x3 + x4 , x3 + x3 − x4 − x4 + x5 , x3 (2)

(1)

(2)

− x4 + x4 , x4 + x4 − x6 , x2 + x2 − x6 , x2 + x2 + x6 (2)

(2)

(1)

(3)

(2)

(2)

(3)

(1)

(3)

(2)

(1)

(2)

(1)

(3)

(2)

(1)

(1)

(2)

(3)

(1)

− x 4 − x 4 , x2 + x 2 − x 3 − x 4 + x 4 , x2 + x 2 − x 3 − x 3 (2)

+ x 4 + x 4 − x 5 , x2 + x 3 − x 4 , x2 − x 3 + x 3 + x 4 − x 5 , (2)

(1)

(2)

x3 + x3 − x5 } − max{c + x6 , c + x2 + x2 − x3 − x3 + x5 , (2)

(3)

(1)

(2)

(3)

c + x2 − x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 (2)

(3)

(2)

(1)

(2)

(3)

(1)

(2)

(1)

(2)

+ x3 − x4 − x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , (2)

(1)

(3)

(2)

(1)

(2)

(4)

(3)

(2)

(1)

x2 + x2 − x6 , c + x2 + x2 + x6 − x4 − x4 , c + x2 + x2 (2)

(4)

(2)

(2)

(1)

(3)

(2)

(3)

(2)

(2)

− x3 − x4 + x4 , c + x2 + x2 − x3 − x3 + x4 + x4 − x5 , (2)

(1)

(4)

(2)

(3)

(1)

(3)

(2)

(2)

c + x2 + x3 − x4 , c + x2 − x3 + x3 + x4 − x5 , c + x3 (1)

(2)

+ x3 − x5 }, (3)

x6

(3)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

= −c + x6 + max{c + x6 , c + x2 + x2 − x3 − x3 + x5 , c + x2

294

KAILASH C. MISRA AND SUCHADA PONGPRASERT (3)

(1)

(2)

(1)

(2)

(1)

(2)

(3)

(1)

(3)

(2)

(3)

− x3 + x3 − x4 + x5 , c + x2 − x3 + x4 , c + x3 + x3 − x4 (2)

(3)

(1)

(2)

(1)

(2)

(4)

(3)

(1)

(2)

(2)

(1)

(2)

(1)

− x4 + x5 , c + x3 − x4 + x4 , c + x4 + x4 − x6 , x2 + x2 (3)

(2)

(2)

− x6 , c + x2 + x2 + x6 − x4 − x4 , c + x2 + x2 − x3 (4)

(2)

(2)

(1)

(3)

(2)

(3)

(1)

(4)

(2)

(3)

(1)

(3)

(2)

(2)

(2)

(2)

− x4 + x4 , c + x2 + x2 − x3 − x3 + x4 + x4 − x5 , c + x2 (2)

(1)

+ x3 − x4 , c + x2 − x3 + x3 + x4 − x5 , c + x3 + x3 (2) ˘ − x5 } − K,

.

As shown in [1, 19], X with maps e˜k , f˜k : X −→ X ∪ {0}, εk , ϕk : X −→ Z, 0 ≤ k ≤ 6 and wt : X −→ Pcl is a Kashiwara crystal where for x ∈ X e˜k (x) = UD(eck )(x)|c=1 , f˜k (x) = UD(eck )(x)|c=−1 , wt(x) =

6 

wtk (x)Λk , where wtk (x) = UD(γk )(x),

k=0

εk (x) = UD(εk )(x), ϕk (x) = wtk (x) + εk (x). In particular, the explicit actions of f˜k , 1 ≤ k ≤ 6 on X is given as follows. (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) f˜1 (x) = (x6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 − 1, x2 , x3 , x4 , (1)

x6 ), ⎧ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) (x , x , x3 , x2 − 1, x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , ⎪ ⎪ ⎪ 6 (1)4 (2) (1) (1) (2) ⎨ x ) if x + x > x + x , 6 2 2 1 3 f˜2 (x) = (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) ⎪ (x , x , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 − 1, x3 , x4 , ⎪ ⎪ ⎩ 6 (1)4 (2) (1) (1) (2) x6 ) if x2 + x2 ≤ x1 + x3 , ⎧ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) (x , x , x3 − 1, x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , ⎪ ⎪ ⎪ 6 (1)4 ⎪ (3) (2) (2) (3) ⎪ x ) if x + x > x + x , ⎪ 6 3 3 2 4 ⎪ ⎪ ⎪ (3) (2) (1) (2) (1) (3) (2) ⎪ x3 + 2x3 + x3 > x2 + x2 + x4 + x4 , ⎪ ⎪ ⎪ ⎨(x(3) , x(4) , x(3) , x(2) , x(2) , x(3) , x(2) − 1, x(2) , x(2) , x(1) , x(1) , x(1) , x(1) , x(1) , 6 4 3 2 5 4 3 6 4 5 1 2 3 4 f˜3 (x) = (1) (3) (2) (2) (3) (2) (1) (1) (2) ⎪ x6 ) if x3 + x3 ≤ x2 + x4 , x3 + x3 > x2 + x4 , ⎪ ⎪ ⎪ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) ⎪ ⎪ (x6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 − 1, x4 , ⎪ ⎪ ⎪ (1) (2) (1) (1) (2) ⎪ ⎪ x ) if x + x ≤ x + x , ⎪ 6 3 3 2 4 ⎪ ⎩ (3) (2) (1) (2) (1) (3) (2) x3 + 2x3 + x3 ≤ x2 + x2 + x4 + x4 , ⎧ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) ⎪ (x6 , x4 − 1, x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , ⎪ ⎪ ⎪ (1) (4) (3) (3) (2) ⎪ ⎪ x6 ) if x4 + x4 > x3 + x5 , ⎪ ⎪ ⎪ (4) (3) (2) (3) (2) (2) (2) ⎪ ⎪ x4 + 2x4 + x4 > x3 + x3 + x5 + x6 , ⎪ ⎪ ⎪ (4) (3) (2) (1) (3) (2) (1) (2) (1) (2) ⎪ x4 + 2x4 + 2x4 + x4 > x3 + x3 + x3 + x5 + x5 + x6 , ⎪ ⎪ ⎪ ⎪ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) ⎪ (x , x , x , x , x , x − 1, x , x , x , x , x , x , x , x ⎪ 6 4 3 2 5 4 3 6 4 5 1 2 3 4 , ⎪ ⎪ ⎪ (1) (4) (3) (3) (2) (3) (2) (2) (2) ⎪ x ) if x + x ≤ x + x , x + x > x + x , ⎪ 6 4 4 3 5 4 4 3 6 ⎪ ⎪ (3) (2) (1) (2) (1) (1) (2) ⎨ x4 + 2x4 + x4 > x3 + x3 + x5 + x6 , f˜4 (x) = (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) ⎪(x6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 − 1, x5 , x1 , x(1) ⎪ 2 , x3 , x4 , ⎪ ⎪ (1) (4) (3) (2) (3) (2) (2) (2) ⎪ ⎪ x6 ) if x4 + 2x4 + x4 ≤ x3 + x3 + x5 + x6 , ⎪ ⎪ ⎪ (3) (2) (2) (2) (2) (1) (1) (1) ⎪ ⎪ x4 + x4 ≤ x3 + x6 , x4 + x4 > x3 + x5 , ⎪ ⎪ ⎪ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) ⎪ ⎪(x6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x(1) − 1, ⎪ 3 , x4 ⎪ ⎪ (1) (2) (1) (1) (1) ⎪ ⎪ x ) if x + x ≤ x + x , ⎪ 6 4 4 3 5 ⎪ ⎪ (3) (2) (1) (2) (1) (1) (2) ⎪ ⎪ x4 + 2x4 + x4 ≤ x3 + x3 + x5 + x6 , ⎪ ⎪ ⎩ (4) (3) (2) (1) (3) (2) (1) (2) (1) (2) x4 + 2x4 + 2x4 + x4 ≤ x3 + x3 + x3 + x5 + x5 + x6 ,

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

295

⎧ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) (x , x , x3 , x2 , x5 − 1, x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , ⎪ ⎪ ⎪ 6 (1)4 (2) (1) (3) (2) ⎨ x ) if x + x > x + x , 6 5 5 4 4 f˜5 (x) = (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) ⎪(x(3) , x , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 − 1, x1 , x2 , x3 , x4 , ⎪ ⎪ ⎩ 6 (1)4 (2) (1) (3) (2) x6 ) if x5 + x5 ≤ x4 + x4 , ⎧ (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) (x6 − 1, x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , ⎪ ⎪ ⎪ ⎪ (1) (3) (2) (4) (3) ⎪ x ) if x + x > x + x , ⎪ 6 6 6 4 4 ⎪ ⎪ (3) (2) (1) (4) (3) (2) (1) ⎪ ⎪ x6 + 2x6 + x6 > x4 + x4 + x4 + x4 , ⎪ ⎪ ⎪ ⎨(x(3) , x(4) , x(3) , x(2) , x(2) , x(3) , x(2) , x(2) − 1, x(2) , x(1) , x(1) , x(1) , x(1) , x(1) , 6 4 3 2 5 4 3 6 4 5 1 2 3 4 f˜6 (x) = (1) (3) (2) (4) (3) (2) (1) (2) (1) ⎪ x6 ) if x6 + x6 ≤ x4 + x4 , x6 + x6 > x4 + x4 , ⎪ ⎪ ⎪ (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) ⎪ ⎪(x(3) ⎪ 6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , ⎪ ⎪ (1) (2) (1) (2) (1) ⎪ ⎪ − 1) if x + x ≤ x + x , x ⎪ 6 6 6 4 4 ⎪ ⎩ (3) (2) (1) (4) (3) (2) (1) x6 + 2x6 + x6 ≤ x4 + x4 + x4 + x4 .

To determine the explicit action of f˜0 (x) we define conditions (F˘1) − (F˘14) as follows. (2) (1) (3) (1) (F˘1) x2 + x2 − x6 ≥ x6 , (2)

(1)

(3)

(2)

(1)

(3)

(2)

(1)

(2)

(1)

(3)

(2)

(3)

(1)

(2)

(1)

(2)

(1)

(3)

(2)

(3)

(1)

(2)

(1)

(3)

(2)

(1)

(3)

(2)

(1)

(2)

(1)

(3)

(2)

(3)

(1)

(2)

(1)

(3)

(2)

(1)

(2)

(2)

(1)

(3)

(2)

(1)

(4)

(3)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

(4)

(2)

(2)

(1)

(3)

(2)

(1)

(3)

(2)

(3)

(2)

(1)

(3)

(2)

(1)

(4)

(2)

(1)

(3)

(2)

(3)

(1)

(3)

(2)

(2)

(1)

(3)

(2)

(3)

(2)

x2 + x2 − x6 ≥ x2 + x2 − x3 − x3 + x5 , x2 + x2 − x6 ≥ x2 − x3 + x3 − x4 + x5 , x2 + x2 − x6 ≥ x2 − x3 + x4 , x2 + x2 − x6 ≥ x3 + x3 − x4 − x4 + x5 , x2 + x2 − x6 ≥ x3 − x4 + x4 , x2 + x2 − x6 ≥ x4 + x4 − x6 , x2 + x2 − x6 ≥ x2 + x2 − x4 − x4 + x6 , x2 + x2 − x6 ≥ x2 + x2 − x3 − x4 + x4 , (2)

(2)

x2 + x2 − x6 ≥ x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x2 − x6 ≥ x2 + x3 − x4 , x2 + x2 − x6 ≥ x2 − x3 + x3 + x4 − x5 , x2 + x2 − x6 ≥ x3 + x3 − x5 , (2) (1) (4) (3) (2) (1) (F˘2) x2 + x2 − x4 − x4 + x6 ≥ x6 , (2)

(1)

(4)

(3)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

(1)

(4)

(3)

(2)

(2)

(3)

(1)

(2)

(1)

(2)

(1)

(4)

(3)

(2)

(2)

(3)

(1)

(2)

(1)

(4)

(3)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

(1)

(4)

(3)

(2)

(2)

(3)

(1)

(2)

(1)

(4)

(3)

(2)

(2)

(1)

(2)

(2)

(1)

(4)

(3)

(2)

(2)

(1)

(3)

(2)

(1)

(4)

(3)

(2)

(2)

(1)

(2)

(4)

(2)

(2)

(1)

(4)

(3)

(2)

(2)

(1)

(3)

(2)

(3)

(2)

(1)

(4)

(3)

(2)

(2)

(1)

(4)

(2)

(1)

(4)

(3)

(2)

(2)

(3)

(1)

(3)

(2)

x2 + x2 − x4 − x4 + x6 ≥ x2 + x2 − x3 − x3 + x5 , x2 + x2 − x4 − x4 + x6 ≥ x2 − x3 + x3 − x4 + x5 , x2 + x2 − x4 − x4 + x6 ≥ x2 − x3 + x4 , x2 + x2 − x4 − x4 + x6 ≥ x3 + x3 − x4 − x4 + x5 , x2 + x2 − x4 − x4 + x6 ≥ x3 − x4 + x4 , x2 + x2 − x4 − x4 + x6 ≥ x4 + x4 − x6 , x2 + x2 − x4 − x4 + x6 > x2 + x2 − x6 , x2 + x2 − x4 − x4 + x6 ≥ x2 + x2 − x3 − x4 + x4 , (2)

(2)

x2 + x2 − x4 − x4 + x6 ≥ x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x2 − x4 − x4 + x6 ≥ x2 + x3 − x4 , x2 + x2 − x4 − x4 + x6 ≥ x2 − x3 + x3 + x4 − x5 ,

296

KAILASH C. MISRA AND SUCHADA PONGPRASERT (2)

(1)

(4)

(3)

(2)

(2)

(3)

(2)

(2)

(1)

(2)

(4)

(2)

(1)

(2)

(1)

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(4)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

(1)

(2)

(4)

(2)

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(3)

(1)

(2)

(1)

(2)

(1)

(2)

(4)

(2)

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(1)

(2)

(1)

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(1)

(3)

(2)

(1)

(2)

(1)

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(1)

(2)

(1)

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(1)

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(1)

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(2)

(1)

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(2)

(1)

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(2)

(1)

(4)

(3)

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(1)

(2)

(4)

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(1)

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(3)

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(1)

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(4)

(2)

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(1)

(4)

(2)

(1)

(2)

(4)

(2)

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(3)

(1)

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(3)

(2)

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(1)

(2)

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(3)

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(3)

(2)

(2)

(2)

(2)

(1)

(3)

(2)

(3)

(2)

(2)

(2)

(1)

(3)

(2)

(3)

(2)

(2)

(1)

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(1)

(3)

(2)

(2)

(1)

(3)

(2)

(1)

(2)

x2 + x2 − x4 − x4 + x6 ≥ x3 + x3 − x5 , (F˘3) x2 + x2 − x3 − x4 + x4 ≥ x6 , x2 + x2 − x3 − x4 + x4 ≥ x2 + x2 − x3 − x3 + x5 , x2 + x2 − x3 − x4 + x4 ≥ x2 − x3 + x3 − x4 + x5 , x2 + x2 − x3 − x4 + x4 ≥ x2 − x3 + x4 , x2 + x2 − x3 − x4 + x4 ≥ x3 + x3 − x4 − x4 + x5 , x2 + x2 − x3 − x4 + x4 ≥ x3 − x4 + x4 , x2 + x2 − x3 − x4 + x4 ≥ x4 + x4 − x6 , x2 + x2 − x3 − x4 + x4 > x2 + x2 − x6 , x2 + x2 − x3 − x4 + x4 > x2 + x2 − x4 − x4 + x6 , (2)

(2)

(3)

(2)

(1)

(3)

(1)

(2)

(1)

(2)

(3)

(1)

(2)

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x2 + x2 − x3 − x4 + x4 ≥ x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x2 − x3 − x4 + x4 ≥ x2 + x3 − x4 , x2 + x2 − x3 − x4 + x4 ≥ x2 − x3 + x3 + x4 − x5 , x2 + x2 − x3 − x4 + x4 ≥ x3 + x3 − x5 , (F˘4) x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x6 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x2 + x2 − x3 − x3 + x5 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x2 − x3 + x3 − x4 + x5 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x2 − x3 + x4 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x3 + x3 − x4 − x4 + x5 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x3 − x4 + x4 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x4 + x4 − x6 , x2 + x2 − x3 − x3 + x4 + x4 − x5 > x2 + x2 − x6 , x2 + x2 − x3 − x3 + x4 + x4 − x5 > x2 + x2 − x4 − x4 + x6 , x2 + x2 − x3 − x3 + x4 + x4 − x5 > x2 + x2 − x3 − x4 + x4 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x2 + x3 − x4 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x2 − x3 + x3 + x4 − x5 , x2 + x2 − x3 − x3 + x4 + x4 − x5 ≥ x3 + x3 − x5 , (F˘5) x2 + x3 − x4 ≥ x6 , x2 + x3 − x4 ≥ x2 + x2 − x3 − x3 + x5 , x2 + x3 − x4 ≥ x2 − x3 + x3 − x4 + x5 , x2 + x3 − x4 ≥ x2 − x3 + x4 , x2 + x3 − x4 ≥ x3 + x3 − x4 − x4 + x5 , x2 + x3 − x4 ≥ x3 − x4 + x4 , x2 + x3 − x4 ≥ x4 + x4 − x6 , x2 + x3 − x4 > x2 + x2 − x6 ,

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL (2)

(1)

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297

x2 + x3 − x4 > x2 + x2 − x4 − x4 + x6 , x2 + x3 − x4 > x2 + x2 − x3 − x4 + x4 , (2)

(2)

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(1)

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(1)

x2 + x3 − x4 ≥ x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x3 − x4 ≥ x2 − x3 + x3 + x4 − x5 , x2 + x3 − x4 ≥ x3 + x3 − x5 , (2) (1) (3) (2) (1) (1) (F˘6) x2 + x2 − x3 − x3 + x5 ≥ x6 , (2)

(1)

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x2 + x2 − x3 − x3 + x5 ≥ x2 − x3 + x3 − x4 + x5 , x2 + x2 − x3 − x3 + x5 ≥ x2 − x3 + x4 , x2 + x2 − x3 − x3 + x5 ≥ x3 + x3 − x4 − x4 + x5 , x2 + x2 − x3 − x3 + x5 ≥ x3 − x4 + x4 , x2 + x2 − x3 − x3 + x5 ≥ x4 + x4 − x6 , x2 + x2 − x3 − x3 + x5 > x2 + x2 − x6 , x2 + x2 − x3 − x3 + x5 > x2 + x2 − x4 − x4 + x6 , x2 + x2 − x3 − x3 + x5 > x2 + x2 − x3 − x4 + x4 , (2)

(2)

(2)

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x2 + x2 − x3 − x3 + x5 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 + x2 − x3 − x3 + x5 ≥ x2 + x3 − x4 , x2 + x2 − x3 − x3 + x5 ≥ x2 − x3 + x3 + x4 − x5 , x2 + x2 − x3 − x3 + x5 ≥ x3 + x3 − x5 , (2) (3) (1) (3) (2) (1) (F˘7) x2 − x3 + x3 + x4 − x5 ≥ x6 , (2)

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x2 − x3 + x3 + x4 − x5 ≥ x2 + x2 − x3 − x3 + x5 , x2 − x3 + x3 + x4 − x5 ≥ x2 − x3 + x3 − x4 + x5 , x2 − x3 + x3 + x4 − x5 ≥ x2 − x3 + x4 , x2 − x3 + x3 + x4 − x5 ≥ x3 + x3 − x4 − x4 + x5 , x2 − x3 + x3 + x4 − x5 ≥ x3 − x4 + x4 , x2 − x3 + x3 + x4 − x5 ≥ x4 + x4 − x6 , x2 − x3 + x3 + x4 − x5 > x2 + x2 − x6 , x2 − x3 + x3 + x4 − x5 > x2 + x2 − x4 − x4 + x6 , x2 − x3 + x3 + x4 − x5 > x2 + x2 − x3 − x4 + x4 , x2 − x3 + x3 + x4 − x5 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 − x3 + x3 + x4 − x5 > x2 + x3 − x4 , x2 − x3 + x3 + x4 − x5 ≥ x3 + x3 − x5 , (2) (3) (2) (1) (F˘8) x3 + x3 − x5 ≥ x6 , (2)

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(1)

x3 + x3 − x5 ≥ x2 + x2 − x3 − x3 + x5 , x3 + x3 − x5 ≥ x2 − x3 + x3 − x4 + x5 , x3 + x3 − x5 ≥ x2 − x3 + x4 ,

298

KAILASH C. MISRA AND SUCHADA PONGPRASERT (2)

(3)

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(2)

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x3 + x3 − x5 ≥ x3 + x3 − x4 − x4 + x5 , x3 + x3 − x5 ≥ x3 − x4 + x4 , x3 + x3 − x5 ≥ x4 + x4 − x6 , x3 + x3 − x5 > x2 + x2 − x6 , x3 + x3 − x5 > x2 + x2 − x4 − x4 + x6 , x3 + x3 − x5 > x2 + x2 − x3 − x4 + x4 , (2)

(2)

(2)

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(1)

x3 + x3 − x5 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x3 + x3 − x5 > x2 + x3 − x4 , x3 + x3 − x5 > x2 − x3 + x3 + x4 − x5 , (2) (3) (1) (2) (1) (1) (F˘9) x2 − x3 + x3 − x4 + x5 ≥ x6 , (2)

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x2 − x3 + x3 − x4 + x5 > x2 + x2 − x3 − x3 + x5 , x2 − x3 + x3 − x4 + x5 ≥ x2 − x3 + x4 , x2 − x3 + x3 − x4 + x5 ≥ x3 + x3 − x4 − x4 + x5 , x2 − x3 + x3 − x4 + x5 ≥ x3 − x4 + x4 , x2 − x3 + x3 − x4 + x5 ≥ x4 + x4 − x6 , x2 − x3 + x3 − x4 + x5 > x2 + x2 − x6 , x2 − x3 + x3 − x4 + x5 > x2 + x2 − x4 − x4 + x6 , x2 − x3 + x3 − x4 + x5 > x2 + x2 − x3 − x4 + x4 , (2)

(2)

x2 − x3 + x3 − x4 + x5 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 − x3 + x3 − x4 + x5 > x2 + x3 − x4 , x2 − x3 + x3 − x4 + x5 > x2 − x3 + x3 + x4 − x5 , x2 − x3 + x3 − x4 + x5 ≥ x3 + x3 − x5 , (2) (3) (1) (1) (F˘10) x2 − x3 + x4 ≥ x6 , (2)

(3)

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x2 − x3 + x4 > x2 + x2 − x3 − x3 + x5 , x2 − x3 + x4 > x2 − x3 + x3 − x4 + x5 , x2 − x3 + x4 ≥ x3 + x3 − x4 − x4 + x5 , x2 − x3 + x4 ≥ x3 − x4 + x4 , x2 − x3 + x4 ≥ x4 + x4 − x6 , x2 − x3 + x4 > x2 + x2 − x6 , x2 − x3 + x4 > x2 + x2 − x4 − x4 + x6 , x2 − x3 + x4 > x2 + x2 − x3 − x4 + x4 , x2 − x3 + x4 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x2 − x3 + x4 > x2 + x3 − x4 , x2 − x3 + x4 > x2 − x3 + x3 + x4 − x5 , x2 − x3 + x4 ≥ x3 + x3 − x5 ,

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL (2)

(1)

(3)

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299

(F˘11) x3 + x3 − x4 − x4 + x5 ≥ x6 , x3 + x3 − x4 − x4 + x5 > x2 + x2 − x3 − x3 + x5 , x3 + x3 − x4 − x4 + x5 > x2 − x3 + x3 − x4 + x5 , x3 + x3 − x4 − x4 + x5 ≥ x2 − x3 + x4 , x3 + x3 − x4 − x4 + x5 ≥ x3 − x4 + x4 , x3 + x3 − x4 − x4 + x5 ≥ x4 + x4 − x6 , x3 + x3 − x4 − x4 + x5 > x2 + x2 − x6 , x3 + x3 − x4 − x4 + x5 > x2 + x2 − x4 − x4 + x6 , x3 + x3 − x4 − x4 + x5 > x2 + x2 − x3 − x4 + x4 , (2)

(2)

x3 + x3 − x4 − x4 + x5 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x3 + x3 − x4 − x4 + x5 > x2 + x3 − x4 , x3 + x3 − x4 − x4 + x5 > x2 − x3 + x3 + x4 − x5 , x3 + x3 − x4 − x4 + x5 > x3 + x3 − x5 , (F˘12) x3 − x4 + x4 ≥ x6 , x3 − x4 + x4 > x2 + x2 − x3 − x3 + x5 , x3 − x4 + x4 > x2 − x3 + x3 − x4 + x5 , x3 − x4 + x4 > x2 − x3 + x4 , x3 − x4 + x4 > x3 + x3 − x4 − x4 + x5 , x3 − x4 + x4 ≥ x4 + x4 − x6 , x3 − x4 + x4 > x2 + x2 − x6 , x3 − x4 + x4 > x2 + x2 − x4 − x4 + x6 , x3 − x4 + x4 > x2 + x2 − x3 − x4 + x4 , x3 − x4 + x4 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x3 − x4 + x4 > x2 + x3 − x4 , x3 − x4 + x4 > x2 − x3 + x3 + x4 − x5 , x3 − x4 + x4 > x3 + x3 − x5 , (F˘13) x4 + x4 − x6 ≥ x6 , x4 + x4 − x6 > x2 + x2 − x3 − x3 + x5 , x4 + x4 − x6 > x2 − x3 + x3 − x4 + x5 , x4 + x4 − x6 > x2 − x3 + x4 , x4 + x4 − x6 > x3 + x3 − x4 − x4 + x5 , x4 + x4 − x6 > x3 − x4 + x4 , x4 + x4 − x6 > x2 + x2 − x6 , x4 + x4 − x6 > x2 + x2 − x4 − x4 + x6 , x4 + x4 − x6 > x2 + x2 − x3 − x4 + x4 ,

300

KAILASH C. MISRA AND SUCHADA PONGPRASERT (2)

(1)

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x4 + x4 − x6 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x4 + x4 − x6 > x2 + x3 − x4 , x4 + x4 − x6 > x2 − x3 + x3 + x4 − x5 , x4 + x4 − x6 > x3 + x3 − x5 , (1) (2) (1) (3) (2) (1) (F˘14) x6 > x2 + x2 − x3 − x3 + x5 , (1)

(2)

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(3)

(1)

(2)

(1)

(4)

(1)

(2)

(3)

(1)

(3)

(2)

(1)

(2)

(3)

(2)

x6 > x2 − x3 + x3 − x4 + x5 , x6 > x2 − x3 + x4 , x6 > x3 + x3 − x4 − x4 + x5 , x6 > x3 − x4 + x4 , x6 > x4 + x4 − x6 , x6 > x2 + x2 − x6 , x6 > x2 + x2 − x4 − x4 + x6 , x6 > x2 + x2 − x3 − x4 + x4 , x6 > x2 + x2 − x3 − x3 + x4 + x4 − x5 , x6 > x2 + x3 − x4 , x6 > x2 − x3 + x3 + x4 − x5 , x6 > x3 + x3 − x5 ,

(1)

ULTRA-DISCRETIZATION OF D6 - GEOMETRIC CRYSTAL

301

Then for x ∈ X we have f˜0 (x) = UD(ec0 )(x)|c=−1 given by

f˜0 (x) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(4) (3) (2) (2) (3) (2) (2) (2) (x(3) 6 + 1, x4 + 1, x3 + 1, x2 + 1, x5 + 1, x4 + 1, x3 + 1, x6 , x4 , (1) (1) (1) (1) (1) (1) x5 , x1 + 1, x2 + 1, x3 , x4 , x6 ) if (F˘1), (3)

(4)

(3)

(2)

(2)

(3)

(3)

(4)

(3)

(2)

(2)

(3)

(3)

(4)

(3)

(4)

(3)

(4)

(3)

(2)

(2)

(3)

(4)

(3)

(2)

(2)

(3)

(4)

(3)

(3)

(4)

(3)

(2)

(3)

(4)

(3)

(2)

(3)

(4)

(3)

(2)

(2)

(3)

(3)

(4)

(3)

(2)

(2)

(3)

(3)

(4)

(3)

(2)

(2)

(3)

(2)

(2)

(3)

(4)

(3)

(2)

(2)

(3)

(2)

(2)

(2)

(2)

(2)

(x6 , x4 + 1, x3 + 1, x2 + 1, x5 + 1, x4 + 1, x3 + 1, x6 + 1, x4 , (1) (1) (1) (1) (1) (1) x5 , x1 + 1, x2 + 1, x3 , x4 , x6 ) if (F˘2), (2)

(2)

(2)

(2)

(2)

(2)

(x6 , x4 + 1, x3 + 1, x2 + 1, x5 + 1, x4 , x3 + 1, x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 , x1 + 1, x2 + 1, x3 , x4 , x6 ) if (F˘3), (3)

(2)

(2)

(3)

(x6 , x4 , x3 + 1, x2 + 1, x5 + 1, x4 + 1, x3 + 1, x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 , x1 + 1, x2 + 1, x3 , x4 , x6 ) if (F˘4), (3)

(2)

(2)

(3)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(x6 , x4 + 1, x3 + 1, x2 + 1, x5 + 1, x4 , x3 , x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 , x1 + 1, x2 + 1, x3 + 1, x4 , x6 ) if (F˘5), (3)

(2)

(x6 , x4 , x3 + 1, x2 + 1, x5 , x4 + 1, x3 + 1, x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 + 1, x1 + 1, x2 + 1, x3 , x4 , x6 ) if (F˘6), (3)

(2)

(x6 , x4 , x3 + 1, x2 + 1, x5 + 1, x4 + 1, x3 , x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 , x1 + 1, x2 + 1, x3 + 1, x4 , x6 ) if (F˘7), (2)

(2)

(3)

(2)

(2)

(3)

(2)

(2)

(2)

(2)

(3)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

(x6 , x4 , x3 , x2 + 1, x5 + 1, x4 + 1, x3 + 1, x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 , x1 + 1, x2 + 1, x3 + 1, x4 , x6 ) if (F˘8), (x6 , x4 , x3 + 1, x2 + 1, x5 , x4 + 1, x3 , x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 + 1, x1 + 1, x2 + 1, x3 + 1, x4 , x6 ) if (F˘9), (x6 , x4 , x3 + 1, x2 + 1, x5 , x4 + 1, x3 , x6 + 1, x4 , (1) (1) (1) (1) (1) (1) x5 + 1, x1 + 1, x2 + 1, x3 + 1, x4 + 1, x6 ) if (F˘10), (x6 , x4 , x3 , x2 + 1, x5 , x4 + 1, x3 + 1, x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 + 1, x1 + 1, x2 + 1, x3 + 1, x4 , x6 ) if (F˘11), (x6 , x4 , x3 , x2 + 1, x5 , x4 + 1, x3 + 1, x6 + 1, x4 , (1) (1) (1) (1) (1) (1) x5 + 1, x1 + 1, x2 + 1, x3 + 1, x4 + 1, x6 ) if (F˘12), (2)

(x6 , x4 , x3 , x2 + 1, x5 , x4 , x3 + 1, x6 + 1, x4 + 1, (1) (1) (1) (1) (1) (1) x5 + 1, x1 + 1, x2 + 1, x3 + 1, x4 + 1, x6 ) if (F˘13), (2)

(x6 , x4 , x3 , x2 + 1, x5 , x4 , x3 + 1, x6 , x4 + 1, (1) (1) (1) (1) (1) (1) x5 + 1, x1 + 1, x2 + 1, x3 + 1, x4 + 1, x6 + 1) if (F˘14).

Theorem 3.1. The map B 6,∞

Ω:

b = (bij )i≤j≤i+5,

1≤i≤6

→ X, (3) (4) (3) (2) (2) (3) (2) (2) (2)  → x = (x6 , x4 , x3 , x2 , x5 , x4 , x3 , x6 , x4 , (1) (1) (1) (1) (1) (1) x5 , x1 , x2 , x3 , x4 , x6 )

defined by

x(l) m

⎧m ⎪ ⎨j=m−l+1 bm−l+1,j , for m = 1, 2, 3, 4 m = j=m−2l+1 bm−2l+1,j , for m = 5 ⎪ ⎩m−1 j=m−2l+1 bm−2l+1,j , for m = 6.

is an isomorphism of crystals.

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KAILASH C. MISRA AND SUCHADA PONGPRASERT

Proof. First we observe that the map Ω−1 : X → B 6,∞ is given by Ω−1 (x) = b where (1)

(2)

(1)

(3)

(2)

(4)

(3)

(3)

(4)

b11 = x1 , b12 = x2 − x1 , b13 = x3 − x2 , b14 = x4 − x3 , b15 = x6 − x4 , (3)

(1)

(2)

(1)

(3)

(2)

(2)

(3)

(1)

(2)

(1)

(2)

(2)

(1)

(1)

(1)

b16 = −x6 , b22 = x2 , b23 = x3 − x2 , b24 = x4 − x3 , b25 = x5 − x4 , (3)

(2)

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(2)

(2)

(4)

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(2)

(1)

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(3)

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(1)

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(1)

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(2)

b26 = x6 − x5 , b27 = −x6 , b33 = x3 , b34 = x4 − x3 , b35 = x6 − x4 , (4)

b36 = x5 − x6 , b37 = x4 − x5 , b38 = −x4 , b44 = x4 , b45 = x5 − x4 , (3)

(1)

b46 = x6 − x5 , b47 = x4 − x6 , b48 = x3 − x4 , b49 = −x3 , b55 = x6 , (2)

(2)

b56 = x5 − x6 , b57 = x4 − x5 , b58 = x3 − x4 , b59 = x2 − x3 , (2)

(1)

(1)

(1)

(1)

(1)

(1)

(1)

b5,10 = −x2 , b66 = x6 , b67 = x4 − x6 , b68 = x3 − x4 , b69 = x2 − x3 , (1)

(1)

(1)

b6,10 = x1 − x2 , b6,11 = −x1 .

Hence the map Ω is bijective. To prove that Ω is an isomorphism of crystals we need to show that for b ∈ B 6,∞ and 0 ≤ k ≤ 6 we have: Ω(f˜k (b)) = f˜k (Ω(b)), Ω(e˜k (b)) = e˜k (Ω(b)), wtk (Ω(b)) = wtk (b), εk (Ω(b)) = εk (b). Hence ϕk (Ω(b)) = wtk (Ω(b)) + εk (Ω(b)) = wtk (b) + εk (b) = ϕk (b). We observe that the conditions for the action of f˜k on Ω(b) in X hold if and only if the corresponding conditions for the action of f˜k on b in B 6,∞ hold for all 0 ≤ k ≤ 6. Suppose (2) (1) (1) (2) Ω(b) = x. If x2 + x2 > x1 + x3 , then b11 + b12 + b22 > b11 + b22 + b23 and f˜2 (x) (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) (1) = (x6 , x4 , x3 , x2 − 1, x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , x6 ) = (2) (1) (1) (2) Ω(f˜2 (b)). If x2 +x2 ≥ x1 +x3 , then b11 +b12 +b22 ≥ b11 +b22 +b23 and e˜2 (x) = (3) (4) (3) (2) (2) (3) (2) (2) (2) (1) (1) (1) (1) (1) (1) (x6 , x4 , x3 , x2 + 1, x5 , x4 , x3 , x6 , x4 , x5 , x1 , x2 , x3 , x4 , x6 ) = ˜ ˜ Ω(˜ e2 (b)). Similarly, we can show Ω(fk (b)) = fk (Ω(b)) and Ω(e˜k (b)) = e˜k (Ω(b)) (2) (1) for k = 0, 1, 3, 4, 5, 6. We also have wt0 (Ω(b)) = wt0 (x) = −x2 − x2 = −b11 − b12 −b22 = −b11 −b12 +b23 +b24 +b25 +b26 +b27 = wt0 (b) for all b ∈ B 6,∞ . Similarly, (3) (4) wtk (Ω(b)) = wtk (b) for 1 ≤ k ≤ 6. Also, ε6 (Ω(b)) = ε6 (x) = max{−x6 , x4 + (3) (3) (2) (4) (3) (2) (1) (3) (2) (1) x4 − 2x6 − x6 , x4 + x4 + x4 + x4 − 2x6 − 2x6 − x6 } = max{−b11 − b12 − b13 − b14 − b15 , −b11 − b12 − b13 − b14 − 2b15 + b22 + b23 + b24 − b33 − b34 − b35 , −b11 −b12 −b13 −b14 −2b15 +b22 +b23 +b24 −b33 −b34 −2b35 +b44 −b55 } = ε6 (b). Similarly, εk (Ω(b)) = εk (b) for 0 ≤ k ≤ 5 which completes the proof.  References [1] Arkady Berenstein and David Kazhdan, Geometric and unipotent crystals, Geom. Funct. Anal. Special Volume (2000), 188–236, DOI 10.1007/978-3-0346-0422-2 8. GAFA 2000 (Tel Aviv, 1999). MR1826254 (3) [2] Mana Igarashi and Toshiki Nakashima, Affine geometric crystal of type D4 , Quantum affine algebras, extended affine Lie algebras, and their applications, Contemp. Math., vol. 506, Amer. Math. Soc., Providence, RI, 2010, pp. 215–226, DOI 10.1090/conm/506/09942. MR2642568 (3) [3] Mana Igarashi, Kailash C. Misra, and Toshiki Nakashima, Ultra-discretization of the D4 (1)

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[4] Mana Igarashi, Kailash C. Misra, and Suchada Pongprasert, D5 -geometric crystal corresponding to the Dynkin spin node i = 5 and its ultra-discretization, J. Algebra Appl. 18 (2019), no. 12, 1950227, 31, DOI 10.1142/S021949881950227X. MR4026787 [5] Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Sci. Publ., River Edge, NJ, 1992, pp. 449–484, DOI 10.1142/s0217751x92003896. MR1187560 [6] Seok-Jin Kang, Masaki Kashiwara, Kailash C. Misra, Tetsuji Miwa, Toshiki Nakashima, and Atsushi Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), no. 3, 499–607, DOI 10.1215/S0012-7094-92-06821-9. MR1194953 [7] Seok-Jin Kang, Masaki Kashiwara, and Kailash C. Misra, Crystal bases of Verma modules for quantum affine Lie algebras, Compositio Math. 92 (1994), no. 3, 299–325. MR1286129 [8] Masaki Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260. MR1090425 [9] M. Kashiwara, On crystal bases of the Q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516, DOI 10.1215/S0012-7094-91-06321-0. MR1115118 [10] Masaki Kashiwara, On level-zero representations of quantized affine algebras, Duke Math. J. 112 (2002), no. 1, 117–175, DOI 10.1215/S0012-9074-02-11214-9. MR1890649 [11] Masaki Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, 223–250. MR2115972 [12] Masaki Kashiwara, Toshiki Nakashima, and Masato Okado, Affine geometric crystals and limit of perfect crystals, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3645–3686, DOI 10.1090/S0002-9947-08-04341-9. MR2386241 [13] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR1104219 [14] V. G. Kac and D. H. Peterson, Defining relations of certain infinite-dimensional groups, ´ Cartan Ast´ erisque Num´ ero Hors S´ erie (1985), 165–208. The mathematical heritage of Elie (Lyon, 1984). MR837201 [15] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498, DOI 10.2307/1990961. MR1035415 (1) [16] Kailash C. Misra and Toshiki Nakashima, An -geometric crystal corresponding to Dynkin index i = 2 and its ultra-discretization, Symmetries, integrable systems and representations, Springer Proc. Math. Stat., vol. 40, Springer, Heidelberg, 2013, pp. 297–318, DOI 10.1007/978-1-4471-4863-0 12. MR3077689 (1) [17] Kailash C. Misra and Toshiki Nakashima, Affine geometric crystal of An and limit of Kirillov-Reshetikhin perfect crystals, J. Algebra 507 (2018), 249–291, DOI 10.1016/j.jalgebra.2018.03.041. MR3807049 (1) [18] Kailash C. Misra and Suchada Pongprasert, D6 -geometric crystal at the spin node, Comm. Algebra 48 (2020), no. 8, 3382–3397, DOI 10.1080/00927872.2020.1737872. MR4115355 [19] Toshiki Nakashima, Geometric crystals on Schubert varieties, J. Geom. Phys. 53 (2005), no. 2, 197–225, DOI 10.1016/j.geomphys.2004.06.004. MR2110832 (1) [20] Toshiki Nakashima, Affine geometric crystal of type G2 , Lie algebras, vertex operator algebras and their applications, Contemp. Math., vol. 442, Amer. Math. Soc., Providence, RI, 2007, pp. 179–192, DOI 10.1090/conm/442/08525. MR2372562 (1) (3) [21] Toshiki Nakashima, Ultra-discretization of the G2 -geometric crystals to the D4 -perfect crystals, Representation theory of algebraic groups and quantum groups, Progr. Math., vol. 284, Birkh¨ auser/Springer, New York, 2010, pp. 273–296, DOI 10.1007/978-0-8176-46974 11. MR2761943 [22] Masato Okado and Anne Schilling, Existence of Kirillov-Reshetikhin crystals for nonexceptional types, Represent. Theory 12 (2008), 186–207, DOI 10.1090/S1088-4165-08-00329-4. MR2403558 [23] Dale H. Peterson and Victor G. Kac, Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci. U.S.A. 80 (1983), no. 6, i, 1778–1782, DOI 10.1073/pnas.80.6.1778. MR699439

304

KAILASH C. MISRA AND SUCHADA PONGPRASERT

Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205 Email address: [email protected] Department of Mathematics, Srinakharinwirot University, Bangkok, Thailand 10310 Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15469

Twisted exterior derivative for universal enveloping algebras I ˇ Zoran Skoda Abstract. Consider any representation φ of a finite-dimensional Lie algebra g ˆ ∗ ) of its dual. Consider by derivations of the completed symmetric algebra S(g ˆ ∗ ) and the exterior algebra Λ(g). We show that the the tensor product of S(g ˜ of that tensor representation φ extends canonically to the representation φ product algebra. We construct an exterior derivative on that algebra, giving rise to a twisted version of the exterior differential calculus with the universal enveloping algebra in the role of the coordinate algebra. In this twisted version, the commutators between the noncommutative differentials and coordinates are formal power series in partial derivatives. The square of the corresponding exterior derivative is zero like in the classical case, but the graded Leibniz rule is deformed.

Contents 1. Preliminaries and basic notation 2. The twisted algebra of differential forms 3. Exterior derivative Acknowledgments References

1. Preliminaries and basic notation 1.1. Viewing the universal enveloping algebras of Lie algebras as noncommutative deformations of symmetric algebras in our earlier article with S. Meljanac [6] we have also deformed the (completed) Weyl algebra of differential operators, found deformed analogues of partial derivatives, and studied the deformations of Leibniz rules, all parametrized by certain datum which comes in many disguises as “orderings”, “representations by (co)derivations”, “realizations by vector fields” and “coalgebra isomorphisms between S(g) and U (g)”(cf. also [1, 9] and [4], Chap. 10). Some noncommutative deformations of this kind are interesting for physical applications, for example the κ-deformed Minkowski space [1, 10]. Physical picture should eventually include full-fledged differential geometry on such spaces, and field 2020 Mathematics Subject Classification. Primary . Key words and phrases. Universal enveloping algebra, exterior calculus, exterior derivative, deformed Leibniz rule, star product, Weyl algebra. c 2021 American Mathematical Society

305

306

ˇ ZORAN SKODA

theories on them; in particular the gauge theories based on connections on noncommutative fiber bundles. Our main motivation is to perform some early steps on the mathematical side of this complex programme. 1.2. In the present article, I will extend this picture to consistently include the exterior calculus in a canonical way, given the datum mentioned above. After the appearance of the first arXiv version of this article, more general non-canonical approaches to a noncommutative differential calculus were found for some special Lie algebras in [5]. We find it remarkable that our canonical extension exists, and that it stems from the unique extension of the smash product structure between the universal enveloping algebra and the space of so-called deformed derivatives to a bigger smash product algebra which includes also the exterior algebra Λ(g). This new observation is the main reason for writing this paper, with a hope that this exterior differential calculus will enable a variant of differential geometry where the base space is represented by the (spectrum of) the enveloping algebra U (g). 1.3. (Universal enveloping algebras as Hopf algebras.) We work over a fixed field k of characteristic 0. The unadorned tensor product ⊗ = ⊗k is over k. Taking a universal enveloping algebra is a functor from Lie to associative k-algebras; hence for any Lie algebra g, the Lie algebra map g → g⊕g, given by x → x⊕x, induces an algebra map Δ : U (g) → U (g ⊕ g) ∼ = U (g) ⊗ U (g), the terminal map g → 0 induces the map  : U (k) → U (0) ∼ = k, and the antihomomorphism g → g, x → −x induces an homomorphism γ : U (g) → U (g)op . U (g) is a Hopf algebra with comultiplication Δ, counit  and antipode γ ([2, 4, 8]). We assume that  the reader is familiar with the Sweedler notation [8] Δ(h) = i h(1)i ⊗ h(2)i = h(1) ⊗ h(2) , with or without an explicit summation index i. 1.4. (Notation on generators, duals; Weyl algebras.) From now on, we fix an n-dimensional Lie k-algebra g with k-basis x ˆ1 , . . . , x ˆn . The basis elements are identified with the generators of the universal enveloping algebra U (g). We introduce s s the structure constants Cij defined via commutation relations [ˆ xi , x ˆj ] = Cij x ˆs . To distinguish the canonical copy of g embedded in the symmetric algebra S(g) from the copy in U (g), we denote the corresponding basis x1 , . . . , xn ; this emphasize that xi -s commute in S(g), while tha generators with hat, x ˆi -s in U (g), do not. The dual basis of g∗ is denoted by ∂ 1 , . . . , ∂ n . Due to an antiisomorphism between the geometric picture with vector fields around unit element of a Lie group and the algebraic picture promoted here, it is natural, following [7], to introduce Weyl algebra An,k in (unusual) contravariant notation as the free associative k-algebra generated by symbols xi , ∂ j whre i, j − 1, . . . , n modulo the ideal generated by elements ∂ i ∂ j −∂ j ∂ i , xi xj −xj xi and ∂ i xj − xj ∂ i − δji where δ is the Kronecker symbol. Given a multiindex I = (i1 , . . . , in ) ∈ n Nn0 , denote xI := xi11 · · · xink , ∂ I := (∂ 1 )i1 · · · (∂ n )in and |I| := k=1 ik . The elements of the form xI ∂ J where I, J run over all multiindices form a basis of An,k and there is an increasing filtration  on An,k ([3]) by the degree of differential operator which is, for an element D = I,J aI,J xI ∂ J ∈ An,k , the maximal k ∈ N0 such that there is J with |J| = k and aI,J = 0. Completion with respect to this filtration is a topological k-algebra denoted Aˆn,k .

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1.5. If A is an associative k-algebra and H a Hopf algebra, then an action $ : H ⊗ A → A, or the equivalent representation ρ : H → Endk (A), is a left  Hopf action (synonym: (A, $) is a left Hopf H-module algebra [8]) if h$(a·b) = (h(1) $  a)(h(2) $ b) (equivalently, ρ(h)(a · b) = ρ(h(1) )(a) · ρ(h(2) )(b)) and h $ 1 = (h)1 for all h ∈ H, a, b ∈ A. Similarly, one defines right Hopf actions. Given a left Hopf action ρ : H → Endk (A), the corresponding smash product algebra A%ρ H is an associative algebra with underlying vector space A ⊗ H and multiplication given by the unique linear extension of the formula  (a%g)(b%h) = aρ(g(1) )(b)%g(2) h, a, b ∈ A, g, h ∈ G, where a%h ∈ A%ρ H denotes a ⊗ h within the smash product. Similarly, for a right Hopf action σ : H → Endop k (A), there is a smash product algebra H%σ A with underlying space H ⊗ A and multiplication  (g%a)(h%b) = gh(1) %σ(h(2) )(a)b. Sometimes, it is useful to present H%σ A by generators and relations. There are inclusions of algebras H → H%σ A ←& A given by H  h → h%1A and A  a → 1H %a; their images generate H%σ A. This means that H%σ A is a quotient of the free product H ∗A (the coproduct in the category of associative algebras). It is not difficult to see that the only additional relations are the commutation relations between elements in A and elements in H. These may be read from the multiplication rule. Hence, H%σ A is the quotient of the free product H ∗A by the ideal generated by all elements of the form  ah − h(1) σ(h(2) )(a), a ∈ A, h ∈ H.  In the case of a left Hopf action ρ, the relations are ha − ρ(h(1) )(a)h(2) . 1.6. A choice of basis x1 , . . . , xn in an n-dimensional k-vector space V (e.g. V = g) induces a manifest isomorphism S(V ) ∼ = k[x1 , . . . , xn ] and of its dual ˆ ∗ ) to the completed power ring k[[∂ 1 , . . . , ∂ n ]]. S(V )∗ ∼ = S(V The symmetric algebra on the dual S(V ∗ ) has a completion (by the filtration ˆ ∗ ) is isomorphic as a vector ˆ ∗ ). Algebra S(V by the degree of polynomial) S(V ∗ space to the linear (algebraic) dual (S(V )) : the nondegenerate pairing inducing this isomorphism is given by xI , ∂ J = |I|!δIJ . In the interpretation of formal power series in partial derivatives as representing a formal differential operator of ˆ ∗ ) and S(V ) is the evaluation of the infinite order, the duality pairing between S(V differential operator at 0. The symmetric algebra S(V ) is a Hopf algebra (even if V is not finite dimensional) in a unique way such that Δ(v) = 1 ⊗ v + v ⊗ 1 for all v ∈ V (equivalently, V may be given the structure of an Abelian Lie algebra a and the canonical algebra isomorphism S(V ) ∼ = U (a) transports the same coalgebra structure). 1.7. Let A be an associative (unital) k-algebra. We denote by Der(A) ⊂ Homk (A, A) the space of k-linear derivations A → A. It is a Lie k-algebra with respect to the usual commutator. Denote the algebra of k-endomorphisms of a kvector space V by Endk (V ). It is straightfoward to check that any homomorphism of Lie algebra φ : g → Der(A), has a unique extension to a left Hopf action U (g) → End(A). Similarly, any antihomomorphism g → Der(A) has a unique extension to a right Hopf action U (g) → Endop (A).

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1.8. If a is finite dimensional Abelian Lie algebra with basis x1 , . . . , xn , there is a Lie representation δ : a → Der(S(a∗ )) (and the completed variant δ : a → ˆ ∗ )) given on generators ∂ 1 , . . . , ∂ n of S(a) by δ(a)(∂ i ) = −a, ∂ i and, in Der S(a particular, δ(xi )(∂ j ) = −δij . If V is the underlying vector space of a, this induces a right Hopf action δ ◦ γ : S(V ) → Endop (S(V ∗ )), a completed variant S(V ) → ˆ ∗ )) and smash product algebras S(V )%δ◦γ S(V ∗ ) → S(V )%δ◦γ S(V ˆ ∗ ). Endop (S(V j j The correspondence xi → xi , ∂ → ∂ extends to an isomorphism between the smash product algebra and the Weyl algebra An,k from 1.4 (the same holds for the completions). An alternative viewpoint are the symplectic Weyl algebras, where instead of the pairing, where the basic datum is a symplectic form on a module over a commutative ring (V ∗ ⊕ V is the module in our case), may be found in [4], Chap. 8. 1.9. ([2]) All derivations of formal (commutative) power series rings, and of ˆ ∗ ) in particular, are continuous in formal topology. Consider a k-derivation S(g P and a formal power series D = D(∂ 1 , . . . , ∂ n ). Then the chain rule P (D) =  n ∂D r r=1 ∂(∂ r ) P (∂ ) holds. The partial derivatives of D are defined algebraically, in the sense of formal power series. In particular, the elements P (∂ r ) for r = 1, . . . , n determine the derivation P . ˆ ∗ )) is a Lie algebra homomorphism, then φ(ˆ 1.10. If φ : g → Der(S(g xi )φ(ˆ xj ) k xj )φ(ˆ xi )(∂ ) = φ([ˆ xi , x ˆj ]) in particular, hence the matrix (∂ ) − φ(ˆ k

xi )(∂ j )) φ = (φji ) := (φ(−ˆ ˆ ∗ ) satisfies the system of formal differential equations with entries in S(g (1)

φlj

∂ ∂ s k (φk ) − φli (φk ) = Cij φs , i, j, k ∈ {1, . . . , n} ∂(∂ l ) i ∂(∂ l ) j

Conversely, by linearity, this system is sufficient for φ to be a Lie homomorphism. ˆ ∗ )), left Hopf actions φ : U (g) → Therefore, homomorphisms φ : g → Der(S(g ∗ i ˆ Der(S(g )) and matrices (φj ) satisfying (1) are in 1-1 correspondence. Moreover, φ ◦ γ is a right Hopf action. Recall that γ(x) = −x for x ∈ g, hence (φ ◦ γ)(x) = φ(−x) and φij = (φ ◦ γ)(ˆ xi )(∂ j ). 1.11. From now on, we assume in addition that the matrix φ is invertible. ˆ ∗ )/ ∪n>0 S n (g∗ ) ∼ ˆ ∗ ) → S(g Consider the projection S(g = k, sometimes interpreted as the ’evaluation at 0 map’ (or the counit for the canonical structure of Hopf algebra on the symmetric algebra). If the image of φij under this projection is the Kronecker δji , we say that φ is close to the unit matrix and symbolically write φji = δij + O(∂). In the remainder of the paper, we assume that φ is invertible and close to the unit matrix. Given φ, these conditions on φ do not depend on the choice of basis of g used in defining φ, hence they are conditions on φ only. ˆ ∗ )) and the right Hopf action 1.12. The left Hopf action φ : U (g) → End(S(g ˆ ∗) ˆ ∗ )%φ U (g) ∼ φ ◦ γ induce the isomorphic smash product algebras S(g = U (g)%φ◦γ S(g which we may call the “φ-twisted Weyl algebra”. Indeed, by the free product deˆ ∗ )∗U (g) = U (g)∗ S(g ˆ ∗) scription from 1.5, both are quotients of the free product S(g by the commutation relations between the generators x ˆi and arbitrary elements in ˆ ∗ ). Using Δ(ˆ xi ) = x ˆi ⊗ 1 + 1 ⊗ x ˆi , we obtain Dˆ xi = x ˆi φ(1)(D) + φ(ˆ xi )(D) = S(g

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x ˆi D − φ(ˆ xi )(D) in one case and x ˆi D = φ(−ˆ xi )(D) + Dˆ xi in another smash product. By linearity of φ, these are clearly the same relations (the underlying general reason is that U (g) is cocommutative, see [7], the discussion following Definition 1). For the generators, when D = ∂ j , these relations read ∂ j x ˆi − x ˆi ∂ j = φji . 1.13. (Realizations in Weyl algebras.) For a finite dimensional g and general φ, the correspondence x ˆi → nk=1 xk φki , ∂ j → ∂ j does not depend on choice of basis, ˆ ∗ ) → Aˆn,k . If and extends uniquely to a homomorphism of algebras U (g)%φ◦γ S(g j the matrix φ = i ) is invertible, this homomorphism is invertible, with inverse (φ n ˆk (φ−1 )ki , ∂ j → ∂ j . given by x ˆi → j=1 x 1.14. It is shown in [6] that, under the assumptions in 1.11, the datum φ is equivalent to specifying a coalgebra isomorphism ξ = ξφ : S(g) → U (g) (example: the symmetrization map ξexp (3) induced by φexp from (4)) which equals the identity when restricted to k ⊕ g ⊂ S(g). However, there is no explicit formula relating φ and ξ in general. The isomorphism ξ enables us to transport the linear operators from S(g) to U (g). Partial derivatives ∂ i transport to the deformed derivatives ∂ˆi = (∂ i ) := ξ ◦ ∂ i ◦ ξ −1 : U (g) → U (g), satisfying the deformed Leibniz rules studied in [6]. This action of generators ∂ i on U (g) together with action of elements in U (g) on U (g) by multiplication, extend ˆ ∗ ) on U (g). In this paper, we try to naturally to an action  of entire U (g)%φ◦γ S(g ˆ Fortunately, there is an alternative avoid working with ξ directly, but we do need ∂. description 1.16 of  in terms of the smash product, generators and matrix φ. 1.15. The U (g)∗ is a topological Hopf algebra by duality with U (g). The transpose (2)

ξ T : U (g)∗ → S(g)∗

to any coalgebra isomorphism ξ : S(g) → U (g) is an algebra isomorphism. Hence, one can transport the topological coalgebra structure along this morphism and ˆ ∗) ∼ obtain a nontrivial topological Hopf algebra structure on S(g = S(g)∗ , stud∗ ˆ ied in the disguise of deformed Leibniz rules for S(g ) in [6]. Instead, we may transport the algebra structure from U (g) to S(g) obtaining the star product ˆ ∗ ) is then the dual to the star f  g = ξ −1 (ξ(f ) ·U(g) ξ(g)). The coproduct on S(g product. An infinite dimensional version of the Heisenberg double construction applied to U (g) results in an algebra, which is shown in [11] to be isomorphic to ˆ ∗ ) whenever the associated matrix φ satisfies the conditions from 1.11. U (g)%φ◦γ S(g Algebra U (g) is filtered by finite dimensional components and the dual is a cofiltered coalgebra. Thesis [12] has exhibited that the Heisenberg double U (g)∗ %U (g) is rigorously defined and has a structure of an internal Hopf algebroid in the symmetric monoidal category of filtered-cofiltered vector spaces. A variant of this Hopf algebroid with some extra completions (providing an ad hoc working setup, less satisfactory from the categorical point of view) has been exhibited a bit earlier ˆ ∗ ) and using explicit calculations with a concrete in [7], in a form of U (g)%φ◦γ S(g choice of φ. In that choice, the coalgebra isomorphism ξ = ξexp : S(g) → U (g) is the symmetrization (or coexponential) map [2, 4, 7, 9], where for any zˆ1 , . . . , zˆk ∈ k

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(not necessarily basis elements) (3)

ξexp (z1 . . . zk ) =

1  zˆσ(1) · · · zˆσ(k) k! σ∈Σ(k)

or, alternatively z l → zˆl for any z ∈ g ⊂ S(g) and l ∈ N. The corresponding φij is (4)

(φexp )ij =

∞  (−1)N BN N i (C )j , N!

N =0

ˆ ∗ ) given where BN are the Bernoulli numbers and C is a matrix of elements in S(g i i k by Cj = Cjk ∂ . This formula has a long history (and direct relations to standard notions in Lie theory, e.g. the linear part of Hausdorff series) is derived over an arbitrary ring of characteristic 0 in [4]. In the case of k = R, the details of the relation to a geometric realization via formal differential operators around the unit element on a Lie group is exhibited in [7], 1.2. 1.16. Under the assumptions 1.11, there is a deformed Fock action of ˆ ∗ ) on the universal enveloping algebra U (g)%φ◦γ S(g ˆ ∗ )) ⊗ U (g) → U (g), : (U (g)%φ◦γ S(g obtained in four steps ([7]). First we embed the second tensor factor U (g) a U (g) ⊗ ˆ ∗ ), then multiply, then use the isomorphism with U (g)%φ S(g ˆ ∗) k → U (g)%φ◦γ S(g and then project the second factor to k via the counit of the symmetric algebra (the ’evaluation at 0 map’ from 1.11); the result is in U (g) ⊗ k ∼ = U (g). This is the standard physics procedure of pushing the partial derivatives to the right across coordinates (this time noncommutative coordinates, x ˆi ) using commutation relations and when all partial derivatives are on the right and coordinates on the left, retaining only the summand without partial derivatives. This way U (g) becomes ˆ ∗ ), the φ-deformed Fock module. Element |0 = a left module of U (g)%φ◦γ S(g 1U(g) is the deformed Fock vacuum |0 g = |0 and we denote the result of the action on the vacuum by x → x|0 . Map ξ is simply the inclusion of algebras −1 k ˆ ∗ ) extending xi → n x S(g) → U (g)%φ◦γ S(g )i , followed by the action on k=1 ˆk (φ the deformed vacuum. ˆ ∗ ) this is essentially the harpoon If, in the first tensor factor, we restrict  on S(g action of the topological dual algebra; this restriction is a topological left Hopf action. If, in the first tensor factor, we restrict  on U (g), action  is simply the multiplication in U (g). Assuming the presentation of the Weyl algebra An,k as the smash product S(V ∗ )%δ◦γ S(V ) for V = k[x1 , . . . , xn ] (see 1.8), the standard Fock module of An,k is a special case of the corresponding deformed Fock module, namely S(V ) = U (a) and a is the Abelian Lie algebra with underlying space V , and we use the deformed Fock space construction for this case, without completions. The action, which we now denote by $, clearly agrees with the standard description as the action of a differential operator ([3]) and the Fock action is |0 = 1S(V ) . For Aˆn,k we use completions of course. The linear map ξ −1 : U (g) → S(g) can be described ˆk φki followed by the action on the ˆi → x ˆφi := x as embedding U (g) → Aˆn,k , x standard Fock vacuum 1S(g) . In particular, we obtain the description of the operator ∂ˆi = ξ ◦ ∂ ◦ ξ −1 = ∂ i  on U (g) by iteratively pushing partial derivatives to the

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right using the commutation relation (5)

∂ i  (ˆ xj u ˆ) = x ˆj (∂ i  u ˆ) + φij  u ˆ, ,

u ˆ ∈ U (g)

and, at the end of the inductive procedure, retaining the summand without partial derivatives. Of course, φij is an infinite series, but commuting with x ˆk can drop the degree of each monomial only by 1 (unless the commutator vanishes), hence we retain at each step only the summands in φ of the degree at most the degree of the noncommutative polynomial to the right of it. See an example 3.2. 1.17. (This paragraph are remarks for deeper understanding and is not used in the rest of the article.) Under the assumptions 1.11, the realization map from 1.13 ˆ ∗ ) and the usual supplies the isomorphism between the smash product U (g)%φ◦γ S(g ˆ completed Weyl algebra An,k , so one may ask why introducing this smash product at all. The action  and the corresponding deformed space make a difference, along with constructions derived from it. This action is Hopf with respect to the ˆ ∗ ) coming from the duality with U (g). The topological Hopf algebra structure on S(g map φ provides a relation between  and the ’harpoon’ action of the formal dual ˆ ∗) ∼ U (g)∗ on U (g). The structure of the topological Hopf algebra on S(g = U (g)∗ together with U (g) considered as a Hopf module over it (actually more than that, a braided commutative Yetter-Drinfeld module making sense in a monoidal category of filtered-cofiltered vector spaces) together induce both a smash product in the appropriate category and an additional structure of an internal Hopf U (g)-algebroid on that internal smash product. This is shown in [12]. Earlier, it was sketched in [11] that this internal smash product is the same as an associative algebra with our smash product above (where the Hopf algebra is U (g) and the Hopf module U (g)∗ , rather than the other way around). A version of the Hopf algebroid (with some extra completions and somewhat ad hoc axioms on completions) is also derived in [7]. If we replace φ by another choice of homomorphism, say ψ, we observe (ξ T -s are from (2)) isomorphisms of internal Hopf algebroids (6)

T −1 idU (g) ⊗(ξφ )

ˆ ∗) U (g)%φ◦γ S(g

/ U (g)%U (g)∗

T idU (g) ⊗ξψ

/ U (g)%ψ◦γ S(g ˆ ∗ ),

where the Heisenberg double smash product Hopf algebroid in the middle is constructed using only the canonical topological Hopf pairing between Hopf algebra U (g) and its canonical dual (toplogical) Hopf algebra U (g)∗ (and filtered cofiltered structures in place [12]). Actions φ and ψ get interchanged along the composition of isomorphisms (6) tensored by the identity on the additional U (g) factor. However, the operator ∂ˆi depends on φ as the image of ∂ i under the composition isomorphism (6) is not ∂ i in general. 2. The twisted algebra of differential forms 2.1. Lemma. Let g be a Lie algebra, A an associative algebra and ρ : g → Der(A) a linear map. (i) If g, h ∈ g and ρ(g)ρ(h) − ρ(h)ρ(g) − ρ([g, h]) vanishes when applied on each of two elements a, b ∈ A then it vanishes on their product ab. (ii) Suppose that there is a family of algebra generators {aλ }λ∈Λ of A and a subset S ⊂ g which spans g, and such that for all g, h ∈ S, ρ(g)ρ(h)(aλ ) − ρ(h)ρ(g)(aλ ) = ρ([g, h])(aλ ),

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Then ρ is a representation of g on A by derivations. (iii) Suppose Aˆ is a topological associative k-algebra containing A as a dense subalgebra. If for each h ∈ H derivation ρ(h) extends (automatically uniquely) to a ˆ then ρ is a representation of k by continuous continuous k-linear map ρ (h) of A, k-derivations iff ρ is a representation. 2.2. (Notation on exterior algebras.) In our constructions, it will be useful to distinguish notationally the generators of two distinct copies of the classical exterior algebra Λ(g): in the first the generators will be denoted by dˆ x1 , . . . , dˆ xn and in the second by dx1 , . . . , dxn (the latter copy first appears in 2.6 and will be denoted Λcl (g)). Both bases correspond to x ˆ1 , . . . , x ˆn under g → Λ(g). Recall the xi )(∂ j ). convention from the introduction: φji := φ(−ˆ ˆ ∗ )) satisfy2.3. Main Theorem. Any Lie homomorphism φ : g → Der(S(g ˜:g→ ing the assumptions from 1.11 uniquely extends to a Lie homomorphism φ ∗ ˆ Der(Λ(g) ⊗ S(g )) satisfying   n  ∂ −1 k s ˜ φ φri . (7) φ(ˆ xi )(dˆ xl ) = − dˆ xk (φ )s ∂(∂ r ) l k,r,s=1

This extension does not depend on the choice of basis of g. Proof. By the Leibniz rule, any k-derivation of an algebra A is determined by its values on the generators of A, hence uniqueness follows even without the ˜ be a Lie homomorphism. For the existence of φ ˜ as a k-linear requirement that φ ˆ ∗ ) and on g are predetermined, it remains to be shown map, since the values on S(g ˆ that the extension by the Leibniz rule to the entire Λ(g) ⊗ S(g) is well defined. The only new nontrivial relation is antisymmetry dˆ xr ∧ dˆ xs = −dˆ xs ∧ dˆ xr . The Leibniz rule gives     ∂ ∂ a b −1 k a ˜ xi )(dˆ φ(ˆ φ φ φ φbi , xr ∧dˆ xs ) = dˆ xk ∧dˆ xs (φ−1 )ka +dˆ x ∧dˆ x (φ ) r k i a ∂(∂ b ) r ∂(∂ b ) s where the right-hand side is evidently antisymmetric under the exchange (r ↔ s). ˜ is automatically a representation (Lie homomorIt remains to show that φ ˆ ∗ ) is continuous ([2]) and satisfies the chain rule, phism). Every derivation of S(g ˆ ∗ ) trivial. which also makes problem of extension of derivations from S(g∗ ) to S(g ∗ ˆ Similar statements hold for Λ(g) ⊗ S(g ). Thus we can apply Lemma 2.1 (ii) to ˜ is a Lie homomorˆ ∗ ) to assert that φ A = Λ(g) ⊗ S(g∗ ) and (iii) to Aˆ = Λ(g) ⊗ S(g phism iff for all i, j, l ∈ {1, . . . , n}, (8)

˜ xi )φ(ˆ ˜ xj )(dˆ ˜ xj )φ(ˆ ˜ xi )(dˆ ˜ xi , x φ(ˆ xl ) − φ(ˆ xl ) − φ([ˆ ˆj ])(dˆ xl ) = 0.

˜ xi ) we calculate, ommiting the summation sign when Using the Leibniz rule for φ(ˆ summing over repeated indices, ˜ xi )φ(ˆ ˜ xj )(dˆ φ(ˆ xl )

∂φs

∂φs

˜ xi )(dˆ = −φ(ˆ xk )(φ−1 )ks ∂(∂ rl ) φrj − dˆ xk φ(ˆ xi )((φ−1 )ks ) ∂(∂ rl ) φrj  s  ∂φ ∂φs −dˆ xk (φ−1 )ks φ(ˆ xi ) ∂(∂ rl ) φrj − dˆ xk (φ−1 )ks ∂(∂ rl ) φ(ˆ xi )(φrj )

We first show that the first two summands mutually cancel. By direct substitu ∂φp  ∂φs tion of (8), the first summand becomes +dˆ xk (φ−1 )kp ∂(∂ rk ) φri (φ−1 )ks ∂(∂ rl ) φrj . By  p ∂D This for D = (φ−1 )ks , the chain rule (see 1.9), φ(ˆ xj )(D) = − nt=1 ∂(∂ p ) φj .

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together with the formula for the derivative of the inverse matrix we find that ∂φs





φ(ˆ xj )((φ−1 )ks ) = (φ−1 )ks ∂(∂rp) (φ−1 )rs φpj , hence the second summand above is 

−dˆ xk (φ−1 )ks

∂φsr p −1 r ∂φsl r φ (φ )s φ . ∂(∂ p ) j ∂(∂ r ) j

The cancelation of the first two terms may be now observed after appropriately renaming dummy indices. By the chain rule, the third summand is +dˆ xk (φ−1 )ks

∂ 2 φsl φp φr ∂(∂ p )∂(∂ r ) i j

and is clearly antisymmetric under exchange (i ↔ j). Therefore, in (8), it cancels ˜ xi )(dˆ ˜ xj )φ(ˆ xl ). It remains to consider the contribuwith the third summand in φ(ˆ tions from the 4th summand, giving ˜ xi )φ(ˆ ˜ xj )(dˆ ˜ xj )φ(ˆ ˜ xi )(dˆ φ(ˆ xl ) − φ(ˆ xl ) =  ∂φrj p ∂(∂ p ) φi

∂φs

= dˆ xk (φ−1 )ks ∂(∂ rl )

∂φs

(1)

−

∂φri p ∂(∂ p ) φj





k r = −dˆ xk (φ−1 )ks ∂(∂ rl ) Cij φk  ˜ xi , x = φ([ˆ ˆj ])(dˆ xl ).

The tensorial notation is suggestive for the covariance properties with respect to the choice of basis. Let the prime indices denote a new basis, B be a matrix of change of   ˆk Bik and the for the dual basis ∂ i = (B −1 )ik ∂ k . basis of g in the sense that x ˆi = x By linearity of the map φ and of k-derivation φ(−ˆ xi ) then 







xi )(∂ j ) = (B −1 )jj  Bii φ(−ˆ xi )(∂ j ) = (B −1 )jj  Bii φji . φji = φ(−ˆ 









This gives also xi = xk Bjk (B −1 )jj  Bii φji φji = xk Bik . For the differentials, the  embedding g → Λ(g) forces basis change dˆ xi = dˆ xk Bik extended by the usual tensoriality of higher exterior powers. For the left hand side of (7), by linearity,   φ(ˆ xi )(dˆ xl ) = Bii Bll φ(ˆ xi )(dˆ xl ). Thus all ingredients in (7) behave tensorially just as expected from the position of indices and the tensorial form is enough for the conclusion. Explicitly, we substitute the above component changes to the ingredients of the right hand side of (7) to obtain         ∂ s dˆ xk Bkk (φ−1 )ks Bss (B −1 )kk φ Bll (B −1 )ss φri Bii (B −1 )rr ,  ∂(∂ r ) l which gives the same change of basis coefficients as for the left hand side, after the contractions of numerical matrices B and B −1 above are accounted for. ˆ ∗ ) of a finite dimen2.4. Corollary. Any representation φ : g → Der(S(g sional Lie algebra g by derivations on completed symmetric algebra of its dual has a canonical extension to a Hopf action of the form ˜ : U (g) → Endk (Λ(g) ⊗ S(g ˆ ∗ )) φ and satisfying (7). In particular, the smash products ∼ ˆ ˆ U (g)%γ◦φ˜ (Λ(g) ⊗ S(g)) = (Λ(g) ⊗ S(g))% ˜ U (g), φ are well-defined.

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2.4.1. These two smash product algebras are canonically isomorphic by the cocommutativity of U (g). We call any of the two the extended algebra of φtwisted differential forms. ’Extended’ is for the additional presence of partial derivatives in the algebra. A different, less canonical, recipe is needed if we want that the subspace of ’forms’ without partial derivatives be a subalgebra (cf. [5]). This is planned to be discussed in a sequel work (part II). Notice that the usage of the antipode γ in the choice of the smash product action is in agreement with the minus sign in the formula φji = φ(−ˆ xi )(∂ j ). 2.5. We would like to describe the extended algebra of φ-differential forms by generators and relations. However, this algebra involves completions, and in particular formal power series in ∂-s and we need to consider it as either topological algebra or algebra with complete cofiltration (as a vector space) and multiplication distributing over formal sums in each argument, see [7], Appendix A.2 and 2.5.1. 2.5.1. A cofiltration on a vector space V is an inverse sequence of epimorphism of its quotients . . . → Vi → Vi−1 → . . . → V0 . Denote the quotient maps πi : V → Vi and bonding epimorphisms πi,i+k : Vi+k → Vi , the identities πi = πi,i+k ◦ πi+k and πi,i+k+l = πi,i+k ◦ πi+k,i+k+l hold. It is useful for us to consider cofiltrations as dual construction to increasing (ascending) filtrations like on U (g) which involve inclusions of filtered components. However, many expositions treat cofiltrations in terms of ’descending filtrations’ of kernels ker π0 ⊃ . . . ⊃ ker πi ⊃ ker πi+1 ⊃ . . . with Vi = V / ker πi and πi.i+k induced by inclusions. These kernels form a basis of neighborhoods of 0 in a topological approach. A thread is a sequence (vr )r∈N0 ∈ Πr Vr such that vr = πr,r+k (vr+k ). The set of threads is a vector space (with a canonical cofiltration) limi Vi = Vˆ equipped with canonical completion map V → Vˆ , v → (πr (v))r . If this map is an isomorphism of vector spaces, we say that the cofiltration is complete and the vector space  complete cofiltered. A possibly infinite expression λ∈Λ vλ of elements in V is a formal sum if for any i ∈ N0 there are only finitely many λ such that πi (vλ ) = 0. A formal sum may be viewed as a useful representative of an element of Vˆ [7, 12]; the completion Vˆ may be viewed as a set of equivalence classes of formal sums. A linear map f : V → W of k-vector spaces equipped with complete cofiltrations distributes over formal sums if for every formal sum λ vλ , the expression    f (v ) is also formal sum and f ( v ) = f (v ). Consider category cfalg⊗ λ λ λ λ λ λ whose objects are associative k-algebras A equipped with complete cofiltration and such that the multiplication μ : A ⊗ A → A distributes over formal sums in each argument (i.e. μ(a, −) and μ(−, a) distribute over formal sums for each a ∈ A); the maps are k-linear maps distributing over formal sums. This is a bit weaker structure than that of complete cofiltered algebras where the multiplication extends ˆ → A; the difference is just like by definition to the completed tensor product, A⊗A between the continuity in each argument and joint continuity. It is reflected by the subscript ⊗ in cfalg⊗ . 2.5.2. Consider now the free associative k-algebra F = kˆ x1 , . . . , x ˆn , dˆ x1 , . . . , dˆ xn , ∂ 1 , . . . , ∂ n on 3n symbols dˆ xi , x ˆi and ∂ i , for i = 1, . . . , n and a subalgebra Fxd generated by x ˆi -s and dˆ xi -s only. We also consider the free product of associative algebras F = Fxd  k[[∂ 1 , . . . , ∂ n ]] where we allow for formal power series in ∂-s which mutually commute, but do not commute with other generators. There are obvious

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maps from F and from F to φ-twisted differential forms; the map from F is not surjective due completions, while the map from F is surjective, but this algebra is canonically completed in a way which is incompatible with quotienting to twisted differential forms (the kernel would not be closed). Let us define a category K as follows. The objects are maps of associative algebras j : F → A where A is an object of cfalg⊗ (2.5.1) such that • j(u∂ i1 · · · ∂ ik+1 ) ∈ ker πk for each k ∈ N0 , u ∈ Fxd and i1 , . . . , ik+1 ∈ {1, . . . , n}. This is a statement about a cofiltration on A. Notice that u multiplies partial derivative symbols from the left. • The following equations hold after applying j (which we omit in writing) (9)

k [ˆ xi , x ˆj ] = Cij x ˆk , E D j xi = 0, ∂ , dˆ

[∂ i , ∂ j ] = 0, 

[dˆ xj , x ˆi ] = dˆ xs (φ−1 )sr

∂ φr ∂(∂ l ) j



[∂ j , x ˆi ] = φji , φli ,

{dˆ xi , dˆ xj } = 0

where the commutator [, ] and the anticommutator {, } are in the sense of associative algebras. Intuitively, these are required relations in a completion of j(F ) ⊂ A. Morphisms in K from j : F → A to j  : F → A are morphisms α : A → A in cfalg⊗ such that j  = α ◦ j. 2.5.3. Theorem. (i) The extended algebra of φ-twisted differential forms is the (codomain of ) the universal initial object in category K. ˆ ∗ ), k ∈ N and gˆ1 , . . . , gˆk ∈ g. Then (ii) Suppose, ω ∈ Λ(g) ⊗ S(g (10)

˜ ◦ γ)(ˆ (φ g1 · · · gˆk )(ω) = [[. . . [ω, gˆ1 ], . . .], gˆk ]. (iii) Degree of a twisted differential form is a well defined nonnegative integer.

Proof (sketch). We use the description 1.5 of the smash product as free ˆ ∗ )) modulo all commutation relations of the form product U (g)  (Λ(g) ⊗ S(g  (11) φ(u)(ω) = u ˆ(1) ωγ(ˆ u(2) ), ˆ ∗ ). It is however sufficient to include the where u ˆ ∈ U (g) and ω ∈ Λ(g) ⊗ S(g relations where u ˆ, ω are within chosen sets of generators. The kernel for the map ˆ ∗ )) is easy to observe. The relations from F to the free product U (g)  (Λ(g) ⊗ S(g (9) enable to write every element in j(F ) as formal sum of the terms with partials on the right, hence we can see cofiltration on the subspace of A consisting of all such elements (for every j) using j(u∂ i1 · · · ∂ ik+1 ) ∈ ker πk . We leave to the reader to show using these facts that the initial object j : F → A lifts to a surjection F → A ˆ and, using this, to conclude the universal property for F → U (g)%γ◦φ˜(Λ(g) ⊗ S(g)). ˆi + x ˆi ⊗ 1, S x ˆi = −ˆ xi , the calculation is (i) Regarding that Δ(ˆ xi ) = 1 ⊗ x ˜ xi )(∂ j ) = −φj ˆ, then dˆ xj or ∂ j for ω, and use identities φ(ˆ easy: substitute x ˆi for u i and (7). For (ii), notice that φ ◦ γ is an antiisomorphism, φ(γ(ˆ gi ))(ν) = φ(−ˆ gi )(ν) = ˆ ∗ ), and proceed by induction. [ν, gˆi ] for all ν ∈ Λ(g) ⊗ S(g 2.6. Theorem. If φ = (φij ) is invertible and close to the identity ( 1.11), the correspondence   (12) x ˆi → xφi := xj φji , ∂ i → ∂ i , dˆ xi →  dˆ xφi := dxj φji j

j

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extends uniquely to an isomorphism ˆ ∗ ˆ U (g)%φ◦γ ˜ (Λ(g) ⊗ S(g )) −→ Λcl (g) ⊗ An,k .

(13)

2.6.1. We here used the notation conventions on exterior algebras 2.2. By abuse of notation, we may however by dxi denote also the preimage of dxi ∈ xk (φ−1 )ki . Notice that Λcl (g) ⊗ Aˆn,k under the isomorphism (12), namely dxi := dˆ j [dxi , x ˆj ] = 0 and [dxi , ∂ ] = 0. It follows that we have a monomorphism Λcl (g) ⊗ ˆ ∗ )) which is in fact an isomorphism because ˆ ∗ )) → U (g)% ˜ (Λ(g)⊗ S(g (U (g)%φ◦γ S(g φ◦γ j ˆ ∗ ˆl generate U (g)%φ◦γ the generators of the form dxi , ∂ , x ˜ (Λ(g) ⊗ S(g )). In this ˆ ∗ )) the intermediate algebra. context, we call Λcl (g) ⊗ (U (g)%φ◦γ S(g ˆ ∗ )) 2.6.2. Sometimes, it is good to consider realizations of Λcl (g)⊗(U (g)%φ◦γ S(g making it isomorphic to Λcl (g) ⊗ Aˆn,k : here the generators of Λ(g) are also dxi and  ˆ ∗ ) is realized by x U (g)%φ◦γ S(g ˆφi = k xk φkj and ∂ i is just ∂ i from Aˆn,k . Hence the intermediate algebra and the isomorphic extended algebra of φ-twisted forms do not depend on φ as algebras (more is true, along the lines in 1.17).   k k 2.6.3. Notation for generators x ˆφi = xφi = k xk φi , dˆ k dxk φi extends to ˆ ∗) ∼ polynomials. The composition U (g) → U (g)%S(g = Aˆn,k is an algebra monomorφ φ phism which agrees with () : u → u . We do not use notation ()φ for derivatives, because our isomorphism sends ∂ i to ∂ i and this does not depend on φ.  Thus dˆ xφi = k dxk φki ∈ Λ(g) ⊗ Aˆn,k . If we commute elements within the image Λ(g) ⊗ Aˆn,k , we get the  same commutators as in (9), but in a realization, e.g. ∂ r φlj . Thus the theorem 2.6 may be interpreted as ˆφj ] = dˆ xφs (φ−1 )sr ∂(∂ [dˆ xφi , x l ) φi a realization of the extended algebra of φ-twisted differential forms in terms of ordinary differential forms and partial derivatives (allowing infinite series).

3. Exterior derivative 3.1. Definition. φ-twisted exterior derivative is the k-linear map given by dˆ = dˆφ :=



ˆ ∗ )) → U (g)%(Λ(g) ⊗ S(g ˆ ∗ )), dˆ xk (φ−1 )kj ∂ˆj : U (g)%(Λ(g) ⊗ S(g

k,j

where (φ−1 )kj acts by multiplication, while ∂ˆj = ∂ j  acts on U (g) tensor factor only, using φ-deformed Fock action  from 1.16. We may write the exterior multiplication following dˆ xk for emphasis; the multiplication is assumed. 3.1.1. Notice that ∂ j and ∂ˆj do not commute, hence (φ−1 )kj and ∂ˆj do not commute in general. By the discussion in 1.17 not only the smash product, but the action  is also independent of φ up to an isomorphism. However, the expression ∂ˆj depends on φ, see 1.17. Thus, dˆφ may depend on φ in general. 3.2. It is clear that this operator does not depend on the choice of basis. If we ˆφi ∈ Aˆn,k then in this realization, the abstract dˆ can also be written realize x ˆi as x as  φ  dˆ = dˆ xk (φ−1 )kj ∂ˆj = dxj ∂ˆj , k,j

j

once we interpret ∂ˆj properly in Λcl ⊗ Aˆn,k , which comes naturally if we work in ˆ ∗ )). This is different from terms of the intermediate algebra 2.6.1, Λcl ⊗ (U (g)%φ S(g

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 the usual exterior derivative d = k dxk ∂ k as ∂ k acts on the factor S(g) in ˆ ∗ )), see 1.4. Of course, via the inverse of the realization map, we Λcl ⊗ (S(g)%δ◦γ S(g could transport d to our setup as well, but there is nothing essentially new. 3.3. Example. Consider the series representation for φij , 1 φij = δji + Aijr ∂ r + Aijrs ∂ r ∂ s + O(∂ 3 ). 2 p i i and Aijrs = 16 Cps Cjr . For the case φ = φexp (see 1.15), formula (4) gives Aijr = 12 Cjr k k ˆ Using the procedure from the end of 1.16 and (5), we obtain ∂ (ˆ xi ) = δi and xi x ˆj ) ∂ˆk (ˆ xi x ˆj x ˆl ) ∂ˆk (ˆ

= =

x ˆi δjk + x ˆj δik + Asij k x ˆi x ˆ j δl + x ˆi x ˆl δjs + x ˆj x ˆj δis + Akjl x ˆi + Akij x ˆl + Akil x ˆj + Akir Arjl + 12 Akijl

ˆ xi ) = dxi , but Thus, d(ˆ (14)

ˆ xi x d(ˆ ˆj ) ˆ ˆj x ˆl ) d(ˆ xi x

= (dxj )ˆ xi + (dxi )ˆ xj + Akij dxk = (dxl )ˆ xi x ˆj + (dxj )ˆ xi x ˆl + (dxi )ˆ xj x ˆl +dxk (Akjl x ˆi + Akij x ˆl + Akil x ˆj + Akir Arjl + 12 Akijl )

Therefore, Leibniz rule is deformed for dˆ in general. In a φ-realization, (14) differs from the application of the classical exterior derivative d on the realization which simply reads d(ˆ xφi x ˆφj ) = (dˆ xφi )ˆ xφj + x ˆφi (dˆ xφj ) φ φ φ φ φ φ d(ˆ xi x ˆj x ˆl ) = (dˆ xi )ˆ xj x ˆl + x ˆφi (dˆ xφj )ˆ xφl + x ˆφi x ˆφj dˆ xφl . 3.4. Theorem. (i) dˆ2 = 0. ˆ and d(dx ˆ s ∧ ω) = −dxs ∧ dω; ˆ ˆ xs ∧ ω) = −dˆ xs ∧ dω (ii) d(dˆ Proof. (i) Regarding that dxm commute with elements in S(g∗ ), (15)

dˆ2 = (dxj ∂ˆj ) ∧ (dxm ∂ˆm ) = dxj ∧ dxm ∂ˆj ∂ˆm

is a contraction of the antisymmetric tensor dxj ∧ dxm and the symmetric tenˆ do not commute with ∂-s nor (φ−1 )k , and sor ∂ˆj ∂ˆm , hence zero. Note that ∂-s j k ˆ dˆ xl does not commute with ∂ . Thus, dˆ2 = dˆ xk ∧ dˆ xl (φ−1 )kj ∂ˆj (φ−1 )lm ∂ˆm and (φ−1 )kj ∂ˆj (φ−1 )lm ∂ˆm is not a symmetric tensor, unlike written in my earlier preprint. ˆ definition dxi = dˆ (ii) The identities follow from the definition od d, xk (φ−1 )ki and the antisymmetry of the product of generators in Λ(g). 3.5. Proposition. dˆ preserves the subalgebra ΛU (g) generated by all dxi and x ˆi . There is an embedding ˆ ∗ )) ˆ ∗ )) ∼ Λcl (g) ⊗ U (g) → Λcl (g) ⊗ (U (g)%φ◦γ S(g = U (g)% ˜ (Λ(g) ⊗ S(g φ◦γ

whose image is ΛU (g). 3.5.1. We say that ΛU (g) is the algebra of U -twisted differential forms. It is a subalgebra of the extended algebra of φ-twisted differential forms, or equivalenty of the “intermediate algebra” 2.6.1. 3.6. Proposition. The usual Fock space action of An,k (and Aˆn,k ) on S(g) ˆ ∗ )) ∼ extends to an action of Λcl (g) ⊗ (U (g)%φ◦γ S(g = Λcl (g) ⊗ Aˆn,k on Λcl (g) ⊗ S(g) by multiplication in the first tensor factor. The φ-deformed Fock space action

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ˆ ∗ ) on U (g) extends to an action of the intermediate  (see 1.16) of U (g)%φ◦γ S(g ˆ ∗ )) on its subalgebra Λcl (g) ⊗ U (g). This action subalgebra Λcl (g) ⊗ (U (g)%φ◦γ S(g restricts to the multiplication on Λcl (g) ⊗ U (g). ˆ ∗) Keep in mind that elements from Λcl (g) commute with the elements in S(g ∗ ˆ and U (g), while the elements in U (g) do not commute with those in S(g ). The cyclic vectors for the two extended Fock spaces are still |0 = 1S(g) and 1U(g) . 3.7. Theorem. Let ω ∈ ΛU (g). Consider the action on usual vacuum |0 = 1S(g) . ˆ (i) Symbolically (ΛU (g))φ |0 = Λcl (g) ⊗ S(g). More explicitly, the linear map sending ω ∈ ΛU (g) to ω φ |0 sends ΛU (g) into Λcl (g) ⊗ S(g). This action on vacuum in φ-realization is an isomorphism of vector spaces (we asume φ close to identity) with inverse given by the φ-deformed Fock action on |0 U(g) , that is idΛcl (g) ⊗ξφ : Λcl (g) ⊗ S(g) → Λcl (g) ⊗ U (g) ∼ = ΛU (g). ˆ (ii) (dω)|0 = d(ω|0 ), where d on the right hand side denotes the usual exterior derivative. ˆ ˆ = ω. (iii) (Poincar´e lemma) If d(ω) = 0 then there is ν ∈ ΛU (g) such that d(ν) Proof. (i) and (ii) follow by direct check. (iii) follows from the classical ˆ the isomorphism in (i), property (ii) and TheoPoincar´e lemma for Λcl (g) ⊗ S(g), rem 3.4, (ii). 3.8. (Star product on ΛU .) Using the isomorphism of vector spaces ξ˜ = ξ˜φ = idΛ(g) ⊗ξφ = Λcl (g) ⊗ S(g) → Λcl (g) ⊗ U (g) ∼ = ΛU (g), we can easily extend the star product  = φ on S(g) (see 1.15) to an associative ˜ ˜ ∧Λcl (g)⊗U(g) ξ(ν)). product ∧ = ∧φ on Λcl (g) ⊗ S(g) given by ω ∧ ν = ξ˜−1 (ξ(ω) ˆ ω ∧ νˆ) = ξ(d( ˜ ξ˜−1 (ˆ ω ) ∧ ξ˜−1 (ˆ ν ))). 3.9. Proposition. If ω ˆ , νˆ ∈ ΛU (g) then d(ˆ Proof. For the deformed derivatives ∂ˆi = (∂ i ) = ξ ◦ ∂ ◦ ξ −1 , hence it is immediate ([6]) that ∂ˆi (ξ(f ) ·U(g) ξ(g)) = ξ(∂ i (f  g)) for all f, g ∈ S(g). By (bi)linearity it is sufficient to prove the statement when ω ˆ = dxi1 ∧ · · · ∧ dxir ξ(f ) and νˆ = dxj1 ∧ · · · ∧ dxjs ξ(g). Then by the definitions of dˆ and of ξ˜ and ˜, we obtain ˆ ω ∧ νˆ) d(ˆ

= = = =

dxk ∧ dxi1 ∧ · · · ∧ dxir ∧ dxj1 ∧ · · · ∧ dxjs (ξ ◦ ∂ i ◦ ξ −1 )(ξ(f ) ·U (g) ξ(g)) ˜ k ∧ dxi ∧ · · · ∧ dxir ∧ dxj ∧ · · · ∧ dxjs ∂ k (f  g)) ξ(dx 1 1 ˜ k ∧ ∂ k (ξ˜−1 (˜ ω ) ∧ ξ˜−1 (˜ ν ))) ξ(dx ˜ ξ˜−1 (˜ ξ(d( ω ) ∧ ξ˜−1 (˜ ν ))).

ˆ ∗ )) 3.10. (Conclusion.) Given a Lie algebra homomorphism φ : g → Der(S(g satisying assumptions 1.11 we exhibited in 2.3 its canonical extension to a Hopf action of U (g) on Λ(g) ⊗ U (g) and a differential dˆφ (defined in 3.1) on the induced smash product algebra 2.4, making it into a complex (by 3.4, (i)), but not a differential graded algebra (the graded Leibniz rule is not satisfied). As an algebra, it does not depend on the choice of φ. This complex has a subcomplex ΛU (g) which, as an algebra, is isomorphic to Λcl (g) ⊗ U (g) and thus deforms the differential graded algebra of polynomial differential forms Λ(V ) ⊗ S(V ) as an algebra and as a complex, but without Leibniz compatibility. The differential on ΛU (g) is related in 3.7 to the usual differential on polynomial differential forms via a φ-deformed

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Fock construction whose action  appears also in the study of Hopf algebroid structure of a Heisenberg double of U (g). The star product on polynomials induced by the coalgebra isomorphism ξφ : S(g) → U (g) extends to the classical complex of polynomial differential forms: the product on ΛU (g) is transferred via Fock action (3.8, 3.9). In a sequel paper, we address noncanonical modifications to our construction and their relation to alternative deformations, e.g. from [5], and to an approach using Drinfeld-Xu 2-cocycle twists of Heisenberg double Hopf algebroids. Acknowledgments The results have been written up as a preprint at IRB, Zagreb in Spring 2008. During a major revision in May 2020, I have been partly supported by the Croatian Science Foundation under the Project “New Geometries for Gravity and Spacetime” (IP-2018-01-7615). References [1] Giovanni Amelino-Camelia and Michele Arzano, Coproduct and star product in field theories on Lie-algebra noncommutative space-times, Phys. Rev. D (3) 65 (2002), no. 8, 084044, 8, DOI 10.1103/PhysRevD.65.084044. MR1899786 [2] N. Bourbaki, Algebra, Chapter 3. [3] S. C. Coutinho, A primer of algebraic D-modules, London Mathematical Society Student Texts, vol. 33, Cambridge University Press, Cambridge, 1995. MR1356713 ˇ [4] Nikolai Durov, Stjepan Meljanac, Andjelo Samsarov, and Zoran Skoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, J. Algebra 309 (2007), no. 1, 318–359, DOI 10.1016/j.jalgebra.2006.08.025. MR2301242 [5] S. Meljanac, S. Kreˇsi´ c-Juri´ c, Noncommutative differential forms on the κ-deformed space, J. Phys. A 42, 365204–365225 (2009) arXiv:0812.4571; Differential structure on κ-Minkowski space, and κ-Poincar´ e algebra, Int. J. Mod. Phys. A 26, 20, 3385–3402 (2011) arXiv:1004.4647. ˇ [6] S. Meljanac, Z. Skoda, Leibniz rules for enveloping algebras and a diagrammatic expansion, www2.irb.hr/korisnici/zskoda/scopr8.pdf (an old/obsolete version at arXiv:0711.0149). ˇ [7] Stjepan Meljanac, Zoran Skoda, and Martina Stoji´ c, Lie algebra type noncommutative phase spaces are Hopf algebroids, Lett. Math. Phys. 107 (2017), no. 3, 475–503, DOI 10.1007/s11005-016-0908-9. MR3606513 [8] Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR1243637 [9] Emanuela Petracci, Universal representations of Lie algebras by coderivations (English, with English and French summaries), Bull. Sci. Math. 127 (2003), no. 5, 439–465, DOI 10.1016/S0007-4497(03)00041-1. MR1991464 [10] S. Meljanac and M. Stoji´ c, New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C Part. Fields 47 (2006), no. 2, 531–539, DOI 10.1140/epjc/s2006-02584-8. MR2242642 ˇ [11] Zoran Skoda, Heisenberg double versus deformed derivatives, Internat. J. Modern Phys. A 26 (2011), no. 27-28, 4845–4854, DOI 10.1142/S0217751X11054772. MR2859802 [12] M. Stoji´ c, Upotpunjeni Hopfovi algebroidi, (Completed Hopf algebroids, in Croatian), PhD thesis, University of Zagreb, 2017. Department of Teacher’s Education, University of Zadar, Franje Tudjmana 24, ´, HR-23000 Zadar, Croatia; and Theoretical Physics Division, Institute Rudjer Boˇ skovic ˇka cesta 54, P.O. Box 180,HR-10002 Zagreb, Croatia Bijenic Email address: [email protected]

Contemporary Mathematics Volume 768, 2021 https://doi.org/10.1090/conm/768/15470

The Heisenberg generalized vertex operator algebra on a Riemann surface Michael P. Tuite Abstract. We compute the partition and correlation generating functions for the Heisenberg intertwiner generalized vertex operator algebra on a genus g Riemann surface in the Schottky uniformization. These are expressed in terms of differential forms of the first, second and third kind, the prime form and the period matrix and are computed by combinatorial methods using a generalization of the MacMahon Master Theorem.

1. Introduction The Heisenberg intertwiner generalized Vertex Operator Algebra (VOA) is an algebra formed from the Heisenberg VOA and all of its modules [DL, BK, TZ]. We consider the partition and all correlation functions for this theory on a genus g Riemann surface in the Schottky parametrization. The partition and n-point correlation functions for the Heisenberg and lattice VOAs are familiar concepts at genus one and have been found on genus two surfaces formed from sewn tori [MT1,MT2]. Here we describe the more general situation of the Heisenberg generalized VOA and compute the partition function and the generating function for all correlation functions on a genus g Riemann surface. Our results imply that we can compute all genus g correlation functions for all VOAs or Super VOAs which can be decomposed into Heisenberg modules at any rank e.g. integral lattice (Super)VOAs. Various specializations of our results have been long anticipated in physics e.g. [Mo, DV]. We also show that the rank 2 Heisenberg VOA genus g partition function is an inverse determinant given by the Motonen-Zograf formula [Mo, Z, McIT]. All of our results are found by combinatorial methods based on a generalization of the MacMahon Master Theorem (MMT) [McM, T] which is reviewed in Section 2. Section 3 reviews some Riemann surface theory and the Schottky uniformization of a genus g surface in a non-standard parameterization suitable for our purposes. We also give detailed formulas for the bidifferential form of the second kind, holomorphic 1-forms (differentials of the first kind) and the period matrix in terms of genus zero data and Schottky sewing parameters [Y, MT1]. Section 4 describes the genus g partition and correlation generating function for the Heisenberg VOA. For convenience we consider the rank 2 case wherein the genus zero correlation generating function is expressed as a permanent. The genus g objects can then be 2020 Mathematics Subject Classification. Primary 17B69, 30F30, 81T40. c 2021 American Mathematical Society

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expressed as sums over multisets of permanents where the multisets label Heisenberg Fock vectors. The MMT implies that the genus g partition function is the inverse of an infinite determinant (similarly to genus two [MT4, MT5]) related to the Motonen-Zograf formula [Mo, Z, McIT]. The Heisenberg VOA correlation generating function is expressed in terms of bidifferential forms. Section 5 generalizes all of these results to the case of Heisenberg generalized VOA where the correlation generating function is expressed in terms of differential forms of the first, second and third kind, the prime form and the period matrix. We conclude with a few important examples. 2. Some generalized MacMahon master theorems We begin with a review of some generalisations of the classic MacMahon Master Theorem (MMT) of combinatorics [McM] described in our earlier paper [T]. Let A “ pAab q be an n ˆ n formal matrix indexed by a, b P t1, . . . , nu. The Permanent of A is defined by perm A :“

n ÿ ź

Aiπpiq ,

πPSn i“1

where the sum is over permutations π P Sn , the symmetric group on n letters. Let i :“ t1rp1q 2rp2q . . . irpiq . . . nrpnq u, řn denote the multiset of size N “ i“1 rpiq formed from the original index set t1, . . . , nu where the index i is repeated rpiq ě 0 times. We let Api, iq denote the N ˆ N matrix indexed by the multiset śn i and define perm Api, iq “ 1 when i is the empty set. Lastly, we let rpiq! :“ i“1 rpiq! which is the order of the symmetric label group of i. Then [McM] Theorem 2.1 (MMT). ÿ perm Api, iq 1 (2.1) “ , rpiq! detpI ´ Aq i where the sum is taken over all multisets i. We review several generalizations of this result [T]. Consider an pn1 `nqˆpn1 `nq matrix with block structure j „ B U (2.2) , V A where A “ pAij q is an n ˆ n matrix indexed by i, j, B “ pBi1 j 1 q is an n1 ˆ n1 matrix indexed by i1 , j 1 , U “ pUi1 j q is an n1 ˆ n matrix and V “ pVij 1 q is an n ˆ n1 matrix. For a multiset i of size N define the pn1 ` N q ˆ pn1 ` N q matrix j „ B U piq (2.3) , V piq Api, iq where, as before, Api, iq denotes the N ˆ N matrix indexed by i, U piq is an n1 ˆ N matrix and V piq is an N ˆ n1 matrix. We find1 [T] 1 In [T], a further parameter β which counts permutation cycles is also discussed. We take β “ 1 throughout the present paper.

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

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Theorem 2.2 (The Submatrix MMT). j „ ÿ 1 r perm B B U piq “ perm , V piq Api, iq rpiq! detpI ´ Aq i k r “ B ` U pI ´ Aq´1 V where pI ´ Aq´1 “ ř for n1 ˆ n1 matrix B kě0 A .

We may extend the sum over permutations in (2.1) to a sum over partial permutations i.e. injective partial mappings from t1, . . . , nu to itself. Let Ψ denote the set of partial permutations of the set t1, . . . , nu and let dom ψ and im ψ denote the domain and image respectively of ψ P Ψ. Let θ “ pθi q, φ “ pφi q be formal n-vectors. We define the pθ, φq-extended Partial Permanent of an n ˆ n matrix A by [T] ÿ ź ź ź pperm θ,φ A :“ (2.4) Aiψpiq θj φk , ψPΨ iPdom ψ

jR im ψ

kR dom ψ

Thus „ pperm θ,φ

A11 A21

j A12 “θ1 φ1 θ2 φ2 ` A11 θ2 φ2 ` A22 θ1 φ1 A22 ` A12 θ1 φ2 ` A21 θ2 φ1 ` A11 A22 ` A12 A21 .

Let Api, iq denote the N ˆ N matrix indexed by i. We also let pperm θ,φ Api, iq denote the corresponding partial permanent with dimension N row vectors p. . . , θi , . . .q and p. . . , φi , . . .q where index i occurs rpiq times in i. We then find [T] Theorem 2.3 (The Partial Permutation MMT). ÿ pperm θφ Api, iq i

rpiq!

´1

T

eθpI´Aq φ “ , detpI ´ Aq

where φT denotes the transpose of the row vector φ. Lastly, we may combine the two generalizations above into one theorem concerning partial permutations of submatrices of the pn1 ` nq ˆ pn1 ` nq block matrix (2.2). Let θ 1 “ pθi1 1 q and φ1 “ pφ1i1 q be n1 -vectors and θ “ pθi q and φ “ pφi q be n-vectors. For a multiset i of„ size N and block matrix (2.3) labelled by j B U piq denote the pΘ, Φq-extended t11 , . . . , n1 u and i, we let pperm Θ,Φ V piq Api, iq 1 1 partial permanent with pn ` N q-vectors Θ :“ pθ11 , . . . , θn1 1 , . . . , θi , . . .q and Φ :“ pφ111 , . . . , φ1n1 , . . . , φi , . . .q respectively. We find [T] Theorem 2.4 (The Submatrix Partial Permutation MMT). j „ ´1 T ÿ 1 eθpI´Aq φ B U piq r “ (2.5) pperm Θ,Φ pperm θ, r B, rφ V piq Api, iq rpiq! detpI ´ Aq i r “ B ` U pI ´ Aq´1 V and n1 -vectors θr and φr given by for n1 ˆ n1 matrix B θr “ θ1 ` θpI ´ Aq´1 V,

φrT “ φ1T ` U pI ´ Aq´1 φT .

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3. Riemann surfaces from a sewn sphere 3.1. Some standard forms on a Riemann surface. Define the indexing sets I “ t´1, . . . , ´g, 1, . . . , gu,

I` “ t1, . . . , gu.

Let Sg be a compact genus g Riemann surface with canonical homology basis αa , βa for a P I` . There exists a unique symmetric bidifferential form of the second kind [Mu, F] ˆ ˙ 1 ωpx, yq “ (3.1) ` regular terms dxdy, px ´ yq2 ű for local coordinates x, y with normalization αa ωpx, ¨q “ 0 for all a P I` . It follows that ¿ (3.2) νa pxq “ ωpx, ¨q, pa P I` q, βa

is the differential of the first kind, a holomorphic 1-form normalized by 2πi δab . The period matrix Ω is defined by ¿ 1 Ωab “ (3.3) νb pa, b P I` q. 2πi

ű αa

νb “

βa

We also define the differential of the third kind for p, q P Sg by żp (3.4) ωpx, ¨q. ωp´q pxq “ q

ωpx, yq can be expressed in terms of the prime form Epx, yq “ Kpx, yqdx´1{2 dy ´1{2 , a holomorphic form of weight p´ 12 , ´ 12 q with (3.5)

ωpx, yq “Bx By plog Kpx, yqq dxdy, ` ˘ where Kpx, yq “ px ´ yq ` O px ´ yq2 and Kpx, yq “ ´Kpy, xq. For the genus p :“ C Y t8u with x, y, p, q P C p we have zero Riemann sphere S0 – C (3.6)

ω p0q px, yq “

dxdy, , px ´ yq2

p0q

ωp´q “

dx dx ´ , x´p x´q

K p0q px, yq “ x ´ y.

3.2. The Schottky uniformization of a Riemann surface. We briefly review the construction of a genus g Riemann surface Sg using the Schottky unip e.g. [Fo, Bo]. For formization where we sew g handles to the Riemann sphere C each a P I, let Ca P C be a circular contour with center wa and radius |ρa |1{2 for some complex parameters wa , ρa . We assume that ρa “ ρ´a and that Ca X Cb “ t u for a ‰ b. Identify z 1 P C´a with z P Ca for each a P I` via the sewing relation [TW] (3.7)

pz 1 ´ w´a qpz ´ wa q “ ρa ,

a P I` .

obius map Define γa P SL2 pCq by the M¨ (3.8)

γa z :“ w´a `

ρa , z ´ wa

a P I.

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

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Notice that γ´a “ γa´1 . The sewing relation (3.7) implies z 1 “ γa z for a P I` so that γa Ca “ ´C´a for all a P I. (3.7) is equivalent to the standard Schottky relation [Fo, Bo] z 1 ´ W´a z ´ Wa “ qa , a P I` , z 1 ´ Wa z ´ W´a ¯ ´ ? ρa 1´4x for2 cpxq “ 1´ 2x ´ 1 and Wa “ where qa “ c ´ pwa ´w 2 ´a q (3.9)

wa `qa w´a . 1`qa

Each

1{2 ´1{2 diagpqγ , qγ q

with |qγ | ă 1 where qγ is called γ P Γ is conjugate in SL2 pCq to the multiplier of γ. In particular, qγa “ qa with attracting (repelling) fixed point W´a (Wa ) for a P I` . The (marked) Schottky group Γ is the free group generated by γa . Every element of Γ may be expressed as a reduced word γa1 . . . γan of length n where p ´ ΛpΓq. Then γ´ai ‰ γai`1 . Let ΛpΓq denote the limit set and let Ω0 pΓq “ C p denote the standard Sg » Ω0 pΓq{Γ is a Riemann surface of genus g. We let D Ă C connected fundamental region with oriented boundary curves Ca . We define the space of Schottky parameters Cg Ă C3g by ! ) 1 1 Cg :“ pw, ρq : |wa ´ wb | ą |ρa | 2 ` |ρb | 2 @ a ‰ b , (3.10) for w, ρ :“ w1 , w´1 , ρ1 , . . . , wg , w´g , ρg . Cg { SL2 pCq for the M¨obius SL2 pCq group.

We define Schottky space as Sg :“

3.3. Some Schottky scheme sewing formulas. In this section we generalize a construction due to Yamada [Y] and developed further in [MT1] for sewing Riemann surfaces. In particular, we describe formulas for the genus g normalized bidifferential of the second kind ω, holomorphic 1-forms νa and the period matrix p0q Ωab for a, b P I` constructed from ω p0q px, yq and ωp´q of (3.6) in the above Schottky sewing formalism. We note that there are classical formulas for these objects in terms of Poincar´e sums that can be derived from the sewing formulas e.g. see (6.6). However, the sewing expansions described here are much more suitable for our later purposes. Lemma 3.1. ωpx, yq “ ω p0q px, yq `

(3.11)

1 ÿ 2πi aPI

¿

ż za

ω p0q px, ¨q ωpy, za q,

Ca pza q

for x, y P D with za “ z ´ wa . Proof. The result follows from (3.1) and the identity ¿ żz ω p0q px, ¨q ωpy, zq “ 0, Cpzq

where C is a simple Jordan curve whose interior region contains Ca for all a P I. For k, l ě 1 and a, b P I we define 1-forms ? k{2 ¿ k{2 kρa ρa ´k p0q ? (3.12) za ω px, za q “ dx, La pk, xq :“ px ´ wa qk`1 2πi k Ca pza q

2 cpxq

“

` ˘ n 1 2n ně1 n n`1 x

ř

is the Catalan series [MT1].



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MICHAEL P. TUITE

and the moment matrix k{2 l{2

Aab pk, lq :“ ´

ρ a ρb ? p2πiq2 kl

¿

¿

x´k y ´l ω p0q px, yq

C´a pxq Cb pyq

$ k{2 l{2 k ’ ρ a ρb & ?p´1q pk ` l ´ 1q! , a ‰ ´b, “ klpk ´ 1q!pl ´ 1q! pw´a ´ wb qk`l ’ %0, a “ ´b,

(3.13)

Let Lpxq “ pLa pk, xqq, Rpyq :“ pL´a pk, yqq and A “ pAab pk, lqq denote the infinite row vector, column vector and matrix doubly indexed by a, b P I and k, lřě 1. Let I denote the infinite doubly indexed identity matrix and pI ´ Aq´1 :“ ně0 An . We then find Proposition 3.2. ωpx, yq for x, y P D is given by ωpx, yq “ ω p0q px, yq ´ Lpxq pI ´ Aq´1 Rpyq,

(3.14)

where pI ´ Aq´1 is convergent for pw, ρq P Cg . Proof. We give a proof using arguments similar to [Y] and Sections 3.2 and 5.2 of [MT1]. Since ωpx, yq is symmetric and applying (3.11) twice we find ¿ ż za 1 ÿ p0q ω p0q px, ¨q ω p0q py, za q ωpx, yq “ω px, yq ` 2πi aPI Ca pza q

ÿ 1 ` p2πiq2 a,bPI

¿

¿

ż za ω

p0q

ż zb px, ¨q ωpza , zb q

ω p0q py, ¨q.

Ca pza q Cb pzb q

Applying (3.12) and noting that Ca pza q „ ´C´a pz´a q we find ωpx, yq “ ω p0q px, yq ´ LpxqpI ` Y qRpyq,

(3.15) for moment matrix

k{2 l{2

(3.16)

Yab pk, lq :“ ´

ρ a ρb ? p2πiq2 kl

¿

¿

x´k y ´l ωpx, yq,

C´a pxq Cb pyq

for a, b P I and k, l ě 1. We note that Yab pk, lq is convergent for pw, ρq P Cg . Define the infinite matrix Y “ pYab pk, lqq. Taking moments of (3.15) we obtain Y “ A ` ApI ` Y qA which can be recursively solved to find ÿ Y “ (3.17) An “ pI ´ Aq´1 ´ I. ně1

Thus (3.15) implies (3.14). Since Y is convergent for pw, ρq P Cg then pI ´ Aq´1 is also.  The matrix A and the determinant detpI ´ Aq defined by ÿ 1 log detpI ´ Aq :“ Tr logpI ´ Aq “ ´ Tr Ak , k kě1 will be of central importance in our later discussions. Theorem 3.3. detpI ´ Aq is non-vanishing and holomorphic on Cg .

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

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Proof. Let Caσ denote the circular Jordan curve of radius σ|ρa | 2 centred at wa with local coordinate za “ z ´ wa for some σ ą 1 i.e. |z´a za | ą |ρa |. Consider the sum of integrals ˙ ˆ ¿ ¿ ÿ 1 ρa Spw, ρq “ . ωpz , z q log 1 ´ ´a a p2πiq2 z´a za aPI σ Cσ C´a a

We first show that Spw, ρq is holomorphic on Cg . Let Δ3g “ tμ : |μi | ă Ri u Ă Cg be a polydisc for local coordinates μ “ pμ1 , . . . , μ3g q. Write ωpz´a , za q “ σ f pz´a , za , μqdz´a dza for z˘a P C˘a . f pz´a , za , μq is holomorphic on Cg which implies that ÿ f pz´a , za , μq “ μn fn pz´a , za q, n

ś is absolutely convergent on Δ3g where μn :“ i μni i for integers ni ě 0. Furthermore, fn pz´a , za q satisfies Cauchy’s inequality e.g. [Gu] M , Rn and M “ maxa supz˘a PC˘a σ supμPΔ |f pza , z´a q|. We then find 3g |fn pz´a , za q| ď

for Rn “ that

ś

i

Rini

|Spw, ρq| ď

ˇ ˙ ˆ ÿ ÿ |μ|n ¿ ¿ ˇˇ ˇ ˇfn pz´a , za q log 1 ´ ρa ˇ dz dz ´a a ˇ ˇ 2 p2πq z´a za aPI n σ Cσ C´a a

˙´1 źˆ ˇ ` ˘ˇ ÿ |μi | 1´ ď M σ 2 ˇlog 1 ´ σ ´2 ˇ |ρa | . Ri i aPI Thus S is absolutely convergent and holomorphic on Cg . Since |z´a za | ą |ρa | we find ¿ ¿ ÿ ÿ ρk 1 a ´k ´k Spw, ρq “ ´ ωpz´a , za qz´a za 2 k p2πiq aPI kě1 σ Cσ C´a a

“ Tr Y “

ÿ

TrpAn q,

ně1

ř using (3.16) and (3.17). Therefore ně1 TrpAn q is holomorphic on Cg and thus it ř  follows that ně1 n1 TrpAn q “ ´ Tr logpI ´ Aq is also. We identify the homology cycle αa with C´a and βa with a path connecting z P Ca to z 1 “ γa z P C´a . The g normalized holomorphic one forms νb , b P I` , can p0q be expressed in terms of Lpxq, Rpxq, A and moments of ωwb ´w´b of (3.6) defined for a P I by $ k{2 ˘ ρa ` ’ ’ & ? ´pw´b ´ wa q´k ` pwb ´ wa q´k , |a| ‰ b, k (3.18) dab pkq :“ k{2 ’ ’ %sgnpaq ρ?a pw´b sgnpaq ´ wa q´k , |a| “ b. k We let db “ pdab pkqq and db “ pd´a b pkqq denote g infinite row and column vectors, respectively, indexed by a P I and k ě 1. We find

328

MICHAEL P. TUITE

Proposition 3.4. νb pxq for x P D and b P I` is given by (3.19)

p0q

νb pxq ´ ωwb ´w´b pxq “ ´db pI ´ Aq´1 Rpxq “ ´Lpxq pI ´ Aq´1 db .

Proof. Consider the identity ¿ żz p0q ωpx, zq ωwb ´w´b “ 0, Cpzq

where C is a simple Jordan curve whose interior region contains Ca for all a P I. Similarly to Lemma 3.1, this implies the following generalization of Corollary 5 of [Y] ˙ ˆż za ¿ 1 ÿ p0q p0q νb pxq ´ ωwb ´w´b pxq “ ωpx, za q ωwb ´w´b ´ sgnpaqδ|a|,b log za . 2πi aPI Ca pza q

The result follows by repeating an approach similar to Proposition 3.2.



We may also obtain the genus g period matrix by generalizing Lemma 5 of [Y] to find Proposition 3.5. The genus g period matrix Ωab for a, b P I` is given by ˙ ˆ pwa ´ wb q pw´a ´ w´b q ´ da pI ´ Aq´1 db , a ‰ b, (3.20) 2πiΩab “ log pw´a ´ wb q pwa ´ w´b q ¸ ˜ ´ρa ´ da pI ´ Aq´1 da . (3.21) 2πiΩaa “ log 2 pwa ´ w´a q 4. The Heisenberg vertex operator algebra on Sg 4.1. Vertex operator algebras. Consider a simple Vertex Operator Algebra with graded vector space V “ ‘ně0 Vn and vertex operators Y pv, zq “ ř (VOA) ´n´1 vpnqz for v P V e.g. [FHL, FLM, Ka, LL, MT3]. We denote the connPZ formal weight of v P Vn by wtpvq “ n. In particular, we highlight the commutator and associativity identities ÿ ˆk ˙ rupkq, Y pv, zqs “ Y pupjqv, zqz k´j . (4.1) j jě0 (4.2)

py ` zqM Y pu, y ` zqY pv, zq “py ` zqM Y pY pu, yqv, zq,

pM " 0q,

We assume that V is of CFT type (i.e. V0 “ C1) with a unique symmetric invertible invariant bilinear form x , y with normalization x1, 1y “ 1 where [FHL, Li] @ D xY pa, zqb, cy “ b, Y : pa, zqc , ¯ ´ ` ˘Lp0q ř a, z ´1 . We refer to x , y for Y : pa, zq “ nPZ a: pnqz ´n´1 :“ Y ezLp1q ´z ´2 as the Li-Zamolodchikov (Li-Z) metric [MT4]. For a V -basis tbu, we let tbu denote the Li-Z dual basis. If a P Vk is quasi-primary, then xapnqb, cy “ xb, a: pnqcy with a: pnq “ p´1qk ap2k ´ n ´ 2q. Thus a: pnq “ ´ap´nq for a P V1 and L: pnq “ Lp´nq.

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Define the genus zero n-point correlation function for vi inserted at yi for i P t1, . . . , nu Z p0q pv, yq :“ Z p0q pv1 , y1 ; . . . ; vn , yn q “ x1, Y pv, yq1y, where x¨, ¨y is the Li-Z metric and Y pv, yq :“ Y pv1 , y1 q . . . Y pvn , yn q. Z p0q pv, yq is a rational function of yi e.g. [FHL, Z, TW]. 4.2. Heisenberg genus zero correlation functions. In order to later apply the MacMahon Master Theorems 2.1–2.4, we consider the rank two Heisenberg VOA M 2 generated by two weight 1 commuting Heisenberg vectors h1 , h2 . Define h˘ :“ ?12 ph1 ˘ ih2 q with non-trivial commutator relation (4.3)

rh` pkq, h´ plqs “ kδk,´l ,

k, l P Z.

M 2 has a Fock basis with elements (4.4)

r` p1q

hpi` , i´ q :“ h` p´1q

h` p´2qr` p2q . . . h´ p´1qr´ p1q h´ p´2qr´ p2q . . . 1,

labelled by a pair of multisets r pi` q

i` “ t1r` p1q 2r` p2q . . . i``

. . .u,

r pi´ q

i´ “ t1r´ p1q 2r´ p2q . . . i´´

. . .u,

where i` occurs r` pi` q times in i` and i´ occurs r´ pi´ q times in i´ . Where no ambiguity arises, we will omit the ˘ subscript in r˘ . The Fock vector has conformal weight ÿ ÿ wtphpi` , i´ qq “ (4.5) i` r pi` q ` i´ r pi´ q . i` Pi`

i´ Pi´

The Fock dual basis with respect to the Li-Z metric has elements ¨ ˛ ź ź ˆ ´1 ˙rpi` q ˆ ´1 ˙rpi´ q 1 ‚hpi´ , i` q. (4.6) hpi` , i´ q “ ˝ i` i´ r pi` q!r pi´ q! i Pi i Pi `

`

´

´

The basic genus zero Heisenberg 2-point function found by applying (4.1) and (4.3) is3 1 (4.7) Z p0q ph` , x` ; h´ , x´ q “ ` . px ´ x´ q2 (4.7) is fundamental to finding the Heisenberg partition and correlation functions on Sg . The genus zero 2n-point function for h˘ inserted at x˘ i for i P t1, . . . , nu is found by using (4.1) for h` pkq with k ě 0 and (4.3) via the recursion formula e.g. [Z, TW] Z p0q ph˘ , x˘ q “

n ÿ

px` 1 j“1

1 {´ Z p0q ph` , x` 2 ; . . . ; h´ , xj ; . . .q, 2 ´ x´ q j

where the h´ insertion at x´ j is deleted. Repeating we find (4.8)

Z p0q ph˘ , x˘ q “ perm

px` i

1 . i, j P t1, . . . , nu. 2 ´ x´ j q

Here, and below, we adopt the standard convention that px ` yqκ :“ for any κ i.e. we formally expand in the second parameter y. 3

` κ ˘ κ´m m y , mě0 m x

ř

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MICHAEL P. TUITE

Defining F p0q ph˘ , x˘ q :“ Z p0q ph˘ , x˘ qdx` dx´ with dx` dx´ :“ we may re-express (4.8) in terms of differential forms with (4.9)

śn

i“1

´ dx` i dxi ,

´ F p0q ph˘ , x˘ q “ perm ω p0q px` i , xj q.

Z p0q ph˘ , x˘ q is a generating function for all genus zero correlation functions for M 2 in the sense that we may extract any genus zero correlation function as the coefficient of an appropriate expansion of Z p0q ph˘ , x˘ q e.g. [MT3, HT]. This follows from the general observation that for a, b P V we have by (4.2) that Z p0q p. . . ; ap´kqb, w; . . .q “ coeff xk´1 Z p0q p. . . ; Y pa, xqb, w; . . .q “ coeff xk´1 Z p0q p. . . ; a, x ` w; b, w; . . .q,

(4.10)

where we may omit any px ` wqM factors when comparing rational functions. ś śrpi` q śrpi´ q ` i` ´1 Z p0q p. . . ; hpi` , i´ q, w; . . .q is the coefficient of q“1 pxi` ,p q i˘ p“1 ´ ¯ ´ i´ ´1 px´ in Z p0q . . . ; h` , x` using Y p1, wq “ 1. i´ ,q q i` ,p ` w; . . . ; h´ , xi´ ,q ` w; . . . We consider below genus zero correlation functions for vectors inserted at wa of (3.7) for a P I. Let y “ x` ´ w´a and z “ x´ ´ wb be local coordinates in the neighborhood of w´a , wb . For the two point function appearing in (4.7) and (4.8) we define ? 1 ´l{2 pab pk, lq “ coeff yk´1 zl´1 A (4.11) “ ´ρ´k{2 ρb klAab pk, lq, a ` ´ 2 px ´ x q for the moment matrix A of (3.13). 4.3. The genus g partition function on M 2 . Let b “ pb1 , . . . , bg q denote an element of a V bg -basis with Li-Z dual b “ pb1 , . . . , bg q and consider the rational genus zero 2g-point correlation function for these vectors inserted at wa of (3.7) Z p0q pb, wq :“ Z p0q pb1 , w´1 ; b1 , w1 ; . . . ; bg , w´g ; bg , wg q. We define the genus g partition function for a VOA V by [TW] ÿ pgq (4.12) ρwtpbq Z p0q pb, wq, ZV pw, ρq :“ bPV bg

where w, ρ :“ w˘1 , ρ1 , . . . , w˘g , ρg and ρwtpbq :“ rameters ρa of (3.7). In general gent coefficients.

pgq ZV pw, ρq

ś

1ďaďg

wtpba q

ρa

for Schottky pa-

is a formal series in ρ but with conver-

Proposition 4.1. The genus g partition function for the Heisenberg VOA M 2 is pgq

ZM 2 pw, ρq “ detpI ´ Aq´1 , for the moment matrix A of (3.13). Proof. Here we sum over the Fock basis vectors ba “ hpia , i´a q labelled by 2g multisets aq ia :“ t. . . irpi . . .u, a

a P I,

i.e. ia occurs rpia q times in ia . It follows from (4.5) and (4.6) that ρwtpbq Z p0q pb, wq “

M pρ, ia q p0q Z p. . . ; hpia , i´a q, wa ; . . .q , rpia q!

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

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where (4.13)

rpia q! :“

źź

rpia q!,

M pρ, ia q :“

aPI ia

ź ź ˆ ´ρia ˙rpia q a

aPI ia

ia

.

Z p0q p. . . ; hpia , i´a q, wa ; . . .q is labelled by ia for a P I and is determined by the coefficient of an appropriate expansion of the generating function (4.8) as described above. Hence p pia , ia q , Z p0q p. . . ; hpia , i´a q, wa ; . . .q “ perm A ř p of (4.11) where A p pia , ia q is an N ˆ N matrix for N “ ř for A aPI ia rpia q. The ia ´ρa {ia multiplicative factors in M pρ, ia q can be absorbed into the permanent to obtain ÿ perm A pia , ia q pgq , ZM 2 pw, ρq “ rpia q! i a

for moment matrix A and where the sum is taken over all multisets ia . We may pgq truncate the formal series ZM 2 pw, ρq to any finite order in ρ and apply the MacMapgq hon Master Theorem 2.1 to find that ZM 2 pw, ρq “ detpI ´ Aq´1 to any finite order in ρ and therefore to all orders since detpI ´ Aq´1 is convergent by Theorem 3.3. Hence the result follows.  pgq

pgq

pgq

From the definition (4.12) we have ZUbV “ ZU ZV V [TW]. Thus we have

pgq ZM 2

“

pgq pZM q2

for any two VOAs U and

so that 1

pgq

ZM “ detpI ´ Aq´ 2 .

(4.14)

pgq

obius (4.14) generalizes results of [MT4, MT5] for genus 2. In general, ZV is M¨ invariant by Proposition 4.3 of [TW]. Thus together with Theorem 3.3 it follows that pgq

Corollary 4.2. ZM 2 pw, ρq “ detpI ´ Aq´1 is holomorphic on Schottky space Sg . 4.4. Montonen-Zograf product formula. detpI ´ Aq can be expressed in terms of an infinite product formula originally discovered by Montonen in 1974 [Mo] (see [DV] also). This formula was subsequently found in the holomorphic part of a Laplacian determinant formula on Sg by Zograf [Z] related to earlier work of D’Hoker and Phong [DP]. γp P Γ (the Schottky group) is called primitive if γp ‰ γ m for any m ą 1 and γ P Γ. Thus γ “ γpm for some primitive γp and some m ě 1 for every γ P Γ. We then have Theorem 4.3. detpI ´ Aq “

źź p1 ´ qγkp q, kě1 γp

where γp is summed over representatives of the primitive conjugacy classes of Γ excluding the identity and qγp is the multiplier of γp .

332

MICHAEL P. TUITE

Proof. The proof ř is a variation of arguments in [Mo] and [DV]. Recall that log detpI ´ Aq “ ´ ně1 n1 Tr An . Using (6.5), of the Appendix, we have 1 ÿ

Tr An “

` ˘ ´1 ´1 Tr Dpμa1 λ´1 a2 qDpμa2 λa3 q . . . Dpμan λa1 q ,

a1 ,...,an PI

for λa , μa of (6.3) where the prime indicates that ´ai ‰ ai`1 for i “ 1, . . . , n ´ 1 and ´an ‰ a1 . Note that the summand trace is over the integer labels of the Dpγq matrices only. From Lemma 6.1 (iii) and (6.4) it follows that4 Tr An “

1 ÿ

˘ ` ´1 ´1 Tr D λ´1 a1 μa1 λ a2 μa2 . . . λ an μan

a1 ,...,an PI

“

1 ÿ

Tr D pγa1 γa2 . . . γan q “

a1 ,...,an PI

ÿ

Tr Dpγq,

γPΓCR n

where ΓCR n is the set of Cyclically Reduced words of length n in Γ i.e. reduced words γa1 . . . γan for which γa´1 ” γ´a1 ‰ γan . For γ P ΓCR we have γ “ γpm for some n 1 primitive cyclically reduced word γp and some m ě 1 where m|n. Every element of Γ is conjugate to a cyclically reduced word and any two cyclically reduced words are conjugate if and only if they are cyclic permutations of each other e.g. Prop. 9 of [C]. Then it follows that there are n{m cyclically reduced words conjugate to γpm . Therefore we find ÿ ÿ 1 ÿ 1 Tr An “ Tr Dpγpm q, n m γ mě1 ně1 p

where γp ranges over representatives of the primitive conjugacy classes of Γ exclud1{2 ´1{2 ing the identity. γp is conjugate in SLp2, Cq´to diagpq , qγp q for multiplier qγp . ¯ γp ř ˘ ` m mk so that From (6.2) we thus find Tr Dkl pγp q “ Tr δkl qγp “ kě1 qγmk p log detpI ´ Aq “ ´

ÿÿ ÿ 1 ÿÿ qγmk “ logp1 ´ qγkp q. p m kě1 γ mě1 kě1 γ p

p

 2

4.5. Genus g correlation functions on M . We define genus g formal npoint correlation differential forms for n vectors v1 , . . . , vn P V inserted at y1 , . . . , yn by [TW] ÿ pgq (4.15) ρwtpbq Z p0q pv, y; b, wqdy wtpvq , FV pv, yq :“ bPV bg

where Z p0q pv, y; b, wq “ Z p0q pv1 , y1 ; . . . ; vn , yn ; b1 , w´1 ; b1 , w1 ; . . . ; bg , w´g ; bg , wg q. pgq Consider FM 2 ph˘ , y ˘ q for Heisenberg generators h˘ inserted at yr˘ for r P t1, . . . , mu which is the generating function for all correlation functions just as (4.9) is at genus zero. Proposition 4.4. The genus g generating function for M 2 is given by pgq

FM 2 ph˘ , y ˘ q “ 4 One

perm ω pyr` , ys´ q , detpI ´ Aq

r, s P t1, . . . , mu,

has to check that the conditions of Lemma 6.1 are satisfied.

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

333

for bidifferential form ωpx, yq of (3.1). Proof. The b sum of (4.15) is taken over hpia , i´a q with summand ρwtpbq Z p0q ph˘ , y ˘ ; b, wq “

M pρ, ia q Xpia q, rpia q!

for Xpia q “ Z p0q p. . . ; h` , yr` ; . . . ; h´ , ys´ . . . ; hpia , i´a q, wa ; . . .q determined by an expansion of the generating function (4.8) given by ff « p p pia q B U Xpia q “ perm p p pia , ia q , V pia q A p of (4.11) and where for r, s P t1, . . . , mu, ia P ia , jb P ib we define for A (4.16) p sq :“ Bpr,

1 , pyr` ´ ys´ q2

p pr, jb q :“ U

jb , pyr` ´ wb qjb `1

Vp pia , sq :“

ia . pys´ ´ w´a qia `1

The additional ´ρiaa {ia and dyr` dys´ factors can be absorbed into the permanent to obtain j „ ÿ 1 B U pia q pgq , perm FM 2 ph˘ , y ˘ q “ V pia q A pia , ia q rpia q! i a

where with L, R of (3.12) we define (4.17) Bpr, sq :“ ω p0q pyr` , ys´ q,

U pr, jb q :“ La pjb , yr` q,

V pia , sq :“ ´Ra pia , ys´ q,

r “ B ` U pI ´ Aq´1 V is given by Applying Theorem 2.2 we find B (4.18)

r sq “ ω p0q pyr` , ys´ q ´ Lpyr` qpI ´ Aq´1 Rpyr´ q “ ωpyr` , ys´ q, Bpr, 

by Proposition 3.2. Thus the result holds. 5. The Heisenberg generalized VOA on Sg

5.1. The Heisenberg generalized VOA M2 . The Heisenberg VOA M 2 , generated by h˘ , has irreducible modules Mα2 “ M 2 b eα for α P C2 where for u b eα P Mα2 (5.1)

h˘ p0qpu b eα q “α˘ pu b eα q,

h˘ pnqpu b eα q “ ph˘ pnquq b eα , n ‰ 0,

where α˘ :“ ?12 pα1 ˘iα2 q. In [TZ] an intertwiner vertex operator Ypubeα , zq which creates u b eα from 1 is defined, similarly for lattice vertex operators [FLM, Ka], by Ypu b eα , zq :“eα Y´ pα, zqY pu, zqY` pα, zqz αp0q , ¸ ˜ ÿ αp˘ nq ¯n , z Y˘ pα, zq :“ exp ¯ n ną0 ř where αpnq :“ 2i“1 αi hi pnq. eα is a twisted group algebra element for the abelian group C2 . The twisted group algebra is associative with 2-cocycle εpα, βq P Cˆ where eα eβ “ εpα, βqeα`β ,

e0 “ 1,

334

MICHAEL P. TUITE

so that εpα, 0q “ εp0, αq “ 1. Associativity implies that the commutator Cpα, βq :“ εpα, βqεpβ, αq´1 , is skew-symmetric and multiplicatively bilinear. For a given commutator, we may choose cocycles such that εpα, ´αq “ 1 for all α P C2 [TZ]. Define the vector space M2 :“ ‘αPC2 Mα2 . For a given commutator Cpα, βq, the vertex operators on M2 form a generalized VOA5 with wt pu b eα q “ wtpuq ` α` α´ P C for the Heisenberg Virasoro vector ω “ h` p´1qh´ [BK, TZ]. In particular, for all u P M 2 and v b eα P Mα2 we obtain the following natural commutator and associativity identities ÿ ˆk ˙ α rupkq, Ypv b e , zqs “ Ypupjqv b eα , zqz k´j , (5.2) j jě0 (5.3) py ` zqN Y pu, y ` zqYpv b eα , zq “py ` zqN YpY pu, yqv b eα , zq,

pN " 0q,

where we identify Ypu b e , zq with Y pu, zq. Let α1 , . . . , αk P C2 and consider the genus zero correlation function 0

1

k

Z p0q peα , zq :“ x1, Ypeα , z1 q . . . Ypeα , zk q1y, abbreviating 1 b eα by eα . Using the standard lattice VOA identity6 [FLM] ´ y ¯α¨β Y` pα, xqY´ pβ, yq “ 1 ´ Y´ pβ, yqY` pα, xq, x where we define α¨β :“ α` β´ `α´ β` for all α, β P C2 . We find that Z p0q peα , zq “ 0 řk řk for t“1 αt ‰ 0 whereas for t“1 αt “ 0 we have ź t u Z p0q peα , zq “ εα (5.4) pzt ´ zu qα ¨α , 1ďtăuďk

where εα :“

(5.5)

k´1 ź t“1

˜ t

ε α,

k ÿ

¸ α

u

.

u“t`1 t

α inserted at For Heisenberg generators h˘ inserted at x˘ i for i P t1, . . . , nu and e řk zt for t P t1, . . . , ku where t“1 αt “ 0, consider the genus zero correlation function

(5.6) 1

k

´ α α Z p0q ph˘ , x˘ ; eα , zq “ x1, Y ph` , x` 1 q . . . Y ph´ , xn qYpe , z1 q . . . Ype , zk q1y.

(5.6) is a generating function for all M2 genus zero correlation functions much as Z p0q ph˘ , x˘ q of (4.8) is for M 2 . Using (4.1), (4.3), (5.1) and (5.2) we find that Z p0q ph˘ , x˘ ; eα , zq “

k ÿ

t α` α Z p0q ph` , x` 2 ; . . . ; e , zq ´ zt

x` t“1 1 n ÿ `

1 {´ p0q α ph` , x` 2 ; . . . ; h´ , xj ; . . . ; e , zq. ` ´ 2Z px ´ x q 1 j j“1

Repeating for each h˘ and using (5.4) we obtain a partial permanent (cf. (2.4)) formula: 5 The 6 See

generalized VOA is of a more general type than those described in [DL]. footnote 3

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

(5.7)

řk

αt “ 0 we have ź t u Z p0q ph˘ , x˘ ; eα , zq “ εα pzt ´ zu qα ¨α pperm θ´ ,θ`

Lemma 5.1. For

335

t“1

1ďtăuďk

1 , 2 px` ´ x´ i j q

where θi˘ “

(5.8)

k ÿ

t α˘ , x˘ ´ zt t“1 i

i P t1, . . . , nu. 1

2

1

2

Let F p0q ph˘ , x˘ ; eα , zq :“ Z p0q ph˘ , x˘ ; eα , zqdx` dx´ dz 2 α for dz 2 α “ αt αt

śk

p0q

dzt ` ´ , much as in (4.9). For genus zero differential of the third kind ωp´q of (3.6) we find t“1

˘ ˘ χ˘ i :“ θi dxi “

k ÿ

p0q

t α˘ ωzt ´z0 px˘ i q,

t“1

řk

χ˘ i

t where is independent of z0 since t“1 α˘ “ 0. Then (5.7) may be rewritten in p0q p0q terms of the genus zero differentials ω px, yq, E p0q px, yq and ωp´q pxq as follows:

Corollary 5.2. (5.9) t

ź

F p0q ph˘ , x˘ ; eα , zq “ εα

E p0q pzt , zu qα

¨αu

` pperm χ´ ,χ` ω p0q px´ i , xj q.

1ďtăuďk

Mα2 has a Fock basis with elements (5.10)

hα pi` , i´ q :“ . . . h` p´i` q

rpi` q

. . . h´ p´i´ qrpi´ q . . . 1 b eα ,

labelled by a pair of multisets i` , i´ and with conformal weight ÿ ÿ 1 i` r pi` q ` i´ r pi´ q ` α ¨ α. wtphα pi` , i´ qq “ 2 i i `

´

A Li-Z metric exists on M leading to a Fock dual basis with elements [TZ] (with cocycle choice εpα, ´αq “ 1) given by 2

(5.11) hα pi` , i´ q “ p´1q

1 2 α¨α

¨ ˛ ź ˆ ´1 ˙rpi` q ˆ ´1 ˙rpi´ q 1 ˝ ‚h´α pi´ , i` q. i i r pi q!r pi q! ` ´ ` ´ i ˘

5.2. Genus g partition functions on M2 . Let αa P C2 for a P I` and consider Mα2 :“ baPI` Mα2a . Similarly to (4.12), we define the genus g partition function for Mα2 by ÿ α pgq (5.12) ρwtpb q Z p0q pbα , wq, ZM 2 pw, ρq :“ α

2 bα PMα a

´a :“ ´αa for a P I` for Z p0q pbα , wq :“ Z p0q p. . . ; ba , w´a ; bα a , wa ; . . .q. Define α a ´a 1 a a α ¨α α α so that 1 b e “ p´1q 2 1be . From (5.4) and (5.5) we find that εα “ 1 and ź ´a a a b x1, . . . Ypeα , w´a qYpeα , wa q . . . 1y “ (5.13) pwa ´ wb qα ¨α , αa

aăb

where the product is taken over a, b P I with ordering ´1 ă 1 ă . . . ´ g ă g.

336

MICHAEL P. TUITE

Proposition 5.3. The genus g partition function for Mα2 is eiπα.Ω.α , detpI ´ Aq

pgq

ZM 2 pw, ρq “ α

ř

where α.Ω.α “

a,bPI` pα

a

¨ αb qΩab for genus g period matrix Ω. a

Proof. In this case we sum over ba “ hα pia , i´a q of (5.10) labelled by 2g multisets with dual Fock basis vectors of (5.11). We find ´ ¯ ź α a M pρ, ia q a 1 a ρwtpb q Z p0q pbα , wq “ Z p0q . . . ; hα pia , i´a q, wa ; . . . . p´ρa q 2 α ¨α rpia q! aPI` ` ˘ a Z p0q . . . ; hα pia , i´a q, wa ; . . . is determined by the coefficient of an appropriate ř 1 1 b p x´w ´ x´w q. expansion of the generating function (5.7) with θ ˘ pxq “ bPI` α˘ b ´b We find that ´ ¯ ź a a b p pia , ia q , Z p0q . . . ; hα pia , i´a q, wa ; . . . “ pwa ´ wb qα ¨α pperm p´ p` A φ ,φ

aăb

p of (4.11) and for a, b P I as in (5.13), A (5.14)

˘ φp˘ a pkq “ coeff xk´1 θ px ` w¯a q “

?

kρ´k{2 a

ÿ

b ¯a α˘ db pkq,

pk P ia q

bPI`

for dab pkq of (3.18). The ´ρiaa {ia multiplicative factors in M pρ, ia q are absorbed p and φp˘ terms using Theorem 2.3 and Proposition 4.1 to obtain into the A ´ ´1 ` T ÿ pperm φ´ ,φ` A pia , ia q eφ pI´Aq pφ q pgq “ F pw, ρq , ZM 2 pw, ρq “F pw, ρq α rpia q! detpI ´ Aq i a

where (5.15) k{2 ÿ b ¯a ρa p˘ φ˘ α˘ db pkq, a pkq :“ ¯ ? φa pkq “ ¯ k bPI`

pk P ia q

(5.16) F pw, ρq :“

ź ˆ aPI`

´ρa pwa ´ w´a q´2

˙ 1 αa ¨αa 2

ź a,bPI` ;a‰b

ˆ

pwa ´ wb qpw´a ´ w´b q pw´a ´ wb qpw´a ´ w´b q

˙ 1 αa ¨αb 2

,

ř a b recalling that α´a “ ´αa . Noting that φ´ pI ´ Aq´1 φ` “ ´ a,bPI` α´ α` da pI ´ ´1 Aq db the result follows on comparison with the expressions for 2πiΩab in (3.20) and (3.21).  5.3. Genus g correlation functions on M2 . We define genus g n-point i correlation differential forms on Mα2 “ baPI` Mα2a for n vectors vi b eβ P M2 inserted at xi by ÿ β pgq (5.17) ρwtpbq Z p0q pv b eβ , x; b, wqdxwtpvbe q , FM 2 pv b eβ , xq :“ α

2 bPMα i

for Z p0q pv, x; b, wq “ Z p0q p. . . ; vi b eβ , xi ; . . . ; ba , w´a ; ba , wa ; . . .q. We describe the genus g generating function for all correlation functions (5.17) generalizing Proposition 4.4 and Corollary 5.2. The result is expressed in terms of the genus g bidifferential ωpx, yq, the normalized 1-form νa pxq, the period matrix Ω,

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

337

the differential of the third kind ωp´q pxq and the prime form Epx, yq of (3.1)–(3.5) as follows: Theorem 5.4. For Heisenberg generators h˘ inserted at yr˘ for r P t1, . . . , mu ř t and eβ , for β t P C2 with nt“1 β t “ 0, inserted at zt for t P t1, . . . , nu we have ź ˘ ` t u pgq (5.18) FM 2 ph˘ , y ˘ ; eβ , zq “εβ Epzt , zu qβ ¨β pperm θr´ ,θr` ω yr` , ys´ α

tău

˜ ˆ exp iπα.Ω.α `

ÿ

a

α ¨β

t

νa

detpI ´ Aq´1 ,

z0

a,t

for a P I` ; t, u P t1, . . . , nu with n ÿ ÿ a t θrr˘ :“ (5.19) α˘ νa pyr˘ q ` β˘ ωzt ´z0 pyr˘ q, aPI`

¸

ż zt

r P t1, . . . , mu,

t“1

for arbitrary choice of z0 in (5.18) or (5.19). Proof. The proof is similar to that of Propositions 4.4 and 5.3. Here the sum in (5.17) is taken over ba “ hαa pia , i´a q with summand ź a M pρ, ia q 1 a Xα pia q, ρwtpbq Z p0q ph˘ , y ˘ ; b, wq “ p´ρa q 2 α ¨α rpia q! aPI` ´ ¯ t for Xα pia q “ Z p0q . . . ; h` , yr` ; . . . ; h´ , ys´ . . . ; eβ , zt ; . . . ; hαa pia , i´a q, wa ; . . . determined by expansions of (5.7) with ˆ ˙ ÿ ˆ ˙ n ÿ 1 1 1 1 b t ` . α˘ ´ β˘ ´ θ ˘ pxq “ x ´ wb x ´ w´b x ´ zt x ´ z0 t“1 bPI` ř for arbitrary choice of z0 (since nt“1 β t “ 0). We find that ź a s ź t u ź a b Xα pia q “εβ pzs ´ wa qα ¨β pzt ´ zu qβ ¨β pwa ´ wb qα ¨α a,s

tău

« ˆ pperm Θ p` p ´ ,Θ

p B p V pia q

aăb

ff

p pia q U p pia , ia q , A

p of (4.11) and B, p U p , Vp of for product indices a, b P I; s, t, u P t1, . . . , nu, with A ˘ ˘ ˘ ˘ p p (4.16) and where Θ “ p. . . , θ pyr q, . . . ψa pkq, . . .q for r P t1, . . . , mu and k P ia with ψpa˘ pkq “ coeff xk´1 θ˘ px ` w¯a q “ φp˘ a pkq `

n ÿ

´ ¯ t pz0 ´ w¯a q´k´1 ´ pzt ´ w¯a q´k´1 , β˘

t“1

for φp˘ of (5.14). Absorbing ´ρiaa {ia and dyr` dys´ factors into the partial permanent we find j „ ÿ 1 B U pia q pgq ˘ β , pperm Θ´ ,Θ` FM 2 ph˘ , y ; e , zq “ Gpw, ρq α V pia q A pia , ia q rpia q! i a

for B, U, V of (4.17) where with F pw, ρq of (5.16) we define a s ź ˆ zs ´ wa ˙α ¨β ź t u 1 2 Gpz, w, ρq :“εβ F pw, ρq (5.20) pzt ´ zu qβ ¨β dz 2 β , zs ´ w´a a,s tău

338

MICHAEL P. TUITE

for a P I` ; s, t, u P t1, . . . , nu. Θ˘ “ p. . . , θr˘ , . . . , ψa˘ pkq, . . .q where θr˘ :“θ ˘ pyr˘ qdyr˘ “

p0q

ÿ

b α˘ ωwb ´w´b pyr˘ q `

n ÿ

p0q

t β˘ ωzt ´z0 pyr˘ q,

t“1

bPI`

¨

˛ ż zt k{2 n ÿ ÿ ρ a b ¯a t α˘ db pkq ` β˘ L¯a pk, ¨q‚, ψa˘ pkq :“ ¯ ? ψpa˘ pkq “ ¯ ˝ k z0 t“1 bPI` and La pk, xq of (3.12). By the general McMahon Master Theorem 2.4 we find pgq FM 2 ph˘ , y ˘ ; eβ , zq α

(5.21)

“ Gpw, ρq

eψ

´

pI´Aq´1 pψ ` qT

detpI ´ Aq

r pperm θr´ ,θr` B,

r sq “ ωpy ` , y ´ q from (4.18) and θr` “ θ ` ` U pI ´ Aq´1 ψ ` given by with Bpr, r s ´ ¯ ÿ p0q ` b r θr “ ωwb ´w´b pyr` q ´ Lpyr` qpI ´ Aq´1 db α` bPI` n ÿ

`

t β`

ˆ ˙ ż zt p0q ` ` ´1 ωzt ´z0 pyr q ´ Lpyr qpI ´ Aq Rp¨q z0

t“1

ÿ

“

a α` νa pyr` q `

n ÿ

t β` ωzt ´z0 pyr` q,

t“1

aPI`

by Propositions 3.2 and 3.4. A similar result holds for θr´ so that we obtain the pperm θr´ ,θr` ω pyr` , ys´ q term in (5.18). The exponent ψ ´ pI ´ Aq´1 pψ ` qT term in (5.21) is given by (5.22) ´

ÿ

a b α´ α` da pI

´ Aq

´1

db ´

n ÿ ÿ

a t α´ β`

t a β´ α`

ż zt

Lp¨qpI ´ Aq´1 da ´

z0

aPI` t“1

ż zt

da pI ´ Aq´1 Rp¨q

z0

aPI` t“1

a,bPI`

´

n ÿ ÿ

n ÿ

t u β´ β`

t,u“1

ż zt

Lp¨qpI ´ Aq´1

x0

ż zu Rp¨q, y0

a b α` terms are combined with the F pw, ρq term to for arbitrary x0 , y0 , z0 . The α´ obtain the iπα.Ω.α exponent in (5.18) as in the proof of Proposition 5.3. Proposition 3.4 implies ˙ ˆ ż zt ż zt ż zt zt ´ wa “´ νa ´ log da pI ´ Aq´1 Rp¨q “ ´ Lp¨qpI ´ Aq´1 da . zt ´ w´a z0 z0 z0 şz ř Thus the αa ¨ β t terms in (5.20) and (5.22) combine to give the a,t αa ¨ β t z0t νa term in (5.18). ¯ ´ 1 1 for Epx, yq “ Kpx, yqdx´ 2 dy ´ 2 of (3.5) so that Let rpx, yq :“ log Kpx,yq x´y řn rpx, yq “ rpy, xq and rpx, xq “ 0. From Proposition 3.2 and since t“1 β t “ 0 we find

´

n ÿ t,u“1

t u β´ β`

ż zt x0

Lp¨qpI ´ Aq´1

ż zu Rp¨q “ y0

n ÿ t,u“1

“

n ÿ t,u“1

t u β´ β`

ż zt ż zu ´ ¯ ωpx, yq ´ ω p0q px, yq x0

y0

t u β´ β` rpzt , zu q “

ÿ tău

β t ¨ β u rpzt , zu q.

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

339

We combine these β t ¨β u terms in (5.22) with those in Gpz, w, ρq together with the t u 1 2 differential dz 2 β to obtain the Epzt , zu qβ ¨β terms in (5.18). Thus the theorem is proved.  pgq

Remark 5.5. FM 2 ph, y; eβ , zq is the generating function for all correlation α functions on M2 for the Fock vectors (5.10) by repeated use of (4.10). Thus for the Virasoro vector, 1 pgq pgq pgq FM 2 pω, zq “ coeff x0 FM 2 ph` , x ` z; h´ , zq “ spzqZM 2 , α α α 6 ´ ¯ 1 for the projective connection spxq :“ 6 limyÑx ωpx, yq ´ px´yq 2 dxdy . Choosing αa “ pα1a , 0q, β t “ pβ1t , 0q in Theorem 5.4 we obtain the generating function for all correlation functions on Mα “ baPI` Mαa for αa P C (on relabelling) as follows: Corollary 5.6. For Heisenberg generators h inserted at yr for r P t1, . . . , mu řn t and eβ for β t P C with t“1 β t “ 0 inserted at zt for t P t1, . . . , nu we have (5.23) pgq

FMα ph, y; eβ , zq “εβ

ź

Epzt , zu qβ

t

β

u

Symm pω, να,β q

¯ ´ şz ř exp iπα.Ω.α ` a,t αa β t z0t νa 1

for a P I` ; t, u P t1, . . . , nu with α.Ω.α “ να,β :“

ÿ

ř

α a νa `

a,bPI` n ÿ

,

detpI ´ Aq 2

tău

αa Ωab αb and where

β t ωzt ´z0 ,

t“1

aPI`

for arbitrary choice of z0 and symmetric (tensor) product of να,β and ω defined by ÿź ź ` ˘ να,β pyq q ωpyr , ys q, Symm ω, να,β :“ ϕ

q

pr,sq

where we sum over all inequivalent involutions ϕ “ pqq . . . prsq . . . of the labels t1, . . . , mu. Example 5.7. Consider the lattice (Super)VOA VL for a rank d even (odd) integral lattice L. Then VL “ ‘λPL Mλd where Mλd “ bdi“1 Mλi and λ “ pλ1 , . . . , λd q P L so that ΘL pΩq pgq ZVL “ , detpI ´ Aqd{2 ř for genus g Siegel lattice theta function ΘL pΩq “ λPbg L eiπ λ.Ω.λ . For even L, this agrees with Proposition 6.9 of [TW] found by the alternative technique of genus g Zhu recursion. pgq

Example 5.8. For αa “ β t “ 0 the correlation function FM ph, yq agrees with Proposition 6.1 of [TW] found from genus g Zhu recursion. Example 5.9. Consider the correlation function for e`1 inserted at xi and e´1 inserted at yj for i, j P t1, . . . , nu on Mm`α “ baPI` Mma `αa for m P Zg and α P Rg . Then ś iπpm`αq.Ω.pm`αq`pm`αq.ζ iăj Epxi , xj qEpyi , yj q e pgq ś , FMm`α pe`1 , x; e´1 , yq “ 1 detpI ´ Aq 2 i,j Epxi , yj q

340

MICHAEL P. TUITE

şxi i“1 yi

řn

νa . Summing over all m P Zg we find ś ÿ pgq iăj Epxi , xj qEpyi , yj q ΘrαspΩ, ζq `1 ´1 ś (5.24) FMm`α pe , x; e , yq “ 1 , detpI ´ Aq 2 i,j Epxi , yj q mPZg ř iπpm`αq.Ω.pm`αq`pm`αq.ζ . for Riemann theta function ΘrαspΩ, ζq :“ mPZg e (5.24) is the correlation generating function for twisted sectors of the rank 2 fermionic SVOA e.g. [R].

where ζa “

6. Appendix p For |x| ą |y| We describe some elementary properties of ω p0q px, yq for x, y P C. we find ω p0q px, yq “ ´bpxqcpyq,

(6.1)

where bpxq, cpyq are infinite row and column vectors with components given by the 1-forms ¿ ? 1 ? bl pxq :“ y ´l ω p0q px, yq “ lx´l´1 dx, l “ 1, 2, . . . , 2πi l C0 pyq ¿ ? 1 ? ck pyq :“ ´ xk ω p0q px, yq “ ´ ky k´1 dy, k “ 1, 2, . . . , 2πi k C8 pxq

for Jordan curves C0 in the neighborhood of 0 and C8 in the neighborhood of 8. Note that C8 pxq „ ´C0 px´1 q implies bk pxq “ ck px´1 q. For γ P SL2 pCq define a matrix Dpγq with components for k, l ě 1 given by [DV] (6.2) 1 ? Dkl pγq :“ p2πiq2 kl plq where By :“

¿

¿

xk y ´l ω p0q px, γyq “

C8 pxq C0 pyq

$b ˇ & l Byplq pγyqk ˇˇ k

%0,

, γp0q ‰ 8,

y“0

γp0q “ 8,

1 Bl l! By l .

Lemma 6.1. Let γ1 , γ2 P SL2 pCq such that γ1 p0q, γ2 p0q, γ1 γ2 p0q ‰ 8. Then (i) bpxqDpγ1 q “ bpγ1´1 xq, (ii) Dpγ1 qcpyq “ cpγ1 yq, (iii) Dpγ1 qDpγ2 q “ Dpγ1 γ2 q. Proof. SL2 pCq invariance of ω p0q px, yq implies ω p0q pγ1´1 x, yq “ ω p0q px, γ1 yq “ ´bpxqcpγ1 yq, for |x| ą |γ1 y| from (6.1). Provided γ1 p0q ‰ 8 we may compute y moments of both sides of this equation to obtain (i). A similar argument leads to (ii). (iii) follows by considering y moments of (ii).  Remark 6.2. Note ` ˘that (iii) does not imply that D is a representation of SL2 pCq e.g. for γ “ 01 10 then I “ Dpγ 2 q ‰ Dpγq2 “ 0.

HEISENBERG GENERALIZED VERTEX OPERATOR ALGEBRA

341

We may now readily express the 1-forms Lpxq, Rpxq and the moment matrix A of (3.12), (3.13). Define λa , μa P SL2 pCq for a P I by ˙ˆ ˙ˆ ˙ ˙ ˆ ´1{2 ˆ 1{2 1 ´wa 0 1 ρa ρa 0 0 , μa :“ (6.3) . λa :“ 1 ´w´a 0 1 0 1 0 1 The sewing condition (3.7) reads λa z 1 “ 1{pλ´a zq “ μa z i.e. the Schottky generators obey γa “ λ´1 a μa .

(6.4)

It is easy to check that the 1-forms Lpxq, Rpyq and the moment matrix A are given by (6.5)

La pk, xq “ bk pλa xq,

Ra pl, yq “ cl pμa yq,

Aab pk, lq “ Dkl pμa λ´1 b q.

Lemma 6.1, (6.1) and (6.4) imply that for integer n ě 1 ÿ ´1 bpλa1 xqDpμa1 λ´1 LpxqAn´1 Rpyq “ a2 q . . . Dpμan´1 λan qcpμan yq a1 ,...,an

“

ÿ

bpxqcpγa1 . . . γan yq “ ´

a1 ,...,an

ÿ

ω p0q px, γyq,

γPΓn

where Γn denotes the elements of the Schottky group consisting of reduced words of length n i.e. for all adjacent generators γ´ai ” γa´1 ‰ γai`1 for i “ 1, . . . , n ´ 1. i Hence Proposition 3.2 implies the classical formula e.g. [Bo] ÿ ωpx, yq “ (6.6) ω p0q px, γyq, γPΓ

where the sum is taken over all the elements of the Schottky group Γ. References B. Bakalov and V. Kac, Generalized vertex algebras, math.QA/0602072. Alexander I. Bobenko, Introduction to compact Riemann surfaces, Computational approach to Riemann surfaces, Lecture Notes in Math., vol. 2013, Springer, Heidelberg, 2011, pp. 3–64, DOI 10.1007/978-3-642-17413-1 1. MR2905610 [C] Daniel E. Cohen, Combinatorial group theory: a topological approach, London Mathematical Society Student Texts, vol. 14, Cambridge University Press, Cambridge, 1989. MR1020297 [DL] Chongying Dong and James Lepowsky, Generalized vertex algebras and relative vertex operators, Progress in Mathematics, vol. 112, Birkh¨ auser Boston, Inc., Boston, MA, 1993. MR1233387 [DP] Eric D’Hoker and D. H. Phong, On determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 104 (1986), no. 4, 537–545. MR841668 [DV] P. Di Vecchia, F. Pezzella, M. Frau, K. Hornfeck, A. Lerda, and S. Sciuto, N -point g-loop vertex for a free bosonic theory with vacuum charge Q, Nuclear Phys. B 322 (1989), no. 2, 317–372, DOI 10.1016/0550-3213(89)90419-7. MR1009740 [F] John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. MR0335789 [FHL] Igor B. Frenkel, Yi-Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64, DOI 10.1090/memo/0494. MR1142494 [FLM] Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR996026 [Fo] Lester R. Ford, Automorphic functions, 2nd ed., Chelsea Publishing Co., New York, 1951. MR3444841

[BK] [Bo]

342

[Gu]

[HT]

[Ka] [Li]

[LL]

[McIT]

[McM] [Mo] [MT1]

[MT2]

[MT3]

[MT4]

[MT5]

[Mu]

[R] [T] [TW] [TZ]

[Y] [Z]

MICHAEL P. TUITE

Robert C. Gunning, Introduction to holomorphic functions of several variables. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990. Function theory. MR1052649 Donny Hurley and Michael P. Tuite, Virasoro correlation functions for vertex operator algebras, Internat. J. Math. 23 (2012), no. 10, 1250106, 23, DOI 10.1142/S0129167X12501066. MR2999051 Victor Kac, Vertex algebras for beginners, 2nd ed., University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1998. MR1651389 Hai Sheng Li, Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 279–297, DOI 10.1016/0022-4049(94)90104-X. MR1303287 James Lepowsky and Haisheng Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, vol. 227, Birkh¨ auser Boston, Inc., Boston, MA, 2004. MR2023933 A. McIntyre and L. A. Takhtajan, Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula, Geom. Funct. Anal. 16 (2006), no. 6, 1291–1323, DOI 10.1007/s00039-006-05827. MR2276541 Percy A. MacMahon, Combinatory analysis, Two volumes (bound as one), Chelsea Publishing Co., New York, 1960. MR0141605 C. Montonen, Multiloop amplitudes in additive dual-resonance models, Nuovo Cim. 19 (1974) 69–89. Geoffrey Mason and Michael P. Tuite, On genus two Riemann surfaces formed from sewn tori, Comm. Math. Phys. 270 (2007), no. 3, 587–634, DOI 10.1007/s00220-006-0163-5. MR2276459 Geoffrey Mason and Michael P. Tuite, Torus chiral n-point functions for free boson and lattice vertex operator algebras, Comm. Math. Phys. 235 (2003), no. 1, 47–68, DOI 10.1007/s00220-002-0772-6. MR1969720 Geoffrey Mason and Michael Tuite, Vertex operators and modular forms, A window into zeta and modular physics, Math. Sci. Res. Inst. Publ., vol. 57, Cambridge Univ. Press, Cambridge, 2010, pp. 183–278. MR2648364 Geoffrey Mason and Michael P. Tuite, Free bosonic vertex operator algebras on genus two Riemann surfaces I, Comm. Math. Phys. 300 (2010), no. 3, 673–713, DOI 10.1007/s00220010-1126-4. MR2736959 Geoffrey Mason and Michael P. Tuite, Free bosonic vertex operator algebras on genus two Riemann surfaces II, Conformal field theory, automorphic forms and related topics, Contrib. Math. Comput. Sci., vol. 8, Springer, Heidelberg, 2014, pp. 183–225. MR3559205 David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkh¨ auser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR688651 A. K. Raina, Fay’s trisecant identity and conformal field theory, Comm. Math. Phys. 122 (1989), no. 4, 625–641. MR1002836 Michael P. Tuite, Some generalizations of the MacMahon master theorem, J. Combin. Theory Ser. A 120 (2013), no. 1, 92–101, DOI 10.1016/j.jcta.2012.07.007. MR2971699 M.P. Tuite and M. Welby, General genus Zhu recursion for vertex operator algebras, arXiv:1911.06596. Michael P. Tuite and Alexander Zuevsky, A generalized vertex operator algebra for Heisenberg intertwiners, J. Pure Appl. Algebra 216 (2012), no. 6, 1442–1453, DOI 10.1016/j.jpaa.2011.10.025. MR2890514 Akira Yamada, Precise variational formulas for abelian differentials, Kodai Math. J. 3 (1980), no. 1, 114–143. MR569537 P. G. Zograf, Liouville action on moduli spaces and uniformization of degenerate Riemann surfaces (Russian), Algebra i Analiz 1 (1989), no. 4, 136–160; English transl., Leningrad Math. J. 1 (1990), no. 4, 941–965. MR1027464

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway, University Road, Galway, Ireland Email address: [email protected]

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CONM

768

ISBN 978-1-4704-5351-0

9 781470 453510 CONM/768

´ et al., Editors Representation Theory • Adamovic

This volume contains the proceedings of the conference on Representation Theory XVI, held from June 25–29, 2019, in Dubrovnik, Croatia. The articles in the volume address selected aspects of representation theory of reductive Lie groups and vertex algebras, and are written by prominent experts in the field as well as junior researchers. The three main topics of these articles are Lie theory, number theory, and vertex algebras.