Representations of Hecke Algebras at Roots of Unity
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Algebra and Applications Volume 15
Series Editors: Alice Fialowski Eötvös Loránd University, Hungary Eric Friedlander Northwestern University, USA John Greenlees Sheffield University, UK Gerhard Hiß Aachen University, Germany Ieke Moerdijk Utrecht University, The Netherlands Idun Reiten Norwegian University of Science and Technology, Norway Christoph Schweigert Hamburg University, Germany Mina Teicher Bar-llan University, Israel Alain Verschoren University of Antwerp, Belgium Algebra and Applications aims to publish well written and carefully refereed monographs with up-to-date information about progress in all fields of algebra, its classical impact on commutative and noncommutative algebraic and differential geometry, K-theory and algebraic topology, as well as applications in related domains, such as number theory, homotopy and (co)homology theory, physics and discrete mathematics. Particular emphasis will be put on state-of-the-art topics such as rings of differential operators, Lie algebras and super-algebras, group rings and algebras, C∗ -algebras, Kac-Moody theory, arithmetic algebraic geometry, Hopf algebras and quantum groups, as well as their applications. In addition, Algebra and Applications will also publish monographs dedicated to computational aspects of these topics as well as algebraic and geometric methods in computer science.
Meinolf Geck r Nicolas Jacon
Representations of Hecke Algebras at Roots of Unity
Meinolf Geck University of Aberdeen Institute of Mathematics Aberdeen AB24 3UE UK
[email protected]
Nicolas Jacon Université de Franche-Comté UFR Sciences et Techniques Route de Gray 16 Besancon 25030 France
[email protected]
ISBN 978-0-85729-715-0 e-ISBN 978-0-85729-716-7 DOI 10.1007/978-0-85729-716-7 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2011929782 Mathematics Subject Classification (2010): 20C08, 20C20, 20C33, 20F55, 20G42, 17B37 © Springer-Verlag London Limited 2011 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
One of the major open problems in the representation theory of finite groups is the determination of the irreducible representations of the symmetric group Sn over a field of characteristic p > 0. Thanks to the work of James [179] in the 1970s, we do have a natural parametrisation of the irreducible representations in the framework of the theory of Specht modules, but explicit combinatorial formulae for their dimensions are not known in general! Note that the analogous problem in characteristic 0 has been solved for a long time, by the work of Frobenius around 1900. In a wider context, this problem is a special case of the problem of determining the irreducible representations of Iwahori–Hecke algebras. These algebras arise naturally in the representation theory of finite groups of Lie type, but they can also be defined abstractly as certain deformations of group algebras of finite Coxeter groups, where the deformation depends on one or several parameters. For the purposes of this introduction, let us assume that all the parameters are integral powers of a fixed element in the base field. If this base parameter has infinite order and the base field has characteristic 0, then we are in the “generic case” where the algebras are semisimple; this case is quite well understood [132], [231]. Also note that, both for historical reasons and as far as applications are concerned, the case where all parameters are equal is particularly important. The main focus in this text will be on the “modular case” where the algebras are non-semisimple. This situation typically occurs over fields of positive characteristic (a familiar phenomenon from the representation theory of finite groups), but it also occurs over fields of characteristic 0 when the base parameter is a root of unity. While leading to a highly interesting and rich theory in its own right, it turns out that the study of the characteristic 0 situation also provides a crucial step for understanding the positive characteristic case, which is most important for applications to finite groups of Lie type. Over the last two decades, there has been considerable progress on the characteristic 0 situation. One of the most spectacular advances is the “LLT conjecture” [208] (where “LLT” stands for Lascoux, Leclerc, Thibon) and its proof by Ariki [7], [10]. This brings deep geometric methods and the combinatorics of crystal/canonical bases of quantum groups into the picture, opening the way for a variety v
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of new theoretical connections and practical applications. Combined with sophisticated computational methods, the theory has now reached the following state: • The classical theory of “Specht modules” has been generalised to Iwahori– Hecke algebras associated to arbitrary finite Coxeter groups, giving rise to natural parametrisations of the irreducible representations. • Explicit descriptions of these parametrisations are now known in terms of socalled “canonical basic sets”. Also, the dimensions of the irreducible representations are known, either by purely combinatorial algorithms (for the classical types) or in the form of explicit tables (for the exceptional types). These results remain valid over fields of characteristic p > 0, as soon as p is larger than some bound depending on the type of the algebra. As far as the parametrisation of the irreducible representations is concerned, the bound is very mild. For example, in the equal-parameter case, it will turn out that it is sufficient to assume that the characteristic is “good” in the sense of the theory of algebraic groups. However, as far as the dimensions of the irreducible representations are concerned, no explicit bound on p is known at the present state of knowledge. But there is a general conjecture – first formulated by James [181] in type A – specifying such a bound. This conjecture has been verified in a number of cases, including algebras of type An for n 9 (see [181]) and all algebras of exceptional type (see Geck, Lux, and Müller [94], [126], [129]). If true, this conjecture would also yield explicit results about the dimensions of the irreducible representations of the symmetric group Sn in characteristic p > 0 where p is such that p2 > n. The purpose of this book is to develop the general theory along the above lines and to show how it is transformed into explicit results. In a sense, this book tries to do for representations of Iwahori–Hecke algebras at roots of unity what the book by Geck and Pfeiffer [132] did for the “generic case”. However, while [132] was essentially self-contained, the situation is more complex here. In fact, in order to obtain our main results, we rely on the following sources: • Ariki’s proof [7] of the LLT conjecture. • Certain deep properties of Kazhdan–Lusztig cells [222], [231] which do not seem to be accessible by elementary methods. • The existence and basic properties of “canonical bases” and “crystals” for the Fock space representations of certain quantised enveloping algebras. The first two ultimately rely on deep geometric theories, an exposition of which would go far beyond the scope of this text. Fortunately, this material is now more readily accessible through a number of books; for example, Kirwan [201], Chriss and Ginzburg [50], Hotta et al. [159], Kiehl and Weissauer [197]. Also note that the geometry only plays a role in the proofs, but not in the formulation of the results! (It is not completely impossible that, some day, more direct and purely algebraic proofs will be found.) Much of what we need about crystal and canonical bases can be found in Ariki’s book [10]; see also Jantzen [185], Kashiwara [191], Lusztig [230]. Our general policy regarding these topics is that we shall introduce the required notation to state the results that we need, but we will not endeavour to give the
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proofs. In this way, we can keep the size of this text within reasonable limits, and yet present some substantial results and applications. The origin of the theory of Iwahori–Hecke algebras lies in the representation theory of finite groups of Lie type, where these algebras arise as endomorphism algebras of certain induced representations. Via some natural functors, a well-defined part of the representation theory of a finite group of Lie type is controlled by the representation theory of Iwahori–Hecke algebras. Thus, the theory and the results that we are going to present in this book form a contribution to the general project of determining the irreducible representations of all non-abelian finite simple groups. Note that such a group is either an alternating group of degree at least 5, or a simple group of Lie type, or one of 26 sporadic simple groups; see Gorenstein et al. [142]. A rough outline of the contents of this book now follows. Chapters 1 and 2 provide a general introduction to the representation theory of Iwahori–Hecke algebras and, thus, may be of some independent interest. The discussion will be based on the Kazhdan–Lusztig theory of “cells” [195], [219]. In Lusztig’s work [220] on characters of reductive groups over finite fields, a crucial role is played by the “a-function”, which associates with every irreducible representation E of a finite Coxeter group a numerical invariant aE . One of the main themes of this book will be to show that these invariants play a similarly important role for “modular” representations. In Theorem 2.6.12, this culminates in the construction of a “cell datum” in the sense of Graham and Lehrer [144], giving rise to a general theory of “Specht modules” for Iwahori–Hecke algebras. (These results originally appeared in [111], [112].) Thus, we now see that the original Specht module theory in type A, due to Dipper and James [62] and Murphy [256], [257] (see also the exposition by Mathas [245]), is the prototype of a picture which applies to all Iwahori–Hecke algebras associated with finite Coxeter groups. In our exposition, we pay a particular attention to treating Iwahori–Hecke algebras of type A as a model case. The required results on Kazhdan–Lusztig cells will be established in a complete and self-contained manner, where no use of geometry is required; see Section 2.8. This treatment of type A is new and entirely independent of the original approach by Dipper, James, and Murphy. In Chapter 3, we study non-semisimple Iwahori–Hecke algebras in the spirit of Brauer’s classical “modular representation theory” involving, in particular, blocks and decomposition numbers. We shall assume that the reader has some familiarity which the basic features of this theory (for a general finite-dimensional associative algebra); this is readily accessible in standard reference texts, like Curtis and Reiner [53] and Feit [83]. In this setting, we define the key concept of a “canonical basic set” in Section 3.2. This concept is independent of the existence of a Graham–Lehrer cell datum, but, in a sense, it captures precisely those features of a cell datum which can be seen by looking only at the decomposition matrix of the algebra. Again, we treat Iwahori–Hecke algebras of type A as a model case. In Section 3.5 we give a new proof of the classification of the modular irreducible representations of these algebras. For this purpose, we have found it convenient to introduce the formal concept of an “abstract Fock datum” in Section 3.4. In another direction, we present a factorisation result for decomposition matrices and formulate a general version of
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James’s conjecture. The exposition in Section 3.7 unifies the original formulation of James [181] with the further developments in [92], [98], [129], [133]. In Chapter 4 we explain the fundamental connection between Iwahori–Hecke algebras and representations of a finite group of Lie type G(Fq ) (where q is a power of a prime number and Fq denotes a finite field with q elements). We begin with a self-contained discussion of the Schur functor and its variations, where we combine the original approach of Dipper [58] with later developments by Cline et al. [51] and Schubert [279]. Following [109], we then show in Theorem 4.4.1 how our results on “cell data” and “canonical basic sets” lead to a natural parametrisation of the modular irreducible representations of G(Fq ) which admit non-zero vectors fixed under a Borel subgroup. This generalises classical results from the characteristic 0 situation (due to Bourbaki, Iwahori, Tits, . . .) to positive characteristic. We also explain how this fits into a (conjectural) classification of all irreducible representations of G(Fq ) in the “non-defining characteristic case”. The determination of canonical basic sets for the classical types Bn and Dn has turned out to be an extremely difficult problem. At the end of Chapter 4 we shall discuss some cases that can be dealt with by elementary methods, based on the work of Dipper, James, and Murphy [66], [68]. The solution in the general case requires completely new methods; this will be achieved as a consequence of the results presented in Chapters 5 and 6. For this purpose, it will be convenient to work in the framework of the theory of Ariki–Koike algebras, which are generalisations of Iwahori–Hecke algebras of type Bn . The main idea of Chapter 5 is to try to generalise as much as possible the combinatorial constructions involved in the discussion of type A in Chapter 3. This leads us to consider in Section 5.7 certain special choices of the parameters which arise from the combinatorics of “FLOTW multipartitions” (where FLOTW stands for Foda, Leclerc, Okado, Thibon, Welsh [88]); these special choices cover, in particular, the equal parameter case for Iwahori–Hecke algebras of type Bn and Dn . As a consequence, in Theorem 5.8.2, we can state the main result concerning the determination of canonical basic sets for this choice of parameters. The methods in Chapter 5 do not allow us to complete the proof of this theorem. The missing piece is a result about the number of irreducible representations of Ariki–Koike algebras which is due to Ariki and Mathas [15] and which relies on the deep work of Ariki [7] on the proof of the LLT conjecture. This will be discussed in Chapter 6. The idea that FLOTW multipartitions are relevant in the modular representation theory of Iwahori–Hecke algebras of classical type first appeared in the work of Jacon [172], [173], [174]. Originally, the base field for the algebras was assumed to be of characteristic 0. The new approach developed in Chapter 5 shows that these results also hold for fields of positive characteristic. e ) and study In Chapter 6 we introduce the quantised enveloping algebra Uq (sl the canonical bases of certain Fock space representations. The associated “crystals” carry some rich combinatorial structure which will be discussed in detail. We can state (without proof) Ariki’s theorem [7] which links the canonical bases of the Fock space representations to the irreducible representations of Ariki–Koike algebras at roots of unity. This allows us to complete the proofs of the main results of the previ-
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ous chapter; see Section 6.3. Since this only covers certain choices of the parameters for Ariki–Koike algebras, we then go further and present some deep results of Uglov [291] on the canonical bases of the Fock space representations. We show how this leads to an explicit description of the “canonical basic sets” for Ariki–Koike algebras at roots of unity – and, hence, of Iwahori–Hecke algebras of classical type – for any choice of the parameters, assuming that the base field is of characteristic 0; see Theorem 6.7.2. We also derive purely combinatorial algorithms for computing decomposition numbers and the dimensions of the irreducible representations (in characterictic 0). Finally, Chapter 7 contains explicit results concerning Iwahori–Hecke algebras of exceptional type H3 , H4 , F4 , E6 , E7 , E8 . We also explain some basic algorithmic methods, including Parker’s M EATA XE. The project of computing the decomposition matrices for these algebras (over fields of characteristic 0) was started almost 20 years ago in [126] and finally completed in [129]; the matrices for type E8 appear here for the first time in print. From these matrices, one can simply read off the corresponding “canonical basic sets”. Acknowledgments. While the idea of writing this book has been around for some time, the actual work began in 2008. Much of the writing of the first chapters was done while the first author enjoyed the hospitality of the Newton Institute (Cambridge, UK) during the special semester on algebraic Lie theory (January to June 2009). We thank the anonymous referees for detailed comments on early versions of the manuscript which, we hope, led to considerable improvements. We are indebted to Gunter Malle, who carefully read all chapters and made numerous and detailed comments, and Cédric Lecouvey for his comments on Chapters 5 and 6. Finally, we wish to thank Springer Verlag for their support and their patiently accepting our repeated excuses for delaying the delivery of the final manuscript. Aberdeen and Besançon, March 2011
Meinolf Geck Nicolas Jacon
Contents
1
Generic Iwahori–Hecke Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Coxeter Groups and Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Representations of H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lusztig’s a-Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Balanced Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Asymptotic Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Introducing Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Examples of Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Cells and Leading Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 2 7 13 20 28 36 43 52
2
Kazhdan–Lusztig Cells and Cellular Bases . . . . . . . . . . . . . . . . . . . . . . . . 59 2.1 The Kazhdan–Lusztig Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.2 A Pre-order Relation on Irr(W ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.3 On Lusztig’s Conjectures, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4 On Lusztig’s Conjectures, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.5 On Lusztig’s Conjectures, III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.6 A Cellular Basis for H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.7 Further Properties of the Cellular Basis of H . . . . . . . . . . . . . . . . . . . . 114 2.8 The Case of the Symmetric Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3
Specialisations and Decomposition Maps . . . . . . . . . . . . . . . . . . . . . . . . . 133 3.1 Grothendieck Groups and Decomposition Maps . . . . . . . . . . . . . . . . . 134 3.2 Canonical Basic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.3 Principal Specialisations and Blocks of Defect 1 . . . . . . . . . . . . . . . . . 153 3.4 Towards Canonical Basic Sets for Classical Types . . . . . . . . . . . . . . . 162 3.5 A Canonical Basic Set for the Symmetric Group . . . . . . . . . . . . . . . . . 171 3.6 Factorisation of Decomposition Maps . . . . . . . . . . . . . . . . . . . . . . . . . 181 3.7 The General Version of James’s Conjecture . . . . . . . . . . . . . . . . . . . . . 190 3.8 Blocks and Bad Specialisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
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4
Hecke Algebras and Finite Groups of Lie Type . . . . . . . . . . . . . . . . . . . . 207 4.1 The Schur Functor and its Variations . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.2 Hom Functors and Harish-Chandra Series . . . . . . . . . . . . . . . . . . . . . . 215 4.3 Unipotent Principal Series Representations, I . . . . . . . . . . . . . . . . . . . 223 4.4 Unipotent Principal Series Representations, II . . . . . . . . . . . . . . . . . . . 232 4.5 Examples and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 4.6 A First Approach to Type Bn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5
Representation Theory of Ariki–Koike Algebras . . . . . . . . . . . . . . . . . . . 261 5.1 The Complex Reflection Group of Type G(l, 1, n) . . . . . . . . . . . . . . . . 262 5.2 Basic Properties of Ariki–Koike Algebras . . . . . . . . . . . . . . . . . . . . . . 266 5.3 Ariki–Koike Algebras as Cellular Algebras . . . . . . . . . . . . . . . . . . . . . 270 5.4 Decomposition Maps for Ariki–Koike Algebras . . . . . . . . . . . . . . . . . 274 5.5 Cyclotomic Ariki–Koike Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 5.6 A Fock Datum for Ariki–Koike Algebras . . . . . . . . . . . . . . . . . . . . . . . 284 5.7 FLOTW Multipartitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 5.8 On Basic Sets for Ariki–Koike Algebras . . . . . . . . . . . . . . . . . . . . . . . 302
6
Canonical Bases in Affine Type A and Ariki’s Theorem . . . . . . . . . . . . . 309 e ) . . . . . . . . . . . . . . . . . . . . . . . . . 310 6.1 The Quantum Affine Algebra Uq (sl 6.2 The Fock Space and Ariki’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 316 6.3 Crystal and Canonical Bases for Highest Weight Modules . . . . . . . . . 325 6.4 Computing Decomposition Matrices of Ariki–Koike Algebras . . . . . 328 6.5 Uglov’s Theory of Fock Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 6.6 Canonical Bases for Fock Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 6.7 Computation of Canonical Basic Sets for Iwahori–Hecke Algebras . 354 6.8 Recent Developments and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 359
7
Decomposition Numbers for Exceptional Types . . . . . . . . . . . . . . . . . . . . 363 7.1 Algorithmic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 7.2 Decomposition Matrices for W of Dihedral Type . . . . . . . . . . . . . . . . 371 7.3 Decomposition Matrices for W of Type F4 . . . . . . . . . . . . . . . . . . . . . . 374 7.4 Decomposition Matrices for Types H3 , H4 , E6 , E7 , E8 . . . . . . . . . . . . 379
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Chapter 1
Generic Iwahori–Hecke Algebras
In this chapter we introduce the main objects of our study: Finite Coxeter groups, generic Iwahori–Hecke algebras, and their representations. The groups and algebras are defined in a purely algebraic way, in terms of generators and defining relations. Thus, a generic Iwahori–Hecke algebra H is seen to be a deformation of the group algebra of a Coxeter group W , where the deformation depends on one or several parameters. We recall the relevant definitions and basic results in Sections 1.1 and 1.2, but we tacitly assume that the reader already has some familiarity with them. Following Lusztig [219], [231], we specify the parameters of H by a “weight function” L : W → Γ , where Γ is a totally ordered abelian group. The typical and most familiar example is the case where Γ = Z, with its natural order. More general choices for Γ are useful for several reasons: first of all, this provides the greatest level of generality and flexibility; furthermore, it brings out more clearly the role that is played by the given total order on Γ . In Section 1.3, the total order is used to define Lusztig’s a-function, which associates with every irreducible representation E ∈ IrrC (W ) an element aE ∈ Γ . The construction relies in an essential way on the generic algebra H and the known connection (via Tits’s deformation theorem) between the irreducible representations of W and those of H. The explicit results summarized at the end of Section 1.3 show the remarkable dependence of the function E → aE on the total order of Γ . The study of the a-function, and its subtle relation with the Kazhdan–Lusztig basis of H, will be one of the main themes of this book. As a first step we will introduce in Section 1.5 an “asymptotic” version of H. Our construction, following [112], is logically independent of Lusztig’s construction of the asymptotic ring J in [223], [231, Chap. 18], but it is, of course, motivated by it. The advantage of our approach is that it does not rely on certain deep properties of the Kazhdan–Lusztig basis of H which are not (yet) known to hold in the general multiparameter case. Instead, we rely on properties of the “balanced representations” in Section 1.4. In Section 1.6, using the asymptotic version of H, we can then give a first definition of the partition of W into left, right, and two-sided “cells” and establish some basic properties of them. This is followed by the discussion of a number of examples and further results in Sections 1.7 and 1.8. M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7 1, © Springer-Verlag London Limited 2011
1
2
1 Generic Iwahori–Hecke Algebras
1.1 Coxeter Groups and Weight Functions We briefly recall the basic definitions concerning Coxeter groups and Iwahori– Hecke algebras. More details and references can be found in [29], [132], [231]. 1.1.1. Let S be a finite set and M = (mst )s,t∈S be a matrix satisfying mss = 1 for all s ∈ S, and mst = mts ∈ {2, 3, 4, 5, . . .} ∪ {∞} for all s = t in S. Such a matrix is called a Coxeter matrix. Then we define a group W = W (M) by a presentation with generators S and defining relations as follows: • s2 = 1 for all s ∈ S; • (st)mst = 1 for all s = t in S with mst < ∞. The pair (W, S) is called a Coxeter system and W is called a Coxeter group. We encode the above presentation in a graph, called the Coxeter graph of W . It has vertices labelled by the elements of S, and two vertices labelled by s = t are joined by an edge if mst 3. Moreover, if mst 4, we label the edge by mst . If the graph is connected, we say that W is an irreducible Coxeter group. If this is not the case, we have a direct product decomposition W = W1 × · · · ×Wd , where Wi = Si and each subset Si ⊆ S corresponds to the vertices in a connected component of the Coxeter graph; furthermore, each Wi is a Coxeter group with generating set Si . The groups Wi will be called the irreducible components of W . Table 1.1 Coxeter graphs of irreducible finite Coxeter groups An−1 n2
Bn n2
I2 (m) m3
H3
E7
0
1
t
2
t
p p p
n−1
1
2
p p p
n−1
t4 t
1
t
2
tm t
1
t
F4
t
Dn
t
n3
t1 @ 2 @t
t
5
1
t
2
t
3
3
4
t
t t
H4
2
3
t4 t
5
t
6
t
1
t
t
p p p
3
4
n−1
t
t0 4
t E6
1
3
5
7
t
2
t
3
t
E8
1
t
t
4
t
1
t
t
5
t
6
7
8
t2 3
t
t2
4
t
5
t
6
t
t
t2
(The numbers on the vertices correspond to a chosen labelling of the elements of S.) Type Order
An−1 n!
Bn 2n n!
Dn 2n−1 n!
I2 (m) 2m
Type Order
H3 120
H4 14400
F4 1152
E6 51840
t
E7 2903040
E8 696729600
t
1.1 Coxeter Groups and Weight Functions
3
There is a complete classification of the finite Coxeter groups. The graphs corresponding to the irreducible finite Coxeter groups, and the group orders, are given in Table 1.1. We say that W is a finite Weyl group or is of crystallographic type if mst ∈ {2, 3, 4, 6}. These are precisely the finite Coxeter groups which arise, for example, in the theory of finite-dimensional semisimple complex Lie algebras, or in the theory of connnected reductive algebraic groups (see also Chapter 4). The standard example is type An−1 , where W can be identified with the symmetric group Sn , generated by the basic transpositions si = (i, i + 1) for 1 i n − 1. This is the Weyl group for the simple Lie algebra sln (C) of n × n-matrices with trace 0, or for the simple algebraic group SLn (k) of n × n-matrices with determinant 1 over any algebraically closed field k. The groups of type H3 , H4 or I2 (m) (m = 5 or m > 7) are non-crystallographic. 1.1.2. Let k be any commutative ring (with 1) and {ξs | s ∈ S} ⊆ k× be a collection of elements such that ξs = ξt whenever s,t ∈ S are conjugate in W . Then, by Bourbaki [29, Chap. IV, §2, Exc. 23], we have a corresponding Iwahori–Hecke algebra Hk = Hk (W, S, {ξs }). This is an associative algebra over k which is free as a k-module, with basis {Tw | w ∈ W }; the multiplication is uniquely determined by the rule Tsw if l(sw) > l(w), Ts Tw = Tsw + (ξs − ξs−1 )Tw if l(sw) < l(w), where s ∈ S and w ∈ W . Here, l : W → Z0 is the length function on W . Recall that, given w ∈ W , we can write w = s1 · · · s p , where si ∈ S. If p is minimal with this property, we say that this is a reduced expression for w; then l(w) = p is called the length of w. In this case, the above rules imply that Tw = Ts1 · · · Ts p . We note that T1 is the identity element of Hk . The elements {ξs } are called the parameters of Hk . We also remark that if W = W1 × · · · × Wd is the decomposition into irreducible components (where Wi = Si as above), then we have Hk ∼ = Hk (W1 , S1 , {ξs }s∈S1 ) ⊗k · · · ⊗k Hk (Wd , Sd , {ξs }s∈Sd ); see [132, Exc. 8.4]. In this way, many questions about Iwahori–Hecke algebras in general can be reduced to the case where (W, S) is irreducible. Example 1.1.3. (a) Assume that ξs = 1 for all s ∈ S. Then the map w → Tw defines an isomorphism of k-algebras from kW (the group algebra of W over k) onto Hk . (b) Assume that ξs = ξt for all s,t ∈ S; this case will be referred to as the equalparameter case. We are automatically in this case when W is of type An−1 , Dn , I2 (m) (m odd), H3 , H4 , E6 , E7 or E8 (since all generators in S are conjugate in W ). (c) Assume that W is finite and irreducible. Then unequal parameters can only occur in types Bn , F4 or I2 (m) (m even). In these cases, the set S falls into two classes under conjugation by W ; see also Example 1.1.11(b) below. 1.1.4. The purpose of this book is to address the following problem.
4
1 Generic Iwahori–Hecke Algebras
Fundamental Problem. Assume that W is finite and k is a field. Then determine the irreducible representations of Hk (W, S, {ξs }). By Example 1.1.3, this includes the problem of determining the irreducible representations of the symmetric group Sn ∼ = W (An−1 ) over fields of positive characteristic. Note that there are many open questions even in this special case; in particular, the dimensions of the irreducible representations are not known! If Hk (W, S, {ξs }) is semisimple, then the above problem is essentially solved; see [132], [231]. So the main focus in this text will be on the non-semisimple situation. The first step consists of noting that any algebra Hk as above can be obtained from a suitable “generic” Iwahori–Hecke algebra by a process of specialisation. For this purpose, we introduce the following notion where, following a suggestion of Bonnaf´e [21], we combine the two settings in [219], [231]. Definition 1.1.5 (Lusztig). Let Γ be an abelian group (written additively). We say that a function L : W → Γ is a weight function if the following condition holds: L(ww ) = L(w) + L(w ) for all w, w ∈ W such that l(ww ) = l(w) + l(w ). Note that L is uniquely determined by the values {L(s) | s ∈ S}. Furthermore, if {cs | s ∈ S} is a collection of elements in Γ such that cs = ct whenever s,t ∈ S are conjugate in W , then there is a (unique) weight function L : W → Γ such that L(s) = cs for all s ∈ S. (This follows from Matsumoto’s lemma; see [132, §1.2].) 1.1.6. Let us assume that a weight function L : W → Γ has been fixed. Let R ⊆ C be a subring and A = R[Γ ] be the free R-module with basis {ε g | g ∈ Γ }. There is a well-defined ring structure on A such that ε g ε g = ε g+g for all g, g ∈ Γ . We write 1 = ε 0 ∈ A. Given a ∈ A we denote by ag the coefficient of ε g , so that a = ∑g∈Γ ag ε g . We apply the general construction in 1.1.2 to the ring A and the collection of elements {vs | s ∈ S} where vs := ε L(s) for s ∈ S. The corresponding algebra will be denoted by H = HA (W, S, L) and called the generic Iwahori–Hecke algebra associated with W, L. Thus, H is an associative algebra which is free as an A-module, with basis {Tw | w ∈ W }; the multiplication is given by Tsw if l(sw) > l(w), Ts Tw = )T if l(sw) < l(w), Tsw + (vs − v−1 w s where s ∈ S and w ∈ W . The element T1 is the identity element. In the setting of 1.1.2, assume that there is a ring homomorphism θ : A → k such that θ (vs ) = ξs for all s ∈ S. Then we can regard k as an A-module (via θ ), and we find that Hk is obtained by extension of scalars from H: Hk (W, S, {ξs }) ∼ = k ⊗A HA (W, S, L). In this situation, we say that θ : A → k is a specialisation and that Hk (W, S, {ξs }) is obtained from H by specialisation (via θ ). For example, if θ1 : A → k is a ring homomorphism such that θ1 (ε g ) = 1 for all g ∈ Γ , then k ⊗A HA (W, S, L) ∼ = kW .
1.1 Coxeter Groups and Weight Functions
5
Example 1.1.7. Let Γ = Z. Then A = R[v, v−1 ] is the ring of Laurent polynomials over R in one indeterminate v := ε . Let L : W → Z be a weight function and set cs = L(s) for s ∈ S. Then the relations in H read as follows, where s ∈ S and w ∈ W : if l(sw) > l(w), Tsw Ts Tw = Tsw + (vcs − v−cs )Tw if l(sw) < l(w). This is the setting of Lusztig [231]; it is particularly relevant for applications to the representation theory of reductive groups over finite fields; see Section 4.2. Remark 1.1.8. For various applications, it will be convenient to set T˙w := ε L(w) Tw for all w ∈ W and us := v2s for all s ∈ S. Then, clearly, {T˙w | w ∈ W } also is an A-basis of H; furthermore, we have the multiplication rules: if l(sw) > l(w), T˙sw ˙ ˙ Ts Tw = us T˙sw + (us − 1)T˙w if l(sw) < l(w), where s ∈ S and w ∈ W . Thus, the introduction of the basis {T˙w | w ∈ W } shows that H is already defined over the subring Z[us | s ∈ S] ⊆ A; that is, all structure constants with respect to this basis lie in Z[us | s ∈ S]. Example 1.1.9. A “universal” weight function is given as follows. For s,t ∈ S, we write s ∼ t if s,t are conjugate in W . Let S ⊆ S be a set of representatives for the equivalence classes of S under this relation. Let Γ0 be the group of all tuples (ns )s∈S where ns ∈ Z for all s ∈ S . (The addition is defined componentwise.) Let L0 : W → Γ0 be the weight function given by sending s ∈ S to the tuple (nt )t∈S , where nt = 1 if t is conjugate to s and nt = 0 otherwise. Let A0 = R[Γ0 ] and H0 = HA0 (W, S, L0 ) be the associated Iwahori–Hecke algebra; we denote the parameters by {v◦s | s ∈ S} in this case. Note that A0 is nothing but the ring of Laurent polynomials in {v◦s | s ∈ S } (and these elements are algebraically independent). Any algebra Hk (W, S, {ξs }) as above is obtained by specialisation from H0 (where we take R = Z). Indeed, since {v◦s | s ∈ S } are algebraically independent, we can certainly find a unital ring homomorphism θ0 : A0 → k such that θ0 (v◦s ) = ξs for all s ∈ S. Thus, Hk (W, S, {ξs }) ∼ = k ⊗A0 H0 (via θ0 ). 1.1.10. As in [219], we shall assume that Γ admits a total ordering which is compatible with the group structure; that is, whenever g, g ∈ Γ are such that g g , we have g + h g + h for all h ∈ Γ . Such an order will be called a monomial order. We usually assume that L(s) 0 for all s ∈ S. (We will see in Lemma 1.1.12 below that this is no severe restriction.) The existence of a monomial order on Γ implies that Γ is torsion free. Furthermore, we can write any 0 = a ∈ A uniquely in the form a = a1 ε g1 + · · · + ad ε gd
where 0 = ai ∈ R, gi ∈ Γ and g1 < . . . < gd .
We denote lt(a) := a1 ε g1 and call this the leading term of a. We also set lt(0) := 0. Then one easily checks that lt(aa ) = lt(a)lt(a ) for any a, a ∈ A. In particular, if a = 0 and a = 0, then lt(aa ) = 0 and so aa = 0; hence, A is an integral domain.
6
1 Generic Iwahori–Hecke Algebras
Finally, since S is a finite set, it is usually sufficient to consider the case where Γ is finitely generated. Consequently, in this case, we have Γ ∼ = Zr for some r 1, which means that A is a ring of Laurent polynomials in r variables. Then specifying a monomial order on Γ amounts to specifying a total order on the monomials in A (compatible with the multiplication). Example 1.1.11. (a) In the set-up of Example 1.1.7, there is a natural monomial order on Γ = Z, and we will usually assume that cs 0 for all s ∈ S. (b) Assume that W is of type Bn , F4 or I2 (m) (m even). Then, in general, L depends on two values a, b ∈ Γ , which are attached to the generators in S: Bn
bt 4 at
I2 (m)
bt m at
m even
at
p p p F4
at
at at
4
bt
bt
The possible choices of monomial orders that are available here can best be seen by taking L = L0 to be the “universal” weight function in Example 1.1.9, where Γ0 = Z2 , b = (1, 0) and a = (0, 1). Then A0 = R[V ±1 , v±1 ] is the ring of Laurent polynomials in the indeterminates V := ε (1,0) and v := ε (0,1) . A familiar monomial order is the pure lexicographic order given by (i, j) lex (i , j )
def
⇔
i i
or
i = i and j j
(i, i , j, j ∈ Z).
More generally, for any α ∈ R, we have a monomial ordering α given by (i, j) α (i , j )
def
⇔
i + α j < i + α j or . i + α j = i + α j and j j
In particular, we see that there are infinitely many monomial orders on Γ0 . For a classification of all orderings on Γ0 , see Tutorial 10 in [206, §1.4] and also [271]. (c) Now assume that W is any finite Coxeter group and L : W → Γ is a weight function. Let be a monomial order on Γ . By analogy to Bonnaf´e and Iancu [26], we say that we are in the asymptotic case if L(s) > 0 for all s ∈ S and if, on any irreducible component of type Bn , F4 or I2 (m) (m even), where L takes values a, b ∈ Γ as above, we have b > ra > 0 for all r ∈ Z1 . As already mentioned above, we will usually assume that L(s) 0 for all s ∈ S. This is justified by the following result, observed by Bonnaf´e [22, Cor. 5.8]. Lemma 1.1.12. Let be a monomial order on Γ . For s ∈ S, set δs = 1 if L(s) 0, and δs = −1 if L(s) < 0. Then there is a well-defined weight function L : W → Γ such that L (s) = δs L(s) (s ∈ S); note that L (s) 0 for all s ∈ S. Let H be the generic Iwahori–Hecke algebra associated with W, L and let {Tw | w ∈ W } be the standard basis of H . Then there is a unique A-algebra isomorphism H → H such that Ts → δs Ts for all s ∈ S. Proof. To show that there is a weight function L as above, we need to check that δs = δt whenever s,t ∈ S are conjugate in W . But, if s,t ∈ S are conjugate, then
1.2 Representations of H
7
L(s) = L(t) and so δs , δt will either both be +1 or both be −1. Now let us show that there is an algebra isomorphism H → H as above. By [132, 4.4.5], the algebra H has a presentation with generators {Ts | s ∈ S} and defining relations Ts2 = T1 + (ε L(s) − ε −L(s) )Ts
(s ∈ S), (s,t ∈ S, s = t, mst < ∞).
T T T ··· = T T T ··· s ts t st mst terms
mst terms
Hence, all we need to check is whether the elements {δs Ts | s ∈ S} satisfy the above relations. Now, we have
(δs Ts )2 = Ts2 = T1 + (ε L (s) − ε −L (s) )Ts = T1 + (δs ε L (s) − δs ε −L (s) )(δs Ts ).
It remains to note that if δs = −1, then δs ε L (s) − δs ε −L (s) = −ε −L(s) + ε L(s) , as required. Now let us check the second type of relations. Let s = t in S be such that mst < ∞. Then the verification reduces to proving that δs δt δs · · · = δt δs δt · · · (with mst factors on both sides). If δs = δt , this is clear. Now assume that δs = δt . In particular, L(s) = L(t) in this case and so mst must be even. But then, on both sides of the above identity, we have mst /2 factors corresponding to s and mst /2 factors corresponding to t. Hence, we get the same result on both sides. Remark 1.1.13. Let θ : A → k be a specialisation into a field k and consider the specialised algebra Hk . Then, in the setting of Lemma 1.1.12, the algebras Hk and H k are isomorphic. Hence, if we have solved our “fundamental problem” in 1.1.4 for H k , then this problem is automatically solved for Hk as well. Thus, indeed, it will be sufficient to consider weight functions such that L(s) 0 for all s ∈ S.
1.2 Representations of H We will assume from now on that W is finite. Let L : W → Γ be a weight function and H be the associated generic Iwahori–Hecke algebra. Let us now turn to the representation theory of W and of H. 1.2.1. We set ZW := Z[2 cos(2π /mst ) | s,t ∈ S] ⊆ R. For example, ZW = Z if W is a finite Weyl group. We shall always assume that ZW ⊆ R (where R ⊆ C is the subring used to define A). Then the field of fractions of R, which will be denoted by K, is a splitting field for W ; see [132, Theorem 6.3.8]. Throughout, we use the following notation for the irreducible representations of W (up to isomorphism): IrrK (W ) = {E λ | λ ∈ Λ }, where Λ is a finite indexing set and each E λ is a K-vector space with a given KW module structure. We also use the notation dλ = dim E λ ,
M(λ ) = an indexing set for a basis of E λ
8
1 Generic Iwahori–Hecke Algebras
(where M(λ ) is ordered in some way so that it makes sense to write down matrices with rows and columns indexed by M(λ )). Finally, we assume that Γ admits a monomial ordering as in 1.1.10. As we have seen, this implies that A is an integral domain. Let K be the field of fractions of A; by extension of scalars, we obtain a finite-dimensional K-algebra HK = K ⊗A H which is known to be split semisimple; see [132, Theorem 9.3.5]. Let Irr(HK ) denote the set of irreducible representations of HK (up to isomorphism). By Tits’s deformation theorem, there is a bijection between this set and IrrK (W ); see [132, 8.1.7] and also Exercises 26 and 27 of Bourbaki [29, Chap. IV, §2]. Thus, we can write Irr(HK ) = {Eελ | λ ∈ Λ }
(dλ = dim Eελ ),
where each Eελ is a K-vector space with a given HK -module structure. The correspondence E λ ↔ Eελ is uniquely determined by the condition trace(w, E λ ) = θ1 trace(Tw , Eελ )
for all w ∈ W ,
where θ1 : A → R is the unique R-linear ring homomorphism such that θ1 (ε g ) = 1 for all g ∈ Γ . Note that, by [132, Theorem 9.3.5], we have
ε L(w) trace(Tw , Eελ ) ∈ ZW [vs | s ∈ S]
for all w ∈ W ;
in particular, these traces lie in A and so it makes sense to apply θ1 to them. It also follows that, for any HK -module V , we have trace(Tw ,V ) ∈ A for all w ∈ W . Remark 1.2.2. The proofs of the statements summarized in 1.2.1, especially the statements concerning splitting fields for W and HK , are by no means easy. In fact, various chapters of [132] are concerned with these questions, where case-by-case arguments (according to the classification of finite Coxeter groups) are required. A number of authors have contributed to the establishment of these results, over an extended period of time; see the bibliographic comments in [132, §5.7 and §9.5]. If one is mainly interested in finite Weyl groups and the equal-parameter case, then more conceptual arguments are available via the geometry of an associated algebraic group; see Springer [282] and Lusztig [216] (see also Example 2.5.7). Example 1.2.3. Let Cl(W ) be the set of conjugacy classes of W . For C ∈ Cl(W ), let wC ∈ C be a representative which has minimal length in C. Then the matrix (a) X(H) := trace(T˙wC , Eελ ) λ ∈Λ ,C∈Cl(W ) is called the character table of H, where we define T˙w := ε L(w) Tw for any w ∈ W , as in Remark 1.1.8. By a result due to Geck and Pfeiffer [132, 8.2.9], X(H) does not depend on the choice of the representatives {wC | C ∈ Cl(W )}; furthermore, there is a unique set of polynomials { fw,C | w ∈ W,C ∈ Cl(W )} ⊆ Z[v2s | s ∈ S] such that (b)
trace(T˙w , Eελ ) =
∑
C∈Cl(W )
fw,C trace(T˙wC , Eελ )
for any λ ∈ Λ and w ∈ W .
1.2 Representations of H
9
The tables X(H) are explicitly known for all W, L; see [132, Chap. 10 and 11] and the references there. When we apply the specialisation homomorphism θ1 : A → R to the entries of X(H), we obtain the classical character table of the finite group W . Example 1.2.4. We shall frequently apply the following “specialisation argument” for representations. Let V be an HK -module and V be a KW -module such that for all w ∈ W , (∗) trace(w,V ) = θ1 trace(Tw ,V ) where θ1 is defined as above; recall that KW ∼ = K ⊗A H, where K is regarded as an A-module via θ1 . For example, (∗) will certainly hold when V ∼ = K ⊗A M and V ∼ = K ⊗A M, where M is an H-module which is finitely generated and free over A. For any λ ∈ Λ , denote by m(V, λ ) the multiplicity of Eελ as an irreducible constituent of V , and denote by m(V , λ ) the multiplicity of E λ as an irreducible constituent of V . Thus, we have trace(Tw ,V ) =
∑ m(V, λ ) trace(Tw , Eελ )
for all w ∈ W ,
∑ m(V , λ ) trace(w, E λ )
for all w ∈ W .
λ ∈Λ
trace(w,V ) =
λ ∈Λ
Applying θ1 and using Tits’s deformation theorem, we obtain that
∑ m(V, λ ) trace(w, E λ ) = ∑ m(V , λ ) trace(w, E λ )
λ ∈Λ
λ ∈Λ
for all w ∈ W .
Since the trace functions associated with the irreducible representations of W are linearly independent, we deduce that m(V, λ ) = m(V , λ ) for all λ ∈ Λ . Example 1.2.5. It is known that every w ∈ W is conjugate to its inverse; see [132, 3.2.14]. Hence, we have trace(w, E λ ) = trace(w−1 , E λ ) for all λ ∈ Λ . A similar property holds on the level of HK ; that is, we have (a)
trace(Tw , Eελ ) = trace(Tw−1 , Eελ )
for all w ∈ W .
This is seen as follows. It is easily checked that the A-linear map h → h defined by Tw = Tw−1 (w ∈ W ) is an anti-involution of H. So we can define the contragredient module Eˆ ελ := HomK (Eελ , K) where Tw acts via Tw : ϕ → ϕ ◦ Tw−1 for ϕ ∈ Eˆελ . For any w ∈ W , we have trace(Tw , Eˆελ ) = trace(Tw−1 , Eελ ) and, hence, θ1 trace(Tw , Eˆελ ) = trace(w−1 , E λ ) = trace(w, E λ ) = θ1 trace(Tw , Eελ ) . By Tits’s deformation theorem, this implies that Eˆελ ∼ = Eελ and so (a) holds. Example 1.2.6. Let sgn denote the sign representation of W , which is given by the group homomorphism sending each w ∈ W to (−1)l(w) . Via tensoring with sgn, we obtain a bijection λ → λ † of Λ such that (a)
† Eλ ∼ = E λ ⊗ sgn
for all λ ∈ Λ .
10
1 Generic Iwahori–Hecke Algebras
This operation can be lifted to representations of HK . Namely, there is a unique A-algebra automorphism † : H → H such that Ts† = −Ts−1 for all s ∈ S; see [132, Exc. 8.2]. By extension of scalars, this induces a K-algebra automorphism of HK , which we denote by the same symbol. Given any finite-dimensional HK -module V , denote by V † the HK -module with the same underlying vector space V , but where h ∈ HK acts via h† . Then, by [132, Prop. 9.4.1], we have † (E λ )ε ∼ = (Eελ )†
(b)
for all λ ∈ Λ .
The trace of Tw on Eελ is determined as follows. There is a unique R-linear ring homomorphism A → A, a → a, ¯ such that ε g = ε −g for all g ∈ Γ . Then we have †
(c)
trace(Tw , Eελ ) = (−1)l(w) trace(Tw , Eελ ) †
for all w ∈ W ;
see [132, Prop. 9.4.1]. Example 1.2.7. Let w0 ∈ W be the longest element. Then Tw20 lies in the centre of HK and, hence, acts by a scalar in every irreducible representation of HK . This scalar can be explicitly described, as follows. Let T := {wsw−1 | s ∈ S, w ∈ W } be the set of all reflections in W . Let S ⊆ S be a set of representatives of the conjugacy classes of W which are contained in T . For s ∈ S , let Ns be the cardinality of the conjugacy class of s; thus, |T | = ∑s∈S Ns . Let ρ λ : HK → Mdλ (K) be a representation afforded by Eελ . Then, by an argument due to Springer (see [132, Theorem 9.2.2]), we have (a)
ρ λ (Tw20 ) = ε 2Nλ Idλ
where Nλ :=
N trace(s, E λ )
s L(s) ∈ Γ dim E λ s∈S
∑
and Idλ denotes the identity matrix of size dλ . Note that, since every s ∈ S has order 2, we have trace(s, E λ ) ∈ Z and so, by a well-known result in the character theory of finite groups, the quantity Ns trace(s, E λ )/ dim E λ also is an integer. Thus, the expression defining Nλ is a well-defined element of Γ . Now let us set P := ε −Nλ ρ λ (Tw0 ). Then P2 = Idλ and so P is a diagonalisable matrix with eigenvalues ±1; in particular, m := trace(P) ∈ Z. Thus, we obtain that trace(Tw0 , Eελ ) = mε Nλ . Applying the specialisation homomorphism θ1 : A → R, we conclude that m = trace(w0 , E λ ) and, hence, (b)
trace(Tw0 , Eελ ) = trace(w0 , E λ ) ε Nλ .
Thus, trace(Tw0 , Eελ ) is explicitly described in terms of character values of W . Remark 1.2.8. We have already remarked in 1.2.1 that trace(Tw , Eελ ) ∈ A for all w ∈ W . So it is natural to ask if it is even possible to find a representation ρ λ afforded by Eελ such that ρ λ (Tw ) ∈ Mdλ (A) for all w ∈ W . This is indeed the case, but the proof requires some deep results on the Kazhdan–Lusztig basis of H and a case-by-case analysis; see [132, 9.3.8] and the references there. (We will recover this result in the context of cellular algebras in Corollary 2.7.14.) However, we can establish a
1.2 Representations of H
11
weak version of this statement by a general argument, and this will be useful in our discussion of cells in Section 1.6. This relies on the following ring-theoretic result. Lemma 1.2.9 (Rouquier [272]). Assume that Γ = Z, so that A is the ring of Laurent polynomials in one indeterminate v = ε . Then the subring f −1 R := f ∈ Z[v, v ], g ∈ Z[v] and g has constant term 1 ⊆ K. g is a principal ideal domain. Proof. We need some standard results from commutative ring theory; a suitable reference is Matsumura [248]. Following Brou´e and Kim [33, §2.B], the first and crucial step is to show that R is a Dedekind domain. For this purpose, by [248, Theorem 11.6], it is enough to show that R is a one-dimensional noetherian domain which is integrally closed in Q(v) (its field of fractions). Now note that R ⊆ Q(v) is the localisation of Z[v, v−1 ] with respect to the multiplicatively closed set M = {g ∈ Z[v] | g has constant term 1}. But the ring Z[v, v−1 ] is known to be noetherian and integrally closed in Q(v). These properties pass on to localisations and, hence, R is noetherian and integrally closed in Q(v). In order to show that R is one-dimensional, we must show that every non-zero prime ideal of R is maximal. So let p be a non-zero prime ideal of R. By [248, Theorem 4.1], p is generated by a prime ideal I ⊆ Z[v, v−1 ] such that I ∩ M = ∅. The prime ideals in Z[v, v−1 ] are explicitly known (see, for example, [132, Exc. 7.9]). Thus, I is either principal (generated by a prime number in Z or by an irreducible polynomial in Z[v]) or generated by two elements and f where > 0 is a prime number and f ∈ Z[v] is a monic polynomial whose reduction modulo is an irreducible polynomial in Fl [v]. But ideals of the latter type have non-empty m intersection with M. Indeed, every irreducible polynomial in Fl [v] divides vl − v m for some m 1. Hence, we have vl − v ∈ I for some m 1. Since v is a unit in m Z[v, v−1 ], this implies that 1 − vl −1 ∈ I ∩ M and so I ∩ M is non-empty, as claimed. Consequently, all the non-zero prime ideals in R are principal. In particular, this shows that every non-zero prime ideal in R is maximal and, hence, R is a Dedekind domain. Finally, in a Dedekind domain, every non-zero ideal is a product of a finite number of prime ideals. We have just seen that every prime ideal in R is principal. Hence, every ideal in R is principal. Corollary 1.2.10. Assume that W is a Weyl group and Γ = Z. Then, for each λ ∈ Λ , there exists a representation π λ : HK → Mdλ (K) afforded by Eελ such that π λ (Tw ) ∈ Mdλ (R) for all w ∈ W . Proof. Since W is a Weyl group, we have ZW = Z and so we can take R = Z. Hence, K = Q(v) is the field of fractions of the ring R, which is a principal ideal domain by Lemma 1.2.9. But then a standard argument (see, for example, [132, 7.3.7]) shows that every irreducible representation of HK can be realised over R.
12
1 Generic Iwahori–Hecke Algebras
1.2.11. We define an A-linear map τ : H → A by τ (T1 ) = 1 and τ (Tw ) = 0 for 1 = w ∈ W . Then we have τ (Ty Tx ) = τ (Tx Ty ) = δx−1 y (Kronecker delta); see [132, 8.1.1]. So τ defines a non-degenerate symmetric bilinear form on H, where {Tw | w ∈ W } and {Tw−1 | w ∈ W } form a pair of dual bases. Thus, H is a symmetric algebra, with trace form τ . Let τK be the canonical extension to a trace form on HK . For λ ∈ Λ , let χ λ denote the character of Eελ ; we have χ λ (h) = trace(h, Eελ ) ∈ A for all h ∈ H. Since HK is split semisimple, the characters {χ λ | λ ∈ Λ } form a basis of the vector space of all trace functions on HK . Hence, by the general theory of symmetric algebras, there is a unique expression (see [132, 7.2.6]): (a)
τK =
∑
λ ∈Λ
λ c−1 λ χ
0 = cλ ∈ K.
where
The elements cλ were called Schur elements in [96]. By [132, Theorem 9.3.5], we have the following important integrality property: (b)
ε L(w0 ) cλ ∈ ZW [vs | s ∈ S]
for all λ ∈ Λ ;
In particular, cλ lies in A. The generic degree corresponding to E λ is defined by (c)
δλ := c−1 λ PW,L ∈ K
where
PW,L :=
∑ ε 2L(w) ,
w∈W
but note that this may no longer be an element of A. Choosing a basis of Eελ , indexed by M(λ ) as above, we obtain a matrix repreλ (h) sentation ρ λ : HK → Mdλ (K). Given h ∈ HK and s, t ∈ M(λ ), we denote by ρst λ the (s, t)-entry of ρ (h). We now have the following Schur relations. In particular, these yield an alternative characterisation of the elements cλ . Proposition 1.2.12 (Schur relations; cf. [132, 7.2.2]). Let {Bw | w ∈ W } be any basis of H and {B∨w | w ∈ W } the corresponding dual basis, such that τ (Bx B∨y ) = δxy for all x, y ∈ W . Given λ , μ ∈ Λ , let s, t ∈ M(λ ) and u, v ∈ M(μ ). Then cλ if λ = μ , s = v, t = u, μ λ ∨ ρ (B ) ρ (B ) = w uv w ∑ st 0 otherwise. w∈W In particular, this implies the orthogonality relations dλ cλ if λ = μ , λ μ ∨ χ (B ) χ (B ) = w ∑ w 0 otherwise. w∈W The origin of the definition of the elements cλ and of the generic degrees δλ lies in the representation theory of finite groups of Lie type. Without going into much detail at this stage, let us briefly describe this connection. 1.2.13. Let us assume that W is of crystallographic type and arises as the Weyl group of a family of finite groups of Lie type
1.3 Lusztig’s a-Invariants
13
S = {G(q) | q any prime power}. Assume that all groups G(q) are of “split type”. Some examples are given by W = W (An−1 ) W = W (Bn ) W = W (E8 )
where
G(q) = GLn (Fq ) for all q,
where where
G(q) = SO2n+1 (Fq ) for all q, G(q) = E8 (Fq ) for all q.
Let Γ = Z and the weight function L be such that L(s) = 1 for all s ∈ S. Then A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . In this situation, it is known that cλ divides PW,L in K[v, v−1 ] and that there is a polynomial Dλ ∈ Q[u] (where u is an indeterminate) such that δλ = Dλ (v2 ) ∈ Q[v]; see [132, 9.3.6]. This polynomial Dλ has the following interpretation. Given a prime power q, let us consider the complex irreducible representations of G(q). Let B(q) ⊆ G(q) be a Borel subgroup and define ρ occurs in the permutation , IrrC (G(q), B(q)) := ρ ∈ IrrC (G(q)) representation C[G(q)/B(q)] the set of (unipotent) principal series representations. Then, by classical results due to Iwahori, Tits, Benson and Curtis, there exists a bijection ∼
IrrC (W ) → IrrC (G(q), B(q)),
E λ → ρqλ ,
such that dim ρqλ = Dλ (q) for all λ ∈ Λ . (See [53, §68], [132, §8.4] or Section 4.3 in this book.) Thus, Dλ (q) is the dimension of an irreducible representation of G(q).
1.3 Lusztig’s a-Invariants In Lusztig’s work [220] on characters of reductive groups over finite fields, a crucial role is played by a certain function which attaches to each irreducible representation E λ of W an invariant aλ ∈ Z0 . In the situation of 1.2.13, this is defined by aλ := max{i 0 | ui divides Dλ }
for any λ ∈ Λ .
(See also 2.2.12 and 4.3.12 for further interpretations of aλ .) One of the main themes of this book will be to show that these a-invariants play a similarly important role for “modular” representations. Throughout this section, let W be a finite Coxeter group and L : W → Γ be a weight function. Let be a monomial order on Γ such that L(s) 0 for all s ∈ S. Let Γ0 := {g ∈ Γ | g 0} and denote by R[Γ0 ] the set of all R-linear combinations of terms ε g where g 0. The notations R[Γ>0 ], R[Γ0 ], R[Γ<0 ] have a similar meaning. We are now ready to introduce the invariants aλ in general.
14
1 Generic Iwahori–Hecke Algebras
Proposition 1.3.1 (Lusztig). Let λ ∈ Λ . Then there exist some aλ ∈ Γ0 and a strictly positive real number fλ ∈ ZW such that
ε 2aλ cλ ∈ R[Γ0 ] and ε 2aλ cλ ≡ fλ mod R[Γ>0 ]. Note that aλ and fλ are uniquely determined by these conditions. We have aλ = min{g ∈ Γ0 | ε g χ λ (Tw ) ∈ K[Γ0 ] for all w ∈ W }. Proof. (Cf. [217, 1.9].) For the following discussion, it will be useful to assume that K ⊆ R. By Example 1.2.5(a), we have χ λ (Tw ) = χ λ (Tw−1 ) for all w ∈ W . Hence, using the orthogonality relations in Proposition 1.2.12 we obtain
∑
χ λ (Tw )2 =
w∈W
∑
χ λ (Tw ) χ λ (Tw−1 ) = dλ cλ .
w∈W
Now we set aλ := min{g ∈ Γ0 | ε g χ λ (Tw ) ∈ K[Γ0 ] for all w ∈ W }. For each w ∈ W , denote by cw,λ ∈ K the constant term of ε aλ χ λ (Tw ); note that cw,λ = 0 for at least one w ∈ W . Let fλ := ∑w∈W c2w,λ . Since all cw,λ are real numbers, not all of which are zero, we conclude that f λ is a strictly positive real number. Now we obtain
ε 2aλ
∑
w∈W
χ λ (Tw )2 ≡
∑
w∈W
2
ε aλ χ λ (Tw )
≡
∑ c2w,λ ≡ fλ
mod K[Γ>0 ].
w∈W
Hence, setting fλ = f λ /dλ , we see that ε 2aλ cλ ∈ fλ + K[Γ>0 ]. Finally, since cλ ∈ ZW [Γ ], we must have fλ ∈ ZW . Remark 1.3.2. Let λ ∈ Λ and w ∈ W . By Proposition 1.3.1, we can write
ε aλ χ λ (Tw ) = cw,λ + K-linear combination of terms ε g where g > 0, where cw,λ ∈ K. These are the “leading coefficients of character values” considered by Lusztig [220, Chap. 5], [225]. From the orthogonality relations in Proposition 1.2.12 (see also [132, Exc. 9.8]), we immediately deduce that fλ dλ if λ = μ , c c = ∑ w,λ w−1 ,μ 0 otherwise. w∈W Thus, the coefficients cw,λ behave as if they were the character values of an algebra with a basis indexed by the elements of W ; see Section 1.5 for a further discussion. Example 1.3.3. Assume that L(s) > 0 for all s ∈ S. Let λ ∈ Λ . (a) If E λ is the unit representation, then aλ = 0 and fλ = 1. (b) If E λ is the sign representation, then aλ = L(w0 ) and fλ = 1. (c) If E λ is neither the unit nor the sign representation, then 0 < aλ < L(w0 ). Here, w0 ∈ W denotes the longest element. (Note that these statements fail if L(s) = 0 for some s ∈ S; see Example 1.3.7 below.) First note that the unit and the sign
1.3 Lusztig’s a-Invariants
15
representation of W correspond to the following representations of HK , respectively: indε : HK → K,
Tw → ε L(w) ,
sgnε : HK → K,
Tw → (−1)l(w) ε −L(w) .
The condition that L(s) > 0 for all s ∈ S implies that 0 < L(w) < L(w0 ) for all w ∈ W \ {1, w0 }. This immediately yields (a) and (b). To prove (c), first note that indε (Tw ) ∈ R[Γ>0 ] for w = 1. This yields that
ε aλ
∑
χ λ (Tw ) indε (Tw−1 ) ≡ ε aλ χ λ (T1 ) mod R[Γ>0 ].
w∈W
Since Eελ ∼ indε , the sum on the left-hand side must be zero by Proposition 1.2.12. = Hence, aλ > 0, as required. On the other hand, by 1.2.1, we have ε L(w) χ λ (Tw ) ∈ R[Γ0 ] for all w ∈ W . This already shows that aλ L(w0 ). Now assume, if possible, that aλ = L(w0 ). Then ε L(w0 ) χ λ (Tw ) ∈ R[Γ>0 ] for all w = w0 and ε L(w0 ) χ λ (Tw0 ) has a non-zero constant term. Using (b), it follows that
ε 2L(w0 )
∑
χ λ (Tw ) sgnε (Tw−1 ) ≡ ±ε L(w0 ) χ λ (Tw0 ) ≡ 0 mod R[Γ>0 ].
w∈W
But, since Eελ ∼ sgnε , we also deduce from Proposition 1.2.12 that the sum on the = left-hand side is zero, which is a contradiction. So we have aλ < L(w0 ), as required. Example 1.3.4. Recall from Example 1.2.6 that we have a “duality” operation λ → † λ † on Λ such that E λ ∼ = E λ ⊗ sgn. By [132, Prop. 9.4.3], we have cλ † = cλ = ε −2Nλ cλ
and aλ † − aλ = Nλ ,
with Nλ as in Example 1.2.7(a).
Here, the map A → A, a → a, ¯ is defined as in Example 1.2.6. Remark 1.3.5. Let W = W1 × · · · ×Wd be the decomposition into irreducible components. Correspondingly, we have IrrK (W ) = {E λ1 · · · E λd | λi ∈ Λi },
where
IrrK (Wi ) = {E λi | λi ∈ Λi }.
Thus, we can identify Λ = Λ1 × · · · × Λd . As already noted in 1.1.2, we have a tensor product decomposition H ∼ = H1 ⊗A · · · ⊗A Hd , where Hi is the generic algebra associated with Wi and the restriction of L to Wi . Hence, we also have λ
Irr(HK ) = {Eελ1 · · · Eε d | λi ∈ Λi }. By [132, Exc. 8.5], this yields that cλ = cλ1 · · · cλd , where λ = (λ1 , . . . , λd ). Consequently, we have aλ = aλ1 + · · · + aλd and f λ = fλ1 · · · fλd . Thus, the determination of aλ and fλ can be reduced to the case where (W, S) is irreducible. Remark 1.3.6. The elements cλ are explicitly known for all types of W ; see [132, Chap. 10 and 11] and the references there. It turns out that they have a quite special
16
1 Generic Iwahori–Hecke Algebras
Table 1.2 The invariants fλ and aλ for type F4 b > 2a > 0 b = 2a > 0 2a > b > a > 0 b = a > 0 b > a = 0 fλ aλ fλ aλ fλ aλ fλ aλ fλ aλ 1 0 1 0 1 0 1 0 6 0 1 12b−9a 2 15a 1 11b−7a 8 4a 6 12b 1 3a 2 3a 1 −b+5a 8 4a 6 0 1 12b+12a 1 36a 1 12b+12a 1 24a 6 12b 1 3b−3a 2 3a 1 2b−a 2 a 12 3b 1 3b+9a 2 15a 1 2b+11a 2 13a 12 3b 1 a 1 a 1 a 2 a 3 0 1 12b+a 1 25a 1 12b+a 2 13a 3 12b 2 3b+a 2 7a 2 3b+a 8 4a 6 3b 1 2b−a 2 3a 1 b+a 1 2a 2 2b 1 6b−2a 1 10a 1 6b−2a 8 4a 2 6b 1 2b+2a 1 6a 1 2b+2a 8 4a 2 2b 1 6b+3a 2 15a 1 5b+5a 1 10a 2 6b 3 3b+a 3 7a 3 3b+a 3 4a 12 3b 3 3b+a 3 7a 3 3b+a 12 4a 12 3b 6 3b+a 6 7a 6 3b+a 24 4a 6 3b 1 b 1 2a 1 b 2 a 6 b 1 7b−3a 1 11a 1 7b−3a 4 4a 6 7b 1 b+3a 1 5a 1 b+3a 4 4a 6 b 1 7b+6a 1 20a 1 7b+6a 2 13a 6 7b 1 3b 1 6a 1 3b 1 3a 12 3b 1 3b+6a 1 12a 1 3b+6a 1 9a 12 3b 1 b+a 2 3a 1 3a 1 3a 3 b 1 7b+a 2 15a 1 6b+3a 1 9a 3 7b 2 3b+a 2 7a 2 3b+a 4 4a 6 3b The notation for IrrK (W ) is defined in [132, Appendix C.3]. Here, a := L(s1 ) = L(s2 ) and b := L(s3 ) = L(s4 ), cf. Table 1.1.
Eλ 11 12 13 14 21 22 23 24 41 91 92 93 94 61 62 121 42 43 44 45 81 82 83 84 161
form. Indeed, one checks that there is a family {Φd | d ∈ I} of monic polynomials in one variable over ZW such that n (a) cλ = fλ ε γλ ∏ Φd ε γλ ,d λ ,d , where γλ ∈ Γ , nλ ,d 0 and 0 = γλ ,d ∈ Γ ; d∈I
(b) all the (complex) roots of Φd are roots of unity; 2 (c) the product of all cλ (for λ ∈ Λ ) lies in Z[u±1 s | s ∈ S], where us := vs (s ∈ S).
Note that the monomials γλ and γλ ,i are not uniquely determined. In fact, depending on the monomial order on Γ , the terms involving those monomials are rearranged so as to produce the relation ε 2aλ cλ ≡ fλ mod R[Γ>0 ] in Proposition 1.3.1. From the explicit knowledge of cλ one can deduce explicit formulae for the invariants aλ and f λ . If L(s) = 0 for all s ∈ S, then cλ = |W |/dλ . Hence, aλ = 0 and fλ = |W |/dλ for all λ ∈ Λ in this case. Now assume that L(s) > 0 for at least some s ∈ S. For W of exceptional type H3 , H4 , E6 , E7 , E8 (where we are automatically in the equal-parameter case), see the tables in [220, Chap. 4] and in the Appendices C and E in [132]. For type F4 , see Table 1.2 (p. 16); note that, by the symmetry of the
1.3 Lusztig’s a-Invariants
17
diagram, we can assume without loss generality that L(s1 ) = L(s2 ) L(s3 ) = L(s4 ). For the types I2 (m), An−1 , Bn and Dn , see the examples below.
Table 1.3 The irreducible representations of HK in type I2 (m) ε : 1W sgnε : sgnε1 : sgnε2 :
σ εj :
→ vs1 , → −v−1 s1 , → vs1 , −v−1 → s1 , −v−1 s1 0 , Ts1 → μ j vs1
→ vs2 , → −v−1 s2 , → −v−1 s2 , vs2 , → vs2 1 Ts2 → 0 −v−1 s2
Ts1 Ts1 Ts1 Ts1
Ts2 Ts2 Ts2 Ts2
λ aλ fλ
1W 0 1
(sgn1 ) a
σj a
(sgn2 ) a
m 2
m 2−ζ j −ζ − j
m 2
sgn ma 1
aλ fλ
0 1
a 1
b
m 2 (b−a)+a
m 2 (a+b)
1
1
(b > a 0) (b > a > 0)
fλ
2
2
2
2
(b > a = 0)
m 2−ζ 2 j −ζ −2 j m 2−ζ 2 j −ζ −2 j
(where b := L(s1 ), a := L(s2 ) and
(b = a > 0) (b = a > 0)
j − j + v−1 v ) μ j := vs1 v−1 s2 + ζ + ζ s1 s2
Example 1.3.7. Let W be of type I2 (m) (m 3); that is, W = s1 , s2 , where s21 = s22 = (s1 s2 )m = 1. In this case, ZW = Z[ζ + ζ −1 ], where ζ ∈ C is a root of unity of order m, chosen such that ζ + ζ −1 = 2 cos(2π /m). By [132, §5.4], we have {1W , sgn, σ1 , σ2 , . . . , σ(m−1)/2 } if m is odd, IrrK (W ) = {1W , sgn, σ1 , σ2 , . . . , σ(m−2)/2 , sgn1 , sgn2 } if m is even, where 1W is the unit and sgn is the sign representation, all σ j are two-dimensional, and sgn1 , sgn2 are two further one-dimensional representations when m is even, in which case we fix the notation such that s1 acts as +1 in sgn1 and as −1 in sgn2 . By [132, §8.3], explicit realisations of the corresponding representations of HK are known; see Table 1.3. Using the formulae for cλ in [132, Theorem 8.3.4], one obtains the invariants aλ and f λ . In Table 1.3, the columns are ordered such that the invariants aλ are increasing from left to right; the columns corresponding to sgn1 , sgn2 must be deleted if m is odd. Let us give a concrete example to see how aλ and fλ are computed as a function of the monomial order . Let m = 4 so that W is the dihedral group of order 8. Let ζ ∈ C be a fourth root of unity; then, by the general formula in [132, 8.3.4], we have
18
1 Generic Iwahori–Hecke Algebras
c σ1 = 4
(v2s1 v2s2 − (ζ + ζ −1 )vs1 vs2 + 1)(v2s1 + (ζ + ζ −1 )vs1 vs2 + v2s2 ) v2s1 v2s2 (2 − (ζ 2 + ζ −2 ))
−2 2 2 2 2 = v−2 s1 vs2 (vs1 vs2 + 1)(vs1 + vs2 )
= ε −2a + ε −2b + ε 2a + ε 2b , where b := L(s1 ) and a := L(s2 ). Hence, if b > a > 0, then all four terms are different, fσ1 = 1 and the “lowest” term is ε −2b ; so aσ1 = b. Similarly, if a > b > 0, then f σ1 = 1 and aσ1 = a. If b > a = 0, then cσ1 = ε −2b + 2 + ε 2b and so fσ1 = 1, aσ1 = b. Finally, if a = b > 0, then cσ1 = 2(ε −2a + ε 2a ) and so fσ1 = 2, aσ1 = a. In the next three examples, we describe the invariants aλ for W of type An−1 , Bn , Dn . In these cases, Irr(W ) is parametrised by suitable partitions or pairs of partitions, where we follow the notational conventions of [220, Chap. 4], [132, Chap. 5]. Recall that a partition is a finite, weakly decreasing sequence of non-negative integers; we often write this in the form λ = (λ1 λ2 . . . λN 0). We say that λ is a partition of n, and write λ n, if |λ | := λ1 + · · · + λN equals n; the numbers λi which are non-zero are called the parts of λ . As is usual, we will not distinguish between two partitions which have identical parts. ∼ Sn . We are automatically in the Example 1.3.8. Let W be of type An−1 , where W = equal-parameter case; write a := L(s) > 0 for s ∈ S. There is a standard labelling IrrK (W ) = {E λ | λ ∈ Λ },
where Λ = {set of all partitions of n}.
For example, the unit and the sign representation are labelled by (n) and (1n ), respectively; see [132, §5.4]. (Here, (1n ) denotes the partition which has n parts equal to 1.) By [132, Prop. 9.4.5], given a partition λ of n, we have fλ = 1
and
aλ = n(λ )a,
where n(λ ) :=
∑
(i − 1)λi ;
1iN
here, we write λ = (λ1 λ2 . . . λN 0) for some N 1. We have
λ† = λ∗
and
aλ ∗ = a
1 λi (λi − 1), 1iN 2
∑
where λ ∗ denotes the conjugate (or transpose) partition; see [132, 5.4.3, 5.4.9]. Example 1.3.9. Let W be of type Bn with weight function L : W → Γ given by b 4 a a a Bn t t t p p p t where a, b 0 There is a standard labelling IrrK (W ) = {E λ | λ ∈ Λ }, where
Λ = {set of all pairs of partitions (λ , μ ) such that |λ | + |μ | = n}. For example, the unit and the sign representation are labelled by ((n), ∅) and (∅, (1n )) respectively; see [132, §5.5]. We have (λ , μ )† = (μ ∗ , λ ∗ ); see [132, 5.5.6].
1.3 Lusztig’s a-Invariants
19
The elements cλ have been determined explicitly by Hoefsmit [157]; see also Lusztig [212, 9.6], and Iancu [169] for alternative proofs. The case where a = b = 0 has already been dealt with in Remark 1.3.6. Now we distinguish three further cases: Case 1. We have a = 0 and b > 0. Then we have a(λ ,μ ) = b |μ |
and
f(λ ,μ ) ∈ {1, 2, . . . , n}.
This directly follows from Hoefsmit’s original formulas for cλ . Case 2. We have a > 0 and b ra for all r ∈ Z0 . By [103, Remark 5.1], we have and f(λ ,μ ) = 1, a(λ ,μ ) = n(λ ) + 2n(μ ) − n(μ ∗ ) a + |μ | b where n(λ ), n(μ ), n(μ ∗ ) are defined using the formula in Example 1.3.8. These statements actually hold as soon as b > (n − 1)a > 0; see [121, Example 3.6]. Case 3. There exists an integer r 0 (which is then unique) such that ra b < (r + 1)a. In this case, we set b := b − ra ∈ Γ0 . Following Lusztig [231, 22.10], we choose a sufficiently large integer N 0 such that we can write
λ = (λ1 λ2 . . . λN+r 0) and
μ = (μ1 μ2 . . . μN 0).
N (λ , μ ) to be the multiset formed by the 2N + r entries Then we define Za,b
(λi + N + r − i)a + b (μ j + N − j)a
(1 i N + r), (1 j N).
Furthermore, let Z 0 be the multiset whose entries are 0, a, 2a, . . . , (N − 1)a, b , a + b , 2a + b , . . . , (N + r − 1)a + b . Then, by Lusztig [231, Prop. 22.14], we have a(λ ,μ ) =
∑
1i2N+r
(i − 1)zi −
∑
(i − 1)z0i ,
1i2N+r
where z1 , z2 , . . . , z2N+r are the entries of Z and z01 , z02 , . . . , z02N+r are the entries of Z 0 , both arranged in decreasing order. Note that the second sum only depends on a, b, N and the whole expression does not depend on N. (It is precisely the extra term arising from Z 0 which guarantees this independence of N.) Furthermore, we have f (λ ,μ ) = 2c for some c 0 where c = 0 if b > 0. Example 1.3.10. Let n 2 and W be of type Bn , as above. Let us now assume that b = 0 and a > 0. Consider the reflection subgroup W˜ := s0 s1 s0 , s1 , . . . , sn−1 ⊆ W , where the generators of W are labelled as in Table 1.1. Let L˜ denote the restriction of L to W˜ n . Then W˜ is a Coxeter group of type Dn , with diagram as given below, and L˜ is a multiple of the length function on W˜ ; see [132, §1.4].
20
1 Generic Iwahori–Hecke Algebras
s1 t sn−1 HHs2 s3 t t p p p t t s0 s1 s0 The irreducible representations of W˜ are classified as follows. Given a pair of partitions (λ , μ ) such that |λ | + |μ | = n, we denote by E [λ ,μ ] the restriction of E (λ ,μ ) ∈ Irr(W ) to W˜ . Then the following hold (see [132, 5.6.1]). • If λ = μ , then E [λ ,μ ] = E [μ ,λ ] is an irreducible representation of Irr(W˜ ). • If λ = μ , then E [λ ,λ ] = E [λ ,+] ⊕ E [λ ,−] , where E [λ ,+] , E [λ ,−] are non-isomorphic irreducible representations of W˜ . (This can only occur if n is even.) Furthermore, all irreducible representations of W˜ arise in this way. The a-invariants of these representations are given as follows. Let E˜ ∈ Irr(W˜ ) and let (λ , μ ) be a pair of partitions as above such that E˜ is a constituent of E [λ ,μ ] . Then, by [132, §10.5], the a-invariant of E˜ is a(λ ,μ ) , where the latter is given by the formula in Case 3 of Example 1.3.9 (with r = 0, b = 0); furthermore, each fE˜ is a power of 2. Dn
Remark 1.3.11. The invariants aλ are inductively determined purely in terms of the characters of W and the parabolic subgroups of W . Indeed, if W = {1}, then IrrK (W ) only consists of the unit representation and this has a-invariant 0. Now assume that W = {1} and that the a-invariants have already been determined for all proper parabolic subgroups of W . For any subset J S, let WJ ⊆ W denote the corresponding parabolic subgroup and write IrrK (WJ ) = {M μ | μ ∈ ΛJ }. Then, for any E λ ∈ IrrK (W ), we can define a λ := max{aμ | μ ∈ ΛJ where J S and M μ ↑ E λ }. Here, we write M μ ↑ E λ if E λ is an irreducible constituent of the representation obtained by inducing M μ from WJ to W . Then, finally, we have if a λ † − a λ Nλ , a λ aλ = otherwise, aλ † − Nλ where Nλ ∈ Γ is given by the formula in Example 1.2.7(a). In the equal-parameter case, this was observed in [132, 6.5.6]; see [115, §4] for the general case.
1.4 Balanced Representations In the previous section we have attached to each irreducible representation E λ of W an invariant aλ ∈ Γ0 . In Proposition 1.3.1, we have seen that ε aλ χ λ (Tw ) ∈ K[Γ0 ] for all w ∈ W , and that aλ is minimal with this property. One might ask if something like this is true not only for the character, but also for the individual entries of a matrix representation afforded by Eελ . It will turn out that this is the case in a very strong sense, as a result of the construction of a cellular basis of H (see Section 2.7). In this section, we provide the foundations for this construction. Throughout, we assume that is a monomial order on Γ such that L(s) 0 for all s ∈ S.
1.4 Balanced Representations
21
1.4.1. We will consider a certain valuation ring O0 in K. Let us denote by K[Γ>0 ] the set of all K-linear combinations of terms ε g , where g > 0; the notation K[Γ0 ] has a similar meaning. Note that 1 + K[Γ>0 ] is multiplicatively closed. Furthermore, every element x ∈ K can be written in the form x = r x ε gx
1+ p , 1+q
where rx ∈ K, gx ∈ Γ and p, q ∈ K[Γ>0 ];
note that, if x = 0, then rx and gx indeed are uniquely determined by x; if x = 0, we have r0 = 0 and we set g0 := +∞ by convention. We set O0 := {x ∈ K | gx 0}
and
m := {x ∈ K | gx > 0}.
One easily checks that O0 is a valuation ring in K, with maximal ideal m; note that O0 ∩ K[Γ ] = K[Γ0 ]
and
m ∩ K[Γ ] = K[Γ>0 ].
We have a well-defined K-linear ring homomorphism O0 → K with kernel m. The image of x ∈ O0 in K is called the constant term of x. Thus, the constant term of x is 0 if x ∈ m; the constant term equals rx if x ∈ O0× . Warning: Note that the algebra H is not defined over O0 ! For example, the coefficients in the quadratic relations for Ts (s ∈ S, L(s) > 0) do not lie in O0 . Example 1.4.2. Let Γ = Z so that A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Then K = K(v) is the field of rational functions in v. In this case (which is the most important one for many applications), O0 is just the localisation of A in the prime ideal generated by v: O0 = f /g f , g ∈ K[v] such that g(0) = 0 ⊆ K(v). Here, the canonical map O0 → K is given by evaluating f /g at v = 0. Definition 1.4.3. Let λ ∈ Λ and consider a matrix representation ρ λ : HK → Mdλ (K) afforded by Eελ . We say that ρ λ is a balanced representation if
ε aλ ρ λ (Tw ) ∈ Mdλ (O0 )
for all w ∈ W .
aλ λ In this case, we denote by cst w,λ ∈ K the constant term of ε ρst (Tw ), for any w ∈ W st and s, t ∈ M(λ ). The coefficients cw,λ will be called the leading matrix coefficients of H. (They will play a major role in our discussion of “cell data” in Section 1.6.)
Remark 1.4.4. (a) Consider the leading coefficients of character values cw,λ in Remark 1.3.2. If ρ λ is a balanced representation afforded by Eελ , then cw,λ =
∑
s∈M(λ )
css w,λ
for all w ∈ W .
22
1 Generic Iwahori–Hecke Algebras
Thus, the leading matrix coefficients in Definition 1.4.3 are refinements of Lusztig’s leading coefficients of character values. (b) Balanced representations and, hence, the corresponding leading matrix coefficients are not uniquely determined (say, by the characters χ λ ). For example, given a balanced representation ρ λ : HK → Mdλ (K), one can just conjugate ρ λ by an invertible matrix with coefficients in K and the resulting new representation will still be balanced. However, we will see that the main constructions derived from balanced representations (for example, the “asymptotic algebra” in Section 1.5 or the relation “L ” in 1.6.10) are independent of any choices. The condition in Definition 1.4.3 is hard to verify, even in concrete examples. Following [103], [112], we will now establish a basic result which provides an efficient criterion for verifying that a given representation is balanced. Proposition 1.4.5 (Cf. [103, §4], [112, §4]). Assume that K ⊆ R. Let λ ∈ Λ and ρ λ : HK → Mdλ (K) be any matrix representation afforded by Eελ . Assume that there exists a symmetric matrix Ω λ ∈ Mdλ (K) such that the following two conditions hold: (a) we have Ω λ ρ λ (Tw−1 ) = ρ λ (Tw )tr Ω λ for all w ∈ W ; (b) all entries of Ω λ lie in O0 and we have Ω λ ≡ Dλ mod m, where Dλ is a diagonal matrix with diagonal coefficients {bs | s ∈ M(λ )} ⊆ K× . Then, ρ λ is balanced and the corresponding leading matrix coefficients satisfy st bt cts w−1 ,λ = cw,λ bs
for all w ∈ W and s, t ∈ M(λ ).
λ (T ) ∈ O for all w ∈ W and all s, t ∈ M(λ )}. Proof. Define γ := min{g ∈ Γ | ε g ρst w 0 λ In order to show that ρ is balanced, we must show that aλ γ . First note that, by the Schur relations (see Proposition 1.2.12), we have for any s, t ∈ M(λ ) λ λ ε 2γ cλ = ∑ ε γ ρst (Tw ) ε γ ρts (Tw−1 ) ∈ O0 . w∈W
By Proposition 1.3.1, we also know that ε 2aλ cλ lies in O0 and has a non-zero constant term. Thus, it remains to show that ε 2γ cλ has a non-zero constant term. (This will actually imply that aλ = γ .) For this purpose, let cˆst w,λ ∈ R be the constant term
λ (T ). Now we multiply the relation Ω λ ρ λ (T λ tr λ γ of ε γ ρst w w−1 ) = ρ (Tw ) Ω by ε and consider constant terms. Using (b), we obtain that st bt cˆts w−1 ,λ = cˆw,λ bs
for all w ∈ W and all s, t ∈ M(λ ).
Now choose s, t ∈ M(λ ) such that cˆst y,λ = 0 for some y ∈ W . Then
ε 2γ cλ ≡
∑ cˆstw,λ cˆtsw−1 ,λ ≡ bs b−1 ∑ t
w∈W
w∈W
st 2 cˆw,λ mod m.
The sum on the right-hand side is a non-zero real number since cˆst y,λ = 0 for some 2 γ y ∈ W . Thus, ε cλ has a non-zero constant term, as desired. As already mentioned
1.4 Balanced Representations
23
above, this implies that aλ = γ . Once this is established, it also follows that cst w,λ = cˆst for all w ∈ W and all s, t ∈ M( λ ), and this yields the remaining assertions. w,λ Note that the relation in (a) will hold for all w ∈ W if it holds for all s ∈ S. This remark, although almost trivial, is nevertheless useful in dealing with examples. Example 1.4.6. Let W be of type I2 (m) (m 3), with generators s1 , s2 such that (s1 s2 )m = 1. Assume that L(s1 ) L(s2 ) 0. We have ZW = Z[ζ + ζ −1 ], where ζ ∈ C is a root of unity of order m. We show that the matrix representations in Example 1.3.7 are balanced. Since these representations are realised over the ring ±1 ZW [v±1 s1 , vs2 ], it follows that the corresponding leading matrix coefficients lie in ZW . Note that the above statements are clear for one-dimensional representations. Now consider a two-dimensional representation σ εj . Let
Ωj =
vs1 μ j (vs2 + v−1 s2 ) vs1 μ j vs1 μ j v2s1 + 1
±1 ∈ M2 (ZW [v±1 s1 , vs2 ]),
j −j where μ j = vs1 v−1 + v−1 s2 + ζ + ζ s1 vs2 . Then Ω j is a symmetric matrix and one immediately checks that
Ω j σ εj (Tw−1 ) = σ εj (Tw )tr Ω j
for all w ∈ W .
(Note that it is enough to do this for w ∈ {s1 , s2 }.) Note that all entries of Ω j lie in O0 . This is clear for the entry v2s1 + 1. Furthermore, we have j −j vs1 μ j = v2s1 v−1 s2 + (ζ + ζ )vs1 + vs2 ,
which lies in O0 since L(s1 ) L(s2 ) 0. Finally, 2 −2 j −j −1 vs1 μ j (vs2 + v−1 s2 ) = vs1 μ j vs2 + vs1 vs2 + (ζ + ζ )vs1 vs2 + 1,
which is still in O0 . Using the above formulae, we obtain that 10 Ωj ≡ mod m if L(s1 ) > L(s2 ) > 0, 01 2+ζ j +ζ−j 0 mod m if L(s1 ) = L(s2 ) > 0. Ωj ≡ 0 1 Hence, in these two cases, the conditions in Proposition 1.4.5 are satisfied and so σ εj is balanced. Now assume that L(s2 ) = 0; that is, vs2 = 1. Then we perform a slight transformation as follows. We set 1 − 12 σˆ εj (Tw ) := P−1 σ εj (Tw )P (w ∈ W ), where P := . 0 1 Then Ωˆ j σˆ εj (Tw−1 ) = σˆ εj (Tw )tr Ωˆ j for all w ∈ W , where Ωˆ j := Ptr Ω j P. Furthermore,
24
1 Generic Iwahori–Hecke Algebras
Ωˆ j ≡
Ωˆ j ≡
20 0 12
if L(s1 ) > L(s2 ) = 0,
mod m
0 2(2 + ζ j + ζ − j ) 1 j −j 0 2 (2 − ζ − ζ )
if L(s1 ) = L(s2 ) = 0.
mod m
Hence, Proposition 1.4.5 applies again and so σˆ εj is balanced. But then σ εj must also be balanced, since the transforming matrix P has all its entries in K. We now turn to the existence of balanced representations in general. This will be based on the following result. Lemma 1.4.7 (Cf. Lusztig [217, 1.7]). Assume that K ⊆ R. For each λ ∈ Λ , there exists a symmetric bilinear form , λ : Eελ × Eελ → K such that the following hold. (a) We have Tw .e, e λ = e, Tw−1 .e λ for all e, e ∈ Eελ and w ∈ W . (b) For every 0 = e ∈ Eελ , there exist some g ∈ Γ and some strictly positive real number b ∈ K such that ε 2g e, eλ ∈ b + m. In particular, the form , λ is non-degenerate. Proof. Let ( , ) be any symmetric bilinear form on Eελ which admits an orthonormal basis, {es | s ∈ M(λ )} say. Then we define e, e λ :=
∑ (Tw .e, Tw .e )
for any e, e ∈ Eελ .
w∈W
We claim that Ts .e, e λ = e, Ts .e λ for all s ∈ S. Indeed, let s ∈ S and split the above sum over w ∈ W into parts according to whether multiplication by s increases or decreases l(w). This yields Ts .e, e λ =
∑ (Tws.e, Tw .e ) + (vs − v−1 s )
w∈W
e, Ts .e λ =
∑ (Tw .e, Tws .e
w∈W
) + (vs − v−1 s )
∑
(Tw .e, Tw .e ),
∑
(Tw .e, Tw .e ).
w∈W :l(ws)
w∈W :l(ws)
Since the first two sums are the same in each case, we conclude that Ts .e, e = e, Ts .e λ , as claimed. Consequently, (a) holds. Now let 0 = e ∈ Eελ . Writing each term Tw .e in the defining formula for e, eλ as a linear combination of the original basis {es }, one readily sees that e, eλ =
∑ x2 ,
where Ne ⊆ K × , Ne = ∅.
x∈Ne
Now write each x ∈ Ne as x = rx ε gx ux , where 0 = rx ∈ K, gx ∈ Γ and ux ∈ 1 + m. Let g := − min{gx | x ∈ Nt }. Then ε 2g e, eλ lies in O0 and has constant term b := ∑x∈M:gx =g rx2 ∈ K. Since all rx are real numbers, there are no cancellations in this sum and so b > 0. Thus, (b) holds.
1.4 Balanced Representations
25
Theorem 1.4.8 (Cf. [103, §4]). For each λ ∈ Λ , there exists a matrix representation ρ λ : HK → Mdλ (K) afforded by Eελ which is balanced. This representation ρ λ can be chosen such that the conditions in Proposition 1.4.5 are satisfied, where bs is a positive real number in K for all s ∈ M(λ ). The corresponding leading matrix coefficients have the following properties: (a) We have cst w,λ ∈ K ∩ R for all w ∈W and s, t ∈ M(λ ). st = cst (b) We have bt cts w,λ bs for all w ∈ W and s, t ∈ M(λ ). In particular, cw,λ = w−1 ,λ ts 0 if and only if cw−1 ,λ = 0. Proof. We can assume that K ⊆ R; see 1.2.1. Let , λ be as in Lemma 1.4.7. Since we are working in characteristic 0, the vector space Eελ has an orthogonal basis, {hs | s ∈ M(λ )} say, with respect to this form. For each s ∈ M(λ ), we have
ε 2gs hs , hs λ ∈ bs + m,
where gs ∈ Γ and bs ∈ K, bs > 0.
Now set fs := ε gs hs for s ∈ M(λ ). Then { fs | s ∈ M(λ )} still is an orthogonal basis, and we have fs , fs λ ∈ bs + m for all s ∈ M(λ ). Let Ω λ be the Gram matrix of , λ with respect to the basis { f s } and Dλ be the diagonal matrix with diagonal coefficients bs (s ∈ M(λ )). Then the conditions in Proposition 1.4.5 are satisfied for the matrix representation ρ λ defined with respect to the basis { fs }. 1.4.9. We summarize the procedure for constructing balanced representations as follows, where we express everything in matrix terms. Let λ ∈ Λ and σ λ : HK → Mdλ (K) be a representation afforded by Eελ . Then set
Ω0λ :=
∑ σ λ (Tw )tr .σ λ (Tw ) ∈ Mdλ (K).
w∈W
Clearly, Ω0λ is a symmetric matrix. By Lemma 1.4.7, we have det(Ω0λ ) = 0
and
Ω0λ σ λ (Tw−1 ) = σ λ (Tw )tr Ω0λ
for all w ∈ W .
In fact, as shown in Lemma 1.4.7, Ω0λ is the Gram matrix of a bilinear form which has no non-zero isotropic vectors. Thus, we can apply the Gram–Schmidt orthogonalisation procedure, which yields an upper triangular matrix Pλ with 1 on the diagonal and such that (Pλ )tr Ω0λ Pλ = Ω1λ , where Ω1λ is a diagonal matrix with the following property. There exist strictly positive real numbers {bs | s ∈ M(λ )} and elements {gs | s ∈ M(λ )} ⊆ Γ such that the diagonal coefficient of Ω1λ at position s ∈ M(λ ) has the form bs ε −2gs + combination of terms ε g where g > −2gs . Let E λ be the diagonal matrix with diagonal coefficients {ε gs | s ∈ M(λ )}. We set
26
1 Generic Iwahori–Hecke Algebras
Ω λ := E λ Ω1λ E λ ∈ Mdλ (O0 ), ρ λ (Tw ) := (E λ )−1 (Pλ )−1 σ λ (Tw ) Pλ E λ
for all w ∈ W .
Then Ω λ ρ λ (Tw−1 ) = ρ λ (Tw )tr Ω λ for all w ∈ W and Ω λ is a diagonal matrix where the diagonal coefficient at position s ∈ M(λ ) is congruent to bs modulo m. Hence, by Proposition 1.4.5, ρ λ is a balanced representation afforded by Eελ . Proposition 1.4.10. For every λ ∈ Λ , let ρ λ : HK → Mdλ (K) be a matrix representation afforded by Eελ which is balanced. Then the corresponding leading matrix coefficients have the following properties. (a) Given λ , μ ∈ Λ , let s, t ∈ M(λ ) and u, v ∈ M(μ ). Then fλ if λ = μ , s = v, t = u, st uv c c = ∑ w,λ w−1 ,μ 0 otherwise. w∈W (b) Let x, y ∈ W . Then
∑ ∑
λ ∈Λ s,t∈M(λ )
ts fλ−1 cst x,λ cy−1 ,λ =
1 0
if x = y, otherwise.
(c) Given λ ∈ Λ and s, t ∈ M(λ ), there exists some w ∈ W such that cst w,λ = 0. Con= 0. In particular, versely, given w ∈ W , there exist some λ , s, t such that cst w,λ λ aλ = min{g ∈ Γ0 | ε g ρst (Tw ) ∈ O0 for all w ∈ W and s, t ∈ M(λ )}.
Proof. The Schur relations in Proposition 1.2.12 yield μ λ δsv δut δλ μ ε aλ +aμ cλ = ∑ ε aλ ρst (Tw ) ε aμ ρuv (Tw−1 ) . w∈W
uv By Definition 1.4.3, the right-hand side is congruent to ∑w∈W cst w,λ cw−1 ,λ modulo m. If λ = μ , the left-hand side is zero. If λ = μ , then the left-hand side is congruent to δsv δut fλ ; see Proposition 1.3.1. This yields (a). Now (b) is a formal consequence of (a). Indeed, by the Artin–Wedderburn theorem, the set of all triples (λ , s, t) has cardinality |W |. Arranging these triples, and the elements of W , in some order, we obtain |W | × |W |-matrices and B := fλ−1 cst A := cst w,λ w−1 ,λ ,
where the triples (λ , s, t) label the rows and the elements of W label the columns. Then (a) means that the product A.Btr is the identity matrix. So Btr .A also is the identity matrix. Writing out the coefficients of Btr .A yields (b). Finally, (c) clearly is a formal consequence of (a) and (b). We now introduce a particular class of matrix representations of H. These are defined using the concept of a W -graph which first appeared in the work of Kazh-
1.4 Balanced Representations
27
dan and Lusztig [195] on “cells” in the equal-parameter case. (We shall say more on these “cells” in Chapter 2.) The following definition works for general weight functions. (In [132, Def. 11.1.1] we made a first attempt to generalise the original definition of [195], but we did not take into account a monomial order on Γ .) Definition 1.4.11. Assume that L(s) > 0 for all s ∈ S. A W -graph for H (with respect to ) consists of the following data: (a) a set X together with a map I which assigns to each x ∈ X a subset I(x) ⊆ S; (b) a collection of elements {msx,y } in A, where x, y ∈ X and s ∈ S are such that s ∈ I(x) and s ∈ I(y). These data are subject to the following requirements. First, we require that vs msx,y ∈ R[Γ>0 ] and
msx,y = msx,y
for all x, y ∈ X, s ∈ I(x) \ I(y),
where a → a¯ (a ∈ A) is as in Example 1.2.6. Furthermore, let V be a free A-module with a basis {ey | y ∈ X}. For s ∈ S, define an A-linear map ρs : V → V by
ρs (ey ) =
⎧ ⎨ vs ey + ⎩
∑
x∈X: s∈I(x) −v−1 s ey
msx,y ex
if s ∈ I(y), if s ∈ I(y).
Then we require that the assignment Ts → ρs defines a representation of H. Example 1.4.12. (a) Assume that W is of type I2 (m), with generators S = {s1 , s2 } such that L(s1 ) L(s2 ) > 0. The irreducible representations of HK are described in Example 1.3.7. The one-dimensional representations certainly are given by W graphs. Now let σ j be one of the two-dimensional representations and define a new representation σ˜ j by sending σ j (Tsi ) to σ j (Tsi )tr for i = 1, 2. Then σ˜ j is given in terms of a W -graph. (Note that σ˜ j is equivalent to σ j .) (b) Let W be of type F4 with generators labelled as in Table 1.1 (p. 2). The weight function L : W → Γ is determined by a := L(s1 ) = L(s2 ) and b := L(s3 ) = L(s4 ). If is a monomial order on Γ such that b > 3a > 0, then the graphs in [132, Fig. 11.3 (pp. 377)] are W -graphs with respect to . Example 1.4.13. Assume that we are in the equal-parameter case where Γ = Z and L(s) = 1 for all s ∈ S, as originally considered by Kazhdan and Lusztig [195]. Then the conditions in Definition 1.4.11 imply that msx,y ∈ R. In this case, Gyoja [150] and Lusztig [224] have shown that every irreducible representation of HK can be realised in terms of a W -graph. (We shall discuss this general existence result in Section 2.7.) For W of exceptional type, such W -graphs have been constructed explicitly: H3 H4 F4 , E6 E 7 , E8
Lusztig [216, §5]; Alvis–Lusztig [3]; Naruse (personal communication January and July 1998); Howlett and Yin [164], [297] (and personal communication 2004).
(The W -graphs for types H3 and H4 are also printed in [132, §11.4].)
28
1 Generic Iwahori–Hecke Algebras
Experiments with the above examples suggest the following connection between W -graphs and balanced representations: Conjecture 1.4.14. Let λ ∈ Λ and assume that a corresponding matrix representation ρ λ : HK → Mdλ (K) is realised in terms of a W -graph. Then ρ λ is balanced. In Section 2.7 we will see that, under some hypothesis on W, L, for any λ ∈ Λ there always exists a balanced representation ρ λ which is afforded by a W -graph.
1.5 The Asymptotic Algebra We shall assume from now on that, for every λ ∈ Λ , we are given a matrix representation ρ λ afforded by Eελ which is balanced; this is possible by Theorem 1.4.8. Let cst w,λ be the corresponding leading matrix coefficients. We now use these coefficients, following [112], to construct a K-algebra. (This construction is independent of Lusztig’s construction of the asymptotic ring J in [223], [231, Chap. 18]; we will see in Corollary 2.3.16 that, up to the change of a sign of basis elements, the two constructions give the same result under some hypothesis on W, L.) Definition 1.5.1 (See [112]). Let J˜ be a K-vector space with a basis {tw | w ∈ W }. We define an element of J˜ and a bilinear product on J˜ by 1J˜ :=
∑ n˜w tw
and
txty :=
w∈D˜
∑ γ˜x,y,z tz−1
(x, y ∈ W ),
z∈W
where
γ˜x,y,z :=
∑
∑
λ ∈Λ s,t,u∈M(λ )
n˜ w :=
∑ ∑
λ ∈Λ s∈M(λ )
tu us f λ−1 cst x,λ cy,λ cz,λ
fλ−1 css w−1 ,λ
(x, y, z ∈ W ),
(w ∈ W ),
D˜ := {w ∈ W | n˜ w = 0}. We will show in several steps that J˜ is a symmetric algebra where 1J˜ is the identity. First, we show that the ingredients in the definition of J˜ only depend on H. Lemma 1.5.2. The constants γ˜x,y,z and n˜ w do not depend on the choice of the balanced representations ρ λ . More precisely, γ˜x,y,z and n˜ w are equal, respectively, to the constant terms of
∑
λ ∈Λ
fλ−1 ε 3aλ χ λ (Tx Ty Tz ) ∈ O0
and
∑
λ ∈Λ
fλ−1 ε aλ χ λ (Tw−1 ) ∈ O0 .
Furthermore, γ˜x,y,z and n˜ w lie in the subring ZW [ fλ−1 | λ ∈ Λ ] ⊆ C.
1.5 The Asymptotic Algebra
29
Proof. Given λ ∈ Λ , consider the expression
∑
ε 3aλ χ λ (Tx Ty Tz ) =
s∈M(λ )
∑
=
λ ε 3aλ ρss (Tx Ty Tz )
s,t,u∈M(λ )
a λ λ λ ε λ ρst (Tx ) ε aλ ρtu (Ty ) ε aλ ρus (Tz ) .
λ (T ), ε aλ ρ λ (T ), By the definition of balanced representations, the terms ε aλ ρst x tu y a λ λ ε ρus (Tz ) all lie in O0 . So the whole sum lies in O0 and its constant term can be computed term by term. Hence,
∑
s,t,u∈M(λ )
tu us −1 3aλ λ fλ−1 cst χ (Tx Ty Tz ) mod m. x,λ cy,λ cz,λ ≡ f λ ε
This yields the required interpretation of γ˜x,y,z . In particular, it shows that γ˜x,y,z only depends on the character of ρ λ , not on the choice of ρ λ itself. A similar argument applies to the defining formula for n˜w : We have
∑
s∈M(λ )
−1 aλ λ f λ−1 css w−1 ,λ ≡ fλ ε χ (Tw−1 ) mod m
for all λ ∈ Λ .
Finally, recall from 1.2.1 that χ λ (Tw ) ∈ ZW [Γ ] for all w ∈ W . Furthermore, the structure constants of H with respect to the basis {Tw } lie in Z[Γ ]. This immediately yields the statement about the ring in which the constants γ˜x,y,z and n˜ w lie. Lemma 1.5.3. We have the following relations:
γ˜x,y,z = γ˜y,z,x
(a)
∑ γ˜x−1 ,y,w n˜w = δxy
(b)
for all x, y, z ∈ W , for all x, y ∈ W ,
w∈W
n˜w = n˜ w−1 and γ˜x,y,z = γ˜y−1 ,x−1 ,z−1
(c)
for all w, x, y, z ∈ W .
Proof. (a) Just note that the defining formula for γ˜x,y,z is symmetrical under cyclic permutations of x, y, z. (b) Using the defining formulae for γ˜x,y,z and n˜ w , the left-hand side evaluates to
us ∑ ∑ ∑ fλ−1 cstx−1 ,λ ctu ∑ ∑ fμ−1 cvv y,λ cw,λ w−1 , μ w∈W
μ ∈Λ v∈M(μ )
λ ∈Λ s,t,u∈M(λ )
=
∑
∑
∑
λ ,μ ∈Λ s,t,u∈M(λ ) v∈M(μ )
tu f λ−1 f μ−1 cst x−1 ,λ cy,λ
vv ∑ cus w,λ cw−1 ,μ
.
w∈W
By Proposition 1.4.10(a), the parenthesised sum evaluates to δuv δsv δλ μ fλ . Inserting this into the above expression yields ∑λ ∈Λ ∑s,t∈M(λ ) fλ−1 cst cts = δxy , where x−1 ,λ y,λ the last equality holds by Proposition 1.4.10(b). (c) Let λ ∈ Λ . By Lemma 1.5.2, we can assume without loss of generality that ρ λ has been chosen such that the additional conditions in Theorem 1.4.8 are satisfied.
30
1 Generic Iwahori–Hecke Algebras
Thus, we have bt cts = cst w,λ bs , where bs , bt are positive real numbers (depending w−1 ,λ ss on λ ). This immediately shows that css w,λ = cw−1 ,λ and so n˜ w = n˜ w−1 . Similarly, −1 −1 ts −1 tu us ut ts su cst bt bu cut bu bs csu x,λ z,λ = cx,λ cy,λ cz,λ . y−1 ,λ cx−1 ,λ cz−1 ,λ = bs bt cy,λ Summing over all λ ∈ Λ and s, t, u ∈ M(λ ), this yields γ˜y−1 ,x−1 ,z−1 = γ˜x,y,z .
In general, the constants γ˜x,y,z are hard to compute. In the following example, we can at least deal with some special case. Example 1.5.4. Assume that L(s) > 0 for all s ∈ S and that 1 ∈ {x, y, z}. Then 1 if x = y = z = 1, γ˜x,y,z = 0 otherwise. Indeed, by Lemma 1.5.3(a), we can choose notation such that z = 1. Now let λ0 ∈ Λ be such that E λ0 is the unit representation of W . Then ρ λ0 (Tw ) = (ε L(w) ) for all w ∈ W . Let M(λ0 ) = {1}. Since aλ0 = 0, it follows that c11 w,λ0 = 0 for w = 1, and = 1. Now let λ ∈ Λ be such that λ = λ . By Example 1.3.3, we have aλ > 0. c11 0 1,λ 0
Since ρ λ (T1 ) is the identity matrix, we conclude that cst 1,λ = 0 for all s, t ∈ M(λ ). ˜ Hence, in the defining formula for γx,y,1 , the terms corresponding to λ are zero. So 11 we obtain γ˜x,y,1 = c11 x,λ cy,λ and this yields the above formula. 0
0
Proposition 1.5.5. J˜ is an associative algebra with identity element 1J˜ . Proof. Let x, y, z ∈ W . We must check that (tx ty )tz = tx (tytz ), which is equivalent to
∑ γ˜x,y,u−1 γ˜u,z,w−1 = ∑ γ˜x,u,w−1 γ˜y,z,u−1
u∈W
for all w ∈ W .
u∈W
Using the defining formula, the left-hand side evaluates to
∑
u∈W
∑
λ ∈Λ s,t,u∈M(λ )
tu us fλ−1 cst x,λ cy,λ cu−1 ,λ
=
∑
λ , μ ∈Λ
∑
s t,u∈M(λ ) s ,t ,u ∈M(μ )
∑
μ ∈Λ s ,t ,u ∈M(μ )
s f μ−1 csu,μt ctz,μu cuw−1 ,μ
tu t u u s fλ−1 f μ−1 cst x,λ cy,λ cz,μ cw−1 , μ
st ∑ cus u−1 ,λ cu,μ
.
u∈W
By Proposition 1.4.10(a), the parenthesised sum evaluates to δut δss δλ μ fλ . Hence, the above expression equals
∑
∑
λ ∈Λ s,t,u,u ∈M(λ )
tu uu u s fλ−1 cst x,λ cy,λ cz,λ cw−1 ,λ .
By a similar computation, the right-hand side evaluates to
1.5 The Asymptotic Algebra
∑
u∈W
∑
λ ∈Λ s,t,u∈M(λ )
∑
=
λ , μ ∈Λ
=
31
tu us f λ−1 cst x,λ cu,λ cw−1 ,λ
∑
∑
λ ∈Λ s,t,t ,u∈M(λ )
∑
μ ∈Λ s t ,u ∈M(μ )
s,t,u∈M(λ ) s ,t ,u ∈M(μ )
∑
us st tu fλ−1 f μ−1 cst x,λ cw−1 ,λ cy,μ cz,μ
f μ−1 csy,μt ctz,μu cuu−1s ,μ
us ∑ ctu u,λ cu−1 ,μ
u∈W
tt t u us fλ−1 cst x,λ cy,λ cz,λ cw−1 ,λ .
We see that both sides are equal, hence J˜ is associative. To show that 1J˜ is the ˜ we let x ∈ W and note that identity element of J,
tx 1J˜ = ∑ n˜w txtw = ∑ ∑ n˜ w γ˜x,w,y−1 ty = ∑ ∑ n˜w γ˜y−1 ,x,w ty = tx w∈W
y∈W w∈W
y∈W w∈W
by Lemma 1.5.3(a) and (b). A similar argument shows that 1J˜ tx = tx . Thus, 1J˜ is the ˜ identity element of J. Proposition 1.5.6. The linear map τ¯ : J˜ → K defined by τ¯ (tw ) = n˜ w−1 is a symmetrising trace such that τ¯ (txty−1 ) = δxy for all x, y ∈ W . Proof. Let x, y ∈ W . Then, using Lemma 1.5.3(b), we obtain
τ¯ (tx−1 ty ) =
∑ γ˜x−1 ,y,w−1 τ¯ (tw ) = ∑ γ˜x−1 ,y,w−1 n˜w−1 = δxy .
w∈W
w∈W
This implies that τ¯ (txty ) = τ¯ (tytx ) for all x, y ∈ W ; hence, τ¯ is a trace function. We also see that {tw | w ∈ W } and {tw−1 | w ∈ W } form a pair of dual bases; hence, τ¯ is non-degenerate. Proposition 1.5.7. For λ ∈ Λ , define a linear map ρ¯ λ : J˜ → Mdλ (K), tw → cst w,λ s,t∈M(λ ) . ˜ and all irreducible repThen ρ¯ λ is an absolutely irreducible representation of J, ˜ resentations of J (up to equivalence) arise in this way. In particular, J˜ is a split semisimple algebra. (Recall that K is any field containing ZW .) We have trace(tw , ρ¯ λ ) ≡
∑
s∈M(λ )
aλ λ css w,λ ≡ ε χ (Tw ) mod m
for all w ∈ W .
Hence, the character of ρ¯ λ is uniquely determined by χ λ (see also Remark 1.4.4). Proof. We must show that ρ¯ λ (txty ) = ρ¯ λ (tx )ρ¯ λ (ty ) for all x, y ∈ W . Now, by the definition of γ˜x,y,z , we have
λ −1 s t t u u s ρ¯ st (tx ty ) = ∑ γ˜x,y,z−1 cst = f c c c −1 ∑ ∑ μ x,μ y,μ z ,μ cstz,λ . z,λ z∈W
z∈W
μ ∈Λ s ,t ,u ∈M(μ )
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1 Generic Iwahori–Hecke Algebras
Using Proposition 1.4.10(a), the right-hand side evaluates to
∑
μ ∈Λ s ,t ,u ∈M(μ )
f μ−1 csx,μt cty,μu δu t δs s δλ μ fλ =
∑
t ∈M(λ )
λ t t ¯ (tx )ρ¯ λ (ty ) st , cst x,μ cy,μ = ρ
as required. To show that ρ¯ λ is absolutely irreducible, we argue as follows. By Proposition 1.5.6, we have a symmetrising trace where {tw | w ∈ W } and {tw−1 | w ∈ W } form a pair of dual bases. Consequently, the relations in Proposition 1.4.10 can be interpreted as “Schur relations” for the representations ρ¯ λ . Thus, we have λ (tw−1 ) = δsv δtu fλ ∑ ρ¯ stλ (tw ) ρ¯ uv
for all s, t, u, v ∈ M(λ ).
w∈W
By a general result about symmetric algebras (see [132, Remark 7.2.3]), the validity of these relations implies that ρ¯ λ is absolutely irreducible. Finally, if λ = μ in μ λ (t ) ρ Λ , then we also have the relations ∑w∈W ρ¯ st w ¯ uv (tw−1 ) = 0. In particular, this λ μ implies that ρ¯ and ρ¯ are not equivalent. Since dim J˜ = |W | = ∑λ ∈Λ dλ2 , we conclude that J˜ is split semisimple, and that λ {ρ¯ | λ ∈ Λ } are the irreducible representations of J˜ (up to equivalence). Remark 1.5.8. Let W = W1 × · · · ×Wd be the decomposition into irreducible components. As already noted in Remark 1.3.5, we have a tensor product decomposition H∼ = H1 ⊗A · · · ⊗A Hd , where Hi is the generic algebra associated with Wi and the restriction of L to Wi . Furthermore, λ
Irr(HK ) = {Eελ1 · · · Eε d | λi ∈ Λi },
where
IrrK (Wi ) = {E λi | λi ∈ Λi }.
For each i and λi ∈ Λi , let ρ λi be a balanced representation afforded by Eελi . Then, using the compatibility of the invariants aλ with the above decomposition (see Remark 1.3.5), we conclude that ρ λ1 · · · ρ λd (where λi ∈ Λi for all i) is a balanced representation for HK . Consequently, we also have a tensor product decomposition J˜ ∼ = J˜ 1 ⊗K · · · ⊗K J˜ d , where J˜ i is the asymptotic algebra defined with respect to Hi and the balanced representations {ρ λi | λi ∈ Λi }. Thus, to describe the structure of J˜ in general, it will be sufficient to do this assuming that (W, S) is irreducible. Definition 1.5.9. Recall that ZW = Z[2 cos(2π /mst ) | s,t ∈ S] and R ⊆ C is a subring such that ZW ⊆ R. A prime ideal p of R is called L-bad if fλ ∈ p for some λ ∈ Λ . We say that R is L-good if there are no L-bad prime ideals in R. Thus, we have R is L-good
⇔
f λ ∈ R× for all λ ∈ Λ
(where R× denotes the multiplicative group of invertible elements in R). We say that a commutative ring k (with 1) is L-good if there is a unital ring homomorphism R → k where R is L-good.
1.5 The Asymptotic Algebra
33
Table 1.4 Conditions for R to be L-good Condition on R (where L(s) > 0 for some s ∈ S) no condition ⎧ × if L(s ) = rL(s ) for some r ∈ {0, 1, . . . , n−1} ⎪ 2 ∈ R 0 1 ⎨ Bn n! ∈ R× if L(s1 ) = 0 ⎪ ⎩ no condition, otherwise Dn 2 ∈ R× F4 , E6 , E7 2, 3 ∈ R× E8 2, 3, 5 ∈ R× × 2 ∈ R if m = 6 and L(s1 ) = L(s2 ) I2 (m) m ∈ R× and 2 cos(2π /m) ∈ R, otherwise (m5) √ H3 2, 5 ∈ R× and 2 cos(2π /5) = 12 (−1 + 5) ∈ R √ 2, 3, 5 ∈ R× and 2 cos(2π /5) = 12 (−1 + 5) ∈ R H4 (The generators in S are labelled as in Table 1.1) Type An−1
Example 1.5.10. Assume that L(s) = 0 for all s ∈ S. Then fλ = |W |/dλ for all λ ; see Remark 1.3.6. Hence, in this case, R is L-good if and only if ZW ⊆ R and |W | ∈ R× . (In fact, in this case, one easily sees that the map w → tw defines an algebra ˜ isomorphism KW ∼ = J.) Now assume that W is irreducible and that L(s) > 0 for at least some s ∈ S. Then, the conditions for R to be L-good are given by Table 1.4. This information is extracted from the knowledge of cλ in all cases; see Remark 1.3.6 and the examples following it. To deal with type I2 (m), note the identities m even m2 2 − ζ 2 j − ζ −2 j ) = ∏ 4 1 j(m−2)/2 m j −j 2 ± (ζ + ζ ) = ∏ 2 1 j(m−2)/2
∏
m odd 2−ζ j −ζ−j = m
∏
2+ζ j +ζ−j = 1
1 j(m−1)/2 1 j(m−1)/2
These are used as follows. Assume, for example, that m is even and L(s1 ) > L(s2 ) > 0. Then, using the formulae in Example 1.3.7 and the above identities, we obtain
∏
λ ∈Λ
fλ =
∏
1 j(m−2)/2
m 2 − ζ 2 j − ζ −2 j
= 4m(m−2)/2−2 .
Hence, for m = 6, we need to require that m ∈ R× . On the other hand, for m = 6, we only need to require that 2 ∈ R× . The other cases are treated similarly. Proposition 1.5.11 (Cf. [111, 2.6], [112, 4.10]). Let R ⊆ C be a subring such that ZW ⊆ R and γ˜x,y,z ∈ R, n˜ w ∈ R for all x, y, z, w ∈ W . (For example, by Lemma 1.5.2, any L-good subring R ⊆ C has these properties.) Then, for each λ ∈ Λ , the balanced representation ρ λ can be chosen such that the following two conditions hold. (a) We have ρ¯ λ (tw ) = cst w,λ ∈ Mdλ (R) for all w ∈ W .
34
1 Generic Iwahori–Hecke Algebras
(b) There exists a symmetric matrix Bλ ∈ Mdλ (R) such that det(Bλ ) = 0
and
Bλ ρ¯ λ (tw−1 ) = ρ¯ λ (tw )tr Bλ
for all w ∈ W ;
furthermore, if p is a prime ideal in R such that det(Bλ ) ∈ p, then p is L-bad. In particular, if R is L-good, then det(Bλ ) ∈ R× . Proof. By Remark 1.5.8, we can assume that W is irreducible; furthermore, we may assume that R is the subring of C generated by ZW and all constants γ˜x,y,z , n˜ w for x, y, z, w ∈ W . In particular, R ⊆ R. Recall that: if W √ is a Weyl group, then ZW = Z; if W is of type H3 or H4 , then ZW = Z[ 12 (−1 + 5)]; if W is of type I2 (m), then ZW = Z[ζ + ζ −1 ], where ζ ∈ C is a root of unity of order m. Thus, except possibly in type I2 (m), the ring ZW is a principal ideal domain and, hence, so is R. So let us first deal separately with W of type I2 (m). We can assume that L(s1 ) L(s2 ) 0. Note that (a) and (b) are clear for one-dimensional representations, where we can just take Bλ = (1). Now consider a two-dimensional representation σ εj . In Example 1.4.6, we have seen that σ εj is balanced and that the corresponding leading matrix coefficients lie in ZW . Hence, (a) holds. Furthermore, let Ω j be as in Example 1.4.6. Then all entries of Ω j lie in ZW [Γ ]. Let B j be the matrix obtained by taking the constant terms of the entries of Ω j . Then the invariance property in (b) holds. Furthermore, if L(s1 ) > L(s2 ) 0, then det(B j ) = 1. If L(s1 ) = L(s2 ) 0, then det(B j ) is non-zero and divides m. So the identities mentioned in Example 1.5.10 show that the additional requirements in (b) hold. We can now assume that W is not of type I2 (m). Then, as remarked above, R is a principal ideal domain. Let J˜ R be the R-span of {tw | w ∈ W }. Since the structure constants of J˜ lie in R, we see that J˜ R is an R-subalgebra of J˜ and J˜ = K ⊗R J˜ R . Then a standard argument (e.g. see [132, Remark 7.3.7]) implies that ρ¯ λ is equivalent to a representation which is realized over R; that is, there exists an invertible matrix Pλ ∈ Mdλ (K) such that Pλ−1 ρ¯ λ (tw )Pλ ∈ Mdλ (R)
for all w ∈ W .
Now use Pλ to transform the original balanced representation ρ λ to a representation
ρ˜ λ : HK → Mdλ (K),
Tw → Pλ−1 ρ λ (Tw )Pλ .
Since all entries of Pλ lie in K, it is clear that ρ˜ λ is also balanced. The corresponding leading matrix coefficients are given by the entries of the matrices Pλ−1 ρ¯ λ (tw )Pλ and, hence, lie in R. Thus, replacing ρ λ by ρ˜ λ , we see that (a) holds. To prove (b), let Bλ1 =
∑ ρ¯ λ (ty )tr .ρ¯ λ (ty ) ∈ Mdλ (R).
y∈W
This matrix clearly is symmetric. Let 0 = e = (e1 , . . . , edλ ) ∈ Rd . Since ρ¯ λ is irreducible, there exists some y ∈ W such that ρ¯ λ (ty )tr = 0 and, hence, the standard scalar product of this vector with itself will be strictly positive. (Recall that R ⊆ R.)
1.5 The Asymptotic Algebra
35
Consequently, we have eBλ1 etr > 0. Thus, Bλ1 is positive-definite and, in particular, det(Bλ1 ) = 0. For any x ∈ W , we have Bλ1 .ρ¯ λ (tx−1 ) =
∑ ρ¯ λ (ty )tr .ρ¯ λ (tytx−1 ) = ∑
y∈W
γ˜y,x−1 ,z−1 ρ¯ λ (ty )tr .ρ¯ λ (tz ).
y,z∈W
Now γ˜y,x−1 ,z−1 = γ˜x−1 ,z−1 ,y = γ˜z,x,y−1 by Lemma 1.5.3. Hence, the right-hand side of the above identity equals
∑
γ˜z,x,y−1 ρ¯ λ (ty )tr .ρ¯ λ (tz ) =
y,z∈W
∑ ρ¯ λ (tztx )tr .ρ¯ λ (tz ) = ρ¯ λ (tx)tr .Bλ1 .
z∈W
Let 0 = n ∈ R be a greatest common divisor of all non-zero coefficients of Bλ1 and Bλ = n−1 Bλ1 ∈ Md (R). Then Bλ is a non-singular symmetric matrix such that the invariance property in (b) holds. It remains to prove the statement about det(Bλ ). This follows by a standard argument on symmetric algebras. Let p be a prime ideal of R such that fλ ∈ p. Note that Bλ ≡ 0 mod p, since 1 is a greatest common divisior of all non-zero coefficients of Bλ . Let k be the field of fractions of R/p and denote by Bλk the matrix obtained by reducing all coefficents modulo p. By reduction modulo p, we also obtain a k-algebra J˜ k = k ⊗R J˜ R and a corresponding matrix representation ρ¯ kλ : J˜ k → Mdλ (k). Hence, we have Bλk = 0
and
Bλk .ρ¯ kλ (tw ) = ρ¯ kλ (tw−1 )tr .Bλk
for all w ∈ W .
Now let us regard kdλ as a J˜ k -module via ρ¯ kλ . Then the above relations imply that the nullspace of Bλk is a proper J˜ k -invariant subspace. Hence, if we can show that ρ¯ kλ is irreducible, then that nullspace must be zero and so Bλk is invertible; that is, det(Bλ ) ∈ p, as required. To show that ρ¯ kλ is irreducible, we argue as follows. By Proposition 1.5.6, we know that J˜ is symmetric and so we have the Schur relations for the matrix coefficients of ρ¯ λ . Reducing these relations modulo p, we obtain fλ mod p if s = v, t = u, λ λ ¯ ¯ ρ (t ) ρ (t ) = −1 w ∑ k,st k,uv w 0 otherwise. w∈W Since fλ ≡ 0 mod p, one easily deduces from this that ρ¯ kλ is (absolutely) irreducible; see [132, Remark 7.2.3]. Conjecture 1.5.12. (a) We have γ˜x,y,z ∈ Z for all x, y, z ∈ W and n˜ w ∈ Z for all w ∈ W . ˜ are orthogonal idempotents in J. ˜ (Recall that (b) The elements {n˜ d td | d ∈ D} 1J˜ = ∑d∈D˜ n˜ d td .) We will see in Section 2.5 that this conjecture holds under certain assumptions on W, L which are satisfied, for example, in the equal-parameter case. In Theorem 1.7.10, we will see that it also holds for certain choices of unequal parameters in type Bn . It would be very interesting and useful to find a general proof of this conjecture which works without any assumptions on W, L.
36
1 Generic Iwahori–Hecke Algebras
1.6 Introducing Cells Lusztig’s work [214, §8] (see also [220, Chap. 4]) on representations of finite groups of Lie type showed that there is a natural partition of IrrK (W ) into “cells” or “families”, where W is a finite Weyl group and L is the length function on W . In a completely different context, “cells” appeared in the work of Joseph on primitive ideals in the enveloping algebra of a complex semisimple Lie algebra; see the survey [187]. Then Kazhdan and Lusztig [195], [219] (see also Chapter 2) constructed a new basis {Cw | w ∈ W } of H (for general W, L) and used this basis to define partitions of W into left, right, and two-sided “cells” (depending on L); furthermore, each “cell” in this sense is shown to carry a representation of H, with a “canonical” basis indexed by the elements in the cell. In 1.6.1 and 1.6.2 we shall give axiomatic versions of these constructions and then apply them to the algebra J˜ with its basis {tw | w ∈ W }. Following [112], this yields a first approach to the theory of “cells”. 1.6.1. Let k be a commutative ring (with 1) and H any associative algebra (with identity) which is free over k with a basis indexed by the elements of our finite Coxeter group W . Let us denote this basis by {cw | w ∈ W } and write cx cy =
∑ hx,y,z cz ,
where
hx,y,z ∈ k.
z∈W
Also assume that the k-linear map H → H defined by cw → cw−1 is an antiautomorphism of H. By analogy with Kazhdan and Lusztig [195], [219], we introduce the following relations on W . Let y, z ∈ W . We write z ←L y if there exists some x ∈ W such that hx,y,z = 0; that is, cz occurs with non-zero coefficient in cx cy (when expressed in the c-basis). Let L be the pre-order1 relation on W generated by ←L ; that is, we have z L y if there exist elements z = y0 , y1 , . . . , ym = y in W such that yi−1 ←L yi for 1 i m. Let ∼L denote the associated equivalence relation; the corresponding equivalence classes are called the left cells of W . Similarly, we can define a pre-order R by considering multiplication by cx on the right. The equivalence relation associated with R will be denoted by ∼R and the corresponding equivalence classes are called the right cells of W . We have z R y
⇔
z−1 L y−1 .
This follows by using the anti-homomorphism cw → cw−1 . Thus, any statement concerning the left pre-order relation L has an equivalent version for the right preorder relation R , via the map cw → cw−1 . Finally, we define a pre-order LR by the condition that z LR y if there exists a sequence z = y0 , y1 , . . . , ym = y such that, for each i ∈ {1, . . . , m}, we have yi−1 L yi 1
A pre-order on a set is, by definition, just a transitive reflexive relation (but we do not necessarily have anti-symmetry). Thus, a partial order is a pre-order which is also anti-symmetric. If is a pre-order, we have an associated equivalence relation given by x ∼ y if and only if x y and y x. Consequently, induces a partial order on the equivalence classes under ∼.
1.6 Introducing Cells
37
or yi−1 R yi . The equivalence relation associated with LR will be denoted by ∼LR and the corresponding equivalence classes are called the two-sided cells of W . Thus, we have defined partitions W=
{left cells} =
{right cells} =
{two-sided cells}.
Obviously, these notions heavily depend on H and on the choice of the basis {cw }. 1.6.2. Let C be a left cell of W (with respect to H). Then C gives rise to a representation of H. Following Kazhdan and Lusztig [195], [219], this is constructed as follows. Let y ∈ W and define the following two k-submodules of H: IL y := cx | x ∈ W and x L yk , ˆIL y := cx | x ∈ W and x L y but x ∼L yk . ˆL By the definition of the relations L and ∼L , it is clear that IL y and Iy are left L L L ideals of H; furthermore, we have Iˆ y ⊆ Iy . Hence, the quotient Iy /Iˆ L y is a left H-module. Note that L IL y = Iy
and
ˆL Iˆ L y = Iy
if y ∼L y .
Hence, if C is the left cell containing y, then the above quotient only depends on C and we shall write ˆL [C]k := IL y /Iy
and
ex := residue class of cx in [C]k (where x ∈ C).
Thus, [C]k is a left H-module with a “canonical” basis {ex | x ∈ C}. Explicitly, the action of cw is given by the formula cw .ex :=
∑ hw,x,y ey ,
where w ∈ W and x ∈ C.
y∈C
ˆR Similarly, we define right ideals IR y , Iy giving rise to right H-modules associated ˆ LR with the various right cells of W . Finally, we can define two-sided ideals ILR y , Iy giving rise to (H, H)-bimodules associated with the various two-sided cells of W . Finally, assume that k is a field and H is semisimple. Then, since the left cells form a partition of W , we obtain a corresponding direct sum decomposition of H: H∼ =
[C]k
(isomorphism of left H-modules).
C left cell
We obtain analogous decompositions with respect to right cells or two-sided cells. Example 1.6.3. Let H = ZW , the group algebra of W where cw = w for all w ∈ W . Then each basis element w is invertible in ZW and, hence, there is only one left cell, only one right cell and only one two-sided cell, namely the whole of W . So these notions are not very interesting in this case.
38
1 Generic Iwahori–Hecke Algebras
The same thing happens when we take H = H where cw = Tw for all w ∈ W . Just note that each basis element Tw is invertible in H. In Chapter 2 we shall consider again the case where H = H, but work with the Kazhdan–Lusztig basis of H. For the remainder of this section, let us assume that H = J˜
and
cw = tw for all w ∈ W .
Note that, in this case, we have hx,y,z = γ˜x,y,z−1 for all x, y, z ∈ W . Also note that, by ˜ Lemma 1.5.3(c), the map tw → tw−1 is an anti-automorphism of J. Definition 1.6.4. The left, right or two-sided cells defined by taking H = J˜ (with ˜ basis {tw | w ∈ W }) are called the left, right or two-sided J-cells of W respectively. Lemma 1.6.5. Let x, y, z ∈ W be such that γ˜x,y,z = 0. Then x ∼L y−1 , y ∼L z−1 , z ∼L x−1 . Proof. By the definition of L , we have the implication: (1) If γ˜x,y,z = 0, then z−1 L y. Now, by Lemma 1.5.3(a), we have γ˜x,y,z = γ˜z,x,y = γ˜y,z,x . Hence, (1) implies: (2) If γ˜x,y,z = 0, then z−1 L y, y−1 L x and x−1 L z. By Lemma 1.5.3(c), we also have γ˜x,y,z = γ˜y−1 ,x−1 ,z−1 . Hence, (1), (2) imply: (3) If γ˜x,y,z = 0, then z L x−1 , x L y−1 and y L z−1 . Combining now (2) and (3), we deduce that x ∼L y−1 , y ∼L z−1 , z ∼L x−1 .
Lemma 1.6.6. If x, y ∈ W are such that x L y, then x ∼L y. (Analogous statements hold for R and LR .) Furthermore, we have w ∼LR w−1 for any w ∈ W . Proof. It is sufficient to prove the first statement for an elementary step in the definition of L ; that is, we can assume that x ←L y. This means that γ˜w,y,x−1 = 0 for some w ∈ W and so Lemma 1.6.5 shows that x ∼L y, as required. Now fix w ∈ W . By Lemma 1.5.3, we have ∑x∈W γ˜w−1 ,w,x n˜ x = 1 and so γ˜w−1 ,w,x = 0 for some x ∈ W . This implies that x−1 L w and so x−1 ∼L w. On the other hand, by Lemma 1.5.3(c), we have γ˜w−1 ,w,x−1 = γ˜w−1 ,w,x = 0. So we can also deduce that x L w and x ∼L w. Thus, we have shown that w ∼L x ∼R w−1 ; in particular, w ∼LR w−1 . Corollary 1.6.7. Let x, y, z ∈ W be such that γ˜x,y,z = 0. Then the elements x±1 , y±1 , ˜ z±1 all lie in the same two-sided J-cell. Proof. This is an immediate consequence of Lemmas 1.6.5 and 1.6.6.
Remark 1.6.8. Let x, y ∈ W . By Lemma 1.6.6, we have x ∼LR y if and only if there is a sequence of elements x = x0 , x1 , . . . , xm = y in W such that, for each i ∈ {1, . . . , m}, ˜ we have xi−1 ∼L xi or xi−1 ∼R xi . It follows that the two-sided J-cells of W are the smallest subsets of W which are at the same time unions of left cells and unions of right cells. Since, furthermore, x ∼R y if and only if x−1 ∼L y−1 , we conclude that ˜ the whole cell structure of W is determined by the knowledge of the left J-cells.
1.6 Introducing Cells
39
˜ Lemma 1.6.9. For any two-sided J-cell F , the subspace J˜ F := tw | w ∈ F K ⊆ J˜ is a two-sided ideal. We have a direct sum decomposition J˜ =
J˜ F .
˜ F two-sided J-cell
˜ Proof. By Lemma 1.6.6, we have J˜ F = ILR w , where w ∈ F ; hence, JF is a two˜ sided ideal in J. Since W is partitioned into two-sided cells, we obtain the above direct sum decomposition. 1.6.10. Let {ρ¯ λ | λ ∈ Λ } be as in Proposition 1.5.11. Recall that each ρ¯ λ is given λ by the leading matrix coefficients cst w,λ associated with a balanced representation ρ afforded by Eελ . Now let w ∈ W and λ ∈ Λ . Then we define E λ L w
def
⇔
cst w,λ = 0 for some s, t ∈ M(λ ).
We claim that L does not depend on the choice of the balanced representations ρ λ . Indeed, if σ λ also is a balanced representation afforded by Eελ , with corresponduv , then the maps J ˜ → Md (K) defined by ing leading matrix coefficients dw, λ λ ρ¯ λ : tw → cst w,λ s,t∈M(λ )
and
uv σ¯ λ : tw → dw, λ u,v∈M(λ )
are equivalent representations of J˜ (since they have the same character; see Proposition 1.5.7). Clearly, this implies that ρ¯ λ (tw ) = 0 if and only if σ¯ λ (tw ) = 0, which yields the desired independence. Once the independence is established, we can assume that the balanced representations ρ λ have been chosen such that the additional conditions in Theorem 1.4.8 are ts satisfied. Hence, for any s, t ∈ M(λ ), we will have cst w,λ = 0 if and only if cw−1 ,λ = 0. This yields the equivalence E λ L w
⇔
E λ L w−1 ,
which will be useful at several places below. Proposition 1.6.11. Let λ ∈ Λ . Then there is some w ∈ W such that E λ L w; ˜ furthermore, all such w belong to the same two-sided J-cell of W . This two-sided ˜ J-cell, therefore, only depends on E λ and will be denoted by Fλ . Thus, we obtain a natural surjective map ˜ IrrK (W ) → {set of two-sided J-cells of W },
E λ → Fλ .
Proof. By Proposition 1.4.10(c), there is some w ∈ W such that E λ L w; conversely, for w ∈ W , there exists some λ ∈ Λ such that E λ L w. So it remains to show that, given λ ∈ Λ , the set {w ∈ W | E λ L w} is contained in a two-sided ˜ J-cell. Equivalently, we must show that the set {w ∈ W | ρ¯ λ (tw ) = 0} is contained ˜ in a two-sided J-cell. This is seen as follows. By the Artin–Wedderburn theorem,
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1 Generic Iwahori–Hecke Algebras
we have a direct sum decomposition J˜ = μ ∈Λ J˜ μ , where J˜ μ is the simple component corresponding to the irreducible representation ρ¯ μ . The ideals {J˜ μ | μ ∈ Λ } ˜ furthermore, we have are precisely the (non-zero) minimal two-sided ideals of J;
{h ∈ J˜ | ρ¯ μ (h) = 0} ⊆ J˜ μ
for any μ ∈ Λ .
Now, the Artin–Wedderburn decomposition will be compatible with the decomposition in Lemma 1.6.9; that is, the given simple component J˜ λ must be contained in J˜ F for a unique F . In particular, {w ∈ W | ρ¯ λ (tw ) = 0} ⊆ F , as required. ˜ Definition 1.6.12. Let F be a two-sided J-cell (or, more generally, a union of two˜ sided J-cells) of W . Then we denote by IrrK (W | F ) the set of all E λ ∈ IrrK (W ) such that E λ L w for some w ∈ F . Remark 1.6.13. Here is an alternative characterisation of the sets IrrK (W | F ). For ˜ any two-sided J-cell F in W , let us define 1F :=
∑
n˜ d td .
˜ d∈D∩F
˜ ˜ in W . Let Z(J) Then we have 1J˜ = ∑F 1F where F runs over all two-sided J-cells ˜ denote the centre of J. Using the decomposition in Lemma 1.6.9, we conclude that ˜ 1F ∈ Z(J),
12F = 1F ,
1F 1F = 0 (if F = F ).
˜ It follows that Thus, the elements {1F } are orthogonal idempotents in Z(J). IrrK (W | F ) = {E λ | λ ∈ Λ is such that ρ¯ λ (1F ) = 0}. Indeed, if ρ¯ λ (1F ) = 0, then we have ρ¯ λ (td ) = 0 for some d ∈ D˜ ∩ F and so E λ ∈ Irr(W | F ). Conversely, let E λ ∈ Irr(W | F ). Now, there certainly exists some two˜ sided J-cell F such that ρ¯ λ (1F ) = 0. Then ρ¯ λ (td ) = 0 for some d ∈ D˜ ∩ F . But then we also have d ∈ F by Proposition 1.6.11 and so F = F as required. Lemma 1.6.14. Let x, y ∈ W . Then we have x ∼LR y if and only if there exist a sequence x = x0 , x1 , . . . , xm = y in W and a sequence λ1 , . . . , λm in Λ such that, for each i ∈ {1, . . . , m}, we have E λi L xi−1 and E λi L xi . Proof. Assume first that x ∼LR y. By definition, there exists a sequence x = x0 , x1 , . . . , xm = y in W such that, for each i ∈ {1, . . . , m}, we have xi−1 ←L xi or xi−1 ←R xi . So it will be sufficient to consider an elementary step where, for some w ∈ W , we have γ˜w,x,y−1 = 0 or γ˜x,w,y−1 = 0. We must show that then there exists some λ ∈ Λ such that E λ L x and E λ L y. Now, if γ˜w,x,y−1 = 0, then the defining formula (see Section 1.5) shows that there tu us exist some λ ∈ Λ and s, t, u ∈ M(λ ) such that cst w,λ = 0, cx,λ = 0 and cy−1 ,λ = 0.
Hence, we have E λ L x and E λ L y−1 . But then, as already remarked in 1.6.10, we also have E λ L y, as required. If γ˜x,w,y−1 = 0, the argument is entirely analogous. Thus, the “only if” part is shown.
1.6 Introducing Cells
41
Conversely, assume that there exists a sequence x = x0 , x1 , . . . , xm = y in W and a sequence λ1 , . . . , λm in Λ such that the above conditions are satisfied. Then we must show that x ∼LR y. Again, it will be sufficient to consider an elementary step in that sequence; that is, we can assume that there exists some λ ∈ Λ such that E λ L x and E λ L y. But then we have x ∼LR y by Proposition 1.6.11. Recall from Example 1.2.6 that we have a “duality” operation λ → λ † on Λ such † that E λ ∼ = E λ ⊗ sgn, where sgn denotes the sign representation of W . Lemma 1.6.15. Let w0 ∈ W be the longest element. Let λ ∈ Λ and w ∈ W . Then E λ L w †
⇔
E λ L ww0 .
Proof. We can assume that K ⊆ R and that the balanced representation ρ λ is chosen as in Theorem 1.4.8, where Ω λ is a certain non-singular symmetric matrix such that (a)
Ω λ ρ λ (Tw−1 ) = ρ λ (Tw )tr Ω λ
for all w ∈ W .
Now, the first thing we need to do is to find a balanced representation afforded by † E λ . But this is easily done using the algebra automorphism † : HK → HK given by † λ † Tw = (−1)l(w) Tw−1 −1 for all w ∈ W . Indeed, by Example 1.2.6, the map Tw → ρ (Tw ) defines a representation afforded by Eελ . Furthermore, the above relation implies †
tr λ l(w) λ λ ρ λ (Tw† )tr Ω λ = (−1)l(w) ρ λ (Tw−1 Ω ρ (Tw−1 ) = Ω λ ρ λ (Tw†−1 ) −1 ) Ω = (−1)
for all w ∈ W . Hence, by Proposition 1.4.5, the representation given by Tw → ρ λ (Tw† ) (w ∈ W ) is balanced. Next, we bring w0 into the picture. It is known that, for any w ∈ W , we have Tw0 = Tw−1 Tww0 (see [132, 1.5.3]) and so l(w) Tww0 Tw−1 . Tw† = (−1)l(w) Tw−1 −1 = (−1) 0
By 1.3.4, we have ρ λ (Tw20 ) = ε 2Nλ Idλ , where Nλ = aλ † − aλ . We deduce that (b)
ε aλ † ρ λ (Tw† ) = (−1)l(w) ε aλ ρ λ (Tww0 ) ε −Nλ ρ λ (Tw0 ) .
We claim that ε −Nλ ρ λ (Tw0 ) ∈ Mdλ (O0 ). To see this, let γ ∈ Γ be such that all entries of the matrix P := ε γ −Nλ ρ λ (Tw0 ) lie in O0 and at least one of them does not lie in m. We will show that γ = 0 (and this proves the claim). Now, we have
ρ λ (Tw0 ) = (Ω λ )−1 ρ λ (Tw0 )tr Ω λ and so P2 = P (Ω λ )−1 Ptr Ω λ . Then recall that ρ λ and Ω λ satisfy the additional conditions in Theorem 1.4.8; in particular, we have Ω λ ≡ Dλ mod m, where Dλ is a diagonal matrix whose diagonal entries bs (s ∈ M(λ )) are positive real numbers. Then we also have (Ω λ )−1 ≡ (Dλ )−1 mod m and so
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1 Generic Iwahori–Hecke Algebras
(P2 )ss ≡
∑
t∈M(λ )
2 b−1 t bs (Pst )
for any s ∈ M(λ ).
Now choose s ∈ M(λ ) such that Pst has a non-zero constant term for some t ∈ M(λ ). Then the above sum is non-zero and so (P2 )ss ≡ 0 mod m. On the other hand, we have P2 = ε 2γ Idλ . It follows that γ = 0, as claimed. Now we can argue as follows. Assume that E λ L w; that is, some matrix entry of ε aλ † ρ λ (Tw ) has a non-zero constant term. But then (b) shows that some matrix entry of ε aλ ρ λ (Tww0 ) has a non-zero constant term and so E λ L ww0 . The reverse implication is then clear. †
˜ ˜ Corollary 1.6.16. Let F be a two-sided J-cell. Then F w0 is a two-sided J-cell and IrrK (W | F w0 ) = {E λ ⊗ sgn | E λ ∈ IrrK (W | F )}. ˜ Proof. In order to show that F w0 is a two-sided J-cell, it is enough to show that if x, y ∈ W are such that x ∼LR y, then xw0 ∼LR yw0 . Now, if x ∼LR y, then there exist a sequence x = x0 , x1 , . . . , xm = y in W and a sequence λ1 , . . . , λm in Λ satisfying the conditions in Lemma 1.6.14. But then Lemma 1.6.15 shows that x0 w0 , x1 w0 , . . . , xm w0 and λ1† , . . . , λm† are also sequences which satisfy the conditions in Lemma 1.6.14. Hence, we have xw0 ∼LR yw0 , as desired. Then the statement about IrrK (W | F w0 ) also follows from Lemma 1.6.15. Example 1.6.17. Assume that L(s) > 0 for all s ∈ S. By Example 1.5.4, we have ˜ γ˜x,y,1 = 0 unless x = y = 1. This shows that {1} is a two-sided J-cell. We claim that IrrK (W | {1}) = {E λ0 },
where E λ0 is the unit representation of W .
Indeed, for any λ = λ0 , we have aλ > 0 (see Example 1.3.3) and, hence, cst 1,λ = 0 for
all s, t ∈ M(λ ). This implies that IrrK (W | {1}) = {E λ0 }, as desired. Furthermore, ˜ and that using Corollary 1.6.16, we conclude that {w0 } is a two-sided J-cell IrrK (W | {w0 }) = {sgn}. Note that these statements fail if L(s) = 0 for some s ∈ S (see Example 1.7.3 below). ˜ ˜ Conjecture 1.6.18. Every left J-cell contains a unique element of D. The situation for this conjecture is similar to that for Conjecture 1.5.12. The following remark establishes a connection between these two conjectures. Remark 1.6.19. Assume that Conjecture 1.6.18 holds. Then we claim that Conjec˜ are orthogonal idempotents ture 1.5.12(b) holds; that is, the elements {n˜ d td | d ∈ D} ˜ we have ˜ To prove this, we first notice that, for any x, y ∈ W and d ∈ D, in J. 1 if x = y ∼L d, (a) d2 = 1 and γ˜x−1 ,y,d n˜ d = 0 otherwise.
1.7 Examples of Cells
43
Indeed, by Lemma 1.5.3(b), we have ∑d ∈D˜ γ˜x−1 ,y,d n˜ d = δxy . Now, if γ˜x−1 ,y,d = 0, then x ∼L d by Lemma 1.6.5. So, by Conjecture 1.6.18, the sum reduces to γ˜x−1 ,y,d n˜ d , where d ∈ D˜ is the unique element such that x ∼L d. Thus, we have γ˜x−1 ,y,d n˜ d = δxy . Now let x = y. Then γ˜x−1 ,x,d = 0 and so we also have x ∼L d −1 by Lemma 1.6.5. On the other hand, by Lemma 1.5.3(c), we already know that ˜ Hence, Conjecture 1.6.18 implies that d = d −1 . Thus, (a) is proved. d −1 ∈ D. ˜ Then td td = ∑z∈W γ˜ −1 tz = ∑z∈W γ˜ −1 tz , where the secNow let d, d ∈ D. d,d ,z z ,d,d ond equality holds by Lemma 1.5.3(a). Hence, (a) yields that td2 = n˜ −1 d td and td td = 0 ˜ are orthogonal idempotents, as claimed. if d = d . Thus, the elements {n˜ d td | d ∈ D}
1.7 Examples of Cells ˜ We consider a number of examples where the two-sided J-cells and the corresponding partition of IrrK (W ) (see Definition 1.6.12) can be explicitly determined. A considerable reduction of the required computations is achieved by the following general result, a first version of which appeared in [103, Theorem 4.10]. Proposition 1.7.1. Let λ ∈ Λ and assume that fλ = 1
ts cst w−1 ,λ = cw,λ ∈ Z
and
for all w ∈ W and s, t ∈ M(λ ).
Then the following hold. (a) For each pair (s, t) ∈ M(λ ) × M(λ ), there exists a unique w ∈ W such that st cst w,λ = 0; we have in fact cw,λ = ±1. Let us denote this element by wλ (s, t). (b) The map M(λ ) × M(λ ) → W , (s, t) → wλ (s, t), is injective. ˜ (c) The two-sided J-cell associated to E λ (see Proposition 1.6.11) is given by Fλ = {wλ (s, t) | s, t ∈ M(λ )} and we have IrrK (W | Fλ ) = {E λ }. ˜ and, for a (d) For a fixed t ∈ M(λ ), the set {wλ (s, t) | s ∈ M(λ )} is a left J-cell ˜ fixed s ∈ M(λ ), the set {wλ (s, t) | t ∈ M(λ )} is a right J-cell. Proof. (a) Let s, t ∈ M(λ ). Using the assumptions on fλ and cst w,λ , the “Schur relations” in Proposition 1.4.10(a) show that 1 = fλ =
∑ cstw,λ ctsw−1 ,λ = ∑
w∈W
w∈W
st 2 cw,λ .
st st Since cst w,λ ∈ Z, there will be a unique w ∈ W such that cw,λ = 0; in fact, cw,λ = ±1. (b) Let s, t, s , t ∈ M(λ ) be such that (s, t) = (s , t ). Assume, if possible, that wλ (s, t) = wλ (s , t ). Then (a) and the relations in Proposition 1.4.10(a) yield that
0=
w∈W
which is a contradiction.
s st s t ∑ cstw,λ ctw−1 ,λ = ∑ cw,λ cw,λ = ±1, w∈W
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1 Generic Iwahori–Hecke Algebras
(c) By (a), we have E λ L wλ (s, t) for all s, t ∈ M(λ ) and so Fλ := {wλ (s, t) | s, t ∈ M(λ )} ⊆ Fλ . Now let us fix s, t ∈ M(λ ) and denote w = wλ (s, t). Consider the relations in Proposition 1.4.10(b): vu ∑ ∑ fμ−1 cuv w,μ cw−1 , μ = 1. μ ∈Λ u,v∈M(μ )
The contribution to this sum coming from the term where μ equals λ is given by
∑
u,v∈M(λ )
vu fλ−1 cuv w,λ cw−1 ,λ =
∑
u,v∈M(λ )
cuv w,λ
2
2 = cst = 1, w,λ
where we used (a) and the injectivity of the map in (b). Hence, we deduce that
∑
∑
μ ∈Λ , μ =λ u,v∈M(μ )
vu f μ−1 cuv w,μ cw−1 , μ = 0.
Now, we can assume that the balanced representations ρ μ satisfy the additional conditions in Theorem 1.4.8. So, for each μ ∈ Λ and u, v ∈ M(μ ), we have vu −1 −1 uv 2 f μ−1 cuv w,μ cw−1 , μ = f μ bv bu cw,μ and this is a non-negative real number. Hence, there are no cancellations in the above sum. So, all terms in that sum must be zero; that is, we have for all u, v ∈ M(μ ) where μ = λ .
cuv w,μ = 0
(∗)
This already shows that w ∈ Fμ for any μ = λ . So all that remains is to show that Fλ ⊆ Fλ . For this purpose, let x, y ∈ W and consider the structure constant γ˜x,y,w . In the defining formula, all terms corresponding to μ = λ will be zero, thanks to (∗). In combination with (a) and Lemma 1.5.3(a) this yields that
γ˜w,x,y = γ˜y,w,x = γ˜x,y,w =
∑
s,t∈M(λ )
tu us fλ−1 cst x,λ cy,λ cw,λ = 0 unless x, y ∈ Fλ .
By the definition of ∼LR , this implies that Fλ ⊆ Fλ , as desired. (d) Let t ∈ M(λ ) be fixed and y = wλ (t, t). Let s ∈ M(λ ) and x = wλ (s, t). We claim that x ∼L y. By Lemma 1.6.6, it is enough to show that x L y and, for this purpose, it is enough to show that γ˜x,y,x−1 = 0. Now, we have
γ˜x,y,x−1 =
∑
∑
μ ∈Λ s ,t ,u ∈M(μ )
f μ−1 csx,μt cty,μu cux−1s ,μ =
∑
s ,t ,u ∈M(λ )
csx,λt cty,λu csx,λu ,
where the second equality holds by (∗) and the assumptions on λ . But then (a) and (b) show that (s , t ) = (s, t) and (t , u ) = (t, t). Hence, we obtain that γ˜x,y,x−1 =
1.7 Examples of Cells
45
2 tt (cst x,λ ) cy,λ = 0, as required. Thus, we have shown that the set {wλ (s, t) | s ∈ M(λ )} ˜ is contained in a left J-cell. Conversely, let z ∈ W and assume that y ∼L z. Then we must show that z = wλ (s , t) for some s ∈ M(λ ). For this purpose, it will be sufficient to consider an elementary step in the definition of ∼L ; recall that ∼L is equivalent to L by Lemma 1.6.6. Thus, we can assume that γ˜x,y,z−1 = 0 for some x ∈ W . By the defining formula, there exists some μ ∈ Λ and s , t , u ∈ M(μ ) such that
csx,μt cty,μ,u cuz−1s ,μ = 0.
Using (∗), we conclude that μ = λ . By the assumptions on λ , we have cuz−1s ,λ = csz,λu . Then (a) and (b) show that x = wλ (s , t ), z = wλ (s , u ) and (t , u ) = (s, t). Hence, we have z = wλ (s , t) as required. Thus, we have shown that the set {wλ (s, t) | s ∈ ˜ ˜ M(λ )} is a left J-cell. The argument for right J-cells is entirely analogous. Example 1.7.2. Let λ ∈ Λ be such that dλ = 1 and fλ = 1. Let us write M(λ ) = {1}. λ For s ∈ S, we have ρ λ (Ts ) = (vs ) or (−v−1 s ). So, for any w ∈ W , we have ρ (Tw ) = 11 ρ λ (Tw−1 ) = (±ε gw ), where gw = gw−1 ∈ Γ . In particular, we have c11 = c ∈Z w,λ w−1 ,λ for all w ∈ W . So we can apply Proposition 1.7.1 and this yields that IrrK (W | Fλ ) = {E λ },
where
Fλ = {wλ (1, 1)}.
The above conditions are satisfied, for example, for the unit and the sign representation of W , where L(s) > 0 for all s ∈ S (see Example 1.3.3). This yields another ˜ (see Example 1.6.17). proof for the fact that {1} and {w0 } are two-sided J-cells
˜ Table 1.5 Two-sided J-cells F and the corresponding sets IrrK (W | F ) for type I2 (m) L(s1 ) = L(s2 ) > 0 {10 } {1W } {(sgn1 ), (sgn2 ), σ εj (all j)} W \ {10 , 1m } {1m } {sgn} (where sgn1 and sgn2 have to be omitted if m is odd) L(s1 ) > L(s2 ) > 0 {10 } {1W } {21 } {sgn1 } W \ {10 , 21 , 1m−1 , 1m } {σ εj (all j)} {sgn2 } {1m−1 } {1m } {sgn} L(s1 ) > L(s2 ) = 0 {10 , 21 } {1W , sgn1 } {σ εj (all j)} W \ {10 , 21 , 1m−1 , 1m } {1m , 1m−1 } {sgn, sgn2 }
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1 Generic Iwahori–Hecke Algebras
Example 1.7.3. Let W be of type I2 (m) (m 3), with generators s1 , s2 such that ˜ (s1 s2 )m = 1. Assume that L(s1 ) L(s2 ) 0. Let us determine the two-sided J-cells for W . We use the following notation. For any k 0, write 1k = s1 s2 s1 · · · (k factors) ˜ and 2k = s2 s1 s2 · · · (k factors); note that 1m = 2m . Then the two-sided J-cells and the corresponding sets IrrK (W | F ) are given in Table 1.5. To prove this, write b := L(11 ) and a := L(21 ), where b a 0. We begin with some general remarks. By Example 1.4.6, the irreducible representations as in Example 1.3.7 are balanced. The representations σ εj all have a-invariant b. We find that
ε b σ ε j (T11 ) =
−1 0 ε b μ j ε 2b
and so
σ εj L 11 .
This already shows that all the representations σ εj belong to IrrK (W | F1 ), where F1 is the two-sided cell containing 11 = s1 . Next, Example 1.7.2 shows that if ˜ where IrrK (W | {10 }) = {1W } b a > 0, then {10 } and {1m } are two-sided J-cells, and IrrK (W | {1m }) = {sgn}. Now we go through the above three cases individually. First assume that a = b > 0. All that remains to show is that if m is even, then the representations sgn1 , sgn2 also belong to IrrK (W | F1 ). Now, sgn1 and sgn2 both have a-invariant b. We have ε b sgnε1 (T21 ) = (−1) and so sgn1 L 21 . Furthermore, we have ε b sgnε2 (T11 ) = (−1) and so sgn2 L 11 . Also note that 2b b ε ε ε b σ εj (T21 ) = σ εj L 21 , and so 0 −1 −1 0 ε b σ εj (T11 ) = σ εj L 11 . and so ε b μ j ε 2b Hence, sgn1 and sgn2 belong to IrrK (W | F1 ), as required. Now assume that b > a > 0. Then all that remains to do is to determine the two˜ sided J-cells to which sgn1 and sgn2 belong. Now sgn1 and sgn2 have a-invariants a and m2 (b − a) + a respectively. We have ε a sgnε1 (T21 ) = (−1) and so sgn1 L 21 . Furthermore, we have sgnε2 (T1m−1 ) = (−1)m/2 ε m(a−b)/2−a and so sgn2 L 1m−1 . Finally, the assumptions in Example 1.7.2 are satisfied and, hence, {21 } and {1m−1 } ˜ are two-sided J-cells such that IrrK (W | {21 }) = {sgn1 } and IrrK (W | {1m−1 }) = {sgn2 }, as required. Finally, assume that b > a = 0. We still have σ j ∈ IrrK (W | F1 ) for all j. It remains to consider what happens with the one-dimensional representations. Both 1W and sgn1 have a-invariant 0. We find that {w ∈ W | 1W L w} = {w ∈ W | sgn1 L w} = {10 , 21 }. Both sgn2 and sgn1 have a-invariant
m 2 b.
We find that
{w ∈ W | sgn2 L w} = {w ∈ W | sgn L w} = {1m−1 , 1m }. Furthermore, since b > 0,
1.7 Examples of Cells
47
ε b σ εj (T10 ) ≡ ε b σ εj (T21 ) ≡
0 0 0 0
mod m.
Then the characterisation of ∼LR in Lemma 1.6.14 shows that {10 , 21 } is a two˜ sided J-cell and that IrrK (W | {10 , 21 }) = {1W , sgn1 }. Applying Corollary 1.6.16 ˜ yields the two-sided J-cell {1m−1 , 1m }. Example 1.7.4. Let W be of type I2 (m), where b := L(s1 ) and a := L(s2 ), as in the previous example. Using similar computations as above, one finds that ⎧ t10 − t11 − t21 + t1m if b = a > 0, ⎨ 1J˜ = ∑ n˜ d td = t10 − t11 − t21 − t23 + (−1)m/2t1m−1 + t1m if b > a > 0, ⎩ d∈D˜ if b > a = 0. t 10 − t 11 − t 23 + t 1m Furthermore, in every irreducible representation of J˜ (arising from the balanced ˜ is represented by a diagonal representations as in Example 1.3.7), each td (d ∈ D) matrix with 0, ±1 on the diagonal. Let us just prove this in the case where b > a > 0. (The other cases are treated by similar methods.) We have ε (T10 ) = 1, 1W
ε m(a+b)/2 sgnε (T1m ) = 1, −1 0 b ε ε σ j (T11 ) ≡ mod m, 0 0
ε a sgnε1 (T21 ) = −1, ε m(b−a)/2+a sgnε2 (T1m−1 ) = (−1)m/2 , 0 0 b ε ε σ j (T23 ) ≡ mod m. 0 −1
˜ Then the above formulae Now set h := t10 − t11 − t21 − t23 + (−1)m/2t1m−1 + t1m ∈ J. ˜ the element h is represented by show that, in every irreducible representation of J, the identity matrix. (Here, we also use the information on the sets Fλ in Example 1.7.3 and the fact that cst w,λ = 0 unless w ∈ Fλ .) Hence, we must have h = 1J˜ . ˜ Remark 1.7.5. We notice that the two-sided J-cells for W of type I2 (m), as determined above, coincide with the two-sided cells in the sense of Kazhdan–Lusztig; see Example 2.1.18. (A similar statement can also be shown for left and right cells.) This seems to be a general phenomenon, and will be further discussed in Section 2.5. 1.7.6. Let n 1 and (W, S) be of type An−1 or Bn , where S = {s1 , . . . , sn−1 } (in type An−1 ) or S = {s0 , s1 , . . . , sn−1 } (in type Bn ), with the labelling in Table 1.1. For any m ∈ {1, . . . , n}, let Wm ⊆ W be the parabolic subgroup generated by s0 , s1 , . . . , sm−1 (where we set s0 = 1 in type An−1 ). Then Wm is of type Am−1 or Bm respectively. Let Λm be an indexing set for IrrK (Wm ). In particular, we have Wn = W and Λn = Λ . A special feature of this case is that, for n 2, the irreducible representations of Wn restrict multiplicity-freely to Wn−1 . More precisely, for each λ ∈ Λn , we have λ n ∼ Eμ for some subset Λ (λ ) ⊆ Λn−1 ; (a) ResW = Wn−1 E μ ∈Λ (λ )
see [132, 6.1.8 and 6.1.9]. We shall consider a weight function Ln : Wn → Γ satisfying the following conditions where a, b ∈ Γ :
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1 Generic Iwahori–Hecke Algebras
Ln (s1 ) = · · · = Ln (sn−1 ) = a > 0, b ∈ {a, 2a, . . . , (n − 1)a} (in type Bn ). Ln (s0 ) = b > 0, Note that, if n 2, then the restriction of Ln to Wn−1 is just Ln−1 . Under these conditions, by Examples 1.3.8 and 1.3.9, we have (b)
cλ ∈ Z[Γ ]
and
fλ = 1
for all λ ∈ Λn .
Note that, since Wn is a Weyl group, we have ZWn = Z. In particular, we can take R = Z as our Ln -good subring of C; then K = Q. Let us denote the generic algebra associated with Wn , Ln by Hn and write Hn,K := K ⊗A Hn . Proposition 1.7.7. In the setting of 1.7.6, assume that Γ = Z. Then there exists a family of representations {σ λ : Hn,K → Mdλ (K) | λ ∈ Λn , n 1} such that the following conditions are satisfied for all n 1 and λ ∈ Λn . (a) The representation σ λ is afforded by Eελ and we have σ λ (Tw ) ∈ Mdλ (R) for all w ∈ Wn , where R is Rouquier’s ring in Lemma 1.2.9. (b) Writing Λ (λ ) = {μ1 , . . . , μr } ⊆ Λn−1 , we have a block diagonal shape ⎡ μ ⎤ σ 1 (Tw ) 0 ... 0 ⎢ ⎥ .. . ⎢ ⎥ σ μ2 (Tw ) . . . 0 λ ⎢ ⎥ σ (Tw ) = ⎢ for all w ∈ Wn−1 . ⎥ .. . . . . ⎣ ⎦ . . . 0 0 ... 0 σ μr (Tw ) Proof. By assumption, we have A = Z[v, v−1 ], where v = ε is an indeterminate. We construct the representations σ λ of Hn,K inductively, as follows. If n = 1, then all irreducible representations of W1 are one-dimensional and there is nothing to prove. Now assume that n 2 and that σ μ has already been constructed for every μ ∈ Λn−1 . Let λ ∈ Λn and write Λ (λ ) = {μ1 , . . . , μr } ⊆ Λn−1 . By a general compatibility property (see [132, 9.1.9]), the relation in 1.7.6(a) also holds for the corresponding representations of Hn−1,K ⊆ Hn,K . Hence, denoting by ei ∈ Hn−1,K μ the central primitive idempotent corresponding to Eε i for 1 i r, we have Eελ = E1 ⊕ · · · ⊕ Er ,
μ Ei := ei Eελ ∼ = Eε i .
where
We will now show that this decomposition already works over R. Indeed, by Corollary 1.2.10, we can find a basis of Eελ such that the corresponding matrix representation π λ : Hn,K → Mdλ (K) satisfies (a). Let V λ be the R-submodule of Eελ spanned by that basis; thus, we can identify Eελ = K ⊗R V λ . By general results on symmetric algebras (see [132, 7.2.7]), the idempotent ei can be written as ei = c−1 μi
∑
w∈Wn−1
μ
trace(Tw , Eε i ) Tw−1 ∈ Hn−1,K .
1.7 Examples of Cells
49
Now we use 1.7.6(b). This shows that f μi = 1 and, hence, cμi ∈ Z[v, v−1 ] is a unit in R. It follows that each ei is an R-linear combination of the basis elements Tw (w ∈ Wn−1 ). Consequently, eiV λ is an R-submodule of V λ and we have a direct sum decomposition V λ = V1 ⊕· · ·⊕Vr , where Vi := eiV λ and Ei ∼ = K ⊗R Vi . Now consider an R-basis of V λ which is compatible with this decomposition. Transforming the original basis to this new one, we obtain a representation π˜ λ : Hn,K → Mdλ (K) such that (a) holds and, in addition, we have ⎡ μ ⎤ σ˜ 1 (Tw ) 0 ... 0 ⎢ ⎥ .. . ⎢ ⎥ 0 σ˜ μ2 (Tw ) . . . λ ⎢ ⎥ π˜ (Tw ) = ⎢ for all w ∈ Wn−1 , ⎥ .. . . . . ⎣ ⎦ . . . 0 0 ... 0 σ˜ μr (Tw ) where σ˜ μi is afforded by Ei for all i. In particular, σ˜ μi and σ μi are equivalent over K. So there exists some invertible matrix Pi ∈ Mdi (K), where di = dim Ei , such that
σ μi (Tw ) = Pi−1 σ˜ μi (Tw ) Pi
for all w ∈ Wn−1 .
Multiplying Pi by a suitable polynomial in Z[v], we may assume that Pi ∈ Mdi (R) and that the greatest common divisior of all non-zero entries of Pi is 1. (Recall from Lemma 1.2.9 that R is a principal ideal domain and, hence, a unique factorisation domain.) Now det(Pi ) is a non-zero element of R. In fact, we claim that det(Pi ) is a unit in R. This follows by a standard argument on symmetric algebras, which we already employed at the end of the proof of Proposition 1.5.11. Let p be a maximal ideal in R; we must show that det(Pi ) ∈ p. For this purpose, let k := R/p be the residue field and consider the specialisation θ : A → k induced by reduction modulo p. Applying θ , we obtain a specialised algebra Hn−1,k and specialised representaμ μ tions σk i and σ˜ k i . Since each cμi is a unit in R, we have θ (cμi ) = 0. As in the proof μ μ of Proposition 1.5.11, this implies that σk i and σ˜ k i are irreducible. The reduction of Pi modulo p defines a non-zero module homomorphism between these representations. Hence, by Schur’s lemma, this homomorphism must be invertible and so det(Pi ) ∈ p, as required. Thus, each Pi is invertible over R. Let P be the block diagonal matrix with diagonal blocks given by P1 , . . . , Pd . Then P is invertible over R and, if we set σ λ (Tw ) := P−1 π˜ λ (Tw ) P for all w ∈ Wn , then σ λ is a matrix representation of Hn,K which satisfies (a) and (b).
Remark 1.7.8. The existence of a compatible family of representations {σ λ } as above was known before. Indeed, the “seminormal representations” constructed by Hoesfmit [157] satisfy these properties; see also Ram [268]. Of course, we could have referred to Hoefsmit’s proof, but the above argument provides an alternative approach and is quite self-contained and much less computational. However, Hoefsmit’s results are actually stronger: they can be used to drop the asumption that Γ = Z. Indeed, in the general case, consider the ring
50
1 Generic Iwahori–Hecke Algebras
R(Γ ) :=
f f ∈ Z[Γ ], g ∈ 1 + Z[Γ>0 ] ⊆ K. g
Then Hoefsmit’s construction [157] yields a family of representations as in Proposition 1.7.7 where, in (a), we have σ λ (Tw ) ∈ Mdλ (R(Γ )) for all w ∈ Wn . (This is immediately seen from the explicit form of the matrices in [157, §2.2].) Corollary 1.7.9. Assume we are in the setting of 1.7.6. Then, for each λ ∈ Λn , there exists a balanced representation ρ λ afforded by Eελ such that the conditions in Proposition 1.7.1 are satisfied. Thus, we have ts cst w,λ = cw−1 ,λ ∈ {0, ±1}
for all w ∈ Wn and s, t ∈ M(λ ).
Proof. Let σ λ be as in Proposition 1.7.7 (or as in Remark 1.7.8 if Γ = Z). Let B ∈ Mdλ (K) be any matrix such that B σ λ (Tw−1 ) = σ λ (Tw )tr B for all w ∈ Wn . Then we claim that B must be diagonal. We prove this by induction on n. If n = 1, then σ λ is one-dimensional and there is nothing to prove. Now assume that n 2 and consider the restriction of σ λ to Hn−1,K . This restriction has a block form as described in Proposition 1.7.7(b). Correspondingly, let us also write B in block form: ⎡ ⎤ B1,1 B1,2 . . . B1,r ⎢ .. ⎥ ⎢ B2,1 B2,2 . . . . ⎥ ⎥. B=⎢ ⎢ . . ⎥ .. ... B ⎣ .. ⎦ r−1,r Br,1 . . . Br,r−1 Br,r Then, for any i, j ∈ {1, . . . , r}, we have Bi, j σ μ j (Tw−1 ) = σ μi (Tw )tr Bi, j for all w ∈ μ Wn−1 . This means that Bi, j defines an Hn−1,K -module homomorphism between Eε j μi and the contragredient dual of Eε . By Example 1.2.5, the latter is isomorphic to μ Eε i . So we must have Bi, j = 0 if i = j. Furthermore, if i = j, then Bi,i is a diagonal matrix by induction. Hence, B is a diagonal matrix, as claimed. We now apply the procedure in 1.4.9 and set Ω0λ := ∑w∈Wn σ (Tw )tr .σ (Tw ). Then λ Ω0 is a symmetric matrix such that Ω0λ σ λ (Tw−1 ) = σ λ (Tw )tr Ω0λ for all w ∈ Wn . We have just seen that this implies that Ω0λ is a diagonal matrix. Hence, the Gram– Schmidt orthogonalisation procedure is not required here and we have Ω0λ = Ω1λ in the notation of 1.4.9. Furthermore, the diagonal coefficient of Ω0λ at position s ∈ M(λ ) has the form bs ε −2ms + combination of terms ε g where g > −2ms , where bs is a positive real number and ms ∈ Γ . Finally, let E λ be the diagonal matrix with diagonal entries {ε ms | s ∈ M(λ )} and set
ρ λ (Tw ) := (E λ )−1 σ λ (Tw ) E λ
for all w ∈ Wn .
1.7 Examples of Cells
51
Then ρ λ is a balanced representation and the corresponding leading matrix coeffi= cst cients satisfy bt cts w,λ bs for all w ∈ Wn and s, t ∈ M(λ ). w−1 ,λ
Since σ λ and, hence, ρ λ are realised over R (or over R(Γ )), we have cst w,λ ∈ Z for all w ∈ Wn and s, t ∈ M(λ ). So it remains to show that bs = bt for all s, t ∈ M(λ ). Now, given s, t ∈ M(λ ), we have the relation (see Proposition 1.4.10) 1 = fλ =
∑
w∈Wn
ts cst w,λ cw−1 ,λ .
But, since all leading matrix coefficients are integers, we conclude that ts −1 st 2 cst cw,λ is a non-negative integer for all w ∈ Wn . w,λ cw−1 ,λ = bs bt −1 st 2 ts Hence, there is a unique w ∈ Wn such that cst cw,λ = 1. So we w,λ cw−1 ,λ = bs bt
−1 ts must have cst w,λ = cw−1 ,λ = ±1 and bs bt = 1, as required.
Theorem 1.7.10 (Cf. [103, §5], [25, §2]). Assume that W = Wn is of type An−1 or Bn , and that L = Ln is a weight function satisfying the conditions in 1.7.6. Then there exists a bijection
M(λ ) × M(λ )
1−1
−→
λ ∈Λ
W,
(s, t) → wλ (s, t),
such that the following hold. (a) Let w ∈ W , λ ∈ Λ and s, t ∈ M(λ ). Then cst w,λ = 0 if and only if w = wλ (s, t). st st Also, if cw,λ = 0, then cw,λ = ±1. Furthermore, wλ (s, t)−1 = wλ (t, s). ˜ (b) The left, right and two-sided J-cells of W are given as described in Proposition 1.7.1. In particular, Irr(W | Fλ ) = {E λ } for all λ ∈ Λ . (c) We have D˜ = {d ∈ W | d 2 = 1} = {wλ (s, s) | λ ∈ Λ , s ∈ M(λ )} and n˜ d = ±1 ˜ Every left J-cell ˜ ˜ for all d ∈ D. contains a unique element of D. In particular, Conjectures 1.5.12 and 1.6.18 hold in this case. Proof. By Corollary 1.7.9, the assumptions of Proposition 1.7.1 are satisfied for every λ ∈ Λ . This yields the existence of the required bijection satisfying (a) and (b). The statement about wλ (s, t)−1 is then clear. In particular, this shows that {w ∈ W | w2 = 1} = {wλ (s, s) | λ ∈ Λ , s ∈ M(λ )}. To prove the remaining assertions in (c), let w ∈ W and write w = wλ (s, t), where λ ∈ Λ and s, t ∈ M(λ ). Using the defining formula for n˜ w and (a), we obtain n˜ w = n˜ w−1 =
∑ ∑
μ ∈Λ u∈M(μ )
st cuu w,λ = δst cw,μ = ±δst .
˜ Using (b), we In particular, D˜ has the required form and n˜ d = ±1 for all d ∈ D. ˜ ˜ then see that every left J-cell contains a unique element of D. In particular, Con-
52
1 Generic Iwahori–Hecke Algebras
jecture 1.6.18 holds. By Remark 1.6.19, this implies that Conjecture 1.5.12(b) also holds. Finally, Conjecture 1.5.12(a) is seen to hold directly from the defining for mulae for γ˜x,y,z and n˜ w , using that fλ = 1 for all λ ∈ Λ . Remark 1.7.11. The original treatment in [103], [25] relied on the Dipper–James– Murphy approach [62], [68] to Hoefsmit’s construction [157] of “seminormal representations”. Here, at least for Γ = Z, we have replaced this reference by our Proposition 1.7.7, which essentially relies on the use of Rouquier’s ring R. Remark 1.7.12. Let W be irreducible of arbitrary type, and L : W → Γ be any weight function such that L(s) 0 for all s ∈ S. Assume that f λ = 1 for all λ ∈ Λ . Then, using the explicit knowledge of aλ and f λ in all cases (see Remark 1.3.6), we see that W must be of type An−1 or Bn where L satisfies the conditions in 1.7.6. Remark 1.7.13. Let W be of type Bn , with generators S = {s0 , s1 , . . . , sn−1 } as above. What happens in the cases which are not covered by the conditions in 1.7.6? To fix some notation, let r ∈ {0, 1, . . . , n − 1}, a ∈ Γ>0 , and consider a weight function L : W → Γ such that L(s0 ) = ra and L(si ) = a for 1 i n − 1. Then one may ˜ conjecture that the left, right and two-sided J-cells of W sets are given by the “combinatorial r-cells” of Bonnaf´e et al. [25, §4]. This is supported by the results of Bonnaf´e [23], Pietraho [264], [265], Taskin [289].
1.8 Cells and Leading Coefficients The main result of this section is a refinement of the “orthogonality relations” for Lusztig’s leading coefficients of character values (see Remark 1.3.2) with respect to ˜ a given left J-cell. This is modelled on the discussion in [220, Chap. 5]. Theorem 1.8.1 (Cf. Lusztig [220, 5.8] in the equal-parameter case). Let C be a ˜ left J-cell in W and λ , μ ∈ Λ . Then the following hold. (a) If λ = μ , then for any s, t ∈ M(λ ) and u, v ∈ M(μ ), we have
∑ cstw,λ cuv w−1 ,μ = 0
and
w∈C
∑ cw,λ cw−1 ,μ = 0.
w∈C
(b) If λ = μ , then for any s ∈ M(λ ), we have ˜ λ ) fλ , ∑ ∑ cstw,λ ctsw−1 ,λ = ∑ cw,λ cw−1 ,λ = m(C,
t∈M(λ ) w∈C
w∈C
λ ) denotes the multiplicity of ρ¯ λ as an irreducible constituent of where m(C, ˜ ˜ the left cell module [C]K of J. λ ) > 0 if and only if there exists some w ∈ C such that cw,λ = 0. In particular, m(C, ˜
1.8 Cells and Leading Coefficients
53
Proof. This is, to a large extent, analogous to the general proof of the Schur relations for symmetric algebras; see, for example, [132, §7.2]. Let P be any matrix of size dλ × dμ , with coefficients in K, and set I(P) :=
∑ ρ¯ λ (tw ) P ρ¯ μ (tw−1 ).
w∈C
We claim that ρ¯ λ (tx ) I(P) = I(P) ρ¯ μ (tx ) for all x ∈ W . Indeed, we have
ρ¯ λ (tx ) I(P) =
∑ ρ¯ λ (txtw ) I(P)ρ¯ μ (tw−1 ) = ∑ ∑ γ˜x,w,y−1 ρ¯ λ (ty) I(P) ρ¯ μ (tw−1 ). w∈C y∈W
w∈C
Suppose that, in the above sum, we have γ˜x,w,y−1 = 0. Then, by Lemma 1.6.5, we must have y ∼L w. Hence, we can first let the two sums runs over all y, w ∈ C and then, applying the argument once more, let w run over all elements of W and y run over all elements in C. Thus, we obtain
ρ¯ λ (tx ) I(P) = ∑ ρ¯ λ (ty ) I(P) ∑ γ˜x,w,y−1 ρ¯ μ (tw−1 ) . w∈W
y∈C
Since γ˜x,w,y−1 = γ˜y−1 ,x,w (see Lemma 1.5.3), we have ∑w∈W γ˜x,w,y−1 tw−1 = ty−1 tx . Hence, the expression on the right-hand side equals I(P) ρ¯ μ (tx ), as required. Now let t ∈ M(λ ), u ∈ M(μ ) and P = Ptu be the matrix with (t, u)-coefficient 1, and coefficient 0 otherwise. Then, for any s ∈ M(λ ) and v ∈ M(μ ), we have Isv (Ptu ) =
μ
∑ ρ¯stλ (tw ) ρ¯uv (tw−1 ) = ∑ cstw,λ cuv w−1 ,μ .
w∈C
w∈C
If λ = μ , then I(P) = 0 for every matrix P and, hence,
∑ cstw,λ cuv w−1 ,μ = 0
for all s, t ∈ M(λ ) and u, v ∈ M(μ ),
w∈C
as desired. This also implies that
∑ cw,λ cw−1 ,μ = ∑
uu ∑ ∑ css w,λ cw−1 ,μ = 0
s∈M(λ ) u∈M(μ ) w∈C
w∈C
if λ = μ .
So let us now assume that λ = μ . Then I(P) must be a scalar multiple of the identity for every matrix P and, hence,
∑ cstw,λ cuv w−1 ,λ = 0
if s = v,
w∈C
∑ cstw,λ cus w−1 ,λ
does not depend on s.
w∈C
These two properties imply that, for a fixed s0 ∈ M(λ ), we have
54
1 Generic Iwahori–Hecke Algebras tt ∑ ∑ css ∑ cttw,λ cttw−1 ,λ = ∑ ∑ cw,s0λt ctsw−10 ,λ . w,λ cw−1 ,λ = ∑
s,t∈M(λ ) w∈C
t∈M(λ ) w∈C
t∈M(λ ) w∈C
Now note that the left-hand side equals ∑w∈C cw,λ cw−1 ,λ ; furthermore, the right hand side is the (s0 , s0 )-coefficient of I(
∑
Ptt ) = ρ¯ λ (z),
where
z :=
t∈M(λ )
∑ twtw−1 ,
w∈C
and this does not depend on s0 . So, it will be sufficient to prove that ˜ λ ) dλ f λ . trace ρ¯ λ (z) = m(C, Now, we have z=
∑ twtw−1 = ∑ ∑ γ˜w,w−1 ,x tx−1 . w∈C x∈W
w∈C
By Lemma 1.5.3, we have γ˜w,w−1 ,x = γ˜x,w,w−1 . Hence, we obtain that
z=
∑ ∑
γ˜x,w,w−1 tx−1 =
x∈W w∈C
˜ μ ) cx,μ tx−1 . ∑ trace(tx , [C]K )tx−1 = ∑ ∑ m(C, x∈W μ ∈Λ
x∈W
Consequently, using the orthogonality relations for Lusztig’s leading coefficients of character values, we find that ˜ μ ) ∑ cx,μ cx−1 ,λ = m(C, ˜ λ ) dλ fλ , trace ρ¯ λ (z) = ∑ m(C, μ ∈Λ
x∈W
as desired.
˜ Example 1.8.2. Let λ ∈ Λ and C be a left J-cell of W such that m(C, ˜ λ ) > 0. Then we claim that C ⊆ Fλ . Indeed, by Theorem 1.8.1(b), there exists some w ∈ C such λ that cw,λ = 0. Hence, we also have css w,λ = 0 for some s ∈ M(λ ) and so E L w. ˜ we This shows that w ∈ Fλ and so Fλ ∩ C = ∅. Since Fλ is a union of left J-cells, must have C ⊆ Fλ , as claimed. Example 1.8.3. Assume that the balanced representations {ρ λ | λ ∈ Λ } are chosen such that the additional conditions in Theorem 1.4.8 are satisfied; that is, we have st bt cts w−1 ,λ = cw,λ bs
for all w ∈ W , λ ∈ Λ and s, t ∈ M(λ ),
˜ in W . Then the identity in where {bs | s ∈ M(λ )} ⊆ R>0 . Now let C be a left J-cell Theorem 1.8.1(b) reads
∑ ∑ bs b−1 t
t∈M(λ ) w∈C
st 2 cw,λ = m(C, ˜ λ ) fλ
for any λ ∈ Λ and s ∈ M(λ ),
and there are no cancellations in the above sum. Hence, we obtain the implication (a)
cst w,λ = 0 for some w ∈ C, λ ∈ Λ , s, t ∈ M(λ )
⇒
m(C, ˜ λ ) > 0.
1.8 Cells and Leading Coefficients
55
Conversely, the formula in Theorem 1.8.1(b) shows that if m(C, ˜ λ ) > 0, then there = 0 for some s ∈ M(λ ). exists some w ∈ C such that cw,λ = 0 and, hence, css w,λ We now obtain the following alternative description of the sets IrrK (W | F ). Lemma 1.8.4. Let λ , μ ∈ Λ . Then Fλ = Fμ if and only if there exist a sequence ˜ λ = λ0 , λ1 , . . . , λm = μ in Λ and a sequence of left J-cells C1 , . . . , Cm such that, for each i ∈ {1, . . . , m}, we have m(C ˜ i , λi−1 ) > 0 and m(C ˜ i , λi ) > 0. Proof. We choose the balanced representations of HK as in Example 1.8.3. Now assume first that F := Fλ = Fμ . There exist some x, y ∈ F such that E λ L x and E μ L y. By the definition of ∼LR and Lemma 1.6.6, there exists a sequence x = x0 , x1 , . . . , xm = y in W such that, for each i ∈ {1, . . . , m}, we have xi−1 ∼L xi or xi−1 ∼R xi . By Proposition 1.4.10(c), there exist λ = λ0 , λ1 , . . . , λm = λ in Λ such that, for each i ∈ {0, 1, . . . , m}, we have csx i,tλi = 0 i
i
for some si , ti ∈ M(λi ).
Note that, by our choice of the balanced representations, we then also have si cti−1
xi ,λi
= 0
for some si , ti ∈ M(λi ).
˜ We define a sequence of left J-cells C1 , . . . , Cm as follows. If xi−1 ∼L xi , then let Ci −1 ˜ be the left J-cell containing xi−1 and xi . If xi−1 ∼R xi , then xi−1 ∼L xi−1 and we let −1 −1 ˜ Ci be the left J-cell containing xi−1 and xi . In both cases, by Example 1.8.3(a), we have m(C ˜ i , λi−1 ) > 0 and m(C ˜ i , λi ) > 0, as required. Conversely, assume that λ = λ0 , λ1 , . . . , λm = λ and C1 , . . . , Cm are sequences ˜ with the required properties. For each i, let Fi be the two-sided J-cell containing Ci . By Theorem 1.8.1(b), there exist elements xi , yi ∈ Ci (1 i m) such that csx i,tλi = 0 and cuy i,vλi i
i
i
i−1
= 0,
where si , ti ∈ M(λi ), ui , vi ∈ M(λi−1 ).
Thus, E λi−1 L yi and E λi L xi for 1 i m. Since xi , yi ∈ Fi , we deduce that E λi−1 ∈ IrrK (W | Fi )
and
E λi ∈ IrrK (W | Fi )
for 1 i m.
But this certainly implies that Fλ = F1 = F2 = · · · = Fm = Fμ , as required.
˜ Example 1.8.5. Let C be a left J-cell. Then we claim that
λ) = m(C, ˜
∑
n˜d cd,λ
˜ d∈D∩C
and
∑
λ ∈Λ
fλ−1 m(C, ˜ λ) =
∑
n˜ 2d .
˜ d∈D∩C
Indeed, using the defining formula for n˜ w (for any w ∈ W ) and the fact that n˜ w = n˜ w−1 (see Section 1.5), we obtain that
∑
˜ d∈D∩C
n˜d cd,λ =
∑ n˜w cw,λ = ∑
w∈C
μ ∈Λ
f μ−1
∑ cw−1 ,μ cw,λ ,
w∈C
56
1 Generic Iwahori–Hecke Algebras
and it remains to use the refined orthogonality relations in Theorem 1.8.1. The second formula is proved similarly. Example 1.8.6. Assume that Conjectures 1.5.12 and 1.6.18 hold. Then γ˜x,y,z ∈ Z and ˜ So the relation in Remark 1.6.19(a) shows that n˜ d ∈ Z for all x, y, z ∈ W and d ∈ D. ˜ ˜ n˜ d = ±1 for all d ∈ D. Now let C be a left J-cell and d ∈ D˜ be the unique element in C. Then the formulae in Example 1.8.5 reduce to (a)
λ ) = n˜ d cd,λ = ±cd,λ m(C, ˜
∑
and
λ ∈Λ
f λ−1 m(C, ˜ λ ) = 1.
Furthermore, for any y ∈ W , we have tytd = ∑x∈W γ˜y,d,x−1 tx = ∑x∈W γ˜x−1 ,y,d tx , where the second equality holds by Lemma 1.5.3. Then the formula in Remark 1.6.19(a) shows that tytd = n˜ d ty if y ∼L d, and tytd = 0 if y ∼L d. Thus, we have (b)
˜ d = ty | y ∈ CK ∼ Jt = [C]K
˜ (isomorphism of left J-modules).
Again, it would be very interesting and useful to find direct and elementary proofs of these statements which do not rely on the above conjectures. 1.8.7. Let us assume that W = Wn is of type An−1 or Bn and that L = Ln is a weight function satisfying the conditions in 1.7.6. Then we have a bijection
M(λ ) × M(λ )
1−1
−→
W,
(s, t) → wλ (s, t),
λ ∈Λ
with the properties in Theorem 1.7.10. In particular, we have Fλ = {wλ (s, t) | s, t ∈ M(λ )}
for any λ ∈ Λ ;
˜ for each t ∈ M(λ ). furthermore, the set Ct := {wλ (s, t) | s ∈ M(λ )} is a left J-cell Let [Ct ]K be the corresponding left cell module, with basis {es | s ∈ M(λ )}, where es = twλ (s,t) for s ∈ M(λ ). Explicitly, the action of J˜ on [Ct ]K is given by tw .es =
∑
s∈M(λ )
γ˜w,wλ (s ,t),wλ (t,s) es
for all w ∈ W and s ∈ M(λ ),
where we used that wλ (s, t)−1 = wλ (t, s). Let ρˆ t : J˜ → Mdλ (K) denote the corresponding matrix representation; we have for all w ∈ W and s, s ∈ M(λ ).
t ρˆ ss (tw ) = γ˜w,w (s ,t),w (t,s) λ λ
(Here, we assume that M(λ ) has been ordered in some way so that it makes sense to write down matrices with rows and columns indexed by M(λ ).)
t (t ) = ±css for all w ∈ W and all s, s ∈ M(λ ). In Lemma 1.8.8. We have ρˆ ss w w,λ particular, the matrix ρ¯ λ (tw ) is zero unless w ∈ Fλ , in which case it has precisely one non-zero entry (which is ±1).
1.8 Cells and Leading Coefficients
57
Proof. If w ∈ Fλ , then ρˆ t (tw ) = 0 by Corollary 1.6.7. Now assume that w ∈ Fλ and write w = wλ (u, v), where u, v ∈ M(λ ). Also write x = wλ (s , t) and y = wλ (t, s). Then the defining equation for γ˜w,x,y in combination with Theorem 1.7.10 yields that t uv vt tu uv ρˆ ss (tw ) = γ˜w,x,y = cw,λ cx,λ cy,λ = ±δus δvs cw,λ , t as required. This also shows that ρˆ ss (tw ) has precisely one non-zero entry.
Proposition 1.8.9. In the setting of 1.8.7, let λ ∈ Λ and t ∈ M(λ ). (a) The cell representation ρˆ t is irreducible and equivalent to ρ¯ λ . In fact, we have
ρˆ t (tw ) = Dt ρ¯ λ (tw ) D−1 t
for all w ∈ W ,
where Dt ∈ Mdλ (K) is a diagonal matrix with ±1 on the diagonal. (b) The balanced representation ρ λ of HK can be chosen such that, in addition to the conditions in Proposition 1.5.11, we have ρ¯ λ (tw ) = ρˆ t (tw ) for all w ∈ W . ˜ Then, in particular, each cst w,λ is equal to γw,x,y for suitable x, y ∈ Fλ . ˜ t , μ ) > 0. By Example 1.8.2, we have Ct ⊆ Fμ Proof. Let μ ∈ Λ be such that m(C and so μ = λ . Now |Ct | = |M(λ )| = dλ = dim E λ and so ρˆ t must be irreducible and equivalent to ρ¯ λ . So there exists an invertible matrix Dt ∈ Mdλ (K) such that ρˆ t (tw ) = Dt ρ¯ λ (tw )D−1 t for all w ∈ W . Taking into account Lemma 1.8.8, a straightforward computation now shows that Dt must be a scalar multiple of a diagonal matrix with ±1 on the diagonal. Thus, (a) is proved. To prove (b) it is sufficient to note that if ρ λ is a balanced representation satisfying the conditions in Proposition 1.5.11, then the representation defined by Tw → Dt ρ λ (Tw )D−1 t is still balanced and it still satisfies the conditions in Proposition 1.5.11. Hence, we just have to replace ρ λ by this new representation. Conjecture 1.8.10. In the setting of 1.8.7, let λ ∈ Λ and t, t ∈ M(λ ). Then we actually have ρˆ t (tw ) = ρˆ t (tw ) for all w ∈ W . This is known to be true when W ∼ = Sn , but the proof requires much more work. (It can be deduced, for example, by combining Lemma 1.8.12 below with the discussion in Remark 2.8.18.) It would be very useful to have a direct proof of Conjecture 1.8.10, valid in all situations described in 1.7.6. Remark 1.8.11. Let λ ∈ Λ and x, x , y, y ∈ Fλ be such that x ∼R x , y ∼R y , x ∼L y, x ∼L y . Then Conjecture 1.8.10 is equivalent to (♥)
γ˜w,x,y−1 = γ˜w,x ,y −1
for all w ∈ W and all x, x , y, y ∈ Fλ as above.
Indeed, by Theorem 1.7.10, we have x = wλ (s, t), y = wλ (s , t), x = wλ (s, t ), y = wλ (s , t ), where s, s , t, t ∈ M(λ ), and it remains to recall the definition of ρˆ t , ρˆ t . Lemma 1.8.12 (Cf. [107, 5.10]). Assume that Conjecture 1.8.10 holds. Then there exists a sign nλ = ±1 (depending only on λ ) such that, for all x, y, z ∈ Fλ , we have
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1 Generic Iwahori–Hecke Algebras
γ˜x,y,z =
nλ 0
if x ∼L y−1 , y ∼L z−1 , z ∼L x−1 , otherwise.
Conversely, if the above formula holds, then Conjecture 1.8.10 is true. Proof. By Lemma 1.6.5, we have γ˜x,y,z = 0 unless x ∼L y−1 , y ∼L z−1 , z ∼L x−1 . Now assume that x, y, z are related in this way. Then x = wλ (s, t), y = wλ (t, u), z = wλ (u, s) for some s, t, u ∈ M(λ ); see Theorem 1.7.10. Let us also set d := wλ (t, t). Then d ∼L d, d = d −1 ∼R y, x ∼R z−1 . So we can apply (♥) and this yields
γ˜x,y,z = γ˜x,d,x−1 = γ˜x−1 ,x,d , where we also used Lemma 1.5.3(a). Furthermore, since every left cell contains a unique element in D˜ (see Theorem 1.7.10(c)), the identity ∑w∈W γ˜x−1 ,x,w n˜ w = 1 in Lemma 1.5.3(b) now reduces to γ˜x−1 ,x,d n˜ d = 1. Hence, we conclude that γ˜x,y,z = n˜d , where d = wλ (t, t). So it remains to show that n˜ d = n˜d for any d ∈ D˜ ∩ Fλ . Let d = wλ (t , t ), where t ∈ M(λ ), and consider the element w = wλ (t, t ). Then w−1 = wλ (t , t) and so w ∼L d , d ∼L w−1 , w ∼R d, d ∼R w−1 . Again, we can apply (♥); arguing as above, this now yields n˜ d = γ˜w−1 ,w,d = γ˜w−1 ,d,w = n˜d , as required. Conversely, assume that the above formula for γ˜x,y,z is true. Then all the non-zero entries of ρˆ t (tw ) have the same sign, for any w ∈ W and any t ∈ M(λ ). Hence, the required identity in Conjecture 1.8.10 is clear by the formula in Lemma 1.8.8.
Chapter 2
Kazhdan–Lusztig Cells and Cellular Bases
The aim of this chapter is to develop a general framework for studying the representation theory of Iwahori–Hecke algebras associated with finite Coxeter groups. The motivating example is the representation theory of the symmetric group Sn . Frobenius showed around 1900 that the irreducible representations of Sn over a field of characteristic 0 are naturally parametrised by the partitions of n. In the 1970s, James [181] developed a “characteristic-free” approach to the representation theory of Sn , where Specht modules and certain bilinear forms on them play a crucial role. Dipper and James [62] extended this theory to Iwahori–Hecke algebras associated with Sn . A considerable simplification was then achieved through the powerful new ideas introduced by Murphy [256], [257]. In fact, what we nowadays call the “Murphy basis” is an example of a “cellular basis” in the formal sense defined later by Graham and Lehrer [144]. Here, we shall construct such a “cellular basis” in the sense of Graham and Lehrer, for the generic algebra H associated with an arbitrary (finite) Coxeter group W . For this purpose, we need two basic ingredients: (1) a basis of H with certain specific multiplicative properties and (2) a suitable partial ordering on Irr(W ). Already Graham and Lehrer identified the Kazhdan–Lusztig basis {Cw | w ∈ W } (see Section 2.1) of H as a natural candidate for (1). However, it is only in some very special examples (in type A or B) that {Cw } itself has the required multiplicative properties. But in any case, this new basis of H provides the necessary tools to define a partial ordering on Irr(W ); see Section 2.2. In order to proceed, we have to rely on certain deep properties of the basis {Cw } for which no elementary proofs are known. Sections 2.3–2.5 are devoted to a discussion of these properties, which appear as conjectures P1–P15 in Lusztig’s book [231]. We can then put all the pieces together and construct, following [111], [112], a “cellular basis” for H; see Sections 2.6 and 2.7. In the final section, we present an elementary treatment of the case where W ∼ = Sn .
M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7 2, © Springer-Verlag London Limited 2011
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2 Kazhdan–Lusztig Cells and Cellular Bases
2.1 The Kazhdan–Lusztig Basis Let W be a finite Coxeter group and L : W → Γ a weight function, where Γ is an abelian group admitting a monomial order such that L(s) 0 for all s ∈ S (as in Chapter 1). Let H = HA (W, S, L) be the corresponding generic Iwahori–Hecke algebra over A = R[Γ ], where R ⊆ C is a subring as in 1.2.1. The main purpose of this section is to introduce the Kazhdan–Lusztig basis {Cw | w ∈ W } of H. This basis first appeared in [195], in the equal-parameter case. Then Lusztig [219] showed that the construction also works in the general multiparameter case. These results are now readily accessible in Lusztig’s book [231], so we will outline the main constructions and formulate the main results but refer to [231] for further details. 2.1.1. Given elements y, w ∈ W , we write y w if y can be obtained by omitting some terms in a reduced expression for w. This defines a partial order relation on W , called the Bruhat–Chevalley order. Here are some properties (see [231, Chap. 2]): (a) Let w ∈ W and s ∈ S. Then sw < w if and only if l(sw) = l(w) − 1. (b) Let y, w ∈ W and s ∈ S be such that sw < w. Then sy sw if sy < y, yw ⇔ y sw if sy > y. Note that (b) provides a recursive description of . 2.1.2. Let w0 ∈ W be the longest element. For any w ∈ W , we can write uniquely Tw Tw0 =
∑ R∗y,w Tyw0 ,
where R∗y,w ∈ Z[Γ ].
y∈W
If w = 1, then R∗1,1 = 0 and R∗y,1 = 0 for all y = 1. Now assume that w = 1 and let s ∈ S be such that sw < w. Then one easily checks the following relation: if sy < y, R∗sy,sw R∗y,w = ∗ ∗ if sy > y. Rsy,sw + (vs − v−1 s )Ry,sw By using 2.1.1 and the above formulae, we obtain (see also [231, 4.5 and 4.7]) (a)
R∗w,w = 1
(b)
R¯ ∗y,w = (−1)l(y)+l(w) R∗y,w .
and
R∗y,w = 0 unless y w,
(Here, a¯ for any a ∈ A is defined in Example 1.2.6.) The above recursion formulae are the same as those for the elements ry,w in [231, 4.4]. Consequently, we have (c)
Tw−1 −1 =
∑ R¯∗y,w Ty
for any w ∈ W .
y∈W
(The relation between the expressions for Tw Tw0 and Tw−1 −1 already appeared in the remarks following [195, Lemma A.4].)
2.1 The Kazhdan–Lusztig Basis
61
2.1.3. We set Γ0 = {g ∈ Γ | g 0} and denote by Z[Γ0 ] the set of all integral linear combinations of terms ε g , where g 0. The notations Z[Γ>0 ], Z[Γ0 ], Z[Γ<0 ] have a similar meaning. Then, by the proof of [231, Theorem 5.2] (see also [228, 7.10]), ∗ | y, w ∈ W } ⊆ Z[Γ ] satisfying the there exists a unique collection of elements {Py,w following conditions: ∗ = 1 and P∗ = 0 unless y w; furthermore, P∗ ∈ Z[Γ ] if y < w. (a) Pw,w <0 y,w y,w (b) For any y, w ∈ W , we have
∑
∗ P¯y,w =
∗ R∗y,z Pz,w .
z∈W : yzw ∗ can be constructed recursively using (a) and (b); see Example 2.1.5. Note that Py,w ∗ are denoted by r , p (Here, the notation is as in [219]; R∗y,w , Py,w y,w y,w in [231].)
Definition 2.1.4 (Kazhdan and Lusztig [195], Lusztig [219]). For w ∈ W , we set Cw :=
∗ Ty ∑ (−1)l(w)+l(y)P¯y,w
∈ H,
y∈W
∗ as in 2.1.3. The elements {C | w ∈ W } form an A-basis of H; to see this with Py,w w just note that, by 2.1.3(a), we have Tw ∈ Cw + ∑y∈W : y<w Z[Γ>0 ]Ty for any w ∈ W . For x, y ∈ W , let us write
CxCy =
∑ hx,y,z Cz ,
where
hx,y,z ∈ A.
z∈W
Example 2.1.5. The formulae in 2.1.2 yield a straighforward algorithm for computing the polynomials R∗y,w . As already mentioned above, the formulae in 2.1.3 can ∗ recursively. Indeed, given y < w, note that then be used to construct Py,w ∗ ∗ P¯y,w − Py,w =
∑
∗ R∗y,z Pz,w .
z∈W : z
Proceeding by induction on l(w) − l(y), all terms on the right-hand side are known. ∗ itself is determined by the additional condition that P∗ ∈ Z[Γ ]. Then Py,w <0 y,w For example, it is clear that C1 = T1 . Now let s ∈ S. Then R∗1,s = vs − v−1 s and so ∗ ∗ − P1,s = P¯1,s
∑
∗ ∗ R∗1,z Pz,s = R∗1,s Ps,s = vs − v−1 s .
z∈W : 1
0). Since P1,s <0 s 1,s 1,s Thus, we obtain if L(s) = 0, Ts Cs = Ts − vs T1 if L(s) > 0. ∗ , let s,t ∈ S be such that In order to see some more complicated polynomials Py,w mst 3 and assume that L(t) L(s) > 0. Then the above precedure yields
Cst = Tst − vt Ts − vs Tt + vs vt T1
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2 Kazhdan–Lusztig Cells and Cellular Bases
and Ctst = Ttst − vt Tst − vt Tts + vt2 Ts vs vt Tt − vs vt2 T1 + −1 2 (vs vt − vs vt )Tt − (vs vt2 − v−1 s vt )T1
if L(s) = L(t), if L(t) > L(s).
With some more effort, it is possible to write down explicit formulae for all basis elements Cw , where W is of type I2 (m); see [231, Chap. 7], [132, Exc. 11.4]. We can now state the following characterisation of the element Cw . This version of the characterisation (which works for finite W ) is due to Lusztig [232]; the proof is very similar to (but the statement as such is different from) the original one in [195], [219] (which relied on the “bar involution” on H). Theorem 2.1.6 (Kazhdan and Lusztig [195], Lusztig [219], [231], [232]). For any w ∈ W , the element Cw is uniquely determined by the following two conditions: Cw ∈ Tw +
∑ Z[Γ>0 ]Ty
Cw Tw0 ∈
and
y∈W
∑ Z[Γ0 ]Ty .
y∈W
Proof. Let us verify that Cw satisfies the above two conditions. The first one is clear by 2.1.3(a); furthermore, using the relations (a), (b) in 2.1.2, we obtain: ∗ Tyw0 Cw Tw0 = ∑ (−1)l(w)+l(y) ∑ (−1)l(y)+l(z)R∗y,z P¯z,w y∈W
=
∑ (−1)
l(w)+l(y)
y∈W
=
∑
z∈W : yzw
∑
∗ ∗ Ry,z Pz,w Tyw0
z∈W : yzw
∗ (−1)l(w)+l(y) Py,w Tyw0 ,
y∈W
where the last equality holds by 2.1.3(b). Thus, we have in fact Cw Tw0 ∈ Tww0 +
∑
Z[Γ<0 ]Tyw0
y∈W : y<w
⊆
∑ Z[Γ0 ]Ty ,
y∈W
as required. Using this expression for Cw Tw0 , one easily deduces the following statement. Let h = ∑x∈W axCx ∈ H, where ax ∈ Z[Γ ] for all x ∈ W . Then we have (∗)
hTw0 ∈
∑ Z[Γ0 ]Ty
⇒
ax ∈ Z[Γ0 ] for all x ∈ W .
y∈W
This immediately implies the uniqueness of Cw . Indeed, assume that C˜w ∈ H also satisfies the desired conditions. Let h := Cw − C˜w . Then we have h∈
∑ Z[Γ>0 ]Ty ⊆ ∑ Z[Γ>0 ]Cy
y∈W
y∈W
and
hTw0 ∈
∑ Z[Γ0 ]Ty.
y∈W
Hence, using (∗), we conclude that, in an expression of h as a linear combination of basis elements {Cx }, all coefficients must be zero and so C˜w = Cw .
2.1 The Kazhdan–Lusztig Basis
63
Remark 2.1.7. As in [195], [219], we set Cw := (−1)l(w)Cw† for all w ∈ W , where † is defined in Example 1.2.6. (The element Cw is denoted by cw in [231].) Using the formula Tw† = (−1)l(w) Tw−1 −1 and the relations in 2.1.2, 2.1.3, we obtain Cw =
∗ ∗ Tz† = ∑ P¯z,w Tz−1 −1 = ∑ ∑ (−1)l(z)P¯z,w
z∈W
z∈W
∗ ∗ P¯z,w R¯ y,z Ty =
y∈W
∗ Ty . ∑ Py,w
y∈W
Furthermore, applying † to the relation CxCy = ∑z∈W hx,y,zCz , we obtain Cx Cy =
∑ (−1)l(x)+l(y)+l(z) hx,y,z Cz
for any x, y ∈ W .
z∈W
We shall write h x,y,z := (−1)l(x)+l(y)+l(z) hx,y,z for any x, y, z ∈ W . Thus, any statement about Cw has an equivalent version for Cw (where typically some signs need to be arranged). For applications to representation theory, it is more convenient to work with Cw ; see, for example, Remark 2.1.12. In this book, we will systematically work with Cw . Theorem 2.1.8 (Kazhdan and Lusztig [195], Lusztig [219], [231, Chap. 6]). For any x, y, z ∈ W , we have hx,y,z = h¯ x,y,z . Furthermore, for s ∈ S and w ∈ W , we have ⎧ ⎪ ⎪ ⎨ CsCw =
⎪ ⎪ ⎩ Csw −
Csw −(vs + v−1 s )Cw l(w)+l(y) s (−1) μy,w Cy ∑
if L(s) = 0, if L(s) > 0 and sw < w, if L(s) > 0 and sw > w,
y∈W :sy
s ∈ Z[Γ ] is such that μ s = μs . ¯ y,w where μy,w y,w
(The analogous formulae for the elements {Cw } are proved in [231, Chap. 6]; then it remains to use the conversion formulae in 2.1.7.) 2.1.9. There is a direct recursive algorithm for simultaneously computing ∗ | y, w ∈ W } {Py,w s {μy,w
and
| s ∈ S, y, w ∈ W such that L(s) > 0 and sy < y < w < sw},
without reference to the polynomials {R∗y,w }. Recall that, first of all, we have (a)
∗ = 1 for all w ∈ W Pw,w
and
∗ = 0 unless y < w; Py,w
see 2.1.3. We shall now list some further properties of these elements. Let y, w ∈ W be such that y < w. Let t ∈ S be such that tw < w. Then we have (b1) (b2) (b3)
∗ ∗ = Pty,tw Py,w ∗ Py,w ∗ Py,w
∗ = vt−1 Pty,w ∗ ∗ = vt Py,tw + Pty,tw −
if L(t) = 0, if L(t) > 0, ty > y,
∑
z∈W :yz
∗ t Py,z μz,tw
if L(t) > 0, ty < y.
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2 Kazhdan–Lusztig Cells and Cellular Bases
Furthermore, for any s ∈ S such that L(s) > 0 and sy < y < w < sw, we have (c1)
s ∗ μy,w − vs Py,w +
∑
∗ s Py,z μz,w ∈ Z[Γ<0 ],
z∈W :y
(c2)
See [231, Chap. 6] and [132, §11.1]. In order to describe a recursion based on the above properties, we need to define an ordering on all pairs of elements (y, w), where y, w ∈ W and y w. This is done as follows: (y , w ) (y, w)
def
⇔
w < w or
w = w and y y .
∗ = 1 and there are The recursion starts with the pair (y, w) = (1, 1). We have P1,1 no μ -polynomials to determine for this pair. Now let (y, w) be such that w = 1 and y w. Assume that Py∗ ,w and the relevant μ -polynomials are already known for all pairs (y , w ) (y, w). Then we proceed as follows. ∗ . If y = w, then P∗ = 1. If y < w, then choose some (1) First we determine Py,w w,w t ∈ S such that tw < w. There are three cases to distinguish:
(i) If L(t) = 0, then (ty,tw) (y, w) and so the right-hand side of (b1) is known by induction. (ii) If L(t) > 0 and ty > y, then (ty, w) (y, w) and so the right-hand side of (b2) is known by induction. (iii) If L(t) > 0 and ty < y, then all terms on the right-hand side of (b3) involve pairs (y , w ), where w < w. In particular, (y , w ) (y, w) for all such pairs and so, by induction, the right-hand side of (b3) is known. s for any s ∈ S such that (2) Now assume that y < w. Then we have to determine μy,w L(s) > 0 and sy < y < w < sw. For this purpose, we set ∗ α := vs Py,w −
∑
∗ s Py,z μz,w .
z∈W :y
(i) For all z appearing in the above sum, we have (y, z) (y, w) and (z, w) (y, w) and, hence, the corresponding terms are known by induction. By (1), ∗ . Thus, α is determined. we also know Py,w (ii) Write α = α+ + α0 + α− , where α+ ∈ Z[Γ>0 ], α− ∈ Z[Γ<0 ] and α0 ∈ Z s are uniquely determined. By (c1) and (c2), we have μy,w = α+ + α0 + α + . s is determined. Thus, μy,w For readers with an interest in “computer algebra” we just mention that it is an excellent programming exercise to implement the above recursion on a computer. For further details see, for example, DuCloux [75] and his C OXETER system, C HEVIE [105], [118], and the references in these articles. The above recursion formulae can actually be used to establish some further ∗ and μ s . We illustrate this with a few examples. properties of Py,w y,w
2.1 The Kazhdan–Lusztig Basis
65
Example 2.1.10. Let y, w ∈ W and s ∈ S. Then we claim that (a)
s ∈ Z[Γ>0 ], vs μy,w
where L(s) > 0 and sy < y < w < sw.
s Indeed, by 2.1.9(c2), this is equivalent to showing that v−1 s μy,w ∈ Z[Γ<0 ]. Multiplying 2.1.9(c1) by v−1 s , we obtain −1 s s ∗ ∗ vs μz,w ∈ Z[Γ<0 ]. Py,z v−1 ∑ s μy,w − Py,w + z∈W :y
s By an inductive argument, we can assume that we already know that v−1 s μz,w ∈ −1 s Z[Γ<0 ] for all z in the above sum. Hence, we also have vs μy,w ∈ Z[Γ<0 ], as required. Assume, furthermore, that we are in the equal-parameter case where Γ = Z and L(s) = 1 for all s ∈ S. Then Z[Γ ] is the ring of Laurent polynomials in one indeterminate v = ε . Let y, w ∈ W and s ∈ S be such that sy < y < w < sw. We have just s ∈ vZ[v]. Hence, we have μ s ∈ Z[v]. Since μ s = μ s , we conclude seen that vμy,w y,w y,w y,w s that μy,w ∈ Z. In fact, we have
(b)
s ∗ ∈ v−1 Z[v−1 ]. μy,w = coefficient of v−1 in Py,w
s ∈ Z, the relation in 2.1.9(c1) reduces to the condition that μ s − Indeed, since μy,w y,w ∗ −1 −1 vPy,w ∈ v Z[v ], which immediately yields the above statement. ∗ Example 2.1.11. Let y, w ∈ W be such that y w and set Py,w := vw v−1 y Py,w . Then the following holds:
(a) If L(s) > 0 for all s ∈ S, then Py,w ∈ Z[Γ0 ] is non-zero, with constant term 1. This is proved as follows (see also [231, Prop. 5.4]). If y = w, then Pw,w = 1 and so (a) holds. Now assume that y < w and choose some t ∈ S such that tw < w. If ty > y, then 2.1.9(b2) yields Py,w = Pty,w and so (a) holds by induction. (Note that y tw by 2.1.1(b) and, hence, ty t(tw) = w.) If ty < y, then 2.1.9(b3) yields t Py,z vtw v−1 vt μz,tw . Py,w = vt2 Py,tw + Pty,tw − ∑ z z∈W :yz
t ∈ Z[Γ>0 ] for all By Example 2.1.10 and induction, we have Py,z ∈ Z[Γ0 ] and vt μz,tw z in the above sum. Hence, we conclude that Py,w ≡ Pty,tw mod Z[Γ>0 ]. Since ty tw by 2.1.1(b), this yields (a) by induction. Note that if L(s) = 0 for some s ∈ S, then the conclusion in (a) no longer holds. For example, if L(s) = 0, then Cs = Ts and so P1,s = 0.
Remark 2.1.12. Assume that L(s) > 0 for all s ∈ S. Then the basis {Cw } gives rise to a W -graph structure on W . Indeed, let us set I(w) := {s ∈ S | sw < w} for w ∈ W . Furthermore, if y, w ∈ W and s ∈ S are such that s ∈ I(y) and s ∈ I(w), we set ⎧ 1 if y = sw, ⎨ s msy,w := −(−1)l(w)+l(y) μy,w if y < w, ⎩ 0 otherwise.
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2 Kazhdan–Lusztig Cells and Cellular Bases
Then we see that the data {I(w)}, {msy,w } give rise to a W -graph structure on the set W , in the sense of Definition 1.4.11. Note that vs msy,w ∈ Z[Γ>0 ] by Example 2.1.10. 2.1.13. Recall that H is a symmetric algebra, where {Tw | w ∈ W } and {Tw−1 | w ∈ W } form a pair of dual bases of H. Since each Cw equals Tw plus a Z[Γ>0 ]-linear combination of basis elements Tz (z ∈ W ), it easily follows that
τ (Cx−1 Cy ) ∈ δxy + Z[Γ>0 ]
(a)
for all x, y ∈ W ;
(see [220, 5.3.3] where this appeared in the equal-parameter case). Now set Dw := Tw +
∑
y∈W : w
∗
Pyw0 ,ww0 Ty
(w ∈ W ),
where w0 ∈ W is the longest element. Then, {Cw | w ∈ W } and {Dw−1 | w ∈ W } form a pair of dual bases; that is, we have
τ (Cx Dy−1 ) = δxy
(b)
for all x, y ∈ W .
In particular, hx,y,z = τ (CxCy Dz−1 ) for all x, y, z ∈ W , a relation which will be used repeatedly in what follows. The relation (b) follows from the following identity:
∑
∗ ∗ (−1)l(w)+l(y) Py,z Pww0 ,zw0 = δyw
for all y w in W ,
z∈W : yzw
which appeared as [195, Theorem 3.1] in the equal-parameter case; see [231, 10.7 and 11.4] or [103, §2] for the general case. Once the above identity is proved, one also obtains the following relation (see [231, 11.6] or [103, 2.6]): s s μww = −(−1)l(w)+l(y) μy,w 0 ,yw0
(c)
for any s ∈ S and y, w ∈ W such that sy < y < w < sw. 2.1.14. The A-linear map H → H, h → h , defined by Tw = Tw−1 (w ∈ W ) is an anti¯∗ involution of H; see Example 1.2.5. Applying to the relation Tw−1 −1 = ∑y∈W Ry,w Ty , ∗ ∗ we find that Ry−1 ,w−1 = Ry,w for all y, w ∈ W . Then, using 2.1.3, it also follows that Cw = Cw−1
and
∗ Py,w = Py∗−1 ,w−1
for all y, w ∈ W .
We can now apply the general definitions concerning “cells” in Section 1.6 to the algebra H = H with its basis {Cw | w ∈ W }. Thus, we obtain pre-order relations L , R , LR on W . Recall, for example, that L is defined as the transitive closure of the relation ←L ; by the multiplicition formulae in Theorem 2.1.6, we have either y = sw, where s ∈ S is such that L(s) = 0 or sw > w, y ←L w ⇔ s = 0, where s ∈ S, L(s) > 0 and sy < y < w < sw. or μy,w Furthermore, we have y R w if and only if y−1 L w−1 . And, finally, LR is the union of L and R .
2.1 The Kazhdan–Lusztig Basis
67
Definition 2.1.15. The left, right or two-sided cells defined, in the sense of 1.6.1, by taking H = H with its basis {Cw | w ∈ W }, are called the left, right or two-sided Kazhdan–Lusztig cells of W respectively. From now on, unless explicitly stated otherwise, the symbols L , R , LR , ∼L , ∼R , ∼LR will always refer to the pre-order relations defined using the Kazhdan–Lusztig basis {Cw } of H. Lemma 2.1.16 (Lusztig [231, 8.6]). Given w ∈ W , define L (w) := {s ∈ S | sw < w and L(s) > 0} and R(w) := L (w−1 ). Then the following hold: (a) If z, y ∈ W are such that z L y, then R(y) ⊆ R(z). (b) If z, y ∈ W are such that z R y, then L (y) ⊆ L (z). In particular, the function w → R(w) is constant on left cells and the function w → L (w) is constant on right cells. Proof. Since the formulation in [231, 8.6] does not include the possibility that L(s) = 0 for some s ∈ S, let us briefly sketch the argument. To prove (a), we may assume that z, y are related by an elementary step in the definition of L ; that is, there is some s ∈ S such that h s,y,z = 0. If L(s) > 0, then the argument is exactly the same as in [231, 8.6], using the fact that t H := Cw | w ∈ W, wt < wA ⊆ H is a left ideal for any t ∈ S such that L(t) > 0; see [231, 8.4]. Now assume that L(s) = 0. Then z = sy by the multiplication formulae in Theorem 2.1.6. Let t ∈ R(y). If sy > y, then R(y) ⊆ R(sy) and so t ∈ R(z), as required. Finally, assume that sy < y; then l(sy) = l(yt). If we had zt > z, then l(syt) = l(y) and so syt = y; see [132, 1.2.6]. Hence, s,t would be conjugate in this case and so L(s) = L(t), which is a contradiction. Thus, we must have zt < z, as required. The proof of (b) is analogous. Example 2.1.17. Assume that L(s) > 0 for all s ∈ S. Let w ∈ W be such that w ∼L 1. Then Lemma 2.1.16 implies that R(w) = R(1) = ∅ and so w = 1. Hence, {1} is a left Kazhdan–Lusztig cell. Similarly, let w ∈ W be such that w ∼L w0 , where w0 ∈ W is the longest element. Then Lemma 2.1.16 implies that R(w) = R(w0 ) = S and so w = w0 . Hence, {w0 } is a left Kazhdan–Lusztig cell. We have the following explicit formula: Cw0 =
∑ (−1)l(w0 )+l(w) ε L(w0)−L(w) Tw .
w∈W
Indeed, if w ∈ W is such that w = w0 , then there exists some s ∈ S such that sw > w. ∗ ∗ = v−1 Hence, the formula in 2.1.9(b2) yields that Pw,w s Psw,w0 . By a simple downward 0 ∗ L(w)−L(w ) 0 , as required. induction on l(w), we conclude that Pw,w0 = ε Example 2.1.18. Let W be of type I2 (m) (m 3); that is, we have W = s1 , s2 , where s21 = s22 = (s1 s2 )m = 1. Let L be a weight function where b := L(s1 ) 0 and a := L(s2 ) 0; here, a = b if m is odd. The relations L , R and LR are determined in [231, 8.8]. (See also [132, Exc. 11.4] for the case a = b.) For any k 0, write 1k = s1 s2 s1 · · · (k factors) and 2k = s2 s1 s2 · · · (k factors); note that 1m = 2m . With this notation, we have:
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2 Kazhdan–Lusztig Cells and Cellular Bases
• If m is odd and a = b > 0, then the left cells are {10 }, {1m }, {21 , 12 , 23 , . . . , 1m−1 }, {11 , 22 , 13 , . . . , 2m−1 }. • If m is even and a = b > 0, then the left cells are {10 }, {1m }, {21 , 12 , 23 , . . . , 2m−1 }, {11 , 22 , 13 , . . . , 1m−1 }. • If m is even and b > a > 0, then the left cells are {10 }, {21 }, {1m−1 }, {1m }, {11 , 22 , 13 , . . . , 2m−2 }, {12 , 23 , 14 , . . . , 2m−1 }. • If m is even and b > a = 0, then the left cells are {10 , 21 }, {1m , 1m−1 }, {11 , 22 , 13 , . . . , 2m−2 }, {12 , 23 , 14 , . . . , 2m−1 }. The two-sided cells and the partial order induced on them are given by {1m } LR W \ {10 , 1m } LR {10 } (a = b > 0), {1m } LR {1m−1 } LR W \{10 , 21 , 1m−1 , 1m } LR {21 } LR {10 } (b > a > 0), {1m , 1m−1 } LR W \ {10 , 21 , 1m−1 , 1m } LR {10 , 21 } (b > a = 0). ˜ Recall that, in Definition 1.6.4, we have introduced the left, right and two-sided J˜ cells of W , using the algebra H = J with its basis {tw | w ∈ W }. In the above example where W is of type I2 (m), the two-sided Kazhdan–Lustzig cells are precisely the ˜ two-sided J-cells determined in Example 1.7.3. If this was known to be true in general, then our task in this book would be considerably simpler! (We will discuss this in more detail in Section 2.5.) To close this section, we will show by a general ˜ argument that, at least, the Kazhdan–Lusztig cells are always unions of J-cells. 2.1.19. We will now bring back into the picture the balanced matrix representations {ρ λ | λ ∈ Λ } and the corresponding leading matrix coefficients cst w,λ ; see Section 1.4. Recall that, for any w ∈ W , we have
ε aλ ρ λ (Tw ) ∈ Mdλ (O0 )
and
aλ λ cst w,λ ≡ ε ρst (Tw ) mod m
for all λ ∈ Λ and s, t ∈ M(λ ). Now consider the expressions for Cw and Dw . Since ∗ Py,w ∈ Z[Γ>0 ] for all y = w, we deduce that λ ε aλ ρst (Cw ) ∈ O0
and
λ ε aλ ρst (Dw ) ∈ O0 ,
λ λ λ ε aλ ρst (Tw ) ≡ ε aλ ρst (Cw ) ≡ ε aλ ρst (Dw ) ≡ cst w,λ mod m,
for all λ ∈ Λ and s, t ∈ M(λ ). Thus, the leading matrix coefficients can be taken with respect to any of the bases {Tw }, {Cw } or {Dw }. ˜ Proposition 2.1.20. Every left Kazhdan–Lusztig cell of W is a union of left J-cells (see Definition 1.6.4). Analogous statements hold for right and two-sided cells. In particular, if x, y, z ∈ W are such that γ˜x,y,z = 0, then the elements x±1 , y±1 , z±1 all lie in the same two-sided Kazhdan–Lusztig cell. ˜ Proof. Let y, z ∈ W belong to the same left J-cell. It will be sufficient to consider an elementary step in the definition of this relation; that is, we can assume that
2.2 A Pre-order Relation on Irr(W )
γ˜x,y,z−1 =
∑
∑
λ ∈Λ s,t,u∈M(λ )
69
tu us f λ−1 cst x,λ cy,λ cz,λ = 0 for some x ∈ W .
We deduce that
∑
u∈M(λ )
us ctu y,λ cz−1 ,λ = 0
for some λ ∈ Λ and s, t ∈ M(λ ).
Using the relations in 2.1.19 we obtain that λ λ λ ε 2aλ ρts (Cy Dz−1 ) ≡ ∑ ε aλ ρtu (Cy ) ε aλ ρus (Dz−1 ) ≡ u∈M(λ )
∑
u∈M(λ )
us ctu y,λ cz−1 ,λ
modulo m. Since the expression on the right-hand side is non-zero modulo m, we conclude that ρ λ (Cy Dz−1 ) = 0 and so Cy Dz−1 = 0. Since τ is non-degenerate, we have τ (CwCy Dz−1 ) = 0 for some w ∈ W . This yields hw,y,z = τ (CwCy Dz−1 ) = 0 and so z L y (in the Kazhdan–Lusztig pre-order). Similarly, we find that λ λ λ us ε 2aλ ρts (DyCz−1 ) ≡ ∑ ε aλ ρtu (Dy ) ε aλ ρus (Cz−1 ) ≡ ∑ ctu y,λ cz−1 ,λ u∈M(λ )
u∈M(λ )
modulo m and, hence, DyCz−1 = 0. Again, we can find some w ∈ W such that τ (Cw DyCz−1 ) = 0. It follows that hz−1 ,w,y−1 = τ (Cz−1 Cw Dy ) = τ (Cw DyCz−1 ) = 0. Hence, we have y−1 R z−1 and so y L z. Thus, we have shown that y, z belong to the same left Kazhdan–Lusztig cell. The arguments for right and two-sided cells are analogous. The last statement (involving γ˜x,y,z ) follows from Corollary 1.6.7.
2.2 A Pre-order Relation on Irr(W ) We have just seen that the weight function L : W → Γ and the monomial order on Γ give rise to the Kazhdan–Lusztig pre-order relations L , R , LR on W . We will now use the two-sided relation LR to define a pre-order relation on the set IrrK (W ) = {E λ | λ ∈ Λ }. Recall that, in Proposition 1.6.11, we constructed a natural surjective map ˜ IrrK (W ) → {set of two-sided J-cells of W },
E λ → Fλ .
By Proposition 2.1.20, we also know that each Fλ is contained in a two-sided Kazhdan–Lusztig cell. This leads us to the following definition. Definition 2.2.1 (Cf. Lusztig [220, 5.15]). Let λ , μ ∈ Λ . Then we define E λ L E μ
def
⇔
x LR y
for all x ∈ Fλ and y ∈ Fμ ,
where LR is the Kazhdan–Lusztig pre-order relation on W ; see 2.1.14. Since each ˜ two-sided J-cell is contained in a two-sided Kazhdan–Lusztig cell, we have
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2 Kazhdan–Lusztig Cells and Cellular Bases
E λ L E μ
⇔
x LR y
for some x ∈ Fλ and some y ∈ Fμ .
Furthermore, we write E λ ∼L E μ if E λ L E μ and E μ L E λ . Thus, E λ ∼L E μ if and only if Fλ , Fμ are contained in the same two-sided Kazhdan–Lusztig cell. We wish to mention right away that the relation ∼L on IrrK (W ) is not yet completely understood nor explicitly known in all cases (and even less so the relation L ). In this section we will, therefore, content ourselves with giving some examples and explaining the open questions. Of particular interest for us will be the relation between L and the function E λ → aλ . We will see that even the first example that one might think of, namely the case where W ∼ = Sn , requires a considerable amount of work; see Example 2.2.13 and Section 2.8. We begin by showing that L and ∼L can be expressed without reference to the map E λ → Fλ . 2.2.2. By the general method described in 1.6.1, each left Kazhdan–Lusztig cell C gives rise to a representation of H. This is constructed as follows. Let [C]A be an A-module with a free A-basis {ew | w ∈ C}. Then the action of H on [C]A is given by (a)
Cw .ex :=
∑ hw,x,y ey ,
where w ∈ W and x ∈ C.
y∈C
(Similarly, right cells give rise to right H-modules and two-sided cells give rise to H-bimodules.) Now let θ : A → k be a ring homomorphism into a field k. Then [C]k := k ⊗A [C]A is a left module for the specialised algebra Hk . In particular, let θ1 : A → K be the ring homomorphism such that θ1 (ε g ) = 1 for all g ∈ Γ , as in 1.2.1. Then we obtain a module [C]1 := K ⊗A [C]A for KW = K ⊗A H. For any λ ∈ Λ , denote by m(C, λ ) the multiplicity of E λ as an irreducible constituent of [C]1 . Then the “specialisation argument” in Example 1.2.4 immediately shows that (b)
m(C, λ ) = multiplicity of Eελ as an irreducible constituent of [C]K ,
where [C]K is the HK -module obtained from [C]A by extending scalars from A to K. Remark 2.2.3. Assume that L(s) > 0 for all s ∈ S. Let C be a left Kazhdan–Lusztig cell and consider the left cell module [C]A . As in Remark 2.1.12, we see that the action of H on [C]A is given by a W -graph, where X = C, I(x) = L (x) (x ∈ C) and ⎧ 1 if x = sy > y, ⎨ s msx,y = −(−1)l(y)+l(x) μx,y if sx < x < y < sy, ⎩ 0 otherwise. Lemma 2.2.4. Let λ ∈ Λ and C be a left Kazhdan–Lusztig cell such that m(C, λ ) > 0. Then we have E λ L w for some w ∈ C, that is, w ∈ C ∩ Fλ . Proof. (Compare with the proof of Theorem 1.8.1.) Consider the identity
∑ Cw Dw−1 = ∑ ∑ hx,y,y Dx−1 .
w∈C
x∈W y∈C
2.2 A Pre-order Relation on Irr(W )
71
(This is proved by multiplying both sides by Cz for some z ∈ W and applying the trace form τ .) Now note that trace(Cx , [C]A ) = ∑y∈C hx,y,y . Taking also into account μ the formula [C]K ∼ = μ ∈Λ m(C, μ ) Eε , we obtain
∑ hx,y,y = ∑ m(C, μ ) χ μ (Cx ) μ ∈Λ
y∈C
for all x ∈ W .
Then the orthogonality relations in Proposition 1.2.12 yield that
χ λ ∑ Cw Dw−1 = ∑ m(C, μ ) ∑ χ μ (Cx ) χ λ (Dx−1 ) = m(C, λ ) dλ cλ . μ ∈Λ
w∈C
x∈W
Multiplying this relation by ε 2aλ and taking constant terms, we deduce that
∑ ∑ cstw,λ ctsw−1 ,λ = m(C, λ ) dλ fλ .
s,t∈M(λ ) w∈C
Since the right-hand side is non-zero by assumption, we conclude that cst w,λ = 0 for some w ∈ C and some s, t ∈ M(λ ), as required. Corollary 2.2.5. Let λ , μ ∈ Λ and C, C be left Kazhdan–Lusztig cells such that m(C, λ ) > 0 and m(C , μ ) > 0. Then E λ L E μ if and only if w LR w for some w ∈ C and some w ∈ C (where LR denotes the Kazhdan–Lusztig pre-order relation). In particular, E λ ∼L E μ if and only if C, C are contained in the same two-sided Kazhdan–Lusztig cell. Proof. First assume that E λ L E μ . By definition, this means that x LR y for all x ∈ Fλ and y ∈ Fμ . Now, by Lemma 2.2.4, there exist elements w ∈ C ∩ Fλ and w ∈ C ∩ Fμ . Hence, we have w LR w , as required. Conversely, assume that w LR w for some w ∈ C and some w ∈ C . Since m(C, λ ) > 0, there exists some w1 ∈ C ∩ Fλ ; see Lemma 2.2.4. Similarly, there exists some w 1 ∈ C ∩Fμ . Hence, we have w1 ∼L w LR w ∼L w 1 and so w1 LR w 1 . As pointed out in Definition 2.2.1, this already implies that E λ L E μ . Remark 2.2.6. Let W = W1 × · · · ×Wd be the decomposition into irreducible components. Correspondingly, we have IrrK (W ) = {E λ1 · · · E λd | λi ∈ Λi },
where
IrrK (Wi ) = {E λi | λi ∈ Λi }.
Thus, as in Remark 1.3.5, we identify Λ = Λ1 × · · · × Λd . Furthermore, we have H∼ = H1 ⊗A · · · ⊗A Hd , where Hi is the generic algebra associated with Wi , L|Wi . The Kazhdan–Lusztig basis of H behaves well with respect to this decomposition, that is, if w = w1 · . . . · wd , where wi ∈ Wi , then Cw = Cw1 · . . . ·Cwd . It follows that E λ L E μ
⇔
E λi Li E μi
for i = 1, . . . , d,
where λ = (λ1 , . . . , λd ) and μ = (μ1 , . . . , μd ). Thus, the determination of L can be reduced to the case where (W, S) is irreducible.
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2 Kazhdan–Lusztig Cells and Cellular Bases
2.2.7. Assume that W, L are such that the following data are explicitly available: • The matrices {ρ λ (Ts ) | s ∈ S} for all λ ∈ Λ . (Recall the algorithm in 1.4.9 for turning any given representation into a balanced representation.) ∗ } and { μ s }. (See the recursive description in 2.1.9.) • All the polynomials {Py,w y,w Since the invariants aλ are also known (see Section 1.3), we can then work out all leading matrix coefficients cst w,λ and the Kazhdan–Lusztig pre-order relations L , R , LR . This, in turn, enables us to explicitly determine the pre-order relation L on IrrK (W ), via the characterisation in Corollary 2.2.5. Now, the above data are available for W of type I2 (m) (any m 3), H3 , H4 , F4 . We will now go through these examples one by one and describe the relation L on IrrK (W ) in each case. Example 2.2.8. Let W be of type I2 (m) (m 3); that is, we have W = s1 , s2 , where s21 = s22 = (s1 s2 )m = 1. Recall from Example 1.3.7 the description of IrrK (W ). By Example 2.1.18, we know the left and two-sided cells. It it also not difficult to determine the cell modules [C]1 and decompose them into irreducibles. Let ψ denote the sum of all the two-dimensional representations. • If m is odd and L(s1 ) = L(s2 ) > 0, then the left cell {10 } affords 1W , {2m } affords sgn, and the two remaining left cells afford ψ . • If m is even and L(s1 ) = L(s2 ) > 0, then {10 } affords 1W , {2m } affords sgn, the first of the two remaining left cells affords ψ ⊕ sgn2 , and the second affords ψ ⊕ sgn1 . • If m is even and L(s1 ) > L(s2 ) > 0, then {10 } affords 1W , {21 } affords sgn1 , {1m−1 } affords sgn2 , {1m } affords sgn, and the two remaining left cells afford ψ . • If m is even and L(s1 ) > L(s2 ) = 0, then {10 , 21 } affords 1W ⊕ sgn1 , {1m , 1m−1 } affords sgn ⊕ sgn2 , and the two remaining left cells afford ψ . Using this information together with the knowledge of LR (see Example 2.1.18) and of aλ (see Example 1.3.7), we find that the pre-order L on IrrK (W ) is “linear” in the sense that, for any λ , μ ∈ Λ , we have E λ L E μ
⇔
a μ aλ .
In particular, E λ ∼L E μ if and only if aλ = aμ . Example 2.2.9. Let W be of type H3 or H4 . Then all generators are conjugate, so we are automatically in the equal-parameter case. Assume that L(s) > 0 for s ∈ S. Then ∗ and μ s . In this way, he explicitly Alvis [2] has computed all polynomials Py,w y,w determined the relations L and LR ; he also found the decomposition of the left cell representations into irreducibles (see [216, §5] for type H3 ). The partial order induced by LR on the set of two-sided cells is, in fact, a total order.1 The equivalence classes of IrrK (W ) under ∼L are explicitly known by [218, §13] and [2, 3.5]. This information, together with the invariants aλ , is listed in [132, Tables C.1 and C.2]. It turns out that, again, the pre-order L is “linear” such that 1
This statement is not contained in [2]; we thank Alvis (personal communication, 2008) for having verified this using his programs for producing the data in [2].
2.2 A Pre-order Relation on Irr(W )
73
E λ L E μ
⇔
aμ aλ
for any λ , μ ∈ Λ . In particular, E λ ∼L E μ if and only if aλ = aμ .
Fig. 2.1 Partial order on two-sided cells in type F4
c 11 c 23 c 42 c 11 c 42 c 91 \ \c8 81 c 3 \ 121 \ c \ \c8 82 c 4 \ \ c 9 4
c @ @
13
81
c AA
c 93 Ac @ @
161
82
c AA
12
c 92 c 43
Ac c 45
c 45
c 24
c 14
c 14
a=b
c 44
b = 2a
c 11 c 23 c 42 c 21 91 c QQ c 83 c 13 81 c c 44 A c A 93 A 161 c QQ c 92 c 43 82 c c 12 A c 84 A A 94 c c 22 c 45 c 24 c 14
c 11 c 23 @ @ c 42
c @ @c 8 3 QQ c 91 44 c # # c 21 # # c 93 c @ 81 @ c 161 QQ c 82 c 92 c c 22 c c 43 c c @ c 94 @ c 84 @ @ c 45 12 c @ @c2 4 13
2a > b > a
c 14 b > 2a
Brackets indicate a two-sided cell with several irreducible components, given as follows: 42 = {21 , 23 , 42 }, 45 = {22 , 24 , 45 }, 13 = {13 , 21 , 83 , 91 }, 12 = {12 , 22 , 84 , 94 }, 121 = {12 , 13 , 41 , 43 , 44 , 61 , 62 , 92 , 93 , 121 , 161 },
161 = {41 , 61 , 62 , 121 , 161 }.
Otherwise, the two-sided cell contains just one irreducible respresentation.
Example 2.2.10. Let W be of type F4 with generators labelled as in Table 1.1 (p. 2). Assume that a := L(s1 ) = L(s2 ) > 0 and b := L(s3 ) = L(s4 ) > 0. (The case where L(si ) = 0 for some i will be considered in Remark 2.4.13.) We may also assume without loss of generality that b a. The pre-order relations L , R , LR have been determined in [105], based on an explicit computation of all the polynomials ∗ and μ s using C HEVIE [118]. The resulting pre-order relations on Irr (W ) Py,w L K y,w are given in Figure 2.1. The notation for the irreducible representations is compatible with that in Table 1.2 (p. 16). For example, 11 is the trivial representation, 14 is the sign representation and 42 is the reflection representation. The pre-order L is not
74
2 Kazhdan–Lusztig Cells and Cellular Bases
“linear” in these cases, but by inspection of the tables we notice that, at least, the following property holds: E λ L E μ
⇒
(with equality only if E λ ∼L E μ ).
a μ aλ
In particular, if E λ ∼L E μ , then aλ = aμ . Remark 2.2.11. The diagrams in Figure 2.1 have a striking symmetry. This is a general phenomenon. Indeed, recall the definition of the bijection λ → λ † on Λ from Example 1.2.6. By Corollary 1.6.16, we have IrrK (W | F w0 ) = IrrK (W | F )† for ˜ every two-sided J-cell F of W . Now, 2.1.13(c) implies that if x, y ∈ W are such that x LR y, then yw0 LR xw0 . It follows that, for any λ , μ ∈ Λ , we have E λ L E μ
⇔
E μ L E λ . †
†
Thus, the pre-order L admits a natural symmetry with respect to λ → λ † . 2.2.12. Assume that W is of crystallographic type and that we are in the equalparameter case where Γ = Z and L(s) = 1 for all s ∈ S. It has recently been shown [113] that then L admits a geometric interpretation, and this actually yields an explicit description of L . It would be beyond the scope of this book to explain this in detail, but we can at least outline the general idea, assuming some familiarity with the theory of algebraic groups and Lusztig’s work [220] (see also Section 4.4). So let G be a connected reductive algebraic group (over C or over F p , where p is a large prime), with Weyl group W . Then, by the Springer correspondence (see [197], [221], [282]), we can naturally associate with every E λ ∈ IrrK (W ) a pair consisting of a unipotent class of G, which we denote by Oλ , and a G-equivariant irreducible local system on Oλ . Thus, we obtain a map IrrK (W ) → {set of unipotent classes in G},
E λ → Oλ .
(The local system on Oλ will not play a role for our purposes here.) We now need the concept of a “special” unipotent class. This is defined as follows. Given λ ∈ Λ , let bλ be the smallest i 0 such that E λ is a constituent of the ith symmetric power of the natural reflection representation of W . Lusztig [215] observed that we always have aλ bλ . We say that E λ is a special representation if aλ = bλ ; let S (W ) := {λ ∈ Λ | aλ = bλ }. Following Lusztig [215], the classes {Oλ | λ ∈ S (W )} are called the special unipotent classes of G (although the word “special” only appeared in later references; see also 4.3.13). By [220, 13.1.1], we have aλ = dim Bu
for any λ ∈ S (W ),
where u ∈ Oλ and Bu is the variety of Borel subgroups in G containing u. Now [113, Cor. 5.6] shows that, for any λ , μ ∈ S (W ), we have
2.2 A Pre-order Relation on Irr(W )
(∗)
E λ L E μ
75
⇔
Oλ ⊆ Oμ := Zariski closure of Oμ .
The map E λ → Oλ is explicitly known in all cases; see Carter [45, §13.3] and the references there. Also, the Zariski closure relation among the special unipotent classes of G is explicitly known in all cases; see Carter [45, §13.4] and Spaltenstein [280]. Hence, (∗) provides an explicit description of the pre-order L for special representations. On the other hand, given any λ ∈ Λ , we have E λ ∼L E λ0
for a unique λ0 ∈ S (W );
see [220, 4.14 and 5.25]. Hence, since the equivalence relation ∼L is explicitly known by Lusztig [220, 4.4–4.13 and 5.25], the relation L is determined once we know it for special representations. Finally, by [220, 5.27], the function λ → aλ is known to be constant on the equivalence classes under ∼L . Hence, by the above characterisation of aλ for λ ∈ S (W ), we also find that, for any λ , μ ∈ Λ , we have E λ L E μ
⇒
a μ aλ
(with equality only if E λ ∼L E μ ).
(In the next two sections, we will say more about the proof of this implication.) Example 2.2.13. In the setting of 2.2.12, let W be of type An−1 . Then W ∼ = Sn and Λ is the set of all partitions of n; see Example 1.3.8. By [220, 4.4], all irreducible representations of W are special. Now W is the Weyl group of G = GLn (over C or over F p , where p is a large prime). Let λ = (λ1 λ2 . . . 0) ∈ Λ . Then the Springer correspondence associates with E λ the unipotent class Oλ consisting of all unipotent matrices in G whose Jordan normal form has blocks of size λ1 , λ2 , . . .; see Springer [282, p. 293], [45, §13.3]. By [104, §2.6] and 2.2.12(∗), we have E λ L E μ
⇔
Oλ ⊆ Oμ
⇔
λ μ,
where denotes the dominance order, which is defined as follows. Write λ = (λ1 λ2 · · · 0) and μ = (μ1 μ2 · · · 0). Then
λ μ
def
⇔
∑
1id
λi
∑
μi
(for all d 1).
1id
It then follows by a completely elementary argument that we have the implication
λ μ
⇒
λ =μ
or aλ > aμ ;
see, for example, [132, Exc. 5.6]. See Corollary 2.8.14 for a much more direct and elementary proof of the above characterisation of L (following [107]). Example 2.2.14. In the setting of 2.2.12, let W be of type Bn . Then Λ is the set of all pairs of partitions (λ , μ ) such that |λ | + |μ | = n; see Example 1.3.9. For any (λ , μ ) ∈ Λ and (λ , μ ) ∈ Λ , let us define
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2 Kazhdan–Lusztig Cells and Cellular Bases
(λ , μ ) (λ , μ )
def
⇔
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
∑
(λi + μi )
∑
(λi + μi ) λd +
1id
λd +
∑
(λi + μi )
1id
1i
∑
(λi + μi ),
1i
(for all d 1)
where λ = (λ1 λ2 . . . 0), λ = (λ1 λ2 . . . 0), μ = (μ1 μ2 . . . 0) and μ = (μ1 μ2 . . . 0). By [220, 4.5], we have: (λ , μ ) ∈ S (W )
⇔
λi + 1 μi λi+1
(for all i 1).
Now W is the Weyl group of G = SO2n+1 (over C or over F p , where p is a large prime). Then, by Spaltenstein [281, §4] and 2.2.12(∗), we have E (λ ,μ ) L E (λ
,μ )
⇔
O(λ ,μ ) ⊆ O(λ ,μ )
⇔
(λ , μ ) (λ , μ )
for (λ , μ ) ∈ S (W ) and (λ , μ ) ∈ S (W ). See [122, §5] for further details. Example 2.2.15. Let again W be of type Bn and consider the reflection subgroup W˜ ⊆ W in Example 1.3.10. Then W˜ is of type Dn and this is the Weyl group of G = SO2n (over C or over F p , where p is a large prime). To be consistent with the notation in Example 1.3.10, the equal-parameter weight function on W˜ will be ˜ Now IrrK (W˜ ) is described in terms of the restrictions of the irreducible denoted by L. representations of W to W˜ . Given E˜ ∈ IrrK (W˜ ) and (λ , μ ) ∈ Λ , we write E˜ | E (λ ,μ )
def
⇔
E˜ is a constituent of the restriction of E (λ ,μ ) to W˜ .
To characterise the special representations in IrrK (W˜ ), it is convenient to define S˜(W ) := {(λ , μ ) ∈ Λ ) | λi μi λi+1 − 1 for all i 1}, where we write λ = (λ1 λ2 . . . 0) and μ = (μ1 μ2 . . . 0). Now let E˜ ∈ IrrK (W˜ ). Then, by [220, 4.6], we have E˜ is special
⇔
E˜ | E (λ ,μ ) for some (λ , μ ) ∈ S˜(W ).
Note that if E˜ is special, then (λ , μ ) on the right-hand side is uniquely determined: just observe that if both (λ , μ ) and ( μ , λ ) belong to S˜(W ), then λ = μ . Now, by Spaltenstein [281, §4] and 2.2.12(∗), we obtain the following result. Let ˜ ) and (λ , μ ) ∈ S˜(W ) ˜ E˜ ∈ IrrK (W˜ ) be special representations. Let (λ , μ ) ∈ S(W E, be such that E˜ | E (λ ,μ ) and E˜ | E (λ ,μ ) . Then if λ = λ = μ = μ , E˜ = E˜ ˜ ˜ ⇔ E L˜ E otherwise, (λ , μ ) (λ , μ ) where (λ , μ ) (λ , μ ) is defined in Example 2.2.15. See [122, §5] for further details.
2.2 A Pre-order Relation on Irr(W )
77
Example 2.2.16. Assume that W is of type E6 , E7 or E8 . The equivalence classes of IrrK (W ) under ∼L are listed in Lusztig [220, 4.11–4.13]; see also [132, App. C]. The Springer correspondence is explicitly described in the tables in [45, 13.3]. (E6 ) We have | IrrK (W )| = 25 and there are 17 equivalence classes under ∼L . The partially ordered set of special unipotent classes is printed in [45, p. 441]. (E7 ) We have | IrrK (W )| = 60 and there are 35 equivalence classes under ∼L . The partially ordered set of special unipotent classes is printed in [45, p. 443]. (E8 ) We have | IrrK (W )| = 112 and there are 46 equivalence classes under ∼L . The partially ordered set of special unipotent classes is printed in [45, p. 445]. Example 2.2.17. Let W be of type Bn and L : W → Γ be a weight function given by Bn
b t
4
a t
a t
p p p
a t
where b > (n − 1)a > 0.
Recall that Λ is the set of all pairs of partitions (λ , μ ) such that |λ | + |μ | = n; see Example 1.3.9. By [122, Example 5.1] (which relies on the series of papers by Bonnaf´e, Geck, Iancu [21], [26], [108], [114], [121]), we have (a)
E (λ ,μ ) L E (λ
,μ )
⇔
(λ , μ ) (λ , μ ).
Here, denotes the dominance order on pairs of partitions, which is defined by ⎧ ⎪ ∑ λi ∑ λi ⎪ ⎪ ⎪ 1id 1id ⎨ def (b) (λ , μ ) (λ , μ ) ⇔ |λ | + ∑ μi |λ | + ∑ μi , ⎪ ⎪ 1id 1id ⎪ ⎪ ⎩ (for all d 1) where λ = (λ1 λ2 . . . 0), λ = (λ1 λ2 . . . 0), μ = (μ1 μ2 . . . 0) and μ = (μ1 μ2 . . . 0). Furthermore, by [121, Cor. 5.5], we have (c)
E (λ ,μ ) L E (λ
,μ )
⇒
a(λ ,μ ) a(λ ,μ ) ,
with equality only if (λ , μ ) = (λ , μ ). For the (infinitely many) remaining open cases in type Bn , at least a conjecture is formulated in [122, 4.11]. Remark 2.2.18. Lusztig’s definition [220, 5.15] of a pre-order relation on IrrK (W ) looks somewhat different from that in Definition 2.2.1, but it is really the same. Let us briefly indicate why this is the case. By the Artin–Wedderburn theorem, the split semisimple algebra HK decomposes as a direct sum of simple two-sided ideals HK = λ ∈Λ HK (λ ), where HK (λ ) is the sum of all left ideals in HK which are isomorphic to Eελ (as left HK -modules). On the other hand, for any y ∈ W , we have and Iˆ LR defined by the general procedure in 1.6.2, with the two-sided ideals ILR y y respect to the basis {Cw | w ∈ W } of HK . Now let T be a two-sided Kazhdan–Lusztig cell and λ ∈ Λ . Then we claim that the following two statements are equivalent:
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2 Kazhdan–Lusztig Cells and Cellular Bases
(a) E λ L w for some w ∈ T. and HK (λ ) ⊆ Iˆ LR for some y ∈ T. (b) HK (λ ) ⊆ ILR y y uv Indeed, if (a) holds, then cst w,λ = 0 and cw−1 ,λ = 0 for some s, t, u, v ∈ M(λ ); see
1.6.10. Hence, by 2.1.19, we have ρ λ (Cw ) = 0 and ρ λ (Dw−1 ) = 0. This yields Cw .HK (λ ) = {0} and Dw−1 .HK (λ ) = {0}. Then the argument in the proof of [220, Lemma 5.2] shows that (b) holds. Conversely, assume that (b) holds. Then the incluinduces an (HK , HK )-bimodule homomorphism ϕ : HK (λ ) → sion HK (λ ) ⊆ ILR y LR LR ˆ Iy /Iy , which is non-zero since HK (λ ) ⊆ Iˆ LR y . Now let us just consider the left ˆ LR HK -module structure. Since T is a union of left cells, the left HK -module ILR y /Iy has a filtration by left cell modules [Ci ]K , where each Ci is a left cell contained in T. Hence, there exists a non-zero HK -module homomorphism Eελ → [Ci ]K for some i. Then m(Ci , λ ) > 0 and so there exists some w ∈ Ci such that E λ L w; see Lemma 2.2.4. Thus, the equivalence of (a) and (b) is proved. Once this is established, we can conclude that there exists some w ∈ W such that (c) E λ L E μ ⇔ LR ˆ LR HK (λ ) ⊆ ILR w , HK ( μ ) ⊆ Iw , HK ( μ ) ⊆ Iw . The condition on the right-hand side is the one used by Lusztig [220, 5.15].
2.3 On Lusztig’s Conjectures, I In the previous section, we have defined a pre-order relation L on IrrK (W ) and we have seen that, in many examples, the following implication holds for any λ , μ ∈ Λ : (♣)
E λ L E μ
⇒
a μ aλ
(with equality only if E λ ∼L E μ ).
This property will turn out to be the key to our main results on representations of Hecke algebras at roots of unity. The somewhat weaker implication (♣ )
E λ ∼L E μ
⇒
aμ = aλ
was a key ingredient in Lusztig’s work [220] on characters of reductive groups over finite fields. Now, a general proof of these apparently simple statements is not yet known. And in those situations where (♣) and (♣ ) are known to hold, the proofs rely on deep results from algebraic geometry, or explicit computations. It is the purpose of this and the following two sections to discuss this in some more detail. Assume first that W is a Weyl group and that we are in the equal-parameter case. Then the proof of (♣ ) in [220, Chap. 5] relies on the theory of primitive ideals in enveloping algebras. Subsequently, Lusztig [225] found a new proof which uses a geometric interpretation of {Cw } and the results in [222], [223]. This interpretation ∗ are non-negative implies, for example, that all coefficients of the polynomials Py,w
2.3 On Lusztig’s Conjectures, I
79
integers. In the general multiparameter case, such a geometric interpretation is not ∗ may be strictly negative; see Example 2.1.5! known – and the coefficients of Py,w In his book [231, Chap. 14], Lusztig has extended the known situation in the equal-parameter case and stated 15 conjectural properties P1–P15 of the basis {Cw } which should hold for any Coxeter group (finite or infinite) and in the general multiparameter case. In [231, Chap. 20], Lusztig shows that (♣) and (♣ ) are formal consequences of P1–P15. Thus, P1–P15 appear to provide the appropriate framework for establishing substantial results concerning the representation theory of H. (See 2.4.1 for a summary of the cases where P1–P15 are known to hold.) 2.3.1. For the convenience of the reader, we state here Lusztig’s conjectures P1– P15 in [231, Chap. 14] in the general framework involving a totally ordered abelian group Γ , and taking into account the possibility that L(s) = 0 for some s ∈ S. Also note that these properties are formulated in [231] with respect to the basis {Cw }, but, using the formulae in Remark 2.1.7, it is a straightforward matter to switch back and forth between Cw and Cw . The following definitions originally appeared in [222], in the equal-parameter case. For a fixed z ∈ W , we set a(z) := min{g ∈ Γ0 | ε g hx,y,z ∈ Z[Γ0 ] for all x, y ∈ W }. Given x, y, z ∈ W , we define cx,y,z−1 ∈ Z by cx,y,z−1 := constant term of ε a(z) hx,y,z ∈ Z[Γ0 ]. ∗ = 0, we define an element Δ (z) ∈ Γ0 and an Furthermore, if z ∈ W is such that P1,z integer 0 = nz ∈ Z by the condition ∗ ε Δ (z) P1,z ≡ nz
mod Z[Γ<0 ];
see [231, 14.1].
∗ (Note that we can only have P1,z = 0 if L(s) = 0 for some s ∈ S; see Example 2.1.11; this is the only place where we explicitly have to mention if L(s) equals zero or not.) ∗ = 0 and a(z) = Δ (z)}. Finally, we set D := {z ∈ W | P1,z
Conjecture 2.3.2 (Lusztig [231, 14.2]). The following properties hold. P1. P2. P3. P4. P5. P6. P7. P8. P9. P10. P11.
∗ = 0, we have a(z) Δ (z). For any z ∈ W such that P1,z If d ∈ D and x, y ∈ W satisfy cx,y,d = 0, then x = y−1 . If y ∈ W , there exists a unique d ∈ D such that cy−1 ,y,d = 0. If x, y ∈ W are such that x LR y, then a(y) a(x). In particular, if x ∼L R y, then a(x) = a(y). If d ∈ D, y ∈ W , cy−1 ,y,d = 0, then cy−1 ,y,d nd = (−1)l(d) . If d ∈ D, then d 2 = 1. For any x, y, z ∈ W , we have cx,y,z = cy,z,x . Let x, y, z ∈ W be such that cx,y,z = 0. Then x ∼L y−1 , y ∼L z−1 , z ∼L x−1 . If x L y and a(x) = a(y), then x ∼L y. If x R y and a(x) = a(y), then x ∼R y. If x LR y and a(x) = a(y), then x ∼LR y.
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2 Kazhdan–Lusztig Cells and Cellular Bases
P12. Let I ⊆ S and WI be the parabolic subgroup generated by I. If y ∈ WI , then a(y) computed in terms of WI is equal to a(y) computed in terms of W . P13. Any left cell C of W contains a unique element d ∈ D. We have cx−1 ,x,d = 0 for all x ∈ C. P14. For any z ∈ W , we have z ∼L R z−1 . P15. If w, w , x, y ∈ W are such that a(x) = a(y), then
∑ hx,w ,z ⊗Z hw,z,y = ∑ hz,w ,y ⊗Z hw,x,z
z∈W
in Z[Γ ] ⊗Z Z[Γ ].
z∈W
We just remark (a) that there are some logical dependences between these properties (for example, “P1 + P3 ⇒ P5” by [231, 14.5]) and (b) that some of these properties seem to be more crucial than others (for example, P4 will appear almost everywhere while P6 will not be needed in the whole discussion below). Remark 2.3.3. In 2.1.14, we have seen that Cw = Cw−1 for all w ∈ W , where h → h is the anti-automorphism of H defined by Tw = Tw−1 . This immediately implies that hx,y,z = hy−1 ,x−1 ,z−1 ,
a(z) = a(z−1 )
and
cx,y,z = cy−1 ,x−1 ,z−1
for all x, y, z ∈ W . Furthermore, we have nz = nz−1 , Δ (z) = Δ (z−1 ), D = D −1 . Remark 2.3.4. P14 holds for finite W by Lemma 1.6.6 and Proposition 2.1.20. (See [220, 5.2] for the equal-parameter case.) The reason why it appears in the above list is that Conjecture 2.3.2 is stated in [231] for arbitrary (possibly infinite) Coxeter groups satisfying a certain boundedness condition. Remark 2.3.5. Assume that we are in the equal-parameter case where Γ = Z and L(s) = 1 for all s ∈ S. Now A is the ring of Laurent polynomials in one indeterminate v = ε . One easily checks that there is a well-defined ring homomorphism α : H → H such that α (v) = −v, α (r) = r for all r ∈ R and α (Tw ) = (−1)l(w) Tw for all w ∈ W . Hence, by the characterisation in Theorem 2.1.6, we must have α (Cw ) = (−1)l(w)Cw . Applying α to the relation CxCy = ∑z∈W hx,y,zCz , we deduce that (a)
hx,y,z (−v) = (−1)l(x)+l(y)+l(z) hx,y,z (v)
for all x, y, z ∈ W .
This also implies that (b)
(−1)l(x)+l(y)+l(z) cx,y,z = (−1)a(z) cx,y,z
for all x, y, z ∈ W .
(These observations are due to Lusztig [222, 3.2].) Remark 2.3.6. Let J be the free Z-module with basis {tw | w ∈ W }. We define an element of J by 1J := ∑d∈D nd td . We define a bilinear product on J by t x ty =
∑ γx,y,z tz−1 ,
z∈W
where
γx,y,z := (−1)l(x)+l(y)+l(z) cx,y,z−1 .
2.3 On Lusztig’s Conjectures, I
81
(Note that this agrees with the notation in [231, 13.6].) Using P1, P4, one deduces that J is an associative ring, where 1J is the identity. Since we will not need this construction here, we refer to Lusztig [231, Chap. 18] for further details. For the ˜ see Proposition 2.3.16 below. identification with our algebra J, Remark 2.3.7. We note that P15 really is a statement about a certain bimodule structure (which appeared in [216], [222, 9.2]). To see this, consider the ring A = R[Γ ] ⊗R R[Γ ] and let E be a free A -module with basis {ez | z ∈ W }. Let H1 = A ⊗A H, where A is embedded into A via a → 1 ⊗ a, H2 = A ⊗A H, where A is embedded into A via a → a ⊗ 1. By P4 and the definition of the Kazhdan–Lusztig pre-order relation LR , there is a left action of H1 on E via
∑
Cw .ex =
(1 ⊗ hw,x,z ) ez
for x, w ∈ W .
z∈W : a(z)=a(x)
Similarly, there is a right action of H2 on E via
∑
ex .Cw =
(hx,w,z ⊗ 1) ez
for x, w ∈ W .
z∈W : a(z)=a(x)
Now let x, w, w ∈ W . Then Cw .(ex .Cw ) =
∑
(hx,w ,z ⊗ 1)Cw .ez =
z∈W a(z)=a(x)
∑
(hx,w ,z ⊗ hw,z,y ) ey .
y,z∈W a(y)=a(z)=a(x)
Here, we recognise the terms appearing on the left-hand side of P15. Similarly, when we expand (Cw .ex ).Cw , we will recognise the terms appearing on the right-hand side of P15. Thus, we conclude that P15 holds if and only if E is an (H1 , H2 )-bimodule. As already mentioned in the introduction to this section, Lusztig [231, Chap. 20] has shown that (♣ ) formally follows from P1–P15. We will now give a somewhat streamlined exposition of this deduction which, eventually, only requires P1, P4. (The stronger property (♣) will be considered in the next section.) For this purpose, we need to relate the functions a(z) and aλ . The following result (which first appeared in [114]) seems to be the only known connection between these two functions which can be proved without assuming any of the properties P1–P15. Lemma 2.3.8. Let λ ∈ Λ and w ∈ W be such that E λ L w. Then a(w) aλ . Proof. By assumption, there exist some s, t ∈ M(λ ) such that cst w,λ = 0. Furthermore, λ (D ) ≡ cst mod m. Now we claim that by 2.1.19, we have ε aλ ρst w w,λ
(a)
λ ρst (Dw ) =
∑
x,y∈W
λ λ c−1 λ ρst (Dx−1 ) ρss (Dy−1 ) hx,y,w−1 .
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2 Kazhdan–Lusztig Cells and Cellular Bases
This is seen as follows. Let x, y ∈ W . Then hx,y,w−1 = τ (CxCy Dw ); see 2.1.13. Furμ −1 μ thermore, τ = ∑μ ∈Λ c−1 μ χ and so hx,y,w−1 = ∑μ ∈Λ cμ χ (CxCy Dw ). This yields hx,y,w−1 =
∑
μ ∈Λ
μ
∑
u,u ,v∈M(μ )
μ
μ
c−1 μ ρuu (Cx ) ρu v (Cy ) ρvu (Dw ).
λ (D λ Now multiply on both sides by ρst x−1 ) ρss (Dy−1 ) and sum over all x, y ∈ W . Using the Schur relations in Proposition 1.2.12, a straightforward computation yields (a). −1 2aλ /(1 + gλ ), where gλ ∈ F[Γ>0 ]. Hence, we obtain Now note that c−1 λ = fλ ε λ ε a(w) ρst (Dw ) =
∑
x,y∈W
fλ−1 a λ λ ε λ ρst (Dx−1 ) ε aλ ρss (Dy−1 ) ε a(w) hx,y,w−1 . 1 + gλ
All terms in the above sum lie in O0 ; see 2.1.19 and also note that a(w) = a(w−1 ) λ (D ) ∈ O . by Remark 2.3.3. Hence the whole sum will lie in O0 and so ε a(w) ρst w 0 λ (D ) ≡ 0 mod m, we conclude that a(w) a , as claimed. Since ε aλ ρst w λ Lemma 2.3.9. Let C be a left Kazhdan–Lusztig cell and λ ∈ Λ be such that m(C, λ ) > 0. (a) If y ∈ W is such that ρ λ (Cy ) = 0, then y R y for some y ∈ C. (b) If z ∈ W is such that ρ λ (Dz−1 ) = 0, then z R z for some z ∈ C. Proof. Since m(C, λ ) > 0, we have that Eελ is an irreducible constituent of [C]K ; see 2.2.2. Now assume that ρ λ (Cy ) = 0; that is, Cy does not act as zero on Eελ . Then Cy cannot act as zero on [C]K either. By the definition of this action, there exist some x, y ∈ C such that hy,x,y = 0. In particular, y R y. Thus, (a) is proved. Now assume that ρ λ (Dz−1 ) = 0. Then Dz−1 cannot act as zero on [C]K . So, by the definition of this action, there exists some z ∈ C such that Dz−1 Cz = 0. Since τ is non-degenerate, there exists some x ∈ W such that τ (Cx Dz−1 Cz ) = 0. This yields hz ,x,z = τ (Cz Cx Dz−1 ) = τ (Cx Dz−1 Cz ) = 0 and so z R z , as required. Lemma 2.3.10. Assume that P4 holds. Let x, y, z ∈ W . Then cx,y,z = ∑
∑
λ s,t,u∈M(λ )
tu us fλ−1 cst x,λ cy,λ cz,λ ,
where the first sum runs over all λ ∈ Λ such that aλ = a(z). λ Proof. We have hx,y,z−1 = τ (CxCy Dz ) and τ = ∑λ ∈Λ c−1 λ χ . This yields
hx,y,z−1 = =
∑ c−1 λ trace
λ ∈Λ
∑
∑
λ ρ (Cx ) ρ λ (Cy ) ρ λ (Dz )
λ ∈Λ s,t,u∈M(λ )
λ λ λ c−1 λ ρst (Cx ) ρtu (Cy ) ρus (Dz ).
−1 2aλ Now note that c−1 /(1 + gλ ), where gλ ∈ F[Γ>0 ]. Hence, we obtain λ = fλ ε
2.3 On Lusztig’s Conjectures, I
ε a(z) hx,y,z−1 =
83
fλ−1 a λ λ (Dz ) . ∑ ∑ 1 + gλ ε λ ρst (Cx ) ε aλ ρtuλ (Cy ) ε a(z)ρus λ ∈Λ s,t,u∈M(λ )
λ (C ), ε aλ ρ λ (C ) and ε aλ ρ λ (D ) lie in O . Let λ ∈ Λ be By 2.1.19, the terms ε aλ ρst x z 0 us tu y λ such that ρus (Dz ) = 0. Let C be a left Kazhdan–Lusztig cell such that m(C, λ ) > 0. Then, by Lemma 2.3.9(b), there exists some w ∈ C such that z−1 R w. By P4, we must have a(w) a(z−1 ) = a(z). Furthermore, by Lemma 2.2.4, there exists some w ∈ C ∩ Fλ . Hence, by P4 and Lemma 2.3.8, we have aλ a(w ) = a(w) a(z). λ (D ) ∈ m for all s, u ∈ M(λ ) and so these terms do But if aλ < a(z), then ε a(z) ρus z a(z) not contribute to ε hx,y,z−1 mod Z[Γ>0 ]. We conclude that
ε a(z) hx,y,z−1 ≡
∑
λ ∈Λ aλ =a(z)
∑
s,t,u∈M(λ )
tu us f λ−1 cst x,λ cy,λ cz,λ
mod m.
This yields the desired formula for cx,y,z . Corollary 2.3.11. Assume that P4 holds. Then {a(z) | z ∈ W } ⊆ {aλ | λ ∈ Λ }.
Proof. Given z ∈ W , let x, y ∈ W be such that cx,y,z = 0. Then Lemma 2.3.10 shows that there exists some λ ∈ Λ such that aλ = a(z). Lemma 2.3.12 (Lusztig [222, 6.1]). Assume that P4 holds. Then P7 also holds. Furthermore, if cx,y,z = 0, then a(x) = a(y) = a(z). Proof. We first show that, for any x , y , z ∈ W , we have (∗)
cx ,y ,z = constant term of ε a(z) τ (Tx Ty Dz ) ∈ Z[Γ0 ].
Indeed, as already noted in Definition 2.1.4, we have Tw = Cw +
∑
w ∈W : w <w
αw,w Cw ,
αw,w ∈ Z[Γ>0 ].
where
Since a(z) = a(z−1 ) and cx ,y ,z ≡ ε a(z) τ (Cx Cy Dz ) mod Z[Γ>0 ], this shows that cx ,y ,z ≡ ε a(z) τ (Tx Ty Dz ) +
∑ ∑
x <x y
αx ,x αy ,y ε a(z) τ (Cx Cy Dz ) mod Z[Γ>0 ].
Since ε a(z) τ (Cx Cy Dz ) ∈ Z[Γ0 ], we see that (∗) holds. Now we argue as follows. Let x, y, z ∈ W and set c := cx,y,z . Assume first that c = 0. Hence, by (∗), we have that ε a(z) τ (Tx Ty Dz ) ∈ Z[Γ0 ] has constant term c. Now, writing Dx in terms of the T -basis (see 2.1.13) and using (∗), we see that ε a(z) τ (Dx Ty Dz ) ∈ Z[Γ0 ] has constant term c and, hence, ε a(z) τ (Ty Dz Dx ) ∈ Z[Γ0 ] has constant term c. Using once more the expression of Dz in 2.1.13, we obtain
ε a(z) τ (Ty Tz Dx ) = ε a(z) τ (Ty Dz Dx ) −
∑
w∈W : z<w
∗
Pww0 ,zw0 ε a(z) τ (Ty Tw Dx ).
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2 Kazhdan–Lusztig Cells and Cellular Bases
Now, since c = cx,y,z = 0, we have hx,y,z−1 = 0 and, hence, z−1 R x. So, by P4, we have a(x) a(z−1 ) = a(z). Combining this with (∗), we deduce that ε a(z) τ (Ty Tw Dx ) ∈ Z[Γ0 ] for all y, w ∈ W . Consequently,
ε a(z) τ (Ty Tz Dx ) ≡ ε a(z) τ (Ty Dz Dx ) ≡ c
mod Z[Γ0 ].
Using (∗), this shows that a(x) a(z). Since we also have a(z) a(x), we conclude that a(x) = a(z) and, hence, cy,z,x = c. Since c = 0, we can repeat the whole argument with cy,z,x and find that cz,x,y = c; furthermore, a(z) = a(y). Thus, if one of the numbers cx,y,z , cy,z,x , cz,x,y is non-zero, then these three numbers are equal to each other and we have a(x) = a(y) = a(z). If all three numbers are zero, they are again equal. Lemma 2.3.13 (Lusztig [231, 14.5]). Assume that P1 holds. Then
∑ (−1)l(d) nd cx−1 ,y,d = δxy
for any x, y ∈ W .
d∈D
∗
Proof. Since Cy−1 Cx = ∑z∈W hy−1 ,x,zCz and τ (Cz ) = (−1)l(z) P1,z , we have
τ (Cy−1 Cx ) =
∗
∑ (−1)l(z) hy−1 ,x,z P1,z
z∈W
=
∑∗
z∈W : P1,z =0
ε a(z) hy−1 ,x,z
∗ (−1)l(z) ε −a(z) P1,z .
By 2.1.13(a), the left-hand side is congruent to δxy modulo Z[Γ>0 ]. Now consider ∗ the right-hand side. By the definition of Δ (z), we have ε −Δ (z) P1,z ≡ nz mod Z[Γ>0 ]. Since P1 is assumed to hold, we have a(z) Δ (z). This yields that if z ∈ D, nz mod Z[Γ>0 ] −a(z) ∗ ε P1,z ≡ 0 mod Z[Γ>0 ] otherwise. Hence, we obtain δxy ≡ τ (Cy−1 Cx ) ≡ ∑d∈D (−1)l(d) cy−1 ,x,d −1 nd mod Z[Γ>0 ]. Finally, by Remark 2.3.3, we have cy−1 ,x,d −1 = cx−1 ,y,d , which yields the desired formula. Proposition 2.3.14. Assume that P1, P4 hold. If λ ∈ Λ and w ∈ W are such that E λ L w, then a(w) = aλ . In particular, (♣ ) holds. Proof. By Lemma 2.3.8, we already know that a(w) aλ . So it will now be sufficient to prove that a(w) aλ . For this purpose, we consider the identity
∑ (−1)l(d) ndCwCd = ∑
d∈D
(−1)l(d) nd hw,d,y Cy .
d∈D,y∈W
Applying ρ λ and multiplying by ε aλ +a(w) , we obtain
∑ (−1)l(d) nd ε aλ +a(w)ρstλ (CwCd ) = ∑
d∈D
d∈D,y∈W
λ (−1)l(d) nd ε a(w) hw,d,y ε aλ ρst (Cy ) .
2.3 On Lusztig’s Conjectures, I
85
Assume that the terms corresponding to d ∈ D, y ∈ W give a non-zero contribution λ (C ) = 0. Let C be a to the sum on the right-hand side; that is, hw,d,y = 0 and ρst y left Kazhdan–Lusztig cell such that m(C, λ ) > 0. By Lemma 2.3.9(a), there exists some z ∈ C such that z R y. On the other hand, since hw,d,y = 0, we have y R w. Furthermore, by Lemma 2.2.4, there exists some w ∈ C ∩ Fλ . Thus, we obtain w, w ∈ Fλ ,
w , z ∈ C,
z R y R w.
˜ Since every two-sided Kazhdan–Lusztig cell is a union of two-sided J-cells (see Proposition 2.1.20) and also a union of left Kazhdan–Lusztig cells, we conclude that w, w , y, z all lie in the same two-sided Kazhdan–Lusztig cell. In particular, since P4 holds, we have a(y) = a(w). Hence, using 2.1.19, the right-hand side of the above identity can be rewritten as ∑ (−1)l(d) nd ε a(y) hw,d,y ε aλ ρstλ (Cy ) d∈D,y∈W
≡
∑
(−1)l(d) nd γw,d,y−1 cst y,λ
mod m.
d∈D,y∈W
By Lemma 2.3.12, we have γw,d,y−1 = γy−1 ,w,d . Hence, Lemma 2.3.13 yields that
∑
st (−1)l(d) nd γw,d,y−1 cst y,λ ≡ cw,λ
mod m.
d∈D,y∈W
Since cst w,λ = 0, we can go back to the left-hand side of the original identity above
λ (C C ) ≡ 0 mod m. Thus, we have and conclude that ∑d∈D (−1)l(d) nd ε aλ +a(w) ρst w d
∑ ∑
d∈D u∈M(λ )
λ λ (−1)l(d) nd ε a(w) ρsu (Cw ) ε aλ ρut (Cd ) ≡ 0 mod m.
λ (C ) ∈ m. So there must be some d ∈ D and some u ∈ M(λ ) such that ε a(w) ρut d Consequently, by Proposition 1.4.10(c) and 2.1.19, we have aλ a(w). Thus, we have shown that a(w) = aλ if E λ L w. Now assume that λ , μ ∈ Λ are such that E λ ∼L E μ . By definition, this means that w ∼LR w , where w, w ∈ W are such that E λ L w and E μ L w . Using P4, we obtain aλ = a(w) = a(w ) = aμ ; that is, (♣ ) holds.
Remark 2.3.15. In the above discussion, we have not found it necessary to use any of the properties P2, P3, P5, P6, P13 in Lusztig’s list. All of these express properties of the elements in D. It seems that these are logically independent of P1, P4, P15. Finally, we show that our algebra J˜ constructed in Section 1.5 really is an incarnation of Lusztig’s asymptotic ring J (see Remark 2.3.6). Proposition 2.3.16 (Cf. [114, §3]). Assume that P1, P4 hold. Then (−1)l(w) nw if w ∈ D, γ˜x,y,z = cx,y,z and n˜w = 0 otherwise,
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2 Kazhdan–Lusztig Cells and Cellular Bases
for all x, y, z, w ∈ W . In particular, Conjecture 1.5.12(a) holds; that is, γ˜x,y,z and n˜ w are integers. Furthermore, we have D˜ = D. ∼ Thus, the map tw → (−1)l(w)tw defines an algebra isomorphism J˜ → K ⊗Z J, where J is Lusztig’s asymptotic ring; see Remark 2.3.6.
Proof. Let x, y, z ∈ W . By P4 and Proposition 2.1.20, we have γ˜x,y,z = 0 unless a(x) = a(y) = a(z). The analogous statement also holds for cx,y,z by Lemma 2.3.12. Thus, in order to show that γ˜x,y,z = cx,y,z , we can assume without loss of generality that a(x) = a(y) = a(z). But, in this case, we have cx,y,z = ∑
∑
λ s,t,u∈M(λ )
tu us fλ−1 cst x,λ cy,λ cz,λ
by Lemma 2.3.10, where the first sum runs over all λ ∈ Λ such that aλ = a(x) = a(y) = a(z). On the other hand, we have
γ˜x,y,z =
∑
∑
λ ∈Λ s,t,u∈M(λ )
tu us fλ−1 cst x,λ cy,λ cz,λ .
But, by Proposition 2.3.14, the leading matrix coefficients appearing in the above expression are zero unless aλ = a(x) = a(y) = a(z). Thus, the desired identity between γ˜x,y,z and cx,y,z is proved. The identity in Lemma 2.3.13 and the fact that P7 holds (see Lemma 2.3.12) now ˜ see the analogous arimply that ∑d∈D (−1)l(d) nd td ∈ J˜ is an identity element in J; gument in the proof of Proposition 1.5.5. Since the identity element of J˜ is uniquely determined, we obtain the desired statement about n˜w .
2.4 On Lusztig’s Conjectures, II The conjectural properties P1–P15 are known to hold in a number of situations (including the equal-parameter case), but a general proof is still missing. There does not even seem to be a general idea of how to prove one of the crucial properties P1, P4, P15 for an arbitrary weight function L. In this section, we first give a summary about the present state of knowledge concerning the validity of P1–P15. This will be followed by a detailed discussion of the case where L(s) = 0 for some s ∈ S. 2.4.1. Here is a summary of the cases where P1–P15 are known to hold. (a) P1–P15 hold for any finite W , assuming that we are in the equal-parameter case where Γ = Z and there is some a > 0 such that L(s) = a for all s ∈ S. (Here, A is the ring of Laurent polynomials in one indeterminate v = ε .) Indeed, as already mentioned, Lusztig [231, Chap. 15] deduces P1–P15 from the following “positivity” properties: ∗ Py,w ∈ Z0 [v, v−1 ] and
h x,y,z ∈ Z0 [v, v−1 ] for all x, y, z, w ∈ W .
2.4 On Lusztig’s Conjectures, II
87
(Recall that h x,y,z = (−1)l(x)+l(y)+l(z) hx,y,z ; see also Remark 2.3.5.) If W is a Weyl group, then these “positivity” properties follow from a geometric interpretation; see Kazhdan and Lusztig [196], Lusztig and Vogan [233] and Springer [283]. If W is of type I2 (m) (any m 2), H3 or H4 , they follow by explicit computations; see Alvis [2] and DuCloux [76]. (b) P1–P15 have been checked by explicit computations for W of type I2 (m) (any m 3) and any weight function such that L(s) > 0 for s ∈ S; see [76], [114, §5]. (c) P1–P15 have been checked by explicit computations (with the help of a computer and C HEVIE [118]) for W of type F4 and any weight function such that L(s) > 0 for s ∈ S; see [105], [114, §5]. (d) P1–P15 hold for W of type Bn and any weight function L : W → Γ given by Bn
b t
4
a t
a t
p p p
a t
where b > (n − 1)a > 0.
See the series of papers by Bonnaf´e, Geck, Iancu [21], [26], [108], [114], [121]. (e) P1–P15 hold if (W, S) is irreducible, Γ = Z and L(s) = 0 for some s ∈ S; see 2.4.8 below. (Here, we are essentially reduced to the equal-parameter case; see also Lusztig [224], [225], where a more general setting is considered.) It is beyond the scope of this book to discuss the proofs of (a)–(d) in any more detail. An elementary proof of P1–P15 for W ∼ = Sn is given in [107]; see also Section 2.8. The geometric arguments used in (a) can be extended to the so-called quasi-split case, in which some choices of unequal parameters occur; see Table 4.1 (p. 227). (The proofs are sketched in [219] and [231, Chap. 16].) Of course, it would be highly desirable to find general proofs (at least for P1, P4, P15) which uniformly work for any W, L. The above results imply the following general statement: Corollary 2.4.2. Let W be any finite Coxeter group and L0 : W → Γ0 be the universal weight function in Example 1.1.9. Let be a monomial order on Γ0 such that we are in the “asymptotic case” as in Example 1.1.11(c). Then P1–P15 hold for W, L0 . For the remainder of this section, we address in some more detail the question of what happens when W is a finite Coxeter group and L(s) = 0 for some s ∈ S. 2.4.3. Let Ω ⊆ W be the parabolic subgroup generated by all t ∈ S such that L(t) = 0. Then we can break down the structure of W as follows. Let W1 ⊆ W be the subgroup generated by S1 := {ω sω −1 | ω ∈ Ω , s ∈ S where L(s) > 0}. Then, by Bonnaf´e and Dyer [24, Theorem 1.1], W1 is a normal subgroup of W such that W1 ∩ Ω = {1}; furthermore, we have a semidirect product decomposition (a)
W = Ω W1 and (W1 , S1 ) is a Coxeter system.
Given w ∈ W , let w = s1 · · · s p (si ∈ S) be a reduced expression. We denote by lΩ (w) the number of i ∈ {1, . . . , p} such that L(si ) = 0, and by l1 (w) the number of i ∈ {1, . . . , p} such that L(si ) > 0. (Note that these two numbers do not depend on the choice of the reduced expression.) By [24, Cor. 1.3], we have (b)
l(w) = lΩ (w) + l1 (w) and l1 |W1 is the length function for (W1 , S1 ).
88
(c)
2 Kazhdan–Lusztig Cells and Cellular Bases
l1 (ω wω −1 ) = l1 (w) for all w ∈ W1 and ω ∈ Ω .
Now let s˜ ∈ S1 and write s˜ = ω sω −1 , where ω ∈ Ω and s ∈ S is such that L(s) > 0. Then one readily checks that L(s) ˜ = L(s); moreover, we have (d)
The restriction L|W1 : W1 → Γ is a weight function, which we denote by L1 .
Indeed, let w ∈ W1 and let w = s˜1 · · · s˜p (s˜i ∈ S1 ) be a reduced expression for w with respect to S1 . For each i, let si ∈ S and ωi ∈ Ω be such that s˜i = ωi si ωi−1 and L(si ) > 0. Writing each ωi as a product of generators t ∈ S such that L(t) = 0, we obtain an expression for w in terms of S which is not necessarily reduced. But we can extract a reduced expression from it, and this reduced expression will contain the factors s1 , . . . , s p and various generators t ∈ S such that L(t) = 0. (By (b), all the factors s1 , . . . , s p must occur since l1 (w) = p.) Thus, we have L(w) = L(s1 ) + · · · + L(s p ). The argument also shows that L(s˜i ) = L(si ) for all i and so L(w) = L(s˜1 ) + · · · + L(s˜p ). Hence, L|W1 : W1 → Γ is a weight function, as required. Example 2.4.4. Assume that (W, S) is irreducible and that {1} = Ω = W . According to the classification in Table 1.1, we are in one of the following cases. (a) Let (W, S) be of type Bn , where the generators are labelled as in the diagram below. Let L(s0 ) = 0 and L(s1 ) = · · · = L(sn−1 ) > 0. Then Ω = {1, s0 } and S1 = {s0 s1 s0 , s1 , s2 , . . . , sn−1 }. The Coxeter system (W1 , S1 ) is of type Dn : s1 t s s0 4 s1 s2 sn−1 s3 sn−1 2 @ Dn Bn t t t p p p t p p p t @t t t s0 s1 s0 (b) Let (W, S) be of type Bn (with generators labelled as above), but now let L(s0 ) > 0 and L(s1 ) = · · · = L(sn−1 ) = 0. Then Ω = s1 , s2 , . . . , sn−1 ∼ = Sn and S1 = {t1 ,t2 , . . . ,tn }, where t1 = s0 and ti = si−1ti−1 si−1 for 2 i n. One easily checks that all the ti commute with each other and so (W1 , S1 ) is of type A1 × · · · × A1 (n factors). (c) Let (W, S) be of type F4 , where the generators are labelled as in Table 1.1. Let L(s1 ) = L(s2 ) = 0 and L(s3 ) = L(s4 ) > 0. Then Ω = s1 , s2 ∼ = S3 and S1 = {s3 , s4 , s2 s3 s2 , s1 s2 s3 s2 s1 }. One easily checks that (W1 , S1 ) is of type D4 , where s3 , s2 s3 s2 and s1 s2 s3 s2 s1 commute with each other, (d) Let (W, S) be of type I2 (m), where m 4 is even. Write S = {s1 , s2 } and let L(s1 ) > 0 and L(s2 ) = 0. Then Ω = {1, s2 and S1 = {s1 , s2 s1 s2 }. One easily checks that (W1 , S1 ) is of type I2 (m/2). Note that, in all of the above cases, L1 is a multiple of the length function of W1 . On the level of H, we have the following result. Proposition 2.4.5 (Cf. Bonnaf´e [22, §2.E]). Assume we are in the setting of 2.4.3. (a) For all ω , ω ∈ Ω and w ∈ W , we have Tω Tw = Tω w ,
Tw Tω = Twω ,
Tω Tω = Tωω ,
Tω −1 = Tω−1 .
2.4 On Lusztig’s Conjectures, II
89
(b) For any s˜ ∈ S1 and w ∈ W1 , we have Tsw ˜ Ts˜Tw = L(s) ˜ − ε −L(s) ˜ )T + ( ε Tsw ˜ w
if l1 (sw) ˜ > l1 (w), if l1 (sw) ˜ < l1 (w).
In particular, H1 := Tw | w ∈ W1 A ⊆ H is a subalgebra, and this is the generic Iwahori–Hecke algebra associated with (W1 , S1 ) and L1 : W1 → Γ . (c) Let {Cw | w ∈ W } be the Kazhdan–Lusztig basis of H. Then Cω Cw = Cω w ,
CwCω = Cwω ,
Cω = Tω
for all ω ∈ Ω and w ∈ W . Furthermore, if w ∈ W1 , then Cw ∈ H1 and this is the Kazhdan–Lusztig basis element constructed within H1 . Proof. (a) Let t ∈ S be such that L(t) = 0. Then Tt Tw = Ttw and Tw Tt = Twt for all w ∈ W (independently of whether l(tw) > l(w) or l(tw) < l(w)). This yields that Tω Tw = Tω w and Tw Tω = Twω for all w ∈ W and ω ∈ Ω . Hence, (a) follows. (b) Let s˜ ∈ S1 and write s˜ = ω sω −1 where ω ∈ Ω and s ∈ S is such that L(s) > 0. Let w ∈ W1 and set w = ω −1 wω . Using 2.4.3(b), we obtain ˜ = l1 (ω −1 sw ˜ ω ) = l1 (sω −1 wω ) = l1 (sw ) = l(sw ) − lΩ (sw ). l1 (sw) We certainly have lΩ (sw ) = lΩ (w ). Using 2.4.3(b), this yields that ˜ − l1 (w) = l(sw ) − l(w ). l1 (sw) Now assume that l1 (sw) ˜ > l1 (w). Then the above relation implies that l(sw ) > l(w ) and so Ts Tw = Tsw . By (a), we have Ts˜ = Tω Ts Tω−1 and Tw = Tω Tw Tω−1 . This yields Ts˜Tw = Tω Ts Tw Tω−1 = Tω Tsw Tω−1 = Tω sw ω −1 = Tsw ˜ , as required. Similarly, if l1 (sw) ˜ < l1 (w), then l(sw ) < l(w ) and so Ts Tw = Tsw + L(s) −L(s) −ε )Tw . Using (a), we deduce that (ε Ts˜Tw = Tω Ts Tw Tω−1 = Tω Tsw + (ε L(s) − ε −L(s) )Tw Tω−1 L(s) = Tω sw ω −1 + (ε L(s) − ε −L(s) )Tω w ω −1 = Tsw − ε −L(s) )Tw , ˜ + (ε as required; note that L(s) ˜ = L(s). Once these relations are established, we see that Ts˜Tw ∈ H1 for all s˜ ∈ S1 and w ∈ W1 . It follows that H1 ⊆ H is a subalgebra. The relations then show that H1 ∼ = HA (W1 , S1 , L1 ), as required. (c) By the formulae in Theorem 2.1.8, Example 2.1.5 and Remark 2.3.3, we have Ct = Tt , Ct Cw = Ctw and CwCt = Cwt for w ∈ W and t ∈ S such that L(t) = 0. This immediately yields the formulae for Cω , Cω Cw and CwCω , where ω ∈ Ω . Now let w ∈ W1 and denote by C˜w the Kazhdan–Lusztig basis element constructed inside H1 . In order to show that C˜w = Cw , we verify that C˜w satisfies the two conditions in Theorem 2.1.6 (with respect to W ). We have C˜w ∈ Tw + ∑y∈W1 Z[Γ>0 ]Ty and so the first condition is satisfied. Now let w˜ 0 ∈ W1 be the longest element (with respect to
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2 Kazhdan–Lusztig Cells and Cellular Bases
S1 ). Then w˜ −1 ˜ 0 ω0 , where ω0 ∈ Ω . By Theo0 w0 ∈ Ω and so we can write w0 = w rem 2.1.6 (applied to W1 ), we have C˜w Tw˜ 0 ∈ ∑y∈W1 Z[Γ0 ]Ty . Since Ty Tω0 = Tyω0 for all y ∈ W1 , we deduce that C˜w Tw0 = C˜w Tw˜ 0 Tω0 ∈ ∑y∈W1 Z[Γ0 ]Tyω0 and so the second condition also holds. Hence, we must have C˜w = Cw , as required. Remark 2.4.6. Define Hω := Tω .H1 = H1 .Tω for ω ∈ Ω . Then Proposition 2.4.5 shows that H=
Hω
and
for all ω , ω ∈ Ω .
Hω .Hω = Hωω
ω ∈Ω
Thus, H is an extended Iwahori–Hecke algebra and the subspaces {Hω | ω ∈ Ω } form an Ω -graded Clifford system in H, in the sense of [53, Def. 11.12]. 2.4.7. Let w ∈ W and write w = ω w1 , where ω ∈ Ω and w1 ∈ W1 . By Proposition 2.4.5(c), we have Tw = Tω Tw1 , Cω = Tω and Cw = Cω Cw1 , where Cw1 is the Kazhdan–Lusztig basis element defined within H1 . Hence, we obtain
∑ (−1)l(ω w1 )+l(y) P¯y,∗ω w1 Ty = Cω w1 = Cω Cw1 = ∑
y1 ∈W1
y∈W
(−1)l(w1 )+l(y1 ) P¯y∗1 ,w1 Tω y1 ,
where Py∗1 ,w1 is defined by the element Cw 1 ∈ H1 . Thus, given any y ∈ W and writing y = ω y1 , where ω ∈ Ω , y1 ∈ W1 , we have ∗ if ω = ω , Py1 ,w1 ∗ = (a) Py,w 0 otherwise. A similar relation can be established for the structure constants hx,y,z . By definition, given x1 , x2 ∈ W1 and ω1 , ω2 ∈ Ω , we have Cω1 x1 Cω2 x2 =
∑
ω3 ∈Ω ,x3 ∈W1
hω1 x1 ,ω2 x2 ,ω3 x3 Cω3 x3 .
Note that, if ω1 = ω2 = ω3 = 1, then hx1 ,x2 ,x3 is a structure constant with respect to the Kazhdan–Lusztig basis in H1 . By the relations in Proposition 2.4.5(c), we have (b)
Cω−1 = Cω −1
and
Cω −1 Cw1 Cω = Cω −1 w1 ω
for any ω ∈ Ω and w1 ∈ W1 . Using these relations, we obtain Cω1 x1 Cω2 x2 = Cω1 Cx1 Cω2 Cx2 = Cω1 ω2 Cω −1 x =
∑
x3 ∈W1
2
hω −1 x 2
1 ω2 ,x2 ,x3
Thus, for any x3 ∈ W1 and ω3 ∈ Ω , we have
hω −1 x ω ,x ,x 1 2 2 3 2 (c) hω1 x1 ,ω2 x2 ,ω3 x3 = 0
1 ω2
Cx2
Cω1 ω2 x3 .
if ω3 = ω1 ω2 , otherwise.
2.4 On Lusztig’s Conjectures, II
91
We see that the structure constants for the Kazhdan–Lusztig basis in H are completely determined by the structure constants inside H1 . 2.4.8. Let a(z), Δ (z) (z ∈ W ) and D be defined as in 2.3.1, with respect to the weight function L : W → Γ . Define a1 (z1 ), Δ 1 (z1 ) (z1 ∈ W1 ) and D1 analogously, with respect to the weight function L1 : W1 → Γ . Then 2.4.7(c) shows that (a)
a(ω z1 ) = a1 (z1 )
for all ω ∈ Ω and z1 ∈ W1 .
∗ = 0 for all z ∈ W . Then ˜ > 0 for all s˜ ∈ S1 , we have P1,z Furthermore, since L1 (s) 1 1 1 2.4.7(a) shows that
(b)
Δ (ω z1 ) = Δ1 (z1 ) (if ω = 1)
and
D = D1 .
Assume now that Γ = Z and that L1 is a multiple of the length function of W1 . In particular, A is the ring of Laurent polynomials in one indeterminate v = ε . Then the “positivity” properties in 2.4.1(a) hold for W1 , L1 . Using Remark 2.1.7 and the formulae in 2.4.7, we conclude that these “positivity” properties also hold for W, L: ∗ Py,w ∈ Z0 [v, v−1 ] and
h x,y,z ∈ Z0 [v, v−1 ] for all x, y, z, w ∈ W .
Taking into account (a) and (b), we can now follow Lusztig’s arguments in [231, Chap. 15] to conclude that P1–P15 hold for W, L. In particular, we see that P1–P15 hold for W, L in all situations described in Example 2.4.4 (where Γ = Z). Proposition 2.4.9 (Bonnaf´e [22, §2.E]). (a) Let x1 , x2 ∈ W1 and ω1 , ω2 ∈ Ω . Then ω1 x1 L ω2 x2 (with respect to L) if and only if x1 L x2 (with respect to L1 ). Similarly, x1 ω1 R x2 ω2 (with respect to L) if and only if x1 R x2 (with respect to L1 ). (b) The left cells of W (with respect to L) are of the form Ω .C1 where C1 is a left cell of W1 (with respect to L1 ). The left cell module [Ω .C1 ]A is isomorphic to the induced module IndH H1 [C1 ]A := H ⊗H1 [C1 ]A . (c) Let x1 , x2 ∈ W1 and ω1 , ω2 ∈ Ω . Then ω1 x1 LR ω2 x2 (with respect to L) if and only if there exists some ω ∈ Ω such that x1 LR ω x2 ω −1 (with respect to L1 ). (d) The two-sided cells of W (with respect to L) are of the form Ω .F1 .Ω , where F1 is a two-sided cell of W1 (with respect to L1 ). Proof. (a) Assume first that ω1 x1 L ω2 x2 (with respect to L). It is enough to consider the case where ω1 x1 ←L ω2 x2 ; that is, there exists some w ∈ W such that hw,ω2 x2 ,ω1 x1 = 0. By 2.4.7(c), this structure constant equals hw1 ,x2 ,x1 (for some w1 ∈ W1 ). Consequently, we have x1 L x2 (with respect to L1 ). Conversely, assume that x1 ←L x2 (with respect to L1 ). Thus, hw1 ,x2 ,x1 = 0 for some w1 ∈ W1 . By 2.4.7(c), hw1 ,x2 ,x1 also is a structure constant for the Kazhdan–Lusztig basis in H and so x1 L x2 (with respect to L). Furthermore, ω1 x1 ∼L x1 and ω2 x2 ∼L x2 . (This immediately follows from the fact that Cω Cw = Cω w for all ω ∈ Ω and w ∈ W ; see Proposition 2.4.5(c).) Hence, we also have ω1 x1 L ω2 x2 (with respect to L). The statement about the relation R is proved using the fact that x L y ⇔ x−1 R y−1 .
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2 Kazhdan–Lusztig Cells and Cellular Bases
(b) The statement about the left cells is an immediate consequence of (a). Now consider the left cell module [Ω .C1 ]A . This module has a basis {eω1 x1 | ω1 ∈ Ω , x1 ∈ C1 }, where the action of Cω w1 (ω ∈ Ω , w1 ∈ W1 ) is given by Cω w1 .eω1 x1 =
∑
ω2 ∈Ω ,x2 ∈C1
hω w1 ,ω1 x1 ,ω2 x2 eω2 x2 .
Using 2.4.7(c), we obtain that Cω w1 .eω1 x1 =
∑
x2 ∈C1
hω −1 w
1 ω1 ,x1 ,x2
1
eωω1 x2 .
On the other hand, by definition, IndH H1 ([C1 ]A ) has a basis {ω1 ⊗ ex1 | ω1 ∈ Ω , x1 ∈ C1 }, where the action of Cω w1 (ω ∈ Ω , w1 ∈ W1 ) is given by Cω w1 .(ω1 ⊗ ex1 ) =
∑
x2 ∈C1
hω −1 w 1
1 ω1 ,x1 ,x2
(ωω1 ⊗ ex2 ).
Hence, [Ω .C1 ]A → IndH H1 ([C1 ]A ), eω1 x1 → ω1 ⊗ ex1 , is an H-module isomorphism (c) For any ω ∈ Ω , the map w1 → ω w1 ω −1 is a Coxeter group automorphism of (W1 , S1 ). Furthermore, by 2.4.7(b), we have Cω−1 = Cω −1 and Cω w1 ω −1 = Cω Cw1 Cω −1 for all w1 ∈ W1 . Hence, for any x1 , x2 ∈ W1 , we have (∗)
x1 L x2
⇔
ω x1 ω −1 L ω x2 ω −1
(with respect to L1 ).
Now let x1 , x2 ∈ W1 and ω1 , ω2 ∈ Ω . Assume first that ω1 x1 LR ω2 x2 (with respect to L). It is enough to consider the case where ω1 x1 L ω2 x2 or ω1 x1 R ω2 x2 (with respect to L). Note that, in the latter case, we have (ω1 x1 ω1−1 )ω1 R (ω2 x2 ω2−1 )ω2 . Hence, using (a), we conclude that x1 L x2 or ω1 x1 ω1−1 R ω2 x2 ω2−1 (with respect to L1 ). Setting ω = 1 or ω = ω1−1 ω2 according to these two cases, and using (∗), we obtain x1 LR ω x2 ω −1 (with respect to L1 ), as required. Conversely, assume that there is some ω ∈ Ω such that x1 LR ω x2 ω −1 (with respect to L1 ). But then, by (a), we have ω x2 ω −1 ∼L x2 ω −1 ∼R x2 and so x1 LR x2 (with respect to L). (d) This immediately follows from (c). Remark 2.4.10. Let F be a two-sided cell of W (with respect to L). By Proposition 2.4.9(d), we have F ∩W1 = ∅. We claim that the following implication holds. (a) If x1 , x2 ∈ F ∩ W1 are such that x1 LR x2 (with respect to L1 ), then we have x1 ∼LR x2 (with respect to L1 ). This is seen as follows. By Proposition 2.4.9(c), since x2 LR x1 (with respect to L), there exists some ω ∈ Ω such that x2 LR ω x1 ω −1 (with respect to L1 ). Hence, since x1 LR x2 (with respect to L1 ), we have x1 LR ω x1 ω −1 (with respect to L1 ). Relation (∗) in the proof of Proposition 2.4.9 shows that then we also have ω x1 ω −1 LR ω 2 x1 ω −2 (with respect to L1 ). Repeating this argument, we obtain that ω i−1 x1 ω −(i−1) LR ω i x1 ω −i (with respect to L1 ), for all i 1. But Ω has
2.4 On Lusztig’s Conjectures, II
93
finite order, and so ω i = 1 for some i 1. We conclude that ω x1 ω −1 LR x1 and, hence, x2 LR x1 (with respect to L1 ). Thus, (a) is proved. 2.4.11. Assume that K ⊆ C is a splitting field for both W1 and W . Then we write IrrK (W1 ) = {E λ1 | λ1 ∈ Λ1 }
and
IrrK (W ) = {E λ | λ ∈ Λ }.
By the argument in Example 1.2.4, we have the following compatibility between specialisation and restriction, where λ ∈ Λ and λ1 ∈ Λ1 : multiplicity of E λ1 in the restriction of E λ to W1 = multiplicity of Eελ1 in the restriction of Eελ to H1,K := K ⊗A H1 . Now, the group Ω acts on W1 and, hence, on IrrK (W1 ). Thus, there is an action of Ω on Λ1 (which we write as λ1 → ω .λ1 ) such that (a)
trace(w1 , E ω .λ1 ) = trace(ω −1 w1 ω , E λ1 )
for all λ1 ∈ Λ1 and w1 ∈ W1 .
Using this notation, Clifford’s theorem ([53, 11.1]) states the following: Let λ ∈ Λ and λ1 ∈ Λ1 be such that E λ1 is a constituent of the restriction of E λ to W1 . Then this restriction is a direct sum of simple modules of the form E ω .λ1 , for various ω ∈ Ω . Since we have an Ω -graded Clifford system as in Remark 2.4.6, there is also a version of Clifford’s theorem on the level of H (see [53, (11.16)]): (b) The restriction of Eελ ∈ Irr(HK ) is a direct sum of simple H1,K -modules of the form Eεω .λ1 , for various ω ∈ Ω . Now let C1 be a left cell of W1 (with respect to L1 ) and ω ∈ Ω . Then, by relation (∗) in the proof of Proposition 2.4.9(c), the set ω C1 ω −1 also is a left cell of W1 (with respect to L1 ). Now, using the formulae in 2.4.7, one sees that hx1 ,x2 ,x2 = hω x1 ω −1 ,ω x2 ω −1 ,ω x3 ω −1
for all x1 , x2 , x3 ∈ W1 .
Hence, the action of Cw1 (w1 ∈ W1 ) on ω C1 ω −1 is the same as the action of Cω −1 w1 ω on C1 . Combining this with (a), we conclude that (c)
m(C1 , μ1 ) = m(ω C1 ω −1 , ω .μ1 )
for all ω ∈ Ω .
With these preparations, we obtain the following corollary. Corollary 2.4.12. Let λ , μ ∈ Λ and λ1 , μ1 ∈ Λ1 be such that E λ1 appears in the restriction of E λ to W1 and E μ1 appears in the restriction of E μ to W1 . Then Fλ1 ⊆ Fλ and Fμ1 ⊆ Fμ . Furthermore, we have E λ L E μ
⇔
E λ1 L1 E ω .μ1
for some ω ∈ Ω .
Proof. Let C1 , C 1 be left cells of W1 such that m(C1 , λ1 ) > 0 and m(C 1 , μ1 ) > 0. By Proposition 2.4.9(b), C := Ω .C1 and C := Ω .C 1 are left cells of W ; furthermore,
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2 Kazhdan–Lusztig Cells and Cellular Bases
H ∼ we have [C]A ∼ = IndH H1 ([C1 ]A ) and [C ]A = IndH1 ([C1 ]A ). Hence, by Frobenius reci procity, we have m(C, λ ) > 0 and m(C , μ ) > 0. Then Corollary 2.2.5 also shows that Fλ1 ⊆ Fλ (since C1 ⊆ C) and Fμ1 ⊆ Fμ (since C 1 ⊆ C ). Now assume that E λ L E μ . By Corollary 2.2.5, this implies that w LR w (with respect to L), for some w ∈ C and some w ∈ C . Let us write w = ω1 w1 and w = ω1 w 1 where w1 ∈ C1 , w 1 ∈ C 1 and ω1 , ω1 ∈ Ω . Then, by Proposition 2.4.9(c), we have w1 LR ω w 1 ω −1 (with respect to L1 ) for some ω ∈ Ω . Using the formula in 2.4.11(c), we conclude that E λ1 L1 E ω .μ1 , as required. Conversely, assume that E λ1 L1 E ω .μ1 , where ω ∈ Ω . By 2.4.11(c), we have m(ω C 1 ω −1 , ω .μ1 ) = m(C 1 , μ1 ) > 0. So Corollary 2.2.5 implies w1 LR ω w 1 ω −1 (with respect to L1 ), for some w1 ∈ C1 and some w 1 ∈ C 1 . Hence, by Proposition 2.4.9(c), we also have w1 LR w 1 (with respect to L). Since w1 ∈ C and w 1 ∈ C 1 , we can use once more Corollary 2.2.5 and conclude that E λ L E μ .
Fig. 2.2 Two-sided cells in type F4 with parameters 0, 0, a, a; see Example 2.4.4(c)
(2,−)
c @ @
(11,−)
c @ @
c (∅, 4)
c {11 , 13 , 23 } (aλ = 0)
c (1, 3) @ @ (2,+) @c c
c {42 , 44 , 83 } (aλ = a) c {91 , 93 } (aλ = 2a)
(∅, 31)
@ c {(11, 2), (1, 21), (∅, 22)} @ @ (11,+) @c c
c {21 , 22 , 41 , 61 , 62 , 81 , 82 , 121 , 161 } (aλ = 3a) c {92 , 94 } (aλ = 6a)
(∅, 211)
@ c (1, 111) c (∅, 1111)
D4 (parameter a > 0)
c {43 , 45 , 84 } (aλ = 7a) c {12 , 14 , 24 } (aλ = 12a) F4 (parameters 0, 0, a, a)
Remark 2.4.13. The above result shows that the relation L on IrrK (W ) is completely determined by the relation L1 on IrrK (W1 ); an example is given in Figure 2.2. Note that the converse is not true, at least not in any straightforward way. For example, using the above notation, assume that λ = μ and ω ∈ Ω is such that λ1 = ω .μ1 = μ1 . Then both sides of the equivalence in Corollary 2.4.3 are trivially true, but we cannot tell whether it is true that E λ1 L1 E μ1 or not. One can show that, in general, we have:
2.4 On Lusztig’s Conjectures, II
(a)
95
E λ1 L1 E μ1
⇔
λ1 = μ1 E λ L E μ
if λ = μ , otherwise.
(See Example 2.2.15 for the case where W1 is of type Dn and W is of type Bn ; see Figure 2.2, where W1 is of type D4 and W is of type F4 . The remaining cases are much easier to deal with; we omit further details.) Proposition 2.4.14 (Cf. [132, 10.5.6], [101, 4.6]). Let λ ∈ Λ and λ1 ∈ Λ1 be such that E λ1 appears in the restriction of E λ to W1 . Then cλ dim E λ = |Ω | cλ1 dim E λ1 ,
aλ = aλ1 ,
fλ dim E λ = |Ω | fλ1 dim E λ1 .
Proof. Let dλ = dim E λ and denote by Idλ the identity matrix of size dλ . Considering a matrix representation ρ λ afforded by Eελ , we have that dλ cλ Idλ =
∑ ρ λ (Tw ) ρ λ (Tw−1 ).
w∈W
(Indeed, the (s, t)-coefficient of the expression on the right-hand side equals
∑ ∑
w∈W u∈M(λ )
λ λ ρs,u (Tw ) ρu,t (Tw−1 ).
By the Schur relations in Proposition 1.2.12, this evaluates to δst dλ cλ , as required.) Now let us write W = {w1 ω | w1 ∈ W1 , ω ∈ Ω }. By Proposition 2.4.5(a), we have Tw1 ω = Tw1 Tω and Tω−1 = Tω −1 . This yields that dλ cλ Idλ =
∑ ∑
w1 ∈W1 ω ∈Ω
= |Ω |
∑
w1 ∈W1
ρ λ (Tw1 )ρ λ (Tω ) ρ λ (Tω−1 )ρ λ (Tw−1 ) 1
ρ λ (Tw1 ) ρ λ (Tw−1 ). 1
Since E λ1 appears in the restriction of E λ from W to W1 , a specialisation argument (see Example 1.2.4) shows that Eελ1 appears in the restriction of Eελ from HK to H1,K . Thus, choosing a suitable basis of Eελ we can assume that, for each w1 ∈ W1 , the matrix ρ λ (Tw1 ) has a block diagonal shape, where one of the blocks equals ρ λ1 (Tw1 ). Let dλ1 = dim E λ1 . Considering the corresponding block in the above identity arising from the Schur relations, we obtain dλ cλ Idλ = |Ω | 1
∑
w1 ∈W1
ρ λ1 (Tw1 )ρ λ1 (Tw−1 ). 1
But then the sum on the right-hand side can be evaluated using the Schur relations for H1,K . This yields the desired identity dλ cλ = |Ω |dλ1 cλ1 . Once this is established, the identities concerning aλ and fλ are immediate consequences.
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2 Kazhdan–Lusztig Cells and Cellular Bases
2.5 On Lusztig’s Conjectures, III Our aim now is to formulate a version of the properties P1–P15 purely in terms of the invariants aλ and our algebra J˜ constructed in Section 1.5. Proposition 2.5.1 (Cf. Lusztig [231, 18.9(b)]). Assume that P1, P4, P15 hold. (a) Let w, w , x, y ∈ W and assume that a(x) = a(y). Then
∑ γ˜x,w ,z−1 hw,z,y = ∑ hw,x,z γ˜z,w ,y−1 .
z∈W
z∈W
(b) Let w, w , y ∈ W and assume that a(w ) = a(y). Then hw,w ,y =
∑
n˜ d hw,d,z γ˜z,w ,y−1 .
z∈W, d∈D˜ a(z)=a(d)
Proof. (a) Consider the identity P15; by P4, we can assume that on both sides the sum is over all z ∈ W such that a := a(z) = a(x) = a(y). Now, we can write
ε a hx,w ,z = cx,w ,z−1 + gx,w ,z ,
where gx,w ,z ∈ Z[Γ>0 ],
ε hz,w ,y = cz,w ,y−1 + gz,w ,y ,
where gz,w ,y ∈ Z[Γ>0 ].
a
Hence, multiplying both sides of P15 by ε a ⊗ 1, we obtain 1⊗
∑
cx,w ,z−1 hw,z,y +
z∈W : a(z)=a
= 1⊗
∑
gx,w ,z ⊗ hw,z,y
z∈W : a(z)=a
∑
cz,w ,y−1 hw,x,z +
z∈W : a(z)=a
∑
gz,w ,y ⊗ hw,x,z .
z∈W : a(z)=a
Finally, Z[Γ ] ⊗Z Z[Γ ] is a free Z-module with basis {ε g ⊗ ε g | g, g ∈ Γ }. Compar ing the coefficients of 1 ⊗ ε g on both sides, we obtain the identity
∑ cx,w ,z−1 hw,z,y = ∑ hw,x,z cz,w ,y−1 .
z∈W
z∈W
The desired identity in (a) now follows from Proposition 2.3.16. ˜ we multiply both sides of the identity in (a) by n˜d and then (b) Taking x = d ∈ D, ˜ sum over all d ∈ D such that a(d) = a(y). This yields
∑ ∑˜
z∈W
d∈D a(y)=a(d)
n˜ d γ˜d,w ,z−1 hw,z,y =
∑ ∑˜
z∈W
n˜ d hw,d,z γ˜z,w ,y−1 .
d∈D a(y)=a(d)
On the right-hand side, we can replace the condition “a(y) = a(d)” by the condition “a(z) = a(d)”, since γ˜z,w ,y−1 = 0 implies a(z) = a(y) by P4 and Proposition 2.1.20. On the other hand, by Lemma 1.5.3(a), we have γ˜d,w ,z−1 = γ˜w ,z−1 ,d . Hence, the lefthand side of the above identity equals
2.5 On Lusztig’s Conjectures, III
∑ ∑˜
z∈W
n˜d γ˜w ,z−1 ,d hw,z,y =
d∈D a(y)=a(d)
97
∑
z∈W
∑
γ˜w ,z−1 ,d n˜ d hw,z,y .
d∈D˜ a(y)=a(d)
Now, if γ˜w ,z−1 ,d = 0, then a(d) = a(w ) by P4 and Proposition 2.1.20. Since a(y) = a(w ), we can omit the condition “a(y) = a(d)” in the above sum. So Lemma 1.5.3(b) yields that the above sum evaluates to hw,w ,y , as required. Corollary 2.5.2. Assume that P1, P4, P15 hold. Then P9, P10, P11 also hold. Proof. To prove P9, let y, w ∈ W be such that y L w and a(y) = a(w). We must show that y ∼L w. It is enough to consider the case where y, w are related by an elementary step of the relation L ; that is, we have hx,y,w = 0 for some x ∈ W . But then Proposition 2.5.1(b) shows that there exist some z ∈ W and d ∈ D˜ such that a(z) = a(d), hx,d,z = 0 and γ˜z,y,w−1 = 0. In particular, by Lemma 1.6.5 and Proposition 2.1.20, y and w belong to the same Kazhdan–Lusztig left cell, as required. Once P9 is established, P10 and P11 easily follow as well; see [231, 14.10, 14.11]. Indeed, to obtain P10, just note that a(z) = a(z−1 ) and that y R w if and only if y−1 L w−1 . Finally, to prove P11, let y LR w be such that a(y) = a(w). By definition, there is a sequence y = y0 , y1 , . . . , ym = w such that, for each i ∈ {1, . . . , m}, we have yi−1 L yi or yi−1 R yi . By P4, we have a(w) = a(ym ) a(ym−1 ) . . . a(y1 ) a(y0 ) = a(y). Since a(y) = a(w), we have a(y) = a(y0 ) = a(y1 ) = . . . = a(ym ) = a(w). Applying P9 or P10 to yi−1 , yi , we obtain yi−1 ∼L yi or yi−1 ∼R yi . Hence, y ∼LR w. 2.5.3. Let us consider the following three statements (♣), (♠), (). These should be regarded as our adaptation of Lusztig’s properties P1–P15 in Conjecture 2.3.2 for the purposes of this book. Note that (♣), (♠), () do not refer to the function a(z) or to γx,y,z , as defined in 2.3.1; these have only played an auxiliary role. (♣) Let λ , μ ∈ Λ . If E λ L E μ , then aμ aλ . In particular, if E λ ∼L E μ , then aλ = aμ . Furthermore, if E λ L E μ and aμ = aλ , then E λ ∼L E μ . (♠) Let w, w , x, y ∈ W be such that x ∼LR y. Then
∑ γ˜x,w ,z−1 hw,z,y = ∑ hw,x,z γ˜z,w ,y−1 .
z∈W
z∈W
(Here, ∼LR refers to the two-sided Kazhdan–Lusztig relation.) ˜ () Every Kazhdan–Lusztig left cell contains a unique element of D.
Let us briefly recall how the first two statements are deduced from P1, P4, P15. To prove (♣), let x ∈ Fλ and y ∈ Fμ . By Proposition 2.3.14, we have a(x) = aλ and a(y) = aμ . So, if E λ L E μ , then x LR y and so aμ = a(y) a(x) = aλ , using P4. If E λ L E μ and aλ = aμ , then x LR y and a(x) = a(y). By Corollary 2.5.2, P11 holds and so x ∼LR y; hence, E λ ∼LR E μ , as required. Finally, if w, w , x, y ∈ W
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2 Kazhdan–Lusztig Cells and Cellular Bases
are as in (♠), then a(x) = a(y) by P4 and so the desired identity holds by Proposition 2.5.1. (One can show that, conversely, (♣) and (♠) imply P1, P4, P15; see [114, 3.8, 4.7].) Finally, if P1, P4, P13 hold, then () holds, since D˜ = D by Proposition 2.3.16. Remark 2.5.4. As in Remark 2.3.7, we note that (♠) really is a statement about a certain bimodule structure. Indeed, let R ⊆ C be an L-good subring and consider the ˜ see Section 1.5. Then J˜ R := tw | w ∈ W R ⊆ J˜ is an R-subalgebra of J˜ algebra J; ˜ and J = K ⊗R J˜ R . By the identification Cw ↔ tw , the natural left H-module structure on H (given by left multiplication) can be transported to a left H-module structure on J˜ A := A ⊗R J˜ R . Explicitly, the action is given by Cw ∗ tx =
∑ hw,x,z tz
for all x, w ∈ W .
z∈W
By the definition of the Kazhdan–Lusztig pre-order LR , we can define a left Hmodule structure on J˜ A by the formula Cw tx =
∑
hw,x,z tz
for all x, w ∈ W .
z∈W : z∼LR x
For any h ∈ H and x ∈ W , the difference h ∗ tx − h tx is an A-linear combination of terms ty , where y LR w and y ∼LR w (in the Kazhdan–Lusztig pre-order). On the other hand, we have a natural right J˜ A -module structure on J˜ A (given by right multiplication). Then these two actions commute if and only if Cw (txtw ) = (Cw tx )tw
for all x, w, w ∈ W .
Writing this out using the defining equations, the above identity is equivalent to
∑
z∈W : z∼LR y
γ˜x,w ,z−1 hw,z,y =
∑
hw,x,z γ˜z,w ,y−1
for all y ∈ W .
z∈W : z∼LR x
Now, by Proposition 2.1.20, we can assume that z ∼LR x for all z on the left-hand side and z ∼LR y for all z on the right-hand side. Thus, if x ∼LR y, then both sides of the above identity are zero. Hence, the above identity holds if and only if (♠) holds. So we conclude (a) (♠) holds if and only if J˜ A is an (H, J˜ A )-bimodule (with the above actions). Since the algebra H is generated by {Cs | s ∈ S} ∪ {T1 }, we also conclude (b) in order to verify (♠), it is sufficient to do this assuming that w = s ∈ S. The following result was proved by Lusztig [223] in the equal-parameter case and in [231, 18.9 and 18.10] in general, assuming that P1–P15 hold. Here, we follow the proof given in [112], which is much less “computational” than that in [223], [231]. Theorem 2.5.5 (Lusztig [231, 18.9]; see also [112, §5]). Assume that property (♠) in 2.5.3 holds. Then there is a unique unital A-algebra homomorphism φ : H → J˜ A
2.5 On Lusztig’s Conjectures, III
99
such that, for any h ∈ H and w ∈ W , the difference φ (h)tw − h ∗ tw is an A-linear combination of terms ty , where y LR w and y ∼LR w. Explicitly, φ is given by
φ (Cw ) =
∑
hw,d,z n˜ d tz
(w ∈ W ).
z∈W,d∈D˜ z∼LR d
Proof. In the setting of Remark 2.5.4, the left H-module structure on J˜ A gives rise to an A-algebra homomorphism
ψ : H → EndA (J˜ A ) such that ψ (h)(tw ) = h tw . Since the left action of H on J˜ A commutes with the right action of J˜ A , the image of ψ lies in EndJ˜ A (J˜ rA ), where the superscript “r” indicates that we consider the right action of J˜ A on itself. Now, we have a natural A-algebra isomorphism
η : EndJ˜ A (J˜ rA ) → J˜ A ,
f → f (1J˜ A ).
We define φ = η ◦ ψ : H → J˜ A . Then φ is an A-algebra homomorphism such that
φ (h) = ψ (h)(1J˜ A ) = h 1J˜ A
for all h ∈ H.
This yields φ (h)tw = (h 1J˜ A )tw = h 1J˜ A tw = h tw or, in other words, the difference φ (h)tw − h ∗tw is an A-linear combination of terms ty , where y LR w and y ∼LR w, as required. Furthermore, we immediately obtain the formula
φ (Cw ) = Cw 1J˜ A =
∑ n˜d Cw td =
d∈D˜
∑
hw,d,z n˜ d tz .
˜ z∼LR d z∈W, d∈D:
Since h1,d,z = δd,z , this yields φ (C1 ) = 1J˜ A ; hence, φ is unital. Finally, assume that φ : H → J˜ A is another homomorphism satisfying the required conditions. But these imply that φ (h)tw = h tw for all w ∈ W and, hence, φ (h) = φ (h)1J˜ A = h 1J˜ A for all h ∈ H. So we have φ = φ as required. Remark 2.5.6. Once Theorem 2.5.5 is established, the further theory of J˜ and H can be developed as in [231, Chap. 18–20], with essentially the same proofs. We just single out the following statement; cf. Lusztig [231, 18.11]: (a) Let θ : A → k be a specialisation, where k is a commutative ring with 1. Let φk : Hk → J˜ k be the induced map. Then ker(φk ) is a nilpotent ideal of Hk . Proof. Let F1 , . . . , FN be the two-sided Kazhdan–Lusztig cells of W , where the labelling is such that if x LR y for all x ∈ Fi and y ∈ F j , then i j. Consequently, each Hk,i := Cw | w ∈ F j , 1 j ik is a two-sided ideal of Hk . Now let h ∈ ker(φk ). Then, by Theorem 2.5.5, h ∗ tw is a k-linear combination of terms ty , where y LR w and y ∼LR w. Recalling the definition of the ∗-action, we deduce that hHk,i ⊆ Hk,i−1 for all i, where we set Hk,0 = {0}. Hence, given N elements h1 , . . . , hN ∈ ker(φk ), then we have h1 · · · hN ∈ Hk,0 = {0}. Thus, (a) is proved.
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2 Kazhdan–Lusztig Cells and Cellular Bases
Example 2.5.7 (Cf. Lusztig [231, 20.1]). The inclusion A ⊆ K induces an algebra homomorphism φ : HK → J˜ K . Since HK is semisimple, Remark 2.5.6(a) shows that φ is an isomorphism. Next, consider the specialisation θ1 : A → K such that θ1 (ε g ) = 1 for all g ∈ Γ . Let φ1 : KW → J˜ be the induced map. Since KW is semisimple, Remark 2.5.6(a) shows that φ1 is an isomorphism. Thus, KW ∼ = J˜ as K-algebras. Finally, the inclusion K ⊆ K induces an algebra isomorphism (φ1 )K : KW → J˜ K . Hence, the composition
ψ = (φ1 )−1 K ◦ φK : HK → KW
is an algebra isomorphism.
(This first appeared in [216] in the equal-parameter case.) Thus, using ψ , one obtains a more natural explanation for the correspondence IrrK (W ) ↔ Irr(HK ) in 1.2.1. But note that the results in 1.2.1 do not rely on the assumption that (♠) holds! Lemma 2.5.8. Assume that (♠) holds. Let x, y, w ∈ W be such that y ∼LR w. Then hx,w,y =
∑
n˜d hx,d,z γ˜z,w,y−1 .
z∈W, d∈D˜ z∼LR d
Proof. The left-hand side of the above identity is the coefficient of ty in the expansion of Cx ∗ tw , and the right-hand side is the coefficient of ty in the expansion of φ (Cx )tw . By Theorem 2.5.5, these two coefficients must be the same. ˜ Lemma 2.5.9. Assume that (♠) holds. Then the left J-cells are precisely the left Kazhdan–Lusztig cells. (Analogous statements hold for right and two-sided cells.) Furthermore, the following implication holds for any y, w ∈ W : y L w
and y ∼LR w
⇒
y ∼L w,
where L , ∼L and ∼LR refer to the Kazhdan–Lusztig relations. ˜ Proof. Recall that, by Proposition 2.1.20, every left (or right or two-sided) J-cell is contained in a left (or right or two-sided respectively) Kazhdan–Lusztig cell. To prove the reverse implications, we begin by showing the following two statements: (a) Let y, w ∈ W be such that y L w and y ∼LR w (with respect to the Kazhdan– ˜ Lusztig relations). Then y, w belong to the same left J-cell (and, hence, y ∼L w). (b) Let y, w ∈ W be such that y R w and y ∼LR w (with respect to the Kazhdan– ˜ Lusztig relations). Then y, w belong to the same right J-cell. To prove (a), we may assume that y, w are related by an elementary step in the Kazhdan–Lusztig pre-order relation L ; that is, we can assume that hx,w,y = 0 for some x ∈ W . But then Lemma 2.5.8 shows that there exist some z ∈ W and d ∈ D˜ such that z ∼LR d, hx,d,z = 0 and γ˜z,w,y−1 = 0. In particular, by Lemma 1.6.5, y and ˜ w belong to the same left J-cell. Thus, (a) is proved. The proof of (b) is analogous. ˜ Thus, Now (a) shows that if y ∼L w, then y, w belong to the same left J-cell. ˜ the left Kazhdan–Lusztig cells coincide with the left J-cells. Using (b), a similar statement holds for right cells. Now consider the two-sided cells. Let y, w ∈ W be
2.5 On Lusztig’s Conjectures, III
101
such that y ∼LR w. Then there is a sequence y = y0 , y1 , . . . , ym = w in W such that, for each i ∈ {1, . . . , m}, we have yi−1 L yi or yi−1 R yi . Since y ∼LR w, all elements yi belong to the same two-sided Kazhdan–Lusztig cell. Hence, by (a) and ˜ (b), all elements yi belong to the same two-sided J-cell. In particular, y, w belong to ˜ the same two-sided J-cell. Example 2.5.10. Assume that (♠) holds. Let C be a left Kazhdan–Lusztig cell of ˜ W . By Lemma 2.5.9, the set C also is a left J-cell. Then we claim that
λ ) = m(C, λ ) m(C, ˜
for all λ ∈ Λ ,
where the left-hand side is defined in Theorem 1.8.1 and the right-hand side is defined in 2.2.2. Indeed, by the argument in the proof of Lemma 2.2.4, we have
∑ ∑ cstw,λ ctsw−1 ,λ = m(C, λ ) dλ fλ .
s,t∈M(λ ) w∈C
˜ λ ) fλ , as required. We By Theorem 1.8.1(b), the left-hand side also equals dλ m(C, can now write the relations in Theorem 1.8.1 in the form if λ = μ , m(C, λ ) fλ ∑ cw,λ cw−1 ,μ = 0 otherwise. w∈C We close this section with some auxiliary results which will be useful at several places below. The proofs of some of these will only require the following weak version of (♣) which we already encountered at the beginning of Section 2.3 (p. 78): E λ ∼L E μ
(♣ )
⇒
aμ = aλ
Lemma 2.5.11. Assume that (♣ ) holds. Let T be a two-sided Kazhdan–Lusztig cell and a ∈ Γ0 be the common value of aλ , where λ ∈ Λ is such that Fλ ⊆ T. Then
ε a hx,y,z ∈ Z[Γ0 ]
and
γ˜x,y,z−1 ≡ ε a hx,y,z mod Z[Γ>0 ]
for all x ∈ W and y, z ∈ T. In particular, γ˜x,y,z−1 ∈ Z. λ Proof. We have hx,y,z = τ (CxCy Dz−1 ) and τ = ∑λ ∈Λ c−1 λ χ . Furthermore, as in the −1 −1 2aλ proof of Lemma 2.3.10, cλ = fλ ε /(1 + gλ ), where gλ ∈ F[Γ>0 ]. This yields
ε a hx,y,z =
f λ−1 a λ λ (Dz−1 ) . ∑ ∑ 1 + gλ ε λ ρst (Cx ) ε aλ ρtuλ (Cy ) ε a ρus λ ∈Λ s,t,u∈M(λ )
Now assume that λ ∈ Λ and s, t, u ∈ M(λ ) are such that all three terms λ ε aλ ρst (Cx ),
λ ε aλ ρtu (Cy ),
λ ε a ρus (Dz−1 )
in the above sum are non-zero. Let C be a left Kazhdan–Lusztig cell such that m(C, λ ) > 0. Then, by Lemma 2.3.9, there exist y , z ∈ C such that y R y and
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2 Kazhdan–Lusztig Cells and Cellular Bases
z R z . Since y, z ∈ T, we deduce that C ⊆ T. By Lemma 2.2.4, there exists some w ∈ C ∩ Fλ . Since C ⊆ T, this implies that Fλ ⊆ T and so a = aλ . Consequently, all of the above three terms lie in O0 . Hence, the whole sum lies in O0 and its constant term can be computed term by term. Thus, we obtain
ε a hx,y,z ≡
∑
∑
λ ∈Λ s,t,u∈M(λ )
tu us cst x,λ cy,λ cz−1 ,λ ≡ γ˜x,y,z−1 mod m.
Since hx,y,z ∈ Z[Γ ], we have γ˜x,y,z−1 ∈ Z and the congruences are modulo Z[Γ>0 ]. Proposition 2.5.12. Assume that we are in the equal-parameter case where Γ = Z and L(s) = 1 for all s ∈ S. Then property (♠) in 2.5.3 is a consequence of (♣ ). Proof. By Remark 2.5.4, it is enough to prove (♠) assuming that w = s ∈ S. Thus, we must show that (a)
∑ γ˜x,w ,z−1 hs,z,y = ∑ hs,x,z γ˜z,w ,y−1
z∈W
for all s ∈ S,
z∈W
where w , x, y ∈ W are such that x ∼LR y (in the Kazhdan–Lusztig pre-order). Let T denote the two-sided Kazhdan–Lusztig cell such that x, y ∈ T. First we note that, by Lemmas 1.6.5 and 1.6.6, we can assume that z ∈ T on both sides of the above identity; furthermore, we can also assume that w ∈ T. Now we argue as follows. If L(s) = 0, then hs,z,y = δzy ; see Theorem 2.1.8. Hence, the left-hand side of (a) reduces to γ˜x,w ,y−1 . Similarly, since hs,x,z = δxz , the right-hand side reduces to γ˜x,w ,y−1 . Hence, the assertion is true in this case. We can assume from now on that L(s) > 0. Case 1: sx < x. Then, by Theorem 2.1.8, we have CsCx = −(vs + v−1 s )Cx and ˜ so the right-hand side of (a) reduces to −(vs + v−1 ) γ . Now let z ∈ W and s x,w ,y−1 assume that the corresponding term on the left-hand side of (a) is non-zero; that is, γ˜x,w ,z−1 = 0 and hs,z,y = 0. By Lemma 1.6.5, this implies that z ∼R x and so sz < z; see Remark 2.1.16. Hence, the left hand side also reduces to −(vs + v−1 s )γ˜x,w ,y−1 . Thus, the identity (a) holds in this case. Case 2: sx > x. Let again z ∈ W and assume that the corresponding term on the left-hand side of (a) is non-zero; that is, we have γ˜x,w ,z−1 = 0 and hs,z,y = 0. Again, this implies that z ∼R x and so sz > z. Hence, the sum on the left-hand side only needs to be extended over all z ∈ W such that sz > z. But then, since we are in the equal-parameter case, we have hs,z,y ∈ Z; see Example 2.1.10(b). Now consider the usual associativity rule in H: the identity Cs (CxCw ) = (CsCx )Cw yields
∑ hx,w ,z hs,z,y = ∑ hz,w ,y hs,x,z
z∈W
for all y ∈ W .
z∈W
Let z ∈ W be such that the corresponding term on the left-hand side is non-zero. Then hx,w ,z = 0 and so z R x; furthermore, hs,z,y = 0 and so y L z. Since x ∼LR y, we deduce that z ∼LR x. Thus, we can assume that z ∈ T in the sum on the left-hand side. Now we use (♣ ). Let a ∈ Γ0 be the common value of aλ , where λ ∈ Λ is such that Fλ ⊆ T. By Lemma 2.5.11, we have
2.5 On Lusztig’s Conjectures, III
103
ε a hx,w ,z ≡ γ˜x,w,z−1 mod Z[Γ>0 ]. Hence, since hs,z,y ∈ Z for all z such that sz > z, we have
ε a ∑ hx,w ,z hs,z,y = ∑ ε a hx,w ,z hs,z,y ∈ Z[Γ>0 ] z∈W
z∈T
and the constant term of this expression equals the left hand side of (a). A similar argument applies to the right-hand side of the above associativity identity: the sum only needs to be extended over all z ∈ T. Furthermore, we have
ε a hz,w ,y ≡ γ˜z,w ,y−1 mod Z[Γ>0 ]. Hence, since sx > x and hs,x,z ∈ Z, we have
ε a ∑ hz,w ,y hs,x,z = ∑ ε a hz,w ,y hs,x,z ∈ Z[Γ>0 ] z∈W
z∈T
and the constant term equals the right hand side of (a). Thus, (a) is proved.
Lemma 2.5.13. Assume that (♣) holds. Then we have γ˜x,y,z ∈ Z for all x, y, z ∈ W and n˜w ∈ Z for all w ∈ W . Proof. If y, z−1 belong to the same two-sided Kazhdan–Lusztig cell, then we have γ˜x,y,z ∈ Z by Lemma 2.5.11. Otherwise, we have γ˜x,y,z = 0 by Proposition 2.1.20. ∗ It remains to consider n˜ w . Let λ0 ∈ Λ be such that E λ0 L w. We have P1,w = (−1)l(w) τ (Cw ). Expressing τ as in the proof of Lemma 2.5.11, we obtain (−1)l(w) ε
−aλ
0
∗
P1,w =
fλ−1 aλ −aλ a λ 0 ε λ ρ (Cw ) . ε ss 1 + g λ λ ∈Λ s∈M(λ )
∑ ∑
Let λ ∈ Λ be such that ρ λ (Cw ) = 0. Then we claim that aλ0 aλ . Indeed, let C be a left Kazhdan–Lusztig cell such that m(C, λ ) > 0. By Lemma 2.3.9(a), we have y R w for some y ∈ C. By Lemma 2.2.4, there also exists some y ∈ C such that E λ L y . In particular, we now have y ∼LR y LR w and so E λ L E λ0 . Since (♣) is assumed to hold, we can conclude that aλ0 aλ , as required. This shows that the above sum lies in O0 and we have (−1)l(w) ε
−aλ
0
∗
P1,w ≡
∑
λ ∈Λ aλ =aλ 0
∑
s∈M(λ )
fλ−1 css w,λ
mod m.
But then the first sum can be extended over all λ ∈ Λ : just note that if css w,λ = 0, then
E λ L w and so E λ ∼L E λ0 ; hence, aλ = aλ0 in this case. So we conclude that (−1)l(w) ε
−aλ
0
∗
P1,w ≡ n˜ w
mod m.
Since the left-hand side lies in Z[Γ ], we deduce that n˜ w ∈ Z, as required.
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2 Kazhdan–Lusztig Cells and Cellular Bases
Example 2.5.14. Assume that (♠), (♣), () are satisfied. Then Conjectures 1.5.12 and 1.6.18 hold. (Indeed, part (a) of Conjecture 1.5.12 holds by Lemma 2.5.13; using Lemma 2.5.9 and (), we see that Conjecture 1.6.18 holds; then part (b) of Conjecture 1.5.12 follows by the argument in Remark 1.6.19.) Now let C be a left Kazhdan–Lusztig cell of W . Then, by Examples 1.8.6 and 2.5.10, we have
∑
λ ∈Λ
fλ−1 m(C, λ ) = 1.
This is a quite powerful statement. (It can also be easily deduced from [231, 21.4].) For example, it directly shows that if f λ = 1 for all λ ∈ Λ , then [C]1 ∈ IrrK (W ). Remark 2.5.15. Following Lusztig [223, 2.8], we can now also give a more direct proof of the fact that φK : HK → J˜ K is an isomorphism. Indeed, let us assume that both (♠) and (♣) hold. For each w ∈ W , choose some λ ∈ Λ such that E λ L w and set aw := aλ . (This does not depend on the choice of λ , thanks to (♣).) Then
ε aw φ (Cw ) = ∑ ε aw n˜ d hw,d,z tz = ∑ ∑ ε aw n˜d hw,d,z tz . ˜ z∼LR d z∈W d∈D:
z∈W, d∈D˜ z∼LR d
Assume that z and d in the above sum are such that hw,d,z = 0. Let μ ∈ Λ be such that E μ L z. Since hw,d,z = 0 and z ∼LR d, we have z LR w and so E μ L E λ . Since (♣) holds, this implies that aw az , with equality only if w ∼LR z. Assume first that w ∼LR z. Then aw = az and Lemma 2.5.11 shows that ε aw hw,d,z lies in Z[Γ0 ] and has constant term γ˜w,d,z−1 . Thus, we have
∑
ε aw n˜d hw,d,z ≡
˜ z∼LR d d∈D:
∑
n˜d γ˜w,d,z−1 mod Z[Γ>0 ],
˜ z∼LR d d∈D:
where w ∼LR z. Now, if γ˜w,d,z−1 = 0, then w, d, z belong to the same two-sided Kazhdan–Lusztig cell; see Proposition 2.1.20. So we can omit the condition z ∼LR d in the above sum. Then Lemma 1.5.3 shows that
∑
˜ z∼LR d d∈D:
n˜d γ˜w,d,z−1 =
∑ n˜d γ˜z−1 ,w,d = δzw .
d∈D˜
Since n˜ d ∈ Z by Lemma 2.5.13, we finally obtain that
ε aw φ (Cw ) = tw + Z[Γ>0 ]-combination of terms tz , where z ∼LR w + Z[Γ ]-combination of terms tz , where z LR w, z ∼LR w. So, for a suitable ordering of the elements of W , the matrix of φ with respect to the basis {ε aw Cw | w ∈ W } of H and the basis {tw | w ∈ W } of J˜ A has a block triangular shape where the determinant of each diagonal block lies in 1 + Z[Γ>0 ]. Hence, the determinant of the whole matrix of φ lies in 1 + Z[Γ>0 ]. In particular, it is non-zero.
2.6 A Cellular Basis for H
105
2.6 A Cellular Basis for H We are now ready to define a new basis of H which will turn out to be a “cellular basis” in the sense of Graham and Lehrer [144]. We recall the basic definitions first. 2.6.1. Let k be a commutative ring (with 1) and H be an associative k-algebra (with identity) which is finitely generated and free over k. Following Graham and Lehrer [144, Def. 1.1], a cell datum for H is a quadruple (Λ , M,C, ∗) satisfying the following conditions. (C1) Λ is a partially ordered set, {M(λ ) | λ ∈ Λ } is a collection of finite sets and λ | λ ∈ Λ , s, t ∈ M(λ )} is a k-basis for H. C = {Cs,t λ )∗ = Cλ for (C2) There is a k-linear anti-involution, h → h∗ , on H such that (Cs,t t,s all λ ∈ Λ and all s, t ∈ M(λ ). (C3) Denote by the partial order on Λ . If λ ∈ Λ and s, t ∈ M(λ ), then λ hCs,t ≡
∑
s ∈M(λ )
rhλ (s , s)Csλ ,t
mod H(≺ λ )
for all h ∈ H,
where rhλ (s , s) ∈ k is independent of t and where H(≺ λ ) is the k-submodule μ of H generated by {Cs ,t | μ ≺ λ ; s , t ∈ M( μ )}. λ } is a cellular basis of H. Assume now If these conditions hold, we say that {Cs,t that this is the case. Given λ ∈ Λ , we can define a corresponding cell representation (or cell module) of H as follows. Let W (λ ) be a free k-module with basis {Cs | s ∈ M(λ )}. Then, using (C3), W (λ ) is seen to be an H-module with action given by
h.Cs =
∑
s ∈M(λ )
rhλ (s , s)Cs
for h ∈ H and s ∈ M(λ ).
This module is equipped with a canonical invariant bilinear form; see the following lemma. Lemma 2.6.2 (Graham and Lehrer [144, 2.4]). Let λ ∈ Λ . Then there is a welldefined symmetric bilinear form , λ : W (λ ) ×W (λ ) → k such that λ λ λ Cu,t Cs,v ≡ Cs ,Ct λ Cu,v
mod H(≺ λ )
for all s, t, u, v ∈ M(λ ).
Furthermore, we have h.Cs ,Ct λ = Cs , h∗ .Ct λ for all s, t ∈ M(λ ) and h ∈ H. Proof. This is a good exercise to see how the axioms are used. By (C3), we have λ λ Cs,v ≡ Cu,t
∑
s ∈M(λ )
rhλ1 (s , s)Csλ ,v mod H(≺ λ ),
λ . where h1 = Cu,t
λ Cλ = Cλ ∗ Cλ ∗ = Cλ Cλ ∗ . On the other hand, by (C2), we have Cu,t s,v v,s v,s t,u t,u λ Cλ and using (C2), we obtain that Applying (C3) to the product Cv,s t,u
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2 Kazhdan–Lusztig Cells and Cellular Bases λ λ Cu,t Cs,v ≡
∑
t ∈M(λ )
λ rhλ2 (t , t)Cu,t mod H(≺ λ ),
λ . where h2 = Cv,s
λ C λ ≡ α C λ mod H(≺ λ ), where α := r λ (u, s) = Hence, we deduce that Cu,t s,v u,v h1 rhλ2 (v, t). Note that rhλ1 (u, s) does not depend on v, and rhλ2 (v, t) does not depend on u. Consequently, α does not depend on u and not on v. Now choose u = v. Then we also see that α is not affected if we exchange the roles of s and t. Thus, we obtain a well-defined symmetric bilinear form , λ , as required. It remains to show that this form has the desired invariance property. Let h ∈ H and s, t ∈ M(λ ). Then
h.Cs ,Ct λ =
∑
s ∈M(λ )
rhλ (s , s) Cs ,Ct λ .
λ and using the defining Now let u, v ∈ M(λ ). Multiplying the above identity by Cu,v formula for , λ , we obtain λ ≡ h.Cs ,Ct λ Cu,v
∑
s ∈M(λ )
λ λ rhλ (s , s)Cu,s Ct,v
On the other, by (C2) and (C3), we have λ ∗ λ λ ≡ Cu,s h∗ ≡ h.Cs,u ∑ rhλ (s , s)Cu,s
mod H(≺ λ ).
mod H(≺ λ )
s ∈M(λ )
and so ∗ λ λ λ h.Cs ,Ct λ Cu,v h Ct,v ≡ ≡ Cu,s ≡
∑
t ∈M(λ )
∑
t ∈M(λ )
λ rhλ∗ (t , t)Cu,s Ctλ ,v
λ λ rhλ∗ (t , t) Cs ,Ct λ Cu,v ≡ Cs , h∗ .Ct λ Cu,v
mod H(≺ λ ),
as required.
Corollary 2.6.3. Let λ ∈ Λ and s, s , t, u ∈ M(λ ). Then rh (s , s) = δus Cs ,Ct λ ,
where
λ h = Cu,t .
Proof. This is clear by Lemma 2.6.2 and the definiton of rh (s , s) in (C3).
Definition 2.6.4. Let Lλ := W (λ )/ rad( , λ ) for any λ ∈ Λ . Then Lλ is a left Hmodule since, by Lemma 2.6.2, the radical of , λ is an H-submodule of W (λ ). Note that we may have Lλ = {0}; this happens if and only if , λ is identically zero. Now we have the following two fundamental results of Graham and Lehrer [144] whose proof we will not give here. (See also Mathas [245, Chap. 2].) Theorem 2.6.5 (Graham and Lehrer [144, 3.4, 3.8]). Assume that k is a field. If , λ = 0, then Lλ is an absolutely irreducible H-module; furthermore,
2.6 A Cellular Basis for H
107
Irr(H) = {Lμ | μ ∈ Λ ◦ },
where Λ ◦ = {λ ∈ Λ | , λ = 0}.
In particular, the algebra H is split. Finally, H is semisimple if and only if Λ = Λ ◦ and , λ is non-degenerate for all λ ∈ Λ . Recall that an algebra is called split if the endomorphism algebra of any irreducible representation consists just of the scalar multiples of the identity. Theorem 2.6.6 (Graham and Lehrer [144, 3.6]). Assume that k is a field. For λ ∈ Λ and μ ∈ Λ ◦ , denote by (W (λ ) : Lμ ) the multiplicity of Lμ as a composition factor of W (λ ). Then and (W (λ ) : Lμ ) = 0 unless λ μ . Thus, the decomposition matrix D = (W (λ ) : Lμ ) λ ∈Λ , μ ∈Λ ◦ has a lower unitriangular shape, if the rows and columns are ordered according to the order relation . (Δ )
(W (μ ) : Lμ ) = 1
Let us now also assume that H is a symmetric algebra, with trace form τ : H → k. λ | λ ∈ Λ , s, t ∈ M(λ )} as above, we have a corresponding Then, given a basis {Cs,t λ | λ ∈ Λ , s, t ∈ M(λ )}. We choose the notation such that dual basis Cˆ := {Cˆs,t λ μ Cˆu,v = τ Cs,t
1 0
if λ = μ , s = v, t = u, otherwise.
To state the following result, note that if V is a left H-module, then Homk (V, k) also is a left H-module where the action is given by h. f (v) = f (h∗ .v) for h ∈ H, f ∈ Homk (V, k) and v ∈ V . Proposition 2.6.7 (Graham [143, 4.12]). Assume that H is symmetric with trace form τ : H → k such that τ (h∗ ) = τ (h) for all h ∈ H. Then, with the above notation, the following hold. ˆ ∗) also is a cell datum for H, where Λ op is the set Λ (a) The quadruple (Λ op , M, C, endowed with the opposite partial order op (that is, λ op μ ⇔ μ λ ). (b) Let λ ∈ Λ and Wˆ (λ ) be the cell module with respect to the cell datum in (a). Then there is an isomorphism of left H-modules Wˆ (λ ) ∼ = Homk (W (λ ), k). (c) If k is a field and H is semisimple, then W (λ ) ∼ = Wˆ (λ ) for all λ ∈ Λ . ˆ ∗) the opposite cell datum to (Λ , M,C, ∗). We shall call (Λ op , M, C, Proof. Let us verify that (C1), (C2), (C3) hold for the quadruple in (a). First note that (C1) is clear and (C2) is easily seen to hold thanks to the assumption on τ . To λ . Let μ ∈ Λ and u, v ∈ M( μ ) prove (C3), let h ∈ H and consider the product hCˆs,t μ λ with be such that Cˆu,v appears with a non-zero coefficient in the expansion of hCˆs,t ˆ Note that this coefficient is given by respect to the basis C. μ μ ∗ μ λ λ λ = τ (h∗Cu,v )Cˆt,s . τ (hCˆs,t )Cv,u = τ (hCˆs,t )Cv,u
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2 Kazhdan–Lusztig Cells and Cellular Bases
Hence, by (C3) for the original cell datum, we must have λ μ ; furthermore, if λ = μ , then the above expression evaluates to δv,t rhλ∗ (s, u). Thus, we have λ hCˆs,t =
∑
u∈M(λ )
λ rhλ∗ (s, u) Cˆu,t
mod H(≺op λ ).
This shows that (C3) holds. The above formula also proves (b). More precisely, if ρ λ : H → Mdλ (k) is the matrix representation afforded by W (λ ) (with respect to its standard basis), then the matrix representation afforded by Wˆ (λ ) (with respect to its standard basis) is given by h → ρ (h∗ )tr (h ∈ H). Finally, to prove (c), assume that k is a field and H is semisimple. Let λ ∈ Λ and Gλ be the Gram matrix of the bilinear form , λ with respect to the standard basis of W (λ ). Then Gλ is invertible by Theorem 2.6.5. On the other hand, the invariance condition in Lemma 2.6.2 implies that Gλ ρ λ (h) = ρ (h∗ )tr Gλ for all h ∈ H. Hence, the two representations are equivalent; that is, W (λ ) ∼ = Wˆ (λ ). We return to the situation where we consider the generic Iwahori–Hecke algebra H = HA (W, S, L) associated with a finite Coxeter group W and a weight function L : W → Γ . Recall that H is defined over A = R[Γ ], where R ⊆ C is a subring such that ZW ⊆ R; furthermore, we assume that there is a monomial order on Γ such that L(s) 0 for all s ∈ S. Let {Cw | w ∈ W } be the associated Kazhdan–Lusztig basis of H; see Section 2.1. Write IrrK (W ) = {E λ | λ ∈ Λ },
dλ = dim E λ ,
and let M(λ ) be an indexing set for a basis of E λ , as in Section 1.2; for each E λ ∈ IrrK (W ), we have a corresponding invariant aλ ∈ Γ0 . In Section 1.5, we used the ˜ leading matrix coefficients cst w,λ to construct the ring J. Definition 2.6.8 (Cf. [111, §3]). Assume that R ⊆ C is L-good in the sense of Definition 1.5.9. Let Λ := Λ and M(λ ) := M(λ ) for all λ ∈ Λ . Let ρ¯ λ and Bλ be as in Proposition 1.5.11. Let us write λ ρ¯ λ (tw ) = cst and Bλ = βst . w,λ s,t∈M(λ ) s,t∈M(λ ) Then, for any λ ∈ Λ and s, t ∈ M(λ ), we define Cλs,t :=
∑ ∑
w∈W u∈M(λ )
λ us βtu cw−1 ,λ Cw
∈ H.
We now show in several steps that (C1), (C2), (C3) hold for these data. Remark 2.6.9. In the defining formula for Cλs,t , we can assume that the first sum runs over all w ∈ Fλ (where Fλ is defined in Proposition 1.6.11). Indeed, if Cw appears with a non-zero coefficient in that sum, then cus = 0 for some u, s ∈ M(λ ), and w−1 ,λ so w−1 ∈ Fλ . But then Lemma 1.6.6 also shows that w ∈ Fλ , as required.
2.6 A Cellular Basis for H
109
Lemma 2.6.10. The elements {Cλs,t | λ ∈ Λ , s, t ∈ M(λ )} form an A-basis of H. In ˜ fact, let y ∈ W and F be the two-sided J-cell containing y. Then Cy is an R-linear combination of elements Cλs,t , where λ ∈ Λ is such that E λ L y. Proof. By the Artin–Wedderburn theorem, |W | = ∑λ ∈Λ |M(λ )|2 . Hence, the above set has the correct cardinality. It is now sufficient to show that the elements {Cλs,t } span H as an A-module. Let us fix y ∈ W . We claim that
∑
Cy =
where (Bλ )−1 = βˆst .
∑
λ ∈Λ s,s ,t∈M(λ )
ˆλ λ fλ−1 css y,λ βs t Cs,t ,
Note that the coefficients in the above sum lie in R, since f λ and det(Bλ ) are invertible in R (since R is L-good and by Proposition 1.5.11(b)). Furthermore, we have E λ L y if Cλs,t occurs with non-zero coefficient in the above sum. Thus, it remains to prove the above identity. For this purpose, we insert the defining formula for Cλs,t into the right-hand side; this yields
∑ ∑
∑
w∈W λ ∈Λ s,s ,u∈M(λ )
=
∑ ∑
fλ−1 css y,λ
∑
w∈W λ ∈Λ s,u∈M(λ )
∑
t∈M(λ )
λ βˆsλ t βtu cus w−1 ,λ Cw
us fλ−1 csu y,λ cw−1 ,λ Cw = Cy
as desired, where the last equality holds by Proposition 1.4.10(b).
Lemma 2.6.11. We have (Cλs,t ) = Cλt,s for all λ ∈ Λ and s, t ∈ M(λ ), where is the anti-involution in 2.1.14. Proof. By 2.1.14, we have Cw = Cw−1 for all w ∈ W . Thus, we obtain (Cλs,t ) =
∑ ∑
w∈W u∈M(λ )
λ us βtu cw−1 ,λ Cw−1 =
∑
w∈W
λ λ B .ρ¯ (tw−1 ) t,s Cw−1 .
By Proposition 1.5.11, we have Bλ .ρ¯ λ (tw−1 ) = ρ¯ λ (tw )tr .Bλ . This yields (Cλs,t ) =
∑
w∈W
λ ρ¯ (tw )tr .Bλ t,s Cw−1 =
∑ ∑
w∈W u∈M(λ )
as required. (Recall that Bλ is symmetric.)
λ λ cut w,λ βus Cw−1 = Ct,s ,
We can now state the main result of this chapter.
Theorem 2.6.12 (Cf. [111, §3] [112, §5]). Assume that R is L-good and that (♠) in 2.5.3 holds. Then the elements {Cλs,t } introduced in Definition 2.6.8 form a cellular basis of H with respect to the anti-involution Tw → Tw = Tw−1 (see 2.1.14), and the partial order L on Λ defined by
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2 Kazhdan–Lusztig Cells and Cellular Bases
def
μ L λ
μ =λ
⇔
E μ L E λ , E μ ∼L E λ ,
or
where L and ∼L are as in Definition 2.2.1. If property (♣) in 2.5.3 also holds, then we have μ L λ ⇒ λ = μ or aμ > aλ .
Proof. Recall that (C1) holds by Lemma 2.6.10; (C2) holds by Lemma 2.6.11. In order to prove (C3), we need to consider a product hCλs,t where h ∈ H and λ ∈ Λ , s, t ∈ M(λ ). It is sufficient to consider the case where h = Cx for some x ∈ W . Now, by the definition of Cλs,t and Remark 2.6.9, we have hCλs,t =
∑ ry Cy
where
ry =
∑
∑
w∈Fλ u∈M(λ )
y∈W
λ us βtu cw−1 ,λ hx,w,y .
Let Tλ be the two-sided Kazhdan–Lusztig cell such that Fλ ⊆ Tλ . Note that if ry = 0, then there is some w ∈ Fλ ⊆ Tλ such that hx,w,y = 0 and so y LR w (in the Kazhdan–Lusztig pre-order). Assume first that ry = 0 and y ∈ Tλ . By Lemma 2.6.10, Cy is a linear combination μ of elements Cu,v , where E μ L y. Hence, since y ∈ Tλ , we conclude that Cy ∈ H( L λ ) and so we do not need to consider these terms in any more detail. Thus, we can now assume that y ∈ Tλ . Then, by Lemma 2.5.8, we have hx,w,y =
∑
n˜ d hx,d,z γ˜z,w,y−1
for any w ∈ Fλ ⊆ Tλ .
z∈W, d∈D˜ z∼LR d
We insert this formula for hx,w,y into the above expression for ry ; this yields ry =
∑
∑
∑
w∈Fλ z∈W, d∈D˜ u∈M(λ )
λ us βtu cw−1 ,λ n˜ d hx,d,z γ˜z,w,y−1
z∼LR d
=
∑
∑
n˜ d hx,d,z
u∈M(λ )
z∈W, d∈D˜ z∼LR d
λ βtu
∑
w∈Fλ
˜ cus γ . −1 −1 w ,λ z,w,y
Now, by 1.6.10 and Proposition 1.6.11, the sum over w ∈ Fλ can be extended to a sum over all w ∈ W . Using the defining equation for γ˜z,w,y−1 , we obtain
∑
w∈Fλ
˜ cus w−1 ,λ γz,w,y−1 = =
∑
∑
∑ cus w−1 ,λ ∑
μ ∈Λ s ,v,v ∈M(μ )
w∈W
μ ∈Λ s ,v,v ∈M(μ )
∑
s v
f μ−1 cz,μ cvy−1s ,μ
∑
w∈W
vs f μ−1 cz,s μv cvv w,μ cy−1 ,μ
vv cus c w−1 ,λ w,μ =
∑
s ∈M(λ )
where the last equality holds by Proposition 1.4.10(a). This yields
cz,s λs cus y−1 ,λ
2.6 A Cellular Basis for H
hCλs,t ≡
∑
111
∑
ry Cy ≡
s ∈M(λ )
z∈W, d∈D˜ z∼LR d
y∈Tλ
∑
n˜ d hx,d,z
cz,s λs
∑ ∑
y∈W u∈M(λ )
λ us βtu cy−1 ,λ Cy
mod H( L λ ). Since the parenthesised sum equals Cλs ,t , we see that hCλs,t ≡
∑
∑
s ∈M(λ ) z∈W, d∈D˜
n˜d hx,d,z cz,s λs Cλs ,t mod H( L λ ).
z∼LR d
Thus, we have shown that, for h = Cx (x ∈ W ), we have
∑
rhλ (s , s) =
z∈W, d∈D˜ z∼LR d
n˜d hx,d,z csz,λs
for all s, s ∈ M(λ );
in particular, this expression does not depend on t, as required.
The model for this theorem, namely the case where W is the symmetric group Sn , will be considered in detail in Section 2.8. Remark 2.6.13. Note that the ingredients for a cellular basis of H (that is, the elements {Cλs,t } and the partial order L ) are defined without reference to (♠); this property is only required for the proof. Remark 2.6.14. Assume that we are in the equal-parameter case. Then we have seen in Proposition 2.5.12 that (♠) is a consequence of the following implication: (♣ )
E λ ∼L E μ
⇒
a μ = aλ .
Thus, in order to prove Theorem 2.6.12 in the equal-parameter case, we only need to assume that (♣ ) holds. Recall that (♣ ) does hold in type I2 (m) (any m 2), H3 , H4 by Examples 2.2.8 and 2.2.9. Furthermore, (♣ ) was already established by Lusztig [220, 5.27] (around 1985) for all finite Weyl groups. Remark 2.6.15. Recall that, by Remark 2.2.11, we have the implication E λ L E μ
⇒
E μ L E λ . †
†
Now, by Examples 1.2.7 and 1.3.4, the following relation holds between aλ and aλ † : aλ † − aλ = Nλ =
N trace(s, E λ )
s L(s). ∑ dim E λ s∈S
It follows that if (♣) is satisfied, then we have the implication
λ L μ
⇒
λ =μ
or Nλ < Nμ .
Thus, Theorem 2.6.12 could be alternatively formulated using Nλ instead of the invariants aλ . Note that Nλ is much easier to define than aλ ; also, Nλ does not
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2 Kazhdan–Lusztig Cells and Cellular Bases
depend on the monomial order on Γ . The idea of using the invariants Nλ appears, in a somewhat different context in Ginzburg et al. [137, §6]; see also Gordon [140]. The following result shows that, for any Iwahori–Hecke algebra associated with a finite Coxeter group, there does exist at least some cellular structure. Corollary 2.6.16. Let k be a commutative ring (with 1) and {ξs | s ∈ S} ⊆ k× a collection of elements such that ξs = ξt whenever s,t ∈ S are conjugate in W . Let Hk = Hk (W, S, {ξs }) be the corresponding Iwahori–Hecke algebra; see 1.1.2. Assume that k is L0 -good (see Definition 1.5.9) for the “universal” weight function L0 λ | λ ∈ Λ , s, t ∈ M(λ )} with in Example 1.1.9. Then Hk admits a cellular basis {Cs,t respect to the anti-involution Tw → Tw = Tw−1 and some partial order on Λ . Proof. Let Γ0 , A0 and H0 be “universal”, as in Example 1.1.9. Choose a monomial order on Γ0 such that, on every irreducible component of type Bn , F4 or I2 (m) (m even), we are in the “asymptotic case” in Example 1.1.11. Then, by Corollary 2.4.2, we know that P1–P15 hold. Hence, as discussed in 2.5.3, the properties (♣) and (♠) also hold and so Theorem 2.6.12 applies. Thus, we obtain a cellular basis {Cλs,t } for H0 , where the partial order on Λ is given by L0 . Since k is L0 -good, there is a ring homomorphism R → k. This extends to a ring homomorphism θ : A0 → k such that θ (v◦s ) = ξs for all s ∈ S , where {v◦s } are the parameters of H0 . Thus, Hk = k ⊗A0 H0 , where k is regarded as an A0 -module via θ . Since the elements {Cλs,t | λ ∈ Λ , s, t ∈ M(λ )} in H satisfy (C1), (C2), (C3), it is λ := 1 ⊗ Cλ | λ ∈ Λ , s, t ∈ M(λ ))} satisfy (C1), (C2), clear that the elements {Cs,t s,t (C3) in Hk . Thus, we have constructed a cell datum for Hk . Example 2.6.17. Let W be of type I2 (4) = B2 , where S = {s1 , s2 } and (s1 s2 )4 = 1. Assume that we are in the equal-parameter case, where Γ = Z and L(s1 ) = L(s2 ) = 1. Then A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Now Theorem 2.6.12 applies where R ⊆ C can be any subring in which 2 is invertible. In order to determine a cellular basis, we need to work out the leading matrix coefficients of the irreducible representations of HK . The two-sided cells are given by {10 }, {14 } and W \ {10 , 14 }, where we use the notation in Example 1.7.3. First consider the representation σ1 . By Example 1.3.7, we have aσ1 = 1 and so −1 0 −1 0 v σ1ε (T11 ) = ≡ mod m, 0 0 2v v2 2 0 0 v v ≡ mod m, v σ1ε (T21 ) = 0 −1 0 −1 0 −1 −v −1 v σ1ε (T12 ) = ≡ mod m, 0 0 2v2 v 0 0 v v2 ≡ mod m, v σ1ε (T22 ) = −2 −v −2 0 −1 0 −1 −v ≡ mod m, v σ1ε (T13 ) = 0 0 0 v2 2 0 0 0 v v σ1ε (T23 ) = ≡ mod m. −2v −1 0 −1
2.6 A Cellular Basis for H
113
A corresponding symmetric matrix is given by 20 Bσ1 = ; see Example 1.4.6. 01 Performing similar (but much simpler) computations for the one-dimensional representations, we obtain the following expressions for Cλs,t (see also [111, Exp. 4.3]): W C11,1 = C10 ,
sgn
C1,1 = C14 ,
Cσ1,11 = −2C11 − 2C13 , Cσ1,21 = −2C12 ,
sgn
Cσ2,11 = −2C22 ,
sgn
σ1 C2,2 = −C21 −C23 .
C1,1 1 = −C21 +C23 , C1,1 2 = −C11 +C13 ,
For the case of unequal parameters, see 2.8.19. Example 2.6.18. Let W be of type I2 (6) = G2 , where S = {s1 , s2 } and (s1 s2 )6 = 1. (a) Assume that we are in the equal-parameter case where L(s1 ) = L(s2 ) > 0. Then, again, Theorem 2.6.12 applies and so we have a cellular basis {Cλs,t }. In this case, we can take for R any subring of C in which 2, 3 are invertible. Expressions for Cλs,t have been worked out in [129, Exp. 2.7] (using computations similar to those in the previous example): sgn
W = C10 , C11,1
C1,1 = C16 ,
sgn
C1,1 1 = C21 −C23 +C25 ,
1 Cσ1,1 = 3C11 + 6C13 + 3C15 , 1 = −3C12 − 3C14 , Cσ1,2
1 Cσ2,1 = −3C21 − 3C24 ,
1 = C21 + 2C23 +C25 , Cσ2,2
sgn
C1,1 2 = C11 −C13 +C15 , Cσ1,12 = C11 −C15 ,
Cσ1,22 = −C12 +C14 , Cσ2,12 = −C21 +C24 , Cσ2,22 = C21 −C25 .
(b) Assume that the monomial order on Γ is such that L(s1 ) > L(s2 ) > 0. By 2.4.1, P1–P15 hold and, hence, by 2.5.3, the hypothesis of Theorem 2.6.12 is satisfied. In this case, we can take for R any subring of C in which 2 is invertible. We find the following expressions for the cellular basis: Cσ1,11 = C11 +C13 ,
Cσ1,12 = C11 −C13 ,
sgn
Cσ2,11 =−C22 −C24 ,
Cσ2,12 = −C22 +C24 ,
sgn
Cσ2,21 = C23 +C25 ,
Cσ2,22 = C23 −C25 .
W = C10 , C11,1
Csgn 1,1 = C16 , C1,1 1 = C21 , C1,1 2 = C15 ,
Cσ1,21 =−C12 −C14 ,
Cσ1,22 = −C12 +C14 ,
Example 2.6.19. Let W be of type I2 (5), where S = {s1 , s2 } and (s1 s2 )5 = 1. We are in the equal-parameter case; assume that L(s1 ) = L(s2 ) > 0. Then Theorem 2.6.12
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2 Kazhdan–Lusztig Cells and Cellular Bases
applies and so we have a cellular basis {Cλs,t }. In this case, we can take for R any √ subring of C in which 5 is invertible and α := 12 (−1 + 5) ∈ R. We find the following expressions for Cλs,t : sgn
W C11,1 = C10 ,
C1,1 = C15 ,
Cσ1,21 = −(2 + α )C12 − (1 + α )C14 ,
Cσ1,22 = (α − 1)C12 + α C14
Cσ2,21 = C21 + (1 + α )C23 ,
Cσ2,22 = C21 − α C23 .
Cσ1,11 = (2 + α )C11 + (3 + 2α )C13 ,
Cσ1,12 = (1 − α )C11 + (1 − 2α )C13
Cσ2,11 = −(2 + α )C22 − (1 + α )C24 ,
Cσ2,12 = (α − 1)C22 + α C24
2.7 Further Properties of the Cellular Basis of H Throughout this section (except for Corollary 2.7.14 at the very end), we assume that we are in the setting of Theorem 2.6.12, where properties (♣) and (♠) in 2.5.3 hold. Thus, we have a cellular basis {Cλs,t } of H, and the partial order on Λ satisfies
μ L λ
⇒
μ =λ
or
aμ > aλ .
Let {W (λ ) | λ ∈ Λ } be the cell modules constructed from the cellular basis; see 2.6.1. By extension of scalars from A to K, we obtain modules for HK which we denote by WK (λ ) (λ ∈ Λ ). Since HK is semisimple, Theorem 2.6.5 shows that Irr(HK ) = {WK (λ ) | λ ∈ Λ }. Proposition 2.7.1. Let λ ∈ Λ . For any h = Cw (w ∈ W ) and s, s ∈ M(λ ), we have
ε aλ rhλ (s , s) ∈ R[Γ0 ],
ss. with constant term equal to cw, λ
In particular, the representation of HK afforded by WK (λ ) (with respect to its standard basis) is balanced, and we have WK (λ ) ∼ = Eελ . Proof. At the end of the proof of Theorem 2.6.12, we obtained the formula rhλ (s , s) =
∑
z∈W, d∈D˜ z∼LR d
n˜d hw,d,z cz,s λs .
Now assume z ∈ W and d ∈ D˜ are such that z ∼LR d and the corresponding terms in the above sum are non-zero; that is, hw,d,z = 0 and cz,s λs = 0. Since cz,s λs = 0, we
have E λ L z. Hence, Lemma 2.5.11 shows that ε aλ hw,d,z lies in Z[Γ0 ] and has constant term γ˜w,d,z−1 . Thus, we have ε aλ rhλ (s , s) ∈ R[Γ0 ] and
2.7 Further Properties of the Cellular Basis of H
ε aλ rhλ (s , s) ≡
∑
∑
˜ z∼LR d z∈W d∈D:
115
n˜d γ˜w,d,z−1 cz,s λs ≡ csw,λ
mod R[Γ>0 ],
where the last congruence follows as in Remark 2.5.15. Once this is established, it is clear that the representation afforded by WK (λ ) is balanced. Furthermore, by Proposition 1.5.7, we also see that WK (λ ) ∼ = Eελ . Corollary 2.7.2. Let λ ∈ Λ and denote by Gλ = gλst s,t∈M(λ ) the Gram matrix of the bilinear form , λ : W (λ ) ×W (λ ) → A. Then
ε aλ gλst ∈ R[Γ0 ]
λ ε aλ gλst ≡ fλ βst mod R[Γ>0 ].
and
Proof. Recall that gλst = gλts = rhλ (s, s), where h = Cλs,t . Hence, using the defining formula for Cλs,t , we obtain
ε aλ gλst =
∑ ∑
w∈W u∈M(λ )
λ us βtu cw−1 ,λ ε aλ rCλw (s, s) .
By Proposition 2.7.1, this expression lies in R[Γ0 ] and has constant term
λ us ss λ us ss β c c = β c c −1 −1 ∑ ∑ tu w ,λ w,λ ∑ tu ∑ w ,λ w,λ = βtsλ fλ , w∈W u∈M(λ )
u∈M(λ )
w∈W
where we used the “Schur relations” in Proposition 1.4.10(a).
Example 2.7.3. Let λ ∈ Λ and consider the representation ρ¯ λ : J˜ → Mdλ (K), where ρ¯ λ (tw ) ∈ Mdλ (R) for all w ∈ W . By first restricting ρ¯ λ to J˜ R and then extending scalars from R to A, we can also regard ρ¯ λ as an A-algebra homomorphism
ρ¯ λ : J˜ A → Mdλ (A). With this convention, the formula at the end of Theorem 2.6.12 means that λ rh (s , s) s ,s∈M(λ ) = ρ¯ λ φ (Cw ) (h = Cw ). By Proposition 2.7.1, we have Eελ ∼ = WK (λ ). Hence, the above formula shows that, for a suitable basis of Eελ , the action of Cw on Eελ is given by the matrix ρ¯ λ (φ (Cw )). We express this by saying that the action of H on Eελ factors through φ . Example 2.7.4. Let λ ∈ Λ and assume that there exists a left Kazhdan–Lusztig cell C such that Eελ ∼ = [C]K . (This is a very special situation, but we will see in Section 2.8 that it holds, for example, when W ∼ = Sn .) Let us write C = {xs | s ∈ M(λ )}. Then we have a corresponding representation ρC : HK → Mdλ (K) such that ρC (Cw ) = hw,xt ,xs s,t∈M(λ )
for all w ∈ W .
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2 Kazhdan–Lusztig Cells and Cellular Bases
By Lemma 2.2.4, there exists some w ∈ C such that E λ L w. Since (♣) is assumed to hold, we can apply Lemma 2.5.11, which yields that ρC is balanced and, for all w ∈ W and s, t ∈ M(λ ), we have
ε aλ hw,xt ,xs ≡ γ˜w,xt ,x−1 mod Z[Γ>0 ]. s
Thus, we can assume that ρ λ = ρC and the leading matrix coefficients are given by cst w,λ = γ˜w,xt ,x−1 s
for all w ∈ W and s, t ∈ M(λ ).
Now, by Lemma 1.5.3, we have γ˜w,xt ,x−1 = γ˜x−1 ,w−1 ,xs = γ˜w−1 ,xs ,x−1 . This means s
t
t
that ρ¯ λ (tw )tr = ρ¯ λ (tw−1 ) for all w ∈ W . Consequently, the conditions in Proposition 1.5.11 are satisfied where we take Bλ to be the identity matrix. The formula for rhλ (s , s) in the proof of Proposition 2.7.1 now reads rhλ (s , s) =
∑
z∈W, d∈D˜ z∼LR d
n˜ d hw,d,z γ˜z,xs ,x−1 = hw,xs ,xs
(h = Cw ),
s
where the last equality holds by Lemma 2.5.8. Thus, we have shown that W (λ ) is nothing but the left cell module [C]A . Furthermore, the action of H on [C]A factors through φ , as in Example 2.7.3. 2.7.5. One important feature of the definition of a “cell datum” is that it behaves well with respect to a specialisation; see [144, (1.8)]. Let θ : A → k be a homomorphism into a field k. Let Hk = k ⊗A H be the corresponding specialised algebra over k. Assume that {Cλs,t } satisfy (C1), (C2), (C3) in H. Then the elements {1 ⊗ Cλs,t } will satisfy (C1), (C2), (C3) in Hk . Hence, a cell datum for H automatically gives rise to a cell datum for Hk . Note that then the cell representations of Hk are given by Wk (λ ) = k ⊗A W (λ ) (λ ∈ Λ ), and the bilinear form , λ on W (λ ) induces the corresponding form , λ ,k on Wk (λ ). In particular, we have the following: (a) Extending scalars from A to K, we obtain a cell datum for HK . As already mentioned (see Proposition 2.7.1), since HK is semisimple, we have Irr(HK ) = {WK (λ ) | λ ∈ Λ },
where
WK (λ ) ∼ = Eελ for all λ ∈ Λ .
(b) In general, given any map θ : A → k as above, we set Lλk = Wk (λ )/rad( , λ ,k ) for λ ∈ Λ . Then Theorem 2.6.5 implies that μ
Irr(Hk ) = {Lk | μ ∈ Λk◦ },
where
Λk◦ := {λ ∈ Λ | , λ ,k = 0}. μ
Furthermore, the composition multiplicities (Wk (λ ) : Lk ) satisfy the conditions (Δ ) in Theorem 2.6.6. Hence, since (♣) is assumed to hold, this means μ for all μ ∈ Λk◦ , (Wk (μ ) : Lk ) = 1 a (Δ ) μ (Wk (λ ) : Lk ) = 0 unless λ = μ or aλ > aμ .
2.7 Further Properties of the Cellular Basis of H
117
Thus, our “fundamental problem” (p. 3) of determining Irr(Hk ) now takes the following more precise form (and this will be addressed in the subsequent chapters).
Fundamental Problem (revised). Given a cell datum for Hk , describe the subset μ Λk◦ ⊆ Λ and determine the dimension of Lk for μ ∈ Λk◦ . Our next result provides an alternative characterisation of the subset Λk◦ ⊆ Λ . In particular, this shows that Λk◦ does not depend on the choices involved in the definition of {Cλs,t }. (Recall that, for example, this definition relies on the balanced representations ρ λ , and these are not uniquely determined.) Proposition 2.7.6. In the setting of 2.7.5, let λ ∈ Λ . Then the following three conditions are equivalent. (a) λ ∈ Λk◦ . (b) θ χ λ (Cw ) = 0 for some w ∈ W such that E λ L w. (c) Cw .Wk (λ ) = {0} for some w ∈ W such that E λ L w. Proof. “(a) ⇒ (b)” If λ ∈ Λk◦ , then , λ ,k = 0 and so Cu ,Ct λ ,k = 0 for some u, t ∈ M(λ ). Now, by Proposition 2.7.1, we have Eελ ∼ = WK (λ ) and so χ λ (h) = trace(h, Eελ ) = trace h,WK (λ ) = ∑ rhλ (s, s) for all h ∈ H. s∈M(λ )
Let h = Cλu,t and apply Corollary 2.6.3. This yields rhλ (s, s) = δus Cs ,Ct λ for all s ∈ M(λ ) and so χ λ (h) = Cu ,Ct λ . Since Cu ,Ct λ ,k = 0, this shows that θ (χ λ (h)) = 0. Finally, by Remark 2.6.9, h is an R-linear combination of elements Cw , where E λ L w. Hence, (b) follows. “(b) ⇒ (c)” As above, we have χ λ (h) = ∑s∈M(λ ) rhλ (s, s) for all h ∈ H. Since Wk (λ ) = k ⊗A W (λ ), this implies θ χ λ (Cw ) = trace Cw ,Wk (λ )
for all w ∈ W ,
where Cw is regarded as an element of Hk on the right-hand side. Hence, if (b) holds, then Cw does not act as zero on Wk (λ ) and so (c) holds. “(c) ⇒ (a)” Let w ∈ W be such that E λ L w and Cw .Wk (λ ) = {0}. By μ Lemma 2.6.10, Cw is an R-linear combination of elements Cu,v , where μ ∈ Λ is μ such that E μ L w. Hence, we also have h.Wk (λ ) = {0}, where h = Cu,v for some μ ∈ Λ and u, v ∈ M( μ ) such that E λ ∼L E μ . In particular, this implies that h.W (λ ) = {0}. By the definition of the action of H on W (λ ), this means that there exist some s, s , t ∈ M(λ ) such that Cλs ,t appears with a non-zero coefficient in the μ
decomposition of hCλs,t = Cu,v Cλs,t . By (C2) and (C3), this implies that λ L μ . Since also E λ ∼L E μ , we conclude that λ = μ .
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2 Kazhdan–Lusztig Cells and Cellular Bases
Thus, we have h.Wk (λ ) = {0}, where h = Cλu,v for some u, v ∈ M(λ ). By the definition of the action of Hk on Wk (λ ) and Corollary 2.6.3, this implies that Cs ,Cv λ ,k = θ rhλ (u, s) = 0 for some s ∈ M(λ ). Thus, , λ ,k = 0 and so (a) holds.
Proposition 2.7.7. Let λ ∈ Λk◦ . Then the following hold. (a) We have θ (χ λ (Cw )) = trace(Cw , Lkλ ) for all w ∈ W such that E λ L w. (b) We have Cw .Lλk = {0} for some w ∈ W such that E λ L w. Proof. (a) Let w ∈ W be such that E λ L w. As in the above proof, θ (χ λ (Cw )) = trace(Cw ,Wk (λ )). Considering a composition series for Wk (λ ), we obtain
θ ( χ λ (Cw )) =
μ
μ
∑ ◦ (Wk (λ ) : Lk ) trace(Cw , Lk ).
μ ∈Λk
Let μ ∈ Λk◦ and assume that the corresponding terms in the sum are non-zero; that μ μ is, we have (Wk (λ ) : Lk ) = 0 and trace(Cw , Lk ) = 0. In particular, this means that μ μ Cw .Lk = {0}. We claim that this implies that E μ L E λ . Indeed, since Cw .Lk = {0}, we also have Cw .Wk (μ ) = {0} and, hence, Cw .W (μ ) = 0. By Lemma 2.6.10, Cw is an R-linear combination of elements Cνu,v , where ν ∈ Λ is such that E ν L w. Hence, we also have h.W (μ ) = {0}, where h = Cνu,v for some ν ∈ Λ and u, v ∈ M(ν ) such that E λ ∼L E ν . Arguing as in the above proof, this implies that μ L ν . Since also E λ ∼L E ν , we conclude that E μ L E λ , as claimed. On the other hand, μ since (Wk (λ ) : Lk ) = 0, we have λ L μ ; see Theorem 2.6.6. In combination with E μ L E λ , this implies that λ = μ . Thus, since (Wk (λ ) : Lλk ) = 1, we have shown that θ (χ λ (Cw )) = trace(Cw , Lλk ). (b) By Proposition 2.7.6, we have θ ( χ λ (Cw )) = 0 for some w ∈ W such that λ E L w. By (a), this implies that trace(Cw , Lλk ) = 0 and so Cw .Lkλ = {0}. Remark 2.7.8. Once a cellular structure for H is available, it also natural to consider the “Jantzen filtration” on cell modules; for recent results and open problems in this direction, see James and Mathas [183], [184], Shan [277] (type A), Bonnaf´e and Jacon [27] and Policzew [266] (exceptional types). Finally, we discuss the existence of W -graph representations, as already briefly mentioned at the end of Section 1.4. We begin with a preliminary result about the ˜ structure and the representations of J. Lemma 2.7.9. In addition to (♣) and (♠), also assume that () holds. Then: ˜ (a) We have γ˜x,y,z ∈ Z and n˜ d = ±1 for all d ∈ D. ˜ (b) The elements {n˜ d td | d ∈ D} are orthogonal idempotents. Furthermore, for each λ ∈ Λ , the balanced representation ρ λ of HK can be chosen ˜ such that the following holds for the corresponding representation ρ¯ λ of J:
2.7 Further Properties of the Cellular Basis of H
119
(c) The conditions in Proposition 1.5.11 hold where ρ¯ λ (tw ) = cst w,λ ∈ Mdλ (ZW ) for all w ∈ W . ˜ the matrix ρ¯ λ (td ) is diagonal with 0, ±1 on the diagonal. (d) For any d ∈ D, Proof. (a), (b) See Example 2.5.14. Once we know that γ˜x,y,z and n˜w are integers, the fact that n˜ d = ±1 follows from the formula in Remark 1.6.19(a). (c), (d) We slightly refine the argument in Proposition 1.5.11. We can assume that (W, S) is irreducible. First let W be of type I2 (m). In the proof of Proposition 1.5.11, we have seen that the representations in Example 1.3.7 satisfy (c). By Example 1.7.4, these representations also satisfy (d). Now assume that W is not of type I2 (m). Then ZW is a principal ideal domain. As in the proof of Proposition 1.5.11, a general argument shows that ρ λ can be chosen such that (c) holds. Let J˜ ZW = tw | w ∈ W ZW and let E¯ λ be a J˜ ZW -module (finitely generated and free over ˜ lie in ZW ) which affords the representation ρ¯ λ . Since the idempotents n˜ d td (d ∈ D) ˜JZ and since ZW is a principal ideal domain, we have a direct sum decomposition W E¯ λ = d∈D˜ E¯dλ , where E¯dλ := n˜ d td .E¯ λ is a ZW -submodule of E¯ λ which is finitely generated and free over ZW . Now choose a ZW -basis of E¯ λ which is adapted to this decomposition and perform a base change (over ZW ) to this new basis. We replace ρ λ by the representation obtained via this base change (as in the proof of Proposition 1.5.11). This new representation is balanced, and it satisfies (c) and (d). 2.7.10. We keep the assumptions of Lemma 2.7.9. We shall consider the effect of Lusztig’s homomorphism φ : H → J˜ A (see Theorem 2.5.5) on Cs , where s ∈ S is ˜ we have CsCd = −(vs + v−1 such that L(s) > 0. For any d ∈ D, s )Cd if sd < d; furthermore, if sd > d, then hs,d,z = 0 unless sz < z, in which case we have s . Thus, the formula in Theorem 2.5.5 can be written as hs,d,z = (−1)l(d)+l(z)+1 μz,d
φ (Cs ) = −(vs + v−1 s )
∑˜ n˜d td
+
d∈D sd
∑
s (−1)l(z)+l(d)+1 n˜ d μz,d tz .
z∈W, d∈D˜ sz
Following Gyoja [150] (where this is discussed in the equal-parameter case, in a somewhat different setting), we define elements of J˜ A as follows: s˜0 :=
∑ n˜d td
d∈D˜ sd
and
s˜1 :=
∑
s (−1)l(z)+l(d)+1 n˜ d μz,d tz .
z∈W, d∈D˜ sz
Thus, φ (Cs ) = −(vs + v−1 s )s˜0 + s˜1 . Now recall that Cs = Ts − vs T1 . Then we have
φ (Ts ) = φ (Cs ) + vs 1J˜ = −v−1 s s˜0 + vs (1J˜ − s˜0 ) + s˜1
for any s ∈ S.
The collection of elements {s˜0 , s˜1 | s ∈ S, L(s) > 0} satisfies the following properties. Lemma 2.7.11 (Cf. Gyoja [150, 2.4]). In the above setting, we have s˜20 = s˜0 ,
s˜0t˜0 = t˜0 s˜0 ,
s˜0 s˜1 = s˜1 ,
for all s,t ∈ S such that L(s) > 0 and L(t) > 0.
s˜1 s˜0 = 0
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2 Kazhdan–Lusztig Cells and Cellular Bases
˜ are orthogonal idempotents Proof. By Lemma 2.7.9(b), the elements {n˜ d td | d ∈ D} 2 ˜ ˜ ˜ in J. This yields that s˜0 = s˜0 and s˜0t0 = t0 s˜0 for all s,t ∈ S. Now consider
l(z)+l(d )+1 s s˜0 s˜1 = . (−1) n ˜ μ n ˜ t t z d d d z,d ∑ ∑ z∈W, d ∈D˜ sz
d∈D˜ sd
In order to show that this equals s˜1 , it will be enough to show that ∑d∈D, ˜ sd
∑˜
∑
s (−1)l(z)+l(d )+1 n˜ d μz,d ˜ d tz td . n
d∈D z∈W, d ∈D˜ sd
Assume, if possible, that this is non-zero. Then for some z, d, d in the above sum, s = 0 and t t = 0. Arguing as above, the latter condition implies that we have μz,d zd s = 0, we have z d . Since we also have z ∼L d. On the other hand, since μz,d L z ∼LR d , we can conclude that z ∼L d ; see Lemma 2.5.9. Hence, () yields that d = d . But we have sd < d and sd > d , which is a contradiction. Hence, the assumption was wrong and so we must have s˜1 s˜0 = 0, as claimed. A version of the following result (for equal parameters, and without taking into account the cellular structure) was first shown by Gyoja [150, §2]; subsequently, Lusztig [224, 3.8] gave a slightly different argument based on his asymptotic algebra. Our proof is a combination of the techniques used in [150] and [224]. Theorem 2.7.12. Recall our standing assumption that (♠) and (♣) hold; also assume that () holds and that L(s) > 0 for all s ∈ S. Then the data in Definition 2.6.8 can be chosen such that the cell modules {W (λ ) | λ ∈ Λ } are afforded by W graphs where the elements {msx,y } ⊆ A (see Definition 1.4.11) all lie in the subring ZW [Γ ] ⊆ A. Proof. By Lemma 2.7.11, the elements {s˜0 | s ∈ S} pairwise commute with each other. Hence, we can define FI := ∏ s˜0 ∏ (1J˜ − s˜0 ) ∈ J˜ for any subset I ⊆ S. s∈I
s∈S\I
These elements have the following properties:
2.7 Further Properties of the Cellular Basis of H
(a)
1J˜ =
∑ FI ,
FI2 = FI
121
(I ⊆ S),
FI FJ = 0
(I = J).
I⊆S
Indeed, by Lemma 2.7.11, the elements {s˜0 | s ∈ S} do not only commute with each other, but they are also idempotents. Hence, each FI is an idempotent (possibly zero). Furthermore, suppose that I = J. If s ∈ I \ J, then the factor s˜0 occurs in FI and the factor 1J˜ − s˜0 occurs in FJ . Hence, we have FI FJ = 0. The argument is analogous if |S| s ∈ J \ I. Finally, notice that 1J˜ = 1J˜ = ∏s∈S s˜0 + (1J˜ − s˜0 ) . Expanding the product yields that 1J˜ = ∑I⊆S FI , as required. Thus, (a) is proved. Now assume that the balanced representations {ρ λ | λ ∈ Λ } are chosen such that ˜ they satisfy the properties in Lemma 2.7.9. In particular, for any λ ∈ Λ and d ∈ D, the matrix ρ¯ λ (td ) is diagonal with 0, ±1 on the diagonal. We conclude that ρ¯ λ (s˜0 ) is a diagonal matrix for any s ∈ S and, hence, ρ¯ λ (FI ) is a diagonal matrix for any I ⊆ S. Since FI is an idempotent, the diagonal coefficients of ρ¯ λ (FI ) will be 0, 1. Since the elements {FI | I ⊆ S} are orthogonal idempotents whose sum is 1J˜ , we conclude that there is a well-defined partition (b)
M(λ ) =
MI (λ )
such that
FI .et = et ⇔ t ∈ MI (λ ).
I⊆S
Let us now extend scalars from R to A. Then E¯ Aλ := A ⊗R E¯ λ is a J˜ A -module but it also becomes an H-module via Lusztig’s homomorphism φ : H → J˜ A . The formula at the end of the proof of Theorem 2.6.12 shows that h ∈ H acts on W (λ ) in the same way as φ (h) ∈ J˜ A acts on E¯Aλ . So let us identify W (λ ) = E¯Aλ . Then, using the formula for φ (Ts ) (s ∈ S) in 2.7.10, we see that the action of H on W (λ ) is given by Ts .et = −v−1 s s˜0 et + vs (1J˜ − s˜0 )et + s˜1 et ,
where t ∈ M(λ ).
We will now show that this formula comes from a W -graph structure on W (λ ). First of all, the definition of a W -graph requires a map from M(λ ) to the set of all subsets of S. We define such a map as follows. Given t ∈ M(λ ), let I(t) be the unique subset I ⊆ S such that t ∈ MI (λ ); see (b). This definition implies that if I = I(t), et (c) FI .et = 0 otherwise. Next we consider the action of Ts , where s ∈ S. If s ∈ I(t), then s˜1 s˜0 = 0 and so s˜1 FI(t) = 0; since et = FI(t) .et , we conclude that s˜1 .et = 0. Furthermore, s˜0 .et = et (since s˜20 = s˜0 and so s˜0 FI(t) = FI(t) ). Hence, we obtain in this case Ts .et = −v−1 s et , as required in the definition of a W -graph. Now assume that s ∈ I(t). Then s˜0 .et = 0 (since s˜0 FI(t) = 0) and so
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2 Kazhdan–Lusztig Cells and Cellular Bases
Ts .et = vs et + s˜1 .et = vs et +
∑
u∈M(λ )
msu,t eu ,
where the terms msu,t ∈ A are such that m¯ su,t = msu,t and vs msu,t ∈ ZW [Γ>0 ] (by the defining formula for s˜1 , Example 2.1.10 and Lemma 2.7.9(a) and (c)). So it remains to show that msu,t = 0 unless s ∈ I(u). But, for any I ⊆ S such that s ∈ I, we have (1J˜ − s˜0 )s˜1 = 0 by Lemma 2.7.11 and so FI s˜1 = 0. Hence, we have
s˜1 .et = ∑ FI s˜1 .et = ∑ FI s˜1 .et ⊆ FI .eu | u ∈ M(λ ), s ∈ I ⊆ SA . I⊆S
I⊆S, s∈I
By (c), the latter submodule is contained in eu | s ∈ I(u)A , as required. So the above formulae show that W (λ ) is afforded by a W -graph. The above result shows that the cell modules {W (λ ) | λ ∈ Λ } arising from our construction of a cellular basis of H are afforded by W -graphs. The following conjecture is a kind of converse to this statement. ¨ Conjecture 2.7.13 (Geck and Muller [129, 4.5]). Assume that L(s) > 0 for all s ∈ S and that, for every λ ∈ Λ , we are given a W -graph affording an H-module V λ such that K ⊗A V λ ∼ = Eελ . Then the data in Definition 2.6.8 can be chosen such that λ {V | λ ∈ Λ } are the corresponding cell modules. In order to prove this conjecture, it would be sufficient to show that every W graph representation of H factors through Lusztig’s homomorphism φ : H → J˜ A . Somewhat related open problems are formulated by Gyoja [150, Remark 2.18]. The final result in this section holds without any assumptions on W, L. Corollary 2.7.14. Let H = HA (W, S, L) be any generic Iwahori–Hecke algebra, where W is finite and the general assumptions in 1.2.1 hold; that is, Γ admits a monomial ordering and A = R[Γ ], where ZW ⊆ R ⊆ C. Then every irreducible representation of HK can be realised over ZW [Γ ]. Proof. Let K be the field of fractions of R and set k := K[Γ ]. Then k certainly is L0 good and so, as in the proof of Corollary 2.6.16, there exists some cell datum for Hk which is obtained by extension of scalars from a cell datum in a “universal” algebra H0 over A0 = K[Γ0 ], where P1–P15 are known to hold. Let {W0 (λ ) | λ ∈ Λ } be the corresponding cell modules of H0 . We have a unique K-linear ring homomorphism θ : A0 → k such that θ (v◦s ) = vs for all s ∈ S, where {v◦s } are the parameters of H0 and {vs | s ∈ S} are the parameters of H. Thus, Hk = k ⊗A0 H0 , where k is regarded as an A0 -module via θ . Now K (the field of fractions of A) equals the field of fractions of k. Since HK = K ⊗k Hk is semisimple, we have Irr(HK ) = {WK (λ ) := K ⊗A0 W0 (λ ) | λ ∈ Λ }; see Theorem 2.6.5 and 2.7.5. Since each W0 (λ ) is realised over ZW [Γ0 ] by Theorem 2.7.12, we conclude that WK (λ ) is realised over the image of ZW [Γ0 ] under θ0 ; that is, over ZW [Γ ], as required.
2.8 The Case of the Symmetric Group
123
2.8 The Case of the Symmetric Group The aim of this section is to give an elementary proof of the properties (♣), (♠) and () in 2.5.3 when W ∼ = Sn . We will then see that the Kazhdan–Lusztig basis {Cw } itself is a cellular basis in this case. Note that even if we were willing to admit from the beginning that P1–P15 hold for W , then there would still be a substantial piece of work to do in order to determine the partial order L in Theorem 2.6.12. We begin with a few general (and more or less well-known) remarks related to longest elements in parabolic subgroups. In 2.8.1–2.8.7, W may be any finite Coxeter group W and L : W → Γ a weight function such that L(s) > 0 for all s ∈ S. (Here, we explicitly exclude the possibility that L(s) = 0 for some s ∈ S.) 2.8.1. Let I ⊆ S and consider the parabolic subgroup WI ⊆ W . Let XI be the set of distinguished left coset representatives of WI in W ; we have XI = {w ∈ W | w has minimal length in wWI }. The map XI ×WI → W , (x, u) → xu, is a bijection and we have l(xu) = l(x)+l(u) for u ∈ WI and x ∈ XI ; see [132, §2.1]. Let HI = Tw | w ∈ WI A ⊆ H be the corresponding parabolic subalgebra of H. For any w ∈ WI , we have Cw ∈ HI and Cw ∈ HI ; hence, {Cw | w ∈ WI } and {Cw | w ∈ WI } are the Kazhdan–Lusztig bases of HI . Lemma 2.8.2. Let wI ∈ WI be the unique element of maximal length. We have CwI = (−1)l(wI ) ε L(wI )
∑ (−1)l(w) ε −L(w)Tw .
w∈WI
Furthermore, the following hold. (a) For any w ∈ WI , we have TwCwI = (−1)l(w) ε −L(w)CwI . (b) We have Cw2 I = (−1)l(wI ) ε −L(wI ) PI CwI , where PI = ∑w∈WI ε 2L(w) ; cf. 1.2.11(c). (c) The set XI wI is a union of left cells in W ; we have XI wI = {w ∈ W | w L wI }. Proof. The formula for CwI already appears in Example 2.1.17. Next, we prove (a), by induction on l(w). First assume that w = s ∈ I. Then we have swI < wI and so the multiplication rule in Theorem 2.1.8 shows that CsCwI = −(vs + v−1 s )CwI . Since Cs = Ts − vs T1 , this yields TsCwI = −v−1 s CwI . If l(w) > 1, then write w = w s, where s ∈ I, w ∈ WI and l(w) = l(w ) + 1. We have Tw = Tw Ts , and so the desired formula follows by induction. Once (a) is established, we compute Cw2 I = (−1)l(wI ) ε L(wI )
ε
l(wI ) L(wI )
= (−1)
∑ (−1)l(w)ε −L(w) TwCwI
w∈WI
∑
ε −2L(w)CwI = (−1)l(wI ) ε −L(wI ) PI CwI .
w∈WI
To obtain the last equality, we used the formula l(wwI ) = l(wI )−l(w) for all w ∈ WI . Thus, (b) is proved. Finally, consider (c). Let w ∈ W be such that w L wI . Then R(wI ) ⊆ R(w); see Lemma 2.1.16. Hence, since R(wI ) = I, we can write w = xwI ,
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2 Kazhdan–Lusztig Cells and Cellular Bases
where x ∈ XI . Conversely, if x ∈ XI , then l(xwI ) = l(x) + l(wI ) and so xwI L wI . (This follows since sw ←L w if s ∈ S, w ∈ W are such that sw > w.) Thus, we obtain XI wI = {w ∈ W | w L wI }. This also shows that XI wI is a union of left cells. Lemma 2.8.3. Let I ⊆ S and IL wI ⊆ H be the left ideal defined by the general procedure in 1.6.2, with respect to the basis {Cw | w ∈ W } of H. Then we have IL wI = CxwI | x ∈ XI A = TxCwI | x ∈ XI A . Proof. By definition, we have IL wI = Cw | w L wI A ; this equals CxwI | x ∈ XI A by Lemma 2.8.2(c). Now set MI := TxCwI | x ∈ XI A . Since IL wI is a left ideal, it L are free of the same rank over A; this . Both M and I is clear that MI ⊆ IL I wI wI L already implies that K ⊗A MI = K ⊗A IL wI . But we also have that H/IwI is free over A; furthermore, H is free as an HI -module and so H/MI is free as an A-module. Hence, we must have HI = IL wI . Lemma 2.8.4. Let I ⊆ S and μ ∈ Λ be such that E μ is a constituent of the induced μ representation IndW WI (sgnI ). Then E L xwI for some x ∈ XI . Proof. Let sgnεI denote the sign representation of HI . By Example 1.3.3, we have sgnεI (Tw ) = (−1)l(w) ε −L(w) for w ∈ WI . So Lemma 2.8.2(a) shows that CwI A ⊆ HI affords sgnεI . Now, the induction of representations is also defined on the level of H; see [132, §9.1]. Hence, by Lemma 2.8.3, we have an isomorphism of left H-modules ∼
H IL wI → IndHI (CwI A ),
TxCwI → Tx ⊗CwI
(x ∈ XI ). μ
By a specialisation argument (see Example 1.2.4), our assumption implies that Eε is L L a constituent of IL wI ,K := K ⊗A IwI . Now, for any w ∈ W , we have trace(Cw , IwI ) = h . Furthermore, by Lemma 2.8.2(a), we can write X w = C ∪ ∑x∈XI w,xwI ,xwI I I 1 ... ∪ Cm , where C1 , . . . , Cm are (pairwise distinct) left cells of W . Consequently, we have trace(Cw , IL wI ) =
∑ ∑ hw,z,z = ∑
1im z∈Ci
trace(Cw , [Ci ]A ) for all w ∈ W .
1im
∼ Since HK is split semisimple, this implies that IL wI ,K = [C1 ]K ⊕. . .⊕[Cm ]K . It follows μ that Eε is a constituent of [Ci ]K for some i and so m(Ci , μ ) > 0; see 2.2.2(b). Hence, by Lemma 2.2.4, there exists some w ∈ XI wI such that E μ L w. Lemma 2.8.5. Let I ⊆ S and μ ∈ Λ be such that ρ μ (CwI ) = 0. Then E μ is a constituent of IndW WI (sgnI ). Proof. As in the above proof, let sgnεI denote the sign representation of HI . Using the formula for CwI in Lemma 2.8.2, we obtain
χ μ (CwI ) = (−1)l(wI ) ε L(wI )
∑
w∈WI
sgnεI (Tw−1 ) χ μ (Tw ),
2.8 The Case of the Symmetric Group
125
where we also used the fact that sgnεI (Tw−1 ) = sgnεI (Tw ) for all w ∈ WI . Now, writing the restriction of χ μ to HI,K as a sum of irreducible characters of HI,K and using the Schur relations in Proposition 1.2.12, we conclude that
χ μ (CwI ) = (−1)l(wI ) ε L(wI ) csgnI m(I, μ ), where m(I, μ ) denotes the multiplicity of sgnεI in the restriction of χ μ to HI,K . By a specialisation argument (see Example 1.2.4), m(I, μ ) also equals the multiplicity of sgnI in the restriction of E μ from W to WI . And by Frobenius reciprocity, this is the same as the multiplicity of E μ as a constituent of IndW I (sgnI ). Thus, it remains to show that χ μ (CwI ) = 0. Now, by Lemma 2.8.2(b), CwI is a non-zero scalar multiple of an idempotent. Hence, ρ μ (CwI ) will be conjugate to a non-zero scalar multiple of a diagonal matrix with 0 and 1 on the diagonal. Since ρ μ (CwI ) = 0, we conclude that χ μ (CwI ) = trace(ρ μ (CwI )) = 0, as required. Corollary 2.8.6. Let I ⊆ S and λ ∈ Λ be such that aλ = L(wI ) and E λ is a constituent of IndW WI (sgnI ). Then wI ∈ Fλ (where Fλ is defined in Proposition 1.6.11). Proof. As in the above proof, χ λ (CwI ) = ±ε L(wI ) csgnI m(I, λ ). Now, we have csgnI =
∑
sgnεI (Tw )sgnεI (Tw−1 ) =
w∈WI
∑
ε −2L(w) .
w∈WI
Since aλ = L(wI ) and L(w) < L(wI ) for w ∈ W , w = wI , we obtain that
ε aλ χ λ (CwI ) ≡ ±m(I, λ )
∑
ε 2(L(wI )−L(w)) ≡ ±m(I, λ ) mod m.
w∈WI
Since we also have ε aλ χ λ (TwI ) ≡ ε aλ χ λ (CwI ) mod m by 2.1.19, we conclude that cwI ,λ = ±m(I, λ ) = 0 and so wI ∈ Fλ . Corollary 2.8.7 (Cf. [107, 4.7, 4.8]). Let I ⊆ S and define I := PI−1 ε L(wI )CxwI CwI y−1 ∈ HK Zx,y
for any x, y ∈ XI .
Then the following hold: I
I =Z I LR LR (a) We have Zx,y x,y ∈ H. Furthermore, Zx,y ∈ IwI , where IwI is defined by the general procedure in 1.6.2, with respect to the basis {Cw | w ∈ W } of H. I ) = 0, then E μ is a constituent of IndW (sgn ). (b) If μ ∈ Λ is such that ρ μ (Zx,y I WI
Proof. (a) By Lemma 2.8.3, CxwI is an A-linear combination of terms Tx1 CwI , where and so C x1 ∈ XI . By 2.1.14, we have CwI y−1 = Cyw wI y−1 is an A-linear combination I of terms CwI Ty−1 , where y1 ∈ XI . Consequently, by Lemma 2.8.2(b), CxwI CwI y−1 is 1 an A-linear combination of terms PI Tx1 CwI Ty−1 , where x1 , y1 ∈ XI . Hence, we have 1
I
I ∈ H and Z I ∈ ILR . Since P = ε −2L(wI ) P , we also see that Z I = Z . Zx,y I I x,y wI x,y x,y
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2 Kazhdan–Lusztig Cells and Cellular Bases
I ) = 0. In the proof of (a), we have seen that Z I is an A(b) Assume that ρ μ (Zx,y x,y linear combination of terms PI Tx1 CwI Ty−1 , where x1 , y1 ∈ XI . Hence, we must have 1 ρ μ (CwI ) = 0 and so the assertion follows from Lemma 2.8.5.
2.8.8. From now on, let W = Sn be the symmetric group where S = {s1 , . . . , sn−1 } and si = (i, i + 1) for 1 i n − 1. The set Λ consists of all partitions of n; we write λ n to denote that λ is a partition of n. Furthermore, we assume that Γ = Z and L(si ) = 1 for 1 i n − 1. By 1.7.6, we have fλ = 1 for all λ n. So R = Z is an L-good subring of C; in particular, A = Z[v, v−1 ], where v = ε is an indeterminate. By Corollary 1.7.9, the balanced representations {ρ λ | λ n} can be chosen such that the corresponding leading matrix coefficients satisfy the following condition: (a)
ts cst w,λ = cw−1 ,λ ∈ {0, ±1}
for all w ∈ W and s, t ∈ M(λ ).
Consequently, we have a bijection (b)
M(λ ) × M(λ )
1−1
−→
(s, t) → wλ (s, t),
W,
λ ∈Λ
satisfying the properties in Theorem 1.7.10; in particular, we have (c)
Fλ = {wλ (s, t) | s, t ∈ M(λ )}
for all λ n.
By Proposition 2.1.20, we already know that Fλ is contained in a two-sided Kazhdan–Lusztig cell. One of our aims is to show that the converse also holds. 2.8.9. For any subset I ⊆ {1, . . . , n − 1}, denote by WI ⊆ W the parabolic subgroup generated by {si | i ∈ I}. We now define, for any partition λ n, a particular parabolic subgroup of W . For this purpose, we set I(λ ) := {1, . . . , n} \ {λ1∗ , λ1∗ + λ2∗ , λ1∗ + λ2∗ + λ3∗ , . . .}, where λ1∗ λ2∗ λ3∗ . . . 0 are the parts of λ ∗ , the conjugate partition of λ . (Thus, WI(λ ) ⊆ W is the Young subgroup Sλ ∗ .) For example, if λ = (1n ), then λ ∗ = (n) and so WI = W . Then Young’s rule shows that, for any μ n, we have (a)
E μ is a constituent of IndW WI(λ ) (sgnI(λ ) )
⇔
κμ ∗ λ ∗ = 0
⇔
μ λ,
where κμ ∗ λ ∗ is a Kostka number and denotes the dominance order on partitions, as defined in Example 2.2.13. (For the first equivalence in (a), see Macdonald [236, p. 115]; the second equivalence is a combination of [236, I.6.5 and I.7.9]. Note also that λ μ if and only if μ ∗ λ ∗ ; see [236, I.1.11].) Consequently, we have (b)
aλ = l(wI(λ ) )
and
wI(λ ) ∈ Fλ
for all λ n,
where we use Corollary 2.8.6 and [132, 5.4.1, 5.4.3] to relate aλ and l(wI(λ ) ). Now we define a two-sided ideal of HK by
2.8 The Case of the Symmetric Group
(c)
127
Nλ := {h ∈ HK | ρ μ (h) = 0 for all μ n such that μ λ }.
We also set Nˆλ := {h ∈ Nλ | ρ λ (h) = 0}. Note that Nλ is the sum of all Wedderburn components of the split semisimple algebra HK which correspond to the irreducible μ representations Eε where μ λ . 2.8.10. We have just seen that wI(λ ) ∈ Fλ for any λ n. In particular, wI(λ ) = wλ (t0 , t0 ) for a unique t0 ∈ M(λ ). By Theorem 1.7.10(b) and Proposition 2.1.20, the set C0 := {wλ (s, t0 ) | s ∈ M(λ )} is contained in a left Kazhdan–Lusztig cell. Hence, by Lemma 2.8.2(c), we have C0 ⊆ XI(λ ) wI(λ ) . Consequently, there is a subset {xs | s ∈ M(λ )} ⊆ XI(λ ) such that wλ (s, t0 ) = xs wI(λ ) for all s ∈ M(λ ). We now set (a)
I(λ )
Zw := Zxs ,xt ,
where
w = wλ (s, t) for s, t ∈ M(λ ).
By Corollary 2.8.7(a), we have Zw = Z w ∈ H. We claim that Zw ∈ Nλ
(b)
for all w ∈ Fλ .
This is seen as follows. Let μ n and assume that ρ μ (Zw ) = 0. We must show that μ λ . Now, by Corollary 2.8.7(b), E μ is a constituent of IndW WI(λ ) (sgnI(λ ) ). By 2.8.9(a), this implies that μ λ , as claimed. Lemma 2.8.11. Let λ ∈ Λ and u, v ∈ M(λ ). Then, for any w ∈ Fλ , we have λ (Zw ) ∈ O0 vaλ ρuv
and
λ vaλ ρuv (Zw ) ≡ ±δsu δtv mod m,
where s, t ∈ M(λ ) are such that w = wλ (s, t). Proof. We have aλ = l(wI(λ ) ) by 2.8.9(b). Hence we obtain λ 2aλ λ (Zw ) = PI(−1 ρuv Cxs wI(λ ) Cw x−1 vaλ ρuv λ) v I(λ ) t 2aλ λ C v ρ C = PI(−1 uv wλ (s,t0 ) wλ (t0 ,t) λ) a λ λ = PI(−1 v λ ρur (Cwλ (s,t0 ) ) vaλ ρrv (Cwλ (t0 ,t) ) . λ) ∑ r∈M(λ )
First of all, this shows that the above expression lies in O0 ; note that PI(λ ) ∈ 1 + m. Furthermore, its constant term can be expressed by the leading matrix coefficients of wλ (s, t0 ) and wλ (t0 , t). Indeed, by 2.1.19 and Theorem 1.7.10, we have λ (Cwλ (s,t0 ) ) ≡ cur vaλ ρur wλ (s,t0 ),λ ≡ ±δsu δrt0 aλ
v
λ ρrv (Cwλ (t0 ,t) ) ≡ crv wλ (t0 ,t),λ
≡ ±δtv δrt0
mod m, mod m.
λ (Z ) ≡ ±δ δ Since PI(λ ) ∈ 1 + m, we obtain vaλ ρuv w su tv mod m, as desired.
Theorem 2.8.12 (Cf. [107, 4.10]). Recall that we are in the setting of 2.8.8, where W = Sn . Then the following hold for any partition λ n.
128
(a) (b) (c)
2 Kazhdan–Lusztig Cells and Cellular Bases
±Cw ∈ Zw + Nˆλ ⊆ Nλ for all w ∈ Fλ , Nλ = Cw | w ∈ Fμ for some μ n such that μ λ K , Nˆλ = Cw | w ∈ Fμ for some μ n such that μ λ K ,
where Nˆλ ⊆ Nλ are the two-sided ideals of HK defined in 2.8.9. Proof. We prove (a) by induction on the dominance order on partitions. The unique minimal element in this order is the partition (1n ). We have F(1n ) = {w0 } (where w0 is the longest element of W ), I(1n ) = {1, . . . , n − 1}, XI(1n ) = {1} and −1 l(w0 ) 2 Zw0 = PI(1 Cw0 = (−1)l(w0 )Cw0 ; n)ε
see Lemma 2.8.2(b).
Hence, (a) holds in this case. Now assume that λ = (1n ) and that (a) holds for all partitions μ n, where μ λ . Let w ∈ Fλ . By 2.8.10(b), we already know that Zw ∈ Nλ . Since Zw = Z w ∈ H (see Corollary 2.8.7(a)), we can write Zw =
∑ ηy Cy ,
where ηy = η y ∈ Z[v, v−1 ] for all y ∈ W .
y∈W
Let y ∈ W be such that ηy = 0; we have y ∈ Fμ for a unique μ n. If μ λ , then ±Cy ∈ Zy + Nˆμ by induction. Furthermore, by 2.8.10(b), we have Zy ∈ Nμ . By definition, it is also clear that Nμ ⊆ Nˆλ . Hence, we conclude that Cy ∈ Nμ ⊆ Nˆλ . So it remains to consider those y where y ∈ Fμ , μ λ . Let us write C := {y ∈ W | ηy = 0 and y ∈ Fμ where μ λ } and set m := max{deg(ηy ) | y ∈ C }. We claim that {y ∈ C | deg(ηy ) = m} ⊆ Fλ
(∗)
and
m = 0.
This is seen as follows. Let y0 ∈ C be such that deg(ηy0 ) = m; then y0 = wμ (u, v), where μ n, u, v ∈ M(μ ) and μ λ . By 2.1.19 and Theorem 1.7.10, we have μ
vaμ ρuv (Cy ) ≡ cuv y,μ ≡ ±δyy0 mod m
for any y ∈ W .
It follows that μ
vaμ +m ρuv (Zw ) ≡
∑
y∈W
μ vm ηy vaμ ρuv (Cy ) ≡ vm ηy0 cuv y0 , μ ≡ 0 mod m.
In particular, this yields that ρ μ (Zw ) = 0 and so μ λ , since Zw ∈ Nλ by 2.8.10(b). Combined with μ λ , we conclude that μ = λ and so y0 ∈ Fλ , which proves the first part of (∗). Now the above congruence reads λ (Zw ) ≡ 0 mod m. vm vaλ ρuv
2.8 The Case of the Symmetric Group
129
But then Lemma 2.8.11 implies that m = 0, as required. Thus, (∗) is proved. Consequently, we can now write Zw ≡
∑
ηy Cy mod Nˆλ ,
where ηy ∈ Z for all y ∈ Fλ .
y∈Fλ
Let s, t ∈ M(λ ) be such that w = wλ (s, t). Since ρ λ (h) = 0 for all h ∈ Nˆλ , we have λ ρuv (Zw ) =
∑
λ ηy ρuv (Cy )
for any u, v ∈ M(λ ).
y∈Fλ
We multiply this identity by vaλ and take constant terms. Using Theorem 1.7.10, Lemma 2.8.11 and 2.1.19, we deduce that ±δsu δtv =
∑
y∈Fλ
uv ηy cuv y,λ = ηy0 cy0 ,λ ,
where
y0 = wλ (u, v).
It follows that ηw = ±cst w,λ and ηy = 0 for all y ∈ Fλ \ {w}, as required. Thus, (a) is proved. Now let Mλ be the K-subspace of HK defined by the right-hand side of the desired identity in (b). We claim that Mλ ⊆ Nλ . Indeed, let w ∈ Fμ , where μ λ . By 2.8.10(b), we have Zw ∈ Nμ . Hence, using (a), we see that Cw ∈ Nμ . Furthermore, by definition, it is clear that Nμ ⊆ Nλ . Hence, we have Cw ∈ Nλ , as claimed. Now notice that dim Mλ = ∑μ λ |Fμ |; furthermore, by 2.8.8(c), we have |Fμ | = |M(μ )|2 . On the other hand, as already noted in 2.8.9, the ideal Nλ is the sum of all Wedderburn components of HK which correspond to the irreducible μ representations Eε where μ λ . Hence, we also have dim Nλ = ∑μ λ |M(μ )|2 and, consequently, Nλ = Mλ . Thus, (b) is proved. This also implies (c) since, by definition, Nˆλ = ∑μ Nμ , where the sum runs over all μ n such that μ λ . Remark 2.8.13. The above proof essentially follows [107, Theorem 4.10]. However, in [107], we referred to the results of Murphy [256], [257] in order to define the ideals Nλ and Nˆλ . The discussion here avoids that reference and, thus, is considerably more self-contained than that in [107]. Corollary 2.8.14. Let λ , μ n. Then we have E μ L E λ if and only if μ λ . In particular, the equivalence classes of IrrK (Sn ) under ∼L are singleton sets. Consequently, the properties (♣) and (♠) (see 2.5.3) hold. Proof. Assume first that μ λ . By 2.8.9(a) (Young’s rule) and Lemma 2.8.4, this implies that xwI(λ ) ∈ Fμ for some x ∈ XI(λ ) . But then we have xwI(λ ) L wI(λ ) and so E μ L E λ ; recall that wI(λ ) ∈ Fλ by 2.8.9(b). Conversely, assume that E μ L E λ . This means that y LR w, where y ∈ Fμ and w ∈ Fλ . By definition, we can find a sequence y = y0 , y1 , . . . , ym = w such that, for each i ∈ {1, . . . , m}, there exist some xi ∈ W such that hxi ,yi ,yi−1 = 0 or hyi ,xi ,yi−1 = 0. Now, by Theorem 2.8.12(b), we have Cym = Cw ∈ Nλ . Since Nλ is a two-sided ideal, we have Cxm Cym ∈ Nλ and Cym Cxm ∈ Nλ . Hence, we have ym−1 ∈ Fμm−1 , where μm−1 λ ; see Theorem 2.8.12(b). We repeat the argument with xm−1 , ym−1 and find that
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2 Kazhdan–Lusztig Cells and Cellular Bases
ym−2 ∈ Fμm−2 , where μm−2 μm−1 λ . Continuing in this way, we eventually obtain that μ = μ0 λ , as required. It is known that this implies aλ aμ , with equality only if λ = μ ; see 2.2.13. Consequently, (♣) holds. But then the weaker property (♣ ) holds and so (♠) also holds; see Proposition 2.5.12. Theorem 2.8.15. The algebra H admits a cellular basis as in Theorem 2.6.12. The data in Definition 2.6.8 can be chosen such that, for λ n and s, t ∈ M(λ ), we have Cλs,t = cst w,λ Cw = ±Cw ,
where
w = wλ (s, t).
Furthermore, the partial order L is given by the dominance order on partitions; we have Nλ = HK (L ) and Nˆλ = HK ( L ). Proof. Since (♣) and (♠) hold, we can apply Theorem 2.6.12 and so H admits a cellular basis where, by Definition 2.6.8, we have Cλs,t =
∑ ∑
w∈W u∈M(λ )
λ us βtu cw−1 ,λ Cw .
ts tr ¯λ ¯λ Now, since cst w,λ = cw−1 ,λ for all s, t ∈ M(λ ), we have ρ (tw−1 ) = ρ (tw ) for all
λ ) the identity matrix. Then the above w ∈ W . Hence, we can take for Bλ = (βst λ st sum reduces to Cs,t = cw,λ Cw , as required. Finally, by Corollary 2.8.14, the partial order L in Theorem 2.6.12 coincides with the dominance order. The identities Nλ = HK (L ), Nˆλ = HK ( L ) now follow from Theorem 2.8.12(b) and (c).
Remark 2.8.16. This result was first stated by Graham and Lehrer [144, Example 1.2], but the argument is very sketchy, especially concerning the ordering L . Some more details are contained in Graham’s thesis [143, Example 4.3] and Williamson’s Honours essay [295]. As far as we are aware, the first elementary proof of the characterisation of L in terms of the dominance order appeared in [107]. In Remark 2.8.18, we show how the signs in Theorem 2.8.15 can be fixed. A completely different construction of a cellular basis (with respect to the above ordering on Λ ) is due to Murphy [256], [257]; the equivalence of the two constructions is shown in [107]. 2.8.17. By Corollary 2.8.14 and Lemma 2.5.9, the Kazhdan–Lusztig cells of W = Sn are given by Theorem 1.7.10. So, for any λ n, the following hold. (a1) The set Fλ = {wλ (s, t) | s, t ∈ M(λ )} is a two-sided Kazhdan–Lusztig cell. (a2) For t ∈ M(λ ), the set {wλ (s, t) | s ∈ M(λ )} is a left Kazhdan–Lusztig cell. (a3) For s ∈ M(λ ), the set {wλ (s, t) | t ∈ M(λ )} is a right Kazhdan–Lusztig cell. In particular, we see that () holds. We also obtain the following result originally due to Kazhdan and Lusztig [195, Theorem 1.4]: for any left Kazhdan–Lusztig cell C of W , we have (b)
[C]1 ∈ IrrK (W )
and
[C]1 ∼ = E λ ⇔ C ⊆ Fλ .
2.8 The Case of the Symmetric Group
131
Indeed, let C be a left cell and λ ∈ Λ be such that m(C, λ ) > 0. Then C ⊆ Fλ by Lemma 2.2.4. Hence, C is equal to a set as in (a2). But then |C| = |M(λ )| = dλ = dim E λ and so we must have E λ ∼ = [C]1 . Finally, assume that C is a left cell such that C ⊆ Fλ . Then the same argument shows that [C]1 ∼ = E λ . (If we had [C]1 ∼ = Eμ, where μ = λ , then C ⊆ Fμ , which is a contradiction.) Thus, (b) is proved. We can now also apply the discussion in Example 2.7.4, which shows that, for each λ n, the balanced representation ρ λ can be chosen such that (c)
W (λ ) = [Cλ ]A ,
where Cλ ⊆ Fλ is a fixed left Kazhdan–Lusztig cell.
See also McDonough and Pallikaros [251], where the above cell modules are identified with the original “Specht modules” of Dipper and James [62]. Remark 2.8.18. Once Theorem 2.8.12, Corollary 2.8.14 and 2.8.17 are established, it is actually not too difficult to show that P1–P15 hold for W = Sn ; see [107, §5], [121, §4]. Furthermore, one can even show a tiny piece of “positivity” by elementary methods; namely, the fact that γx,y,z 0 for all x, y, z ∈ W ; see [107, Theorem 5.10]. (Recall that γx,y,z = (−1)l(x)+l(y)+l(z) cx,y,z ; see Remark 2.3.6.) The argument relies on basic properties of the “Knuth–Robinson–Schensted correspondence” and the Kazhdan–Lusztig “star operations”; see Kazhdan and Lusztig [195, §5], Knuth [205, §5.1.4] and Ariki [8]. We will not go into any more detail here, as these results are not needed for the further discussions in this book. Let us just explain how the signs in Theorem 2.8.15 can be fixed, assuming that P1 and P4 hold and that γx,y,z 0 for all x, y, z ∈ W (which is also known to be the case by 2.4.1(a)). This is done as follows. Choosing the balanced representation ρ λ as in Proposition 1.8.9, each coefficient cst w,λ is equal to a structure constant γ˜w,x,y for suitable x, y ∈ Fλ . Now, by Proposition 2.3.16, Remark 2.3.5 and Proposition 2.3.14, we have
γ˜w,x,y = (−1)l(w)+l(x)+l(y) γw,x,y = (−1)a(y) γw,x,y
and
a(y) = aλ .
Since γw,x,y 0 and cst w,λ ∈ {0, ±1}, we deduce that aλ cst w,λ = (−1) ,
where
w = wλ (s, t).
λ ) be equal to Arguing as in the proof of Theorem 2.8.15, we now let Bλ = (βst a λ λ (−1) times the identity matrix. Then we obtain Cs,t = Cw , as required.
For a further discussion of the combinatorics involved in the above constructions (Knuth–Robinson–Schensted correspondence, etc.), we refer the reader to the references cited in Remark 2.8.18. In a somewhat different context, we will have more to say about the combinatorics of Young tableaux in Section 3.5. 2.8.19. Having dealt with W = Sn , it is natural to ask what happens with the other cases in 1.7.6. So, let W be of type Bn and L : W → Γ a weight function given by b 4 a a a Bn t t t p p p t
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2 Kazhdan–Lusztig Cells and Cellular Bases
where a, b > 0 and b ∈ {a, 2a, . . . , (n − 1)a}. Recall that Λ is the set of all pairs of partitions (λ , μ ) such that |λ | + | μ | = n; see Example 1.3.9. As in the proof of Theorem 2.8.15, one sees that, for any (λ , μ ) ∈ Λ and s, t ∈ M(λ , μ ), we have (λ , μ )
Cs,t
= ±Cw ,
where
w = w(λ ,μ ) (s, t).
However, property (♠) is not known in general, so we cannot conclude that the above elements form a cellular basis of H. Complete results are available in the asymptotic case, where b > (n − 1)a > 0; denote by Lasy the weight function in this case. Then P1–P15 hold for W, Lasy by the series of papers by Bonnaf´e, Geck, and Iancu [21], [26], [108], [114], [121]. Furthermore, as already mentioned in Example 2.2.17, we have (λ , μ ) Lasy (λ , μ )
⇔
(λ , μ ) (λ , μ ).
Arguing as in 2.8.17, we obtain the following result originally due to Bonnaf´e and Iancu [26, Prop. 7.9]: for any left Kazhdan–Lusztig cell C of W (with respect to Lasy ), we have [C]1 ∈ IrrK (W )
and
[C]1 ∼ = E λ ⇔ C ⊆ F(λ ,μ ) .
Furthermore, for (λ , μ ) ∈ Λ , the balanced representation ρ (λ ,μ ) can be chosen such that W (λ , μ ) = [C(λ ,μ ) ]A , where C(λ ,μ ) ⊆ F(λ ,μ ) is a fixed left Kazhdan–Lusztig cell. A completely different construction of a cellular basis is due to Dipper, James and Murphy [68]; but, by [124], the above cell modules in the asymptotic case are naturally isomorphic to the “Specht modules” of [68]. See also Chlouveraki, Gordon and Griffeth [49] for further realisations of these modules. The construction of [68] has been further generalised to Ariki–Koike algebras; see Dipper, James and Mathas [67] (and also Graham and Lehrer [144, §5] for a slightly different approach). We will describe these results on Ariki–Koike algebras in Section 5.3.
Chapter 3
Specialisations and Decomposition Maps
In Chapter 2 we constructed a “cell datum” for the Iwahori–Hecke algebra Hk = Hk (W, S, {ξs }) associated with a finite Coxeter group W . This yields a natural parametrisation of Irr(Hk ) in terms of a certain subset Λk◦ ⊆ Λ , where Λ is an indexing set for Irr(W ). The main theme of this chapter is that Λk◦ can be characterised in an entirely different way, using Brauer’s classical theory of decomposition numbers. This alternative approach is important for (at least) three reasons: • It applies in more general situations, e.g. not only for Iwahori–Hecke algebras associated with finite Coxeter groups, but also for Ariki–Koike algebras. • It does not rely on the validity of the deep properties (♠) and (♣) in 2.5.3. For instance, we will be able to determine a subset Λk◦ as above for type Bn with unequal parameters, where (♠) and (♣) are not yet known to hold in general. • Brauer’s decomposition numbers are independent of any “cellular structure”, and efficient geometric and algorithmic methods are available for their computation. After discussing some foundations in Section 3.1, we define in Section 3.2 a key concept for this book, the so-called “canonical basic sets”. This may be seen as an axiomatic version of those features of a “cellular algebra” which do not explicitly refer to an underlying “cellular basis”. We then have to deal with the various types of specialisations of our ground ring A (which typically is a ring of Laurent polynomials with integer coefficients). A primary role will be played by those specialisations which arise from height 1 prime ideals. This leads us to consider so-called “principal specialisations” in Section 3.3. In this setting, one can show that canonical basic sets for so-called “blocks of defect 1” have a particularly simple description. In Section 3.4, we develop a general strategy for determining canonical basic sets for algebras of classical type. A model case is given by W ∼ = Sn , and this will be treated in detail in Section 3.5. In Section 3.6 we prove a basic factorisation result which shows that any decomposition map always passes through a “principal specialisation”. James’s conjecture and its generalisations predict that “nothing happens” in the second step under certain precise conditions; see Section 3.7. Finally, Section 3.8 is devoted to the proof of a remarkable result which characterises the partition of IrrK (W ) by Kazhdan–Lusztig cells in terms of certain “blocks”. M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7 3, © Springer-Verlag London Limited 2011
133
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3 Specialisations and Decomposition Maps
3.1 Grothendieck Groups and Decomposition Maps We begin with a brief discussion of decomposition maps in a general setting appropriate for our applications to Iwahori–Hecke algebras, following [132, Chap. 7]. 3.1.1. Let A be an integral domain and H be an associative A-algebra (with identity 1H ) which is finitely generated and free over A. Let K be the field of fractions of A and θ : A → k be a ring homomorphism into a field k such that k is the field of fractions of θ (A). By extension of scalars, we obtain a K-algebra H K := K ⊗A H (via the inclusion A ⊆ K) and a k-algebra H k := k ⊗A H (where k is regarded as an A-module via θ ). We wish to relate the representations of H k with those of H K . Denote by R0 (H k ) the Grothendieck group of finite-dimensional representations of H k . Recall that R0 (H k ) is the abelian group generated by expressions [V ], one for each finite-dimensional H k -module V (up to isomorphism), with relations [V ] = [V ] + [V ] for each short exact sequence {0} → V → V → V → {0} of H k modules. Two H k -modules V,V give rise to the same element in R0 (H k ) if and only if V,V have the same composition factors, counting multiplicities. It follows that R0 (H k ) is free abelian, with basis given by Irr(H k ), the set of simple H k -modules (up to isomorphism). Let R+ 0 (H k ) be the submonoid of R0 (H k ) consisting of elements [V ] where V is an H k -module. Let X be an indeterminate over k and Maps(H, k[X]) be the k-algebra of maps from H to k[X]. Following [132, §7.4], [134], we define x k : R+ 0 (H k ) → Maps(H, k[X]) by sending [V ] ∈ R+ 0 (H k ) to the map which assigns to each h ∈ H the characteristic polynomial of 1⊗h ∈ H k in its action on V . Considering Maps(H, k[X]) as a monoid with respect to multiplication, the map xk is a monoid homomorphism. These definitions apply, in particular, to the inclusion A ⊆ K. Hence, we also have a monoid homomorphism xK : R+ 0 (H K ) → Maps(H, K[X]). Theorem 3.1.2 (Cf. [98, §2], [132, 7.4.3], [134, 2.11]). In the setting of 3.1.1, assume that A is integrally closed in K and that H k is split. Then the following hold: (a) The image of xK : R+ 0 (H K ) → Maps(H, K[X]) lies in Maps(H, A[X]). Thus, by restriction, we obtain a monoid homomorphism xA : R+ 0 (H K ) → Maps(H, A[X]). (H ) → R+ (b) There exists a unique additive map dθ+ : R+ K 0 0 (H k ) such that the following diagram is commutative: xA - Maps(H, A[X]) R+ (H ) 0
dθ+
K
? (H R+ k) 0
xk
tθ ? - Maps(H, k[X])
where tθ : Maps(H, A[X]) → Maps(H, k[X]) is the natural map obtained by applying θ to the coefficients of the polynomials in A[X]. (c) We have dθ ([K ⊗A N]) = [k ⊗A N] for any H-module N which is finitely generated and free over A.
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135
The assumption that A is integrally closed in K is used in the proof of Theorem 3.1.2(a). The assumption that H k is split implies that xk is injective (a result originally due to Brauer and Nesbitt; see [132, 7.3.2, 7.3.3]). This is needed to establish the uniqueness part in Theorem 3.1.2(b). For further details, see [132, §7.4]. Example 3.1.3. Assume that A is a discrete valuation ring; that is, A is a noetherian local ring of positive dimension whose maximal ideal is principal; see [248, §11]. Let k be the residue field and θ : A → k be the canonical map. This is the usual setting in the representation theory of finite groups, where H = AG is the group algebra of a finite group G; e.g. see Curtis and Reiner [53, §16] and Feit [83, §I.17]. Note that a valuation ring is always integrally closed in its field of fractions. Furthermore, since A is a principal ideal domain, a standard argument (see, for example, [132, 7.3.7]) shows that, for every finite-dimensional H K -module V , there exists an H-module N, which is finitely generated and free over A, such that V ∼ = K ⊗A N. See Du et al. [73, §1] for a more general setting where A is a regular local ring of Krull dimension at most 2. Definition 3.1.4. Let dθ be the canonical extension of dθ+ to a group homomorphism dθ : R0 (H K ) → R0 (H k ). Then dθ is called the decomposition map associated with the specialisation θ : A → k. It is determined by the equation dθ ([V ]) =
∑
dV,M [M]
V ∈ Irr(H K ),
for any
M∈Irr(H k )
where each dV,M is a non-negative integer. The matrix Dθ = dV,M V ∈Irr(H ), M∈Irr(H K
k)
is called the decomposition matrix associated to θ . Remark 3.1.5. Assume that H k is split and that kˆ ⊇ k is a field extension. Then H kˆ is split and we have a bijection (see [132, 7.3.4]): (a)
Irr(H k ) → Irr(H kˆ ),
M → kˆ ⊗k M.
So there is no serious loss of generality involved in our standing assumption that k should be the field of fractions of θ (A). More generally, assume that A is contained in a bigger integral domain Aˆ and θ ˆ extends to a homomorphism θˆ : Aˆ → kˆ such that kˆ is the field of fractions of θˆ (A) ˆ Let Kˆ be the field of fractions of and θˆ |A = ι ◦ θ for a suitable embedding ι : k → k. ˆ Assume that H k is split and that either K = Kˆ or H K is split. Also assume that A is A. ˆ Then, formally, we have two integrally closed in K and Aˆ is integrally closed in K. decomposition maps dθ : R0 (H K ) → R0 (H k ) and dθˆ : R0 (H Kˆ ) → R0 (H kˆ ). By (a), the extensions of scalars from K to Kˆ and from k to kˆ (via ι ) give rise to bijections Irr(H K ) −→ Irr(H Kˆ ),
1−1
V → Vˆ := Kˆ ⊗K V ;
1−1
M → Mˆ := kˆ ⊗k M.
Irr(H k ) −→ Irr(H kˆ ),
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3 Specialisations and Decomposition Maps
By the characterisation of decomposition maps in Theorem 3.1.2(b), we have (b)
dV,M = dˆVˆ ,Mˆ
for all V ∈ Irr(H K ) and M ∈ Irr(H k ),
where Dθ = dV,M is the decomposition matrix associated with θ and Dθˆ = dˆVˆ ,Mˆ is the decomposition matrix associated with θˆ . Thus, the decomposition numbers of H remain unchanged under “controlled” extensions of θ . Example 3.1.6. Let us keep the assumptions of Theorem 3.1.2. There exists a valuation ring Aˆ ⊆ K such that A ⊆ Aˆ and A ∩ p = ker(θ ), where p is the unique maximal ˆ (See, for example, [138, §5]; note that, in general, Aˆ will not be a discrete ideal of A. ˆ Then θ : A → k extends to a homovaluation ring.) Let kˆ be the residue field of A. morphism θˆ : Aˆ → kˆ and there is an inclusion ι : k → kˆ such that θˆ |A = ι ◦ θ . Thus, we are in the setting of Remark 3.1.5 where Kˆ = K; note that any valuation ring is integrally closed in its field of fractions. Now let V ∈ Irr(H K ) and M ∈ Irr(H k ). Since Aˆ is a valuation ring, we can find an ˆ ˆ such that K ⊗ ˆ N ∼ (A ⊗A H)-module N, which is finitely generated and free over A, A = V (see [138, 5.4]). Then, by Theorem 3.1.2(c) and 3.1.5(b), we have (a)
dV,M = dˆVˆ ,Mˆ = multiplicity of Mˆ as a composition factor of kˆ ⊗Aˆ N.
If, for example, A is a Dedekind domain, then the localisation of A in ker(θ ) will be a discrete valuation ring, in which case we can take Aˆ to be that localisation. In our further study of Iwahori–Hecke algebras, this will not always be the case. 3.1.7. In a number of interesting situations, the decomposition matrix Dθ turns out to have a triangular shape with 1 on the diagonal. We formalise this as follows. In the setting of Theorem 3.1.2, assume that we are given a partial order on Irr(H K ) such that the following three conditions are satisfied: • Given M ∈ Irr(H k ), let Sθ (M) := {V ∈ Irr(M K ) | dV,M = 0}. Then the set Sθ (M) contains a unique minimal element (with respect to ), which we denote by VM . • The map Irr(H k ) → Irr(H K ), M → VM , is injective. • We have dVM ,M = 1 for all M ∈ Irr(H k ). If this holds, let Bθ := {VM | M ∈ Irr(H k )}. Thus, we obtain a bijection 1−1
Irr(H k ) ←→ Bθ ⊆ Irr(H K ), and Bθ will be called a canonical basic set for H k (with respect to ). Note that, if a canonical basic set exists, then it is uniquely determined by the above conditions. In this situation, let us choose a labelling Bθ = {V1 , . . . ,Vr } such that “Vi V j ⇒ i < j”. Then Irr(H k ) = {M1 , . . . , Mr }, where Vi = VMi for 1 i r. Hence, the matrix D◦θ := dVi ,M j )1i, jr will have a square lower triangular shape with 1 on the diagonal.
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137
3.1.8. In the setting of Theorem 3.1.2, the Brauer graph of H (with respect to θ ) is defined as follows: it has vertices labelled by Irr(H K ) and there is an edge joining two vertices, labelled by V = V say, if there exists some M ∈ Irr(H k ) such that dV,M = 0 and dV ,M = 0. Considering the connected components of this graph, we obtain a partition of Irr(H K ) into pieces called the θ -blocks of Irr(H K ). Let Irr(H K ) = Irr1 (H K ) Irr2 (H K ) · · · Irrr (H K ) be the partition into θ -blocks. For each i, let Irri (H k ) be the set of all M ∈ Irr(H k ) such that dV,M = 0 for some V ∈ Irri (H K ). Then we also have a partition Irr(H k ) = Irr1 (H k ) Irr2 (H k ) · · · Irrr (H k ). Ordering the rows and columns of Dθ accordingly, we obtain a block shape ⎛ ⎞ Dθ ,1 0 . . . 0 ⎜ . ⎟ ⎜ 0 Dθ ,2 . . . .. ⎟ ⎜ ⎟, Dθ = ⎜ . ⎟ . . . . . ⎝ . . . 0 ⎠ 0 . . . 0 Dθ ,r where Dθ ,i has rows and columns labelled by the elements of Irri (H K ) and Irri (H k ) respectively. Thus, in order to determine Dθ , we can proceed block by block. Of course, this remark will only be useful if there is more than one block and if there are alternative characterisations of blocks which do not rely on using dθ . In Lemma 3.1.10, we provide a basic criterion in term of central characters. (In Section 3.8, we shall discuss further characterisations in terms of central idempotents.) Definition 3.1.9. Assume that A is integrally closed in K and H K is split semisimple. Denote by Z(H) the centre of H. Let V ∈ Irr(H K ) and z ∈ Z(H). Then z also lies in the centre of H K . Hence, by Schur’s lemma, z acts as multiplication by a scalar on V , which we denote by ωV (z). By [132, 7.3.8], we have ω V (z) ∈ A. Thus, we obtain an A-algebra homomorphism ω V : Z(H) → A, called the central character of V . Lemma 3.1.10. In the setting of Theorem 3.1.2, assume that H K is split semisimple and let V,V ∈ Irr(H K ). If V,V belong to the same θ -block, then θ (ω V (z)) = θ (ω V (z)) for all z ∈ Z(H). (The converse holds under additional assumptions on A; see Example 3.8.5.) Proof. It is enough to consider the case where V,V are directly linked in the Brauer graph of H; that is, we have dV,M = 0 and dV ,M = 0 for some M ∈ Irr(H k ). Then, using the set-up and the notation in Example 3.1.6, we have dV,M = multiplicity of Mˆ as a composition factor of N¯ := kˆ ⊗Aˆ N, dV ,M = multiplicity of Mˆ as a composition factor of N¯ := kˆ ⊗ ˆ N , A
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3 Specialisations and Decomposition Maps
ˆ such that where N, N are (Aˆ ⊗A H)-modules (finitely generated and free over A) ∼ ∼ K ⊗Aˆ N = V and K ⊗Aˆ N = V . Now let z ∈ Z(H). Then z acts as ω V (z) times the identity on V and, hence, also as ω V (z) times the identity on N; recall that ω V (z) ∈ A. Consequently, the central element z¯ ∈ H kˆ (obtained by extending scalars ¯ Similarly, z¯ acts from A to kˆ via θˆ |A ) acts as θ (ω V (z)) times the identity on N. ¯ as θ (ω V (z)) times the identity on N . The condition dV,M = 0 implies that Mˆ is ˆ ¯ So z¯ must act as θ (ω V (z)) times the identity on M. a composition factor of N. Similarly, the condition dV ,M = 0 implies that z¯ acts as θ (ω V (z)) times the identity ˆ Thus, we conclude that θ (ω V (z)) = θ (ω V (z)), as required. on M. Example 3.1.11. Let us keep the assumptions of Lemma 3.1.10. Then, by the Artin– Wedderburn theorem, the split semisimple algebra H K is a direct sum of simple two-sided ideals where the direct factors correspond to the various irreducible representations of H K (see [53, §3B]). Consequently, we can write uniquely 1H K =
∑
eV ,
V ∈Irr(H K )
where each eV acts as the identity on V and as zero on any simple module not isomorphic to V ; furthermore, eV is an idempotent (that is, we have eV2 = eV ) in the centre of H K . Assume now that V ∈ Irr(H K ) satisfies the following condition: (∗)
α eV ∈ H
for some α ∈ A such that θ (α ) = 0.
Then we claim that {V } is a θ -block of Irr(H K ). Indeed, assume that V ∈ Irr(H K ) belongs to the same θ -block as V , where V ∼ = V . Let z := α eV ∈ Z(H). We have ω V (z) = α and ω V (z) = αω V (eV ) = 0. This implies that θ (ω V (z)) = θ (α ) = 0 and θ (ω V (z)) = 0, contradicting Lemma 3.1.10. 3.1.12. Recall the basic set-up from Section 1.2. Let W be a finite Coxeter group with generating set S. Let L : W → Γ be a weight function where Γ admits a monomial order such that L(s) 0 for all s ∈ S. Let R ⊆ C be a subring such that ZW ⊆ R; let K be the field of fractions of R. We shall now also assume that • the ring R is noetherian and integrally closed in K; • the group Γ is finitely generated. As discussed in 1.1.10, these conditions imply that A := R[Γ ] is the ring of Laurent polynomials in finitely many variables over R; in particular, A is noetherian. Let H = HA (W, S, L) be the corresponding generic Iwahori–Hecke algebra with parameters {vs | s ∈ S}, where vs = ε L(s) for s ∈ S. Let K be the field of fractions of R and K be the field of fractions of A. Also recall from 1.2.1 that HK = K ⊗A H is a split semisimple algebra which is abstractly isomorphic to the group algebra of W over K. We have Irr(HK ) = {Eελ | λ ∈ Λ }, where Λ is an indexing set for IrrK (W ). Let θ : A → k be a ring homomorphism, where k is a field such that k is the field of fractions of θ (A). We denote the corresponding specialised algebra by Hk , or by Hk,θ if it is necessary to specify θ . We can now use our results on “cell data” from Chapter 2 to address the validity of the assumptions in Theorem 3.1.2.
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139
Table 3.1 Conditions for R to be L0 -good (extracted from Table 1.4, p. 33) Type An−1 , Bn G2 , Dn F4 , E6 , E7 E8 I2 (m) (m5,m=6)
H3 H4
Condition on R no condition 2 ∈ R× 2, 3 ∈ R× 2, 3, 5 ∈ R× m ∈ R× and 2 cos(2π /m) ∈ R √ 2, 5 ∈ R× and 2 cos(2π /5) = 12 (−1 + √5) ∈ R 2, 3, 5 ∈ R× and 2 cos(2π /5) = 12 (−1 + 5) ∈ R
Lemma 3.1.13. In the above setting, assume that R is L0 -good (see Table 3.1) where L0 is the universal weight function in Example 1.1.9. Then the following hold. (a) The algebra Hk is split. (b) For every finite-dimensional HK -module V , there exists some H-module N which is free of finite rank over A such that V ∼ = K ⊗A N. Proof. Since R is L0 -good, so is the ring A. Consequently, by Corollary 2.6.16, the λ } with respect to some partial order on Λ . algebra H admits a cellular basis {Cs,t By 2.7.5, the algebra Hk also admits a cellular basis. So Theorem 2.6.5 shows that Hk is split. Thus, (a) is proved. Furthermore, since HK is semisimple, it is enough to prove (b) assuming that V is irreducible. Let {N λ | λ ∈ Λ } be the cell modules arising from the cell datum for H. As remarked in 2.7.5(a), we have Irr(HK ) = {K ⊗A N λ | λ ∈ Λ }. Thus, we have V ∼ = K ⊗A N λ for some λ . (It may be true that Hk is always split, even if R is not L0 -good. However, at the moment, we do not see a general argument for proving this.) Theorem 3.1.14. Let H = HA (W, S, L) be a generic Iwahori–Hecke algebra as in 3.1.12; also assume that R is L0 -good. Then, given any specialisation θ : A → k (where k is the field of fractions of θ (A)), the assumptions of Theorem 3.1.2 are satisfied and so we have a well-defined decomposition map dθ : R0 (HK ) → R0 (Hk ) such that dθ ([K ⊗A N]) = [k ⊗A N] for any H-module N which is free of finite rank over A. Furthermore, dθ is surjective. We shall now write the equations determining dθ in the form: dθ ([Eελ ]) =
∑
dλ ,M [M],
where
dλ ,M ∈ Z0 .
M∈Irr(Hk )
Proof. By Lemma 3.1.13(a), Hk is split. Furthermore, since R is integrally closed in K, the ring A (which is a ring of Laurent polynomials in finitely many variables) will be integrally closed in K; see, for example, [132, Exercise 7.2]. Thus, we can apply Theorem 3.1.2, which yields the existence of dθ .
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3 Specialisations and Decomposition Maps
It remains to prove that dθ is surjective. We use again the fact that H admits a cell datum; see the proof of Lemma 3.1.13. Let {N λ | λ ∈ Λ } be the corresponding cell modules. Since HK is semisimple, we have Irr(HK ) = {NKλ := K ⊗A N λ | λ ∈ Λ }; see 2.7.5. Furthermore, we have an induced cell datum for Hk and so Irr(Hk ) = μ {Lk | μ ∈ Λk◦ }, where Λk◦ is a certain subset of Λ . By the definition of dθ , we have dθ ([NKλ ]) = [Nkλ ], where we write Nkλ := k ⊗A N λ . Thus, the entries of the decompoμ sition matrix Dθ are given by the multiplicities (Nkλ : Lk ) defined in Theorem 2.6.6. The properties of these multiplicities then show that the square matrix with entries μ (Nkλ : Lk ) (where λ , μ ∈ Λk◦ ) has a triangular shape with 1 on the diagonal (for a suitable ordering of the rows and columns). Consequently, dθ is surjective. Example 3.1.15. Without going into too much detail at this point, we mention three situations where decomposition maps and blocks of H have interesting applications. In all three cases, assume that R is L0 -good and Γ = Z. Thus, A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . (a) Let θ : A → k be such that char(k) = p > 0 and θ (v) = ±1. Then Hk is the group algebra of W over k. We will see in Example 3.6.5 that dθ essentially equals the usual p-modular decomposition map of W in the theory of finite groups. √ (b) Let e 1 and ζ2e := exp(π −1/e) ∈ C, the “canonical” primitive root of unity of order 2e. Let k = K[ζ2e ] and consider the specialisation θe : A → k such that θe (v) = ζ2e and θe (r) = r for all r ∈ R. The corresponding decomposition map will be called the Φe -modular decomposition map. The study of dθe is one of the central themes of this book; see Chapter 4 for applications. (c) Let W be of type Bn ; then R := ZW = Z is L0 -good. Assume that L(s) = 1 for all s ∈ S. (Note that R is not L-good.) By reduction modulo 2, we obtain a specialisation θ : A → k = F2 (v). Here is the matrix Dθ for type B2 = I2 (4): Eλ 1W sgn1 σ1 sgn2 sgn
1 0 0 0 0
dλ ,M 0 0 1 0 1 1 0 1 0 0
0 0 0 0 1
θ -blocks: {1W }, {sgn1 , σ1 , sgn2 }, {sgn}.
We notice that the θ -blocks coincide with the sets IrrK (W | F ) computed in Example 1.7.3! That this is true for any n has been shown by Gyoja [151]. So it seems that there are close relations between decomposition maps and “cells”. We will discuss this in more detail in Section 3.8. Example 3.1.16. By 3.1.8, we have a partition of Irr(HK ) into θ -blocks. Since Irr(HK ) is indexed by Λ , we can also interpret this as a partition of Λ into θ blocks. Now recall that there is a unique A-algebra automorphism † : H → H such that Ts† = −Ts−1 for all s ∈ S; see 1.2.6. We now show that, in the setting of Theorem 3.1.14, † induces a symmetry of the Brauer graph of H. This is seen as follows. By extension of scalars, † induces a K-algebra automorphism of HK and a k-algebra automorphism of Hk , which we denote by the same symbol. As in 1.2.6, given any
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141
finite-dimensional HK -module V , denote by V † the HK -module with the same underlying vector space V , but where h ∈ HK acts via h† . A similar operation is defined for finite-dimensional Hk -modules. These operations induce maps †∗K : R0 (HK ) → R0 (HK )
and
†∗k : R0 (Hk ) → R0 (Hk ).
Then we have the following commutative diagram: dθ - R+ (Hk ) R+ 0 (HK ) 0
†∗K
? (H R+ K) 0
dθ
†∗k ? - R+ (Hk ) 0
(Indeed, note that xK is compatible with †∗K , in the sense that xK ([V ])(h† ) = xK ([V † ])(h) for any [V ] ∈ R+ 0 (HK ) and h ∈ H. A similar remark applies to xk . Then the characterisation of dθ in Theorem 3.1.2(b) yields the commutativity of the above diagram.) Thus, † induces a symmetry of the Brauer graph of H: two vertices labelled by λ = μ are connected by an edge if and only if the vertices labelled by λ † , μ † are connected (where λ → λ † is defined in 1.2.6). In particular, the map λ → λ † induces a permutation of the θ -blocks of Λ . In Chapter 1 we have seen that H carries a symmetrising trace τ : H → A. In particular, we have a polynomial cλ ∈ A for each λ ∈ Λ . The following result allows us to easily find “trivial” blocks. (The notion “defect 0” will become clear later.) Proposition 3.1.17 (Blocks of defect 0; see [132, 7.5.11]). In the setting of Theorem 3.1.14, let λ ∈ Λ be such that θ (cλ ) = 0. Then {Eελ } is a θ -block, and the corresponding decomposition matrix is the (1 × 1)-matrix with entry 1. Proof. Since HK is a symmetric algebra, we have the following explicit formula for the central idempotent eλ ∈ HK corresponding to Eελ (see [132, 7.2.7]): eλ = c−1 λ
∑ trace(Tw , Eελ ) Tw−1 .
w∈W
Now, since θ (cλ ) = 0, the condition (∗) in Example 3.1.11 is satisfied. Hence, {Eελ } is a θ -block. The statement about the decomposition matrix is then clear, since dθ is known to be surjective (see Theorem 3.1.14). 3.1.18. In the setting of Theorem 3.1.14, a general semisimplicity criterion for symmetric algebras (see [132, 7.4.7]) shows that we have the following equivalence: Hk is semisimple
⇔
Now recall from Remark 1.3.6 that P˜W,L := ∏ cλ ∈ Z[u±1 s | s ∈ S] λ ∈Λ
θ (cλ ) = 0 for all λ ∈ Λ .
(where us := v2s for s ∈ S).
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3 Specialisations and Decomposition Maps
Hence, Hk is semisimple if and only if θ (P˜W,L ) = 0. Assume now that Hk is semisimple. Then Proposition 3.1.17 applies to all λ ∈ Λ and so, up to reordering the columns, Dθ is the identity matrix. In particular, we obtain natural bijections Irr(Hk )
1−1
←→
Irr(HK )
1−1
←→
IrrK (W ),
where the first is induced by dθ and the second is given by Eελ ↔ E λ . (This is one version of Tits’s deformation theorem which we already saw in Section 1.2.)
Table 3.2 Degrees of finite Coxeter groups Type An−1 Bn Dn I2 (m) H3
degrees d j 2, 3, 4, . . . , n 2, 4, 6, . . . , 2n 2, 4, 6, . . . , 2(n − 1), n 2, m 2, 6, 10
Type H4 F4 E6 E7 E8
degrees d j 2, 12, 20, 30 2, 6, 8, 12 2, 5, 6, 8, 9, 12 2, 6, 8, 10, 12, 14, 18 2, 8, 12, 14, 18, 20, 24, 30
Example 3.1.19. In the setting of Theorem 3.1.14, assume that Γ = Z. Then A = R[v, v−1 ] is the ring of Laurent polynomials in one variable v = ε . By Remark 1.3.6, each cλ ∈ A is a Laurent polynomial in v all of whose (complex) roots are roots of unity. So, given θ : A → k, we have Hk not semisimple
⇒
θ (v) ∈ k is a root of unity or R is not L-good.
In particular, if R is L-good, then Hk is semisimple unless θ (v) is a root of unity. Now assume that we are in the equal parameter case where L(s) = 1 for all s ∈ S. By a classical result due to Solomon (see, for example, [44, §9.4]), we have PW :=
∑ ul(w) = ∏
w∈W
(1 + u + u2 + · · · + ud j −1 )
(u := v2 ),
1 j|S|
where d1 , . . . , d|S| are the so-called degrees of W ; we have |W | = d1 · · · d|S| . The degrees for all types of W are known (see [29]) and given in Table 3.2. Note that PW = c1W where 1W denotes the unit representation of W . It is known that, for all λ ∈ Λ , we have cλ ∈ R[u, u−1 ] and cλ divides PW in K[u, u−1 ]; see [132, 9.3.6] (if W is of crystallographic type) and [132, 8.3.4 and App. E] (otherwise). Thus, assuming that R is L-good, we conclude that Hk is semisimple
⇔
θ (PW ) = 0.
Example 3.1.20. Let W be of type Bn with generators S = {s0 , . . . , sn−1 } labelled as in Table 1.1 (p. 2). Let L0 : W → Γ0 be the “universal” weight function, where
3.2 Canonical Basic Sets
143
Γ0 = Z2 . Then R = ZW = Z is L0 -good and A = Z[V ±1 , v±1 ] is the ring of Laurent polynomials in two independent indeterminates V := ε (1,0) and v := ε (0,1) ; see Example 1.1.11(b). Thus, in H, we have the quadratic relations Ts20 = T1 + (V −V −1 )Ts0
and
Ts2i = T1 + (v − v−1 )Tsi
(i = 1, . . . , n − 1).
In this case, the analogue of the above polynomial PW is given by PW,L :=
∑ ε L(w) = ∏
w∈W
(1 + u j−1U)
1 jn
∏
(1 + u + u2 + · · · + ui−1 ),
1in
where U := V 2 and u := v2 ; see [132, 10.5.1]. Hoefsmit [157] obtained explicit formulae for all cλ . (See also Lusztig [212, 9.6] and Iancu [169] for alternative proofs.) We still have cλ ∈ Z[U ±1 , u±1 ], but it is no longer true that each cλ divides PW,L . In fact, Hoefsmit’s formulae show that each cλ is a product of terms of the form U + u±r (where 0 r n − 1) and Φd (u) (where 2 d n); here, Φd (v) ∈ Z[u] denotes the dth cyclotomic polynomial. Furthermore, each of these terms does occur in some cλ . Hence, we conclude that Hk is semisimple
⇔
θ (PW,L ) = 0 and
∏
θ (U + u j−1 ) = 0.
1 jn
This semisimplicity criterion for type Bn with general parameters was first established by Dipper and James [66, Theorem 5.5]. For similar descriptions in types I2 (m) and F4 with unequal parameters, see Sections 7.2 and 7.3. Hence, in all cases, we have an efficient criterion to determine whether a specialised algebra Hk is semisimple or not.
3.2 Canonical Basic Sets We consider a generic Iwahori–Hecke algebra H = HA (W, S, L) and a specialisation θ : A → k such that the conditions in Theorem 3.1.14 are satisfied and, hence, we have a corresponding decomposition map dθ : R0 (HK ) → R0 (Hk ). The following definition is fundamental to the whole discussion in the subsequent chapters of this book. It provides the key technical notion which will allow us to obtain a new approach to the classification of Irr(Hk ), via dθ : R0 (HK ) → R0 (Hk ). In Proposition 3.2.7 below, we will reconcile this approach with that in Chapter 2. Definition 3.2.1 (Geck and Rouquier [134], Geck [110, Def. 4.13]). Consider the decompositon matrix Dθ = (dλ ,M ) associated with θ : A → k; see Theorem 3.1.14. Recall that is a monomial order on Γ such that L(s) 0 for all s ∈ S. By Section 1.3, we have a function Irr(W ) → Γ0 , E λ → aλ . Assume that the three conditions in 3.1.7 are satisfied for the partial order on Irr(HK ) defined by
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3 Specialisations and Decomposition Maps
μ
Eελ Eε
def
λ =μ
⇔
or
aλ < a μ .
Explicitly, in the present context, this means: • Given M ∈ Irr(Hk ), let Sθ (M) := {λ ∈ Λ | dλ ,M = 0}. Then the function Sθ (M) → Γ , λ → aλ , reaches its minimum (with respect to ) at exactly one element of Sθ (M), which we denote by λM . • The map Irr(Hk ) → Λ , M → λM , is injective. • We have dλM ,M = 1 for all M ∈ Irr(Hk ). If this holds, let Bθ := {λM | M ∈ Irr(Hk )}. Thus, we obtain a bijection 1−1
Irr(Hk ) ←→ Bθ ⊆ Λ , μ
and Bθ = {Eε | μ ∈ Bθ } is called a canonical basic set for Hk ; see 3.1.7. We shall also call Bθ ⊆ Λ itself a canonical basic set. Note again that if a canonical basic set exists, then it is uniquely determined by the above conditions.
Remark 3.2.2. The above definition is an axiomatic version (without reference to any underlying “cell datum”) of the properties in Theorem 2.6.6. Indeed, assume that we have a canonical basic set Bθ as above. Then there is a unique labelling Irr(Hk ) = {M μ | μ ∈ Bθ }
where
λM μ = μ
( μ ∈ Bθ ).
Given λ ∈ Λ and μ ∈ Bθ , denote the decomposition number dλ ,M μ by dλ ,μ . Then the conditions in Definition 3.2.1 translate to: (Δ a )
dμ ,μ = 1 dλ , μ = 0
for all μ ∈ Bθ , unless λ = μ or aλ > aμ .
Thus, the decomposition matrix Dθ = dλ ,μ )λ ∈Λ , μ ∈Bθ has a lower unitriangular shape, if the rows and columns are ordered according to increasing values of aλ . Remark 3.2.3. Suppose that, in some way, the decomposition matrix Dθ has been determined. (For example, by using the algorithmic methods described in Section 7.1.) The rows of Dθ are labelled by the set Λ . Let r be the number of columns of Dθ . Then Irr(Hk ) contains r simple modules, which we denote by M1 , . . . , Mr . Assume that we are also given a monomial order on Γ such that L(s) 0 for all s ∈ S. Then a canonical basic set (if it exists) is found as follows. Consider the invariants aλ and order the rows of Dθ according to increasing value of aλ . For each index i ∈ {1, . . . , r}, find the first row in Dθ (from top to bottom) which has a non-zero entry in the ith column; let λi ∈ Λ be the label of that row. Performing this procedure for all i, we obtain a subset Bθ := {λ1 , . . . , λr } ⊆ Λ . If there exists some i ∈ {1, . . . , r} and some λ ∈ Λ such that λ = λi , aλ = aλi and dλ ,Mi = 0, then the first condition in Definition 3.2.1 is not satisfied. Similarly, if
3.2 Canonical Basic Sets
145
λi = λ j for some i = j, then the second condition is not satisfied. Finally, if dλi ,Mi = 1 for some i, then the third condition is not satisfied. If none of the above happens, then Bθ is a canonical basic set for Hk . Thus, if Dθ is known, the question of showing the existence of a canonical basic set, and of explicitly determining it, is solved by a purely mechanical procedure. Example 3.2.4. Consider the partitions of Irr(HK ) and Irr(Hk ) into θ -blocks as in 3.1.8. Correspondingly, we have a partition Λ = Λ1 · · · Λr , where Λi labels the irreducible representations in the θ -block Irri (HK ). Then it is clear that, for a fixed i, the three conditions in Definition 3.2.1 can be formulated with respect to Λi and Irri (Hk ). Just note that if M ∈ Irri (Hk ), then the set Sθ (M) is automatically contained in Λi , by the definition of blocks. Hence, a canonical basic set exists for H if and only if each θ -block admits a canonical basic set. Furthermore, in this case, the canonical basic set for H is the union of the canonical basic sets for all θ -blocks. Assume, for example, that λ ∈ Λ is such that θ (cλ ) = 0. By Proposition 3.1.17, the set {Eελ } is a θ -block whose decomposition matrix is the (1 × 1)-matrix with entry 1. Consequently, λ belongs to the canonical basic set for H (if it exists). Example 3.2.5. Let W be of type I2 (m) (m 3), with generators s1 , s2 such that (s1 s2 )m = 1. Let ζ ∈ C be a root of unity of order m such that ζ + ζ −1 = 2 cos(2π /m). Let A = R[Γ ], where m ∈ R× and ζ + ζ −1 ∈ R ⊆ C. (Thus, R is Lgood.) The irreducible representations of HK are described in Example 1.3.7. Let θ : A → k be a specialisation where k is the field of fractions of θ (A). In the following, we assume that ξ := θ (vs1 ) = θ (vs2 ). The corresponding decomposition matrices are determined by M¨uller [253, §3] (see also Section 7.2 in the appendix). It turns out that non-trivial θ -blocks (i.e., blocks which are not singleton sets), only exist if ξ 2 is a root of unity of order e 2, where e divides m. Furthermore: (a) If e = 2, then {1W , sgn, sgn1 , sgn2 } is a θ -block (where sgn1 and sgn2 are omitted if m is odd). (b) If e > 2, then {1W , sgn, σ j(e) } is a θ -block, where the index j(e) is uniquely determined by the condition that ξ 2 = θ (ζ )± j(e) . The decomposition matrices of these blocks are, respectively, given by Eλ • 1W (sgn1 ) (sgn2 ) sgn
dλ ,M 1 1 1 1
Eλ • 1W • σ j(e) sgn
dλ ,M 1 0 1 1 0 1
Now let be a monomial order on Γ such that L(s1 ) > 0 and L(s2 ) > 0. Then each block admits a canonical basic set, which is marked by “•” in the above matrices. (Indeed, the table in Example 1.3.7 shows that, regardless of the exact values of L, we always have a1W < asgn1 aσ j asgn2 < asgn if L(s1 ) L(s2 ) > 0. A similar statement holds if L(s2 ) L(s1 ) > 0, with the roles of sgn1 and sgn2 interchanged. It remains to apply the procedure described in Remark 3.2.3.)
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3 Specialisations and Decomposition Maps
Example 3.2.6. Let W be of type B3 with generators s0 , s1 , s2 labelled as in Table 1.1. Assume that we are in the setting of Example 3.1.20, where A = Z[V ±1 , v±1 ] is the ring of Laurent polynomials in two independent indeterminates V, v and, in H, we have the quadratic relations Ts20 = T1 + (V −V −1 )Ts0
and
Ts2i = T1 + (v − v−1 )Tsi
(i = 1, 2).
√ Consider a specialisation θ : A → k such that θ (V ) = −1 and θ (v) = −1 ∈ k, where char(k) = 2. The matrix Dθ is given as follows (recall that, here, Λ is the set of all pairs of partitions such that the total sum of their parts equals 3): E (λ , μ ) (3, ∅) (21, ∅) (111, ∅) (2, 1) (11, 1) (1, 2) (∅, 3) (1, 11) (∅, 21) (∅, 111)
1 0 1 0 0 1 0 1 0 0
d(λ ,μ ),M 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 1
0 0 0 0 0 1 0 1 1 0
alex (λ , μ ) (0, 0) (0, 1) (0, 3) (1, 0) (1, 1) (2, −1) (2, 2) (3, −3) (3, 1) (3, 6)
The column labelled alex (λ , μ ) contains the invariants a-invariants with respect to the monomial order on Γ0 given by the lexicographic order (see Example 1.1.11(b)). Note that the rows of Dθ are already ordered according to increasing value of alex (λ ,μ ) . Using Remark 3.2.3, we find that a canonical basic set exists and is given by Bθ = { (3, ∅), (21, ∅), (2, 1), (1, 2) }. Note also that, here, we are in the “asymptotic case” and so there is a cellular basis of H as in Theorem 2.6.12, with respect to the dominance order on Λ ; see 2.8.19. Now let us choose a different monomial order: consider the reverse lexicograhic order on Λ (where (0, 1) > (r, 0) > (0, 0) for all r ∈ Z0 ). The corresponding ainvariants are printed in the column labelled arevlex (λ , μ ) in the table below. Hence, reordering the rows of Dθ according to increasing value of arevlex (λ ,μ ) , we obtain E (λ , μ ) (3, ∅) (∅, 3) (2, 1) (1, 2) (21, ∅) (∅, 21) (11, 1) (1, 11) (111, ∅) (∅, 111)
1 0 0 1 0 0 0 1 1 0
d(λ ,μ ),M 0 0 0 1 1 1 0 0 1 0 0 0 1 1 0 0 0 0 0 1
0 0 0 1 0 1 0 1 0 0
arevlex (λ ,μ ) (0, 0) (1, 0) (0, 1) (1, 1) (−1, 2) (2, 2) (0, 3) (1, 3) (−2, 6) (3, 6)
We see that a corresponding canonical basic set exists and is given by
3.2 Canonical Basic Sets
147
Bθ = { (3, ∅), (∅, 3), (1, 2), (2, 1) }. This example shows that, in the general multiparameter case, Bθ heavily depends on the chosen monomial order on Γ . On the other hand, canonical basic sets do not always exist. Consider, for example, the decomposition matrix in Example 3.1.15(c). Since sgn1 , sgn2 and σ1 all have aλ = 1, the procedure in Remark 3.2.3 fails for the second and third columns of that matrix! Note that R is not L-good in this case. A general existence result for canonical basic sets is proved in [110, Theorem 6.6], following earlier work of Geck [97] and Geck and Rouquier [134]. Here, we obtain a new proof as an immediate consequence of the results in Chapter 2. This also shows how the condition that R should be L-good naturally comes into play. Proposition 3.2.7. Assume that R is L-good and that the conditions (♠) and (♣) in 2.5.3 hold. Thus, H admits a cellular basis {Cλs,t } as in Theorem 2.6.12. Via θ : A → k, we obtain a cell datum for Hk . So we have cell modules {Wk (λ ) | λ ∈ Λ } μ for Hk and there is a natural parametrisation Irr(Hk ) = {Lk | μ ∈ Λk◦ }; see 2.7.5. Then the following hold. μ (a) Let M ∈ Irr(Hk ) and μ ∈ Λk◦ be such that M ∼ = Lk . Let λ ∈ Λ . Then we have μ
dλ ,M = (Wk (λ ) : Lk ),
μ
where (Wk (λ ) : Lk ) is defined in Theorem 2.6.6.
(b) The subset Bθ := Λk◦ ⊆ Λ is a canonical basic set with respect to . Thus, using the notation of Remark 3.2.2, we have for all μ ∈ Λk◦ , unless λ = μ or aλ > aμ .
dμ ,μ = 1 dλ ,μ = 0
Furthermore, we have dλ ,μ = 0 unless λ = μ or Nλ < Nμ . Proof. By Proposition 2.7.1, for any λ ∈ Λ , the HK -modules WK (λ ) and Eελ are isomorphic. Hence, we have [Eελ ] = [WK (λ )] in R0 (HK ). Since W (λ ) is an H-module which is free of finite rank over A, we also have dθ ([WK (λ )]) = [Wk (λ )] by Theorem 3.1.14. This yields the formula for dλ ,M in (a). Now consider the conditions in μ Definition 3.2.1. Let M ∈ Irr(Hk ); then M ∼ = Lk for a unique μ ∈ Λk◦ . The relations (Δ ) in Theorem 2.6.6 show that μ ∈ Sθ (M), dμ ,M = 1, and that λ L μ for any λ ∈ Sθ (M). Since, by Theorem 2.6.12, we also have the implication
λ L μ
⇒
λ =μ
or
aλ > a μ ,
it follows that μ is the unique element of Sθ (M) at which the function Sθ (M) → Γ , λ → aλ , reaches its minimum. Thus, we have μ = λM . Consequently, the three conditions in Definition 3.2.1 are satisfied, where Bθ = Λk◦ . The statements in (b) concerning the invariants Nλ then follow as in Remark 2.6.15.
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3 Specialisations and Decomposition Maps
Example 3.2.8. Let us keep the assumptions of Proposition 3.2.7. The algebra H is symmetric (see 1.2.11) and, clearly, we have τ (Tw ) = τ (Tw−1 ) = τ (Tw ) for all w ∈ W . Hence, τ satisfies the additional condition in Proposition 2.6.7. So we have an oppositive cell datum for H; let {Wˆ (λ ) | λ ∈ Λ } be the corresponding cell modules. Since HK is semisimple, we have by Propositions 2.6.7 and 2.7.1: (a)
Wˆ K (λ ) ∼ = WK (λ ) ∼ = Eελ
for all λ ∈ Λ .
μ Furthermore, by Theorem 2.6.5, we have a parametrisation Irr(Hk ) = {Lˆ k | μ ∈ λˆ k◦ }, ◦ where Λˆ k ⊆ Λ is defined with respect to the opposite cell datum. We claim that:
(b)
Λk◦ = {μ † | μ ∈ Λˆ k◦ }
(with μ → μ † as in Example 1.2.6).
This is seen as follows. Since the two sets have the same cardinality, it is enough to prove one inclusion. Let μ ∈ Λˆ k◦ . By Proposition 3.2.7, it will be sufficient to show μ that μ † ∈ Bθ . Let us denote M := (Lˆ k )† ∈ Irr(Hk ). The commutative diagram in Example 3.1.16 shows that dμ † ,M = dμ ,M † = dμ ,Lˆ μ = 1 k
and so
μ † ∈ Sθ (M).
Now let λ ∈ Sθ (M). Using Theorem 2.6.6, Proposition 3.2.7(a) and once more the commutative diagram in Example 3.1.16, we obtain μ (Wˆ k (λ † ) : Lˆ k ) = dλ † ,Lˆ μ = dλ † ,M † = dλ ,M = 0 and so k
λ † L,op μ .
This yields μ L λ † and so λ L μ † (see Remark 2.2.11). Thus, we have aλ aμ † with equality only if λ = μ † . It follows that μ † ∈ Bθ . Thus, (b) is proved. Example 3.2.9. Let (W, S) be of type An−1 . Then W ∼ = Sn and S = {s1 , . . . , sn−1 }, where si corresponds to the basic transposition (i, i + 1) ∈ Sn for all i. We are automatically in the equal parameter case; let Γ = Z and L(si ) = 1 for all i. Now A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . By Example 1.3.8, Λ is the set of all partitions λ n. We have fλ = 1 for all λ n, so any subring R ⊆ C is L-good. By Theorem 2.8.15, the Kazhdan–Lusztig basis {±Cw } gives rise to a cell datum for H. Hence, for any specialisation θ : A → k, we have a natural parametrization μ
Irr(Hk ) = {Lk | μ ∈ Λk◦ }
with Λk◦ defined as in 2.7.5.
By Proposition 3.2.7, the set Λk◦ is a canonical basic set. We have also seen in Theorem 2.8.15 that the order L on Λ is given in terms of the dominance order on partitions. Thus, for λ n and μ ∈ Λk◦ , we have (Δ )
dμ ,μ = 1
and dλ ,μ = 0 unless λ μ .
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149
Similar statements first appeared (with rather different proofs) in the pioneering work of Dipper and James [62]. Note that the above discussion does not yet give us an explicit description of the set Λk◦ ; this problem will be solved in Theorem 3.5.14. Let us just consider the example where W ∼ = S5 is of type A4 and Λ consists of all partitions of 5. Let θ : A → k be a specialisation such that θ (v2 ) = −1 = 1. Then, by James [181, p. 254], the matrix Dθ is given by Eλ (5) (41) (32) (311) (221) (2111) (11111)
1 0 0 1 0 0 1
dλ ,M 0 1 0 0 0 1 0
aλ 0 1 2 3 4 6 10
0 0 1 1 1 0 0
The invariants aλ are computed using the formula in Example 1.3.8. Applying Remark 3.2.3, we find that Bθ = {(5), (41), (32)} is a canonical basic set. Example 3.2.10. Let (W, S) be of type Bn and assume that we are in the “asymptotic case” as in 2.8.19, with weight function Lasy : W → Γ given by Bn
b t
4
a t
a t
p p p
a t
where b > (n − 1)a > 0.
Recall that Λ is the set of all pairs of partitions (λ , μ ) such that |λ | + |μ | = n. We have f(λ ,μ ) = 1 for all (λ , μ ) ∈ Λ , so any subring R ⊆ C is Lasy -good. As discussed in 2.8.19, the Kazhdan–Lusztig basis {±Cw } gives rise to a cell datum for H. Hence, for any specialisation θ : A → k, we have a natural parametrisation (λ , μ )
Irr(Hk ) = {Lk
| (λ , μ ) ∈ Λk◦ }
with Λk◦ defined as in 2.7.5.
By Proposition 3.2.7, the set Λk◦ is a canonical basic set. We have also seen in 2.8.19 that the order Lasy on Λ is given in terms of the dominance order on pairs of partitions. Thus, for (λ , μ ) ∈ Λ and (λ , μ ) ∈ Λk◦ , we have (Δ )
d(λ ,μ ),(λ ,μ ) = 1
and d(λ ,μ ),(λ ,μ ) = 0
unless (λ , μ ) (λ , μ ).
Similar statements first appeared (with rather different proofs and in terms of the opposite order; see Example 3.2.8) in the work of Dipper, James and Murphy [68]. As in Example 3.2.9, this discussion does not yet give us an explicit description of the set Λk◦ . This problem has been solved by Ariki [8], [9], [10]; see also Ariki and Mathas [15] and Ariki and Jacon [11]. We will obtain this result in Theorem 6.7.14. We shall also consider weight functions which do not correspond to the “asymptotic case”; this will be essential for applications to finite groups of Lie type in Section 4.4. To state the following, quite useful technical result, recall the definition of the set Fλ = {w ∈ W | E λ L w} for λ ∈ Λ ; see Proposition 1.6.11. Also recall that if (♠) holds, then Fλ is a two-sided Kazhdan–Lusztig cell; see Lemma 2.5.9.
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3 Specialisations and Decomposition Maps
Lemma 3.2.11. In the setting of Proposition 3.2.7, let λ ∈ Λ . Then λ ∈ Λk◦ if and only if dλ ,M = 0 and Cw .M = {0} for some M ∈ Irr(Hk ) and some w ∈ Fλ . Proof. Assume first that λ ∈ Λk◦ . Then, setting M := Lλk , we have dλ ,M = 0 by 2.7.5 and Cw .M = {0} for some w ∈ Fλ by Proposition 2.7.7. Conversely, assume that dλ ,M = 0 and Cw .M = {0}, where M ∈ Irr(Hk ) and w ∈ Fλ . Let μ ∈ Λk◦ be such μ μ that M ∼ = Lk . Then Proposition 3.2.7 shows that (Wk (λ ) : Lk ) = dλ ,M = 0. Hence, μ M∼ = Lk is a composition factor of Wk (λ ) and so Cw .Wk (λ ) = {0}. Since w ∈ Fλ , it follows that λ ∈ Λk◦ ; see Proposition 2.7.6. 3.2.12. As in Section 2.4, let us now consider the case where L(s) 0 for all s ∈ S and the parabolic subgroup Ω = t ∈ S | L(t) = 0 ⊆ W is non-trivial. Then, as in Proposition 2.4.5, we have a semidirect product decomposition W = Ω W1 (where W1 is a certain reflection subgroup of W ) and two generic algebras H1 ⊆ H (where H1 is defined with respect to the restriction of L to W1 ). We shall write HK,1 = K ⊗A H1 and Hk,1 = k ⊗A H1 . Our aim is to show that, under appropriate assumptions, the canonical basic sets for the specialised algebras Hk and Hk,1 determine each other; see Theorem 3.2.14 below. As in 2.4.11, we write IrrK (W1 ) = {E λ1 | λ1 ∈ Λ1 }
and
IrrK (W ) = {E λ | λ ∈ Λ }.
For any ω ∈ W and w1 ∈ W1 , we have Tω−1 Tw1 Tω = Tω −1 w1 ω . Thus, conjugation by a fixed Tω defines a K-algebra automorphism of HK,1 . Given any HK,1 -module V1 , we can define a new HK,1 -module structure on V1 by composing the original action (ω ) with the above automorphism. We denote that new HK,1 -module by V1 . Thus, we (ω ) have V1 = V1 as K-vector spaces, but h1 ∈ HK,1 acts on V (ω ) in the same way as −1 Tω h1 Tω acts on V . This notation is compatible with the action λ1 → ω .λ1 defined in 2.4.11, that is, we have (a)
λ (ω ) Eεω .λ1 ∼ = Eε 1
for all λ1 ∈ Λ1 and ω ∈ Ω .
In this set-up, we have the following compatibility property. (b) Assume that (♠) holds. Then Fω .λ1 = ω Fλ1 ω −1 for any λ1 ∈ Λ1 and ω ∈ Ω . Indeed, let C1 be a left Kazhdan–Lusztig cell of W1 (with respect to L|W1 ) such that m(C1 , λ1 ) > 0. As discussed in 2.4.11, the set ω C1 ω −1 also is a left Kazhdan– Lusztig cell and we have m(ω Cω −1 , ω .λ1 ) = m(C1 , λ1 ) > 0. So, by Lemmas 2.2.4 and 2.5.9, we have C1 ⊆ Fλ1 and ω C1 ω −1 ⊆ Fω .λ1 . The first inclusion implies that we also have ω C1 ω −1 ⊆ ω Fλ1 ω −1 . Hence, (b) holds. 3.2.13. In the above setting, let us now consider decomposition maps. We will require that R is good for both L and L|W1 . Note that this implies that |Ω | ∈ R× . (Indeed, let λ ∈ Λ be such that dim E λ = 1; then the formula in Proposition 2.4.14 shows that |Ω | divides fλ .) Now consider the decomposition maps dθ : R0 (HK ) → R0 (Hk )
and
dθ1 : R0 (HK,1 ) → R0 (Hk,1 )
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151
associated with θ : A → k; let Dθ = (dλ ,M ) and D1θ = (dλ11 ,M1 ) be the corresponding decomposition matrices. Recall that conjugation by Tω (where ω ∈ Ω ) defines an HK,1 -algebra automorphism, and this gives rise to an action of Ω on Irr(HK,1 ). Similar remarks apply, of course, to Hk,1 -modules: given an Hk,1 -module M1 , we can define a new Hk,1 -module by composing the original action with the k-algebra automorphism h1 → Tω−1 h1 Tω (h1 ∈ Hk,1 ). As above, this new Hk,1 -module will be (ω ) denoted by M1 . The characterisation of decomposition maps in Theorem 3.1.2 immediately yields the following compatibility property: (a)
d1
(ω )
ω .λ1 ,M1
= dλ11 ,M1
for all λ1 ∈ Λ1 and M1 ∈ Hk,1 .
Now, the Ω -graded Clifford system for H in Remark 2.4.6 certainly gives rise to an Ω -graded Clifford for Hk . Hence, there is also a version of Clifford’s theorem for Hk -modules (see [53, (11.16)]). (b) Let M ∈ Irr(Hk ) and M1 ∈ Irr(Hk,1 ) be a constituent of the restriction of M to Hk,1 . Then that restriction is the direct sum of simple Hk,1 -modules which are (ω ) all of the form M1 for various ω ∈ Ω . In particular, this shows that the restriction of M ∈ Irr(Hk ) to Hk,1 is semisimple. More generally, the restriction of representations from HK to HK,1 and from Hk to Hk,1 certainly give rise to maps on the level of Grothendieck groups: ResK : R0 (HK ) → R0 (HK,1 )
and
Resk : R0 (Hk ) → R0 (Hk,1 ).
Then the characterisation in Theorem 3.1.2 shows that the following diagram is commutative: dθ - R+ (Hk ) R+ 0 (HK ) 0 ResK
? R+ (H K,1 ) 0
dθ1
Resk ? - R+ (Hk,1 ) 0
With these preparations, we can now state the following theorem. Theorem 3.2.14 (Cf. [101, 5.5]). In the setting of 3.2.12 and 3.2.13, assume that (♠) and (♣) hold for W, L and for W1 , L|W1 . Using the corresponding cellular struc◦ tures of H and of H1 , we obtain subsets Λk◦ ⊆ Λ and Λk,1 ⊆ Λ1 such that μ
Irr(Hk ) = {Lk | μ ∈ Λk◦ }
and
μ
◦ Irr(Hk,1 ) = {Lk 1 | μ1 ∈ Λk,1 };
see 2.7.5. Then these two subsets determine each other, as follows: ◦ Λk,1 = {λ1 ∈ Λ1 | E λ1 appears in the restriction of some E λ where λ ∈ Λk◦ }, ◦ }. Λk◦ = { λ ∈ Λ | the restriction of E λ contains some E λ1 where λ1 ∈ Λk,1 ◦ , then ω .λ ∈ Λ ◦ for all ω ∈ Ω . In particular, if λ1 ∈ Λk,1 1 k,1
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3 Specialisations and Decomposition Maps
◦ and that λ ∈ Λ is such that E λ1 is a constituent Proof. First assume that λ1 ∈ Λk,1 λ of the restriction of E to W1 . We must show that λ ∈ Λk◦ . This is seen as follows. ◦ , let By 2.4.11, Eελ1 is a constituent of the restriction of Eελ to Hk,1 . Since λ1 ∈ Λk,1 us apply Lemma 3.2.11: so there exists some M1 ∈ Irr(Hk,1 ) and some w1 ∈ Fλ1 such that dλ1 ,M1 = 0 and Cw1 .M1 = {0}. Then the commutativity of the diagram in 3.2.13 shows that there exists some M ∈ Irr(Hk ) such that dλ ,M = 0 and M1 is a constituent of the restriction of M to Hk,1 . The latter condition implies that we also have Cw1 .M = {0}. Furthermore, Fλ1 ⊆ Fλ ; see Corollary 2.4.12. So we can apply Lemma 3.2.11 again and conclude that λ ∈ Λk◦ , as required. Conversely, let λ ∈ Λk◦ . As a first step, we show that there exists at least some ◦ such that E λ1 is a constituent of the restriction of E λ to W . This is seen λ1 ∈ Λk,1 1 as follows. By Lemma 3.2.11, we have dλ ,M = 0 and Cw .M = {0} for some M ∈ Irr(Hk ) and some w ∈ Fλ . Writing w = ω w1 , where ω ∈ Ω and w1 ∈ W1 , we have Cw = Tω Cw1 ; see Proposition 2.4.5. Since Tω is invertible, we also have Cw1 .M = {0}. Thus, Cw1 does not act as zero on the restriction of M to Hk,1 . By Clifford’s theorem, that restriction is semisimple; see 3.2.13(b). Hence, there must exist some M1 ∈ Irr(Hk,1 ) such that Cw1 .M1 = {0} and M1 is a constituent of the restriction of M to Hk,1 . By the commutativity of the diagram in 3.2.13, we will have dλ11 ,M1 = 0
for some λ1 ∈ Λ1 such that Eελ1 is a constituent of the restriction of Eελ to HK,1 . Now, we have Eελ1 ∼ = WK (λ1 ) by Proposition 2.7.1. Hence, the fact that dλ11 ,M1 = 0 implies that M1 is a composition factor of Wk (λ1 ); see Theorem 3.1.2(c). So, since Cw1 .M1 = {0}, we conclude that Cw1 .Wk (λ1 ) = {0}. Consequently, we also have Cw1 .W (λ1 ) = {0} and so Cw1 .Eελ1 = {0}. But then Lemmas 2.3.9 and 2.2.4 show that y1 LR w1 (with respect to L|W1 ), where y1 ∈ Fλ1 . On the other hand, by Corollary 2.4.12, we have Fλ1 ⊆ Fλ . So Remark 2.4.10 implies that w1 ∈ Fλ1 . ◦ , as required. Now we can apply Lemma 3.2.11 again and conclude that λ1 ∈ Λk,1 By Clifford’s theorem, as stated in 3.2.13(b), it now remains to show that, for any λ1 ∈ Λ1 and ω ∈ Ω , we have ◦ λ1 ∈ Λk,1
⇒
◦ ω .λ1 ∈ Λk,1 .
◦ , To prove this implication, we use again Lemma 3.2.11. This shows that if λ1 ∈ Λk,1 1 then we have dλ ,M = 0 and Cw1 .M1 = {0} for some M1 ∈ Irr(Hk,1 ) and some w1 ∈ 1
1
(ω )
Fλ1 . Now let ω ∈ Ω . Then, by 3.2.13(a) and the definition of M1 , we have d1
(ω )
ω .λ1 ,M1
= dλ11 ,M1 = 0
and
(ω )
Tω Cw1 Tω−1 .M1
= {0}.
Now, we have Tω Cw1 Tω−1 = Cω w1 ω −1 ; see 2.4.7. Furthermore, by 3.2.12(b), we have ω Fλ1 ω −1 = Fω .λ1 . Hence, another application of Lemma 3.2.11 shows that ω .λ1 ∈ ◦ , as desired. Λk,1 Example 3.2.15. Let (W, S) be of type Bn with generators and weight function L : W → Z given by
3.3 Principal Specialisations and Blocks of Defect 1
Bn
0 t
4
1 t
1 t
153
p p p
1 t
As in Example 2.4.4, we have Ω ∼ = Z/2Z and (W1 , S1 ) is of type Dn ; furthermore, L(s) ˜ = 1 for all s˜ ∈ S1 . By 2.4.1(a), Lusztig’s conjectures P1–P15 hold for W1 , L|W1 . Hence, they will also hold for W, L; see 2.4.8. So, by 2.5.3, the properties (♠) and (♣) hold for H and for H1 . By the table in Definition 1.5.9, every subring R ⊆ C in which 2 is invertible is good with respect to L and with respect to L|W1 . Recall now that Λ is the set of all pairs of partitions (λ , μ ) such that |λ |+|μ | = n; see Example 1.3.9. Given (λ , μ ) and (λ , μ ) in Λ , we write (λ , μ ) ∼ (λ , μ ) if (λ , μ ) = (λ , μ ) or (λ , μ ) = (μ , λ ). Let [λ , μ ] denote the equivalence class of (λ , μ ) and let Λ¯ be the set of all equivalence classes. Then we have
Λ1 = {[λ , μ ] ∈ Λ¯ | λ = μ } ∪ {(λ , ±) | [λ , λ ] ∈ Λ¯ }; see Example 1.3.10. Now assume that Λk◦ ⊆ Λ is known and let Λ¯ k◦ be the set of all [λ , μ ] ∈ Λ¯ such that (λ , μ ) ∈ Λk◦ or (μ , λ ) ∈ Λk◦ . Then Theorem 3.2.14 shows that ◦ Λk,1 = {[λ , μ ] ∈ Λ¯ k◦ | λ = μ } ∪ {(λ , ±) | [λ , λ ] ∈ Λ¯ k◦ }.
(Note that (λ , μ ) ∈ Λk◦ if and only if (μ , λ ) ∈ Λk◦ .) Thus, once we have found an explicit description for the subset Λk◦ ⊆ Λ , then we immediately obtain an explicit de◦ ⊆ Λ . In this way, the problem of determining a canonscription for the subset Λk,1 1 ical basic set in type Dn is reduced to an analogous problem in type Bn (with weight function L as above). This result first appeared in [101, §6]. Remark 3.2.16. Assume that (W, S) is irreducible. We have seen in 2.3.8 that (♠) and (♣) hold if the properties P1, P4 and P15 in Lusztig’s Conjecture 2.3.2 hold. By the information summarized in 2.4.1, the only case where these are not yet known to hold is where W is of type Bn and we are neither in the equal parameter case nor in the “asymptotic case” nor in the case considered in Example 3.2.15. In Sections 5.8 and 6.7, we will nevertheless be able to construct explicitly canonical basic sets in type Bn for any choice of L, under some assumptions on the target field of θ .
3.3 Principal Specialisations and Blocks of Defect 1 We keep the basic setting of Section 3.1 where H = HA (W, S, L) is a generic Iwahori–Hecke algebra associated with a finite Coxeter group W ; see 3.1.12. Now we discuss the important class of specialisations θ : A → k, where the localisation of A in ker(θ ) is a discrete valuation ring. In this case, we are as close as possible to the classical setting of Brauer’s modular representation theory (see, e.g., [53, §16]). 3.3.1. The product decomposition of the elements cλ in Remark 1.3.6 leads us to consider the following set-up. Assume that there exists a subset P ⊆ A such that the following three conditions hold: (P1) For each λ ∈ Λ , we have cλ = f λ ε γλ ∏Φ ∈P Φ nλ ,Φ , where γλ ∈ Γ and nλ ,Φ 0.
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3 Specialisations and Decomposition Maps
(P2) Each Φ ∈ P is non-constant (i.e., Φ ∈ R) and generates a prime ideal in A such that the quotient ring A/(Φ ) is integrally closed in its field of fractions. (P3) For Φ , Φ ∈ P, we have (Φ ) = (Φ ) unless Φ = Φ . For each Φ ∈ P, the natural map A → A/(Φ ) gives rise to a specialisation θΦ : A → kΦ , where kΦ is the field of fractions of A/(Φ ). One easily checks that the localisation A(Φ ) of A in the prime ideal generated by Φ is a discrete valuation ring; see [132, Example 7.3.6]. The corresponding valuation vΦ : K × → Z is defined by sending a non-zero a ∈ A to the integer n 0 such that a ∈ (Φ n ) but a ∈ (Φ n+1 ). Example 3.3.2. Let W be a finite Weyl group and assume that we are in the equal parameter case where Γ = Z and L(s) = 1 for all s ∈ S. Then A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Since ZW = Z, we can assume that R ⊆ Q = K. For any integer d 1, denote by Φd (v) ∈ Z[v] the dth cyclotomic polynomial. Thus, Φd (v) is irreducible in Q[v] and we have vd − 1 = ∏d |d Φd (v). Also note that:
Φd (v2 ) = Φ2d (v)
if d is even;
Φd (v ) = Φd (v)Φ2d (v) and Φ2d (v) = ±Φd (−v) 2
if d is odd.
Now let u := v2 . Then, as discussed in Example 3.1.19, each cλ lies in Z[u, u−1 ] and divides the polynomial PW in Q[u, u−1 ]. Using Solomon’s formula for PW (see Example 3.1.19), we see that each cλ can be written as cλ = fλ u−aλ
∏
Φd (u)nλ ,d
(nλ ,d 0),
d∈E(W )
where E(W ) is the set of all integers d 2 such that d divides some degree of W . Thus, the three conditions in 3.3.1 hold for P := {Φ2d (v) | d ∈ E(W )} ∪ {Φd (v) | d ∈ E(W ) odd}. Note that, for Φ = Φd (v) ∈ P, we have A/(Φ ) ∼ = R[ζd ] where ζd ∈ C is a root of unity of order d, and it is well-known (see, e.g., [53, (4.5)]) that this is integrally closed in its field of fractions. Example 3.3.3. We generalise the setting in Example 3.3.2 as follows. Let W be any finite Coxeter group, Γ = Z and L : W → Z be any weight function such that L(s) 0 for all s ∈ S. Recall that R is assumed to be noetherian and integrally closed in K. As before, A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . By Remark 1.3.6, each cλ still is a Laurent polynomial in v all of whose (complex) roots are roots of unity. Then the set P := {Φ ∈ A | Φ is K-cyclotomic and divides cλ for some λ ∈ Λ } satisfies the conditions in 3.3.1. Here, by a “K-cyclotomic polynomial”, we mean a non-constant monic polynomial in R[v] which is irreducible √ in K[v] and all of whose complex roots are roots of unity. For example, if K = Q[ 5], then
3.3 Principal Specialisations and Blocks of Defect 1
√ Φ5,a (v) = v2 + 12 (1 + 5)v + 1
and
155
√ Φ5,b = v2 + 12 (1 − 5)v + 1
are K-cyclotomic polynomials such that Φ5,a (v)Φ5,b (v) = Φ5 (v) ∈ Q[v]. (These appear in P for W of type H3 or H4 .) We also set E(W, L) := {d 1 | Φd (v2 ) ∈ A is divisible by some Φ ∈ P}. Note that d 2 for all d ∈ E(W, L). (Indeed, if 1 ∈ E(W, L), then Φ = v ± 1 ∈ P and we could conclude that HkΦ ∼ = KW is not semisimple, which is a contradiction.) Also note that if W, L and R are as in Example 3.3.2, then E(W, L) = E(W ). Example 3.3.4. Assume that W is of type Bn and we are in the setting of Example 3.1.20, where R = Z and A is the ring of Laurent polynomials in two independent variables V, v. Then the three conditions in 3.3.1 hold for the set P = {Φ2d (v) | 2 d n} ∪ {Φd (v) | 2 d n, d odd} ∪ {V 2 +v±2r | 0 r n − 1}. √ Note that, if Φ = V 2 + v±2r ∈ P then A/(Φ ) ∼ = Z[ −1, v±1 ], while if Φ = Φd (v) ∈ P then A/(Φ ) ∼ = Z[ζd ,V ±1 ], where ζd ∈ C is a root of unity of order d. In both cases, A/(Φ ) is integrally closed in its field of fractions (see, e.g., [132, Exc. 7.2]). We are now ready to introduce the following general notion. Definition 3.3.5. Assume that R is L0 -good. Let θ : A → k be a specialisation where k is the field of fractions of θ (A). Let O = {a/b ∈ K | a, b ∈ A and θ (b) = 0} be the localization of A in ker(θ ). We say that θ is a principal specialisation if O is a discrete valuation ring. In this case, a general argument shows that for every finite-dimensional HK module V there exists some HO -module which is free of finite rank over O such that V∼ = K ⊗O N; see, for example, [132, 7.3.7]. We can also define a decomposition map dO,θ : R0 (HK ) → R0 (Hk ) using O instead of A; note that θ has a natural extension to a map O → k. Then Remark 3.1.5 shows that dθ = dO,θ . We usually denote by Φ ∈ O a generator for the unique maximal ideal of O, and replace a subscript “θ ” by “Φ ”. Thus, we write dΦ instead of dθ , and DΦ denotes the decomposition matrix. Furthermore, the θ -blocks will be called Φ -blocks. Note that we can (and usually will) assume without loss of generality that Φ ∈ A. We denote by vΦ : K × → Z the valuation associated with O. Thus, any 0 = x ∈ K can be written uniquely in the form x = Φ n u, where n = vΦ (x) and u ∈ O × . Furthermore, we have O = {x ∈ K × | vΦ (x) 0} ∪ {0}. We assume for the remainder of this section that θ is a principal specialisation, with corresponding valuation ring O and maximal ideal generated by Φ ∈ A. For later reference, we shall now state the following result which hold thanks to this assumption; this is a very useful technical tool both for computational and for theoretical purposes. (See, for example, Proposition 3.3.12 and Theorem 3.7.16.)
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3 Specialisations and Decomposition Maps
Proposition 3.3.6 (Geck and Rouquier; see [132, 7.5.3]). Given M ∈ Irr(Hk ), we define a function PM : Z(H) → K by the formula PM (z) :=
∑ c−1 λ dλ ,M ωλ (z)
λ ∈Λ
(z ∈ Z(H)),
where ωλ : Z(H) → A denotes the central character of Eελ . Then we have PM (z) ∈ O for all z ∈ Z(H). In particular, taking z = 1, we have ∑λ ∈Λ c−1 λ dλ ,M ∈ O. (The proof relies on the fact that idempotents can be lifted from Hk to HO .) Remark 3.3.7. Recall the definition of the character table X(H) and of the polynomials { f w,C } in Example 1.2.3. These polynomials can be used to construct a basis of Z(H). Indeed, let us define (a)
zC =
∑ ε −2L(w) fw,C T˙w−1
for any C ∈ Cl(W ).
w∈W
Then, by a result due to Geck and Rouquier (see [132, 8.2.4]), the elements {zC | C ∈ Cl(W )} form an A-basis of Z(H). Using the above formula, the orthogonality relations in Proposition 1.2.12 can be rewritten as follows:
if λ = μ , c λ ˙ (b) ∑ χ (TwC ) ωμ (zC ) = 0λ otherwise. C∈Cl(W ) (See also [132, Exc. 9.5].) Multiplying the defining formula for PM (zC ) by χ μ (T˙wC ) and summing over all C, we obtain (c)
dλ ,M =
∑
χ λ (T˙wC ) PM (zC )
for all λ ∈ Λ and M ∈ Irr(Hk ).
C∈Cl(W )
Example 3.3.8. There is a recursive formula for computing the polynomials fw,C ; see [132, 8.2.7]. Hence, in concrete examples, one can work out the central elements zC , but explicit closed formulae only seem to be known in some special cases. Here is one such special case. Let C ∈ Cl(W ) be such that C ∩ S = ∅; that is, C consists of reflections. Then, by Iancu [170, §3], we have the explicit formula
L(w)−L(w ) C ε if w ∈ C, zC = ∑ ε −L(wC )−L(w) T˙w , that is, fw,C = 0 if w ∈ C. w∈C The elements {zC | C ∈ Cl(W ) such that C ∩ S = ∅} appear to be suitable generalisations of the Jucys–Murphy elements in types A, B; see Example 3.4.8 and 3.5.1. Definition 3.3.9. Let λ ∈ Λ and recall that 0 = cλ ∈ A. The integer dλ := vΦ (cλ ) 0 is called the Φ -defect of λ . The Φ -defect of a Φ -block Λ1 ⊆ Λ is defined by d(Λ1 ) := max{dλ | λ ∈ Λ1 }.
3.3 Principal Specialisations and Blocks of Defect 1
157
As in the representation theory of finite groups (where these notions had their origin; see [53, §56]), the defect of a block is a rough measure of how complicated the representation theory (i.e., the decomposition matrix, etc.) of that block is. Note that saying that λ ∈ Λ has Φ -defect 0 simply means that θ (cλ ) = 0. Thus, Proposition 3.1.17 describes the decomposition matrices of Φ -blocks of defect 0. There is also a sophisticated theory of blocks of defect 1, due to Brauer in the framework of the modular representation theory of finite groups; e.g. see [53, §62]. It was shown in [92] that this theory can be adapted to the framework of decomposition maps of Iwahori–Hecke algebras. Here, we shall present a simplified version of this adaptation, taking into account from an early point our results on cell data from Chapter 2. The discussion, closely following the original work of Brauer as presented in [138, §11], will rely on some subtle matrix-theoretic arguments. The following remarks are a preparation for this discussion. 3.3.10. Recall from Section 1.3 that Irr(HK ) = {Eελ | λ ∈ Λ }. Choosing a basis of Eελ , we obtain a matrix representation ρ λ : HK → Mdλ (K), where dλ := dim Eελ . As before, let M(λ ) be an indexing set for that basis. Given h ∈ H and s, t ∈ M(λ ), the λ (h). As already mentioned above, since (s, t)-entry of ρ λ (h) will be denoted by ρst O is a discrete valuation ring, we can choose the basis of Eελ such that λ ρst (Tw ) ∈ O
for all w ∈ W and s, t ∈ M(λ ).
Then denote by ρ¯ λ : Hk → Mdλ (k) the representation of Hk obtained by extension of scalars via the map O → k. In general, ρ¯ λ will not be irreducible. But we can always choose the basis to be adapted to a composition series of ρ¯ λ . More precisely, let Irr(Hk ) = {M1 , . . . , Mr }, where dim Mi = li for 1 i r. Choosing a basis of Mi , we obtain a matrix representation σi : Hk → Mli (k). Then, as in [138, (11.9)], the above basis of Eελ can be chosen such that the matrices ρ¯ λ (Tw ) (where Tw now also is a basis element of Hk ) will have an upper block triangular shape ⎛ ⎜ ⎜ ρ (Tw ) = ⎜ ⎜ ⎝ ¯λ
σi1 (Tw ) 0 .. . 0
⎞ ... ∗ .. ⎟ . σi2 (Tw ) . . . ⎟ ⎟ ⎟ .. .. . . ∗ ⎠ . . . 0 σit (Tw ) ∗
for all w ∈ W ,
where i1 , . . . , it ∈ {1, . . . , r}. Thus, the decomposition number dλ ,Mi is given by dλ ,Mi = number of indices in the list (i1 , . . . , it ) which are equal to i. In what follows, we will assume that the above choices have been made. Lemma 3.3.11 (Brauer; cf. [138, 11.8], [92, 9.1]). Let λ , μ ∈ Λ be such that dλ = dμ = 1. Assume there exist s, t ∈ M(λ ) and u, v ∈ M(μ ) such that μ
λ ρst (h) ≡ ρuv (h) mod Φ
for all h ∈ H.
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3 Specialisations and Decomposition Maps
Then we have δst cλ − 2δλ μ δsv δtu cλ + δuv cμ ≡ 0 mod Φ 2 . μ
λ (T ) − ρ (T ) ≡ 0 mod Φ and Proof. Our assumption implies that we have ρst w uv w μ λ ρst (Tw−1 ) − ρuv (Tw−1 ) ≡ 0 mod Φ for all w ∈ W . For each fixed w, we multiply together these two congruences and then sum over all w ∈ W ; the result is μ μ ∑ ρstλ (Tw ) − ρuv (Tw ) ρstλ (Tw−1 ) − ρuv (Tw−1 ) ≡ 0 mod Φ 2. w∈W
We expand the terms in the sum on the left-hand side and consider the resulting four sums. Using the Schur relations in Proposition 1.2.12, we can evaluate these sums and obtain that δst cλ − 2δλ μ δsv δtu cλ + δuv cμ ≡ 0 mod Φ 2 , as required. Proposition 3.3.12 (Brauer; cf. [138, 11.9], [92, 9.3]). Let Λ1 ⊆ Λ be a Φ -block of defect 1 and Irr1 (Hk ) the corresponding subset of Irr(Hk ). Assume that char(k) = 2. Let M ∈ Irr1 (Hk ). Then the set SΦ (M) := {λ ∈ Λ1 | dλ ,M = 0} contains precisely two elements, and we have dλ ,M ∈ {0, 1} for all λ ∈ Λ1 . Proof. First we show that dλ ,M 1 for all λ ∈ Λ1 . Assume, if possible, that dλ ,M 2. In the setting of 3.3.10, saying that dλ ,Mi 2 (where M ∼ = Mi ) means that two diagonal blocks in ρ¯ λ (Tw ) are actually equal for all w ∈ W . Thus, there exist s = t λ (T ) = ρ λ (T ) or, in other words, ρ λ (T ) ≡ ρ λ (T ) mod Φ ¯ tt in M(λ ) such that ρ¯ ss w w ss w tt w for all w ∈ W . Applying Lemma 3.3.11 with λ = μ yields that cλ −2δst δst cλ +cλ ≡ 2cλ ≡ 0 mod Φ 2 . Since char(k) = 2, this implies that Φ 2 | cλ and so dλ 2, which is a contradiction. So all decomposition numbers with respect to Λ1 are 0 or 1. Next consider the set SΦ (M). Assume, if possible, that SΦ (M) contains only one element, λ0 ∈ Λ1 say. We have just proved that then dλ0 ,M = 1. Hence, the sum in Proposition 3.3.6 reduces to c−1 λ0 and so we conclude that λ0 has defect 0. So, by Proposition 3.1.17, {λ0 } would form a Φ -block by itself, which is a contradication. Consequently, SΦ (M) contains at least two elements, λ = μ say. Recall that ρ λ and ρ μ are chosen as in 3.3.10. Then saying that dλ ,Mi = 0 and dμ ,Mi = 0 (where M∼ = Mi ) means that σi occurs as a diagonal block both in ρ¯ λ and in ρ¯ μ . In particular, λ (T ) = ρ¯ μ (T ) or, in other words, there are s ∈ M(λ ) and u ∈ M(μ ) such that ρ¯ ss w uu w μ λ ρss (Tw ) ≡ ρuu (Tw ) mod Φ for all w ∈ W . Applying Lemma 3.3.11 yields that cλ + cμ ≡ 0 mod Φ 2 (since λ = μ ). Finally assume, if possible, that SΦ (M) contains a third element, ν say. Then we can repeat the above argument with λ , ν and with μ , ν instead of λ , μ . So we conclude that cλ ≡ −cν mod Φ 2 and cμ ≡ −cν mod Φ 2 . These congruences would again imply that 2cλ ≡ 0 mod Φ 2 , which is impossible since char(k) = 2. Thus, we have shown that SΦ (M) contains precisely two elements. Theorem 3.3.13 (Geck [92, §9], [97, §4], Geck and Rouquier [134, 4.4]). Let be a monomial order on Γ such that L(s) 0 for all s ∈ S. Assume that R is L-good and H admits a cell datum as in Theorem 2.6.12. Also assume that char(k) = 2. Let Λ1 be a Φ -block of defect 1 and Irr1 (Hk ) be the corresponding subset of Irr(Hk ). (a) Suppose that |Λ1 | = n 2. Then there is a unique labelling Λ1 = {λ1 , . . . , λn } such that λn L λn−1 L · · · L λ1 (with L defined as in Theorem 2.6.12).
3.3 Principal Specialisations and Blocks of Defect 1
159
(b) The Brauer graph of Λ1 is a tree, as follows:
λ1 t
λ2 t
λn−1 t
p p p
λn t
(c) The edges in the above tree are in bijection with Irr1 (Hk ). Thus, | Irr1 (Hk )| = n − 1 and we can write Irr1 (Hk ) = {M1 , . . . , Mn−1 }, where M j labels the edge joining λ j and λ j+1 . With this notation, we have
1 if i = j or i = j + 1, dλi ,M j = 0 otherwise. Moreover, assuming that (♣) holds, then aλ1 < aλ2 < · · · < aλn and BΦ ,1 = {λ1 , . . . , λn−1 } is a canonical basic set for Λ1 . μ
Proof. Our assumptions imply that Irr(Hk ) = {Lk | μ ∈ Λk◦ }, with Λk◦ ⊆ Λ as in 2.7.5; furthermore, as in Proposition 3.2.7, for λ ∈ Λ and M ∈ Irr(Hk ), we have μ μ dλ ,M = (Wk (λ ) : Lk ), where μ ∈ Λk◦ is such that M ∼ = Lk . μ ◦ ◦ Let Λk,1 ⊆ Λk◦ be the subset such that Irr1 (Hk ) = {Lk | μ ∈ Λk,1 }. Let M ∈ ◦ Irr1 (Hk ). The following remark tells us how we can identify the element μ ∈ Λk,1 μ such that M ∼ = Lk . By Proposition 3.3.12, the set SΦ (M) = {λ ∈ Λ | dλ ,M = 0} has precisely two elements. We claim that SΦ (M) = {λ1 , λ2 },
(∗)
λ where λ2 L λ1 and M ∼ = Lk 1 .
◦ be such that M ∼ Lλ1 . By Theorem 2.6.6, (W (λ ) : Lλ1 ) = 1 Indeed, let λ1 ∈ Λk,1 = k k 1 k and so λ1 ∈ SΦ (M). Let λ2 be the second element of SΦ (M). Then λ2 L λ1 since (Wk (λ2 ) : Lkλ1 ) = 0; see again Theorem 2.6.6. Thus, (∗) is proved. We use (∗) to eliminate certain configurations on the Brauer graph of Λ1 . First assume, if possible, that this Brauer graph contains a piece as follows:
(I)
λ1 t
λ2 t
λ3 t
where
λ1 L λ2 , λ3 L λ2 .
Let M ∈ Irr1 (Hk ) label the edge between λ1 and λ2 . Then SΦ (M) = {λ1 , λ2 } and λ (∗) shows that M ∼ = Lk 2 . Similarly, let M ∈ Irr1 (Hk ) label the edge between λ2 and λ λ3 . Then SΦ (M ) = {λ2 , λ3 } and (∗) shows, again, that M ∼ = M = Lk 2 . Thus, M ∼ but SΦ (M) = SΦ (M ), which is a contradiction. Next assume, if possible, that our Brauer graph contains a piece as follows: (II)
λ1 t
λ2 t
λ3 t
where
λ2 L λ1 , λ2 L λ3 .
Then we use the automorphism † and its compatibility with blocks and decomposition numbers; see 3.1.16. Thus, Λ1† = {λ † | λ ∈ Λ1 } also is a Φ -block and its Brauer graph contains a piece as follows:
λ1† t
λ2† t
λ3† t
where
λ2 L λ1 , λ2 L λ3 .
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3 Specialisations and Decomposition Maps
Note also that cλ † equals cλ (up to a monomial ε γ for some γ ∈ Γ ); see Example 1.3.4. Hence, Λ1† is a block of Φ -defect 1. By Remark 2.2.11, we have λ1† L λ2† and λ3† L λ2† . Hence, we have found a piece in Λ1† that looks like (I), which is contradiction. We now prove (a) and (b). Note that (∗) implies that, if λ1 = λ2 are connected by an edge in the Brauer graph of Λ1 , then SΦ (M) = {λ1 , λ2 } for some M ∈ Irr1 (Hk ) and so λ1 L λ2 or λ2 L λ1 . Thus, the labels on any two vertices connected by an edge in the Brauer graph of Λ1 are always related by L . Then the impossibility of having any piece which looks like (I) or (II) means that, along every path in the Brauer graph of Λ1 , the labels of the vertices are either strictly increasing or strictly decreasing (with respect to L ). First of all, this shows that the Brauer graph of Λ1 does not have any closed cyles; that is, it is a tree. Second, it shows that there is no bifurcation point. Indeed, if there was a bifurcation point, then there would exist four labels λ0 , λa , λb , λc ∈ Λ1 with no edges among λa , λb , λc but with each of these connected to λ0 . But then at least one path passing through λ0 would contain a piece which looks like (I) or (II), which is a contradiction. This completes the proof of (a) and (b). Finally, consider (c). Let 1 j n−1. Then SΦ (M) = {λ j , λ j+1 } for some M ∈ Irr1 (Hk ), by (∗) and the definition of the Brauer graph. We claim that M is uniquely determined (up to isomorphism). Indeed, if SΦ (M ) = {λ j , λ j+1 } and M ∼ = M , then the two columns of the decomposition matrix of Λ1 corresponding to M and M would be equal, contradicting the fact that dΦ is surjective (see Theorem 3.1.14). Thus, each edge determines a unique M ∈ Irr1 (Hk ). Every M ∈ Irr1 (Hk ) arises in this way, since M must label some edge in the Brauer graph. Furthermore, each M ∈ Irr1 (Hk ) labels a unique edge, since SΦ (M) only has two elements. Hence, the edges in the tree in (b) correspond bijectively to Irr1 (Hk ). Thus, (c) is proved. The final statement about the invariants aλ and BΦ ,1 is now also clear. Remark 3.3.14. The fact that the Brauer graph of a Φ -block of defect 1 is an open polygon and that its decomposition matrix is given as in Theorem 3.3.13(c) can be shown without reference to (♠) or (♣); see [92, Theorem 9.6]. The references to (♠) and (♣) are needed to establish the ordering in Theorem 3.3.13(a). Example 3.3.15. Assume that we are in the equal parameter case where Γ = Z and L(s) = 1 for all s ∈ S. In particular, A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Let e 2 and supppose that K contains all roots of unity of order e. We consider the Φe -modular specialisation √ θe : A → K, v → ζ2e := exp(π −1/e) ∈ K; see Example 3.1.15(b). Note that θe is a principal specialisation where Φ = v − ζ2e . We shall denote extension of scalars from A to K (via θe ) simply by a subscript “(e)”. Thus, for example, H(e) := K ⊗A H is the corresponding specialised algebra. The assumptions of Proposition 3.2.7 are satisfied. Hence, we have a cell datum for H (see Theorem 2.6.12) and there is a natural parametrisation μ
◦ }, Irr(H(e) ) = {L(e) | μ ∈ Λ(e)
◦ ⊆ Λ is a canonical basic set. where Λ(e)
3.3 Principal Specialisations and Blocks of Defect 1
161
The corresponding decomposition matrix will be denoted by D(e) ; it has rows la◦ . belled by Λ and columns labelled by Λ(e) (a) Assume that W is a finite Weyl group and let e = h be the Coxeter number of W ; see [132, §1.5] and note that h is the largest degree of W . It is shown by Bleher, Geck and Kimmerle [20, §6] that there is precisely one Φh -block of defect 1 and all other Φh -blocks have defect 0. The Brauer tree of the block of defect 1 is given by
λ (0) t
λ (1) t
λ (|S|−1) t
p p p
λ (|S|) t
where λ (i) labels the ith exterior power of the reflection representation of W . (Thus, λ (0) corresponds to the unit representation and λ (|S|) to the sign representation.) (b) Now assume that (W, S) is of exceptional type H3 , H4 , F4 , E6 , E7 or E8 . In each of these cases, the partition into Φe -blocks has been determined in [132, Prop. 11.5.13]; the blocks of positive Φe -defect are explicitly listed in [132, Appendix F]. For example, in type H3 , we find the following table: e
d
Φe -blocks of defect d
2
3
{1r , 3s , 3 s , 5r , 5 r , 3s , 3s , 1 r }
3
1
{1r , 5r , 4 r }
+ dual
5
1
{1r , 4 r , 3s }
+ dual
6
1
{1r , 5r , 5 r , 1 r }
10
1
{1r , 3 s , 3s , 1 r }
(Here, “dual” indicates a block obtained by applying the map λ → λ † ; see 3.1.16.) Thus, for e ∈ {3, 5, 6, 10}, all blocks have defect 0 or 1. In the above table (and, more generally, in all tables in [132, App. F]), the representations belonging to a block of defect 1 are ordered according to increasing value of aλ . Hence, by Theorem 3.3.13, we can immediately write down the corresponding decomposition matrix. For example, the block for e = 10 (the Coxeter number) has a Brauer tree as described in (a). There is a further Φ2 -block of defect 3. Its decomposition matrix has been determined by M¨uller [253, §4]; it is given by the first of the two matrices in Table 3.3. Taking into account the invariants aλ (which M¨uller did not do), one can easily determine the corresponding canonical basic set; see Remark 3.2.3. In the second matrix, the canonical basic set is marked by “•”. Remark 3.3.16. Given λ , μ ∈ Λ , it may be conjectured that dλ = dμ whenever Eελ , μ Eε belong to the same Φ -block. Under the assumption that W is a finite Weyl group and we are in the equal parameter case, this is shown by a general argument in [92, Prop. 7.4], using an analogous result for blocks of finite groups of Lie type [91, 1.4]. For W of exceptional type, this property can also be verified by inspection using the explicit knowledge of the Φe -blocks; see [132, Appendix F]. It would be interesting to find a general argument purely in the framework of Iwahori–Hecke algebras. In general, the determination of the decomposition matrix of a Φ -block of defect 2 is a hard problem. For W of exceptional type, this problem can be attacked
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3 Specialisations and Decomposition Maps
Table 3.3 A canonical basic set in type H4 e=2 1 r 1r 5 r 5r 3s 3s 3 s 3s
1 1 1 1 1 1 1 1
dλ ,M . . 1 1 1 . 1 .
. . 1 1 . 1 . 1
e=2 aλ • 1r 0 • 3 s 1 • 3s 1 5r 2 5 r 5 3s 6 3s 6 1 r 15
1 1 1 1 1 1 1 1
dλ ,M . 1 . 1 1 1 . .
. . 1 1 1 . 1 .
(Here, and in all subsequent tables, the dot stands for 0)
by computational methods; see Section 7.1 for further details. A solution for W of classical type An−1 , Bn and Dn has been achieved through quite remarkable (and, perhaps, rather unexpected) connections with something completely different: the theory of canonical bases for quantized envelopping algebras. A first step in this direction will be taken in the following section.
3.4 Towards Canonical Basic Sets for Classical Types In this section, we lay the theoretical foundations for studying canonical basic sets for Iwahori–Hecke algebras of classical type Xn ∈ {An−1 , Bn , Dn }. The essential feature that we can exploit here is that we naturally have an infinite chain W (X1 ) ⊆ W (X2 ) ⊆ W (X3 ) ⊆ · · · and so we may try to use inductive arguments. This will be formalised in the concept of an “abstract Fock datum”; in this form, it will also apply to the Ariki–Koike algebras in Chapter 5. 3.4.1. Let H be an A-algebra as in 3.1.1. Let K be the field of fractions of A and θ : A → k be a ring homomorphism into a field k such that k is the field of fractions of θ (A). We assume that A is integrally closed in K; furthermore, we assume that the K-algebra H K is split semisimple and the k-algebra H k is split. We have an associated decomposition map dθ : R0 (H K ) → R0 (H k ) as in Theorem 3.1.2. We wish to develop a method for constructing “good” approximations to the columns of the corresponding decomposition matrix Dθ . By definition, an abstract Fock datum for H consists of a family {(H m , zm ) | 1 m n} satisfying the following conditions. (F1) H 1 := 1H A ⊆ H 2 ⊆ . . . ⊆ H n = H and each H i is an A-subalgebra of H. (F2) Each H m is finitely generated and free over A; furthermore, H m is free as a left H m−1 -module. (F3) For each m, we have zm ∈ H m and zm belongs to the centre of H m . (F4) For each m, the K-algebra H K,m := K ⊗A H m is split semisimple and the kalgebra H k,m := k ⊗A H m is split. For each m, we write Irr(H K,m ) = {V λ | λ ∈ Λ m }, where Λ m is a finite indexing set.
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163
For any λ ∈ Λ m , denote by ωλ the central character of V λ ; see Definition 3.1.9. Since zm ∈ Z(H m ), we have ωλ (zm ) ∈ A. So we can define
ω¯ λ (zm ) := θ (ωλ (zm ))
for all m and λ ∈ Λ m .
(m)
For m 1, denote by IndK the induction of representations from H K,m−1 to H K,m (where H 0 = {0} by convention). Thus, for any H K,m−1 -module V , we have (m)
IndK
V := H K,m ⊗H K,m−1 V.
This operation can be refined as follows. For μ ∈ Λ m−1 and λ ∈ Λ m , let (m) μ V .
m(λ , μ ) := multiplicity of V λ as an irreducible constituent of IndK Then, for a fixed ξ ∈ k, we define
ξ -IndK (V μ ) := (m)
λ
V λ ⊕ . . . ⊕V λ , m(λ , μ ) times
where the sum runs over all λ ∈ Λ m such that ω¯ λ (zm ) − ω¯ μ (zm−1 ) = ξ . By linearity, (m) we can define ξ -IndK (V ) for any H K,m -module V . Definition 3.4.2. Given an abstract Fock datum for H as above, we define a family of representations E of H K inductively as follows. For m = 1, let E1 = {1}, where 1 denotes the trivial representation of H K,1 = 1H K . Then set (m) Em = ξ -IndK (V ) | V ∈ Em−1 and ξ ∈ k \ {0} for 2 m n. Let us also set E := En . Thus, starting from 1, the representations in E are obtained by successive applications of the refined induction operators defined above. Remark 3.4.3. Let 1 m n and Y ∈ Em . For λ ∈ Λ m , denote by m(Y, λ ) the multiplicity of V λ as an irreducible constituent of Y . Then the inductive definition of Em immediately shows that, for any λ , λ ∈ Λ m , we have the implication m(Y, λ ) > 0 and m(Y, λ ) > 0
⇒
ω¯ λ (zm ) = ω¯ λ (zm ).
Thus, we can define ξ (Y ) ∈ k by the condition that ω¯ λ (zm ) = ξ (Y ) for all λ ∈ Λ m such that m(Y, λ ) > 0. We also define dY := m(Y, λ ) λ ∈Λ . The following key m result shows how these vectors are related to the decomposition matrix Dθ of H. Theorem 3.4.4. In the above setting, assume that A is the ring of Laurent polynomials in finitely many indeterminates over a Dedekind domain. Then, for each Y ∈ E , the vector dY is a linear combination of the columns of Dθ , where the coefficients are non-negative integers. In particular, we have | Irr(H k )| rank m(Y, λ ) Y ∈E ,λ ∈Λ (Λ := Λ n ).
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3 Specialisations and Decomposition Maps
Proof. The assumptions on A imply that A is a “regular” ring; see Matsumura [248, Theorem 19.5]. Consequently, one can apply the general construction in [248, Exc. 14.4]. This yields a discrete valuation ring Aˆ ⊆ K such that A ⊆ Aˆ and ˆ (See also [132, Exc. 8.7] for p ∩ A = ker(θ ), where p is the maximal ideal of A. ˆ some further details.) Let k be the residue field of O. Since p ∩ A = ker(θ ), there is an embedding ι : k → kˆ and so θ extends to a map θˆ : Aˆ → kˆ such that θˆ |A = ι ◦ θ . Then, by Remark 3.1.5, the decomposition matrices associated with θ and with θˆ are the same, up to possibly permuting some columns. Thus, replacing A by Aˆ and ˆ we can assume without loss of generality that A is a discrete valuation ring; k by k, in particular, θ : A → k is a principal specialisation in the sense of Section 3.3. Let Dθ = dλ ,M λ ∈Λ , M∈Irr(H ) k
be the decomposition matrix of H. We shall now also need some results about projective H-modules. Since H K is split semisimple, a result of Heller (see [53, 30.18]) shows that the Krull–Schmidt–Azuyama theorem holds for H-modules which are finitely generated and free over A. (Such H-modules will be called H-lattices; if M is an H-lattice, we denote MK := K ⊗A M and Mk := k ⊗A M.) Furthermore, idempotents can be lifted from H k to H (see [53, Exc. 6.16]). Thus, the general results from “Brauer’s modular representation theory” are valid without passing to the completion of A. In particular, the following statement holds (see Feit [83, §I.17], [132, 7.5.2], and also 4.1.11 in Chapter 4). Let P be a projective indecomposable H-lattice. Then Pk is a projective indecomposable H k -module and, hence, M := Pk /rad(Pk ) ∈ Irr(H k ); furthermore, in R0 (H K ), we have [PK ] =
∑ dλ ,M [V λ ]
“Brauer reciprocity”.
λ ∈Λ
Thus, the vector containing the multiplicities of the various simple modules of H K as irreducible constituents of PK is a column of the matrix Dθ . Consequently, given any projective H-lattice P (not necessarily indecomposable), the corresponding vector of multiplicities will be a sum of some columns of Dθ . Thus, we see that it is enough to show that, for each Y ∈ E , there exists a projective H-lattice P such that Y ∼ = PK . We shall prove this by using the inductive definition of E = En ; note that all of the above remarks hold not only for H, but also for any subalgebra H m where m 1. First let m = 1. Then E1 = {1} and, clearly, 1 is obtained by scalar extension from a projective H 1 -lattice. Now assume that m 2 and let Y ∈ Em . By definition, we have (m) Y = ξ -IndK (Y ) for some Y ∈ Em−1 and some ξ ∈ k. So, arguing by induction, we already know that there exists a projective H m−1 -lattice P such that Y ∼ = PK . Now, the process of inducing representations also works over A, and (F2) implies that H P˜ := IndH mm−1 P ) = H m ⊗H m−1 P
is a projective H m -lattice.
∼ P˜K ; hence, we will try to find the required projective H We have IndK (Y ) = m ˜ For this purpose, consider the Brauer graph of lattice P as a direct summand of P. (m)
3.4 Towards Canonical Basic Sets for Classical Types
165
H m and the corresponding θ -blocks. Thus, as in 3.1.8, we have a partition Irr(H K,m ) = Irr1 (H K,m ) · · · Irrr (H K,m ), Now, by Brauer reciprocity, the irreducible constituents (over K) of a projective indecomposable H m -lattice all belong to the same θ -block. Thus, we have a canonical decomposition P˜ = P1 ⊕ · · · ⊕ Pr , where Pi is the sum of all indecomposable direct summands of P˜ whose irreducible constituents (over K) belong to Irri (H K,m ). Consequently, by Lemma 3.1.10, for each i ∈ {1, . . . , r} there exists a well-defined ξi ∈ k such that for all λ ∈ Λ m such that V λ is a constituent of PK,i .
ω¯ λ (zm ) = ξi
Thus, setting P := i Pi , where the sum runs over all i ∈ {1, . . . , r} such that ξi − ξ (Y ) = ξ , we have Y ∼ = PK , as required. Proposition 3.4.5. In the setting of Theorem 3.4.4, assume that dθ : R0 (H K ) → R0 (H k ) is surjective and that we are given a partial order on Irr(H K ). Furthermore, assume that there exists a subset Λ ⊆ Λ and a set of representations {Pμ | μ ∈ Λ } of H K such that the following conditions hold. (a) We have | Irr(H k )| |Λ |. (b) For each μ ∈ Λ , some positive multiple of Pμ lies in E . (c) For each μ ∈ Λ , we have the following relation in R0 (H K ): [Pμ ] = [V μ ] +
∑
λ ∈Λ :V λ V μ
mλ μ [V λ ],
where mλ μ ∈ Z0 for all λ .
Then | Irr(H k )| = |Λ | and Bθ := {V μ | μ ∈ Λ } is a canonical basic set for H k with respect to , in the sense of 3.1.7. Proof. For any μ ∈ Λ , let dμ denote the vector which contains the multiplicities of the various V λ (λ ∈ Λ ) as constituents of Pμ . By (b) and Theorem 3.4.4, there exists a positive integer pμ such that pμ dμ is a sum of some columns of Dθ . Now, the assumption that dθ is surjective means that the elementary divisiors of the matrix Dθ are all 1. This implies that dμ must already be a sum of some columns of Dθ . Using (c), we see that there exists a column of Dθ which has coefficient 1 in the row labelled by μ ; furthermore, the coefficient in a row labelled by any λ ∈ Λ is 0 unless λ μ . Let us denote this column of Dθ by dμ . By (c), the vectors {dμ | μ ∈ Λ } are linearly independent. Hence, Dθ has at least |Λ | columns. Using (a), we conclude that Dθ has exactly |Λ | columns. It follows that Dθ consists precisely of the column vectors {dμ | μ ∈ Λ }. Consequently, we can write Irr(H k ) = {M μ | μ ∈ Λ } and Dθ = (dλ ,μ )λ ∈Λ , μ ∈Λ , where (Δ )
dμ ,μ = 1 dλ ,μ = 0
for all μ ∈ Λ , unless λ = μ or V λ V μ .
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3 Specialisations and Decomposition Maps
Thus, the conditions in 3.1.7 are satisfied for the set Bθ := {V μ | μ ∈ Λ }.
Remark 3.4.6. In the above discussion of Fock data, we have worked with only one central element zm in each of the subalgebras H m , for m = 1, . . . , n. Of course, one could also work with collections of central elements {zm,i | i ∈ Im } ⊆ Z(H m ) at each stage. The refined induction operator in 3.4.1 would then have to be defined as follows. For ξ := {ξi j | i ∈ Im , j ∈ Im−1 } ⊆ k, we let
ξ -IndK (V μ ) := (m)
λ
V λ ⊕ . . . ⊕V λ , m(λ , μ ) times
where the sum runs over all λ ∈ Λ m such that ω¯ λ (zm,i ) − ω¯ μ (zm−1, j ) = ξi j for all i ∈ Im and j ∈ Im−1 . With this modification, it is clear that Theorem 3.4.4 and Proposition 3.4.5 continue to hold without any further change. 3.4.7. We return to our basic set-up as in 3.1.12, where H = HA (W, S, L) is a generic Iwahori–Hecke algebra over A = R[Γ ] and R is assumed to be L0 -good. Let θ : A → k be a specialisation into a field k, where k is the field of fractions of θ (A). By Theorem 3.1.14, we have a corresponding decomposition map dθ : R0 (HK ) → R0 (Hk ); let Dθ be the decomposition matrix. In order to exhibit an abstract Fock datum for H, let us consider a chain of parabolic subgroups W1 = {1} ⊆ W2 ⊆ · · · ⊆ Wn = W and the corresponding chain of parabolic subalgebras H1 = T1 A ⊆ H2 ⊆ · · · ⊆ Hn = H. Then (F1) and (F2) hold by general results on parabolic subalgebras; see [132, §9.1]. Since R is L0 -good, (F4) holds by Lemma 3.1.13. So it remains to choose suitable central elements zm ∈ Z(Hm ) for 1 m n. There are several candidates for zm ; perhaps the most natural idea is to use properties of the longest element in Wm . Example 3.4.8. Let n 1 and (W, S) be of type An−1 or Bn where S = {s1 , . . . , sn−1 } (in type An−1 ) or S = {s0 , s1 , . . . , sn−1 } (in type Bn ), with the labelling in Table 1.1. Having fixed this labelling, there is a natural chain of parabolic subgroups. Indeed, as in 1.7.6, for any m ∈ {1, . . . , n}, let Wm ⊆ W be the parabolic subgroup generated by s0 , s1 , . . . , sm−1 (where we set s0 = 1 in type An−1 ). Then Wm is of type Am−1 or Bm respectively. In particular, we have Wn = W . We set L1 := Ts0
and
Lm := Tsm−1 · · · Ts2 Ts1 Ts0 Ts1 Ts2 · · · Tsm−1
for 2 m n.
(Note that Ts0 is the identity element in type An−1 .) The elements Lm will be called the Jucys–Murphy elements of Hn ; see Hoefsmit [157], Mathas [245, §3.3] and Ram [268, (3.16), (4.20)]. We shall set zm := L1 + L2 + · · · + Lm ∈ Hm
for 1 m n.
The following result shows that zm ∈ Z(Hm ) for all m. Lemma 3.4.9. Let w0 ∈ Wn be the longest element. Then the following hold.
3.4 Towards Canonical Basic Sets for Classical Types
(a) (b) (c) (d)
167
If (W, S) is of type An−1 , then Tw20 = L1 L2 · · · Ln ∈ Z(Hn ). If (W, S) is of type Bn , then Tw0 = L1 L2 · · · Ln ∈ Z(Hn ). The elements L1 , . . . , Ln commute with each other. We have zn = L1 + · · · + Ln ∈ Z(Hn ).
Proof. (a) We use induction on n. If n = 1, then W1 = {1} and Tw0 = L1 = T1 ; so the assertion is clear. Now assume that n 2. By the description of w0 in [132, Exp. 1.5.4], we have w0 = w 0 sn−1 sn−2 · · · s1 , where w 0 is the longest element in Wn−1 = s1 , . . . , sn−2 ∼ = Sn−1 . By induction, Tw2 = L1 L2 · · · Ln−1 and so 0
L1 L2 · · · Ln = Tw Tw Tsn−1 · · · Ts2 Ts1 Ts1 Ts2 · · · Tsn−1 = Tw Tw0 Ts1 Ts2 · · · Tsn−1 . 0
0
0
Now, for any i ∈ {1, . . . , n − 1}, we have w0 si = sn−i w0 and so Tw0 Tsi = Tsn−i Tw0 . Applying this relation repeatedly, we obtain Tw0 Ts1 Ts2 · · · Tsn−1 = Tsn−1 Tw0 Ts2 · · · Tsn−1 = · · · = Tsn−1 · · · Ts2 Ts1 Tw0 and so L1 L2 · · · Ln = Tw Tsn−1 · · · Ts2 Ts1 Tw0 = Tw20 , as desired. Finally, we have al0 ready noted in Example 1.2.7 that Tw20 ∈ Z(Hn ). (b) Again, we use induction on n. If n = 1, then W1 = Ts0 has order 2 and Tw0 = L1 = Ts0 ; so the assertion is clear. Now assume that n 2. By the description of w0 in [132, Exp. 1.4.6], we have w0 = w 0tn−1 ,
where
tn−1 := sn−1 sn−2 · · · s1 s0 s1 · · · sn−2 sn−1
and w 0 is the longest element in Wn−1 = s0 , s1 , . . . , sn−2 . Note that Ln = Ttn−1 . By induction, Tw 0 = L1 L2 · · · Ln−1 and so L1 L2 · · · Ln = Tw 0 Ttn−1 = Tw0 , as required. Finally, by [132, Exp. 1.4.6], w0 lies in the centre of Wn and, hence, Tw0 ∈ Z(Hn ). (c) Let hm := L1 · · · Lm ∈ Hm for 1 m n. By (a) and (b), we have hm ∈ Z(Hm ). It follows that the elements h1 , . . . , hn commute with each other. Furthermore, each hm equals a power of a standard basis element of Hm and, hence, is invertible in Hm . So, since hm = hm−1 Lm for m 2, the elements L1 , . . . , Ln also commute with each other. This argument also shows that Lm commutes with Tsi for 0 i m − 2. (d) Again, we use induction on n. If n = 1, then z1 = L1 and the assertion is clear by (a) and (b). Now assume that n 2. We have zn = zn−1 + Ln . By induction, zn−1 ∈ Z(Hn−1 ) and so zn−1 commutes with Tsi for 0 i n − 2. As already noted in the proof of (c), the element Ln also commutes with Tsi for 0 i n − 2. Hence, it remains to show that zn commutes with Tsn−1 . Now, it is clear that Tsn−1 commutes with all elements in Hn−2 ; in particular, Tsn−1 will commute with L1 , . . . , Ln−2 . So, finally, it will be enough to show that Tsn−1 commutes with Ln−1 + Ln . Now, we have Ln = Tsn−1 Ln−1 Tsn−1 and so Tsn−1 Ln = Ts2n−1 Ln−1 Tsn−1 = Ln−1 Tsn−1 + (v − v−1 )Ln , Ln Tsn−1 = Tsn−1 Ln−1 Ts2n−1 = Tsn−1 Ln−1 + (v − v−1 )Ln .
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3 Specialisations and Decomposition Maps
This yields Tsn−1 Ln − Ln Tsn−1 = Ln−1 Tsn−1 − Tsn−1 Ln−1 and so Ln−1 + Ln com mutes with Tsn−1 , as required. For our applications to Iwahori–Hecke algebras, the following version of Proposition 3.4.5 will be useful. Proposition 3.4.10. In the setting of 3.4.7, let zm ∈ Z(Hm ) for 1 m n and consider the corresponding set of representations E . Assume that it is known that a canonical basic set exists for Hk ; see Definition 3.2.1. Furthermore, assume that there exists a subset E ⊆ E such that the following conditions hold. (a) We have | Irr(Hk )| |E |. (b) For each Y ∈ E , there exists some λY ∈ Λ such that, in R0 (HK ), we have [Y ] = m(Y, λY )[EελY ] +
μ
∑
μ ∈Λ : aμ >aλ
m(Y, μ ) [Eε ],
Y
where m(Y, μ ) ∈ Z0 for all μ and m(Y, λY ) = 0. (c) We have λY = λY for Y = Y in E . Then | Irr(Hk )| = |E | and Λ := {λY | Y ∈ E } is the canonical basic set for Hk . Proof. Recall that A = R[Γ ] is the ring of Laurent polynomials in finitely many indeterminates. Furthermore, we can assume without loss of generality that either K = R or that K is a finite extension of Q and that R is the ring of algebraic integers in K. In both cases, the general assumption on A in Theorem 3.4.4 is satisfied. Now let Λ := {λY | Y ∈ E } ⊆ Λ . By (c), the map E → Λ , Y → λY , is a bijection and so we can also write E = {Yμ | μ ∈ Λ }, where λYμ = μ for all μ ∈ Λ . For any μ ∈ Λ , let dμ denote the vector which contains the multiplicities of the various Eελ (λ ∈ Λ ) as constituents of Yμ . Using (b) we see that there exists a column of Dθ which has a non-zero coefficient in the row labelled by μ ; furthermore, the coefficient in a row labelled by any λ ∈ Λ is 0 unless λ = μ or aλ > aμ . Let us denote this column of Dθ by dμ . As in the proof of Proposition 3.4.5, we can conclude that Dθ consists precisely of the column vectors {dμ | μ ∈ Λ }. Furthermore, we can write Irr(Hk ) = {M μ | μ ∈ Λ } and Dθ = (dλ ,μ )λ ∈Λ , μ ∈Λ , where (Δ )
0 dμ ,μ = dλ ,μ = 0
for all μ ∈ Λ , unless λ = μ or aλ > aμ .
Now, we are assuming that Hk admits a canonical basic set. Hence, comparing (Δ ) with the relations (Δ a ) in Remark 3.2.2, we conclude that we must have dμ ,μ = 1 for all μ ∈ Λ , and Λ is the canonical basic set. We close this section by explaining a general method for computing | Irr(Hk )|. Proposition 3.4.11 ([132, 7.5.7]). Recall from Example 1.2.3 the definition of the character table X(H). Then, in the setting of Theorem 3.1.14, we have | Irr(Hk )| = rank of X(H)θ ,
3.4 Towards Canonical Basic Sets for Classical Types
169
where X(H)θ is the matrix obtained from X(H) by applying θ to each entry. Proof. By Theorem 3.1.14, the columns of the corresponding decomposition matrix Dθ are linearly independent over k. Hence we can apply [132, Cor. 7.5.7], which yields that | Irr(Hk )| can be expressed by the rank of the above matrix. This has been used in [132, Prop. 11.5.13] to determine the numbers | Irr(Hk )| where W is of exceptional type and k = C. We shall now apply this to W ∼ = Sn . 3.4.12. Let (W, S) be of type An−1 and keep the notation of Example 3.2.9. In particular, A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Following Dipper and James [62], given a specialisation θ : A → k, we set e := min{i 2 | 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0}
(u := v2 ).
(We let e = ∞ if 1+ θ (u)+ θ (u)2 +· · ·+ θ (u)i−1 = 0 for all i 2.) Thus, if θ (u) = 1, then e is the multiplicative order of θ (u) ∈ k× . On the other hand, if θ (u) = 1 and char(k) = > 0, then e = ; finally, if θ (u) = 1 and char(k) = 0, then e = ∞. By the discussion in Example 3.1.19, we have Hk is semisimple
⇔
e > n.
(Recall that the degrees of W ∼ = Sn are 2, 3, . . . , n.) Lemma 3.4.13 (Cf. [131, 4.3]). In the setting of 3.4.12, let Λe denote the set of all partitions of n which have no part divisible by e. Then we have | Irr(Hk )| |Λe |, where equality holds if char(k) = 0. Proof. First we deal with the case where e = ∞. Then Hk is semisimple by 3.4.12. So | Irr(Hk )| = | Irrk (Sn )| is the number of all partitions of n; see 3.1.18. On the other hand, all partitions of n are in Λe in this case. Thus, the statement is true if e = ∞. We can assume from now on that e < ∞. Now, by Proposition 3.4.11, the problem is equivalent to computing the rank of the specialised matrix X(H)θ . We shall use Starkey’s rule (see [99], [132, 9.2.11], [285]), which shows that the character table X(H) is obtained by the multiplying ordinary character table of the group W ∼ = Sn by the matrix R := rλ μ λ ,μ n , where rλ μ
⎧ ⎨ |Cλ ∩Wμ | (u − 1)−l(μ ) (uλi − 1) ∏ |Wμ | = i1 ⎩ 0
if Cλ ∩Wμ = ∅, otherwise.
Here, l(μ ) denotes the number of non-zero parts of μ ; furthermore, Cλ denotes the conjugacy class of elements of W ∼ = Sn of cycle type λ and Wμ ⊆ W denotes the parabolic subgroup corresponding to the Young subgroup Sμ ⊆ Sn . In particular, all entries of X(H) lie in the ring B := Z[u, u−1 ]. Now assume first that char(k) = 0 and let q := θ (u). Then e 2 is the multiplicative order of q ∈ k× and so we have, for any integer m 1:
170
(∗)
3 Specialisations and Decomposition Maps
θ (um − 1) = 0
⇔
e divides m. ¯ = |Λe |, where R¯ := θ (rλ μ ) . Now, we must show that rank(R) λ ,μ n This is seen as follows. Let λ be a partition of n which is not in Λe ; that is, some part λi of λ is divisible by e. Then (∗) shows that θ (uλi − 1) = 0 and so θ (rλ μ ) = 0 ¯ |Λe |. Next note that R has a triangular shape for all μ n. Thus, we have rank(R) for a suitable ordering of the rows and colums, since rλ μ = 0 unless Cλ ∩ Sμ = ∅; that is, λ is a refinement of μ . So it will be enough to show that θ0 (rλ λ ) = 0 for all ¯ = |Λe |, as required. λ ∈ Λe . But this is clear by (∗). Thus, we have rank(R) Finally, consider the case where char(k) = > 0. Let Φe ∈ Z[u] be the eth cyclotomic polynomial and Be := Z[u, u−1 ]/(Φe ); let θe : B → Be be the canonical map. Then, by the definition of e, we have θ (Φe (u)) = Φe (q) = 0 and so there is a factorisation θ |B = ϕ ◦ θe , where ϕ : Be → k is a ring homomorphism. Clearly, the rank of X(H)θ is less than or equal to the rank of X(H)θe (over the field of fractions of Be ). Since the field of fractions of Be has characteristic 0, we already know that the rank of X(H)θe is given by |Λe |, as required. In the above proof, we employed in a special situation the technique of factorising a specialisation into a field of positive characteristic through a specialisation into a field of characteristic 0. This idea will be discussed in a wider context in Section 3.6. Example 3.4.14. Let n = 4. The character table of H is given as follows: ⎡ ⎤ 1 −1 1 1 −1 ⎢ 3 u − 2 −2u + 1 −u + 1 u ⎥ ⎢ ⎥ 2 −u 0 ⎥ X(H) = ⎢ (see [132, §10.2]). ⎢ 2 u − 1 u2 + 1 ⎥ ⎣ 3 2u − 1 u − 2u u2 − u −u2 ⎦ 1 u u2 u2 u3 Then, Starkey’s rule (see the above proof) yields the product decomposition ⎤ ⎡ (u−1)2 (u−1)2 (u−1)3 ⎡ ⎤ 1 u−1 2 4 6 24 1 −1 1 1 −1 ⎢ ⎥ (u2 −1)(u−1) ⎥ u2 −1 u2 −1 0 u+1 ⎢ 3 −1 −1 0 1 ⎥ ⎢ ⎥ ⎢ 2 2 2 4 ⎢ ⎥ ⎢ (u2 −1)(u+1) ⎥ , ⎥ · ⎢ 0 0 (u+1)2 2 0 2 −1 0 X(H) = ⎢ ⎥ 0 ⎢ ⎥ 4 8 ⎥ ⎣ 3 1 −1 0 −1 ⎦ ⎢ 2 3 ⎥ ⎢0 0 u +u+1 u −1 0 ⎦ ⎣ 3 3 1 1 1 1 1 u3 +u2 +u+1 0 0 0 0 4 where the first factor is the character table of W ∼ = S4 and the second factor is the matrix R. The partitions labelling the rows and columns of these matrices are ordered as follows: (1111), (211), (22), (31), (4). Thus, for example, the first row of X(H) corresponds to the sign representation and the first column to the conjugacy class containing the identity element. Now consider a specialisation θ : A → k where char(k) = 0 and e = 2; that is, θ (u) = −1. Then the second, third and fifth rows of R specialise to 0 while the diagonal entries of the first and fourth rows, labelled by (1111) and (31), remain non-zero. Thus, we have | Irr(Hk )| = rank(X(H)θ ) = 2.
3.5 A Canonical Basic Set for the Symmetric Group
171
3.5 A Canonical Basic Set for the Symmetric Group We are now ready to determine a canonical basic set for W ∼ = Sn . This was first achieved by Dipper and James [62]. Here, we will follow a different approach based on the ideas in the previous section. This will also provide a model for dealing with algebras of type Bn and, more generally, Ariki–Koike algebras in Chapter 5. 3.5.1. Let (W, S) be of type An−1 and keep the notation of Example 3.2.9. In particular, W ∼ = Sn and A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . A Fock datum as in 3.4.1 is obtained as follows. For any m ∈ {1, . . . , n}, consider the parabolic subgroup Wm := s1 , . . . , sm−1 ∼ = Sm and denote the corresponding generic algebra by Hm . Then we have a chain of parabolic subgroups and parabolic subalgebras W1 = {1} ⊆ W2 ⊆ · · · ⊆ Wn = W
and H1 = T1 A ⊆ H2 ⊆ · · · ⊆ Hn = H.
For 1 m n, let Λm be the set of all partitions λ m. In Example 3.4.8, we already defined central elements zm ∈ Hm . Recall that zm = L1 + · · · + Lm , where L1 = T1
and Lm = Tsm−1 · · · Ts2 Ts1 Ts1 Ts2 · · · Tsm−1
for 2 m n.
Since the above definition involves Ts1 Ts1 , the formula can be rewritten as follows: Lm = Tsm−1 · · · Ts2 T1 + (v − v−1 )Ts1 Ts2 · · · Tsm−1 = Tsm−1 · · · Ts3 Ts2 Ts2 Ts3 · · · Tsm−1 + (v − v−1 )Tsm−1 ···s2 s1 s2 ···sm−1 . We repeat the argument by writing Ts22 = T1 + (v − v−1 )Ts2 , and so on. Also note that si · · · sm−2 sm−1 sm−2 · · · si = (i, m) for 1 i < m. Hence, eventually, we find that Lm :=
Lm − T1 = v−1 (T(1,m) + T(2,m) + · · · + T(m−1,m) ) ∈ Hm . v2 − 1
It will turn out that it is actually more convenient to work with Lm instead of Lm . (See also the comments in [245, Note 7, p. 54].) We shall set z m := L1 + L2 + · · · + Lm ∈ Hm
for 1 m n.
Since zm ∈ Z(Hm ) for all m, it is clear that we also have z m ∈ Z(Hm ) for all m. Observe that vz n = ∑w∈C Tw , where C is the conjugacy class containing S; hence, zC = z n is the central basis element corresponding to C as in Example 3.3.8. In (m) what follows, we assume that the operators ξ -IndK are defined with respect to the (m) elements {z m }. Our next task is to find explicit formulae for ξ -IndK . 3.5.2. We introduce some useful combinatorial notions. Let n 0 and λ n. Writing λ = (λ1 λ2 . . . 0), the Young diagram of λ is defined by [λ ] := {(a, b) | a 1 and 1 b λa } ⊆ Z1 × Z1 .
172
3 Specialisations and Decomposition Maps
The elements of [λ ] are called the nodes of λ . The content of a node γ = (a, b) ∈ [λ ] is defined by c(γ ) := b − a. A Young diagram [λ ] will be visualised by an array of boxes in left-justified rows, with λ1 boxes in the first row, λ2 boxes in the second row and so on. The set of border nodes of λ is defined by Nλ := {(a, λa ) | a 1 and λa > 0} ⊆ [λ ]. By definition, a λ -tableau is a filling of the boxes of [λ ] by the numbers 1, . . . , n. Thus, more formally, a λ -tableaux is a bijection t : [λ ] → {1, . . . , n}. We say that t is row-standard if the sequence t(a, 1), t(a, 2), . . . is strictly increasing for each a; we say that t is column-standard if the sequence t(1, b), t(2, b), . . . is strictly increasing for each b. We say that t is standard if t is row-standard and column-standard. Here are some examples, where n = 5 and λ = (3, 2): 1 3 5 2 4 (λ -diagram)
1 4 5 2 3
(standard tableau)
(row-standard tableau)
Here, Nλ = {(1, 3), (2, 2)}. Now let n 1 and recall that, by 1.7.6, we have λ ∼ n ResW Wn−1 (E ) =
Eμ
for some subset Λ (λ ) ⊆ Λn−1 .
μ ∈Λ (λ )
This subset can now be described more precisely as follows. We say that γ ∈ [λ ] is a removable node if [λ ] = [μ ] ∪ {γ } for some partition μ n − 1; alternatively, we say that λ can be obtained from μ by adding the node γ . Then the “branching rule” (see [132, 6.1.8]) can be stated as follows:
Λ (λ ) = {μ n − 1 | [λ ] = [μ ] ∪ {γ } for some removable node γ ∈ Nλ }. Thus, Λ (λ ) consists of all partitions of n − 1 which can be obtained by removing a border node from Λ . Lemma 3.5.3. Let n 1, λ n and μ ∈ Λ (λ ). Then, writing [λ ] = [μ ] ∪ {γ }, where γ = (a, λa ) ∈ Nλ is removable, we have
ωλ (z n ) − ωμ (z n−1 ) =
v2c(γ ) − 1 , v2 − 1
where
c(γ ) = λa − a.
Proof. Let {σ λ | λ ∈ Λm , 1 m n} be a family of representations as in Proposition 1.7.7. Given λ ∈ Λn , let Λ (λ ) = {μ1 , . . . , μr } ⊆ Λn−1 ; then σ λ (Tw ) (w ∈ Wn−1 ) is a block diagonal matrix where the blocks are given by the matrices σ μl (Tw ) for 1 l r. By Lemma 3.4.9(a), we have Tw20 = Tw2 Ln , where w 0 is the longest ele0 ment in Wn−1 . Using Example 1.2.7, we obtain
3.5 A Canonical Basic Set for the Symmetric Group
⎡ v
2Nλ
Idλ = σ
λ
⎢ ⎢
173
v2Nμ1 Id1
0
0 .. .
v2Nμ2 Id2 .. .
0
...
(Tw20 ) = ⎢ ⎢ ⎣
⎤ ... 0 ⎥ .. .. ⎥ λ . . ⎥ · σ (Ln ), ⎥ .. . 0 ⎦ 0 v2Nμr Idr
where dl = dim E μl for l = 1, . . . , r. This already shows that σ λ (Ln ) is a diagonal matrix with eigenvalues v2(Nλ −Nμl ) , where each of these occurs with multiplicity equal to dl for 1 l r. Now, by Example 1.3.4, we have Nλ = aλ † − aλ
Nμl = aμ † − aμl .
and
l
Furthermore, write [λ ] = [μl ] ∪ {γl }, where γl = (al , λal ) ∈ Nλ is removable. Then the formulae for the a-invariants in Example 1.3.8 directly imply that aλ − aμl = al − 1
and
aλ † − aμ † = λal − 1. l
This yields Nλ −Nμl = (λal −1) − (al −1) = λal −al = c(γl ). So σ λ (Ln ) has eigenvalues v2c(γl ) where each of these occurs with multiplicity equal to dl for 1 l r. Using that Ln = (v2 − 1)Ln + T1 , we see that σ λ (Ln ) is diagonal with eigenvalues (v2c(γl ) − 1)/(v2 − 1). On the other hand, since z n = z n−1 + Ln , we also have ⎡ ⎢ ⎢ ωλ (z n ) Idλ = ⎢ ⎢ ⎣
ωμ1 (z n−1 ) Id1
0
0 .. .
ωμ2 (z n−1 ) Id2 .. .
0
...
⎤ ... 0 ⎥ .. .. ⎥ . . ⎥ + σ λ (Ln ). ⎥ .. ⎦ . 0 0 ωμr (z n−1 ) Idr
Hence, for each l, the difference ωλ (z n ) − ωμl (z n−1 ) is an eigenvalue of σ λ (Ln ) which appears in the diagonal block corresponding to σ μl in the above matrices; so this eigenvalue equals (v2c(γl ) − 1)/(v2 − 1), as desired. 3.5.4. Let θ : A → k be a specialisation such that k is the field of fractions of θ (A). As in 3.4.12, we set e := min{i 2 | 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0} (We let e = ∞ if no such i exists.) For any integer b ∈ Z, we set ⎧ 1 + θ (u) + θ (u)2 + · · · + θ (u)b−1 ⎨ 1 [b]θ := ⎩ −θ (u)b (1 + θ (u) + θ (u)2 + · · · + θ (u)−b−1 ) Then we note that, for any b, b ∈ Z, we have [b]θ = [b ]θ
⇔
b ≡ b mod e.
(u := v2 ).
if b > 0, if b = 0, if b < 0.
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3 Specialisations and Decomposition Maps
Furthermore, set ξι := 1 + θ (u) + · · · + θ (u)ι −1 ∈ k for 0 ι e − 1. Then, for any b ∈ Z, we have [b]θ = ξι , where ι ∈ {0, 1, . . . , e − 1} is such that b ≡ ι mod e. Now let n 1 and μ n − 1. Using the formula in Lemma 3.5.3, we can write
ω¯ λ (z n ) − ω¯ μ (z n−1 ) = [c(γ )]θ ,
[λ ] = [μ ] ∪ {γ }.
where
With these preparations, we can now state the following proposition. Proposition 3.5.5 (Cf. Lascoux, Leclerc and Thibon [208, 3.3]). Given any partition μ n − 1 (where n 1) and any integer ι ∈ {0, 1, . . . , e − 1}, we have (n) ξι -IndK Evμ ∼ =
Evλ , ι
λ n such that μ →λ ι
where we write μ → λ if [λ ] = [μ ] ∪ {γ } and γ ∈ Nλ is a removable node such that c(γ ) ≡ ι mod e. Proof. By 3.5.2, Frobenius reciprocity and a general compatibility of induction with specialisations (see [132, §9.1]), the restriction formula in 3.5.2 translates to (n) IndK Evμ ∼ =
λ n such that
Evλ . μ ∈Λ (λ )
Now let λ n be such that μ ∈ Λ (λ ). Then [λ ] = [μ ] ∪ {γ }, where γ ∈ [λ ]; see (n) μ 3.5.2. Using Lemma 3.5.3, we obtain ξι -IndK Ev ∼ = λ Evλ , where λ runs over all partitions of n such that [λ ] = [μ ] ∪ {γ } for some removable γ ∈ Nλ satisfying [c(γ )]θ = ξι . By 3.5.4, the latter condition is equivalent to c(γ ) ≡ ι mod e. The above result provides the basis for a purely combinatorial description of the sets E1 , E2 , E3 , . . .. Our aim will now be to understand how this works in detail. Example 3.5.6. Given e 2, we define a directed graph as follows. The vertices are given by the elements of the set Λ∞ := n0 Λn , and the edges are defined by ι the relation μ → λ in Proposition 3.5.5. Then the sets of representations En are completely determined by this graph, which may be called the LLT-graph. (e) Let In be the set of all sequences η = (ι1 , ι2 , . . . , ιn ) such that ι1 = 1 and ι j ∈ {0, 1, . . . , e − 1} for all j 2. Given η , we obtain a sequence of representations: Y(ι1 ) := 1,
(2) Y(ι1 ,ι2 ) := ξι2 -IndK Y(ι1 ) ,
...,
(n) Yη := ξιn -IndK Y(ι1 ,...,ιn−1 ) . (e)
Hence, with this notation, we have En = {Yη | η ∈ In } \ {0}. For example, let e = 2. In Figure 3.1, we show the part of the LLT-graph corresponding to all partitions of n 5. In the graph, each partition is represented by its Young diagram where the boxes are filled by c(x) mod e; furthermore, we just write λ instead of Evλ and l · λ instead of Evλ ⊕ · · · ⊕ Evλ (l terms).
3.5 A Canonical Basic Set for the Symmetric Group
175
Fig. 3.1 Part of the LLT-graph for e = 2 0 1
1
@ R
0 1 0
1
R @
A1 U A
1
0 1 0 1 0
E1 E2 E3 E4 E5
= = = = =
B B
1
1
BBN
0 1 0 1 1
0
0
0A
U A
0 1 0 1
0 1 0 1 0
1
R @
0 1 1
0 1 0
0 1
B B BN
0
0 @ R @
0 1 1 0
0 1 0 1
1 ?Q s Q
0 1 1 0
@ 0 0 @ 0 @0 J1
@ @
B0 J R @
@ ^ R BN J
0 1 0 1 0
0 1 0 1 0
0 1 1 0 0
0 1 0 1 1 CCW
0 Q s Q
0 1 1 0 1
0 1 0 1 0
{Y(0) = (1)}, {Y(0,1) = (2) ⊕ (11)}, {Y(0,1,0) = (3) ⊕ (111), Y(0,1,1) = 2 · (21)}, {Y(0,1,0,1) = (4) ⊕ (31) ⊕ (211) ⊕ (1111), Y(0,1,1,0) = 2 · (31) ⊕ 2 · (22) ⊕ 2 · (211)}, {Y(0,1,0,1,0) = (5) ⊕ (32) ⊕ 2 · (311) ⊕ (221) ⊕ (11111), Y(0,1,0,1,1) = Y(0,1,1,0,1) = 2 · (41) ⊕ 2 · (2111), Y(0,1,1,0,0) = 4 · (32) ⊕ 4 · (311) ⊕ 4 · (221)}.
The LLT-graph is a remarkable combinatorial object; see Lascoux, Leclerc and Thibon [208], Kleshchev [202], [203] and Grojnowski [146] for further reading. Remark 3.5.7. Let μ n and s be a μ -tableau. We define a corresponding sequence (e) ηe (s) ∈ In as follows. For j ∈ {1, . . . , n}, let c j be the content of the node which is filled by the number j in s. Then ηe (s) := (ι1 , . . . , ιn ), where ι j ≡ c j mod e for 1 j n. This sequence will be called the e-residue sequence of s. (e) Now let η be an arbitrary sequence in In and consider the corresponding representation Yη ∈ En . Then the following two statements are equivalent: μ
(a) Ev is an irreducible constituent of Yη . (b) There exists a standard μ -tableau s such that η = ηe (s). To see this, note that a standard μ -tableau s is uniquely characterised by a sequence of partitions μ(1) , . . . , μ(n) = μ such that, for each j ∈ {1, . . . , n}, we have
μ( j) j
and
[μ( j) ] = [μ( j−1) ] ∪ γ j ,
where s(γ j ) = j.
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3 Specialisations and Decomposition Maps
Thus, μ(n−1) is obtained from μ = μ(n) by removing the unique node γn which is filled by the number n; then μ(n−2) is obtained from μ(n−1) by removing the unique node γn−1 which is filled by the number n − 1, and so on. The inductive definition of the sets En immediately yields the equivalence of (a) and (b). Now the plan is to construct some distinguished representations in En and to apply Proposition 3.4.10. This construction involves the following type of partitions. Definition 3.5.8. Let e 2 and λ n. We write λ = (1m1 , 2m2 , 3m3 , . . .), where the notation means that, for any i 1, there are mi parts equal to i. Then we say that λ is e-regular if mi < e for all i. 3.5.9. The construction of the desired distinguished representations involves a recursive procedure which associates with every e-regular partition λ n a “canonical” (e) standard λ -tableau te (λ ) and a corresponding sequence ηe (λ ) ∈ In . This procedure is a variation of (but not identical to) the ladder method used by Lascoux, Leclerc and Thibon [208, §6.2] (see also James and Kerber [182, 6.3.51, p. 283]). First, we need some notation. Let λ n be e-regular; let λ1 λ2 · · · λl > 0 be the non-zero parts of λ . Denote Nλ = {γ1 , . . . , γl },
where
γi = (i, λi ) ∈ [λ ] for 1 i l.
For b ∈ Z, let Nλ (b) be the set of all γi ∈ Nλ such that c(γi ) ≡ b mod e. Now let r 1 be the index of the first node in Nλ which can be removed. Then λ1 = · · · = λr > λr+1 ; let cr ∈ {0, 1, . . . , e−1} be such that c(γr ) ≡ cr mod e. Thus, γr is a removable node in Nλ (cr ). Now consider the set Nλ (cr −1). If N (cr −1) = ∅, let i0 1 be the minimal element of Nλ (cr − 1). If N (cr − 1) = ∅, set i0 = n + 1. Then note that r < i0 . Indeed, if Nλ (cr − 1) = ∅, this is clear since i0 = n + 1. On the other hand, if Nλ (cr − 1) = ∅, we have c(γ j ) ≡ cr + (r − j) mod e for 1 j r. Since λ is e-regular, we have r < e and so c(γ j ) ≡ cr − 1 mod e for 1 j r. This means that r < i0 , as claimed. Thus, we have shown that the set {i ∈ {1, . . . , l} | i < i0 and γi ∈ Nλ (cr ) is removable} is non-empty. Let i be the maximal element of this set. (We have r i < i0 .) We start building the desired tableau te (λ ) by assigning n to the node γi . We then remove the node γi from the diagram of λ and obtain a partition λ n − 1. Next we claim that λ is also an e-regular partition. Assume, if possible, that this is not the case. Since λ is e-regular, this could only happen if λi = λi+1 + 1 and λi+1 = λi+2 = · · · = λi+e−1 > λi+e . Then c(γi ) ≡ cr , c(γi+1 ) ≡ cr − 2, c(γi+2 ) ≡ cr − 3, . . . , c(γi+e−1 ) ≡ cr − e ≡ cr where all congruences are modulo e. Hence, we have γ j ∈ Nλ (cr − 1) for i j i + e − 1. The definition of i0 and the fact that i < i0 now imply that i + e i0 . But then γi+e−1 ∈ N (cr ) is removable and i + e − 1 < i0 , contradicting the choice of i. Thus, the assumption was wrong and so λ is e-regular. We can now repeat the above procedure with λ , find the appropriate node to be removed, fill that node by n − 1, and continue. Eventually, after n iterations, we
3.5 A Canonical Basic Set for the Symmetric Group
177
have filled all the nodes in [λ ] and, thus, constructed a standard λ -tableau te (λ ). Then we define ηe (λ ) to be the e-residue sequence of te (λ ). Here is an example where e = 3 and λ = (44321) 14. In the Young diagram of λ , each node γ is filled by c(γ ) mod e.
λ:
0 2 1 0 2
1 2 0 0 1 2 2 0 1
r = 2, c2 = 2, i0 = 4 Nλ (0) = {1, 3} Nλ (1) = {4} Nλ (2) = {2, 5}
λ :
0 2 1 0 2
1 2 0 0 1 2 0 1
So, the node γ2 = (2, 4) ∈ [λ ] will be removed and, in t3 (λ ), the corresponding node will be filled by n = 14. We repeat the argument with λ = (43321) 13: now, r = 1, c1 = 0, Nλ (0) = {1, 3}, Nλ (1) = {2, 4}, Nλ (2) = {5} and i0 = 5. Hence, the nodes γ1 = (1, 4) and γ3 = (3, 3) belong to Nλ (0) and are removable. Since 1 < 3 < i0 , the node γ3 will be removed and, in t3 (λ ), the corresponding node will be filled by n − 1 = 13. Continuing in this way, we eventually find that t3 (λ ) =
1 3 7 12 2 5 10 14 4 8 13 6 11 9
η3 (λ ) = (0, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 0, 2).
3.5.10. In order to state the crucial Lemma 3.5.11 below, it will be convenient to extend the notion of Young diagrams and tableaux to compositions. Recall that a composition of n is a finite sequence μ = (μ1 , . . . , μr ) of non-negative integers such that ∑1ir μi = n. The height of μ is defined as max{i | μi = 0} if μ = ∅, and as zero otherwise. As before, the set [μ ] := {(a, b) | 1 a r, 1 b μa } is called the Young diagram of μ and a bijection s : [μ ] → {1, . . . , n} will be called a μ -tableau. We also say that μ is the shape of s. A μ -tableau s is called row-standard if the sequence s(a, 1), s(a, 2), . . . is strictly increasing for each a. (Note that, now, it does not really make sense to define the notion of a column-standard tableau.) Now fix an integer e 2. For each γ = (a, b) ∈ [μ ], the number c(γ ) := b − a is called the content of γ . Furthermore, the unique ι ∈ {0, 1, . . . , e − 1} such that c(γ ) ≡ ι mod e is called the e-residue of γ and denoted by rese (γ ). As before, the e-residue sequence of a μ -tableau s is defined by ηe (s) = (ι1 , . . . , ιn ), where ι j is the e-residue of the node which is filled by the number j in s. Finally, we also extend the definition of the dominance order. Let μ , μ be compositions of n; then
μ μ
def
⇔
∑
1id
μi
∑
μi
for all d ∈ {1, . . . , r},
1id
where μ = (μ1 , μ2 , . . . , μr ) and μ = (μ1 , μ2 , . . . , μr ). (We can assume that μ , μ have the same number of parts by adding zeroes at the end if necessary.)
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3 Specialisations and Decomposition Maps
We can now state the following result, which contains the combinatorial essence of our modified “ladder method” in 3.5.9. It is inspired by (but slightly stronger than) a similar result by Mathas [245, Lemma 3.40]; see also Murphy [255, §3]. Lemma 3.5.11 (Jacon [174, §4]). Let λ n be an e-regular partition and μ be a composition of n. Assume that there exists a row-standard μ -tableau s such that ηe (s) = ηe (λ ). Then μ λ . Proof. Let us write t := te (λ ) and ηe (λ ) = (ι1 , . . . , ιn ). We shall argue by downward induction on d, where d is defined as follows. If t = s, then set d := n + 1; otherwise, let d ∈ {1, . . . , n} be minimal such that t−1 (d) = s−1 (d). Thus, if d = n + 1, then there is nothing to prove. Now assume that d n. Let at (d) be the row index of t−1 (d) and as (d) be the row index of s−1 (d). Then at (d) = as (d), by the minimality of d. We claim that at (d) < as (d). This is seen as follows. Let us successively remove the n − d nodes which are filled by n, n − 1, . . . , d + 1 from t = te (λ ). By the inductive construction in 3.5.9, the resulting tableau will be te (λ˜ ) for an e-regular partition λ˜ d. Similarly, let us successively remove the n − d nodes which are filled by n, n − 1, . . . , d + 1 from s. The result will be a row-standard μ˜ -tableau s˜ , for some composition μ˜ of d. Furthermore, we have ηe (te (λ˜ )) = ηe (μ˜ ). Let λ˜ d − 1 be the partition obtained by removing the node filled by d in te (λ˜ ) and let μ˜ be the composition of d − 1 obtained by removing the node filled by d in s˜ . Since d is minimal such that t−1 (d) = s−1 (d), we have λ˜ = μ˜ . Thus, both λ˜ and μ˜ are obtained by adding a node with eresidue ιd to the diagram of λ˜ . But, by the construction of te (λ˜ ) and λ˜ , all border nodes of λ˜ with row index strictly less than at (d) have e-residues which are not congruent to ιd − 1 modulo e. Hence, the only way to add a node of e-residue ιd to λ˜ is to add it to a row of index at (d) or bigger. Thus, since at (d) = as (d), we must have at (d) < as (d), as claimed. So, in the tableau s, we have the following configuration where “∗” stands for a filling of the node by a number in {1, . . . , d − 1} and where the node filled by n1 has the same coordinates as the node filled by d in t: row at (d) :
∗
∗
row as (d) :
∗
∗
·· ·· · ·
··· ···
∗ n1 n2
·· ·· ·· ·· · · · · ∗ m1 m2
··· ···
d < n1 < n2 < · · · < n p
np
·· · mq
d = m1 < m2 < · · · < mq
(The nodes filled by “∗” also appear in the tableau t; note that the case p = 0 is allowed here, which just means that the row indexed by at (d) only consists of the nodes filled by “∗”.) Since ηe (s) = ηe (t), the e-residues of the nodes filled by m1 = d and n1 are equal. Consequently, for any i, the node filled by ni has the same e-residue as the node filled by mi . We now perform some modifications on s in order to obtain a new composition ν of n and a corresponding row-standard ν -tableau u. This is done as follows. We keep all nodes and their fillings in those rows of s which are
3.5 A Canonical Basic Set for the Symmetric Group
179
not indexed by at (d) or as (d). Thus, the modification will only take place in the two rows indexed by at (d) and as (d): • in row at (d), we replace ni by min{ni , mi } for 1 i min{p, q}; • in row as (d), we replace mi by max{ni , mi } for 1 i min{p, q}; • if q > p, we move all nodes filled by m p+1 , . . . , mq from row as (d) to row at (d). Denote the resulting tableau by u and let ν be the shape of u. One easily checks that u is row-standard. (Note that this is just a statement about the reshuffling of two sequences of integers n1 , . . . , n p and m1 , . . . , mq according to the above rules; see Remark 3.5.12.) Taking into account the above remarks concerning e-residues, it is clear that ηe (u) = ηe (s) = ηe (λ ). Furthermore, the row of u indexed by at (d) has at least as many nodes as the row of s indexed by at (d); since at (d) < as (d), the definition of the dominance order immediately shows that μ ν . Hence, if ν = λ , then we are done. Otherwise note that, by construction, we have t−1 (i) = u−1 (i) for 1 i d. Hence, if we let d ∈ {1, . . . , n} be minimal such that t−1 (d ) = u−1 (d ), then d > d. So, by induction, we have ν λ and, hence, μ λ , as required. Remark 3.5.12. The above proof uses the following (very elementary) statement: Let n1 < n2 < · · · < n p and m1 < m2 < · · · < mq be two finite increasing sequences of integers. Let p := max{p, q} and q := min{p, q}. For 1 i q , let n i := min{ni , mi } and m i := max{ni , mi }. If q > p, let n i := mi for p + 1 i p . Then we have n 1 < n 2 < · · · < n p and m 1 < m 2 < · · · < m q . (Proof left to the reader.) Example 3.5.13. (a) To illustrate the above proof, let λ = (422) 8 and e = 3. Furthermore, let μ = (32111) 8 and s be the row-standard μ -tableau given below: 1 3 6 8
te (λ ) = 2 5
η3 (λ ) = (0, 2, 1, 1, 0, 2, 2, 0)
4 7
s=
1 3 7 2 5 4 8 6
We have η3 (s) = η3 (λ ). Now d = 6 is the smallest number which appears in different nodes of t = t3 (λ ) and of s; we have at (6) = 1 and as (6) = 5. With the notation of the above proof, n1 = 7 and m1 = 6 where p = q = 1. So we just swap 6 and 7 in the above tableau. Let u be the resulting tableau; its shape is still ν = μ . Then d = 7 is the smallest number which appears in different nodes of t and of u; we have at (7) = 3 and au (7) = 5. With the notation of the above proof, m1 = 7 where p = 0 and q = 1. So we just move 7 from the fifth row to the third row. The resulting composition is ν = (3221); we have μ ν λ and we can go on by induction. (In more complicated examples, even if we start with two partitions, it may happen that genuine compositions appear in intermediate steps of the construction.) (b) Let n = 5 and θ : A → k be such that θ (v2 ) = −1 = 1, as in Example 3.2.9. Then e = 2. The 2-regular partitions of 5 are (5), (41), (32). We find that
η2 (5) = (0, 1, 0, 1, 0),
η2 (41) = (0, 1, 0, 1, 1),
η2 (32) = (0, 1, 1, 0, 0).
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3 Specialisations and Decomposition Maps
The corresponding representations Yηe (λ ) in E5 are computed in Example 3.5.6. Now, if λ is 2-regular and E μ is a constituent of Yηe (λ ) , then there exists a standard μ -tableau s such that η2 (s) = η2 (λ ); see Remark 3.5.7. So Lemma 3.5.11 shows that μ λ , which we can see explicitly in the formulae for E5 in Figure 3.1. Theorem 3.5.14 (Dipper and James [62]). Recall that we are in the setting of Example 3.2.9, where W ∼ = Sn . As in 3.5.4, let e := min{i 2 | 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0}
(u := v2 ).
(We let e = ∞ if no such i exists.) Then we have
Λk◦ = {λ n | λ is e-regular}
(cf. Definition 3.5.8).
Proof. First, it is known that the number of e-regular partitions of n is equal to the number of partitions of n with no part divisible by e. (This is one of the classical identities among partition counting functions; see Andrews [4, Cor. 1.3] and James and Kerber [182, Lemma 6.1.2]. For an alternative argument, see also James [181, 2.22, 2.27].) Hence, by Lemma 3.4.13, we already have that | Irr(Hk )| number of e-regular partitions λ n. We set E = En . Let λ n be e-regular and consider the tableau te (λ ) in 3.5.9. Let ηe (λ ) be its e-residue sequence and denote Yλ := Yηe (λ ) ∈ E . We show that the set E := {Yλ | λ n is e-regular} ⊆ E satisfies the conditions in Proposition 3.4.10. Recall from Example 3.2.9 that H admits a cell datum and, hence, a canonical basic set. Furthermore, the condition in Remark 3.5.7(b) is satisfied for λ itself and so Evλ is an irreducible constituent of Yλ . μ Now let μ n be any partition such that Ev is an irreducible constituent of Yλ . Then Remark 3.5.7 shows that there exists a standard μ -tableau s such that ηe (s) = ηe (λ ) and so Lemma 3.5.11 implies that μ λ . Thus, in R0 (HK ), we have [Yλ ] = m(Yλ , λ )[Evλ ] +
∑
μ n: μ λ
m(Yλ , μ ) [Evμ ],
where m(Yλ , μ ) ∈ Z0 for all μ and m(Yλ , λ ) = 0. It is known that if μ λ , then aλ aμ , with equality only if λ = μ ; see 2.2.13. Hence, the three conditions in Proposition 3.4.10 are satisfied for E , as claimed. Thus, E is a canonical basic set for Hk . Finally, Proposition 3.2.7 shows that E = {Yλ | λ ∈ Λk◦ }. As mentioned earlier, the original proof of Dipper and James [62] is different from the one given here. For further variations, see Murphy [257], Mathas [245] and Xi [296]. Note that some authors (for example, Mathas [245, §3.4]) obtain a parametrisation of Irr(Hk ) in terms of “e-restricted” partitions. Since a partition is e-restricted if and only if the conjugate partition is e-regular, the two versions are seen to be equivalent by passing to the opposite cell datum and using Example 3.2.8.
3.6 Factorisation of Decomposition Maps
181
3.6 Factorisation of Decomposition Maps We keep the basic setting of the previous sections where H = HA (W, S, L) is a generic Iwahori–Hecke algebra associated with a finite Coxeter group W . Our aim now is to try to get a general overview of all possible specialisations and the associated decomposition maps. The basic tool for doing this is the factorisation result in Theorem 3.6.3 below. In the case where Γ = Z, this will allow us to deal with any specialisation in a two-step procedure where, in a first step, we specialise to a field of characteristic 0 (via a principal specialisation) and then possibly specialise further into a field of positive characteristic. A first version of this result appeared in [92]; a systematic and more conceptual framework for establishing such factorisation results was developed by Geck and Rouquier [133]. We will then go on to discuss various questions related to specialisations of the parameters to roots of unity in a field of characteristic zero. We will see that these questions can be essentially reduced to the study of the Φe -modular decomposition maps introduced in Example 3.1.15(b). To give an idea of what kind of specialisations we will have to deal with in general, we mention the following example—which is, in any case, the most important as far as applications are concerned (see also, for example, Brou´e and Kim [33]). Example 3.6.1. Assume that Γ = Z so that A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Let θ : A → k be a specialisation into a field k such that k is the field of fractions of θ (A). Now, if R is a field, then A is a principal ideal domain; furthermore, k has characteristic 0. More generally, assume that R is the ring of algebraic integers in a finite field extension K ⊇ Q. Then the kernel of θ is a prime ideal in A, and one easily shows that the non-zero prime ideals in A are given as follows (e.g. see [33, §2.B]): f A, where f ∈ A is a non-constant, irreducible and non-invertible polynomial. In this case, the field of fractions A/ f A has characteristic 0. • pA, where p is a non-zero prime ideal in R; thus, pA consists of all p-linear combinations of terms vn (n ∈ Z). In this case we have A/pA ∼ = F[v, v−1 ] where F = R/p is a finite field of characteristic p > 0 where p ∈ p. • pA + f A, where p is a prime ideal in A and f ∈ A is non-constant, irreducible and non-invertible modulo p. In this case, pA + f A is maximal and the residue field has characteristic p > 0 where p ∈ p. •
In particular, we see that dim A = 2. The prime ideals in the first and second cases have height 1; those in the third case have height 2. Note that if ker(θ ) has height 1, then the localisation of A in ker(θ ) will be a discrete valuation ring (see [248, §11]); in particular, θ is a principal specialisation. In the case where ker(θ ) has height 2, the situation is more complicated. 3.6.2. Recall our general assumption that R is L0 -good. Let θ : A → k and θ : A → k be specialisations where k is the field of fractions of θ (A) and k is the field of fractions of θ (A). Let us assume that
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3 Specialisations and Decomposition Maps
ker(θ ) ⊆ ker(θ ).
(a)
Then we can factorise θ = ϕ ◦ θ , where ϕ is a ring homomorphism from A := θ (A) ⊆ k to k . (Explicitly, given a ∈ A , write a = θ (a) for some a ∈ A and set ϕ (a ) := θ (a); this does not depend on the choice of a since ker(θ ) ⊆ ker(θ ).) Clearly, the image of ϕ equals the image of θ ; hence, k also is the field of fractions of ϕ (A ). Theorem 3.6.3 below will show that the factorisation θ = ϕ ◦ θ induces a factorisation of the corresponding decomposition maps. In order for this to work, an additional requirement on ϕ is needed, which can be formulated as follows. (b)
ϕ can be extended to a subring B ⊆ k which is integrally closed in k.
We will say that θ lies above θ if the above two conditions, (a) and (b), are satisfied. The two standard situations in our applications are: (1) the case where B = θ (A) itself is integrally closed in k and (2) the case where ϕ can be extended to k. Theorem 3.6.3 (Geck [92, §4], [98, 2.6], Geck and Rouquier [133, 2.12]). Assume that θ lies above θ , in the sense defined in 3.6.2. Then there exists a unique additive + map dθθ : R+ 0 (Hk ) → R0 (Hk ) such that the following diagram is commutative: dθ+ - R+ (Hk ) R+ (H ) K 0 0 Q 3 Q θ dθ+ QQ s + dθ R0 (Hk ) Thus, we also have a factorisation of the corresponding decomposition matrices as Dθ = Dθ .Dθθ , where Dθθ is a matrix all of whose entries are non-negative integers. Following James [181] (who considered Hecke and q-Schur algebras of type A), the matrix Dθθ will be called the adjustment matrix with respect to θ and θ . Proof. Let ϕ˜ : B → k be an extension of ϕ as in 3.6.2(b). Clearly, k also is the field of fractions of the image of ϕ˜ . Since B is integrally closed in k, we can apply + Theorem 3.1.2. Thus, ϕ˜ induces a decomposition map dϕ+˜ : R+ 0 (Hk ) → R0 (Hk ) such that the following diagram is commutative: xB - Maps(HB , B[X]) R+ 0 (Hk ) dϕ+˜
? (H R+ k ) 0
xB,k
tB,ϕ˜ ? - Maps(HB , k [X])
where tB,ϕ˜ : Maps(HB , B[X]) → Maps(HB , k [X]) is the natural map obtained by applying ϕ˜ to the coefficients of the polynomials in B[X]. We set dθθ := dϕ+˜ . Then it remains to show that dθ+ = dθθ ◦ dθ+ . For this purpose, we first note that the natural map H → HB induces monoid homomorphisms ιB : Maps(HB , B[X]) → Maps(H, B[X]) and ιk : Maps(HB , k [X]) → Maps(H, k [X]). Hence, we have a commutative diagram:
3.6 Factorisation of Decomposition Maps
Maps(HB , B[X])
183
ιB
tB,ϕ˜ ? Maps(HB , k [X])
ιk
- Maps(H, B[X]) tϕ˜ ? - Maps(H, k [X])
Noting that xk = ιB ◦ xB and xk = ιk ◦ xB,k , we obtain the following diagram in which each square is commutative: dθ+ dϕ+˜ - R+ (Hk ) - R+ (Hk ) R+ 0 (HK ) 0 0 xA ? Maps(H, A[X])
tθ
xk ? - Maps(H, B[X])
tϕ˜
xk ? - Maps(H, k [X])
Now, the composition of the two maps of the bottom row is tθ : Maps(H, A[X]) → Maps(H, k [X]). Then the characterisation in Theorem 3.1.2(b) shows that the composition of the two maps of the top row must be dθ+ , as required. Example 3.6.4. Assume we are in the setting of 3.3.1, where we have factorisations of the elements cλ in terms of a certain subset P ⊆ A. Condition (P2) in 3.3.1 implies that whenever we have a specialisation θ : A → k such that θ (Φ ) = 0 for some Φ ∈ P, then θ lies above θΦ in the sense of 3.6.2. So Theorem 3.6.3 yields a facΦ torisation of decomposition matrices Dθ = DΦ .DΦ θ , where Dθ is the corresponding adjustment matrix. Example 3.6.5. Assume that K ⊇ Q is a finite extension and R is a discrete valuation ring in K with residue field F of characteristic p > 0. By the classical modular representation theory of finite groups, we have a corresponding decomposition map d p : R0 (KW ) → R0 (FW ), where KW and FW are the group algebras of W over K and F respectively; see Example 3.1.3. Let D p denote the decomposition matrix of d p . We now use Theorem 3.6.3 to relate d p to decomposition maps of H. For simplicity, let us assume that Γ = Z and L(s) = 1 for all s ∈ S (equal parameter case). √ Also assume that R is large enough so that our standard root of unity ζ2p = exp(π −1/p) lies in R. Then consider the Φ p -modular decomposition map d(p) : R0 (HK ) → R0 (HK ) associated with the specialisation θ p : A → K such that θ p (v) = ζ2p , as in Example 3.3.15. Let D(p) denote the decomposition matrix of d(p) . The rows of both D p and D(p) are labelled by the set Λ . We claim that D p = D(p) .D ,
where D is a suitable “adjustment matrix”;
that is, the entries of D are non-negative integers. This can be interpreted as saying that the Φ p -modular decomposition matrix D(p) of H is a first approximation to the usual p-modular decomposition matrix D p of W . This is proved as follows.
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3 Specialisations and Decomposition Maps
First, following [92, Prop. 4.6], consider the Φ1 -modular specialisation θ1 : A → K such that θ1 (v) = −1. Thus, HK = KW ; note also that θ1 (A) = R is integrally closed in K. Furthermore, let θF,1 : A → F be the composition of θ1 with the natural map R → F. Let DF,1 denote the corresponding decomposition matrix. Clearly, we have ker(θ1 ) ⊆ ker(θF,1 ), and so Theorem 3.6.3 applies. This yields a factorisation DF,1 = D1 .D , where the adjustment matrix D is associated with the decomposition map d : R0 (KW ) → R0 (FW ) induced by the natural map R → F. Hence, we have D = D p and so DF,1 = D1 .D p . Now, since HK = KW is semisimple, D1 is the identity matrix (for a suitable ordering of the rows and columns); see 3.1.18. Thus, we have DF,1 = D p . Second, following [98, Exp. 3.2], consider the Φ p -modular specialisation θ p : A → K, v → ζ2p ; again, we have θ p (A) = R. Now, since F has characteristic p, the image of ζ2p in F is −1, and so ker(θ p ) ⊆ ker(θF,1 ). So we can apply Theorem 3.6.3 and this yields a factorisation DF,1 = D(p) .D , where D is a suitable adjustment matrix. Combining this with the identity DF,1 = D p , we conclude that D p = DF,1 = D(p) .D , as required. Let us consider some concrete exanples. Let W be of type B2 = I2 (4) and p = 2. The matrix D(2) is given by Example 3.2.5; since W is a 2-group, it is well known that D2 has only one column, with entries dim E λ (λ ∈ Λ ). Hence:
D(2) :
Eλ 1W sgn1 sgn2 sgn σ1
dλ ,M 1 . 1 . 1 . 1 . . 1
D2 :
Eλ 1W sgn1 sgn2 sgn σ1
dλ ,M 1 1 1 1 2
D =
1 2
.
So D2 and D(2) do not even have the same size! But even if D p and D(p) do have the same size, they will not necessarily be equal. For example, let W ∼ = S4 be of type A3 and p = 2. Then, by [181, p. 253], we have
D(2) :
Eλ (4) (31) (22) (211) (1111)
dλ ,M 1 . 1 1 . 1 1 1 1 .
D2 :
Eλ (4) (31) (22) (211) (1111)
dλ ,M 1 . 2 1 1 1 2 1 1 .
D =
1 1
. 1
.
James [181] systematically computed D p and D(p) for W ∼ = Sn and 1 p n 10. The results of these computations led him to formulate the following conjecture.
Conjecture 3.6.6 (First version of James’s conjecture [181]). In the setting of Example 3.6.5, assume that W is of type An−1 so that W ∼ = Sn . If p2 > n, then D p = D(p) ; that is, the Φ p -modular decomposition matrix of H equals the usual p-modular decomposition matrix of Sn .
3.6 Factorisation of Decomposition Maps
185
Note that, by Theorem 3.5.14, both D p and D(p) have rows labelled by the set of all partitions of n and columns labelled by the p-regular partitions of n. In general, the problem of determining D p for W ∼ = Sn is completely open. On the other hand, as we will see in Section 6.2, the analogous problem for D(p) is solved,1 thanks to the work of Ariki [7], [10]. We will discuss the relevance of the condition “p2 > n”, and generalisations of James’s conjecture, in the next section. We now consider two further applications of the factorisation of decomposition maps which will be useful later on. The following result shows that canonical basic sets behave well with respect to such factorisations. Lemma 3.6.7. Recall that is a monomial ordering on Γ such that L(s) 0 for all s ∈ S. In the setting of Theorem 3.6.3, assume that the following conditions hold: (a) we have | Irr(Hk )| = | Irr(Hk )| and (b) there exists a canonical basic set Bθ ⊆ Λ for Hk (see Definition 3.2.1). Then Bθ also is a canonical basic set for Hk . Proof. Let D◦θ be the submatrix of Dθ with rows labelled by Bθ ; similarly, let D◦θ be the submatrix of Dθ with rows labelled by Bθ . By Theorem 3.6.3, we have Dθ = Dθ .Dθθ
and
D◦θ = D◦θ .Dθθ .
As in Remark 3.2.2, we have a canonical labelling of Irr(Hk ) by the set Bθ , and we can write Dθ = dλ ,μ λ ∈Λ , μ ∈B . θ
Assume now that the elements of Bθ are ordered according to increasing value of aλ . Then, since Bθ is a canonical basic set for Hk , the matrix D◦θ is square and triangular with 1 along the diagonal. By condition (a), both D◦θ and Dθθ are square matrices; furthermore, the entries of these matrices are non-negative integers. An elementary argument about matrices then shows that there is an ordering of the set Irr(Hk ) (that is, the indexing set of the columns of D◦θ and of the rows of Dθθ ) such that both D◦θ and Dθθ are triangular with 1 along the diagonal. This ordering defines a labelling Irr(Hk ) = {M μ | μ ∈ Bθ }. We can now also write and Dθθ = aμ ,ν μ ,ν ∈B , Dθ = dλ ,μ λ ∈Λ , μ ∈B θ
θ
where dν ,ν = aν ,ν = 1 for all μ ∈ Bθ . Now fix ν ∈ Bθ and assume that λ ∈ Λ is such that dλ ,ν = 0. The above factorisation yields an equation dλ ,ν =
∑
μ ∈Bθ
dλ ,μ aμ ,ν = dλ ,ν aν ,ν +
∑
μ ∈Bθ \{ν }
dλ , μ a μ , ν .
1 This situation is reminiscent of that in the representation theory of simple algebraic groups over fields of positive characteristic; see Lusztig [227]; in this case, the formulation of the analogous problem in characteristic 0 involves the theory of quantised enveloping algebras.
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3 Specialisations and Decomposition Maps
Since all the terms in the above expression are non-negative integers, there are no cancellations and so dλ ,ν = 0. Since Bθ is a canonical basic set for Hk , it follows that λ = ν or aλ > aν . We have already seen that dν ,ν = 1 for all ν ∈ Bθ . Thus, Bθ is a canonical basic set for Hk . The following example illustrates one of the most difficult cases in type Bn that we will have to deal with. Example 3.6.8. Let W be of type Bn with weight function L : W → Z given by Bn
2c+1 t
4
2 t
2 t
· · ·
2 t
where c 0.
Now A is the ring of Laurent polynomials in one indeterminate v = ε . Let q ∈ Z be a prime power and be a prime such that q. Let θ : A → k be a specialisation such that char(k) = > 0 and θ (u) = q1k , where u := v2 . Now recall that IrrK (W ) is labelled by the set Λ of all pairs of partitions (λ , μ ) such that |λ | + |μ | = n. By the end of this book, we will have determined a canonical basic set Bθ ⊆ Λ for Hk . This involves the following steps. First of all, it is not yet known if the hypotheses of Proposition 3.2.7 (that is, the conjectural properties (♠) and (♣) in 2.5.3) are known to hold. We shall see in Example 4.5.15 (using an argument involving the representation theory of the finite unitary groups over Fq ) that there does exist a canonical basic set Bθ ⊆ Λ for Hk . However, this is only an existence result. The difficulty in obtaining an explicit description of Bθ depends heavily on the parameter e := min{i 2 | 1 + q + q2 + · · · + qi−1 ≡ 0 mod }. If e = 2 or e is odd or e is divisible by 4, then an explicit description of Bθ will be obtained by elementary methods; see the discussion in Example 4.4.16. If e = 2e , where e 3 is odd, then Proposition 3.6.7 will allow us to reduce the problem to characteristic 0, where completely new methods are available (involving among others the theory of canonical bases); see Example 4.5.15 for details. Finally, we wish to show that the study of decomposition maps associated with specialisations where the parameters are sent to some roots of unity (in a field characteristic 0) can be reduced to the study of the Φe -modular decomposition maps as defined in Example 3.1.15(b). For W ∼ = Sn , a statement of this kind explicitly appeared in Mathas [245, Cor. 6.24]. Owing to its nature, the discussion will be somewhat technical. We begin with some preparations. 3.6.9. As in Remark 1.1.8, let us set T˙w := ε L(w) Tw for all w ∈ W and us := v2s for all s ∈ S. Then the structure constants of H with respect to the basis {T˙w | w ∈ W } lie in the subring R[us | s ∈ S] ⊆ A. Now let α : A → A be a ring isomorphism such that α (R) = R and α (us ) = us for all s ∈ S. Extending scalars via α , we obtain a new algebra Hα . Since α (us ) = us for all s ∈ S, this new algebra is canonically isomorphic to H. So we will identify Hα with H. By Theorem 3.1.14, the composition of α with the inclusion A ⊆ K gives rise to a group homomorphism dα : R0 (HK ) → R0 (HK )
3.6 Factorisation of Decomposition Maps
187
which, by 3.1.18, is an isomorphism inducing a bijection of Irr(HK ) onto itself. Thus, there exists a permutation λ → λ α of Λ such that dα ([Eελ ]) = Eελ
α
for all λ ∈ Λ .
This permutation is uniquely determined by the condition that α for all w ∈ W . trace(Tw , Eελ ) = α trace(Tw , Eελ ) Proposition 1.3.1 and the orthogonality relations in Proposition 1.2.12 imply that
α (cλ ) = cλα
and
aλ = a λ α
for all λ ∈ Λ .
The permutation λ → λ α is easily determined using the above condition on character values. Assume that (W, S) is irreducible. Then λ → λ α is given as follows. (a) If α (r) = r for all r ∈ ZW and α (vs ) = vs (s ∈ S), then λ → λ α is the identity. This follows from the fact that all values trace(Tw , Eελ ) lie in ZW [Γ ]; see 1.2.1. (b) If α (r) = r for all r ∈ ZW and α (vs ) = −vs (s ∈ S), then λ = λ α unless λ labels one of the following exceptional representations listed in [132, 9.2.3]: α
4 r ←→ 4r α α 16rr ←→ 16r , 16 r ←→ 16 rr α 512 a ←→ 512a α α 4096z ←→ 4096x , 4096 x ←→ 4096 z . √ (c) Let W be of type H3 or H4 , where ZW = Z[ 12 (−1 + 5)]. If α (vs ) = vs (s ∈ S) √ √ and α ( 5) = − 5, then λ → λ α is given by algebraic conjugation as specified in the tables in [132, Appendix C]. (d) Let W be of type I2 (m) (m 5, m = 6). Then ZW = Z[2 cos(2π /m)] ⊆ Q and so there are several maps α permuting the two-dimensional representations σ j in Example 1.3.7. H3 H4 E7 E8
3.6.10. Let θ : A → k be a specialisation where k is the field of fractions of θ (A). Let α : A → A be a ring isomorphism as in 3.6.9; we have just seen that this induces a permutation λ → λ α of Λ . Now θ = θ ◦ α is another specialisation where k is the field of fractions of θ (A). Formally, we have two specialised algebras Hk,θ and Hk,θ . However, since θ (us ) = θ (us ) for all s ∈ S, these two algebras are the same, so we can just denote them by Hk . Using Theorem 3.6.3, we obtain a factorisation dθ = dθ ◦ dα . Hence, we have dλ ,M = dλα ,M
for all λ ∈ Λ and M ∈ Irr(Hk ),
where Dθ = dλ ,M and Dθ = dλ ,M are the decomposition matrices associated with θ and θ respectively. 3.6.11. Let Γ = Z so that A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Assume that R is the ring of algebraic integers in some finite extension K ⊇ Q. Let
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3 Specialisations and Decomposition Maps
θ : A → k be a specialisation such that K⊆k
and k = K[θ (v)],
where θ (v) ∈ k is some root of unity.
We wish to relate the decomposition map dθ to the Φe -modular decomposition map de (for a suitable e) in Example 3.1.15(b). First of all, by enlarging K if necessary, we can assume without loss of generality that k = K and θ (v) ∈ R; see Remark 3.1.5. Let d 1 be the order of θ (v) and set e := d/2 (if d is even) or e := d (if d is odd). Thus, (−1)d θ (v) is a root of unity of order 2e. Consequently, since we are in characteristic 0, there exists a field automorphism αK ∈ Aut(K) such that αK (ζ2e ) = (−1)d θ (v). Now, we have two specialised algebras: HK,θe (where K is regarded as an A-module via θe ) and HK,θ (where K is regarded as an A-module via θ ). Since αK (ζ2e ) = ±θ (v), the algebra HK,θ can be identified with that obtained from HK,θe by scalar extension from K to K via αK . Thus, αK induces a bijection Irr(HK,θe ) → Irr(HK,θ ),
M → M ,
which induces a group isomorphism αK∗ : R0 (HK,θe ) → R0 (HK,θ ). Since R is the ring of integers in K, we have αK (R) = R. Now note that θ (R) ⊆ K and so θ induces a field automorphism θK ∈ Aut(K). Again, since R is the ring of integers in K, we have θK (R) = R. We can now define a ring homomorphism α : A → A such that α (v) = (−1)d v and α (r) = θK−1 (αK (r)) for all r ∈ R. Thus, α is a ring isomorphism as in 3.6.9, and we have αK ◦ θe = θ ◦ α . Then, using Remark 3.1.5 and 3.6.10, we have αK∗ ◦ dθe = dαK ◦θe = dθ ◦α = dθ ◦ dα , which means that the diagram dθ e - R0 (HK,θ ) R0 (HK ) e dα
? R0 (HK )
dθ
is commutative. Then let (e) D(e) = dλ ,M λ ∈Λ , M∈Irr(H
K,θe )
αK∗ ? - R0 (HK,θ )
and
Dθ = dλ ,M λ ∈Λ , M ∈Irr(H
K,θ )
be the decomposition matrices associated with θe and θ respectively. The commutativity of the above diagram means that (e)
dλ ,M = dλα ,M
for all λ ∈ Λ and M ∈ Irr(HK,θe ).
Thus, we have found the desired relation between the decomposition matrices Dθ and D(e) : up to a permutation of the rows, these two matrices are actually equal (where the columns are matched via the bijection M → M ). Corollary 3.6.12. In the setting of 3.6.11, let be a monomial order such that L(s) 0 for all s ∈ S. Assume that there is a canonical basic set Bθe with respect to θe and . Thus, we can write
3.6 Factorisation of Decomposition Maps
Irr(HK,θe ) = {M μ | μ ∈ Bθe }
189 (e)
(e)
and dλ ,μ := dλ ,M μ
(λ ∈ Λ , μ ∈ Bθe ).
Define Bθ := {μα | μ ∈ Bθe }. Then the following hold. (a) The set Bθ is a canonical basic set with respect to θ and . Thus, we can also write Irr(HK,θ ) = {M ν | ν ∈ Bθ } and dλ ,ν := dλ ,M ν for λ ∈ Λ and ν ∈ Bθ . (b) With the notation in (a), the identity in 3.6.11 reads (e)
dλα ,μα = dλ ,μ
for all λ ∈ Λ and μ ∈ Bθe . (e)
Proof. (a) Let M ∈ Irr(HK,θe ). Since dλ ,M = dλα ,M for all λ ∈ Λ , we deduce that Sθ (M ) = {λ α | λ ∈ Sθe (M)}. Now, we have already remarked in 3.6.9 that aλ = aλα for all λ ∈ Λ . Hence, if the function Sθe (M) → Γ , λ → aλ , reaches its minimum at a uniquely determined λM , then the function Sθ (M ) → Γ , λ → aλ , reaches its minimum at a uniquely α . It follows that there is a canonical basic set determined λM , and we have λM = λM with respect to θ , and we have Bθ = {λ α | λ ∈ Bθe }. The statement in (b) is then an immediate consequence of the identity in 3.6.11. Example 3.6.13. The essence of the above discussion is that, in most cases, we actually have Dθ = D(e) and Bθ = Bθe . By 3.6.11, this applies whenever W is a finite Weyl group with no component of type E7 or E8 . Here are two examples where one can see the effect of a non-trivial permutation λ → λ α . (a) Let W be of type I2 (m) (m 3) and L(s) = 1 for s ∈ S (equal parameter case). Consider an R-linear specialisation θ : A → K such that θ (v) is a root of unity of order d 2. Define e as above and assume that e > 2 and e divides m. Then, by Example 3.2.5(b), there is a non-trivial θ -block given by {1W , sgn, σ j(e) }, where j(e) is determined by the condition that θ (v)2 = ζ ± j(e) . Recall that ζ ∈ C is a root of unity of order m such that ζ + ζ −1 = 2 cos(2π /m). Consequently, if θ = θe , then we find that j(e) = m/e. On the other hand, if θ = θe , then it may happen that j(e) = m/e. But then there will be an isomorphism α : A → α A as in 3.6.11 such that j(e) ←→ m/e. Thus, when speaking about the blocks and decomposition numbers for W of type I2 (m), it is important to say precisely which roots of unity are involved. Similar remarks apply to W of type H3 and H4 . (b) Let W be of type E7 and e = 5. By Table F.5 in [132, Appendix F], there is a Φ5 -block of defect 1 given by {7 a , 378 a , 512a , 168 a , 27 a }. On the other hand, in [92, Theorem 12.6], we find that {7 a , 378 a , 512 a , 168 a , 27 a } is a Φ5 -block of defect 1. This apparant contradiction is explained by the fact that, in [92], we used the Rlinear specialisation A → K, v → −1, to establish a bijection Irr(HK ) ↔ Irr(W ), whereas in [132] we used the specialisation v → 1. Thus, the two situations are transformed into each other using the ring isomorphism α : A → A such that α (v) = −v and α (r) = r for all r ∈ R. As mentioned in 3.6.11(b), we have 512 a = 512αa in this case (and α fixes all other labels in Λ ). A similar remark applies to several further blocks which appear in [92, §12] or [132, Appendix F].
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3 Specialisations and Decomposition Maps
Remark 3.6.14. Assume that W is a Weyl group and Γ = Z so that A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Let θ : A → k and θ : A → k be two specialisations such that k is the field of fractions of θ (A) and of θ (A). Assume that θ (v) and θ (v) are roots of unity of the same order in k× . Then one may conjecture that Dθ = Dθ (for a suitable matching of the columns). This was first stated by James [181, §4] for the case where W ∼ = Sn (see also Mathas [245, 6.38]). If char(k) = 0, this easily follows (for any type of W ) from the identity in 3.6.11. (Indeed, let λ → λ α be the permutation associated with θ and λ → λ α be the one associated with θ . By the construction in 3.6.11, we have α (v) = α (v) = (−1)d v, where d is the order of θ (v). Hence, since W is a Weyl group, we have λ α = λ α for all λ ∈ Λ ; see 3.6.9(a) and (b).) However, if char(k) = > 0, the above proofs will not work, as, in positive characteristic, two roots of unity of a given order are not necessarily conjugate under a field automorphism. See, however, the recent work of Brundan and Kleshchev [39, §6], where an entirely new approach to this problem is found (at least, as far as algebras of type An−1 and Bn are concerned).
3.7 The General Version of James’s Conjecture In the previous section, we have seen a first version of James’s conjecture for W ∼ = Sn . Our aim now is to try to understand how the hypothesis “p2 > n” involved in that conjecture comes about, to formulate a general version for any type of W , and to establish some partial results supporting it. To begin with, let us state the full original version of James’s conjecture. 3.7.1. Let W ∼ = Sn be of type An−1 . Let Γ = Z and L(s) = 1 for all s ∈ S. Then A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Let θ : A → k be a specialisation where k is the field of fractions of θ (A). Let Dθ be the corresponding decomposition matrix. Then set e := min{i 2 | 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0}
(u := v2 ).
(We let e = ∞ if 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0 for all i 2.) If e > n or e = ∞, then Hk is semisimple; see 3.4.12. Hence, in this case, Dθ is the identity matrix, for a suitable ordering of the columns; see 3.1.18. Now assume that e < ∞. Then consider the Φe -modular specialisation such that v → ζ2e ∈ C, as in Example 3.3.15. Let D(e) be the corresponding decomposition matrix. By Theorem 3.5.14, both Dθ and D(e) have rows labelled by the set Λ consisting of all partitions of n and columns labelled by the subset Λk◦ ⊆ Λ consisting of all e-regular partitions of n. Now we can state the following conjecture. Conjecture 3.7.2 (James [181, §4]). In the above setting, assume that char(k) = > 0 and e > n. Then we have Dθ = D(e) .
3.7 The General Version of James’s Conjecture
191
Note that the “first version” in Conjecture 3.6.6 is the special case where θ is such that θ (u) = 1; we then have e = . We also remark that James not only considers the algebra H, but also the (slightly bigger) q-Schur algebra [64]. We shall see that an appropriate setting for understanding and generalising the above conjecture is given by the factorisation result in Theorem 3.6.3. In this framework, a general version for any type of W first appeared in [92]. Taking into account our results on cell data from Chapter 2, we will give a more elegant formulation of that general version, which formally looks exactly like James’s original one. 3.7.3. Throughout, we assume that R is L-good and L(s) 0 for all s ∈ S. Furthermore, we assume that there is a subset P ⊆ A such that the three conditions in 3.3.1 are satisfied. In particular, for Φ ∈ P, we have a corresponding principal specialisation θΦ : A → kΦ , where kΦ is the field of fractions of A/(Φ ). 3.7.4. Assume that (♠) in 2.5.3 holds; then we have a cellular basis {Cλs,t } as in Theorem 2.6.12. Let Φ ∈ P and θ : A → k be any specialisation such that k is the field of fractions of k and θ (Φ ) = 0. (Thus, θ lies above θΦ ; see Example 3.6.4). Let {W (λ ) | λ ∈ Λ } be the cell representations defined in terms of the given cellular basis of H. Let Gλ = (gλst )s,t∈M(λ ) be the Gram matrix of the canonical bilinear form on W (λ ); see Lemma 2.6.2. By specialisation, we obtain matrices λ GΦ := θΦ (gλst ) s,t∈M(λ )
Gλk := θ (gλst ) s,t∈M(λ ) .
and
λ = 0} and Λ ◦ := {λ ∈ Λ | Gλ = 0}, we have Then, setting ΛΦ◦ := {λ ∈ Λ | GΦ k k μ
Irr(HkΦ ) = {LΦ | μ ∈ ΛΦ◦ } μ
μ
μ
Irr(Hk ) = {Lk | μ ∈ Λk◦ },
and μ
μ
where dim LΦ = rank(GΦ ) and dim Lk = rank(Gk ); see Theorem 2.6.5 and 2.7.5. μ μ Here, we just write LΦ instead of LkΦ . Now, since θ lies above θΦ , we can factorise θ = ϕ ◦ θΦ , where ϕ : A/(Φ ) → k λ. is a ring homomorphism. Hence, Gλk is obtained by applying ϕ to the entries of GΦ λ λ First of all, this implies that if GΦ = 0, then Gk = 0. Furthermore, we certainly have λ ). Thus, we conclude that rank(Gλk ) rank(GΦ
Λk◦ ⊆ ΛΦ◦
and
μ
μ
dim Lk dim LΦ
for all μ ∈ Λk◦ .
Recall from Example 3.6.4 that we also have a factorisation of decomposition maps Dθ = DΦ .DθΦ , where DθΦ is the corresponding adjustment matrix. Let us write DΦ θ = dμν μ ∈Λ ◦ , ν ∈Λ ◦ . Φ
k
The following observation shows that if the inclusion and the inequalities above are all equalities, then we can even say something more precise about the shape of DθΦ . (A statement of this kind first appeared in [172, §3.1.B].)
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3 Specialisations and Decomposition Maps
Proposition 3.7.5. In the setting of 3.7.4, assume that | Irr(HkΦ )| = | Irr(Hk )|. Then Λk◦ = ΛΦ◦ and, for a suitable ordering of the elements of Λk◦ = ΛΦ◦ , the adjustment matrix DθΦ is lower triangular with 1 on the diagonal. μ μ ◦ Furthermore, DΦ θ is the identity matrix if dim Lk = dim LΦ for all μ ∈ Λk . Proof. Our assumption implies that |ΛΦ◦ | = |Λk◦ |. Since we already know that Λk◦ ⊆ ΛΦ◦ , we deduce that these two sets are actually equal. Let us set Λ ◦ := ΛΦ◦ = Λk◦ . Now consider the factorisation of decomposition maps Dθ = DΦ .DθΦ . Let D◦θ be the submatrix of Dθ obtained by deleting all rows corresponding to labels λ ∈ Λ ◦ . Similarly, we define the submatrix D◦Φ of Dφ . Then D◦θ , D◦Φ and DΦ θ are square matrices with rows and columns labelled by Λ ◦ , and we have D◦θ = D◦Φ .DΦ θ. The cell datum for H comes with a partial order L on Λ . If we choose a total order on Λ ◦ which refines L , then Theorem 2.6.6 shows that both D◦θ and D◦Φ are lower triangular with 1 along the diagonal. Hence, the same is true for DθΦ as well. It follows that, for any μ ∈ Λ ◦ , we have equations μ
μ
dim LΦ = dim Lk + μ
∑
ν ∈Λ ◦ : ν
Lμ
dμν dim Lνk ,
where DθΦ = (dμν )μ ,ν ∈Λ ◦ .
μ
Hence, if dim Lk = dim LΦ for all μ ∈ Λ ◦ , then dμν = 0 whenever ν L μ . Since Φ we already know that Dθ has a lower triangular shape with 1 along the diagonal, we deduce that DΦ θ is the identity matrix, as required.
We now have the following general finiteness result. Note that it does not yield any explicit numerical bound. The general version of James’s conjecture will be concerned precisely with specifying a numerical bound. Proposition 3.7.6 (Cf. [92, 5.5], [98, 2.7]). In the setting of 3.7.4, assume that A is noetherian and dim A = 2. Then, for a fixed Φ ∈ P, the set ! M(Φ ) := p ∈ Spec(A) ! p = ker(θ ) where θ lies above θΦ and DθΦ = Id is finite (where Id denotes the identity matrix of the appropriate size). μ
μ
Proof. For μ ∈ ΛΦ◦ , let rμ := dim LΦ = rank(GΦ ). So there is an (rμ ×rμ )-submatrix H μ of Gμ such that θΦ (det(H μ )) = 0. Let d := ∏μ ∈ΛΦ◦ det(H μ ) ∈ A; note that θΦ (d) = 0. We claim that ! M(Φ ) ⊆ p ∈ Spec(A) ! (Φ ) ⊆ p and d ∈ p . Indeed, let p = ker(θ ), where θ lies above Φ and d ∈ p. We show that p ∈ M(Φ ). Now, since θ (d) = 0, we also have θ (det(H μ )) = 0 for all μ ∈ ΛΦ◦ . In particular, μ this implies that Gk = 0 and so μ ∈ Λk◦ . Since we already know that Λk◦ ⊆ ΛΦ◦ , we conclude that ΛΦ◦ = Λk◦ . Furthermore, the fact that θ (det(H μ )) = 0 implies that μ μ μ μ det(Hk ) = 0 and so dim Lk = rank(Gk ) rμ . Since we already know that dim Lk μ rμ , we conclude that dim Lk = rμ for all μ ∈ ΛΦ◦ . So Proposition 3.7.5 implies that DΦ θ is the identity matrix and, hence, p ∈ M(Φ ). Thus, the above claim is proved.
3.7 The General Version of James’s Conjecture
193
Now set B := A/(Φ ). Then the prime ideals of A containing (Φ ) correspond bijectively to the prime ideals of B. Hence, we have M(Φ )
1−1
←→
! m ∈ Spec(B) ! 0 = θΦ (d) ∈ m .
Finally, we use our assumption that A is noetherian and dim A = 2. First note that (Φ ) is a height 1 prime ideal in A. Hence, B = A/(Φ ) is a one-dimensional noetherian ring. Consequently, every prime ideal of B containing θΦ (d) = 0 is minimal with that property. But then it is well known that, in a noetherian ring, the set of prime ideals containing a given element has ! many elements; see, for only finitely example, [248, Exc. 4.12]. Hence, the set m ∈ Spec(B) ! 0 = θΦ (d) ∈ m is finite. Example 3.7.7. In the setting of 3.7.4, let Γ = Z; then A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Assume that K ⊆ C is a finite Galois extension of Q and that R is the ring of algebraic integers in K. Then R is a Dedekind domain and so A is noetherian and dim A = 2. As in Example 3.3.3, the set P consists of K-cyclotomic polynomials. Let us fix Φ ∈ P. Then B = A/(Φ ) is the ring of algebraic integers in some finite extension of Q. In the above proof, we have seen that all p ∈ M(Φ ) correspond to maximal ideals in B. Hence, A/p is a finite field for every p ∈ M(Φ ). The finiteness statement in Proposition 3.7.6 then means that there exists a bound N > 0 (depending only on W, L) such that DΦ θ = Id whenever θ : A → k lies above θΦ and char(k) = > N. A statement of this kind first appeared in [92, Prop. 5.5]; see also [98, 2.7]. This shows that, in order to determine the matrices Dθ for all possible specialisations θ : A → k, it will be enough to solve the following three problems: (1) Determine DΦ for all Φ ∈ P. (2) Find the bound N. (3) Determine Dθ where 0 < char(k) N. Note that, although the proof of Proposition 3.7.6 does not give an explicit formula2 for the bound N, it nevertheless leads to a straightforward algorithm for computing the set M(Φ ) in concrete examples. This algorithm involves the following steps: μ
• Compute the Gram matrices Gμ (μ ∈ ΛΦ◦ ) and determine rμ = rank(GΦ ). • For each μ ∈ ΛΦ◦ , find an (rμ × rμ )-submatrix H μ of Gμ such that θΦ (H μ ) = 0. • Let d := ∏μ ∈ΛΦ◦ det(H μ ) ∈ A and denote by d¯ = 0 the image of d in B = A/(Φ ). Then find all maximal ideals m B such that d¯ ∈ m. This algorithm has been used by Geck and M¨uller [129] to determine M(Φ ) for W of exceptional type G2 , F4 , E6 , E7 , E8 (in the equal parameter case). It turns out that, in these cases, N can be taken to be the largest prime divisior of W . 2
As far as the analogous problem for simple algebraic groups over fields of positive characteristic is concerned (see the footnote p. 185), Fiebig [84], [85] recently found an explicit bound.
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3 Specialisations and Decomposition Maps
We are now almost ready to state the general version of James’s conjecture. We need one further piece of notation. Definition 3.7.8. Let Φ , Φ ∈ P. We write Φ ∼ Φ if Φ = α (Φ ), where α : A → A is a ring isomorphism as in 3.6.9; that is, we have α (R) = R and α (vs ) = ±vs for all s ∈ S. In this case, the decomposition matrices DΦ and DΦ are the same up to reordering the rows according to the permutation λ → λ α ; see 3.6.10. Now let θ : A → k be a specialisation such that k is the field of fractions of θ (A). (a) We say that Φ ∈ P is θ -isolated if
θ (Φ ) = 0
but
θ (Φ ) = 0 for all Φ ∈ P such that Φ ∼ Φ .
(b) We say that Φ ∈ P is strongly θ -isolated if
θ (Φ ) = 0
but
θ (Φ ) = 0 for all Φ = Φ ∈ P.
(Slightly different versions of (a) and (b) appeared in Geck and Rouquier [133, 3.3].)
Conjecture 3.7.9 (General version of James’s conjecture; cf. Geck [92, 5.6], [98, 3.4], Geck and Rouquier [133, §5.2]). Recall our basic assumptions 3.7.3; also assume that (♠) holds; see 3.7.4. Let θ : A → k and Φ ∈ P be such that Φ is θ -isolated; see Definition 3.7.8. Then
ΛΦ◦ = Λk◦
and
μ
μ
dim LΦ = dim Lk
for all μ ∈ ΛΦ◦ .
Also, the adjustment matrix DθΦ is the identity matrix (see Proposition 3.7.5).
We will see in Example 3.7.13 below that this contains James’s original conjecture as a special case. First we need some preparations. 3.7.10. Assume we are in the setting of Example 3.3.3, where A = R[v, v−1 ] and L : W → Z is a weight function such that L(s) 0 for all s ∈ S. In order to avoid unnecessary technical complications, we shall assume that (∗)
1 ∈ ∑ Z L(s). s∈S
(A consequence of (∗) is that, for any ring isomorphism α : A → A as in Definition 3.7.8, we have α (v) = ±v and, hence, α (Φd (v2 )) = Φd (v2 ) for any d 1.) Let θ : A → k be a specialisation such that char(k) = > 0. We set Eθ := {d 2 | θ (Φ ) = 0 for some Φ ∈ P such that Φ | Φd (v2 )}. First we claim that Eθ is non-empty unless Hk is semisimple. This is seen as follows. Assume that Hk is not semisimple. Then there is some λ ∈ Λ such that θ (cλ ) = 0; see 3.1.18. Since R is L-good, this implies that θ (Φ ) = 0 for some Φ ∈ P. Since
3.7 The General Version of James’s Conjecture
195
Φ is K-cyclotomic, we have Φ | Φd (v2 ) for some d 1. In particular, d ∈ E(W, L) and so d 2; see Example 3.3.3. Thus, we have Eθ = ∅, as claimed. Now, if Hk is semisimple, then we set e = ∞. Otherwise, we let e be the minimal element of Eθ and Φ ∈ P be such that θ (Φ ) = 0 and Φ | Φe (v2 ). This definition of e should be regarded as the appropriate generalisation of the definition of e for W of type An−1 in 3.7.1. Next we need some standard facts about cyclotomic polynomials. 3.7.11. In the above setting, let u := v2 and q := θ (u). (a) Assume that Φd (q) = Φd (q) = 0, where d d. Then d = di for some i 0. To prove this, write d = ma , where a 0 and m. Now Φd (u) certainly divides a a Φm (u ) and so Φn (q ) = 0. Since the map α → α is a ring homomorphism of k, we conclude that Φm (q) = 0. A similar argument applies to d : writing d = m b , where b 0 and m , we conclude that Φm (q) = 0. Assume, if possible, that m = m . Then umm −1 = Φm (u)Φm (u)p(u), where p(u) ∈ Z[u]. Differentiating with respect to u and then setting u = q would yield mm = 0 in k, which is a contradiction. Hence, we have m = m and so d = ma and d = m b . This implies (a). (b) Let d 1. Then Φd (q) = 0 if and only if Φd (q) = 0. This is seen as follows. It is well known that
Φd (u) Φd (u) Φd (u ) = Φd (u)
if d, if | d.
Since the map α → α is a ring homomorphism of k, we conclude that Φd (u ) = Φd (u) (in k[u]). Hence Φd (u) = Φd (u)m (in k[u]), where m = or m = − 1. Thus, we also have Φd (q) = Φd (q)m , as required. Lemma 3.7.12. In the above setting, assume that Hk is not semisimple and let e 2 be as in 3.7.10. Let Φ ∈ P be such that θ (Φ ) = 0 and Φ | Φe (v2 ). Then the following hold. (a) Φ is θ -isolated if and only if ei ∈ E(W, L) for all i 1. (b) If Φ is θ -isolated and 2e, then Φ is strongly θ -isolated. (c) If L(s) = 1 for all s ∈ S and |W |, then Φ is strongly θ -isolated. Proof. As above, we write u := v2 and q := θ (u). (a) First assume that ei ∈ E(W, L) for all i 1. Let d ∈ E(W, L) and assume that θ (Φ ) = 0 for some Φ ∈ P such that Φ | Φd (v2 ). Then Φd (q) = 0 and so 3.7.11(a) shows that d = ei for some i 0. Our assumption implies that i = 0 and so d = e. Hence, both Φ and Φ divide Φe (v2 ). If e is even, then Φe (v2 ) = Φ2e (v); thus, Φ and Φ both divide Φ2e (v). If e is odd, then Φe (v2 ) = Φe (v)Φ2e (v), where Φ2e (v) = ±Φe (−v). Thus, replacing Φ by Φ (±v) and Φ by Φ (±v) for appropriate signs, we can assume that both Φ and Φ divide Φ2e (v). (Note that Φ ∼ Φ (±v) and Φ ∼ Φ (±v).) But, since all
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roots of unity in K of a given order are algebraically conjugate, there will be a ring isomorphism α : A → A such that α (R) = R and α (Φ ) = Φ . Hence, Φ ∼ Φ . Thus, we have shown that Φ is θ -isolated, as required. To prove the converse assume, if possible, that Φ is θ -isolated and that there is some i 1 such that ei ∈ E(W, L). Then 3.7.11(b) shows that Φei (q) = 0. Hence, there will be some Φ ∈ P such that θ (Φ ) = 0 and Φ | Φei (v2 ). Since Φ is θ isolated, we must have Φ ∼ Φ . In particular, this implies that both Φ and Φ divide Φe (v2 ). (Here, assumption (∗) in 3.7.10 is used.) But, if i 1, then Φei (u) and Φe (u) are coprime in Q[u] and, hence, also coprime in K[v], which is a contradiction. (b) Let Φ ∈ P and assume that θ (Φ ) = 0. We must show that Φ = Φ . Let d ∈ E(W, L) be such that Φ | Φd (v2 ). By 3.7.11(b), we have d = ei for some i 0. By (b), we must have i = 0. Thus, both Φ and Φ divide Φe (v2 ) and, hence, also v2e − 1. Assume, if possible, that Φ = Φ . Then we can write v2e − 1 = ΦΦ f for some f ∈ R[v]. Differentiating with respect to v and then applying θ would yield that θ (2e) = 0; that is, | 2e, which is contrary to our assumption. (c) In this case, one easily checks that E(W, L) consists of all integers d 2 such that d divides some degree of W ; see Example 3.3.2 (for W of crystallographic type) and [132, 8.3.4 and Appendix E] (otherwise). Since the product of all degrees is the order of W , the assertion follows from (a) and (b). Example 3.7.13. Let W be a finite Weyl group and assume that we are in the equal parameter case where L(s) = 1 for all s ∈ S, as in Example 3.3.2. Since ZW = Z, we can assume that R ⊆ Q = K. Assume that Hk is not semisimple. First, one easily checks that the definition of e in 3.7.10 can be rewritten as follows: e = min{i ∈ E(W ) | 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0}
(u := v2 );
recall that E(W ) is the set of all integers d 2 such d divides some degree of W . Let Φ ∈ P be such that θ (Φ ) = 0 and Φ | Φe (v2 ); by the description of P in Example 3.3.2, we have Φ = Φ2e (v) (if e is even) or Φ ∈ {Φe (v), Φ2e (v)} (if e if odd). By Lemma 3.7.12(a), we have
Φ is θ -isolated if and only if e does not divide any degree of W . A general condition of this kind first appeared in [133, §5]; see also [129, §3]. (a) Consider the special case where W ∼ = Sn is of type An−1 . By Example 3.1.19, the degrees of W are 2, 3, . . . , n. Hence, we have E(W ) = {2, 3, . . . , n} and so
Φ is θ -isolated if and only if e > n. Thus, Conjecture 3.7.2 really is a special case of Conjecture 3.7.9. (b) Now let W be of type F4 or E6 . In these cases, R ⊆ Q is L-good if 2, 3 ∈ R× . So we are only considering specialisations θ : A → k where char(k) = 5. Comparing with the list of degrees of W in Example 3.3.2, we see that Φ is automatically θ isolated. In these cases, Conjecture 3.7.9 was verified (assuming that q = θ (v)2 lies in the prime field of k) by Geck and Lux [126] (type F4 ) and Geck [94] (type E6 ). These results for groups of exceptional type provided a major motivation for formulating the general version of James’s conjecture above.
3.7 The General Version of James’s Conjecture
197
(c) Using the algorithm already mentioned in Example 3.7.7, the general version of James’s conjecture has been systematically verified by Geck and M¨uller [129] for W of exceptional type G2 , F4 , E6 , E7 , E8 . Here is an entirely different instance of James’s conjecture which was described by Leclerc and Miyachi [210]. Example 3.7.14. Let W be of type Bn and assume that we are in the setting of Example 3.3.4, where A = Z[V ±1 , v±1 ] with two independent variables V, v and Ts20 = T1 + (V −V −1 )Ts0
and Ts2i = T1 + (v − v−1 )Tsi
for 1 i n − 1.
Now let t be an indeterminate over F2 = Z/2Z and k = F2 (t). Let 1 r n−1. Then the natural map Z → F2 extends to a specialisation θr : A → k such that V → t r and v → t. Furthermore, consider the principal specialisation defined by Ψr := V 2 +v2r ∈ P. We have θr (Ψr ) = 0, and one easily checks that Ψr is strongly θr -isolated. So, r according to Conjecture 3.7.9, the adjustment matrix DΨ θr should be the identity matrix. This has been shown to be the case by Leclerc and Miyachi [210, Prop. 14]. In their notation, Dθr is the matrix associated with the decomposition map d2r,+ in 2 r [210, Prop. 14], and DΦ θr corresponds to the map dr,− . Leclerc and Miyachi interpret DΨr in terms of the constructible representations defined by Lusztig [231, Chap. 22]. This is related to the discussion in Example 3.8.10; see [106, §6] for further details. Apart from partial results in special situations where much is known about decomposition numbers (see, for example, Fayers [78], [79], [81] and Geck and M¨uller [129], or the “defect 1” case treated in [92]), James’s conjecture – in all its versions – is completely open. (See also [80], [258], [259], [260] and [290, Chap. VII] for further attempts at this conjecture.) We will now present a general argument for proving at least one piece of that conjecture; namely, the assertion that ΛΦ◦ = Λk◦ . For W ∼ = Sn , this has already been established in Theorem 3.5.14. In general, by Proposition 3.7.5, it will be sufficient to prove that | Irr(HkΦ )| = | Irr(Hk )|. Lemma 3.7.15. Let θ : A → k and Φ ∈ P be such that θ (Φ ) = 0, so that θ lies above θΦ . Assume that, for any M ∈ Irr(HkΦ ), the function PM : Z(H) → K (see Proposition 3.3.6) takes its values in Ap , the localisation of A in p = ker(θ ). Then we have | Irr(Hk )| = | Irr(HkΦ )|. Proof. By 3.7.4, we already know that | Irr(H k )| | Irr(HkΦ )|. So we only need to prove the reverse inequality. Write DΦ = dλ ,M . Then, by Remark 3.3.7(c), we have dλ ,M =
∑
χ λ (T˙wC ) PM (zC ) for all λ ∈ Λ and M ∈ Irr(HkΦ ).
C∈Cl(W )
These identities can be written as a matrix equation DΦ = X(H).Ptr , where P has entries PM (zC ) (M ∈ Irr(HkΦ ) and C ∈ Cl(W )). Now, the entries of X(H) lie in A and those of DΦ are rational integers. Hence, our assumption on the values of PM
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3 Specialisations and Decomposition Maps
implies that all entries of the above matrices lie in Ap . So we can apply the canonical extension of θ to Ap and obtain an identity among the specialised matrices over k. Since DΦ and its specialisation have rank equal to | Irr(HkΦ )| (see Theorem 3.1.14), we conclude that the rank of X(H)θ is greater than or equal to | Irr(HkΦ )|. Hence, by Proposition 3.4.11, we also have | Irr(Hk )| | Irr(HkΦ )|, as required. Theorem 3.7.16 (Cf. Geck and Rouquier [100], [133, Theorem 3.3]). If Φ ∈ P is strongly θ -isolated, then | Irr(HkΦ )| = | Irr(Hk )|. Hence, in the setting of 3.7.4, we have ΛΦ◦ = Λk◦ . Proof. Assume that Φ is strongly θ -isolated. We must show that, for any M ∈ Irr(HkΦ ), the condition on the values of PM in Lemma 3.7.15 holds. Now, we have PM (zC ) =
∑ c−1 λ dλ ,M ωλ (zC )
λ ∈Λ
(C ∈ Cl(W )),
where the numbers dλ ,M are the entries of DΦ and ωλ (zC ) ∈ A. Recall from 3.3.1 that, for each λ ∈ Λ , we can write cλ = Φ nλ ,Φ c λ ,
where
c λ := fλ vγλ
∏
(Φ )nλ ,Φ .
Φ ∈P\{Φ }
Note that θ (c λ ) = 0 for all λ ∈ Λ , since Φ is strongly θ -isolated and R is L-good. Now we can argue as follows. Setting f = ∏λ c λ and n := max{nλ ,Φ | λ ∈ Λ }, we have f Φ n PM (zC ) ∈ A. So we can write f Φ n PM (zC ) = Φ m a, where m 0 and a ∈ A is such that Φ a. Since θ (c λ ) = 0 for all λ ∈ Λ , we have Φ f . We conclude that we must have m−n = vΦ (PM (zC )), where vΦ is the discrete valuation associated with Φ . But, by Proposition 3.3.6, we have PM (zC ) ∈ A(Φ ) and, hence, m − n 0. Finally, since θ (c λ ) = 0 for all λ ∈ Λ , we also have θ ( f ) = 0. So we conclude that PM (zC ) = f −1 Φ m−n a ∈ Aker(θ ) , as required. Finally, if we are in the setting of 3.7.4 and | Irr(HkΦ )| = | Irr(Hk )|, then Proposition 3.7.5 shows that we also have ΛΦ◦ = Λk◦ . We just mention that similar arguments can be used to show that if Φ ∈ P is strongly θ -isolated, then the partition of Λ into Φ -blocks coincides with the partition into θ -blocks. (See [92, Prop. 7.6] for further details.)
3.8 Blocks and Bad Specialisations The main result of this section will show that the relation ∼L on IrrK (W ), which is defined in terms of the Kazhdan–Lusztig cells of W (see Section 2.2), has a ringtheoretic interpretation in terms of certain “blocks” of HK (in the general sense of the representation theory of associative algebras). These blocks are in some way related to specialisations of A which arise from the “L-bad” prime ideals in ZW , in the sense of Definition 1.5.9. (So far, such specialisations have been excluded from our discussion as we generally assumed that R should be an L-good subring of C.)
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199
We begin with some remarks on block idempotents. We have already encountered idempotents at various places above but, because they will play a crucial role in this section, let us collect here the basic facts and definitions in a general setting. (As before, the symbol H will eventually denote a generic Iwahori–Hecke algebra H = HA (W, S, L). Since we will have to deal with algebras over various different base rings, we shall use the symbol H˙ in the following general discussion.) 3.8.1. Let H˙ be an associative ring with identity. An element e ∈ H˙ is called an idempotent if e2 = e. Two idempotents e, f are called orthogonal if e f = f e = 0; an idempotent e = 0 is called primitive if e cannot be expressed as the sum of two non˙ A central idempo˙ the centre of H. zero, orthogonal idempotents. Denote by Z(H) ˙ ˙ tent is an idempotent e ∈ H which lies in Z(H). We say that e is a centrally primitive idempotent if e = 0 is a central idempotent and if e is primitive as an idempotent of ˙ We set Z(H). ˙ := { set of all centrally primitive idempotents in H˙ }; Bl(H) ˙ are also called block idempotents. Note that two block idemthe elements of Bl(H) potents b = b are automatically orthogonal. Indeed, b = bb + (b − bb ) is a decomposition of b as a sum of two orthogonal central idempotents; since b is centrally primitive, we have bb = 0 or bb = b. Similarly, b = bb + (b − bb ) is a decomposition of b as a sum of two orthogonal central idempotents; since b is centrally primitive, we have bb = 0 or bb = b . Hence, if we had bb = 0, then b = bb = b , which is a contradiction. So we have bb = 0, as claimed. 3.8.2. A useful method for finding the block idempotents in H˙ is as follows. Assume that we are given a decomposition 1H˙ = ∑e∈P e, where P is a finite set ˙ (We do not assume that the of pairwise orthogonal, primitive idempotents in H. idempotents in P are central.) Given e, e ∈ P, we write e ↔ e if there exists a sequence e = e0 , e1 , . . . , em = e in P such that, for each i ∈ {1, . . . , m}, we have ˙ i = {0} or ei He ˙ i−1 = {0}. Also note that ei−1 He e H˙ e ∼ = HomH˙ (H˙ e, H˙ e )
˙ for any idempotents e, e ∈ H.
Thus, e H˙ e = {0} if and only if there exists a non-zero homomorphism H˙ e → H˙ e . Now let P = P1 · · · Pr be the partition of P into equivalences classes under ↔. Then the following hold (see, for example, [48, 2.1.4]): ˙ = {e1 , . . . , er } Bl(H)
where
ei :=
∑e
for all i.
e∈Pi
Thus, the block idempotents of H˙ are obtained by summing up the idempotents in each equivalence class of P under ↔. Furthermore, for every central idempotent ˙ there exists a subset J ⊆ {1, . . . , r} such that b = ∑i∈J ei . b ∈ H, 3.8.3. Now assume that H˙ is finitely generated and free over an integral domain A0 . Let K be a field which contains the field of fractions of A0 and assume that the Kalgebra H K = K ⊗A0 H˙ is split semisimple. As in Example 3.1.11, we have a central
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3 Specialisations and Decomposition Maps
idempotent eV ∈ H K corresponding to any V ∈ Irr(H K ) where eV acts as the identity on V and as zero on any simple module not isomorphic to V . It follows that 1H K =
∑
eV
Bl(H K ) = {eV | V ∈ Irr(H K )}.
and
V ∈Irr(H K )
˙ then b ∈ Z(H K ). Fur˙ ⊆ Z(H K ). Hence, if b ∈ Bl(H), Now, we certainly have Z(H) thermore, eV b = eV ⇔ eV b = 0 ⇔ b.V = {0} for any V ∈ Irr(H K ). It follows that b=
∑
eV
where
V ∈Irr(H K |b)
Irr(H K | b) := {V ∈ Irr(H K ) | eV b = 0};
Conversely, if S ⊆ Irr(H K ) is any subset such that ˙ ∈ H ∈ H˙ for any S S , S = ∅, e and e V V ∑ ∑ V ∈S
V ∈S
then ∑V ∈S eV is a block idempotent of H (see also [48, 2.1.6]). Thus, we obtain: Irr(H K ) =
" ˙ b∈Bl(H)
Irr(H K | b)
and
1H˙ =
∑
b.
˙ b∈Bl(H)
Definition 3.8.4. In the setting of 3.8.3, let V,V ∈ Irr(H K ). We say that V,V belong ˙ to the same H-block (or just to the same A0 -block if H˙ is clear from the context), if ˙ such that V,V ∈ Irr(H K | b). there exists some b ∈ Bl(H) Example 3.8.5. Let us go back to the setting in Section 3.1. Let H, K and θ : A → k be as in 3.1.1. Assume that A is integrally closed in K; furthermore, assume that H k is split and H K is split semisimple. Hence, by Theorem 3.1.2, we have a well-defined decomposition map dθ : R0 (H K ) → R0 (H k ). As discussed in 3.1.8, the corresponding Brauer graph gives rise to the partition of Irr(H K ) into θ -blocks. Now let A0 ⊆ K be the localisation of A in the prime ideal ker(θ ) and set H˙ := A0 ⊗A H. Then we are in the set-up of 3.8.3. Assume that A0 is a discrete valuation ring and let V,V ∈ Irr(H K ). Then the following three conditions are equivalent: (a) V and V belong to the same θ -block (see 3.1.8); (b) V and V belong to the same A0 -block (see Definition 3.8.4); (c) θ (ω V (z)) = θ (ω V (z)) for all z ∈ Z(H) (see Lemma 3.1.10). This is part of the classical representation theory of associative algebras; see, for example, [83, Cor. I.17.9], [132, 7.5.10]. (As already discussed in the proof of Theorem 3.4.4, it is not necessary to assume that A0 is complete.) In the following discussion of Iwahori–Hecke algebras, we shall apply Definition 3.8.4 to an algebra H˙ over a base ring A0 which is not a local ring. 3.8.6. Now let again W be a finite Coxeter group and H = HA (W, S, L) be the generic Iwahori–Hecke algebra associated with a weight function L : W → Γ such that
3.8 Blocks and Bad Specialisations
201
L(s) 0 for all s ∈ S; see 3.1.12. Here, A = R[Γ ], where R ⊆ C is a subring such that ZW ⊆ R. Also recall our notation IrrK (W ) = {E λ | λ ∈ Λ }
and
Irr(HK ) = {Eελ | λ ∈ Λ },
where K is the field of fractions of R and K is the field of fractions of A. As in Remark 1.7.8, consider the subring
! # f ! f ∈ Z[ Γ ], g ∈ 1 + Z[Γ>0 ] ⊆ K. R(Γ ) := ! g Since the structure constants of H lie in Z[Γ ], we have an R(Γ )-subalgebra $ % HR(Γ ) := ∑ aw Tw | aw ∈ R(Γ ) for all w ∈ W ⊆ HK . w∈W
Hence, setting A0 = R(Γ ) and H˙ := HR(Γ ) , we are in the set-up of 3.8.3. Consequently, we can speak of the HR(Γ ) -blocks (or just the R(Γ )-blocks) of Irr(HK ). The following remarkable result was first proved by Rouquier [272] in the equalparameter case, following earlier work of Gyoja [151]. A version for the case of unequal parameters appeared in [106, §3]. Theorem 3.8.7. In the setting of 3.8.6, assume that the properties (♣), (♠), ( ) in 2.5.3 hold. Let λ , μ ∈ Λ . Then E λ ∼L E μ
μ
Eελ , Eε belong to the same R(Γ )-block,
⇔
where the relation ∼L on IrrK (W ) is defined in Definition 2.2.1. Proof. The basic idea is to reduce everything to a problem inside the asymptotic ˜ For this purpose, consider Lusztig’s homomorphism φ : H → J˜ A , where algebra J.
φ (Cw ) =
∑
hw,d,z n˜ d tz
(w ∈ W );
z∈W, d∈D˜ z∼LR d
see Theorem 2.5.5. Here, we already used that (♠) holds. Since (♣) also holds, we ˜ see Lemma 2.5.13. Hence, have γ˜x,y,z ∈ Z for all x, y, z ∈ W and n˜ d ∈ Z for all d ∈ D; we can use the above formula to define an R(Γ )-algebra homomorphism
φR(Γ ) : HR(Γ ) → J˜ R(Γ ) where J˜ R(Γ ) := ∑w∈W rwtw | rw ∈ R(Γ ) for all w ∈ W . The first crucial observation is that, by Remark 2.5.15, this homomorphism is an isomorphism of R(Γ )algebras. Consequently, we have Bl(J˜ R(Γ ) ) = {φR(Γ ) (b) | b ∈ Bl(HR(Γ ) )}.
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3 Specialisations and Decomposition Maps
Recall that the simple HK -modules afford balanced representations {ρ λ | λ ∈ Λ } ˜ Extending and these give rise to the irreducible representations {ρ¯ λ | λ ∈ Λ } of J. λ ˜ scalars to K, we obtain irreducible representations ρ¯ K : JK → Mdλ (K) such that ρ¯ Kλ (tw ) = ρ¯ λ (tw ) for all w ∈ W . As in Example 2.7.3, we see that, for a suitable basis of Eελ , the action of Cw (w ∈ W ) on Eελ is given by the matrix ρ¯ Kλ (φR(Γ ) (Cw )). Thus, for any block idempotent b ∈ Bl(HR(Γ ) ), we have b.Eελ = {0}
ρ¯ Kλ (φR(Γ ) (b)) = 0.
⇔
So, since φR(Γ ) (b) ∈ Bl(J˜ R(Γ ) ), we have μ
Eελ , Eε belong to the same R(Γ )-block (of HK )
⇔
ρ¯ Kλ , ρ¯ Kμ belong to the same R(Γ )-block (of J˜ K ).
On the other hand, recall from Definition 2.2.1 that E λ ∼L E μ if and only if the ˜ associated two-sided J-cells are contained in the same two-sided Kazhdan–Lusztig cell. Now, since (♠) holds, the two-sided Kazhdan–Lusztig cells coincide with the ˜ two-sided J-cells; see Lemma 2.5.9. Hence, we have E λ ∼L E μ if and only if Fλ = Fμ . Thus, it remains to show the following equivalence: (∗)
Fλ = F μ
μ ρ¯ Kλ , ρ¯ K belong to the same R(Γ )-block.
⇔
To prove this, recall from Remark 1.6.13 that we have a central idempotent 1F =
∑
n˜d td
˜ for any two-sided J-cell F;
˜ d∈D∩F
˜ furthermore, we have Fλ = Fμ if and only if there exists some two-sided J-cell F λ μ ¯ ¯ such that ρ (1F ) = 0 and ρ (1F ) = 0. Consequently, in order to prove (∗), it is sufficient to prove the following two statements: ˜ the element n˜ d td is a primitive idempotent in J˜ R(Γ ) . (a) For each d ∈ D, ˜ (b) The elements {1F | F two-sided J-cell} are the block idempotents of J˜ R(Γ ) . Note that, since ( ) holds, we already know that the elements in (a) are pairwise ˜ see Lemma 2.7.9. Now, to orthogonal idempotents and that n˜ d = ±1 for all d ∈ D; prove (a), assume that n˜d td = e + e , where e, e ∈ J˜ R(Γ ) are orthogonal idempotents and e = 0. Then we must show that e = 0. For this purpose, we shall compute τ¯ (e) in two different ways, where τ¯ is the symmetrising trace in Proposition 1.5.6. First, writing e = ∑w∈W rwtw , where rw ∈ R(Γ ), we find that
τ¯ (e) =
∑ rw τ¯ (tw ) = ∑ rd n˜d ∈ R(Γ ) = ∑ (±rd ) ∈ R(Γ ). d∈D˜
w∈W
d∈D˜
On the other hand, by the defining formula for n˜ w , we have
τ¯ (tw ) = n˜ w =
∑
λ ∈Λ
fλ−1 trace(ρ¯ λ (tw ))
for all w ∈ W
3.8 Blocks and Bad Specialisations
203
and, hence, τ¯ (e) = ∑λ ∈Λ fλ−1 trace(ρ¯ Kλ (e)). Now, since e is an idempotent, the trace of ρ¯ Kλ (e) is a non-negative integer for all λ ∈ Λ , and it is non-zero for at least some λ . Since the elements fλ are strictly positive real numbers, we conclude that τ¯ (e) is a strictly positive real number. Thus, we have shown that τ¯ (e) ∈ R(Γ ) ∩ R>0 . But, by the definition of R(Γ ), we have R(Γ ) ∩ R = Z. Hence, we deduce that τ¯ (e) is a strictly positive integer; in particular, τ¯ (e) 1. If e = 0, then the same argument would apply to e and so τ¯ (e ) 1. This would yield the contradiction 1 = τ¯ (n˜ d td ) = τ¯ (e) + τ¯ (e ) 2. Thus, we must have e = 0, as required. To prove (b), we use the method in 3.8.2. We must show that two given ele˜ if and only if d ↔ d . To prove the ments d, d ∈ D˜ lie in the same two-sided J-cell “only if” part, it is sufficient to consider an elementary step of ↔; that is, we have td J˜ R(Γ ) td = {0}. Then td twtd = 0 for some w ∈ W and so 0 = td twtd =
∑ γ˜d,w,x−1 txtd = ∑
x∈W
γ˜d,w,x−1 γ˜x,d ,y−1 ty .
x,y∈W
Hence, there exist some x, y ∈ W such that γ˜d,w,x−1 = 0 and γ˜x,d ,y−1 = 0. By Corol˜ lary 1.6.7, this implies that d, w, x, d , y all lie in the same two-sided J-cell, as re ˜ ˜ quired. Conversely, assume that d, d ∈ D belong to the same two-sided J-cell. ˜ such that d ∈ C0 and d ∈ C . Let λ , μ ∈ Λ be Let C0 and C be the left J-cells such that m(C0 , λ ) > 0 and m(C , μ ) > 0. Then we have Fλ = Fμ by Example 1.8.2. So Lemma 1.8.4 shows that there exist a sequence λ = λ0 , λ1 , . . . , λm = μ ˜ in Λ and a sequence of left J-cells C1 , . . . , Cm such that, for each i ∈ {1, . . . , m}, we have m(Ci , λi−1 ) > 0 and m(Ci , λi ) > 0. Consequently, m(Ci , λi−1 ) > 0 and m(Ci−1 , λi−1 ) > 0 for 1 i m. Let us also set Cm+1 := C and recall that μ = λm . Then [Ci−1 ]K and [Ci ]K have an irreducible constituent in common and so HomJ˜ ([Ci−1 ]K , [Ci ]K ) = {0}
for 1 i m + 1.
Now, for each i, let di ∈ D˜ be the unique element contained in Ci . Then we have ˜ d ∼ Jt [Ci ]K ; see Example 1.8.6. It follows that i = ˜ d , Jt ˜ d ) = {0} ˜ d ∼ HomJ˜ (Jt tdi−1 Jt i = i i−1
for 1 i m + 1.
Note that then we also have tdi−1 J˜ R(Γ ) tdi = {0} for 1 i m + 1 and so d = d0 ↔ dm+1 = d , as required. Thus, (a) and (b) are proved. Remark 3.8.8. By going once more through the above proof, we notice that the same argument works when we replace R(Γ ) by any noetherian ring Θ , where Z[Γ ] ⊆ Θ ⊆ K, such that the following conditions hold: 1 + Z[Γ>0 ] ⊆ Θ × and Θ ∩ ∑ f λ−1 mλ | mλ ∈ Z for all mλ ∈ Z ⊆ Z. λ ∈Λ
(This observation appeared in [106, §3].) Indeed, the first condition makes sure that φΘ : HΘ → J˜ Θ is an isomorphism of Θ -algebras. The second condition is required ˜ (See (a) in in the proof of the fact that n˜ d td is a primitive idempotent for each d ∈ D.
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3 Specialisations and Decomposition Maps
the above proof.) Recall that we needed to evaluate τ¯ (e), where e ∈ J˜ Θ is a non-zero idempotent. First, writing e = ∑w∈W rwtw , where rw ∈ Θ , we see that τ¯ (e) ∈ Θ . Then, using τ¯ (e) = ∑λ ∈Λ fλ−1 trace(ρ¯ Kλ (e)), we see that τ¯ (e) = ∑λ ∈Λ fλ−1 mλ is a positive real number, where mλ ∈ Z for all λ ∈ Λ . Hence, the second condition implies that τ¯ (e) ∈ Z. Now we can continue to argue as above. Thus, we obtain that E λ ∼L E μ
⇔
μ
Eελ , Eε belong to the same Θ -block.
We will illustrate this in Example 3.8.10 below.
Table 3.4 Character table and Schur elements for H(G2 ) v 1W sgnv1 sgnv2 sgnv σ1v σ2v
T1 1 1 1 1 2 2
Ts1 u u −1 −1 u−1 u−1
Ts2 u −1 u −1 u−1 u−1
Ts1 s2 Ts1 s2 s1 s2 Ts1 s2 s1 s2 s1 u2 u4 u6 2 −u u −u3 2 −u u −u3 1 1 1 u −u2 −2u3 −u −u2 2u3
cλ (u + 1)2 (u4 + u2 + 1) 3u−1 (u + 1)2 3u−1 (u + 1)2 −6 u (u + 1)2 (u2 + u2 + 1) 6u−1 (u2 − u + 1) 2u−1 (u2 + u + 1)
Example 3.8.9. Since H is a symmetric algebra, there is an explicit formula for the block idempotents of HK in terms of the values of the irreducible characters. (This was already used in the proof of Proposition 3.1.17.) Hence, using the notation and the formulae in Remark 3.3.7, we find that eλ = c−1 λ = c−1 λ = c−1 λ
∑ ε −2L(w) χ λ (T˙w ) T˙w−1
w∈W
∑ ∑
ε −2L(w) fw,C χ λ (T˙wC ) T˙w−1
w∈W C∈Cl(W )
∑
χ λ (T˙wC ) zC .
C∈Cl(W )
Now let Θ ⊆ K be a subring as in Remark 3.8.8 and consider the partition of Irr(HK ) into Θ -blocks. By the discussion in 3.8.3, this partition is found by determining the non-empty subsets S ⊆ Λ which are minimal with respect to the property that (∗)
∑
λ ∈S
λ ˙ c−1 λ χ (TwC ) ∈ Θ
for all C ∈ Cl(W ).
The following simple rules are now obvious where we assume that ZW = Z: (a) If fλ = 1, then {Eελ } is a Θ -block. (Indeed, in this case, we have cλ ∈ 1+Z[Γ>0 ] and so eλ ∈ HΘ .)
3.8 Blocks and Bad Specialisations
205
(b) Let b ∈ Bl(HΘ ) and a := max{aλ | Eελ ∈ Irr(HK | b)}. Then ∑λ fλ−1 dim E λ ∈ Z, where the sum runs over all λ ∈ Λ such that aλ = a and Eελ ∈ Irr(HK | b). (Indeed, just consider the coefficient of zC in (∗) where C = {1}.) In any case, since the polynomials {cλ } and the character table X(H) are known, we now have a practical algorithm for determining the Θ -blocks. For example, let W be of type G2 = I2 (6) with generators s1 , s2 such that (s1 s2 )6 = 1. Assume that we are in the equal parameter case where Γ = Z and L(s1 ) = L(s2 ) = 1. Then A is the ring of Laurent polynomials in one indeterminate v = ε ; let u := v2 . The character table of H and the polynomials {cλ } are given in Table 3.4. Let Θ = R (the original version of Rouquier’s ring). By (a), we have the v v R-blocks {1W } and {sgnv }. Let S := Irr(HK ) \ {1W , sgnv }. Then all irreducible representations in S have aλ = 1. Furthermore,
∑
E λ ∈S
fλ−1 dim E λ =
1 1 2 2 + + + . 3 3 6 2
We notice that σ2v does not form an R-block by itself. Hence, there is no proper subset of S such that the corresponding sum of terms fλ−1 dim E λ would still lie in Z. Hence, we conclude that S is an R-block. Example 3.8.10. Let Γ = Z so that A = R[v, v−1 ] is the ring of Laurent polynomials in one indeterminate v = ε . Let W be of type Bn and consider a weight function L : W → Z such that we are not in one of the cases in 1.7.6; that is, we have r 4 1 1 1 t t t p p p t where r ∈ {0, 1, . . . , n − 1}. Bn Assume that (♣), (♠), ( ) hold for W, L. This is the case, for example, for r = 1 (the equal parameter case) and for r = 0 (the case relevant for dealing with type Dn ). Since ZW = Z, we can assume that R ⊆ K = Q. By Example 1.3.9, we know that fλ ∈ {1, 2, 4, 8, 16, . . .}
for all λ ∈ Λ .
So the subring R = Z is not L-good but it is L0 -good; see Table 3.1 (p. 139). Now consider the specialisation θ2 : A → k = F2 (v) given by reducing the coefficients of the polynomials in A modulo 2. We obtain a well-defined decomposition map d2 : R0 (HK ) → R0 (Hk );
see Theorem 3.1.14.
Let Θ ⊆ K be the localisation of A = Z[v, v−1 ] in the prime ideal generated by 2. Then Θ satisfies the conditions in Remark 3.8.8 and so we conclude that E λ ∼L E μ if μ and only if Evλ , Ev belong to the same Θ -block. Now, since Θ is a discrete valuation ring, the Θ -blocks coincide with the θ2 -blocks defined in terms of the Brauer graph associated with d2 ; see Example 3.8.5. Thus, we have shown the following result. Theorem 3.8.11 (Gyoja [151]). Let W be of type Bn and L : W → Z be a weight function as in Example 3.8.10, where (♣), (♠) and ( ) are assumed to hold. Let λ , μ ∈ Λ . Then we have
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3 Specialisations and Decomposition Maps μ
Eελ ∼L Eε
⇔
μ
Eελ , Eε belong to the same θ2 -block.
In other words: the relation ∼L on IrrK (W ) is determined by the Brauer graph associated with the “bad” specialisation θ2 : A → k = F2 (v), which brings us back to a statement announced in Example 3.1.15(c). One can even go further and also interpret the decomposition matrix of d2 in terms of left cells; see [151], [106, §6]. The special feature of the above example is that all f λ are powers of one fixed prime number. Thus, Rouquier’s ring R(Γ ) really is the optimal choice for dealing with the general situation where more than one prime number will be involved. Remark 3.8.12. One can associate a “generic” algebra H to any finite complex reflection group G. Although many results from the theory of finite Coxeter groups extend to this more general setting, it is not yet known how to define “cells” a` la Kazhdan–Lusztig. But it does make sense to consider blocks of Irr(HK ) with respect to a suitable version of Rouquier’s ring R. And it turns out that these blocks have many properties that one would expect to hold if there was a suitable theory of Kazhdan–Lusztig cells for complex reflection groups. We refer the reader to Brou´e and Kim [33], Brou´e, Malle and Michel [35], Chlouveraki [48], Malle [238], Malle and Rouquier [241] for further details, background and references on this area.
Chapter 4
Hecke Algebras and Finite Groups of Lie Type
In this chapter we show how the theory of Iwahori–Hecke algebras is applied to the representation theory of a finite group of Lie group G. Examples of such groups are GLn (Fq ) (the finite general linear group) or Sp2n (Fq ) (the finite symplectic group). Any G as above is naturally defined over a finite field of characteristic p > 0. We shall work with representations of G over a sufficiently large field k of characteristic 0, where either = 0 or > 0 but = p. The remaining case where = p is treated by different methods and will not be considered here. (For further details and references on this case, see Curtis and Reiner [53, §72], Cabanes and Enguehard [43, Chap. 6], Donkin [71] and Lusztig [227].) So, from now on, = p. If = 0, then it is well-known that the irreducible representations of G can be partitioned into Harish-Chandra series. A particular case is given by the “unipotent principal series representations” of G, which are precisely those irreducible representations of G which admit non-zero vectors fixed by a Borel subgroup of G. By classical results due to Bourbaki, Iwahori, Tits, etc., these representations are in natural bijection with the irreducible representations of the Weyl group W of G. It is remarkable that, with suitable and quite natural modifications, there are analogous versions of these statements for “modular” representations; that is, for the case where > 0. In Section 4.1 we give a self-contained account of the basic theory of Schur functors and Dipper’s Hom functors. This provides the theoretical foundation for the discussion of “modular” Harish-Chandra series of G; a survey of this theory (mostly without proofs) will be given in Section 4.2. We then discuss the unipotent principal series representations in more detail. In Section 4.3 we show how generic Iwahori–Hecke algebras enter the picture. Here, we rely to some extent on the theory of algebraic groups (which underlies the definition of G). We then come to a highlight of this chapter: the “modular” analogue of the classification of the unipotent principal series representations; see Theorem 4.4.1. This is one of the main applications of our results on canonical basic sets from Chapter 3. Finally, in Sections 4.5 and 4.6, we discuss a number of examples and open problems. In particular, we explain how Theorem 4.4.1 fits into a (conjectural) classification of all irreducible representations of G in characteristic > 0.
M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7 4, © Springer-Verlag London Limited 2011
207
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4 Hecke Algebras and Finite Groups of Lie Type
4.1 The Schur Functor and its Variations This section prepares the ground for the study of Harish-Chandra series. We assume that the reader has a basic familiarity with the representation theory of finitedimensional associative algebras (Artin–Wedderburn theorem, projective modules and their relation with simple modules, etc.; e.g. see the relevant chapters of Curtis and Reiner [53] or Feit [83].) Throughout, let A be a finite-dimensional associative algebra over a field k. (In our applications, A will be the group algebra of a finite group.) We assume that A has an identity element and that all A-modules are unital; furthermore, all A-modules will be tacitly assumed to be finite-dimensional. Let M be a (finite-dimensional) A-module and H := EndA (M)◦ be the opposite algebra of its endomorphism algebra. In this set-up, we have the Hom functor FM : A-mod → H-mod,
V → HomA (M,V ),
where H acts on FM (V ) via h. f = f ◦ h (for h ∈ H, f ∈ FM (V )) and FM sends a map ϕ : V → V to the map FM (ϕ ) = ϕ∗ : FM (V ) → FM (V ), f → ϕ ◦ f . Let Irr(A) = set of irreducible representations of A (up to isomorphism), Irr(H) = set of irreducible representations of H (up to isomorphism). Our aim is to establish a relation, via FM , between Irr(H) and the set Irr(A | M) := {V ∈ Irr(A) | V appears in M/rad(M)}. If M is projective, then FM is an example of a Schur functor. In this case, FM induces a bijection between the two sets; see Proposition 4.1.3 and Remark 4.1.4. We will then show that the same conclusion holds under a weaker assumption on M. The original reference is Dipper [58], [59], but we have incorporated here some variations proposed by Cline, Parshall and Scott [51] and Schubert [279]. Apart from the use of standard properties of projective modules, Hom spaces and tensor products, our exposition will be self-contained. Remark 4.1.1. To obtain a one-sided inverse to FM , consider the functor GM : H-mod → A-mod,
E → M ⊗H E,
where M is considered as a right H-module in a natural way. (We have m.h = h(m) for m ∈ M and h ∈ H.) A straightforward application of the “Change of Rings Lemma” (see [53, 2.19]) yields that E ∼ = FM GM (E) for all E ∈ H-mod. Thus, GM is a right inverse to FM . Lemma 4.1.2. Let M = M1 ⊕ · · · ⊕ Mn be a direct sum decomposition of M. Then, for each i, the following holds: dimk HomA (Mi ,V ) = dimk HomH (FM (Mi ), FM (V )) for any V ∈ A -mod.
4.1 The Schur Functor and its Variations
209
Proof. We have a natural map Φi : HomA (Mi ,V ) → HomH FM (Mi ), FM (V ) ,
α → α∗ = ( f → α ◦ f ).
First we note that this map is injective. Indeed, if Φi (α ) = α∗ = 0, then α ◦ f = 0 for all f ∈ FM (Mi ). Now let ei : M → Mi be the canonical projection. Then ei ∈ FM (Mi ) and so α (ei (m)) = 0 for all m ∈ M. Thus, we have α = 0, as claimed. Now, since HomA and FM preserve direct sums, we have dimk FM (V ) = dimk HomA (M,V ) = ∑i dimk HomA (Mi ,V ), ∑i dimk HomH FM (Mi ), FM (V ) (by the injectivity of Φi ) = dimk HomH FM (M), FM (V ) . However, the Hom space in the last line certainly is isomorphic to FM (V ). Hence, all maps Φi must be surjective.
Proposition 4.1.3. Assume that M is projective. Then FM induces a bijection 1−1
FM : Irr(A | M) −→ Irr(H). Proof. Let M = M1 ⊕ · · · ⊕ Mn be a decomposition of M into indecomposable direct summands. Being projective, each Mi has a unique simple factor module; we set Yi = Mi /rad(Mi ) for all i. It is well known that Yi ∼ = Y j if and only if Mi ∼ = M j ; we write i ∼ j in this case. Then Irr(A | M) = {Yi | i ∈ I}, where I ⊆ {1, . . . , n} is a set of representatives under the relation ∼. Now let Yi ∈ Irr(A | M). We show that FM (Yi ) is a simple H-module. Indeed, let 0 = f ∈ FM (Yi ). Since Yi is simple, the homomorphism f : M → Yi must be surjective. Hence, since M is projective, the induced map f∗ : FM (M) → FM (Yi ) (given by composition with f ) also is surjective. This means that, as an H-module, we have FM (Yi ) = H. f and so FM (Yi ) is simple, as claimed. Thus, FM induces a map from Irr(A | M) to Irr(H). We show that this map is injective. So assume that FM (Yi ) ∼ = FM (Y j ). Since there is a non-zero map Mi → Yi , we also have a non-zero map FM (Mi ) → FM (Yi ) by ∼ Lemma 4.1.2. Composing this with the given isomorphism FM (Yi ) = FM (Y j ) yields that HomH FM (Mi ), FM (Y j ) = {0}. But then, again by Lemma 4.1.2, we have HomA (Mi ,Y j ) = {0} and so Yi ∼ = Y j , as required. Finally, we show that each E ∈ Irr(H) is isomorphic to FM (Yi ) for some i. Let V := GM (E) ∈ A-mod. By Remark 4.1.1, we have E ∼ = FM (V ). So, since E = {0}, there is a non-zero homomorphism M → V . Hence, if V is simple, then V ∈ Irr(A | M) and we are done. Otherwise, let U ⊆ V be a non-trivial submodule. Since M is projective, the functor FM sends an exact sequence of A-modules to an exact sequence of H-modules. Consequently, E will be isomorphic to FM (U) or to FM (V /U). By an inductive argument, we will eventually find some simple subquo tient V of V such that FM (V ) ∼ = E. As before, this implies that V ∈ Irr(A | M).
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4 Hecke Algebras and Finite Groups of Lie Type
Remark 4.1.4. Assume that M is projective and let A-modM be the full subcategory of A-mod consisting of all V ∈ A-mod such that • every non-zero submodule of V has a composition factor in Irr(A | M) and • every non-zero quotient module of V has a composition factor in Irr(A | M). Then one can show that FM induces an equivalence of abelian categories between A-modM and H-mod, with inverse being given by GM . For further details and references, see Green [145, §6.2] and Brundan, Dipper and Kleshchev [38, §3.1]. The above results relied on the assumption that M is projective. The following result will allow us to obtain similar conclusions under a weaker assumption on M. Lemma 4.1.5 (Schubert [279]; see also Cline, Parshall and Scott [51]). Assume that there is a surjective homomorphism of A-modules β : P → M, where P is projective, such that the following holds: (S)
ker(β ) ⊆ ker(ϕ )
for every ϕ ∈ HomA (P, M).
Then M is a projective A/I-module, where I denotes the annihilator of M in A. Proof. Denote (as usual) by IP the A-submodule of P spanned by all terms a.p, where a ∈ I and p ∈ P. Then IP ⊆ ker(β ) and so β induces a surjective homomorphism of A/I-modules β¯ : P/IP → M. Also note that P/IP is a projective A/Imodule. Hence, it will be enough to show that β¯ is injective or, in other words, that ker(β ) ⊆ IP. To see this, we use the following well-known characterisation of P as a finitely generated projective module: there exist elements p1 , . . . , pn ∈ P and f 1 , . . . , fn ∈ HomA (P, A) such that p = ∑ni=1 f i (p).pi for all p ∈ P. (This readily follows from the fact that P is a direct summand of a direct sum of copies of A.) Now we argue as follows. Let z ∈ ker(β ). We want to show that z ∈ IP. For this purpose, let i ∈ {1, . . . , n} and m ∈ M be fixed. Then the map P → M defined by sending p ∈ P to f i (p).m is an A-module homomorphism. By condition (S), we have fi (z).m = 0. Since this holds for all m ∈ M, we conclude that fi (z) ∈ I. Thus,
we see that z = ∑ni=1 f i (z).pi ∈ IP, as required. Remark 4.1.6. Schubert [279] shows that the converse of Lemma 4.1.5 is also true (but we do not need this result here). One easily checks that (S) is equivalent to (D)
ker(β ) is invariant under any endomorphism in EndA (P).
This condition is the one that appeared in Dipper’s original approach [58]. The following observation provides an efficient tool for checking that the condition in Lemma 4.1.5 is satisfied. Together with its application to Harish-Chandra series (see Theorem 4.2.9), the “if” part first appeared in Geck, Hiss and Malle [120, §2]; see also [102, §2]. (The “only if” part is due to Dipper [58, (2.8)].) Lemma 4.1.7. Condition (S) in Lemma 4.1.5 holds if dimk HomA (P, M) = dimk HomA (M, M).
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211
Conversely, if (S) holds, then dimk HomA (P,V ) = dimk HomA (M,V ) for any V ∈ A -mod which is isomorphic to a factor module of M. Proof. Let V ∈ A -mod be isomorphic to a factor module of M. Consider the k-linear map β ∗ : HomA (M,V ) → HomA (P,V ), α → α ◦ β . Since β is surjective, this map is injective. Now let V = M and assume that the two Hom spaces have the same dimension. Then β ∗ will be surjective. So, if ϕ ∈ HomA (P, M), then ϕ = α ◦ β for some α ∈ HomA (M, M) and, hence, ker(β ) ⊆ ker(ϕ ); that is, condition (S) holds. Conversely, assume that (S) holds. We must show that β ∗ is surjective. Let ϕ ∈ HomA (P,V ). In the proof of Lemma 4.1.5, we have seen that ker(β ) = IP. Now our assumption on V implies that I annihilates V and so IP ⊆ ker(ϕ ). Hence, ϕ induces a map ϕ¯ ∈ HomA (M,V ) such that ϕ = ϕ¯ ◦ β . Thus, β ∗ is surjective, as claimed.
Proposition 4.1.8 (Dipper [58, 2.28]). Let β : P → M be a surjective homomorphism of A-modules where P is projective. Assume that condition (S) in Lemma 4.1.5 is satisfied for β . Then the Hom functor FM : A -mod → H -mod induces a bijection 1−1
FM : Irr(A | M) −→ Irr(H). Proof. Let A := A/I, where, as before, I denotes the annihilator of M in A. To have a separate notation, let us write M instead of M when we consider M as an A -module. By Lemma 4.1.5, M is a projective A -module. So we can apply Proposition 4.1.3 and there is an induced bijection FM : Irr(A | M ) −→ Irr(H ), 1−1
where H := EndA (M )◦ .
Now note that a k-linear map from M into itself is an A-module homomorphism if and only if it is an A -homomorphism. Hence, we have an equality H = H . Furthermore, let V be a simple A-module. If V is a composition factor of M, then I is contained in the annihilator of V and so V can also be regarded as an A -module; let us denote this A -module by V . Then we have an equality FM (V ) = HomA (M,V ) = HomA (M ,V ) = FM (V ). It also follows that we can naturally identify Irr(A | M) and Irr(A | M ). Thus, we
see that FM induces a bijection Irr(A | M) → Irr(H), as required. Remark 4.1.9. There are alternative ways to make sure that FM induces a bijection as in Proposition 4.1.8. For example, instead of the above condition (S) in Lemma 4.1.5, it would be enough to assume that H is a symmetric algebra and that the set of composition factors of M/rad(M) equals the set of composition factors of soc(M). For further details see [117, §2] and [102, §2]. Here, we chose to rely on the condition (S) in Lemma 4.1.5 because it is rather straightforward to verify, thanks to Lemma 4.1.7. Note, however, that the alternative conditions discussed in [102] yield further structural properties of M. Corollary 4.1.10 (Dipper [58, 2.32]). Let M = M1 ⊕ · · · ⊕ Mn be a decomposition of M into indecomposable direct summands. Assume that we are in the setting of
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4 Hecke Algebras and Finite Groups of Lie Type
Proposition 4.1.8, so that we have a bijection 1−1
FM : Irr(A | M) −→ Irr(H). Also assume that EndA (Y ) = k idY for all Y ∈ Irr(A | M). (a) Each Mi has a unique maximal submodule; let Yi := Mi / rad(Mi ). We have Yi ∼ = Y j if and only if Mi ∼ = M j ; we write i ∼ j in this case. Then Irr(A | M) = {Yi | i ∈ I}, where I ⊆ {1, . . . , n} is a set of representatives under the relation ∼. (b) Let V ∈ A -mod be isomorphic to a factor module of M. Then the dimension of HomA (Mi ,V ) equals the multiplicity of Yi as a composition factor of V . (c) The H-module FM (Mi ) is projective indecomposable, and its unique simple factor module is isomorphic to FM (Yi ). (d) The Cartan matrix of H records the multiplicities of the various simple modules in Irr(A | M) as composition factors of M. Proof. We use the notation in the proof of Proposition 4.1.8 where I is the annihilator of M and A = A/I. In particular, we can identify Irr(A | M) = Irr(A | M ) and H = H where M is M regarded as an A -module. (a) Since Mi is an indecomposable A-module, the same is true when we regard Mi as an A -module. But, by Lemma 4.1.5, the latter module is projective. Hence, all the statements follow from general properties of projective indecomposable modules. (b) Since V is isomorphic to a factor module of M, the annihilator of V contains I. Hence, we can regard V as an A -module. It remains to note that HomA (Mi ,V ) = HomA (Mi ,V ) and Mi is a projective indecomposable A -module. (c) The decomposition M = M1 ⊕ · · · ⊕ Mn gives rise to a decomposition H = FM (M) = FM (M1 ) ⊕ · · · ⊕ FM (Mn ). Hence, each FM (Mi ) is a projective H-module. Since Yi is a factor module of Mi and since FM (Yi ) is simple, we also have a surjective homomorphism FM (Mi ) → FM (Yi ) by Lemma 4.1.2. Thus, it remains to show that FM (Mi ) is indecomposable. To see this, one could of course refer to the “Fitting correspondence” (e.g. see [53, §6]). Alternatively, one can use Lemma 4.1.2, which provides an isomorphism of vector spaces EndA (Mi ) = HomA (Mi , Mi ) ∼ = HomH (FM (Mi ), FM (Mi )) = EndH (FM (Mi )). We have seen that this isomorphism is given by Φi : α → α∗ = ( f → α ◦ f ). Hence, in the present context, Φi even is a ring homomorphism and so the above two Endspaces are isomorphic as k-algebras. It remains to use the fact that the endomorphism ring of a module is a local ring if and only if the module is indecomposable. (d) This follows from (b) and (c), where we take V = Y j .
For our applications to finite groups of Lie type, it will be essential to relate the above results to a situation where k is the residue field of a discrete valuation ring in a field of characteristic 0. This will be done within the following general setting. 4.1.11. Let us assume that k is the residue field of a discrete valuation ring O with field of fractions K; furthermore, assume that A can be identified with Ak := k ⊗O
4.1 The Schur Functor and its Variations
213
A , where A is an O-algebra which is finitely generated and free over O. In order to avoid any technical complications, we assume that Ak is split and AK := K ⊗O A is split semisimple.1 An A -module which is finitely generated and free over O will be called an A lattice. For any A -lattice V˜ , we shall write V˜K := K ⊗O V˜ and V˜k := k ⊗O V˜ . We are now in the classical setting of “modular representation theory”. Thus, we have a decomposition map dA : R0 (AK ) → R0 (Ak ), where R0 ( · ) denotes the appropriate Grothendieck group; see also Example 3.1.3. Recall that dA is defined as follows. Given ρ ∈ Irr(AK ), let V˜ be an A -lattice such that ρ ∼ = V˜K . Then dA ([ρ ]) := [V˜k ] ∈ R0 (Ak ). In order to separate the notation here from that used for Iwahori–Hecke algebras in Chapter 3, we shall write dA ([ρ ]) =
∑
Y ∈Irr(Ak )
ρ : Y A [Y ]
for any ρ ∈ Irr(AK );
the numbers ρ : Y A are called the decomposition numbers of A and Dec(A ) = ρ : Y A ρ ∈Irr(A ),Y ∈Irr(A ) K
k
is called the decomposition matrix of A . This matrix also has an interpretation in terms of projective modules, as follows. Let P˜ be a projective indecomposable A lattice. Then P˜k is a projective indecomposable Ak -module and Y := P˜k /rad(P˜k ) ∈ Irr(Ak ); furthermore, in R0 (AK ), we have [P˜K ] =
∑
ρ ∈Irr(AK )
ρ : Y A [ρ ]
“Brauer reciprocity”.
(This reciprocity relation already played an essential role in our discussion of “abstract Fock data”; see Theorem 3.4.4.) 4.1.12. We shall need some compatibility relations between Hom spaces and scalar extension. Let V˜ , V˜ be A -lattices. Since K is the field of fractions of O, we have dimK HomAK (V˜K , V˜K ) = dimK K ⊗O HomA (V˜ , V˜ ) . On the other hand, we have a natural injection k ⊗O HomA (V˜ , V˜ ) → HomAk (V˜k , V˜k ), which is an isomorphism if V˜ is projective. Thus, we obtain dimK HomAK (V˜K , V˜K ) dimk HomAk (V˜k , V˜k ), where equality holds if V˜ is projective. Note also that V˜ is projective if and only if V˜k is projective. (For all this, see [53, §30].) Usually, it is also assumed that O is complete, but this assumption is unnecessary thanks to our assumption that AK is split semisimple. Indeed, idempotents can be lifted from A = Ak to A by [53, Ex. 6.16] and the Krull–Schmidt–Azuyama theorem holds for A -lattices by [53, (30.18)].
1
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4 Hecke Algebras and Finite Groups of Lie Type
Lemma 4.1.13 (Dipper [58, 4.7]). Let M˜ be an A -lattice and assume that there ˜ where P˜ is finitely exists a surjective homomorphism of A -modules β˜ : P˜ → M, generated and projective, such that (D)
ker(β˜ )K and M˜ K have no irreducible constituents in common.
Then condition (S) in Lemma 4.1.5 holds for the induced map β˜k : P˜k → M˜ k . Further˜ ◦ ; note that H is an O-algebra which is finitely generated more, let H := EndA (M) and free over O. The natural injective map Hk → EndAk (M˜ k )◦ is an isomorphism. Proof. First note that P˜k is projective and the induced map β˜k is surjective. Now consider the map β˜k∗ : HomAk (M˜ k , M˜ k ) → HomAk (P˜k , M˜ k ), α → α ◦ β˜k . Since β˜k is surjective, this map is injective. Hence, we conclude (∗)
dimk HomAk (M˜ k , M˜ k ) dimk HomAk (P˜k , M˜ k ).
By 4.1.12, the right-hand side equals dimK HomAK (P˜K , M˜ K ) and the left-hand side is greater than or equal to dimK HomAK (M˜ K , M˜ K ). Thus, we deduce that equality holds in (∗) if and only if dimK HomAK (M˜ K , M˜ K ) = dimK HomAK (P˜K , M˜ K ). Since P˜K is semisimple, the latter equality holds thanks to condition (D). Hence, we also have equality in (∗) and so (S) holds for β˜k by Lemma 4.1.7. Finally, consider the natural injective map Hk := k ⊗O H → EndAk (M˜ k )◦ . Going once more through the above argument, we see that the two spaces have the same dimension and so the map is also surjective.
˜ ◦ . Since H is finitely Now let us fix an A -lattice M˜ and set H := EndA (M) generated and free over O, we can apply the discussion in 4.1.11 to H and so there is a decomposition map dH : R0 (HK ) → R0 (Hk ). Note also that, by 4.1.12 and Lemma 4.1.13, we have HK ∼ = EndAK (M˜ K )◦
Hk ∼ = EndAk (M˜ k )◦ .
and
Theorem 4.1.14 (Dipper [58, 4.10]). In the above setting, assume that there exists ˜ where P˜ is finitely generated a surjective homomorphism of A -modules β˜ : P˜ → M, ˜ ◦ and projective, such that condition (D) in Lemma 4.1.13 holds. Let H = EndA (M) as above. Then the functors FK := FM˜ K and Fk := FM˜ k induce bijections 1−1 FK : Irr(AK | M˜ K ) −→ Irr(HK ),
1−1 Fk : Irr(Ak | M˜ k ) −→ Irr(Hk ),
and the decomposition matrix of H is contained in the decomposition matrix of A . More precisely, for any ρ ∈ Irr(AK | M˜ K ), we have dH [FK (ρ )] = ∑ ρ : Y A [Fk (Y )]. Y ∈Irrk (Ak |M˜ k )
4.2 Hom Functors and Harish-Chandra Series
215
Proof. By condition (D), we have dimK HomAK (M˜ K , M˜ K ) = dimK HomAK (P˜K , M˜ K ) and so condition (S) holds for the induced map β˜K by Lemma 4.1.7. Hence, FK induces a bijection as above. An analogous statement holds for Fk by Lemma 4.1.13. It remains to prove the statement about the decomposition numbers. Let M˜ k = M1 ⊕ · · · ⊕ Mn be a decomposition into indecomposable direct summands and write Irr(Ak | M˜ k ) = {Yi | i ∈ I}, where Yi := Mi /rad(Mi ); see Corollary 4.1.10(a). Let us fix ρ ∈ IrrK (AK | M˜ K ); then ρ is a direct summand of M˜ K . We begin by establishing a reinterpretation of the decomposition numbers ρ : Yi A . By [53, (23.7)], there is a pure A -submodule U ⊆ M˜ such that, if we set V˜ := ˜ M/U, then V˜ is an A -lattice such that V˜K ∼ = ρ ; furthermore, V˜k will be isomorphic ˜ ˜ to a factor module of Mk . Having chosen V in this particular way, we can now apply Corollary 4.1.10(b) and this yields that dimk HomAk (Mi , V˜k ) equals the multiplicity of Yi as a composition factor of V˜k . By the definition of dA , this multiplicity is ρ : Yi A . Using also Lemma 4.1.2, we conclude that ρ : Yi A = dimk HomHk Fk (Mi ), Fk (V˜k ) . By Corollary 4.1.10(c), Fk (Mi ) is a projective indecomposable Hk -module, and its unique simple factor module is isomorphic to Fk (Yi ). Hence, we obtain that ρ : Yi A = multiplicity of Fk (Yi ) as a composition factor of Fk (V˜k ). Thus, by the definition of dH , it remains to show that (∗) dH [FK (ρ )] = [Fk (V˜k )] ∈ R0 (Hk ). ˜ V˜ ). Note Now, we have FK (ρ ) ∼ = HomAK (M˜ K , V˜K ) ∼ = X˜K , where X˜ = HomA (M, ˜ ˜ that X is an H -lattice (where, as before, ϕ ∈ H acts via f → f ◦ ϕ for f ∈ X). ∼ ˜ ˜ Hence, (∗) will follow if we can show that Fk (Vk ) = Xk . Since we certainly have an injection X˜k → HomAk (M˜ k , V˜k ), it will be enough to show that (∗ )
dimK HomAK (M˜ K , V˜K ) = dimk HomAk (M˜ k , V˜k ).
But, since P˜ is projective, we have dimK HomAK (P˜K , V˜K ) = dimk HomAk (P˜k , V˜k ); see 4.1.12. This implies (∗ ) by Lemma 4.1.6, since (S) holds for β˜K and β˜k .
The above proof is somewhat different from the original one given by Dipper [58]; yet another proof (due to Plesken) is contained in [153, 5.1.2].
4.2 Hom Functors and Harish-Chandra Series We shall now apply the general theory of the previous section to the representation theory of finite groups of Lie type. This essentially relies on the existence of a suitable BN-pair (or Tits system) in any such group. So let G be a finite group con-
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4 Hecke Algebras and Finite Groups of Lie Type
taining two subgroups B, N which form a split BN-pair of characteristic p (where p is a prime number). Hence, the following properties hold: • G is generated by B and N. • H := B ∩ N is normal in N and the quotient W := N/H is a finite group generated by a set S of elements of order 2. The group W is called the Weyl group of G. • ns Bns = B if s ∈ S and ns is a representative of s in N. • ns Bn ⊆ Bns nB ∪ BnB for any s ∈ S and n ∈ N. • n∈N nBn−1 = H. • We have B = UH, where U is a normal p-subgroup of B and H is abelian of order prime to p. • Chevalley’s commutator relations hold; see [53, §69] and [43, Chap. 2]. If G is a finite group of Lie type, then the above conditions hold; see Steinberg’s lectures [287], or the remarks in [45, §1.18, §2.6]. Let k be a field which is a splitting field for G and all its subgroups. Let Irrk (G) denote the set of simple kG-modules (up to isomorphism). All kG-modules will be tacitly assumed to be finite-dimensional. Let 0 be the characteristic of k. Recall that we assume throughout that either = 0 or > 0 is a prime different from p. In this section, we describe (mostly without proofs) the classification of Irrk (G) in terms of Harish-Chandra series; this provides the background and the motivation for the subsequent study of the unipotent principal series representations. For = 0, all this is a classical story; see, for example, [45, Chap. 9], [53, §70] and [55, Chap. 6]. Here, we present a streamlined version of the expositions in [61], [120] and [154]. 4.2.1. We recall some basic facts about Harish-Chandra induction and restriction.2 Let W be the Weyl group of G and S be the set of simple reflections determined by B. For each subset J ⊆ S, we have a corresponding (standard) parabolic subgroup PJ , with standard Levi decomposition PJ = UJ LJ . The subgroup UJ can be characterised as the largest normal p-subgroup of PJ . Now let PG := {nPJ | J ⊆ S, n ∈ N}
and
LG := {nLJ | J ⊆ S, n ∈ N}.
(Here, for any subgroup A ⊆ G, we write nA = nAn−1 .) Now let L ∈ LG and P ∈ PG . We say that L is a Levi complement in P if L ⊆ P and P = UP L, where UP denotes the largest normal p-subgroup of P. Note that L itself is a finite group with a split BN-pair of characteristic p. Now we have functors RG L⊆P : kL-mod → kG-mod
and
∗ G RL⊆P :
kG-mod → kL-mod,
which are called Harish-Chandra induction and Harish-Chandra restriction respectively. Given a kL-module X and a kG-module Y , these are defined by G ˜ RG L⊆P (X) := IndP (X)
2
and
∗ G RL⊆P (Y )
:= eUP .Y,
These are special cases of the more general concept of induction and restriction with respect to subquotients in finite groups; see Cabanes and Enguehard [43, Part 1] for an up-to-date exposition.
4.2 Hom Functors and Harish-Chandra Series
217
where X˜ denotes the “inflation” of X from L to P (via the natural map P → L with kernel UP ) and IndG P denotes the usual induction of representations; furthermore, eUP ∈ kUP is the idempotent given by eUP := |UP |−1 ∑u∈UP u. (Since UP is normalised by L, we have leUP = eUP l for all l ∈ L and so there is a natural kL-module structure on ∗RG L⊆P (Y ).) These two functors are left and right adjoint to each other: (a)
∗ G ∼ HomkG RG L⊆P (X),Y = HomkL X, RL⊆P (Y ), ∗ G ∼ HomkG Y, RG L⊆P (X) = HomkL RL⊆P (Y ), X .
Now let (M, Q) be another pair where M ∈ LG is a Levi complement in Q ∈ PG . Assume that P ⊆ Q and L ⊆ M. Then L ∈ LM is a Levi complement in P ∩ M ∈ PM . In this situation, we have the following transitivity: M G M ∗ G ∼ G (b) RG RL⊆P (Y ) ∼ = ∗RL⊆P∩M ∗RM⊆Q (Y )). L⊆P (X) = RM⊆Q RL⊆P∩M (X)) and G The Mackey formula gives a direct sum decomposition of ∗RG M⊆Q RL⊆P (X)), where, again, M ∈ LG is a Levi complement in Q ∈ PG . For each n ∈ N, we have nL ∩ M ∈ LM and this is a Levi complement in nP ∩ M ∈ PM ; similarly, nL ∩ M ∈ LnL is a Levi complement in nL ∩ Q ∈ PnL . By “transport of structure”, we can define an nL-action on X by nln−1 : x → l.x (l ∈ L, x ∈ X). This knL-module will be denoted by nX. Then the Mackey formula says that (c)
∗ G ∼ RM⊆Q RG L⊆P (X) =
∗ nL n RM nL∩M⊆nP∩M RnL∩M⊆nL∩Q ( X) ,
n∈D(Q,P)
where D(Q, P) ⊆ N is a set of (Q, P)-double coset representatives in G. The above three properties are rather straightforward generalisations of similar properties of ordinary induction and restriction; this was worked out in detail by Dipper and Fleischmann [61]. Finally, we have the following independence property, which was first established by Howlett and Lehrer [163] and Dipper and Du [60]: (d)
∼ G RG L⊆P (X) = RM⊆Q (X )
if M = nL and X ∼ = nX for some n ∈ N.
In particular, if L ∈ LG is a Levi complement in both P ∈ PG and P ∈ PG , then ∼ G RG L⊆P (X) = RL⊆P (X). Therefore, we will now omit P from the notation and simply ∗ G write RG L (X) and RL (Y ). Definition 4.2.2. A kG-module Y is said to be cuspidal if ∗RG L (Y ) = {0} for all L ∈ LG such that L G. We shall need the following two technical results. Lemma 4.2.3. Let Y ∈ Irrk (G). Let L ∈ LG be minimal such that ∗RG L (Y ) = {0}. Then all composition factors of ∗RG (Y ) are cuspidal. L Proof. Let X ∈ Irrk (L) be any composition factor of ∗RG L (Y ). To show that X is cuspidal, let M ∈ LG be such that M ⊆ L and assume that ∗RLM (X) = {0}. Then we
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4 Hecke Algebras and Finite Groups of Lie Type
also have ∗RLM (Y ) = {0} and so ∗RG M (Y ) = {0}, by the transitivity of Harish-Chandra restriction. By the minimality of L, this implies M = L, as required.
Lemma 4.2.4. Let Y ∈ Irrk (G). Let L, M ∈ LG , X ∈ Irrk (L), Z ∈ Irrk (M) and assume that the following conditions are satisfied: ∗ G • L is minimal such that ∗RG L (Y ) = {0} and X is a composition factor RL (Y ). • Z is cuspidal and Y is isomorphic to a submodule or to a factor module of RG M (Z).
Then there is some n ∈ N such that L = nM and X ∼ = nZ. Proof. Let PX be a projective cover of X. Since X is a composition factor of ∗RG L (Y ), (Y ) = {0}. So, by the adjointness of Harish-Chandra inwe have HomkL PX , ∗RG L duction and restriction, Y is isomorphic to a factor module of RG (P ). Let P Y be L X (P ) is projective. (Indeed, a projective cover of Y . Now one easily shows that RG L X since PX is a direct summand of kL and since RG L respects direct sums, we have that G RG L (PX ) is a direct summand of RL (kL). Now assume that L is a Levi complement G in P ∈ PG ; then the definition shows that RG L (kL) = IndUP (k), which is projective since |UP | is invertible in k.) But then we conclude that PY is a direct summand of G RG L (PX ). On the other hand, since Y is a composition factor of RM (Z), there is a G non-zero homomorphism PY → RM (Z). Thus, we obtain G HomkG RG L (PX ), RM (Z) = {0}. Using adjointness and the Mackey formula, we deduce that n n = {0} HomkL PX , RLnM∩L ∗RnM M∩L ( Z)
for some n ∈ N.
Now note that nZ also is a cuspidal knM-module. Hence, we must have nM ∩ L = nM; that is, nM ⊆ L. But, since Y is a submodule or a factor module of RG M (Z), we have G HomkG (Y, RG M (Z)) = {0} or HomkG (RM (Z),Y ) = {0}. Using the independence in 4.2.1, the same is true with (M, Z) replaced by (nM, nZ). So, by adjointness, we have ∗RG (Y ) = {0} and then the minimality of L shows that L = nM. Returning to the nM above Hom space, we now also see that HomkL (PX , nZ) = {0}. Since PX has a unique
simple quotient (namely, X), we must have X ∼ = nZ. Definition 4.2.5. Let LG◦ be the set of all pairs (L, X), where L ∈ LG and X ∈ Irrk (L) is cuspidal. For (L, X) ∈ LG◦ , the set L ∈ LG is minimal such that ∗RG L (Y ) = {0} Irrk (G | (L, X)) := Y ∈ Irrk (G) and X is a composition factor of ∗RG L (Y ) is called the Harish-Chandra series associated with (L, X). (By 4.2.1(d) and adjointness, this does not depend on the choice of P ∈ PG such that P = UP L.) Given (L, X) and (L, X ) in LG◦ , we write (L, X) ∼N (L , X ) if there exists some n ∈ N such that L = nL and X ∼ = nX. This defines an equivalence relation on LG◦ . Theorem 4.2.6 (Hiss [154]). We have a disjoint union
4.2 Hom Functors and Harish-Chandra Series
Irrk (G) =
219
Irrk (G | (L, X)).
(L,X)∈LG◦ /∼N
Furthermore, given Y ∈ Irrk (G) and a pair (L, X) ∈ LG◦ , we have Y ∈ Irrk (G | (L, X))
⇔
Y is isomorphic to a factor module of RG L (X)
⇔
Y is isomorphic to a submodule of RG L (X).
Proof. First we show the equivalence of the above three statements. Assume that Y ∈ Irrk (G | (L, X)). Let X ∈ Irrk (L) be a submodule of ∗RG L (Y ). By Lemma 4.2.3, X is cuspidal and, by adjointness, Y is isomorphic to a factor module of RG L (X ). We now apply Lemma 4.2.4 with (M, Z) = (L, X ). So there is some n ∈ N such that L = nL and X ∼ = nX . Hence, by the independence in 4.2.1, Y is isomorphic to a ∼ G factor module of RG L (X) = RL (X ). A similar argument shows that, if X ∈ Irrk (L) is G ∗ G a factor module of RL (Y ), then Y is isomorphic to a submodule of RL (X) ∼ = RG L (X ). Conversely, assume that Y is isomorphic to a submodule or to a factor module of RG L (X). By adjointness, X is a submodule or a factor module (in particular, a composition factor) of ∗RG L (Y ). It remains to show that L is minimal such that ∗RG (Y ) = {0}. Choose M ∈ L to be minimal such that M ⊆ L and ∗RG (Y ) = {0}. G L M Let Z ∈ Irrk (M) be a composition factor of ∗RG M (Y ); by Lemma 4.2.3, Z is cuspidal. Then Lemma 4.2.4 (with the roles of L, X and M, Z reversed) shows that M = nL. Since M ⊆ L, we conclude that M = L, as required. By Lemma 4.2.3, it is clear that every Y ∈ Irrk (G) belongs to some HarishChandra series. Finally, assume that (L, X) and (L , X ) are two pairs in LG◦ such that Y belongs to the corresponding Harish-Chandra series. By the above equiva lences, Y is isomorphic to a factor module of RG L (X ). Applying Lemma 4.2.4 with (M, Z) = (L , X ) we obtain (L, X) ∼N (L , X ), as desired.
Definition 4.2.7. Let (L, X) ∈ LG◦ . Let P ∈ PG be such that P = UP L and assume that RG L is defined with respect to P. Then ◦ H (L, X) := EndkG RG L (X) is called the Hecke algebra associated with (L, X). (Here, the superscript ◦ denotes the opposite algebra.) Note that, by the independence in 4.2.1(d), we obtain an isomorphic algebra if we choose another subgroup P ∈ PG such that P = UP L. Example 4.2.8. Consider a pair (H, η ) ∈ LG◦ , where H = N ∩ B and η is a onedimensional representation of H. (Every irreducible representation of H is automatically cuspidal.) By Theorem 4.2.6, Y ∈ Irrk (G) belongs to the Harish-Chandra series associated with (H, η ) if and only if Y is isomorphic to a factor module of G ˜ G G ∼ ), where η˜ denotes the RG H (η ). By the definition of RH , we have RH (η ) = IndB (η “inflation” of η from H to B. Hence, by Frobenius reciprocity, we have Y ∈ Irrk (G | (H, η ))
⇔
Yη = {0},
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4 Hecke Algebras and Finite Groups of Lie Type
where Yη := {y ∈ Y | h.y = η (h)y for all h ∈ H}. The representations of G belonging to Irrk (G | (H, η )) for some η ∈ Irrk (H)) are called the principal series represen∼ tations of G. If η = kH (the trivial representation of H), then RG H (kH ) = k[G/B], the permutation representation of G on the cosets of B. In this case, we denote Irrk (G | (H, kH )) simply by Irrk (G | B) and call it the set of unipotent principal series representations of G. The corresponding Hecke algebra is given by Hk := H (H, kH ) = EndkG (k[G/B])◦ . Returning to the general case, our next aim is to establish a relation between Irrk (G | (L, X)) and H (L, X), for a fixed (L, X) ∈ LG◦ . We place ourselves in the setting of Section 4.1, where A = kG and M = RG L (X). Then we have the Hom functor FRG (X) : kG-mod → H (L, X)-mod, Y → HomkG RG L (X),Y ). L
In the following, we will simply write Fk instead of FRG (X) . L
Theorem 4.2.9 (Geck, Hiss and Malle [120]). Let us fix (L, X) ∈ LG◦ (and a subgroup P ∈ PG such P = UP L). Then the Hom functor Fk induces a bijection 1−1
Fk : Irrk (G | (L, X)) −→ Irr(H (L, X)). Furthermore, the statements in Corollary 4.1.10 hold for M = RG L (X). Proof. Let PX be a projective cover of X. Then RG L (PX ) is a projective kG-module and the natural map PX → X induces a surjective homomorphism of kG-modules G β : RG L (PX ) → RL (X). By the results in Section 4.1, it will be enough to verify the condition on Hom spaces in Lemma 4.1.7; that is, we have to prove G G G (∗) dimk HomkG RG L (X), RL (X) = dimk HomkG RL (PX ), RL (X) . Now, by adjointness and the Mackey formula, we have ∗ G G G ∼ HomkG RG L (X), RL (X) = HomkG X, RL RL (X) n ∼ HomkL X, ∗RLnL∩L ∗RnLL∩L (nX) . = n∈D(P,P)
Note again that nX is a cuspidal knL-module for any n ∈ N. Hence, we only need to consider terms where nL ∩ L = L; that is, n ∈ D(P, P) ∩ NG (L). Thus, we obtain G ∼ HomkG RG L (X), RL (X) =
HomkL (X, nX).
n∈D(P,P)∩NG (L) G We can repeat this argument to determine HomkG (RG L (PX ), RL (X)); this yields
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221
G ∼ HomkG RG L (PX ), RL (X) =
HomkL (PX , nX).
n∈D(P,P)∩NG (L)
But then we note that PX has a unique simple quotient (namely, X) and so we have HomkL (PX , nX) ∼ = HomkL (X, nX)
for any n ∈ N.
Consequently, the terms corresponding to a given n in the above two direct sum decompositions have the same dimensions. Thus, (∗) is proved.
The desired bijection induced by Fk now follows from Proposition 4.1.8. Proposition 4.2.10. Let (L, X) ∈ LG◦ . Then RG L (X) is a semisimple kG-module if and only if H (L, X) is a semisimple algebra. In this case, we have ∼ RG L (X) =
E∈Irr(H (L,X))
YE ⊕ · · · ⊕YE ,
dimk E times
where YE denotes the unique simple module in Irrk (G | (L, X)) such that Fk (YE ) ∼ = E. Proof. The endomorphism algebra of a semisimple module certainly is a semisimple algebra. The problem is to prove the reverse implication. So assume that H (L, X) is semisimple and let RG L (X) = M1 ⊕ · · · ⊕ Mn be a decomposition into indecomposable direct summands. We must show that each Mi is simple. By Theorem 4.2.9, the statements in Corollary 4.1.10 hold for RG L (X). Hence, setting Yi := Mi /rad(Mi ) ∈ Irrk (G | (L, X)) for 1 i n, we have Mi ∼ = M j if and only if Yi ∼ = Y j . Furthermore, the Cartan matrix of H (L, X) records the multiplicities of the various Y j as composition factors of the modules Mi . Since H (L, X) is assumed to be semisimple, that Cartan matrix is the identity matrix. Consequently, none of the composition factors of rad(Mi ) belongs to Irrk (G | (L, X)). Now assume, if possible, that there is some i such that rad(Mi ) = {0}. Let Z ⊆ Mi be a simple submodule. As we have just seen, Z ∈ Irrk (G | (L, X)). On the other hand, Z is a simple submodule of RG L (X) and so, by Theorem 4.2.6, we have Z ∈ Irrk (G | (L, X)), which is a contradiction. Finally, the statement about the decomposition of RG L (X) is now clear, since (X)) is isomorphic to the left regular representation of H (L, X).
Fk (RG L Proposition 4.2.11. Let (L, X) ∈ LG◦ . Then dimk H (W, L) = |W (L, X)|, where W (L, X) := {n ∈ (NG (L) ∩ N)L | nX ∼ = X}/L. The group W (L, X) is called the inertia group of X. Proof. Arguing as in the proof of Theorem 4.2.9, we obtain G ∼ H (L, X) = HomkG RG HomkL (X, nX), L (X), RL (X) = n∈D(P,P)∩NG (L)
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4 Hecke Algebras and Finite Groups of Lie Type
and so dimk H (L, X) = |{n ∈ D(P, P) ∩ NG (L) | nX ∼ = X}|. It remains to note that, by general results on double coset representatives of parabolic subgroups (see the
proof of [45, 9.2.4]), we have (NG (L) ∩ N)L/L ∼ = D(P, P) ∩ NG (L). Theorem 4.2.12 (Howlett and Lehrer [161] ( = 0); Geck, Hiss and Malle [120] ( > 0)). Let us fix (L, X) ∈ LG◦ and let W (L, X) be defined as in Proposition 4.2.11. Then there is a Coxeter system (W1 , S1 ) and a finite group Ω acting on (W1 , S1 ) such that W (L, X) = Ω W1 ; furthermore, H (W, L) has a basis {T˙w | w ∈ W (L, X)} where the multiplication is given by the following rules: (a) For all w ∈ W and w ∈ Ω , we have T˙w T˙w = μ (w, w )T˙ww and T˙w T˙w = μ (w , w)T˙w w where μ : W (L, X) × W (L, X) → k× is a 2-cocycle. (b) There are elements {ps˜ | s˜ ∈ S1 } ⊆ k \ {0, 1} such that T˙sw if l1 (sw) ˜ > l1 (w), ˜ T˙s˜T˙w = ˙ if l1 (sw) ˜ < l1 (w), ps˜Tsw ˜ + (ps˜ − 1)T˙w for all s˜ ∈ S1 and w ∈ W1 ; here, l1 denotes the length function on W1 . The above relations form a presentation of H (L, X) as a k-algebra. (The formulation of this result in [120] is slightly different from the original formulation in [161]; the two formulations are reconciled in [1].) We will not attempt to give a proof here, which would be rather long and would require a number of further preparations. For = 0, this is discussed in detail by Carter [45, Chap. 10]; for > 0, see Cabanes and Enguehard [43, Chap. 3] for an up-to-date exposition. Once the above result is established, one can also show the following proposition, whose proof is similar to the one given in [45, Prop. 10.9.1] (in the case where = 0). Proposition 4.2.13 (Howlett and Lehrer [162] ( = 0); Geck and Hiss [117] ( > 0)). Define a linear map τ : H (L, X) → k by τ (T˙1 ) = 1 and τ (T˙w ) = 0 for w = 1. Then τ is a symmetrizing trace on H (L, X). Remark 4.2.14. One may conjecture that the cocycle μ in Theorem 4.2.12(a) is always trivial. (For = 0, this conjecture already appears in [161].) By Howlett and Kilmoyer [160], this is the case when L = H (where H = B ∩ N) and X = η is onedimensional. It is also known to be true when = 0 and G arises as the fixed point set of a connected reductive algebraic group under a Frobenius map (see Lusztig [220, Chap. 8] and [95].) The question is open for > 0 in general. Remark 4.2.15. Assume that > 0 (and l = p). Then the classification of the cuspidal irreducible representations of G is an open problem. (The analogous problem in characteristic 0 is solved by Lusztig [220].) For G = GLn (Fq ), it is true that every cuspidal irreducible representation in characteristic can be lifted to a representation in characteristic 0; see Dipper [59, Cor. 5.23]. Similar results hold for finite classical groups and special characteristics; see Gruber and Hiss [149]. But, otherwise, there may exist cuspidal representations in characteristic which cannot be lifted to characteristic 0. (By Example 4.5.3 below, this already happens for the Suzuki groups.) See [117], [119], [120], [148], [153] for further examples and references.
4.3 Unipotent Principal Series Representations, I
223
4.3 Unipotent Principal Series Representations, I We keep the general setting of the previous section where k is a field of characteristic 0 ( = p), but now exclusively consider the Harish-Chandra series Irrk (G | B) = {Y ∈ Irrk (G) | Y admits non-zero vectors fixed by B}; see Example 4.2.8. Recall that the corresponding Hecke algebra is given by Hk := H (H, kH ) = EndkG (k[G/B])◦ . The aim of this section is to show how generic Iwahori–Hecke algebras enter the picture. This will allow us to apply the machinery developed in Chapter 3. 4.3.1. The structure of Hk was determined by Iwahori [171]. (In this case, the proof of Theorem 4.2.12 simplifies drastically; see [132, §8.4].) We have W (H, kH ) = W , the Weyl group of G. Then Hk has a basis {T˙w | w ∈ W } such that if l(sw) > l(w), T˙sw ˙ ˙ Ts Tw = q¯s T˙sw + (q¯s − 1)T˙w if l(sw) < l(w), for all s ∈ S and w ∈ W ; here, the parameters {q¯s } are given by q¯s = qs 1k , where qs = |BsB/B| ˙ for s ∈ S; see [132, 8.4.6]. As an endomorphism of k[G/B] ∼ = RG H (kH ), ˙ the map Tw : k[G/B] → k[G/B] is explicitly given as follows (see [132, 8.4.1]): ˙ T˙w (xB) = sum of all cosets yB ∈ k[G/B] such that x−1 y ∈ BwB (where, as usual, w˙ denotes a representative of w ∈ W in N). The parameters {q¯s } satisfy the condition that q¯s = q¯t whenever s,t ∈ S are conjugate in W (see [132, 1/2 Exc. 4.7]). Since k is large enough, we can find square roots q¯s ∈ k (s ∈ S) such 1/2 1/2 −1/2 that q¯s = q¯t whenever s,t ∈ S are conjugate in W . Setting Ts := q¯s T˙s (s ∈ S) and Tw := Ts1 · · · Tsn whenever w = s1 · · · sn (si ∈ S) is a reduced expression, we obtain Ts Tw =
Tsw 1/2 −1/2 Tsw + (q¯s − q¯s )Tw
if l(sw) > l(w), if l(sw) < l(w),
for all s ∈ S and w ∈ W . Thus, we see that Hk is obtained by specialisation from a suitable generic Iwahori–Hecke algebra associated with W (see Example 1.1.9). This statement will be made more precise in 4.3.9. Note that q¯s = 0 for all s ∈ S, but it may happen that q¯s = 1 for some s ∈ S. But then (in accordance with Theorem 4.2.12) we can further break down the structure of Hk as follows. Let Ω ⊆ W be the subgroup generated by all s ∈ S such that q¯s = 1. Let S1 := {ω sω −1 | ω ∈ Ω , s ∈ S such that q¯s = 1} and W1 ⊆ W be the subgroup generated by S1 . Then, by 2.4.3, we have W = Ω W1 and (W1 , S1 ) is a Coxeter system. Furthermore, the subspace Hk,1 = T˙w | w ∈ W1 k is a subalgebra of Hk and we have the following multiplication rules (see Proposition 2.4.5):
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4 Hecke Algebras and Finite Groups of Lie Type
(a) We have T˙w T˙w = T˙ww and T˙w T˙w = T˙w w for all w ∈ W and w ∈ Ω . (Thus, the cocycle μ in Theorem 4.2.12(a) is trivial in this case.) (b) For s˜ ∈ S1 and w ∈ W1 , we have if l1 (sw) ˜ > l1 (w), T˙sw ˜ T˙s˜Tw = ˙ ps˜Tsw if l1 (sw) ˜ < l1 (w); ˜ + (ps˜ − 1)T˙w here, l1 is the length function on W1 and, for s˜ ∈ S1 , the parameter ps˜ is given by ps˜ = q¯s if s˜ = ω sω −1 , where ω ∈ Ω and s ∈ S is such that q¯s = 1. The following result provides a useful criterion for checking if Hk is semisimple. (It is implicitly contained in Hiss [153, Kap. 6] and Puig [267].) Lemma 4.3.2. We have the following equivalences: Hk is semisimple ⇔ k[G/B] is a semisimple kG-module ⇔ [G : B]1k = 0. Proof. The first equivalence is a special case of Proposition 4.2.10. The second equivalence is clear if = 0. So now asume that > 0. If divides [G : B], then the trivial module kG is not a direct summand of k[G/B]. (This is a well-known fact which holds for the permutation representation of any finite group on the cosets of a subgroup.) Hence, k[G/B] is not semisimple. Conversely, assume that [G : B]. Let ψ : Hk → k be the linear map obtained by taking the trace of an element of Hk in its action on k[G/B]. By the description of T˙w in 4.3.1, we see that ψ (T˙1 ) = [G : B]1k = 0 and ψ (T˙w ) = 0 for w = 1. Hence, ψ is a non-zero scalar multiple of the standard symmetrizing trace on Hk . Consequently, the associated bilinear form Hk × Hk → k, (ϕ , ϕ ) → ψ (ϕ .ϕ ), is non-degenerate. Now it is a general fact that if a finite-dimensional associative algebra carries a non-degenerate bilinear form which is associated with the trace function of a representation, then that algebra must be semisimple. Hence, k[G/B] is semisimple by the first equivalence.
Example 4.3.3. In the setting of 4.3.1, we see that there are unique one-dimensional representations ind : Hk → k and sgn : Hk → k such that ind(T˙s ) = q¯s
and
sgn(T˙s ) = −1
for all s ∈ S.
We have ind = sgn if q¯s = −1 for all s ∈ S. We claim that Fk (kG ) ∼ = ind. Indeed, identifying RG H (kH ) = k[G/B], we have Fk (kG ) = HomkG (k[G/B], kG ). This space has dimension 1; it is spanned by the function f1 : k[G/B] → k which takes constant value 1 on all cosets. Now let s ∈ S. Then T˙s . f1 = f1 ◦ T˙s and so (T˙s . f1 )(xB) = ( f1 ◦ T˙s )(xB) = ∑yB f1 (yB), where the sum runs over all cosets yB ∈ ˙ Thus, (T˙s . f1 )(xB) = |BsB/B|1 ˙ G/B such that x−1 y ∈ BsB. k = q¯s and we have shown ˙ that Ts . f1 = q¯s f1 = ind(T˙s ) f1 , as required. It is much more difficult to describe a simple kG-module Y such that Fk (Y ) ∼ = sgn. Note that if Hk is semisimple, then Fk (StG,k ) ∼ = sgn, where StG,k = {m ∈ k[G/B] | T˙w (m) = (−1)l(w) m for all w ∈ W }
4.3 Unipotent Principal Series Representations, I
225
is the Steinberg representation [286]. Example 4.3.4. Assume that the Weyl group W of G has order 2. Examples are: G = GL2 (Fq )
(general linear group of type A1 , q any prime power),
G = GU3 (Fq )
(general unitary group of type 2A2 , q any prime power), √ 2 f +1 (Suzuki group of type 2B2 , q = 2 for some f 0), √ 2 f +1 (Ree group of type 2G2 , q = 3 for some f 0).
G = Suz(q2 ) G = Ree(q2 )
(See Carter [44] for the construction of these groups; see also Example 4.3.6 below.) We have LG = {G, H}, where, as above, H = B ∩ N. Hence, all irreducible representations of G are either cuspidal or lie in the principal series. Let us consider the series Irrk (G | B) = Irrk (G | (H, kH )), where kH is the trivial representation of H. Let W = {1, s}. By 4.3.1, we have W (H, kH ) = W and so Hk = H (H, kH ) = T˙1 , T˙s , where T˙s2 = q¯s T˙1 + (q¯s − 1)T˙s ; note that [G : B] = 1 + qs in this case. For the groups in the above list, we have qs = q, q3 , q4 , q6 respectively. As we have seen in Example 4.3.3, there are two one-dimensional representations ind and sgn of H . If q¯s = −1, then ind = sgn and Irr(H ) = {ind, sgn}; if q¯s = −1, then ind = sgn and Irr(H ) = {ind}. Thus, we have two essentially different cases: ∼ • If q¯s = −1, then Hk is semisimple and RG H (kH ) = kG ⊕ StG , where StG,k is the Steinberg module which has dimension qs in this case; see [286]. Thus, we have Irrk (G | B) = {kG , StG,k }; furthermore, Fk (kG ) ∼ = ind and Fk (StG,k ) ∼ = sgn. • If q¯s = −1, then Hk is not semisimple and RG (k ) is indecomposable, with a H H unique simple submodule and a unique simple factor module, both isomorphic to kG . Thus, we have Irrk (G | B) = {kG } and Fk (kG ) ∼ = ind. In particular, we see that the series Irrk (G | B) only depends on whether q¯s ∈ k× has order 2 or not. We will establish a general statement of this kind for any G in Theorem 4.4.8 below. In order to proceed, it will be useful to assume that our group G is realised as the fixed point set of an algebraic group under a Frobenius map. Definition 4.3.5. Let F p be an algebraic closure of F p = Z/pZ. Let G be a connected reductive algebraic group over F p and F : G → G be a homomorphism of algebraic groups such that some power of F is the Frobenius map relative to a rational structure on G over some finite subfield of F p . Then the fixed point set GF := {g ∈ G | F(g) = g} is a finite group which is called a finite group of Lie type. A split BN-pair in G is obtained by taking B = BF (where B ⊆ G is an F-stable Borel subgroup) and N = NG (T0 )F (where T0 ⊆ B is an F-stable maximal torus). It is beyond the scope of this text to discuss the theory of algebraic groups in more detail. (See [104] for an introduction.) But we shall give some typical examples which the reader may keep in mind throughout this chapter.
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4 Hecke Algebras and Finite Groups of Lie Type
Example 4.3.6. (a) Let G = GLn (F p ) be the general linear group of n × n-matrices. Given a power q of p, we have the “standard” Frobenius map Fq : G → G, (ai j ) → aqij . Let Fq ⊆ F p be the unique subfield with q elements. Then GFq = GLn (Fq ), the general linear group over Fq . The subgroup B ⊆ G consisting of all upper triangular matrices is an Fq -stable Borel subgroup, where the diagonal matrices form an Fq stable maximal torus T0 ⊆ B. We have | GLn (Fq )| = qn(n−1)/2 (q − 1)(q2 − 1)(q3 − 1) · · · (qn − 1). (b) Let again G = GLn (F p ) and consider the n × n permutation matrix Qn with entry 1 at the positions (1, n), (2, n − 1), . . ., (n, 1); note that Qn = Q−1 n . Define an automorphism of algebraic groups α : G → G by α (A) := Qn · (Atr )−1 · Qn . Then α commutes with Fq and α 2 is the identity. Hence, F := Fq ◦ α also is a Frobenius map with respect to some Fq -rational on G (see [104, Exc. 4.4]). Since F 2 = Fq2 , we have GF ⊆ GLn (Fq2 ). Now the restriction of Fq to GLn (Fq2 ) is an automorphism of ¯ Then we obtain order 2, which we denote by A → A. GF = GUn (Fq ) := {A ∈ GLn (Fq2 ) | A¯ tr · Qn · A = Qn }, the general unitary group with respect to the hermitian form defined by Qn . (The reason for choosing Qn as above is that then the Borel subgroup B ⊆ G and the torus T0 in (a) are both Fq -stable and F-stable; see [104, 4.2.6].) We have |GUn (Fq )| = qn(n−1)/2 (q + 1)(q2 − 1)(q3 + 1) · · · (qn − (−1)n ). (c) Let G = Sp4 (F2 ) be the four-dimensional symplectic subgroup, defined with respect to the symplectic form given by the matrix Q4 in (b). By [104, 3.3.5], we have G = U, U , where U ⊆ G is the subgroup consisting of all upper unitriangular matrices and U ⊆ G is the subgroup consisting of all lower unitriangular matrices. Explicitly, we have U = {u(t1 ,t2 ,t3 ,t4 ) | ti ∈ F2 }, where ⎤ ⎡ 1 t1 t3 +t1t2 t4 +t1 t3 ⎥ ⎢ 0 1 t2 t3 ⎥. u(t1 ,t2 ,t3 ,t4 ) := ⎢ ⎦ ⎣ 0 0 1 t1 0 0 0 1 Similarly, U consists of the transposes of the above matrices. By [104, 3.3.6], there is a unique homomorphism of algebraic groups α : G → G such that ⎡ ⎤ 1 t2 t4 t32 + t2t4 ⎢ 0 1 t 2 t4 + t 2 t2 ⎥ 1 1 ⎥ u(t1 ,t2 ,t3 ,t4 ) → ⎢ ⎣ 0 0 1 ⎦ t2 0 0 0 1 and u(t1 ,t2 ,t3 ,t4 )tr is sent to the transpose of the above matrix. Now α is bijective and α 2 is the standard Frobenius map F2 on G. Let r = 2 f ( f 0) and consider
4.3 Unipotent Principal Series Representations, I
227
the standard Frobenius map Fr : G → G (where F1 = id by convention). Then α commutes with Fr and, setting F := α ◦ Fr , we obtain F 2 = F2 ◦ Fr2 = F2r2 . Let q > 0 be such that q2 = 2r2 . The finite group GF is called the Suzuki group (of type 2B ) and denoted by Suz(q2 ); we have 2 |Suz(q2 )| = q4 (q2 − 1)(q4 + 1),
where
q=
√
2 f +1
2
( f 0).
See [104, 4.6] for further details and references. The groups GLn (Fq ), GUn (Fq ) and Suz(q2 ) provide excellent examples for testing the general theory that we are now going to develop. From now on, assume that G = GF , where G and F are as in Definition 4.3.5. Table 4.1 Weight functions in the quasi-split case W
|γ |
W
A2m−1
2
Bm
A2m
2
Bm
Dn
2
Bn−1
D4
3
G2
E6
2
F4
L
W
1
s
4
2
s
2
s
p p p
2
3
4
2
s
2
p p p
2
2
s
4
1
1
s
p p p
1
3
6
1
2
2
s s
1
s
s
s s
1
s
4
s
s
s s s
|γ |
W
B2
2
A1
G2
2
A1
F4
2
I2 (4)
L 4
s
6
s
4
s
8
2
s
|γ | denotes the order of γ ; if γ =id, then L(s)=1 (s∈S).
4.3.7. Let B ⊆ G be an F-stable Borel subgroup and T0 be an F-stable maximal torus contained in B. Then the groups B and N := NG (T0 ) form a BN-pair in G and the fixed point sets B := BF and N := NF form a BN-pair in G satisfying the properties mentioned at the beginning of Section 4.2. Let W = NG (T0 )/T0 be the Weyl group of G, with set of simple reflections S. Then F induces an automorphism γ : W → W such that γ (S) = S. Now the Weyl group W of G can be identified with Wγ = {w ∈ W | γ (w) = w}. The set S of simple reflections of W can be identified ¯ here, S¯ denotes the set of orbits of S under the action of γ with the set {wI | I ∈ S}; and wI denotes the longest element in the parabolic subgroup of W generated by I. (For all this, see [104, 1.7.1].) Let δ 1 be minimal such that F δ is the Frobenius map relative to a rational structure on G over a finite subfield k0 ⊆ F p . Define q > 0 to be the unique real number such that |k0 | = qδ . (If G is simple modulo its centre, then δ = 1 and q is a power of q, except √ when √G is a Suzuki or Ree group, in which case δ = 2 and q is an odd power of 2 or 3.) Then we have (a)
|BsB/B| ˙ = qcs ,
where
cs := l(wI ) for s = wI ∈ S;
here, l is the length function of W with respect to S; see Steinberg [287, pp. 131, 190]. Thus, we obtain the well-defined weight function
228
(b)
4 Hecke Algebras and Finite Groups of Lie Type
L: W → Z
L(s) = cs for s ∈ S.
such that
We have in fact [G : B] = ∑w∈W qL(w) . The weight functions arising in this way are what Lusztig calls the quasi-split case in [231, Chapter 16]; see Table 4.1 for a list of the various possibilities when W is irreducible. 4.3.8. We now place ourselves in the setting of 4.1.11, where > 0 and k is the residue field of a discrete valuation ring O with field of fractions K ⊆ C; let A := OG. We shall assume that K is a sufficiently large algebraic extension of Q of finite degree; in particular, K will be a splitting field for G and all its subgroups. Finally, we shall assume that is good for G in the sense of the theory of algebraic groups (see, for example, [45, p. 28]). Recall that this means that is good for each simple factor involved in G; the conditions for the various simple types are as follows. An : Bn ,Cn , Dn : G2 , F4 , E6 , E7 : E8 :
no condition, = 2, = 2, 3, = 2, 3, 5.
Let R ⊆ K be the subring obtained by adjoining the inverses of all bad primes for G to the ring of algebraic integers in K; note that R ⊆ O. Comparing with the conditions in Table 1.4 (p. 33), we see that R is L-good, where L : W → Z0 is the weight function in 4.3.7. Let Γ = Z and H = HA (W, S, L) be the generic Iwahori–Hecke algebra over the ring of Laurent polynomials A = R[v, v−1 ] in an indeterminate v. Thus, the algebra H has an A-basis {Tw | w ∈ W } and, for s ∈ S and w ∈ W , we have if l(sw) > l(w), Tsw Ts Tw = L(s) −L(s) Tsw + (v − v )Tw if l(sw) < l(w), where |BsB/B| ˙ = qL(s) for all s ∈ S. 4.3.9. Since K is assumed to be sufficiently large, we can find a square root q1/2 ∈ R. Let θ : A → O be the canonical ring homomorphism induced by the inclusion R ⊆ O and sending v to q1/2 . Then the corresponding specialised algebra HO,θ is naturally isomorphic to HO := EndOG (O[G/B])◦ . (Note that Iwahori’s description in 4.3.1 works over any commutative ground ring instead of k; see [132, §8.4].) We obtain induced maps θK : A → K (by composing θ with the inclusion O ⊆ K) and θk : A → k (by composing θ with the natural map O → k). Let HK and Hk denote the corresponding specialised algebras. We have HK ∼ = EndKG (K[G/B])◦
and
Hk ∼ = EndkG (k[G/B])◦ .
Thus, the endomorphism algebras that we need to consider can all be naturally identified with specialisations of the generic algebra H. Let K = K(v) be the field of fractions of A. Since R is L-good, we obtain the decomposition maps (see Section 3.1) dθK : R0 (HK ) → R0 (HK )
and
dθk : R0 (HK ) → R0 (Hk ).
4.3 Unipotent Principal Series Representations, I
229
4.3.10. We can now express the classification of IrrK (G | B) in terms of H. Consider the decomposition map dθK : R0 (HK ) → R0 (HK ). Being isomorphic to the endomorphism algebra of a semisimple module, HK is semisimple. Hence, by 3.1.18, the map dθK induces a bijection between Irr(HK ) and Irr(HK ). Thus, writing IrrK (W ) = {E λ | λ ∈ Λ } and Irr(HK ) = {Evλ | λ ∈ Λ } as in Chapter 1, we have λ | λ ∈ Λ }, Irr(HK ) = {EK
where
λ dθK ([Evλ ]) = [EK ] ∈ R0 (HK ).
With this notation, Proposition 4.2.10 shows that IrrK (G | B) = {ρ λ | λ ∈ Λ }
and K[G/B] ∼ =
λ ∈Λ
ρλ ⊕ · · · ⊕ ρλ ,
dimK E λ times
λ . Furtherwhere ρ λ ∈ IrrK (G | B) is determined by the condition that FK (ρ λ ) ∼ = EK more, using the Schur elements cλ in 1.2.11, we have
dimK ρ λ =
[G : B] ; θK (cλ )
see [132, §8.4]. Thus, we have established the classical fact that there is a natural bijection IrrK (G | B) ↔ IrrK (W ). We note that exactly the same arguments apply whenever Hk is semisimple. In this case, we can write again Irr(Hk ) = {Ekλ | λ ∈ Λ },
where
dθk ([Evλ ]) = [Ekλ ] ∈ R0 (Hk ).
Furthermore, we have Irrk (G | B) = {Y λ | λ ∈ Λ }
and
k[G/B] ∼ =
λ ∈Λ
Y λ ⊕ · · · ⊕Y λ ,
dimK E λ times
where Y λ ∈ IrrK (G | B) is determined by the condition that Fk (Y λ ) ∼ = Ekλ . For further reading on the semisimple case, see Exercises 22–27 of Bourbaki [29, Chap. IV, §2], Curtis, Iwahori and Kilmoyer [52], Curtis and Reiner [53, §68], Carter [45, Chap. 10], Geck and Pfeiffer [132, §8.4] and Howlett–Lehrer [162]. Remark 4.3.11. Taking q and as fixed, recall from 4.3.8 and 4.3.9 that we chose a square root q1/2 ∈ R and a discrete valuation ring O ⊆ K (with residue field of characteristic ). These choices can be made such that if we set (a)
e := min{i 2 | 1 + q + q2 + · · · + qi−1 ∈ rad(O)},
then (b)
q1/2 − ζ2e ∈ rad(O),
where
√ ζ2e = exp(π −1/e) ∈ R ⊆ K
is the “standard” 2eth root of unity. Indeed, if δ = 1, then q is an integer and so
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4 Hecke Algebras and Finite Groups of Lie Type
e = min{i 2 | 1 + q + q2 + · · · + qi−1 ≡ 0 mod }, independently of any choices made. Hence, if p is any maximal ideal of R containing , then Φe (q) ∈ p. It follows that q − ζ ∈ p for some primitive eth root of unity ζ . Since all these roots of unity are algebraically conjugate, we can choose p such that 2 ∈ p. Then we can take as O the localisation of R in p. Replacing q1/2 by q − ζ2e −q1/2 if necessary, we can make sure that (b) holds. If δ = 2, then we first define e := {i 2 | 1 + q2 + q4 + · · · + q2(i−1) ≡ 0 mod }. Now note that Φe (q2 ) = Φ2e (q) (if e is even) or Φe (q2 ) = Φ2e (q)Φe (q) (if e if odd); see Example 3.3.2. Hence, as above, we see that there is a maximal ideal p 2 ∈ p, where e = 2e (if e is even) or e ∈ {e , 2e } of R such that ∈ p and q − ζ2e (if e is odd). Again, let O be the localisation of R in p. Replacing q1/2 by −q1/2 if necessary, we see that (b) holds. Then one also checks that e satisfies the condition in (a); that is, e 2 is minimal such that 1 + q + q2 + · · · + qe−1 ∈ rad(O), as required. Having fixed the above choices, let us now consider the Φe -modular specialisation θe : A → K such that θe (v) = ζ2e (and θe (r) = r for all r ∈ R); denote by H(e) the corresponding specialised algebra. By (b), we have ker(θe ) = (v − ζ2e ) ⊆ ker(θk ). Further note that θe (A) = R is integrally closed in K. Hence, θk lies above θe in the sense of 3.6.2. So Theorem 3.6.3 yields that (c)
dθk factors through the decomposition map dθe : R0 (HK ) → R0 (H(e) ).
This remark will play an essential role in the subsequent discussion; see, for example, the proof of Theorem 4.4.8 and Example 4.5.15 below. In the following section, we shall need the following ingredients from Lusztig’s theory of representations of finite groups of Lie type. 4.3.12. Let ρ ∈ IrrK (G | B). Via the bijection IrrK (G | B) ↔ IrrK (W ) in 4.3.10, ρ corresponds to a unique E λ ∈ IrrK (W ). In Chapter 1, using the generic algebra H, we have attached a numerical invariant aλ ∈ Z0 to Eλ ; thus, we can define aρ := aλ . In fact, one can attach a numerical invariant (generalising aλ ) to any ρ ∈ IrrK (G). This requires the following definitions. Let O be an F-stable conjugacy class of G consisting of unipotent elements. Then the fixed point set OF splits into conjugacy classes under the action of the finite group G. Let u1 . . . , ur ∈ OF be representatives of the G-conjugacy classes contained in OF . For each j, let A(u j ) be the group of connected components of the centraliser of u j in G. Since F(u j ) = u j , there is an induced action of F on A(u j ) which we denote by the same symbol. We define the average value of ρ ∈ IrrK (G) on OF by (a)
AV(O, ρ ) :=
∑
[A(u j ) : A(u j )F ] trace(u j , ρ ).
1 jr
(Note that AV(O, ρ ) does not depend on the choice of the representatives u j .) Assuming that p, q are large enough, Lusztig [229] has shown that, given ρ ∈ IrrK (G), there exists a unique F-stable unipotent class Oρ such that • AV(Oρ , ρ ) = 0 and
4.3 Unipotent Principal Series Representations, I
231
• given any F-stable unipotent class O such that AV(O, ρ ) = 0, then O = Oρ or dim O < dim Oρ . The class Oρ is called the unipotent support of ρ . The assumptions on p, q have subsequently been removed by Geck and Malle [128]. Thus, every ρ ∈ IrrK (G) has a well-defined unipotent support Oρ . Using this concept, we associate with every ρ ∈ IrrK (G) a numerical invariant aρ by setting aρ := dim Bu , where Bu is the variety of Borel subgroups of G containing an element u ∈ Oρ . (This variety already appeared in 2.2.12; this is not a coincidence, but we refer to [220] and [229] for further details.) Now let E λ ∈ IrrK (W ) and consider the corresponding unipotent principal series representation ρ = ρ λ ∈ IrrK (G | B); see 4.3.10. Then it is known that (b) aρ (defined as above using the unipotent support of ρ ) equals aλ (defined using the Schur element cλ as in Corollary 1.3.1). This statement is contained in [229, §10] (see also [128, Theorem 3.7]). It relies on the fact that the map ρ → Oρ has an alternative description in terms of the Springer correspondence and induction of representations of Weyl groups. 4.3.13. By the fundamental work of Deligne and Lusztig [54], we have a virtual representation RT,θ of G for any pair (T, θ ) where T is an F-stable maximal torus T of G and θ ∈ IrrK (TF ); thus, [RT,θ ] ∈ R0 (KG). The construction involves the theory of -adic cohomology of certain varieties on which the finite group G acts; see also the exposition by Carter [45, Chap. 7]. (Here, one has to identify K with a subfield of Q .) It is known that any ρ ∈ IrrK (G) appears in RT,θ for at least some pair (T, θ ); see [45, 7.5.8]. The irreducible representations of G in the set (a)
UnipK (G) := {ρ ∈ IrrK (G) | ρ appears in RT,1 for some T}
are called unipotent representations; here, 1 stands for the trivial representation of TF . Since RT0 ,1 ∼ = K[G/B] (see [45, 7.2.4]), we have (b)
IrrK (G | B) ⊆ UnipK (G).
If G = GLn (Fq ), then equality holds (see [55, 15.8]); but in general, the inclusion is strict. The importance of UnipK (G) lies in the fact that, in a well-defined sense, the classification of IrrK (G) can be reduced to the classification of unipotent representations for G itself and smaller groups (see Lusztig [220, 4.23] and [226] for details). By [220, Chap. 13] and [128, §4], the set of unipotent classes (c)
{Oρ | ρ ∈ UnipK (G)}
(where Oρ denotes the unipotent support) turns out to be the set of special unipotent classes defined by Lusztig [215, §9]; see also 2.2.12. These special unipotent classes are in bijection with the two-sided Kazhdan–Lusztig cells (see Section 2.1) of the Weyl group W of G with respect to the equal-parameter weight function.
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4.4 Unipotent Principal Series Representations, II In the previous section we saw that Irrk (G | B) is naturally in bijection with Irrk (W ) whenever the algebra Hk is semisimple. The aim of this section is to formulate and prove the “modular” analogue of this statement. For G = GLn (Fq ), this is originally due to Dipper and James; see [58] and [62]. Here, we follow the general approach in [109] based on the “canonical basic sets” discussed in Chapter 3. Recall our basic set-up: G = GF is a finite group of Lie type; see Definition 4.3.5. Furthermore, as in 4.3.8, we have p = > 0 and k is the residue field of a discrete valuation ring O in K ⊆ C. Here, K is a sufficiently large algebraic extension of Q; in particular, K and k will be splitting fields for G and all its subgroups. We are now ready to state the main results about the series Irrk (G | B) in the case where k[G/B] is not semisimple. Note that the statement does not refer to any properties of the Hecke algebra HO ; these will only be used in the proof. Theorem 4.4.1 (Cf. [109]). Recall that is assumed to be good for G. As in 4.1.11, denote by ρ : Y OG (where ρ ∈ IrrK (G) and Y ∈ Irrk (G)) the entries of the decomposition matrix Dec(OG). • Given Y ∈ Irrk (G | B), let Sˆk (Y ) := {ρ ∈ UnipK (G) | ρ : Y OG = 0}. Then the function Sˆk (Y ) → Z0 , ρ → aρ (defined using the unipotent support), reaches its minimum at exactly one element of Sˆk (Y ), which we denote by ρY . • The map Irrk (G | B) → UnipK (G), Y → ρY , is injective and its image is contained in IrrK (G | B). • We have ρY : Y OG = 1 for all Y ∈ Irrk (G | B). Thus, setting Bk (G) := {ρY | Y ∈ Irrk (G | B)}, we obtain a natural bijection 1−1
Irrk (G | B) ←→ Bk (G) ⊆ IrrK (G | B) ⊆ UnipK (G). In analogy to Definition 3.2.1, Bk (G) is called a canonical basic set for Irrk (G | B). Proof. The natural map O → k does not only induce the decomposition map dOG : R0 (KG) → R0 (kG), but also a decomposition map dHO : R0 (HK ) → R0 (Hk ). Now the idea is to translate the above statements from G to HO and then to use the known results on canonical basic sets from Section 3.2. We begin by showing that dHO can be “lifted” to the generic algebra H. This is done by an argument entirely analogous to that in Example 3.6.5. Indeed, since the specialisation θk : A → k is the composition of θ with the natural map O → k, it is clear that θk lies above θK in the sense of 3.6.2. Thus, by Theorem 3.6.3, there is a factorisation dθk = dHO ◦ dθK . We have already seen in 4.3.10 that dθK induces a bijection between Irr(HK ) and Irr(HK ). Identifying R0 (HK ) and R0 (HK ) via this bijection, we see that dθk is identified with the decomposition map dHO . Thus, using the notation in 4.3.10, we have
4.4 Unipotent Principal Series Representations, II
(1)
233
dθk ([Evλ ]) = dHO ([EKλ ]) = dHO [FK (ρ λ )] ∈ R0 (Hk )
for all λ ∈ Λ .
Next we show that we are in a situation where Dipper’s Theorem 4.1.14 applies. Write B = U H, where U is the largest normal p-subgroup of B and H = TF0 . Since the order of U is prime to , the permutation module O[G/U] is projective and, since U ⊆ B, we have a natural surjective map
β˜ : P˜ → M˜
P˜ := O[G/U] and M˜ := O[G/B].
where
Extending scalars from O to K, we find that K[G/U] ∼ = K[G/B] ⊕ ker(β˜ )K . ∼ Furthermore, K[G/B] ∼ = RG = RG H (KH ) and K[G/U] H (KH), where KH denotes the ∼ (left) regular representation of H. Since KH = η ∈IrrK (H) η , we deduce that ker(β˜ )K ∼ =
RG H (η ),
KH =η ∈IrrK (H)
and so an easy application of the Mackey formula yields G HomKG (K[G/B], ker(β˜ )K ) ∼ HomKG RG = H (KH ), RH (η ) = {0}. KH =η ∈IrrK (H)
Thus, condition (D) in Lemma 4.1.13 is satisfied for the above map β˜ . Hence, Dipper’s Theorem 4.1.14 applies and so the functor Fk := Fk[G/B] induces a bijection 1−1
Fk : Irrk (G | B) −→ Irr(Hk ); furthermore, the matrix Dec(HO ) (associated with dHO : R0 (HK ) → R0 (Hk )) is contained in Dec(OG). Combining this with (1), we obtain (2)
dθk ([Evλ ]) =
∑
ρ λ : Y OG [Fk (Y )]
for all λ ∈ Λ .
Y ∈Irrk (G|B)
Next we show that, for any Y ∈ Irrk (G | B), we have (3)
Sˆk (Y ) ⊆ IrrK (G | B).
This is seen as follows. Let ρ ∈ Sˆk (Y ) and V˜ be an OG-lattice such that V˜k ∼ = ρ . Let PY be a projective cover of Y . Thus, since ρ : Y OG = 0, we have HomkG (PY , V˜k ) = {0}. Since Y is isomorphic to a factor module of M˜ k and, hence, of P˜k , we conclude that PY is a direct summand of P˜k . So we also have HomkG (P˜k , V˜k ) = {0}. Using 4.1.12 and the fact that P˜ is projective, we deduce that HomKG (P˜K , V˜K ) = {0}. Now recall from the above computation that ker(β˜ )K is a direct sum of terms RG H (η ), ∼ R ( η ) . Consequently, where KH = η ∈ IrrK (H). By [45, 7.2.4], we have RG = T0 ,η H
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all irreducible constituents of ker(β˜ )K are non-unipotent (see [45, 7.3.8]). It follows that ρ must be a constituent of K[G/B], as required. Now we can complete the proof as follows. Given Y ∈ Irrk (G | B), let E := Fk (Y ) ∈ Irr(Hk ). Then (2) and (3) mean that Sˆk (Y ) = {ρ λ | λ ∈ Sθk (E)}, with Sθk (E) as in Definition 3.2.1. Hence, taking into account 4.3.12(b), we see that all the required statements concerning Bk (G) will hold if and only if Hk admits a canonical basic set Bθk as in Definition 3.2.1. We have seen in Proposition 3.2.7 that such a canonical basic set for Hk exists if the properties (♠) and (♣) in 2.5.3 are satisfied for H = HA (W, S, L). Also recall from 2.5.3 that (♠) and (♣) can be deduced from P1, P4, P15 in Conjecture 2.3.2. Now, by 2.4.1, these properties do hold in the “equal-parameter case”, where γ = id and L is the length function on W = W . In [231, Chapter 16], Lusztig also sketches a proof of P1, P4, P15 for the
general quasi-split case; hence, (♠) and (♣) will hold in this case as well.3 Remark 4.4.2. The above proof relies on the assumption that the properties (♠) and (♣) hold for W, L. But note that these are only used in the very last steps of the proof, after the translation of the original problem to a problem concerning Iwahori–Hecke algebras is achieved. In particular, the statements (1), (2) and (3) in the above proof do not rely at all on (♠) or (♣). Remark 4.4.3. One may define the set of “modular” unipotent representations by Unipk (G) := {Y ∈ Irrk (G) | ρ : Y OG = 0 for some ρ ∈ UnipK (G)}. We shall see in Remark 4.5.6 that the statement of Theorem 4.4.1 is “natural” in the sense that it fits into a general (conjectural) bijection between UnipK (G) and Unipk (G), which itself is part of a global conjecture concerning all of Irrk (G). μ
Corollary 4.4.4. Write Irr(Hk ) = {Lk | μ ∈ Λk◦ } as in Proposition 3.2.7, where Λk◦ ⊆ Λ is the canonical basic set associated with θk : A → k. Then we have Irrk (G | B) = {Y μ | μ ∈ Λk◦ }, μ where Y μ is determined by the condition that Fk (Y μ ) ∼ = Lk . Furthermore, we have
dλ ,μ = ρ λ : Y μ OG
for all λ ∈ Λ and μ ∈ Λk◦ ,
where Dθk = dλ ,μ λ ∈Λ , μ ∈Λ ◦ is the decomposition matrix of H with respect to θk . k
Proof. This is just a reformulation of what we established in the proof of Theorem 4.4.1: under the bijection IrrK (G | B) ↔ Λ in 4.3.10, the subset Bk (G) corresponds to a canonical basic set for Irr(Hk ), in the sense of Definition 3.2.1. The 3 Since not all details of the proof of P1, P4, P15 for the quasi-split case are worked out in [231, Chapter 16], the incredulous reader should add the assumption to the theorem that if W is of type Bn and L(s) > 0 (s ∈ S), then γ = id and L = l; the other cases where γ = id are covered by 2.4.1.
4.4 Unipotent Principal Series Representations, II
235
equality of decomposition numbers is expressed in equation (2) of the above proof. (It is a consequence of Dipper’s Theorem 4.1.14.)
Example 4.4.5. Let G = GLn (Fq ). In this case, W = W ∼ = Sn and Λ consists of all partitions λ n. Then, by Theorem 3.5.14, we have μ
Irr(Hk ) = {Lk | μ ∈ Λk◦ },
where
Λk◦ = {μ n | μ is e-regular}
and e = min{i 2 | 1 + q + q2 + · · · + qi−1 ≡ 0 mod }. Hence, by Corollary 4.4.4, we also have a parametrisation of Irrk (G | B) by the e-regular partitions of n. In fact, much more is known in this case: Dipper and James [64], [65] describe the whole decomposition matrix of G in terms of q-Schur algebras, and Dipper [59, Theorem 5.21] gives a complete description of the partition of Irrk (G) into HarishChandra series. For further reading on GLn , see [38], [56], [57], [58], [59], [62], [64], [72], [119, §7], [180]. Example 4.4.6. Assume that | qL(s) − 1 for all s ∈ S. Then Hk ∼ = kW , the group algebra of W over k. By Example 3.6.5, the map dθk : R0 (HK ) → R0 (Hk ) can be naturally interpreted as the usual -modular decomposition map for the finite group W . Hence, Corollary 4.4.4 shows that the decomposition matrix Dec(OW ) embeds into Dec(OG). For G = GLn (Fq ), this (and a much more general statement) was first proved by Dipper [56]. For G in general, see [86, 4.4] and [92, 4.6]. Here is a simple crtiterion for identifying the subset Irrk (G | B) ⊆ Irrk (G). Lemma 4.4.7. Let Y ∈ Irrk (G) and assume that Y ∈ Irrk (G | B). Then ρ : Y OG = 0 for all ρ ∈ UnipK (G) such that ρ ∈ IrrK (G | B). Furthermore, we have that a
divides
∑ ρ λ : Y OG dimK ρ λ ,
λ ∈Λ
where a (a 0) is the largest power of dividing [G : B]. Proof. Using the notation in Corollary 4.4.4, we have Y ∼ = Y μ for some μ ∈ Λk◦ . Now let ρ ∈ UnipK (G). Then, by (2) and (3) in the proof of Theorem 4.4.1, we have if ρ ∼ dλ ,μ = ρ λ where λ ∈ Λ , ρ : Y OG = 0 otherwise. Thus, the first condition holds. Now consider the algebra HO and the decomposition map dHO : R0 (HK ) → R0 (Hk ). By (1) in the proof of Theorem 4.4.1, the matrix (dλ ,μ ) is the decomposition matrix associated with dHO . Further note that, since O is a discrete valuation ring and HO is a symmtric algebra, where HK is split semisimple, the statement of Proposition 3.3.6 also holds for HO and dHO . (See [132, 7.5.3].) Thus, since the Schur elements of HK are given by θK (cλ ) (λ ∈ Λ ), we have ∑ θK (cλ )−1 dλ ,μ ∈ O for any μ ∈ Λk◦ . λ ∈Λ
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Finally, by the dimension formula in 4.3.10, we have dimK ρ λ = [G : B]/θK (cλ ). It follows that 1 dλ ,μ dimK ρ λ ∈ O, ∑ [G : B] λ ∈Λ
which yields the required statement.
The next result shows that, in a well-defined sense, the classification of Irrk (G | B) in Theorem 4.4.1 only depends on “e”, as defined in Remark 4.3.11.
Theorem 4.4.8 (Cf. [109]). In the setting of Theorem 4.4.1, assume that W is irreducible and that does not divide |W| · |γ |. Recall that q1/2 − ζ2e ∈ rad(O), where e 2 is minimal such that 1 + q + q2 + · · · + qe−1 ∈ rad(O); see Remark 4.3.11. Consider the Φe -modular specialisation θe : A → K, v → ζ2e , and denote by H(e) the corresponding specialised algebra. Then, under the bijection IrrK (G | B) ↔ Λ ◦ ⊆ Λ for H . in 4.3.10, the set Bk (G) corresponds to the canonical basic set Λ(e) (e) In this sense, Bk (G) only depends on e.
Proof. We already know that, under the bijection IrrK (G | B) ↔ Λ , the set Bk (G) corresponds to the canonical basic set Λk◦ ⊆ Λ ; see Corollary 4.4.4. So we must ◦ show that Λk◦ = Λ(e) . Now, we have seen in Remark 4.3.11(c) that dθk factors through the decomposition map dθe : R0 (HK ) → R0 (H(e) ). Hence, by Proposition 3.7.5 (or Lemma 3.6.7), it will be enough to show that | Irr(Hk )| = | Irr(H(e) )|. Assume first that [G : B]. Then Lemma 4.3.2 applies and so Λk◦ = Λ . Since | Irr(Hk )| | Irr(H(e) )|, we deduce that Λk◦ = Λe◦ = Λ in this case, as required. Thus, we can now assume that | [G : B] and, hence, Hk is not semisimple. Note that this implies that > 2 and q ≡ 1 mod rad(O). (For, otherwise, we would have q ≡ 1 mod rad(O) and so Hk ∼ = kW would be semisimple since |W |.) To deal with this case, we shall use Theorem 3.7.16. This requires some preparations. As mentioned in 4.3.7, we have [G : B] = PW,L (q), where PW,L ∈ Z[u] is the “weighted” Poincar´e polynomial PW,L := ∑w∈W uL(w) . (Here, we set u = v2 .) Now the results in Example 3.3.2 generalise to the quasi-split case. More precisely, by Steinberg [287, pp. 190–191] (or [45, p. 75]), we have a factorisation PW,L =
1 − ζ j ud j , 1 jr 1 − ζ j u
∏
where d1 , . . . , dr are the degrees of W and ζ1 , . . . , ζr are certain roots of unity such |γ | that ζ j = 1 for all j. In particular, PW,L ∈ Z[u] is a product of cyclotomic polyno˜ mials. Let E(W, L) be the set of all natural numbers d 2 such that Φd (u) divides PW,L . From the above factorisation of PW,L , we deduce that (∗)
˜ d ∈ E(W, L)
⇒
d divides |W| · |γ | (and, hence, d).
4.4 Unipotent Principal Series Representations, II
237
Now recall from 4.3.10 that dimK ρ λ = [G : B]/θK (cλ ) for all λ ∈ Λ . Then a standard argument (see, for example, Step 1 in the proof of [132, Cor. 8.3.6]) shows that the Schur elements cλ divide PW,L in K[v, v−1 ]. Hence, with the notation in 3.3.1 (see also Example 3.3.3), we conclude that P :={Φ ∈ A | Φ is K-cyclotomic and divides cλ for some λ ∈ Λ } ={Φ ∈ A | Φ is K-cyclotomic and divides PW,L )}. Since K is assumed to be sufficiently large, every polynomial in P will be of the form v − ζ , where ζ ∈ K is a root of unity. Hence, we have ˜ L)}. P = {v − ζ ∈ K[v] | Φd (ζ 2 ) = 0 for some d ∈ E(W, ˜ L); by (∗), Now, since | [G : B] = PW,L (q), we have | Φd (q) for some d ∈ E(W, we have d. On the other hand, by the definition of e, we have | Φe (q). Hence, 3.7.11(a) shows that e = di for some i ∈ Z. As noted earlier, q ≡ 1 mod rad(O) and so e is the order of the image of q in k× ; in particular, e. Thus, we must have d = e and so Φ := v − ζ2e ∈ P. Using (∗), we can now conclude that Φ is strongly θk -isolated in the sense of Definition 3.7.8. (Indeed, if γ = id and W is of type B2 , G2 or F4 , this is verified directly using the formulae in Example 4.4.9 below; for all the other cases, see Lemma 3.7.12(b).) So we can apply Theorem 3.7.16, and this
yields that | Irr(Hk )| = | Irr(H(e) )|, as desired. Example 4.4.9. Let W be of type B2 , G2 or F4 , and γ : W → W be the non-trivial automorphism of order 2. This gives rise to a weight function L on W ∼ = Wγ as in Table 4.1. Using the notation in the above proof, we have (1 − u2 )(1 + u4 ) (type 2B2 ), (1 − u)(1 + u) (1 − u2 )(1 + u6 ) (type 2G2 ), PW,L = (1 − u)(1 + u) (1 − u2 )(1 + u6 )(1 − u8 )(1 + u12 ) PW,L = (type 2F 4 ). (1 − u)(1 − u)(1 + u)(1 + u) ˜ ˜ Hence, we find that E(W, L) = {8} (type 2B2 ), E(W, L) = {4, 12} (type 2G2 ) and 2 ˜ E(W, L) = {4, 8, 12, 24} (type F 4 ). In the first two cases, GF is a Suzuki or Ree group of type 2B2 or 2G2 respectively. Here, W = Wγ is of type A1 and so Λ = {(2), (11)} (where (2) labels the unit and (11) labels the sign representation). Hence, we have (see also Example 4.3.4) Λ if qL(s) ≡ 1 mod , ◦ Λk = {(2)} otherwise. PW,L =
If GF is a Ree group of type 2F 4 , then W = Wγ is of type I2 (8) and the set Λk◦ is described in Table 7.6 (p. 373). Remark 4.4.10. In the proof of Theorem 4.4.8 we have seen that Φ = v − ζ2e is strongly θk -isolated in the sense of Definition 3.7.8. Hence, if the general version
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4 Hecke Algebras and Finite Groups of Lie Type
of James’ Conjecture 3.7.9 was known to hold, then we could not only conclude that Irr(H(e) ) and Irr(Hk ) have the same cardinality, but also that the decomposition matrices of H corresponding to θe and to θk are equal. By Example 3.7.7 we know at least that this stronger conclusion holds whenever is “large enough”. The statement of Theorem 4.4.8 holds under somewhat weaker assumptions on , but then the proof requires some case-by-case considerations and the results discussed in Section 4.6 concerning type Bn . The stronger statement is as follows. Theorem 4.4.11 (Cf. [109, 3.4]). In the statement of Theorem 4.4.8, instead of the assumption that does not divide |W| · |γ |, it is sufficient to assume that is good for G and that does not divide the order of the following parabolic subgroup of W : if θk (q) = 1, s ∈ S | L(s) > 1 and θk (q)L(s) = 1 Ωk := {1} if θk (q) = 1. In particular, if L(s) = 1 for all s ∈ S, then it is enough to assume that is good for G. Proof. As in the proof of Theorem 4.4.8, it is enough to show that | Irr(Hk )| = | Irr(H(e) )|. Assume first that W = Wγ is of exceptional type. If does not divide |W| · |γ |, then we already know that the desired equality holds. Hence, for each type of W, there is only a finite number of additional cases to be considered. These additional cases can be handled by explicit computations with the character tables of HK , using Proposition 3.4.11. In the case where γ = id, details of such computations are given in [100, §3]. Similar methods apply to the cases where W is of type D4 or E6 , and W = Wγ is of type G2 or F4 respectively. The cases where γ = id and W is of type B2 , G2 , F4 are already covered by Theorem 4.4.8. So it remains to consider W of type An−1 , Bn or Dn . If W is of type An−1 , where γ = id, the desired equality holds by Theorem 3.5.14. If W is of type Bn or Dn , then the results in Section 4.6 show that | Irr(Hk )| only depends on e and γ and, hence, the desired equality holds; see the explicit consideration of these cases in Examples 4.4.13–4.4.15 below. Now assume that W is of type An−1 and γ has order 2, so that W = Wγ is of type Bm (for suitable m). If e = 2, then the discussion in Example 4.4.16 below shows again that the desired equality holds. Finally, if e = 2, then θk (q) = −1 and so Ωk ∼ = Sm . Hence, the condition on implies that |W |. Consequently, in the setting of Section 4.6, we have e˜ = > m. Then Proposition 4.6.10 shows that | Irr(Hk )| and | Irr(H(e) )| are both given by the number of partitions of m.
Example 4.4.12. Let G be of exceptional type G2 , F4 , E6 , E7 or E8 . Let F be a split Frobenius map so that γ = id. Then the assumptions of Theorem 4.4.11 are satisfied whenever is good for G. Hence, the information in Table 7.1 (pp. 367) and Table 7.3 (p. 373) yields explicit descriptions for the canonical basic sets Λk◦ in these cases. The twisted exceptional groups GF = 3 D4 (Fq ) and 2 E 6 (Fq ) are covered by the information in Table 7.4 (p. 373) and Table 7.8 (p. 376).
4.4 Unipotent Principal Series Representations, II
239
To close this section, we shall now consider the various cases in Table 4.1 (p. 227) where W is of type Bn . Recall that then IrrK (W ) is parametrised by the set Λ consisting of all pairs of partitions (λ , μ ) such that |λ | + |μ | = n. Let e := min{i 2 | 1 + q + q2 + · · · + qi−1 ≡ 0 mod }. The Examples 4.4.13–4.4.16 contain a summary of results whose proofs rely on the theories developed later in Section 4.6 and Chapters 5 and 6. Example 4.4.13. Let G = Sp2n (Fq ) (the finite symplectic group) or G = SO2n+1 (Fq ) (the odd-dimensional orthogonal group); see [104, 4.2.6] and Grove [147]. Then W = W is of type Bn and the associated weight function L : W → Z is given by Bn
1 t
4
1 t
1 t
· · ·
1 t
That is, we are in the equal-parameter case. Since 2 is L-bad in this case, we shall assume that char(k) = > 2. The set Λk◦ ⊆ Λ is determined as follows. If e is odd, then the assumptions of Example 4.6.4 below are satisfied and so Λk◦ consists of all (λ , μ ) ∈ Λ such that both λ and μ are e-regular. If e is even, then θk (q) ∈ k× is a root of unity of order e and so Λk◦ is given by the “FLOTW bipartitions” of Theorem 5.8.5 in Chapter 5. Example 4.4.14. Let G = SO◦2n (F p ) be the connected component of the identity of the even-dimensional orthogonal group; see [104, 1.3.16] and Grove [147]; thus, the Weyl group W of G is of type Dn . Assume that F is a split Frobenius map with respect to Fq . Then G = GF is the split orthogonal group; see Case 1 in [104, 4.2.6]. In this case W = W and Λk◦ is determined as in Example 3.2.15, by using the canonical basic set for a group of type Bn with weight function Bn
0 t
4
1 t
1 t
· · ·
1 t
Since 2 is L-bad in this case, we shall assume that char(k) = > 2. If e is odd, then we can again apply Example 4.6.4 below, and so this canonical basic set (for type Bn ) consists of all (λ , μ ) ∈ Λ such that both λ and μ are e-regular. If e is even, then θk (q) ∈ k× is a root of unity of order e, and so the canonical basic set is given by the “FLOTW bipartitions” of Theorem 5.8.15 in Chapter 5. As explained in Example 3.2.15, this then yields an explicit description of Λk◦ (for W of type Dn ); see Theorem 5.8.19 in Chapter 5. Example 4.4.15. Now let G = SO◦2(n+1) (F p ) and assume that F is a non-split Frobenius map with respect to Fq . Then G = GF is the non-split orthogonal group; see Case 3 in [104, 4.2.6]. In this case, W = Wγ is of type Bn with weight function Bn
2 t
4
1 t
1 t
· · ·
1 t
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4 Hecke Algebras and Finite Groups of Lie Type
Since 2 is L-bad in this case, we shall assume that char(k) = > 2. Again, if e is odd, we can apply Example 4.6.4 below and so Λk◦ consists of all (λ , μ ) ∈ Λ such that both λ and μ are e-regular. If e 4 is even, then θk (q) ∈ k× is a root of unity of order e, and so Λk◦ is given by the “FLOTW bipartitions” of Theorem 5.8.13. Finally, assume that e = 2; that is, θk (q) = −1. To obtain a description of Λk◦ , we use the fact that θk lies above the Φe -modular specialisation θe : A → K; see Remark 4.3.11(c). Let H(e) be the corresponding specialised algebra. We shall see in Remark 6.7.12 that a canonical basic set B(e) ⊆ Λ for H(e) exists and that there is an explicit combinatorial description of B(e) (using the theory of canonical bases for quantised enveloping algebras). Furthermore, Theorem 4.6.13 will show that | Irr(Hk )| = | Irr(H(e) )|. So Lemma 3.6.8 implies that Λk◦ = B(e) . Example 4.4.16. Let G = GUN (Fq ). Recall that G = GF , where G = GLN (Fq ) and F : G → G is the Frobenius map described in Example 4.3.6(b). We are in the “quasi-split case” where W ∼ = Wγ is of type Bn ; here, N = 2n + c, = SN and W ∼ where c ∈ {0, 1}. The associated weight function L : W → Z is given by Bn
2c+1 t
4
2 t
2 t
· · ·
2 t
There are no L-bad primes, so there is no restriction on char(k) = . The following distinction of cases works for arbitrary values of c 0 (and not just c ∈ {0, 1}). If e = 2, then θk (q)2 = 1 and θk (q)2c+1 = −1. In this case, Λk◦ is given by Proposition 4.6.10 below. Now assume that e = 2; in particular, this means that = 2. If e = 2e , where e 3 is odd, then θk (q)2 ∈ k× is a root of unity of order e and so Λk◦ is given by the “FLOTW bipartitions” of Theorem 5.8.9 (c = 0) and Theorem 5.8.11 (c = 1) in Chapter 5. (For c 2, see Example 4.5.15 below.) Finally, assume that e is not twice an odd number. Then θk (q2c+1 + q2 j ) = 0 for all j ∈ Z. (Indeed, if θk (q)2c+1 = −θk (q)2 j , then θk (q)2c−2 j+1 = −1 and so θk (q) ∈ k× would be a root of unity whose order is twice an odd number, contrary to our assumption.) Thus, setting e˜ = e (if e is odd) or e˜ = e/2 (if e is even), we ˜ see have that Λk◦ consists of all (λ , μ ) ∈ Λ such that both λ and μ are e-regular; Proposition 4.6.3 below. We remark that, with the single exception of the case e = 2 in Example 4.4.15, the results cited in the above examples actually imply the existence of a canonical basic set for Hk (type Bn ), without reference to the validity of (♠) and (♣) in 2.5.3.
4.5 Examples and Conjectures The results obtained in the previous section raise the question as to what extent the restriction to unipotent principal series representations is really necessary. The difficulty in answering this question is that very little is known on decomposition numbers in general. We shall now formulate some conjectures which hint at a more
4.5 Examples and Conjectures
241
general picture and then illustrate these with several examples. Note, however, that it is not the place here to provide a comprehensive survey of the state of knowledge on modular representations of finite groups of Lie type; throughout this section, we shall give references for further reading. The starting point is the following result on “basic sets” for G. Let T be an Fstable maximal torus of G. Since TF is abelian, IrrK (TF ) is an abelian group under taking tensor products; so we can speak of the order of θ ∈ IrrK (TF ). Theorem 4.5.1 (Geck and Hiss [116], [93]). Assume that is good for G and does ◦ ) . (Here, Z denotes the centre of G and (Z /Z ◦ ) not divide the order of (ZG /ZG F G G G F ◦ denotes the largest quotient of ZG /ZG on which F acts trivially). Let ρ appears in some RT,θ where . E (G) := ρ ∈ IrrK (G) θ ∈ IrrK (TF ) has order prime to Then |E (G)| = | Irrk (G)| and {dOG ([ρ ]) | ρ ∈ E (G)} is a Z-basis of R0 (kG). Note that we always have UnipK (G) ⊆ E (G). If is not good for G, we have |E (G)| = | Irrk (G)| in general. For further information on the cardinalities of the sets |E (G)| in this case, see [117, 6.6]. The following conjecture is somewhat more precise than its precursors in [90], [109], [117]. The present version appeared in the first author’s talk at Michel Brou´e’s 60th birthday conference (ENS Paris, October 2006). Note that, implicitly, it relies on a basic result of Brou´e and Michel [37] concerning O-blocks of G (see Theorem 4.5.4 for a special case).
Conjecture 4.5.2 (Cf. Geck [90], [109]; Geck and Hiss [117]). Assume that is ◦) . good for G and does not divide the order (ZG /ZG F (a) Given Y ∈ Irrk (G), let Sˆk (Y ) := {ρ ∈ E (G) | ρ : Y OG = 0}. Then the function Sˆk (Y ) → Z0 , ρ → aρ (defined using the unipotent support), reaches its minimum at exactly one element of Sˆk (Y ), which we denote by ρY . (b) The map Irrk (G) → E (G), Y → ρY , is a bijection. (c) We have ρY : Y OG = 1 for all Y ∈ Irrk (G).
The model case where this conjecture is known to hold is G = GLn (Fq ); see Dipper and James [64]. (To be more precise, [64] contains all the essential ingredients to conclude that the conjecture, as formulated above, holds in this case.) It is also true for finite unitary groups and other finite groups of Lie type of low rank (where explicit computations are possible); see [109], [117], [152] and the references therein. Note the formal analogy with Definition 3.2.1; in fact, it was the above conjecture which motivated our work on “canonical basic sets” for Iwahori–Hecke algebras. √ 2 f +1 for some f 0; see ExamExample 4.5.3. Let G = Suz(q2 ), where q = 2 ple 4.3.6(c). Let us verify Conjecture 4.5.2 in this case. Using the notation in [104,
242
4 Hecke Algebras and Finite Groups of Lie Type
Table 4.2 Irreducible representations of Suz(q2 ) in characteristic 0
ρ 1G W W
dim ρ 1 √1 q(q2 − 1) 2
aρ 0 (unit representation) 1 (cuspidal unipotent)
√1 q(q2 − 1) 2 q4
1 (cuspidal unipotent)
StG {Xl } q4 + 1 √ 2 {Ym } (q − 1)(q2 − q 2 + 1) √ {Zn } (q2 − 1)(q2 + q 2 + 1)
4 0 0 0
(Steinberg representation) (non-unipotent, various l) (non-unipotent, various m) (non-unipotent, various n)
§4.6], the irreducible representations of G over K are given in Table 4.2. Now let be a prime, = 2. The set E (G) consists of the unipotent representations and some of the representations Xl , Ym , Zn (for certain values of l, m, n). The decomposition numbers of G are easily determined. (Note that the Sylow -subgroups are always cyclic; see Burkhardt [42] for further details). For ρ ∈ E (G) \ {StG }, we have that dOG ([ρ ]) is the class of an irreducible kG-module (which we denote by ρk ). For the Steinberg representation, the following hold: then dOG ([StG ]) is the class of an irreducible kG-module. • If | q2 − 1, √ • If | q2 + q 2 + 1, then dOG ([StG ]) = [1G ] + [Wk ] + [W k ] + [ψ ], where ψ ∈ (and can be lifted to Ym for suitable values of m). Irrk (G) is cuspidal √ • If | q2 − q 2 + 1, then dOG ([StG ]) = [1G ] + [ψ ], where ψ ∈ Irrk (G) is cuspidal of dimension q4 − 1 (and cannot be lifted to characteristic 0). Thus, the decomposition numbers of the unipotent representations are given by ρ 1G W W StG
dim ρ aρ 1 0 √1 q(q2 − 1) 1 2 √1 q(q2 − 1) 2 q4
1 .
dec. numbers . . . 1 . .
1
.
.
1
.
4
a
b
b
1
where a, b ∈ {0, 1} according to the above cases. In particular, we see that Conjecture 4.5.2 holds in this case. We also see the appearance of “new” cuspidal representations over k which cannot be lifted to characteristic 0 (that is, which cannot be obtained by extension of scalars via the natural map O → k from an OG-lattice). This is not an isolated phenomenon; see Examples 4.5.9 and 4.5.18. (One can produce many more such examples using the techniques in [119], [120]). The above conjecture admits, of course, a block-wise version. In this context, it is particularly interesting to consider those O-blocks which contain unipotent representations. Note that the work of Bonnaf´e and Rouquier [28] provides an almost complete reduction to this case, similar to Lusztig’s [220] “Jordan decomposition of characters” in characteristic 0. The union of these blocks is described by the following basic result.
4.5 Examples and Conjectures
243
Theorem 4.5.4 (Brou´e and Michel [37]). The set ρ appears in some RT,θ where E,1 := ρ ∈ IrrK (G) the order of θ is a power of is a union of O-blocks for G. ◦ )F . 4.5.5. Assume that is good for G and does not divide the order of (ZG /ZG Consider the union of O-blocks E,1 described in the above theorem. Let
Unipk (G) := {Y ∈ Irrk (G) | ρ : Y OG = 0 for some ρ ∈ UnipK (G)}. Then the restriction of Theorem 4.5.1 to E,1 shows that |UnipK (G)| = |Unipk (G)| and {dOG ([ρ ]) | ρ ∈ UnipK (G)} is a Z-basis for the subgroup of R0 (kG) generated by the classes of the simple modules in Unipk (G). Furthermore, E,1 ∩ E = UnipK (G). Now Conjecture 4.5.2 reads: (a) Given Y ∈ Unipk (G), let Sˆk (Y ) := {ρ ∈ UnipK (G) | ρ : Y OG = 0}. Then the function Sˆk (Y ) → Z0 , ρ → aρ (defined using the unipotent support), should reach its minimum at exactly one element of Sˆk (Y ), which we denote by ρY . (b) The map Unipk (G) → UnipK (G), Y → ρY , should be a bijection. (c) We should have ρY : Y OG = 1 for all Y ∈ Unipk (G).
Remark 4.5.6. Theorem 4.4.1 shows that the statements (a) and (c) in 4.5.5 hold for all Y ∈ Irrk (G | B) ⊆ Unipk (G). Hence, in combination with Theorem 4.5.4, the properties (a) and (c) in Conjecture 4.5.2 are seen to hold for all Y ∈ Irrk (G | B). Furthermore, the map Y → ρY is injective; see Theorem 4.4.1. In this sense, Conjecture 4.5.2 holds “as far as Irrk (G | B) is concerned”. Remark 4.5.7. We note that, conversely to Remark 4.5.6, the statements in Theorem 4.4.1 are formal consequences of the statements (a), (b) and (c) in 4.5.5. (This immediately follows from (3) in the proof of Theorem 4.4.1.) Hence, by the last part of the proof of Theorem 4.4.1, we also see that the existence of a canonical basic set for Hk is a consequence of (a), (b) and (c) in 4.5.5. Remark 4.5.8. Assume that the statements (a), (b) and (c) in 4.5.5 hold. In particular, this shows that, for any Y,Y ∈ Unipk (G), we have the implication ρY ,Y OG = 0
⇒
Y = Y
or
aρY > aρY .
One may conjecture that, in fact, the following stronger implication holds: (∗)
ρY ,Y OG = 0
⇒
Y = Y
or
OρY OρY .
It follows from the results in [113] that (∗) holds for all Y,Y ∈ Irrk (G | B), assuming that G is of “split” type (that is, γ = id and W = W ). By Example 4.5.10 below, (∗) also holds for all Y,Y ∈ Unipk (G), where G = GLn (Fq ) or GUn (Fq ).
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4 Hecke Algebras and Finite Groups of Lie Type
Table 4.3 Decomposition numbers of UnipK (Sp4 (q)) where 2 = | q + 1
ρ dim ρ aρ decomposition numbers 1G 1 0 1 . . . . . θ10 12 q(q − 1)2 1 . 1 . . . . sgn1 21 q(q2 + 1) 1 1 . 1 . . . 1 . . 1 . . sgn2 21 q(q2 + 1) 1 σ1 21 q(q + 1)2 1 . . . . 1 . q4 4 1 a 1 1 . 1 StG (If =3 and 9 q+1, then a=1; otherwise, a=2)
Example 4.5.9. Let G = Sp4 (Fq ). We have |G| = q4 (q − 1)2 (q + 1)2 (q2 + 1). Let = 2 be a prime dividing the order of G (but not dividing q), and consider the union of O-blocks E,1 . This set contains six unipotent representations, five of which lie in the principal series (and are labelled by the corresponding representations of W of type B2 ) and one of which (denoted θ10 ) is cuspidal; see Srinivasan [284]. If | q2 + 1, then the Sylow -subgroups are cyclic and the corresponding decomposition numbers are known by Fong and Srinivasan [87]. One checks that Conjecture 4.5.2 holds in this case (for the union of blocks E,1 ). If | q − 1, then dOG ([ρ ]) is the class of an irreducible kG-module for any ρ ∈ UnipK (G); see White [293], [294]. Hence, Conjecture 4.5.2 holds in this case as well. Now assume that | q + 1. Then the decomposition numbers are given in Table 4.3; see White [293], [294] (who showed that a 1) and Okuyama and Waki [261, Theorem 2.3] (who determined the precise value of a). Again, we see that Conjecture 4.5.2 holds. By Example 3.2.5, we have Irrk (G | B) = {kG , σ¯ 1 }, where σ¯ 1 ∈ Irrk (G) labels the fifth column of the above matrix. (We have dOG ([σ1 ]) = [σ¯ 1 ].) The dimension of the modular irreducible representation corresponding to the last column of the above matrix is given by (q − 1)2 (q2 + 1) (q − 1)2 (q2 + 12 q + 1)
if a = 2, if a = 1.
One can check that this modular irreducible representation is cuspidal. Example 4.5.10. Let G = GLn (F p ) and F : G → G be such that GF = GLn (Fq ) or GUn (Fq ); see Example 4.3.6. In both cases, there is a natural labelling UnipK (G) = n {ρ λ | λ n}; see [55, §15.4]. For example, ρ (n) is the unit and ρ (1 ) is the Steinberg λ representation. The unipotent support (see 4.3.12) of ρ is given as follows: Oρ λ = Oλ
and
dim Bu = aλ
(u ∈ Oλ ),
where Oλ denotes the conjugacy class of G consisting of unipotent matrices of Jordan type λ and aλ is given by the formula in Example 2.2.13. (This easily follows from the description of the map ρ → Oρ in terms of the Springer correspondence, already mentioned in the remarks following 4.3.12(b).)
4.5 Examples and Conjectures
245
Now let be a prime not dividing q, and let us verify that Conjecture 4.5.2 holds as far as E,1 is concerned. This can be done in a uniform way, as follows. Kawanaka [193] has associated with every unipotent class Oμ (where μ n) a so-called generalised Gelfand–Graev representation (GGGR for short). These GGGRs are obtained by inducing certain representations from unipotent subgroups of G. Denote by Γ μ the GGGR associated with Oμ . Then, by [194, Theorem 2.4.1], we have ∗ Γμ ∼ = ρ μ ⊕ (direct sum of unipotent representations ρ λ where Oλ Oμ ) ⊕ (direct sum of non-unipotent irreducible representations);
here, μ ∗ denotes the transpose partition. Since a GGGR is obtained by induction ∗ μ from a unipotent subgroup, we have Γ μ ∼ = ΨK , where Ψ μ is a finitely generated projective OG-module. Let ϒ μ be the unique indecomposable direct summand of μ Ψ μ such that ρ μ appears with multiplicity 1 in ϒK . Then ϒ μ is a finitely generated projective indecomposable OG-module; let Y μ be the corresponding irreducible kGmodule. By Brauer reciprocity (see 4.1.11), we have ρ λ : Y λ OG = 1 λ
μ
ρ : Y OG = 0
for all λ n, unless λ = μ or Oλ Oμ .
It follows that Unipk (G) = {Y μ | μ n} and that Conjecture 4.5.2 holds as far as E,1 is concerned, as claimed; note also that Oλ Oμ ⇒ aμ < aλ . (This whole argument μ first appeared in [90, §2.5].) Let us also denote by ϒuni the projection of ϒ μ onto the space of unipotent representations; that is, we have μ ϒμ ∼ = ϒuni ⊕ (direct sum of non-unipotent irreducible representations).
While the above description works uniformly for both GLn and GUn , the situation is different for Harish-Chandra series (already over K). Indeed, if GF = GLn (Fq ), then we have IrrK (G | B) = UnipK (G) (see [55, Prop. 15.8]). On the other hand, if GF = GUn (Fq ), then the subset IrrK (G | B) ⊆ UnipK (G) corresponds to the set of all partitions λ n such that, in the Young diagram of λ , the number of boxes which have an odd hook length equals the number of boxes which have an even hook length; see [212, 9.6]. The picture is much more involved as far as modular HarishChandra series and decomposition numbers are concerned; see the references in Example 4.4.5 (for GLn ) and [117], [119], [120] (for GUn ). The references [89], [156] and [262] illustrate some of the difficulties that occur already for GU3 . Example 4.5.11. Let n = 3 and assume that | q + 1; then e = 2. (a) If G = GL3 (Fq ), then the decomposition numbers in the union of blocks E,1 are given by Table 4.4; see James [181, p. 253]. By Example 4.4.5, we have Irrk (G | B) = {Y (3) ,Y (21) }; note also that UnipK (G) = IrrK (G | B) in this case. One can check that Y (111) is non-cuspidal. (b) If G = GU3 (Fq ), then the decomposition numbers in the union of blocks E,1 are given by Table 4.5; see Geck [89] (who showed that a 2 for = 2), Okuyama
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4 Hecke Algebras and Finite Groups of Lie Type
Table 4.4 Decomposition numbers of UnipK (GL3 (q)) where | q + 1
λ dim ρ λ aλ (3) 1 0 (21) q(q + 1) 1 (111) q3 3
dec. num. 1 . . . 1 . 1 . 1
λ dim ϒuni 2 (q + 1)(q − q + 1) q(q + 1) q3
Table 4.5 Decomposition numbers of UnipK (GU3 (q)) where | q + 1
λ dim ρ aρ dec. num. (3) 1 0 1 . . (21) q(q − 1) 1 . 1 . (13 ) q3 3 1 a 1 If = 2 and 4 | q−1, then a = 1; in all other cases, a = 2.
λ dim ϒuni 2 (q + 1)(q − q + 1) aq3 + q2 − q q3
and Waki [262] (who determined the precise value of a for = 2) and Hiss [156] (who dealt with = 2). We have IrrK (G | B) = {ρ (3) , ρ (111) } and Irrk (G | B) = {Y (3) } (see Example 4.3.4). Furthermore, both Y (21) and Y (111) are cuspidal. Here is a further conjecture which would provide a general source for cuspidal representations in positive characteristic.
Conjecture 4.5.12 (Geck [90, 2.6.11], [92, 6.6]). Assume that is good for G and ◦ ) . Let ρ ∈ E (G) be cuspidal. Then d does not divide the order of (ZG /ZG F OG ([ρ ]) is the class of an irreducible kG-module (which is automatically cuspidal).
(This applies, in particular, to all cuspidal ρ ∈ UnipK (G).) By Example 4.5.3, this conjecture holds for G = Suz(q2 ). A more striking example is given by the cuspidal unipotent representation of G = GUn (Fq ), where n = r(r +1)/2 for some r 0; see [119, Theorem 6.10]. (But note that G = GUn (Fq ) can have many more cuspidal representations in characteristic ; see, for example, [120, §4].) The conjecture also holds for classical groups, where = 2 (see [127]) and other groups of Lie type of low rank where explicit computations are possible. Remark 4.5.13. Let Y ∈ Irrk (G). Following Hiss [155], we say that Y is supercuspidal if Y does not occur as a composition factor in RG L (X) for any X ∈ kL-mod where L ∈ LG is such that L = G. Thus, clearly, if Y is supercuspidal, then Y is cuspidal. Hiss [155, Conj. 3.1] conjectures that a supercuspidal Y ∈ Irrk (G) can be lifted to characteristic 0. He also shows (see [155, Prop. 3.3]) that the truth of this conjecture is a consequence of the truth of Conjecture 4.5.12.
4.5 Examples and Conjectures
247
◦) . 4.5.14. Assume that is good for G and does not divide the order of (ZG /ZG F Fix L ∈ LG and assume that there is some cuspidal ψ ∈ UnipK (L) for which Conjecture 4.5.12 holds; that is, there is an OL-lattice X˜ such that X˜K ∼ = ψ and
(∗)
X˜k is an irreducible (and, automatically, cuspidal) kL-module.
Then, by Theorem 4.2.9, we have bijections IrrK (G | (L, X˜K ))
−→
1−1
Irr(H (L, X˜K )),
Irrk (G | (L, X˜k )
−→
1−1
Irr(H (L, X˜k )),
˜ ◦ and H (L, X˜k ) := EndkG RG ˜ ◦ where H (L, X˜K ) := EndKG RG L⊆P (XK ) L⊆P (Xk ) ; here, P ∈ PG is such that P = UP L. Now let ˜ ◦ ˜ := EndOG RG H (L, X) L⊆P (X) . ∼ K ⊗O H (L, X). ˜ Now, by the argument in [102, Clearly, we have H (L, X˜K ) = Lemma 5.22], we have W (L, X˜K ) = W (L, X˜k ). By [102, Prop. 5.21], this implies that we also have ˜ H (L, X˜k ) ∼ = k ⊗O H (L, X) ˜ embeds into Dec(OG) (as in Dipper’s Theorem 4.1.14). and that Dec(H (L, X)) As shown by Lusztig [211, §5] (see also [220, Chap. 8]), the statement of Theorem 4.2.12 can be considerably strengthened in this situation. Namely, we have Ω = {1} and W (L, X˜K ) = W1 is itself a Coxeter group; furthermore, there is a weight function L1 : W1 → Z, where L1 (s) ˜ > 0 for all s˜ ∈ S1 , such that H (L, X˜K ) ∼ = K ⊗A HA (W1 , S1 , L1 )
(via θK : A → K, as defined in 4.3.9)
where HA (W1 , S1 , L1 ) is the associated generic Iwahori–Hecke algebra over A = R[v, v−1 ]. In fact, by the argument in [102, Cor. 5.26], we actually have ˜ ∼ H (L, X) = O ⊗A HA (W1 , S1 , L1 )
(via θ : A → O, as defined in 4.3.9).
The various possibilities for W,W1 , L1 have been determined by Lusztig and are listed explicitly in [213, Table II, p. 35]; see also Carter [45, p. 464]. Now assume that Hk admits a canonical basic set. (Recall that this is the case when (♠) and (♣) in 2.5.3 are satisfied for HA (W1 , S1 , L1 ).) Then we can argue as in the proof of Theorem 4.4.1 to obtain the following conclusions. • Given Y ∈ Irrk (G | (L, X˜k )), let Sk (Y ) := {ρ ∈ IrrK (G | (L, X˜K )) | ρ : Y OG = 0}. Then the function Sk (Y ) → Z0 , ρ → aρ (defined using the unipotent support), reaches its minimum at exactly one element of Sk (Y ), which we denote by ρY . • The map Irrk (G | (L, X˜k )) → IrrK (G | (L, X˜K )), Y → ρY , is injective. • We have ρY : Y OG = 1 for all Y ∈ Irrk (G | (L, X˜k )).
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4 Hecke Algebras and Finite Groups of Lie Type
Thus, we obtain a natural injection Irrk (G | (L, X˜k )) → IrrK (G | (L, X˜K )). Since we also have bijections IrrK (G | (L, X˜K ))
1−1
←→
Irr(H (L, X˜K ))
1−1
←→
IrrK (W1 ),
we obtain a natural injection Irrk (G | (L, X˜k )) → IrrK (W1 ). By an argument similar to that in Theorem 4.4.8, the image of this injection will only depend on e, if is sufficiently large. We omit further details. Example 4.5.15. Let G = GUN (Fq ) and assume that N = 2n + 12 c(c + 1), where c 0 and n 1. Then we have a Levi subgroup L ∈ LG such that L∼ )n × GU 1 c(c+1) (Fq ). = (F× q2 2
By Lusztig [212, §9], UnipK (L) contains a unique cuspidal representation, denoted by ψ . The inertia group W (L, ψ ) = W1 is of type Bn , with weight function given by Bn
2c+1 t
4
2 t
2 t
· · ·
2 t
(see Lusztig [213, Table II, p. 35]); the situation in Example 4.4.16 corresponds to the special case where c ∈ {0, 1}. It is known that Conjecture 4.5.12 holds for ψ ; see [119, Theorem 6.10]. Hence, we are in the setting of 4.5.14 but note that it is not (yet) known if (♠) and (♣) hold for all values of r. However, by Example 4.5.10 we know that the properties (a), (b) and (c) in 4.5.5 hold. Hence, by Remark 4.5.7, we can conclude that a canonical basic set for Hk exists. We can now distinguish cases as in Example 4.4.16. In order to obtain a canonical basic set in the “non-elementary” case where e = 2e and e 3 is odd, we argue as follow. We have seen in Remark 4.3.11 that θk lies above the Φe -modular specialisation θe : A → K. Let H(e) be the corresponding specialised algebra. In Theorem 6.7.11 (Chapter 6) we shall establish the existence of a canonical basic set B(e) ⊆ Λ for H(e) and obtain an explicit combinatorial description of B(e) (using the theory of canonical bases for quantised enveloping algebras). Furthermore, Theorem 4.6.13 will show that we have | Irr(Hk )| = | Irr(H(e) )|. So Lemma 3.6.8 implies that the subset B(e) ⊆ Λ also is a canonical basic set for Hk . Thus, we obtain an explicit description of the injection Irrk (G | (L, X˜k )) → IrrK (W1 ). Example 4.5.16. Let G be a finite symplectic or orthogonal group over Fq where W is of type BN or DN . By Lusztig [212, 8.11], for suitable values of N and n there exist an L ∈ LG and a cuspidal ψ ∈ UnipK (L) such that the corresponding inertia group W (L, ψ ) = W1 is of type Bn , with weight function given by b 4 1 1 1 Bn t t t t · · · where b is determined by N and n. For example, if W is of type BN , where N = n + c2 + c for some c 0, then there is a pair (L, ψ ) as above such that b = 2c + 1; see Lusztig [213, Table II, p. 35]. Similarly, if W is of type DN , where N = n + c2 for some c 1, then there is a pair (L, ψ ) as above such that b = 2c.
4.5 Examples and Conjectures
249
However, in these cases, it is not known if Conjecture 4.5.12 holds for ψ ; furthermore, it is not known if (♠) and (♣) hold for W1 , L1 . By Theorem 6.7.9, we know at least that a canonical basic set exists for the corresponding algebra H(e) (see Remark 4.3.11). By Theorem 4.6.13, we also know that | Irr(Hk )| = | Irr(H(e) )|. Example 4.5.17. Let G = E8 (Fq ) and L ∈ LG be of type D4 . By Lusztig [220, p. 359], UnipK (L) contains a unique cuspidal representation, denoted by ψ . The inertia group W (L, ψ ) = W1 is of type F4 , with weight function given by F4
1 t
1 t
4
4 t
4 t
(see Lusztig [213, Table II, p. 35]). It is known that Conjecture 4.5.12 holds for ψ ; see [130, Prop. 5.1] (for q odd) and [96, Beisp. 4.5.4] (for q a power of 2). Hence, we are in the setting of 4.5.14; the properties (♠) and (♣) hold by 2.4.1(c). Consequently, we obtain a natural injection Irrk (G | (L, X˜k )) → IrrK (W1 ). The image of this injection can be extracted from the matrices in Table 7.9 (p. 377). Note that, by Bremke [30], these matrices coincide with the decomposition matrices of Hk . If L ∈ LG has type E6 or E7 , then UnipK (L) also contains cuspidal representations, where the inertia groups are of type G2 (with weights 1, 9) and A1 (with weight 15) respectively. It is known that Conjecture 4.5.12 holds for the cuspidal unipotent representations in type E6 , at least if q is not a power of 2 or 3; see [117, Theorem 7.4]. The corresponding decomposition matrices and canonical basic sets are contained in Table 7.5 (p. 373). The analogous question for type E7 is open. Our final example shows that some things can go wrong when is a bad prime. Table 4.6 Decomposition matrix of the union of blocks E2,1 of Ree(q2 )
ρ 1G ξ6 ξ5 ξ7 ξ8 ξ9 ξ10 StG ρθ ,1 ρθ ,2
dim ρ 1 √ 1 2 − q 3 + 1) √ q(q − 1)(q 2 3 √ 1 √ q(q − 1)(q2 + q 3 + 1) 2 3 √ 1 √ q(q − 1)(q2 + q 3 + 1) 2 3 √ 1 √ q(q − 1)(q2 − q 3 + 1) 2 3 √1 q(q2 − 1)(q2 + 1) 3 √1 q(q2 − 1)(q2 + 1) 3 q6
q4 − q2 + 1 q2 (q4 − q2 + 1)
aρ 0 1
1 .
decomposition numbers . . . . . . 1 . . . . .
1
.
1
1
.
.
.
.
1
.
.
1
1
.
.
.
1
.
.
.
1
.
.
.
1
.
.
.
.
1
.
.
1
.
.
.
.
.
1
.
6 0 2
1 1 1
1 . 1
2 1 1
1 . 1
. . .
. . .
1 . 1
√ 2 f +1 Example 4.5.18. Let G = Ree(q2 ) (type 2G2 ), where q = 3 for some f 0; see Example 4.3.4. We have |G| = q6 (q2 − 1)(q6 + 1). Let = 3 be a prime which divides the order of G. If = 2, then the Sylow -subgroups of G are cyclic and
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4 Hecke Algebras and Finite Groups of Lie Type
the corresponding decomposition numbers are determined by Hiss [153, §D.2]. One checks that Conjectures 4.5.2 and 4.5.12 hold in this case. Now let = 2 (which is a “bad” prime for G); in this case, the Sylow 2-subgroups are elementary abelian of order 8. The union of blocks E2,1 contains eight unipotent representations and two non-unipotent principal series representations which are constituents of RT,θ , where T ⊆ G is an F-stable maximal torus contained in an F-stable Borel subgroup and θ ∈ Irr(TF ) is unique of order 2; the two constituents are denoted by ρθ ,1 and ρθ ,2 . Following previous work of Fong, the decomposition matrix of E2,1 was completely determined by Landrock and Michler [207]; see Table 4.6. Note that the six unipotent representations ξ5 , ξ6 , . . . , ξ10 are all cuspidal! We notice that: • Theorem 4.5.1 and, consequently, Conjecture 4.5.2 fail: the classes {dOG ([ρ ]) | ρ ∈ UnipK (G)} are not linearly independent. (A similar thing also happens, for example, for groups of type D4 when = 2; see [130, Prop. 5.1]. So the failure is not related to the fact that Ree(q2 ) is a “very twisted” group.) • The statement of Conjecture 4.5.12 fails for the cuspidal representations ξ5 and ξ7 . (No other example for a failure of that statement is known.) So, it seems that the assumption that should be a good prime is really necessary. We also see from the decomposition matrix that the modular irreducible representation labelling the last column cannot be lifted to characteristic 0. Since this representation is not in the principal series, it must be cuspidal.
4.6 A First Approach to Type Bn In the previous sections we have seen that Iwahori–Hecke algebras of type Bn with various choices of unequal parameters occur in the theory of modular HarishChandra series of finite groups of Lie type. The study of the representation theory of these algebras has turned out to be an extremely difficult problem. However, certain cases can be dealt with by elementary methods, and these will be discussed in this section. This will also serve as a motivation for the following chapters. Let W be of type Bn with weight function L : W → Γ given by Bn
b t
4
a t
a t
· · ·
a t
where a, b ∈ Γ0 .
We label the generators of W by s0 , s1 , . . . , sn−1 as in Table 1.1 (p. 2). Recall that IrrK (W ) = {E (λ ,μ ) | (λ , μ ) ∈ Λ }, where Λ is the set of all pairs of partitions (λ , μ ) such that |λ | + |μ | = n. Let H be the generic Iwahori–Hecke algebra of W over A = R[Γ ], where R ⊆ C is L-good. We have the quadratic relations Ts20 = T1 + (ε b − ε −b )Ts0
and
Ts2i = T1 + (ε a − ε −a )Tsi
(i = 1, . . . , n − 1).
4.6 A First Approach to Type Bn
251
Let θ : A → k be a specialisation into a field k such that k is the field of fractions of θ (A). The basic references for the study of Hk are Dipper and James [66] and Dipper, James and Murphy [68], [69]. We have nothing original to add to the excellent exposition in these articles. So we will not reproduce them here but merely state the results that we need and then explain in detail the implications on canonical basic sets. Note that these implications require some work since Dipper, James and Murphy did not take into account any monomial ordering4 on Γ . Already the article [66] shows that, independently of any monomial ordering on Γ , the representation theory of Hk essentially depends on two parameters. Let U := ε 2b and u := ε 2a . Then, as in type A, the first parameter is given by e˜ := min{i 2 | 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0}. (We let e˜ = ∞ if 1 + θ (u) + θ (u)2 + · · · + θ (u)i−1 = 0 for all i 2; we use the symbol e˜ because e is already used as in Remark 4.3.11.) Furthermore, the difficulty in studying representations of Hk will depend very much on whether f n (θ ) :=
∏
θ (U + ui )
−(n−1)in−1
equals zero or not. We shall see that the results of [66] actually yield canonically basic sets for any monomial ordering on Γ provided that θ (u) = 1 or fn (θ ) = 0. (The cases where θ (u) = 1 and fn (θ ) = 0 require completely new methods; see Chapters 5 and 6.) In order to explain this, we first need to prepare some notation. 4.6.1. The elements s1 , s2 , . . . , sn−1 generate a parabolic subgroup of W which we can (and will) identify with the symmetric group Sn . Let r ∈ {0, 1, . . . , n} and consider the parabolic subgroup (a)
r
W = s1 , . . . , sr−1 , sr+1 , . . . , sn−1 ∼ = Sr × Sn−r ;
here, at the extreme cases, we have 0W = nW = s1 , s2 , . . . , sn−1 = Sn . Let rΛ be the set of all pairs of partitions (λ , μ ) ∈ Λ such that |λ | = r (and, hence, |μ | = n − r). Given (λ , μ ) ∈ rΛ , we have corresponding irreducible representations E λ ∈ IrrK (Sr ) and E μ ∈ IrrK (Sn−r ). Taking the outer tensor product we obtain an irreducible representation of Sr × Sn−r which, via (a), defines an irreducible representation of rW . Thus, we can write (b)
IrrK (rW ) = {E λ E μ | (λ , μ ) ∈ rΛ }.
Let r H ⊆ H be the parabolic subalgebra corresponding to rW . Then r H can be identified with the generic Iwahori–Hecke algebra associated with Sr × Sn−r and the weight function r L which assigns a to each generator of rW . Let r Dθ be the decomposition matrix of r H with respect to the specialisation θ : A → k. Since r H is the tensor product of two algebras, a completely general argument shows that Dθ 4
In retrospective, one might say that their results (especially those in [68]) are most directly related to the asymptotic case where b > (n − 1)a > 0; see also the remarks in 2.8.19.
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4 Hecke Algebras and Finite Groups of Lie Type
is the Kronecker product of the decomposition matrices of the two algebras which are tensored together. First of all, by Theorem 3.5.14, this implies that Dθ has rows labelled by rΛ and columns labelled by the set rΛ θ consisting of all (λ , μ ) ∈ rΛ such that both λ and μ are e-regular. ˜ Let us write r Dθ = rd (λ ,μ ),(λ ,μ ) (λ ,μ )∈rΛ , (λ ,μ )∈rΛ . θ
Then, furthermore, the relations in Example 3.2.9 imply that, given (λ , μ ) ∈ rΛ and (λ , μ ) ∈ rΛ θ , we have rd (λ ,μ ),(λ , μ ) = 1, (c) rd unless λ λ and μ μ . (λ ,μ ),(λ ,μ ) = 0 With this notation, we can now state the following result. Theorem 4.6.2 (Dipper and James [66, 5.8]). Assume that fn (θ ) = 0 and let Dθ be the decomposition matrix of H with respect to θ : A → k. Then Dθ is a block diagonal matrix with diagonal blocks given by r Dθ for 0 r n. In particular, this shows that the irreducible representations of HK which are labelled by all pairs of partitions in a set rΛ (where r is fixed) form a union of θ blocks. Dipper and James [66, 4.7] actually show that Hk is Morita equivalent to the direct sum of algebras 0rn r Hk . Using this theorem, we can now establish the following result, without reference to the properties (♠) and (♣) in 2.5.3. Proposition 4.6.3 (Cf. [101, 6.8], [125, 3.1]). Assume that fn (θ ) = 0. Then, independently of the monomial ordering on Γ , we have that ˜ Bθ := {(λ , μ ) ∈ Λ | both λ and μ are e-regular} is a canonical basic set for Hk . Proof. Let us first deal with the case where a = 0. Then θ (u) = 1 and so either e˜ = ∞ or e˜ = char(k) = > 0. Since R is L-good, we must have > n in the second case; see Table 1.4 (p. 33). Hence, in both cases, we have e˜ > n. Since also fn (θ ) = 0, it follows that Hk is semisimple; see Example 3.1.20. Hence, by 3.1.18, Dθ is the identity matrix and so Bθ = Λ is a canonical basic set. Let us now assume that a > 0. By Theorem 4.6.2 and the discussion in 4.6.1, the matrix Dθ has columns labelled by Bθ and, writing Dθ = d(λ ,μ ),(λ ,μ ) (λ ,μ )∈Λ , (λ ,μ )∈B , θ
we have for any (λ , μ ) ∈ Λ and (λ , μ ) ∈ Bθ d(λ ,μ ),(λ ,μ ) = 1, d(λ ,μ ),(λ ,μ ) = 0 unless |λ | = |λ |, |μ | = | μ |, λ λ , μ μ .
4.6 A First Approach to Type Bn
253
Hence, in order to prove that Bθ is a canonical basic set, it is enough to show that the following implication holds, for any (λ , μ ) ∈ Λ and (λ , μ ) ∈ Bθ : |λ | = |λ |, | μ | = | μ |, (†) ⇒ (λ , μ ) = (λ , μ ) or a(λ ,μ ) > a(λ ,μ ) . λ λ , μ μ Thus, we are reduced to a purely combinatorial statement. We claim that, in fact, (†) holds for all pairs of partitions (λ , μ ), (λ , μ ) ∈ Λ . To prove this, we use the formulae for the a-invariants in Example 1.3.9. Assume first that b ra for all r ∈ Z1 . Then we have a(λ ,μ ) = n(λ ) + 2n(μ ) − n(μ ∗ ) a + | μ | b. Now, if λ λ and μ μ , then n(λ ) n(λ ) (with equality only if λ = λ ) and n(μ ) n(μ ) (with equality only if μ = μ ); see Example 2.2.13. Since we also have μ μ if and only if μ ∗ μ ∗ (see [236, (I.1.11)]), the above formula immediately shows that a(λ ,μ ) a(λ ,μ ) , with equality only if (λ , μ ) = (λ , μ ). Now assume that there exists some r ∈ Z0 such that ra b < (r + 1)a, and set b = b − ra. As in Case 3 of Example 1.3.9, we fix a large integer N 0 and N (λ , μ ) and Z N (λ , μ ). Writing associate with (λ , μ ) and (λ , μ ) the multisets Za,b a,b N Za,b (λ , μ ) = {z1 , z2 , . . . , z2N+r },
where
z1 z2 . . . z2N+r ,
N Za,b (λ , μ ) = {z1 , z2 , . . . , z2N+r },
where
z1 z2 . . . z2N+r ,
we have a(λ ,μ ) − a(λ ,μ ) =
∑
(i − 1)(zi − zi ).
1i2N+r
Now note that, in order to prove (†), it is sufficient to do this in the case where λ = λ or μ = μ . If λ λ and μ = μ , then one can further reduce to the case where λ is obtained from λ by increasing one part by 1 and by decreasing another part by 1; see [236, (I.1.16)]. A similar statement holds if λ = λ and μ μ . Thus, N (λ , μ ) is given by it will be sufficient to consider the case where the multiset Za,b N Za,b (λ , μ ) = {z1 , . . . , zi−1 , zi + a, zi+1 , . . . , z j−1 , z j − a, z j+1 , . . . , z2N+r },
where 1 i < j 2N + r; note, however, that this is just an equality as multisets. In order to be able to compare the a-invariants, we need to arrange the entries (written as above) in decreasing order. So we distinguish the following four cases: (I) zi−1 zi + a (if i > 1) and z j − a z j+1 (if j < 2N + r). (II) i > 1 and zi−1 < zi + a and z j − a z j+1 (if j < 2N + r). (III) zi−1 zi + a (if i > 1) j < 2N + r and z j − a < z j+1 . (IV) i > 1 and j < 2N + r and zi−1 < zi + a and z j − a < z j+1 . In each of these cases, we must show that a(λ ,μ ) > a(λ ,μ ) . Now, if (I) holds, then the N (λ , μ ) (written as above in terms of the entries of Z N (λ , μ )) already entries of Za,b a,b
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4 Hecke Algebras and Finite Groups of Lie Type
are arranged in decreasing order; that is, we have zi = zi + a, zj = z j − a and zk = zk for k = i, j. This yields that a(λ ,μ ) − a(λ ,μ ) = (i − 1)(zi − zi ) + ( j − 1)(z j − zj ) = ( j − i)a > 0, as required. If (II) holds, then all entries are arranged in decreasing order, except for the two entries zi−1 , zi + a. Now recall from Example 1.3.9 that the entries {zk } N (λ , μ ) have the form of z = n a + δ b , where n ∈ Z of Za,b 1 and δk ∈ {0, 1}. k k k k Furthermore, if b > 0, then all entries are distinct; if b = 0, then it may happen that two entries are equal but no entry is repeated more than twice. These conditions imply that zi−2 zi + a (if i > 2). Hence, all we need to do is to swap the entries zi−1 and zi + a; that is, we have zi−1 = zi + a, zi = zi−1 , zj = z j − a and zk = zk for all k = i − 1, i, j. This yields that a(λ ,μ ) − a(λ ,μ ) = (i − 2)(zi−1 − zi−1 ) + (i − 1)(zi − zi ) + ( j − 1)(z j − zj ) = (i − 2)(zi−1 − zi − a) + (i − 1)(zi − zi−1 ) + ( j − 1)a = (zi + a) − zi−1 + ( j − i)a > 0, as required. Next, assume that (III) holds. Arguing as above, we see that all we need to do is to swap the entries z j − a and z j+1 ; that is, we have zi = zi + a, zj = z j+1 , zj+1 = z j − a and zk = zk for k = i, j, j + 1. This yields that a(λ ,μ ) − a(λ ,μ ) = (i − 1)(zi − zi ) + ( j − 1)(z j − zj ) + j(z j+1 − zj+1 ) = −(i − 1)a + ( j − 1)(z j − z j+1 ) + j(z j+1 − z j + a) = z j+1 − (z j − a) + ( j − i)a > 0, as required. Finally, assume that (IV) holds. In this case, we need to swap the entries zi−1 , zi + a and the entries z j − a, z j+1 ; that is, we have zi−1 = zi + a, zi = zi−1 , zj = z j+1 , zj+1 = z j − a and zk = zk for k = i − 1, i, j, j + 1. This yields that a(λ ,μ ) − a(λ ,μ ) = (i − 2)(zi−1 − zi − a) + (i − 1)(zi − zi−1 ) + ( j − 1)(z j − z j+1 ) + j(z j+1 − z j + a) = (zi + a) − zi−1 + z j+1 − (z j − a) + ( j − i)a > 0, as required. Thus, (†) is proved and so Bθ is a canonical basic set, as claimed.
Example 4.6.4. Assume that the following three conditions hold: (a) char(k) = 2, (b) e˜ is odd, (c) θ (U) = θ (u) j for some j ∈ Z. We claim that fn (θ ) = 0. Indeed, if fn (θ ) = 0, then θ (U) = −θ (u)i for some i and so θ (u) j−i = −1, using (c). By (a), this would imply that θ (u) = 1 is a root of unity of even order and so e˜ would have to be even, contradicting (b). Thus, fn (θ ) = 0 as
4.6 A First Approach to Type Bn
255
claimed. Now Proposition 4.6.3 applies and so there exists a canonical basic set for ˜ Hk , given by the set of all (λ , μ ) ∈ Λ such that both λ and μ are e-regular. 4.6.5. Assume that θ (u) = 1 and fn (θ ) = 0. Then we must have θ (U) = −1 and so, in Hk , we have the quadratic relations (Ts0 + T1 )2 = 0 and Ts2i = T1
(i = 1, . . . , n − 1).
Let Ω ⊆ W be the parabolic subgroup generated by s1 , . . . , sn−1 ; we have Ω ∼ = Sn . The corresponding parabolic subalgebra of Hk is nothing but the group algebra of Ω and will be denoted by kΩ ⊆ Hk . Now consider an auxiliary weight function L : W → Γ such that L (s0 ) = b and L (si ) = 0 for 1 i n − 1. Let H be the corresponding generic Iwahori–Hecke algebra over A. Thus, in H , we have the quadratic relations Ts20 = T1 + (ε b − ε −b )Ts0 and Ts2i = T1 for 1 i n − 1, as in Example 2.4.4(b). We define t0 := s0 and ti = siti−1 si for 1 i n − 1. Then the submodule H1 = Tt0 , Tt1 , . . . , Ttn−1 A ⊆ H is the generic Iwahori–Hecke algebra of a Coxeter group of type A1 × · · · × A1 (n factors). Furthermore, as in Remark 2.4.6, the submodules {Hω := H1 Tω | ω ∈ Ω }
form an Ω -graded Clifford system in H .
Now note that we can canonically identify the specialised algebras Hk and Hk ; furthermore, the Clifford system in H certainly gives rise to an Ω -graded Clifford system in Hk = Hk with respect to the subalgebra Hk,1 = Tt0 , . . . , Ttn−1 k ⊆ Hk . Hence, as already noted in 3.2.13(b), the restriction of any irreducible representation of Hk to Hk,1 is a direct sum of irreducible representations. But Hk,1 is commutative, hence every irreducible representation of Hk,1 is one-dimensional. Since (Ts0 + T1 )2 = 0, it follows that Ts0 = Tt0 ∈ Hk,1 acts as minus the identity in every irreducible representation of Hk . We now obtain the following proposition. Proposition 4.6.6 (Dipper and James [66, 5.4]). In the setting of 4.6.5, every irreducible representation of kΩ ∼ = kSn has a unique extension to an irreducible representation of Hk ; furthermore, all irreducible representations of Hk arise in this way. Hence, by Theorem 3.5.14, Irr(Hk ) is naturally parametrised by the e-regular ˜ partitions of n, where e˜ = ∞ (if char(k) = 0) or e˜ = (if char(k) = > 0). Proof. By 4.6.5, Ts0 acts as minus the identity in every irreducible representation of Hk . Since Hk is generated by Ts0 together with Ts1 , . . . , Tsn−1 , it is clear that the restriction of an irreducible representation of Hk to kΩ remains irreducible. Conversely, if ρ : kΩ → Md (k) is an irreducible representation, then we obtain a representation ρˆ : Hk → Md (k) by setting ρˆ (Ts0 ) := −Id (where Id denotes the identity matrix of size d) and ρˆ (Tsi ) := ρ (Tsi ) for 1 i n − 1. (One immediately checks that the defining relations for Hk are satisfied.) Clearly, ρˆ is irreducible.
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4 Hecke Algebras and Finite Groups of Lie Type
Furthermore, this is the only possible way to extend ρ from kΩ to Hk , since Ts0 has to act as minus the identity.
Example 4.6.7. In the setting of 4.6.5, assume that a = 0; that is, L equals the auxiliary weight function L and so H = H. As already noted, H1 is the generic Iwahori–Hecke algebra of type A1 × · · · × A1 (n factors). Let ind1 : H1 → A be the one-dimensional representation such that ind1 (Tti ) = ε b for 0 i n − 1. Specialising ε b to −1 ∈ k, we obtain the representation indk,1 : Hk,1 → k. By 4.6.5, we have Irr(Hk,1 ) = {indk,1 }. The a-invariant of ind1 is 0; hence, {ind1 } is the canonical basic set for Hk,1 . Now the standard construction of IrrK (W ) in [132, 5.5.4] (using (λ , μ )
Clifford theory) shows that, for any (λ , μ ) ∈ Λ , the restriction of Eε to HK,1 contains indK,1 if and only if μ = ∅. Hence, Theorem 3.2.14 implies that Bθ = {(λ , ∅) | λ n} ⊆ Λ is a canonical basic set for Hk . Note that (♠) and (♣) hold for H and H1 by 2.4.8 and 2.5.3. 4.6.8. In the setting of 4.6.5, assume now that a > 0. In order to describe a canonical basic set in this case as well, we need some further preparations. We consider the induction of representations from Ω ∼ = Sn to W . Recall that IrrK (Ω ) = {E ν | ν n}. ν By general results on Lusztig’s a-invariants, an induced representation IndW Ω (E ) is ( λ , μ ) a direct sum of various E ∈ IrrK (W ) where a(λ ,μ ) aν ; furthermore, there will be equality for at least one pair (λ , μ ). (These results can be found in [231, Chap. 20] or [103, §3].) Now, by Example 1.3.8, we have aν = n(ν )a and fν = 1. Also note that b ∈ Za. (Indeed, if we had b = ra for some r ∈ Z, then = −1 = θ (U) = θ (u)r = 1 and so char(k) = 2, contradicting the assumption that R is L-good.) So we are in Case 3 of Example 1.3.9; in particular, f(λ ,μ ) = 1 for all (λ , μ ) ∈ Λ . In this sutuation, there is actually a unique J(ν ) ∈ Λ such that aJ(ν ) = aν = n(ν )a and ν ∼ J(ν ) ⊕ IndW Ω (E ) = E
direct sum of various E (λ ,μ ) ∈ IrrK (W ) where (λ , μ ) ∈ Λ and a(λ ,μ ) > aJ(ν ) = aν ;
see [231, 22.25] or [103, 3.8]. There is a purely combinatorial description of the map ν → J(ν ). To explain this, write b = ar + b , where r 0 and 0 < b < a. As in Example 1.3.9, choose a large integer N and let Z 0 be the multiset whose entries are 0, a, 2a, . . . , (N − 1)a, b , a + b , 2a + b , . . . , (N + r − 1)a + b . Let z01 , z02 , . . . , z02N+r be the entries of Z 0 arranged in decreasing order. Now write ν = (ν1 ν2 . . . νt 0) (where t n) and consider the multiset 0 0 , zt+2 , . . . , z02N+r }. Z := {z01 + ν1 a, z02 + ν2 a, . . . , zt0 + νt a, zt+1
Then, with the notation of Example 1.3.9 (Case 3), there is a unique pair of partitions N (λ , μ ). We leave it as an exercise for the reader to check (λ , μ ) ∈ Λ such that Z = Za,b that, indeed, J(ν ) = (λ , μ ). (See [231, 22.17] and [125, §3] for further details.)
4.6 A First Approach to Type Bn
257
Example 4.6.9. Assume that b > (n − 1)a > 0. Then r + 1 n and we see that the n largest entries of Z 0 are z01 = (N + r − 1)a + b , z02 = (N + r − 2)a + b , . . . , z0n = (N + r − n)a + b . N (λ , μ ) immediately shows that Hence, the definition of Za,b
J(ν ) = (ν , ∅)
for all ν n.
So in this case, the statement of Proposition 4.6.10 below agrees with [68, 7.3]. Note that this is not necessarily true for other values of a, b. For example, assume that a = 2 and b = 1 (hence, r = 0 and b = 1). Then for n = 4 we obtain J(4) = ((4) , ∅), J(31) = ((3) , (1)), J(22) = ((2) , (2)), J(211) = ((21) , (1)), J(1111) = ((11) , (11)). Proposition 4.6.10 (Cf. [125, 3.4]). In the setting of 4.6.5, assume that a > 0. Then Bθ := {J(ν ) | ν n is e-regular} ˜ ⊆Λ is a canonical basic set for Hk , where the map ν → J(ν ) is described in 4.6.8. Recall that, in this case, we have e˜ = ∞ (if char(k) = 0) or e˜ = (if char(k) = > 0). Proof. Using the notation in 4.6.1, let n H ⊆ H be the parabolic subalgebra generated by Ts1 , . . . , Tsn−1 . This is the generic Iwahori–Hecke algebra associated with Sn and the weight function which assigns a to each generator of Sn . We write Irr(n HK ) = {Eελ | λ n}. As already noted, the specialised algebra n Hk is isomorphic to kSn . We have a corresponding decomposition matrix (dλ ,ν ), with rows labelled by all the partitions λ n and columns labelled by the e-regular ˜ partitions ν n; see Theorem 3.5.14. Given any e-regular ˜ partition ν n, we define an n HK -module Qν such that [Qν ] =
∑ dλ ,ν [Eελ ]
(in R0 (n HK )).
λ n
Now consider the induction of representations from n HK to HK . We simply denote this operation by Ind and define for any e-regular ˜ partition ν n. Pν := Ind Qν (λ ,μ )
Let dν denote the vector which contains the multiplicities of the various Eε Irr(HK ) as constituents of Pν . We claim that
∈
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4 Hecke Algebras and Finite Groups of Lie Type
(∗)
dν is a linear combination of the columns of the decomposition matrix Dθ of H where the coefficients are non-negative integers.
Indeed, as in the proof of Proposition 3.4.10, we can assume without loss of generality that A is the ring of Laurent polynomials in finitely many indeterminates over a Dedekind domain. Then (∗) follows by an argument completely analogous to that in the proof of Theorem 3.4.4, using Brauer reciprocity. Let us now consider in more detail the vector dν . By Example 3.2.9, we have dν ,ν = 1 and dν ,ν = 0 unless ν = ν or aν > aν (where ν n is an arbitrary partition). This yields that [Qν ] = [Eεν ] +
∑
ν n : aν >aν
dν ,ν [Eεν ].
Now apply Ind and take into account the results in 4.6.8; also recall that Irr(HK ) = (λ ,μ ) {Eε | (λ , μ ) ∈ Λ }. Using the general compatibility of induction with specialisations (see [132, §9.1]), we deduce that [Pν ] = [Ind(Qν )] = [Ind(Eεν )] + J(ν )
= [Eε
∑
ν n : aν >aν (λ , μ )
] + sum of various [Eε
dν ,ν [Ind(Eεν )]
] where a(λ ,μ ) > aJ(ν ) = aν .
˜ partitions of n (see Proposition 4.6.6), Since | Irr(Hk )| equals the number of e-regular we can now argue as in the proof of Proposition 3.4.5 to conclude that Bθ is a canonical basic set for Hk .
With no assumptions on θ (u) or fn (θ ), we have the following result which gives at least an upper bound on the number of irreducible representations of Hk . Theorem 4.6.11 (Dipper, James and Murphy [68, 6.9]). We have ˜ | Irr(Hk )| |{(λ , μ ) ∈ Λ | both λ and μ are e-regular}|. By Theorem 4.6.2, equality holds if fn (θ ) = 0. In general, the inequality will be strict; see Example 3.2.6 where n = 3, e˜ = 2 and | Irr(Hk )| = 4, but there are six pairs of partitions (λ , μ ) ∈ Λ such that both λ and μ are 2-regular. Thus, if fn (θ ) = 0, then the inequality in the above theorem will be strict in general. As already mentioned, completely new methods will be required to study these questions, and this will be done in Chapters 5 and 6. Here is a brief outlook. 4.6.12. Assume that θ (u) = 1 and fn (θ ) = 0. Thus, there is some ξ ∈ k× such that
θ (u) = ξ = 1
and
θ (U) = −ξ d ,
where d ∈ Z.
Note that then e˜ 2 is the order of ξ ∈ k× . Let us set Tˆs0 := −ε b Ts0
and
Tˆsi = ε a Tsi
for 1 i n − 1.
4.6 A First Approach to Type Bn
259
Then, in Hk , the usual type Bn braid relations hold for Tˆs0 , . . . , Tˆsn−1 , and we have the following quadratic relations (see also Lemma 1.1.12): (Tˆs0 − ξ c1 )(Tˆs0 − ξ c2 ) = 0, where c1 = d and c2 = 0, (Tˆsi − ξ )(Tˆsi + 1) = 0 for 1 i n − 1. Thus, we see that Hk can be identified with an Ariki–Koike algebra as in Section 5.2 (where l = 2 in the notation of 5.2.1). The following two chapters contain a detailed study of the representation theory of these algebras. For example, this will yield a purely combinatorial formula for the cardinality of the set Irr(Hk ); see Theorem 6.2.22. Let us just mention here the following consequence of that formula. Theorem 4.6.13 (Ariki and Mathas [15, Theorem A]). In the setting of 4.6.12, assume that e˜ < ∞. Then | Irr(Hk )| only depends on e˜ and the congruence class of d modulo e. ˜ Let e˜ 2 and d mod e˜ be fixed. Then Ariki and Mathas actually describe a generating function for the cardinalities | Irr(Hk )| (as n varies); see [15, Theorem D]. There are also versions of the above results for the case where e˜ = ∞, but we will not pursue these here. (This is related to the situation considered in Example 3.7.14.)
Chapter 5
Representation Theory of Ariki–Koike Algebras
The aim of this chapter is to introduce a natural generalisation of Iwahori–Hecke algebras of type An−1 and Bn : the Ariki–Koike algebra or Hecke algebra associated with the complex reflection group of type G(l, 1, n). As we will see, many properties of Iwahori–Hecke algebras of finite Coxeter groups still hold for this class of algebras: they are finite dimensional, symmetric and even cellular. As a consequence, we also have a theory of cell modules which gives much information on the representation theory, both in the semisimple and in the modular case. In addition, the symmetric structure will allow us to define an a-invariant associated with each simple module of certain specialisations of Ariki–Koike algebras: the cyclotomic specialisations. One of the main goals of this chapter and the next one will be to study the representation theory of Ariki–Koike algebras with respect to this symmetric structure and to the a-invariants attached to the simple modules. This chapter is organized as follows. We begin by reviewing the definition of the complex reflection group of type G(l, 1, n). The Ariki–Koike algebra is then defined in the second part as a deformation of this group algebra. In the next sections, we study a cellular structure and present a useful result by Dipper and Mathas showing that an Ariki–Koike algebra is always Morita equivalent to a direct sum of a tensor product of Ariki–Koike algebras of special kinds. In these first parts, several proofs will be omitted, as there are already good references on these subjects. Section 5.5 is devoted to the study of certain specialisations of generic Ariki– Koike algebras: the cyclotomic Ariki–Koike algebras. These algebras will allow us to define an a-invariant for each simple module over G(l, 1, n). We can then formally introduce the notion of canonical basic sets in the context of Ariki–Koike algebras. Then, our main aim will be to prove the existence of canonical basic sets as for the symmetric group in Chapter 3. We define an operator of “i-induction” in the same spirit as in the case of Iwahori–Hecke algebras. Using this we construct an abstract “Fock datum” for our algebras which will eventually lead to the existence of canonical basic sets. At the end of this chapter we can state Theorem 5.8.2, which yields an explicit description of these basic sets for Ariki–Koike algebras and, in certain special cases, also for type Bn . However, the completion of the proof of this theorem will rely on the deep methods and results to be discussed in Chapter 6. M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7 5, © Springer-Verlag London Limited 2011
261
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5 Representation Theory of Ariki–Koike Algebras
5.1 The Complex Reflection Group of Type G(l, 1, n) We here define and study the complex reflection group of type G(l, 1, n) and its Hecke algebra in the same spirit as in Section 1.1. Further details and references can be found in [32], [48], [269]. 5.1.1. In the classification of finite groups generated by complex reflections by Shephard and Todd [278], there are a single infinite family of groups G(l, p, n) and exactly 34 further exceptional groups. In this chapter, we are interested in a special case of the “infinite series”: the class of complex reflection groups of type G(l, 1, n). It may be seen as the wreath product (Z/lZ)Sn . There is a presentation of G(l, 1, n) by generators S = {si | i = 0, 1, . . . , n − 1} and relations • s2i = 1 for all i = 1, . . . , n − 1, • sl0 = 1, and the braid relations symbolised by the following braid diagram: s0 t
4
s1 t
s2 t
· · ·
sn−1 t
That is to say: • si s j = s j si , if |i − j| > 1, • si+1 si si+1 = si si+1 si if 1 i n − 2, • s0 s1 s0 s1 = s1 s0 s1 s0 . Note that the three last relations correspond to the type-B braid relations. We also remark that if we set l = 1 then we obtain the Weyl group of type An−1 and G(2, 1, n) is nothing but the Weyl group of type Bn . It is easy to see that the order of G(l, 1, n) is l n n!. In addition, we may realize this group as the subgroup of GLn (C) consisting of matrices such that • the entries are either 0 or lth roots of unity; • there is exactly one non-zero entry in each row and each column. The identification is obtained as follows. Let Ei, j be the n × n matrix with 1 in the (i, j) position and all other entries 0. Then s0 is identified with ζl E1,1 + ∑ j=1 E j j and √ si with Ei,i+1 + Ei+1,i + ∑ j=i,i+1 E j j for i = 1, . . . , n − 1, where ζl := exp(2π −1/l). Finally, note that the subgroup generated by {si | i = 0, 1, . . . , n − 2} is isomorphic to G(l, 1, n − 1) and that the elements of {si | i = 1, . . . , n − 1} generate a natural subgroup Sn isomorphic to a Weyl group of type An−1 . Consequently, for each composition λ of n, we have an associated Young subgroup Sλ of G(l, 1, n). For m ∈ Z>0 , we set Wm := G(l, 1, m). 5.1.2. We introduce generalisations of some combinatorial objects we have a lready met in type An−1 : the multicompositions and the multipartitions. An l-composition (or a multicomposition) of rank n is an l-tuple of compositions λ = (λ 1 , . . . , λ l ) such that |λ 1 | + |λ 2 | + . . . + |λ l | = n. We denote λ |=l n. If all the λ i are partitions, we
5.1 The Complex Reflection Group of Type G(l, 1, n)
263
say that λ = (λ 1 , . . . , λ l ) is an l-partition of n (or a multipartition) and we denote λ l n. For example, the set of 3-partitions of rank 3 is as follows: {(3, ∅),(∅, 3), (2.1, ∅), (∅, 2.1), (1.1.1, ∅), (∅, 1.1.1), (2, 1), (1, 2), (1.1, 1), (1, 1.1)} 5.1.3. In all our discussion, the parameter l in G(l, 1, n) will generally be fixed. So we shall denote the group of type G(l, 1, n) simply by Wn from now on. Let R be a commutative ring with unity such that Z[ζl ] ⊆ R ⊂ C and let K denote the field of fractions of R. Then K is a splitting field for Wn (see [17] and [18] for a study of the splitting fields for all complex reflection groups). We shall now explain, following [239, Section 3] (for example), how the irreducible representations of Wn may be obtained from the ones of the Weyl group of type An−1 . The construction is similar to the case of the Weyl group of type Bn , as discussed in [132, (5.5.4)]. For k = 1, . . . , l, we consider the l linear representations of Wn which are uniquely defined as follows: σk : S → K s0 → ζlk−1 s j → 1 ( j = 1, . . . , n − 1). Note that, for each l-composition λ := (λ 1 , . . . , λ l ) of n, we have an associated Young subgroup Sλ = Sλ 1 × . . . × Sλ l of Wn . Let (n1 , . . . , nl ) be such that ∑1il ni = n and for 1 i l let λ i be a partition of ni so that we have λ = (λ 1 , . . . , λ l ) l n. Let Wnλ be the subgroup Wn1 × . . . ×Wnl of Wn obtained from the natural embedding GLn1 (C) × . . . × GLnl (C) ⊂ GLn (C). Via the natural projection Wni → Sni induced by the semi-direct product, each repi resentation E λ of Sni may be seen as a representation for Wni . Consequently, we can define the following KWn -module: n (E λ (E λ ⊗ σ2 ) . . . (E λ ⊗ σl )), E λ := IndW Wλ 1
2
l
n
where denotes the outer tensor product. It turns out that this procedure gives us a complete set of non-isomorphic simple modules: IrrK (Wn ) = {E λ | λ l n}. (We refer to [236, Chapter 1, Appendix B] for further details.) Example 5.1.4. There are exactly 2l irreducible representations of Wn over K with dimension 1. We have already met the σi . One can define l other representations uniquely as follows: sgni : S → K s0 → ζl i−1 s j → −1 ( j = 1, . . . , n − 1)
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5 Representation Theory of Ariki–Koike Algebras
for i = 1, . . . , l. The simple module E λ affords ∅ if j = i, j • the representation σi if λ = (n) if j = i, ∅ if j = i, • the representation sgni if λ j = (1n ) if j = i. Note that if we set l = 1 (and respectively l = 2), the above remarks are consistent with the fact that the simple modules of the Weyl group of type An−1 (and respectively of type Bn ) are parametrised by the set of partitions (and respectively of bipartitions) of rank n. 5.1.5. We define some additional combinatorial notions. Let λ = (λ 1 , . . . , λ l ) be an l-composition of n. The Young tableau [λ ] of λ is the subset [λ ] := {(a, b, c) | a 1, c ∈ {1, . . . , l} and 1 b λac } ⊆ Z>0 × Z>0 × {1, . . . , l}. The elements of [λ ] are called the nodes of λ . The set of border nodes of λ is defined by Nλ := {(a, λac , c), | a 1, c ∈ {1, . . . , l} and λac > 0} ⊆ [λ ]. As in the case of partitions, it is convenient to visualize [λ ] as an l-tuple of arrays of boxes in the plane, as in the following example. Assume now that λ is an l-partition. We say that γ ∈ [λ ] is a removable node if [λ ] = [μ] ∪ {γ } for some l-partition μ of rank n + 1. We denote γ = [λ ]/[μ]. In this case, γ is called an addable node of μ. The set of removable nodes of λ is denoted by R(λ ) and the set of addable nodes of λ by A (λ ). Example 5.1.6. Let λ = (2.1.1, 3.3, 1) 3 11. The Young tableau of λ is the tableau ⎛ ⎞ ⎝
,
,
⎠
The set of border nodes is here given by Nλ = {(1, 2, 1), (2, 1, 1), (3, 1, 1), (1, 3, 2), (2, 3, 2), (1, 1, 3)}. Furthermore, we obtain R(λ ) = {(1, 2, 1), (3, 1, 1), (2, 3, 2), (1, 1, 3)}, A (λ ) = {(1, 3, 1), (2, 2, 1), (4, 1, 1), (1, 4, 2), (3, 1, 2), (1, 2, 3), (2, 1, 3)}. 5.1.7. As Wn−1 can be seen as a subgroup of Wn , we can define functors of induction Wn n IndW Wn−1 and restriction ResWn−1 between the category of finitely generated KWn−1 modules and the category of finitely generated KWn -modules. The proposition below describes the branching rule for complex reflection groups of type G(l, 1, n). The proof is inspired from the paper [239, Lemma 3.3].
5.1 The Complex Reflection Group of Type G(l, 1, n)
265
Proposition 5.1.8. For all λ l n, we have (E λ ) IndWn+1 n W
E μ,
μ
where the sum is taken over all multipartitions μ such that [μ] = [λ ] ∪ {γ } for γ ∈ A (λ ). In addition, we have λ n ResW Wn−1 (E )
Eμ,
μ
where the sum is taken over all multipartitions μ such that [μ] = [λ ] \ {γ } for γ ∈ R(λ ). Proof. Let λ l n and denote by V the KWn -module E λ (E λ ⊗ σ2 ). . .(E λ ⊗ σl ) so that we have W W IndWn+1 (E λ ) = IndWn+1 λ (V ). n 1
2
l
n
W λ ×W Now we have IndWnλ 1 (V ) = 1 jl (V σ j ) and, thus, we are reduced to describe n W IndWn+1 1 jl V σ j ). For 1 j l, using the already known branching rule λ ×W ( n
1
in type An−1 , we conclude that the induction of each V σ j from Wnλ ×W1 to Wn1 × . . . ×Wn j +1 × . . . ×Wnl is the direct sum of all the modules of the form E λ (E λ ⊗ σ2 ) . . . (E μ ⊗ σ j ) . . . (E λ ⊗ σl ), 1
2
j
l
where μ j is obtained from λ j by adding an addable node (see 3.5.2). We can then conclude using the definition of the E λ and the transitivity of Ind. This gives the formula for the induction. The second part of the proposition follows using Frobenius reciprocity. 5.1.9. More generally, one defines the group G(l, p, n) as the group of n × n matrices such that • the entries are either 0 or lth roots of unity, • there is exactly one non-zero entry in each row and each column, • the (l/p)-th power of the product of the non-zero entries is 1. The Weyl group of type Dn is a particular case of these groups (corresponding to the case l = p = 2). In fact, keeping the notation of the presentation of Wn above, one can realize the group G(l, p, n) as the subgroup of index p in Wn generated by a = s0p , a0 = s0 s1 s0 and ai = si for 1 i n − 1. Note that the reflections ai for i = 1, . . . , n − 1 again generate the symmetric group Sn . It is possible to obtain a complete set of irreducible representations for these groups using Clifford theory. We refer to [6], [135], [136], [239] for details.
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5 Representation Theory of Ariki–Koike Algebras
5.2 Basic Properties of Ariki–Koike Algebras In this section, we define a deformation of the group algebra of the complex reflection group Wn , which may be seen as an analogue of the Iwahori–Hecke algebra for Wn . This class of algebras has been independently introduced by Ariki and Koike [13] and Brou´e and Malle [34]. We study several basic properties of them here. 5.2.1. Let l ∈ Z>0 and n ∈ Z1 . Let k be any commutative ring with 1. Let ξ , ξ1 , ξ2 , . . ., ξl be elements in k and assume that ξ is invertible in k. The Ariki–Koike algebra Hk,n := Hk,n (ξ ; ξ1 , . . . , ξl ) is the unital associative k-algebra with a presentation by generators {Ti | i = 0, . . . , n − 1} and relations • (Ti − ξ )(Ti + 1) = 0 for i = 1, . . . , n − 1, • (T0 − ξ1 )(T0 − ξ2 ) . . . (T0 − ξl ) = 0, and the braid relations symbolised by the following diagram: T0 t
4
T1 t
T2 t
· · ·
Tn−1 t
That is to say: • Ti T j = T j Ti if |i − j| 1, • Ti Ti+1 Ti = Ti+1 Ti Ti+1 if 1 i n − 2, • T0 T1 T0 T1 = T1 T0 T1 T0 . Example 5.2.2. (a) Assume that ξ = 1 and that ξ j = ζl for j = 1, . . . , l. Then the map si → Ti for i = 0, 1, . . . , n − √ 1 defines an isomorphism of k-algebras from kWn onto Hk,n (where ζl := exp(2π −1/l).) (b) Let l = 1 and assume that k contains a square root ξ 1/2 of ξ ; then, Hk,n is an Iwahori–Hecke algebra of type An−1 with parameter ξ 1/2 . Indeed, if we put T i := ξ −1/2 Ti , we obtain the relation (T i − ξ 1/2 )(T i + ξ 1/2 ) = 0 for 1 i n − 1. 1/2 1/2 (c) Let l = 2. Assume that k contains square roots ξ1 and ξ2 of ξ1 and ξ2 and 1/2 of ξ ; then, Hk,n is an Iwahori–Hecke algebra of type Bn . Indeed, a square root ξ −1/2 Ti for i 1 and T 0 = (ξ2 ξ1 )−1/2 T0 , we obtain the relations if we put Ti := ξ j−1
1/2 −1/2
(T 0 − ξ1 ξ2
−1/2 1/2 ξ2 )
)(T 0 − ξ1
and
(T i − ξ 1/2 )(T i + ξ 1/2 ) = 0
for i = 1, . . . , n − 1. Thus, Hk,n is an Iwahori–Hecke algebra of type Bn with param1/2 −1/2 and ξ 1/2 . eters ξ1 ξ2 The above examples show that an Ariki–Koike algebra may be viewed as an analogue of an Iwahori–Hecke algebra for the complex reflection group Wn .
5.2 Basic Properties of Ariki–Koike Algebras
267
Definition 5.2.3. For m = 1, 2, . . . , n, we define the following elements: Lm := ξ 1−m Tm−1 . . . T1 T0 T1 . . . Tm−1 . They are called the Jucys–Murphy elements for the Ariki–Koike algebra Hk,n . (See also Example 3.4.8.) Remark 5.2.4. One can easily check that, for all 1 i n−1 and 1 j n, we have the following relations involving the Jucys–Murphy elements (see [63, Sections 2.1, 2.2], [10, Lemma 13.2]): 1. Li and L j commute. 2. Ti and L j commute if i = j − 1, j. 3. Ti commutes with Li Li+1 and with Li + Li+1 . 4. For all a ∈ k, and i = j, Ti commutes with ∏1m j (Lm − a). A basis for Hk,n has been given by Ariki and Koike. Theorem 5.2.5 (Ariki and Koike [13]). The algebra Hk,n is a free k-module with basis
c1 L1 . . . Lcnn Tw | w ∈ Sn and 0 cm l − 1 for m = 1, 2, . . . , n . In particular, we have dimk (Hk,n ) = l n n!. We refer to [10, Cor. 13.9] for the proof. Several remarks must be added. First, we see that the subalgebra of Hk,n generated by T0 , T1 , . . . Tn−2 is isomorphic to Hk,n−1 . In the general case where l ∈ Z>0 , we see that the elements T1 , . . . , Tn−1 generate an Iwahori–Hecke algebra of type An−1 which will be denoted by Hk,n (Sn ). Consequently, if w ∈ Sn and if si1 si2 . . . sir is a reduced expression for w, we can (and we do) write Tw := Ti1 Ti2 . . . Tir ∈ Hk,n (Sn ) ⊆ Hk,n . 5.2.6. Now, it will be convient for us to work with √ a “generic Ariki–Koike algebra”. Let R ⊂ C be a subring such that ζl := exp(2π −1/l) ∈ R. Assume that u, V1 , . . . , Vl are independent indeterminates. Then let Hn := HA,n (u;V1 , . . . ,Vl ),
where
A := R[u±1 ,V1 , . . . ,Vl ].
We also denote ε := (u,V1 , . . . ,Vl ). Let K be the field of fractions of R and let K be the field of fractions of A. Under these hypotheses, by [10, Cor. 13.9], we have the following theorem. Theorem 5.2.7 (Ariki and Koike [13]). The algebra HK,n := K ⊗A Hn is split semisimple. 5.2.8. Let us consider the specialisation θ : A → K such that
θ (u) = 1
and
θ (Vi ) = ζli−1
(i = 1, . . . , l).
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5 Representation Theory of Ariki–Koike Algebras
Then, the specialized algebra HK,n := K ⊗A Hn is isomorphic to the group algebra KWn . As K is a splitting field for Wn (see 5.1.3) and as the characteristic of K is zero, the algebra KWn is split semisimple. As a consequence, Tits’s deformation theorem applies: the irreducible representations of HK,n (up to isomorphism) are in bijection with IrrK (Wn ); see [132, (8.1.7)]. By 5.1.3, we can write Irr(HK,n ) = {Eελ | λ l n}. Example 5.2.9. The 2l irreducible representations of HK,n over K of dimension 1 are defined uniquely as follows:
σiε : HK,n → K T0 → Vi T j → u ( j = 1, . . . , n − 1),
sgnεi : HK,n → K T0 → Vi Tj → −1 ( j = 1, . . . , n − 1).
By Example 5.1.4, the simple module Eελ affords ∅ if m = i, ε m • the representation σi if λ = (n) if m = i, ∅ if m = i, • the representation sgnεi if λ m = (1n ) if m = i. 5.2.10. We can define a structure of symmetric algebras on Ariki–Koike algebras. The existence of such a structure is fundamental in our work. Let τ : Hn → A be the A-linear map determined by 1 if c1 = . . . = cn = 0 and w = 1, c1 c2 cn τ (L1 L2 . . . Ln Tw ) = 0 otherwise. The proof of the fact that Ariki–Koike algebras are symmetric algebras was first established by Bremke and Malle in [31, Theorem 2.8], but for a different definition of the trace. The proof of the following result was then completed by Malle and Mathas in [240, Section 3]. Theorem 5.2.11 (Bremke and Malle [31], Malle and Mathas [240]). The trace τ defined above is a trace function which makes Hn into a symmetric algebra. It is clear that this symmetrising form is a generalisation of the form constructed for Hecke algebras of type An−1 . The fact that it also generalises the already-defined trace in type Bn in 1.2.11 follows from a result by Malle and Mathas [240, Proposition 2.2]. 5.2.12. Given the trace form τ , we have corresponding Schur elements cλ := cλ (u;V1 , . . . ,Vl ) ∈ A indexed by the l-partitions of n such that
τ=
∑ c−1 λ χλ ,
λ l n
where χλ denotes the character associated with the simple HK,n -module Eελ .
5.2 Basic Properties of Ariki–Koike Algebras
269
Explict expressions for the Schur elements cλ have been conjectured by Malle in [237]. The proof was then obtained by Geck, Iancu and Malle [123, Theorem 1.3]. To give their expressions, we need to introduce several notations which will be also be useful in the next sections. Let λ := (λ 1 , . . . , λ l ) be an l-composition. For j = 1, . . . , l, we denote by h j the height of the partition λ j . The height of λ is then the number hλ := max{h j , j = 1, . . . , l}. It is zero if and only if λ is the empty l-composition. Let h hλ . The ordinary symbol B of size s of λ (first introduced in [237, Section 2]) is the collection of finite subsets B := (B1 , B2 , . . . , Bl ) such that, for each i = 1, . . . , l, we have Bi = (Bis , . . . , Bi1 ),
Bij = λ ji − j + s for all j = 1, . . . , s.
where
(We set λ ji := 0 if j > hi .) This symbol is usually written as an l-row tableau, where for i = 1, . . . , l, starting from the bottom, the cth row contains the sequence Bc as in the following example. Example 5.2.13. Let λ := (2.1, 3.3.1, 4) 3 14. Then we have hλ = 3. Then the ordinary symbol of size 3 is B := (B1 , B2 , B3 ) such that B1 := (4.2.0), B2 := (5.4.1), B3 := (6.1.0). We can write it as follows: ⎞ ⎛ 016 B = ⎝ 1 4 5 ⎠. 024 We associate with a symbol as above the following elements:
∏
δB (u,V1 , . . . ,Vl ) :=
(uα Vi − uβ V j ),
1i jl (α ,β )∈Bi ×B j α >β if i= j
θB (u,V1 , . . . ,Vl ) :=
∏
(ukVi −V j ),
1i, jl α ∈Bi 1kα
νB (u,V1 , . . . ,Vl ) :=
∏
(Vi −V j )h θB (u,V1 , . . . ,Vl ),
1i< jl
h l + n(l − 1), σB := 2 2
l(h − 1) + 1 l(h − 2) + 1 τB := + +..., 2 2 |B| :=
∑
1il α ∈Bi
α.
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5 Representation Theory of Ariki–Koike Algebras
The Schur elements are then given in the following proposition. This is a reformulation (given in [33, Proposition 3.7]) of the form of the Schur elements conjectured in [237]. Proposition 5.2.14 (Geck, Iancu and Malle [123]). Let λ be an l-partition of rank n and let B be an ordinary symbol of size h of this l-partition. Then cλ (u,V1 , . . . ,Vl ) = (u − 1)
∏
Vi
−n
(−1)σB uτB −|B|+n
1il
νB (u,V1 , . . . ,Vl ) . δB (u,V1 , . . . ,Vl )
This does not depend on the choice of h hλ . Note that Mathas has given another way to compute these Schur elements in [246, Theorem 5.18].
5.3 Ariki–Koike Algebras as Cellular Algebras We have already seen that Ariki–Koike algebras may be viewed as natural generalisations of the Iwahori–Hecke algebras of type An−1 and Bn . It is then natural to ask for generalisations of the fundamental properties satisfied by the Iwahori–Hecke algebras of finite Coxeter groups. We are first interested on the cellularity. The aim of this part is thus to give a cellular basis for Ariki–Koike algebras and to study the consequences of the existence of such a structure for the representations of our algebras. We here work with the generic Ariki–Koike algebra Hn as in 5.2.6. 5.3.1. We first need to introduce several combinatorial notions. They essentially generalise the notation for partitions introduced in 3.5.2. Let λ be an l-composition of n. A λ -tableau is a bijection t : [λ ] → {1, . . . , n}. We say that t is column standard if the sequence t(1, b, c), t(2, b, c), . . . is strictly increasing for each relevant b and c. It is called row-standard if the sequence t(a, 1, c), t(a, 2, c), . . . is strictly increasing for each relevant a and c. It is called standard if it is both row-standard and column-standard and if λ is an l-partition. The set of λ -tableaux will be denoted by Tab(λ ). The set of row-standard λ tableaux will be denoted by Rstd(λ ). The set of standard λ -tableaux will be denoted by Std(λ ). We also denote by Std(n) the set of all standard λ -tableaux such that λ l n. One can visualize a λ -tableau as an array of boxes where each box is filled by the associated integer as in the following example. Example 5.3.2. Let λ = (2.1, 3.2, 1) 3 9. The first of the following two λ -tableaux is standard, whereas the second is row-standard but not standard.
2 7 8 3 5 8 1 5 1 4 , , 6 , , , 7 4 6 3 9 2 9
5.3 Ariki–Koike Algebras as Cellular Algebras
271
5.3.3. Let λ be an l-partition of rank n. Then note that a row-standard λ -tableau is equivalent to the datum of the sequence of l-compositions λ t [i] |=l i such that [λ t [i]] = t−1 {1, . . . , i} for all i = 1, . . . n. With this notation, each λ t [i+1] is obtained from λ t [i] by adding the node t−1 (i + 1). In addition, t is standard if and only if all the l-compositions λ t [i] are actually l-partitions. We can define a partial order, called the dominance order, on the set of lcompositions of rank n. This generalises the dominance order on compositions defined in 3.5.10. Let λ = (λ 1 , . . . , λ l ) |=l n and μ = (μ 1 , . . . , μ l ) |=l n be multicompositions of heights hλ and hμ respectively. Then we set λ μ if and only if for all c = 1, . . . , l and N = 1, . . . , max(hλ , hμ ) we have
∑ ∑
1ic 1 jN
λ ji
∑ ∑
μ ij .
1ic 1 jN
(Here we set λij := 0 if i is strictly greater than the height of λ j .) Let λ |=l n. Then one can define a partial order on the set of row-standard λ tableaux as follows. Let t be a row-standard λ -tableau. For (t, s) ∈ Rstd(λ )2 , we set t s if for all i = 1, . . . , n, we have λ t [i] λ s [i] for i = 1, . . . , n. We write t s if t s or t = s (see Definition 2.2.13). Example 5.3.4. Let λ = (2, 3.1) 2 6 and consider the row- standard λ -tableaux
3 5 6 2 3 6 and s= 1 5 , . t= 1 2 , 4 4 Then, λ t [1] = (1, ∅), λ t [2] = (2, ∅), λ t [3] = (2, 1), λ t [4] = (2, 1.1), λ t [5] = (2, 2.1), λ t [6] = (2, 3.1) and λ s [1] = (1, ∅), λ s [2] = (1, 1), λ s [3] = (1, 2), λ s [4] = (1, 2.1), λ s [5] = (2, 2.1), λ s [6] = (2, 3.1). Thus we have t s. For each l-composition λ of rank n, there is a row-standard λ -tableau tλ of a special kind, the canonical λ -tableau: it is the λ -tableau such that tλ (a, b, c) =
∑ |λ j | + ∑ λic + b j
for each (a, b, c) ∈ [λ ].
i
It is easy to see that for all λ -tableaux t we have tλ t. Note that tλ is standard if and only if λ is an l-partition. Example 5.3.5. Let λ = (3.2, 3.2, 1). We have
1 2 3 6 7 8 tλ = , , 11 . 4 5 9 10 Note that Sn acts naturally on the set of λ -tableaux by permuting the entries as follows: Sn × Tab(λ ) → Tab(λ )
→ tσ (σ , t)
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5 Representation Theory of Ariki–Koike Algebras
such that t σ (a, b, c) = σ (t(a, b, c)) for all (a, b, c) ∈ [λ ]. We now recall the construction of the cellular basis for Ariki–Koike algebras by Dipper, James and Mathas [67, Theorem 3.26]. This one is particularly well adapted to the theory of cyclotomic q-Schur algebras (quasi-hereditary covers of Ariki–Koike algebras; see [67], [274]). Another cellular basis is given in the original article of Graham and Lehrer [144, Section 5]. Definition 5.3.6. Let λ = (λ 1 , . . . , λ l ) be an l-composition of n. We set m−1
a1 = 0
and
am =
∑ |λ i |
for m = 2, . . . , l.
i=1
Let a = (a1 , . . . , al ). Then we define l
ai
La = ∏ ∏ (Lm −Vi ). i=1 m=1
Finally, we set xλ =
∑
Tw
and
mλ = xλ La .
w∈Sλ
Example 5.3.7. Assume that l = 3, n = 6 and λ = (2.1, ∅, 3). Then we have a = (0, 3, 3) and we obtain that La = (L1 −V2 )(L2 −V2 )(L3 −V2 )(L1 −V3 )(L2 −V3 )(L3 −V3 ). 5.3.8. Note that, by Remark 5.2.4, we have mλ = La xλ . Let λ be an l-composition of rank n. Under the action of Sn on Tab(λ ), the row stabiliser of tλ is the Young subgroup Sλ . Let s ∈ Rstd(λ ). Then the element d(s) ∈ Sn is uniquely defined by the equality s = d(s)tλ . The set D λ := {d(s) ∈ Sn | s ∈ Rstd(λ )} is a complete set of left coset representatives for the action of Sλ on Sn . In fact, they are the elements of minimal length in these cosets and thus, each y ∈ Sn can be uniquely written as d(s)yλ for s ∈ Rstd(λ ) and yλ ∈ Sλ with l(x) = l(d(s)) + l(yλ ). The elements of D λ are called the distinguished left coset representatives of Sλ in Sn . Hence, we have a natural bijection between Rstd(λ ) and the distinguished left coset representatives of Sλ in Sn . We can now define the elements which will give the cellular structure for our algebra. To do this, we denote by ∗ : Hn → Hn the A-linear anti-automorphism of Hn determined by Ti∗ = Ti
for all i ∈ {0, 1, . . . , n − 1}.
Thus, for all w ∈ Sn , we have Tw∗ = Tw−1 . Moreover, each Jucys–Murphy element L j is self-dual with respect to this anti-automorphism.
5.3 Ariki–Koike Algebras as Cellular Algebras
273
Definition 5.3.9. Let λ be an l-composition of n and let (s, t) ∈ Rstd(λ )2 . Then, we define ∗ . ms,t = Td(s) mλ Td(t) Remark 5.3.10. The elements defined above do not correspond to the ones defined in [67], [247]. This comes from the fact that we have chosen to deal with left modules in this book, contrary to [67], [247], where right modules are used throughout. One of the main theorems of this section is the following one; we refer to [67, Theorem 3.26] for its proof. Theorem 5.3.11 (Dipper, James and Mathas [67]). We set
M = ms,t | (s, t) ∈ Std(λ )2 , λ l n . Then the quadruple (Π l (n), Std(n), M , ∗) is a cell datum for Hn in the sense of 2.6.1. Here, Π l (n) is the set of all l-partitions equipped with the partial order defined as follows: λ μ ⇐⇒ λ μ. As a consequence of the cellularity, note that if the algebra Hk,n is semisimple then it is split semisimple. Note also that, as a special case, this theorem implies the existence of a cellular basis for Hecke algebras of type An−1 and Bn ; see the remarks in 2.8.16, 2.8.17 and 2.8.19. Remark 5.3.12. With respect to the cellular basis of Dipper, James and Mathas (Theorem 5.3.11), we have a family of cell modules (see 2.6.1) which we denote as follows: {Sλ | λ l n}. These modules are called the Specht modules of Hn ; they have been extensively used in the literature. (They are, for example, systematically used in the book of Ariki [10].) Recall that K denotes the field of fractions of A. Then by Theorem 5.2.7, HK,n is split semisimple so the set of HK,n -modules SKλ := K ⊗A Sλ with λ l n provides a complete set of non-isomorphic simple HK,n -modules. By [10, Theorem 13.21], for all λ l n, we have Eελ SKλ . 5.3.13. As in the case of Iwahori–Hecke algebras, it is possible to define an opposite cell datum for Hn (see Proposition 2.6.7). In the present situation, this requires a bit more work in order to establish the required properties of the trace form in Theorem 5.2.11. Lemma 5.3.14. We have τ (h∗ ) = τ (h) for all h ∈ Hn . Consequently, we have an opposite cell datum (Π l (n)op , Std(n), Mˆ, ∗) as defined in Proposition 2.6.7. Proof. For all i = 1, . . . , n − 1 and m = 1, . . . , n, one can check that we have the following relations:
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5 Representation Theory of Ariki–Koike Algebras
⎧ ⎨ Lm Ti Ti .Lm = (u − 1)Lm + Lm−1 Tm−1 ⎩ Lm+1 Tm − (u − 1)Lm+1
if i = m − 1, m, if i = m − 1, if i = m.
Hence, if w ∈ Sn and if 0 cm l − 1 for m = 1, 2, . . . , n, then a non-identity element Tw Lncn . . . Lc11 is a linear combination of non-identity basis elements from
c1 L1 . . . Lcnn Tw | w ∈ Sn , 0 cm l − 1 for m = 1, 2, . . . , n . This implies that we have
τ (Tw Lncn . . . Lc11 ) =
1 0
if c1 = . . . = cn = 0 and w = 1, otherwise.
This concludes the proof, as Tw∗ = Tw−1 for all w ∈ Sn and L∗j = L j for all j.
5.4 Decomposition Maps for Ariki–Koike Algebras We now introduce the decomposition maps and the decomposition matrices for Ariki–Koike algebras. First, as in 3.1.12, we need to be more specific about the subring R of C. We thus shall now assume that R is noetherian and integrally closed in K. Recall that K is the field of fractions of A and that HK,n := K ⊗A Hn is split semisimple. 5.4.1. Let k be an arbitrary field and let θ : A → k be a specialisation such that k is the field of fractions of θ (A). Then the algebra Hk,n := k ⊗A Hn is cellular and thus it is split (see Theorem 2.6.5). By Theorem 3.1.2 one can consider the associated decomposition map dθ : R0 (HK,n ) → R0 (Hk,n ). We shall write the equations determining dθ in the form dθ (Eελ ) =
∑
dλ ,M [M],
where
dλ ,M ∈ Z0
M∈Irr(Hk,n )
We then have an associated decomposition matrix Dθ := (dλ ,M )λ l n, M∈Irr(Hk,n ) . As in 3.1.18, one sees that Dθ is the identity matrix (up to the ordering of the columns) if and only if Hk,n is semisimple. The following theorem proved in [5] gives a criterion of semisimplicity for the algebra Hk,n . We refer to [10, Proposition 13.10] for its proof. Theorem 5.4.2 (Ariki [5]). Let θ : A → k be a specialisation such that k is the field of fractions of θ (A). Set ξ := θ (u) and ξi := θ (Vi ) for i = 1, . . . , l. Then the specialised algebra Hk,n is split semisimple if and only if
5.4 Decomposition Maps for Ariki–Koike Algebras
∏
∏
−n
(ξ d ξi − ξ j )
∏
275
(1 + ξ + ξ 2 + . . . + ξ i−1 ) = 0.
1in
The main problem of the rest of this chapter (and of the following one) will be to study the decomposition maps in the same spirit as in Chapter 3. To do this, a useful result by Dipper, James and Mathas will allow us to restrict ourselves to certain specialisations. When l = 2, the following theorem is due to Dipper and James [66, Theorem 4.17] and for general l, a special case has been proved by Du and Rui [74, Theorem 4.14]. In the general case, the theorem is proved in [70, Theorem 1.1]. Theorem 5.4.3 (Dipper and Mathas [70]). Let θ : A → k be a specialisation such that k is the field of fractions of θ (A). Set ξ := θ (u) and for i = 1, . . . , l, set ξi := θ (Vi ). We set V = {ξ1 , . . . , ξl } and we assume that we have a partition V := V1 V2 . . . Vs for subsets Vi of V (i = 1, . . . , s) such that f (ξ , V ) =
∏
∏
∏
1α <β s (Vi ,V j )∈Vα ×Vβ −n
(ξ a ξi − ξ j )
is non-zero in k. Then Hk,n is Morita equivalent to the algebra
Hk,n1 (ξ ; V1 ) ⊗k Hk,n2 (ξ ; V2 ) ⊗k . . . ⊗k Hk,ns (ξ ; Vs ).
n1 +...+ns =n n1 ,...,ns 0
5.4.4. Let us explain the consequences of the above theorem. Let θ : A → k be a specialisation such that θ (Vi ) = ξi ∈ k for all i = 1, . . . , l and θ (u) = ξ ∈ k. Let Dθ be the associated decomposition matrix. Then there exists a partition {V1 , . . . ,Vl } = V1 V2 . . . Vs such that, for all i = 1, . . . , s, the sets Vi := θ (Vi ) satisfy the conditions of the above theorem. For i = 1, . . . , s, set Ai := R[u±1 , Vi ], HiAi ,ni := HAi ,ni (u, Vi ) and θi : Ai → ki , where ki denotes the field of fractions of θ (Ai ) and where ni ∈ {1, . . . , n}. Then we have associated decomposition maps dθi ,ni and matrices Dθi ,ni . The above theorem implies that, for a suitable ordering of the rows and columns, Dθ has the form of a block diagonal matrix where each block is given by a matrix Dθ1 ,n1 ⊗ . . . ⊗ Dθs ,ns for (n1 , . . . , ns ) ∈ Zs0 such that n1 + . . . + ns = n. As a consequence, to compute the decomposition matrices for Ariki–Koike algebras, it will be sufficient to consider the case where each of the parameters ξ1 , . . . , ξl is a power of ξ . Hence, it will also be sufficient to know a labelling of the simple modules in this case to solve that problem in general. As in Chapter 3, we will try to solve the problem of labelling for simple modules using the notion of canonical basic sets.
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5 Representation Theory of Ariki–Koike Algebras
Remark 5.4.5. It is also interesting to obtain a complete description of the blocks of Ariki–Koike algebras. This has been solved by Lyle and Mathas in [235].
5.5 Cyclotomic Ariki–Koike Algebras We shall now introduce a certain one-parameter specialisation of Hn . This will be used to define the a-invariants for the simple modules of HK,n . At the end of this section, we can then give a formal definition of canonical basic sets for Ariki–Koike algebras. √ 5.5.1. We set as before ζl = exp(2π −1/l). In addition, we fix a sequence of integers (r1 , . . . , rl ) ∈ Zl . Let R be as in 5.1.3. Let y be an indeterminate and let r ∈ Z>0 . We consider the Ariki–Koike algebra HR[y±1 ],n defined over the ring R[y±1 ] with the following choice of parameters:
ξ = yr We have
∏
∏
−n
j
ξ j = yr ζlj−1
and
(ξ d ξi − ξ j )
∏
for j = 1, . . . , l.
(1 + ξ + ξ 2 + . . . + ξ i−1 ) = 0,
1in
so the algebra HK(y),n is a (split) semisimple by Theorem 5.4.2. Following the terminology of [34], such an algebra is obtained from the generic Ariki–Koike algebra Hn by a “cyclotomic specialisation”:
θy : A → K(y) j V j → yr ζl j−1 for j = 1, . . . , l, u → yr . This justifies the name cyclotomic Ariki–Koike algebras. We define a sequence m = (m1 , . . . , ml ) of rational numbers by setting m j := r j /r for all j = 1, . . . , l. This sequence will be referred to a weight sequence. To the datum of a weight sequence and a positive integer r is uniquely associated a cyclotomic Ariki–Koike algebra. Let us now study in detail the representations of these algebras. Because of the semisimplicity of HK(y),n , we have a set
λ E y | λ l n which gives a compete set of non-isomorphic simple HK(y),n -modules. Following 5.2.6, these modules can be seen as reductions of the simple HK,n -modules Eελ via the specialisation θy . Remark 5.5.2. If we set r = 1 and m1 = 1 and m j = 0 for all j = 2, . . . , l, then the associated cyclotomic Ariki–Koike algebra is known as the spetsial Ariki–Koike
5.5 Cyclotomic Ariki–Koike Algebras
277
algebra. Such algebras play a central role in the program “Spets”, whose goal is to give to complex reflection groups the role of Weyl groups of some as yet mysterious structures (see [35], [238]). 5.5.3. We can now define the a-invariants for the simple HK,n -modules by studying the Schur elements of the cyclotomic Ariki–Koike algebras. Let λ be an l-partition of rank n. Fix m = (m1 , . . . , ml ) ∈ Zl , r ∈ Z>0 and consider the associated cyclotomic Ariki–Koike algebra HK(y),n as in 5.5.1. Let cλ be the Schur element of the simple HK(y),n -module Eyλ . This is a non-zero Laurent polynomial in y which is the specialisation through θy of the Schur element of Eελ given in Proposition 5.2.14. Then, we define a(m,r) (λ ) := min{s ∈ Z | ys cλ ∈ R[y]}, where, by convention, a(m,r) (∅) := 0. Note that this a-invariant depends on the choice of the weight sequence m and on r. To give an explicit expression, it is convenient to introduce several additional combinatorial notions. Remark 5.5.4. In [33], [46], [47], [48], the a-invariant is defined as the valuation of the associated Schur elements. This corresponds to the opposite of “our” a-invariant. Our definition is in agreement with [35], [132], [238]. 5.5.5. Let m = (m1 , . . . , ml ) ∈ Ql be an l-tuple of positive rational numbers. Keeping the above notations, we will define the shifted m-symbol and the ordinary m-symbol of an l-partition λ following [33]. Let β = (β1 , . . . , βk ) be a sequence of integers and let s be a non-negative rational number. We denote by [s] the integer part of s; that is, the maximal integer k such that s − k 0. We set if 0 s < 1, (β1 , . . . , βk ) β (s) := (s − [s], s − [s] + 1, . . . , s − 1, β1 + s, . . . , βk + s) if s > 1. It is a sequence of rational numbers with exactly k + [s] elements. Let now λ be an l-composition and for i = 1, . . . , l, let hi be the heights of the compositions λ i . For i = 1, . . . , l we consider the sequence of hi non-negative integers
β i = (λhi i − hi + hi , . . . , λ ji − j + hi , . . . , λ1i − 1 + hi ). Note that this is a sequence of strictly increasing integers if and only if λ is an lpartition. If s is an integer greater than hλ , note that the ordinary symbol of size s ∈ Z>0 which we have already defined in 5.5.3 is the family of sequences B = (B1 , . . . , Bl ),
where
B j = (β j (s − h j )) for j = 1, . . . , l.
Now, for i = 1, . . . , l we put hci = hi − mi
and
hcλ = max(hc1 , . . . , hcl ).
Let s be an integer such that s hcλ . The shifted m-symbol of λ of size s is the family of sequences
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5 Representation Theory of Ariki–Koike Algebras
B = (B1 , . . . , Bl )
such that
B j = (β j (s − hc j )) for j = 1, . . . , l. j
j
Each sequence B j contains exactly s + [m j ] elements (Bs+[m j ] , . . . , B1 ). Example 5.5.6. Let m = (3, 0, 2, 1) and let λ = (2.2, 4.4, 2, 5.5.1) 4 25. Then, we have hc1 = −1, hc2 = 2, hc3 = −1, hc4 = 2, hcλ = 2. For s = 3, the shifted m-symbol of λ of size s is then ⎛ ⎞ 0278 ⎜0 1 2 3 6 ⎟ ⎟. B=⎜ ⎝0 5 6 ⎠ 012367 Let m = (2, 1/2, 0) and let λ = (2.2, 4.1, 5.1) 3 15. Then, we have 3 hc2 = , 2
hc1 = 0,
hc3 = 2,
hcλ = 2.
For s = 3, the shifted m-symbol of λ of size s is then ⎛ ⎞ 0 2 7 ⎠. B = ⎝ 1/2 5/2 13/2 0 1 2 56 5.5.7. Continuing with the general discussion in 5.5.5, let now s be an integer such that s hcλ . Let ρ = [max(m1 , . . . , ml )]. j j For j = 1, . . . , l, we define a sequence B j = (B ρ +s , . . . , B 1 ) such that, for i = 1, . . . , ρ + s, we have B j = λ j − i + m j + ρ + s. i
i
The ordinary m-symbol of λ of size s is then the family of sequences := (B 1 , . . . , B l ). B of size s and the ordinary symbol B of size s + ρ Note that the ordinary m-symbol B are connected as follows: j j j B j = (Bρ +s + m j , Bρ +s−1 + m j , . . . , B1 + m j ).
The ordinary m-symbol and the shifted m-symbol (both of size s) of λ are connected as follows: j j B j = (m j , . . . , ρ + m j − [m j ] − 1, Bs+[m j ] + ρ , . . . , B1 + ρ ).
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279
Example 5.5.8. Let m = (3, 0, 2, 1) and let λ = (2.2, 4.4, 2, 5.5.1). Then the ordinary m-symbol of size 3 is ⎛ ⎞ 1 2 3 5 10 11 ⎜ ⎟ = ⎜ 2 3 4 5 6 9 ⎟. B ⎝0 1 2 3 8 9 ⎠ 3 4 5 6 9 10 Let m = (2, 1/2, 0) and let λ = (2.2, 4.1, 5.1). Then the ordinary m-symbol of size 3 is ⎛ ⎞ 0 1 2 4 9 = ⎝ 1/2 3/2 5/2 9/2 17/2 ⎠ . B 2 3 4 7 8 The following result comes from [174, Proposition 3.2] and is largely inspired by the computations in [33, Proposition 3.18]. Proposition 5.5.9. Let m := (m1 , . . . , ml ) ∈ Ql be a weight sequence and r ∈ Z>0 . Set ρ = [max(m1 , . . . , ml )] and let B be the ordinary symbol of size h := ρ + s of λ := (B 1 , . . . , B l ) be the ordinary m-symbol of size s associated with s ∈ Z>0 . Let B with λ . Then, we have a(m,r) (λ ) = r. f (n, h, m) + ∑ min {a, b} − ∑ min {k, m j } , 1i jl (a,b)∈B i ×B j a>b if i= j
1i, jl a∈B i 1ka
where l f (n, h, m) = n ∑ m j − τB + |B| − n − h j=1
∑
+
∑
min {mi , m j }
1i< jl
min {k, m }. j
1i, jl α ∈B i 1kmi
Proof. From the definition of the a-invariants and the formulae for the Schur elements in Proposition 5.2.14, we get that l a(m,r) (λ ) = r. n ∑ m j − τB + |B| − n − h j=1
+
∑
1i jl (α ,β )∈Bi ×B j α >β if i= j
We have
∑
1i< jl
min{α + m , β + m j } − i
min {mi , m j }
∑
1i, jl α ∈Bi 1kα
min{k + mi , m j } .
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5 Representation Theory of Ariki–Koike Algebras
∑
min {k + mi , m j } =
1i, jl α ∈Bi 1kα
∑
∑
min {k, m j } −
1i, jl α ∈Bi 1kα +mi
min {k, m j }.
1i, jl α ∈Bi 1kmi
Using the definitions of the ordinary symbol and of the ordinary m-symbol, we get the formula. We can also check that this is in agreement with the convention that a(m,r) (∅) = 0. 5.5.10. In this part we give a simple formula for a(m,r) (λ ) with λ l n. Let B be the shifted m-symbol of λ of size s and let
κ1 κ2 . . . . κt be the elements of B written in decreasing order (allowing repetition), where t = ls + ∑1il [mi ]. We set
κm (λ ) = (κ1 , . . . , κt )
t
and
nsm (λ ) = ∑ (i − 1)κi . i=1
Now we can state the following proposition. Proposition 5.5.11. Let λ be an l-partition, m = (m1 , . . . , ml ) be a weight sequence and r ∈ Z>0 . For all s hcλ , we have a(m,r) (λ ) = r.(nsm (λ ) − nsm (∅)). Proof. The proof is by induction on n. When n = 0, the result is obvious. Assume that n > 0 and that λ is an l-partition of rank n. Then there exists a removable node R = (α , β , γ ) of [λ ]. Let μ l n be such that [μ] = [λ ] \ {R}. If s hcλ , we also have s hcμ ; so, by induction, we have a(m,r) (μ) = r.(nsm (μ) − nsm (∅)). Note that
l
f (n + 1, h, m) =
∑ m j + f (n, h, m).
j=1
λ ) (or B(μ)) Let B( be the ordinary m-symbol of size s associated with λ (or μ). We denote by δ1 δ2 . . . . δlh λ ) written in decreasing order (with repetition), where h := s + ρ . the elements of B( We have min{a, b} = ∑ (i − 1)δi . ∑ 1i jl λ)j λ )i ×B( (a,b)∈B( a>b if i= j
Similarly, we denote by
1ilh
5.5 Cyclotomic Ariki–Koike Algebras
281
δ1 δ2 . . . . δlh the elements of B(μ) written in decreasing order (with repetition). We have lh
∑
1i jl j i ×B(μ) (a,b)∈B(μ) a>b if i= j
min {a, b} = ∑ (i − 1)δi . i=1
Let k be the maximal integer such that γ
δk = λα − α + mγ + ρ + s. Then we have
lh
lh
i=1
i=1
∑ (i − 1)δi = ∑ (i − 1)δi + (k − 1).
In addition, we have that
∑
min {k, m j } −
1i, jl λ )i a∈B( 1ka
∑
1i, jl i a∈B(μ) 1ka
min {k, m j } =
∑
min{δk , m j } =
1 jl
∑
mj
1 jl
because, by definition, we have
δk 1 − hγ + mγ + m j + hγ − mγ m j
for all 1 j l.
We use all these identities together to obtain that a(m,r) (λ ) = a(m,r) (μ) + r.(k − 1). On the other hand, from the relation between the shifted m-symbols and the ordinary m-symbols, we also have that nsm (λ ) = nsm (μ) + (k − 1), which completes the proof.
Example 5.5.12. The above proposition shows that the preorder induced by a(m,r) does not depend on r ∈ Z>0 . Example 5.5.13. Assuming that l = 1, the cyclotomic Ariki–Koike algebra HK(y),n associated with any weight sequence is nothing but the Hecke algebra of type An−1 with a = r in the setting of Example 1.3.8. We recover that the a-invariant of λ is here given by a.n(λ ). Example 5.5.14. Assume that l = 2. Let a ∈ Z>0 and b ∈ Z0 . We set m1 = b/a, m2 = 0 and r = a. Consider the cyclotomic Ariki–Koike algebra HK(y),n to the weight sequence m = (m1 , m2 ) and r. Then, using the notations of 5.5.1, we have
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5 Representation Theory of Ariki–Koike Algebras
(r1 , r2 ) = (b, 0). Thus, HK(y),n is nothing but the Hecke algebra of type Bn with L : W → Z given by b 4 a a a t t t p p p t Bn Let (λ 1 , λ 2 ) be a bipartition of n and consider its shifted m-symbol B = (B1 , B2 ) of size N. The elements appearing in B correspond to the element of the multiset N (λ 1 , λ 2 ) divided by a. Hence, we recover the formula for the a-invariant in type Za,b Bn in Example 1.3.9, Case 3. Example 5.5.15. Let s ∈ {1, . . . , l} be such that ms = max(m1 , . . . , ml ). Let ∅ if j = s, λ = (λ 1 , . . . , λ l ), where λj= (n) if j = s. Then Eελ affords the representation σsε of Example 5.2.9. Proposition 5.5.11 gives us that a(m,r) (λ ) = 0 for all r ∈ Z>0 . From this, we deduce the following useful result. (For this purpose, we extend the definition of the dominance order to sequences of rational numbers.) Proposition 5.5.16. Assume that λ and μ are two l-partitions of rank n such that
κm (μ) κm (λ ). Then we have
a(m,r) (λ ) > a(m,r) (μ).
Proof. Let s be an integer such that s hcμ and s hcλ and set t := ls + ∑1il mi . μ μ Set κm (μ) = (κ1 , . . . , κt ) and κm (λ ) = (κ1λ , . . . , κtλ ). Assume that κm (μ) κm (λ ). By Proposition 5.5.11, we need to show that nsm (λ ) > nsm (μ). We have nsm (λ ) =
∑
1it
(i − 1)κiλ =
∑ ∑
κiλ .
2 jt jit
Now, because κm (λ ) and κm (μ) have the same rank and κm (μ) κm (λ ), we have μ ∑ jit κiλ ∑ jit κi for all 2 j t. This proves that nsm (λ ) nsm (μ). As κm (λ ) = κm (μ), we also have nsm (λ ) = nsm (μ), which concludes the proof using Proposition 5.5.11. Example 5.5.17. Let m = (0, 2, 1), r = 1 and let λ = (1, 4.1, 2). Let s = 2. The shifted m-symbol of λ of size s and the shifted m-symbol of ∅ of size s are ⎛ ⎞ ⎛ ⎞ 014 012 ⎝0 1 2 3⎠ B = ⎝0 1 3 7⎠ and 02 01 respectively. Then we have
5.5 Cyclotomic Ariki–Koike Algebras
283
nsm (λ ) = 7 × 0 + 4 × 1 + 3 × 2 + 2 × 3 + 1 × 4 + 1 × 5 = 25, nsm (∅) = 3 × 0 + 2 × 1 + 2 × 2 + 1 × 3 + 1 × 4 + 1 × 5 = 18. Hence, we obtain a(m,r) (λ ) = 7. Example 5.5.18. Let n ∈ Z>1 and assume that m = (m1 , . . . , ml ) is a weight sequence such that mi − m j > n − 1 for all i < j. Let λ l n and write its shifted m-symbol of size s: B = (B1 , . . . , Bl ). For all c = 1, . . . , l, the elements of Bc are the λac − a + s + mc for a = 1, 2, . . . , s + [mc ]. Let c > c. Then an arbitrary element of Bc writes λac − a + s + mc for a = 1, 2, . . . , s + [mc ]. Let a ∈ {1, 2, . . . , s + [mc ]} and a ∈ {1, 2, . . . , s + [mc ]} be such that λac = 0 and λac = 0. Then, it is an easy combinatorial exercice to show that (λac − a) − (λac − a ) 1 − n. As a consequence, we obtain that λac − a + s + mc > λac −a +s+mc . Now assume that λ and μ are both l-partitions of n such that μ λ . Then using the discussion above, one can show that κm (μ) κm (λ ). One way to see this is to use the same arguments as in [274, Prop. 6.4]. It is indeed sufficient to prove this statement for l-partitions such that there exist no ν l n satisfying μν λ . The result then follows from the definition of κm . We thus remark that for this choice of weight sequence we have μλ
⇒
a(m,r) (λ ) > a(m,r) (μ).
But for general m this property is not true! Having defined a partial order on the set of simple modules for Wn , we can now define the notion of basic set in the context of Ariki–Koike algebras.
Definition 5.5.19. Consider the decomposition matrix Dθ = (dλ ,M ) associated with θ : A → k. Let (r1 , . . . , rl ) ∈ Zl and r ∈ Z>0 . For all i = 1, . . . , l, we set mi := ri /r and we define m := (m1 , . . . , ml ). By 5.5.3, we have a function IrrK (Wn ) → Q, E λ → a(m,r) (λ ). Assume that the three conditions in 3.1.7 are satisfied for the partial order μ
Eελ Eε
def
⇔
λ =μ
or a(m,r) (λ ) < a(m,r) (μ).
Explicitly, in the present context, this means: • Given M ∈ Irr(Hk,n ), let Sθ (M) := {λ l n | dλ ,M = 0}. Then the function Sθ (M) → Q, λ → a(m,r) (λ ), reaches its minimum (with respect to ) at exactly one element of Sθ (M), which we denote by λM . • The map Irr(Hk,n ) → Π l (n), M → λM , is injective (where Π l (n) := {λ l n}). • We have dλM ,M = 1 for all M ∈ Irr(Hk,n ).
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5 Representation Theory of Ariki–Koike Algebras
If this holds, let Bθ := {λ M | M ∈ Irr(Hk,n )}. Thus, we obtain a bijection 1−1
Irr(Hk,n ) ←→ Bθ ⊆ Π l (n), μ
and Bθ = {Eε | μ ∈ Bθ } is called a canonical basic set for Hk,n ; see 3.1.7. We shall also call Bθ ⊆ Π l (n) itself a canonical basic set. Note again that if a canonical basic set exists, then it is uniquely determined by the above conditions.
Remark 5.5.20. The above definition shows that the canonical basic set depends on the choice of the weight sequence m. In the following, we will only deal with specific values of m which are sufficient for our applications on Iwahori–Hecke algebras; see 5.7.1. It should be interesting to study the existence of canonical basic sets for Ariki–Koike algebras in full generality. Remark 5.5.21. According to 5.4.4, to study the decomposition matrices for Ariki– Koike algebras, it is sufficient to consider specialisations θ : A → k such that θ (u) and all θ (Vi ) are integral powers of one fixed element in k. In the following sections, we will thus only deal with specialisations θ which satisfy these assumptions.
5.6 A Fock Datum for Ariki–Koike Algebras The goal of this section is to introduce a family of operators of “i-induction” similar the the ones constructed in Section 3.5 in type An−1 . We then study the “branching” rule induced by these elements. 5.6.1. We consider the Ariki–Koike algebra Hn where A := R[u±1 ,V1 , . . . ,Vl ]. Recall that K denotes the field of fractions of K and that
Irr(HK,n ) = Eελ | λ l n . We now wish to define a Fock datum for Hn as introduced in 3.4.1. Let θ : A → k be a specialisation such that
θ (u) = 1
and
θ (V j ) = θ (u)s j
for some s j ∈ Z (1 j l);
see Remark 5.5.21. Let e > 1 be the order of θ (u) ∈ k× and write ηe := θ (u). Thus, if e < ∞, then ηe is a primitive root of unity of order e. We can define central elements in a similar way as in 3.5.1 by setting zm = L1 + . . . + Lm , where the Li are the Jucys–Murphy elements given in 5.2.3. The fact that zm ∈ Z(Hn ) directly follows from Remark 5.2.4.
5.6 A Fock Datum for Ariki–Koike Algebras
285
(We have excluded the case where θ (u) = 1 because it is somewhat special in this context; it has been studied by Mathas in [244]. See Proposition 4.6.6, where the analogous problem in type Bn is solved.) Proposition 5.6.2. The family {(Hm , zm ) | 1 m n} defines an abstract Fock datum in the sense of 3.4.1. Proof. First, we have H1 ⊆ H2 ⊆ . . . ⊆ Hn so that (F1) is satisfied. By Theorem 5.2.5, the algebra Hn is finitely generated and free over A. In addition, we have
Hn =
Hn−1 Lni Td ,
0il−1 d∈D n−1
where D n−1 is the set of distinguished left coset representatives of Sn−1 in Sn . Hence, Hn is free as a left Hn−1 -module. So (F2) is satisfied. The property (F3) has already been established. Finally, the facts that HK,n is split semisimple and that Hk,n is split follow from the cellularity of these algebras. Hence, (F4) holds. (n)
Given the above Fock datum, we can define operators ζ -IndK with respect to the central elements {zm } for all ζ ∈ k. The aim of the following parts is to study these operators. (n)
5.6.3. Let us denote by IndK the induction of representations from HK,n−1 to HK,n (where HK,0 = {0} by convention). Thus, for any HK,n−1 -module V , we have (n)
IndK (V ) := HK,n ⊗HK,n−1 V. Let n 1 and λ l n. Then we already know by Proposition 5.1.8 that the restriction of E λ to Wn−1 is the direct sum of all E μ , where μ runs over all the l-partitions of n − 1 such that [μ] = [λ ] \ {γ } with γ ∈ R(λ ). By Frobenius reciprocity and a general criterion of compatibility of induction with specialisations, we obtain (n)
μ
IndK (Eε )
λ ∈Λ (μ)
Eελ
for any μ l n − 1,
where Λ (μ) is the set of all λ l n such that [λ ] = [μ] ∪ {γ } and γ ∈ A (μ). By the discussion in 3.4.1, we can now define for all ζ ∈ k certain refined induction (n) operators ζ -IndK . We want to give explicit formulae for these operators. To do this, we need to study the elements ωλ (zm ) for all l-partitions λ . Our strategy will be to use the cellular structure of Ariki–Koike algebras on Hn using the following result of [184, Proposition 3.7]. Lemma 5.6.4 (James and Mathas [184]). Let λ l n. Let t be a standard λ -tableau and let i be an integer such that 1 i n. Assume that t−1 (i) = (a, b, c). Then for each s ∈ Std(λ ) there exists as ∈ R such that Li .mt,tλ = Vc ub−a mt,tλ + ∑ as ms,tλ . st
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5 Representation Theory of Ariki–Koike Algebras
Proof. Assume first that t = tλ so that mt,tλ = mλ . Let j := min{t(α , β , c) | α 1, 1 β λac }. Then we have j i. We denote Ti, j = Ti−1 Ti−2 . . . T j ∈ Sλ (with Ti, j = 1 if j = i) and Tj,i = Ti,∗j . As Ti, j commutes with La (see Definition 5.3.9), we obtain that Li .mλ = Li La xλ = u( j−i) Ti, j L j La T j,i xλ = u( j−i) Ti, j (L j −Vc )La T j,i xλ +Vc u( j−i) La Ti, j T j,i xλ . If we set b := (a1 , . . . , ac−1 , ac + 1, ac+1 , . . . , ar ), we obtain by definition Ti, j (L j − Vc )La T j,i xλ = Ti, j Lb T j,i xλ . Note that we have Ti, j Lb T j,i xλ ∈ Hn mλ Hn . By 2.6.1(C3), this implies that Ti, j Lb T j,i xλ ∈ Hn (λ ). We claim that we have in fact Ti, j Lb T j,i xλ ∈ Hn (λ ). This can be seen as follows. If c = 1, consider the l-partition ν such that for all i = 1, . . . , l (observe that λ c = ∅): ⎧ ⎪ . . . .1) if i = c, c − 1, ⎪ ⎪ (1.1. ⎪ ⎪ ⎪ |λ i | times ⎪ ⎪ ⎨ if i = c − 1, ( 1.1. . . . .1 ) νi = ⎪ c−1 |+1 times | λ ⎪ ⎪ ⎪ ⎪ ⎪ ( 1.1 . . . .1 ) if i = c. ⎪ ⎪ ⎩ |λ c |−1 times
Then we have Ti, j Lb T j,i xλ ∈ Hn mν Hn ⊂ Hn (ν). If c = 1, then one can set ⎧ ⎪ (1.1. . . . .1) ⎪ ⎪ ⎪ ⎪ i ⎪ |λ | times ⎪ ⎪ ⎨ ( 1.1. . . . .1 ) νi = ⎪ 2 |λ |−1 times ⎪ ⎪ ⎪ ⎪ ⎪ ( 1.1 . . . .1 ) ⎪ ⎪ ⎩
if i = 1, 2, if i = 2, if i = 1.
|λ 1 |+1 times
This leads to the same property: Ti, j Lb T j,i xλ ∈ Hn (ν) by the definition of cellular algebras. Note that in both cases λ is not bigger than ν with respect to the dominance order. Hence, we obtain that Ti, j Lb Tj,i xλ ∈ Hn (λ ) and thus we can deduce that Li .mλ ≡ Vc u( j−i) Ti, j T j,i La xλ mod Hn (λ ). Now the element u( j−i) Ti, j Tj,i is a Jucys–Murphy element in type An−1 . The action of such element on xλ can be deduced from [245, Theorem 3.32]. We obtain that Li .mλ ≡ ub−aVc u( j−i) mλ mod Hn (λ ).
5.6 A Fock Datum for Ariki–Koike Algebras
287
This completes the proof in the case t = tλ . Assume now that t = tλ . Then there exists an integer k such that s = sk t and s t. In this case, one can show mt,tλ = Tk ms,tλ (e.g. see [245, Corollary 3.4]). The lemma follows using exactly the same arguments as in [245, Lemma 3.29]. Lemma 5.6.5. Let n 1, μ l n − 1 and λ ∈ Λ (μ). Then writing [λ ] = [μ] ∪ {γ }, where γ = (a, b, c) is a removable node of λ (or, equivalently, an addable node for μ), we have ωλ (zn ) − ωμ (zn−1 ) = Vc ub−a . Proof. Let t be a standard λ -tableau. By Lemma 5.6.4, for each s ∈ Std(λ ) there exists as ∈ R such that zn .mt,tλ = ∑ Vcub−a mt,tλ + ∑ as ms,tλ . (a,b,c)∈[λ ]
st
Now, by Remark 5.3.12, we have Eελ S λK and S Kλ has a basis given by the elements mt with t ∈ Std(λ ). The action of zn is given by (see 2.6.1) zn .mt = ∑ Vc ub−a mt . (a,b,c)∈[λ ]
We thus obtain that ωλ (zn ) = ∑(a,b,c)∈[λ ] Vc ub−a , as required.
We can now use the same strategy as in 3.5.4 to deduce the following result (see [10, Lemma 13.37]). Proposition 5.6.6 (Ariki [10]). Let i ∈ {0, 1, . . . , e − 1} and n 1. Then we have (n)
μ
ηei -IndK (Eε )
Eελ
for any μ l n − 1,
i
λ l n such that μ→λ i
where we write μ → λ if [λ ] = [μ] ∪ {x} and x = (a, b, c) is an addable node of μ such that b − a + sc ≡ i mod e. Proof. Let n 1, μ l n − 1 and λ ∈ Λ (μ). Using the specialisation θ of 5.6.1 and Lemma 5.6.5, we obtain ω λ (zn ) − ω μ (zn−1 ) = ηeb−a+sc . The proof is now entirely similar to that of Proposition 3.5.5, and follows from the definition of the operators (n) ηei -IndK . 5.6.7. If x = (a, b, c) is a node of an l-composition λ , we define b − a + sc mod e to be the residue of this node, we denote i := re (x) and we say that x is an i-node. In the following, we will often identify an element of Z/eZ with its representant in {0, 1, . . . , e − 1}. If, moreover, λ is an l-partition, we denote by Ai (λ ) the set of addable i-nodes of λ and by Ri (λ ) the set of removable i-nodes of λ . We have A (λ ) =
i∈Z/eZ
Ai (λ )
and
R(λ ) =
i∈Z/eZ
Ri (λ ).
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5 Representation Theory of Ariki–Koike Algebras i
Hence, in the above proposition, we have μ → λ if and only if [λ ] = [μ] ∪ {γ } for (n) (n) γ ∈ Ai (λ ). Similarly, we will often simply write i -IndK instead of ηei -IndK .
Fig. 5.1 Part of the LLT-graph for l = 2, e = 2, s1 = 0 and s2 = 1. ∅ ∅ 1
0
@ R
∅ 1
0
∅ 1 0
0
C C
0 ∅
C0 1 C C WC
∅ 1 0
0
A @ A @ A 1 @1 A @ A @ AU R @ 0 1 ∅
1
0 1
∅
Example 5.6.8. In the same spirit as Example 3.5.6, one can define the LLT-graph as follows. The vertices are given by the all l-partitions and the edges are defined by i the relation μ → λ in Proposition 5.6.6. 5.6.9. Following 3.4.1, we can now define a family of representations E of HK,n inductively as follows. For m = 1, let E1 = {1}, where 1 denotes the trivial representation of HK,1 = 1HK,1 K . Then set
(m) Em = ξ -IndK (V ) | V ∈ Em−1 and ξ ∈ k \ {0} for 2 m n. As in type An−1 , the LLT-graph allows us to understand the elements of E1 , E2 , (e) . . . for Ariki–Koike algebras. Let In be the set of all sequences I = (i1 , i2 , . . . , in ) such that i j ∈ {0, 1, . . . , e − 1} for all j 1. We obtain a sequence of representations: Y(i1 ) := 1,
(2) Y(i1 ,i2 ) := i2 -IndK Y(i1 ) ,
...,
(n) YI := in -IndK Y(i1 ,...,in−1 ) . (e)
Hence, with this notation, we have En = {YI | I ∈ In } \ {0}. Example 5.6.10. For example, let e = 2, s1 = 0 and s2 = 1. Figure 5.1 shows the part of the LLT-graph graph corresponding to all partitions of n 5. We have E1 = {Y(0) = (1, ∅),Y(1) = (∅, 1)}, E2 = {Y(1,0) = (∅, 1.1) ⊕ (∅, 2) ⊕ (1, 1);Y(0,1) = (1.1, ∅) ⊕ (2, ∅) ⊕ (1, 1)}, where we just write λ instead of Eελ .
5.7 FLOTW Multipartitions
289
We end this section with a formula for the computation of the multiplicities of the simple modules in an i-induction process. Corollary 5.6.11. Let λ l n and μ l n − 1, where n 1. Let i ∈ {0, . . . , e − 1} and s ∈ Z>0 . If Eελ appears with non-zero multiplicity as a direct summand in (n) μ (i -IndK )s (Eε ), then this multiplicity is s!. μ
Proof. Assume that Eελ appears with non-zero multiplicity in (i -IndK )s (Eε ). This means that there exist addable i-nodes γ1 , . . . , γs for μ such that [λ ] = [μ] ∪ {γ1 , . . . , γs }. Depending on the order we choose to add the γi , we see that we have exactly s! possibilities to obtain λ from μ. Note, indeed, that as all the γ j are i-nodes, adding these nodes in any order to λ still gives well-defined multipartitions. This concludes the proof. (n)
5.7 FLOTW Multipartitions We continue with our programme to establish the existence of canonical basic sets for Ariki–Koike algebras. We have already introduced a Fock datum for these algebras. Recall that we assume throughout that θ (u) = 1 and that there exists (s1 , . . . , sl ) ∈ Zl such that θ (V j ) = θ (u)s j for all j = 1, . . . , n. As before, we write ηe = θ (u), where e > 1 is the order of θ (u) ∈ k× . Thus, if e < ∞, then ηe is a primitive root of unity of order e. Note that each s j is only determined up to adding integer multiples of e (if e < ∞). We shall now also assume that s := (s1 , . . . , sl ) ∈ Sel where Sel := {s = (s1 , . . . , sl ) ∈ Zl | 0 s j − si < e for all i < j}. Under these conditions, we wish to prove the existence of canonical basic sets in the sense of Definition 5.5.19. To do this, we use a similar strategy as for the symmetric group. This will require the following ingredients: • We must fix a weight sequence m which induces the partial order m on the set of l-partitions. • We must find analogues of the set of e-regular partitions in our setting. 5.7.1. In this part, we will make a special choice for the weight sequence. Let v = (v1 , . . . , vl ) ∈ Ql be such that 0 < v j − vi < e
for all i < j.
Then we set m := (s1 − v1 , . . . , sl − vl ) ∈ Ql and we consider a positive integer r such that r.m ∈ Zl . We then have an a-function a(m,r) which depends on the choices of r, s and v. This choice for the weight sequence may look mysterious at the moment. Let us explain why we introduce such
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5 Representation Theory of Ariki–Koike Algebras
parameters. Recall that the a-invariants for finite Coxeter groups were introduced in Section 1.3 with respect to a choice of an arbitrary weight function L : W → Γ . For Ariki–Koike algebras, these a-invariants were defined in a slightly different way, by studying the Schur elements of a cyclotomic Ariki–Koike algebra associated with the datum of m. The above choice of m (see also 6.6.5) is what we need to deduce in full generality the canonical basic sets for type Bn from the properties that we will obtain at the level of Ariki–Koike algebras (see also Remark 5.7.4 below). Example 5.7.2. Assume that l = 1, e ∈ Z>0 and consider the Hecke algebra of type An−1 with weight function L such that L(s) = a ∈ Z>0 for all simple reflections s. Assume in addition that we have a specialisation θ such that θ (u) = ηe . We can then set s1 = v1 = 1 and r = a. In this case, the generic Hecke algebra of type An−1 is nothing but the cyclotomic Ariki–Koike algebra associated with m = (0) and r. Thus, a(m,r) is the usual a-invariant in type An−1 . Example 5.7.3. Let W be the Weyl group of type Bn with weight function L : W → Z given by b 4 a a a t t t p p p t where a > 0 and b 0. Bn Let H be the corresponding generic Iwahori–Hecke algebra over the ring R[v, v−1 ], where v is an indeterminate; let u = v2 . We consider a specialisation θ : R[v, v−1 ] → k and set η := θ (u). In Section 4.6, we have already considered the cases where η a = 1 or where η b = −η ad for all d ∈ Z. So now we can assume that η a = 1 and that there exists an integer d ∈ Z such that η b = −η ad . Assume that η a is a primitive root of unity of order e > 1 and set ηe := η a . Furthermore, assume that d can be chosen such that the following condition holds: ()
d − e < b/a < d.
We then set v1 = d − b/a and v2 = e. Then we have 0 < v2 − v1 < e. We then also set s1 = d, s2 = e. Note that we do not have (s1 , s2 ) ∈ Se2 in general! Nevertheless, in all cases, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised algebra Hk of type Bn . In addition, if we set r = a and m = (b/a, 0) = (s1 − v1 , s2 − v2 ), then a(m,r) is the usual a-function in type Bn ; see Example 5.5.14. Remark 5.7.4. The above choice of m comes from a generalisation of a very natural choice for the cyclotomic Ariki–Koike algebra (see [174, Section 4.1]). Indeed, set r = l and for all j = 1, . . . , l, m j = s j − v j , where v j = je/l. Then we have an associated cyclotomic Ariki–Koike algebra HR[y±1 ],n . We now consider the specialisation θy : R[y±1 ] → k such that θy (y) is an l-root of ηe . Then the specialised algebra through this specialisation is nothing but Hk,n := Hk,n (ηe ; ηes1 , . . . , ηl sl ). Thus, the weight sequence m defines a cyclotomic Ariki–Koike algebra which can be specialised to Hk,n . 5.7.5. We now need to define a partial order on the set of l-compositions. Let s ∈ Zl . Let λ and μ be both l-compositions of rank n; we write
5.7 FLOTW Multipartitions
λ m μ
291 def
⇐⇒
λ = μ or κm (λ ) κm (μ),
where we recall that m := (s1 − v1 , . . . , sl − vl ) for v = (v1 , . . . , vl ) ∈ Ql such that for all i < j we have 0 < v j − vi < e (see 5.5.10) and s ∈ Zl . Example 5.7.6. Assume that l = 1 and that m ∈ Z; then, by the definition of m , we have λ m μ if and only if λ μ. Proposition 5.7.7. Let s ∈ Zl . If λ and μ are two l-partitions such that λ m μ and λ = μ, then we have a(m,r) (λ ) < a(m,r) (μ). Proof. This follows from Proposition 5.5.11.
In this chapter, we will mainly use the partial order m in the case where s ∈ Sel . We now consider the second problem of finding an appropriate generalisation of the notion of e-regular partitions. This is given by the set of FLOTW l-partitions which we now define. Definition 5.7.8 (Foda, Leclerc, Okado, Thibon and Welsh [88, 2.11]). Let l ∈ Z>0 and e ∈ Z>1 . Recall that we defined Sel := {s = (s1 , . . . , sl ) ∈ Zl | 0 s j − si < e for all i < j}. Assume that s = (s1 , . . . , sl ) ∈ Sel . Then, the multipartition λ = (λ 1 , . . . , λ l ) is called an FLOTW l-partition if and only if: 1. λ is a cylindrical multipartition; that is, for every k = 1, . . . , l − 1 we have λik k+1 λi+s for all i 1 (the partitions are taken with an infinite numbers of empty k+1 −sk 1 for all i 1. parts) and λil λi+e+s 1 −sl 2. For all r > 0, among the residues appearing at the right ends of the length r rows of λ , at least one element of {0, 1, . . . , e − 1} does not appear. the set of FLOTW l-partitions associated with s ∈ Sel and by We denote by Φ(s,e) Φ(s,e) (n) the set of FLOTW l-partitions of rank n.
Remark 5.7.9. The second condition of Definition 5.7.8 can be alternatively stated as follows. For all r > 0, the map γ := (a, b, r) ∈ N (λ ) → re (γ ) ∈ Z/eZ is not surjective (see Definition 5.1.5 and 5.6.7). 5.7.10. Of course, this definition may not be very natural at the moment. However, we will really see in the next chapter why this is the kind of multipartition that we have to study in our framework. For instance, we can just note that if we take l = 1, the first condition above is empty and the second one is equivalent to the definition of “e-regularity” for partitions. Thus, the FLOTW l-partitions may been seen as generalisations of e-regular partitions.
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5 Representation Theory of Ariki–Koike Algebras
Example 5.7.11. Let l = 3, e = 4, s1 = 0 and s2 = s3 = 1; then we consider the 3-partition (3.1.1, 3.1, 1) of 10. The Young diagram with residue is as follows: ⎞ ⎛ 0 1 2 1 2 3 ⎝ 3 , , 1 ⎠. 0 2 Then note that the set of residues appearing at the right ends of the part with length 1 is {0, 1, 2, 3}. Hence, the second condition of FLOTW 3-partitions is violated and (3.1.1, 3.1, 1) is not an FLOTW 3-partition. Example 5.7.12. Here is the list of FLOTW 2-partitions when e = 4, s1 = 0 and s2 = 1 when n 4: Φ((0,1),4) (1) = {(∅, 1), (1, ∅)}, Φ((0,1),4) (2) = {(∅, 2), (1, 1), (1.1, ∅), (2, ∅)}, Φ((0,1),4) (3) = {(∅, 3), (1, 1.1), (1, 2), (1.1, 1), (1.1.1, ∅), (2, 1), (2.1, ∅), (3, ∅)}, Φ((0,1),4) (4) = {(∅, 4), (1, 2.1), (1, 3), (1.1, 1.1), (1.1, 2), (2, 1.1), (2, 2), (2.1, 1),
(2.1.1, ∅), (2.2, ∅), (3, 1), (3.1, ∅), (4, ∅)}. In the rest of the chapter, we will show that these kinds of multipartitions enjoy similar properties as the ones given in Section 3.5 for the set of e-regular partitions. To do this, we need some preparatory results. Recall that Nλ denotes the set of border nodes of λ . We also denote by Nλ (k) the set of border nodes of λ with residue k ∈ Z/eZ. Lemma 5.7.13. Let e ∈ Z>0 . Assume that s ∈ Sel and let λ = (λ 1 , . . . , λ l ) ∈ Φ(s,e) be a non-empty FLOTW l-partition. We set
lmax := max{λ11 , λ12 , . . . , λ1l }. Then, there exists a removable k-node γ1 , for some k, on a part λ ji11 = lmax which satisfies the following property. If γ2 ∈ Nλ (k − 1) is on a part λ ji22 , then
λ ji11 > λ ji22 . Proof. Let λ l1 , . . . , λ lr be the partitions of λ such that λ1l1 = . . . = λ1lr = lmax are the parts of maximal length. Let k1 , . . . , kr be the residues of the removable nodes γ1 ,. . . ,γr on the parts of length lmax . We want to show that there exists 1 i r such that there is no node with residue ki − 1 on the border of a part with length lmax . Assume that, for each 1 i r, there exists a node on the border of a part of length lmax with residue ki − 1; see the first of the two diagrams below. Then, there exists a partition λ lt1 , for some 1 t1 r, with a k1 − 1-node on the border of a part of length lmax . Let kt1 be the residue of the removable node on the border of the part
5.7 FLOTW Multipartitions
293
of λ lt1 with length lmax . We have t1 = 1, otherwise the nodes on the border of the parts with length lmax on λ lt1 would describe the set {0, . . . , e − 1}. This violates the second condition of FLOTW l-partition. ... ...
λ lt1 =
... ...
... ... k1 − 1 ... ... kt1
... ...
λ lt2 =
... ...
... ...
... ... kt1 − 1 ... ... kt2
... ...
We use the same strategy for the residue kt1 . There exists λ lt2 , for some 1 t2 r, with a kt1 − 1-node on the border of a part of length lmax ; see the second of the above two diagrams. Let kt2 be the residue of the removable node on the border of the part of λ lt2 with length lmax . We have t2 = t1 (for the same reasons as above) and t2 = 1, otherwise the nodes on the border of the parts with length lmax on λ lt2 and on λ lt1 would describe all the set {0, . . . , e − 1}. Continuing in this way, we finally obtain that there exists 1 tr r such that / {1,t1 ,t2 , . . . ,tr−1 }. tr ∈ This is a contradiction, since tr ∈ {1, . . . , r} and for all i = j, we have ti = t j . So, there exists 1 i r such that there is no ki − 1-node on the border of the parts with maximal length. This concludes the proof. 5.7.14. We now associate a canonical standard tableau and a corresponding eresidue sequence with each FLOTW l-partition. This generalises the work we have already done when l = 1 in 3.5.9. First, we need some preparatory results which will help us to show the existence our sequence. In particular, we introduce a partial order on the set of i-nodes of a given l-composition λ . Let γ , γ be two i-nodes of an l-composition λ . We write either b − a + sc < b − a + sc def γ ≺(s,e) γ ⇐⇒ or b − a + sc = b − a + sc and c > c . Note that this partial order becomes a total order if λ is an l-partition. This order will play a crucial role in the next chapter, where it is used to construct an action of the quantum group of affine type A on the vector space generated by all the l-partitions. It will also be helpful in our context. For the moment one can remark the following point. Assume that we have an l-tuple of integers (v1 , . . . , vl ) such that, for all i < j, we have 0 < v j − vi < e. Set m := (s1 − v1 , . . . , sl − vl ). Then, if γ and γ are two nodes of the same l-partition, we have
γ ≺(s,e) γ
⇐⇒
b − a + mc < b − a + mc .
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5 Representation Theory of Ariki–Koike Algebras
This comes from the fact that, in this case, there exists m ∈ Z such that b − a + sc = b − a + sc + me. Then we have γ ≺(s,e) γ if and only if m −1 if c < c and m 0 if c c . Note that if b − a + sc = b − a + sc and c = c , then γ and γ are not comparable (which does not happen if λ is an l-partition). The following proposition shows some sort of compatibility of this order with m . Proposition 5.7.15. Let λ be an l-composition. Let k ∈ Z>0 and let i ∈ Z/eZ. Assume that γ1 , . . . , γk , γ1 ,. . . , γk are i-nodes satisfying the following conditions: • μ and μ given as follows are well-defined l-compositions: [μ] = [λ ] ∪ {γ1 , . . . , γk },
[μ ] = [λ ] ∪ {γ1 , . . . , γk };
• for all j = 1, . . . , k, we have γ j ≺(s,e) γ j or γ j = γ j . Then we have μ m μ. Proof. For all j = 1, . . . , k, set (a j , b j , c j ) := γ j and (aj , bj , cj ) := γ j . Then, by assumption, we have
b j − a j + mc j bj − aj + mc j
for all j,
because the nodes γ j and γ j have all the same residues. We set κm (λ ) = (κ1 , . . . , κr ), where r is sufficiently large. Then we have κm (μ) = (κ1 , . . . , κr ), where there exist l1 , . . . , lk such that κli = κli + 1 for i = 1, . . . , k and κs = κs otherwise. Similarly, we have κm (μ ) = (κ1 , . . . , κr ), where there exist l1 , . . . , lk such that κl = κli + 1 for i
i = 1, . . . , k and κs = κs otherwise. Now by the definition of κm , for all i = 1, . . . , k, we have κli κli . This property implies that κm (μ) κm (μ ). Moreover, if μ = μ ,
then there exists j such that γ j = γ j . We obtain that b j − a j + mc j < bj − aj + mc j and thus κm (μ) κm (μ ). Thus, we can conclude that μ m μ. The proof of the following proposition is entirely similar. Proposition 5.7.16. Let λ be an l-composition. Let k ∈ Z>0 and let i ∈ Z/eZ. Assume that γ1 , . . . , γk , γ1 ,. . . , γk are i-nodes satisfying the following conditions: • μ and μ given as follows are well-defined l-compositions: [λ ] = [μ] ∪ {γ1 , . . . , γk }, [λ ] = [μ ] ∪ {γ1 , . . . , γk }; • for all j = 1, . . . , k, we have γ j ≺(s,e) γ j or γ j = γ j . Then we have μ m μ .
In the case where λ is an FLOTW l-partition, the following result establishes a connection between the above order and the order ≺(s,e) on two removable or addable nodes with the same residue. For the proof, we will need the following basic property: if γ = (a, b, c) is a node of an l-partition λ , we have
5.7 FLOTW Multipartitions
295
λac =
b if γ is removable, b − 1 if γ is addable.
. Lemma 5.7.17. Let e ∈ Z>0 . Assume that s ∈ Sel and let λ = (λ 1 , . . . , λ l ) ∈ Φ(s,e) Let γ1 = (a1 , b1 , c1 ) be a removable or addable i-node of λ and γ2 = (a2 , b2 , c2 ) be a removable or addable i-node of λ . Assume that λac11 < λac22 then γ1 ≺(s,e) γ2 .
Proof. First, assume that γ2 ≺(s,e) γ1 . We want to show that λac11 λac22 . First, we consider the case where c1 < c2 . We then have b1 − a1 + sc1 b2 − a2 + sc2 . Hence, we obtain a1 b1 − b2 + a2 + sc1 − sc2 , which implies that 1 b1 − b2 + a2 + sc1 − sc2 b1 − b2 + a2 and that
λac11 λbc11−b2 +a2 +sc
1
−sc2 .
By the characterisation of the FLOTW l-partitions, we deduce
λac11 λbc12−b2 +a2 . If b1 − b2 0, then we obtain λac11 λac22 , and the case b1 > b2 also gives the result because then b1 = λac11 or b1 = λac11 − 1 and b2 = λac22 or b2 = λac22 − 1. The case c1 = c2 is trivial. We finally have to consider the case where c1 > c2 . Because γ1 and γ2 have the same residue modulo e, we have b1 − a1 + sc1 b2 − a2 + sc2 + e. Hence, we obtain a1 b1 − b2 + a2 + sc1 − sc2 + e, which implies that 1 b1 − b2 + a2 + sc1 − sc2 + e b1 − b2 + a2 and that
λac11 λbc11−b2 +a2 +sc
1
−sc2 +e .
By the characterisation of the FLOTW l-partitions, we deduce
λac11 λbc12−b2 +a2 . Then the result follows exactly as above.
Lemma 5.7.18. Let n ∈ Z>0 . Assume that s ∈ Sel and let λ = (λ 1 , . . . , λ l ) ∈ Φ(s,e) (n) be a non-empty FLOTW l-partition. Let γ1 be a removable k-node, for some k ∈ Z/eZ, which is chosen as in Lemma 5.7.13. Assume that
γs := (as , bs , cs ) ≺(s,e) . . . ≺(s,e) γ2 := (a2 , b2 , c2 ) ≺(s,e) γ1 := (a1 , b1 , c1 )
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5 Representation Theory of Ariki–Koike Algebras
are the removable k-nodes of λ . Let γ = (a, b, c) be the maximal element of Nλ (k − 1) with respect to ≺(e,s) . Let u = max{i ∈ {1, . . . , s}, bi > b}. Then, the multipartition λ such that [λ ] = [λ ] {γ1 , . . . , γu } is in Φ(s,e) (m) for some m < n. Proof. The rank of λ is strictly smaller than the rank of λ by Lemma 5.7.13. We now have to check the two conditions of the FLOTW l-partitions for λ . c
c +1
1. First, we have to check that if λa jj = λa jj+sc exists t u such that
−sc j for some j j +1 c j +1 γt = (a j + sc j +1 − sc j , λa j +sc +1 −sc , c j + 1) j j
u, then there so that γ j and γt
must both be removed to obtain λ . Observe that c +1
λa jj+sc
j +1
−sc j
c
− (a j + sc j +1 − sc j ) + sc j +1 = λa jj − a j + sc j . c +1
Thus, the residue of the node η := (a j + sc j +1 − sc j , λa jj+sc +1 −sc , c j + 1) is k. j j Moreover, this is a removable one, otherwise we would have c +1
c
λa jj+si+1 −si +1 > λa jj+1 , contradicting our assumption that λ is an FLOTW l-partition. Since we have c c +1 c +1 λa jj = λa jj+sc +1 −sc and λa jj+sc +1 −sc > b, by construction, the node η is a γt for j j j j t u. The result follows. 2. We now check the second condition of FLOTW l-partitions. The only problem may happen when there exists t ∈ {1, . . . , u} such that the set of residues of the nodes on the border of the parts of λ with length λactt − 1 is equal to the following set: {0, . . . , e − 1} \ {k − 1}. We want to show that, among the residues on the border of the parts of λ with length λactt − 1, k does not occur. There exists at least one k-node on the border of a part of λ with length λactt − 1. Such a k-node must be a removable one. Indeed, otherwise, we would have a k − 1-node on the border of a part of λ with length λactt − 1, which contradicts our assumption. We have λactt − 1 > b because there is no k − 1-node on parts with length λactt − 1. So, all the k-nodes on the border of parts with length λactt − 1 must be removed. Then, the set of residues of the nodes on the border of parts of λ with length λactt − 1 is equal to {0, . . . , e − 1} \ {k}. Thus, λ is an FLOTW l-partition.
5.7.19. Let us explain the consequences of the above combinatorial lemmas. First assume that we have a standard λ -tableau t. Then one can associate with this a sequence of elements in Z/eZ called the e-residue sequence η(s,e) (t) and which
5.7 FLOTW Multipartitions
297
generalises the one we have already defined in Chapter 3 when l = 1. This is defined as follows: η(s,e) (t) = (re (t −1 (1)), re (t −1 (2)), . . . , re (t −1 (n))). The result above gives us a way to define a remarkable e-residue sequence attached to each FLOTW l-partition. Assume that s ∈ Sel and let λ = (λ 1 , . . . , λ l ) ∈ Φ(s,e) be a non-empty FLOTW l-partition. We will construct a canonical standard tableau t(s,e) (λ ) and an associated e-residue sequence using the result described above. We first take a removable node γ1 as in Lemma 5.7.13. If we have several choices for it, we take the maximal node with respect to ≺(s,e) . Set k := re (γ1 ). Let γ1 , γ2 ,. . . , γs be removable k-nodes of λ as in the previous lemma. For convenience, we now assume that these nodes are ordered such that
γs (s,e) γs−1 (s,e) . . . (s,e) γ1 . We say that these nodes are the admissible nodes for λ and that k is the admissible residue. From this, we start building the tableau t(s,e) (λ ) by assigning the number n to γ1 then n − 1 to γ2 , . . . , n − s + 1 to γs . Let λ be the l-partition such that [λ ] = [λ ] ∪ {γ1 , γ2 , . . . , γs }. and the rank of λ is strictly smaller than the rank of λ by Lemma Then λ ∈ Φ(s,e)
5.7.18. We can apply the same procedure as above for λ . This allows us to define a standard λ -tableau t(s,e) (λ ) by induction. Then we can define η(s,e) (λ ) to be the e-residue sequence of t(s,e) (λ ). Here is an example. We assume that l = 2, s1 = 0, s2 = 1 and e = 4. The FLOTW 2-partitions are the 2-partitions (λ 0 , λ 1 ) which satisfy: 2 and λ 2 λ 1 for all i ∈ Z ; • λi1 λi+1 >0 i i+3 • for all k > 0, among the residues appearing at the right ends of the length k rows of λ , at least one element of {0, 1, 2, 3} does not occur.
We consider the 2-partition λ = (2.2, 2.2.1) with the following diagram with residues: ⎞ ⎛ 1 2 0 1 ⎝ , 0 1 ⎠. 3 0 3 Then λ is an FLOTW 2-partition. At the beginning the only candidates for γ1 are the nodes (2, 2, 1) and (2, 2, 2). However, we have re (2, 2, 2) = 1, and there is a part of length 2 with a node on the border which has k − 1 = 0 (mod e) as a residue. So, we have to take γ1 = (2, 2, 1) and k = 0. Note that we can remove this node, as 3-nodes on the border have maximal length 1. There is no other removable 0-node in λ . We can check that the 2-partition obtained by removing this node (2.1, 2.2.1) is an FLOTW 2-partition. One has to assign the number 9 to the node (2, 2, 1) in the standard λ -tableau t(λ ). Now, the removable nodes on the part with maximal length have 1 as a residue and there is no 0-node, so we assign 7 and 8 to the nodes (2, 2, 2) and (1, 2, 1)
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5 Representation Theory of Ariki–Koike Algebras
respectively. Repeating the same procedure, the associated standard λ -tableau is ⎛ ⎞ 1 6 2 7 t((0,1),4) (λ ) = ⎝ , 3 8 ⎠ 4 9 5 and η((0,1),4) (λ ) = 1, 0, 0, 3, 3, 2, 1, 1, 0. Note that, for all m ∈ {1, . . . , 9}, the 2partitions λ [m] such that [λ [m]] = t((0,1),4) (λ )−1 {1, . . . , m} are FLOTW bipartitions:
λ [1] = (∅, 1), λ [2] = (1, 1), λ [3] = (1, 1.1), λ [4] = (1.1, 1.1), λ [5] = (1.1, 1.1.1), λ [6] = (1.1, 2.1.1), λ [7] = (2.1, 2.1.1), λ [8] = (2.1, 2.2.1), λ [9] = (2.2, 2.2.1). This is in fact a general property of our construction, as explained in the next result. Lemma 5.7.20. Let λ be an FLOTW l-partition of n. For all m ∈ {1, . . . , n} define λ [m] to be the l-partition such that [λ [m]] = t(s,e) (λ )−1 {1, . . . , m}. Then λ [m] is a well-defined FLOTW l-partition. In addition, η(s,e) (λ [m]) is the subsequence of the m first elements of η(s,e) (λ ) and t(s,e) (λ [m]) is the restriction of t(s,e) (λ ) to [λ [m]] ⊆ [λ ]. Proof. First, note that, for all i ∈ {1, . . . , n}, the l-composition λ [i] is an l-partition because t(s,e) (λ ) is standard (note that λ [n] = λ ). Now, assume that k is the admissible residue for λ and that γ1 , . . . , γs are all the admissible k-nodes for this l-partition written in decreasing order with respect to ≺(s,e) . By definition, the lpartition obtained from λ by removing γ1 , . . . , γs is λ [n − s] and it is an FLOTW l-partition by Lemma 5.7.18. Thus, by induction, it suffices to show that, for all m ∈ {n − s, . . . , n}, the l-partition λ [m] is an FLOTW l-partition. Assume to the contrary that λ [m] is not an FLOTW l-partition. One can assume that m is maximal for this property so that λ [m + 1] is an FLOTW l-partition. Write γn−m = (a, b, c) so that [λ [m]] ∪ {γn−m } = [λ [m + 1]]. Then there are two cases to consider: • λ [m] violates the first condition to be an FLOTW l-partition. Assume that c = l − 1. This means that λ [m + 1]ca = λ [m + 1]c+1 a+sc+1 −sc and that we have to remove γm from λ [m +1] to obtain λ [m]. But in this case, the node γ := (a+sc+1 −sc , λ [m + 1]c+1 a+sc+1 −sc , c + 1) has the same residue as γn−m does and by definition γn−m (s,e) γ . In addition, γ is a removable node, otherwise we have λ [m + 1]c+1 a+sc+1 −sc = c > λ [m + 1]c λ [m + 1]c+1 = λ [m + 1] which contradicts the fact that a a+1 a+sc+1 −sc +1 λ [m + 1] is an FLOTW l-partition. If c = l − 1, then the result follows using the same argument. • Assume that λ [m] violates the second condition to be an FLOTW l-partition. The only possible way to have this is when the set of residues appearing at the right ends of the part with length λ [m]ca of λ [m] is equal to the set {0, 1, . . . , e − 1}. In this case, as λ is an FLOTW l-partition, the set of residues appearing at the right
5.7 FLOTW Multipartitions
299
ends of the parts with length λ [m]ca of λ is {0, 1, . . . , e − 1} \ {k − 1}. Hence, there exists a node γ = (a , b , c ) such that λ [m]ca = λ [m]ca − 1 and re (γ ) = k. This node is removable, otherwise (a + 1, b , c ) would be a k − 1-node of λ with b = λ [m]ca . Hence, by Lemma 5.7.17, we obtain that γ ≺(e,s) γn−m . One can easily see that this node is admissible and so γ = γt for some t < n − m, which is a contradiction because it still belongs to [λ [m]]. Now note that k is the admissible residue for λ [m], which completes the proof.
The following result is the analogue of Lemma 3.5.11 for FLOTW l-partitions. We use the partial order m on l-compositions defined in 5.7.5. Lemma 5.7.21. Let λ l n be an FLOTW l-partition and let μ an l-composition of n. Assume that there exists a row-standard μ-tableau s such that η(s,e) (s) = η(s,e) (λ ). Then λ m μ if λ = μ. Proof. Let us write t := t(s,e) (λ ) and assume that μ = λ . We set ds := n + 1 if t = s. Otherwise, ds ∈ {1, . . . , n} is the smallest integer such that t−1 (ds ) = s−1 (ds ). We now argue by downward induction on ds . If ds = n + 1, there is nothing to prove. We now assume that if u is a row-standard ν-tableau such that η(s,e) (u) = η(s,e) (λ ) and du > ds , then λ m ν if λ = ν. Set d := ds , (a1 , b1 , c1 ) := t−1 (d) and (a2 , b2 , c2 ) := s−1 (d). Then we claim that we have b1 − a1 + mc1 > b2 − a2 + mc2 . This is seen as follows. We remove successively the nodes filled by n, . . . , n − d + 1 from t. By Lemma 5.7.20, the resulting tableau will be t(s,e) (λ ) for an FLOTW lpartition λ . Similarly, if we remove the nodes filled by n, . . . , n − d + 1 from s we tableau s) by Lemma 5.7.20. If obtain a row-standard μ s. We have η (λ ) = η ( (s,e)
(s,e)
we remove the node filled by d in t(s,e) (λ ), we obtain a λ -tableau and if we remove -tableau. Because of the minimality of d, the node filled by d in s, we obtain a μ = λ . Hence, both λ and μ are obtained by adding a node γ1 = we have in fact μ (a1 , b1 , c1 ) and γ2 = (a2 , b2 , c2 ) respectively, with the same residue on the same lpartition. By definition, we have γ2 ≺(s,e) γ1 . As re (γ1 ) = re (γ2 ) and by the definition of ≺(s,e) , this implies that b1 − a1 + mc1 > b2 − a2 + mc2 , as claimed. So, in the tableau s, we have the following configuration, where “∗” stands for a filling of the node by a number in {1, . . . , d − 1} and where the node filled by n1 has the same coordinates as the node filled by d in t:
row a1 of sc1 :
∗
∗
row a2 of sc2 :
∗
∗
··· ···
∗ n 1 n2
∗ m1 m2
··· ···
d < n1 < n2 < . . . < n p ,
np
mq
d = m1 < m2 < . . . < mq .
Since η(s,e) (s) = η(s,e) (λ ), the e-residues of the nodes filled by m1 and n1 are equal. Consequently, for any relevant i, the node filled by ni has the same e-residue as the node filled by mi . We now perform some modifications on s in order to obtain a new
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5 Representation Theory of Ariki–Koike Algebras
l-composition ν of n and a corresponding row-standard ν-tableau u. This is done as follows. We keep all nodes and their fillings in the rows of s except for the a1 th row of the c1 th component and the a2 th row of c2 th component. For these two rows we modify as follows: • in the a1 th row of the c1 th component, we replace ni by min{ni , mi } for 1 i min{p, q}; • in the a2 th row of the c2 th component, we replace mi by max{ni , mi } for 1 i min{p, q}; • if q > p, we move all nodes filled by m p+1 , . . . , mq from the a2 th row of the c2 th component to the a1 th row of the c1 th component. We denote the resulting tableau by u. It is row-standard (see Remark 3.5.12) and we have η(s,e) (u) = η(s,e) (s) = η(s,e) (λ ). Furthermore, the a1 th row of the c1 th component of u has at least as many nodes as the a2 th row of the c2 th component of s. As we have b1 − a1 + mc1 > b2 − a2 + mc2 , if μ = ν, we obtain that ν m μ. Note that, by construction, we have t−1 (i) = u−1 (i) for 1 i d. Hence, if du ∈ {1, . . . , n} is minimal such that t−1 (du ) = u−1 (du ), then du > d. So, we can conclude by induction. Example 5.7.22. To illustrate the above proof, take s1 = 0, s2 = 1, e = 3 and consider the FLOTW 2-partition (3.2, 2) 2 7. Let (v1 , v2 ) be rational numbers such that 0 < v2 − v1 < 3. The Young diagram of λ with residue is
0 1 2 , 1 2 . 2 0 We have t((0,1),3) (λ ) =
1 3 6 , 4 7 2 5
and
η((0,1),3) (λ ) = 0, 2, 1, 1, 0, 2, 2.
We consider the following row-standard tableau: ⎞ ⎛ 1 4 6 s = ⎝ 2 , 5 ⎠. 3 7 We have η((0,1),3) (s) = η((0,1),3) (λ ). The smallest number appearing in different nodes of t := t((0,1),3) (λ ) and s is 3. With the notation of the above proof, we have m1 = 3, where q = 1 and p = 0. Hence, the tableau u is just ⎞ ⎛ 4 6 1 3 u=⎝ , 5 ⎠. 2 7 We continue and take du = 5. With the above notation, we have p = 0 and q = 1 with m1 = 5. So we have to move 5 from the second row of the second component
5.7 FLOTW Multipartitions
301
to the second row of the first component, and we obtain the tableau ⎛ ⎞ 4 6 1 3 ⎠. u1 = ⎝ , 2 5 7 The shape of that tableau is (2.1, 2.0.1). We now have du1 = 6, p = 0, q = 1 and m1 = 6. So we move 6 from the first row of the second component to the first row of the first component. We obtain the tableau ⎛ ⎞ 4 1 3 6 ⎠. , u2 = ⎝ 2 5 7 The shape of that tableau is (3.2, 1.0.1). We have du2 = 7 and so we move 7 from the third row of the second component to the first row of the second component and we obtain t. As a conclusion, we have (3.2, 2) m (3.2, 1.0.1) m (2.2, 2.0.1) m (2.1, 2.1.1) m (1.1.1, 2.1.1). Using the above key result, we can now give the main results of this section. Proposition 5.7.23. Let s = (s1 , . . . , sl ) ∈ Sel and let (v1 , . . . , vl ) be such that 0 < v j −vi < e if i < j. Set m = (s1 −v1 , . . . , sl −vl ) and let r ∈ Z>0 be such that r.m ∈ Zl . Let λ l n be an FLOTW l-partition. Then there exists Yλ ∈ E (see 5.6.9) such that [Yλ ] = m(Yλ , λ )[Eελ ] +
μ
∑
m(Yλ , μ)[Eε ],
μl n : a(m,r) (μ)>a(m,r) (λ )
where m(Yλ , μ) ∈ Z0 for all μ l n and m(Yλ , λ ) = 0. Proof. Let λ l n be an FLOTW l-partition. Consider the e-residue sequence η(s,e) (λ ) of 5.7.19. Then, by Lemma 5.7.21, the module Yλ := Yηe (λ ) satisfies [Yλ ] = m(Yλ , λ )[Eελ ] +
∑
λ m μ : λ =μ
μ
m(Yλ , μ)[Eε ]
and the results follow now from the compatibility of the a-invariant with the order m (see Proposition 5.7.7). Proposition 5.7.24. In the setting of Proposition 5.7.23, for all FLOTW l-partitions λ l n and for all l-partitions μ l n, the number m(Yλ , λ ) divides m(Yλ , μ). Proof. Assume that we have
η(s,e) (λ ) = i1 , . . . , i1 , i2 , . . . , i2 , . . . , is , . . . , is , a1
a2
as
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5 Representation Theory of Ariki–Koike Algebras
where, for all j = 1, . . . , s − 1, we have i j = i j+1 . We show by induction on s that m(Yλ , λ ) = ∏1is (ai !). Let λ be the l-partition obtained from λ by removing the nodes filled by n, n − 1, . . . , n − as + 1 in t(s,e) (λ ). By Lemma 5.7.20, we have
η(s,e) (λ ) = i1 , . . . , i1 , i2 , . . . , i2 , . . . , is−1 , . . . , is−1 , a1
a2
as−1
and by induction we may assume that we have m(Yλ , λ ) = ∏1is−1 ai !. Now, by (n) Corollary 5.6.11, Euλ appears with multiplicity as in (is -IndK )as (Euλ ). It thus sufμ
fices to show that the multiplicity of Euλ in any of the (is -IndK )as (Eu ), where m(Yλ , μ ) = 0, is zero. Assume to the contrary that we can remove as nodes with residue i from λ to obtain an l-partition μ which satisfies m(Yλ , μ ) = 0. Using Proposition 5.7.16, we see that μ m λ , which is a contradiction to Proposition 5.7.23. Thus, the formula for m(Yλ , λ ) is proved. Now, by Proposition 5.6.11, the integers m(Yλ , μ) are multiples of ∏1is ai !. So the result follows. (n)
5.8 On Basic Sets for Ariki–Koike Algebras We are now almost in position to prove the existence of canonical basic sets for our choice of weight sequence. We keep the setting of Section 5.7. Recall that we only consider the case where m = (s1 − v1 , . . . , sl − vl ), s = (s1 , . . . , sl ) ∈ Sel and v = (v1 , . . . , vl ) is such that 0 < v j − vi < e if i < j. At the end of the section, we will discuss the consequences for Iwahori–Hecke algebras of type Bn and Dn . To show the existence of basic sets for Hk,n , we need the following fundamental result. Theorem 5.8.1 (Ariki, Ariki–Mathas, Foda–Leclerc–Okado–Thibon–Welsh). In the above setting, the set Irr(Hk,n ) has cardinality equal to the number of FLOTW (n). l-partitions Φ(s,e) The analogous result for the case of the symmetric group was established in Lemma 3.4.13. Unfortunately, contrary to that case, there is no elementary known proof of Theorem 5.8.1. This will be solved in the next chapter using Ariki’s theorem and an interpretation of the FLOTW l-partitions in the context of crystal graph theory for quantum groups; see Theorem 6.3.2. In the case where char(k) = 0, the following result appeared in [174, Theorem 4.1]. We can now prove this result for any field k. (This is a new result and appears here for the first time.)
Theorem 5.8.2. Let k be a field, (s1 , . . . , sl ) ∈ Sel for e ∈ Z>1 . Assume that we have a specialisation θ : A → k such that θ (u) is a primitive root of unity of order e and θ (V j ) = θ (u)s j for all j = 1, . . . , l. Set r ∈ Z>0 and m = (s1 − v1 , . . . , sl − vl ), where
5.8 On Basic Sets for Ariki–Koike Algebras
303
v = (v1 , . . . , vl ) is such that 0 < v j − vi < e if i < j. With respect to the partial ors der induced by a(m,r) , the algebra Hk,n := Hk,n (ηe ; ηes1 , . . . , ηe l ) admits a canonical basic set Φ(s,e) (n) in the sense of Definition 5.5.19.
Proof. Recall that A is the ring of Laurent polynomials in finitely many indeterminates. Furthermore, we can assume without loss of generality that either K = R or that K is a finite extension of Q and that R is the ring of algebraic integers in K. In both cases, the general assumption on A in Theorem 3.4.4 is satisfied. We will use Proposition 3.4.5. First, recall that by our assumptions in Section 5.4, we have a well-defined decomposition map dθ : R0 (HK,n ) → R0 (Hk,n ); this map is surjective by the same argument as in Theorem 3.1.14. Now condition (a) of Proposition 3.4.5 is satisfied by Theorem 5.8.1. Next, recall the notation of Proposition 5.7.23. For each λ ∈ Φ(s,e) (n), define the following representation of HK,n :
Pλ = Eελ ⊕
a(m,r) (μ)>a(m,r) (λ )
m(Yλ , μ) μ E . m(Yλ , λ ) ε
This is well defined by Proposition 5.7.24. Furthermore, by Proposition 5.7.23, these elements satisfy conditions (b) and (c) of Proposition 3.4.5. As a consequence, Hk,n admits a canonical basic set given by the set Φ(s,e) (n). Remark 5.8.3. In the case where e = ∞ and s = (s1 , . . . , sl ) is such that s1 s2 . . . sl , the algebra Hk,n still admits a canonical basic set with respect to a(m,r) . This canonical basic set is given by the set of FLOTW l-partitions Φ(s, f ) (n) with
f sufficiently large; that is, the l-partitions λ = (λ 1 , . . . , λ l ) are such that, for all k+1 . The proof is easily adapted from the proof k = 1, . . . , l − 1, we have λik λi+s k+1 −sk in the case where e ∈ Z>1 . We now examine the consequences of the above result for the Iwahori–Hecke algebras of type Bn with n 2. First, let us look at the “equal-parameter case”. 5.8.4. Let W be of type Bn and consider the weight function L : W → Z such that L(t) = L(si ) = 1 for all i = 1, . . . , n − 1. Bn
1 t
4
1 t
1 t
· · ·
1 t
We will use the notation and results established in Example 5.7.3. Let k be such that char(k) = 2. We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = ηe , a primitive root of unity of order e where e ∈ Z>1 . By Example 5.7.3, one can assume that there exists d ∈ Z such that ηe = −ηed . Since char(k) = 2, this implies that e must be even. Then we have to take d = 1 + e/2 so that the condition () in Example 5.7.3 is satisfied. We then set l = 2, r = 1, s = (1 + e/2, e), v = (e/2, e). Note that s ∈ Se2 , so we can use Theorem 5.8.2 to compute the canonical basic set for Hk,n .
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5 Representation Theory of Ariki–Koike Algebras
The weight sequence is m = s − v = (1, 0). Note that the cyclotomic Ariki–Koike algebra associated with m = (1, 0) and r = 1 is the Hecke algebra of type Bn associated with L and thus that the a-invariant a(m,r) is the usual a-invariant in type Bn associated with L; see Example 5.5.14. In addition, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn over k. Theorem 5.8.2 thus implies the following result. Theorem 5.8.5. In the setting of 5.8.4, the algebra Hk,n admits a canonical basic . set Φ((1+e/2,e),e) Remark 5.8.6. Note that we have (λ 1 , λ 2 ) ∈ Φ((1+e/2,e),e) if and only if the two following conditions are satisfied: 2 1 1. We have λi1 λi+e/2−1 and λi2 λi+1+e/2 . 2. We associate with each node γ = (i, j, c) (where i denotes the row, j the column and c ∈ {1, 2} the component of γ ) of the Young diagram of (λ 1 , λ 2 ) its residue j − i mod e if c = 2, re (γ ) := j − i + 1 + e/2 mod e if c = 1.
Then, for all r > 0, the set of residues appearing at the right ends of the length r rows of (λ 1 , λ 2 ) is a proper subset of Z/eZ. Example 5.8.7. In the special case when e = 2, the associated basic set can be described in an easy way without the use of the notion of residues; that is, we have if and only if the three following conditions are satisfied: (λ 1 , λ 2 ) ∈ Φ((1+e/2,e),e) 1 , 1. λi1 λi2 and λi2 λi+2 2. λ 1 and λ 2 are 2-regular, 1 = 0. 3. for all i ∈ Z>0 , we cannot have λi2 = λi+1
Indeed, if 1. is satisfied, one can easily check that 2. and 3. are equivalent to the second condition of FLOTW bipartitions. 5.8.8. We still assume that W is a Weyl group of type Bn . Now we consider the weight function L : W → Z such that L(t) = 1 and L(si ) = 2 for all i = 1, . . . , n − 1. (This case, and the following one in 5.8.10, occurs in the representation theory of the finite unitary groups; see Example 4.4.16.) Bn
1 t
4
2 t
2 t
· · ·
2 t
Let k be a field of arbitrary characteristic. We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = η f , a root of unity of order f > 1. Assume that f = 2e, where e 3 is odd and set ηe := η 2f . Then ηe is a primitive root of unity of order e. Let e+1 = −η f and the condition () in Example 5.7.3 is d = (e + 1)/2. Then η 2d f = ηf satisfied. We set l = 2, r = 2, s = ((e + 1)/2, e) and v = (e/2, e). Note that we have s ∈ Se2 , so we can use Theorem 5.8.2 to compute the canonical basic set for Hk,n .
5.8 On Basic Sets for Ariki–Koike Algebras
305
The weight sequence that we have to consider is thus m = s − v = (1/2, 0). Note that the cyclotomic Ariki–Koike algebra associated with m = (1/2, 0) and r = 2 is the Iwahori–Hecke algebra of type Bn associated with L and thus that the a-invariant a(m,r) is the usual a-invariant in type Bn associated with L; see Example 5.5.14. In addition, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn over k, where ηe = η 2 . Theorem 5.8.2 thus implies the following result. Theorem 5.8.9. In the setting of 5.8.8, the algebra Hk,n admits a canonical basic set Bθ = Φ(((e+1)/2,e),e) . 5.8.10. We still assume that W is a Weyl group of type Bn . We now consider the weight function L : W → Z such that L(t) = 3 and L(si ) = 2 for all i = 1, . . . , n − 1. Bn
3 t
4
2 t
2 t
· · ·
2 t
Let k be a field of arbitrary characteristic. We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = η f , a primitive root of unity of order f > 1. As in the previous example, assume that f = 2e, where e 3 is odd, and set ηe := η 2f . Then ηe is a primie+3 = −η 3f and the tive root of unity of order e. Now let d = (e+3)/2. Then η 2d f = ηf condition () in Example 5.7.3 is satisfied. We set l = 2, r = 2, s = ((e+3)/2, e) and v = (e/2, e). Note that we have s ∈ Se2 , so we can use Theorem 5.8.2 to compute the canonical basic set for Hk,n . The weight sequence that we have to consider is thus m = s − v = (3/2, 0). Note that the cyclotomic Ariki–Koike algebra associated with m = (3/2, 0) and r = 2 is the Iwahori–Hecke algebra of type Bn associated with L and thus that the a-invariant a(m,r) is the usual a-invariant in type Bn associated with L; see Example 5.5.14. In addition, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn over k, where ηe = η 2 . Theorem 5.8.2 thus implies the following result. Theorem 5.8.11. In the setting of 5.8.10, the algebra Hk,n admits a canonical basic . set Bθ = Φ(((e+3)/2,e),e) 5.8.12. We still assume that W is a Weyl group of type Bn . We now consider the weight function L : W → Z such that L(t) = 2 and L(si ) = 1 ∈ Z>0 for all i = 1, . . . , n − 1. Bn
2 t
4
1 t
1 t
· · ·
1 t
We assume that k is a field such that char(k) = 2. We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = ηe , a primitive root of unity of order e, where e ∈ Z>1 . By Example 5.7.3, one can assume that there exists d ∈ Z such that ηe2 = −ηed . Hence, e is even and we have to set d = e/2 + 2 so that the condition () in Example 5.7.3 is satisfied. We then set l = 2, r = 1, s = (2 + e/2, e), v = (e/2, e). Note that s ∈ Se2 only if e = 2. So we have to assume that e = 2 in order to use Theorem 5.8.2.
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5 Representation Theory of Ariki–Koike Algebras
The weight sequence is m = s − v = (2, 0). Note that the cyclotomic Ariki–Koike algebra associated with m = (2, 0) and r = 1 is the Iwahori–Hecke algebra of type Bn associated with L and thus that the a-invariant a(m,r) is the usual a-invariant in type Bn associated with L; see Example 5.5.14. In addition, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn over k. Theorem 5.8.2 thus implies the following result. Theorem 5.8.13. In the setting of 5.8.12, when e = 2, the algebra Hk,n admits a . canonical basic set Φ((2+e/2,e),e) Note that the case where e = 2 is discussed in Example 4.4.15 and Remark 6.7.12. 5.8.14. We assume that W is a Weyl group of type Bn . Finally, we consider the weight function L : W → Z such that L(t) = 0 and L(si ) = 1 for all i = 1, . . . , n − 1. Bn
0 t
4
1 t
1 t
· · ·
1 t
We assume that k is a field such that char(k) = 2. We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = ηe , a primitive root of unity of order e, where e ∈ Z>1 . By Example 5.7.3, one can assume that there exists d ∈ Z such that ηe0 = −ηed . Hence, e is even and we have to set d = e/2 so that the condition () in Example 5.7.3 is satisfied. We then set l = 2, r = 1, s = (e/2, e), v = (e/2, e). Note that we have s ∈ Se2 . The weight sequence is m = s − v = (0, 0). Note that the cyclotomic Ariki–Koike algebra associated with m = (0, 0) and r = 1 is the Iwahori– Hecke algebra of type Bn associated with L and thus that the a-invariant a(m,r) is the usual a-invariant in type Bn associated with L. In addition, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn over k. Theorem 5.8.2 thus implies the following result. Theorem 5.8.15. In the setting of 5.8.14, the algebra Hk,n admits a canonical basic . set Bθ = Φ((e/2,e),e) Remark 5.8.16. Note that we have (λ 1 , λ 2 ) ∈ Φ((e/2,e),e) if and only if the two following conditions are satisfied: 2 1 1. We have λi1 λi+e/2 and λi2 λi+e/2 . 2. We associate with each node γ = (i, j, c) (where i denotes the row, j the column and c ∈ {1, 2} the component of γ ) of the Young diagram of (λ 1 , λ 2 ) its residue j − i mod e if c = 2, re (γ ) := j − i + e/2 mod e if c = 1.
Then, for all r > 0, the set of residues appearing at the right ends of the length r rows of (λ 1 , λ 2 ) is a proper subset of Z/eZ. Example 5.8.17. In the special case when e = 2, the associated basic set can be described in an easy way without the use of the notion of residues. We have (λ 1 , λ 2 ) ∈ Bθ if and only if
5.8 On Basic Sets for Ariki–Koike Algebras
307
2 and λ 2 λ 1 ; 1. λi1 λi+1 i i+1 1 2. λ and λ 2 are 2-regular; 3. for all i ∈ Z>0 , we cannot have λi2 = λi1 = 0.
Indeed, if 1. is satisfied, one can easily check that 2. and 3. are equivalent to the second condition of FLOTW bipartitions. 5.8.18. We now also obtain an explicit description of the canonical basic set for type Dn , first established in [173]. We place ourselves in the setting of Example 3.2.15. Let W be a Weyl group of type Bn with function L as in 5.8.14; that is, L(t) = 0 and L(si ) = 1 for all i = 1, . . . , n − 1. Let W˜ ⊆ W be the reflection subgroup of type Dn and L˜ : W˜ → Z0 be the weight function such that W˜ = L|W1 ; note that L˜ takes value ˜ be the generic Iwahori–Hecke algebra associated 1 on each generator of W˜ . Let H −1 ˜ ˜ with W , L over the ring R[v, v ]. Let u := v2 . Let k be a field such that char(k) = 2 and let θ : R[v, v−1 ] → k be a specialisation. As before, write ηe = θ (u), where e 1 denotes the order of θ (u) ∈ k× . If e = 1 or ˜ k is given by Example 3.2.15 and if e ∈ Z>1 is odd, then a canonical basic set for H the results in Section 4.6. Theorem 5.8.19. In the setting of 5.8.18, and using the notation of Example 3.2.15, ˜ k admits a canonical basic set given by assume that e ∈ Z>1 is even. Then H Bθ = {[λ 1 , λ 2 ] | λ 1 = λ 2 , (λ 1 , λ 2 ) ∈ Φ((e/2,e),e) (n)}
∪ {(λ , ±) | λ is (e/2)-regular of rank n/2}
(if n is even).
Proof. By Example 3.2.15 combined with Theorem 5.8.15, the canonical basic set in type Dn is given by the set (n)} Bθ = {[λ 1 , λ 2 ] | λ 1 = λ 2 , (λ 1 , λ 2 ) ∈ Φ((e/2,e),e) ∪ {(λ , ±) | (λ , λ ) ∈ Φ((e/2,e),e) (n)}
(if n is even).
Now every bipartition of the form (λ , λ ) has the property 1. in Remark 5.8.16. Furthermore, one easily checks that property 2. is satisfied if and only if λ is (e/2)regular. Remark 5.8.20. By Remark 5.8.16, we have (λ 1 , λ 2 ) ∈ Φ((e/2,e),e) if and only if 2 1 (λ , λ ) ∈ Φ((e/2,e),e) .
Recall that the validity of the main results of this section depends on Theorem 5.8.1. In Chapter 6 we introduce the objects that will be needed to give a proof of this result.
Chapter 6
Canonical Bases in Affine Type A and Ariki’s Theorem
The main purpose of this chapter is to go further in the study of the representation theory of Ariki–Koike algebras. To do this, we will need to take an excursion to another area of representation theory: the theory of canonical bases and crystal bases for Kac–Moody algebras and quantum algebras. In fact, we will focus on a particular e ). A fundamental result in this theory is the case: the quantum affine algebra Uq (sl existence, proved independently by Kashiwara and Lusztig, of a “canonical basis” e )-modules: the irreducible highest weight modules. We for certain irreducible Uq (sl shall see that this theory has a deep interpretation in the modular representation theory of Iwahori–Hecke algebras. Indeed, the matrices associated with the canonical bases for these modules give the decomposition matrices for Ariki–Koike algebras in characteristic 0. This result has first been conjectured for Iwahori–Hecke algebras of type A by Lascoux, Leclerc and Thibon [208] and proved for the wider class of Ariki–Koike algebras by Ariki in [7]. Of course, it is not the purpose of this book to give a complete survey on the theory of quantum algebras and canonical bases; we refer to [158], [185], [188], [190], [191] and [230] for details on this area. We will also not give the proof of Ariki’s theorem, as there is already a book on this subject [10]. The main aim of this chapter will be to present an explicit way to compute the canonical bases for irreducible highest weight modules following [174]. As a consequence, using Ariki’s theorem and some deep results by Uglov, we will be able to complete the description of the canonical basic sets in type Bn and Dn . The chapter is organized as follows. The first two parts give a brief introduce ) and the theory of tion to the representation theory of the quantum algebra Uq (sl e ) on the crystal and canonical bases. In particular, we study the action(s) of Uq (sl Q(q)-vector space generated by the set of multipartitions, namely the Fock space. This will allow us to give an explicit realisation of the irreducible highest weight e )-modules. The end of the second part is devoted to Ariki’s theorem and the Uq (sl connection of the theory of canonical bases with the modular representation theory of Iwahori–Hecke algebras. In the third section, we describe a fundamental object e )-module: the crystal graph. attached with each irreducible highest weight Uq (sl M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7 6, © Springer-Verlag London Limited 2011
309
310
6 Canonical Bases in Affine Type A and Ariki’s Theorem
We then study the “canonical bases” for irreducible highest weight modules and present an explicit combinatorial algorithm for computing them. These results will give a way to compute the decomposition matrices of Ariki–Koike algebras. Ariki’s theorem will allow us to complete the proof of a result which was necessary in Chapter 5 to obtain the canonical basic sets in certain cases for types Bn and Dn . For the remaining cases, a much deeper study of the theory of Fock spaces will be necessary. This will be done in the remaining parts, where Uglov’s theory of canonical bases for Fock spaces is studied. The chapter ends with the determination of the canonical basic sets in the remaining cases. We then close with some comments on recent developments and problems connecting Iwahori–Hecke algebras, the theory of canonical bases for quantum groups and some other class of algebras: the Cherednik algebras and the Khovanov–Lauda–Rouquier algebras.
e ) 6.1 The Quantum Affine Algebra Uq (sl e ) following [10, Chap. 3]. We begin by recalling the definition of the algebra Uq (sl Then we study several aspects of its representation theory. 6.1.1. Let e ∈ Z>0 and let h be the Q-vector space with basis {h0 , . . . , he−1 , D}. Let {Λ0 , . . . , Λe−1 , ∂ } be a basis of h∗ defined with respect to the pairing , : h∗ × h → Q, such that, for 0 i, j e − 1, we have Λi , h j = δi, j ,
Λi , D = ∂ , hi = 0,
∂ , D = 1,
where we set δi, j = 0 if i = j and δi,i = 1. In the following, if i ∈ Z, we will put Λi := Λi mod e . In addition, for l ∈ Z>0 and s = (s1 , . . . , sl ) ∈ Zl , we set Λs := ∑1il Λsi . For any integer i such that 0 i e − 1, we define the following elements in h∗ :
αi := −Λi−1 + 2Λi − Λi+1 + δi,0 ∂ . Definition 6.1.2. The elements of {Λi | 0 i e − 1} are called the fundamental e. weights and the elements of {αi | 0 i e − 1} are called the simple roots for sl We put P :=
e−1 i=0
ZΛi ⊕ Z∂ ,
P+ :=
e−1 i=0
Z0Λi ⊕ Z∂ ,
Q :=
e−1
Zαi .
i=0
Then P is called the weight lattice, P+ the set of dominant weights and Q the root e . The matrix (ai j )0i, je−1 such that ai j = αi , h j for 0 i, j e − 1 is lattice of sl e. called the Cartan matrix of sl
e ) 6.1 The Quantum Affine Algebra Uq (sl
311
We will need the following standard notation. For k ∈ Z0 , we denote [k] =
qk − q−k q − q−1
For 0 k m, we set
and
[k]! := [k][k − 1] . . . [1].
[m]! m , := k [m − k]![k]!
with [0]! = 1 by convention. Following [192], we obtain the following definition of the quantum affine algebra. e ) is the unital associative Q(q)Definition 6.1.3. The quantum affine algebra Uq (sl algebra with • generators: ei , fi , ti , ti−1 (0 i e − 1) and d, d−1 ; • relations: dd−1 = d−1 d = 1,
titi−1 = ti−1ti = 1,
ti f j ti−1 = q−ai j f j ,
ei f j − f j ei = δi j
[d, ei ] = δi0 ei ,
ti t j = t j ti , ti − ti−1 q − q−1
[d, f i ] = −δi0 fi ,
ti e j ti−1 = qai j e j ,
(0 i, j e − 1), [d,ti ] = 0;
and the following relations known as “Serre relations”, where i = j: 1−a −k k 1 − ai j ei i j e j eki = 0, ∑ (−1) k 0k1−a ij
∑
(−1)
0k1−ai j
k
1 − ai j k
1−ai j −k
fi
f j fik = 0.
e ) the subalgebra of Uq (sl e ) generated by ei , fi , t ±1 Moreover, we denote by Uq (sl i with 0 i e − 1. We have an analogue of the Poincar´e–Birkhoff–Witt theorem for this algebra. e ) generated Indeed, Let Uq (n+ ) (or Uq (n− ), or Uq (h)) be the subalgebra of Uq (sl by the ei (or fi , or ti±1 ) with 0 i e − 1. Then we have an isomorphism of Q(q)vector spaces (see [10, Theorem 6.2]): Uq (n+ ) ⊗ Uq (h) ⊗ Uq (n− )
e ) → Uq (sl
n+ ⊗ h ⊗ n−
→ n+ hn− .
e ) which turns it into a Hopf algebra. Remark 6.1.4. There is a coproduct Δ on Uq (sl This is given by the following formulae, where 0 i e − 1:
312
6 Canonical Bases in Affine Type A and Ariki’s Theorem
Δ (ei ) = ei ⊗ ti−1 + 1 ⊗ ei , Δ ( fi ) = f i ⊗ 1 + ti ⊗ fi , Δ (ti ) = ti ⊗ ti , Δ (ti−1 ) = ti−1 ⊗ ti−1 , Δ (d) = d ⊗ 1 + 1 ⊗ d. 6.1.5. For 0 i e − 1 and k ∈ Z0 , we define the divided powers of ei and fi as follows: ek fk (k) (k) and fi = i . ei = i [k]! [k]! e ) and k ∈ Z0 , we also set Moreover, if x ∈ Uq (sl x − x−1 {x} = , q − q−1
x = 1, 0
{x}{q−1 x} . . . {q−(k−1) x} x . = k [k]!
e ). This is the Q[q, q−1 ]-subalgebra Uq (sl e ) We now introduce a Q-form for Uq (sl Q t (k) (k) i e ) generated by the divided powers e , f and by of Uq (sl , ti , where k ∈ Z0 i i k e )-module, we denote by MQ the associated and 0 i e − 1. If M is an Uq (sl Uq (sle )Q -module. e )-modules and then focus on a particular 6.1.6. We briefly recall the theory of Uq (sl e )-module and class of Uq (sle )-modules: the integrable modules. Let M be a Uq (sl e+1 let Λ = ∑0ie−1 aiΛi + d ∂ ∈ P for (a0 , . . . , ae−1 , d) ∈ Z . The Q(q)-vector space generated by the set {m ∈ M | d.m = dm and ti m = qai m for all i ∈ {0, 1, . . . , e − 1}} is called the Λ -weight space and it is denoted by MΛ . If MΛ = 0, we say that Λ is a weight for M and the non-zero elements in MΛ are called weight vectors of weight Λ . We say that a non-zero element u ∈ MΛ is a highest weight vector with e )-module M is a weight Λ if ei u = 0 for all i ∈ {0, 1, . . . , e − 1}. We say that a Uq (sl cyclic module with highest weight Λ if there exists a highest weight vector mΛ with e ).mΛ . Similarly, we say that a U (sl e )-module M is weight Λ such that M = Uq (sl q cyclic with highest weight Λ if there exists a highest weight vector mΛ with weight e ).mΛ . In this case, by the Poincar´e–Birkhoff–Witt theorem, Λ such that M = Uq (sl the weight space MΛ has dimension 1. If
Λ=
∑
aiΛi + d ∂ ∈ P+ ,
0ie−1
then the positive integer l = ∑0ie−1 ai is called the level of M. For each Λ ∈ e )-module with highest weight Λ , P, there exists a unique simple and cyclic Uq (sl
e ) 6.1 The Quantum Affine Algebra Uq (sl
313
e ) is denoted by V (Λ ). which we denote by V (Λ ). The restriction of V (Λ ) to Uq (sl We have that (see [10, Lemma 6.9]): e )-modules if and only if Λ 1 = Λ 2 , • V (Λ 1 ) V (Λ 2 ) as Uq (sl e )-modules if and only if Λ 1 − Λ 2 ∈ Z∂ . • V (Λ 1 ) V (Λ 2 ) as U (sl q
e )-module M is an integrable module if and Definition 6.1.7. We say that a Uq (sl only if we have:
• M = Λ ∈P MΛ , • for all Λ ∈ P, MΛ is a finite-dimensional vector space, • for all i ∈ {0, 1, . . . , e − 1}, ei and fi act locally nilpotently on M; that is, for any m ∈ M, there exists N ∈ Z>0 such that eNi .m = f iN .m = 0. e )-modules M such that Uq (n+ )u is finite dimenThe category of integrable Uq (sl sional for all u ∈ M is denoted by Oint . 6.1.8. By the work of Lusztig (see [191, Section 3.3] or [10, Lemma 6.9]), we have the following properties: e )-module V (Λ ) is irreducible and is • For each Λ ∈ P+ , the highest weight Uq (sl in Oint . • Each irreducible highest weight module in Oint is isomorphic to a V (Λ ) with Λ ∈ P+ . e )-module M in Oint is • The category Oint is semisimple: each integrable Uq (sl + isomorphic to a direct sum of V (Λ ) with Λ ∈ P . 6.1.9. We give a brief introduction to the theory of crystal bases, crystal graphs and canonical bases following [10, Sections 4–9] (see also [189], [230].) Let M be an e )-module and let Λ ∈ P. Let x ∈ MΛ and i ∈ {0, 1, . . . , e − 1}. Set integrable Uq (sl N := Λ , hi . Then one can show that x can be written uniquely in the form x=
∑
(s)
fi xs ,
smax(0,−N)
where ei xs = 0, ti xs = qΛ ,hi +2s xs and only finitely many xs are non-zero (see [10, Section 4.2].) Definition 6.1.10. Keeping the above notations, we define e i x =
∑
smax(0,−N)
(s−1)
fi
xs
and
f i x =
∑
(s+1)
fi
xs
smax(0,−N)
for i ∈ {0, 1, . . . , e − 1}. The operators e i and f i are called the Kashiwara operators. To formulate the following definition, we set a(q) A := ∈ Q(q) a(q), b(q) ∈ Q[q], b(0) = 0 . b(q)
314
6 Canonical Bases in Affine Type A and Ariki’s Theorem
e )Definition 6.1.11. A pair (L, B) is called a crystal basis for the integrable Uq (sl e )-module M ) if it satisfies the following conditions: module M (and for the Uq (sl 1. L is a free A-module such that Q(q) ⊗A L M. 2. B is a basis of the Q-vector space L/qL. 3. L = Λ ∈P LΛ and B = Λ ∈P BΛ , where LΛ = L ∩ MΛ and BΛ = B ∩ (LΛ /qLΛ ). 4. e i L ⊂ L, f i L ⊂ L for all 0 i e − 1. 5. e i B ⊂ B ∪ {0}, f i B ⊂ B ∪ {0}. 6. For any i ∈ {0, 1, . . . , e − 1} and u, v in B, we have u = e i v if and only if v = f i u. One can associate with any crystal basis a coloured oriented graph as follows. e )-module M. Definition 6.1.12. Let (L, B) be a crystal basis for an integrable Uq (sl The crystal graph G (or the crystal) associated with (L, B) is the coloured oriented graph with • vertices: the elements in B; i • edges: for u and v in B we have an arrow u → v coloured by i ∈ {0, 1, . . . , e − 1} if and only if v = f i u. An example of such a graph is given in Example 6.2.18. The following theorem shows the existence of a crystal basis for every irreducible e )-modules. We refer to [190, Theorem 2] for its proof. highest weight Uq (sl Theorem 6.1.13 (Kashiwara, Lusztig). Let Λ ∈ P+ and set L(Λ ) = ∑ A f i1 f i2 . . . f ik mΛ ,
B(Λ ) = f i1 f i2 . . . f ik mΛ mod qL(Λ )) \ {0}. Then (L(Λ ), B(Λ )) is a crystal basis for V (Λ ). Moreover, any crystal basis of V (Λ ) coincides with (L(Λ ), B(Λ )) up to a scalar multiple. e )-modules and if (L1 , B1 ) and (L2 , B2 ) are 6.1.14. If M1 and M2 are integrable Uq (sl crystal bases of M1 and M2 respectively, then we say that (L1 , B1 ) and (L2 , B2 ) are e )-modules Ψ : M1 → M2 such isomorphic if there exists an isomorphism of Uq (sl that the following conditions hold: • Ψ induces an isomorphism of A-lattices Φ : L1 → L2 . • We have Φ (B1 ) = B2 , where Φ is the induced isomorphism of vector spaces Φ : L1 /qL1 → L2 /qL2 . Note that if (L1 , B1 ) and (L2 , B2 ) are isomorphic, the associated bijection
Φ : B1 {0} → B2 {0} (with Φ (0) = 0) has the the following property. For all i ∈ {0, . . . , e−1} and b1 ∈ B1 , we have Φ ( f i b) = f i Φ (b) and Φ (
ei b) = e i Φ (b). This implies that we have an arrow
e ) 6.1 The Quantum Affine Algebra Uq (sl
315
i
b → b in the crystal graph G1 associated with (L1 , B1 ) if and only if we have an i arrow Φ (b) → Φ (b ) in the crystal graph G2 associated with (L2 , B2 ). If a bijection between B1 and B2 satisfies such a property, then we say that there is an isomorphism of crystal graphs between G1 and G2 . The following result shows the compatibility of the crystal bases theory with the e )-modules in Oint . The first assertion implies the existence of a direct sum of Uq (sl e )-modules in Oint by Theorem 6.1.13 and the second one crystal basis for the Uq (sl the uniqueness up to isomorphism. We refer to [190, Theorem 3] for its proof. Proposition 6.1.15 (Kashiwara). e ) modules in Oint equipped with crystal bases 1. Let M1 and M2 be integrable Uq (sl (L1 , B1 ) and (L2 , B2 ). Then (L1 , B1 ) ⊕ (L2 , B2 ) := (L1 ⊕ L2 , B1 B2 ) is a crystal basis for M = M1 ⊕ M2 . e )-module in Oint equipped with a crystal basis 2. Let M be an integrable Uq (sl (L, B). Then there exists an isomorphism between M and a direct sum Λ V (Λ ), highest weight modules, which induces where the V (Λ ) are certain irreducible an isomorphism between (L, B) and Λ (L(Λ ), B(Λ )). 6.1.16. Now, the theorem below, independently proved by Kashiwara and Lusztig, shows that the basis B(Λ ) of L(Λ )/qL(Λ ) can be lifted to obtain a basis of V (Λ ). e ). This is To state the theorem, we first need to introduce an involution on Uq (sl given as follows: q = q−1 ,
ei = ei ,
fi = fi ,
ti = ti−1 ,
∂ =∂
for 0 i < e. This induces an involution on V (Λ )Q with Λ ∈ P+ by setting u.mΛ = u.mΛ
e ) . for all u ∈ Uq (sl Q
Theorem 6.1.17 (Kashiwara [190], Lusztig [230]). Let Λ ∈ P+ , let (L(Λ ), B(Λ )) be the crystal basis for V (Λ ). The module V (Λ )Q has a unique Q[q, q−1 ]-basis B(Λ ) = {G(b) | b ∈ B(Λ )} such that for all b ∈ B(Λ ) we have G(b) = G(b)
and
G(b) ≡ b mod qL(Λ ).
This basis is called the Kashiwara–Lusztig canonical basis (or the global crystal e )-module V (Λ ) and for the U (sl e )-module V (Λ ). basis) for the Uq (sl q Note that the specialisation at q = 1 of the above canonical basis leads to a canon e ). ical basis for the irreducible highest weight modules of U (sl
316
6 Canonical Bases in Affine Type A and Ariki’s Theorem
6.2 The Fock Space and Ariki’s Theorem The aim of this section is to give a realisation of the irreducible highest weight e )-modules. To do this, we construct an action of Uq (sl e ) on a remarkable Uq (sl Q(q)-vector space called the Fock space. We then deduce the structure of the desired modules and give the associated crystals. Finally, we present the connections with the representation theory of Ariki–Koike algebras given by Ariki’s theorem. Definition 6.2.1. Let l ∈ Z>0 and let s ∈ Zl . The Fock space Fs is the Q(q)-vector space with a distinguished basis |λ , s with λ l n. In other words, we denote Fsn =
Q(q)|λ , s
for n ∈ Z0 ,
λ l n
Fs =
Fsn .
n∈Z0
The positive integer l is called the level of the Fock space Fs and s is called the l-charge or the multicharge of Fs . 6.2.2. Let s ∈ Zl and let i ∈ {0, . . . , e − 1}. Recall the definition of addable and removable i-nodes for an l-partition in 5.1.5. Recall also from 5.7.14 the total order that we have already introduced on the set of addable and removable i-nodes of an l-partition. Let γ , γ be two removable or addable i-nodes of λ . We denote either b − a + sc < b − a + sc , def γ ≺(s,e) γ ⇐⇒ or b − a + sc = b − a + sc and c > c . Remark 6.2.3. The version of the order ≺(s,e) used in [291] is slightly different. This is given as follows: we have γ <(s,e) γ if we have b − a + sc < b − a + sc or if b − a + sc = b − a + sc and c < c . One can recover all the results in [291] by reversing the order of components of λ on the basis vectors |λ , s of the Fock space. Our convention is in agreement with [88]. Example 6.2.4. Fix n ∈ Z>1 and assume that s ∈ Zl is such that si − s j > n − 1 − e for all i < j. Let γ = (a, b, c), γ = (a , b , c ) be two removable i-nodes of the same l-partition λ l n. Assume that c < c. Then, as already observed in Example 5.5.18, we have b − a − (b − a) 1 − n. We thus have b − a + sc − (b − a + sc ) > −e. As γ and γ have the same residue, this implies that b − a + sc − (b − a + sc ) 0. We conclude that we have γ ≺(s,e) γ . Hence, for this choice of s, we have γ ≺(s,e) γ if and only if c < c or if c = c and a < a (note, however, that this is only valid for l-partitions of rank n). This order on i-nodes is connected with the one used in Ariki’s book [10, Definition 10.7] (but now for general choice of λ l n and n ∈ Z>0 !). It can be seen as an “asymptotic” version of our choice of order. Let λ and μ be two l-partitions of rank n and n + 1 such that there exists an i-node γ such that [μ] = [λ ] ∪ {γ }. We define the following numbers:
6.2 The Fock Space and Ariki’s Theorem
317
Ni (λ , μ) ={addable i − nodes γ of λ such that γ (s,e) γ } − {removable i − nodes γ of μ such that γ (s,e) γ }, Ni≺ (λ , μ) ={addable i − nodes γ of λ such that γ ≺(s,e) γ } − {removable i − nodes γ of μ such that γ ≺(s,e) γ }, Ni (λ ) ={addable i − nodes of λ } − {removable i − nodes of λ }, Nd (λ ) ={0 − nodes of λ }. e ) on the Fock space. These numbers allow us to define an action of Uq (sl Theorem 6.2.5 (Jimbo, Misra, Miwa and Okado [186, Prop. 3.5]). Let s ∈ Zl . e )-module with action Then Fs is an integrable Uq (sl ei |λ , s =
∑
≺ (μ,λ )
q−Ni
re ([λ ]/[μ])=i
ti |λ , s = qNi (λ ) |λ , s,
|μ, s,
fi |λ , s =
∑
(λ ,μ)
qNi
re ([μ]/[λ ])=i
|μ, s,
d|λ , s = −(Δ (s) + Nd (λ ))|λ , s,
where 0 i e − 1 and Δ (s) ∈ Q is defined in [291, Section 2.1] (recall also that re (γ ) denotes the residue of the node γ ). e )-module Proof. One checks that the above formulae give a well-defined Uq (sl structure and that Fs is in Oint , directly from the definition of the action. Remark 6.2.6. The action of d is not really important for our purpose, as we will e ). mainly deal with the action of Uq (sl The following proposition generalises the definition of Ni (λ , μ) and Ni≺ (λ , μ) and gives the action of the divided powers on the standard basis of the Fock space. It is largely inspired from [245, Proposition 6.16] and it can be seen as a quantisation of Proposition 5.6.11. Proposition 6.2.7. Let λ be an l-partition, i ∈ {0, . . . , e − 1} and j ∈ Z>0 . 1. We have
(λ ,μ)
f i |λ , s = ∑ qNi ( j)
|μ, s,
where the sum is taken over all the |μ, s such that μ is obtained from λ by adding j nodes with residue i. We have addable i − nodes γ Ni (λ , μ) := ∑ of μ such that γ (s,e) γ γ ∈[μ]/[λ ] removable i − nodes γ . − of λ such that γ (s,e) γ 2. We have
≺ (λ ,μ)
ei |μ, s = ∑ q−Ni ( j)
|λ , s,
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
where the sum is taken over all the |λ , s obtained from μ by removing j nodes with residue i. We have addable i − nodes γ Ni≺ (λ , μ) := ∑ of μ such that γ ≺(s,e) γ γ ∈[μ]/[λ ] removable i − nodes γ . − of λ such that γ ≺(s,e) γ Proof. We only give the proof of the first assertion. The second one is completely analogous. We argue by induction on j ∈ Z>0 . If j = 1, this is the definition of the action in Theorem 6.2.5. So now assume that j > 1. Let μ be an l-partition such that μ = λ ∪ {γ1 , . . . , γ j } for i-nodes γ1 , . . . , γ j . We can assume that γ1 (s,e) γ2 (s,e) . . . (s,e) γ j . Let μs , s = 1, . . . , j, be the l-partitions such that [μ]/[μs ] = γs . By induction, the ( j−1) |λ , s is qNi (λ ,μs ) . Hence, the coefficient of |μ, s in coefficient of |μs , s in fi ( j−1) |λ , s is ∑ qNi (λ ,μs )+Ni (μs ,μ) . fi fi 1s j
Now, we define the following numbers: addable i − nodes η Ni (λ , μ, γs ) := of μ such that η (s,e) γs removable i − nodes η , − of λ such that η (s,e) γs where s = 1, . . . , j. We have Ni (λ , μs ) =
∑
Ni (λ , μs , γt ) +
∑
Ni (λ , μ, γt ) +
1ts−1
∑
Ni (λ , μs , γt )
s+1t j
∑
(Ni (λ , μ, γt ) + 1) 1ts−1 s+1t j = Ni (λ , μ) − Ni (λ , μs , γs ) + j − s, Ni (μs , μ) = Ni (λ , μs , γs ) − s + 1. =
( j−1)
Hence, the coefficient of |μ, s in fi fi ( j−1)
( j)
(λ ,μ)
|λ , s is qNi
∑
q j+1−2s . Now, we
1s j ( j)
(λ ,μ)
have f i fi |λ , s = [ j]q fi . Thus, the coefficient of |μ, s in f i |λ , s is qNi This concludes the proof.
.
6.2.8. An interesting consequence of the above constructions is that we will have e )-modules. To be several ways to realise the same irreducible highest weight Uq (sl more precise, we introduce the extended affine symmetric group. As this object will play a crucial role in the next parts, we give several useful comments on this group.
6.2 The Fock Space and Ariki’s Theorem
319
r of type A(1) is by definition the Coxeter Let r ∈ Z>0 . The affine symmetric group S r−1 group with presentation by: • Generators: σi with i = 0, 1, . . . , r − 1. • Relations: σi σi+1 σi = σi+1 σi σi+1 for all i = 0, . . . , r − 1, σi σ j = σ j σi if i − j ≡ 1 mod r, σi2 = 1 for all i = 0, . . . , r − 1, where the subscripts are understood to be modulo r. The extended affine symmetric r is the semi-direct product S
r τ such that τ Z and τσi = σi+1 τ for group S all i = 0, . . . , r − 1. Hence it has a presentation by: • Generators: τ , σi with i = 0, 1, . . . , r − 1. • Relations: σi σi+1 σi = σi+1 σi σi+1 σi σ j = σ j σi σi2 = 1 τσi = σi+1 τ
for all i = 0, . . . , r − 1, if i − j ≡ 1 mod r, for all i = 0, . . . , r − 1, if i = 0, . . . , r − 1,
where, again, the subscripts are understood to be modulo r. Note that the subgroup r generated by the elements σi , i = 1, . . . , r − 1 can be identified with the symof S metric group Sr of rank r. There is another way to define this group. Let Pr := Zr and let {yi | i = 1, . . . , r} r can then be seen as the semi-direct prodbe the standard basis of Pr . The group S uct Pr Sr , where the relations of the semidirect product are given by σi y j = y j σi for j = i, i + 1 and σi yi σi = yi+1 . In other words, we have a presentation by: • Generators: σi with i = 1, . . . , r − 1 and yi with i = 1, . . . , r. • Relations:
σi σi+1 σi σi σ j σi2 yi y j σi y j
= σi+1 σi σi+1 = σ j σi =1 = y j yi = y j σi
σi yi σi = yi+1
for all i = 1, . . . , r − 1, if i − j ≡ 1 mod r, for all i = 1, . . . , r − 1, for all i, j = 1, . . . , r, for all i = 1, . . . , r − 1, j = 1, . . . , r and j ≡ i, i + 1 mod r, for all i = 1, . . . , r − 1.
For a = (a1 , . . . , ar ) ∈ Zr , we denote ya = ya11 . . . yar r . One can go from one presentation to the other using the following formulae:
τ = σ1 . . . σr−1 yr ,
σ0 = σr−1 . . . σ2 σ1 σ2 . . . σr−1 y−1 1 yr .
r is a Coxeter group and, therefore, it has an associated Bruhat– As noted above, S r is not a Coxeter group. However, one Chevalley order and a length function. But S r : let
r to S can extend the Bruhat–Chevalley order and the length function from S 2 2 (w1 , w2 ) ∈ Sr ; then there exist unique elements (k1 , k2 ) ∈ Z , (x1 , x2 ) ∈ Sr such
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
that w1 = τ k1 x1 and w2 = τ k2 x2 . We then define w1 < w2 if k1 = k2 and x1 < x2 and we set l(w1 ) := l(x1 ). Hence the length of w is 0 if and only if w ∈ τ . e )-modue Fs defined in Definition 6.2.1. Let s ∈ 6.2.9. We now return to the Uq (sl l Z , then one easily check that the vector |∅, s is a highest weight vector for the e ) on Fs . Hence, the module action of Uq (sl e )|∅, s V (Λs )s := Uq (sl e )-module. The U (sl e )-module V (Λs )s := is an irreducible highest weight Uq (sl q e )|∅, s is an irreducible highest weight U (sl e )-module with weight Λs Uq (sl q (modulo ∂ ). Thus, it is isomorphic to V (Λs ) (see 6.1.6). Let us give an interesting consequence of this result. Consider e a fixed positive integer. Then we have an l on Zl by setting for any s = (s1 , . . . , sl ) ∈ Zl action of S
σc .s = (s1 , . . . , sc−1 , sc+1 , sc , sc+2 , . . . , sl ) for c = 1, . . . , r − 1 and yc .s = (s1 , . . . , sc−1 , sc + e, sc+1 . . . , sl ) for c = 1, . . . , r. A fundamental domain for this action is given by (s1 , . . . , sl ) ∈ Zl | 1 s1 . . . sl e , which is contained in the set Sel that we have already met in Section 5.7 (see in particular Definition 5.7.8.) In particular, note that if s1 = (s11 , . . . , s1l ) and s2 = (s21 , . . . , s2l ) are l-tuples of integers in the same orbit with respect to this action, then 1 2 e )-modules with V (Λs1 )s and V (Λs2 )s are both irreducible highest weight Uq (sl the same weight (modulo ∂ ). As a consequence, we have an isomorphism V (Λs1 )s → V (Λs2 )s , 1
2
such that the canonical basis of the first module is mapped to the canonical basis of the latter by this isomorphism (see 6.1.6). e )-module, by The6.2.10. Let s ∈ Zl . As the Fock space Fs is an integrable Uq (sl orem 6.1.15, it is equipped with an associated crystal basis and a crystal graph. We begin this section with the description of this graph. We then deduce the crystal e )-module. graph of an arbitrary irreducible highest weight Uq (sl Let λ be an l-partition. We can consider its set of addable and removable i-nodes. Let wi (λ ) be the word obtained first by writing the addable and removable i-nodes of λ in increasing order with respect to ≺(s,e) and then by encoding each addable i i (λ ) = A p Rq node by the letter A and each removable i-node by the letter R. Write w for the word derived from wi by deleting as many subwords of type RA as possible. wi (λ ) is called the i-word of λ and w
i (λ ) the reduced i-word of λ . The addable
i (λ ) are called the normal addable i-nodes. The removable i-nodes in i-nodes in w
i (λ ) are called the normal removable i-nodes. If p > 0, let γ be the rightmost w
i . The node γ is called the good addable i-node. If q > 0, the addable i-node in w
6.2 The Fock Space and Ariki’s Theorem
321
i is called the good removable i-node. Note that this leftmost removable i-node in w notion depends on the order ≺e,s and, thus, on the choice of s ∈ Zl . These definitions are due to Kleshchev [204] (when l = 1) who used them to describe the branching rules for symmetric group in positive characteristic. Example 6.2.11. We consider the 2-partition λ = (2.2, 2.1.1). Set e = 4 and s = (0, 1); then λ has the following diagram with residues: ⎛ ⎞ 1 2 ⎝ 0 1 , 0 ⎠. 3 0 3 2 = AA and the 2-node (3, 1, 1) is a good We have w2 = AARA. Hence, we have w addable 2-node with respect to ≺(s,e) . There is no normal removable 2-node (and thus no removable good i-node). If we now take s = (0, 5), then λ has the same diagram with residues. However, the word w2 is not the same because the order 2 = ≺(s,e) has been modified: in this case, we have w2 = AAAR. We then have w AAAR and the 2-node (4, 1, 2) is a good addable 2-node with respect to ≺(s,e) . The node (1, 2, 2) is a good removale 2-node. Using the definition of the Kashiwara operators, we now get the following result. Theorem 6.2.12 (Jimbo, Misra, Miwa and Okado [186, §3]). Let λ be an lpartition. Then, we have |λ \ {γ }, s if γ is a good removable i-node, e i |λ , s = 0 if λ has no good removable i-node,
fi |λ , s = |λ ∪ {γ }, s if γ is a good addable i-node, 0 if λ has no good addable i-node, where the action is given modulo qL(Λs ). In other words, the crystal graph Ge (Λs )s of Fs is the coloured oriented graph with • vertices: the symbols |λ , s with λ l n and n ∈ Z0 ; i • arrows: |λ , s → |μ, s if and only if μ is obtained by adding to λ a good addable i-node. Definition 6.2.13. Let s ∈ Zl . The set of Uglov l-partitions Φ(s,e) is defined recursively as follows. • We have ∅ := (∅, ∅, . . . , ∅) ∈ Φ(s,e) . • If λ ∈ Φ(s,e) with λ = ∅, there exist i ∈ {0, . . . , e − 1} and a good removable i-node γ such that, if we remove γ from λ , the resulting l-partition is in Φ(s,e) . We also set Φ(s,e) (n) for the elements in Φ(s,e) with rank n. The following result is a direct consequence of Theorems 6.1.13 and 6.2.12.
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
e )Proposition 6.2.14. The crystal graph of the irreducible highest weight Uq (sl s s ◦ s module V (Λs ) is the connected component (Ge (Λs ) ) of the graph Ge (Λs ) containing |∅, s. Hence, this is the graph with • vertices: the symbols |λ , s with λ ∈ Φ(s,e) ; i
• arrows: |λ , s → |μ, s if and only if μ is obtained by adding to λ its good addable i-node. In particular, (L(Λs ), Φ(s,e) ) is the crystal basis of V (Λs )s . Remark 6.2.15. Note that the residue of a node does not change if we replace s = (s1 , . . . , sl ) by s+ = (s1 + e, . . . , sl + e), nor does the order between two arbitrary nodes with the same residue. As the notion of Uglov l-partitions only depends on these data, we deduce that Φ(s,e) = Φ(s+ ,e) . Example 6.2.16. Fix n ∈ Z>1 and assume that s ∈ Zl is such that si − s j n − 1 − e for all i < j. In this case, we say that s is a very dominant multicharge. Let λ be an l-partition and let γ be a good removable i-node with respect to ≺(s,e) . Let γ = (a , b , c ) be any removable or addable i-nodes of the same l-partition λ l n. Then we have seen in Example 6.2.4 that γ ≺(s,e) γ if and only if c < c or if c = c and a < a. Thus, in the definition of good removable i-node, one can replace ≺(s,e) with this order (for the l-partitions of rank less than n). In this case, the Uglov l-partitions (of rank n) are known as the Kleshchev l-partitions. They coincide with ∗ ∗ the ones in [10, Section 11.2] up to transformation (λ 1 , . . . , λ l ) → (λ 1 , . . . , λ l ). Note that this class of l-partitions only depends on the equivalence classes of the s j modulo e. We denote them by Φ (s mod e,e) . A non-recursive definition of them in the case l = 2 is derived from results of Ariki, Kreiman and Tsuchioka [14]. l , and s ∈ Zl , we have an isomorphism 6.2.17. By the discussion in 6.2.9, if w ∈ S s w.s between V (Λs ) and V (Λw.s ) . One can check that this implies that the crystal graphs (Ge (Λs )s )◦ and (Ge (Λw.s )w.s )◦ of V (Λs )s and V (Λw.s )w.s respectively are isomorphic in the sense of 6.1.14. However, they are really different in general. Indeed, the l-partitions labelling these graphs, Φ(s,e) and Φ(w.s,e) , are distinct (see Example 6.2.18), even if there is a bijection
ϕs,w.s : Φ(s,e) → Φ(w.s,e) induced by the isomorphism of crystal graphs. This bijection may be described by induction as follows. • We have ϕs,w.s |∅, s = |∅, w.s. • Let λ ∈ Φ(s,e) (n) for n ∈ Z>0 ; then there exist i ∈ {0, 1, . . . , e − 1}, a good removable i-node γ for λ (with respect to the order ≺(s,e) ; see 6.2.10) and λ ∈ Φ(s,e) (n − 1) such that [λ ] = [λ ] ∪ {γ }. Now μ := ϕs,w.s (λ ) is known by induction. Then there exist a good addable i-node γ for μ (with respect to the order ≺(w.s,e) ) and μ ∈ Φ(s,e) (n) such that [μ] = [μ ] ∪ {γ }. We then have ϕs,w.s (λ ) = μ.
6.2 The Fock Space and Ariki’s Theorem
323
A non-recursive combinatorial description of ϕs,w.s has been given in [177, Theorem 4.6] for l = 2 and [178, Theorem 5.4.2] in the general case. We illustrate the above phenomenon in the next example.
Fig. 6.1 Part of the crystal graph (G2 (Λ(0,0,1) )(0,0,1) )◦ (∅, ∅, ∅)
HH
0
(1, ∅, ∅) 1
(2, ∅, ∅) 0
(4, ∅, ∅)
(3, 1, ∅)
(∅, ∅, 1)
@0 R @
@ 0 R @
(1, 1, ∅)
@1 R @
(3, ∅, ∅)
1 0
(∅, ∅, 2)
@ 1 R @
(2, ∅, 1) 0
H1H j
(2, 1, ∅)
@ 1 R @
A1 U A
(3, ∅, 1) (2.1, ∅, 1)
(2, 2, ∅)
0
@1 R @
(1, ∅, 2)
Q 0 Q1 ? Q s
(1, 1, 2)
(∅, ∅, 3)
@0 R @
(1, ∅, 3)
(∅, ∅, 4)
Fig. 6.2 Part of the crystal graph (G4 (Λ(1,0,4) )(1,0,4) )◦ (∅, ∅, ∅)
HH
0
(∅, ∅, 1) 1
(∅, ∅, 2) 0
(∅, ∅, 3)
1 0 (∅, ∅, 4)
(1, ∅, ∅)
@0 R @
@ 0 R @ (1, ∅, 1)
(∅, 1, 1)
@1 R @
@ 1 R @
(∅, ∅, 2.1) (∅, 1, 2) 0
@ 1 R @
H1H j
A1 U A
0
(2, ∅, 1)
@1 R @
Q 0 Q1 ? Q s
(1, ∅, 2)
(∅, ∅, 3.1) (∅, ∅, 3.1)(1, ∅, 2.1) (∅, 1, 2.1) (2.1, ∅, 1) (2, ∅, 2)
@0 R @ (1, ∅, 3)
Example 6.2.18. In Figure 6.1 we show part of the crystal graph (G2 (Λ(0,0,1) )(0,0,1) )◦ of the irreducible highest weight module (V (2Λ0 + Λ1 )(0,0,1) ) with e = 2. To simplify the notation, we identify the symbols |λ , s with λ . The 3-partitions appearing in this graph are the Uglov 3-partitions of Φ((0,0,1),2) of rank less than or equal to 4.
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
Now consider the graph (G4 (Λ(1,0,4) )(1,0,4) )◦ of the irreducible highest weight module V (2Λ0 + Λ1 )(1,0,4) with e = 2. Here, V (2Λ0 + Λ1 )(1,0,4) is isomorphic to V (2Λ0 + Λ1 )(0,0,1) (see 6.1.6) and the associated graph for l-partitions of rank at most 4 is given in Figure 6.2. We see that the sets of Uglov l-partitions are different in these cases. 6.2.19. Let s ∈ Zl . By the above results, the module V (Λs )s generated by the empty l-partition is an irreducible highest weight module. Hence, it admits a canonical basis labelled by the set of Uglov l-partitions: q Bs = Gq (μ, s) | μ ∈ Φ(s,e) . For all μ ∈ Φ(s,e) (n), we have Gq (μ, s) =
∑ dλ ,μ (q)|λ , s ∈ V (Λs )sQ .
λ l n
e ) -module (Fs )Q . The isomorphism between U (sl e ) and Consider the Uq (sl Q Q⊗ −1 Uq (sle ) /ti −1 ; i = 0, . . . , e −1 induces a structure of U (sle )-module Q[q,q
]
Q
on (Fs )Q . The associated module is then denoted by (Fs )Q,1 and the submodule generated by the empty l-partition by V (Λs )sQ,1 . It is in fact an irreducible highest e )-module. A basis of V (Λs )s is naturally obtained by specialising weight U (sl Q,1
the canonical basis of V (Λs )s at q = 1. This basis is also called the canonical basis of V (Λs )Q,1 . It is the set Bs = G1 (μ, s) | μ ∈ Φ(s,e) .
For all μ ∈ Φ(s,e) (n), we have G1 (μ, s) =
∑ dλ ,μ (1)|λ , s.
λ l n
6.2.20. We now come back to the representation theory of Ariki–Koike algebras. Recall one of the main problems of this chapter: the determination of the decomposition numbers for Ariki–Koike algebras. By Theorem 5.4.3 and the discussion in 5.4.4, we can restrict ourselves to specialisations θ : A → k such that θ (u) = ηe , a primitive root of order e ∈ Z>1 and θ (Vi ) = ηesi for i ∈ {1, 2, . . . , l} with s = (s1 , . . . , sl ) ∈ Zl . The specialised algebra Hk,n is, in general, non-semisimple and we have an associated decomposition matrix: Dθ = (dλ ,M )λ l n,M∈Irr(Hk,n ) . For any simple Hk,n module M with n ∈ Z0 , we define the following elements of e )-module (Fs )Q,1 : the U (sl B(M, s) =
∑ dλ ,M |λ , s.
λ l n
6.3 Crystal and Canonical Bases for Highest Weight Modules
325
Then we consider the subspace of (Fs )Q,1 generated by all these elements: M s := B(M, s) | M ∈ Irr(Hk,n ), n ∈ Z0 Q , where we denote B(M, s) = |∅, s if n = 0. The following fundamental result asserts that these elements correspond to the canonical basis elements of the irreducible highest weight modules.
Theorem 6.2.21 (Ariki [7], [10, Theorem 12.5]). In the above setting, assume that e ) e )-module M s is equal to the U (sl the characteristic of k is zero. Then, the U (sl s module V (Λs )Q,1 . In addition, we have Bs = B(M, s) | M ∈ Irr(Hk,n ), n ∈ Z0 .
As a consequence, the problem of computing the decomposition matrices for Ariki–Koike algebras in characteristic 0 is equivalent to the problem of determining e )-modules. In fact, in posthe canonical basis for irreducible highest weight Uq (sl itive characteristic, the first part of this theorem is still true as shown by Ariki and Mathas. The precise statement is as follows. Theorem 6.2.22 (Ariki and Mathas [15, Theorems A, C and 2.3]). Without any e )-module M s still is equal to the condition on the characteristic of k, the U (sl s U (sle )-module V (Λs )Q,1 . In particular, we have | Irr(Hk,n ))| = |Φ(s,e) (n)|. Thus, the cardinality of the set Irr(Hk,n ) only depends on s, e and n, but not on the particular field k. This theorem gives us a first result concerning the problem of computing the number of simple modules for Hk,n . Remark 6.2.23. Let s := (s1 , . . . , sl ) ∈ Zl and consider the Ariki–Koike algebra s l and define s := (s , . . . , s ) Hk,n := Hk,n (ηe ; ηes1 , . . . , ηe l ) as above. Let now w ∈ S 1 l l l on Z given in 6.2.9. Then it is easy to check such that s = w.s for the action of S s
s
that we have Hk,n = H k,n , where H k,n := Hk,n (ηe ; ηe 1 , . . . , ηe l ). From this and Theorem 6.2.22, we deduce that the cardinality of the set Irr(Hk,n ) actually only depends l on Zl . on e, n and the class of s modulo the action of S We will now be interested in the description of Φ(s,e) (n).
6.3 Crystal and Canonical Bases for Highest Weight Modules We will now be interested in the explicit description of the crystal basis and the e )-module. The aim canonical basis of an arbitrary irreducible highest weight Uq (sl
326
6 Canonical Bases in Affine Type A and Ariki’s Theorem
will be to give an explicit algorithm for computing this remarkable basis. To do this, we first need to study the l-partitions labelling the crystal basis: the Uglov l-partitions. This is done in this section. l domain for 6.3.1. Recall that we have an action of lSl on Z and that a fundamental this action is given by (s1 , . . . , sl ) ∈ Z | 1 s1 . . . sl e ⊂ Sel , where
Sel := {s = (s1 , . . . , sl ) ∈ Zl | 0 s j − si < e for all i < j}. This means that if t ∈ Zl then there exists s ∈ Sel with V (Λs )s V (Λt )t . Recall that there is an isomorphism of crystal graphs between (Ge (Λs )s )◦ and (Ge (Λt )t )◦ . In particular, there is a bijection between the sets of l-partitions labelling these graphs: Φ(s,e) and Φ(t,e) ; see 6.2.17. We will first be interested in the explicit determination of Φ(s,e) when s ∈ Sel . To do this, we will use some combinatorial notions and results that we have studied in Chapter 5. The goal of this section is to show that Φ(s,e) is exactly the set of l-partitions we have already met at the end of Chapter 5: the FLOTW l-partitions (see Definition 5.7.8). Theorem 6.3.2 (Foda, Leclerc, Okado, Thibon and Welsh [88, 2.10]). Assume that s ∈ Sel . Then, the multipartition λ = (λ 1 , . . . , λ l ) belongs to Φ(s,e) if and only if λ is an FLOTW multipartition in the sense of Definition 5.7.8. To prove this result, we need to establish several combinatorial properties. Lemma 6.3.3. Assume that s ∈ Sel and let λ = (λ 1 , . . . , λ l ) ∈ Φ(s,e) . Let γ1 = (a1 , b1 , c1 ) be a removable i-node of λ and let γ2 = (a2 , b2 , c2 ) be an addable inode of λ . Assume that b2 = b1 + 1 (that is, λac22 = λac11 ). Then we have γ1 ≺(s,e) γ2 . Proof. Assume to the contrary that we have γ2 ≺(s,e) γ1 . The case c1 = c2 leads clearly to a contradiction. Assume first that c1 < c2 . If a2 = 1, then
λac22 − a2 + sc2 = b2 − 1 + sc2 = b1 + sc2 = λac11 + sc2 > λac11 − a1 + sc1 which is a contradiction. Hence, we can assume that a2 > 1, as in the proof of Lemma 5.7.17, we obtain:
λac11 λac12+sc
2
−sc1
λbc12−b2 +a2 .
Hence, we have λac11 λac22−1 . Then we obtain λac22 λac22−1 . Thus, we have λac22 = λac22−1 , which is a contradiction since γ2 is an addable node. The case c1 > c2 is similar. Lemma 6.3.4. Assume that s ∈ Sel . Let λ = (λ 1 , . . . , λ l ) ∈ Φ(s,e) be a non-empty FLOTW l-partition. Then there exist i ∈ {0, 1, . . . , e − 1} and a good removable i-node γ for λ . Moreover, if we remove γ from λ , the resulting l-partition is an FLOTW l-partition.
6.3 Crystal and Canonical Bases for Highest Weight Modules
327
Proof. We take a removable i-node γ = (a, b, c) of λ for some i ∈ Z/eZ as in Lemma 5.7.13. By definition, for all removable or addable i-nodes γ = (a , b , c ) distinct from γ , we have that λac > λac . This implies by Lemma 5.7.17 that γ ≺(s,e) γ . As a consequence, γ is a normal removable i-node for λ . Hence, there exists a good removable i-node for λ . Assume now that γ is a good removable i-node for λ . Let μ be the l-partition such that [μ] = [λ ] \ {γ }. We must show that μ is an FLOTW l-partition. To do this, we have to check the two conditions to be an FLOTW lpartition. Assume to the contrary that μ is not an FLOTW l-partition. 1. If μ violates the first condition to be an FLOTW l-partition, this means that: c+1 and that γ = (i) there exist j > 0, c ∈ {1, . . . , l − 1} such that λ jc = λ j+s c+1 −sc c ( j, λ j , c) is a good removable i-node for λ ; 1 (ii) or there exist j > 0 such that λ jl = λ j+s and that γ = ( j, λ jl , l) is a good 1 −sl +e removable i-node for λ . c+1 Assume that λ is as in case (i). The node γ = ( j + sc+1 − sc , λ j+s , c + 1) c+1 −sc c c is then a removable node. Otherwise, as λ j > λ j+1 because γ is removable, we c+1 c have λ j+1 < λ j+s and this contradicts the first condition of FLOTW lc+1 −sc +1
c+1 − ( j + sc+1 − sc ) + sc+1 = λ jc − j + sc . This partition. Note also that λ j+s c+1 −sc implies in particular that γ is a removable i-node. Moreover, we have γ ≺(s,e) γ and there cannot be any removable or addable i-node η such that γ ≺(s,e) η ≺(s,e) γ . This implies that γ is not a good removable i-node. This is a contradiction. Case (ii) is similar. 2. Assume that μ violates the second condition to be an FLOTW l-partition. This implies that there exists a good removable i-node γ = ( j, λ jc , c) for λ and that the set of residues of the nodes on the border of the parts of λ with length λ jc − 1 > 0 is equal to the set {0, 1, . . . , e − 1} \ {i − 1}. Let γ be (one of) the i-node ( j , λ jc , c ) such that λ jc = λ jc − 1. This node must be a removable node otherwise there exists an i − 1-node on the border of the parts of λ with length λ jc − 1. Assume that there exists an addable i-node η = ( j , λ jc + 1, c ) such that
γ ≺(s,e) η ≺(s,e) γ . By Lemma 5.7.17, we have λ jc λ jc λ jc . By the hypothesis on the set of residues of the parts of λ with length λ jc − 1, we must have λ jc = λ jc . Using Lemma 6.3.3, we obtain a contradiction. This implies that we have no addable i-node η such that γ ≺(s,e) η ≺(s,e) γ . By the definition of good i-nodes, γ cannot be a good removable i-node, which is a contradiction.
Lemma 6.3.5. Let s ∈ Sel and let λ = (λ 1 , . . . , λ l ) ∈ Φ(s,e) (n) be an FLOTW lpartition. Assume that there exist i ∈ {0, 1, . . . , e − 1} and a good addable i-node γ for λ . Then the l-partition λ ∪ {γ } is an FLOTW l-partition. Proof. We must show that μ = λ ∪ {γ } is an FLOTW l-partition. To do this, we have to check the two conditions to be an FLOTW l-partition. The proof is entirely similar to the proof of Lemma 6.3.4.
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
We are now ready to prove the main result of this section. 6.3.6. Proof of Theorem 6.3.2. Let n ∈ Z0 . We show that λ belongs to Φ(s,e) (n) if and only if λ is an FLOTW l-partition. This is done by induction on n. If n = 0, the result is obvious. Let n > 0 and let λ be an FLOTW l-partition. By Lemma 6.3.4, there exist i ∈ {0, 1, . . . , e − 1} and a good i-node γ such that μ := λ \ {γ } is an FLOTW l-partition. By induction, μ is in Φ(s,e) (n − 1). Hence, λ ∈ Φ(s,e) (n). Conversely, let λ ∈ Φ(s,e) (n); then there exist i ∈ {0, 1, . . . , e − 1}, a good removable i-node γ and an l-partition μ ∈ Φ(s,e) (n − 1) such that λ = μ ∪ {γ }. By induction, μ is an FLOTW l-partition. By Lemma 6.3.5, λ is an FLOTW l-partition. Note that, from now, we can use the notation Φ(s,e) instead of Φ(s,e) for the FLOTW l-partitions (see Definition 5.7.8).
6.3.7. As a corollary, we can now derive a proof for Theorem 5.8.1 of Chapter 5. This is indeed a direct consequence of Proposition 6.3.2 and Theorem 6.2.22. This was the missing piece to complete the proof of the main Theorem 5.8.2 and of Theorems 5.8.5, 5.8.9, 5.8.11, 5.8.13, 5.8.15 and 5.8.19.
6.4 Computing Decomposition Matrices of Ariki–Koike Algebras We shall now discuss the problem of obtaining an explicit algorithm for the come )-modules. As a putation of the canonical basis of irreducible highest weight Uq (sl consequence of Ariki’s theorem, this will lead to an algorithm for the computation of decomposition matrices for Ariki–Koike algebras. 6.4.1. Now that we have an explicit description of the elements of Φ(s,e) (n) in the case where s ∈ Sel , we can use the results in Chapter 5 to present an algorithm for computing the elements of the canonical basis. To do this, we will need the partial order m defined in 5.7.5. Here, we have m = (s1 − v1 , . . . , sl − vl ), where v = (v1 , . . . , vl ) is an arbitrary l-tuple of rational numbers chosen such that, for all i < j, we have 0 < v j − vi < e. Assume that λ ∈ Φ(s,e) (n). Recall that we have associated with λ a certain residue sequence η(s,e) (λ ). Assume that
η(s,e) (λ ) = i1 , . . . , i1 , i2 , . . . , i2 , . . . , is , . . . , is , a1
a2
as
where, for all j = 1, . . . , s − 1, we have i j = i j+1 . We denote (as ) (as−1 ) (a ) fis−1 . . . f i1 1 |∅, s.
A(λ , s) := f is
From the definition of the action of the operators f i on the set of l-partitions, |μ, s appears with non-zero coefficient in the expansion of A(λ , s) if and only if there exists a row-standard μ-tableau s for an l-partition μ such that ηe (s) = ηe (λ ).
6.4 Computing Decomposition Matrices of Ariki–Koike Algebras
329
Theorem 6.4.2 (Jacon [174, Prop. 4.6]). Let s ∈ Sel ; recall that this means that s = (s1 , . . . , sl ) ∈ Zl , where 0 s j − si < e for all i < j. Let λ be an FLOTW lpartition. Then we have A(λ , s) = |λ , s +
∑
λ m μ, μ=λ
cλ ,μ (q)|μ, s,
where cλ ,μ (q) ∈ Z[q, q−1 ]. Proof. By the above discussion together with Lemma 5.7.21, we have (as ) (as−1 ) (a ) fis−1 . . . fi1 1 |∅, s
f is
= cλ ,λ (q)|λ , s +
∑
λ m μ,λ =μ
cλ ,μ (q)|μ, s,
where cλ ,μ (q) ∈ Z[q, q−1 ] is non-zero. So, we have to show that cλ ,λ (q) = 1. Let ν be the l-partition obtained by removing the as greatest removable is -nodes γ j with respect to ≺(s,e) (for j = 1, . . . , as .) By construction of the sequence η(s,e) (λ ), ν is a FLOTW l-partition. Hence, we have by induction (a
)
(a )
fis−1s−1 . . . fi1 1 |∅, s = |ν, s +
∑
νm α ,α =ν
cν,α (q)|α , s.
Let α be an l-partition obtained by removing as arbitrary removable is -nodes from λ and assume that α = ν. We have that α m ν by Proposition 5.7.16. Hence, we have cν,α (q) = 0. This implies that if cν,α (q) = 0 then the element |λ , s appears with non(a ) zero coefficient in the expansion of fis s |α, s if and only if α = ν. By Proposition 6.2.7, it is now sufficient to prove that Nis (ν, λ ) = 0. There is no addable is -node of λ greater than the γl with respect to ≺(s,e) and there is no removable is -node of ν greater than the nodes γl with respect to ≺(s,e) . Hence, the result follows by the definition of Nis (ν, λ ). Corollary 6.4.3. Let s ∈ Sel . Then A(λ , s) | λ ∈ Φ(s,e) is a basis of V (Λs )sQ . Proof. By Theorem 6.4.2, the A(λ , s) are linearly independent. Now we fix n ∈ Z0 and we consider the Q(q)-vector space (V (Λs )sQ )n := V (Λs )sQ ∩ Fsn . We have V (Λs )sQ = ⊕n∈Z0 (V (Λs )sQ )n . Let n ∈ Z0 . It is known that the set Gq (λ , s) | λ ∈ Φ(s,e) (n) is a basis of (V (Λs )sQ )n . This comes from a general property of the canonical basis (see [10, Theorem 7.3 (1)]), but this can also be seen as a direct consequence of Theorem 6.2.21. Hence, we have dim(V (Λs )sQ )n = |Φ(s,e) (n)|,
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
and this implies that A(λ , s) | λ ∈ Φ(s,e) (n) is a basis of (V (Λs )sQ )n .
The proof of the following result is inspired from the proof of [245, Theorem 6.21] and is a generalisation of a result by Lascoux, Leclerc and Thibon [208] (corresponding to the case l = 1). Theorem 6.4.4 (Lascoux–Leclerc–Thibon, Jacon). Let s ∈ Sel . Then, we have Gq (μ, s) = |μ, s +
∑
μm λ ,μ=λ
dλ ,μ (q)|λ , s
for all μ ∈ Φ(s,e) ,
where dλ ,μ (q) ∈ qZ[q] if μ m λ and μ = λ . Proof. By Corollary 6.4.3, for all μ ∈ Φ(s,e) (n), there exist Laurent polynomials (γμ,ν (q))ν∈Φ(s,e) (n) in Q[q, q−1 ] such that Gq (μ, s) =
∑
ν∈Φ(s,e) (n)
γμ,ν (q)A(ν, s).
As both the canonical basis elements and the A(ν, s) are bar invariant, we have that γμ,ν (q) = γμ,ν (q−1 ) for all ν ∈ Φ(s,e) (n) by Corollary 6.4.3. We consider an arbitrary total order m which is a refinement of our partial order m . Let λ be the minimal element with respect to m such that γμ,λ (q) = 0. That is, we have γμ,ν (q) = 0 unless λ m ν. Then the coefficient of |λ , s in Gq (μ, s) is ∑ν∈Φ(s,e) (n) γμ,ν (q)cν,λ (q) (see Theorem 6.4.2 for the definition of the Laurent polynomials cν,λ (q).) By hypothesis and by Theorem 6.4.2, this coefficient is equal to γμ,λ (q). But by the definition of the canonical basis, it is in qZ[q] if λ = μ. As it is also bar invariant we obtain that μ = λ and γμ,μ (q) = 1. In addition, we have γμ,ν (q) = 0 unless μ m ν. We now argue by induction on m . If μ ∈ Φ(s,e) (n) is maximal with respect to m then we have Gq (μ, s) = A(μ, s) and we are done. We now assume that the theorem is true for all ν ∈ Φ(s,e) (n) such that μ m ν and ν = μ. By Theorem 6.4.2 and by the above discussion, there exist bar-invariant Laurent polynomials αμ,ν (q) such that Gq (μ, s) = A(μ, s) + ∑ αμ,ν (q)Gq (ν, s). νm μ,ν=μ
If ν = μ and αμ,ν (q) = 0, we have μ m ν. Indeed, assume to the contrary that ν = μ, αμ,ν (q) = 0 and that μ and ν are not comparable with respect to m . We can assume that ν is minimal with respect to m with this property. We have ν m μ. The coefficient of |ν, s in Gq (μ, s) is cμ,ν (q) + ∑ν m μ,ν =μ αμ,ν (q)dν,ν (q). This last term is equal to cμ,ν (q) + αμ,ν (q) by induction hypothesis. But we also have cμ,ν (q) = 0 by hypothesis. Thus, we have αμ,ν (q) = 0 because it is both bar-invariant and in qZ[q]. This is a contradiction. Hence the theorem is proved. We conclude this section with a necessary and sufficient condition to be an FLOTW l-partition.
6.4 Computing Decomposition Matrices of Ariki–Koike Algebras
331
Definition 6.4.5. Assume that s ∈ Sel .We say that an l-partition λ is (s, e)-regular if there exists a standard λ -tableau t(λ ) satisfying the following property. For all l-partition μ and standard μ-tableau s such that η (s) = η (t(λ )), we have λ m μ. Corollary 6.4.6. Assume that s ∈ Sel . Then λ ∈ Φ(s,e) if and only if it is (s, e)regular. Proof. If λ ∈ Φ(s,e) then it is (s, e)-regular by the discussion in 6.4.1 and Theorem 6.4.2. Conversely, assume that λ is an (s, e)-regular l-partition and denote by t(λ ) the λ -tableau as in Definition 6.4.5. Set
η (t) = i1 , . . . , i1 , i2 , . . . , i2 , . . . , is , . . . , is . a1
a2
as
There exist Laurent polynomials bλ ,μ (q) ∈ Z[q, q−1 ] such that (as ) (as−1 ) (a ) f is−1 . . . f i1 1 |∅, s
fis
= bλ ,λ (q)|λ , s +
∑
λ m μ,μ=λ
bλ ,μ (q)|μ, s,
with bλ ,λ (q) = 0. Now by Corollary 6.4.3, there exist Laurent polynomials eλ ,μ (q) ∈ Q[q, q−1 ] such that (as ) (as−1 ) (a ) fis−1 . . . f i1 1 |∅, s
fis
∑
=
μ∈Φ(s,e)
eλ ,μ (q)A(μ, s).
Let ν ∈ Φ(s,e) be (one of) the minimal element (with respect to m ) satisfying eλ ,ν (q) = 0. Then, by Proposition 6.4.2, we have ν = λ . Remark 6.4.7. Assume that l = 1 and that s ∈ Z. Then by 5.7.10 we have λ ∈ Φ(s,e) if and only if λ is e-regular. Then, by Example 5.7.6, the “only if” part of Corollary 6.4.6 had already been established in Lemma 3.5.11. Remark 6.4.8. If l = 2, then a similar characterization of the Kleshchev 2-partitions conjectured by Dipper, James and Murphy has been obtained in [11, Corollary 4.3]. We can now derive an algorithm for the computation of the canonical basis. 6.4.9. We fix n ∈ Z0 , l ∈ Z>0 , e ∈ Z>0 and s ∈ Sel . The aim of the following algorithm is to compute the canonical basis Gq (μ, s) | μ ∈ Φ(s,e) (n) of V (Λs )sQ . It was first described in [175, Section 3]. Step 1. For each μ ∈ Φ(s,e) (n), we denote
η(s,e) (μ) = i1 , . . . , i1 , i2 , . . . , i2 , . . . , is , . . . , is . a1
a2
as
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
Then, we compute the elements A(μ, s) of 6.4.1 using the action of the quantum group described in Proposition 6.2.7: (as ) (as−1 ) (a ) f is−1 . . . f i1 1 |∅, s.
A(μ, s) := fis
These elements have a triangular decomposition with respect to m by Theorem 6.4.2. We obtain a basis of (V (Λs )sQ )n {A(μ, s) | λ ∈ Φ(s,e) (n)}. Since fi = fi for all i = 0, . . . , e − 1, we have A(μ, s) = A(μ, s). Step 2. Let ν ∈ Φ(s,e) (n) be a maximal element with respect to m . Then, by the definition of the canonical basis, we have Gq (ν, s) = A(ν, s). Step 3. Let μ ∈ Φ(s,e) (n). The elements Gq (λ , s) with μ m λ are known by induction. There exist polynomials αλ ,μ (q) such that Gq (μ, s) = A(μ, s) −
∑
μm λ ,μ=λ
αμ,λ (q)Gq (λ , s).
We want to compute αμ,λ (q) for all λ ∈ Φ(s,e) (n). By Theorem 6.4.2, we have ()
A(μ, s) = |μ, s +
∑
μm λ ,μ=λ
cμ,λ (q)|λ , s.
Now, since Gq (λ , s) = Gq (λ , s) and A(λ , s) = A(λ , s) for all λ ∈ Φ(s,e) (n), we must have αμ,λ (q) = αμ,λ (q−1 ) for all λ in Φ(s,e) (n). Let ν l n be one of the minimal l-partitions with respect to m such that cμ,ν (q) ∈ / qZ[q]. If ν does not exist, then, by unicity, we have Gq (μ, s) = A(μ, s). Otherwise, by the existence of the canonical basis, we have ν ∈ Φ(s,e) (n). Assume now that we have cμ,ν (q) = ai qi + ai−1 qi−1 + . . . + a0 + . . . + a−i q−i , where (a j ) j∈{−i,−i+1,...,i−1,i} is a sequence of elements in Z and i is a positive integer. Then, we define
αμ,ν (q) = a−i qi + a−i+1 qi−1 + . . . + a0 + . . . + a−i q−i . We have αμ,ν (q−1 ) = αμ,ν (q). Then, in (), we replace the element A(μ, s) by A(μ, s) − αμ,ν (q)Gq (ν, s), which is bar-invariant, and we repeat this step until we have Gq (μ, s) = A(μ, s). We finally obtain elements which verify Theorem 6.1.17; that is, the canonical basis elements.
6.4 Computing Decomposition Matrices of Ariki–Koike Algebras
333
Note that we also obtain an algorithm for computing the canonical basis of the e )-module with weight Λs by specializing the above elements at q = 1. U (sl 6.4.10. The algorithm in 6.4.9 is only concerned with the case where the multicharge s is in Sel . We here explain how we can deduce an algorithm for the computation of the canonical basis associated with an arbitrary multicharge following [12, Section 4.2]. We have seen in 6.2.9 that there is an action of the (extended) affine symmetric l on Zl such that S l contains a fundamental domain for this action. group S e r such For any t : = (t1 , . . . ,tl ) ∈ Zl , there exist s := (s1 , . . . , sl ) ∈ Sel and w ∈ S that t = w.s. Since t and s yield the same dominant weight, we have an isomorphism e )-modules from V (Λs )s to V (Λt )t . We can assume that φs,t (|∅, s) = φs,t of Uq (sl |∅, t. For each λ ∈ Φ(s,e) , recall that we have already defined an element: (as ) (as−1 ) (a ) fis−1 . . . fi1 1 |∅, s ∈ V (Λs )s .
A(λ , s) := fis
Moreover, there exist coefficients γλ ,μ (q) ∈ Z[q, q−1 ] such that Gq (λ , s) =
()
∑
μ∈Φ(s,e)
γλ ,μ (q)A(μ, s),
which can be explictly obtained using our previous algorithm. Now, we fix λ ∈ Φ(s,e) and set (a ) (a ) A(λ , t) = fis s · · · f i1 1 |∅, t ∈ V (Λt )t . Then we have φs,t (A(λ , s)) = A(λ , t). By the uniqueness of the crystal basis proved by Kashiwara, we also have φs,t (Gq (λ , s)) = Gq (ϕs,t (λ ), t), where ϕs,t is the bijection Φ(s,e) to Φ(t,e) as in 6.2.17. By applying φs,t to (), we obtain Gq (ν , t) =
∑
μ∈Φ(s,e)
γϕ −1 (ν ),μ (q)A(μ, t), s,t
for ν ∈ Φ(t,e) , and it follows that
∑
μ∈Φ(s,e)
γλ ,μ (q)A(μ, t) | λ ∈ Φ(s,e)
is the canonical basis for the module V (Λt )t . Remark 6.4.11. Another algorithm has been recently proposed by Fayers [82, Sece )-modules. tion 4] for computing the canonical bases of the highest weight Uq (sl In this case, this module is realiszed as the tensor product of level-one Fock spaces. This is connected with the case where the multicharge is very dominant (see Example 6.2.16). 6.4.12. Let us give the consequences of the previous results on the representation theory of Ariki–Koike algebras. We assume that the characteristic of k is zero. Let e ∈ Z>0 and let s ∈ Zl . We consider a specialisation
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
θ :A→k such that θ (Vi ) = ηesi for all i = 1, . . . , l and θ (u) = ηe is a primitive root of unity of s order e ∈ Z>1 . The specialised algebra Hk,n (ηe ; ηes1 , . . . , ηe l ) is non-semisimple and we have an associated decomposition matrix Dθ . For studying this matrix, without loss of generality, one can assume that s ∈ Sel . By Ariki’s theorem, D(s,e) := Dθ is e )-module with weight Λs . Taking this the matrix of the canonical basis of the U (sl into account, using 5.4.4 and 6.4.9, we deduce the following theorem.
Theorem 6.4.13 (Lascoux–Leclerc–Thibon, Uglov, Jacon). There is an explicit and purely combinatorial algorithm for computing the decomposition matrices of Ariki–Koike algebras when the characteristic of k is zero.
6.5 Uglov’s Theory of Fock Spaces In this section we survey Uglov’s theory [291] concerning canonical bases for Fock spaces. The determination of canonical basic sets for all choices of parameters requires some new information that we are not able to obtain directly from the theory exposed in Section 6.2. This section is inspired by the thesis of Yvonne [298]. 6.5.1. The first new object that we need to define is the affine Hecke algebra of type A. This can be seen as a deformation of the extended affine symmetric group which we introduced in the first section. We then study several actions of this algebra. In the next section, this will yield the definition of a new space: the semi-infinite qr be the affine wedge product which will play a crucial role in the following. Let H Hecke algebra of type A over K = Q(q). As a K -algebra, Hr is presented by: • Generators: Ti (with 0 i r − 1), τ , τ −1 . • Relations: for all 0 i, j r − 1, Ti Ti+1 Ti = Ti+1 Ti Ti+1 Ti T j = T j Ti if i − j = 1 mod r (Ti − q−1 )(Ti + q) = 0
τ Ti = Ti+1 τ , r can be written as a product where the indices are read modulo r. Each w ∈ S r with k ∈ Z and σi . . . σit ∈ Sr is a decomposition of minimal τ k σi1 . . . σit ∈ H 1 r is then length. We then set Tw = τ k Ti1 . . . Tit . As a K -vector space, a basis of H given by the set
r Tσ | σ ∈ S
6.5 Uglov’s Theory of Fock Spaces
335
2; and the multiplication is characterized by the following property. Let (σ , σ ) ∈ S r then if l(σ σ ) = l(σ ) + l(σ ), Tσ Tσ = Tσ σ −1 (Tσi − q )(Tσi + q) = 0 if 0 i r − 1. r defined as the unique Q-algebra autoThere is a canonical involution x → x of H morphism such that q = q−1 ,
Tw = Tw−1 −1
r. for all w ∈ S
By the work of Bernstein, we can obtain another system of generators as follows. Let p ∈ Pr := Zr . Recall the elements yp defined in 6.2.8. We write p = p+ − p− , where p+ , p− are in Pr+ := {(p1 , . . . , pr ) ∈ Zr | p1 . . . pr }. Then one can show that the element X p := Typ− T −1 + is well defined. Moreover, for all p, q in Pr , we yp have that X p X q = X p+q = X q X p . r . We refer to [291, Then the elements X p with p ∈ Pr and T1 ,. . . ,Tr−1 generate H Lemma 3.13] and [200] for a proof of this assertion. 6.5.2. We now study two actions of the extended affine symmetric group on Pr . These actions will then be used to construct actions of the affine Hecke algebra. Let r on Pr depending on l and defined as follows. l ∈ Z>0 . We can put a left action of S For all (b1 , . . . , br ) ∈ Pr , we have σi .(b1 , . . . , br ) = (b1 , . . . , bi−1 bi+1 , bi , bi+2 , . . . , br ) for all i = 1, . . . , r − 1, y j .(b1 , . . . , br ) = (b1 , .., b j−1 , b j + l, b j+1 , . . . , br ) for all j = 1, . . . , r. A fundamental domain for this action is Bl := {(b1 , . . . , br ) ∈ Pr | l b1 . . . . bl 1} . r )b the isotropy group of b for the action of S r on Pr . It Let b ∈ Bl . We denote by (S is clear that we have (Sr )b ⊂ Sr . In fact, if we set Ib := {1 i r − 1 | bi = bi+1 } , r /(S r ) has a unique representative r )b = σi | i ∈ Ib . Each element of S we have (S b r )b be the set of these representatives. with minimal length. Let (S r )b and u ∈ (S r ) such that r , there exist unique elements y ∈ (S • For all w ∈ S b w = yu and we have l(w) = l(y) + l(u). r )b such that p = y.b. • For all p ∈ Pr , there exist unique elements b ∈ Bl and y ∈ (S r on Pr as follows. For Let e ∈ Z>0 . By analogy, one can define a right action of S all (a1 , . . . , ar ) ∈ Pr , we have (a1 , . . . , ar ).σi = (a1 , . . . , ai+1 , ai , . . . , ar ) for all i = 1, . . . , r − 1, (a1 , . . . , ar ).y j = (a1 , .., a j + e, . . . , ar ) for all j = 1, . . . , r. A fundamental domain for this action is
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
Ae := {(a1 , . . . , ar ) ∈ Pr | 1 a1 . . . . ar e} . r ) as above. For a ∈ Ae , we denote by (S r )a r )a and a (S One can define groups (S the isotropy group of a for the right action of Sr on Pr . It is clear that we have r )a ⊂ Sr . Recall that Ia = {1 i r − 1 | ai = ai+1 }; then we have (S r )a = (S σi | i ∈ Ia . Each element of (Sr )a \ Sr has a unique representative of minimal r ) be the set of these representatives. length. Let a (S r ) and u ∈ (S r )a such that • For all w ∈ W , there exist unique elements y ∈ a (S w = uy and we have l(w) = l(u) + l(y). r ) such that p = a.y. • For all p ∈ Pr , there exist unique elements a ∈ Ae and y ∈ a (S 6.5.3. We turn now to the affine Hecke algebra. We define a left and then a right acr on two K -vector spaces PrL := p∈P K |p) generated by the symbols tion of H r |p) with p ∈ Pr and PrR := p∈Pr K (p| generated by the symbols (p| with p ∈ Pr . r generated by the Tw • For b ∈ Bl , we consider the parabolic subalgebra Hb of H r )b . We denote by K−q the left Hb -module with dimension 1 such with w ∈ (S r that Ti with i ∈ Ib acts by multiplication with −q. We then consider the left H module Hr ⊗Hb K−q . As a K -vector space, it has a basis given by the Ty ⊗ 1, r )b . Now, there is an isomorphism of K -vector spaces where y ∈ (S (•)
r ⊗H K−q H b Ty ⊗Hb 1
→
r )b ) → (y ∈ (S
b y∈S r
K |y.b),
|y.b).
r )b . We From now, we can (and we do) identify |y.b) with Ty ⊗Hb 1 for y ∈ (S r on the space thus have an action of H br K |y.b). By linearity and by y∈S b r ) , we obtain a left action of H r on the space PrL := the definition of (S r )b K |y.b). One can check that this action is explicitly given as b∈Bl y∈(S follows (see [291, Proposition 3.12] or [298, Proposition 1.29]). For all p ∈ Pr , and i = 0, 1, . . . , r − 1, we have ⎧ if pi < pi+1 ⎨ |σi .p) if pi = pi+1 Ti .|p) = −q|p) ⎩ |σi .p) − (q − q−1 )|p) if pi > pi+1 and
τ .|p) = |τ .p).
This action is thus obtained using the identification induced by (•). r generated by the Tw with • By analogy, let Ha be the parabolic subalgebra of H w ∈ (Sr )a . We denote by Kq−1 the right Ha -module with dimension 1 such that r is then Ti with i ∈ Ia acts by multiplication with q−1 . A basis of Kq−1 ⊗Ha H a given by the elements 1 ⊗Ha Ty with y ∈ Sr . Using this, we obtain a right action r on Kq−1 ⊗Ha H r . Now consider the isomorphism of vector spaces: of H
6.5 Uglov’s Theory of Fock Spaces
337
r Kq−1 ⊗Ha H 1 ⊗Ha Ty
→
r ) → (y ∈ a S
b y∈S r
K (a, y|,
(a.y|.
r . We thus From now, we can (and we do) identify (a.y| with 1 ⊗Ha Ty for y ∈ a S R obtain by linearity a right action of Hr on Pr = a∈Ae r K (a.y|. It is y∈a S explicitly given as follows. ⎧ if pi < pi+1 , ⎨ (p.σi | −1 if pi = pi+1 , (p|.Ti = q (p| ⎩ (p.σi | − (q − q−1 )(p| if pi > pi+1 , (p|.τ = (p.τ |. 6.5.4. Using the above constructions, we will now be able to define a new space which will have strong connections with our main objects of interest. Let e ∈ Z>0 , l ∈ Z>0 , a ∈ Ae and b ∈ Bl . We set r ⊗H K−q , Λ (a, b) = Kq−1 ⊗Ha H b
Λ (b) =
Λ (a, b),
a∈Ae
ΛqrV =
Λ (b).
b∈Bl
This last K -vector space is called the q-wedge product. The main aim of this section will be to give a basis of Λ (b) for all b ∈ Bl and thus of ΛqrV . To do this, we need some preparatory results. We define a linear map
Φb : PrR → Λ (b) r such that as follows. Let p ∈ Pr . Then there exist unique elements a ∈ Ae and x ∈ a S −1 l(w ) −1 l(w ) b b p = a.x. Define [p]b := (−q ) 1 ⊗Ha Tx ⊗Hb 1 = (−q ) (p| ⊗Hb 1, where b of maximal length. We can set Φb ((p|) = [p]b . One can wb is the element in S r easily show that Ker(Φb ) = ∑ j∈Ib PrR (T j + q). We define ++ := {(p1 , . . . , pr ) ∈ Pr | i ∈ Ib ⇒ pi > pi+1 } . Pr,b
Then we have the following result. ++ Proposition 6.5.5 (Uglov). The set [p]b | p ∈ Pr,b is a basis of Λ (b) (as a K vector space). ++ is a generating set for Λ (b). To do Proof. We first show that [p]b | p ∈ Pr,b r can this, it suffices to show that an arbitrary element 1 ⊗Ha Tx ⊗Hb 1 with x ∈ S e be written as a linear combination of this set. Let x ∈ Sr and let a ∈ A . We set r ) be such that p = a.y. Then we have u := x.y−1 ∈ (S r) p := a.x ∈ Pr . Let y ∈ a (S a and l(x) = l(u) + l(y). It follows that
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
1 ⊗Ha Tx ⊗Hb 1 = 1 ⊗Ha Tu Ty ⊗Hb 1 = Tu ⊗Ha Ty ⊗Hb 1 = q−l(u) 1 ⊗Ha Ty ⊗Hb 1. Set p = (p1 , . . . , pr ); then we have 1 ⊗Ha Tx ⊗Hb 1 = (−q)l(wb ) [p]b . As we have Ker(Φb ) = ∑ j∈Ib PrR (Tj + q), we obtain that [p]b =
0 if pi = pi+1 , −q−1 [p.σi ]b if pi < pi+1 ,
which leads to our claim.
++ is free over K . Assume that we We now show that the set [p]b | p ∈ Pr,b ++ have ∑p∈P++ ap [p]b = 0. For p ∈ Pr,b , we have ap ∈ K and there is a finite r,b
++ subset of Pr,b such that ap = 0 unless p is in this subset. It follows that we have ∑p∈P++ ap (p| ∈ Ker(Φb ) = ∑ j∈Ib PrR (Tj + q). Now, we consider the element r,b
l(y)−l(wb ) T . This is an element of the Kazhdan–Lusztig basis Cwb := ∑y∈(S y r )b (−q) for Sr ; see Example 2.1.17 (where ε = q−1 ). Thus, it satisfies Cwb = Cwb . It is an easy computation to show that, for all i ∈ Ib , we have (Ti + q)Cwb = 0. Hence, we obtain that ∑ ap (p|.Cwb = 0 = ∑ ap (p|.Cwb = 0; ++ p∈Pr,b
that is,
∑
++ p∈Pr,b
++ r )b p∈Pr,b ,y∈(S
(−q)−l(y)+l(wb ) ap (p|.Ty−1 = 0.
By the definition of the right action of the affine Hecke algebra on PrR , we deduce that ∑ (−q)−l(y)+l(wb ) ap (p.y| = 0, ++ r )b p∈Pr,b ,y∈(S
++ and which implies that all the ap are zero as all the elements (p.y| with p in Pr,b y ∈ (Sr )b are distinct.
6.5.6. We can now give a basis of ΛqrV . To do this, we need to introduce some more notation. Let k = (k1 , . . . , kr ) ∈ Pr . For i = 1, . . . , r, there exist unique elements ci ∈ {1, . . . , e}, di ∈ {1, . . . , l} and mi ∈ Z such that ki = ci + e(di − 1) + elmi . We then associate with k the following elements: c := c(k) = (c1 , . . . , cr ), d := d(k) = (d1 , . . . , dr ), m := m(k) = (m1 , . . . , mr ).
6.5 Uglov’s Theory of Fock Spaces
339
In addition, there exist elements which are uniquely defined as follows: a := a(k) ∈ r ) and v := v(k) ∈ (S r )b such that c = a.u and Ae , b := b(k) ∈ Bl , u := u(k) ∈ a (S m a m r ) (where y ∈ S r ) and p(k) := a.x ∈ Pr . d = v.b. We also set x(k) := uy v ∈ (S r )b . The element wb ∈ Sr is the longest element in the parabolic subgroup (S Example 6.5.7. Assume that we have e = 4, l = 3, r = 4 and k = (1, −11, −17, 2). Then we obtain c = (1, 1, 3, 2),
d = (1, 1, 2, 1),
m = (0, −1, −2, 0),
b = (2, 1, 1, 1), u = σ3 , v = σ2 σ1 , x = σ3 y
(0,−1,−2,0)
a = (1, 1, 2, 3),
σ2 σ1 , p(k) = (−5, 1, −3, 2).
Consider now k = (12, 5, 2, 1). Then, with the same values for e, l and r, we obtain in this case c = (0, 1, 2, 1),
d = (1, 2, 1, 1),
b = (2, 1, 1, 1),
m = (1, 0, 0, 0),
u = σ3 σ2 σ1 σ3 y(−1,0,0,0) ,
a = (1, 1, 2, 4), v = σ1 .
We obtain x = σ3 σ2 σ3 and p(k) = (1, 4, 2, 1). Note that, in this situation, we have Ib = {2, 3} and if i ∈ Ib , we have pi > pi+1 for p(k) = (p1 , p2 , p3 , p4 ). This phenomenon can be explained by looking at the proof of the next theorem. 6.5.8. For k ∈ Pr , we now define the following element of Λ (a, b) ⊂ ΛqrV : (••)
uk = uk1 ∧ . . . ∧ ukr = (−q−1 )l(wb ) 1 ⊗Ha Tu X −m (Tv−1 )−1 ⊗Hb 1;
this is called a q-wedge. If k ∈ Pr is in the set Pr++ := {k = (k1 , . . . , kr ) | ki > ki+1 for all i ∈ {1, . . . , r − 1}}, then we say that uk is an ordered q-wedge. The following result gives a basis of our space ΛqrV . As the proof is quite long, we only give a sketch of it following [298, Proposition 2.8]. A different proof can be found in [291, Proposition 3.21]. Theorem 6.5.9 (Uglov). The set {uk | k ∈ Pr++ } is a basis of ΛqrV . ++ } and consider the map Proof. We set S := {(b, p) ∈ Bl × Pr | p ∈ Pr,b
Ψ : Pr++ → S,
k → (b(k), p(k)).
++ We show that this is well defined (that is, p(k) ∈ Pr,b if k ∈ Pr++ ) and that this is a bijection. This can be done by purely combinatorial arguments. We then show that for all k ∈ Pr++ we have uk = [p(k)]b(k) . We can then conclude using Proposition 6.5.5 and the definition of ΛqrV .
6.5.10. The theorem shows that the set of q-wedges is not linearly independent. However, each q-wedge can be expressed uniquely in terms of ordered q-wedges. The way to express them is given by the following rules. First we assume that r = 2.
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
Then any q-wedge of the form uk1 ∧ uk2 can be expressed as a linear combination of ordered q-wedges using the following rules (R1), (R2), (R3) and (R4). Let k1 k2 and, for i = 1, 2, put ki = ai + e(l − bi ) − elmi , where ai ∈ {1, . . . , e}, bi ∈ {1, . . . , l} and mi ∈ Z. We define α := (a2 −a1 ) mod el and β := e(b1 −b2 ) mod el. Then the relations (R1), (R2), (R3) and (R4) are given as follows. (R1) if α = 0 and β = 0:
uk1 ∧ uk2 = −uk2 ∧ uk1 ;
(R2) if α = 0 and β = 0: uk1 ∧ uk2 = −q−1 uk2 ∧ uk1 + (q−2 − 1)
∑ q−2m uk2−α −elm ∧ uk1 +α +elm
m0
− (q
−2
− 1)
∑ q−2m+1uk2 −elm ∧ uk1+elm ;
m1
(R3) if α = 0 and β = 0: uk1 ∧ uk2 =quk2 ∧ uk1 + (q2 − 1)
∑ q2m uk2 −β −elm ∧ uk1 +β +elm
m0
+ (q2 − 1)
∑ q2m−1 uk2 −elm ∧ uk1 +elm;
m1
(R4) if α = 0 and β = 0: uk1 ∧ uk2 =uk2 ∧ uk1 + (q − q−1 )
q2m+1 + q−2m−1 ∑ q + q−1 uk2−β −elm ∧ uk1 +β +elm m0
+ (q − q−1 )
q2m+1 + q−2m−1 uk2 −α −elm ∧ uk1 +α +elm q + q−1 m0
+ (q − q−1 )
q2m+2 − q−2m−2 uk2 −β −α −elm ∧ uk1 +β +α +elm q + q−1 m0
+ (q − q−1 )
q2m − q−2m uk2 −elm ∧ uk1 +elm , −1 m1 q + q
∑ ∑ ∑
where the summations are taken over the set of ordered q-wedge. The proofs for these relations can be obtained by induction using (••). We refer to [298, Section 2.5] for a complete proof of them. Thus, for r ∈ Z0 , an arbitrary q-wedge can be expressed as a linear combinaison of ordered q-wedges using the relations (R1)– (R4) in every adjacent pair of the factors. We have the following remarkable result which will be useful in the following and which is a direct consequence of the above relations.
6.5 Uglov’s Theory of Fock Spaces
341
Proposition 6.5.11 (Uglov). Let (a, b) ∈ Z2 be such that a b. Then we have ub ∧ ua ∧ ua−1 ∧ . . . ∧ ub = 0
and
ua ∧ ua−1 ∧ . . . ∧ ub ∧ ua = 0.
6.5.12. Let a ∈ Ae and b ∈ Bl . We denote Ia :=
∑ (Ti − q−1 ).Hr
and
Ib :=
i∈Ia
∑ Hr .(Ti + q).
i∈Ib
r /Ib . Using We have an isomorphism of vector spaces between Λ (a, b) and Ia \ H this, we can define an involution on the q-wedge product. To do this, we will use the r . As for all i = 1, . . . , r − 1, we have T −1 = Ti + (q − q−1 ), canonical involution of H i r . the elements Ti + q and Ti − q−1 are invariant with respect to the involution of H As a consequence, the submodules Ia and Ib of Hr are invariant with respect to this involution. This implies that the natural involution of the affine Hecke algebra induces an involution on Λ (a, b) and hence on ΛqrV . For p := (p1 , . . . , pr ) ∈ Pr++ we set υ (p) := {(i, j) ∈ Z>0 | 1 i < j r, pi = p j }. Let k ∈ Pr be such that uk ∈ Λ (a, b). Then we can show that (see [291, Proposition 3.23, Remark 3.24]) (◦)
uk = (−q)υ (b) (q−1 )υ (a) ukr ∧ . . . ∧ uk1 .
We thus obtain an involution on the space ΛqrV . Using the relations (R1)–(R4), we can write uk = ∑ ak,l (q)ul . l∈Pr++
The involution can be encoded in a matrix: A := (ak,l (q))k,l∈Pr++ .
Remark 6.5.13. If we define Lr+ = k∈Pr++ Q[q]uk , then one can show the exis tence of a unique basis G(k) | k ∈ Pr++ of ΛqrV , such that the following two conditions are satisfied: • for all k ∈ Pr++ , we have G(k) = G(k), • for all k ∈ Pr++ , we have G(k) = uk mod qLr+ . (The proof is analogous to the one of Theorem 6.6.11.) The elements of the basis {G(k) | k ∈ Pr++ } are called the canonical basis elements of ΛqrV . The existence of such a basis comes from the unitriangularity of the matrix of the involution. There exists a matrix Δ := (dk,l (q))k,l∈Pr++ such that G(k) = ∑l∈Pr++ dk,l .(q)ul for all k ∈ Pr++ . It is possible to show that the coefficients dk,l (q) can be expressed in terms of r (see [291, Theorem 3.26] and [292]). Kazhdan–Lusztig polynomials for S e ) on the space 6.5.14. We now show that it is possible to define an action of Uq (sl r Λq V which is “compatible” with the involution defined by (••) in 6.5.12. For a
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
statement S, we here denote δS = 1 if S is true and 0 if S is false. First, we have an e ) on Λ 1V . To define it, we first define an action of Uq (sl e ) on Ve,l := action of Uq (sl q −1 e l K [X, X ] ⊗ K ⊗ K (where we denote K := Q(q)). We fix a basis (v1 , . . . , ve ) of K e and a basis (w1 , . . . , wl ) of K l and define for 0 i e − 1, 1 c e and 1 d l: ti (X m ⊗ vc ⊗ wd ) = qδi≡c mod e −δi+1≡c mod e (X m ⊗ vc ⊗ wd ), ei (X m ⊗ vc ⊗ wd ) = δi+1≡c mod e (X m−δi,0 ⊗ vc−1 ⊗ wd ), fi (X m ⊗ vc ⊗ wd ) = δi≡c mod e (X m+δi,0 ⊗ vc+1 ⊗ wd ), d(X m ⊗ vc ⊗ wd ) = −mX m ⊗ vc ⊗ wd , e ) on Λ 1V (where we set v0 := ve and ve+1 := v1 ). This gives an action of Uq (sl q using the isomorphism of vector spaces Ve,l → Λq1V,
X m ⊗ vc ⊗ wd → uc+e(d−1)+elm .
e ) to obtain an action of Uq (sl e ) on (Λ 1V )⊗r Now we use the coproduct Δ of Uq (sl q (see Remark 6.1.4). Explicitly, this action is given as follows: ti (uk1 ⊗ . . . ⊗ ukr ) = ti (uk1 ) ⊗ ti (uk2 ) ⊗ . . . ⊗ ti (ukr ), ei (uk1 ⊗ . . . ⊗ ukr ) = f i (uk1 ⊗ . . . ⊗ ukr ) = d(uk1 ⊗ . . . ⊗ ukr ) =
r
∑ uk1 ⊗ . . . ⊗ uk j−1 ⊗ ei (uk j ) ⊗ ti−1 (uk j+1 ) ⊗ . . . ⊗ ti−1 (ukr ),
j=1 r
∑ ti (uk1 ) ⊗ . . . ⊗ ti (uk j−1 ) ⊗ fi (uk j ) ⊗ uk j+1 ⊗ . . . ⊗ ukr ,
j=1 r
∑ uk1 ⊗ . . . ⊗ uk j−1 ⊗ d(uk j ) ⊗ uk j+1 ⊗ . . . ⊗ ukr .
j=1
Now ΛqrV can be seen as the quotient of (Λq1V )⊗r by the relations (R1)–(R4). Using e ) on Λ rV , which is this we check that we obtain a well-defined action of Uq (sl q explictly given as follows: ti (uk1 ∧ . . . ∧ ukr ) = ti (uk1 ) ∧ ti (uk2 ) ∧ . . . ∧ ti (ukr ), ei (uk1 ∧ . . . ∧ ukr ) = fi (uk1 ∧ . . . ∧ ukr ) = d(uk1 ∧ . . . ∧ ukr ) =
r
∑ uk1 ∧ . . . ∧ uk j−1 ∧ ei (uk j ) ∧ ti−1 (uk j+1 ) ∧ . . . ∧ ti−1 (ukr ),
j=1 r
∑ ti (uk1 ) ∧ . . . ∧ ti (uk j−1 ) ∧ fi (uk j ) ∧ uk j+1 ∧ . . . ∧ ukr ,
j=1 r
∑ uk1 ∧ . . . ∧ uk j−1 ∧ d(uk j ) ∧ uk j+1 ∧ . . . ∧ ukr .
j=1
6.5 Uglov’s Theory of Fock Spaces
343
6.5.15. We now use the above results to construct a new space called the semiinfinite q-wedge product. We will then see later the connection with the Fock space. First, we fix s ∈ Z. Let t > r be two positive integers. We define a map
ΛqrV → Λqt V, v → v ∧ us−r ∧ us−r−1 ∧ . . . ∧ us−t+1 . We now consider the inductive limit
Λ s = lim(ΛqrV ), −→
which is called the semi-infinite q-wedge product with charge s. For v ∈ ΛqrV , we denote by v ∧ us−r ∧ us−r−1 ∧ us−r−2 ∧ . . . the image of the elements v under the canonical map ΛqrV → Λ s . Such an element is called a semi-infinite q-wedge. We denote by P(s) the sequence of integers k = (ki )i1 such that ki = s − i + 1 if i is sufficiently large. From the definition, it is clear that the set of semi-infinite q-wedges uk = uk1 ∧ uk2 ∧ . . . with k = (ki )i1 ∈ P(s) is a generating set for Λ s . Let P++ (s) be the elements k = (ki )i1 in P(s) such that ki > ki+1 for all i 1. If k ∈ P++ (s), the semi-infinite q-wedge uk ∈ Λ s is said to be ordered. Proposition 6.5.16 (Takemura and Uglov [288, 4.1]). The set {uk | k ∈ P++ (s)} is a basis of Λ s . Proof. Let u ∈ Λ s . Then, there exists v ∈ ΛqrV such that u = v ∧ us−r ∧ us−r−1 ∧ us−r−2 ∧ . . . Now, by Theorem 6.5.9, v is a linear combination of ordered q-wedges. Hence, u is a linear combination of elements of the form uk1 ∧ uk2 ∧ . . . ∧ ukr ∧ us−r ∧ us−r−1 ∧ us−r−2 ∧ . . . with k1 > . . . > kr . Such an element is zero if kr s−r by Proposition 6.5.11. Otherwise, it is a semi-infinite ordered q-wedge. This shows that the semi-infinite ordered q-wedges span Λ s . The fact that the semi-infinite ordered q-wedges are linearly independent follows directly from the definition of the inductive limit together with Theorem 6.5.9. Hence, one can express an arbitrary semi-infinite q-wedge as a sum of semiinfinite ordered q-wedges using the formulae (R1)–(R4). 6.5.17. We give an alternative indexation for the elements of the basis of Λ s which will be useful to connect Λ s with the Fock space. This indexation uses the notion
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
of abaci. Let us define this new object. A 1-runner abacus is a subset A of Z such / A for large enough k ∈ Z>0 . This can be represented as follows: that −k ∈ A and k ∈ each k ∈ A corresponds to a the position of a bead on a horizontal abacus which is full of beads on the left and empty on the right (see Figure 6.3). An l-abacus is an l-tuple of abaci (A1 , . . . , Al ). It is identified with the subset {(k, c) | k ∈ Ac , 1 c l} ⊂ Z × {1, . . . , l} (see Figure 6.4).
Fig. 6.3 A 1-runner abacus
Fig. 6.4 A 3-runner abacus
Now, let λ = (λ 1 , . . . , λ l ) l n and s = (s1 , . . . , sl ) ∈ Zl be such that ∑1il si = s. We associate with this datum an l-abacus: Aλs := {(λi
(b)
− i + sb + 1, b) | i 1, 1 b l}.
If we denote by A l the set of l-abaci, we check that the map
Π l (n) × Zl → A l ,
(λ , s) → Aλs
is a bijection. Consequently, each semi-infinite ordered q-wedge can be labelled by a symbol |λ , s, where λ is an l-partition of rank n and s = (s1 , . . . , sl ) ∈ Zl is such that ∑1il si = s. This is done as follows. Let uk = uk1 ∧ uk2 ∧ . . . be a semiinfinite ordered q-wedge. For i = 1, 2, . . ., we put ki = ai + e(l − bi ) − elmi , where (b) (b) ai ∈ {1, . . . , e}, bi ∈ {1, . . . , l} and mi ∈ Z. For b = 1, . . . , l, let k1 > k2 > . . . be the semi-infinite sequence obtained by ordering the elements of the set {ai − emi | bi = b} in strictly decreasing order. Then the set (b)
{(ki , b) | i 1, 1 b l}
6.5 Uglov’s Theory of Fock Spaces
345
defines an l-abacus which corresponds to an l-partition λ and an l-tuple s under (b) the above bijection. Hence, there is a unique sb ∈ Z such that ki = sb − i + 1 for 1 sufficiently large i and we can set τl (uk ) = |λ , s., where λ = (λ , . . . , λ l ) and where (b) (b) for i > 0, λi = ki − sb + i − 1 with s = (s1 , . . . , sl ). This defines a bijection τl between the set of semi-infinite ordered q-wedges and the set of symbols |λ , s, where λ l n and s = (s1 , . . . , sl ) ∈ Zl is such that ∑1il si = s.
Fig. 6.5 A 1-runner abacus representing the sequence k = (15, 12, 8, 7, 3, 1, −2, −4, −5, −6, . . .)
Fig. 6.6 A 3-runner abacus representing the sequence k = (15, 12, 8, 7, 3, 1, −2, −4, −5, −6, . . .) for e = 4
In a less formal way, the infinite decreasing sequence (k1 , k2 , . . .) can be pictured as a set of coloured beads on an infinite runner. Each element k j corresponds to the position of a bead on a horizontal abacus which is full of beads on the left and empty on the right (see Figure 6.5). On the other hand, using the decomposition ki = ai + e(l − bi ) − elmi , the same sequence can be represented by an l-abacus; that is, as a set of coloured beads on l infinite runners (see Figure 6.6).
Fig. 6.7 A 3-runner abacus representing the sequence k = (15, 12, 8, 7, 3, 1, −2, −4, −5, −6, . . .) for e = 4 with another indexation of the beads.
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
Fig. 6.8 The 3-runner abacus obtained after moving the beads to obtain s
In this representation, the coloured beads are labelled by the integers ki (with (b) i = 1, 2, . . .). We can alternatively label them by the integers ki (with i = 1, 2, . . . and b = 1, 2, . . . , l) as in Figure 6.7. Let λ be the l-partition and s = (s1 , . . . , sl ) be the l-tuple of integers associated to (b) (b) k. For i = 1, 2, . . . and b = 1, 2, . . . , l, we have λi = ki − sb + i − 1. Thus, one can now easily determine λ by counting the number of non coloured beads at the left of each coloured bead on each runner. The multicharge s can be read in the l-abacus by moving the beads to the left as much as possible (see Figure 6.8). Continuing the example, we obtain λ = (6.1, 2.2, 4.1) and s = (−2, 2, 3). e ) on 6.5.18. We now show the existence of an action of the quantum group Uq (sl the semi-infinite q-wedge product. This is derived from the action of Uq (sle ) on ΛqrV that we have constructed previously. The action of the Chevalley operator fi on an arbitrary semi-infinite ordered q-wedge uk is given as follows: fi (uk1 ∧ uk2 ∧ . . .) :=
∑ ti (uk1 ) ∧ . . . ∧ ti (uk j−1 ) ∧ fi (uk j ) ∧ uk j+1 ∧ . . .
j1
for k ∈ P++ (s). Note that, by 6.5.14 and by Proposition 6.5.11, there are only a finite number of q-wedges in the above sum. Now, we want to give the action of the ei and the ti . They are a bit more involved to describe and we will need to introduce more notations to present them. In [291, Section 4.2], this requires in particular the notion of weight of a q-wedge. As noted in [298, (3.2.1)], we can bypass this by arguing as follows. First, for m ∈ Z, we set |elm := uelm ∧ uelm−1 ∧ uelm−2 ∧ . . . Note that each semi-infinite ordered q-wedge uk can be written as v ∧ |elm for v ∈ Λ rV with r = s − elm. However, this decomposition is not unique, as such a q-wedge can also be written as v ∧ |elm for m < m and v ∈ Λ t V for t = s − elm . We now set ti |elm := ql δi,0 |elm
and
ti (v ∧ |elm) := ti (v) ∧ |elm,
which is well defined. The action of ei is given as follows:
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347
ei (v ∧ |elm) := ei (v) ∧ ti−1 |elm. Finally, we set d|elm := −elm sgn(m) where
⎧ ⎨ 0 1 sgn(m) = ⎩ −1
m(m + 1) |elm, 2 if m = 0, if m > 0, if m < 0,
and we have d(v ∧ |elm) := d(v) ∧ |elm, e )-module. which is well defined. One can then see that this makes Λ s into a Uq (sl It is now a natural question to ask what becomes of these formulae when we use the indexation of the basis by the set of l-partitions. Recall that the Fock space associated with s = (s1 , . . . , sl ) ∈ Zl is the Q(q)-vector space generated by the symbols |λ , s with λ l n: Q(q)|λ , s. Fs := n0 λ l n
Now, we have seen that one can use the indexation by l-partitions for the standard basis of Λ s . Using this, the Fock space Fs can be seen as a subset of Λ s where ∑1il si = s, and from now on we will make this identification. Proposition 6.5.19 (Uglov [291, §4.2]). For all s = (s1 , . . . , sl ) ∈ Z such that we e )-submodule of Λ s . Moreover, the inhave ∑1il si = s, the space Fs is a Uq (sl e ) on Fs is given in Theorem 6.2.5. duced action of Uq (sl What the above result says is that the action constructed above generalises the action constructed in Section 6.2. Now, one can now ask if we can give a definition for the canonical basis of the whole space of semi-infinite q-wedge product.
6.6 Canonical Bases for Fock Spaces In this section we define an involution on the Fock space which will be used to obtain the definition of the canonical basis. To do this, we use the semi-infinite q-wedge product that we introduced in the last section. 6.6.1. Let s = (s1 , . . . , sl ) ∈ Zl . We define an involution on the Fock space. First, recall that we have already introduced an involution on the space ΛqrV in 6.5.12. If k = (k1 , k2 , . . .) ∈ Ps++ , we set deg(uk ) = min{i 1 | k j = ki − ( j − i) for all j > i}. Then our involution on Λ s is defined as follows.
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Proposition 6.6.2 (Uglov). Let k = (k1 , k2 , . . .) ∈ Ps++ . The formula uk := uk1 ∧ . . . ∧ ukr ∧ ukr+1 ∧ ukr+2 ∧ . . . , where r deg(uk ), gives a well-defined involution on Λ s . Proof. This comes from the fact that in Λqt V , if t > r deg(uk ), we have uk1 ∧ . . . ∧ ukr ∧ ukr+1 ∧ . . . ∧ ukt = uk1 ∧ . . . ∧ ukr ∧ ukr+1 ∧ . . . ∧ ukt ;
see [291, Lemma 4.10].
An easy, but quite long computation [291, Proposition 3.31] shows us that this definition is compatible with the action of the quantum group and the involution defined in 6.1.14 in the following sense. e ) and u ∈ Λ s , we have we .u = we .u. Proposition 6.6.3 (Uglov). For all we ∈ Uq (sl Moreover, if s = (s1 , . . . , sl ) is such that s = ∑1il si , then, for all u ∈ Fs we have u ∈ Fs . Let us now fix s ∈ Zl such that s = ∑1il si , then the above discussion shows the existence of an involution on the Fock space by restriction: u ∈ Fs
→
u ∈ Fs .
For all λ l n, there exist coefficients aλ ,μ (q) ∈ Z[q] such that |λ , s =
∑ aλ ,μ (q)|μ, s.
μl n
We set A(s) := (aλ ,μ (q))λ l n,μl n . The following proposition gives us a first property of this matrix. Lemma 6.6.4. For all λ l n, we have aλ ,λ (q) = 1. Proof. Let uk = uk1 ∧ uk2 ∧ . . . be a semi-infinite ordered q-wedge of degree r. We define kr := (k1 , . . . , kr ) ∈ Pr++ . Then using the notation of 6.5.12, by (◦) in 6.5.12 together with Proposition 6.6.2, we have uk := (−q)υ (b(k )) (q−1 )υ (a(k )) ukr ∧ . . . ∧ uk1 ∧ ukr+1 ∧ ukr+2 ∧ . . . r
r
Using the relations (R1)–(R4), we can express ukr ∧ . . . ∧ uk1 ∧ ukr+1 ∧ ukr+2 ∧ . . . as a sum of semi-infinite ordered q-wedges. We are only interested in the coefficient r r of uk in this expression. This coefficient is precisely (−q)−υ (b(k )) (q)υ (a(k )) , as required. 6.6.5. We now wish to study the matrix of the involution A(s). The main aim will be to prove the unitriangularity of this matrix, with the partial order m that we have defined in 5.7.5. This is a generalisation of the main result of [176, Theorem 4.7].
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Recall that we have fixed a sequence s = (s1 , . . . , sl ) ∈ Zl . Let v = (v1 , . . . , vl ) ∈ Ql be such that for all i < j we have 0 < v j − vi < e and set m := (s1 − v1 , . . . , sl − vl ). Recall the following partial order. Let λ and μ be both l-compositions of rank n. As in 5.5.10, we define def
λ m μ
⇐⇒
λ = μ or κm (λ ) κm (μ).
In Chapter 5, we used this partial order to study the canonical basic sets for Ariki– Koike algebras in the case where s ∈ Sel . Here, we consider a general multicharge s ∈ Zl . Using the property of unitriangularity that we prove here, we will be able to deduce the existence of a canonical basis for the Fock space Fs , where s ∈ Zl . Before doing this, we study the form of the elements appearing in the expansion of an element |λ , s with λ l n in terms of elements |μ, s with μ l n. More precisely, we examine the forms of the l-partitions appearing in this expansion from the l-partition λ . First, we introduce some more notation. Let uk be a semi-infinite (possibly non ordered) q-wedge. Let u k be the semi-infinite ordered q-wedge obtained from uk by reordering the ki in strictly decreasing order. The bijection τl then allows us to associate with u k a symbol |λ , s such that λ l n and s = (s1 , . . . , sl ). We denote by π the linear map from the vector space generated by all semi-infinite q-wedges to s∈Zl ,s1 +...+sl =s Fs such that
π (uk ) = |λ , s. In particular, τl and π coincide on the set of semi-infinite ordered q-wedge. Note that the map π is not injective. 6.6.6. Let uk = uk1 ∧ uk2 ∧ . . . be an arbitrary semi-infinite q-wedge and assume that this is non-ordered. Then there exists i ∈ Z>0 such that ki < ki+1 . The relations (R1)– (R4) then show how to express uk in terms of semi-infinite q-wedges uk with ki > . Let us denote π (u ) = |λ , s and π (u ) = |λ , s. Using the representations of ki+1 k k uk and uk by l-abaci, we can see that uk is obtained from uk by moving two beads representing ki and ki+1 to two beads representing ki and ki+1 and lying in the same runners as ki and ki+1 . As a consequence, the l-partition λ is obtained from λ by removing a “ribbon” R of size N (or a rim N-hook; see [182, Section 2.7]) from a component b and adding a ribbon R of size N to a component b . In a more formal way, there exist positive integers r, s, p and t such that
λ (b) = (λ1 , . . . ., λr−1 , λr+1 − 1, . . . , λs − 1, λr − (N − (r − s)), λs+1 , . . .),
(b)
(b)
(b)
(b )
(b )
(b )
λ (b ) = (λ1 , . . . , λ p−1 , λt
(b)
(b)
(b )
+ (N − (p − t)), λ p
(b)
(b )
(b )
+ 1, . . . , λt−1 + 1, λt+1 , . . .).
In fact, the two l-partitions λ and λ are both obtained by adding the ribbons R and R to the components b and b of the same l-partition μ which is characterized as follows. We have |μ, s = π (ur ), where ur = uk1 ∧ uki−1 ∧ ukˆ i ∧ ukˆ i+1 ∧ uki+2 ∧ . . .
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
} such that the beads Here, kˆ i and kˆ i+1 are the minimal integers of {ki , ki+1 , ki , ki+1 ˆ ˆ associated with ki and ki+1 lie at the components b and b of the l-abacus of ur and kˆ i+1 < kˆ i .
6.6.7. Let B be the shifted m-symbol of μ of size k (which is assumed to be sufficiently large) and write all the elements appearing in this symbol by {γ1 , . . . , γh }. These elements are of the form
μ cj − j + sc − vc + k,
where c ∈ {1, . . . , l},
j ∈ {1, . . . , [k + sc − vc ]}.
Note that κm (μ) is the sequence of elements of this set written in decreasing order. By definition of κm and of the semi-infinite q-wedge, there exist x and y such that κm (λ ) is the sequence obtained by reordering the elements of {γ1 , . . . , γh } \ {γx } ∪ {γx + N} in decreasing order and κm (λ ) is the sequence obtained by reordering the elements {γ1 , . . . , γh } \ {γy } ∪ {γy + N} in decreasing order, where N is a certain positive integer. Lemma 6.6.8. Keeping the above notation, assume that λ = λ ; then we have γx > γy . Proof. We use the same argument as in [174, pp. 581–583]. Keeping the notation of the previous section, we set kˆ i =: a + e(l − b) − elm and kˆ i+1 =: a + e(l − b ) − elm . Then, by the discussion in 6.5.17, the elements of κm (μ) are closely related to the (k1 , . . . , ki−1 , kˆ i , kˆ i+1 , ki+2 , . . .). Indeed, we have that
γx = a − em − vb − 1 + k and γy = a − em − vb − 1 + k. Hence, one has to show that (◦◦)
a − em − vb > a − em − vb .
For j = i, i + 1, we set k j =: a j + e(l − b j ) − elm j and we define α := (ai+1 − ai ) mod el and β := e(bi+1 − bi ) mod el. We assume that λ = λ . Then, regarding the relations (R1)–(R4) in 6.5.10, we have several cases to consider: • There exists an m > 0 such that ki := ki+1 − elm and ki+1 = ki + elm. • We have β = 0 and there exists an m 0 such that ki = ki+1 − β − elm and = ki + β + elm. ki+1 • We have α = 0 and there exists an m 0 such that ki = ki+1 − α − elm and = ki + α + elm. ki+1 • We have β = 0, α = 0 and there exists an m 0 such that ki = ki+1 − β − α − elm = ki + β + α + elm. and ki+1
In each case, we easily check that the inequality (◦◦) holds using the fact that bi < bi+1 if and only if 0 < vbi < vbi+1 < e. Note that, as a consequence, we obtain that λ m λ by the definition of m .
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6.6.9. We are now in a position to prove the unitriangularity of the matrix A(s). Let λ l n and set up := τl−1 (|λ , s). We set uk = uk1 ∧ uk2 ∧ . . . := up . We have by definition that π (uk ) = |λ , s. Now, after several applications of the discussion in 6.6.6 and 6.6.7, we obtain that if uk is a semi-infinite ordered q-wedge appearing in the expansion of up , then we have λ m λ , where τl (uk ) = |λ , s. Combining this with Lemma 6.6.4, we obtain the following proposition. Proposition 6.6.10. Let s = (s1 , . . . , sl ) ∈ Zl and let v := (v1 , . . . , vl ) be such that 0 < vi −v j < e if j > i. Set m = (s1 −v1 , . . . , sl −vl ) and let λ l n. Then, aλ ,μ (q) = 0 implies λ m μ. Moreover, we have aλ ,λ (q) = 1. The above results will allow us to define the canonical basis for the Fock space. To do this, we set Q[q]|λ , s. L [s] := λ l n
Then we obtain the following theorem.
Theorem 6.6.11 (Leclerc and Thibon [209, 4.1] (l = 1), Uglov [291,§4] (l 1)). Let s = (s1 , . . . , sl ) ∈ Zl . There exists a unique basis Bq (μ, s) | μ l n of the Fock space Fs such that Bq (μ, s) = Bq (μ, s)
Bq (μ, s) = |μ, s mod qL [s].
and
Proof. Let v := (v1 , . . . , vl ) be such that 0 < vi − v j < e if j > i. Set m = (s1 − v1 , . . . , sl −vl ). We construct the elements B(λ , s) by induction on m . First, assume that μ l n is maximal with respect to m . As the matrix A(s) is unitriangular by Proposition 6.6.10, we have |μ, s = |μ, s and we can set Bq (μ, s) = |μ, s. Let λ l n. By induction, we assume that we have already constructed the canonical basis elements Bq (λ [1], s), . . . , Bq (λ [i − 1], s) for all the l-partitions λ [ j] satisfying λ m λ [ j] and λ = λ [ j] with j = 1, . . . , i − 1. In addition, we assume that they satisfy aν, j (q)|ν, s Bq (λ [ j], s) = |λ [ j], s + ∑ λ [ j]m ν,λ [ j]=ν
for certain coefficients aν, j (q) in qZ[q] with j = 1, . . . , i − 1. We set λ [i] := λ . We want to show that there exist am,i (q) in qZ[q] with 1 m < i such that Bq (λ [i], s) = |λ [i], s +
∑
am,i (q)|λ [m], s.
1m
First, as A(s) is unitriangular by Proposition 6.6.10 and by the above induction hypothesis, there exist coefficients βm (q) in Z[q, q−1 ] with 1 m < j such that |λ [i], s − |λ [i], s = ∑1m
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
polynomials am,i (q) ∈ qZ[q] with 1 m < i such that βm (q) = am,i (q) − am,i (q−1 ) for all 1 m < i. It is readily checked that the element Bq (λ [i], s) = |λ [i], s + ∑1m
∑
μm λ ,μ=λ
bλ ,μ (q)|λ , s,
where bλ ,μ (q) ∈ qZ[q] if μ m λ and μ = λ . 6.6.13. Let s ∈ Zl . Consider the submodule V (Λs )s of Fs generated by the symbol |∅, s. By 6.1.16, it is equipped with a canonical basis Gq (μ, s) | μ ∈ Φ(s,e) indexed by the set a Uglov l-partitions, which has been explicitly computed in 6.4.9 and 6.4.10. We prove the compatibility of this basis with the one just defined. Theorem 6.6.14 (Uglov). For all μ ∈ Φ(s,e) , we have Gq (μ, s) = Bq (μ, s). Proof. By Proposition 6.6.3, the restriction of the bar involution to the irreducible highest weight module V (Λs )s corresponds to the involution defined in 6.1.16. In addition, the elements Gq (μ, s) have the same congruence property with respect to qL [s] as do the Bq (μ, s). By the uniqueness of these elements, we can conclude. Hence, for all μ l n, there exist elements bλ ,μ (q) with λ l n such that Bq (μ, s) =
∑ bλ ,μ(q)|λ , s
λ l n
and we have bλ ,μ (q) = dλ ,μ (q)
for all μ ∈ Φ(s,e) (n) and λ l n.
So we may now change the notation and define dλ ,μ (q) := bλ ,μ (q) for all μ l n / Φ(s,e) (n). Consequently, the matrix of the canonical basis and λ l n such that μ ∈ D(s,e) (q) = (dλ ,μ (q))λ l n,μ∈Φ(s,e) (n) for V (Λs )s is a submatrix of the matrix of the canonical basis of the Fock space Δ (s,e) (q) = (dλ ,μ (q))λ l n,μl n . One can obtain the first from the latter by keeping only the rows indexed by the elements in Φ(s,e) (n). Example 6.6.15. Assume that s = (0, 0), e = 2 and l = 2. Then the matrix of the canonical basis for the associated Fock space can be obtained first by computing
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353
the matrix of the involution A(s) and then by following the proof of the existence of such basis (Theorem 6.6.11). We have ⎛ ⎞ 1 . . . . (2, ∅) ⎜ q 1 . . . ⎟ (1.1, ∅) ⎜ ⎟ ⎟ Δ(s,e) (q) = ⎜ ⎜ q2 . 1 . . ⎟ (∅, 2) ⎝ q q q 1 . ⎠ (∅, 1.1) (1, 1) . . . . 1 and the matrix of the canonical basis for V (Λs )s is given by ⎞ ⎛ 1 . (2, ∅) ⎜ q . ⎟ (1.1, ∅) ⎟ ⎜ ⎟ D(s,e) (q) = ⎜ ⎜ q2 . ⎟ (∅, 2) . ⎝ q . ⎠ (∅, 1.1) (1, 1) . 1 Let (v1 , v2 ) ∈ Q2 be such that 0 < v2 − v1 < 2. Set, as usual, m = (s1 − v1 , s2 − v2 ) = (−v1 , −v2 ). Then, one can check that the minimal elements with respect to m appearing with non-zero coefficient in the each column of D(s,e) (q) are (2, ∅) and (1, 1). This is in agreement with the above discussion and Theorem 6.6.12 because we have Φ(s,e) (2) = {(2, ∅), (1, 1)}. Note that this set corresponds to the set of FLOTW 2-partitions because (s1 , s2 ) ∈ S22 in this case; see Definition 5.7.8. 6.6.16. We have already seen how one can compute the matrices of the canonical basis for the irreducible highest weight modules. It is now natural to ask if this is possible to provide such an algorithm for computing the matrices of the canonical basis for the whole Fock space. Such an algorithm can be obtained by computing the matrix of the involution A(s). The most direct way to do that is to use the relations (R1)–(R4). Unfortunately, it is in practice very difficult (and not efficient!) to use these relations. In particular, if l 0 and if s1 s2 . . . sl , the number of q-wedges involved in such a computation is too high to be done in practice. Yvonne [300] provides a faster algorithm for computing A(s) which does not require the relations (R1)–(R4). The idea is to compute an invariant basis of Λ s with respect to the involution. Such a basis is obtained by studying the action of three algebras on e ), the algebra Uq (sl l ) and finally a Heisenberg the vacuum vector: the algebra Uq (sl algebra. This result gives us an invariant basis for each weight subspace of the Fock spaces and a transition matrix T (q) between the standard basis and this invariant basis. The matrix of the involution of this weight space follows. This result generalizes earlier works of Leclerc and Thibon [209], corresponding to the case l = 1.
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
6.7 Computation of Canonical Basic Sets for Iwahori–Hecke Algebras We now come back to the problem of parametrising the canonical basic sets for Iwahori–Hecke algebras of type B. First, we generalise our previous results on the canonical basic sets for Ariki–Koike algebras and then focus on Iwahori–Hecke algebras of type Bn . 6.7.1. As in Section 5.8, let e ∈ Z>1 and let s ∈ Zl . Assume that k is a field of characteristic 0. Let θ : A → k be a specialisation such that θ (u) = ηe , a primitive root of unity of order e > 1. Assume in addition that, for all j = 1, . . . , l, we have s θ (V j ) = ηe j for s = (s1 , . . . , sl ) ∈ Zl . We get a specialised algebra Hk,n := Hk,n (ηe ; ηes1 , . . . , ηesl ). Let v = (v1 , . . . , vl ) ∈ Ql be such that i < j implies 0 < v j − vi < e. As before, we set m j := s j − v j ∈ Q for all j = 1, . . . , l, m := (m1 , . . . , ml ) and consider D(s,e) := Dθ the associated decomposition matrix. We want to study the existence of a canonical basic set with respect to the a-function a(m,r) (where r is an arbitrary positive integer). In Theorem 5.8.2, we have already proved the existence of such a canonical basic set, but only in the case where s ∈ Sel . It is true that if s ∈ Zl then there l such that s = w.s . We then have D(s,e) = D(s ,e) , because exist s ∈ Sel and w ∈ S the associated specialised algebras are isomorphic. However, the partial orders in duced by the a-invariants a(m,r) (where m = (s1 − v1 , . . . , sl − vl )) and a(m ,r) (where m = (s 1 − v1 , . . . , s l − vl )) are really different in the two cases; see Example 6.7.3. Hence, the question of existence of a canonical basic set in the general setting is not trivially deduced from the case where s ∈ Sel . Theorem 6.7.2. Assume that we are in setting of 6.7.1. Then, with respect to the order on simple HK,n -modules induced by a(m,r) , the algebra Hk,n admits a canonical basic set Bθ in the sense of Definition 5.5.19. We have Bθ = Φ(s,e) (n), the set of Uglov l-partitions.
Proof. By Ariki’s theorem, the decomposition matrix D(s,e) is equal to the matrix D(s,e) (1). Now by Theorem 6.6.12 together with Proposition 6.6.14, for all μ ∈ Φ(e,s) (n), we have Gq (μ, s) = |μ, s +
∑
μm λ , μ=λ
dλ ,μ (q)|λ , s.
Now note that the rational numbers v j for i < j satisfy 0 < v j − vi < e. Thus, by Proposition 5.7.7, μ m λ and μ = λ implies a(m,r) (λ ) > a(m,r) (μ). We deduce that the decomposition matrix is unitriangular with respect to the a-invariant and that the associated canonical basic set is given by the set Φ(s,e) (n).
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355
What the above result shows is that the simple modules of an Ariki–Koike algebra may be labelled by several sets depending on the choice of the weight sequence m we take. This is illustrated in the following example. Example 6.7.3. Let l = e = n = 2 and s = (0, 1); then the decomposition matrix of the associated Ariki–Koike algebra is ⎛ ⎞ 1 . (2, ∅) ⎜ . 1 ⎟ (∅, 2) ⎜ ⎟ ⎟ D((0,1),2) := ⎜ ⎜ 1 1 ⎟ (1, 1) . ⎝ 1 . ⎠ (1.1, ∅) . 1 (∅, 1.1) We set v = (0, 1) and r = 1. Then m = (0, 0). The a(m,r) -invariants of the 2-partitions associated with this choice of multicharge are given as follows: a(m,r) (2, ∅) = a(m,r) (∅, 2) = 0,
a(m,r) (1, 1) = 1,
a(m,r) (1.1, ∅) = a(m,r) (∅, 1.1) = 2. We have an associated canonical basic set given by the set of Uglov 2-partitions:
Φ((0,1),2) (2) = {(2, ∅), (∅, 2)}. Now assume that s = (2, 1) and take as above v = (0, 1), then the decomposition matrix D((2,1),2) is the same as above, but the a(m,r) -invariants are now a(m,r) (2, ∅) = 0, a(m,r) (∅, 2) = 3, a(m,r) (1.1, ∅) = 1,
a(m,r) (1, 1) = 2,
a(m,r) (∅, 1.1) = 6.
Consequently, we also have an associated canonical basic set given by the set of Uglov 2-partitions: Φ((2,1),2) (2) = {(2, ∅), (1, 1)}. Finally, assume that s = (0, 3) and take as above v = (0, 1), then the decomposition matrix D((0,3),2) is the same as above, but the a(m,r) -invariants are now a(m,r) (2, ∅) = 3,
a(m,r) (∅, 2) = 0,
a(m,r) (1.1, ∅) = 6,
a(1, 1) = 2,
a(m,r) (∅, 1.1) = 1.
Consequently, we also have an associated canonical basic set given by the set of Uglov 2-partitions: Φ((0,3),2) (2) = {(∅, 2), (1, 1)}. l we have D(s,e) (1) = D(w.s,e) (1). As noted We have already seen that for all w ∈ S by Yvonne [298], the following example shows that, in general, this is not the case for the matrix Δ (s,e) .
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
Example 6.7.4. Assume that e = 2 and l = 2. The matrices of the canonical basis Δ((4,−3),2) (q) and Δ((1,−0),2) (q) for the Fock spaces F(4,−3) and F(1,0) can be computed using the algorithm described in [300]. The following tables are taken from [298]. We obtain: ⎛ ⎞ 1 . . . . . . . . . (∅, 3) ⎜ q 1 . . . . . . . . ⎟ (∅, 2.1) ⎜ 2 ⎟ ⎜ q q 1 . . . . . . . ⎟ (1, 2) ⎜ ⎟ ⎜ . q . 1 . . . . . . ⎟ (∅, 1.1.1) ⎜ ⎟ ⎜ . q2 q q 1 . . . . . ⎟ (1.1, 1) ⎟ Δ((4,−3),2) (q) = ⎜ ⎜ . . q . . 1 . . . . ⎟ (3, ∅) , ⎜ ⎟ ⎜ . . q2 . q q 1 . . . ⎟ (2.1, ∅) ⎜ ⎟ ⎜ . . . q q2 . q 1 . . ⎟ (1.1.1, ∅) ⎜ ⎟ ⎝ . . . . . . . . 1 . ⎠ (2, 1) (1, 1.1) . . . . . . . . . 1 ⎛
1 ⎜ . ⎜ ⎜ . ⎜ ⎜q ⎜ ⎜ . Δ((1,0),2) (q) = ⎜ ⎜ . ⎜ 2 ⎜q ⎜ ⎜ . ⎜ ⎝ . .
. 1 q q q2 . . . . .
. . 1 . q q2 . . . .
. . . 1 q . q q2 . .
. . . . . . . . 1 . q1 . . q . . . . .
. . . . . . 1 q . .
. . . . . . . . . . . . . . 1 . . 1 . .
⎞ . (∅, 3) .⎟ (∅, 2.1) ⎟ .⎟ (1, 2) ⎟ .⎟ (∅, 1.1.1) ⎟ .⎟ ⎟ (1.1, 1) . .⎟ ⎟ (3, ∅) .⎟ ⎟ (2.1, ∅) .⎟ ⎟ (1.1.1, ∅) . ⎠ (2, 1) (1, 1.1) 1
2 on Z2 . Note that (1, 0) and (4, −3) are in the same orbit for the action of S However, Δ ((1,0),2) (q) is not equal to Δ((4,−3),2) (q). In fact, there are not the same number of zeros in the matrices Δ((1,0),2) (q) and Δ((4,−3),2) (q) and thus we have Δ((1,0),2) (1) = Δ((4,−3),2) (1) even after arbitrary permutation of rows/columns! This phenomenon was first observed by Yvonne. However, we have D((1,0),2) (1) = D((4,−3),2) (1). 6.7.5. We finally give the explicit characterisation of the canonical basic sets in type Bn in characteristic 0. We assume that k is a field such that char(k) = 0 and that we have a weight function L : W → Z such that L(t) = b > 0 and L(si ) = a > 0 for all i = 1, . . . , n − 1. Bn
b t
4
a t
a t
· · ·
a t
Let v be an indeterminate and set u := v2 . We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = η ∈ k× (where u = v2 ). We assume that η a is a primitive root of unity of order e, where e ∈ Z>1 , and we set ηe := η a . By Example 5.7.3, one may assume that there exists d ∈ Z such that η b = −η ad . Note that this condition ensures that b/a is different from d + p.e for all p ∈ Z. Taking the notation of the
6.7 Computation of Canonical Basic Sets for Iwahori–Hecke Algebras
357
previous section and using Example 5.7.3, we set l = 2, r = a and we choose d such that ( ) d − e < b/a < d. We set s = (d, e) and v = (v1 , v2 ) = (d − b/a, e). Then note that we have v2 − v1 = b/a − d + e. Hence, we have 0 < v2 − v1 < e. In addition, note that the cyclotomic Ariki–Koike algebra associated with m = (s1 − v1 , s2 − v2 ) = (b/a, 0) and r = a is the Iwahori–Hecke algebra of type Bn associated with L. This implies that the a-function a(m,r) is the usual a-function in type B associated with L; see Example 5.5.14. Finally, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn , where ηe := η a . We can then apply Theorem 6.7.2 and deduce the following result.
Theorem 6.7.6 (Cf. [125, Theorem 5.4]). Assume that L : W → Z is such that L(t) = b > 0 and L(si ) = a > 0 for all i = 1, . . . , n − 1 and that char(k) = 0. Then, in the setting of 6.7.5, the algebra Hk,n admits a canonical basic set Bθ = Φ((d,e),e) (n), the set of Uglov l-partitions; see Definition 6.2.13.
Remark 6.7.7. By Remark 6.2.15, we have Φ((d,e),e) (n) = Φ((d−e,0),0) (n). Let us now describe more precisely the canonical basic sets in some cases. 6.7.8. Let W be a Weyl group of type Bn and consider a weight function L : W → Z such that L(t) = b > 0 and L(si ) = 1 for all i = 1, . . . , n − 1. Bn
b t
4
1 t
1 t
· · ·
1 t
We assume that k is a field such that char(k) = 0. We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = ηe , a primitive root of unity of order e > 1. By Example 5.7.3, one can assume that there exist d ∈ Z such that ηeb = −ηed . Thus e is even. We have to take d = b + e/2 so that condition ( ) in 6.7.5 is satisfied. We set l = 2, r = 1, s = (b + e/2, e) ∈ Se2 , v = (e/2, e). The weight sequence that we have to consider is thus m = s − v = (b, 0). Note that the cyclotomic Ariki– Koike algebra associated with m = (b, 0) and r = 1 is the Iwahori–Hecke algebra of type Bn associated with L and thus that the a-invariant a(m,r) is the usual a-invariant in type Bn associated with L; see Example 5.5.14. Furthermore, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn . Theorem 6.7.9. In the setting of 6.7.8, the algebra Hk,n admits a canonical basic set Bθ = Φ((b+e/2,e),e) (n). 6.7.10. We still assume that W is a Weyl group of type Bn . We now consider a weight function L : W → Z such that L(t) = 2c + 1 (where c 0) and L(si ) = 2 for all i = 1, . . . , n − 1.
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
Bn
2c+1 t
4
2 t
2 t
· · ·
2 t
We assume that k is a field such that char(k) = 0. We consider a specialisation θ : R[v, v−1 ] → k such that θ (u) = η f , a primitive root of unity of order f > 1. By Example 5.7.3, one can assume that there exist d ∈ Z such that η 2c+1 = −η 2d f . f Thus, f is even and we can set f = 2e for an integer e. Then ηe := η 2f is a primitive root of unity of order e, where e is odd. We assume that e > 1. We have to take d = (2c + 1 + e)/2 so that condition ( ) in 6.7.5 is satisfied. We set l = 2, r = 2, s = ((2c+1+e)/2, e) ∈ Se2 , v = (e/2, e). The weight sequence that we have to consider is thus m = s − v = ((2c + 1)/2, 0). Note that the cyclotomic Ariki–Koike algebra associated with m = ((2c + 1)/2, 0) and r = 2 is the Iwahori–Hecke algebra of type Bn associated with L and thus that the a-invariant a(m,r) is the usual a-invariant in type Bn associated with L; see Example 5.5.14. In addition, the Ariki–Koike algebra Hk,n (ηe ; ηes1 , ηes2 ) is isomorphic to the specialised Iwahori–Hecke algebra of type Bn , where ηe := η 2 . Theorem 6.7.11. In the setting of 6.7.10, the algebra Hk,n admits a canonical basic set Bθ = Φ(((2c+1+e)/2,e),e) (n). Remark 6.7.12. We are now also in position to solve the problem of computing the canonical basic set in type Bn when L : W → Z is such that L(t) = 2 and L(si ) = 1 for all i = 1, . . . , n − 1. The case where θ : R[v, v−1 ] → k is such that θ (u) = −1 was missing in Theorem 5.8.13. Assume that k is a field of characteristic 0. We can here argue exactly as in 5.8.12 and we obtain that, in this situation, the Iwahori–Hecke algebra Hk,n admits a canonical basic set Bθ = Φ((3,2),2) (n). When char(k) = p, this last set is also a canonical basic set, as already remarked in Example 4.4.15. 6.7.13. We finally consider the asymptotic case in type Bn . In this case, we can just apply Theorem 6.7.6 and obtain the description of the canonical basic sets in characteristic 0. The proof is due to Ariki [9]. However, we can have a stronger statement in this setting. Theorem 6.7.14. Let k be any field. Let L : W → Z be such that L(t) = b > 0 and L(si ) = a > 0 for all i = 1, . . . , n − 1, and assume that b > (n − 1)a > 0. Let e be the order of θ (u)a ∈ k× . Assume that e ∈ Z>1 and that θ (u)b = −θ (u)ad for some d ∈ {1, . . . , e} (see Example 5.7.3). Then, Hk,n admits a canonical basic set given by the set of Kleshchev bipartitions (see Example 6.2.16): Bθ = Φ ((d,0) mod e,e) (n). Proof. First assume that char(k) = 0. We can then use Theorem 6.7.6: the algebra Hk,n admits a canonical basic set Bθ = Φ((d,e),e)(n) , where d ∈ Z is such that d − e < b/a < d. Now note that we have d − e > n − 1 − e; hence, by Example 6.2.16, the multicharge (d, e) is very dominant. We thus have Φ((d,e),e) (n) = Φ ((d,0) mod e,e) (n), the set of Kleshchev bipartitions. This completes the proof in the case where char(k) = 0.
6.8 Recent Developments and Conjectures
359
Now assume that char(k) = p > 0. By Example 3.2.10, we already know that a canonical basic set for Hk,n exists. So we can try to use Lemma 3.6.7. Now, since e > 1 is the order of θ (u) in k× , we have Φe (θ (u)) = 0, where Φe (u) ∈ Z[u] denotes the eth cyclomotic polynomial. Then the specialisation θ : R[v, v−1 ] → k lies above a suitable specialisation θ0 : R[v, v−1 ] → k0 , where k0 has characteristic 0 and Φe (θ0 (u)) = 0; that is, θ0 (u) ∈ k0× is a primitive root of unity of order e. Furthermore, Theorem 6.2.22 shows that | Irr(Hk,n )| = | Irr(Hk0 ,n )|. Thus, all the assumptions of Lemma 3.6.7 are satisfied. So we can deduce that Φ ((d,0) mod e,e) (n) also is a canonical basic set for Hk,n , as claimed.
6.8 Recent Developments and Conjectures In this final section we discuss several new developments connected to the topics of this book. 6.8.1. Both finite Coxeter groups and the groups of type G(l, p, n) are particular cases of finite complex reflection groups. Let K be a finite abelian extension of Q and let V be a finite-dimensional K-vector space. We say that s ∈ GL(V ) is a pseudoreflection if it is a non-trivial element which acts trivially on a hyperplane. A finite subgroup W of GL(V ) generated by pseudo-reflections is called a complex reflection group. As noted before, a classification for the irreducible complex reflection groups (i.e., complex reflection groups which act irreductibly on V ) has been obtained by Shephard and Todd [278]. They are given by: • the family of groups G(l, p, n) (with l 1, p|l and n 1) that we have already introduced in 5.1.9, • a list of 34 exceptional groups. Complex reflection groups generalise both the main groups that we are interesting in in this book: the finite Coxeter groups and the groups of type G(l, 1, n). One can also associate with these groups some Hecke algebras. These were introduced and studied by Brou´e, Malle, Michel, Rouquier; see [35], [36], [238], [241]. It is natural to consider the problem of existence of canonical basic sets for the wider class of complex reflection groups and their Hecke algebras. 6.8.2. Let R be an integral domain, q be a unit in R and let x = (x1 , . . . , xd ) be a sequence of elements in R. Following Ariki [6], the cyclotomic Hecke algebra of type G(r, p, n) over R with parameters q and x has a presentation by generators a0 , . . . , an and defining relations
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6 Canonical Bases in Affine Type A and Ariki’s Theorem
(a0 − x1 ) . . . (a0 − xd ) = 0, (ai − q)(ai + 1) = 0, 1 ≤ i ≤ n, a1 a3 a1 = a3 a1 a3 , ai ai+1 ai = ai+1 ai ai+1 , 2 ≤ i ≤ n − 1, (a1 a2 a3 )2 = (a3 a1 a2 )2 , a1 ai = ai a1 , 4 ≤ i ≤ n, ai a j = a j ai , 2 ≤ i < j ≤ n, j ≥ i + 2, r−2
a0 a1 a2 = (q−1 a1 a2 )2−r a2 a0 a1 + (q − 1) ∑ (q−1 a1 a2 )1−k a0 a1 = a1 a2 a0 , a0 ai = ai a0 , 3 ≤ i ≤ n.
i=1
This class of algebras has been defined in [6]. Several aspects of the representation theory of these algebras can be deduced using Clifford theory as in type Dn (see 5.1.9). In the semisimple case, the simple modules are in one-to-one correspondence with the simple modules of the group algebra. These modules have been constructed in [6]. The modular case has been studied by Hu [165], [166], [167] and Genet and Jacon [136]. Morita equivalences for this class of algebras have been obtained by Hu and Mathas [168], in the same spirit as Theorem 5.4.3. Finally, it is possible to prove the existence of basic sets and the unitriangularity of the decomposition matrices using Clifford theory and the existence of basic sets for Ariki–Koike algebras; see Genet and Jacon [136]. The parametrisation of the basic sets is derived from the definition of FLOTW l-partitions. 6.8.3. Let us come back to the general case of a complex reflection group W . Let S be the set of pseudo-reflections in W and let c be a map from S to C which is constant on conjugacy classes. Following Etingof and Ginzburg [77], the rational Cherednik algebra Ht,c attached to W with parameters c and t ∈ C is defined as the quotient of the smash product of CW and the tensor algebra of V ⊕ V ∗ by the relations [x, x ] = 0,
[y, y ] = 0,
[y, x] = ty, x −
∑ cs αs , yx, αs∨ s
s∈S
for all x, x ∈ V ∗ , y, y ∈ V . Here, y, x is the canonical pairing between V and V ∗ , the element αs is a generator of Im(s|V ∗ − 1) and αs∨ is a generator of Im(s|V − 1) such that αs , αs∨ = 2. The representation theory of these algebras turns out to be very different if t = 0 or t = 0, but, quite remarkably, both are closely related to the modular representation theory of Hecke algebras. First, let us consider the case t = 0. Then the representations of the associated algebras have been studied by Gordon [139] and a conjecture by Gordon and Martino [141] gives connections between these representations when W is a Weyl group and the theory of two-sided cells for finite Weyl groups. A further conjecture by the same authors gives a relation between blocks of the Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. This conjecture has been proved for W = G(l, 1, n) (see Martino [242]) and W = G(l, p, n) (see Bellamy [16] and Martino [243]).
6.8 Recent Developments and Conjectures
361
6.8.4. Assume that t = 0. Then, by Ginzburg et al. [137], there is a functor, called the KZ functor, from a certain category of Ht,c -modules, the category O, to the category of finitely generated modules over the Hecke algebra of W . To each irreducible representation of W , one can attach a certain “standard module” in this category O with a simple head. When W = G(l, 1, n), we denote these simple modules by L(λ ) with λ l n. In the associated Grothendieck group, the image of L(λ ) through the KZ functor corresponds to the class of the associated Specht module (see Shan [276, Lemma 3.1]). This functor also provides a proof for the existence of canonical basic sets for Ariki–Koike algebras in characteristic 0, for certain choices of m as in Remark 5.7.4; see Ginzburg et al. [137] and Chlouveraki, Gordon and Griffeth [49]. In addition, one can also define some parabolic restriction and induction functors for these algebras; see Bezrukavnikov and Etingof [19]. Using this, one can construct a crystal structure on the set of simple modules L(λ ) which turns out to be isomorphic to the one of the Fock space; see Shan [276, Theorem 6.3]. Furthermore, connections with the representation theory of q-Schur algebras, and even category O, have been given by Rouquier [274]. Finally, conjectures by Yvonne [299] and Rouquier [274] assert that the decomposition matrices for these q-Schur algebras should be given by the canonical basis of Fock spaces for a particular choice of the multicharge. 6.8.5. Ariki’s theorem gives us a deep connection between the representation theory of Ariki–Koike algebras and the theory of canonical bases for quantum groups by saying that the decomposition matrices are equal to the evaluation at q = 1 of e )-modules D(s,e) (1). It is the matrices for the canonical bases of irreducible Uq (sl natural to ask if the matrices D(s,e) (q) themselves have an interpretation in the representation theory of Hecke algebras. An answer for this problem has been recently given by Brundan, Kleshchev and Wang [39], [40], [41]. We refer to Kleshchev [203] for a complete survey on these results. The main point is that the Ariki–Koike algebras are isomorphic to (sign-modified) “cyclotomic Khovanov–Lauda–Rouquier” algebra defined in [198], [199], [275]. As these algebras are naturally Z-graded, this shows the existence of a natural grading on Ariki–Koike algebras. Such results have been previously predicted by Turner [290] and Rouquier [273]. It is then possible to show that the Specht modules in Remark 5.3.12 are graded with respect to this gradation. As a consequence, one can define some “q-decomposition numbers” for these Specht modules, which are defined with respect to this grading and which correpond to the usual decomposition numbers after evaluation at q = 1. The main result of Brundan and Kleshchev [40] shows that the graded decomposition numbers correspond precisely to the coefficient of the matrix of the canonical e )-modules for a particular choice of the multicharge. This choice basis for the Uq (sl corresponds to the case where the Fock space can be seen as a tensor product of Fock spaces with level 1. It thus gives a generalisation and a q-analogue of Ariki’s theorem. An interpretation for general multicharge is still missing. In another direction, an interpretation of the “q-decomposition numbers” has been given by Shan [277] when l = 1 in terms of the Jantzen filtrations. This gives a proof of a conjecture by Lascoux, Leclerc and Thibon [208].
Chapter 7
Decomposition Numbers for Exceptional Types
One of the declared aims of this book has been to determine the decomposition matrices and canonical basic sets associated with (principal) specialisations of Iwahori– Hecke algebras. In this chapter we present these results for algebras of exceptional type. These are the culmination of work by various authors over a number of years: F4 (equal parameters) E6 F4 (some unequal parameters) E7 E8 (partial results) I2 (m), H3 , H4 F4 (any parameters) E8 (completed)
Geck and Lux [126]; Geck [94]; Bremke [30]; Geck [96], M¨uller [253]; M¨uller [253]; M¨uller [254]; McDonough and Pallikaros [249], [250]; Geck and M¨uller [129].
In most cases, the proofs rely on computer calculations based on the ideas in Parker’s M EATA XE [263], [270] and the G AP part of the C HEVIE system; see [118], [252]. We begin by explaining some fundamental computational methods in Section 7.1. We will then mainly be interested in the case where we consider a Φe -modular specialisation as in Example 3.1.15(b); that is, A is the ring of Laurent polynomials in one variable and this variable is specialised to a root of unity in C. In Section 7.2, we briefly discuss the case where W is of dihedral type; here, everything is easily computed “by hand”. In Section 7.3, we consider various choices of parameters for W of type F4 . Section 7.4 contains tables for W of type H3 , H4 , E6 , E7 and E8 . The decomposition matrices for W of type E7 and E8 appear here for the first time in print. In order to keep the volume of data to be printed within a reasonable size, we shall also rely on the information contained in the tables in Appendix F of [132], concerning the distribution of the irreducible representations into Φe -blocks and the Brauer trees of Φe -blocks of defect 1. So, here, we will only consider the decomposition matrices of blocks of defect at least 2.
M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7 7, © Springer-Verlag London Limited 2011
363
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7 Decomposition Numbers for Exceptional Types
7.1 Algorithmic Methods Let W be a finite Coxeter group, H = HA (W, S, L) a corresponding generic Iwahori– Hecke algebra and θ : A → k be a specialisation, where k is a field. Assume that R is L0 -good so that we have a decomposition map dθ : R0 (HK ) → R0 (Hk ). As in Section 3.2, let us write dθ ([Eελ ]) =
∑
dλ ,M [M]
(λ ∈ Λ ).
M∈Irr(Hk )
As explained in Remark 3.2.3, if we can manage to compute the decomposition numbers dλ ,M , then we can also determine a corresponding canonical basic set as in Definition 3.2.1. In principle, the numbers dλ ,M are computed as follows. By Lemma 3.1.13, there exists an H-module N λ which is finitely generated and free over A, such that dθ ([Eελ ]) = [k ⊗A N λ ]. Hence, dλ ,M is the multiplicity of M in a composition series of k ⊗A N λ . In order to turn this theoretical result into some efficient practical method, the first thing we need are realisations of the modules N λ in terms of explicitly given representing matrices on the generators Ts (s ∈ S). Suitable candidates for such realisations are provided by W -graph representations; see Definition 1.4.11. For W of exceptional type, such W -graphs are explicitly known by Examples 1.4.12 and 1.4.13. (They are available in electronic form through Michel’s development version of the C HEVIE system; see [118], [252].) By specialisation, we obtain corresponding matrix representations of Hk . Our next task is to decompose these representations into irreducible ones. 7.1.1. We recall the basic principles of Parker’s M EATA XE algorithm [263]. This is a general computational method for deciding whether a finite-dimensional representation of an associative algebra over a field is irreducible, and for finding an explicit non-trivial submodule when it is not. So let H be a finite-dimensional associative algebra over a field k. Let ρ : H → Md (k) be a matrix representation of H. Let H ◦ be the opposite algebra of H. Then we also have a representation ρ ◦ : H ◦ → Md (k) such that a → ρ (a)tr (a ∈ H). Thus, we have an H-module V = kd (via ρ ) and an H ◦ -module V ◦ = kd (via ρ ◦ ). Concretely, if H is generated (as an algebra) by, say h1 , . . . , hn ∈ H, then ρ and ρ ◦ are of course determined by specifying the representing matrices for h1 , . . . , hn . For any matrix B ∈ Md (k), we denote by ker(B) := {v ∈ kd | B.v = 0} the “nullspace” of B. For any 0 = v ∈ kd , we denote by v H ⊆ V the smallest Hsubmodule of V containing v. Similarly, v H ◦ ⊆ V ◦ will denote the smallest H ◦ submodule of V ◦ containing v. Given algebra generators for H, these subspaces are effectively computed using the “spinning algorithm”; see [234, §1.3] for technical details. Now the following result is the theoretical basis of the M EATA XE: Norton’s Irreducibility Criterion. Let h ∈ H be fixed. Then V is irreducible if kd = v H = v H ◦ for every 0 = v ∈ ker(ρ (h)) and every 0 = v ∈ ker(ρ ◦ (h)).
7.1 Algorithmic Methods
365
(See [234, 1.3.3] for a proof; note that the converse is trivially true.) Algorithmically, the critical point is the choice of the element h ∈ H to be used in the above criterion. The ideal case is when one succeeds in finding an element h ∈ H such that dimk ker(ρ (h)) = 1. For, in this case, it will be enough to determine only two subspaces under the “spinning algorithm” (corresponding to non-zero vectors in the one-dimensional nullspaces) in order to decide whether ρ is irreducible or not. However, in general, this will not be possible, and then one has to run through all vectors in the nullspaces. Hence, this only provides a practical method when k is a finite field. By testing various “standard words” in the generators of H, the M EATA XE algorithm tries to find an element h ∈ H such that ker(ρ (h)) has, at least, small dimension; some statistics about this can already be found in [263]. In any case, this algorithm has turned out to be extremely efficient in applications. It runs for representations over a finite field k of dimension up to a few thousand; see the remarks at the end of [234, §1.3] for further details and references. A sophisticated computer implementation (together with detailed descriptions) is due to Ringe [270]. In this system, the C HOP function takes as input the representing matrices ρ (h1 ), . . . , ρ (hn ) (where h1 , . . . , hn are algebra generators) and combines the above ideas into a recursive procedure to determine a composition series for ρ . The output consists of explicit lists of matrices specifying the action of the generators h1 , . . . , hn on the various composition factors. It is this function C HOP that we use in our applications to Iwahori–Hecke algebras. 7.1.2. We return to the problem of computing decomposition numbers for a generic Iwahori–Hecke algebra H = HA (W, S, L). Let us now place ourselves in the setting of 3.3.1 where we have factorisations of the elements cλ in terms of a certain subset P ⊆ A. Furthermore, let be a monomial order on Γ such that L(s) 0 for all s ∈ S and assume that H admits a cellular basis {Cλst } as in Theorem 2.6.12, where (♠) and (♣) hold. Let Φ ∈ P and consider the corresponding principal specialisation θΦ : A → kΦ , where kΦ is the field of fractions of A/(Φ ). Suppose that we can find a specialisation θ : A → k such that
θ (Φ ) = 0, Φ is strongly θ -isolated, and k is a finite field. Using the methods described in 7.1.1, we can find the irreducible representations of Hk and the corresponding decomposition numbers. The remaining task, then, is to see how results about decomposition numbers related to specialisations into finite fields can be “lifted” to principal specialisations, whose target fields will be of characteristic 0, in most cases. As in 3.7.4, we have μ
Irr(HkΦ ) = {LΦ | μ ∈ ΛΦ◦ }, where ΛΦ◦ ⊆ Λ is a canonical basic set for HkΦ , μ
Irr(Hk ) = {Lk | μ ∈ Λk◦ },
where Λk◦ ⊆ Λ is a canonical basic set for Hk .
Let DΦ = dλΦ,μ and Dθ = dλθ ,μ be the decomposition matrices associated with the specialisations θΦ and θ respectively. These are given by the equations dΦ ([Eελ ]) =
μ
∑◦ dλΦ,μ [LΦ ]
μ ∈ΛΦ
and
dθ ([Eελ ]) =
μ
∑ ◦ dλθ ,μ [Lk ].
μ ∈Λk
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7 Decomposition Numbers for Exceptional Types
By Example 3.6.4 and Theorem 3.7.16, we have Φ ΛΦ◦ = Λk◦ and Dθ = DΦ .DΦ θ , where Dθ is an “adjustment matrix”.
Now if the general version of James’ conjecture was known to hold, then we could conclude that DθΦ is the identity matrix and so DΦ = Dθ . At the present state of knowledge, we need to use some ad-hoc arguments to obtain this conclusion. We shall illustrate some of these arguments in the examples below, but refer the reader to [96], [129], [253] for further details. The final result is as follows.
¨ Theorem 7.1.3 (Cf. Geck and Muller [129]). Let (W, S) be of exceptional type H3 , H4 , F4 , E6 , E7 or E8 ; let Γ = Z and L(s) = 1 for all s ∈ S (equal parameter case). Assume we are in the setting of Example 3.3.15, where we consider a Φe -modular specialisation θe : A → K (e 2). Thus, we have a natural parametrisation μ
◦ }, Irr(H(e) ) = {L(e) | μ ∈ Λ(e)
◦ ⊆ Λ is a canonical basic set. where Λ(e)
μ
For blocks of defect at least 2, the sets Λ(e) and the dimensions of L(e) for μ ∈ Λ(e) are given in Table 7.1. (The remaining blocks are covered by the tables in [132, Appendix F], as discussed in Example 3.3.15(b).)
The basic strategy of the proof is outlined as follows. 7.1.4. Assume we are in the general setting of 7.1.2. Now let Γ = Z and R ⊆ C be the subring generated by ZW and all fractions fλ−1 , where λ ∈ Λ . Then R is L-good and A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Let e√ 2 and consider the Φe -modular specialisation θe : A → C, where θe (v) = exp(π −1/e). Let D(e) be the corresponding decomposition matrix and write (e) D(e) = dλ ,μ λ ∈Λ , μ ∈Λ ◦ . (e)
◦ and the entries of D . Choosing a suitable prime number Our task is to find Λ(e) (e) > 0, we can certainly find a specialisation θ : A → F such that θ (Φ2e ) = 0, where Φ2e ∈ Z[v] denotes the 2eth cyclotomic polynomial. For example, let be any prime such that ∈ R× and 2e divides − 1; then define θ by sending v to an element of order 2e in F× . Choosing large enough, we can also assume that Φe is strongly θ -isolated; see Lemma 3.7.12. (For example, if L(s) = 1 for all s ∈ S, then this is the case if does not divide |W |.) Thus, by Theorem 3.7.16, we have ◦ Λ(e) = Λk◦
and
(e)
Dθ = D(e) .Dθ .
◦ once we have determined D and Λ ◦ . To In particular, this shows that we obtain Λ(e) θ k ◦ =Λ solve the latter problem, we use the methods described in 7.1.1. The sets Λ(e) θ
7.1 Algorithmic Methods
367
Table 7.1 The canonical basic sets Λζ◦e for type H3 , H4 , F4 , E6 , E7 , E8
H3 e = 2 {1r (1), 3 s (2), 3s (2)} H4 e = 2 {1r (1), 4t (4), 4t (4), 9s (5), 9s (5), 25r (17)} e = 3 {1r (1), 16rr (16), 25r (8), 10r (1), 40r (16)}, {4t (4), 4t (4), 16r (8), 16t (4), 16t (4)} e = 4 {1r (1), 9s (8), 9s (8), 25r (8), 24t (8), 24t (8), 25 r (1)} e = 5 {1r (1), 9s (9), 16rr (6), 8r (1), 24s (9)}, {4t (4), 16r (16), 36rr (16), 24t (4), 48rr (16)} e = 6 {1r (1), 4t (4), 4t (4), 25r (16), 18r (1), 24s (4), 24s (4)} e = 10 {1r (1), 4t (4), 9s (9), 36rr (22), 24s (1), 30s (4), 40r (9)} F4 e = 2 {11 (1), 21 (2), 23 (2), 91 (5)}, e = 3 {11 (1), 21 (1), 23 (1), 41 (1)},
{42 (4)} {42 (4), 81 (4), 83 (4), 161 (4)}
e = 4 {11 (1), 42 (4), 91 (4), 61 (1), 121 (4)} e = 6 {11 (1), 21 (2), 23 (2), 81 (5), 83 (5)} E6 e = 2 {1 p (1), 6 p (6), 20 p (14), 15q (14), 30 p (10), 60 p (46} e = 3 {1 p (1), 6 p (5), 20 p (14), 15 p (10), 15q (1), 30 p (25), 64 p (10), 60 p (5), 60s (14), 80s (25)} e = 4 {1 p (1), 6 p (6), 15 p (15), 15q (8), 81 p (60), 10s (1), 80s (6), 90s (15)} e = 6 {1 p (1), 6 p (6), 20 p (13), 15 p (14), 30 p (11), 60 p (32), 24 p (11), 60s (14), 80s (13), 60 p (1), 30 p (6)} E7 e = 2 {1a (1), 7 a (6), 27a (14), 35b (14), 105 a (78), 189 b (56), 315 a (126)}, {56 a (56), 120a (64), 280b (216)} e = 3 {1a (1), 21a (21), 35b (34), 120a (98), 105b (7), 168a (35), 210a (91), 280b (14), 210b (49), 420a (196)}, {7 a (7), 21 b (14), 56 a (49), 15 a (1), 105 a (35), 70 a (21), 280 a (196), 336 a (91), 512a (98), 84 a (34)} e = 4 {1a (1), 56 a (56), 105b (48), 210a (154), 189a (35), 405a (147), 70a (21), 315a (120)}, {7 a (7), 15 a (8), 105 a (105), 189 c (84), 280b (168), 378 a (21), 210 b (27)}, {27a (27), 21a (21), 35b (8), 216 a (168), 210b (7), 105c (84), 378a (105)}, {21 b (21), 120a (120), 189 b (48), 35 a (35), 70 a (1), 315 a (147), 336a (154), 405 a (56)} e = 6 {1a (1), 7 a (7), 21 b (13), 21a (21), 35b (27), 15 a (14), 105 a (77), 105b (43), 168a (43), 210a (92), 70 a (42), 280 a (90), 315 a (13), 84a (14), 210b (27), 420a (92), 210 b (1), 420 a (77), 280a (21), 315a (7)}
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7 Decomposition Numbers for Exceptional Types
Table 7.1 The canonical basic sets Λζ◦e for type H3 , H4 , F4 , E6 , E7 , E8 (cont’d) E8 e = 2 {1x (1), 8z (8), 35x (27), 84x (48), 50x (42), 210x (202), 560z (246), 700x (126), 1400z (792), 1050x (651), 1400x (378), 4200x (1863)}, {112z (112), 160z (160), 400z (288), 1344x (1184), 2240x (1056), 3360z (2016)} e = 3 {1x (1), 35x (35), 28x (28), 84x (48), 50x (1), 210x (147), 300x (70), 700x (518), 1344x (497), 175x (28), 350x (322), 1050x (35), 1400x (1225), 2240x (322), 4096z (1036), 4200x (147), 700xx (48), 3200x (497), 4200y (518), 4480y (1225)}, {8z (8), 112z (104), 160z (56), 560z (384), 400z (8), 448z (56), 1400z (848), 840z (448), 1400z z (104), 4096x (1896), 4200z (384), 5600z (1896), 7168w (848)} e = 4 {1x (1), 35x (34), 112z (77), 50x (16), 210x (176), 567x (280), 400z (96), 175x (1), 350x (70), 1050x (336), 1575x (946), 525x (168), 3360z (1654), 2800z (1302), 2835x (34), 6075x (280), 3150y (77), 4480y (176), 5670y (946)}, {8z (8), 560z (560), 1344x (784), 840z (56), 1400z z (832), 4536z (2360), 4200 z (1008), 2240 x (1400)}, {28x (28), 160z (160), 300x (300), 972x (512), 840x (28), 700x x (512), 1344w (160), 840 x (300)}, {56z (56), 1008z (1008), 1400z (1400), 3240z (832), 2240x (8), 4200z (2360), 3200 x (784), 4536 z (560)}, {84x (84), 700x (616), 2268x (1652), 4200x (1848), 2100x (448), 2016w (84), 5600w (1652), 4200 x (616)} e = 5 {1x (1), 28x (28), 84x (83), 567x (539), 1344x (722), 972x (166), 2268x (1729), 4096z (1078), 168y (1), 1134y (28), 2688y (722), 4536y (1729), 4096 z (539), 972 x (83)} e = 6 {1x (1), 8z (8), 35x (35), 28x (28), 84x (40), 50x (41), 210x (210), 560z (279), 300x (225), 700x (86), 56z (56), 448z (85), 1400z (489), 175x (85), 350x (266), 1050x (660), 1400x (259), 840z (259), 1400zz (40), 840x (41), 4200x (1906), 2100x (1036), 2400z (266), 4200z (279), 5600z (489), 420y (1), 1680y (56), 4200y (660), 4480y (8), 4536y (225), 5670y (28), 4200 x (35), 1400 x (210)}, {112z (112), 160z (160), 400z (288), 1344x (1072), 2240x (768), 3360z (2128), 3200x (2128), 1344w (288), 7168w (1072), 3360 z (160), 2240 x (112)} e = 8 {1x (1), 35x (34), 160z (160), 567x (373), 175x (174), 1400x (992), 1575x (1042), 525x (152), 2835x (1668), 6075x (3516), 2016w (174), 5600w (1042), 7168w (992), 2835 x (1), 6075 x (373), 1400 x (34), 1575 x (160)} e = 10 {1x (1), 8z (8), 28x (28), 84x (75), 567x (531), 448z (372), 1008z (449), 1400z (786), 972x (897), 2268x (502), 4536z (2406), 1400y (449), 3150y (372), 4200y (897), 4480y (786), 4536 z (75), 2268 x (531), 448 z (1), 1008 z (28), 1400 z (8)} e = 12 {1x (1), 35x (35), 112z (76), 50x (50), 210x (99), 400z (349), 1050x (651), 1400x (974), 525x (449), 3360z (1386), 2800z (1202), 1400y (99), 2688y (651), 4536y (974), 2100y (449), 3360 z (349), 2800 z (76), 1050 x (50), 1400 x (1), 210 x (35)} μ
◦ . Each list corresponds to a block of defect 2; the numbers in brackets give dim L(e) for μ ∈ Λ(e)
7.1 Algorithmic Methods
369
are printed in Table 7.1 and the matrices Dθ are the ones printed in the tables in the subsequent sections. It turns out that, in each case, we actually have D(e) = Dθ , but the proof of this statement requires considerably more work and will not be discussed in detail here. We shall just give a few examples in order to illustrate some of the difficulties involved in the proof. Example 7.1.5. Assume we are in the setting of 7.1.4. Then the factorisation Dθ = (e) D(e) .Dθ means that the columns of Dθ are obtained by possibly adding together some columns of D(e) . The triangular shape of Dθ and D(e) then immediately yields the following criterion: (a) Write Dθ = dλ ,μ and assume that, for any μ = μ in Λk◦ such that aμ < aμ , there exists some λ ∈ Λ such that dλ ,μ < dλ ,μ . Then D(e) = Dθ . (The proof is an easy exercise about matrices whose proof is left to the reader.) If (a) is satisfied, we say that the decomposition matrix Dθ is strongly triangular. One can of course also formulate a block-wise version of this criterion. Hence, in this case, we know that D(e) = Dθ , without any further computation! For example, in type F4 with equal parameters, all the decomposition matrices have this property; see Section 7.3. Unfortunately, this is not always satisfied but the exceptions are relatively rare for (W, S) of exceptional type. Example 7.1.6. Let W be of type H3 . Let us apply the procedure in 7.1.4, where e = 2. So, choosing a suitable prime number , we find the following decomposition matrix Dθ by the method described in 7.1.1: • 1r • 3 s • 3s 5r • 4r • 4 r 5 r 3s 3s 1 r
aλ 0 1 1 2 3 3 5 6 6 15
1 1 1 1 . . 1 1 1 1
. 1 . 1 . . 1 1 . .
Dθ . . 1 1 . . 1 . 1 .
. . . . . . . . 1 . . 1 . . . . . . . .
1 . 1 . . . . . 1 1
. 1 . 1 . . 1 1 . .
D(2) ? . . . . . . 1 . . 1 . . . 1 . . . 1 1 . . . . . 1 . . . . .
1 1 . . . . . 1 . 1
. 1 . 1 . . 1 1 . .
D(2) ? . . . . . . 1 . . 1 . . . 1 . . . 1 1 . . . . . 1 . . . . .
We see that this matrix is not “strongly triangular” in the sense of Example 7.1.5: Either the second or the third column could be subtracted from the first column. Thus, there are potentially two candidates for a non-trivial adjustment matrix; the corresponding candidates for D(2) are printed above. How can we exclude these possibilities? Let us assume, if possible, that D(2) is the second of the above matrices. This would show that the Φ2 -modular reduction of 3 s is irreducible. Then one has to show by some direct methods that this is not true. (A similar argument applies if D(2) was the third of the above matrices.) Thus, even though Dθ is not “strongly triangular”, the knowledge of Dθ considerably restricts the possibilities for the adjustment matrix and it shows precisely to which representations of HK some special arguments need to be applied.
370
7 Decomposition Numbers for Exceptional Types
As in the above example, a special argument is typically needed to show that the Φe -modular reduction of some irreducible representation of HK is not irreducible. Here, the general techniques of the M EATA XE algorithm can still be helpful. The following is another simple method to find non-trivial submodules. Lemma 7.1.7. Let ρ : HK → Md (K) be a representation such that ρ (Tw ) ∈ Md (A) for all w ∈ W . Then there exists a matrix B ∈ Md (A) such that det(B) = 0
and
B.ρ (Tw−1 ) = ρ (Tw )tr .B for all w ∈ W .
Let θ : A → k be any specialisation into a field k. Let Bk be the matrix obtained by applying θ to all entries of B. If Bk = 0 and det(Bk ) = 0, then the specialised representation ρk : Hk → Md (k) is not irreducible. Proof. Consider the contragredient representation ρˆ : HK → Md (K) by ρˆ (Tw ) := ρ (Tw−1 )tr for w ∈ W . By Example 1.2.5, ρ and ρˆ are equivalent. So there exists an invertible matrix B1 ∈ Md (K) such that B1 .ρ (Tw−1 ) = ρ (Tw )tr .B1 for all w ∈ W . Let 0 = d ∈ A be such that B := dB1 has all entries in A. Then B satisfies the above conditions. Assume now that ρk is irreducible. Then ρˆ k will also be irreducible. So Schur’s Lemma implies that Bk = 0 or Bk is invertible. Example 7.1.8. Let us return to Example 7.1.6, where e = 2 and W is of type H3 . The representation of HK labelled by 3 s is given by ⎤ ⎤ ⎡ ⎡ ⎡ −1 ⎤ v 0 0 v 0 0 −v 1 0 0 ⎦ , Ts2 → ⎣1 −v−1 −α¯ ⎦ , Ts3 → ⎣ 0 v 0⎦ , Ts1 → ⎣0 v 0 0 v 0 0v 0 −α¯ −v−1 √ √ where α = 12 (−1 + 5) and α¯ = 12 (−1 − 5). We find that a matrix B satisfying the condition in Lemma 7.1.7 is given by ⎤ ⎡ 0 v + v−1 −1 B = ⎣ −1 v + v−1 α¯ ⎦ . 0 α¯ v + v−1 Now apply the specialisation θ2 : A → C. We obtain ⎡ ⎤ 0 −1 0 Bθe = ⎣−1 0 α¯ ⎦ and so rank(Bθe ) = 2. 0 α¯ 0 Hence, the Φ2 -modular reduction of 3 s is not irreducible. A similar computation shows that the Φ2 -modular reduction of 33 is not irreducible. Thus, we conclude that, in the setting of Example 7.1.6, the adjustment matrix is the identity.
7.2 Decomposition Matrices for W of Dihedral Type
371
7.2 Decomposition Matrices for W of Dihedral Type In this section, let W be of type I2 (m) (m 3), with generators s1 , s2 such that (s1 s2 )m = 1. Let ζ ∈ C be a root of unity of order m such that ζ + ζ −1 = 2 cos(2π /m). Let L : W → Γ be a weight function and A = R[Γ ], where m ∈ R× and ζ + ζ −1 ∈ R ⊆ C. (Thus, R is L-good.) Then, given any specialisation θ : A → k, where k is a field, the algebra Hk is split and we have a corresponding decomposition map dθ : R0 (HK ) → R0 (Hk ). In 7.2.1, we briefly describe how dθ can be computed. Then we present a number of particular cases which are relevant in the representation theory of finite groups of Lie type. 7.2.1. By Example 1.3.7, we have an explicit description of the irreducible representations of HK , as follows: ε : 1W
Ts1 →
vs1 ,
Ts2 →
vs2 ,
:
Ts1 →
Ts2 →
sgnε1 :
−v−1 s1 ,
−v−1 s2 ,
Ts1 →
vs1 ,
Ts2 →
sgnε
sgnε2 :
σ εj :
Ts1 → Ts1 →
−v−1 s1 ,
−v−1 s1
0
μ j vs1
Ts2 →
,
Ts2 →
−v−1 s2 ,
vs2 , vs2
1
0 −v−1 s2
,
j − j + v−1 v ; here, 1 j (m − 2)/2 (if m is even) where μ j = vs1 v−1 s2 + ζ + ζ s1 s2 and 1 j (m − 1)/2 (if m is odd). We notice that all these representations are realised over A. Given a matrix representation σ ε of HK such that all coefficients of σ ε (Tw ) (w ∈ W ) lie in A, we denote by σ¯ the representation of Hk obtained by extension of scalars from A to k (via θ ). Thus, we have dθ ([σ ε ]) = [σ¯ ] ∈ R0 (Hk ). In the following, we shall write ξsi = θ (vsi ) for i = 1, 2 and ζ¯ = θ (ζ ).
(a) If σ ε is one-dimensional, then σ¯ clearly is irreducible. The only question is whether some of the one-dimensional representations become equal or not. We have: ¯ = sgn ¯ 1 = sgn ¯ 2. If ξs21 = −1 and ξs22 = −1, then 1¯ W = sgn If ξs21 = −1 and ξs22 = −1, then 1¯ W = sgn ¯ 2 = sgn ¯ = sgn ¯ 1. 2 2 ¯ ¯ 1 = sgn ¯ = sgn ¯ 2. If ξs1 = −1 and ξs2 = −1, then 1W = sgn 2 2 ¯ If ξs = −1 and ξs = −1, then 1W , sgn, ¯ sgn ¯ 1 , sgn ¯ 2 are pairwise different. 1
2
(Here, in each case, sgn1 and sgn2 only appear if m is even; also, the second and the third cases only occur if m is even.) (b) Let σ εj be one of the two-dimensional representations, where 1 j (m − 2)/2 (if m is even) or 1 j (m − 1)/2 (if m is odd). Then σ¯ j is reducible if and only if σ¯ j (Ts1 ) and σ¯ j (Ts2 ) have a common eigenvector. A straightforward ζ¯ ± j or ξs1 = computation shows that σ¯ j is reducible if and only if ξs1 = +ξs−1 2 −ξs2 ζ¯ ± j . (Note that ξs1 = −ξs2 ζ¯ ± j can only occur if m is even and ξs1 = ξs2 .) We have:
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7 Decomposition Numbers for Exceptional Types
¯ If ξs1 = +ξs−1 ζ¯ ± j , then [σ¯ j ] = [1¯ W ] + [sgn]. 2 ¯ 1 ] + [sgn ¯ 2] If ξs1 = −ξs2 ζ¯ ± j , then [σ¯ j ] = [sgn
(assuming m is even).
(c) Finally, let j, j be such that σ¯ j and σ¯ j are irreducible. Then we claim that [σ¯ j ] = [σ¯ j ] if j = j . To see this, we compute that trace(σ εj (Ts1 s2 )) = ζ j + ζ − j . Since R is assumed to be L-good, the characteristic of k is zero or a prime which does not divide m. Hence, the roots of unity {ζ¯ , ζ¯ 2 , . . . , ζ¯ m−1 } are all distinct in k and so we will have trace(σ¯ j (Ts1 s2 )) = ζ¯ j + ζ¯ − j = ζ¯ j + ζ¯ − j = trace(σ¯ j (Ts1 s2 )) if j = j . Consequently, we also have [σ¯ j ] = [σ¯ j ] if j = j . The above information completely determines the blocks and their decomposition matrices in all cases. Example 7.2.2. Let Γ = Z (with its natural order); then A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Let L : W → Z be a weight function such that L(s1 ) = c ∈ {1, 2, 3, . . .}
and
L(s2 ) = 1.
Let e 2 and assume that R ⊆ C contains all roots of unity of order e. We consider the Φe -modular specialisation √ θe : A → K, v → ζ2e := exp(π −1/e) ∈ K; see Example 3.1.15(b). Using 7.2.1, we can compute the corresponding decomposition matrix. See Table 7.2 for the case where c = 1 (equal parameter case). Here, and in the tables below, we use the following conventions. First, there is a table describing the Φe -blocks (where blocks of defect 0 are not printed). In each block, the representations are ordered according to increasing value of aλ . For a block of defect 1, this information is sufficient to determine the decomposition matrix; see Theorem 3.3.13. Hence, we only need to further print the decomposition matrices of blocks of defect at least 2. In each case, one sees (independently of the theoretical results in Chapter 2) that there is a canonical basic set (marked by “•”).
Table 7.2 Φe -blocks and decomposition numbers for type I2 (m) (equal parameters) e
d
Φe -blocks of defect d
2
1
{1W , sgn}
(m odd)
2
2
{1W , sgn1 , sgn2 , sgn}
(m even)
1
{1W , σm/ j , sgn}
j = 3, 4, . . . , m and j divides m
e=2 • 1W (sgn1 ) (sgn2 ) sgn
0 1 1 m
dλ ,M 1 1 1 1
Tables 7.3–7.6 show analogous results for W of type G2 and I2 (8) and those weight functions which appear in the context of the representation theory of finite groups of Lie type (see 4.3.7 and Example 4.5.17).
7.2 Decomposition Matrices for W of Dihedral Type
373
Table 7.3 Φe -blocks and decomposition numbers for type G2 with equal parameters e
d
Φe -blocks of defect d
2
2
{1W , sgn1 , sgn2 , sgn}
3
1
{1W , σ2 , sgn}
6
1
{1W , σ1 , sgn}
e=2 • 1W sgn1 sgn2 sgn
0 1 1 6
dλ ,M 1 1 1 1
Table 7.4 Φe -blocks and decomposition numbers for type G2 with parameters 3, 1 e
d
Φe -blocks of positive defect d
2
2
{1W , sgn1 , sgn2 , sgn}
3
1
{1W , σ2 , sgn}, {sgn1 , σ1 , sgn2 }
6
2
{1W , sgn1 , σ2 , sgn2 , sgn}
12
1
{1W , σ1 , sgn}
e = 2 dλ ,M • 1W 0 1 sgn1 1 1 sgn2 7 1 sgn 12 1
e=6 • 1W 0 • sgn1 1 σ2 3 sgn2 7 sgn 12
dλ ,M 1 . . 1 1 1 1 . . 1
Table 7.5 Φe -blocks and decomposition numbers for type G2 with parameters 9, 1 e
d
Φe -blocks of defect d
2
2
{1W , sgn1 , sgn2 , sgn}
3
1
{1W , σ2 , sgn}, {sgn1 , σ1 , sgn2 }
6
2
{1W , sgn1 , σ1 , sgn2 , sgn}
12
1
{sgn1 , σ1 , sgn2 }
15
1
{1W , σ2 , sgn}
18
1
{1W , sgn2 }, {sgn1 , sgn}
24
1
{sgn1 , σ2 , sgn2 }
30
1
{1W , σ1 , sgn}
e = 2 dλ ,M • 1W 0 1 sgn1 1 1 sgn2 25 1 sgn 30 1
e=6 • 1W 0 • sgn1 1 σ1 9 sgn2 25 sgn 30
dλ ,M 1 . . 1 1 1 1 . . 1
Table 7.6 Φe -blocks and decomposition numbers for type I2 (8) with parameters 4, 2 e
d
Φe -blocks of defect d
4
2
{1W , sgn1 , σ2 , sgn2 , sgn}
8
2
{1W , sgn1 , σ3 , sgn2 , sgn}
12
1
{1W , σ2 , sgn}
24
1
{1W , σ1 , sgn}
e=4 • 1W 0 sgn1 1 • σ2 2 sgn2 5 sgn 12
dλ ,M 1 . 1 . 1 1 . 1 . 1
e=8 • 1W 0 • sgn1 1 σ3 2 sgn2 5 sgn 12
dλ ,M 1 . . 1 1 1 1 . . 1
374
7 Decomposition Numbers for Exceptional Types
7.3 Decomposition Matrices for W of Type F4 In this section, let (W, S) be of type F4 , with generators and diagram given by s1 s2 4 s3 s4 F4 t t t t We will consider various principal specialisations θ : A → k, where k is a field of characteristic 0, and describe the corresponding blocks and decomposition numbers. There are two essentially different cases: the one where Γ = Z and the one where Γ = Z2 . In each case, one sees (independently of the theoretical results in Chapter 2) that there is a canonical basic set (marked by “•” in the tables). The notation for Irr(W ) is taken from [132, Table C.3 (p. 413)]. Example 7.3.1. Let Γ = Z (with its natural order) so that A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Let L : W → Z be given by L(s1 ) = L(s2 ) = 1
and
L(s3 ) = L(s4 ) = c,
where
c ∈ {1, 2, 4}.
(These are the ones that arise in the context of representations of finite groups of Lie type; see 4.3.7 and Example 4.5.17). Thus, Ts2i = T1 + (v − v−1 )Tsi Ts2i
−c
= T1 + (v − v )Tsi c
(i = 1, 2), (i = 3, 4).
Let e 2 and assume that R = K ⊆ C is a subfield which contains all roots of unity of order e. We consider the Φe -modular specialisation √ θe : A → K, v → ζ2e := exp(π −1/e) ∈ K; see Example 3.1.15(b). Note that θe is a principal specialisation where Φ = v − ζ2e . The Φe -blocks and the corresponding decomposition matrices are contained in Table 7.7 (c = 1), Table 7.8 (c = 2) and Table 7.9 (c = 4). These results are due to Geck and Lux [126] and Bremke [30]. Example 7.3.2. Let Γ = Z2 and R = Q, so that A = Q[V ±1 , v±1 ] is the ring of Laurent polynomials in two independent indeterminates V := ε (1,0) and v := ε (0,1) . Let L : W → Γ be given by L(s1 ) = L(s2 ) = (0, 1) and L(s3 ) = L(s4 ) = (1, 0), Thus, Ts2i = T1 + (v − v−1 )Tsi
(i = 1, 2),
Ts2i = T1 + (V −V −1 )Tsi
(i = 3, 4).
We choose the usual lexicographic order as a monomial order on Γ . (Thus, we are in the asymptotic case; see Example 1.1.11.) From the list of the Laurent polynomials cλ in [132, Table 11.1 (p. 379)], we find that the three conditions in 3.3.1 hold, where P = {Φ3 (V ), Φ3 (v), Φ4 (V ), Φ4 (v), Φ6 (V ), Φ6 (v), Φ12 (V ), Φ12 (v),
Φ4 (V v±1 ), Φ4 (V 2 v±1 ), Φ4 (V ±1 v2 ), Φ8 (V v±1 ), Φ12 (V v±1 )}.
7.3 Decomposition Matrices for W of Type F4
375
Table 7.7 Φe -blocks and decomposition numbers for type F4 with equal parameters e
d
Φe -blocks of defect d
2
2
{42 , 43 , 44 , 45 }
4
{11 , 23 , 21 , 91 , 81 , 83 , 12 , 92 , 62 , 13 , 93 , 61 , 82 , 84 , 94 , 24 , 22 , 14 }
2
{11 , 21 , 23 , 12 , 13 , 41 , 22 , 24 , 14 }, {42 , 81 , 83 , 43 , 44 , 16, 82 , 84 , 45 }
3
1
{21 , 43 , 24 }, {23 , 44 , 22 }
2
{11 , 42 , 91 , 41 , 61 , 12, 94 , 45 , 14 }
6
2
{11 , 21 , 23 , 81 , 83 , 92 , 93 , 12, 82 , 84 , 22 , 24 , 14 }
8
1
{11 , 91 , 16, 94 , 14 }
12
1
{11 , 42 , 62 , 45 , 14 }
4
e=2 • 11 0 • 23 1 • 21 1 • 91 2 81 3 83 3 12 4 92 4 62 4 13 4 93 4 61 4 82 9 84 9 94 10 24 13 22 13 14 24
1 . . . 1 1 1 . 1 1 . 1 1 1 . . . 1
dλ ,M . . 1 . . 1 1 1 . 1 1 . . . 1 1 . . . . 1 1 . . . 1 1 . 1 1 1 . . 1 . .
. . . 1 1 1 . 1 1 . 1 1 1 1 1 . . .
e = 2 dλ ,M • 42 1 1 43 4 1 44 4 1 45 13 1
e=4 • 11 0 • 42 1 • 91 2 41 4 • 61 4 • 12 4 94 10 45 13 14 24
1 . 1 . 1 . . . .
dλ ,M . . . 1 . . 1 1 . . 1 . . 1 1 1 1 . . 1 1 . . . . . 1
. . . . . 1 1 1 .
e=3 • 11 0 • 21 1 • 23 1 12 4 13 4 • 41 4 22 13 24 13 14 24
1 1 1 . . 1 . . .
dλ ,M . . 1 . . 1 1 . . 1 1 1 . 1 1 . . .
. . . . . 1 1 1 1
e=3 • 42 1 • 81 3 • 83 3 43 4 44 4 • 16 4 82 9 84 9 45 13
1 1 1 . . 1 . . .
dλ ,M 0 . 1 . . 1 1 . . 1 1 1 . 1 1 . . .
. . . . . 1 1 1 1
e=6 • 11 0 • 21 1 • 23 1 • 81 3 • 83 3 • 92 4 • 93 4 • 12 4 82 9 84 9 22 13 24 13 14 24
dλ ,M 1. . . . . . . .1. . . . . . . .1. . . . . 11 . 1 . . . . 1.1.1. . . .1.1.1. . . .1.1.1. 1 . . 11 . . 1 . . . . 1 . 11 . . .1.1.1 . . . . . .1. . . . . .1. . . . . . . . .1
(As in Example 3.3.4, one sees that in each case the ring A/(Φ ) is integrally closed in its field of fractions.) Let Φ ∈ P and consider the corresponding principal specialisation θ : A → kΦ ; see Section 3.3. The Φ -blocks and corresponding decomposition matrices are given in Table 7.10. These tables are extracted from the results obtained by McDonough and Pallikaros [249], [250].
376
7 Decomposition Numbers for Exceptional Types
Table 7.8 Φe -blocks and decomposition numbers for type F4 with parameters 1, 1, 2, 2 e
d
Φe -blocks of defect d
2
4
{11 , 23 , 42 , 91 , 13 , 21 , 44 , 93 , 81 , 61 , 12, 62 , 41 , 92 , 43 , 82 , 12 , 94 , 22 , 45 , 24 , 14 }
3
2
{11 , 23 , 13 , 21 , 41 , 12 , 22 , 24 , 14 }, {42 , 83 , 44 , 81 , 16, 43 , 82 , 84 , 45 }
4
1
{42 , 43 }, {44 , 45 }
6
2
{11 , 13 , 91 , 93 , 16, 92 , 12 , 94 , 14 }, {23 , 83 , 61 , 62 , 84 , 24 }
1
{83 , 16, 84 }
3
{11 , 23 , 42 , 21 , 91 , 13 , 44 , 81 , 61 , 41 , 12, 43 , 82 , 12 , 22 , 94 , 45 , 24 , 14 }
8
1
{21 , 92 , 84 , 14 }, {11 , 83 , 93 , 22 }
10
1
{23 , 93 , 82 , 14 }, {11 , 81 , 92 , 24 }
12
1
{11 , 91 , 16, 94 , 14 }, {23 , 83 , 12, 84 , 24 }
18
1
{11 , 42 , 62 , 45 , 14 }
e=2 • 11 0 • 23 1 • 42 2 • 91 3 13 3 21 3 44 5 93 6 • 81 6 61 7 • 12 7 62 7 41 7 92 10 43 11 82 12 12 15 94 15 22 15 45 20 24 25 14 36
dλ ,M 1. . . . . .1. . . . . 11 . . . 1111 . . 1. . . . . . .1. . . . 11 . . . 1111 . . 1 . 111 . 1 . . 11 . . 121 . 1 1 . . 11 . . . .1. . . . 1111 . .1. .1 1 . 111 . . . . .1. . . 1111 . .1. . . . .1. .1 . . . . .1 . . . .1.
e=3 • 11 0 • 23 1 13 3 • 21 3 • 41 7 12 15 22 15 24 25 14 36
dλ ,M 1. . . 11 . . .1. . 1.1. 1111 . .1. .1.1 . . 11 . . .1
e=4 • 11 0 • 13 3 • 91 3 • 93 6 16 7 92 10 12 15 94 15 14 36
dλ ,M 1. . . .1. . 1.1. .1.1 . . 11 1.1. 1. . . .1.1 .1. .
e=3 • 42 2 • 83 3 44 5 • 81 6 • 16 7 43 11 82 12 84 15 45 20
dλ ,M 1. . . 11 . . .1. . 1.1. 1111 . .1. .1.1 . . 11 . . .1
e=4 • 23 1 • 83 3 61 7 62 7 84 15 24 25
dλ ,M 1 . 1 1 . 1 . 1 1 1 1 .
e=6 • 11 0 • 23 1 • 42 2 • 21 3 • 91 3 • 13 3 • 44 5 81 6 61 7 41 7 • 12 7 43 11 82 12 12 15 22 15 94 15 45 20 24 25 14 36
dλ ,M 1. . . . . . . .1. . . . . . . .1. . . . . 1. .1. . . . 111 . 1 . . . . . . . .1. . . 1 . . . 11 . 1 . 111 . . . 11 . . 1 . 1 . .1. .1. . . . 11 . 1 . . 1 1 . . 11 . . . . 1 . . . 111 . . .1. . . . . . . . . 11 . . 1 . . 1 . 11 . . . . . . .1 . . . .1. . . . . . . . .1.
7.3 Decomposition Matrices for W of Type F4
377
Table 7.9 Φe -blocks and decomposition numbers for type F4 with parameters 1, 1, 4, 4 e 2 3 4 6 8 10 12 14 18 20 24 30 e=2 • 11 0 • 23 1 13 3 • 42 4 • 91 7 44 7 21 9 93 10 • 81 12 41 13 61 13 62 13 • 12 13 82 18 22 21 92 22 43 25 94 27 45 34 12 39 24 49 14 60
d 4 2 2 1 3 1 2 1 2 1 2 1 1 1 1 1
Φe -blocks of defect d {11 , 23 , 13 , 42 , 91 , 44 , 21 , 93 , 81 , 41 , 61 , 62 , 12, 82 , 22 , 92 , 43 , 94 , 45 , 12 , 24 , 14 } {11 , 23 , 13 , 21 , 41 , 22 , 12 , 24 , 14 }, {42 , 83 , 44 , 81 , 16, 82 , 43 , 84 , 45 } {11 , 13 , 83 , 91 , 21 , 93 , 16, 22 , 92 , 94 , 84 , 12 , 14 } {83 , 16, 84 } {11 , 23 , 13 , 42 , 44 , 21 , 93 , 81 , 41 , 62 , 12, 82 , 22 , 92 , 43 , 45 , 12 , 24 , 14 } {13 , 14 }, {93 , 94 }, {91 , 92 }, {11 , 12 }, {44 , 45 }, {42 , 43 } {23 , 83 , 61 , 62 , 84 , 24 } {23 , 44 , 22 }, {21 , 43 , 24 } {11 , 42 , 91 , 41 , 61 , 12, 94 , 45 , 14 } {42 , 81 , 43 }, {44 , 82 , 45 }, {23 , 41 , 24 } {11 , 13 , 83 , 21 , 93 , 16, 22 , 92 , 84 , 12 , 14 } {13 , 82 , 94 , 24 }, {23 , 91 , 81 , 12 } {11 , 81 , 92 , 24 }, {23 , 93 , 82 , 14 }, {13 , 44 , 61 , 43 , 12 } {11 , 91 , 16, 94 , 14 } {23 , 83 , 12, 84 , 24 } {11 , 42 , 62 , 45 , 14 }
dλ ,M 1. . . . . .1. . . . 1. . . . . . 11 . . . 1111 . . . 11 . . . . .1. . . 1111 . . 1 . 111 . . . .1. . 1 . . 11 . 1 . . 11 . . 121 . 1 1 . 111 . . .1. . . . . 1111 . .1. .1 . . 1111 . .1. .1 . . . .1. . . . . .1 . . . .1.
e=3 • 11 0 • 23 1 13 3 • 21 9 • 41 13 22 21 12 39 24 49 14 60
dλ ,M 1. . . 11 . . .1. . 1.1. 1111 .1.1 . .1. . . 11 . . .1
e=3 • 42 4 • 83 5 44 7 • 81 12 • 16 13 82 18 43 25 84 29 45 34
e=4 dλ ,M • 11 0 1 . . . . . . . • 13 3 . 1 . . . . . . • 83 5 1 1 1 . . . . . • 91 7 1 . 1 1 . . . . 21 9 . . . 1 . . . . • 93 10 . 1 1 . 1 . . . • 16 13 . . 1 1 1 1 . . 22 21 . . . . 1 . . . • 92 22 . . . 1 . 1 1 . • 94 27 . . . . 1 1 . 1 84 29 . . . . . 1 1 1 12 39 . . . . . . 1 . 14 60 . . . . . . . 1
dλ ,M 1. . . 11 . . .1. . 1.1. 1111 .1.1 . .1. . . 11 . . .1
e = 10 • 11 0 • 42 4 • 91 7 41 13 • 61 13 • 12 13 94 27 45 34 14 60
dλ ,M 1. . . . .1. . . 111 . . . .1. . 1 . 11 . . 11 . 1 . . 111 . . . .1 . . .1.
e = 12 dλ ,M • 11 0 1 . . . . . • 13 3 . 1 . . . . • 83 5 1 . 1 . . . • 21 9 1 . . 1 . . • 93 10 . 1 1 . 1 . • 16 13 1 . 1 . 1 1 22 21 . 1 . . 1 . 92 22 1 . . 1 . 1 84 29 . . . . 1 1 12 39 . . . 1 . . 14 60 . . . . 1 .
e=6 dλ ,M • 11 0 1 . . . . . . . • 23 1 . 1 . . . . . . • 13 3 . . 1 . . . . . • 42 4 1 1 . 1 . . . . • 44 7 . . . . 1 . . . 21 9 1 . . 1 . . . . • 93 10 . 1 1 . 1 1 . . • 81 12 1 1 . 1 . . 1 . 41 13 . 1 . . . 1 . . 62 13 . 1 1 1 . 1 . . 12 13 . 1 . . 1 1 1 . • 82 18 . . 1 . 1 1 . 1 22 21 . . 1 . . . . 1 92 22 . 1 . 1 . 1 1 . 43 25 . . . . . . 1 . 45 34 . . 1 . . 1 . 1 12 39 . . . 1 . . . . 24 49 . . . . . 1 . . 14 60 . . . . . . . 1 e=8 • 23 1 • 83 5 61 13 62 13 84 29 24 49
dλ ,M 1 . 1 1 . 1 . 1 1 1 1 .
378
7 Decomposition Numbers for Exceptional Types
Table 7.10 Φ -blocks and decomposition numbers for type F4 with parameters v, v,V,V
Φ
d
Φ -blocks of defect d
Φ3 (v), Φ6 (v)
1
{12 , 24 , 14 }, {43 , 84 , 45 }, {81 , 16, 82 }, {21 , 41 , 22 }, {42 , 83 , 44 }, {11 , 23 , 13 }
Φ3 (V ), Φ6 (V )
1
{13 , 22 , 14 }, {44 , 82 , 45 }, {83 , 16, 84 }, {23 , 41 , 24 }, {42 , 81 , 43 }, {11 , 21 , 12 }
Φ4 (v) Φ4 (V )
1
{12 , 14 }, {43 , 45 }, {92 , 94 }, {91 , 93 }, {42 , 44 }, {11 , 13 }
2
{21 , 81 , 61 , 62 , 82 , 22 }
1
{13 , 14 }, {44 , 45 }, {93 , 94 }, {91 , 92 }, {42 , 43 }, {11 , 12 }
2
{23 , 83 , 61 , 62 , 84 , 24 }
Φ12 (v)
1
{21 , 81 , 12, 82 , 22 }
Φ12 (V )
1
{23 , 83 , 12, 84 , 24 }
Φ4 (V v) Φ4 (V v−1 )
1
{23 , 44 , 22 }, {21 , 43 , 24 }
2
{11 , 42 , 91 , 41 , 61 , 12, 94 , 45 , 14 }
1
{22 , 45 , 24 }, {23 , 42 , 21 }
2
{13 , 44 , 93 , 41 , 62 , 12, 92 , 43 , 12 }
Φ4 (V 2 v)
1
{11 , 81 , 92 , 24 }, {23 , 93 , 82 , 14 }
Φ4 (V 2 v−1 )
1
{13 , 82 , 94 , 24 }, {23 , 91 , 81 , 12 }
Φ4 (V v2 )
1
{11 , 83 , 93 , 22 }, {21 , 92 , 84 , 14 }
1
{13 , 83 , 91 , 21 }, {22 , 94 , 84 , 12 }
1
{11 , 91 , 16, 94 , 14 }
Φ4
(V −1 v2 )
Φ8 (V v) Φ8
(V v−1 )
1
{13 , 93 , 16, 92 , 12 }
Φ12 (V v)
1
{11 , 42 , 62 , 45 , 14 }
Φ12 (V v−1 )
1
{13 , 44 , 61 , 43 , 12 }
Φ4 (v) • 21 • 81 61 62 82 22
aλ (3, −3) (3, 0) (3, 1) (3, 1) (3, 6) (3, 9)
dλ ,M 1 . 1 1 . 1 . 1 1 1 1 .
Φ4 (V v) aλ dλ ,M • 11 (0, 0) 1 . . . . • 42 (1, 0) . 1 . . . • 91 (2, −1) 1 1 1 . . 41 (3, 1) . . 1 . . • 61 (3, 1) 1 . 1 1 . . 1 1 . 1 • 12 (3, 1) 94 (6, 3) . . 1 1 1 45 (7, 6) . . . . 1 14 (12, 12) . . . 1 .
Φ4 (V ) • 23 • 83 61 62 84 24
aλ (0, 1) (1, 1) (3, 1) (3, 1) (7, 1) (12, 1)
dλ ,M 1 . 1 1 . 1 . 1 1 1 1 .
Φ4 (V v−1 ) aλ dλ ,M • 13 (0, 3) 1 . . . . (1, 3) . 1 . . . • 44 • 93 (2, 2) 1 1 1 . . (3, 1) . . 1 . . 41 • 62 (3, 1) 1 . 1 1 . • 12 (3, 1) . 1 1 . 1 92 (6, −2) . . 1 1 1 43 (7, −3) . . . . 1 12 (12, −9) . . . 1 .
7.4 Decomposition Matrices for Types H3 , H4 , E6 , E7 , E8
379
7.4 Decomposition Matrices for Types H3 , H4 , E6 , E7 , E8 In this section, we present the Φe -modular decomposition matrices of blocks of defect at least 2 for the exceptional type H3 , H4 , E6 , E7 , E8 . Information regarding the block distribution, including defect 0 representations and Brauer trees for blocks of defect 1, already appears in the tables in [132, Appendix F]. Throughout, let Γ = Z (with its natural order) and L(s) = 1 for all s ∈ S (equal parameter case). In particular, A = R[v, v−1 ] is the ring of Laurent polynomials in v = ε . Let e 2 and assume that R = K ⊆ C is a subfield which contains all roots of unity of order e. We consider the Φe -modular specialisation √ θe : A → K, v → ζ2e := exp(π −1/e) ∈ K; see Example 3.1.15(b). Note that θe is a principal specialisation where Φ = v − ζ2e . We shall denote extension of scalars from A to K (via θe ) simply by a subscript “(e)”. Thus, for example, H(e) := K ⊗A H is the corresponding specialised algebra. The assumptions of Proposition 3.2.7 are satisfied. Hence, we have a cell datum for H (see Theorem 2.6.12) and there is a natural parametrisation μ
◦ Irr(H(e) ) = {L(e) | μ ∈ Λ(e) },
◦ where Λ(e) ⊆ Λ is a canonical basic set.
As in Section 3.2, the map θe : A → K gives rise to a decomposition matrix D = dλ μ λ ∈Λ , μ ∈Λ ◦ . (e)
μ
By Proposition 3.2.7 (see also Remark 3.2.2), dλ μ equals the multiplicity of L(e) as a composition factor in the Graham–Lehrer cell module of H(e) labelled by λ . In Example 3.3.15, we have already discussed the example where W is of type H3 ; we recall the results in Table 7.11. Here, the rows of the decomposition matrix marked with “•” correspond to the canonical basic set, the first column contains standard labels for Irr(W ) and the second column contains the invariants aλ . Table 7.11 Φe -blocks and decomposition numbers for type H3 e
d
Φe -blocks of defect d
2
3
{1r , 3s , 3 s , 5r , 5 r , 3s , 3s , 1 r }
3
1
{1r , 5r , 4 r }
+ dual
5
1
{1r , 4 r , 3s }
+ dual
6
1
{1r , 5r , 5 r , 1 r }
10
1
{1r , 3 s , 3s , 1 r }
e=2 • 1r 0 • 3 s 1 • 3s 1 5r 2 5 5 r 3s 6 3s 6 1 r 15
1 1 1 1 1 1 1 1
dλ ,M . 1 . 1 1 1 . .
. . 1 1 1 . 1 .
Tables 7.12–7.15 show similar results for W of type H4 , E6 , E7 and E8 , where we omit the tables concerning the block distribution, and where we only print decomposition matrices for blocks of defect at least 2. These results are all due to M¨uller and the first-named author; see [94], [96], [129], [253], [254].
380
7 Decomposition Numbers for Exceptional Types
Table 7.12 Φe -blocks of defect 2 and decomposition numbers for type H4 e=2 • 1r 0 • 4t 1 • 4t 1 • 9s 2 • 9s 2 • 25r 4 36rr 5 6s 6 6s 6 10r 6 18r 6 30s 6 30s 6 36 rr 15 25 r 16 9 s 22 9s 22 4t 31 4t 31 60 1r
dλ ,M 1. . . . . .1. . . . . .1. . . . . 11 . . .1. .1. . 11 . . 1 111111 1. .1. . 1. . .1. . . . 11 . 1. . . .1 . . 21 . 1 . 2 . . 11 111111 . 11 . . 1 . . 11 . . .1. .1. .1. . . . . .1. . . 1. . . . .
e=4 • 1r 0 • 9s 2 • 9s 2 • 25r 4 8rr 6 • 24t 6 • 24t 6 • 25 r 16 9 s 22 9s 22 1 r 60 e=3 • 4t 1 • 4t 1 • 16r 3 8rr 6 • 16t 6 • 16t 6 16 r 18 4t 31 4t 31
dλ ,M 1. . . . . . 11 . . . . . 1.1. . . . 1111 . . . . . .1. . . . . 111 . . .1.1.1. . . . 1111 . . . . . 11 . . . .1.1 . . . . . .1
dλ ,M 1. . . . .1. . . 111 . . . .1. . 1 . 11 . . 11 . 1 . . 111 . . .1. . . . .1
e=6 • 1r 0 • 4t 1 • 4t 1 • 25r 4 • 18r 6 • 24s 6 • 24s 6 25 r 16 4t 31 4t 31 1 r 60
e=3 • 1r 0 • 16rr 3 • 25r 4 8r 6 • 10r 6 • 40r 6 25 r 16 16 rr 18 1 r 60
dλ ,M 1. . . . . . .1. . . . . . .1. . . . 1111 . . . 1 . . 11 . . . . 11 . 1 . .1.1. .1 . . . 1111 . . . . . .1 . . . . .1. . . . .1. .
dλ ,M 1. . . . .1. . . 111 . . . .1. . 1 . 11 . . 11 . 1 . . 111 . . . .1 . . .1.
e=5 • 1r 0 • 9s 2 • 16rr 3 6s 6 • 8r 6 • 24s 6 16 rr 18 9 s 22 1 r 60
e = 10 • 1r 0 • 4t 1 • 9s 2 • 36rr 5 • 24s 6 • 30s 6 • 40r 6 36 rr 15 9s 22 4t 31 1r 60 dλ ,M 1. . . . .1. . . 111 . . . .1. . 1 . 11 . . 11 . 1 . . 111 . . . .1 . . .1.
dλ ,M 1. . . . . . .1. . . . . . .1. . . . 1111 . . . 1 . . 11 . . .1.1.1. . . 11 . . 1 . . . 1111 . . . . . .1 . . . . .1. . . . .1. .
e=5 • 4t 1 • 16r 3 • 36rr 5 16t 6 • 24t 6 • 48rr 6 36 rr 15 16 r 18 4t 31
dλ ,M 1. . . . .1. . . 111 . . . .1. . 1 . 11 . . 11 . 1 . . 111 . . . .1 . . .1.
Table 7.13 Φe -blocks of defect 2 and decomposition numbers for type E6 e=2 • 1p . • 6p 1 • 20 p 2 15 p 3 • 15q 3 • 30 p 3 • 60 p 5 24 p 6 81 p 6 10s 7 20s 7 60s 7 90s 7 81 p 1. 60 p 11 24 p 12 15 p 15 15 q 15 30 p 15 20 p 2. 6 p 25 1 p 36
dλ ,M 1. . . . . .1. . . . . 11 . . . 1.1. . . 1. .1. . . 11 . 1 . . .1. .1 . .1.1. 1111 . 1 . . . .1. .1.1. . . . .1.1 . 12 . 11 1111 . 1 . .1. .1 . .1.1. 1.1. . . 1. .1. . . 11 . 1 . . 11 . . . .1. . . . 1. . . . .
e=3 • 1p 0 • 6p 1 • 20 p 2 • 15 p 3 • 15q 3 • 30 p 3 • 64 p 4 • 60 p 5 24 p 6 10s 7 20s 7 • 60s 7 • 80s 7 60 p 11 24 p 12 64 p 13 15 p 15 15 q 15 30 p 15 20 p 2. 6 p 25 1 p 36
dλ ,M 1. . . . . . . . . 11 . . . . . . . . 111 . . . . . . . .1.1. . . . . . . .1.1. . . . . .1. . .1. . . . . 111 . 11 . . . . 11 . 1111 . . . .1. . .1. . . .1. . . . .1. . . . .1. .1. . . 11111 . 111 . . 1 . 1 . 111 . 1 11 . 1 . . . 111 . . .1. . . .1. . . . 1 . . 1111 . . . . . . 11 . . 1. . . . . . .1. . . . . . . .1.1 . . . . 1 . . 11 . . . . .1. .1. . . . . .1. . . . .
e=4 • 1p 0 • 6p 1 • 15 p 3 • 15q 3 • 81 p 6 • 10s 7 • 80s 7 • 90s 7 81 p 1. 15 p 15 15 q 15 6 p 25 1 p 36
dλ ,M 1. . . . . . . .1. . . . . . . .1. . . . . 11 . 1 . . . . . 11 . 1 . . . 1. .1.1. . . 1 . 11 . 1 . . .1.1. .1 . . . . 1 . 11 . . . . . . .1 . . . 1 . 11 . . . . . . .1. . . . . .1. .
e=6 • 1p 0 • 6p 1 • 20 p 2 • 15q 3 • 30 p 3 • 60 p 5 • 24 p 6 • 60s 7 • 80s 7 • 60 p 11 24 p 12 15 q 15 • 30 p 15 20 p 2. 6 p 25 1 p 36
dλ ,M 1. . . . . . . . . . .1. . . . . . . . . 111 . . . . . . . . 1. .1. . . . . . . . 11 . 1 . . . . . . 1 . 11 . 1 . . . . . . .1. . .1. . . . . . .1.1.1. . . . . 1 . 111 . 1 . . . . . . . 1 . 111 . . . . .1. . .1. . . . . . . . .1.1. . . . . . .1.1.1 . . . . . . . . 111 . . . . . . . . . .1 . . . . . . . . .1.
7.4 Decomposition Matrices for Types H3 , H4 , E6 , E7 , E8
381
Table 7.14 Φe -blocks of defect 2 and decomposition numbers for type E7 e=2 • 1a 0 • 7 a 1 • 27a 2 21 b 3 21a 3 • 35b 3 15 a 4 • 105 a 4 • 189 b 5 105b 6 168a 6 210a 6 189 c 7 35 a 7 70 a 7 280 a 7 • 315 a 7 189a 8 405a 8 378 a 9 84a 10 210b 10 420a 10 105c 12 84 a 13 210 b 13 420 a 13 378a 14 105 c 15 189 a 15 405 a 15 35a 16 70a 16 280a 16 315a 16 189c 20 105 b 21 168 a 21 210 a 21 189b 22 15a 25 105a 25 21 a 30 35 b 30 21b 36 27 a 37 7a 46 1 a 63
dλ ,M 1. . . . . . 11 . . . . . 121 . . . . 111 . . . . 111 . . . . 1111 . . . 1. .1. . . 121 . 1 . . 122111 . 1121 . 1 . . 12 . 11 . 233111 . 122111 . 1111 . . . . .1. .1. 234112 . 1221111 122111 . 1341121 . 131121 . . 11 . 1 . . . 11 . 11 2342121 1121 . 1 . . . 11 . 1 . . . 11 . 11 2342121 . 131121 1121 . 1 . 122111 . 1341121 1111 . . . . .1. .1. 234112 . 1221111 122111 . 1121 . 1 . . 12 . 11 . 233111 . 122111 . 1. .1. . . 121 . 1 . . 111 . . . . 1111 . . . 111 . . . . 121 . . . . 11 . . . . . 1. . . . . .
e=2 • 56 a 3 • 120a 4 • 280b 7 216 a 8 336 a 10 336a 13 216a 15 280 b 16 120 a 25 56a 30
dλ ,M 1. . 11 . .11 ..1 11 1 11 1 ..1 .11 11 . 1. .
e=6 • 1a 0 • 7 a 1 • 21 b 3 • 21a 3 • 35b 3 • 15 a 4 • 105 a 4 • 105b 6 • 168a 6 • 210a 6 • 70 a 7 • 280 a 7 • 315 a 7 • 84a 10 • 210b 10 • 420a 10 105c 12 84 a 13 • 210 b 13 • 420 a 13 105 c 15 70a 16 • 280a 16 • 315a 16 105 b 21 168 a 21 210 a 21 15a 25 105a 25 21 a 30 35 b 30 21b 36 7a 46 1 a 63
dλ ,M 1. . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . 111 . . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . 11 . . 1 . . . . . . . . . . . . . . . 1. . . .1. . . . . . . . . . . . . . .1.1. .1. . . . . . . . . . . . . 111 . 11 . 1 . . . . . . . . . . . . 111 . 1 . 1 . 1 . . . . . . . . . . . . 111 . . 1 . . 1 . . . . . . . . . . 1. . .1. . . . .1. . . . . . . . . . . .1. .1. .1.1. . . . . . . . . 11 . 1 . 1111 . . 1 . . . . . . . . . . .1. . .1. . . .1. . . . . . 1 . 1 . 11 . 11 . 1 . . . 1 . . . . . . . 1 . . . 1 . 11 . 11 . . 1 . . . . . . . . . . . . .1. .1. . . . . . . . . . . .1.1. . . . . .1. . . . . . . . . 1 . . 11 . 1 . 111 . 1 . . . . . 1 . . . . 1 . 1 . 11 . . 1 . 1 . . . .1. . . . . . . . . . . .1. . . . . . . . . . . . . .1. . .1.1. . . . . . . . . . . . . . 1 . . . 1 . 11 . . . 1 . . . . 11 . . . 1 . 11 . 1 . 1 . . . . . . . . 1 . . . 111 . 1 . . 1 . . . . . . . 1 . . . . 1 . 1 . 11 . 1 . . . . . . . . . . . . 1 . . 1 . 111 . . . . . . . . . . . . .1. .1. . . . . . . . . . . . . . . . . . . . 111 . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . .1.1. .1 . . . . . . . . . . . .1. . .1. .1 . . . . . . . . . . . . . . . . . . .1 . . . . . . . . . . . . . . . .1. . .
e=3 • 1a 0 • 21a 3 • 35b 3 • 120a 4 • 105b 6 • 168a 6 • 210a 6 • 280b 7 84a 10 • 210b 10 • 420a 10 512 a 11 105c 12 336a 13 35a 16 70a 16 280a 16 15a 25 105a 25 56a 30 21b 36 7a 46
dλ ,M 1. . . . . . . . . .1. . . . . . . . 1.1. . . . . . . 11 . 1 . . . . . . . . . 11 . . . . . 1 . 11 . 1 . . . . .1.1. .1. . . 1 . 111111 . . 1.1. .1.1. . . 1 . 111 . . 1 . . . . 1 . 11 . . 1 11 . 1111111 . . . . . . 11 . . . . . . . 111 . 1 . . . . .1. . . . .1. . . . . .1. . . . . . 1 . . 11 1. . . . . .1. . . . . . 11 . 11 . . . . .1. . .1. . . . .1. .1. . . . . .1. . . . .
e=3 • 7 a 1 • 21 b 3 • 56 a 3 • 15 a 4 • 105 a 4 35 a 7 • 70 a 7 • 280 a 7 • 336 a 10 • 512a 11 • 84 a 13 210 b 13 420 a 13 105 c 15 280 b 16 105 b 21 168 a 21 210 a 21 120 a 25 21 a 30 35 b 30 1 a 63
dλ ,M 1. . . . . . . . . 11 . . . . . . . . 1.1. . . . . . . .1.1. . . . . . 111 . 1 . . . . . . . . .1. . . . . . .1. .1. . . . . .1.1.1. . . . 1 . . 1 . 11 . . 111111111 . . 1 . 11 . . . . 1 1 . 1 . 11 . . 1 . . . . . 1 . 111 . .1. . . . .1. . 11 . 11 . . 111 1. . . . . . .1. . . . 11 . . . 11 . . . . . 1 . 11 . . . .1.1. .1. . . . . .1. . . . . . .1. . . . .1 . . .1. . . . . .
382
7 Decomposition Numbers for Exceptional Types
Table 7.14 Φe -blocks of defect 2 and decomposition numbers for type E7 (cont’d) e=4 • 1a 0 • 56 a 3 • 105b 6 • 210a 6 • 189a 8 • 405a 8 336 a 10 35a 16 • 70a 16 • 315a 16 189b 22 120 a 25 21b 36
dλ ,M 1. . . . . . . .1. . . . . . 111 . . . . . .1.1. . . . . . . 11 . . . . 111 . 1 . . . . . 111 . . . . . .1. . . 1.1. . .1. . .1. .1.1 . . 1 . . . 11 . . . . . . .1 . . . . . .1.
e=4 • 21 b 3 • 120a 4 • 189 b 5 • 35 a 7 • 70 a 7 • 315 a 7 • 336a 13 189 a 15 • 405 a 15 105 b 21 210 a 21 56a 30 1 a 63
dλ ,M 1. . . . . . . .1. . . . . . 111 . . . . . . . .1. . . . 1.1.1. . . . 11 . . 1 . . . . . 1 . 11 . . . .1. .1. . . 1 . . 111 . .1.1. .1 . . . . . . 11 . . . . . . .1 . . . .1. . .
e=4 • 7 a 1 • 15 a 4 • 105 a 4 • 189 c 7 • 280b 7 • 378 a 9 • 210 b 13 105 c 15 216a 15 21 a 30 35 b 30 27 a 37
dλ ,M 1. . . . . . 11 . . . . . . .1. . . . . . 11 . . . 1.1.1. . . . 1111 . 11 . . 1 . 1 . . .1.1. . . . . 111 . . . . .1. .1. . . .1 . . . . . .1
e=4 • 27a 2 • 21a 3 • 35b 3 • 216 a 8 • 210b 10 • 105c 12 • 378a 14 280 b 16 189c 20 15a 25 105a 25 7a 46
dλ ,M 1. . . . . . .1. . . . . 1.1. . . . 11 . 1 . . . 1 . 111 . . .1. . .1. . 1 . 1 . 11 . . . 11 . 1 . . . . . 11 . .1.1. . . . . . . .1 . . . .1. .
Table 7.15 Φe -blocks of defect 2 and decomposition numbers for type E8 e = 10 • 1x 0 • 8z 1 • 28x 3 • 84x 3 • 567x 6 • 448z 7 • 1008z 7 • 1400z 7 • 972x 10 • 2268x 10 • 4536z 13 • 1400y 16 • 3150y 16 • 4200y 16 • 4480y 16 • 4536 z 23 972 x 30 • 2268 x 30 • 448 z 37 • 1008 z 37 • 1400 z 37 567 x 46 28 x 63 84 x 63 91 8 z 1 x 120
dλ ,M 1. . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . 11 . 1 . . . . . . . . . . . . . . . . . 11 . 1 . . . . . . . . . . . . . . . 1. .1.1. . . . . . . . . . . . . . . .1.1.1. . . . . . . . . . . . . . 1 . 11 . . 1 . . . . . . . . . . . . . . .1. . . .1. . . . . . . . . . . . . . . 1 . 11 . 1 . . . . . . . . . . . . . 1 . 1 . 11 . 1 . . . . . . . . . . . . . . .1. .1.1. . . . . . . . . . . . .1. . . .1.1. . . . . . . . . . . . . . .1.1. .1. . . . . . . . . . . . . 1 . 11 . . . 1 . . . . . . . . . . . . . . . 1 . 1111 . . . . . . . . . . . . . . . . .1.1. . . . . . . . . . . . .1.1. .1.1. . . . . . . . . . . . . . .1. .1.1. . . . . . . . . . . . .1. . . .1.1. . . . . . . . . . . . . . . 111 . . 1 . . . . . . . . . . . . . . . . 1 . 11 . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . .1.1.1 . . . . . . . . . . . . . . . . . . .1 . . . . . . . . . . . . . . . . .1. .
e = 12 • 1x 0 • 35x 2 • 112z 3 • 50x 4 • 210x 4 • 400z 6 • 1050x 8 • 1400x 8 • 525x 12 • 3360z 12 • 2800z 13 • 1400y 16 • 2688y 16 • 4536y 16 • 2100y 20 • 3360 z 24 • 2800 z 25 • 1050 x 32 • 1400 x 32 525 x 36 400 z 42 50 x 52 • 210 x 52 112 z 63 35 x 74 1 x 120
dλ ,M 1. . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . 111 . . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . 11 . 1 . . . . . . . . . . . . . . . 1. .1.1. . . . . . . . . . . . . . . . . 1 . 11 . . . . . . . . . . . . . 1.1. .1.1. . . . . . . . . . . . . .1. . . . .1. . . . . . . . . . . . . . . . 111 . 1 . . . . . . . . . . . . 1 . 1 . . 11 . 1 . . . . . . . . . . . . . 1 . . . . . 11 . . . . . . . . . . . . . .1. .1. .1. . . . . . . . . . . . . . 1 . 11 . . 1 . . . . . . . . . . . . . .1.1. . .1. . . . . . . . . . . . . . 1 . . 11 . 1 . . . . . . . . . . . . . . 11 . 11 . 1 . . . . . . . . . . . . . . .1. .1.1. . . . . . . . . . . . . . . 1 . 11 . 1 . . . . . . . . . . . . . . .1.1. . . . . . . . . . . . . . . . . . 1 . 11 . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . .1. . . .1. .1 . . . . . . . . . . . . . . . . 1 . 11 . . . . . . . . . . . . . . . . . . .1 . . . . . . . . . . . . . . . . . .1.
7.4 Decomposition Matrices for Types H3 , H4 , E6 , E7 , E8
383
Table 7.15 Φe -blocks of defect 2 and decomposition numbers for type E8 (cont’d) e=2 dλ ,M • 1x 0 1 . . . . . . . . . . . • 8z 1 . 1 . . . . . . . . . . • 35x 2 . 11 . . . . . . . . . 28x 3 1 . 1 . . . . . . . . . • 84x 3 1111 . . . . . . . . • 50x 4 . 1 . . 1 . . . . . . . • 210x 4 . 1 . . . 1 . . . . . . • 560z 5 . 22 . 111 . . . . . 300x 6 . . 2 . . . 1 . . . . . 567x 6 1221 . 11 . . . . . • 700x 6 11111111 . . . . 56z 7 . 1 . 1 . . . . . . . . 448z 7 1 . 11 . . 11 . . . . 1008z 7 12311121 . . . . • 1400z 7 . 221111 . 1 . . . 175x 8 1 . . 1 . . . 1 . . . . 350x 8 . 12 . 1 . 1 . . . . . • 1050x 8 11211 . 1 . . 1 . . • 1400x 8 . 2212121 . . 1 . 1575x 8 122211111 . . . 3240z 9 12432131111 . 972x 10 . . 11 . . 1 . . 1 . . 2268x 10 123221211 . 1 . 840z 10 . . . 1 . . . . 1 . . . 1400z z 10 . 1111 . 1 . . 11 . 525x 12 11211 . 11 . . . . 840x 12 . . . 11 . 11 . . 1 . • 4200x 12 11331 . 2111 . 1 700x x 13 1 . . 1 . . . . . 1 . . 2100x 13 1222111 . 11 . . 4536z 13 . . 222 . 211111 2835x 14 . . 11 . . 1 . . 1 . 1 6075x 14 125421411211 2400z 15 12422131 . 11 . 4200z 15 11332 . 32 . 111 5600z 15 135431421111 70y 16 1 . 1 . 1 . . . . . . . 168y 16 . . . . 1 . . 1 . . . . 420y 16 . . . . 1 . . . . . 1 . 1134y 16 . . 2 . 2 . 21 . . 1 . 1400y 16 12322 . 22 . . 1 . 1680y 16 12221121 . 1 . .
3150y 16 4200y 16 4536y 16 5670y 16 2100y 20 2400 z 21 4200 z 21 5600 z 21 2835 x 22 6075 x 22 4536 z 23 840 x 24 4200 x 24 700x x 25 2100 x 25 840 z 28 1400z z 28 972 x 30 2268 x 30 3240 z 31 175 x 32 350 x 32 1050 x 32 1400 x 32 1575 x 32 525 x 36 56 z 37 448 z 37 1008 z 37 1400 z 37 300 x 42 700 x 42 567 x 46 560 z 47 50 x 52 210 x 52 28 x 63 84 x 63 35 x 74 8 z 91 1 x 120
. . 121 . 22 . . 11 . . 121 . 2 . . 211 12522141 . 111 123421212111 12222121 . 11 . 12422131 . 11 . 11332 . 32 . 111 135431421111 . . 11 . . 1 . . 1 . 1 125421411211 . . 222 . 211111 . . . 11 . 11 . . 1 . 11331 . 2111 . 1 1. .1. . . . .1. . 1222111 . 11 . . . . .1. . . .1. . . . 1111 . 1 . . 11 . . . 11 . . 1 . . 1 . . 123221211 . 1 . 12432131111 . 1. .1. . .1. . . . . 12 . 1 . 1 . . . . . 11211 . 1 . . 1 . . . 2212121 . . 1 . 122211111 . . . 11211 . 11 . . . . .1.1. . . . . . . . 1 . 11 . . 11 . . . . 12311121 . . . . . 221111 . 1 . . . . .2. . .1. . . . . 11111111 . . . . 1221 . 11 . . . . . . 22 . 111 . . . . . .1. .1. . . . . . . .1. . .1. . . . . . 1.1. . . . . . . . . 1111 . . . . . . . . . 11 . . . . . . . . . .1. . . . . . . . . . 1. . . . . . . . . . .
e=2 dλ ,M • 112z 3 1 . . . . . • 160z 4 . 1 . . . . • 400z 6 1 . 1 . . . • 1344x 7 . 1 . 1 . . • 2240x 10 . . . 11 . 1296z 10 1 . . 1 . . • 3360z 12 . 1 . 1 . 1 2800z 13 11111 . 3200x 15 . . . 1 . 1 448w 16 . 11 . . . 1344w 16 . . 1 . 1 . 2016w 16 . . . . . 1 5600w 16 . 1 . 211 3200 x 21 . . . 1 . 1 3360 z 24 . 1 . 1 . 1 2800 z 25 11111 . 2240 x 28 . . . 11 . 1296 z 30 1 . . 1 . . 1344 x 37 . 1 . 1 . . 400 z 42 1 . 1 . . . 160 z 52 . 1 . . . . 112 z 63 1 . . . . . e=5 dλ ,M • 1x 0 1 . . . . . . . . . . . . . • 28x 3 . 1 . . . . . . . . . . . . • 84x 3 1 . 1 . . . . . . . . . . . • 567x 6 . 1 . 1 . . . . . . . . . . • 1344x 7 . . 111 . . . . . . . . . • 972x 10 1 . 1 . 11 . . . . . . . . • 2268x 10 . . . 1 . . 1 . . . . . . . • 4096z 11 . 1 . 11 . 11 . . . . . . • 168y 16 1 . . . . 1 . . 1 . . . . . • 1134y 16 . 1 . . . . . 1 . 1 . . . . • 2688y 16 . . . . 11 . 1 . . 1 . . . • 4536y 16 . . . . . . 11 . . . 1 . . • 4096 z 26 . . . . . . . 1 . 1111 . • 972 x 30 . . . . . 1 . . 1 . 1 . . 1 2268 x 30 . . . . . . . . . . . 11 . 1344 x 37 . . . . . . . . . . 1 . 11 567 x 46 . . . . . . . . . 1 . . 1 . 28 x 63 . . . . . . . . . 1 . . . . 84 x 63 . . . . . . . . 1 . . . . 1 1 x 120 . . . . . . . . 1 . . . . .
384
7 Decomposition Numbers for Exceptional Types
Table 7.15 Φe -blocks of defect 2 and decomposition numbers for type E8 (cont’d) e=3 • 1x 0 • 35x 2 • 28x 3 • 84x 3 • 50x 4 • 210x 4 • 300x 6 • 700x 6 • 1344x 7 • 175x 8 • 350x 8 • 1050x 8 • 1400x 8 • 2240x 10 • 4096z 11 525x 12 840x 12 • 4200x 12 • 700x x 13 2100x 13 • 3200x 15 70y 16 168y 16 420y 16 1400y 16 1680y 16 2688y 16 • 4200y 16 • 4480y 16 2100y 20 3200 x 21 840 x 24 4200 x 24 700x x 25 2100 x 25 4096 z 26 2240 x 28 175 x 32 350 x 32 1050 x 32 1400 x 32 525 x 36 1344 x 37 300 x 42 700 x 42 50 x 52 210 x 52 28 x 63 84 x 63 35 x 74 1 x 120
dλ ,M 1. . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . . . .1. . . . . . . . . . . . . . . . . 11 . 1 . . . . . . . . . . . . . . . . 1 . . 11 . . . . . . . . . . . . . . . . 11 . . 1 . . . . . . . . . . . . . . . 1 . 1 . 11 . . . . . . . . . . . . . .1. . .1.1. . . . . . . . . . . . 1111 . 1111 . . . . . . . . . . . . . . . .1. . .1. . . . . . . . . . . .1. . . . . . .1. . . . . . . . . . . . . . . . 11 . . 1 . . . . . . . . . .1. .1. . . . . .1. . . . . . . . . . . . 1 . 1 . 1 . . 11 . . . . . . 1 . 1 . . 1 . 11 . 1 . 111 . . . . . . .1. . . . .1. . . . . . . . . . . . . . . . . .1. . . . .1. . . . . . . . 1 . . 21111 . 11111 . . . . 1 . . 11 . 1 . 1 . . 1 . . . . 1 . . . . . 1 . . 11 . 1 . 1 . . . 1 . . . . . 11 . 1111111 . . . 11 . . 1 . . . . . . . .1. . . . . . . . . . . . . 1 . . 11 . 1 . . . . . . . . . 1 . . . . . 1 . . 11 . . 1 . . . . . 1 . . . . . . . . . 11 . . . . . . . 11 . . . . . . . . . . . . . . 1 . . 11 . . . . . 1111112 . 11 . 1 . . 1111 . . . 11 . . 1111111 . 111 . 11 . . . 1 . . 1 . . . 11 . 1111 . . . 1 . . . . . .1.1. . . . .1. .1. . 1 . 1 . 1 . 1 . 1 . 11 . . 11111 . . . . . . . . . . .1. . . . . . .1. . 11 . . 11 . . 11 . . . 12 . 111 11 . 11 . 1 . . . . . . . . . 11 . . . . . . . . 1 . . 1 . . . 111 . 1 . . . . . . 1 . . . . 11 . . 111 . 111 . . 1 . . . . . . . 1 . . . . 1 . . 11 . .1. . . . . . . . . . . .1. . . . . . . . . . . . .1. . .1. . . . . . . 1 . . . . . . . . . . . . . . . 11 . . . . . . . . . .1. . . . .1. . .1 . . . . . . . . .1. . . . . . .1. . . . . . 1 . 1 . . 1 . 1 . . . 1111 . . . . . . . 1 . . . . 1 . . . 11 . . . . . . . . . . . . . .1. . .1. .1. 1. . .1. . . . . . . . . . .1. . . . . . . . . . . .1.1. . .1. . . . . . . . . . . . .1. . . . . . . . . . . . . .1. . . . . .1. . . .1. . . . . . . . . . . . . .1. . . . . . . . . . . .1. . . . . . . . . . . . . . .
e=3 • 8z 1 • 112z 3 • 160z 4 • 560z 5 • 400z 6 56z 7 • 448z 7 • 1400z 7 • 840z 10 • 1400z z 10 • 4096x 11 3360z 12 2800z 13 2400z 15 • 4200z 15 • 5600z 15 448w 16 1344w 16 5600w 16 • 7168w 16 2400 z 21 4200 z 21 5600 z 21 3360 z 24 2800 z 25 4096 x 26 840 z 28 1400z z 28 56 z 37 448 z 37 1400 z 37 400 z 42 560 z 47 160 z 52 112 z 63 8 z 91
dλ ,M 1. . . . . . . . . . . . 11 . . . . . . . . . . . . 11 . . . . . . . . . . 2111 . . . . . . . . . 1 . . 11 . . . . . . . . . .1. . . . . . . . . . 1. .1.1. . . . . . . 1111 . . 1 . . . . . . 1. .1. . .1. . . . . . . . 1111 . 1 . . . . 2 . 121111 . 1 . . . 1 . 11111 . 11 . . . . .1. . .1. .1. . . . .1. . . .1.1. . . 1 . 112111111 . . 1 . 111111 . 1 . 1 . . . . . . . .1. . . . . 1 . 1111 . 1 . . 1 . . 2 . 1121 . 2 . 111 . 2121221111111 . . . . .1.1. . .1. 211111 . 1 . . 111 1 . 1 . 11 . 1 . 1111 111 . 11 . . . . 111 . . . . . 1 . . . . . 11 1 . 1 . 21 . 1 . . 211 . . . .1. .1. .1. . 111 . . . . . . . 1 . 1 . . . . .1. . . . . . . . .1.1. . . . .1. . . . . . 11 . . 1 . 1 . 1 1. . .1. . . . .1. . . . . . 21 . . 1 . 1 . . . . . . .1. .1. . . . . . . .1. . .1. . . . . . . .1. . . . . . . .
7.4 Decomposition Matrices for Types H3 , H4 , E6 , E7 , E8
385
Table 7.15 Φe -blocks of defect 2 and decomposition numbers for type E8 (cont’d) e=4 dλ ,M e=4 dλ ,M • 56z 7 1 . . . . . . . • 1x 0 1 . . . . . . . . . . . . . . . . . . • 35x 2 11 . . . . . . . . . . . . . . . . . • 1008z 7 . 1 . . . . . . • 112z 3 111 . . . . . . . . . . . . . . . . • 1400z 7 . . 1 . . . . . • 50x 4 . 1 . 1 . . . . . . . . . . . . . . . • 3240z 9 . 111 . . . . • 210x 4 . 1 . . 1 . . . . . . . . . . . . . . • 2240x 10 . . 111 . . . • 567x 6 . 11 . 11 . . . . . . . . . . . . . • 4200z 15 . 1 . 1 . 1 . . • 400z 6 11111 . 1 . . . . . . . . . . . . • 3200x 21 1 . . . . 11 . • 175x 8 1 . 1 . . . 11 . . . . . . . . . . . • 4536z 23 . . . 1 . 111 840z 28 1 . . . . . 1 . • 350x 8 . . . . . 1 . . 1 . . . . . . . . . . • 1050x 8 1211111 . . 1 . . . . . . . . . 1400z z 28 . . . 11 . . 1 1344x 37 . . . . . . 11 • 1575x 8 . . 1 . 111 . . . 1 . . . . . . . . 560 z 47 . . . . . . . 1 1296z 10 . . . . . 1 . . 1 . 1 . . . . . . . . 8 z 91 . . . . 1 . . . • 525x 12 . . 1 . . 1 . . . . . 1 . . . . . . . • 3360z 12 111 . 112 . . . 1 . 1 . . . . . . e=4 dλ ,M • 2800z 13 . . . . 111 . . . 1 . . 1 . . . . . • 2835x 14 12111121 . 1 . . 1 . 1 . . . . • 28x 3 1 . . . . . . . • 6075x 14 . 11 . 132 . 111111 . 1 . . . • 160z 4 . 1 . . . . . . 70y 16 . . . . . . . . 1 . . . . . . . . . . • 300x 6 . . 1 . . . . . 420y 16 . 1 . 1 . . . . . 1 . . . . 1 . . . . • 972x 10 . 111 . . . . 1134y 16 . . . . . 1 . . 11 . 1 . . . 1 . . . • 840x 12 . . 111 . . . 2688y 16 . 1 . . . 1 . . 11 . . 1 . 11 . . . • 700x x 13 11 . . . 1 . . • 3150y 16 1111 . 121 . 1 . 11 . 111 . . • 1344w 16 . 1 . 1 . 11 . • 4480y 16 . 1 . 1112 . . 1 . . 1111 . 1 . • 840 x 24 1 . . . . 1 . 1 • 5670y 16 . . . . . 12 . 1 . 1 . 11 . 1 . . 1 700x x 25 . . . 11 . 1 . 2100y 20 . . . . . 1 . . 1 . . 1 . 1 . 1 . . . 972 x 30 . . . . . 111 2835 x 22 11 . 1 . . 21 . 1 . . 1 . 2111 . 300 x 42 . . . . . . . 1 6075 x 22 . . . . . 12 . 11 . 11113111 160 z 52 . . . . . . 1 . 3360 z 24 . . . . . . 21 . . . . 1 . 11111 28 x 63 . . . . 1 . . . 2800z 25 . . . . . . 1 . . . . . . 1 . 1 . 11 1296 z 30 . . . . . . . . 1 . . . . . . 1 . . 1 e=4 dλ ,M 175 x 32 1 . . . . . 11 . . . . . . . . 1 . . • 8z 1 1 . . . . . . . 350 x 32 . . . . . . . . 1 . . . . . . 1 . . . • 560z 5 . 1 . . . . . . 1050 x 32 . . . 1 . . 11 . 1 . . . . 2111 . • 1344x 7 . 11 . . . . . 1575x 32 . . . . . . 1 . . . . . . . . 1111 • 840z 10 . . 11 . . . . 525 x 36 . . . . . . . . . . . 1 . . . 11 . . • 1400z z10 11 . . 1 . . . 400 z 42 . . . 1 . . 11 . . . . . . 1 . 11 . • 4536z 13 . 11 . 11 . . 567 x 46 . . . . . . . . . . . . . . 1111 . 3200x 15 . . 11 . 1 . . 50 x 52 . . . 1 . . . . . . . . . . 1 . . . . • 4200 z 21 . . . . 111 . 210 x 52 . . . . . . . . . . . . . . 1 . . 1 . • 2240 x 28 1 . . . 1 . . 1 112 z 63 . . . . . . . 1 . . . . . . 1 . 1 . . 3240 z 31 . . . . 1 . 11 35 x 74 . . . . . . . 1 . . . . . . 1 . . . . 56 z 37 . . . 1 . . . . 1 x 120 . . . . . . . 1 . . . . . . . . . . . 1008 z 37 . . . . . . 1 . 1400 z 37 . . . . . . . 1
e=4 dλ ,M • 84x 3 1 . . . . . . . • 700x 6 11 . . . . . . • 2268x 10 . 11 . . . . . • 4200x 12 1111 . . . . • 2100x 13 . . 1 . 1 . . . 448w 16 . . . . 1 . . . • 2016w 16 1 . . 1 . 1 . . • 5600w 16 . . 111 . 1 . • 4200 x 24 . . . 1 . 111 2100 x 25 . . . . 1 . 1 . 2268 x 30 . . . . . . 11 700 x 42 . . . . . 1 . 1 84 x 63 . . . . . 1 . .
e=8 dλ ,M • 1x 0 1 . . . . . . . . . . . . . . . . • 35x 2 11 . . . . . . . . . . . . . . . • 160z 4 . . 1 . . . . . . . . . . . . . . • 567x 6 . 111 . . . . . . . . . . . . . • 175x 8 1 . . . 1 . . . . . . . . . . . . • 1400x 8 11 . 1 . 1 . . . . . . . . . . . • 1575x 8 . . 11 . . 1 . . . . . . . . . . • 525x 12 . . . 1 . . . 1 . . . . . . . . . • 2835x 14 1 . . . 11 . . 1 . . . . . . . . • 6075x 14 . . . 1 . 111 . 1 . . . . . . . • 2016w 16 . . . . 1 . . . 1 . 1 . . . . . . • 5600w 16 . . . . . . 1 . . 1 . 1 . . . . . • 7168w 16 . . . . . 1 . . 11 . . 1 . . . . • 2835 x 22 . . . . . . . . 1 . 1 . 11 . . . • 6075 x 22 . . . . . . . 1 . 1 . 11 . 1 . . 175 x 32 . . . . . . . . . . 1 . . 1 . . . • 1400 x 32 . . . . . . . . . . . . 1111 . • 1575 x 32 . . . . . . . . . . . 1 . . 1 . 1 525 x 36 . . . . . . . 1 . . . . . . 1 . . 567 x 46 . . . . . . . . . . . . . . 111 160 z 52 . . . . . . . . . . . . . . . . 1 35 x 74 . . . . . . . . . . . . . 1 . 1 . 1 x 120 . . . . . . . . . . . . . 1 . . .
386
7 Decomposition Numbers for Exceptional Types
Table 7.15 Φe -blocks of defect 2 and decomposition numbers for type E8 (cont’d) e=6 dλ ,M • 1x 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 8z 1 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 35x 2 . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 28x 3 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 84x 3 111 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . • 50x 4 11 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . • 210x 4 . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . • 560z 5 . 111 . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . • 300x 6 . . 1 . 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . • 700x 6 111 . 1111 . 1 . . . . . . . . . . . . . . . . . . . . . . . • 56z 7 . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . • 448z 7 111 . 1 . . 1 . . . 1 . . . . . . . . . . . . . . . . . . . . . • 1400z 7 . 1111 . 1111 . . 1 . . . . . . . . . . . . . . . . . . . . • 175x 8 11 . . 11 . . . . . . . 1 . . . . . . . . . . . . . . . . . . . • 350x 8 . . . 1 . . . . . . 1 . . . 1 . . . . . . . . . . . . . . . . . . • 1050x 8 . 1111 . . 1 . . . . . . . 1 . . . . . . . . . . . . . . . . . • 1400x 8 . 1 . 1 . 111 . 1 . . 1 . . . 1 . . . . . . . . . . . . . . . . • 840z 10 . 1 . 1 . . . 1 . . . . . . 1 . . 1 . . . . . . . . . . . . . . . • 1400z z10 111 . 21 . 1 . 1 . 1 . 1 . 1 . . 1 . . . . . . . . . . . . . . 525x 12 . 1 . 1 . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . • 840x 12 11 . . 11 . 1 . 1 . 1 . . . . . 1 . 1 . . . . . . . . . . . . . • 4200x 12 . 1112 . . 211 . 11 . . 1 . . 1 . 1 . . . . . . . . . . . . 700x x 13 . . . . . . . . . . . . . . . 1 . . 1 . . . . . . . . . . . . . . • 2100x 13 . . . 1 . . . . 1 . 1 . 1 . 1 . . . . . . 1 . . . . . . . . . . . • 2400z 15 . . . 1 . . . . . . 1 . 1 . 1 . 1 . . . . 11 . . . . . . . . . . • 4200z 15 . 1 . 111 . 1 . 1 . . 11 . 11 . 1 . 1 . . 1 . . . . . . . . . • 5600z 15 . 1 . 11 . . 111 . . 2 . 1 . . 1 . . 11 . . 1 . . . . . . . . 168y 16 . . . . . 1 . . . 1 . . . . . . . . . 1 . . . . . . . . . . . . . • 420y 16 1 . . . 11 . . . 1 . 1 . 1 . . . . 11 . . . . . 1 . . . . . . . • 1680y 16 . . . . . . . . . . 1 . . . 1 . . . . . . 11 . . . 1 . . . . . . • 4200y 16 . . . . 2 . . 1 . 1 . 1 . 1 . 1 . . 2 . 1 . . 1 . . . 1 . . . . . • 4480y 16 . 1 . . 11 . 1 . 2 . 111 . . 11111 . . 11 . . . 1 . . . . • 4536y 16 . . . . 1 . . . 11 . . 1 . . . . . 1 . 11 . . 1 . . . . 1 . . . • 5670y 16 . . . 1 . . . 1 . 1 . . 1 . 1 . 11 . . 11111 . . . . . 1 . . 2100y 20 . . . . . . . . . 1 . . 1 . . . . . . . . 1 . . 1 . . . . . . . . 2400 z 21 . . . . . . . . . . . . . . 1 . . 1 . . . 11 . 1 . 1 . . . 1 . . 4200 z 21 . . . . 1 . . 1 . 1 . 1 . . . . . 1111 . . 11 . . 11 . 1 . . 5600 z 21 . . . . . . . . . 1 . . 1 . . . 1 . 1 . 11112 . . . 111 . . 840 x 24 . . . . . 1 . . . 1 . . . 1 . . 1 . 11 . . . 1 . 1 . . 1 . . . . • 4200 x 24 . . . . 1 . . . . 1 . . . 1 . . . . 2 . 1 . . 21 . . 11111 . 700x x 25 . . . . 1 . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . 2100 x 25 . . . . . . . . . . . . . . . . . . . . . 11 . 1 . 1 . . 11 . . 840 z 28 . . . . . . . . . . . . . . . . 1 . . . . . 11 . . . . 1 . 1 . . 1400z z 28 . . . . 1 . . . . 1 . 1 . 1 . . . . 21 . . . 1 . 1 . 11 . . 1 . 175 x 32 . . . . . . . . . . . 1 . . . . . . 11 . . . . . 1 . . 1 . . . . 350 x 32 . . . . . . . . . . . . . . . . . . . . . . 1 . . . 1 . . . 1 . . 1050 x 32 . . . . . . . . . . . . . . . . . . 1 . . . . 1 . . . 11 . 11 . • 1400 x 32 . . . . . . . . . 1 . . . . . . . 1 . 1 . . . 11 . . . 1 . 1 . 1
525 x 36 ......... . ... . .... . . ... . 1 . . .1 . 1 . . 56 z 37 ......... . ... . .... . . ... . . . 1. . . . . . 448 z 37 ......... . ...1....1 . ...1 . 1 . .1 . . 1 . 1400 z 37 .........1... . ....1 . ...11 . . .11111 300 x 42 ......... . ... . ....1 . ... . . . . . . 1 . 1 . 700 x 42 .........1... . ....11...1 . 1 . .1 . . 11 560 z 47 ......... . ... . .... . . ...1 . . . .1 . 111 50 x 52 ......... . ... . .... . 1... . . 1 . .1 . . . . 210 x 52 ......... . ... . .... . . ... . . . . . . . . . 1 28 x 63 ......... . ... . .... . . ... . . . . . . . 1 . . 84 x 63 ......... . ... . ....1 . ... . . 1 . .1 . . 1 . 35 x 74 ......... . ... . .... . . ... . . . . . . . . 1 . 8 z 91 ......... . ... . .... . . ... . . . . .1 . . . . 1 x 120 ......... . ... . .... . . ... . . 1 . . . . . . .
e=6 dλ ,M • 112z 3 1 . . . . . . . . . . • 160z 4 . 1 . . . . . . . . . • 400z 6 1 . 1 . . . . . . . . • 1344x 7 11 . 1 . . . . . . . • 2240x 10 1 . 111 . . . . . . • 3360z 12 . 1 . 1 . 1 . . . . . • 3200x 15 . . . 1 . . 1 . . . . • 1344w 16 . . 1 . 1 . . 1 . . . • 7168w 16 . . . 1111 . 1 . . 3200 x 21 . . . . . 1 . . 1 . . • 3360 z 24 . . . . . . 1 . 11 . • 2240 x 28 . . . . 1 . . 11 . 1 1344 x 37 . . . . . . . . 111 400 z 42 . . . . . . . 1 . . 1 160 z 52 . . . . . . . . . 1 . 112 z 63 . . . . . . . . . . 1
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Index
a-invariants, 13, 277 addable node, 264 adjustment matrix, 182, 183 affine Hecke algebra of type A, 334 affine symmetric group, 319 Ariki–Koike algebra, 259, 266 asymptotic case, 6, 149 balanced representation, 21 block, 200 block idempotents, 199 border nodes, 172, 264 Brauer graph, 137 Bruhat–Chevalley order, 60 canonical λ -tableau, 271 canonical basic set, 136, 144, 232, 284 Cartan matrix, 310 cell datum, 105 cell module, 105 cell representation, 105 cellular basis, 105 central character, 137 central idempotent, 199 centrally primitive idempotent, 199 character, 12 character table, 8 C HOP function, 365 Clifford system, 90 Clifford’s theorem, 93, 151 column standard, 270 complex reflection group, 359 composition, 177 constructible representations, 197 content, 172 contragredient module, 9 Coxeter group, 2
crystal, 314 crystal basis, 314 crystal graph, 314 crystallographic type, 3 cuspidal, 217 cyclic module, 312 cyclotomic Ariki–Koike algebras, 276 cylindrical multipartition, 291 decomposition map, 135 decomposition matrix, 107, 135, 213 decomposition numbers, 213 defect, 156 degrees, 142 discrete valuation ring, 135 distinguished left coset representatives, 272 divided powers, 312 dominance order, 75, 77, 177, 271 dominant weights, 310 e-regular, 176 e-residue, 177 e-residue sequence, 175 equal-parameter case, 3 exceptional representations, 187 extended affine symmetric group, 319 extended Iwahori–Hecke algebra, 90 FLOTW l-partition, 291 Fock datum, 162 Fock space, 316 fundamental weights, 310 generalised Gelfand–Graev representation, 245 generic degree, 12 generic Iwahori–Hecke algebra, 4 good addable i-node, 320
M. Geck, N. Jacon, Representations of Hecke Algebras at Roots of Unity, Algebra and Applications 15, DOI 10.1007/978-0-85729-716-7, © Springer-Verlag London Limited 2011
399
400 good removable i-node, 321 Harish-Chandra induction, 216 Harish-Chandra restriction, 216 Harish-Chandra series, 218 Hecke algebra, 219 height, 177, 269 highest weight vector, 312 Hom functor, 208, 220 i-word, 320 inertia group, 221 integrable module, 313 irreducible components, 2 isomorphism of crystal graphs, 315 Iwahori–Hecke algebra, 3 ˜ J-cells, 38 James’s conjecture, 185, 190 Jucys–Murphy elements, 156, 166, 267 Kashiwara operators, 313 Kashiwara–Lusztig canonical basis, 315 Kazhdan–Lusztig basis, 60 Kazhdan–Lusztig cells, 67 Kleshchev l-partitions, 322 Kostka number, 126 L-bad, 32 l-charge, 316 l-composition, 262 L-good, 32 l-partition, 263 ladder method, 176 leading matrix coefficients, 21 left cells, 36 level, 312 level of the Fock space, 316 Levi complement, 216 lies above, 182 LLT-graph, 174 Mackey formula, 217 monomial order, 5 multicharge, 316 multicomposition, 262 multipartition, 263 nodes, 264 normal addable i-nodes, 320 normal removable i-nodes, 320 opposite cell datum, 107 oppositive cell datum, 148
Index ordered q-wedge, 339 ordinary m-symbol, 277 ordinary symbol, 269 orthogonality relations, 12 partition, 18 Φe -modular decomposition map, 140 Φe -modular specialisation, 160 pre-order, 36 principal series representations, 13, 220 principal specialisation, 155 pseudo-reflection, 359 q-wedge, 339 q-wedge product, 337 quantum affine algebra, 311 quasi-split case, 87, 228 reduced i-word, 320 reflections, 156 removable node, 264 residue, 287 residue sequence, 296 right cells, 36 root lattice, 310 row-standard, 270 Schur elements, 12 Schur functor, 208 Schur relations, 12 semi-infinite q-wedge, 343 semi-infinite q-wedge product, 343 shape, 177 shifted m-symbol, 277 simple roots, 310 Specht modules, 273 special representation, 74 special unipotent classes, 74, 231 specialisation, 4 spetsial Ariki–Koike algebra, 277 split BN-pair, 216 Springer correspondence, 74 standard, 270 Starkey’s rule, 169 Steinberg representation, 225 strongly triangular, 369 supercuspidal, 246 Suzuki group, 227 symmetric algebra, 12, 107 tableau, 270 θ -blocks, 137 θ -isolated, 194 Tits’s deformation theorem, 8, 142
Index two-sided cells, 37 Uglov l-partitions, 321 unipotent principal series representations, 220 unipotent representations, 231 unipotent support, 231
401 weight function, 4 weight lattice, 310 weight sequence, 276 weight space, 312 weight vectors, 312 Weyl group, 3, 216
very dominant multicharge, 322 W -graph, 27, 364 weight, 312
Young diagram, 171, 177 Young tableau, 264 Young’s rule, 126
