Representations of SU(2,1) in Fourier Term Modules (Lecture Notes in Mathematics, 2340) 303143191X, 9783031431913

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Lecture Notes in Mathematics  2340

Roelof W. Bruggeman Roberto J. Miatello

Representations of SU(2,1) in Fourier Term Modules

Lecture Notes in Mathematics Volume 2340

Editors-in-Chief Jean-Michel Morel, Ecole Normale Supérieure Paris-Saclay, Paris, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Rupert Frank, LMU, Munich, Germany Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK László Székelyhidi Germany

, Institute of Mathematics, Leipzig University, Leipzig,

Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

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Roelof W. Bruggeman • Roberto J. Miatello

Representations of SU(2,1) in Fourier Term Modules

Roelof W. Bruggeman Mathematisch Instituut Universiteit Utrecht Utrecht, The Netherlands

Roberto J. Miatello FaMAF-CIEM Universidad Nacional de Córdoba Córdoba, Argentina

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-031-43191-3 ISBN 978-3-031-43192-0 (eBook) https://doi.org/10.1007/978-3-031-43192-0 Mathematics Subject Classification: Primary: 11F70, Secondary: 11F55, 22E30 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

To Ida and Isabel

Preface

If one is asked what is an automorphic form, one can answer in various ways:     = (cz + d)k f (z) (a) A function f on . z ∈ C : Im z > 0 that satisfies .f az+b cz+d   ab for all . ∈ SL2 (Z), with some further properties. cd (b) A function f on .SL2 (R) that satisfies .f (γ z) = f (z) for all .γ in a cofinite discrete subgroup . ⊂ SL2 (R), with some further properties. (c) A function f on some semisimple Lie group G that satisfies .f (γ z) = f (z) for all .γ in a cofinite discrete subgroup of G, with some further properties. (d) A function f on .G(k)\G(A) for some reductive algebraic group .G over a number field k with adele ring .A, with some further properties. The automorphic forms that we aim at here fit between (b) and (c). We look at functions on the Lie group .SU(2, 1) that are invariant under a cofinite discrete subgroup . ⊂ SU(2, 1). The group .SU(2, 1) has minimal dimension among the semisimple Lie groups with non-commutative unipotent subgroups.  1 x  In (b), we have the unipotent subgroup .N = : x ∈ R . In (a) 01 and (b), the Fourier expansion of automorphic forms is based on the expansion  

of functions on .  ∩ N N in terms of characters of N . For .SU(2, 1), the characters of a unipotent subgroup do not suffice. We need to take into account more representations. Miatello and Wallach [32] studied Poincaré series as a method to construct automorphic forms on semisimple Lie groups of real rank one. They use only Poincaré series based on characters of maximal unipotent subgroups. We would like to construct Poincaré series based on other representations of unipotent subgroups. This makes it desirable to understand the functions on G that transform on the left according to a given representation of a unipotent subgroup as well. These functions generate .(g, K)-modules for G.

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Preface

We put ourselves to the task to understand these modules in the case of .SU(2, 1). That led to this monograph. The theorems in the introduction give an overview of the results. In the last chapter, we apply these results to automorphic forms on .SU(2, 1). In [5] we apply the results in this monograph to show the completeness of the Poincaré series for the spaces of square integrable automorphic forms. Acknowledgements The authors wish to thank the anonymous referees for their very useful remarks and suggestions. Utrecht, the Netherlands Córdoba, Argentina July 2023

Roelof W. Bruggeman Roberto J. Miatello

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Overview of Chaps. 2–4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 8

2

The Lie Group SU(2,1) and Subgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Realization of the Group SU(2, 1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Symmetric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Maximal Compact Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Irreducible Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Unipotent Subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Characters and Stone-von Neumann Representation . . . . . . . . . 2.3.2 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Discrete Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Fourier Expansions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 13 15 16 19 20 22 23 29 31

3

Fourier Term Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 (g, K)-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Special Cyclic Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Explicit Differentiation of K-finite Functions . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Large Fourier Term Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Abelian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Non-abelian Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Normalization of Standard Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Central Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Eigenfunction Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 One-Dimensional K-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Kernels of Downward Shift Operators . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Kernels of Upward Shift Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 33 39 44 48 49 51 57 59 63 65 67 71

ix

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Contents

3.5

Special Fourier Term Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Principal Series and Logarithmic Modules . . . . . . . . . . . . . . . . . . . 3.5.2 Submodules Characterized by Boundary Behavior. . . . . . . . . . . 3.5.3 Intertwining Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 75 77 82

4

Submodule Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Lattice Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Isomorphism Classes of Irreducible Representations . . . . . . . . 4.1.3 Submodules Determined by Shift Operators. . . . . . . . . . . . . . . . . . 4.2 Principal Series and Related Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Kernels of Shift Operators and Submodules . . . . . . . . . . . . . . . . . . 4.2.2 Submodules of Principal Series Modules . . . . . . . . . . . . . . . . . . . . . 4.2.3 Characterization by Sets of K-types . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Logarithmic Submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Intersection of Kernels for the Principal Series . . . . . . . . . . . . . . . 4.3 Submodule Structure of Abelian Fourier Term Modules . . . . . . . . . . . . . 4.3.1 Structure Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Submodule Structure of Non-abelian Fourier Term Modules . . . . . . . . 4.4.1 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Families with Fixed K-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Intersection of Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Special Submodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Modules with Regular Behavior at 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.7 Structure Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.8 Intertwining Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Unitary Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Invariant Sesquilinear Forms and Unitarizability . . . . . . . . . . . . . 4.5.2 Principal Series Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Other Irreducible Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 86 87 89 91 92 93 108 110 111 114 117 120 121 122 128 131 134 141 149 157 159 160 161 163

5

Application to Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Growth Conditions and Fourier Expansions . . . . . . . . . . . . . . . . . . 5.1.2 Holomorphic Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Families of Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fourier Term Operators for Modules of Automorphic Forms . . . . . . . . 5.3.1 Irreducible Modules of Square Integrable Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Families of Modules of Automorphic Forms with Exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 General Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Fourier Term Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Summary of Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 172 174 176 179 179 184 187 187 189 190

Contents

xi

A Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 A.1 Modified Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 A.2 Whittaker Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Chapter 1

Introduction

Automorphic forms on a semisimple Lie group G are .-invariant functions on G for some discrete subgroup that satisfy some additional conditions: finiteness under the action of a maximal compact subgroup, a relation for the action of the center of the enveloping algebra and the condition of polynomial growth at the cusps. See for instance Harish Chandra [19] or Langlands [31], who develop the theory of Eisenstein series. Poincaré series are constructed analogously to Eisenstein series. They are well known in the context of holomorphic automorphic forms on the complex upper half-plane; for instance Petersson [36]. Neunhöffer [33] and Niebur [34] gave real-analytic Poincaré series (for the trivial K-type), and Miatello and Wallach [32] considered Poincaré series for Lie groups of real rank one. This leads to meromorphic families of automorphic forms that are allowed to have some exponential growth at the cusps. At some first order singularities of the family the residue is a cusp form. Not all cusp forms can be obtained in this way. (See Gelbart, Piatetskii-Shapiro [14].) There are cusp forms that have zero Fourier coefficients for all non-trivial characters .χ of the unipotent subgroups .Nc for all cusps .c. Such cusp forms cannot be obtained as residues of the Poincaré series studied in [32]. The real Lie group .SU(2, 1) is the smallest rank one Lie group with a nonabelian unipotent subgroup. This has the consequence that automorphic forms have expansions in which, besides Fourier terms based on a character of the unipotent group, other terms occur corresponding to the Stone-von Neumann representation of the Heisenberg group. These are the Fourier-Jacobi expansions. For Poincaré series corresponding to representations of the Heisenberg group we need to understand the exponentially increasing terms in such Fourier-Jacobi expansions. A very special case of automorphic forms are the holomorphic ones for which the function on G corresponds to a holomorphic function on the symmetric space .G/K. For .SU(2, 1) we can go back to 1883, when Picard [37] studied a class of algebraic curves, now called Picard curves. He showed that the isomorphism classes of such

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. W. Bruggeman, R. J. Miatello, Representations of SU(2,1) in Fourier Term Modules, Lecture Notes in Mathematics 2340, https://doi.org/10.1007/978-3-031-43192-0_1

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2

1 Introduction

curves correspond to the points of the quotient .\SU(2, 1)/K for a specific discrete subgroup. In this way the coefficients in a general formula for the Picard curves can be viewed as holomorphic automorphic forms on .SU(2, 1). In this work we study, for a fixed Iwasawa decomposition .SU(2, 1) = G = N AK, the spaces .C ∞ (\G)K of K-finite functions that are invariant under a lattice ∞ . ⊂ N . Elements of .C (\N)K can be expanded in a general type of Fourier expansion, which is the sum of two parts: the abelian part, with terms parametrized by characters of N that are trivial on ., and the non-abelian part, with terms parametrized by an orthogonal collection of realizations of the Stone-von Neumann representation in .L2 (\N). We use .N as a general symbol, running over characters of N, and over realizations of the Stone-von Neumann representation. Thus, we arrive at operators FN : C ∞ (\G)K → FN ⊂ C ∞ (G)K ,

.

(1.1)

where .FN consists of the functions in .C ∞ (G)K that transform on the left according to .N. The Fourier term operators .FN are intertwining operators of .(g, K)-modules. We call their images Fourier term modules. In Section 8 we define all .FN explicitly. Automorphic forms on G for the discrete subgroups . that we consider belong to .(g, K)-modules in .C ∞ (\G)K . We consider only automorphic forms on which the center of the universal enveloping algebra of .g acts as a character. We denote this character by .ψ. Automorphic forms have a Fourier expansion at each cusp .c of .. For each cusp .c of . we get intertwining operators .FcN from the module of automorphic forms to .FN . These intertwining operators preserve the infinitesimal ψ character .ψ. So .FcN has values in the submodule .FN ⊂ FN in which the center of the universal enveloping algebra acts according to .ψ. ψ The main aim of this work is to carry out a detailed study of the modules .FN , of their structure as .(g, K)-modules, and of the boundary behavior of the functions in these modules. For .N = N0 corresponding to the trivial character of N, we deal with functions ψ ψ on .N\G. The module .F0 = FN0 is, for most infinitesimal characters .ψ, the direct sum of principal series modules. We parametrize the subgroup A by .a(t) with .t ∈ (0, ∞), and use the behavior in ψ t to define, for all .N = N0 , two important submodules of .FN . The Fourier terms of ψ ψ cusp forms are in modules of this type. The submodule .MN ⊂ FN is defined by the behavior as .t ↓ 0 by prescribing a convergent expansion. Under conditions on .ψ, ψ elements of .MN can be used to define Poincaré series. For the Fourier expansion of ψ such Poincaré series (and their meromorphic continuation) we will need both .MN ψ and .WN .

1.1 Overview of Chaps. 2–4

3 ψ

Various authors have studied the modules .WN . One says that an irreducible ψ representation admits a Whittaker model if it can be realized in .WN for a nontrivial character .N; see, for instance, [24], [41], Proposition 9.2 in [25, p. 317], and [15]. For .SU(2, 1) the non-trivial characters of N are the generic characters. A representation that does admit a Whittaker model is called generic. Gelbart and Piatetskii-Shapiro, in their study [14] of lifting from .U1,1 to .U2,1 and L-functions, exhibit by adelic methods many non-generic representations (called by them hypercuspidal). The study of non-abelian Fourier terms is important in the theory of automorphic forms, in particular, since for ‘non-generic’ representations abelian Fourier terms turn out to be zero (except possibly for the N-trivial term). This is connected with the Gelfand-Kirillov dimension of modules. (See [27] and [47]; see also [14], and [40].) Explicit Fourier expansions of automorphic forms on .SU(2, 1) have been given by Koseki, Oda [26], and Ishikawa [22, 23]; also by Bao, Kleinschmidt, Nilsson, Persson and Pioline [2], who studied holomorphic Eisenstein series and Eisenstein series on .G/K, respectively. ψ

In this work we determine the module structure of all the modules .FN on .G = SU(2, 1). In the reducible cases we see considerable differences in structure between the reducible principal series, the Fourier term modules arising from a character of N , and the modules arising from infinite-dimensional representations of N. The structure of reducible principal series representations can be found in many places, for instance in [1, 10, 28]. As far as we know, the results on the other types of modules are new. The unipotent subgroup N is a Heisenberg group, which enables us to work out ψ explicitly the full algebraic structure of the Fourier term modules .FN . The wish to work explicitly leads at several places to computations of a size that are hard to carry out by hand and are more suitable for symbolic computation. We explain and carry out these computations in the Mathematica notebook [6], which we consider to be a substantial complement to this work. We refer to the relevant sections in the text. At many other places we checked with Mathematica computations carried out by hand. They can be found in the notebook as well, but are often not indicated in the text.

1.1 Overview of Chaps. 2–4 Chapter 2 summarizes results on .SU(2, 1) and the subgroups in the Iwasawa decomposition .G = SU(2, 1) = NAK. These results are known, sometimes in wider generality and in other notations than we need here.

4

1 Introduction ψ

Chapters 3 and 4 contain the results on the modules .FN , which are contained in ∞ .C (σ \G)K for a standard lattice .σ (Definition 2.2) in the unipotent subgroup N. We use the following main division: ψ

• Abelian Fourier term modules. The module .Fβ consists of functions transforming on the left according to a character .χβ of N, .β ∈ C, as defined in (2.42). • Non-abelian Fourier term modules. The group N is non-abelian. The Stone-von ψ ψ Neumann representation of N leads to modules .F,c,d .The elements of .F,c,d are described by use of theta-functions on N. See Sect. 3.3.2. The parameter . ∈ σ 2 Z=0 determines a character of the center of N, and the parameter .c ∈ Z mod 2 determines a shift in the theta functions. The “metaplectic parameter” .d ∈ 1+2Z determines a character of the double cover of the group .M ⊂ K normalizing NA. See Sects. 2.3.2 and 3.3.2. One can parametrize the characters .ψ of the center of the enveloping algebra ZU (g) by characters of AM (with .M ⊂ K normalizing N A). The characters .ξ of M correspond to integers .jξ , the characters of A to complex numbers .ν. The Weyl group W of type .A2 , isomorphic to the symmetric group .S3 , acts on the elements 2 2 .(j, ν) ∈ C , and the orbits of W in .C correspond bijectively with the characters .ψ of .ZU (g). For .SU(2, 1) only the intersection of these orbits with .Z × C is relevant. This intersection, denoted .OW (ψ), can have from zero to six elements. For our purpose, this intersection .OW (ψ) is relevant only when it is non-empty. See Table 3.11, p. 62, for a further discussion. .

ψ

ξ,ν

N -trivial Fourier Term Modules The N -trivial modules .F0 contain modules .HK in the principal series, discussed in Sect. 3.5.1. If .OW (ψ) does not contain elements of the form .(j, 0), then by Proposition 3.27 ψ

F0 =

.



ξ,ν

(1.2)

HK .

(jξ ,ν)∈OW (ψ) ξ,ν

If .OW (ψ) contains elements of the form .(j, 0), the principal series modules .HK ψ with .(jξ , ν) in .OW (ψ) do not suffice to obtain the whole of .F0 . In that case we need also terms containing a logarithm; see Propositions 3.29 and 4.4. ψ

Submodules Determined by Boundary Behavior Inside the submodules .WN and ψ ψ .M N of .FN defined above, we have submodules parametrized by .(jξ , ν) ∈ OW (ψ) with .Re ν ≥ 0.   ξ,ν • .WN , consisting of functions for which .t → f na(t)k has exponential decay as .t ↑ ∞. (We use the Iwasawa decomposition, and the parametrization .t → a(t) of A by .(0, ∞).)   ξ,ν • .MN , consisting of functions for which .t → f na(t)k has the form .t → t 2+ν h(t) with h extending holomorphically from .(0, ∞) to .C.

1.1 Overview of Chaps. 2–4

5

We will now state the main theorems in this work, summarizing the results, proved in Chaps. 3 and 4, on the algebraic structure of the .(g, K)-modules .F ψ . This structure will lead to applications on the precise form of the Fourier expansion of automorphic forms, to be discussed in Chap. 5. Fourier Term Modules Under Generic Parametrization We consider first a character .ψ of .ZU (g) represented by elements .(j, ν) such that .ν ≡ j mod 2, or .(j, ν) = (0, 0). In the terminology used in Table 3.11, p. 62, this is called generic parametrization. For such .ψ the set .OW (ψ) often consists of two elements .(j, ν) and .(j, −ν). It may happen that .3ν ≡ j mod 2. Then .OW (ψ) is a full Weyl group orbit with six elements. We put   OW (ψ)+ = (j, ν) ∈ OW (ψ) : Re ν ≥ 0 .

(1.3)

.

The irreducible representations of the maximal compact subgroup K are the .(p + 1)-dimensional representations .τph discussed in Sect. 2.2.1. The parameters satisfy .p ∈ Z≥0 , .h ≡ p mod 2. Theorem A Let the character .ψ of .ZU (g) correspond to generic parametrization, and let .β ∈ C∗ . Then (i) For each .(j, ν) ∈ OW (ψ)+ the submodules .Wβ

ξ,ν

ξ,ν

and .Mβ

ξ,ν

ψ

of .Fβ are

ξ,−ν

irreducible .(g, K)-modules, isomorphic to .HK and also to .HK . ξ,ν ξ,ν The K-types .τph in .Wβ and in .Mβ satisfy .|h − 2jξ | ≤ 3p. They occur in both modules with multiplicity  ξ ,νone. ξ ,ν  ψ j (ii) .Fβ = ⊕ Mβj . (j,ν)∈OW (ψ)+ Wβ ψ

For the non-abelian modules .Fn we need the following integral quantity. m0 (j ) =

.

  1 1 sign () d − 2j − . 6 2

(1.4)

We put   + OW (ψ)+ : m0 (j ) ≥ 0 . n = (j, ν) ∈ OW (ψ)

.

(1.5)

Theorem B Let the character .ψ of .ZU (g) correspond to generic parametrization. Let .n be a parameter triple .(, c, d). Then (i) .Fn is non-zero if and only if .m0 (j ) ∈ Z≥0 for some .(j, ν) ∈ OW (ψ)+ . β ξ,ν ξ,ν (ii) For each .(j, ν) ∈ OW (ψ)+ and .Mn of .Fn are n the submodules .Wn ξ,ν ξ,−ν irreducible .(g, K)-modules isomorphic to .HK and to .HK . h The K-types .τp in these modules have multiplicity one, and satisfy the inequality .|h − 2jξ | ≤ 3p.  ξj ,ν ξj ,ν ψ + Wn ⊕ M . (iii) .Fn = n (j,ν)∈OW (ψ)n ψ

6

1 Introduction

These two theorems show that under generic parametrization the .(g, K)-modψ ules .FN are the direct sum of modules with isomorphism type of a principal series representation. We obtain these results in Sects. 3.1–3.5; the proof is completed on p. 81. These sections contain also explicit information on the functions on G in these modules. Integral Parametrization A character .ψ of .ZU (g) corresponds to integral parametrization if the elements .(j, ν) ∈ OW (ψ) satisfy .ν ≡ j mod 2. ψ

Under integral parametrization the structure of the N-trivial modules .F0 stays ξ,ν as indicated in (1.2). However, the principal series modules .HK become reducible. Much more generally than only for .SU(2, 1), one knows that all irreducible .(g, K)ξ,ν modules occur as subquotients of some .HK (see Harish Chandra [17]), and even as submodules (see Casselman and Miliˇci´c [7]). In Sect. 4.2.2 we describe the total submodule structure of all reducible principal series modules, giving explicit illustrations of the sets of K-types. ψ ξ,ν ξ,ν For the other modules .FN the submodules .WN and .MN become reducible ψ as well. However, the way they fit together in .FN differs remarkably from the Ntrivial case: “They coincide wherever they can.” To formulate this more precisely (in parts ii) of Theorems C and D), we denote by .Vh,p the subspace of K-type .τph in the .(g, K)-module V . Theorem C Let the character .ψ of .ZU (g) correspond to integral parametrization, and let .β ∈ C∗ . Then ξ,ν

ψ

ξ,ν

(i) The .(g, K)-submodules .Mβ and .Wβ of .Fβ are reducible for each element + h .(ξ, ν) ∈ OW (ψ) . The K-types .τp in these modules have multiplicity one, and satisfy .|h − 2jξ | ≤ 3p. ξ,ν

ξ  ,ν 

(ii) If a K-type .τph occurs in .Mβ and in .Mβ ξ,ν

ξ  ,ν 

for .(jξ , ν), (jξ  , ν  } ∈ OW (ψ)+ ,

then .Mβ;h,p = Mβ;h,p ; and similarly for the .W-modules. 



ψ ξ,ν ξ,ν (iii) .Fβ = . ⊕ (j,ν)∈OW (ψ)+ Mβ (j,ν)∈OW (ψ)+ Wβ  ξj ,ν (iv) The intersection . (j,ν)∈OW (ψ)+ Mβ is the unique irreducible submodule of

 ξj ,ν ξ,ν . is the unique (j,ν)∈OW (ψ)+ Mβ and the intersection . (j,ν)∈OW (ψ)+ Wβ

ξ,ν irreducible submodule of . (j,ν)∈OW (ψ)+ Wβ . In the terminology discussed in Sect. 4.1.2, the irreducible modules in iv) are of large



ψ ψ ξ,ν ξ,ν discrete series type. The modules .Mβ = (j,ν) Mβ and .Wβ = (j,ν) Wβ are isomorphic to each other. The intersections of summands determine all submodules. Most of the results in Theorem C are stated in Lemma 4.10. The proof is completed on p. 119. ψ Theorem D concerns the non-abelian modules .Fn under integral parametrization. This case is more complicated than the three earlier ones. This is reflected in the complexity of the theorem.

1.1 Overview of Chaps. 2–4

7

ξ,ν

ξ,ν

The modules .Mn and .Wn may coincide in some or in all K-types. So a decomposition as in (iii) of Theorem C cannot hold. An analogous decomposition ξ,ν holds if we define other submodules .Vn to take the role of .M in the decomposiξ,ν tion. See (4.55). The definition of .Vn is not intrinsic, but it serves to give us some hold on the complications. ξ,ν The modules .Mn stay important, since their elements have a nice behavior in .a(t) ∈ A as .t ↓ 0. This is used in the construction of Poincaré series. Theorem D Let the character .ψ of .ZU (g) correspond to integral parametrization, and let .n = (, c, d) be a non-abelian parameter triple. Then (i) For each .(ξ, ν) ∈ OW (ψ)+ n the modules .Vn , .Wn and .Mn are reducible, and are in general non-isomorphic. These modules contain, with multiplicity one, the K-types .τph satisfying .|h− 2jξ | ≤ 3p. ξ,ν

ξ,ν

ξ,ν

ξ,ν

ξ  ,ν 

(ii) Let .X denote any of .V, .W or .M. If a K-type .τph occurs in .Xn and in .Xn ξ  ,ν 

for .(jξ , ν), (jξ  , ν  ) ∈ OW (ψ)+ , then .X = Xn;h,p . 

n  n;h,p ψ ξ,ν ξ,ν + Vn + Wn (iii) .Fn = . ⊕ (j,ν)∈OW (ψ)n (j,ν)∈OW (ψ)n ξ,ν

ψ ψ ψ ξ,ν Denote .Vn = Vn and define .Wn and .Mn similarly. We put (j,ν)∈OW (ψ)+ n     + .jr = max j : (j, ν) ∈ OW (ψ) and .jl = min j : (j, ν) ∈ OW (ψ)+ . ψ

ψ

(iv) The modules .Mn and .Vn intersect non-trivially in the following cases. + (a) If .OW (ψ)+ n = OW (ψ) , then .Mn = Vn . ψ ψ + (b) If .OW (ψ)+ .M n = OW (ψ) , and . > 0, then n;h,p = Vn;h,p for all K-types h .τp that satisfy the additional condition . h − 2jr ≤ 3p. ψ

ψ

+ .M = Vn;h,p for all K-types (c) If .OW (ψ)+ n = OW (ψ) , and . < 0, then n;h,p h .τp that satisfy the additional condition . h − 2jl ≤ 3p. ψ

ψ

ψ

ψ

(v) The modules .Mn and .Wn have a non-trivial intersection in the following cases. (a) Suppose that . > 0. If .m0 (jl ) ≥ 0 and .m0 (j ) < 0 for other .(j, ν) ∈ ψ ψ OW (ψ)+ , then .Mn;h,p = Wn;h,p for all K-types .τph satisfying . h − 2jl ≤ 3p, and . h − 2j > 3p for .(j, ν) ∈ OW (ψ)+ , .j = jl . (b) Suppose that . < 0. If .m0 (jr ) ≥ 0 and .m0 (j ) < 0 for other .(j, ν) ∈ ψ ψ OW (ψ)+ , then .Mn;h,p = Wn;h,p for all K-types .τph satisfying . h−2jr ≤ 3p and . h − 2j > 3p for .(j, ν) ∈ OW (ψ)+ , .j = jr . The structure of the non-abelian Fourier term modules under integral parametrization shows a lot of variation. It needs the long Sect. 4.4 to establish this. The proof of Theorem D is completed on p. 150.

8

1 Introduction ξ,ν

ξ,ν

ξ,ν

Unlike what we saw in the abelian case, the modules .Mn , .Wn and .Vn are not isomorphic to each other for many combinations of .(ξ, ν) and .n = (, c, d). ψ ψ ψ The structure of .Vn is the same as the structure of .Wβ ∼ = Mβ in the generic ψ

ψ

abelian case. The submodule structure of .Wn and .Mn shows a lot of variety, which we explicitly illustrate in Sect. 4.4.7. It turns out that all isomorphism classes of irreducible .(g, K)-modules except the finite-dimensional ones occur in some ξ,ν module .Wn . In Sect. 4.4.8 we draw conclusions from Theorems A–D, giving for each isomorphism type of irreducible .(g, K)-modules the ways it can be embedded in the ψ modules .FN . See Table 4.14 and Remark 4.36. We thank N. Wallach for drawing our attention to his paper [52], which motivated the discussion around Table 4.14. This gives an extension in the case of .SU(2, 1) of the results in Wallach’s paper, now including the non-abelian Fourier term modules. The paper [52] gives similar results for abelian Fourier terms in a much wider context. The results in the theorems above are discussed and proved under an assumption discussed in Sect. 2.4, which leads to lattices in N of a simple form (the lattices .σ in Definition 2.2). In this way, we need only a simple type of theta functions for the non-abelian Fourier term modules. In Sect. 5.4 we show that this assumption is not a genuine restriction for the structure of Fourier term modules. All lattices contained in N are isomorphic under conjugation in G to more general lattices .τ,σ with .τ ∈ C, .Im τ > 0, and .σ ∈ Z≥1 . Proposition 5.18 tells that the Fourier term modules based on the lattices .τ,σ are isomorphic as .(g, K)-modules to the Fourier term modules based on .σ = i,σ , describing explicit isomorphisms. All results based on the right translation by elements of K or by right differentiation by elements of the Lie algebra .g of G can be transferred to the more general lattices .τ,σ .

1.2 Automorphic Forms In Chap. 5 we apply our results to automorphic forms. Our main motivation to ψ study the modules .FN arose from the wish to get a grip on the full Fourier-Jacobi expansion of automorphic forms, with both abelian and non-abelian terms. In our ψ work [5] on Poincaré series we need to understand the modules .MN , that consist, in general, of functions with exponential growth. In Sect. 5.1 we recall the definition of automorphic forms, and we also consider automorphic forms with moderate exponential growth, of which Poincaré series are examples. Automorphic forms generate .(g, K)-modules. This leads to intertwining operaψ tors .FcN from the module of automorphic forms to the modules .FN , where .c denotes a cusp of the discrete group .. Since the operators .FcN are intertwining operators, the algebraic structure of the module of automorphic forms under consideration and the ψ structure of .FN put restrictions on the Fourier terms that can be non-zero. For this purpose, the information in Theorems A–D is important. Furthermore, the growth of

1.2 Automorphic Forms

9

automorphic forms at the cusps is inherited by the Fourier terms, which puts further conditions on the Fourier terms. For automorphic forms, with at most polynomial ψ growth at the cusps, this implies that all Fourier terms .FcN f are in .WN if .N = N0 . If .N corresponds to a non-trivial character this gives the usual Whittaker model. For non-abelian .N we get a generalized Whittaker model. The structure of modules of automorphic forms as a .(g, K)-module has an influence on the form of the Fourier expansion of the automorphic forms occurring in it. Section 5.3.1 shows this for modules of square integrable automorphic forms. We compare the results with Ishikawa’s expansions in [22, 23]. We see that abelian Fourier terms with a non-trivial character are zero for modules of holomorphic and antiholomorphic discrete series type, and that there are restrictions on the parameters . and d in .n = (, c, d). Section 5.3.2 considers modules of automorphic forms for which only one Fourier term is allowed to have exponential growth, and discusses the interaction between, on the one hand, the form of the Fourier expansions, and, on the other hand, the structure of the module as a .(g, K)-module. In Sect. 5.1.2 we identify the spaces of automorphic forms corresponding to scalar-valued holomorphic automorphic forms on the symmetric space .G/K, and give the form of the Fourier expansions of these functions. Eisenstein series and Poincaré series occur in meromorphic families depending on the spectral parameter .ν. In such a situation it is convenient not to focus on modules of automorphic forms, but to use the actual coefficients expressing the ψ Fourier terms in a suitable basis of .FN . See Sect. 5.2. In Sects. 5.1–5.3 we work under an assumption on the discrete group . so that its Fourier terms determine functions on .σ \N for a standard lattice .σ ; see Sect. 2.4. In Sect. 5.4 we show that this simplifying assumption can be removed. In Sect. 5.4.3 we state the results that are valid for general lattices and general cofinite discrete subgroups with cusps, and indicate their proofs. In a sequel to this paper we use results from this monograph to extend the results of [32] in the case of .G = SU(2, 1). We study Poincare’ series for .SU(2, 1), attached to characters and to Stone-von Neumann representations of N. We show that the special values of these series together with their residues span all automorphic forms in .L2 (\G). See [5].

Chapter 2

The Lie Group SU(2,1) and Subgroups

The aim of this work is to understand the modules involved in the Fourier expansions of functions on .\SU(2, 1) for cofinite discrete subgroups .. Section 2.4 discusses discrete subgroups and Fourier expansions. This is a preparatory chapter. We choose a realization of the Lie group .SU(2, 1) inside .SL3 (C). The Iwasawa decomposition NAK of the group .SU(2, 1) involves a non-abelian unipotent subgroup N and a maximal compact subgroup K. The representations of these subgroups are considered, in Sect. 2.2 for K, and in Sect. 2.3 for N . All this is known; we give a summary, as a preparation for the later chapters.

2.1 Realization of the Group SU(2, 1) We choose a realization of the group .SU(2, 1), and we describe the Iwasawa decomposition, the Bruhat decomposition, and the symmetric space. The Lie group .SU(2, 1) is the group of matrices .g ∈ SL3 (C) that preserve a given hermitian form of signature .(2, 1). Different hermitian forms give isomorphic realizations of .SU(2, 1) as a real Lie group. We use the hermitian form .(x, y) = y¯ t I2,1 x, with ⎞ 10 0 = ⎝0 1 0 ⎠ , 0 0 −1 ⎛

I2,1

.

(2.1)

which leads to   G := SU(2, 1) = g ∈ SL3 (C) : g¯ t I2,1 g = I2,1 .

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. W. Bruggeman, R. J. Miatello, Representations of SU(2,1) in Fourier Term Modules, Lecture Notes in Mathematics 2340, https://doi.org/10.1007/978-3-031-43192-0_2

(2.2)

11

12

2 The Lie Group SU(2,1)

This defines G as a semi-simple Lie group of dimension 8 with real rank one. It is the same realization as used by Ishikawa [22]. We choose the Iwasawa decomposition .G = NAK, with the maximal compact subgroup   K = k(η, α, β) : η, α, β ∈ C, |η| = 1, |α|2 + |β|2 = 1 , ⎛ ⎞ ηα ηβ 0 . k(η, α, β) = k(−η, −α, −β) = ⎝−ηβ¯ ηα¯ 0 ⎠ ; 0 0 η−2

(2.3)

the unipotent subgroup   N = n(b, r) : b ∈ C, r ∈ R , ⎛ 1 + ir − . ⎜ n(b, r) = n(Re b, Im b, r) = ⎝ −b¯ ir −

|b|2 2

|b|2 2

b −ir + 1 b¯

|b|2 2 |b|2 2

b 1 − ir +

⎞ ⎟ ⎠;

(2.4)

and the connected component of I in an .R-split torus ⎛ .

  A = a(t) : t > 0 ,

t+t −1 ⎜ 2

a(t) = ⎝ 0

0 1 0

⎞

t−t −1 2 ⎟

0 ⎠.

(2.5)

 ¯ 1) . n(b, r)n(b1 , r1 ) = n b + b1 , r + r1 + Im (bb

(2.6)

t−t −1 2

t+t −1 2

The group product for N is .

The commutative group AM with   M = m(ζ ) : |ζ | = 1 ⊂ K , ⎛ ⎞ ζ 0 0 . m(ζ ) = ⎝ 0 ζ −2 0⎠ = k(ζ −1/2 , ζ 3/2 , 0) , 0 0 ζ

(2.7)

normalizes N :

a(t)m(ζ )n(b, r)m(ζ )−1 a(t)−1 = n ζ 3 tb, t 2 r) .

.

(2.8)

  The center of G is .Z(G) = m(ω) : ω3 = 1 . At some places it is convenient to use .n(x, y, r) = n(x + iy, r), with three real parameters.

2.1 Realization of the Group SU(2, 1)

13

Other Realizations The realization in (2.2) has the advantage that K has the simple ⎞ ⎛ ∗∗0 form . ⎝∗ ∗ 0⎠ . Isomorphic realizations are obtained by replacing the matrix 00∗ ¯ t I2,1 U or by .J = −U¯ t I2,1 U with .U ∈ GL3 (C). Then we .I2,1 in (2.1) by .J = U obtain the isomorphic Lie group .U −1 GU. For instance, with ⎛

UT

.

⎞ 1 0 −i 1 = √ ⎝0 2 0 ⎠ 2 10 i

(2.9)

we have ⎞ 0 0 −i = ⎝0 2 0 ⎠ , i 0 0 ⎛

U¯ T I21 UT

.

(2.10)

and in the realization .UT−1 GUT of .SU(2, 1) the subgroup .UT−1 N AUT consists of upper triangular matrices, but the group .UT−1 KUT has a more complicated description. In Sect. 2.4 we will discuss more realizations. Rational Structure We can view G as the group .GR of real points of an algebraic group .G over .Q. This can be done by viewing .SL3 as an algebraic group over .Q(i), and obtaining .G as an algebraic subgroup of the Weil restriction .RQ(i)/Q SL3 . See [54, §1.3]. The group of rational points .GQ can be identified with the subgroup of .g ∈ G that have matrix coefficients in .Q(i). With other realizations of .G, we can follow the same approach. This may lead to other rational structures.

2.1.1 Symmetric Space The symmetric space corresponding to .SU(2, 1) is the quotient .G/K. We use the realization as the upper half-plane model:   X = (z, u) ∈ C2 : |u|2 < Im z .

.

(2.11)

As an analytic variety it is diffeomorphic to .NA, which is visible in the left action

 n(b, r)a(t) · (z, u) = t 2 z + 2tbu + 2r + i|b|2 , tu + i b¯ ,

.

(2.12)

14

2 The Lie Group SU(2,1)

 which satisfies .n(b, r)a(t) · (i, 0) = t 2 i + 2r + i|b|2 , i b¯ . The group K leaves the point .(i, 0) fixed. To describe the action of .g ∈ G given in matrix form, one uses .UT as in (2.9) and writes UT−1 gUT

.

⎞ ⎛   a11 a12 b1 Ab = = ⎝a21 a22 b2 ⎠ c d c1 c2 d

(2.13)

with .A ∈ M2 (C), b a column vector, c a row vector, and d a complex number. Then

t

−1

 g · (z, u) = c(z, u)t + d A(z, u)t + b  −1 

a11 z + a12 u + b1 , a21 z + a22 u + b2 . = c1 z + c2 u + d

.

(2.14) The action of general elements .k ∈ K is complicated, except for elements of the form .m(ζ ) or .w ⎛ ⎞ −1 0 0 w = ⎝ 0 −1 0⎠ = k(1, −1, 0) , 0 0 1 .  −1 −iu 

 w · (z, u) = , , m(ζ ) · (z, u) = z, ζ −3 u . z z

(2.15)

The space .X inherits the complex structure of .C2 . The action of G preserves this complex structure. Boundary The boundary of .X can be described as   ∂X = {∞}  (z, u) ∈ C2 ; |u|2 = Im z ,

.

(2.16)

where .∞ is the limit of .(z, u) as .Im z → ∞ while .Re z and u stay bounded. The point .∞ ∈ ∂X is fixed by the parabolic subgroup .NAM ⊂ G. We can write .∂X as the disjoint union 

∂X = {∞  N · 0, 0) .

.

(2.17)

We have .(0, 0) = w · ∞. Relation (2.17) leads to the Bruhat decomposition G = NAM  NwNAM = NAM  N wAMN .

.

(2.18)

2.2 Maximal Compact Subgroup

15

With the notation



 h(c) = a |c| m c/|c|

c ∈ C∗ ,

.

(2.19)

each element of G can be written uniquely as either .g = nh(c), .n ∈ N and .c ∈ C∗ , or .g = n1 w h(c)n2 with .n1 , n2 ∈ N and .c ∈ C∗ . To go from g in the big cell .NwMNA of the Bruhat decomposition to the Iwasawa decomposition we use the following lemma. Lemma 2.1 For .b ∈ C, .r ∈ R, .t > 0, and .c ∈ C∗ : w h(c)n(b, r)a(t) = n(b , r )a(t )k , with

.

b =

−cb , c¯2 D

r =

−r , |c|2 |D|2

t =

t , |c| |D|

D = 2ir + t 2 + |b|2 , ⎛ ⎞ c(D¯ − 2t 2 )/|c| |D| −2ctb/|c| |D| 0 ⎠ ¯ D¯ k = ⎝ 0 2ct ¯ b/c c(D ¯ − 2t 2 )/cD¯ ¯ 0 0 cD/|c| |D| Proof We find this relation by applying .w h(c) n(b, r) a(t) to .(i, 0) ∈ X. That gives the values of .b , .r and .t , and then of .k by a computation. We carried out the computation in [6, §1b]. 

2.2 Maximal Compact Subgroup We discuss the structure of the maximal compact Lie subgroup .K ⊂ G chosen in (2.3), and describe its irreducible representations. We give explicit realizations of these representations in polynomial functions on K. We have .k(η, α, β) = k(η, 1, 0)k(1, α, β) = k(1, α, β)k(η, 1, 0), in the notation of (2.3). The group K has the subgroups   Z(K) = k(η, 1, 0) : |η| = 1 ,

.

  K0 = k(1, α, β) : |α|2 + |β|2 = 1 . (2.20)

Z(K) is the center of K; it is isomorphic to .U(1). The group .K0 is isomorphic to SU(2). Since .k(−1, 1, 0) = k(1, −1, 0) the intersection .Z(K) ∩ K0 is a group of order 2. Hence we have the isomorphisms

. .

K ∼ =

.

  U(1) × SU(2) Z/2Z ,

(2.21)

16

2 The Lie Group SU(2,1)

by ⎛ ⎞ ⎞ ⎛ ⎞ ⎛  η0 0 ηα ηβ 0 α β0  ¯ α¯ 0⎠ → ⎝−ηβ¯ ηα¯ 0 ⎠ , . ⎝ 0 η 0 ⎠ , ⎝−β 0 0 η−2 0 01 0 0 η−2 with kernel generated by ⎛ ⎞ ⎛ ⎞  −1 0 0 −1 0 0  , . ⎝ 0 −1 0⎠ , ⎝ 0 −1 0⎠ 0 0 1 0 0 1 where .±1 is embedded diagonally in the product. See (1), (2) on p. 185 of [49]. The presence of .U(1) shows that K is not simply connected. It has an  infinite covering. The characters of K are induced by characters of the center . k(η, 1, 0) : |η| =  1 of K that are trivial on .K0 . These characters are of the form ⎞ ηα ηβ 0 ¯ ηα¯ 0 ⎠ → η−2j , .ξj : k(η, α, β) = ⎝−η β 0 0 η−2 ⎛

(2.22)

with .j ∈ Z, and extend to characters of K. On the subgroup .M ⊂ K these characters take the form ξj : m(ζ ) = diag(ζ, ζ −2 , ζ ) → ζ j .

.

If a character .ξ of K is given, we use .jξ ∈ Z for the corresponding integral parameter.

2.2.1 Irreducible Representations The isomorphism classes .τp of irreducible representations of .K0 are parametrized by their dimension .p+1, with .p ∈ Z≥0 . The irreducible representations of the center of K are the characters parametrized by .h ∈ Z. With the product description (2.21) the isomorphism classes .τph of irreducible representations of K are parametrized by 2 .(h, p) ∈ Z , .p ≥ 0, .h ≡ p mod 2 :  h .τp

ζ 1/2 u 0 0 ζ −1



= (ζ 1/2 )−h τp (u)

|ζ | = 1 , u ∈ SU(2) .

(2.23)

We use here that each .k ∈ K can be written as .k = k(ζ 1/2 , α, β) = k(−ζ 1/2 , −α, −β).

2.2 Maximal Compact Subgroup

17

Realizations of Irreducible Representations of .SU(2) To realize the representations of .K0 ∼ = SU(2) in .C ∞ (K) provided with the action by right translation we start p with polynomial functions . r,q on .K0 ∼ = SU(2) determined by the identity (ax + c)(p−q)/2 (bx + d)(p+q)/2  p a b  . = r,q x (p−r)/2 , c d r

(2.24)

where r and q are in .p + 2Z, with absolute value at most .p. See [46, Chapter 6], p p where the function .tr,q in (2), Section 6.3.2, is a multiple of . r,q . The generating function shows that right translation by each element of .SU(2)  p preserves the space . r,q C r,q , where we let both r and q run from .−p to p in steps of 2. Right translation        ab aη b/η ab η 0 = .

→ cd cη d/η cd 0 η−1 multiplies . r,q by .η−q . With use of the generating function we can also check   η 0 p multiplies each . r,q by .η−r . Since left and that left translation by . 0 η−1 the right translations commute, the action of .SU(2) by right translations preserves  p eigenspaces for left translation. Hence we get p invariant subspaces . q C r,q . There are .p + 1 realizations of .τp in .C ∞ (K). In Sect. 2.2.2 we give the Lie algebra action on these spaces, which shows their irreducibility and the isomorphism between them. p

Realizations of Irreducible Representations of K In view of (2.23) we put .

h

p r,q

 1/2  ζ u 0 p = ζ −h/2 r,q (u) . 0 ζ −1

(2.25)

This is well defined for .h ≡ p mod 2, independently of the choice of .ζ 1/2 . We need .p ≡ r ≡ q mod 2 and .|r|, |q| ≤ p. ⎛ ⎞ η 0 0 p Left translation by .⎝0 η−1 0⎠ acts on . h r,q as multiplication by .η−r . Right 0 0 1  p translation by elements of K preserves the spaces . q C h r,q . This gives .p + 1 h of .τ h in .C ∞ (K). The function . 2j 0 is the character .ξ of K realizations .τr,p j p 0,0 in (2.22).

18

2 The Lie Group SU(2,1)

Table 2.1 Multiplication relations for polynomial functions on K

For η ≡ 1 mod 2, ε, ζ ∈ {1, −1} : η

p

1ε,ζ h r,q = Aε,ζ (p, r)

h+η

+ Bε,ζ (p, r, q)

h+η

Aε,ζ (p, r) = Bε,ζ (p, r, q) =

p + εr + 2 , 2(p + 1) 

p+1

r+ε,q+ζ p−1

r+ε,q+ζ ,

ε(ζp−q) 2(p+1)

if εr ≤ p − 2 ,

0

if εr = p .

p

Polynomial Functions The functions . h r,q are polynomial functions in the matrix elements of .k ∈ K. For .k = k(η, α, β) as in (2.3) we have, with .j ∈ Z and .h ≡ 1 mod 2: 2j

00,0 (k) = η−2j ,

h

11,1 (k) = η−h α¯ ,

.

h

h

1−1,1 (k) = η−h β ,

h

11,−1 (k) = η−h β¯ ,

(2.26)

1−1,−1 (k) = η−h α .

Manipulations with the generating function in (2.24) lead to the multiplication relations in Table 2.1. p

2 The polynomial functions . h r,q form  an orthogonal basis of .L (K). We normalize the Haar measure on K so that . dk = 1. We will not need an explicit K h p    formula for . r,q K , but will use the relation

p! p p  h r,q 2K = p+r  p−r   h p,q 2K . ! ! 2 2

.

(2.27)

This can be checked by use of the relation

.

L(Z21 )ϕ1 , ϕ2

 K



+ ϕ1 , L(Z12 )ϕ2 K = 0 ,

where .Z12 = W1 − iW2 and .Z21 = W1 + iW2 , in the complexified Lie algebra kc = C ⊗R k. See Table 2.2 below.

.

2.2 Maximal Compact Subgroup

19

Table 2.2 Actions of .kc by left and right differentiation p

p

p

R(Ci ) h r,q = L(Ci ) h r,q = −ih h r,q , p

R(W0 ) h r,q = −iq

h

p

r,q ,

p

p

L(W0 ) h r,q = −ir

p

L(Z12 ) h r,q

p

r,q ,

p

R(Z21 ) h r,q = (q − p) h r,q+2 ,  p (r − p − 2) h r−2,q p L(Z21 ) h r,q = 0 p R(Z12 ) h r,q

h

if r ≥ 2 − p , if r = −p ,

h p r,q−2 ,

= (q + p)  p (r + p + 2) h r+2,q = 0

if r ≤ p − 2 , if r = p .

2.2.2 Lie Algebra A basis of the real Lie algebra .k of K is ⎞ i 0 0 Ci = ⎝0 i 0 ⎠ , 0 0 −2i . ⎞ ⎛ 0 10 W1 = ⎝−1 0 0⎠ , 0 00 ⎛

⎞ i 0 0 W0 = ⎝0 −i 0⎠ , 0 0 0 ⎞ ⎛ 0i 0 W2 = ⎝ i 0 0⎠ . 000 ⎛

(2.28)

The element .Ci spans the Lie algebra of the center of K, the three remaining elements span the Lie algebra of .K0 . The exponentials are exp(tCi ) = k(eit , 1, 0) , .

exp(tW0 ) = k(1, eit , 0) ,

exp(tW1 ) = k(1, cos t, sin t) ,

exp(tW2 ) = k(1, cos t, i sin t) .

(2.29)

The element ⎞ i 0 0 1 3 .Hi = W0 − Ci = ⎝0 −2i 0⎠ ∈ k 2 2 0 0 i ⎛

(2.30)

spans the Lie algebra of M, and .

exp(tHi ) = m(eit ) .

(2.31)

20

2 The Lie Group SU(2,1)

Actions of the Lie Algebra on the Polynomial Functions The action of .X ∈ k by right differentiation of functions in .C ∞ (K) is given by

 Xf (g) = ∂t f g exp(tX) |t=0 .

.

(2.32)

This is extended .C-linearly to an action of .kc . We write .R(X)f for .Xf in discussions where other actions by differentiation occur as well, for instance the action by left differentiation. The left differentiation is the right action of .k given by

 L(X)f (g) = ∂t f exp(tX)g t=0

.

(g ∈ G, X ∈ k) .

(2.33)

Table 2.2 gives the left and right actions of the Lie algebra on basis elements. Some of these relations are easily seen from the definition, for instance we have p seen that .exp(tW0 ) acts on . h r,q under left translation as multiplication by .e−irt , and under right translation as multiplication by .e−iqt . This gives the actions of .W0 . In (2.25) we see that the center of K acts by .k(η, 1, 0) → ζ −h ; this leads to the action of .L(Ci ) = R(Ci ). The actions of .Z12 and .Z21 take more computations, carried out in [6, §4b]. In the formulas for .R(Z21 ) and .R(Z12 ) we have the factor .q ∓ p, which becomes zero if q has the value .±p for which . q2 ± 1 threatens to be outside the range of q. For the left differentiation these values of q have to be treated separately. Parameters and We have seen that h is determined by the eigenvalue  Eigenvalues h . The parameter p is determined by the action of the Casimir −ih of .Ci in . r τr,p element of the Lie algebra of .K0 :

.

CK = W02 + W12 + W22 = W02 − 2iW0 + Z12 Z21

.

(2.34)

 h as multiplication by .−p(p + 2). It acts on . r τr,p

2.3 Unipotent Subgroup The group .SU(2, 1) is the smallest semisimple Lie group of rank one with unipotent subgroups that are not commutative. The representation theory of the unipotent subgroup N is important for the Fourier expansion of automorphic forms and for Poincaré series. Our main aim in this section is to give an orthonormal basis of .L2 (σ \N) for a class of standard lattices .{σ : σ ∈ Z≥1 }. This requires a discussion of the Stone-von Neumann representation and of its realizations by means of theta functions. The material in this section is known in more generality. See Thangavelu [45] and Igusa [21, Chap I].

2.3 Unipotent Subgroup

21

The group N in (2.4) with multiplication as in (2.6) is a realization of the Heisenberg group. It fits into the exact sequence 1 −→ Z(N ) −→ N −→ R2 −→ 0 ,

.

(2.35)

  where .Z(N ) = n(0, r) : r ∈ R is the center of .N. The homomorphism .N → R2 is given by .n(x, y, r) → (x, y). The group .Aut(N ) of continuous automorphisms of N is isomorphic to the semi-direct product .GL2 (R) R2 .Here .R2 corresponds to the group of interior ab ∈ GL2 (R) corresponds to the outer automorautomorphisms. Furthermore, . cd phism n(x, y, r) → n(ax + by, cx + dy, (ad − bc)r) .

.

(2.36)

Conjugation by .a(t)m(eih ) ∈ AM corresponds to the automorphism of N given by the matrix   t cos 3h −t sin 3h . . t sin 3h t cos 3h Lie Algebra A basis of the real Lie algebra .n of N is ⎛i

⎞ 0 − 2i X0 = ⎝ 0 0 0 ⎠ , i i 2 0 −2 . ⎞ ⎛ 0i 0 X2 = ⎝ i 0 −i ⎠ . 0i 0 2

⎞ 0 10 X1 = ⎝−1 0 1⎠ , 0 10 ⎛

(2.37)

The sole non-zero commutators of these elements are  .

X1 , X2 ] = −[X2 , X1 ] = 4X0 .

(2.38)

The exponential map gives exp(tX0 ) = n(0, t/2) , .

See [6, §3c].

exp(tX2 ) = n(it, 0) .

exp(tX1 ) = n(t, 0) ,

(2.39)

22

2 The Lie Group SU(2,1)

Comparison We use the multiplication relation on N given in (2.6). In [45] Thangavelu uses the multiplication relation   1 [x, y, t] [u, v, s] = x + u, y + v, t + s + (uy − vx) . 2

.

(2.40)

The isomorphism T : n(x, y, r) → [x, 2y, −r]

.

(2.41)

relates both realizations of the Heisenberg group.

2.3.1 Characters and Stone-von Neumann Representation There are two types of unitary irreducible representations of N, namely the unitary characters, and the Stone-von Neumann representations, which are infinite dimensional. Characters of N are trivial on the center .Z(N ), so they are characters of .R2 . They have the form ¯

χβ : n(b, r) → e2π iRe (βb)

.

(2.42)

with .β ∈ C. All other irreducible unitary representations of N have infinite dimension. See [45, Theorem 1.2.4]. The center .Z(N) acts as multiplication by a non-trivial character of .Z(N). For each non-trivial central character the corresponding Stone-von Neumann representation represents the unique isomorphism class of irreducible representations with that central character. Schrödinger Representation The Schrödinger representation is a realization of the Stone-von Neumann representation in .L2 (R). It is characterized by a non-trivial character n(0, r) → eiλr

.

of .Z(N ), parametrized by .λ ∈ R∗ .

 The Schrödinger representation .πλ n(x, y, r) applied to .ϕ in the space .S(R) ⊂ L2 (R) of Schwartz functions on .R is given by

 πλ n(x, y, r) ϕ(ξ ) = eiλ(r−2ξ x−xy) ϕ(ξ + y) .

.

(2.43)

The Schwartz space is invariant under these transformations. The operators .πλ (n) extend to .L2 (R), and determine .πλ as a unitary representation of N in .L2 (R).

2.3 Unipotent Subgroup

23

Comparison Thangavelu [45, (1.2.1)] uses the representation πμT [x, y, r]ϕ(ξ ) = eiμ(r+ξ x+xy/2) ϕ(ξ + y) .

.

(2.44)

With .U ϕ(ξ ) = ϕ(ξ/2) and the isomorphism T in (2.41) we have



 T T (n(x, y, r) U ϕ = U πλ n(x, y, r) ϕ . π−λ

.

(2.45)

This shows that both representations are equivalent. Automorphisms For each automorphism .A ∈ Aut(N ) the representation .n → πλ (An) is equivalent to some Schrödinger representation .πλ . So there exists a unitary map .UA : L2 (R) → L2 (R) and a number .λ ∈ R∗ such that UA πλ (An) = πλ (n)UA .

.

 If the automorphism A corresponds to

.

t 0 0t

(2.46)

 ∈ GL2 (R), then we can take

UA ϕ(ξ ) = tϕ(tξ ) and .λ = t 2 λ.

.

Lie Algebra Action Application of (2.39) gives the derived representation: dπλ (X0 )ϕ = .

i λϕ, 2

dπλ (X1 )ϕ(ξ ) = −2iλ ξ ϕ(ξ ) ,

(2.47)

dπλ (X2 )ϕ = ϕ . This derived action is well defined if .ϕ is a Schwartz function.

2.3.2 Theta Functions The Stone-von Neumann representation can be realized in spaces generated by theta functions on N modulo a lattice, i.e., a discrete subgroup such that the quotient .\N is compact. Definition 2.2 We denote by .σ the lattice generated by n(1, 0, 0) ,

.

n(0, 1, 0) ,

 So .σ = n(b, r) : b ∈ Z[i] , r ∈

n(0, 0, 2/σ )

(σ ∈ Z≥1 ) .

(2.48)



2 σZ

. We call the lattices .σ standard lattices.

Since the commutator of the generators .n(1, 0, 0) and .n(0, 1, 0) is .n(0, 0, 2), we need the restriction that .σ ∈ Z≥1 . Any lattice is isomorphic to a standard lattice by an element in .Aut(N ).

24

2 The Lie Group SU(2,1)

Let .σ ∈ Z≥1 and .λ ∈ R=0 . The central character of the Schrödinger representation .πλ is .n(0, 0, r) → eiλr . To have it trivial on .σ ∩ Z(N ) we take σ .λ = 2π  with . ∈ 2 Z=0 . The space of Schwartz functions .S(R) is dual to the space .S (R) of tempered distributions. Under this duality the Schrödinger representation .π2π  on .S(R) corresponds to .π−2π  on .S (R). The relations

 π−2π  n(1, 0, 0) δa = e4π ia δa

 π−2π  n(0, 1, 0) δa = δa−1 .

 π−2π  n(0, 0, 2/σ ) δa = e−4π i/σ δa

(2.49)

imply for .c ∈ Z mod 2 that the distribution μ,c =



.

δk+c/2

(2.50)

k∈Z

is invariant under .π2π  (σ ). This motivates the definition   ,c (ϕ)(n) = ,c (ϕ; n) = π2π  (n)ϕ, μ,c

.

(2.51)

as a function on .σ \N . Writing this out explicitly gives for .ϕ ∈ S(R) the following theta function: 

  2π i r−x(c/+2k+y)  c ϕ .,c ϕ; n(x, y, r) = e (2.52) +k+y . 2 k∈Z

The decay of Schwartz functions ensures absolute convergence of the series and of all its derivatives with respect to the coordinates .x, y and .r. It transforms via the central character determined by .λ = 2π , and some computations show that it is left-invariant under multiplication by elements of .σ . Actually, we have

 ,c ϕ; n(1/2, 0, 0)n = e−π ic/ ,c (ϕ; n) , .

 ,c ϕ; n(0, 1/2, 0)n = ,c+1 (ϕ; n) .

(2.53)

 So we have .,c (ϕ) ∈ C ∞ σ \N . Since .n(x, y, r)n(t, 0, 0) = n(x + t, y, r − ty) we obtain





 X1 ,c ϕ; n(x, y, r) = ∂x − y∂r ,c ϕ; n(x, y, r) . 

= −4π i,c ϕ1 ; n(x, y, r)

(2.54)

2.3 Unipotent Subgroup

25

with .ϕ1 (ξ ) = ξ ϕ(ξ ). Proceeding in a similar way we get



 X2 ,c ϕ; n(x, y, r) = ,c ϕ ; n(x, y, r) , .

X0 ,c (ϕ; n) = π i,c (ϕ; n) .

(2.55)

Comparison with (2.47) shows that .ϕ → ,c (ϕ) induces for each .c ∈ Z, .0 ≤ c < 2||, an intertwining operator between .π2π  and the subspace spanned by theta functions .,c (ϕ), inducing a unitary injection .L2 (R) → L2 (σ \N). Thus we have .2|| orthogonal realizations of .π2π  in the space of functions on .σ \N. See [6, §5c]. Let us choose the Haar measure dn on N as .dn = dx dy dr for .n = n(x, y, r), with the Lebesgue measure on .R in each of the coordinates. With this choice, .σ has covolume . σ2 . By taking apart the summations in (2.52) we find 

 ,c (ϕ),  ,c (ψ)

.

σ \N

=

2 σ

(ϕ, ψ)R

0

if  =  , c ≡ c mod 2 , otherwise .

(2.56)

So the .2|| realizations of the Stone-von Neumann representation are mutually

2|| → L2 (σ \N) induced orthogonal, and we have injective linear maps . L2 (R) by  (ϕc )c mod 2 →

.

σ 2



,c (ϕc ) .

(2.57)

c mod 2

The image is contained in the subspace .L2 (σ \N) determined by the central character corresponding to .. It is known that this map is a unitary isomorphism; below we indicate an argument. The conclusion is that the orthogonal complement of .L2 (σ \N)0 in .L2 (σ \N) is described by theta functions. Argument for Unitarity Let .F ∈ C ∞ (σ \N) be orthogonal to .,c (ϕ) for all c and .ϕ. To see that this implies that F vanishes we consider

 f (x, y) = e2π ixy F n(x, y, 0) .

.

(2.58)

Then .f (x + 1, y) = f (x, y), and .f (x, y + 1) = e4π ix f (x, y). The assumption implies that 

1

0=



1

.



 F n(x, y, 0) ,c (ϕ) n(x, y, 0) dy dx

x=0 y=0

 =

1



1

x=0 y=0

f (x, y) e−2π ixy

 k

e2π ix(c/+2k+y) ϕ c/2 + k + y) dy dx

26

2 The Lie Group SU(2,1)



1

=





1

f (x, y) x=0 y=0

 =

1

 =

1

x=0

k



x=0 k

e2π ix(c/+2k) ϕ c/2 + k + y) dy dx

 f (x, y + k) e2π icx ϕ c/2 + k + y dy dx

1 y=0

e2π icx



∞

f (x, y) ϕ(y + c/2) dy dx .

y=−∞

This holds for all .c ∈ Z and all Schwartz functions .ϕ. In particular, replacing .ϕ by a translate depending on c, we get for all .c ∈ Z and all Schwartz functions .ϕ  0 =

1

e2π icx hϕ (x) dx ,

.

 hϕ (x) =

x=0 ∞

f (x, y) ϕ(y) dy . y=−∞

The function .hϕ is continuous, 1-periodic, and all its Fourier coefficients vanish. So hϕ (x) = 0 for all .x ∈ R. The Schwartz functions .ϕ are dense in .L2 (R) so for each .x ∈ R the function .y → f (x, y) is zero. Hence F vanishes. .

Hermite Basis Normalized Hermite functions provide us with a suitable basis of Schwartz functions to use in the theta functions. See [45, §1.4]. The Hermite polynomials .Hm are determined by the identity e−ξ Hm (ξ ) = (−1)m ∂ξm e−ξ . 2

2

.

The normalized Hermite functions are the following Schwartz functions: h,m (ξ ) = 2(1−m)/2 ||1/4 (m!)−1/2 Hm

.

  2 4π|| ξ e−2π||ξ

(2.59)

for . ∈ R=0 and .m ∈ Z≥0 . For any given . ∈ R=0

.

h,m , h,m

 R

= δm,m .

(2.60)

The Hermite polynomials satisfy the relation .Hm+1 = 2ξ Hm − 2m Hm−1 . This leads to relations between Hermite function .h,m , .h,m+1 , and .h,m−1 . The derived Schrödinger representation is described in Table 2.4. Since .ϕ → ,c (ϕ) is an intertwining operator for the action of N, we have the corresponding relations for theta functions built with normalized Hermite functions. In the direct sum decomposition L2 (σ \N) =

L2 (σ \N )

.

∈(σ/2)Z

2.3 Unipotent Subgroup

27

according to the central character, we have orthonormal bases  .

σ/2 ,c (h,m ) : c mod 2, m ∈ Z≥0



(2.61)

for each summand with . = 0. The character .χβ is trivial on .σ for .β ∈ Z[i]. An orthonormal basis of .L2 (σ \N)0 is  .

 σ/2 χβ : β ∈ Z[i] .

(2.62)

√ The factor . σ/2 is caused by the choice to use .dn = dx dy dr as the Haar measure on .σ \N , with .n = n(x, y, r). Automorphisms of N and Theta Functions The map .n(b, r) → n(−ib, r) is an outer automorphism of N leaving invariant the lattice .σ ⊂ N. It is given by .n → m(i)nm(−i). So the inclusions .σ ⊂ N ⊂ NAM provide us with a group of automorphisms of .σ . Proposition 2.3 Let .m ∈ Z≥0 , .σ ∈ Z, and . ∈ σ2 Z=0 . The automorphism .n(b, r) → n(ib, r) of N induces the linear transformation determined by

 .,c (h,m ) n(−ib, r) =



m 2||−1 

 − i sign () eπ icc / ,c (h,m ) n(b, r) √ 2|| c =0

(2.63)  in the  .2||-dimensional space of theta functions with basis . ,c (h,m ) : 0 ≤ c < 2|| . We give the proof in two lemmas. Lemma 2.4 Let . ∈

σ 2 Z=0 ,

and .c ∈ {0, 1, . . . , 2|| − 1}. For each .ϕ ∈ S(R)

2||−1 





 1 eπ icc / ,c f ϕ n(b, r) , ,c (ϕ) n(−ib, r) = √ 2||

.

(2.64)

c =0

where

.

  f ϕ (ξ ) = 2|| ϕ(2ξ ˆ ).

(2.65)

 Proof We write out .,c (ϕ) n(y − ix, r) with (2.52) and apply the following consequence of the Poisson summation formula  .

k∈Z

e2π iβk ϕ(α + k) = e−2π iαβ

 k∈Z

e2π iαk ϕ(k ˆ − β) .

(2.66)

28

2 The Lie Group SU(2,1)

That leads to the relation 

 ,c (ϕ) n(y − ix, r) = e2π i(r−xy) e2π i(c/2−x)k ϕ(k ˆ + 2y)

.

k∈Z

= e2π i(r−xy)

2||−1  



e2π i(c/2−x)(c +2k)

c =0 κ∈Z



 ϕˆ c + 2(k + y)

2||−1 



 1 eπ icc / ,c f ϕ n(x + iy, r) , =√ 2|| c =0



which ends the proof. Lemma 2.5 For . ∈

σ 2 Z=0

and .m ∈ Z

f h,m =

.



m − i sign () h,m .

(2.67)

Proof The case .m = 0 can be checked by an explicit computation, based on Table 2.3, p. 28. We start with the third relation in Table 2.4, p. 29, and take the Fourier transforms of all terms:  √  √ ˆ ,m (ξ ) = 2π || mhˆ ,m−1 (ξ ) − m − 1hˆ ,m+1 (ξ ) . .2π iξ h Replacing .ξ by .2ξ we formulate this in terms of .f h,m :   √ √ √ 4π i 2||f h,m = 2|| π mf h,m−1 − m − 1f h,m+1 .

.

Using (2.67) for .h,m and .h,m−1 , and taking the second relation in Table 2.4 into account, we see that (2.67) is valid for .h,m+1 as well. 

Table 2.3 Some normalized Hermite functions

m

.h,m (ξ )

0

.

1 2 3

√ 2 2||1/4 e−2π ||ξ √ 3/4 ξ e−2π ||ξ 2 .4 π||

1/4 8π||ξ 2 − 1) e−2π ||ξ 2 .|| !

 2π 3/4 8π||ξ 2 − 3 ξ e−2π ||ξ 2 .2 3 ||

2.4 Discrete Subgroups

29

Table 2.4 Derived Schrödinger representation on Hermite functions dπ2π  (X0 )h,m = π i h,m , dπ2π  (X1 )h,m = −4π i ξ h,m     m m+1 h,m−1 + h,m+1 , = −2i sign () π || 2 2     m m+1 dπ2π  (X2 )h,m = h,m = 2 π || h,m−1 − h,m+1 . 2 2

2.4 Discrete Subgroups For the study of Fourier expansions of .-invariant functions on G we fix in this section the class of discrete subgroups that we use. Let . be a cofinite discrete subgroup of G with cusps. By a cusp .c of . we mean here a parabolic subgroup .Pc such that its unipotent radical .Nc ⊂ Pc intersects . in a lattice in .Nc . We use the name cusp also for the unique point in the boundary of the symmetric space .X fixed by .Pc ; see Sect. 2.1.1. All cusps are conjugate to each other by elements of .G = SU(2, 1). For each cusp .c there are .g ∈ G such that −1 . Conjugation by elements of . results in finitely many .-orbits .Pc = gN AMg of cusps.  We recall the standard lattices  .σ in N with .σ ∈ Z≥1 , of the form .σ = n(b, r) : b ∈ Z[i], r ∈ 2σ −1 Z (see Definition 2.2). If .∞ is a cusp of ., then we want that . ∩ N be conjugate to a standard lattice by an element of NAM. More generally, we require that there exists for each cusp .c of . an element .gc ∈ G and an integer .σ (c) ≥ 1 such that . ∩ Nc = gc σ (c) gc−1 . We call this the .Z[i]-condition at the cusps. This condition needs to be checked only for cusps in a system of representatives of the .-orbits of cusps. In the last section, Sect. 5.4, we will actually show that, with some additional effort, the results in this work are applicable to all cofinite discrete subgroups of G that have cusps. The most obvious example of a discrete subgroup satisfying the above condition is

 −1 = G ∩ SL2 Z[i] .

.

(2.68)

From (2.4) we see that .−1 ∩ N consists of the elements .n(b, r) with b in the ideal −1 with (1 + i) in .Z[i] and .r√∈ Z. So .∞ is a cusp of .−1 , and . ∩ N = g∞ 4 g∞ π i/4 ). .g∞ = h[i + 1] = a( 2) m(e The group .−1 is an example of what is called a Picard modular group. Namely, if .O is the ring of integers in any imaginary quadratic number field, and .GJ = U −1 GU is a realization such that .J = U¯ t I21 U has entries in .O, then the discrete subgroup . = GJ ∩ SL3 (O) is a Picard modular group. .

30

2 The Lie Group SU(2,1)

Zink [58] has shown that the number of .-orbits of cusps of a Picard modular group is equal to the class number of the imaginary quadratic number field. Since .Q(i) has class number 1, the group .−1 satisfies the .Z[i]-condition at the cusps, −1 since we have already shown that √ . ∩ N = g∞ 4 g∞ .  On the other hand, the field .Q −3 has class number 1 as well; however, one can show that the Picard modular group

⎞ 010 H = ⎝1 0 0⎠ 001 ⎛



−3 = GH ∩ SL3 Z[e2π i/3 ] ,

.

(2.69)

does not satisfy the .Z[i]-condition at the cusps. This group has been studied for a long time, since the quotient .−3 \GH /KH of the symmetric space parametrizes the Picard curves .y 3 = p4 (x) with .p4 a polynomial of degree 4; see for instance, Picard [37], Shimura [43], Shiga [42], Holzapfel [20]. Finis [11] gave explicit computations of Hecke eigenvalues of holomorphic automorphic forms on .−3 of low weight. For more recent work concerning holomorphic automorphic forms for .−3 , see Cléry and van der Geer [8], Bergström and van der Geer [3], and references in these papers. In Sect. 5.4 we will show how we can apply the main results in this work to Picard modular groups for number fields other than .Q(i). Remark In Sect. 2.1 we mentioned the realization .UT−1 GUT with .UT in (2.9). This is related to the realization used by Francsics and Lax [12], with ⎛ −i √

2

0

√1 2

⎞

⎜ ⎟ UFL = ⎝ 0 1 0 ⎠ . −1 −i √ 0 √

.

2

(2.70)

2

They give an explicit domain for the associated Picard modular group

fundamental  −1 FL = G ∩ UFL SL3 Z[i] UFL , the inspection of which confirms that it has only one cusp. The group .FL satisfies the .Z[i]-condition at the cusps.

.

Arithmeticity If .SU(2, 1) is realized as the group .GR of real points of an algebraic group .G over .Q, the group .GZ , and all subgroups of .GR that are commensurable to .GZ , are called arithmetic. (Subgroups . and . of a group G are commensurable if there exists an element .g ∈ G such that . ∩ gg −1 has finite index in . and in −1 .) .gg All Picard modular groups and their subgroups of finite index are arithmetic. However, at least in the case of the related group .PU(2, 1) there are constructions of discrete subgroups that are not arithmetic; see for instance Parker [35].

2.4 Discrete Subgroups

31

2.4.1 Fourier Expansions We consider Fourier expansions of functions on G that are invariant under a standard lattice in N. Let .C ∞ (σ \G)K with .σ ∈ Z≥1 denote the space of smooth functions on G that are .σ -invariant on the left and K-finite on the right. If .f ∈ C ∞ (σ \G)K , then the function .n → f (nak) lies in .C ∞ (σ \N) ⊂ L2 (σ \N) for each .a ∈ A, .k ∈ K, and can be expanded in terms of the orthonormal basis in (2.61) and (2.62): f (nak) =



χβ (n) fβ (ak)

.

β∈Z[i]

+ σ fβ (ak) = 2

where

f,c,m (ak) =

σ 2









,c (h,m ; n) f,c,m (ak) ,

∈(σ/2)Z=0 c mod 2 m∈Z≥0

σ \N



σ \N

χβ (n ) f (n ak) dn , .

(2.71)

,c (h,m ; n ) f (n ak) dn .

(2.72)

Proposition 2.6 The Fourier expansion converges absolutely for each function .f ∈ C ∞ (σ \N)K , and also for all derivatives uf with u in the universal enveloping algebra .U (n). Proof We know that the Fourier series converges in the sense of .L2 (σ \N ). In the coordinates .(x, y, r) ↔ n(x, y, r) on N we have the elliptic operator

1 L = X20 + X21 + X22 = ∂x − y ∂r )2 + (∂y + x ∂r)2 + ∂r2 . 4

.

This operator acts on .fβ with eigenvalue .−4π 2 |β|2 and on .,c (h,m ) with eigenvalue .−(4 + 8m + π||)||. Since f is smooth we can apply partial integration as many times as we want. In this way we obtain better and better estimates, uniform for ak in compact sets. Now we apply Sobolev theory; see for instance [30, Appendix 4]. The Sobolev inequality [30, p. 393] bounds the supremum norm in terms of the second Sobolev norm, and the basic estimate [30, p. 401] bounds the second Sobolev norm in terms of the .L2 -norm of f and .Lf. This gives pointwise convergence. We can differentiate .n → f (nak) as many times as we want, and also interchange differentiation and taking Fourier terms. 

32

2 The Lie Group SU(2,1)

For .f ∈ C ∞ (σ \G)K and .β ∈ Z[i] we have χβ (n) fβ (ak) =

.

=

σ 2 σ 2

 

σ \N

σ \N

 χβ n(n )−1 f (n ak) dn χβ (n )−1 f (n nak) dn ,

since .χβ is a (unitary) character. So with σ .Fβ f (g) := 2

 σ \N

χβ (n) f (ng) dn ,

(2.73)

we have .χβ (n)fβ (ak) = Fβ f (nak). Since we integrate over a compact set, the action of .g by right differentiation commutes with the operator .Fβ . We see also that the action of K by right translation commutes with .Fβ . So .Fβ is an intertwining operator of .(g, K)-modules. (See, e.g. Section 3.3 in [51].) Fβ : C ∞ (σ \G)K → Fβ

.

(2.74)

where .Fβ consists of the K-finite elements of .C ∞ (G) that transform on the left according to the character .χβ of .N. For . ∈ σ2 Z=0 , .c mod 2, we define the operator .F,c in .C ∞ (σ \G)K by F,c f (nak) =



.

,c (h,m ; n) f,c,m (ak) ,

(2.75)

m≥0

with .f,c,m as in (2.72). Here we cannot use a simple integral like in (2.73). Proposition 3.15 will show that the operator .F,c is an intertwining operator of .(g, K)-modules as well. There, we will also describe the .(g, K)-module .F,c such that .F,c Cc∞ (σ \G)K ⊂ F,c . The Fourier expansion can be rewritten as f (g) =



.

β∈Z[i]

Fβ f (g) +





F,c f (g) .

(2.76)

∈(σ/2)Z=0 c mod 2

Remark 2.7 Let . ⊂ G be a cofinite discrete subgroup satisfying the .Z[i]condition at the cusps introduced above and let .f ∈ C ∞ (\G)K . For each cusp c : g → f (g g) is in .C ∞ ( .c the function .f c σ (c) \G)K , with a Fourier expansion as in (2.76). The operator .f → f c commutes with the action of .g by right differentiation.

Chapter 3

Fourier Term Modules

The aim of this chapter is the study of the spaces in which Fourier term operators have their image. Since we restrict the consideration to K-finite automorphic forms, the images of the Fourier term operators are in a space in which the Lie algebra .g and the group K act. In Sect. 3.1 we discuss .(g, K)-modules and in Sect. 3.2 we discuss how to carry out explicit computations of the Lie algebra action in modules of tensor form with respect to the Iwasawa decomposition. In Sects. 3.3–3.4 we apply this to the ψ ψ modules .FN and .FN . In Sect. 3.5 we consider submodules of .FN . ψ The Fourier term modules .FN can be divided in two classes determined by the infinitesimal character .ψ. See the concepts of generic parametrization and integral parametrization in Table 3.11. If .ψ is in the class of generic parametrization, the structure of the Fourier term modules is relatively simple, and we can prove Theorems A and B in the Introduction. See Sect. 3.5.2, p. 81.

3.1 (g, K)-Modules After fixing notations for the Lie algebra .g and its universal enveloping algebra, we turn to .(g, K)-modules. We take advantage of the relatively simple structure of .SU(2, 1). In the study of the structure of .(g, K)-modules we focus on the highest weight subspaces in the K-types occurring in the module. Of course, the subspace of the .(g, K)-modules spanned by these highest weight subspaces is not invariant under the action of .g. However, there is a remnant of the action: namely, four “shift operators” going from the highest weight subspace in a K-type to the highest weight subspaces in neighboring K-types. Irreducibility of .(g, K)-modules can be investigated with

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. W. Bruggeman, R. J. Miatello, Representations of SU(2,1) in Fourier Term Modules, Lecture Notes in Mathematics 2340, https://doi.org/10.1007/978-3-031-43192-0_3

33

34

3 Fourier Term Modules

the use of these shift operators; see Sect. 3.1.1. We use the shift operators throughout Chaps. 3 and 4 in the study of Fourier term modules. The real Lie algebra .g of .SU(2, 1) is a real form of the complex Lie algebra of type .A2 . As a vector space it is the sum .g = n ⊕ a ⊕ k of the Lie algebras of the subgroups .N, A and K. We gave a basis .{X0 , X1 , X2 } for .n in (2.37), and a basis .{Ci , W0 , W1 , W2 } of .k in Sect. 2.2.2. The group A in the Iwasawa decomposition has Lie algebra .a = R Hr , ⎛ ⎞ 001 .Hr = ⎝0 0 0⎠ . 100

(3.1)

  Inside K is the group .M = m(ζ ) : |ζ | = 1 with Lie algebra .m = R Hi , ⎞ i 0 0 3 1 W0 − Ci = ⎝0 −2i 0⎠ . .Hi = 2 2 0 0 i ⎛

(3.2)

It satisfies .etHi = m(eit ). There are two Cartan subalgebras of .gc = C ⊗R g of interest. The Cartan subalgebra .ac ⊕ mc is adapted to the Iwasawa decomposition. We may choose .X1 − iX2 and .X1 + iX2 in .nc as the root vectors for the choice of simple positive roots, .α1 and .α2 , respectively. On the basis .{Hr , iHi } of .ac ⊕ mc we have the values in Table 3.1. The element .X0 ∈ n is a root vector for .α1 + α2 . There is also the Cartan subalgebra .h = C Ci ⊕ C W0 contained in .kc , with root vectors as indicated in Table 3.2. The universal enveloping algebra .U (g) is the associative .C-algebra generated by .g. The center .ZU (g) of .U (g) is isomorphic to a polynomial algebra in two variables, which can be chosen as indicated in Table 3.3. The first generator C of degree two is Table 3.1 The roots .α1 and on .ac ⊕ mc

.α2

.H

.α1 (H)

.α2 (H)

.Hr

1 .−3

1 3

.iHi

Table 3.2 Root vectors for .h

.Z .Z12 .Z23 .Z13 .Z21 .Z32 .Z31

= W1 − iW2 = 12 (−W1 − iW2 + X1 + iX2 )

 = 12 iHi − 2iW0 + Hr + 2iX0 = W1 + iW2

 = 12 − W1 + iW2 + X1 − iX2

 = 12 − iHi + 2iW0 + Hr − 2iX0

.

 Ci , Z

.

 W0 , Z

0

.2iZ12

.3iZ23

.−iZ23

.3iZ13

.iZ13

0 .−3iZ32

.−2iZ21

.−3iZ31

.−iZ31

.iZ32

3.1 (g, K)-Modules

35

Table 3.3 Generators of the center of the enveloping algebra C = H2r − 4Hr −

1 2 H + 4X0 Hi − 8X0 W0 3 i

+ 4X20 − 2X1 W1 + X21 − 2X2 W2 + X22 1 = − C2i + 2iCi − W02 + 2iW0 − Z12 Z21 + 4Z13 Z31 + 4Z23 Z32 , 3 i 3 = − C3i + iCi W02 + iZ12 Ci Z21 + 2iZ13 Ci Z31 − 6iZ13 W0 Z31 9 − 6Z13 Z21 Z32 + 2iZ23 Ci Z32 + 6iZ23 W0 Z32 + 6Z23 Z12 Z31 − 2C2i + 2Ci W0 + 24Z13 Z31 + 24Z23 Z32 + 8iCi .

the Casimir element. As a second generator we use the element .3 of degree three given in [16, Proposition 3.1]. In [6, §3] the Lie algebra relations are defined for symbolic manipulation. We carry out various checks, among them the centrality of C and .3 . (g,K)-modules A .(g, K)-module has the structure of a .g-module and of a K-module in which the actions of .g and K are compatible, and in which each vector is K-finite. See, e.g., [51, Section 3.3] for a discussion in a more general context. The group .K ⊂ SU(2, 1) is connected, and the .g-action determines the action of K. Nevertheless the action of K gives additional information; not every .g-module can be made into a .(g, K)-module. Let V be any .(g, K)-module. For each .v ∈ V the finite-dimensional representation generated by Kv is a direct sum of irreducible representations, each one isomorphic to some representation .τph discussed in Sect. 2.2.1. We denote by .Vh,p the submodule of V consisting of a sum of copies isomorphic to .τph . This submodule is characterized by the eigenvalues .−ih of .Ci ∈ k and .−p(p + 2) of the Casimir element .CK in (2.34). We can characterize .Vh,p as the intersection of the kernels of the elements Ci + ih and CK + 2p + p2 in U (k) .

.

If we know for a given element .v ∈ V a finite set of .(hi , pi )’s such that .v ∈ i Vhi ,pi it is possible to give the projection of v onto a given factor .Vh,p as the image uv by an element .u ∈ U (k) (actually a polynomial in .Ci and .CK ). However, in general it is impossible to find an element in .U (k) that works for all .v ∈ V . The best one can do is to give a sequence of elements that work for an increasing collection of finite sets. Inside the component .Vh,p we have a further decomposition into weight spaces .Vh,p,q , characterized as the kernel of .W0 + iq. In this way Vh,p =

.

|q|≤p, q≡p mod 2

Vh,p,q .

(3.3)

36

3 Fourier Term Modules

The actions of .Z21 , Z12 ∈ kc shift the weight by .2. We have seen this in the description of the action on basis functions on K in Table 2.2, p. 19; in a general .(g, K)-module this follows from the commutation relations. The occurrence in the module V of a K-type .τph is completely determined by the highest weight space .Vh,p,p . We use the term weight for the eigenvalues of .iW0 in representations of .k. The highest weight subspace in .Vh,p is the kernel of .Z21 . The commutation relations imply that the action of each of the basis elements .Z23 , Z32 , .Z13 , and .Z31 sends elements in .Vh,p,p to elements in the sum of spaces    .Vh ,p  ,q  with .|h − h| = 3, .|p − p| = 1, .|q − p| = 1. The action of elements of .U (g) enables us to project the result to one of these spaces. The resulting operators are well known in the theory of semi-simple Lie groups. Here we call them shift operators. Proposition 3.1 There are shift operators in each .(g, K)-module V

.

S13 : Vh,p,p → Vh+3,p+1,p+1 ,

S1−3 : Vh,p,p → Vh−3,p+1,p+1 ,

3 : Vh,p,p → Vh+3,p−1,p−1 , S−1

−3 S−1 : Vh,p,p → Vh−3,p−1,p−1 ,

defined in terms of the action of .U (g) in V as indicated in Table 3.4. 3 and .S −3 commute. The operators .S13 and .S1−3 commute, and the operators .S−1 −1 Proof We define the upward shift operators .S13 and .S1−3 by the action of the Lie algebra elements .Z31 and .Z23 , respectively. Since .[Z31 , Z23 ] = 12 Z21 raises the weight by 2, and since the action of .Z12 on highest weight spaces is zero, the operators .S13 and .S1−3 commute, provided that we show that the operators .S1±3 send .Vh,p,p to .Vh±3,p+1,p+1 . To check that .S13 Vh,p,p ⊂ Vh+3,p+1,p+1 we use that for all K-types 





 Vh ,p ,p = ker Ci + ih ∩ ker W0 + ip ∩ ker CK + p (p + 2) .

.

(3.4)

We start with .v ∈ Vh,p,p , which is in this intersection for .(h , p ) = (h, p), and also satisfies .Z21 v = 0. For .w = S13 v = Z31 v we have  Ci w = Ci Z31 v = Z31 Ci v + Ci , Z31 v

.

= −ihw − 3iZ31 v = −i(h + 3)w ,

Table 3.4 Description of the shift operators 3

.S1

: Vh,p,p → Vh+3,p+1,p+1

.Z31

.S1

−3

: Vh,p,p → Vh−3,p+1,p+1

.Z23

3 .S−1

: Vh,p,p → Vh+3,p−1,p−1

.Z32

p(p + 1)−1 Z32 + (2(p + 1))−1 Z31 Z12

−1 .Z13 − 2(p + 1) Z12 Z23

−1 = .p(p + 1)−1 Z13 − 2(p + 1) Z23 Z12 .=

−3

.S−1

: Vh,p,p → Vh−3,p−1,p−1

−1 + 2(p + 1) Z12 Z31

3.1 (g, K)-Modules

37

 W0 w = Z31 W0 v + W0 , Z31 v = −ipw − iZ31 v = −i(p + 1)w , 

CK w = − 2iW0 + W02 w + Z12 Z21 Z31 v    = −(p2 + 4p + 3)w + Z12 Z31 0 + Z21 , Z31 v

 = −(p2 + 4p + 3)w + Z12 0 = −(p + 1)(p + 3)w . This shows that .w = S13 v ∈ Vh+3,p+1,p+1 . We can check that .S1−3 Vh,p,p ⊂ Vh−3,p+1,p+1 in a similar way. 3 and .S −3 are based on .Z and .Z , respecThe downward shift operators .S−1 32 13 −1 tively, but in a more complicated way than for the upward shift operators. Let .v ∈ Vh,p,p . So it has the properties indicated above. We put .u = Z32 v. Then  Ci u = Z32 Ci v + Ci , Z32 v = −i(h + 3)u ,  W0 u = Z32 W0 v + W0 , Z32 v = −i(p − 1)u .

Preliminary computations suggest that .u Vh+3,p±1,p−1 . If this is true, then

=

up−1 + up+1 with .up±1

∈

CK u = −(p − 1)(p + 1) up−1 − (p + 1)(p + 3) up+1 .

.

It turns out that the two equations for .up+1 and .up−1 have the unique solutions   1 1 CK u + (p + 1)(p + 3)u = u + Z12 Z31 v , 4(p + 1) 2(p + 1)  −1  1 Z12 Z31 v . CK u + (p2 − 1)u = − = 4(p + 1) 2(p + 1)

up−1 = .

up+1

(3.5)

So we take 3 S−1 v = up−1 = Z32 v +

.

1 Z12 Z31 v . 2(p + 1)

(3.6)

We check the computations in [6, §7a], and also for the other case −3 S−1 v = Z13 v −

.

1 Z12 Z23 v , 2(p + 1)

(3.7)

and for the relation −3 3 −3 3 S−1 S−1 v = S−1 S−1 v.

.

This completes the proof.

(3.8) 

38

3 Fourier Term Modules

The proof has given us explicit descriptions in Table 3.4 of the shift operators. In the case of the downward shift operators this description depends on the K-type of the element to which the operator is applied. Corollary 3.2 Let V be a .(g, K)-module, and let .v ∈ Vh,p,p . The submodule U (g) v of V is equal to the space .U (k)P v where P runs over all compositions of shift operators.

.

Proof The statement is clear if we let P run through the products of the elements Z31 , .Z23 , .Z32 and .Z13 . The relations in Table 3.4 allow us to rewrite the products in terms of shift operators. 

.

Definition 3.3 Let V be a .(g, K)-module. We call a vector .v ∈ Vh,p,p minimal if −3 3 .v = 0, .S −1 v = 0 and .S−1 v = 0. Lemma 3.4 Let .v ∈ Vh,p,p be a minimal vector. Then we have: (i) .2(p + 1)Z32 v = −Z12 Z31 v and .2(p + 1)Z13 v = Z12 Z23 v. −3 3 3 S −3 v = 1 (p − h)(p + 1)v. (ii) .S−1 S1 v − (p + 1)S−1 1 2 (iii) For the Casimir element in Table 3.3, p. 35: Cv =

.

 p + 2 −3 3 h2 + p2 + 2h + 2p v + 4 S S v. 3 p + 1 −1 1

1

(iv) For the central element .3 ∈ ZU (g) in Table 3.3: 3 v =

.

1 h(h + 3p + 12)(h − 3p + 6) v 9 (p + 2)(h − 3p + 6) −3 3 +2 S−1 S1 v . p+1

Proof Part (i) is a direct consequence of the description of the downward shift operators in Table 3.4, p. 36. For the remaining statements we use Mathematica to carry out computations. See [6, §7b]. For (ii) we write out the description of the shift operators in Table 3.4, using that .S1±3 v ∈ Vh±3,p+1 . Taking the relations in the universal enveloping algebra into account, this gives −3 3 3 S−1 S1 v − (p + 1)S−1 S1−3 v

.

  1 (p − h)(p + 1)v − Z23 2(p + 1)Z32 + Z12 Z31 v 2 1 3 = (p − h)(p + 1)v − 2(p + 1) Z23 S−1 v. 2

=

3 vanishes on v this gives the formula. Since .S−1

3.1 (g, K)-Modules

39

The computations for (iii) and (iv) are carried out with Mathematica in a similar way. 

3.1.1 Special Cyclic Modules To study the submodule structure of a .(g, K)-module it suffices to consider the highest weight spaces in the K-types in the module, and the shift operators between these highest weight subspaces. Definition 3.5 By a special cyclic module we mean a .(g, K)-module V with the following properties. (a) The module V is generated by a minimal vector in .Vh0 ,p0 , with .h0 ≡ p0 mod 2, .p0 ∈ Z≥0 . (b) The Casimir element C acts in .Vh0 ,p0 as multiplication by .μ2 ∈ C. (c) All K-types occurring in V have multiplicity 1, and there are .A, B ∈ Z≥0 ∪{∞} such that

.V = kc Za31 Zb23 Vh0 ,p0 ,p0 . (3.9) 0≤a 0. Then V has a minimal vector .v(a1 , b1 ) with .(a1 , b1 ) = (0, 0). .

Proof Suppose that .α+ (a, b) = 0. Then relation (3.10) implies that α− (a, b) α+ (a − 1, b) = 0 .

.

So at least one of the situations in Fig. 3.2 occurs. If .α− (a, b) = 0 the vector v(a, b) is a minimal vector. Otherwise, the square relation (3.10) leads to the relation 3 .S −1 v(a − 1, b) = 0. Proceeding in this way the process cannot go on longer when we arrive in the situation of Fig. 3.3. So a minimal vector is reached. −3 If .S−1 v(a, b) = 0 we proceed similarly, with reflected figures.  .

Fig. 3.2 Squares with factors for downward shift operators

42

3 Fourier Term Modules

Fig. 3.3 Propagation of the 3 kernel of .S−1

Fig. 3.4 A connected collection of zero squares

Let .S(a, b) be the square in the .(a, b)-plane with .(a, b) at the top, and .(a −1, b − 1) at the bottom, like in Fig. 3.1. We call it a zero square if at least one of .α+ (a, b) and .α− (a, b − 1) on the right is zero. Then also at least one of the .α’s in the left part of the figure is zero, by (3.10). If .S(a, b) is a zero square, then at least one of the adjoining squares on the right, .S(a + 1, b) and .S(a, b − 1) is also a zero square. The same holds on the left. So zero squares do not come singly, but form connected regions, like sketched in Fig. 3.4. The region may reach one of the boundary lines .b = 0 or .a = 0, or it may extend to infinity, parallel to a boundary line. The factors .α± (a  , b ) corresponding to common edges of adjoining squares are zero. Lemma 3.10 If .v(a, b) with .(a, b) = (0, 0) is a minimal vector in a special cyclic module .V , then V is reducible. Proof Let .W ⊂ V be the submodule generated by .v(a, b). If we can find a product of shift operators that sends .v(a, b) to a non-zero multiple of .v(0, 0), then .W = V . So we have to show that, if starting from .v(a, b) we cannot reach .v(0, 0) by shift operators, then W is a non-trivial invariant subspace. The upward shift operators are non-zero on K-types corresponding to an upward edge starting at any point in the figure. For the downward shift operators we cannot follow a downward edge marked with 0. 3 v(a, b) = 0 and .S −3 v(a, b) = 0. So going downward directly from We have .S−1 −1 .(a, b) yields zero. We have to look for a path that ultimately goes down to .(0, 0). The

3.1 (g, K)-Modules

43

Fig. 3.5 Paths leaving the component C of zero squares. The point .(a  , b ) is marked with a dot

square with top .(a, b) is a zero square, which is in a maximal connected component C of zero squares. Let us first assume that the square above .(0, 0) is not in C. Then the path we are looking for has to leave C at a point .(a  , b ) on the lower boundary .Cl of C. Figure 3.5 indicates the possibilities. We indicate the boundary edges near .(a  , b ) by . = 0, and draw the adjoining zero squares. For the interior edges ending at .(a  , b ) we conclude that they carry a 0. This implies that the path from .(a, b) to .(0, 0) cannot leave the interior of C along this edge. There is the possibility that the lower boundary of C is empty. Then the square above .(0, 0) is a zero square, and both edges down to .(0, 0) carry a zero. Then the path cannot reach .(0, 0). (Edges marked with . = are part of the lower boundary of C. We have already considered that case.)  By Lemmas 3.7, 3.9 and 3.10 we now have equivalence between reducibility, existence of non-trivial minimal vectors, and vanishing of at least one .α± . The next step is to find more relations for the .α’s than those given by the square relation (3.10). Lemma 3.11 In an irreducible special cyclic module, all coefficients .α± (a, b), with 0 ≤ a < A, .0 ≤ b < B, are determined by the parameter set .[μ2 ; h0 , p0 ; A, B].

.

44

3 Fourier Term Modules

Proof We start with parts (b) and (c) of Definition 3.5, which imply that .v(a, b) satisfies .Cv(a, b) = μ2 v(a, v). We write

.

1 C = −W02 − C2i − Z12 Z21 + 4(s31 + s23 ) , 3 s31 = Z31 Z13 ,

(3.11)

s23 = Z23 Z32 .

The terms in .U (k) all have .v(a, b) as an eigenvector. So .(s31 + s23 )v(a, b) = μa,b v(a, b) with an explicitly known eigenvalue .μa,b . With Mathematica it is no problem to compute .μa,b , but we do not need an explicit value. A not too complicated calculation shows that there is .d ∈ Z such that −3 3 3 (s31 + s23 )v(a, b) = S−1 S1 v(a, b) + S−1 S1−3 v(a, b) + d v(a, b) . 

= α− (a + 1, b) + α+ (a, b + 1) v(a, b) + d v(a, b) . (3.12)

So .α− (a + 1, b) + α+ (a, b + 1) is equal to a well-defined number .Ca,b . With (3.10) we now have two relations: α− (a, b) α+ (a − 1, b) = α+ (a, b) α− (a, b − 1) , .

α+ (a − 1, b) + α− (a, b − 1) = Ca−1,b−1

(3.13)

Since V is assumed to be irreducible all factors .α± are non-zero. The values of α+ (a − 1, b) and .α− (a, b − 1) determine the values of .α+ (a, b) and .α− (a, b) completely. We need only start the induction. To the minimal vector .v(0, 0) we apply (iii) in Lemma 3.4 that gives the value of .α− (1, 0). Then (ii) in the same lemma also gives .α+ (0, 1). This suffices to start the induction.  .

End of Proof of Theorem 3.6 Lemma 3.11 implies the statement of the theorem. 

3.2 Explicit Differentiation of K-finite Functions From general .(g, K)-modules we turn to .(g, K)-modules contained in the space C ∞ (G)K of smooth K-finite functions on .G. It is a .(g, K)-module for the actions of .g and K by right differentiation and right translation. The aim of this section is to establish explicit formulas for the differentiation of elements of .C ∞ (G)K , and to implement these formulas as Mathematica routines. We use an approach that is known for general semisimple Lie groups, specially for functions with a prescribed left behavior under N. Then only the differentiation on A remains to be carried out: the radial parts of the differentiation operators.

.

3.2 Explicit Differentiation

45

By the Iwasawa decomposition any element .F ∈ C ∞ (G)K has the form F (nak) =



.

p

Fh,p,r,q (na) h r,q (k) ,

(3.14)

h,p,r,q

with component functions .Fh,p,r,q ∈ C ∞ (N A). The summation variables run over integers such that .h ≡ p ≡ r ≡ 1 mod 2, .|r|, |q| ≤ p. Only finitely many component functions are non-zero. The aim in this section is to describe explicitly the action of .gc . The action of any .X ∈ kc involves only the basis elements in Table 2.2, p. 19. We have to describe the action of the basis elements .Z13 , .Z32 , .Z23 and .Z31 . The procedure is known. Our task is just to work it out explicitly. We will carry out the following steps: 1. We consider the action of .k ∈ K by conjugation on the basis elements of .L2 (K). 2. We relate the action by right differentiation to the following action of .X ∈ g



 M(X)F (nak) = ∂t F naetX k t=0 .

.

(3.15)

3. We describe .M(X) on functions of the form .nak → h(na) (k) in terms of the right differentiation of h and the left differentiation of .. Conjugation by Elements of K A direct computation in [6, §8a] gives for .k = k(η, α, β) ∈ K

.

¯ 3 Z23 , kZ13 k −1 = αη3 Z13 − βη

kZ31 k −1 = αη ¯ −3 Z31 − βη−3 Z32 ,

kZ23 k −1 = βη3 Z13 + αη ¯ 3 Z23 ,

¯ −3 Z31 + αη−3 Z32 . kZ32 k −1 = βη (3.16)

The factors are polynomial functions on .K. From (2.26) we have the following:

.

αη±3 =

∓3

1−1,−1 (k) ,

βη±3 =

∓3

αη ¯ ±3 =

∓3

11,1 (k) ,

¯ ±3 = − ∓3 11,−1 (k) . βη

1−1,1 (k) ,

(3.17)

Right Differentiation and Interior Differentiation The right differentiation by .X ∈ g in .nak ∈ NAK 

XF (nak) = R(X)F (nak) = ∂t F naketX t=0

.

(3.18)

and the interior differentiation

 M(X)F (nak) = ∂t F naetX k t=0

.

(3.19)

46

3 Fourier Term Modules

Table 3.5 Decomposition of elements of .gc in .nc ⊕ ac ⊕ kc

Z13 Z31 = Z13 Z23 Z32 = Z23

= iX0 + 12 Hr + 2i (Hi − 2W0 ) = −iX0 + 12 Hr − 2i (Hi − 2W0 ) = 12 (X1 + iX2 ) − 12 (W1 + iW2 ) − 12 (W1 − iW2 ) = 12 (X1 − iX2 )

are related by XF (nak) = M(kXk −1 )F (nak) .

(3.20)

.

This relation extends to .X ∈ gc by linearity.   We apply this with .X ∈ Z31 , Z13 , Z32 , Z23 , and get R(X)F (nak) =



.

(3.21)

ϕij (k) M(Zij )F (nak)

ij

with functions .ϕij on K indicated in (3.16). By the Iwasawa decomposition we have the relations in Table 3.5. In this way, p we get for F of the form .F (nak) = b(na)(k), with .b ∈ C ∞ (NA) and . = h r,q a formula for interior differentiation.





 M(X) b(na) (k) = RN A (X)b (na) (k) + b(na) LK (X) (k) .

.

(3.22)

We have given the left differentiation L on K in Table 2.2, p. 19. To carry out the in Table 2.1, multiplication by .ϕij (k) in (3.21) we use the multiplication formulas  p. 18. In this way we have reduced the action of the elements of . Z31 , Z13 , Z32 , Z23 to known relations and right differentiation on NA. Implementation In this way, the action by right differentiation of (a basis of) .gc on functions in the form (3.14) can be described in terms of right differentiation on N A of the components .Fh,p,r,q in (3.14). Carrying out such computations we gladly leave to a computer, since errors slip in easily into computations by hand. Of course, it requires great care to write the routines. The version in Section 8 of [6] gives results that we checked in various ways. These routines are the basis for essential computations in this work. In the notebook we give explanations of the way we build the routines. p

Example A computation in [6, §8e] gives for .F (nak) = b(na) h r,p (k)

p  Z31 b h r,q =

.

  1 (2 + p + r) (2Hr − 4iX0 )b 8(p + 1)  p+1 + (h + 2p − r)b h+3 r+1,q+1

(3.23)

3.2 Explicit Differentiation

47

 p+1 − 2(2 + p − r) (X1 − iX2 )b h+3 r−1,q+1   − (p − q) (4iX0 − 2Hr )b + (4 − h + 2p + r)b 

 p−1 + 2(p − q) (X1 − iX2 )b h+3 r−1,q+1 ,

h+3

p−1

r+1,q+1

with the action of .nc ⊕ ac by right differentiation on the function b on .NA. A similar formula is available for all elements in the Lie algebra. For the elements p in .kc the formula is easy, since it involves only . h r,q . For .Z13 , .Z32 and .Z23 the general structure of the formula is the same as in (3.23). Application to Shift Operators If we take .q = p in (3.23) the last two terms become zero, and we arrive weight vector in .Vh+3,p+1,p+1 . Thus, we obtain the

at a highest p  description of .S13 b h r,q , in accordance with Proposition 3.1. The same works for .S1−3 . See Table 3.6 for the resulting description of the shift operators. ±3 are based on .Z32 and .Z13 . In these cases we The downward shift operators .S−1 h±3 just delete the contributions of the K-type .τp−1 . In Proposition 3.1 the projection to a given K-type was given by an element of .U (g) depending on the K-type of the argument. It would be inefficient to do that in the actual computations.

Table 3.6 The action of the shift operators on vectors in .C ∞ (G)K . See [6, §8f] 



 p  8(p + 1) S13 b h r,p = (2 + p + r) 2Hr − 4iX0 + h + 2p − r b ·

h+3

p+1

r+1,p+1



− 2(2 + p − r) X1 − iX2 b

h+3

 p+1 r−1,p+1 ,



 p  p+1 8(p + 1)S1−3 b h r,p = 2(2 + p + r) X1 + iX2 b h−3 r+1,p+1

 p+1 + (2 + p − r) 2Hr + 4iX0 + 2p + r − h b h−3 r−1,p+1 ,

h p 

 3 b r,p = p 2Hr − 4iX0 − 4 + h − 2p − r b 4(p + 1) S−1 ·

h+3

p−1

r+1,p−1 

p−1 + 2p X1 − iX2 b h+3 r−1,p−1 , p−1 −3 h p  b r,p = −2p(X1 + iX2 )b h−3 r+1,p−1 4(p + 1) S−1

 p−1 + p 2Hr + 4iX0 − (4 + h + 2p − r) b h−3 r−1,p−1 .

48

3 Fourier Term Modules

3.3 Large Fourier Term Modules We turn to the submodules of .C ∞ (G)K of our main interest: the modules in which the operators .Fβ and .F ,c on .C ∞ ( σ \G)K in (2.73) and (2.75) take their values. The automorphic forms that we will discuss in Chap. 5 are .-invariant, K-finite and eigenfunctions of the differential operators corresponding to elements of the center of the enveloping algebra of .g. Since the operators .Fβ and .F ,c are, by Proposition 3.15, intertwining operators of .(g, K)-modules, the Fourier terms .Fβ and .F ,c of automorphic forms are elements of submodules .C ∞ (G)K that transform under left translation by elements of N according to the Fourier term order and transform under differentiation by elements of .ZU (g) according to a character .ψ. ψ ψ The resulting Fourier term modules .Fβ and .F ,c are discussed in the next section. In this section we do not impose a condition on the action of .ZU (g), and consider ψ ψ the larger modules .Fβ and .F ,c containing all modules .Fβ and .F ,c , respectively. We call these modules large Fourier term modules. We adapt the Mathematica routines for the shift operators to the large Fourier term modules, and arrive at explicit descriptions. In the generic abelian case, .Fβ with .β ∈ Z[i]  {0}, the upward shift operators turn out to be injective. In the non-abelian case the modules .F ,c turn out to be a infinite direct sum of submodules .F ,c,d . Proposition 3.12 (Large Fourier Term Modules) Fourier term module .Fβ ⊂ C ∞ (G)K determined by (i) Let .β ∈ C. The abelian  the condition .F ng = χβ (n) F (g) on .F ∈ C ∞ (G)K is a .(g, K)-submodule of ∞ .C (G)K . (ii) Let . ∈ σ2 Z =0 and let .c ∈ Z mod 2 . We define the non-abelian Fourier term module .F ,c as the vector space spanned by functions of the form

 p f na(t)k =  ,c (ϕ; n) f (t) h r,q (k) ,

.

(3.24)

where .ϕ ∈ S(R) runs over (finite) linear combinations of normalized Hermite functions .h ,m with .Z≥0 , where .f ∈ C ∞ (0, ∞), and where the integers .h, p, r, q satisfy .h ≡ p ≡ r ≡ q mod 2, .|r| ≤ p, .|q| ≤ p. The module ∞ .F ,c is a .(g, K)-submodule of .C (G)K . Proof The invariance under the .(g, K)-action in (i) follows from the fact that the actions on the right and on the left commute. In (ii) the invariance under the action of .k and K is clear. The action of the remaining basis elements .Zij can be worked out by interior differentiation, with the approach in Sect. 3.2. Since A normalizes N, the action of .n on the functions on N A leads to an action of .n on theta functions. The relations (2.54) and (2.55) show that the differentiation produces linear combinations of theta functions with the same parameters . and c. 

3.3 Large Fourier Term Modules

49

The Fourier term operators on .C ∞ ( σ \G)K take values in these large Fourier term modules. These modules are large, since we can take .f ∈ C ∞ (0, ∞) arbitrarily. When we apply the explicit differentiation

of the previous paragraph we use that if a function b on NA is of the form .b na(t) = u(n) f (t), then we have, with .exp(xHr ) = a(ex ), the relation

 Hr b na(t) = u(n) t f  (t) .

(3.25)

.



 For .X ∈ n the action of .X satisfies .Xb na = (aXa −1 )u (n) f (a). With (2.39) this leads to



 Xj b na(t) = t (Xj u)(n) f a(t) (j = 1, 2) , . (3.26)



 2 X0 b na(t) = t (X0 u)(n) f a(t) .

3.3.1 Abelian Case Proposition 3.13 All (g, K)-modules Fβ with β = 0 are isomorphic. Proof Any element β ∈ C∗ can be written as β = ζ −3 t with |ζ | = 1 and t > 0. Conjugation by x = a(t)m(ζ ) as described in (2.8) transforms χ1 into χβ . So, the left translation (L(x)F )(g) = F (xg)

.

gives a bijective linear map F1 → Fβ . Since left and right translations commute, Lx is an intertwining operator.  In the abelian case we have u(n) = χβ (n), and



  X1 χβ (n) f (t) = 2π iRe (β) χβ (n) f (t) , X0 χβ (n) f (t) = 0 ,

 . X2 χβ (n) f (t) = 2π iIm (β) χβ (n) f (t) ,

 Hr χβ (n) f (t) = t χβ (n) f  (t) .

(3.27)

These relations can be applied to work out the differentiation relations. In particular we get the description of the shift operators in Table 3.7. Kernel Relations The component functions of a highest weight function in a given K-type τph are parametrized by r, which runs over the finitely many values satisfying |r| ≤ p, r ≡ p mod 2. If the context allows it, we will write  .

r

instead of

 r≡p(2), |r|≤p

.

(3.28)

50

3 Fourier Term Modules

Table 3.7 Shift operators acting in Fβ , with β ∈ Z[i]  {0}. See [6, §9b]

8(p + 1) S13 χβ f

h

p

r,p

·



h+3



 = χβ (2 + p + r) 2tf  + (h + 2p − r)f p+1

r+1,p+1

 p+1 − 4π i(2 + p − r)β¯ tf h+3 r−1,p+1 , 

p  8(p + 1)S1−3 χβ f h r,p = χβ 4π iβ(2 + p + r)tf

h−3

p+1

r+1,p+1 

 p+1 + (2 + p − r) 2tf  + (2p + r − h)f h−3 r−1,p+1 , 

 p  3 χβ f h r,p = χβ p 2tf  + (−4 + h − 2p − r)f 4(p + 1) S−1 ·

h+3

p−1

r+1,p−1

 p−1 r−1,p−1 ,  p  −3

χβ f h r,p = χβ − 4π ipβtf 4(p + 1) S−1 ¯ + 4π i βptf

h+3

h−3

p−1

r+1,p−1 

  h−3 p−1 + p 2tf − (4 + h + 2p − r)f r−1,p−1 .

A generating element of the highest weight space Fβ;h,p,p has the form 

 p F na(t)k = χβ (n) fr (t) h r,p (k) ,

.

(3.29)

r

with components fr ∈ C ∞ (0, ∞). The description of the shift operators shows that if F is in the kernel of a shift operator, then there are relations between components fr and fr+2 , which we call kernel relations. See Table 3.8. Proposition 3.14 Let β ∈ C∗ . (i) The upward shift operators S13 and S1−3 : Fβ;h,p,p → Fβ;h±3,p+1,p+1 are injective. 3 or S −3 vanish have (ii) For each K-type τph the subspaces of Fβ;h,p,p on which S−1 −1 infinite dimension. Proof In the kernel relations for S13 in Table 3.8 we see that if S13 F = 0 for F as in (3.29), we have f−p = 0. Furthermore, since fr+2 is expressed in terms of fr and its derivative, we conclude that all fr vanish, and hence F = 0. If S1−3 F = 0 we proceed similarly, now starting with fp . The proof of i) clearly breaks down for β = 0. 3 in ii) we can pick f ∞ For the kernel of S−1 −p arbitrarily in C (0, ∞). This −3 determines the higher components. For S−1 we start with any fp in C ∞ (0, ∞). 

3.3 Large Fourier Term Modules

51

Table 3.8 Kernel relations in Fβ;h,p,p , with β ∈ Z[i]{0}. The condition p ≥ 1 for the downward ±3 shift operators is in accordance with the fact that S−1 vanishes on one-dimensional K-types. See [6, §9c] S13 :

2tfp + (h + p) fp = 0 ¯ − r) tfr+2 (2 + p + r)(2tfr + (h + 2p − r)fr = 8π i β(p for − p ≤ r ≤ p − 2 , ¯ −p = 0 ; βf

S1−3 :

βfp = 0 ,  (p − r)(2tfr+2 + (2p + r + 2 − h)fr+2 = −2π iβ(2 + p + r) tfr

for − p ≤ r ≤ p − 2 ,  2tf−p 3 S−1

:

2tfr

+ (p − h)f−p = 0 ;

¯ r+2 + (h − 2p − r − 4)fr = −8π i βtf for − p ≤ r ≤ p − 2 and p ≥ 1 ;

−3 S−1 :

 2tfr+2 − (2 + h + 2p − r)fr+2 = 2π iβt fr

for − p ≤ r ≤ p − 2 and p ≥ 1 .

3.3.2 Non-abelian Case In .F ,c the components are linear combinations of functions on NA of the form

 na(t) →  ,c h ,m ; n f (t)

.

with .f ∈ C ∞ (0, ∞) and .m ∈ Z≥0 . In a given module .F ,c only m varies, and we abbreviate .ϑm =  ,c (h ,m ). We obtain from the substitution rules in Table 2.4, p. 29, and (2.54), (2.55). RN A (Hr )(ϑm f ) = ϑm tf  ,

.

RN A (X0 )(ϑm f ) = π i ϑm t 2 f , 

 RN A sign ( )X1 + iX2 (ϑm f ) = −2i 2π| |(m + 1) ϑm+1 tf , 

 RN A sign ( )X1 − iX2 (ϑm f ) = −2i 2π| |m ϑm−1 tf . The sign of . plays a role in these relations, hence also in the resulting differentiation formulas and in the expressions for the shift operators in Table 3.9. We will often write ε = sign ( ) ,

.

δx =

1+x for x = ±1 . 2

(3.30)

52

3 Fourier Term Modules p

Table 3.9 Shift operators acting in .F ,c , for .F = ϑm f (t) h r,p . .ε and .δx as in (3.30). See [6, §10b]

 8(p + 1)S13 F = ϑm (2 + p + r) 2tf  + (h + 2p − r + 4π t 2 )f ·

h+3

p+1

r+1,p+1

 p+1 + 4iεϑm−ε (2 + p − r) 2π | |(m + δ−ε ) t f h+3 r−1,p+1 ,  8(p + 1)S1−3 F = −4iεϑm+ε (2 + p + r) 2π | |(m + δε ) t f ·

h−3

p+1

r+1,p+1

 + ϑm (2 + p − r) 2tf  − (h − 2p − r + 4π t 2 )f ·

h−3

p+1

r−1,p+1 ,

 4(p + 1) 3 S−1 F = ϑm 2tf  − (4 − h + 2p + 4 − 4π t 2 )f p ·

h+3

p−1

r+1,p−1  − 4iεϑm−ε 2π | |(m + δ−ε ) t f

h+3

p−1

r−1,p−1 ,

 4(p + 1) −3 p−1 S−1 F = 4iεϑm+ε 2π | |(m + δε ) t f h−3 r+1,p−1 p

 p−1 + ϑm 2tf  − (4 + h + 2p − r + 4π t 2 )f h−3 r−1,p−1 .

Submodules In Table 3.9 we can check that the quantity d := 3 sign ( )(2m + 1) + h − 3r ∈ 1 + 2Z

.

(3.31)

is preserved by the action of the shift operators. The action of the elements in .k and right translation by elements of K preserve this quantity as well. So we can split .F ,c into invariant submodules

F ,c =

.

F ,c,d ,

(3.32)

d≡1 mod 2

where .F ,c,d consists of finite linear combinations of functions of the form

 p na(t)k →  ,c h ,m ; n f (t) h r,q (k)

.

with .(6m + 3) sign ( ) + h − 3r = d. Notation We use .N as a general notation, that can be specified as .Nβ or .N ,c,d when used in the text. In subscripts we write .FNβ = Fβ , and analogously for .N ,c . We use .n as an abbreviation of .( , c, d).

3.3 Large Fourier Term Modules

53

Metaplectic Action The splitting of .F ,c as a direct sum can be understood in greater generality (see Weil [55], or Ishikawa [22, pp. 489, 490]) by an action of the double cover of M in the module .F ,c . Let us make this more explicit. The one-parameter group of automorphisms u(v) : n → m(eiv )nm(e−iv ), with .v ∈ R, induces on the Lie algebra the automorphism determined by

.

u(v)X1 = (cos 3t)X1 + (sin 3t)X2 , .

u(v)X2 = (cos 3t)X2 − (sin 3t)X1 ,

u(v)X0 = X0 ,

(3.33)

and hence u (0) :

.

X1 → 3X2 ,

X2 → −3X1 ,

X0 → 0 .

(3.34)

On the other hand, .B = 8π1i ∂ξ2 + 2π i ξ 2 defines an operator in the Schwartz space .S(R). With use of Table 2.4, p. 29, we see that Bh ,m =

.

i (2m + 1) sign ( )h ,m . 2

(3.35)

(Checked in [6, §5g].) Hence there is a group homomorphism from .R to the unitary operators in .L2 (R) such that .evB h ,m = ei(m+1/2) sign ( )v h ,m . With (2.47) we find for Schwartz functions .ϕ the relations B dπ2π (X1 ) ϕ − dπ2π (X1 )Bϕ = −dπ2π (X2 )ϕ , .

B dπ2π (X2 )ϕ − dπ2π (X2 )Bϕ = dπ2π (X1 )ϕ ,

(3.36)

B dπ2π (X0 ) ϕ − dπ2π (X0 )Bϕ = 0 . Comparison with (3.34) shows that for .X ∈ n 

 ∂v e−3vB dπ2π (X)e3vB v=0 = dπ2π u (0)X .

.

(3.37)

Integrating this, we obtain for .n ∈ N

 e−3vB π2π (n)e3vB = π2π m(eiv )nm(e−iv ) .

.

(3.38)

We note that the right-hand side depends only on .v ∈ R mod 2π Z. However, e2π B h ,m = −h ,m , and .e−3(2π )B = −1. So .m(eiv ) → evN is not defined on M, but can be viewed as a function on the double cover .M˜ of M. ˜ from .R mod 4π Z to the operators on .F ,c We define a group homomorphism .m given on basis elements by

.

 



 p p m(v) ˜  ,c (ϕ) · f · h r,q =  ,c e3vB ϕ · f · L m(eiv ) h r,q .

.

54

3 Fourier Term Modules

Fig. 3.6 K-types .τph depicted in the .(h/3, p)-plane. The arrows indicate the change of K-type given by all four shift operators. Repeated application of shift operators leaves invariant the set of thick points, which satisfy . h3 ≡ p mod 2

It turns out that the elements . ,c (h ,m ) · f · eigenvalue i

e3i(m+1/2) sign ( )v eiv(h−3r)/2 = e 2 v

h p r,q

are eigenvectors of .m(v) ˜ with



.

(6m+3) sign ( )+h−3r



= eivd/2 ,

with d as in (3.31). So the decomposition in (3.32) is the decomposition into eigenspaces for this action of the double cover of M. We call d the metaplectic parameter. K-types The K-types .τph occurring in .(g, K)-modules can be pictured as points in the .(h/3, p)-plane satisfying . h3 ≡ p mod 2, .p ∈ Z≥0 . The shift operators change the K-types by .(h/3, p) → (h/3 ± 1, p ± 1) (occurrences of .± are not coupled). See Fig. 3.6. h with .r ≡ p mod 2, .|r| ≤ p. The K-type .τph can occur in the realizations .τp,r The definition of .F ,c,d gives the condition that .d = (6m + 3) sign ( ) + h − 3r. Since .m ∈ Z≥0 this imposes the requirement that

3r ≥ 3 + h − d if > 0 ,

.

3r ≤ −3 + h − d if < 0 . This imposes the following condition on the K-types:

.

h − 3p ≤ d − 3

if > 0 ,

h + 3p ≥ d + 3

if < 0 .

(3.39)

See Fig. 3.7. This restriction on the K-types is special for the modules .F ,c,d ; in the abelian modules .Fβ all K-types can occur.

3.3 Large Fourier Term Modules

55

Fig. 3.7 K-types allowed in .F ,c,d in the .(h/3, p)-plane; for . > 0 on the left, and for . < 0 on the right

We get a more complicated decomposition into components of the highest weight element .F ∈ Fn;h,p,p . (Notation as in (3.3).) F =



.

ϑm(h,r) fr

h

p

r,p ,

(3.40)

r

where .r ≡ p mod 2, .|r| ≤ p, with the additional condition

.

 r ≥ r0 (h) r ≤ r0 (h)

if ε = 1 , if ε = −1 ,

(3.41)

and the following quantities, depending implicitly on d r0 (h) =

.

h−d +ε, 3

m(h, r) =

1 ε (d − h + 3r) − . 6 2

(3.42)

The K-type .τph does not occur in .Fn if .r0 (h) > p if .ε = 1, and if .r0 (h) < −p if .ε = −1. The kernel relations for the shift operators in .Fn depend on the quantity .r0 (h). We will work them out when we need them. Fourier Term Operators The operator .F ,c : C ∞ ( σ \G)K → F ,c in (2.75) can be split up according to the decomposition (3.32): F ,c F =



.

F ,c,d F .

(3.43)

d≡1 mod 2

The sum is finite for each .F ∈ C ∞ ( σ \G)K . F ,c,d F (nak) =

.



 h pr,q (k)  ,c h ,m ; n    h pr,q 2 m,h,p,r,q  

 σ p  ,c h ,m ; n F (n ak  ) h r,q (k  ) dk  dn , · 2 n ∈ σ \N k  ∈K (3.44)

56

3 Fourier Term Modules

where the sum runs over integers satisfying .m ≥ 0, .(6m + 3) sign ( ) + h − 3r = d, h ≡ p ≡ r ≡ p mod 2, .|r| ≤ p, .|q| ≤ p.

.

Proposition 3.15 The Fourier term operators Fβ : C ∞ ( σ \G)K → Fβ .

(β ∈ Z[i]) ,

F ,c : C ∞ ( σ \G)K → F ,c

( ∈

F ,c,d : C ∞ ( σ \G)K → F ,c,d

σ Z =0 , c mod 2 ) , 2

(3.45)

(d ≡ 1 mod 2) ,

are intertwining operators of .(g, K)-modules. Proof In Sect. 2.4.1 the intertwining property for .Fβ was already noted. p For .f (nak) = fN A (na) h r,q (k), we consider .F ,c given by F ,c f (nak) =



.

 ,c (h ,m ; n)

m≥0

σ 2

 n ∈ σ \N

 ,c (h ,m ; n ) fN A (n a) dn

p

· h r,q (k) .

 It suffices to check that .Z F ,c f = F ,c (Zf ) for all .Z in a basis of .gc . For basis elements in .kc this is directly clear. For other basis elements we use the discussion in Sect. 3.2, which reduces the question to the action by interior differentiation, between NA and .K. In Table 3.5 we give the decomposition .Z = Zn + Za + Zk corresponding to the Iwasawa decomposition of .G. The action of .M(Z ) is the same for .F ,c f and .f. We have to look at k

1 2 RN A (X1 ) = t ∂x − y∂r ) t ∂r 2 .

RN A (X2 ) = t ∂y + x∂r) RN A (Hr ) = t∂t RN A (X0 ) =

(3.46)

in the coordinates .(x, y, r, t) ↔ n(x, y, r)a(t). To see this we use (2.39) and the fact that .exp(xHr ) = a(ex ). The action of .Hr is the same for .F ,c f and .f. For the function .f : n → fN A (na) with fixed .a ∈ A we have F ,c f (na) =



.

   ,c (h ,m ; n) f,  ,c (h ,m )

σ \N

m≥0

where  .

 f1 , f2

σ \N

σ = 2



σ \N

f1 (n)f2 (n) dn .

3.3 Large Fourier Term Modules

57

Partial integration gives for .j = 0, 1, 2  .

 Xj f1 , f2

σ \N

  = − f1 , Xj f2

σ \N

.

This gives the desired formula .X0 F ,c f = F ,c X0 f, since .dπ2π (X0 )h ,m = π i h ,m ; see (2.47). Let .ε = sign ( ). For the other basis elements we obtain with Table 2.4, p. 29:    ,c (h ,m ) (εX1 ∓ iX2 )f,  ,c (h ,m )

.

σ \N





= − ,c (h ,m ) f, (εX1 ± iX2 ) ,c (h ,m )

σ \N

 ⎧ √ ⎨ f, −4i π| |(m + 1)  ,c (h ,m+1 )

σ \N  = − ,c (h ,m )  ⎩ f, −4i √2π| |m  ,c (h ,m−1 )



σ \N

± = +, ± = −;

(εX1 ∓ iX2 )  ,c (h ,m ) g,  ,c (h ,m ) \N σ  √

 −4i 2π| |m  ,c (h ,m−1 ) ± = +, = f,  ,c (h ,m ) \N · √ σ −4i π| |(m + 1)  ,c (h ,m+1 ) ± = − . Taking the sum over .m ∈ Z≥0 we obtain equality. In this way we conclude that .F ,c is an intertwining operator. The subspaces .F ,c,d in the decomposition (3.32) are invariant .(g, K)-modules. Hence the operators .F ,c,d are intertwining operators. 

3.3.3 Normalization of Standard Lattices In the proof of Proposition 3.13 we used the left translation .(Lh f )(g) = f (hg) with h ∈ AM to get an isomorphism of .(g, K)-modules between large abelian modules .Fβ . Here we consider left translations that preserve . σ -invariance on the left. .

Proposition 3.16 Let . σ be a standard lattice. The normalizer   NormP ( σ ) = p ∈ NAM : p σ p−1 = σ

.

(3.47)

is the semi-direct product of the groups    NormN ( σ ) = n(β/σ, ρ ∈ N : β ∈ Z[i], ρ ∈ R ,

.

(3.48)

58

3 Fourier Term Modules

and   NormM ( σ ) = m(ζ ) ∈ M : ζ 12 = 1 .

(3.49)

.

Proof Suppose that .p = n(β, ρ)a(τ )m(ζ ) normalizes . σ . Then .pn(b, r)p−1 = n(τ ζ 3 , r  ) for some .r  ∈ R. The projection .N → N/Z(N ) sends . σ to .Z[i], and conjugation by p descends to .b → τ ζ 3 b in .C. This leaves .Z[i] invariant only if 3 12 = 1. .τ ζ ∈ Z[i]. So .τ = 1 and .ζ We consider the action by conjugation of .p = n(β, ρ) on basis elements of . σ :

 pn(1, 0)p−1 = n ζ 3 , −2Im (β/ζ 3 ) ,

 . pn(i, 0)p−1 = n iζ 3 , 2Re (β/ζ 3 ) ,



 pn 0, 2/σ p−1 = n 0, 2/σ .

(3.50)

This gives the requirement that .σβ ∈ Z[i]. A direct check shows that these elements normalize . σ .  Conjugation by .n(0, ρ) or by .m(ζ ) where .ζ is a third root of unity, induce the identity on . σ . In Table 3.10 we describe the action of elements in .NormP ( σ ) on basis vectors for the large Fourier term modules. Left translation by an element of .NormN ( σ ) is absorbed in the character .χβ in the abelian case, and handled with (2.53) in the non-abelian case. Left translation by .m(ζ ) has the effect m(ζ )n(b, r)a(t)k = n(ζ 3 b, r) a(t) m(ζ )k .

.

p

p

h  . In the non-abelian case we need We use that .L(Hi ) h r,q = i h−3r r,q 2 Proposition 2.3 and arrive at the following description.

Table 3.10 Action of elements of .NormP ( σ ) on basis vectors in .Fβ and .F ,c . For the case marked with .∗ see (3.51) .F

p

= χβ f h r,q

.F

p

=  ,c (h ,m )f h r,q

.n

= n(0, ρ) = n(1/σ, 0) .n = n(i/σ, 0)

=F = χβ (n)F .Ln F = χβ (n)F

= = e2π ic/σ F h p .Ln F =  ,c+2 /σ (h ,m )f r,q

= m(e2π i/3 ) .m = m(i)

= eπ i(h−3r)/3 F π i(h−3r)/4 χ h p .Lm F = e −iβ f r,q

.∗

.n

.m

.Ln F

.Ln F

.Ln F

.Ln F

.Lm F

e2π iρ F

3.4 Central Action

Lm(i)  ,c (h ,m )f

59

h

p r,q

.

m eπ i(h−3r)/4 i sign ( ) = √ 2| |   e−π icc /  ,c (h ,m ) f ·

h

(3.51)

p

r,q .

c mod 2

The left translation gives injective morphisms of .(g, K)-modules. In most cases left translation by the element in the table gives in the abelian case isomorphisms ∼ =

Fβ → Fβ , except for .Lm(i) : Fβ → F−iβ . In the non-abelian case the quantity 2π i/3 ) we get .d = 3 sign ( ) (2m + 1)h − 3r is preserved. For .n(1/σ, ρ)and .m(e isomorphisms .F ,c,d → F ,c,d . The other cases yield isomorphisms of the direct sum .⊕c mod 2 F ,c,d into itself. .

3.4 Central Action ψ

Inside the large Fourier term modules .FN there are many submodules .FN determined by the condition .(u − ψ(u)) f = 0 for all .u ∈ ZU (g), for a given character .ψ of the algebra .ZU (g). We start the study of these Fourier term modules by parametrizing the characters of .ZU (g), and next translating the condition that .ZU (g) acts by a character into a system of coupled linear differential equations. The coupling makes these ‘eigenfunction equations’ too hard to allow us to solve them explicitly, except in special cases. Nevertheless, the explicit availability of these eigenfunction equations is put to work in the last three subsections. We get information on the set of K-types that are present in Fourier term modules, and some information on the multiplicity; see Propositions 3.23 and 3.25. We arrive at these results by Proposition 3.21 and 3.24, which give necessary conditions for shift operators to have a non-trivial kernel on a given K-type. ψ Let .ψ be a character of .ZU (g). For .N = Nβ of .N = Nn we define .FN as the .(g, K)-submodule of functions .F ∈ FN that satisfy .uF = ψ(u)F for all .u ∈ ZU (g). This submodule is much smaller than .FN ; we call it a Fourier term module. Since .ZU (g) is a polynomial algebra in the Casimir element C and the element .3 of degree 3, the character .ψ is determined by its values .ψ(C), ψ(3 ) ∈ C. Example Let us consider the function in .F0 given by

 ϕ na(t)k = t 2+ν

.

2j

00,0 (k)

(3.52)

60

3 Fourier Term Modules

with .ν ∈ C, .j ∈ Z. Application of the formulas in Table 3.7, p. 50, or a computation in [6, §11c], gives

ν + j  2+ν 2j +3 1 S13 ϕ = 1 + 1,1 , t 2 .  1 2 −3 3 S−1 S1 ϕ = ν − (j + 2)2 ϕ . 8

(3.53)

2j

Since .ϕ has K-type .τ0 , it is a minimal vector in .F0 . With (iii) and (iv) in Lemma 3.4 we obtain: 1 λ2 (j, ν) := ν 2 − 4 + j 2 , 3

2 1  λ3 (j, ν) := (j + 3) ν − 9 (j − 6)2

Cϕ = λ2 (j, ν) ϕ , .

3 ϕ = λ3 (j, ν) ϕ ,

(3.54)

Parametrization by Weyl Group Orbits The functions .λ2 and .λ3 on .C2 are invariant under the transformations 1

 1 (3ν − j ), (j + ν) , 2 2 .  1  1 S2 : (j, ν) → − (3ν + j ), (ν − j ) . 2 2 S1 : (j, ν) →

(3.55)

This is in agreement with a theorem of Harish Chandra for general reductive Lie groups, stating that characters of the center of the enveloping algebra correspond bijectively with the Weyl group orbits of invariant polynomial functions on a Cartan subalgebra. See, e.g., [51], §3.2, specially Theorems 3.2.3 and 3.2.4. In our notations we use the Cartan subalgebra .ac ⊕ mc , and identify .C2 with its dual space, by letting the simple roots in (3.1) satisfy the correspondence .α1 ↔ (−3, 1), .α2 ↔ (3, 1). Then .(j, ν) ∈ C2 corresponds to the linear form that satisfies Hr → ν ,

.

Hi → ij .

A linear form on the Lie algebra corresponds to a (possibly multi-valued) character of the group. For the linear form corresponding to .(j, ν) ∈ C2 :

ν iy j e a(ex )m(eiy ) = exp(xHr + yHi ) → exν+y(ij ) = ex .

.

Since .m(e2π i ) = 1, only the .(j, ν) ∈ Z × C ⊂ C2 correspond to a character of the group AM. The condition .j ∈ Z would not be necessary if we were to work with the universal covering group of .SU(2, 1). This is a point where it is important to work with .(g, K)-modules, and not just .g-modules. The Weyl group W for .SU(2, 1) corresponds to the group of linear transformations of .C2 generated by .S1 and .S2 in (3.55). It is isomorphic to the symmetric group

3.4 Central Action

61

S3 . The general theory tells us that the characters of .ZU (g) are parametrized by the orbits of W in .C2 . Not all of these characters can occur in a .(g, K)-module.

.

Proposition 3.17 Let V be a .(g, K)-module in which .ZU (g) acts as multiplication by the character .ψ. Then there exist elements .(j, ν) ∈ Z × C such that ψ(C) = λ2 (j, ν),

.

ψ(3 ) = λ3 (j, ν) .

(3.56)

If .(j1 , ν1 ) ∈ C2 also satisfies this relation, then .(j1 , ν1 ) is in the orbit of .(j, ν) under the Weyl group W . We call .j ∈ Z and .ν ∈ C spectral parameters, and call .(j, ν) a spectral pair. Proof We pick a non-zero element in a highest weight subspace of V of some K−3 3 type, and apply downward shift operators until we have eigenvector of .S−1 S1 , say with eigenvalue .ϑ. Inserting .λ2 (j, ν) and .λ3 (j, ν) as eigenvalues of C and .3 we get two relations between j , .ν and .ϑ. Solving these relations with Mathematica, [6, §11d], leads to six solutions for .(j, ν). Among these solutions there are two solutions of the form j =

.

h − 3p , 2

 ν = ± a complicated expression in p, h and ϑ .

Since .h ≡ p mod 2 this shows that .(j, ν) with the desired properties exist. For the solutions of .λn (j1 , ν1 ) = λn (j, ν) for .n = 2, 3 we find precisely the orbit .W (j, ν), illustrated in Fig. 3.8.  In this way we have concluded directly for .SU(2, 1) how the characters of .ZU (g) that occur in .(g, K)-modules are parametrized. The result is in accordance with the general theory.

Fig. 3.8 Weyl group orbit of a point in .C2

62

3 Fourier Term Modules

Several Types of Parametrization By .W we denote the collection of the W -orbits in C2 that intersect .Z × C. We use the symbol .ψ in two ways. It may denote an element of .W, and it may denote the corresponding character of .ZU (g). If .(j, ν) ∈ C2 we denote the set .W (j, ν) ∈ W by .ψ[j, ν].

.

We put for .ψ ∈ W     OW (ψ) = (j, ν) ∈ ψ : j ∈ Z , O1W (ψ) = j : (j, ν) ∈ OW (ψ) ,   (j, ν) ∈ ψ : sign ( )(2j − d) + 3 ≤ 0 , . OW (ψ)n = OW (ψ) ,c,d =   O1W (ψ)n = j : (j, ν) ∈ OW (ψ)n . (3.57)

 In general .OW ψ[j, ν] ) has 2 elements, .(j, ν) and .(j, −ν). If the number of elements in .OW (ψ) is at most 2 we speak of simple parametrization. The set

.OW ψ[j, ν] has more than 2 elements if .3ν ≡ j mod 2, in which case we speak of multiple parametrization. We also make a distinction between integral parametrization and generic parametrization, summarized in the scheme in Table 3.11. The corresponding subsets of .W are indicated by the symbols .Wsp , .Wgp , .· · · indicated in the table. The points .(j, ν) giving integral parametrization form a lattice in .R2 minus the origin. See Fig. 3.9 for an illustration. Lemma 3.18 Let .ψ ∈ Wgmp . Then the elements of .O1W (ψ) represent all three classes in .Z/3Z. Proof The element .ψ[j, ν] corresponds to generic multiple parametrization if .ν =  1 1 3 (j + 2a) for some .a ∈ Z and .a ≡ j mod 3. Then .OW (j, ν) = j, a, −a − j }. One checks that these three elements represent pairwise different elements of .Z/3Z.  Table 3.11 Several types of parametrization of characters .ψ[j, ν] of .ZU (g). For each case we indicate the symbol corresponding to the set of .ψ ∈ W with that type of parametrization

≡ j mod 2 j =ν=0 .Simple parametrization   .OW (j, ν) ≤ 2 .3ν

.or

.Generic .Wsp

≡ j mod 2 .ν ≡ j mod 2 ν ≡ j mod 2 .and (j, ν) = (0, 0) .Multiple parametrization Wmp   .OW (j, ν) > 2 .3ν

.and

parametrization Wgp

Integral parametrization .Wgmp

.Wip

3.4 Central Action

63

Fig. 3.9 Points for integral parametrization, depicted in the .(j, ν)-plane (j horizontal, .ν vertical.) We have chosen the scaling such that the lines fixed by .S1 (given by .ν = j ), fixed by .S2 (given by .ν = −j ), and fixed by .S1 S2 S2 (given by .ν = 0) intersect each other in angles of size .π/3. The thick points indicate the position of the root system .A2

3.4.1 Eigenfunction Equations ψ

The definition of the Fourier term modules .FN imposes relations of the form .CF = ψ(C)F and .3 F = ψ(3 )F . To make these relations more explicit we note that ψ ψ .F N is the direct sum of the subspaces .FN;h,p of the K-types occurring in it, and ψ

ψ

that the space .FN;h,p is known if we know the highest weight subspace .FN;h,p,p . A ψ

highest weight element in .FN;h,p,p has the form

  p F na(t)k = ur (n) fr (t) h r,p (k)

.

(3.58)

r

with basis functions .ur on .N, component functions .fr in .C ∞ (0, ∞), and the polynomial basis functions on K discussed in Sect. 2.2. The summation parameter runs over .r ≡ p mod 2, .|r| ≤ p, with the additional condition (3.41) in the nonabelian case. The prescribed action of C and .3 imposes further relations for the components, which we call the eigenfunction equations. To specify the eigenfunction equations it is convenient to distinguish the following Fourier term modules: • N -trivial modules: .N = N0 . • Generic abelian modules: .N = Nβ , .β = 0. • Non-abelian modules: .N = N ,c,d with . ∈

σ 2 Z =0 , .c

mod 2 , .d ∈ 1 + 2Z.

64

3 Fourier Term Modules

Except for special cases, the eigenfunction equations are complicated and ask for computer help. We use the differentiation routines in Sect. 3.2 and adapt them, in [6, §11a], to the use with elements of the enveloping algebra .U (g). For each of the three cases indicated above we develop routines to give the eigenfunction equations. We choose to write them with .(j, ν) ∈ Z × C as the parameters. Hence the equations will differ if we go to another element of .OW (ψ[j, ν]). In [6, §11efg] we derive the eigenfunction equations, and keep in §11h routines for later use, to avoid recomputation of the relation every time that we need them in later sections. Lemma 3.19 (N -trivial Fourier Terms) The components .fr of the element .F ∈ ψ F0 in (3.58) must satisfy 0=

t 2 fr





− 3tf +

.

 (h − 3r)2 j2 2 −ν +4− fr , 12 3

(3.59)

0 = (h − 3r − 2j )(h − 3r − 3ν + j )(h − 3r + 3ν + j ) fr . Proof A Mathematica computation in [6, §11e] shows that

.

 h p C − λ2 (j, ν) fr r,q r p

gives an expression in which the factor of . h r,q depends only on .fr and its derivatives. We get an uncoupled system of differential equations for .fr . Up to a factor these are the differential equations in the first line of (3.59). The equation for .3 also gives an uncoupled system of differential equations. Subtraction of . 21 (h − 2r + 6) times the first equation gives the relation in the second line.  The three factors in the second equation are permuted by the action of the Weyl group on .(j, ν). This is like it should be, since the eigenvalues of C and .3 are invariant under the action of the Weyl group. The second equation shows that non-zero solutions are possible only for certain values of .r. ψ

Lemma 3.20 (Abelian Fourier Terms) The components .fr of the element .F ∈ Fβ in (3.58) must satisfy the following relations, where .r ≡ p mod 2, .|r| ≤ p: 0 = t 2 fr − 3tfr +

.

 (h − 3r)2

− ν2 + 4 −

 j2 − 4π|β|2 t 2 fr 3

12 ¯ + 2π i(p − r) β t fr+2 − 2π i(p + r) β t fr−2 ,  0 = (h − 3r − 2j )(h − 3r − 3ν + j )(h − 3r + 3ν + j )  + 216π 2 |β|2 r t 2 fr

3.4 Central Action

65

   − 27π i(p − r) β¯ 2t 2 fr+2 + (3r − h − 2) t fr+2    − 27π i(p + r) β 2t 2 fr−2 + (h − 3r − 2) t fr−2 . Proof The computation in [6, §11f] is more complicated than in the N-trivial case, since there are more terms. Moreover, neighboring components are coupled. We have to rearrange the sum of individual terms in such a way that all contain the same p factor . h r,p . After that we carry out a simplification by subtracting a multiple of  the first relation from the second one. The relation should be valid for all r between .−p and .p. The terms with .fr±2 contain the factor .p ∓ r, which masks components that do not exist. Substitution of .β = 0 in Lemma 3.20 gives (3.59). Non-abelian Fourier Terms The computations in [6, §11g] give eigenfunction ψ equations in non-abelian modules .Fn that are complicated, and copying them here seems not to make sense. In Table 3.9, p. 52, we managed to describe the shift operators uniformly for . > 0 and . < 0. For the eigenfunction equations it is simpler to consider separate formulas for .ε = sign ( ) = 1 and .−1. The final form is in [6, §11h].

3.4.2 One-Dimensional K-types In general, the coupling between the equations for the components is an obstruction to get explicit solutions. The one-dimensional K-types are an exception, since then there is only one component function. ψ[j,ν] The functions in .FN;h,0 have the form

F na(t)k) = u(n) f (t) h 00,0 (k)

.

with the function .u ∈ C ∞ (N ) determined by .N, and .f ∈ C ∞ (0, ∞). In all cases the eigenfunction equations give a second order differential equation for f and a second equation depending only on .f, imposing conditions on .h. The computations in [6, §12] show that in all cases this condition is (h − 2j )(h − 3ν + j )(h + 3ν + j ) = 0 .

.

(3.60)

So .h = 2j is one possibility. The other factors are relevant only under multiple parametrization. (See Table 3.11 on p. 62.) Each element .j  ∈ O1W (ψ[j, ν]) gives  .h = 2j as a possibility.

66

3 Fourier Term Modules

3.4.2.1

N-trivial Modules

From Lemma 3.19 we get the differential equation t 2 f  − 3tf  + (4 − ν 2 )f = 0 .

.

(3.61)

It has a two-dimensional solution space spanned by .t → t 2+ν and .t → t 2−ν if 2 2 .ν = 0, and by .t → t and .t → t log t if .ν = 0.

3.4.2.2

Generic Abelian Modules

ψ[j,ν]

For .Fβ

with .h = 2j, we find for .β = 0 that f (t) = t 2 jν (2π|β|t)

.

(3.62)

where .jν is a solution of the modified Bessel differential equation τ 2 jν (τ ) + τ jν (τ ) − (τ 2 + ν 2 ) jν (τ ) = 0 .

.

(3.63)

We get a two-dimensional solution space spanned by modified Bessel functions, discussed in Sect. A.1.

3.4.2.3

Non-abelian Modules ψ[j,ν]

In [6, §12c] we obtain in the same way for .F ,c,d a component function of the form f (t) = t wκ,ν/2 (2π| |t 2 ) ,

.

(3.64)

where .wκ,s is a solution of the Whittaker differential equation 1

κ s 2 − 1/4  (3.65) + wκ,s 4 τ τ2

 with parameters .κ = −m − j sign ( ) + 1 /2, .m ∈ Z≥0 , and .s = ν/2. So here as well we have a two-dimensional solution space spanned by known functions. In Sect. A.2 we discuss facts concerning Whittaker functions. We note that the definition of .F ,c,d implies that  wκ,s =

.

m=

.

−

sign ( ) 1 (d − 2j ) − , 6 2

1 κ = − sign ( ) (d + j ) . 6

(3.66)

3.4 Central Action

67 2j

In (3.39) we arrived at a condition that is, for .τph = τ0 , equivalent to . sign ( )d ≥ ψ[j,ν]

2 sign ( )j + 3. It is equivalent to .m ≥ 0. So .Fn;2j  ,0 is non-trivial with dimension 2

 if and only if .j  is in the set .O1W ψ[j, ν] n . (See (3.57).) ψ

Summary Let .β ∈ C. The space .Fβ;2j,0 is non-trivial if and only if .j ∈ O1W (ψ). ψ

Let .n = ( , c, d). The space .Fn;2j,0 is non-trivial if and only if .j ∈ O1W (ψ)n . These 2j

spaces of K-type .τ0 have dimension 2. Remark For higher-dimensional K-types we arrive, for generic abelian and nonabelian modules, at coupled systems of differential equations for which it is hard to get explicit solutions, except in special cases. ψ

Aim We now proceed the investigation of the modules .FN by deriving conditions under which the shift operators may have a non-trivial kernel.

3.4.3 Kernels of Downward Shift Operators ψ

We consider modules .FN with .N = Nβ or .N = Nn , .n = ( , c, d). It suffices to consider the kernels of downward shift operators on K-types .τph with .h ≡ p mod 2 and .p ∈ Z≥0 . ±3 : FN;h,p,p → FN;h±3,p−1,p−1 Proposition 3.21 Let .ψ ∈ W. If the kernel of .S−1 is non-trivial then ψ

h ∓ 3p = 2j

.

ψ

(3.67)

for some .j ∈ O1W (ψ). (The occurrences of .± in the statement of the proposition are coupled.) This ±3 proposition gives only a necessary condition for .S−1 to have a non-trivial kernel ψ

on .FN;h,p,p .

Proof For .p = 0 both downward shift operators are zero. That is relevant here only ψ if .FN;h,0,0 is non-zero. The eigenvalue equations for the generators C and .3 of 1 .ZU (g) imply that this occurs only if .h = 2j for some .j ∈ O (ψ). W For .p ≥ 1 as well, we use the eigenvalue equations of C and .3 , but we have to choose how to apply these equations. This requires different choices for the N-trivial case, the generic abelian case, and the non-abelian case. Supporting computations are in [6, §13].

68

3 Fourier Term Modules ψ

Elements of .FN;h,p,p have the form

 F na(t)k =



.

p

ur (n) fr (t) h r,p (k) ,

(3.68)

r:(−p,p)

with .ur = χβ if .N = Nβ , and .ur = ϑm(h,r) if .N = Nn . We first consider the N-trivial case. Lemma 3.19 shows that .h − 3r = 2j for 3 in Table 3.8, p. 51, shows how some .(j, ν) ∈ OW (ψ). The kernel relation for .S−1  .fr depends on .fr . (We have .β = 0.) Inserting this in the eigenfunction relations shows that (h − 3p − 3ν + j )(h − 3p + 3ν + j ) = 0

.

for non-zero .fr . So .h = 3p + 2j  for .j  ∈ O1W (ψ). This is what the lemma states 3 . For .S −3 we proceed similarly. for .S−1 −1 In the generic abelian case we also use the kernel relations in Table 3.8, but now 3 satisfy a with .β = 0 and .ur (n) = χβ in (3.68). The components .fr of .F ∈ ker S−1 relation expressing .fr+2 in terms of .fr . So, if the element in the kernel is non-zero then .f−p has to be non-zero. We use the eigenfunction relations in Lemma 3.20 for .r = p for some choice of .(j, ν) such that .ψ = [(j, ν)]. The occurrences of .f−p−2 in the eigenfunction relation are masked by the factor .p + r = 0. For .f2−p we use the relation obtained from the kernel relations. We obtain two quantities that have to be zero. A suitable linear combination is (h − 3p − 2j )(h − 3p + j − 3ν)(h − 3p + j + 3ν)f−p .

.

So indeed .h = 3p + 2j  for some .j  ∈ O1W (ψ) as a necessary condition. The case −3 of .S−1 goes similarly. In the non-abelian case the description of an element of .Fn;h,p,p in terms of its components is more complicated. We use the description and notations in (3.42). 3 the relation We find for F in the kernel of .S−1 ⎧  ⎨ √ −i 2  4 2π| |m(h,r) .fr = (h − 2p − r − 2 + 4π t )fr−2 + 2tfr−2 · i ⎩ √ 

4 2π| |(m(h,r)+1)

if > 0 if < 0 (3.69)

under the condition .m(h, r) ≥ 1 if . > 0 and .m(h, r) ≥ 0 if . < 0. So non-zero elements of the kernel are determined by the component .f−p if .m(h, −p) ≥ 0, or by the component .fm(r0 ) with .r0 > p and .m(h, r0 ) = 0. There are many cases to consider, worked out in [6, §13c]. There are two easy cases: .ε = 1, .r0 = p, and .ε = −1, .r0 = −p. Then there is only one non-zero component. The second

3.4 Central Action

69

coordinate of the eigenfunction equations gives the condition (h − 3p − 2j ) (h − 3p − 3ν + j ) (h − 3p + 3ν + j ) = 0 .

.

(3.70)

This implies the necessary condition in the proposition. The remaining cases are .ε = 1, .r0 < p, with .r = −p; .ε = 1 .−p ≤ r0 < p, with .r = r0 ; and .ε = −1, .r0 > −p, with .r = −p. In the eigenfunction equation occur .fr and .fr+2 . We use (3.69) to replace .fr+2 and its derivative by expressions concerning .fr . Then we take a suitable linear combination of the two coordinates of the eigenfunction equations, and observe that it gives (3.70) in all cases. −3 The case of the operator .S−1 requires also the consideration of many cases, all of which we work out in [6, §13c].  The proposition has the consequence that non-trivial kernels of downward shift operators occur only in K-types corresponding to points on at most three lines in the .(h/3, p)-plane. Figure 3.10 illustrates this in a case of generic multiple parametrization. Definition 3.22 To each .j ∈ Z we associate the set .Sect(j ) of K-types .τph that satisfy .h ≡ p mod 2 and .|h − 2j | ≤ 3p. The set .Sect(l) corresponds in the .(h/3, p)-plane to the lattice points satisfying h ≡ p mod 2 that are on or between the lines .h = 2j − 3p and .h = 2j + 3p in (3.67). If .ψ ∈ Wsp there is one sector .Sect(j ), with .{j } = O1W (ψ).

.

ψ[j,ν]

Fig. 3.10 Points in the .(h/3, p)-plane corresponding to K-types in .FN with possibly non3 (lines with slope 1) and .S −3 (lines with slope .−1), for .(j, ν) = (−3, 1/3) trivial kernels of .S−1 −1 (generic multiple parametrization)

70

3 Fourier Term Modules

ψ

Fig. 3.11 Points in the .(h/3, p)-plane corresponding to K-types that can occur in .FN under generic multiple parametrization. We use the same value .(j, ν) = (−3, 1/3) as in Fig. 3.10

If .ψ ∈ Wgmp there are three sectors, corresponding to the elements of .O1W (ψ). These three sectors have no lattice point in common, as illustrated in Fig. 3.11. If 1 .ψ ∈ Wip the two or three sectors .Sect(j ) with .j ∈ O (ψ) do have lattice points in W common. Proposition 3.23 Let .ψ ∈ W. ψ

(i) The K-types occurring in .FN are contained in ⎧ ⎨ j ∈O1 (ψ) Sect(j ) .  W ⎩ Sect(j ) 1 j ∈OW (ψ)n

if N = Nβ , β ∈ C , if N = Nn .

(3.71)

(ii) Let .ψ ∈ Wsp , .j ∈ O1W (ψ) if .N = Nβ and .j ∈ O1W (ψ)n if .N = Nn . Then, for all .p ∈ Z≥0 ψ

3 S−1 Fβ;2j +3p,p,p = {0} ,

.

ψ

−3 S−1 Fβ;2j −3p,p,p = {0} . ψ

(3.72) ψ

Proof We take a point .v = 0 in .FN;h,p,p for a K-type .τph that occurs in .FN . If .p ≥ 1 the point .(h/3, p) cannot be on both lines .h − 3p = 2j and .h + 3p = 2j 3 v or in Proposition 3.21 for the same .j ∈ O1W (ψ). So there is a non-zero vector .S−1 −3 h±3 .S −1 v, in the K-type .τp−1 . If the point .(h/3, p) is in a sector .Sect(j ), then the next point is in .Sect(j ) as well. The process can be continued until we reach a minimal vector. That can be a nonzero vector in a one-dimensional K-type, with .p = 0. In Sect. 3.4.2 we saw that this 2j determines at most three K-types, namely .τ0 with .j ∈ O1W (ψ), in the non-abelian 1 case .j ∈ OW (ψ)n .

3.4 Central Action

71

A minimal vector can also occur in a K-type with .p ≥ 1. Then we have by Proposition 3.21 h = 3p + 2j1 = −3p + 2j2

.

with two different elements .j1 , j2 ∈ O1W (ψ). In the non-abelian case at least one of .j1 and .j2 is in .O1W (ψ)n , since otherwise the intersection point would be in the region ruled out in Fig. 3.7. A point .(h/3, p) outside the union of the sectors stays outside the sectors, since ±3 the application of .S−1 sends it to a point .(h/3, p) + (±1, −1). A non-zero vector in ψ

FN;h,p,p stays non-zero under this process. So we would end in a point on the .(h/3)-

.

axis that is not of the form . 31 h = 23 j with .j ∈ O1W (ψ), which yields a contradiction to the results concerning one-dimensional K-types. This ends the proof of (i). ψ ±3 For (ii) we consider a non-zero vector .v ∈ FN;2j ±3p,p,p with .S−1 v = 0 for 1 1 .j ∈ O (ψ) (and .j ∈ O (ψ)n in the non-abelian case). Then .(h/3 ± 1, p − 1) is W W outside the sector .Sect(j ). Under simple parametrization, that is a single sector, and  we get a contradiction with (i).

3.4.4 Kernels of Upward Shift Operators We turn to the possibility that upward shift operators in Fourier term modules may have a non-zero kernel. Proposition 3.24 ψ

(i) If .ψ ∈ Wgp , then the upward shift operators in .FN are injective. (ii) Let .ψ ∈ Wip . The operators ψ

ψ

S13 : FN;h,p,p → FN;h+3,p+1,p+1 ,

.

S1−3 : FN;h,p,p → FN;h−3,p+1,p+1 ψ

ψ

are injective if .N = Nβ , .β = 0. The operator .S13 , respectively .S1−3 , may have a non-zero kernel for .N = N0 or .Nn if there are .j, j  ∈ O1W (ψ), .j = j  , such that (a) .τph ∈ Sect(j ), (b) .h + 3p + 6 = 2j  , respectively .h − 3p − 6 = 2j  , (c) if .N = N ,c,d , then . > 0 (respectively . < 0) and .j  ∈ O1W (ψ)n .

72

3 Fourier Term Modules

Proof Proposition 3.14 implies the statement in the generic non-abelian case .N = Nβ , .β = 0. In the other cases, the proof is based on the eigenfunction equations for C and .3 , like for Proposition 3.21. In the non-abelian case we have to handle a lot of separate cases. By Proposition 3.23 we need consider only those K-types .τph for which there exists .(j, ν) ∈ OW (ψ) such that .|h − 2j | ≤ 3p. We will show that the presence of ψ ±3 f = 0 implies that a non-zero element .f ∈ FN;h,p,p with .S−1 (h ± 6 ± 3p − 3ν + j ) (h ± 6 ± 3p + 3ν + j ) = 0 .

.

(3.73)

(All occurrences of .± are coupled.) This relation implies that .h±6±3p is equal to .2j  for some element .(j  , ν  ) of the Weyl group orbit of .(j, ν), illustrated in Fig. 3.8. In fact, .j  = 3ζ ν+j 2 , .ζ ∈ {1, −1}. So .h + 3p ≡ 2j  mod 6. On the other hand, the fact that .τph is in .Sect(j ) means that .h = 2j + 3(a − b) with .a, b ∈ Z≥0 and .a + b = p. So .2j ≡ h + 3p mod 6 as well. Hence .j ≡ j  mod 2. Under generic parametrization this implies .j = j  (with the use of Lemma 3.18). However .|h−2j | ≤ 3p is not compatible with .h±3±3p = 2j . This rules out generic parametrization, and we are left with .ψ ∈ Wip . N -trivial Case The eigenfunction equations in Lemma 3.19 show that components ψ of elements of .F0 satisfy an uncoupled system of differential equations. So we can consider elements of the simple form

p F na(t)k) = fr (t) h r,p (k)

.

with .p ≡ r mod 2, .|r| ≤ p, and .h = 2j + 3r. ±3 We carry out some computations in [6, §14a]. From .S−1 F = 0 we see that .fr is ∓(h+j )/3−p . Insertion of this function into the eigenfunction equations a multiple of .t with .(j, ν) ∈ OW (ψ) gives relation (3.73). The discussion following (3.73) concludes the discussion of the N-trivial case.

Non-abelian Case Now we consider

  p .F na(t)k = ϑm(h,r) (n) fr (t) h r,p (k) , r

with .r = −p, 2 − p, . . . , p, with the restriction that only terms with .m(h,

r) ≥ 0 can contribute. We have .m(h, r ±2) = m(h, r)±ε by (3.31), and also .m h, εp ≥ 0 (otherwise all components of F vanish). We use the notation .ε = sign ( ). We carry out several computations, in [6, §14b].

3.4 Central Action

73

The kernel relations for the upward shift operator .S13 give the following relations between the components: fr = .

·

  

1  i(p + r) (2 + h + 2p − r + 4π t 2 fr−2 + 2tfr−2 4(2 + p − r)t ⎧ ⎨√ 1 if ε = 1 and m(r) ≥ 1 , 2π| |m(r) ⎩√

−1 2π| |m(r+1)

if ε = −1 and m(r) ≥ 0 , (3.74)

valid for .r ≥ 2 + max(r0 , −p) if .ε = 1, and for .r ≤ min(r0 , p) if .ε = −1. If ε = 1 and .r0 < −p, or if .ε = −1, the lowest all

component .fp−p determines  other components. A computation shows that .S13 ϑm f−p (t) h −p,p = 0 implies

.

ψ

f−p = 0, unless .ε = 1 and .m = 0. This implies that the kernel of .S13 on .Fn;h,p,p is zero, unless .ε = 1 and .−p ≤ r0 ≤ p. In this remaining case, the important component is .fr0 . If .r0 < p we use a kernel relation to express .fr0 +2 in .fr0 and its derivative, and insert this in the eigenfunction equations for .r = r0 . A Mathematica computation shows that this gives for .fr0 = 0 the relation

.

(h + 6 − 3p − 2j )(h + 6 − 3p − 3ν + j )(h + 6 − 3p + 3ν + j ) = 0 .

.

(3.75)

The factor .h + 6 − 3p − j cannot be zero, since .τph ∈ Sect(j ). The other two factors give (3.73), and we have .h + 6 + 3p = 2j  for .j  ∈ O1W (ψ), .j  = j . Since h−d .r0 (h) = 3 +1, by (3.42), the requirement .r0 ≥ −p implies that .h ≥ −3p +d −3. 2j 

Hence .2j  = h + 6 + 3p ≥ d + 3. The K-type .τ0 satisfies .2j  − 3 · 0 ≥ d + 3, and does not satisfy the condition in (3.39). So .j  ∈ O1W (ψ)n . If .r0 = p the kernel relation leads to a differential equation with explicit solutions 2 spanned by .fr0 (t) = t −(h+p)/2 e−2π t . Inserting this solution in the eigenfunction ψ equations shows that we have an element of .Fn with .ψ represented by .j = h−3p 2 h + 2. We note that .τp ∈ Sect(j ) for this choice of .(j, ν). It turns out and .ν = h+p 2 that .(j  , ν  ) = S1 (j, ν) satisfies .j  = h + 6 + 3p, hence (3.75) holds in this case as well. The same argument as for .−p ≤ r0 < p shows that .j  ∈ O1W (ψ)n . For the shift operator .S1−3 we proceed similarly. The kernel relations are fr = .

·

 

1  i(p − r) (2 − h + 2p + 4 − 4π t 2 )fr+2 + 2tfr+2 4(2 + p + r) t ⎧ −1 ⎨√ if ε = 1 and m(r) ≥ 0 2π | |(1+m(r)) ⎩√

1 2π | |m(r)

if ε = −1 and m(r) ≥ 1 . (3.76)

Now the highest non-zero component determines the other components.

74

3 Fourier Term Modules

Like in the case of .S13 there cannot be a non-trivial kernel if .ε = 1, or if .ε = −1 and .r0 > p. We have to consider the case .ε = −1 and .−p ≤ r0 ≤ p. In the case of a kernel element with one component we should have .f−p equal 2 to a multiple of .t → t (h−p)/2 eπ t . This satisfies the eigenfunction equations with 1 1 h .j = 2 (h + 3p) and .ν = 1 + 2 (p − h), for which .τp ∈ Sect(j ). It turns out that    .h − 6 − 3p = 2j with .(j , ν ) = S2 (j, ν). In the other case we substitute into the eigenfunction equations for .r = r0 the expression for .fr0 −2 that follows from the kernel relation. That leads to the relation (h − 6 − 3p − 2j )(h − 6 − 3p + j − 3ν)(h − 6 − 3p + f + 3ν) = 0 ,

.

and then to (3.73). So in this case as well .h − 6 − 3p = 2j  for .j  ∈ O1W (ψ). Further we have 2j  = h − 3p − 6 = 3r0 + d + 3 − 3p − 6 ≤ 3p + d − 3p − 3 = d − 3 ,

.

in contradiction to the requirement that .2j  ≥ d + 3; see (3.39).



ψ subspaces .FN;h,p,p

have dimension 2 if .τph ∈ Proposition 3.25 Let .ψ ∈ Wgp . The 1 Sect(j ) for some .j ∈ OW (ψ), and dimension 0 otherwise. ψ

Proof Under simple parametrization we have .dim FN;2j,0,0 = 2, by the summary at the end of Sect. 3.4.2. Repeated application of the upward shift operators brings us from the one-dimensional K-type to all K-types corresponding to points in .Sect(j ) for the sole .j ∈ O1W (ψ). The injectivity in i) in Proposition 3.24 gives multiplicity at least 2 for all K-types that occur. From any K-type we can go down to the K-type 2j .τ 0 by application of downward shift operators on highest weight spaces on which they are injective. So all multiplicities are equal to .2. Under generic multiple parametrization, the K-types correspond to points in three disjoint sectors, to each of which we can apply the same reasoning. 

3.5 Special Fourier Term Modules ψ

We turn to the structure of the modules .FN , under the assumption of generic parametrization, in which these modules are the direct sum of a finite number of ψ .(g, K)-modules. The study of modules .F N under integral parametrization will be carried out in the next chapter. ξ,ν In this section we discuss first the principal series modules .HK , which are ψ submodules of .F0 , for the character .ψ = ψ[jξ , ν] of .ZU (g). In the other modules .FN we distinguish submodules by their behavior on A. We define in this way ξ,ν ξ,ν modules .WN with exponential decay as .t ↑ ∞, and modules .MN with nice

3.5 Special Fourier Term Modules

75 ξ,ν

ξ,ν

behavior as .t ↓ 0. Under generic parametrization the modules .HK , .WN and ξ,ν .M N are isomorphic. We discuss a few intrinsically defined intertwining operators.

3.5.1 Principal Series and Logarithmic Modules ψ[j,ν]

For any choice of .(j, ν) ∈ Z × C the module .F0 in the principal series

ξ,ν

HK = .



h p ϕr,q



contains the following module

p

C hϕr,q (ν) ,

h,p,q

ν; na(t)k = t

(3.77)

2+ν h

p r,q (k) ,

with .ξ = ξj the character of M corresponding to .j ∈ Z. The sum is over integers satisfying .h ≡ p ≡ q mod 2, .|q| ≤ p, .|r| ≤ p, and .h = 2j + 3r. The element 2jϕ 0 is a solution of the differential equation in Sect. 3.4.2.1. . 0,0 ξ,ν

The module .HK depends on the choice of .(j, ν) in .OW (ψ[j, ν]). The module ψ[j,ν] .F depends only on the Weyl group orbit .OW (ψ[j, ν]), and contains (in 0 general) several principal series modules. Specialization of the results in Table 3.7, p. 50, gives the shift operators. 2+p±r p+1 (4 ± h + 2ν + 2p ∓ r) h±3ϕr±1,p+1 (ν) , 8(1 + p) p p−1 h p ϕr,p (ν) = (2ν ± h − 2p ∓ r) h±3ϕr±1,p−1 (ν) . 4(p + 1)

S1±3 hϕr,p (ν) = p

.

±3 S−1

(3.78)

Under the assumption of generic parametrization all upward operators are ξ,ν injective. See Proposition 3.24; or alternatively, check it with (3.78). So .HK is a special cyclic module (Definition 3.5). Its type is . λ2 (j, ν); 2j, 0; ∞, ∞ . Proposition 3.21 shows that all downward shift operators that stay in the sector .Sect(j ) are also injective. With Lemma 3.8 this implies the following result. Proposition 3.26 Under the condition of generic parametrization the representations in the principal series are irreducible special cyclic modules. p

We note that the basis vectors . hϕr,q form holomorphic families depending on .ν ∈ C. Proposition 3.27 Let .ψ ∈ Wgp such that .ν = 0 for all .(j, ν) ∈ OW (ψ). Then ψ

F0 =

.

(j,ν)∈OW (ψ)

ξ ,ν

HKj .

(3.79)

76

3 Fourier Term Modules

Proof The eigenfunction equations in Lemma 3.19 lead to solutions that have the form .t → t 2+ν on A with the values of .ν occurring as second coordinate in .OW (ψ). Comparison with the dimension statements in Proposition 3.25 shows that we have a basis of solutions, provided .ν = 0 does not occur in .OW (ψ).  If .ν = 0 occurs in .OW (ψ) we get solutions .t 2 and .t 2 log t. This leads to a basis of solutions, but the logarithmic solution does not occur in a principal series module. ξ,ν

ξ,−ν

Proposition 3.28 If .ψ[j, ν] ∈ Wgp , then .HK and .HK

are isomorphic.

Proof Since .λ2 (j, −ν) = λ2 (j, ν) this follows from Theorem 3.6 and the type  ξ,ν λ2 (j, ν); 2j, 0; ∞, ∞ of .HK . 

.

This isomorphism is well-known. See, e.g., Zhang [57, §5.5]. It can be chosen such that p

p

i0 = i0 (j, ν) : hϕr,q (ν) → c(h, p, r, ν) hϕr,q (−ν) , .

c(h, p, r, ν) =

(1 + (1 +

p−ν 2 p+ν 2

+ +

p−ν h−r 4 )(1 + 2 p+ν h−r 4 )(1 + 2

+ +

r−h 4 ) r−h 4 )

(3.80) .

This is well defined for generic parametrization. Some checks are in [6, §15b]. The map .i0 (j, ν) is meromorphic in .ν; the singularities occur only for some .(j, ν) corresponding to integral parametrization. Remark Under generic multiple parametrization there are more principal series modules corresponding to the same Weyl group orbit. If .j = j  in the same Weyl ξ,ν ξ  ,ν  group orbit then .HK and .HK are not isomorphic, since the parameter .h0 is given by 2j for principal series representations. Logarithmic Modules In the case of .ν = 0 the differential relations in (3.59) admit solutions with components of the form .c1 t 2 log t + c2 t 2 . To describe these solutions we use the intertwining operator .

 1

ψ[j,ν] ξ,ν . 1 − i0 (j, ν) : HK → F0 2ν

(3.81)

Since .c(h, p, r, 0) = 1 for .(j, 0) corresponding to generic parametrization, this operator is well defined for .ν = 0. In particular, the value of .c(h, p, q, 0) equals .1. Hence ν →

.

 1 p 1 − c(h, p, r, ν) hϕr,p (ν) =: 2ν

h p λq,r (ν)

(3.82)

extends holomorphically to .ν = 0 with a term with .t 2 log t in its components at ξ,ν → Fψ[j,ν] for .ν = 0. In this way we obtain an injective intertwining operator .H 0 ψ[j,ν] ξ,ν .ν in a neighborhood of 0 in .C. We call the image .L . 0 ⊂ F0

3.5 Special Fourier Term Modules

77

 Proposition 3.29 Let .j ∈ {0} ∪ 1 + 2Z . j,ν ∼ ξ,ν (i) .L0 = HK for .ν = 0 in a neighborhood of 0. ψ[j,0] ξ,0 ξ,0 = HK ⊕ L0 . (ii) .F0

3.5.2 Submodules Characterized by Boundary Behavior ψ

The modules .FN are generated by highest weight functions in the K-types of the form given in (3.58): 

 F na(t)k =

.

p

ur (n) fr (t) h r,p (k) .

r:(−p,p) ψ

Since .FN consists of K-finite vectors, each of its elements is determined by finitely many component functions .fr = fr (h, p). Definition 3.30 (Boundary Behavior) (i) A function f on .(0, ∞) has .ν-regular behavior at 0 if f (t) = t 2+ν h(t) ,

(3.83)

.

where h is the restriction to .(0, ∞) of an entire function. (ii) A function f on .(0, ∞) has exponential decay at .∞ if there exists .a > 0 such that f (t)  e−at

.

as t → ∞ .

(3.84)

(iii) An element .F ∈ FN;h,p,p , has the property in i), respectively ii), if all its component functions have this property. Examples p

j,ν

(i) All . hϕr,q (ν) in the principal series module .HK have .ν-regular behavior at .0. ψ[j,ν] (ii) The function .μ0,0 with .β = 0 β (j, ν) ∈ Fβ

 2 μ0,0 β j, ν; na(t)k = χβ (n) t Iν (2π|β|t)

.

2j

00,0 (k)

(3.85)

has .ν-regular behavior at .0. This follows from the expansion (A.2). This ψ[j,ν] function is an element of .Fβ;2j,0,0 ; see Sect. 3.4.2.2. (iii) Similarly we find with .m0 = 6ε (d − 2j ) − 21 ∈ Z≥0 and .ν ∈ Z≤−1 the function



 2 μ0,0 ,c,d j, ν; na(t)k =  ,c h ,m0 ; n t Mκ,ν/2 (2π| |t )

.

2j

00,0 (k) , (3.86)

78

3 Fourier Term Modules

 where .κ = −m0 − 12 εj +1 . See the expansion (A.9) for the .ν-regular behavior at .0. ψ[j,ν] (iv) The function .ωβ0,0 (j, ν) in .Fβ with .β = 0, given by

 ωβ0,0 j, ν; na(t)k = χβ (n) t 2 Kν (2π |β|t)

.

2j

00,0 (k)

(3.87)

has exponential decay at .∞. See (A.5). ψ[j,ν] (v) The function .ωn0,0 (j, ν) in .F ,c,d , with .m0 and .κ like in iii), given by 

 0,0

j, ν; na(t)k =  ,c h ,m0 ; n t Wκ,ν/2 (2π | |t 2 ) ω ,c,d

.

2j

00,0 (k)

(3.88)

has exponential decay at .∞. See (A.14). Proposition 3.31 The properties of .ν-regular behavior at 0, exponential decay, and polynomial growth at .∞ are preserved under the action of .g and .K. Proof Since the actions of K and of .k do not change the component functions, it suffices to show that the actions of .Z31 , Z23 , Z32 , Z13 ∈ gc preserve these properties. On each K-type these elements can be related to shift operators. Tables 3.7 and 3.8 pp. 50, 51 describe the action of the shift operators in the module .FN . Inspection shows that the operations on the components are linear combinations of .b → b, .b → t b, .b → t 2 b, and .b → t b . If .b(t) = t 2+ν h(t) with h extending as a holomorphic function on .C, these operations change h by .h → t c h, .c = 1, 2, 3, or by .h → t h + (2 + ν)h. So the shift operators preserve the property of .ν-regular behavior at .0. For the property of exponential decay at .∞ we use the convolution representation ψ theorem of Harish Chandra; Theorem 1 on p. 18 of [18]. One writes .F ∈ FN in the form 

 .F nak = F (nakg −1 )α(g) dg (3.89) G

with .α ∈ Cc∞ (G). So .kg −1 in the integral runs over a compact set, and we can write .kg −1 = n1 a1 k1 where .n1 , .a1 , and .k1 run over compact sets in .N, .A, and −1 = (n an a −1 ) aa k , with .a(t)a = a(t ) where .K, respectively. Then .nakg 1 1 1 1 1 .t/b ≤ t1 ≤ tb for some .b > 1 depending on .α. Right differentiation by an element of .g can be carried out on  



.XF na(t)k = F na(t)kg1−1 Xα(g1 ) dg1 . (3.90) G

If .g1 varies through a compact set then the Iwasawa components .n1 , .a(t1 ) and .k1 in na(t)kg1 = n1 a(t1 )k1 vary through compact sets. In particular there is .b > 1 such that .t/b ≤ t1 ≤ tb. This preserves the estimate of exponential decay. In a similar  way, we see that also polynomial growth is preserved.

.

3.5 Special Fourier Term Modules

79

Proposition 3.31 suggests the following definitions: Definition 3.32 We put   ψ ψ WN = F ∈ FN : F has exponential decay at ∞ .

(3.91)

.

Notation It is convenient to use the following subsets of Weyl group orbits. For ψ ∈ W we put:

.

  OW (ψ)+ = (j, ν) ∈ OW (ψ) : Re ν ≥ 0 , .

(3.92)

+ OW (ψ)+ n = OW (ψ)n ∩ OW (ψ) .

The restriction of the projection map .OW (ψ) → O1W (ψ) to .OW (ψ)+ → O1W (ψ) is a bijection. ψ

Definition 3.33 We define .MN as the .(g, K)-submodule of linear combinations of ψ functions .F ∈ FN that have .ν-regular behavior at 0 for some .(j, ν) ∈ OW (ψ)+ . Remark Let .ψ ∈ W. The principal series modules .HK with .(jξ , ν) ∈ OW (ψ)+ ψ ξ,ν ξ,ν are submodules of .M0 . For other .N, we will define, in (3.94), .MN and .WN as ξ,ν

ψ[jξ ,ν]

submodules of .MN

ψ[jξ ,ν]

, respectively .WN

, with similar properties. ψ

ψ

Lemma 3.34 Let .N be .Nβ with .β = 0, or .Nn . If .ψ ∈ Wgp then .MN ∩ WN = {0}. ψ

ψ

ψ

Proof The intersection .MN ∩ WN is a .(g, K)-submodule of .FN . Consider a nonψ ψ zero element of .MN;h,p,p ∩WN;h,p,p . Then .τph ∈ Sect(j ) for some .j ∈ O1W (ψ), by Proposition 3.25. Proposition 3.21 implies that if .p ≥ 1 at least one of the downward shift operators is injective. Thus, we get a non-zero element in the intersection of ψ h±3 K-type .τp−1 . Proceeding in this way we arrive at a non-zero element in .MN;2j,0,0 ∩ ψ

ψ

0,0 WN;2j,0,0 . We know an explicit basis .ωN (j, ν), .μ0,0 N (j, ν) of the space .FN;2j,0,0 ; 0,0 see (3.85)–(3.88). Of these, only .ωN (j, ν) has exponential decay at .∞, and it has no .ν-regular behavior at 0. A non-zero element with both properties does not exist. 

Basis Families Let .N = Nβ with .β = 0, or .N = Nn . We put for .a, b ∈ Z≥0 , (a, b) = (0, 0),

.

3 a −3 b 0,0 ψ[j,ν] S1 μa,b μN (j, ν) ∈ MN;2j +3(a−b),a+b,a+b , N (j, ν) = S1 . b 0,0

a

ψ[j,ν] a,b ωN (j, ν) = S13 S1−3 ωN (j, ν) ∈ WN;2j +3(a−b),a+b,a+b .

(3.93)

80

3 Fourier Term Modules

Lemma 3.35 Let .N = Nβ , .β =

0, or .N = Nn . Take .(j, ν) ∈ Z × C, put .ψ = ψ[j, ν], and assume that . 6ε d − 12 − 3ε j ∈ Z≥0 if .N = Nn . (i) The functions in (3.93) form meromorphic families in .ν. The families .ν → a,b ωN (j, ν) are even and holomorphic in .C. In the abelian case, the families .ν → μa,b β (j, ν) are holomorphic on .C and a,b satisfy .μa,b β (j, −n) = μβ (j, n) for .n ∈ Z. In the non-abelian case, the family

ν → μa,b n (j, ν) may have singularities at points of .Z≤−1 . If a singularity occurs at .−ν0 ∈ Z≤−1 , then it has first order, with a multiple of .μa,b n (j, ν0 ) as its residue. (ii) The components of the functions in (3.93) are linear combinations of special functions of the following type .

.

for ωβa,b (j, ν) :

t → t c Kν+k (2π|β|t)

c ∈ Z≥2 , k ∈ Z≥0 ,

for μa,b β (j, ν) :

t → t c Iν+k (2π|β|t)

c ∈ Z≥2 , k ∈ Z≥0 ,

for ωna,b (j, ν) :

t → t c Wκ+k,ν/2 (2π | |t 2 )

c ∈ Z≥1 , k ∈ Z≥0 ,

for μa,b n (j, ν) :

t → t c Mκ+k,ν/2 (2π | |t 2 )

c ∈ Z≥1 , k ∈ Z≥0 ,

 with .κ = −m0 (j ) − 12 εj + 1 . ψ a,b (iii) Let .a, b ∈ Z≥0 . Then .ωN (j, ν) ∈ WN . If .(j, ν) ∈ OW (ψ)+ , then ψ a,b .μ N (j, ν) ∈ MN . (iv) If .ψ ∈ Wgp , then the spaces ξ,ν

WN

:=

a,b U (k) ωN (j, ν)

a,b≥0 .

ξ,ν

MN

:=

U (k) μa,b N (j, ν)

provided ν ∈ Z≤−1 .

(3.94)

a,b≥0 ψ

ξ,ν

ψ

are .(g, K)-submodules of .FN . In particular, .WN ⊂ WN , and if .(jξ , ν) ∈ ψ ξ,ν OW (ψ)+ , then .MN ⊂ MN . ξ,ν

ξ,ν

Remark We postpone the definition of the modules .WN and .MN under integral parametrization till Lemma 4.10 and Definition 4.20. Proof The statements in (i) are valid for .a = b = 0, as can be checked in Appendix A; see in particular (A.4), (A.11), (A.3) and (A.10). The properties are preserved under application of the shift operators. In the proof of Proposition 3.31 we gave the action of the shift operators on the component functions. We apply this repeatedly to the special function in the cases when .(a, b) = (0, 0). Then we apply

3.5 Special Fourier Term Modules

81

the contiguous relations in (A.7) and (A.19) to see that we stay in the linear space spanned by the functions indicated in (ii). This gives also (iii). Under generic parametrization the upward shift operators are injective. So a,b the elements .μa,b N (j, ν) and .ωN (j, ν) are non-zero and linearly independent. ψ[j,ν] . Proposition 3.25 (on the dimensions) implies that together they span .FN a,b (j, ν) and .μa,b (j, ν) to a linear So the downward shift operators send both .ωN N 







a ,b (j, ν) and .μaN,b (j, ν). The downward shift operators also combination of .ωN ξ,ν ξ,ν preserve linear combinations as indicated in ii). So they preserve .WN and .MN . ξ,ν In the definition of .MN we imposed the condition .ν ∈ Z≤−1 , thus avoiding the  complications that may be caused by singularities.

Proposition 3.36 For all .p ∈ Z≥0

.

p,0

−3 S−1 ωN (j, ν) = 0

for ν ∈ C ,

p,0

−3 S−1 μN (j, ν) = 0

for ν ∈ C  Z≤−1 .

3 S−1 ωN (j, ν) = 0 3 μN (j, ν) = 0 S−1

0,p 0,p

(3.95)

Proof Under simple parametrization this follows from (ii) in Proposition 3.23. The p,0 0,p families .xN (j, ·) and .xN (j, ·) are holomorphic in their domain, and that property  is preserved under differentiation. Proof (of Theorems A and B) The role of .m0 (j ) in Theorem B is based on the 2j discussion in Sect. 3.4.2 of the functions in the K-type .τ0 , which have the form 2j 0 (k), with a normalized Hermite function .h .nak →  ,c (h ,m ; n) ft ) ,m with 0,0 .m ∈ Z≥0 . The eigenfunction equations show that K-type .m = m0 (j ) in the case of a one-dimensional K-type. ψ ψ Lemma 3.35 gives elements of .WN and .MN in terms of basis functions, both in the generic abelian case and in the non-abelian case. Since the modified Bessel functions or Whittaker functions are linearly independent, this shows that the Ktypes with .|h − 2j | ≤ 3p occur in both modules with multiplicity at least one. Proposition 3.25 implies that the multiplicities are exactly one. Proposition 3.21 tells that the downward shift operators vanish on boundaries of ξ,ν ξ,ν the sectors .Sect(j ). Hence there are no K-types in .WN or .MN that do not satisfy .|h − 2jξ | ≤ 3p. Proposition 3.24 gives the injectivity of the upward shift operators. ξ,ν ξ,ν So .MN and .WN are special cyclic modules as in Definition 3.5, with parameter  ξ,ν set . λ2 (jξ , ν); 2jξ , 0; ∞, ∞]. Hence they are isomorphic to .HK by Theorem 3.6 and Proposition 3.26.  ψ

ψ

Remark 3.37 Among these modules, .FN and .WN are defined in an intrinsic way; ψ ψ .F N as the codomain of the Fourier term operator .FN , with the submodule .WN determined by the condition of exponential decay.

82

3 Fourier Term Modules

Under generic parametrization and the additional condition .ν ∈ Z≤−1 for ψ ξ,ν ξ,ν Mn , we define inside .WN the special Fourier term modules .WN generated by ξ,ν 0,0 + .ω N (jξ , ν) with .ν ∈ OW (ψ) . The modules .MN are intrinsically defined by the ψ condition of .ν-regular behavior at 0. The definition of .MN in (3.33) is much less intrinsic. The restriction to .Re ν ≥ 0 is motivated by (i) in Proposition 3.36. .

3.5.3 Intertwining Operators Under generic parametrization, we have obtained various irreducible simple modules that are isomorphic by Theorem 3.6, namely .

ξ,ν

HK ,

ξ,−ν

HK

,

ξ,ν

Mβ ,

ξ,ν

Wβ ,

ξ,ν

ξ,ν

Mn ,

Wn ,

(3.96)

 where we take .β = 0, and .n = ( , c, d) such that .m0 = 6ε d − 2j − 21 ∈ Z≥0 . The isomorphisms are given by intertwining operators determined up to a non-zero 2j factor. We may fix them by prescribing them in the K-type .τ0 , for instance by 0,0 letting the basis vectors .xN correspond to each other. In this way we can also build, under generic parametrization, an injective morphism ξ,ν

ξ,ν

ξ,−ν

WN → MN ⊕ MN

.

(3.97)

based on (A.4) and (A.11). We discuss three types of intertwining operators that are defined in a more intrinsic way. Left Translations by Elements Normalizing the Lattice In Sect. 3.3.3 we discussed that left translation by elements normalizing the lattice . σ gives intertwining operators of .(g, K)-modules between large Fourier term modules .FN . Hence they ψ preserve the modules .FN . The description in terms of basis elements in Table 3.10, 0,0 p. 58, preserves the functions f on A. Hence they preserve the families .ωN and 0,0 a,b a,b .μ N , and by the intertwining property also the derivatives .ωN and .μN . So they preserve the special Fourier term modules as well. ξ,ν

Evaluation at Zero A basis element of K-type .τph in .MN has the form F =



.

p

ur (n) t 2+ν hr (t) h r,q .

r

For each r, the function .hr is entire, and .ur is a basis element on N , either a character or a theta function. The summation variable satisfies .|r| ≤ p, .r ≡ p mod 2, and

3.5 Special Fourier Term Modules

83

some further condition in the non-abelian case. Evaluation at zero is the operator ξ,ν ξ,ν E0 : MN → HK induced by

.

E0 F =



.

p

t 2+ν hr (0) h r,q .

(3.98)

r

So we replace all .ur by 1, and .hr (t) by its value at .t = 0. Proposition 3.38 Evaluation at zero is an intertwining operator of .(g, K)-modules. Proof Clearly .E0 commutes with the action of .k and .K. So it suffices to check the operation on a basis of a complementary space of .k in the Cartan decomposition, or for the shift operators on highest weight vectors. It is not too hard to do this by hand, on the basis of Tables 3.7, 3.9, pp. 50, 52, and the relations in (3.78). A check is in [6, §16].  An Inverse Operator In [15] Goodman and Wallach define, in a much more general context than .SU(2, 1), a linear form on the analytic vectors in principal series representations given by an infinite sum of differential operators. This induces a ξ,ν ξ,ν family of intertwining operators .HK → Mβ , which is inverse to evaluation at zero up to a factor that depends meromorphically on .ν. Kunze-Stein Operators An interesting family of intertwining operators is given by the Kunze-Stein operators. See Kunze, Stein [29], or Schiffmann [41]. These operators turn up in the computation of Fourier coefficients of Poincaré series. Here we mention their definition, but do not go into computations. The Kunze-Stein operators act on functions .F ∈ C ∞ (G)K that satisfy an estimate

 F na(t)k  t 2+ε

(3.99)

.

for some .ε > 0, uniformly in n (and k). For .β ∈ C and .η ∈ NwAMN (the big cell in the Bruhat decomposition) the abelian Kunze-Stein operator is given by  (Sβ (η)F )(g) =

.

n ∈N

χβ (n ) F (ηn g) dn ,

(3.100)

and for .n = ( , c, d) the non-abelian Kunze-Stein operator is

 (Sn (η)F ) nak =







 ,c (h ,m ; n)

m,h,p,r,q

n ∈N

.





· F (ηn ak )

h p (k  ) dk  dn r,q

k  ∈K

 ,c (h ,m ; n )

h p (k) r,q   h  pr,q 2

(3.101) .

84

3 Fourier Term Modules

The sum is over .m, h, p, r, q ∈ Z satisfying .m ≥ 0, .h ≡ p ≡ r ≡ q mod 2, |r| ≤ p, .|q| ≤ p, and . sign ( ) (6m + 3) + h − 3r = d. (These operators are similar to the Fourier term operators in Proposition 3.15.) Under the condition (3.99) ξ,ν the integrals converge absolutely. Applied to .F ∈ MN with .Re ν > 0 we get holomorphic families of intertwining operators

.

 Sβ (η) :

ξ,ν MN

.

ξ,ν

→

ξ,−ν

HK

ξ,ν Wβ ξ,ν

Sn (η) : MN → Wn .

,

if β = 0 , if β = 0 ,

(3.102)

Chapter 4

Submodule Structure

In the previous chapter we saw that under generic parametrization (see Table 3.11, ξ,ν ξ,ν p. 62) the special Fourier term modules, like .MN and .WN , are isomorphic if they determine the same element .[jξ , ν] ∈ W. Under integral parametrization this is no longer the case. Then these modules have non-trivial submodules, and the lattice of submodules is not determined only by an element of .W. The main purpose of this ψ chapter is to determine the submodule structure of all modules .FN , and to give a proof of Theorems C and D in the Introduction. By a general theorem, all irreducible .(g, K)-modules occur as a submodule of a principal series module. So the irreducible modules that we will find in Sect. 4.2, discussing the principal series, represent all isomorphism classes of irreducible ψ .(g, K)-modules. In comparison the generic abelian modules .F β in Sect. 4.3 have a simpler submodule structure. In Sect. 4.4 we will see that the submodule structure of ψ the non-abelian modules .F,c,d is complicated, and depends not only on the spectral parameters, but also on the parameters . and d. Section 4.5 gives the irreducible .(g, K)-modules with a unitary structure.

4.1 Preliminaries The investigation of the submodule structure will be done in separate sections for the N-trivial case, the generic abelian case, and the non-abelian case. Here we carry out preparations that will be used in all these cases. The subquotient theorem of Harish Chandra states that all isomorphism classes of irreducible .(g, K)-modules can be realized as subquotients of some representation ξ,ν .H K in the principal series; see e.g. [51, Theorem 3.5.6]. A result by CasselmanMiliˇci´c [7] implies that we even get all irreducible .(g, K)-modules as submodules of ξ,ν some .HK . In Sect. 4.2 we will give the irreducible submodules of representations © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. W. Bruggeman, R. J. Miatello, Representations of SU(2,1) in Fourier Term Modules, Lecture Notes in Mathematics 2340, https://doi.org/10.1007/978-3-031-43192-0_4

85

86

4 Submodule Structure

in the principal series. Thus we will have the complete list of isomorphism classes of irreducible .(g, K)-modules. In Sect. 4.1.2 below we will give this list.

4.1.1 Lattice Points The modules in (3.96) are in the class of irreducible principal series modules. All ξ,ν other isomorphism classes are represented in some .HK with parameters .(jξ , ν) corresponding to integral parametrization. Under integral parametrization we deal with Weyl group orbits in the lattice   L = (j, ν) ∈ Z2 : j ≡ ν mod 2 .

.

(4.1)

See Fig. 3.9, p. 63. The origin .(0, 0) in this lattice does not correspond to integral parametrization. A fundamental set for the action of the Weyl group W on L is the closure of the positive Weyl chamber .L+ = {(j, ν) ∈ L : ν ≥ |j | . (It is positive for the choice of .α1 and .α2 in (3.1) as simple positive roots.) The walls of .L+ are the lines .j = ν ∈ Z≥1 and .−j = ν ∈ Z≥1 . We use the convention to denote elements of .L+ by .(j+ , ν+ ), with .j+ ≡ ν+ mod 2, and .ν+ ≥ |j+ |. Furthermore, we denote elements of the adjacent Weyl chambers by .(jr , νr ) ∈ S1 L+ and .(jl , νl ) ∈ S2 L+ . Hence .jr ≥ 1, .0 ≤ νr ≤ jr , and .jl ≤ −1, .0 ≤ νl ≤ −jl . On the walls we have .(j+ , ν+ ) = (jr , νr ) = (j, j ), .j ∈ Z≥1 , and .(j+ , ν+ ) = (jl , νl ) = (−j, j ), .j ∈ Z≥1 (Fig. 4.1). Fig. 4.1 Notation for elements of a Weyl group orbit

4.1 Preliminaries

87

If .(j+ , ν+ ), .(jr , νr ) and .(jl , νl ) are in the same Weyl group orbit we have the relation  + , , ν+ +j 2  −3ν+ −j+ ν+ −j+  . , 2 (jl , νl ) = S2 (j+ , ν+ ) = 2

(jr , νr ) = S1 (j+ , ν+ ) = .

 3ν+ −j+ 2

(4.2)

Under these relations we have the following identities: jr + jl + j+ = 0 ,

.

νr − ν+ + νl = 0 .

(4.3)

4.1.2 Isomorphism Classes of Irreducible Representations We give in Sect. 4.2.2 a list of isomorphism classes of irreducible .(g, K)-modules that are embedded in a principal series representation. As discussed in the introduction to this section, this is the complete list. All the irreducible .(g, K)-modules turn out to be special cyclic modules; see Definition 3.5. Here we classify the isomorphism classes into four types, .I I , .I F , .F I , and .F F , according to the action of the upward shift operators, which in Table 3.4, p. 36, are given by the action of elements of the complexified Lie algebra .gc . The first letter refers to .S13 , and the second letter to .S1−3 . This letter is I if the shift operator acts injectively in the module; otherwise it is F . Most classes have a K-type of dimension 1. If the dimension of the minimal K-type is larger than 1, we add a subscript .+. To completely determine the isomorphism class we add the choice of a ξ ,ν spectral pair .(j, ν) such that the irreducible module occurs in the module .HKj in the principal series. In many cases this choice is unique. The following list gives all isomorphism classes of irreducible .(g, K)-modules. We indicate also which of the classes admit a unitary structure, to be discussed in Sect. 4.5. • Irreducible principal series – .I I (j, ν) with .j ∈ Z, .Re ν ≥ 0, .ν ≡ j mod 2, or .(j, ν) = (0, 0). The principal ξ ,ν ξ ,−ν series modules .HKj and .HKj are in this isomorphism class. A unitary structure occurs in the following cases: Re ν = 0, unitary irreducible principal series. ν ∈ R, .0 < ν < 2 if .j = 0, or .0 < ν < 1 if j is odd, complementary series.

. .

• Discrete series types – .I I+ (j+ , ν+ ) with .j+ ∈ Z, .ν+ ∈ Z≥0 , .j+ ≡ ν+ mod 2, .ν+ ≥ |j+ |. Large discrete series type.

88

4 Submodule Structure

– .I F (jr , νr ) with .jr ∈ Z≥2 , .νr ≡ jr mod 2, .0 ≤ νr ≤ jr − 2. Holomorphic discrete series type. – .F I (jl , νl ) with .jl ∈ Z≤−2 , .νl mod jl mod 2, .0 ≤ νl ≤ |jl | − 2. Antiholomorphic discrete series type. All modules of discrete series type admit a unitary structure. • Langlands representations – .I F + (jr , −νr ) and .I F (jr , −jr ) with .jr ∈ Z≥1 , .νr ≡ jr mod 2, and .1 ≤ νr ≤ jr − 2 for .I F + (jr , νr ); .νr = jr for .I F (−jr ). – .F I + (jl , −νl ) and .F I (jl , jl ) with .jl ∈ Z≤−1 , .νl ≡ jl mod 2, and .1 ≤ νl ≤ |jl | − 2 for .F I + (jl , −νl ); .νl = jl for .F I (jl , jl ) . – .F F (j+ , −ν+ ) with .j+ ∈ Z, .ν+ ≡ j+ mod 2, .ν+ ≥ |j+ | + 2. Finitedimensional irreducible modules. The Langlands representations that admit a unitary structure are .F F (2, 0), and all classes .I F (jr , −1), .F I (jl , −1). We call the representations with .ν = −1 thin representations. These unitarizable representations occur in cohomology groups of .\G. See Ishikawa’s discussion in [23, §3]. Occurrence in Principal Series Modules These isomorphism classes are represented by one or more non-trivial submodules of a principal series representation and by one or more quotients of principal series representations. We will see this ξ ,ν explicitly in Sect. 4.2.2. The classes .I I (j, ν) are represented by .HKj , which is a (trivial) submodule and a (trivial) quotient of itself. Discrete Series Type Discrete series representations have a spectral pair .(j, ν) in the interior of a Weyl chamber in Fig. 3.9, p. 63. They are characterized by being represented in .L2 (G). Limits of discrete series correspond to .(j, ν) on a wall between Weyl chambers. These classes are not represented in .L2 (G). We put these classes together into a discrete series type. We distinguish holomorphic and antiholomorphic discrete series type. These concepts are interchanged if we change the complex structure of the symmetric space .G/K (or of the space into which .G/K is embedded). We prefer to keep both names, in accordance with [49, §7]. Langlands Representations All Langlands representations are non-tempered. They ξ,ν occur as quotients of a unique principal series module .HK with .ν ≥ 1, and are often called Langlands quotients.

4.1 Preliminaries

89

4.1.3 Submodules Determined by Shift Operators From Propositions 3.21 and 3.24 we obtain the information that kernels of shift operators can occur only in K-types .τph that correspond to points in the .( h3 , p)-plane on specific lines. This allows us to draw conclusions concerning submodules. Proposition 4.1 Let .j ∈ Z and let V be a .(g, K) module such that V =



.

(4.4)

V2j +3(a−b),a+b .

a,b≥0

(i) For .c ∈ Z and u a linear form on .R2 let .Xu,c be the subspace of V given by Xu,c =



.

Vh,p .

(4.5)

(h/3,p)∈Sect(j ) : u(h,p)≥c

The linear space .Xu,c is a proper .(g, K)-submodule of V in the following cases: 3 (a) .u(h, p) = − h3 − p, .c ≤ − 2j 3 , and .S1 Vh,p,p = {0} for all .(h/3, p) ∈ Sect(j ) such that .u(h, p) = c. −3 (b) .u(h, p) = h3 −p, .c ≤ − 2j 3 , and .S1 Vh,p,p = {0} for all .(h/3, p) ∈ Sect(j ) such that .u(h, p) = c. 3 (c) .u(h, p) = − h3 +p, .c ≥ 2j 3 , and .S−1 Vh,p,p = {0} for all .(h/3, p) ∈ Sect(j ) such that .u(h, p) = c. −3 (d) .u(h, p) = h3 + p, .c ≥ 2j 3 , and .S−1 Vh,p,p = {0} for all .(h/3, p) ∈ Sect(j ) such that .u(h, p) = c.

(ii) Let .Y ⊂ V be a submodule such that the subspace .Yh,p of K-type .τph is nonzero. If .Sβ3α Yh,p,p = 0 for .α, β ∈ {1, −1}, then .Yh+3α,p+β = 0. Remarks (1) In the proposition we do not assume that the module is generated by a minimal element, or that the Casimir element acts as multiplication by a scalar. We will ψ apply the proposition to the Fourier term modules .FN , in which the Casimir operator acts as a scalar. (2) The proposition enables us to reduce the study of submodules for the action of .g to the consideration of shift operators. We will use this result repeatedly in this chapter. Part (i) tells us that lines on which a shift operator vanishes determine submodules. Part (ii) tells that all submodules are visible in the vanishing of shift operators. Proof Part (ii) follows from the fact that .Yh,p = U (k)Yh,p,p . We turn to part (i)(c), sketched in Fig. 4.2. We use here, and later on, the convention that a downward arrow to a point .(h/3, p) indicates that application of the corresponding downward shift operator “stops at that point”: the space .Vh,p,p

90

4 Submodule Structure

Fig. 4.2 Illustration for (i)(c) in Proposition 4.1

Fig. 4.3 Illustration for (i)(d) in Proposition 4.1

is in the kernel of that shift operator. In the picture are two lines of points where the 3 is zero, and one line of points where .S −3 is zero. shift operator .S−1 −1 The action of the Lie algebra sends an element of .Vh,p to an element in the sum |  of spaces .Vh ,p with . |h−h 3 , |p − p | ∈ {−1, 0, 1}. The critical K-types correspond to points on the line in the interior of the sector. Let .(h/3, p) be such a point, and suppose that for .v ∈ Vh,p,q there exists .u ∈ U (g) such that .uv ∈ Vh+3,p−1 . The elements .Z12 and .Z21 in .kc change the weight in a K-type by one, and are injective except on the lowest or highest weight spaces, respectively. Using this we can reduce the situation to .v ∈ Vh,p,p and .uv ∈ Vh+3,p−1,p−1 . β γ We write u as a linear combination of elements in .U (k)Zα31 Z23 Z13 Zδ32 . A 3 v = 0 to terms contribution with .δ ≥ 0 can be reduced by the assumption that .S−1 with lower values of .δ. (We use the description of the shift operators in Table 3.4, p. 36.) Repeating this we arrange .δ = 0. Further, we note that .Z31 , .Z23 and .Z13 preserve the K-types with .u(h, p) ≥ c. (For .Z13 we use again Table 3.4.) This shows that .uv is contained in W . So under condition (c) we get invariance of .Xu,c . Under condition (d) we proceed in the same way, with a reversed role of .Z32 and .Z13 (Fig. 4.3). For (b) the module .Xu,c has K-types in the region between the lines with slope 1. For .c = 2j 3 both lines coincide, and .Xu,c is a non-trivial subspace of V . An upward arrow to a point .(h/3, p) indicates that the space .Vh,p,p is in the kernel

4.2 Principal Series and Related Modules

91

Fig. 4.4 Illustrations for (i)(b) in Proposition 4.1

Fig. 4.5 Illustration for (i)(a) in Proposition 4.1

of the corresponding upward shift operator. In the picture on the right the points (h/3, p) = (2j/3, 0) + a(1, 1) correspond to spaces .Vh,p,p on which both shift 3 and .S −3 vanish (Fig. 4.4). operators .S−1 1 Like above we reduce consideration to .v ∈ Vh,p,p with .u(h, p) = c, for which we know that .S1−3 v = 0. Decomposing u as above, we reduce to .γ = δ = 0 by 3 , and of .S −3 , does not decrease the value the observation that application of .S−1 −1 of .u(h, p). Since .S1−3 v = 0 we can remove all terms with .β ≥ 1. The proof for assumption (b) is completed by the observation that the action of .Z31 does not decrease the value of .u(h, p) (Fig. 4.5). For assumption (a) we proceed similarly with a reversed role of .Z31 and .Z23 .

.

Conventions in the Figures The conventions explained in the proof will be used for many figures in the .(h/3, p)-plane in the sequel.

4.2 Principal Series and Related Modules Under the assumption of integral parametrization, the variety of submodule structures of principal series modules is large: in Sect. 4.2.2 we will meet twelve different submodule structures. Most isomorphism classes in the list in Sect. 4.1.2 occur in only one principal series module. The type .I I + (j+ , ν+ ) forms the sole exception.

92

4 Submodule Structure ψ

In most cases of integral parametrization, the Fourier term module .F0 is the direct sum of a number of non-isomorphic principal series modules. If the spectral pair is on a wall of a Weyl chamber we need also the logarithmic submodules, discussed in Sect. 4.2.4. The last subsection, Sect. 4.2.5, is of a more technical nature. It is relevant not ψ only for the modules .F0 , but also for other Fourier term modules.

4.2.1 Kernels of Shift Operators and Submodules ξ,ν

K-types The K-types .τph occurring in .HK have multiplicity one. These K-types correspond to points in the sector .Sect(j ) in the .(h/3, p)-plane indicated in Fig. 4.6. Proposition 4.2 Let (j, ν) ∈ L and (jj , νj ) = Sj (j, ν), and let τph ∈ Sect(j ). Then hϕ p (ν), with h = 3r + 2j , is in the kernel of a shift operator precisely in the cases r,p in Table 4.1.

ξ,ν

Fig. 4.6 Points corresponding to the K-types occurring in HK . The downward shift operator 3 , respectively S −3 , is zero on the highest weight in K-types indicated by line with slope 1, S−1 −1 respectively −1 Table 4.1 Conditions for vanishing of shift operators on the function hϕ p (h−2j )/3,p (ν). We use

If

For

S13

j2 ≥ j + 3 (ν ≤ −j − 2)

h 3

+ p = 32 j2 − 2

j1 = 12 (3ν − j ) and j2 = − 12 (3ν + j )

S1−3

j1 ≤ j − 3

h 3

− p = 32 j1 + 2

3 S−1

j2 < j

h 3

− p = 32 j2

(ν ≤ j − 2)

(ν ≥ −j + 2)

all (j, ν) −3 S−1

j1 > j all (j, ν)

or (ν ≥ j + 2)

h 3

or

h 3

− p = 23 j

+ p = 32 j1 h 3

+ p = 32 j

4.2 Principal Series and Related Modules

93 p

ξ,ν

Proof We use that h = 2j + 3r for hϕr,p (ν) ∈ HK . p For all values of (j, ν) the element hϕ±p,p (ν) is in the kernel of the downward ±3 shift operator S−1 . The corresponding points in Fig. 4.6 are on the right, respectively left, boundary of the sector Sect(j ). This gives the second lines of the possibilities for h3 ∓ p in the last column. ±3 h p ϕ±p,p (ν) vanishes also if In (3.78) we see that S−1 

1 h−p .0 = ±h + 2ν − 2p ∓ r = 2 ν ± 13j ± 3

 .

3 this gives (We used that r = 13 (h − 2j ).) For S−1

.

2 1 h − p = −ν − j = j2 . 3 3

This represents   a line with slope 1 in Fig. 4.6 intersecting the horizontal axis in the point 23 j2 , 0 . If j2 < j this line has points in the sector minus the right boundary ξ,ν 3 vanishes. This line. So, if j2 < j it gives new spaces in HK;h,p,p on which S−1

−3 we obtain the line 13 h + p = 23 j1 , which gives the third box in Table 4.1. For S−1 −3 if j1 > j . This gives the bottom box in the leads to new kernel elements of S−1 table. p For the vanishing of S1±3 hϕr,p (ν) we find in (3.78) the condition

4 ± h + 2ν + 2p ∓ r = 0 .

.

This yields the vanishing of S13 for points in the sector on the line h3 + p = 23 j2 − 2 if j2 ≥ j + 3. This gives the first box in the table. We obtain the second box in a similar way. For general (j, ν) the upward shift operators are injective. So if the line that we find coincides with a boundary line of the sector Sect(j ) we get new information. Hence we need no strict inequalities in the first two boxes.

ξ,ν

Remark For two different points in a Weyl group orbit in Wip the space HK and ξ  ,ν 

HK have intersection zero. Nevertheless the boundary lines of Sect(j  ) have a j,ν significance for HK . On points in Sect(j ) ∩ ∂Sect(j  ) at least one shift operator vanishes. On points in ∂Sect(j ) ∩ ∂Sect(j  ) at least two shift operators vanish.

4.2.2 Submodules of Principal Series Modules Table 4.1 determines half-planes in .R2 that induce subsets of L for which there is ξ,ν a line intersecting .Sect(j ) on which a shift operator vanishes in .HK . There are non-empty intersections of two of such half-planes, but no non-empty intersection

94

4 Submodule Structure

Fig. 4.7 Points .(j, ν) in the lattice L and the number of ξ,ν genuine submodules of .HK

of three half-planes. There are several possibilities, indicated in Fig. 4.7. If a point is ξ,ν in only one of the half-planes, then .HK has one genuine submodule. (By a genuine submodule we mean a non-zero submodule that is not the total module.) If .(j, ν) is in the intersection of two half-planes, there are several possibilities. If cases (i)(a) and (i)(d) in Proposition 4.1 are combined, or cases (i)(b) and (i)(d), then the halfplanes are bounded by two parallel lines in the .(h/3, p)-plane, leading to two or three genuine submodules, depending on the relative position, indicated by 2 or .3 in Fig. 4.7. In the other combinations the half-planes are bounded by intersecting lines. This results in three genuine submodules, indicated by 3 in the figure. In this way we obtain twelve regions in the lattice to investigate. We consider these regions separately. For each we give two illustrations. On the left we sketch the corresponding subset of the lattice L. On the right we give a sketch of the .(h/3, p)-plane for one point .(j, ν), using the conventions indicated at the end of Sect. 4.1. If we move the point on the left, then the lines where the shift operators vanish move as well. As long as we stay in the interior of a Weyl chamber the submodule structure keeps its general structure. Different structures arise when we cross a wall. We give parameters .(h0 , p0 , A, B) as in Definition 3.5 for the irreducible submodules and the irreducible quotients. We use the notation for the spectral parameters indicated in Sect. 4.1.1. In particular we assume in each case that .(j+ , ν+ ), .(jr , νr ) and .(jl , νl ) are related by (4.2). From the sketch in the .(h/3, p)plane elementary geometrical arguments lead to the parameters. We indicate the argument in two cases, Sects. 4.2.2.1 and 4.2.2.3, and leave it to the reader in the other cases. In all cases we describe the submodule lattice in a composition diagram. Afterwards, we summarize the occurrence of irreducible modules in principal series modules in Table 4.3 and Fig. 4.23. We stress that the structure of principal series is known. See for instance Zhang [57, Chap. 5]. In the context of this work, the principal series modules are contained in the N-trivial Fourier term module. In the next sections we will apply the same methods to the other Fourier term modules.

4.2 Principal Series and Related Modules

4.2.2.1

95

Spectral Pairs (j+ , ν+ ) Satisfying ν+ ≥ |j+ | + 2

Figure  4.8 sketches the corresponding region in the .(j, ν)-plane, and the region in the . h3 , p -plane corresponding to the K-types. There are four genuine submodules, indicated in Fig. 4.9. The downward shift operators are zero on the boundary points of the region a. The upward shift operators preserve the region a. Hence this region is irreducible, and the parameters A and B (introduced in Definition 3.5) have value .∞. We note that the picture sketches the . h3 -p-plane. The parameters .h0 and .p0

determine the lowest point . h30 , p0 . This point is the intersection of the line with     slope 1 through . 2j3 l , 0 and the line with slope .−1 through . 2j3r , 0 . Hence  .

h0 , p0 3



 =

   1  2jl jl + jr jr − jl 2jr  1  2jr 2jl  = . , + − , 2 3 3 2 3 3 3 3

Fig. 4.8 The region .ν+ ≥ |j+ | + 2 in Sect. 4.2.2.1. Illustrated for .(j+ , ν+ ) = (2, 6)

ξj ,ν+

Fig. 4.9 Genuine submodules of the modules .HK + to the sets of K-types indicated in the sketch

in Sect. 4.2.2.1. The letters in the table refer

96

4 Submodule Structure

This gives .h0 = jl + jr and .p0 = 13 (jr − jl ). The table in Fig. 4.9 rewrites this in a simpler way, with use of the relations in (4.2). If one uses the Mathematica notebook, then the routines in [6, §17a] are handy for this computation, and for similar ones. The first parameter .μ2 in Definition 3.5 is the eigenvalue of the Casimir operator, j2

ξ ,ν

2 − 4 + + in .H + + . The irreducible which is known to be .λ2 (j+ , ν+ ) = ν+ K 3 submodule determined by a has type .I I + (j+ , ν+ ). The union of a and b determines an invariant submodule that is not irreducible, since it contains a. Its parameters .h0 , .p0 , A and B can be read off from the sketch in an analogous way. The same holds for the union of a and c. The union of a, b and c determines an invariant submodule as well. It is not a special cyclic .(g, K)-module. ξ .ν It has two minimal vectors. The quotient .HK+ + mod (abc) is irreducible of type .F F (j+ , −ν+ ). We find in addition the following irreducible subquotients

(ab)



.

(ac)



(a) ∼ = (abc) (a) ∼ = (abc)

 

(ac)

type F I + (jl , −νl )

(ab)

type I F + (jr , −νr )

This leads to the following composition diagram.

(4.6)

.

Conventions in the Decomposition Diagrams Each arrow in the diagram denotes an inclusion of submodules, producing a subquotient of the principal series representation. We indicate the isomorphism class of the subquotient, above or below the arrow. The left-most arrows correspond to irreducible submodules, the right-most arrows correspond to irreducible quotients of the principal series module. The other arrows correspond to other subquotients.

4.2.2.2

Spectral Pairs (j+ , j+ ) Satisfying j+ ∈ Z≥1

See Fig. 4.10. In this case .(jr , νr ) = (j+ , ν+ ). The irreducible submodule has type I I + (j+ , j+ ), with the following parameters.

.

h0 = jl + j+ = −j+

.

p0 =

j+ − jl = ν+ 2

A = B = ∞.

(4.7)

4.2 Principal Series and Related Modules

97

Fig. 4.10 The region .ν+ = j+ ∈ Z≥1 in Sect. 4.2.2.2. Illustrated for .(j+ , ν+ ) = (1, 1)

Fig. 4.11 The region .1 ≤ νr ≤ jr − 2 in Sect. 4.2.2.3. Illustrated for .(jr , νr ) = (7, 3)

The unique subquotient has type .I F (jr , −jr ). Composition diagram: .

4.2.2.3

(4.8)

Spectral Pairs Satisfying 3 ≤ νr ≤ jr − 2

See Fig. 4.11. The sketch on the right shows two irreducible submodules, described in Table 4.2. Number 2 is of type .I F (jr , νr ). The parameters .h0 and .p0 are clearly given by .(h0 , p0 ) = (2jr , 0). In this case there is an upper boundary line, corresponding to the vanishing of the shift operator .S1−3 on the line through

98

4 Submodule Structure

Table 4.2 Two irreducible submodules. Submodule 1 has type .I I + (j+ , ν+ ), and submodule 2 is of type .I F (jr , νr ). The unique subquotient has type .I F + (jr , −νr ) 1

.h0

= jr + jl = −j+

.p0

=∞ .h0 = 2jr .A

.

jr −jl 3

= ν+

=∞ .p0 = 0

.A

2

=

.B

=∞

.B

=

jr −j+ 3

− 1 = νl − 1

+ 2, 0 with slope 1. To determine the parameter B we need to intersect this

line with the line through . 2tj3 r , 0 with slope .−1. The intersection point is at height    2j+ 1 1 2jr − +2 = (jr − j+ ) − 1 . . 2 3 3 3 2j+ 3

Submodule 2 has type .I F (jr , νr ), and submodule 1 has type .I I + (j+ , ν+ ). The j ,ν quotient of .HKr r by the sum of these two irreducible modules is irreducible of type .I F+ (jr , −νr ). Composition diagram:

(4.9)

.

A special case occurs if .νr = jj − 2. Then the K-types in the submodule in the holomorphic discrete series form a boundary line of the sector .Sect(jr ). See Fig. 4.12. 4.2.2.4

Spectral Pairs Satisfying jr ∈ 2Z≥1 , νr = 0

  See Fig. 4.13. In this case .(jl , νl ) = (j+ , ν+ ) = 12 jr , 12 jr . In the figure on the right there are no lattice points in the region between the lines with slope 1     ξ ,0 through . 23 jl , 0 and . 23 j+ + 2, 0 . The principal series module .HKr is a direct sum of two irreducible submodules, of types .I I + (−j+ , j+ ) = I I + (−jr /2, jr /2) and .I F (jr , 0), respectively. 1 h0 = jr + jl = − j+ A=∞ . 2 h0 = 2jr A=∞

l p0 = jr −j 3 = ν+ B=∞ p0 = 0 + −1= B = jr −j 3

(4.10) jr 2

−1

4.2 Principal Series and Related Modules

99

Fig. 4.12 The region .3 ≤ νr = jr − 2. Illustrated for .(jr , νr ) = (7, 5)

Fig. 4.13 The region .jr ∈ 2Z≥2 , .νr = 0, in Sect. 4.2.2.5. Illustrated for .(jr , νr ) = (6, 0)

Composition diagram:

.

(4.11)

100

4 Submodule Structure

Fig. 4.14 The region of .(jr , −νr ) for .1 ≤ νr ≤ jr − 2. Illustrated for .(jr , −νr ) = (7, −3)

4.2.2.5

Spectral Pairs (jr , −νr ) Satisfying 1 ≤ νr ≤ jr − 2

See Fig. 4.14. There is one irreducible submodule, of type .I F + (jr , −νr ). h0 = j+ + jr = −jl .

A = ∞

jr − j+ = νl 3 j+ − jl − 1 = νr − 1 B = 3

p0 =

(4.12)

The two irreducible subquotients have types .I I + (j+ , ν+ ) and .I F (jr , νr ). Composition diagram:

.

(4.13)

A special case occurs if .νr = 1. Then .jl = j+ − 3, and the irreducible submodule has K-types forming one line. See Fig. 4.15.

4.2 Principal Series and Related Modules

101

Fig. 4.15 The region of .(jr , −1) for .jr ≥ 3 odd, in Sect. 4.2.2.6. Illustrated for .jr = 7

Fig. 4.16 The region of .(jr , −jr ) for .jr ≥ 1, in Sect. 4.2.2.6. Illustrated for .jr = 4

4.2.2.6

Spectral Pairs (jr , −jr ) Satisfying jr ≥ 1

See Fig. 4.16. We have .(j+ , ν+ ) = (jr , νr ). There is one irreducible submodule, which has type .I F (jr , −jr ).

.

h0 = 2jr

p0 = 0

A = ∞

B=

jr − jl − 1 = jr − 1 3

(4.14)

The irreducible quotient has type .I I + (j+ , ν+ ). Composition diagram: .

(4.15)

102

4 Submodule Structure

We note that in the pictures the modules in the isomorphism class .I F (jr , −jr ) look the same as the modules in the class .I F (2jr , 0). To see that we have different isomorphism classes we compare the full parameter sets, with .j ∈ Z≥0 . jr Type 2j I F (2j, 0) . j I F (j, −j )

4.2.2.7

λ2 1 (2j )2 − 4 3 4 2 3j − 4

h 0 p0 A B 4j 0 ∞ j − 1

(4.16)

2j 0 ∞ j − 1

Spectral Pairs (j+ , −ν+ ) Satisfying ν+ ≥ |j+ | − 1

See Fig. 4.17. There is one irreducible submodule with finite dimension equal to .(A + 1)(B + 1). h0 = 2j+ .

A =

p0 = 0

jr − 3 − j+ = νl − 1 3

B =

j+ − (jl + 3) = νr − 1 3

(4.17)

The irreducible quotient has type .I I + (j+ , ν+ ). Composition diagram:

.

(4.18)

Fig. 4.17 The region of .(j+ , −ν+ ) for .ν+ ≥ |j+ | + 2, in Sect. 4.2.2.7. Illustrated for .j+ = 2, = −6

.−ν+

4.2 Principal Series and Related Modules

103

Fig. 4.18 The region of .(jl , jl ) for .jl ≤ −1, in Sect. 4.2.2.8. Illustrated for .jl = −4

4.2.2.8

Spectral Pairs (jl , jl ) Satisfying jl ≤ −1

See Fig. 4.18. We note the similarity with Sect. 4.2.2.6. There is one irreducible submodule of type .F I (jl , jl ). h0 = 2jl

.

p0 = 0

A =

jr − 3 − jl = |jl | − 1 3

B=∞

(4.19)

The irreducible quotient has type .I I + (j+ , ν+ ). Composition diagram: (4.20)

.

4.2.2.9

Spectral Pairs (jl , −νl ) Satisfying 1 ≤ νr ≤ −jl − 2

See Fig. 4.19. We note the similarity to Sect. 4.2.2.5. There is one irreducible submodule, of type .F I + (jl , −νl ). h0 = jl + j+ = −jr .

jr − j+ − 1 = νl − 1 A = 3

p0 =

j+ − jl = νr 3

(4.21)

B = ∞

The irreducible subquotients have types .I I + (j+ , ν+ ) and .F I (jl , νl ). Composition diagram:

.

(4.22)

104

4 Submodule Structure

Fig. 4.19 The region .jl + 2 ≤ −νl ≤ −1, in Sect. 4.2.2.9. Illustrated for .jl = −7, .νl = 3

Fig. 4.20 The region .jl ≤ −2 even, .νl = 0, in Sect. 4.2.2.10. Illustrated for .jl = −8, .νl = 0

4.2.2.10

Spectral Pairs (jl , 0) with jl ∈ 2Z≤−1

See Fig. 4.20. Compare Sect. 4.2.2.4. ξ ,0 In this case .(jr , νr ) = (j+ , ν+ ). The module .HKl is a direct sum of two irreducible submodules, of types .I I + (j+ , ν+ ) and .F I (jl , 0). 1 h = jr + jl = −j+ A=∞ . 2 h0 = 2jl l A = j+ −3−j = νr − 1 3

l p0 = jr −j 3 = ν+ B=∞ p0 = 0 B=∞

(4.23)

4.2 Principal Series and Related Modules

105

Fig. 4.21 The region .1 ≤ νl ≤ −jl − 2, in Sect. 4.2.2.11. Illustrated for .(jl , νl ) = (−5, 3)

Composition diagram:

(4.24)

.

4.2.2.11

Spectral Pairs (jl , νl ) with 1 ≤ νl ≤ −jl − 2

See Fig. 4.21. Compare Sect. 4.2.2.3. There are two irreducible submodules, of types .I I + (j+ , ν+ ) and .F I (jl , νl ). 1 h0 = jr + jl = −j+ A=∞ . 2 h0 = 2jl l A = j+ −3−j = νr − 1 3

l p0 = jr −j 3 = ν+ B=∞ p0 = 0 B=∞

(4.25)

The type of the irreducible subquotient is .F I + (jl , −νl ). Composition diagram:

.

(4.26)

106

4 Submodule Structure

Fig. 4.22 The region .−j+ = ν+ ≥ 1, in Sect. 4.2.2.12. Illustrated for .(j+ , ν+ ) = (−3, 3)

4.2.2.12

Spectral Pairs (−νl , νl ) with νl ≥ 1

See Fig. 4.22. Compare Sect. 4.2.2.2. In this case .(jl , νl ) = (j+ , ν+ ). There is one irreducible submodule, of type .I I + (−ν+ , ν+ ) = I I + (j+ , −j+ ). h0 = j+ + jr = −j+

p0 =

.

jr − j+ = ν+ 3

A=∞

B=∞

(4.27)

The irreducible quotient has type .F I (jl , −νl ) = F I (jl , jl ). Composition diagram: .

(4.28)

Observations First we consider the cases in which .(j, ν) is in the interior of a Weyl chamber. (This means that .(j, ν) determines a regular character of .ZU (g).) Then ξj ,ν .H has for .ν ≥ 1 one or two irreducible submodules and one irreducible quotient, K and for .ν ≤ −1 one irreducible submodule and one or two irreducible quotients. See Collingwood [10, p. 46], where these observations for a regular character of .ZU (g) are placed in a wider context. ξ ,ν If .(j, ν) is on the wall between Weyl chambers, then .HKj is the direct sum of two irreducible submodules if .ν = 0, and otherwise has an an irreducible submodule ξ ,ν V such that .HKj /V is the irreducible quotient. 4.2.2.13

Unique Embedding

In Table 4.3 we summarize the irreducible isomorphism types occurring in reducible principal series modules. We see that almost all types of irreducible .(g, K)-modules under integral paraξ,ν metrization occur as a submodule in only one principal series module .HK . Only for

4.2 Principal Series and Related Modules

107

Table 4.3 Isomorphism types under integral parametrization, and their embeddings in principal series representations. See Fig. 4.1 for the notation of spectral pairs in a Weyl group orbit Type .I I + (j+ , ν+ )

.ν+

It occurs in

.

It occurs in HKr

.I I + (j+ , −j+ )

.j+

.I F (jr , 0)

.I F + (jr , −νr )

.μ2

=

4 2 3 jr

−4

.

It occurs in HK+

.μ2

=

1 2 3 jl

−4

= 43 jl2 − 4

.F F (j+ , −ν+ )

.∞

.−j+

.ν+

.∞

.∞

.2jr

.0

.∞

.νl

−1

.=

jr −νr 2

.νl

−1

.=

jr 2

.2jr

≤ νr ≤ jr − 2

It occurs in

.0

.∞

It occurs in

.νr

−1

.2jr

.0

.∞

.jr

−1

.2jl

.0

.νr

−1

.∞

l = .− jl +ν 2

.2jl

.0

.νr

−1

= .− j2l .−jr

.νr

.νl

−1 .∞

−1

−1

.∞

r = . jr −ν 2

.2jl

.0

.|jl |

.2j+

.0

.

−1

−1

.∞

ξ ,j HKl l , Fig. 4.18

≥ |j+ | + 2

It occurs in

.∞

ξ ,−ν HKl l , Fig. 4.19

∈ Z≤−1

It occurs in

.νl

ξ ,0 HKl , Fig. 4.20

+ 2 ≤ −νl ≤ −1

It occurs in

.−jl

ξ ,ν HKl l , Fig. 4.21

= −2jr = −j+ ∈ 2Z≤−1

It occurs in

−1

ξ ,−j HKr r , Fig. 4.16

≤ νl ≤ −jl − 2

It occurs in

−1

ξ ,−ν HKr r , Fig. 4.14

∈ Z≥1

.ν+ .

.∞

ξ ,0

.jl .

.ν+

It occurs in HKr , Fig. 4.13

.jl .

.F I (jl , jl )

.−j+

ξ ,ν HKr r , Fig. 4.11

∈ 2Z≥1

.jr

.

.∞

, Fig. 4.22

≤ νr ≤ jr − 2

It occurs in

.jl

.F I + (jl , −νl )

.μ2

ξ ,−j+

.1

.

.∞

ξ ,0 HKr , Fig. 4.13

It occurs in

. .F I (jl , 0)

= −ν+ ∈ Z≥1

.

.1

.F I (jl , νl )

.ν+

ξ ,0

.j+

.

.−j+

ξ ,j HK+ + , Fig. 4.10

It occurs in HKl , Fig. 4.20

. .I F (jr , −jr )

= ν+ ∈ Z≥1

.

.jr

= 13 jr2 − 4

B

, Fig. 4.11

It occurs in

.

A

ξ ,ν HKl l , Fig. 4.21

.

.1

.I F (jr , νr )

.μ2

ξ ,νr

It occurs in

.p0

ξ ,ν HK+ + , Fig. 4.8

.

. .I I + (j+ , j+ )

≥ |j+ | + 2

.h0

ν+ −j+ 2

−1

.

j+ +ν+ 2

−1

ξ ,−ν HK+ + , Fig. 4.17

the large discrete series type there are several embeddings: three for large discrete series, and two for limits of large discrete series. See Collingwood, [9], pp. 115– 119 in [10, §5.3], for the unique embedding property in a wider context. See also Fig. 4.23.

108

4 Submodule Structure

ξ ,ν

Fig. 4.23 Lattice points .(j, ν) ∈ L and embeddings of isomorphism types in .HKj . The limits of large discrete series are on the walls between the large discrete series and the holomorphic and antiholomorphic discrete series. The horizontal walls carry the limits of antiholomorphic and holomorphic discrete series. The thin representations are the Langlands representations that are unitarizable (with .ν = −1). The point .(j, ν) = (0, −2) corresponds to the trivial representation. The other unitarizable modules occur for .ν ≥ −1. The circled dots indicate the roots. This Weyl group orbit corresponds to the character of .ZU (g) represented by .ρ = (0, 2), which is half the sum of the positive roots

For the spectral pair in the notation .I I + (j+ , ν+ ) we have chosen to use the dominant Weyl chamber. For all other isomorphism classes in the list in Sect. 4.1.2 the spectral pair .(j, ν) is uniquely determined by the unique principal series module in which it occurs as a submodule. We note that the Weyl group orbit .(jl , ±νl ), .(j+ , ±ν+ ), .(jr , ±νr ) corresponds to a unique character of the ring .ZU (g). The choice of an element of this orbit determines a character of AM, and hence of N AM. Induced up to G, this provides a specific principal series module.

4.2.3 Characterization by Sets of K-types Let us take .jl < j+ < jr . The corresponding spectral pairs .(jl , ±νl ), .(j+ , ±ν+ ), and .(jr , ±νr ) form one Weyl group orbit, determining a character .ψ of .ZU (g). Each of the six corresponding principal series modules contains one or two irreducible .(g, K)-modules. We consider the sets of K-types occurring in each of these irreducible modules. These K-types correspond to points in the union Sect(jl ) ∪ Sect(j+ ) ∪ Sect(jr ) .

.

In Fig. 4.17 we see that the K-types .τph in a representation in the class .F F (j+ , −ν+ ) satisfy 2jl h 2j+ jl in both cases. This gives . ϕrl ,p (νl ) ∈ K0;h,p in case 1, and . ϕrl ,p (−νl ) ∈ K0;h,p in case 2. (Computations of the Weyl group action in [6, §17a].) ξ ,ν We handle elements of .HKr r in an analogous way. We need to consider p 3 h .S −1 ϕrr ,p (±νr ), and have .S1 (jr , νr ) = (j+ , ν+ ) and .S1 (jr , −νr ) = (jl , νl ). This gives elements in .K0;h,p in cases 2 and 3. ξ ,ν Finally we consider elements of .HK+ + . We have

S2 (j+ , ν+ ) = (jl , νl ) ,

S1 (j+ , ν+ ) = (jr , νr ) ,

S2 (j+ , −ν+ ) = (jr , −νr ) ,

S1 (j+ , −ν+ ) = (jl , −νl ) .

.

With Table 4.1, p. 92, this gives the possibilities in Table 4.4. If .j+ is equal to .jl or .jr , we have to deal with only one intersection point of boundaries of sectors; one case does not exist, and the two other cases coincide. The same reasoning leads to two elements in the intersection. p

Table 4.4 Combinations for .j = j+ and .ϕ± = hϕr+ ,p (±ν+ ) 1

.±

.3h

−p

.3h

.+

= j2 < j

3 .S−1 ϕ+

.−

2

3

.jl .jl

= j1 < j

.j2

+p

.j+

−3

=j

.S−1 ϕ+

=0

=j

−3 .S−1 ϕ−

=0 =0

=0

.jr

= j1 > j

−3 .S−1 ϕ+

= jr > j

.jr

= j2 > j

.j1

.jl

= j2 < j

.S−1 ϕ+

.−

.jl

= j1 < j

.j2

.−

.j+

= jr > j

.+

.+

=0

3

.j+

=j

3 .S−1 ϕ+

.j+

=j

3 .S−1 ϕ−

= jl < j

=0

.jr

= j1 > j

−3 .S−1 ϕ+

=0

.jr

= j2 > j

.j1

=0

= jl < j

114

4 Submodule Structure

Non-zero elements of different principal series modules are linearly independent. We have to see whether logarithmic modules (Proposition 4.4) may be in the intersection of kernels. A computation in [6, §17c] shows that logarithmic solutions

do not give elements of .K0;h,p .

4.3 Submodule Structure of Abelian Fourier Term Modules ψ

We turn to the submodule structure of generic abelian modules .Fβ with .β = 0. ξ,ν

Under integral parametrization, we still have to define the submodules .Mβ

and

ξ,ν .W β .

These modules differ from those in the N-trivial case in two aspects: (1) In these modules there are only irreducible submodules of type .I I + . (2) The modules ξ  ,ν  ξ,ν .H and .HK in the principal series have zero intersection if .(j, ν) and .(j  , ν  ) K ξ,ν

ξ  ,ν 

are different elements in the same Weyl group orbit. We will see that .Wβ ∩ Wβ ξ  ,ν 

ξ,ν

and .Mβ ∩ Mβ

are non-zero modules.

Preliminaries There are some facts that hold for both the abelian and the nonabelian case. a,b In Lemma 3.35 we obtained the families .ν → ωN (j, ν) and .ν → μa,b N (j, ν) a,b that are holomorphic in .ν ∈ C (for .ω), or in .C  Z≤−1 (for .μn ). In the cases .a = 0 or .b = 0, Proposition 3.36 states a vanishing result for the downward shift operators, which stays valid in the case of integral parametrization. p These families may be compared to the basis families .ν → hϕr,p (ν) for the p h principal series. An important difference is that . ϕr,p (ν) is explicitly known, a,b whereas the families .ωN and .μa,b N have a much more complicated description. We have to look for situations in which we can obtain a relatively simple description. One of such situations occurs when the intersection of the kernels of the downward shift operators is non-zero. Proposition 3.21 and Lemma 4.5 show that this happens for a given K-type only for one character of .ZU (g). Families with a Fixed K-type In (3.93) we defined the families .ωβa,b and .μa,b β by repeated application of the upward transfer operators to .ωβ0,0 and .μ0,0 β . Proposition 3.14 implies that these families are non-zero for all .ν. p Lemma 4.7 In the decomposition .F = χβ r fr h r,p the following holds. . p,0 ωβ (j, ν) has lowest component f−p (t) = t p+2 Kν (2π |β|t) ,

.

. p,0 μβ (j, ν) has lowest component f−p (t) = t p+2 Iν (2π |β|t) , . 0,p ωβ (j, ν) has highest component fp (t) = t p+2 Kν (2π |β|t) , . 0,p μβ (j, ν) has highest component fp (t) = t p+2 Iν (2π |β|t) .

4.3 Submodule Structure of Abelian Fourier Term Modules

115

. We use .= to indicate equality up to a non-zero factor. −3 3 x Determining Components Proposition 3.36 implies that .S−1 β and .S−1 xβ are identically zero for .x = ω and .x = μ. The kernel relations in Table 3.8, p. 51, imply that the components in the lemma determine all other components. p,0

0,p

a,b Proof (of Lemma 4.7) The families .xN are defined inductively. For .a = b = 0 relations (3.85) and (3.87) give the component .f0 (t) = t 2 jν (2π |β|t), with .jν = Iν or .Kν . The description of the shift operators in Table 3.7, p. 50, implies that the relevant upward shift operator acts on the lowest or highest component as multiplication by a non-zero multiple of t.



Intersection of Kernels Proposition 4.8 Let .β = 0, and let .τph with .p ≥ 1 be a K-type that occurs in 3 .Fβ . We denote by .Kβ;h,p the intersection of the kernels of .S −1 : Fβ;h,p,p → −3 : Fβ;h,p,p → Fβ;h−3,p−1,p−1 . Fβ;h+3,p−1,p−1 and .S−1 (i) The dimension of .Kβ;h,p is equal to 2. A basis is

kIβ;h,p =

p

χβ (iβ/|β|)(r+p)/2 t 2+p I|h−r|/2 (2π |β|t) h r,p ,

r≡p(2), |r|≤p

.

kK β;h,p =

p

χβ (−iβ/|β|)(r+p/2 t 2+p K|h−r|/2 (2π |β|t) h r,p .

r≡p(2), |r|≤p

(4.40) ψ

(ii) (a) .Kβ;h,p is a subspace of the module .Fβ where .ψ = ψ[−h, p] ∈ Wip . (b) Put .j1 = 12 (h − 3p), .j2 = 12 (h + 3p). There are unique .ν1 , ν2 ∈ Z≥0 such that .(jn , νn ) ∈ OW (ψ)+ for .n = 1, 2. With these values we have: . p,0 . 0,p kIβ;h,p = μβ (j1 , ν1 ) = μβ (j2 , ν2 ) , .

. p,0 . 0,p kK β;h,p = ωβ (j1 , ν1 ) = ωβ (j2 , ν2 ) .

(4.41)

Remarks The K-types .τph discussed here are the sole higher-dimensional K-types for which we obtained reasonably simple descriptions of elements of .Fβ;h,p,p . Corollary 4.12 describes one more instance with explicit expressions similar to those in (4.41). Such K-types correspond to points .(h/3, p) on the intersection of two sectors, here denoted .Sect(j1 ) and .Sect(j2 ). p Proof We consider an element .F = χβ r fr h r,p in .Kβ;h,p ⊂ Fβ;h,p,p . Its 3 and .S −3 in Table 3.8, p. 51. These components .fr satisfy the kernel relations for .S−1 −1 two relations lead to a second order differential equation for .fr . A computation in [6, §18a] shows that this differential equation is related to the modified Bessel

116

4 Submodule Structure

differential equation (A.1) with .ν = (h − r)/2, and .fr (t) = t 2+p j (2π |β|t), where we can take j equal to .I(h−r)/2 or .K(h−r)/2 . The function .Kν is even in the parameter .ν. The same holds for .Iν for integral values of .ν. To determine the relation between the coefficients for various values of r in these linear combinations, we use again the kernel relations. With use of the contiguous relations in (A.7) we get in [6, §18b] a two-dimensional solution space for the coefficients, with basis as indicated in (4.40). In this way we have two explicit linearly independent elements of .Kβ;h,p . In [6, §18c] we see that both functions are eigenfunctions of .ZU (g) with character .ψ = ψ[−j, p]. To complete the proof of (i) and (ii)(a), we still have to show that the dimension of .Kβ;h,p is two. p,0 The lowest component of .μβ (j1 , ν1 ) is equal to .t p+2 Iν1 (2π |β|t) by Lemma 4.7. The lowest component of .kIβ;h,p is .t 2+p I|h+p|/2 (2π |β|t). By (iii) in Lemma 4.5, these two functions are proportional. The other relations in (4.41) follow in the same way.

For the dimension assertions we use the following induction result. Lemma 4.9 Let h, p, .ψ, .(j1 , ν1 ) and .(j2 , ν2 ) as in the proposition. ψ If .dim Fβ;h,p,p > 2 then at least one of the following statements holds: .

ψ

ψ

dim Fβ;h+3,p−1,p−1 > 2 ,

dim Fβ;h−3,p−1,p−1 > 2 .

Proof The lemma states that the property of having a dimension higher than 2 propagates to at least one lower K-type. Use of the downward shift operators reduces the proof to a computation. We know two linearly independent elements 0,p−1

ωβ

.

0,p−1

(j2 , ν2 ) , μβ

ψ

(4.42)

ψ

(4.43)

(j2 , ν2 ) ∈ Fβ;h+3,p−1,p−1 ,

and two linearly independent elements p−1,0

ωβ

.

p−1,0

(j1 , ν1 ) , μβ

(j1 , ν1 ) ∈ Fβ;h−3,p−1,p−1 . ψ

To prove the lemma we assume that .F ∈ Fβ;h,p,p is not in .Kβ;h,p , and want to show that at least one of ψ

3 F− = S−1 F ∈ Fβ;h+3,p−1,p−1

.

and

−3 F+ = S−1 F ∈ Fβ;h−3,p−1,p−1 (4.44) ψ

is not a linear combination of the two elements in (4.43) or (4.42). Let us suppose that .F− is a linear combination of basis functions in (4.43). −3 These functions are in the kernel of .S−1 , and hence determined by their highest component. It suffices to show that the component b of .F− of order .p − 1 vanishes.

4.3 Submodule Structure of Abelian Fourier Term Modules

117

3 F , it can be expressed On the other hand, since b is the highest component of .S−1 in the highest two components of F .

p  ¯ p + 2tfp−2 4π i βtf .b = + (h − 3p − 2)fp−2 . (4.45) 4(p + 1)  This enables us to express .fp−2 in .fp−2 , .fp and the function b. We substitute this in the eigenfunction equations for the highest component of F . We have to take .(j, ν) in the eigenfunctions equations equal to .(j2 , ν2 ), and use (iii) in Lemma 4.5. The second eigenfunction relation takes the form .

− 216π iβ(p + 1)tb = 0 .

Hence .b = 0. In a similar way we obtain that .F+ = 0 from the assumption that it is a linear combination of the functions in (4.42). Both computations are in [6, §18d].

To finish the proof of Proposition 4.8 we show more than is needed for the present proposition, namely that for .ψ = ψ[−h, p] .



ψ

dim Fβ;h ,p ,p = 2 or 0 ,

for all K-types τph .

(4.46)

For .p = 0 we know (4.46) from Sect. 3.4.2: the dimension is 2 for .h = 23 j with .j ∈ O1W (ψ) and 0 otherwise. The injectivity of the upward shift operators (Proposition 3.14) shows that the dimension is at least 2 whenever .Fβ;h ,p ,p = {0}. When we apply a downward shift operator, the dimension cannot decrease if that  shift operator is injective. So from a given K-type .τph we can go down to a K

±3 type .τph −1 without decreasing the dimension as long as one of the downward shift ψ

operators is injective on .Fβ;h ,p ,p . The only possibility of a change in dimension occurs if both downward shift operators have a non-trivial kernel. Lemma 4.9 shows that a dimension larger than 2 cannot drop to 2. Thus we get (4.46).

4.3.1 Structure Results ξ,ν

ξ,ν

We still have to define the modules .Wβ and .Mβ under integral parametrization, which implies that .ν ∈ Z. We restrict ourselves to the case .ν ∈ Z≥0 . Lemma 4.10 Let .ψ ∈ Wip . For each .(j, ν) ∈ OW (ψ)+ we put

ψ ξ,ν Mβ = U (k) μa,b β (j, ν) ⊂ Mβ , a,b≥0 .

ξ,ν

Wβ

=

a,b≥0

ψ

U (k) ωβa,b (j, ν) ⊂ Wβ .

(4.47)

118

4 Submodule Structure ψ

ξ,ν

ξ,ν

(i) These spaces are .(g, K)-submodules of .Fβ , and .Mβ ∩ Wβ = {0}. ξ,ν

ξ,ν

The K-types .τph in .Wβ and .Mβ have multiplicity one if .(h/3, p) ∈ Sect(jξ ), and do not occur in these modules otherwise. (ii) Suppose that .(h/3, p) ∈ Sect(j ) ∩ Sect(j  ) for .(j, ν), (j  , ν  ) ∈ OW (ψ)+ . ξ  ,ν 

ξ ,ν

ξ  ,ν 

ξ ,ν

j j j j Then .Wβ;h,p,p = Wβ;h,p,p and .Mβ;h,p,p = Mβ;h,p,p (iii) We have

ψ

Fβ =

.



ξ,ν

Wβ



⊕



(j,ν)∈OW (ψ)+

ξ,ν

Mβ

.

(4.48)

ξ,ν

(4.49)

(j,ν)∈OW (ψ)+

(iv) ψ

Mβ =

.

ξ,ν

Mβ ,

ψ

Wβ =

(j,ν)∈OW (ψ)+

Wβ .

(j,ν)∈OW (ψ)+

ξ,ν

Proof The space .Wβ is invariant under .k. To see that it is invariant under .g, it suffices to consider the shift operators on a highest weight vector in a K-type. We use the definition in Table 3.4, p. 36 and apply (ii) in Lemma 3.4 to see that a,b a  ,b .ω β (jξ , ν) is sent to a linear combination of elements .u ωβ (jξ , ν) with .u ∈ U (k). By Proposition 3.36 the point corresponding to the K-type cannot leave the sector ξ,ν ξ,ν .Sect(jξ ). Hence .W β is a .(g, K)-module. The same reasoning works for .Mβ . ξ,ν

ξ,ν

Now consider an element .f ∈ Wβ ∩ Mβ of a given K-type .τph . Using a downward path in the .(h/3, p)-plane given by injective downward shift operators ψ we ultimately arrive at a minimal vector .v ∈ Wβ , on which both downward shift operators vanish. In that situation we know that minimal vector explicitly, from Sect. 3.4.2 (.p = 0) or (4.40) (.p ≥ 1). Since the I-Bessel functions and the K-Bessel functions with the same parameter are linearly independent, this minimal vector vanishes. We used a path of injective downward shift operators, and conclude that .f = 0. The upward shift operators are injective by Proposition 3.24. So all K-types ξ,ν ξ,ν corresponding to points of .Sect(jξ ) occur in .Wβ and in .Mβ with multiplicity at least 1. By (4.46) the multiplicities are exactly one. This gives (i). Consider .jl , jr ∈ O1W (ψ) (in the conventions of (4.2)). Proposition (4.8) (ii)(b) ξl ,νl ξr ,νr = Wβ;h,p,p for .(h/3, p) equal to the lowest point of the implies that .Wβ;h,p,p intersection .Sect(jl ) ∩ Sect(jr ). From any other K-type corresponding to a point of .Sect(jl ) ∩ Sect(jr ) we can go down to the K-type by a path of injective downward shift operators. This gives the analogous equality for the K-types in the intersection. This corresponds to the triangular region above 2 in Fig. 4.26. For .jl < j+ < jr we are in the rectangular regions above 1 and 3. In  ξ+ ,ν+ ξl ,νl = Wβ;h,p,p for .(h/3, p) ∈ Sect(jl ) ∩ the same way the equalities .Wβ;h,p,p   ξ+ ,ν+ ξr ,νr = Wβ;h,p,p for .(h/3, p) ∈ Sect(j+ ) ∩ Sect(j+ )  Sect(jr ), and .Wβ;h,p,p

4.3 Submodule Structure of Abelian Fourier Term Modules

119

Fig. 4.26 Three points at which both downward shift operators vanish

 Sect(jr )  Sect(jl ). The upward shift operators are injective by Proposition 3.24. . So relations .ωβa,b (j1 , ν1 ) = ωβa,b (j2 , ν2 ) are preserved if we increase a or b. This ξ,ν

shows that in all K-types the spaces .Wβ;h,p,p are the same for all .(jξ , ν) such that .(h/3, p) ∈ Sect(jξ ). The same reasoning goes through for .M. This gives (ii), and implies (iii). ψ ψ ψ We turn to the submodules .Wβ and .Mβ of .Fβ in Definitions 3.32 and 3.33. From the inclusions

ψ ψ ξ,ν ξ,ν . Wβ ⊂ Wβ , Mβ ⊂ Mβ , (j,ν)∈OW (ψ)+

(j,ν)∈OW (ψ)+



we obtain (iv) by comparing multiplicities of K-types in (4.48).

Proof (of Theorem C) Lemma 4.10 gives most of the statements of Theorem C. We still have to prove part (iv) of the theorem. That implies the reducibility in (i). ξ ,ν The intersection .∩(j,ν)∈OW (ψ)+ Wβj is of course an invariant submodule. From the maximal weight in the minimal K-type in the intersection we can reach all Ktypes in the intersection by injective upward shift operators, and we can go back by injective downward shift operators. Since the highest weight vectors in a subspace of a given K-type generate the whole subspace, this suffices for irreducibility.

Illustration Figure 4.27 gives an illustration of the submodule structure. ψ

Remark 4.11 The status of the submodules of the module .Fβ under integral parametrization is the same as in Remark 3.37. The difference concerns the ξ,ν intersections of the special modules in (4.47). The modules .Wβ coincide in all ξ,ν

K-types that they have in common; the same holds for the modules .Mβ . On the order hand, all principal series modules have zero intersection. With (4.47) we get the following addition to (ii)(b) in Proposition 4.8. Corollary 4.12 Consider .(jl , νl ), .(j+ , ν+ ), .(jr , νr ) in one Weyl group orbit, with jl < j+ < jr , in the notation of (4.2). Let .(j1 , ν1 ) = (jl , νl ), .(j2 , ν2 ) = (jr , νr ). The minimal K-type .τph in the intersection .Sect(j1 ) ∩ Sect(j+ ) ∩ Sect(j2 ) has descriptions .h = 2j1 + 3p = 2j2 − 3p = 2j+ + 3(a − b), .p = a + b with .a, b ∈ Z≥0 . .

120

4 Submodule Structure

ψ

Fig. 4.27 Sketches of the submodule structure in the generic abelian cases, applying to .Wβ and ψ

to .Mβ . On top the case .(jl , j+ , jr ) = (−15, 3, 12), at the bottom the cases .(−6, 3, 3) (left) and (right). We use the conventions in (4.2). The dots indicate the minimal K-type of the irreducible submodule

.(−4, −4, 8)

In this situation we have the following addition to (4.41): . I μa,b β (j+ , ν+ ) = kβ;h,p ,

.

. ωβa,b (j+ , ν+ ) = kK β;h,p .

(4.50)

4.4 Submodule Structure of Non-abelian Fourier Term Modules Under integral parametrization, the non-abelian case is more complicated than the ψ ψ generic abelian case. The modules .Mn and .Wn have in some cases a non-zero intersection. For this reason we also use modules based on the unusual Whittaker functions .Vκ,s in (A.12). Furthermore, the families .ωna,b and .μa,b n may have zeros. As a consequence, we need more complicated families to describe submodules. The aim of this section is to prove Theorem D, and to determine the submodule ψ structure of the non-abelian modules .Fn . Proposition 4.16 is of independent interest. It discusses a situation for which we have a reasonably simple description of elements of .Fn;h,p,p with .p > 1. In Sect. 4.4.2 we deal with the problem that the families .ωna,b (j, ν) and .μa,b n (j, ν) may have zeros at certain values of .ν, which have to be divided out. Another problem

4.4 Submodule Structure of Non-abelian Fourier Term Modules ψ

121

ψ

is that the modules .Wn and .Mn may have a non-zero intersection. We define ψ another module .Vn with exponentially increasing functions. This module has zero ψ intersection with .Wn . The next three subsections follow the same approach as in the generic abelian case. Section 4.4.6 considers a problem not present in the abelian case: how does ψ ψ ψ the module .Mn sit in the direct sum .Wn ⊕ Vn ? After that we are ready to prove Theorem D in Sect. 4.4.7, and to determine the different submodule structures of ψ ψ ψ .Wn , .Vn and .Mn . Section 4.4.8 looks back on Chaps. 3 and 4, by summarizing, for all irreducible ψ .(g, K)-modules, in which way they can be embedded in modules .F N .

4.4.1 Notations and Conventions ψ

For the non-abelian modules .Fn we need several notations. We used some of them in an earlier section, to be recalled here. The type of realization of the Stone-von Neumann representation is indicated by 1 .n = (, c, d) with . ∈ 2 Z=0 , .c mod 2, .d ≡ 1 mod 2. We abbreviate . sign () = ε. ψ The K-types that occur in .Fn have to satisfy the condition (3.39), which can be written as   3p − 3 ≤ ε h − d .

.

(4.51)

The decomposition of .F ∈ Fn;h,p,p into component functions has the form   p F na(t)k = ϑm(h,r) (n) fr (t) h r,p (k) ,

.

(4.52)

r

where .ϑm is an abbreviation of .,c (h,m ) with the convention that .ϑm = 0 if m ∈ Z m0 (j )

.

−p p+1

.

p  p+1 p

.− .ε

= −1

.

p,0

.ω ˜n

+

p  p+1 p



j +ν   p 2

+

 + j −ν 2   j +ν  j −ν

j −ν  2

p,0

.μ ˜n

p+ 2 −ip p+ 2 √ (p+1) 2π ||(m0 (j )+p) 0,p

.S−1 x ˜n

0,p−1



/x˜n  j +ν









3

−ip p− 2 p− j −ν 2 √ (p+1) 2π || (m0 (j )+p)

=1

.

.ε

= −1, 1 ≤ p ≤ m0

p− j −ν ip p− j +ν 2 √ 2 . (p+1) 2π || (m0 (j )−p+1)

.ε

= −1, p > m0 (j )

.−

.ε

p p+1

 p − j +ν p+ 2   p j +ν .− p+1 p − 2

.

p,0

.υ ˜n

p p+1



0,p

.ω ˜n

ν−j  2

0,p

.υ ˜n

0,p

.μ ˜n

Remarks 1. We recall the use of .m0 (j ) and .r0 (h) in this proposition and the accompanying tables. In the general situation of Table 4.5 the quantities .m0 = m0 (j ) and .r0 = r0 (h) have the following significance:

126

4 Submodule Structure

Table 4.10 Relation between .m0 = m0 (j ) and .r0 = r0 (h). By (ac) we indicate that the value of imposes no restriction on the components with .|r| ≤ p

.r0

1 .−1

0,p

p,0

.x ˜n

.ε .

.x ˜n

m0 ≥ 0 ⇔ r0 ≤ p m0 ≥ p ⇔ r0 ≤ −p (ac) .m0

Table 4.11 Factor for upward shift operators in simple cases

.m0

≥ 0 ⇒ r0 ≥ p (ac)

.

m0 ≥ 0 ⇔ r0 ≥ −p m0 ≥ p ⇔ r0 ≥ p (ac) p,0

p+1,0

0,p

0,p+1

→ xn √ √ .i 2π|| m0 − p √ √ .−i 2π| m0 + 1 + p .xn

.ε

= 1, p < m0

.ε

= −1

→ xn √ √ .−i 2π || 1 + m0 + p √ √ .i 2π || m0 − p .xn

.ε

=1

.ε

= −1, p < m0

≥ 0 ⇒ r0 ≥ p(ac)

2j

• .m0 (j ) ≥ 0 is equivalent to the occurrence of the K-type .τ0 in .Fn . • .r0 (h) determines which components .fr of an element of .Fn;h,p,p can be non  zero, namely .max r0 (h), −p ≤ r ≤ p if .ε = sign () = 1, and .−p ≤ r ≤ min r0 (h), p if .ε = −1. p,0

In Proposition 4.14 we have that .h = 2j + 3p for .x˜n (j, ν) and .h = 2j − 3p 0,p for .x˜n (j, ν). This interpretation leads to the scheme in Table 4.10. 2. Proposition 4.14 may be compared to Lemma 4.7 in the generic abelian cases. Since in the abelian case the upward shift operators are injective, there was in Sect. 4.3 no need to introduce families .x˜ p,0 and .x˜ 0,p by dividing out zeros. Proof (of Proposition 4.14) In the case when .p = 0 we take .x˜n0,0 (j, ν) equal to 0,0 .xn (j, ν). Then the assertions in (i) hold for .x ∈ {ω, μ} by Sect. 3.4.2.3. For .x = υ we need (4.55) and (A.12). We proceed by induction on the variable p. Most steps can be carried out by hand with the description of the upward shift operators in Table 3.9, p. 52. We prefer to carry out all steps with Mathematica. See [6, §20]. p,0 0,p In many steps the determining component of .xn , respectively .xn , is multiplied by a simple non-zero factor, as indicated in Table 4.11. This gives in many cases p,0 part (i) of the proposition for the action of .S13 on .x˜n and for the action of .S1−3 on 0,p .x ˜n . Since  p −1  p −1 p+1,0 p,0 p+1  p −1 p+1 3 p,0 .S1 x ˜n = ϕ+ S13 xn = ϕ+ xn = ϕ+ ϕ+ x˜n , and similarly for .S1−3 x˜n , the entries in Table 4.8 give the quotients of successive p p values of .ϕ± . Hence the factors .ϕ± are essentially known. See [6, §20a]. The remaining cases, with .p ≥ m0 (w), are more complicated. The lowest p,0 p+1,0 component .xn has order .r0 (h) and for .xn the order increases to .r0 (h + 3) = 0,p

4.4 Submodule Structure of Non-abelian Fourier Term Modules

127 p,0

r0 (h) + 1. This has the consequence that we need also the component of .xn of 3 in Table 4.6. order .r0 (h) + 2. This can be determined by the kernel relation for .S−1 p,0

The lowest component of .S13 xn can now be expressed in the lowest component of F . It is necessary to write it in the form .t m0 +1 wκ,ν/2 (2π||t 2 ) for .wκ,ν equal to one of the three Whittaker functions .Wκ,ν/2 , .Vκ,ν/2 and .Mκ,ν/2 . The computations in this 0,p case and in the case of .xn are in [6, §20b]. Proposition 3.36 and Eq. (4.56) give the vanishing of the holomorphic families p,0 p,0 p,0 p −3 0,p 3 .S and .S−1 xn on .C  Z≤−1 . Since .x˜n = xn /ϕ+ is holomorphic on −1 xn 0,p

p,0

3 x˜ C  Z≤1 , the vanishing of .S−1 n follows. We proceed similarly for .x˜n . The action of the downward shift operators in Table 4.9 is obtained by computations similar to those for the upward shift operator. We have to use that there is one downward shift operator for which the image is zero, by Proposition 3.36, and Eq. (4.56) for .x = υ, to get a relation between the components. See [6, §20cd]. p,0 0,p The minimal component of .ω˜ n (j, ν) and the maximal component of .ω˜ n (j, ν) 1 c are of the form .t Wκ,s with .c ∈ Z, .κ ∈ 2 Z and .ν ∈ C. The recursive relations between the components used in the proof of Lemma 4.13, together with (A.18), imply that the other components are linear combinations of functions of the same form, and hence have exponential decay at .∞ according to (A.14).



.

p,0

p,0

Relation (A.13) implies that each .μ˜ n (j, ν) is a linear combination of .ω˜ n (j, ν) 0,p 0,p and .υ˜ n (j, ν), and analogously for .μ˜ n (j, ν). In general both coefficients in the linear combination are non-zero. If .ν ≡ j mod 2 and .ν ≥ 0, then the following special relations occur. Lemma 4.15 Let .n = (, c, d), put .ε = sign (), and let .j ≡ p ≡ ν (mod 2), p, ν ∈ Z≥0 . Suppose that .m0 (j ) ≥ 0. . p,0 p,0 (i) .μ˜ n (j, ν) = υ˜ n (j, ν) in the following cases:   • .ε = 1 and .−j + ν ≤ 2 max p, m0 (j ) , • .ε = −1 and .j + ν ≤ 2m0 (j ). . 0,p 0,p (ii) .μ˜ n (j, ν) = υ˜ n (j, ν) in the following cases:

.

• .ε = 1 and .−j + ν ≤ 2m0 (j ),  • .ε = −1 and .j + ν ≤ 2 max p, m0 (j ) . . p,0 p,0 (iii) .μ˜ n (j, ν) = ω˜ n (j, ν) in the following cases:   • .ε = 1 and .j + ν ≤ −2 − 2 max p, m0 (j ) , • .ε = −1 and .−j + ν ≤ −2 − 2m0 (j ). . 0,p 0,p (iv) .μ˜ n (j, ν) = ω˜ n (j, ν) in the following cases: • .ε = 1 and .j + ν ≤ −2 − 2m0 (j ),   • .ε = −1 and .−j + ν ≤ −2 − 2 max p, m0 (j ) . . By .= we denote equality up to a non-zero factor.

128

4 Submodule Structure

We note that the conditions in (i) and (ii) and the corresponding conditions in (iii) and (iv) exclude each other. . Proof With (A.16) we have .Mκ,s = Vκ,s if and only if . 12 + κ + s ∈ Z≤0 . We p,0 p,0 apply this to the determining components of .μ˜ n (j, ν) and .υ˜ n (j, ν) as given in Table 4.7, p. 124, to get .j ≡ ν mod 2 and j + ν ≤ 2m0 (j )

.

if ε = −1 ,

−j + ν ≤ 2m0 (j )

if ε = 1 and p ≤ m0 (j ) ,

−j + ν ≤ 2p

if ε = 1 and p ≥ m0 (j ) .

This can be reformulated as the conditions in (i). For (ii) we proceed analogously. . Equation (A.16) gives also .Mκ,s = Wκ,s if and only if . 12 − κ + s ∈ Z≤0 . We use . p,0 p,0 again Table 4.7 to get for .μ˜ n (j, ν) = ω˜ n (j, ν) the conditions .

  − j + ν ≤ −2 m0 (j ) + 1   j + ν ≤ −2 m0 (j ) + 1 j + ν ≤ −2(p + 1)

if ε = −1 , if ε = 1 and p ≤ m0 (j ) , if ε = 1 and p ≥ m0 (j ) .



This gives (iii) and, analogously, (iv).

4.4.3 Intersection of Kernels The intersection of kernels of downward shift operators was considered for the abelian cases in Propositions 4.6 and 4.8. Here again, we use the notations and conventions of Lemma 4.5. Proposition 4.16 Let .τph be a K-type occurring in .Fn . Denote by .Kn;h,p the −3 3 intersection of the kernels of .S−1 : Fn;h,p,p → Fn;h+3,p−1,p−1 and of .S−1 : Fn;h,p,p → Fn;h−3,p−1,p−1 . We define kW n;h,p =

p

ϑm(h,r) t p+1 cW (r) Wκ(r),s(r) (2π||t 2 ) h r,p ,

r

.

kVn;h,p

=

p

ϑm(h,r) t p+1 cV (r) Vκ(r),s(r) (2π||t 2 ) h r,p ,

(4.57)

r

where .r ≡ p mod 2, .|r| ≤ p, .m(h, r) ≥ 0, and where .ϑm is an abbreviation for ,c (h,m ). We use .m(h, r) as indicated in Table 4.5, p. 122, and

.

1 κ(r) = −m(h, r) − ε s(h, r) − , 2

.

s(r) = s(h, r) =

h−r . 4

(4.58)

4.4 Submodule Structure of Non-abelian Fourier Term Modules

129

The coefficients are  cW (r) = i m(h,r) m(h, r)! ,

.

(−1)m(h,r) , cV (r) = √ m(h, r)!

(4.59)

for .r ≡ p mod 2 such that .m(h, r) ≥ 0. (i) (a) If .|r0 (h)| > p, then .Kn;h,p has dimension 2, and is spanned by .kW n;h,p and V .k n;h,p . (b) If .|r0 (h)| ≤ p, then .kVn;h,p spans .Kn;h,p . (ii) The subspace .Kn;h,p of the large Fourier term module .Fn is contained in the ψ module .Fn , where .ψ = ψ[−h, p]. With the notation of Lemma 4.5, we have the following equalities up to a non-zero factor: (a) If .m0 (j1 ) ≥ 0 and .m0 (j2 ) ≥ 0, then . 0,p . p,0 kW ˜ n (j1 , ν1 ) = ω˜ n (j2 , ν2 ) , n;h,p = ω .

. 0,p . p,0 kVn;h,p = υ˜ n (j1 , ν1 ) = υ˜ n (j2 , ν2 ) ,

(4.60)

(If .r0 (h) = −εp, then .kW n;h,p ∈ Kn;h,p .) (b) If .ε = 1 and .0 ≤ m0 (j1 ) < p (and hence .m0 (j2 ) < 0), then . p,0 kVn;h,p = υ˜ n (j1 , ν1 ) .

.

(4.61)

(c) If .ε = −1 and .0 ≤ m0 (j2 ) < p (and hence .m0 (j1 )) < 0), then 0,p

kVn;h,p = υ˜ n (j2 , ν2 ) .

.

(4.62)

Remarks (1) The Whittaker functions .Wκ,s and .Vκ,s are well defined and linearly independent for all values of the parameters. In Proposition 4.27 we will define .kM n;h,p based on .Mκ,s in a similar way. (2) The condition .|r0 (h)| > p for dimension 2 in (i) is stricter than the conditions on .m0 (j1 ) and .m0 (j2 ) in (ii)(a). This part is valid if one of the .m0 ’s is equal to p and the other equal to 0. In that case .kW n;h,p is not an element of .Kn;h,p . (3) This proposition is analogous to Proposition 4.8 in the abelian case. In Proposition 4.24 we will discuss a result analogous to Corollary 4.12. Proof The proof has to consider many cases, some of them will be considered in Lemma 4.17. We use the kernel relations for the downward shift operators. That reduces the question to an equality of the determining components.

130

4 Submodule Structure

Suppose that F =

.

ϑm(h,r) fr

h

p

r,p

r

is an element of .Kn;h,p . Then it has to satisfy the kernel relations in Table 4.6, p. 123. The computations are mostly done with Mathematica. In [6, §21a] we check that the components are indeed given by Whittaker functions. We first consider the case that only one component can be non-zero. That happens if .m(h, εp) = 0, with component .fεp . Then the kernel relations impose a linear differential equation for .fεp which implies that it is of the form . fεp (t) = t p+1 Vκ(εp),s(h,εp) (2π||t 2 ) .

.

(4.63)

. Hence .F = kVn;h,p spans .Kn;h,p in the case of one component. Thus we have (i)(b) if .m(h, εp) = 0. In all other cases there are more components to consider. We take r such that .fr and .fr+2 can occur in the sum. We combine the kernel relations to get a second order differential equation that implies that .fr (t) = t p+1 gr (2π||t 2 ) where g is a solution of the Whittaker differential equation (A.8) with parameters .κ(r) and .s(h, r). So each component is in a well-defined two-dimensional subspace of .C ∞ (0, ∞). The kernel relations involve differentiations and multiplications by powers of t. The first six contiguous relations in (A.18) and (A.19) imply that if one component has the form .t p+1 Wκ,s (2π||t 2 ), then all other components can be expressed in W -Whittaker functions. Analogously for V -Whittaker functions. So we look for expressions of the form given in (4.57), and try to determine how the coefficients are related. For that purpose we need also contiguous relations in which the parameters are shifted by . 21 . The complicated computations are in [6, 21b] and lead to recursive relations for the coefficients, for which (4.59) gives solutions. If .|r0 (h)| ≤ p there is one more kernel relation. The case .r0 (h) = εp has already been discussed. If .ε = 1 the find that the W -Whittaker function does not satisfy the relation. So in this case we are left with .kVn;h,p . Let .ε = −1 and .−p < r0 (h) ≤ p. In this case as well, only .kVn;h,p satisfies the kernel relations. This completes the proof of (i). For (ii) we consider the identifications up to a non-zero factor. The results then ψ imply that .Kn;h,p ⊂ Fn . It suffices to compare the determining components in Table 4.7 with the corresponding component of the functions in (4.57). We have to check the resulting relations for Whittaker functions, which we consider in Lemma 4.17 below. Part (i) of the lemma deals with the determining components for (ii)(a). We work with .Wκ,s and .Vκ,s , which are even in s. Part (ii) of the lemma gives the proof of (ii)(b) and (ii)(c). In the proof of (ii)(b) and (ii)(c) we use that .ν1 = ±(2p + j1 ) and .ν2 = ±(2p + j2 ). This is in agreement

4.4 Submodule Structure of Non-abelian Fourier Term Modules

131

with the kernel conditions in Table 4.9, and seems to make the proof circular. However, it is also a consequence of .h = 2j1 + 2p = 2j2 − 3p together with

the relations in Lemma 4.5. Lemma 4.17 We use the notations of Proposition 4.16. (i) The condition .m0 (j1 ) ≥ 0, .m0 (j2 ) ≥ 0 is equivalent to .εr0 (h) ≥ p. These equivalent conditions imply equality of the parameters in the Whittaker differential equation. εj1 + 1 , 2 εj2 + 1 κ(p) = −m0 (j2 ) − , 2

κ(−p) = −m0 (j1 ) − .

s(−p) = ±ν1 /2 , (4.64) s(p) = ±ν2 /2 .

(ii) The condition .m0 (j2 ) < 0 ≤ m0 (j1 ) is equivalent to .ε = 1 and .−p < r0 (h) ≤ p. Under these equivalent conditions . t m0 (j1 )+1 V−p−(j1 +1)/2,(h+p)/4 (2π ||t 2 ) = t p+1 Vκ(r0 (h)),s(r0 (h)) (2π||t 2 )

.

2 . = t 2+j1 +2p+m0 (j1 ) eπ t .

(iii) The condition .m0 (j1 ) < 0 ≤ m0 (j2 ) is equivalent to .ε = −1 and .−p ≤ r0 (h) < p. Under these equivalent conditions . t m0 (j2 )+1 V−p+(j2 −1)/2,(h−p)/4 (2π ||t 2 ) = t p+1 Vκ(r0 (h)),s(r0 (h)) (2π||t 2 )

.

2 . = t 2−j2 +2p+m0 (j2 ) eπ||t .

Proof The equivalences follow from .m0 (j2 ) − m0 (j1 ) = −εp, and .r0 (h) = p − 2εm0 (j1 ) = −p − 2εm0 (j2 ). The equality of the two sets of parameters .κ and s in (i) can be checked by a computation, using Tables 4.5 and 4.9, and Lemma 4.5. The parameters of the Whittaker functions in (ii) are not equal. Working them out, we arrive at values of the parameters to which we can apply the specialization in (A.17). This leads for both functions to an explicit expression in t, with different

non-zero factors. The computations are carried out in [6, 21c].

4.4.4 Dimension ψ

Lemma 4.18 In the notations of Proposition 4.16: If dim Fn;h,p,p > 2 then at least one of the following statements holds: .

ψ

dim Fn;h+3,p−1,p−1 > 2 ,

ψ

dim Fn;h−3,p−1,p−1 > 2 .

132

4 Submodule Structure

Proof In principle this is the same situation as in Lemma 4.9. However, in the nonabelian case more possibilities have to be considered. In all cases the question is reduced in such a way that a computation can provide the answer. ψ We know that dim Fn;h,p,p is at least two, by the presence of two linearly p,0

p,0

independent elements: bω = υ˜ n (j1 , ν1 ) and bυ = ω˜ n (j1 , ν1 ) if m0 (j1 ) ≥ 0, 0,p 0,p and bω = υ˜ n (j2 , ν2 ) and bυ = ω˜ n (j2 , ν2 ) if m0 (j2 ) ≥ 0. If both m0 (j1 ) ≥ 0 and m0 (j2 ) ≥ 0, the two choices of bω and bυ are proportional. ψ ψ Similarly, dim Fn;h+3,p−1,p−1 ≥ 2 if m0 (j2 ) ≥ 0, and dim Fn;h−3,p−1,p−1 ≥ 2 if m0 (j1 ) ≥ 0. Since the K-type τph occurs, we cannot have both m0 (j1 ) < 0 and m0 (j2 ) < 0. We choose bω+ = ω˜ n

(j1 , ν1 ) ,

bυ+ = υ˜ n

(j1 , ν1 )

bω− = ω˜ n

(j2 , ν2 ) ,

bυ− = υ˜ n

(j2 , ν2 )

p−1,0

.

0,p−1

p−1,0 0,p−1

  m0 (j1 ) ≥ 0 ,   m0 (j2 ) ≥ 0 .

(4.65)

If m0 (j1 ) < 0, then Fn;h−3,p−1,p−1 = {0}, and we take bω+ = bυ+ = 0. Similarly, we take bω− = bυ− = 0 if m0 (j2 ) = 0. ψ ψ To prove the lemma we suppose that dim Fn;h,p,p > 2, and that Fn;h−3,p−1,p−1 ψ

and Fn;h+3,p−1,p−1 are either zero or spanned by bω± and bυ± . We take F ∈ ψ

∓3 Fn;h,p,p that is not a linear combination of bω and bυ , and put F ± = S−1 F ∈ ψ ψ

Fn;h∓3,p−1,p−1 . See Fig. 4.28. We can write these derivatives as F ± = cω± bω± + cυ± bυ± .

.

(4.66)

That should lead to a contradiction. If both F + and F − are zero, then F ∈ Kh,p , hence F is a linear combination of bω and bυ . So we assume that at least one of the derivatives is non-zero. We have many cases, requiring essentially two different approaches.

Fig. 4.28 Shift operator ∓3 F relation F ± = S−1

4.4 Submodule Structure of Non-abelian Fourier Term Modules

133

Case 0 < m0 (j1 ) < p and ε = 1 Then m0 (j2 ) < 0, and F − = 0. Computations in [6, §22a]. 3 F =0 The function F has components of order r between r0 (h) and p. Since S−1 the function F is determined by its lowest component fr0 . The lowest component of F + has order r0 + 1 and a computation shows that it is equal to a non-zero multiple of

2tfr0 − (4 + h + 2p − r0 + 4π t 2 )fr0 .

.

. −3 In this case bω is not an element of Kh,p = C bυ , and S−1 bω = bω+ . See Proposition 4.16. So we can subtract a multiple of bω from F to arrange that . F = bυ+ . With Table 4.7 we get . 2tfr0 − (4 + h + 2p − r0 + 4π t 2 )fr0 = t 0+1 V−p+1−(j1 +1)/2,ν  /2 (2π||t 2 ) . (4.67)

.

This gives expressions for the derivatives of fr0 in terms of fr0 and V -Whittaker functions. In the eigenfunction equations for r = r0 also terms with fr0 +2 are present. 3 F = 0 we can express f Since S−1 r0 +2 in terms of fr0 by the kernel relation for 3 S−1 . Substitution of all these expressions into the eigenfunction equations gives two linear combination of V -Whittaker functions. The asymptotic behavior in (A.15) shows that the implicit factor in (4.67) has to be zero. Hence fr0 and F have to vanish. Case m0 (j1 ) = 0 and ε = 1 So m0 (j2 ) < 0 and F − = 0. Computations in [6, §22b]. The function F has in this case only one component, fr0 = fp . So there is no need to determine fr0 +2 . The further computation is similar to the previous one. Cases 0 ≤ m0 (j2 ) < p and ε = −1 Then m0 (j1 ) < 0 and F + = 0. We proceed as in the previous cases. See [6, §22cd]. Cases m0 (j1 ) ≥ 0, m0 (j2 ) ≥ 0 Now both F + and F − may be nonzero. We can write F ± = cω± bω± + cυ± bυ± .

.

(4.68)

The components of F can have order −p ≤ r ≤ p, and the components of F ± can have order between 1 − p and p − 1. The determining components of bω± and bυ± have order ±(1 − p). + − of F + , and the lowest component f1−p We compute the highest component fp−1 ±  of F − . The function f∓(p−1) is a linear combination of f∓p , f∓(p−2) and f∓(p−2) . Since we know the determining components of the functions in the right-hand side

134

4 Submodule Structure

 of (4.68), we can solve for f∓(p−2) . Substitution into the eigenfunction equations for r = −p and for r = p leads to a relation involving Whittaker functions only, which shows that all four coefficients in (4.68) have to vanish. We carry out the actual computations for ε = 1 and for ε = −1 separately, in [6, §22ef]. ψ

Conclusion In all cases we conclude that the presence of F ∈ Fn;h,p,p linearly independent of bω and bυ would lead to derivatives F + and F − , of which at least one is linearly independent of bω± and bυ± .

ψ

Proposition 4.19 For ψ ∈ Wip the dimension of Fn;h,p,p is equal to 2 for all K  types τph that satisfy h − 2j  ≤ 3p for some j ∈ O1W (ψ)+ n . All other K-types do ψ

not occur in Fn . ψ

Proof By Sect. 3.4.2.3 we have dim Fn;h,p,p = 2 for (h/3, p) = (2j/3, 0) for p,0

p,0

all j ∈ O1W (ψ)n . Proposition 4.14 provides us with ω˜ n (j, ν) and υ˜ n (j, ν) in ψ Fn;2j +3p,p,p for each p ≥ 1 and each j ∈ O1W (ψ)n . These elements are linearly ψ

independent. Similarly, the spaces Fn;2j −3p,p,p have dimension at least two for j ∈ O1W (ψ)n . At least one of the upward shift operators is injective, by (ii) in ψ Proposition 3.24. Hence all spaces Fn;h,p,p indicated in the lemma have dimension at least two.   For any point (h/3, p) ∈ Sect(j ) there is a path to the base point 2j/3, 0 , corresponding to downward shift operators. Along this path the dimension of ψ Fn;h,p,p cannot decrease by Lemma 4.18. So all K-types mentioned in the lemma have multiplicity exactly equal to 2. Starting from points outside the sectors with base points (2j  /3, 0), we obtain ψ a path to a point (h/3, 0) on the horizontal axis, for which dim Fn;h,0,0 = 0. This

concludes the proof.

4.4.5 Special Submodules ψ

ψ

ψ

Inside .Fn we have the special submodules .Wn and .Mn , defined by their behavior as .t ↑ ∞ and .t ↓ 0 on .na(t)k. See Definitions 3.32 and 3.33. We define for .ψ ∈ Wip ψ ξ,ν ξ,ν ξ,ν subspaces .Wn , .Vn and .Mξ,ν of .Fn , for .j ∈ OW (ψ)+ and n . We show that .W ψ ψ ξ,ν ξ,ν ξ,ν .V are .(g, K)-modules, and that .W ⊂ Wn and .M ⊂ Mn . We define ψ ψ ξ,ν .Vn ⊃ Vn , with properties similar to .Wn .

4.4 Submodule Structure of Non-abelian Fourier Term Modules

135

Definition 4.20 Let .ψ ∈ Wip . We define for .(j, ν) ∈ OW (ψ)+ n the following Kmodules

p,0 0,p ξ,ν .Wn = U (k) (S13 )a (S1−3 )b ω˜ n (j, ν) + U (k) (S13 )a (S1−3 )b ω˜ n (j, ν) , p≥0 a,b≥0 ξ,ν

Vn =



p≥0 a,b≥0 ξ,ν

Mn =



p,0 0,p U (k) (S13 )a (S1−3 )b υ˜ n (j, ν) + U (k) (S13 )a (S1−3 )b υ˜ n (j, ν) ,

p,0 0,p U (k) (S13 )a (S1−3 )b μ˜ n (j, ν) + U (k) (S13 )a (S1−3 )b μ˜ n (j, ν) .

p≥0 a,b≥0

Remarks (i) We use this complicated description, since in the non-abelian case the upward a+p,b shift operators are not always injective. So .ωn (j, ν) may be zero in p,0 −3 b 3 a situations where .(S1 ) (S1 ) ω˜ n (j, ν) is non-zero. (ii) In the definition we speak of K-modules. We still have to show that these spaces are invariant under the action of .g. ψ[ξ,ν] ξ,ν ξ,ν (iii) The spaces .Wn are contained in .Wn , and similarly for .Mn . ψ ψ Lemma 4.15 implies that the modules .Wn and .Mn may have non-zero p,0 0,p intersection. This makes it useful to use also the families .υ˜ n and .υ˜ n . We ψ have not yet defined .Vn . Lemma 4.21 The .(k, K)-modules in Definition 4.20 are .(g, K)-modules. Proof We use .x p,0 and .x 0,p as a general notation. The shift operators on these −3 p,0 3 x p,0 = 0, and analoelements satisfy .S13 x p,0 ∈ C x p+1,0 , .S−1 x ∈ C x p−1,0 , .S−1 0,p gously for .x . See Propositions 3.1, 3.36, and (4.56). Each element of .U (g) can be written as a linear combination of .uZa31 Zb23 Zc32 Zd13 , with .u ∈ U (k), and .a, b, c, d ∈ Z≥0 . Applied to .x p,0 this can be rewritten, with the expressions in Table 3.4. p. 36,   −3 d  p,0 ) x , which as a linear combination of elements of the form .u (S13 )a (S1− )b (S−1

is in the space under consideration. The image of .x 0,p is handled analogously. ψ

ψ

We turn to the .(g, K)-modules .Wn and .Mn in Definitions 3.32 and 3.33. Lemma 4.22 Let .ψ ∈ Wip .

.

(j,ν)∈OW (ψ)+ n

ξ ,ν

Wnj

ψ

⊂ Wn ,

ξ ,ν

Mnj

ψ

⊂ Mn .

(4.69)

(j,ν)∈OW (ψ)+ n

Proof Let x stand for .ω or .μ, and .X for .W or .M. Let .(j, ν0 ) ∈ OW (ψ)+ n. ψ[j,ν)] for all .ν in a neighborhood of .ν0 . This The elements .xn0,0 (j, ν) are in .Xn property is preserved by the upward shift operators. However, they might give the result zero. In Proposition 4.14 we form the elements .x˜ p,0 and .x˜ 0,p by dividing out such zeros. They are determined by their minimal or maximal component, which

136

4 Submodule Structure

has exponential  decay for  .x = ω and a-regular behavior at 0 for .x = μ, with a ≥ 1 + min p, m0 (j ) + ν0 ≥ 0. The other components are determined by the kernel relations in Table 4.6, p. 123. This clearly preserves exponential decay. Application of the kernel relations in the case .x = μ is problematic. At each transition .r → r ± 2 we may loose a factor t. Let us look at the family .ν → μp,0 (j, ν), obtained by differentiation; see (3.93). It is .C ∞ in .g ∈ G and .ν in a neighborhood of .ν0 and holomorphic in .ν, and has .ν-regular behavior at 0, according to Proposition 3.31. Its components have the form .t → t ν hr (t, ν) where .hr extends holomorphically to .C times a neighborhood on .ν0 . There may be common zeros p,0 to be divided out in the recursion leading to .μ˜ n (j, ν). So we have .hr (t, ν) = p,0 kr (t, ν) (ν − ν0 )a , with .a ∈ Z≥0 depending on p, but not on r. Then .μ˜ n (j, ν0 ) is a multiple of the function with components .t → t ν0 kr (t, ν0 ), and has .ν0 -regular 0,p behavior at 0. The same approach can be followed for .μ˜ n (j, ν0 ). ψ ψ

Since .Mn and .Wn are .(g, K)-modules, the lemma follows.

.

We define analogously the following .(g, K)-module. ψ

Vn =

.

ξ ,ν

Vnj .

(4.70)

(j,ν)∈OW (ψ)+ n

Lemma 4.23 Let .ψ ∈ Wip . ξ ,ν ψ (i) .Wn = (j,ν)∈OW (ψ)+n Wnj . h (ii) Let .(j, ν), (j  , ν  ) ∈ OW (ψ)+ n . If the K-types .τp corresponds to a point ξ  ,ν 

j (h/3, p) ∈ Sect(j ) ∩ Sect(j  ), then .Wn;h,p,p = Wn;h,p,p .

ξ,ν

.

Fix .(j, ν) ∈ OW (ψ)+ n. ξ,ν

ξ,ν

(iii) The K-types .τph occurring in .Wn and in .Vn correspond to the points .(h/3, p) ∈ Sect(j ), and have multiplicity one. (iv) We define for .a, b ∈ Z≥0 the families .υ˜ na,b and .ω˜ a,b by  .x ˜na,b (j, ν)

=

b S1−3 x˜ a,0 (j, ν)  3 a 0,b S1 x˜ (j, ν)

if  > 0 , if  < 0 ,

(4.71)

for .x = υ or .ω. ξ,ν Put .h = 2j + 3(a − b), .p = a + b, The space .Vn;h,p,p is spanned by .

ξ,ν

υ˜ na,b (jξ , ν), and .Wn;h,p,p is spanned by .ω˜ na,b (jξ , ν). ξ,ν

ξ,ν

(v) .Wn ∩ Vn = {0}. Proof The kernel relations in Table 4.6, p. 123, and the contiguous relations p,0 0,p in (A.19) imply that the components of .ω˜ n and .ω˜ n are linear combinations of functions .t c Wκ+m,ν/2 (2π||t 2 ) with .c, m ∈ Z. Application of the upward and ψ downward shift operators stay within the space of functions in .Fn with components

4.4 Submodule Structure of Non-abelian Fourier Term Modules

137

of this form. The asymptotic behavior (A.14) implies that these functions have ψ ξ,ν exponential decay at .∞. Hence .Wn ⊂ Wn . ξ,ν For the modules .Vn we have similar descriptions, now with V -Whittaker functions instead of W -Whittaker functions. Lemma A.1 implies that non-zero j,ν elements of .U (g)Vn cannot have polynomial growth. This gives (v). ξj ,ν ξj ,ν We get in particular .dim Wn;h,p,p ≥ 1 and .dim Vn;h,p,p ≥ 1 for each .(h/3, p) ∈ Sect(j ). With Proposition 4.19 we conclude that these dimensions are exactly equal ψ ψ to 1. This gives (iii). In this way we get also .dim Wn;h,p,p = 1 and .dim Vn;h,p,p = ξ ,ν

ψ

j 1 for .(h/3, p) ∈ Sect(j ). Hence .Wn;h,p,p = Wn;h,p,p , and similarly for .V. This implies (ii) and (i). In Proposition 4.14 we constructed the non-zero families .x˜na,0 (j, ν) and 0,b .x ˜n (j, ν) by dividing out zeros of .xna,0 (j, ν) and .xn0,b (j, ν). The injectivity of the shift operators .S1∓3 for .± > 0 (Proposition 3.24) leads to non-zero families .x ˜na,b (j, ν), which span the corresponding highest weight spaces by the multiplicity one of the K-types.



This result gives immediately the following identifications, completing (ii) in Proposition 4.16. Proposition 4.24 Let .τph , .Kn;h,p , .kVn;h,p and .kW h, be as in Proposition 4.16. ψ

(i) .kVn;h,p spans the space .Vn;h,p,p ⊂ Kn;h,p . ψ

(ii) If .r0 (h) > p, then .kW n;h,p spans the space .Wn;h,p,p ⊂ Kn;h,p . Remark This result is analogous to Corollary 4.12 in the generic abelian case. Here we cannot formulate the result in terms of the families .υna,b and .ωna,b , since these may be zero at the relevant parameter values. Vanishing of Shift Operators We turn to the determination of the lines in the ξ,ν ξ,ν (h/3, p)-plane corresponding to K-types in .Wn and .Vn on which one of the shift operators vanishes.

.

Lemma 4.25 Let .ψ ∈ Wip , and let .(j, ν) ∈ OW (ψ)+ n with .n = (, c, d). ξ,ν

(i) Theupward  shift operators in .Vn are injective. (ii) Let . h3 , p ∈ Sect(j ). The upward shift operator S1±3 : Wn;h,p,p → Wn;h±3,p+1,p+1

.

ξ,ν

ξ,ν

is zero if and only the following conditions are satisfied: • .± > 0, • there is .j  ∈ O1W (ψ)O1W (ψ)n such that .m0 (j  ) < 0, and .h−2j  ±3p±6 = 0. (The occurrences of .± are coupled.)

138

4 Submodule Structure

Fig. 4.29 Upward shift operators can vanish on the dotted line inside the sectors

Remark The number of such elements .j  is at most equal to 2. We have in the notations of (4.2): set of j  ε j 1 j = ll < j+ {j+ , jr } . j = j+ = jr {jr } −1 j = jr > j+ {jl , j+ } j+ = jl {jl }

(4.72)

±3 can occur if .ε = Proof Proposition 3.24 shows that a non-zero kernel of .S−1  sign () = ±1 for K-types on a line .h − 2j ± 3p ± 6 = 0 with .j  ∈ O1W (ψ) such that .m0 (j  ) < 0, which means .j  ∈ O1W (ψ)  O1W (ψ)n . In Fig. 4.29 these points are on the dashed lines inside the sector. Let .ε = 1, hence .S1−3 is injective. Consider a point .(h/3, p) on the line .h = ψ 2j  − 6 − 3p, and functions .yj ∈ Fn;h−3j,p+j,p+j related by .S1−3 yj = yj +1 . Suppose that .S13 y0 = 0. The upward shift operators commute (Proposition 3.1). So we have .S13 y1 = 0, and also .S13 y−1 = 0 if .(h/3, p) is in the sector .Sect(j ), and .y−1 corresponds to a point on the dashed line. This shows that the kernels of .S13 on K-types corresponding to points on the dotted line are related by the injective map .S1−3 . See Fig. 4.30.   This brings us to the intersection point . h30 , p0 of the line .h = 2j  − 6 − 3p with the right boundary line .h = 2j + 3p of the sector .Sect(j ). Table 4.8, p. 125, p ,0 p ,0 3 υ 3 ω shows that .S−1 ˜ n 0 (j, ν) is non-zero for all .ν, and that .S−1 ˜ n 0 (j, ν) = 0 for   .ν = ± 2p0 + 2 + j . So we have (i) in the case .ε = 1.   For part (ii) we consider Fig. 4.31. One step up from . h30 , p0 is the point  h1   . 3 , p1 , which is the lowest point in .Sect(j )∩Sect(j ). We can apply Lemma 4.5 to 1| . this situation, with j in the role of .j1 and .j  in the role of .j2 . This gives .ν = |h1 +p 2

.

h1 + p1 1 = 2j + 4p1 ) = j + 2p0 + 2 . 2 2

4.4 Submodule Structure of Non-abelian Fourier Term Modules

139

Fig. 4.30 Propagation of non-trivial kernel of .S13

Fig. 4.31 Illustration of the use of Lemma 4.5

ξ,ν

So indeed .S13 vanishes on .Wn;h0 ,p0 ,p0 , and hence on all K-types corresponding to points in .Sect(j ) on the line .h = 2j  − 6 − 3p. This gives (ii) in the case .ε = 1. The case of .S1−3 goes analogously, now using .ε = −1.

ψ

Lemma 4.26 Let .ψ ∈ Wip , and let .τph be a K-type occurring in .Fn . ±3 is zero on .Vn;h,p,p if .h = 2j ± 3p for some (i) The downward shift operator .S−1 1 .j ∈ O (ψ), and injective otherwise. W ψ ±3 (ii) The downward shift operator .S−1 is zero on .Wn;h,p,p if .± < 0 and .h = 2j  ± 3p for some .j  ∈ O1W (ψ)n , and injective otherwise. ψ

(The occurrences of .± are coupled.)

140

4 Submodule Structure

Fig. 4.32 Intersection of boundary lines of sectors

ψ

ψ

Proof The spaces .Wn;h,p,p and .Vn;h,p,p have dimension at most 1, so injectivity and vanishing are the only possibilities. By (ii)(a) in Proposition 4.14 the operator ±3 .S −1 vanishes on K-types corresponding to points on the line .h = 2j ±3p if .(j, ν) ∈ OW (ψ)+ n. Proposition 3.21 shows that we also have to consider  h  points in .Sect(j ) on lines 2 1   .h ∓ 3p = 2j for .j ∈ O (ψ)  O (ψ)n . Let . W W 3 , p be the intersection point of the lines .h ± 3p = 2j and .h ∓ 3p = 2j  . See Fig. 4.32. Table 4.5, p. 122, shows that .m0 (j ) − m0 (j  ) = − 13 (j − j  ) sign (). Since   .m0 (j ) < 0 ≤ m0 (j ) we need .∓ > 0. Furthermore .±(j − j ) = 3p, and .m0 (j ) =  m0 (j ) + p < p. We use this in the application   ofTable 4.9,  p. 125, with .(j, ν) ∈ 1    OW (ψ)+ n . Lemma 4.5 shows that .ν = 2 h ∓ p = j ∓ 2p . We obtain 0,p

p,0

case +

3 S−1 ω˜ n (j, ν) = 0 ,

3 S−1 υ˜ n (j, ν) = 0 ,

case –

−3 ω˜ n (j, ν) = 0 , S−1

−3 S−1 υ˜ n (j, ν) = 0 .

.

0,p

0,p

(4.73)

We can extend these properties of points upward along the line .h ∓ 3p = 2j  ψ ∓3 as long as the shift operator .S−1 is injective on the corresponding K-types in .Fn , in analogy with the approach in the proof of Lemma 4.25. This injectivity does not hold if we meet a line .h = 2j  ∓ 3p with another .j  ∈ O2W (ψ). See Fig. 4.33.

4.4 Submodule Structure of Non-abelian Fourier Term Modules

141

Fig. 4.33 Intersection with boundary of a third sector

Since .j  and .j  are on different sides of j we have .m0 (j  ) ≥ m0 (j ) ≥ 0, and 1  .j ∈ O (ψ)n . There we can start the reasoning again. This gives (i) and (ii).

W

4.4.6 Modules with Regular Behavior at 0 The discussion in the previous subsections looks rather satisfactory, except for the ψ fact that .Vn is not naturally defined. It depends on the choice of the unusual V -Whittaker function .Vκ,s in (A.12), depending on a choice of a branch of the continuation of .Wκ,s . The function .Mκ,s is much more natural, it leads to functions with .ν-regular behavior at 0; see Definition 3.30. It has the disadvantage of being proportional to .Wκ,s or to .Vκ,s for some combinations of the parameter values that are relevant in the non-abelian case. ψ[j,ν] In this subsection we establish results for .Mn under integral parametrization. We restrict ourselves to values .ν ∈ Z≥0 . Propositions 4.27 and 4.31 consider for ψ ψ .Mn questions studied in Proposition 4.16, concerning subspaces in .M n;h,p,p with .p ≥ 1 on which both downward shift operators vanish. We extend in Lemma 4.30 the multiplicity one result in (iii) in Lemma 4.23. Proposition 4.33 determines the ψ ψ conditions under which .Mn;h,p,p has a non-zero intersection with .Vn;h,p,p or with ψ

Mn;h,p,p .

.

Intersection of Kernels of Downward Shift Operators Proposition 4.16 describes the intersection of the kernel .Kn;h,p of both downward shift operators on a moreψ dimensional K-type .τph in .Fn . It gives two types of information: it tells how kernel p,0

elements arise as values of families .xn and .x 0,p for .x = υ˜ or .ω, ˜ and moreover it gives an explicit expression of basis elements of .Kn;h,p . ψ

Here we consider the intersection of .Kn;h,p with .Mn , and determine which p,0 0,p of the families .μ˜ n and .μ˜ n have values in this intersection. In some cases an ψ ψ explicit description of a basis of .Kn;h,p ∩ Mn is possible. We recall that .Mn is the

142

4 Submodule Structure

Fig. 4.34 Position of .(h/3, p) in the three combinations of .(j1 , j2 ) in Proposition 4.27: Combination 1: .j1 = jl < j2 = j+ < jr , combination 2: .j1 = jl < j2 = jr , combination 3: .jl < j1 = j+ < j2 = jr . We use the conventions in (4.2)

ψ

submodule of .Fn generated by .ν-regular behavior at 0 for some .ν with .Re ν ≥ 0. See Definitions 3.30 and 3.33. The point .(h/3, p) corresponds to the K-type .τph in the intersection of the boundaries of the sectors .Sect(j1 ) and .Sect(j2 ) for elements .j1 , j2 ∈ OW (ψ)+ , determined by .h = 2j1 + 3p = 2j2 − 3p with .p ≥ 1. The description of ψ .Kn;h,p ∩ Mn depends on the choices of the combination .(j1 , j2 ). In general there are three choices, depicted in Fig. 4.34, corresponding to points .(j1 , ν1 ) and .(j2 , ν2 ) in the interior of Weyl chambers in Fig. 3.9, p. 63. If .(j1 , ν1 ) and .(j2 , ν2 ) are on walls of Weyl chambers, the three combinations reduce to one, which we arbitrarily take under combination 2. The values of .h = j1 + j2 and .p = 13 (j2 − j1 ) depend on the combination. We use a subscript 1, 2, or 3 when needed. With Lemma 4.5 we can check that .h < −p for combination 1, .|h| ≤ p for combination 2, and .h > p for combination 3. Proposition 4.27 We use the notations of Proposition 4.16. We put kM n;h,p =

r

.

c (r) = −e M

p

ϑm(h,r) cM (r) t p+1 Mκ(r),|s(r)| (2π||t 2 ) h r,p ,

π i(m(h,r)−κ(r))

   12 + |s(r)| − κ(r)   , √ m(h, r)! 2|s(r)| !

(4.74)

where the sum runs over .r ≡ p mod 2, .|r| ≤ p, .m(h, r) ≥ 0. The function .kM n;h,p is an element of .Kn;h,p that spans the intersection .Kn;h,p ∩ ψ

Mn if one of the following conditions is satisfied: (a) .m0 (j1 ) ≥ 0 and .m0 (j2 ) ≥ 0, and one of both is larger than p. . . p,0 0,p Under these conditions, .kM ˜ n (j1 , ν2 ) = μ˜ n (j2 , ν2 ). If, in n;h,p = μ   V addition, .ε h − r0 (h) ≥ 0, then .kM n;h,p = kn;h,p . (b) .ε = 1, .0 ≤ m0 (j1 ) ≤ p, and .r0 (h) ≤ h. . p,0 V M Under these conditions, .kM ˜ n (j1 , ν1 ). n;h,p = kn;h,p , and .kn;h,p = μ

4.4 Submodule Structure of Non-abelian Fourier Term Modules

143

(c) .ε = −1, .0 ≤ m0 (j2 ) ≤ p, and .r0 (h) ≥ h. . 0,p V M Under these conditions, .kM ˜ n (j2 , ν2 ). n;h,p = kn;h,p , and .kn;h,p = μ We prepare the proof by two lemmas. The first lemma gives elements that we ψ know to be in .Kn;h,p ∩ Mn under some conditions. These conditions can be formulated in terms of .m0 (j1 ) and .m0 (j2 ). In particular, .m0 (j1 ) ≥ 0 if and only ξ ,ν if .Mn1 1 is non-zero, and similarly for .m0 (j2 ). It can also be formulated in terms of .r0 = r0 (h) = p − 2εm0 (j1 ) = −p − 2εm0 (j2 ). Lemma 4.28 We use the notations indicated above. p,0

ξ ,ν1

(i) The element .μ˜ n (j1 , ν1 ) is in .Kn;h,p ∩ Mn1 comb. 1 ε=1 . ε = −1 2, 3 ε = ±1

under the following conditions.

conditions m0 (j1 ) ≥ p r0 ≤ −p m0 (j1 ) ≥ 0 r0 ≥ −p m0 (j1 ) ≥ 0 εr0 ≤ p

0,p

ξ ,ν2

(ii) The element .μ˜ n (j2 , ν2 ) ∈ Kn;h,p ∩ Mn2

under the following conditions.

comb. conditions 1, 2 ε = ±1 m0 (j2 ) ≥ 0 εr0 ≤ p . 3 ε = 1 m0 (j2 ) ≥ 0 r0 ≤ p ε = −1 m0 (j2 ) ≥ p r0 ≥ p p,0

ξ ,ν

1 1 Proof The element .μ˜ n (j1 , ν1 ) ∈ Mn;h,p,p is in the kernel of the shift operator

−3 2 2 3 , and .μ S−1 ˜ n (j2 , ν2 ) ∈ Mn;h,p,p is in the kernel of the shift operator .S−1 . We use Table 4.9 on p. 125 to determine the behavior of the other downward shift operators on these functions.  ν1 if ε = 1 and p > m0 (j1 ) , p,0 μ˜ n (j1 , ν1 ) ∈ Kn;h,p ⇔ 2p + j1 = ±ν1 otherwise ; .  ν2 if ε = −1 and p > m0 (j2 ) , 0,p μ˜ n (j2 , ν2 ) ∈ Kn;h,p ↔ 2p − j2 = ±ν2 otherwise . (4.75) 0,p

.

ξ ,ν

Computations based on Lemma 4.5 lead to the following scheme. j1 1 jl . 2 jl 3 j+

j2 2p + j1 − ν1 2p + j1 + ν1 2p − j2 − ν2 2p − j2 + ν2 j+ −2νl 0 0 2ν+ jr 0 ν+ − j+ 0 ν+ + j+ jr 0 2ν+ −2νr 0

(4.76)

144

4 Submodule Structure p,0

For combinations 2 and 3 this implies that .μ˜ n (j1 , ν1 ) ∈ Kn;h,p if .m0 (j1 ) ≥ 0. p,0 Under combination 1 we have .νl ≥ 1, and to have .μ˜ n (j1 , ν1 ) ∈ Kn;h,p we need an additional condition: .ε = 1, or .m0 (j1 ) ≥ p. This gives (i). For .μ˜ 0,p (j2 , ν2 ) we proceed similarly, now using that .νr ≥ 1 under combination 3. This gives (ii).

Lemma 4.29 In the notation introduced above, we define .kM n;h,p as in Proposition 4.27. ψ

(i) .kM n;h,p spans the space .Kn;h,p ∩ Mn . V (ii) If .ε(h − r0 ) ≥ 0, then .kM n;h,p = kn;h,p . ψ

Proof Suppose that .g ∈ Kn;h,p ∩ Mn . Then .g = αkVn;h,p + βkW n;h,p , with .β = 0 if .|r0 (h)| ≤ p, by (i) in Proposition 4.16. The component of order r of g has the form

t p+1 αC V (r) Vκ(r),s(r) + βcW (r) Wκ(r),s(r) ,

.

with the notations in (4.58) and (4.59). Here r runs over .r ≡ p mod 2, .|r| ≤ p such that .m(h, r) ≥ 0. Each component has to have .ν-regular behavior at 0 for .Re ν ≥ 0, and hence should be a multiple of .t p+1 Mκ(r),|s(r)| . The functions .Wκ,s and .Vκ,s are even in s, so going over to .|s(r)| ∈ 12 Z≥0 is the sensible thing to do. By (A.13) we have Mκ(r),|s(r)| = A(r) Vκ(r),|s(r)| + B(r) Wκ(r),|s(r)| ,   −eπ iκ(r)  1 + 2|s(r)| A(r) =  ,   12 + |s(r)| − κ(r) .     −ieπ i κ(r)−|s(r)|  1 + 2|s(r)| . B(r) =    12 + |s(r)| + κ(r)

(4.77)

Since . 12 + |s(r)| − κ(r) = 1 + |s(r)| + εs(r) ≥ 1 for all r in the sum, we have .A(r) = 0. The factor .B(r) may be zero for the values that we use here. ψ For the assumed .g ∈ Kn;h,p ∩ Nn we get coefficients .cM (r) such that for all relevant values of r αcV (r) = cM (r)A(r) ,

.

βcW (r) = cM (r)B(r) .

(4.78)

The first relation implies that if .α were zero, then all coefficients .c(r) would vanish. So if the supposed g exists as a non-zero function, then we normalize it so that .α = 1, and put cM (r) = cV (r)/A(r) .

.

(4.79)

4.4 Submodule Structure of Non-abelian Fourier Term Modules

145

This leads to the relation β =

.

cV (r) B(r) , cW (r) A(r)

(4.80)

valid for all r occurring in the sum. The factor .B(r) is the sole factor that may vanish. So if .B(r) = 0 for one relevant value of r, then .B(r  ) = 0 for all .r  occurring in the sum, and .β = 0. In that case the hypothetical function g is a multiple of .kVn;h,p , which we know explicitly. The factor .B(r) vanishes if and only if .

1 ε + |s(r)| + κ(r) = |s(r)| − εs(r) − m(h, r) = |s(r)| − εs(r) − (r − r0 ) ∈ Z≤0 . 2 2

For all r with .m(h, r) ≥ 0 we have  B(r) = 0 ⇔

.

− 2ε (r − r0 ) ≤ 0

if εs(r) ≤ 0 ;

r ≥ r0 and r0 ≤ h

if ε = 1 ,

r ≤ r0 and r0 ≥ h

if ε = −1 .

 ⇔

if εs(r) ≥ 0 ,

− h) ≤ 0

ε 2 (r0

The condition .ε(r − r0 ) ≥ 0 is just the condition .m(h, r) ≥ 0. If .ε = 1 we have .max(−p, r0 ) ≤ r ≤ p for all r relevant for the sum. That rules out combination 3, and gives the condition .r0 ≤ h for the other combinations. For .ε = −1 combination 1 cannot occur, and the other cases go similarly. Hence we . find .g = kVn;h,p under the following conditions. ε = 1 ε = −1 1 r0 ≤ h . 2 r0 ≤ h r0 ≥ h r0 ≥ h 3

(4.81)

The other possibility is that .β = 0, and .B(r) = 0 for all r occurring in the sum. Then we should have .r0 > h if .ε = 1, and .r0 < h if .ε = −1. A computation gives β = ieπ i(m(h,r)−2|s(r)|)/2

.

(|s(r)| + εs(r) + m(h, r)) . m(h, r)! (|s(r)| − εs(r) − m(h, r))

Using that .ε(r0 − h) > − and .ε(r − r0 ) ≥ 0 implies that .|s(r)| = −εs(r) we obtain β = 

.

which does not depend on r.

ie−π i|h−r0 |/4  , |h − r0 |/2 − 1 !

(4.82)

146

4 Submodule Structure

Whether .β = 0 or not, we conclude that the hypothetical element .g ∈ Kn;h,p ∩ ψ Mn is a multiple of .kVn;h,p + βkW n;h,p , and has the expansion (4.74) with coefficients ψ

cM (r) = cV (r)/A(r). So the element .kM n;h,p indeed spans the space .Kn;h,p ∩ Mn

.



Proof (of Proposition 4.27) By Lemma 4.29 we know that .kM n;h,p spans the ψ

intersection of .Kn;h,p with .Mn whenever it is defined. If .|r0 | > p it is a linear p,0 0,p combination of .kVn;h,p and .kW ˜ n (j1 , ν1 ) and .μ˜ n (j2 , ν2 ) n;h,p . The identification of .μ then are automatic. They can be confirmed by comparison of a determining component with help of (i) in Lemma 4.17. The identification with .kVn;h,p under an additional condition follows from (ii) in Lemma 4.29. . V By (i) in Proposition 4.16 we should have .kM n;h,p = kn;h,p if .|r0 | ≤ p. Hence we need the condition .ε(h − r0 ) ≥ 0. Comparison of the determining component gives p,0 0,p the proportionality with .μ˜ n (j1 , ν1 ), respectively .μ˜ n (j2 , ν2 ).

Lemma 4.30 Let .ψ ∈ Wip . For each K-type .τph  ψ . dim M n;h,p,p

=

1

if (h/3, p) ∈ Sect(j ) for some j ∈ OW (ψ)+ n ,

0

otherwise . ξ,ν

ξ  ,ν 

In particular, if the K-type .τph occurs in both .Mn and .Mn ∈

ξ,ν OW (ψ)+ n , then .Mn h,p

=

ξ  ,ν  Mn;h,p .

(4.83)

for .(jξ , ν), .(jξ  , ν  )

Proof The spaces .Mn;h,p,p with .(jξ , ν) ∈ OW (ψ)+ n are non-trivial, see Definition 4.20. ψ The dimension of .Mn;h,p,p does not change if we go to a lower K-type by application of an injective downward shift operator. A path given by successive applications of injective downward shift operators can stop at a K-type for which both downward shift operators have a non-trivial kernel. This may occur at a 2j ξ,ν one-dimensional K-type .τ0 . Then the dimension of .Mn;h,p,p is one if .(j, ν) ∈ OW (ψ)+ n , and zero otherwise. The path may also stop at a K-type studied in ψ Proposition 4.27. That proposition implies that .dim Mn;h,p,p ≥ 1. Moreover, ξ,ν

ψ

ψ

dim Mn;h,p,p ≤ 2 by Proposition 4.19. Any element of .Mn;h,p,p should have components with .ν-regular behavior at 0. A determining component is a solution of a Whittaker differential equation, which has only a one-dimensional space of solutions with .ν-regular behavior at 0.



.

With Lemma 4.30 we know that, analogously to (4.71), we have  .μ ˜ a,b n (j, ν)

ξ,ν

=

b S −3 μ˜ a,0 (j, ν)  13 a 0,b S1 μ˜ (j, ν)

if  > 0 , if  < 0 ,

for .a, b ∈ Z≥0 spanning .Mn;h,p,p for .h = 2j + 3(a − b), .p = a + b.

(4.84)

4.4 Submodule Structure of Non-abelian Fourier Term Modules

147

Identifications With Propositions 4.16 and 4.27 we have explicit descriptions of p,0 0,p elements .x˜n (j1 , ν1 ) and .x˜n (j2 , ν2 ), with .x = υ, .ω, or .μ, if they happen to be in the kernel of both downward shift operators. In the case that we call combination 2 (.j1 = jl , .j2 = jr ) there may be a third element to be considered, like we did in Corollary 4.12 in the abelian cases. There we could use the notation .xβa,b (j+ , ν+ ). In the non-abelian case, the upward shift operators are not always injective. We need to use the construction in Proposition 4.14. Proposition 4.31 Let .ψ ∈ Wip . We use the notations of Proposition 4.16, and the further notations introduced at the start of this subsection (p. 142). We consider combination 2, with .j1 = jl < j+ < j2 = jr , and take .h = h2 = 2j1 + 3p2 = 2j2 − 3p2 , .p = p2 = 13 (jr − jl ). Assume that .m0 (j+ ) ≥ 0. ξ ,ν

+ + is contained in .Kn;h,p,p , then it is • If the one-dimensional space .Wn;h,p,p spanned by .kW . n;h,p

ξ ,ν

+ + is contained in .Kn;h,p,p , then it is spanned • If the one-dimensional space .Vn;h,p,p V by .kn;h,p .

ξ ,ν

+ + is contained in .Kn;h,p,p , then it is • If the one-dimensional space .Mn;h,p,p M spanned by .kn;h,p .

ξ ,ν

+ + = 1 by Proof We let .X denote any of .W, .V, and .M. We know that .dim Xn;h,p,p

ξ ,ν

ξ ,ν

+ + l l = Xn;h,p,p by (iv) Lemma 4.30. If .ε = 1, then .m0 (jl ) = m0 (j+ ) + p1 , and .Xn;h,p,p

ξ ,ν

+ + is contained in .Kn;h,p , then the in Lemmas 4.23 and 4.30. Suppose that .Wn;h,p,p

ξ ,ν

l l , and by Proposition (4.27) we conclude that .kX same holds for .Wn;h,p,p n;h,p spans

ξ ,ν

ξ ,ν

l l l Xn;h,p,p = Xnl h,p,p (with the obvious notation .kX n;h,p . For .ε = −1 we proceed

.

ξ ,ν

ξ ,ν

+ + r r similarly, now by the identification .Xn;h,p,p = Xn;h,p,p .



Remark 4.32 The proof of Proposition 4.31 is based on a rather unspecified ξl ,νl ξ ,νl = Xnl ;h,p,p . It is not too hard to specify an element of identification of .Xn;h,p,p ξ ,ν

+ + Xn;h,p,p (in the notations used in the proof). We discuss this for the case .ε = 1. We

.

3 p3 0,0 start with .x 0,0 (j+ , ν+ ) ∈ X0,0 n;2j+ ,0,0 . The element .(S1 ) xn (j+ , ν+ ) may be zero. p ,0

ξ ,ν

+ + , by Proposition 4.14 gives a non-zero element .x˜n 3 (j+ , ξ+ ) ∈ Xn;2j + +3p3 ,p3 ,p3

working with the family .xn0,0 (j+ , ν), and dividing out common zeros in .ν whenever possible. For .ε = 1 the upward shift operator .S1−3 is injective, by Proposition 3.24. ξ+ ,ν+ p ,0 This produces a non-zero element .(S1−3 )p1 x˜n 3 (j+ , ν+ ) spanning .Xn;h,p,p .

The Whittaker function .Mκ,s may be a multiple of the basis solutions .Wκ,s and .Vκ,s of the Whittaker differential equation; see (A.16). In the case of integral parametrization this brings the need to get an overview of the K-types for which ψ ψ ψ .M n;h,p,p might be equal to .Wn;h,p,p or to .Vn;h,p,p .

148

4 Submodule Structure

Proposition 4.33 Let .ψ ∈ Wip , with .(jl , νl ), (j+ , ν+ ), (jr , νr ) ∈ OW (ψ)+ according to the conventions in 4.2. ψ Let .τnh be a K-type occurring in .Mn . ψ

ψ

(i) The space .Mn;h,p is equal to the space .Vn;h,p in the following cases: (a) .m0 (jl ) ≥ 0 and .m0 (jr ) ≥ 0. (b) .ε = 1, .m0 (jr ) < 0, and .(h/3, p) ∈ Sect(jr ). (c) .ε = −1, .m0 (jl ) < 0, and .(h/3, p) ∈ Sect(jl ). ψ

ψ

(ii) The space .Mn;h,p is equal to the space .Wn;h,p in the following cases: (a) .m0 (jr ) ≤ m0 (j+ ) < 0 and .(h/3, p) ∈ Sect(jl )  Sect(j+ ). (b) .m0 (jl ) ≤ m0 (j+ ) < 0 and .(h/3, p) ∈ Sect(jr )  Sect(j+ ). ψ

Remarks The K-types occurring in .Mn correspond to the points in the union of the sectors .Sect(j ) with .j ∈ {jl , j+ , jr } for which .m0 (j ) ≥ 0. So in case (i)(a) we ψ ψ ψ have .Mn = Vn . In the cases not mentioned under (i) and (ii) the space .Mn;h,p is ψ

ψ

not equal to one of .Vn;h,p or .Wn;h,p . In the pictures in Sect. 4.4.7, pp. 151–158, all possibilities are illustrated. Proof Lemma 4.15 is the basis for the proof. Part (i) gives information for .(h/3, p) on a boundary line of the sector .Sect(j ). It suggests that the quantity .Q(j, ν) = 2m0 (j ) + εj − ν is crucial. In the proof we will use many times that .j → m0 (j ) is a strictly decreasing function if .ε = 1 and a strictly increasing function if .ε = −1. If we view the formula for .m0 as describing a function on .R, then the derivative is .− 3ε . Let .ε = 1. If .Q(j, ν) ≥ 0, then (i)(a) in Lemma 4.15 shows that .Mn;h,p = Vn;h,p for all points .(h/3, p) on the right boundary of the sector .Sect(j ). For .ε = 1 the shift operator .S1−3 is injective by Proposition 3.24. Hence .Mn;h,p = Vn;h,p for all K-types corresponding to points .(h/3, p) ∈ Sect(j ). Still assuming that .ε = 1, let us suppose that .m0 (j ) ≥ 0 for all .j ∈ {jl , j+ , jr }. For .(jr , νr ) we know that .νr ≤ jr . Hence .Q(jr , νr ) ≥ 2 · 0 + jr − νr ≥ 0. Furthermore, the relations for .m0 and the relations in Lemma 4.5 imply that .Q(jl , νl ) = Q(j+ , ν+ ) = 2m0 (jr ) ≥ 0. A check for .Q(j+ , ν+ ) goes as follows:   1 1 2m0 (j+ ) + j+ − ν+ = 2 m0 (jr ) − (j+ − jr ) + j+ − (jr − jl ) 3 3 1 1 1 = 2m0 (jr ) + j+ + jr + jl = 2m0 (jr ) . 3 3 3

.

ψ

ψ

(See [6, 23a] for further checks.) Thus, we get .Mn;h,p = Vn;h,p for all K-types corresponding to points in .Sect(jr ) ∪ Sect(j+ ) ∪ Sect(jl ). This gives (i)(a) in the case of .ε = 1. The case of .ε = −1 goes analogously, working with the left boundary of a sector, and using the injectivity of .S13 . We have .m0 (jl ) = 2m0 (jl ) − jl − νl ≥ 0, and check that .Q(j+ , ν+ ) = Q(jr , νr ) = 2m0 (jl ).

4.4 Submodule Structure of Non-abelian Fourier Term Modules

149

We turn to the case when at least one of the .m0 (j ) is negative. For .ε = 1, this ψ means that .m0 (jr ) < 0, and .m0 (jl ) ≥ 0; otherwise .Mn = {0}. In this case, we have .Q(jl , νl ) = Q(j+ , ν+ ) < 0, and we need to take into account the role of p in Lemma 4.15. For points .(h/3, p) = (2jl /3 + a, a) on the ξl ,νl ξl ,νl = Vn;h,p,p if and only if .νl − jl ≤ 2a. right boundary of .Sect(jl ) we have .Mn;h,p,p The lowest of these points occurs for .a0 = lowest value that

νl −jl 2 .

We note with Lemma 4.5 for this

2jl + 6a0 = 2jl + 3νl − 3jl = −jl + (jr − j+ ) = 2jr .

.

Since .2jl + 3a0 = 2jr − 3a0 , the point .(h/3, p) is at the intersection of the right boundary of .Sect(jl ) and the left boundary of the sector .Sect(jr ). Since .ε = 1 the shift operator .S1−3 is injective (Proposition 3.24), we conclude that all points ξl ,νl ξl ,νl .(h/3, p) in .Sect(jl ) ∩ Sect(jr ) satisfy .M n;h,p = Vn;h,p . If .m0 (j+ ) ≥ 0 we have also to apply the same reasoning (and an analogous ξl ,νl ξl ,νl = Vn;h,p for K-types corresponding to points in computation) to get .Mn;h,p .Sect(j+ ) ∩ Sect(jr ). This gives (i)(b). For .ε = −1 we proceed analogously to get (i)(c). (Computations in [6, §23b].) We turn to (ii) in the proposition, for .ε = 1. For the base point of the sector 1 1 .Sect(jl ), Lemma 4.15 gives the relation . (jl + νl ) ≤ −1 − m0 (jl ). Since .− (jl + 2 2 νl ) = 13 (j+ − jl ) = p1 ≥ 0, we get the relation .m0 (jl ) ≤ −1 + p1 , which implies .m0 (j+ ) ≤ −1. Hence we can restrict our attention to .Sect(jl ). For the points .(2jl /3 + a, a) on the right boundary line of .Sect(jl ), Lemma 4.15 gives the l +ν condition .b + 1 ≤ − j 2 l = p1 . So we get all points that are not in the sector ξl ,νl ξl ,νl −3 .Sect(j+ ). By the injectivity of .S 1 we conclude that .Mn;h,p = Wn;h,p for all Ktypes corresponding to points of .Sect(jl )  Sect(j+ ). This gives (ii)(b). For .ε = −1 we obtain (ii)(a) in an analogous way. (Computations in [6, §23c].)

ψ

Remark 4.34 In Lemmas 4.25 and 4.5 we determine the lines of K-types in .Vn ψ and .Wn on which shift operators vanish. We do not need to repeat that work for ψ ψ ψ ψ .Mn . For K-types such that .M n;h,p,p = Vn;h,p,p we can use the results for .Vn , ψ

ψ

and similarly for K-types where .Mn and .Wn agree. On the other K-types a shift ψ ψ operator vanishes on .Mn;h,p,p if and only if it vanishes on both .Wn;h,p,p and ψ

Vn;h,p,p .

.

4.4.7 Structure Results ψ

ψ

ψ

In the non-abelian case the submodule structure of .Wn , .Vn and .Mn depends strongly on the question for which .j ∈ O1W (ψ)+ the condition .m0 (j ) ≥ 0 is satisfied. That leads to many combinations that we will consider in detail. First we prove the last main theorem stated in the introduction.

150

4 Submodule Structure ψ

Table 4.12 Isomorphism types of the sole irreducible submodule of .Wn . The main spectral parameters .j+ and .ν+ are in .Z, and satisfy .ν+ ≡ j+ mod 2, .ν+ ≥ |j+ |. They determine .(jl , νl ) and .(jr , νr ) according to (4.2). See [6, §24] for some computations .m0 (j ) . .= .>

.>

.>

.
0 for all a, b ∈ Z≥0 (A + a + x) (B + b + x)

if and only if A, B ∈ Z≥1 and |x| < min(A, B) ,

.

or A, B ∈

  1 + Z and |x| < max 21 , min(A, B) . 2

Proof First we consider .A, B ∈ Z. Then we need to have .p(a + 1, b)/p(a, b) = A+a−x A+a+x > 0 for all .a, b ∈ Z≥0 . If .A = 0 this does not hold for .a = 0. So we need .A ≥ 1. Then .A + a > 0 for all .a ∈ Z≥0 , and .A + a + |x| > 0. Then we need also .A + a − |x| > 0, hence .|x| < A. If this condition is satisfied the quotient .(A + a − x)/ (A + a − x) is indeed positive. Similarly we arrive at the condition .|x| < B. Now let .A, B ∈ 12 + Z. Then .A + a does not take the value 0, and from the requirement that .(A + a − x)/ (A + a + x) > 0 we arrive at the condition that .|x| < |A + a| for all .a ≥ 0. If .A > 0 this leads to the necessary condition .|x| < A. If .A < 0 we take .a = −A − 12 to get the necessary condition .|x| < 12 . We check that .(A + a − x)/ (A + a + x) is indeed positive if this condition holds. For the other quotient we arrive at the condition .|x| < max( 12 , B). Both

conditions together give the condition in the lemma.

4.5.3 Other Irreducible Modules ξ,ν

Irreducible modules occur in principal series representations .HK , as the whole ξ,ν of .HK under general parametrization, and as a genuine submodule under integral parametrization. So for the types .I I + , .I F , .F I and .F F we assume that .(j, ν) ∈ Wip . Equation (4.90) defines a sesquilinear form with help of the meromorphic family ξ,ν ξ,−ν .i0 of morphisms .H of .(g, K)-modules defined in (3.80). At values K → HK .ν = ν0 in .Z it need not be an isomorphism. It may even have a singularity. Replacing .i0 by .α(ν) i0 for a suitable analytic function .α we may remove the singularity at .ν0 . For an irreducible module the resulting sesquilinear form is unique up to a constant in .C∗ .

164

4 Submodule Structure

So the outcome of a check whether the form can be made positive definite on the ξ,ν determines the unitarizability of this submodule of .HK in which we are  interested,   module. For .ν = 0 we work with . ·, · ups instead of . ·, ·)cmpl . 4.5.3.1

Isomorphism Types I I + ξ,ν

The isomorphism class .I I + (j, ν) can be represented by a module .V ⊂ HK with 2 .(j, ν) ∈ Z , .j ≡ ν mod 2 and .ν ≥ max(|j |, 1). The module V has parameters   . λ2 (j, ν); −j, ν; ∞, ∞ . See Fig. 4.8. p The basis vectors occurring in V are . hϕr,p (ν) with .p = ν + a + b, .h = −j + 3(a − b) and .r = 13 (h − 2j ) = a − b − j with .a, b ∈ Z≥0 . The factor c in (4.91) takes the value c(p, r, ν) =

.

a! b! . (a + ν)! (b + ν)!

(4.92)

This is well defined for .a, b ∈ Z≥0 , and positive. So the isomorphism class I I + (j, ν) is unitarizable. See [6, §25a] for computations.

.

4.5.3.2

Isomorphism Types I F and F I

In Table 4.3, p. 107, we listed the isomorphism classes .I F (j, ν) and .F I (j, ν) and ξ,ν parameters of a submodule of .HK representing this class. From this table we collect, and reformulate with (4.2), the information in Table 4.14. ξ,0

Cases b and f In cases b and f the irreducible module is contained in .HK . By restriction of . ·, · ups we get a positive definite sesquilinear form.

Table 4.14 Isomorphism classes of types .I F and .F I .Isomph.

class

.a

.I F (j, ν)

.b

.I F (j, 0)

.c

.I F + (j, −ν)

.d

.I F (j, −j )

.e

.F I (j, ν)

.f

.F I (j, 0)

.g

.F I + (j, −ν)

.h

.F I (j, j )

.1

≤ν ≤j −2

A

0

.∞

0

.∞

2j

≤ν ≤j −2

j +3ν . 2

≥1

.2j

0

≤ ν ≤ −j − 2

2j

.0

∈ 2Z≤−1

.2j

.0

≤ ν ≤ |j | − 2

j −3ν . 2

ν+j .− 2

.ν

≤ −1

2j

.0

.−j

.1

.j

.j .1

.p0

2j

∈ 2Z≥1

.j

.1

.h0

.j

j −ν . 2

−1

B .

j −ν 2 −1 j . −1 2

.∞

.ν

.∞

.j

−1 −1

j +ν 2 −1 j .− − 1 2

.∞

−1

.∞

.−

−1

.∞

.∞

4.5 Unitary Structure

165

Case a We take 



ca (p, q, ν ) =

.

sin π ν +j 2 

sin π ν −j 2

c(p, r, ν  ) .

(4.93)

With some relations for gamma functions and goniometrical functions this meromorphic function in .ν  can be written as 

ca (p, q, ν ) = (−1)

.

p−r

 −ν  −p+r+j    1+ 2  ν  −p+r+j  (1 +  2



−ν  +p+r+j  2 ν  +p+r+j  2

.

(4.94)

We write .p = p0 + a + b, .h = h0 + 3(a − b), and .r = 31 (h0 − 2j ) + a − b, and obtain   j −ν   a + j −ν 2 ! 2 −b−1 ! ca (p, r, ν) =    j +ν  a + j +ν 2 ! 2 −b−1 ! . (4.95) (1 + a + B)! (B − b)! = . (B + a + ν + 1)! (ν + B − b)! The factors depending on a are positive for all .a ∈ Z≥0 ; for the factors with b we need .0 ≤ b ≤ j −ν 2 − 1 = B. This means that .ca induces a positive definite sesquilinear form. Case c cc (p,r, ν  ) =



sin π j −ν 2

.

= (−1)

(p−r)/2

π



−1

−1

c(p, r, ν  )   −ν  −p+r+j       1+  1 + −ν +p−r−j 2 2  −ν  +p+r+j   1+ 2

−ν  +p+r+j  2

,

(4.96) which specializes to b+1 j −ν b!

cc (p, r, −ν) = (−1)

.

i

  (B − b)! a + (ν + j )/2 !   . π a + (j − ν)/2 )!

(4.97)

Only if .ν = 1 this determines a positive definite sesquilinear form. The irreducibility of the module implies that the sesquilinear form is unique up to a multiple. Hence a positive definite sesquilinear form is impossible if .ν > 1. Or, alternatively, suppose that q is an invariant sesquilinear form on the submodule of

166

4 Submodule Structure ξ,ν

type .I F + (j, −ν) of .HK with .ν ≥ 2. Then it satisfies for .ϕ0 = h0 −4ϕ p0 +1 .ϕ1 = r0 −1,p0 +1 (−ν) the relation

h0ϕ p0 (−ν), ,r0

and

    3 q S1−3 ϕ0 , ϕ1 + q ϕ0 , S−1 ϕ1 = 0 .

.

We use that .S1−3 is given by .Z23 , and .Z23 = Z32 . With Table 3.4, p. 36, and (3.78) we obtain explicit factors .cu and .cd such that     cu ϕ1 , ϕ1 + cd ϕ0 , ϕ0 = 0 .

(4.98)

.

It turns out that the product .cu cd is positive for .ν ≥ 2. So there cannot be a positive definite invariant sesquilinear form. The Other Cases In [6, §25c] we handle the other cases in a similar way.

4.5.3.3

Isomorphism Types F F

We use .ν ≤ |j | − 2 to describe the isomorphism class .F F (j, −ν) with parameters  .

λ2 (j, −ν); 2j, 0; ν−j 2 − 1,

 j +ν −1 . 2

See Fig. 4.17. With cf (p, r, ν) =

.

c(p, r, ν) ν−j sin π j +ν 2 sin π 2

this leads to cf (p, r, ν) =

.

(−1)a+b  a+ π2

ν+j  2 !

 ν−j 2

  −1−a ! b+

ν−j  2 !

 ν+j 2

 − 1 − b !.

ν+j See [6, §25d]. This is well defined for .0 ≤ a ≤ ν−j 2 − 1 and .0 ≤ b ≤ 2 − 1. The factor .(−1)a+b shows that the sesquilinear form is positive-definite only if .A = B = 0, hence .j = 0 and .ν = −2. The isomorphism class .F F (0, −2) contains the trivial representation.

Chapter 5

Application to Automorphic Forms

The first three chapters have given us explicit information concerning the Fourier ψ term modules .FN . Now we apply this information to automorphic forms and to their Fourier expansions. In Sect. 5.1 we allow automorphic forms to have faster growth at the cusps than just the usual polynomial growth. This has consequences for the Fourier expansions. We also discuss the relation with holomorphic automorphic forms on the symmetric space. Eisenstein series and Poincaré series form families of automorphic forms depending on the spectral parameter .ν. Such families are discussed in Sect. 5.2. In Sect. 5.3 we discuss how the structure of .(g, K)-modules of automorphic forms influence the form of Fourier expansions. In Sect. 5.3.1 we give Fourier expansions for generating vectors in modules of square integrable automorphic forms. In Sect. 5.3.2 we show by an example how the structure of a module of automorphic forms is determined by the Fourier expansion, and we show also that in turn this structure determines which Fourier terms can be non-zero. Up till here we considered Fourier term modules based on standard lattices .σ and discrete subgroups of G satisfying the .Z[i]-condition at the cusps. We show in Sect. 5.4 how these restrictions can be removed.

5.1 Automorphic Forms Let . be a cofinite discrete subgroup of G. Harish Chandra gives in [19, Chap I, §2] a definition of an automorphic form as a function .f : G → V where V is a finite-dimensional vector space, with commuting representations .σ of K and .χ of .ZU (g), such that

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. W. Bruggeman, R. J. Miatello, Representations of SU(2,1) in Fourier Term Modules, Lecture Notes in Mathematics 2340, https://doi.org/10.1007/978-3-031-43192-0_5

167

168

(a) (b) (c) (d)

5 Application to Automorphic Forms

f is left-invariant under translation by elements of .; R(u)f = χ (z)f for all .u ∈ ZU (g); .R(k)f = σ (k)f for all .k ∈ K; f has polynomial growth in terms of a norm depending on the matrix elements of .g ∈ G ⊂ SL3 (C). .

In (a)–(d) we adapted the formulation of Harish Chandra to the notation used in this work. In particular, we interchanged the actions on the left and on the right. We vary the concept of Harish Chandra slightly: 1. We work with functions in .C ∞ (\G) with values in .C. 2. We replace the representation .χ of .ZU (g) by a character .ψ of .ZU (g). This is a genuine restriction. 3. We require that the translates .R(k)f with .k ∈ K generate a finite-dimensional space. 4. We work with several types of growth conditions at the cusps of .. 5. We assume that . satisfies the .Z[i]-condition at the cusps; see 2.4. Cusps have been discussed in Sect. 2.4. For each cusp .c there are elements gc ∈ G such that the function .f c (g) = f (gc g) is in a space .C ∞ (σ (c) \G)K for some .σ (c) ∈ Z≥1 . For .f ∈ C ∞ (\G)K and each cusp .c of ., we introduced in Remark 2.7 the function .f c ∈ C ∞ (σ (c) \G)K by .f c (g) = f (gc g). The .-invariance allows us to consider the Fourier expansion only in a set .R of representatives of the .-orbits of cusps. For each .c ∈ R we have a Fourier expansion of .f c :

.

fc =



.

FN f c ,

(5.1)

N

where the abelian terms .Fβ f c ∈ Fβ are parametrized by .β ∈ Z[i], and where the non-abelian terms are in .F ,c,d with . ∈ σ (c) 2 Z=0 , .c ∈ Z mod 2 and .d ∈ 1 + 2Z. So the Fourier terms .FN f c (called .Fc f in the introduction) are in the large Fourier modules .FN defined in Proposition 3.12. We will define several spaces of automorphic forms using the following growth conditions: Definition 5.1 Let .f ∈ C ∞ (G). • f has exponential growth if there are .a, b ∈ R>0 such that    b f na(t)k = O eat

.

as t ↑ ∞

(5.2)

uniformly in .n ∈ N and .k ∈ K. We call b the order of the exponential growth. • f has polynomial growth if there is .a > 0 such that     f na(t)k = O t a

.

uniformly in .n ∈ N and .k ∈ K.

as t ↑ ∞

(5.3)

5.1 Automorphic Forms

169

• f has quick decay if for all .a > 0     f na(t)k = O t −a

.

as t ↑ ∞

(5.4)

uniformly in .n ∈ N and .k ∈ K. • f has exponential decay if there are .a, b > 0 such that    b f na(t)k = O e−at

.

as t ↑ ∞

(5.5)

uniformly in .n ∈ N and .k ∈ K. We call b the order of exponential decay. The use of .O-statements in these growth conditions shows that they are understood with “at most” added implicitly. We define a decreasing sequence of spaces of automorphic forms that differ in the properties of the functions .f c . Definition 5.2 Let . be a cofinite discrete subgroup of G satisfying the .Z[i]condition at the cusps in Sect. 2.4, and let .ψ ∈ W. • Unrestricted automorphic forms. The space .Au (; ψ) consists of the functions ∞ .f ∈ C (G) that satisfy (i) .f (γ g) = f (g) for all .γ ∈ . (ii) f is K-finite. (iii) .uf = ψ(u)f for all .u ∈ ZU (g). • Automorphic forms with moderate exponential growth. The space .A! (; ψ) consists of those .f ∈ Au (; ψ) for which there exists for each .c ∈ R a finite set .Ec of Fourier term orders such that fc −



.

FN f c

(5.6)

N∈Ec

has polynomial growth. • Automorphic forms. The space .A(; ψ) consists of those .f ∈ Au (; ψ) such that c .f has polynomial growth for each .c ∈ R. (See Definition 5.1.) • Square integrable automorphic forms. The space .A(2) (; ψ) consists of those u .f ∈ A (; ψ) for which  .

  f (g)2 dg < ∞ .

(5.7)

\G

• Cusp forms. The space .A0 (; ψ) consists of those .f ∈ A(; ψ) for which .f c has quick decay for each .c ∈ R. (See Definition 5.1.) Examples. Bao, Kleinschmidt, Nilsson, Persson and Pioline discuss Eisenstein series of trivial K-type in Sections 2–4 of [2], for a group closely related to the group

170

5 Application to Automorphic Forms

FL on p.30. These Eisenstein series form meromorphic families .s → E(s) with  E(s) ∈ A ψ[0, s − 1] 0,0 for all s at which the family does not have a singularity. Eisenstein series can be defined in other K-types as well, and provide us with non-trivial examples of automorphic forms for all .. We indicate the definition in the case of . = −1 in (2.68). For a K-type .τph occurring in the principal series

. .

ξ,ν

module .HK with .jξ ∈ 4Z we take .r = 13 (h − 2ξj ). In Table 3.10 we see that   p p h p = hϕr,q (nak). Hence . hϕr,q is left invariant under .(−1 )P = . ϕr,q m(i)nak  ∩ N AM. For .Re ν > 2 the Eisenstein series  h p .E∞ (ν; g) = ϕr,p (ν; γ g) (5.8) γ ∈P \

  converges absolutely and provides us with a non-trivial element of .A ; ψ[jξ , ν] . There is a meromorphic continuation in .ν. For other groups . the construction can be used as well. p Poincaré series are defined similarly, with a sum over .N \, and . hϕr,q replaced ψ[j,ν] by an element of a module .MN . They converge absolutely for .Re ν > 2, and may be zero. If non-zero, they are elements of .A! (; ψ[jξ , ν])  A(; ψ[jξ , ν]) for general values of .ν. See [5]. Cusp forms tend to be more elusive. Reznikov [40, §5] shows, on the basis of a Kuznetsov formula, that there are infinitely many generic cusp forms with K-type 0 .τ on .\SU(2, 1) for suitable discrete subgroups .. Generic means that the cusp 0 form has non-zero Fourier terms of the form .c Fβ f c for some .β = 0. Most statements in the following proposition are well-known. Proposition 5.3 Let .ψ ∈ W. (i) For each K-type .τph the space .A(γ ; ψ)h,p has finite dimension. (ii) All spaces of automorphic forms in Definition 5.2 are .(g, K)-modules. (iii) The Fourier terms of automorphic forms have the following properties for every cusp .c: ψ

(a) If .f ∈ A0 (; ψ) then .F0 f c = 0 and .Ff c ∈ WN for .N = N0 . ψ (b) If .f ∈ A(2) (; ψ), then .FN f c ∈ WN for .N = N0 , and .F0 f c can be in ξ,ν .H K if .ψ contains an element .(jξ , ν) with .Re ν < 0. ψ ψ (c) If .f ∈ A(; ψ), then .F0 f c ∈ F0 and .FN f c ∈ WN for .N = N0 . ψ (d) If .f ∈ A! (; ψ), then .FN f c ∈ WN , except for finitely many combinations ψ c .(c, N) of a cusp .c ∈ R and a Fourier term order .N, for which .FN f ∈ F . N ψ (e) If .f ∈ Au (; ψ) then .FN f c ∈ FN for all .N and .c. Proof The dimension statement in (i) is valid for all semisimple Lie groups. It follows from Theorem 1, [19, p 8]. Properties (i) and (ii) are preserved by the right action of .g and K. So .Au (; ψ) is a .(g, K)-module. The proof of Proposition 3.31 can be easily extended to the

5.1 Automorphic Forms

171

condition of quick decay. This implies that .A0 (; ψ) and .A(; ψ) are .(g, K)modules. The Fourier term operators are intertwining operators of .(g, K)-modules. This handles the case of moderate exponential growth. Right translation defines a unitary action of G on .L2 (\G). This induces an action of .g in .C ∞ (G)K . So the condition of square integrability is also preserved by the .(g, K)-action. A Fourier term of order 0 has a factor on A that is in general a linear combination of .t 2+ν and .t 2−ν , and is a linear combination of .t 2 and .t 2 log t if .ν = 0. These functions have polynomial growth. For quick and exponential decay, we need .F0 f = 0. ψ Let .N = N0 . Definition 3.32 implies that elements of .WN have exponential ψ decay. The question is whether other elements of .FN can have polynomial growth. ψ ψ ψ We have .FN = WN ⊕ MN , except in the non-abelian case under integral ψ ψ ψ ψ parametrization, in which case we use .Fn = Wn ⊕ Vn . Suppose that .F ∈ Mβ ψ

or .F ∈ Vn has polynomial growth and is non-zero. We can restrict our attention to one K-type and use that polynomial growth is preserved under .R(gc ) to consider 3 F or .S −3 F only the case that F is a highest weight vector in that K-type. If .S−1 −1 ±3 is non-zero, then .S−1 F has also polynomial growth. This reduction stops when −3 3 .S −1 F = 0 and .S−1 F = 0. Then we may be in a one-dimensional K-type, for which the explicit formulas in Sect. 3.5.2 lead to a contradiction. The other possibilities are handled with use of Propositions 4.8 and (4.16), or with Lemma A.1. This implies the statements in part (iii) except if .f ∈ A(2) (; ψ). We have 

  f (g)2 dg < ∞ ,

.

\G

with .dg = dn t −5 dt dk for .g = na(t)k ∈ NAK. This implies that for each .c ∈ R 



∞

.

n∈σ (c) \N

t=T



 c  f na(t)k 2 dk dt dn < ∞ t5 k∈K

for all sufficiently large .T0 . The orthonormality of the basis of .L2 (σ (c) \N) implies the corresponding property for all Fourier terms, and the orthogonality of the h p ∈ L2 (K) implies this property for the component functions on A in the . r,q ψ expansion over K. For .N = N0 this implies quick decay, and hence .FN f c ∈ WN . For .F0 f c the component on A in .G = NAM can be only a multiple of .t 2+ν with .Re ν < 0.



172

5 Application to Automorphic Forms

5.1.1 Growth Conditions and Fourier Expansions Proposition 5.3, part (iii), gives properties of the Fourier expansion of the different types of automorphic forms. In the case of .SL2 (R) one can define corresponding spaces by prescribing the form of the Fourier expansions. The crucial fact is that for .SL2 (R) an absolutely convergent Fourier series with exponentially decreasing terms represents an exponentially decreasing function. This is well-known in the context of holomorphic automorphic forms for .SL2 (R). The proof works for general realanalytic automorphic forms; see e.g. [4, Lemma 4.3.7]. The proof uses that there are only finitely many terms with approximately the same decay. In the case of .SU(2, 1) the decay of the non-abelian terms of order .N ,c,d is mainly determined by the value of . . Since .(c, d) can run over infinitely many values, we cannot characterize the spaces of automorphic forms in Definition 5.2 on the basis of the properties in part (iii) of Proposition 5.3. We can use the following theorem. Theorem 5.4 Let .ψ be a character of .ZU (g). If an element .f ∈ C ∞ (σ \G)K satisfies (a) .uf = ψ(u)f for all .u ∈ ZU (g), (b) f has polynomial growth, (c) .F0 f = 0, then f has quick decay. Proof (Indication of the Proof) Harish Chandra [18] gives this deep result for a general semisimple Lie group. One may also consult Gan [13, p 84–89]. The result is a consequence of Lemma 10, [18, p 11]. The proof is in §7, see specially Lemma 20. Harish Chandra defines spaces of functions .LF (λ) depending on a linear form .λ on .a and on a set F of roots. For .SU(2, 1) we have .dim a = 1. Going through the definitions in [18, Ch I, §3], we identify .L∅ (λ0 ) = L as a space containing those .f ∈ C ∞ (σ \G)K for which .uf has polynomial growth of an order specified by .λ for each .u ∈ U (g), and .L{α} (λ0 ) = L as a space containing the elements of .C ∞ (σ \G)K for which all .uf have quick decay. (The latter space does not depend on .λ0 .) Lemma 10 then tells that the map .f → f − F0 f sends .L(λ) to .L.

There are Fourier series for which we can follow the approach known for .SL2 (R). We give a simple example: Proposition 5.5 Let .σ = 1, .ν ∈ C, .j ∈ Z. Suppose that .f ∈ C ∞ (σ \G)2j,0,0 has an absolutely convergent Fourier expansion of the form f (g) =



2 −1 

.

0,0 a ,c ω ,c,3+2j (j, ν; g) .

(5.9)

∈(1/2)Z≥1 c=0

Then .f has exponential decay, and satisfies .uf = ψ[j, ν](u) f for all .u ∈ ZU (g).

5.1 Automorphic Forms

173

Proof The behavior under the action of .ZU (g) follows from the absolute convergence of the Fourier expansion of all derivatives of f . We have to show the exponential decay. In Table 4.5 we see that .m0 (j ) = 0. Equation (3.88) implies that   0,0 j, ν; na(t)k =  ,c (h ,0 ; n) t Wκ,ν/2 (2π t 2 ) ω ,c,3+2j

.

2j

00,0 (k) ,

(5.10)

with .κ = − j +1 2 . The restriction on the number of normalized Hermite functions .h ,m occurring in the expansion is essential. √ We use that the theta functions have norm . 2 in .L2 (1 \N) and that the basis function . 2j 00,0 has norm 1 in .L2 (K) to get  q(t) :=

.

τ,1 \N

   2  2   |a ,c |2 t 2 Wκ,ν/2 (2π t 2 ) . f na(t)k  = 2 K

(5.11)

,c

With the asymptotic behavior (A.14) this implies that for a sufficiently large .t0 there exists .c1 > 0 such that q ≥ c1



.

−j

|a ,c |2 −1−j t0 e−2π t0 . 2

(5.12)

,c

The terms of the convergent series tend to zero, hence  2 j/2 a ,c = O l (1+j )/2 t0 eπ t0 .

(5.13)

.

The theta series . ,c (h ,0 ) are bounded uniformly in . , and . as well. Hence for .t ≥ 2t0

2j 0 0,0

is bounded

   2 f na(t)k)  |a ,c | t 2π t 2 )−(1+j )/2 e−π t

.

,c





j/2 π t02 −(1+j )/2 −j

l (1+j )/2 t0

e



t

e−π t

2

,c





(2 ) (t0 /t)j/2 e−π (t

2 −t 2 ) 0

 t −j/2

∞ 

4n t −j/2 e−2π n(t/2)t

n=1

∈(1/2)Z≥1





e−(π t

2 −δ)n

 t −j/2 e−(π t

2 −δ)

n≥1

for each small .δ > 0. So f has exponential decay.



174

5 Application to Automorphic Forms

5.1.2 Holomorphic Automorphic Forms Holomorphic automorphic forms are not functions on G but on the symmetric space G/K, which can be realized as .X in (2.11). Here we discuss how to relate scalarvalued holomorphic automorphic forms on .X to spaces of automorphic forms on G. Holomorphic automorphic forms for .−1 in (2.68) have been studied by Resnikoff and Tai [38, 39].

.

Proposition 5.6 Let .h ∈ 2Z. The linear map .h : C ∞ (G)h,0 → C ∞ (X) defined by     (h f ) na(t) · (i, 0) = t h/2 f na(t)

(5.14)

.

is a bijection intertwining the representation L of G on .C ∞ (G)h,0 and the right representation .|−h/2 of G on .C ∞ (X) given by  h/2  F |−h/2 g(z, u) = j g, (z, u) F (g · (z, u) , .     j g, (z, u) = c (z, u)t + d

(5.15)

  Ab in (2.13). c d The function .h f is holomorphic on .X if and only if .R(Z31 )f = 0.

where we use the block description .UT−1 gUT =

Proof (Sketch of the Proof) Since .p → p·(i, 0) gives a bijection .NA → X, the map h is a bijection. The space .C ∞ (H )h,0 is invariant under the right representation L, since left translation commutes with right differentiation. One can also check that the action .g → |−h/2 g is a right representation. So the intertwining property can be checked on generators. That is fairly easy for elements of NAM. In view of the Bruhat decomposition (2.18) we still need to check the relation for the   generator .w. To do that we use Lemma (2.1), and the fact that . h 00,0 k(η, α, β) depends only on .η.   For the differentiation we write .f n(x, y, r)a(t) h 00,0 (k) in terms of .F = h f . We express .R(Z31 )f in terms of the right differentiation on NA by application of the routines in Sect. 3.2. We write the result in terms of the coordinates on N A and then go over to derivatives of F with respect to the coordinates z and u on .X. This gives:

.

.

− 2iρ 1−h/4 ∂z¯ F

h+3

  11,1 + ρ 1−h/4 2u∂z¯ F + i∂u¯ f

h+3

1−1,1 .

(5.16)

We note that .ρ = Im (z) − |u|2 is positive on .X, and conclude that .R(Z31 )f = 0 is equivalent to the holomorphy of F . Computations are carried out in [6, §26a].



5.1 Automorphic Forms

175

Definition 5.7 The space .Mw () of holomorphic automorphic forms for . with weight .w ∈ Z consists of the holomorphic functions F on .X that satisfy for all .γ ∈     w F γ · (z, u) = j γ , (z, u) F (z) .

(5.17)

.

Proposition 5.8 Let .h ∈ 2Z. The linear map .h in Proposition 5.6 gives a bijection .

  f ∈ A ; ψ[−h − 3, 1] h,0 : R(Z31 )f = 0 −→ M−h/2 () .

(5.18)

If these spaces are non-zero then .h ∈ 2Z≤−2 , or .h = 0. In the latter case .M0 () consists of the constant functions. If .h ≤ −4, the corresponding Fourier expansions are of the following form 0 f c = a0c hϕ0,0 (−h/2 − 1).   h 0,0 c h a ,c ω ,c,h+3 + 2 , − 2 − 2) ,

.

(5.19)

c mod 2 ∈ σ (c) 2 Z≥1



(h f c )(z, u) = a0c +



c mod 2 ∈ σ (c) 2 Z≥1

·



−1/4 eπ i z c a ,c √ π (2π )h/4

e 2 π (c/ +2k) e−π 1

2



u+c/ +2k

(5.20)

2 .

k∈Z

Proof Proposition 5.6 gives a bijection between .M−h/2 () and the space of .f ∈ C ∞ (G)h,0 that are left-invariant under . and satisfy .S13 f = R(Z31 )f = 0. We express the function f in terms of the holomorphic function .F = h f , and apply the generators C and .Δ3 of .ZU (g) to f . This gives Cf =

.

 h2 3

+ 2h f ,

Δ3 f =

1 h(h + 12)(h + 6) f . 9

(5.21)

These to the character .ψ[−h − 3, 1] of .ZU (g). So .f ∈  eigenvalues correspond  Au ; ψ[−h − 3, 1] h,0 . See [6, §26b]. All Fourier terms .f c have to satisfy .S13 f c = 0. This has drastic consequences for the non-zero Fourier terms. By Proposition 3.14 all generic abelian Fourier terms have to vanish. For each cusp .c the Fourier term .F0 f c must be in a principal series module 0 (−h/2 − or in a logarithmic module. In (3.78) we see the possibility . hϕ0,0 2) = t −h/2 h 00,0 , which corresponds to the constant function 1 on .X. A direct computation shows that the logarithmic modules cannot contribute.

176

5 Application to Automorphic Forms ψ[−h−3,1]

Next we look for elements of .F ,c,d;h,0 ψ[−h−3,1] .Mn

in the kernel of .S13 . It suffices

ψ[−h−3,1] .Wn .

to consider and We inspect Figs. 4.35–4.46. The sole 0,0 (jl , νl ) with possibilities occur in Figs. 4.42 and 4.44, giving multiples of .ω ,c,d 0,0 > 0, .m0 (jl ) ≥ 0 > m0 (j+ ). The requirement that .S13 ω ,c,d (jl , νl ) = 0 is equivalent to .j+ = jl + 3, which gives .2 + jl + νl = 0. With the formula for .m0 in Table 4.5 (p. 122) we see that .m0 (jl ) − m0 (jp ) = 1. Since .m0 (jl ) ≥ 0 > m0 (j+ ), this implies .m0 (jl ) = 0. We need that .ψ[jl , νl ] = ψ[−h − 3, 1], and .νl = −jl − 2 ≥ 0; see Fig. 4.1 for the notational convention that we use here. This leads to

.

(jl , νl ) = S2 S1 (−h − 3, 1) =

.

h

h 2 , −2 − 2



, and jl =

h . 2

Furthermore we need .0 ≤ νl ≤ |jl | − 2. For .h ≤ −4 we can have non-abelian terms 0,0 (h/2, −h/2 − 2) in the Fourier expansion. The non-abelian terms form with .ω ,c,h+3 a convergent series of the form considered in Proposition 5.5. Hence this part of the expansion has exponential decay at all cusps. Together with the constant term this gives polynomial growth of all .f c , and .f ∈ A(; ψ[−h − 3, 1]). The Fourier expansions of f now have the form indicated in (5.19). We have to carry out the transformation to .h in (5.20), using (3.88), (A.17), (2.52), and

Table 2.3. See [6, §26c].

5.2 Families of Automorphic Forms Eisenstein series and Poincaré series occur in families that are parametrized by ν, so that the Fourier term are meromorphic functions of .ν. To formulate this in terms of Fourier coefficients we have to choose basis families. The families in Proposition 5.10 were convenient in our paper [5].

.

Definition 5.9 Let U be a connected open set in .C, let .j ∈ Z and let .τph be a Ktype such that .|h − 2j | ≤ 3p. A holomorphic family of automorphic forms for ∞ .(j, h, p) on U is an element .f ∈ C (U × G) such that .g → f (ν, g) is an element   ! of .A , ψ[j, ν] h,p,p for each .ν ∈ U , and such that .ν → f (ν, g) is holomorphic on U for each .g ∈ G. A meromorphic family of automorphic forms has, locally 1 in .ν, the form .(ν, g) → ϕ(ν) f (ν, g) where f is a holomorphic family and .ϕ is a non-zero holomorphic function. Remarks (1) The restriction to automorphic forms with moderate exponential growth is nonessential. Prescribing the first spectral parameter j , the K-type .τph , and the highest weight p in the K-type is practical. One obtains more general families as a .U (k)-linear combination of families of this type. (2) Meromorphically continued Eisenstein series are examples.

5.2 Families of Automorphic Forms

177

Families of Modules of Automorphic Forms Let .ν → f (ν) be such a family of automorphic forms. For an open dense subset .U0 ⊂ U the spectral pairs .(j, ν) with .ν ∈ U0 correspond to simple parametrization. The .(g, K)-module .M(ν) generated ξ ,ν by .f (ν) is isomorphic to .HKj for .ν ∈ U0 . On .U0 ∪{0} we can chose a holomorphic ξ ,ν

family of isomorphisms .M(ν) → HKj . At .ν = 0 ∈ U0 we have to deal with the possibilities of logarithmic modules in Proposition 3.29. Now we take .ν0 ∈ U such that .(j, ν0 ) corresponds to integral parametrization. At .ν = ν0 ∈ U0 the structure of .M(ν) changes in an essential way. It may become reducible, and families in some K-types in .M(ν) may have a zero at .ν0 . Fourier Terms The Fourier term operators are given by integration over compact sets. Hence if f is a holomorphic family of automorphic forms, then .ν → FN f (ν)c ψ[j,ν] is a holomorphic family of elements in the modules of type .FN . If f is a meromorphic family of automorphic forms on U for .(j, h, p), then all its Fourier terms can be written in the form

p p d0c (ν) hϕr,p (ν) + c0c (ν) hϕr,p (−ν) if N = N0 , c . FN f (ν) = (5.22) a,b a,b c c dN (ν)μN (j, ν) + cN (ν)ωN (j, ν) otherwise , c and .d c on U . We take .r = 1 (h − 2j ), integers with meromorphic functions .cN N 3 .a, b ≥ 0 such that .h = 2j + 3(a − b), .p = a + b, and use the families in (3.77) and (3.93). We define families of Fourier terms as in Definition 5.9; we need only replace ψ[j,ν] ! .A (; ψ)h,p,p by .F N;h,p,p . c and If f is a holomorphic family of automorphic forms, then the coefficients .cN c .d N are holomorphic on .U  Z. The families used in (5.22) do not always form a ψ[j,ν] basis of .FN;h,p,p . That may cause singularities of the coefficients that are not due to singularities of the family f . When dealing with meromorphic families this is often no problem. However if we are interested in values or residues at integral points it is better to use an adapted basis if .N = N0 , for instance the basis in the following proposition.

Proposition 5.10 Let .(j, h, p) be as above, and let .N = Nβ , .β ∈ Z[i]  {0}, or N = N ,c,d with .m0 (j ) ≥ 0.

.

(i) There is a holomorphic family .ν → N;h,p (j, ν) of Fourier terms such that ξ ,ν

j , and there is also a for each .ν ∈ C we have .C N;h,p (j, ν) = WN;h,p,p meromorphic family .ν → MN;h,p (j, ν) with at most first order singularities in

ξ ,ν

j Z≤−1 such that .MN;h,p (j, ν) spans .MN;h,p,p if .ν ∈ Z≤−1 . At .ν ∈ Z≤−1 , the

.

ψ[j,ν]

values and residues are elements of .FN;h,p,p .

ψ[j,ν]

(ii) The elements .N;h,p (j, ν) and .MN;h,p (j, ν) form a basis of .FN;h,p,p , except if .± > 0 , .±j ≤ −1 , .ν ≥ 0, .ν ≡ j mod 2, .0 ≤ m0 (j )
0 can be treated in an analogous way.

5.3 Fourier Term Operators for Modules of Automorphic Forms

185

Exponential Growth The basis families in Proposition 5.10 can be chosen such that   Mβ  ;2j,0 (j, ν) na(t)k = χβ  (n) t 2 Iν (2π |β  |t) 2j 00,0 (k) , .   ϒn ;2j,0 (j, ν) na(t)k =   ,c ,d  (h  ,m0 (j ) ; n) tVκ0 (j ),ν/2 (2π | |t 2 )

2j

00,0 (k) . (5.32)

See (3.86) and (4.55). The asymptotic results (A.6) and (A.15) show that the Fourier term .FN f has exponential growth. Proposition 5.15 Let .M(ν) be the .(g, K)-module of automorphic forms generated by .f (ν). ξ ,ν

(i) If .ν ∈ U  {ν0 }, then .M(ν) is isomorphic to the principal series module .HKj . ξ,ν (ii) .M(ν0 ) is isomorphic to the module .Mβ  0 if .N = Nβ  (abelian case), and to ξ,ν0

the module .Vn

if .N = Nn (non-abelian case). ψ[j,ν]

Proof The operator .FN : M(ν) → FN is an intertwining operator of ψ[j,ν] .(g, K)-modules (Proposition 3.15). The module .F  can be written as a sum N of submodules as in Theorems A (ii), B (iii), C (iii) and D (iii). So we get a sum ψ[j,ν] W W X .FN = F  ⊕ F  where the intertwining operator .F  has values in .W  , and N N N N X .F  has values in N

ψ[j,ν] .X  N

=

M[ψ(j,ν)]

if N = Nβ  ,

ψ[j,ν] Vn

if N = n .

(5.33)

Furthermore, by the explicit form of the Fourier term .FcN f (ν) for the generator .f (ν) ξ ,ν

we conclude that .FX has values in .XNj  (in an analogous notation) and that .FW N N ξ ,ν

has values in .WNj  .

ξ,ν

ξ,ν

Now we use Proposition 3.24 and Lemma 4.25 to see that .WN and .XN are 2j

generated by an element in the one-dimensional K-type .τ0 , except possibly for ξ ,ν

Wnj

.

ξ ,ν

(non-abelian case). Hence the non-zero intertwining operator .FX : M(ν) → N

XNj  is an isomorphism of .(g, K)-modules. This implies part (ii) of the proposition. ξ ,ν

In the case .ν = ν0 all the relevant modules are isomorphic to .HKj . See ξ ,ν

ξ ,ν

Sect. 3.5.3, which can be easily extended to .XNj  = Vnj This gives part (i) of the proposition.

in the non-abelian case.



Restrictions on Fourier Term Operators We turn to the other Fourier term operators ξ ,ν ξ ,ν ξ ,−ν FW and .FN if .N = N . They have values in .WNj , or in .HKj + HKj if N .N = N0 . Proposition 5.15 tells us what is the structure of the .(g, K)-module .M(ν). ξ ,ν ξ ,ν ξ ,−ν Comparison with the structure of the receiving module .WNj or .HKj + HKj determines the dimension .D(N) of the space of intertwining operators. .

186

5 Application to Automorphic Forms

Proposition 5.16 Let .M(ν) be the .(g, K)-module generated by .f (ν), for .ν ∈ U . ξ ,ν

(i) The dimension .D(N0 ) of the space of intertwining operators .M(ν) → HKj + ξ ,−ν

HKj

is equal to 2, except if .ν = ν0 . Then .D(N0 ) = 1. ξ ,ν

(ii) The dimension .D(N) of the space of intertwining operators .M(ν) → WNj , with .N = N0 , is equal to 1, except in the following cases, in which .D(N) = 0. (a) . > 0 and .m0 (j ) < ν0 − j . (b) . < 0 and .m0 (j ) < ν0 + j . ξ ,ν

ξ ,ν

ξ ,−ν

Proof If .ν = ν0 all relevant modules are isomorphic to .HKj (or to .HKj ⊕ HKj if .N = N0 ). This gives .D(N) = 1 if .N = N0 , and .D(N0 ) = 2. For .ν = ν0 there may be restrictions, which we now consider case by case. ψ[j,ν] The module .MN has the structure indicated in the upper case in Fig. 4.27, p. 120, for the generic abelian case. A case by case inspection in Sect. 4.4.7 shows ξ ,ν that this same structure holds also in the non-abelian case. The submodule .XNj  ∼ = M(ν0 ) corresponds to the submodule given by the middle sector with base point .(2j+ /3, 0) = (2j/3, 0). ξ ,ν The structure of .HKj 0 is sketched in Fig. 4.8 on the right. It has the same ξ ,ν0

structure as .M(ν0 ), so the space of intertwining operators .M(ν0 ) → HKj

has

ξj ,−ν0 .H . K

dimension 1. In Fig. 4.17, p. 102, we find a sketch of the structure of The vanishing of upward shift operators on certain K-types makes it impossible ξ ,−ν to have non-zero intertwining operators .M(ν0 ) → HKj 0 . This gives part (i) of the proposition. ξ ,ν In the generic abelian case the modules .Wβj 0 (for all .β ∈ Z[i]  {0}) have the ξ ,ν same submodule structure as .Mβj 0 ∼ = M(ν0 ). So we get a one-dimensional space of intertwining operators. ξj ,ν0 For the non-abelian modules .W ,c,d we have to go through the cases considered ψ[j,ν ]

in Sect. 4.4.7. In the sketches of the submodule structure of .W ,c,d 0 we have to look at the sector with base point .(2j+ /3, 0) = (2j/3, 0). Since we have assumed that .(j, ν0 ) = (j+ , ν+ ) satisfies .ν0 > |j |, we are in the situation that the three values .m0 (jl ), .m0 (j+ ) and .m0 (jr ) are different. Furthermore, we need to have .m0 (j+ ) ≥ 0, otherwise the Fourier term operator .F ,c,d does not appear in the Fourier expansion. That gives part (ii) (a). Figure 4.35, p. 151 The conditions are .m0 (jl ) ≥ 0, .m0 (j ) ≥ 0 and .m0 (jr ) > 0. ξj ,ν0 ξj ,ν0 The module .W ,c,d is isomorphic to .V ,c,d . So no obstructions are present. ξ ,ν

j 0 is not Figure 4.37 If . > 0 and .m0 (jr ) < 0 ≤ m0 (j ), the module .W ,c,d ξj ,ν0 ∼ isomorphic to .V ,c,d = M(ν0 ). The vanishing of the upward shift operators .S13 on some K-types allows only the zero intertwining operator.

5.4 General Lattices

187

Figure 4.40 For . < 0 and .m0 (jl ) < 0 ≤ m0 (j ), the reflected figure occurs. ξj ,ν0 Now the other upward shift operator .S1−3 is zero on some K-types in .W ,c,d , and .D(N ,c,d ) = 0. Table 4.5, p. 122, and relations (4.2) imply jl = .

3 1 ν0 − j , 2 2

3 1 jr = − ν0 − j , 2 2

m0 (jl ) = m0 (j ) + sign ( ) (j + ν0 ) ,

(5.34)

m0 (jr ) = m0 (j ) + sign ( ) (j − ν0 ) . So if . > 0, the condition .m0 (jr ) < 0 ≤ m0 (j ) is equivalent to .0 ≤ m0 (j ) < ν0 −j . ψ[j,ν] Since .m0 (j ) < 0 implies .F ,c,d = {0} this gives (ii)(a). In the case . < 0 we get statement (ii)(b) in a similar way.



5.4 General Lattices Up to this point, we have studied automorphic forms for discrete groups . ⊂ G that satisfy the .Z[i]-condition in Sect. 2.4. That implies that for all cusps .c the functions c ∞ .f are in .C (σ (c) \G)K for a standard lattice .σ (c) as in Definition 2.2. In this way, our results are not directly applicable to discrete subgroups that do not satisfy the .Z[i]-condition at the cusps. For instance, the Picard modular group .−3 in (2.69) is not yet covered. In this last section we show that most of the results in this work are valid without the .Z[i]-condition at the cusps, at the cost of some complications in the description.

5.4.1 Theta Functions It is known that any lattice in N is isomorphic to a standard lattice .σ . To see this we start with the observation that the image of N in .N/Z(N ) ∼ = C is generated by an .R-basis .{ω1 , ω2 } of .C. Conjugation by an element of N can be used to transform a general lattice into a lattice with generators .n(ω1 , 0), .n(ω2 , 0), and .n(0, ρ). Conjugation by an element of AM gives all transformations .n(b, r) → n(ub, |u|2 r) with .u ∈ C∗ . In this way we conjugate any lattice to a lattice generated by  τ n √ ,0 , Im τ

.

 1 ,0 , n √ Im τ

 2 , n 0, σ

(5.35)

with .τ in the complex upper half-plane .H and .σ ∈ Z≥1 . We call this lattice .τ,σ . So .σ = i,σ .

188

5 Application to Automorphic Forms

The element .τ arises from the choice of an  .R-basis  of .C. Another choice of the ab aτ +b basis amounts to replacing .τ by . cτ +d with . ∈ SL2 (Z). We may take .τ in cd √ a strict fundamental domain for .SL2 (Z)\H. The use of the denominators . Im τ has the advantage that the commutator of the second and the first generator equals .n(0, 2), and that the volume of .τ,σ \N does not depend on .τ . √  For Picard modular groups associated to the number field .Q −d with d √ √ positive and square-free we can take .τ = − 12 + 2i d if .d ≡ 3 mod 4, and .τ = i d otherwise. If .τ ∈ H, then the automorphism  x Im τ − y Re τ   y Aτ n(x, y, r) = n ,√ ,r √ Im τ Im τ

.

(5.36)

of N induces an isomorphism .Aτ : τ,σ → i,σ , which is the identity for .τ = i. For general .τ one cannot obtain .Aτ by means of conjugation by an element of N AM. The formulas for the theta functions . ,c (ϕ) in (2.52) on .i,σ are simpler than those obtained by working directly with more general lattices. For the lattice .τ,σ we might use the isomorphism .Aτ in (5.36) to transport the definitions. It turns out to be convenient to transform the Schwartz functions as well. Proposition 5.17 We put U ,τ ϕ(ξ ) = (Im τ )1/4 e−2π i ξ

.

2 Re τ

 ϕ ξ Im τ 1/2 ) .

(5.37)

This defines an isomorphism of the space .S(R) of Schwartz functions, extending unitarily to .L2 (R). The theta functions τ

.

   ,c (ϕ; n) :=  ,c U ,τ ϕ; Aτ (n)

(5.38)

are in .C ∞ (τ,σ \N) if . ∈ (σ/2)Z=0 and .c ∈ Z/2 . The linear operator induced by .jτ :  ,c (ϕ) → τ  ,c (ϕ) is an intertwining operator for the action of .n by right differentiation. Proof It is clear that .U ,τ sends Schwartz functions to Schwartz functions, and extends to a unitary isomorphism of .L2 (R). The uniqueness of the Stone-von Neumann representation implies that there are operators U in .L2 (R) that satisfy U π2π (n) = π2π (Aτ n) U .

.

(5.39)

A computation shows that we can take .U = U ,τ ; [6, §28b]. We recall the description

5.4 General Lattices

189

   ,c (ϕ; n) = π2π (n)ϕ, μ ,c

i

.

in (2.51). It implies τ

.

   ,c (ϕ; n) = U ,τ dπ2π (n)ϕ, μ ,c .

(5.40)

 From .dR(X)  ,c (ϕ) =  ,c dπ2π (X)ϕ), we conclude that .jτ is an intertwining operator for the action of .n.

For the right action on functions on N the functions .τ  ,c behave exactly in the same way as the functions .i  ,c . This does not hold for the action by left translation on functions on N, or for the action of NAM by conjugation. This means that Proposition 2.3 is special for .i,σ .

5.4.2 Fourier Term Modules Functions in .C ∞ (τ,σ \G)K have a Fourier expansion similar to the expansion discussed in Sect. 2.4.1. The abelian terms in this expansion are functions on G transforming on the left according to the characters .χβ of N that satisfy .χβ (n) = 1 for .n ∈ τ,σ . Explicitly,  i  Z + τZ . β ∈ Bτ := √ Im τ

(5.41)

.

All Fourier term modules .Fβ with .β = 0 are isomorphic, by Proposition 3.13. These (g, K)-modules have the structure described in Sect. 4.3. The module .F0 is the same for all .τ . For the non-abelian terms we use spaces .τF ,c spanned by the elements

.

p

na(t)k → τ  ,c (ϕ; n) f (t) h r,q (k)

.

like in part (ii) of Proposition 3.12. Proposition 5.18 The spaces .τF ,c are .(g, K)-modules for the action on the right. The linear map .τJ ,c : F ,c → τF ,c induced by p

p

J ,c :  ,c (ϕ) ⊗ f ⊗ h r,q → τ  ,c (ϕ) ⊗ f ⊗ h r,q

τ

.

(5.42)

is an isomorphism of .(g, K)-modules. Proof The intertwining property of .(g, K)-modules follows from the explicit description in Sect. 3.2 of the action by right differentiation. The action on the p factors f and . h r,q is the same for .τF ,c and for .F ,c . The action on the theta functions is an intertwining operator by Proposition 5.17.



190

5 Application to Automorphic Forms

These results allow us to work with the more general lattices .τ,σ . As an example we consider the Fourier expansions in Proposition 5.8. There we considered the h  0,0 h functions .ω ,c,h+3 2 , − 2 − 2 in the non-abelian Fourier term and arrived at the corresponding functions   −1/4 eπ i z  1 π (c/ +2k)2 −π u+c/ +2k 2 2 e (z, u) → √ e π (2π )h/4

.

(5.43)

k∈Z

on the symmetric space .X. If we work with a lattice .τ,σ we use the isomorphism .τJ ,c in Proposition 5.18 to obtain the Fourier term h  0,0 h na(t)k → τJ ,c,h+3 ω ,c,h+3 2 , − 2 − 2; na(t)k ) (5.44) .   = τ  ,c h ,0 ; n t W−(1+j )/2,−1−j/2 (2π t 2 ) h 00,0 (k) . The corresponding holomorphic function on .X is (z, u) →

(Im τ )1/4 (2π )−h/4 eπ i z π 1/2 1/4  2 1/2 2 · e−(π iτ/2) (c/ +2k) e−π (u+(c/ +2k)(Im τ ) ) k∈Z

.

(Im τ )1/4 2 = 1/2 1/4 (2π )−h/4 e−π u +π i z π √  2 2 · e−2π Im (τ )α −2π i Re (τ )α −4π uα Im τ .

(5.45)

α≡c/(2 ) mod 1

5.4.3 Summary of Fourier Expansions In this final subsection we formulate the results on Fourier expansions without the restriction to standard lattices and to the .Z[i]-condition at the cusps.

5.4 General Lattices

191

Proposition 5.19 (Lattices and Fourier Term Modules)   (i) There is a collection .L = τ,σ : τ ∈ H, σ ∈ Z≥1 of lattices in N such that for each .τ,σ ∈ L the elements .f ∈ C ∞ (\G)N have an absolutely convergent Fourier expansion f (g) =



.

β∈Bτ

Fβ f (g) +



Fn f (g) .

(5.46)

n

Bτ is the lattice in .C consisting of those .β ∈ C such that .χβ (n) = 1 for all σ () .n ∈ . The parameter .n = ( , c, d) runs over . ∈ 2 Z=0 , .c ∈ Z mod 2 , .d ∈ 1 + 2Z. The abelian Fourier terms .Fβ f are given by (2.73). The non-abelian Fourier terms .F ,c,d are given by (3.44), with the theta functions . ,c replaced by more general theta functions .τ  ,c in (5.38) associated to the lattice .τ,σ . (ii) The Fourier terms .FN f in (5.46) are elements of a Fourier term module .τFN ⊂ C ∞ (τ,σ \G)K . The module .τFN is a .(g, K)-module isomorphic to a Fourier term module .FN for the standard lattice .σ = i,σ . The module .τFβ is equal to the Fourier term module .Fβ for all .β ∈ Bτ . All modules .Fβ with .β ∈ C∗ are isomorphic to each other. In the non-abelian case .N = N ,c,d , the module .τF ,c,d is isomorphic as a .(g, K)-module to .F ,c,d . ψ (iii) The isomorphisms can be used to define submodules .τFN in which .ZU (g) acts ψ ξ,ν according to the character .ψ, and submodules .τ WN and .τ WN having the ψ ξ,ν property of exponential decay at .∞, and submodules .τMN and .τMN having the property of .ν-regular behavior at 0. .

Proof The lattices .τ,σ in (i) are given in (5.35). They depend on .σ ∈ Z≥1 and on τ with .Im τ > 0. The choice of .τ is not unique. The characters .χβ of N are defined in (2.42). For part (ii) we use Proposition 3.13 in the abelian cases, and the isomorphism τ . J ,c in Proposition 5.18 in the non-abelian cases. The submodules in part (iii) are the image under .τJ ,c of the corresponding ψ ψ submodules of .FN = iFN . The .ν-regular behavior at 0 and the exponential decay are determined by the functions .f ∈ C ∞ (0, ∞) in the decomposition .na(t)k → w(n) f (t) (k), with . a polynomial function on K, and w a character of N or a theta function .τ  ,c (h). In the non-abelian cases the functions f are preserved by ψ ψ the isomorphism .τJ ,c . We take .τ W ,c,d = τJτ W ,c,d , and similarly for the other τ submodules. In the abelian cases the modules . Fβ are equal to the modules .Fβ .

.

Proposition 5.20 (Discrete Subgroups and Automorphic Forms) (i) Let . be a cofinite discrete subgroup of G with cusps. Let .R be a set of representatives of the .-orbits of cusps.

192

5 Application to Automorphic Forms

For each cusp .c ∈ R there exist .τ (c) ∈ H, .s(c) ∈ Z≥0 , and .gc ∈ G such that  ∩ Nc = gc τ (c),σ (c) gc−1 . For .f ∈ C ∞ (\G)K the functions .f c : g → f (gc g) with .c ∈ R are elements of .C ∞ (c \G)K and have a Fourier expansion as in (5.46). (ii) Definition 5.2 of the different spaces of automorphic forms goes through for .. ψ The Fourier terms satisfy .FN f c ∈ τ (c)FN . The description in Proposition 5.3 of the properties of the Fourier terms for each of the spaces of automorphic forms extends to the general situation. .

Proof The collection .L of lattices has been chosen in such a way that all lattices  ∩ Nc can be conjugated by an element of G to a lattice in .L. This gives part (i). The assertions in (ii) concerning abelian Fourier terms follow from the fact that τ ψ = Fψ . For the non-abelian cases we use that the isomorphism .τJ . F ,c preserves β β growth conditions and the .ν-regular behavior at 0.

.

All results depending on the structure of the Fourier term modules under right differentiation by elements of .g and right translation by elements of K are valid in the general situation. This includes Theorems A–D in the Introduction, almost all results in Chap. 3, the results in Chap. 4 and the results in Sects. 5.1–5.3. The explicit basis families for Fourier term modules are valid directly in the abelian cases, and can be defined by application of .τJ ,c in the non-abelian cases. This holds in particular for the families .N;h,p , .MN;h,p and .ϒN;h,p in Proposition 5.10. Results that are based on left translation or conjugation have to be considered separately for each value of .τ . In Sect. 3.3.3 we considered the normalizer in NMA of standard lattices .σ = i,σ . Proposition 5.21 (Normalization of Lattices) (i) Let .τ,σ ∈ L. The normalizer .NormP (τ,σ ) is contained in NM with .τ,σ as a subgroup of finite index. For each .p ∈ NormP (τ,σ ) the left transformation .L(p) in .C ∞ (τ,σ \G)p induces automorphisms of .(g, K)-modules  Pq (p) ∈ Aut





τ

Fβ ,

β∈Bτ : |β|=q

.

P ,d (p) ∈ Aut

 

τ



(5.47)

F ,c,d

c mod 2

for each .q ≥ 0 occurring as absolute value of elements of .Bτ , and for each pair ( , d) with . ∈ σ2 Z=0 and .d ∈ 1 + 2Z. (ii) For . as in (i) and .c ∈ R we put .Nc =  ∩ Nc and .Pc =  ∩ Nc Ac Mc . The group .Nc has finite index in .Pc . .

5.4 General Lattices

193

For each .γ ∈ Pc we have .gc−1 γ gc ∈ NormP (τ (c),σ (c) ). The Fourier terms of .f c determine elements of the modules in (5.47) that are fixed by the automorphisms .Pq (gc−1 γ gc ) and .P ,d (gc−1 γ gc ), respectively. Proof Elements .p ∈ NAM normalize N. Let .p = n1 m1 a(t1 ) ∈ NMA with t1 ∈ (0, 1). A computation shows that conjugation of .λ ∈ τ,σ by powers of p gives a sequence of points of .τ,σ tending to .1 ∈ N . This contradicts the discreteness of .τ,σ in N. So the normalizer in NAM of .τ,σ is contained in NM. The compactness of M implies that .τ,σ has finite index in the normalizer. Let .p ∈ NormP (τ,σ ) and .f ∈ C ∞ (τ,σ \G), then .pτ,σ p−1 = τ,σ implies that .L(p)f ∈ C ∞ (τ,σ \G). Left translation commutes with the right action. So .L(p) is an intertwining operator of .(g, K)-modules. Write .p ∈ NormP (τ,σ ) as .p = n1 m1 , .n1 ∈ N and .m1 ∈ M. If .f ∈ τFβ = Fβ then .

 .

 L(p)f (nak) = f (n1 m1 nm1 a m1 k) = χβ (n1 ) χβ (m1 nm−1 1 ) g(a m1 k) .

So .L(p)f ∈ Fβ  , where .|β  | = |β| by (2.42) and (2.8). This gives (5.47) in the abelian cases. In the non-abelian cases the basis functions have the form   p na(t)k → τ  ,c hm ; n ϕ(t) h r,q (k) .

.

The value of .m ∈ Z≥0 is coupled to the metaplectic parameter d for fixed h and r. The function .n → n1 m1 nm−1 1 does not change the central character of the theta τ function. Hence .τ  ,c (hm ; n1 m1 nm−1 1 ) is a linear combination of .  ,c (hm ; n)  with .c running through .Z/2 Z. (Since h and r do not change, the Schwartz function in the theta function stays unchanged.) So (5.47) holds in the non-abelian cases as well. For part (ii) we start with the that the group . acts on the cusps by   observation conjugation. If .γ ∈ , then .γ  ∩ Nc γ −1 =  ∩ Nγ c . So if .γ ∈ Pc we have .γ c = c, and   gc−1 γ gc τ (c),σ (c) gc−1 γ −1 gc = gc−1  ∩ γ −1 Nc γ −1 gc = gc−1  ∩ Nγ c gc−1  = gc−1  ∩ Nc gc−1 = τ (c),σ (c) .

.

Hence .gc−1 γ gc normalizes .τ (c),σ (c) . f and .c as in (ii) the relation .L(γ )f = f for .γ ∈ Pc induces  For −1 c c .L gc γ gc )f = f . Now we apply (i).



Appendix A

Special Functions

In the description of Fourier term modules we use modified Bessel functions and Whittaker functions. Here we collect some facts concerning these special functions.

A.1 Modified Bessel Functions The modified Bessel differential equation is x 2 j ′′ (x) + x j ′ (x) − (x 2 + ν 2 )j (x) = 0 ,

.

(A.1)

for functions j on .(0, ∞). See, e.g., [53]. The exponents near .t = 0 are .ν and .−ν. The exponent .ν leads to the following modified Bessel function Iν (x) =

Ʃ

.

m≥0

(x/2)ν+2m . m! Г(ν + m + 1)

(A.2)

So .Iν (x) = (x/2)ν h(x), where h is the restriction of an entire function with value 1 at .x = 0. For .ν ∈ C \ Z the functions .Iν and .I−ν span the solution space. This is not the case if .ν ∈ Z: In (x) = I−n (x)

.

for n ∈ Z .

(A.3)

The solution Kν (x) =

.

π I−ν (x) − Iν (x) 2 sin π ν

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. W. Bruggeman, R. J. Miatello, Representations of SU(2,1) in Fourier Term Modules, Lecture Notes in Mathematics 2340, https://doi.org/10.1007/978-3-031-43192-0

(A.4)

195

196

A Special Functions

extends holomorphically to a function of .ν ∈ C. It satisfies .K−ν = Kν . It is linearly independent of .Iν . This independence can be seen in the behavior near .x = 0. The expansion of .Iν (x) near zero starts with a non-zero multiple of .x ν , or a multiple of −ν if .ν ∈ Z ν .x ≤−1 . The expansion of .Kν (x) has always non-zero multiples of .x and −ν of .x if .ν /∈ Z, and a logarithmic term if .ν ∈ Z. The linear independence is also visible in the asymptotic behavior as .x ↑ ∞. The function .Kν is characterized by its exponential decay as .x ↑ ∞; in fact it has an asymptotic expansion / .

Kν (x) ∼

(1 ) (1 ) π −x Ʃ (−1)m 2 − ν m 2 + ν m e , 2x m! (2x)m

(A.5)

m≥0

whereas ex Ʃ .Iν (x) ∼ √ 2π x m≥0

(1 2

−ν

) (1 m 2

+ν

) m

m! (2x)m

(A.6)

.

See [53, 7.23]. Contiguous Relations Section 3.71 in [53] gives relations for .Kν and .Iν : Kν−1 (x) − Kν+1 (x) = − .

2ν Kν (x) , x

Iν−1 (x) − Iν+1 (x) =

2ν Iν (x) , x

Kν−1 (x) + Kν+1 (x) = −2Kν′ (x) , Iν−1 (x) + Iν+1 (x) = 2Iν′ (x) . (A.7)

See [6, §A1].

A.2 Whittaker Functions The Whittaker differential equation for functions on .(0, ∞) is y ′′ (τ ) =

.

(1 4

−

κ s 2 − 1/4 ) + y(τ ) . τ τ2

(A.8)

It has parameters .κ, s ∈ C. See e.g. [44, (1.6.2)]. The exponents at .τ = 0 are . 12 + s and . 12 − s. The exponent . 12 + s leads to the solution ( ) Ʃ 12 + s − κ τ n s+1/2 −τ/2 ( ) n . (A.9) .Mκ,s (τ ) = τ e 1 + 2s n n! n≥0

A

Special Functions

197

It is of the form .τ ‫ →׀‬τ s+1/2 h(τ ) with an entire function h with value 1 at 0. If .s ∈ C \ 12 Z the functions .Mκ,s and .Mκ,−s span the solution space. At values 1 .s ∈ 2 Z≤−1 the function .Mκ,s may have a first order singularity. If a singularity occurs at .ν = −ν0 ∈ Z≤−1 , then the residue is (−1)ν0 −1 .

( 1−ν0 2

−κ

2 ν0 ! (ν0 − 1)!

) ν0

Mν0 /2,κ .

(A.10)

The solution given for .s /∈ 12 Z by Wκ,s (τ ) =

.

Ʃ ∓Mκ,±s (τ ) π sin 2π s ± Г(1/2 ∓ s − κ) Г(1 ± 2s)

(A.11)

extends as a holomorphic function of s, and satisfies .Wκ,−s = Wκ,s . This solution is characterized by its exponential decay as .τ ↑ ∞. It is convenient to have another solution that is invariant under .s ↔ −s. We make the choice to use Vκ,s (τ ) = W−κ,s (−τ ) (this implies a choice of a branch) .

πi Ʃ ±e±π is Mκ,±s (τ ) = . sin 2π s ± Г(1/2 ∓ s + κ) Г(1 ± 2s)

(A.12)

Unlike .Mκ,s and .Wκ,s , this is not a commonly used notation. The expression in (A.12) gives .Vκ,s as a meromorphic linear combination of .Mκ,s and .Mκ,s and even in s. In [6, §A2e] we carry out a check that it is actually holomorphic in s. The functions .Wκ,s and .Vκ,s form a basis of the solution space for all .±s ∈ C. We have the following meromorphic relation with .Mκ,s . (See [6, §A2a].) (

−i e−π is Wκ,s (τ ) Г(1/2 + s + κ) ) 1 Vκ,s (τ ) . − Г(1/2 + s − κ)

Mκ,s (τ ) = eπ iκ Г(1 + 2s) .

(A.13)

Exponential Decay and Growth We have as .τ ↑ ∞ ( s 2 − (κ − 1/2)2 Wκ,s (τ ) ∼ τ κ e−τ/2 1 + τ ( 2 ) 2 ) 2 ) s − (κ − 1/2) (s − (κ − 3/2)2 + · · · ,. 8τ 2

.

(A.14)

198

A Special Functions

( (κ + 1/2)2 − s 2 Vκ,s (τ ) ∼ −e−π iκ τ −κ eτ/2 1 + τ )( ) ( 2 2 ) (κ + 1/2) − s (κ + 3/2)2 − s 2 + · · · . + 8τ 2

(A.15)

We use (4.2.22) in [44] for .Wκ,s , and (4.1.21) to get the asymptotic behavior of .Vκ,s . Check in [6, §A2b]. The families .Wκ,s and .Vκ,s are linearly independent for all choices of the parameters. Lemma A.1 Let f be a linear combination of functions .t ‫ →׀‬t 1+c Vκ+k,s (2π ut 2 ), where c runs over a finite subset of .Z≥0 and k over a finite subset of .Z. The quantities .u > 0, .κ ∈ R and .s ∈ C are fixed. If .f (t) = O(t A ) as .t ↑ ∞ for some .A ∈ R, then it is zero. Proof The contiguous relations in (A.19) allow us to express .Vκ,s as a linear combination of .Vκ1 ,s with .κ1 running over .κ − 1, κ, κ + 1. Using this repeatedly we arrive at a finite sum Ʃ . cj Vκ+j,s (2π ut 2 ) = O(T A ) , j

with possibly another value of A. Let .jc be the minimal value of j for which .cj /= 0. The estimate (A.15) implies that .cjc is zero. Proceeding in this way we arrive at .f = 0. Linear Dependence If .s /∈ 12 Z≤−1 we obtain from (A.13): 1 − κ + s ∈ Z≤0 , 2 1 ⇔ + κ + s ∈ Z≤0 . 2

Mκ,s ∈ C Wκ,s ⇔ .

Mκ,s ∈ C Vκ,s

(A.16)

For .s0 ∈ C \ Z≤−1 the function .Mκ,s0 (τ ) spans the space of solutions of the Whittaker differential equation with parameters .κ and .s0 that are of the form .τ s0 +1/2 times an entire function. Behavior at Zero Let .Re s0 ∈ 12 Z≤0 and . 12 + s0 + κ ∈ Z≥1 . Then the leading term in the expansion of .Wκ,s0 (τ ) as .τ ↓ 0 is a non-zero multiple of .τ s0 +1/2 if .s0 > 0, and a non-zero multiple of .τ 1/2 log τ if .s0 = 12 . The expansion of .Vκ,s0 (τ ) as .τ ↓ 0 has the same properties under the conditions .Re s0 ∈ 12 Z≥0 and . 12 + s0 − κ ∈ Z≥1 . Specializations For special combinations of the parameters these Whittaker functions have expressions in simpler functions. (See [6, §A2c].)

A

Special Functions

199

Wκ,±(κ−1/2) (τ ) = τ κ e−τ/2 = Mκ,κ−1/2 (τ ) , .

Vκ,±(κ+1/2) (τ ) = −e−π iκ τ −κ eτ/2

(A.17)

= −e−π iκ Mκ,−κ−1/2 (τ ) . The relations with .Mκ,κ±1/2 are valid as holomorphic functions in .κ. The function Mκ,s may have a singularity as a function of .(κ, s) at these points.

.

Contiguous Relations We will need several of the relations in Section 2.5 of [44]. See also [6, §A2d]. κ) 1 Wκ,s (τ ) − Wκ+1,s (τ ) , 2 τ τ (1 κ ) (κ + 1/2)2 − s 2 . V ′ (τ ) = − V Vκ+1,s (τ ) , (τ ) + κ,s κ,s 2 τ τ ) (1 (1 κ ) ′ Mκ,s − Mκ,s (τ ) + + κ + s τ −1 Mκ+1,s (τ ) ; (τ ) = 2 τ 2 ( (τ − 2κ)Wκ,s (τ ) = Wκ+1,s (τ ) + (κ − 1/2)2 − s 2 )Wκ−1,s (τ ) , ′ Wκ,s (τ ) =

.

(1

−

(A.18)

(τ − 2κ)Vκ,s (τ ) = (s 2 − (κ + 1/2)2 )Vκ+1,s (τ ) − Vκ−1,s (τ ) , ) (1 − κ + s Mκ−1,s (τ ) − + κ + s)Mκ+1,s (τ ) ; 2 2 (A.19) Wκ+1/2,s (τ ) = (s − κ)Wκ−1/2,s (τ ) + τ 1/2 Wκ,s−1/2 (τ ) ,

(τ − 2κ) Mκ,s (τ ) =

.

(1

Vκ+1/2,s (τ ) = (κ + s)−1 Vκ−1/2,s (τ ) −

i τ 1/2 Vκ,s−1/2 (τ ) , κ +s

(s + κ) Mκ+1/2,s (τ ) = (κ − s) Mκ−1/2,s (τ ) + 2sτ 1/2 Mκ,s−1/2 (τ ) ; (A.20) Wκ+1/2,s (τ ) = −(κ + s) Wκ−1/2,s (τ ) + τ 1/2 Wκ,s+1/2 (τ ) , .

Vκ+1/2,s (τ ) =

i 1 Vκ−1/2,s (τ ) + τ 1/2 Vκ,s+1/2 (τ ) , κ −s s−κ

(2s + 1) Mκ+1/2,s (τ ) = (2s + 1) Mκ−1/2,s (τ ) − τ 1/2 Mκ,s+1/2 (τ ) .

(A.21)

References

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Indices

Index arithmetic group, 30 antiholomorphic discrete series type, 88 automorphic form, 169 —, holomorphic, 175 — with moderate exponential growth, 169 —, square integrable, 169 —, unrestricted, 169 automorphism group of N , 21

big cell, 15 boundary of symmetric space, 14 Bruhat decomposition, 14

Cartan subalgebra, 34 Casimir element, 20, 35 characters of M ⊂ K, 16 cohomological representation, 88 combinations 1, 2 and 3, 142 complementary series, 87, 162 component function, 45, 50 composition diagram, 96 Z[i]-condition at the cusps, 29 contiguous relations, 196, 199 cusp, 29 cusp form, 169

determining component, 115 discrete series, 88 downward shift operator, 37, 67

eigenfunction equations, 63 Eisenstein series, 169 evaluation at zero, 83 exponential decay, 77, 169 exponential growth, 168 exponentials, 19, 21

Fourier expansion, 31, 32 —, absolute convergence, 31 Fourier term module, 4. 48, 59 —, abelian, 4 —, generic abelian, 63 —, large, 48, 48 —, non-abelian, 4, 63 —, N -trivial, 4, 63 Fourier term operator, 2, 32, 55

generic abelian Fourier term module, 63, 114 generic parametrization, 62

Haar measure, 18, 25 Heisenberg group, 21 Hermite polynomial, 26 highest weight space in a K-type, 36 holomorphic automorphic form, 175 holomorphic discrete series type, 88 holomorphis family of — automorphic forms, 176 — Fourier terms, 177

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 R. W. Bruggeman, R. J. Miatello, Representations of SU(2,1) in Fourier Term Modules, Lecture Notes in Mathematics 2340, https://doi.org/10.1007/978-3-031-43192-0

205

206 integral parametrization, 62 interior differentiation, 45 invariant sesquilinear form, 160 Iwasawa decomposition, 12, 15

Indices quick decay, 169

realization of SU(2, 1), 11 ν-regular behavior at 0, 77 right differentiation, 20

kernel relations, 50, 123 Kunze-Stein operator, 83

Langlands representation, 88 large abelian Fourier term module, 48 large discrete series type, 6, 87 large Fourier term module, 48 large non-abelian Fourier term module, 48 lattice, 23 left differentiation, 20 Lie algebra, 19, 21 limits of discrete series, 88 logarithmic module, 76, 110

maximal compact subgroup, 12 maximal component, 122 meromorphic family of — automorphic forms, 176 — Fourier terms, 177 metaplectic parameter, 4, 54 minimal component, 122 minimal vector, 38 moderate exponential growth —, automorphic form with, 169 modified Bessel differential equation, 66, 195 modified Bessel function, 195 (g, K)-module, 35 multiple parametrization, 62 multiplication relations for functions on K, 18

non-abelian Fourier term module, 63 normalized Hermite function, 26 norm in L2 (K), 18

parameter set of a special cyclic module, 39 Picard modular group, 29 Poincaré series, 170 polynomial growth, 168 positive Weyl chamber, 86 principal series, 4, 75 —, irreducible, 87 —, unitary irreducible, 87

Schrödinger representation, 22 Schwartz function, 22 sector of lattice points in (h/3, p)-plane, 69 sesquilinear form, 160 shift operator, 36, 47 —, in abelian case, 50 —, downward, 37 —, in non-abelian case, 52 —, upward, 36 shift parameter for theta functions, 4 simple parametrization, 62 simple positive roots, 34 special cyclic module, 39 spectral parameters, spectral pair, 61 split torus, 12 standard lattice, 23 Stone-von Neumann representation, 22 subquotient theorem, 85 symmetric space, 13

theta function, 24 thin representation, 161 type of irreducible representation, 87

unique embedding in principal series, 107 unipotent subgroup, 12 unitary principal series, 162 unitarizable, 160 universal enveloping algebra, 34 upper half-plane model of symmetric space, 13 unrestricted automorphic form, 169 upward shift operator, 36, 71

walls of positive chamber, 86 weight in a K-type, 36 weight space in a K-type, 35 Weil restriction, 13 Weyl group, 60 Whittaker differential equation, 196

Indices

207

List of Notations A(Г; ψ), A∗ (Г; ψ) A ⊂ G 12 Aτ 188 Aut(N ) 21 a 34 a(t) ∈ A 12 Bτ

169

189

Ci ∈ k 19 C ∈ ZU (g) 35 CK 20 C ∞ (G)K 44 C ∞ (Λσ \G)K 31 c shift parameter 24 cM (r) 142 cV (r), cW (r) 129

D(N) 185 d ∈ 1+2Z metaplectic param. 4, 52

E0

M ⊂ K ⊂ G 12, 34 M(X)f 45 Mw (Г) 175 Mκ,s 196 m(ζ ) ∈ M 12 m(h, r) 55, 121 m0 = m0 (j ) 122

H 187 Hi 19 Hr 34 H 30 ξ,ν HK 4, 75 h 34 h(c) 15 hℓ,m 26

83

Fβ 32 Fℓ,c 48 Fℓ,c,d = Fn 52 ψ ψ Fβ , Fn 4, 59 FN 2 τF , τF , τF 189 β ℓ,c N Fβ 32 Fℓ,c 32 Fℓ,c,d = Fn 55 FN 2 F F , F I 87 f c 32 fℓ,c,m 31 fβ 31

G realization of SU(2, 1) 11 g 34 gc 34 gc 29

Iν 195 I2,1 11 I I , I F 87 i0 76

Jℓ,c , JN 189 j = jξ 16 jτ 188 j+( , jr , jl ) 86 j g, (z, u) 174 K ⊂ G 12, 15 K0 ⊂ K 15 Kν 195 K0;h,p , Kβ;h,p , Kn;h,p 112, 115, 128 k 19 kc 18 kIβ;h,p , kK 115 β;h,p V kW n;h,p , kn;h,p M kn;h,p 142 k(η, α, β) ∈ K dk 18

128 12

L 191 ξ,ν L0 76 j,0 ˜ L0 110 L, L+ 86 L(X) 20 Lh 57 ℓ parametr. char. of Z(N ) 4, 24 ψ

MN 79 ξ,ν MN 4, 80, 135

N 52 N 12 Nc 29 NormP (Λσ ), NormP (Λ) 57, 192 n 21, 34 n = (ℓ, c, d) 52 n(x + iy, r) = n(x, y, r) ∈ N 12 dn 25 OW (ψ), O1W (ψ) 4, 62 OW (ψ)n , O1W (ψ)n 62 OW (ψ)+ 5 OW (ψ)+ 5, 79 n Pc

29

R(X)f = Xf 20, 45 r0 (h) 55, 121 S(R) 22 SU(2, 1) 11 SN (η) 83 Sect(j ) 69 S1 , S2 60 ±3 S±1 36 Tk+ , Tk− UT 13 U (g) 34 Uℓ,τ 188 ψ

Vn

136

161

208

Indices

ξ,ν

Vn 7, 135 Vκ,s 197 v(a, b) in Sect. 3.1.1

40

W 62 Wsp , Wmp , · · · 62 ψ WN 79, ξ,ν WN 4, 80, 117, 135 W0 , W1 , W2 ∈ k 19 W 60 Wκ,s 197 w ∈ G 14 X 13 X0 , X1 , X2 ∈ n

21

Zij ∈ gc 18, 34 Z(G) 12 Z(N ) 21 ZU (g) 34

κ0 = κ0 (j )

122

p

60

29,30

Pq (p), Pℓ,d (p)

Δ3 ∈ ZU (g) δ(·) 51

35

σ (c)

ε = sign (ℓ)

51, 121

τp τph

22

ψ[ξ, ν], ψ[j, ν] 60

62

ΩN;h,p 177 a,b ωN (j, ν) 78, 79 ω˜ na,b (j, ν) 124, 136

16

πλ 22 dπλ 23

Г 167 Г−1 , Г−3 , ГFL

Фr,q 17 h Фp 17 r,p hϕ p 75 r,q

χβ

MN;h,p 177 μa,b 77, 79 N (j, ν) μ˜ a,b 124, 146 n (j, ν) ν spectral parameter ν+ , νr , νl 86

34

17

ϒN;h,p 178 υna,b (j, ν) 123 υ˜ na,b (j, ν) 124, 136

Λσ = Λi,σ 23 Λτ,σ 187 λ2 (j, ν), λ3 (j, ν) hλp 76, 110 r,q

ξ = ξj

α1 , α2

h τr,p

Ɵℓ,c 24 τƟ 188 ℓ,c ϑm 51, 121

29

16 5, 16

192

|w 174 ∞ ∈ ∂X 14 (·, ·)ups 162 (·, ·)cmpl 162 . = 115 Ʃ 49, 121 r Vh,p subspace of K-typeτph 6, 35 Vh,p,q ⊂ Vh,p weight q subspace 35

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