Modular Representation Theory of Finite and p-Adic Groups 9789814651806

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MODULAR REPRESENTATION THEORY OF

FINITE AND p -ADIC GROUPS

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LECTURE  NOTES  SERIES Institute for Mathematical Sciences, National University of Singapore Series Editors: Chitat Chong and Wing Keung To Institute for Mathematical Sciences National University of Singapore ISSN: 1793-0758 Published Vol. 20 Mathematical Horizons for Quantum Physics edited by H Araki, B-G Englert, L-C Kwek & J Suzuki Vol. 21 Environmental Hazards: The Fluid Dynamics and Geophysics of Extreme Events edited by H K Moffatt & E Shuckburgh Vol. 22 Multiscale Modeling and Analysis for Materials Simulation edited by Weizhu Bao & Qiang Du Vol. 23 Geometry, Topology and Dynamics of Character Varieties edited by W Goldman, C Series & S P Tan Vol. 24 Complex Quantum Systems: Analysis of Large Coulomb Systems edited by Heinz Siedentop Vol. 25 Infinity and Truth edited by Chitat Chong, Qi Feng, Theodore A Slaman & W Hugh Woodin Vol. 26 Notes on Forcing Axioms by Stevo Todorcevic, edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Vol. 27 E-Recursion, Forcing and C*-Algebras edited by Chitat Chong, Qi Feng, Theodore A Slaman, W Hugh Woodin & Yue Yang Vol. 28

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Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore

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30

MODULAR REPRESENTATION THEORY OF

FINITE AND p -ADIC GROUPS Editors

Wee Teck Gan Kai Meng Tan National University of Singapore, Singapore

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Library of Congress Cataloging-in-Publication Data Modular representation theory of finite and p-adic groups / edited by Wee Teck Gan (NUS, Singapore), Kai Meng Tan (NUS, Singapore). pages cm. -- (Lecture notes series, institute for mathematical sciences, national university of singapore ; volume 30) NUS represents National University of Singapore. Includes bibliographical references and index. ISBN 978-9814651806 (hardcover : alk. paper) 1. Group theory. 2. p-adic groups. 3. Representations of groups. I. Gan, Wee Teck, editor. II. Tan, Kai Meng, editor. QA174.2.M63 2015 512'.23--dc23 2015000042



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00a-contents

CONTENTS

Foreword

vii

Preface

ix

Modular Representations of Finite Reductive Groups Marc Cabanes

1

`-Modular Representations of p-Adic Groups (` 6= p) Vincent S´echerre

47

p-Modular Representations of p-Adic Groups Florian Herzig

73

Representation Theory and Cohomology of Khovanov–Lauda–Rouquier Algebras Alexander S. Kleshchev

109

Cyclotomic Quiver Hecke Algebras of Type A Andrew Mathas

165

v

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FOREWORD

The Institute for Mathematical Sciences (IMS) at the National University of Singapore was established on 1 July 2000. Its mission is to foster mathematical research, both fundamental and multidisciplinary, particularly research that links mathematics to other efforts of human endeavor, and to nurture the growth of mathematical talent and expertise in research scientists, as well as to serve as a platform for research interaction between scientists in Singapore and the international scientific community. The Institute organizes thematic programs of longer duration and mathematical activities including workshops and public lectures. The program or workshop themes are selected from among areas at the forefront of current research in the mathematical sciences and their applications. Each volume of the IMS Lecture Notes Series is a compendium of papers based on lectures or tutorials delivered at a program/workshop. It brings to the international research community original results or expository articles on a subject of current interest. These volumes also serve as a record of activities that took place at the IMS. We hope that through the regular publication of these Lecture Notes the Institute will achieve, in part, its objective of reaching out to the community of scholars in the promotion of research in the mathematical sciences. November 2014

Chitat Chong Wing Keung To Series Editors

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PREFACE

A month-long program on “Modular Representations of Finite and p-Adic Groups” was held during April 2013 at the Institute for Mathematical Sciences (IMS) in Singapore. The program was organised by Joseph Chuang (City University, London, UK), Karin Erdmann (University of Oxford, UK), Wee Teck Gan (NUS), Florian Herzig (Toronto, Canada), Kay Jin Lim (NUS), Alberto M´ınguez (Jussieu, France) and Kai Meng Tan (NUS). The goal of the program is to bring together leading researchers in the areas of modular representation theory of finite groups and p-adic groups to discuss the latest developments in the field, chart out new directions for research and explore possible collaboration. It was well attended by about 50 participants worldwide. By “modular representations”, one means the representations of a group G on a vector space over a field of nonzero characteristic `, when ` divides the (pro-)order of G. It is with such `-modular representations that the program is concerned. The groups G which feature in the program include the finite groups of Lie type, their Weyl groups and related Lie or Hecke algebras, as well as the reductive p-adic groups which are infinite topological groups that play an important role in the Langlands program and number theory. While the modular representation theory of finite groups have been pursued for about a hundred years, the case of p-adic groups is a relatively young field, where it began its life about 20 years ago. There is so far relatively little interaction between the two areas and it is the hope of the program to stimulate such interactions. The program began with six short lecture series (aka tutorials) given by leading experts, followed by a two-week conference. The notes of five of these tutorials are collected in this volume. The chapter by M. Cabanes introduces the finite groups of Lie type and discusses recent results and outstanding conjectures about their modular representation theory. The understanding of the representation theory of these finite groups of Lie ix

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Preface

type provides a crucial starting point for the study of p-adic groups. The chapters by V. S´echerre and F. Herzig (with notes prepared by K. Koziol) give an exposition of the state-of-the-art in the `-modular representation theory of p-adic groups when ` 6= p and ` = p, respectively. They convey the exciting developments in the `-modular and p-modular Langlands program. Finally, the chapters by A. Kleshchev and A. Mathas touch upon the representation theory of the increasingly popular and important Khovanov– Lauda–Rouquier (KLR) algebras. Specifically, Kleshchev discusses the basic representation theory of these KLR algebras and their homological properties, whereas the chapter by Mathas is devoted to the emerging graded representation theory in the special case of type A. These KLR algebras arise from the introduction of the idea of “categorification” in representation theory. The last tutorial, given by R. Rouquier, is an introduction to this revolutionary idea of “categorification”. Unfortunately, due to circumstances beyond our control, the content of Rouquier’s talks is not included in this volume. We take this opportunity to thank all the participants of the program for contributing to its success. We especially want to thank the six tutorial lecturers for their excellent presentations and for taking the time and effort to write up their notes for publication. We hope that the chapters contained in this volume will serve as a useful reference for researchers and students in these and related areas. Finally, we would like to convey our deepest appreciation to IMS for generously and graciously providing both financial and administrative support for our program, without which the program would not have been successful. November 2014

Wee Teck Gan and Kai Meng Tan National University of Singapore Singapore Volume Editors

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MODULAR REPRESENTATIONS OF FINITE REDUCTIVE GROUPS

Marc Cabanes Institut de Math´ematiques de Jussieu Universit´e Paris Diderot Bˆ atiment Sophie Germain 75205 Paris Cedex 13, France [email protected]

We report on the main results about linear representations of finite reductive groups or finite groups of Lie type. Following the historical order, we comment on representations in the defining characteristic, ordinary characters and representations in non-defining characteristic.

Contents 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Representations Group Algebras Coxeter Groups Iwahori-Hecke Algebras BN-Pairs and Simple Groups Reductive Groups Finite Reductive Groups Rational G-Modules and Weights Defining Characteristic: The Simple Modules Steinberg’s Tensor Product and Restriction Theorems Weyl Modules and Lusztig’s Conjecture More Modules Harish Chandra Philosophy Deligne-Lusztig Theory Unipotent Characters and Lusztig Series `-Blocks Some Derived Equivalences Decomposition Numbers Some Harish Chandra Series 1

3 4 5 8 10 13 17 19 20 21 22 23 25 28 30 32 35 37 41

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References

43

INTRODUCTION This survey deals with linear representations of a class of finite groups strongly related to linear algebra itself, finite reductive groups. Finite reductive groups also provide almost all finite simple groups and their essential central extensions. They are therefore omnipresent, explicitly or not, in finite group theory. Finite reductive groups are finite analogues of reductive algebraic groups whose structure is in turn close to that of Lie groups. This also explains the terminology “groups of Lie type” which is often used. Let us mention also the term “Chevalley groups”, which pays tribute to Chevalley’s construction of those finite groups from integral forms of semi-simple complex Lie algebras [Ch55]. The later approach is also described in [Cart1], though the prevailing approach is now to see them as fixed point subgroups GF under a Frobenius endomorphism F : G → G of some algebraic group G, an approach due to Steinberg and allowing to take advantage of the geometry of algebraic groups. The pace of this survey is intended as quite slow, giving details necessary to understand most definitions. This should suit beginners or more experienced readers from other branches of representation or group theory. We tried to comment on some examples, mainly in type A (general linear groups, finite or not), and we strongly recommend Bonnaf´e’s book [Bn] where much of our matter, and more, is thoroughly explored for SL2 . In Part I, we describe roughly the main features of representation theory, that is mainly the study of module categories for rings that are mostly finite groups algebras or close analogues. This also leads to the study of associated categories, most famously the derived and the homotopic categories. We did not comment on the already spectacularly successful methods of categorification, referring instead to Rouquier’s series of talks. Part II comments on the constructions and structure of those groups both from the elementary point of view of split BN-pairs and of algebraic groups. Part III deals with linear representations most naturally associated with finite reductive groups since performed over the field Fp defining the ambient algebraic group. The classical theorems of Chevalley and Steinberg are supplemented by more recent contributions by Lusztig, though it seems that some questions remain as mysterious as they were 25 years ago.

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Part IV, by far the longest, comments on both the theory by DeligneLusztig leading to a very precise description of ordinary characters of finite groups of Lie type and more recent investigations on representations in characteristics different from p. The first is presented in many references (see [Sri], [DiMi], [Cart2]) and we quickly recall the main results. The second part was largely initiated by the papers of Fong-Srinivasan ([FoSr82], [FoSr89]) relating ordinary characters of finite classical groups GF with blocks of F` (` 6= p). Many contributions followed by Dipper (decomposition numbers [Dip85a], [Dip85b]), and Geck-Hiss-Malle (`-modular Harish Chandra theory [GeHiMa94], [GeHiMa96]). Brou´e in [Br90] gave several conjectures, the one on Jordan decomposition of characters being later proved by Bonnaf´e-Rouquier (see [BnRo03]). We report on all those subjects, which leads us to many theorems on `-blocks, decomposition numbers and modular Harish Chandra series. I. MODULAR REPRESENTATIONS We refer to the first chapter of [Be] or the short book [Sch] for most of the information we need. 1. Representations The framework is the one of representations of non commutative rings A that are mainly finite dimensional algebras over a commutative field K. The finite dimensional representations can be seen as objects of the category A-mod of finitely generated A-modules. It may also be that A is an O-free algebra of finite rank over a local ring O, in which case we require that the objects of A-mod to be O-free of finite rank. We do not recall here the terminology of simple, indecomposable, or projective A-modules. The Jacobson radical of A is denoted by J(A), the group of invertible elements of A is denoted by A× . Simple modules are equivalently called irreducible representations, and Irr(A) denotes their set of isomorphism types. When A is over a field K, we denote by K0 (A) the Grothendieck ring of A, which may be seen as the commutative group ZIrr(A) endowed with the multiplication induced by tensor product of modules over K. The multiplicity of a simple A-module S as a composition factor in the Jordan-H¨ older series of an A-module M is denoted by [M : S]. We also use the notion of blocks as minimal indecomposable two-sided ideal direct summands of A. Note that this also makes sense when A is an O-free algebra of finite rank over a local ring O.

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Our modules are left modules but we need also the notion of bimodule. It is defined as follows. We have two rings A, B, and T is an A-module on the left, and a B-module on the right, with both actions commuting, that is a(tb) = (at)b for any a ∈ A, t ∈ T , b ∈ B. This of course coincides with the notion of an A × B opp -module where B opp is the opposite ring of B. An object T of A × B opp -mod defines a functor B-mod → A-mod by M 7→ T ⊗B M . The notion of a Morita equivalence is a typical example of such a functor (see [Be] Sect. 2.2). In connection with representations of finite groups, one may be led to ∂i

consider the category Cb (A) of bounded complexes . . . M i −→ M i+1 . . . of A-modules (∂ i+1 ∂ i = 0 for all i ∈ Z, and only finitely many i’s are such that M i 6= {0}) and the related homotopic category Hob (A).

2. Group Algebras We are mostly interested in the cases where A is a group algebra RG of a (multiplicative) finite group G over a commutative ring R. Recall that this P is the R-free R-module of all sums g∈G rg g (for (rg )g∈G any family of elements of R indexed by G) with R-bilinear multiplication extending the one of G. For us, an R-linear representation of G is an R-free RG-module M of finite rank or equivalently a group morphism G → GLn (R). When H ≤ G is a subgroup, restriction ResG H has an adjoint on both G sides IndH which is the functor RH-mod → RG-mod associated with RG seen as a bimodule with translation actions of G on the left and H on the right, i.e. IndG H (N ) = RG ⊗RH N whenever N is an RH-module. When R = K is a field where |G| inverts and with primitive |G|-th roots Q of 1, then KG is a split semi-simple algebra ∼ = i Matdi (K). If moreover the characteristic of K is 0, then the isomorphism type of a KG-module M is given by its trace character χM : G → K sending g to the trace of its action on M . Irreducible characters are the ones of simple KG-modules, and one denotes by Irr(G) the corresponding set of functions on G. This has values in Z[ω|G| ] where ω|G| is a primitive |G|-th root of 1, it is therefore ∼ independent of K. Note that Irr(KG) −→ Irr(G). For most of character theory, see [Isa]. When R is a field k of characteristic a prime divisor of |G| with a primitive |G|p0 -th root of 1, the algebra kG/J(kG) is split semi-simple. Then Irr(kG) is often denoted as IBr(G) and identifies with the central functions on Gp0 (p-regular elements of G) obtained by Brauer’s method of lifting p0 -roots of 1 in k into an extension of Zp (see [Sch] Sect. 3.1).

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When p is a prime number and G is a finite group, a p-modular system (O, K, k) is a triple where O is a complete discrete valuation ring containing the |G|-th roots of 1, free of finite rank over Zp , with K denoting the field of fractions of O (a finite extension of Qp ) and k = O/J(O) is its (finite) residue field. The decomposition of OG into blocks OG = ⊕i Bi (sometimes called the p-blocks of G) gives the decomposition of kG into blocks kG = ⊕i Bi ⊗k and induces a partition of both Irr(G), by Irr(G) = Irr(KG) = ti Irr(Bi ⊗ K), and IBr(G) by IBr(G) = Irr(kG) = ti Irr(Bi ⊗ k). The principal block of G in characteristic p is defined as the one not in the kernel of the one-dimensional trivial representation of G (where each element of G acts by 1). II. THE GROUPS

3. Coxeter Groups 3.A. Definitions References for what follows are the corresponding chapters of [Bou], [GePf]. Definition 3.1. A Coxeter graph on a set S is a non-oriented graph without loops, with nodes the elements of S, and valued edges carrying a number mst ∈ {3, . . . , ∞} (of course mst = mts ) for any s, t ∈ S. One omits mst when mst = 3 and one completes the matrix (mst )s,t∈S by putting mst = 2 whenever s 6= t and there is no edge between s and t. The associated Coxeter group is the group generated by the elements of S subject to the relations: (quadratic) s2 = 1 for all s ∈ S. (braid) sts · · · = tst . . . (with mst terms on each side) for all s 6= t ∈ S with mst 6= ∞. One denotes by lS : W → N the length function with regard to the generating set S. Note that the relation for s 6= t and mst = 2 specifies that s and t commute. So the connected components of the Coxeter graph gives a partition of S into pairwise commuting sets.

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Example 3.2. (a) The graphs m

•−−−−• or •−−−−• for m ≥ 4 give rise to the dihedral groups of order 2m (6 in the first case, infinite in the case when m = ∞). (b) Another important example is the one of the graph of so-called type An as follows (n ≥ 1 nodes) •−−−−•−−−−•−− · · · −−•−−−−• whose associated group is isomorphic with the symmetric group Sn+1 by the map sending the i-th generator si in the above list (from left to right) to the transposition (i, i + 1) (Moore’s presentation of the symmetric group). Several other basic properties are as follows. Fact 3.3. S injects in W and mst is the order of the product st in W . Fact 3.4. If W is finite, it has a single element wS ∈ W of maximal lS . 3.B. Parabolic subgroups, finite Coxeter groups One calls parabolic subgroups of W the subgroups generated by a subset of S. If I ⊆ S, one denotes WI := . The following is a consequence of the so-called exchange condition governing the way to obtain minimal decompositions (see [Bou] IV.1.5). Fact 3.5. The map I 7→ WI is a bijection between subsets of S and parabolic subgroups of W . It satisfies WI ∩ WJ = WI∩J for all I, J ⊆ S. Moreover S ∩ WI = I and the restriction of lS to WI is lI . An abstract group W with a subset S of involutions is called a Coxeter group if W is isomorphic through the canonical map with the group presented by S subject to the relations of Definition 3.1 for mst being defined as the order of the product st (which requires that S generates W ). A Coxeter group is said to be irreducible if, and only if, it comes from a connected Coxeter graph. Theorem 3.6. An irreducible Coxeter graph gives rise to a finite group W if and only if it is among the following (the index n recalls the number of nodes):

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Dn

4

F4 •−−−−•− −−−•−−−−• 5

H3 •−−−−•−−−−•

7

4

An •−−−−•−−−−•−− · · · −−•−−−−• •+ •−− · · · −−•−−−−• •

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BCn •−−−−•−−−−•−− · · · −−•−−−−• En

6

G2 •−−−−• or

•−−−•+ •−− · · · −−•−−−• •

for n = 6, 7, 8

m

I2 (m) •−−−−• for m = 5 or ≥ 7 5

H4 •−−−−•−−−−•−−−−•

On the other hand a typical graph producing an infinite group is as follows *•−−−•−−−· · · .. e A • . (n + 1 nodes). n •−−−•−−−· · · 3.C. Reflection representations Coxeter groups have a reflection representation. It is defined as follows. Let us start from the set S and integers mst for s 6= t ∈ S. One then defines a real vector space V = ⊕s∈S Res whose basis is in bijection with S by s 7→ es . One defines on it a symmetric bilinear form sending (es , es ) to 1 and (es , et ) to − cos(π/mst ) when s 6= t. Denote by Isom(V ) ≤ GLR (V ) the corresponding orthogonal group. Then it is easy to see that the map sending s to the reflection through the vector es is a group morphism W → Isom(V ). A more remarkable fact is that it is injective. Another fact is that W is finite if and only if the above bilinear symmetric form is definite positive (see [Bou] Sect. V.4.8). The latter is a key fact in the proof of the above Theorem 3.6. An important generalization of the finite case above is when V is a finite dimensional complex vector space and W ≤ GLC (V ) is a finite subgroup generated by elements r ∈ W such that the image of r − IdV is a line (“pseudo-reflections”, remembering that r has finite order hence is semi-simple). Such a W is called a finite reflection group, a good survey is provided by [GeMa]. A defining property is that the action of W on S(V ), the ring of polynomials on V is such that the invariant subring S(V )W is isomorphic to a polynomial ring (Chevalley, see [Bou] V.5.3).

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4. Iwahori-Hecke Algebras A deformation of the group algebra of a Coxeter groups, Iwahori-Hecke algebras were defined in connection with representations of finite reductive groups. 4.A. Generalities All that follows can be taken from [GePf]. Definition 4.1. If R is a commutative ring, q ∈ R and W is a Coxeter group (with subset S and integers mst ), one denotes by HR,q (W ) the Ralgebra with generators (As )s∈S subject to the relations: (quadratic) (As + 1)(As − q) = 0 for any s ∈ S, (braid) As At · · · = At As . . . (mst terms on each side) for any distinct s, t ∈ S. Note that the choice q = 1 gives us the group algebra RW . More generally, one has freeness over R for an R-basis indexed by W . Theorem 4.2. If w = s1 . . . slS (w) is a minimal expression of w as a product of elements of S, the product As1 . . . Ask is 6= 0 and depends only on w. One denotes it by Aw := As1 . . . Ask (with A1 := 1). Then HR,q (W ) = ⊕w∈W RAw . Many properties ensue, among them the relation with parabolic subgroups of W , or with scalar extension. 4.B. Semi-simplicity in characteristic zero The first assertion below is due to Gyoja-Uno ([GyUn89], see also a generalization with several parameters in [GeP f ]), the second is due to Tits. Theorem 4.3. Assume R = C and W is finite. Let q ∈ C. P (a) HC,q (W ) is semi-simple if, and only if, q · w∈W q lS (w) 6= 0. (b) (Deformation theorem). If HC,q (W ) is semi-simple, then HC,q (W ) ∼ = CW . P Note that the so-called Poincar´e polynomial PW (X) = w∈W X lS (w) di +1 |S| has a factorization PW (X) = Πi=1 X X−1−1 where the di are the exponents of W , that is the degrees of the generators of the invariants of the action of W on the symmetric powers of the reflection representation. So we see that complex Iwahori-Hecke algebras over W are isomorphic to the group

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algebra CW except for finitely many values of q that are either 0 or a root of unity. Variants of Iwahori-Hecke algebras have been defined in the case of complex (finite) reflection groups (see [BrMaM93]). 4.C. The Kazhdan-Lusztig polynomials Let (W, S) be a Coxeter group, possibly with W finite. If w = s1 . . . slS (w) is a minimal expression of w as a product of elements of S, then the set [1, w] ⊆ W defined by all sub-products si1 . . . sit for 1 ≤ i1 < · · · < it ≤ lS (w), is independent of the minimal expression chosen for w. One then defines the Bruhat order ≤ on W by x ≤ y if and only if x ∈ [1, y]. See [GePf] Ex. 1.7 for this. See also Sec. 6.B below for an interpretation in terms of Zariski closure in reductive groups. Let q 1/2 be an indeterminate whose powers are denoted by (q 1/2 )m = m/2 q . Let R = Z[q −1/2 , q 1/2 ] the ring of Laurent polynomials in those indeterminates. Let H be the Iwahori-Hecke algebra HR,q (W ) (see Definition 4.1 above). Note that q being invertible, each generator As (s ∈ S) is invertible with (As )−1 = q −1 As + (q −1 − 1)A1 , so that any Aw (w ∈ W ) is also invertible. Definition 4.4. Let H → H, x 7→ x be the Z-linear map sending q m/2 to q −m/2 and Aw to (Aw−1 )−1 . This is a ring endomorphism. Here is the existence theorem for Kazhdan-Lusztig’s polynomials (see [KL79]). Theorem 4.5. There exists a unique set of polynomials Pv,w ∈ Z[q] for v ≤ w with Pw,w = 1, Pv,w has degree ≤ (lS (w) − lS (v) − 1)/2 for v < w P 0 := −lS (w)/2 and such that Cw q v≤w Pv,w Aw satisfies 0 = C0 Cw w

for any w ∈ W . It has long been conjectured that the above polynomials have coefficients in N. This was proved by Kazhdan-Lusztig for finite W – and also affine en above – by showing the relation with cohomology of Schubert type like A varieties BwB/B (see Sec. 6B below). The positivity in the general case was proved recently by Elias-Williamson [EW12]. The polynomials allow Kazhdan-Lusztig to define certain subsets of the group W , called cells, and associated cell representations of the IwahoriHecke algebra. See more generally the notion of cellular algebras [GraL].

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0 The new basis {Cw } of Theorem 4.5 also allows a more explicit version of the isomorphism in Theorem 4.3 above. More general Iwahori-Hecke algebras can be given a family of parameters (qs )s indexed by elements of S modulo W -conjugacy. See [Lus03] Ch. 14 for several conjectures about those algebras – some of them having been checked since then.

5. BN-Pairs and Simple Groups BN-pairs, first axiomatized by Tits, are a basic trait common to all groups of Lie type, be them algebraic, p-adic or finite. 5.A. The axioms and basic properties Definition 5.1. A group G is said to have a BN-pair if it possesses two subgroups B and N , and a subset S ⊆ N/B ∩ N such that • T := B ∩N is normal in N and one has s2 = 1 in the group W := N/T for any s ∈ S, • G = and W = , • for all s ∈ S, sBs 6= B, • for all s ∈ S, w ∈ N/T , one has BsBwB = BswB ∪ BwB. The groups B, T , W are often called the Borel subgroup, a maximal torus and the Weyl group of G, respectively. Fact 5.2. The elements of S are the only non-trivial elements of W such that B∪BsB is a subgroup of G (this is why they can be considered implicit in the definition of the “BN”-pair). Fact 5.3. (W, S) is a Coxeter group. Fact 5.4. Bruhat decomposition. The double cosets BwB for w ∈ W are pairwise disjoint and are all the elements of B\G/B: G = tw∈W BwB. Fact 5.5. The set of subgroups of G containing B is in bijection with the set of subsets of S by I 7→ PI := BWI B = BB. We get in particular PI ∩ PJ = PI∩J for any I, J ⊆ S. 5.B. Split BN-pairs and Levi decompositions The group G with a BN-pair is said to be split whenever the group B is a semi-direct product B = U o T . Some authors sometimes add the

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condition that T is abelian. An important property of the same nature is the following. Assume G has a split BN-pair B = U T with finite W . Definition 5.6. For I ⊆ S, denote by wI the element of maximal length of (WI , I) and UI := U ∩ wI U wI . Also B − = wS BwS and LI := BWI B ∩ B − WI B − ⊆ PI = BWI B. The latter is called the standard Levi subgroup associated to I. One says that PI satisfies a Levi decomposition if UI C PI and PI = UI o LI . 5.C. Examples, finite and p-adic Example 5.7. The case of GLn (F). Let F be a field, n ≥ 1 an integer, and G := GLn (F). Let B (resp. U , resp. T ) be the subgroup of upper triangular (resp. unipotent upper triangular, resp. diagonal) matrices in G. Let N be the subgroup of monomial matrices in G (invertible matrices with only one non-zero element on each row and on each column). Then N/T ∼ = Sn (permutation matrices) and the permutations (1, 2), . . . , (n − 1, n) used in Example 3.2.(b) above give a subset S ⊆ N/T with properties of a BN-pairs. Whenever I ⊆ S generates WI ∼ = Sn1 × · · · × Snk , one gets for PI =   A1 ∗ ∗ ∗  0 A2 ∗ ∗    BWI B the subgroup of matrices in the form   with Ai ∈  0 0 ... ∗  0 0 0 Ak GLni (F). We also have the Levi decomposition PI = UI o LI where UI (resp. LI ∼ = GLn1 (F) × · · · × GLnk (F)) is the subgroup of matrices in the     A1 0 0 0 I n1 ∗ ∗ ∗  0 A2 0 0   0 In2 ∗ ∗      form  (resp.    with Ai ∈ GLni (F)).  0 0 ... ∗   0 0 ... 0  0

0

0 Ink

0 0 0 Ak

Example 5.8. A p-adic example. Let O be a finite extension of some Zp , with fraction field K and residual field k = O/J(O). The group G = SLn (K) has the following BN-pair (Iwahori-Matsumoto). Reduction of matrix entries modulo J(O) gives a surjective group morphism SLn (O) → SLn (k). Let B ≤ SLn (O) be the inverse image of the upper triangular subgroup, a Borel subgroup, of SLn (k). Let N be the subgroup of monomial matrices in G. Then T := B ∩ N is the subgroup of

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diagonal matrices with coefficients in O and determinant 1. The quotient en−1 when n ≥ 3 (see Sec. W = N/T is an affine Coxeter group of type A e 3.B above), A1 = I2 (∞) if n = 2. Note that the proper parabolic subgroups for this BN-pair (sometimes called “parahoric” subgroups of G = SLn (K)) are finite unions of double cosets BwB since they are associated to finite subgroups of W . 5.D. Simple groups Recall that a simple group is any group having no other normal subgroup than itself and the trivial subgroup. A perfect group is any group G generated by its commutators, i.e. G = [G, G]. We have the following important criterion of simplicity. Theorem 5.9. Let G a group with a BN-pair B, N , S and associated Weyl group (W, S). Assume the hypotheses: • (W, S) is irreducible, • B is solvable and G is perfect. Then G/ ∩g∈G gBg −1 is simple non-abelian. The proof is remarkably easy. It suffices to show that any normal subgroup of G is either G or a subgroup of B. So let H E G. Then BH is a subgroup of G containing B, therefore BH = BWI B for some I ⊆ S (see Fact 5.5 above). Let us show that if s ∈ S \ I and w ∈ I, then they commute. We have sws−1 = sws ⊆ sBHs ⊆ sBsH ⊆ BH ∪ BsBH

since H E G,

⊆ BWI B ∪ BsBWI B ⊆ B(WI ∪ sWI )B by the axioms of BN-pairs. The uniqueness of the Bruhat decomposition (see Fact 5.4) then implies that sws−1 or s ∈ WI , and therefore sws−1 ∈ WI in both cases. Then sws−1 ∈ PI ∩ P{s,w} = P{w} = B ∪ BwB by Fact 5.5. Again by Bruhat decomposition, we get now sws−1 = w. In view of the irreducibility hypothesis we must have I = S or ∅, that is HB = G or H ≤ B. If HB = G, then G/H ∼ = B/B ∩ H, so the last assumptions of the theorem imply that this factor group is at the same time solvable and equal to its derived subgroup. So it is indeed trivial and H = G. So BN-pairs can be seen as a way to construct simple groups, finite or not. Understandably, finite BN-pairs have been classified (see [FoSe74], [HeKaSe72]) as a (small) part of the classification of finite simple groups.

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This in turn can be seen as a (very difficult) converse of the above, see below Sec. 7.C. 6. Reductive Groups We fix F an algebraically closed field. One considers F-varieties essentially as locally closed subvarieties of affine varieties. Algebraic groups over F are affine F-varieties such that multiplication (x, y) 7→ xy and inversion x 7→ x−1 are F-algebraic morphisms. Much of their abstract theory was done by Borel (see the reference [Bo] and the textbooks [MT], [Spr]). The group GLn (F) is clearly such an algebraic group (n ≥ 0 an integer), and in fact all algebraic groups over F are closed subgroups of some GLn (F). The latter is related with the existence of linear representations of algebraic groups, that is algebraic morphisms G → GLn (F) (see [MT] 5.5). An important property of algebraic groups is the notion of unipotent and semi-simple elements, along with the Jordan decomposition of elements of G, x = us = su where u, resp. s, is sent to a unipotent (resp. semi-simple) element by any linear representation of G. 6.A. Reductive groups Definition 6.1. The unipotent radical Ru (G) of an algebraic group G is its largest normal connected subgroup whose elements are all unipotent. The group G is said to be reductive if and only if it is connected and Ru (G) = 1. It is said to be semi-simple if in addition its center Z(G) is finite (this is also equivalent to being perfect, i.e. [G, G] = G, and connected). When F is the field of complex numbers, the above notion coincides with the one of semi-simple Lie groups. Those are classified essentially by use of Lie algebras and the classification of root systems (see [Hum1]). Some similar results can be obtained for reductive groups over algebraically closed fields F (Chevalley, 1956), leading to presentations by generators and relations (Steinberg, 1962) and identification for non exceptional Dynkin types to classical groups. We recall below some main notions and steps for this classification, mainly because they are also crucial to the description of linear representations of both reductive groups and their finite analogues. 6.B. Borel subgroups, tori and root data In what follows G is a reductive groups over F.

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A Borel subgroup B ≤ G is any connected solvable closed subgroup maximal for inclusion. They form a single G-conjugacy class. Definition 6.2. A torus of G is any closed subgroup isomorphic (as an algebraic group) to a direct product of copies of GL1 (F) = F× . Maximal tori of G are a single G-conjugacy class. It is now customary to fix a maximal torus T in G and introduce several notions attached to this. Each Borel subgroup can be written B = Ru (B) o T for a (non-unique) maximal torus of G. Let X(T) = Hom(T, F× ) (morphisms of algebraic groups), which is a lattice isomorphic to Zr whenever T ∼ = (F× )r . Similarly one defines Y (T) = × Hom(F , T) and gets a pairing X(T) × Y (T) → Z ∼ = Hom(F× , F× ), (x, y) 7→ hx, yi = x ◦ y. Definition 6.3. A root subgroup is any minimal unipotent subgroup normalized by T. Each root subgroup associated to T is isomorphic to F by some map F 3 a 7→ u(a) ∈ G and (consequently) there exists α ∈ X(T) such that t.u(a).t−1 = u(α(t)a) for all a ∈ F, t ∈ T. The set of α ∈ X(T) giving rise to such root subgroups is denoted by Φ(G, T) and called the set of roots of G with respect to T. One also defines a set Φ(G, T)∨ ⊆ Y (T) and a bijection Φ(G, T) → Φ(G, T)∨ with α 7→ α∨ actually obtained as follows. For α ∈ Φ(G, T) and associated root subgroup Uα ≤ G, there is some morphism ϕα : SL2 (F) → G sending the unipotent upper triangular matrices into Uα , the diagonal ∨ × ∨ matrices into   T. Then α ∈ Y (T) = Hom(F , T) is defined by α (λ) = λ 0 ϕα for any λ ∈ F× . This α∨ is called the coroot associated with 0 λ−1 the root α. It is well-defined thanks to the fact that ϕα above is unique up to T-conjugacy. The relation with the usual notion of (crystallographic) root system is that Φ(G, T) ⊆ X(T) ⊆ X(T)⊗Z R endowed with the above pairing X(T) × Y (T) → Z and α 7→ α∨ allow to define reflections sα : X(T) ⊗Z R → X(T) ⊗Z R associated with any α and to prove that they preserve the set Φ(G, T). A positive subsystem Φ(G, T)+ is obtained by selecting the α such that the corresponding root subgroups Uα are subgroups of a given Borel subgroup B. Recall that a positive subsystem of a root system always contains a

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unique basis ∆ of the root system (this allows to define many notions related with this positivity phenomenon, such as the notion of highest positive root). Such a basis ∆ determines the Dynkin type of the root system and the corresponding reflections are a generating set S := {sδ | δ ∈ ∆} for the group W ≤ GL(X(G, T) ⊗Z R) generated by all reflections above. The pair (W, S) is a Coxeter system (see Sec. 3 above). The (infinite) group G has a BN-pair for the Borel subgroup B and N = NG (T). The quotient group N/T identifieswith the  group W described 0 1 above (associate with α ∈ ∆ the class of ϕα ). −1 0 Example 6.4. (a) The case of G = GLn (F) is as follows. Let T ∼ = (F× )n n be the subgroup of diagonal matrices. Then X(T) = ⊕i=1 Zei where ei is the morphism sending that n-tuple (t1 , . . . , tn ) to ti . With that notation, for any i 6= j in {1, . . . , n}, α := ei −ej sends (t1 , . . . , tn ) to ti (tj )−1 and this is an element of Φ(G, T) corresponding to the subgroup Uα of matrices in u such that u − In has only zeros except possibly the element at row i and column j. The positive roots corresponding to the Borel subgroup B ≤ G of upper triangular matrices correspond to pairs (i, j) with j > i. The associated basis is {e1 − e2 , . . . , en−1 − en }. The type of Coxeter system is An−1 (see Theorem 3.6 above) corresponding with the above ordering of roots. (b) The case of G = Sp2n (F). Let us denote by M 7→ M T the transposition of matrices. Let Jn ∈ GLn (F) be the matrix whose only non-zero elements are 1’s on the second diagonal, that is the (i, j) element is δn+1−i,j . Let Sp2n (F) ≤ GL2n (F) be defined by the equation     0 −Jn 0 −Jn M MT = . One can choose as maximal torus the Jn 0 Jn 0 subgroup T of diagonal matrices with diagonals of type −1 (t1 , . . . , tn , t−1 n , . . . , t1 ),

and one obtains  as UT where U is the subgroup of ma a Borel subgroup X XJn S where X ∈ Un (F) (unipotent upper tritrices of type 0 Jn (X −1 )T Jn angular matrices of GLn (F)) and S ∈ Symn (F) (symmetric n × n matrices). It is not difficult to single out “one-parameter” unipotent subgroups normalized by T. They come in two types corresponding with the roots α sending the above diagonal matrix to ti tj for some 1 ≤ i, j ≤ n (α = ei +ej , first type) and to ti t−1 for some i 6= j (α = ei − ej , second type). Then j e1 − e2 , . . . , en−1 − en , 2en gives a basis of the root system and the

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corresponding Coxeter diagram is of type BCn 4

•−−−−•−−−−•−− · · · −−•−−−−• Relation with the Bruhat order. It is easy to prove the following interpretation of the Bruhat order ≤ in W (G, T) in terms of Zariski closure of double cosets, namely ByB = ∪1≤x≤y BxB. Following the model of Lie algebras, one is often led to consider the Borel subgroups “opposite” the one already used. In the above case, noting that U = , one defines U− := . Note that U− = wS UwS where wS is the element of W of maximal length with regard to the basis above. The group B− := TU− is a Borel subgroup. The product BwS BwS = BB− = UTU− is called the big cell and is open, hence dense in G (use the closure property recalled above). 6.C. Classification in terms of root data One defines a notion of root datum which stands for quadruples (X, Φ, Y, Φ∨ ) where X, Y are lattices endowed with a perfect pairing over Z, and Φ ⊆ X, Φ∨ ⊆ Y are subsets endowed with a bijection α 7→ α∨ satisfying axioms producing root systems in the same fashion as seen above in the case of reductive groups where a maximal torus has been chosen. Then the classification theorem of Chevalley ensures that the choice of an algebraically closed field F and a root datum (X, Φ, Y, Φ∨ ) selects a reductive group G over F such that one of its maximal torus T is such that (X, Φ, Y, Φ∨ ) ∼ = (X(T), Φ(G, T), Y (T), Φ∨ (G, T)). Such a G is unique up to an isomorphism which itself is unique up to T-conjugacy. The group is said to be of simply-connected type whenever it is semisimple and Y = ZΦ∨ . From the isomorphism theorem it is clear enough that, once Φ a root system and F are chosen, then there is only one group Gsc which is of simply connected type. Any semi-simple G with same Φ and F is a central quotient Gsc → G (see [Cart2] p 25). The group is said to be of adjoint type whenever it is semi-simple and X = ZΦ. This is also equivalent to having trivial center. Any semi-simple G with same Φ and F satisfies G/Z(G) ∼ = Gad .

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Example 6.5. For the root system of type An−1 , one has Gsc = SLn (F) and Gad = PGLn (F). In type Cn , Gsc is the symplectic group Sp2n (F) (see Example 6.4.b above) and Gad is its quotient by ±Id2n . In certain types, the simply connected covering is less naturally found. In type Bn , Gad is SO2n+1 (F) (special orthogonal group for the quadratic form defined by the sum of squares of coordinates in F2n+1 ) but Gsc is a spin group Spin2n+1 (F) defined by means of the Clifford algebra. 7. Finite Reductive Groups Let p be a prime and F the algebraic closure of the field with p elements. 7.A. Definition and Lang’s theorem Let f ≥ 1 and q := pf . Let Frob : GLn (F) → GLn (F) be the raising of matrix entries to the q-th power. This turns the Frobenius automorphism of the field into a so-called Frobenius endomorphism of the algebraic group GLn (F). Definition 7.1. A finite reductive group is any group of fixed points GF := {g ∈ G | F (g) = g} where F : G → G is an algebraic endomorphism such that, for at least one k ≥ 1, F k is the restriction to G of a Frobenius endomorphism Frob : GLn (F) → GLn (F) with Frob(G) = G. The study of finite reductive groups is made easier by the following important theorem due to Lang and generalized by Steinberg (see [CaE] 7.1). Theorem 7.2. In the above setting, if S is a closed connected F -stable subgroup of G, one has S = {g −1 F (g) | g ∈ S}. In particular this implies the existence of F -stable Borel subgroups and F -stable maximal tori in G. Fact 7.3. Finite reductive groups GF have BN-pairs of type BF , N F where B, N = NG (T) make the BN-pair of the reductive group G and T ≤ B are F -stable. 7.B. Examples Taking the examples of 6.4 above, and with k = 1 in Definition 7.1 above, one finds the finite groups GLn (q) and Sp2n (q).

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The case k = 2 can account for finite unitary groups GUn (q), the latter a subgroup of GLn (q 2 ). For this we take F = Frob ◦ σ where Frob is the raising of matrix entries to the q-th power and σ : GLn (F) → GLn (F) is the automorphism of order two sending a matrix to its transpose-inverse. Cases where k 6= 1 is needed in Definition 7.1 are often called twisted. The above case where Frob is composed with a so-called graph automorphism also occurs in type Dn and E6 . In types B2 and F4 with p = 2, and in type G2 with p = 3, one may build an endomorphism F : G → G which behaves like a Frobenius composed with a graph automorphism on certain root subgroups (changing the root and applying the Frobenius to the parameter) and like a graph automorphism on others (changing the root but not the parameter), see [Cart1] Sect. 12.3, 12.4, [Cart2] 1.19. Then F 2 is a Frobenius endomorphism. The 2 2 corresponding groups GF sc are denoted by B2 (q) or F4 (q) for q a power of 2 2, and G2 (q) for q a power of 3. 7.C. Classification: Finite simple groups, quasi-simple groups The classification of finite simple groups is the collective work of dozens of mathematicians and was completed at the start of the 80s. Theorem 7.4. The finite simple groups are either • any cyclic group of prime order • any alternating group An with n ≥ 5 • a finite group with a BN-pair of type G/Z(G) where G is a finite reductive group, or the Tits group [2 F4 (2),2 F4 (2)] • one of the 26 sporadic groups The members of the third item above are called simple groups of Lie type, see [GLS] for an in-depth analysis of their properties. As is well-known, perfect groups G (i.e. such that [G, G] = G) have a unique maximal central extension ˆ→G G ˆ maximality being (a surjective morphism with central kernel and perfect G), here defined by the property that any other would be covered by that one. The kernel of the above universal map is called the Schur multiplier, it is finite when G is. Finite quasi-simple groups are defined as perfect groups G such that G/Z(G) is simple. They are central quotients of the maximal central extensions of non-abelian finite simple groups. Apart from a finite number of exceptions, the situation for simple groups of Lie type parallels

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the one of semi-simple groups G. Namely, the maximal quasi-simple groups are the finite reductive groups E = (Gsc )F for G of irreducible Weyl group (or root system). Then the simple quotient is of course E/Z(E). Note that the latter is not necessarily a finite reductive group, see the example below. Example 7.5. PSLn (q) := SLn (q)/Z(SLn (q)) while SLn (q) = (Gsc )F for ¯q . G of type An−1 over F = F III. REPRESENTATIONS IN DEFINING CHARACTERISTIC 8. Rational G-Modules and Weights In this section, we consider a reductive group G over an algebraically closed field F and we assume chosen T ≤ B a maximal torus and Borel subgroup of G. We recall the set of roots Φ(G, T) (see Sec. 6.C above). We denote U = Ru (B) and we recall U− . We consider the rational representations of a reductive group G. Definition 8.1. A rational representation of G is any rational group morphism G → GLF (M ) where M is a finite dimensional F-vector space. Note that M is then a FG-module for the (infinite dimensional) group algebra FG, and we denote simply by g.m or gm the outcome of the action of g ∈ G on m ∈ M . Fact 8.2. One has ResT M = ⊕λ∈X(T) Mλ where Mλ = {m ∈ M | t.m = λ(t)m for all t ∈ T, m ∈ M }. The latter are called the weight subspaces of M . Recall the exponential notation for fixed points: for instance M T = M0 = {m ∈ M | t.m = m for any t ∈ T}. Fact 8.3. If M 6= {0}, then M U 6= {0} Fact 8.4. FG.M U = FU− .M U . The character of M , ch(M ) keeps track of the dimensions of the weight subspaces. Definition 8.5. One defines formal symbols eλ indexed by elements of X(T) and satisfying eλ eµ = eλ+µ . The formal character of M is X ch(M ) = dimF (Mλ )eλ . λ∈X(T)

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Let us recall the pairing X(T) × Y (T) → Z that we denote below as h−, −i. Definition 8.6. The dominant weights are defined by X + (T) := {λ ∈ X(T) | hλ, α∨ i ≥ 0 for all α ∈ Φ(G, T)+ }. 9. Defining Characteristic: The Simple Modules We keep the same notations as in the section above. The following is due to Chevalley. Theorem 9.1. The (isomorphism types of) simple rational representations of G are in bijection with dominant weights. One denotes by L(λ) the simple rational FG-module associated with λ ∈ X + (T). It is characterized by the property that L(λ)U = L(λ)λ and is a line. Moreover L(λ)µ 6= {0} implies that λ − µ ∈ NΦ(G, T)+ . Example 9.2. The case ofG = SL  2 (F). In this case we take for T the t 0 diagonal torus of matrices for t ∈ F× and Borel subgroup TU 0 t−1   1a where U = { | a ∈ F}. Then X(T) ∼ = Z by the map associating to 01   t 0 m ∈ Z the map 7→ tm . The dominant weights correspond to N. 0 t−1 Consider the natural action of G onF[X, Y ] = S(F2 ) where P (X, Y ) is ac sent to P (aX + bY, cX + dY ) by ∈ SL2 (F). For p − 1 ≥ m ≥ 0, let bd L0 (m) ⊆ F[X, Y ] be the subspace of homogeneous polynomials of degree m. Then L0 (m)U is clearly the line FX m = L0 (m)m while L0 (m) = FU− .X m (this is where the hypothesis that m ≤ p − 1 is used). So we get part of the parametrization of Theorem 9.1. Note that L0 (m) has dimension m + 1 in that case. For the whole parametrization of simple rational representations, see Theorem 10.3 below. Note that when m is any integer ≥ 0, then FG.X m = FU− .X m provides indeed a simple rational representation of G, that we should rename L(m). This is the parametrization of Theorem 9.1 by dominant weights. But we no longer have L(m) = L0 (m) in general and the dimension can be smaller than m + 1, (though computable, see Theorem 10.2 below). Example 9.3. L(0) = F the trivial module. In the notations of Example 6.4.a above, L(e1 ) for GLn (F) is the natural representation, of dimension n.

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10. Steinberg’s Tensor Product and Restriction Theorems We now turn to relations of simple rational modules with simple modules in the same characteristic for finite reductive groups. The theorems are by Steinberg (see [St63]) and are the main source of information on representations of finite reductive group in the defining characteristic. We keep G a reductive group over F which is assumed to be of characteristic the prime p. We keep T ≤ B ≤ G a maximal torus and Borel with corresponding definition of root system Φ(G, T) with basis ∆. Definition 10.1. If q is a power of p, we denote Xq+ (T) := {λ ∈ X(T) | 0 ≤ hλ, δ ∨ i ≤ q − 1 for any δ ∈ ∆}. We now assume that G ≤ GLn (F) is a subgroup invariant under Frob the raising of matrix entries to the p-th power. If M is the underlying vector space of a rational representation of G, and i ≥ 0, one denotes by M [i] the same vector space but where the action of G is twisted by Frobi . With the new action being denoted by ∗, we have g ∗ m = Frobi (g).m for any g ∈ G, m ∈ M. The following is Steinberg’s tensor product theorem. Theorem 10.2. Let λ ∈ X + (T) written as λ = Xp+ (T). Then

P

i≥0

pi λi where λi ∈

L(λ) ∼ = ⊗i≥0 L(λi )[i] . Assume F = (Frob)f where q = pf . The following gives the simple FGF -modules. Theorem 10.3. For any λ ∈ Xq (T) the module L(λ) restricts into an irreducible representation of GF . Moreover, if Y (T) = ZΦ∨ , then all simple kGF -modules are of that type. Definition 10.4. When Y (T) = ZΦ∨ , one defines the Steinberg module ∨ of GF as ResG GF L(λ) where λ ∈ X(T) is defined by hλ, α i = q − 1 for all α ∈ ∆. The Steinberg FGF -module is projective and has dimension |UF |, the order of the Sylow p-subgroup of GF .

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11. Weyl Modules and Lusztig’s Conjecture We return to the study of FG-modules L(λ) for λ ∈ X + (T0 ). We assume in this section that G = Gsc , that is Y (T0 ) = ZΦ∨ . Weyl G-modules V (λ) are a construction borrowed from the characteristic zero case. In the case of characteristic zero the formal characters of those modules are known thanks to Weyl’s so-called character formula (see [Hum1] Sect. 24). So it is enough to determine the multiplicities [V (λ) : L(µ)] for any λ, µ ∈ X + (T0 ) to have the characters of the simple modules L(λ). 11.A. Weyl modules Let g be the semi-simple Lie algebra over C with root system Φ. Any λ ∈ X + (T0 ) can be considered as a dominant weight of g, so it gives rise to a simple g-module with highest weight λ, V (λ)C , which corresponds to L(λ) for the simply-connected algebraic group GC of root system Φ over C. Then, by a construction of Chevalley (see [Hum1] Ch. VII), it is possible to choose a lattice V (λ)Z stable under the action of a Lie subalgebra gZ such that V (λ) := F ⊗Z V (λ)Z has a compatible algebraic action of G = Gsc . The formal character of V (λ) is known from the one of V (λ)C which in turn is given by Weyl’s character formula (see [Hum2] Sect. 3.2). 11.B. Affine Weyl group action on X(T) and Lusztig’s conjecture Let α0 be the highest root of Φ with regard to the basis ∆. Let ρ ∈ X(T0 ) be the half sum of all positive roots, so that hρ, α∨ i = 1 for all α ∈ ∆. Definition 11.1. Let Wa be the subgroup of GL(X(T)) generated by W (T) and the translations by elements of pΦ = {pα | α ∈ Φ}. This is a Coxeter (affine) group for the generators {sδ | δ ∈ ∆} ∪ {r0 } where r0 is the reflection in the hyperplane {x ∈ X(T) | hx, α0∨ i = p}. Let Wdom := {w ∈ Wa | −w(ρ) − ρ ∈ X + (T)}. Since Wa is a Coxeter group for the indicated set of generating reflections, one has an associated Bruhat order ≤, length map l : Wa → N, and Kazhdan-Lusztig polynomials Pv,w ∈ Z[x] (v ≤ w in Wa ). Lusztig’s conjecture, formulated as a “Problem” in [Lus80], is as follows Problem 11.2. Assume p > hα0∨ , ρi and let w ∈ Wdom with hα0∨ , −w(ρ)i ≤ p(p − h + 2) where h is the Coxeter number of Φ (see for instance [GePf] p. 29), then

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ch(L(−w(ρ) − ρ)) =

X

23

(−1)l(w)−l(v) Pv,w (1) ch(V (−v(ρ) − ρ)).

v∈Wdom v≤w

The above, combined with Andersen-Humphreys’ “linkage principle” and Jantzen’s “translation principle ” (see [Jn] II.6 and II.7), allows to express any ch(L(λ)) (λ ∈ X + (T0 )) in terms of ch(V (µ))’s as long as p > h. The latter ch(V (µ)) is known in turn by Weyl’s character formula. It has been proved by Andersen-Jantzen-Soergel (see [AJS94]) with additional restriction on p. Theorem 11.3. Lusztig’s problem has a positive answer for large p. In this work the bound on p is not explicit. An explicit bound exponential in the rank was given by Fiebig [F12]. Using Soergel’s bimodules and Juteau-Mautner-Williamson’s theory of parity sheaves, G. Williamson has shown that part of Lusztig’s conjecture for p > h is equivalent to absence of p-torsion in certain cohomology of Schubert varieties. In [W13], Williamson gives a method to find such torsion numbers. In particular Fibonacci numbers Fn and Fn+1 are shown to be such numbers for G = SL4n+7 (F), which readily excludes that Lusztig’s conjecture could be proved for p ≥ f (h) with f a linear function (for SLn ). It is expected that any polynomial bound can be excluded. 12. More Modules We show here some relations between simple modules for finite reductive groups in defining characteristic and representations of modular IwahoriHecke algebras (Green-Tinberg-Sawada, see [CaE] Ch. 6). This leads to define certain indecomposable modules with interesting properties (see [Gre78]). We are back to a general finite reductive group GF where G is a reductive group over F of characteristic p. Then we have seen in Fact 7.3 that G = GF has a split BN-pair with subgroups B = BF , T = TF , N = NG (T)F . Recall that U = Ru (B) is such that U = UF is a Sylow psubgroup of G. Note that an immediate corollary of Bruhat decomposition is that G = tn∈N U nU (disjoint union).

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12.A. Modular Iwahori-Hecke algebra The group G acts by translation on the set G/U of left cosets with regard to U , so the vector space Y := FG/U is a FG-module. It is isomorphic with the induced module IndG U F where F denotes here the trivial FU -module. Definition 12.1. Let HF (G, U ) = EndFG (Y ). This is a finite dimensional P algebra with basis (an )n∈N defined by an (U ) = x∈U/U ∩nU n−1 xnU (where we have given the image of the class U in G/U ⊆ FG/U , the image of the other classes being then easily determined). Proposition 12.2. (i) The simple HF (G, U )-modules are 1-dimensional. (ii) HF (G, U ) is a self-injective algebra (i.e. the regular representation is injective). 12.B. Fixed point functor and simple modules We now define the following classical functor HY from (left) FG-modules to right HF (G, U )-modules. If M is an FG-module, HY (M ) = HomFG (Y, M ) ∼ = MU. The second equality results from Y = IndG U F and Frobenius reciprocity. Note that this second equality implies that HY (M ) 6= {0} whenever M 6= {0} since U is a p-group for p the characteristic of F. Using self-injectivity from Proposition 12.2 in a crucial way, one proves the following. Theorem 12.3. (Green) One has a decomposition Y = ⊕ai=1 Yi with each Yi an indecomposable module. Each Yi had a simple head and simple socle (though not isomorphic in general). Moreover if 1 ≤ i, j ≤ a, then Yi ∼ = Yj ⇔ i = j ⇔ hd(Yi ) ∼ = hd(Yj ) ⇔ soc(Yi ) ∼ = soc(Yj ) ⇔ ∼ ∼ hd(HY (Yi )) = hd(HY (Yj )) ⇔ soc(HY (Yi )) = soc(HY (Yj )). Corollary 12.4. ([CE] 1.25.(i)) Let FG-modY be the full subcategory of the category of FG-modules whose objects are the FG-modules in the form e(Y m ) for m ≥ 1 and e ∈ EndFG (Y m ). Then HY induces an equivalence between FG-modY and the category of right HF (G, U )-modules. 12.C. The p-blocks The following is due to Humphreys (see [CaE] 6.18).

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Theorem 12.5. If S = GF /Z(GF ) is simple non-abelian, then FS = FSt ⊕ B0 (FS) a sum of the one-dimensional block corresponding to the Steinberg module and the principal block (see Sec. 2 above) of S. Remark 12.6. So we see that simple groups of Lie type have only two blocks in the defining characteristic. Certain groups have even less, that is just one (principal) block. This is the case for the simple Mathieu groups M22 and M24 with regard to the prime p = 2. On the other hand, groups with just one p-block are never a simple group of Lie type. It can even be ¯pG proved that for p an odd prime and G a finite group, the group algebra F is just one block if and only if G has a normal p-subgroup P C G such that CG (P ) ≤ P (see [Ha85] Th. 1).

IV. OTHER CHARACTERISTICS We take G = GF a finite reductive group with G over F of characteristic p. We let k be an algebraically closed field of characteristic ` 6= p. The goal is to study the category kG-mod. The case ` = 0 is very developed thanks to the work of Deligne-Lusztig and subsequent work of Lusztig, Shoji and others. Much less is known in the case of a positive ` but Harish Chandra philosophy can be used and provides a partition of simple modules. 13. Harish Chandra Philosophy We assume GF is equipped with its BN-pair BF , TF , N , etc... as in Fact 7.3. A Levi decomposition is any GF -conjugate of some standard Levi decomposition of a standard parabolic subgroup PI = BWI B = Ru (PI ) o LI (see Sec. 5.B above) where I is an F -stable subset of ∆, the basis of Φ(G, T) associated with B. We then write PF = Ru (P)F o LF where P = gPI g −1 , L = gLI g −1 , for some g ∈ GF . 13.A. Harish Chandra induction and restriction For PF = Ru (P)F o LF a Levi decomposition, we define X e = |Ru (P)F |−1 x ∈ kGF , x∈Ru (P)F

an idempotent commuting with the elements of LF . One defines two functors.

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Definition 13.1. Let F F RG L : kL -mod → kG -mod

defined by

F

N 7→ kG e ⊗kLF N and ∗

F F RG L : kG -mod → kL -mod

M 7→ M

Ru (P)F

defined by

= eM = ekGF ⊗kGF M.

The first is clearly the inflation of LF -modules to PF -modules followed ∗ G by induction from PF to GF . The two functors RG L and RL are exact and ∗ G are clearly adjoint to each other on both sides. Note that RG G = RG is the identical functor. G L We have a transitivity formula RG H = RL ◦ RH whenever Q = Ru (Q)H is another F -stable Levi decomposition in G with Q ≤ P and H ≤ L (so that Q ∩ L = (Ru (Q) ∩ L). H is a Levi decomposition in the reductive group L). G We have a Mackey formula for the compound ∗ RG L ◦ RL0 : Theorem 13.2. Whenever P0 = Ru (P0 )L0 is another F -stable Levi decomposition, one has X ∗ G ∗ L0 RL ◦ RG RL L0 = L∩gL0 g −1 ◦ adg ◦ RL0 ∩g −1 Lg g

where g ranges over a representative system of the double cosets F PF \GF /P0 . In order to simplify our notation we have omitted the parabolic subgroup used to define RG L . In fact we have invariance with regard to the choice of P, for a given L. The following is due simultaneously to DipperDu and Howlett-Lehrer ([DipDu93], [HowL94], see also [CaE] 3.10): Theorem 13.3. Whenever P0 = Ru (P0 )L is another F -stable Levi decomposition with same Levi subgroup L, and one denotes X e0 := |Ru (P0 )F |−1 x ∈ kGF x∈Ru (P0 )F

one has an isomorphism kGe → kGe0 by the map sending x to xe0 .

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Corollary 13.4. Assume P0 = Ru (P0 )L and P = Ru (P)L as above are two F -stable Levi decompositions with same Levi subgroup L, then P and P0 give the same RG L. Remark 13.5. While at least ∗ RG L as fixed point functor can be defined easily in characteristic p, it would no longer be independent of the parabolic used to define it. See [CaE] Ex. 5 p. 99. 13.B. Cuspidality and partition of simple modules Definition 13.6. A non-zero kGF -module M is said to be cuspidal if F and only if ∗ RG L (M ) = 0 for any L 6= G. A simple kG -module is said to be supercuspidal if and only if its projective cover is cuspidal. Note that by exactness of the ∗ R functors, a module is cuspidal if and only if all its simple composition factors are cuspidal. This shows that supercuspidal simple modules are cuspidal. It is also easily seen that a simple kGF -module M is cuspidal (resp. supercuspidal) if and only if it is not a submodule (resp. a composition factor) of any RG L N for L 6= G and N a kLF -module. Given a simple kGF -module S one may define the minimal pair (L, N ) such that there is a non-zero map RG L (N ) → S (for instance with minimal |LF | + dimk (N )). Mackey formula, see Theorem 13.2 allows to show that N is simple cuspidal and that the pair (L, N ) is unique up to GF -conjugacy. This gives a partition a (13.7) Irr(kGF ) = Irr(kGF , L, N ) (L,N )

indexed by conjugacy classes of the above “cuspidal pairs” (L, N ) where Irr(kGF , L, N ) is the set of simple kGF -module with a nonF zero map RG L (N ) → S. When (L, N ) is such a pair, Irr(kG , L, N ) is parametrized by the irreducible representations of the endomorphism algebra EndkGF (RG L (N )). For a field k of characteristic zero the latter is a Iwahori-Hecke algebra of the type studied by Howlett-Lehrer and Lusztig (see [HowL80], [Lus76] Sect. 5). In characteristic ` > 0, the knowledge is less complete. The following is due to Geck-Hiss-Malle (see [GeHiMa96] 2.4.(a)): Theorem 13.8. If P = Ru (P)L is an F -stable Levi decomposition of an F -stable parabolic subgroup of G, and N is a simple cuspidal kLF -modules, then as a kGF -module RG L (N ) = Y1 ⊕ · · · ⊕ Ya

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where the Yi ’s are indecomposable and • for each i, soc(Yi ) ∼ = hd(Yi ) and they are simple modules • for any 1 ≤ i < j ≤ a, one has soc(Yi ) ∼ 6 soc(Yj ) (and therefore = also Yi ∼ 6 Yj ). = The case of GF = GLn (q) (q a power of p) has been worked out early on by Dipper (see [Dip85b]). One has the following (see also [CaE] 19.11). We assume that k is the residual field of a valuation ring O, whose fraction field (of characteristic 0) is denoted by K. Theorem 13.9. Take G = GLn (q). Any simple cuspidal kG-module is the reduction modulo J(O) of a G-stable O-lattice inside a simple cuspidal KG-module. Remark 13.10. Let us go back to an arbitrary finite reductive group G. Let us explain briefly how each simple kG-module is a composition factor of at least one RG L π where π is a supercuspidal simple kL-module. If a simple kG-module S is not supercuspidal, this means that ∗ RG L PS 6= {0} for some proper standard Levi L ≤ G and where PS denotes the projective cover of 0 S. Then there is a simple kL-module S 0 such that HomL (∗ RG L PS , S ) 6= {0}, G 0 or equivalently by adjunction HomG (PS , RL S ) 6= {0}, which means that S 0 0 is a composition factor of the module RG L S . By induction hypothesis, S is L 00 a composition factor of some RM S for a supercuspidal simple kM -module 00 and therefore by exactness S is a composition factor of RG M S . So indeed [ Irr(kG) = Irr(kG, L, π) (L,π)

where (L, π) ranges over the (G-conjugacy classes of) pairs of a standard Levi L of G and π is a simple supercuspidal kL-module, and where Irr(kG, L, π) denotes the set of simple composition factors of RG L π. However, in contrast to the union (13.7) above, it is not clear whether the above union is always disjoint. See however Theorem 19.3 below. 14. Deligne-Lusztig Theory In the next two sections we give the basic definitions introduced by DeligneLusztig’s seminal paper ([DeLu76]) and some of the main consequences on the parametrization of irreducible characters of finite reductive groups (see also [Lus84]). We keep GF a finite reductive group associated with an F -stable reductive group G (see Definition 7.1 above).

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14.A. Deligne-Lusztig varieties and functors A key innovation brought by Deligne-Lusztig consists in generalizing Harish Chandra functors RG L to Levi subgroups that are F -stable but not necessarily present in the Levi decomposition of an F -stable parabolic subgroup. Let us recall that Levi subgroups of G are simply G-conjugates of standard Levi subgroups LI , or more intrinsically centralizers CG (S) of tori (non maximal in general) of G (see for instance [DiMi] 1.22). Example 14.1. A subclass of the class of F -stable Levi subgroups is the one of F -stable maximal tori. Assume as before that a pair T0 ≤ B of a maximal tori and Borel subgroup, both F -stable, have been chosen (see Sec. 7.A above). Any other maximal torus has to be in the form gT0 g −1 . It is F -stable if and only if g −1 F (g) ∈ NG (T0 ). Since any element of G can be written in the form g −1 F (g) (see Theorem 7.2 above), we may have maximal tori gT0 g −1 such that g −1 F (g) is in any class of NG (T0 )/T0 . Arguing this way one finds easily that gT0 g −1 7→ g −1 F (g)T0 induces a bijection between GF -conjugacy classes of F -stable maximal tori and the so-called F -classes of the group W = NG (T0 )/T0 (the equivalence relation is defined by w ∼F w0 in W if and only if there exists v ∈ W such that w = v −1 w0 F (v)). The class of the maximally split torus T0 just corresponds to 1 ∈ W . On ´etale topology and `-adic cohomology groups, see [CaE] A.3, [Cart2], [Sri]. One chooses a prime ` 6= p. Definition 14.2. If P = Ru (P) o L is a Levi decoposition with F (L) = L, one defines the Deligne-Lusztig variety YL,G := {gRu (P) | g −1 F (g) ∈ Ru (P)} ≤ G/Ru (P) which is stable by left translation under GF and right translation under LF . The ´etale topology on YL,G allows to define the so-called groups of `-adic cohomology with compact support Ri Γc (YL,G ) (for i ∈ Z), some K-vector spaces with inherited actions of GF on the left and LF on the right. Their alternating sum gives an element of K0 (KGF × LF opp ). By tensor product one then gets two adjoint functors F F ∗ G F F RG L : K0 (KL ) → K0 (KG ) and RL : K0 (KG ) → K0 (KL ).

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It can be shown that those functors are independent of the prime ` 6= p chosen to define `-adic cohomology (identifying simple KGF -modules with complex characters of GF ). When the parabolic subgroup P is F -stable, then YL,G is just the finite set GF /Ru (P)F and the above functors are the images in the Grothendieck groups of the Harish Chandra functors denoted the same in the preceding section. As in the case of the Harish Chandra functors, one has a transitivity formula associated with inclusions Q ≤ P of parabolic subgroups, see [DiMi] ∗ G 11.5. One may also ask if RG L and RL are independent of the parabolic P used to define YL,G . This is related to the Mackey formula that can be expected for that functor, see Theorem 13.2. It is known from [DeLu76] when L is a torus (see also [DiMi] 11.15.(i)). For a general proof leaving out a small number of cases (q = 2 in exceptional types), see [BnMi11]. 15. Unipotent Characters and Lusztig Series 15.A. Unipotent characters Definition 15.1. The irreducible components of various RG T (1) for T an F -stable maximal torus of G and 1 the trivial representation of TF make a subset E(GF , 1) of Irr(GF ) called the set of unipotent characters. It is relatively easy to show that ∼

E(GF , 1) ← − E(GF ad , 1) through natural maps. More strongly, one can show that the set of unipotent characters “does not depend on q” but only on the type of the Dynkin diagram of G along with the action of F on it. Their degrees are polynomials in q. Example 15.2. Let us discuss that case of type An−1 . In this case E(GF , 1) is in bijection with the set Pn of partitions of n. The GF -classes of F stable maximal tori of G are in bijection w 7→ Tw with elements w of the symmetric group Sn (see Example 14.1 above). If ρ ∈ Irr(Sn ) denotes P 1 G a character of the symmetric group, then n! w∈Sn ρ(w)RTw (1) is ± an F element of E(G , 1) and this induces a bijection Irr(Sn ) → E(GF , 1) in that case (see for instance [DiMi] 15.8). It is in turn well-known that the irreducible characters of Sn , like its conjugacy classes, are parametrized by Pn .

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15.B. Lusztig series An important feature of character theory of finite reductive groups is the notion of dual groups. If G is a group over F, its dual is also one, so this notion is quite intrinsic and slightly different from Langland’s dual (a group over C). Here G∗ is the reductive group over F whose root datum is (Y, Φ∨ , X, Φ) if the one of G was (X, Φ, Y, Φ∨ ). One has a Frobenius endomorphism F ∗ : G∗ → G∗ inducing the same action as F on the groups X and Y . Then it is elementary to check that GF -classes of pairs (T, θ) where T is an F -stable maximal torus of G and θ ∈ Irr(TF ), are in bijection with G∗ F -classes of pairs (T∗ , s) where T∗ is an F -stable maximal torus of G∗ and s ∈ (T∗ )F . This, along with a deep disjunction theorem on the generalized characters RG T (θ) due to Deligne-Lusztig (see [DiMi] 13.3) allows to prove the following Theorem 15.3. There exists a partition Irr(GF ) = ts E(GF , s) where s ranges over the G∗ F -classes of semi-simple elements of G∗ F . Moreover, one has a bijection (sometimes called a Jordan decomposition of characters) ∼

E(GF , s) ← − E(CG∗ (s)F , 1) where E(CG∗ (s)F , 1) denotes the set of irreducible components of characters of CG∗ (s)F induced from a unipotent character of the finite reductive group CG∗ (s)◦F . The “Jordan decomposition” of characters has many additional properties. In particular the character degrees are multiplied by |G∗ F : CG∗ (s)F |p0 when going from right to left. This is a consequence of a more general compatibility with generalized characters of type RG T θ for T a torus. This compatibility also ensures uniqueness of the Jordan decomposition when Z(G) is connected. Moreover the relation with RG L functors is such that ∗ := ∗ when L CG∗ (s) is a Levi subgroup of G (hence connected) dual to some F -stable Levi subgroup L ≤ G, the above bijection is induced, up to a sign, by ζ 7→ RG sζ) where sˆ is the linear character of LF induced by s L (ˆ (an element in the center of L∗ = CG∗ (s)) and duality.

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16. `-Blocks Let us assume now that GF is a finite reductive group of characteristic p, that ` is a prime 6= p and that (O, K, k) is an `-modular system for GF and its subgroups in the sense of section 2 above. Along with (G, F ), we consider its dual group (G∗ , F ), see Sec. 15.B. 16.A. `-blocks and Lusztig series Definition 16.1. When s is a semi-simple `-regular element of G∗ F , one denotes E` (GF , s) := ∪t∈sCG∗ (s)F` E(GF , t) where t actually ranges over the semi-simple elements of G∗ F with `-regular part s. The following theorem is due to Brou´e-Michel (see [BrMi89] and [CaE] 9.12). Theorem 16.2. There is a sum of blocks of the group ring OGF , denoted by B` (GF , s) such that Irr(B` (GF , s) ⊗O K) = E` (GF , s). Remark 16.3. Relations between the functor RG L and `-modular questions can be expected from the definition of RG , and actually a decisive argument L for the above is given by the character formula of Deligne-Lusztig (see [DiMi] 12.2) whose proof indeed relies on projectivity arguments for OGF modules (see for instance [Sri] 6.4, [DeLu76] 3.5). A related more elementary statement is as follows (Hiss, see [CaE] 9.12): Proposition 16.4. Any `-block B 0 in B` (GF , s) satisfies Irr(B 0 ⊗O K) ∩ E(GF , s) 6= ∅. From general theory, we know that irreducible Brauer characters in characteristic ` generate the subspace of the space KIrr(GF ) of central functions on GF that vanish outside `-regular elements. Here E(GF , s) seems to play a similar role within E` (GF , s) and relates strongly with irreducible Brauer characters (see Theorem 18.3 below).

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16.B. Jordan decomposition of characters as a Morita equivalence Bonnaf´e-Rouquier’s theorem on Jordan decomposition (2003, see [BnRo03]) is the deep proof of a result conjectured by Brou´e. Theorem 16.5. Assume s ∈ G∗ F is a semi-simple `-regular element. Let L∗ be an F -stable Levi subgroup of G∗ such that CG∗ (s) ≤ L∗ . Then considering an F -stable Levi subgroup L ≤ G dual to L∗ , one has a Morita equivalence between the module categories ∼

B` (LF , s)-mod − → B` (GF , s)-mod such that the induced bijection between ordinary characters over K is inF duced by Lusztig’s functor E` (LF , s) 3 ζ 7→ ±RG L (ζ) ∈ E` (G , s). The proof in [BnRo03] involves a thorough analysis of Deligne-Lusztig varieties and their finite coverings, along with deep considerations on the derived category of kGF -mod, even though the theorem is a Morita equivalence (see also [CaE] 10–12). More derived equivalences will be given in the next section. 16.C. Unipotent blocks and generalized Harish Chandra theory Taking the minimal case in Theorem 16.5 where L∗ can be taken to be CG∗ (s), Bonnaf´e-Rouquier’s theorem suggests strongly that the study of `-blocks of GF reduces to the study of blocks of some B` (GF , s) where s is central in G∗ , and in turn easily to blocks of some B` (GF , 1). The latter are called the unipotent blocks of the group GF (see Proposition 16.4). The cardinality of any finite reductive group GF is a polynomial expression of q, with coefficients in Z (see tables in [Car2] Sect. 2.9), often called the polynomial order of (G, F ). In this polynomial P(G,F ) (x), the prime divisors 6= x are cyclotomic polynomials. If S is an F -stable subgroup of G whose polynomial order is a power of the d-th cyclotomic polynomial φd , then S is a torus. It is natural to study those “φd -tori” like `-elements of a finite group (` a prime). This leads to an analogue of Sylow’s theorem, due to Brou´e–Malle, see ([BrM92], [CaE] 13.18). If S is a φd -torus, then CG (S) is an F -stable Levi subgroup, called a “d-split” Levi subgroup of G. More important, Brou´e-Malle-Michel ([BrMaM93], see also [FoSr86]) have shown the existence of a generalized dHarish Chandra theory where d-split Levi subgroups replace the standard Levi subgroups used in the Harish Chandra theory described in Sec. 13 above.

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The following relation with unipotent `-blocks has been shown by Cabanes-Enguehard (see [CaE] 22.9). Let us denote by e(q, `) the order of q in (Z/`Z)× . Theorem 16.6. Assume ` ≥ 7. One has B` (GF , 1) = ⊕L,ζ BL,ζ where the BL,ζ ’s are `-blocks of GF and (L, ζ) ranges over the pairs where L is an e(q, `)-split Levi subgroup of G and ζ ∈ E(LF , 1) is such that ∗ L RL0 (ζ) = 0 for any proper e(q, `)-split Levi subgroup of L. Moreover Irr(K ⊗ BL,ζ ) ∩ E(GF , 1) is the set of irreducible components of the generalized character RG L (ζ). Remark 16.7. When ` divides q − 1, the partition of unipotent characters into `-blocks correspond with the one induced by Harish Chandra theory. Example 16.8. Let us illustrate the above theorem in the case of GF = GLn (q) (the theorem is then due to Fong-Srinivasan, see [FoSr82]). We assume GLn (F) endowed with its usual Frobenius endomorphism F raising matrix entries to the q-th power. Let us abbreviate e = e(q, `). A typical e-split Levi subgroup of GLe (F) is the maximal torus Ce of type the cycle of order e in the Weyl group Se with respect to the diagonal torus (see Example 14.1 above). Note that (Ce )F is a cyclic group of order q e − 1. Then we can define the e-split Levi subgroups Lm ≤ GLn−me (F) × GLe (F)m ≤ GLn (F) for 0 ≤ me ≤ n with Lm ∼ = GLn−me (F) × (Ce )m . It can be shown that the pairs (L, ζ) as in Theorem 16.6 are the pairs (Lm , ζ) where 0 ≤ me ≤ n and ζ ∈ E(Lm , 1) correspond to an element of E(GLn−me (q), 1) whose associated partition of n − me (see Example 15.2 above) is a so-called e-core. (Recall from the combinatorics of partitions that a partition λ1 ≥ λ2 ≥ · · · ≥ λk > 0 is an e-core if and only if, for all i, λi − i − e ∈ {λj − j | j = 1, . . . , k} or λi − i − e ≤ −k, see for instance [JaKe] 2.7 or [CaE] 5.11.) The above theorem describes the splitting of B` (GF , 1) into `-blocks from the point of view of ordinary (unipotent) characters. In fact much more is known about the blocks of GF . From a similar point of view, a splitting of an arbitrary B` (GF , s) into blocks and the ordinary characters of those blocks are known (see [CaE99] for ` ≥ 7, and [KM13] for the remaining cases). Those references give also information on the so-called defect `-subgroup of GF associated to each block.

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The notion of defect group of an `-block is due to R. Brauer and has been reformulated by J.A. Green (see [N]). In the following definition, let B be a block of the ring OG of a finite group G, with (O, K, k) an `-modular system for G. Definition 16.9. The defect subgroup of B is any minimal subgroup D ≤ G such that the OG-bimodule map B ⊗OD OG → B, x ⊗ y 7→ xy splits. It is a single G-conjugacy class of `-subgroups of G. Defect subgroup can be seen as measuring the non-simplicity of the algebra B ⊗O k. It is {1} when B has projective module such that its reduction modulo J(O) is simple. On the other extreme, the principal block (see Sec. 2 above) has defect the Sylow `-subgroup of G. The Steinberg module gives a p-block with defect {1} in defining characteristic (”` = p”) for any group GF . The `-block of GLn (q) defined by a pair (Lm , ζ) as in Remark 16.8 above has defect any Sylow `-subgroup of GLme (q) (Fong-Srinivasan 1982, see for instance [CaE] 22.10, [FoSr82]). 16.D. Cyclic defect When the defect group D of a block B is cyclic, B ⊗O k has only a finite number of indecomposable modules and the whole category B ⊗O k-mod can be described by a combinatorial object called the Brauer tree of the block. This is a tree whose edges (not nodes !) correspond with simple B ⊗O k-modules. The projective cover of a given simple module corresponding with the edge  can be described in terms of the edges adjacent to . The determination of Brauer trees occurring for blocks of finite reductive groups was started by Fong-Srinivasan and is now close to completion thanks to recent work of Craven, Dudas and Rouquier (see [Cr12] and its references). The methods make use of modules in characteristic ` defined by `-adic cohomology of Deligne-Lusztig varieties. Craven has proposed in [CR12] 1.3 a new conjecture on the cohomology of those varieties. 17. Some Derived Equivalences In [B90], M. Brou´e issued several conjectures on derived equivalences between module categories of blocks of finite groups. In many cases that have been verified, not only a derived equivalence but a homotopic equivalence has been checked. This is a stronger result and since the homotopic category is easier to describe we concentrate on this case. The homotopic category Hob (A) of some A-mod (or any abelian category) is obtained from the

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category Cb (A-mod) of bounded complexes of A-modules by keeping the same objects and replacing the additive groups of morphisms HomA (C, C 0 ) between complexes of A-mod with their quotients HomA (C, C 0 )/I(C, C 0 ) where I(C, C 0 ) is the subgroup of morphisms homotopic to the null morphism. When A is a semi-simple ring, for instance a division ring, every bounded complex C = (C i , ∂ i )i is isomorphic with its homology H(C) = (Ker(∂ i )/∂ i−1 (C i−1 ))i , seen as a complex with morphisms all equal to zero. If B and B 0 are two blocks of finite group algebras OG and OG0 and one has an equivalence Hob (B) ∼ = Hob (B 0 ) this clearly implies an equivalence Hob (B ⊗O K) ∼ = Hob (B 0 ⊗O K) and therefore a bijection Irr(B) → Irr(B 0 ) between irreducible characters with signs attached to each element of Irr(B). Brou´e’s conjectures often consist in asking if some given correspondence between characters is coming from such an equivalence of homotopic categories. 17.A. Chuang-Rouquier’s theorems We have seen in Example 16.8 that the unipotent blocks of GLn (q) are indexed by pairs (m, ζ) where 0 ≤ me ≤ n and ζ ∈ E(GLn−me (q), 1) is associated with a partition of n − me with no e-hook. In [ChR08] 7.18, Chuang-Rouquier proved the following. Theorem 17.1. For i = 1, 2, let Bi be the unipotent `-block of GLni (q) associated with (mi , ζi ). Assume m1 = m2 , then Hob (B1 ) ∼ = Hob (B2 ). This theorem is in fact an application of the study of the series of Grothendieck groups of modular representations of symmetric groups, through actions of Lie algebras sl2 on those categories - or “categorifications”. This approach proves a similar theorem as the above for blocks of symmetric groups (also conjectured by Brou´e in the form of a derived

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equivalence) and for cyclotomic Iwahori-Hecke algebras (see [ChR08] 7.2 and 7.12). 17.B. Alvis-Curtis duality as a self-equivalence Let us consider the BN-pair of G = GF (see Sec. 7.3 above). In particular we have the Weyl group (W, S) and the parabolic subgroups PI = UI o LI associated with subsets I ⊆ S. Keep (O, K, k) an `-modular system for G. Alvis-Curtis duality is the map X ∗ G DG := (−1)|I| RG LI ◦ RLI : K0 (KG) → K0 (KG). I⊆S

Alvis-Curtis showed that there is a bijection χ 7→ χ∗ from Irr(G) to itself, such that DG (χ) = ±χ∗ for any χ ∈ Irr(G) (see [DiMi] 8). Theorem 17.2. There is an equivalence ∼

Hob (OG) −→ Hob (OG) ∼

such that the induced equivalence K0 (KG) −→ K0 (KG) is Alvis-Curtis duality DG . A version of the above with derived categories instead of homotopic categories was first conjectured by Brou´e and proved by Cabanes-Rickard (2001, see [CaE] 4.19), the version with the homotopic categories is due to Okuyama (2006, see [Ca09]). The methods used there are fairly elementary, through a complex related with the coefficient system on subsets of S such that XI = OGeI ⊗PI eI OG and XI → XJ for I ⊆ J is the map sending x ⊗PI y to x ⊗PJ y. The independence Theorem 13.3 above is used in a crucial way. 18. Decomposition Numbers Given the better knowledge we have of ordinary characters of our groups, it is reasonable to expect information on modular characters from them. The coefficients expressing this are the so-called decomposition numbers. 18.A. Decomposition matrices Let us recall the notion of a decomposition matrix, relating Brauer’s modular characters with the ordinary characters. Let G be a finite group, let ` be a prime and (O, K, k) be an `-modular system for G (see Sec. 2 above). The decomposition matrix Dec(OG) =

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(dχφ )χ,φ of G in characteristic ` is defined as a matrix whose rows are indexed by Irr(G) and columns by IBr(G) and the element dχ,φ ∈ O is defined by X χ(g) = dχ,φ φ(g) φ∈IBr(G)

for any `-regular element g of G.

  χ···   

S .. . dχ,S .. . .. .

  ···   = Dec(OG).  

Each row may be seen as representing the multiplicities of simple kGmodules in the reduction modulo J(O) of an OG-lattice inside a fixed simple KG-module of character χ ∈ Irr(G). Each column can be seen as PS ⊗O K for PS the projective cover of a simple kG-module S. This could of course be defined in the same fashion for any O-algebra, O-free of finite rank, but a group algebra OG has the property that its decomposition matrix with regard to ` has always more rows than column when ` divides the order of G. From those definitions it is not difficult to see that the partitions of Irr(G) and IBr(G) induced by blocks of OG allow to write the decomposition matrix “diagonally” (note that the matrices are not square) as   Dec(B1 )   Dec(B2 ) Dec(OG) =  . .. . In other words dχ,φ = 0 when χ ∈ Irr(G) and φ ∈ IBr(G) select distinct blocks of OG. One also sees easily that if two blocks B of OG, resp. B 0 of OG0 , are Morita equivalent B ∼M B 0 , then we have induced Morita equivalences B ⊗O K ∼M B 0 ⊗O K and B ⊗O k ∼M B 0 ⊗O k, hence bijections between simple modules Irr(B) → Irr(B 0 ) and IBr(B) → IBr(B 0 ), yielding equality Dec(B) = Dec(B 0 ) between decomposition matrices.

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18.B. Triangular decomposition matrices: Symmetric groups The work on decomposition matrices of finite reductive groups was initiated by Dipper for general linear groups (see [Dip85a] and [Dip85b]). The model is the corresponding older theorem on symmetric groups. We have (see [JaKe] 6.3.60): Theorem 18.1. There are orderings of the rowsand columns of the decom D1 position matrix of Sn such that Dec(OSn ) = where D1 is square D2 lower unitriangular. 18.C. Triangular decomposition matrices: Finite reductive groups Concerning now finite reductive groups in non-defining characteristics, Theorem 16.5 and the above generalities imply that we can somehow reduce the study of blocks to the study of the sum of unipotent blocks B` (GF , 1) (see Theorem 16.2). Let us denote by Decuni the corresponding part of the `-decomposition matrix. Building on many examples we have the following conjecture (Geck, see [GeHi97] 3.4). Conjecture Under mild conditions on the prime ` 6= p, one has  18.2.  D1 Decuni = where D1 is square lower unitriangular. D2 Note that a consequence of the unitriangularity of Dec(G), once the subset of Irr(G) corresponding to D1 has been chosen (a so-called basic set), then we have a unique injection IBr(G) ,→ Irr(G). The following is due to Geck-Hiss [GeHi91] (see also [CaE] 14.4) and somehow relates with Proposition 16.4 above. Theorem 18.3. If Z(G) is connected and ` ≥ 7, then putting first the lines of Decuni corresponding to unipotent characters, one gets a square submatrix D1 with determinant ±1. In other words E(GF , 1) has same cardinality as the set of Brauer characters of B` (GF , 1) and their restrictions to `-regular classes generate the same space of class functions. Apart form Dipper’s work ([Dip85a] and [Dip85b]), Geck’s conjecture was proved for any prime ` 6= p by Geck for unitary groups, by GeckHiss-Miyachi for type E6 , by K¨ohler for type F4 , by Hiss-Shamash for type G2 . When GF is a classical group and the prime ` is linear (i.e. a Sylow

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`-subgroup of GF is included in a Levi subgroup of type A), then GruberHiss have proved the unitriangularity property of Geck’s conjecture (see [GrHi97]). Note that, combined with the fact that `-solvable groups also satisfy unitriangularity of the whole decomposition matrix (with D1 an identity matrix, by a theorem of Fong-Swan, 1961), this implies that actually many finite groups satisfy this statement for almost all primes. 18.D. The q-Schur algebra The strategy followed by Dipper and formalized in subsequent work by Dipper-James consists in choosing certain projective OGF -modules of type RG L (ΓL ) where L is a split Levi subgroup whose root system has only types A and ΓL is a so-called Gelfand-Graev representation of LF . Gelfand-Greav representations of GF are induced from one-dimensional representations of the maximal unipotent subgroup UF of GF (hence their projectivity), see [DeLu76] 10 or [DiMi] 14.29. This leads in turn to study the Iwahori-Hecke algebra HO (Sn , q) of type A and parameter q over O, and certain ideals xλ HO (Sn , q) defined as follows. Definition 18.4. For λ = (λ1 ≥ λ2 ≥ · · · ≥ λk > 0) ` n, let Sλ ∼ = Sλ1 × · · · × Sλk the corresponding parabolic subgroup of Sn and xλ = P l(w) Aw ∈ H := HO (Sn , q) in the notations of Theorem 4.2 w∈Sλ (−q) above. The q-Schur algebra of Sn and parameter q ∈ O× is Y SO (n, q) := EndH ( xλ H). λ`n

Dipper-James have shown that SO (n, q) ⊗O K and SO (n, q) ⊗O k have the same number of simple modules, hence a square decomposition matrix DS , which is also proved to be unitriangular (see [DipJam89]). Their proof of Geck’s conjecture shows namely that the part of Decuni corresponding to the unipotent characters of GLn (q) is precisely the decomposition matrix of SO (n, q). 18.E. Genericity Several theorems show that for a given type of group GF , the decomposition matrix of the unipotent blocks Decuni is as generic as one can expect. For instance it is shown in [DipJam89] and [GrHi97] that the entries of Decuni are bounded independently of q and ` for GF a group as considered there. A conjecture by G. James is about the case of GF = GLn (q). Recall that e(q, `) denotes the multiplicative order of q modulo `. James’ con-

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jecture asserts that if e(q, `)` < n, then Decuni only depends on e(q, `). This was proved for n ≤ 10 by James [Ja90] but Williamson’s recent work on Lusztig’s conjecture implies that counter-examples to James’ statement exist (for quite large but explicit n, see [W13] Sect. 6). 19. Some Harish Chandra Series 19.A. The case of GLn (q) We describe here the partition of IBr(B` (GF , 1)) in terms of Harish Chandra series for GF = GLn (q). In what follows, denote G := GLn (q). Recall the set E(GF , 1) in bijection with partitions of n by the bijection Irr(Sn ) → E(GF , 1), ρ 7→ χρ := ±

1 X ρ(w)RG Tw (1) n! w∈Sn

(see Example 15.2 above). We choose an actual parametrization of Irr(Sn ) n by partitions of n, λ 7→ [λ] such that [λ] is present in both IndS Sλ∗ 1 and Sn IndSλ  where λ∗ is the transpose partition of λ and  is the signature character — note that this is the transpose of the convention in [JaKe] Sect. 2. We then get a parametrization λ 7→ χ[λ] of E(G, 1) by the set Pn of partitions of n. The unitriangularity of the decomposition matrix of B` (G, 1) then implies that we get in turn a parametrization λ 7→ φλ of IBr(B` (G, 1)) by Pn . Assume then that λ ` n. Theorem 19.1. φλ represents a cuspidal kG-module if and only if n = e(q, `)`a for an integer a ≥ 0 and λ = (n). In that case φ(n) is the reduction F modulo ` of some ±RG T (θ) where θ = θ` is a regular character of T (the stabilizer of θ in NG (T)F is TF ) of multiplicative order a power of ` and TF is cyclic of order q n − 1. The following description of Harish Chandra series (see 13.B above) is due to Dipper-Du (see [DipDu93], [CaE] 19.20). We abbreviate e(q, `) = e. Let us decompose each λi in λ = (λ1 ≥ P (−1) (j) (−1) λ2 ≥ · · · ) ` n as λi = λi + e( j≥0 `j λi ) with 0 ≤ λi < e, and P (j) (j) 0 ≤ λi < ` for any j ≥ 0. Denoting mj = i λi for any j ≥ −1, we associate now λ := (1m−1 , (e(m0 ) , . . . , (e`j )mj ) ` n (the partition with 1 repeated m−1 times and e`j repeated mj times).

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Theorem 19.2. Let GLn (q) ⊇ Lλ ∼ = GL1 (q)m−1 ×Πj≥0 GLe`j (q)mj a standard Levi subgroup endowed with the cuspidal Brauer character ρ defined in Theorem 19.1 for each component. Then φλ is in the Harish Chandra series associated with the cuspidal pair (Lλ , ρ). 19.B. Other types For other finite reductive groups GF , less complete results are known regarding indexation of Irr(kGF ) - a problem related with triangularity of the decomposition matrix in characteristic ` - and Harish Chandra series over the field k of characteristic `. See [GeHiMa96] and its references for the state of the art at the time of that paper. Most modules involved are produced by inducing Gelfand Graev representations of Levi subgroups, or their generalization due to Kawanaka. Some new methods were used recently by Dudas to produce other modules using `-adic cohomology (see [Du13]), thus giving decomposition numbers unknown until then, especially for groups of small rank. 19.C. Supercuspidality We have defined in Definition 13.6 above the notion of supercuspidality, a strengthening of the notion of cuspidality. This leaves open the question of existence and disjointness of series modeled on the case of cuspidality (see (13.7) above). A simple kGF -module M would define a pair (L, ρ) where L is a standard Levi subgroup of GF , ρ is a supercuspidal element of Irr(kL) and [RG L ρ, M ] 6= {0}. The existence of such a pair was seen in Remark 13.10, its uniqueness (i.e. that the union in Remark 13.10 is disjoint) is less clear. However, we have Theorem 19.3. (Hiss [Hi96]) Uniqueness above is ensured for G = GLn (q). The proof goes roughly as follows. Let us recall first the relation between Lusztig series and Harish Chandra series of ordinary characters in this case of GF = GLn (q). If s ∈ G∗ F = GLn (q), there is a single GF -class of pairs (Ls , ζs ) such that Ls is a split Levi subgroup of an F -stable parabolic of G, G F ζs ∈ Irr(LF s ) is cuspidal and RLs ζ has components in E(G , s). Indeed, Ls is ∗ ∗ dual to the smallest split Levi subgroup Ls of G containing the centralizer of s (which implies that s is regular in L∗ ) and ζs ∈ E(LF s , s) is the character corresponding with the trivial character of the torus CL∗s (s)F (of type the

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Coxeter element of L∗s ) through Jordan decomposition of characters (see Theorem 15.3). Now, if (LF , ρ) is a supercuspidal pair as mentioned before and RG Lρ intersects the sum of blocks B` (GF , s) for some `-regular semi-simple element of G∗ F (see Theorem 16.2 above), one shows that L = Ls and ρ is the restriction of ζs to p-regular elements. This proves the claimed uniqueness (disjunction) by the fact that RG L functor preserves series s (itself an easy consequence of the transitivity of those functors). First ρ belongs to the sum of blocks B` (LF , s0 ) for some `-regular semi-simple s0 ∈ L∗F . Since ρ is cuspidal, by the classification recalled in Theorem 19.1 above (and suitably generalized to arbitrary series, see [Dip85b] 3.5) its Brauer character is the restriction to `-regular elements of some ±RL T θ where θ corresponds by duality to some semi-simple element s0 t ∈ T∗F ≤ L∗F with t = (s0 t)` and s0 t is regular in L∗ . On the other hand the projective cover Pρ , seen as an OLF -module, has a charG acter in ZE` (LF , s0 ). By the preservation of series by RG L functors, RL Pρ is in the sum of blocks B` (GF , s), so we can assume s0 = s. Moreover, thanks to Theorem 16.4 above, the character of Pρ has non-zero projection on E(LF , s). By supercuspidality, those components have to be cuspidal as ordinary characters. Then E(LF , s) contains a cuspidal ordinary character, which readily implies that L = Ls in the notation above and ζs is a com0 0 ponent of the character of Pρ . On the other hand ζs = ±RL T θ where θ 0 correspond with s by duality. Now (st)`0 = s implies θ`0 = θ as a multiplicative character of TF . Then ±RL T θ has same restriction to `-regular elements of LF as ζs and this gives our claim about ρ. In [H96], all supercuspidal pairs (L, ρ) with ρ in the “unipotent block” B` (L, 1) are determined in groups GF of classical type and under mild restrictions in exceptional types. Acknowledgments This survey is based on four lectures given at IMS of National University of Singapore in April 2013. The author thanks Professors Wee Teck Gan and Kai Meng Tan for the invitation and the kind atmosphere during the whole stay. References [AJS94]

H. H. Andersen, J. C. Jantzen, and W. Soergel. Representations of quantum groups at a p-th root of unity and of semisimple groups in

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characteristic p: independence of p. Ast´erisque, 220, 1994. D. Benson. Representations and Cohomology I: Basic Representation Theory of finite Groups and associative Algebras, Cambridge, 1991. [Bn] C. Bonnaf´e. Representations of SL2 (Fq ). Springer, 2011. [BnMi11] C. Bonnaf´e and J. Michel. Computational proof of the Mackey formula for q > 2, J. Algebra, 327 (2011), 506–526. [BnRo03] C. Bonnaf´e and R. Rouquier. Cat´egories d´eriv´ees et vari´et´es de ´ Deligne-Lusztig, Publ. Math. Inst. Hautes Etudes Sci., 97 (2003), 1–59. [Bo] A. Borel. Linear algebraic groups. Springer, 1991. [Bou] N. Bourbaki. Groupes et alg`ebres de Lie, IV, V, VI. Masson, 1968. [Br90] M. Brou´e. Isom´etries parfaites, types de blocs, cat´egories d´eriv´ees. Ast´erisque, 181–182 (1990), 61–92. [BrM92] M. Brou´e and G. Malle. Th´eor`emes de Sylow g´en´eriques pour les groupes r´eductifs sur les corps finis. Math. Ann., 292 (1992), 241–262. [BrMaM93] M. Brou´e, G. Malle, and J. Michel. Generic blocks of finite reductive groups. Ast´erisque, 212 (1993), 7–92. [BrMi89] M. Brou´e and J. Michel. Blocs et s´eries de Lusztig dans un groupe r´eductif fini, J. reine angew. Math., 395 (1989), 56–67. [Ca09] M. Cabanes. On Okuyama’s theorems about Alvis-Curtis duality. Nagoya Math. J. 195 (2009), 1–19. [CaE99] M. Cabanes and M. Enguehard. On blocks of finite reductive groups and twisted induction. Adv. Math. 145 (1999), 189–229. [CaE] M. Cabanes and M. Enguehard. Representation theory of finite reductive groups, Cambridge, 2004. [Cart1] R. W. Carter. Simple groups of Lie type. Wiley, 1972. [Cart2] R. W. Carter. Finite groups of Lie type, Conjugacy classes and complex characters. Wiley, 1985. [CartL76] R.W. Carter and G. Lusztig. Modular representations of finite groups of Lie type. Proc. London Math. Soc. 32 (1976), 347–384. [Ch55] C. Chevalley. Sur certains groupes simples, Tohoku Math. J., (2) 7, (1955), 1–66. [ChR08] J. Chuang and R. Rouquier. Derived equivalences for symmetric groups and sl2 -categorification, Ann. of Math., 167 (2008), 245–298. [Cr12] D. Craven. Perverse equivalences and Brou´e’s conjecture II: The cyclic case. Preprint, (2012). [DeLu76] P. Deligne and G. Lusztig. Representations of reductive groups over finite fields, Ann. of Math., 103 (1976), 103–161. [DiMi] F. Digne and J. Michel. Representations of finite groups of Lie type, Cambridge, 1991. [Dip85a] R. Dipper. On the decomposition numbers of the finite general linear groups, Trans. Amer. Soc., 290 (1985), 315–344. [Dip85b] R. Dipper. On the decomposition numbers of the finite general linear groups II, Trans. Amer. Soc., 292 (1985), 123–133. [DipDu93] R. Dipper and J. Du. Harish-Chandra vertices, J. reine angew. Math., 437 (1993), 101–130. [DipJam89] R. Dipper and G. James. The q-Schur algebra, Proc. London Math.

[Be]

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Soc., 59 (1989), 23–50. O. Dudas. A note on decomposition numbers for groups of Lie type of small rank, J. Algebra, 388 (2013), 364–373. [EW12] B. Elias and G. Williamson. The Hodge theory of Soergel bimodules. preprint, ArXiv:1212.0791. [F12] P. Fiebig. An upper bound on the exceptional characteristics for Lusztig’s character formula. J. reine angew. Math., 673 (2012), 1–31. [FoSe74] P. Fong and G.M. Seitz. Groups with a (B,N)-pair of rank 2, I. Invent. Math., 21 (1973), 1–57; II, ibid., 24 (1974), 191–239. [FoSr82] P. Fong and B. Srinivasan. The blocks of finite general and unitary groups, Invent. Math., 69 (1982), 109–153. [FoSr86] P. Fong and B. Srinivasan. Generalized Harish-Chandra theory for unipotent characters of finite classical groups, J. of Algebra, 104 (1986), 301–309. [FoSr89] P. Fong and B. Srinivasan. The blocks of finite classical groups, J. reine angew. Math., 396 (1989), 122–191. [FoSr90] P. Fong and B. Srinivasan. Brauer trees in classical groups, J. Algebra, 131 (1990), 179–225. [GeHi91] M. Geck and G. Hiss. Basic sets of Brauer characters of finite groups of Lie type. J. reine angew. Math., 418 (1991), 173–188. [GeHi97] M. Geck and G. Hiss. Modular representations of finite groups of Lie type in non-defining characteristic, in Finite reductive groups, (M. Cabanes ed.), Prog. Math., 141 (1997), Birkh¨ auser, pp 195–249. [GeHiMa94] M. Geck, G. Hiss and G. Malle. Cuspidal unipotent Brauer characters, J. Algebra, 168 (1994), 182–220. [GeHiMa96] M. Geck, G. Hiss and G. Malle. Towards a classification of the irreducible representations in non-describing characteristic of a finite group of Lie type, Math. Z., 221 (1996), 353–386. [GeMa] M. Geck and G. Malle. Reflection groups. (Handbook of algebra. Vol. 4, 337–383 ) Elsevier/North-Holland, 2006. [GePf] M. Geck and G. Pfeiffer. Characters of finite Coxeter groups and Iwahori-Hecke algebras. Oxford, 2000. [GLS] D. Gorenstein, R. Lyons, and R. Solomon. The classification of the finite simple groups. Number 3. Part I. Chapter A. AMS, 1998. [GraL96] J.J. Graham and G.I. Lehrer. Cellular algebras. Invent. Math. 123 (1996), 1–34. [Gre78] J.A. Green. On a theorem of Sawada, J. London Math. Soc., 18 (1978), 247–252. [GrHi97] J. Gruber and G. Hiss. Decomposition numbers of finite classical groups for linear primes, J. reine angew. Math., 485 (1997), 55–91. [GyUn89] A. Gyoja and K. Uno. On the semisimplicity of Hecke algebras. J. Math. Soc. Japan, 41-1 (1989), 75–79. [Ha85] M.E. Harris. On the p-deficiency class of a finite group. J. Algebra, 94 (1985), 411–424. [HeKaSe72] C. Hering, W. Kantor, and G.M. Seitz. Finite groups with a split BN-pair of rank 1, I. J. Algebra, 20 (1972), 435–475. [Du13]

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G. Hiss. Supercuspidal representations of finite reductive groups. J. Algebra, 184 (1996), 839–851. [HowL80] B. Howlett and G. Lehrer. Induced cuspidal representations and generalized Hecke rings, Invent. Math., 58 (1980), 37–64. [HowL94] B. Howlett and G. Lehrer. On Harish-Chandra induction for modules of Levi subgroups, J. Algebra, 165 (1994), 172–183. [Hum1] J. E. Humphreys. Introduction to Lie algebras and representation theory. Springer, 1972. [Hum2] J. E. Humphreys. Modular representations of finite groups of Lie type, Cambridge, 2006. [Isa] I. M. Isaacs. Character theory of finite groups. Academic, 1976. [Ja90] G. James. The decomposition matrix of GLn (q) for n ≤ 10, Proc. London Math. Soc., 60 (1990), 225–265. [JaKe] G. James and A. Kerber. The representation theory of the symmetric group. Addison-Wesley, 1981. [Jn] J.C. Jantzen. Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107. AMS, 2003. [KL79] D. Kazhdan and G. Lusztig. Representations of Coxeter groups and Hecke algebras. Invent. Math., 53 (1979), 165–184. [KM13] R. Kessar and G. Malle. Quasi-isolated blocks and Brauer’s height zero conjecture. Ann. of Math. (2) 178 (2013), 321–384. [Lus76] G. Lusztig. Irreducible representations of finite classical groups, Invent. Math., 58 (1976), 37–64. [Lus80] G. Lusztig. Some problems in the representation theory of finite Chevalley groups, Proc. Sympos. Pure Math., 37 (1980), 313–317. [Lus84] G. Lusztig. Characters of reductive groups over a finite field, Ann. Math. Studies, 107, Princeton, 1984. [Lus03] G. Lusztig. Hecke algebras with unequal parameters, CRM Monograph Series 18, AMS, 2003. [MT] G. Malle, D. Testerman. Linear algebraic groups and finite groups of Lie type. Cambridge, 2011. [N] G. Navarro. Characters and blocks of finite groups. Cambridge, 1998. [Sch] P. Schneider. Modular representation theory of finite groups. Springer, 2013. [Spr] T. A. Springer. Linear algebraic groups, Birkh¨ auser, second edition, 1998. [Sri] B. Srinivasan. Representations of finite Chevalley groups. Springer Lecture Notes in pure Mathematics, Springer, 1979. [St63] R. Steinberg. Representations of algebraic groups. Nagoya Math. J., 22 (1963), 33–56. [W13] G. Williamson. Schubert calculus and torsion. (Sep 2013) arXiv: 1309.5055.

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`-MODULAR REPRESENTATIONS OF p-ADIC GROUPS (` 6= p)

Vincent S´echerre Universit´e de Versailles Saint-Quentin-en-Yvelines Laboratoire de Math´ematiques de Versailles ´ 45 avenue des Etats-Unis 78035 Versailles cedex, France [email protected]

In these notes, we give an overview of the representation theory of p-adic reductive groups with coefficients in fields of characteristic different from p. We emphasize the case of GL(n) and its inner forms.

Contents 1 Lecture 1

49

1.1 Notation and preliminaries

49

1.2 Parabolic functors

50

1.3 Cuspidal representations

51

2 Lecture 2

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2.1 Supercuspidal support

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2.2 Decomposition of RR (G)

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3 Lecture 3

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3.1 Reduction to the (super)unipotent case

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3.2 Classification of (super)unipotent representations, I

62

3.3 Classification of (super)unipotent representations, II

63

3.4 Comments

65

4 Lecture 4

66

4.1 Reduction mod `

66

4.2 The local Langlands correspondence References

68 69

47

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Introduction The representation theory of reductive p-adic groups with coefficients in the field of complex numbers has been developed since the 1960’s (see Cartier’s introduction [11]). It has inherited certain techniques coming from harmonic analysis on reductive groups over Archimedean fields, making a large use of the fact that the representations have complex coefficients (Harish-Chandra [16], Langlands’ classification [25]). Then in the 1980’s, Bernstein and Zelevinski developed a fully algebraic approach [2, 3, 4]. The arithmetic of modular forms has required to develop the representation theory of reductive p-adic groups with coefficients in fields — or even rings — other than complex numbers. Provided that p is invertible in the coefficient ring, a large part of Bernstein-Zelevinski’s algebraic approach can be reproduced (Vign´eras [27, 28]). In these lectures, I will assume that the coefficient ring is an algebraically closed field with characteristic different from p. For bibliographic references, see Vign´eras [27], Blondel [5]. In Lecture 1, I define parabolic induction and restriction, and the notions of cuspidal representation and cuspidal support. An important aspect of the theory of `-modular representations is that there is a difference between the two notions of cuspidal and supercuspidal representations (Example 1.11). This leads to the notion of supercuspidal support, then to the problem of classifying irreducible representations with a given supercuspidal support. In Lecture 2, I discuss the case of the group GLn and its inner forms. I explain how, thanks to the theory of types developed by Bushnell and Kutzko, one can prove the uniqueness of supercuspidal support for irreducible representations of these groups (Theorem 2.1). In Lecture 3, I present the classification of all irreducible representations of GLn (F) in terms of multisegments, generalizing Zelevinski’s classification of complex irreducible representations. First, the theory of types allows one to reduce to the classification of all unipotent irreducible representations. Then one defines a map: Z : m 7→ Z(m) that associates a unipotent irreducible representation to any multisegment. The definition of this map and the proof that it is injective relies on the theory of generic representations of GLn (F) (see also Remark 3.11 for inner forms). Proof of surjectivity requires a counting argument that relies on results of [1, 12] on the classification of simple modules over an affine Hecke algebra of type A at a root of unity.

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In Lecture 4, I introduce the operation of reducing mod ` an irreducible `-adic representation of G having a stable lattice. Then I present the properties of the map Z and of the local Langlands correspondence with respect to reduction mod `. 1. Lecture 1 1.1. Notation and preliminaries In all these lectures, we fix a locally compact non-Archimedean field F of residue characteristic denoted p; we write O for its ring of integers, p for the maximal ideal of O and q for the cardinality of its residue field. We also fix an algebraically closed field R of characteristic not dividing q. Let G be a connected reductive group defined over F, and let G = G(F) be the group of its F-points. When endowed with the topology coming from that of F, the group G is locally compact, and its neutral element 1 has a basis of neighborhoods made of compact open pro-p-subgroups (that is, all of whose open subgroups have index of the form pr , r > 0). Example 1.1: If G = GLn , then G is the group GLn (F). The identity matrix has a basis of neighborhoods made of the compact open pro-psubgroups Ki = 1 + Mn (pi ) for i > 1. Definition 1.2: A smooth R-representation of G is a pair (π, V) made of a vector space V over R together with a group homomorphism: π : G 7→ GL(V) such that, for all v ∈ V, there is a compact open subgroup of G fixing v. In these lectures, all representations will be smooth R-representations. Therefore we will often write representation for smooth R-representation. Given two smooth R-representations (π, V) and (σ, W) of G, a homomorphism from (π, V) to (σ, W) is an R-linear map f : V → W such that f ◦ π(g) = σ(g) ◦ f for all g ∈ G. The space of all such maps will be denoted HomG (π, σ). This defines the abelian category, denoted RR (G), of smooth R-representations of G. Definition 1.3: A smooth R-representation (π, V) of G is said to be admissible if, for any open subgroup H of G, the space VH of H-fixed vectors of V is finite-dimensional.

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A smooth R-character (or character for short) of G is a group homomorphism from G to R× with open kernel. Given a representation π and a character χ of G, we write πχ for the twisted representation g 7→ π(g)χ(g). A first very important fact is that one can define R-valued Haar measures on G (see [27], I.2). Such a measure is a nonzero R-linear form on: C∞ c (G, R), the space of compactly supported and locally constant R-valued functions on G, which is invariant under right translations (it is then automatically invariant under left translations). Such a measure is unique up to a nonzero scalar. To define a Haar measure, let us fix a compact open pro-p-subgroup H ⊆ G and define the measure of any compact open subgroup K of G by: meas(K) =

(K : K ∩ H) ∈ R, (H : K ∩ H)

which is well defined since the denominator is a p-power and p is invertible in R. We have here our first important difference between the complex theory (when R is the field C of complex numbers) and the modular theory (when R has characteristic ` > 0). Unlike the complex case, meas(K) may be 0 in the modular case. Smooth R-representations of K need not be semi-simple, and the functor V 7→ VK of K-fixed vectors need not be exact. 1.2. Parabolic functors Let P be a parabolic subgroup of G, together with a Levi decomposition P = MN defined over F, where N is the unipotent radical of P. Write P = P(F), M = M(F) and N = N(F). Attached to the parabolic subgroup P there is a complex character δP of M defined by: δP (m) = (mKm−1 : mKm−1 ∩ K)/(K : mKm−1 ∩ K) for all m ∈ M, where K is an arbitrary compact open pro-p-subgroup of N (the complex number δP (m) does not depend on the choice of K). These values are integer powers of q. Thus for m ∈ M, there is a v(m) ∈ Z such that δP (m) = q v(m) . Let us make a choice of a square root of q in R, denoted √ q, and write: p √ δP : m 7→ ( q)v(m) ∈ R× .

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Let (σ, W) be a smooth representation of M. Write iG M,P (W) for the space of locally constant functions f : G → W such that: p f (mng) = δP (m)σ(m)f (g), m ∈ M, n ∈ N, g ∈ G. The representation of G on this space by right translation is smooth, denoted iG M,P (σ) and called the parabolic induction of (σ, W) to G along P. The functor: iG M,P : RR (M) → RR (G) has a left adjoint r G M,P , the parabolic restriction functor from G to M along P. Given (π, V) a smooth representation of G, write V(N) for the subspace of V spanned by π(n)v − v for all n ∈ N and v ∈ V. Then r G M,P (π) is the natural representation of M on the quotient V/V(N) twisted by the inverse √ of the character δP . Remark 1.4: G G (1) The functors iG M,P , r M,P are exact, r M,P preserves the property of being of finite type, and iG M,P preserves admissibility (see [27], II.2.1). G (2) More difficult: the functors iG M,P and r M,P both preserve the property of having finite length ([27], II.5.13).

We have the following theorem (see [27], II.2.18 for more details), known as the Geometric Lemma. Theorem 1.5: Given parabolic subgroups P = MN and Q = LU of G, G there is a formula describing the functor r G L,Q ◦ iM,P . If R is the field of complex numbers, the functor iG M,P has a right adjoint, − which is r G with P the parabolic subgroup of G opposite to P with − M,P respect to M (this is known as the second adjointness property; see [7]). In the modular case, this is not known in general, but partial results can be found in [13]. However, there is a version for admissible representations (see [27], II.3.8). 1.3. Cuspidal representations Definition 1.6: A representation (π, V) of G is cuspidal if the following equivalent conditions are satisfied: (1) The space r G M,P (V) is zero for all proper parabolic subgroups P = MN ( G.

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(2) The space HomG (V, iG M,P (W)) is zero for all smooth R-representations (σ, W) of M and all proper parabolic subgroups P = MN ( G. Theorem 1.7: (See [27, 22]) Given (π, V) an irreducible representation of G, there are a parabolic subgroup P = MN ⊆ G and an irreducible cuspidal representation (σ, W) of M such that π embeds in iG M,P (σ). Moreover, the cuspidal pair (M, σ) is unique up to G-conjugacy. Definition 1.8: The G-conjugacy class of (M, σ) is called the cuspidal support of (π, V), denoted cusp(π, V). √ The cuspidal support depends on the choice of q ∈ R× that we have made. Problem 1: Classify all irreducible representations of G having given cuspidal support. There ia another characterization of cuspidality. Write Z for the centre of G. Proposition 1.9: An irreducible representation (π, V) is cuspidal if and only if, for all v ∈ V and all smooth linear form ξ : V → R, the function g 7→ ξ(π(g)v) has support whose image in G/Z is compact. Corollary 1.10: (See [27]) All irreducible representation of G are admissible and have a central character. When R = C, Proposition 1.9 is crucial. It is one of the key properties used for the Bernstein decomposition of the category RC (G) into blocks with respect to the notion of (inertial) cuspidal support. This is related to the fact that an irreducible cuspidal representation of G with central character ω is projective in the full subcategory of RC (G) made of smooth R-representations having central character ω. As observed by Vign´eras, this is no longer true in the modular case, since irreducible cuspidal representations of G may occur as subquotients of proper parabolically induced representations. Example 1.11: Assume G = GL2 (Q5 ) and ` = 3 (where Qp is the field of p-adic numbers). Let B be the subgroup of upper triangular matrices. The representation of G on the space V of R-valued locally constant functions on B\G is indecomposable and has length 3. Its unique irreducible subrepresentation is the trivial character of G, and its unique irreducible quotient is g 7→ |det g|, where x 7→ |x| denotes the absolute value of F giving value

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q −1 to any uniformizer. The remaining (infinite-dimensional) subquotient is cuspidal. Definition 1.12: An irreducible representation of G is supercuspidal if for all proper P = MN and all irreducible representation (σ, W) of M, it does not occur as a subquotient of iG M,P (σ). All supercuspidal representations of G are cuspidal, but the converse need not be true (see Example 1.11). Remark 1.13: When R = C, any cuspidal representation is supercuspidal. Remark 1.14: (1) Assume (π, V) is a supercuspidal irreducible representation of G having a projective cover Pπ of finite type. Then Pπ is cuspidal (by Frobenius reciprocity). This implies that π does not occur as a subquotient of iG M,P (σ) for any smooth (σ, W). (2) It is not known in general whether or not a supercuspidal irreducible representation of G has a projective cover of finite type. This is known for G = GLn (F), n > 1 ([14]). Proposition 1.15: For all irreducible representation (π, V) of G, there are a parabolic subgroup P = MN and a supercuspidal irreducible representation (σ, W) of M such that π occurs as a subquotient of iG M,P (σ). Let us denote by scusp(π, V) the set of all possible such pairs (M, σ), called the supercuspidal support of (π, V). Problem 2: Given an irreducible representation (π, V) of G, is scusp(π, V) made of a single G-conjugacy class? The answer to Problem 2 is known only for the groups GLn (F), n > 1 and their inner forms (see Lecture 2). See also [19] for the unitary group U(2, 1) with respect to an unramified quadratic extension. 2. Lecture 2 From now on, we will assume that G is GLn , n > 1 or possibly one of its inner form, so that G is of the form GLm (D) with D a central division F-algebra of degree d2 , with n = md. Let α = (m1 , . . . , mr ) be a family of positive integers of sum m. Denote by Mα the subgroup of GLm (D) of invertible matrices which are diagonal

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by blocks of size m1 , . . . , mr respectively (it is isomorphic to GLm1 (D) × · · · × GLmr (D)) and by Pα the subgroup of GLm (D) generated by Mα and the upper triangular matrices. Write iα for the functor of parabolic induction associated with Mα and Pα , and r α for its left adjoint. If π1 , . . . , πr are smooth representations of GLm1 (D), . . . , GLmr (D) respectively, write: π1 × π2 × · · · × πr = iα (π1 ⊗ π2 ⊗ · · · ⊗ πr ). If (M, σ) is a cuspidal pair, then up to conjugacy we have M = Mα for some α as above and σ has the form σ1 ⊗ · · · ⊗ σr where σi is a cuspidal irreducible representation of GLmi (D). Therefore the GLm (D)-conjugacy class of (M, σ) will be identified with the formal sum σ1 + · · · + σr . 2.1. Supercuspidal support Theorem 2.1: (See [28, 22]) Assume G is GLn (F) or one of its inner forms. For all irreducible representations (π, V) of G, the set scusp(π, V) is a single G-conjugacy class. We need to introduce Bushnell-Kutzko’s theory of types [9]. This is a monumental machinery, initially developed by Bushnell and Kutzko in order to prove that any complex irreducible cuspidal representation of GLn (F) is compactly induced from an irreducible representation of a compact mod centre, open subgroup of GLn (F). More precisely: Theorem 2.2: (See [9, 27, 23]) Assume G is GLn (F) or one of its inner forms. Then there is a family of pairs (J, λ) made of a compact open subgroup J ⊆ G and an irreducible representation λ of J with the following properties: (1) For any irreducible cuspidal representation ρ of G, there is a pair (J, λ), unique up to G-conjugacy, such that λ embeds in the restriction of ρ to J. (2) If ρ is an irreducible representation of G containing a pair (J, λ), then it is cuspidal. (3) If two irreducible cuspidal representations ρ, ρ0 both contain (J, λ), then there is an unramified character χ : G → R× (that is, trivial on all compact subgroups of G) such that ρ0 ' ρχ. (4) Given (J, λ), any irreducible subquotient of indG J (λ), the compact induction of λ to G, is isomorphic to a quotient of indG J (λ).

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(5) Given (J, λ), the representation λ extends to its G-normalizer b J; comb pact induction from J to G induces a bijection between the set of representations of b J extending λ and that of isomorphism classes of irreducible cuspidal representations of G containing λ. Example 2.3: Write GLn (q) for the group of invertible n × n matrices with entries in the residue field of F, and let σ be an irreducible cuspidal representation of GLn (q). Inflate σ into an irreducible representation of K = GLn (O) that is trivial on K1 = 1 + Mn (p), still denoted σ. Then the pairs of the form (K, σ) obtained this way fullfil all the properties 1 to 5 for irreducible cuspidal representations of G = GLn (F) having nonzero K1 -fixed vectors (such irreducible representations are said to have level 0). Moreover, if σ b is a representation of KZ (where Z is the centre of G) extending σ and if ρ is the representation of G compactly induced form σ b, then the space ρK1 of K1 -fixed vectors of ρ – which is naturally a representation of K/K1 ' GLn (q) – is isomorphic to σ. Moreover, if ρ is an irreducible representation of G containing (K, σ), then it is supercuspidal if and only if σ is supercuspidal as a representation of GLn (q). Example 2.4: We give a positive level example for G = GLn (F). Let us fix a uniformizer $ of F, a character ψ : F → R× trivial on p but not on O, and a character ω : F× → R× trivial on 1 + p. Define a compact open subgroup:   p O ··· O  .. . . . . ..  . . . .   I1 = 1 +  .  ..  .. . O p ··· ··· p of G. Given t ∈ O× , define a character θ = θt of I1 by: θ(1 + x) = ψ(x1,2 + · · · + xn−1,n + t$−1 xn,1 ),

1 + x ∈ I1 .

Write J = O× I1 and let λ = λt be the character of J defined by λ(xg) = ω(x)θ(g) for all x ∈ O× and g ∈ I1 . The F-algebra E generated by:   0 t · idn−1 ∈ Mn (F) β= $ 0 is a totally ramified extension of degree n and uniformizer β. The group E× normalizes the pair (J, λ), and the G-normalizer of the latter is b J = E× J.

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Given z ∈ R× , there is a unique character z λ = z λt of b J extending λ such that: z λ(β)

=z

and its compact induction z ρ = z ρt is a supercuspidal irreducible representation (of level 1/n) of G. It is obtained from 1 ρ by twisting by the unramified character g 7→ z val(det(g)) , where val denotes the valuation of F giving value 1 to any uniformizer. Two pairs (J, λt ), (J, λu ) with t, u ∈ O× are G-conjugate if and only if −1 tu ∈ 1 + p. Remark 2.5: This construction shows that, given ` and n > 1, there is an irreducible mod ` supercuspidal representation of GLn (F). Note that it may happen that all level 0 mod ` cuspidal representations of GLn (F) are non-supercuspidal (this is the case, for instance, if n = q = 2 and ` = 3). The pairs (J, λ) appearing in Theorem 2.2 are called the maximal simple types of G. The proof of Theorem 2.1 requires a larger family of types, called the semisimple types of G (see [10, 23] for a precise definition). We will only give the following crucial fact about these types. Let α = (n1 , . . . , nr ) be a family of positive integers of sum n. For i ∈ {1, . . . , r}, let (Ji , λi ) be a maximal simple type of GLni (F). Then there exists a semisimple type (J, λ) of G = GLn (F) such that the compact induction indG J (λ) is isomorphic to the parabolic induction: GLn1 (F)

indJ1

GLnr (F)

(λ1 ) × · · · × indJr

(λr ).

(2.1)

For instance, if m1 = · · · = mr = 1 and if λi is the trivial character of Ji = O× for all i, then one can choose for λ the trivial character of the standard Iwahori subgroup I of G. Let us sketch the proof of Theorem 2.1 for the group G = GLn (F). Let (π, V) be an irreducible representation of G. Step 1. — We first reduce to the case where π is cuspidal, by applying an appropriate parabolic restriction functor r α (such that r α (π) is cuspidal) and by using the Geometric Lemma 1.5. Step 2. — We prove uniqueness up to inertia: there are a unique positive integer r dividing n and an irreducible supercuspidal representation ρ of GLn/r (F) such that any pair in scusp(π, V) is G-conjugate to a pair of the form: (GLn/r (F) × · · · × GLn/r (F), ρχ1 ⊗ · · · ⊗ ρχr )

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where the χi ’s are unramified characters of GLn/r (F). This step requires type theory and uniqueness of supercuspidal support for irreducible representations of the groups GLm (q u ), where mu divides n. Step 3. — We finally prove that any pair in scusp(π, V) is G-conjugate to a pair of the form: (GLn/r (F) × · · · × GLn/r (F), ρ ⊗ ρ|det | ⊗ · · · ⊗ ρ|det |r−1 ) where x 7→ |x| denotes the absolute value of F (normalized so that |$| = q −1 ) and ρ is uniquely determined up to a twist by some power of | det | (note that ρ|det |r is isomorphic to ρ). This step requires the theory of Whittaker models [27] , which allows one to prove the following crucial fact: if χ1 , . . . , χr are unramified characters of GLn/r (F) such that ρχ1 ×· · ·×ρχr has a cuspidal subquotient, there is an i ∈ {1, . . . , r} such that, for all k ∈ Z, there is a j ∈ {1, . . . , r} such that ρχi | det |k ' ρχj (see [22], 8.2). This proof uses strong results that are known so far for GLn (F) and its inner forms only. They are of two kinds: (a) results from the theory of types, and (b) results from the representation theory of finite reductive groups. We give more details on Step 2 in the case where π is a level 0 cuspidal representation. We first introduce a useful tool. Given a smooth R-representation (σ, W) of G, let us form the space WK1 of K1 -fixed vectors of W. Since K1 is normal in K = GLn (O), there is a representation of K on WK1 , and K1 acts trivially. This gives us a representation of the quotient K/K1 , which naturally identifies with GLn (q). This defines an exact functor from RR (G) to the category of R-representations of GLn (q). Let P = MN be a standard parabolic subgroup (that is, containing all upper triangular matrices) of G. We write M(q) for the standard Levi subgroup of GLn (q) that corresponds to M. By restricting functions from G to K, one can see from the Iwasawa decomposition G = PK that the functor W 7→ WK1 transforms the parabolic induction functor iG M,P into the Harish-Chandra induction functor from M(q) to GLn (q), denoted R. Assume that π occurs as an irreducible subquotient of ρ1 × · · · × ρr where ρi is an irreducible supercuspidal representation of GLni (F), and with n1 + · · · + nr = n. Then π K1 , which is nonzero, is a subquotient of:
K (n ) 1 (nr ) (ρ1 × · · · × ρr )K1 ' R ρ1 1 1 ⊗ · · · ⊗ ρK r where K1 (n) stands for the K1 -group of GLn (F), n > 1. This implies that

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all the ρi ’s have level 0. Following Example 2.3, write: GLn (F)

σ ), π ' indG KF× (b

ρi ' indK(nii)F× (b σi ), i ∈ {1, 2, . . . , r},

where K(n) = GLn (O), n > 1. By taking the K1 -fixed vectors, we get that σ is a subquotient of the Harish-Chandra induction of σ1 ⊗ · · · ⊗ σr . This implies (see Marc Cabanes’s lectures) that σ1 ' σ2 ' · · · ' σr (denoted σ0 ) and n1 = n2 = · · · = nr (denoted n0 ). We get the result by choosing r = n/n0 and ρ = ρ1 . Remark 2.6: The level 0 case is easy because it is the smallest possible level for an irreducible representation, and because the compatibility of the functor of K1 -fixed vectors with parabolic induction follows from the Iwasawa decomposition. The positive level case is more difficult, and requires the use of endo-classes [8, 6]. 2.2. Decomposition of RR (G) A (super)cuspidal pair of G is a pair (M, ρ) made of a Levi subgroup M ⊆ G and an irreducible (super)cuspidal representation ρ of M. Definition 2.7: Two cuspidal pairs (M, ρ) and (M0 , ρ0 ) in G are inertially equivalent if there is an unramified character χ of M such that (M0 , ρ0 ) is G-conjugate to (M, ρχ). Let S(G) denote the set of all inertial classes of supercuspidal pairs of G. Let us fix Ω ∈ S(G) and choose (M, ρ) ∈ Ω with: M = GLn1 (F) × · · · × GLnr (F), ρ = ρ1 ⊗ · · · ⊗ ρr , where ρi is a supercuspidal irreducible representation of GLni (F). For each i = 1, . . . , r choose a pair (Ji , λi ) for ρi as in Theorem 2.2 and write: GLn1 (F)

U (Ω) = indJ1

GLnr (F)

(λ1 ) × · · · × indJr

(λr ).

According to (2.1), this representation can be described as the compact induction of a semisimple type. For instance, if n1 = · · · = nr = 1 and all ρi are the trivial character of F× , then U (Ω) is the compact induction indG I (1) of the trivial character of an Iwahori subgroup I of G.

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Theorem 2.8: (See [24]) (1) For all irreducible representations (π, V) of G and all Ω ∈ S(G), one has: scusp(π, V) ∈ Ω

⇔

π is a subquotient of U (Ω).

(2) For all smooth representations (π, V) of G and all Ω ∈ S(G), let V(Ω) denote the maximal subrepresentation of V all of whose irreducible subquotients have supercuspidal support in Ω. Then: M V= V(Ω). Ω∈S(G)

(3) If (π, V),(σ, W) are smooth representations of G, then HomG (V, W) decomposes canonically as the product of all HomG (V(Ω), W(Ω))’s for Ω ∈ S(G). (4) The full subcategory RR (Ω) made of all smooth representations (π, V) of G such that V = V(Ω) is indecomposable. The strategy of the proof is very different from Bernstein’s proof for complex representations. It uses type theory as well as a decomposition theorem with respect to the supercuspidal support for representations of GLn (q). Example 2.9: Let (π, V) be a smooth level zero R-representation of G, that is, V is generated by VK1 . As a representation of GLn (q), VK1 decomposes as a direct sum: M VK1 ([L, σ]) [L,σ]

where [L, σ] ranges over all possible supercuspidal supports of GLn (q) and where VK1 ([L, σ]) is the maximal subrepresentation of VK1 all of whose irreducible subquotients have supercuspidal support [L, σ]. Write V[L, σ] for the subrepresentation of V generated by VK1 ([L, σ]). Then V decomposes as the direct sum of the V[L, σ]’s. Now write: L = GLn1 (q) × · · · × GLnr (q), σ = σ1 ⊗ · · · ⊗ σr , where σi is a supercuspidal irreducible representation of GLni (q). For each i = 1, . . . , r, inflate σi to an irreducible representation of Ji = K(ni ) = GLni (O), still denoted σi . By Example 2.3, the pair (Ji , σi ) is a level 0 maximal simple type. Choose a supercuspidal irreducible representation ρi

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of GLni (F) containing the pair (Ji , σi ). The representation ρ = ρ1 ⊗ · · · ⊗ ρr is a supercuspidal irreducible representation of a standard Levi subgroup M of G. Let Ω be the inertial class of the supercuspidal pair (M, ρ). The process [L, σ] 7→ Ω is well defined and induces a bijection between supercuspidal supports of GLn (q) and inertial class of level 0 supercuspidal pairs of G. Moreover, one has V(Ω) = V[L, σ]. By type theory, one can prove that the endomorphism algebra of U (Ω) is a finite tensor product of affine Hecke algebras of type A. Together with part (2) of Theorem 2.8, this implies that RR (Ω) is indecomposable. Problem 3: What is the structure of RR (Ω) for Ω ∈ S(G)? Find a progenerator PΩ in RR (Ω) and compute EndG (PΩ ). For G = GLm (D), type theory provides candidates for the PΩ ’s. Theorem 2.10: (See [14]) Let ρ be an irreducible supercuspidal representation of G = GLn (F). Let Ω be its inertial class. Write n(ρ) for the number of unramified characters χ of G such that ρχ ' ρ and v for the `-adic valuation of q n(ρ) − 1. There is a progenerator PΩ of RR (Ω), with: v

EndG (PΩ ) ' R[X, X−1 , T]/(T` ). When Ω has level 0, Guiraud [15] proved that there is a progenerator PΩ , and that the computation of its endomorphism algebra reduces to the case where n1 = · · · = nr and ρ1 = · · · = ρr (with the notation of the beginning of the paragraph). 3. Lecture 3 In this lecture, our goal is the classification of all irreducible representations of G = GLm (D) having given (super)cuspidal support. For complex representations, this has been done by Zelevinski [32] for GLn (F), and by Tadi´c [26] for its inner forms. 3.1. Reduction to the (super)unipotent case Assume G = GLn (F) for simplicity. The indecomposable block corresponding to the inertial class Ω1 of the pair (F×n , 1⊗n F× ) is called the unipotent block. An irreducible representation (π, V) of G is unipotent if it has supercuspidal support in Ω1 , that is, if π is a subquotient of U (Ω1 ) ' indG I (1) where I denotes the standard Iwahori subgroup of G.

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An irreducible representation (π, V) of G is superunipotent if cusp(π) ∈ Ω1 , that is, if VI is nonzero. Theorem 3.1: (See [23]) Let s be a cuspidal support in G. It writes uniquely as s1 + · · · + st such that (a) sj , sk have inertially equivalent terms in common if and only if j = k, and (b) any two terms in sj are inertially equivalent, for all j. Then: (1) The map (π1 , . . . , πt ) → π1 × · · · × πt induces a bijection: cusp−1 (s1 ) × · · · × cusp−1 (st ) → cusp−1 (s). (2) Assume that the terms of s are supercuspidal. Then the same map induces a bijection: scusp−1 (s1 ) × · · · × scusp−1 (st ) → scusp−1 (s). We are thus reduced to describe cusp−1 (Ωρ ) and scusp−1 (Ωρ ) where Ωρ is the inertial class of (GLn/r (F)r , ρ⊗r ) and ρ a (super)cuspidal irreducible representation of GLn/r (F), r dividing n. Theorem 3.2: (See [23]) Let ρ be a cuspidal irreducible representation of GLn/r (F). (1) There are a finite extension F0 /F of degree dividing n and a bijective map from cusp−1 (Ωρ ) to the set of superunipotent representations of GLr (F0 ). (2) Assume that ρ is supercuspidal. Then there is a bijective map from scusp−1 (Ωρ ) to the set of unipotent representations of GLr (F0 ). Moreover, these bijections preserve the (super)cuspidal support in the following sense. Given an unramified character χ of G, it writes χ = ξ ◦ det where ξ is an unramified character of F× . Then write χ0 for the unramified character ξ 0 ◦ det of G0 = GLr (F0 ), where ξ 0 is the unramified character of F0× whose value at a uniformizer of F0 is the same as that of ξ at a uniformizer of F. We get a bijective map χ 7→ χ0 between unramified characters of G and unramified characters of G0 . Then an irreducible representation π ∈ cusp−1 (Ωρ ) has cuspidal support ρχ1 + · · · + ρχr if and only if it corresponds to a superunipotent representation π 0 of G0 having cuspidal support χ01 + · · · + χ0r . There is a similar statement for the supercuspidal support in the case where ρ is supercuspidal. Therefore we are reduced to classify (super)unipotent representations of GLn (F).

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3.2. Classification of (super)unipotent representations, I We write HR (G, I) for the Hecke-Iwahori R-algebra, that is the space of functions f : G → R with compact support and such that f (xgx0 ) = f (g) for all g ∈ G, x, x0 ∈ I, endowed with the convolution product with respect to the Haar measure on G giving measure 1 to I. For all smooth representation (π, V), the space: VI ' HomG (C∞ c (I\G, R), V) is made into a right module over the Iwahori-Hecke algebra HR (G, I) by the formula: X v∗f = f (g)π(g −1 )v for v ∈ VI and f ∈ HR (G, I), where g ranges over a set of representatives of I\G in G. If R has characteristic ` different from 0, p, the functor V 7→ VI from smooth representations of G to right modules over HR (G, I) need not be exact (more precisely, this functor is exact if and only if ` does not divide q − 1). Thus C∞ c (I\G, R) need not be projective as a representation of G, but it has the following crucial property. Theorem 3.3: (See [28]) (1) The representation C∞ c (I\G, R) is quasi-projective, that is, for any surjective G-homomorphism C∞ c (I\G, R) → V, the restriction HR (G, I) → VI is also surjective. (2) The functor V → VI induces a bijection between the isomorphism classes of superunipotent representations of G and the isomorphism classes of simple right modules over HR (G, I). But this functor kills all unipotent non-superunipotent representations. In order to deal with these representations as well as the superunipotent ones, Vign´eras has introduced the following affine Schur algebra. Write: M V= C∞ c (P\G, R) P

where P ranges over all standard parahoric subgroups (that is, I ⊆ P ⊆ K) and their conjugates by the G-normalizer of I. The endomorphism algebra SR (G, I) = EndG (V) is called the affine Schur algebra (see [30]).

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Fix a Haar measure µ on G and let HR (G, µ) denote the space C∞ c (G, R) endowed with the convolution product with respect to µ. Any smooth Rrepresentation (π, V) is given a structure of left HR (G, µ)-module by: Z f · v = f (g)π(g)v dµ(g). G

Let J = JR (G) be the ideal of HR (G, µ) that annihilates the representation C∞ c (I\G, R). Theorem 3.4: (See [30]) (1) There is a functor from RR (G) to the category of right SR (G, I)-modules inducing an equivalence between the full subcategory of RR (Ω1 ) made of all representations that are killed by J and the category of right SR (G, I)-modules. (2) This induces a bijection between the isomorphism classes of unipotent representations of G and the isomorphism classes of simple modules over SR (G, I). (3) There is an integer N > 1 such that J N annihilates the whole block RR (Ω1 ). 3.3. Classification of (super)unipotent representations, II We now classify unipotent representations of GLn (F) by multisegments. Definition 3.5: (1) A segment is a pair (ξ, n) ∈ R× × Z>0 . (2) A multisegment is a formal finite sum of segments. (3) The integer n is called the length of the segment (ξ, n), and the length of a multisegment is the sum of the lengths of its segments. Given a segment (ξ, n), write Z(ξ, n) for the character: g 7→ ξ −val(det(g)) of the group GLn (F). For a multisegment m = (ξ1 , n1 ) + · · · + (ξr , nr ) of length n > 1, we want to define an irreducible subquotient Z(m) of: I(m) = Z(ξ1 , n1 ) × · · · × Z(ξr , nr ). Let U be the subgroup of upper triangular unipotent matrices of GLn (F). Let us fix a smooth nontrivial character ψF of F, and set ψ(u) = ψF (u1,2 + · · · + un−1,n ) for all u ∈ U.

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Definition 3.6: An irreducible representation (π, V) of G is generic if the space HomU (π, ψ) is nonzero. (This notion does not depend on the choice of ψF .) Given a multisegment m of length n > 1 as above, write µm for the partition of n conjugate to (n1 > n2 > · · · > nr ). Proposition 3.7: (1) µm is the unique maximal partition (for the dominance order) of n such that the parabolic restriction r µm (I(m)) contains a generic irreducible subquotient. (2) Such a generic irreducible subquotient is unique, and occurs with multiplicity 1. Write Z(m) for the unique irreducible subquotient of I(m) such that the parabolic restriction r µm (Z(m)) contains a generic irreducible subquotient. Theorem 3.8: (See [22]) The map m 7→ Z(m) is a bijection between multisegments of length n and isomorphism classes of unipotent representations of GLn (F). This bijection does not depend on the choice of

√

q ∈ R× .

Proof: For injectivity, µm can be recovered by the uniqueness property in Proposition 3.7(1). Then m can be recovered by looking at the generic irreducible subquotient in r µm (Z(m)). For surjectivity, one uses a counting argument that is based on the classification of all simple modules over HR (G, I) by aperiodic multisegments [1, 12]. Now write ` for the characteristic of R and define an integer e > 0 by:  0 if ` = 0, e= k−1 the smallest k > 2 such that 1 + q + · · · + q = 0 if ` > 0. Then the multisegment m writes uniquely as: XXX m=a+ c(ξ, n, u) · (ξ, n)u u>0 n>1 ξ

where ξ ranges over a set of representatives of R× /q Z in R× and:

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(1) (ξ, n)u is the multisegment: (ξ, n) + (ξq, n) + · · · + (ξq e`

u

−1

, n)

where e is as above, and c(ξ, n, u) is an integer in {0, 1, . . . , ` − 1}; (2) a is q-aperiodic, which means that for all segments (ζ, d) and all v > 0, the multisegment (ζ, d)v does not occur in a (when ` = 0, any multisegment is q-aperiodic thus a = m). u

Given an integer u > 0, the induced representation 1 × ν × · · · × ν e` −1 (where ν is the absolute value x 7→ |x|) possesses a unique cuspidal irreducible subquotient, denoted ρu . Given ξ ∈ R× , we also write χξ,u for the unramified character Z(ξ, e`u ) of GLe`u (F). Theorem 3.9: (See [22]) The cuspidal support of Z(m) is: XXX scusp(Z(a)) + n · c(ξ, n, u) · ρu χξ,u . u>0 n>1 ξ

We finally have the following decomposition theorem. Theorem 3.10: (See [22]) Let m be a multisegment. Then the semisimplification of I(m) writes: X Z(m) + dm,n · Z(n) n

where n ranges over all multisegments and dm,n ∈ Z>0 , with the following property: if dm,n 6= 0, then µn is smaller than µm (for the dominance order). Remark 3.11: When G is a non-split inner form of GLn (F), there is no theory of generic representations for G. In order to define the irreducible representation Z(m) of G, we introduce the notion of residually generic representation (see [22]) by using the functor W 7→ WK1 defined in Paragraph 2.1 (and more general functors coming from type theory to deal with positive level representations). 3.4. Comments When R is the field of complex numbers, the map m 7→ Z(m) gives Zelevinski’s classification of all irreducible representations having nonzero Iwahorifixed vectors. When the segments of m are put in a suitable order, the representation Z(m) can be characterized as the unique irreducible subrepresentation of I(m).

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There is also a Langlands classification m 7→ L(m) where L(m) is uniquely determined as the unique irreducible quotient of: J(m) = L(ξ1 , n1 ) × · · · × L(ξr , nr ) when the segments are put in a suitable order, and where L(ξ, n) is the unique generic irreducible representation with the same cuspidal support as Z(ξ, n). These two classifications are exchanged by the Zelevinski involution. 4. Lecture 4 In this section, G is the group GLn (F). 4.1. Reduction mod ` Let us fix a prime number ` 6= p and an algebraic closure Q` of the field of `-adic numbers. The residue field F` of its ring of integers Z` is an algebraic closure of a finite field of characteristic `. Definition 4.1: An irreducible Q` -representation (π, V) of G is said to be integral if V has a G-stable lattice L, that is a free Z` -module generated by a Q` -basis of V. Then L ⊗ F` is a smooth F` -representation of finite length of G, and its semisimplification does not depend on the choice of L; it is denoted r` (π) (see [31] and [27], II.5.11). × We fix a square root of q in Z× ` ⊆ Q` . By reducing mod the maximal ideal of Z` , it gives us a square root of q in F× ` . We write MS(R) for the semigroup of multisegments made of segments (ξ, n) with ξ ∈ R× and ZR for the bijection from MS(R) to the set of all isomorphism classes of unipotent representations. A multisegment m = (ξ1 , n1 ) + · · · + (ξr , nr ) ∈ MS(Q` ) is integral if the ξi ’s belong to Z× ` . If m as above is integral, define: r` (m) = (ξ 1 , n1 ) + · · · + (ξ r , nr ) ∈ MS(F` ) × where ξ denotes the image of ξ ∈ Z× ` in F` .

Lemma 4.2: (1) Let m ∈ MS(Q` ). Then Z(m) is integral if and only if m is integral.

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(2) Let π be an integral unipotent Q` -representation of G. Then all irreducible subquotients of r` (π) are unipotent. Proof: For 2, let m = (ξ1 , n1 ) + · · · + (ξr , nr ) ∈ MS(Q` ) be an integral multisegment such that π is isomorphic to Z(m). By construction, π is an irreducible subquotient of: I(m) = ZQ` (ξ1 , n1 ) × · · · × ZQ` (ξr , nr ). Since reduction mod ` commutes with parabolic induction, all irreducible subquotients of r` (π) are subquotients of: r` (I(m)) = ZQ` (ξ 1 , n1 ) × · · · × ZQ` (ξ r , nr ) = I(r` (m)). The result follows. Theorem 4.3: Let m ∈ MS(Q` ) be an integral multisegment. Then: r` (ZQ` (m)) = ZF` (r` (m)) +

X

a(m, n) · ZF` (n)

n

where n ranges over all multisegments and a(m, n) ∈ Z>0 , with the property: if a(m, n) is nonzero, then µn is smaller than µm (for the dominance order). Example 4.4: Assume G = GL2 (Q5 ) and ` = 3 (see Example 1.11). Thus we have q = 5. The unipotent Q` -representation corresponding to (1, 2) ∈ MS(Q` ) is the trivial Q` -character of G, whose reduction mod ` is the trivial F` character of G, that corresponds to the multisegment (1, 2) ∈ MS(F` ). Now consider m = (1, 1) + (q, 1) ∈ MS(Q` ). We write St for the Steinberg Q` -representation of G and π for the cuspidal subquotient of the F` representation V of Example 1.11. We have the following diagram: (1, 1) + (q, 1)

ZQ

`

/ StZ (√q, 2) Q`

r`

r`

 (1, 1) + (−1, 1)



ZF

/ π + Z (−1, 2) F` `

and π = ZF` ((1, 1) + (−1, 1)) is unipotent but not superunipotent.

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4.2. The local Langlands correspondence Let us fix a separable closure F of F. Its residue field k is a separable closure of the residue field k of F. The Galois group ΓF = Gal(F/F) acts on k such that there is a surjective group homomorphism: γF : ΓF → Gal(k/k). The group ΓF is profinite, and IF = Ker(γF ) is a closed subgroup of ΓF . Let FrobF ∈ Gal(k/k) be the Frobenius automorphism x 7→ xq . Then write: WF = {g ∈ ΓF | γF (g) ∈ FrobZ F }. It is called the Weil group of F. There is a unique topology on WF such that: (1) the topology of IF induced by WF and that induced by ΓF coincide; (2) the subgroup IF is open in WF . Note that this is not the topology on WF induced by ΓF (for which (2) is not satisfied). Remark 4.5: The group WF is locally compact, and its neutral element has a basis of neighborhoods made of compact open pro-p-subgroups. There is a notion of smooth R-representation of WF , just as for the group G. There is also a notion of reduction mod ` for (integral) finite-dimensional Q` -representations of WF . Now write: Gn0 (F) for the set of isomorphism classes of n-dimensional irreducible C-representations of WF , and: An0 (F) for the set of isomorphism classes of irreducible cuspidal C-representations of GLn (F). The local Langlands correspondence [20, 17, 18] asserts that there exists a unique family of bijections: πn0 : Gn0 (F) → An0 (F),

n>1

satisfying certain specific conditions that we do not give explicitly here.

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Now fix an isomorphism of fields α : C → Q` . By extension of scalars, any smooth complex representation of WF , G gives rise to a smooth Q` representation of WF , G. It also gives bijections: 0 α πn

: Gn0 (F, Q` ) → An0 (F, Q` ),

n>1

depending on α, because πn0 for n even depends on the choice of

√

q ∈ C× .

Theorem 4.6: (See [29]) (1) For any ρ ∈ An0 (F, Q` ), the reduction r` (ρ) is irreducible and cuspidal. (2) The map: r` : An0 (F, Q` )int → An0 (F, F` ) is surjective, where An0 (F, Q` )int denotes the subset of integral representations in An0 (F, Q` ). (3) A representation σ ∈ Gn0 (F, Q` ) is integral if and only if α πn0 (σ) is integral. (4) Assume σ, σ 0 ∈ Gn0 (F, Q` ) are integral. Then: r` (σ) = r` (σ 0 )

⇔

r` (α πn0 (σ)) = r` (α πn0 (σ 0 )).

(5) r` (σ) is irreducible if and only if r` (α πn0 (σ)) is supercuspidal. (6) This induces a bijection α π 0n between isomorphism classes of irreducible n-dimensional F` -representationsof WF and isomorphism classes of irreducible supercuspidal F` -representations of G. Acknowledgments The content of these notes is based on four lectures given at IMS of National University of Singapore in April 2013. I thank the organizers and IMS for providing such an opportunity. References 1. S. Ariki – On the decomposition numbers of the Hecke algebra of G(m, 1, n), J. Math. Kyoto Univ. 36 (1996), no. 4, p. 789-808. 2. J. Bernstein – Le centre de Bernstein, in Representations of reductive groups over a local field, Travaux en Cours, p. 1-32. Hermann, Paris, 1984. Written by P. Deligne. 3. J. Bernstein – Representations of p-adic groups, Lectures at Harvard University, 1992. Written by K. E. Rumelhart. 4. J. Bernstein & A. Zelevinski – Induced representations of reductive p-adic ´ groups. I, Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), no. 4, p. 441-472.

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5. C. Blondel – Basic representation theory of reductive p-adic groups, in p-adic representations, theta correspondence and the Langlands-Shahidi method, 2013, Science Press. ´cherre & S. Stevens – Smooth representations of 6. P. Broussous, V. Se GL(m, D), V: endo-classes, Documenta Math. 17 (2012), p. 23-77. 7. C. J. Bushnell – Representations of reductive p-adic groups: localization of Hecke algebras and applications, J. London Math. Soc. (2) 63 (2001), no. 2, p. 364-386. 8. C. J. Bushnell & G. Henniart – Local tame lifting for GL(N ). I. Simple ´ characters, Inst. Hautes Etudes Sci. Publ. Math. 83 (1996), p. 105-233. 9. C. J. Bushnell & P. C. Kutzko – The admissible dual of GL(N ) via compact open subgroups, Princeton University Press, Princeton, NJ, 1993. 10. C. J. Bushnell & P. C. Kutzko – Semisimple types in GLn , Compositio Math. 119 (1999), no. 1, p. 53-97. 11. P. Cartier – Representations of p-adic groups: a survey, Proceedings of Symposia in Pure Mathematics, vol. 33 (1979), part 1, p. 111-155. 12. N. Chriss & V. Ginzburg – Representation theory and complex geometry, Birkh¨ auser Boston Inc., Boston, MA, 1997. 13. J.-F. Dat – Finitude pour les repr´esentations lisses de groupes p-adiques, J. Inst. Math. Jussieu 8 (2009), no. 2, p. 261-333. 14. J.-F. Dat – Th´eorie de Lubin-Tate non ab´elienne `-enti`ere, Duke Math. J. 161 (2012), no. 6, p. 951-1010. 15. D. Guiraud – On semisimple `-modular Bernstein-blocks of a p-adic general linear group, J. Number Theory 133 (2013), 3524-3548. 16. Harish-Chandra – Harmonic analysis on reductive p-adic groups, Notes by G. van Dijk. Lecture Notes in Mathematics, vol. 162. Springer-Verlag, Berlin-New York, 1970. 17. M. Harris & R. Taylor – The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies 151, Princeton University Press. With an appendix by Vladimir G. Berkovich. 18. G. Henniart – Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139 (2000), no. 2, p. 439-455. 19. R. Kurinczuk – `-modular representations of unramified p-adic U(2, 1), Algebra Number Theory 8 (2014), No. 8, 1801-1838. 20. G. Laumon, M. Rapoport & U. Stuhler – D-elliptic sheaves and the Langlands correspondence, Invent. Math. 113 (1993), no. 2, p. 217-338. ´cherre – Unramified `-modular representations of 21. A. M´ınguez & V. Se GLn (F) and its inner forms, Int. Math. Res. Notices 8 (2014), p. 2090-2128. ´cherre – Repr´esentations lisses modulo ` de GLm (D), 22. A. M´ınguez & V. Se Duke Math. J. 163 (2014), p. 795-887. ´cherre – Types modulo ` pour les formes int´erieures de 23. A. M´ınguez & V. Se GLn sur un corps local non archim´edien, to appear in Proc. Lond. Math. Soc. (2014). With an appendix by V. S´echerre and S. Stevens. Preprint available at http://lmv.math.cnrs.fr/annuaire/vincent-secherre/. ´cherre & S. Stevens – Blocks of the category of `-modular smooth 24. V. Se representations of GLm (D). Preprint available at:

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http://lmv.math.cnrs.fr/annuaire/vincent-secherre/. 25. A. Silberger – The Langlands quotient theorem for p-adic groups, Math. Ann. 236 (1978), no. 2, p. 95-104. ´ – Induced representations of GL(n, A) for p-adic division algebras 26. M. Tadic A, J. Reine Angew. Math. 405 (1990), p. 48-77. ´ras – Repr´esentations l-modulaires d’un groupe r´eductif p-adi27. M.-F. Vigne que avec l 6= p, Progress in Mathematics, vol. 137, Birkh¨ auser Boston Inc., Boston, MA, 1996. ´ras – Induced R-representations of p-adic reductive groups, Se28. M.-F. Vigne lecta Math. (N.S.) 4 (1998), no. 4, p. 549-623. With an appendix by Alberto Arabia. ´ras – Correspondance de Langlands semi-simple pour GL(n, F ) 29. M.-F. Vigne modulo ` 6= p, Invent. Math. 144 (2001), no. 1, p. 177-223. ´ras – Schur algebras of reductive p-adic groups I, Duke Math. 30. M.-F. Vigne J. 116 (2003), no. 1, p. 35-75. ´ras – “On highest Whittaker models and integral structures, 31. M.-F. Vigne in Contributions to Automorphic forms, Geometry and Number theory: Shalikafest 2002, John Hopkins Univ. Press, 2004, p. 773-801. 32. A. Zelevinski – Induced representations of reductive p-adic groups. II. On ir´ reducible representations of GL(n), Ann. Sci. Ecole Norm. Sup. (4) 13 (1980), no. 2, p. 165-210.

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p-MODULAR REPRESENTATIONS OF p-ADIC GROUPS∗

Florian Herzig Department of Mathematics University of Toronto 40 St. George Street, #6290 Toronto, ON, M5S 2E4 Canada [email protected]

These notes are an introduction to the p-modular (or “mod-p”) representation theory of p-adic reductive groups. We will focus on the group GL2 (Qp ), but we try to provide statements that generalize to an arbitrary p-adic reductive group G (for example, GLn (Qp )).a

1. Motivation We fix two primes p and `. The motivation for studying the representation theory of p-adic groups comes from Local Langlands Conjectures. 1.1. The case ` 6= p In this case, we are in the setting of the classical Local Langlands Correspondence, which can be stated (roughly) as follows: Let n ≥ 1. We then have an injective map     continuous representations of  irreducible, admissible            Gal(Qp /Qp ) on n-dimensional representations of GLn (Qp ) ,−→     on Q` -vector spaces, Q` -vector spaces, up to         isomorphism up to isomorphism (1.1) The statement above is a bit imprecise; one needs to impose that Frobenius acts semisimply on the left-hand side. Usually, the left-hand side is ∗ Notes transcribed by Karol Koziol a We thank Vytas Paˇ sk¯ unas for his help

with Section 10. 73

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enlarged by replacing it by the set of Frobenius-semisimple Weil-Deligne representations, so that one obtains a bijection. This correspondence can be uniquely characterized by a list of properties (equivalence of L- and εfactors of pairs, compatibility with contragredients, etc.). In particular, for n = 1, the correspondence reduces to local class field theory. The correspondence was first established by Harris-Taylor ([13]) and Henniart ([14]) in 2000, and more recently by Scholze ([23]). 1.2. The case ` = p In this case, we would like to have a p-adic analog of (1.1), to be dubbed the “p-adic Local Langlands Correspondence.” Additionally, we would like to have a “mod-p version;” that is, a “mod-p Local Langlands Correspondence,” which would be compatible with the p-adic correspondence by reduction of lattices on both sides. When ` = p, we have “more” Galois representations, due to the fact that Gal(Qp /Qp ) and GLn (Qp ) have compatible topologies. Breuil proposed to replace the right-hand side above with certain (not necessarily irreducible) Banach space representations of GLn (Qp ) over Qp (or, more precisely, a fixed finite extension of Qp ). For the mod-p correspondence, one takes admissible representations of GLn (Qp ) over Fp . When n = 2, such a correspondence has been made precise and proven by work of Breuil, Colmez, Paˇsk¯ unas and others (see [7], [10], [12], [19], [22], and the references therein). However, for n > 2, there are fewer tools one has to attack this problem. For an overview of what is known, see [8]. The goal of these notes will be to describe the irreducible representations of GL2 (Qp ) over Fp , or more generally, an algebraically closed field E of characteristic p. The classification was first obtained by Barthel-Livn´e ([3], [4]), and completed by Breuil ([6]). One of the main differences between the mod-p theory and the classical theory (i.e., the theory of complex representations) is the absence of an Fp -valued Haar measure, which limits the techniques at one’s disposal. 2. p-Adic Groups We begin with the basics. Let p be a prime number, and let Zp denote the ring of p-adic integers; it is a discrete valuation ring with uniformizer p. We have Zp /pr Zp ∼ = Z/pr Z,

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so in particular, the residue field of Zp is Fp . We let Qp = Frac(Zp ) be the field of p-adic numbers. The topology of Qp is defined by a fundamental system of neighborhoods of 0: Zp ⊃ pZp ⊃ p2 Zp ⊃ . . . ⊃ pr Zp ⊃ . . . The basis for the topology is then given by the cosets of these neighborhoods. With this topology, Zp and Qp are topological rings. We note that the sets pr Zp are all compact and open. We take n ≥ 2, and set G = GLn (Qp ). This is again a topological group, and a fundamental system of neighborhoods of 1 is given by GLn (Zp ) ⊃ 1 + pMn (Zp ) ⊃ 1 + p2 Mn (Zp ) ⊃ . . . ⊃ 1 + pr Mn (Zp ) ⊃ . . . . We again note that the subgroups 1 + pr Mn (Zp ) are compact and open. We define K := GLn (Zp ), K(r) := 1 + pr Mn (Zp ). The group K is a maximal compact subgroup of GLn (Qp ), and we have K/K(r) ∼ = GLn (Zp /pr Zp ). In particular, K/K(1) ∼ = GLn (Fp ). When n = 2, we define the following closed subgroups of GL2 (Qp ):  B :=

 ∗∗ , 0∗

 T :=

 B :=

 ∗0 , ∗∗

 ∗0 , 0∗

  1∗ U := , 01

 U :=

 10 , ∗1

where a ∗ indicates an arbitrary entry. The group T is a maximal torus of GL2 (Qp ), B is a Borel subgroup, and U its unipotent radical. We have factorizations as follows: B = T n U,

B = T n U.

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3. Smooth Representations In what follows, we let Γ be a closed subgroup of GLn (Qp ) with the subspace topology, or a finite group with the discrete topology. These assumptions ensure that the identity element of Γ has a neighborhood basis of compact open subgroups. We take E to be an algebraically closed field of characteristic p (e.g., Fp ). This will serve as the coefficient field for all representations we consider. Given a representation π of Γ, we will often identify π with its underlying vector space. Moreover, given a vector v ∈ π, we denote the action of γ ∈ Γ on v by γ.v. If S is a subset of π and H is a submonoid of Γ, we let hH.SiE be the smallest subspace of π containing S and stable by the action of H. Definition 3.1: A representation π of Γ on an E-vector space is said to be smooth if [ π= πW , open subgroups W W

where π denotes the subspace of vectors fixed by a subgroup W . Equivalently, π is smooth if the above equality holds with W running over compact open subgroups. We remark that the smoothness condition is equivalent to saying that every vector is fixed by an open subgroup. This, in turn, is equivalent to saying that the action map Γ × π −→ π (γ, v) 7−→ γ.v is continuous, where π is given the discrete topology. Definition 3.2: If π is any representation of Γ (not necessarily smooth), we define [ π ∞ := πW ⊂ π open subgroups W

to be the largest smooth subrepresentation of π. We now proceed to define two types of induction functors. Assume first that H is a closed subgroup of Γ, and let σ be a smooth representation of H. We define IndΓH (σ) := {f : Γ −→ σ : f (hγ) = h.f (γ) for all h ∈ H, γ ∈ Γ}∞ ,

(3.1)

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where the action of Γ is given by (g.f )(γ) = f (γg) for every γ, g ∈ Γ. This procedure gives a smooth representation of Γ, and we call this functor induction. Assume now that W is an open subgroup of Γ. Since Γ is a topological group, this implies that W is also closed in Γ. Let τ be a smooth representation of W , and define    f (wγ) = w.f (γ) for all w ∈ W, γ ∈ Γ indΓW (τ ) := f : Γ −→ τ : .  W \supp(f ) is compact (3.2) We note that supp(f ) will be a union of W -cosets, so the quotient W \supp(f ) makes sense. Additionally, since W is open, the condition of W \supp(f ) being compact is equivalent to W \supp(f ) being finite. We define an action of Γ on indΓW (τ ) as above, and again obtain a smooth representation (without having to take smooth vectors!). We call this functor compact induction. For γ ∈ Γ, x ∈ τ , we will denote by [γ, x] ∈ indΓW (τ ) the function satisfying supp([γ, x]) = W γ −1 and [γ, x](γ −1 ) = x. We have [γw, x] = [γ, w.x] for w ∈ W , and moreover, the action of g ∈ Γ on [γ, x] is given by g.[γ, x] = [gγ, x]. Remark 3.3: If the index (Γ : H) is finite, then the subgroup H is open if and only if it is closed. In this case, the functors IndΓH (−) and indΓH (−) coincide. Remark 3.4: Since the elements of the compact induction indΓW (τ ) have finite support modulo W , we have a Γ-equivariant isomorphism indΓW (τ ) ∼ = E[Γ] ⊗E[W ] τ, as in the representation theory of finite groups. The following result is central in representation theory. Theorem 3.5: (Frobenius Reciprocity) Let H be a closed subgroup of Γ and W an open subgroup of Γ. Assume that π is a smooth representations of Γ, σ a smooth representation of H, and τ a smooth representation of W . We then have natural isomorphisms:

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∼ HomH (π|H , σ). (1) HomΓ (π, IndΓH (σ)) = (2) HomΓ (indΓW (τ ), π) ∼ = HomW (τ, π|W ). Proof: (1) Given a homomorphism ϕ ∈ HomΓ (π, IndΓH (σ)), we obtain an element of HomH (π|H , σ) by post-composing ϕ with the natural Hequivariant evaluation map IndΓH (σ) f

−→ 7−→

σ f (1).

(2) Again, given a homomorphism ϕ ∈ HomΓ (indΓW (τ ), π), we obtain an element of HomW (τ, π|W ) by pre-composing ϕ with the W -equivariant map θ, given by θ

−→ 7−→

τ x

indΓW (τ ) [1, x].

The remaining details are left as an exercise. Remark 3.6: Note that any element of the compact induction indΓW (τ ) can be written as r X

[γi , xi ] =

i=1

r X

γi .[1, xi ] =

i=1

r X

γi .θ(xi ),

i=1

where γi ∈ Γ, xi ∈ τ . This shows that θ(τ ) generates indΓW (τ ) as a Γrepresentation. × Example 3.7: Let n = 2, and G = GL2 (Qp ). We let χ1 , χ2 : Q× p −→ E be two smooth characters (i.e., smooth, one-dimensional representations) of Q× p , and denote by

χ1 ⊗ χ2 : 

B  α0 γ δ

−→

E×

7−→

χ1 (α)χ2 (δ)

the character of B obtained by inflation from T . Taking the induction of χ1 ⊗ χ2 from B to G, we obtain a smooth G-representation IndG (χ1 ⊗ χ2 ), B called a principal series representation. Note that this representation is infinite-dimensional: it is not hard to verify that we have a vector space isomorphism

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p-Modular Representations of p-Adic Groups ∼

{f ∈ IndG (χ1 ⊗ χ2 ) : supp(f ) ⊂ BU } B

−→

f

7−→

79

C ∞ (Qp , E) c   1x x 7−→ f , 01

where Cc∞ (Qp , E) denotes the space of locally constant, compactly supported, E-valued functions on Qp . We also remark that this vector space isomorphism can actually be upgraded to an isomorphism of Brepresentations, as the left-hand side is B-stable. We now proceed to discuss some distinguished features of representation theory in characteristic p. Definition 3.8: A pro-p group is a topological group which is compact and Hausdorff, and possesses a fundamental system of neighborhoods of the identity consisting of normal subgroups of p-power index. Example 3.9: (1) A finite p-group with the discrete topology is obviously a pro-p group. (2) The ring of p-adic integers Zp is a pro-p group (using that Zp /pr Zp ∼ = r Z/p Z). (3) Using the fact that K(r)/K(r + 1) ∼ = Mn (Fp ) for r > 0 (exercise), one sees that K(1) is a pro-p group. This implies in particular that any E-valued Haar measure µ of G vanishes on K(r) for all r > 0, hence µ = 0. (4) Let I(1) denote the preimage in K = GLn (Zp ) of the unipotent (pSylow) subgroup   1 ∗ ∗  ..  0 . ∗ ⊂ GLn (Fp ). 0 0 1 This subgroup is pro-p by the same argument as in (3), and is called the pro-p-Iwahori subgroup of K. The following lemma is very useful in mod-p representation theory. Lemma 3.10: Any nonzero smooth representation τ of a pro-p group H has H-fixed vectors; that is, τ H 6= 0. Proof: Forgetting the E-vector-space structure on τ , we can reduce to the case where τ is an Fp -linear representation of H. Fix a vector x ∈ τ − {0};

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since the action of H on τ is smooth, StabH (x) is open, and hence of finite index in H (by compactness). The orbit of x under H is then finite, and therefore hH.xiFp is a finite dimensional Fp -vector-space (isomorphic to Fdp , say). Now, since H is pro-p, the image of H −→ GLd (Fp ) will be a p-group. As the p-Sylow subgroups of GLd (Fp ) are all conjugate to   1 ∗ ∗  ..  0 . ∗ , 0 0 1 the image of H (after conjugation) will fix the first basis vector. We may now begin to analyze the mod-p representations of G. Definition 3.11: We shall call the irreducible mod-p representations of GLn (Fp ) weights. Corollary 3.12: (1) We have a canonical bijection {irreducible smooth representations of K over E} 1:1

←→ {irreducible representations of GLn (Fp ) over E}. (2) If π is a nonzero smooth representation of G = GLn (Qp ), then π|K contains a weight. Proof: (1) Since K(1) is open and normal in K and K/K(1) ∼ = GLn (Fp ), we obtain irreducible smooth representations of K by inflation from K/K(1). Conversely, let V be a smooth irreducible representation of K. As K(1) is pro-p, we obtain V K(1) 6= 0 by Lemma 3.10, and V K(1) is Kstable (again using normality of K(1) in K). Hence, V = V K(1) , and V is a representation of K/K(1). (2) Choosing x ∈ π K(1) − {0}, we have (by abuse of notation) hK.xiE = hGLn (Fp ).xiE . As this space is finite dimensional, it contains an irreducible subrepresentation of K. Example 3.13: Using Corollary 3.12 above and the decomposition Q× p = Z Z× × p , we obtain a bijection p

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× {smooth characters χ : Q× p −→ E } χ

1:1

←→ 7−→

81

× × Hom(F× p ,E ) × E (χ|Z× , χ(p)). p

4. Weights From now on, we focus on the case n = 2. This will allow us to do computations very explicitly. We first describe the weights of K = GL2 (Zp ). For any group X defined over Qp , we denote by Xp denote the analogous group over Fp ; for example  ×  Fp Fp Gp := GL2 (Fp ), Bp := , 0 F× p  Tp :=

 F× p 0 , 0 F× p

Up :=

  1 Fp , 0 1

etc.

Proposition 4.1: Up to isomorphism, the weights of Gp = GL2 (Fp ) are precisely the following: F (a, b) := Syma−b (E 2 ) ⊗ detb , where 0 ≤ a − b ≤ p − 1, 0 ≤ b < p − 1. The action of Gp on E 2 is the standard one, given by the inclusion Gp ,−→ GL2 (E). We may describe the action of Gp on F (a, b) explicitly as follows. We have an isomorphism F (a, b) ∼ = E[X, Y ](a−b) , where E[X, Y ](a−b) is the space   of homogeneous polynomials of degree a − b, with the action of αβ ∈ Gp on f ∈ E[X, Y ](a−b) given by γ δ    αβ .f (X, Y ) = f (αX + γY, βX + δY )(αδ − βγ)b . γ δ   We remark that this definition makes sense for αγ βδ ∈ GL2 (E), so that E[X, Y ](a−b) becomes a representation of the algebraic group GL2 (E). For GLn (Fp ) with n > 2, there is no simple known explicit way to describe the irreducible mod-p representations. However, they still arise by restricting certain irreducible representations of the algebraic group GLn (E) to GLn (Fp ). Proof: (Idea of proof (Proposition 4.1)) The proof of irreducibility relies on several observations:

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(1) The space F (a, b)Up is one-dimensional, and F (a, b)Up = EX a−b .

(4.1)

The group Tp acts on this space by the character ηa ⊗ ηb , where ηa (x) = xa for x ∈ F× p. (2) We have hGp .X a−b iE = F (a, b). To show this, it suffices to consider the action of U p , and then the claim follows from a computation with
a Vandermonde determinant (that is, one checks that the elements γ1 10 .X a−b for 0 ≤ γ ≤ a−b span F (a, b)). The first observation along with Lemma 3.10 implies that the Up -invariants of any nonzero subrepresentation of F (a, b) must contain X a−b . Thus, this subrepresentation must be all of F (a, b) by the second observation. This shows that the representations F (a, b) are all irreducible. By considering the action of Tp on F (a, b)Up , we see that the representations F (a, b) are all pairwise inequivalent, except for possibly the representations F (b, b) and F (p − 1 + b, b). Since dimE F (b, b) = 1 and dimE F (p − 1 + b, b) = p, we conclude that all of the F (a, b) are indeed distinct. Now, the number of p-modular representations of the finite group Gp is equal to the number of p-regular conjugacy classes in Gp . Using rational canonical form, it follows that the latter number is exactly p(p − 1), which shows that the representations F (a, b) with 0 ≤ a − b ≤ p − 1 and 0 ≤ b < p−1 form a full system of representatives for the p-modular representations of Gp . Lemma 4.2: We have F (a, b)U p ∼ = ηa ⊗ ηb as Tp -representations, where VU p denotes the U p -coinvariants of a Gp representation V . Proof: This follows from observation (1) of the previous proof, and the identity (VU p )∗ ∼ = (V ∗ )U p

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∼ F (a, b) ⊗ det−a−b and, by conjugation, (note that we have F (a, b)∗ = F (a, b)U p = EY a−b ). Concretely, we have F (a, b)U p = F (a, b)/hEX i Y a−b−i : 0 ≤ i < a − bi. Corollary 4.3: The natural map obtained by composing F (a, b)Up ,−→ F (a, b) − F (a, b)U p is a Tp -linear isomorphism. Using the above results, we can classify the weights appearing in principal series representations. × Proposition 4.4: Fix two smooth characters χ1 , χ2 : Q× p −→ E . Then G dimE HomK (V, IndB (χ1 ⊗ χ2 )|K ) ≤ 1

for all weights V . , then there is precisely one V such that equality holds, 6= χ2 |Z× If χ1 |Z× p p and dimE V > 1. If χ1 |Z× = χ2 |Z× , then two choices of V such that equality p p holds, and either dimE V = 1 or dimE V = p. Proof: The proof essentially follows from Frobenius Reciprocity. Note first × must factor as that, by Corollary 3.12, the characters χi |Z× : Z× p −→ E p χi |Z×

/ E× >

p

Z× p

ηi

F× p × for some ηi : F× p −→ E . Now, the Iwasawa Decomposition says that we have a factorization of the form

G = BK, from which it follows that IndG (χ1 ⊗ χ2 )|K ∼ (χ1 |Z× ⊗ χ2 |Z× ) = IndK B B∩K p p (this is a special case of the Mackey Decomposition). Both of these facts are left as an exercise for GL2 (Qp ). Hence, we obtain HomK (V, IndG (χ1 ⊗ χ2 )|K ) B

Mackey

∼ =

K HomK (V, IndB∩K (χ1 |Z× ⊗ χ2 |Z× )) p p

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Frobenius

∼ =

action factors mod p

∼ =

U p acts trivially

∼ =

⊗ χ2 |Z× ) HomB∩K (V |B∩K , χ1 |Z× p p HomB p (V |B p , η1 ⊗ η2 ) HomTp (VU p , η1 ⊗ η2 ).

Lemma 4.2 shows that HomK (V, IndG (χ1 ⊗ χ2 )|K ) is one-dimensional if B VU p ∼ = η1 ⊗η2 , and zero-dimensional otherwise. A quick computation verifies that if η1 6= η2 , there is one such weight V (satisfying dimE V > 1), and two otherwise (satisfying dimE V = 1 or p). 5. Hecke Algebras Throughout this discussion, fix a weight V , and π a smooth representation of G = GL2 (Qp ). Since the representation π|K contains a weight (cf. Corollary 3.12), it is natural to consider the multiplicity space HomK (V, π|K ) of the weight V in π. By Frobenius Reciprocity, we have HomK (V, π|K ) ∼ = HomG (indG K (V ), π), and the latter space comes equipped with a right action of H(V ) := EndG (indG K (V )) by pre-composition. The following proposition gives a more concrete description of the algebra H(V ). Proposition 5.1: We have    ϕ(k1 gk2 ) = k1 ◦ ϕ(g) ◦ k2   H(V ) ∼ . = ϕ : G −→ EndE (V ) : for all k1 , k2 ∈ K, g ∈ G    K\supp(ϕ)/K is finite The composition product on the left-hand side becomes the convolution product on the right-hand side: for ϕ1 , ϕ2 : G −→ EndE (V ), we have X (ϕ1 ∗ ϕ2 )(g) = ϕ1 (gγ)ϕ2 (γ −1 ). γ∈G/K

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Proof: (Idea of proof) By Frobenius Reciprocity, we have H(V ) ∼ = HomK (V, indG K (V )|K ) ⊂ Map(V, Map(G, V )) ∼ = Map(G, Map(V, V )), where the last isomorphism is the natural map. Since the right-hand side of Proposition 5.1 is naturally a subspace of Map(G, Map(V, V )), one simply needs to check that the conditions on H(V ) cut out the same subspace (see, for example, [18], Proposition 12 for more details). It is then straightforward to verify the final statement. Remark 5.2: Taking V = 1K , the trivial representation of K, we obtain H(1K ) ∼ = Cc (K\G/K, E), the usual double coset algebra of compactly supported, K-biinvariant, Evalued functions on G. Remark 5.3: If ϕ ∈ H(V ), f ∈ HomK (V, π|K ), and v ∈ V , then the right action of H(V ) on HomK (V, π|K ) is given by X (f ∗ ϕ)(v) = g −1 .f (ϕ(g).v). (5.1) g∈K\G

To further analyze the structure of the algebra H(V ), we will need to know about the double coset space K\G/K: Lemma 5.4: (Cartan Decomposition)  r  G p 0 G= K K. 0 ps r,s∈Z r≤s

This decomposition holds more generally (cf. [25], Section 3.3.3); in the case G = GL2 (Qp ), it can be proven using the theory of elementary divisors. Theorem 5.5: (1) For every r, s ∈ Z with r ≤ s,there exists a unique function  rTr,s ∈ r H(V ) such that supp(Tr,s ) = K p0 p0s K and such that Tr,s p0 p0s ∈ EndE (V ) is a linear projection. (2) The set {Tr,s }r,s∈Z,r≤s forms a basis for H(V ).

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(3) We have an algebra isomorphism H(V ) ∼ = E[T1 , T±1 2 ], where T1 := T0,1 and T2 := T1,1 . In particular, H(V ) is commutative. Proof: (Idea of proof) (1) As an example, consider the following identity in G:       α β 10 10 α pβ = , pγ δ 0p 0p γ δ where α, δ ∈ Z× p , β, γ ∈ Zp . Since the action of G on V factors through GL2 (Fp ), we have         αβ 10 10 α0 ◦ϕ =ϕ ◦ 0 δ 0p 0p γ δ for any ϕ ∈ H(V
). Taking α = δ =U 1, γ = 0 shows that the image of the 1 0 p operator ϕ 0 p is
contained in V . Likewise, taking α =1 δ0
= 1, β = 0 1 0 shows that ϕ 0 p factors through VU p . Hence, the map ϕ 0 p : V −→ V factors as   V 

VU p

0 ϕ 1 0p

/V O ? / V Up

Taking β = γ = 0, we see that the dotted arrow is Tp -linear. Thus, by Corollary 4.3, we see that there is a one-dimensional
space of such maps. We
take T0,1 to be the function supported on K 10 p0 K, such that T0,1 10 p0 ∼ is (the map associated to) the inverse of the obvious map V Up −→ VU p of
Corollary 4.3. It follows that T0,1 10 p0 is a projection. More generally, for r < s, the argument is the same as above. For r = s, it is more elementary, and left as an exercise. (2) This follows from the Cartan decomposition and the above proof of (1). (3) It is clear that the elements {Tr,s }r,s∈Z,r≤s form a basis, and one can show that this basis is related to {Ti1 Tj2 }i≥0,j∈Z by a unitriangular change of coordinates (for a suitable ordering of both sides). To see this presentation of the Hecke algebra more naturally, we may use the mod-p Satake transform. We let

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HT (VU p ) := EndT (indTT ∩K (VU p )), and note that Proposition 5.1 remains valid for HT (VU p ) if (G, K, V ) is replaced with (T, T ∩K, VU p ). The Satake transform SG : H(V ) −→ HT (VU p ) is given explicitly by   X SG (ϕ)(t) := prU ◦  ϕ(ut) , u∈U ∩K\U

where ϕ ∈ H(V ), t ∈ T , and prU : V − VU p is the natural map. It is straightforward to check that with this definition SG is well-defined and an algebra homomorphism. With more work one shows that SG is injective and that     x0 −1 im SG = ψ ∈ HT (VU p ) : ψ = 0 if yx 6∈ Zp . (5.2) 0y As T is abelian, one easily obtains the following analog of Theorem 5.5: HT (VU p ) has basis {τr,s = τ1s−r τ2r }r,s∈Z , where τr,s is determined by  r   r  p 0 p 0 supp(τr,s ) = (T ∩ K) , τ = 1, r,s 0 ps 0 ps and τ1 := τ0,1 , τ2 := τ1,1 . From (5.2), we get H(V ) ∼ = im SG = E[τ1 , τ2±1 ]. In fact, it is not hard to verify that SG (T1 ) = τ1 and SG (T2 ) = τ2 . We remark that the Satake transform exists more generally for any p-adic reductive group (see [16] and [15]). × Proposition 5.6: Suppose χ1 , χ2 : Q× p −→ E are two smooth characters, G and let f : V ,−→ IndB (χ1 ⊗ χ2 )|K be a nonzero K-linear map. Then

f ∗ T1 = χ2 (p)−1 f f ∗ T2 = χ1 (p)−1 χ2 (p)−1 f. Proof: This is a calculation using equation (5.1). Alternatively, the map f induces a T -linear map f : indTT ∩K (VU p ) − χ1 ⊗ χ2 (see the proof of Proposition 4.4). The map f then factors as IndG (f ) ◦ F0 , B where

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G T F0 : indG K (V ) −→ IndB (indT ∩K (VU p ))

is the map introduced in the proof of Theorem 6.1. Since F0 ◦ ϕ = IndG (SG (ϕ)) ◦ F0 , B one reduces the proof to the analogous, but much simpler, problem for T : f ◦ τ1 = χ2 (p)−1 f f ◦ τ2 = χ1 (p)−1 χ2 (p)−1 f . 6. Comparison Isomorphisms Using the results of the previous section, we can now describe comparison isomorphisms between compact and parabolic induction. × We fix two smooth characters χ1 , χ2 : Q× p −→ E , let V be a weight, and suppose (χ1 ⊗ χ2 )|K f : V ,−→ IndG B is a nonzero K-linear homomorphism. Frobenius Reciprocity provides us with a nonzero G-linear map G fe : indG K (V ) −→ IndB (χ1 ⊗ χ2 ).

By Proposition 5.6, if ϕ ∈ H(V ), we have fe ◦ ϕ = χ0 (ϕ)fe, where χ0 : H(V ) −→ E is the algebra homomorphism defined by χ0 (T1 ) = χ2 (p)−1 χ0 (T2 ) = χ1 (p)−1 χ2 (p)−1 . The universal property of tensor products implies that we have a G-linear map G 0 f : indG K (V ) ⊗H(V ) χ −→ IndB (χ1 ⊗ χ2 ).

Theorem 6.1: The map f is an isomorphism if dimE V > 1. Proof: (Idea of proof) One can even prove a “universal” version of this theorem: consider the G-linear map G T F0 : indG K (V ) −→ IndB (indT ∩K (VU p ))

obtained as in the proof of Proposition 4.4 from the T ∩ K-linear map VU p −→ indTT ∩K (VU p ), which corresponds to the identity map in

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EndT (indTT ∩K (VU p )) under Frobenius Reciprocity. A calculation shows that G F0 is H(V )-linear with respect to SG , i.e., that F0 ◦ ϕ = IndB (SG (ϕ)) ◦ F0 for all ϕ ∈ H(V ). As SG (T1 ) = τ1 is invertible in HT (VU p ) and the actions of G and H(V ) commute, F0 induces a G-linear and H(V )[T−1 1 ]-linear map G T −1 F : indG K (V )[T1 ] −→ IndB (indT ∩K (VU p )).

(6.1)

It now suffices to show F is an isomorphism, and then specialize T1 and T2 (that is, we apply the functor − ⊗H(V )[T−1 ] χ0 to both sides of the 1

isomorphism (6.1) above to recover the map f ). To show F is surjective, one shows that the elements F (T−n 1 ([1, x])), where n ≥ 0 and x ∈ V Up − {0}, generate the right-hand side of (6.1) as a G-representation (this argument uses the Bruhat Decomposition and the fact that dimE V > 1). For the injectivity of F , one can use an elegant argument of Abe which uses the Satake transform (and which works even if dimE V = 1). See [18], Theorems 16 and 32 for more details. Remark 6.2: Notice that any pair (V, χ0 ) subject to dimE V > 1 and χ0 (T1 ) 6= 0 arises in this theorem. Hence, for any such pair, indG K (V ) ⊗H(V ) χ0 is a principal series representation. Corollary 6.3: If dimE V > 1, the weight f (V ) generates IndG (χ1 ⊗ χ2 ) B as a G-representation. Proof: We have already seen that the image of V inside indG K (V ) generates the compact induction as a G-representation. By Theorem 6.1, we have indG (V ) − indG (V ) ⊗H(V ) χ0 ∼ = IndG (χ1 ⊗ χ2 ), K

K

B

and the claim follows. × be two smooth characters, and Corollary 6.4: Let χ1 , χ2 : Q× p −→ E × . Then = 6 χ | suppose χ1 |Z× 2 Zp p G (χ1 ⊗ χ2 ) π = IndB

is an irreducible G-representation. Proof: By Proposition 4.4, π contains a unique weight V and it satisfies dimE V > 1. Moreover, V generates π as a G-representation by Corollary 6.3. Suppose σ is a nonzero subrepresentation of π; then σ contains the unique weight V , and thus has to be the whole of π.

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7. The Steinberg Representation In this section we discuss the Steinberg representation, which arises when one takes χ1 = χ2 = 1 (the trivial character) in the definition of principal series. Definition 7.1: We define the Steinberg representation St by the following short exact sequence: (1 ⊗ 1) −→ St −→ 0, 0 −→ 1G −→ IndG B where the first map arises from Frobenius Reciprocity. More concretely, using (3.1), we can identify St as the quotient {locally constant functions on B\G ∼ = P1 (Qp )}/{constant functions}. Theorem 7.2: The representation St is irreducible. Proof: (Idea of proof) Recall that the subgroup I(1), defined in Example 3.9(4), is a pro-p subgroup of G. One can prove in a straightforward way that dimE StI(1) = 1 (one needs to show that any I(1)-invariant function G in the quotient is the image of an I(1)-invariant function in IndB (1 ⊗ 1); see [4], Lemma 27 for more details). This implies that St contains a unique weight, which is of multiplicity 1. Let V = F (p − 1, 0) = Symp−1 (E 2 ), and note that dimE V = p. We thus obtain the exact sequence 0 −→ HomK (V, 1G |K ) −→ HomK (V, IndG (1 ⊗ 1)|K ) −→ HomK (V, St|K ). B G (1 ⊗ 1)|K ) = 1 (by Since HomK (V, 1G |K ) = 0 and dimE HomK (V, IndB Proposition 4.4), we obtain dimE HomK (V, St|K ) ≥ 1, which shows that V is the aforementioned unique weight of St. By Corollary 6.3, V generates IndG (1 ⊗ 1) as a G-representation, and therefore generates St. The result B now follows as in the proof of Corollary 6.4 above.

Remark 7.3: The sequence defining St does not split. To see this, just note that the unique nontrivial weight of IndG (1 ⊗ 1) generates IndG (1 ⊗ 1) as B B a G-representation, by Corollary 6.3. × Remark 7.4: If χ : Q× is a smooth character, we can twist the p −→ E exact sequence defining St to obtain G 0 −→ χ ◦ det −→ IndB (χ ⊗ χ) −→ St ⊗ (χ ◦ det) −→ 0.

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By what we have proved for the representation St, we know that the representations χ ◦ det and St ⊗ (χ ◦ det) are irreducible, and contain unique weights (of dimensions 1 and p, respectively). Moreover, the Hecke eigenG values on these representations are the same as those of IndB (χ ⊗ χ), and are given by T1 7−→ χ(p)−1 ,

T2 7−→ χ(p)−2 .

8. Change of Weight We now fix two weights V, V 0 , and consider the module of intertwiners given by G 0 H(V, V 0 ) := HomG (indG K (V ), indK (V )).

It naturally has the structure of an (H(V 0 ), H(V ))-bimodule by pre- and post-composition. Proposition 8.1: (1) We have  

  ϕ(k1 gk2 ) = k1 ◦ ϕ(g) ◦ k2  H(V, V 0 ) ∼ . = ϕ : G −→ HomE (V, V 0 ) : for all k1 , k2 ∈ K, g ∈ G    K\supp(ϕ)/K is finite The bimodule structure on the right-hand side is given by convolution (as in Proposition 5.1). (2) We have H(V, V 0 ) 6= 0 if and only if VU p ∼ = VU0 p as Tp -representations. (3) If V ∼ 6 V 0 and VU p ∼ = = VU0 p , then there exists ϕ ∈ H(V, V 0 ) satisfying  r  supp(ϕ) = K p0 p0s K if and only if r < s. Proof: The proof is exactly the same as for Proposition 5.1. In case (3) of the above proposition, the only possible choices for V and V 0 are V = F (b, b) and V 0 = F (p − 1 + b, b) for 0 ≤ b < p − 1 (or vice versa). Therefore, there exist G-linear maps ϕ+

indG K (V

)o

/

0 indG K (V )

−

ϕ

satisfying supp(ϕ− ) = supp(ϕ+ ) = K to a scalar.

1 0 0p




K. These maps are unique up

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For the next proposition, it will be convenient to make the identifications 0 ∼ H(V ) ∼ = E[T1 , T±1 2 ] = H(V ),

and call both algebras H. (More naturally, both algebras are identified with the same subalgebra of HT (VU p ) = HT (VU0 ) by the Satake transform.) p

Proposition 8.2: (1) The maps ϕ− , ϕ+ are H-linear and commute with each other. (2) We have (up to a scalar) ϕ+ ◦ ϕ− = T21 − T2 . Proof: (Sketch of proof) The mod-p Satake transform gives an injective map H(V, V 0 ) ,−→ HT (VU p , VU0 ) with the same formula as before. This p map is compatible with convolution. Then (1) follows from the fact that HT (VU p ) = HT (VU0 ) is commutative, while (2) is a calculation. p

Corollary 8.3: If χ0 : H −→ E is an algebra homomorphism such that χ0 (T21 − T2 ) 6= 0, then we obtain a G-linear isomorphism G 0 ∼ 0 0 indG K (V ) ⊗H χ = indK (V ) ⊗H χ .

Proof: By Proposition 8.2, part (1), the maps ϕ− , ϕ+ induce G-linear maps between the two representations. Their composite is χ0 (T21 − T2 ) 6= 0, so we obtain an isomorphism. × Proposition 8.4: Let χ1 , χ2 : Q× be two smooth characters, and p −→ E suppose χ1 6= χ2 . Then

π = IndG (χ1 ⊗ χ2 ) B is an irreducible G-representation. Proof: If χ1 |Z× 6= χ2 |Z× , then the claim follows from Corollary 6.4. We p p therefore may assume χ1 |Z× = χ2 |Z× and p p χ1 (p) 6= χ2 (p).

(8.1)

By Proposition 4.4, π contains two weights V, V 0 of the form F (b, b) and F (p − 1 + b, b) with 0 ≤ b < p − 1 (or vice versa). Let us label these weights so that V = F (b, b) and V 0 = F (p − 1 + b, b). Corollary 6.3 then implies that V 0 generates π as a G-representation, since dimE V 0 = p > 1.

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Now let σ be a nonzero G-subrepresentation of π. We claim that σ must contain V 0 . Indeed, if this were not the case, we would necessarily have a K-linear injection V ,−→ σ|K (as σ|K has to contain a weight). This implies the existence of a nonzero G-linear map indG K (V ) −→ σ, which descends to 0 indG K (V ) ⊗H χ −→ σ

for some χ0 : H −→ E, as HomK (V, π|K ) is one-dimensional. By Proposition 5.6, χ0 : H −→ E is the character given by χ0 (T1 ) = χ2 (p)−1 ,

χ0 (T2 ) = χ1 (p)−1 χ2 (p)−1 .

Hence, we have χ0 (T21 − T2 ) = χ2 (p)−1 (χ2 (p)−1 − χ1 (p)−1 ) 6= 0, by equation (8.1). Therefore, Corollary 8.3 shows that we have a nonzero G-linear map 0 0 indG K (V ) ⊗H χ −→ σ,

which, by Frobenius Reciprocity, gives a nonzero map V 0 ,−→ σ|K . Since σ contains V 0 and V 0 generates π as a G-representation, we obtain σ = π, and therefore π is irreducible. 9. Classification of Representations We are now in a position to give a classification of smooth irreducible representations of G = GL2 (Qp ), at least under an admissibility assumption. We remark that this assumption is in fact unnecessary (see [5], [3], [6]). For the following definition and proposition, suppose that Γ is a closed subgroup of G. Definition 9.1: A smooth Γ-representation π is called admissible if dimE π W < ∞ for all open subgroups W of Γ. Proposition 9.2: Let π be a smooth Γ-representation. Then π is admissible if and only if dimE π W < ∞ for one open pro-p subgroup W of Γ.

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Proof: Let W be a fixed pro-p subgroup such that dimE π W < ∞, and let 0 0 W 0 be an arbitrary open subgroup. Since dimE π W ≤ dimE π W ∩W , it is 0 enough to show dimE π W ∩W < ∞. We may therefore assume that W 0 is an open subgroup of W (and hence of finite index). This gives 0 π W = HomW 0 (1W 0 , π|W 0 ) ∼ = HomW (indW W 0 (1W 0 ), π|W ).

It thus suffices to show that HomW (M, π|W ) is finite-dimensional for any finite-dimensional smooth W -representation M . We argue by induction on dimE M . By Lemma 3.10, we have a short exact sequence 0 −→ 1W −→ M −→ M/1W −→ 0. Applying HomW (−, π|W ) yields 0 −→ HomW (M/1W , π|W ) −→ HomW (M, π|W ) −→ HomW (1W , π|W ) ∼ = πW . The last term is finite-dimensional by assumption and the first term is finitedimensional by induction. Hence the middle term is finite-dimensional. The proof of the following proposition is left as an exercise (see also Lemmas 23 and 24 in [18]). Proposition 9.3: Let π be a smooth representation of G. (1) The representation π is admissible if and only if dimE HomK (V, π|K ) < ∞ for any weight V . (2) If π is admissible, then π possesses a central character. Corollary 9.4: All principal series representations and all representations × of the form St ⊗ (χ ◦ det), with χ : Q× a smooth character, are p −→ E admissible. Proof: This follows from part (2) of the previous proposition, Proposition 4.4, and (the proof of) Theorem 7.2. Definition 9.5: Let π be an irreducible, admissible G-representation. We say π is supersingular if for any weight V the action of T1 on HomK (V, π|K ) is nilpotent (or, equivalently, if all eigenvalues of T1 are zero). The following gives a coarse classification of representations of G. Theorem 9.6: (Barthel-Livn´ e) Every irreducible, admissible representation of G = GL2 (Qp ) falls into one of the following four families:

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the principal series IndG (χ1 ⊗ χ2 ) with χ1 6= χ2 , B the characters χ ◦ det, twists of the Steinberg representation St ⊗ (χ ◦ det), the supersingular representations.

The four families are disjoint, and the characters appearing in cases (1) to (3) are uniquely determined. Proof: Let π be an irreducible, admissible representation, and let V be a weight of π. As π is admissible and H(V ) is commutative, the (finitedimensional) weight space HomK (V, π|K ) contains a common eigenvector f : V ,−→ π|K for H(V ), with eigenvalues given by an algebra homomorphism χ0 : H(V ) −→ E. If χ0 (T1 ) = 0 for all such pairs (V, χ0 ), then π is supersingular. We may therefore assume without loss of generality that χ0 (T1 ) 6= 0. In this case, we get a nonzero (and in fact surjective) G-linear map 0 indG K (V ) ⊗H(V ) χ − π.

We consider several possibilities: G 0 ∼ • If dimE V > 1, then indG K (V ) ⊗H(V ) χ = IndB (χ1 ⊗ χ2 ) for some choice of χ1 , χ2 (cf. Remark 6.2), and we obtain that π is either an irreducible principal series representation or a twist of a Steinberg representation. • If dimE V = 1 and χ0 (T21 − T2 ) 6= 0, then by Corollary 8.3, we have G 0 0 0 ∼ indG K (V ) ⊗H(V ) χ = indK (V ) ⊗H(V ) χ , for some p-dimensional weight 0 V . We may now proceed as in the previous case. • If dimE V = 1 and χ0 (T21 −T2 ) = 0, then (after twisting π by a character of the form χ ◦ det) we may assume V = 1K and χ0 (T1 ) = χ0 (T2 ) = 1. The claim now follows from the following fact (cf. [3], Theorem 30 or [1], Proposition 4.7): G 0 ss ∼ ss ∼ (indG K (1K ) ⊗H(1K ) χ ) = (IndB (1 ⊗ 1)) = 1G ⊕ St.

To show that the four families are disjoint, we analyze their weights and G Hecke eigenvalues. Note firstly that if π is any subquotient of IndB (χ1 ⊗χ2 ) and V is any weight of π, we have VU p ∼ ⊗ χ2 |Z× = χ1 |Z× p p as Tp -representations. This implies that the eigenvalues of H(V ) on the space HomK (V, π|K ) are given by χ0 (T1 ) = χ2 (p)−1 ,

(9.1)

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χ0 (T2 ) = χ1 (p)−1 χ2 (p)−1 .

(9.2)

The condition χ0 (T1 ) = 0 distinguishes the supersingular representations (family (4)). The irreducible principal series representations (family (1)) are distinguished by the condition 1 < dimE V < p or χ0 (T21 − T2 ) 6= 0. Finally, the characters of G (family (2)) are determined by dimE V = 1 and χ0 (T21 −T2 ) = 0, while the twists of the Steinberg representation (family (3)) are determined by dimE V = p and χ0 (T21 − T2 ) = 0. Moreover, equations (9.1) and (9.2) above (along with knowledge of VU p ) imply how to uniquely recover the characters χ1 , χ2 (respectively, χ) from χ0 . Definition 9.7: An irreducible, admissible G-representation is called supercuspidal if it is not a subquotient of a principal series representation. Corollary 9.8: If π is an irreducible, admissible representation of G = GL2 (Qp ), then π is supercuspidal if and only if π is supersingular. Remark 9.9: The above description of irreducible, admissible representations in terms of parabolic induction, supersingular representations, and (generalized) Steinberg representations generalizes to arbitrary p-adic reductive groups. Also, Corollary 9.8 still holds. See [17] for the case of GLn , [1] for the case of a general split, connected, reductive group, and [2] for the general case.

10. Supersingular Representations Almost all of the arguments we have considered up to this point apply with little change to the group GL2 (F ), where F is a finite extension of Qp . In this section, however, we will crucially use the fact that F = Qp to give a more precise description of the classification of Theorem 9.6. Our main goal will be to prove the following theorem, due to Breuil. We follow an argument due to Paˇsk¯ unas [21] and Emerton [11]. Theorem 10.1: (Breuil) The irreducible, admissible, supersingular representations of G = GL2 (Qp ) are exactly 0 indG K (V ) ⊗H(V ) χ ,

where V is a weight and χ0 (T1 ) = 0.

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By Theorem 9.6 above, any irreducible, admissible, supersingular rep0 0 resentation of G is a quotient of indG K (V ) ⊗H(V ) χ with χ (T1 ) = 0, so it is enough to show that this quotient is irreducible and admissible. In order to 0 do this, we shall use a slightly different model for indG K (V ) ⊗H(V ) χ , given as follows. Let Z denote the center of G. We
inflate V to a representation of KZ by decreeing that the element p0 p0 acts trivially. Proposition 10.2: We have an isomorphism ∼ EndG (indG KZ (V )) = E[T], where (via an isomorphism analogous to that of
Proposition 5.1) T is
the function supported on the double coset KZ 10 p0 KZ, such that T 10 p0 ∈ EndE (V ) is a linear projection. Proof: The proof is analogous to the proof of Theorem 5.5. × × Now let η : Q× p −→ E be a (smooth) character which is trivial on Zp , and which satisfies η(p)−2 = χ0 (T2 ). Then one easily checks that we have a surjective G-linear map G indG K (V ) − indKZ (V ) ⊗ η ◦ det

[g, v]K 7−→ [g, v]KZ ⊗ η(det g), where the subscript denotes the group with respect to which the element [g, v] is equivariant. Note that the action of T1 on the left is compatible with the action of T · η(p)−1 on the right, while the action of T2 is compatible with the action of the scalar η(p)−2 = χ0 (T2 ). It is not hard to check that this induces an isomorphism ∼

0 indG K (V ) ⊗H(V ) χ −→

indG KZ (V ) ⊗ (η ◦ det). (T − χ0 (T1 )η(p))

Therefore, to prove Theorem 10.1 it is enough to show indG KZ (V )/(T) is irreducible and admissible as a G-representation. Proposition 10.3: The representation indG KZ (V )/(T) is nonzero. Proof: (Idea of proof) As the operator T is not invertible (by Proposition 10.2), it suffices to show that T is injective on indG KZ (V ). Consider the special case where V = 1K . We then have indG KZ (1K ) = Cc (KZ\G, E),

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so that the compactly induced representation is equal to the space of finitely supported E-valued functions on KZ\G. In this context, it is convenient to use the Bruhat-Tits tree X of GL2 (Qp ) (pictured below for p = 2; see [24] for an overview).

The vertices of X correspond to homothety classes of Zp -lattices contained in Q⊕2 p . The group GL2 (Qp ) acts transitively on the set of such lattices, and the stabilizer of the class of Zp ⊕ Zp is exactly KZ. Hence, the representation indG KZ (1K ) is exactly the set of compactly supported functions on the vertices of X. In this setup, the operator T is simply the “sum  over neighbors” map; that is, (Tf )(x) = f (y), where y runs through the neighbors of x. Now let f ∈ indG KZ (1K ) be an arbitrary nonzero function, and let T be the convex hull of supp(f ); it is a finite subtree of X. If we let x denote an extremal vertex of T and y ∈ T a neighbor of x, then we have (Tf )(y) = 0, which shows T is injective. We introduce some additional notation for the proof of Theorem 10.1. Let red : K = GL2 (Zp ) − Gp = GL2 (Fp ) denote the “reduction-modulo-p” map, and define I := red−1 (Bp ),

I(1) := red−1 (Up ).

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We let  Π :=

 01 ; p0

we then have ΠI(1)Π−1 = I(1)   p0 2 Π = . 0p We also set   p0 t := , 01

  01 s := , 10

(10.1) (10.2)

  11 X := − 1 ∈ E[I ∩ U ]. 01

Note that Π = st. We will also need the technique of weight cycling mentioned above. Let π denote the quotient indG KZ (V )/(T), and write V = F (a, b). We denote by f the composition of natural maps V −→ indG KZ (V ) − π. Note that f is injective, as V generates indG KZ (V ), and f ∗ T = 0, by definition of π. The map f allows us to identify V with a K-subrepresentation of π. Let us fix a nonzero element v ∈ V Up ⊂ π I(1) . By equation (4.1), we see that I acts on v by the character ηa ⊗ ηb (using the identification I/I(1) ∼ = Tp ). By (10.1) above, the element v 0 := Π.v also lies in π I(1) , and I acts on v 0 by the character ηb ⊗ ηa . Hence, by Frobenius Reciprocity, we get a nonzero K-linear map indK I (ηb ⊗ ηa ) [1, 1]

j

−→ 7−→

π v0 .

Additionally, using Lemma 4.2 and Frobenius Reciprocity, we have a complex i

0 0 −→ V = F (a, b) −→ indK I (ηb ⊗ ηa ) −→ V := F (b + p − 1, a) −→ 0.

See also [9], Theorem 7.1 for an alternate derivation of this complex. Since dimE F (a, b) = a − b + 1 and dimE F (b + p − 1, a) = p + b − a, the complex above is exact. Lemma 10.4: We have j ◦ i = f ∗ T (up to a scalar).

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Proof: Calculation. The lemma holds equally well for any smooth representation π. In our situation we therefore have j ◦ i = 0, so we see that the map j factors through the projection to V 0 : / / V0 indK I (ηb ⊗ ηa ) j

 x π This implies that the K-subrepresentation of π generated by v 0 is isomorphic to V 0 . The upshot of this discussion is that Π acts on π I(1) , exchanging v and v 0 (by (10.2) above), and we have hK.v 0 iE ∼ = F (b + p − 1, a).

hK.viE ∼ = F (a, b),

Note that these two weights are nonisomorphic. Lemma 10.5: (Iwahori Decomposition) We have a factorization     × 1 0 1 Zp Zp 0 I = (I ∩ U ) (I ∩ T ) (I ∩ U ) = , 0 Z× pZp 1 0 1 | {z } | {z } | {z } p I+

I0

I−

where the factors may be taken in any order. In addition, we have tI + t−1 ⊂ I + ,

tI 0 t−1 = I 0 ,

tI − t−1 ⊃ I − .

Proof: This decomposition is a general fact about reductive groups over local fields. For the case of GL2 (Qp ), one can prove this directly: for example, one has (for α, δ ∈ Z× p , β, γ ∈ Zp )       α β 1 βδ −1 α − pβγδ −1 0 1 0 = . pγ δ 0 1 0 δ pγδ −1 1 Proposition 10.6: (1) Let M := hI + tN .(Ev ⊕ Ev 0 )iE ⊂ π be the subspace of π generated by the orbit of I + tN on the vectors v and v 0 . Then M is stable by and irreducible for the action of the monoid   Zp − {0} Zp I 0 I + tN = . 0 Z× p (2) We have V ⊂ M .

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Given this proposition, we may prove irreducibility of the representation π = indG KZ (V )/(T). Proof: (Proof of Theorem 10.1 (irreducibility), assuming Proposition 10.6) By Proposition 10.6, we have    Zp − {0} Zp 0 + N M = hI I t .V iE = .V , 0 1 E since I 0 normalizes I + . Moreover, since t.M ⊂ M , we have an increasing chain of subspaces M ⊂ t−1 .M ⊂ t−2 .M ⊂ . . . . Using the Iwasawa decomposition (cf. proof of Proposition 4.4) and the fact that the center Z acts by scalars on π, we obtain    [ Qp − {0} Qp t−n .M = .V 0 1 E n≥0

Z acts by scalars

=

V is a weight

=

Iwasawa

hB.V iE hBK.V iE

=

hG.V iE

=

π.

Now, let σ ⊂ π|B be a nonzero B-subrepresentation. Then, by the above computation, there exists an integer n ≥ 0 and m ∈ M − {0} such that t−n .m ∈ σ. As σ is B-stable, we have m ∈ tn .σ = σ, which implies M ⊂ σ by Proposition 10.6. Hence, we obtain [ π= t−n .M ⊂ σ, n≥0

which shows that π is irreducible, even as a B-representation. Thus, we have shown that it suffices to prove Proposition 10.6 to show the irreducibility of π. We let N := hI + t2N .viE ⊂ M,

N 0 := hI + t2N .v 0 iE ⊂ M,

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and set r := a − b. The proof of Proposition 10.6 will follow from the following lemmas. Lemma 10.7: (1) We have tX t−1 = X p . (2) We have v = (∗)X r t.v 0 v 0 = (∗)X p−1−r t.v, where (∗) is some element of E × . Proof: (1) Since E has characteristic p, we have        p 11 1p 11 tX t−1 = t − 1 t−1 = −1= − 1 = X p. 01 01 01 (2) Note that, if X r−i Y i is an element of V = F (a, b) = E[X, Y ](r) with 0 ≤ i ≤ r, we have X .X r−i Y i = X r−i (Y + X)i − X r−i Y i = iX r−i+1 Y i−1 + Q(X, Y ), where Q(X, Y ) is a homogeneous polynomial of degree r, such that the degree of Q(X, Y ) as a polynomial in Y is strictly less than i − 1. Applying this r times, we obtain ( 0 if i 6= r, X r .X r−i Y i = r r!X if i = r. Hence, as det s = −1, we have X r t.v 0 = X r sΠ.v 0 = X r s.v = (−1)b r!v, and by symmetry, we obtain X p−1−r t.v = (−1)a (p − 1 − r)!v 0 . Note that the condition 0 ≤ r ≤ p − 1 implies r! and (p − 1 − r)! are nonzero in E. Lemma 10.8: We have +

N I = Ev, +

+

(N 0 )I = Ev 0 . +

Remark 10.9: Note that N I and (N 0 )I are both nonzero, since I + is a pro-p group.

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Proof: We prove the claim for N ; the proof for N 0 is identical. By Lemma 10.7, we have v = (∗)X r tX p−1−r t.v = (∗)X r+p(p−1−r) t2 .v, where (∗) denotes an unspecified nonzero constant. Iterating this, we obtain v = (∗)X en t2n .v, where en = (r + p(p − 1 − r))(1 + p2 + . . . + p2n−2 ). Since v ∈ π I(1) , we have X .v = 0; hence, X en t2n .v 6= 0, but X en +1 t2n .v = 0. Let Nn := hI + t2n .viE ⊂ N . As I + ∼ = Zp , we have  Zp 11 I+ = = (1 + X )Zp . 01 Since we may express the action of I + in terms of the operator X , we have an isomorphism E[X ]/(X en +1 ) 1

∼ = 7−→

Nn t2n .v +

that is compatible with the action of X . Note that NnI = Nn [X ], the X -torsion elements. The X -torsion elements of E[X ]/(X en +1 ) are one-dimensional (spanned by X en ), and correspond to the subspace EX en t2n .v = Ev of Nn . 2n Now, since t2n .v = (∗)X p e1 t2n+2 .v ∈ Nn+1 , we obtain a series of inclusions [ N0 ⊂ N1 ⊂ . . . ⊂ N = Nn . n≥0

This, combined with the fact that NnI + N I = Ev.

+

= Ev shown above, proves that

Remark 10.10: The identity I + = (1 + X )Zp relies crucially on the fact that Zp is pro-cyclic. There is no analogous such statement for the ring of integers in a finite extension of Qp . Lemma 10.11: The subspaces M, N, and N 0 are I-stable. Proof: Recall that M = hI + tN .(Ev ⊕ Ev 0 )iE . Therefore, by the Iwahori Decomposition (Lemma 10.5), we have I.M = hI + I 0 I − tN .(Ev ⊕ Ev 0 )iE .

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By Lemma 10.5, we have I 0 I − tn ⊂ tn I 0 I − for every n ≥ 0, and moreover, the space Ev ⊕ Ev 0 is I-stable by definition. Hence I.M ⊂ hI + tN .(Ev ⊕ Ev 0 )iE = M. The same proof applies to N and N 0 . Lemma 10.12: We have t.v ∈ N 0 and t.v 0 ∈ N . Proof: By Lemma 10.7, we have t.v = (∗)X pr t2 .v 0 ∈ N 0 . The other claim follows by symmetry. Lemma 10.13: The K-subrepresentation generated by v is contained in M ; that is, V = hK.viE ⊂ M. Proof: The Bruhat Decomposition for Gp = GL2 (Fp ) states Gp = Bp t Bp sBp , which we inflate to a decomposition of K = GL2 (Zp ): K = I t IsI. Since I acts on v by a character, we obtain hK.viE = hI.viE + hIsI.viE = Ev + hIs.viE = Ev + hItΠ−1 .viE = Ev + hIt.v 0 iE . By Lemma 10.12, t.v 0 is contained in N ⊂ M , and by Lemma 10.11, It.v 0 is contained in M . Lemma 10.14: The space M decomposes as M = N ⊕ N 0. Proof: We clearly have N + N 0 ⊂ M ; we prove the opposite inclusion. This follows easily from I + tN = I + t2N ∪ I + t2N t,

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along with Lemma 10.12 and that N and N 0 are I + t2N -stable. Assume now that N ∩ N 0 6= 0. On the one hand, I + is a pro-p group, + and therefore we obtain (N ∩ N 0 )I 6= 0; on the other hand, Lemma 10.8 + + + gives (N ∩ N 0 )I = N I ∩ (N 0 )I = Ev ∩ Ev 0 = 0, a contradiction. We are now ready to finish the proof of Proposition 10.6. Proof: (Proof of Proposition 10.6) Part (2) of the proposition follows from Lemma 10.13. We prove part (1). It follows from Lemma 10.5 that M is I 0 I + tN -stable. Let M 0 be a nonzero submodule of M , stable by I 0 I + tN . By Lemmas 10.8 and 10.14, we have +

M I = Ev ⊕ Ev 0 . +

Note that (M 0 )I 6= 0. We consider two cases. • If 0 < r < p − 1, then I 0 acts on v and v 0 by distinct characters. This + implies that either v or v 0 is contained in (M 0 )I ⊂ M 0 , and the relation v = (∗)X r t.v 0 (resp. v 0 = (∗)X p−1−r t.v) implies that both vectors must be contained in M 0 . Hence M 0 = M . + • If r = 0 or r = p − 1, then λv + µv 0 ∈ (M 0 )I ⊂ M 0 for some λ, µ ∈ E, not both zero. We assume λµ 6= 0, else we may proceed as above to conclude M 0 = M . Lemma 10.7 shows that applying X p−1 t to λv + µv 0 gives an element of M 0 , equal to either (∗)v or (∗)v 0 . Hence, we proceed as above and conclude M 0 = M . All that remains is to show π is admissible, which will occupy the remainder of these notes. To do this, we’ll use the following lemmas. Lemma 10.15: Any quotient of an admissible I + -representation is admissible. Proof: (Sketch of proof) This comes down to the fact that EJI + K ∼ = EJX K is noetherian, where EJI + K denotes the completed group ring of I + . Suppose that M is an admissible I + -representation. As M is smooth, M is naturally an EJX K-module, where X = ( 10 11 ) − 1, as above. Hence M ∗ := HomE (M, E) inherits an EJX K-module structure as well. It is easy to show that ∼

M ∗ /X M ∗ −→ (M [X ])∗ . Therefore M is admissible as an I + -representation if and only if M [X ] = + M I is finite dimensional (by Proposition 9.2), if and only if M ∗ /X M ∗ is

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finite-dimensional. As M ∗ is separated for the X -adic topology (exercise), the latter condition is equivalent to M ∗ being finitely generated as an EJX Kmodule, by a variant of Nakayama’s Lemma (see [20], Theorem 8.4). Thus, if M − N with M admissible, then N ∗ ,−→ M ∗ and N ∗ is finitely generated, as EJX K is noetherian. Let σ := N + Π.N 0 . Then σ is G+ -stable by Proposition 4.12 of [21], where G+ := {g ∈ G : det g ∈ p2Z · Z× p }. Lemma 10.16: The G+ -representation σ is admissible. Proof: Consider the exact sequence of I-representations 0 −→ Π.N 0 −→ σ −→ N/(N ∩ Π.N 0 ) −→ 0.

(10.3)

We know that (Π.N 0 )I(1) ∼ = (N 0 )I(1) is finite-dimensional by Lemma 10.8. On the other hand, N is I + -admissible; hence N/(N ∩ Π.N 0 ) is I + admissible by Lemma 10.15, and thus also I(1)-admissible. It follows from (10.3) that σ I(1) is finite-dimensional, and we are done by Proposition 9.2.

Proof: (Proof of Theorem 10.1 (admissibility)) Note that (G : G+ ) = 2 and that G = hG+ , Πi. It follows that σ + Π.σ = π, as this is a nonzero G-subrepresentation and we have already shown that π is irreducible. Similarly, σ ∩ Π.σ equals either 0 or π, hence π = σ ⊕ Π.σ or π = σ. In either case, we have dimE π I(1) < ∞ by Lemma 10.16. (In fact, σ ∩ Π.σ = 0 by Corollary 6.5 in [21].)

We remark that this is not Breuil’s original proof; his method relies on computing I(1) (indG KZ (V )/(T))

using explicit calculations with the Bruhat-Tits tree X. References 1. Abe, N., “On a classification of irreducible admissible modulo p representations of a p-adic split reductive group.” Compos. Math., 149, (2013), no. 12, 2139-2168. 2. Abe, N., Henniart, G., Herzig, F., and Vign´eras, M.-F., “A classification of irreducible admissible mod p representations of p-adic reductive groups.” Preprint (2014); Available at http://arxiv.org/abs/1412.0737

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3. Barthel, L. and Livn´e, R., “Irreducible Modular Representations of GL2 of a Local Field.” Duke Math J., 75, (1994), no. 2, 261-292. 4. Barthel, L. and Livn´e, R., “Modular Representations of GL2 of a Local Field: The Ordinary, Unramified Case.” J. Number Theory, 55, (1995), no. 1, 1-27. 5. Berger, L., “Central characters for smooth irreducible modular representations of GL2 (Qp ).” Rend. Semin. Mat. Univ. Padova, 128, (2012), 1-6. 6. Breuil, C., “Sur quelques repr´esentations modulaires et p-adiques de GL2 (Qp ) I.” Compos. Math., 138, (2003), 165-188. 7. Breuil, C., “Sur quelques repr´esentations modulaires et p-adiques de GL2 (Qp ) II.” J. Inst. Math. Jussieu, 2, (2003), 1-36. 8. Breuil, C., “The emerging p-adic Langlands programme.” Proceedings of I.C.M. 2010, Vol. II, (2010), 203-230. 9. Carter, R.W. and Lusztig, G., “Modular Representations of Finite Groups of Lie Type.” Proc. London Math. Soc., 32, (1976), 347-384. 10. Colmez, P., “Repr´esentations de GL2 (Qp ) et (ϕ, Γ)-modules.” Ast´erisque 330, (2010), 281-509. 11. Emerton, M., “On a class of coherent rings, with applications to the smooth representation theory of GL2 (Qp ) in characteristic p.” Preprint (2008); Available at http://www.math.uchicago.edu/∼emerton/pdffiles/frob.pdf 12. Emerton, M., “Local-global compatibility in the p-adic Langlands programme for GL2/Q .” Preprint (2010); Available at http://www.math.uchicago.edu/∼emerton/pdffiles/lg.pdf 13. Harris, M. and Taylor, R., “The geometry and cohomology of some simple Shimura varieties.” Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton, NJ, (2001). 14. Henniart, G., “Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique.” Invent. Math., 113, (2000), no. 2, 439-455. 15. Henniart, G., and Vign´eras, M.-F., “A Satake isomorphism for representations modulo p of reductive groups over local fields.” To appear in J. Reine Angew. Math., (2013). 16. Herzig, F., “A Satake isomorphism in characteristic p.” Compos. Math., 147, (2011), no. 1, 263-283. 17. Herzig, F., “The classification of irreducible admissible mod p representations of a p-adic GLn .” Invent. Math., 186, (2011), no. 2, 373-434. 18. Herzig, F., “The mod p representation theory of p-adic groups.” Notes for a graduate course at The Fields Institute, typed by C. Johansson. Notes available at http://www.math.toronto.edu/∼herzig/modpreptheory.pdf 19. Kisin, M., “Deformations of GQp and GL2 (Qp ) representations.” Ast´erisque, 330, (2010), 511-528. 20. Matsumura, H., “Commutative Ring Theory.” Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, United Kingdom, (1989). 21. Paˇsk¯ unas, V., “Extensions for supersingular representations of GL2 (Qp ).” Ast´erisque, 331, (2010), 317-353. 22. Paˇsk¯ unas, V., “The image of Colmez’s Montreal functor.” Publ. Math. Inst. ´ Hautes Etudes Sci., 118 (2013), 1-191.

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23. Scholze, P., “The Local Langlands Correspondence for GLn over p-adic fields.” Invent. Math., 192 (2013), no. 3, 663-715. 24. Serre, J.-P., “Arbres, amalgames, SL2 .” Ast´erisque, 46, (1977). 25. Tits, J., “Reductive Groups over Local Fields.” Automorphic Forms, Representations and L-function, Proc. Symp. Pure Math. XXXIII, (1979), 29-69.

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REPRESENTATION THEORY AND COHOMOLOGY OF KHOVANOV–LAUDA–ROUQUIER ALGEBRAS

Alexander S. Kleshchev Department of Mathematics University of Oregon Eugene, OR 97403, USA [email protected]

This expository chapter is based on the lectures given at the program ‘Modular Representation Theory of Finite and p-Adic Groups’ at the National University of Singapore. We are concerned with recent results on representation theory and cohomology of KLR algebras, with emphasis on standard module theory.

Contents 1 Set Up and Motivation 1.1 KLR algebras 1.2 Some motivation 2 Basic Representation Theory of KLR Algebras 2.1 Semiperfect and Laurentian algebras 2.2 Formal characters 2.3 Crystal operators and extremal words 2.4 Khovanov–Lauda–Rouquier categorification 3 Standard Module Theory 3.1 Convex orders and cuspidal systems 3.2 Standard modules 3.3 Restrictions of proper standard modules 3.4 Classification of irreducible modules 3.4.1 The case π 6= (ρn ) 3.4.2 The case π = (ρ) 3.4.3 The case π = (ρn ) 3.5 Reduction modulo p 3.6 PBW bases and canonical bases 3.7 Cuspidal modules and dual PBW bases

109

110 110 113 117 117 119 120 123 124 124 126 127 129 130 131 132 132 133 134

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4 Homological Properties of KLR Algebras 4.1 Finiteness of global dimension 4.2 Standard modules 4.3 Homological properties of standard modules 5 Projective Resolutions of Standard Modules 5.1 Minimal pairs 5.2 Projective resolutions 6 Type A 6.1 Set up 6.2 Basic algebra Bn 6.3 Skew shapes 6.4 The elements ψλ,µ 6.5 Irreducibles and PIMs for cuspidal blocks 6.6 Basic algebras of cuspidal blocks 6.7 Resolutions for cuspidal blocks 6.8 Resolving proper standard and costandard modules References

136 136 137 139 140 140 142 144 144 146 148 150 154 155 158 160 162

1. Set Up and Motivation This expository chapter is based on the lectures given at the program ‘Modular Representation Theory of Finite and p-Adic Groups’ at the National University of Singapore. We are concerned with recent results on representation theory and cohomology of KLR algebra, with emphasis on standard module theory, as developed in [21], [13], [27], [17], [5]. Some proofs are given, but often we just review or illustrate the results. Other topics in the theory of KLR algebras are nicely reviewed in [1]. 1.1. KLR algebras In this chapter we will be mainly concerned with KLR algebras of finite Lie type. So let C = (cij )i,j∈I be a Cartan matrix of finite type. As in [10, §1.1], let (h, Π, Π∨ ) be a realization of the Cartan matrix C, so we have simple roots {αi | i ∈ I}, simple coroots {αi∨ | i ∈ I}, and a bilinear form (·, ·) on h∗ such that cij = 2(αi , αj )/(αi , αi ) for all i, j ∈ I. We normalize (·, ·) so that (β, β) = 2 if β is a short root. The fundamental dominant weights {Λi | i ∈ I} have the property that hΛi , αj∨ i = δi,j , where h·, ·i is the natural pairing between h∗ and h. We have the set of P dominant weights P+ = i∈I Z≥0 · Λi and the positive part of the root L lattice Q+ := i∈I Z≥0 αi . For α ∈ Q+ , we write ht(α) for the sum of its coefficients when expanded in terms of the αi ’s.

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111

Denote A := Z[q, q −1 ] for an indeterminate q. For n ∈ Z≥0 , define [n]q :=

n Y q n − q −n ! , [n] := [m]q . q q − q −1 m=1

Given in addition α ∈ Q+ and a simple root αi , we let dα := (α, α)/2 and set qα := q dα , [n]α := [n]qα , [n]!α := [n]!qα , qi := qαi [n]i := [n]αi , [n]!i := [n]!αi . Sequences of elements of I will be called words. The set of all words is denoted hIi. If i = i1 . . . id is a word, we denote |i| := αi1 + · · · + αid ∈ Q+ . We refer to |i| as the content of the word i. For any α ∈ Q+ we denote hIiα := {i ∈ hIi | |i| = α}. If α is of height d, then the symmetric group Sd with simple permutations s1 , . . . , sd−1 acts transitively on hIiα from the left by place permutations. Let F be an arbitrary field. Define the polynomials {Qij (u, v) ∈ F [u, v] | i, j ∈ I} in the variables u, v as follows. Choose signs εij for all i, j ∈ I with cij < 0 so that εij εji = −1. Then set:  if i = j; 0 Qij (u, v) := 1 (1.1) if cij = 0;  εij (u−cij − v −cji ) if cij < 0. Fix α ∈ Q+ of height d. The KLR-algebra Rα is an associative graded unital F -algebra, given by the generators {1i | i ∈ hIiα } ∪ {y1 , . . . , yd } ∪ {ψ1 , . . . , ψd−1 } and the following relations for all i, j ∈ hIiα and all admissible r, t: P 1i 1j = δi,j 1i , i∈hIi 1i = 1; α

yr 1i = 1i yr ;

yr yt = yt yr ;

(1.2)

(1.3) (1.4)

ψr 1i = 1sr i ψr ;

(1.5)

(yt ψr − ψr ysr (t) )1i = δir ,ir+1 (δt,r+1 − δt,r )1i ;

(1.6)

ψr2 1i = Qir ,ir+1 (yr , yr+1 )1i

(1.7)

ψr ψt = ψt ψr

(|r − t| > 1);

(1.8)

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(ψr+1 ψr ψr+1 − ψr ψr+1 ψr )1i =δir ,ir+2

Qir ,ir+1 (yr+2 , yr+1 ) − Qir ,ir+1 (yr , yr+1 ) 1i . yr+2 − yr

(1.9)

The grading on Rα is defined by setting: deg(1i ) = 0,

deg(yr 1i ) = (αir , αir ),

deg(ψr 1i ) = −(αir , αir+1 ).

These algebras were defined in [14, 15, 31]. It is pointed out in [15] and [31, §3.2.4] that up to isomorphism the graded F -algebra Rα depends only on the Cartan matrix and α. Fix in addition a dominant weight Λ ∈ P+ . The corresponding cycloΛ tomic KLR algebra Rα is the quotient of Rα by the following ideal: hΛ,α∨ i i

JαΛ := (y1

1

1i | i = i1 . . . id ∈ hIiα ).

(1.10)

For a graded algebra R, denote by R-Mod the abelian category of all graded left R-modules, denoting (degree-preserving) homomorphisms in this category by homR . We write ∼ = for the isomorphism in this category, and ' for the isomorphism in the category of usual modules. Let R-mod denote the abelian subcategory of all finite dimensional graded left R-modules and R-proj denote the additive subcategory of all finitely generated projective graded left R-modules. We also consider the Grothendieck groups [R-mod] and [R-proj]. We view [R-mod] and [R-proj] as A -modules via q m [M ] := [q m M ], where q m M denotes the module obtained by shifting the grading up by m: (q m M )n = P Mn−m . More generally, given a formal Laurent series f (q) = n∈Z fn q n L with coefficients fn ∈ Z≥0 , f (q)V denotes n∈Z q n V ⊕fn . If V is a locally finite dimensional graded vector space (i.e. the dimension of each graded comP ponent Vn is finite), its graded dimension is dimq V := n∈Z (dim Vn )q n . Given M, L ∈ R-mod with L irreducible, we write [M : L]q for the corP n responding graded composition multiplicity, i.e. [M : L]q := n∈Z an q , n where an is the multiplicity of q L in a graded composition series of M . Given α, β ∈ Q+ , we set Rα,β := Rα ⊗ Rβ . There is an injective nonunital algebra homomorphism Rα,β ,→ Rα+β , 1i ⊗ 1j 7→ 1ij , where ij is the concatenation of i and j. The image of the identity element of Rα,β P under this map is 1α,β := i∈hIiα , j∈hIiβ 1ij . We consider the induction and restriction functors: Indα,β := Rα+β 1α,β ⊗Rα,β ? : Rα,β -Mod → Rα+β -Mod, Resα,β := 1α,β Rα+β ⊗Rα+β ? : Rα+β -Mod → Rα,β -Mod,

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which preserve the categories of finite dimensional and finitely generated projective modules. For M ∈ Rα -Mod, N ∈ Rβ -Mod, denote M ◦ L N := Indα,β M  N. Then [R-proj] := α∈Q+ Rα -proj and [R-mod] := L R -mod are Q -graded A -algebras, with multiplication coming + α∈Q+ α from the induction product ◦. Example 1.11: For m ≥ 1 and i ∈ I, the KLR algebra Rmαi is the nil-Hecke algebra NHm , which is given by generators y1 , . . . , ym and ψ1 , . . . , ψm−1 and relations: yi yj = yj yi , ψi yj = yj ψi for j 6= i, i + 1, ψi yi+1 = yi ψi + 1, yi+1 ψi = ψi yi + 1, ψi2 = 0, together with the usual type Am braid relations for ψ1 , . . . , ψm−1 . It is well known that the nil-Hecke algebra is a matrix algebra over its center; see e.g. [32, §2] or [19, §4] for recent expositions. Moreover, writing w0 for the longest element of Sm , the degree zero element m−1 em := y2 y32 · · · ym ψw0

(1.12)

m(m−1)/2

is a primitive idempotent, hence P (αim ) := qi Rmαi em is an indecomposable projective Rmαi -module. The degree shift has been chosen so that irreducible head L(αim ) of P (αim ) has graded dimension [m]!i . Thus Rmαi ∼ = [m]!i P (αim ) as a left module. 1.2. Some motivation The first reason why representation theory of KLR algebras is interesting is that it can be used to categorify quantum groups. One way to make this statement more precise is as follows. Let f be the quantized enveloping algebra over the field Q(q) associated to C with standard generators {θi | L i ∈ I}, cf. [25]. It is naturally Q+ -graded: f = α∈Q+ fα . Khovanov and Lauda showed that there is a unique Q+ -graded algebra isomorphism ∼

γ : f → Q(q) ⊗A [R-proj], θi 7→ [Rαi ],

(1.13)

where Rαi is the left regular module over the algebra Rαi . If C is symmetric and F has characteristic zero, Rouquier [32] and Varagnolo and Vasserot [34] have shown further that γ maps the canonical basis of f to the basis for [R-proj] arising from the isomorphism classes of graded self-dual indecomposable projective modules. Taking a dual map to γ yields another algebra isomorphism ∼

γ ∗ : Q(q) ⊗A [R-mod] → f ∗ .

(1.14)

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If C is symmetric and F has characteristic zero, this sends the basis for [R-mod] arising from isomorphism classes of graded self-dual irreducible Rα -modules to the dual canonical basis for f . For some further details concerning Khovanov–Lauda–Rouquier categorification see §2.4. Another motivation for studying representation theory of KLR algebras is the following fact first proved in [3]: cyclotomic KLR algebras of finite and affine types A are explicitly isomorphic to blocks of cyclotomic Hecke algebras. The main reason this is interesting is that now we can transport the grading from KLR algebras to cyclotomic Hecke algebras, and the resulting grading on Hecke algebras turns out to be very important, see for example [4, 6, 9]. As yet another illustration, we now construct explicitly the irreducible modules for all semisimple cyclotomic Hecke algebras (both degenerate and non-degenerate). This is of course just a version of Young’s orthogonal form, but the reader might appreciate how much simpler the construction via KLR algebras is. We give the necessary definitions. Until the end of this subsection we assume that the Cartan matrix C is either of type A∞ (this is equivalent to (1) working with sufficiently large finite type A) or of affine type Ae−1 (above we only defined the KLR algebras for finite Lie types, but the definition for (1) Ae−1 is really the same). When C = A∞ , we set e = 0 so that in both finite and affine types A we can identify the set I with Z/eZ. Fix an ordered tuple κ = (k1 , . . . , kl ) ∈ I l such that Λ = Λk1 + · · · + Λkl . An l-multipartition of d is an ordered l-tuple of partitions Pl µ = (µ(1) , . . . , µ(l) ) such that m=1 |µ(m) | = d. We refer to µ(m) as the mth component of µ. Let Pdκ be the set of all l-multipartitions of d. Of course, Pdκ only depends on l, and not on κ, but as soon as we consider contents of nodes of multipartitions, the dependence on κ becomes essential. The Young diagram of the multipartition µ = (µ(1) , . . . , µ(l) ) ∈ P κ is {(a, b, m) ∈ Z>0 × Z>0 × {1, . . . , l} | 1 ≤ b ≤ µ(m) a }.

The elements of this set are the nodes or boxes of µ. More generally, a node is any element of Z>0 × Z>0 × {1, . . . , l}. Usually, we identify the multipartition µ with its Young diagram and visualize it as a column vector

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of Young diagrams. For example, ((3, 1), ∅, (4, 2)) is the Young diagram

∅

To each node A = (a, b, m) we associate its content, which is an element of I = Z/eZ defined as follows cont A := contκ A = km + (b − a) (mod e) . P Define the weight of µ to be wt(µ) := A∈µ αcont A ∈ Q+ . For α ∈ Q+ , denote Pακ := {µ ∈ P κ | wt(µ) = α}. We call a partition separating if for any two nodes A = (a1 , a2 , m), B = (b1 , b2 , n) of µ, we have that contκ A = contκ B implies that A and B are on the same diagonal of the same component, i.e. m = n and a2 − a1 = b2 − b1 . Let µ = (µ(1) , . . . , µ(l) ) ∈ Pdκ . A µ-tableau T = (T(1) , . . . , T(l) ) is obtained by inserting the integers 1, . . . , d into the boxes of µ, allowing no repeats. The group Sd acts on the set of µ-tableaux from the left by acting on the entries of the tableaux. Let Tµ be the µ-tableau in which the numbers 1, 2, . . . , d appear in order from left to right along the successive rows, working from top row to bottom row. Set iT = iκ,T = iT1 . . . iTd ∈ I d ,

(1.15)

where iTr is the content of the node occupied by r in T for all 1 ≤ r ≤ d. A µ-tableau T is called standard if its entries increase from left to right along the rows and from top to bottom along the columns within each component of T. Let St(µ) be the set of standard µ-tableaux. Let α ∈ Q+ be of height d and fix a separating multipartition µ ∈ Pακ . L Consider a formal vector space S(µ) := T∈St(µ) F · vT on basis {vT | T ∈ St(µ)} labeled by the standard µ-tableaux and concentrated in degree zero. Λ Define the following action of the generators of Rα on S(µ):  vsr T if sr T is standard, 1i vT = δi,iT vT , ys vT = 0, ψr vT = (1.16) 0 otherwise. Theorem 1.17: Suppose that µ is separating. The formulas (1.16) define a Λ (graded) action of Rα on S(µ). Moreover, S(µ) is an irreducible Rα -module, ∼ and S(µ) = 6 S(ν) whenever µ 6= ν.

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Proof: To prove the first statement we need to observe that the defining relations of Rα hold for the linear operators defined by (1.16). The relations (1.3)–(1.5) are clear. To see that (1.6) holds it suffices to observe that in a standard tableau, r and r + 1 can never occupy boxes on the same diagonal of the same component. As µ is separating, it follows that iTr 6= iTr+1 for all T ∈ St(µ), which implies (1.6). To see (1.7), if r and r + 1 occupy adjacent nodes in T, then sr T is not standard, and in this case we get ψr2 vT = 0 = QiTr ,iTr+1 (yr , yr+1 )vT as required. On the other hand, if r and r + 1 do not occupy adjacent nodes in T, then sr T is standard, ciTr ,iTr+1 = 0, and ψr2 vT = vT = QiTr ,iTr+1 (yr , yr+1 )vT , again as required. The relation (1.8) holds trivially. Finally, to check the relation (1.9), it is enough to notice that we never have iTr = iTr+2 for a standard µ-tableau T under the assumption that µ is separating. To see that S(µ) is irreducible, note first that S 6= T for standard µtableaux S and T implies that iT 6= iS , so acting with the idempotents 1i yields projections to each 1-dimensional subspace F · vT spanned by the basis elements vT . So to prove the irreducibility of S(µ) it suffices to show that for any standard µ-tableaux T and S, there exists a series of admissible transpositions which takes T to S, which means that there exist 1 ≤ k1 , . . . , kl < d such that skl skl−1 . . . sk1 T = S and skm skm−1 . . . sk1 T is standard for all m = 1, . . . , l. The existence of such a sequence follows from the following: Claim. For any standard µ-tableau T there exists a series of admissible transpositions which takes T to Tµ . To prove the Claim, let A be the last box of the last row of µ. In Tµ , the box A is occupied by d. In T, the box A is occupied by some number k ≤ d. Note that in T, the numbers k + 1 and k do not lie on adjacent diagonals. So we can apply an admissible transposition to swap k and k + 1, then to swap k + 1 and k + 2, etc. As a result, we get a new standard µ-tableau in which A is occupied by d. Next, remove A together with d, and apply induction. The pair (Λ, d) ∈ P+ × Z≥0 is separating if all miltipartitions µ ∈ Pdκ are separating. This notion is well-defined, since it does not depend on the choice of κ = (k1 , . . . , kl ) such that Λ = Λk1 + · · · + Λkl . If (Λ, d) is separating, then all multipartitions µ ∈ Pdκ have different contents, and the L Λ Λ algebra α∈Q+ ,ht(α)=d Rα is a semisimple algebra, with each Rα being zero or simple. We have mentioned above that by the main result of [3], this al-

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L gebra is isomorphic to a cyclotomic Hecke algebra α∈Q+ ,ht(α)=d HαΛ . This cyclotomic Hecke algebra is semisimple if and only if (Λ, d) is separating. Thus in all cases where a cyclotomic Hecke algebra is semisimple, Theorem 1.17 yields an easy construction of all its irreducible representations via the isomorphism of [3]. 2. Basic Representation Theory of KLR Algebras 2.1. Semiperfect and Laurentian algebras We begin with some generalities on graded algebras. All gradings will be Z-gradings. Let H be a graded algebra over a ground field F . All modules, ideals, etc. are assumed to be graded, unless otherwise stated. In particular, rad V (resp. soc V ) is the intersection of all maximal (graded) submodules (resp. the sum of all irreducible (graded) submodules) of V . All idempotents are assumed to be degree zero. We denote by N (H) the (graded) Jacobson radical of H. For modules U and V , we write homH (U, V ) for homogeneous H-module L homomorphisms, and set HOMH (U, V ) := n∈Z HOMH (U, V )n , where HOMH (U, V )n := homH (q n U, V ) = homH (U, q −n V ). We define extdH (U, V ) and EXTdH (U, V ) similarly. If U is finitely generated, then HOMH (U, V ) = HomH (U, V ), where HomH (U, V ) denoted the homomorphisms in the ungraded category. We have a similar fact for Extd provided U has a resolution by finitely generated projective modules, in particular if U is finitely generated and H is Noetherian. For an H-module V denote by Z(V ) the largest submodule of V with the trivial zero degree component, i.e. Z(V )0 = 0. Define V := V /Z(V ). Lemma 2.1: [28] Let V be an irreducible graded H-module, and W be an irreducible H0 -module. (i) If Vn 6= 0 for some n ∈ Z, then Vn is irreducible as an H0 -module. (ii) The graded H-module X := H ⊗H0 W is irreducible, and X0 ∼ = W as H0 -modules. (iii) If V0 6= 0, then we have V ∼ = H ⊗H 0 V 0 . Proof: (i) is clear. (ii) First of all note that X0 = (H ⊗H0 W )0 ∼ = (H ⊗H0 W )0 ∼ = H0 ⊗H0 W ∼ = W 6= 0.

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Note that H ⊗H0 W is generated as an H-module by its degree zero part 1 ⊗ W , hence X is also generated by its degree zero part X0 . Moreover, X0 is irreducible as an H0 -module, so X is generated by any non-zero vector in X0 . Now to prove the irreducibility of X it suffices to take any homogeneous vector v, say of degree n, and prove that H−n v 6= 0. Well, otherwise Hv is a graded submodule of X which avoids X0 , a contradiction. (iii) By (i) we have that V0 is an irreducible H-module, and we have that the H-module H ⊗H0 W is isomorphic to V , because it is irreducible by (ii) and surjects onto V . Now we assume that H is (graded) semiperfect, i.e. every finitely generated (graded) H-module has a (graded) projective cover. By [7], this is equivalent to H0 being semiperfect, and is also equivalent to the fact that the following two properties hold: (1) H/N (H) is (graded) semisimple Artinian; (2) idempotents lift from H/N (H) to H. We fix a complete irredundant set of irreducible H-modules up to isomorphism and degree shift: {L(π) | π ∈ Π}, and for each π ∈ Π, we fix a projective cover P (π) of L(π). By the semiperfectness of H, we have H/N (H) is (graded) left Artinian, so the set Π is finite. Moreover, if EndH (L(π)) is finite dimensional over F then by the graded version of the Wedderburn-Artin Theorem [28, 2.10.10] the irreducible module L(π) is finite dimensional. Finally, if EndH (L(π)) = F for all π ∈ Π, i.e. if H is Schurian, then H/N (H) is a finite direct product of (graded) matrix algebras over F and we have M (dimq L(π))P (π). HH = π∈Π

A graded algebra H is called Laurentian if each of its graded components Hn is finite dimensional and Hn = 0 for n  0. In this case dimq H as well as dimq V for any finitely generated H-module are Laurent series. Lemma 2.2: Let H be a Laurentian algebra. Then: (i) H has only finitely many irreducible modules up to isomorphism and degree shift; (ii) all irreducible H-modules are finite dimensional; (iii) H is semiperfect.

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Proof: (i) Since H0 is finite dimensional, it has only finitely many irreducible modules. It now follows from Lemma 2.1 that up to a degree shift, H has only finitely many irreducible graded modules. (ii) Let V be an irreducible H-module. Then each Vn is irreducible over H0 by Lemma 2.1. So each Vn is finite dimensional. On the other hand, since V is cyclic and H is Laurentian, V has to be bounded below. Now, V also has to be bounded above, since it is irreducible and H is Laurentian. (iii) follows from [7, Theorem 3.5] since H0 is semiperfect being finite dimensional. 2.2. Formal characters Fix α ∈ Q+ with ht(α) = d. The results of the previous subsection apply to the KLR algebra Rα , since it is easily seen to be Schurian, see e.g. [14, Corollary 3.19], and is also Laurentian for example in view of the following Basis Theorem: Theorem 2.3: [14, Theorem 2.5], [31, Theorem 3.7] For each element w ∈ Sd fix a reduced expression w = sr1 . . . srm and set ψw := ψr1 . . . ψrm . The elements {ψw y1m1 . . . ydmd 1i | w ∈ Sd , m1 , . . . , md ∈ Z≥0 , i ∈ hIiα } form an F -basis of Rα . There exists a homogeneous algebra anti-involution τ : Rα −→ Rα ,

1i 7→ 1i ,

ψs 7→ ψs (2.4) L for all i ∈ hIiα , 1 ≤ r ≤ d, and 1 ≤ s < d. If M = d∈Z Md is a finite dimensional graded Rα -module, then the graded dual M ~ is the graded Rα -module such that (M ~ )n := HomF (M−n , F ), for all n ∈ Z, and the Rα action is given by (xf )(m) = f (τ (x)m), for all f ∈ M ~ , m ∈ M, x ∈ Rα . For every irreducible module L, there is a unique choice of the grading shift so that we have L~ ∼ = L [14, §3.2]. When speaking of irreducible Rα -modules we often assume by fiat that the shift has been chosen in this way. For i ∈ hIiα and M ∈ Rα -mod, the i-word space of M is Mi := 1i M. We have the word space decomposition: M M= Mi . i∈hIiα

yr 7→ yr ,

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We say that i is a word of M if Mi 6= 0. Note from the relations that ψr Mi ⊂ Msr i . Define the (graded formal) character of M as follows: X chq M := (dimq Mi )i ∈ A hIiα . i∈hIiα

The character map chq : Rα -mod → A hIiα factors through to give an injective A -linear map chq : [Rα -mod] → A hIiα , see [14, Theorem 3.17]. Let i = i1 . . . id and j = id+1 . . . id+f be two elements of hIi. Define the quantum shuffle product: X i ◦ j := q −e(σ) iσ(1) . . . iσ(d+f ) ∈ A hIi, where the sum is over all σ ∈ Sd+f such that σ −1 (1) < · · · < σ −1 (d) and σ −1 (d + 1) < · · · < σ −1 (d + f ), and X e(σ) := ciσ(k) ,iσ(m) . k≤dσ −1 (m)

This defines an A -algebra structure on the A -module A hIi, which consists of all finite formal A -linear combinations of elements i ∈ hIi. In view of [14, Lemma 2.20], we have chq (M1 ◦ · · · ◦ Mn ) = chq (M1 ) ◦ · · · ◦ chq (Mn ).

(2.5)

2.3. Crystal operators and extremal words The theory of crystal operators has been developed in [14], [22] and [11] following ideas of Grojnowski [8], see also [16]. We review necessary facts for the reader’s convenience. Let α ∈ Q+ and i ∈ I. By Example 1.11, Rnαi is a nil-Hecke algebra with unique irreducible module L(αin ) with dimq L(αin ) = [n]!i . We have functors Rα−α

ei : Rα -mod → Rα−αi -mod, M 7→ ResRα−αi

i

,αi

◦ Resα−αi ,αi M,

fi : Rα -mod → Rα+αi -mod, M 7→ Indα,αi M  L(αi ). If L ∈ Rα -mod is irreducible, we define f˜i L := head(fi L),

e˜i L := soc(ei L).

A fundamental fact is that f˜i L is again irreducible and e˜i L is irreducible or zero. We refer to e˜i and f˜i as the crystal operators. These are operators on

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B∪{0}, where B is the set of isomorphism classes of irreducible Rα -modules for all α ∈ Q+ . Define wt : B → P, [L] 7→ −α if L ∈ Rα -mod. Theorem 2.6: [22] The set B with the operators e˜i , f˜i and the function wt is the crystal graph of the negative part Uq (n− ) of the quantized enveloping algebra of g of Lie type C. For any M ∈ Rα -mod, we define εi (M ) := max{k ≥ 0 | eki (M ) 6= 0}. Then εi (M ) is also the length of the longest ‘i-tail’ of words of M , i.e. the maximum of k ≥ 0 such that jd−k+1 = · · · = jd = i for some word j = j1 . . . jd of M . Proposition 2.7: [14], [22] Let L be an irreducible Rα -module, i ∈ I, and ε = εi (L). (i) e˜i f˜i L ∼ = L and if e˜i L 6= 0 then f˜i e˜i L ∼ = L; (ii) ε = max{k ≥ 0 | e˜ki (L) 6= 0}; (iii) Resα−εαi ,εαi L ∼ = e˜εi L  L(αiε ). Let i ∈ I. Consider the map θi∗ : hIi → hIi such that for j = j1 . . . jd ∈ hIi, we have  j1 , . . . , jd−1 if jd = i; ∗ θi (j) = (2.8) 0 otherwise. We extend θi∗ by linearity to a map θi∗ : A hIi → A hIi. Let x be an element of A hIi. Define εi (x) := max{k ≥ 0 | (θi∗ )k (x) 6= 0}. A word ia1 1 . . . iab b ∈ hIi, with a1 , . . . , ab ∈ Z≥0 , is called extremal for x if
ab = εib (x), ab−1 = εib−1 ((θi∗b )ab (x)) , . . . , a1 = εi1 (θi∗2 )a2 . . . (θi∗b )ab (x) . A word ia1 1 . . . iab b ∈ hIiα is called extremal for M ∈ Rα -mod if it is an extremal word for chq M ∈ A hIi, in other words, if eai22 . . . e˜aibb M ). ab = εib (M ), ab−1 = εib−1 (˜ eaibb M ) , . . . , a1 = εi1 (˜ The following useful result, which is a version of [2, Corollary 2.17], describes the multiplicities of extremal word spaces in irreducible modules. We denote by 1F the trivial module F over the trivial algebra R0 ∼ = F. Lemma 2.9: Let L be an irreducible Rα -module, and i = ia1 1 . . . iab b ∈ hIiα be an extremal word for L. Then dimq Li = [a1 ]!i1 . . . [ab ]!ib , and ∼ f˜ab f˜ab−1 . . . f˜a1 1F . L= ib

ib−1

i1

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Moreover, i is not an extremal word for any irreducible module L0 ∼ 6 L. = Proof: Follows easily from Proposition 2.7, cf. [2, Theorem 2.16]. Corollary 2.10: Let M ∈ Rα -mod, and i = ia1 1 . . . iab b ∈ hIiα be an extremal word for M . Then we can write dimq Mi = m[a1 ]!i1 . . . [ab ]!ib for some a ab−1 m ∈ A . Moreover, if L ∼ . . . f˜ia11 1F and L~ ∼ = f˜ibb f˜ib−1 = L, then we have [M : L]q = m. Proof: Apply Lemma 2.9, cf. [2, Corollary 2.17]. Now we establish some useful ‘multiplicity-one results’. The first one shows that in every irreducible module there is a word space with a one dimensional graded component: Lemma 2.11: Let L be an irreducible Rα -module, and i = ia1 1 . . . iab b ∈ hIiα Pb be an extremal word for L. Set N := m=1 am (am − 1)(αim , αim )/4. Then dim 1i LN = dim 1i L−N = 1. Proof: This follows immediately from the equality dimq 1i L [a1 ]!i1 . . . [ab ]!ib , which comes from Lemma 2.9.

=

The following result shows that any induction product of irreducible modules always has a multiplicity one composition factor. Proposition 2.12: Suppose that n ∈ Z>0 and for r = 1, . . . , n, we have (r)

a

(r)

a

α(r) ∈ Q+ , an irreducible Rα(r) -module L(r) , and i(r) := i1 1 . . . ikk ∈ Pn (r) hIiα(r) is an extremal word for L(r) . Denote at := r=1 at for all 1 ≤ t ≤ k. Then j := ia1 1 . . . iakk is an extremal word for L(1) ◦ · · · ◦ L(n) , and the graded multiplicity of the ~-self-dual irreducible module ak−1 N ' f˜iakk f˜ik−1 . . . f˜ia11 1F

in L(1) ◦ · · · ◦ L(n) is q m , where P P (u) (t) m := − 1≤t0 . Then by Lemma 3.10, we have n(n−1)/2

that the module qρ

L◦n ρ is ~-self-dual. The result follows.

3.3. Restrictions of proper standard modules We recall the Mackey Theorem of Khovanov and Lauda [14, Proposition 2.18]. Given x ∈ Sn and γ = (γ1 , . . . , γn ) ∈ Qn+ , we denote xγ := (γx−1 (1) , . . . , γx−1 (n) ) ∈ Qn+ . X s(x, γ) := − (γm , γk ) ∈ Z. 1≤mx(k)

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Writing Rγ for Rγ1 ,...,γn , there is an obvious natural algebra isomorphism ϕx : Rxγ → Rγ permuting the components. Composing with this isomorphism, we get a functor x

Rγ -mod → Rxγ -mod, M 7→ ϕ M. Making an additional shift, we get a functor x

Rγ -mod → Rxγ -mod, M 7→ x M := q s(x,γ) (ϕ M ).

(3.12)

Theorem 3.13: Let γ = (γ1 , . . . , γn ) ∈ Qn+ and β = (β1 , . . . , βm ) ∈ Qm + with γ1 + · · · + γn = β1 + · · · + βm =: α. Then for any M ∈ Rγ -mod we have that Resβ Indγ M has filtration with factors of the form
; ... ; γn ; ... ; βm x(α) Resαγ11,...,α Indαβ11,...,α n ; ... ; α1 ,...,αn 1 ; ... ; αn ,...,αn M 1

1

m

m

1

m

1

m

with α = (αba )1≤a≤n, 1≤b≤m running over all tuples of elements of Q+ such Pm Pn that b=1 αba = γa for all 1 ≤ a ≤ n and a=1 αba = βb for all 1 ≤ b ≤ m, and x(α) is the permutation of mn which maps 1 2 n (α11 , . . . , αm ; α12 , . . . , αm ; . . . ; α1n , . . . , αm )

to 1 n (α11 , . . . , α1n ; α21 , . . . , α2n ; . . . ; αm , . . . , αm ).

We use the Mackey Theorem to study restrictions of proper standard modules: Proposition 3.14: Let π, σ ∈ Π(α). Then: ∼ Lσ . ¯ (i) Res|σ| ∆(σ) = ¯ (ii) Res|π| ∆(σ) 6= 0 implies π ≤ σ. ¯ Proof: Write π = (m1 , . . . , mN ), σ = (n1 , . . . , nN ). Let Res|π| ∆(σ) 6= 0. It ∼ ¯ suffices to prove that π ≥l σ or π ≤r σ implies that π = σ and Res|π| ∆(σ) = Lσ . We may assume that π ≥l σ, the case π ≤r σ being similar. We apply induction on ht(α). Pick the minimal a with ma 6= 0. Let π 0 = (0, . . . , 0, ma+1 , . . . , mN ) ∈ Π(α − ma ρa ) and σ 0 = (0, . . . , 0, na+1 , . . . , nN ) ∈ Π(α − na ρa ).

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¯ By Theorem 3.13, Res|π| ∆(σ) has filtration with factors of the form ma ρa ;|π 0 | Indκ1 ,...,κc ;γ V, where ma ρa = κ1 + · · · + κc , with κ1 , . . . , κc ∈ Q+ \ {0}, and γ is a refinement of |π 0 |. Moreover, the module V is obtained by twisting and degree shifting as in (3.12) of a module obtained by restriction of nN 1 to a parabolic which has κ1 , . . . , κc in the beginnings of Ln ρ1  · · ·  LρN the corresponding blocks. In particular, if V 6= 0, then for each b = 1, . . . , c we have that Resκb ,ρk −κb Lρk 6= 0 for some k = k(b) with nk 6= 0. Let 1 ≤ b ≤ c. If Resκb ,ρk −κb Lρk 6= 0, then by the definition of cuspidal modules, κb is a sum of roots  ρk . Moreover, since π ≥l σ and nk 6= 0, we have that ρk  ρa . Thus κb is a sum of roots  ρa . Using Lemma 3.1, we conclude that c = ma and κb = ρa = ρk(b) for all b = 1, . . . , c. Hence na ≥ ma . Since π ≥l σ, we conclude that na = ma , and a ¯ ¯ 0 Res|π| ∆(σ) ' L◦m ρa  Res|π 0 | ∆(σ ).

Since ht(α − ma ρa ) < ht(α), we can apply the inductive hypothesis. 3.4. Classification of irreducible modules We continue to work with a fixed convex preorder  on Φ+ . In this subsection we prove the following theorem: Theorem 3.15: For a given convex preorder, there exists a unique cuspidal system {Lρ | ρ ∈ Φ+ }. Moreover: ¯ (i) For every root partition π, the proper standard module ∆(π) has an irreducible head; denote this irreducible module L(π). (ii) {L(π) | π ∈ Π(α)} is a complete and irredundant system of irreducible Rα -modules up to isomorphism. (iii) L(π)~ ∼ = L(π). ¯ ¯ (iv) [∆(π) : L(π)]q = 1, and [∆(π) : L(σ)]q 6= 0 implies σ ≤ π. ∼ (v) Res|π| L(π) = Lπ and Res|σ| L(π) 6= 0 implies σ ≤ π. (vi) L◦n ρ is irreducible for all ρ ∈ Φ+ and all n ∈ Z>0 . The rest of §3.4 is devoted to the proof of Theorem 3.15, which goes by induction on ht(α). To be more precise, we prove the following statements for all α ∈ Q+ by induction on ht(α): (1) For each ρ ∈ Φ+ with ht(ρ) ≤ ht(α) there exists a unique up to isomorphism irreducible Rρ -module Lρ which satisfies the property (Cus) of Definition 3.2. Moreover, Lρ also satisfies the property (vi) of Theorem 3.15 if ht(nρ) ≤ ht(α).

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¯ (2) The proper standard modules ∆(π) for all π ∈ Π(α), defined as in (3.7) using the modules from (1), satisfy the properties (i)–(v) of Theorem 3.15. The induction starts with ht(α) = 0, and for ht(α) = 1 the theorem is also clear since Rαi is a polynomial algebra, which has only the trivial irreducible (graded) representation Lαi . The inductive assumption will stay valid throughout §3.4. 3.4.1. The case π 6= (ρn ) In the following proposition, we exclude the case where the proper standard module is of the form L◦n ρ . The excluded cases will be dealt with in §§3.4.2 and 3.4.3. Proposition 3.16: Let π = (m1 , . . . , mN ) ∈ Π(α), and suppose that there are 1 ≤ k 6= l ≤ N such that mk 6= 0 and ml 6= 0. (i) (ii) (iii) (iv) (v)

¯ ∆(π) has an irreducible head; denote this irreducible module L(π). If π 6= σ, then L(π) 6' L(σ). L(π)~ ∼ = L(π). ¯ ¯ [∆(π) : L(π)]q = 1, and [∆(π) : L(σ)]q 6= 0 implies σ ≤ π. Res|π| L(π) ' Lπ and Res|σ| L(π) 6= 0 implies σ ≤ π.

¯ Proof: (i) and (v) If L is an irreducible quotient of ∆(π) = Ind|π| Lπ , then by adjointness of Ind|π| and Res|π| and the irreducibility of the R|π| module Lπ , which holds by the inductive assumption, we conclude that Lπ is a submodule of Res|π| L. On the other hand, by Proposition 3.14(i) the ¯ multiplicity of Lπ in Res|π| ∆(π) is 1, so (i) follows. Note that we have also proved the first statement in (v), while the second statement in (v) follows from Proposition 3.14(ii) and the exactness of the functor Res|π| . (iv) By (v), Res|σ| L(σ) ∼ = Lσ 6= 0. Therefore, if L(σ) is a composition ¯ ¯ 6= 0 by exactness of Res|σ| . By Proposifactor of ∆(π), then Res|σ| ∆(π) tion 3.14, we then have σ ≤ π and (iv). (ii) If L(π) ' L(σ), then we deduce from (iv) that π ≤ σ and σ ≤ π, whence π = σ. (iii) follows from (v) and Lemma 3.11.

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3.4.2. The case π = (ρ) We now assume that α = ρk ∈ Φ+ . There is a trivial root partition (ρk ) ∈ Π(α). Proposition 3.16 yields |Π(α)| − 1 irreducible Rα -modules, namely the ones which correspond to the non-trivial root partitions π ∈ Π(α). We define the cuspidal module Lα to be the missing irreducible Rα -module, cf. Lemma 3.4. Then, of course, we have that {L(π) | π ∈ Π(α)} is a complete and irredundant system of irreducible Rα -modules up to isomorphism. We now prove that Lα satisfies the property (Cus) and is uniquely determined by it: Lemma 3.17: Let α = ρk ∈ Φ+ . If β, γ ∈ Q+ are non-zero elements such that α = β + γ and Resβ,γ Lα 6= 0, then β is a sum of roots less than α and γ is a sum of roots greater than α. Moreover, this property characterizes Lα among the irreducible Rα -modules uniquely up to isomorphism and degree shift. Proof: We prove that β is a sum of roots less than α, the proof that γ is a sum of roots greater than α being similar. Let L(π)  L(σ) be an irreducible submodule of Resβ,γ Lα , so that π = (m1 , . . . , mN ) ∈ Π(β) and σ = (n1 , . . . , nN ) ∈ Π(γ). Let a be minimal with ma 6= 0. Then Resρa ,β−ρa L(π) 6= 0, and hence Resρa ,γ+β−ρa Lα 6= 0. If we can prove that ρa is a sum of roots less than α, then by convexity, ρa is a root less than α, whence, by the minimality of a, we have that β is a sum of roots less than α. So we may assume from the beginning that β is a root and L(π) = Lβ . Moreover, we may assume that β is the maximal positive root for which Resβ,γ Lα 6= 0. Now, let l be the minimal with nl 6= 0. Then we have a non-zero map Lβ  Lρl  V → Resβ,κ,γ−ρl Lα , for some 0 6= V ∈ Rγ−ρl -mod. By adjunction, this yields a non-zero map f : (Indβ,ρl Lβ  Lρl )  V → Resβ+ρl ,γ−ρl Lα . If ρl = γ, then we must have β ≺ γ, for otherwise Lα is a quotient of the proper standard module Lβ ◦ Lγ , which contradicts the definition of the cuspidal module Lα . Now, since α = β + ρl , we have by convexity that β ≺ α ≺ γ, in particular β ≺ α as desired. Next, let ρl 6= γ, and pick a composition factor L(π 0 ) of Indβ,ρl Lβ  Lρl , which is not in the kernel of f . Write π 0 = (m01 , . . . , m0N ) ∈ Π(β + ρl ). By the assumption on the maximality of β, we have ρc  β whenever m0c > 0.

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Thus β + ρl is a sum of roots  β. Lemma 3.1 implies that ρl  β, and so ¯ by adjointness, Lα is a quotient of the proper standard module Lβ ◦ ∆(σ), which is a contradiction. The second statement of the lemma is clear since, in view of Proposition 3.16(v) and Lemma 3.1, the irreducible modules L(π), corresponding to non-trivial root partitions π ∈ Π(α), do not satisfy the property (Cus). 3.4.3. The case π = (ρn ) Assume now that α = nρk for some ρk ∈ Φ+ and n ∈ Z>1 . Lemma 3.18: The induced module L◦n ρk is irreducible. Proof: In view of Proposition 3.16, we have the irreducible modules L(π) for all root partitions π ∈ Π(α), except for π = σ := (ρnk ), for which ¯ ∆(σ) = L◦n ρk . By Lemma 3.1, σ is the unique minimal element of Π(α). By Proposition 3.16(v), we conclude that L◦n ρk has only one composition factor L appearing with certain multiplicity c(q) ∈ A , and such that L 6∼ = L(π) for all π ∈ Π(α) \ {σ}. Finally, by Corollary 2.13, we conclude that L◦n ρk ' L. The proof of Theorem 3.15 is now complete. 3.5. Reduction modulo p In this subsection we work with two fields: F of characteristic p > 0 and K of characteristic 0. We use the corresponding indices to distinguish between the two situations. Given an irreducible Rα (K)-module LK for a root partition π ∈ Π(α) we can pick a (graded) Rα (Z)-invariant lattice LZ as follows: pick a homogeneous word vector v ∈ LK and set LZ := Rα (Z)v. The lattice LZ can be used to reduce modulo p: ¯ := LZ ⊗Z F. L ¯ depends on the choice of the lattice LZ . In general, the Rα (F )-module L ¯ However, we have chq L = chq LK , so by linear independence of characters of irreducible Rα (F )-modules, composition multiplicities of irreducible ¯ are well-defined. In particular, we have well-defined Rα (F )-modules in L decomposition numbers ¯ dπ,σ := [L(π) : LF (σ)]q

(π, σ ∈ Π(α)),

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which depend only on the characteristic p of F , since prime fields are splitting fields for irreducible modules over KLR algebras. Lemma 3.19: Let LK be an irreducible Rα (K)-module and let i = ia1 1 . . . iab b be an extremal word for LK . Let N be the irreducible ~-selfdual ¯ : N ]q = 1. Rα (F )-module defined by N := f˜iakk . . . f˜ia11 1F . Then [L Proof: Reduction modulo p preserves formal characters, so the result follows from Corollary 2.10. Proposition 3.20: Let π, σ ∈ Π(α). Then dπ,σ 6= 0 implies σ ≤ π. In particular, reduction modulo p of any cuspidal module is an irreducible cuspidal ¯ ρ ' Lρ,F . module again: L Proof: By Theorem 3.15(v), which holds over any field, we conclude that ¯ ρ is isomorphic to Lρ,F up to a degree shift. any composition factor of L Now use Lemma 3.19. 3.6. PBW bases and canonical bases We now return to the algebra f and recall some results on its PBW bases. For a fixed convex order on Φ+ , Lusztig used a certain braid group action to define root vectors {rρ |ρ ∈ Φ+ } in f . The corresponding dual root vectors rρ∗ := (1 − qρ2 )rρ

(ρ ∈ Φ+ )

(3.21)

are invariant under b∗ . For π = (m1 , . . . , mN ) ∈ Π(α), we set rπ :=

rρm11 rρmNN . . . , [m1 ]!ρ1 [mN ]!ρN

rπ∗ := q sh(π) (rρ∗1 )m1 · · · (rρ∗N )mN .

(3.22)

Theorem 3.23: [25] Let α ∈ Q+ . Then {rπ | π ∈ Π(α)} and {rπ∗ | π ∈ Π(α)} are a pair of dual bases for the free A -modules (fA )α and ∗ (fA )α respectively. One can use the b∗ -invariance of the dual root vectors together with the Levendorskii-Soibelman formula [23, Proposition 5.5.2] or [26, Proposition 1.9] to deduce: b∗ (rπ∗ ) = rπ∗ + (a Z[q, q −1 ]-linear combination of rσ∗ for σ < π).

(3.24)

b(rπ ) = rπ + (a Z[q, q −1 ]-linear combination of rσ for σ > π).

(3.25)

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In view (3.24)–(3.25) and Lusztig’s Lemma, there exist unique bases {bπ |π ∈ ∗ Π(α)} and {b∗π | π ∈ Π(α)} for (fA )α and (fA )α , respectively, such that b(bπ ) = bπ ,

bπ = rπ + (a qZ[q]-linear combination of rσ for σ > π), (3.26)

b∗ (b∗π ) = b∗π ,

b∗π = rπ∗ + (a qZ[q]-linear combination of rσ∗ for σ < π). (3.27)

These are the canonical and dual canonical bases, respectively (cf. [24] in simply-laced types or [33] in non-simply-laced types). 3.7. Cuspidal modules and dual PBW bases We continue to work with a fixed convex order ≺ on Φ+ . Suppose that we are given elements ∗ {Eρ∗ ∈ (fA )ρ | ρ ∈ Φ+ }.

(3.28)

If π = (m1 , . . . , mN ) is a root partition, define the corresponding dual PBW monomial ∗ Eπ∗ := q sh(π) (Eρ∗1 )m1 . . . (Eρ∗N )mN ∈ fA .

We say that (3.28) is a dual PBW family if the following properties are satisfied: (i) (‘convexity’) if β  γ are positive roots then Eγ∗ Eβ∗ − q −(β,γ) Eβ∗ Eγ∗ is an A -linear combination of elements Eπ∗ with π < (β, γ) ∈ Π(β + γ); ∗ )α for all α ∈ Q+ ; (ii) (‘basis’) {Eπ∗ | π ∈ Π(α)} is an A -basis of (fA (iii) (‘orthogonality’) (Eπ∗ , Eσ∗ )

= δπ,σ

N Y

((Eρ∗k )mk , (Eρ∗k )mk );

k=1

(iv) (‘bar-triangularity’) b∗ (Eπ∗ ) = Eπ∗ + an A -linear combination of dual PBW monomials Eσ∗ for σ < π. The following result shows in particular that the elements Eρ∗ of the dual PBW family are determined uniquely up to signs (for a fixed preorder ): Lemma 3.29: Assume that (3.28) is a dual PBW family. Then: (i) The elements of (3.28) are b∗ -invariant.

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∗ (ii) Suppose that we are given another family {0 Eρ∗ ∈ (fA )ρ | ρ ∈ Φ+ } of ∗ b -invariant elements which satisfies the basis and orthogonality properties. Then Eρ∗ = ± 0 Eρ∗ for all ρ ∈ Φre +.

Proof: (i) The convexity of ≺ implies that for ρ ∈ Φ+ the root partition (ρ) ∈ Π(ρ) is a minimal element of Π(ρ). So the bar-triangularity property (iv) implies that the elements of a dual PBW family are b∗ -invariant. (ii) We apply induction on ht(ρ), the induction base being clear. By the basis property of dual PBW families, we can write X 0 ∗ Eρ = cEρ∗ + cπ Eπ∗ (c, cπ ∈ A ). (3.30) π∈Π(ρ)\{(ρ)}

Fix for a moment a root partition π ∈ Π(ρ)\{(ρ)}. By the orthogonality property of dual PBW families and non-degeneracy of the form (·, ·), the element 1 E∗ Xπ := (Eπ∗ , Eπ∗ ) π satisfies (Eσ∗ , Xπ ) = δσ,π for all σ ∈ Π(ρ). So pairing the right hand side of (3.30) with Xπ yields cπ . On the other hand, by the inductive assumption, Eπ∗ = ±0 Eπ∗ . So using the orthogonality property for the primed family in (ii), we must have (0 Eρ∗ , Xπ ) = 0 for all π ∈ Π(ρ) \ {(ρ)}. So cπ = 0. Thus 0 ∗ Eρ = cEρ∗ . Furthermore, the elements 0 Eρ∗ and Eρ∗ belong to the algebra ∗ fA and are parts of its A -bases, whence 0 Eρ∗ = ±q n Eρ∗ . Since both 0 Eρ∗ and ∗ Eρ are b∗ -invariant, we conclude that n = 0. ∗ Proposition 3.31: The following set of elements in fA

{Eρ∗ := γ ∗ ([Lρ ]) | ρ ∈ Φ+ } is a dual PBW family. Proof: Under the categorification map γ ∗ , the graded duality ~ corresponds to b∗ , so γ ∗ ([L]) is b∗ -invariant for any ~-self-dual Rα -module L. Moreover, under γ ∗ , the induction product corresponds to the product ∗ in fA , so the convexity condition (i) follows from Theorem 3.15(iv) and ¯ Lemma 3.9. Now, note that Eπ∗ = γ ∗ ([∆(π)]), so the conditions (ii) and (iv) follow from Theorem 3.15(iv) again. It remains to establish the orthogonality property (iii). Let π = (m1 , . . . , mN ). Under γ ∗ , the coproduct r corresponds to the map on the Grothendieck group induces by Res. So using (2.14), we get
¯ (Eπ∗ , Eσ∗ ) = (Eρ∗1 )m1 ⊗ · · · ⊗ (Eρ∗N )mN , γ ∗ ([Res|π| ∆(σ)]) .

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¯ By Proposition 3.14, Res|π| ∆(σ) = 0 unless π ≤ σ, and for π = σ we have ◦mN 1 ¯ Res|π| ∆(σ) = L◦m ρ1  · · ·  LρN .

Since the form (·, ·) is symmetric, the orthogonality follows from the preceding remarks. It is shown in Lusztig [25] and [33] that {rρ∗ | ρ ∈ Φ+ } is a dual PBW family. Since the dual PBW families are unique up to a sign by Lemma 3.29, it follows that γ ∗ ([Lρ ]) = ±rρ∗ for all ρ ∈ Φ+ . In fact: Proposition 3.32: For every ρ ∈ Φ+ we have that γ ∗ ([Lρ ]) = rρ∗ = b∗(ρ) is a dual canonical basis element. Proof: By (3.27), we have rρ∗ = b∗(ρ) is a dual canonical basis element. Now, in view of the commutativity of the triangle (2.15), to show that Eρ∗ = b∗(ρ) , it suffices to know that for an arbitrary element b∗ of the dual canonical basis, there exists at least one word i ∈ hIi such that the coefficient of i in ι(b∗ ) evaluated at q = 1 is positive. But this follows from Lemma 2.16. 4. Homological Properties of KLR Algebras We now review some ‘standard homological properties’ of KLR algebras of finite Lie type. We continue to work with a fixed convex order ≺ on Φ+ . We mainly follow [5], to where we refer the reader for detailed proofs. 4.1. Finiteness of global dimension First of all, we record a key fundamental fact: Theorem 4.1: If the Cartan matrix C is of finite type, the KLR algebra Rα (C) has global dimension equal to ht(α) (as a graded algebra). The finiteness of the global dimension of Rα (C) (as a graded algebra) was first proved by Kato [13] for the case char F = 0 and C of finite ADE types. For an arbitrary F and C of finite BCFG types McNamara [27] computed the global dimension explicitly as ht(α). In [5, Appendix], it was verified that the methods of [27] also lead to the same answer for finite ADE types over any field, not surprisingly the case E8 being the most difficult. Still for C of finite type, the algebras Rα (O) are affine quasi-hereditary. This is shown in [19] for finite type A and in [18] for other finite Lie types.

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From this we have the following slight generalization: if O is a commutative ring of finite global dimension, then Rα (O, C) also has finite global dimension, even as an ungraded algebra. Finally, it can be checked that for a fixed C, the algebras Rα (F, C) have finite global dimension for all α ∈ Q+ if and only if C is of finite type. 4.2. Standard modules Throughout this subsection, ρ ∈ Φ+ is a fixed positive root. Recall the cuspidal module Lρ . The proof of the following result relies on the finiteness of the global dimension of Rα . Theorem 4.2: [27, §4] Let d ≥ 1. Then dimq EXTdRρ (Lρ , Lρ ) =



qρ2 if d = 1; 0 if d ≥ 2.

This theorem allows one to extend Lρ by qρ2 Lρ , then by qρ4 Lρ , etc. to get in the limit the modules ∆(ρ) with the following properties: Theorem 4.3: [5, Theorem 3.4, Corollary 3.4] There is a short exact sequence 0 −→ qρ2 ∆(ρ)−→∆(ρ) −→ Lρ −→ 0.

(4.4)

Moreover: (i) ∆(ρ) is a cyclic module, and in the Grothendieck group we have 1 [∆(ρ)] = [Lρ ]; (4.5) 1 − qρ2 (ii) ∆(ρ) has irreducible head isomorphic to Lρ ; (iii) we have that EXTdRρ (∆(ρ), V ) = 0 for d ≥ 1 and any finitely generated Rρ -module V with all composition factors ' Lρ ; (iv) ENDRρ (∆(ρ)) ∼ = F [x] for x in degree 2dρ . (v) The functor HOMRρ (∆(ρ), −) defines an equivalence from the category of finitely generated graded Rρ -modules with all composition factors ' Lρ to the category of finitely generated graded F [x]-modules (viewing F [x] as a graded algebra with deg(x) = 2dρ ). Remark 4.6: In ADE types, there is a more elementary construction of 0 ∆(ρ). For any α ∈ Q+ of height d, let Rα be the subalgebra of Rα generated by {1i | i ∈ hIiα } ∪ {y1 − y2 , . . . , yd−1 − yd } ∪ {ψ1 , . . . , ψd−1 }.

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Denote by L0ρ the restriction of Lρ from Rρ to Rρ0 . Then ∆(ρ) ∼ = Rρ ⊗Rρ0 L0ρ . By (3.21) and (4.5), the module ∆(ρ) categorifies the root vector rρ . ¯ Compare this to Proposition 3.32, which shows that ∆(ρ) = Lρ categorifies ∗ the dual root vector rρ . Next, we explain how to category the divided powers rρm /[m]!ρ for all m ∈ Z≥0 . For this we need to compute the endomorphism algebra of ∆(ρ)◦m . Choose a non-zero homogeneous vector vρ of minimal degree in ∆(ρ). It generates ∆(ρ) as an Rρ -module. The proof of the following lemma is based on the Mackey Theorem and splitting coming from Theorem 4.3(iii). Lemma 4.7: [5, Lemma 3.6] Let w ∈ S2n be the permutation mapping (1, . . . , n, n + 1, . . . , 2n) to (n + 1, . . . , 2n, 1, . . . , n). There is a unique R2ρ module homomorphism τ : ∆(ρ) ◦ ∆(ρ) → ∆(ρ) ◦ ∆(ρ) of degree −2dρ such that τ (1ρ,ρ ⊗ (vρ ⊗ vρ )) = ψw 1ρ,ρ ⊗ (vρ ⊗ vρ ). Now pick a non-zero endomorphism x ∈ ENDRρ (∆(ρ))2dρ . By Theorem 4.3(iv) we have that ENDRρ (∆(ρ)) = F [x], so x is unique up to a scalar. Now we have commuting endomorphisms x1 , . . . , xm ∈ ENDRmρ (∆(ρ)◦m )2dρ with xr := id◦(r−1) ◦x ◦ id◦(m−r) . Moreover, the endomorphism τ from the previous lemma yields τ1 , . . . , τm−1 ∈ ENDRmρ (∆(ρ)◦m )−2dρ with τr := id◦(r−1) ◦τ ◦ id◦(m−r−1) . Now [5, Lemmas 3.7–3.9] yield: Theorem 4.8: For a unique choice of x ∈ ENDRρ (∆(ρ))2dρ , there is an algebra isomorphism ∼

NHm → ENDRmρ (∆(ρ)◦m )op ,

yi 7→ xi , ψj 7→ τj .

By the theorem, we can view ∆(ρ)◦m as an (Rmρ , N Hm )-bimodule. Finally define the divided power module ∆(ρm ) := qρm(m−1)/2 ∆(ρ)◦m em

(4.9)

where em ∈ NHm is the idempotent (1.12). ∼ [m]! ∆(ρm ) Lemma 4.10: [5, Lemmas 3.7–3.9] We have that ∆(ρ)◦m = ρ as an Rmρ -module. Moreover ∆(ρm ) has irreducible head L(ρm ), and in the

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Grothendieck group we have [∆(ρm )] =

1 [L(ρm )]. (1 − qρ2 )(1 − qρ4 ) · · · (1 − qρ2m )

The lemma shows that in the Grothendieck group [∆(ρm )] corresponds the to the divided power rρm /[m]!ρ under the Khovanov–Lauda–Rouquier categorification. More generally, for a root partition π = (m1 , . . . , mN ), define the standard module mN 1 ∆(π) := ∆(ρm 1 ) ◦ · · · ◦ ∆(ρN ).

(4.11)

Theorem 4.12: [5, Theorem 3.11] For a root partition π = (m1 , . . . , mN ), the module V0 := ∆(π) has an exhaustive filtration V0 ⊃ V1 ⊃ V2 ⊃ · · · such ¯ ¯ that V0 /V1 ∼ and all other sections of the form q 2m ∆(π) for m > = ∆(π) ∼ 0. Moreover, ∆(π) has irreducible head = L(π), and in the Grothendieck group: mk N Y .Y ¯ [∆(π)] = [∆(π)] (1 − qρ2rk ). k=1 r=1

4.3. Homological properties of standard modules Now that we have constructed the standard modules, we list some of their homological properties. Throughout the subsection, α ∈ Q+ is fixed. Theorem 4.13: [13, Theorem 4.12], [5, Theorm 3.12] Let π ∈ Π(α). (i) If EXTdRα (∆(π), V ) 6= 0 for some d ≥ 1 and a finitely generated Rα module V , then V has a composition factor ' L(σ) for σ  π. (ii) We have for all d ≥ 0 and σ ∈ Π(α): ¯ dimq EXTdR (∆(π), ∇(σ)) = δd,0 δπ,σ . α

We say that an Rα -module V has a ∆-filtration, written V ∈ Fil(∆), if there is a finite filtration V = V0 ⊃ V1 ⊃ · · · ⊃ Vn = 0 such that Vi /Vi+1 ' ∆(π (i) ) for each i = 1, . . . , n − 1 and some π (i) ∈ Π(α). If V ∈ Fil(∆), then by Theorem 4.13(ii), the (graded) multiplicity of ∆(π) in a ∆-filtration of V is well-defined (i.e. independent of the ∆-filtration) and is equal to that ¯ (π ∈ Π(α)). [V : ∆(π)]q = dimq HOMR (V, ∇(π)) α

Theorem 4.14: [5, Theorem 3.13] Let V be a finitely generated Rα ¯ module. Then V ∈ Fil(∆) if and only if EXT1Rα (V, ∇(σ)) = 0 for all σ ∈ Π(α).

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An immediate corollary is the following version of the ‘BGG reciprocity’. Note that the projective cover P (π) of the irreducible module L(π) exists in view of the general theory described in §2.1. Corollary 4.15: [13, Remark 4.17], [5, Corollary 4.17] Let π, σ ∈ Π(α). ¯ Then P (π) ∈ Fil(∆) and [P (π) : ∆(σ)]q = [∆(σ) : L(π)]q . This implies the following important dimension formula. A more elementary proof of this formula is given in [18]. Corollary 4.16: [5, Corollary 3.15] We have that X ¯ dimq Rα = (dimq ∆(π))(dimq ∆(π)) π∈Π(α)

=

2 ¯ (dimq ∆(π)) . QN Qmk 2r k=1 r=1 (1 − qρk ) π=(m1 ,...,mN )∈Π(α)

X

The following corollary yields a description of the standard modules ¯ ∆(π) and ∆(π) in spirit of standardly stratified algebras, cf. [13, Corollary 4.18]. Corollary 4.17: [5, Corollary 3.15] Let π ∈ Π(α), and X X X X ¯ K(π):= imf, K(π):= σ6π f ∈HOMRα (P (σ),P (π))

imf.

σ6≺π f ∈HOMRα (P (σ),P (π))

∼ ¯ ¯ Then ∆(π) ∼ = P (π)/K(π) and ∆(π) = P (π)/K(π). 5. Projective Resolutions of Standard Modules We now explain how the standard modules ∆(ρ) fit into some short exact sequences, giving an alternative way to deduce their properties. This bounds the projective dimension of standard modules, and allows us to construct some projective resolutions of standard modules. As usual, we work with a fixed convex order ≺ on Φ+ and denote by ρ an arbitrary positive root. 5.1. Minimal pairs A two-term root partition π = (β, γ) ∈ Π(ρ) is called a minimal pair for ρ, if it is a minimal element of Π(ρ) \ {(ρ)}. Equivalently, a minimal pair for ρ is a pair (β, γ) of positive roots with β + γ = ρ and β  γ such that there exists no other pair (β 0 , γ 0 ) of positive roots with β 0 + γ 0 = ρ and β  β 0  ρ  γ 0  γ. Let MP(ρ) denote the set of all minimal pairs for ρ.

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For π = (β, γ) ∈ MP(ρ), it follows from Theorem 3.15 and the min¯ imality of π that all composition factors of rad ∆(π) are ' Lρ . Since (β,γ) ~ ¯ Lγ ◦ Lβ by Lemma 3.9, there ∆(π) = Lβ ◦ Lγ and (Lβ ◦ Lγ ) ∼ = q are short exact sequences 0 −→ q −(β,γ) M ~ −→ Lβ ◦ Lγ −→ L(π) −→ 0, 0 −→ q

−(β,γ)

L(π) −→ Lγ ◦ Lβ −→ M −→ 0,

(5.1) (5.2)

~ ¯ where M := q −(β,γ) (rad ∆(π)) is a finite dimensional module with all composition factors ' Lρ . It turns out that one can be much more precise. Let Φ = Φ+ t −Φ+ be the set of all roots. For any β, γ ∈ Φ, define the number

pβ,γ := max{m ∈ Z | β − mγ ∈ Φ}. Theorem 5.3: [5, Theorem 4.7, Corollary 4.3] Let π = (β, γ) ∈ MP(ρ). (i) There are short exact sequences 0 −→ q pβ,γ −(β,γ) Lρ −→ Lβ ◦ Lγ −→ L(π) −→ 0, 0 −→ q

−(β,γ)

L(π) −→ Lγ ◦ Lβ −→ q

−pβ,γ

Lρ −→ 0.

(5.4) (5.5)

(ii) In the Grothendieck group we have that h i   Resργ,β Lρ = [pβ,γ + 1]q Lγ  Lβ . Moreover, Resγ,β Lρ is uniserial with socle q pβ,γ Lγ  Lβ . We now explain how minimal pairs allow one to get a generalization of high weight theory which was first developed in [21] for the so-called Lyndon convex orders. Fix an arbitrary minimal pair mp(ρ) ∈ MP(ρ) for each non-simple positive root ρ ∈ Φ+ . Dependent on this choice, we recursively define a word iρ ∈ hIiρ and a bar-invariant Laurent polynomial κρ ∈ A for all ρ ∈ Φ+ as follows. For i ∈ I set iαi := i and καi := 1; then for non-simple ρ ∈ Φ+ suppose that (β, γ) = mp(ρ) and set iρ := iγ iβ ,

κρ := [pβ,γ + 1]q κβ κγ .

(5.6)

For example, in simply-laced types we have that κρ = 1 for all ρ ∈ Φ+ ; this is also the case in non-simply-laced types for multiplicity-free positive P roots, i.e. roots ρ = i∈I ci αi with ci ∈ {0, 1} for all i. Finally for a root mN 1 partition π = (β1 ≥ · · · ≥ βl ) = (ρm 1 , . . . , ρN ) let iπ := iβ1 · · · iβl ,

κπ :=

N Y

k [mk ]!ρk κm ρk .

k=1

(5.7)

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The following lemma shows that the words iπ distinguish irreducible modules, generalizing [21, Theorem 7.2(ii)]. Lemma 5.8: [5, Lemma 4.5] Let α ∈ Q+ and π, σ ∈ Π(α). Then dimq L(π)iσ = 0 if σ 6 π, and dimq L(π)iπ = κπ . 5.2. Projective resolutions Let ρ ∈ Φ+ be a non-simple positive root, (β, γ) be a minimal pair for ρ, and m := ht(γ). Let w ∈ Sn be the permutation (1, . . . , n) 7→ (n − m + 1, . . . , n, 1, . . . , n − m), so that ψw 1γ,β = 1β,γ ψw . It is proved in [5, Lemma 4.9] that there is a unique homogeneous homomorphism ϕ : q −(β,γ) ∆(β) ◦ ∆(γ) → ∆(γ) ◦ ∆(β)

(5.9)

such that ϕ(1β,γ ⊗(v1 ⊗v2 )) = ψw 1γ,β ⊗(v2 ⊗v1 ) for all v1 ∈ ∆(β), v2 ∈ ∆(γ). Theorem 5.10: [5, Theorem 4.10] For (β, γ) ∈ MP(ρ) there is a short exact sequence ϕ

0 −→ q −(β,γ) ∆(β) ◦ ∆(γ) −→ ∆(γ) ◦ ∆(β) −→ [pβ,γ + 1]q ∆(ρ) −→ 0. Let us again fix a choice of minimal pairs mp(ρ) ∈ MP(ρ) for each ρ ∈ Φ+ of height at least two, and recall κρ and κπ from (5.6). Let ˜ ∆(ρ) := κρ ∆(ρ),

˜ ∆(π) := κπ ∆(π).

(5.11)

˜ For simply laced C we have ∆(ρ) = ∆(ρ). We want to construct a ˜ projective resolution P∗ (ρ) of ∆(ρ) for each ρ ∈ Φ+ . Then more generally, given α ∈ Q+ and π = (β1  β2  · · ·  βl ) ∈ Π(α), the total complex of the ‘◦’-product of the complexes P∗ (β1 ), . . . , P∗ (βl ) gives a projective ˜ resolution P∗ (π) of ∆(π). The resolution P∗ (ρ) is going to be of the form P∗ (ρ) :

˜ 0 → Pn−1 (ρ) −→ . . . −→ P1 (ρ) −→ P0 (ρ) −→ ∆(ρ) → 0,

where n = ht(ρ). The construction of P∗ (ρ) is recursive. For i ∈ I we have ˜ i ) = Rα , which is projective already. So we just set P0 (αi ) := Rα and ∆(α i i Pd (αi ) := 0 for d 6= 0 to obtain the required resolution. Now suppose that ρ ∈ Φ+ is of height at least two and let (β, γ) := mp(ρ), a fixed minimal pair for ρ. We may assume by induction that the projective resolutions

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P∗ (β) and P∗ (γ) are already defined. Taking the total complex of their ‘◦’product using [35, Acyclic Assembly Lemma 2.7.3], we obtain a projective ˜ ˜ resolution P∗ (β, γ) of ∆(β) ◦ ∆(γ) with M Pd (β, γ) := Pd1 (β) ◦ Pd2 (γ), d1 +d2 =d


∂d := id ◦∂d2 − (−1)d2 ∂d1 ◦ id d

1 +d2 =d

: Pd (β, γ) → Pd−1 (β, γ).

˜ ˜ Similarly we obtain a projective resolution P∗ (γ, β) of ∆(γ) ◦ ∆(β) with M Pd (γ, β) := Pd1 (γ) ◦ Pd2 (β), d1 +d2 =d

∂d := ∂d1 ◦ id +(−1)d1 id ◦∂d2


d1 +d2 =d

: Pd (γ, β) → Pd−1 (γ, β).

There is an injective homomorphism ˜ ˜ ˜ ˜ ϕ˜ : q −(β,γ) ∆(β) ◦ ∆(γ) ,→ ∆(γ) ◦ ∆(β) defined in exactly the same way as the map ϕ in (5.9), indeed, it is just a direct sum of copies of the map ϕ from there. Applying [35, Comparision Theorem 2.2.6], ϕ˜ lifts to a chain map ϕ˜∗ : q −(β,γ) P∗ (β, γ) → P∗ (γ, β). Then we take the mapping cone of ϕ˜∗ to obtain a complex P∗ (ρ) with Pd (ρ) := Pd (γ, β) ⊕ q −(β,γ) Pd−1 (β, γ), ∂d := (∂d , ∂d−1 + (−1)d−1 ϕ˜d−1 ) : Pd (ρ) → Pd−1 (ρ). In view of Theorem 5.10 and [35, Acyclic Assembly Lemma 2.7.3] once ˜ again, P∗ (ρ) is a projective resolution of ∆(ρ). Let us describe P∗ (ρ) more explicitly. First, for i ∈ I and the empty tuple σ, set iαi ,σ := i. Now suppose that ρ is of height n ≥ 2 and that (β, γ) = mp(ρ) with γ of height m. For σ = (σ1 , . . . , σn−1 ) ∈ {0, 1}n−1 , let |σ| := σ1 + · · · + σn−1 , σ m := (σm+1 , . . . , σn−1 ). Define iρ,σ ∈ hIiρ and dρ,σ ∈ Z≥0 recursively from  iγ,σm if σm = 0, iρ,σ := iβ,σ>m iγ,σm + dγ,σm + dγ,σ 1,

Ts Ts+1 Ts = Ts+1 Ts Ts+1 , Tr Lt = Lt Tr if t 6= r, r + 1, where 1 ≤ r < n, 1 ≤ s < n − 1 and 1 ≤ t ≤ n. By definition, Hn is generated by L1 , T1 , . . . , Tn−1 but we prefer including L2 , . . . , Ln in the generating set. Let Sn be the symmetric group on n letters. For 1 ≤ r < n let sr = (r, r + 1) be the corresponding simple transposition. Then {s1 , . . . , sn−1 } is the standard set of Coxeter generators for Sn . A reduced expression for w ∈ Sn is a word w = sr1 . . . srk with k minimal and 1 ≤ rj < n for 1 ≤ j ≤ k. If w = sr1 . . . srk is reduced then set Tw = Tr1 . . . Trk . Then Tw

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is independent of the choice of reduced expression by Matsumoto’s Monoid Lemma [110] since the braid relations hold in Hn ; see, for example, [104, Theorem 1.8]. Arguing as in [10, Theorem 3.3], it follows that Hn is free as a Z-module with basis (1.1.2) { La1 1 . . . Lann Tw | 0 ≤ a1 , . . . , an < ` and w ∈ Sn } . Consequently, Hn is free as a Z-module of rank `n n!, which is the order of the complex reflection group G`,n = Z/`Z o Sn of type G(`, 1, n). Definition 1.1.1 is different from Ariki and Koike’s [10] definition of the cyclotomic Hecke algebras of type G(`, 1, n) because we have changed the commutation relation for Tr and Lr . Ariki and Koike [10] defined their algebra to be the unital associative algebra generated by T0 , T1 , . . . , Tn−1 subject to the relations Q` 0 (Tr + v −1 )(Tr − v) = 0, l=1 (T0 − Ql ) = 0, T0 T1 T0 T1 = T1 T0 T1 T0

Ts Ts+1 Ts = Ts+1 Ts Ts+1 ,

Tr Ts = Ts Tr if |r − s| > 1. We have renormalised the quadratic relation for the Tr , for 1 ≤ r < n, so that q = v 2 in the notation of [10]. Ariki and Koike then defined L01 = T0 and set L0r+1 = Tr L0r Tr for 1 ≤ r < n. In fact, if v − v −1 is invertible in Z then Hn is (isomorphic to) the Ariki-Koike algebra with parameters Q0l = 1 + (v − v −1 )Ql for 1 ≤ l ≤ `. To see this set L0r = 1 + (v − v −1 )Lr in Hn , for 1 ≤ r ≤ n. Then Tr L0r Tr = (v − v −1 )Tr Lr Tr + Tr2 = L0r+1 , which implies our claim. Therefore, over a field, Hn is an Ariki-Koike algebra whenever v 2 6= 1. On the other hand, if v 2 = 1 then Hn is a degenerate cyclotomic Hecke algebra [13, 79]. We note that the Ariki-Koike algebras with v 2 = 1 include as a special the group algebras ZG`,n of the complex reflection groups G`,n , for n ≥ 0. One consequence of the last paragraph is that ZG`,n is not a specialization of Hn . This said, if F is a field such that Hn and F G`,n are both split semisimple then Hn ∼ = F G`,n . On the other hand, the algebras Hn always fit into the spetses framework of Brou´e, Malle and Michel [17]. The algebras Hn with v 2 = 1 are the degenerate cyclotomic Hecke algebras of type G(`, 1, n) whereas if v 2 6= 1 then Hn is an Ariki-Koike algebra in the sense of [10]. Our definition of Hn is more natural in the sense that many features of the algebras Hn have a uniform description in both the degenerate and non-degenerate cases: • The centre of Hn is the set of symmetric polynomials in L1 , . . . , Ln (Brundan [19] in the degenerate case when v 2 = 1 and announced when v 2 6= 1 by Graham and Francis building on [42]).

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• The blocks of Hn are indexed by the same combinatorial data (Lyle and Mathas [96] when v 2 6= 1 and Brundan [19] when v 2 = 1). • The irreducible Hn -modules are indexed by the crystal graph of b e ) (Ariki [3] when the integral highest weight module L(Λ) for Uq (sl 2 v 6= 1 and Brundan and Kleshchev [23] when v 2 = 1). • The algebras Hn categorify L(Λ). Moreover, in characteristic zero the projective indecomposable Hn -modules correspond to the canonical basis of L(Λ). (Ariki [3] when v 2 6= 1 and Brundan and Kleshchev [23] when v 2 = 1.) • The algebra Hn is isomorphic to a cyclotomic quiver Hecke algebra of type A (Brundan and Kleshchev [21]). In contrast, the Ariki-Koike algebras with v 2 = 1 do not share any of these properties: their center can be larger than the set of symmetric polynomials in L1 , . . . , Ln (Ariki [3]); if ` > 1 then they have only one block (Lyle and Mathas [96]); their irreducible modules are indexed by a different set (Mathas [103]); they do not categorify L(Λ) and no non-trivial grading on these algebras is known. In this sense, the definition of the Ariki-Koike algebras from [10] gives the wrong algebras when v 2 = 1. Definition 1.1.1 corrects for this. Historically, many results for the cyclotomic Hecke algebras Hn were proved separately in the degenerate (v 2 = 1) and non-degenerate cases (v 2 6= 1). Using Definition 1.1.1 it should now be possible to give uniform proofs of all of these results. In fact, in the cases that we have checked uniforms arguments can now be given for the degenerate and non-degenerate cases.

1.2

Quivers of type A and integral parameters

Rather than work with arbitrary cyclotomic parameters Q1 , . . . , Q` , as in Definition 1.1.1, we now specialize to the integral case using the Morita equivalence results of Dipper and the author [32] (when v 2 = 6 1) and Brundan 2 and Kleshchev [20] (when v = 1). First, however, we need to introduce quivers and quantum integers. Fix e ∈ {1, 2, 3, 4, . . . } ∪ {∞} and let Γe be the quiver with vertex set Ie = Z/eZ and edges i −→ i + 1, for i ∈ Ie , where we adopt the convention that eZ = {0} when e = ∞. If i, j ∈ Ie and i and j are not connected by an edge in Γe then we write i — / j. When e is fixed we write Γ = Γe and I = Ie . Hence, we are considering either the linear quiver Z (e = ∞) or a

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cyclic quiver (e < ∞): 2

4 3

2 5

0

2

...

1 0

e=2

1

e=3

0

1

e=4

0

1

e=5

···

In the literature the case e = ∞ is often written as e = 0, however, we prefer e = ∞ because then e = |Ie |. There are also several results that hold when e > n — using the “e = 0 convention” this condition must be written as e > n or e = 0. We write e ≥ n to mean e ∈ {n, n + 1, n + 2, . . . } ∪ {∞}. To the quiver Γe we attach the symmetric Cartan matrix (cij )i,j∈I , where   2, if i = j,    −1, if i → j or i ← j, cij = −2, if i  j,     0, otherwise, b e be the Kac-Moody algebra of Γe [66] Following [66, Chapter 1], let sl with simple roots { αi | i ∈ I }, fundamental weights { Λi | i ∈ I }, positive L L weight lattice P + = i∈I NΛi and positive root lattice Q+ = i∈I Nαi . Let (·, ·) be the usual invariant form associated with this data, normalised so that (αi , αj ) = cij and (Λi , αj ) = δij , for i, j ∈ I. Fix a sequence κ = (κ1 , . . . , κ` ) ∈ Z` , the multicharge, and define Λ = Λ(κ) = Λκ1 + · · · + Λκ` , where a = a + eZ ∈ I for a ∈ Z. Then Λ ∈ P + is dominant weight of level `. The integral cyclotomic Hecke algebras defined below depend only on Λ, however, the bases and our combinatorics often depends upon the choice of multicharge κ. Recall that Z is an integral domain. For t ∈ Z × and k ∈ Z define the t-quantum integer [k]t by ( t + t3 + · · · + t2k−1 , if k ≥ 0, [k]t = −1 −3 2k+1 −(t + t + · · · + t ), if k < 0. When t is understood we simply write [k] = [k]t . Unpacking the definition, if t2 6= 1 then [k] = (t2k − 1)/(t − t−1 ) whereas [k] = ±k if t = ±1. The quantum characteristic of v is the smallest element of e ∈ {2, 3, 4, 5, . . . } ∪ {∞} such that [e]v = 0, where we set e = ∞ if [k]v 6= 0 for all k > 0. Definition 1.2.1. Suppose that Λ = Λ(κ) ∈ P + , for κ ∈ Z` , and that v ∈ Z has quantum characteristic e. The integral cyclotomic Hecke

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algebra of type A of weight Λ is the cyclotomic Hecke algebra HnΛ = Hn (Z, v, Q1 , . . . , Qr ) with Hecke parameter v and cyclotomic parameters Qr = [κr ]v , for 1 ≤ r ≤ `. When v 2 6= 1 the parameter Qr corresponds to the Ariki-Koike parameters Q0r = v 2κr , for 1 ≤ r ≤ `, where we use the notation of §1.1. As observed in [57, §2.2], translating the Morita equivalence theorems of [32, Theorem 1.1] and [20, Theorem 5.19] into the current setting explains the significance of the integral cyclotomic Hecke algebras. Theorem 1.2.2 (Dipper-Mathas [32], Brundan-Kleshchev [20]). Every cyclotomic quiver Hecke algebra Hn is Morita equivalent to a direct sum of tensor products of integral cyclotomic Hecke algebras. Brundan and Kleshchev treated the degenerate case when v 2 = 1 using very different arguments to those in [32]. With the benefit of Definition 1.1.1 the argument of [32] now applies uniformly to both the degenerate and non-degenerate cases. The Morita equivalences in [20, 32] are described explicitly, with the equivalence being determined by orbits of the cyclotomic parameters. See [20, 32] for more details. In view of Theorem 1.2.2, it is enough to consider the integral cyclotomic Hecke algebras HnΛ where v ∈ Z × has quantum characteristic e and Λ ∈ P + . This said, for most of Section 1 we consider the general case of a not necessarily integral cyclotomic Hecke algebra because we will need this generality in §4.2. 1.3

Cellular algebras

We recall Graham and Lehrer’s cellular algebra framework [48]. This will allow us to define Specht modules for Hn as cell modules. Significantly, the cellular algebra machinery endows the Specht modules with an associative bilinear form. Here is the definition. Definition 1.3.1 (Graham and Lehrer [48]). Suppose that A is a Z-algebra that is Z-free and of finite rank as a Z-module. A cell datum for A is an ordered triple (P, T, C), where (P, B) is the weight poset, T (λ) is a finite set for λ ∈ P, and a C: T (λ) × T (λ) −→ A; (s, t) 7→ cst , λ∈P

is an injective map of sets such that:

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(GC1 ) { cst | s, t ∈ T (λ) for λ ∈ P } is a Z-basis of A. (GC2 ) If s, t ∈ T (λ), for some λ ∈ P, and a ∈ A then there exist scalars rtv (a), which do not depend on s, such that X cst a = rtv (a)csv (mod ABλ ), v∈T (λ) Bλ

where A = h cab | µ B λ and a, b ∈ T (µ) iZ . (GC3 ) The Z-linear map ∗ : A −→ A determined by c∗st = cts , for all λ ∈ P and all s, t ∈ T (λ), is an anti-isomorphism of A. A cellular algebra is an algebra that has a cell datum. If A is a cellular algebra with cell datum (P, T, C) then the basis { cst | λ ∈ P and s, t ∈ T (λ) } is a cellular basis of A with cellular algebra anti-isomorphism ∗. K¨onig and Xi [86] have given an equivalent definition of cellular algebras that does not depend upon a choice of basis. Goodman and Graber [45] have shown that (GC3 ) can be relaxed to the requirement that (cst )∗ ≡ cts (mod ABλ ) for some anti-isomorphism ∗ of A. The prototypical example of a cellular algebra is a matrix algebra with its basis of matrix units, which we call a Wedderburn basis. As any split semisimple algebra is isomorphic to a direct sum of matrix algebras it follows that every split semisimple algebra is cellular. The cellular algebra framework is, however, most useful in studying non-semisimple algebras that are not isomorphic to a direct sum of matrix rings. In general, a cellular basis can be thought of as an approximation, or weakening, of a basis of matrix units. (This idea is made more explicit in [108].) The cellular basis axioms determine a filtration of the cellular algebra, via the ideals ABλ . As we will see, this leads to a quick construction of the irreducible representations. For λ ∈ P, let ADλ = h cab | µ D λ and a, b ∈ T (µ) iZ . Then it follows from Definition 1.3.1 that ADλ is a two-sided ideal of A. Fix λ ∈ P. The cell module C λ is the (right) A-module with basis { ct | t ∈ T (λ) } and where a ∈ A acts on C λ by: X rtv (a)cv , for t ∈ T (λ), ct a = v∈T (λ)

where the scalars rtv (a) ∈ Z are those appearing in (GC2 ). It follows immediately from Definition 1.3.1 that C λ is an A-module. Indeed, if s ∈ T (λ) then C λ is isomorphic to the submodule (cst + ABλ )A of A/Aλ via the map ct 7→ cst + Aλ , for t ∈ T (λ). The cell module C λ comes with a symmetric bilinear form h , iλ that is uniquely determined by (1.3.2) hct , cv iλ cab ≡ cat cvb (mod ABλ ) ,

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for a, b, t, v ∈ T (λ). By (GC2 ) of Definition 1.3.1, the inner product hct , cv iλ depends only on t and v, and not on the choices of a and b. In addition, hxa, yiλ = hx, ya∗ iλ , for all x, y ∈ C λ and a ∈ A. Therefore, (1.3.3) rad C λ = { x ∈ C λ | hx, yiλ = 0 for all y ∈ C λ } is an A-submodule of C λ . Set Dλ = C λ / rad C λ . Then Dλ is an A-module. The following theorem summarizes some of the main properties of a cellular algebra. The proof is surprisingly easy given the strength of the result. In applications the main difficulty is in showing that a given algebra is cellular. If M is an A-module and D is an irreducible A-module, let [M : D] be the decomposition multiplicity of D in M . Theorem 1.3.4 (Graham and Lehrer [48]). Suppose that Z = F is a field. Then: a) If µ ∈ P then Dµ is either zero or absolutely irreducible. b) Let K = { µ ∈ P | Dµ 6= 0 }. Then { Dµ | µ ∈ K } is a complete set of pairwise non-isomorphic irreducible A-modules. c) If λ ∈ P and µ ∈ K then [C λ :Dµ ] 6= 0 only if λ D µ. Moreover, [C µ :Dµ ] = 1. If µ ∈ K let P µ be the projective cover of Dµ . It follows from Definition 1.3.1 that P µ has a filtration in which the quotients are cell modules such that C λ appears with multiplicity [C λ :Dµ ]. Consequently, an analogue of Brauer-Humphreys reciprocity holds for A. In particular, the Cartan matrix of A is symmetric.

1.4

Multipartitions and tableaux

A partition of m is a weakly decreasing sequence λ = (λ1 , λ2 , . . . ) of nonnegative integers such that |λ| = λ1 + λ2 + · · · = m. An (`-)multipartition of n is an `-tuple λ = (λ(1) , . . . , λ(`) ) of partitions such that |λ(1) | + · · · + |λ(`) | = n. We identify the multipartition λ with its diagram, which is (l) the set of nodes JλK = { (l, r, c) | 1 ≤ c ≤ λr for 1 ≤ l ≤ ` }. In this way, we think of λ as an ordered `-tuple of arrays of boxes in the plane and we talk of the components of λ. Similarly, by the rows and columns of λ we will mean the rows and columns in each component. For example, if λ = (3, 12 |2, 1|3, 2) then   . λ = JλK = 

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A node A is an addable node of λ if A ∈ / λ and λ∪{A} is the (diagram of) a multipartition of n + 1. Similarly, a node B is a removable node of λ if B ∈ λ and λ \ {B} is a multipartition of n − 1. If A is an addable node of λ let λ + A be the multipartition λ ∪ {A} and, similarly, if B is a removable node let λ − B = λ \ {B}. Order the nodes lexicographically by ≤. The set of multipartitions of n becomes a poset under dominance where λ dominates µ, written as λ D µ, if l−1 X k=1

|λ(k) | +

i X j=1

(l)

λj ≥

l−1 X k=1

|µ(k) | +

i X

(l)

µj ,

j=1

for 1 ≤ l ≤ ` and i ≥ 1. If λ D µ and λ 6= µ then write λ B µ. Let Λ PnΛ = P`,n be the set of multipartitions of n. We consider PnΛ as a poset ordered by dominance. Fix λ ∈ PnΛ . A λ-tableau is a bijective map t : JλK −→ {1, 2, . . . , n}, which we identify with a labelling of (the diagram of) λ by {1, 2, . . . , n}. For example,     1 2 3 6 7 9 10 11 9 12 13 6 8 1 3 5 8 12 13 7 2 4  and  10   4 5 11 are both λ-tableaux when λ = (3, 12 |2, 1|3, 2). A λ-tableau is standard if its entries increase along rows and down columns in each component. For example, the two tableaux above are standard. Let Std(λ) be the set of standard λ-tableaux. If P is any set S of multipartitions let Std(P) = λ∈P Std(λ). Similarly set Std2 (P) = { (s, t) | s, t ∈ Std(λ) for λ ∈ P }. If t is a λ-tableau set Shape(t) = λ and let t↓m be the subtableau of t that contains the numbers {1, 2, . . . , m}. If t is a standard λ-tableau then Shape(t↓m ) is a multipartition for all m ≥ 0. We extend the dominance ordering to Std(PnΛ ), the set of all standard tableaux, by defining s D t if Shape(s↓m ) D Shape(t↓m ), for 1 ≤ m ≤ n. As before, write s B t if s D t and s 6= t. Finally, define the strong dominance ordering on Std2 (PnΛ ) by (s, t) I (u, v) if s D u and t D v. Similarly, (s, t) I (u, v) if (s, t) I (u, v) and (s, t) 6= (u, v). It is easy to see that there are unique standard λ-tableaux tλ and tλ such that tλ D t D tλ , for all t ∈ Std(λ). The tableau tλ has the numbers (1) 1, 2, . . . , n entered in order from left to right along the rows of tλ , and then (2) (`) tλ , . . . , tλ . Similarly, tλ is the tableau with the numbers 1, . . . , n entered

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(`)

(2)

(1)

in order down the columns of tλ , . . . , tλ , tλ . If λ = (3, 12 |2, 1|3, 2) then the two λ-tableaux displayed above are tλ and tλ , respectively. Given a standard λ-tableau t define permutations d(t), d0 (t) ∈ Sn by λ t d(t) = t = tλ d0 (t). Then d(t)d0 (t)−1 = d(tλ ) with `(d(t)) + `(d0 (t)) = `(d(tλ )), for all t ∈ Std(λ). Let ≤ be the Bruhat order on Sn with the convention that 1 ≤ w for all w ∈ Sn . Independently, Ehresmann and James [59] showed that if s, t ∈ Std(λ) then s D t if and only if d(s) ≤ d(t) and if and only if d0 (t) ≤ d0 (s). A proof can be found, for example, in [104, Theorem 3.8]. Finally, we will need to know how to conjugate multipartitions and tableaux. The conjugate of a partition λ is the partition λ0 = (λ01 , λ02 , . . . ) where λ0r = # { s ≥ 1 | λs ≥ r }. That is, we swap the rows and columns of λ. The conjugate of a multipartition λ = (λ(1) | . . . |λ(`) ) is the multipartition λ0 = (λ(`)0 | . . . |λ(1)0 ). Similarly, the conjugate of a λ-tableau t = (t(1) | . . . |t(`) ) is the λ0 -tableau t0 = (t(`)0 | . . . |t(1)0 ) where t(k)0 is the tableau obtained by swapping the rows and columns of t(k) , for 1 ≤ k ≤ `. Then λ D µ if and only if µ0 D λ0 , and s D t if and only if t0 D s0 . 1.5

The Murphy basis of HnΛ

Graham and Lehrer [48] showed that the cyclotomic Hecke algebras (when v 2 6= 1) are cellular algebras. In this section we recall another cellular basis for these algebras that was constructed in [31] when v 2 = 6 1 and in [13] when v 2 = 1. When ` = 1 these results are due to Murphy [113]. First observe that Definition 1.1.1 implies that there is a unique anti-isomorphism ∗ on Hn that fixes each of the generators T1 , . . . , Tn−1 , L1 , . . . , Ln of Hn . It is easy to see that Tw∗ = Tw−1 , for w ∈ Sn . Fix a multipartition λ ∈ PnΛ . Following [31, Definition 3.14] and [13, §6], if s, t ∈ Std(λ) define mst = Td(s)−1 mλ Td(t) , where mλ = uλ xλ , |λ(1) |+···+|λ(l) |

uλ =

Y

Y

1≤l 1 then αr (t)αr0 (tsr ) = αr0 (t)αr (tsr0 ), for 1 ≤ r, r0 < n. As the reader might guess, the conditions on the scalars αr (t) in Definition 1.6.6 correspond to the quadratic relations (Tr − v)(Tr + v −1 ) = 0 and the braid relations for T1 , . . . , Tn−1 . The simplest example of a seminormal coefficient system is
Z 1 − v −1 cZ r+1 (t) + vcr (t)
αr (t) = , Z cZ r+1 (t) − cr (t) whenever 1 ≤ r < n and t, t(r, r + 1) ∈ Std(PnΛ ). Another seminormal coefficient system is given in (1.7.1) below.

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Seminormal coefficient systems arise because they describe the action of Hn on a seminormal basis. More precisely, we have the following: Theorem 1.6.7 (Hu-Mathas [57]). Suppose that Z = K is a field and that Hn is content separated and that { fst | (s, t) ∈ Std2 (PnΛ ) } is a seminormal basis of Hn . Then {fst } is a cellular basis of Hn and there exists a unique seminormal coefficient system α such that fst Tr = αr (t)fsv +

1 + (v − v −1 )cZ r+1 (t) fst , Z (t) (t) − c cZ r r+1

where v = t(r, r + 1). Moreover, if s ∈ Std(λ) then Fs = γ1s fss is a primitive ∼ Fs Hn is irreducible for all λ ∈ P Λ . idempotent and S λ = n

Proof. [Sketch of proof] By definition, {fst } is a basis of Hn such that fst∗ = fts for all (s, t) ∈ Std2 (PnΛ ). Therefore, it follows from (1.6.5) that {fst } is a cellular basis of Hn with cellular automorphism ∗. It is an amusing application of the relations in Definition 1.1.1 to show that there exists a seminormal coefficient system that describes the action of Tr on the seminormal basis. See [57, Lemma 3.13] for details. The uniqueness of α is clear. We have already observed in (1.6.5) that Fs = γ1s fss , for s ∈ Std(λ), so it remains to show that Fs is primitive and that S λ ∼ = Fs Hn . By what we have already shown, Fs Hn is contained in the span of { fst | t ∈ Std(λ) }. On the P other hand, if f = t rt fst ∈ Fs Hn and rv 6= 0 then rv fsv = f Fv ∈ Fs Hn . P It follows that Fs Hn = t Kfst , as a vector space. Consequently, Fs Hn is irreducible and Fs is a primitive idempotent in Hn . Finally, S λ ∼ = Fs Hn by Lemma 1.6.2 since Hn is content separated.  Corollary 1.6.8 ([57, Corollary 3.17]). Suppose that α is a seminormal coefficient system and that s B t = s(r, r + 1), for tableaux s, t ∈ Std(PnΛ ) and where 1 ≤ r < n. Then αr (s)γt = αr (t)γs . Consequently, if the seminormal coefficient system α is known then fixing γt , for some t ∈ Std(λ), determines γs for all s ∈ Std(λ). Conversely, these scalars, together with α, determines the seminormal basis. Corollary 1.6.9 (Classifying seminormal bases [57, Theorem 3.14]). There is a one-to-one correspondence between the ∗-seminormal bases of Hn and the pairs (α, γ) where α = { αr (s) | 1 ≤ r < n and s ∈ Std(PnΛ ) } is a seminormal coefficient system and γ = { γtλ | λ ∈ PnΛ }.

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Finally, the seminormal basis machinery in this section can be used to classify the semisimple cyclotomic Hecke algebras Hn , thus re-proving Ariki’s semisimplicity criterion [2], when v 2 6= 1, and [13, Theorem 6.11], when v 2 = 1. Theorem 1.6.10 (Ariki [2] and [13, Theorem 6.11]). Suppose that F is a field. The following are equivalent: a) Hn = Hn (F, v, Q1 , . . . , Q` ) is semisimple. b) Hn is content separated. Y Y c) [1]v [2]v . . . [n]v (v 2d Qr + [d]v − Qs ) 6= 0. 1≤r 1,

ψr ys = ys ψr ,

if s 6= r, r + 1,

ψr yr+1 e(i) = (yr ψr + δir ir+1 )e(i), yr+1 ψr e(i) = (ψr yr + δir ir+1 )e(i),

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ψr2 e(i) =

(2.2.3)

  (yr+1 − yr )(yr − yr+1 )e(i),       (yr − yr+1 )e(i),

if ir  ir+1 , if ir → ir+1 ,

(yr+1 − yr )e(i), if ir ← ir+1 ,     0, if ir = ir+1 ,    e(i), otherwise, and (ψr ψr+1 ψr − ψr+1 ψr ψr+1 )e(i) is equal to   (yr + yr+2 − 2yr+1 )e(i), if ir+2 = ir  ir+1 ,    −e(i), if ir+2 = ir → ir+1 , (2.2.4)  e(i), if ir+2 = ir ← ir+1 ,     0, otherwise, for i, j ∈ I β and all admissible r and s. Part of the point of these definitions is that Rβ is a Z-graded algebra with degree function determined by deg e(i) = 0,

deg yr = 2

and

deg ψs e(i) = −cis ,is+1 ,

for 1 ≤ r ≤ n, 1 ≤ s < n and i ∈ I n . F Suppose that n ≥ 0. Then I n = β I β is the decomposition of I n into a disjoint union of Sn -orbits. Define M (2.2.5) Rn = Rβ . β∈Q+

Set β = i∈I β e(i), for β ∈ Q+ . Then Rβ = eβ Rn eβ is a two-sided ideal of Rn and (2.2.5) is the decomposition of Rn into blocks. That is, Rβ is indecomposable for all β ∈ Q+ . Khovanov and Lauda [74, 75] and Rouquier [121] define quiver Hecke algebras for quivers of arbitrary type. In the short time since their inception a lot has been discovered about these algebras. The first important result is that these algebras categorify the negative part of the corresponding quantum group [22, 74, 122, 132]. P

Remark 2.2.6. We have defined only a special case of the quiver Hecke algebras from [74, 121]. In addition to allowing arbitrary quivers, Khovanov and Lauda allow a more general choice of signs. Rouquier’s definition, which is the most general, defines the quiver Hecke algebras in terms of a matrix Q = (Qij )i,j∈I with entries in a polynomial ring Z[u, v] with the properties that Qii = 0, Qij is not a zero divisor in Z[u, v] for i = 6 j and Qij (u, v) = Qji (v, u), for i, j ∈ I. For an arbitrary quiver Γ,

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Rouquier [121, Definition 3.2.1] defines Rβ (Γ) to be the algebra generated by ψr , ys , e(i) subject to the relations above except that the quadratic and braid relations are replaced with ψr2 e(i) = Qir ,ir+1 (yr , yr+1 )e(i) and (ψr ψr+1 ψr − ψr+1 ψr ψr+1 )e(i) is equal to ( Qi ,i (yr ,yr+1 )−Qi ,i (yr ,yr+1 ) r r+1 r r+1 , if ir+2 = ir , yr+2 −yr 0,

otherwise.

The assumptions on Q ensure that the last expression is a polynomial in the generators. In general, yr e(i) is homogeneous of degree (αir , αir ), for 1 ≤ r ≤ n and i ∈ I n . Under some mild assumptions, the isomorphism type of Rβ is independent of the choice of Q by [121, Proposition 3.12]. We leave it to the reader to find a suitable matrix Q for Definition 2.2.1. For the rest of these notes for w ∈ Sn we arbitrarily fix a reduced expression w = sr1 . . . srk , with 1 ≤ rj < n. Using this fixed reduced expression for w define ψw = ψr1 . . . ψrk . 2.2.7. Example As the ψ-generators of Rn do not satisfy the braid relations the element ψw will, in general, depend upon the choice of reduced expression for w ∈ Sn . For example, by (2.2.4) if e 6= 2, n = 3 and w = s1 s2 s1 = s2 s1 s2 then ψ1 ψ2 ψ1 e(0, 2, 0) = ψ2 ψ1 ψ2 e(0, 2, 0) + e(0, 2, 0), by (2.2.4). Therefore, these two reduced expressions determine different elements of Rn . ♦ Khovanov and Lauda [74, Theorem 2.5] and Rouquier [121, Theorem 3.7] proved the following. Theorem 2.2.8 (Khovanov-Lauda [74] and Rouquier [121]). Suppose that β ∈ Q+ . Then Rβ (Z) is free as an Z-algebra with homogeneous basis { ψw y1a1 . . . ynan e(i) | w ∈ Sn , a1 , . . . , an ∈ N and i ∈ I β } . Li [92, Theorem 4.3.10] has constructed a graded cellular basis of Rn . In the special case when e = ∞, Kleshchev, Loubert and Miemietz [85] give a graded affine cellular basis of Rn , in the sense of K¨onig and Xi [87]. In these notes we are not directly concerned with the quiver Hecke algebras Rn . Rather, we are more interested in certain cyclotomic quotients of these algebras. Definition 2.2.9 (Brundan-Kleshchev [21]). Suppose that Λ ∈ P + . The cyclotomic quiver Hecke algebra of type Γe and weight Λ is the (Λ,α ) quotient algebra RnΛ = Rn /hy1 i1 e(i) | i ∈ I n i.

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We abuse notation and identify the KLR generators of Rn with their images in RnΛ . That is, we consider the algebra RnΛ to be generated by ψ1 , . . . , ψn−1 , y1 , . . . , yn and e(i), for i ∈ I n , subject to the relations in Definition 2.2.1 and Definition 2.2.9. From this point onwards, fix Λ ∈ P + . When Λ is a weight of level 2, the algebras RnΛ first appeared in the work of Brundan and Stroppel [26] in their series of papers on the Khovanov diagram algebras. In full generality, the cyclotomic quotients of Rn were introduced by Khovanov-Lauda [74] and Rouquier [121]. Brundan and Kleshchev initiated the study of the cyclotomic quiver Hecke algebras RnΛ , for any Λ ∈ P + . Although we will not use this here we note that, rather than working algebraically, it is often easier to work diagrammatically by identifying the elements of RnΛ with certain diagrams. In these diagrams, the endpoints of the strings are labeled by {1, 2, . . . , n, 10 , 20 , . . . , n0 } and the strings themselves are coloured by I n . For example, following [74], the KLR generators can be identified with the diagrams: i1 i2

i1

in

e(i) =

ir−1irir+1

ψr e(i) =

in

1

s − 1s

s

n

ys e(i) =

.

Multiplication of diagrams is given by concatenation, read from top to bottom, subject to the relations above that are also interpreted diagrammatically. As an exercise, we leave it to the reader to identify the two relations in Definition 2.2.1 that correspond to the following ‘local’ relations on strings inside braid diagrams: i

j

i

=

j

i

i i±1 i

i

+ δij

and

i i±1 i

=

i i±1 i

±

.

(For the second relation, e 6= 2.) For more rigorous definitions of such diagrams, and non-trivial examples of their application, we refer the reader to the papers [53, 81, 92, 97]. 2.2.10. Example (Rank one algebras) Suppose that n = 1 and Λ ∈ P + . hΛ,α i Then R1Λ = hy1 , e(i) | y1 e(i) = e(i)y1 and y1 i e(i) = 0, for i ∈ Ii, with deg y1 = 2 and deg e(i) = 0, for i ∈ I. Therefore, there is an isomorphism of graded algebras M R1Λ ∼ Z[y]/y (Λ,αi ) Z[y], = i∈I (Λ,αi )>0

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where y = y1 is in degree 2. Armed with this description of RnΛ it is now straightforward to show that HnΛ ∼ ♦ = RnΛ when Z is a field and n = 1. 2.3

Nilpotence and small representations

In this section and the next we use the KLR relations to prove some results about the cyclotomic quiver Hecke algebras RnΛ for particular Λ and n. By Theorem 2.2.8 the algebra Rn is infinite dimensional, so it is not obvious from the relations that the cyclotomic Hecke algebra RnΛ is finite dimensional — or even that RnΛ is non-zero. The following result shows that yr is nilpotent, for 1 ≤ r ≤ n, which implies that RnΛ is finite dimensional. Lemma 2.3.1 (Brundan and Kleshchev [21, Lemma 2.1]). Suppose that 1 ≤ r ≤ n and i ∈ I n . Then yrN e(i) = 0 for N  0. (Λ,α

)

Proof. We argue by induction on r. If r = 1 then y1 i1 e(i) = 0 by Definition 2.2.9, proving the base step of the induction. Now consider yr+1 e(i). By induction, we may assume that there exists N  0 such that yrN e(j) = 0, for all j ∈ I n . There are three cases to consider. Case 1. ir+1 — / ir . By (2.2.3) and (2.2.2), N N yr+1 e(i) = yr+1 ψr2 e(i) = ψr yrN ψr e(i) = ψr yrN e(sr · i)ψr = 0,

where the last equality follows by induction. Case 2. ir+1 = ir ± 1. Suppose first that e 6= 2. This is a variation on the previous case, with a twist. By (2.2.3) and (2.2.2), again 2N −1 2N −1 2N yr+1 e(i) = yr+1 yr e(i) + yr+1 (yr+1 − yr )e(i) 2N −1 2N −1 2 = yr yr+1 e(i) ± yr+1 ψr e(i) 2N −1 = yr yr+1 e(i) ± ψr yr2N −1 e(sr · i)ψr 2N −1 N = yr yr+1 e(i) = · · · = yrN yr+1 e(i) = 0. 2 The case when e = 2 is similar. First, observe that yr+1 e(i) = (2yr yr+1 − 2 2 yr − ψr )e(i) by (2.2.3). Therefore, arguing as before, 3N −2 3N N yr+1 e(i) = yr (2yr+1 − yr )yr+1 e(i) = · · · = yrN (2yr+1 − yr )N yr+1 e(i) = 0.

Case 3. ir+1 = ir . Let φr = ψr (yr − yr+1 ). Then φr ψr e(i) = −2ψr e(i) by (2.2.2), so that (1 + φr )2 e(i) = e(i). Moreover, (1 + φr )yr (1 + φr )e(i) = (yr + φr yr + yr φr + φr yr φr )e(i) = yr+1 e(i),

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where the last equality uses (2.2.2). Now we are done because
N N yr+1 e(i) = (1 + φr )yr (1 + φr ) e(i) = (1 + φr )yrN (1 + φr )e(i) = 0, since φr commutes with e(i) and yrN e(i) = 0 by induction.



We have marginally improved on Brundan and Kleshchev’s original proof of Lemma 2.3.1 because the argument above gives an upper bound for the nilpotency index of yr . In general, this bound is far from sharp. For a better estimate of the nilpotency index of yr see [57, Corollary 4.6] (and [53] when e = ∞). See [67, Lemma 4.4] for another argument that applies to cyclotomic quiver Hecke algebras of arbitrary type. Combining Theorem 2.2.8 and Lemma 2.3.1 we have: Corollary 2.3.2 (Brundan and Kleshchev [21, Corollary 2.2]). Suppose Z is an integral domain. Then RnΛ is finite dimensional. As our next exercise we classify the one dimensional representations of RnΛ when Z = F is a field. For i ∈ I let i+ n = (i, i + 1, . . . , i + n − 1) and ± n i− = (i, i − 1, . . . , i − n + 1). Then i ∈ I . If (Λ, αi ) = 0 then e(i± n n n ) = 0 by Definition 2.2.9. However, if (Λ, αi ) 6= 0 then using the relations it is easy + to see that RnΛ has unique one dimensional representations Di,n = F d+ i,n − and Di,n = F d− i,n such that d± d± i,n e(i) = δi,i± n i,n

± and d+ i,n yr = 0 = di,n ψs ,

for i ∈ I n , 1 ≤ r ≤ n and 1 ≤ s < n and such that deg d± i,n = 0. In Λ particular, this shows that e(i± ) = 6 0 and hence that R = 6 0. If e 6= 2 n n ± then { Di,n | i ∈ I and (Λ, αi ) 6= 0 } are pairwise non-isomorphic irreducible + − − representations of RnΛ . If e = 2 then i+ n = in so that Di,n = Di,n . Proposition 2.3.3. Suppose that Z = F is a field and that D is a one ± dimensional graded RnΛ -module. Then D ∼ hki, for some k ∈ Z and = Di,n i ∈ I such that (Λ, αi ) 6= 0. Proof. Let d be a non-zero element of D so that D = F d. Then d = P n j∈I n de(j) so that de(i) 6= 0 for some i ∈ I . Moreover, de(j) = 0 if and only if j = i since otherwise de(i) and de(j) are linearly independent elements of D, contradicting assumption that D is one dimensional. Now, deg dyr = 2 + deg d, so dyr = 0, for 1 ≤ r ≤ n, since D is one dimensional. Similarly, dψr = de(i)ψr = 0 if ir = ir+1 or ir = ir+1 ± 1 since in these cases deg e(i)ψr 6= 0. It remains to show that i = i± n and that (Λ, αi1 ) 6= 0. First, since 0 6= d = d e(i) we have that e(i) 6= 0 so that (Λ, αi1 ) 6= 0 by Definition 2.2.9.

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To complete the proof we show that if i 6= i± n then d = 0, which will give a contradiction. First, suppose that i = i for some r, with 1 ≤ r < n. r r+1
Then d = de(i) = d ψr yr+1 − yr ψr e(i) = 0 by (2.2.2), which is not possible, so ir = 6 ir+1 . Next, suppose that ir+1 = 6 ir ± 1. Then d = de(i) = dψr2 e(i) = dψr e(sr · i)ψr = 0 because D is one dimensional and de(j) = 0 if j 6= i. This is another contradiction, so we must have ir+1 = ir ± 1 for 1 ≤ r < n. Therefore, if i 6= i± 6 2, n > 2 and ir = ir+2 = ir+1 ± 1 for some r. n then e = Applying the braid relation (2.2.4), d = de(i) = ±d · (ψr ψr+1 ψr − ψr+1 ψr ψr+1 )e(i) = 0, ± a contradiction. Hence, D ∼ hdeg di, completing the proof. = Di,n 2.4



Semisimple KLR algebras

Now that we understand the one dimensional representations of RnΛ we consider the semisimple representation theory of the cyclotomic quiver Hecke algebras. These results do not appear in the literature, but there are few surprises here because everything we do can be easily deduced from results that are known. The main idea is to show by example how to use the quiver Hecke algebra relations. In this section we fix e > n and Λ ∈ P + such that (Λ, αi,n ) ≤ 1, for all i ∈ I, and we study the algebras RnΛ . Notice that these conditions ensure that HnΛ is semisimple by Corollary 1.6.11. Recall from §1.8 that it = (it1 , . . . , itn ) is the residue sequence of t ∈ Std(PnΛ ), where itr = cZr (t) + eZ. We caution the reader that if t is a standard tableau then the contents cZr (t) ∈ Z and the residues itr ∈ I are in general different. If i ∈ I then a node A = (l, r, c) is an i-node if i = κl + c − r + eZ. Therefore, extending the definitions of §1.4, we can now talk of addable and removable i-nodes. Lemma 2.4.1. Suppose that e > n and (Λ, αi,n ) ≤ 1, for all i ∈ I. Let s, t ∈ Std(PnΛ ). Then s = t if and only if is = it . Λ Proof. Observe that if i ∈ I and µ ∈ Pm , where 0 ≤ m < n, then µ has at most one addable i-node since (Λ, αi,n ) ≤ 1. Hence, it follows easily by induction on n that s = t if and only if is = it . 

Lemma 2.4.1 also follows from Theorem 1.6.10 and Corollary 1.6.11. Let IΛn = { it | t ∈ Std(PnΛ ) } be the set of residue sequences of all of the standard tableaux in Std(PnΛ ). By the proof of Lemma 2.4.1, if i = it ∈ IΛn

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and ir+1 = ir ± 1 then r and r + 1 must be in either in the same row or in the same column of t. Hence, we have the following useful fact. Corollary 2.4.2. Suppose that e > n and that (Λ, αi,n ) ≤ 1, for all i ∈ I, and that i ∈ IΛn with ir+1 = ir ± 1. Then sr · i ∈ / IΛn . When Λ = Λ0 the next result is due to Brundan and Kleshchev [21, §5.5]. More generally, Kleshchev and Ram [83, Theorem 3.4] prove similar results for quiver Hecke algebras of simply laced type. Proposition 2.4.3 (Seminormal representations of RnΛ ). Suppose that Z = F is a field, e > n and that Λ ∈ P + with (Λ, αi,n ) ≤ 1, for all i ∈ I. Then for each λ ∈ PnΛ there is a unique irreducible graded RnΛ module S λ with homogeneous basis { ψt | t ∈ Std(λ) } such that deg ψt = 0, for all t ∈ Std(λ), and where the RnΛ -action is given by ψt e(i) = δi,it ψt ,

ψt yr = 0

and

ψt ψr = ψt(r,r+1) ,

where we set ψt(r,r+1) = 0 if t(r, r + 1) is not standard. Proof. By Lemma 2.4.1, if s, t ∈ Std(λ) then s = t if and only if is = it . Moreover, itr+1 = itr ± 1 if and only if r and r + 1 are in the same row or in the same column of t. Similarly, itr 6= itr+1 for any r. Consequently, since ψt = ψt e(it ) almost all of the relations in Definition 2.2.1 are trivially satisfied. In fact, all that we need to check is that ψ1 , . . . , ψn−1 satisfy the braid relations of the symmetric group Sn with ψr2 acting as zero when itr+1 = itr ± 1, which follows automatically by Corollary 2.4.2. By the same reasoning if t(r, r + 1) is standard then deg e(it )ψr = 0. Hence, we can set deg ψt = 0, for all t ∈ Std(λ). This proves that S λ is a graded RnΛ -module. It remains to show that S λ is irreducible. If s, t ∈ Std(λ) then s = P λ t d(s) = td(t)−1 d(s), so ψs = ψt ψd(t)−1 ψd(s) . Suppose that x = t rt ψt is a non-zero element of S λ . If rt 6= 0 then ψt = r1t xe(it ), so it follows that ψs ∈ xRnΛ , for any s ∈ Std(λ). Therefore, S λ = xRnΛ so that S λ is irreducible as claimed.  Hence, e(i) 6= 0 in RnΛ , for all i ∈ InΛ . This was not clear until now. We want to show that Proposition 2.4.3 describes all of the graded irreducible representations of RnΛ , up to degree shift. To do this we need a better understanding of the set IΛn . Okounkov and Vershik [117, Theorem 6.7] explicitly described the set of all content sequences (cZ1 (t), . . . , cZn (t)) when ` = 1. This combinatorial result easily extends to higher levels and so suggests a description of IΛn .

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If i ∈ I n and 1 ≤ m ≤ n set i↓m = (i1 , . . . , im ). Then i↓m ∈ I m and = { i↓m | i ∈ IΛn }.

Lemma 2.4.4 (cf. Ogievetsky-d’Andecy [116, Proposition 5]). Suppose that e > n and (Λ, αi,n ) ≤ 1, for all i ∈ I. Let i ∈ I n . Then i ∈ IΛn if and only if it satisfies the following three conditions: a) (Λ, αi1 ) 6= 0. b) If 1 < r ≤ n and (Λ, αir ) = 0 then {ir −1, ir +1}∩{i1 , . . . , ir−1 } = 6 ∅. c) If 1 ≤ s < r ≤ n and ir = is then {ir − 1, ir + 1} ⊆ {is+1 , . . . , ir−1 }. Proof. Suppose that t ∈ Std(PnΛ ) and let i = it . We prove by induction on r that i↓r ∈ IΛr . By definition, i1 = κt + eZ for some t with 1 ≤ t ≤ `, so (a) holds. By induction we may assume that the subsequence (i1 , . . . , ir−1 ) satisfies properties (a)–(c). If (Λ, αir ) = 0 then r cannot be in both the first row and in the first column of any component of t, so t has an entry in the row directly above r or in the column immediately to the left of r — or both! Hence, there exists an integer s with 1 ≤ s < r such that its = itr ± 1. Hence, (b) holds. Finally, suppose that ir = is as in (c). As the residues of the nodes in different components of t are disjoint it follows that s and r are in same component of t and on the same diagonal. In particular, r is not in the first row or in the first column of its component in t. As t is standard, the entries in t that are immediately above or to the left of r are both larger than s and smaller than r. Hence, (c) holds. Conversely, suppose that i ∈ I n satisfies properties (a)–(c). We show by induction on m that i↓m ∈ IΛm , for 1 ≤ m ≤ n. If m = 1 then i↓1 ∈ IΛ1 by property (a). Now suppose that 1 < m < n and that i↓m ∈ IΛm . By Λ induction i↓m = is , for some s ∈ Std(Pm ). Let ν = Shape(s). If i ∈ I then (Λ, αi,n ) ≤ 1, so the multipartition ν can have at most one addable i-node. On the other hand, reversing the argument of the last paragraph, using properties (b) and (c) with r = m + 1, shows that ν has at least one addable im+1 -node. Let A be the unique addable im+1 -node of ν. Then i↓(m+1) = it Λ where t ∈ Std(Pm+1 ) is the unique standard tableau such that t↓m = s and t(A) = m + 1. Hence, i ∈ IΛm+1 as required.  By Proposition 2.4.3, if i ∈ IΛn then e(i) 6= 0. We use Lemma 2.4.4 to show that e(i) = 0 if i ∈ / IΛn . First, a result that holds for all Λ ∈ P + . Lemma 2.4.5. Suppose that Λ ∈ P + and that e(i) 6= 0, for i ∈ I n . Then (Λ, αi1 ) 6= 0. Moreover, {ir − 1, ir + 1} ∩ {i1 , . . . , ir−1 } = 6 ∅ whenever (Λ, αir ) = 0 for some 1 < r ≤ n.

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Proof. By Definition 2.2.9, e(i) = 0 whenever (Λ, αi1 ) = 0. To prove the second claim suppose that (Λ, αir ) = 0 and ir ± 1 ∈ / {i1 , . . . , ir−1 }. We may assume that ir 6= is for 1 ≤ s < r. Applying (2.2.3) r-times, 2 e(i) = ψr−1 e(i) = ψr−1 e(i1 , . . . , ir , ir−1 , ir+1 , . . . , in )ψr−1

= · · · = ψr−1 . . . ψ1 e(ir , i1 , . . . , ir−1 , ir+1 , . . . , in )ψ1 . . . ψr−1 = 0, where the last equality follows because (Λ, αir ) = 0.



Proposition 2.4.6. Suppose that 1 ≤ m ≤ n and that (Λ, αi,m ) ≤ 1, for all i ∈ I. Then y1 = · · · = ym = 0. Moreover, if i ∈ I n then e(i) 6= 0 only if i↓m ∈ IΛm . Proof. We argue by induction on r to show that yr = 0 and e(i) = 0 if (Λ,α ) i↓r ∈ / IΛr , for 1 ≤ r ≤ m. If r = 1 this is immediate because y1 i1 e(i) = 0 by Definition 2.2.9 and (Λ, αi1 ) ≤ 1 by assumption. Suppose then that 1 < r ≤ m. We first show that e(i) = 0 if i↓r ∈ / IΛr . By induction, Lemma 2.4.4 and Lemma 2.4.5, it is enough to show that e(i) = 0 whenever there exists an integer 1 ≤ s < r such that is = ir and {ir − 1, ir + 1} 6⊆ {is+1 , . . . , ir−1 }. We may assume that s is maximal such that is = ir and 1 ≤ s < r. There are three cases to consider. Case 1. r = s + 1. By (2.2.2), e(i) = (ys+1 ψs − ψs ys )e(i) = ys+1 ψs e(i), since ys = 0 by induction. Using this identity twice, reveals that e(i) = ys+1 ψs e(i) = 2 2 ys+1 e(i)ψs = ys+1 ψs e(i)ψs = ys+1 ψs2 e(i) = 0, where the last equality comes from (2.2.3). Therefore, e(i) = 0 as we wanted to show. Case 2. s < r − 1 and {ir − 1, ir + 1} ∩ {is+1 , . . . , ir−1 } = ∅. By the maximality of s, ir ∈ / {is+1 , . . . , ir−1 }. Therefore, arguing as in the proof of Lemma 2.4.5, there exists a permutation w ∈ Sr such that e(i) = ψw e(i1 , . . . , is , ir , is+1 , . . . , ir−1 , ir+1 , . . . , in )ψw . Hence, e(i) = 0 by Case 1. Case 3. s < r − 1 and {ir − 1, ir + 1} ∩ {is+1 , . . . , ir−1 } = {j}, where j = ir ± 1. Let t be an index such that it = j = ir ± 1 and s < t < r. Note that if there exists an integer t0 such that it = it0 and s < t < t0 < r then we may assume that is ∈ {it+1 , . . . , it0 −1 } by Lemma 2.4.4(c) and induction. Therefore, since s was chosen to be maximal, t is the unique integer such

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that it = j and s < t < r. Hence, arguing as in Case 2, there exists a permutation w ∈ Sr such that e(i) = ψw e(. . . , is−1 , is+1 , . . . , it−1 , is , it , ir , it+1 , . . . , ir−1 , ir+1 , . . . )ψw . For convenience, we identify e(i1 , . . . , is , it , ir , . . . , in ) with e(i, j, i), where i = is = ir and j = i ± 1. Then, by (2.2.4),
e(i, j, i) = ± ψ1 ψ2 ψ1 − ψ2 ψ1 ψ2 e(i, j, i) = ±ψ1 ψ2 e(j, i, i)ψ1 ∓ ψ2 ψ1 e(i, i, j)ψ2 = 0, where the last equality follows by Case 1. Combining Cases 1–3, if e(i) 6= 0 then {ir − 1, ir + 1} ⊆ {is+1 , . . . , ir−1 } whenever there exists an integer s such that is = ir and 1 ≤ s < r. Hence, as remarked above, induction, Lemma 2.4.5 and Lemma 2.4.4 show that e(i) 6= 0 only if i↓r ∈ IΛr . To complete the proof of the inductive step (and of the proposition), it remains to show that yr = 0. Using what we have just proved, it is enough to show that yr e(i) = 0 whenever i↓r ∈ IΛr . If ir−1 = ir ± 1 then, by induction and (2.2.3), 2 yr e(i) = (yr − yr−1 )e(i) = ±ψr−1 e(i) = ±ψr−1 e(sr−1 · i)ψr−1 = 0,

/ IΛr by Corollary 2.4.2. where the last equality follows because (sr−1 · i)↓r ∈ If ir−1 6= ir ± 1 then ir−1 — / ir by Lemma 2.4.4 since i↓r ∈ IΛr . Therefore, 2 yr e(i) = yr ψr−1 e(i) = ψr−1 yr−1 ψr−1 e(i) = 0 since yr−1 = 0 by induction. This completes the proof.  Before giving our main application of Proposition 2.4.6 we interpret this result for the cyclotomic quiver Hecke algebras of the symmetric groups. 2.4.7. Example (Symmetric groups) Suppose that Λ = Λ0 , n ≥ 0 and set f = min{e, n}. Then (Λ, αi,f −1 ) ≤ 1 for all i ∈ I. Therefore, Proposition 2.4.6 shows that yr = 0 for 1 ≤ r < f and that e(i) 6= 0 only if i↓(f −1) ∈ IΛf −1 . In addition, we also have ψ1 = 0 because if i ∈ I n then ψ1 e(i) = e(s1 · i)ψ1 = 0 because if i↓(f −1) ∈ IΛf −1 then (s1 · i)f −1 ∈ / IΛf −1 . Translating the proof of Proposition 2.4.6 back to Lemma 2.4.1, the reason why ψ1 = 0 is that if i = it is the residue sequence of some standard tableau t ∈ Std(PnΛ ) then i1 = 0 and i2 6= 0, so that s1 · i ∈ / InΛ is not a residue sequence and, consequently, ψ1 e(i) = e(s1 · i)ψ1 = 0. By the same reasoning, ψ1 6= 0 if Λ has level ` > 1. ♦

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We now completely describe the structure of the KLR algebras RnΛ when e > n and Λ ∈ P + such that (Λ, αi,n ) ≤ 1, for all i ∈ I. For (s, t) ∈ Std2 (PnΛ ) λ define est = ψd(s)−1 e(iλ )ψd(t) , where iλ = it . Theorem 2.4.8. Suppose that e > n and Λ ∈ P + with (Λ, αi,n ) ≤ 1, for all i ∈ I. Then RnΛ is a graded cellular algebra with graded cellular basis { est | (s, t) ∈ Std2 (PnΛ ) } with deg est = 0 for all (s, t) ∈ Std2 (PnΛ ). Proof. By Proposition 2.4.6, yr = 0 for 1 ≤ r ≤ n and e(i) = 0 if i ∈ / InΛ . In particular, this implies that ψ1 , . . . , ψn−1 satisfy the braid relations for the symmetric group Sn because, by Lemma 2.4.4, if i ∈ IΛn then (i, i ± 1, i) is not a subsequence of i, for any i ∈ I. Therefore, RnΛ is spanned by the elements ψv e(i)ψw , where v, w ∈ Sn and i ∈ IΛn . Moreover, if j ∈ I n then e(j)ψv e(i)ψw = 0 unless j = v · i ∈ IΛn . Therefore, RnΛ is spanned by the elements { est | (s, t) ∈ Std2 (PnΛ ) } as required by the statement of the theorem. (Note that s and t must have the same shape because is and it are in the same Sn -orbit and the multiset of contents determines the shape; compare with Theorem 1.8.1) Hence, RnΛ has rank at most P 2 = `n n!, where this combinatorial identity comes from Λ | Std(λ)| λ∈Pn Theorem 1.6.7. Let K be the algebraic closure of the field of fractions of Z. Then RnΛ (K) ∼ = RnΛ (Z) ⊗Z K. By the last paragraph, the dimension of RnΛ is at most `n n!. Let rad RnΛ (K) be the Jacobson radical of RnΛ (K). For each multipartition λ ∈ PnΛ , Proposition 2.4.3 constructs an irreducible graded Specht module S λ . By Lemma 2.4.1, if λ, µ ∈ PnΛ and d ∈ Z then Sλ ∼ = S µ hdi if and only if λ = µ and d = 0. By the Wedderburn theorem, X `n n! ≥ dim RnΛ (K)/ rad RnΛ (K) ≥ (dim S λ )2 Λ λ∈Pn

=

X

| Std(λ)|2 = `n n!.

Λ λ∈Pn

Hence, we have equality throughout, so { est | (s, t) ∈ Std2 (PnΛ ) } is a basis of RnΛ (K). As the elements {est } span RnΛ (Z), and their images in RnΛ (K) are linearly independent, so {est } is a basis of RnΛ (Z). It remains to prove that {est } is a graded cellular basis of RnΛ . The orthogonality of the KLR idempotents implies that est euv = δtu esv . Therefore, {est } is a basis of matrix units for RnΛ . Consequently, RnΛ is a direct sum of matrix rings, for any integral domain Z, and {est } is a cellular basis of RnΛ . Finally, we need to show that est is homogeneous of degree zero. This will follow if we show that deg ψr e(i) = 0, for 1 ≤ r < n and i ∈ IΛn . In fact,

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this is already clear because if i ∈ IΛn then ir 6= ir+1 , by Lemma 2.4.4, and if ir+1 =ir ± 1 then ψr e(i)=0 by Corollary 2.4.2 and Proposition 2.4.6.  By definition, est euv = δtv esv . Let Matd (Z) be the ring of d × d matrices over Z. Hence, the proof of Theorem 2.4.8 yields the following. Corollary 2.4.9. Suppose that Z is an integral domain e > n and that Λ ∈ P + with (Λ, αi,n ) ≤ 1, for all i ∈ I. Then M RnΛ (Z) ∼ Matsλ (Z), = Λ λ∈Pn

where sλ = # Std(λ) for λ ∈ PnΛ . Another consequence of Theorem 2.4.8 is that the KLR relations simplify in the semisimple case — giving a non-standard presentation for a direct sum of matrix rings. Corollary 2.4.10. Suppose that Z is an integral domain, e > n and that Λ ∈ P + with (Λ, αi,n ) ≤ 1, for all i ∈ I. Then RnΛ is the unital associative Z-graded algebra generated by ψ1 , . . . , ψn−1 and e(i), for i ∈ I n , subject to the relations P e(i)(Λ,αi1 ) = 0 e(i)e(j) = δij e(i), i∈I n e(i) = 1, ψr e(i) = e(sr · i)ψr

e(i) = 0 if ir = ir+1 ,

ψr ψs = ψs ψr , if    (ψr+1 ψr ψr+1 − 1)e(i), ψr ψr+1 ψr e(i) = (ψr+1 ψr ψr+1 + 1)e(i),   ψ ψ ψ e(i), r+1

r

r+1

ψr2 e(i) = e(i)

|r − s| > 1, if ir+2 = ir → ir+1 , if ir+2 = ir ← ir+1 , otherwise,

for all i, j ∈ I and admissible r and s. Moreover, RnΛ is concentrated in degree zero. n

The reader is encouraged to check the details here. Note that these relations, together with the argument of Proposition 2.4.6, imply that e(i) 6= 0 if only if i ∈ IΛn . In particular, the combinatorics of tableau content sequences is partially encoded in the failure of the braid relations for the ψr . As a final application, we prove Brundan and Kleshchev’s graded isomorphism theorem in this special case. Corollary 2.4.11. Suppose that Z = K is a field, e > n, and that Λ ∈ P + with (Λ, αi,n ) ≤ 1, for all i ∈ I. Then RnΛ ∼ = HnΛ .

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Proof. By Corollary 2.4.10 and Theorem 1.6.7, there is a well-defined homomorphism Θ : RnΛ −→ HnΛ determined by e(is ) 7→ Fs and    Z Z  1 T − cr+1 (s)−cr (s) F , if α (s) 6= 0, r s r Z −1 1+(v−v )cr+1 (s) ψr e(is ) 7→ αr (s) 0, otherwise, for s ∈ Std(PnΛ ) and 1 ≤ r < n. Using Theorem 2.4.8, or Proposition 2.4.3, it follows that Θ is an isomorphism.  We emphasize that it is essential to work over a field in Corollary 2.4.11 because Corollary 2.4.9 says that RnΛ is always a direct sum of matrix rings. If n > 1 this is only true of HnΛ when it is defined over a field. These results suggest that RnΛ should be considered as the “idempotent completion” of the algebra HnΛ obtained by adjoining idempotents e(i), for i ∈ I n . We will see how to make sense of the idempotents e(i) ∈ HnΛ for any i ∈ I n in Theorem 3.1.1 and Lemma 4.2.2 below. 2.5

The nil-Hecke algebra

Still working just with the relations we now consider the shadow of the nilHecke algebra in the cyclotomic KLR setting. For the affine KLR algebras the nil-Hecke algebras case has been well-studied [74,121]. For the cyclotomic quotients (in type A) the story is similar. For this section fix i ∈ I and set β = nαi and Λ = nΛi . Following (2.2.5), set RβΛ = e(i)RnΛ e(i), where i = iβ = (in ). Then RβΛ is a direct summand of RnΛ and, moreover, it is a non-unital subalgebra with identity element e(i). As e(i) is the unique non-zero KLR idempotent in RβΛ , ψr = ψr e(i) and ys = ys e(i). Therefore, RβΛ is the unital associative graded algebra generated by ψr and ys , for 1 ≤ r < n and 1 ≤ s ≤ n, with relations y1n = 0,

ψr2 = 0,

ψr yr+1 = yr ψr + 1, ψr ψs = ψs ψr

if |r − s| > 1,

yr ys = ys yr , yr+1 ψr = ψr yr + 1, ψr ys = ys ψr

if s 6= r, r + 1,

ψr ψr+1 ψr = ψr+1 ψr ψr+1 . The grading on RβΛ is determined by deg ψr = −2 and deg ys = 2. Some readers will recognize this presentation as defining as a cyclotomic quotient of the nil-Hecke algebra of type A [88]. Note that the argument from Case 3 of Lemma 2.3.1 shows that yr` = 0 for 1 ≤ r ≤ `. Let λ = (1|1| . . . |1) ∈ PβΛ . Then the map t 7→ d(t) defines a bijection between the set of standard λ-tableaux and the symmetric group Sn .

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For convenience, we identify the standard λ-tableaux with the set of (nonstandard) tableaux of partition shape (n) by concatenating their components. In other words, if d = d(t) then t = d1 d2 ··· dn , where d = d1 . . . dn is the permutation written in one-line notation. If v, s ∈ Std(λ) then write s  · v if s  v and (d(v)) = (d(s)) + 1. To make this more explicit write t ≺v m if t is in an earlier component of v than m — that is, t is to the left of m in v. The reader can check that s · v if and only if there exist integers 1 ≤ m < t ≤ n such that s = v(m, t), m ≺v t and if m < l < t then either l ≺v m or t ≺v l. 2.5.1. Example Suppose that n = 6. Let v = 4 6 5 3 1 2 and take t= 3. Then  3 6 5 4 1 2, 4 6 3 5 1 2, 4 6 5 2 1 3, 4 6 5 1 3 2 is the set of λ-tableaux { s | s = v(3, r)  · v for 1 ≤ r ≤ n }.

♦

We can now state the main result of the section. Proposition 2.5.2. Suppose that β = nαi and Λ = nΛi , for i ∈ I. S λ with homogeneous basis Then there is a unique graded RβΛ -module n { ψs | s ∈ Std(λ) } such that deg ψs = 2 − 2(d(s)) and  ψs(r,r+1) , if s  s(r, r + 1) ∈ Std(λ), ψs ψr = 0, otherwise,   ψu − ψu , ψv yt = 1≤k 0 and that e = pf , where f > 1. Then F cannot contain an element v of quantum characteristic e, so Theorem 3.1.1 says nothing about the quiver Hecke algebra RnΛ (F ). As a first consequence of Theorem 3.1.1, by identifying HnΛ and RnΛ we can consider HnΛ as a graded algebra. Corollary 3.1.2. Suppose that Λ ∈ P + and Z = F is a field. Then there is a unique grading on HnΛ such that deg e(i) = 0, deg yr = 2 and deg ψs e(i) = −cis ,is+1 , for 1 ≤ r ≤ n, 1 ≤ s < n and i ∈ I n . Brundan and Kleshchev prove Theorem 3.1.1 by constructing family of isomorphisms RnΛ −→ HnΛ , together with their inverses, and then painstakingly checking that these isomorphisms respect the relations of both algebras. Their argument starts with the well-known fact that HnΛ decomposes into a direct sum of simultaneous generalized eigenspaces for the Jucys-Murphy elements L1 , . . . , Ln . These eigenspaces are indexed by I n , so for each i ∈ I n there is an element e(i) ∈ HnΛ , possibly zero, such that e(i)e(j) = δij e(i). We describe these idempotents explicitly in Lemma 4.2.2 below. Translating through Definition 1.1.1, Brundan and Kleshchev’s isomorphism is given by e(i) 7→ e(i) and X X

1 yr 7→ v −ir Lr − [ir ]v e(i), and ψs 7→ Ts + Ps (i) e(i), Qs (i) n n i∈I

i∈I

for 1 ≤ r ≤ n, 1 ≤ s < n, i ∈ I n and where Pr (i) and Qr (i) are certain rational functions in yr and yr+1 that are well-defined because (Lt −[it ]v )e(i) is nilpotent in HnΛ , for 1 ≤ t ≤ n; see [21, §3.3 and §4.3]. Here we are abusing notation by identifying the KLR generators with their images in HnΛ . The inverse isomorphism is given by e(i) 7→ e(i), X X

Lr 7→ v ir yr + [ir ]v e(i) and Ts 7→ ψs Qs (i) − Ps (i) e(i), i∈I n

for 1 ≤ r ≤ n, 1 ≤ s < n and i ∈ I n .

i∈I n

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Rouquier [121, Corollary 3.20] has given a more direct proof of Theorem 3.1.1 by first showing that the (non-cyclotomic) quiver Hecke algebra Rn is isomorphic to the (extended) affine Hecke algebra of type A. Following [57], we sketch another approach to Theorem 3.1.1 in §4.2 below. The following easy but important application of Theorem 3.1.1 was a surprise (at least to the author!). Corollary 3.1.3. Suppose that Z = F is a field and that v, v 0 ∈ F are two elements of quantum characteristic e. Then HnΛ (F, v) ∼ = HnΛ (F, v 0 ). Proof.

By Theorem 3.1.1, HnΛ (F, v) ∼ = RnΛ (F ) ∼ = HnΛ (F, v 0 ).



Consequently, up to isomorphism, the algebra HnΛ depends only on e, Λ and the field F . Therefore, because HnΛ is cellular, the decomposition matrices of HnΛ depend only on e, Λ and p, where p is the characteristic of F . In the special case of the symmetric group, when Λ = Λ0 , this weaker statement for the decomposition matrices was conjectured in [104, Conjecture 6.38]. When F = C it is easy to prove Corollary 3.1.3 because there is a Galois automorphism of Q(v), as an extension of Q, that interchanges v and v 0 . It is not difficult to see that this automorphism induces an isomorphism HnΛ (F, v) ∼ = HnΛ (F, v 0 ). This argument fails for fields of positive characteristic because such fields have fewer automorphisms.

3.2

Graded Specht modules

As we noted in §2.1, if we impose a grading on an algebra A then it is not true that every (ungraded) A-module has a graded lift, so there is no reason to expect that graded lifts of Specht modules S λ exist. Of course, graded Specht modules do exist and this section describes one way to define them. Recall from §1.5 that the ungraded Specht module S λ , for λ ∈ PnΛ , has basis { mt | t ∈ Std(λ) }. By construction, S λ = mtλ HnΛ . Brundan, Kleshchev and Wang [25] proved that S λ has a graded lift essentially by declaring that mtλ should be homogeneous and then showing that this induces a grading on the Specht module S λ = mtλ RnΛ . Partly inspired by [25], Jun Hu and the author [54] showed that HnΛ is a graded cellular algebra. The graded cell modules constructed from this cellular basis coincide exactly with those of [25]. Perhaps most significantly, the construction of the graded Specht modules using cellular algebra tech-

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niques endows the graded Specht modules with a homogeneous bilinear form of degree zero. Following Brundan, Kleshchev and Wang [25, §3.5] we now define the degree of a standard tableau. Suppose that µ ∈ PnΛ . For i ∈ I let Addi (µ) be the set of addable i-nodes of µ and let Remi (µ) be its set of removable i-nodes. Definition 3.2.1. If A is an addable or removable i-node of µ define: dA (µ) = # { B ∈ Addi (µ) | A > B } − # { B ∈ Remi (µ) | A > B } , dA (µ) = # { B ∈ Addi (µ) | A < B } − # { B ∈ Remi (µ) | A < B } , di (µ) = # Addi (µ) − # Remi (µ).

If t is a standard µ-tableau then its codegree and degree are defined inductively by setting codege t = 0 = dege t if n = 0 and if n > 0 then codege t = codege t↓(n−1) + dA (µ) and dege t = dege t↓(n−1) + dA (µ), where A = t−1 (n). If e is fixed write codeg t = codege t and deg t = dege t. Implicitly, all of these definitions depend on the choice of multicharge κ. The definition of the degree and codegree of a standard tableau is due to Brundan, Kleshchev and Wang [25], however, the underlying combinatorics dates back to Misra and Miwa [111] and their work on the crystal graph  e ). and Fock space representations of Uq (sl Recall that we have fixed an arbitrary reduced expression for each permutation w ∈ Sn . In §1.4 for each standard tableau t ∈ Std(λ) we have defined permutations d (t), d(t) ∈ Sn by tλ d (t) = t = tλ d(t).

Definition 3.2.2 ([54, Definitions 4.9 and 5.1]). Suppose that µ ∈ PnΛ . Deµ 1 n fine non-negative integers dµ 1 , . . . , dn and dµ , . . . , dµ recursively by requiring µ µ µ 1 k that dµ + · · · + dµ = codeg(t↓k ) and d1 + · · · + dk = deg(tµ ↓k ), for 1 ≤ k ≤ n. µ

d1

dn

dµ

dµ

Now set iµ = itµ , iµ = it , yµ = y1 µ . . . ynµ and y µ = y1 1 . . . ynn . For (s, t) ∈ Std2 (µ) define   = ψd (s) e(iµ )yµ ψd (t) and ψst = ψd(s) e(iµ )y µ ψd(t) , ψst where  is the unique (homogeneous) anti-isomorphism of RnΛ that fixes the KLR generators.

3.2.3. Example Suppose that e = 3, Λ = Λ0 +Λ2 and µ = (7, 6, 3, 2|4, 3, 1), with multicharge κ = (0, 2). Then   1 2 3 4 5 6 7 19 20 21 22  8 9 10 11 12 13  23 24 25  tµ =   14 15 16  26 17 18

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The reader may check that e(iµ ) = e(01201202012011200120121200). We have shaded the nodes in tµ when they have column index divisible by e and when they have residue 2 = restµ (19). This should convince the reader that y µ = y32 y62 y8 y10 y11 y13 y15 y16 y21 y25 . With analogous shadings,   9 13 17 20 22 24 26 1 4 6 8  10 14 18 21 23 25  2 5 7 . tµ =   11 15 19  3 12 16 Hence, reading right to left, yµ = y32 y4 y7 y11 y15 y19 . Note that restµ (9) = 0. ♦

Theorem 3.2.4 (Hu-Mathas [54, Theorem 5.8]). Suppose that Z = F is a field. Then { ψst | (s, t) ∈ Std2 (PnΛ ) } is a graded cellular basis of RnΛ  with ψst = ψts and deg ψst = deg s + deg t, for (s, t) ∈ Std2 (PnΛ ). 3.2.5. Example Let β = nαi and Λ = nΛi , for some i ∈ I, so that RβΛ is the nil-Hecke algebra RβΛ of §2.5. Let λ = (1|1| . . . |1). Then the definitions 2 give y λ = y1n−1 . . . yn−2 yn−1 . Hence, the basis {ψst } of RβΛ coincides with that of Corollary 2.5.3. ♦ 3.2.6. Example As in Example 2.2.7, in general, the basis element ψst depends on the choices of reduced expressions that we have fixed for the permutations d(s) and d(t). For example, let Λ = 2Λ0 + Λ1 , κ = (0,  µ = (1|1|1) and consider  the standard µ-tableaux  1, 0) and  tµ = 1  2  3 and tµ = 3  2  1 . Then d(tµ ) = 1 and d(tµ ) = (1, 3) = s1 s2 s1 = s2 s1 s2 has two different reduced expressions. Let ψtµ tµ = ψ1 ψ2 ψ1 e(iµ )y µ ψ1 ψ2 ψ1 and ψˆtµ tµ = ψ2 ψ1 ψ2 e(iµ )y µ ψ2 ψ1 ψ2 . Then the calculation in Example 2.2.7 implies that ψˆtµ tµ = ψtµ tµ + ψtµ tµ + ψtµ tµ + ψtµ tµ . This is probably the simplest example where different reduced expressions lead to different ψ-basis elements, but examples occur for almost all RnΛ . This said, in view of Proposition 2.4.3, ψst is independent of the choice of reduced expressions for d(s) and d(t) whenever e > n and (Λ, αi,n ) ≤ 1, for all i ∈ I. The ψ-basis can be independent of the choice of reduced expressions even when RnΛ is not semisimple. For example, this is always the case when e > n and  = 2 by [55, Appendix], yet these algebras are typically not semisimple. ♦

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Using the theory of graded cellular algebras from §2.1, Theorem 3.2.4 allows us to construct a family { SFλ | λ ∈ PnΛ } of graded Specht modules for HnΛ . By [54, Corollary 5.10] the graded Specht modules attached to the ψ-basis coincide with those constructed by Brundan, Kleshchev and Wang [25]. When e > n and (Λ, αi,n ) ≤ 1, for i ∈ I, it is not hard to show that these Specht modules coincide with those we constructed in Proposition 2.4.3 above. Similarly, for the nil-Hecke algebra considered in §2.5, the graded Specht module SFλ , with λ = (1|1| . . . |1), is isomorphic to the graded module constructed in Proposition 2.5.2. Moreover, on forgetting the grading SFλ coincides exactly with the ungraded Specht module S λ F constructed in §1.5, for λ ∈ PnΛ . If λ ∈ PnΛ the graded Specht module SFλ has basis { ψt | t ∈ Std(λ) }, with deg ψt = deg t. The reader should be careful not to confuse ψt ∈ SFλ with ψd(t) ∈ RnΛ ! By Theorem 3.2.4 we recover [22, Theorem 4.20]: dim q HnΛ =

X (s,t)∈Std(λ)

q deg s+deg t =

X


2 dim q SFλ .

Λ λ∈Pn

In essence, Theorem 3.2.4 is proved in much the same way that Brundan, Kleshchev and Wang [25] constructed a grading on the Specht modules: we proved that the transition matrix between the ψ-basis and the Murphy basis of Theorem 1.5.1 is triangular. In order to do this we needed the correct definition of the elements y µ , which we discovered by first looking at the one dimensional two-sided ideals of HnΛ (which are necessarily homogeneous). We then used Brundan and Kleshchev’s Graded Isomorphism Theorem 3.1.1, together with the seminormal forms (Theorem 1.6.7), to show that e(iµ )y µ 6= 0. This established that the basis of Theorem 3.2.4 is a graded cellular basis. Finally, the combinatorial results of [25] are used to determine the degree of ψ-basis elements. Following the recipe in §2.1, for µ ∈ PnΛ define DFµ = SFµ / rad SFµ , where rad SFµ is the radical of the homogeneous bilinear form on SFµ . This yields the classification of the graded irreducible HnΛ -modules. The main point of the next result is that the labelling of the graded irreducible HnΛ -modules agrees with Corollary 1.5.2. Corollary 3.2.7 ([22, Theorem 5.13], [54, Corollary 5.11]). Suppose that Λ ∈ P + and that Z = F is a field. Then { DFµ hdi | µ ∈ KnΛ and d ∈ Z } is a complete set of pairwise non-isomorphic graded HnΛ -modules. Moreover, µ µ (DFµ )~ ∼ = DF and DF is absolutely irreducible, for all µ ∈ KnΛ .

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The graded decomposition numbers are the Laurent polynomials X µ λ (3.2.8) dF [SFλ : DFµ hdi)] q d , λµ (q) = [SF : DF ]q = d∈Z

for λ ∈ PnΛ and µ ∈ KnΛ . Write S λ = SFλ , Dµ = DFµ and dλµ (q) = dF λµ (q) when F is understood. By definition, dλµ (q) ∈ N[q, q −1 ] is a Laurent polynomial with non-negative coefficients. Let dq = (dλµ (q))λ∈PnΛ ,µ∈KnΛ be the graded decomposition matrix of HnΛ . The KLR algebra Rn is always Z-free, however, it is not clear whether the same is true for the cyclotomic KLR algebra RnΛ . To prove this you cannot use the Graded Isomorphism Theorem 3.1.1 because this result holds only over a field. Using extremely sophisticated diagram calculus, Li [92] proved the following. Theorem 3.2.9 (Li [92]). Suppose that Λ ∈ P + . Then the quiver Hecke algebra RnΛ (Z) is free as a Z-module of rank `n n!. Moreover, RnΛ (Z) is a graded cellular algebra with graded cellular basis { ψst | (s, t) ∈ Std2 (PnΛ ) }. Therefore, RnΛ is free over any commutative ring and any field is a splitting field for RnΛ . Moreover, the graded Specht modules, together with their homogeneous bilinear forms, are defined over Z. The integrality of the graded Specht modules can also be proved using Theorem 3.6.2 below. The next result lists some important properties of the ψ-basis. Proposition 3.2.10. Suppose that (s, t) ∈ Std2 (PnΛ ) and that Z is an integral domain. Then: a) [54, Lemma 5.2] If i, j ∈ I n then ψst = δi,is δj,it e(i)ψst e(j). b) [55, Lemma 3.17] Suppose that ψst and ψˆst are defined using different reduced expressions for the permutations d(s), d(t) ∈ Sn . Then there exist auv ∈ Z such that X ψˆst = ψst + auv ψuv , (u,v)I(s,t) u

s

v

where auv 6= 0 only if i = i , i = it and deg u+deg v = deg s+deg t. c) [56, Corollary 3.11] If 1 ≤ r ≤ n then there exist buv ∈ Z such that X ψst yr = buv ψuv , (u,v)I(s,t)

where buv 6= 0 only if iu = is , iv = it and deg u + deg v = deg s + deg t + 2. Part (a) follows quickly using the relations in Definition 2.2.1 and the definition of the ψ-basis. In contrast, parts (b) and (c) are proved by using

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Theorem 3.1.1 to reduce to the seminormal basis. With part (c), it is fairly easy to show that buv = 6 0 only if u D s. The difficult part is showing that buv 6= 0 only if v D t. Again, this is done using seminormal bases. Finally, we note that Theorem 3.2.9 implies that e(i) 6= 0 in RnΛ if and only if i ∈ IΛn = { it | t ∈ Std(PnΛ ) }, generalizing Proposition 2.4.6. In fact, if F is a field and HnΛ (F ) ∼ = RnΛ (F ) then it is shown in [54, Lemma 4.1] that the non-zero KLR idempotents are a complete set of primitive (central) idempotents in the Gelfand-Zetlin algebra Ln (F ) and that Ln (F ) = hy1 , . . . , yn , e(i) | i ∈ I n i. It follows that Ln (F ) is a positively graded commutative algebra with one dimensional irreducible modules indexed by IΛn , up to shift. It would be interesting to find a (homogeneous) basis of Ln (F ). The author would also like to know whether RnΛ is projective as a graded Ln -module. 3.3

Blocks and dual Specht modules

This section shows that the blocks of HnΛ are graded symmetric algebras and it sketches the proof of an analogous statement that relates the graded Specht modules and their graded duals. Theorem 1.8.1 describes the block decomposition of HnΛ so, by Theorem 3.1.1, it gives the block decomposition of RnΛ . As in (2.2.5), set X RβΛ = RnΛ eβ , where eβ = e(i). i∈I β

It follows from Definition 2.2.1 that eβ is central in RnΛ , so RβΛ = eβ RnΛ eβ + + is a two-sided ideal of RnΛ . Let Q+ | eβ 6= 0 } in RnΛ . n = Qn (Λ) = { β ∈ Q Λ Λ λ β Λ λ Similarly, let Pβ = { λ ∈ Pn | i ∈ I } = { λ ∈ Pn | β = β }. Combining Theorem 3.2.9, Theorem 3.1.1 and Corollary 1.8.2 we obtain the following. L RβΛ is the Theorem 3.3.1. Suppose that Λ ∈ P + . Then RnΛ = β∈Q+ n Λ decomposition of Rn into indecomposable two-sided ideals. Moreover, RβΛ is a graded cellular algebra with cellular basis { ψst | (s, t) ∈ Std2 (PβΛ ) } and weight poset PβΛ . By virtue of Theorem 3.2.9, the block decomposition of RnΛ holds over Z, even though we cannot think about the blocks as linkage classes of simple modules in this case. Compare with Theorem 2.4.8 in the semisimple case. Suppose that A is a graded Z-algebra. Then A is a graded symmetric algebra if there exists a homogeneous non-degenerate trace form τ : A −→ Z,

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where Z is in degree zero. That is, τ (ab) = τ (ba) and if 0 6= a ∈ A then there exists b ∈ A such that τ (ab) 6= 0. The map τ is homogeneous of degree d if τ (a) 6= 0 only if deg a = −d. Fix β ∈ Q+ . The defect of β is the non-negative integer  1 1 (Λ, Λ) − (Λ − β, Λ − β) . def β = (Λ, β) − (β, β) = 2 2 Notice that def β = def Λ β depends on Λ. If λ ∈ PnΛ set def λ = def β λ (see Corollary 1.8.2). If λ ∈ P1,n is a partition then def λ is equal to its e-weight; see, for example, [36, Proposition 2.1] or the proof of [89, Lemma 7.6]. Definition 3.2.1, and the definition of defect, readily implies the following combinatorial relationships between degrees, codegrees and defects. Lemma 3.3.2. Suppose that λ ∈ PnΛ . a) [25, Lemma 3.11] If A ∈ Addi (λ) then dA (λ) + 1 + dA (λ) = di (λ) and def(λ+A) = def λ + di (λ) − 1. b) [25, Lemma 3.12] If s ∈ Std(λ) then deg s + codeg s = def λ. 0 In Definition 3.2.2 we defined two sets of elements {ψst } and {ψst } Λ in Rn . Just as there are two versions of the Murphy basis, {mst } and {m0st }, that are built from the trivial and sign representations of HnΛ [106], respectively, there are two versions of the ψ-basis. By [54, Theorem 6.17], 0 { ψst | (s, t) ∈ Std2 (PnΛ ) } is a second graded cellular basis of HnΛ with 0 weight poset (PnΛ , E) and with deg ψst = codeg s + codeg t. We warn the reader that we are following the conventions of [55], rather than the notation of [54]. See [55, Lemma 3.15 and Remark 3.12] for the translation. 0 The bases {ψst } and {ψuv } of RnΛ are dual in the sense that if (s, t), (u, v) ∈ 2 Std (PβΛ ) then, by [56, Theorem 6.17],

(3.3.3)

0 ψst ψts 6= 0

0 and ψst ψuv 6= 0

only if it = iu and u D t.

Let τ be the usual non-degenerate trace form on HnΛ [20,100]. We can write P τ = d τd , where τd is homogeneous of degree d ∈ Z. Let τβ = τ−2 def β be the homogeneous component of τ of degree −2 def β. By [56, Theorem 6.17], 0 if (s, t) ∈ Std2 (PnΛ ) then τβ (ψst ψts ) 6= 0, so (3.3.3) implies the following. Theorem 3.3.4 (Hu-Mathas [54, Corollary 6.18]). Let β ∈ Q+ n. Then RβΛ a graded symmetric algebra with homogeneous trace form τβ of degree −2 def β. It would be better to have an intrinsic definition of τβ for RnΛ (Z). Webster [134, Remark 2.27] has given a diagrammatic description of a

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trace form on an arbitrary cyclotomic KLR algebra. It is unclear to the author how these two forms on RnΛ are related. The ψ 0 -basis is a graded cellular basis of HnΛ so it defines a collection of graded cell modules. For λ ∈ PβΛ , the dual graded Specht module Sλ is the corresponding graded cell module determined by the ψ 0 -basis. The dual Specht module Sλ has basis { ψt0 | t ∈ Std(λ) }, with deg ψt0 = codeg t, and X dim q Sλ = q codeg t . t∈Std(λ)

We can identify Sλ hcodeg tλ i with (ψt0λ tλ + Hn0Cλ )HnΛ , where Hn0Cλ is 0 the two-sided ideal of HnΛ spanned by ψst where (s, t) ∈ Std2 (µ) for some multipartition µ such that λ B µ. Similarly, we can identify S λ hdeg tλ i with (ψtλ tλ + HnBλ )HnΛ . By (3.3.3) there is a non-degenerate pairing { , } : S λ hdeg tλ i × Sλ hcodeg tλ i −→ Z given by {a + HnBλ , b + Hn0Cλ } = τβ (ab? ). Hence, Lemma 3.3.2 implies: Corollary 3.3.5 (Hu-Mathas [54, Proposition 6.19]). Suppose that λ ∈ PnΛ . Then S λ ∼ = Sλ~ hdef λi and Sλ = (S λ )~ hdef λi. This result holds for the Specht modules defined over Z by Theorem 3.2.9 or by [81, Theorem 7.25]. There is an interesting byproduct of the proof of Corollary 3.3.5. In the ungraded setting the Specht module S λ is isomorphic to the submodule of HnΛ generated by an element mλ Twλ m0λ ; see [33, Definition 2.1 and Theorem 2.9]. By [54, Corollary 6.21], mλ Twλ m0λ is homogeneous. In fact, ψtλ tλ ψwλ ψt0λ tλ = mλ Twλ m0λ and ψtλ tλ ψwλ ψt0λ tλ RnΛ ∼ = S λ hdef λ + codeg tλ i. 3.4

Induction and restriction

Λ The cyclotomic Hecke algebra HnΛ is naturally a subalgebra of Hn+1 , Λ Λ and Hn+1 is free as a Hn -module by (1.1.2). This gives rise to the usual induction and restriction functors. These functors can be decomposed into the “classical” i-induction and i-restriction functors, for i ∈ I, by projecting onto the blocks of these two algebras. As we will see, these functors are implicitly built into the graded setting. Recall that I = Z/eZ and Λ ∈ P + . For each i ∈ I define X Λ en,i = e(j ∨ i) ∈ Rn+1 . j∈I n

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Λ The relations for Rn+1 in Definition 2.2.1 imply that en,i is an idempotent P P Λ and that i∈I en,i = i∈I n+1 e(i) is the identity element of Rn+1 . Let Rep(RnΛ ), and Rep(RβΛ ) for β ∈ Q+ , be the category of finite dimensional (graded) RnΛ -modules, respectively, RβΛ -modules. Similarly, let Proj(RnΛ ) and Proj(RβΛ ) be the categories of finitely generated projective modules for these algebras.

Lemma 3.4.1. Suppose that i ∈ I and that Z is an integral domain. Then Λ there is a (non-unital) embedding of graded algebras RnΛ ,→ Rn+1 given by e(j) 7→ e(j ∨ i),

yr 7→ en,i yr

and

ψs 7→ en,i ψs ,

for j ∈ I n , 1 ≤ r ≤ n and 1 ≤ s < n. This map induces an exact functor Λ Λ i-Ind : Rep(RnΛ ) −→ Rep(Rn+1 ); M 7→ M ⊗RnΛ en,i Rn+1 . L Moreover, Ind = i∈I i-Ind is the graded induction functor from Rep(RnΛ ) Λ to Rep(Rn+1 ).

Proof. The images of the homogeneous generators of RnΛ under this embedding commute with en,i , so this map defines a non-unital degree Λ preserving homomorphism from RnΛ to Rn+1 . This map is an embedding by Theorem 3.2.9. The remaining claims follow because, by definition, en,i P Λ is an idempotent and i∈I en,i is the identity element of Rn+1 .  The i-induction functor i-Ind functor is obviously a left adjoint to Λ the i-restriction functor i-Res, which sends a Rn+1 -module M to i-Res M = M en,i ∼ = Hom RnΛ (en,i RnΛ , M ). A much harder fact is that these functors are two-sided adjoints. Theorem 3.4.2 (Kashiwara [71, Theorem 3.5]). Suppose i ∈ I. Then (i-Res, i-Ind) is a biadjoint pair. Kashiwara proves this theorem by constructing explicit homogeneous adjunctions. He does this for the i-induction and i-restriction functors for any cyclotomic quiver Hecke algebra defined by a symmetrizable Cartan matrix. As we do not need this result we feel justified in stating it now, even though its proof builds upon Kang and Kashiwara’s result that the cyclotomic quiver Hecke algebras of arbitrary type categorify the integrable highest weight modules of the corresponding quantum group [67]; compare with Proposition 3.5.12 and Corollary 3.5.27 below. This biadjointness property is also a consequence of Rouquier’s Kac-Moody categorification

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axioms [121, Theorem 5.16]. Theorem 3.4.2 was conjectured by KhovanovLauda [74]. Recall from (2.1.4) that ~ defines a graded duality on Rep(RnΛ ). Similarly, define # to be the graded functor given by (3.4.3)

M # = Hom RnΛ (M, RnΛ ),

for M ∈ Rep(RnΛ ),

where the action of RnΛ on M # is given by for f ∈ M # , h ∈ RnΛ and m ∈ M . L Λ We consider ~ and # as endofunctors of Rep(RnΛ ) = β Rep(Rβ ) and L Λ Λ Proj(Rn ) = β Proj(Rβ ). As noted in [22, Remark 4.7], Theorem 3.3.4 implies that these two functors agree up to shift. (f · h)(m) = h? f (m),

Lemma 3.4.4. As endofunctors of Rep(RβΛ ), there is an isomorphism of functors # ∼ = h2 def βi ◦ ~. Proof.

By Theorem 3.3.4, RβΛ ∼ = (RβΛ )~ h2 def βi. If M ∈ Rep(RβΛ ) then
M # = Hom RβΛ (M, RβΛ ) = Hom RβΛ M, (RβΛ )~ h2 def βi
∼ = Hom RβΛ M, Hom Z (RβΛ , Z) h2 def βi
∼ = Hom Z M ⊗RβΛ RβΛ , Z h2 def βi ∼ = M ~ h2 def βi,

where the third isomorphism is the standard adjointness of tensor and hom. As all of these isomorphisms are functorial, the lemma follows.  As M is finite dimensional, (M ~ )~ ∼ = M for all M ∈ Rep(HnΛ ). Hence, ∼ (M ) = M by Lemma 3.4.4. Therefore, ~ and # define self-dual equivalences on the module categories Rep(RnΛ ) and Proj(RnΛ ). # #

Proposition 3.4.5. Suppose that β ∈ Q+ and i ∈ I. Then there are functorial isomorphisms Λ ~ ◦ i-Res ∼ ) −→ Rep(RnΛ ), = i-Res ◦ ~ : Rep(Rn+1 # ◦ i-Ind ∼ = i-Ind ◦# : Proj(R Λ ) −→ Proj(R Λ ). n

n+1

∼ i-Res ◦~ is immediate from the Proof. The isomorphism ~ ◦ i-Res = definitions. For the second isomorphism, recall that if P ∈ Proj(RβΛ ) then Hom RnΛ (P, M ) ∼ = Hom RnΛ (M, RnΛ ) ⊗RnΛ M , for any RnΛ -module M . Now, Λ Λ Λ Λ Λ (en,i Rn+1 )# = Hom Rn+1 (en,i Rn+1 , Rn+1 )∼ , = en,i Rn+1

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the last isomorphism following because e?n,i = en,i . Therefore, Λ ∼ Hom RΛ (P, en,i R Λ ) i-Ind(P # ) = Hom RnΛ (P, RnΛ ) ⊗RnΛ en,i Rn+1 = n+1 n
Λ Λ ∼ Λ Λ Hom P, Hom (e R , R ) = n,1 R n

Rn+1

n+1

n+1

Λ Λ ∼ Λ (P ⊗RnΛ en,i Rn+1 , Rn+1 ) = Hom Rn+1

∼ = (i-Ind P )# , where the second last isomorphism is the usual tensor-hom adjointness.  It follows from Proposition 3.4.5 and Lemma 3.4.4 that the functors ~ and i-Ind, and # and i-Res, commute only up to shift. This difference in degree shift is what makes Lemma 3.5.13 work below. We next describe the effect of the i-induction and i-restriction functors on the graded Specht modules, for i ∈ I. This result generalizes the well-known (ungraded) branching rules for the symmetric group [59, Example 17.16] and the cyclotomic Hecke algebras [12, 109, 125]. Recall the integers dA (λ) and dA (λ) from Definition 3.2.1. Theorem 3.4.6. Suppose that Z is an integral domain and λ ∈ PnΛ . a) [56, Main theorem] Let A1 < A2 · · · < Az be the addable i-nodes of λ. Then i-Ind S λ has a graded Specht filtration 0 = I0 ⊂ I1 ⊂ · · · ⊂ Iz = i-Ind S λ , such that Ij /Ij−1 ∼ = S λ+Aj hdAj (λ)i, b) [25, Theorem 4.11] Let B1 > B2 > · · · > By be the removable i-nodes of λ. Then i-Res S λ has a graded Specht filtration 0 = R0 ⊂ R1 ⊂ · · · ⊂ Ry = i-Res S λ , such that Rj /Rj−1 ∼ = S λ−Bj hdBj (λ)i, for 1 ≤ j ≤ y. c) [56, Corollary 4.7] Let A1 > A2 > · · · > Az be the addable i-nodes of λ. Then i-Ind Sλ has a graded Specht filtration 0 = I0 ⊂ I1 ⊂ · · · ⊂ Iz = i-Ind Sλ , such that Ij /Ij−1 ∼ = Sλ+Aj hdAj (λ)i, for 1 ≤ j ≤ z. d) Let B1 < B2 < · · · < By be the removable i-nodes of λ. Then i-Res S λ has a graded Specht filtration 0 = R0 ⊂ R1 ⊂ · · · ⊂ Ry = i-Res Sλ , such that Rj /Rj−1 ∼ = Sλ−Bj hdBj (λ)i, for 1 ≤ j ≤ y. Observe that parts (a) and (c), and parts (b) and (d), are equivalent by Corollary 3.3.5 (and Lemma 3.3.2).

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Part (b) is proved using the fact that the action of HnΛ on the ψ-basis is compatible with restriction. Part (a), which was conjectured by Brundan, Kleshchev and Wang [25, Remark 4.12], is proved by extending elegant ideas of Ryom-Hansen [125] to the graded setting using [54]. 3.5

Grading Ariki’s Categorification Theorem

The aim of this section is to prove the Ariki-Brundan-Kleshchev Categorib e )-modules fication Theorem [3] that connects the canonical bases of Uq (sl Λ with the simple and projective indecomposable Rn -modules in characteristic zero. Our argument runs parallel to Brundan and Kleshchev’s with the key difference being that we use the representation theory of HnΛ , and in particular the graded branching rules, to construct a bar involution on the Fock space. In this way we are able to show that the canonical basis is categorified by the basis of simple HnΛ -modules if and only if the graded decomposition numbers are polynomials. As a consequence, Ariki’s categorification theorem [3] lifts to the graded setting. Throughout this section we assume that the Hecke algebra HnΛ is defined over a field F. In the end we will assume that F is a field of characteristic zero, however, almost all of the results in this section hold over an arbitrary b e ) until we actually field. We delay introducing the quantum group Uq (sl need it because we want to emphasize the role that the quantum group is playing in the representation theory of HnΛ . For the time being fix an integer n ≥ 0. Let A = Z[q, q −1 ] be the ring of Laurent polynomials in q over Z. Let [Rep(HnΛ )] and [Proj(HnΛ )] be the Grothendieck groups of the categories Rep(HnΛ ) and Proj(HnΛ ), respectively. If M is a finitely generated HnΛ -module let [M ] be its image in [Rep(HnΛ )]. Abusing notation slightly, if M is projective we also let [M ] be its image in [Proj(HnΛ )]. Consider [Rep(HnΛ )] and [Proj(HnΛ )] as A-modules by letting q act as the grading shift functor: [M hdi] = q d [M ], for d ∈ Z. Definition 3.5.1. Suppose that µ ∈ KnΛ . Let Y µ be the projective cover of Dµ in Rep(HnΛ ). Importantly, the module Y µ is graded. Since Y µ is indecomposable, the grading on Y µ is uniquely determined by the surjection Y µ  Dµ , for µ ∈ KnΛ . We use the notation Y µ because these modules are special cases of the graded lifts of the Young modules constructed in [105]; see [55, §5.1] and [99, §2.6]. (The symbol P µ is usually reserved for the projective

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indecomposable modules of the cyclotomic Schur algebras [20, 31, 55, 128].) By definition, the Grothendieck groups [Rep(HnΛ )] and [Proj(HnΛ )] are free A-modules that come equipped with distinguished bases: M M [Rep(HnΛ )] = A[Dµ ] and [Proj(HnΛ )] = A[Y µ ], Λ µ∈Kn

Λ µ∈Kn




respectively. Recall from (3.2.8) that dq = dλµ (q) is the graded decomposition matrix of HnΛ . If λ ∈ PnΛ and µ ∈ KnΛ then in [Rep(HnΛ )], X X [S λ ] = dλτ (q)[Dτ ] and [Y µ ] = dσµ (q)[S σ ], Λ τ ∈Kn λDτ

Λ σ∈Pn σDµ

where the second formula comes
from Corollary 2.1.6. By Theorem 2.1.5(c), d the submatrix dK = (q) λµ Λ of the graded decomposition matrix dq q λ,µ∈Kn is invertible over A with inverse
K −1 eK = eλµ (−q) λ,µ∈KΛ . q = (dq ) n

(The reason why we consider eλµ (−q) as a Laurent polynomial in −q is explained after Corollary 3.5.27 below.) Hence, if λ ∈ KnΛ then X [Dλ ] = eλµ (−q)[S µ ]. Λ µ∈Kn

Consequently, { [S µ ] | µ ∈ KnΛ } is a second A-basis of [Rep(HnΛ )]. The set of projective indecomposable HnΛ -modules {[Y µ ]} is the only natural basis of the split Grothendieck group [Proj(HnΛ )]. Somewhat artificially, but motivated by the formulas above, for µ ∈ KΛ define X Xµ = eλµ (−q)[Y λ ] ∈ [Proj(HnΛ )]. Λ λ∈Kn

Then { Xµ | µ ∈ KΛ } is an A-basis of [Proj(HnΛ )]. We will use the bases { [S µ ] | µ ∈ KΛ } and { Xµ | KΛ } of [Rep(HnΛ )] and [Proj(HnΛ )], respectively, to construct new distinguished bases of these Grothendieck groups. The bar involution on A = Z[q, q −1 ] is the unique Z-linear map such that q d = q −d , for d ∈ Z. In particular, dim q M ~ = dim q M , for any RnΛ module M . A semilinear map of A-modules is a Z-linear map θ : M −→ N such that θ(f (q)m) = f (q)θ(m), for all f (q) ∈ A and m ∈ M . A sesquilinear map f : M × N −→ A, where M and N are A-modules, is a function that is semilinear in the first variable and linear in the second. Let h , i : [Proj(HnΛ )] × [Rep(HnΛ )] −→ A be the sesquilinear pairing (3.5.2)

h[P ], [M ]i = dim q Hom HnΛ (P, M ),

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for P ∈ Proj(HnΛ ) and M ∈ Rep(HnΛ ). This pairing is naturally sesquilinear because Hom HnΛ (P hki, M ) ∼ = Hom HnΛ (P, M h−ki), for any k ∈ Z. The functors ~ and #, of (2.1.4) and (3.4.3), induce semilinear automorphisms of the Grothendieck groups [Rep(HnΛ )] and [Proj(HnΛ )]: [P ]# = [P # ],

and

[M ]~ = [M ~ ]

for M ∈ Rep(HnΛ ) and P ∈ Proj(HnΛ ). The next result is fundamental. Λ Lemma 3.5.3. Suppose that [P ] ∈ [ProjΛ A ] and [M ] ∈ [RepA ]. Then

h[P ], [M ]~ i = h[P ]# , [M ]i. Proof.

Applying the definitions, and tensor-hom adjointness,


h[P ], [M ]~ i = dim q Hom RnΛ (P, M ~ ) = dim q Hom RnΛ P, Hom RnΛ (M, F) = dim q Hom RnΛ (P ⊗RnΛ M, F) = dim q (P ⊗RnΛ M )~ = dim q P ⊗RnΛ M = dim q Hom RnΛ (P # , RnΛ ) ⊗RnΛ M = dim q Hom RnΛ (P # , M ) = h[P ]# , [M ]i. For the second last line, note that Hom RnΛ (Q, M ) ∼ = Hom RnΛ (Q, RnΛ ) ⊗RnΛ M whenever Q is projective.  Lemma 3.5.4. Suppose that λ, µ ∈ KnΛ . Then h[Y λ ], [Dµ ]i = δλµ = hXλ , [S µ ]~ i. Proof. The first equality is immediate from the definition of the sesquilinear form h , i because Y λ is the projective cover of Dλ , for λ ∈ KnΛ . For the second equality, using the fact that ~ is semilinear, X hXλ , [S µ ]~ i = eσλ (−q)h[Y σ ], [S µ ]~ i Λ σ∈Kn

=

X

eσλ (−q) dµτ (q)h[Y σ ], [Dτ ]i

Λ σ,τ ∈Kn

=

X

dµσ (q) eσλ (−q) = δλµ ,

Λ σ∈Kn

K −1 where the last equality follows because eK . q = (dq )



Lemma 3.5.5. Suppose that µ ∈ KnΛ . Then [Y µ ]# = [Y µ ], [Dµ ]~ = [Dµ ], X X (Xµ )# = Xµ + aσµ (q)Xσ and [S µ ]~ = [S µ ] + aµτ (q)[S τ ], Λ σ∈Kn σBµ

for some Laurent polynomials aσµ (q), aτ µ (q) ∈ A.

Λ τ ∈Kn µBτ

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Proof. That [Dµ ]~ = [Dµ ] is immediate by Corollary 3.2.7, whereas [Y µ ]# = Y µ because Y µ is a direct summand of HnΛ — alternatively, use Lemma 3.5.4 and Lemma 3.5.3. If µ ∈ KnΛ then, by Theorem 2.1.5,  X ~ X [S µ ]~ = dµν (q)[Dν ] = dµν (q) [Dν ] Λ ν∈Kn µDν

= [S µ ] +

Λ ν∈Kn µDν

X  X Λ τ ∈Kn µBτ

 dµν (q) eντ (−q) [S τ ]

Λ ν∈Kn µDνDτ

as claimed. Note that dµµ (q) = 1 = eµµ (−q). P Finally, we can compute (Xµ )# by writing Xµ = µ eµλ (−q)[Y λ ] and then using essentially the same argument to show that (Xµ )# can be written in the required form. Alternatively, use Lemma 3.5.4 and Lemma 3.5.3.  Λ The triangularity of the action of ~ and # on [RepΛ A ] and [ProjA ], respectively, has the following easy but important consequence.

Proposition 3.5.6. Suppose that F is a field. Then there exist unique bases { Bµ | µ ∈ KnΛ } and { B µ | µ ∈ KnΛ } of [Proj(HnΛ )] and [Rep(HnΛ )], respectively, such that (Bµ )# = Bµ , (B µ )~ = B µ X X Bµ = Xµ + bσµ (q)Xσ and B µ = [S µ ] + bµτ (q)[S τ ] Λ σ∈Kn σBµ

Λ τ ∈Kn µBτ

for polynomials bµσ (q), bσµ (q) ∈ δσµ + qZ[q]. Proof. The existence and uniqueness of these two bases follows immediately from Lemma 3.5.5 by Lusztig’s Lemma [95, Lemma 24.2.1]. We give a variation of Lusztig’s argument for the basis {B µ }. Fix a multipartition µ ∈ KnΛ , for some n ≥ 0, and suppose that B µ and B˙ µ are two elements of [Rep(HnΛ )] with the required properties. By assumption the element B µ − B˙ µ is ~-invariant and we can write X B µ − B˙ µ = b˙ µτ (q)[S τ ], µBτ

for some polynomials b˙ µτ (q) ∈ qZ[q]. As the left-hand side is ~-invariant and b˙ µτ (q) ∈ q −1 Z[q −1 ], Lemma 3.5.5 forces B µ = B˙ µ . To prove existence, we argue by induction on dominance. If µ is minimal in KnΛ then we can take B µ = [S µ ] = [Dµ ] by Lemma 3.5.5. If µ ∈ KnΛ is

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not minimal with respect to dominance then set B˙ µ = [Dµ ]. Then X (B˙ µ )~ = B˙ µ and B˙ µ = [S µ ] + b˙ µτ (q)[S τ ], Λ τ ∈Kn µBτ

for some Laurent polynomials b˙ µτ (q) ∈ Z[q, q −1 ]. If b˙ µτ (q) ∈ qZ[q], for all µ B τ , then B µ = B˙ µ has all of the required properties. Otherwise, pick any multipartition µ B ν that is maximal with respect to dominance such that b˙ µν (q) ∈ / qZ[q]. By induction, there exists an element B ν with all of the required properties. Replace B˙ µ with the element B˙ µ − pµν (q)B ν , where pµν (q) is the unique Laurent polynomial such that pµν (q) = pµν (q) and b˙ µν (q) − pµν (q) ∈ qZ[q]. Then (B˙ µ )~ = B˙ µ and the coefficient of [S ν ] in B˙ µ belongs to qZ[q]. Continuing in this way, after finitely many steps we will construct an element B µ with the required properties.  Corollary 3.5.7. Suppose that λ, µ ∈ KΛ . Then X hBµ , B λ i = bλσ (q)bσµ (q) = δλµ . σ∈KΛ λDσDµ

Proof. If σ, τ ∈ KnΛ then hXσ , [S τ ]~ i = δστ by Lemma 3.5.4. Therefore, since the form h , i is sesquilinear and B λ~ = B λ , X X hBµ , B λ i = hBµ , B λ~ i = bσµ (q) bλτ (q)hXσ , [S τ ]~ i σDµ λDτ

=

X

bλσ (q) bσµ (q).

λDσDµ

In particular, (Bµ , B λ ) ∈ δλµ + q −1 Z[q −1 ]. On the other hand, # hBµ , B λ i = hBµ , B λ i = hBµ , B λ~ i = hBµ , B λ i

by Lemma 3.5.3. Therefore, hBµ , B λ i = δλµ as this is the only bar invariant polynomial in δλµ + q −1 Z[q −1 ].  Applying Lemma 3.4.4 to Proposition 3.5.6 we obtain. Corollary 3.5.8. Suppose that µ ∈ KΛ . Then (q − def µ Bµ )~ = q − def µ Bµ and (q def µ B µ )# = q def µ B µ . In order to link the bases {Bµ } and {B µ } with the representation theory b e ). A self-contained of HnΛ we need to introduce the quantum group Uq (sl account of much of what we need can be found in Ariki’s book [5]. See also [95, §3.1] and [22].

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b e ) associated with the quiver Γe is the Q(q)The quantum group Uq (sl algebra generated by { Ei , Fi , Ki± | i ∈ I }, subject to the relations: Ki − Ki−1 Ki Kj = Kj Ki , Ki Ki−1 = 1, [Ei , Fj ] = δij , q − q −1 Ki Ej Ki−1 = q cij Ej , Ki Fj Ki−1 = q −cij Fj , s { X 1 − cij 1−c −c (−1)c Ei ij Ej Eic = 0, c q 0≤c≤1−cij s { X 1 − cij 1−c −c (−1)c Fi ij Fj Fic = 0, c q 0≤c≤1−cij qdy Qm where c q = JdK!/JcK!Jd − cK! and JmK! = k=1 (q k − q −k )/(q − q −1 ), for b e ) is a Hopf algebra with coproduct integers c < d, m ∈ N. Then Uq (sl determined by ∆(Ki ) = Ki ⊗ Ki , ∆(Ei ) = Ei ⊗ Ki + 1 ⊗ Ei and ∆(Ki ) = Fi ⊗ 1 + Ki−1 ⊗ Fi , for i ∈ I. Λ The combinatorial Fock space FA is the free A-module with basis S Λ the set of symbols { |λi | λ ∈ P }, where P Λ = n≥0 PnΛ . For future use, S Λ Λ Λ let KΛ = n≥0 KnΛ . Set FQ(q) = FA ⊗A Q(q). Then, FQ(q) is an infinite Λ dimensional Q(q)-vector space. We consider { |λi | λ ∈ P } as a basis Λ of FQ(q) by identifying |λi and |λi ⊗ 1Q(q) . Recall the integers dA (λ), dB (λ) and di (λ) from Definition 3.2.1. Λ Theorem 3.5.9 (Hayashi [52, 111]). Suppose that Λ ∈ P + . Then FQ(q) b e )-module with Uq (sl b e )-action determined by is an integrable Uq (sl X X A Ei |λi = q dB (λ) |λ−Bi and Fi |λi = q −d (λ) |λ+Ai, B∈Remi (λ)

and Ki |λi = q

di (λ)

|λi, for all i ∈ I and λ ∈

A∈Addi (λ)

PnΛ .

b e ) on the Fock space Remark 3.5.10. A slightly different action of Uq (sl is used in many places in the literature, such as [5, 89, 104]. As is already b e )evident, and will be made precise in Proposition 3.5.16 below, the Uq (sl action on the Fock space is closely related to induction and restriction for b e )-action on the Fock space used the graded Specht modules. The Uq (sl in [5, 89, 104] corresponds to the action of the induction and restriction functors on the dual graded Specht modules. Equivalently, this difference b e )-action arises because, ultimately, we will work with an action in the Uq (sl b of Uq (sle ) on the Grothendieck groups of the finitely generated RnΛ -modules, whereas these other sources consider the corresponding adjoint action on the projective Grothendieck groups.

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Hayashi [52] considered only the special case when Λ = Λ0 , however, b e ) because this implies the general case using the coproduct of Uq (sl Λκ Λκ Λ ∼ FQ(q) = FQ(q)1 ⊗ · · · ⊗ FQ(q)`

b e )-modules. The crystal and canonical bases of F Λ , which were as Uq (sl Q(q) first studied in [65, 111, 131], play an important role in what follows. A self-contained proof of Theorem 3.5.9, stated with similar language, can be found in Ariki’s book [6, Theorem 10.10]. Λ An element x ∈ FQ(q) has weight wt(x) = Γ if Ki x = q (Γ,αi ) x, for i ∈ I. In particular, if 0` = (0|0| . . . |0) ∈ P Λ is the empty multipartition of level ` then Ki |0` i = q (Λ,αi ) |0` i, for i ∈ I, so that |0` i has weight Λ. More generally, if β ∈ Q+ then writing λ = µ + A it follows by induction that (3.5.11)

if λ ∈ PβΛ then di (λ) = (Λ − β, αi ), for all i ∈ I.

Therefore, wt(|λi) = Λ − β by Theorem 3.5.9. Set di (β) = (Λ − β, αi ). b e )vΛ be the irreFor each dominant weight Λ ∈ P + let L(Λ) = Uq (sl ducible integrable highest weight module of highest weight Λ, where vΛ is a highest weight vector of weight Λ. By Theorem 3.5.9, |0` i is a highest Λ vector of weight Λ in FQ(q) . In fact, it follows from Theorem 3.5.9 that b e )|0` i. L(Λ) ∼ = Uq (sl For example, see [5, Theorem 10.10]. Henceforth, we set vΛ = |0` i. To compare the Grothendieck groups [Rep(HnΛ )] and [Proj(HnΛ )] with the Fock space we need to consider all n ≥ 0 simultaneously. Define M M [RepΛ [Rep(HnΛ )] and [ProjΛ [Proj(HnΛ )]. A] = A] = n≥0

n≥0

Λ Λ Λ Set [RepΛ Q(q) ] = [RepA ] ⊗A Q(q) and [ProjQ(q) ] = [ProjA ] ⊗A Q(q).

Proposition 3.5.12. Suppose that Λ ∈ P + . Then the i-induction and b i-restriction functors of [RepΛ Q(q) ] induce a Uq (sle )-module structure on b e )-modules, [ProjΛ ] and [RepΛ ] such that, as Uq (sl Q(q)

Q(q)

Λ ∼ ∼ [ProjΛ Q(q) ] = L(Λ) = [RepQ(q) ].

Proof. Recall that dq is the graded decomposition matrix of HnΛ and dTq is its transpose. Abusing notation slightly, by simultaneously using these

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matrices for all n ≥ 0, define linear maps [ProjΛ Q(q) ]

dTq

Λ FQ(q)

dq

cq

[RepΛ Q(q) ] P P µ where dTq ([Y µ ]) = λ dλµ (q)|λi, dq (|λi) = µ dλµ (q)[D ] and where cq = dq ◦ dTq is the Cartan map. As vector space homomorphisms, dTq is injective and dq is surjective. As defined these maps are only vector space homomorphisms, however, we claim that both maps can be made b e )-module homomorphisms. into Uq (sl The i-induction and i-restriction functors are exact, for i ∈ I, because they are exact when we forget the grading [49, Corollary 8.9]. Therefore, they send projective modules to projectives and they induce endomorphisms Λ of the Grothendieck groups [RepΛ Q(q) ] and [ProjQ(q) ]. By Theorem 3.4.6, [i-Res S λ ] =

X

q dB (λ) [S λ−B ],

B∈Remi (λ)

[i-Ind S λ h1 − di (λi] =

X

q dA (λ)+1−di (λ) [S λ+A ]

A∈Addi (λ)

=

X

A

q −d

(λ)

[S λ+A ],

A∈Addi (λ)

where the last equality uses Lemma 3.3.2(a). Identifying Ei with i-Res, and Fi with q i-Ind Ki−1 , the linear maps dq and dTq become well-defined b e )-module homomorphisms by Theorem 3.5.9. As Uq (sl b e )-modules, Uq (sl Λ [RepΛ ] and [Proj ] are both cyclic because they are both generated Q(q) Q(q) 0 0 0 T 0 by [Y` ] = [S` ] = [D` ]. By definition, dq ([Y` ]) = vΛ and dq (vΛ ) = [S`0 ], so ∼ Uq (sl b e )vΛ is irreducible. the proposition follows because L(Λ) =  b e ) be Lusztig’s A-form of Uq (sl b e ), which is the A-subalgebra Let UA (sl (k) b of Uq (sle ) generated by the quantised divided powers Ei = Eik /JkK! (k) b e) and Fi = Fik /JkK!, for i ∈ I and k ≥ 0. Theorem 3.5.9 implies that UA (sl Λ Λ acts on the A-submodule FA of FQ(q) ; compare with [89, Lemma 6.2] and with [104, Lemma 6.16]. Therefore, by Proposition 3.5.12, [RepΛ A] b e )-modules. Moreover, if we set LA (Λ) = UA (sl b e )vΛ and [ProjΛ ] are U ( sl A A

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b e )-module homomorphisms [ProjΛ ] ,→ LA (Λ)  [RepΛ ]. then there are UA (sl A A b In particular, LA (Λ) ∼ = [ProjΛ A ] as UA (sle )-modules by Proposition 3.5.12. Lemma 3.5.13. Suppose that i ∈ I. The involution ~ commutes with the Λ actions of Ei and Fi on [RepΛ A ] and on [ProjA ]. Proof. By Proposition 3.4.5, there are isomorphisms i-Res ◦~ ∼ = ~ ◦ i-Res and i-Ind ◦# ∼ = # ◦ i-Ind. In particular, the actions of Ei = i-Res and ~ commute. Fix β ∈ Q+ . Recall from after (3.5.11) that di (β) = (Λ − β, αi ). Identifying Fi with the functor q ◦ i-Ind ◦Ki−1 = q 1−di (β) i-Ind on Rep(HβΛ ), there are isomorphisms Fi ◦ ~ ∼ by Lemma 3.4.4, = q i-Ind Ki−1 ◦ q −2 def β # 1−d (β)−2 def β ∼ i-Ind ◦# =q i ∼ = q 1−di (β)−2 def β # ◦ i-Ind ∼ = q −2 def(β+αi ) # ◦ q di (β)−1 ◦ i-Ind

by Proposition 3.4.5,

∼ = ~ ◦ q −1 i-Ind Ki ∼ = ~ ◦ Fi ,

by Lemma 3.4.4.

Hence, Ei and Fi commute with ~ on [RepΛ A]

[RepΛ A]

by Lemma 3.3.2(a), and [ProjΛ A ] as claimed.



[ProjΛ A]

In contrast, Ei and Fi on and do not commute with #. Λ We want to relate the Cartan pairing h , i on [ProjΛ A ] × [RepA ] with the representation theory of LA (Λ). Define a non-degenerate symmetric bilinear Λ form ( , ) on the Fock space FA by def λ (3.5.14) (|λi, |µi) = δλµ q , for λ, µ ∈ P Λ . wt(y) By Theorem 3.5.9, (Ki x, y) = q (x, y) = (x, Ki y), for weight vectors Λ x, y ∈ FA and i ∈ I. By restriction, we also consider ( , ) as a bilinear form on LA (Λ). Lemma 3.5.15. The bilinear form ( , ) on LA (Λ) is characterised by the three properties: (vΛ , vΛ ) = 1, (Ei x, y) = (x, Fi y) and (Fi x, y) = (x, Ei y), for all i ∈ I and x, y ∈ LA (Λ). Proof. By definition, (vΛ , vΛ ) = 1. If i ∈ I then in order to check that Ei and Fi are biadjoint with respect to ( , ) it is enough to consider the cases when x = |λi and y = |µi, for λ, µ ∈ P Λ . By Theorem 3.5.9, (Fi |λi, |µi) = 0 = (|λi, Ei |µi) unless µ = λ + A for some A ∈ Addi (λ). On the other hand, if A ∈ Addi (λ) and µ = λ + A then, using Lemma 3.3.2(a) for the second equality, A A (Fi |λi, |µi) = q def µ−d (λ) = q def λ+di (λ)−1−d (λ) = q def λ+dA (µ) = (|λi, Ei |µi).

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A similar calcuation shows that (Ei |λi, |µi) = (|λi, Fi |µi), for all λ, µ ∈ PnΛ . As vΛ is the highest weight vector in the irreducible module LA (Λ), these three properties uniquely determine the bilinear form ( , ) on LA (Λ) by induction on weight.  By restriction, the next result categorifies the pairing ( , ) on LA (Λ). Λ Proposition 3.5.16. Let x ∈ [ProjΛ A ] and y ∈ FA with wt(λ) = β. Then

hx, dq (y)i = q − def β (dTq (x# ), y). Proof. As ( , ) is bilinear and h , i is sesquilinear it is enough to verify this identity when x = Xµ and y = |λi, for µ ∈ KΛ and λ ∈ PβΛ . Then X dλσ (q)hXµ , [Dσ ]i hx# , dq (y)i = hXµ , [S λ ]i = Λ σ∈Kβ

X X

=

dλσ (q)eστ (−q)hXµ , [S τ ]~ i

Λ σ∈KΛ τ ∈Kβ β

X

=

dλσ (q)eσµ (−q),

Λ σ∈Kβ

where the last equality uses Lemma 3.5.4. For the right hand side, X

(dTq (x# ), y) = dTq (Xµ# ), |λi = eσµ (−q) dTq ([Y σ ]), |λi Λ σ∈Kβ

=

X  X Λ ν∈Pn νDµ


dτ σ (q)eσµ (−q) |τ i, |λi

Λ σ∈Kn νDσDµ

= q def β hx, dq (y)i, by (3.5.14) and calculation above. The proof is complete.



Now we can prove the results that we are really interested in. Λ Corollary 3.5.17. Let P ∈ Proj(HnΛ ), y ∈ Rep(Hn+1 ),and i ∈ I. Then

hi-Ind x, yi = hx, i-Res yi and hi-Res x, yi = hx, i-Ind yi. Proof. By Theorem 3.4.2, (i-Res, i-Ind) is a biadjoint pair so the corollary follows directly from the definition of the Cartan pairing in (3.5.2). As it is non-trivial to show that i-Res is left adjoint to i-Ind we prove this at the level of Grothendieck groups. Write x˙ = dTq (x) and y = dq (y) ˙ where x, ˙ y˙ ∈ LA (Λ)

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and wt(y) ˙ = Λ − β. Then hi-Res x, yi = 0 unless wt(x) = Λ − (β + αi ). To improve readability, identify x and x˙ = dTq (x) below. Then,
hi-Res x, yi = q − def β (Ei x)# , y˙ by Proposition 3.5.16,
def β ~ =q Ei x , y˙ , by 3.4.4 and 3.5.13,
def β ~ =q x , Fi y˙ , by Lemma 3.5.15,
def β−2 def(β+αi ) # =q x , Fi y˙ , by Lemma 3.4.4, = q def β−def(β+αi ) hx# , Fi yi,

by Proposition 3.5.16,

= hx, i-Ind yi, where the last equality uses Lemma 3.3.2 and the identification of Fi and q i-Ind ◦Ki−1 on [RepΛ  A ], via Proposition 3.5.12. b e ) such that Let τ be the unique semilinear anti-isomorphism of Uq (sl −1 −1 −1 τ (Ki ) = Ki , τ (Ei ) = qFi Ki and τ (Fi ) = q Ki Ei , for all i ∈ I. Then the biadjointness of induction and restriction with respect to the Cartan pairing translates into the following more Lie theoretic statement. Λ Corollary 3.5.18. Suppose that x ∈ [ProjΛ A ] and y ∈ [RepA ]. Then b e ). hux, yi = hx, τ (u)yi, for all u ∈ UA (sl

b e) The bar involution of A extends to a semilinear involution of UA (sl −1 determined by Ki = Ki , Ei = Ei and Fi = Fi , for all i ∈ I. Similarly, define a bar involution on LA (Λ) by b e ) and x ∈ LA (Λ). v Λ = vΛ and ux = u x, for u ∈ UA (sl As noted in [22, §3.1], it follows from the relations that τ ◦ = ◦ τ −1 . As in [22, §3.3], the Shapovalov form on L(Λ) is the sesquilinear map hx, yi = q def β (x, y), for x, y ∈ L(Λ) with wt(y) = Λ − β, for β ∈ Q+ . As our notation suggests, the Shapovalov form is categorified by the Cartan pairing. Corollary 3.5.19. Suppose that x ∈ [ProjΛ A ] and y ∈ LA (Λ). Then hdTq (x), yi = hx, dq (y)i. Proof. By Lemma 3.5.15, the pairing , ) on LA (Λ) is unique symmetric bilinear map on LA (Λ) that is biadjoint with respect to Ei and Fi and such that (vΛ , vΛ ) = 1. This implies that the Shapovalov form is the unique sesquilinear form on LA (Λ) such that hvΛ , vΛ i = 1 and hux, yi = b e ). Hence, the result follows from hx, τ (u)yi, for x, y ∈ LA (Λ) and u ∈ UA (sl Corollary 3.5.18. 

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b e )vΛ is the standard A-form of the irreducible The module LA (Λ) = UA (sl b Uq (sle )-module L(Λ). The costandard A-form of L(Λ) is the dual lattice LA (Λ)∗ = { y ∈ L(Λ) | (x, y) ∈ A for all x ∈ LA (Λ) } = { y ∈ L(Λ) | hx, yi ∈ A for all x ∈ LA (Λ) } . Λ b We can now identify both [ProjΛ A ] and [RepA ] as UA (sle )-modules. b e )-modules, Corollary 3.5.20. Suppose that Λ ∈ Q+ . Then, as UA (sl Λ ∼ Λ ∼ [ProjA ] = LA (Λ) and [RepA ] = LA (Λ)∗ . Proof. The first isomorphism we noted already after Proposition 3.5.12. The second isomorphism follows from Corollary 3.5.19 and Lemma 3.5.4.  Λ By Lemma 3.5.13, the action of Fi on [RepΛ A ] and [ProjA ], for i ∈ I, commutes with ~. In the language of [22, §3.1], ~ is a compatible barinvolution. As is easily proved by induction on weight, every integrable b e )-module has a unique bar-compatible involution, so UA (sl Λ dq (y) = dq (y)~ for all y ∈ FQ(q) (3.5.21) . µ Λ Proposition 3.5.6 implies that { B | µ ∈ K } is Kashiwara’s upper global basis at q = 0 [69], or Lusztig’s dual canonical basis [94, §14.4], of L(Λ). By Corollary 3.5.8, q − def µ Bµ is bar invariant and, thinking of ( , ) Λ as a pairing from [ProjΛ A ] × [RepA ] to A, we have − def µ λ # (q Bµ , B ) = hBµ , B λ i = hBµ , B λ i = δλµ , by Proposition 3.5.16 and Corollary 3.5.19. Hence, { q − def µ Bµ | µ ∈ KΛ } is the canonical basis, or the lower global basis, of L(Λ). The equivalence of parts (a)–(d) of the next result could have been b e ). For (e), however, we need the work of proved without introducing Uq (sl Misra and Miwa [111], and Kashiwara’s theory of crystal bases [69, 70], to connect the crystal bases of the Fock space with those of L(Λ).

Proposition 3.5.22. Suppose that F is an arbitrary field and that n ≥ 0. Then the following are equivalent: a) For all µ ∈ KnΛ , B µ = [Dµ ]. b) For all λ, µ ∈ KnΛ , eλµ (−q) ∈ δλµ + qN[q]. c) For all µ ∈ KnΛ , Bµ = [Y µ ]. d) For all λ, µ ∈ KnΛ , dλµ (q) ∈ δλµ + qN[q]. e) For all λ ∈ PnΛ and µ ∈ KnΛ , dλµ (q) ∈ δλµ + qN[q]. P Proof. In the Grothendieck groups, [Dµ ] = [S µ ] + µBτ eµτ (−q)[S τ ] P and [Y µ ] = Xµ + µBσ dσµ (q)Xσ , where in the sums σ, τ ∈ KnΛ . Moreover,

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by Lemma 3.5.5, [Y µ ]# = [Y µ ] and [Dµ ]~ = [Dµ ], for all µ ∈ KnΛ . By definition, dλµ (q) ∈ N[q, q −1 ] and eλµ (−q) ∈ Z[q, q −1 ]. Hence, parts (a) and (b), and parts (c) and (d), are equivalent by Proposition 3.5.6. Moreover, K −1 eK , dµµ (1) = 1 = eµµ (−q) and the Laurent polynomials dλµ (q) q = (dq ) and eλµ (−q) are non-zero only if λ D µ by Theorem 1.3.4, so parts (b) and (d) are also equivalent. Certainly, (e) implies (d) so to complete the proof it is enough to show that (a) implies (e). Suppose that (a) holds so that Bµ = Dµ , for all µ ∈ KΛ . To prove that (e) holds we need the machinery of crystal bases [69, 70] in the special Λ case of the Fock space FQ(q) . We will refer the reader to the literature for the definitions and results that we need. Following [70, §2] define rings A = Q[q, q −1 ] = Q ⊗Z A, A0 = A(q) and A∞ = A(q−1 ) , so that A0 and A∞ are the rational functions in Q(q) that are regular at 0 and ∞, respectively. Set M M L0 (Λ) = A0 B µ = A0 [S µ ] µ∈KΛ

µ∈KΛ

Λ µ and B0 (Λ) = { [S µ ] + qL0 (Λ)
| µ ∈ K }. As {B } is the upper crystal basis, the pair L0 (Λ), B0 (Λ) is an upper crystal base at q = 0 for L(Λ) as defined by Kashiwara [69, §2]. Similarly, in the Fock space define M F0Λ = A0 |λi and C0Λ = { |λi + qF0Λ | λ ∈ P Λ } . λ∈P Λ

Misra and Miwa [111] showed that (F0Λ , C0Λ ) is an upper crystal basis Λ for FQ(q) . (As we discuss below they also explicitly describe the crystal Λ Λ graph of FQ(q) .) By [69, Theorem 7], the Fock space FQ(q) has a unique λ Λ basis { C | λ ∈ P }, Kashiwara’s upper global basis, such that and C λ ≡ |λi (mod qF0Λ ) , for λ ∈ P Λ .

Let F•Λ = FAΛ , F0Λ , F0Λ and L• (Λ) = LA (Λ), L0 (Λ), L0 (Λ)~ , where FAΛ = A ⊗A0 F0Λ and LA (Λ) = A ⊗A0 L0 (Λ). Then F•Λ and L• (Λ) are Λ balanced triples in the sense of [70, §2]. The Λ-weight space of FQ(q) b e )is Q(q)vΛ so, up to a scalar, the decomposition map dq is the unique Uq (sl (3.5.23)

Cλ = Cλ

Λ module homomorphism dq : FQ(q) −→ [RepΛ Q(q) ]. By [69, Proposition 5.2.1], Λ the image of F• under dq is a balanced triple contained in L(Λ). In fact, we have dq (F•Λ ) = L• (Λ) by [69, Proposition 5.2.2] because dq sends vΛ = |0` i to [S 0` ]. Consequently, if λ ∈ P Λ then dq (|λi) ∈ L0 (Λ). That is, M M [S λ ] = dq (|λi) ∈ A0 [S µ ] = A0 [Dµ ]. µ∈KΛ

µ∈KΛ

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P As [S λ ] = µ dλµ (q)[Dµ ] it follows that dλµ (q) ∈ N[q, q −1 ] ∩ A0 = N[q]. Moreover, because of (3.5.21), dq sends canonical basis elements in F0Λ to canonical basis elements in L0 (Λ), or to zero. It follows that ( B λ = [Dλ ], if λ ∈ KΛ λ dq (C ) = 0, otherwise. By (3.5.23), |λi − C λ ∈ qF0Λ for all λ ∈ P Λ . Consequently, if λ ∈ / KΛ then [S λ ] = dq (|λi) = dq (|λi − C λ ) ∈ dq (qF0Λ ) = qL0 (Λ). Hence, dλµ (q) ∈ δλµ + qN[q], for all λ ∈ P Λ and all µ ∈ KΛ . Thus, (e) holds and the proposition is proved.  Remark 3.5.24. The difference between the upper and lower crystal bases, or the dual canonical and canonical bases, can be interpreted as changing between the bases of Specht modules and dual Specht modules. The global bases and their crystal lattices are: P λ upper q = 0 B µ ≡ [S µ ] (mod λ∈KΛ A0 [S ]) P lower q = ∞ cq (q − def µ Bµ ) ≡ [Sm(µ)0 ] (mod λ∈KΛ A∞ [Sλ ]) where m is an involution on KnΛ that generalises the well-known Mullineux map for the symmetric groups. See Theorem 3.6.6 below. Remark 3.5.25. As mentioned in Remark 3.5.10, a different action on the Fock space is commonly used in the literature. With respect to the Cartan pairing, as in Corollary 3.5.18, this action is the adjoint of the action in Theorem 3.5.9. As a consequence, the papers that use a different b e )-action also use a different coproduct for Uq (sl b e ), as they have to if Uq (sl they want Kashiwara’s tensor product rule to connect the crystal bases at different levels for a fixed Λ ∈ P + . In the dual set up, # categorifies the bar involution on L(Λ), { Bµ | µ ∈ KΛ } is the canonical basis, or lower global crystal basis at q = 0 for L(Λ) and { q def µ B µ | µ ∈ KΛ } is the dual canonical basis. It is natural to ask when the conditions of Proposition 3.5.22 are satisfied. This is a difficult open problem. The next result implies that the conditions of Proposition 3.5.22 hold whenever F is a field of characteristic zero. We can now state Ariki’s celebrated Categorification Theorem. By b e ) ⊗ Q becomes the Kac-Moody specializing q = 1 the quantum group UA (sl b algebra U (sle ). Let L1 (Λ) be the irreducible integrable highest weight b e )-module of high weight Λ. The canonical bases of L1 (Λ) are obtained U (sl

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by specializing q = 1 in the canonical bases of LA (Λ). Forgetting the ∼ grading in the results above, RepΛ , where RepΛ = = ProjΛ = L1 (Λ) ∼ Q Q Q L L Λ Λ Λ n Rep(H n ) ⊗Z Q and ProjQ = n Proj(H n ) ⊗Z Q. Theorem 3.5.26 (Ariki’s Categorification Theorem [3,23]). Suppose that F is a field of characteristic zero. Then the canonical basis of L1 (Λ) coincides with the basis of (ungraded) projective indecomposable HnΛ -modules { [Y µ ] | µ ∈ KΛ } of ProjΛ . Q This theorem was proved by Ariki [3, Theorem 4.4] when v 2 6= 1 and by Brundan and Kleshchev when v 2 = 1 [23, Theorem 3.10]. For a detailed proof of this important result when v 2 = 6 1 see [5, Theorem 12.5]. For an overview and historical account of Ariki’s theorem see [44]. Combining Theorem 3.5.26 with Proposition 3.5.22 we obtain the main result of this section. Corollary 3.5.27 (Brundan and Kleshchev [22, Theorem 5.14]). Suppose that F is a field of characteristic zero. Then the canonical basis of LA (Λ) coincides with the basis { q − def µ [Y µ ] | µ ∈ KΛ } of [ProjΛ Q(q) ]. In Λ Λ particular, dλµ (q) ∈ δλµ + qN[q], for all λ ∈ P and µ ∈ K . When Λ is a weight of level 2 and e = ∞ this was first proved by Brundan and Stroppel [26, Theorem 9.2]. For extensions of this result to cyclotomic quiver Hecke algebras of arbitrary type see [67, 91, 122, 134]. Corollary 3.5.27 implies that the graded decomposition numbers dλµ (q) = [S λ : Dµ ]q = bλµ (q) are parabolic Kazhdan-Lusztig polynomials. Explicit formulas are given in [55, Appendix A] and [99, Lemma 2.46]. For the canonical basis {Bµ } it is immediate that the Laurent polynomials bλµ (q) ∈ Z[q] are polynomials, for λ ∈ PnΛ and µ ∈ KnΛ , however, it is a deep fact that their coefficients are non-negative integers. In contrast, it is immediate that dλµ (q) ∈ N[q, q −1 ] but it is a deep fact that the graded decomposition numbers are polynomials rather than Laurent polynomials. Thus, the difficult result changes from positivity of coefficients in the ungraded setting, to positivity of exponents in the graded setting. In fact, it is also true when F = C that the inverse graded decomposition numbers eλµ (q) = bλµ (−q) are polynomials in q with non-negative integer coefficients. This is perhaps best explained by passing to the Koszul dual of the corresponding graded cyclotomic Schur algebras [7,55,128] using [55,99]; see [55, Lemma 2.15] where this is stated explicitly. Brundan and Kleshchev’s proof of Corollary 3.5.27 is quite different to the one given here. They have to work quite hard to define triangular

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bar involutions on LA (Λ) whereas we have done this by exploiting the representation theory of HnΛ . One benefit of Brundan and Kleshchev’s approach is that they have an explicit description of the bar involution Λ on FA . In contrast, we have no hope of working with our bar involution unless we already know the graded decomposition matrices. On the other hand, the approach here works for an arbitrary multicharge κ. To complete the proof of Corollary 3.5.27, Brundan and Kleshchev lift Grojnowski’s approach [49] to the representation theory of HnΛ to the graded setting. As a result they obtain graded analogues of Kleshchev’s modular branching rules [18, 77, 78]. Under categorification, these branching rules correspond to the action of the crystal operators on the crystal graph of L(Λ); see [22, Theorem 4.12]. By invoking Ariki’s theorem they deduce an analogue of Corollary 3.5.27, although with a possibly different labelling of the irreducible modules. Finally, they prove that the labelling of the irreducible HnΛ -modules coming from the branching rules agrees with the labelling of Corollary 1.5.2; compare with [6, 9]. We have not yet given an explicit description of the labelling of the irreducible HnΛ -modules because we defined KnΛ = { µ ∈ PnΛ | Dµ 6= 0 }. Extending Definition 3.2.1, if µ ∈ PnΛ and given nodes A < C define dC A (µ) = # { B ∈ Addi (µ) | A 0 and λ, µ ∈ PnΛ then aλµ (q) = δλµ if ep > n. A natural strengthening of this conjecture is that the adjustment matrix of RβΛ is trivial whenever def β < p. For the symmetric groups, the condition def β < p exactly corresponds to the case when the defect group of the block RβΛ is abelian. The James conjecture is known to be true for blocks of weight at most 4 [38, 39, 60, 119]. Moreover, for every defect w ≥ 0 there exists a Rouquier block of defect w for which the James conjecture holds [61]. Starting from the Rouquier blocks, there was some hope that the derived

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equivalences of Chuang and Rouquier [28] could be used to prove the James conjecture for all blocks. Notwithstanding all of the evidence in favour of the James conjecture, it turns out that the conjecture is wrong! Again, Williamson [135, §6] has cruelly (but ultimately kindly) produced counter-examples to the James conjecture. At the same time he also found counter-examples to the Lusztig conjecture [93] for SLn . These examples rely upon Williamson’s recent work with Elias that gives generators and relations for the category of Soergel bimodules [34]. As of writing, the smallest known counter-example to the James conjecture occurs in a block of defect 561 in F839 S467874 . Williamson has not revealed which Specht modules his counter-examples appear in, so the size of Gram matrix that needs to be computed in order to verify this example is not known. The Gram matrices of the Specht modules will be significantly larger, and harder to compute, than the one dimensional intersection form that Williamson reduces to (using a chain of deep results in geometric representation theory), and then calculates, using elementary techniques (and a computer). ♦ Williamson’s counter-examples to the James and Lusztig conjectures suggest that there is no block theoretic criterion for the adjustment matrix of a block to be trivial, except asymptotically where the Lusztig conjecture is known to hold [1]. With hindsight, perhaps this is not so surprising because the condition given in Corollary 1.7.6 for a Specht module to be irreducible is rarely a block invariant. The failure of the James and Lusztig conjectures suggests that we should, instead, look for necessary and sufficient conditions for the RnΛ (F )-modules DZµ ⊗ F to be irreducible, for µ ∈ KnΛ . Some steps towards such a criterion are made in Conjecture 4.4.1 below. Brundan and Kleshchev [22, §5.6] remarked that aF λµ (q) ∈ N in all of the examples that they had computed. They asked whether this might always be the case. The next examples show that, in general, aF / N. λµ (q) ∈ 3.7.12. Example (Evseev [35, Corollary 5]) Suppose that e = 2, Λ = Λ0 and let λ = (3, 22 , 12 ) and µ = (19 ). Take F = F2 to be a field of 2 characteristic 2 and let aFq 2 = (aFλµ (q)) be the corresponding adjustment matrix. 2 As part of a general argument Evseev shows that aFλµ (q) ∈ / N. In fact, it is F2 −1 not hard to see directly that aλµ (q) = q+q . Comparing the decomposition matrix for F2 S9 given by James [59] with the graded decomposition matrices F2 F2 when e = 2 given in [104], shows that dQ λµ = 0, dλµ = 2, and that aλµ (1) = 2.

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Now DFµ2 = DFµ2 e(iµ ) is one dimensional, so any composition factor of SFλ2 that is isomorphic to DFµ2 hdi, for some d ∈ Z, must be contained in SFλ2 e(iµ ). There are exactly six standard λ-tableau with residue sequence iµ , namely: deg t 1 1 1 1 1 −1 t

1 6 9 2 7 3 8 4 5

1 4 9 2 5 3 8 6 7

1 4 5 2 7 3 8 6 9

1 2 3 4 7 5 8 6 9

1 4 7 2 5 3 6 8 9

1 4 7 2 5 3 8 6 9

As Dµ is one dimensional, and concentrated in degree zero, it follows that 2 2 aFλµ = dFλµ (q) = q + q −1 . We can see a shadow of the adjustment matrix entry in the Gram matrix of SZλ e(iµ ), that is equal to 0

0 0 0 0 0 0 0 0 0

0 0

0 0 0 0 0 4

0 0 0 0 0 −2

0 0 0 0 0 2

0 0 4  −2  . 2 0



The elementary divisors of this matrix are 2, 2, 0, 0, 0, 0, with the 2’s in degrees ±1. Therefore, the graded dimension of DFλ2 e(iµ ) decreases by q+q −1 in characteristic 2. ♦ 3.7.13. Example Motivated by the runner removable theorems of [27, 63] and Example 3.7.12, take e = 3, F = F2 , λ = (3, 24 , 13 ) and µ = (114 ). (The partitions λ and µ are obtained from the corresponding partitions in Example 3.7.12 by conjugating, adding an empty runner, and then conjugating again.) Again, we work over F2 and consider the corresponding adjustment matrices. F2 Calculating with Specht [102] we find that dQ λµ = 0 and that dλµ = 2. Once again, it turns out that there are six λ-tableaux with 3-residue sequence iµ , with five of these having degree 1 and one having degree −1. (Moreover, the Gram matrix of S λ e(iµ ) is the same as the Gram matrix given in 2 2 Example 3.7.12.) Hence, as in Example 3.7.12, aFλµ (q) = q + q −1 = dFλµ (q). As the runner removable theorems compare blocks for different e over the same field we cannot expect to find an example of a non-polynomial adjustment matrix entry in odd characteristic in this way. Nonetheless, it seems fairly certain that non-polynomial adjustment matrix entries exist for all e and all p > 0. Evseev [35, Corollary 5] gives three other examples of adjustment matrix entries that are equal to q + q −1 when e = p = 2. All of them have similar analogues when e = 3 and p = 2. Finally, if we try adding further empty runners to the partitions λ and µ, so that e ≥ 4, then the corresponding adjustment matrix entry is zero (all of these partitions have weight 4). ♦

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Seminormal bases and the KLR grading

In this final section we link the KLR grading on RnΛ with the semisimple representation theory of HnΛ using the seminormal bases. We start by showing that by combining information from all of the KLR gradings for different cyclic quivers leads to an integral formula for the Gram determinants of the ungraded Specht modules. 4.1

Gram determinants and graded dimensions

In Theorem 1.7.3 we gave a “rational” formula for the Gram determinant of the ungraded Specht modules S λ , for λ ∈ PnΛ . We now give an integral formula for these determinants and give both a combinatorial and a representation theoretic interpretation of this formula. Suppose that the Hecke parameter v from Definition 1.1.1 is an indeterminate over Q and consider an integral cyclotomic Hecke algebra HnΛ over the field Z = Q(v) where Λ ∈ P + such that e > n and (Λ, αi,n ) ≤ 1, for all i ∈ I. Then HnΛ is semisimple by Corollary 1.6.11. Definition 4.1.1. Suppose that λ ∈ PnΛ . For e ≥ 2 and i ∈ Ien define X dege,i (λ) = dege t, t∈Stdi (λ) t

where Stdi (λ) = { t ∈ Std(λ) | i = i }. Set dege (λ) = P a prime integer p > 0 set Degp (λ) = k≥1 degpk (λ).

P

i∈Ien

dege,i (λ). For

By definition, dege (λ), Degp (λ) ∈ Z. For e > 0 let Φe (x) ∈ Z[x] be the eth cyclotomic polynomial in the indeterminate x. Theorem 4.1.2 (Hu-Mathas [57, Theorem C]). Suppose that Λ ∈ P + , e > n and that (Λ, αi,n ) ≤ 1, for all i ∈ I. Let λ ∈ PnΛ . Then Y det Gλ = Φe (v 2 )dege (λ) . e>1 λ

Consequently, if v = 1 then det G =

Y

pDegp (λ) .

p prime

Proving this result is not hard: it amounts to interpreting Definition 1.6.6 in light of the KLR degree functions on Std(λ). There is a power of v in the statement of this result in [57]. This is not needed here because we have renormalised the quadratic relations in the Hecke algebra given in Definition 1.1.1.

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The Murphy basis is defined over Z[v, v −1 ]. Therefore, det Gλ ∈ Z[v, v −1 ] and Theorem 4.1.2 implies that dege (λ) ≥ 0 for all λ ∈ PnΛ and e ≥ 2. In fact, [57, Theorem 3.24] gives an analogue of Theorem 4.1.2 for the determinant of the Gram matrix restricted to S λ e(i), suitably interpreted, and the following is true: Corollary 4.1.3 ([57, Corollary 3.25]). Suppose that e ≥ 2, λ ∈ PnΛ and i ∈ Ien . Then dege,i (λ) ≥ 0. The definition of the integers dege,i (λ) is purely combinatorial, so it should be possible to give a combinatorial proof of this result perhaps using Theorem 3.4.6. We think, however, that this is probably difficult. Fix an integer e ≥ 2 and a dominant weight Λ ∈ P + and consider the Hecke algebra HnΛ over a field F . If λ ∈ PnΛ then, by definition, X dλµ (q) Chq Dµ ∈ A[I n ]. Chq S λ = Λ µ∈Kn

Let ∂ : A[I n ] −→ Z[I n ] be the linear map given by ∂(f (q) · i) = f 0 (1)i, where f 0 (1) is the derivative of f (q) ∈ A evaluated at q = 1. Then P ∂ Chq S λ = i dege,i (λ) · i. The KLR idempotents are orthogonal, so µ dim q Di = dim q Diµ since (Dµ )~ ∼ = Dµ . Therefore, ∂ Chq Dµ = 0. Hence, applying ∂ to the formula for Chq S λ shows that X X X (4.1.4) dege,i (λ) · i = ∂ Chq S λ = d0λµ (1) dim Diµ · i. i∈I n

Λ i∈I n µ∈Kn

Consequently, dege,i (λ) = µ d0λµ (1) dim Diµ . So far we have worked over an arbitrary field. If F = C then dλµ (q) ∈ N[q], by Proposition 3.5.6, so that d0λµ (1) ≥ 0. Therefore, dege,i (λ) ≥ 0 as claimed. (In fact, by Theorem 3.7.6, the right-hand side of (4.1.4) is independent of F , as it must be.) Theorem 1.7.4 shows that taking the p-adic valuation of the Gram determinant of S λ leads to the Jantzen sum formula for S λ . Therefore, (4.1.4) suggests that X X (4.1.5) [Jk (S λ d0λµ (1)[Dµ C )] = C ], P

k>0

µBλ

where we use the notation of Theorem 1.7.4. That is, Theorem 4.1.2 corresponds to writing the Jantzen sum formula as a non-negative linear combination of simple modules. In fact, we have not done enough to prove (4.1.5). (One way to do this would be to establish analogous statements for the Gram determinants of the Weyl modules of the cyclotomic Schur algebras [31].)

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Nonetheless, (4.1.5) is true, being proved by Ryom-Hansen [124, Theorem 1] in level one and by Yvonne [138, Theorem 2.11] in general. A better interpretation of (4.1.4) is in terms of grading filtrations [15, §2.4]. L Let R˙nΛ = Hom RnΛ (Y, Y ), where Y = µ∈KnΛ Y µ is a progenerator for RnΛ . Then R˙Λ is a graded basic algebra for R Λ and the functor n

n

Fn : Rep(RnΛ ) −→ Rep(R˙nΛ ); M 7→ Hom RnΛ (Y, M ),

for M ∈ Rep(RnΛ ),

is a graded Morita equivalence; see, for example, [55, §2.3-2.4]. Recall
that the matrix cq = cλµ (q) = dTq ◦ dq is the Cartan matrix of RnΛ . By Corollary 2.1.6, cλµ (q) = dim q Hom RnΛ (Y λ , Y µ ) so that X dim q R˙nΛ = cλµ (q) ∈ N[q, q −1 ]. Λ λ,µ∈Kn

For the rest of this section assume that F = C. Then cλµ (q) ∈ N[q] by Corollary 3.5.27. Therefore, dim q R˙nΛ ∈ N[q] so that R˙nΛ is a positively graded Lz algebra. Let M˙ = d=a M˙ d be a R˙nΛ -module. The grading filtration of M˙ is the filtration M˙ = Ga (M˙ ) ⊇ Ga+1 (M˙ ) ⊇ · · · ⊇ Gz (M˙ ) ⊃ 0, where M Gd (M˙ ) = M˙ k . k≥d

Then Gr (M˙ ) is a graded R˙nΛ -module precisely because R˙nΛ is positively graded. The grading filtration of a Rn -module M is the filtration given by λ λ Gr (M ) = F−1 n (Gr (Fn (M ))), for r ∈ Z. By Corollary 3.6.7, S = G0 (S ) λ and Gr (S ) = 0 for r > def λ. P (r) (r) For λ ∈ PnΛ and µ ∈ KnΛ write dλµ (q) = r≥0 dλµ q r , for dλµ ∈ N. Lemma 4.1.6. Suppose that F = C and λ ∈ PnΛ . If 0 ≤ r ≤ def λ then M
⊕d(r) Gr (S λ )/Gr+1 (S λ ) ∼ Dµ hri λµ . = Λ µ∈Kn

Proof. This is an immediate consequence of the definition of the grading filtration and Corollary 3.5.27.  Comparing this with (4.1.5) suggests that Jr (S λ ) = Gr (S λ ), for r ≥ 0. Of course, there is no reason to expect that Jr (S λ ) is a graded submodule of S λ . Nonetheless, establishing a conjecture of Rouquier [89, (16)], Shan has proved the following when Λ is a weight of level 1. Theorem 4.1.7 (Shan [126, Theorem 0.1]). Suppose that F is a field of characteristic zero, Λ = Λ0 , and that λ ∈ PnΛ . Then Jr (S λ ) = Gr (S λ ) is (r) a graded submodule of S λ and [Jr (S λ )/Jr+1 (S λ ) : Dµ hsi] = δrs dλµ , for all Λ µ ∈ Kn and r ≥ 0.

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Shan actually proves that the Jantzen, radical and grading filtrations of graded Weyl modules coincide for the Dipper-James v-Schur algebras [30]. This implies the result above because the Schur functor maps Jantzen filtrations of Weyl modules to Jantzen filtrations of Specht modules. There is a catch, however, because Shan remarks that it is unclear how her geometrically defined grading relates to the grading on the v-Schur algebra given by Ariki [7] and hence to the KLR grading on RnΛ . As we now sketch, Theorem 4.1.7 can be deduced from Shan’s result using recent work. Since Shan’s paper cyclotomic quiver Schur algebras have been introduced for arbitrary dominant weights [7,55,128], thus giving a grading on all of the cyclotomic Schur algebras introduced by Dipper, James and the author [31]. The key point, which is non-trivial, is that the module categories of the cyclotomic quiver Schur algebras are Koszul. When e = ∞ this is proved in [55] by reducing to parabolic category O for the general linear groups, which is known to be Koszul by [14, 15]. Using similar ideas, Maksimau [99] proves that Stroppel and Webster’s cyclotomic quiver Schur algebras are Koszul for arbitrary e by using [123] to reduce to affine parabolic category O. As the module categories of the cyclotomic quiver Schur are Koszul, an elementary argument [15, Proposition 2.4.1] shows that the radical and grading filtrations of the graded Weyl modules of these algebras coincide. By definition, the analogue of Lemma 4.1.6 describes the graded composition factors of the grading (=radical) filtrations of the graded Weyl modules — compare with [55, Corollary 7.24] when e = ∞ and [99, Theorem 1.1] in general. The graded Schur functors of [55, 99] send graded Weyl modules to graded Specht modules, graded simple modules to graded simple RnΛ -modules (or zero), grading filtrations to grading filtrations and Jantzen filtrations to Jantzen filtrations. Combining these facts with Shan’s work [126] implies Theorem 4.1.7 when Λ = Λ0 . We note that the v-Schur algebras were first shown to be Koszul by Shan, Varagnolo and Vasserot [127]. It is also possible to match up Shan’s grading on the v-Schur algebras with the gradings of [7, 128] using the uniqueness of Koszul gradings [15, Proposition 2.5.1]. As these papers use different conventions, it is necessary to work with the graded Ringel dual. The obstacle to extending Theorem 4.1.7 to arbitrary weights Λ ∈ P + is in showing that the Jantzen and radical (=grading) filtrations of the graded Weyl modules of the cyclotomic quiver Schur algebras coincide. As the cyclotomic quiver Schur algebras are Koszul it is possible that this is straightforward. It seems to the author, however, that it is necessary to generalize Shan’s arguments [126] to realize the Jantzen filtration geometrically

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using the language of [123]. 4.2

A deformation of the KLR grading

Following [57], especially the appendix, we now sketch how to use the seminormal basis to prove that RnΛ ∼ = HnΛ over a field (Theorem 3.1.1). The aim in doing this is not so much to give a new proof of the graded isomorphism theorem. Rather, we want to build a bridge between the KLR algebras and the well-understood semisimple representation theory of the cyclotomic Hecke algebras. In §4.3 we cross this bridge to construct a new graded cellular basis {Bst } of HnΛ that is independent of the choices of reduced expressions that are necessary in Theorem 3.2.4. Throughout this section we consider a cyclotomic Hecke algebra HnΛ defined over a field F that has Hecke parameter v ∈ F × of quantum characteristic e ≥ 2. As in §1.2, Λ ∈ P + is determined by a multicharge κ ∈ Z` . We set up a modular system for studying HnΛ = HnΛ (F ). Let x be an indeterminate over F and let O = F [x](x) be the localization of F [x] at the principal ideal generated by x. Let K = F (x) be the field of fractions of O. Let HnO be the cyclotomic Hecke algebra with Hecke parameter t = x + v, a unit in O, and cyclotomic parameters Ql = xl + [κl ]t , for 1 ≤ l ≤ `. Then HnK = HnO ⊗O K is a split semisimple algebra by Theorem 2.4.8. Moreover, by definition, HnΛ = HnΛ (F ) ∼ = HnO ⊗O F , where we consider F as an O-module by letting x act as multiplication by 0. As the algebra HnK is semisimple, it has a seminormal basis {fst } in the sense of Definition 1.6.4. With our choice of parameters, the content functions from (1.6.1) become 2(c−b) l cZ x + [κl + c − b]t = t2(c−b) xl + [cZr (s)]t r (s) = t 2 Λ if s(l, b, c) = r, for 1 ≤ k ≤ n. Then, Lr fst = cZ r (s)fst , for (s, t) ∈ Std (Pn ). By Corollary 1.6.9, the basis {fst } determines a seminormal coefficient system α = { αr (t) | t ∈ Std(PnΛ ) and 1 ≤ r < n } and a set of scalars { γt | t ∈ Std(PnΛ ) }. For i ∈ I n let Std(i) = { s ∈ Std(PnΛ ) | is = i } be the set of standard tableaux with residue sequence i. Define X (4.2.1) fiO = Ft . t∈Std(i)

By definition, fiO ∈ HnK but, in fact, fiO ∈ HnO . This idempotent lifting result dates back to Murphy [112] for the symmetric groups. For higher

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levels it was first proved in [108]. In [57] it is proved for a more general class of rings O. Lemma 4.2.2 ([57, Lemma 4.4]). Suppose that i ∈ I n . Then fiO ∈ HnO . We will see that fiO ⊗O 1F is the KLR idempotent e(i), for i ∈ I n . P Notice that 1 = i fiO and, further, that fiO fjO = δij fiO , for i, j ∈ I n , by Theorem 1.6.7. As detailed after Theorem 3.1.1, Brundan and Kleshchev construct their ∼ isomorphisms RnΛ −→ HnΛ using certain rational functions Pr (i) and Qr (i) in F [y1 , . . . , yn ]. The advantage of working with seminormal forms is that, at least intuitively, these rational functions “converge” and can be replaced with “nicer” polynomials. The main tool for doing this is the following result, generalizing Lemma 4.2.2. Let Mr = 1 − t−1 Lr + tLr+1 , for 1 ≤ r < n. Then Mr fst = MrZ (s)fst , Z Z where MrZ (s) = 1 − t−1 cZ r (s) + tcr+1 (s). The constant term of Mr (s) is 2cZr (s)−1 Z Z equal to v [1 − cr (s) + cr+1 (s)]v 6= 0. Consequently, Mr acts invertibly on fst whenever s ∈ Std(i) and 1 − ir + ir+1 6= 0 in I = Z/eZ. This observation is part of the proof of part (a) of the next result. Similarly, set Z Z Z ρZ r (s) = cr (s) − cr+1 (s). Then ρr (s) is invertible in O if ir 6= ir+1 . Corollary 4.2.3 (Hu-Mathas [57, Corollary 4.6]). Suppose that 1 ≤ r < n and i ∈ I n . X 1 O 1 Fs ∈ HnO . a) If ir 6= ir+1 + 1 then fi = Mr MrZ (s) s∈Std(i) X 1 1 b) If ir 6= ir+1 then Fs ∈ HnO . fiO = Lr − Lr+1 ρZ r (s) s∈Std(i)

Mr fiO ,

The invertibility of when ir 6= ir+1 + 1, allows us to define analogues of the KLR generators of RnΛ in HnO . The invertibility of (Lr − Lr+1 )fiO is needed to show that these new elements generate HnO . Define an embedding I ,→ Z; i 7→ ˆı by letting ˆı be the smallest nonnegative integer such that i = ˆı + eZ, for i ∈ I. Definition 4.2.4. Suppose that 1 ≤ r < n. Define elements ψrO = P O O O i∈I n ψr fi in Hn by  −1 t2ˆır O  if ir = ir+1 ,  (Tr + t ) Mr fi , O O −2ˆ ı O r ψr fi = (Tr Lr − Lr Tr )t fi , if ir = ir+1 + 1,   (T L − L T ) 1 f O , otherwise. r r r r Mr i P O −2ˆ ır −1 If 1 ≤ r ≤ n then define yr = i∈I n t (Lr − [ˆır ])fiO .

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We now describe an O-deformation of cyclotomic KLR algebra RnΛ . This is a special case of one of the main results of [57], which allows greater flexibility in the choice of the ring O. Theorem 4.2.5 (Hu-Mathas [57, Theorem A]). As an O-algebra, the algebra HnO is generated by the elements { fiO | i ∈ I n } ∪ { ψrO | 1 ≤ r < n } ∪ { yrO | 1 ≤ r ≤ n } subject only to the following relations: Y (y1O − xl − [κl − i1 ])fiO = 0, 1≤l≤` κi ≡i1 (mod e)

fiO fjO = δij fiO ,

O i∈I n fi

P

yrO fiO = fiO yrO ,

= 1,

ψrO fiO = fsOr ·i ψrO ,

yrO ysO = ysO yrO ,

O ψrO yr+1 fiO = (yrO ψrO + δir ir+1 )fiO ,

O yr+1 ψrO fiO = (ψrO yrO + δir ir+1 )fiO ,

ψrO ysO = ysO ψrO ,

if s 6= r, r + 1,

ψrO ψsO

if |r − s| > 1,

(ψrO )2 fiO

=

=

ψsO ψrO ,  h1+ρ (i)i  (yr r     h1+ρ r (i)i   (yr h1−ρr (i)i (y  r+1

h1−ρr (i)i

O − yr+1 )(yr+1

− −

− yrO )fiO ,

O )fiO , yr+1 yrO )fiO ,

if ir  ir+1 , if ir → ir+1 , if ir ← ir+1 ,

   0,    f O , i

if ir = ir+1 , otherwise,

O O fiO is equal to ψ O − ψ O ψrO ψr+1 and ψrO ψr+1  h1+ρ (i)i r h1+ρ r+1 (i)i h1+ρ (i)i h1−ρ (i)i r  (y + yr+2 r − yr+1 r − yr+1 r )fiO ,   r  −t1+ρr (i) f O , i fiO ,  




 

0,

where ρr (i) = ˆır − ˆır+1 and

hdi yr

=

t2d yrO

−1

+t

if ir+2 = ir  ir+1 , if ir+2 = ir → ir+1 , if ir+2 = ir ← ir+1 , otherwise,

[d], for d ∈ Z.

The statement of Theorem 4.2.5 is slightly different to [57, Theorem A] because we are using a different choice of modular system (K, O, F ) and because Definition 1.1.1 renormalises the quadratic relations for the generators Tr of HnO , for 1 ≤ r < n. The strategy behind the proof of Theorem 4.2.5 is quite simple: we compute the action of the elements defined in Definition 4.2.4 on the seminormal basis and use this to verify that they satisfy the relations in the

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theorem. To bound the rank of the algebra defined by the presentation in Theorem 4.2.5 we essentially count dimensions. By specializing x = 0, we obtain Theorem 3.1.1 as a corollary of Theorem 4.2.5. To give a flavour of the type of calculations that were used to verify that the elements in Definition 4.2.4 satisfy the relations in Theorem 4.2.5, for s ∈ Std(i) and 1 ≤ r < n define  αr (s)t2ˆır   , if ir = ir+1 ,    MrZ (s) −2ˆ ır (4.2.6) βr (s) = αr (s)ρZ , if ir = ir+1 + 1, r (s)t   Z  α (s)ρ (s)  r  r , otherwise. MrZ (s) Then Theorem 1.6.7 easily yields the following. Lemma 4.2.7. Suppose that 1 ≤ r < n and that (s, t) ∈ Std2 (PnΛ ). Set i = is , j = it , u = s(r, r + 1) and v = t(r, r + 1). Then 1 ψrO fst = βr (s)fut − δir ir+1 Z fst . ρr (s) Moreover, if s(l, b, c) = r then
yrhdi fst = t−1 t2(c−b+d−ir ) xl + [cZk (s) + d − ˆır ] fst , for 1 ≤ r ≤ n and d ∈ Z. Armed with Lemma 4.2.7, and Definition 1.6.6, it is an easy exercise to verify that all of the relations in Theorem 4.2.5 hold in HnO . For the quadratic relations, Lemma 4.2.7 implies that (ψrO )2 fst = 0 if s ∈ Std(i) and ir = ir+1 whereas if ir 6= ir+1 then (ψrO )2 fst = βr (s)βr (u)fst , where u = s(r, r + 1). The quadratic relations in Theorem 4.2.5 now follow using (4.2.6) and Lemma 4.2.7. For example, suppose that ir → ir+1 and s ∈ Std(i). Pick nodes (l, b, c) and (l0 , b0 , c0 ) such that s(l, b, c) = r and s(l0 , b0 , c0 ) = r + 1. Then, using Lemma 4.2.7 and Definition 1.6.6, (ψrO )2 fst = t−2ˆır+1 βr (s)βr (u)fst = t−2ˆır+1 MrZ (u)fst . h1+ρ (i)i

O On the other hand, by Lemma 4.2.7, (yr r − yr+1 ) acts on fst as multiplication by the same scalar. It follows that X 1 X 1 O (ψrO )2 fiO = (ψrO )2 fss = (yrh1+ρr (i)i − yr+1 ) fss γs γs s∈Std(i)

=

(yrh1+ρr (i)i

−

s∈Std(i)

O yr+1 )fiO

when ir → ir+1 . These calculations are perhaps not very pretty, but nor are they are difficult. As indicated by Remark 2.2.6, the quadratic relations appear in, and simplify, the proof of the deformed braid relations.

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A distinguished homogeneous basis

One of the advantages of Theorem 4.2.5 is that it allows us to transplant questions about the KLR algebra RnΛ into the language of seminormal bases. Definition 1.6.6 defines ∗-seminormal bases, which provide a good framework for studying the semisimple cyclotomic Hecke algebras. The algebra HnΛ comes with two cellular algebra automorphisms, ∗ and ?, where ? is the unique anti-isomorphism fixing the homogeneous generators of Definition 2.2.9 and ∗ is the unique anti-isomorphism fixing the generators of Definition 1.1.1. In general, these automorphisms are different. Definition 4.3.1 (Hu-Mathas [57, §5]). A ?-seminormal coefficient system is a collection of scalars β = { βr (t) | t ∈ Std(PnΛ ) and 1 ≤ r ≤ n } such that βr (t) = 0 if v = t(r, r + 1) is not standard, if v ∈ Std(PnΛ ) then βr (v)βr (t) is given by the product of the particular β-coefficients in (4.2.6), and βr (t)βr+1 (tsr )βr (tsr sr+1 ) = βr+1 (t)βr (tsr+1 )βr+1 (tsr+1 sr ), and if |r − r0 | > 1 then βr (t)βr0 (tsr ) = βr0 (t)βr (tsr0 ) for 1 ≤ r, r0 < n. Exactly as in Corollary 1.6.9, a ?-seminormal coefficient system determines a ?-seminormal basis {fst } that, like Definition 1.6.4, consists of elements fst ∈ Hst such that fst? = fts , for (s, t) ∈ Std2 (PnΛ ). The left (and right) the action of ψrO on fst is exactly as in Lemma 4.2.7 but where the coefficients come from an arbitrary ?-seminormal coefficient system β. Definition 4.3.1 gives extra flexibility in choosing a ?-seminormal basis. By [57, (5.8)] there exists a ?-seminormal basis {fst } such that the ψ-basis O of Theorem 3.2.4 lifts to a ψ O -basis {ψst } with the property that X O (4.3.2) ψst = fst + ruv fuv , (u,v)I(s,t)

for some ruv ∈ K. In this way we recover Theorem 3.2.4 and with quicker proof than the original arguments in [54]. Perhaps most significantly, by working with HnO we can improve upon the ψ-basis. Theorem 4.3.3 (Hu-Mathas [57, Theorem 6.2, Corollary 6.3]). Suppose O that (s, t) ∈ Std2 (PnΛ ). There exists a unique element Bst ∈ HnO such that X O −1 Bst = fst + pst )fuv , uv (x Λ (u,v)∈Std2 (Pn ) (u,v)I(s,t)

2 O Λ where pst uv (x) ∈ xK[x]. Moreover, { Bst | (s, t) ∈ Std (Pn ) } is a cellular O basis of Hn .

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The existence and uniqueness of this basis essentially come down to Gaussian elimination, although for technical reasons it is necessary to work O over the xO-adic completion of O. Proving that {Bst } is a cellular basis is more involved and, ultimately, this relies on the uniqueness properties of the B O -basis elements. O As the B O -basis is determined by a ?-seminormal basis, the basis {Bst } Λ behaves well with respect to the KLR grading on Hn . The main justification for using this seminormal basis as a proxy for choosing a “nice” basis for HnΛ , apart from the fact that it works, is that Theorem 2.4.8 shows that the natural homogeneous basis of the semisimple cyclotomic quiver Hecke algebras is a ?-seminormal basis. In characteristic zero the non-zero polynomials pst uv (x) satisfy (4.3.4)

1 0 < deg pst uv (x) ≤ 2 (deg u − deg s + deg v − deg t),

whenever (u, v) I (s, t) by [57, Proposition 6.4]. Moreover, if s, t, u, v are s t all standard tableaux of the same shape then pst uv (x) = pu (x)pv (x), where 1 1 s t 0 < deg pu (x) ≤ 2 (deg u−deg s) and 0 < deg pv ≤ 2 (deg v−deg t), whenever u B s and v B t, respectively. O As the basis {Bst } is defined over O we can reduce modulo the ideal xO O to obtain a basis {Bst ⊗O 1K } of HnΛ = HnΛ (K). This basis is hard to O compute and we do not know if the elements of {Bst ⊗O 1K } are homogeneous in general. Nonetheless, it is possible to construct a homogeneous basis {Bst } O of HnΛ from {Bst }. If λ ∈ PnΛ then define Btλ tλ to be the homogeneous O component of Btλ tλ ⊗ 1K of degree 2 deg tλ . More generally, for s, t ∈ Std(λ) we define Bst = Ds? Btλ tλ Dt , where Ds , Dt ∈ HnΛ are certain homogeneous elements in HnΛ . In characteristic zero, Bst is essentially the homogeneous O ⊗ 1K of degree deg s + deg t, and all other components component of Bst are of larger degree. For any field, by (4.3.2) and Theorem 4.3.3, X (4.3.5) Bst = ψst + auv ψuv , (u,v)I(s,t)

for some auv ∈ K that are non-zero only if iu = is , iv = it and deg u + deg v = deg s + deg t. Therefore, the B-basis resolves the ambiguities of Proposition 3.2.10(b). More importantly, we have the following. Theorem 4.3.6 (Hu-Mathas [57, Theorem 6.9]). Suppose that K is a field. Then { Bst | (s, t) ∈ Std2 (PnΛ ) } is a graded cellular basis of RnΛ with weight poset (PnΛ , D), cellular algebra automorphism ? and with deg Bst = deg s + deg t, for (s, t) ∈ Std2 (PnΛ ). Moreover, if (s, t) ∈ Std2 (PnΛ ) then Bst + HnBλ depends only on s and t and not on the choice of reduced expressions for the permutations d(s), d(t) ∈ Sn .

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By construction, the basis {Bst } depends on the field F . If F is a field of positive characteristic then Bst depends upon the choice of the elements Ds and Dt , which are uniquely determined modulo the ideal HnBλ . If e < ∞ and ` > 1 then {Bst } can depend on the choice of multicharge. 4.4

A simple conjecture

O The construction of the basis {Bst } of HnO in Theorem 4.3.3, together with the degree constraints on the polynomials pst uv (x) in (4.3.4), is reminiscent of the Kazhdan-Lusztig basis [73]. There is no known analogue of the Kazhdan-Lusztig bar involution in this setting. On the other hand, we do require that the basis elements Bst are homogeneous, which might be an appropriate substitute for being bar invariant in the graded setting. Partly motivated by this analogy with the Kazhdan-Lusztig basis, we now define analogues of cell representations for the B-basis. The basis {Bst } of Theorem 4.3.6 is a graded cellular basis so it defines a new homogeneous basis { Bt | t ∈ Std(λ) } of the graded Specht module S λ . ˙B Let the pre-order B on Std(λ) be the transitive closure of the relation  P Λ ˙ where tB v if there exists a ∈ Rn such that Bt a = s rs Bs with rv = 6 0. (So B is reflexive and transitive but not anti-symmetric.) Let ∼B be the equivalence relation on Std(λ) determined by B so that t ∼B v if and only if t B v B t. For example, tλ B t B tλ , for all t ∈ Std(λ). Let Std[λ] be the set of ∼B -equivalence classes in Std(λ). The set Std[λ] is partially ordered by B , where T B V if t B v for some t ∈ T and v ∈ V. Write T B v if t B v for some t ∈ T and T B v if T B v and v ∈ / T. λ λ Define ST to be the vector subspace of S with basis { Bv | T B v }. λ Similarly, let ST be the vector space with basis { Bv | T B v }. The λ λ definition of B ensures that ST and ST are both graded HnΛ -submodules λ λ λ λ λ of S and that ST ( ST . Therefore, STλ = ST /ST is a graded HnΛ module. By choosing any total order on Std[λ] that extends the partial order B , it is easy to see that S λ has a filtration with subquotients being precisely the modules STλ , for T ∈ Std[λ]. For λ ∈ PnΛ let Tλ = { t ∈ Std(λ) | t ∼B tλ }. In view of (3.7.2), if s, t ∈ Std(λ) and hBs , Bt i = 6 0 then s ∼B tλ ∼B t so that s, t ∈ T λ . λ λ Therefore, dim D ≤ |T |. Of course, if λ ∈ / KnΛ then this bound is not λ λ sharp because D = 0 whereas |T | ≥ 1.

Conjecture 4.4.1. Suppose that F is a field of characteristic zero and that λ ∈ PnΛ . Then STλ is an irreducible HnΛ -module, for all T ∈ Std[λ].

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As discussed in [57, §3.3], and is implicit in (4.1.4), by fixing a composition series for S λ and using a Gaussian elimination argument, it is possible to construct a basis {Ct } of S µ such that (1) each module in the composition series has a basis contained in {Ct }, and (2), if t ∈ Std(λ) then Ct = ψt plus a linear combination of “higher terms” with respect to some total order on Std(λ). This defines a partition of Std(λ) = T1 t · · · t Tz (disjoint union), where the tableaux in the set Tk are in bijection with a basis of the kth composition factor. Therefore, there exists an equivalence relation on Std(λ), together with an associated composition series, such that the analogue of Conjecture 4.4.1 holds for this equivalence relation. Our conjecture attempts to make this equivalence relation on Std(λ) explicit and canonical. P If T ⊆ Std(λ) define its character to be chq T = t∈T q deg t · it ∈ A[I n ]. The point of this definition is that chq T is a purely combinatorial invariant of T . As two examples, Chq S λ = chq Std(λ) and Chq STλ = chq T. Proposition 4.4.2. Suppose that Conjecture 4.4.1 holds when F = C. µ µ a) Suppose that µ ∈ KnΛ . Then DCµ ∼ = STµ and Chq DC = chq Tµ . Λ b) If λ ∈ Pn and T ∈ Std[λ] then there is a unique pair (ν T , dT ) in KnΛ × N such that chq T = q dT Chq DCν T = q dT chq Tν T . Moreover, X dλµ (q) = q dT . T∈Std[λ] ν T =µ

6 0 since µ ∈ KnΛ . The irreducible Proof. By Corollary 3.2.7, DCµ = µ µ module DC is generated by Btµ + rad SCµ = ψtµ + rad SCµ , so DCµ ∼ = ST µ since both modules are irreducible by Conjecture 4.4.1. Hence, (a) follows. For part (b), STλ ∼ = DCν hdi, for some ν ∈ KnΛ and d ∈ Z, because STλ is irreducible by Conjecture 4.4.1. Therefore, Chq STλ = q d Chq DCν . The uniqueness of (ν T , dT ) = (ν, d) ∈ KnΛ × Z now follows from Theorem 3.7.1. Moreover, d ≥ 0 by Corollary 3.5.27. As every composition factor of SCλ is isomorphic to STλ , for some T ∈ Std[λ], the formula for dλµ (q) is now immediate.  Proposition 4.4.2 shows that Conjecture 4.4.1 encodes closed formulas for the characters and graded dimensions of the irreducible HnΛ -modules and for the graded decomposition numbers of HnΛ . For this result to be useful we need to first verify Conjecture 4.4.1 and then to explicitly determine the equivalence relation ∼B . Our last result is a step in this direction. Lemma 4.4.3. Suppose that s, t ∈ Std(λ) and that t = s(r, r + 1) such that isr+1 6= isr ± 1, where 1 ≤ r < n and λ ∈ PnΛ . Then s ∼B t.

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Proof. By assumption, either s B t or t B s. Without loss of generality we assume that s B t. It follows from (4.3.5), and Theorem 3.6.2, that Bs ψr = ψt +

X u

au ψu = Bt +

X

bu B u ,

u

where au , bu ∈ F are non-zero only if `(d(u)) < `(d(s)). Therefore, s B t. If isr+1 6= itr then e(is )ψr2 = e(is ) by (2.2.3), so s ∼B t. Now consider the more interesting case when isr+1 = isr or, equivalently, isr = itr . Using (2.2.2), Bt yr+1 = Bs ψr −

X u

X
bu Bu yr+1 = Bs (yr ψr + 1) − bu Bu yr+1 . u

In view of Proposition 3.2.10(c), Bs appears on the right-hand side with coefficient 1. Hence, t B s implying that s ∼B t as claimed.  Finally, we remark that it is easy to check that Conjecture 4.4.1 is true in the trivial cases considered in Example 3.7.7 and Example 3.7.8. With considerably more effort, using [26, Lemma 9.7] and results of [55, Appendix], it is possible to verify the conjecture when Λ ∈ P + is a weight of level 2 and e > n. In all of these cases, the conjecture can be checked because Bst = ψst , for all (s, t) ∈ Std2 (PnΛ ). The B-basis, and hence Conjecture 4.4.1 and all of the results in this section (except that in positive characteristic we can only say that dT ∈ Z in Proposition 4.4.2, rather than dT ∈ N), make sense over any field. We restrict our conjecture to fields of characteristic zero because it would be foolhardy to venture into the realms of positive characteristic without strong evidence. This said, whether or not our conjecture for the B-basis is true, we are convinced that, in all characteristics, there exists a “canonical” graded cellular basis {Cst } of RnΛ such that the analogous version of Conjecture 4.4.1 holds for the ∼C equivalence classes. To put it another way, the results of [57, §3.3] show that the KLRtableau combinatorics is rich enough to give closed combinatorial formulas for both the graded decomposition numbers and the graded dimensions of the irreducible representations of HnΛ . We believe that over any field the graded Specht modules have a distinguished homogeneous basis that “canonically” determines these combinatorial formulas.

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