Finite Reductive Groups: Related Structures and Representations

Progress in Mathematics Volume 141

Series Editors Hyman Bass Joseph Oesterle Alan Weinstein

Finite Reductive Groups: Related Structures and Representations Proceedings of an International Conference held in Luminy, France

Marc Cabanes Editor

Birkhauser Boston • Basel • Berlin

Marc Cabanes UFR de Mathematiques Universite Paris VII 75005 Cedex France

Library of Congress Cataloging In-Publication Data Finite reductive groups : related structures and representations / Marc Cabanes, editor. p. cm. -- (Progress in mathematics; v. 141) English and French. Papers from the Luminy conference held in Oct. 1994. Includes bibliographical references. ISBN 0-8176-3885-7 (hardcover : alk. paper). -- ISBN 3-7643-3885-7 (hardcover : alk. paper) 1. Representations of groups--Congresses. 2. Finite groups-Congresses. l. Cabanes, Marc, 1959II. Series: Progress in mathematics (Boston, Mass.) ; vol. 141. 96-15820 QA176.F56 1996 CIP 512' .2--dc20

Printed on acid-free paper

© 1997 Birkhiiuser Boslon

Birkhiiuser

ir

Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhiiuser Boston for libraries and other users registered with the Copyright Clearance Center (Ccq, provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhiiuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S A ISBN 0-8176-3885-7 ISBN 3-7643-3885-7 Typeset from disk by Tgcniques, Brighton, MA Printed and bound by Quinn-Woodbine, Woodbine, NJ Printed in the U.SA 98765 432 1

Contents Preface

vii

Introduction Paul Fong

ix

q-Analogue of a Twisted Group Ring Susumi Ariki

1

Formule des traces sur les corps finis Anne-Marie Aubert

15

Heights of Spin Characters in Characteristic 2 Christine Bessenrodt and J¢ffi B. Olsson

51

Sur certains elements reguliers des groupes de Weyl et les varietes de Deligne-Lusztig associees Michel Broue et Jean Michel.

73

Local Methods for Blocks of Reductive Groups over a Finite Field Marc Cabanes and Michel Enguehard 141 Splitting Fields for Jordan Subgroups Arjeh M. Cohen and Pham Huu Tiep

165

A Norm Map for Endomorphism Algebras of Gelfand-Graev Representations Charles W. Curtis and Toshiaki Shoji

185

Modular Representations of Finite Groups of Lie Type in Non-Defining Characteristic Meinol! Geck and Gerhard Hiss

195

Centers and Simple Modules for Iwahori-Hecke Algebras Meinol! Geck and Raphael Rouquier

251

Quantum Groups, Hall Algebras and Quantized Shuffles James A. Green

273

Fourier Transforms, Nilpotent Orbits, Hall Polynomials and Green Functions Gus I. Lehrer

291

VI

Contents

Degres relatifs des algebres cyclotomiques associees aux groupes de reflexions complexes de dimension deux Gunter Malle

311

Character Values of Iwahori-Hecke Algebras of Type B Gotz Pfeiffer

333

The Center of a Block Lluis Puig ,

361

Unipotent Characters of Finite Classical Groups Toshiaki Shoji

373

A propos d'une conjecture de Langlands modulaire Marie-France Vigneras

415

Preface This book is an outgrowth of the intense work in the field of finite reductive groups that has taken place over the last several years. Thirty mathematicians participated in the conference "Autour des groupes reductifs finis, representations" at Luminy, October 3-7, 1994, after which it became clear that a volume of the mathematics discussed there in talks and more informal conversations would greatly benefit the mathematical community. Given the enthusiastic participation of the contributors to this endeavor, and the high quality of the refereeing process, I believe we have accomplished our aim. I would like to thank the Societe Mathematique de France for providing the site of the meeting, the director Jean-Paul Brasselet of the CIRM, his assistant Annie Zeller-Meier, and all the personnel of this charming mas provent;al. The meeting was supported by grants from the CNRS, SMF, and the Commission of the European Union. It was a real pleasure for me to organize the conference and gather the artides for this book. I thank again all the authors, and particularly Paul Fong for his Introduction. Further thanks to the staff at Birkhaiiser for its competence and patience. Marc Cabanes Editor

Introduction Paul Fang

This volume of talks and papers presented at the Luminy Conference in October 1994 tells of much which is beautiful and imaginative in contemporary work in the representation theory of groups and related structures. The framework for this mathematics is formed by two areas: the modular representation theory of finite groups of Richard Brauer and the representation theory of finite reductive groups of George Lusztig. The one theory addresses all abstract finite groups, the other addresses the paramount examples of finite groups. The theories are notable achievements in mathematics. What has been fascinating and delightful to witness in recent years are not only the continuing developments of the theories, but the confluence of the two. On the one hand, the study of Brauer blocks of finite reductive groups has led to deep questions on the underlying varieties and cohomology complexes in the Deligne-Lusztig theory. On the other hand, the wealth of structure in blocks in finite reductive groups has led to wonderful conjectures in the Brauer modular representation theory of abstract finite groups whose very formulation would not have been possible without these examples. A greater and higher unity is at work here, and its shape is still to be discovered. The modular representation theory of a finite group G begins with the group algebras of G over coefficient domains 0, K, and k, where 0 is a valuation ring, K is its field of quotients, and k is its residue class field. Here K is of characteristic 0, k is of prime characteristic p, and K and k are splitting fields for the irreducible representations of all subgroups of G. The decomposition of OG and kG into algebra direct sums of indecomposable ideals give rise to block algebras and associated grouIrtheoretic invariants such as defect groups. That something new can still be said on an object as basic as a block algebra in this general context is unexpected and welcome. The paper [Pu] gives a connection between the center of a block algebra OGe and its associated generalized decomposition matrix. Other algebras have come to the fore in recent years. The paper [Gr] gives a categorical setting for two such algebras, one being the positive part of a quantum enveloping algebra in Lusztig's work, the other being Ringel's twisted Hall algebra. The

x

P. Fang

readers of this volume need no reminders about the key roles of such algebras in representation theory. The representation theory for a Coxeter group or a finite reductive group has particular significance - the problems here lie at the core of the subject and the solutions illuminate the general theory. One basic problem is giving an intrinsic description of the distribution of the irreducible characters of the group into blocks, a prototype being the Nakayama conjecture for the blocks of the symmetric group Sn. A combinatorial description of the blocks of a covering group Sn of Sn was recently given by Bessenrodt and Olsson for the prime p = 2. The sequel [Be-Ol] contained in this volume is a study of the arithmetic invariants called heights of the irreducible characters in blocks of Sn' Coxeter groups give rise to Hecke algebras, and these algebras play prominent roles in a number of the papers here. The Hecke algebra 'H(W, q) of a Coxeter group W is a q-deformation of the group algebra of W. In particular, if W is a representation group in the sense of Schur for G = W/Z(W), then a suitable quotient algebra of 'H(W,q) can be viewed as a q-deformation of the group algebra of G. The study of such q-deformations is initiated in the paper [Ar]. A character table for 'H (W, q) can be meaningfully defined since elements T wand T Wi of 'H(W, q) are conjugate by a unit of the algebra if the elements wand w' are conjugate in Wand have minimal length in their conjugacy class. These are results of Ram, Geck and Pfeiffer. The paper [Pf] makes this explicit in the case W is of type B n . Finite reductive groups are groups G = G F , where G is a connected reductive algebraic group defined over an algebraic closure of a finite field IF q of characteristic p, and F is a Frobenius endomorphism of G with an associated rational structure over IF q • The valuation ring o for the representation theory of G is an £-adic ring over the ring Zt of £-adic integers, where £ is a prime different from p. The panoply of structures needed for the representation theory of G is well-known Harish-Chandra theory, Gelfand-Graev representations, Shintani theory, and above all Deligne-Lusztig theory. Yet important questions remain open. One concerns the values of irreducible characters. These can be computed if the following conjecture by Lusztig is true: the two bases of class functions on G consisting of almost characters of G and of characteristic functions of F-stable character sheaves on G coincide up to scalar multiples when p is almost good. Explicit scalar multiples are computed in the paper [Sh] for unipotent almost characters when G is a classical group and p -=J 2. The paper [Au] also touches on relations of such bases via trace formulas. The values of characters on unipotent elements of G ties up with the Green functions of G. The

Introduction

xi

paper [Le] concerns values of Green functions in the context of Fourier transforms. The paper [Cu-Shj proves Shintani descent for the irreducible components of the Gelfand-Graev representations, of GF in terms of irreducible components of Gelfand-Graev representations " of G F '" using the endomorphism algebras of, and ,'. The modular representation theory of G = GF over the field k of characteristic £ different from the defining characteristic p of G poses a number of challenging problems. The distribution of the irreducible characters into blocks and a description of the defect groups of blocks have been found for blocks containing a unipotent character of G. But the general case remains open. What is missing here is a precise statement on the compatibility of the Jordan decomposition of irreducible characters of finite reductive groups with generalized Deligne-Lusztig induction. The paper rCa-En] gives a definitive analysis of the problem. The Harish-Chandra theory of cuspidal representations, so important for the theory of representations of Gover K, has analogues for representations of Gover k. The paper [Ge-Hi] reports on some significant progress here. BroUl~ has posed a most intriguing conjecture for an arbitrary finite group G having a block aGe with an abelian defect group D. The conjecture asserts the existence of a Rickard equivalence between aGe and aNc(D)f, where f is the Brauer correspondent of e. When G is a finite reductive group G, the conjecture should hold for deep geometrical reasons coming from the etale cohomology of Deligne-Lusztig varieties X w for suitable regular elements w of the Weyl group W of G. The desiderata are discussed in [Br-Mi]. One is that the endomorphism algebra of the cohomology complex be a specialization of a cyclotomic Hecke algebra associated to Cw(w). Such algebras were defined for complex reflection groups by Broue and Malle, who also gave a host of invariants for these algebras. When the reflection group is a Weyl group of a connected reductive algebraic group G, these invariants have fundamental meaning for the representations of GF. Their significance for the wider class of complex reflection groups is not yet known. The paper [Maj gives the invariants for a class of such groups. Hecke algebras have been mentioned in several contexts already. The paper [Ge-Roj gives yet another one. The concept of a generic group was introduced by Broue, Malle and Michel as an intermediary between a root datum for a connected reductive algebraic group and a finite reductive group - in a sense generic groups are finite reductive groups without the parameter q. The paper [Ge-Ro] investigates whether there is a similar generic aspect to the irreducible representations of a Hecke algebra over k.

xii

P. Fang

The two remaining papers in this volume deal with algebraic groups defined over fields of characteristic zero, Lie groups in the one case and p-adic groups in the other. It goes without saying that these areas, vast worlds in themselves, impinge considerably on the subjects here. The paper [Co-'Ii] considers rationality questions of Jordan subgroups of simple complex adjoint Lie groups, while paper [Vi] considers matters connected with a possible modular local Deligne-Langlands conjecture.

q-Analogue of a Twisted Group Ring Susumi A riki

1. Introduction

As is well known, group rings of Coxeter groups have their q-analogue, which are called Hecke algebras. Its natural generalization seems to be q-deformation of twisted group rings. But few examples are found. Since a twisted group ring is a direct summand of the group ring of its representation group in the sense of projective representations, if the group ring of the representation group has a q-deformation, we can obtain a q-analogue of the twisted group ring by taking a direct summand of it, or equivalently, by taking a quotient of it. For example, if a Coxeter group is a representation group of a certain quotient group of it, we can find an example of a q-analogue of a twisted group ring. (In the above, we consider group rings over a field of characteristic zero.) To summarize, we consider the following situation. Let W be an irreducible finite Coxeter group. If W is a representation group of G = W/Z (Z is a subgroup contained in the center), then we have an example of q-analogue of the twisted group ring for G as the quotient of the Hecke algebra of W. In the first half of this paper, we consider the above case. The answer turns out to be Coxeter groups of type I~4m), E 8 , and H 4 . (Theorem 1) In the second half, we consider the simplest example, namely, the q-analogue of the twisted group ring of dihedral groups. Then, we can find its representation by generators and relations, and hence we can define it over any field. Thus we can consider its modular representation theory. Its block structure and all indecomposable modules are determined by a simple argument if parameters are not roots of unity and the base field is of odd characteristic. We also know when it becomes a symmetric algebra. The results are stated in Theorem 2.

2. Preliminaries We first review the theory of projective representations. Let G be a group, F be a field, an<j [aj E H 2 (G, F X ). The twisted group ring is,

s.

2

Ariki

by definition, FOG = ffixECFx such that its multiplication rule is given by xy = a(x,y)xy. We can assume a(x, 1) = a(I,x) = 1. Since considering a FOG-module is equivalent to considering a set of matrices {p(x )}xEC satisfying p(x)p(y) = a(x, y)p(xy), we identify a FOG-module with a projective representation of G whose obstruction class to linearizability is [a]. Let G" be a covering group of G, i.e., there is a central extension 1 ~ Z ~ G" ~ G ~ 1. Then p can be viewed as a projective representation of G" whose obstruction class is p" ( [a]) E H 2 (G" , F x ). If Imp" = {I}, the representation theory of various FOG's reduces to that of a single FG". If we consider the exact sequence,

Imp" = {I} is nothing but the surjectivity of the connecting homomorphism. Among such covering groups, we pick up , minimal' ones, namely we call G" a representation group of G if ~ is bijective. In important cases, Hom(Z,F X ) and H 2 (G,F X ) are finite groups or vector spaces over a field. In these cases, we can replace the condition by the injectivity of ~ and the coincidence of order or dimension. The injectivity of ~ is equivalent to Z C [G", G"]. Now, a theorem of I. Schur is as follows.

Theorem([4, pp.97]). Let G be a finite group, and F an algebraically closed field. Then a finite covering G" is a representation group of G if and only if Z c [G",G"j and 1Z 1=1 H 2 (G,F X ) I. We note that we need F to be algebraically closed in order to conclude that 1Z 1=1 Hom(Z,F X ) I. Schur also showed the existence of representation groups of finite order for G finite, F = C. In this note, we assume that F = C, since we then can avoid consideration of any modification of the list of H 2 (W,F X ) and the character tables of projective representations for exceptional type Weyl groups, which will be used to solve the problem below. As was explained in the introduction, the following question arises naturally.

Problem. Let W be an irreducible finite Coxeter group, and Z a subgroup of the center of W. Then, when is W a representation group of G = W/Z ? This will be treated in the next section.

3

q-Analogue of a Twisted Group Ring

3. Classification Since W is an irreducible Coxeter group, the center is represented by real scalar matrices on the reflection representation. Hence the order of the center is at most 2, and W has a nontrivial center if and only if its exponents are all odd. In that case, it is generated by (SI'" SI)h/2, where SI ... , Sl are Coxeter generators and h is the Coxeter number. ([1, pp. 123, Corollary 3]) By looking at the values of linear characters of W at (SI ... SI)h/2, we have the following lemma.

Lemma 1. Let W be an irreducible finite Coxeter group with nontrivial center. Then Z c [W, W] if and only if the type of W is B 2m , D 2m , E s , F 4 , I~4m), H 4 .

We list the exponents of exceptional groups for the reader's convenience. E6

1,4,5,7,8,11

F4

1,5,7,11

E7 Es

1,5,7,9,11,13,17 1,7,11,13,17,19,23,29

H3

1,5,9

H4

1,11,19,29

If Z = {I}, H 2 (W, eX) must be {I}, hence by the following list taken from [3], we know that W is a representation group of W if and only if the type of W is AI, A2,I~2m+l). In the rest of this section, we treat the case Z i- {I}. (In the list, we omit G 2 since G 2 = I~6).)

H 2 (W,e

X )

A

:={1}

A

I(2m+l)

2, 2

At:::::3' B 2, E I=6,7,S, I~2m), H 3 , H 4

C2 C 2 x C2 C2 X C2

1,

B 3 ,DI 25 ,F4 X

C2

B1 24 ,

D4

To know the classification, it is enough to check if H 2 (W/ z, eX) := C 2 for the cases in Lemma 1.

Proposition 1. Let W be any finite group whose center Z has order 2 such that Z c [W, W]. Assume that H 2(W/Z, eX) := C2 and that all linear characters of W take rational values. Let E~,q = HP(W/Z,HQ(C2,e X )) =} HP+q(W,e X ) be the Lyndon-HochschildSerre spectral sequence. Then, (1) HI (W/Z, eX) := HI (W, eX),

S. Ariki

4

(2) Ke1'cIg,1 = {I}, (3) Eg,2 = {I}, E~2 = {l}, (4) E~'o = {l}, E~o = {l}, (5) I Ej,1 \::; \ H 1(W,C X) \, \ E~;} \ ::; I Hl(W,C X) \, (6) 1 H 2(W,C X) 1 ::; 1 H 1(W,C X) I· Proof. (1) Since Z those of W/Z.

c [W, W],

all linear characters of W come from

In particular, K e1'cIg' 1 = {I}. (3) We prove E~,2

= {I} by induction on 1'.

Assume that E~,2 = {I}. Since E~~1 is a subquotient of E~,2, we have EO,2 = {I} and EO,2 = {I} r+l 00' (4) Since Eg,1 = HI (C 2, C X ) ':::::. C 2, and ImcIg,1 = Eg,1 / Ke1'cIg,1 ':::::. Eg,1 by (2), we have ImcIg,1 ':::::. C 2. On the other hand, we have assumed E~'o = H 2(W/Z,C X) is the cyclic group of order 2. Thus,

E 2,0 3

2,0 E2,0 = K e1' (d 2 : 2

Im(d~,l : Eg,l

E 4 ,-I) --+ --+

2

E~'O)

= E 2,0/Imdo,1 = {I} 2

2

.

2,0 '"" E 2,0 (1' > 3) proves E 2,0 '"" E 2,0 = {I} E r+l r 00 3 .

(5) El,1 3

=

Ker(d~,l:E~,l_Ei'o)

Im(d 1,2:E 1,2_E~,I) z z

':::::.

Ke1'd 1,1 shows that 1E 1,1 1 ex is such that t(gxg- 1 ) = t(x) for x E Wo, t(l) = 1, t(x- 1 ) = t(X)-l, then t~(~tJ))a(x,y) for a standard cocycle a is again standard.) Another method to obtain a character table is as follows. Assume that the ordinary character table of a representation group W* is given. Take a section s: W ----> W* such that s(x) and s(y) are conjugate if x and yare conjugate in W. Then a(x,y) = ..\(s(x)s(Y)S(xy)-l) for each ..\ E H om(Z, eX) is a class function cocycle of finite order, and we can obtain the character table for a-characters from the ordinary character table of W*. Lemma 2. Let W be a finite group whose center is {1,z} (z2 = 1). Let a be a class function cocycle of finite order. (1) If the column (Xj(z)h::;j::;h in the character table of acharacters satisfies I Xj (z) I= I Xj (1) I, then z is represented by -module a scalar linear transformation on each irreducible

caw

Vi·

(2) If there is no such column in the character table of a-characters, then z is represented by a nonscalar linear transformation on each irreducible

caw -module Vi.

Proof. (1) Since a is of finite order, z2 is a root of unity. Hence z is of finite order. Thus I Xj (z) I= I Xj (1) I implies that z is represented by a scalar transformation. (2) If the non-identity central element z is a-regular, then I Xj(z) 1= I Xj(1) I for all j, contradiction. Thus z is not a-regular and tr(z, Vi) = 0, since a is a class function cocycle. Hence z cannot be scalar transformation. • Lemma 3. Let 1 groups.

---->

Z

---->

W .!!... G

---->

1 be a central extension of finite

(1) If there is an irreducible projective representation of W such that all element in Z are represented by scalar linear transformations, then the obstruction class [aj E H 2(W, eX) of this representation belongs to Im(p* : H 2(G,C X ) ----> H 2(W,C X )).

q-Analogue of a Twisted Group Ring

7

(2) If there is an irreducible projective representation of W such that an element in Z is represented by a nonscalar transformation, then the obstruction class does not belong to Imp*. Proof. (1) Let (p, V) be such a representation. Since p(Z) are scalar transformations, we can choose p(z) (z E Z) such that P(Zl)P(Z2) = P(ZlZ2). Take a section s : G ----> Wand fix a set of linear transformations {p(S(W))}wEC. We can take p satisfying p(zs(w)) = p(z)p(s(w)), since multiplying scalar to each group element does not change the cohomology class. Then,

Namely p(gdp(g2) = (3(p(gd,p(g2))p(glg2) and hence p*([{3]) = [a]. (2) If there exists [(3] E H 2(G, iCX) such that p*([{3]) = [a], we can choose {P(g)}9EW such that

Thus p(z) (z E Z) commutes with all p(g). By irreducibility, p(z) is a scalar transformation. •

Corollary. (1) H 2(W/Z,iC X )

'::: C2 if the type ofW is I~4m), E 8 , H 4 . (2) If the type ofW isF4, H 2(W/Z,iC X ) has order 4.

Proof. A conclusion of Lyndon-Hochschild-Serre's spectral sequence is that if 1 ----> K ----> H ----> L ----> 1 is exact and A is an H-module, then 1----> H1(L,A K ) ----> H1(H,A) ----> H1(K,A)L

----> H 2(L, A K

)

----> H 2(H, A)

is exact. Hence, in our case, we have an exact sequence 1----> H1(W/Z,iC X ) " : ' H1(W,iC X

)

----> H1(Z,iC X )

----> H 2(W/Z,iC X ) ----> H 2(W,iC X ).

S. Ariki

8

Now we study case by case. For I~4m), W/Z is the dihedral group of type I~2m). Hence the second cohomology group is C2. For Es and H 4 , we check the character tables given by A.O. Morris ([5]) and E. W. Read ([6]). In these papers, ordinary character tables of representation groups are given. Thus the second method to have the character table of a-characters is used. By applying Lemmas 2 and 3, we have proved (1). (2) is similarly obtained by looking at the character table given by E.W. Read ([7]). In the paper, he has constructed a standard cocycle for each cohomology class, and the first method to have the character table of a-characters is used. • We have now proved the following theorem.

Theorem 1. Let W be an irreducible finite Coxeter group with non trivial center Z. Then W is a representation group of G = W / Z if and only if the type of W is I~4m), Es, H 4. 4. Dihedral Case As was explained in the introduction, in the situation of Theorem 1, we can proceed to the construction of a q-analogue of the twisted group ring. We study the simplest case here. The Hecke algebra of type I~4m) is the algebra over iC (u, v) defined by generators and relations as follows.

The complete set of irreducible representations are given by four one dimensional representations and Pk : Sf-> Tf->

where (4m is a 4m-th primitive root of unity and 1 :S k :S 2m - 1. We can show that Pk((ST)2m) = (-l)kh Hence, Pk'S for odd k are

q-Analogue of a Twisted Group Ring

9

the complete set of irreducible representations for the q-analogue of the twisted group ring of the dihedral group whose type is I~2m). In other words, the desired quotient of the Hecke algebra is the quotient by the two sided ideal generated by (ST)2m + 1. Hence we reach the following definition of the q-analogue of the twisted group ring of dihedral groups.

Definition. Let A be an integral domain,

U, v E A x. Sj(A) is the algebra over A defined by generators S, T and their relations

(S-u)(S+u- 1) = 0, (T-v)(T+v- 1) = 0, (ST)2m = (Ts)2m =-1.

Proposition 2.

(1) Sj(A) is free of rank 4m as an A-module. (2) If f : A -+ B be an algebra homomorphism between integral domains such that parameters correspond. Then, Sj(B) ~ Sj(A) &Jf B.

Proof. We first assume that A = Z[U±1, v±1]. Since

1

etc., we know that 2::::'0- A(TS)i + A(TS)iT is stable under right multiplication by S, T. Thus it coincides with Sj(A). Further, since Pk for odd k are nonequivalent irreducible representations of Sj(iC(u, v)), the semi-simple quotient has dimension 4m and we can conclude that {(TS)i, (T S)iT}0$i::;2m-1 have no nontrivial linear relation over A. Hence we have (1). Since there is a natural surjective homomorphism Sj(B) -+ Sj(A) &Jf B, we have by (1) that they are isomorphic. In particular, Sj(B) is free as an B-module. Thus (1) for general A is also obtained. (2) for general A is now obvious. •

In the following, we study the modular representation theory of this algebra. Let F be a field of an odd characteristic p. We assume that u, v are not roots of unity. We put m = pT mo , g.c.d(p, mo) = 1. ( is a primitive 4mo-th root of unity. We also assume that F contains (. Replacing (4m by (, we define P2k+1 (0 S k S mo - 1). Proposition 3. -=J -1, (u/v)2m -=J -1, then {P2k+1 lOS k S mo -I} is the complete set of irreducible representations. (2) Ifuv = (2k o+1, (u/v)2m -=J -1, then

(1) If (uv)2m

10

S. Ariki {P++, p__ }U {P2k+l 10 :s k :s mo - 1, k # k o , 2mo - 1 - ko} is the complete set of irreducible representations, where P++ : S I-t U, T I-t V and p__ : S I-t _u- 1 , T I-t _v- 1 . (3) If (uv )2m # -1, u/v = _(2k o +l, then {p+_,p_+}U {P2k+l 10:S k:S mo -1,k # ko,2mo -1- ko} is the complete set of irreducible representations, where P+_ : S I-t U, T I-t _v- 1 and p_+ : S I-t _u- 1 , T I-t V.

Proof. (1) Let V be an irreducible module. We put a = ST and take an irreducible F[a]-submodule. It is of the form Fe and ae = (2k+l e for some k (O:S k:S 2mo-l). Then .lj(F)0P[a]Fe --+ V is surjective. Since .lj(F) has no one dimensional representation, P2k+l must be irreducible. Assume that P2k+l and P2!+1 are equivalent. By explicit calculation, we have k = I or I = 2mo - 1 - k, and we find a non zero intertwiner (2k+l(u_u-l)+(v_v-l) (4k+2_1

_

1

]

ek+l(u_u-1)+(4k+2(v_v-1) from P2k+l 1 (4k+2_1 to P2( 2m o-l-k)+1' Since these are irreducible, they are isomorphic. (2) If P2k+l is reducible, it contains P++ or p__ . In the former case, the existence of simultaneous eigenvector leads to uv = (-2k-l. Similarly, the latter case leads to uv = (2k+l. Hence P2k+l (k # k o, 2mo -1- k o ) are irreducible, and the rest is actually reducible. As in the proof of (1), P2k+l is equivalent to P2( 2m o-l-k)+1 if they are irreducible. (3) It is similarly proved as (2). •

P -

[

We put ek = 2;'0 :E;:~-l (-(2k+l)pr i ap r i and P k = .lj(F)ek (0 :s k :s 2mo -1). It is of dimension 2p r. It is easy to see that ekel = 8k/ek, :E ek = 1. In particular, Pk is a projective module. To understand the radical series of Pk , we need the following Lemma 4.

Lemma 4. Let A be a finite dimensional F -algebra, M be a finitely generated A-module. If M has a composition series M :l M 1 :l .. , :l M n :l 0 such that JMi :l M i +2 for a left ideal J C radA and MilMi+2 is uniserial. Then M is uniserial.

Proof. We show that radiM = Mi. Assume that it holds for i. Then

Since MilMi+2 is uniserial, we have radi+l M = Mi+l, and induction proceeds. In particular, ra~ M / radi + 1 M is simple for all i. Hence if there is a composition series, it must coincide with the radical series.•

q-Analogue of a Twisted Group Ring

11

Lemma 5. Pk is as above. Then, (1) There is a sequence of submodules Pk :l N 1 :l ... :l Npr_l :l 0 such that NdNi+l is equivalent to P2k+l' (2) Pk is an uniserial module. In particular, it is indecomposable. Proof. (1) Put N i = .lj(F)(a2mo + l)i ek . Then Npr = 0, and dim(NdNi+1 ) 2 because F[a] = F[a 2mo ] 0 F[a P"]. Since dimPk = 2pr, the equality holds. Since a P" and a 2mo act as ((2k+l)p" and -1 on (a 2mo + l)iekmodNi+l' a acts as (2k+l. Thus we have NdNi+l c::: P2k+l· (2) Put J = .lj(F)(a 2mo + 1). Since Pk/JPk c::: P2k+l and J acts as 0 on irreducible modules, we have J C rad.lj(F). If P2k+l is irreducible, then NdNi+1 is simple and Pk is uniserial. If P2k+l is reducible, then one can check P2k+ 1 is uniserial and can apply Lemma 4. •

:s

Lemma 6. If a finite dimensional F -algebra has only one irreducible module, and the indecomposable projective module is uniserial, then it is a symmetric algebra. Proof. Let A be such an algebra, and let P be the indecomposable projective module. Then A c::: EndA(pffJ m ) for some m. We show that A = EndA(P) c::: F[x]/(x n ) where n is the length of P. Since the head of P and (radA)P is the same, we have a surjection P ---> (radA)P/(radA)2 P, which factors through (radA)P. We set x : P ---> (radA)P C P. We note that its image is (radA)P. Thus x(radA)ip = (radA)i+l P. Now we can prove

by downward induction. Since any element in the left hand side induces an element in HomA(P/(radA)P, (radA)iP/(radA)i+l P), by subtracting a scalar multiple of Xi, we can apply the induction hypothesis. Thus we have the explicit form of A. Since any linear map ¢ : F[x]/(x n ) ---> F which does not vanish on the unique minimal ideal has the property ¢(ab) = ¢(ba) and nonvanishing property on any non-zero one-sided ideal, A has a symmetric algebra structure. Since A is a matrix algebra over A, ¢ 0 Tr enjoys the same property that ¢ itself has, and hence A becomes a symmetric algebra. •

Theorem 2. Let F be a field of odd characteristic p, containing a primitive 4mo-th root of unity where m = pr mo and mo is coprime to

12

S. Ariki

p. Let .f)(F) be the q-analogue of the twisted group ring of the dihedral group of type I~2m). We assume that parameters u, v are not roots of unity. Then, (1) All blocks are oftheformBk = PkffiP2mo-l-k (0 ~ k ~ rna-I). (2) .f) (F) is a Frobenius algebra. (3) All blocks of .f)(F) are uniserial algebras. (4) .f)(F) has finite representation type. All indecomposable modules are homomorphic image of indecomposable projective modules. (5) .f)(F) is a symmetric algebra if and only if both (uv)2m f= -1 and (u/v)2m f= -1 hold.

Proof. (1) By Lemma 5(2), Bk and Bl (k f= l) do not have a common simple module, but Pk and P2mo-l-k have simple modules in common. Hence Bk'S are two-sided ideals and indecomposable as an algebra. Thus these are block algebras. (2) Let {a(i),r(i)} be the dual basis of {(TS)i, (TS)iT}. Then the action of S, T with respect to these bases on the F-dual of the right regular representation and the left regular representation coincide. (3) A direct consequence of (2) and Lemma 5(2). (4) Theorem 62.25 ([2]) asserts that if a finite dimensional F-algebra is uniserial, then it has finite representation type and all indecomposable modules are homomorphic image of indecomposable projective modules. Thus we have the result.

(5) If (uv?m = -1 or (u/v)2m = -1, then by Lemma 5(1), Pk/radPk is not equivalent to SOCPk for some k. Thus .f)(F) cannot be symmetric. If otherwise, then all block algebras are symmetric algebras by Lemma 5(1) and Lemma 6(2). Thus .f)(F) is symmetric. • References 1. N. Bourbaki, Groupes et algebres de Lie, V, Hermann, 1968. 2. C.W. Curtis, 1. Reiner, Methods of Representation Theory with applications to finite groups and orders, vol.II, Wiley-Interscience, 1987. 3. S. Ihara, T. Yokonuma, On the second cohomology groups ( Schur multipliers) of finite reflection groups, J. Fac. Sci. Univ. Tokyo Sect I 11 (1965), I55-I7!. 4. G. Karpilovsky, Projective representations of finite groups, Marcel Dekker, 1985.

q-Analogue of a Twisted Group Ring

13

5. A.a. Morris, Projective characters of exceptional Weyl groups, J. Algebra 29 (1974), 567-586. 6. E.W. Read, The linear and projective character of the finite reflection group of type H 4 , Quart. J. Math. Oxford Ser. 25 (1974a), 73-79. 7. E.W. Read, The projective characters of the Weyl group of type F 4 , J. London Math. Soc. 8 (1974b), 83-93.

Division of Mathematics Tokyo University of Mercantile Marine Etchujima 2-1-6, Koto-ku, Tokyo 135, Japan Received January 1995

Formule des traces sur les corps finis Anne-Marie Aubert

o.

Introduction

James Arthur a recemment mis en lumiere de nombreuses analogies entre des objets attaches it un groupe reductif p-adique et des objets attaches aux divers R-groupes (cf [Ar2, en particulier Remarks (2) p. 118]). Nous nous interessons ici au cas d'un groupe reductif fini, i.e., du groupe G F des points fixes sous un endomorphisme de Frobenius F d'un groupe G algebrique reductif connexe sur une cloture algebrique d'un corps fini ll"q et defini sur ll"q. Le role des R-groupes est joue ici par les groupes de ramification

WGdM,a) = {r E NGdM)/M F I a = a} T

associes it des paires (M, a) formees d'un sous-groupe de Levi F--stable M d'un sous-groupe parabolique F-stable de Get d'une representation irreductible cuspidale a de M F; ces groupes sont des extensions centrales de groupes de Coxeter. On dispose de deux bases orthonormees de l'espace C(G F ) des fonctions centrales (i. e., invariantes par G F -conjugaison) sur G F : celle des fonctions caracteristiques des classes de conjugaison et celles des caracteres irreductibles; si h et 12 sont deux fonctions centrales sur G F , la formule des traces pour la fonction h x 12 de G F x G F traduit simplement l'egalite des expressions du produit scalaire usuel dans G F de h et 12 (note (h, h)GF) it l'aide la premiere base (appelee "cote geometrique") et de la seconde base (appelee "cote spectral"), voir (2.3).11 est aussi possible d'en definir Ie terme "elliptique" en s'inspirant des travaux d'Arthur pour les groupes p-adiques : on considere Ie sous-espace Ccusp(GF) de C(G F ) engendre par les fonctions cuspidales au sens d'Arthur (i.e., les fonctions f telles que h = *Rf(f) = 0 pour tout sous-groupe de Levi F-stable L f= G d'un sous-groupe parabolique F -stable de G; ici * Rf designe Ie foncteur de restriction de Harish-Chandra); Ie terme elliptique de la formule des traces, que nous noterons J;tt, est egal au produit scalaire des projections sur Ie sous-espace Ccusp(GF) de h et de 12 (note (h, 12)GF,ell)'

16

A.-M. Aubert

Nous fixons un tore maximalement deploye To de G et nous posons

WGF = NGF(To)jTr Soit [, l'ensemble des sous-groupes L de G tels que L :l To est sous-groupe de Levi rationnel d'un sous-groupe parabolique rationnel P :l To de G. Le terme total de la formule des F traces JG = (!I,12)GF s'exprime en fonction des termes elliptiques des elements L de [, de la maniere suivante : (0.1)

(voir tho 3.19); l'expression (0.1) est l'analogue sur les corps finis de l'expression du terme J(j) de [Ad, prop. 6.1]. L'expression spectrale du terme elliptique Je?t se presente comme une version simplifiee du terme elliptique de la formule des traces pour les groupes p-adiques: elle fait intervenir des triplets T = (M, a, r), avec r E WGF(M,a)ell, ou WGdM,a)ell designe l'ensemble des elements "elliptiques" de WGdM,a), i.e., contenus dans aucun des WLdM,a) pour L E [, avec L -=I- G (voir def. 4.11 et tho 4.16), qui jouent le role des triplets elliptiques d'Arthur; en particulier (voir cor. 4.19), si Xp et Xp' sont deux representations irreductibles du groupe G F qui interviennent dans l'induite parabolique de (M, a), associees respectivement awe representations irreductibles pet p' du groupe WGF(M,a), on a

(analogue de l'expression (1*) de [Ar2]). F Au contraire, l'expression geometrique du terme elliptique Je?J fait intervenir beaucoup plus de termes que dans le cas des groupes padiques (elements semi-simples non reguliers, elements unipotents) et s'apparenterait plutot it la formule des traces globale (voir tho 6.15); toutefois si l'on se restreint it des fonctions centrales h et 12 telles que h (g) = h (gs) et h(g) = 12(gs) pour tout g E G F de partie semi-simple gs, on obtient (!I,12)GF,ell

= I: {T}

IWGF(T)I- 1 (ITFI- 1

I:

h(t)12(t)) ,

tETF

ou T parcourt les classes de G F -conjugaison de tores maximaux Fstables elliptiques de G, expression qui est analogue au produit scalaire elliptique d'Arthur (voir [Ar2, (6.7)]).

Formule des traces sur les corps finis

17

Dans Ie cas des groupes reductifs finis it centre connexe, on dispose de la theorie des faisceaux-caracteres, due it Lusztig ([L5]); les fonctions caracteristiques de ces derniers forment une nouvelle base de C(GF), dont Lusztig a conjecture qu'elle est identique, it multiplication par des scalaires pres, it celle, definie par lui dans [L3], des caracteres fantomes de GF; Shoji a demontre la conjecture de Lusztig (voir [S]) dans Ie cas ou la caracteristique de IF q est presque bonne (voir [L6, 1.12]). Nous admettrons cette conjecture et nous supposerons que q est assez grand (de maniere it disposer de la "formule de Mackey" pour les foncteurs de Lusztig, due it Deligne, ainsi que de l'egalite au signe pres (demontree par Lusztig en [L6, prop. 9.2]) de l'induction de Lusztig des fonctions centrales et de l'induction, au niveau des fonctions caracteristiques, des faisceaux-caracteres cuspidaux). Nous sommes alors en mesure de "repartir" les caracteres fantomes de GF en series (voir Ie §5), de maniere parallele it la repartition classique (rappelee au §4) des caracteres irreductibles de GF en "series de Harish-Chandra". Ces series de caracteres fantomes conduisent it un "cote fantome" de la formule des traces, dont Ie terme elliptique est Ie "reflet sur les caracteres fantomes" du terme spectral elliptique (voir §7). Dans la derniere partie, nous etendons un resultat de Waldspurger [W, lem. 5.U] et nous montrons que l'espace des fonctions centrales cuspidales est en bijection avec l'espace engendre par les fonctions caracteristiques des elements "anisotropes" de G F (voir def. 8.1 et 8.3); une preuve identique it celle de [W, cor. 5.12] montre alors que la dimension du sous-espace des fonctions uniformes de l'espace engendre par les fonctions caracteristiques d'elements anisotropes unipotents est egale au nombre de classes de conjugaison de tores elliptiques maximaux de G (cf. cor. 8.6).

1. Definition de la formule des traces Nous reprenons sous une forme legerement differente la premiere partie de l'article de Marie-France Vigneras [V]. Soit H un groupe fini. Nous noterons IHlle cardinal de H et F(H, C) l'ensemble des fonctions sur H it valeurs complexes. Nous considerons la "representation reguliere" R H du groupe H x H sur F(H, C) definie par

NollS notons Tr (R H ) son caractere.

18

A.-M. Aubert

Soient h E F(H,C) et 12 E F(H,C). Nous notons h X 12 E F(H x H, C) la fonction definie par (h x h)(h 1 , h2) = h (hd h(h 2). Nous allons calculer de deux manieres l'expression suivante

Cote geometrique : Si h E H, nous notons CH(h) Ie centralisateur dans H de h. En utilisant Ie fait que les fonctions caracteristiques des elements de H forment une base de l'espace F(H, C), nous voyons que la valeur en (hI, h2) de Tr (R H) est egale a ICH(hdl = ICH(h2)1 si hI et h2 sont Hconjugues et est egale a zero dans Ie cas contraire. Soit f(H) l'ensemble des classes de conjugaison dans H. Soit h E H. Nous notons C1H(h) E f(H) la classe de H-conjugaison de h et, pour f E F(H, C), nous posons f(C1H(h)) f(h').

L

On a H

Tr (R ) (1) =

1~12

L

ICH(h)1 h(C1H(h)) h(C1H(h))

CIH (h)Er(H)

HI1 2

I

'L.J " h(h)

12 (x -1 hx).

hEH xEH

(1.1) Nous notons JtJ,arn l' expression (1.1).

Cote spectral : Soit C(H) l'espace des fonctions centrales (i.e., invariantes par conjugaison) sur H. Si f E F(H, C) et 1/; E C(H), nous posons

1/;(1)

:=

~

L

1/;(h) f(h).

(1.2)

hEH

Soit Irr(H) 1'ensemble des caracteres irreductibles de H. Nous notons X Ie caractere de H defini par X(h) := X(h) = X(h- 1 ). Nous posons

Js~ec:=

L XElrr(H)

x(1d x(12).

(1.3)

Formule des traces sur les corps finis

19

Puisque

XEIrr(H)

(1.4) nous obtenons

l:

x(Jd x(12)·

xEIrr(H)

La formule des traces exprime l'egalite des termes J:'om et J:{,ec' 2. Interpretation de la formule des traces Boit ( , ) H Ie produit scalaire usuel sur C(H) :

Nous supposons dorenavant que les fonctions h et 12 sont des fonctions centrales sur H. Boit 12 la fonction definie par 12(h) 12 (h). La formule (1.1) devient

Boit h E H. Boit 1{! la fonction centrale sur H qui vaut ICH(h)1 sur C1H(h) et 0 ailleurs. Boit i E {1,2}. On a Ji(h) = (Ji,I{!)H et (1;;"I{!JH = ICH(hi)l. Comme .= " (1;, 1;;')H H

f

t

~

CIH(h;)Er(H)

(1 H 1 H) hi'

I

h; ,

hi H

on obtient (on retrouve ici la formule (1.1)) :

(iI, 1{!)H (12, 1{!)H (1{!,I{!)H

A.-M. Aubert

20

Soit

if:

la fonction definie par

(2.1) Comme

(if:, if:) H =

1,

DOUS

obtenons

L

(h, if:)H (12, if:)H.

(2.2)

CIH(h)Er(H)

ParalleIement, on a x(fI)

L

=

(h,XI)H Xl,

(fI, X) H et x(h) et

=

(X, h) H· Comme

L

12

Xl Elrr(H)

(X2, h)H X2

X2Elrr(H)

on obtient (on retrouve ici la formule (1.5)) : H = (f1,h)H J spec =

'L.J "

(f1,x)H(x,h)H.

XElrr(H)

Pour simplifier les notations nous poserons dorenavant J :'orn (fl, h)H. La formule des traces s'ecrit alors

L XElrr(H)

.

,

cote geornetrique

(1/JI,X)H (1/J2,X)H.

.

cote spectral

(2.3) 3. La formule des traces pour les groupes reductifs sur les corps finis Soient JF q un corps fini de caracteristique p et G un groupe algebrique reductif connexe sur iFq , defini sur JF q , muni d'un endomorphisme de Frobenius F. Nous notons G F Ie groupe des points rationnels de G. Nous fixons un tore maximalement deploye T de G. Soit WGF := G NGdT)jTF. Soit.c l'ensemble des sous-groupes L de G tels que L est un sous-groupe de Levi F-stable d'un sous-groupe parabolique F-stable P :J T de G. Nous noterons U Ie radical unipotent de P. Pour tout M E .c, nous notons .c(M) l'ensemble des L E .c qui contiennent M et nous posons

=.c

(3.1)

21

Formule des traces sur les corps finis

Rr

Nous notons Ie foncteur "induction de Harish-Chandra" de la categorie des CL F -modules it gauche dans celle des CG F-modules it gauche, dMini par Rr(E) := (C G F jU F ) Q9CLF E, pour tout CL F-module it gauche E.

Rr

Rr Rr

Soit * Ie foncteur adjoint du precedent. Les foncteurs et * sont independants du sous-groupe parabolique P (cf. [LS]), et l'on a

Au niveau des fonctions centrales, les foncteurs induction et restriction de Harish-Chandra sont decrits de la maniere suivante. Soient I E C(G F ) et !' E C(L F ). On a (3.2)

Rr(J')(g) = I;FI

ou }, designe l'extension de

*Rr(J)(l) =

!' it p F , et

I~FI

L

l(lu), pour 1 ELF.

(3.3)

uEUF

Si L E .c, et I E C(G F ), nous noterons h la fonction centrale sur LF egale it *Rr(J). La fonction h E C(LF) est l'analogue sur les corps finis du "terme constant" pour la fonction I (cf. par exemple [Ar2, p. 96-97]). Definition 3.4. Nous dirons qu'une lonetion IE C(G F ) est cuspidale

si *Rr(J)

=

0 pour tout L

E.c tel que L f= G.

Nous notons Ccusp(GF) l'ensemble des fonctions de C(G F ) qui sont cuspidales et pr~:p Ie projecteur sur Ie sous-espace Ccusp(GF). Un caractere irreductible de G F est dit cuspidal s'il est cuspidal comme fonction centrale sur G F (voir def. 3.4). Nous notons Irrcusp(GF) l'ensemble des caracteres irreductibles cuspidaux de G F .

Remarques 3.5. (1) Une fonction I E C(G F ) est cuspidale si et seulement si a(Jd = 0 pour tout L E .c et tout a E Irr(L F ). La notion de fonction cuspidale introduite ici est done l'analogue de la notion de fonction cuspidale d'Arthur (voir [Ar2, p. 95-96]).

22

A.-M. Aubert (2) On a Irrcusp(GF) C Ccusp(GF), mais tout element de Ccusp(GF) n'est pas combinaison Iineaire de caracteres irreductibles cuspidaux (par exempIe, si G := GL 2 , la fonction Id - St est cuspidaIe, ou Id et St designent respectivement le caractere identite et le caractere de Steinberg de GL 2 (lF'q)). Pour Ia commodite du Iecteur, nous rappelons quelques resultats de

[Au2]. On a (3.6)

Soit [.c]wGF un systeme de representants des classes de WGFconjugaison dans .c. On a Ia decomposition orthogonale

EB

Rr (Ccusp(LF)),

(3.7)

LE[L:JwGF

et Ie projecteur sur Ie terme Rr(Ccusp(LF)) est

(3.8) Comme Ie fait Kazhdan en [K,], nous posons

EB

Rr (Ccusp (L

F

(3.9)

)).

LE[Clw

GF L, (z)

---->

Sn ~ Sn

---->

1

where (z) is a central subgroup of order 2 in SnThose irreducible characters of Sn, which have (z) in their kernel, will be referred to as ordinary chamcters. The other irreducible characters of Sn are referred to as spin chamcters. It is well-known that the ordinary characters of Sn are labelled canonically by the partitions A = (£'1, £2, ... ,£m) of n; thus

The length £( A) of A is defined as m. The set of partitions of n is denoted P(n) and for A E P(n), [AJ denotes the corresponding ordinary character lsupported by a grant from the EC (Network on Algebraic Combinatorics).

52

C. Bessenrodt and J.B. Olsson

of Sn (resp. of Sn). We also write A f- n instead of A E P(n). For A E P(n), H>, denotes the product of all the hook lengths of A; then the hook formula for the degree of [A] is

[A](l)

=

n!/H>,

(see [5], 2.3.21). Let p be a prime number. The distribution of the ordinary characters into p-blocks is described by a theorem, which is still called the Nakayama Conjecture (see [5], 6.1.21). If A E P(n), let A(p) denote its p-core, obtained from A by removing successively allp-hooks from A ([5], 2.7.16). Then for A, Ji E P(n), [A] and [Ji] are in the same p-block B of Sn if and only if A(p) = Ji(p). In this situation IAI - IA(p)1 is a multiple of p, say IAI-IA(p) I = pw. The integer w is an invariant of the block B, called the weight w(B) of B. Let sgn = [In] denote the sign character of Sn and Sn' An irreducible character X of Sn is called self-associate if X . sgn = X· Otherwise X is called non self-associate and X and Xl = X . sgn are called a pair of associate characters. The associate classes of spin characters of Sn are labelled canonically by the partitions of n into distinct parts, A = (£1, £2,"" £m), £1 > £2 > .. '£m > 0, £1 + £2 + ... + £m = n. We let D(n) denote the set of such partitions and divide D(n) into two subsets as follows:

{A {A

= =

(£1,'" ,£m) E D(n) (£1,'" ,£m) E D(n)

In In -

m even} m odd}.

To each A E D+(n), corresponds a self-associate spin character (A) and to each A E D-(n), corresponds a pair (A) and (A) I of associate spin characters. For A E D(n) we let H>, denote the product of the bar lengths of A, (see [7], [3]). Then the bar formula for the degree of (A) is (A)(I) = 2[n-~(>')]n!/H>, (see [3], Theorem (10.7)). For odd primes p a result analogous to the Nakayama Conjecture holds for spin characters ([4], [2]); instead of removing p-hooks you have to remove p-bars. In this case a p-block cannot contain ordinary and spin characters at the same time. The weight of a block of spin characters is defined analogously to the weight of blocks of ordinary characters. In this paper we consider the case p = 2, where the characters of a 2-block B of Sn may be considered as the ordinary characters in a

53

Heights of Spin Chameters

unique 2-block B of Sn' Then B also contains some spin characters. The distribution of spin characters into 2-blocks was described in [1] (see Theorem 1.1 for the exact statement). The weight w(B) of a 2block of Sn is defined as w(B), where B is the 2-block of Sn contained in B. We consider the following questions: (I) What is the minimal power of 2 dividing the degree of spin characters of Sn and what are the labels of these characters? (II) What are the possible heights of the spin characters in a given 2-block B of Sn and what are the labels of characters of minimal height? To these questions we remark the following: The first part of (I) was answered by Wagner [12] and in [8] the power of 2 dividing a spin character degree was computed. In Section 2 we give a complete answer to question (I) including an essentially different proof of Wagner's result: The minimal power of 2 in a spin character degree is 2t where t = [n-~(n)]. Here s(n) is the number of summands in the 2-adic decomposition of n. The number of spin characters of Sn with a minimal 2-power in their degree depend in a complicated way on n. Thus there is no result analogous to MacDonald's beautiful result for the ordinary characters (Theorem 2.1 below). The second author has proved that the number of characters of a given height in a p-block of Sn depends only on the weight of the block for any prime p. A similar statement can be made for p-blocks of spin characters of Sn for odd primes p. Here there is a modification in that also the sign of the p-bar-core of the block plays a role, i.e. whether the p-bar-core is in V+ or in V-. But in any of the cases the heights of characters lie between 0 and (w - 'Lai)/(p - 1) where w = 'Laipi i

i

is the p-adic decomposition of w. The maximal height occurs in some but not all blocks for p = 2. For instance a 2-block of Sn of weight 8 does not contain a character of height 7. In Section 3 we prove that also the number of spin characters of a given height in a 2-block B of Sn depends only on the weight w = w(B) of B. Moreover the heights of spin characters in a 2-block of weight w range between

It is also possible to describe explicitly the labels of spin characters of

the minimal possible height. Such characters always exist whereas there may be no spin characters of the maximal possible height.

C. Bessenrodt and J.B. Olsson

54

In the final section we check some conjectures concerning block invariants for the 2-blocks of Sn. A recent conjecture of Robinson [10] is verified for all p-blocks of Sn' We give a brief survey of the results in characteristic 2 which are needed here. It follows from the Nakayama Conjecture that the 2-blocks of Sn (and thus of Sn) are labelled canonically by the 2-cores of integers t satisfying t == n (mod n). It is easy to see that the only 2-cores at all are the partitions /'i,k, k 2:: 0, where /'i,k = (k, k - 1, ... , 1) is a "triangular" partition of k(k+I)/2. The main result of [1] is as follows. For a partition A = (1\, ... 1 £rn) E V(n) we set dbl(A)= ([£1;1],

[£d] , [£2;1],

[£;], ... ,[£rn;I], [£;]) ,

the doubling of A. Then Theorem 1.1

alJ)

Let A E V(n). Then (A) and [dbl(A)] belong to the

same 2-block of Sn. For the study of the powers of 2 dividing spin character degrees a theory of 4-cores and 4-quotients for partitions A E V( n) plays a role, which is similar to that of p-cores and p-quotients in the study of ordinary character degrees. Given A E V(n) it is possible to define its 4-core A(4) (which is a partition of the form (4t + 1, 4t - 3, ... ,5,1) or (4t + 3, 4t -1, ... ,7,3)) and its 4-quotient A(4) (see [1]). The 4-quotient is a partition and the following relation is satisfied: IAI = IA(4)1

+ 2IA(4)1·

Moreover, dbl(A(4)) = (dbl(A))(2), which implies that IA(4)1 is the weight of the 2-block containing (A). From A(4) and A(4) one may easily recover the partition A. Suppose P = A(4) is the 4-quotient of >., say P = (i 2rn ,+E,) (written exponentially) with Ci E {O, I}. Then we set Po = (irn,) and Pe = W,)· Let Ao (resp. Ae ) denote the partition consisting of all odd (resp. even) parts of A. Then Ae = 2pe. There is a combinatorial process associating to each partition Q with odd distinct parts a new partition /1( Q); this is described in [1, §3], [8, §4] and [9, §7]. With this notation, Po = /1(A o). There is for instance an explicit formula (Theorem 3.1 below) for the height of (A) in the 2-block containing it based on the partitions Po and Pe, involving the 2-powers in the character degrees [Po](I) and (Pe)(I).

55

Heights of Spin Characters 2. On the 2-Part of Spin Character Degrees First we fix some further notation. Let n E IN. Then V2 (n) 2-adic valuation of n O(n) {>. = (£1, ... ,em) f- n I £i odd for i Mo(n) {p f- n I v2I:[p](I)) = O} mo(n) IMo(n)1

= 1, ... ,l}

For a partition n of n, we set

s(n)

=

s(ln\),

2-core of n, and we write n E M o as an abbreviation of n E Mo(lnl). For a partition>' = (£1,'" ,em) E V(n), we set n(2)

4-quotient of >. (see [1, § 3]),

>. (4) n(>.)

=

[n~m].

=

For later use we recall

Theorem 2.1 (Macdonald (6J) s

If n

=

L2

i=1

k "

k l > k 2 > ... > k., then mo(n)

= 2k, +...+k•.

In fact, the set M o(n) can be described explicitly using the 2-core tower (see [9, §6]). Using the notation of the theorem, a partition n belongs to M o(n) if and only if there is exactly one 2-core (1) in the ki - th layer of the 2-core tower of n, for i = 1, ... , s, and all other 2-cores in the 2-core tower of n are 0. It is the aim of this section to give a description of the set of partitions labelling spin characters of minimal 2-part in their degree. First we consider only partitions in distinct odd parts.

Proposition 2.2 Let n > 1, s = s(n) and let>. E V(n) n O(n). Let K = dbl(>')(2)' so IKI = k(k;l) for some k E INa. Then we have n

"2 n-I

-2n+2 -2-

!!±l 2

[ n+(I"~+3)/2]

if k = 0 if k = 1 or 2 if k = 3 if k = 5 otherwise

c.

56

Proof. Let p = >.0\ r = have

Ipl, so n =

Bessenrodt and J.B. Olsson

111":1 + 4r. By [9, 7.12] or [8, 4.8] we

where Using

s :::; s(lI":)

+ s(r),

we obtain

Hence V2( (>')(1)) 2:: n

+ 111":1 - 2s(lI":) 2

It is easy to check that for k E {a, 1,2,3, 5} the stated expressions

follow. For k = 4 or k > 5 one has s(lI":)
1, s Then

=

s(n) and>'

E

V(n) n O(n).

Definition 2.4 For n E IN, set

Proposition 2.5 For n E IN we have

Mj(n) n O(n) =

=

forn=3 {(3)} {>. E V(n) n O(n) 11>'1 - 41>.(4) I :::; 3, S(>.(4)) = 1, >.(4) E M o} { for n> 3

Heights of Spin Characters

57

Proof. We use the notation of Proposition 2.2 and its proof. Assuming that V2( (A)(l)) =Jn~s] + 1 holds, one immediately obtains k 2 by Proposition 2.2. ore precisely, for k = 0 s = 1, for k = lone has s = 2, and for k = 2 s = 2 or 3. Checking the inequalities of the proof of Proposition 2.2, one finds that p E M o has to be satisfied. Moreover, s = 2 and k = 2 only occurs for n = 3, A = (3), and the other cases are equivalent to r being a 2-power and k 2. This gives the sets on the right hand side above. Conversely, in all these cases the required equality holds.

s:

s:

Theorem 2.6 Let n E IN, s = s(n).

(a) If A E V+(n), then

v2((A)(1))

~

n-s+1] 2 . [

(b) If A E V-(n), then

Proof. Let

A = L 2iAi with Ai E (V n O)(IAil) or Ai = 0. i2:0

Then

n = L2ilAii i2:0

and

d2(A)

=

L

d2(Ai) + (n -

L

IAil)

i

d2(A) v2(H>,) [9, 7.7] or [8, 4.3]. Notice also that €(A) Li €(Ai) == Li IAil (mod 2), since the partitions Ai have only odd parts.

where

Furthermore, by [9, 7.8] or [8, 4.4]

Now by the Bar Formula

C. Bessenrodt and J.B. Olsson

58 (a) If A E 'O+(n), then n ==

E 1\1 (mod2),

and we obtain

V2( (A)(1)) = n - s + 2]n(Ai) - d2(Ai)) -

,

~(n -

IAil) .

L

,

By Corollary 2.3

if

IAil > 1.

Since s S

Hence

Ei S(Ai), we thus obtain

V2((A)(1))

~

n;s+

~

CAil-2S(Ai)_[IAil-SYi)+1]+1)

1";1>1

~

n- s

-2-

1

+ "2 1{I Ai l > 1 IIAil ¢ S(Ai) (mod2)}1 +1{IAil> 1 IIAil == S(Ai) (mod2)}1

Now, if there is no contribution from some IAil > 1, then A is the partition corresponding to the 2-adic decomposition of n, and hence S == n (mod 2) as A E '0+ (n), so [n-?!] = n2s. Thus in any case V2((A)(1)) ~ [n-~+!]. (b) If A E 'O-(n), then n ¢

Ei IAil (mod2) and we have -

1

V2((A)(1)) = n - S + L(n(Ai) - d2(Ai)) - "2(n - L

,

>

n-s-1 2

,

IAil + 1)

1

+ "2!{!Ai! > 1 IIAil ¢ S(Ai) (mod2)}1

+1{IAil > 1 IIAil == S(Ai) (mod2)}1 by similar reasoning as in (a).

Heights of Spin Characters

59

Again, if there is no contribution from some 1,\1 > 1, then A corresponds to the 2-adic decomposition of n, and thus n-~-l = [n 2s] as

A E V-(n). Hence v2((A)(1));:::

[n 2s]

for all A E V-(n).

For the following we have to introduce some further notations. For n E IN we set

V(n) I v2((A)(1)) = [n - ;(n)]}

Mo(n)

{A

Mt(n)

{A E V+ (n) 1V2 ((A) (1))

Mo(n)

{A E V- (n) 1 V2 ((A) (1)) = [n - ; (n) ] }

mo(n) mt(n) mo(n)

IMo(n)1

E

=

[n - s ~n)

+ 1] }

IMt(n)1 IMo(n)1

Attention Mt(n) is not the set Mo(n) n V+(n)! s

Furthermore, if n =

L2

k

;,

kl > k 2 > ... > k s , is the 2-adic de-

i=1

composition of n, we let 82(n) = (2 k1 , ... , 2k .) E V(n) denote the corresponding partition of n. Theorem 2.7 Let n E IN, s = s(n), and let

E

be a sign. We set

vg (n) = {A = L 2i Ai E 1Y (n) I 3!io : IAio I > 1; and this

Aio satisfies:

i~O

S(Aio)

:s: 2, Aio E M I , S = l{Ai =I- 0}1 + S(Aio) -

Then we have {82(n)} if n == s (mod2) { Vt(n)

if n"l- s (mod2) if n == s (mod2) Vo(n) if n"l- s (mod2)

Mo(n)

Vo(n) { {8 2(n)}

U

Mo(n)

{82 ( n)} U

Va (n)

I}

60

C. Bessenrodt and J.B. Olsson

Proof. This follows from the proof of the previous theorem and Proposition 2.5.

Remark 2.8 By different methods, A. Wagner [12] has shown that for a field F of characteristic =I 2, the degree of any projective representation of Sn over F is divisible by 2 [n-;(nlJ. He has also noticed that the complex representation labelled by the 2-adic decomposition of n is divisible by exactly this 2-power. 3. Heights of Spin Characters in 2-Blocks We now want to study the height of irreducible spin characters in their 2-blocks. The relationship between the 2-combinatorics for dbl(>.) and the 4combinatorics for >. in described in detail in [1]. As in § 1, we denote by P = >.(4) the 4-quotient of>., say P = (i 2rn.+,.) with Ei E {O, I}, and we set Po = (irni ) and Pe = (i'i). Let >'0 resp. >'e denote the partition consisting of all odd resp. even parts of >.. Then >'e = 2pe and Po = /1(>'0) in the notation of [9, 7.11]. Furthermore, the spin character (>.) belongs to a 2-block of weight w = w(>.) = 21Pol + IPel· Finally, we define h(>.) = h( (>')) to be the height of the spin character (>.) in its 2-bloek of

Sn'

We now have:

Theorem 3.1 Let>. E V(n), w = w(>'), Po,Pe as defined above. Then

where (

r Pe

) =

{I0

if IPe I odd and Pe otherwise

EV-

Proof. As a 2-block of Sn of weight w is of defect v2(2· (2w)!), we have

h(>.)

1)2 ( (>')(1)) - 1)2(2 . nl) v2((2w)!) -

+ 1)2(2 . (2w)!)

d2(>') + n(>.)

61

Heights of Spin Characters

With Ao defined as above, we have by [9, 7.5 and 7.6] or [8, 4.1 and 4.2]: Furthermore,

[n -;(A)] = [IAol + IAel - ;(Ao) -l(Ae)]

n(A) =

n(Ao) + n(po) +

[I~el] + ,(Pe)

as is easily checked. As

1/2((2w)!)

=

2w - s(w)

=

w + 1/2 (1:'1)

+ 1/2((2IPol)!) + 1/2(IPel!),

we thus obtain

k(A)

=

1/2((2IPol)!) - d2(A o) + n(Ao) + 1/2(IPel!) - d2(Pe) +n(Pe)1/2 (1:'1)

+ w -IPel +

[I~el] + ,(Pe)

By [9, 7.12] we know

d2(A o) - n(Ao) =

IPol + d2(po),

hence

k(A) =

1/2((2IPol)!) - IPol +2/po/

1/2(IPol!) -

d2 (po)

+ 1/2((Pe)(1)) + 1/2 (,:'1)

+ [I~el] + ,(Pe)

d2 (po)

+ 1/2((Pe)( 1)) + 1/2 (1:'1) + 21Pol

+ [I~el] + ,(Pe) 1/2([Po] (1)) + 1/2((Pe)(l)) proving the assertion.

+ 1/2 (I:',) + 21Pol +

[I~el] + ,(Pe) ,

c.

62

Bessenrodt and J.B. Olsson

The main point of this formula is that it does not depend on the 2core of the 2-block, but only on the 4-quotient of >.. Thus in conjunction with the corresponding result for 2-blocks of Sn, it implies the following reduction result:

8 be a 2-block of Sn of weight w, and let principal 2-block of S2w. Then ki (8) = ki (80 ) for all i E INa.

Theorem 3.2 Let

80

be the

Based on the results of the previous section, we now want to investigate the spin characters of minimal height in a 2-block. Theorem 3.3 Let n E IN,

>. E V(n), w = w(>'), s = s(w).

(a) If>' E 1J+(n), then

(b) If>'

E

V-(n), then

~ [2W 2- S] .

h(>')

(c) If>' has 'I-quotient p ..... (Po,Pe) = (0, 82(w)), then h(>') In this case, >. E V(s)(n), where e(s) = {

=

= [2W2-S].

if s is even if s is odd.

Proof. We use the same notation as before, so by Theorem 3.1 we have

h(>') = V2([Po](1)) +v2((Pe)(1»)

+v2(1:'1) +2IPol+ [I~el] +,(Pe)

> v2((Pe)(1))+s(Pe)+s(Po)-s(w)+21Pol +

[IP;I] +,(Pe)

(a) By [1, 3.3], >. E v+(n) if and only if w is even and Pe E V+, or w is odd and Pe E V-. Using w == (mod 2), we obtain in the first case by Theorem 2.6:

IPel

h(>.) 2:

[IPel- S~Pe) + 1] + 8+ s(Pe) + s(Po) _ s + 21Pol + I~el

Heights of Spin Characters

where

63

8= {01 otherwise if Pe ¢ Mt

Hence

h()..) >

[2 1Pe I + s(Pe) [2W

>

+ 2S(~o) - 2s + 41Pol + 1]

+ S(Pe) + ~S(Po) - 2s + 1]

2W + S(Po) - S + [ 2

~ [2W -2S + In the second case,

( since S S s(Po) + s(Pe))

1]

IPel

with

1]

is odd and Pe E V-, so ,(Pe)

8= {Io

if Pe ¢ M otherwise

=

1 and hence

o

Similarly as above we get this time

(b) Again we use [1, 3.3], and consider first the case where (mod 2) is even and Pe E V-. Here, similarly as above,

h()..) >

W

==

IPel

[lpel- S(Pe)] + 8 + s(Po) + s(Pe) - S + 21Pol + I~I 2

> [2W - S] 2 In the second case,

h()..) >

IPel

is odd and Pe E V+, so

[IPel- S~Pe) + 1] + 8 + S(Pe) + s(Po) _ S + 21Pol + Ipel 2

> [2W - S] 2

1

64

C. Bessenrodt and J.B. Olsson

as before. (c) The first assertion is easily checked using the formula given in Theorem 3.1. The second one is immediate from the fact that the number of even parts in>' equals the number of parts of Pe = 82 (w). Again, by going through the sequence of inequalities in the proof above, we can describe the set of spin characters of minimal height in a 2-block in detail. First we need some further definitions. Let 8 be a 2-block of Sn of weight w. Then we set

Kt(8)

o

{>.

E 'D(n) I w(>')

{>.

E V+(n)

= w, h(>.) =

I w(>.) = w,

[2W -2 S(W)]}

h(>.) =

[2W - s~w) +

I]}

K(8) n'D-(n) IKo(8)1, kt(8) = IKt(8ll, ko(8) = IKo(8ll

K (8)

ko(8)

Furthermore, for a sign e we let if n is even if n is odd.

be a 2-block of Sn of weight w = 2::=1 2w " > ... > w., and let e be a sign. Set

Theorem 3.4 Let W2

'D~(n, w) Vf(n,w)

{>. {>.

8

E 'De(n) I w = w(>'), E

'De(n) I w = w(>'),

>'e

=

WI

>

2pe, Pe E M~(w)} (Po,Pe),

>.(4) .....

3w; > 0 : Po E M o(2 Wi -

l

)

,Pe E MK(w - 2Wi )}

Then we have: if e(s) = e 'D6(n, w) U 'DHn, w) if e(s) =J e

K e(8) = { 'D6(n, w) o

and

K (8) = { 'Dt(n,w) U'Do(n,w) U'D1(n,w) if s is even o

'Do(n, w)

if s is odd

Proof. This follows by a careful analysis of the inequalities in the proof of the preceding Theorem. We omit the details.

Heights of Spin Chameters

65

Remark 3.5 (i) By definition, Ko(B) ~ Ko(B), but note that Kt(B) ~ Ko(B) if and only if s(w) is even. (ii) If e(w) = e(s) = e, then KQ(B) = P. = K, + 282 (w)}, where K, is the 4-core of the spin characters in B. (iii) If w is odd and s(w) even, then note that in the VI contribution of Ko(B) above, !or any Wi > 0 the partition Pe = 82 (w - 2W ,) is the only element in Mo(w - 2w ,) = Mt(w - 2Wi ). Corollary 3.6 Let B be a 2-block of W2>'"

>W

Sn

of weight w =

2::=1

2Wi , WI >

S '

(i) Ifw and s(w) = s are both even, then

kt(B)

mt(w)

ko(B)

mo(w) + Lmo(w - 2W')2W,-1

= 1 s

i=1

s

mo(w) + Lmo(w - 2W')2 Wi - 1 i=1

kt(B) + ko(B)

ko(B)

(ii) If w is even, s(w) = s odd, then s

+ Lmt(w -

2W')2W,-1 = mt(w)

kt(B)

mt(w)

ko(B)

mo(w) = mo(w) = ko(B)

w

+2

kl

(iii) Ifw is odd, s(w) = s even, then

kt(B)

mo(w) = mo(w)

ko(B)

mt(w)

s-I

W -

2

i=1

ko(B)

1

+ L2w ;-1 = mt(w-) +-w-l

mt(w) +mo(w) + - 2 -

(iv) Ifw and s(w) = s are both odd, then s-I

kt(B) = mo(w) + L mo(w - lWi)2Wi - 1 i=1 s-I

mo(w) + L mo(w - 2Wi )2W,-1 i=1

C. Bessenrodt and J. B. Olsson

66

Proof. This follows from the preceding theorem, Theorem 2.8 and Macdonald's Theorem 2.1.

Before proceeding, let us look at some examples to illustrate the results above. Examples 3.7 (i) Let B be the 2-block of weight w the minimal spin character height is [2W-;(w)] = 4.

=

5 in

513 ,

Then

To compute Ko(B) we need M o(5) and M o(2), which are easy to calculate:

{A E V+(5) I v2(('x)(I)) = 2} {A E V-(5) I v2(('x)(I)) = I} {(2), (12)}

Mt(5) M (5) M o(2)

o

Hence Kt(B) = {(8,3,2)}, Ko(B) here Ko(B) = Kt(B) U Ko(B).

=

(ii) Let B be the 2-block of weight w spin character height is [2W-;(W)] = 5. We first compute:

Mt(6) M o(1) M (2)

o

= =

o o

{(4,2)}, M (6) {(I)}, M o(2) {(2)}, M (4)

{(5)} {(4,1)}

{(1O,3),(1l,2),(7,3,2,1)},

=

6 in

515 ,

Here the minimal

{(6), (3,2, I)} {(2), (12)} {(4)}

With this we obtain

Kt(B) = {(8,4,3)} Ko(B) = {(12,3),(6,4,3,2),(8,7),(11,4),(7,4,3,1)} Again, Ko(B)

=

Kt(B) U Ko(iJ).

We want to conclude this section by considering spin characters of maximal height in their 2-block. We have the following upper bound for the height:

Theorem 3.8 Let'x E V(n), w = w(,x), s = s(w). Then

Heights of Spin Characters

67

Proof. Using the notation introduced before, we have by Theorem 3.1

h(A)

= v2([Po](1)) + v2((Pe)(1)) + ,(Pe) + 21Pol + V2 (I~I) +

:s

IPol- S(Po)

since ,(Pe)

= 0 if Pe

[I~el]

+ IPel- S(Pe) + 21Pol + s(Po) + s(Pe) - s +

is a 'I-core. As w

h(A)

:s w + [W] "2 -

[I~el],

= 21Pol + IPel, this gives s=

[3W 2-

2S] .

Keeping the notation from above, we can describe explicity for which

A E V(n) the bound above is attained:

Theorem 3.9 Let A E V(n), W = W(A), s = s(w), e = e(w).

Then h(A) = [3W 22S] if and only if Po is a 2-core and one of the following holds:

(i) Pe is a 'I-core (in this case, A E Ve(n)).

= (4k + 1,4(k -1) + 1, ... ,5,2, 1) or Pe = (4k + + 3, ... ,7,3,2) for some k E lN o (in this case, A E

(ii) w is odd and Pe

3,4(k - 1) V+(n)).

Proof. That Po has to be a 2-core is immediate from the inequality in the proof above. Now v2((Pe)(1)) + ,(Pe) = IPel - s(Pe) if and only if Pe is a 'I-core or v2((Pe)(1)) = IPel- s(Pe) -1 and ,(Pe) = 1. By [8, p. 245] this happens exactly in the cases stated in (ii) above. The assertions on the parity of A follow easily from the fact that the number of even parts in A is the length of Pe. It is clear that the conditions on Po, Pe given above lead to a partition A with h(A) = [3W 22S] .

B be a 2-block of Sn of weight w, s = s(w). Then B contains a spin character of height [3W 22S] if and only if w = 2Doo + Dol

Corollary 3.10 Let

or w is odd and w = 2Do o+ Dol + 2, where Doo, Dol are triangular numbers.

68

C. Bessenrodt and J.B. Olsson

More precisely, B contains a non-selfassociate spin character of height r~w22S] if and only if w is odd and of the form w = 2Do o + Do], where 10, Dol triangular numbers. If this is the case, the number of pairs of non-selfassociate spin characters in B of height [3w22s1 equals the number of decompositions of w as w = 2Do o + Do] with 0, Do] triangular numbers.

Examples 3.11 (i) w = 4 is the smallest weight that can not be written in the form above; in such a block the maximal spin height is 4.

(ii) For w = 7 we have 7 = 2·3 + 1 = 2· 1 + 3 + 2, leading to A = (9,3,2,1) E '0- (15) and A = (6,5,4) E '0+ (15) as the partitions of maximal spin height [3W 22S] = 7 in a 2-block of weight 7 in S15. Remark 3.12 Note that by the bounds obtained in this section, there is only an interval of length [~] - [~] for possible spin character heights in a 2-block of weight w. 4. Applications

Using the results of [1] and the results of the preceding sections we want to prove that the following conjectures hold for the 2-blocks of Sn (see [9]); below, B is always a p-block of the finite group G and 8(B) is its defect group. Conjecture 4.1 (Brauer) k(B)

:s: 18(B)I.

Conjecture 4.2 (Brauer's Height 0 Conjecture) k(B) only if 8(B) is abelian. Conjecture 4.3 (Olsson) ko(B)

= ko(B) if and

:s: 18(B) : 8(B)'1

All these conjectures are known to hold for the p-blocks of Sn if pis an odd prime (see [9]). For dealing with the case p = 2, we first recall a result from [1]: Theorem 4.4 Let B be a 2-block of Sn of weight w, and let B be the 2-block of Sn containing B. Then k(B) = k(B)

+ p(w) + p-(w)

= k(2,w)

+ p(w) + p-(w)

where k(2,w) is the number of 2-quotients of weight w, i.e. the number of pairs of partitions (AO,A]) with IAol + IA]I = w.

69

Heights of Spin Characters

Corollary 4.5 Let B be a 2-block of Sn, then k(B)

:s: 18(B)I.

Proof. Let B cF =f 0 sont tous dans une

CF =f 0, on a (, = 1, on voit que la formule 2.4 s'ecrit

CFq m(2N-(a-,+A-'»,(g). ,EUch(CF)

On voit que n'interviennent dans cette formule que les caracteres unipotents de la serie principale (ceux tels que RI>CF =f 0). On sait que ces caracteres sont en bijection X f-+ avec les caracteres de W telle que CF = X(I). D'ou.

'x


3.4. I'element west central dans B(w). 3.5. Question. L'injection naturelle de V( w) dans V definit-elle un morphisme injectif P( w) ~ P ?

Remarquons que si tel est Ie cas, alors

3.6. les fleches naturelles induisent Ie diagramme commutatif suivant, ou les fleches horizontales sont injectives, et les suites verticales sont

Elements reguliers et varietes de Deligne-Lusztig

93

exaetes :

{I}

{I}

1

1

{I} 1

(1t' )

'-+

P(w)

1 (w) 1

'-+

B(w)

(w)

'-+

W(w)

'-+

P

'-+

B

'-+

W

1

1 1

1

{I}

1

1

{I}

1

{I}

Demonstration. Seule l'injectivite du morphisme B(w) --+ B n'est pas immediate. Mais si un element B( w) s'envoie sur 0 dans B, il s'envoie par composition sur 0 dans W, donc aussi sur 0 dans W(w). n provient donc d'un element de P( w), qui s'envoie sur 0 dans B, donc est nul. • Remarque. L'exactitude des suites courtes verticales ne depend evidemment pas de la reponse it la question 3.5.

On designe encore par 1t' l'image dans P du chemin defini plus haut. clair que 1t' est Ie meme element que celui defini au §1 ci-dessus.

n est

Soit D( w) Ie diagramme "it la Coxeter" associe au groupe W( w) comme decrit dans [BrMa]. D'apres [BrMaRo]' on sait (sauf peut-etre dans Ie cas ou W( w) ~ G 31 , ou ceci est encore conjectural) que D( w) symbolise it la fois une presentation de B( w) et une presentation de W (w) par generateurs et relations, presentations compatibles avec la surjection canonique B( w) --+ W( w), ou l'ensemble S( w) des sommets de D( w) represente un systeme de generateurs de B( w) qui s'envoient sur des pseudo-reflexions de W (w), et les liens de D( w) representent les relations (homogenes) entre les elements de S( w) (cf. [BrMaRo] pour plus de details). On appelle "bon systeme de gemErateurs" de B( w) tout systeme de generateurs de B( w) qui s'envoient sur des pseudo-reflexions de W( w) et satisfont aux conditions imposees par Ie diagramme D( w). On note B(w)1(w) Ie sous-monolde de B(w) forme des mots en les elements de S(w). Le resultat suivant est une consequence de [BrMaRo].

3.7. Theoreme. (1) On a Z(P)

Z(P(w))

(1t'), et le diagramme commutatif

M. Broue et J. Michel

94

suivant, ou les suites verticales sont exactes

{I} 1 (1r)

{I} ~

1

---t

Z(P(w))

'-+

Z(B(w))

'-+

Z(W(w))

1 (w)

~

1

t---

Z(P)

+-'

Z(B)

+-'

Z(W)

1

1 (w)

{I} 1

1

1

1

1

{I}

1

{I}

{I}

(2) On a 1rEB~

n existe

et

£s(1r)=2N.

un bon systeme de generateurs S( w) de B( w) tel que

(a) 1r E B(w)~(w)'

(b) £S(w)(1r) = N(w)

+ N(w)*.

Dans Ie paragraphe suivant, nous montrons essentiellement que l'on peut aussi choisir S de sorte que w E B~.

B. Racines de 1r et elements reguliers L'un des buts de ce paragraphe est d'etudier les racines de l'element central 1r, i.e., les elements w E B+ dont une certaine puissance est egale it 1r. En un certain sens, ces elements peuvent etre vus comme precisant la notion d'element regulier. Nous demontrerons en effet (3.14) que si w d = 1r, alors l'image w de w dans West un element d-regulier de W. • La reciproque est fausse (par exemple parce qu'on peut trouver des elements d-reguliers de longueur superieure it 2N/ d). Cependant, nous demontrons (3.11) que tout element d-regulier est conjugue it l'image dans W d'une racine d-ieme de 1r. En outre, l'annexe 1 donne une liste de racines d-iemes de 1r pour tous les nombres reguliers d pour tous les groupes de Coxeter finis irreductibles. •

La notion suivante joue un role essentiel.

3.8. Definition. On dit que west une bonne racine d-ieme de 1r si (1) west une racine d-ieme de 1r, i.e., w d = 1r, (2) pour tout entier m ~ d/2, w m est reduit.

Elements reguLiers et varietes de Deligne-Lusztig

95

ExempLes. • L'element Wo est une bonne racine carree de 1t'. • Dans Ie cas du type A 3 : ()---()--(), l'element de Coxeter sut stu

est une bonne racine quatrieme de 1t', mais l'element de Coxeter stu, bien que racine quatrieme de 1t', n'est pas une bonne racine. La proposition suivante donne une autre definition possible des bonnes racines d-iemes de 1t'. Rappelons que l'on designe par B;;'d l'ensemble des elements de B+ qui ont meme longueur que leur image dans W, et que de tels elements sont dits niduits. La restriction a B;;'d de la surjection naturelle B+ -+ West une bijection. On designe par w f-+ w son inverse.

3.9. Proposition. Soit w E W. Les proprietes suivantes sont equivaLentes : (i) On a w d = 1 et pour tout entier m tel que m S; d/2, on a £(w m ) = 2Nm/d. (ii) west une bonne racine d-ieme de 1t'. Demonstration. 11 est clair que (ii) implique (i). Demontrons la reciproque. W

1. Supposons d'abord que d est pair. Il suffit alars de verifier que = Wo, ce qui resulte immediatement de l'egalite £(w d / 2 ) = N.

d 2 /

2. Supposons maintenant que d est impair. Soit d' := (d - 1)/2. On definit t et u par la condition Wo = tw d' = w d' u. Puisque £( w d') = 2Nd'/d, on a £(t) = £(u) = £(w)/2. Comme wd'utw d' = w5 = 1t', il suffit de verifier que ut = w. Pour cela, puisque £(u) + £(t) = £(w), il suffit de verifier que ut = w. Mais de la definition de t et u on deduit t = wow- d' et u = w- d' Wo, d'ou, ut = w- d' w- d' = w. • La notion suivante d"'element de Springer" fournit des exemples a priori de bonnes raeines de 1t'. On peut se reporter a l'annexe 1 pour trouver une liste de bonnes racines d-iemes de 1t' pour tous les nombres reguliers d et tous les groupes de Coxeter finis irreductibles. Soit m > d/2, et d impair. On pose d' := (d - 1)/2. Comme dans la demonstration de 3.9, on definit t et u par la condition Wo = w d' U = tw d', et on voit que ut = w. Soit alors B' Ie sous-groupe de Borel defini par la condition B.F d' ~ B' ~ B.Fd'+I. w

On a B 4 B', et aussi B' puisque twm-d'-l -# woo

t

m-d'-l

w -+

B.F m , d'ou une contradiction

La proposition suivante est immediate it demontrer.

•

113

Elements reguliers et varieUs de Deligne-Lusztig 5.3. Proposition. Soit w une racine d-ieme de

'IT.

La variete

X?)

s'identifie Ii une sous-varieU de X~Fd) par Ie morphisme suivant (qui est invariant par l'action Ii gauche de G F ) : Ii (Bo,BI, ... ,Bn ) E X?), on associe l'element (B o ,... ,B n _ 1 ,Bo.F, ... ,Bn_1.F, ... ,Bo.Fd-1 ,... ,B n _ 1 .F d- 1 ,B n .F d- 1 )

de la varieU X~Fd) . On rappelle (cf. §2 ci-dessus) que Ie monolde B+ opere sur

X~Fd).

Pour x E B+, on designe par D~Fd) (ou plus simplement par D x ) cette operation. On designe par B1 le sous-monoi·de de B+ qui stabilise la sous.'t' vane e

X(F) w

,

. l.e.,

W

Nous allons mettre en evidence des elements remarquables du mono·ide B1 . W

5.4. Lemme. Soit x un diviseur (Ii gauche) de w (ainsi Ie conjugue X W de w par x dans B est en fait un element de B+ conjugue Ii w par permutations circulaires). Alors D~Fd) envoie

X?)

dans X~;).

•

Demonstration. C'est evident (voir aussi 2.1 ci-dessus). Il est clair que 5.5. la restriction de

D?d)

Ii

X?)

est egale Ii F.

On en deduit la construction suivante d'elements de B1w

•

5.6. Proposition. (1) Soient (Xl, X2, ... ,X n ) et (YI, Y2, ... ,Yn) deux suites d'elements de B+ telles que pour tout j (0::; j ::; n - 1), on a YjXj = Xj+IYj+l, et YnXn = XIYI = w. Posons x := XIX2 ... Xn et Y := YIY2 ... Yn. Alors

(a) x et Y centralisent w, i.e., wx = xw et wY = yw, (b) xy = yx = W n , (c) x et Y appartiennent Ii B1w • (2) On note B~ l'ensemble des elements x comme ci-dessus. note B w Ie sous-groupe de B engendre par B~. Alors

(a) B~ est un sous-monoide de B1 , (b) Bw = {w-nx I (n E N)(x E B£)}.

On

114

M. Broue et J. Michel

Exemple (G. Lusztig). L'exemple ci-dessous a ete fourni aux auteurs

en

~:::~d:r::s1:~::::4a~o~ri~ilne de l,a ~r~::::~: :::::d::::~ieme So

.93

82

de 1r definie par w := SlS0S2S0S3S0. (Remarquons que, pour mieux faire apparaitre la trialite qui se reflete dans les decompositions des elements que nous allons considerer, nous n'utilisons pas ici les memes notations que dans nos tables pour Ie diagramme de Coxeter de D 4 ). Nous allons definir trois elements s', 1', u' E B~ par la methode decrite dans 5.6 ci-dessus. 1. Definition de u'.

On pose

et

•

Xl:= S2, Yl := SlS0S2S3S0,

•

X2:= S3, Y2 := SlS0S3S2S0,

d'ou. d'ou.

YIXI = SlS0S3S0S2S0,

d'ou. d'ou. d'ou.

YIXI = SOSlS0S2S0S3,

Y2X2 =

w,

u' := u := XIX2 = S2S3 .

2. Definition de t'.

On pose •

Xl:= SlS0S2S0S3, Yl := So,

•

X2:= Sl, Y2 := SOSlS2S0S3,

•

X3:= S2, Y3 := SOS2S1S0S3,

•

X4:= So, Y4 := Sl S 0 S 2 S 0 S 3,

d'ou. l' t ' W -1

Y2 X 2 = SOS2S0S1S0S3, Y3X3 = SOSlS0S2S0S3,

:= XIX2X3X4 = SlS0S2S0S3S1S2S0 = SOlSlS2S0W. -1

= So

On pose t

:=

SlS2S0'

3. Definition de s'.

On pose •

X1:= Sl, Y1 := SOS2S0S3S0,

•

X2:= So, Y2 := S2S0S3S0S1,

•

X3:= S3, Y3 := S2S0S3S1S0,

•

X4:= Sl, Y 4 := S2S0S1S3S0,

•

X5:= S2S0S3S0S1, Y5 := So,

•

X6:= SOS2 S 0 S 3 S 0, Y6 := Sl,

d'ou. d'ou. d'ou. d'ou. d'ou.

YIXI = SOS2S0S3S0S1, Y2X2 = S2S0S3S0S1S0, Y3X3 = S2S0S1S0S3S0, Y 4 X 4 = S2S0S3S0S1 SO, Y5X5 = SOS2 S 0 S 3 S 0 S 1,

d'ou. S' X1X2X3X4X5X6 SlS0S3S1S2S0S3S0S1S0S2S0S3S0 -1 -1 2 SlS0S3S1S0 Sl W. n pose S := S W -2 = SlS0S3S1S0-1 Sl-1 .

a

'

On peut verifier que s't'u' = w 4 = 1r et donc que stu = W et Ie systeme S(w) := {s, t, u} satisfait aux relations definies par Ie diagramme

80a~ (cf. [BrMa] ou [BrMaRo] pour la definition de ce

diagramme).

115

Elements reguliers et varietCs de Deligne-Lusztig

B. Conjectures Si west une racine d-ieme de 1r, on sait (cf. 3.14) que son image w dans West un element d-regulier. Nous utilisons les notations introduites dans Ie paragraphe 3 ci-dessus. En particulier, on note W( w) Ie centralisateur de w, groupe engendre par des pseudo-reflexions dans son action sur l'espace V(w). Si D(w) est un diagramme "it la Coxeter" pour W(w) (on peut se reporter it l'annexe 2 pour une liste complete des diagrammes associes it tous les centralisateurs d'elements reguliers des groupes de Weyl), on note S( w) l'ensemble des generateurs representes par les sommets de D( w), on note S( w) l'ensemble de leurs antecedents dans Ie groupe des tresses generalise B( w) (cr. §3), et on designe par B( w)+ Ie monoide engendre par S( w ).

5.7. Conjectures. Soit w une racine d-ieme de

1r.

Soit

n=2N/d

Rf(Xw , Qi):=

ffi

Hn(X w , Qi)

n=O

Ie module gradue de la cohomologie R-adique de X w module gradue.

,

vu comme QiGF-

(1) Les composants de Rf(Xw,Qi) sont disjoints comme G F modules:

_

(2) On a B w = B(w). (3) II existe une presentation de W(w) par un diagramme "Ii la Coxeter" tel que les operations definies par les elements du monoide correspondant B( w)+ engendrent 1'algebre des QiGFendomorphismes Rf(Xw , iji) comme decrit ci-dessous. Pour Z E B(w)+, on note D z l'operation de z sur Rf(Xw,Qi)' existe une specialisation de u de la forme

n

-2i1r/d )n. j { u.,j t-+ e 2i1rj/e. ( e x'

}

(.Esab)(O:::;j:::;e.-l),

(symbolisee par u t-+ (d,x)) OU • x est une indeterminee, • n.,j est un nombre rationnel positif, qui ne depend que de W et w, telle que

M. Broue et J. Michel

116

(a) I'application Tz ...... D z (z E B( w)+ ) definit un isomorphisme entre l'algebre de Heeke specialisee via la specialisation definie ci-dessus, notee Q/H(d,q)(W( w)), et l'algebre commutante En~,GF(Rr(Xw, Qt)) ,. (b) pour tout h E Q/H(d,q)(W(w)), on a

n

ou t(d,q) est la specialisation correspondante de la forme t u (definie en 4.15). (4) Pour 0 S; n S; 2Njd, Ie (QtGF,Q/H(d,q)(W(w)))-bimodule Hn(Xw,Qt) se decompose en la somme suivante de caraeteres irreduetibles distinets :

'If'

ffi

Hn(Xw, Qt) =

&;

'P(d,q) ,

If'EIrr(W(w))(n)

ou

{Irr(W(w))(n)}o~n:9N/d

est une partition de l'ensemble

Irr( W (w )) des caraeteres irreduetibles de W (w) qui ne depend

que de W et de w (et pas de q). Remarques. (1) II resulte des travaux de Lusztig (cf. [Lu3]) que les caracteres unipotents de G F sont "generiques", i. e., sont parametres par un ensemble Uch(G) qui ne depend que du type de (G,F) (le "groupe generique" G, selon la terminologie de [BrMaMi]). II est naturel de conjecturer que, en ce sens, l'application

Irr(W( w)) {

-+

'P ......

Uch(G) ,

'If"

est generique, i.e., ne depend pas de q. La conjecture (5) ci-dessus fournit alors une version generique IHl n (w) des groupes de cohomologie etale de la variete X w (it defaut d'une version generique de cette variete elle-meme ... ), par une formule independante de q (et, bien entendu, de

£) : n IHl (w) :=

L

'If' 0 'P(d,x)·

If'EIrr(W(w))(n)

(2) II resulte des conjectures (1) et (3) ci-dessus que Ie caractere de

n

Elements reguliers et varietCs de Deligne-Lusztig

117

(vu comme (QfGF,Q/H(d,q)(W(w)))-bimodule virtuel) se decompose comme suit en une somme directe de caracteres irreductibles de multiplicite 1 :

EB

tr(- I : l ) - l t Hn (Xw ,Qf)) =

CG FC;'qm(2N-(a-,+A-,))/d,(g).

,EUch(GF) On voit que n'interviennent dans cette formule que les caracteres unipotents tels que RW>GF I: o. On sait (cf. [BrMaMi], theorem 3.2) qu'il existe au moins une bijection 'P ...... ,

GF I: 0, et une famille de signes (E'f' )'f'EIrr(W( w)), tels

j (vx)) ,

M. Broue et J. Michel

126

d'ou y.i(vx) E B~d' On applique ¢l-j ala derniere relation, et on I pose x' = ,pi-lx, y' = i- y . On obtient ainsi x'y' = wet y,.vi x ' = (y'¢x,)j¢-j E

B~d'

On voit donc que l'element w'¢ := y'¢x' est conjugue de w¢ par permutations circulaires et verifie bien la conclusion du lemme 6.9. •

c.

Algebres de Heeke "avec automorphisme"

Les resultats du §4, E se generalisent en remplal;ant les racines de 1r par les ¢-racines, comme nous allons l'indiquer. Le contexte ici est celui du §4, plus la donnee d'un automorphisme ¢ de V, d'ordre fini 8, qui stabilise (dans son operation par conjugaison) l'ensemble des generateurs de W correspondant a l'ensemble N(D) des sommets du diagramme D. L'automorphisme ¢ opere donc sur B+, Bet sur l'algebre de Hecke generique 1i u . II n'est pas difficile de generaliser 4.14 comme suit. On note Irr(W ¢) un ensemble forme d'une extension reelle a W), correspondants aux valeurs propres respectives CI, C2, ... ,Cr. La famille des paires {(d l , CI), ... ,( dr, cr)} ne depend que de ((V, W), ¢» (cf. par exemple [Sp], 6.1). Les dj sont les degres, et les Cj

sont les facteurs correspondants de la paire ((V, W),¢).

Nom

Diagramme

Automorphisme

2A r

0-0···0

Si +-+ 8 r +l-i

Degres Fa----Q)

d r 2d )

,

0

(stu )5

..... ">-

u

G 28 G30

F4

H4

">

b 0,10,18,28

s:60

Q(0)

2

~

(stuV)15

5 does not divide the order of (Z(G*)/zo(G*))F. (For more details, see the introduction to Section 3.) Denote by e the order of q (mod i). Theorem 3.3 shows that the i-blocks are parametrized by the GF-conjugacy classes of certain e-cuspidal pairs (L, (). Those pairs generalize the unipotent case described by [BMM]. We show that the defect groups write as a semidirect product D = Z.S where Z is the unique maximal abelian normal subgroup of D, Z = Z(Cc;(Z))[, and S is a Sylow i-subgroup of a finite reflection group acting on Z. The "local" method to determine

142

M. Cabanes and M. Enguehard

the blocks and their defect groups consists essentially in including a given i-block of G F in a "maximal subpair" (see [AB]). Thus we combine (and partly develop) some techniques belonging to block theory (see Proposition 3.2) and to the theory of reductive groups over a finite field (Sections 1 and 2). We also draw heavily on the preparatory material of [CE.l], e.g. the decomposition G = Ga.G b . Nevertheless our proof applies to reductive groups of any type.

Notations. When H is an algebraic group, HO denotes its identity component. We use the notations CH(X) and ZO(H) for the identity components of the centralizer of X in H and of the center of H. If H is reductive and connected, then Had = H/Z(H) is its adjoint group. If K is an algebraically closed field of characteristic zero, and G is a finite group, then Irr(G) is the set of irreducible characters of the group algebra K[GJ. 1. Reductive Groups over a Finite Field

1.1. The groups

The groups GF we consider in this paper are defined as follows. Let be the algebraic closure of a finite field q. Let G be a connected reductive affine algebraic group over and let F: G ----> G be an endomorphism which is a Frobenius morphism for an q-rational structure of G. We abbreviate this by saying that (G, F) is "a connected reductive group defined over q". The group of points of G on q is denoted by GF. Such a group (G, F) may be defined by a root datum with Frobenius endomorphism F, say ((X, R, Y, RV ), F). Here X = X(T) and Y = Y (T) are free -modules in duality over , and are identified respectively with the group of characters and the group of one-parameter subgroups of some maximal F-stable torus T of G. Then R ~ X (resp. R V ~ Y) is the set of roots (resp. coroots) of G relative to T. The sets Rand R V are in bijection by (r f---> rV). Two root data with Frobenius endomorphism ((X, R, Y, RV), F) and ((X*, R*, Y*, (R*t), F*) are said to be in duality if ((X*,R*,Y*, (R*)V),F*) is isomorphic to ((Y, RV , X, R), F). Then the corresponding reductive groups (G, F) and (G*, F*) are said to be in duality and the set of GF-conjugacy classes of maximal tori of (G, F) is in bijection with the analogous set in (G*, F*) in such a way that tori T and T* in corresponding classes may be used to define the given duality between (G, F) and (G*, F*) (see rCa, Chapter 4] for more details and properties of reductive groups in duality). In this situation we usually identify

Blocks of Reductive Groups

143

((X(T*), R*, Y(T*), (R*t), F*) with ((Y(T), RV , X(T), R), F). Furthermore an F-stable Levi subgroup L of G that contains T defines a Levi subgroup L* of G *, such that (L, F) and (L*, F) are in duality : T* is a maximal torus of L * and the sets of roots of L relative to T and of L * relative to T* are in bijection by r f-+ r v. One obtains finally a bijection between the set of G F -conjugacy classes of F-stable Levi subgroups of G and the set similarly defined in G*, see [DM.2, p.113]. Let us recall the following notations [CE.1, 1.1]. Notation 1.1. If (G, F) is a connected reductive group defined over then let ,(G, F) be the set of prime integers £ satisfying all of the following conditions: £ does not divide q, £ is good for G, £ i=- 2, £ does not divide I(Z(G)/zo(G))FI· q,

Let f(G,F) = (')'(G,F)n,(G*,F))\{3} when Gad F has a component isomorphic to some 3D 4 (qm) ; let f(G,F) = ,(G,F) n,(G*,F) otherwise. Concerning the condition that £ does not divide I(Z(G)/ZO(G))FI, we mention the following. Let A be a finite abelian group and F an endomorphism of A. Then £ does not divide IAFI if and only if, for every section S = B / B 1 with F -stable subgroups Band B 1 , £ does not divide ISFI (assume S = SF and use IBI = IBFI.I[B, PII). Lemma 1.2. Assume G is a central product of closed connected Fstable subgroups Hand J. If ZO (G) ~ H, then f( G, F) ~ f(H, F) . Proof. A prime is good for a central product if and only if it is good for every component. First, Z(H)/ZO(H) is a section of Z(G)/ZO(G) since Z(H) ~ Z(G) and ZO(H) ~ ZO(G). Concerning dual groups, the quotient Z(G*)/ZO(G*) is isomorphic, with F-action, to the kernel of the simply connected covering (G sc -+ [G, G]) rCa, 4.5.8]. One has [G, G] = [H, H].[J, J] and G sc = H sc x J sc ' hence Z(G*)/ZO(G*) ~ Z(H*)/ZO(H*) x Z(J*)/ZO(J*) and therefore Z(H*)/ZO(H*) is an F-stable section of Z(G*)/ZO(G*). The lemma follows .

•

1.2. Polynomial orders We use the notion of polynomial order of a connected reductive group (G,F) defined over q [BM], and denote it by P(G,F)' Let now E be a non-empty set of positive integers. We call ¢Esubgroups of G the F-stable tori S such that P(S,F) is a product of

M. Cabanes and M. Enguehard

powers of the cyclotomic polynomials ¢m for m E E. Their centralizers in G are called E-split Levi subgroups of G. Note that, if S is an Fstable torus of G, then S contains a unique maximal ¢E-subgroup; we denote it by ScPE' see [BM, 3.1.(2)]. The case when E is a singleton has remarkable properties [BM, 3.4]. Proposition 1.3. Let (G, F) be a connected reductive group defined over q, and let (G*, F) be dual to (G, F). (i) The bijection between the sets of G F -conjugacy classes of F -stable Levi subgroups of G and of (G*)F_conjugacy classes of F-stable dual Levi subgroups of G* induces a bijection between the sets of conjugacy classes of E-split Levi subgroups of G and G* for each non-empty set E of positive integers. (ii) Denote Eq,t = {d; £ divides ¢d(q)}. Let £ E r(G,F). IfS is a ¢Eq,l-subgroup ofG, then CG(S) = Cb(Sf) and CG(S)F = CG(Sf)F. Conversely, if L is an F -stable Levi subgroup of G such that L = Cb(Z(L)f), then L is Eq,t-split.

Proof. (i) Just replace e by E in the proof of [CE.I, 1.41. (ii) The proof of [CE.I, 2.2.(ii)] applies.

•

1.3. On type A The induction method used in Section 4 essentially reduces the study of blocks to the case of type A. We need some ad hoc results in this situation. We freely use the notions of "rationally irreducible components", corresponding to the orbits of F on the irreducible components of a reductive group G with F'robenius endomorphism F, and associated "rational type", see [CE.I, § 1.1]. Definition 1.4. If (G,F) is of rational type xw(Anw,€wqmw), then a maximal F-stable torus T of G of polynomial order p(ZO(G),F)IIw(x mw - €w)n w will be called a diagonal torus of G. All diagonal tori are GF-conjugate. If a Levi subgroup L contains a diagonal torus T of G, then T is a diagonal torus of L. An F-stable maximal torus of polynomial order p(ZO(G),F)IIw ((xmw(nw+l) _€)nw+l))/(x mw _ €w)) will be called a Coxeter torus of G. All Coxeter tori of G are G F _ conjugate. Notation 1.5. Let (G, F) be a connected reductive group defined over q' Let £ be a prime not dividing q and good for G. As in fCE.1, 2.3} define G a as the central product of (G) and of the rationally irreducible components oJ[G, G] whose rational type is of the form (An, €qm) with

zo

Blocks of Reductive Groups

145

i dividing qm - E:. Let G b be the central product of the components of [G, G] not included in Ga. Then G = Ga.G b (central product). Moreover (i) If H is an F -stable Levi subgroup of G a , then H = H a . (ii) 1fT is a maximal F-stable torus ofG such that ~ ZO(G), then T is a Coxeter torus of TG a .

Tf

Proofs. (i) This is clear from the definition of Ga. (ii) One may clearly assume G = G a is of irreducible rational type (An, E:qm) with E:qm == 1 (mod i). The polynomial order of a maximal torus is of the form P{T,F) = p{ZO{G),F)(X m - E:)-lIIi(xn,m - E: n,). If ~ ZO(G), then (T/zo(G))F is t. This readily implies that in P{T /ZO{G),F) = (p{ZO{G),F») -1 P{T,F) = (X m - E: )-1 IIi(xn,m - E: n,) there • is just one ni.

Tf

We need an elementary property of pairs of Levi subgroups of groups of type A sharing a maximal torus which is diagonal in one and Coxeter in the other. Proposition 1.6. Let (G, F) be a connected reductive group defined Over q of irreducible rational type (An-I, r) where r = f-qb, f- E {-I, I}. Let E be a non-empty set of positive integers. Denote ¢E(X) = IlmEE¢m(X). Let C be an F-stable Levi subgroup of G of rational type IIk(Ank_l,rak) (nk 2: 1, L.knkak = n) and let T be a diagonal torus of C. Let K = CG (T )F. It is shown in [FS.2, Appendix] that for some classical groups G, the equality (2.l.R) holds for all M, L, p, and A. The bijections A f-+ X~ A defined by Lusztig satisfy the projection of (2.l.R) on the spaces ~f

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uniform functions, that is (2.l.R) for M = T a torus, I-L = 1 and any A [1.2, 5.1]. In the next proposition, we draw several consequences. Proposition 2.3. Let (G,F) and s E (G*)~ such that CG.(s)F = CG·(s)F. Let L be some F-stable Levi subgroup ofG, of dual L* with s E L*. Let A f-+ X~A be a bijection from £(Cr..(s)F,l) to £(LF,s) such that the equaliti~s (2.l.R) hold whenever M is a torus. Let A E £(Cr.. (s)F, 1).

(a) Let p be the characteristic of q and let L = G. Then X~\ (1) = A(l).I(G*)F: CG.(s)Flpl. ' F (b) Let L = G. If t E Z(G*)F and i E Irr(G ) is defined by t and the . G' G dualzty, then Xst,A = tXs,A' (c) IfCG.(s) ~ L*, then X~A = cGcLRrx~A' (d) Let x E NG(L)F and y E Nc~.(s)(L*)F having the same image in NW(G)(W(L))/W(L). Then XX~,A = X~,YA' Proof. The equality of (a) stems from the expression of the regular representation in terms of Deligne-Lusztig characters, see for instance [DM.2, 13.24]. As for (b), it is clear that the condition (2.l.R) for L = G and M a torus implies that X~ A and Ex? A have the same inner product with all Deligne-Lusztig char'acters R!;8. But this implies that they are equal since, by [1.2, 5J], [1.1, 4.23] applies to any series £(GF,s) such that CG.(s)F = CG.(s)F. The assertion (c) is proved by a similar argument. In (d), assuming T ~ L ~ G and T* ~ L* ~ G* in duality, one identifies W(G) := W(G, T) with W(G*, T*) and W(L) := W(L, T) with W(L*, T*). Note that, by conjugacy of maximal tori in a reductive group, NG(L)/L is isomorphic in a natural way with NW(G)(W(L))/W(L). This gives sense to the claimed equality, The characters XX~A and X~YA are identified by their scalar products with Deligne-Lusztig charact~rs. Then with usual notations (see for instance [DM.2, 13.13]) one has XR~(s) = R~s(s) for any maximal F-stable torus S of Cr.. (s). Now (XX~,A' R~(S))LF = (X~,A' R~-lS(S))LF = Rc~.(s)l) (Y A'Rc~.(s)l) (L R L ()) • (A'y-1s c~.(s)F = s c~.(s)F = XS,YA' S S LF. 2.2. Generalized Harish-Chandra theory Definition 2.4. Fix d :::: 1 an integer. Let (G, F) be a connected reductive group defined over a finite field. An irreducible character X of G F is said to be d-cuspidal if and only if for every proper d-split Levi subgroup L of G, one has *Rr( = o. If L is a d-split Levi subgroup of

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149

G and ( E Irr(L F ) is d-cuspidal, then (L, () is called a d-cuspidal pair of (G,F). When d = 1, the above is just the usual concept of cuspidality in "Harish-Chandra theory" for reductive groups over a finite field, see for instance [DM.2, chapter 6]. Notation 2.5. Let (G, F) be a connected reductive group defined over q' When L 1 , L 2 are F-stable Levi subgroups of G and Xi E Irr(LQ, one writes (L 1 , xd ~ (L 2, X2) if and only if L 1 ~ L2 and (X2, RL~Xl)L2F =f. O. The equality (2.1.R) on (L, M, A, ,) implies that (M, X~!,) (L, X~..) is equivalent to (C M, (s), Il) ~ (Cr., (s), A). The following relates d-cuspidality with Jordan decomposition.


X~,A (see Definition 2.1) satisfying (2.1.R) for any torus M. Let, E £(CG,(s)F, 1) and X = X~" If X is d-cuspidal, then (i) , is d-cuspidal and (ii) ZO(C G, (S))¢d = ZO(G*)¢d' Assume now that (2.1.R) holds for any d-split Levi subgroup M, L = G and any irreducible characters A, Il. Then (i) and (ii) imply that X is d-cuspidal. Proof. If ZO(C G, (S))¢d =f. ZO(G*)¢d' then CG' (ZO(C G, (s))¢J is a proper d-split Levi subgroup of G*. For a dual Levi subgroup L of G, one has *R~X =f. 0 by Proposition 2.3.(c), hence X is not d-cuspidal. If X is d-cuspidal, then one has *IV; X = 0 whenever T is an Fstable maximal torus of G such that T¢d S?; Z(G). By (2.1.R), this implies that *Ri~'(S), = 0 for all F-stable maximal torus of CG,(s) such that (T*)¢d S?; Z(G*). Hence, is d-cuspidal. Moreover there exists a torus T* such that *Ri~'(S), =f. 0, so that (T*)¢d = ZO(G*)¢d and that implies ZO(C G, (S))¢d = ZO(G*)¢d' If X is not d-cuspidal, then there exists a proper d-split Levi subgroup M of G, in duality with M* containing s, and some Il E £(C M,(s)F,l), such that (M,X~!,) ~ (G,X). By (2.1.R) we get

eo

(s)

that *Ret', (s)' =f. O. On the other hand, the subgroups of the form C M' (s) = M* n CG' (s) for d-split M* with s E M* are precisely the d-split Levi subgroups of (C G' (s), F). So either , is not d-cuspidal, or there is a proper d-split M* in G* such that C M' (s) = C G' (s).

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But the latter may happen only if ZO(CG.(s)) contains a cPd-subgroup which is non central in G *. • The following is standard in the case d = l. Proposition 2.7. Let (G, F) be a connected reductive group defined over q . Letd:::: 1. Lets E (G*)F semi-simple and such that CG.(s)F = CG • (s Assume that (2.1.R) holds for any pair (L, M) of d-split Levi subgroups of G and any (A, p,). We then have the following: (i) If X E £(G F , s), then there exists a d-cuspidal pair (L, () of (G, F) such that (L,() ~ (G,X). Moreover (L,() is unique up to G F _ conjugacy. (ii) If(G,x), (L,() are as in (i) above, then

t.

*Rrx = (*RrX,()LF

L

9(.

9EN GF (L)/N GF (L,()

Proof. Let, E £(C G.(s)F,l) such that X = X?" and let (L*s,A) be a d-cuspidal pair of (CG.(s),F) such that (L*s,A) ~ (CG.(s),,). One has L*s = Cc~.(s)(ZO(L*s)J and L: is uniquely defined modulo CG.(s)F_conjugacy. Let T* be a maximal F-stable torus in L:, T a dual torus in G and Lad-split Levi subgroup of G in duality around (T, T*) with C G• (ZO(L:)J. Then ( := X~,A is defined and, by Proposition 2.6 and (2.l.R), (L, () is a d-cuspidal pair such that (L,() ~ (G,X). By our hypothesis and [BMM, 3.A], (V,(/) ~ (G,X) is equivalent to the existence of a dual F-stable Levi subgroup V* in G* such that s E V*, (I = X~/A' (AI E £(CI,/.(s)F,l)) and, up to rational conjugacy, (L:,A) ~ (CI,/.(S),A I). This implies that ZO(CI,/.(S))d ~ ZO(L:)d up to G*F-conjugacy. By Proposition 2.6, ZO(L:)d = ZO(L*)d' If (V,(/) is also a d-cuspidal pair of (G,F), then ZO(V*)d ~ ZO(L*)d up to G*F-conjugacy so that L ~ V up to G F -conjugacy. If (L, (i) are d-cuspidal pairs such that (i X~ A and (L,(i) ~ (G,X) (i = 1,2), then (L:,Ai) are d-cuspid~l' pairs such that (L:, Ai) ~ (C G• (s), I)' Therefore Al and A2 are conjugate under Nc~.(s)F(L:). By conjugacy of maximal tori in L*, Al and A2 are conjugate under some y E (N c~. (s)F (T*) n N(L*)) and PT· (y) = w E W(C G• (s), T*t. Since L: is dsplit in CG.(s), one has Nc~.(s)F(L;) = Nc~.(s)F(ZO(L;)J ~ N(G.)F(ZO(L:)J = N(G.)F(L*). With notations of Proposition 2.3.(d), w E NW(G)(W(L))/W(L). By the equality in Proposi-

Blocks of Reductive Groups

151

tion 2.3.(d), (I and (2 are conjugate under NGF(L). Thus one has (Rr(l, X)GF = (Rr(2, X)GF. • 3. Blocks of Reductive Groups over a Finite Field We state below our main theorems on blocks of reductive groups over a finite field. The hypotheses of Theorem 3.3 and Theorem 3.5 bring two main restrictions. The first restriction is on the series [(G F , s) for which our theorems are stated. This stems from the hypothesis (2.1.R) in which we assume commutation between Lusztig's map Rr and a "Jordan decomposition" of [(G F, s), this only when CGo (s)F = CGo (s)F. When CGo (s) is not connected, one may expect that some analogue of the Jordan decomposition satisfies similar properties (see [1.2; DM.1]), and that the method described below can be adapted. Also, our results are about i-blocks of G F for i E r(G, F) (see Notation 1.1). Bad primes actually provide exceptions, even for unipotent blocks : one has numerous examples of cuspidal unipotent characters of Gad F that are not of i-defect zero for a bad prime i di viding q - 1 (see the tables of degrees of unipotent characters in [Ca; 1.1]). The case treated in [CE.2] is one where the series does not necessarily satisfy CGo (s)F = CGo (s)F and i is good but not in r(G, F). 3.1. Some general results on blocks Let G be a finite group, let i be a prime number and 0 a local complete finite extension of (. Let K denote its fraction field. Assume that K is algebraically closed. We consider i-blocks as primitive idempotents in the center Z(O[G]) ofthe group algebra O[G]. Let k = OjJ(O) and a f-+ a the reduction map modulo J(O) from O[G] to k[G]. We recall the Brauer morphism associated with an i-subgroup Q of G (see [AB, 2.4]) : BrQ : (O[G])Q --> k[Cc(Q)], L9EC >'g9

f-+

L9ECc(Q) >'g9·

The definitions and main properties of subpairs (Q, b), inclusion of subpairs, defect groups, self-centralizing subpairs and their canonical character are taken from [AB], [B], [Bd]. The decomposition ofthe unit 1 E Z(O[G]) as the sum ofthe blocks induces an orthogonal decomposition of the space of central functions CF(G, K) with respect to the usual inner product ( , )c. For b a block of G and f E CF(G, K), one denotes by b.f the corresponding projection of f. One also has the partition Irr(G) = UbIrr(b). If X E Irr(G), then one denotes by bc(X) the i-block of G such that X E

152

M. Cabanes and M. Enguehard

Irr(bc(x))· The following statement is well-known, see [B, 6], [Br.l, 1.C].

Proposition 3.1. If (P, bp) is a subpair in G, then bp is a block of Nc(P,bp). Let (Q,bQ) be a maximal bp-subpair in Nc(P,bp). Then (P,bp) 2, then the required Jordan subgroup J j

=

{ea Ju I a

E IFp , u E

c::=

z;m

has inverse

W} c::= p~+2m

in go = SLq(C) (here and below, go = Inn (£)). Moreover, Cg(J) = J and G = Ng(J) c::= J. (Sp(W) ·Z2)' The additional factor Z2 is generated by the outer automorphism T : X f-t _t X of £. Now we return to the case p = 2. For a fixed square root i of -1 and a E IF2 we set 'a a= Z = i a = 1.

{I

0,

In particular, i a • i b = (_l) ab i a+b. The Jordan subgroup J c::= z~m has inverse image m

j = {±ial+bIJu I u = l:(ajej

+ bjlj ) E W}

c::=

2:+ 2m

j=l

go SLq(C). Furthermore, Cg(J) = J x Z2 and G = Ng(J) (J x Z2) . Sp(W). Again, the additional factor Z2 is generated by the

in

169

Splitting Fields for Jordan Subgroups outer automorphism T : X f-t with the following basis of 1:-: Iu

=

_I

X of 1:-. It is more convenient to work

(_l)B(u,U)i B(u,u)+IJu , u E W\{O}.

In what follows we shall often write W# instead of W\{O}. It can be verified (see e.g. §1.3 of [Ko~I']) that each automorphism 9 E N of I:- acts rationally in the basis {Iu I u E W#}, namely,

where the maps h: W ~ lF 2 and '{J E Sp(W) satisfy the condition

h(u + v)

=

h(u)

+ h(v) + C(u, v) + C('{J(u), '{J(v)).

Here we are denoting, for u, v E W

C(u, v)

1

= B(u, u)B(v, v) + B(v, u) + (u, v)(B(u, u) + B(v, v)).

The outer automorphism T also acts rationally in the basis {Iu I u E W#}, namely, T(Iu) = (_l)B(u,u)+IIu' For x E End (I:-) and M an x-invariant subspace of 1:-, we denote by Spec (M, x) the set of eigenvalues of x in its action on M. Observe that, for any u E W, Spec (I:-, I u ) ~ Qi. (2) Furthermore, from (1) it follows that if (u,v) =0, if (u,v) = 1, where

D(u, v) = B(u, u)B(v, v)

+ B(u + v, u + v) . {I + B(u, u) + B(v, v)} + 1.

Our computations show that the Q-subspace (Iu I u E W#)iQ is a Q-form for G. Consequently, Q is the unique minimal splitting field for G.

2.2 I:- is of type C2m-l or D 2m-l, m ~ 3. Let W, (','), I u be as in §2.1. One can embed I:- in the Lie algebra {, of type A 2m-l,

in such a way that

I:-

= (Iu I u E W,

q(u)

= 1)(:

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A. M. Cohen and P. H. Tiep

(see §2.1 of [KoT]). Here q is a quadratic form on W associated with (', '), i.e., q(u) + q(v) + q(u + v) = (u, v) for any u, v E W. The corresponding Jordan subgroup J remains the same as in §2.1: J c::: z~m. Furthermore, q is of type - if £, is of type C 2m-l, and + if £, is of type D 2m-l. Finally,

NAut(.C)(J) = Nrnn(l)(J) n Aut (£,) c::: J. O(q). (Remark that if £, is of type D 4 , then its triality outer automorphisms do not normalize the Jordan subgroup J of order 26 .) Therefore, the Q-subspace (I" I u E W, q(u) = l)iQl is a Q-form, and Q is the unique minimal splitting field for N Aut (C) (J).

2.3 £, is of type En (n ~ 2) or D n + 1 (n ~ 4), and J = z~n. Here we realize £, as the Lie algebra of skew-symmetric matrices of order m, where m = 2n + 1 or m = 2n + 2 according as £, is of type En or D n + 1 . If E ij = (bi,kbj,lh~k,l~m, 1 :S i,j :S m, are the "standard matrix basis", and E(i,j) = E ij - E ji , then £,

= (E(i,j) 11 :S i,j:S m)c.

As shown in §2.2 of [KoT], each automorphism 9 E Ng(J) acts rationally in the basis {E(i,j) 11 :S i < j :S m} of £'. Namely, there exist maps f: {l, 2, ... , m} -+ F 2 and

..Ku for any u E W. Since the elements K u, u E W#, are linearly independent over C, one has V = (>..Ku I u E W#)JK. Furthermore, [V, V] ~ V and [>..Ku, >"KvJ = LXEW# >..Au,v(x), >..Kx , where Au,v(x) is defined in (7). Consequently, JK contains all the coefficients >..Au,v(x), u,v,x E W#. In particular, JK 3 >"Au,-u(x) = >"'l9(E 4 (u,x) _ E- 4 (u,x))

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A. M. Cohen and P. H. Tiep

for all u, x =I O. Choosing u such that (u, x) = 1/4 and (u, x) = 1/2, respectively, one finds >'19(c: - c 1 ), >.19(c: 2 - c 2 ) E II{. We arrive at the conclusion that OC contains

i.e., OC contains

1Ko,

as claimed.

All the above computations are done with respect to a specific embedding of the finite group G = (SP2m(P) . Z2) in PSLpm(C) . Z2. However, in case m = 1 the following statement holds:

z;m .

Proposition 4. Let p be an odd prime and assume that G = Z2. SL 2 (p) has a faithful action on a complex Lie algebra I: of dimension p~ - 1 with nontrivial multiplication. Then I: is a simple Lie algebra of type A p - 1 and IKo = Q( cos(21r /p)) is the unique minimal splitting field for G. Proof. 1) First we show that I: ~ A p - 1 • From the faithfulness of the action of G on I: and the equation dim I: = p2 - 1 it follows that this action is absolutely irreducible. But 0 =I [1:,1:] is a G-submodule of 1:, so [1:,1:] = 1:. In particular, the Killing form ~ of I: is nonzero. As the kernel of ~ is also a G-submodule of 1:, the form ~ is in fact nondegenerate. Hence, I: is semisimple. Decompose I: into a sum of simple components: I: = 1: 1 EB··· EB I: m . As the action of G on I: is irreducible, G permutes the m components I: i transitively. Observe that dim I: i 2: 3, so m ::; (p2 - 1)/3. In particular, the subgroup J ~ Z; of G leaves each component 1:; fixed. If p = 3, then obviously I: ~ A 2 . Assume that p 2: 5. Observe then that the simple Lie algebra I:i admits no outer automorphisms of order p. Hence J consists of only inner automorphisms of 1:. In particular, for a fixed element x E J\ {I} the fixed point subalgebra C.c(x) contains a Cartan subalgebra 'H. of 1:. Note that J acts fixed-point-freely on 1:. Thus, if we choose y E J such that J = (x, y), then y acts fixed-point-freely on C.c(x). This implies that the Lie algebra C.c(x) is nilpotent. Therefore C.c(x) = 'H. and rank (I:) = dim C.c(x) = p - 1. As a consequence, we get m = rank (I:)/rank (I: i ) ::; p - 1. If m 2: 2, then G/ J ~ SL 2 (p) has no subgroups of index m, a contradiction. So m = 1, I: is a simple Lie algebra of rank p - 1 and dimension p2 - 1. In other words, I: ~ Ap - 1. 2) Denote by X the G-character afforded by 1:. Then X(x) = -1 for all x E J. It is shown in Proposition 4.3.10 of [KoT] that there exists a unique subgroup J c 9 = Inn (I:) ~ PSLp(C) acting on I: with such a character. Furthermore, due to [Sah], we have H 1(SL 2 (p), W) = 0,

177

Splitting Fields for Jordan Subgroups

where W = IF; denotes the natural module for SL 2 (p). Therefore, one can choose a basis {I" I u E W#} and define a G-invariant Lie structure {X, Y} = X . Y - Y . X such that (5) holds. Here (.,.) is some nondegenerate symplectic structure on W, and we identify J with {w w E W} and SL 2(p) with {ep ep E Sp(W)}. In particular, with respect to L '}, the group G has IKo as the unique minimal splitting field, as we have shown above. We would like to show that the initial Lie bracket [".J on I: is defined in the same way as {" .}, and so IKo is also the unique minimal splitting field for G under the Lie multiplication [".J (but the 1

1

multiplications

[.,.J and L·} need not coincide).

We shall reconstruct the initial Lie structure [" .J using the basis {I" u E W#}. Since G consists of only inner automorphisms of 1:, [X, YJ = X· Y - y. X for some G-invariant associative structure X· Yon I:tfJ (E) with E· X = X . E = X, and Vg E G g(E) = E.

1

Recall that the action of G on {I,,} is given by (5). Our goal is to show that there exists a non-degenerate symplectic structure (,1,) on Wand a rescaling of the matrices I" such that, for all u, v E W,

I u . I v --

",("lv)/2 I

c

u+v·

From (5) it follows that

w(I" . Iv) = w(I,,) . w(Iv) = e(w,,,+v) I" . Iv for any w E W, so I" . Iv = O:",vI,,+v for some O:",v E C. Similarly,

hence (I,,)P = (3"E for some (3" E C. But for any ep E SL2(p) we have

therefore (3" = (3 for all u E W#. Suppose that (3 = O. Then I" is an ad-nilpotent element in 1:. On the other hand, if we choose x E W# such that (u,x) = 0, then I" belongs to the Cartan subalgebra Cdi), a contradiction. So (3 =I O. Replacing I" by (3-1/ PI", One gets (I,,)P = E for any u E W. Now we may suppose that h" = O:k(I,,)k for 1 ::; k ::; p and O:k E C', 0:1 = O:p = 1. Here, the coefficients O:k are the same for all u E W#, as SL2(p) acts transitively on the set {I" I u E W#}. Furthermore,

A. M. Cohen and P. H. Tiep

178

Applying the automorphism tj; with t.p

=

(~ ~_I), t.p(u) = SU, SElF;,

one gets

(8) Setting l

= 1 in (8) we have

ak/ak+1

= asaks/a(k+l)s. Hence

i.e., (as)P = 1 for any s E IFp . Replacing each I u by a2Iu, we still have (Iu)P = E, but the collection aI, a2, ... , a p will possess the property al = a2 = a p = 1. Now taking k = l = 1 in (8) we obtain a2s = a;. Finally, with setting l = 1 and s = 2 in (8) we get ak/ak+1

= a2k/a2k+2 =

(ak/ a k+I)2,

that is, ak = ak+l. This means that ak = 1 and so (Iu)k = hu for all k. Next we choose u, v E W such that W = (u, v) and write down hu . Iv = Ikhu+v for k = 1,2, ... , p, Ik E te, IP = 1. Then

On the other hand, applying tj; with t.p : U

f--->

U, V

f--->

U

+ v to the identity

hu . Iv = Ikhu+v, one finds hu . I u+v = 'kI(k+I)u+v. Therefore, Ik+1 = Ik/I, Ik = If and If = 1. If II = 1, then Ik = 1 for all k, and so Ix . I y = I x+y for any x, yEW. The last identity means that the Lie multiplication on £ is trivial, a contradiction. Hence, II i- 1. In this case

one can define an SL 2 (p)-invariant symplectic form (·1·) on W such that C I = e(ulv)/2. Then for any k one has hu . Iv = e( ku lv)/2 hu+v. From this it is not difficult to derive that Ix . I y = e(x!y)/2Ix+y , as required. This settles Proposition 4. 0

Remark. The proof of Proposition 4 shows that the number of conjugacy classes of embeddings of G = Z; . SL2(p) in Inn (12) = PSLp(C) is equal to the number of SL 2(p)-invariant nontrivial symplectic structures on W = IF;, i. e., equal to p - 1. (It is clear that the nontrivial symplectic forms on W are all scalar multiples of each other.) All these embeddings afford the same character, as G has a unique absolutely irreducible real character of degree p2 - 1. Moreover, they are algebraically conjugate. Namely, let e' be another pth primitive root of 1. Then one can use e' instead of e in the definition of the matrices D, P (d. §2.1), and in the definition of the action of G (see the beginning of §3). In this way,

Splitting Fields for Jordan Subgroups

we arrive at the associative structure I u • Iv

179

=

e'(u,v)/2Iu+v' Thus, these

p - 1 embeddings are in bijective correspondence with the elements of

Gal (Q(e)/Q). Remark also that there are (p - 1)/2 Aut (.c)-conjugacy classes of embeddings of G in Aut (.c) ~ PSLp(C) . Z2. Indeed, the complex conjugation e f---> C l leaves all the coefficients Au,v(x) (see (7)) fixed. Therefore, these Aut (.c)-conjugacy classes are in bijective correspondence with the elements of Gal (OCo/Q).

4. The case of exceptional types and p odd To conclude the proof of our theorem, it remains to consider Jordan subgroups J ~ of 9 = Aut (.c), where .c is of type F 4 , E 6 or E 8 , with p = 3,3,5 in the respective cases (d. Table 1). Our arguments will follow the same scheme, so below we shall treat only the case J ~ Z~, .c = E8 • We shall write e = exp(27ri/5), '13 = e-c l , 0C0 = Q(e+c l ) = Q( y'5). It is known that G = Ng(J) is a split extension of J by SL 3 (5). Following the proof of Proposition 4, one can establish

Z;

Lemma 5. Assume that G ~ Z~· SL 3 (5) is a subgroup of Aut (.c), where .c is a complex Lie algebra .c of dimension 248 with nonzero multiplication. If G has an absolutely irreducible action on .c, then .c is a simple Lie algebra of type E 8 , J = 05(G) is a Jordan subgroup of 9 = Inn (.c), and G coincides with N g (J). D Unlike the case of Proposition 4, here the group G has just two irreducible characters of degree 248, and both of them can be realized by embeddings of G in g, see [Tiep]. Consider one of these embeddings and the orthogonal decomposition (3) corresponding to J. Fix a non-identity element g E J. Then by Lemma 1 the fixed point subalgebra C = C[, (g) is a semisimple Lie algebra of type 2A 4 . So C = C1 EBC2, where C', s = 1,2, are simple Lie algebras of type A 4 . On the other hand, by (3), C is a direct sum of six Cartan subalgebras of .c, say 'H j , j = 1, ... ,6, of (3). The proof of the following assertion will be omitted:

Lemma 6. The normalizer N c ( (g)) acts on C as a subgroup of type (Zg x Zg) ·GL 2(5) of Aut (C) ~ (PSL 5(C) ,Z2)/Z2. This action induces a subgroup H ~ Z~· SL 2(5) such that (05(H), g) = J, and H acts faithfully and irreducibly on each component C', s = 1,2. In particular, 05(H) is a Jordan subgroup of g' = Inn (C'), and H coincides with N g s(05(H)).

o First we claim that OCo is a splitting field for G. For, consider the

A. M. Cohen and P. H. Tiep

180

lKo-subspace V

= EB;~I Vi of 1:, where

Vi = {x E H j I Spec (I:,adx)

l'(wv) = II(W' ). Thus Bw,Bv = ,Bw'v = ,Bw for some 0 I- , E k. Hence, by (2.3)(c), BwBv = Bw,B; = Bw,(o:B I + f3B v) = (I-lo:)B wv + ,-If3Bw,Bv = (I-lo:)B wv + f3B w. 0

,-I

,-I

Lemma 2.6 Letw,w',w" E R such that Tw,,(BwB w') Il' (w) - I' (w') I and equality holds only if w" = ww' .

I- O.

Thenl'(w") 2:

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Proof. Using Lemma 2.5 and (2.3)(b), one shows by induction on l'(w) and l'(w') that BwBw' = IBww'

L

+

,xBx,

xER,I'(xl>ll'(w)-I'(w'll

with 0 =J I E k and IX E k. The two assertions now follow from the observation l'(ww') ~ Il'(w) -l'(w')I. 0 Proposition 2.7 The bilinear form H x H ~ k defined by (h, h') f--+ Tl(hh'), for h,h' E H, is symmetric, non-degenerate and associative. More precisely, we have

for all w,w' E W(L,X). In particular, H is a symmetric algebra.

Proof. Let w, w' E R such that Tl(BwB w') =J O. Lemma 2.6 implies that 1 = ww'. Next, let x, x' E C, w, w' E R such that Tl (BxwBx'w') =J O. The multiplication rules in (2.3)(a) show that BxwBx'w' = IBxx,Bx'-lwx,Bw' for some I E k. Thus BxwBx'w' lies in H l B xx" and thus the condition Tl (BxwBx'w') =J 0 implies that xx' = 1. It then follows from the first part of the proof that X,-lWX' = W,-l, and hence that x'w' = (XW)-l. Let w E R and write w = V1V2··· Vr as a reduced expression in standard generators Vi. By (2.3)(c), B;. = aiBl + !3i B v, with ai,!3i E k, ai =J 0, i = 1, ... , r. Putting w' := WVr, we obtain with (2.3)(b), BwBw-l = Bw,Bvr B vr Bw'-l. It follows from (2.3)(c) and (b) that BwBw-l = arBw,Bw,-l + !3rIBwBwl-l for some I E k. The first assertion of the proposition implies that Tl (BwBw-l) = arTl (Bw,Bw,-l), and so r

Tl (BwBw-l)

=

II ai, ;=1

by induction. This is non-zero and equal to Tl (Bw-l B w) by exactly the same argument. Finally, let x E C, w E R and let )..' denote the 2-cocycle appearing in the multiplication rules (2.3)(a). Then one checks that

Similarly,

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Since B x and Bx-l commute, we have X(x, X-I) = X(x- I , x) and the result follows. It follows that the bilinear form on H defined in the statement of the theorem is associative, non-degenerate and symmetric, and so gives H the structure of a symmetric algebra. 0 2.8. This result, together with the "Property A" mentioned in (2.2), shows that we are in a similar situation as the one studied by J. A. Green in [25]. In fact, Green considered the permutation module on the cosets of a subgroup of a given group, and assumed that the analogue of the above "Property A" holds and that the endomorphism algebra is a quasi-Frobenius algebra (which is a weaker condition than being a symmetric algebra). M. Linckelmann [36] has pointed out to us that the assumption that the module under consideration is a permutation module is in fact unnecessary in order to obtain the same conclusions as in [25], Theorems 1,2. Linckelmann and Cabanes independently drew our attention to a paper of Cabanes [5], where a more general situation is studied. Applying the following result with A = kG and Y = Rf(X) (for a cuspidal pair (L, X)) yields, first of all, a new proof of [23], Theorem 2.4(a), (b), and shows, secondly, that the head and the socle of each indecomposable direct summand of Rf(X) are isomorphic to each other (which we could not prove using the methods in [loco cit.]). The result is as follows. Theorem 2.9 ([25, 5, 36]) Let Y be an A-module, for some associative k-algebra A, and E := EndA(Y). Assume that the sets of composition factors of the head and the socle of Yare equal ("Property A" for Y ) and that E is symmetric. Let Fy : mod-A ~ mod-E be defined by Fy(V) := HomA(Y, V). Then the following holds. (a) Let Y' be an indecomposable direct summand ofY. Then the head and the socle of Y' are simple and isomorphic to each other. (b) Two indecomposable direct summands of Yare isomorphic if and only if their heads (or, their socles) are. (c) The functor Fy induces a bijection between the set of composition factors of the socle of Y and the set of isomorphism classes of simple E-modules.

Proof. For the convenience of the reader, we briefly sketch the main ingredients of the proof. To a large extent, we could follow Green's

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argument word by word. See [25], (1.3), where formal properties of Fy are established on which the subsequent arguments in [loco cit.] are built up. These formal properties do not depend on the fact that A is a group algebra and Y is a permutation module. A slight complication will arise from the fact that, in our situation, we can not assume that Y is self-dual. We therefore proceed as in [5]. There, Cabanes defines a full subcategory mody-A of mod-A as follows. A finitely generated A-module V is in mody-A, if and only if V is the image of some endomorphism of yn, for some n, where yn denotes the direct sum of n copies of Y. Cabanes then shows in [5], Theorem 2 and Corollary 3, that Fy induces an equivalence of additive categories between mody-A and mod-E, and that a module V in mody-A is indecomposable (as A-module), if and only if Fy(V) is indecomposable. "Property A" for Y implies that the composition factors of the head of Y belong to mody-A. It is clear that the indecomposable direct summands of Y also belong to mody-A. These summands of Yare sent by Fy to the projective indecomposable modules of E. The equivalence of additive categories now implies (b) and (c). Finally (a) follows from the fact that the same property holds for the projective indecomposable 0 modules of the symmetric algebra E. Let (L, X) be a cuspidal pair and Y = Rt(X). As already remarked above, the methods in [23] did not yield a proof of the fact that the socle of an indecomposable direct summand of Y is isomorphic to its head. Since the dual of X is cuspidal, it nevertheless follows from [23], Theorem 2.4(a), that the socles of such direct summands are simple. This theorem was proved by showing that Dipper's hypothesis [9], (2.6), holds for Y, which means that Y has a projective cover {3 : P ~ Y such that the kernel of (3 is invariant under all endomorphisms of P. We shall meet such modules Y later on in (4.6). For those modules it need no longer be true that they satisfy "Property A" nor that their indecomposable direct summands have simple socles. Let E = Endkc(Y). It was shown in [23], Theorem 2.4(c), that the Cartan matrix of E records the multiplicities of the simple modules in the head of Y as composition factors of the various indecomposable direct summands of Y. Simple examples show that this property does not follow from the fact that E is symmetric. It certainly follows from the fact that Y satisfies Dipper's hypothesis. The modules Rt(X) for cuspidal pairs (L, X) satisfy Dipper's hypothesis, they posess "Property A", and their endomorphism rings are symmetric. The combination of all of these properties indicates the importance of these modules for the representation theory of finite groups

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of Lie type. We would now like to propose the following: Definition 2.10 Let Y E mod-kG be an indecomposable module. We say that Y is a Harish-Chandra module if Y is a direct summand of RT,(X), for some Levi subgroup L ofG and some cuspidal simple module X E mod-kL.

By definition, all cuspidal simple kG-modules are Harish-Chandra modules. Here are some more basic properties of this class of modules. Proposition 2.11 Let Y E mod-kG be a Harish-Chandra module of

G, belonging to the (L, X)-Harish-Chandra series. (a) The head and the sode of Y are simple, and they are isomorphic to each 0 ther. (b) The composition factors of the heart ofY belong to (L', X')-HarishChandra series of G with L strictly con tained in L'. (c) Endkc(Y) is a symmetric k-algebra.

Proof. Let H be as in (2.3) above. Then L, X, and H satisfy all assumptions needed in (2.6), as already remarked earlier. This proves (a). The statement in (b) is essentially shown in [30], see the remarks in [22], (2.2). Finally, (c) follows from the fact that Endkc(RT,(X)) is symmetric. D 2.12. Combining the preceding results, we obtain a classification of the set of isomorphism classes of simple kG-modules by triples (L, X, ¢) where (L, X) is a cuspidal pair in G and ¢ is an irreducible character of H, the endomorphism algebra of RT,(X) (see [23], (2.5)). Given a simple kG-module Y the associated pair (L, X) was already defined in (2.2) above. Now Y is a composition factor ofthe head of RT,(X) hence it is isomorphic to the head of some indecomposable direct summand of RT,(X), by Theorem 2.7. This indecomposable direct summand, in turn, corresponds to a projective indecomposable module for H (Fitting's Lemma) and hence to a simple H-module, giving rise to an irreducible character ¢. In order to achieve this classification we must at first classify the simple cuspidal modules for all Levi subgroups L of G and then determine the irreducible characters of the associated endomorphism algebras. In Section 4 we show how this can be done in special cases for finite classical

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groups. In Section 7, we work out explicitly the case of "unipotent" simple modules in the exceptional group E 6 (q). We now have a general plan for studying modular representations in terms of their distribution into Harish-Chandra series. On the other hand these representations are also partitioned into blocks and, using the decomposition matrix, we also have a link with the ordinary character theory about which, in general, much more is known than for the modular case. This will be studied in the next section. 3. Blocks and basic sets

3.1. We will now change our notation and assume that G is a connected reductive algebraic group over the algebraic closure of the finite field with p elements. We let q be a power of p and assume that G is defined over JFq , with corresponding Frobenius map F : G ----+ G. Let E. be a prime not equal to p and K ~ Qf a sufficiently large field generated (over Q) by roots of unity. In the following, the term "character" will always refer to an ordinary character associated with a representation of G F over K. We assume throughout that the centre of G is connected. Let G* be a group dual to G, also defined over JFq • For every semisimple element s E G*F we then have a corresponding geometric conjugacy class of characters of G F denoted by C. (see, for example, [8], Definition 13.16). If it is necessary to indicate the underlying group, we shall write c.( G F ). We fix a discrete valuation ring 0 in K with residue class field k of characteristic P.. We also assume that k is sufficiently large. Thus (K, 0, k) is a splitting E.-modular system for G. Brauer characters and blocks will always be taken with respect to our fixed prime number E. i- p. The restriction of any class function f to the set of E.-regular elements of the group will be denoted by j. Thus, if p is an ordinary character then p is a Brauer character. It will be convenient to extend such functions by zero to the E.-singular elements. 3.2. In this situation, many applications of the ordinary character theory of G F to the modular situation are related to the question of the existence of a basic set. Let B be a union of E.-blocks of G F . Recall that a set of Brauer characters in B is called a basic set if it is linearly independent and if every Brauer character in B is an integral linear combination of the elements in that set. We say that a basic set is ordinary if it consists of the restrictions of some ordinary characters to the E.-regular elements of G F . In this case, we consider these ordinary characters themselves as elements of the basic set.

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The importance of the existence of an ordinary basic set for B lies in the fact that, first of all, its cardinality gives the exact number of Brauer characters in that union of blocks. (More precisely, the cardinality of the intersection of the basic set with the set of ordinary characters in any block contained in B gives the number of Brauer characters in that block.) Secondly, in order to determine the decomposition matrix of Bit will be sufficient to compute the decomposition numbers for the ordinary characters in the basic set and the relations expressing the remaining ordinary characters in B on the £-regular elements as linear combinations of the characters in the basic set. (Of course, all this is valid for any finite group and any prime, and these notions can be taken either with respect to all characters of that group or only for those characters in a fixed union of blocks. Note that, in general, it is an open question whether ordinary basic sets always exist. ) 3.3. The distribution of the ordinary characters of G F into blocks is compatible with the distribution into geometric conjugacy classes. In fact, if s E G*F is semisimple of order prime to £ then we have a corresponding union of £-blocks Bs defined by Brow§ and Michel [4]. An ordinary character belongs to Bs if and only if it is contained in Est where t E G*F is semisimple of £-power order and commutes with s. If it is necessary to indicate the underlying group, we shall write again Bs(G F ). A general result about the existence of ordinary basic sets was obtained in [20]. It states that, if £ is good for G then the characters in Es give rise to an ordinary basic set for Bs (without any further assumption on G or q). The proof is based on properties of the twisted induction and a counting argument relating classes of semisimple elements of £-power order in G F and G*F. (See also [19], II, for an extension to the case where the centre of G is not necessarily connected.) Closely related with the notion of an ordinary basic set is the question about the shape of that part of the decomposition labelled by the rows corresponding to the ordinary characters in the basic set. We wish to formulate a conjecture for the case when £ is good. We will only formulate it for the special case where s = 1. A general version would be given in terms of the (conjectural) Jordan decomposition of characters and blocks (cf. [30]). Let W be the Weyl group of G (corresponding to some maximallysplit torus of G). Then F induces an automorphism, of W of finite order. The unipotent characters of G F are partitioned into sets M(F), one for each ,-stable family F of W. With every irreducible representation of W there is associated an integer, namely the exponent of the lowest power

Rt

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of q (regarded as a formal parameter) dividing the corresponding generic degree. This function is constant on the families of W. Let a(F) be this constant value for a given family F. (For all this, see [38], Chap. 4.) Conjecture 3.4 Assume that £ is good. Let F 1 , ••• ,Fr be the ,-stable families of the Weyl group, ordered such that a(F1 ) :S a(F2 ) :S ... :S a(Fr ). Let D be the part of the decomposition matrix of B 1 with rows labelled by the characters in £1 .

• The irreducible Brauer characters in B 1 can be labelled such that D has the following shape.

o D=

* where D 1 , D 2 , ... are square matrices with rows labelled by the characters in M(F1 ), M(F2 ), ••• , and each D; is in fact the identity matrix of the appropriate size.

• If, moreover, p is a cuspidal unipotent character of G F then the row of D labelled by p has only one non-zero entry, that is, p is an irreducible Brauer character. The first part of the conjecture is known to be true for the cases where GF = GLn(q) (see [12] and the references there), GF = GUn(q) (see [17]), and some explicitly worked-out cases of small rank, like G 2 (q) and 3 D 4 (q) (see [30] and the references there). The second part can also be checked in the small rank cases. Some more specific results for classical groups will be given below and, in the case of "linear" characteristic, in the next section. In Section 6, we will also check that this is true for G F = E 6 (q). The importance of having a decomposition matrix of triangular shape lies in the fact that we then have a canonical labelling of the irreducible Brauer characters in terms of the ordinary characters of the basic set. Note that if a different ordering of the same basic set also results in a triangular decomposition matrix, we obtain the same labelling. (This remark follows, for example, from the Bruhat decomposition of GLn(C).)

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We believe that a clue to a conceptual proof of the above conjecture will be provided by a suitable adaptation and improvement of N. Kawanaka's theory of generalized Gelfand-Graev representations. The first part should follow from a study of a refinement of these representations, as sketched in the survey article [35], (2.2.8), (2.4.5). The second part should then follow from a study of Harish-Chandra induction of generalized Gelfand-Graev representations, along the lines employed by G. Malle and the first author in [24]. We hope to be able to obtain results in this direction in the not too far future. 3.5. A group G as in (3.1) will be called classical if it is simple modulo its centre, or a torus, if its root system is of type Am' B m , em or D m (for m 2: 1), and if GF is not of type 3D4 . Lusztig has shown that each geometric conjugacy class of characters of GF contains at most one cuspidal character; moreover, the condition for the existence of a cuspidal unipotent character only depends on m (see [6], (13.7)). In some places it will be necessary to specify exactly the group GF we are considering. For this purpose, we give the following list. (a) GUn(q) (b) SPn(q) (c) CSPn(q) (d) SOn(q)

(any q, n 2: 1) (q a power of 2, n 2: 2 even) (q odd, n 2: 2 even) (q odd, n 2: 1 odd)

If GF is one of these groups, we will also write GF = Gn(q). We will now collect some known results on the existence of cuspidal unipotent Brauer characters. The following two results give an affirmative answer to the second part of Conjecture 3.4 in some special cases.

Theorem 3.6 (Geck-Hiss-Malle [22], Theorem 6.10): Let GF = GUn(q), where n is of the form n = s(s + 1) / 2, so that G F has a cuspidal unipotent character X. Then the i-modular reduction of X is an irreducible Brauer character, for all primes i not diViding q. Theorem 3.7 (Geck-Malle [24], Theorem 4.4): Let q be odd and G be a classical group. Assume that G is of split type, and that q is a large enough power of a large enough prime p (see (24], Section 2, for a discussion of this condition). Suppose that G F has a cuspidal unipotent character x. Then the 2-modular reduction of X is an irreducible Brauer character. 3.8. Next we consider the question of finding all cuspidal unipotent Brauer characters of GF . This is a much harder problem. First of all,

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there is an example in the exceptional group G F = G 2 (q) of a cuspidal unipotent 5-modular representation which is not a composition factor in the 5-modular reduction of any ordinary cuspidal representation. In the classical groups, we do not know whether or not such things can happen. Moreover, there can exist many cuspidal unipotent Brauer characters even if G F does not have any cuspidal unipotent ordinary characters at all. Finally, the situation may depend strongly on whether we consider a group of split type or of non-split type. The following two results will illustrate this in the case where G is of type An-I, i.e., G F = GLn(q) or G F = GUn(q). In these cases, the unipotent characters have a natural labelling by partitions of n. (The trivial character corresponds to the partition (n), and the Steinberg character to (1 n).) As already mentioned in the remarks following Conjecture 3.4, the decomposition matrix D of the unipotent characters has a lower unitriangular shape if the partitions of n are ordered reversed lexicographically. Thus, we also have a natural labelling of the irreducible Brauer characters in B I by partitions on n. Let us write 1{Jp. for the irreducible Brauer character corresponding to the partition J.l.

Theorem 3.9 (Geck-Hiss-l\lalle [22], Theorem 7.6): Let G F = GLn(q) and J.l be a partition of n. Then 1{Jp. is non-cuspidal for all partitions i- (In). Moreover, l{J(In) is cuspidal if and only if n = e£i for some i ~ 0, where e denotes the multiplicative order of q modulo £. J.l

Thus, we have complete answers on the existence of cuspidal unipotent Brauer characters and their appropriate labelling for G F = GLn(q).

Theorem 3.10 (Geck-Hiss-Malle [22], Proposition 6.8, and [23], Theorem 4.12): Let G F = GUn(q) and J.l be a partition of n. Denote by J.l' the partition conjugate to J.l. If 1{Jp. is cuspidal then J.l' must be 2-regular, i.e., no two parts of J.l' are equal. Moreover, if f!. > nand f!. divides q + 1 then all Brauer characters 1{Jp., where J.l' is 2-regular, are cuspidal. 3.11. Let us finally discuss an interesting subclass of the cuspidal representations, the so-called supercuspidal representations. These were introduced by M.-F. Vigneras in the study of f!.-modular representations of p-adic groups [48]. In our situation, an irreducible f!.-modular representation (and its Brauer character) is called supercuspidal, if it does not occur as a composition factor of a representation obtained by Harish-Chandra induction from any proper Levi subgroup. In particular, a supercuspidal representation is cuspidal. It is easy to see that an irreducible f!.-modular representation is supercuspidal if and only if its projective cover is cuspidal. Thus, if there exists a supercuspidal unipotent f!.-modular representation, there has to exist an ordinary cuspidal unipotent representation.

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It turns out that there are only very few supercuspidal unipotent representations in classical groups. Theorem 3.12 (Hiss [28]): Let G be a classical group and let f!. be odd. Assume that G F has a cuspidal unipotent ordinary character X. If X lies in a block with a cyclic defect group, the reduction modulo f!. of X is irreducible and supercuspidal. All unipotent supercuspidal Brauer characters arise in this way.

We believe that G F has no supercuspidal unipotent 2-modular representations if G is a classical group. There is another situation where we can give very precise results on the cuspidal unipotent Brauer characters and the decomposition matrix of the unipotent characters of G F . This will be discussed in detail in the next section. 4. Cuspidal unipotent representations and decomposition matrices 4.1. Here we consider the problem of finding the unipotent cuspidal Brauer characters of a classical group GF = Gn(q) as in (3.5). This can be solved completely for primes f!. satisfying suitable additional conditions. To describe these, we introduce the notion of a linear prime for G F (see also [16]).

Definition 4.2 Let G be a classical group (3.5). We say that f!. is linear for G F , if f!. is good for G, and iffor every split F-stable Levi subgroup L of G the following holds: If L F has a cuspidal unipotent ordinary character, this lies in a block of defect 0 in L F jZ(L F ).

The solution of the above problem will enable us to reduce the computation of the decomposition numbers (in linear characteristics) for the unipotent characters of G F to the determination of the decomposition numbers of a certain algebra, which may be viewed as an analogue to the q-Schur algebra introduced by Dipper and James in [12]. In his thesis [26], Gruber was able to reduce this further to the analogous problem for the ordinary q-Schur algebra. In other words, the decomposition numbers for a classical group G F (so far we have to exclude the orthogonal groups in even dimension) in linear characteristics f!. can be calculated from those for various general linear groups (see Theorem 4.13 below for a precise statement). We give the following characterizations of linear primes for the various types of classical groups.

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Lemma 4.3 Let d, e and f denote the orders of (-q), q respectively q2 modulo £. We then have:

(a) If d is even, £ is odd. (b) If e£ is odd, then d is even. (c) e is odd if and only if £ divides q! - 1. (d) Every prime not dividing q is linear for GLn(q). A prime £ is linear for GUn(q), if d is even, and £ is linear for a group G F not of type A, if e£ is odd. Proof. Parts (a)-(c) are elementary and part (d) follows from an in0 spection of the degrees of the cuspidal unipotent characters. In the terminology of Fong and Srinivasan [16], £ is a linear prime for q, if £ is odd and divides q! - 1. Theorem 4.4 (Geck-Hiss-Malle [23], Theorem 4.11): Let G F be one of the groups from (3.5)(a)-(d), and let £ be a linear prime for G F .

If G F has a cuspidal unipotent character, this lies in an £-block of defect 0, and thus its reduction modulo £ is irreducible and cuspidal. All cuspidal unipotent Brauer characters arise in this way. In his thesis [26], Gruber has extended this result to the orthogonal groups in even dimensions. We believe that an analogous result holds without the restriction to unipotent blocks. The situation is more complicated for the exceptional groups of Lie type. 4.5. We now present some new consequences of Theorem 4.4. For this purpose we first introduce some notation. Let G F be any finite group of Lie type, £ a prime not dividing the characteristic of the underlying field. Recall that (K, 0, k) denotes a splitting £-modular system for G F. Given an OGF-lattice with character X, we write X = 'l/J + J.l, where J.l is a sum of unipotent characters and 'l/J has no unipotent constituent. Then X has a uniquely determined pure submodule Y with character'l/J. The OGF-lattice XjY is called the unipotent quotient of X. It is easy to see that taking unipotent quotients commutes with direct sums and with Harish-Chandra induction (see [27], Lemma 6.1). If G F arises from an algebraic group with connected centre, there is a unique Gelfand-Graev representation (over 0) of G F , afforded by a projective OGF-lattice. Its unipotent quotient is called the Steinberg lattice of G F over 0 (see [12]).

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Let Y be any OG F -lattice. Following Dipper [9], Definition 4.1, we introduce the decomposition matrix D y of Y as follows. The columns of D y correspond to the isomorphism classes of indecomposable direct summands of Y. The rows of D y correspond to the isomorphism classes of the irreducible constituents of the KGF-module Y 00 K. If V, X represent a row respectively column of D y , then the (V, X)-entry in D y is the multiplicity of V in X 00 K. It follows from Fitting's lemma, and the Brauer reciprocity law applied to the O-algebra E := EndoGF(Y), that D y is equal to the decomposition matrix of E in the usual sense (see [9], Corollary 4.4). 4.6. Now let G F be again one of the groups from (3.5), and let P. be a

linear prime for G F . Put 8 := 2 if G F is a unitary group and 8 := 1, otherwise. Every Levi subgroup L F of G F = Gn(q) is ofthe form L F = Ga(q) x L o where L o is a Levi subgroup of GLm(qO), and n = a + 2m. Fix a Levi subgroup La := Ga(q) X GL 1 (qo)m, such that Ga(q) has a cuspidal character. Since P. is linear for GF, this is a defect 0 representation for Ga(q)jZ(Ga(q)), and we let X a denote the OGa(q)-lattice affording it (and on which Z(Ga(q)) acts trivially. For every partition A = (AI, ... , Ar ) f- m let L>. denote the standard Levi subgroup of GLm(qO) isomorphic to

GL>'l (l) x ... x GL>'k(l)· For each A f- m let X>. denote the Steinberg lattice of L>. over O. Put

and

X a,>. := X a 0 X>.. Note that Xa,(lm) 00 K is a cuspidal unipotent K La-module. Note also that X a and X>. are unipotent quotients of projective indecomposable modules of Ga(q) respectively L>.. Hence X a,>. is the unipotent quotient of a projective indecomposable OLa,>.-module. Finally, put Ya :=

EB Rr

a ,>. (Xa

,>.) ,

>.f-m

and E a := EndoGF(Ya)· Let {3 : P ~ Ya denote the projective cover of Ya. Since Ya is the unipotent qu~tient of a projective OG F -module, the kernel of {3 is invariant

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under all endomorphisms of P, in other words, Dipper's hypothesis [9], (2.6), is satisfied. This implies that direct summands of the kGF-module Y a 00 k and kG F-homomorphisms between them are liftable. In particular, EndkGF(Ya 00 k) ~ E a 00 k. For an easy proof of these facts, due to Plesken, see [29], Satz 5.1.2.

Theorem 4.7 Let 0 :::; al < a2 < ... < a r :::; n denote the integers such that n - ai is even and such that Ga,(q) has a cuspidal unipotent character. For a = ai, 1 :::; i :::; r, let D a denote the decomposition matrix of the V-algebra E a, and let Ua denote the set of unipotent characters lying in the ordinary Harish-Chandra series determined by (La, Xa,(lm)). Then the elements of U a correspond naturally to the irreducible representations of E a 00 K. Under this bijection, D a is also the matrix of decomposition numbers of the elements in Ua' Furthermore, D a is a square matrix, and the decomposition matrix D of the unipotent characters of G F is a block diagonal matrix of the form

o D=

o Proof. If two unipotent characters are in the same €-block, then they are in the same Harish-Chandra series. This follows from the FongSrinivasan classification of €-blocks [15, 16] and the assumption on e. Thus D is a block diagonal matrix with diagonal entries D~ corresponding to Ua' Since D is a square matrix by [20], the same is true for the matrices D~. The ordinary irreducible constituents of Ya 00 K are exactly the elements of Ua. Thus, by Fitting's lemma, the latter correspond naturally to the irreducible representations of E a 00 K. It remains to prove that D~ = D a for 1 :::; i :::; r. If a = n, then G = Ga(q) = La and X a = Ya affords an irreducible defect zero representation. In this case E a = V and D a = (1) = D~. We thus may assume that a < n in the following. Let Ba denote the union of €-blocks containing Ua' Every X a,>' is the unipotent quotient of a projective indecomposable VLa.>.-module. Since direct products and Harish-Chandra induction commute with unipotent quotients, every indecomposable direct summand of RTa ,>. (X a ,>.) and hence of Y a

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is the unipotent quotient of an indecomposable projective OGF-module lying in Ba . To finish the proof we must show that the unipotent quotient of every projective indecomposable module in Ba occurs as a direct summand of Ya . Since a < n, there is no cuspidal irreducible kG F -representation in Ba , by the results in [23], Theorem 4.11. This implies (see [27], Lemma 4.1(c)) that every projective indecomposable kGF-module in Ba is a direct summand of some Rf(Z 00 k), where Z is a projective OLF-lattice such that Z 00 k is the projective cover of a cuspidal irreducible unipotent kLF-module of some Levi subgroup L F . Again by [23], Theorem 4.11, such an L F is of the form L F = L b,>., and the unipotent quotient of Z is of the form X b,>., where 1 :::; j :::; n and A is a suitable partition of (n - b)/2. By Harish-Chandra theory for ordinary characters, the ordinary unipotent constituents of Rf(Z) lie in Bb , and so a = b. The proof is complete. 0

Corollary 4.8 Let Gn(q) be as in Theorem 4.7, and let £ be linear for Gn(q). Then the £-decomposition numbers of the unipotent chamcters of Gn(q) are bounded independently of q, namely by max{x(1) I X E IrrK(W F)}, where W F is the Weyl group of G F = Gn(q). Proof. The unipotent quotient of every projective indecomposable OGn(q)-module is a direct summand of some Rfa ,>. (X a ,>.). The decomposition numbers in question are therefore bounded above by the maximal multiplicity of some irreducible character in some Rfa ,>. (X a ,>.). This multiplicity can be calculated in the Weyl group of type B(n-a)/2, which is a subgroup of the Weyl group of Gn(q). 0 4.9. We next introduce an analogue of the q-Schur algebra. Let R be a commutative ring with 1 and let Q, q E R with q invertible. The Weyl group of type B m generated by fundamental reflections corresponding to the Dynkin diagram

---0--0 Sm-l

will be denoted by W m . Let'H := 'HR(Wm ) be the Iwahori-Hecke algebra corresponding to Wmover R with parameters Q (corresponding to t) and q (corresponding to the Si)' Let Sm denote the subgroup of W m generated by the fundamental reflections si,1 :::; i :::; m - 1. For a partition A f- m let S>. denote the standard parabolic (Young) subgroup of Sm corresponding to A.

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The analogue of the q-Schur algebra is defined as follows. Definition 4.10 For .\ I- m let Y>. :=

LWES" (-q)-l(w)Tw'

Then put

SR(Q,q,Bm):= End7-l(EB y>.H). >.f-m

The relevance of these algebras comes from the fact that they describe the unipotent representations of some of the classical groups. To show this, we relate the algebras SR( Q, q, B m ) and E a defined in the previous subsection. From now on we resume the notation from the previous section. Let n = a + 2m. Let Z := Za := Rt. (Xa,(l~))' Then the endomorphism algebra Ha(Wm) of Za is an Iwahori-Hecke algebra of type B m with parameters Qa and l, where Qa depends on a and the type of GF . Let SO(Qa,l,B m) be defined with respect to 11.:= Ha(Wm). Let modZ-OGF denote the full subcategory of mod-OGF whose objects are the OGF-submodules of Z. Consider the functor

defined by Fz(X) := HomOGF(Z, X) (see also Theorem 2.9). The restriction to modZ-OGF of the functor introduced by Dipper in [9] is naturally isomorphic to the one introduced here (see [9], Remark 2.22). As we have already observed in (4.6), Dipper's hypothesis [9], (2.6), is satisfied for the projective cover P of Z. Since Z 1810 K is a submodule of P 1810 K, it is P 1810 K-torsionless in the sense of [9], Definition 2.11, i.e., there is a non-trivial homomorphism from P 1810 K to every non-trivial submodule of Z 1810 K. This implies, of course, that Z is P-torsionless. Since Z 1810 k is Harish-Chandra induced from a cuspidal kLa-module, the set of composition factors of the head of Z 1810 k equals the set of composition factors of its socle (2.2). This implies that Z 1810 k is P 1810 k-torsionless. Theorem 4.11 With the above notations, E a ~ SO(Qa, l, B m). Furthermore, for R E {K, k}, the functor HomRGF (Z 1810 R, -) induces an isomorphism between E a 1810 Rand SR(Qa 1811, l18l1, B m). Proof. For.\ I- m we let Vy>.Z denote the smallest pure submodule of Z containing y>.Z. The elements Y>. lie in the parabolic subalgebra H(Sm) of 11. spanned by the elements Tw for w E Sm. Since y>.H(Sm) is a pure right ideal in H(Sm) (see [12], Lemma 2.8), y>.H is pure in H. By

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[9], Corollary 4.14, it follows that F z (Vy>.Z) = y>.H. Thus F z induces a homomorphism

for all .\, /.l f-- m. By [9], Theorem 4.17, this is an isomorphism. Moreover, F z preserves composition of morphisms. This implies (see [9], (4.18)) that SO(Qa, qO, Em) ~ EndOGF(EEhl-m Vy>.Z). It remains to observe that Vy>.Z ~ Ria' (X a ,>.). Let

and 11.>.:= EndoLa,,(Z>.) ~ EndoL,(Rf~l~)(X(l~))). Then 11.>. may be considered as a subalgebra of 11. in a natural way via the induction functor Ria,' ,and y>. E 11.>.. We thus have y>.Z ~ y>.Ria" (Z>.) ~ Ria,' (y>.Z>.). It follows from [12], Lemma 3.7, that X a ,>. ~ Vy>.Z>.. Since induction is an exact functor and preserves O-torsion modules, we get Vy>.Z ~ JRia,, (y>.Z>.) ~ Ria,' (Vy>.Z>.) ~ Ria)Xa ,>.). The last assertion is proved in exactly the same way, using the fact that the hypotheses of [9] are satisfied for all the coefficient rings. 0 4.12. Let C F be one of the groups considered in the previous subsection, and let £ be a linear prime for C F . By the considerations at the beginning of that subsection, in order to find the £-decomposition numbers of the unipotent characters of C F , it suffices to determine the £-decomposition matrices for the algebras SO(Qa, qO, Em) for a E S. We close this section by announcing a result, due to Jochen Gruber, which completely reduces the solution of this problem to the case of the general linear groups. For this purpose we introduce some more notation. We write 1.\1 for the sum of the parts of a partition.\. The £-decomposition number in GLn(q) which corresponds to the unipotent character X>. and the Brauer character 'Pp. is denoted by d>.,p.(q). We let the notation be as in 4.7. Recall that the characters in Ua (see Theorem 4.7) are labelled as X = X(>'1,>'2) by bipartitions (.\1, .\2) of m via ordinary Harish-Chandra theory. Theorem 4.13 (Gruber [26]): Let C F be one of the groups from (3.5). Suppose that £ is a linear prime for C F . Write n = a + 2m, where a is such that Ca(q) has a cuspidal unipotent character. Then the diagonal block D a of the decomposition matrix is equal to a

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block diagonal matrix with entries Da,j,

Da,o

0

0

Da,l

0::::; j::::; m

= (n - a)/2:

o

Da =

0

Da,m-l

0

o

Da,m

Furthermore, Da,j is equal to the matrix of £-decomposition numbers of = (n - a)/2. More precisely, there is a labelling 'P = 'P(1'1,1'2) of the Brauer characters in Sa by bipartitions (Ill, 1l2) of m such that the multiplicity of 'P(1'\,1'2) in X(>'I,>'2) equals 0, ifl)1J1 =f lilli, and equals d>'\,1'1 (l)d>'2,1'2 (l), otherwise.

GLj(l) x GLm-j(l), 0::::; j ::::; m

Gruber gives two proofs of this theorem. The first one uses Green correspondence and is independent of the above results on the generalized q-Schur algebras. The idea of the second proof is to extend the results of Dipper and James in [13] on the representation theory of the IwahoriHecke algebra 'H(Wn ) to the algebra So(Qa,l, B m ). It should be noted that the results in [13] already give those parts of the decomposition matrices D a which correspond to columns labelled by bipartitions of (n - a)/2 which are e'-regular, where e' is the smallest integer l' such that £ divides 1 + q + ... + qr-l. We also mention that Gruber has similar results for the even-dimensional orthogonal groups.

Corollary 4.14 Suppose that £ > 2 and that the order of q modulo £ is odd. Then the ordinary and modular irreducible representations of So( Qa, l , B m ) are indexed by bipartitions of m. The decomposition matrix of SO(Qa, l, B m ) is a square lower unitriangular matrix. It should be interesting to compare the algebras SO(Qa, l, B m ) with the generalized q-Schur algebras introduced by Du and Scott in [14]. We now have two orderings of the unipotent characters of the unitary group which give triangular decomposition matrices, namely the one in [17] and the one indicated by Theorem 4.13. By the remarks following Conjecture 3.4, the induced labellings for the irreducible Brauer characters by ordinary unipotent characters are the same. This observation can be used to show that [22]' Conjecture 9.2, is true. All of the above results are concerned with the case where £ is a good prime for G. In the next section, we shall collect some fundamental

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results about character sheaves. This will prepare the ground for the study of unipotent Brauer characters in bad characteristic in Section 6.

5. Character sheaves 5.1. Let C and K be as in (3.1). Moreover, let x I-t X be the automorphism of K defined by mapping each root of unity to its inverse ("complex conjugation" in K). Representations and characters will always be taken over K. The term "scalar product" will refer to the usual hermitian product defined on class functions from C F to K. As before we shall assume throughout that the centre of C is connected. As in (3.1), denote by Es the geometric conjugacy class of characters corresponding to the semisimple element s E C*F. Instead of the ordinary characters of C F we shall work with a basis of class functions consisting of almost characters. Recall that the almost characters of C F are obtained by applying an almost diagonal transformation, given in terms of certain non-abelian Fourier transforms, to the basis consisting of ordinary irreducible characters; see [38], (4.24.1), (13.6). Note that almost characters are only defined up to multiplication by roots of unity. We choose one representative from each equivalence class of scalar multiples of almost characters. The resulting set of class functions, denoted by A(CF ), forms an orthonormal basis of the space of all class functions on C F . Since the transformation from irreducible characters to almost characters takes place inside the individual geometric conjugacy classes of characters, we also obtain a partition of A(C F ) into a disjoint union of pieces As (C F ), one for each class of semisim pIe elements s E C*F. 5.2. The theory of character sheaves was developed by Lusztig in a series of papers [39]. There are some basic properties of character sheaves which are believed to hold in general but which, at present, are only proved under some mild restrictions on p and q. We shall now describe these properties. (Recall that the centre of C is assumed to be connected. ) With each F-stable character sheaf A we can associate a class function XA on C F , called characteristic function of A, normalized so that its inner product is 1 (see [39], (25.1)). This normalization determines XA only up to multiplication by a root of unity, but for our purposes it will not be important to specify this scalar precisely. We assume chosen such a normalization, for all A in a complete set of representatives of

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isomorphism classes of F-stable character sheaves on G. The resulting set of class functions forms an orthonormal basis for the space of all class functions on G F (see [39], Corollary 25.7). It has been conjectured by Lusztig that this basis is the same, up to scalar multiples, as the basis A(G F ) of almost characters of GF defined in (5.1).

Assumption A. The above conjecture of Lusztig holds for our group G. Shoji [44]' [45], (3.2), (4.1), has shown that this is the case if G is simple modulo its centre and p is an almost good prime for G (see also

[41]). We say that L is a regular subgroup of G if L is an F-stable Levi complement of some (not necessarily F-stable) parabolic subgroup of G. If L is a regular subgroup then there are operations "induction" and "restriction" for character sheaves with similar properties as the ordinary Harish-Chandra induction considered in the previous section (see [39], (3.8), (4.1), and (8.1), (8.2)). Correspondingly, we also have the notion of a cuspidal character sheaf (see [39], (6.9)). Using our Assumption A we can translate this notion to almost characters and say that an almost character is cuspidal if it corresponds to the characteristic function of an F-stable cuspidal character sheaf of G.

Assumption B. Up to scalar multiplication, the "induction" of cuspidal character sheaves from a regular subgroup L corresponds, on the level of class functions, to the operation of twisted induction Rf, of cuspidal almost characters. Lusztig [42], (9.2), has shown that this is the case if p is almost good and q is a sufficiently large power of p. (See also [loco cit.] for a precise statement.) The assumption on q has been removed by Shoji in [47], Theorem 4.2. Note that, if f is a class function on L F , then Rf,(f) only depends on the GF -conjugacy class of the pair (L, f), as in the case of the usual induction of a class function from a subgroup. (This follows, for example, from the character formula in [8], (12.2).) Recall that an element g EGis called isolated if the connected centralizer Cc(gs)O of its semisimple part gs is not contained in any proper regular subgroup of G (see [39], (3.11)). This notion is related with the notions of cuspidal character sheaves and "cleanness" by [39], (3.12), (7.7). Namely, a cuspidal character sheaf has support on the Zariski closure of a set E ~ G which is the inverse image of an isolated conjugacy class of GjZ(G) in G. If the character sheaf is F-stable then the set E will be F-stable, too. The assumption that G is "clean" means that the restriction of this character sheaf to E \ E is zero. Using Assumption A we can reformulate this in terms of almost characters.

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Assumption C. The support of a cuspidal almost character (i.e. the set of elements in GF on which it is has a non-zero value) is contained in a set E F as above, where E is the inverse image of an F-stable isolated conjugacy class of GIZ(G) in G. Lusztig [39], (23.1), has shown that this is true if p is almost good. We shall assume that the above assumptions hold for G and all regular subgroups of G. Combining the remarks above we see that this is the case in the following situation. • The centre of G is connected, and G is simple modulo its centre. • The prime p is almost good for G (i.e., no condition if G is of classical type, and p good, if G is of exceptional type). 5.3. We shall say that (L, 'l/J) is a cuspidal pair in G if L is a regular subgroup of G and 'l/J E A(L F) is a cuspidal almost character. Let us fix such a pair (L, 'l/J). The almost character 'l/J corresponds to a cuspidal character sheaf; we let N denote its stabilizer in Nc(L) and W = NIL. Let n EN and w the image of n in W. Choose z E G such that Z-l F(z) = n- 1 . Then L w = ZLZ-l also is a regular subgroup, and we have a corresponding cuspidal almost character 'l/Jw E A(L~). Note that the pair (L w,'l/JW) is only well-defined up to conjugation in G F. If 'l/J is supported on the set E then 'l/Jw is supported on E w = zEz- 1 (see [39], (10.6)). Now let X be any almost character of GF with non-zero multiplicity in Rf,('l/J); we then have that

W

This is contained in [39], (10.4.5) combined with (10.6.1), modulo a reformulation in terms of almost characters and twisted induction. We will denote by A(L, 'l/J) the set of all X E A( GF) which have non-zero multiplicity in Rt,J'l/JW), for some w E W. Using (a) we see that (b) the class functions in A(L, 'l/J) span the same space as the class functions Rt,J'l/JW), for wE W.

We can now state some results which are crucial for our applications to basic sets in the next section (and, for many others as well). They can be formulated entirely within the framework of almost characters and twisted induction, and yet their proofs essentially depend on the

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theory of character sheaves. These results, combined with those in (5.3), indicate that the sets A(L, '1/;) fit into the framework of a general HarishChandra theory for almost characters. This is worked out in detail in [1]' where the reader will also find a Mackey formula and explicit formulas for the scalar products of characters obtained by twisted induction. Note that part (b) of the following proposition gives a characterization of cuspidal almost characters entirely in terms of the Rf, map. This is also established in [41]' Proposition 2.2, by using a quite different proof but with slightly more restrictive assumptions on p. Proposition 5.4 Assume that the conditions in (5.2) hold.

(a) For every almost character X of G F there exists some cuspidal pair (L,'I/;) such that X has non-zero multiplicity in Rf,('I/;). If(L','I/;') is another such pair then Land L' are conjugate in G. (b) An almost character X of G F is cuspidal if and only if it is orthogonal to all RT, (f), where L is a proper regular subgroup in G and f is a class function on L F . (c) Let (L,'I/;) be a cuspidal pair. Then Rf,('1/;) is not identically zero. Moreover, if(M, 0) is anothercuspidal pairthen Rf,('1/;) and Rif(O) are either equal or orthogonal to each other. (Note that in the statement of (c), and in the sequel, such an equality of class functions may only be achieved after a suitable renormalization of the almost characters involved. In general, we will only have equalities up to non-zero scalar multiples of absolute value one.) Proof. The first statement in (a) is a reformulation, in terms of almost characters and twisted induction, of the results in [39], (10.4), (10.6). The uniqueness statement is contained in [39], (7.6). Now we consider (b). First assume that X is an almost character orthogonal to all Rf,(f) with L -I G. Then, in particular, X is orthogonal to all RT,('I/;) where (L,'I/;) is a cuspidal pair and L -I G. Using (a) we conclude that X must be cuspidal itself. Conversely, assume that X is a cuspidal almost character with non-zero multiplicity in RT,(f) for some regular subgroup L -I G and some class function f on G. Since the almost characters of L F form a basis of the space of class functions we may assume that f E A(L F ). Moreover, using (5.3)(a) and transitivity of twisted induction (see [8], (11.5)), we may assume that f is cuspidal. Hence (G, X) and (L, f) are two cuspidal pairs such that X has nonzero multiplicity in the corresponding "induced" functions. Using the

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uniqueness statement in (a), we conclude that L = G. This contradiction completes the proof of (b). Finally, let us prove (c). At first we show that Rf, ('ljJ) i= O. Let E be the supporting set for 'ljJ and fix an element s E GF which is the semisimple part of some element in E F . Let H := Gc(s)O and H uni the set of uni potent elements in H. Then the values of Rf, ('ljJ) at elements su (with u E H [ni) can be expressed in terms of generalized Green functions associated with the group H. We shall use the orthogonality relations for these functions to show that

x (s):= L

Rf, ('ljJ)(su)Rf, ('ljJ)(su)

i= O.

UEH:m

This would then clearly imply that Rf,('ljJ) cannot be identically zero. Now let us consider the above sum. Using the character formula [39], (8.5), we can write X(s) = Lx.x,c(x,x')f(s,x,x') where x,x' E G F , are such that X-I sx, X - I SX' E E, c(x, x') is a positive rational number, involving the cardinalities of L:, L:, etc. (where Lx = xLx- 1 n H etc.), and f(s, X, x') = QLx.H, ... (U)QLx"H, ... (U).

L

UEH:m

Here, QLx,H, ... denotes an appropriate generalized Green function. (See also the argument in [39], (9.5).) Now the orthogonality relations [39], (9.11) (see also [39], (25.6.1) to take into account complex conjugation), imply that f(s, x, x') is either zero or else the objects (Lx, .. .), respectively, (Lx" ... ), indexing the generalized Green functions are conjugate in H F . In the latter case, f(s. X, x') is given by the right hand side of the equation in [loco cit.] and, hence, is a strictly positive rational number. Putting these values together we find that X (s) itself is a strictly positive number. Hence there exists at least one U E H[ni such that Rf,('ljJ)(su) i= O. Now assume that Rf,('ljJ) and Rtf(O) are not orthogonal to each other. By assumption B, this can be reformulated into a statement about the characteristic functions of two induced complexes having non-zero scalar product. By [39], (9.9), (and using once more [39], (25.6.1), to take into account complex conjugation), this implies that there exists some n E GF such that nLn- 1 = M and ad(n)(O) = 'ljJ. But this certainly implies that Rf,('ljJ) and Rtf(O) are equal (using the character formula in [8], (12.2)). D

5.5. We shall now collect some explicit results about cuspidal almost characters which can be extracted from Lusztig's results in [39],

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Sections 18-23. (The assumptions in (5.2) are assumed to hold; in particular, the centre of G is connected.) Assume that G is simple modulo its centre. Let G --> Gad be the adjoint quotient of G. We now consider the various types individually. (a) Let G be of type An (n 2: 1). Then GF does not have any cuspidal almost characters at all. This is contained in [39], of course, but it also follows easily from Proposition 5.4 and the fact that every irreducible character of GF is uniform (d. [8], (15.4)). (b) Let G be of type B n , Cn or D n . Then every geometric conjugacy class As (for semisimple s E G*F) contains at most one cuspidal almost character. As explained in [44], (5.20.2), this implies that every cuspidal almost character X of GF is of the form X' . .\ where X' is a cuspidal almost character pulled back from G~d and .\ is a linear character of GF. For later reference, we point out explicitly some more details for groups of low rank (i.e., for those which are involved in an exceptional group). If G has type B 3 , C3 , D s , D 6 or D 7 then G does not have any cuspidal almost character at all. If G has type B 2 ~ C2 or D 4 then there is precisely one cuspidal unipotent almost character, and every other cuspidal almost character is obtained from this one by multiplying with a linear character of GF. They all have the same supporting set I: which is the preimage of a class of 2-singular elements (if P -I 2), respectively, unipotent elements (if P = 2). For these facts, see the proof of [39], (19.3). (c) Let G be of type G2 , F4 or E s . Then GF = G~d X Z(G)F and we may assume that G itself is simple. In this case, all cuspidal almost characters are unipotent, and there are precisely 4, 7, respectively 13. Proofs and information about the supporting sets I: can be found in [39], (20.6), [44], (7.2) (III,IV) (for G2 ), in [39], (21.3), [44]' (7.2) (I,ll) (for F4 ), and in [39], (21.2) , [45], (5.1) (for E s ). For later reference, we record in the following table the number of cuspidal almost characters such that the supporting set I: consists of £-regular elements (for bad primes £ -I p). Type £=2 £=3 £=5 G2 2 3 F4 3 5 9 7 9 Es,p -13 9 9 Es,p = 3 (d) Let G be of type E 6 or E 7 . Then every cuspidal almost character of GF is of the form X' ·'\ where X' is a cuspidal unipotent almost character

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of G~d and .\ is a linear character of G F (see [45], (4.2.1)). In each case there are 2 cuspidal unipotent almost characters which are supported on the same set 1::. Let (] be the isolated class in Gad of which 1:: is the inverse image. For type E 6 , the class C is either unipotent (if p = 3) or consists of 3-singular elements (if p =f 3). For type E 7 , the class C also is either unipotent (if p = 2) or consists of 2-singular elements (if p =f 2). This information is contained in the proofs of [39], (20.2). Note that this is also valid in the case where (G, F) is of non-split type E 6 , by [39], (20.4). (e) The above explicit results imply that, if (L, 'l/J) is a cuspidal pair in G with 'l/J unipotent then L is like G, i.e., simple modulo its centre. (This observation will be used frequently in several proofs by induction on dim G below.) Indeed, since G is simple modulo its centre, the Dynkin diagram of its root system is connected. The almost character 'l/J is unipotent hence constant on the cosets of the centre of L F . We can therefore view it as a cuspidal almost character of L~d' Since a simple group of adjoint type An (n ~ 1) does not have any cuspidal almost characters at all (see (a)) we conclude, using the classification of Dynkin diagrams, that Lad corresponds to a subdiagram which itself is connected. Hence the group L is like G. We can also extract the following useful results about unipotent almost characters.

Lemma 5.6 Assume that the conditions in (5.2) hold and that G is simple modulo its centre. Let X be a cuspidal unipotent almost character of G F and 1:: the supporting set for X. If 1:: contains £-regular elements then £ cannot divide the determinant of the Cartan matrix of the root system of G.

Proof. Every unipotent character of G F has the centre in its kernel, hence X (which is a linear combination of unipotent characters) is constant on the cosets of Z( Gl and may be viewed as a cuspidal unipotent almost character of G~d' Let d be the determinant of the Cartan matrix of the root system of G, and assume that £ divides d. We now check, case by case, that 1:: consists of £-singular elements. If G is of type An then n = 0, see (5.5)(a). Hence d = 1 and there is nothing to prove. If G is of type B n , Cn or D n then £ = 2. Hence p =f 2. By [40], (7.8) and (7.11), a cuspidal unipotent almost character is supported on a set 1:: which consists of 2-singular elements. (Note: The results in [loco cit.]

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are proved under some more restrictive conditions on p and q; these are not necessary any more, see Shoji's work [46].) If G is of type G2 , F4 or E s then d = 1 hence there is nothing to prove. If G is of type E 6 then £ = 3. Hence p -=J 3. By (S.S)(d), all cuspidal almost characters of G F are supported on a set E which consists of 3singular elements. Finally, if G is of type E 7 then £ = 2. Hence p -=J 2, and the analogous argument as in the case E 6 works as well. D A regular subgroup L in G will be called split if it is a Levi complement in some F-stable parabolic subgroup of G. Proposition 5.7 Assume that the conditions in (5.2) hold and that G is simple modulo its centre. Let X be a unipotent almost character of GF . Then there exists a unique cuspidal pair (L, 'ljJ), with L split and 'ljJ unipotent, such that X E A(L,'ljJ). Moreover, the stabilizer group W (cf (5.3)(a)) equals the group W(L) := Nc(L)j L, and the number of elements in A(L,'ljJ) equals the number of F-conjugacy classes ofW(L).

Proof. We know, by Proposition S.4(a), that there exists some cuspidal pair (L, 'ljJ) such that X E A(L, 'ljJ); moreover, L is unique up to conjugation in G. Since twisted induction preserves geometric conjugacy classes (see [37], Cor. 6) we must have that 'ljJ is unipotent. By (S.S), L is like G and corresponds to a unique subdiagram of the Dynkin diagram of G. This proves that the first component in the pair (L, 'ljJ) can be chosen to be split and then is unique. Now assume that G is of classical type. Then L also is of classical type and, hence, L F has at most one cuspidal unipotent almost character (see (S.S)(b)). So, in this case, 'ljJ clearly is unique. We also trivially have that W = W(L) (d. [44]' (S.16.1)). It remains to show the assertion about the cardinality of A(L, 'ljJ). Now (S.3)(b) and (S.4)(c) imply that this ('ljJW) , for w E W(L).f number equals the number of different functions If L w and LWf are not conjugate in G F then Rfj'ljJW) and Rtf ('ljJw ) are orthogonal to each other (see the proof of (S.4)(c)). Hence, since each L w has a unique cuspidal unipotent almost character the above number equals the number of GF -conjugacy classes of regular subgroups conjugate (in G) to L. This is nothing but the number of F-conjugacy classes of W(L), as required. Now assume that G is of exceptional type. Here, we use a counting argument, as follows. Firstly, the possibilities for the cuspidal pairs

Rt

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(L, 'l/J) in G F with L split and 'l/J unipotent are known by (5.5). In particular we see that, if L -I G then L is a torus (in which case 'l/J is the trivial character) or L corresponds to a sub diagram B 2 , D 4 (in which case there is only one 'l/J), or E 6 , E 7 (in which case there are exactly two 'l/J's). If L = G then the possibilities are listed in (5.5)(c,d). Next, for each group G and each cuspidal pair (L, 'l/J) with L -I G, we shall need the number of G F -conjugacy classes in the set {zLz- 1 I z E G, zLz- 1 F -stable}. These numbers are given in terms of F-conjugacy classes in the relative Weyl groups W(L) = Nc(L)/ L, and they can be computed explicitly in CHEVIE [21], for example. Then we fix a proper split regular subgroup L admitting (at least) one cuspidal almost character 'l/J. The number of pairs (L w , 'l/JW), with 'l/J and w varying, giving rise to different class functions Rfw ('l/JW) is bounded above by the number of possibilities of L w times the number of cuspidal unipotent almost characters of L~, hence by the number of F-conjugacy classes of W(L) times 1 (if L is a torus or of type B 2 , D4 ) respectively times 2 (if L is of type E 6 , E 7 ). Let n(L) be this total number. Then (5.3) (b) and (5.4) (c) show that the cardinality of U", A(L, JjJ) is bounded above by n(L). Now summing as well over the various possibilities for L, we obtain an upper bound for the total number of unipotent almost characters of G F . In each case, we find that this bound is in fact the exact number of unipotent almost characters (which is given in the tables in [6], (13.9)). The required numerical data, for G of type G 2 , F4 etc., are given as follows.

G2 : L = torus (6 classes, one 'l/J each); L = G (4 'l/J's); IA1(GF)1 = 10. F4 : L = torus (25 classes, one 'l/J each); L of type B 2 (5 classes, one 'l/J each); L = G (7 'l/J's); IA 1 (G F )1 = 37. E 6 : L = torus (25 classes, one 'l/J each); L of type D 4 (3 classes, one 'l/J each); L = G (2 'l/J's); IA1(GF)1 = 30.

E 7 : L = torus (60 classes, one 'l/J each); L of type D4 (10 classes, one 'l/J each); L of type E 6 (2 classes, two 'l/J's each); L = G (2 'l/J's); IA1(GF)1 = 76. E 8 : L = torus (112 classes, one 'l/J each); L of type D4 (25 classes, one 'l/J each); L of type E 6 (6 classes, two 'l/J's each); L of type E 7 (2 classes, two 'l/J's each); L = G (13 'l/J's); IA1(GF)1 = 166. The conclusion is that, first of all, we always have that the stabilizer group W equals W (L). In particular, it does not depend on the

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cuspidal almost character 'IjJ. This implies that, if w, w' E Ware such that Lw = LWI then also 'ljJw = 'ljJw ' (see the precise construction in [39], (10.6)). It follows that different 'IjJ's for a fixed L give rise to disjoint sets A(L,'IjJ). Hence, given X, the corresponding pair (L,'IjJ) indeed is unique. Moreover, the cardinality of each such set A( L, 'IjJ) equals the number of 0 F-conjugacy classes of W(L). This completes the proof.

6. On the number of unipotent Brauer characters 6.1. We keep the assumptions in (3.1). Our results on basic sets and Harish-Chandra series of Brauer characters explained above, as well as the conjecture in (3.4), are only concerned with the case where £ is a good prime. Already in [20], we mentioned the example where C F = C 2 (q) and £ = 2, in which case the unipotent characters do not give rise to an ordinary basic set of Brauer characters for HI' The reason for this was the fact that each unipotent character of C 2 (q) is a linear combination of Deligne-Lusztig characters Rrj 1 and certain class functions one of which is zero on all 2-regular eleme~ts. This implied that the restrictions of the unipotent characters to the 2-regular elements satisfy a non-trivial linear relation. Similar things also happen for £ = 3. We wish to show now how these questions can be approached in a conceptual way in the framework of the theory of character sheaves, by using the results that we have collected in Section 5. So, from now on, we consider a group C which satisfies the Assumptions A,B,C in (5.2). It is the purpose of this section to determine explicitly the number of unipotent irreducible Brauer characters of C F , for all bad primes £ not dividing q. 6.2. To formulate our first result about Brauer characters we introduce the following notion. We say that CF is £-uniform if, for all regular subgroups L c;: C, there exists no non-central semisimple element s E L*F of £-power order such that As(L F ) contains a cuspidal almost character. Here are some examples. (a) If £ is a good prime for C then C F is £-uniform. Indeed, assume that X E As(C F ) is cuspidal where s has £-power order. Since £ is good, L' := Cc.(s) is a regular subgroup in C* (see [20], Lemma 2.1). Let L c;: C be a regular subgroup dual to L'. Then Rt, induces an isometry between Es(L F ) and Es(CF) (see [37]). In particular, X equals (up to some non-zero scalar) Rt,U) for some class function f on L F . Using Proposition 5.4(b), we conclude that L = C, hence L' = C* and so, s is a central element. The same argument applies to every regular subgroup

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of L, hence C F is £-uniform. (b) If C is simple modulo its centre and of classical type of low rank then C F is 2-uniform. More precisely, this is the case if C has type B n , en (n ::; 3) or D n (n ::; 7). This follows easily using the information provided in (S.S)(b). (c) If C is simple modulo its centre and of exceptional type then C F is £-uniform. Indeed, all groups of classical type which can possibly occur as regular subgroups in Care £-uniform by (a), (b). We now consider in turn C of type C 2 , F 4 , E 6 , E 7 and E 8 • At each step it will be sufficient to check that the condition for £-uniformity holds with L = C. The relevant information can now be extracted from (S.S)(c), (d). In the following, we work in the K -vector space VI (C F ) of class functions on CF generated by the irreducible Brauer characters in B I (C F ). It is a well-known result from the representation theory of finite groups that this vector space is also generated by the characters p, where p runs over the ordinary characters in B I (C F ). We can now state:

Theorem 6.3 Assume that C satisfies the conditions in (5.2) and that C F is £-uniform. Then every Brauer character in BI(C F ) is a K -linear combination of the restrictions of the unipotent characters to the £-regular elements of C F . Proof. (The following proof is very similar to that of [20], Theorem 3.1.) As remarked earlier, VI (C F ) is generated by the elements p, where p runs over the ordinary characters of B I (C F ). Instead of ordinary characters, we may as well consider almost characters. To abbreviate notation, we shall denote by A t ( CF ) the union of all geometric conjugacy classes A s ( CF ) with s of £-power order. Then we are reduced to showing that every X (for X E A t ( C F )) can be written as a linear combination of the restrictions of the functions in Al (C F ) to the £-regular classes. We proceed by induction on dim C. If X is cuspidal then X must lie in A s ( C F ), for some central element s E oF of £-power order, since C F is £-uniform. Let As be the linear character of C F "dual" to s. Then X' := X . A.-I is a cuspidal almost character in Al (C F ) and we have X = X'. Hence we are done in this case. If X is not cuspidal then, by Proposition S.4(a), we Carl find a cuspidal pair (L, 'ljJ) with L =f C and such that X has non-zero multiplicity in Rf,('ljJ). Moreover, using (S.3)(a) and the fact that twisted induction

c

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preserves geometric conjugacy classes, we can write

x=

~ awRtj'l/JW ) ,

for various w E W, aw E K and 'l/Jw E Al(L~).

W

By induction, each '0w is a K-linear combination of the restrictions of the functions in Al (L~) to the €-regular elements. Now this restriction process commutes with twisted induction (see [8], (12.6); this was also one of the main ingredients in the proof of [20], (3.1)). Hence we deduce that X is a K-linear combination of class functions R~(e), for various proper regular subgroups M in G and almost characters e E Al (M F ). Using once more that R~ preserves geometric conjugacy classes yields that R~(e) is a K-linear combination of elements in AI(GF ), and using once more that restriction to €-regular elements commutes with twisted induction, we finally conclude that X indeed is a K-linear combination of the restrictions of the functions in Al (G F ) to the €-regular elements. This completes the proof. D Theorem 6.4 Assume that G satisfies the conditions in (5.2) and that G is simple modulo its centre. Let

Then any two different elements in A; have different restrictions to the €regular elements, and these restrictions are linearly independent elements in VI (GF). If, moreover, GF is €-uniform, then they do form a basis for

VI(GF). Proof. We proceed in several steps. (a) Let us fix one cuspidal pair (L, 'l/J) with 'l/J E Al (L F ) and '0 -=J O. Let 1:: ~ L be the supporting set for 'l/J as in the remarks preceding Assumption (5.2C). This set is the inverse image of some isolated class C in L / Z (L ). Since 'l/J is constant on the cosets of Z (L land '0 -=J 0 we deduce that the class C must consist of €-regular elements. (b) We keep the assumptions in (a) and define

where z runs over the central elements in L' F of €-power order (we shall denote this set by Z(L')f) and '\z is the linear character of L F dual to z. We then claim that

'l/J' ='0' = IZ(L')[I'0·

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The second equality is clear since all linear characters .\Z have value 1 on all £-regular elements. To prove the first we must show that, if x E L F is such that 'l/J'(x) =f 0 then x is £-regular. The condition 'l/J'(x) =f 0 is equivalent to 'l/J(x) =f 0 and Lz .\z(x) =f O. The first inequality implies that x E E F . Since C only contains £-regular elements (see (a)) we conclude that the £-part of x lies in Z (L)F. On the other hand, the orthogonality relations for the characters of a finite abelian £-group and the second inequality Lz .\z (x) =f 0 imply that the £-part of x lies in the intersection of the kernels of the .\z, hence in L:er (where Lder denotes the derived subgroup). Taken these two statements together we see that the £-part of x lies in the centre of L:er . Now Lemma 5.6 (note that L also is simple modulo its centre) shows that this £-part must be trivial, since ;j; =f 0 and the order of Z (Lder)F divides the determinant of the Cartan matrix of the root system of L. Hence x is an £-regular element, as required. (c) Let (L, 'l/J) be a cuspidal pair as in (a). Then the character 'l/J' as in (b) is a sum of almost characters, exactly one of which, 'l/J, is unipotent and all the others, 'l/J . .\Z for Z =f 1, are non-unipotent (see [8], (13.30)). Since twisted induction preserves geometric conjugacy classes we conclude that the scalar product between RT, ('l/J) and RT, ('l/J . .\z), for Z =f 1, is zero. Hence the scalar product between RT, ('l/J) and RT, ('l/J') is the same as the scalar product of RT, ('l/J) with itself, which is non-zero by Proposition 5.4(c). Consequently, RT, ('l/J') has non-zero scalar product with itself. In particular, this class function is non identically zero. Using the fact that restriction to £-regular commutes with twisted induction, we also conclude that RT,(;j;) =f o. (d) Now let (L i ,'l/Ji) (1 ~ i ~ t) be cuspidal pairs in G such that RT,; ('l/Ji) E A; for all i. Assume, in addition, that RT,i (;j;i) =f RT,j (;j;j) if i =f j. Suppose we have a linear combination

where ai E K. We must show that aj = 0 for all j. For each i, let ad IZ (L7) fI and 'l/J: = ;j;; be constructed as in (a). Then we have the relation a~ RT,1 ('l/J~) + ... + a~RT,t ('l/J~) = O.

a; :=

Now fix j and consider the scalar product of RT,j ('l/Jj) with the term (for some i) in that relation. Assume that this scalar product is non-zero. By the definition of 'l/J;, 'l/Jj, there exist elements Zi E Z(L7){ and Zj E Z(Lj){ such that the scalar product between RT,)'l/Jj· and RT,,('l/Ji· .\z') is non-zero. Now Proposition 5.4(c) implies that these two

RT" ('l/JD

.\zJ

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class functions are equal (both 'l/Ji . .\z, and 'l/Jj . .\Zj are cuspidal almost characters). Hence their restrictions to the £-regular elements are equal. So we must have that i = j. Thus, if we take the scalar product of Rtj ('l/Jj) with both sides of the above linear relation we see that, on the left hand side, only the term with index j remains. This must be zero, so we conclude that aj = 0 (since Rt/'l/Jj) # 0, see (c)). (e) We have shown in (d) that the different restrictions of the class functions in A; to the £-regular elements are linearly independent. It remains to show that different elements in A; indeed have different restrictions to the £-regular elements. Let (L, 'l/J) and (M,O) be cuspidal pairs giving rise to elements in A; such that Rt(;j;) = R~(e). Using (b) we find that Rt('l/J') is a non-zero multiple of R~(e') where 'l/J',O' are defined as in (b). Now the scalar product of Rt ('l/J) with Rt ('l/J') is the same as the scalar product with itself, hence is non-zero (see (c)). So the scalar product with R~(O') also is non-zero. Hence there exists some z E Z(M*){ such that Rt('l/J) and R~ (0· .\z) have a non-zero scalar product. By Proposition 5.4(c) we conclude that Rt('l/J) = R~f(O· .\z). But this can only happen if z = 1 since 'l/J is unipotent and O· .\z is non-unipotent for z # 1. So we have that Rt('l/J) = R~(O). This completes the proof. (f) Finally, if GF is £-uniform then Theorem 6.3 implies that the restrictions of the unipotent almost characters to the £-regular elements generate VI(G F). Using (5.3)(a) and (5.4)(a), we conclude that VI(G F) is also generated by the restrictions of the class functions in A;. Since they are also linearly independent, they do form a basis for VI(GF), as required. 0 Remark 6.5 Let G be as in (6.4) and assume that G F is £-uniform. (a) Let £ be a good prime. This implies that, if (L, 'l/J) is any cuspidal pair in G then ;j; # 0 (using the characterization of the supporting set E in Assumption 5.2C). By Theorem 6.4, we conclude that the set A; contains all unipotent almost characters of GF . Hence the number of irreducible Brauer characters in HI (G F ) is equal to the number of unipotent characters of GF . In [20], (3.1), it is shown that every Brauer character in this union of blocks is an integral linear combination of the restrictions of the unipotent characters to the £-regular elements. Thus, we obtain a new proof for the fact that the unipotent characters even give rise to a basic set. (In [20], this was proved using a counting argument.) (b) Let £ be a bad prime and G of exceptional type. By (6.4), we know that every Brauer character in HI (G F ) is a K-linear combination

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of the restriction of the unipotent characters to the £-regular elements, and we know how many of these restrictions are linearly independent (see the tables in (6.6) below). But the question remains whether or not we can select an ordinary basic set from the unipotent characters. We hope to be able to settle this by explicitly computing the necessary parts of the character table of G F , in the framework of CHEVIE [21]. 6.6. We close this section with a list giving the number of irreducible Brauer characters in B 1 (G F ), for the cases where G is simple modulo its center, satisfying the conditions in (5.2).

• If £ is a good prime, this number equals the number of unipotent characters of G F , and these even form an ordinary basic set (see

[20]). • If G is of classical type En, en or D n and £ = 2, this number equals the number of unipotent classes of G F . If p is sufficiently large and G is of split type, then there also exists an ordinary basic set (see [19], III). • If G is of exceptional type, these numbers are given as follows. (We omit the entry if £ is good.)

£=2 £=3 £=5 9 8 F4 28 35 E6 27 28 E7 64 72 150 162 E 8 , p -=J 3 131 E 8 , p = 3 133 162 Type

G2

For G of type G 2 these numbers are correct without any assumption on p or q, since they can also be computed from the explicit knowledge of the character table, see [29]. For the other types, these results depend on the validity of the conditions in (5.2). Note also that the result for type E 6 is independent of whether F is of split or non-split type. We shall give some indications for the proofs in the cases where G is of exceptional type. In the following, we consider only unipotent almost characters. The strategy is given by Theorem 6.4. We then use a counting argument as in the proof of Proposition 3.7. We give one example, the other being entirely analogous. Let G be of type E 7 and £ = 2. We

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consider the possibilities for cuspidal pairs (L, 'l/J) with 'l/J unipotent and 0: If L = T torus then ;j; is the trivial character restricted to the 2-regular elements of T F . If L is of type D 4 then ;j; = a by (5.5)(b). If L is of type E 6 there are two possibilities for 'l/J, and we have that ;j; -I a (see (5.5)(d)). If L = G then there are again two possibilities for 'l/J, and in both cases we have ;j; = a (see (5.5)(d)). Thus, we need only count the pairs (L, 'l/J) where L is a torus in G or of type E6 • The numbers of G F -conjugacy classes of such pairs are given in the proof of Proposition 5.7. Also, if (L, 'l/J) and (L', 'l/J') are two pairs where Land L' are not GF-conjugate then Rf('l/J) -I Rf,('l/J'). Hence, we sum over the above possibilities and find that there are 60 + 2 . 2 = 64 unipotent irreducible Brauer characters.

;j; -I

7. A worked example: The exceptional groups of type E 6

7.1. As before, let G be a connected reductive group defined over IF q , and such that G has connected centre and is simple modulo its centre. Throughout this section we consider exclusively the case where G is of split type E 6 . We also keep the notation introduced in (3.1). The purpose of this section is to work out explicitly properties of the e-modular Brauer characters in the unipotent blocks B 1 of GF (d. (3.3)), by using the various methods described before. We will assume throughout that e > 3 is a good prime. (Recall that the bad primes for G are 2 and 3.) Then the unipotent characters form an ordinary basic set for B 1 (see [20]). Information about these characters, their degrees, their distribution into families respectively ordinary Harish-Chandra series is given in [6], p.480. There are 17 families of unipotent characters (14 with 1 element, 2 with 4 elements, and 1 with 8 elements). We order them according to the scheme described in (3.4) (see Tables 1 and 2 below), and we let D denote that part of the decomposition matrix of B 1 which has rows labelled by unipotent characters. (We do not yet fix an ordering of the irreducible Brauer characters nor of the characters inside a given family.) For d;::: 1 let d E Z[X] denote the d-th cyclotomic polynomial. We then have

Given q and e we let e be the multiplicative order of q modulo e. Next we assume that e divides IGFI, so that e E {I, 2, 3, 4, 5, 6, 8, 9, 12}. Then edivides e(q) and this is the only term with this property in the above factorization of IGFI, unless e = 1 and e= 5.

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7.2. If e(q) is the only term in the above factorization of ICFI divisible by £ then we are in the "generic" situation defined and studied in [3]. In these cases, the defect groups of the unipotent blocks are abelian and the block distribution of the unipotent characters is known and given in terms of twisted induction of e-cuspidal characters from e-split regular subgroups in C (see [lac. cit.] for more details). Let us consider the case where e = 1. Then we also assume that £ > 5. In this case, the reduction map X f--+ X defines a bijection between the unipotent characters of C F and the irreducible Brauer characters in B 1 • Hence D is the identity matrix. Moreover, the correspondence X f--+ X defines a bijection between the ordinary Harish-Chandra series of unipotent characters (as given in the table in [6], p.480) and the modular Harish-Chandra series of irreducible Brauer characters in B 1 . (See [29] and [43].) Now assume that e E {5, 8, 9, 12}. Then the Sylow £-subgroups of C F are cyclic. In these cases, the decomposition matrices are given in terms of Brauer trees. All unipotent characters are real valued except the two cuspidal unipotent characters which are complex conjugate to each other. Therefore, the Brauer trees are straight lines with the two cuspidal unipotent characters lying on opposite sides of one node. Thus, the decomposition matrices indeed have a lower unitriangular shape, and the two cuspidal unipotent characters remain irreducible as Brauer characters (they are leaves on the tree). Also, the distribution of the Brauer characters into Harish-Chandra series is known in these cases. (For all this and more precise information about these trees, see [31]). Thus, in the above cases, our matrix D is explicitly known. We are now left with examining the cases where e = 2,3,4 or 6. We will not be able to determine the matrix D completely. Our aim is to prove that the two cuspidal unipotent characters remain irreducible as Brauer characters (in all cases), and to determine the Harish-Chandra series of irreducible Brauer characters (in the case e = 3). Note that, in [23], we already found the numbers of irreducible unipotent Brauer characters in the various Harish-Chandra series, for e odd. 7.3. We shall say that an ordinary character of C F lies in B 1 if it is a sum of the ordinary characters contained in B 1 . An ordinary character in B 1 will be called projective if it is a sum of the characters of the projective indecomposable modules (PIM's for short) corresponding to the various irreducible Brauer characters of B 1 . Using Brauer reciprocity, we can think of the entries of our matrix D as the multiplicities of the unipotent characters in the characters of the PIM's corresponding to the irreducible Brauer characters in B1 . We shall now describe two construc-

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tions which yield ordinary characters which are projective regardless of which particular prime £ we consider. (a) One source is given by Harish-Chandra induction of ordinary Gelfand-Graev representations from (split) Levi subgroups L) of G F where J is a subset of the set of fundamental reflections of the Weyl group W of G (d. Section 2). We fix the following labelling of the nodes of the Dynkin diagram of G: I

3

•

•

4

5

6

•

A complete set of associate classes of subsets J as above is given as {I}, {I,4}, {2,4}, {I,4,6}, {I,3,6}, {2,3,5,6}, {I,3,5,6}, follows: {4,5,6}, {2,3,4,6}, {I,2,3,5,6}, {3,4,5,6}, {I,2,3,4,6}, {2,3,4,5}, {I, 3, 4, 5, 6}, {I, 2, 3, 4, 5}, {I, 2, 3, 4, 5, 6}. Now let J be one of these subsets. We consider the Harish-Chandra induction of the Gelfand-Graev character f LJ of L). This gives a projective character of G F . The multiplicities of the unipotent characters of G F in (f L J ) are given by the multiplicities of the irreducible characters of J W in the induced sign representations from the parabolic subgroups Wj of W (see [7], (70.24)). The resulting matrix of scalar products is given in Table 1. The labels for the unipotent characters in the first column are taken from the table in [6], p.480; the second column gives the number of the family in which the unipotent character lies. Furthermore, Wi denotes the restriction of (f L J ) to 8 1 , where J is the i-th subset in J the above list. We have put . instead of O.

n,

Rt

Rt

(b) From now on, we shall also assume that q is a power of a good prime for G. Then another source for projective characters is given by Kawanaka's generalized Gelfand-Graev representations (GGGR's) [35]. These are representations induced from unipotent subgroups of G F hence they yield projective characters, one for each unipotent class of GF. By [loco cit.], (2.3.2), the characters of the various GGGR's are uniform functions (for our group GF), and they can be explicitly written as linear combinations of Deligne-Lusztig characters Rfi,(J' Since also the multiplicities of the unipotent characters in the Deligne-Lusztig characters are known, we can determine explicitly the matrix of scalar products between unipotent characters and characters of GGGR's; it is given in Table 2. There, * denotes an entry 2 0 and. stands for O. The columns of this matrix correspond to the GGGR's of the various special unipotent classes of G. The ordering is given as follows. There is a bijective cor-

237

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Table 1: Harish-Chandra induced Gelfand-Graev characters (1,0) (6,1) (20,2) (15,5) (30,3) (15,4) D 4 ,1 (64,4) (60,5) (24,6) (81,6) (80,7) (90,8) (20,10) (60,8) D 4 ,T (10,9)

E 6 10] Ed 02 ] (81,10) (60,11) (24, 12) (64,13) (15,17) (30,15) (15,16) D 4 ,E (20,20) (6,25) (1,36)

:F 1 2 3 4 4 4 4 5 6 7 8 9 9 9 9 9 9 9 9 10 11 12 13 14 14 14 14 15 16 17

WI W2 W3 W4 W5 W6 W7 1 6 1 20 5 1 15 5 1 1 30 10 3 1 1 15 5 2 1

wa W9 WIO Wl1

64 60 24 81 80 90 20 60

24 25 10 36 40 45 10 30

8 11 4 15 20 21 4 16

4 4 2 9 12 15 4 8

2 5 1 6 10 9 1 9

1 2 1 1 3 1 6 3 6 2 1 5 3

1 2 2 1 3 1 2 1

10

5

3

1

2

1

1

1

81 60 24 64 15 30 15

45 35 14 40 10 20 10

24 21 8 24 6 13 7

18 14 6 20 6 11 5

12 13 5 14 3 8 5

9 9 3 11 3 7 4

4 3 5 6 4 4 2 1 2 6 4 8 1 1 3 4 4 5 3 3 2

2 3 1 4 1 3 2

1 3 1 2

20 15 11 10 8 7 5 4 6 6 5 4 4 3 3 2 2 3 1 1 1 1 1 1 1 1 1

4 2 1

WI2 WI3 WI4 WI5 WI6 Wl7

1 1 1

1

1

1 1

1 1

1

2 2

2 1 2 1

1 1 1

3 1 1

3 2 1

2 1 1

1 2 1 1

3 2 1

1 1 1 1 1

1 1 1

1

respondence between families :F and special unipotent classes C, given in terms of the Springer correspondence (see [38], (13.1)). Then f 4 , f 5 (resp. f 17 , f 18 ) are the GGGR's attached to the special unipotent class corresponding to the two 4-element families, flO, f ll , f l2 correspond to the 8-element family, and all others correspond to the remaining 1element families. We see that we have obtained a first approximation to a square triangular matrix. We can now state our first result. Theorem 7.4 Let G be of type E 6 as in (7.1). Assume that q is a power of a good prime for G and that £ > 3. Then the restrictions of

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238

Table 2: Generalized Gelfand-Graev characters (1,0) (6,1) (20,2) (15,5) (30,3) (15,4) D 4 ,1 (64,4) (60,5) (24,6) (81,6) (80,7) (90,8) (20,10) (60,8) D 4 ,T (10,9) E 6 [0] E 6 [02] (81,10) (60,11) (24,12) (64,13) (15,17) (30,15) (15,16) D 4 ,E (20,20) (6,25) (1,36)

:F 1 2 3 4 4 4 4 5 6 7 8 9 9 9 9 9 9 9 9 10 11 12 13 14 14 14 14 15 16 17

[1 [2 [3 [4 [5 [6 [7 [8 [9 [10[11 [12 [13 [14 [15 [16 [17 [18 [19 [20 [21

1

* * * * * * * * * * * * * * * * * * * * * * * * * * * * *

1

* * * * * * * * * * * * * * * * * * * * * * * * * * * *

1

* * * * * * * * * * * * * * * * * * * * * * * * * * *

1 1 1 1

* * * * * * * * * * * * * * * * * * * * * * *

1

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

1 1

* * * * * * * * * * * * * * * * * * * *

1

* * * * * * * * * * * * * * * * * * *

* * * * *

1 2 1

* * * * * * * * * * *

* * * * * * * * * * *

1 1 1 1 1

* * * * * * * * * * *

1

* * * * * * * * * *

*

1

* * * * * * * *

* * * * * * * *

1

* 1 * * * *

1 1 1 1

* * * * 1 * * * * * 1 * * * * * *

1

the two cuspidal unipotent characters to the £-regular elements of G F are irreducible Brauer characters.

Proof. If e = 1 then £ divides l(q) and, possibly, 5(q). Checking the table of character degrees in [6], pA80, we see that the two cuspidal unipotent characters are of £-defect a hence clearly remain irreducible as Brauer characters. The same argument applies in the cases where e E {2, 4, 5, 8} and £ > 3. If e E {9,12} then we can apply the results on Brauer trees already mentioned in (7.2). So it remains to consider the cases where e = 3 and e = 6. The idea of the proof is to show, at first, that our matrix D has in fact a lower unitriangular shape and

Representations in Non-defining Characteristic

239

that, secondly, the rows corresponding to the two cuspidal unipotent characters have only one non-zero entry. Building a lower unitriangular matrix of scalar products means that we have to order the unipotent characters in such a way that, for each unipotent character X, we can find a projective character,

'2 >'3 (D 4 ,1) (D 4 , r) (D 4 , c)

1 1

1 1

1

This follows since Rf(cp) (where L is of type D 4 and cp is the unique cuspidal unipotent character of LF, which is of €-defect 0), f 5 , f ll , f 18 already yield a lower unitriangular matrix of scalar products. Then use the fact that the decomposition numbers for a block with a cyclic defect group are or 1. We can now label >'1, >'2, >'3 by (D 4 , 1), (D 4 , r), (D 4 , c) respectively, and we see that >'1, >'2 lie in the unique modular BarishChandra series corresponding to a Levi subgroup of type D 4 , and that >'3 is cuspidal. All the other unipotent characters lie in the principal block of G F . This block now contains 24 irreducible Brauer characters, 5 of which are cuspidal, 10 are in the principal series, 5 correspond to the Levi subgroup of type A 2 and 4 to that of type A 2 x A 2 . In the course of the proof of Theorem 7.4 we have already shown that the restrictions of the cuspidal unipotent Brauer characters to the €-regular elements are irreducible Brauer characters. We obtain the matrix of scalar products with projective characters displayed in Table 3. No.9, 10, 11, 12, 19, 23 may split up further while the others indeed are characters of unipotent quotients of PIM's. For their construction see the remarks below. (a) The columns labelled by ps give the projective characters corresponding to the irreducible Brauer characters in the principal series. These columns are simply the columns of the decomposition matrix of the endomorphism algebra of the RGF-permutation module on the cosets of a Borel subgroup of G F (by a result of Dipper [9]). This algebra is a specialization of the generic Iwahori-Becke algebra H(E6 ) associated with the Weyl group W of G; its decomposition numbers have been computed in [17]. (b) The columns labelled by A 2 correspond to the irreducible Brauer characters in the unique Barish-Chandra series associated with a Levi subgroup of type A 2 • Consider a Levi subgroup L of type D 5 . Its Sylow €-subgroups are cyclic hence the decomposition numbers are given in terms of Brauer trees, see [16]. We find that, already in LF, there are 5 irreducible Brauer characters in the Barish-Chandra series associated with a Levi subgroup of type A 2 . We consider the Barish-Chandra induction of the corresponding projective characters, and these give us the 5 columns above. (They cannot split up any further: The only possibil-

°

243

Representations in Non-defining Characteristic

Table 3: Projective characters for e = 3 No. 1 2 3 4 5 6 7 8 9 1011 12 13 14 15 Series ps ps ps ps ps ps ps ps A~ c c A~ ps A 2 ps 1 (1,0) (6,1) 1 1 (20,2) 1 1 1 (15,5) 1 1 (30,3) 1 1 (15,4) 1 1 (64,4) 1 1 1 1 1 (60,5) 1 1 1 1 1 1 (24,6) 1 1 1 1 E 610j E 6 [02] 1 1 (10,9) 1 1 (60,8) 1 1 1 1 1 1 1 1 1 (20,10) 1 1 I (80,7) 1 1 1 1 1 1 1 1 1 (60,11) 1 1 1 1 1 1 * * (24, 12) 1 1 * * (64,13) 1 1 1 2 1 1 1 * * 1 1 1 (15,17) 1 * * (30,15) I 1 1 2 * * (15,16) 1 1 1 * * (20,20) 1 1 1 * * 1 1 1 1 (6,25) 1 * * 1 (1,36) 1

* *

16 17 18 192O 21 22 2324 C A~ A~ A 2 C C

A2 A2 A2

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

*

ity would be to subtract projective characters corresponding to Brauer characters in the principal series, and this is not possible, as can be seen from the entries in the above matrix.) (c) Now all the remaining columns must correspond to Brauer characters which are cuspidal or belong to the unique Barish-Chandra series associated with a Levi subgroup of type A 2 x A 2 • In the latter case, we simply say that they lie in the A~-series. The projective characters 'lis (induced from A 2 x A 2 ), 'lin (induced from A 2 x A 2 x Ad and W15 (induced from A 5 ) show that the irreducible Brauer characters corresponding to no. 9, 12, 20 must lie in the A~-series. Note that 'lin is a direct summand of 'lis, hence we can subtract the first from the latter; we shall denote this new character by w~. (It is the projective character no. 9 in the above table.) Then the projective characters corresponding to the irreducible Brauer characters in the A~-series are summands of w~ or 'lin.

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244

(d) The Brauer characters corresponding to no. 10, 11 clearly are cuspidal, by Theorem 7.4. The corresponding projective characters in the above table are obtained by splitting up the GGGR r 12 (see Lemma 7.5 and the argument in the proof of Theorem 7.4). Note that this splitting can only happen if (10,9) is not a constituent in the two summands. The entries marked by a star are given by evaluating non-constant polynomials in q, hence they only give rather large bounds. Since e divides the index of every proper split Levi subgroup of GF we conclude, using [22], Theorem 4.2, that no. 24 (the modular Steinberg character, i.e., the irreducible Brauer character whose PIM is the restriction of the Gelfand-Graev representation to 8 1 ) must correspond to a cuspidal Brauer character. In order to find the decision for the remaining Brauer characters we use a refinement of this argument: Let

O. One has av : : : : bv modp. Let us assume that p is an injection for modules of dimension at most n and assume that M and N have dimension n + 1. If there is a V E S which is a submodule of M and N, then the modules M IV and N IV are isomorphic, since their dimension is less than n. Hence, we may ':Y assume that for every V E S, av = 0 or bv = O. Let M' = EBvV p and N' = EBvV~. Again, M' and N' have dimension less than n; hence M' ::: N',i.e., M' = N' = 0 and M = N = 0, which gives a contradiction.

o Lemma 2.6 The subgroup Ro(H) of Ro(K'H) consists of those elements f such that PKI(f)(h) E K[X] for all hE H.

Proof. It follows from the construction of Ro(H) that, for f E Ro(H) and h E H, p(f)(h) E K. Let now f E Ro(K'H) such that p(f)(h) E K for all h E H. We can clearly assume that H is semisimple, since p(f)(h+r) = p(f)(h) for h E Hand r E J(H). We can also assume that H is simple. Let VI,·" , v.. be a complete set of representatives of isomorphism classes of simple K' H -modules. Then the Galois group of K' over K acts transitively on {ch( [Vi])}, hence p(f) = Q' 1:::=1 ch( [Vi]) for some Q' E K. Since p is injective, by Proposition 2.5, we have f = Q' 1:::=1 [Vi] for some integer Q' and the lemma is proved. 0 We put Irr(H) modules.

=

{ch(,;)V])} where V runs over the simple H-

Proposition 2.7 The map lK @ ch : K @ Ro(H) - CF(H) is an injection, Le., the elements of Irr(H) are linearly independent. If H is semisimple, then the map above is an isomorphism. Proof. One can assume that H is simple and the proposition follows 0 then from Lemma 2.4. Assume now that for every simple H-module V, the canonical map K - Z(EndH(V)) is an isomorphism, i.e., EndH(V) is a central Kalgebra; we say that H is a quasicentral K-algebra.

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258

Proposition 2.8 Assume that H is quasicentral. (1) The form (., .) induces a perfect pairing between Ko(H) and Ro(H):

if V, VI are two simple H -modules and Pv is a projective cover of V, then we have (~v [Pv ], m~, [VI]) = 8[V],[v'j. (2) If K ' is an extension of K, then the mapst~: Ro(H)

and

K

tK~

-

: Ko(H)

---+

---+

-

Ro(K'H)

Ko(K'H) are isomorphisms.

(3) If K ' is a neutralizing field for H, then Ro(H) = Ro(K'H) and rank Ro(H) = rank Ro(K' H). Proof. Since HomH(PV , V') ':::::' HomH(V, V') and dimK HomH(V, V') = m~8[V],[v'l we have ([PvL [V']) = m~8[V],[v'l. This implies that the pairing induced by (0,.) is perfect. The other statements are clear. 0

There is an easy characterization of quasicentral algebras: Proposition 2.9 The following statements are equivalent:

(1) the K -algebra H is quasicentral; (2) Ro(H)

= Ro(K'H) for any extension K' of K;

(3) rank Ro(H) = rank Ro(K'H) for any extension K' of K;

(4) there is a finite Galois extension L of K which is a neutralizing field for H and such that rank Ro(H) = rank Ro(LH). Proof. Let L be a finite Galois extension of K which is a neutralizing field for H. Let V be a simple H-module and D = EndH(V), Then, V@KL is a direct sum of [Z(D) : K] non-isomorphic simple LH-modules. If Z(D) #- K, it implies rank Ro(H) < rank Ro(LH). Hence, (4)::::} (1). By Proposition 2.8, (1) implies (2). Finally, (2) ==? (3) ==? (4) is clear. 0

2.3 The Brauer-Cartan square

From now on, V is an integrally closed local domain, K its field of fractions and k its residue field. We assume K and k are perfect. Let H be an V-algebra, free and finitely generated as an V-module. Lemma 2.10 The image of ch : Ro(KH)

---+ CF(KH) is contained in the V-submodule CF(H) and the image of PK : R6(KH) ---+ Maps (KH, K[X]) is contained in the V-submodule Maps(H, V[X]).

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Proof. Let V be a simple K'H-module where K' is a finite extension of K, with ch(V)(x) E K for all x E KH. Then, for h E H, the characteristic polynomial p(V) (h) divides (in K' [X]) the characteristic polynomial p( K H) (h) = p( H) (h) associated to the regular representation. Since p(H)(h) E O[X], the roots of p(V)(h) are algebraic over 0, hence ch(V)(h) is algebraic over O. Since ch(V)(h) E K and 0 is integrally closed in K, this implies ch(V)(h) E O. Similarly, the coefficients of p(V)(h) are algebraic over 0 and are in K, hence are in O. 0

The following proposition establishes the existence of the decomposition map: Proposition 2.11 There exists a unique map do : Ro(K H) - Ro(kH)

which makes the following diagram commutative: Ro(KH)

ld

:J

R;(KH) ~

:J

1 kt(kH)

o

Ro(kH)

~

Maps(H,O[X])

lt

k

Maps(kH, k[X])

Proof. Note first that the unicity follows from the injectivity of Pk (cf Proposition 2.5). Let K' be a finite extension of K such that K' H is split. Let 0' be a valuation ring, 0 C 0' c K' with maximal ideal I such that In 0 = m, the maximal ideal of 0, and residue field k'. Let V be a simple K' H-module with [V] E Ro(K H). Since finitely generated torsionfree O'-modules are free [Go, §5.2], there exists an 0'H-Iattice V' such that K'V' :::: V. Then, Pk,([k'V']) is the reduction mod I of PK([V]) E Maps(O' H, O'[X]). Since PK([V]) is actually in Maps(OH, O[X]) byassumption, we have also Pk,([k'V']) in Maps(kH, k[X]), hence [k'V'] E Ro(kH) by Lemma 2.6 and we put d([V]) = [k'V']. 0

Note that the decomposition map exists not only for 0 integrally closed but, more generally, when the image of PK : kt(K H) Maps(K H, K[X]) is contained in the O-submodule Maps(H, O[X]). The following proposition is a direct consequence of the definition: Proposition 2.12 Let p be a prime ideal of 0 such that kp = (Op)jp

is perfect and 0 jp is integrally closed. Then, the following diagram is commutative: Ro(KH) ~ Ro(kH)

1

r

Ro(kpH)

Ro(kpH)

do.

dOl.

260

M. Geck and R. Rouquier

Suppose now that K H is a quasicentral K -algebra. We define the map e : Ko(kH) ----+ Ko(KH) as the map dual to d with respect to the pairing (0, .), i.e., for TJ E Ko(kH) and X E Ro(KH), we have

(TJ, d(x)h = (e(TJ), X)K. Let us now give an alternative definition of e, without using d. Let 0 be a strict henselisation of 0 [Ray, Chapitre VIlI]: this is an henselian (local) ring, local extension of 0, faithfully flat as an 0module, with residue field k a separable closure of k. Furthermore, since o is integrally closed, the ring 0 is an integral domain [Ray, Chapitre IX, corollaire 1]. Let now 0' be a valuation ring, local extension of 0, contained in the field of fractions K of 0, with residue field k'. Since 0 is henselian, every idempotent of kH can be lifted to an idempotent of OH, hence every projective kH-module can be lifted to a projective OH-module. Let P be a projective kH-module and V a KH-module. There exists an O'H-lattice M such that V@K:::: M@K and a projective 0' H -module Q such that Q @ k' :::: P @ k'. We have

([P]' d([V]))); = ([Q @ k'], [M @ k'])k' = ([Q], [M])o' = ([KQ], [K M]) j{. (Note that similarly, ([P],l @ 1.i,).i: = (e[[P]), f @ 1K )K . 1.i: for f E CF(H)). Hence, e([P]) = [KQ] (viewed in Ko(KH)). Furthermore, one has dOCK 0 e([P]) = cd [P]) , hence, the following diagram ("BrauerCartan square") is commutative:

Ro(KH)

lCK Ko(KH) With the additional assumption that the algebra K H is semi-simple, one has Ko(KH) = Ro(KH) and we recover the usual Brauer-Cartan triangle (cf [Se, §15]). 3. On the number of simple modules

We keep the assumptions above: We have 0 an integrally closed local domain with residue field k perfect and field of fractions K perfect and H an O-algebra, free and finitely generated as an O-module. The following proposition generalizes a theorem of Hattori [CuRe, Theorem 32.5] about the injectivity of the Cartan map. In Hattori's

Centers and Simple Modules for Iwahori-Hecke algebras

261

theorem, it is assumed that H/[H, H] is free as an O-module (and 0 is assumed to be a discrete valuation ring). Proposition 3.1 Assume that kH is quasicentral. If the image of the

canonical map tk : CF(H) - CF(kH) contains ch(Ro(kH)), then the map e is injective (hence, given two projective H -modules M and N, we have K M ~ K N if and only if M ~ N). In particular, the decomposition map d has finite cokernel and the algebra K H has at least as many simple modules as the algebra kH. Proof. Let us prove first that e is injective. Replacing 0 by a strict henselisation of 0, we can assume that kH is split. Let P, Q be two non-zero projective kH-modules with no common direct summand such that e([P] - [Q]) = O. Assume that P has minimal dimension with this property. Let V be a simple kH-module and put 'P = [V]. By assumption, ch'P is the image of some f E CF(H). So we have

< [P], 'P > ·h = ([P]' ch'Ph = ([P], f . 1kh = (e[P], f)K . h. As e([P]) = e([Q]), we get < [P] - [Q], 'P >k ·h = O. If k has characteristic zero, this implies that [P] = [Q]. Assume then that the characteristic p of k is positive. Then, the multiplicities of a projective cover Pv of V in P and Q as a direct summand are equal modulo p. Since P and Q have no common direct summand by assumption, it implies that the multiplicities of Pv in P and Q are both divisible by p. So, there exists Po and Qo two projective kH-modules with P ~ Pg and Q ~ Qb; hence e([Po] - [Qo]) = O. Since Po has strictly smaller dimension than P, it implies Po ~ Qo, hence P ~ Q, which is impossible. This completes the proof that ker e = O. Now, by definition of e as dual of d, it follows that d has finite cokernel. Since d: Ro(KH) - Ro(kH), it is clear that K H has at least as many simple modules as kH. 0 Lemma 3.2 Assume that the canonical map tk : CF(H) -

CF(kH) is surjective and that K H is quasicentral. Then kH also is a quasicentral algebra.

Proof. Let k' be a finite Galois extension of k neutralizing for H. Let 0' be a local domain, local extension of 0 with residue field k' and field of fractions K'. By Lemma 2.3, the canonical map CF(O' H) - CF(k' H) is also surjective, hence Proposition 3.1 proves that do' has finite cokernel. As KH is quasicentral, one has Ro(K'H) = Ro(KH) by Proposition 2.9; hence the image by the decomposition map of Ro(K' H) is contained in Ro(kH). Since the decomposition map do' : Ro(K'H) _

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262

R.o(k'H) has finite cokernel, we get rankR.o(kH) kH is quasicentral, by Proposition 2.9.

=

rankR.o(k'H); hence 0

Assume V is noetherian and has Krull dimension 2. Let p be a height one prime ideal of V with Vjp integrally closed (i.e., Vjp is a discrete valuation ring) and kp = (Vp)jp perfect. For q a height one prime ideal of V, let kq be a finite separable extension of kq = (Oq)jq neutralizing for kqH and (\ a discrete valuation ring, (unrami!ied) extension of Vq with residue fie~d kq and field of fractions

C\

K q ; let denote the completion of (\ and K q its field of fractions. Here is now the crucial result: Theorem 3.3 Assume that K H is quasicentral and that the canonical

map CF(H)

---+

CF(kH) is surjective.

(1) The map ld9d ojp : k@Ro(kpH) ---+ k@Ro(kH) is an isomorp,hism if and only if the restriction of the bilinear form (', .) -'- to Ko(Op) x 0, CF(H) has values in V.

i- p of V, the algebra kqH is semisimple, then the two equivalent statements in (1) hold. In particular, the number of simple kH -modules is equal to the number of simple kpH -modules.

(2) If for every height one prime ideal q

Proof. Note that kpH is quasicentral since kp is perfe,ct, by Lemma 3.2. Let q be a height one prlme ideal o~ V. Note that Oq n K = Vq (the intersection is taken in K q ). Since Oq is a c?mplete discrete valuation ring, iderr~potents can be lifted from kqH to OqH. Hence, the canonical map Ko(OqH) ---+ Ko(kqH) is an isomorphism. We have the following commutative diagram:

CF(H)

lt

o /,

CF((Vjp)H)

surj.

CF(kH)

-------

r

CF((Vjp)H)

We have

CF((Vjp)H)

= im(tojp)

+ (mjp)CF((Vjp)H)

(note that CF((Vjp)H) is a direct summand of Homojp((Vjp)H,Vjp) as Vjp-modules since Vjp is a discrete valuation ring); hence im(tojp) = CF((Vjp)H) by Nakayama's lemma. This proves that the canonical

263

Centers and Simple Modules for lwahori-Hecke algebras map CF(H)

---+

CF( (0 /p)H) is surjective, hence the bilinear form (', .) ~

0.

A

restricted to Ko(OpH) x CF(H) has values in 0 if and only if (', ·h. restricted to Ko(kpH) x CF((O/p)H) has values in O/p. Since (0,,) induces a perfect pairing between Ko(kpH) and Ro(kpH), the submodule ch Ro (kpH) of CF( (0/p )H) is pure if and only if the form (', ·h. restricted to Ko(kpH) x CF((O/p)H) has values in O/p. By Proposition 3.1, the decomposition map dojp : Ro(kpH) ---+ Ro(kH) has finite cokernel; hence the map 1k @ dojp is an isomorphism if and only if it is injective, i.e., if and only if chRo(kpH) is a pure submodule of CF((O/p)H). This completes the proof of (1). Assume that for q i p, the algebra kqH is semi~imple. Then, the algebra kqH is split semisimple; hence, the ~lgebra OqH is isomorphic to a Adirect produ?t of matrix algebras over Oq. So, the canonical map

Ko(OqH)

---+

Ko(KqH) is an isomorphism and the form (', .)~ restricted

A

Kq

A

to Ko(KqH) x CF(H) has values in Oq and finally the form(·, ')K restricted to Ko(KH) x CF(H) has values in nq;ipOq. Composing the canonical map K o( map Ko(KH)

t\!:l)

---+

K

K o( pH) with the inverse of the canonical

Ko(KpH) (note that KH is quasicentral), we get that the bilinear form (".) ~ restricted to Ko(OpH) x CF(H) has values in ---+

A

0.

A

nq;ip Oq n Op, hence in 0 since 0 is a Krull ring [Bki2, Chapitre VII, §1, theoreme 4]. 0

4. Center and class functions for symmetric algebras Let 0 be a commutative ring and H an O-algebra, free and finitely generated as an O-module. Let T E CF(H). We say that T is a symmetrizing form for H (ef [Br]) if the induced map f :H

---+

Homo(H, 0), h

f--->

(hi

f--->

T(hh l))

is an isomorphism. More concretely, this means that, if B is an O-basis of H, then the determinant of the matrix (T(hhl)h.hIEB is a unit inO. When such a symmetrizing form exists, we say that the algebra H is

symmetric. Assume now that T is a symmetrizing form for H. To simplify the notations, for h E Hand f E Homo(H,O), we put h* = f(h) and 1* = f-1(/). Note that f induces an isomorphism of O-modules

M. Geck and R. Rouquier

264

Z(H) --+ CF(H): If A is a commutative O-algebra, then the canonical map CF(H) --+ CF(AH) is surjective if and only if the canonical map Z(H) --+ Z(AH) is surjective. If B is an O-basis of H, then the dual basis {bVhEB is defined by the requirement T(blb~) = 8b1b2 for b1 , b2 E B. For f E Homo(H, 0), one has 1* = LbEB f(b)b v . Let M be an H-module. We have a map Tr : Endo(M) --+ EndH(M) given by Tr(J)(m) = Lbf(bVm). bEB

We have Higman's lemma, following [Br] : Lemma 4.1 The module M is projective if and only if there is f E Endo(M) such that Tr(J) is the identity.

Assume 0 is an integrally closed integral domain with field of fractions K perfect. Let us assume that K H is quasicentral. Proposition 4.2 Let X E Irr( K H) and let cx be the scalar by which X· E Z(KH) acts on a simple KH-module V affording a multiple ofx.

Then the following hold: (1) The element cx lies in O. (2) The module V is projective if and only if cx =I- O. In particular, the algebra K H is semisimple if and only if cx =I- 0 for all X E Irr(K H). (3) Assume K H semisimple. Then, X· = cxe x where ex denotes the central primitive idempotent corresponding to X (i.e., X'ex =I- 0). Moreover, T=

L -

XE1rr(KH)

1

-X· cx

Proof. Let V be a simple module with character a multiple of X E Irr( K H). The polynomial X - cx divides p( H) (X'): the roots of this polynomial are algebraic over 0, hence cx is algebraic over O. Finally, Cx E K, hence Cx lies in the integral closure of 0 in K, i. e., in O. There is a finite extension K' of K such that K' H is split and it is clear that if the parts (2) and (3) of the proposition hold for K' H, they hold for KH. Hence, we can assume that KH is split. Let V be a simple KH-module with character X E Irr(KH). Note that if i is a primitive idempotent of EndoW), then Tr(i) = cx1v. Hence, if Cx =I- 0, it follows from Lemma 4.1 that V is projective.

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265

Now, if V is a projective module and ex the central primitive idempotent of KH such that exV =I- 0, then the algebras exKH and (l-e x )KH are symmetric algebras with symmetrizing form TlexKH and TI(l-ex)KH and we have K H = exKH EB (1 - ex)KH. It is then clear that for the remaining part of the proposition, we can assume that K H is simple and let X be the unique irreducible character of K H. Then Z(KH) = K· 1, hence X* = cx' It impliesx = CXT. If i is a primitive idempotent of KH, we have X(i) = 1, hence Cx =I- 0 and the proof is complete. 0 Let now k be the residue field of 0 which we assume to be perfect. Proposition 4.3 [Ge3] The algebra kH is semisimple if and only if for every X E Irr( K H), we have 1k cx =I- O. If kH is semisimple, then K H is

semisimple and kH is quasicentral. Proof. Suppose that kH is semisimple. Then, we have dim(kH) :S Lv(dim VjmV)2 where V runs over a complete set of representatives of simple kH-modules. Since KH is quasicentral, we have dim(KH) ~ Ls(dimSjms)2 where S runs over a complete set of representatives of simple KH-modules. We have d([KH]) = [kH], hence d(LS[SjmS]) = Lv Qv[Vjmv] where S (resp. V) runs over a complete set of representatives of simple KH-modules (resp. kH-modules) and Qv > 0 for all V. It follows that Ls(dim SjmS)2 ~ Lv(dim VjmV)2 and we have equality if and only if for all S, there exists V such that d(S) = d(V) and ms = mv. Now,

L s

(dimS)2 :S dim(KH) = dimkH:S (dim V)2

ms

mv

Hence we have equalities everywhere above, i.e., kH is quasicentral and KH is semisimple. Futhermore, for X E Irr(KH), then d(X) E Irr(kH). By Proposition 4.2(2), we get cxh = Cd(x) =I- o. Suppose now 1k cx =I- 0 for all X E Irr(KH). We have d(X)(x) = 0 for x E J(kH). Since T = Lx ...!...x, we get f(x) = 0 for x E J(kH), where e" f = 1k @T. Hence, J(kH) is an ideal of kH which is in the kernel of f: since f is a symmetrizing form for kH, it implies J(kH) = 0 and kH is semisimple. 0 Let P be a projective H-module. Let e be an idempotent of Mn(H), for some n, such that eHn ~ P. Let 1J(P) : Z(H) ---+ 0 be the restriction to Z(H) of Tr(e)*. We have 1J(P)(z) = T(Tr(e)z) = z*(Tr(e)) = ([P], z*)o. Note that it implies that 1J(P) depends only on P.

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Assume in addition K H semisimple and split. Given P a projective kH-module, J. Muller suggested considering the map 'l/J( P) : Z (K H) -+ K defined by

L

'l/J(P) =

1

(e([P]), X) -w x Cx

XElrr(KH)

where W x is the one dimensional representation of Z(KH) acting (as multiplication by scalars) on a simple K H-module with character X. Proposition 4.4 The map 'l/J(P) restricts to a map Z(H)

h

@

'l/J(P)

=

-+

0 and

TJ(P). In particular, 'l/J(P)(l)

(e([P]), X) E O.

L

=

XE/rr(KH)

Cx

Proof. Let 0' be an henselisation of 0; we have K n 0' = 0, where the intersection is taken in the field of fractions of 0', since 0' is faithfully flat over O. Hence, to prove the proposition, we can assume that 0 is henselian. There exists an idempotent e of Mn(H) such that keHn ~ P. Then, TJ(eHn) is the restriction to Z(H) of Tr(e)*. We have wx,(X*) = 8x,x'cx for X,X' E Irr(KH), hence

TJ(eH n) =

L x(Tr(e)) x

and h

@

'l/J(P) = h

@

Cx

W

x = 'l/J(P)

TJ(eHn) = TJ(P).

o

Note that if H = OG is a group algebra with its usual symmetrizing form, K having characteristic zero and k characteristic p > 0, then the integrality property above is equivalent to the statement that the dimension of a projective kG-module is divisible by the order of a Sylow p-subgroup of G. When H is an Iwahori-Hecke algebra associated to a Weyl group, with equal parameters and k has characteristic zero, this result was proven in [Ge4, Proposition 2.1] using algebraic groups.

5. Iwahori-Hecke algebras

We fix a finite Weyl group W with a corresponding set SeW of simple reflections. Let {Us}sES be a set of indeterminates such that Us = Ut whenever s, t E S are conjugate in W, and A = Z[u s , U,;-l]sES be the ring

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of Laurent polynomials in these indeterminates. The generic IwahoriHeeke algebra 1i is the A-free A-algebra with A-basis {Tw}wEW and relations: if l(ww l ) = l(w) + l(w l ), TwTw' = Tww' { (Ts - us)(Ts + 1) = 0 for s E S where w f---> l (w) is the length function on W with respect to the generating set S. The algebra 1i is symmetric, with respect to the form T : 1i ---+ A defined by T(T1) = 1 and T(Tw ) = 0 for w i- 1. The elements in the dual basis of {Tw} are given by T':: = ind(Tw)-lTw-" where ind : 1i ---+ A is the 1-dimensional representation of 1i defined by ind(Ts ) = Us for s E S. Thus, we can apply the results of the previous section to the pair (1i, T). 5.1 Centers of Iwahori-Hecke algebras For each conjugacy class C of W, we denote by Cmin the set of elements of minimal length in C, and we choose one element We E Cmin . For each class C and each w E W, there exists an element f W,e E A (called class polynomial in [Ge-Pm uniquely determined by the property that

'P(Tw) =

L e

fw,e'P(Twc )

for all 'P E CF(1i).

(It is shown in [Ge-P~ that, if w, Wi E Cmin then Tw and Tw ' are conjugate in 1i. In particular, every class function of 1i has the same values on Tw and Tw" Hence the definition of fw,e is independent of the choice of We E Cmin .) For each class C we define a function fe : 1i ---+ A by

fe: Tw f---> fw,e

(w E W).

By inverting the defining formula for fw,e above, we see that fe is in fact a central function. Hence, given any 'P E CF(1i), one has

i.e., the set Ue} is a basis of the A-module CF(1i). Using the correspondence between central functions on H and central elements in H, we conclude that the elements {ze := fe} form an A-basis of the centre of 1i. Explicitly, we have ze =

L wEW

ind(Tw)-l fe(Tw)Tw-l.

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Note that the class polynomials have the following properties. Let C, C I be conjugacy classes in W, and w E C I . Then

fw,c = 8c ,c'

if wE C:r,in-

Moreover, if we specialize all parameters Us to 1 then the function fc specializes to the indicator function of the conjugacy class C and, hence, the element zc specializes to the class sum of C in AW. Thus, the elements {zc} indeed are "generic" analogues of the class sums. Lemma 5.1 Let B be a commutative A-algebra and z =

LWEW awTw E Z (BH). Let C be a conjugacy class in W. Then the following hold:

(1) If W, Wi E Cmin then aw = aw,. (2) Ifw E C does not have minimal length, then there exists an element Wi E C and an element s E S such that l(w) = l(w l), l(swls) = l(w l ) - 2 and aw = aw, = (1/u s )a sw's + (1 - 1/u s)a sw"

Proof. The coefficient aw of Tw in z is given by T(zT~). Now assume that Tvv and T~ (for v, w E W) are conjugate by a unit, say h E BH. Using that T is a central function and that z is a central element, we deduce that

av = T(zTvV ) = T(zhT~h-l) = T(h-lzhT~) = T(zT~) = aw. Now let w,w l E C and x E W such that l(w) = l(wl),W I = xwx- 1 , and l(xw) = l(x) + l(w). As in [Ge-P~, we can compute in BH that TxTw = Tw,Tx , hence Tw and Tw' are conjugate in BH. A similar relation will also hold with W, Wi, x replaced by their inverses. Thus, T~ and T~, are conjugate in BH. Using [Ge-Pf, Theorem 1.1], we conclude that T~ andT~, are conjugate in BH, for all w, Wi E Cmin . Hence, (1) is proved using the above argument. Now let w,w" E Wand s E S such that w" = sws and l(w") ~ l(w). As in [Ge-P~ we see that, if l(w) = l(w"), the element Tw is conjugate to T sws . If l(w") = l(w) - 2 then Tw is conjugate to usTsws + (us -1)Tws . Again, similar relations hold with w replaced by w- 1 . Thus, T~ will be conjugate either to Ts"ws or to (1/u s)Ts"ws + (1 - 1/u s)Ts"w. Now, by [Ge-Pf, Theorem 1.1], there exist Wi E W such that l(w l ) = l(w), Tw' is conjugate to Tw and l(swls) = l(w l ) - 2. Hence, the argument above 0 implies (2). Theorem 5.2 Let B be a commutative A-algebra. Then, the set {IB @

zc} (where C runs over the conjugacy classes ofW) forms a B-basisfor

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the centre of B1i. In particular, the centre of B1i is free as a B-module of rank equal to the number of conjugacy classes in Wand the canonical morphism B @ Z(1i) -+ Z(B1i) is an isomorphism. Proof. Since fwc,G' = oC,G', the elements IB @ Zc are linearly independent in B1i. So we must show that they generate Z(B1i). The strategy for the following proof is taken from [Di-Ja]. Let z = LWEW awTw E Z(B1i) (where aw E B). Assume that z =I- 0 and let w E W be of minimal possible length such that aw =I- o. Then w lies in some conjugacy class C and we claim that w E Cmin . This can be seen as follows. Assume, if possible, that w does not have miminal length in C. By Lemma 5.1(2), there exist some Wi E Wand s E S such that aw = aw, = (l/u s )a sw's + (1 - l/u s )a sw'. Since l(w) = l(w' ) and l(sw's) = l(w' ) - 2, both swls and ws l have length strictly smaller than w. By the minimality of w, we conclude that asw,s = asw , = 0 and, hence, also aw = 0, a contradiction. Thus, a w =I- 0 for some w E Cmin . Moreover, Lemma 5.1(1) shows that a w = a w , for all Wi E Cmin . We now consider the element Zl := z - Lc awcind(Twc )(IB @ zc) E Z(B1i). The above mentioned properties of the elements fw,c show that the coefficient of T w in Zl is zero, for any element w of minimal length in any conjugacy class of W. Thus, Zl = 0 and we are done. 0 5.2 Number of simple modules for Iwahori-Hecke algebras Consider the following polynomials associated to irreducible finite Weyl groups: n

II [i]x

QA n

i=l n-1

II [2]x.y(x

QBn

i

+ y)[i]x

i=O

n-1

2[n]x

QDn

II [2i]x

i=l

QE6 QE 7 QEB QF4 QC 2

=

6[2]x [5]x [6]x [8]x [9]x [12]x 6[2]x [6]x[8]x [1O]x[12]x [14]x[18]x 30[2]x [8]x [12]x[14]x [18]x[20]x [24]x [30]x 6[6]x[6]y[2] xy 2[2]x2y[2]xy[2]x2y2[2]x3y3(x + y2)(X 2 + y)(x + y) . .(x2 + y2)(X3 + y3) 2[2]x[2]y[3]xy(x2 + xy + y2)

where [i]q = 1 + q + ...

+ qi-1.

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M. Geck and R. Rouquier

The following proposition gives a criterion for a multi-parameter Iwahori-Hecke algebra to be semisimple. It generalizes [Gy-Uno] which deals with the equal parameters case and characteristic zero fields. Assume W is irreducible and let {Sl' S2} be a set of representatives of conjugacy classes of elements in S, with Sl corresponding to a long root in type B n. Let B = Z[JU:;, JU:;-I]SES = A[vu:;]sEs and K = Q(JU:;)sES the field of fractions of B. Note that, by [Di-Mi, theoreme 3.1], the algebra K1i is quasicentral (actually, by [Ge2], the algebra K1i is even split, but we won't need it here). Proposition 5.3 [Ge3] Let k be a perfect field which is a B-algebm. Then, the algebm k1i is semisimple if and only if 1k . Qw( u sl , U S2 ) =I- O. Proof. A criterion to decide when the specialized Iwahori-Hecke algebra k1i is semisimple is given by Proposition 4.3. One has to check that B[{J...} _ ] = B[-QI ]. Let us define Pw = Lw ind(Tw). For X an e"

XElrr(K'H)

W

irreducible character of H, we put D x = Pw / ex: this is the generic degree of X. The generic degrees are given in [Cal and one checks easily the property above. 0 To simplify the exposition, we assume now that there is a set of integers (not all zero) {as}SES (with greatest common divisor 1) such that Us = t a , where t is an indeterminate. In this case, B = Z[0, 0-\ and the polynomial Qw is a product of cyclotomic polynomials in t, up to a power of t. Let £ be a prime and q an integer with £ ;f'q. Let e ~ 1 be minimal with l+q+·· .+qe-I = Omod£. If d is an integer, we put F@F by setting r(xi) = Xi @ 1 + 1 @Xi for all i E I; here the multiplication on F@F is to be defined by Lusztig's rule (1.2a).

L3. Define elements Ya (a E (I)) in F* = HomA (F, A) by the equations (Ya,Xb) = oab(b E (I)), where (-, -) : F* x F ---> A denotes the natural pairing. Let G be the A -submodule of F* generated by the Ya (a E (I)). Since the Ya are A-linearly independent, they form an A-basis of G.

J. A. Green

278

We give G the structure of NI-graded A-module G setting G v = LaEI(v) A . Ya for all v E NI.

= L~ENI G v , by

Since r is coassociative, it induces on F* an associative, A-bilinear multiplication by the rule (YY' , x)

= (Y @ Y' , r (x) )

(2.1a)

for all y,y' E F* and x E F. The pairing (-,-) : (F*@F*) x (F@ F) -+ A on the right side of (4.2a) is defined by (YI @ Y2, Xl @ X2) = (YI' Xl) (Y2' X2), for all YI, Y2 E F* and all Xl, X2 E F. It is clear that G is an NI-graded A-algebra with respect to this multiplication, since r(Fv ) ~ Lv'+vll=v Fv' @ FVIl for all v E NI. Now let Bi(i E 1) be arbitrary non-zero elements of A. Define the A-algebra map 'IjJ : F -+ G by setting

'IjJ(Xd = BiYi,

for all i E I.

(2.1b)

Lusztig proves ([L], p. 4) that the A-bilinear form (-, -) : F x F defined by

(X, x')

= 'IjJ(x')(x),

for all x, x' E F

-+

G

(2.1c)

is symmetric and satisfies axiom L3, with (Ui, ud = B i for all i E I. Thus F (with generators xi(i E 1), and with r as defined above) becomes an object of £(A,v,I,.). If L is any object of £(A, v,I, '), with generators ui(i E 1), then there is a morphism () : F -+ L in £(A, v,I, '), namely the A-algebra map () which takes Xi -+ Ui, all i E I. We refer to F as a free object of £(A, v,I, .). Notice that any two free objects F, F' of £(A, v,I,·) are isomorphic in the category £(A, v,I,·), even though (Ui, ud = Bi may differ from (u~, uD = B~ for some (or all) i E I. 2.2. Given L in £(A, v, I, '), with generators ui(i E I), comultiplication r and bilinear form (-, - ), then the argument of [1], p. 5 shows that J = rad( -, -) is a two-sided ideal of L, and that (a) J = LVENI Lv n J, and (b) r(J) ~ J @ L + L @ J. From these it follows that LO = L/J is an object of £(A,v,I,·), with generators u? = Ui + J(i E 1), and with comultiplication rO and bilinear form (-, _)0 inherited in obvious ways from L. The natural A-algebra epimorphism L -+ LO is a morphism in £(A,v,I,·) It is also clear that (-, _)0 has zero radical.

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279

Definition. An object of £(A, v, I,,) is non-degenerate, if its bilinear form has zero radical. In §2.5 below, we shall prove that any two non-degenerate objects of £(A, v, I,.) are isomorphic. Proposition 2.3. Let v E NI, and let a, b E I(v). Then there exists an element Ma,b(t) E Z[t, t- l ] (t is an indeterminate) such that for any A, v, I" (as in 1.2) and for any L E £(A, v,I,') with generators Ui (i E I) there holds (2.3a)

where Bv(L) = I1iEI(Ui,Ui)"'i. This will be proved in §4.5 below (or see [G], §3.6). The point of this proposition, is that the Laurent polynomials Ma,b(t) depend only on the datum (I, '), and not on A nor v.

Remarks. (1) Ma,b(t) is uniquely defined by Proposition 2.3. For if we apply (2.3a) to the case where L is the "free" object F of £(A,v,I,·) defined in 2.1, taking B i = 1 for all i E I, and taking A = Q(t), v = t (t indeterminate), we see that Ma,b(t) is equal to (x a, Xb). (2) An explicit formula for Ma,b(t) is given in (4.4e) below (see also [G], 3.6(4)). Corollary 2.4. Let L be an object of £(A, v,I, '), as in §1.2. Let x = LaE(I) CaU a be an element of L (the sum is over a family of elements Ca E A (a E (I)), with Ca = 0 for all but a finite number of a E (I)). Then x lies in rad( -, -) if and only if LaEI(v) CaMa,b(v) = 0 for all b E I(v), and all v E NI. Proof. By definition, x E rad( -, -) if and only if (x, y) = 0 for all y E L, i.e. if and only if (X,Ub) = 0 for all monomials ub(b E (I)). If b E I(v) then by L3(a), (u a , Ub) = 0 for all words a = al'" ap which lie in I(J.L) for some J.L in NI, J.L i= v. So x E rad( -, -) if and only if LaEI(v) ca(u a,Ub) = 0 for all b E I(v), and all v E NI. But formula (2.3a) shows that this is the same as the condition given in the corollary, since by L3(b), Bv(L) = I1iEI(Ui,Ui)"'i is a non-zero element of the integral domain A. 0 Proposition 2.5. Any two non-degenerate members of £(A, v, I,,) are isomorphic. Proof. Let L (resp. L') be non-degenerate members of £(A, v, I, '), with generators ui(resp.uD (i E I). According to Corollary 2.4, an element x = La CaU a of L(ca E A) is zero if and only if

L:aEI(v) CaMa,b(V) = 0 for all b E 1(v), and all v E N1. But the same condition is necessary and sufficient for the element x' = L:aEI(v) cau~ of L' to be zero. So the rule x 1-+ x' defines a bijective map L --+ L', which is clearly an isomorphism in £(A, v, 1, .). D

Proposition 2.6. Let L be an object of £(A, v,I, '), and let F be a ''free'' object as defined in 2.1. Then there exist morphisms F --+ L --+ FO; in other words every object of£(A,v,I,·) is "sandwiched" between F and FO. Proof. The (trivial) existence of the morphism F --+ L has already been remarked in §2.1. The morphism L --+ FO is the composite of the natural map L --+ LO with the isomorphism LO --+ FO coming from Proposition 2.5. D 3. Two Examples 3.1. Lusztig's Algebra f Definition. (Lusztig, [L, p.2]). A Cartan datum is a datum (1,,) such that 1 is finite, and the following two conditions are satisfied (a) i· i E {2,4,6,·· .}, for all i E 1, and (b) 2(i. j)/(i. i) E {O, -1, -2, -3,·· '}, for all i

-I j in 1. Notice that if (1, .) is a Cartan datum, then the matrix b. = (aij), where

aij = 2(i . j)/(i . i), is a symmetrizable, generalized Cartan matrix in the sense of Kac [K,pp. 1,16]. Let Q be the rational number field, and Q(t) the field of rational functions over Q in an indeterminate t. Let (1,.) be a Cartan datum. Then Lusztig's algebra f (see[L], p. 5) is, in the terminology of Section 2, a non-degenerate object of the category £(Q(t), t,I,·); by Proposition 2.4, this description determines f as Q(t)-algebra, up to isomorphism. In §4.1 below, we observe that f can also be viewed as subalgebra of a "quantized shufRe algebra" . Lusztig uses the algebra f in his definition of the quantum group U associated to a Kac-Moody Lie algebra of type (1,') ([L], Chapter 3). He shows ([L], Chapter 33) that f is isomorphic to the positive part U+ of U, Le. that f is the Q(t)-algebra on generators (Ji = x?, satisfying the "quantum Serre relations" (see [L], p. 11) as defining relations.

Notation. v,'f, f, (Ji in [L], correspond respectively to t, F, FO and (or x? = x + J) in the present paper.

Xi

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281

3.2. Hall Algebras

In this paragraph we shall review briefly C. Ringel's theory of the Hall algebras which are connected to quantum groups. Throughout, k is a finite field, and R is a finitely-generated, hereditary k-algebra. We denote by R-fin the full subcategory of R-mod, whose objects are those R-modules X which are finite as sets: IXI < 00. Definitions. Let P be the set of all isomorphism classes in R-fin, and let I(c;;. P) be the set of all simple isomorphism classes in R-fin. If A E P, we write U>. for a member of A. Notice that {Uili E I} is a complete set of simple modules in R-fin. The Grothendieck group of R-fin is identified with ZI in the usual way, so that the Grothendieck class of X is taken to be the dimension vector dim X, and this is regarded as an element of ZI (see [R3], p. 8, or [G], Section 1). Ringel ([R2], Section 2) introduces a bilinear form (-, -) : ZI x ZI -+ Z by the rule (i,j) = e(Ui' Uj)(i,j E I), where for any X, Y E Rfin,

Because R is hereditary, e(X, Y) depends only on the Grothendieck classes dim X and dim Y. We extend the notation (i,j), by defining (0, (3) to be e(Uco U(3), for any 0, (3 in P. Finally we make a symmetrized version of (-, -), denoted 'R by defining O'R(3 = (0, (3)+((3, 0) for all 0, (3 in P (see [R3], p. 8). Let A be an integral domain of characteristic zero, which contains the rational field Q, and also contains an element v such that v 2 = Ikl. The Ringel-Hall algebra H = HA,v(R) is by definition a free Amodule on a set of symbols u>. (A E P) as A-basis, and with A-bilinear multiplication given by

UaU(3 =

L v(a,(3) . g~(3 . U>.

(3.2a)

>'EP

for all 0, (3 E P, where for each triple 0, (3, A of elements of P, g~,(3 denotes the number of R-submod ules S of U>. such that U>. / S ~ Ua and S ~ U(3. It is easily checked that HA,v(R) is an associative Aalgebra with identity element Uo (0 E P is the class of zero R-modules). Moreover HA,v(R) is NI-graded, if we give each basis element u>. the degree dimU>. (see [R2] , Section 2.).

J. A. Green

282 Next define a comultiplication r : H

---t

H

@

H by the rule (3.2b)

for all A E P, where an = IAutR(Un )1, for alln E P. The main result of [G] is that the map r is multiplicative, provided we define the product in H @ H by the rule

for all p, a, p', a' E P; here a . p' = a . R p' as defined above. But (3.2c) is exactly the Lusztig rule (1.2a), for the N1-graded algebra HA,v(R), with respect to the datum (1, ·R). Moreover if we define the symmetric, non-degenerate A-bilinear form (-, -) on HA,v(R) by setting (u a , u,6) = 8a ,6 . ul", for all 0:, (3 E P, then it is an immediate consequence of (3.2a) and (3.2b) that

(x,yz)

=

(r(x),y@z)

(3.2d)

for all x, y, z E H = HA,v(R). Therefore HA,v(R) satisfies all the conditions Ll, L2, L3 to be an object of £(A, v, I, 'R), except that HA,v(R) may not be generated by the elements ui(i E /). So define CA,v(R) to be the A-subalgebra of HA,v(R) generated by the ui(i E 1). It is clear that C = CA,v(R) is a sub-coalgebra of HA,v(R), i.e. that r(C) ~ C @ C. It follows that CA,v(R) is an object of the category £(A, v, I, ·R). Ringel calls CA,v(R) the composition algebra of R (see [Rl], p. 396). 3.3. In this paragraph we assume that (1,,) is a Cartan datum, and indicate briefly the connection between Lusztig's algebra f, which is an object of the category £(Q(t), t, I, '), and a "generic composition algebra" associated to (I, .). For details, see [G], §§3.4, 3.5, or [R4]. Given any finite field k and any Cartan datum (1, '), it is possible to find a finite-dimensional, hereditary k-algebra Rk such that the simple modules in Rk-mod (= Rk-fin) may be indexed by I, and in such a way that the symmetrized Ringel form 'R (see §3.2) coincides with the form given in (1,.) (see [R4] , Part III, Section 4). If A ~ Q is an integral domain containing an element Vk such that v~ = Ikl, we have seen that C A,Vk (Rk) is an object of £(A, Vk,!, .); thus f and CA,Vk (R k ) both belong to categories £(A,v,!,'), but for fwe take A,v to be Q(t),t, while for CA,Vk(R k ) we take A,v to be A,Vk' In order to compare these algebras, we first replace f, which is the non-degenerate object

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283

of the category £(Q(t), t, I, .), by the non-degenerate object FO of the category £(A, t,I, .), where A is the subring Q[t, t-1j of Q(t). So we start with the free associative A-algebra F = A(Xi : i E I); then FO = F jJ, where J = rad( -, -) (see §§2.1, 2.2). It is easy to see that f

9:'

Q(t) @A FO .

(3.3a)

For each finite field k, there is a unique ring homomorphism Xk : F -+ CA,Vk(Rk) such that Xk(t) = Vk, and n(Xi) = U~k) for all i E I (here U~k) denotes the basis element of the Ringel-Hall algebra HA,Vk (Rk), corresponding to the simple Rk-module Ui(k»). Let K be any set of finite fields k, such that the set {Ikl : k E K} is infinite. Then it is clear that for any polynomial p(t) in Z[t], and for any Laurent polynomial p(t) in A = Q[t, t- 1],

If P(Vk) = 0 for all k E K, then p(t) = O.

(3.3b)

Now define the ring-homomorphism X: F -+ IlkEKCA,vk(Rk) by X = IlkEK n· From (3.3b) follows that the element X(t) satisfies no nontrivial polynomial relation over Z, and so we may identify X(t) with t, X(t-l) with t- 1 , and regard 1m X as an A-algebra generated by the elements X(Xi) = Ui, for all i E I. Denote 1m X by CA(I, .); this is Ringel's generic composition algebra (see [Rl], p. 398). A distinctive contribution of the present theory, which one proves easily by Proposition 2.4 (see [G], (3.4a)), is that Ker X lies in the radical J of the bilinear form of the object F of £(A, t,I, .). Since FO = FjJ by definition, we deduce from X an epimorphism of A-algebras CAU,·) -+ FO. Hence by applying the functor Q(t)@A -, together with (3.3a), we get an epimorphism ofQ(t)-algebras X: C(I,.) -+ f, where C(1,') = Q(t)@ACA(1,')' But Ringel has shown (see [R2], Proposition 2, or [R4], Part III, Section 2) that the elements U~k) of HA,Vk (Rk) satisfy the quantum Serre relations. Hence there is a map of Q(t)-algebras taking U+ 9:' f -+ C(1, .). It is easy to see that Ringel's map must be the inverse of X, and that, therefore, both maps are isomorphisms. 4. The Quantized Shuffle Algebra 4.1. Let (1, .), A, v be as in §1.2. In §2.1, we considered the free NIgraded A-algebra F = A(Xi : i E I) = L~ENI Fv , and the A-submodule G of F* which is generated by the elements Ya (a E (I)) in F* = HomA(F,A) defined by the equations (Ya,Xb) = 8ab(b E (I)), where

J. A. Green

284

(-, -) : F* x F --+ A denotes the natural pairing. G has NI-grading G = L~ENI Gv, where for each v E NI, G v = LaEI(v) A· Ya. We shall see from the multiplication formula (4.3f) below that G is closed to the multiplication on F* induced by the comultiplication r on F, and that with this multiplication, G becomes an NI-graded A-algebra. Notice that G contains elements Yi(i E 1), but in general these do not generate G as A-algebra, and the element Ya(a = al ... ap E (I)) is in general not equal to the monomial Ya, ... Yap (see 4.4). Definition. G is called a quantized shuffie algebra of type (1, .). It depends, of course, on A and v as well as on the datum (1, .). In the cases where (1,.) is the "zero datum" (Le. i . j = 0 for all i, j E 1), or (1,.) is arbitrary and v = 1, this is the algebra introduced by Rimhak Ree in [Rh], p. 211 (see also [Re], p. 24). In these cases, the Lusztig product rule (1.2a) for F@F reduces to the standard product, and the multiplication in G is commutative. In general, multiplication in G is non-commutative. Notation. In this section, the datum (1,.) is kept fixed; the notation G(A, v, I,·) will be used when it is desired to mention A and v explicitly. 4.2 Lusztig's algebra f as a subalgebra of a quantized shuffle algebra We take arbitrary non-zero elements B i E A, for all i E I, and define the A-algebra map7/;: F --+ G and the A-bilinear form (-, -) on F, as in §2.1. It follows from (2.1c) that the radical of (-, -) is the same as the kernel of 7/;. Therefore7/;: F --+ G induces an isomorphism FO = FIKer7/; --+ Im7/;, hence FO is isomorphic as A-algebra to the A-subalgebra of G generated by the elements BiYi(i E 1). The isomorphism in question takes x? = x + rad (-, - ) to BiYi, for all i E I. In [1] Chapter 1 Lusztig assumes that (I,.) is a Cartan datum, and takes A = Q(t) (t indeterminate; notice that Lusztig uses v as an indeterminate), v = t and B i = (1 - Ci-i)-l(i E 1). Thus the Q(t)-algebra f is isomorphic to the subalgebra g of G = G(Q(t), t, I,·) generated by the elements (1- Ci-i)-I·Yi' i E I. Of course, g is equally generated, as Q(t)-algebra, by the Yi, i E I. However much of [L] is concerned with properties of a certain A-order Af of f (see [L], p.13), and to transfer this to g one must use the isomorphism f --+ g described above, which takes (}i I--; (1- t-i-i)-l . Yi, i E I. 4.3 Multiplication formula for G Let d be an integer ~ 2, and let a(l), ... , a(d) be words (Le. elements of (I)) of lengths nl, ... ,nd, respectively. We want to calculate the

Quantum Groups, Hall AIgebms and Quantized Shuffies

285

product Ya(l) ... Ya(d). This element of F* is determined as soon as we know (Ya(l)" ·Ya(d),X a) for all a E (/)), and we see from (2.1a) (by an induction on d) that, for any word a E (/)), (4.3a) where rd : F ---+ F0 d is defined as in 1.8. We may use (1.8b) (replacing the object L of £(A, v, /,.) by F) to calculate rd(x a ). This gives

L v a. a = L v .

(Ya(l) ... Ya(d), x a ) =

P

(Ya(l)

P

(Ya(l)' Xa\PJ ... (Ya(d), XalPd) ,

@ ... @

Ya(d), XalPl

@ ... @

xalPJ

P

(4.3b)

P

where the sum is over all d-tuples P = (PI,".' Pd) of subsets of !!(n = l(a)) such that!! = PIU· ··UPd (disjoint union). Clearly the P summand in (4.3b) is zero unless

alPI = a(l), ... , alPd = a(d) .

(4.3c)

Conditions (4.3c) imply that the orders of the sets PI,' .. ' Pd are nl, .. . nd, respectively, hence that n = nl + ... + nd. Therefore (Ya(l)" 'Ya(d),X a ) is zero unless l(a) = nl + ···nd; this proves that Ya(l) ... Ya(d) is an element of G. In fact if for each h E g we have a(h) = ahl ... ahnh (with ahl,···, ahnh E 1) and Ph = { 7Thl < ... < 7Thnh}' then conditions (4.3c) determine the word a completely, namely it is the word oflength nl +... + nd which has ahl, ... , ahnh in positions 7Thl, .•. ,7Thnh respectively, for all h E g.

Definition (1). The word a just described will be denoted P(a(l), ... ,a(d)), and called the P-shuffie of a(l), ... a(d). The PshufRe of a(l), ... , a(d) is defined for each d-tuple P = (PI'.'.' Pd) of subsets of !!(n = nl + ... + nd) such that (4.3d)

Definition (2). The integer a . P defined in (1.8c), where a = P(a(l), , a(d)), will be called the exponent of the P-shufRe P(a(l), , a(d)). In the notation above, (4.3e)

286

J. A. Green

summed over all quadruples (h, a, k, T) such that h, kEg, a E {I, ... ,nh}, T E {I, ... ,nd, with h > k and 7rha < 7rkT' Going back to (4.3b), which gives the coefficient (Ya(l)'" Ya(d) , xa) of Ya in Ya(l) ... Ya(d), we have a multiplication formula for the quantized shuffie algebra G: if a(l), ... ,a(d) are words of lengths nl, ... ,nd respectively, then

Ya(l) ... Ya(d)

" a·P YP(a(I), ... ,a(d» , = 'LV

(4.3f)

P

the sum being over all d-tuples P = (H, ... Pd ) of subsets of n(n = ni + ... nd) satisfying conditions (4.3d); for each such P, the exponent a· P is given by (4.3e).

Remarks. (1) (4.3c) shows that if a(l), ... , a(d) have respective weights v(l), ... ,v(d), then the weight of a is v(l)+ ... +v(d); therefore all the shufRes P(a(l), ... , a(d)) appearing in (4.3f) have this weight. It follows that G is an NI-graded algebra. (2) There are (n~~!·.·.·.~~t)\ d-tuples P satisfying conditions (4.3d), and so this is the number of terms in the sum (4.3f). However the words P(a(l), ... , a(d)) may not all be distinct (see example (2) below). (3) In the "classical" shufRe algebra of Ree,

Ya(l) ... Ya(d) = LYP(a(I), ...,a(d» P

(see [Rh], p.211, or [Re], p.24). (4) In the definition of the exponent a . P of the shufRe P(a(l), ... , a(d)), the sum (4.3e) is composed of those products aha' akr, such that (a) aha comes earlier than akT in P(a(l), ... , a(d)), while (b) the word a(h) comes later than the word a(k) in the sequence a(l), ... , a(d). So for example if d = 2, a(l) = ijk and a(2) = pq, then for P = ({2,4,5}, {1,3}) the shufRe P(a(1),a(2)) is the word piqjk. To calculate the exponent a· P, imagine that the "shufRe" piqjk is obtained from the "unshufRed" word a(1)a(2) = ijkpq by moving the letters i,j,k (without altering their relative order) past the letters p, q as necessary, so as to reach the order piqjk. Each time a letter u is moved past a letter t, the exponent a . P receives a contribution t· u(= u· t). In the present case, k must be moved past both p and q; then j must also be moved past p and q; finally i must be moved past p. The total contribution is p. k + q. k + p. j + q. j + p. i = a . P.

Quantum Groups, Hall Algebras and Quantized Shuffies

287

Examples. (1) If i, j, k are any elements of I, then YiYjk = Yijk + Vj·iYjik + vj·i+k.iYjki. For if a(l) = i, a(2) = jk, the three shuffles of a = a(l)a(2) = ijk are ijk, jik and jki, coming from P = ({I}, {2, 3}), ({2}, {I, 3}) and ({3}, {I, 2}), respectively. The corresponding exponents a· Pare 0, j . i and j . i + k· i. Now use (4.3f). (2) Let i E I and n E N. Denote by i(n) the word ii··· i, with n i's. If n 2: 1, then (4.3f) gives YiYi(n-1) = (n)Yi(n), where (n) = 1 + Vi .i + V2i .i + ... + u(n-1)i.i. Repeated application of this gives the formula yi = (1)(2) ... (n)Yi(n)' 4.4. The elements Ma,b(t) We consider next the case where all the words a(I), ... , a(d) have length 1. Then a(h) = ah E I, for all h E d.. Denote the word a1 ... ad bya.

Conditions (4.3d) require that P = ({1rd, ... ,{1rd}), where (1r1, ... ,1rd) is a permutation of (1, ... ,d). Denote by 1r the element of the symmetric group Sd such that 1r(h) = 1rh, for all h E d.. The shufRe P(a1"'" ad) is the word of length d having ah in position 1r(h) for all h E d.. In other words, it is a 0 1r- 1 , where we let Sd act on the set I d of all words of length d by the rule i 1 ... id

0

S = i s(1)''' is(d) ,

for all i 1 ... id E I d ,

S E Sd.

(4.4a)

By definition 4.3(2), the exponent of this shufRe is e( 1r- 1 : a), where for any s E Sd we put (4.4b) summed over all (h, k) Ed. x d. such that h > k and S-1 (h) < S-1 (k). Therefore (4.3f) reads, in our present case,

Yal ... Yad _-

"'"" ~

v e( s:a) Yaos

(4.4c)

SESd

or, if we collect together all the terms Yaos for which a 0 s has a given value b E I d ,

Yal '" Yad =

L Ma,b(V)Yb ,

(4.4d)

bEld

where Ma,b(V) is the value at t = v of the following element of Z[t, r 1] M a,b (t) = "'"" te(s:a) ~ aos=b

,

(4.4e)

J. A. Green

288 the sum being over all s E 3 d such that a 0 s

= b.

Remarks. (1) From (4.4a) we see that a 0 s has the same weight as a, for all s E 3 d • Therefore by (4.4e) Ma,b(t) is zero unless a and b have the same weight. The Appendix gives tables of Ma,b(t) for all a, b E I(v), for some "small" v E NI.

(2) It is an easy exercise (which we leave to the reader) to prove from (4.4e) that Ma,b(t) = Mb,a(t), for any a, bE I(v), v E NI. 4.5. Proof of Proposition 2.3 Let v E NI, and let a, b E I(v). We shall prove that for any A, v and for any Lin £(A, v, I,·) with generators ui(i E 1) there holds (4.5a) where Bv(L) = I1iEI(Ui, Uit i , and Ma,b(t) is the element of Z[t, t-Ij given by (4.4e). The argument is just a variant of the argument in 4.4. Let d = l(a) = l(b) = L:iEI Vi and let a = al ... ad and b = bI .. , bd. If d = 1, then a = b E I, and trivially Ma,a(t) = 1, hence (4.5a) holds. Assume now that d 2 2. From the axiom L3(c), and making use of the assumption that the bilinear form (-, -) on L is symmetric, we deduce (by induction on d) that (u a, Ub) = (rd(u a ), Ubl @···@UbJ. So by (1.8b) we have

(U a,Ub) =

L va.P(Ualpp UbJ .. · (UaIPd, UbJ ,

(4.5b)

P

and the sum may be restricted to those d-tuples P = ({ 7l'Il, ... , {7l'd}), such that (7l'I, ... ,7l'd) is a permutation of (l, ... ,d), and a7l"l = bI , ... a7l"d = bd. Denote by s the element of 3 d such that s(h) = 7l'h for all h E Q. Then we have a 0 s = b (see (4.4a)) and a· P = e(s : a) (see (4.4b)). On the other hand, for any P as just described, the product (UaIPpUbl)",(uaIPd,UbJ = (UbpUbl)"'(Ubd,UbJ = Bv(L). Therefore (4.5b) gives (u a, Ub) = (L:aos=b ve(s:a») Bv(L) = Ma,b(V)' Bv(L), and (4.5a) is proved. 0

Appendix. Tables of Ma,b(t) In these tables, i, j, k are distinct elements of I. When a table is shown for a given weight v = L:iEI Vii, the table shows the Laurent polynomial Ma,b(t) in the row indexed by a E I(v) and the column

Quantum Groups, Hall Algebras and Quantized Shuffies

289

indexed by b E I(v).

~

I(v) = {i}

Mi,i(t) = 1 .

Iv=2il

I(v) = {ii}

Mii,ii(t) = 1 + t i .i

Iv=i+jl

I(v) = {ij,ji}

ij

ji

ij~

ji~

I(v)

Iv =

2i+ j

I

I(v)

= {iii}

= {iij,iji,jii}

iji t i ·j (1

iij

1 + t i ·i

1 + t 2i -j +i·i ti-j (1 + t i .i )

t i·j (1

+ t i .i ) ei.j (1 + t i .i )

I v = i + J' + k I ijk

ikj

ijk

1

t j ·k

ikj

t j ·k

jii

+ t i .i )

+ t i .i ) ti-i (1 + t i .i ) 1 + t i ·i ei.j (1

I(v) = { ijk, ikj,jik,jki, kij, kji} jik ti-j

jki ti-j+i.k

kij t i.k+j .k

kji ti.j+i.k+j.k

ti-j+j·k

ti.j+i.k+j.k

t i ·k ti.j+i.k+j.k

ti.j+i.k

t i .k

ti.j+j.k

t j ·k

1 t i' j

ti-j

jik

ti-j

1 ti-j+j·k

jki

ti.j+i.k

ti.j+i.k+j.k

1 t i .k

kij

t i ·k+ j .k

t i .k

ti.j+i.k+j.k

1 ti.j+j.k

kji

ti-j+i-k+j.k

ti.j+i-k

t i.k+ j .k

t j ·k

ii.k+j.k

1

290

J. A. Green

References [G] Green, J.A., Hall algebras, hereditary algebras and quantum groups, Inv. Math. 120 (1995), 361-377. [G1] Green, J.A., ShufRe algebras, Lie algebras and quantum groups, Textos de Matematica, Serie B, No.9, pp. 29, Departamento de Matematica, Universidade de Coimbra, 1995. [K] Kac, V.G., Infinite Dimensional Lie Algebras, Cambridge University Press, Cambridge, 1990. [1] Lusztig, G., Introduction to Quantum Groups, Birkhiiuser, Boston MA,1993. [Rh] Ree, R., Lie elements and an algebra associated with shufRes, Annals of Math. 68 (1958), 210--220. [Re] Reutenauer, C., Free Lie Algebras, Oxford University Press, Oxford 1993. [Rl] Ringel, C.M., From representations of quivers via Hall and Loewy algebras to quantum groups, Contemporary Math. 131 (1992),38140l. [R2] Ringel, C.M., Hall algebras revisited, in: Israel Mathematica Conference Proceedings, Vol. 7 (1993), 171-176. [R3] Ringel, C.M., The Hall algebra approach to quantum groups, Notas de Cursos, Escuela Latinoamericana de Matematicas (ELAM), Aportaciones Matematicas, Sociedad Matematica Mexicana, 1993. [R4] Ringel, C.M., Green's theorem on Hall algebras, in: Proceedings of ICRA VII, 1994, U.N.A.M. Mexico D.F. [Ro] Rosso, M., Groupes quantiques et algebres de battage quantiques, Comptes Rendus de l'Acad. Sci. Paris, 320 (1995), Serie I, 145~ 148.

119 Cumnor Hill Oxford OX2 9JA England Received January 1995

Fourier transforms, Nilpotent Orbits, Hall Polynomials and Green Functions Gus 1. Lehrer 1. Introduction Let G be a connected reductive group defined over the finite field IF q, and let 15 be its Lie algebra. Let F : G --+ G be the corresponding Frobenius endomorphism and write also F : 15 --+ 15 for the induced Frobenius map on 15. We are concerned here with the space C(Q;F) of Ad GF -invariant functions: Q;F --+ C. Let Q;nil be the nilpotent variety of 15.

1Ft

1.1 Definition. Let 'IjJ : --+ ex be a fixed non-trivial additive character of IF q • Define the Fourier transform F : C(Q;F) --+ C(Q;F) by Ff(X) = IQ;FI-! 2: 'IjJ((X,Y))f(Y) where (X,Y) is an Ad GYE~F

invariant non-degenerate bilinear form on 15, which is assumed to be defined over IF q (i. e. which commutes with F).

In [Kal], Kawanaka studied the restriction to 15;;\ of F f for functions which are supported on 15;;\. This work was extended and generalised by Lusztig in [Lu]. They showed that for a large class of functions f, this restriction is a multiple of Df (see [Lu, (8.5)]), where D is the "duality involution" (d. [Ll]) defined as follows. If P = LU is an F-stable Levi decomposition of the rational parabolic subgroup P of G and .c = Lie L, recall (d. [Ll, §3]) that we have mutually adjoint maps

f

Tl Tl

The maps p~ and are respectively called Harish-Chandra induction and truncation. They are defined with the aid of, but are independent of P.

1.2 Definition. Let B ~ T be an F -stable Borel subgroup and maximal torus of G respectively. For each rational parabolic subgroup P ~ B, let .cp be the Lie algebra of the (unique) Levi subgroup of P which contains T. Then define (for f E C(Q;F))

G. 1. Lehrer

292

Vf =

L

(_1)n(P)p~pTfpf

P~B

where n(P) is the semisimple rank of P.

In this note we wish to consider the following two (related) problems.

Problem A. Determine explicitly the Fourier transform of an arbitrary nilpotently supported function in C(lB F ).

Problem B. Determine explicit formulae for the Green functions of GF • The relationship between problems A and B arises as follows. Springer [Sp] has shown that the Green functions may be defined by (1.3) Qw(N) = c(w)q~ICGF(N)I(.1'~sw,~N) (N E lB;;I' r = rank(G)) where w is an element of the Weyl group W of G with respect to T, c is the alternating character of Wand Sw is a regular semisimple element of type w in lB F (see [L1, §6]). As in [L1], we use the notation ~x to denote the characteristic function of Ox, the Ad G F -orbit of X E lB F . It follows from the properties ofF ([L1, §4]) that (1.4) Hence problem B reduces to the following special case of problem A:

Problem B'. Determine the values on regular semisimple elements of the Fourier transforms of nilpotently supported functions in C(lB F ).

The key tool in our approach to Problem A is the following result ([L1, §4]).

1.5 Theorem. Let.c be as in the preamble to (1.2). Then we have the Fourier transform .1'£ : C(.c F ) ---- C(.c F ) obtained using the restriction to .c of (, ) and the same character'IjJ : ex as .1': We have

1Ft

(i) .1'£Tf = Tf.1'~. (ii) .1'~p~ = p~.1'£. (iii) .1'V = V.1'. In the present work we reduce problem A for the characteristic function of a regular nilpotent orbit to a specific conjecture (see (4.3) below) concerning the adjoint action of a maximal unipotent subgroup of G on lB. This conjecture is proved here for groups of type A; it is also known

Fourier Transforms and Nilpotent Orbits

293

for some of the exceptional groups. We use it here, together with the techniques mentioned above, to give complete solutions of problems A and B when lB = gIn. The author is grateful to D. Jackson for his careful reading of the manuscript and useful comments and to the referee for pointing out an error in the original version of this work.

2. The basic strategy We set out below the steps in our strategy. These show clearly how "Hall polynomials" enter. For N E lB;:;I' we have a "generalised Gelfand-Graev function" iN in C(lB F ). This was defined originally by Kawanaka [Ka2], but we follow Lusztig's treatment [Lu, §2]. Recall the definition of iN: let (N, N', H) be an sl2-triple in lB F and write lB = E9 lB i for the ad(H)---€igenspace decomposition of lB, where lB i is iEZ

the i---€igenspace of ad (H); write lB?:j =

E9 lB i (j

E Z). Let ,X : lB?: 1

--+

i?:j

Wq

be the linear form defined by 'x(X) = (N', X); then {X, Y} = Y]) defines a non-singular symplectic form on lB I; if J 1 is a totally isotropic subspace of lB 1 , write (following Kawanaka [Ka2]) lB>1.5 = J 1 EEl lB?:2. Define WN E C(lB~1.5) by WN(X) = 1jJ(,X(X)); then-iN = Ind~>l5 (WN). It is straightfo~ward to check that iN depends only on the Ad C F -orbit of N. With this notation, our strategy is as follows. ,X( [X,

Step 1. Show that for certain "special" nilpotent elements N, including distinguished N,

(2.1)

DiN =

L

aN' ,N~N'

0N'?:ON

where there are few terms on the right hand side and the aN',N are known (only special N' should occur). Step 2. For special N, extend Lusztig's formula [Lu, (2.5)] for FiN to obtain an explicit formula (2.2)

FiN

=

L CO,N~O

(CO,N E

q

o

the sum being over the Ad C F -orbits 0 in lB F . This step involves the geometry of Ad G-orbits in lB.

294

G. 1. Lehrer

Step 3. Combine steps 1 and 2 to obtain an explicit formula for F~N: the relation V''(N = L aN' ,N~N' may be inverted to give ~N = °N,?ON

L

bN',NV,N', whence from (1.5) we obtain

°N,?ON

Step 4. For any element N E 0) of n. The order relation on nilpotent orbits is the usual "dominance" relation for partitions; we write A :-s; J.L for this relation. Partitions A also parameterize the conjugacy classes of standard Levi subalgebras of ~. For A f- n, denote by £>. the corresponding Levi subalgebra, by 0>. (= O>.(~)) the corresponding nilpotent orbit and by 6 its characteristic function. Thus ~(n) is the function ~rn of §5.

6.1 Lemma. Let A = (AI 2: ... 2: Ap > 0) f- n. Write 6(£>.) (= ~(>.tl~(>'2) ... ~(>.p)) for the chamcteristic junction of the regular nilpotent orbit in £>.. Then

pt, (6(£>.)) =

L g~(q)~1lIl-?>'

305

Fourier Transforms and Nilpotent Orbits

where g~(q) = g()\J)(A2) ...(A p)(q) in the notation of [G, p.412] is a (Hall) polynomial in q, with g~ ::/= O. This follows easily from the definition of consequence.

pt

and has the following

6.2 Corollary. We have, for oX f- n,

6

=

L

h~(q)pt (~Jl(.cJl))

Jl?A where the

h~(q)

are mtional functions in q.

It is clear that the only denominators which occur in inverting the relation (6.2) are products of the g~(q), which are non-zero. Now the semisimple orbits of ~F are parameterised by polynomials

Write P n for this set of polynomials (identified with semisimple orbits). For f E P n , write 01 for the corresponding semisimple orbit, E(J) for its sign, ~I for its characteristic function and 11 for the corresponding function 1a of (5.1).

6 be the chamcteristic function of the corresponding nilpotent orbit of ~F. Then

6.4 Theorem. Let oX f- n and let

F6 =

L

h~(q)d;:l

Jl~A ?Jlp(~»)

Jl=(Jl1?···

L

E(J)ptb/)

IEP~

where dJl = (q - 1)p(Jl)q(3n-2p(Jl))/2, PJl is the set of semisimple orbits of.c: (identified with polynomials f = hh··· fp(Jl))' E(J) is the corresponding sign and 11 is the function on.c: defined in (5.1). Proof. Apply F(= FQ'J) to (6.2). We obtain

(6.4.1)

F6 =

L

h~(q)FPt(~Jl(.cJl))·

Jl?A But by [L1, (4.5)], FPt notation Fn = Fgl n ' we see

=

p~~F£~; hence using the transparent

306

G. I. Lehrer

But by (5.1), we have FJ.L'~(J.L')

(6.4.3)

= (q - 1)-lq-(3J.L,-2)/2 L

c(lih!,

!,E'P",

If J.L has p(J.L) parts, we see that (6.4.4) FJ.Ll (~(J.LI))FJ.L2(~(J.L2))··· = (q _1)-p(J.L)q-(3n-2 p(J.L))/2 L

cUh!·

!E'P"

Theorem (6.4) follows. D We turn now to the Green functions. These are defined, consistently with the notation of Green [G] and of (1.3), by 6.5 Definition. For partitions A,

II

of n, define

where Sv is regular semisimple of type other notation is as above. To compute the

Q~

II,

N>. is nilpotent of type A and

explicitly we shall require

6.6 Lemma. For any pair of partitions J.L, 'Yr." for the Gelfand-Graev function of £J.L)

II

of n, we have (writing

where J.L 2: II indicates that II is a refinement of J.L (i. e. J.L is obtained R

from II by combining some of the parts of II) and k(J.L, II) is the number of factorisations f = II h···fp(J.L) of a regular polynomial f of type II (i. e. a polynomial over JF q which factorises as a product of distinct irreducible polynomials whose degrees are the parts of II) where the degree of fi is J.Li (J.L = (J.L I 2: ... 2: J.Lp(J.L))). Proof. We have

(Fl5pt('YJ.L),~sJI5 = (pt(Fr.,,('Yr.,,)),~sJI5

(by [L1, (4.5)])

= (Fr." ('Yr.,,), TE" (~sJ)r."

[L1, (3.2)]

= (Fr." ('Yr.,,), Rest (~sJ )r."

[L1, (3.9)]

= q-n/2LCJ.L(R)(~~",Rest~s~)r." by (4.10) R

Fourier Transforms and Nilpotent Orbits

307

where the sum is over the regular orbits R of £~ and cJ.L(R) = ICL" (X)FI for any X E R. But OSv n £J.L = 0 unless j.L 2: II; if this condition is R

satisfied, then (~~", Rest ~sJ.c" = CJ.L(R)-l for each of the k(j.L, II) orbits R in £J.L such that OR C OSv and is zero otherwise. The lemma follows. D We are now in a position to prove 6.7 Theorem. Let A, II be partitions ofn. The Green function is given by Q~(q) = ICc(N).{1 h~(q)C~l k(j.L, II) J.L?>' J.L?V

Q~(q)

L R

where CJ.L = \CL" (NJ.L)F\, NJ.L is regular nilpotent in £J.L' the h~(q) are "Hall functions" defined in (6.2), the relations 2: and 2: denote domiR

nance and refinement respectively and k(j.L, II) is defined in (6.6) above. Proof. We have

(F6, ~sv) = ('OF(6), 'O~sv)

= (c(II)(F'O(6),~sv) (by [L1, (3.10)]). But from (6.2) we see that

L

'0(6) =

h~(q)'OIBPt (~J.L)

J.L?>'

L

=

h~(q)pt('O.c,,(~J.L))'

J.L?>' Moreover 'OJ.L(~J.L) = C~l')'.c" by (3.6). Hence (6.7.2)

'06 =

L

h~(q)C~l pt h.c,,)·

J.L?>' Combining (6.7.1) and (6.7.2) we obtain

(F6,~sJ = c(lI)

L

h~(q)C~l(FPth.c,,),~sJ. J.L?>' By (6.6), the right side of (6.7.3) is equal to

(6.7.3)

c(lI)

L

h~(q)c~lq-n/2k(j.L, II).

J.L?>' J.L?V R

The theorem now follows by substituting this expression for (F6, into (6.5).

~sv

[

G. I. Lehrer

308 References

[DLM] F. Digne, G.!. Lehrer and J. Michel, The characters of the group of rational points of a reductive group with non-connected centre, J. reine angew.Math. 425(1992), 155-192. [DM] F. Digne and J. Michel, Representations of reductive groups over finite fields, Cambridge D.P., Cambridge (1991). [G] J .A. Green, The characters of the finite general linear groups, Trans. A.M.S. 80(1955), 402-447. [Ka1] N. Kawanaka, Fourier transforms of nilpotently supported invariant functions on a finite simple Lie algebra, Invent. Math. 69(1982),411-435. [Ka2] N. Kawanaka, Generalised Gelfand-Graev representations of exceptional simple algebraic groups over a finite field, Invent. Math. 84(1986), 575-616. [Ke] S.V. Keny, Existence of regular nilpotent elements in the Lie algebra of a simple algebraic group in bad characteristics, J. Algebra, 108(1987), 194-20l. [K] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85(1963), 327-404. [L1] G.!. Lehrer, The space of invariant functions on a finite Lie algebra, Trans. A.M.S. 348 (1996) 31-50. [L2] G.!. Lehrer, On the values of characters of semisimple groups over finite fields, Osaka J. Math. 15(1978), 77-99. [L3] G.!. Lehrer, Rational tori, semisimple orbits and the topology of hyperplane complements, Comment. Math. Helvetici 67(1992), 226-25l. [Lu] G. Lusztig, A unipotent support for irreducible representations, Adv. Math. 94(1992), 139-179. [LS] G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. Land. Math. Soc. 19(1979),41-52. [Me] !.G. MacDonald, Symmetric functions and Hall polynomials, Clarendon Press, Oxford 1979. [Sh] T. Shoji, Geometry of orbits and Springer correspondence, Soc. Math. France Asterisque 168(1988), 61-140. [SpL] T.A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173-207. [Sp2] T.A. Springer, Generalisation of Green's polynomials, Proc. Symp. Pure Math. A.M.B. 21(1972), 159-153. [SpSt] T.A. Springer and R. Steinberg, Conjugacy classes, in Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes

Fourier Transforms and Nilpotent Orbits

309

in Math. 131, Springer Verlag (1970), 167-266. [Stl] R. Steinberg, Regular elements in algebraic groups, Publ. Math. I.H.E.S. 25(1965), 49-80. [St2] R. Steinberg, Lectures on Chevalley groups, Yale University (1967). School of Mathematics and Statistics University of Sydney Sydney, Australia, 2006 Received February 1995

Degres relatifs des algebres cyclotomiques associees aux groupes de reflexions complexes de dimension deux l Gunter Malle

1. Introduction

Selon Ie Fundamental Theorem de [3] il existe un lien etroit entre la decomposition du foncteur de Lusztig Rr()..) pour un groupe reductif fini G, avec).. caractere unipotent d'un sous-groupe de Levi L de G, et Ie groupe W = Wc(L, )..), un groupe de reflexions complexes, appeIe groupe de Weyl cyclotomique. M. Broue et l'auteur ont conjecture dans [2] que ce lien devrait etre explique par les algebres cyclotomiques 1t(W) pour les groupes de reflexions W, introduites dans loco cit. Plus precisement, l'algebre commutante d'un certain complexe definissant Ie caractere de Lusztig Rr()..) devrait etre une specialisation de 1t(W). Dans ce cas, en vertu de [2, §1], les degres relatifs de 1t(W), multiplies par Ie degre de Rr ()..) , donneraient les degres des composantes irreductibles de Rr()..). Les degres relatifs sont connus, au moins modulo certaines conjectures, pour presque toutes les algebres cyclotomiques associees a des groupes de reflexions finis, et la consequence indiquee ci-dessus a ete verifiee dans [10] pour les cas imprimitifs, et dans [2] pour quelques cas primitifs. NollS determinons ici les degres relatifs des series infinies de groupes de reflexions imprimitifs de dimension 2, confirmant ainsi une conjecture enoncee dans [10, 2.20]. Puis nous indiquons comment trouver les degres relatifs des 19 groupes de reflexions primitifs de dimension 2. A l'aide des formules explicites, nous verifions les consequences des conjectures de [2] concernant les degres. En outre, nollS determinons dans quels cas les groupes primitifs ont des degres unipotents associes, qui possedent les proprietes etudiees dans [10] dans Ie cas des groupes imprimitifs. Pour quatre groupes primitifs, les degres relatifs avaient deja ete determines dans [2, §5]. D'autre part, Ie cas de G6 (avec les notations de [L1]) est traite dans [7]. On trouve Ill. aussi des matrices pour les I Je tiens aremercier Anne-Marie Aubert et Jean Michel pour une lecture attentive d'une version anterieure, et la Fondation Alexander von Humboldt pour son soutien financier.

312

G. Malle

representations irreduetibles de 1-l(W), pour W

= Gi

ou. 4 :S i :S 15.

2. Notations et rappels 2A. Diagrammes et Algebres Cyclotomiques Soit W un groupe de reflexions fini, c'est it dire admettant un plongement W '----> GLn(C) tel que l'image de West engendree par des elements qui tous centralisent un sous-espace de dimension n - 1 (ces elements sont appeles rejiexions complexes). Nous disons que West irreductible (resp. primitif) si l'image de W agit de maniere irreductible (resp. primitive). Si West irreduetible, il est clair qu'il faut au moins n reflexions pour engendrer W. D'autre part on sait que n + 1 reflexions suffisent toujours. Pour les groupes de reflexions reels, on a une presentation canonique de W comme groupe de Coxeter sur n reflexions. De telles presentations existent aussi dans Ie cas general. Si West encore engendre par n reflexions, alors la presentation peut etre decrite par un diagramme ressemblant it un diagramme de Coxeter (et trouve par Coxeter); en general, la presentation comprend n + 1 refl.exions et on doit ajouter une relation circulaire, voir [2J et [5J. Les diagrammes correspondants sont appeles diagmmmes cyclotomiques. Ces diagrammes ont des proprietes favorables; par exemple dans presque tous les cas on retrouve les sous-groupes paraboliques de W en enlevant des nceuds du diagramme cyclotomique. Dans Ie cas des groupes de reflexions reels, la presentation de Coxeter conduit it une definition possible de l'algebre d'Iwahori-Hecke 1-l(W) associee it W. Les relations sont soit des relations d'ordre pour un generateur, soit des relations de tresses entre deux generateurs avec meme nombre d'elements des deux cotes. Pour obtenir une presentation de 1-l(W), on garde les relations de tresses et deforme les relations d'ordre. Un procede similaire fonctionne pour les groupes de reflexions complexes, en partant des presentations derivees des diagrammes cyclotomiques (voir [2, §3]). Plus precisement, soit W un groupe de reflexions irreductible, et W

= (s

E

5

I Sd == 1 pour s E 5, r == 1 pour r s

E R}

(2.1)

la presentation de W definie dans [2] ou [5], ou. S designe l'ensemble des generateurs et R l'ensemble des relations de tresses entre les s E S.

Definition 2.2 Soient {Us,i I s E S, 1 :S i :S ds } algebriquement independants sur Q teis que Us,i = Ut,i si s '" t dans W, et u = (Us,i), u- 1 = (u;,I). L'algebre cyclotomique 1-l(W, u) generique sur Il[u, u- 1], attachee Ii La presentation (2.1) de W, est engendree par des elements

313

Degres relatifs

T., s E S, satisfaisant aux memes relations de tresses r E R que les s E S et avec

(Ts - u s, I)'" (Ts - u s, d s ) = 0

pour s E S.

L'importance de ces algebres vient du fait qu'il existe un lien conjectural avec la decomposition des caracteres de Lusztig Itl(>,) pour les groupes reductifs finis, voir [2, §1]. Cette conjecture a ete demontree dans certains cas simples ou. West cyclique, voir [8] et [6]. Le cas general est ouvert. Quelques consequences de la conjecture peuvent quand meme etre verifiees. Par exemple, la conjecture suivante a ete demontree dans presque tous les cas (voir [2, Satz 4.7] et [1]).

Conjecture 2.3 Soit W un groupe de refiexions complexes. Alors l'algebre cyclotomique 1t(W, u) est libre de rang IWI sur Il[u, u- I ]. En particulier pour tout corps K algebriquement clos contenant Il[u], l'algebre = K 0Z[uj 1t(W, u) est isomorphe Ii l'algebre de groupe KW.

1t K (W)

En [2, Folg. 6.7J on a aussi montre que les proprietes de rationalite des caracteres irreductibles des 1t(W, u) coincident avec celles des caracteres unipotents des groupes reductifs finis qui leur sont associes par la conjecture enoncee dans loco cit.

2B. Degres Relatifs Soient R un anneau integre commutatif de corps de fractions K, et 1t une R-algebre R-libre de dimension finie telle que 1t K := K 0R 1t est semi-simple scindee. Les formes symetrisantes sur 1t K sont donnees par les combinaisons K-lineaires LXElrr('HK) 8x X de caracteres irreductibles de 1t K avec 8x ::/= 0 pour tout X E Irr(1t K ). Soit t 1t -+ Rune forme lineaire avec t(hh') = t(h'h) pour toutes h, h' E 1t. On a alors t =

L

8x (t) X

xElrr('HK)

et 8x (t) est appele Ie degre relatif de X par rapport at. Vne R-base (1"i)I 1 and Qi f---> (i for a primitive r-th root of unity r ) the group algebra of the wreath product C I §n' ( one obtains from r This specialization gives rise to a bijection of the irreducible characters by Tits' Deformation Theorem ([3], (68.17)). For r = 2 and Ql = -1 the algebra H~2) is the Iwahori-Hecke algebra of type Bn with parameters qs; = q, i = 1, ... , n - 1, and qt = Qo. In this case we denote Qo simply by Q. From Section 10 on we restrict our investigations to this case. Finally note that for r = 1 and Qo = 1 the algebra H~r) is the Iwahori-Hecke algebra H n of type An-I'

HA

4. Tuples of Partitions and Further Notation We have to generalize some of the notation from Section 2 before we proceed. An r-tuple of partitions of n is a sequence [ = hO, ... , [r-l) of r partitions with L~':J hi I = n. A 2-tuple of partitions is also called a double partition. The r-tuple of partitions [ is represented by the r-tuple of diagrams [i, i = 0, ... , r - 1 . This diagram also is denoted by [. If 0: and [ are r-tuples of partitions with o:i ~ [i for i = 0, ... , r - 1, then we write 0: ~ f. In this case the set theoretic difference [ \ 0: is a skew diagram. A strip [ \ 0: does not contain any 2 x 2-blocks, and a hook is a connected strip. For a hook [ \ 0: we have o:i = [i for all 0 :::; i :::; r - 1 except i = rh \ 0:). This means, rh \ 0:) denotes the position of the component of [, which (completely) contains the hook [\ 0:. A tableau 11' of shape [ is an r-tuple (11'0, ... , lI'r-l) of ordinary tableaux lI'i, where lI'i is of shape [i for i = 0, ... ,r - 1. Such a tableau is called semistandard tableau, if each component lI'i is an (ordinary) semistandard tableau, and it is called standard tableau, if it furthermore contains each of the numbers 1, ... ,n exactly once. Let k :::; n and let 0: be an r-tuple of partitions of n - k with 0: ~ [. A (skew) standard tableau 11' of shape [\ 0: maps each box of the diagram [ \ 0: to one of the numbers n - k + 1, ... ,n, such that these numbers strictly increase along the rows and along the columns of each component of 11'. The number of standard tableaux of shape [ (resp. [\ 0:) is denoted by (resp. r\O). Denote by STh) the set of all standard tableaux of shape [.

r

Lemma 4.1 Let k :::; n and let [ be an r-tuple of partitions of n. The map STh) ---;

U

ST(o:) x STh \

0:)

h\ol=k

11'

t---+

(11'1,11'2),

which decomposes each standard tableau 11' of shape [ into a part 11'1, which contains the numbers 1, ... , n - k, and a part 11'2, which contains the numbers n - k + 1, ... ,n, is a bijection.

Proof. Since 11' is a standard tableau, the boxes of 11' which contain the numbers 1, ... , n - k form a diagram 0: ~ [ for some r-tuple of partitions 0: of n - k.

IwahoT'i-Hecke Algebras of Type B

339

Conversely, a standard tableau of shape 0: and a standard tableau of shape 1 \ 0: are uniquely combined to a standard tableau of shape I. 0 If 11' is a standard tableau, define 7(11' : m) for m ~ n such that the number m is contained in the component lI'T(ll':m) of 11'. The content of m in 11' is defined to be c(lI' : m) = i - j if m lies in the i-th row and in the j-th column of this component. The axial distance between m1 and Tn2 in 11' is defined as r(m1' m2) = c(lI' : m2) - c(lI' : md. The symmetric group §n acts on the set of all tableaux of shape 1 and of content [In] by permuting the entries. So the image 11'( i, i + 1) of 'll' under 8i = (i, i + 1) arises from 11' by exchanging the entries i and i + 1 in 11'. Here the image lI'(i, i + 1) of a standard tableau 11' is a standard tableau, if and only if the numbers i and i + 1 are neither contained in the same row nor in the same column of a component of 11'.

5. Wreath Products and the Weyl Group of Type Bn Let X n

=

C r / §n be the wreath product of the cyclic group C r of order

r and the symmetric group §n on n points.

The elements of X n are of the form x = (j; a) = (iI, ... , fn; a) with = 1, ... , n and a E §n. For an element x = (j; a) E X n and a k-cycle /'i, = (j, j/'i" ... ,j/'i,k-1) of a the product

1; E C r , i

is an element of C r = (c) and is called cycle product of x and /'i,. The cycle structure of x is an r-tuple of partitions 1r which is defined as follows. For this let 1r j , j = 0, ... , r - 1, be the partition which contains a part k for each k-cycle /'i, of a with g((j;a),/'i,) = d. Then 1r = (1r 0 , ... ,1rr - 1) is an r-tuple of partitions and the conjugacy classes of X n are described in the same way as the conjugacy classes of the symmetric group.

Proposition 5.1 Two elements of X n are conjugate if and only if the have the same cycle structure. Proof. [10] (4.2.8).

o

Thus the conjugacy classes of Cr/§n are parameterized by the r-tuples of partitions of n. The Weyl group W n = W(B n ) of type B n is generated by S = {t, 81, ... , 8n -1} subject to the relations implied by the following Dynkin diagram.

340

G. Pfeiffer

--

sn-1

The elements Sl, ... ,Sn-1 generate a subgroup of W n which is isomorphic to §n. The sign changes t i defined by to = t and t i = Siti-1Si for i = 1, ... , n - 1, generate an elementary abelian normal subgroup of order 2n in W n . For each k :::; n the subgroup (tn-k, Sn-k+1, ... , Sn-1) of W n is isomorphic to W k and the subgroup (t, S1, ... , Sn-k-1, tn-k, Sn-k+1,"" Sn-1) is isomorphic to Wn - k X Wk. W n is isomorphic to the wreath product C2 I §n' Hence the general results for wreath products apply to this group. In particular the conjugacy classes of W n are parameterized by the double partitions of n. A standard representative of minimal length in the conjugacy class of W n which belongs to the double partition 7[" = (7["0,7["1) of n is constructed as follows. Let l = l (7["1) and define a sequence a of sums by O'i =

0 O'i-1 { O'i-1

+ 7["l-i+1 + 7["?-/

for i = 0, for i = 1, ... ,l, and for i = l + 1, ... l + l(7["O).

The representative arises from the word SOS1 ... Sn-1Sn by replacing Si by t i for all i E a such that i < a/ and by omitting Si for all remaining i E a. Note that always 0 and n are in a, whence the undefined symbols So and Sn are always replaced or omitted. If, for instance, 7[" = ([3], [4, 1]) is a double partition of 8, then a = (0,1,5,8) with l = 2. This yields tohs2S3S4S6S7 = tS1tS1S2S3S4S6S7 as a representative of minimal length for this class. A presentation of the wreath product Cr I §n is obtained by specializing q t---+ 1 and Qi t---+ (i for a primitive r-th root of unity ( in the Definition 3.1 of the cyclotomic algebra H~r). We get the same presentation by replacing the relation t2 = 1 by the relation tr = 1 in the above presentation of W(Bn ) by the Dynkin diagram. The group Cr I §n is a complex reflection group, i. e., it has a complex representation in which the generators are pseudo reflections (n x n-matrices with n - 1 eigenvalues 1). In the classification of the irreducible finite complex reflection groups by Shephard and Todd [17] Cr I §n is denoted by G(r, 1, n).

6. Characters of Wreath Products The irreducible characters of X n = Cr I §n are parameterized by the rtuples of partitions of n. The character xC> corresponding to an r-tuple

341

Iwahori-Hecke Algebras of Type B a = (aD, ... ,ar -

1

) of partitions of n can be described as follows (d. [10] chapter 4). For this let

1

and i 2: n - k

353

+ 2 we

obtain (by the braid relations)

and for j < k

Therefore

(Tn_kT",)k = (Tn-kSn-k+1 ... Sn_2)k-1 Sn-1

where the (k -1 )-th power equals Tn equals Tn - 1 by definition.

...

k ••.

Sn-k+1 Tn-kSn-k+1 ... Sn-1,

T n - 2 by induction and the rest D

In the case where 1\0: is a hook we define one further quantity d~ as the content of a box immediately underneath the hook 1\0:. Note that e~ and ~ depend on the actual choice of I and 0: and not only on the skew diagram I \ 0:.

Lemma 11.5 If I \ 0: is a hook, then e~

= n + d~.

Proof. A hook of length k always occupies exactly one box in a row of k successive diagonals. Hence k

cb) - c(o:) = ~)~ + i) = k~ + k(k + 1)/2, i=l

and the assertion follows from the definition of e~.

D

Lemma 11.6 We have

Proof. Let

n-1

P= By (11.4) we have Tt:J. on V'Y\o by (11.3).

( II

i=n-k

'Y Q T(i+1) ) l/k eo'

= (Tn_kT",)k

and pk is the only eigenvalue of Tt:J.

G. Pfeiffer

354

Therefore the eigenvalues of Tn-kT" are of the form ~j p for a k-th root of unity ~ and certain j = 0, ... , k - 1. Hence XJ\o (Tn-kT,,) = Z P for some sum Z of k-th roots of unity. p specializes to (IT(-lr(i+l))l/k, and by (11.2) we have

Z( IT

(_If(i+l))l/k

i=n-k (-l)I~(_1)Th\O)

{

O

for hooks 1\0:, otherwise.

Since p -=I- 0 we have Z = 0 and therefore XJ\O(Tn-kT,,) = 0, if 1\0: is not a hook. Otherwise all boxes of the diagram I \ 0: lie in the same component r( I \ 0:) of the diagram I and n-l

II

i=n-k

QT(i+l)

=

Q~h\o)'

Then p = qe~QTh\o) and p specializes to (-1) T h\o). Therefore Z (-1)1~, and by (11.5) we have e~ = n +~. 0 The recursion formula in this case can now be stated as follows.

Proposition 11.7 Let h E H~~k' Then

XJ(hTn-kT"J =

L

QT(-y\O) (_l)l;;qn+d;;X;(h),

h\ol=k

where the sum is taken over all double partitions 0: of n - k ! such that 1\0: is a hook. Here d~ denotes the content of a box directly underneath the hook 1\0:.

Proof. The assertion follows from the decomposition of the character XJ = Lo X;XJ\o in (9.3) and the computation of XJ\O(Tn_kT,,) in (11.6).

o 12. The Character Table of the Iwahori-Hecke Algebra of Type B n We combine the formulas of the preceding two sections into a theorem which completely describes the character table of the Iwahori-Hecke algebra of type B n . For this let X~[ 1,[ ]) = 1.

355

Iwahori-Hecke Algebras of Type B

Theorem 12.1 Let"( = ("(0,"{1) and 7[" = (7["0,7["1) be double partitions of n and let w E Wn be an element of minimal length in the class of element with cycle structure 7[". Furthermore let E = 1, if 7["0 = [ ], and = 0 otherwise. In the case E = 0 let k = 7["? for l = l(7["°), otherwise let k = 7["~. Finally let p be the double partition of n - k, which results from 7[" by removing the part k. Then the value XJ (7[") of the character XJ on T w is given by

E

if E = 0, and otherwise.

Here the sum is taken over all double partitions 0: of n - k, such that "( \ 0: is a strip in the case E = 0 and a hook in the case E = 1.

Proof. The standard representative W 1r of the class with label 7[" which is constructed in Section 4 consists of blocks of the form t~ Si+ 1 ... Si+ j -1' Such a block corresponds for E = 1 to a part j of 7["1 and for E = 0 to a part j of 7["0. In [8] it is shown, that these elements are of minimal length in their conjugacy class. The corresponding element TW1C E H~2) therefore has the form

for some T wp E H~~k whose character values are known by induction. In the case E = 1 we have 7["0 = [ ] and k = 7["~. Since Sn-k+1 . .. Sn-1 = TKo the assertion follows here by (11.7) and h = T wp E H~~k' Otherwise k = 7["~1rO) and the assertion follows by (10.3) and h = T wp '

o The result is illustrated by character table of the Iwahori-Hecke algebra of type B 2 in Table 1 and that of the Iwahori-Hecke algebra of type B 3 in Table 2. The rows in these tables correspond to the irreducible characters and the columns correspond to the conjugacy classes of the Weyl group. For each class a representative of minimal length is given.

13. Concluding Remarks Proposition (8.2) has an interesting consequence.

356

G. Pfeiffer

(1 2 ,0) 1 1 2 1 1 1

H(B2)

(P,O) (1,1) (0, 12 ) (2,0) (0,2)

(1, 1)

(0,1 2 )

(2,0)

(0,2)

t Q Q -1

tsts Q2 -2qQ

s

ts -Q

q-1

-1

1

-1

0 1

Q

q'IQ2 q2

q q

qQ -q

-1

-1

Table 1: The Iwahori-Hecke algebra of type B 2 . H(B 3 )

(1 3 ,0) (1 2 , 1) (1, p) (0, 13 ) (21,0) (1,2) (2,1) (0,21 ) (3,0) (0,3) H(B 3 ) (cont. ) (1 3 ,0) (1 2 ,1) (1, p) (0,1 3 ) (21,0) (1,2) (2,1) (0,21) (3,0) (0,3)

(F,O) 1 1 3 3 1 2 3 3 2 1 1

(P,l)

(1, p)

to Q 2Q -1 Q-2

totl Q2 Q2 _ 2qQ -2qQ + 1

-1

(0,

1

q2Q2 + Q2 -2qQ+ rl q2Q2 _ 2qQ q2 + 1 q'IQ2 q2

2Q Q-2 2Q -1

-2 Q

-1

(1,2)

(2,1)

(0,21)

tos l -Q -Q

tos2 -Q qQ - Q+ 1 -Q - q + 1

t ot l s2 _Q2

1

-1 0

1 1 qQ-Q -q qQ -q+ 1 qQ -q

qQ-Q qQ - q + 1 qQ - Q - q -q+ 1 qQ -q

qQ qQ

_q2Q _q2Q

0 (fQ2 q3

1~)

(21,0)

totl t2

Sl

Q~

-1

_3 q2Q2 3q2Q -1

q-2 q-2

2q3Q3 3q4Q _3 q4Q2 -2 q3 q6Q'!>

q -1 2q -1 2q -1 q -1 q q

-qC'

-1

(3,0)

(0,3)

SlS2

tos l S2 Q

1 -q+ 1 -q + 1

0 0 -1

1 -q q'2 _ q q2 _ q

-qQ

-q q2 q2

q q2Q _q2

0 0

Table 2: The Iwahori-Hecke algebra of type B 3 •

Iwahori-Hecke Algebras of Type B

357

Corollary 13.1 For Tn- I = Sn-I ... SITS I ... Sn-I we have 2n-2r-1

II

II(Tn- 1 - qiQj) i=O j=O

= O.

Proof. Tn- I acts diagonal in every irreducible representation of H~r) and its eigenvalues are of the form qc(ll':n)+n-IQj by (8.2). The content c(1I' : n) of n in a standard tableau 11' of shape I for an r-tuple of partitions I of n lies between -(n-1) and n-l. Therefore 0::; c(1I' ; n)+ (n-1) ::; 2n - 2. (For r = 1 and n = 1,2,3 further restrictions apply here.) D Corollary 13.2 HZ is an epimorphic image of a specialization of the cyclotomic algebra of type r (2(n-k)+I)).

Bi

Proof. The generators Si, i = n - k + 1, ... , n - 1 of HZ satisfy the defining relations (ii), (v) and (vi) of the cyclotomic algebra of type r (2(n-k)+I)) in Definition (3.1). The generator Tn- k of HZ commutes with all Si, i > n - k + 1, (iv), and

Bi

Tn-kSn-k+ITn-kSn-k+1 = Tn-kTn-k+1 = Sn-k+ITn-kSn-k+ITn-k. By (13.1) Tn -

k

= Tn-k+ITn-k

also satisfies the relation

2(n-k) r-I

II II (Tn- k -

qiQj)

=

O.

i=O j=O

Bi

Therefore all defining relations of a cyclotomic algebra of type r (2(n-k)+I)) with parameters q and qiQj, i = 0, ... , 2(n - k), j = 0, ... , r -1 are satD isfied. Moreover H~~k commutes with HZ.

Lemma 13.3 Let hI E H~~k and h 2 E HZ. Then h Ih 2 = h 2h l . Proof. Because of the relations in Definition (3.1) it is sufficient to show that Tn- k commutes with the generators of H~~k' Let 1 ::; j < n - k. Then we have (by the braid relations)

Sn-k . . , SISj+ITS I

Sn-k

Sn-k'" SITSj+IS I

Sn-k = Tn-kSj .

Sn-k . .. S2 TS ITS I

Sn-k

Sn-k'" SITS ITS2

Sn-k

Moreover TTn- k

=

Tn-kT.

D

G. Pfeiffer

358

This explains to what extent it is appropriate to talk in Section 11 about a restriction to a subalgebra of type B x B. For r = 1 (and Qo = 1) the algebra H~r) is the Iwahori-HeckeAlgebra H n of type An-I. In this case the proof of (10.3) together with the ordinary Littlewood-Richardson rule (2.2) yields the formula for the character table of H n from [13]. Theorem 13.4 Let 7f and 'Y be partitions of n and let w E §n be an element of minimal length in the class of elements with cycle structure 7f. Moreover let k = 7fi for some i ~ l(7f) and let p = [7fI, ... ,7fi-l, 7fi+I, ... , 7fl(1l")]. Then the value xJ(7f) of the character XJ on Tw is given by

XJ(7f) =

L

(q _1)C2-1(_1)12l-12-c;:X~(p),

l-y\ol=k

where the sum is over all partitions a of n - k such that 'Y \ a is a strip. The proof of (11.7) also works for r > 2. After a further generalization of the Littlewood-Richardson rules in Section 7 for wreath products Cr I §n the proof of (10.3) also works for elements of the form hT" in a cyclotomic algebra H~r) with r > 2. Then both character formulas would be valid in general in cyclotomic algebras H~r). For r > 2, however, these formulas are not sufficient to determine a complete "character table" of this algebra. Here one first has to answer the question, for which elements h E H~r) in analogy to Theorem (1.1) the characters should be evaluated, and how the character values of other elements can be obtained from these. References

[1] S. Ariki and K. Koike, A Hecke algebra of (Z/rZ) I Sn and construction of its irreducible representations, Adv. Math. 106 (1992), 216-243. [2] M. BroUl~ and G. Malle, Zyklotomische Heckealgebren, Representations unipotentes generiques et blocs des groupes reductifs finis, Asterisque, vol. 212, 1993, pp. 119---189. [3] C. W. Curtis and 1. Reiner, Representation theory of finite groups and associative algebras, Wiley, New York, 1962. [4] M. Geck, On the character values of Iwahori-Hecke algebras of exceptional type, Proc. London Math. Soc. (3) 68 (1994), 51-76.

Iwahori-Hecke Algebras of Type B

[5]

359

, Beitrage zur Darstellungstheorie von Iwahori-Hecke Algebren, Aachener Beitriige zur Mathematik, vol. 11, Verlag der Augustinus Buchhandlung, Aachen, 1995.

[6] M. Geck, G. HiB, F. Lubeck, G. Malle, and G. Pfeiffer, CHEVIE - A system for computing and processing generic chamcter tables,

Applicable Algebra in Engineering, Communication and Computing 7 (1996), 175-210. [7] M. Geck and J. Michel, "Good" elements in conjugacy classes of Coxeter groups, and an application - computation of the chamcter table of the Iwahori-Hecke algebra of type E s , to appear J. London Math. Soc.

[8] M. Geck and G. Pfeiffer, On the irreducible chamcters of Hecke algebras, Adv. Math. 102 (1993), 79-94. [9] P. N. Hoefsmit, Representations of Hecke algebras of finite groups with BN pairs of classical type, Ph.D. thesis, University of British Columbia, Vancouver, 1974. [10] G. D. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Math., vol. 16, Addison-Wesley, 1981.

[11] A. Kerber, Algebraic combinatorics via finite group actions, BI-Wissenschaftsverlag, Mannheim, 1991. [12] G. Pfeiffer, Character tables of Weyl groups in GAP, Bayreuther Math. Schr. 47 (1994), 165-222. [13]

, Young chamcters on Coxeter basis elements of IwahoriHecke algebms and a Murnaghan-Nakayama formula, J. Algebra 168 (1994),525-535.

[14]

, Charakterwerte von Iwahori-Hecke-Algebren von klassischem Typ, Aachener Beitriige zur Mathematik, vol. 14, Verlag der Augustinus Buchhandlung, Aachen, 1995.

[15] A. Ram, A Frobenius formula for the characters of the Hecke algebras, Invent. Math. 106 (1991), 461-488. [16] M. Schonert et aI., GAP 3.1-Groups, Algorithms and Programming, Lehrstuhl D fUr Mathematik, RWTH Aachen, 1992.

360

G. Pfeiffer

[17] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304. [18] J. R. Stembridge, On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math. 140 (1989), 353-396. Department of Mathematics University of St Andrews Fife KY16 9SS Scotland [email protected] Received December 1994

The Center of a Block Lluis Puig

1. Introduction 1.1. When Richard Brauer, facing the difficulty of determining the irreducible modular characters of a block b, introduced in [1] the notion of basic set, was he aware that the invariant behind the corresponding generalized decomposition matrices is just the center of b, endowed with the ideal generated by the ordinary characters? Here, we will not try to answer this question but only to provide a proof of this fact, which, as far as we know, has never been published. 1.2. As usual, let p be a prime number, k an algebraically closed field of characteristic p, 0 a complete discrete valuation ring of characteristic zero having k as the residue field (Le., k = OjJ(O) = 6, where J(O) is the radical of 0) and K is the field of fractions of O. Actually, most of our statements remain true without assuming that 0 is complete and k algebraically closed. Let G be a finite group and b an idempotent of Z(OG), and assume that K is a splitting field for the pair (G, b), that is to say, that KGb is isomorphic to a direct product of matrix algebras over K. 1.3. Let us call the generalized decomposition matrix of the pair (G,b) the square matrix DC,b formed by the ordinary and generalized decomposition numbers. That is, denoting by IrrK:(G, b) the set of irreducible ordinary characters of G associated with b (i.e., the characters of the simple KGb-modules and, for any p-element u of G, by Irrk(GC(U), Bru(b)) the set of irreducible modular characters of Gc(u) associated with the idempotent Bru(b) of Z(kGc(u)) (i.e., the modular characters of the simple kGc(u)Bru(b)-modules), where Br u: (OG)(u) --+ kGc(u) is the Brauer homomorphism mapping L:XEC AxX E (OG)(u) on L:xECG(u) >'xx, the matrix DC,b is formed by

the coefficients d

Zo(B)

and

Z°(f): ZO(B)

----->

----->

ZO(A),

B maps

(2.11.1)

but, in general, there is no evident relationship between Z(A), Z(B) and f. However, Proposition 2.2 implies that 2.11.2. If ~o: ZO(B) -----> ZO(A) is a bijective V-linear map, there is at most one map ~: Z(B) -----> Z(A) such that ~O(z· (}O) = ~(z)· ~O({}O) for any z E Z(B) and any (}o E ZO(B), and then ~ is an V-algebra isomorphism. Indeed, if J.L 0 is a generator of ZO(B) and such a map ~ exists, ~O(J.L0) has to be a generator of ZO(A) since ~o is surjective and therefore ~(z) is determined by ~O(z, J.L 0) for any z E Z(B); moreover, in that case, if z and z' belong to Z(B) we have ~(z

+ z')· ~O(J.L0) = ~o(z· J.L 0) + ~o(z' . J.L 0) = (~(z) + ~(z')) . ~ 0(J.L 0) (z' . J.L 0)) = ~(z) . ~o(z' . J.L 0) = (~(z)~(z')) . ~0(J.L0) (2.11.3)

~(zz') . ~O(J.L0) = ~O(z·

and therefore ~ is an V-algebra isomorphism. Let us show a significant situation where such a map ~ exists; here we consider both £,dA) and £,dB) endowed with their usual scalar product 2.11.4. Assume that K 00 A and K 00 B are split semisimple Kalgebras. Then, if ~o: ZO(B) -----> ZO(A) is a bijective V-linear map which induces a group isomorphism A: £,dB) -----> £,dA), there is an V-algebra isomorphism~: Z(B) -----> Z(A) such that ~O(z·{}O) = ~(z)· ~0({}0) for any z E Z(B) and any (}o E ZO(B), if and only if A is an isometry. Indeed, the existence of A forces ~O(Z~h(B)) = Z~h(A) and the existence of ~ implies that, for any X E Irr,dB) there is (E Irr,dA) such that ~o(VXo) = V(o (cf. Proposition 2.7), so that, denoting respectively by Xx and Xc the elements of £,dB) and £,dA) such that ch B (10 Xx) = XO and ch A (10 Xc) = (0, we have A(Xx ) = Xc or -Xc. Conversely, if A is an isometry then, for any X E Irr,dB) there is ( E Irr,dA) such that ~O(VXO) = V(o; now, it is quite clear that (cf. 2.8.2)

The Center of a Block

369

and it suffices to apply Theorem 2.9 to get an O-algebra isomorphism ~: Z(B) s:! Z(A) fulfilling the above conditions. 3. A Basis for the Cocenter of a Block 3.1. Let us come back to the situation of the Introduction. Now, according to Theorem 2.9 and Remark 2.10, in order to prove Theorem 1.6, it suffices to exhibit the generalized decomposition matrix DC,b as the matrix of the O-linear map ch ocb in the canonical O-basis of o 0z .cdOGb) = 0 0z .cdG, b) and a suitable O-basis of ZO(OGb). First of all, we will describe an O-basis of Zo(OGb), which is O-free since

Zo(OGb) = b· Zo(OG)

(3.1.1 )

and it is clear that Zo(OG) has no torsion. 3.2. Recall that a pointed element U E over the interior G-algebra OGb is a pair formed by an element U of G and a conjugacy class c of primitive idempotents of the algebra of fixed elements (OGb)(u) , called point of U over OGb (cf. Definition 1.1 in [5]). We say that U E is a local pointed element or that c is a local point of U over OGb if U is a p-element and Bru(c) i:- {O} (cf. 1.3); in that case, since Bru(c) is a conjugacy class of primitive idempotents of kCc(u), c determines an irreducible modular character CjZO(C), where ZO(C) is the identity component of the center Z(C) of C. Let £ be a local system on E which is the inverse image of ,C' I:8J £' under the natural map C -> CjC der X CjZO(C), where CCder is the derived subgroup of C, and £' is a C-equivariant irreducible local system on C (under the conjugation action of C) and ,C' E S(C j CCder). A pair (E, £) is called a cuspidal pair ([L3, 2.4]) if the following condition is satisfied; for any parabolic subgroup P £; C with unipotent radical Up and any element 9 E PjUp we have H~(7rpl(g) n E,£) = 0, where trp : P -> PjUp is the natural map and {j = dim (EjZO(C)) - dim (class of gin PjUp). Let L be a Levi subgroup of a parabolic subgroup P of C and let

375

Unipotent Characters

i be the set of character sheaves on L. In [L4, I, 4.1], the notion of induction ind~ of character sheaves was introduced. In particular, for each A E L, ind~A is a semisimple perverse sheaf on G, and each irreducible direct summand is a character sheaf. A character sheaf on G is said to be cuspidal if it is not contained in ind~A for any P S;; G and any A E L. Since p is almost good, it is known by the main result of [L4] that, for any cuspidal pair (E, f), the (shift of) intersection cohomology complex IC(.t,£)[dimE], extended to the whole of G by 0 on G -.t, is a cuspidal character sheaf on G. All the cuspidal character sheaves on G are obtained in this way. Furthermore any character sheaf of Gis obtained as a direct summand of ind~Ao for some parabolic subgroup P of G and a cuspidal character sheaf A o on L. 2.3. We now assume that G has a fixed F q-structure with Frobenius map F : G -> G. A complex K E DG is said to be F-stable if F* K ~ K. For an F-stable complex K, with a given isomorphism cP : F* K ~K, we define a characteristic function XK,cp : G F -> Ql by XK,cp(X)

=

L( _l)i Tr (cp, 1i~(K)),

where 1i~(K) denotes the stalk at x E G F of i-th cohomology sheaf 1i i (K) of K, and cp is the induced linear map on 1i~(K). If K is a G-equivariant perverse sheaf, XK,cp gives rise to a class function on G F . Let L be an F-stable Levi subgroup of a (not necessarily F-stable) parabolic subgroup P of G, and let (E, £) be a cuspidal pair on L. We assume that (E,£) is F-stable, i.e., F(E) = E and F*£ ~ £. We fix an isomorphism CPo : F* £ ~£. Let K = ind ~Ao for A o = Ie (.t, £ )[dim E]. Then as the following construction shows, K has a natural mixed structure (ef. [L4, II, 8.1]). Let E reg be the open set of E consisting of gEE such that Z~(gs) c L, where gs is the semisimple part of g, and let Y = UgEG gEregg- l . We consider the diagram

E

o

-----+

'

Y

~

-----+

-

Y

~

-----+

Y,

where

Y=

{(g,xL) E G x GIL I x-lgx E E reg }

,

Y = {(g, x) o:(g,x) = x-Igx,

E

G x G Ix

-I

gx E Ereg},

{3(g, x) = (g,xL),

7r(g, xL) = g.

376

T. Shoji

Now Y is a smooth, irreducible subvariety of C, and 7T: is a principal covering of Y with group W = Nc(L)/L. Hence there is a canonical local system £ on Y such that (3*£ = 0:*£, and K is shown to be isomorphic to IC(Y, 7T:*£)[dim Y], extended to the whole of C by 0 on C - Y. Since all the data in the diagram are F-stable, one can define a natural mixed structure

£g on the stalk £g of £ induced from 'Po : F* £ ~£ has all the eigenvalues of the form root of 1 times

q(dimL-dimE)/2.

Let 'P : F* K ~K be the mixed structure of K induced from 'Po as above. For each F-stable character sheaf A E G which is isomorphic to an irreducible component of K, let 'PA : F* A~A be its mixed structure. We shall see how 'PA is determined from 'P. Let VA = HomMc(A, K). Then VA is a finite dimensional Q/-vector space, and it becomes an irreducible left A-module under the composition of maps, (B, v) f-+ Bov, (B E A, v E VA). For each v E VA, we denote by F*(v) the corresponding homomorphism F* A -> F* K. If 'PA : F* A~A is given, a bijective map 0A : VA -> VA is defined as OA( v) = 'P 0 F* (v) 0 'PAl. 'PAis an A-semilinear map in the sense that 0A (Bv) = L(B)a A (v) for BE A, v E VA, where L : A -> A is the automorphism of the algebra A defined by L( B) = 'P 0 F* (v) 0 'P- l . We now assume that (2.7.2) A is isomorphic to the group algebra QdWE ], and WE = W = Nc(L)jL.

Then F acts naturally on W, and by [L4, II, 10.2]' A has the canonical basis {Bw I W E W} satisfying the following properties; BwBwl = BWWI and L(Bw ) = BF-l(w), (w, Wi E W). Now, = F-l : W -> W is a Coxeter group automorphism of finite order, and one can define a semidirect product W VA is A-semilinear and bijective, W-module VA can be extended to an irreducible W.. Let {3u. = ((3i) be the type of u. Then for any a E Ac(u), a· (3u. corresponds to the type of Ua, where U a is the twisted element in C F obtained by U a = ,U,-l for, E G such that F(/) = iL, (iL E Zc(u) is a representative of a).

,-I

3.3. In the case where G = SON, let F' be a non-split Frobenius map on G, and take an F'-stable unipotent class C in G. Then C is Fstable and we choose a split element u E C F . Let {3 = ((3i) be the type of u. Now we have a coset decomposition Ac(u) = Ac(u) IJ Ac(u)a, and for each C E Ac(u)a, c· (3 gives the type of the element u' E CF'. 80, if we fix such a E Ac(u) - Ac(u), the map {3' ~ a . (3' gives the bijection between the set of GF-conjugacy classes in C F and G F'_ conjugacy classes in C F '. We choose a specific ao E Ac(u) such that ao = aio where io is the maximal odd number such that rio =I- O. We define a split element in CF' as follows.

(3.3.1) ([81,3.7]) Assume that G is of type D n and F' is a non-split Frobenius map. Then a split element u' E C F ' is defined by the type {3' = ao . {3, where {3 is the type of split element u E C F , and ao is as above. Note that this defines a unique class of split elements in PS0 2n . We define a split element in S02n, as in 3.2, by the type {3' = ao . {3. 80, there exists two classes of split elements in CF', one is positive and another is negative. 3.4. Next we consider the case where G ~ G I X G 2 with G i = SPN or SON. If F stabilizes each factor, u = (Ul' U2) E C F , (Ui E G i ) is said to be split if Ui E G[ is split. Assume that F permutes two factors.

382

T. Shoji

Then F(Gt) = G2, and F 2(Gt) = G 1. In this case u = (U1, U2) is said to be split if U2 = F( ut) and U1 is a split element of Gf. Finally we consider the case where G is a quotient of G1 x G2 as above by its c~ntral subgroup. Then we define split elements as the image of split elements in G 1 x G 2 . 3.5. Let G be a group as in (3.2.1). We return to the setting in 2.8, and consider a cuspidal pair (17, £) in L. Note that L = LIZo(L) is also a group as in (3.2.1). Let (C, F) be as in 2.8. We may regard (C, F) as the cuspidal pair in L. Then the local system £ on 17 is obtained as the inverse image of F under the map 17 ~ 17IZo(L) ::: C. The mixed structure of £ is determined by the mixed structure 'Po : F* F ~F. We fix a split element Uo E C F and define 'Po by the requirement that it induces on the stalk Fu. the map q(dim L-dim C)/2 times identity. The mixed structure 'P1 : F* £ ~£ induced by 'Po satisfies the property

(2.7.1). Let A E G be a component of K = ind ~Ao corresponding to the pair (C', £') as in 2.8. The mixed structure 'P A : F* A~A is determined by the choice of an extension VA of VA. We choose VA so that it gives rise to a good extension of irreducible W-module VA in the sense of 3.1. Then the mixed structure of £' is described as follows. Lemma 3.6 Let (C', £') be as above. The mixed structure F* £' ~£' induced from 'P A : F* A~A the satisfies the following property: let

u E C,F be a split element. We assume that u is compatible with Uo, i. e., u and Uo are simultaneously positive or negative on each component G i . Then the induced map £~ ~ £~ is qrn/2 times identity, where m = dimG - dimC' - dimZo(L).

In fact this is shown by modifying the arguments in [L5, 3.4]. W acts on the Q/-space H~(K), and the Frobenius action 'Pu. on it induced by 'P : F* K ~K makes H~(K) into W-module. To prove the lemma, it is enough to show that 'Pu. acts on the submodule of H;;d(K) isomorphic to VA as qrn/2 times O'A, where d = dimC' + dimZO(L) (see 2.8.). Let Zu. = {xP E GIP I x-lux E CUp}. We define a local system F on Zu. so that the inverse image of F under the map Zu. = {x E G I x-lux E CUp} ~ Zu.,x f--t xP equals the inverse image of F under the map Zu. ~ C,X f--t C-component of x-lux E CUp. Then by [L4, V, (24.2.5)], H~(K) ::: H~+r(zu., F), where r = dim supp A = dim Gdim L + dim 17. This isomorphism is compatible with the Frobenius action. Also, H~+r(zu., F) becomes a W-module via this isomorphism. Hence we are reduced to showing the following:

Unipotent Chamcters

383

(3.6.1) If :F is provided with the mixed structure defined by the isomorphism '1,'" , Ap}, T = {Jl1,Jl2,'" ,Jlp}, where Ai = ai + i,Jlj = {3j + (j -1). Then A(E) = (~) E ;, and E f--t A(E) gives the bijection W~ -. ;'. In the case of type D n , E E W~ is parametrized by an unordered pair of partitions (a, (J), which is counted twice if a = {3. We express a,{3 as a: O:S: a1 :S: a2:S:"':S: a p , {3: O:S: {31 :S: {32:S:"':S: {3p for some p ~ 1, and define S = {A1, A2, ... , Ap}, T = {Jl1, Jl2, ... , Jlp} by Ai = ai + (i - 1), Jlj = {3j + (j - 1). Then E f--t A(E) gives the bijection W~ -. ~. For the symbols with higher defects in n, ;t, we have the following relations (d. [LI, 3.2]).

(t ~ 0) AO,A1,"',Ap+2t ) ( 0, 1, ... , 2t + 1, Jl1 + 2t, ... , Jlp + 2t if.4t

'*'n

if. 1

~'*'n-4t2,

A1"",AP+4t) ( Jl1,'" ,Jlp

(t ~ 1) ( f--t

A1,A2,···,A p+4t ) 0,1"" ,4t-2,Jl1+4t-I,··· ,Jlp+4t-I

393

Unipotent Chamcters

In the case of classical groups, there exists at most one unipotent cuspidal character. The condition for the existence is that n = d2 + d (resp. n = 4d 2 ) for some d ~ 1 if C is of type B n or C n (resp. D n ). In that case the symbol corresponding to the unipotent cuspidal character is given by

(0, 1, 2,~ .. ,2d) ~d+l (C: Ao __ (0,1, .. '_' 4d - 1) E '*' (C'.

(5.1.3) A o =

E

if. 4d

n

type B n or Cn, n = d2 type D n , n

+ d)

= 4d 2 )

A o is the unique element of maximal defect in n or ;t. A o is called a cuspidal symbol. We shall define a family in n or ;. Two symbols A, A' belong to a same family if A, A' are represented by (~), (~:) such that S U T = S' U T' and that S n T = S' n T'. A family containing Ao is called s cuspidal family. Each family F contains a canonical representative in ;' or ~ according as F c n ~ F c ;t, and the corresponding irreducible representation of W n or W n is called a special representation associated to the family F. In the case where F = F o is the cuspidal family containing A o, the symbol corresponding to the special representation E of W n (resp. W n ) is given by

A(E) = ( A(E) =

0,2,·" ,2d ) E ;', 1,3,," ,2d-1

(0,2, ... ,4d - 2) E ~. 1,3"" ,4d - 1

5.2. For each symbol A of rank n, the corresponding unipotent character of C F is denoted by PA. If A = A(E) for E E W", PA = PE is the irreducible representation appearing in the decomposition of Ind~~ 1 corresponding to E. In [L2], the notion of almost characters of C F was introduced. It is defined as a linear combination of irreducible characters of C F in a combinatorial way (ef. [L2, 4.24.1]). In particular, in the case where F is of split type, for each symbol A there corresponds an almost character R A which is a linear combination of unipotent characters PAl with A' belonging to the same family as A. Conversely, PAis expressed as a linear combination of RA'. In the special case where PA = PE, E is the special representation of the cuspidal family F o given in 5.2, we have the following formula (cf.[L2, 4.15]).

(5.2.1)

PE

1

= 2k

L AEFo

R A,

T. Shoji

394 where k

= d (resp.

k

= 2d -

1) if G is of type B n or Cn (resp. D n ).

5.3. We consider the set of character sheaves G.c o , where Lo = Ql is the constant sheaf on T. Then by the main result of [L4], G.c o is parametrized by symbols n (resp. ;t) if G is of type B n or Cn (resp. Dn ). We denote by AA the character sheaf corresponding to A E n or ;t. AA is cuspidal if and only if A is a cuspidal symbol. Let L, P be as in 4.2. We assume that L.c o contains a cuspidal character sheaf. Then the semisimple rank of L is equal to t 2 + t (resp. 4t 2 ) if G is of type B n or Cn (resp. D n ) for some integer t ~ O. Moreover, if t ~ 1, W = Nc(L)/ L is isomorphic to Wn-(t2+t) (resp. W n - 4t 2), respectively. Hence, by (5.1.1), (5.1.2), we have natural bijections

according as G is of type B n , Cn or of type D n . We denote this correspondence by E t---+ A(E), (E E W!\). Let Ao be the unique cuspidal character sheaf in L.c o . Then any character sheaf appearing in the decomposition K = ind ~Ao belongs to G.c o . As discussed in 2.9, each direct summand of K is parametrized by W!\, which is given by E t---+ A E . This parametrization is compatible with the parametrization via symbols, i.e., we have A E = AA(E) for E E W!\. Note that the parametrization of unipotent characters via symbols also satisfies this property with respect to the usual Harish-Chandra induction Ind~~ Po for a unipotent cuspidal character Po of LF. 5.4. Here we review the combinatorics developed in [L3] to describe the generalized Springer correspondence of classical groups. For an even integer N ~ 2, let WN be the set of all pairs (~), where A is a finite subset of {O, 1, 2, ... ,} B is a finite subset of {I, 2, 3, ... ,} subject to the condition that (i) A, B contain no consecutive integers, (ii) IA.I + IBI = odd, and (iii) l:aEA a + l:bEB b = ~N + ~(IAI + IBI)(IAI + IBI - 1); these pairs are taken modulo the equivalence relation generated by the shift operation (~) ~ (~:) if A' = {O} U (A + 2), B' = {I} U (B + 2). Next, for any integer N ~ 3, let wN be the set of (unordered) pairs (~), where A and B are finite subsets of {O, 1,2, ... } satisfying the same condition (i) as above and the condition (iii)' l:aEA a + l:bEB = ~N+~((IA.I+IBI-1)2-1); these pairs are taken modulo the equivalence relation generated by the shift operation (~) ~ (~:) if A' = {O} U (A + 2), B' = {O} U (B + 2). Note that, in this case (iii)' implies that IAI + IBI == N (mod 2). Now for each 8 = (~) E WN (resp. wN), we define the defect d( 8) of 8 by d(8) = IAI - IBI, (resp. d(8) = the absolute value of IAI- IBI)·

Unipotent Characters

395

We denote by \It'Jv (resp. we have a partition (5.4.1)

\l1 N

=

\It''jy)

II

the set of () such that d(()) = d. Then

II

\l1~ =

\l1'Jv,

\l1''jy

d>2

dEZ d: odd

d-=N

(2)

On the other hand, by [L3, 12.2, 13.2] the structure of \It'Jv, \It~ is reduced to the case where d = 1, Le., we have the following natural bijections;

\It'Jv ..::;\It;'" -d( d-l)

(~) ~ Cl,3, ... ,2d-3~UB+(2d-l))

if d

~

1

(5.4.2)

(~) ~ CO, 2, ... , -2d~U A + (2 - 2d))

ifd~-l.

It follows from this, by changing the variable j = d - 1 if d ~ 1, j = -d if d ~ -1, that we have a bijection

(5.4.3)

For the case

\It;"',

we have (assuming that

(d

~

IAI

~

IBI)

1)

(5.4.4)

(~) ~ ({O,2,4, ... ,2d-~}UB+(2d-2)). It follows from this,

(N: odd)

(5.4.5)

(N: even).

5.5. Two elements (), ()' of \It N (resp. \It;"') are said to be similar if they are represented by () = (~), ()' = (~:) with AUB = A'UB', and AnB =

T. Shoji

396

A' n B'. We denote by \If N/ ~ (resp. \If N / ~) the set of similarity classes in \If N (resp. \IfN)' respectively. In each similarity class, there exists a unique element called a distinguished element, (see. [L3, 11.5]). The similarity class has the close relationship with unipotent classes of classical groups, which will be given as follows. Let X N and Xl. be as in 3.2. Then we have a natural bijection (d. [L3, 11.6, 11.7]), (5.5.1) The correspondence is given as follows. First we consider the case XN. Let>. = (Fl, 2 r2 , . . . ) E XN. Let 2m be an even integer such that 2m ~ rl +r2+' .. , and let Zl ~ Z2 ~ ... ~ Z2m be the sequence obtained from>. by adding 0 to the sequence of>. exactly 2m-(rl +r2+" ) times. Let zi < z~ < ... < z~m be the sequence defined by z~ = Zi + (i - 1). This sequence contains exactly m even numbers 2Yl < 2Y2 < ... < 2Ym and m odd numbers 2y~ + 1 < 2y~ + 1 < ... < 2y:" + 1. We set

+ 2 < y~ + 3 < ... < Y:" + (m + I)} = {Yl + 1 < Y2 + 2 < ... < Ym + m}.

A = {O < y~ B

Then (~) E \If N is a distinguished element, and>' ........ (~) induces the required bijection. Next consider the case of Xl.. Let>. = (1 r " 2r2 , • •• ) E Xl.. Let M be an integer, M ~ rl + r2 + ... ,M == N (mod 2), and let Zl ~ Z2 ~ ... ~ ZM be the sequence obtained from>. by adding 0 to the sequence of>. exactly M - (rl + r2 + ... ) times. Let zi < z~ < ... < zk be the sequence defined by z~ = Zi + (i - 1). This sequence contains exactly [M/2] even numbers 2Yl < 2Y2 < ... < 2Y[M/2], and [(M + 1)/2] odd numbers 2y~ + 1 < 2y~ + 1 < ... < 2Y[(M+l)/2] + 1. We set

+ 1, ... , Y[(M+l)/2] + ([(M + 1)/2] = {YI' Y2 + 1"" ,Y[M/2] + [M/2] - I}.

A = {y~, y~ B

I)},

Then (~) E \If N is a distinguished element, and>' ........ (~) induces the required bijection. 5.6. Let N c be the set of pairs (C',£') in G, where C' is a unipotent class and £' is an irreducible G-equivariant local system on C'. Then by [L3, §11], N c is in bijection with \IfN (resp. \IfN) if G = SPN (resp. G = SON)' The cuspidal pairs in G are described as follows. First consider the case where G = SPN (cf. [L3, §13]). Then G contains a cuspidal pair if and only if N = d( d - 1) for some odd

Unipotent Chamcters

397

(possibly negative) integer d. If N satisfies the above condition, the cuspidal pair (Co, £0) is uniquely determined. The unipotent class Co has the Jordan blocks given by the partition>. = (2,4, ... ,) and the irreducible character Po E Ae(u)"', (u E Co) corresponding to £0 is given by (PO(a2), po(a4), ... ) = (+1, -1, ... ) under the notation in 3.2. The element in \f!'Jy corresponding to (Co'£o) is given by (0,2, ..:.:.2d-2) if

d ~ 1 and by C,3, ...~1-2d) if d :::; -1. Next consider the case G = SON (d. [L3, §14]). Then G contains a cuspidal pair if and only if N = d 2 for some integer d ~ 1. If N satisfies the above condition, the cuspidal pair (Co, £0) is uniquely determined. The unipotent class Co has the Jordan blocks given by the partition >. = (1,3, ... ) and the irreducible character Po E Ae(u)''', (u E Co) corresponding to £0 is given by (PO(a1),po(a3), ... ) = (+1, -1, ... ). The element in \f!~d corresponding to (Co'£o) is given by (0,2, ..:.-.2d-2). 5.7. By the generalized Springer correspondence, N e is in bijection with the set of character sheaves of G obtained by decomposing the complex K = ind ~Ao as in 2.8, where A o is a cuspidal character sheaf of L as given in 2.8. Then L has the same type as G, and it follows from 5.6 that L has the semisimple rank hU + 1) (resp. ~(P -1), 2 ) for some integer j ~ 0 if G is of type Cn (resp. B n , D n ). Hence W = Ne(L)/L is isomorphic to W n - tj (j+1) (resp. Wt(N-P))' if G = SP2n (resp. G = SON), (the case j > 0) and to We (the Weyl group of G) if j = O. Thus the generalized Springer correspondence is given by the following bijection.

h

(5.7.1)

\f! N

+-----+

Ne

+-----+

II W:- t

j (j

+ 1)

j?-o

(5.7.2) j>1 (2)

j=N

Following [L3, §12, §13], we shall give a combinatorial description of the generalized Springer correspondence. First consider the case G = SP2n' We construct a bijection

e ·. W m

ll ----+ ,T.1

'¥2m

as follows. Let E = E a ,/3 E W~, where (a, /3) is a pair of partitions of n. We express a, {3 as in 5.1, i.e., 0 :::; ao :::; a1 :::; ... :::; ap,O :::; {31 < {32 :::; ... :::; {3p for some P ~ 1. We set

< a1 + 2 < ... < a p + 2p} {{31 + 1 < b2 + 3 < ... < {3p + 2p - I}.

A = {ao B =

T. Shoji

398

Then (~) E lJI~m and this gives a bijection 8. We need also a variant of 8. We define a bijection t8 : w~ ----; IJI~ by t8(Eo ,{3) = 8(E{3,o). Now in view of (5.4.2) and (5.4.3), the generalized Springer correspondence is decribed by giving the bijection between 1JI}y -de d-l) and Wn-~d(d-l)' We have (5.7.3) ([L3, Th. 12.3]) The generalized Springer correspondence between WL~d(d-l) and IJI}Y-d(d-l) is given by 8 for m = n -1d(d -1) if d ~ 1, and by t8 if d::; -1, (see Remark 5.8 below). Next consider the case G = SON. We construct a bijection

as follows. Let E o ,{3 E W~ be as before. So, we have 0 ::; 0:0 ::; 0:1 ::; ... ::; O:p, 0 ::; {31 ::; {32 ::; ... ::; {3p for some integer p ~ 1. We set A B

= {o:o < 0:1 + 2 < = {{31 < f32 + 2
. = (2,4, ... ,2j). (Here we use the change of the variable j = d - 1 if d ~ 1 , odd and j = -d if d ::; -1, odd as in 5.4.) Let C be the class of G induced from Co and let C be the class of G generated by Co. Then C corresponds to (2,4, ... , 2j - 2, 2j + 4) and C corresponds to

Unipotent Chamcters

399

(1 4 , 2, 4, ... , 2j). Assume first that j is even (Le., d 2: 1). Then under the correspondence N c +-+ 1J1 2n , C and C determine unique elements - ·+1 (), () E lJ1~n as follows.

ii =

(0,2, ... , 2j - ~ 2j - 2, 2j + 2) ,

() = (0,2,4, ... , 2j

+ 4) .

2,4

c:.)

Now, under the bijection (5.4.2), ii and () correspond to and (°2244 ) in lJ1~n_j(j+1)" By 9.2 and 9.5 in [L3], those two elements corresp~nd to the unit representation and the sign representation of W 2 • This agrees with the map On the other hand, if we assume j is odd, (Le., d ~ -1), then C and C determine unique elements ii, () E 1J12~ as follows.

e.

ii=( 1,3,5, ... ,2j - -5, 2j -

3, 2j

1,3 ) ( 1,3,5, ... ,2j+3 .

+ 1) ' 2

Then by (5.4.2), ii and () corresponds to (°3 ) and (\3) in lJ1~n_j(j+1)' By 9.2 and 9.5 in [L3], (j and () should correspond to the unit representation and the sign representation of W 2 , which agrees with the map As discussed in the proof of Theorem 13.2, the general case follows from this.

teo

6. The main results 6.1. In this section, we consider a classical group G of split type. We assume that the center of G is connected and that G is simple modulo center, of type B n , Cn or D n . As discussed in section 5, unipotent characters, almost characters of G F and character sheaves in 8 Co are parametrized by symbols A in n or ;t, which we denote by PA, RA and AA, respectively. Now, by the main result of [82], Lusztig's conjecture holds for G, Le., we have RA = (AXAII,'1 M1

< ... < >'[M/2]) < M2 < ... < M[M/2]

.T,l

E '.i'd2+d

with >'[M/2] = d+2[M/2], M[M/2] = d+2[M/2J-2, where M = L~=l rii. Hence, if we take ()' E R' n 1J1~~+d corresponding to E~, the largest entry of the upper row in ()' is equal to >'[M/2] or M[M/2]. We note that >'[M/2] appears in the upper row. In fact, if >'[M/2] appears in the lower row, we have (3~ = d + t + 2 or O:T = d - t + 1 according as j' = t + 1 or j' = -to This contradicts (6.12.3). Now let R" be the similarity class in IJ1 d2+d corresponding to C~, and take ()" E R" that

n 1J1~:+d' We note

(6.12.6) j' = j". If we set E' = Eo' ,{3', E" = Eo" ,{3", then O:T t if j' > 0 and {3~ = {3~ = d + t + 1 if j' < O.

o:~ = d -

406

T. Shoji

In fact, if j' = t + 1, then 0:;- = d - t, and so we have O:d+t = d - t. But if j" < 0, then {3~ = d + t + 1, and {3d+t = d + t + 1. This again contradicts (6.12.3). The same argument works for the case j' = -to Hence we have j' = j". The latter statement is straightforward from this. In the following, we express the common number j' = j" by j. We define E:,E~ E W[(d2-dl-C t2 - t )}/2 by removing 0:;- = o:~ from the partition 0:' or 0:" if j > 0, and by removing {3~ = {3~ from the partition {3' or {3" if j < O. Let us define e:, e~ E wdt~J as the ones corresponding to E:, E~, respectively. We also define unipotent classes C:, C:' in SPd2-d by removing the entry 2d from the partition ofCb, C~, respectively. So, C: O. Dans cet article, nous allons donner deux nlsultats: 4) La reduction modulo 1 des representations irreductibles lisses l-entieres de WF sur R de dimension n; voir (1.21)*. 5) La classification des modules simples de l'algebre de Hecke affine de GL(n, F) sur R, dans le cas generique, ou regulier, ou si q = 1 dans R*, ou si n = 2, ou si n = 3; voir (2.5).

On verifiera que 1) ** est vrai dans le cas moderement ramifie, en toute caracteristique 1 #- p. Nous allons donner une preuve elementaire et sans elegance de 5) (en manquant de peu Ie cas n = 4). La raison de publier cette classification inachevee et sans elegance est la comparaison avec celIe de Grojnowski [Gro] lorsque R = C, et q une racine de l'unite d'ordre > 1, et celIe de Dipper et James pour q = 1 dans R* [DJ2]. On est alors amene it. la conjecture suivante , qui tient lieu de 2) en caracteristique > O. Conjecture. La classification des HR(n, q)-modules simples ne depend de (R, q) que via l'ordre c(q, R*) de q dans R*.

Les HR(n, q)-modules simples classifient les representations irreductibles de support cuspidal egal au support supercuspidal, a

= Xl + ... + Xn,

pour des caracteres non ramifies Xl, ... Xn. Ils sont classes par certains "bons" (a, N). Les (a, N) qui ne sont pas bons, correspondent aux representations de support supercuspidal a different du support cuspidal dans tous les cas que nous connaissons. L'analogue de 3) dans Ie cas modulaire est encore inconnu, meme conjecturalement. Ceci provient de notre ignorance en ce qui concerne la classification des representations irreductibles de support cuspidal different de leur support supercuspidal. On note v : W F ----> R* Ie caractere non ramifie usuel (1.1). En admettant 1) pour 1 #- p, dans Ie cas generique ou regulier ou si n = 2, * Nous n'avions que Ie cas moderement ramifie, mais Henniart a vu que Ie resultat etait general. ** On a conjecture en [Vig2j que 1) est vrai pour 1 #- p.

Conjecture de Langlands modulaire

417

(cas ou l'on sait nlsoudre 2) et 3)), on obtient la conjecture de DeligneLanglands, it. savoir:

6) Boit n > 0 un entier. Les classes d'isomorphisme des paires (a, N) formees

(a) d'une representation semi-simple (V,a) de dimension n de WF sur R, (b) d'un endomorphisme nilpotent N E EndR V tel que a(w)Na(w)-1 = v(w)N, sont en bijection avec IrrR GL(n, F). Les representations supercuspidales de GL(n, F) doivent etre en bijection avec les representations irreductibles de W F de dimension n (N = 0). La representation semi-simple a doit correspondre au support supercuspidal de la representation de GL(n, F) associee it. une paire (a, N). Les representations generiques de GL(n, F) doivent etre en bijection avec les paires (a, 0). La bijection pour Ql doit respecter les representations l-entieres, et etre compatible avec la reduction modulo l [Vig2], dans un sens qui devra etre precise. Les representations cuspidales non supercuspidales de GL(n, F) sur Fl doivent etre en bijection avec les a qui se relevent en des representations irreductibles sur Ql. En 1, nous demontrons 4); en 2, nous donnons une demonstration elementaire de 5); en 3, nous demontrons 6), avec les reserves indiquees ci-dessus. Ces resultats ont ete exposes en 1993 it. l'Universite de Tel-Aviv, en 1994 au C.1.R.M. et it. l'E.M.l. L'auteur remercie particulierement J. Bernstein, D. Barbasch, G. Henniart, 1. Grojnowski, et W. Zink, qui l'ont beaucoup aidee par leurs travaux ou leurs remarques constructives.

1.

Representations modulaires de WF.

Toutes les representations de W F seront supposees de dimension finie, et triviales sur un sous-groupe ouvert. On note IrrR WF l'ensemble des classes d'isomorphisme des representations irreductibles de W F de dimension finie, it. coefficients dans R. 1.1 Rappels sur W F •

On refere it. [Sel] ou it. [Fr], [De], [Ta]. On fixe un isomorphisme VF: WF/h ----; Z

418

M. -F. Vignems

du quotient de WF par Ie groupe d'inertie IF. Le groupe d'inertie est un groupe profini, qui admet un unique pro-p-groupe de Sylow PF, Ie groupe de ramijication sauvage, de quotient

pour Ie systeme projectif fourni par les normes. Le groupe de Weil WF est Ie produit semi-direct de PF et de WFIPF ([Iw] Lemma 4 Sect 1, cette propriete m'a ete signalee par W. Zink). La theorie du corps de classes fournit un isomorphisme de F* sur Ie quotient abelien separe maximal de WF, compatible avec les decompositions de F*, F * '" - pZ F 0*F,

F;

oil. PF est une uniformisante de F, est identifie avec Ie groupe des racines de l'unite dans F* d'ordre premier a q, et PF est l'ideal maximal de l'anneau des entiers OF de F. Si ElF est une extension finie, la sous-extension moderement mmijiee maximale E mr = EPF est l'ensemble des elements de E invariants par PF, et la sous-extension non ramijiee maximale E nr = ElF est l'ensemble des elements de E invariants par IF. Une extension finie non ramifiee est cyclique, determinee par son degre sur F; on note Ff IF l'extension non ramifiee de degre f. Supposons ElF galoisienne. Alors E mr I Fest galoisienne [Fr 8 th.l], et Emrl E nr est cyclique, d'ordre e premiera p, egal a l'indice de ramification e de E mr IF, divisant qf - 1, si E nr = F f . il existe (qf - 1) I e extensions moderement ramifiees galoisiennes d'indice de ramification e, et de degre residuel f non isomorphes; elles sont de la forme Ff(c l / e ) oil. c E F f est une uniformisante, avec Ff(c l / e ) = Ff(C'l/e) si et seulement si c' E c(Fj)e [Fr 8 prop1]. Le groupe de Galois G = Gal(E I F) admet une filtration en sousgroupes

ou G l = Gal(EIEmr ) est Ie p-groupe de ramification sauvage de l'extension ElF, ou G2 = Gal(EIEnr ), donc G2/G l = Gal(EmrIEnr ) est cyclique d'ordre premier a p, et G IG2 = Gal(EnrlF) est cyclique.

Conjecture de Langlands modulaire

419

1.2 Rappels sur les representations de Wp. L'isomorphisme de la theorie du corps de classes permet d' identifier les caracteres de W p (representations de dimension 1) et ceux de F*. Si E / F est une extension finie, la restriction d'un caractere de W p a WE correspond a la norme E* ---+ F*. V ne representation de W pest dite non ramifiee si elle est triviale sur I p . Si elle est irreductible, c'est un caractere car Vp : W p / I p ---+ Z est un isomorphisme. Le groupe WRF* des caracteres non ramifies de W p sur Rest isomorphe a R* par l'application qui identifie r E R* et Ie caractere non ramifie

On note encore I/r Ie caractere non ramifie (trivial sur OF) de F* correspondant. Si E / Fest une extension finie, la restriction WRF* ---+ WRE* correspond a l'elevation a la puissance f : r ---+ r f , oil f est Ie degre residuel de E/ F; elle est surjective car Rest algebriquement clos. Elle est bijective si E / Fest une extension totalement ramifiee. Le groupe de Weil W p est dense dans Ie groupe de Galois Galp de Fs/F. Vne representation de Wp qui se prolonge a Galp est dite galoisienne. Le noyau d'une representation irreductible galoisienne est un sous-groupe d'indice fini distingue de Galp, donc de la forme GalE pour une extension galoisienne finie E / F. Son image est alors finie, et inversement une representation de Wp d'image finie est galoisienne. En effet, il est connu [De 4.10 page 542] et facile de voir que:

1.3 Lemme. Soit G une extension de Z par un groupe profini I, et a E IrrR G. Il existe un caractere 7j; de G trivial sur I tel que l'image de a7j; est finie. 1.4 Corollaire. Chaque orbite de WRF* dans Irr R W p contient une representation galoisienne. 1.5 Relevement

a la caracteristique o.

Le groupe /.1'ZI des racines de I'unite d'ordre premier

al

contenues

dans Z~ est isomorphe a F~, par reduction modulo l (i.e. modulo l'ideal maximal A de Zl). Vn caractere non ramifie Wp ---+ F~ se releve donc canoniquement en un caractere non ramifie W p ---+ /.1'ZI' V ne representation a E IrrF1 W p se releve a Ql si et seulement si une representation de sa WF1 F* -orbite se releve a Ql. Or cette orbite contient une representation galoisienne (1.4). II est bien connu que Ie

M. -F. Vigneras

420

groupe de Galois d'une extension galoisienne finie E / Fest resoluble (il faut voir que Ie groupe de ramification sauvage est resoluble [SeI IV §2 Cor. 5 page 76]). Soit l un nombre premier. Un groupe fini G est dit l-resoluble, s'il admet une filtration en sous-groupes 1 = Go C G 1 C ... C G r = G

avec G i - 1 distingue dans Gi pour tout 1 ::; i ::; r, telle que les quotients Gi/G i - 1 soient des l-groupes ou des groupes d'ordre premier it l. Lorsque les quotients sont abeliens, Ie groupe est dit resoluble. Un groupe resoluble est l-resoluble pour tout l. On sait que [Se2 16.3, 17.6]: 1.6 Rappel du theoreme de Fong-Swan. Si G est un groupe fini l-resoluble, alors tout FIG-module simple se releve en un QIG-module simple.

Corollaire. Une representation a E IrrF1 Wp se releve en une representation dans IrrQ1 W p, pour tout l.

1.7

1.8 Representation l-entiere.

Soit G un groupe, et V une representation de dimension n de G sur Ql. Un ZIG-reseau de Vest un Zl-module libre LeV, de rang n, stable par G. On dit alors que Vest l-entiere. Exemples. Si G est fini, Vest toujours l-entiere [Se2 15.2, tho 32]. Si G est profini, l'action de G est triviale sur un sous-groupe distingue d'indice fini de G, donc Vest toujours l-entiere. Un caractere de G est l-entier si et seulement si ses valeurs appartiennent it

Z;.

Si G = Wp, V n'est pas toujours l-entiere. L'action de Wp sur V s'identifie it un homomorphisme a : W p -> GL(n, Q l ). En composant a avec Ie determinant, on obtient un caractere

deta: W p

->

Q;.

Si V E IrrQ1 W pest l-entiere, Ie determinant est l-entier. On deduit de (1.4) que la reciproque est vraie car tout a E IrrQ1 Galp de dimension nest l-entiere, et pour 7j; E WQ1 F*, a7j; est l-entiere si et seulement si 7j; est l-entier, si et seulement si det (a7j;) = 7j;n det a est l-entier.

Conjecture de Langlands modulaire

1.9 Lemme. a E IrrQ ! Wp est l-entiere determinant est l-entier.

421 st

et seulement

St

son

1.10 Principe de Brauer. On peut n§d uire modulo A un Zl W p-reseau L de (V, a). On obtient une representation L I AL de W p sur Fl. Le principe de Brauer pour un groupe fini implique que la classe d'isomorphisme de la semisimplification de L I AL ne depend pas du choix de L si a est galoisienne [Se2 15.2 theoreme 32]. Le meme principe reste valable si I'on multiplie a par un caractere l-entier. Le principe de Brauer est done vrai pour Wp. La classe d'isomorphisme ria de la semi-simplification de la reduction modulo A d'un ZIWp-reseau de a E IrfQ! Wp est appele la reduction de a modulo l. 1.11 Representation moderement ramifiee. Vne representation de Wp ou de I p est dite moderement ramifiee lorsqu'elle est triviale sur Ie groupe de ramification sauvage Pp. Cette propriete est respectee par multiplication par un caractere non ramifie. On peut done considerer les WRF* -orbites moderement ramifiees dans IrrR Wp. Si l = p, comme Pp est un pro-p-groupe, la representation triviale est la seule representation irreductible de Pp sur F p, la restriction d'une representation irreductible de Wp it Pp est semi-simple, done toute representation de IrrF p W pest moderement ramifiee. Vne representation de Wp est dite monomiale, si elle est de la forme indwF,H X ou X est un caractere d'un sous-groupe H de Wp. Le produit d'une representation monomiale de Wp par un caractere est encore monomial. Le noyau d'une representation moderement ramifiee galoisienne de dimension finie est de la forme GalE pour une extension galoisienne moderement ramifiee finie ElF. Le groupe de Galois G = Gal(EIF) contient un groupe abelien distingue H (Ie sous-groupe d'inertie) de quotient G I H abelien. Vn tel groupe est dit metabelien. Rappelons la propriete suivante [CR 52.2 page 357], que l'on deduit de la theorie de Clifford:

1.12 Rappel. Une representation irreductible a E IrrR G d'un groupe fini metabelien G est monomiale. Vne representation irreductible moderement ramifiee galoisienne est done monomiale. Les caracteres moderement ramifies des extensions finies moderement ramifiees et leur induites a Wp sont bien connus dans Ie cas complexe. II y a quelques differences dans Ie cas de

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caracteristique positive, comme nous allons Ie voir ci-dessous.

1.13 Caracteres moderement ramifies. Un caractere moderement ramifie de WF s'identifie it un caractere de F* trivial sur 1 + OF. C'est Ie produit f..t = lIr X d'un caractere non ramifie lIr et d'un caractere X : F; ---. R*. On rappelle que X est regulier sur F q si et seulement s'il verifie l'une des deux proprietes equivalentes suivantes: - si X = f..tN, ou N : F;n ---. F;d est la norme, et f..t un caractere de F;d' alors d = n, - si r = X(x) est l'image d' un generateur x de F;n, alors q r, r , ... , r qn-l sont d'ISt'Inc t s.

Exemple. L'inclusion F;n ---. X est regulier si et seulement si r

F:

est un caractere regulier sur F q ;

= X (x) est de degre n sur F q'

On rappelle qu'un caractere moderement ramifie de l'extension non ramifiee de degre nest dit regulier sur F si: X = f..tN, ou N : F~ ---. F; est la norme, et f..t un caractere de F;'j, alors d = n. C'est equivalent pour Ie caractere X de WFn correspondant it: Ie stabilisateur de X dans WF est WFn • F~ / F

Une representation moderement ramifiee de IF s'identifie it une representation du groupe commutatif IF / PF = !!!!!- F;n. Si elle est irreductible, c'est un caractere. Un caractere moderement ramifie de IF s'identifie it un caractere X.. F*qn---' R*

regulier sur F q. On l'identifie it un caractere moderement ramifie F~ ---. R* trivial sur PF, regulier sur F, ou it son image par la theorie du corps de classes. Cette identification respecte la regularite. La reduction modulo l ne respecte pas la regularite. La representation induite it WF

est galoisienne et moderement ramifiee. Inversement soit a E IrrR WF moderement ramifiee. La restriction de a a IF est semi-simple. On choisit un caractere contenu dans alh' 11 s'identifie it un caractere X : F;n ---. R* regulier sur F q, et

Conjecture de Langlands modulaire

423

pour un certain caractere non ramifie l/r. Le caractere I/rX est unique it conjugaison pres dans W p: donc r est unique, et la CalF q -orbite de X est unique. 1.14 Classification des representations moderement ramifiees. L 'application (r, X) ---. a(r, X) donne une pammetrisation des representations a E Irr R W p moderement mmifiees par R* x X q ou X q est l'ensemble des CalF q -orbites de camcteres reguliers sur F q. 1.15 Description de a(x). Soit X = f-LN provenant via la norme d'un caractere f-L : R* regulier sur F q. Alors N : a(f-L) est irreductible, et a(x) est de longueur m = n/d, de quotients a(f-L)l/~, 1 ~ i ~ m - 1, ou x E R* est tel que x d engendre le groupe des racines m-iemes de l'unite dans R* .

F;n ---. F;d

F;d ---.

Preuve. Soit X : F~n ---. R* un caractere et r E R*. Meme si X n'est pas n~gulier sur F q , on peut definir a(X), a(r,x). Le second se ramene au premier par produit par un caractere non ramifie. Le caractere l/r etant la restriction it W P n du caractere l/x pour tout x E R* tel que x n = r, on a a(r,x) = I/xa(X). On va decrire a(x), en commen 0, et tout y E< x >, I'arc oriente ou Ie segment oriente debutant en y, de longueur m (y, m) = (y

yx

...

yx m- l )

est appele un €-type simple. La multiplicite d'un point z du €-cercle dans (y, m) est Ie nombre d'(~lements de la suite (yxi)O~i~m_1 egaux it z. Une somme finie E(Yi, mi) de €-types simples est un €-type de longueur m = E mi· La multiplicite t = L t i d'un point z du €cercle dans Ie €-type est la somme des multiplicites ti de z dans les composantes simples (Yi, mi). Le support avec multiplicite du €-type, est la fonction t :< x >---- N qui associe it un point du €-cercle sa multiplicite. Le support du €-type est Ie support de t. Le €-type est generique si € = 00 ou si son support ne contient pas Ie €-cercle, il est regulier si les multiplicites sont 1 ou O.

€-cycle, € > 1. Un €-cycle est un €-type de la forme LYE<x> (y, m). II ne depend que de m. Un €-type ne contenant pas de €-cycle est dit bon. II existe un seul €-cycle regulier LYE <x> y. Un €-type de longueur < € ou generique, est toujours bon. Lorsque € = 1, les I-types de longueur n sont en bijection avec les partitions de n. Dans Ie cas ou € > n, les €-types s'identifient aux oo-types, dont la classification bien connue [ZI]' s'identifie it la classification des modules simples de I'algebre de Hecke affine Hc(n, q) (1.3.14), et it la classification des representations complexes du groupe de Weil-Deligne de dimension n, de partie semi-simple triviale (2.2). (E, x)-type On etend la definition de €-type it la situation ou E est un ensemble avec poids d : E ---- N*, muni d'une bijection x : s ---- sx, qui respecte Ie poids d(sx) = d(s), sEE. Pour SEE, on note €(s, x) Ie plus petit entier i > 0 tel que sx i = S. Pour tout entier m > 0, et tout SEE,

(s,m) = (s

sx

...

sxm -

l )

Conjecture de Langlands modulaire

427

est un (E, x)-type simple de longueur d(s)m. Une somme finie E(Si, mi) est un (E, x)-type de longueur E d(si)mi' Soit s < x > l'orbite de sEE par < x >. Les (s < x >, x)-types sont en bijection avec les E(S, x)-types. Si toutes les orbites ont Ie meme nombre d\~lements E, I'ensemble des (E, x )-types s'identifie au produit de E / < x > et de l'ensemble des E-types.

Exemples. 1) (IrrR Wp, v)-type. R est un corps algebriquement clos, F est un corps local non archimedien de corps residuel d'ordre q inversible dans R*, d'ordre E = E(q, R*), v = v q est Ie caractere non ramifie (1.1) qui agit sur IrrR Wp par multiplication, la fonction d sur Irr R W pest la dimension des representations. 2) (C*, x)-types. x E C* agit sur C* par multiplication, la fonction d sur C* est identiquement egale a 1. Si E = E(X, C*) est l'ordre de x dans C*, les (C*, x)-types s'identifient au produit de C* / < x > et de I'ensemble des E-types. (E,x)-cycles, x =I id. Un (E,x)-cycle est une somme EyE<x>(s) (sxY,m), il ne depend que de (s,m). Un (E,x)-type ne contenant pas de (E, x )-cycle est dit bon. I-cycles dans R*. Si R est un corps algebriquement clos de caracteristique l > 0, un I-cycle dans R* est une somme de l copies d'un l-type (y, m) = (y ----. y ----. ... ----. y), Y E R*, m entier 2 1. Un I-type ne contenant pas de I-cycle est dit bon.

2.2 Representations du groupe de Weil-Deligne. Les (IrrR Wp, v)-types de dimension n sont en bijection avec les classes d'isomorphisme des paires (a, N) formees (a) d 'une representation semi-simple (V, a) de dimension n de W p sur R, (b)

d'un endomorphisme nilpotent N

E

EndR V

tel que

v(w)a(w)N = Na(w), wE Wp. Preuve. 1) Soit (a, N) comme en a) et b). Montrons qu'on peut lui associer un (IrrR Wp, v)-type de dimension n. Le groupe < N > agit sur les composants irreductibles de a. La representation a est la somme directe de ces orbites. Chaque orbite de < N > est de la forme T + TV + ... + TV m - 1 , et correspond it un (IrrR W p, v)-type simple (T,m) := T ----. TV ----. '" ----. TV m - 1. La somme des (IrrR Wp,v)-types simples correspondant aux orbites de < N > est un (IrrR Wp, v)-type de dimension n. 2) Inversement, soit (T, m) un (IrrR Wp, v)-type simple. Montrons que l'on peut lui associer une paire (a, N) qui est une section de

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M.-F. Vigneras

I'application ci-dessus. A un (IrrR WF, v)-type, on associera la somme directe des paires definies par ses types simples. La representation semi-simple est a = EBO~i~m-1 TV i . Notons Wi I'espace de TV i . Pour 0 :::; i :::; m - 2, on choisit un isomorphisme WF-equivariant N i : Wi ----- WHI · Soit N m- I = O. L'endomorphisme nilpotent est N = EBO dans I'ensemble des composants irreductibles de a. On se ramene ainsi au cas ou Ie type est simple (T,m), et a = a'. Notons que GLR(a) ~ [JPEIrrRWFGL(t(p),R), ou t : Irr R WF ----- N est Ie support avec multiplicite de (T, m). En conjuguant N' par un element de GLR(a), on peut supposer que Ker N = Ker N' = Wm-I, puis en divisant par Wm-I, et par induction sur m, on peut supposer que les filtrations des noyaux

o ~ Ker N m - I

~

...

~ Ker N ~ 0

pour N et N' sont egales. Pour une decomposition de a = i EBO::;i::;m-l TV en somme directe compatible cette filtration, N et N' sont comme en 2). Leurs composantes N i = NIri sont egales it un scalaire multiplicatif pres ri E R*. Par conjugaison par (R*)m i [JO::;i::;m-1 GLR(TV ) C GLR(a), on peut supposer que N = N'. Remarque. Lorsque c:( a, v) > n, un endomorphisme M E EndR V tel que a(w)Ma(w)-1 = v(w)M, est nilpotent. C'est faux sinon. Par exemple, si c: = c:(q,R*) :::; n, (V, a) = EB~':J(Rei,vi), alors I'endomorphisme M E EndR V tel que M(ei) = eHI, pour tout o :::; i :::; c: - 2 et M(ee-d = eo, verifie bien la condition ci-dessus mais n'est pas nilpotent. 2.3 Modules simples de Hc(n, x). [Groj] Soit Hc(n, x) l'algebre de Heeke affine assoeiee Ii GL(n) et Ii x E C*,x -11 [livre 1.3.14]. Les classes d'isomorphisme des Hc(n,x)-modules simples sont classees par les bons (C*, x)-types de longueur n. Le corps C n'intervient pas vraiment, I'ordre c:(x, C*) de x joue Ie role primordial. 2.4 Conjecture pour HR(n, x). Soit n > 0 un entier, R un corps algebriquement clos, et x E R*. II existe une bijection entre les classes d'isomorphisme des HR(n,x)-modules simples et les bons (R*,x)-types de longueur n.

Conjecture de Langlands modulaire

429

Cette conjecture est un theoreme de Dipper et James [DJ2] pour l'algebre de Hecke HR(n, x). Cette conjecture ramene la classification des HR(n, x)-modules simples a celIe des e = e(X, R*)-types. La bijection devrait etre naturelIe dans Ie sens que Ie bon type associe a un module simple devrait provenir d'un poids "minimal". Une variante plus faible est que la bijection conserve Ie support (2.9).

2.5 Theoreme. La conjecture pour HR(n, x) est vmie si e = 1 ou e ~ n, ou si n = 2,3. Pour obtenir (2.5), on va decrire en (2.6) les e-types generiques, ou reguliers, ou de longueur n ::; 4, de support donne. Si e = 1, les e-types de longueur n sont les partitions de n (2.1), on supposera donc e > 1. On comparera ensuite avec les HR(n, x)-modules simples.

2.6 Exemples de e-types Reduction aux composantes connexes. Deux points y, y' du ecercle < x > sont dits lies si y = y'x±l. La determination des e-types se ramene a celle des e-types de support connexe (Ie support est de la forme {y, YX, ... , yx r }, r < e). Un e-type de support donne X est une somme de e-types de support X*, OU les X* sont les composantes connexes de X (deux elements du support d'un e-type appartenant a deux composantes connexes distinctes ne peuvent pas appartenir a un meme composant simple (y, m)). Nous supposons desormais que Ie support est connexe, de la forme {I = xo, ... , x r }, r < e. Cas gimerique: Ie support n'est pas Ie cercle. Les e-types de support donne generique ne dependent pas de e, que l'on peut choisir e = 00. Cas regulier : Ies multiplicites sont egales a 1. Dans Ie cas regulier generique, on a 2n - 1 types de longueur n. Dans Ie cas regulier non generique, de support Ie e-cercle, de longueur n = e, on en a 2n -1 types, dont 2n - 2 bons, et un cycle [1 + x + '" + xn-I]. En effet, un type E(Yi, mi) de support regulier est caracterise par Ie sous-ensemble Y des Yi. Le support est un ensemble it n elements. Dans Ie cas generique, Y est un sous-ensemble contenant 1 du support: Ie nombre de types est donc Ie nombre 2n - 1 de parties d'un ensemble a n - 1 elements. Dans Ie cas non generique, Y est un sous-ensemble non vide quelconque du support. Le nombre de e-types est Ie nombre 2n - 1 de parties non vides d'un ensemble it n elements. Les bons etypes correspondent aux parties propres (differentes de la partie vide ou du support).

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430

Nous donnons maintenant la liste des types de longueur n :'::: 4, pour € > 1, classes selon leur support (1 a, (x)b, (x 2 )C, ... ) avec multiplicite.

n = 2 : Ie support est regulier ou generique; • support generique, non regulier 12 : 1 seul type 1 + 1, • support regulier (1, x): 2 types 1 ---. x et 1 + x pour € > 2 (cas generique) . • 3 = 2 + 1 types, dont 2 bons 1 ---. x, x ---. 1, et un cycle [1 + xl si € = 2 (cas non generique).

n = 3 : Ie support est regulier ou gem3rique, sauf pour (12, x), € = 2. • support generique non regulier (1 3 ): 1 seul type, • support non regulier (1 2 ,x): 2 types 1 + (1 ---. x), 1 + 1 + x si € > 2 (cas generique) . •4

et [1

= 3+1 types, dont 3 bons 1+(1 ---. x), 1+(x ---. 1), 1 ---. x ---. 1,

+ x] + 1, si € =

2 (cas non generique).

• support regulier (l,x,x 2 ) .4 types si € > 3 (cas generique): 1 ---. x ---. x 2 , (1 ---. x) + x 2 , 1 + (x ---. x 2 ), 1 + x + x 2 • • 7 = 6 + 1 types dont 6 bons (les trois premiers types ci-dessus, et tourner sur Ie cercle) et un cycle [1 + x + x 2 l, si € = 3 (cas non generique).

n = 4: Ie support est regulier ou generique, sauf pour (1 3,x), (1 2 , (X)2) si € = 2, et (1 2,x,x2) si € = 3, modulo multiplication par un element du €-cercle. • support generique non regulier (1 4 ): 1 seul type. • support non regulier (13, x) : 2 types 1+ 1 + (1 ---. x), 1+ 1+ 1+ x si € > 2 (cas generique) . • 3 bons types 1 + 1 + (1 ---. x), 1 + 1 + (x ---. 1), 1 + (1 ---. x ---. 1), et 1 pas bon 1 + 1 + [1 + x], si € = 2 (cas non generique). • support non regulier (1 2 , (X)2): 3-types (1 ---. x) + (1 ---. x), 1 + x + (1---. x), 1 + x + 1 + x si € > 2 (cas generique), .10 = 6+4 types dont 6 bons (1 ---. x)+(1 ---. x), (x ---. 1)+(x---. 1), (1 ---. x ---.1) +x, (x ---. 1 ---. x) + 1, 1 ---. x ---. 1 ---. x, x ---.1 ---. x---. 1, et 4 pas bons [(x ---. 1) + (1 ---. x)], [1 + x] + (1 ---. x), [1 + x] + (x ---. 1), [1 + x] + [1 + x], si € = 2 (cas non generique). • support non regulier (1 2,x,x2) : 4 types 1 + (1 ---. x ---. x 2), 1 + 1 + (x ---. x 2), 1 + (1---. x) + x 2, 1 + 1 + x + x 2 si € > 3 (cas generique).

Conjecture de Langlands modulaire

431

• 9 = 8 + 1 types, dont 8 bons (1 ---. x ---. x 2 ---. 1), (1 ---. x ---. 2 x ) + 1, (x ---. x 2 ---. 1) + 1, (x 2 ---. 1 ---. x) + 1, (1 ---. x) + (x 2 ---. 1), (x + x 2) + 1 ---. 1), (1 ---. x) + x 2 + 1, (x 2 ---. l)x + 1, et un type [1 + x + x 2 ] + 1 qui n'est pas bon, si c = 3 (cas non regulier, non generique) . • support non regulier (1, (X)2,X 2) : 5 types: (1 ---. x) + (x ---. 2 x ), (1 ---. x ---. x 2)+x, l+x+(x ---. x 2), (1---. x)+x+x 2, 1+x+x+x2 si c > 3 (cas generique). Le cas c = 3 se ramene au cas precedent. • support regulier (1,x,x 2,x3): 8 types tous bons (rajouter aux types de support (1, x, x 2), soit +x 3 soit ---. x 3) : 1 ---. x ---. x 2 ---. x 3, 1 ---. x ---. x 2 + x 3, 1 ---. x + x 2 ---. x 3, 1 ---. x + x 2 + x 3, 1 + x ---. x 2 ---. x 3, 1 + x ---. x 2 + x 3, 1 + x + x 2 ---. x 3, 1 + x + x 2 + x 2, si c > 4 (cas generique) . • 15 = 14 + 1 types dont Ie cycle [1 + x + x 2 + x 3] et 14 bons types obtenus par rotation des autres types ci-dessus: les 4 x 3 = 12 types z ---. zx ---. zx 2 ---. zx 3, Z ---. zx ---. zx 2+ zx 3, X + zx + zx 2 ---. zx 3 pour z appartant au c-cercle, et les 2 types 1 ---. x+x 2 ---. x 3, X ---. X2+X 3 ---. X. Nous voulons maintenant etendre certains resultats de la theorie des Hc(n, q)-modules simples [Rog1, Rog2], pour q E C* d'ordre c(q, C*) = 00, aux HR(n, q)-modules simples, pour q E R* d'ordre fini c(q, R*).

2.7 Rappel sur la structure de HR(n, q). Soit R un anneau commutatif, et ql/2 E R*. L'algebre de Hecke HO = H'R(n, q) est engendree comme algebre par les elements T i , 1 :::; i :::; n - 1, verifiant [livre 1.2.14]:

pour 1 :::; i,j < n, Ii - jl > 1, 1 :::; k < n - 1. La base canonique de H'R(n, q) [livre 1.2.1:1.] est formee des T w = Til'" T ik pour W E Sn de decomposition reduite w = Sil ... Sik' OU Si = (i, i + 1), 1 :::; i < n. Comme ql/2 E R*, l'algebre de Hecke affine H = HR(n, q) admet la presentation suivante [livre 1.2.14]. Elle est engendree par HO et par une sous-algebre commutative

liees par les relations:

432

M.-F. Vigneras

2.8 Rappel sur Ie module I(X)' Soit X: A ---. R un earaetere; Ie module

I(X) = H

@A,x

R

est universel dans Ie sens: pour tout H-module M, alors

est la partie X-isotypique MX de M. Un H-module simple eontient X, si et seulement si e'est un quotient de I(X).

I(X) est Ie prolongement it H de la representation reguliere de HO tel que T 1 soit un veeteur propre de valeur propre X pour A. Done I(X) a une base (fw, wE Sn) telle que

Si < est l'ordre de Bruhat dans Sn, on a la formule : pour a E A, wE Sn, il existe des elements uniques ay,w,x E A tels que [Rogl page 446]:

aTw = Tww-1(a)

+ LTyay,w,a; y<w

en l'appliquant it iI, on obtient a!w = x(w-1a)!w+ Ly<w x(ay,w,a)!y' On deduit:

L 'action de A sur I(X) est triangulable, de diagonale les conjugues de X par Sn' Donc les suites de Jordan-Holder de I(X) et de I(X') sont disjointes si X, X' E HomR(A, R) ne sont pas conjugues par Sn. L'aetion de A sur un H-module M n'est pas semi-simple en general. On notera

La dimension de I(X)~X est egale it l'ordre du stabilisateur de X dans Sn, pour tout w E Sn. Un earaetere

Conjecture de Langlands modulaire

433

se releve toujours en un caractere

On peut par exemple fixer un isomorphisme de F~ dans Z~, et assoder aux ri leurs images (ce n'est pas en general un relevement ayant de bonnes proprietes). La reduction du module I Zl (J.l) est Ie module I Fl (X)· Ceci implique que certains resultats pour IR(X) sont vrais s'ils Ie sont en caracteristique zero, et d'utiliser [Rogl-2]. Les techniques de [Rog 1, th.2.3 page 448] impliquent:

Les suites de Jordan-Holder de I(X) et de I(X') sont egales si X, X' E HomR(A, R) sont conjugues par Sn. 2.9 Support. Un caractere X : A -> R est determine par la suite X(X i ) = ri E R*, 1 :::; i :::; n. On notera aussi X = (ri), I(X) = I(ri)' Le support avec multiplicites de X est l'ensemble des nombres r E R* qui interviennent dans (n) avec leur multiplicites. Le groupe Sn opere naturellement sur A et sur ses caracteres. Deux caracteres de A sont conjugues par Sn si et seulement s'ils ont Ie meme support avec multiplicites. Nous supposons desormais que R est un corps algebriquement dos. Si M est un H-module de dimension finie, la trace de A dans M, appelee Ie caractere-poids de M, est une somme de caracteres de A appeles les poids de M,

Le caractere-poids de I(X) est LWESn XW-1. Tout H-module simple contient une droite stable par A, done est quotient d'un I(X). Done les poids d'un H-module simple Mont Ie meme support avec multiplicites, appeIe Ie support avec multiplicites de M. Les H-modules simples de support avec multiplicites donne sont les sous-quotients simples de I(X) pour un caractere X de meme support avec multiplicites. La conjecture (2.4) doit respecter Ie support avec multiplicites.

2.10 Rappel sur les automorphismes de H. a) En caracteristique # 2, la multiplication par l'unique caractere non trivial (voir (2.18)) induit une involution * sur les H-modules simples, qui envoie un poids X = (ri) sur Ie poids obtenu en lisant les ri de droite a gauche (rn-i)' * c'est l'involution de Zelevinski; j'espere donner plus de details sur une interpretation geometrique de cette involution dans un arti-

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M.-F. Vignems

b) La contragediente induit une bijection d'ordre 2 sur les Hmodules simples, qui envoie un poids (ri) sur son inverse (ri l ). c) Soit r E R*; l'application

induit un automorphisme de H, donc une bijection sur les H-modules simples qui envoie un poids (r;) sur un multiple (rri). Notons E = E(q,R*) l'ordre de q dans R*. Via la conjecture (2.4), les automorphismes a), b), c) correspondent a des bijections sur les Etypes. Les deux bijections suivantes qui restectent les E-types simples doivent correspondre it b) et it c): - b') on remplace un E-type simple par Ie E-type simple obtenu en inversant ses elements, et en les lisant de droite a gauche: (y, m) ---. (ql-my-l, m).

- c') la multiplication par r E R*: (y,m) ---. (rY,m), L'analogue de l'action de l'involution de Zelevinski sur les E-types est difficile. Si E = 2, elle consiste aremplacer un type simple par Ie type simple obtenu en Ie lisant de droite a gauche: (z,m) ---. (qm-lz,m). Elle echange Ie caractere signe et Ie caractere trivial [livre III.3.14]. Dans Ie cas generique, elle est decrite dans [MW page 149].

2.11 Cas n = 2 dimension 2,

Soit X de support (rl' r2). Le module I(X) est de

- irreductible si et seulement si rl =I- q±lr2, - reductible semi-simple si et seulement si q = 1, r2 - sans multiplicites sauf si q = -1, r2 = -rl.

= rl,

Preuve. L'algebre HR(2, q) est engendree par T = T 1,xt 1,xiI; on pose S2 = {l,s}; Ie module I(X) a une R-base !I,fs. La representation I(X) est reductible si et seulement si elle contient une droite stable, et reductible semi-simple si elle contient deux droites distinctes stables. Les deux vecteurs C=fs-q!I, C'=fs+fl,

sont propres pour T de valeurs propres -1 et q. Lorsque q = -1, C = C' est l'unique vecteur propre de T. Le vecteur C est propre pour de prochain en collaboration avec Cary Rader, cette involution a aussi une interpretation par la cohomologie des faisceaux sur l'immeuble, voir mon preprint 1995 : Cohomology of sheaves on the building and R-representations.

Conjecture de Langlands modulaire

435

A si et seulement si rl = qr2, Ie vecteur C' est propre pour A si et seulement si r2 = qrl. Dans les deux cas, la valeur propre est xs. Remarque. Lorsque rl =I- r2, I(X) a une base forme de vecteurs propres pour A: II de valeur propre X et

de valeur propre xs. La multiplication it droite par As(X) dans HO induit un H-homomorphisme

I(xs)

-+

I(X)·

L'eIement As(X) est inversible dans l'algebre commutative HO si et seulement si r2 =I- rl q±l . 2.12 Corollaire: Irreductibilite de I(X) Soit X = (ri) un caractere de A. Un H -module M contenant Ie poids X, contient aussi Ie poids XSi si ri =I- q±lri+l, 1 :::; i < n. I(X) est irreductible si et seulement si ri =I- q±lrj pour tout 1 :::; i,j < n. On note I(X) si q =I- l.

= I(ri)'

Alors I(1n) est irroouctible si et seulement

Preuve.Soit M un H-module simple contenant xw- l = (rD, wE Sn. Pour tout i, 1 :::; i < n, la representation I(r~, r~+l) est irreductible, done xw- l Si est aussi un poids de M. On en deduit que ch M = l EWESn xw- , done M = I(X). La condition sur (ri) est verifiee, si et seulement s'il n'existe qu'un seul type Eri de support (ri)' Done (2.11) est compatible avec la conjecture (2.4). 2.13 Reduction aux composantes connexes. Le groupe cyclique < q > engendre par q opere sur R* par multiplication. Comme pour les (R*, q)-types (2.6), deux elements r, r' de R* sont dits lies si r = q±lr'. Le support d'un caractere X : A -+ Rest union de ses composantes connexes, comme en (2.6). Par (2.12), un H-module contient un poids X tel que les ri = X(Xi ) dans une meme composante connexe sont adjacents. On dit que X est mnge. On associe it un poids range X la suite de composition (dj)l~j~k de n, tels que les (ri)nj+l~i~nHl' 0:::; j < k, sont les composantes connexes de (ri)l~i~n, ou I'on pose no = 0, nl = d l , n2 = d l +d2, ... , nk = n.

436

M.-F. Vignems

L'algebre A s'identifie au produit tensoriel ®j=IAj des sousalgebres

et X au produit de caracteres Xj : A j -> R. On dit que les Xj sont les composantes connexes du caractere X = (Xj). La sous-algebre H(d j ) de H engendree par les Ti,X j , 1 :::; i,j :::; n, i =I- nl, ... ,nk, s'identifie au produit tensoriel ®j=IHR(dj,q). On note I(Xj) Ie HR(dj,q)-module defini par Xj. Si N est un H(dj)-module, Ie H-module H ®H(d j ) Nest dit induit par N. L'associativite du produit tensoriel montre que I(X) est Ie Hmodule induit par ®j=II(Xj). Notons W(d j ) C Sn Ie sous-groupe engendre par les Si, i =Inl, ... , nk, isomorphe au produit n~=1 Sdj" Si M est un H-module de dimension finie, dont les poids sont conjugues it X par Sn, (2.12) montre que Ie H(dj)-module

Mres =

L

Mto,

X'EXWE

est non nul. On l'appelle la restriction de M.

Le module M est induit par M res : l'application naturelle

est un isomorphisme. La preuve [Rog2 propA.1 page 248] est la meme en toute caracteristique. Elle utilise que 1) H = ffiwEW/W(dj)TwH(dj) ou west l'element de plus grande longueur de WW(d j ) (croissant sur chaque bloc (nj+1, nj+I), 0:::; j < k) [livre IILO.9]; donc la dimension de Nest egale it [W : W(d j )] fois la dimension de M si M est induit par N. 2) Le caractere-poids ChM se deduit du caractere-poids ch Mred , en rempla R un camctere mnge de composantes connexes Xj. L'induction induit une bijection entre les sousquotients simples de ®I(Xj) et les sous-quotients simples de I(X). Son inverse est la restriction.

Conjecture de Langlands modulaire

437

Preuve. Un sous-quotient M de I(X) est induit par M res , par ce qui precede. Si M est simple, M res est un H(d;)-module simple, i.e. un produit tensoriel Mred = @j=IMj de HR(d j , q)-modules simples M j . Les Mj sont des sous-quotients simples de I(Xj)' Inversement, pour de tels M j , Ie H-module induit par @j=IMj est un H-module simple M. II suffit de voir que M res est simple. Elle contient l'image de @j=IMj dans M par l'injection naturelle, et lui est egale pour une raison de meme dimension. On a done M res ~ @j=IMj . La classification des H-modules simples est ainsi ramenee it celie des H-modules simples de support connexe contenant 1. Cette reduction est parallele it la classification des E-types, la somme de deux E-types de support connexes disjoints correspondant it l'induction. Nous cherchons done les sous-quotients simples de I(X) pour un caractere X de support (l,q, ... ,qr), r < E. Pour q = 1, un seul caractere X = (In) est de cette forme.

2.15 Cas q = 1. Les HR(n, I)-modules simples de poids X = (In) sont classes par les partitions de n si la camcteristique de Rest 0, et par les partitions l-regulieres de n si la caracteristique de Rest l.

Preuve. On a

Les sous-H'k(n,I)-quotients simples sont stables par HR(n, 1), et sont classes par les partitions de n [JK 6.1.12 page 243] indiquees ci-dessus.

Exemple. Supposons n = 2. Par (2.11), si l = 2, on a un seul module simple, Ie caractere trivial, et une seule partition 2-reguliere (2). Si l =I- 2, il y a deux modules simples, les caracteres triviaux et signes, et deux partitions l-regulieres (2) et (1,1). Supposons n = 3. II y a une seule partition 2-reguliere (3), il y a deux partitions 3-regulieres (3), (2,1, 1), et trois partitions l-regulieres (3), (2, 1, 1), (1, 1, 1) si l > 3. La conjecture (2.4) est done vraie pour q = 1. II reste it etudier Ie cas ou Ie support (l,q, ... ,qr), 1 ~ r

de X est connexe et contient 1 =I- q.

< E,

438

M.-F. Vignems

2.16 Cas generique. Le support de X ne recouvre pas Ie cercle (i.e. r < € - 1). Notons C Ie €-cercle prive du point q€-l, muni de I'ordre total

Les suites de points C sont munies de I'ordr€ lexicographique. On va decrire une bijection entre les €-types de support dans C, et les suites de points dans C, que nous appelons bien ordonnees. La definition d'une suite bien ordonnee est due a Zelevinski. Deux €-types simples (z, m) et (z', m') de support contenu dans C sont dits lies si l'un n'est pas contenu dans I'autre, et si leur union (ensembliste) est connexe. Si de plus Z < z', on dit que (z,m) precede (z',m'). Par exemple, si z, zq E C, alors z precede zq, et zq et z ne precedent pas z (ici l'hypothese generique est fondamentale). On dit qu'une suite de €-types simples (Zj, mjh::;j::;k de support dans C est ordonnee si a) Zj+l -I Zjqm j pour j = 1, ... k - 1, b) pour tout i < j, (Zi, mi) ne precede pas (Zj, mj). On associe it une suite de points dans Ie €-cercle, I'uniqu€ suite de €-types simples definissant la meme suite de points, et verifiant a). On associe it une suite (Zj, mj h::;j::;k, de €-types simples, Ie €-type

L~=l(Zj,mj).

Vne suite de points dans C est dite ordonnee, si la suite en €-types simples associee est ordonnee; elle est dite bien ordonnee si la suite en €-types simples associee est maximale parmi les suites de points, de suite en €-types simples ordonnee, et de meme €-type. L 'application qui Ii une suite associe son €-type, induit une bijection entre les suites bien ordonnees de support dans C, et les €-types de support dans C.

Preuve. Soit t : C -> N. On range par ordre croissant, les suites de points dont le support avec multiplicite est egal it t. Chacune donne une suite de €-types simples. On ne garde que les suites ordonnees. En verifiant que chaque €-type dont le support avec multiplicite est egal it t, est le €-type associe it une suite ordonnee, il est clair que l'on obtient la bijection voulue. Considerons l'ensemble des suites (Zj, mjh::;j::;k definissant un €type donne de support t. Si k = 1, la suite est evidemment bien ordonnee. Par induction, supposons que la suite (Zj, mjh<j::;k est bien ordonnee. Montrons que l'on peut intercaler (Zl' md de sorte que la suite obtenue reste bien ordonnee. On n'a pas it la fois Zl = Zjqm j

Conjecture de Langlands modulaire

439

et zlqffi 1 = Zj, puisque ces egalites impliquent Zl > Zj et Zj > Zl *. Si Zl i= Zjqffi j , alors (Zj, mj), (Zl, ml) verifie a); si zlqffi 1 i= zj, alors (zl,ml),(Zj,mj) verifie a). Supposons quej est Ie premier indice 2: 2 tel que (zj,mj) precede (zl,ml). On a Zj < Zl, et (zl,ml),(Zj,mj) est bien ordonne. Si j = 2, on place (zl,ml) it. gauche de (z2,m2). Si j > 2, et si Zl i= Zj_Iqffi j - 1 on place (zl,ml) it. gauche de (zj,mj). Si j > 2, Zl = Zj_lqffi j - 1 , et si j' est Ie plus grand indice 2: 2 tel que Zl = Zj'_lqffij'-l on place (zl,ml) it. gauche de (zj'-l,mj'-l). Exemples. Pour c > 2, t de support 1, q avec t(l) = 2, t(q) = 1, les suites de support t sont par ordre croissant (1,1, q), (1, q, 1), (q, 1, 1). Les suites en c-types simples correspondantes sont (1, (1, q)), ((1, q), 1), (q, 1, 1). Les c-types simples 1 et (1, q) ne sont pas lies, les c-types simples q, 1 sont lies, et 1 precede q. Les 3 suites sont ordonnees, seulement les deux dernieres sont bien ordonnees. Le caractere X s'identifie it. la suite (ri = X(X i )), 1 ~ i ~ n de points dans C. On peut appliquer les definitions ci-dessus aux conjugues dans X par Sn' Un module irreductible de support dans C possede un plus haut poids unique, appele son poids maximal. On peut remplacer "maximal" par "minimal", en appliquant l'involution de Zelevinski. Les demonstrations complexes de Zelevinski [Zl] ou de Rogawski [Rog 2] restent valables et donnent les resultats ci-dessous.

Poids maximal. 1) Un caractere X de support dans C est Ie poids maximal H -module simple si et seulement si X est bien ordonne. 2) Deux H -modules simples avec Ie meme poids maximal sont isomorphes.

L'application qui associe it. un H-modules simple de support dans C son poids maximal, puis Ie c-type associe, induit donc une bijection entre les H -modules simples de support dans C, et les c -types de support dans C. La conjecture (2.4) est donc vraie dans Ie cas generique. On note M(X) Ie module simple de poids maximal X, et si (Zj, mjh~j~k est la suite de c-types simples definie par X, on note

*

Contre exemple: ce n'est pas possible pour (1, c -1) + (qe-l), dont Ie support n'est pas generique.

440

M. -F. Vigneras

la representation induite par Ie produit tensoriel des HR(mj, q)modules simples de poids maximal (Zj, mj). Le caractere-poids de 7r(X) se calcule comme en (2.13). La proposition suivante permet d'exprimer un sous-quotient irreductible de I(X) comme une combinaison lineaire de tels modules, donc de calculer son caractere-poids facilement. Remplacer deux c-types simples lies par leur union (z, m) U (z', m') et leur intersection, (z, m) n (z', m'), est appele une operation elementaire. Description de 7r(X). 1) Les poids ordonnes de M(X) sont les poids ordonnes de meme c-type simple que X. 2) La multiplicite de M(X) dans 7r(X) est egale Ii 1, et les sousquotients irreductibles de 7r(X) sont les M(X') OU X' se deduit de X par une operation elementaire. Par exemple, (z, m) x (z', m') est irreductible si les types ne sont pas lies, et il est de longueur deux, avec deux sous-quotients irreductibles distincts s'ils sont lies, l'un etant ((z, m) U (z', m')) x ((z, m) n (z', m')). 2.17 Cas regulier . C'est Ie cas ou X est distinct de tous ses conjugues par Sn, ou ce qui est equivalent les multiplicites dans X sont egales it. 1, Ie support de X est (1, q, . .. qn-I), n:::; c. Donc X est generique, sauf si n = c. Proposition. I(X), n :::; c, est sans multiplicites, de longueur 2n si n = c, de longueur 2n - 1 si n < c.

-

2

Preuve. Le cas n = 2 (2.11) donne un vecteur propre As, (X') de A dans I(X), de valeur propre X'Si pour tout 1 :::; i < c, et pour tout caractere X' conjugue it. X. Si w = Sl ... St est une decomposition reduite, Ie produit dans HO

est non nul [Rogl 2.1 page 448]. C'est donc un vecteur propre pour A, de valeur propre XW-I. Les Aw(X) forment une R-base de I(X), et

HAw(X) = HO Aw(X). Soit M un H-module simple contenant X. Donc M est l'unique quotient irreductible f : I(X) --+ M. Les poids de M sont de multiplicite 1, et XW-I, wE Sn, est un poids de M si et seulement si f(Aw(X)) i= O. L'image de H Aw(X) est alors egale a M, et l'on doit avoir Al (X) C HAw(X), puisque M contient X. Comme Ho = HAl (X), ceci signifie Ho = HAw(X) = HO Aw(X), Le. Aw(X) est inversible it. gauche dans HO, i.e. As, (X), ... ,A Sl (X( S2 ... Sr) -I) sont tous inversibles it. gauche. Par Ie cas n = 2, ceci est equivalent it. rHI i= riq±1 pour tous les Si qui interviennent. Dans Ie cas general, on se trouve devant un probleme

441

Conjecture de Langlands modulaire

combinatoire n§solu par Zelevinski. On associe it X un signe 1](qi) pour chaque qi contenu dans Ie support C de X. Il est egal it. +1 si qi-l appartient it. C, et precede qi dans la suite donnant X, et il est egal it. -1 sinon. Les w E Sn tels que A w (X) est inversible it gauche dans HD, sont ceux tels que X et xw-l donnent la meme suite de signes [ZI theorem 2.2 page 177]. Ce sont les poids deMo

Dans Ie cas generique, pour tout X regulier de support (1, ... , qn-l ), on a 1](1) = -1 car q-l n'appartient pas au support. Les n - 1 signes rest ant sont possibles, on peut toujours fabriquer une suite dont les elements sont les qi, de fa 3 on associe aux modules simples generiques de caracterepoids respectifs

les types bien ordonnes (1 ---+ q ---+ q2), q2 + q + 1, q2 + (1 ---+ q), (q---+ q2) + 1, en suivant (2.16). Si c(q) = 3, on associe aux caracteres triviaux qi(1,q,q2) les c-types qi(l---+ q ---+ q2), et aux caracteres signes qi(q2,q, 1) les c-types (qi-l ---+ qi) +qi+l pour i = 0,1,2 modulo 3, par (2.18). 2.22 Le cas n

Lorsque n

=4 .

= 4, on a trois cas nouveaux

(1 3 ,q) := (1,1, 1,q), (1 2 , (q)2) := (1, l,q,q), e = 2; (1 2,q,q2):= (1, 1,q,q2), e = 3.

Sauf dans Ie premier cas ou ne savons pas classer les modules simples de support avec multiplicites (1 3 ,q), nous allons demontrer que la conjecture (2.4) est vraie. Nous avons donne tous les details, les resultats sont indiques avec e. On note M 1 x M 2 Ie module induit d'un module M 1 Q9 M~ de HR(nl' q) x HR(n2, q), comme en (2.13). a) 1(1 3 ,q) = 1(1 2,q) X 1 a 4 sous-quotients, (l,q,l) x 1 de multiplicite 2 de caractere-poids 2(1,1, q, 1)

+ 2(1, q, 1, 1),

M.-F. Vigneras

444

et deux autres sous-quotients 2(1, l,q) x 1, 2(q, 1, 1) x 1, de caracterespoids

6(1,1,1, q)

+ 2(1, 1, q, 1),

6(q, 1, 1, 1)

+ 2(1, q, 1, 1).

Ces trois representations sont probalement irreductibles. precisement:

• Poure = 2, et pour tout n

~

2, les trois HR(n

Plus

+ l,q)-modules

(l,q,l) x 1(l n - 2), 2(1,I,q) x 1(l n - 2), 2(q,I,I) x 1(l n - 2), sont probablement irreductibles. Les e-types correspondant sont necessairement (1 --+ q --+ 1) + I n - 2, (1 --+ q) + In-I, (q --+ 1) + In-I. b) 1(1 2 , (q)2) = 1(1 2,q) X q a 4 sous-quotients (l,q, 1) x q de multiplicite 2, de caractere-poids

(1, q, 1, q)

+ 2(1, q, q, 1) + (q, 1, q, 1)

et 2( 1, 1, q) x q, 2( q, 1, 1) x q de caracteres-poids

2(1, q, 1, q)+4(1, 1, q, q)+2(q, 1, 1, q), 2(q, 1, 1, q)+2(q, 1, q, 1)+4(q, q, 1, 1). Il est clair que les deux caracteres de poids (1, q, 1, q), (q, 1, q, 1) sont sous-quotients de l'induite (1, q, 1) x q du caractere (1, q, 1) Q9 q. Par un argument deja vu, un sous-quotient qui contient (l,q,q, 1) Ie contient avec une multiplicite ~ 2, done (l,q, 1) x q est de longueur 3, avec un sous-quotient 'ITo de caractere-poids 2(I,q,q, 1). Les automorphismes de H (2.10) montrent qu'il existe un module simple 'ITI de de caracterepoids 2(q, 1, 1, q). Ceci implique que 2(1, 1, q) x q ou 2(q, 1, 1) x q admet 'IT I comme sous-quotient. Mais les caracteres-poids de ces modules se deduisent l'un de l'autre par l'involution de Zelevinski, et 2(1, q, q, 1), done 'IT I , est fixe par l'involution de Zelevinski, done 'IT I est sous-quotient de chacun. Nous resumons: 1(1 2 , (q)2) a une filtration par - les deux modules de caractere-poids:

2(1, q, 1, q)

+ 4(1, 1, q, q),

2(q, 1, q, 1)

+ 4(q, q, 1, 1),

avec multiplicite 1, - les modules simples 2(1, q, q, 1), 2(q, 1, 1, q), avec multiplicite 2, - les caracteres (1, q, 1, q), (q, 1, q, 1). On deduira de la filtration de 1(1 2, (q)2) provenant du cas generique regulier par la reduction (q2 = 1) (voir c)), que les caracteres sont de multiplicite 4.

Conjecture de Langlands modulaire

445

• Donc 1(1 2 , (q)2) est de longueur 11 avec 6 sous-quotients irn§ductibles non isomorphes, ayant chacun un unique poids (avec une multiplicite ~ 0), -les deux caracteres (1, q, 1, q), (q, 1, q, 1) de multiplicite 4, associes aux bons c-types simples (1,4) := (1 ........ q ........ 1 ........ q), (q,4) := (q ........ 1 ........ q ........ 1), - les deux modules simples 7fo,7fl de caractere-poids 2(1, q, q, 1), 2(q, 1, 1, q), de multiplicite 2. II semble que Ie bon c-type (1 ........ q ........ 1) +q doit correspondre it un sous-quotient simple de (l,q, 1) x q, donc it 7f o . Pour la meme raison, (q ........ 1 ........ q) + 1 doit correspondre it 7fl, -les deux modules simples de caractere-poids 4(1,1, q, q), 4(q, q, 1, 1) de multiplicite 1. La decomposition en c-types simples des caracteres contient un c-cycle 1 + q, et suggere que les c-types correspondant sont respectivement 2(1 ........ q), 2(q ........ 1). c) Dans Ie cas generique regulier c > 4, la representation 1(1, q, q2, q3) est sans multiplicite de longueur 8, les caracteres-poids de ses sous-quotients irreductibles etant les suivants (2.16-17), et ceux deduits par l'involution de Zelevinski:

(1, q, q2, q3), (1, q2, q, q3)

+ (q2, 1, q, q3) + (1, q2, q3, q) + (q2, 1, q3, q) + (q2, q3, 1, q) (1, q, q3, q2) + (1, q3, q, q2) + (q3, 1, q, q2), (q3 , 1, q2, q) + (1, q3, q2, q) + (q3, q2, 1, q).

Ils sont associes aux types bien ordonnes (2.16) (1 ........ q q2 ........ q3), (q2 ........ q3) + (1 ........ q), q3 + (1 ........ q ........ q2), q3 + q2 + (1 q), et aux types bien ordonnes deduits par l'involution de Zelevinski (2.10): q3 + q2 + q + 1, q3 + (q ........ q2) + 1, (q2 ........ q3) + q + 1, (q ........ q2 ........ q3) + 1. Dans Ie cas regulier c = 4, on deduit de (2.17) que la representation 1(1,q,q2,q3) est sans IDultiplicite de longueur 14: - les 4 caracteres qi(l, q, q2, q3), i = 0,1,2,3, associes it. qi(l ........ q ........ q2 ........ q3), - les 4 modules simples de caracteres poids qi((l, q, q3, q2) + (1, q3, q, q2)), i = 0,1,2,3, associes it. qi((l ........ q ........ q2) + q3), - les 4 modules simples de caracteres poids qi((q3,1,q2,q) + (q3,q2, 1,q)), i = 0,1,2,3, associes it. (q3 ........ 1) + (q ........ q2), - les 2 modules simples de caracteres-poids qi((1,q2,q,q3) + (q2, q, q3)+(1, q2, q3, q)+(q2, 1, q3, q)), i = 0,1, associes it. qi(1+q+(q2 ........ q3)). Par reduction (q2 = 1), on obtient une filtration de 1(1 2 , (q)2) par les 4 modules de caractere-poids (1, q, 1, q), 4(1,1, q, q) +

446

M. -F. Vigneras

(l,q,l,q), 2(1,q,q,1) + (q,l,q,l), 2(q,1,1,q) + (l,q,l,q), et les 4 modules deduits par l'involution de Zelevinski. En comparant avec la filtration de 1(1 2 , (q)2), c = 2, obtenue en b), on voit que les 3 modules de caractere-poids 2(1, q, 1, q) + 4(1,1, q, q), 2(q, 1, 1, q) + (1, q, 1, q), 2(1, q, q, 1) + (1, q, 1, q) ont un sous-quotient qui est Ie caractere (1, q, 1, q) donc la multiplicite de (1, q, 1, q) est 4. Ceci termine la demonstration de b). d) 1(1 2,q,q2) a une filtration obtenue par reduction (q3 = 1) du cas generique regulier, de caractere-poids (1, q, q2, 1), (1,q2,q, 1) + (q2, 1,q, 1) + 2(q2, 1, 1,q) + (1,q2, 1,q), (1, q, 1, q2) + 2(1, 1, q, q2), (1,q2, 1,q) + 2(1, 1,q2,q), et ceux obtenus par l'involution de Zelevinski. On deduit par reduction it. partir du cas generique, que 1 x (q, q2, 1) a un sous-quotient de dimension 3, de caractere-poids

II est simple, car il ne peut avoir de sous-quotient de dimension 1. En utilisant les involutions, on voit que les sous-quotients ci-dessus de caracteres-poids

sont simples. On deduit par reduction it. partir du cas generique, que (1, q) x (q2, 1) a un sous-quotient de caractere-poids

Cette representation est irreductible, car la decomposition de 1(1 2 , q) dans Ie cas generique, montre qu'un module qui contient Ie poids (q2, 1, 1, q) contient necessairement (q2, 1, 1, q) +(1, q2, 1, q)+( q2, 1, q, 1). On a donc: • 1(1 2 , q, q2) est de longueur 10, Ie caractere signe et Ie caractere trivial apparaissant avec multiplicite 2, et les 6 sous-quotients simples restant sont non isomorphes. On associe au - caractere trivial (1, q, q2, 1), Ie c-type (1 ---+ q ---+ q2 ---+ 1), - module simple de caractere-poids (1,q2,q,1) + (q2, 1,q, 1) + 2(q2, 1, 1,q), Ie c-type (1 ---+ q) + (q2 ---+ 1) selon la regIe qu'un sousquotient simple de (l,q) x (q2,1) doit contenir ce type,

447

Conjecture de Langlands modulaire

- module simple de caractere-poids (q, 1,q2, 1) + 2(q,q2, 1, 1), Ie Etype 1 + (q -+ q2 -+ 1), selon la regIe ci-dessus, - module simple de caractere-poids (l,q, 1,q2) + 2(1, 1,q,q2), Ie Etype (1 -+ q) + 1 + q2, qui correspond it. la decomposition en E-types simples. Les autres sous-quotients simples se deduisent de ceux-ci par l'involution de Zelevinski. 3.

Comparaison gaJoisien-GL(n).

Le cas moderement ramifie du point du vue galoisien correspond au cas de niveau 0 [livre III.3.3] du point de vue GL(n). La comparaison se ramEme it. la construction de Green-Deligne-Lusztig-Dipper-James [DJ1] des representations supercuspidales de GL( n, F q), grace it. la theorie des types de niveau 0 de Howe-Moy-Bushnell-Kutzko [livre 111.2-3]. 3.1 Correspondance avec GL(n, F q). La construction de Green-Deligne-Lusztig X -+ 1T(X) definit une bijection des GalF q -orbites des caracteres X : -+ Q~ reguliers sur F q, sur les classes d'isomorphisme des representations irreductibles supercuspidales 1T(X) de GL(n,F q ) sur QI. Si l =I- p, Dipper et James [livre IlI.2.3, IlI.2.8-9] ont montre que la reduction modulo l de 1T(X) ne depend que de la reduction nx modulo l de X; elle est toujours irreductible, tandis que la representation irreductible rla(x) de W F sur QI associee it. X, peut-etre reductible (1.17). Si /-L : F;d -+ F~ est Ie caractere rElgulier sur F q tel que rlX = /-LN pour la norme N : F;n -+ F;d' alors

F;n

La representation rlX peut ne plus etre supercuspidale, car elle est un sous-quotient de l'induite parabolique X~l

La bijection a(x)

-+

1T(/-L),

md = n.

1T(X) est compatible avec la reduction modulo l.

Remarque. Si l = p, Ie phenomeme inverse se produit: rpX reste regulier sur F q, la representation rpa(x) reste irreductible, tandis que r p1T(X) est reductible. Il serait interessant de comprendre ce que ceci reflete.

M. -F. Vigneras

448

3.2 Correspondance irreduetible galoisien GL(n), dans Ie cas moderement ramifie.

~

supercuspidal

Soit R un corps algebriquement clos de caracteristique 1 i= p. Pour r E R* et pour X : F;n ........ R* regulier sur F q, notons a(r, X) la representation irreductible moderement ramiMe de dimension n construite en (1.14), et n(r, X) =

indcL(n,F),F·cL(n,oF)

vra(x)·

la representation irreductible cuspidale de GL(n, F) de niveau 0, i.e. ayant un vecteur invariant par 1 + PFM (n, 0 F) [livre III.3.3]. Supposons R = QI et r E

Z;.

Theoreme. L 'application a(r, X) ........ n(r, X),

r E

Z;,

X: F;n ........

Q;,

X regulier sur F q, induit une bijection entre les representations lentieres moderement ramifiees de IrlQl W F de dimension n, et les representations l-entieres, de niveau 0, cuspidales de IrrQI GL(n, F). Cette bijection est compatible avec la reduction modulo l, et induit une bijection entre - les representations moderement ramifiees de IrrFI W F de dimension n, et les representations de niveau 0 supercuspidales de IrrFI GL(n, F), ainsi qu'entre - les representations semi-simples moderement ramifiees de W F sur F I , de dimension n, qui se relevent en des representations irreductibles sur QI, et les representations de niveau 0 cuspidales de IrrFI G L( n, F).

La preuve consiste a lire la classification et de la reduction modulo 1 des objets en question, faite ici pour les representations du groupe de Weil (1.14, 1.18) et dans (livre III.3.1O, III.3.16) pour les representations de GL(n, F). La bijection a(r, X) ........ n(r, X) se lit sur la classification. Considerons maintenant la reduction modulo l. Dans Ie cas galoisien ou GL(n, F), la reduction modulo 1 de la representation ne depend que des reductions modulo 1 des parametres (r, X). On a (1.18):

avec les notations de (3.1), en notant encore r son image dans

F;,

et

449

Conjecture de Langlands modulaire

rl'rr(r, X) = 7l"(r, r/x) est im§ductible, cuspidale, et sous-quotient de

X:'l /I~7l"(r, p,),

md = n.

Si la conjecture de Deligne-Langlands sur Q/ (Introduction 6), envoie a = a(r, X) identifie it. la paire (a,O) sur 7l" = 7l"(r, X), alors la conjecture sur F/ envoie (na,O) sur r/7l". L'im§ductibilite de ria est equivalente it. la supercuspidalite de r/7l". Le cas general n'est pas encore bien compris : tout repose sur les "caracteres simples ()" qui semblent plus sauvages que simples. Ils correspondent aux representations irreductibles de Fp; la construction des representations cuspidales it. partir de () est refletee par la construction galoisienne (1.23). Les resultats de [livre IlLS] indiquent que la conjecture de Deligne-Langlands: irreductible galoisien ~ supercuspidal GL (n), est probablement vraie dans Ie cas modulaire. N ous l'admettons dans (3.3). Il reste alors it. verifier cette conjecture lorsque Ie support (la representation semi-simple galoisienne, Ie support supercuspidal pour GL(n)) est fixe.

3.3 La correspondance modulaire pour un support donne. Comment classifier les representations de Irr R GL(n, F) ? Dans Ie cas R de caracteristique 0 on classifie separement - les cuspidales [BK], - les representations de support cuspidal donne [ZI]. Dans Ie cas R de caracteristique > 0, on a classifie les cuspidales et les supercuspidales (livre). II reste it. classifier les representations de support cuspidal, ou supercuspidal donne. La conjecture de DeligneLanglands etendue a F/, donnee en 6) dans l'introduction, est supposee classer les representations de support supercuspidal donne (correspondant it. la representation semi-simple), par les valeurs possibles de l'endomorphisme nilpotent. Nous n'avons qu'un resultat tres partiel (qui admet la conjecture de Deligne-Langlands sur Q/). Proposition. La conjecture de Deligne-Langlands modulaire est vraie, dans Ie cas generique c:(q, R*) > n, ou regulier c:(q, R*) = n, ou si n = 2.. Preuve. Dans Ie cas generique, ou banal, cuspidal = supercuspidal = Z-projectif pour tous les sous-groupes de Levi de GL(n, F) [livre IlLS. IS]. Les arguments de la classification de Zelevinski [ZI] des representations irreductibles de GL(n, F) de support supercuspidal LPi donne sont valables (comme en (2.16)).

450

M. -F. Vigneras

Dans Ie cas n§gulier, cuspidal = supercuspidal = Z-projectif pour tous les sous-groupes de Levi de GL(n, F), sauf pour les representations cuspidales de Steinberg (Xdet) 0St(I, n) E IrrFI GL(n, F) (St(I, n) est l'unique sous-quotient g€merique de la representation induite du caractere trivial du groupe triangulaire superieur, X : F* --+ est un caractere) [livre III.5.I4-I5]. La classification de Zelevinski est encore valable, sauf pour les representations de IrrFI GL(n, F) dont Ie support supercuspidal est egal it. celui d'une representation (Xdet) 0 St(I, n). On se ramene facilement au cas ou X est trivial. Mis it. part la representation de Steinberg cuspidale St(I, n), que l'on associe au seul c-cycle 1 + q + ... + qE-1 qui n'est pas bon it. scalaire multipliclatif pres (2.6), ces representations ont un support cuspidal egal it. leur support supercuspidal. Comme c(q, Fi) = n => 1, les HFI (n, q)-modules simples sont en bijection avec les representations de IrrR GL(n, F) ayant un vecteur invariant par Ie sous-groupe d'Iwahori. Les HFI (n, q)-modules simples ont ete classes en (2.17). Pour n = 2 et c(q, F1) = 1 (pour c(q, F1) > 1, on est dans Ie cas generique ou regulier si n = 2), la conjecture resulte de [VigIl.

F;

Remarque. En general, Ie probleme de la description des composants irreductibles d'une induite d'une representation irreductible cuspidale est ouvert, meme pour n = 3. 3.4 Exemples.

Ces exemples indiquent que les c-types qui ne sont pas bons proviennent des representations cuspidales non supercuspidales (voir aussi Ie cas regulier, non generique precedent). a) Decomposition de 1 x 1 x 1/ X .•• X I/E-1. Outre les sous-quotients irreductibles de support cuspidal=support supercuspidal, figure Ie sousquotient irreductible St ou St est Ie sous-quotient cuspidal de 1x 1/ X .•. X I/E-1 [livre III.3.I5]. II correspond au c-type pas bon I+I+q+ +qE-1. La conjecture de Deligne-Langlands pour Ie support 1+1+1/+ +I/E-1, sera un corollaire de la conjecture sur les HR(n, q)-modules simples. b) Decomposition de 1 x 1 x 1/ X 1/, c = 2. Outre les sous-quotients irreductibles de support cuspidal=support supercuspidal, figure ceux de support cuspidal St + St ou St +1 + 1/, et lorsque I = 2 Ie sousquotient cuspidal St4 de St x St. Dans tous les cas, on peut analyser St x St. On obtient: - si I =I- 2, il existe deux sous-quotients irreductibles non isomorphes, de support cuspidal St + St. On leur associe les c-types pas bons (q --+ 1)+(1 --+ q) pour celui qui n'a pas de modele de Whittaker, et I+q+I+q pour l'autre.

Conjecture de Langlands modulaire

451

- si I = 2, il existe un sous-quotient irreductible de support cuspidal St + St. On lui associe Ie E-type pas bon (q - 1) + (1 - q). On associe a St4 Ie E-type pas bon 1 + q + 1 + q. Si X est Ie caractere trivial de poids (1, q), ou signe de poids (q, 1), la representation St xX est irreductible. Ce sont les deux sous-quotients irreductibles de support cuspidal St + 1 + 1/. On leur associe les E- types pas bons 1 + q + (1 - q) ou 1 + q + (q - 1). On en deduit, avec les resultats precedents que: La conjecture de Deligne-Langlands est vraie pour Ie support L:~I I/i-I, n:S; 4. Nous reviendrons sur ce type de questions dans un article ulterieur. Bibliographie

[Bourbaki A 8] N. Bourbaki,Algebre, chapitre 8, Modules et anneaux semi-simples, Hermann, Paris 1958. [CR] C. W. Curtis, I. Reiner, Representation Theory of finitf groups and associated algebras, Wiley, 1988. [De] P. Deligne, Les constantes des equations fonctionnelles des fonctions L, Springer Lecture Notes 349. [DJ1] R. Dipper, G.D. James, Identification of the Irreducible Modular Representations of GLn(q), J. Algebra, 104, (1986), 266288. [DJ2] R. Dipper, G.D. James, Representations of Heeke algebras of general linear groups. Proc. London Math. Soc., 52 (1986), 20-52. [FrJ A. Frohlich, Local fields, Algebraic Number Theory, edited by J. W.S. Cassels and A. Frohlich, Thomson book Company Inc., 1967. [Gro] I. Groj nowski, Representations of affine Heeke algebras (and affine quantum GL n ) at roots of unity, Internat.-Math.-Res.Notices 1994,no.5,215ff., approx. 3pp. (electronic). [He] G. Henniart, La conjecture de Langlands locale pour GL(3), Mem. Soc. Math. France, 11/12 (1988), 497-544. [Iw] K. Iwasawa, On Galois groups of local fields, Trans. A.M.S., 80 (1955), 448-469. [JK] G.D. James, A. Kerber, The representation theory of the symmetric group, Addison Wesley, London, 1981. [KZ] H. Koch, W. Zink, Zur Korrespondenz von Darstellungen der Galoisgruppen un der zentralen Divisionsalgebren uber lokalen Korpern (der zahme Fall), Math. Nachr., 98 (1980), 83-119. [KM] P. Kutzko, A. Moy, On the local Langlands conjecture in prime dimensions, Annals of Math. (1985) 495-517.

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M.-F. Vigneras

[livre] M.-F. Vigneras, Representations d'un groupe reductif p-adique sur F I , Ii- p ( et cuspidales de GL(n)), Preprint 1987-1995. [LRS] G. Laumon, M. Rapoport, U. Stuhler, D-elliptic sheaves and the Langlands correspondence, Invent. Math., 113 (1993), 217338. [MW] C. Moeglin, J.-L. Waldspurger, Sur l'involution de Zelevinsky, J. reine angew. Math., 372 (1986), 136-177. [Moy] A. Moy, Local constants and the tame Langlands correspondance, Amer. J. Math., 108 (1986), no.4, 863-930. [Rol] J.D. Rogawski, On modules over the Hecke algebra of a p-adic group, Invent. Math., 79 (1985), 443-465. [Ro2] J.D. Rogawski, Representations of GL(n) over a p-adic field with an Iwahori-fixed vector, Israel J. Math., 54 (1986), 242256. [Sel] J.P. Serre, Representations lineaires des groupes finis, Hermann, Paris, 1967. [Se2] J.P. Serre, Corps locaux, Hermann, Paris, 1968. [Ta] J. Tate, Number Theoretic background, Proc. Symp. Pure Math., Vol. 33 (1979), 3-26. [Vigl] M.-F. Vigneras, Representations de GL(2, F) en caracteristique I, F corps p-adique, pi-I, Compositio Math., 72 (1989), 33-66. [Vig2] M.-F. Vigneras, Sur la conjecture locale de Langlands pour GL(n, F) sur FI' C. R. Acad. Sci. Paris, 1-318 (1994) p.905908. [Zl] A.V. Zelevinski, Induced representations of reductive p-adic groups II, Ann. scient. Ecole Norm. Sup. 4e serie, tome 13 (1980), 165-210. [Zink] W. Zink, Representation theory of local division algebras, J. reine angew. Math. 428, (1992) 1-44. U.R.A. 748 , U.F.R. de Math, Tour 45-55, 5 Hage, 2, place Jussieu F-75251 Paris Cedex 05, France Received January 1995

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