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
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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<1 E 1,1 I 2
3
-
2
.
E~,l = H 1(W/Z,H 1(C2,C X)) ':::::. H1(W/Z,{±I}) is the set of linear characters of W /Z taking rational values, which is the set of all linear characters by assumption. Thus, E~,1 ':::::. H 1(W/Z,CX) = H 1(W,CX) by (1). We conclude that 1Ej,1 I::; I H 1(W,C X) I·
(6) This is a direct consequence of (3), (4), (5),
•
q-Analogue of a Twisted Group Ring
5
Corollary. If an irreducible finite Goxeter group W has a nontrivial center Z and W is a representation group of W / Z, then the type of W cannot be B 2m (m 2: 2), D 2m (m 2: 2). Proof. The criterion (6) of Proposition 1 proves this corollary.
•
To finish the classification, we have to determine H 2 (W/Z,C X ) explicitly for W of type E g , P4, I~4m), H 4 . For this purpose, we use the character tables of a-characters. In the following, we quickly recall the character theory of projective representations. Let W be a finite group, P be an algebraically closed field, paw a twisted group ring. x E W is called a-regular if g x g-l = x for all 9 E Gw (x). We denote by WO the set of a-regular elements. The dimension of the center equals the number of a-regular classes. Hence, for P = C, the number of irreducible paW-modules equals the number of a-regular classes. Let VI, ... , Vh be irreducible CaW-modules, G l , ... , Gh be aregular classes. Xi (g) = tr(g, Vi)'s are called a-characters. Since ei = L:9EWO a(g, g-l )-lXi(g-l)g are mutually orthogonal idempotents, we have the orthogonal relations
lit?
h
L n(gdn(gjl) = Dija(gi, gil) I Gw(gd I k=l
(We fix gi E Gi for each i.) A consequence of Conlon's theorem ([4, 6.2]) is that we can choose a to be a class function cocycle of finite order. Actually it asserts stronger statement that we can choose a to be of finite order and standard in the sense that x-I = x-I for all x E W and g X g-l = gxg- l for all a-regular x and all 9 E W. We particularly have that the center of paw has a basis consisting of class sums of aregular classes, and thus a-characters are class functions. Namely a is a class function cocycle. If a is a class function cocycle of finite order, then the orthogonality relations are simplified as
S. Ariki
6 h
L n(gi)n(gj) = Oij I Cw(gill . k=l
The square matrix (Xj(gi)) for a class function cocycle of finite order is called a character table. We can read from the character table whether there is an a-regular central element or not, by using the orthogonality relations. (We remark that there are several choices of a to make a- characters class functions. For example, if t : W ----> 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,
Nous obtenons alors la decomposition orthogonale
(3.10) On a
Rr
0
(3.11)
*Rr
et 1
L MEL: L
fLdM,L) IWMFI R~
0
*R~, (3.12)
Formule des traces sur les corps finis
23
ou Il£ est la fonction de Mobius de l'ensemble
.c; en part iculier , (3.13)
et les projecteurs prGF,L
F
sont autoadjoints pour Ie produit scalaire
( , )GF. La fonction de Mobius Il£ a ete calculee par Fleischmann-Janiszczak (voir [FJ]) et Deriziotis-Rolt (voir [DR]).
Notation 3.14. Si L
.c, i
E
la fonction centrale sur Nous posons
LF
E {1, 2} et
egale
Ii
a (fi)L.
E
C(G F ), nOus noterons Ii,L
(3.15) (3.16) et
(3.17) Remarque 3.18. Puisque pr~:p est un projecteur et est autoadjoint, on
a
(h, h)GF,ell = (h, pr~:p(h))GF = (pr~:p(fd, h)GF.
Theoreme 3.19. On a
Demonstration. Il resulte de la decomposition (3.7) que les projecteurs F
pr G ' L
F
"fient ven
L
prGF,L
F
= IdC(GF).
LE[£]W GF Nous en deduisons que
JG
F
=
(h,
L LE£
L LE[£]w
F F pr G ' L (h))GF
GF IN wGF (L)I IWGFI
F F (fl,pr G ' L (h))GF.
(3.20)
A.-M. Aubert
24
Nous appliquons la formule (3.8) :
En utilisant la remarque 3.18, il s'ensuit
• Par definition de la fonction de Mobius, on a (3.21)
Remarque. On peut retrouver l'egalite (3.21) en appliquant la formule (3.13) et la remarque 3.18 :
4. Description du cote spectral de la formule des traces Soient M E.c et a E Irrcusp(MF). L'ensemble Irrcusp(MF) vajouer ici Ie meme role que l'ensemble {II 2 (M(F))} chez Arthur (cf. [Ar2, p. 83 et 92]). Nous notons NGF(M) Ie normalisateur de M dans G F . Si i = (M, a), nollS noterons WGF (i) Ie groupe de ramification de Met de a, i.e.,
Formule des traces sur les corps finis
25
Le groupe WGdi) est une extension d'un groupe de Coxeter (cf. [HL]), analogue au groupe W a defini en [Ar2, p. 85]. Lusztig (cf. [L2]) et Howlett et Lehrer (cf. [HL]) ont etudie l'algebre de Heeke
1tG(i): = EndGF (R~(a)) des operateurs d'entrelacement de la representation induite R~(a) et demontre que cette algebre est isomorphe a l'algebre de groupe CWGdi) de WGdi), tordue par un cocycle. Lusztig a demontre que Ie cocycle est trivial lorsque Ie centre de G est connexe (cf. [L3, 8.6]) ; Geck a recemment etendu Ie resultat de Lusztig a un groupe reductif connexe quelconque (cf. [G]). La situation est ici beaucoup plus simple que pour les groupes padiques et c'est Ie groupe WGdi) lui-meme qui va jouer Ie role du R-groupe R a et de son extension Ra (cf. [Ar2, p. 86 et 87]). Nous appellerons paires cuspidales dans G les paires i = (M,a) avec M E .c et a E Irrcusp(MF) et nous noterons Pcusp(G) l'ensemble des classes de G F -conjugaison des paires cuspidales dans G. Pour tout i = (M,a) E Pcusp(G), nous poserons l1 (G F ) := {X E Irr(G F ) I (X, R~(a))GF -I- O}. i
Les ensembles l1 i (G F ) avec i E P cusp (G) forment une partition de l'ensemble Irr(G F ) des caracteres irreductibles de G F ; ils jouent Ie role des ensembles l1 a (G(F)) d'Arthur (cf. [Ar2, p. 83 et prop. 1.1]). Fixons une paire cuspidale i = (M, a) de G. II existe une famille d'isometries ou L E .c(M), telles que (1) pour tout L
E.c
et tout i
E
L
G
G
RL
0
Ii = Ii
Pcusp(G), on a WGF(i)
0
IndWLF(i) ,
ou Ind:~:(W designe Ie foncteur d'induction ordinaire de WLdi) au groupe WGF(i). (2) La famille (If )LE.c est stable sous l'action par conjugaison de
WGF. (3) If'A envoie Ie caractere trivial du groupe trivial WMF(i) sur a. Pour simplifier les notations, nous poserons Xp := I iG (p),
pour tout P E Irr(WGdi)).
(4.1)
26
A.-M. Aubert
Definition 4.2. La fonction
Res:~;gi(
C(WGF(i)) est cuspidale si .c(M) tel que L f= G. E
Nous notons
• CcusP(WGF(i)) I'ensemble des fonctions de C(WGF(i)) qui sont cuspidaIes, WGF(i) I ' • prcusp e proJecteur sur Ie sous-espace C cusp (TXT yy GF (")) ~ . D'apres [Au2], on a
(4.3) Soit [.c(M)]WGP(i) un systeme de repn§sentants des classes de WGF(i)conjugaison dans .c(M). On a Ia decomposition orthogonale
C(WGF(i)) =
EB
Ind:~:(~i?(CcusP(WLF(i))),
(4.4)
LE[L:(M)]wGF(i)
et Ie projecteur sur Ie terme
Ind:~:gi(CcusP(WLdi)))est
Nous posons (4.6) LE[C(M)]W L,
GF
Remarque. Le sous-espace IiG(Cind(WGF(i))) de C(G F ) est I'analogue de I'espace Cind(IIa(G(F))) dMini par Arthur it Ia page 90 de [Ar2]. Nous obtenons Ia decomposition orthogonale
Si
Formule des traces sur les corps finis
27
On a d'autre part:
R
In d WaP(i) WLP(i)
0
GF LF
'
et pr i
WaP(i) eSWLP(i)
est egal it
1
'"
~ IL.c(M) NEL:L(M)
INwaP(i)(L)1
(N , L)
ITF (')1 n'NF Z
In dWaP(i) R WaP(i) WNP(i) 0 eSWNP(i)
ou IL.c(M) est la fonction de Mobius de l'ensemble .c(M). En particulier pr~f:(i) est egal it
'"
~
G) IN
IL.c(M)(N,
IWNF(i)1 W P(i) ' (N)I IndW~p(i)
0
WaP(i) ResWNP(i)
WaP(t)
NE[.c(M)]WaF(i)
et les projecteurs pr;F ,L
F
sont autoadjoints pour Ie produit scalaire
( , )WaP(i)'
Soient
._ ( WaP(i)(,,) . - prcusp '1'1,
WaP(i) ( )) prcusp
(4.8)
Proposition 4.9. Boit i une paire cuspidale dans G. On a GF
pr cusp
0
G
Ii
= IG i
0
WaP(i)
prcusp
.
Demonstration. Soit i = (M,a). Soit
G (prcusp
0
F IG)( WaP(i) (
=
on a pr~:p (If (
(* R LG
0
0
IL) i (
0,
= 0.
IiG ) (
= IL( i Res WaP(i) w L P(i) (
on a If(
pr~:p(IiG(
= (IiG
0
WaP(i))) pr cusp (
'
28
A.-M. Aubert
Nous en deduisons que
•
Corollaire 4.10. Soient i une paire cuspidale dans G et
Demonstration. Cela resulte clairement de la proposition 4.9.
•
Nous allons considerer les triplets T
:= (M, a, r),
avec M E £, a E Irrcusp(MF) et r E WGF(M, a).
Nous notons T(G) l'ensemble des classes de WGF-conjugaison de tels triplets.
Definition 4.11. Soit i une paire cuspidale. Nous dirons qu'un element r du groupe W GF (i) est elliptique si r n'est contenu dans aucun sous-groupe de WGF(i) de la forme WLF(i) avec L E £(M) et L -=I- G.
Nous notons W GF (i)ell l'ensemble des elements elliptiques de WGF(i). Remarque 4.12. Soit i = (M,a). Le groupe NG(M)/M opere sur Ie centre Z(M) de M. Un element r de NG(M)/M est contenu dans un groupe de la forme NL(M)/M avec L E £ et L -=I- G si et seulement si r a une valeur propre egale it 1. L'ensemble WGF(i)ell est done egal it l'ensemble des elements r de WGF(i) tels que det(l - r)Z(M) -=I- 0; autrement dit WGF(i)ell est l'analogue de l'ensemble ReT, reg defini par Arthur en [Ar2, (2.5)].
Le groupe W GF (i) est reunion disjointe de W GF (i)ell et de l'ensemble WGF(i)':=
U
WLF(i)
LeC(M) L;tG
(les groupes WLF(i) jouent Ie role des groupes R~ d'Arthur). NollS poserons Tell(G) := {T = (M,a,r) E T(G) IrE WGF(M,a)eU}'
(4.13)
29
Formule des traces sur les corps finis
Soit i = (M, a) une paire cuspidale. Rappelons que I;VGF(i) designe la fonction centrale sur WGdi) qui vaut ICWGF(i)(r) I sur ClwGF(i)(r) et o ailleurs. Si T = (M, a, r) E T(G), nous posons (4.14) Remarquons que la fonction i~F est cuspidale si et seulement si
T
E
Teu(G). Base de l'espace Ccusp(GF) : Proposition 4.15. Les fonctions i~ pour T parcourant Tell (G) forment une base orthonormee de l'espace Ccusp(GF). Demonstration. On utilise la proposition 4.9 et la definition 4.11.
•
Remarque. Nous decrirons une autre base orthogonale de cet espace en (5.7).
Theoreme 4.16. On a
L
(h, i~)GF (h, i~)GF ,
(4.17)
TET(G)
(4.18)
Demonstration. La premiere egalite vient du fait que les fonctions i~ pour T parcourant T(G) forment une base orthonormee de l'espace C(G F ) (puisque JiG est une isometrie). La seconde egalite resulte alors de la proposition 4.15. •
Le resultat suivant est l'analogue de l'expression (1*) de [Ar2].
Corollaire 4.19. Soient p E Irr(WGdi)) et p'
E
Irr(WGdi)).
On a alors
p(r) p'(r).
Demonstration. C'est une consequence du theoreme 4.16 et du fait que wG JG , .) . • ( Xp, I-G) T GF = (p,l r F(i)) WGF(i) (car i est une . lsometne
A.-M. Aubert
30
T
Comme Ie fait Arthur (cf. [Ar2, (2.3) et (3.1)]), posons, pour tout E T(G) et tout f E C(G F ),
8 G (T,1)
ou i
= (M,a)
p(r) xp(f),
:=
(4.20)
et T = (M,a,r). Soit r E WGF(i). Puisque
p(r) p, pEIrr
on a, pour tout
f
(w
GF
(i))
E C(G F ),
11 s'ensuit (par Ie theoreme 4.16) : (h,12)GF,el! =
L
r=(M,a,r)ETell(G)
IC
1
(r)1 8G(T, h) 8 G (T, h)·
WGP(M,a)
5. Series de Harish-Chandra fantomes La "philosophie de Harish-Chandra" consiste it definir une partition d'un ensemble d'objets irreductibles associes it un groupe reductif G en "series" definies par des "donnees cuspidales" formees d'un sousgroupe L de G et d'un objet irreductible "cuspidal" associe it L. Nous avons rencontre de telles series au §3 du present article. D'autres exemples existent en theorie des representations complexes des groupes reductifs finis (voir [BMM]), en theorie des representations modulaires des groupes reductifs finis (voir [H], [GHMl], [GHM2] et [Aul, l.B]), en theorie des representations complexes des groupes reductif p-adiques (voir [BZ]) ou encore en theorie des faisceaux-caracteres (voir [L7, §4], Lusztig utilise lit Ie terme "blocs" au lieu de series et la "donnee cuspidale" est un triplet (L, c, E), OU L est un sous-groupe de Levi d'un sous-groupe parabolique de G, c une L-orbite nilpotente dans l'algebre de Lie de L et E un systeme local L-equivariant irreductible cuspidal sur c). Soit comme precedemment G un groupe algebrique reductif connexe sur iFq , defini sur IF'q, muni d'un endomorphisme de Frobenius F. Nous supposons desormais que Ie centre de G est connexe. Nous prenons
31
Formule des traces sur les corps finis
maintenant comme objets irreductibles les "caracteres fantomes" (en anglais "almost characters") de G F . Les caracteres fantomes de G F ont ete introduits par Lusztig; ils forment une base de l'espace C(G F ), qui s'obtient it partir de la base des caracteres irreductibles en appliquant une matrice de transformation de Fourier non-abelienne ([L3, §4]). Les fonctions caracteristiques des faisceaux-caracteres definissent aussi une base de C(G F ), et Lusztig a conjecture que ces deux bases sont les memes, it multiplication par des scalaires pres. L'objet de cette partie est de definir une partition de l'ensemble des caracteres fantomes de G F en "series de Harish-Chandra" . Le premier pas dans cette direction consiste it definir la notion de "caractere fantome cuspidal". II y a en fait deux manieres de proceder : soit dire qu'un caractere fantome est cuspidal s'il est cuspidal comme fonction centrale sur G F (par la definition 3.4), soit dire qu'il est cuspidal si Ie faisceau-caractere qui lui correspond par la conjecture de Lusztig est cuspidal (notion introduite par Lusztig dans [L5]). II se trouve que ces definitions conduisent it des theories de Harish-Chandra differentes.... Nous allons dans un premier temps nous interesser it la deuxieme definition. Soit A(GF) l'ensemble des caracteres fantomes de G F . Nous dirons que L est un sous-groupe regulier de G si L est un sous-groupe de Levi F-stable d'un sous-groupe parabolique (non necessairement Fstable) de G. En nous inspirant du formalisme "generique" developpe dans [BMM, §1], nous noterons (f, WGFo) Ie groupe reductif fini generique associe it la donne radicielle f = (X, R, X', R V ) de groupe de Weyl W G et it l'endomorphisme Fo (d'ordre fini) de f G tel que F = qFo et (fL, WLwLFo) son sous-groupe de Levi generique associe it la donne radicielle fL = (X,RL,X',Rt), avec w E WG et tel que R't est un sous-systeme parabolique de R V qui est wLFo-stable (voir [BMM, p. 10-12]). Rappelons la definition de l'induction de Lusztig ([DL], [11]). Soit L un sous-groupe regulier de G et soit P un sous-groupe parabolique de G de sous-groupe de Levi L. Nous notons U Ie radical unipotent de P et nous considerons la variete algebrique YL,u = {g(UnFU) E G/(UnFU) I g-lF(g) E F(U)}. Le groupe fini G F x L F agit sur YL,u par (go, lo) : 9 ~ gOglol et l'on a une action induite de GF x L F sur H~(YL,U). Si E est un L F module, l'ensemble des points fixes par L F de H~(YL,U) @ E, note F (H~(YL,U) @ E)L , definit un G F -module et l'on pose
Rf,u(E) :=
L (_l)i (H~(YL,U) i
@
E)L
F
(representation virtuelle de G F ). Ceci s'etend par linearite en un homomorphisme Rf u de l'ensemble des representations virtuelles de L F dans l'ensemble des representations virtuelles de G F . On en deduit une application Rfu de C(L F ) dans C(GF). Nous notons *Rfu l'application adjoint~ de Rf,u' ' Si L est un sous-groupe de G et si 9 E G, nous posons 9L := gLg-l et L9 := g-1 Lg. Si L et L' sont deux sous-groupes reguliers de G, nous notons S(L, L') l'ensemble des elements 9 de G tels que L n 9L' contient un tore maximal de G et [LF\S(L, L')F /L'F] un systeme de representants de LF\S(L, L')F /L'F Nous faisons les hypotheses suivantes.
Hypotheses: (HI) La conjecture de Lusztig citee ci-dessus est vraie pour Ie groupe G. (H 2 ) Au niveau des fonctions centrales, l'induction des faisceauxcaracteres cuspidaux correspond, a multiplication par un scalaire pres, a l'induction de Lusztig Rf des caracteres fant6mes absolument cuspidaux. (H 3 ) Soient L et L' deux sous-groupes reguliers de G. On a dans l'espace C(L'F) des fonctions centrales sur L'F, pour tout choix de U et de U' : *Rf,u
0
Rf"u'
L
Rtn9L',un9u'
0
Ad(g)
0
*Rt:nL9 U'nU9'
9E[LF\S(L,L'V IL'F] Validite des hypotheses (HI)' (H 2 ) et (H 3 )
:
• Shoji a demontre l'hypothese (HI) dans Ie cas ou la caracteristique p est presque bonne (voir [L6, 1.12] pour un rappel de la definition) et ou Ie groupe G est a centre connexe (voir
[S]). • L'hypothese (H2) est vraie si p est presque bonne et, soit q est suffisamment grand (voir [L6, tho 9.2]), soit G est a centre connexe (voir [8]). • Deligne a demontre l'hypothese H3 (appelee formule de Mackey) dans Ie cas ou p et q sont suffisamment grands. Pour les fonctions centrales qui sont combinaisons lineaires de caracteres unipotents, la formule de Mackey est vraie sans condition sur q (voir [BMM, tho 1.35 (1)]).
Formule des traces sur les corps finis
33
L'hypothese H3 implique que les applications Rt,u et *Rt,u, au niveau des fonctions centrales, sont independantes de U (voir par exemple [DM2, prop. 6.1]); nous les noterons dorenavant simplement Rt et *Rt.
Definition 5.1. Nous dirons qu'une fonction centrale 'Ij; sur G F est absolument cuspidale si *Rt ('Ij;) = 0 pour tout sous-groupe regulier propre L de G. Nous notons Cacusp(GF) l'espace des fonctions centrales absolument cuspidales sur G F et Aacusp(GF) l'ensemble des caracteres fant6mes de G F qui appartiennent a Cacusp(GF). Remarque. Toute fonction centrale absolument cuspidale est cuspidale; mais il existe des fonctions centrales cuspidales qui ne sont pas absolument cuspidales : par exemple, soit G: = G L n et soit T Ie tore de Coxeter de G, la fonction R¥(1) est cuspidale, mais n'est pas absolument cuspidale.
Si un faisceau-caractere est cuspidal (definition de [L5, (3.10)]), alors sa fonction caracteristique est une fonction centrale absolument cuspidale (d'apres [L5, tho 6.9(bJ]). Les fonctions caracteristiques des faisceaux-caracteres cuspidaux sur G F forment une base (orthogonale) de l'espace Cacusp(GF) : ce resultat a ete demontre par Lusztig dans Ie cas ou Ie groupe G est simplement connexe, presque simple et q suffisamment grand (voir [L6, 9.5] et [L8, prop. 2.2]); Geck et HiB en donnent une preuve differente avec des hypotheses legerement plus faibles en [GH, prop. 5.4 (b)]. Nous appelons "donnees absolument cuspidales pour Ie groupe G" les paires 8 = (L, 'Ij;) formees d'un sous-groupe regulier L de G et d'un caractere fant6me absolument cuspidal 'Ij; de L F . Nous noterons Dacusp(G) l'ensemble des classes de GF -conjugaison de donnees absolument cuspidales pour G. Nous posons (5.2) Des resultats voisins de ceux qui suivent ont ete recemment demontres par Geck et HiB (voir [GH, prop. 5.4 et 5.7]).
Proposition 5.3. Soient 8 = (L, 'Ij;) et 8' = (L',1//) deux donnees absolument cuspidales pour G F . Alors (Rt('Ij;), Rt,('Ij;'))GF est egal a IWG F(8)1, si 8 et 8' sont G F conjugues et est egal a0 sinon.
A.-M. Aubert
34
Demonstration. Soit co,o' := (Rf('l/J), Rf,('l/J'))cF. La formule de Mackey ci-dessus montre que
L L
CO,O'
('l/J,Rtn9L' o Ad(g)
0
*Rt:nL9('l/J'))LF
9E[LF\S(L,L'V /L'F]
('l/J, R~L'
0
Ad(g) (1{i'))LF ,
9E(LF\S(L,U)F /L'FJ 9L'CL
car 'l/J' E Aacusp(L'F). Comme 'l/J E Aacusp(LF), nous en deduisons que Co,o' = 9E[LF\S(L,L'jF /L,FJ 9L'=L
Pour que Ie terme co,o' soit non nul il faut donc qu'il existe 9 E G F tel que 9L' = L et Ad(g)('l/J') = 'l/J,
a cause de l'orthogonalite des fonctions caracteristiques des faisceauxcaracteres (voir [L5, (25.7)]). Si CO,O' L = L' et que 'l/J = 1{i'. On a Co,o
=
-I 0, nous pouvons supposer que
ILFI- 1 !{ 9 E G F I 9L = L et ad(g)('l/J) = 'l/J}I ILF!-l INGd L, 'l/J)I·
• La definition suivante generalise celle des tores elliptiques. Definition 5.4. Nous dirons qu 'un sous-groupe regulier de G est elliptique s'il n'est contenu dans aucun sous-groupe de Levi F -stable d'un sous-groupe parabolique F-stable propre de G.
Nous posons Dacusp,ell(G) := {8 = (L,'l/J) E Dacusp(G)!
Lest elliptique}. (5.5)
Proposition 5.6. Soit 8 = (L, 'l/J) E Dacusp(G). On a
• Rf('l/J) E Ccusp(GF) si et seulement si 8 E Dacusp ,ell(G),. • Rf('l/J) E Cind(G F ) si et seulement si 8 rJ. Dacusp,ell(G).
35
Formuie des traces sur ies corps finis Demonstration.
(1) Supposons que Lest elliptique. Soit N un sous-groupe de Levi F-stable d'un sous-groupe parabolique F-stable de G. Par application de la for mule de Mackey, on a
(R~L
0
Ad(g)) (1J;),
gEINF\S(N.L)F jLFJ LCN9
puisque 1J; E Cacusp(LF). Mais, comme Lest elliptique, si L C N9 alors N = G et la fonction Rt (1J;) est donc bien cuspidale. (2) Supposons maintenant que L n'est pas elliptique. 11 existe donc un sous-groupe de Levi F-stable N d'un sous-groupe parabolique F-stable de G tel que N i- G et LeN. Choisissons un tel sous-groupe N minimal. Alors L est un sous-groupe de Levi elliptique de N et R~ (1J;) est une fonction cuspidale sur M F (d'apres (1)). Comme Rt(1J;) = R~ (R~(1J;)), la decomposition (3.10) implique que la fonction Rt(1J;) n'est pas cuspidale. • (5.7) Base de l'espace Ccusp(GF) : Les fonctions Rt(1J;) pour 8 = (L,1J;) parcourant Dacusp,ell(G) forment une base orthogonale de l'espace Ccusp(GF). Rappels sur les fonctions de F --classes : Si H est un groupe fini muni de l'action d'un groupe (F) cyclique fini, engendre par un element F, on dit que deux elements hI et h2 de H sont F -conjugues si les elements hI F et h2 F de H)
(h,h)H.F :=
1", THT L..J
-
h(hF)h(hF).
(5.8)
hEH
Si K est un sous-groupe de H stable par F, nous noterons Res~:~ la restriction des fonctions de F -classe et nous definissons une induction
36
A.-M. Aubert
des fonctions de F -classe par
H.F) (hF) (IndK.Ff
:=
1 -IHI
I f(xhF(x- )).
"" L..;
(5.9)
xEH x(hF)x-1EK·F
ana
(5.10) Soit L = (fL, WLwLFo) un sous-groupe regulier de G. Nous posons WG(L) := NG(L)/L. Pour tout element W de WG(L), nous choisissons un representant n de W dans NG(L) et un element z de G tels que I I Z-I F(z) = n- , et nous posons L w := zLz- . Le groupe L w est aussi un sous-groupe regulier de G et l'on a L w = (fL, WLwwLFo). Remarque. Si Ie groupe G n'est pas deploye, un sous-groupe regulier de G n'est pas toujours conjugue dans G a un sous-groupe de Levi F-stable d'un sous-groupe parabolique F-stable de G. On dit que deux elements W et w'de WG(L) sont wLF-conjugues s'il existe x E WG(L) tel que W = X-IW(wLF)(x). Le nombre de classes de GF -conjugaison de sous-groupes reguliers de G conjugues dans GaL est egal au nombre de classes de wLF-conjugaison du groupe WG(L), i.e., a IWG(L)WLFI. Soit 8 = (L, 'lj;) une donnee absolument cuspidale pour G. Nous notons 'lj;w Ie caractere fantome absolument cuspidal de L{:; qui correspond a 'lj;. Nous posons 8w := (L w , 'lj;W). Soit WG(8) := NG(L, 'lj;)/L. Si
L
On deduit immediatement de (5.11) que Rtw('lj;W) =
L
ep(wwLFo) R~,
(5.12)
WLFo
OU ep est une extension de 'P au groupe engendre par WG(8) et par wLF. La formule de decomposition de l'induit d'un faisceau-caractere [L5, (10.4.45)] et l'hypothese (H 2 ) montrent que les R~ sont (a multiplication par un scalaire pres) egaux aux fonctions caracteristiques des faisceaux-caracteres. L'hypothese (HI) montre qu'ils sont egaux (a multiplication par un scalaire pres) aux caracteres fantomes de G F . Nous les identifions desormais aux caracteres fantomes de G F . Soit {8} E Dacusp(G). Nous noterons A{8}(G F ) l'ensemble des X E A(G F ) tels que (X, Rtj'lj;W))GF #- 0 pour un w E WG (8).
37
Formuie des traces sur ies corps finis
Theoreme 5.13. Les ensembles A{8}(G F ), avec {8} E Dacusp(G), forment une partition de A(G F ) et l'on a
(5.14) Demonstration. Cela resulte clairement de la proposition 5.3 et de
(5.12).
• 6. Application au produit scalaire elliptique
La proposition 5.3 montre que les fonctions Rt('l/J), oil (L, 'l/J) parcourt un systeme de representants de Dacusp(G), forment une base orthogonale de C(GF). On a donc, en ecrivant it et 12 dans cette base et en utilisant la proposition 5.3 :
(it, 12)GF
L
IWG F(8)1- 1
(it, Rt('l/J))GF (h,Rt('l/J))GF .(6.1)
{8=(L,,p)}EDacusp(G)
Le resultat suivant decrit l'analogue sur les corps finis du produit scalaire elliptique d'Arthur defini en [Ar2, (6.5)]. Theoreme 6.2. On a
L
(I1,L, 'l/J)LF (12,L, 'l/J)LF,
(6.3)
,pEAacusp (LF)
ou L parcourt les classes de G F -conjugaison de sous-groupes reguliers elliptiques de G ,. JG
F
=
L
IWG F(L)I- 1
{L}
L
(I1,L, 'l/J)LF (12,L, 'l/J)LF,
(6.4)
,pEAacusp (LF)
ou L parcourt les classes de G F -conjugaison de sous-groupes reguliers de G. Demonstration. Par definition d u produit scalaire elliptique ( , ) G F ,ell
(voir 3.16) et en utilisant la proposition 5.6, on obtient
L {8=(L,,p) }ED:~lusP (G)
IWGF (8)1- 1 (it, Rt('l/J))GF (12, Rt('l/J))GF .
A.-M. Aubert
38
Grace it la proposition 5.3, on en deduit que
(h,h)GF,ell =
" L.J
INGF(8)1 ILFI G IGFI IN F(8'II(h,Rt(1/'J))GF(h, Rd1/'J))GF
c5={ (L,,p))
L {c5=(L,,p)}
G
i~:',
,
(h,Rt(1/'J))GF (h,Rt(1/'J))GF,
ou l'on somme sur les donnees absolument cuspidales pour G. II s'ensuit que
(h,h)GF,eIl
=
L
L
IWGF(L)I- 1
{L}
(/J,Rt(1/'J))GF (h,Rt(1/'J))GF,
,pEAacu,p (LF)
ou L parcourt les classes de G F ~conjugaison de sous-groupes reguliers elliptiques de G et avec la notation 3.14. Un raisonnement analogue conduit it la seconde egalite du theoreme.
•
Nous noterons Unif(G F ) l'espace des fonctions uniformes sur G F , i.e., Ie sous-espace de C(G F ) engendre par les caracteres de DeligneLusztig R¥((}). Soit pr~~f Ie projecteur sur ce sous-espace. Si h E C(G F ) et h E C(G F ), nous posons F F GF = (h,h)unif:= (prUnif G (h), prUnif(h G ) )GF, (6.5) J Unif et GF
Jell,unif = (h, h)ell,unif .- ((pr~~f
0
pr~:p) (h), (pr~~f
0
pr~:p) (h))GF.
Remarque. Pour tout f E C(G F ), on a F GF 0 prcusp GF ) (f ) = (G (prUnif prcusp
0
(6.6)
GF )(f) . prUnif
Nous allons definir un "cote fantome" de la formule des traces par analogie avec Ie cote spectral. Soient h E C(G F ) et h E C(G F ). En ecrivant h et h dans la base de C(G F ) formee des caracteres fantomes nous obtenons
(6.7) v
cote fantome
39
Formule des traces sur les corps finis
Proposition 6.8.
On a F
Jell,unif G
=
~ IWGF(T)
""'
1-1
{T} =
~ ""'
(h,T,'Ij; )TF (h,T,'Ij; )TF
1bEA(TF)
L
(6.9)
1
IWGF(T)I- (h,T, hThF,
{T}
ou T parcourt les classes de G F -conjugaison de tores maximaux Fstables elliptiques de G ,.
J~~ =
L
=
L
L
IWGF(T)!-l
{T}
(fl.T,'Ij;hF (h,T,'Ij;hF
1bEA(TF)
(6.10)
1
IWGF(T)I- (h,T, h,ThF,
{T}
ou T parcourt les classes de G F -conjugaison de tores maximaux Fstables de G. Demonstration. La preuve est analogue a celle du theoreme 6.2, en considerant seulement les sous-groupes reguliers de G qui sont des tores et en utilisant (6.7). •
Si g E G, nous noterons g = gsgu sa decomposition de Jordan en partie semi-simple et partie unipotente, et C~ (g) la composante neutre de son centralisateur dans G. Nous appellerons p-constantes les fonctions f de C(G F ) qui verifient f(g) = f(gs) pour tout g E G F . Si h et h sont des fonctions p-constantes, les elements unipotents "ne comptent pas" et l'on trouve une formule identique au produit scalaire elliptique defini par Arthur dans Ie cas des groupes p-adiques (cf. [Ar2, (6.5)]. Corollaire 6.11. Si h C(G F ), on a (h, h)GF,ell
=
L
et h
sont des fonctions p-constantes de
1
IWGF(T)I- (ITFI-
{T}
1
L
h(t) h(t)),
(6.12)
tETF
ou T parcourt les classes de G F -conjugaison de tores maximaux Fstables elliptiques de G ,. (h, h)GF
=L {T}
1
IWGF(T)I- (ITFI-
1
L tETF
h(t) h(t)),
(6.13)
40
A.-M. Aubert
ou T parcourt les classes de G F -conjugaison de tores maximaux Fstables de G. Demonstration. Les fonctions p-constantes etant uniformes (cela se deduit facilement de [DM2, prop. 12.6 et 12.13]), on a (fI, 12)GF,ell (fl' h)ell,unif. Mais, d'apres (6.7), on a
L
(fl,T,'Ij;h F (f2,T,'Ij;h F
1bEA(TF)
II suffit alors d'utiliser que, si fest p-constante, * R5f (f) = Res~; (f) (voir [DM2, prop. 12.6]). • Remarque. L'egalite (6.13) s'ecrit
IGFI (fl, h)GF =
L (L
h(t) h(t)),
{T} tETF
ou T parcourt les tores maximaux F-stables de G. Lorsque fI = 12, on retrouve un resultat de Kawanaka (voir [e, prop. 7.6.8] ou [Le, cor. (1.16)]). Pour des fonctions centrales quelconques, on doit tenir compte des elements unipotents. Boit Q¥ la fonction de Green definie par Deligne et Lusztig dans [DL] par Q¥(u) := R¥(u) pour un element unipotent u de G F . On obtient l'expression suivante du produit scalaire elliptique uniforme. Theoreme 6.14. On a
til EC2;Ct)unip U2
Ec2; (t)unip
ou T parcourt les classes de G F -conjugaison de tores maximaux Fstables elliptiques de G. Demonstration. On utilise (6.7) et la for mule du caractere
(*R~ f) (t)
=
I~r~~),
L uEC'b (t )unip
Q~'b(t)(u) f(tu) .•
41
Formuie des traces sur ies corps finis
Nous allons considerer maintenant Ie cas du produit scalaire elliptique (non necessairement uniforme) de deux fonctions centrales quelconques. Si u est un element unipotent de G F et v un element unipotent de L F , nous posons
(rf
est independant du radical unipotent U du sous-groupe parabolique de sous-groupe de Levi L choisi) et, si (L, 'lj;) est une paire absolument cuspidale, ( ) . - IC~L(gs)FI-I Q C~(9S) C~d9s),1/IX u .-
On a alors
En utilisant la premiere expression de (h, 12)GF ,ell dans la demonstration du theoreme 6.2, on obtient immediatement Ie resultat suivant : Theoreme 6.15. On a
L
IG F I- 2 IL F I- 2
II (g) 12(g)
1/IEAacusp(LF)
XIEC F
x2 ECF
Xl1g1,sXlELF
x;:192,sX2ELF
ou L parcourt ies classes de G F -conjugaison de sous-groupes reguiiers elliptiques de G.
A.-M. Aubert
42
7. Formules des traces en miroir : Cas deploye
Dans cette partie, nous supposons que Ie groupe G est deploye et nous nous restreignons au cas des caracteres fant6mes unipotents. Nous allons utiliser Ie parametrage en series A<5(G F ) des caracteres fant6mes de G F defini a la partie 5. Soit 8 E Dacusp(G). D'apres [L6, 9.3. (d)], il existe un representant (L, 'lj;) de 8 tel que Lest sous-groupe de Levi d'un sous-groupe parabolique F-stable de G. Nous supposons que 'lj; est unipotent. Lusztig a demontre que Ie groupe WG(L) est un groupe de Coxeter (voir [L4, 9.2 (b)]). Geck et RiB ont prouve l'egalite WG(L,'lj;) = WG(L), lorsque 'lj; est unipotent (voir [GR, prop. 5.7]). Les caracteres fant6mes appartenant a la serie associee a (L, 'lj;) sont done en bijection avec les caracteres irreductibles du groupe WGF(L). On a, pour tout wE WGF(L), Rrw('lj;W)
L
=
'P(w) R~,
'PElrr(WcF(L))
ou
R~ = IWG~(L)I
L
'P(w) Rrw ('lj;W).
wEWcF(L)
Si M est un sous-groupe regulier de G qui contient L, on a
R~(R~M)
=
R
G WcF(L)
IndwMF(L)('PM)
et *R~(R~)
=
R
M W
F(L)
,
ResW~F(L)('P)
(7.1) ou 'PM E Irr(WMF(L)) et 'P E Irr(WGF(L)). Pour tout sous-groupe regulier M de G qui contient L, on definit une isometrie
Ir: Zlrr(WMF(L))
-----+
ZA<5(M)
par Ir('P) := R~. D'apres (7.1), on a
De maniere parallele a la definition 4.2, nous dirons qu'une fonction ¢ E C(WGF(L)) est "cuspidale" si Res~~:i~)(¢) = 0 pour tout sousgroupe regulier M de G tel que M :) L et M oF G et nous noterons pr~;p(L) Ie projecteur sur Ie sous-espace des fonctions cuspidales de C(WGF(L)). Vne preuve analogue a celIe de la proposition 4.9 (grace a (7.1)) montre que F G I <5G = I <5G 0 P rcusp Wc(L) ' (7.2) prcusp 0
43
Formule des traces sur les corps finis
Nous allons considerer les triplets v := (L, 'Ij;, w), avec L sous-groupe regulier de G, 'Ij; caractere fantome unipotent absolument cuspidal de L F et w E WGF(L). Nous notons Fant(G) l'ensemble des classes de W GF-conjugaison de tels triplets. De maniere parallele it la definition 4.11, nous dirons qu'un element w du groupe W GF (L) est "elliptique" si w n'est contenu dans aucun sousgroupe de WGF(L) de la forme WMF(L) avec M sous-groupe regulier propre de G tel que M:) L. Nous poserons Fantell(G) := {v = (L,'Ij;,w) E Fant(G) Si v
I west elliptique.}
= (L, 'Ij;, w) E Fant(G), nous posons
La fonction i~t est cuspidale si et seulement si v E Fantell(G). Soit Unip(GF) Ie sous-espace de C(GF) engendre par les caracteres fantomes unipotents. Proposition 7.3. Les fonctions i~ pour v parcourant Fantell(G) forment une base orthonormee de l'espace Ccusp(GF) n Unip(GF).
Demonstration. On utilise (7.2) et la definition d'element elliptique de WGF(L).
•
Theoreme 7.4. Si!I E Unip(G F ) et hE Unip(G F ), on a
Jr:.~
=
(!I, h)GF
=
L
(!I, i~)GF (12, i~)GF ,
(7.5)
lIEFant(G)
lIEFantell (G)
Demonstration. La premiere egalite vient du fait que les fonctions i~ pour v parcourant Fant(G) forment une base orthonormee de l'espace C(G F ) (puisque If est une isometrie). La seconde egalite resulte alors de la proposition 7.3. • 8. Classes de conjugaison anisotropes
Nous suivons la termimologie de Waldspurger (voir [W], voir aussi [GM]).
44
A.-M. Aubert
Definition 8.1. Nous dirons qu 'un element unipotent u de G F est anisotrope si tout tore deploye de e~(u) est contenu dans le centre de
G. Exemples d'elements unipotents anisotropes. On dit qu'un element unipotent u de G est "distingue" si e~(u) ne contient pas de tore non central (cf. [e, def. p. 173]), ou, ce qui est equivalent, si u n'est contenu dans aucun sous-groupe de Levi propre de G. Les elements unipotents distingues de G F sont donc anisotropes. En particulier, les elements unipotents reguliers de GF sont anisotropes.
Soit 9 un element de GF. Nous notons £(g) l'ensemble des elements de £ qui contiennent e~(g). Soit £(g)min l'ensemble des elements minimaux de £(g), pour la relation d'inclusion. Proposition 8.2.
(a) Si M E £(g)min, la composante neutre Z~ du centre de M contient un tore deploye maximal de e~ (g). (b) Deux elements de £(g)min sont conjugues sous e~(g)F. Demonstration.
(a) Soit M E £(g)min' Soit S un tore maximal deploye de Z~. On aM = eG(S). Supposons S contenu dans un tore deploye T de e~(g). Alors L := eG(T) est un sous-groupe de Levi rationnel d'un sous-groupe parabolique rationnel de G et L contient g. Mais, d'apres [BT, (4.16)(b)], comme L est Ie sous-groupe de Levi rationnel d'un sous-groupe parabolique rationnel de G, on a MeL. Par minimalite de M, on a M = L et donc T = S. (b) L'application T ~ eG(T) definit une bijection de l'ensemble des tores maximaux rationnels deployes de e~(g) sur £(g)min' Mais les tores maximaux rationnels deployes de e~ (g) sont conjugues sous e~(g)F, donc aussi les elements de £(g)min . • Definition 8.3. Nous dirons qu 'un element semi-simple s de G F est anisotrope si eG(s) n'est contenu dans aucun sous-groupe de Levi Fstable d'un sous-groupe parabolique F-stable de G.
Nous dirons que 9 E G est anisotrope si g8 et
gu
sont anisotropes.
Soit 9 E G F . Pour tout ME £(g)min, l'element 9 est contenu dans M et est anisotrope dans M. Soit C la classe de conjugaison de 9 dans G F . Nous posons £(C) .- {xMx- 1 I x E G F , ME £(g)}.
Formule des traces sur les corps finis
45
Soit .c(C)min l'ensemble des elements minimaux de .c(C). D'apres la proposition 8.2 (b), les elements de de .c(C)min sont conjugues sous G F . On definit une application de l'ensemble Cl(G F ) des classes de G F -conjugaison d'elements de G F dans l'ensemble des classe de G F -conjugaison de sous-groupes de Levi rationnels de sous-groupes paraboliques rationnels de G F en associant .c(C)min aCE Cl(G F ). Lemme 8.4. Soit Q = MV une decomposition de Levi F-stable d'un sous-groupe parabolique F -stable de G et soit CM une classe de conjugaison dans M F. Alors
ou m s est la partie semi-simple d'un element m de CM et ou l'on somme sur les classes de conjugaison C dans G F teUes que 1m (CG(m s ) n VF)nCI#o. Demonstration. Pour toute classe de conjugaison C dans G F
:
Soit m un element de CM. Nous notons m s sa partie semi-simple. L'element m appartient a C~ (ms ) (cf. par exemple [DM2, prop. 2.5]). Comme C~ (ms ) est un sous-groupe reductif connexe de G de rang maximal (cf. [DM2, prop. 2.3 (ii)]) , Ie groupe C~(ms) n Q est un so us-groupe parabolique de C~(ms), de decomposition de Levi (C~(ms)nM) (C~(ms)nV). On a C~(ms)nM = C~(ms), et C~(ms) est donc un sous-groupe de Levi de C~(ms). En utilisant [DM2, prop. 7.1], on voit que F
G (Ie ) (mu).
Si C
=
ClGF (g) avec 9 E G F , nous obtenons
Comme les fonctions I~r forment une base orthogonale de l'espace des fonctions centrales sur G F , toute fonction centrale 'Ij; s'ecrit
46
A.-M. Aubert
•
et Ie resultat s'en deduit. Soient C(GF)unip l'espace des fonctions centrales sur G F
a support
unipotent et C(GF)unip,anis l'espace engendre par les fonctions 19t:F(u) avec u E G F unipotent anisotrope. Waldspurger a montre que si G est un groupe classique on a la decomposition suivante : C(GF)unip =
EB
R~(C(MF)uniP,aniJ,
{M}wc
ou M decrit un systeme de representants des classes de W G conjugaison dans £ (cf. [W, lem. 5.11]). Nous etendons ici ce resultat a l'espace C(G F ) tout entier et a G reductif connexe quelconque. Si C et C' sont deux classes de conjugaison dans G, nous ecrirons C :::; c' si C C C'. Ceci definit un ordre partiel sur l'ensemble des classes de conjugaison de G. Si Q = MV est un decomposition de Levi d'un sous-groupe parabolique de G, si u est un element unipotent de M et u' un element unipotent de G, alors la variete uV n ClG(u') est vide sauf si ClG(u):::; ClG(u') (cf. [DLM, prop. 5.8] et [R]). Theoreme 8.5. On a la decomposition suivante C(G
F
)
=
EB
R~(C(MF)anis)'
{M}wc
ou M decrit un systeme de representants des classes de W Gconjugaison dans £. Demonstration. Le principe de la preuve est Ie meme que dans [W, lem. 5.11]. 11 suffit de montrer que la matrice exprimant les fonetions R~(l~~), pour C M classe anisotrope dans M F dans la base 19F de C(G F ) (ou C decrit les classes G F -conjugaison) est triangulaire pour un ordre convenable, avec des coefficients non nuls sur la diagonale. Nous savons calculer precisement cette matrice (a l'aide du lemme 8.4). Si m EMF, on a
R~(l~:F(m)) =
L
ICG(ms)nVFI-
1
F 1m (CG(ms)nV ) n
CI 19F,
C
ou m s est la partie semi-simple de m et ou l'on somme sur les classes de conjugaison C dans G F telles que 1m (CG(ms) n V F ) n CI -=I- O.
47
Formule des traces sur les corps finis
Analysons la variete m (GG(m s ) n VF)nG. Soient x E G F et 9 E G F tels que x-1gx Em (GG(m s ) n V F ). 11 existe alors v dans GG(m s ) n V F tel que x-1gx = mv. On a (x-1gx)s = x-1gsx, (x-1gx)u = x-1gux, (mv)s = m s et (mv)u = muv (par unicite de la decomposition de Jordan et car v et m s commutent). II en resulte que x-1gux = muvet x-1gsx = ms. En particulier, pour que la variete m (GG(m s ) n V F ) n GlGF (g) soit vide, il faut que les parties semi-simples respectives de 9 et de m soient GF--conjuguees et que la variete m u VFnGlGF(gu) soit non vide. Pour que la variete m uV n GlGF (gu) soit non vide, il faut que GlG(v) ::; GlGF(gu). La matrice en question est donc triangulaire par blocs, les blocs etant indexes par les classes de G-conjugaison et les lignes et colonnes de chaque bloc diagonal etant indexes par les classes de GF -conjugaison en lesquelles se scinde une classe de G-conjugaison donnee. Nous allons montrer que les blocs diagonaux sont en fait de taille un. Supposons que les elements unipotents m u et gu sont Gconjugues et que la variete m (GG(m s ) n V F ) n GlGF(g) est non vide. D'apres ce qui precede, les elements semi-simples m s et gs sont alors GF_conjuguees et la variet€ m u V F n GlGF(gu) est non vide. Soit y E m u V F n GlGF(gu). L'element m u est donc G-conjugue a y. Comme y E m uV, l'element m u de M F est V F --conjugue a y (cf. [DLM, lem. (5.12)]); par consequent m u et gu sont GF -conjugues. Les blocs diagonaux sont de taille un et non nuls, donc inversibles.
•
Corollaire 8.6. La dimension du sous-espace des fonctions uniformes de C(GF)unip,anis est egale au nombre de classes de conjugaison de tores elliptiques maximaux de G. Demonstration. La preuve est identique
cor. 5.12].
a celle
de Waldspurger [W, •
REFERENCES [Arl] [Ar2] [Au1]
[Au2]
J. Arthur, A local trace formula, Publ. I.H.E.S. 73 (1991), 1-96. J. Arthur, On elliptic tempered characters, Acta Math. 171 (1993),73-138. A.-M. Aubert, Series de Harish-Chandra de Modules et Correspondance de Howe Modulaire, J. of Algebra 165 No.3 (1994), 576-601. A.-M. Aubert, Systemes de Mackey, Rapport de Recherches du LMENS 93-15 (1993).
48
[BZ]
[BMM]
[C] [DL] [DH]
[DLM]
[DM2] [FJ]
[G] [GB]
[GHM1] [GHM2]
[GM] [B]
[HL]
[K]
[Le]
A.-M. Aubert
J.N. Bernstein et A. Zelevinski, Induced representations of reductive p-adic groups I, Ann. Sci. Ec. Norm. Super. 10 (1977),441-472. M. Broue, G. Malle et J. Michel, Generic Blocks of Finite Reductive Groups, Asterisque 212 (1993), 7-92. R. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley-Interscience, 1985. P. Deligne et G. Lusztig, Representations of reductive groups over finite fields, Ann. of Maths 103 (1976), 103-16l. D.1. Deriziotis et D.F. Holt, The Mobius function of the lattice of closed subsystems of a root system, Comm. in Algebra 21(5) (1993), 1543-1570. F. Digne, G.1. Lehrer et J. Michel, The characters of the group of rational points of a reductive group with nonconnected centre, J. reine angew. Math. 425 (1992), 155192. F. Digne et J. Michel, Representations of finite groups of Lie type, Cambridge University Press, Cambridge, 1990. P. Fleischmann et I. Janiszczak, The Number of Regular Semisimple Elements for Chevalley Groups of Classical Type, J. of Algebra 155 (1993), 482-528. M. Geck, A note on Harish-Chandra induction, manuscripta math. 80 (1993), 393-40l. M. Geck et G. Hiss, Modular Representations of Finite Groups of Lie type in Non-defining Characteristic, ce volume. M. Geck, G. Hiss et G. Malle, Cuspidal unipotent Brauer characters, J. of Algebra 168 (1994), 182-220. M. Geck, G. Hiss et G. Malle, Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type, Math. Z. 221 (1996), 353-386. M. Geck et G. Malle, Cuspidal unipotent classes and cuspidal Brauer characters, J. London Math. Soc. it paraitre. G. Hiss, Harish-Chandra series of Brauer characters in a finite group with a split BN -pair, J. London Math. Soc. 48 (1993), 219-228. R.B. Howlett et G.1. Lehrer, Induced cuspidal representations and generalized Hecke rings, Invent. math. 58 (1980), 37-64. D. Kazhdan, Cuspidal geometry ofp-adic groups, J. Analyse Math. 47 (1986), 1-36. G.1. Lehrer, Rational tori, semi-simple orbits, and the topol-
Formule des traces sur les corps finis
[L1] [L2] [L3] [L4] [L5]
[L6] [L 7] [L8] [LS] [R] [S]
[V]
[W]
49
ogy of hyperplanes complements, Comment. Math. Helvetici 67 (1992), 226 - 2516. G. Lusztig, On the finiteness of the number of unipotent classes, Inventiones math. 34 (1976). G. Lusztig, Representations of finite Chevalley groups, Am. Math. Soc. CBMS 39 (1977). G. Lusztig, Characters of reductive groups over a finite field, Ann. Math. Stud., Princeton, 1984. G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. math. 75 (1984), 205-272. G. Lusztig, Character Sheaves, Adv. in Math., 56 (1985), 193-237; 57 (1985), 226-265; 57 (1985), 266-315; 59 (1986),1-63; 61 (1986), 103-155.. G. Lusztig, Green functions and character sheaves, Annals of Math. 131 (1990), 355-408. G. Lusztig, A unipotent support for irreducible representations, Adv. in Math. 94 (1992), 139-179. G. Lusztig, Remarks on computing irreducible characters, J. of the Amer. Math. Soc. 5 (1992),971-986. G. Lusztig et N. Spaltenstein, Induced unipotent classes, J. of London Math. Soc. (2) 19 (1979), 41-52. R.W. Richardson, Conjugacy classes in parabolic subgroups, Bull. Land. Math. Soc. 6 (1974), 21-24. T. Shoji, Character sheaves and almost characters of reductive groups I, II, Adv. in Math 111, No 2 (1995), 244-313, 314-354. M.F. Vigneras, An elementary introduction to the local trace formula of J. Arthur. The case of finite groups, Jubilaum band DMV (1991), B.G. Teubner Stuttgart. J.-L. Waldspurger, Quelques questions sur les integrales orbitales et les algebres de Heeke, Bull. Soc. Math. France 124 (1996), 1-34.
Ecole Normale Superieure Departement de Mathematiques et d'Informatique 45 rue d'Ulm, F-75005 Paris, France
[email protected] Received January 1995
Heights of Spin Characters in Characteristic 2 Christine Bessenrodt1 and 10m B. Olsson 1 Abstract Based on our earlier description of the distribution into 2-blocks of the spin characters of the covering groups of symmetric groups we compute the heights of such characters in the blocks containing them. We also give a complete set of labels for the spin characters of minimal height in a 2-block. Another related topic treated here is the determination of the minimal power of 2 dividing a spin character degree and the explicit description of the labels of spin characters with this minimal power of 2 in their degree. Also. an upper bound for the heights of spin characters in 2-blocks is derived, and the labels of spin characters attaining this bound are described. As an application of our results we show that the 2-blocks of the covering groups of symmetric groups provide further evidence for some important representation theoretical conjectures. 1. Introduction and Preliminaries
It was proved by Schur [11] in 1911 that the finite symmetric groups Sn have covering groups Sn of order 21Snl = 2· n!. This means that there is a non-split exact sequence 1 ----> (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":) <
Since
9, and thus
rn+ ~1I":1/21 = [n + (III":~ + 3)/2] ,
the assertion follows. Corollary 2.3 Let n E IN, n > 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 <:;; B be the 2-block of Sn contained in B, and let w be its weight. It is known that k(B) = k(2, w) :s: 18(B)1 (see [9]). Hence by the theorem above we obtain k(B) = k(B)
+ p(w) + p(w) :s: 2k(2,w) :s: 218(B)1
= 18(B)I·
With the same notations as above, we know by Theorem 3.3 that the irreducible spin characters in B are of height 2:: [2tu-;(tu)] , and that there always is an irreducible spin character of exactly this height in B. Hence there are irreducible spin characters of height 0 in B if and only if [2tu-;(tu)] = 0, i.e. exactly if w :s: 1. This immediately implies Corollary 4.6 Brauer's Height 0 Conjecture holds for all 2-blocks of Sn, for all n E IN. Corollary 4.7 Olsson's Conjecture holds for all 2-blocks of Sn, for all n E IN. Recently, Robinson [10] has put forward a new conjecture on the height of an irreducible character: Conjecture 4.8 (Robinson) Let B be a p-block of the finite group G, D = 8(B) its defect group and X an irreducible character in B. If D is non-abelian, then h(X) < logplD : Z(D)I· We provide further evidence for this conjecture by proving Theorem 4.9 Robinson's Conjecture holds for all p-blocks of Sn, for all primes p and all n E IN. Proof. Let B be a p-block of Sn of weight w, and let D be its defect group. If p is an odd prime, then the height of an irreducible character X in B satisfies h( ) < w - Lai
X -pI
'
where w = Laipi is the p-adic decomposition of w [9, 11.9 and 13.8]. The defect group D of B is isomorphic to a Sylow p-subgroup of Spw,
70
c.
Bessenrodt and J.B. Olsson
which is nonabelian if and only if w 2:: 3. Hence the inequality above holds. If p = 2, then the height of an ordinary irreducible character in B is bounded by w - s(w) [9, 11.9], and the height of an irreducible spin character in B is bounded by [3W-;S(W)] by Theorem 3.8. On the other hand, by [12] we have IZ(D)I = 2 if D is non-abelian, so log21D : Z(D:II = v2((2w)!) = 2w - s(w). Hence if D is nonabelian, which is exactly the case if w 2:: 2, then again the inequality above is satisfied for the irreducible characters in B. Acknowledgement. A major part of this work was done during a stay at the Mathematisches Forschungsinstitut Oberwolfach in August 1994. The authors gratefully acknowledge support by a grant from the EC (Network on Algebraic Combinatorics). Thanks are due to A. O. Morris for drawing our attention to the reference [12]. References
[1] C. Bessenrodt, J. B. Olsson, The 2-blocks of the symmetric groups, Institut fiir Experimentelle Mathematik, Essen, Preprint No. 17 (1993). [2] M. Cabanes, Local structure of the p-blocks of (1988), 519--543.
Sn"
Math. Z. 198
[3] P. N. Hoffman, J. F. Humphreys, Projective Representations of the Symmetric Groups, Clarendon Press, Oxford 1992. [4] J. F. Humphreys, Blocks of projective representations of the symmetric group, J. London Math. Soc. 133 (2) (1986),441-452. [5] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, 1981. [6] I.G. Macdonald, On the degrees of the irreducible representations of symmetric groups, Bull. London Math. Soc. 3 (1971), 189--192. [7] A.O. Morris, J. B. Olsson, On p-quotients for spin characters, J. Algebra 15 (1988), 51-82. [8] J. B. Olsson, Frobenius symbols for partitions and degrees of spin characters, Math. Scand. 61 (1987),223-247.
Heights of Spin Characters
71
[9] J. B. Olsson, Combinatorics and Representations of Finite Groups, Vorlesungen aus dem Fachbereich Mathematik der Universitat GH Essen, Heft 20, 1993. [10] G. R. Robinson, Local Structure, Vertices and Alperin's Conjecture. Preprint 1994 [11] I. Schur, tIber die Darstellung der symmetrischen und der alternierende Gruppe durch gebrochene lineare Substitutionen, J. reine ang. Math. 39 (1911) 155-250 (ges. Abhandlungen 1, 346441, Springer-Verlag 1973). [12] A. Wagner, An observation on the degrees of projective representa-
tions of the symmetric and alternating group over an arbitrary field, Arch. Math. 29 (1977), 583-589.
C. Bessenrodt Fakultat fUr Mathematik, Otto-von-Guericke-Universitat Magdeburg PSF 4120, 39016 Magdeburg, Germany J.B. Olsson
Matematisk Institut, Kobenhavns Universitet Universitetsparken 5, 2100 Copenhagen 0, Received December 1994
Sur certains elements reguliers des groupes de Weyl et les varietes de Deligne-Lusztig associees Michel Brow§ et Jean Michel
Sommaire 1. Varietes de Deligne-Lusztig A. Contexte et notations B. Les varietes de De1igne-Lusztig 2. La variete X.,. A. Operation de B+ sur X.,. B. Valeurs de caracteres - Conjectures 3. Bons elements reguliers A. Elements reguliers et groupes des tresses associes B. Racines de 11" et elements reguliers 4. Groupes de reflexions complexes et algebres de Hecke associees A. Generalites B. Caracteres et degres fantomes C. Groupes de tresse D. Algebres de Hecke E. Valeurs de caracteres sur les racines de 11" 5. Varietes associees aux racines de 11" A. Quelques proprietes B. Conjectures C. Valeurs de caracteres et applications 6. Le cas non-deploye A. Generalites B. Elements reguliers et
74
M. Broue et J. Michel
Introduction Cet article est une etape vers la comprehension de la forme particuliere que prennent, dans Ie cas des groupes reductifs finis G F , les conjectures generales enoncees dans [Br] pour les blocs it groupes de defaut abeliens des groupes finis "abstraits". Apres avoir verifie l'aspect de ces conjectures port ant sur les caracteres des groupes reductifs finis (d. [BrMaMi]) , il reste it etablir la version, beaucoup plus profonde et difficile, port ant sur les equivalences de categories derivees concernees. Certaines de ces equivalences devraient en particulier etre fournies par les cohomologies etales de faisceaux convenables sur les varietes de Deligne-Lusztig X w associees it certains elements reguliers w du groupe de Weyl W de G (ef. [Br Mal, §1). Ceci implique des proprietes particulieres de la cohomologie etale de ces varietes, en particulier la disjonction des divers groupes de cohomologie Hn(X w , Ql), vus comme G F -modules. Nous precisons ici ce que sont les "bons" elements reguliers. II est connu que Ie centre du groupe des tresses pures associe it W (nous supposons ici W irreductible) est cyclique, engendre par un element positif (sur Ie systeme de generateurs distingue) que nous notons 11". Les bons elements reguliers sont les images dans W des racines d-iemes de 11" (quand elles existent). Nous etudions les racines de 11" au §3. Nous y montrons en particulier que 11" a une racine d-ieme si et seulement si d est un nombre regulier pour W. Grace it un theoreme de Deligne (ef. [De2]), on peut definir sans ambigulte une variete de Deligne-Lusztig X w associee it un element w du monoi"de des tresses positives. C'est ce que nous faisons au §5. Si west une racine d-ieme de 11", dont l'image w dans West un element regulier, les conjectures enoncees dans [BrMa] predisent que l'algebre des endomorphismes de la cohomologie de X w est isomorphe it une certaine specialisation de "l'algebre de Heeke cyclotomique" associee au centralisateur Cw( w) de w dans W, de sorte que l'endomorphisme de Frobenius agisse comme l'element "Tw " de cette algebre de Heeke. Cette hypothese suffirait it peu pres pour demontrer les resultats numeriques exposes au §5, C. Cependant, pousses it la fois par des idees (et des exemples) de Lusztig et par l'analogie avec Ie cas des groupes finis "abstraits" et Ie role qui y est joue par les modules de permutation (ef. [Ri]) , nous avons essaye de formuler des conjectures un peu plus precises sur l'operation du groupe de tresse generalise associe au groupe de reflexions complexe C w (w) sur la variete X w (d.
Elements reguliers et varietes de Deligne- Lusztig
75
[BrMaRo]). C'est la raison qui nous a pousse it introduire au prealable (§2) la variete X1I" qui correspond en fait au cas d = 1 et qui, it la difference de la variete traditionnelle Xl = G F / BF, admet une action du groupe des tresses (cf. §2, A). Son etude est peut-etre aussi difficile que celle des varietes X w . L'algebre commutante de G F dans sa cohomologie etale devrait etre l'algebre de Hecke classique, mais nous ne savons pas encore Ie demontrer. Nous n'avons, bien siir, pas plus de resultats effectifs sur Ie cas general des varietes X w pour w racine de 11". Cependant, une formulation precise des conjectures, et l'etude de certaines algebres de Hecke generalisees (§4), nous permettent de donner des resultats (nous devrions ecrire des "predictions", puisque ces resultats dependent des conjectures) sur les parametres lies it la cohomologie de X w . C'est ainsi que nous calculons (d. §5, C) les valeurs propres de l'endomorphisme de Frobenius F associees it chaque caractere unipotent de GF concerne, ainsi que les valeurs a priori des parametres des algebres de Hecke cyclotomiques concernees, confirmant a posteriori les valeurs decrites dans [BrMa]' et etayant ainsi "numeriquement" nos conjectures. Le paragraphe 4 occupe une place particuliere dans ce memoire. On y calcule les valeurs des caracteres des algebres de Hecke classiques sur les images des racines de 11" - dont nous avons besoin au §5 pour mener it bien les calculs mentionnes ci-dessus. Mais ces calculs necessitent aussi des informations sur les valeurs des caracteres des algebres de Hecke generalisees associees aux groupes de reflexions complexes ([BrMa], [ArKo]' [Ar], [BrMaRo]). Nous avons donc choisi de nous placer dans un cadre plus general, celui des groupes de reflexions complexes et de leurs algebres de Hecke generiques. Nous sommes d'ailleurs persuades que les proprietes des varietes X w que nous recherchons ici sont en fait un cas particulier d'un phenomene beaucoup plus general, OU les groupes de Weyl peuvent etre remplaces par des groupes de reflexions complexes (d. par exemple [Mal]) l'histoire ne dit pas encore par quoi sont remplaces les groupes algebriques. Nous prenons donc aussi date pour cette generalisation en redigeant Ie §4 dans ce cadre general. Nous confessons ici n'avoir - hormis Ie travail fondamental de G. Lusztig sur les varietes associees aux elements de Coxeter ([Lu2]), certains resultats recents sur les varietes associees it certains autres elements reguliers dans Ie cas An et dans Ie cas D 4 ([DiMi2]) - que bien peu d'exemples it fournir it l'appui de nos conjectures. La verification de leurs consequences "numeriques" (cf. [BrMaMi], [BrMa], ainsi que
M. Broue et J. Michel
76
notre §5, C) est certes un argument de poids en leur faveur. Mais plus convaincantes encore nous semblent etre la coherence qu'elles montrent avec d'autres aspects ou domaines des Mathematiques, et les directions de recherche qu'elles nous ouvrent. 1. Varietes de Deligne-Lusztig
A. Contexte et notations Soit G un groupe algebrique reductif connexe defini sur une cloture algebrique ifp du corps a p elements, et soit F: G --t G un endomorphisme de Frobenius definissant sur G une structure rationnelle sur Ie sous-corps Fq a q elements de ifp . On fait operer F a droite sur G. Pour simplifier l'exposition nous supposons G deploye. Nous donnerons quelques indications sur Ie cas general dans Ie dernier paragraphe. On designe par T Ie tore maximal (defini comme dans [DeLu]) de G, et par Y Ie Z-module des co-caracteres de T. On note W Ie groupe de Weyl de G, vu comme groupe engendre par des reflexions dans son action sur l'espace vectoriel complexe V := C 129 Y; on note VIR := lR 129 Y . On note A l'ensemble des hyperplans de reflexion de W dans V, et on pose
M:= V-
U H. HEA
On note III Ie systeme de racines de W, et on designe par 2N son cardinal. On a donc IAI = N. On note C Ia chambre fondamentale de W, et on note S l'ensemble des reflexions par rapports aux murs de C. Pour s, t E S, on note m(s, t) l'entier de Cartan correspondant. On sait que les relations
("Is, t E S),
s2 = 1,
et
stst ...
tsts' ..
~
~
m{s,t) termes
m{s,t) termes
constituent une presentation de W. Pour Xo un point de C, on note p: V
P
--t
V/W
(la surjection canonique),
7l"l(M, xo) (Ie groupe fondamental de M en xo), B := 7l"l(M/W,p(xo)) (le groupe fondamental de M/W en p(xo)), :=
Elements reguliers et varietes de Deligne- Lusztig
77
et on appelle Bet P respectivement Ie groupe des tresses et Ie groupe des tresses pures associes a G. Comme W opere librement sur M, on a la suite exacte courte
{I}
(*)
--+
P
--+
B
--+
W
--+
{I}.
Pour tout mur de C, dont la reflexion associee est notee s, on designe par /. la classe d'homotopie de chemins dans M, d'origine Xo et d'extremite s( xo), qui contient la ligne brisee de sommets successifs Xo, Xo + ixo, s(xo) + ixo, s(xo). L'image s de /. dans M/W est un lacet de point base p(xo). On pose 5 := {s I (s E S)}. L'assertion suivante est due a Deligne (d. [Del], 4.4). 1.1. Le groupe Best engendre par 5, et les relations
tsts .. .
stst·· .
'-v---'
'-v---'
m(.,!) termes
m(.,!) termes
en constituent une presentation. La fleche B
(*) est definie par l'application s
f-+
--+
W de la suite exacte
S.
On designe par B+ Ie sous-monolde de B forme des mots en les puissances positives des elements de 5. On designe par 11': [0,1] --+ M Ie lacet d'origine Xo defini par 11'(8) = e27ri9xo. Comme nous l'a fait remarquer R. Rouquier, il est facile de verifier que 11' definit un element central dans B (encore designe par 11'). On peut en fait demontrer (d. par exemple [Del] ou [BrSa]) Ie theoreme suivant. 1.2. Theoreme. Supposons l'action de W sur V irreductible.
(I) L 'element 11' est un generateur du centre de P, et la suite exacte (*) induit la suite exacte {I}
--+
Z(P)
--+
Z(B)
--+
Z(W)
--+
{I}.
(2) On a 11' E B+, et la longueur de 11' sur Ie systeme de generateurs 5 est egale 2N (nombre de racines de W). (3) Si Wo designe un element de B+ de longueur N sur 5 et d'image Wo (l'element de plus grande longueur) dans W, on a 11' = w6.
a
M. Broue et J. Michel
78
B. Les varietes de Deligue-Lusztig On rappelle que, si B designe la variete des sous-groupes de Borel de G, Ie quotient de B x B par l'action de G est canoniquement indexe par W (cf. par exemple [DeLu], 1.2). Si B et B' sont deux elements de B te1s que l'orbite contenant (B, B') correspond awE W, on dit que B et B' sont en position relative w et on note B ~ B' . La propriete suivante est bien connue ([DeLu], 1.2).
1.3. Lemme. Supposons que w = WIW2 avec £(w) = £(Wl)
+ £(W2).
(1) Si (B, B') est un couple de sous-groupes de Borel tel que B ~ B', il existe un unique sous-groupe de Borel B 1 tel que B ~ B 1 etBl~B'.
(2) Reciproquement, si (B, B 1 , B2) est un triplet de sous-groupes de Borel tel que B ~ B 1 et B 1 ~ B 2, alors B ~ B 2. Pour w E W, on note B(w) la variete des paires de sous-groupes de Borel (B,B') de G telles que B ~ B', munie des deux projections (B,B') t--4 Bet (B,B') t--4 B' (done vue comme (B x B)-schema). Soit ::D la categorie des (B x B)-schemas. Le produit A· B := A x B B fait de ::D une categorie monoldale. Le lemme 1.3 fournit alors la premiere assertion du lemme suivant.
1.4. Lemme. (1) Si w = WIW2 avec £(w) = £(Wl)
+ £(W2),
on a un isomorphisme
B(w)....:::.....B(wd· B(W2). (2) La jamille des isomorphismes ainsi definis est telle que, pour W = WIW2W3 avec £(w) = £(wI) + £(W2) + £(W3), Ie diagramme suivant est commutatij :
B(w)
----+
B(WIW2)· B(W3)
----+
B(Wl)· B(W2)· B(W3)
1 B(Wl) . B(W2W3)
1
Comme Pierre Deligne nous l'a fait remarquer, il resulte alors de [De2], 1.11, que
1.5. Theoreme (P. Deligue). n existe une fUche B+ --t ::D, b t--4 B(b), et des isomorphismes B(b 1 b 2 )"":::""'B(bI) . B(b 2), qui coincident avec les donnees de 1.4 dans Ie cas ou best reduit, et qui sont tels que Ie diagramme B(b)
----+
B(b 1 b 2 )·B(b 3 )
----+
B(bI)· B(b 2) . B(b 3)
1 B(b 1 )· B(b 2b 3)
1
79
Elements reguliers et varieUs de Deligne-Lusztig soit commutatif.
On dit qu'un element de B+ est reduit si son image dans West reduit - ou encore, si sa longueur sur S est egale it la longueur sur S de son image dans W. On note B;';,d l'ensemble des elements reduits de B+. On sait que la projection B+ ---+ W induit une bijection de B;';,d surW. Si W E B;';,d a pour image w dans W, on note B ~ B' la relation B~B'.
des elements reduits de B+ et soit b:= WI W2 .,. W n . Alors la variete B(b) est isomorphe (par un unique isomorphisme) it la variete des (n + 1)-uplets de sous-groupes de Borel (B o, B l , ... , B n ) de G tels que Soient
Wl,W2, ... ,W n
munie des deux projections sur Ie premier et Ie dernier termes. Remarque. On voit ainsi que la definition de la variete B(b) est inspiree par la methode de desingularisation de certaines varietes due a H.C. Hansen et M. Demazure (ef. [DeLu], et aussi [DiMi2]), et decrit une variete isomorphe it celle definie par Lusztig dans [Lu1], p. 25.
Remarquons que, puisque G est suppose deploye, si (B, B') est un couple de sous-groupes de Borel tel que B ~ B', on a aussi B.F ~ B'.F. Plus generalement, si (Bo,Bl, ... ,B n ) E B(b) pour b E B+, on a (Bo.F,Bl.F, ... ,Bn.F) E B(b).
1.6. Definition. Soit b E B+. On dbigne par X~F) (ou plus simplement Xb) et on appelle "varieU de Deligne-Lusztig associee b" l'image reciproque par B(b) ---+ B x B du graphe de I'endomorphisme de Frobenius F. Ainsi, si b = SlS2'" Sn (ou Sj E S pour 1 j n), la varieU X~F) est isomorphe (par un unique isomorphisme) la varieU des (n + 1)uplets de sous-groupes de Borel (B o, B l , ... , B n ) de G tels que
a
:s: :s: a
Chaque variete de Deligne-Lusztig est munie d'une action it gauche (par conjugaison) du groupe fini G F des points rationnels de G, et d'une action (it droite) du monolde engendre par F, qui commutent. Pour un element reduit W de B+, d'image w E W, on designe aussi par X w la variete correspondante (ce sont ces varietes qui sont habituellement appelees "varietes de Deligne-Lusztig").
80
M. Broue et J. Michel
2. La variete X.,. Nous etudions dans ce paragraphe une variete qui doit etre comprise comme un "substitut naturel" a la variete triviale Xl consideree habituellement pour definir la serie principale - i.e., essentiellement, l'ensemble (fini) des sous-groupes de Borel rationnels. Notre variete X.,. est a priori beaucoup plus compliquee que Xl. Mais, non seulement I' algebre des G F -endomorphismes de sa cohomologie est (conjecturalement) isomorphe a l'algebre de Heeke, mais, a la difference de Xl, Ie monolde B+ opere sur X 7r (et pas seulement sur sa cohomologie). De plus, notre variete X.,. est une espece de "modele" pour les varietes X w associees aux racines de 11" que nous introduisons plus loin. Soit Wo l'element de plus grande longueur de W. On note Wo l'element de longueur N de B+ d'image Wo dans W. Rappelons (cf. 1.2 ci-dessus) qu'on pose 11" := et que 11" est un generateur du centre du groupe des tresses pures P. Soit 11" = SIS2··· S2N une decomposition de 11" sur S. On voit par 1.6 que X.,. s'identifie a l'ensemble des (2N + l)-uples (B o,B I , ... ,B2 N) 51 52 52N d e sous-groupes d e Bore I d e G tels que B o --t B I --t B 2 ••• --t B 2 N et B 2 N = Bo.F.
w5,
A. Operation de B+ sur X.,. Dans ce paragraphe, on definit une operation naturelle (a droite) de B+ sur X.,. qui commute aux operations de G F . Cette operation est en quelque sorte "Ie double" de l'action de B+ sur X wo definie par Lusztig dans [LuI], p. 24. Pour definir une operation de B+ sur X.,., nous allons definir une famille d'operateurs a droite (D w ) EB+ sur la variete X.,. qui commuW
red
tent aux operations de G F , qui verifie la propriete suivante : Si
W, WI
et
W2
E B;';,d sont tels que
W
= WI W2,
alors D w
=D
W1
D W2 •
Ceci montrera en effet l'existence d'une application S f-+ D 5 qui est multiplicative et satisfait aux relations de tresses, done definira un morphisme de B+ dans Ie monolde des G F -endomorphismes de X.,..
1. Definissons l'operateur D w pour
E B;';,d
E B;';,d. On sait que W divise 11" : il existe b E B+ tel que Comme 11" est central, on a aussi 11" = bw. Plus generalement, soient b, c E B+. On definit un morphisme
Soit 11"
W
W
= wb.
D b(be) : X be de la maniere suivante.
--t
X eb
Elements reguliers et varieUs de Deligne-Lusztig
81
2.1. Posons b = SIS2'" Sm et c = t l t 2 ··· t n avec Sj, tk E S. Considerons (B o , B I , ... , B m , CI , C 2 , ••• , Cn) E Xbc, ou C n = Bo.F et
On pose alors Co := B m et (B o , B I , ... , B m , CI , C 2 , .•• ,Cn).D~bcJ :=
(Co, CI , ... , Cn, BI.F, ... , Bm.F). On pose D w := Dt J • L'assertion suivante est evidente et sa demonstration est laissee au lecteur.
2.2. Lemme. (1) Avec les notations precedentes, on a
(ou Fest vu comme endomorphisme de Xbc). (2) En particulier, on a D.,. = F. 2. Verifions que les D w sont multiplicatifs
Plus generalement, on verifie facilement sur la definition 2.1 ci-dessus que 2.3. pour b l , b 2 , b 3 E B+, on a :
En particulier, si D w = D W1 • D w ,·
W, WI, W2
E B~d sont tels que
W
=
WI W2,
on a
Comme indique precedemment, l'assertion ci-dessus suffit a prouver que l'application W f-+ D w se prolonge en un morphisme de monoides de B+ dans End(X.,.), encore notee b f-+ Db.
B. Valeurs des caracteres - Conjectures Plus de rappels et de notations. Nous introduisons d'abord quelques notations (cf. [Lu3] pour plus de details), qui seront largement utilisees par la suite.
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1. Pour x une indeterminee, on designe par 'tix(W) l'algebre de Heeke generique de W, i.e., l'algebre sur Z[x, x-I] definie par generateurs et relations de la maniere suivante : • • •
la famille de generateurs (T. )'ES est indexee par Ie systeme de generateurs distingue de W, les elements T. verifient les relations de tresses definies par Ie diagramme de Coxeter associe it S, pour tou t s E S, on a (T. - x)( T. + 1) = 0 .
Pour tout anneau commutatif unitaire A, on pose
Rappelons les proprietes suivantes de l'algebre 'tix(W). Structure de Z[x, x- l ]-module. Si (SI, S2, ... ,s/) est une decomposition reduite d'un element w de W, Ie produit T' l T' 2 ••• T' I ne depend que de w et on Ie designe par T w . La famille T w est une base de 'tix(W) comme Z[x, x- l ]-module. Forme lineaire eanonique. La forme lineaire t x : 'tix(W)
--+
Z[x, X-I] definie par
lsi w = 1, tx(Tw ) = { 0 . SInon, definit sur 'tix(W) une structure de Z[x,x- l J-algebre symetrique.
Isomorphisme de Lusztig. 11 existe un isomorphisme d'algebres
Le choix de cet isomorphisme definit alors une bijection X f-+ Xx entre l'ensemble Irr(W) des caracteres irreductibles de W et l'ensemble Irr('tix(W)) des caracteres irreductibles de l'algebre Q( JX)'tix(W). Degres generiques. Pour tout X E Irr(W), on definit les elements bx (x) par la formule
tx =
L
bx(x)Xx.
xElrr(W)
Alors 151 (x )-1 est Ie polynome de Poincare Pw( x) de W, et pour tout X E Irr(W), bx(x)Pw(x) est un element de Q[x] appele Ie degre generique de X et note Deg x (x ).
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On definit les entiers ax et Ax comme respectivement la valuation (ordre de la racine nulle) et Ie degre du polynome Degx(x). Nous allons considerer des representations a coefficients dans Qt. Si 0' E Qt n'est ni nul ni une racine de Pw(x), on sait que la specialisation x f-+ 0' definit une Qralgebre Q/Ha(W) qui est encore isomorphe a l'algebre de groupe Qt W. On designe par Xa la specialisation correspondante de Xx, et on a donc une bijection X f-+ Xa entre Irr(W) et l'ensemble des caracteres irreductibles de Q/Ha(W) (noter que Q/Hl(W) = QtW et Xl = X)· 2. Pour tout element w E W, on note R w Ie caractere du QtGFmodule virtuel L:n( -ltH n(X w ,Qt)· Soit X E Irr(W) un caractere irreductible de W sur Qt. On note 1
R x :=
IWI
L
X(w)R w
wEW
Ie "caractere-fantome" associe. 3. Soit, E Uch(G F ) un caractere unipotent de G F . • On designe par C-y la racine de l'unite associee aux valeurs propres de F : pour tout entier n et pour tout b E B+, les valeurs propres de F sur les composants isotypiques de type , de Hn(Xb, Qt) sont produit de C-y par une puissance entiere de ql/2. • On note d'autre part a-y et A-y respectivement la valuation et Ie degre du degre generique de " note Deg-y(x) (on rappelle que
Deg-y(x) E Q[x]). Pour tout g E G F et tout entier naturel m, on peut calculer IX~F~ I, i.e., la trace de gFm sur la somme alternee L:n(-1)nH~(X1I',Qt)de la cohomologie £-adique a support compact de X1I" a l'aide de la formule suivante (d. [DiMi1], III, et aussi [Lu3], 2.10, ou [DiMi2], Prop.1.2) :
II resulte de [Lu3], 5.12.1, que Xq~(T1I')
= X(1)qm(2N-(a x+A x».
Les entiers ax et Ax ne dependent que de la famille contenant Ie caractere X, et les caracteres X tels que RX>cF =f 0 sont tous dans une
<"
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meme famille (determinee par ,). On a donc (ax + Ax) = (a, + A,) pour tout caractere unipotent, tel que RX>CF =f O. Comme de plus on a L:xEIrr(W)X(1)Rx = R I , et que, si <"RI>CF =f 0, on a (, = 1, on voit que la formule 2.4 s'ecrit
<"
IX~r 1=
L <" R I>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 <,x,RI>CF = X(I). D'ou.
'x
<"
2.5. Proposition. On a
L
IX~F~ 1=
X(I)qm(2N-(a x +A x »,x(g)
L
=
XEIrr( W)
'x(g)Xq(T;').
XEIrr( W)
En "faisant m = 0" dans la formule precedente, et en utilisant la dualite de Poincare, on obtient :
2.6. Corollaire. Pour tout g E G F, on a
L( -Ittr(g; Hn(X,., Qt)) = L n
x(1hx(g)·
xEIrr(W)
Autrement dit, L:n(-I)nHn(x,.,Qt) a meme caractere que R I . Posons Rf(X,.,Qt) ;= EB~~~N Hn(X,.,Qt). Pour tout w E B~d' on note encore D w l'endomorphisme de Rf(X,.,Qt) defini par D w . Le theoreme suivant justifie Ie fait que nous considerions X,. comme un "substitut" ala variete XI.
2.7. Theoreme. L 'application Tw
f-+ D w definit un morphisme de l'algebre de Heeke Q/Hq(W) dans l'algebre des QtGF -endomorphismes de H:(X,.,Qt)
Demonstration de 2.7. II suffit de verifier que, pour tout s E S et tout entier n, l'operateur de H~(X,., Qt) defini par D. (et encore note D.) verifie les relations (D.
+ I)(D s
-
q) = O.
Nous imitons [LuI], 3.10 (b). La strategie est de ramener la question au cas des groupes de rang 1. 1. Preliminaire: Sous-groupes paraboliques opposes. Deux sous-groupes paraboliques P et pi de G sont dits opposes si pnp ' est un sous-groupe de Levi ala fois de P et de Pl. Nous rappelons ci-dessous quelques proprietes liees a cette definition.
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2.8. Etant donnes un sous-groupe parabolique P de G et un sous-groupe de Levi L de P, il existe un unique sous-groupe parabolique P' oppose Ii P tel que L = P n P'.
En effet, soit T un tore maximal de L, et soit III Ie systeme de racines de G correspondant. Pour 0' E Ill, on note Ua Ie sousgroupe radiciel associe. 11 est facile de voir que Ie seul sous-groupe parabolique contenant Let oppose it P est Ie groupe engendre par L et les sous-groupes radiciels Ua tels que Ua ct P. Ainsi, l'application qui associe it L, sous-groupe de Levi de P, l'unique sous-groupe parabolique P' oppose it P tel que L = P n P' induit une bijection entre l'ensemble des sous-groupes de Levi de P et l'ensemble des sous-groupes paraboliques opposes it P. Comme Ie radical unipotent Ru( P) de P agit fidelement et transitivement sur l'ensemble des sous-groupes de Levi de P, il agit de meme fidelement et transitivement sur l'ensemble des sous-groupes paraboliques opposes itP. 2.9. Soit (P, P') un couple de sous-groupes paraboliques de G opposes, et soit L := pnP'. L 'application B f-+ BnL (resp. B' f-+ B' nL) definit une bijection entre l'ensemble des sous-groupes de Borel de P (resp. de P') et I'ensemble des sous-groupes de Borel de L, et les sous-groupes de Borel B et B' de G sont opposes si et seulement si les sous-groupes de Borel B n L et B' n L de L Ie sont.
Pour s E 5, notons p. la classe de conjugaison sous G des sousgroupes paraboliques de G engendres par deux sous-groupes de Borel B, B' tels que B ~ B'. Tout sous-groupe de Borel Best contenu dans un unique sous-groupe parabolique de la classe p., que nous noterons P.(B). 2.10. Pour S E 5, posons s' := Was. Si les sous-groupes de Borel B et B' sont opposes, alors les sous-groupes paraboliques p.( B) et p., (B') Ie sont aussi.
En effet, soit Ill+ l'ensemble de racines positives defini par B dans Ie systeme de racines III de G attache au tore maximal T = B n B', et soit 0' la racine correspondant it s. On voit que P.(B) = (B,U- a ) et P.,(B') = P.,(woB) = WO(P.,(B)) = WO((B,U_wo a )) = (B',U a ). 2. Reduction au cas de rang 1. En choisissant la decomposition 11" = W6, on peut identifier X .... it l'ensemble des couples de sous-groupes de Borel (B, B') tels que B ~
B' ~ B.P.
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2.11. Posons s' := wOs. L'application f : (B,B ' )
f-+
(Ps(B),Ps,(B ' ))
envoie X 7r dans la variete X,..,s definie par X,..,s
:=
{(P, PI) I PEPs, pi E PSi, p' oppose it P et it P.F}.
En effet, cela resulte de ce que Ps(B.F) = Ps(B).F, puisque s est fixe par F (car G est deploye), et de la remarque 2.10 ci-dessus. 2.12. Les fibres de f sont stables par D s .
En effet, (B.Ds,B'.D s ) est defini comme l'unique couple de sousI
I
s B .D s swo S B ' . D s s----; Wo groupes de Bore I te Is que B ----; ----; B' et B ' ----; B.F. Tout sous-groupe de Borel B 1 tel que B ~ B 1 est contenu dans Ps(B), donc B.D s E Ps(B) et B'.D s E Ps,(B ' ). Soit (P, PI) E X,..,s. Notons L := P n pi, et soit u ' E Ru(P I) l'unique element tel que u'L = pi n P.F (on a donc u' P = P.F), et soit I u E Ru(P.F) l'unique element tel que L.F = U(U L). On designe par F ' l'endomorphisme de G (operant it droite) defini par g f-+ (uu')-'(g.F). Ainsi, F ' est un endomorphisme de Frobenius, qui definit sur G une structure rationnelle sur Ie meme corps que F, et qui stabilise L.
2.13. (1) L 'application f: X,.. ----; X,.. ,s est surjective, et la fibre f-l(P,P ' ) de f en (P,P ' ) E X,..,s est isomorphe Ii la variete X~FI)(L) du groupe L relative Ii 1'endomorphisme de Frobenius F'. (2) Get isomorphisme est compatible avec l'action de l'endomorphisme D s . Demonstration de 2.13. (1) Soit (P,P I) E X,..,s, et soit L := P n Pl. Pour tout couple (BL, BD de sous-groupes de Borel opposes de L, les sous-groupes de Borel B := BLRu(P) et B ' := B~Ru(PI) de G sont opposes (cf. 2.9 ci-dessus), et on a f(B, B ' ) = (P, Pi). Verifions que les sous-groupes de Borel B.F de P.F et B ' de pi sont opposes (dans G) si et seulement si les sous-groupes de Borel (B n L ).F' et B ' n L de L sont opposes. Ceci etablira bien les deux premieres assertions de 2.13. On a B = (B n L)Ru(P). Comme u-1.F- 1 E Ru(P), on a aussi
B
= u- ' .r ' (B n L)Ru(P)
donc B.F
= u-\(B n L).F)Ru(P.F).
I D'autrepart,onaB I =(B I nL)Ru(p ' ) = u 1-1 ( B , nL)Ru(p). Onvoit donc que B.F et B ' sont opposes si et seulement si les sous-groupes
Elements reguliers et varietes de Deligne-Lusztig
87
de Borel u-\BL.F) et u'B~ de u'L Ie sont, i.e., si et seulement si les sous-groupes de Borel BL.F' et B~ sont opposes. (2) Pour etablir la compatibilite avec D., on remarque que B~Ru(P) est en position SWQ avec Bl,Ru( P') (ce qui se voit dans Ie systeme de racines du tore BLnB~), et de meme B~.FRu(P') est en position s'wQ avec BL.FRu(P.F). Via l'isomorphisme decrit ci-dessus, l'application D. envoie donc (B L , BU sur (BL BL.F), ce qui est bien l'action de D. sur la variete
X;:.
•
3. Le cas de rang 1. 2.14. Lemme. Soit L un groupe reduetij de type AI. Soit s l'unique element non trivial du groupe de Weyl de L. Alors la variete X.,. de L possede deux groupes de cohomologie non nulle, H;(X.,.,!Qt) 0'11. L F opere par sa representation de Steinberg et 0'11. D. agit par multiplication par -1, et H:(X.,.,!Qt) 0'11. L F opere trivialement et 0'11. D. agit par la multiplication par q.
Demonstration de 2.14. On sait (pour une reference, voir [DiMi2], 1.18 et 1.19) que les seuls groupes de cohomologie non nuls sont ceux de degres 2 et 4, avec les valeurs annoncees comme G F -modules, et que F = D; agit par trivialement sur H;(X.,.,!Qt) et par multiplication par l sur H: ( X.,. , Qt). Donc D. a pour valeurs propres ± 1 et ±q. Pour preciser les signes, nous appliquons la formule de Lefschetz it un operateur de la forme gD. OU g ELF. Pour voir que nous pouvons l'appliquer, nous prolongeons d'abord l'operateur gD. it la variete X de tous les couples de sous-groupes de Borel par la formule gD.(B,B') = (9B',9B.F); cette variete est complete et non singuliere, et gD. ayant une puissance egale it une puissance de F, est fini et a des points fixes isoles et transversaux; sa trace sur la somme alternee des cohomologies de X est donc egale it son nombre de points fixes. La variete X.,. s'identifie ici it l'ouvert forme des couples de sous-groupes de Borel (B, B') tels que B =f B', B.F =f B'. Son complementaire dans X peut lui-meme se decouper en l'union de deux ouverts (du complementaire) formes des couples B = B' =f B.F (resp. B =f B' = B.F), qui sont echanges par gD., et du ferme forme des couples de sous-groupes de Borels rationnels. Les longues suites exactes ouvert-ferme correspondant it la stratification ci-dessus montrent que la formule de Lefschetz a lieu pour X.,. si elle a lieu pour la variete (discrete, de dimension 0) des couples de sous-groupes de Borel rationnels, ce qui est trivial. 11 est facile de verifier que l'ensemble des sous-groupes de Borel B tels 2 que g ~ Bet 9 B.F = B s'identifie it l'ensemble des points fixes de X.,. 1 sous gD. par B f-+ (B, 9- B). Si on prend g = 1 on voit que D. n'a
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pas de points fixes, ce qui, puisque la dimension de la representation de Steinberg est q, montre qu'il existe 6 E {I, -I} tel que les valeurs propres de D, sont -6 et q6. Si on prend pour 9 un element unipotent non trivial (donc regulier), il existe q + 1 sous-groupes de Borel tels que 2 9 B.F = B dont un au plus d'entre eux contient g. Puisque Ie caractere de la representation de Steinberg est nul sur g, on voit que 6 = 1. • Utilisons maintenant la suite spectrale de Leray associee au morphisme j:
EP,q = HP(X(') iili) 2 c,..' 'Hqj,iili ."J'.l ) :::} Hp+q(X c ,.. ,"J'.l .
(£)
Comme, par la formule du changement de base it support compact, on a en tout point y E xi'):
l'operateur D, induit un operateur encore note D, sur 'H qj!Q l tel qu'en chaque fibre on ait (D, - q)(D, + 1) = 0 d'apres Ie lemme 2.14, donc (D, - q)(D, + 1) = O. L'operateur D, deduit par passage it la limite dans (£) a donc la meme propriete, d'ou. Ie resultat. En fait, Ie theoreme 2.7 devrait etre precise par les resultats suivants, qui ne sont pour l'instant que conjecturaux. Nous utiliserons dans ce qui suit la notation suivante : si M est un Ql'Ha-module, de caractere Xa, on note 1vl* := Hom(M,Ql) son dual, vu comme module-ilJ1t'H a (module it droite sur Ql'H a ), et on designe par X* son caractere. 2.15. Conjectures. Soit n=2N
Rf(X,..,Ql):=
EB
Hn(X,..,Q l )
n=O
Ie module gradue de la cohomologie R.-adique de X,.., vu comme QlG F module gradue. (1) L 'algebre des QlG F -endomorphismes de Rf(X,.., Ql) est egale l'algebre engendree par les operations D w pour w E B+. (2) La correspondance Tw f-t D w definit un isomorphisme de I'algebre de H ecke Ql'H q(W) sur cette algebre commutante, et la representation de Ql'Hq(W) sur Rf(X,..,Ql) definie par cet isomorphisme est equivalente sa representation "speciale" sur
a
a
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Ie module de base G F / B F (ou B designe Ie sous-groupe de Borel deploye de G). (3) Le (Q1GF,Q/Hq(W))-bimodule Rr(X.,..,Ql) ainsi dejini se decompose comme suit en une somme directe de modules irreductibles de multipliciU 1 :
Rr(X.,.., Ql) =
EB 'x
(5)
X; .
xEIrr(W)
(4) Les compos ants de degre impair de Rf(X.,..,Ql) sont nuls, et ceux de degre pair sont disjoints. En particulier, on a
(5) (Precision suggeree par G. Lusztig) Pour tout entier pair 2n (O ~ n ~ N), Ie (Q1G F , Q/Hq(W))-bimodule H2n (X.,.., Ql) se decompose en la somme suivante de caracteres irreductibles distincts:
EB
2n
H (X.,.., Ql) =
IX 0 X; ,
XElrr(W)(n)
0'11 Irr(W)( n) designe I'ensemble des caracteres irreductibles X de W tels que ax
+ Ax = 2n.
Remarque. II est vraisemblable que les assertions (1) sont impliquees par la conjecture suivante
a (4)
ci-dessus
N H 2m (X;n ) . palre . 2 .16. L a partle EB mm"","-<'-l de La cohomoLogie de : O X.,.. est disjointe, comme Q1GF -module, de sa partie impaire cn m =NH 2m - 1 (X ;n) U'm=l ""'''-<'-l •
Indiquons sommairement pourquoi . • Pour tout W E B;;'d' W =f 1, l'operateur D w n'a pas de point fixe dans X.,... En admettant qu'il en resulte que
la representation de Q/Hq(W) sur Rr( X.,.., Ql) definie (d. 2.7) par Tw f-+ D w est equivalente a la representation "speciale" de l'algebre de Hecke Q/Hq(W) sur Ql[G F /B F ]. Ainsi, l'image de Q/Hq(W) dans sa representation sur Rr(X.,..,Ql) a meme dimension que Q/Hq(W). Ceci demontre les assertions (1) it (3) de 2.15.
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• Puisque Ie caractere de Rr( X.,., (iii) est combinaison lineaire a coefficients positifs de caracteres de G F (ef. 2.6), on voit que 2.16 implique l'assertion (4) de 2.15. 3. Bons elements reguliers
Ce paragraphe est entierement consacre aux elements reguliers des groupes de Coxeter finis et aux elements qui leur correspondent dans les groupes des tresses associes. Pour tout ce paragraphe, W designe donc maintenant, plus generalement, un groupe de Coxeter fini irreductible, et V designe Ie complexifie de l'espace vectoriel reel de la representation naturelle de W. Les notations introduites au §1, A et B, pour les groupes des tresses associes aux groupes de Weyl, sont utilisees ici sans etre redefinies. Enfin, Ie systeme de racines de West defini comme par exemple dans [Hi], chap. I, §3.
A. Elements reguliers et groupes des tresses associes
3.1. Definition. Si d est un entier, on appelle element d-regulier de
W un element w qui admet un vecteur propre pour la valeur propre e2i1r / d qui n'appartient Ii aucun des hyperplans de refiexion de W. Remarque. On peut aussi appeler "(-regulier" (cE. [Sp]) un element w qui admet un vecteur propre pour la valeur propre ( qui n'appartient a aucun des hyperplans de reflexion de W. On peut alors noter que si West un groupe de Weyl, les elements d-reguliers cOIncident avec les elements (-reguliers pour ( racine primitive d-ieme que 1conque de l'unite, • mais que, par contre, dans Ie groupe de Coxeter de type H 3 la c1asse des elements 5-reguliers est differente de celle des elements e4i1r /5 -reguliers.
•
De plus, la definition et la precedente remarque s'appliquent aussi au cas plus general ou West un groupe engendre par des pseudo-reflexions (d. §4 ci-dessous ). Les elements d-reguliers (e 2i1r / d -reguliers) sont ceux qui correspondent aux "racines d-iemes de 11''' (ef. §3, B ci-dessous). On choisit dans ker( w - e2i1r / d ld) un vecteur regulier XQ (i. e., un element qui n'appartient a aucun des hyperplans de reflexion de W).
Elements reguliers et varieUs de Deligne-Lusztig
91
On introduit les notations suivantes :
V(w)
:=
ker(w - e2i 7l"/dld) ,
W( w) := Gw( w)
(Ie centralisateur de w dans W) ,
On sait (d. [Sp], 4.2) que W(w) est un groupe de reflexions dans son action sur V( w), et que west d'ordre d. De plus, on sait (d. [DeLo], 2.9, ou les tables de [BrMa]) que W( w) est irreductible dans son action sur V(w). Les couples (V(w), W(w)) sont les "groupes cyclotomiques" etudies dans [BrMa]. On note
A(w)
l'ensemble des hyperplans de reflexion de W(w) sur V(w),
M(w) := V(w) - UHE.A(w)H, P(w) := 7r1(M(w),xo) (Ie groupe fondamental de M(w) en xo), p: V(w) --+ V(w)/W(w) (Ia surjection canonique), B(w)
:=
7r1(M(w)/W(w),p(xo)) (Ie groupe fondamental de M(w)/W(w) en p(xo)) ,
on note N(w) Ie cardinal de A(w), et on note N(w)* Ie nombre de pseudo-reflexions de W( w) agissant sur V( w). La propriete suivante a aussi ete demontree, simultanement et independamment, par Denef et Loeser ([DeLo]), et par Lehrer ([Le], 5.8).
3.2. Proposition. Sous les hypotheses et avec les notations precedentes, on a A(w) = {H n V(w) I (H E An· Demonstration. Nous commenc;ons par demontrer un lemme elementaire sur les sous-groupes finis de GL(V) non contenus dans SL(V) (d. par exemple [Le], 1.5). 3.3. Lemme. Soit G un sous-groupe fini de GL(V) non contenu dans SL(V). Soit
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des reflexions et donc n'est pas contenu dans SL(V). Ce groupe W H' est normalise par w, donc on deduit de 3.3 que
(WHI
=f {I})
~ (CwH , (w)
=f {I}).
Mais CwH , (w) = W( w)H', et par consequent ce groupe n'est pas trivial si et seulement si H' E A( w). On voit donc que A( w) est l'ensemble des hyperplans Hide V( w) te1s que W H' ne soit pas trivial, donc (puisque W H' est engendre par des reflexions) te1s qu'il existe un hyperplan de reflexion H de W contenant H'. La proposition 3.2 resulte alors du fait que, par hypothese, V( w) n'est contenu dans aucun hyperplan de reflexion de W. •
Remarque. Comme note dans [DeLo] et [Le], la proposition precedente et sa demonstration s'etendent au cas plus general ou West un groupe engendre par des pseudo-reflexions (et meme au cas ou west un element regulier qui normalise W sans necessairement lui appartenir), grace au theoreme fondamental de Steinberg ([St1], 1.5) qui affirme que Ie fixateur d'un sous-espace ("sous-groupe parabolique") est aussi engendre par des pseudo-reflexions. On designe par 11": [0,1] --t M(w) Ie lacet d'origine XQ defini par 2 e 71"i9 xQ , et on designe par w: [0,1] --t M(w) Ie chemin de XQ a W.XQ defini par w(8) = e 27ri9 / d xQ. On note encore 11" et w respectivement les elements de P( w) et B( w) ainsi definis. On voit que, dans Ie groupe B( w), on a
11"(8) =
Comme nous l'a fait remarquer R. Rouquier, il est facile de verifier que
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
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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
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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.
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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
3.10. Definition. On appelle d-eLements de Springer Les elements reguLiers de W possedant La propriete suivante : iL existe un vecteur propre v E V correspondant Ii La vaLeur propre e2i1r / d, tel que La racine a E
O. On sait d'apres [Sp], 4.10, que pour chaque nombre regulier d de W il existe un d-element de Springer.
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M. Broue et J. Michel
3.11. Proposition. Soit d un nombre regulier pour W et soit w un d-element de Springer. Alors west une bonne racine d-ieme de 1r. Demonstration. On pose ( := eZi1r / d . D'apres 3.9 ci-dessus, nous devons verifier que pour tout m S; d/2, on a £(w m ) = 2Nm/d. Suivant l'argument de Springer ([Sp], 4.10), on a , donc les nombres
3.12. Theoreme. Si west une racine d-ieme de
1r,
alors son image
1r,
d est un nombre
dans West un element d-regulier de W.
3.13. Corollaire. S'il existe une racine d-ieme de regulier pour W.
Nous commen<;ons par demontrer 3.12 dans un cas particulier. Le cas general sera alors une consequence immediate du theoreme 3.17 ci-dessous.
3.14. Proposition. Soit w une bonne racine d-ieme de
1r.
Alors w
est un element d-regulier de W.
Demonstration. Premiere etape.
A chaque orbite 0 de w sur
,wnl+n2-1(a) E
ou nl + ... + nZk(O) = 101. En d'autres termes, nous mettons les racines consecutives (pour l'action de w), et de meme signe, dans Ie meme paquet. 11 est clair que £( w) = 2::0 k( 0).
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Pour x E W, soit 0, on a nj 2: d' pour i impair. On applique maintenant ce resultat it -0, et on obtient la meme conclusion pour nj et i pair. Done, pour tout i < k( 0) - 1, on a nj + nj+l 2: 2d' 2: d - l. En fait nj + nj+l = d - 1 est impossible. En effet, si f3 := wn,+ ...+ni-,-la, on aurait f3 = w df3 = wn,+ ...+n,+n'+'a, ce qui montre que f3 est it la fois posit if et negatif. 11 en resulte que, pour tout i < k(O) -1, on a nj + nj+l 2: d, d'ou k(O) ~ lIOI/dj. Puis que few) = 2N/d = 1
°
L
101 = 1
o
0
0
L'inegalite entre Ie terme Ie plus it gauche et Ie terme Ie plus it droite n'est possible que si pour chaque orbite, on a 101 = dk(O). On a done demontre : 3.15.
• • •
le cardinal 101 de chaque orbite est multiple de d, k(O)=IOI/d, les nj pour i impair sont tous egaux Ii d' ou sont tous egaux Ii d - d', et les nj pour i pair sont tous egaux Ii d - nl.
Remarque. Comme west d'ordre d, on voit que 101 divise d, et par consequent pour toute orbite on a 101 = d et k(O) = l. Deuxieme etape.
°
Nous pouvons demontrer maintenant que west regulier. Soit V( w) Ie sous-espace propre de w pour la valeur propre ( := e Zj1r / d • Le projecteur p de V sur V (w) parallelement aux autres sous-espaces propres de w est p = 1 + C1w + CZw z + ... (l-dwd-l .
°
Soit une orbite de w sur l'ensemble des racines, et soit a E
98
M. Broue et J. Michel
alors que a,w(a), ... ,wnt-1(a) E
wnt(a). Ainsi, nous avons demontre qu'aucune racine n'a de projection nuUe sur V(w), i.e., que west regulier. •
3.16. Definition. On dit que deux elements de B;"d sont "conjugues par permutations circulaires" s'ils sont equivalents par la cloture transitive de la relation Z definie sur B;"d par
wZw'
si et seulement si il existe
x,y E B;"d tels que
{w,= xy et w =yx.
Remarque. II est clair que deux elements conjugues par permutations circulaires sont conjugues dans B. La reciproque est fausse en general: dans Ie cas du type A z : ()--O, les elements s et t sont conjugues •
t
dans Ie groupe des tresses, mais pas conjugues par permutations circulaires.
3.17. Theoreme. I Toute racine d-ieme de
1r est conjuguee par permutations circulaires d 'une bonne racine d-ieme de 1r.
Demonstration. Nous introduisons tout d'abord quelques notations, ainsi que les resultats de [Del] et [Cha] que nous utilisons. Pour x, y E B+, on ecrit xly et on dit que x divise y (it gauche) s'il existe Z E B+ tel que xz = y. II resulte de [Del], 1.19 que 3.18. pour x E B+, il existe un unique element de plus grande longueur de B;"d qui divise x (Ii gauche).
Soit x E B+.
On dit qu'une decomposition x = VI ... Vk est la VI, ... ,Vk E B;"d' et si pour tout i, Vi est l'element de plus grande longueur de B;"d qui divise Vi ... Vk. Ainsi, chaque element de B+ a une unique forme normale. On designe par v(x) Ie nombre de termes dans la forme normale de x. Nous allons utiliser les proprietes suivantes de la fonction v, qui sont dues it Charney ([Cha], Prop. 3.1 et 3.3).
forme normale de x, si
1 Nous remereions F. Digne pour l'aide qu'il nous a apportee dans la demonstration de ee theoreme.
Elements reguliers et varietes de Deligne-Lusztig
99
3.19. On a
(1) v(xy) ~ max(v(x),v(y)), (2) si VI ... Vk est la forme normale de x, si y E B~d et v(xy) = v(x), alors VkY E B~d' Indications sur la demonstration. (1) est clair dans [Cha], 3.1 et 3.3. Pour (2), avec les notations de loco cit., Prop. 3.3., il s'agit du fait que Tli f3i est minimal, et cela resulte de ce que IkTJi f3i = Jii1i f3i = ~ puisque a*aJ1k = JikJii1if3i' • Notons la consequence suivante de 3.19, (1), qui nous sera utile par la suite. Elle resulte du fait que v( 1r) = 2 (puisque 1r = w5).
3.20. Pour tout diviseur x de 1r, on a v(x) S; 2. Nous sommes maintenant en mesure de demontrer Ie theoreme 3.17. II resulte du lemme suivant, applique pour n = d. 3.21. Lemme. Si w n 11r, alors west conjugue par permutations circulaires d 'un element w' tel que w' Ln/2J appartient a B~d' Demonstration. Elle se fait par recurrence sur n. Pour n = 0 il n'y a rien it demontrer. On remarque aussi que si n est impair, la conclusion cherchee est identique it la conclusion cherchee pour (n -1), done resulte de l'hypothese de recurrence. On peut done supposer que n est pair et on pose n = 2j. Par l'hypothese de recurrence, west conjugue par permutations cirI culaires it un element w' tel que W,i - E B~d' Done, quitte it remplacer w par w' on peut supposer que w i - I E B~d' On peut supposer v(w i ) > 1 (sinon wi E B~d et la demonstration est terminee). D'apres 3.20 ci-dessus, on voit qu'alors v(w i ) = 2. Soit (wi-Ix)y la forme normale de wi (notons que xy = w). Notre hypothese s'ecrit (w i - I xy)(W i - I xy)I1r, et il resulte de 3.19, (1) que
d'ou v((wi-Ix)y)
= v((wi-Ix)y(Wi-Ix)) = 2.
De 3.19, (2), on deduit alors y(wi-1x) E B~d' Mais, puis que xy = w, on a y(wi-Ix) = (yx)i. On voit done que l'eIement w' := yx est conjugue de w par permutations circulaires et verifie W ,i E B~d' ce qui demontre Ie lemme 3.21. •
M. Broue et J. Michel
100
Remarques et Questions. • Comme l'element 1r et Ie monolde B+ peuvent etre de£inis pour tous les groupes de reflexions complexes irreductibles (cf. [BrMaRo]), il est naturel de poser la question suivante : Ie theoreme 3.12 est-il encore vrai pour un groupe de reflexions complexe ? • Deux racines d-iemes de 1r sont-elles toujours conjuguees par un element de B ? Noter que (grace it 3.17) pour Ie cas des groupes de Coxeter, il suffit de repondre it la question pour Ie cas ou les deux racines sont bonnes. • Soit h Ie nombre de Coxeter de W. Comme deux elements de Coxeter sont conjugues par permutations circulaires (cf. [Bou] ch. v, §6, 1, demonstration du lemme 1), et comme il existe un element de Coxeter dont l'image reciproque c dans B~d soit une racine h-ieme de 1r (cf. [Bou], chap. 5, §6, ex. 2), on voit que toute image reciproque dans B~d d'un element de Coxeter est une racine h-ieme de 1r.
4. Groupes de reflexions complexes et algebres de Hecke associees A. Generalites Une bonne partie du contenu de ce paragraphe s'applique, non seulement aux groupes de Coxeter W et it leurs centralisateurs d'elements reguliers W( w), mais egalement it tout groupe fini lineaire irreductible engendre par des pseudo-reflexions ("groupe de reflexions complexe"). Pour ce que nous utiliserons sur ces groupes, Ie lecteur pourra se reporter it [BrMaRo], et aussi (par exemple) it [Bou], [OrSo], fOrTe], [Sp]. Nous changeons ici provisoirement de notation: V designe toujours un espace vectoriel complexe (de dimension r), mais W dtEsigne maintenant un sous-groupe nni de GL(V) engendre par des pseudo-reflexions et agissant irreductiblement sur V. Un sous-groupe parabolique de West par definition Ie sous-groupe des elements de W qui operent trivialement sur un sous-espace de V. Par un theoreme de Steinberg ([Stl], 1.5), on sait qu'un sous-groupe parabolique de West encore engendre par des pseudo-reflexions. On note A l'ensemble des hyperplans de reflexion de W, et on pose N := IAI. On note N* Ie nombre de pseudo-reflexions de W. On remarque que si West un groupe de Coxeter (ou, plus generalement, si toute pseudo-reflexion de West d'ordre 2), on aN = N*. Pour HE A, on designe par WH l'ensemble des elements de W qui
Elements reguliers et varietes de Deligne-Lusztig
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operent trivialement sur H (ainsi, W H est un sous-groupe parabolique minimal de W), et on pose eH := IWHI. On designe par SH l'element de W H de determinant e2i1r /eH. On designe par Cw(WH) Ie centralisateur de WH dans W et on pose
L'entier WH est egal au cardinal de l'orbite de H sous W, et aussi egal au cardinal de la c1asse de conjugaison de SH. Soit S(V) l'algebre symetrique de V, soit R = S(V)W la sous-algebre des invariants de W, soit R+ l'ideal de R forme des elements de degre positif. On pose S(V)w := S(V)/R+S(V). Rappelons quelques proprietes des degres et des codegres de W (d. [BrMaRo]). • Les degres d l ,d2 , ... ,d r de (V, W) possedent les proprietes suivantes : Ie polynome de Poincare du C-espace vectoriel gradue (V 16) S(V)w)W est
et on a i=r
2)di -l)=L(eH-l)= L wH(eH-l)=N*. i=1 HEA HEA/W
• Les codegres di', d2,... ,d; de (V, W) possedent les proprietes suivantes : Ie polynome de Poincare du C-espace vectoriel gradue (v* 16) S(V)w) West
et on a L~~~(d:
+ 1) =
LHEA 1 = LHEA/WWH = N.
• On a N + N* = L~~~(di + di) = LHEA/WwHeH. • Le centre Z de West d'ordre IZI = pgcd{d l ,d2 , ... ,dr }. • L'ordre de West IWI = d}d 2 ·•• dr. Enfin, la representation de W sur Vest definie sur Ie corps !Q(xv) engendre par les valeurs du caractere Xv de cette representation (d. par exemple [Be]' 7.1.1). On pose K := !Q(xv).
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M. Broue et J. Michel B. Caracteres et degres fantomes
Soit X E Irr(W) un caraetere absolument irreduetible de W. On note
Vx un espace de representation correspondant . • Les exposants (el(x),e2(x), ... ,e x(I)(X)) sont definis par la condition suivante : Ie polyn6me de Poincare du module gradue
(Vx I8i S(V)w)w (appele degre fantome de X) est
•
Les coexposants (er(X), e2(x),· .. , e~(i)(X)) sont les exposants de
V;.
On pose
n=x(l) N*(X):=
L
n=X(I)
en(x)
et
N(X):=
n=l
L
e~(x)·
n=l
On voit donc en particulier que, si Xv designe Ie caraetere de la representation naturelle de W, on a N = N(xv) et N* = N*(xv). Les entiers N(X) et N*(X) se calculent it partir des sous-groupes paraboliques minimaux, grace it la proposition 4.1 ci-dessous. Sa premiere partie est facile et laissee au lecteur. La deuxieme assertion est une reformulation d'un result at de Gutkin (cf. [Gu]).
4.1. Proposition. Pour tout H E A et tout entier n, soit mH,n La muLtipLiciU de e2nirrjeH comme element du spectre de SH dans La representation associee a X. ALors
(1) N(ResttH(x)) = l:~~~H-l n.mH,n(X), (2) N(x) = l:HEA N(ResttH(x)). En utilisant Ie fait que mH,n(X*) = mH,eH-n(X), on deduit facilement de 4.11es formules suivantes, qui nous seront utiles dans la suite.
4.2. Corollaire. On a
n=eH-l
L L eHmH,n(X) = N(X) + N*(X), HEA L eHmH,o = X(l)(N + N*) - (N(X) + N*(X))· n=l
HEA Le cas des caracteres Lineaires La propriete suivante est demontree dans [Co], (1.8), Proposition.
Elements T/;guliers et varietes de Deligne-Lusztig
103
4.3. Proposition. Les applications restriction induisent un isomorphisme
Hom(W,eX)~ (II HOm(WH,e X )) W , HEA
ou le terme de droite designe le sous-groupe des points fixes par l'action de W dans le produit direct DH EA Hom(WH, ex ). En particulier, pour tout H E A, il existe un caractere lineaire det V,H de W possedant la propriete suivante : pour toute pseudo-refiexion pEW, on a detv H(p) ,
={
detv(p) si pest conjugue . 1 smon.
On a
detv
II
=
a un
element de W H,
detV,H,
HEA/W
= 1, N*(detV,H) = wH(eH -1) et N*(ResltH (detV,H)) = eH N(detV,H) = WH
N(detv)
=
2: 1 = N
et
et
N(Restt(detV,H))
N * ( det v)
HEA
=
2: (e
H -
1)
-
1,
= N*.
HEA
Nombres reguliers et expos ants Springer a demontre (d. [Sp], 4.5) que si west un element regulier de W pour la valeur propre (, alors Ie spectre de w dans une representation de caractere X est
On voit donc en particulier que, dans ce cas, Ie determinant de w dans une representation de caractere X est
Comme detxdet x' = 1, on en deduit
4.4. Proposition. Si d est un nombre regulier pour W et X un caractere de W, d divise N(X) + N*(X). En appliquant la proposition 4.4 ci-dessus au cas ou X = detV,H, on obtient
4.5. Corollaire. Pour tout nombre regulier d et tout hyperplan de refiexion H, d divise WHe H.
M. Broue et J. Michel
104
C. Groupes des tresses Soit D Ie diagramme "a la Coxeter" associe a W (cf. [BrMaRo]; pour Ie cas des groupes cyclotomiques, on peut se reporter it [BrMa] ou it l'annexe 2 ci-dessous). On note S l'ensemble des sommets de D. Soit B Ie groupe defini par S et par les relations de tresses du diagramme D. Il existe un morphisme surjectif B f-+ W possedant les proprietes suivantes. (a) Pour tout s E S, son image s est une pseudo-reflexion telle que si H := ker(s - 1), et si eH := \WHI, alors s est Ie generateur SH de WH de determinant e 2i1r /eH. On pose e. := eH. (b) L'ensemble S image de S, soumis aux relations de tresses de D et aux relations se, = 1, est une presentation de W. On designe par P Ie noyau de l'homomorphisme surjectif B On a donc la suite exaete
{I}
(4.6)
-+
P
-+
B
-+
W
-+
-+
W.
{I}.
Les groupes B et Pont une interpretation topologique (encore conjeeturale dans certains cas) analogue a celle presentee plus haut pour Ie cas des groupes de Weyl ou des groupes de Weyl cyclotomiques (cf. [BrMaRo]). On les appelle respectivement le groupe des tresses et le groupe des tresses pures associes a W. On note B+ Ie sous-monolde de B forme des mots en les puissances positives des elements de S. Abelianises de B et de W
On designe par sah l'image de S dans l'''abelianise'' Bah := B/[B, B] de B. On laisse au leeteur Ie soin de decrire sah comme un ensemble quotient de S en utilisant les relations fournies par Ie diagramme D (par exemple, dans Ie cas ou D n'a que des relations de tresses impliquant deux elements au plus, sah peut etre vu comme Ie quotient de Spar la cloture transitive de la relation "s et t verifient une relation de tresse d'ordre impair").
4.7. On a
Bah
=
II
(s)
,ou (s)
~ Z.
sES ab L'abelianise wah .- W/[W, W] de West isomorphe a TIsEsab(Z/e.Z) (cf. 4.3 ci-dessus). L'ensemble sah est en bijection naturelle avec l'ensemble A/W des classes de conjugaison des hyperplans de reflexion. L 'element
1r
On a la propriete suivante des centres des groupes consideres (cf. [BrMaRo]).
Elements reguliers et varietes de Deligne-Lusztig
105
4.8. Theoreme. (1) Le centre Z(B) est cyclique d'ordre infini et engendre par un element {3 E B+, dont La longueur sur S est
£({3) = (N (2) La suite exacte
{l}
--+
4.6
+ N*)/IZ(W)I·
induit une suite exacte
Z(P)
--+
Z(B)
--+
Z(W)
--+
{I}.
Donc le centre de P est cyclique et engendre par un element B+ dont la longueur sur S est £(1r) = (N + N*).
1r E
On peut calculer l'image 1r ab de 1r dans l'abelianise B ab de B. Vne verification facile sur les tables de [BrMaRo] montre
4.9. Proposition. On a 1r
ab
II (s)""
=
e, .
sES· b
Pour tout H E A, on designe par SH un element de B qui est conjugue it un element s E S et s'envoie sur l'element SH de W (si les conjectures sur l'interpretation topologique de B enoncees dans [BrMaRo] sont verifiees, on peut choisir SH comme Ie generateur de la monodromie autour de H dans M/W qui s'envoie sur SH).
4.10. Corollaire. On a 1r
ab
=
II
II (sIf)e H .
(SIf)"'He H =
HEA/W
HEA
Remarque. Soit BH := (SH) Ie groupe des tresses associe it WHo On a alors (avec des notations evidentes) PH = Z(PH) = (1rH) ou 1rH := s';{ , et Ie corollaire 4.10 peut s'ecrire 1r ab
=
II 1rH. HEA
M. Broue et J. Michel
106
D. Algebres de Heeke Les definitions suivantes ont ete donnees dans [BrMa] (d. aussi [ArKo] et [Arl). Soit
un ensemble de L:HEA/W eH indeterminees. On pose
U
-1 := (-1 UH,j ) (HEA/W)(0:Sj:SeH -1)
.
et pour s E S, on pose u.,j := UH,j pour j = 0,1, ... , eH - 1, ou H designe l'hyperplan de refl.exion de s. On note N(D) l'ensemble des sommets du diagramme D.
4.11. Definition. L'algebre de Heeke generique H u associee Ii D est la Z[u, u-I]-algebre engendree par une famille (T')'EN('D) telle que • •
les elements T. satisfont aux relations de tresses definies par D, on a (T. - u.,o)(T. - U.,I)··· (T. - u.,e,-I) = O.
On remarque que l'application s f--t T. definit un morphisme B et on note Tw l'image d'un element wEB par ce morphisme.
-+
H~,
Par la specialisation
(4.12)
d v : u.,j
f--t
detv(s)j
= e2i1rj /e,
(Vs E S) (0 ~ j ~ e. - 1),
l'algebre H u devient l'algebre de groupe de W sur une extension cyclotomique de Z. L'enonce suivant est bien connu dans Ie cas des groupes de Coxeter. Il est maintenant demontre pour toutes les series infinies de groupes de refl.exions complexes, comme consequence de [ArKo], [BrMa], et [Ar]. 11 a ete verifie pour quatre des groupes cyclotomiques exceptionnels (d. [BrMa], Satz 4.7).
Theoreme-Conjeeture 4.13. (1) H u est un module libre et de rang IWI sur Z[u, u- I ]. (2) L 'algebre iQ( u)H u est semi-simple, et elle se deploie sur une extension de iQ(u) de laforme K(exP(ml~(~)I)'u,v,), ou
(a) m := ppcm{eH I (H E An, (b) vest un ensemble de monomes en des puissances fractionnaires des u.,j dont les denominateurs divisent IZ(W)I.
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Elements reguliers et varietfs de Deligne- Lusztig
(3) Il existe P(u) E K[u] tel que les representations irreductibles de l'algebre K(exP(mI1(~)I),u,v)Hu sont realisables dans l'anneau
K[exp( mI1~~)I)' u, u-l, v, l/P(u)]. On dit qu'un caractere de K( exp( mI1~~)I)' u, v)H u est generiquement rationnel s'il est it valeurs dans K(exp( mI1(~)1 ))(u). Exemples. Les caracteres lineaires de l'algebre de Hecke generique sont les caracteres
. . { Tt det(sj)u. H u --+ iQ[u] . , , Tt
f-t
1 si t et s ne sont pas conjugues,
f-t
Ut,j SI t
.
. , et s sont conJugues.
Ainsi les caracteres lineaires sont tous generiquement rationnels. Dans toute la suite de ce paragraphe, et dans le but de simplifier la presentation de nos calculs, nous nous pla<;ons dans un corps suffisamment gros, defini comme suit . • On choisit un ensemble d'indeterminees
telles que lWI v H,j --
UH ,J' .
Pour tout diviseur n de IWI, on pose alors U1H/n. := ,} aussi Vs,j := VH,j si H = ker(s - 1). 2i7r • On pose (:= exp(ppcm{mIZ(W)I, IWI})'
vHIWI/n ,J
On pose
Nous nous pla<;ons dorenavant sur le corps K((, v), extension galoisienne de iQ( u), et sur lequel 1'algebre H u se deploie. La propriete suivante est une consequence immediate de 4.13 cidessus. 4.14.
Corollaire. Le choix de la specialisatiOIL-_{qui prolonge celle
definie en 4.12)
d v : Vs,j
f-t
e2i1rj/IWle.
definit une bijection X f-t Xv entre l'ensemble Irr(W) des caracteres absolument irreductibles de W et l'ensemble Irr(H u ) des caracteres absolument irreductibles de iQ(v)H u , telle que le diagramme suivant soit commutatij.
K((, v)H u ~ K((, v)
dvl K(()W
dv
~
1
K(()
108
M. Broue et J. Michel
La conjecture suivante est verifiee pour les groupes de Coxeter, les groupes G(d, 1,r) ([BreMa]), et dans un petit nombre de cas pour les aut res groupes de reflexions. Pour toutes les series infinies (d. [Ar] et [Mal], et aussi [BreMa]), ou pour tous les groupes exceptionnels de rang 2 (d. [Ma2]), la forme t u it considerer apparait c1airement. Par analogie avec Ie cas des groupes de Coxeter, on designe par B~d l'ensemble des elements de B+ qui ont meme longueur sur S que leur image dans W sur S. 4.15. Theoreme-Conjecture. Il exi.'Jte une unique forme lineaire t u : H u -+ Z[u, u- 1 ] avec le.'J propriete.'J .'Juivante.'J :
(a) La forme t u e.'Jt centrale, i.e., on a tu(hh') = tu(h'h) pour tou.'J h, h' E H u .
(b) On a t u (1) = 1, et pour tout w E B~d' w
-#
1, on a
(c) L 'algebre H u , munie de la forme t u , e.'Jt une algebre .'Jymetrique. E. Valeurs de caracteres sur les racines de 11" On utilise les notations introduites precedemment. Pour X E Irr(W), on designe par mH,j la multiplicite de UH,j comme element du spectre SpecxJTSH ) de T SH dans une representation de caractere Xv. On introduit l'element suivant de Q[v] : Z
(11")'Xy'-
II
j=eH - I
HEA/W
j=O
II
(l/x(I))mH,jWHeH UH,j ,
et, pour dun nombre regulier pour W, on pose Zxy(1I")I/d:=
II
j=eH -1
HEA/W
j=O
II
u~:rl))mH,j(WHeH/d)
Remarquons que, d'apres 4.5 ci-dessus, on a bien
ZXy(1I")I/d
E Q[v].
4.16. Proposition. (1) On a
Xv(T.,..) = x(1 )e- 2i1rN (x)/X(I) ZXy (11"),
(2) Si w e.'Jt une racine d-ieme de 11", d'image
w
dan.'J W, on a
Elements reguliers et varietes de Deligne-Lusztig
109
Demonstration. Puisque (1) est un cas particulier de (2), nous demontrons (2). D'apres 4.10, on a j=eH-l
detXy(T.... )
=
II II HEA/W
u';;j,jWHe H = ZXy(1r)x(1) .
j=O
Comme 1r est central dans B, Ie spectre de T.... dans une representation de caraetere Xv est
ou ~ est une racine x(l)-ieme de l'unite. Comme T!. = T.... , on voit que Ie spectre de Tw dans une representation de caractere Xv est
ou les
~k
sont des racines (dX(l))-iemes de l'unite, d'ou k=X(I)
Xv(Tw )=
L
~kZxy(1r)I/d.
k=l
En utilisant 4.1 ci-dessus, on voit aisement que par la specialisation d v : Vs,j f-+ e2i7rj /\W\e. , ZXy(1r)l/d s'envoie sur e2i7rN (x)/dx(1).
Comme Xv(Tw ) s'envoie sur X( w), on en deduit que L:::~(1) ~k = e-2i7rN(x)/dx(1)X(w) , d'ou Ie resultat. •
4.17. Corollaire. Pour tout X E Irr(W), on a Xv(T.... ) E K((, u). En d'autres termes, pour tout H E .A et tout entier j (0 ::; j ::; eH - 1), X(l) divise mH,jWHeH. Demonstration. Rappelons que, d'apres nos choix de v et de (, Ie corps K((,v) est une extension galoisienne de Q(u). Puisque Xv(T.... ) est un monome en v, on voit que pour tout (J' E Gal(K((, v)/K((, u)), on a "(Xv )(T.... ) = ("Xv(T.... ) ou (" est une racine de l'unite. Mais par la specialisation d v , les deux termes de l'egalite precedente deviennent l'entier positif X(l). On voit donc que (" = 1. • Soit x une indeterminee. Considerons l'algebre H x sur Z[x,x- 1 ] definie it partir de H u par la specialisation
Us
0
f-+
X
{ Us,j
f-+
e2 ' 7r}/e.
pour 1 ::; j ::; e s - 1.
M. Broue et J. Michel
110
Ainsi, H x est engendree par un systeme (T. ).ES satisfaisant aux relations de tresses et aux relations
(T. - x)(l Grace
+ T. + ... + T:,-l)
= O.
a l'identite suivante, ou on pose U := xT- 1
(T - x)(l
+ T + ... + T e - 1 )
= _x-(e-l)Te(U _ 1)(x e-
1
+ x e- 2U + ... + xU e- 2 + U e- 1 ),
on voit que H x est isomorphe a l'algebre H(l ,x) (la notation serajustifiee plus loin) definie a partir de H u par la specialisation
u. {
0
f-+
U ' .
f-+
',]
1 xe2i1rj/e,
pour 1
:s j :s e. -
1.
On voit que H(l,x) est engendree par un systeme (U.).ES satisfaisant aux relations de tresses et aux relations (U. - l)(X e,-1 + Xe,-2U. + ... + xU:,-2 + U:,-l) = O. On choisit une indeterminee x 1 / 1w1 , et on note X f-+ Xx et X f-+ X(l,x) les bijections entre caracteres absolument irreductibles de W et de H x et H(l,X) definies respectivement par les specialisations
d
x :
v.,O
f-+
x
{ v.,j
f-+
e
1
IW1
/
I~i:,
pour 1
:s j :s e. -
1,
et v.,o
f-+
1
dO,x): { v.,j
f-+
x
l/iwi ~ eWe,
pour 1
:s j :s e. -
1.
La proposition suivante est une consequence facile de 4.2 ci-dessus.
4.18. Proposition. Si west une racine d-ieme de
1r,
on a
Xx(Tw ) = X(w)x(N+N"-N(X~l~O(X»)/d X(l,x)(Uw)
=
X(w)x(N(xHN"(x»/dX(l).
En particulier, on a
Xx(T.... ) = X(l)x
N+N0 N(x)+N°(x) xCi)
X(l,x)(U.... ) = X(l)x(N(xHN"(x))/X(l). On sait deja, comme consequence de 4.17 ci-dessus, que
X(l) divise N(X) On obtient en outre
+ N*(X)·
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Elements reguliers et varietes de Deligne-Lusztig
4.19. Corollaire. Si Xx est generiquement rationnel et si X( w) alors dX(l) divise N(X) + N*(X)·
-# 0,
Supposons que West un groupe de Coxeter. L'algebre H x est alors l'algebre de Heeke generique ordinaire de W (d. §2, B). De plus, comme tous les caracteres d'un groupe de Coxeter sont reels, on a N(X) = N*(X). On deduit alors de 4.18
4.20. Corollaire. Si West un groupe de Coxeter, et si west une racine d-ieme de 1r, on a
xx(Twl = X(w)x
2N-(2N(X)/X(i))
d
•
En particulier,
Xx(T1r )
= X(1)x 2N -(2N(x)!x(I)).
Or Lusztig demontre (d. [Lu3], 5.12.2) que Xx(T1r ) Comparant avec la formule ci-dessus, on voit que
4.21. ax
= X(1)x 2N -(a x +A x ) .
+ Ax = 2N(X)/X(1).
Remarque. Supposons que V n'a que des liens simples. Soit w un element d-regulier, et soit X E Irr(W) generiquement rationnel et tel que X( w) -# 0. Alors pour toute reflexion p de W on a
N d
(1 + X(P)) EZ. X(l)
. X(l) + X(s) En effet, pUlsque Xx(Ts ) = x 2 N(X)
X(l) - X(s) 2 ' on voit que
= N X(l); X(s) .
5. Varietes de Deligne-Lusztig associees aux racines de
1r
Nous revenons ici aux notations anterieures des §1 et 2 : en particulier
West un groupe de Weyl. Nous conjecturons (d. 5.7 ci-dessous) que les varietes de DeligneLusztig associees aux racines de 1r possedent des proprietes remarquables, qui generalisent celles de X 1r • Ces conjectures sont motivees par des conjectures generales sur les representations de groupes finis "abstraits" sur des anneaux £-adiques
M. Broue et J. Michel
112
(cf. [Br]). Elles precisent celles enoncees dans [Br] et [BrMa] (ou peu de precision etait donnee sur "Ie bon choix" de w). Elles doivent etre vues comme des generalisations des resultats de Lusztig sur les varietes associees aux elements de Coxeter (cf. [Lu2]), et de ses resultats partiels sur la variete X wo (cf. [Lul], 3.10., (b)). On notera que beaucoup des proprietes de la cohomologie des varietes associees aux racines de 1r se ramenent au cas des bonnes racines, grace au theoreme 3.17 ci-dessus, et au lemme suivant (cf. par exemple [DeLu], 1.6, case 1). 5.1. Lemme. Suppo~on~ que b et b ' ~ont deux element~ de B;;'d qui ~ont conjugues par permutations circulaire~. Alors le~ groupe~ de cohomologie Hn(Xb,Qf) et Hn(Xb"Qf) ~ont isomorphe~ comme QfG F module~.
A. Quelques proprietes Nous commenl;ons par etablir un certain nombre de proprietes de ces varietes. 5.2 Proposition. Suppo~on~ que w IX~ml = a pour 1::; m < d.
e~t
une racine d-ieme de
1r. Alor~
Demonstration. D'apres 3.17 et 5.1 ci-dessus, on peut supposer que west une bonne racine d-ieme. On note alors w son image dans W, et on a X w = X w (cf. §1.B ci-dessus). Soit m un entier naturel, et soit (B, B.F) E xt;m. On a donc B.Fm = B. On veut demontrer qu'un tel element ne peut pas exister
si m < d.
Si m ::; d/2, puisque west bonne, on a (B, B.Fm) E X~m), ce qui est en contradiction avec l'hypothese B.Fm = B. Supposons d > m > d/2, et d pair. On a alors B ~ B.F d/ 2 . Mais on m-d/2 a aussi B.F d/ 2 w -+ B.Fm, ce qui est contradictoire avec l'hypothese puisque Wo -# w m - d / 2 . Finalement, supposons d > 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).
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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)) =
C
n
ou c
L
=
c
ou Deg,'P (x) designe Ie degre generique associe au caractere unipotent ,
c.
Valeurs de caracteres et applications
Nous reprenons les notations (et une partie des raisonnements) du §2.B ci-dessus, auquel nous renvoyons Ie lecteur. La formule suivante (d. [DiMi2], Prop.1.2, et aussi [Lu], 2.10, ou [DiMil], III), generalise 2.4. Pour tout entier positif m et tout 9 E GF , on a
(5.8)
IX~Fml =
L (L
,EUch(GF)
<"RX>GFC;'xqm(Tw)) ,(g).
XEIrr(W)
11 resulte de 4.20 ci-dessus (et de [Lu3], 5.12.1) que
Xqm(Tw ) = X(w)qm(2N-(a x+A,,:l)/d. Le meme raisonnement qu'au §2, B montre alors que, comme
L
X(w)R x = R w
,
XEIrr(W) la for mule 5.8 s'ecrit
L
IX~Fml =
<"R w >G 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 = c
<"
M. Broue et J. Michel
118 5.9. Pour toute bijection r.p
f-t
1'1' et toute famille de signes
(C'f')'f'EIrr(W(w)) comme ci-dessus, pour tout entier m et tout 9 E G F ,
on a
L
I
IX~Fm =
c'f'r.p(I)(~ qm(2N -(a~,,+A~,,))jd,Ag).
'f'Elrr(W( w))
Nous allons voir que la formule 5.9 ci-dessus permet d'etablir un certain nombre de consequences utiles d'une forme affaiblie des conjectures
5.7. Nous ferons reference aux hypotheses suivantes (qui sont toutes impliquees par les conjectures 5.7).
5.10. Hypotheses.
(H.l) La cohomologie paire HP(Xw, Qf) := EBn H 2n (Xw , Qf) et la co2n homologie impaire Hi(Xw , Qf) := EBn H - 1 (Xw , Qf) sont disjointes comme QfG F -modules. (H.2) Il existe une bijection r.p f-t 1'1' de Irr( W( w)) sur 1'ensemble Uch( G F , w) des caraeteres unipotents 1 de GF tels que RW>GF I: 0, et une famille de signes (E'f' )'f'EIrr(W( w)), tels
<"
que
HP(Xw,Qf) =
L
r.p(I)/'f'
{'f'I€,,=1} i
H (XW1 Qf) =
L
r.p(lh'f'·
{'f'I€,,=-1}
La bijection precedente et la famille de signes correspondantes sont supposees independantes de q (suivant les notations de [BrMaMi], la bijection provient d'une bijection de Irr(W(w)) sur l'ensemble Uch(G, (11' w, 1))). (H.3) II existe une specialisation de u de la forme
(d ,x ) ..= { Ua,j
f-t
-2irrjd )n, j} e 2irrjje, ( e x ' (aES.b)(O~j~e,-l),
ou • x est une indeterminee, • na,j est un nombre rationnel positif, qui ne depend que de Wet w, telle que 1'application T x f-t D x (x E B( w)+) definit un isomorphisme entre l'algebre de Heeke specialisee Q/H(d,q)(W(w)) et l'algebre commutante EndolGF(Rr(Xw , Qf))'
Elements reguliers et varietes de Deligne-Lusztig
119
Grace it la dualite de Poincare entre Rfc(Xw,Qt) et Rf(Xw,lQ!t), les hypotheses (H. 1) et (H.2) ci-dessus impliquent l'existence d'une bijection'P 1-+ BIf' de Irr(W( w)) sur l'ensemble des caracteres irreductibles de Endij,GF (Rfc(Xw , Qt)), telle que
L
IX~pm 1=
EIf"If'(g)B~(Fm).
If'EIrr(W(w» En comparant avec la formule donnee dans 5.9 ci-dessus, on voit donc que, pour tout entier m 2: 0,
On en deduit (utilisant encore une fois la dualite de Poincare) Ie lemme suivant. 5.11. Lemme. Supposons (H.l) et (H.2) ci-dessus.
(1) F est central dans l'algebre commutante Endij,GF(Rf(Xw , lQ!t)). (2) Notons alf' := a,,,, et AIf' := A,,,, et designons par AIf' la valeur propre de F sur une representation de caraciere intervenant dans un des groupes de cohomologie Hn(Xw,lQ!t) (n ::::: 2Njd).
'If'
On a
A
''''
=
f
"''''
q(a",+A",)/d.
Remarque. Le lemme precedent, joint a la proposItIOn 5.2 cidessus, permet de determiner la structure de l'algebre commutante Endij, GF (Rf( X w, Qt)), dans Ie cas ou Ie groupe W (w) est cyclique - toujours en supposant la disjonction des cohomologies paires et impaires de X w' • On verifie d'abord que les nombres AI f .f q(a~",+A~", )/d sont " - ",,,, tous distincts. • On en deduit que l'algebre commutante
n
est engendree par F. Le theoreme suivant permet (dans notre cadre conjectural) de calculer explicitement les valeurs propres de F, ainsi que les parametres (xn"j) de l'algebre cyclotomique 'H(d,x)'
M. Broue et J. Michel
120
5.12. Theoreme. On suppose les hypotheses 5.10 satisfaites. (1) On a la valeur suivante des racines de I 'unite associees aux valeurs propres de F :
('I' = e2i1r (a",+A",)/d' w'I'(w).
(2) On
a
(3) Pour tout H E A et 0 :::; j :::; eH - 1, on pose
On a
Demonstration. On note 'P ...... 'P(d,x) ...... 'Pu la bijection Irr(W(w)) ::: Irr(H(d,x)) ::: Irr(H u ) ou. Ie premier isomorphisme est defini par la specialisation x ...... e 2i1r / d (le compose des deux isomorphismes correspond donc it la specialisation dv - d. §4 ci-dessus). D'apres Ie lemme 5.11, on a 'P(d,x)(D w
)
= 'P(l)('I'x(a",+A",)/d.
Par la specialisation x ...... e 2i1r / d , on voit que
('I' = e- 2i1r (a",+A",)/d'W'l'(W) , d'ou. les deux premieres assertions de 5.12. Comme (d. 4.9 ci-dessus) on a aussi j=eH-l
'P(d,x)(D 1r )
= 'P(1)
II
HEA(w)/W(w)
II (e-
2i1r
(a",+A",)/d x
)
WHeHnH j
' ,
j=O
on voit en particulier que
d'ou. on deduit bien la troisieme assertion de 5.12.
•
La formule explicite suivante pour les valeurs propres de F resulte immediatement du theoreme 5.12 et du lemme 5.11 ci-dessus.
Elements reguliers et varietCs de Deligne-Lusztig
121
5.13. Corollaire. On suppose les hypotheses 5.10 satisfaites. Soit'P E Irr(W(w)). La valeur propre de F sur la representation de caraciere I'i' intervenant dans R w est
6. Le cas non dep]oye A. Generalites On utilise les notations introduites dans ce qui precede: Soit G un groupe algebrique reductif connexe defini sur une cloture algebrique iFp du corps a p elements, et soit F: G --4 G un endomorphisme de Frobenius, definissant sur G une structure rationnelle sur Ie sous-corps lFq a q elements de iFp . On fait operer F a droite sur G. On designe par T Ie tore maximal de G, et par Y Ie Z-module des co-caracteres de T. On note W Ie groupe de Weyl de G, vu comme groupe engendre par des reflexions dans son action sur l'espace vectoriel complexe V := C 0 Y . On pose VIR := IR 0 Y. On designe par q
{I}
--4
P
--4
B
--4
W
--4
{I}.
Pour wE W, on pose ¢w :=
W(
122
M. Broue et J. Michel
fondamentale de W - en d'autres termes, l'ensemble S( <1» des reflexions fondamentales de W( <1» est en bijection naturelle avec l'ensemble des classes de conjugaison de dans l'ensemble S des reflexions fondamentales de W. On note B( stabilise l'ensemble des formes normales, montrent que 6.1. Ie monoide et Ie groupe des tresses associes Ii W( <1»
coincident avec les ensembles des points fixes de respectivement dans Ie monoide et Ie groupe des tresses de W :
B(
= CB+(
et B(rp)
= CB(9).
L'automorphisme rp opere sur l'abelianise Bab de B, et on voit (grace ci-dessus) que l'abelianise de B(rp) cOIncide avec Ie groupe des points fixes de rp sur Bab :
a 4.7
On considerera, comme dans [BrMaMi], la classe W de modulo W dans Ie produit semi-direct W ><1 (rp). De meme, on considerera les classes B+ et Brp dans Ie produit semi-direct B ><1 (rp). Le fait que rp n'est pas necessairement trivial est reflete dans la notation utilisee pour les varietes de Deligne-Lusztig : pour b E B+, on note X~~) (ou plus simplement X bql ) la "variete de Deligne-Lusztig associee a b", definie comme au §1 (d. 1.6) grace au theoreme 1.5. Si b := WI W2 ... W n est un element de B+, decompose sous forme de produit d'elements de B~d' X~~) est isomorphe (par un unique isomorphisme) a la variete des (n + l)-uplets de sous-groupes de Borel (B o, Bl, ... , B n ) de G tels que
On pose qI(WI, W2,' .. , w n ) := (qlWI, qlW2,' .. , qlw n ) , et on remarque que l'endomorphisme de Frobenius definit un morphisme
Ainsi, FO definit un endomorphisme de X bql .
Elements reguliers et varietes de Deligne-Lusztig
123
En particulier, on en deduit une definition de la variete X"'
B. Elements reguliers et ¢J-racines de
1r
On s'interesse aux elements de W
¢J
L'ensemble des resultats du §3 ci-dessus peut etre etendu au cas 1, en generalisant convenablement la notion de racine de 1r.
I:
6.2. Definition. Pour tout entier d, on appelle ¢J-racine d-ieme de 1r tout element w¢J E B+¢ tel que (W¢)d = 1r¢Jd. Les resultats du §3 ci-dessus s'etendent it cette situation plus generale. En particulier
6.3. il existe une ¢-racine d-ieme de 1r sz et seulement si d est un nombre ¢J-regulier pour W. Remarque. La question de savoir si 6.3 s'etend au cas des groupes de reflexions complexes est ouverte. On definit la longueur d'un element w¢J E W ¢J comme la longueur de w:
£(w¢J) := £(w). Ainsi, la longueur de w¢J est egale au nombre de racines positives de W transformees en une racine negative par w¢. On a alors la generalisation suivante de 3.9.
6.4. Proposition-Definition. Boit w¢J E W ¢J. Boit w la pre-image de w dans B~d' Boit dEN. Les proprietes suivantes sont equivalentes : (i) On a (w¢J)d = ¢Jd, et pour tout entier m tel que m ~ d/2, on a £((w¢J)m) = 2Nm/d. (ii) On a (W¢J)d = 1r¢Jd, et pour tout entier m tel que m ~ d/2, on a (w¢J)m E B~d¢Jm.
124
M. Broue et J. Michel
Dans ce cas, on dit que w¢i est une bonne ¢i-racine d-ieme de 1r. Demonstration. Il suffit de remplacer w m par (w¢i)m dans la demon-
stration de 3.9.
•
L'exemple des elements dont Springer montre l'existence dans [Sp], 6.8, prouve que pour tout nombre ¢i-regulier d, il existe une bonne ¢i-racine d-ieme de 1r.
6.5.
Proposition. Soit w¢i E W ¢i admettant un vecieur propre regulier v pour la valeur propre e Zirr / d tel que <1>+ est l'ensemble de racines a telles que 2R( O. Alors w¢ est une bonne ¢i-racine d-ieme de 1r.
Demonstration. La demonstration suit les lignes de celle de 3.1l. Cependant, w¢i n'etant pas necessairement d'ordre d, ses orbites sur <1> ne sont pas necessairement toutes d'ordre d (considerer par exemple Ie cas du type Z A 5 , ou l'element 3-regulier w¢i est d'ordre 6, et a des orbites d'ordre 6). Neanmoins, la demonstration de 3.11 montre que dans une orbite 0, il y a exactement mlOljd racines positives qui sont envoyees sur une racine negative par (w¢i) m, et par consequent on a bien £((wcP )m) = 2Nmjd. On generalise la proposition 3.14 comme suit.
6.6. Proposition. Soit w¢i une bonne ¢-racine d-ieme de w¢i est un ¢i-element d-regulier de W.
1r.
Alors
Demonstration. La demonstration suit celle de 3.14, en y rempla<;ant w par w¢i. Pour toute orbite de w¢ sur <1>, on definit les entiers k( 0) et nl, ... ,nZk(O) comme dans la premiere partie de la demonstration de 3.14, et on demontre de maniere analogue l'assertion 3.15 (on utilise ici Ie fait que f3:= (w¢itd ... +ni-l-la est de meme signe que (w¢i)df3, puisque (w¢i)d = ¢id et que ¢i normalise l'ensemble des racines simples). La deuxieme partie se fait de maniere identique. •
°
La "conjugaison par permutation circulaires" (d. 3.16) se generalise comme suit.
6.7. Definition. On dit que deux elements de B;"d ¢i sont "conjugues par permutations circulaires" s'ils sont equivalents par la cloture transitive de la relation 2 definie sur B;"d par
(w¢ )2 (w' ¢i) si et seulement si il existe x, Y E B;"d w¢i = xy¢i et tels que
{ w' ¢i
= y(¢x)¢i.
Elements reguliers et varietes de Deligne-Lusztig
125
Remarquons que les egalites precedentes peuvent s'ecrire
w<jJ = x(y<jJ)
et
w' <jJ = (y<jJ)x.
De meme que dans Ie §3, on demontre alors Ie theoreme suivant.
6.8. Theoreme. Toute <jJ-racine d-ieme de 'IT est conjuguee par permutations circulaires d'une bonne <jJ-racine d-ieme de 'IT. Indications sur la demonstration. La demonstration est quasiidentique it celIe du cas particulier 3.17. On utilise les resultats de [Del] et de [Cha] rappe1es en 3.18 et 3.19. L'etape cruciale est Ie lemme suivant. 6.9. Lemme. Si (w<jJ)n<jJ-n divise 'IT, alors w<jJ est conjugue par permutations circulaires d'un element w' <jJ tel que (w' <jJ) Ln/2J E B+ ",Ln/2J . red If' Demonstration de 6.9. On remarque que pour tout entier j ::; 1 on a w 1>w ... 1>j - 1 W = (w <jJ )i <jJ - i . La demonstration se fait par recurrence sur n. Pour n = 0 il n'y a rien it demontrer. On remarque aussi que si n est impair, la conclusion cherchee est identique it la conclusion cherchee pour (n - 1), donc resulte de l'hypothese de recurrence. On peut donc supposer que n est pair et on pose n = 2j. Par l'hypothese de recurrence, west conjugue par permutations circulaires it un element w' tel que w' <jJi- 1 E B~d<jJi-l. Donc, quitte it remplacer w par w' on peut supposer que (w<jJ )i- 1 <jJ-U-l) E B~d . Posons v := (w<jJ )i- 1 <jJ-U-l). On peut supposer que v(v. q,j-1 w) > 1 (sinon (v. q,j -1 w)<jJi = (w <jJ)i E B~d <jJi, et la demonstr ation est terminee). D'apres 3.20 ci-dessus, on voit qu'alors v(V.1>j-l w) = 2. Soit (vx).y la forme normale de (v. q,j-l w). Notre hypothese s'ecrit
donc en particulier on a 2
= v( vxy)
::; v(vxy. q,j (vx)) ::; v(vxy. q,j (vxy)) ::; 2.
11 resulte donc de 3.19, (1) que v((vx).y) = v((vx).y.1>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)
dv:
Vs,j
1---7
.
el~i;.
definit une bijection X 1---7 Xv entre l'ensemble Irr(W ¢) et l'ensemble Irr(1i u ¢), telle que Ie diagramme suivant soit commutatij (oiL ( designe une racine de l'unite d'ordre convenable).
K«()W¢ ~ K«() On peut alors calculer la valeur des elements de Irr(1i u ¢) sur les ¢-racines d-iemes de 1r, generalisant ainsi la proposition 4.16 et ses corollaires. Les notations utilisees sont celles de §4. De plus, et par abus de notation, pour X E Irr(W ¢), on note encore X sa restriction a w. L'elements) zXv (1r) et les entiers N(X), ax et Ax sont alors bien definis (exactement comme au §4, ou ils sont definis pour tous les elements de Irr(W)).
Elements reguliers et varietfs de Deligne-Lusztig 6.11. Proposition. Si w¢ est une ¢-racine d-ieme de dans W ¢, on a
127 1r,
d'image w¢
En particulier, dans Ie cas ou West un groupe de Coxeter, et ou on specialise l'algebre generique en l'algebre de Hecke "ordinaire" 1i x , on obtient comme dans Ie §4 (en utilisant aussi 4.21) 6.12. Si w¢ est une ¢-racine d-ieme de
Xx(Tw ¢)
1r,
on a
= X( w¢ )x(2N -(2N( yJ)/x{1))/d = X( w¢ )x(2N -(a,,+A,,:l)/d . D. Varietes associees aux ¢-racines de
1r
Les proprietes et conjectures enoncees dans Ie §5 se generalisent aux varietes de Deligne-Lusztig Xw
1r.
On deflnit de meme un plongement
invariant par l'action (a gauche) de GF, et permet de definir Ie sousmonoidesuivant de CB+(¢d):
Si w¢ = xy¢ et si (w¢)X = y
On laisse au lecteur Ie soin de formuler la generalisation de la proposition 5.6, et de definir Ie sous-monoi"de B+ '" de B+ x w¢ . La formule w'+' D w = F (d. 5.5) est ici remplacee par
M. Broue et J. Michel
128
Les calculs de caracteres des algebres de Hecke faits au §C ci-dessus permettent de generaliser les resultats de §5, C. Pour tout element w¢ E W ¢, on note R w ¢ Ie caractere du QiG F module virtuel Ln( -1 tHn(X w , Qi)' Soit X E Irr(W¢). On note
Ie "caracthe-fantome" associe. On utilise la forme generale de la formule 2.4 : pour tout entier positif m multiple de 8 et tout 9 E GF,
Comme en §5, C, et puisque ~XElrr(W¢) X(w¢)R x alors de 6.12 que 1X.~~m 1=
<"
L "YEUch(C F
=
R w , on deduit
R w¢>c F(;,/oqm(2N-(a-,+A-,))/d,(g).
)
Generalisant les notations introduites au §3, on note W( w¢) Ie centralisateur de w¢ dans W, i.e.,
W(w¢) := {x E W
I (w(¢x)w- 1
= xn·
Soit (d. [BrMaMi], theorem 3.2) 'P ...... ''f' une bijection de Irr(W(w¢)) sur l'ensemble des caracteres unipotents tels que Rw¢>cF I: 0, it laquelle est associee une famille de signes (C'f')'f'Elrr(W(w¢)l telles que <''f',Rw¢>cF = c'f''P(l). On a la generalisation suivante de 5.9.
<"
6.15. Pour toute bijection 'P ...... ''f' et toute famille de signes (C'f')'f'Elrr(W(w
L
C'f''P(1)(';:/o qm(2N-(a-,,,+A-,,,))/d''f'(g).
'f'Elrr(W(w¢))
Sous des hypotheses analogues it 5.10, on calcule les parametres des algebres cyclotomiques associees comme en 5.12, et on demontre (d. 5.13) que
129
Elements reguliers et varietes de Deligne-Lusztig
6.16. pour 'P E Irr(W(w¢i)), la valeur propre de FO sur la representation de caractere '''' intervenant dans R w ", est
Annexe 1 : Les bonnes racines de
'IT
Nous remercions Fran~ois Digne et Gunter Malle pour leurs contributions Ii l'etablissement de cette liste. Pour chaque nombre regulier, nous donnons une bonne ¢i-racine dieme de 'IT.
Ici 2 A r est defini par l'automorphisme ¢i : Si ...... Sr+1-i. Nous commen<;ons par : • c = S1 S2 ... S Lr/2J SrSr-1 ... S Lr/2J +1, un bon element de Coxeter (d. [Bou], ch. v, §6, ex. 2); c est d'ordre r + l. • c'" = SIS2" . SL(r+I)/2J , un bon element de Coxeter ¢i-tordu (c",¢i est d'ordre 2(r + 1) quand r est pair et 2r quand r est impair). • w = SI S2 ... S Lr/2J SrSr-1 ... S Lr/2J, un bon element regulier d'ordre r. Quand r est pair (C",¢i)2 = c et west ¢i-stable alors que quand rest impair (c",¢i f = w et c est ¢i-stable. Nous obtenons la liste suivante de bons elements reguliers: + 1 (d = (r + 1)/i), et wi OU i divise r (d
• A r : ci OU i divise r
si west d-regulier on a W( w) ::: • •
Bt~ J'
= r/i);
2A r , r pair: (c",¢i)i ou i divise r + 1 (d = 2(r + 1)/i), et wi¢i ou i divise r (d = r/i). 2A r , r impair: (c",¢i)i ou i divise r (d = 2r/i), et ci¢i ou i divise r+l(d=(r+l)/i). Pour 2 A r , si w¢ est un ¢i-element d-regulier on a si d est impair si d == 2 (mod 4) si d == 0 (mod 4)
Remarques. Le fait que la liste des nombres reguliers donnee ci-dessus est complete peut se deduire de [Sp]. Pour verifier que les elements
M. Broue et J. Michel
130
fournis sont bons, nous les presentons comme des permutations de r + 1 lettres :
• c=(1,2,···,lr/2J+1,r+1,r,···,lr/2J+2) w = (1,2, ... , l(r + 1)/2J,r + 1,r, ... , l(r + 1)/2J + 2). Sous cette forme il est facile de verifier que c(r+l)/2 = Wo, et wow(r-l)/2 est de longueur (r + 1)/2 si r est impair, et que w r/ 2 = Wo et woc r/ 2 est de longueur r /2 quand r est pair.
•
A1.2. B r
:
C=8-(}" 0 t
092
093
Sr
Soit C = t8385 ... 828486' .. , un bon element de Coxeter. Alors c i ou i divise 2r sont des bons elements reguliers; on a d = 2r / i et si d est pair SInon
A1.3. Dr, 2D r et 3D 4
:
9'2 e>---CHJ···O
Ici 2D rest defini par I' automorphisme ¢ qui echange 81 et 82, et par T : 81 1---7 82 1---7 84 1---7 81' Nous commen<;ons par: • c = 838587 ... 81828486"', un bon element de Coxeter. Remarquons que c est ¢-stable et d'ordre 2r - 2. • c¢ = 818384'" 8r, un bon element de Coxeter ¢i-tordu (c¢¢i est d'ordre 2r). • w = (c¢¢i)2 = 818283828483 ... 8 r 8 r - l = 8182 ... 8r8283 ... 8 r -1 = 818384 ... 8r8283 ... 8 r , un bon element regulier d'ordre r. Nous obtenons la liste suivante de bons elements reguliers : • Dr: ci ou i divise 2r-2 (d = (2r-2)/i), et wi OU i divise r (d = r/i). • 2Dr: ci¢i OU i divise 2r - 2 (d = (2r - 2)/i), et (c
si d divise r
W(w¢i) ::: SInon Si d est pair, on a:
nt
d)
si d 12r (et si 2r/d est impair, dans Ie cas 2D r )
2r
W(w¢) '" {
a
B(d)
L2(r -l) J d
sman
131
Elements reguliers et varietCs de Deligne-Lusztig
Remarques. Ici aussi, un bon moyen de verifier que w et c sont bons consiste it les presenter comme permutations, via Ie plongement Dr C B r C A 2r - 1 . 11 est alors facile de comparer leur puissance avec woo
Pour 3D4 , nous utilisons C r = 8183, un bon element de Coxeter Ttordu (d'ordre 12). On a (C r T)3 = w. D'autre part, C est T-stable. Nous obtenons ainsi la liste suivante de bons element reguliers : • ciT OU i divise 6 (d = 6/i), 2 • CrT (on a (C r T)4 = C T) (d = 12). Si w¢ est un ¢-element d-regulier on a: d 3,6 12 2 type de W( w¢)
lei 2 E 6 est defini par l'automorphisme ¢ qui echange et
81
et
86,
et
83
85.
Pour E 6 , les bons element reguliers sont : c i OU i divise 12 et C = 818486828385, un bon element de Coxeter (d = 12/i). L'element c est ¢-stable. • wi OU i divise 9 et w = 8183848284868584 (d = 9/i). • W 'i OU i divise 8 et w' = 838584818683858482 (d = 8/i). L'element w' est ¢-stable. Si west un element d-regulier on a: d 2 3 4 6 8,9,12 type de W(w) F4 A/ A/ B/,3 •
Pour 2 E 6 , nous obtenons la liste : c i ¢ OU i divise 12 (d = 12/i). (cq,¢)i OU i divise 9 et cq, = 82818384, un bon element de Coxeter ¢-tordu (on a (cq,¢f = w) (d = 18/i). • w ' ¢(d=8). Si w¢ est un ¢-element d-regulier on a: d 1 2 3 4 8,12 18 type de W( w¢)
• •
9"
A1.5. E 7
• •
:
()--()--()....o
ci OU i divise 18 et c = 81848682838587 (d = 18/i). wi OU i divise 14 et w = 818384828487868584 (d = 14/i). Si west un element d-regulier on a:
M. Broue et J. Michel
132
d type de W(w)
7,14
9
9,18
82
Al.6.Es:~ ci ou i divise 30 et c ='.sIS4S6SSS2S3SSS7 (d = 30ji). wi ou i divise 24 et w = S1S3S4S2S4SSS7S6SSS4 (d = 24ji). 'i OU i divise 20 et w' = SSS7S6SSS4S2S3S1S4S3SSS4 (d = 20ji). • W Si west un element d-regulier on a: 3,6 4 5,10 15,30 20,24 d 2 3 S type de W(w) E s A4 £4 A 2 • •
Al.7. F4 et 2F4
:
0--C=J-0 stu
v
lei 2 F4 est defini par ¢ qui echange
S
et v, et t et u.
Pour F4 nous obtenons la liste : ci OU i divise 12 et c = sutv (d = 12ji). wi OU i divise 8 et w = svtutu (d = 8ji). L'element west ¢-stable. Si west un element d-regulier on a: d 23,6 4 8,12 type de W(w)
• •
Pour 2F4 , (c¢¢)i OU i divise 3 et c¢ = su (on a (C¢¢)2 = c) (d = 24ji). (w¢¢)i OU i divise 3 et w¢ = stut (d = 12ji). Noter que (w¢¢f est un element regulier d' ordre 6 qui n'est pas Ie carre d'un bon element de Coxeter. • w i ¢ OU i divise 8 (d = 8ji). Noter que (C¢¢)3 n'est pas ¢-stable, done est un element regulier d'ordre 8 different de w¢. Si w¢ est un ¢-element d-regulier on a: 1,2 4 8 12 24 d
• •
type de W(w¢)
h(8)
K2
A/
C6
C12
Al.8. h(e) et 2 h(e) : ()--!-() (Ceci couvre les cas B 2 , 2 B 2 , 8
lei
• •
2I2 (e)
G 2 , 2G 2 )
t
est defini par ¢ qui echange
S
et t.
Pour h(e): ci ou i divise e et c = st (d = eji). e 2 Wo (d = 2); on a Wo = c / si e est pair. Si west d-regulier on a W( w) ~ C e sauf si d = 2 auquel cas si e est impair sInon
133
Elements reguliers et varieUs de Deligne-Lusztig
Pour 2h(e): • (cc/J¢)i ou i est un diviseur impair de 2e et Cc/J = s (d = 2e/i). • ¢(d=l). Si west un ¢-element d-regulier on a W( w¢) ~ C e sauf si d auquel cas si e est pair
=2
sInon
A1.9.H3:~ stu
• ci ou i divise 10 et c = sut (d = 10/i). • wi OU i divise 6 et w = ststu (d = 6/i). Si west un element d-regulier on a: d 2 3,6 5,10 type de W(w) A1.lO. H 4
:
cY-o-o-o stu v
ci OU i divise 30 et c = sutv (d = 30/i). wi ou i divise 20 et w = ststuv (d = 20/ i). • W 'i ou i divise 12 et w' = ststutstuv (d = 12/ i). Si west un element d-regulier on a: d 2 3,6 4 5,10 12,20 type de W(w) A25
• •
15,30
Annexe 2 : Groupes de Coxeter tordus Nous presentons ici une table qui regroupe des informations connues sur les "groupes de Coxeter tordus", i.e., les couples ((V, W), ¢) formes d'un groupe de Coxeter fini irreductible W (vu dans sa representation naturelle reelle V) et d'un automorphisme ¢ de V qui stabilise l'ensemble des generateurs distingues de W (cf. §6 ci-dessus). Seules les deux dernieres colonnes necessitent une explication.
• Corps. Nous indiquons ici Ie plus petit corps K sur lequella representation du produit semi-direct W >l (¢) est definie. • Degnfs et facteurs. Comme l'operation de ¢ sur chaque composante de degre donne de l'espace 5(V)w des invariants de W dans l'algebre symetrique de V est semi-simple, on peut trouver une famille {51' 52, ... ,5r } d'elements
M. Broue et J. Michel
134
homogenes de degres respectifs (d l , d2 , ••• ,d r ), avec les proprietes suivantes (cf. [St2], 2.1). (1) La famille {51, 52, ... , 5 r } est algebriquement independante, et engendre l'algebre 5(V)w. (2) Elle est formee de vecteurs propres pour l'action de ¢>, 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<:teurs 3
r+1
1 -1
( -1)T+1
2 81
82
8r
~s. .. ·0 ~
2D r
81
3D 4
83
81
2(r - 1)
2 4
r
1 -1
1 1
81 +-+ 82
Sr
83
2 4 6 1 j
81 t--+ 82 t--+ 84 t--+ S 1
4
o---o=c:r-o
S +-+
v,t
+-+
Q
P
1
8.
2 2F4
Corps
6 8
12
Q(V2)
1 -1 1 -1
u
t
9
82
2E6
2
o--
83
8.
8.
S 1 +-+ S 6, 83 +-+
s5
5 6 8
~
Q
1
86 2
2[2 (e)
9 12
1 -1 1 1 -1
s
+-+
t
e
1 -1
Q( cos( 7l"Ie))
t
Annexe 3 : Groupes de Coxeter cyclotomiques
Les tables suivantes regroupent des informations utiles sur tous les "groupes cyclotomiques", i. e., les centralisateurs d'elements reguliers des groupes de Coxeter finis irreductibles. L'ensemble des informations est essentiellement une synthese de celles donnees dans [Be], [BrMa], [BrMaRo], [OrSo], [Sp] . • La colonne "S& T" fournit Ie nom donne au groupe par Shephard et Todd et utilise traditionnellement depuis. • La colonne "Nom" donne Ie nom donne au diagramme dans [BrMa].
135
Elements reguliers et varietes de Deligne-Lusztig
• La colonne "Diagramme" fournit Ie diagramme introduit dans [BrMa]. Nous reproduisons ici, pour Ie confort du lecteur, les conventions permettant d'interpreter ces diagrammes. correspond it la presentation
(])--!---(l)
Le diagramme
s
Sd
t
= t d = 1 et
ststs···
'---v---" e factors
~
Le diagramme
=~ tstst· .. e factors
correspond it la presentation
t
S5
sGJQt
Le diagramme
sa
= tb =
U
Le diagramme
= t 3 = 1 et
= 1 et
stu stu ... = tustus· ..
'-v--'
'-v--'
e factors
e factors
?9
~ v
= tsts .
correspond it la presentation
~u
C
stst
t
= 'ustust .... -v--' e factors
correspond it la presentation
w
uv = vu , sw = ws, vw = wv, sut = uts = tsu , svs = vsv, tvt = vtv, twt = wtw, wuw
= uwu.
• Les "degres" et "codegres" ont ete introduits dans Ie §4, A, c!dessus. • Chaque sommet du diagramme represente un generateur d'un sous-groupe parabolique minimal W H associe it un hyperplan de reflexion H (cf. 4, A). La colonne "Orbites d'hyperplans" fournit, pour chaque sommet du diagramme, Ie cardinal WH de la classe de conjugaison de H sous W (qui joue un role fondamental dans les calculs du §4, E). • La colonne "Corps" fournit Ie corps K engendre par les valeurs du caractere de la representation de reflexion du groupe considere. • La colonne "IZWI" fournit Ie cardinal du centre de W, et la colonne {3 fournit Ie generateur du centre du monolde de tresse B+ (cf. [BrMaRo], ou 4.8 ci-dessus).
-""' ~
<.l
~
....; ...., ""'
S&T
'I
G4
I:l:l
Gs
~
G I6 Gn G 32 G(d,l,r) d>2 -
Gs GID G 26 G(2d, 2, r) d2:2 G2D
Gg <0
M
......
Degres
0--0···0
Ar
' ';l"' ~
Diagramme
Nom
2,3, ... , r
@-----<])
A(4) 2
0-----(D
A(S) 2
@-----@
A(3)
@---@----@
,
,
t
'2
@~
B(4,3) 2
0--:])
B(2,3)
e-J>----Q)
d r 2d )
,
0<4) 2
0,2
8 :4
Q(j)
2
(8t)3
8,12
0,4
8:6
Q( i)
4
(8t)3
20,30
0, 10
8: 12
Q(e 2i "./S)
10
(8t)3
6,9,12
0,3,6
8: 12
Q(j)
3
(8tu)4
12,18,24,30
0,6,12,18
8 :40
Q(j)
6
(8tuV)S
[1,2, ... ,rJd
[0,1, ... ,r-1Jd
Q(e 2i "./d)
d
6, 12
0,6
8 :4, t:4
Q(j)
6
(8t)2
12,24
0, 12
8:6, t:8
Q(e 2i "'/12)
12
(8t)2
,
6,12,18
0,6,12
8:9, t:12
Q(j)
6
(8tu)3
tor, 82:
r(r;l)
d
(t8283 ... 8r
r
t
t
~ • .. 0 8283
I~3,3\5)
4,6
t
,
t
(81 ... 8 rr+ 1
'r
'3
B(3,3) 2
3
1
v
u
@-J-O···O t
Q
t
@---@-@---@
r
2
f3
t
,
B(d)
81:
IZWI
t
,
4
0,1, ... ,r - 1
r(r+l)
Corps
t
,
A(3)
+1
Orbites d 'hyperplans
'r
'2
A(3) 2
3
Codegres
34
@--L.a>
,
r
12,30
0,18
8:20
8,24
0, 16
8:6, t: 12
Q(ei'" /d)
r
d(2I\r) t=(8~8283 ... 8r)
V5)
6
('8t)S
Q(e i "./4 )
8
(8t)3
Q(j,
t
0-:)
,
.!I
[1,2, ... ,r-1J2d t:r,82: r (r-l)d rd [O,I, ... ,r-1J2d excepter=2: t :2, 82 : d l 8; : d
t
~ 2M
S&T
Nom
Diagramme
G(e,e,2)I() e>3 2 e G I2 G 22 G 23
Degres
~ ,
2,e
Codegres O,e - 2
t
6,8
,ca~
M2
H3
Corps
IZWI
e pair ~:e/2, t:e/2 Q(cos(271"/e)) e /\ 2 e ImpaIr s:e
~">-
f3
~
"> ;:l
.....
(st)e/(eI\2)
'""'l
">-
,ca~
f{2
Orbites d'hyp erp lans
12,20
0,10 0,28
s: 12 s:30
Q(yC2) Q(i,0)
2
""~
(stu )4
"> "'l
4
'"">
(stu )5
..... <::
'" .....
o-Lo--o t
2,6,10
0,4,8
s: 15
Q(0)
2
(stu )5
;1.
CK>=O-O v t u
2,6,8,12
0,4,6,10
s:12, t:12
Q
2
(stuv)6
'".....
cY-o-o-o , t v
2,12,20,30
">">-
u
G 28 G30
F4
H4
">
b 0,10,18,28
s:60
Q(0)
2
~
(stuV)15
<.C;.
u
G31
G35
L4
E6
~ ~ ,. ,, ~ , , ,. ~ ,. ,, , 1
G 36
E7
, 1
G37
E8
t
v
, 1
I
t:-<
8,12,20,24
0,12,16,28
s:60
Q(i)
4
..
>:
(stuwv)6
'".....
w
"
<.C;.
2,5,6,8,9,12
0,3,4,6,7,10
s:36
Q
1
(Sl ... S6 )12
2,6,8,10,12,14,18
0,4,6,8,10,12,16
s:63
Q
2
(Sl .. . S7)9
s: 120
Q
2
(Sl ... s8 )15
"
"
"
;:l ">
"
"
"
"
"
2,8,12,14,18,20,24,30 0,6,10,12,16,18,22,28
I
......
~
-.]
138
M. Broue et J. Michel REFERENCES
[Ari]
S. Ariki, Representation theory of a Hecke algebra ofG(r,p,n), J. Algebra 177 (1995), 164-185. [ArKo] S. Ariki et K. Koike, A H ecke algebra of (Z / rZ) I Sn and construction of its irreducible representations, Advances in Math. 106 (1994), 216-243. [Be] D. Benson, Polynomial Invariants of Finite Groups, London Math. Soc. Lecture Note Series 190, Cambridge University Press, Cambridge, 1993. N. Bourbaki, Groupes et algebres de Lie, chap. 4, 5 et 6, Hermann, [Bou] Paris, 1968. K. Bremke et G. Malle, Reduced words and a length function for [BreMa] G(e, 1, n), submitted (1995). [BrSa] E. Brieskorn et K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245-271. [Br] M. Broue, Isometries parfaites, types de blocs, categories derivees, Asterisque 181-182 (1990), 6]-92. M. Broue et G. Malle, Zyklotomische Heckealgebren, Asterisque 212 [BrMa] (1993), 119-189. [BrMaMi] M. Broue, G. Malle, J. Michel, Generic blocks of finite reductive groups, Asterisque 212 (1993), 7-92. [BrMaRo] M. Broue, G. Malle, R. Rouquier, On Complex Reflection Groups and their associated Braid Groups, Representations of Groups (B.N. Allison and G.H. Cliff, eds.), Canadian Mathematical Society, Conference Proceedings, vol. 16, Amer. Math. Soc., Providence, 1995, pp. 1-13. R. Charney, Artin groups of finite type are biautomatic, Math. Ann. [Cha] 292 (1992), 671-683. A. M. Cohen, Finite complex reflection groups, Ann. scient. Ec. Norm. [Co] Sup. 9 (1976), 379-436. [Del] P. Deligne, Les immeubles des groupes de tresses generalises, Invent. Math. 17 (1972), 273-302. P. Deligne, Action du groupe des tresses sur une categorie, Invent. Math. [De2] (1995) (to appear). P. Deligne et G. Lusztig, Representations of reductive groups over finite [DeLu] fields, Annals of Math. 103 (1976), 103-16l. J. Denef and F. Loeser, Regular elements and monodromy of discrimi[DeLo] nants of finite reflection groups, Indag. Mathern. 6 (2) (1995),129-143. F. Digne et J. Michel, Fonctions £. des varietes de Deligne-Lusztig et [DiMi1] descente de Shintani, Memoires de la S.M.F., vol. 20, 1985. _ _ _ , Cohomologie de certaines varietes de Deligne-Lusztig attachees [DiMi2] Ii des elements reguliers, Preprint (1994). E.A. Gutkin, Matrices connected with groups generated by mappings, [Gu] Func. Anal. and Appl. (Funkt. Anal. i Prilozhen) 7 (1973), 153-154 (81-82). H. Hiller, Geometry of Coxeter Groups, Research Notes in Mathematics [Hi] vol. 54, Pitman, Boston, 1982. G. Lehrer, Poincare Polynomials for Unitary Reflection Groups, Invent. [Le] Math. (1995) (to appear). G. Lusztig, Representations of finite Chevalley groups, C.B.M.S. Re[LuI] gional Conference Series in Mathematics, vol. 39, A.M.S., Providence, 1977. _ _ _ , Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 [Lu2] (1976), 101-159. _ _ _ , Characters of reductive groups over a finite field, Annals of [Lu3] Mathematical Studies, no 107, Princeton University Press, Princeton, New Jersey, 1984. G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgrup[Mal]
Elements n£guliers et varieth de Deligne-Lusztig
[Ma2] [OrSo] fOrTe] [Ri] [Sp]
[Stl] [St2]
139
pen, J. Algebra 177 (1995), 768-826. ___ , Degds relatifs des algebres cyclotomiques associees aux groupes de re.flexions complexes de dimension deux (1995) (to appear). P. Orlik et L. Solomon, Unitary reflection groups and cohomology, Invent. Math. 59 (1980), 77-94. P. Orlik et H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin-Heidelberg, 1992. J. Rickard, Splendid equivalences, Proc. ofthe London Math. Soc. (1995) (to appear). T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159-198. R. Steinberg, Differential equations invariant under finite reflection groups, Trans. Am. Math. Soc. 112 (1964), 392-400. ___ , Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, number 80, American Mathematical Society, Providence, 1968.
Michel Broue Institut Universitaire de France, U.FoR. de Mathematiques de l'Universite Paris 7 Denis-Diderot, et Institut de Mathematiques de Jussieu (UMR 9994 du CNRS) e-mail: [email protected] Jean Michel UoFoR. de Mathematiques, Universite Paris 7 Denis-Diderot, 2 Place Jussieu, F-75251 Paris Cedex 05, France e-mail: [email protected] Received February 1995
Local Methods for Blocks of Reductive Groups over a Finite Field Marc Cabanes and Michel Enguehard
Introduction The aim of this paper is to show for general blocks of reductive groups over a finite field some analogues of the results in [CE.l] about unipotent blocks. This includes the distribution of ordinary characters into blocks (Theorem 3.3) and the structure of defect groups (Theorem 3.5), thus yielding the main results of [FS.1; FS.2; Br.1; CE.1]. It should be noted that, by a theorem of BroUl~ [Br.2, 2.3], most blocks are in some sense "equivalent" to a unipotent block. This theorem provides an important idea one must keep in mind, but in addition to the restrictions on the blocks concerned, BroUl~'s equivalence (a perfect isometry) does not imply the isomorphism of defect groups (Remark 3.6) but just the equality of their orders. On the other hand, our statements do include a restriction on the "rational series" we consider, and an assumption on the "Jordan decomposition" of ordinary characters of reductive groups over a finite field (2.1.R), part of which is yet conjectural in the present state ofthe theory (see Definition 2.1). Let G be a connected reductive algebraic group with a Frobenius endomorphism F defining a rational structure over the finite field q' Let (G*, F) be a dual group. We study the i-blocks of GF for i a prime not dividing q. In order to avoid technicalities in this introduction, assume Z(G) is connected and i > 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
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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
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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
Proof. Denote by e the order of q mod i. Then Eq,t is the set of all ei B (8 2: 0). As i divides (E:qb - 1), ¢ets divides (f-X b - 1) for some s
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and this implies that ¢e divides (fX b - 1) (recall that £ =f. 2). Then, for s ~ t, ¢et' divides ((fXb)O -1) if, and only if, ¢et' divides ((fXb)Ol -1). This in turn occurs if and only if £t ~ O:t or ¢et'/ol divides (fX b - 1). Hence bk = (ak)t (the £-part of ak), and mk is prime to £. The order of the center of a simply-connected covering of [K, K] is TIkmk. • We recall a useful property of defect groups in reductive groups over a finite field. Notation 1.8. [C, 2.2] If A is an abelian normal subgroup of a group G, then one writes A <J G if and only if, for any subgroup H <:;;; G such that A <:;;; H, A is the unique maximal abelian normal subgroup of H. Proposition 1.9. Let (G, F) be a connected reductive group defined over q. Let £ be a prime:::: 5 and not dividing q or £ = 3 E r(G, F). Assume G = G a and let T be a diagonal torus of G. Let Q be a Sylow £-subgroup ofNGF(T), then Tf <J Q. Proof. Note that if G = G a , the diagonal torus T satisfies T = CG(TJ by [CE.I, 2.4] and Te is thus a maximal ¢e-subgroup ofG. For £ :::: 5, Proposition 1.9 is therefore a consequence of [C, 4.4]. Moreover the relation Tf <J Q fails only when some non trivial £-element v of NG(T)F satisfies [v, [v, Tf]] = {I} [C, 2.3]. By [C, 4.5], this has to happen in a rationally irreducible component H of G of type (A 2 , cqd) for some c = ±I, with w := vT of order £ = 3 in a symmetric group of degree 3 acting on S = Tn H [C, 4.1]. We may assume ZO(G) <:;;; H. Then 3 E r(H, F) (Lemma 1.2). Denote by 7r: H -+ Had the reduction map mod. Z(H). By a direct computation in Had = PGL(n)d, and since 7r(C H (h)) = CHad (7r(h)) for any semisimple element h E H [Ca, 3.5.3], one sees that CH(v) n S <:;;; Z(H). We have also CH(v)F = CH(v)F by [CE.I, 2.1.(iii)]' so the relation [v, [v, Sf]] = {I} implies [v, sf] <:;;; Z(H)F. Then w, a cycle of order 3, acts trivially on 7r(sf). But 7r(sf) = 7r(S)[ [CE.I, 2.1.(ii)]' so w fixes all the rational 3-elements of the diagonal torus of Had. This is easily seen to be impossible in Had F = PGL(3, cqd) (recall that 3 divides cqd - 1 since G = G a relatively to £ = 3) .
•
2. The Irreducible Characters 2.1. Lusztig series and Jordan decomposition of characters
When (G, F) is a connected reductive group defined over q, we consider the irreducible representations, characters and central functions of GF with values in an algebraically closed field K of characteristic zero.
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When L ~ G is an F-stable Levi subgroup of G, let us recall the existence of Lusztig map Rr : CF(L F , K) ---t CF(G F , K) and its adjoint (for the usual inner product) "Rr : CF(GF,K) ---t CF(LF,K), between the spaces of central functions on G F and L F (see [DM.2, §§1112] for a systematic account). Let us recall also the partition of the set of irreducible characters of G F into Lusztig's rational series Irr(G F ) = U£(GF,s) where s ranges over the semi-simple elements of (G*)F modulo (G*)F_conjugacy, see [1.2,§2; DM.2, 14.4:1]. When s = 1, the elements of £(G F , 1) are called unipotent characters. One has the so-called "Jordan decomposition" of characters. For any semi-simple element s of G*F, Lusztig has shown the existence of a bijection 'Y f-+ X~, from £(CGo (s)F, 1) to £(G F , s), [1.2, 5.1]. In what follows, we shall impose CG• (s) F = CG s) F (otherwise the bijection holds with an appropriate definition of £(C G• (s 1)). Certain natural conditions force the uniqueness ofthe bijection, see [DM.I, 7.1]. We define a property of compatibility between such a "Jordan decomposition" and Lusztig's functors. 0
(
t,
Definition 2.1. Let (G, F) be a connected reductive group defined over q. Let s be a semi-simple element of (G*)F such that CG• (s)F = CG.(s)F. For any pair (L, L*) of dual Levi subgroups by duality between maximal tori T and T* of G and G * and such that s E T*, there exists a bijection A f-+ X~ A from £(cr.. (s)F, 1) to £(L F , s). When T ~ M ~ Land T* ~ M* ~ L* are in duality and A E £(Cr.. (s)F, 1), p, E £(CMo (s)F, 1), we will denote by (2.l.R) the following equality
(We put as usual CG = (_I)q-rank(G), see [DM.2, 8.3].)
Remark 2.2. If L* is a Levi subgroup of G * and s is semi-simple in L*, then CGo (s)nL* = Cr.. (s) (write L* = CG• (8*) = CGo (8*) for a torus 8* and note that 8* ~ CG• (s)). A consequence is that, if s is a rational semi-simple element of G* which satisfies the condition CG.(s)F = C G• (s)F, then it satisfies it in any F-stable Levi subgroup containing s. By [CE.I, 2.1.(ii)]' this applies to CG.(t) when t E CG'(S)f(G,F) : one gets CG• «s, t»F = CG• «s, 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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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.
<
Proposition 2.6. Let (G, F) a connected reductive group defined Over a finite field and s E (G*);: such that CG,(s)F = CG,(s)F. Let d 2: l. Assume there is a family of bijections A --> 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-
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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
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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)
Proposition 3.2. Let 5, H, H, HI, ... , H m be subgroups of a jinite group. Assume the following: • H
Blocks of Reductive Groups (ii) Let
0 E Irr(b)
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denote the canonical character of b. By Clifford
fr
-
theory, ResHO = u L()E(H/H).()o 0 where u divides IH: HI [NT, 3.5.12],
00 E Irr(H), and HI H acts on Irr(H) by conjugacy. Since S permutes the O's and HI H is of order prime to i, one may assume that 00 is S-fixed. Then Brs(bH(OO)) =f. 0 by (i). The fact that He ~ H also implies Z(H)e ~ H, so Z(H)e ~ Z(H). Moreover Z(H)e acts trivially on the representation space of 0, so each of the O's also has Z(H)e in its kernel. The fact that u is prime to i gives O(l)e = O(l)e = IH : Z(H)le. Therefore 0 is the canonical character in a block of defect Z(H)e in H, and Z(H)e = Z(Hk Blocks with central defect groups are easy to express from one of their irreducible characters [NT, 3.6.22, 5.8.14]. One has b =
IH:~(lk)"1
IH:Wk)"
LhEH", 0(h-1)h and bH(O) = I LhEH", 0(h-1)h for all o. Now, our claim is equivalent to O(HS) S?; J(O). It suffices to check that (Res~O)(HS) S?; J(O), or equivalently Brs(L()E(H/H).()o bH(O)) =f. O. Assume Brs(L()E(H/H).()o bH(O)) = 0 and multiply this by Brs(bH(OO)). One obtains Brs(L()E(H/H).()o bH(O)bH(OO)) = 0 since Brs is an algebra morphism on (O[H])s. But the bH(O)'S are distinct since they are defined by distinct canonical characters. We get Brs(bH(OO)) = 0, a contradiction. • 3.2. Blocks of reductive groups over a finite field and their ordinary characters
In Theorem 3.3, we describe the i-blocks of G F by means of e-cuspidal pairs of (G, F). Theorem 3.5 gives the structure of defect groups. The proofs are given in Section 4. Theorem 3.3. Let (G, F) be a connected reductive group defined over LeU E r(G, F) (see Notation 1.1) and e be the order ofq mod i. Let s E (G*)f, be semi-simple and such that CG.(s)F = CG.(s)F. Assume
q.
that (2.l.R) holds for any pair (L, M) of e-split Levi subgroup of G and any (A, p,). Let (L, () be an e-cuspidal pair such that L is in duality with a Levi subgroup L* ofG*, s E L* and (E £(LF,s). Then (1) There is an i-block of G F , denoted by bGF(L,(), such that Irr(bGdL,()) nE(GF,s) = {X; (Rr(,X)GF =f. O}. (2) If (V, is another e-cuspidal pair such that bGF (V, is defined, then bGF (V, (I) = bGF (L, () if and only if (V, nand (L, () are G F _ conjugate. (3) Let t E CG.(s)f, let'Y E £(C G.(st)F,l). Let G(t) be a Levi subgroup of G in duality with C G•(t). Then X~'1" E Irr(bGdL, ())
n
n
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if, and only if, for some X E Irr(G F ), one has (Rr(, X)GF =j:. 0 and G G(t) ) (RG(t)Xs,"Y ,x GF =j:. O.
Concerning maximal tori and Weyl groups, we use the following notations.
Notation 3.4. When T ~ G is a maximal torus in a connected reductive group, one denotes by PT: NG(T) ---+ W(G, T) the reduction map mod T. If H is a connected subgroup of G normalized by T, then we denote by W(GIH, T) the subgroup of W(G, T) generated by reflections associated with the roots of G relative to T that are orthogonal to all the roots of H relative to T. Theorem 3.5. We keep the hypotheses and notations of Theorem 3.3. In addition, let M* = C G• (ZO(Ct. (s))%'). Let T* be a maximal Fstable torus ofCt.(s). Then T* ~ M* ~ L* are all Levi subgroups. Let T ~ M ~ L be in duality with them. Then there is a subgroup N such that
(*) N ~ NG(T), PT(N) = W(C G • (s)ICt. (s), T*) (see Notation 3.4), F(N) = N, and N n T = T 2 • Take any N satisfying (*) and any Sylow f.-subgroup S in N F . Denote Z = ZO(M)f. Then (1) S ~ PT(S) and the latter is a Sylow f.-subgroup of W(CG.(s)ICt.(s), T*)F, (2) Z.S is a semi-direct product with Z <J Z.S (see Notation 1.8), (3) ZS is a defect group of bGF(L, (). Remark 3.6. We keep the same notations and hypotheses as above. The defect groups of bGF (L, () are isomorphic to the semi-direct product of Z = ZO(M)f by a Sylow f.-subgroup of W(C G • (s)ICt. (s), T*)F, the action being defined by restriction of the natural one of W (G *, T*) ~ W(G, T) on T. Let (G(s), F) be a connected reductive group defined over q and in duality with C G• (s) around dual tori T and T*. One may assume that G(s) is defined by a root datum ((X(T), R s , Y(T), R(C G • (s), T*)), F), but G(s) is not necessarily a subgroup of G. Let L(s) be the e-split Levi subgroup of G(s) containing T and in duality with Ct. (s). Then A E £( Ct. (s)F, 1) defines an element of £(L(s)F, 1) that we still call A and which is e-cuspidal. Theorems 3.3 and 3.5 apply to (L(s), A) in G(s) (one has f. E r(G(s), F) by [CE.l, 1.2]). We claim that the defect group of the unipotent block bG(sV (L(s), A) is isomorphic to the defect group of bGF (L, (). By Proposition 1.3, Ct. (s) = Cc~. (s)(ZO(Ct. (s))%'). Hence the defect group of bG(sV(L(s), A) is the semi-direct product of
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ZO(L(s))f by a Sylow R-subgroup of W(C~*(s)ICi.*(s),T*)F (Theorem 3.5). In those descriptions of the defect groups of bGF (L, () and bG(s)F(L(s), A), the Weyl group acting is the same with same action on T F . By rCa, 4.4.5] one has IZO(M)fl = IZO(M*)fl, IZO(Ci.* (s))fl = IZO(L(s))fl· We show in Lemma 4.1 that CM*(s) = Ci.* (s) and ZO(M*)f = ZO(Ci.* (s))f. Hence one has W(L(s), T) W(Ci.*(s), T*) ~ W(M*, T*) = W(M, T), which implies Z(M) ~ Z(L(s)). Therefore ZO(L(s))f = ZO(M)f. This isomorphism of defect groups gives further evidence of the conjecture made by several authors that the Jordan decomposition of characters induces a Morita equivalence between block algebras (see [Br.2, 2.3]), since, by a recent theorem of Puig proved under certain additional hypotheses, any two Morita equivalent block algebras must have isomorphic defect groups. Concerning perfect isometries (see [Br.2, § 1]), our theorems easily imply the following. Suppose T* ~ C~*(s) ~ K* where K* is an F-stable Levi subgroup of G*. Let T ~ K where the latter is an Fstable Levi subgroup of G in duality with K*. Then a defect group of bGF(L, X~.x) may be taken in KF, it is a defect group of bKF (L n K, x~\nK)), 'and the functor those blocks (use [Br.2, 2.1]).
R~
defines a perfect isometry between
Remark 3.7. The statement (3) of Theorem 3.3 might be stated in terms of e-cuspidal pairs of G(t), as in [CE.l, 4.4.(iii)]. 4. Proof of the Theorems
4.1. Three main steps We fix s E G* and (L, () as in Theorem 3.3. The first inclusion step already allows to denote bGF (X) bGF(L,() for any X occuring in Rr(, see [AB, 3.4].
=
Step 1. If X E Irr(G F ) and (L, () :S (G, X) (see Notation 2.5), then ({I},b GF(X)) ~ (ZO(L)f,bLF(()). Proof of Step 1. One has L = Ca(ZO(L)f) and L F = CGF(ZO(L)f) by Proposition 1.3.(ii), so the subpairs above make sense. The inclusion • follows from [CE.l, 4.2] and Proposition 2.7 (ii). Let now L* be an F-stable Levi subgroup of G*, containing s, in the dual class of the G F-conj ugacy class of L. Let T* ~ Ci.* (s) be a maximal F -stable torus, then s E T* and T* is a maximal torus of G *. There is a dual torus Tin L and a duality between Land L *, defined by
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root data relative to T, T*. Then, by Proposition 2.6, ( = X~ A' where >. is e-cuspidal unipotent in £(Ct- (S)F, 1), and ZO(L*)e = ZO(Ct_ (s) )e' Defining M* := CG-(ZO(Ct_(s))[), this is a Levi subgroup of G by [CE.l,2.1.(ii)]. Since T* ~ M* ~ L*, we can define an F-stable Levi subgroup M of G, dual to M* and such that T ~ M ~ L. Let Z = ZO(M)f. With these definitions one has Lemma 4_1. (i) M and M* are Eq,e-split and M = CG(Z). (ii) CM- (s) = Ct- (s). (iii) ZO(M*)f = ZO(Ct-(s))f and it is of order \Z\. (iv) ZO(M)e = ZO(L)e' (v) Denote ~ = X~A E £(M F , s). One has ( = cL<~MRt:~.
Proof_ (i) follows from Proposition 1.3.(i)-(ii). (ii) By (i) and (ii) of Proposition 1.3, L* is e-split and L* = CG-((ZO(L*)J[) = CG-((ZO(Ct-(s))J[) ~ M* ~ T*. We also have C M-(s) = Cr.- (s) since both are the connected centralizers in C G- (s) of ZO(Ct- (s))f. (iii) We have f E r(M*,F) ([CE.l, 1.2]) , so ZO(Ct-(s))f ~ ZO(M*) ~ ZO(C M_(s)), and therefore by (ii) ZO(Ct_ (s))f = ZO(M*)f. (iv) We also get ZO(M*)e ~ ZO(Ct_ (s))e = ZO(L*)e while ZO(L*) ~ ZO(M*), so ZO(M*)e = ZO(L*)e' By [Ca, 4.4.5] and the definition of polynomial order in [CE.l, § 1.2]' this equality implies that zo (M)e and zo (L )e have same polynomial order. They are therefore equal since one has an inclusion. (v) Since ( = X~A and Ct- (s) ~ M*, one has ( = CLCMRt:~ by Proposition 2.3.(c). • Here is the second step on subpair inclusions : Step 2. (ZO(L)f, bLd()) ~ (Z, bMdO) and the latter is selfcentralizing with canonical character ~. Proof of Step 2. By Lemma 4.1.(i) and Proposition 1.3, M F = CGdZ), so the above subpairs make sense. The claimed inclusion is clearly equivalent to the inclusion ({1}, bLF(()) ~ (Z, bMF(~)) in L F . To prove it, we use a slight generalization of [CE.l, 4.2]. As in [BMi, 2.3], denote by Ps (resp. Ps ,£) the orthogonal projection associated with the subset £(M F , s) ~ Irr(M F ) (resp. Ut£(M F , t) ~ Irr(MF) where t runs over the t E (M*)F such that te = s). Let A be a subgroup of Z, H = Ct(A), and XH E Irr(H F ) be such that (M,~) :S (H, XH)' Then [CE.l, 4.2] assumes the equality *R~XH = (~, *R~XH)MF LgENH(M)F /NH(M,~)Fg~. We do not have
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this equality in general, but it is easily seen that the proof of [CE.l, 4.2] applies as soon as the image by Ps,f of this equality holds (this is due to the fact that Ps,f is the projection on a union of blocks by [BMi, 2.2], and therefore commutes with the decomposition map). Now the group H is a Levi subgroup [CE.l, 2.1.(ii)] and XH = X~A by Prop~ sition 2.3.(c). In fact, Proposition 2.3.(c) implies that R~ 'is an isometry from £(M F , s) onto £(H F , s) and its inverse is given by Pso*R~ on £(H F , s) (see also [G, 1.5]). Now, PS,f*R~X~A = P s*R~X~A = cMcH~, whence our claim. There remains to check that (Z, bMF (0) is self-centralizing. One may apply [CE.l, 4.3] to A since e E r(C L>(s), F) by [CE.l, 1.2]. So A is in a block of defect group {I} of the quotient C L>(S)F /Z(C L>(S))F, hence A(I)f = IC L>(S)F /Z(C L>(s))Flf = IC L>(S)F /ZO(C L>(s))Flf. By Proposition 2.3.(a), one has ~(I)f = A(lkl(M*)F : C M>(s)Flf. Using (ii) and (iii) of Lemma 4.1, one gets ~(I)f = IM F /Zlf. As s is of order prime to e, Z is in the kernel of ~, so the above equality shows that ~ is of central defect group Z. Thus Z = Z(MF)f and the pair (Z, bMd~)) is self-centralizing with canonical character ~ as claimed. • The third step allows to reach a maximal subpair. As a first information on the defect groups, we have the following. Lemma 4.2. Let M, Z = ZO (M)f, ~ = X~A be as before. Then PT1(W(Ca>(s)ICL>(s), T*))F normalizes (Z,bMdO). So the defect groups of bGdL, () have order at least IZI.IW(C a >(s)IC L>(s), T*)Flf.
Proof of Lemma 4.2. Recall the identification W(G, T) = W(G*, T*). Take v E NGdT) and v* E (Nc~>(s)(T*))F such that PT*(v*) = PT(V) E W(Ca>(s)ICL>(s), T*)F. Then PT(V) fixes each root of CL>(s) relative to T*, so v normalizes ZO(CL>(s))f and M*. Therefore v permutes the roots of M, and it normalizes M and Z. We have VX~A = X~v> A by Proposition 2.3.(d). Since v* fixes the roots of C M>(s), it acts on [CM> (s), c M>(s)] by an interior aut~ morphism. Hence v* E (CG> ([C M>(s), C M>(s)])[C M>(s), C M>(S)])F. The natural F-isomorphism sends v*CG> ([C M>(s), C M>(sm into ([C M >(s), C M>(S)]ad)F. Then v* fixes A since restriction induces a bijection between the sets of irreducible unipotent characters of C M >(S)F and of ([CM>(s),CM>(s)]ad)F [CE.l, 3.1]. This proves the first assertion. On the other hand, we have C > (s) n M* = C M >(s) = C L>(s) by Remark 2.2 and Lemma 4.1.(ii), so W(C > (s)IC L>(s), T*) n W(M*, T*) = 1 and PT1(W(Ca>(s)ICL>(s), T*)) n M = T. Now, applying Proposition 3.1 to the self-centralizing subpair (Z, bMd~)), we get that the defect groups of bGF (L, () have order at least
a
a
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IZI.lpT1(W(CCo (s)ICI'.o (s), T*))F TFle. gives the second statement of the lemma.
By Lang's theorem, this
•
By [C, 4.2]' there is a subgroup N' ~ NG(T) such that NG(T) = T.N', F(N') = N' and N' n T F = Tf. Now, N' n PT1(W(CCo(s)ICI'.o(s),T*)) clearly satisfies condition (*) of Theorem 3.5.
Step 3. Take any N satisfying condition (*) of Theorem 3.5 and any Sylow f.-subgroup S of N F . Then S normalizes Z and S ~ PT(S), a Sylow f.-subgroup of W(Cco (s)ICI'.o(s), T*)F. Moreover, Z <J ZS (Notation 1.8) and there is a maximal subpair in G F of the form (ZS, bzs ) such that (Z, bMF (~))
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Z <J Z.vS. Using Lemma 4.2, this implies (Z,bMF(~))
4.2. The "minimal" case Assume now that Z(D) ~ Ga. We prove Step 3 in that case. The condition Z(D) ~ G a imposes strong constraints to the situation. Lemma 4.3. Assume Z(D) ~ Ga. Then (i) D ~ Gal L ~ M ~ Gb, (ii) Ci.o (S) n T*(G*)a = T* and it is a diagonal torus ofCTo(Go)Js), (iii) T*(G*)anM* =CTo(Go)JT*¢E q ) T*(G*)a nM*, (iv) Z
=
andT* is a Coxetertorus of
(T n G a ):, S ~ G a F and Z <J ZS (see Notation 1.8).
Proof. By [CE.I, 2.3 (ii)], D ~ G a , so one obtains Z ~ G a , M = C G (Z) ~ Gb, and G*b ~ M* ~ L*. This yields (i). The pair (Ci.o (s), A) is unipotent e-cuspidal in CGo (s) (Proposition 2.6). This takes place in the central product C(GO)Js )C(GO)b (s), so one has a unipotent e-cuspidal pair ((G*)a n Ci.o (s), Aa ) in C(GO)Js) = C(Go)JS)a (Notation 1.5.(i)) and therefore (G*)anCi.o (s) is a diagonal torus in C(GO)a (s) [CE.I, 3.3.(i)]. It contains hence equals T* n (G*)a, whence also (ii). This implies Z(M* n T*(G*)a): = (T*): by definition of M*, hence M* n T*(G*)a = CTO(Go)J(T*)¢Eq ) by Proposition 1.3 (ii). By Lemma 4.1.(iii), one has (T* n (G*)a): ~ ZO(M*). Thanks to Notation 1.5.(ii), this implies that T* n (G*)a is a Coxeter torus of M* n (G*)al whence (iii). Using the inclusion (T* n (G*)a)[ ~ ZO(M*), the fact that T* n (G*)b is connected and centralizes M* n T* (G*)a, along with the definition of (G*)a and (G*)b, one gets (T*): = (T* n (G*)a)f(T* n (G*)b): ~ ZO(M* n T*(G*)a) and I(T*):I :s IZO(M* n T*(G*)a):I. On the other hand, M* n T*(G*)a and M n TG a are in duality, so [Ca, 4.4.5] ITfi :s IZO(M n TGa):I. Also T ~ ZO(M n TG a ) and therefore T[ = ZO(M n TGa)f. Then (T n G a )[ centralizes M, so Z = (T n Ga)f. As 1* ~ (G*)b, then W(CGo (s)ICi.o (s), T*) ~ W(T*(G*)a, T*) and N ~ TG a . Thus N n G a
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Let us check Z <J Z 8. For groups of type A every prime number is good, so CTO(GO)a (s) is a Levi subgroup of T*(G*)a containing T* [CE.l, 2.1.(ii)]. So, one may define G(s), the F-stable Levi subgroup of TG a containing T and in duality with CTO(GO)a (s). Then N ~ G(s). Moreover, defining Ga(s) := G(s) n G a , this is a Levi subgroup of G a and T.Ga(s) = G(s). Then T (resp. Tn G a ) is diagonal in TGa(s) (resp. Ga(s)) by (ii) and Z8 ~ Ga(s) as we have just established. Then (Tn Ga)f = Z ~ Z8 in NGa(s)F(T). Now Proposition 1.9 gives Z <J Z8 as claimed, since f. E f(G a , F) ~ f(Ga(s), F) (Lemma 1.2 and [CE.l, 1.2]). • Lemma 4.4. Assume Z(D) ~ G a , then there is a subpair (Z8, bzs) such that (Z, bMF(~)) <J (Z8, bzs).
Proof. Thanks to Lemma 4.3.(ii), (iii), we may apply Proposition 1.6 and Corollary 1.7 in T*(G*)a with C = CTO(GO)a (s) and K = M* n T* (G*)a both containing T*. This describes the type of M* nT*(G*)a and the action of W(T*C(Go)a (s), T*)F on the roots of M*nT*(G*)a. Let IIwM w be the decomposition of [M, M] n G a into its rationally irreducible components, as in Proposition 1.6. We apply Proposition 3.2 with if = M F , b = bMF(O, H = ZO(Ga)F.IIwM~.GbF, the Hi's being the subgroups M~ and Gb F . By Corollary 1.7, Z(Mw ) is an f.'-group for any component M w , and so is Z(Gb)F. Now it follows from Lang's theorem and [St, 4.5] that if j H is an f.'-group. As 8 ~ G a by Lemma 4.3.(iv), 8 centralizes Gb. By Lemma 4.2, 8 normalizes M and bMF(~), so 8 permutes the Mw's and we may conclude once we have checked Ns(M w) ~ Cs(M w) for all w. By Proposition 1.6, if x E Ns(M w) then PT(X) fixes all the coroots of M w, so it fixes all the roots of M w. Since M w is semi-simple, this implies that each element of N s (M w ) acts on M w in the same way as an element of Tn M w. So Ns(M w) ~ ((T n Mw)CG(Mw))f. By [St, 4.5] again, ((T n Mw)CG(Mw))F j(T n Mw)FCG(Mw)F is a quotient of Z(M w ), so it is of order prime to f. and therefore Ns(M w) ~ (T n Mw)fCG(Mw)f. But (T n Mw)f ~ Z(M) (Lemma 4.3.(iv)), so Ns(M w) ~ Cs(M w). Proposition 3.2 applies and we get our claim by the definition of normal inclusion.
•
Step 3 now amounts to proving that (Z8,bzs) is maximal. Since Z <J Z8 (Lemma 4.3.(iv)), we may apply [C, 3.1.(1)], it suffices to prove that if (E, bE) satisfies (Z, bMF(O) <J (E, bE), then lEI ::; IZ 81. As (Z, bM F (~)) is self-centralizing with canonical character ~ (Step 2) and E ~ G a (Lemma 4.3.(i)), then En M F = Z and
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EjZ injects in N(TGa)F(M, ~)jMF. One has NG(M)jM ~ NG(M) n NG(T)jNM(T) by conjugacy of maximal tori. Let W E W(G, T) be the image modulo T of some i-element of N (TGa)F (M, 0 n NG(T). Then w fixes ~ = X~A' so it fixes the Lusztig series defined by s. This implies that w stabilizes the intersection with T* of the M*conjugacy class of s. By [DM.2, O.12.(iv)], w E W(T*(G*)a, T*) n (W(M*, T*)CW(G*,T*)(S))F. As s is of order prime to i, so is C G*(s)jC~* (s) ~ CW(G* ,T)(S)jW(C~* (s), T*) [CE.l, 2.1.(i)]. Hence wE (W(M*, T*)W(C~*(s), T*))F. By Remark 2.2, W(C~* (s), T*) n W(M*, T*) = W(C M * (s), T*). Using projection on W(T*(G*)a, T*), one sees that E j Z injects in the group of F-fixed points of NW(T*C('G*la (s),T*)(W(T*CM*n(G*)a (s), T*))jW(T*CM*n(G*)a (s), T*). But W(T*CM*n(G*)Js), T*) = {I} by Lemma 4.1.(ii) and Lemma 4.3.(ii). Then EjZ injects in W(T*C(G*)a (s), T*V, a subgroup of W(C~* (s)ICi.* (s), T*)F (Lemma 4.3.(ii) again). On the other hand, lSI is the order of a Sylow i-subgroup of W(C~*(s)ICi.*(s),T*)F by what we have already proved of Step 3. So lEI :S IZSI and the proof of Step 3 is complete. 4.3. End of the proof Steps 1, 2, 3 have allowed us to define blocks bGF (L, () such that (L, () :S (G, X) implies X E Irr(bGF (L, ()). With this definition of bGF (L, (), Theorem 3.5 follows by Step 3. Let us now prove the injectivity of (L, () f-t bGF (L, (). If (L, () and (L', (') are two e-cuspidal pairs defining the blocks bGF (L, () and bGdL',(') which satisfy Steps 1, 2, 3, then we have ({I},b GdL,()) ~ (Z,bMdO) <J (ZS,bzs) and ({I},b GF(L',(')) ~ (Z',b(M/)F(f)) <J (Z'S',bz's') with evident notations. If bGF(L,() = bGdL',('), then, by conjugacy of maximal subpairs, there is x E G F such that X(Z'S', bz's') = (ZS, bzs). But then X(Z', b(M,)F(e))<J(ZS, bzs), while Z <J ZS. So XZ' ~ Z. Whence by symmetry x Z' = Z and therefore X(Z',b(M')F(f)) = (Z,~d~)) (see [AB, 3.4]). By Step 2, this implies X(M',n = (M,~). Then xL' = L by Lemma 4.1.(iv) and X(' = ( by (v) of the same Lemma, whence Theorem 3.3.(2). The equality in Theorem 3.3.(1) also follows since we already had an inclusion. There now remains to check that Irr(bGF(L, ()) is as stated in Theorem 3.3.(3). By [BMi, 2.1], Irr(bGF(L,()) ~ Ut£(GF,st), the union being on the i-elements t E C~* (sV. Note that, for such at, C~* (st) = C~~* (s)(t) is a Levi subgroup of C~* (s) (Remark 2.2 and [CE.l, 2.1 (ii)]). Assume t E T* and consider X' = X~'''Y E £(G F , st)
M. Cabanes and M. Enguehard
162
with 'Y E £(CGo(st)F, 1). Then X?~t) E £(G(t)F,s) and Rg(t)x~~t) is a non-zero element of £(G F , s). So, the following Lemma completes the proof of Theorem 3.3.(3).
Lemma 4.5. Keep hypotheses and notations as above. Then for any X E £(GF,s), such that (Rg(t)X?~t),X)GF =I- 0, one has bGF(x~,1') = bGF(X)· Proof. The proof is very similar to the one in [CE.l] about unipotent characters. Since CGo (t) is a Levi subgroup of G* [CE.l, 2.1 (ii)], let G(t) ;;;? T be an F-stable Levi subgroup of G in duality with CGo (t). By Proposition 2.3.(b) and (c), one has X~,1' = CGCG(t)Rg(t)X~~) and = lxG(t) Then d1XG(t) = d1XG(t) where d 1 is the "decomXG(t) 8t,1' 8,1' . 8t,1' 8,1' position map" which multiplies the central functions with the characteristic function of the elements of order prime to £. Using the compatibility between d 1 and the Lusztig map [BMi, 3.5], one obtains dlX~,1' = dl(Rg(t)x?~t)). By [G, Theorem A] the d1X for X E £(G F , s) are linearly independent. Applying Brauer's second Main Theorem [NT, 5.4.2]' this yields bGF(x~1') = bGF (X) whenever G(t) G ( RG(t)X8,1'
°
•
,x ) GF ../.r .
References
[AB] J. L. Alperin, M. Brow§, Local methods in block theory, Ann. Math. 110 (1979), 143-157. [B] R. Brauer, On blocks and sections in finite groups, I, Amer. J. Math. 89 (1967), 1115-1136. [Br.l] M. Broue, Les £-blocs des groupes GL(n, q) et U(n, q2) et leurs structures locales, Asterisque 133-134 (1986), 159-188. [Br.2] M. Broue, Isometries parfaites, types de blocs, categories derivees, Asterisque 181-182 (1990), 61-92. [BM] M. Broue, G. Malle, TMoremes de Sylow generiques pour les groupes reductifs sur les corps finis, Math. Ann. 292 (1992), 241-262. [BMi] M. Broue, J. Michel, Blocs et series de Lusztig dans un groupe reductif fini, J. reine angew. Math. 395 (1989), 56-67. [BMM] M. Broue, G. Malle, J. Michel, Generic blocks of finite reductive groups, Asterisque 212 (1993) 7-92. [C] M. Cabanes, Unicite du sous-groupe abelien distingue maximal dans certains sous-groupes de Sylow, C. R. Acad. Sci. Paris 1-318 (1994), 889-894.
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[CE.1] M. Cabanes, M. Enguehard, On unipotent blocks and their ordinary characters, Invent. Math. 117 (1994),149-164. [CE.2] M. Cabanes, M. Enguehard, On (q-1)-blocks ofG(q), in preparation. [Cal R. w. Carter, Finite Groups of Lie type: Conjugacy classes and Complex Characters, Wiley, 1985. [DM.1] F. Digne, J. Michel, On Lusztig's parametrization of characters of finite groups of Lie type, AsUrisque, 181-182, 113-156. [DM.2] F. Digne, J. Michel, Representations of finite groups of Lie type, Cambridge Press, 1991. [FS.1] P. Fong, B. Srinivasan, The blocks of finite general and unitary groups, Invent. Math. 69 (1982), 109-153. [FS.2] P. Fong, B. Srinivasan, The blocks of finite classical groups, J. reine angew. Math. 396 (1989), 122-19l. [G] M. Geck, Basic Sets of Brauer Characters of Finite Groups of Lie Type II, J. London Math. Soc. 47 (1993), 255-268. [L.1] G. Lusztig, Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton 1984. [L.2] G. Lusztig, On the representations of reductive groups with disconnected center, AsUrisque 168 (1988), 157-166. [NT] H. Nagao, Y. Tsushima, Representations of Finite Groups, Academic, 1989. [St] R. Steinberg, Endomorphisms of linear algebraic groups, A.M.S. Memoirs 80 (1968). Universite Paris 7, UFR de Mathematiques 2, Place J ussieu, F-75005 Paris, France Received February 1995
Splitting Fields for Jordan Subgroups Arjeh M. Cohen and Pham Huu Tiep*
Abstract Let J be a Jordan subgroup of a simple complex adjoint Lie group g. Then N = Ng(J) is a Lie primitive subgroup of g. In this paper, the minimal field k for which there exists a k-form of g containing N is determined. 1. Introduction
Let g be a simple complex Lie group. An elementary abelian finite subgroup J of g is said to be a Jordan subgroup if the following conditions hold: (i) N = Ng (J) is a finite group; (ii) J is a minimal normal subgroup of N; (iii) if J' is another finite subgroup of g with properties (i), (ii) and such that J <;;; J', N <;;; Ng(J'), then N = Ng(J'). This notion was introduced by A. V. Alekseevskii, who classified all Jordan subgroups of simple complex Lie groups in [Ale]. For the reader's convenience we reproduce this nice result in Table 1. The interrelation between Jordan subgroups of Lie groups g and the s~called orthogonal decompositions of corresponding Lie algebras {, has been investigated in detail by A. V. Borovik, V. P. Burichenko, A. 1. Kostrikin and P. H. Tiep; for more about this, see [KoT]. The definition of Jordan subgroups can be carried over to algebraic groups over fields of characteristic p > 0 without any change. As shown by Borovik [Bor 1]' the classification of Jordan subgroups then remains unaltered, except for the natural restriction exp( J) =j:. p. Thus, the group J together with its normalizer N embeds in finite groups gq of Lie type corresponding to g (under some conditions regarding the fields of definition IF q). Furthermore, if we take the definition field IF q to be minimal with respect to containing J, then Ngq(J) usually turns out to be a local maximal subgroup of gq (see [CLSS]). 'Supported by the Alexander von Humboldt Foundation.
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As can be seen from [Asch]' [Bor 2]' and [CLSS], Jordan subgroups play a role in the classification of • finite Lie primitive subgroups of complex Lie groups (here Lie primitive means that the subgroup is maximal among all proper closed Lie subgroups); • (local) maximal subgroups of groups of Lie type, algebraic or finite. Namely, the normalizers of Jordan subgroups represent a subclass of the class of maximal (closed) finite subgroups with abelian socle. The two other classes are those with non-abelian simple socle and with the socle equal to A5 x Aa (in the latter case, the group must be of type E s , see [Bor 2]).
Table 1. Jordan subgroups of simple complex Lie groups y Apn_l' p a prime En, n ~ 3 C 2n-l, n ~ 2 D 2n - l , n ~ 3 Dn+i, n ~ 4
G2 F4
E6 Es Es D4 ·Z 3 (Q is not simple)
J
Cg(J)/J 1 1 1 1 1 1 1 Z33 1 ZlO 2
Ng( J) /Cg(J)
Z'l.n z~n 2 z2n 2 z2n 2 z2n 2 Z32 Z33 Z33 Z35 Z52 Z32
Z~· Z3
8£3(2)
8P2n(P) &.!n+l
Oin(2) Otn(2) &.!n+2
8£3(2) 8£3(3) 8£3(3) 8£3(5) 8£5(2)
ef. § 2.1,3 2.3 2.2 2.2 2.3 2.4 4 4 4
2.5 2.4
The aforementioned activity on classifying maximal finite subgroups of complex Lie groups is now believed to be nearly complete, due to work of A. M. Cohen, R. L. Griess Jr., P. B. Kleidmann, A. J. E. Ryba, JP. Serre, D. B. Wales, R. A. Wilson and others (cf. [CoW2]). One of the open questions left is to determine the minimal splitting field for a given maximal finite subgroup G of y. We shall be concerned with the situation where y is interpreted as the automorphism group Aut (.c) of a given complex simple Lie algebra.c. Often the subgroup Inn (.c) of Aut.c consisting of all inner automorphisms of.c will suffice for defining J.
Definitions. Given a complex algebraic group y containing G, a lKform ofy for G is a lK-form yoc of y such that G is contained in YOC(lK), the group of lK-rational points of Yoc.
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Let £, be a complex simple Lie algebra, 9 = Aut (£,), and G a finite subgroup of g. A subfield lK of C is said to be a splitting field for G in £', if there exists a lK-subspace V of £, (viewed as lK-space) with the following properties: (i) V is invariant under G; (ii) V 00C C = £'; and (iii) V is closed under the Lie multiplication of £'. The resulting Lie algebra V over lK is called a lK-form of £, for G. According to a result of T. A. Springer ([Spr]), there is a unique minimal (with respect to inclusion) splitting field lK for each Lie primitive finite subgroup G of 9 (to be more precise: for each finite subgroup G whose centralizer in 9 is contained in Z(Q)). The elegant proof of this result does not tell us what the minimal splitting field is, nor what the corresponding lK-form goc is. The different lK-forms of £, and those of 9 are in bijective correspondence, since Aut 9 ~ 9 and the (pointed) first order Galois cohomology set H 1 ( Gal (lK, lK), g), where Gal (lK, lK) denotes the Galois group of the algebraic closure lK of lK over lK, describes the different lK-forms of £', d. [Serlo Therefore, we can study the lK-forms of £, to determine the minimal splitting field and the corresponding lK-form of 9 for a given subgroup G. For the various finite Lie primitive subgroups of 9 of type G2 , the minimal splitting fields have been fully determined (see [CoW] and [CNP]). In this paper we determine them for all Jordan subgroups of simple complex Lie groups.
Theorem. Let £, be a complex simple Lie algebra, 9
= Aut (£,), J a Jordan subgroup of g, and let p = exp(J). Then Q(cos(27r/p)) is the unique minimal splitting field for Ng( J) in g. 2. The case p = 2
The aim of this section is to prove the theorem stated in the introduction for the case where p = 2. Thus, we establish that Q is a splitting field for the normalizer Ng(J) of any Jordan 2-subgroup J of 9 = Aut (£'). We distinguish cases according to Alekseevskii's classification (see Table 1).
2.1 £, is of type A2m-l, m :2: 1. First we recall the construction of the Jordan subgroup J for the case where £, is of type Apm_l with p
A. M. Cohen and P. H. Tiep
168
an arbitrary prime, d. [KoT]. Let q = pm, W = IF~ viewed as a 2mdimensional vector space over IFp with the symplectic form
= TrlFq IlFp (ad - be)
(u, v)
for u = (a,b), v = (e,d), a,b,e,d E IFq . For such vectors u,v we also set B(u, v) = TrlFq IlFp (-be). Put e = exp(27ri/p) and write
D=
1 0 0 0 e 0 0 0 e2
0 0 0
0
eP -
0 0
p= l
Choosing a basis {el,"" em, iI,
Ju = Dal pb l 0
0 0 1 0 0 1
0 1 0 0 0 0
0 0
1 0
, 1m} of W,
one can set
0 Dam pbm
for u = L~l (a;e; + b;f;). It is shown in §1.l of [KoT] that, for an appropriate choice of the basis {el,"" 1m}, one has T
Ju
•
J
Furthermore, the set {Ju I u E
-
v -
cB(u,v)J
W\ {O}}
(1)
u+v·
c
forms a basis for the Lie algebra
£. If p image
> 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
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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
=
(_I)f(i)+f(j)E(
Furthermore, [E(i,j), E(k, l)]
={
E(~, l)
if I{i,j} n {k,l}1 = 0 or 2, if i =j:. j = k =j:. l =j:. i.
Hence the Q-subspace (E( i, j) I 1 :S i < j :S m)iQl is a Q-form, and Q is the unique minimal splitting field for Ny( J) in g.
2.4 £, is of type G2 or D 4 , and J c::: z~. If £, = D 4 , one has to take 9 containing not only Inn (£,), but also a triality automorphism of £'. For more about the Jordan subgroup J and its normalizer in 9 see [Gri]. Recall (see §3.1 of [KoT]) that, if J c::: Z; is a Jordan subgroup of g, a Lie group of type G 2 , D 4 , F4 , E 6 or E s , one can associate to J the following orthogonal decomposition (3)
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of £'. Each component 1ik is a Cartan subalgebra of £, and coincides with the fixed point subalgebra C.c(J')
=
{x E £, I Vg E J',g(x)
=
x}
for a unique maximal subgroup J' of J. If we again set G = Ng(J), then one easily sees that G is just the group of all elements from 9 that preserve decomposition (3). We shall need the following assertion:
Z;
Lemma 1. Let p be a prime and J::::: be a Jordan subgroup of the Lie group 9 = Aut (£,) of type G 2 , D 4 , F4 , E 6 or E s , respectively. Then, for any non-identity element 9 from J, the fixed point subalgebra C.c(g)
=
{x E £, I g(x)
= x}
is a semisimple Lie algebra of type 2A 1 , 4A 1 , 2A 2 , 3A 2 or 2A4 , respectively.
Proof. For E s , the element 9 must have class 5A in the notation of [CoG], and so the result can be read off from the table 4 in [CoG]. Although a similar proof can be given for the other cases, we shall give a less case dependent proof here. Set C = C.c(g). From the definition of C one can show that the kernel of the Killing form KC corresponding to C is contained in the kernel of the Killing form K associated with £'. But K is non-degenerate, so KC is also non-degenerate. This means that C is a semisimple Lie algebra. Considering decomposition (3) corresponding to J one sees that C contains at least one component 1ik of (3), which is a Cartan subalgebra of £'. Because 1ik is also a Cartan subalgebra of C, the ranks of C and £, coincide. Furthermore, standard computations with the characters of J show that the dimension of C is equal to
(p
+ 1). rank(£') = / + 1
p +p+ 1
dim(£').
Finally, the simple components of the Lie algebra C are isomorphic to each other, since C is acted on irreducibly by Nc( (g)). These data regarding C suffice to determine its type. D Now we suppose that J::::: Z~ and £, is of type G 2 or D 4 . Note that, if is of type D 4 , some triality automorphism of £, centralizes the Jordan subgroup J of order 23 (compare with §2.2). We show that Q is the unique £,
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minimal splitting field for G. Observation (2) suggests considering the Q-subspace V = EBk=1 Vk of .c, where Vk = {x E Hk I Spec(.c, ad x) ~ Qi}.
The fact that V is a G-invariant Q-subspace follows from the following simple observation:
Lemma 2. Let.c be a semisimple Lie algebra over C. with Cartan subalgebra H. If OC is a subfield of C. and A E c., A f= 0, then the set
u = {x E H I Spec (.c, ad x)
~ AOC}
is an N Aut (.C) (H) -invariant OC-subspace of dimension equal to dim H.
Proof. The commutativity of H implies that U is a OC-subspace. Let R denote the root system corresponding to H. Then, for x E H,
xEU
¢:}
V 0: E R o:(x) E AK
(4)
This equivalence shows that U is invariant under N Aut (.C) (H). But R spans a Q-space of dimension dim H in H*. So dim U = dim H. 0 It remains to check that V is multiplicatively closed: [V, V] ~ V, or equivalently, [Vk, Vi] ~ V for all k, l. If k = l, then [Vk, Vi] = O. Suppose that k f= l. Then there exists a non-identity element 9 E J such that Hk and HI both are contained in the subalgebra C = C.c.(g). By Lemma 1, C is a semisimple Lie algebra of type r AI, where r = 2 or r = 4. One has a decomposition of C into a sum of simple components C = L:=1 C', each of them being a Lie algebra of type Al invariant under J. On the other hand, C is a sum of three Cartan subalgebras, namely Hk, HI and some other H m . Without loss of generality we suppose k = 1, l = 2, m = 3. Because of (3), we also get an orthogonal decomposition
for each C', s = 1, ... , r. Moreover, H t = EB:=IH:, t = 1,2,3. Now we set ~. = {x E I Spec(C',adx) E Qi}.
H:
The root systems corresponding to H:, s = 1, ... , r, together span the same Q-space as the root system corresponding to H t . This remark and (4) show that
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173
By §2.1, EB~=I v;,' is a Q-form. In particular, [v;,', v;,n ~ V. Furthermore, it is obvious that [v;,', v;,n = 0 whenever s =j:. s'. Now we have r
[VI, V2 ]
=
2:
[EB~=I VI" EB~/=I V/] ~
[Vt, V/] ~
V,
8,8'=1
as required. 2.5 £ is of type E s and J c::: Z~. The fact that Q is the unique minimal splitting field for Ng(J) has been established by J. G. Thompson [Tho]. Another way to see it is to follow the scheme of arguments of §2.4 by EBf~I'Hk correspond-
• considering the orthogonal decomposition £ ing to J, and
=
• the Q-space V = EBf~IVk, Vk = {x E 'Hk then observing that
I Spec (12,adx)
~ Qi};
• the sum of each pair of Cartan subalgebras 'Hk, 'HI is contained in the Lie algebra C.c(g) of type D s , 1 =j:. g E J (in the Lie algebra C.c(g, g', gil) of type 8A I , 1 =j:. g, g', gil E J, if you like the Lie algebra Al more!), and that • Q is the unique minimal splitting field for D s by §2.2 (for 8A I by §2.1).
3. The case
Apm_I
for p odd
z;m
Here we are concerned with the Jordan subgroup J c::: of 9 = Aut (12) of the Lie algebra £ = slq(C), q = pm, p an odd prime. We also use the notation of the beginning of §2.1. The normalizer G = Ng(J) c::: J . (Sp(W) . Z2) is in fact split; namely, one can identify J with the subgroup {w E Aut (12) I w E W}, where
w( Ju )
= E(w,u) Ju,
u, W E W,
and
Sp(W)· Z2 =
= ±1,
{
Vu,v E W, (
= J-L
with the subgroup {tj; E Aut (12) I
u
-
c{B(
II f""'
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A. M. Cohen and P. H. Tiep
First we replace the basis {Ju} by Then, in view of (1), one has:
= c (u,v)/2Iu+v,
I u· I v
Taking
f)
=c- c
1
IU
{Iu
W'(Iu )
=
C
E W#} with
(w,u)I u,
r.p'(I) u
=
Iu
=
c B (u,u)/2Ju '
IJ;
(5)
= 2isin(27fjp), we set, for u E W,
=
K u
L
f)
c(u,v) Iv.
vEW#
Since
bv ,oq2, we find
LUEW c(u,v)
WEW, wKu
=
L
f)
LUEW
K u = O.
Furthermore, for
c(w+u,v) Iv = K w+ u ,
vEW#
and, for r.p E Sp(W) . Z2, rj;(Ku )
L
= IJ;
L
c(u,v) I
vEW#
c(I"'I'
vEW#
=
(6)
IJ;
Finally, for u, v E W we have: K u ' K v
= f)2 (-10 + q2c 2(v,u) I 2(u-v»)
-
f) (Ku
+ K v ),
from which we obtain, [Ku , KvJ
L
=
Au,v(x)Kx ,
xEW#
where Au,v(x)
=
f)(c 2(v,u)+2(u-v,x) _ c 2(u,v)+2(v-u,x) _ c 2(v,u)
+ c 2(u,v»).
(7)
Indeed, q 2f)2 (c2(V,U) I 2(u-v) f) (c 2(V'U)
L xEW
_
c
c
2(u,v) I
c(2(u-v),x) K
x
2(v-u)
)
- c 2(u,v)
L
c(2(v-u),x) Kx)
xEW
As Au,v(x) E Q(c) and Au,v(x) = Au,v(x), we obtain that Au,v(x) belongs to the field lKo = Q(cos(27fjp)) = Q(c) nR We have shown that lKo is a splitting field for G.
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175
We can actually show that JK a is the minimal splitting field for any subgroup H of G with the property: J
= q2 =
deg X + 1,
.c 1 = {x E.c I Vs E S, s(x) = J.l(s)x} is spanned by the vector K a. We shall need the following simple statement: Lemma 3. Let JK ~ L be fields of characteristic 0, G a finite group, V a JKG-module of finite rank. Assume that V lL = V 0JK L has a composition factor U which is realizable over K Then V has a direct summand, which is isomorphic to U.
Proof. Any G-module over JK and L is completely reducible. So V = VI EB· .. EB Vn for some simple JKG-modules VI, ... ,Vn- Furthermore, V;lL = W i1 EB··· EB Wit, for some simple LG-modules W ij , 1 :S i:S n, 1 :S j :S t i . One may suppose that W ll ~ UlL and U is a JKG-module. We come to the situation in which Vi and U are simple JKG-modules and ~lL and UL have a common composition factor (namely W ll ). By Corollary 9.7 [Isa], VI ~ U. 0 By Lemma 3, the JKS-module V has a direct summand, which affords the character J.l. But dim.c 1 = 1, so this summand is spanned over JK by >..Ka for some >.. E C'. This means that V contains >..Ka. Hence V contains u(>"Ka) = >..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,
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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)
<;;;;
191Ko}·
Then V is G-invariant. It remains to check that V is multiplicatively closed: [V, V] <;;;; V, or equivalently, [Vk, VI] <;;;; V for all k, i. If k = l, then [Vk, VI] = O. Suppose that k =I i. Without loss of generality, one can suppose that k = 1 and l = 2. Because of (3), we also have an orthogonal decomposition
cs =
EB~=I
H:
for each CS, 8 = 1,2. This decomposition is nothing else but the orthogonal decomposition (3) corresponding to the Jordan subgroup 05(H) of gs. Moreover, H t = EB;=IH: for t = 1, ... , 6. Now we set ~s
= {x
E
H: I Spec (CS, adx)
E
191Ko}.
Remark that the root systems (of type A4 ) corresponding to H:, 8 = 1,2, together span the same Q-space as the root system (of type E 8 ) corresponding to H t . This remark and the equivalence (4) show that
By the results of §3, EB~=I ~s is a lKo-form. In particular, [~S, ~J] <;;;; V. Furthermore, it is obvious that [~S, ~n = 0 whenever 8 =I 8'. Now we have 2
[VI, V2] = [EB;=I vt, EB;'=I Vl]
<;;;;
L
[vt, Vl]
<;;;;
V,
5,5'=1
as required. Finally, in order to show that IKo is a minimal splitting field, suppose that OC is a splitting field for G in I: and that V is a OC-form for G. We shall need the following simple statement: Lemma 7. Assume that a finite group A acts irreducibly on a complex vector space V of dimension n. Let Q <;;;; OC <;;;; C and suppose A fixes two nonzero OC-subspaces U, U' in V. Then U' = )..U for some).. E C*. Proof. We know that OC-representations of A on U and U' are equivalent over C. So by the Deuring-Noether theorem they are equivalent over oc. This means that there exist a basis {e I, ... , en} of U and a basis {h, ... , fn} of U' such that the action of any a E A is written by the same matrix (a) in {ei} and {til. Write
181
Splitting Fields for Jordan Subgroups
for X E GLn(C). Then X(a)X- I = (a) for all a E A. By Schur's lemma, the matrix X is a scalar: X = )..E for some).. E C'. In particular,
U' = )..U.
0
Consider the Cartan subalgebra 'H = 'HI contained in C and set U = {x E 'H I Spec (.c,adx) ~
nq.
Then U is obviously a JKM-module, where M = Nc('H). By Lemma 3 of J which acts trivially on 'H, we applied to the subgroup (g, h) ~ see that V n'H is another nonzero JKM-module in 'H. Furthermore, M acts irreducibly on 'H. Therefore, by Lemma 7 there exists)" E C' such that V n'H = )..U. Put 'H s = 'H n cs and
Zg
us = {x E 'H s I Spec (CS, ad x) s
~
JK},
= 1,2. As above, one has U = UI EB U 2 . Hence,
in particular, V nCI =I O. It is now obvious that V nC I is a JK-form for the 8L2(5) distinguished by Lemma 6. Applying Proposition group H = 4, we arrive at the conclusion that JK ;;;2 lKo, Our theorem is completely proved.
Zg.
Acknowledgments. Part of this work has been done during the Canadian Mathematical Society Annual Seminar "Representations of Groups: Lie, Finite, Quantum" held at Banff, June 1994. The authors' thanks go to the organizers of the Seminar for their support. The second author is also grateful to Prof. Dr. G. O. Michler and the Volkswagen Foundation for supporting his participation in the CMS Annual Seminar 1994. References [Ale] A. V. Alekseevskii, Jordan finite commutative subgroups of simple complex Lie groups, Funktsional Anal. i Prilozhen. 8 (1974), no. 4, 1 - 4; English transl. in Functional Anal. Appl. 8 (1974), 277 - 279. [Asch] M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984),469 - 514. [Bor 1] A. V. Borovik, Jordan subgroups of simple algebraic groups, Algebra i logika 28 (1989), no. 2, 144 - 159; English transl. in Algebra and Logic 28 (1989), no. 2, 97 - 108.
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[Bor 2] A. V. Borovik, The structure of finite subgroups of simple algebraic groups, Algebra i logika 28 (1989), no. 3, 249 - 279; English trans!' in Algebra and Logic 28 (1989), no. 3, 163 - 183. [CLSS] A. M. Cohen, M. W. Liebeck, J. Saxl and G. M. Seitz, The local maximal subgroups of exceptional groups of Lie type, finite and algebraic, Proc. London Math. Soc. (3) 64 (1992), no. 1, 21 - 48.
[CNP] A. M. Cohen, G. Nebe and W. Plesken, Cayley Orders, to appear in Compositio Math., 1995. [CoG] A. M. Cohen and R. L. Griess, On finite simple subgroups of the complex Lie group of type E 8 , Proc. Symp. Pure Math. 47 (1987), Pt. 2,367 - 405. [CoW] A. M. Cohen and D. B. Wales, Finite subgroups ofG 2 (C), Comm. Algebra, 11 (1983),441 - 459. [CoW2] A. M. Cohen and D. B. Wales, Finite simple subgroups of semisimple complex Lie groups - a survey, pp. 77-96 in "Groups of Lie Type and their Geometries", eds. W.M. Kantor and L. Di Martino, LMS Lecture Notes, no. 207, Cambridge University Press, 1995. [Gri] R. L. Griess, Code loops and a large finite group containing triality for D 4 , Rend. Circ. Mat. Palermo (2) Suppl. no. 19 (1988), 79 - 98. [Isa] I. M. Isaacs, 'Character Theory of Finite Groups', Academic Press, London, 1976, 303 pp. [KoT] A. I. Kostrikin and Pham Huu Tiep, 'Orthogonal Decompositions and Integral Lattices', Walter de Gruyter, Berlin a.o., 1994, 535 pp. [Sah] C.-H. Sah, Cohomology of split group extensions, J. Algebra 29 (1974),255 - 302. [Ser] J-P. Serre, Cohomologie Galoisienne, Lecture Notes in Math., 5, Springer-Verlag, Berlin, 1964. [Spr] T. A. Springer, private communication, Zeist, 1994. [Tho] J. G. Thompson, A simple subgroup of E 8 (3), Finite Groups Symposium, N. Iwahori, Japan Soc. for Promotion of Science, 1976, pp. 113 - 116.
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[Tiep] Pham Huu Tiep, Invariant lattices of type E s and their automorphism groups, Algebra i Analiz, 4 (1992), no. 5, 227 - 256; English transl. in St. Petersburg Math. J. 4 (1993), no. 5, 1029 - 1054.
Arjeh M. Cohen Fak. Wisk. & Inf., Eindhoven University of Technology Postbox 513, 5600 MB Eindhoven, The Netherlands email:[email protected] Pham Huu Tiep Institut fur Experimentelle Mathematik, Universitiit Essen Ellernstr. 29, 45326 Essen, Germany Received February 1995
A Norm Map for Endomorphism Algebras of Gelfand-Graev Representations Charles W. Curtis and Toshiaki Shoji
Abstract
e
Let be a connected, reductive algebraic group defined Over a finite field, with Frobenius endomorphism F : ~ In a previous article by the first author [3], the irreducible representations, in an algebraically closed field K of characteristic zero, of the Hecke algebra 'H of a Gelfand-Graev representation, of the finite reductive group F were determined. They are parametrized by pairs (T,O), with T an F-stable maximal torus of e, and 0 an irreducible character of the finite torus T F . Each irreducible representation fr.o can be factored, fr,o = (j 0 fr, with fr a homomorphism of algebras, independent of 0, from'H to the group algebra KT F , and (j the extension of 0 to an irreducible representation of the group algebra of the torus T F . The main results may be summarized as follows. Let F' = Fm for a positive integer m, and let " be a Gelfand-Graev representation of the finite reductive group C F ' with Hecke algebra 'H'. A construction of a homomorphism of algebras ~ : 'H' ~ 'H is given, based on further study of the maps {fr}. A condition on m for ~ to be surjective is derived, and a correspondence of primitive idempotents and irreducible characters from 'H to 'H' is obtained in case ~ is surjective. The homomorphism ~ is Characterized as the unique linear map ~ : 'H' ~ 'H with the property that for each F-stable maximal torus T, one has
e
e.
e
fr
0
~
= NT 0
f~,
with fr : 'H ~ KT F and f!r : 'H' ~ KT F ' the homomorphisms of algebras mentioned above, and NT the extension of the norm map NT : T F ' ~ T F to a homomorphism from the group algebra of T F ' to the group algebra of T F .
Let
e denote a connected reductive group defined over a finite field e and e be finite re-
F q , with Frobenius endomorphism F, and let
F
F
'
ductive groups of fixed points, with F' = Fm, for some m ~ 1. The theory of Shintani descent establishes, in certain cases, a correspondence
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C. W. Curtis and T. Shoji
from the set of F-stable irreducible complex valued characters of GF ' to the set of all irreducible characters of GF , based on a norm map which sets up a bijection between the set of F-conjugacy classes of G F ' and the set of all conjugacy classes of G F . In this article, a correspondence of characters is obtained, under certain conditions, from the set of irreducible components of a Gelfand-Graev representation " of GF ' to the set of irreducible components of a Gelfand-Graev representation, of GF , based on a homomorphism of algebras, which we call a norm map, from the endomorphism algebra, or Hecke algebra, of " to the Hecke algebra of f. We begin with some preliminary results. All representations and characters are taken in the algebraically closed field QR as in [4]. We shall use the notation K for this field in what follows. Let H be a Hecke algebra of a Gelfand-Graev representation, of G F , indexed by some element Z E H1(F, Z(G)) as in ([5], Chapter 14). For each F-stable maximal torus T of G, let
be the homomorphism of algebras defined in ([3], Theorem 4.2). Let {TW}WEW/~F be a set of representatives, to remain fixed, of the G F_ conjugacy classes of F-stable maximal tori. Lemma 1 The homomorphism of algebras
IT h
:H
---.
T
IT KT F , T
where T runs over the set {Tw }, is injective.
For the proof, let DT h(h) = DT h(h' ), for h, h' E H. Let () run through the set of characters of T F , and let Bdenote the extension of () to a representation of KTF. Then
IT B
0
T,flETF
h(h) =
IT B
0
h(h' ).
T,flETF
This means that h,fI(h) = h.fI(h' ) for each of the irreducible representations h,fI = Bo h of H, and since these form a complete set of irreducible representations of H by ([3], Theorems 3.1 and 4.2), one has h = h', as required.
Norm Map for Endomorphism Algebras
187
Lemma 2 Let r be the character of the Gelfand-Graev representation" f let T, T be F -stable maximal tori, and let 0, Of be irreducible characters of T F and T fF , respectively. The following statements are equivalent. (i) ho = h, ,0' ; (ii) XT,O = XT' ,0', where XT,O and XT' ,0' are the irreducible characters of G F corresponding to the representations h,o and h',oJ of 'H, respectively; (iii) there exists an irreducible character X of G F such that (X, RT,o) # 0, (X, RTJ,o') # 0, and (X, r) # 0. As XT,O is the irreducible character corresponding to the representation ho of 'H, the equivalence of (i) and (ii) follows from the theory of Hecke algebras. The equivalence of (ii) and (iii) is a consequence of the uniqueness of the character XT,O, proved in Theorem 2.1 of [3].
Definition 1 For 0, Of in TF, put 0 '" Of whenever the pairs (T,O) and (T, Of) satisfy anyone of the conditions stated in the preceding Lemma. The equivalence class of a character 0 ofT F is denoted by [OJ. Lemma 3 For each F -stable maximal torus T and equivalence class [OJ of characters of T F, put
for all Of in [OJ. Then (KT F) [0] is a subalgebra of KT F containing 1. Moreover, imh = n[o](KTF)[oj, where the intersection is taken over all equivalence classes of characters ofT F . Let a E imh, so that a = h(h) for some element h E H Let then ho(h) = hfl'(h) by Definition 1, and hence O(a) = Of(a) by the factorization property of the homomorphisms h,o. Therefore a E n(KTF)[oJ, and it follows that
o'" Of;
imh ~ n(KTF)IO]' Both algebras above are commutative and semisimple subalgebras of KT F . In this situation, it is an easy exercise to show that anyone of their irreducible representations is the restriction of an irreducible representation of KT F . Using this fact, we prove next that the number of irreducible representations of imh is equal to the number of equivalence
C. W. Curtis and T. Shoji
188
classes [0]. For each such representation 'P, the remark above implies that 'P = 81imfT for some 0 E TF. Moreover, 81fT = 811fT for all 01 E [0] since imh ~ n(KT F )[II]' Thus a map exists from irreducible representations 'P of imh to equivalence classes [OJ, and it is immediate to check that it is bijective. The final step begins by noting that each class [0] defines an irreducible representation of n(KT F )[II] by restriction, and that all irreducible representations of n(KTF occur in this way. Therefore the number of irreducible representations of the algebra n(KT F ) [II] is less than or equal to the number of classes [OJ, completing the proof. Let m ~ 1 be a positive integer, let F' = Fm, and let be a GelfandGraev representation of the finite reductive group GF', with character fl, indexed by some element Zl E H1(Fm, Z(G)). Denote the Hecke algebra of by HI. For the time being, Zl and z can be chosen independently of each other. For each F-stable maximal torus T of G, let N TF ,/TF : T F' -+ T F be the norm map ([4], 5.3). We shall denote it by NT in what follows, and let NT denote its extension to a homomorphism of algebras NT : KT F' -+ KT F . Both NT and its extension NT are surjective, by [4], §5. Finally, let PI' : HI -+ KT F' be the homomorphism of algebras defined in Theorem 4.2 of [3].
,I
,I
Lemma 4 Let T be an F-stable maximal torus of G, and NT: T F' T F the norm map. Then NT 0 f!r(H
I )
-+
~ h(H).
For each equivalence class [1]] of characters of T F', let [1]]F denote the set of F-stable characters of T F ' belonging to the class [1]], and put (KTF')[TJ]F = {a E KT F' : ;j(a) = ij*(a)},
with 1]1,1]* E [1]t. Noting that (KTF')[TJJF = KT F' if [1]]F is empty and . general, we have (KT F') [TJ] ~ (KT F') [TJ]F m f!r(H
I )
~
n[TJ] (KTF')[TJ]F,
by Lemma 3. We shall prove that -
F'
NT(n[TJ](KT )[TJ]F ) ~ h(H).
Let a E n[TJ] (KTF')[TJ]F; then ijl(a) = ij*(a) for all F-stable characters of T F ' such that 1]1 ,...., 1]*. By Lemma 3, the result will follow if we can prove that 1]1, 1]*
Norm Map Jor Endomorphism Algebras
189
for all characters 0' and 0* of T F such that 0' ,..., 0*. For this, in turn, it is sufficient to prove that rl ,..., r", for the F -stable characters rl = 0' 0 NT and ",* = 0* 0 NT. As 0' ,..., 0*, one has XT,fI' = XT,fl o. By the proof of Theorem 2.1 of [3], we have XT,fI' = Xs',z and XT,flo = XSo,z for semisimple elements s', s* of o C*F belonging to the same rational conjugacy class. By ([5], 5.21.5 and 5.21.6), the CF-conjugacy classes of pairs (T,O') and (T, 0*) correspond to the semisimple classes (s') and (s*) of C*P- in such a way that the conjugacy classes of pairs (T, 0' 0 NT) and (T, 0* 0 NT), under the action of C F' = C Fm , correspond to the semisimple classes of C*p-m containing the elements s' and s*, respectively. By the proof of Theorem 2.1 of [3], in order to prove that 0' 0 NT ,..., 0* 0 NT, we have to prove that the characters X~' z' and X~o z' of C F' coincide, and this will follow if s' and s* om are conjugate' in C*F :This is is immediate since s' and s* are already o conjugate in C*F , and completes the proof of the Lemma. Let h' E 'H'. For each F-stable maximal torus T in the set {Tw }, there exists, by Lemma 4, an element hT E 'H such that NT 0 JH h') = h(hT ). The next lemma shows that it is possible to match up the elements hT with a single element of 'H, in the following sense.
Lemma 5 Let h' E 'H'. For each F -stable maximal torus in the representative set T w choose hT E'H such that NT 0 JHh') = h(hT ). Then there exists h E 'H such that h(h) = h(hT ) Jor all T. !
For each F-stable maximal torus in the set {Tw }, let {IT,d, 1 ::; i ::; rT, be a set of representatives of the set of all irreducible representations h,fI of 'H. Then, as T ranges over the set {Tw }, the maps {h,d range over the set of all irreducible representations of 'H, with repetitions when h,fli = h"fI' for different pairs. We shall prove: ]
The argument is similar to the proof of Lemma 4. The hypothesis h,fli = JTI,fl implies that Xs,z = Xs',z, where s and s' are rational semisimple o elements of C*F corresponding to the pairs (T,Oi) and (T', OJ). By the o proof of Theorem 2.1 of [3], the elements s and s' are conjugate in C*F . As in the proof of Lemma 4, it follows that the irreducible characters X~ Zl and X~, Zl of C F' coincide, and hence the corresponding irreducible representati~ns J!I'"floNT and IT','fI'j oNT' of 'H' coincide. We now have: l ]
C. W. Curtis and T. Shoji
190
Similarly, h'IJ,(h (hi), T,) = f~'IJ'oN , 1 j T' I
and the result follows. The maps {Oi ° h} run through the set of all irreducible representations of'H, and it has been shown that Oioh(hT ) = Ojoh,(hT,) whenever the representations Oi ° h and OJ ° h, coincide. By the Frobenius-Schur Theorem, it follows that there exists h E 'H such that
for all T in the set {Tw }, and all ()i, 1 ::; i ::; rT. For each character () of T F , one has 0 ° h = Oi ° h for some ()i. Therefore
00 h(h) =
Oi
° h(h) =
Oi
° h(hT ) = 0° h(hT ).
As this holds for all characters () of T F , we conclude that h(h) = h(h T ) for all T, completing the proof. Theorem 1 Let '}-f' and'}-f be the Hecke algebms of Gelfand-Gmev repFm
and C F , resentations " and, of the finite reductive groups C F ' = C respectively. There exists a homomorphism of algebms ~ : 'H' -+ 'H, which is chamcterized as the unique linear map from 'H' to 'H with the property that ho~
= NTof~
for all F -stable maximal tori T of
c.
Let T run through the set of representatives Tw , as in the previous discussion. Let hi E '}-f'. By Lemmas 4 and 5, there exists h E '}-f such that h(h) = NT ° f~(h').
II T
II T
II T
For each choice of hi E '}-fl, the solution h of the preceding equation is uniquely determined, by Lemma 1. Upon setting ~(h') = h for each pair of solutions of the equation, it is clear that ~ is a linear map from '}-f' to '}-f satisfying the intertwining property stated in the Theorem. It is also clear from the equation that ~ is a homomorphism of algebras, since the maps h, f!r, and NT have this property. The uniqueness of ~ as a linear map with the intertwining property follows from Lemma 1, completing the proof. As an example, we shall describe the map ~ in case C = S£2, defined over a finite field F q of odd characteristic, with F the usual Frobenius,
Norm Map for Endomorphism Algebras
191
and F' = F 2 . We use the notation, and formulas for the homomorphisms Jr, from [3], §5. Let X be a fixed nontrivial additive character of F q, and f the Gelfand-Graev character of C F = SL 2 (F q ) induced from the linear defined by the additive character X, as in [3]. Let character 'ljJ on f' be the Gelfand-Graev character of C F ' = SL2(F q2) induced from the linear character 'ljJ' of Ut' defined by the additive character X' on F q2, with X' = X 0 Tr and Tr the trace map from F q2 to F q • Let 'H' and 'H be the Hecke algebras of f' and f, respectively, with standard bases c'±I, c'N' A' E F q 2 and C±I, C>., A E F q , as in [3]. The homomorphism ~ is defined as follows:
ut
~(C~I) = CI, ~(c~,) =
L d>.,NC>. + d_I,>.'C-I + dI,>.'CI, >.
with coefficients given by
and, for
~ =
±1, d~I,N
=
Q8x',>.'c
In these formulas, N denotes the norm map from F q2 to F q , x ---. x the nontrivial element of the Galois group of F q2 Over F q , and {jx,y the Kronecker delta. The fact that the homomorphism ~ is given by the preceding formulas is proved by checking the intertwining formulas in the statement of Theorem 1. These, in turn, are proved using Chang's identities for exponential sums ([2]' Lemma 1.2; see also [3], Lemma 5.6, where the connection between these identities and the Davenport-Hasse formulas for Gauss sums is pointed out.) We next turn to the question of when the homomorphism ~ is surjective. Keep the preceding notation, and let ~ be as in Theorem 1, with F' = F m for some nonnegative integer m. In order to prove that ~ is surjective, it is enough to show the injectivity of the induced map ~. : 'H. ---. 'H'., where 'H. is the dual space of 'H. By Theorem 1, ~. is given by the map Jr,o ---. Jr,O' , where ()' = () 0 NT for each basis element Jr,o of 'H.. So we only have to show that if fTI, ol =I Jr2,02 then Jrl,O~ =I Jr2,02' The set of basis elements Jr,o is in bijective correspondence with the set of semisimple conjugacy classes of C· F •. Let 81 and S2 be semisimple elements in C· F • corresponding to the pairs (Tl , ()1) and (T2, ()2), respectively. We will find a condition on m such that if SI and Fm S2 are not conjugate in C*F' then they are not conjugate in C· • •
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Lemma 6 Let C = Z(C)/Z(C)O, and let a be the map on C induced by the natural action of F on Z(C). Assume that (m,ICI) = 1, and that m == 1 (mod o(a)) where o(a) is the order of a. Then for each s E C*Fo, the set of representatives of the C*Fo -conjugacy classes which om are conjugate to sunder C* belong to distinct conjugacy classes in C*F . For s E C*Fo, put A(s) = Zco(s)/Zco(s)o. By Lemma 2.3.1 in Asai's paper [1], A(s) can be viewed as a subgroup of the character group of C, with corresponding F-action. Let m satisfy the conditions in the statement of the Lemma; then m is prime to IA(s)l, and the F-twisted classes in A(s) coincide with the Fm-twisted classes in A(s). Now let A(s) = A(s) < a > be the semidirect product of A(s) with the cyclic group < a >, where we denote by a the restriction of F to A(s). As m is prime to IA(s)l, the map x ----> x m gives a bijection on the conjugacy classes of A(s). Since this map leaves A(s)a invariant, it induces a bijection on the set A(s)a/~ ~ A(S)/~F' Note that for each x E A(s), we have (xa)m = xF(x)··· Fm-l(x)a. For each c E A(s)/~F> choose, E C* such that ,-IF(r) = c where c is a representative of , in Zco (s), and put Se = 'S,-I. Then the elements Se o are the representatives of the C*F classes in the C* -class of s. But then m , - I pm(r) = cF(C) ... pm-l(C). So Se represents a class in C*Fo corresponding to ba = (ca)m in A(s)a/ ~. Since the map xa ----> (xa)m is a bijection on A(s)a/~ = A(s)am/~, this shows that the elements Se are also representatives of the C*Fmo -classes in the C* -class of s, completing the proof of the Lemma.
Theorem 2 The homomorphism ~ : 'H' ----> 'H defined in Theorem 1 is surjective whenever m satisfies the conditions stated in Lemma 6. The proof of the Theorem is immediate from Lemma 6 and the remarks preceding the statement of the Lemma. Let {e~',II'} and {eT,II} denote the primitive idempotents in the commutative semisimple algebras 'H' and 'H, associated with the irreducible representations {ff, ,II'} and {fT,II} of 'H' and 'H, respectively, as in [3], §3. When the homomorphism ~ is surjective, its action on the primitive idempotents of 'H is given as follows.
Theorem 3 Assume the homomorphism ~ described in Theorem 1 is surjective. The primitive idempotents e' of 'H' not in the kernel of ~ all have the form e' = e~,lIoNT for some F -stable maximal torus T, and an
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irreducible character 0 of TF . Moreover, each of them is mapped by ~ to a primitive idempotent of 'H, in such a way that ~ (e~,l)oNT) = eT,1)
for pairs (T,O) as above. Let e' be a primitive idempotent of 'H' not in the kernel of ~, associated with an irreducible representation l' of 'H', and assume that ~ is surjective, as in the hypothesis of the Theorem. Then it is a standard result that ~(e') is a primitive idempotent of 'H, and hence ~(e') = eT,I), for some pair (T, 0) as in the statement of the Theorem. Moreover, from the formulas de' = 1'(d)e', dE 'H', we obtain ~(c')~(e') = 1'(c')~(e'),
and hence l' = h,1) o~. We now have, by the factorization property of h,1) and the intertwining formula for ~ stated in Theorem 1, the result that
1'(c')
=
h.1)(~(c'))
=
if
0
h(~(c'))
for all d E 'H'. It follows that e'
=
=
if
e~,l)oNT'
0
NT
0
f~(c')
=
f~,l)oNT(c')
completing the proof.
References
[1] T. Asai. Twisting operators on the space of class functions of finite special linear groups, Proc. Symp. Pure Math. Vol. 47, 99-148, Amer. Math. Soc., Providence R.I., 1987. [2] B. Chang, Decomposition of the Gelfand-Graev characters of GL 3 (q), Comm. Algebra 4 (1976),375-401. [3] C. W. Curtis, On the Gelfand-Graev representations of a reductive group over a finite field, J. Algebra 157 (1993), 517-533. [4] P. Deligne and G. Lusztig, Representations of reductive groups over finite fieldS, Ann. of Math. 103 (1976), 103-161. [5] F. Digne and J. Michel, Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts 21, Cambridge University Press, London/New York, 1991.
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Charles W. Curtis Department of Mathematics, University of Oregon Eugene, Oregon 97403, U.S.A. and Toshiaki Shoji Department of Mathematics, Science University of Tokyo Noda, Chiba 278, Japan Received February 1995
Modular Representations of Finite Groups of Lie Type in Non-defining Characteristic Meinol! Geck and Gerhard Hiss
1. Introduction
Let us consider a connected reductive algebraic group G, defined over the finite field IF q with corresponding Frobenius morphism F. We are concerned here with properties of finite-dimensional modules for the finite group G F over a sufficiently large field k of characteristic £, where £ is a prime not dividing q. On the one hand, it is possible to study the simple kGF-modules directly in terms of their distribution into Harish-Chandra series. This works in the general framework of the theory of finite groups with a split B N -pair, and is developed in Section 2. (Our approach here is different from the original one in [23].) As a result we obtain a classification of the simple kGF-modules in terms of triples (L, X, ¢) where L is a split Levi subgroup, X is a simple cuspidal kLF-module and ¢ is an irreducible character of the endomorphism algebra of the Harish-Chandra induction of X to G F . We propose to call any indecomposable direct summand of such an induced module a Harish-Chandra module of G F . We show that Harish-Chandra modules have some very remarkable properties: They have a simple head and a simple socle which are isomorphic to each other, and their endomorphism algebras are symmetric algebras. Thus they have a similar nature as the modules studied by J.A. Green in [25]. On the other hand, the concept of the decomposition matrix provides a link between modular and characteristic 0 representations. Our point of view is that the ordinary characters of GF are "sufficiently well understood". Then knowing the decomposition matrix is equivalent to knowing the irreducible Brauer characters corresponding to the various simple kGF-modules. In all known examples (e.g., GLn(q)) the irreducible ordinary and the irreducible Brauer characters can be arranged in such a way that the decomposition matrix has a lower unitriangular shape. This then provides a canonical labelling of the irreducible Brauer characters. In Section 3 we formulate a precise conjecture in this direction, under the condition that £ is a good prime for G. Let us assume that the decomposition matrix has a lower unitrian-
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gular shape. Then we can label the irreducible Brauer characters of C F in two possible ways: By the triple (L, X, ¢) arising from the HarishChandra theory and by the ordinary character corresponding to it via the triangularity of the decomposition matrix. The resulting "identification" problem first arose (and was solved) in the work of Dipper and James on GLn(q). We propose to formulate a "weak identification problem" as follows. Assume given an irreducible Brauer character in correspondence with some ordinary character of C F : Then determine in which Harish-Chandra series it lies. In particular, determine whether or not it is cuspidal. Even this weaker form of the problem is not solved in general. We believe that its solution would be a major step in the modular representation theory of finite groups of Lie type. This paper arose from an attempt to study in more detail the situation for the union B1 of unipotent blocks of C F . To start with, we collect in Section 3 some known results, due to G. Malle and the authors, on the existence of cuspidal unipotent Brauer characters for finite classical groups. We also describe a result on "supercuspidal" representations, due to the second named author. The motivation for studying this class of representations came from similar investigations by M.F. Vigneras in the case of ~adic groups. In Section 4, we present a far reaching extension of a result in [23], which gave a complete classification of the cuspidal unipotent Brauer characters in the case where C F is a classical group and P. is a "linear" prime (see (4.3) below for the exact conditions). Our extension describes the whole decomposition matrix of the unipotent characters in terms of the decomposition matrix of a certain endomorphism algebra whose construction is entirely analogous to that of the q-Schur algebra of GLn(q) studied by Dipper and James. We state without proof SOme further properties of these algebras, due to J. Gruber [26]. His results imply that the decomposition numbers for the unipotent characters of a classical group C F (the results for the orthogonal groups in even dimension are not complete yet) for linear primes P. can be calculated from those for various general linear groups (Theorem 4.13). All of the above results are only concerned with the case where P. is a good prime for C. Now Lusztig's theory of character sheaves and Shoji's proof of Lusztig's conjecture on character sheaves provide us with new tools in studying such questions in the bad prime case. We collect some basic results of the theory of character sheaves in Section 5. Here, we have tried as much as possible to formulate these results in terms of almost characters (instead of the "geometric" language of the original articles of Lusztig and Shoji). Our hope is that this formulation will
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have some independent interest inasmuch as it might help readers to become acquainted with some fundamental implications of the theory of character sheaves to the ordinary character theory of G F . These results will be used in Section 6 to determine the exact number of unipotent irreducible Brauer characters (modulo some mild restrictions on q). For good primes £ this was already done by the authors in [20], by determining explicitly a "basic set" of Brauer characters; for the case where £ = 2 and G is a classical group, see [19], III. Hence, here we can restrict ourselves to the cases where G is an exceptional group and £ = 2,3,5. Our methods also provide a new proof of one part of the main result on basic sets in [20] (see the remarks in (6.5)). In Section 7 we study the example where G is of split type E 6 and q has multiplicative order 3 modulo £. We show how the various methods and results of the previous sections can be used to solve the "weak identification problem" in this example: We prove that the decomposition matrix has a lower unitriangular shape, and we determine exactly in which modular Harish-Chandra series the irreducible unipotent Brauer characters of GF lie. We also prove that the two cuspidal unipotent Brauer characters of G F remain irreducible as Brauer characters, for all good primes £. 2. Harish-Chandra series
2.1. In this section we study representations of finite groups of Lie type in the general framework of the theory of groups with a BN-pair. Our results are concerned with the distribution of the irreducible representations into Harish-Chandra series and the study of endomorphism algebras of representations obtained by Harish-Chandra induction of cuspidal representations from Levi subgroups. If the field over which the representations are taken has characteristic 0, this is a classical theory (see, for example, [6]). Here, we are going to present some ofthe main results obtained by G. Malle and the authors (see [23], Sections 2,3), from a new point of view which takes the investigation of those endomorphism algebras as a starting point and derive structural properties of the Harish-Chandra induced representations from this. In order to achieve this we have to show that these endomorphism algebras are symmetric. Again, if the ground field has characteristic 0, this is a classical result (see [6]); in the general case, it is new. Throughout this section, G denotes a finite group with a split BNpair of characteristic p satisfying the commutator relations (see [6], Chapter 2). Let k be a sufficiently large field of characteristic £ i- p. (Note
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that we do not exclude the case £ = 0.) We shall write Y E mod-kG to denote that Y is a finitely generated right kG-module. Let W be the Weyl group of G and S = {Si liE I} be the corresponding set of standard generators for W. With each subset J ~ I there is associated a parabolic subgroup P j of G with a corresponding Levi decomposition Pj = UjL j , where Uj is the largest normal ~subgroup of P j and L j is a Levi complement (defined as in [loco Cit.]). We say that a subgroup L of G is a Levi subgroup of G if there exists some subset J as above and an element n EN such that L = nLjn- l . Note that Levi subgroups are again finite groups with a split BN-pair of characteristic p satisfying the commutator relations. 2.2. Let us recall the operations of Harish-Chandra induction and restriction with respect to a fixed Levi subgroup L of G. Choose a parabolic subgroup P of G such that P = UpL, where Up is the largest normal ~subgroup of P. If X E mod-kL then Rf (X) is defined to be the kGmodule obtained by first lifting X to kP via the canonical map P -+ L and then inducing this module from P to G. On the other hand, if Y E mod-kG then •Rf(Y) is the kL-module obtained by taking the fixed points of Up on Y, on which L acts since Up is normal in P. Howlett-Lehrer [33] and Dipper-Du [10] have shown that these operations indeed are independent of the chosen parabolic subgroup P. In complete analogy with the classical characteristic 0 case, we say that a kG-module Y is cuspidal if •Rf(Y) = 0 for all proper Levi subgroups L of G. (This definition, for £ > 0, first appeared in [11].) We say that (L, X) is a cuspidal pair if L is a Levi subgroup of G and X E mod-kL is cuspidal and simple (and taken up to isomorphism). It follows from the transitivity of Harish-Chandra induction and restriction that, given a simple module Y E mod-kG, there exists a cuspidal pair (L, X) such that X is a composition factor of' Rf(Y). The second author has shown [30] that such a pair is in fact uniquely determined up to conjugation by elements in N. In this situation, we say that Y belongs to the (L, X)-Harish-Chandra series of G. If we take a complete set of representatives of the N-conjugacy classes of cuspidal pairs in G we obtain a partition of the set of isomorphism classes of simple kGmodules into Harish-Chandra series, one for each cuspidal pair in that set of representatives (see [loco cit.]). Under the assumption that Harish-Chandra induction from mod-kL to mod-kG is independent of the parabolic subgroup used to define it, it is also shown in [loco cit., Theorem 5.8] that the set of simple kG-modules in the (L, X)-Harish-Chandra series can be equivalently characterized as the set of composition factors of the head of Rf(X), and also as the set
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of composition factors of the socle of Rf(X). Since this assumption has been shown to be always satisfied, we see that the sets of composition factors of the head and of the socle of Rf(X) are the same, for each cuspidal pair (L, X). We will refer to this fact as "Property A" for Rf(X). 2.3. We shall now fix one cuspidal pair (L, X) and consider the endomorphism algebra H = H(L,X) := Endkc(Rf(X)),
where we use the convention that endomorphisms act from the left. At first we recall the results in [23], Section 3, describing the structure of H, and then we will show how this can be used to obtain information about the direct summands of Rf(X) and the simple kG-modules in the (L, X)-series of G. This will provide new proofs for the main result (Theorem 2.4) of [lac. cit.], Section 2. Let W(L,X) be the stabilizer of X in (Nc(L) nN)L/L. This group has a semidirect product decomposition W(L,X) = R(L,X)C(L,X) where R(L,X) is a normal subgroup (which is in fact a Coxeter group) and C(L,X) is a complement to R(L,X) in W(L,X). For each w E W (L, X) there is an associated element B w E H, and these elements form a basis of H. In particular, the dimension of H equals the cardinality of W(L,X). (This follows from (3.5), from results of Dipper and Fleischmann [11] describing, in a more general setting, properties of such endomorphism algebras.) Note that in [23] we worked with left rather than right modules. It is easy to adjust the definition of the Bw's to the dual situation. Of course, the main results of [23] remain true if left modules are replaced by right modules throughout. We are going to use some of the main results of Dipper's paper [9], which are formulated for right modules and endomorphisms acting from the left. This is the reason why we changed our notation in this work. Following the original proofs of Howlett-Lehrer in the case of characteristic zero [32], in [23], Section 3, we obtained rules for multiplying together two such basis elements of H. Let {va I a E ~/} be the set of standard generators for the Coxeter group R(L,X) defined in [lac. cit.], (3.2). The multiplication rules may now be summarized as follows. (a) Twisting by a cocycle. There exists a 2-cocycle X : W(L, X) x W(L,X) ~ P such that
BwBx = X(w,x)B wx
BxBw = X(x,w)B xw for all w E W(L, X) and x E C(L, X). (See [lac. cit.], Theorem 3.12(i), (ii).) and
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(b) Homogeneous relations. If W E R(L, X) and W = Val'" Va, is a reduced expression then there exists a non-zero scalar I E k such that
(See [loco cit.], Lemma 3.4 and Proposition 3.6.) (c) Quadratic relations. For each standard generator V of R(L, X) there exist constants 0 i- 0: E k and f3 E k such that
(See [loco cit.], Proposition 3.7.) The homogeneous and quadratic relations imply that the k-subspace HI of H defined by HI := (Bw I W E R(L, X))k is in fact a subalgebra of H of dimension IR(L, X)I. Under additional assumptions, the cocycle )..' in (a) and hence the constants appearing in the homogeneous relations in (b) can be shown to be trivial. One such assumption is that Rr,(X) has an indecomposable direct summand with multiplicity 1, or that the simple module X can be extended to its inertia subgroup in (Nc(L) nN)L (see [loco cit.], Corollary 3.13). In these cases the above subalgebra HI is an Iwahori-Hecke algebra associated with the Coxeter group R(L, X) and with standard generators B va , for a E S. (See [2], Exercise IV, 2.23; one can also normalize the basis elements B w so as to obtain the familiar relations in terms of basis elements Tw , see [23], Theorem 3.12.) It may be conjectured that HI always is an Iwahori-Hecke algebra. For more results about the structure and the possible values which the parameters in the quadratic relations can take, we refer to [23], (3.14ff). Here, we shall content ourselves by summarizing the above results as follows.
Proposition 2.4 For each x E C(L, X), let H x := HI . B x . Then the family of k-subspaces {Hx I x E C(L, of H forms a C(L, X)-graded Clifford system in H (in the sense of !'lj, Definition 11.12). That is, the following conditions are satisfied.
xn
(a) If x, y E C(L, X) then Hxy is generated (as a k-vector space) by all products hxh y, for x E Hx and hy E Hy.
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(b) For each x E C(L,X) the element B x is invertible in H and we have H x = BxHI = HIB x .
(c) As a k-vector space, we have that H = over all x E C(L, X).
EBx H x
where the sum is
This is easily verified using the multiplication rules in (2.3). The basis element B x in (b) is invertible since by (2.3)(a) we have BxBx-1 = A'(x,x-I)B I. The importance of this description is that it allows us to use the results in [7], Section nc, about a "Clifford theory" for the irreducible representations of such an algebra. Our next aim is to show that H is a symmetric algebra. For this purpose we introduce, for each w E W (L, X), the k-linear map Tw : H --t k defined by
Tw(
L
O:w,Bw') = O:w
(where O:w' E k).
w'EW(L,X)
The proof below is similar to the one given in [6], Proposition 10.9.1, for the case that the characteristic of k is zero. Let R := R( L, X) and C := C(L,X). For w E R, let l'(w) denote the length of w with respect to the Coxeter group R. Lemma 2.5 Let w E R and v be a standard generator of R. Then
for some" 8, ,',8' E k, ",'
I- o.
Proof. This follows from the multiplication rules (2.3)(b) and (c). We give a proof only for the first equation. Suppose first that I' (wv) > I' (w). Then, by (2.3)(b), BwBv = ,Bwv for some 0 1-, E k. Suppose now that l'(wv) < l'(w). In this case put w' := wv. Then w = w'v and II(W'V) = l'(w) > 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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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
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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,
°
En. f
(15,17) (30, 15) (15,16) (D 4, E)
17
1 1
W15
1 1
f
18
1 1
If e = 6 then (15,17) has £-defect 0, if e = 3 then (D 4,E) lies in a different block than the other three characters (and than the two cuspidal unipotent characters), see [3], Tables 1, 2, case 5. Hence, in both cases, we only need to consider three characters, and we see that we can reorder the above matrix of scalar products such that we obtain a lower unitriangular shape. Note also that the two cuspidal unipotent characters are not constituents of the chosen projective characters. Now let :F = {(15,5), (30,3), (15,4), (D 4 , From Tables 1 and 2 we find that W4 W5 f 5 (15,5) 1 1 (30,3) 1 1 1 (15,4) 1 (D 4 ,1)
In.
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As before, either e = 3 and (D 4 , 1) lies in a different block or e = 6 and (15,5) has £-defect 0. In both cases, we obtain a lower unitriangular matrix of scalar products for the remaining characters. But now f 5 (which is needed if e = 6) has the two cuspidal unipotent characters as constituents. In order to replace f 5 by another projective character we argue by using Barish-Chandra induction from a Levi subgroup L of type D 4 . Let e = 6. Then the Sylow £-subgroups of L F are cyclic hence the decomposition numbers of the unipotent characters of L F are given in terms of Brauer trees, see [16]. We see that the unique cuspidal unipotent character 'P of L F is a leaf on such a tree and is joined to the execptional vertex which corresponds to non-unipotent characters of L F . We conclude that the Barish-Chandra induction of 'P is the unipotent part of a projective character of G F . But its decomposition into unipotent characters is simply given by (D 4 , 1) + (D 4 , c) + 2(D 4 , r). This gives us a suitable projective character which, together with W4 and W5, yields the desired unitriangular shape, and none of these projective characters contains the two cuspidal unipotent characters as constituents. It remains to consider the "big" family containing the two cuspidal unipotent characters. From Tables 1 and 2 we find that f 12 flO WlO 'lin f 11 1 1 1 (20, 10) 2 1 (90,8) 1 1 1 (80,7) 1 1 1 (60,8) 1 (D 4 , r) 1 (10,9) 1
E6 [B] E6 [B2]
If e = 6 then (20,10), (10,9), (90,8) have defect 0; if e = 3 then (90,8) has defect and (D 4 , r) lies in a different block. Arguing in a similar way as above we would obtain a lower unitriangular shape with the additional requirement on scalar products with the two cuspidal unipotent characters, if we were able to split up f 12 into a sum of two projective characters one of which contains E 6 [B] and the other contains E6 [B2] as a constituent. That this is indeed possible follows using the fact that all ordinary characters in our basic set (i.e., the unipotent characters) and all projective characters considered are real valued except the two cuspidal unipotent characters (which are complex conjugate to each other), 0 and Lemma 7.5 below.
°
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Lemma 7.5 Let 9 be any finite group, p any prime and B a union of p-blocks of g. Assume that the ordinary characters XI,···, Xn+2 (n 2 0) form an ordinary basic set for B and that WI, ... , Wn are projective characters such that the following conditions hold. (a) The character Xn+1 is the complex conjugate of Xn+2, and both of them have multiplicity 0 in WI, ... , Wn' (b) The n x n-matrix of scalar products (Xi, Wj) (1::; i,j ::; n) has full rank. Then there exist two projective characters W~+I and W~+2 such that = Oij for i,j E {n + 1,n + 2}.
(Xi, wj)
(This is a general result about finite groups, and its rather simple proof will be omitted.) 7.6. Let us consider in more detail the case where e = 3. (Note that we then automatically have that £ > 5.) Since the unipotent characters form a basic set for BI we know that BI contains exactly 30 irreducible Brauer characters. By Section 2 they are distributed into modular Barish-Chandra series. In [23], Table 5.4, we have shown that there are 10 such Barish-Chandra series, corresponding to Levi subgroups L of type 0, A 2 , A 2 X A 2 , D 4 and E 6 . If L -=I- C then L F has a unique cuspidal unipotent Brauer character, and the corresponding series contains, respectively, 13, 5, 4, 2 irreducible Brauer characters. Bence there are 6 cuspidal unipotent Brauer characters of C F left. In the course of the proof of Theorem 7.4, we have shown that the unipotent characters of C F can be ordered such that our matrix D has a lower unitriangular shape. This then defines a canonical labelling of the irreducible Brauer characters in B I , in terms of the unipotent characters. We denote by AX the irreducible Brauer character corresponding to the unipotent character X. Our aim is to determine exactly in which Barish-Chandra series the irreducible Brauer characters AX lie. (Note that, for e odd, e -=I- 3, the distribution into Harish-Chandra series is known by the results on Brauer trees in [31].) The characters (81,6), (81,10), (90,8) have defect 0 hence they remain irreducible as Brauer characters, and these Brauer characters do lie as well in the principal series. The characters {(D 4 , 1), (D 4 , c), (D4 , r)} form an ordinary basic set for a single block contained in BI whose defect is cyclic (see [3]; this was already used in the proof of Theorem 7.4). Its decomposition matrix is
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M. Geck and G. Hiss
given as follows.
>'1 >'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.
M. Geck and G. Hiss
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
°
°
Representations in Non-defining Characteristic
245
hence £ E {5, 7, 11 }. Since e = 3 we can only have that £ = 7. But then 6 + 3a + 2b - 3c = 7 which is impossible. (g) We have now found all cuspidal irreducible Brauer characters in B I . It follows that no. 20 and no. 21 must correspond to irreducible Brauer characters in the A~-series. Let us determine the character
M. Geck and G. Hiss
246
which is the correct one. Note finally that Conjecture 3.4 would predict that the projective character no. 20 can be subtract from no. 19. But again, we were not able to decide this. Acknowledgements. We thank Jochen Gruber for useful discussions on the material of Section 4 on decomposition numbers and also for allowing us to announce some of the results of his PhD-thesis [26]. We also thank Frank Lubeck for helpful comments on character sheaves.
References
11] A.M. AUBERT, Formule des traces sur les corps finis, this volume. [2] N. BOURBAKI, Groupes et algebres de Lie, Chap. IV, V, VI, Hermann, Paris, 1968.
13] M. BROUE, G. MALLE, AND J. MICHEL, "Repn?sentationsunipotentes generiques et blocs des groupes reductifs finis", Asterisque 212 (1993), 7-92.
14] M. BROUE AND J. MICHEL, "Blocs et series de Lusztig dans un groupe reductif fini", J. reine angew. Math. 395 (1989), 56--67. [5] M. CABANES, "A criterion for complete reducibility and some applications", in: Representations Lineaires de Groupes Finis, Luminy, 16-21 mai 1988, Asterisque 181-182 (1990),93-112. [6] R.W. CARTER, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, 1985. [7] C.W. CURTIS AND I. REINER, Methods of representation theory, vols. 1, 2, Wiley, 1981, 1987. [8] F. DIGNE AND J. MICHEL, Representations of finite groups of Lie type, London Math. Soc. Students Texts 21, Cambridge University Press, 1991. [9] R. DIPPER, "On quotients of Horn-functors and representations of finite general linear groups I", J. Algebra 130 (1990), 235-259. [10] R. DIPPER AND J. Du, "Harish-Chandra vertices", J. reine angew. Math. 437 (1993), 101-130.
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[11] R. DIPPER AND P. FLEISCHMANN, "Modular Barish-Chandra theory I", Math. Z. 211 (1992), 49-71. [12] R. DIPPER AND G.D. JAMES, "The q-Schur algebra", Proc. London Math. Soc. 59 (1989), 23-50. [13] R. DIPPER AND G.D. JAMES, "Representations of Becke algebras of type B n ", J. Algebra 146 (1992), 454-481. [14] J. Du AND L. SCOTT, "Lusztig conjectures, old and new, I", J. reine angew. Math. 455 (1994), 141-182. [15] P. FaNG AND B. SRINIVASAN, "The blocks of finite general linear and unitary groups", Invent. Math. 69 (1982), 109-153. [16] P. FaNG AND B. SRINIVASAN, "The blocks of finite classical groups", J. reine angew. Math. 396 (1989), 122-191. [17] M. GECK, "On the decomposition numbers of the finite unitary groups in non-defining characteristic", Math. Z. 207 (1991), 83-89. [18] M. GECK, "The decomposition numbers of the Becke algebra of type E 6 ", Math. Compo 61 (1993),889-899. [19] M. GECK, "Basic sets of Brauer characters of finite groups of Lie type, II", J. London Math. Soc. 47 (1993), 255-268; III, Manuscripta Math. 85 (1994), 195-216. [20] M. GECK AND G. HISS, "Basic sets of Brauer characters of finite groups of Lie type", J. reine angew. Math. 418 (1991), 173-188. [21] M. GECK, G. HISS, F. LUBECK, G. MALLE, AND G. PFEIFFER, "CHEVIE-A system for computing and processing generic character tables", AAECC 7 (1996), 175-210. [22] M. GECK, G. HISS, AND G. MALLE, "Cuspidal unipotent Brauer characters", J. Algebra 168 (1994), 182-220. [23] M. GECK, G. HISS, AND G. MALLE, "Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type", Math. Z. 221 (1996), 353-386. [24] M. GECK AND G. MALLE, "Cuspidal unipotent classes and cuspidal Brauer characters", J. London Math. Soc. 53 (1996), 63-78.
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[25] J .A. GREEN, "On a theorem of Sawada", J. London Math. Soc. 18 (1978),247-252. [26] J. GRUBER, Cuspidale Untergruppen und Zerlegungszahlen klassischer Gruppen, (PhD-Thesis, Heidelberg, 1995). [27] G. HISS, "Harish-Chandra series of Brauer characters in a finite group with a split BN-pair" J. London Math. Soc. 48 (1993), 219-228. 1
[28] G. HISS, "Supercuspidal representations of finite reductive groups", J. Algebra 184 (1996), 839~851. [29] G. HISS, Zerlegungszahlen endlicher Gruppen vom Lie- Typ in nicht-definierender Charakteristik (Habilitationsschrift, RWTH Aachen, 1990). [30] G. HISS, "Decomposition numbers of finite groups of Lie type in non-defining characteristic", in: G. O. Michler and C. M. Ringel, Eds., Representation Theory of Finite Groups and Finite-Dimensional Algebras, Birkhiiuser, 1991, pp. 405-418. [31] G. HISS, F. LUBECK, AND G. MALLE, "The Brauer trees of the exceptional Chevalley groups of type E 6 ", Manuscripta Math. 87 (1995), 131-144. [32] R. B. HOWLETT AND G. I. LEHRER, "Induced cuspidal representations and generalized Heeke rings", Invent. Math. 58 (1980), 37-64. [33] R. B. HOWLETT AND G. I. LEHRER, "On Harish-Chandra induction for modules of Levi subgroups", J. Algebra 165 (1994), 172-183. [34] G.D. JAMES, "The decomposition matrices of GLn(q) for n ::; 10", Proc. London Math. Soc. 60 (1990), 225-265. [35] N. KAWANAKA, "Shintani lifting and Gelfand-Graev representations", Proc. Symp. Pure Math., Amer. Math. Soc. 47 (1986), 575616. [36] M. LINCKELMANN, Letter to the authors, April 25th, 1994. [37] G. LUSZTIG, "On the finiteness of the number of unipotent classes", Invent. Math. 34 (1976), 201-213.
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[38] G. LUSZTIG, Characters of reductive groups over a finite field, Ann. Math. Studies 107, Princeton U. Press, 1984. [39] G. LUSZTIG, "Character sheaves", Adv. Math. 56 (1985), 193-237; II, 57 (1985), 226-265; III, 57 (1985), 266-315; IV, 59 (1986), 163; V, 61 (1986), 103-155. [40] G. LUSZTIG, "On the character values of finite Chevalley groups at unipotent elements", J. Algebra 104 (1986), 146-194. [41] G. LUSZTIG, "Remarks on computing irreducible characters", J. Amer. Math. Soc. 5 (1992),971-986. [42] G. LUSZTIG, "Green functions and character sheaves", Annals of Math. 131 (1990),355-408. [43] L. PUlG, "Algebres de source de certains blocs de groupes de Chevalley", in: Representations Lineaires de Groupes Finis, Luminy, 16-21 May 1988, Asterisque 181-182 (1990), 221-236. [44] T. SHOJI, "Character sheaves and almost characters of reductive groups", Adv. Math. 111 (1995),244-313. [45] T. SHOJI, "Character sheaves and almost characters of reductive groups, II", Adv. Math. 111 (1995),314-354. [46] T. SHOJI, "Unipotent characters of finite Chevalley groups", this volume. [47] T. SHOJI, "On the computation of unipotent characters of finite classical groups", AAECC 7 (1996), 165-174. [48] M.-F. VIGNERAS, "Sur la conjecture locale de Langlands pour GL(n, F) sur F'z, C. R. Acad. Sci. Paris Ser. I Math. 318 (1994), 905-908. Meinolf Geck Lehrstuhl D fUr Mathematik RWTH Aachen 52062 Aachen, Germany
Gerhard Hiss IWR der Universitiit Heidelberg 1m Neuenheimer Feld 368 69120 Heidelberg, Germany
[email protected]
[email protected]
Received December 1994
Centers and Simple Modules for Iwahori-Hecke Algebras Meinol! Geck and Raphael Rouquier
1. Introduction
The work of Dipper and James on Iwahori-Hecke algebras associated with the finite Weyl groups of type An has shown that these algebras behave in many ways like group algebras of finite groups. Moreover, there are "generic" features in the modular representation theory of these algebras which, at present, can only be verified in examples by explicit computations. This paper arose from an attempt to provide a conceptual explanation of these phenomena, in the general framework of the representation theory of (symmetric) algebras. We will study relations between the center of such algebras and properties of decomposition maps, and we will use this to obtain a general result about the "genericity" of the number of simple modules of Iwahori-Hecke algebras. Usually, the formalism of decomposition maps is developed for algebras over a complete discrete valuation ring. However, in our applications to Iwahori-Hecke algebras, we have to make sure that this also works over the ring of Laurent polynomials in one indeterminate over the integers. Roughly speaking, this will be achieved by using the theory of Henselian rings (see [Ray]). In Section 2, we describe such a general setting for decomposition maps of algebras over integrally closed ground rings (see Proposition 2.11). Furthermore, we extend the standard results on the "Brauer-Cartan triangle" to the case of orders in non-semisimple and non-split algebras, by using enlargements of the usual Grothendieck groups. As a formal consequence of the definition, we get a factorization property of decomposition maps (see Proposition 2.12). Previo~sly, this factorization was only established using strong additional assumptions on the realizability of representations (d. [Gel], (2.4), (5.3)). Let H be an algebra over a local integrally closed domain 0 with residue field k. Then we have a canonical map from central functions on This paper is a contribution to the DFG research project on "Algorithmic Number Theory and Algebra". The second author thanks the Lehrstuhl D fUr Mathematik, RWTH Aachen, for its hospitality and support during part of this work.
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M. Geck and R. Rouquier
H to central functions on kH (induced by reduction modulo the maximal ideal of 0). In Proposition 3.1, generalizing a theorem of Hattori, we show that the surjectivity of this map implies that the decomposition map has finite cokernel and that the Cartan matrix of kH has non-zero determinant. For a symmetric algebra, this surjectivity is equivalent to the surjectivity of the reduction map Z(H) ---. Z(kH). In Theorem 5.2 we prove that this surjectivity holds for Iwahori-Hecke algebras, by constructing a basis of the center from [Ge-Pfj. This part is inspired by the work of [Di-Ja], where the type An was considered. We believe that this stability of the center under reduction is an important similarity between group algebras and Iwahori-Heeke algebras. In another direction we show that, under suitable hypothesis, the number of simple modules of the algebra kH is "generic", in the following sense. Assume that the ground ring 0 has Krull dimension 2. Fix a height 1 prime ideal p and let k p be the quotient field of O/p. Then Theorem 3.3 gives a condition on p which implies that the number of simple modules of kpH equals the number of simple modules of kH. In our applications to Iwahori-Hecke algebras, 0 will be the localization of the ring of Laurent polynomials over Z in one indeterminate. The choice of height 1 and height 2 prime ideals yield algebras kH and kpH, where k is a finite field of characteristic £ and kp is a cyclotomic field of characteristic 0 (see [Gel] for more details). In Theorem 5.4 we check that the above hypotheses are satisfied whenever the prime £ is not too small (e.g., does not divide the order of the underlying finite Weyl group). Hence the number of simple modules of kH is determined by the algebra kpH, i.e., it is "generic". This is one step in an attempt to prove the more general conjecture of [Get], (5.6), that even the decomposition maps themselves are "generic". 2. Decomposition maps It is the purpose of this section to develop the basic theory of decom-
position maps for algebras over integrally closed rings. Much of what follows is inspired by [Bra-Ne] and [Be]. 2.1 Grothendieck groups and bilinear forms
Let 0 be a commutative local ring and H an O-algebra, finitely generated and free as an O-module. We denote by Ko(H) the Grothendieck group of the category of finitely generated projective left H-modules and by Ro( H) the Grothendieck group of the category of finitely generated H-modules which are free as O-modules (such modules are called
Centers and Simple Modules for Iwahori-Hecke algebras
253
H-lattices). The imbedding of the first category into the second one induces a map ("Cartan map") Co : Ko(H) ---. Jlo(H). We denote by R!;(H) the subset of Jlo(H) given by the classes of the H-Iattices. Note that R!;(H) generates Jlo(H). In what follows, all modules are supposed to be finitely generated. There is a bilinear form (-, ·)0: Ko(H) x Jlo(H) ---. Z defined by ([P]' [V])o
=
rankoHomH(P, V)
for P a projective H-module and V an H-Iattice (where [P] and [V] denote the classes of P and V in Ko(H) and Jlo(H) respectively). The fact that HomH (P, V) is free over 0 follows from the existence of an integer n such that PIHn, since then HomH(P, V) is a direct summand of HomH(Hn, V) ~ vn as an O-module. Let us now prove that this form is well defined. If [P] = [PI] + [P2 ], then HomH(P, V) ~ HomH(PJ, V) EB HomH(P2 , V). If 0 ---. VI ---. V ---. V2 --t 0 is an exact sequence of Hlattices, then 0 ---. HomH(P, VI) ---. HomH(P, V) ---. HomH(P, V2 ) ---.0 is exact because P is projective. These two facts show that (-, ·)0 is indeed well defined. Let us denote by CF(H) = Homo(H/[H,H],O) the module of class functions (where [H, H] denotes the O-submodule of H generated by the commutators hh' - h'h, h, h' E H). We introduce now a bilinear form (., ·)0: Ko(H) x CF(H) ---. 0 as follows: Let P be a projective H-module. There exists an integer n such that P is a direct summand of Hn. Let e be the corresponding idempotent in EndH(Hn). The latter space can be canonically identified with the space Mn(H) of n x n-matrices over H. Let Tr(e) E H be the trace of e. It is straightforward to check that the image of Tr(e) in H/[H,H] depends only on the class [P] of Pin Ko(H) and that the corresponding map Ko(H) ---. H/[H, H] is additive. If f E CF(H) then we define ([P], 1)0 = f(Tr(e)). Assume that f is the character ch([V]) of an H-Iattice V (where we denote by ch : Jlo(H) ---. CF(H) the character map). One has HomH(eHn, V) ~ evn as O-modules, hence ([P], [V]) = rankoeVn. But ([P]' 1)0 = f(Tr(e)) = 10· rankoeVn, hence ([P], I) = ([P], [V]) . 10 . This proves the following : Lemma 2.1 One has (x, ch(y))
=
(x, y) . 10 for x E Ko(H) and y E
Jlo(H). We define a semi-group morphism Po from R!;(H) to the set Maps(H, o [X]) of maps H ---. O[X] (with operation given by pointwise multiplication of maps) as follows:
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M. Geck and R. Rouquier
Given an H-Iattice M and x E H, we let Po([M])(x) be the characteristic polynomial of x acting on the free O-module M. Let B be a commutative local O-algebra given by tB : 0 ----+ B. If M is an O-module, we denote by BM the B-module B0 0 M. Without specification, tensor products are taken over 0, i.e., B 0 M means B 00 M. There are canonical maps t~O : Ko(H) ----+ Ko(BH), t~ : Ro(H) ----+ Ro(BH), t~F : CF(H) ----+ CF(BH) and ttt : Maps(H,O[X]) ----+ Maps(BH, B[X]) induced by extension of scalars. The following lemma gives the compatiblities with extension of scalars: Lemma 2.2 Let x E Ko(H), Y E Ro(H) and
t FID B
0 Co
=
CB
0
tKo B,
t ~F 0 ch
f
E
CF(H). Then
= ch 0 t~ ,
Proof. Only the second assertion does not follow directly from the definitions. If x = [P] and y = [V], then (tB(X), tB(Y))B = rankB(eV n ) 00 B, where n is such that PIHn and e is the corresponding idempotent of EndH(Hn). Since eV n is a free O-module, one has rankB(eV n ) 00 B = rankoeV n, hence (tB(x), tB(Y))B = (x, Y)o. 0 Lemma 2.3 Assume B is fiat over O. Then, the map IB 0 t~F : B 0
CF(H)
----+
CF(BH) is an isomorphism.
Proof. From the exact sequence 0 ----+ [H,H] ----+ H ----+ H/[H,H] ----+ 0, one gets the exact sequence 0 ----+ [BH, BH] ----+ BH ----+ B 0 (H/[H, H]) ----+ O. Hence, B 0 (H/[H, H]) ':::: BH/[BH, BH] and finally HomB(BH/[BH, BH], B) ':::: B 0 Homo(H/[H, H], 0). 0 2.2 Algebras over a field
Let us first recall without proof some classical results about simple algebras (cf [Bkil]). Assume that 0 = K is a field. We have the following commutative diagram:
H - mod
---->
Rt(H)
---->
Rt(H/J(H))
r
(H/ J(H)) - mod
roo
p
---->
Maps(H, K[X])
r
Maps(H/J(H), K[X])
where J(H) denotes the radical of H. Hence, in order to study Rt(H) and its image in Maps(H, K[X]) , we can assume that J(H) = O. Now, the algebra H is semisimple, i.e., is isomorphic to a finite direct product of simple algebras. So, let us assume that H is simple. Let V
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be a simple H-module, D = EndH(V) and n = dimD(V). Then, H is isomorphic to the ring Mn(DO) of (n x n)-matrices over the skewfield DO opposite to D. Let m be the integer such that [D : Z(D)] = m 2 . Let Trd : H ---. Z(D) be the reduced trace of the central simple Z(D)-algebra H. It has the property that if L is a neutralizing field for H, i.e., such that Z(D) C LCD and H ®Z(D) L ~ Mmn(L), then the usual trace Mmn(L) ---. L is given by Trd ® 1£. Denote by chK(V) and chz(D)(V) the character of V respectively viewed as a module over the K-algebra H and as a module over the Z(D)-algebra H. We have chK(V)
= Trz(D)/KchZ(D) (V)
and
chz(D)(V)
= m Trd
where TrZ(D)/K : Z(D) ---. K denotes the trace map of the K-algebra Z(D), i.e., the character of the module Z(D) for the K-algebra Z(D). Similarly, we have PK(V) = NZ(D)[Xl/K[Xj(PZ(D)(V))
and
PZ(D)(V) = Prd
m
where NZ(D)[Xj/K[Xj : Z(D)[X] ---. K[X] is the norm map of the K[X]algebra Z(D)[X] and where Prd : H ---. Z(D)[X] is the reduced characteristic polynomial map and PK(V) and PZ(D) (V) are the characteristic polynomial maps of V respectively viewed as a module over the K-algebra H and as a module over the Z(D)-algebra H. Lemma 2.4 The following statements are equivalent for H a simple
K -algebra with simple module V and D = EndH(V): (1) the extension Z(D) of K is separable, (2) TrZ(D)/K -=I- 0,
(3) Trz(D)/KTrd -=I- 0,
(4) the algebra H ®K Z(D) is semisimple, (5) the algebra D ®K Z(D) is semisimple, (6) the H ®K Z(D)-module V ®K Z(D) is semisimple.
If Z(D) is a separable extension of K, then the algebra H ®K Z(D) is isomorphic to a direct product of [Z(D) : K] central simple Z(D)algebras and the module V ®K Z(D) is isomorphic to the direct sum of [Z(D) : K] non-isomorphic simple modules.
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Let us go back to the case where H is any finite dimensional algebra over K. From now, we assume that K is perfect (one could as well work with the weaker assumption that given any simple H-module V, then Z(EndH(V)) is a separable extension of K). A basis for Ro(H) (resp. Ko(H)) is given by the images of a complete set of representatives of isomorphism classes of simple (resp. projective indecomposable) modules. Hence, one has an isomorphism Ro(H) -+ Ko(H) given by sending the class of a simple module to the class of one of its projective covers. An irreducible character of H is defined as the character of a simple H-module and we denote by Irr(H) the set of irreducible characters of H. If K' is a field extension of K, then the canonical maps t~? : Ko(H) -+ Ko(K' H) and t~ : Ro(H) -+ Ro(K'H) are injective. There exists a finite Galois extension K' of K such that K' H is split, i.e., such that for every simple K'H-module V, the canonical map K' -+ EndK1H(V) given by multiplication is an isomorphism; we call such a field K' a neutralizing field for H. Let V be a simple H-module. Then, there are non-isomorphic simple K'H-modules VI, ... ,Vs and an integer mv (the Schur index of V) such that K'V ::: (VI EB ... EB Vs)m v . Note that we have [Kv : Z(Kv )] = m~ where K v = EndH(V). Let V' be another simple H -module, with K'V' ::: (V{ EB ... EB V;, )m v ' where the modules V;' are simple and V;' i- V; for i -=I- j. Then, if V i- V', we have Vi i- V;, for all i, j. Let Pv be a projective cover of V. Then, HOmK'H(K'Pv,K'V') ::: K' lSi HomH(PV , V') ::: K' lSi HomH(V, V') ::: HOmK'H(K'V,K'V'). Hence, K'Pv is a projective cover of K'V, i.e., denoting by Pi a projective cover of Vi, we have
K' Pv ::: (PI EB··· EB ps)m v . We then define Ro(H) as the subgroup of Ro(K'H) with basis {,;v [V]} where V runs over the simple H-modules (cf [Se, §12.1]). Similarly, we define Ko(H) as the subgroup of Ko(K' H) with basis {,;v [Pv ]} where V runs over the simple H-modules. Note that the group Ro(H) (resp. Ko(H)) is a subgroup of finite index ofRo(H) (resp. Ko(H)). In particular, rankRo(H) = rankRo(H) and rank Ko(H) = rankKo(H). It is clear that CK,(', '/K and (., ')K extend to maps Ko(H) -+ Ro(H), Ko(H) x Ro(H) -+ Z and Ko(H) x CF(H) -+ K compatible with the extension to K'. Furthermore, we define R{;(H) as Ro(H) n Rt(K'H).
Proposition 2.5 Recall that K is assumed to be perfect. Then the map PK : R{;(H) -+ Maps(H, K[X]) is an injection.
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Proof. (cf [Bra-Ne, Lemma 2]) It is enough to prove the lemma in the case KH split, which we assume now. Let M and N be two H-modules such that p([M]) = p([N]). By replacing M and N with their associated semisimple modules, one can assume that M and N are semisimple. Let S be a set of representatives of isomorphism classes of simple H-modules. For V E S, let av and bv be the multiplicities of Vasa composition factor of M and N. Since ch([M]) = ch([N]), one has (aV-bv)-1K = O. If 0 has characteristic zero, this implies av = bv , hence [M] = [N]. Otherwise, let p be the characteristic of K, p > 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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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.
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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
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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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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
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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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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
Centers and Simple Modules for Iwahori-Hecke algebras
269
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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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 O. Then, the number of simple k1i-modules is equal to the number of simple k'1imodules. More precisely, the map 1k @ do,: k @ Ro(k'1i) ---+ k @ Ro(k1i) is an isomorphism. Proof. This is a direct consequence of Theorem 3.3; we need to check its assumptions. The canonical map Z(1i) ---+ Z(k1i) is an isomorphism by
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Theorem 5.2, hence the canonical map CF(1i) -+ CF(k1i) is surjective, since 1i is a symmetric algebra (ef §4). Let now q be a height one prime ideal of 0, q =I- p. The algebra ((Oq)j q) 1i is semisimple if and only if Qw ~ q, according to Proposition 5.3. If £ E q, then Qw ~ q. If ~ E q, then ~ = ~lT, for some integer r, r > 0 since q =I- p. But
Remark 5.5 The decomposition map dOl is the map denoted d~ in [Gel]. In [Gel], it is conjectured that do : Ro(k'1i) -+ Ro(k1i) maps actually classes of simple modules to classes of simple modules.
References
[Bkil]
N. Bourbaki, Algebre, Chapitre VIII, Hermann 1958.
[Bki2]
N. Bourbaki, Algebre commutative, Chapitres V son 1985.
[Bra-Ne]
R Brauer and C. Nesbitt, On the modular representations of groups of finite order, Univ. of Toronto Sudies Math. Ser. 4 (1937).
[Br]
M. BroUl§, On representations of symmetric algebras: an introduction, preprint ETH Zurich, 1990.
[Cal
RW. Carter, Finite groups of Lie type: conjugacy classes and complex characters, Wiley, New York, 1985.
[CuRe]
C.W. Curtis and I. Reiner, Methods of representation theory, volume 1, Wiley, New York, 1981.
[Di-Ja]
R Dipper and G.D. James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. 54, 57-82 (1987).
[Di-Mi]
F. Digne et J. Michel, Fonctions L des varieles de DeligneLusztig et descente de Shintani, Memoire SMF 20, 1985.
[Gel]
M. Geck, Brauer trees of Hecke algebras, Comm. Algebra 20, 2937-2973 (1992).
a VII, Mas-
M. Geck and R. Rouquier
272 [Ge2]
M. Geck, On the character values of Iwahori-Hecke algebras of exceptional type, Proc. London Math. Soc. 68, 51-76 (1994).
[Ge3]
M. Geck, Beitriige zur Darstellungstheorie von Iwahori-Hecke Algebren, Aachener Beitriige zur Mathematik 11, Verlag der Augustinus Buchhandlung, Aachen, 1995.
[Ge4]
M. Geck, The decomposition numbers of the Hecke algebra of type £6, Math. of Camp. 61, 889-899 (1993).
[Ge-Pf]
M. Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras, Advances in Math. 102, 79-94 (1993).
[Go]
D.M. Goldschmidt, Lectures on character theory, Publish or Perish, Berkeley, 1980.
[Gy-Uno]
A. Gyoja and K. Uno, On the semi-simplicity of Hecke algebras, J. Math. Soc. Japan 41, 75-79 (1989).
[Ray]
M. Raynaud, Anneaux locaux henseliens, Springer Lecture Notes in Mathematics 169, Berlin-Heidelberg, 1970.
[Se]
J.-P. Serre, Representations lineaires des groupes finis, Hermann, 1978.
Meinolf Geck Lehrstuhl D fUr Mathematik RWTH Aachen, Templergraben 64, 52062 Aachen, Germany email: [email protected] and Raphael Rouquier U.F.R. de Mathematiques de l'Universite Paris 7 Denis-Diderot et UMR 9994 du CNRS 2 Place J ussieu F-75251 Paris Cedex 05, France email: [email protected] Received January 1995
Quantum Groups, Hall Algebras and Quantized Shuffles l James A. Green 2
Introduction This paper is concerned with a class of algebras whose origin IS III George Lusztig's book [L]. Lusztig introduces an algebra f over the field Q(t) of rational functions over Q in an indeterminate t, which he shows later is isomorphic to the positive part U+ of a certain quantum group. Lusztig defines the algebra f by structural properties. We propose below (Section 1) a category L:(A, v, I,,) of algebras which is based on Lusztig's definition of f; Section 2 contains some generalities on L:(A, v, I, .). In language appropriate to such categories, f may be characterized as a non-degenerate member of the class L:(Q(t), t, I,,) (see §3.1). An abstract definition needs at least two interesting examples to justify its existence! Fortunately we have a second example: it was proved in [G] that the "composition subalgebra" of the Hall algebra associated to a hereditary algebra over a finite field is also a member of a suitable L:(A, v, I,') - for this purpose, we should use Claus Ringel's "twisted" Hall algebra. These things are reviewed in Section 3; the remarkable connections which Ringel has found between Hall algebras and quantum groups can be explained, at least in part, by the fact that composition algebras and the algebras U+ share this common structure. Section 4 contains an interpretation of this work in terms of a "quantized" version of the "shuffle algebra" introduced some years ago by Rimhak Ree [Rh]. Lusztig's algebra f (and hence also U+) can be expressed as a subalgebra of such a quantized shuffle algebra. This is a purely formal re-writing of Lusztig's definition, but it leads to methods I This article is an enlarged version of a lecture given at the meeting "Autour des groupes reductifs finis; representations" held at CIRM, LuminyMarseille, 3-7 October 1994. 2 Added in proof. Since submitting I have learned that the quantum shuffle algebra and its application to quantum groups has also been described by M. Rosso in [Ro].
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of calculation which may be useful. I am much indebted to Pierre Cartier for suggesting that the shuffle algebra should be relevant to the category L:(A, v, I, .). 1. The Category L:(A, v, I,') 1.1. Let I be a set, ZI the free Abelian group with I as basis, and . : ZI x ZI ----+ Z a symmetric bilinear form on ZI. The pair (/,.) will be called a datum - later we shall use the special kind of datum which Lusztig calls a "Cartan datum," but for the moment this restriction is unnecessary. A datum is completely described by the symmetric integer matrix (i· j)i,jEf. Let NI = {v = LiEf viilvi EN}; this sub-monoid of ZI will provide an index set for gradings. 1.2. Definition. Let (/,.) be a datum. Let A be a (commutative) integral domain of characteristic zero (we assume A ;:2 Z), and v an invertible element of A. Then we say that an A-algebra L belongs to the class L:(A, v, I, '), or is an object of the category L:(A, v, I, '), if the following conditions LI, L2, L3 are satisfied. Ll. L = L~ENf is an NI-graded, associative A-algebra generated by elements Ui E L i (i E 1). Assume also La = A.l, where 1 is the identity element of A. We refer to the Ui (i E 1) as the generators of L. L2. There is a comultiplication r: L that
----+
L@L (@ stands for @A) such
(a) r(ui) = Ui @ 1 + 1 @ Ui for all i E I (so that the generators Ui are primitive elements of the co-algebra A), and (b) r is multiplicative, when L @ L is given the structure of A-algebra by using the following non-standard product (Lusztig's rule, see [L, p. 3])
(1.2a) for all homogeneous elements Xl, X2, Yl, Y2 in L (if z E Lv, we write Izi = v). We refer to r as the comultiplication of L. L3. There is a symmetric, A-bilinear form (-, -) : L x L (possibly degenerate) such that (a) (L,.., Lv) = 0 for all
J..L
=I- v in NI,
(b) (1,1) = 1, and (Ui, Ui) =I- 0 for all i E I, and (c) (x, yz) = (r(x), Y ® z) for all x, Y, z E L.
----+
A
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We refer to (-, -) as the bilinear form on L. 1.3. A morphism between objects L, L' of L:(A, v, I, '), is, by definition, an A-algebra map () : L --. L' such that ()(Ui) = u; for all i E I, where u; (i E /) are the generators of L'. Clearly, if such a morphism () exists, it is unique. 1.4. Remarks. (1) The bilinear form on L @ L which features on the right side of L3(c) is given by the (standard) rule (x @ y, x' @ y') = (x,x')(y,y'), for all x,y,x',y' E L. (2) If L E L:(A, v,I, .) then clearly the comultiplication r : L --. L @ L is completely determined, since L2(a) gives the values of r on the generators Ui of L. (3) r is coassociative. (To prove this, use the argument in [L], p. 3.) (4) The bilinear form (-, -) on L is uniquely determined, by L3, as soon as the elements (Ui, Ui) (i E /) are given. See Proposition 2.3, below. 1.5. Notation. It will be useful to think of the set I as an "alphabet," and to denote by (1) the set of all words (Le. finite sequences) a = al'" ap with al,"', ap E I. The length l(a) of the word a = al ap is Pi we allow the empty word 0 of length zero. The weight of a = al ap is the element /.I(a) = /.I of NI such that, for each i E I, /.Ii is the number of 7f E {I, ... ,p} for which a1l" = i. It is clear that a word of weight /.I has length p = LiEf /.Ii· Denote this sum LiEf /.Ii by tr /.I (cf [L], p. 2.) For given /.I E NI, we write 1(/.1) for the set of all words a = al ... ap of weight /.I. We have (1) = UVENfI(/.I), disjoint union. Example. If /.I {iij, iji, jii}.
= 2i + j (i,j being distinct elements of /), then 1(/.1)
=
1.6. Now suppose that L is an object of the category L:(A, v, I, .), with generators ui(i E /). For each word a = al'" ap E (I), write U a for the monomial u a , ••• u ap (if a = 0, take U a = 1). It is clear from axiom Ll that, for each /.I E NI, the homogeneous subspace Lv is the A-span of the set {u a Ia E I (/.I)} (in general there will be A-linear relations between these monomials u a ). Example. L 2i +j = A . U~Uj
+ A . UiUjUi + A . UjU~.
1.7. Suppose that a = a I ... an E (/) has length n. Then for any subset P = {7f1 < ... < 7fp } of the set 11 = {I, ... , n}, we define alP = a1l"l ... a1l"p' so that alP is a word of length IFI = p. Let L be an object of L:(A,v, I, '), with generators ui(i E /) and comultiplication r.
J. A. Green
276 By L2 we have
the terms in this non-commuting product are taken in the order 7f = 1, ... ,no This last product can be expanded as a sum Lp~.!! z(P), where for each subset P of !l we define z(P) = Zl ... Zn, with Z1r = u a" @ 1 if 7f E p, and Z1r' = 1 @ u a", if 7f' E pI = !l\P. Multiply out the product z(P) = Zl ... ZnJ using the Lusztig rule (1.2a). If 7f E P and 7f' E P', then Z1r,Z1r = (1 @x a",) (x a" @ 1) = va""a" (x a" @ x a",) = va",.a"(x a,, @ 1)(1 @ Xa",) = va",.a"Z1rZ1r" By repeated use of this equation, we may rearrange the factors in z(P) = Zl ... Zn to give (1.7b) where
a· P =
L a1r , . a1rJ
sum over all
(7f' ,7f)
E p' x P such that
7f'
<
7f •
(1.7c) Example. Suppose a = ijklm and P
= {2, 4}. Then
Z(P) = (1 @Ui)(Uj @ 1)(1 @Uk)(UI @ 1)(1 @u rn ) = vi-j+H+k.! . UjUI @ UiUkUrn . Apply (1.7b) to (1.7a). We get a formula which will be useful later.
If a E (I) has length l(a) = n, then for any L in .C( A, v,I, .) there holds r(u a ) =
L va'P(Ualp@ualP')'
(1.7d)
P~.!!
where the integer a . P is given by (1.7c). Equation (1.7d) may be generalized as follows. Define maps L ---. L 0d = L @ ... @ L (d factors) inductively for all integers d ~ 2 by r2 = r, and rd = (rd-l @ l)r if d ~ 3. Then rd(ui) = ui@l@l@ .. ·@l+l@ui@l@· .. @l+l@l@l@",@ui, for all i E I. 1.8.
rd
:
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277
Also rd is multiplicative, when L0d is given the structure of A-algebra, using the natural extension of the Lusztig product rule (1.2a), namely
for all homogeneous XI, ... ,Xd, YI, ... Yd E L; the sum L:lx>.I·IYILI in (1.8a) is over all (,x, J.L) Ed. x d. such that ,x > J.L. We may now extend the argument in 1.7 to calculate rd(u a ), where a = al ... an is a given word of length n. The result, as the reader may verify rather easily, is
rd(u a ) =
L v a.
P
(UalPl
@ ...
@ualPJ ,
(1.8b)
P
where the sum is over all d-tuples P = (PI, ... , Pd ) of subsets of!l such that !l = PI U ... U Pd (disjoint union), and for each such P
a·P
=
La7r' 'a
7r ,
(1.8c)
summed over all (7l",7l') E !l x !l such that (a) 7l" < 7l', and (b) 7l" E Ph,7l' E Pk with h > k.
2. The Structure of £(A, v,I,·) 2.1. Let (I, '), A, v be as in §1.2, and let F = A(Xi : i E I) be the free associative A-algebra in free generators (i.e. non-commuting indeterminates) xi(i E I). An argument of Lusztig (see [L], pp. 2-4) shows that F may be regarded as an object of £ (A, v, I, .). We recall this argument. F = L:~ENI F v is an N1-graded A-algebra on the generators xi(i E 1); the homogeneous subspace Fv being (freely) generated as A-module by the monomials Xa = Xal ... x ap , a E l(v).
L1.
L2. Since the Xi are free A-algebra generators of F, we may define an A-algebra map r : F ---> 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.
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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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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.
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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
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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 <5;;1' obtain an expression (2.4)
~N
=
L
h£,N,N£(q)p~(~~,)
£,N£ where h£,N,N £ (q) are "Hall-like" polynomials and N £ is a special nilpotent element in the split Levi subalgebra .c. Step 5. By (1.5)(ii) we then have (2.5)
F~N
L
=
h£,N,N£(q)p~(Fd~~£))
£,N£ and the right side is known by step 3. This provides the general solution to problem A. For the Green functions, one requires (F~N, ~sw)' Step 6. From Step 5 and using results from [11] one then has
(F~N, 6w)~
=
L
h£,N,N£(q)(F£~~£, Res~(~sw))£'
£,N£ From Step 3, we have a formula F£~~£ = L d£,oV£~o, where the sum
.c
o
is over the Ad LF -orbits 0 in F and the d£,o are known. It follows that (F£~~£, Res~(~sw))£ = L d£,o(V£~o, Res~(~sw))£ .
°
n.c
F is the union of But Res~(~sw) = Li ~i,£, where, if OSw orbits Oi,£, ~i,£ is the characteristic function of Oi,£' It follows after a little calculation that
(2.6)
(F~N, ~sw)~ =
L £,N£
h£,N,N£(q)d£,(sw)c(w)ICLF(Sw)I-
1
LF_
Fourier Transforms and Nilpotent Orbits
L
where d£,(sw) =
295
C(sw),N,b'j:.',Nk(r., w) (see (2.3)) and k(r., w)
°N,?ON
is the number of LF-orbits in r. F whose GF-orbit contains Sw. Thus we obtain the following closed form for the Green functions of G. (2.7) Qw(N) = qT/2 h£,N,N£(q)ICLF(Sw)I-1k(r., w) cfsw),N,b'j:.',N· £,N£ 0N,?ON
L
L
It is therefore clear that to compute the Green functions explicitly, one requires only the coefficients aN',N (generally easy in practice) and the values of Green functions at special nilpotent elements as well as a knowledge of the "Hall polynomials" h£,N,N£(q). We carry out this program completely for G of type A below (when only the case of regular N occurs) solving both problems A and B. We also deal with the general case if N is regular. En route, we indicate some geometric problems and conjectures which arise in the general case. The formula we obtain for Green's polynomials is different from but equivalent to Green's original one. This approach to Green functions was pioneered by Springer in [Spl, Sp2].
3. Ordinary Gelfand-Graev functions and regular nilpotent orbits In this section we discuss the case where N (of §2) is regular. Let B = TU be an F-stable Levi decomposition of the Borel subgroup B of G and let U = Lie U. Then U has Fq-basis {eala E 11>+}, where 11>+ is the set of positive roots of G with respect to T (we assume G to be F-split). If {1/Jala E II} (where II is the set of simple roots in 11>+) is any set of additive characters, the formula (3.1)
IlJ(
L aEW+
aaea) =
II 1/Ja(aa) aEil
defines an Ad U F -invariant function on
UFo
3.2 Definition. If 1/Ja i= 1 for all a E II, the function, = In~ (IlJ) is called a Gelfand-Graev function on (5F. Such functions have been studied extensively in the group context and we shall therefore be a little sketchy with the proofs of some of the following results.
296
G. 1. Lehrer
3.3 Lemma.
(i) The distinct Gelfand-Graev functions correspond to the orbits
of T F on £-tuples cP = (¢oIQ E II) (£ = Inl = semisimple JFq-rank of G). (ii) The orbits of (i) are in bijective correspondence with HI (F, Z) ~ £-1 (Z)/(ZTF) (where £ : T --t T is defined by £(t) = t- 1F(t) and Z = Z(G)). (iii) If G has connected centre, there is a unique Gelfand-Graev function. (iv) In general, if t z E T represents z E H I (F, Z), then , z = Inc1fl(cPz) is the Gelfand-Gmev function corresponding to z, where cPz(u)
¢ (
= cPo(Ad(t;I)U)
(u E ilF ) and cPo (
L: aoeo) =
oE+
L: ao) N as in §1).
oEIT
These statements are all standard (d. [L2] or [DLMD. The regular nilpotent orbits in (5F are also parameterised by HI (F, Z). 3.4 Lemma. With notation as in (3.3), let Uo
= L: eo. Then we oEIT
have that {Ad(tz)uo[z E H 1(F, Z)} is a set of representatives for the regular nilpotent Ad G F -orbits in (5F.
We write Ad(tz)uo.
~:n
for the characteristic function of the orbit containing
3.5 Lemma. If G has connected centre, then there is just one Gelfand Gmev function" and we have (r,,)~F = IZ(G)OFlql.
The proof is entirely analogous to the group case, which may be found in [St2]. 3.6 Lemma.
(i) With notation as in (3.3), D,z is supported on the regular nilpotent set of (5F.
(ii) If G has connected centre, D, = IZO(G)Flql~rn. Proof. (i) is proved as in [DLM, §3]. (ii) By (i) D, = ~rn for some constant c. If St~ is the Steinberg function of (5 (see [LID then by [Ll,(3.6)] (r, St~)~ = 1. Since D is an isometry and D(St~) = 1~, it follows that (D(,), 1~)~ = c(~rn, 1~) = c(lzo(G)Flql)-1 = 1 since for
297
Fourier Transforms and Nilpotent Orbits
any regular nilpotent element X E ~F we have ICcdX)1 = Izo(G)Fll. The result follows. D 3.7 Corollary. We have, with the above notation
L
V,z =
z/EHl(F,Z)
where CZ',z = IZo (G)F Iq£ in case G has connected centre, and CZ',z is "known" in general (cf. [DLM]) and depends only on the product z'z.
4. Determination of F, Suppose, is a Gelfand-Graev function on ~F. 4.1 Proposition. (cf. Lusztig [Lu]) There is a unique regular nilpotent element X o E ~F satisfying the following (i) X o E U- = L Fqc cr crE+
(ii) If So = X o + lB F (where lB = Lie B) then
=
F,
L
I~FI-l/2
c(O)ISo n Ol~o
OE\5F /C F
where ~o is the characteristic function of 0 and C(O) is the cardinality of the centraliser in G F of any of the elements of O.
Proof. We have (4.1.1)
F,(X)
=
I~FI-l/2
L
,¢((X, Y)),(Y)
YE\5F
=
I~FI-l/2
L L
,¢((X, Y))Ind~(¢)(Y)
YE\5F
= I~FI-l/2
L
,¢((X, Y))IUFI- 1
YE\5F
= I~FI-l/2IUFI-l
¢(Ad(g) Y)
9EC F
Ad(g) YEil F
L
L
'¢((Ad(g) X, Ad(g) Y))¢(Ad(g)Y)
gEC F YEAd(g)-lil F
= I~FI-l/2IUFI-l
L L
'¢((Ad(g) X, Y))¢(Y).
gECF YEil F
But the function Y 1-+ '¢( (Ad(g) X, Y) )¢(Y) is a character of the abelian group UF . Hence the inner sum is zero unless (4.1.2)
¢(Y)
= '¢( (-Ad(g) X, Y)) (all Y
EU
F
).
298
G. 1. Lehrer
But ¢(Y) =
I1 crEIT
Wcr(a cr ) if Y =
L
acre cr . Now there are unique
crE~+
elements Xcr E IF; (0: E II) such that Wcr(a) = w(xcra) (all a E lF q ). Hence the condition (4.1.2) is equivalent to (4.1.3)
L
(-Ad(g) X,
acre cr ) =
crEW+
(all Y
=
L
acre cr E UF
L
xcra cr
crEIT
).
crE~+
It follows that the inner sum above is zero unless the linear form Y ~ (-Ad(g) X, Y) (on UF ) is as specified by (4.1.3). But this means that -Ad(g) X is determined modulo the subspace of (!) perpendicular to U, which by [L1, §3] is lB. Now there is an element X o E (U-)F (unique by the above) such that (-Xo, Y) = L xcra cr (as in (4.1.3)). Moreover clearly X o (and crEIT
- X o) is regular nilpotent (the coefficient of e_ cr for 0: E II must be non-zero). It follows that the inner sum in (4.1.1) is zero unless
(4.1.4)
Ad(g) X E X o + lB
= So
Hence from (4.1.1) we have
which amounts to the stated formula.
D
4.2 Proposition. Let (X, Y, H) be an sl2 -triple in (!), with X (and hence Y) regular. Let lB be the Lie algebra of the unique Borel subgroup B of G such that Y E Lie B. Define S = X + lB. Then
(i) S consists of regular elements of (!). (ii) Each regular Ad G-orbit in (!) intersects S. (iii) If U is the unipotent radical of B, U acts freely on S. Proof. The statements (i) and (ii) are respectively Lemmas 10 and 11 of [K, §4]. Kostant's proof remains valid in positive characteristic, provided that one has 4.2.1 If II is a set of simple roots of (!) (with respect to some Cartan subalgebra) and ecr is a basis element of (!)cr (the root subalgebra corresponding to 0:) for 0: E II, then L ecr is regular nilpotent. crEIT
But (4.2.1) is proved in [Ke], whence we have (i) and (ii).
Fourier Transforms and Nilpotent Orbits
299
The fact that U acts on S follows from the commutator formula in ~ (observe that since U is generated by simple root subgroups one need only check that these act on S). Finally a straightforward calculation shows that the centraliser in U of any element of S is trivial, i.e. that the action is free. 0
4.3 Conjecture. In the situation of (4.2) (and with some constraints on the characteristic) the intersection of S with any regular orbit is a single U -orbit (hence an affine space AN, where N is the number of positive roots).
4.4 Theorem. Conjecture (4.3) holds if G is a group of type A n - 1 over any field K. Proof. If suffices to prove the statement for G = GL n , since a little reflection shows that (4.3) is "isogeny invariant" - i.e. if (4.3) holds for G, then it holds for any group in the isogeny class of G . Thus we take ~ = gIn, the Lie algebra of n x n matrices over K and let {Eij 11 :::; i, j :::; n} be the standard basis (Eij is the matrix with 1 in 1 position (i,j) and zeros elsewhere). We may take X o = L::1 -Ei+l,i. Then S = {Xo + Li:::;j aijEijlaij E K,1 :::; i :::; j :::; n}. The assertion of (4.3) amounts (given (4.2)) to
4.4.1 If two matrices in S have the same characteristic polynomial, they are conjugate under U, where U is the group of upper unitriangular matrices. For any Y E ~ write Cy(t) = det (In - Y). This is a polynomial of the form t n + a1 t n - l + ... + an' If we identify the set of all such polynomials with An, the morphism 'YS : S ----> An defined by 'YS (Y) = Cy(t) has fibres which are precisely the intersections S n 0, where 0 is a regular orbit. Clearly 'YS is surjective. Now dim U = N = ~n(n - 1). Moreover it follows from (4.2) (iii) that each U-orbit on S has dimension N. Since U acts on the fibres of 'YS, the fact that each fibre is a single U -orbit, which is the assertion of (4.4.1), will follow from
4.4.2 For any polynomial f
=
tn
+ a1t n - 1 + ... + an
E An, we have
'Ys1(J) ~ AN.
1
We now prove (4.4.2). Let Y n = L::1 Ei+1,i + L7=1(t-ajj)Ejj + Li<j -aijEij =tI - X, where X = X o + Li:::;j aijEij is a typical element of S. Then clearly Cx(t) = det Y n . Expanding detYn by the last row, we obtain
300
G. 1. Lehrer
4.4.3 detYn = tdetYn _ 1 -anndetYn - 1 +an-l,ndetYn-2+an-2,ndetYn-3+ - al n where Yi is the leading i x i submatrix of Y n .
... + a2ndetYI
Now for any choice of {aij 11 :::; i, j :::; n - I} the polynomials {detYn- 1 , detYn -2 ... ,detY1 , I} form a linear basis of the space of polynomials of degree (:::;)n -1. Hence given f(t) = t n +al t n - 1 +... +a n E An, the equation detYn = f(t) has a unique solution (aij)I::;i::;j::;n for any given set of values of {aij 11 :::; i :::; j :::; n - I}. Hence 1 (f(t)) ~ AN, proving (4.4.2) and hence (4.4). D
1s
A general proof of (4.3) will probably require a map like 1s which may involve a characteristic p theory of polynomial invariants for
~.
Suppose now that in (4.2), G is defined over IF q and that (X, Y,H) is F-stable. Then for each F-stable regular G-orbit 0 C ~ it follows from (4.2) that S n 0 is a union of U -orbits. We shall require
4.5 Hypothesis. With notation as above, there is an F -stable U- orbit in S n 0 for each F-stable regular G-orbit 0 in ~. Of course (4.5) is an immediate consequence of (4.3), which asserts that snO consists of a single U -orbit. Moreover since U is connected, it is clear that (4.5) is equivalent to
(4.5)'. For each F-stable regularG-orbit 0, (SnO)F is non-empty. 4.6 Remark. We have seen that (4.3), and therefore (4.5) holds for G of type A. D. Jackson has also verified it when G is of type G 2. 4.7 Proposition. Suppose that in (4.2), G is defined over JF q and that (X, Y, H) is F-stable. Assume (4.5). Then for each F-stable regular orbit 0 C ~, there is precisely one regular G F -orbit O(F, S) in ~F such that (i) (S n O)F = O(F, S) n S. (ii) O(F, S) n S is a single U F -orbit; hence its cardinality is qN. (iii) SF = 11 O(F,S) n S, where the union is over the F-stable o regular orbits in ~. Proof. It follows from (4.2) that (SnO)F is a union of UF-orbits, and is non--empty for each F-stable 0 by (4.5)'. If n(O) is the number of U F-orbits in (S n O)F then we have (4.7.1)
ISFI
=L o
1(0 n S)FI
=L
n(O)qN.
0
But the number of terms in the sum (Le. the number of F -stable regular orbits) is (by [L3, (2.5)]) equal to qT, where r = rank~. Hence
Fourier Transforms and Nilpotent Orbits ISFI =
(~n(o)) qN
for all O. But ISFI
301
2: qr+N, with equality if and only if n(O) = 1
= I~FI = qr+N, whence n(O) = 1 for each
O.
It follows that for each F-stable regular orbit 0, (SnoV is a single U F -orbit in ~F. All statements are immediate consequences of this.D
4.8 Corollary. Suppose G (in (4.7)) is such that centmlisers of regular elements are connected (thus G has connected centre; if G' is simply connected, this suffices). Then for each regular Ad G F -orbit 0 in ~F, we have Isnol = qN. 4.9 Theorem. Suppose G is a connected reductive group over IF q which satisfies (4.5) and let 'Yz be a Gelfand-Gmev function on ~F (z E H1(F, Z(G)) (cf. (3.3)). For each F-stable regular orbit 0 C ~, denote by Oz the Ad G F -orbit in ~F which arises from (4.7) with S = So (as in (4.1}). Then F'Yz
= q-r/2 L
c(O)~o.
o where c(O) = IGCF(X)I for any X E OF and the sum is over the F-stable regular orbits in ~.
Proof. We have from (4.1), F'Yz = I~FI-I/2
L
c(O')ISo
n O'I~o'.
O'E\5F /CF
But from (4.7), IsonO'1 result follows.
= 0 unless 0' = Oz
and ISo nOzl
= qN.
The D
4.10 Corollary. Suppose G satisfies the hypotheses of (4.8). Then each F -stable regular Ad G -orbit R in ~ intersects ~F in a unique Ad G F -orbit, which we denote by OR. Denote the chamcteristic function of OR by ~R. If'Y is the (unique) Gelfand-Graev function on ~F, then F'Y = q-r/2 C(R)~R
L R
where c(R) = IGcF(X)1 for any element X E OR.
5. The Fourier transform of a regular nilpotent orbit In this section we assume that G satisfies the hypotheses of (4.10), Le. that centralisers of regular elements are connected. Then there is just one Gelfand-Graev function and just one regular nilpotent orbit in ~F. Let its characteristic function be denoted by ~~n or ~rn if there is no risk of confusion.
G. 1. Lehrer
302
5.1 Theorem. With the above notation we have
<7
where the sum is over the semisimple orbits a in ~F and '"'(<7 is defined by unless Ys E a '"'(<7(Y) = { 0 if Ys Ea. '"'(C
(5.1.1)
= c- l q-r/2
(c = IZO(G)Fll)
L
c(O)D~o
(using (4.9))
o where the sum is over the regular Ad G F --orbits 0 in ~F and c(O) denotes ICCF(X)I for any X E o. We turn our attention to the computation of D~o. 5.1.2 Lemma. Let a = {Sl,··· ,Sa} be a semisimple Ad G F -orbit in ~F. For each i E {I,··. ,a} let L i = CC(Si), £i = Lie (L i ). LetOR be the regular orbit corresponding to a. Then OR n £i is the union of the regular L[ -orbits of the form Sk +Oirn where Sk E Z(£i) and Oirn is the regular nilpotent L[ -orbit on £[ . Proof. £i nOR clearly contains the orbits described. Conversely, any element X of OR has Jordan decomposition X = Sk + N, some k. But if X E £i, then Sk E £i and N E £i and N is conjugate under Ad G to a regular nilpotent element of £i. Hence N is regular nilpotent in £i. The only semisimple elements centralising N therefore lie in Z(£i), whence the result. [J 5.1.3 Corollary. For Z E Z(~F) define translation TZ : C(~F) --+ = f(X - Z) (J E C(~F), X E ~F). Let ~R be the characteristic function of OR in (5.1.2). Then
C(~F) by TZ(J)(X)
Res~,~R =
L
TS;(~~)
j:£,=£;
where ~~ is the characteristic function of the regular nilpotent set in
£[. This is clear from (5.1.2); note that Sk E Z(£i) implies that £k = £i since C l15 (Sk) = £k 2 £i and the two varieties have the same dimension.
303
Fourier Transforms and Nilpotent Orbits
5.1.4 Lemma. For any function g E C(~F) and nilpotent element N E we have
£f,
D(J5g(Si
+ N)
= c(a)D£i(Res~ig)(Si
+ N).
This follows directly from [L1, (3.15)]. Using (5.1.3) and (3.6) we see moreover that
L
D£iRest~R =
(5.1.5)
C(R)-ITS/Y£i
j:£j=£i
where c(R) IGcdX)1 for any element X of OR and I£i is the Gelfand-Graev function on £i, since TZ clearly commutes with D(J5. The formula (5.1.5) also makes use of the fact that for X E OR n £i, Gc(X) -:; L i . 5.1.6 Corollary. D£i (Res~i~R) is supported on {Sj N E (£f)nil}.
+ NI£j
= £i,
5.1. 7 Lemma. With the above notation, a
D~R = c(a)c(R)-1 LTSil£i i=l
where c(a) is the sign of the semisimple orbit a and c(R) is the cardinality of the centraliser for the corresponding regular orbit R.
Proof. (of (5.1.7)). By (5.1.4) and (5.1.5) the two sides of (5.1.7) coincide when evaluated on any element with semisimple part Si (some i).
Let ¢ be the function on the right side of (5.1.7). Since the summands of ¢ have disjoint support, we may write
¢
~ o(8)c(8)-' ~ (j,t;", TS'~"')
where the outer sum is over the distinct £i. Thus ¢ =
L ¢i, £i
¢i E C(£f) and the ¢i have disjoint support. Hence
L XE(J5F
2
1¢(X)1 =
L L
l¢i(X)1
2
£i XE£[ = (by (5.1.5))
L
L
£i XE£[
=L
L I~R(X)12 £i XE£[
ID£i(Rest~R)(X)12
where
304
G. I. Lehrer
since Dr.; is an isometry on C(£[). But from (5.1.2) it follows that
L L ~
Thus
L XEe F
I~R(X)12 =
I~R(X)12
=
L
I~R(X)12.
XEe F
XEr.r
L
1¢(X)1
2
XEe F
L
=
ID~R(X)12
XEe F
Since ¢(X) and D~R(X) coincide on the support of ¢, it follows that = 0 for X outside this support. D Observe that L~=l TS;'Yr.; = ICY as defined in the statement of (5.1). Hence (5.1. 7) may be expressed in the form D~R(X)
(5.1.8) where a is the semisimple orbit corresponding to the regular orbit OR. Combining (5.1.1) with (5.1.8), we obtain e -1 -r/2""",, ( ) F <"rn=C q ~E:a,cy CY
which is the formula of (5.1).
D
5.2 Remark. Observe that for consistency of notation (e.g. with (5.1.8) and (3.5)) if G is a torus, then the Gelfand-Graev function I of ~ is defined by I(X) = IGFI if X = 0, while I(X) = 0 if X ::/= o.
6. The case of gIn In this section we give complete solutions to problems A and B for ~
= glnCiFq). Here it is well known that the nilpotent orbits are parameterized by partitions A = (AI 2: A2 2: ... 2: Ap > 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
Lemme 2.4 Soit (T;)1~i~k une base quasi-symetrique. Le vecteur(8x (t))x des degres relatifs est l'unique solution sur K du systeme lineaire d'equations si i = 1, (2.5) si i ::/= 1. La conjecture 2.3 peut etre etendue comme suit: Pour W un groupe de reflexions et 1t I'algebre cyclotomique associee, il existe une base quasi-symetrique de 1t formee d'elements de type Tw = TS1 ••• Tsr pour W = S1 ... Sr E W des mots de longueur assez petite telle que la forme symetrique correspondante se specialise en la forme symetrique standard de ZW. Pour Ie moment il n'est pas clair quelles bases sont quasisymetriques et donnent naissance a une forme du type decrit. A cause de cela, nous allons faire un choix experimental de base dans les calculs suivants. Les degres relatifs pour cette base quasi-symetrique devraient alors avoir des specialisations en les degres de certains caracteres unipotents contenus dans des 1 peuvent etre vus comme sous-groupes distingues d'indice p de G(e, 1, 2). Le diagramme cyclotomique correspondant a G(e, 1,2) est B 2(e)
.
.
0=0
(3.1)
Le resultat suivant montre que pour connaitre les degres relatifs de G(e, p, 2), p impair, il suffit de les determiner dans Ie cas p = 1 (voir aussi [10, (5.12)] pour des formules explicites) : Proposition 3.2 (Ariki [1, Prop. 1.6]) Soit p impair. Alors il existe une specialisation des parametres u 1-+ ii de l'algebre 1t(B~e), u) telle que 1t(G(e,p, 2), v) est une sous-algebre de 1t(B~e), ii) de far;on naturelle. Cette reduction ne marche pas dans Ie cas ou p est pair, car alors l'algebre cyclotomique generique comme nous la definissons a un parametre de plus. Par exemple, si e est pair, Ie groupe de reflexions reel 12 (e) S:! G(e, e, 2) est un sous-groupe de G(e, 1,2), mais on retrouve seulement l'algebre de Iwahori-Hecke de 12 (e) avec parametres egaux comme sous-algebre de 1t(B~e)). Les cas ou p est pair seront traites dans la section 3B. Les representations irreductibles de l'algebre cyclotomique 1t(B~e)) construite selon la definition 2.2 apartir du diagramme (3.1) sont donnees par
pour i nJ.
= 1,2, 1 '5: j '5: e, et
• '1J
ItJk' IL
-+ K2x2
,
pour 1 '5: j < k '5: e, ou K := Q(XI, X2, Zl, ... , ze). On verifie facilement que l'ensemble {(SIS2?m+l, SI(SIS2?m
10'5: m '5: e; I} U {~(SIS2?m I p+m < e}
(3.3) forme un systeme de representants des classes de conjugaison de G(e, 1, 2). En evaluant les caracteres Xij (resp. Xjk) des representations E4j (resp. Rjk) sur les elements ci-dessus on obtient la table des caracteres suivante : Theoreme 3.4 Les degres relatifs de 1t(B~e), (Xl, X2; Zl, ... , Ze)) par rapport Ii chaque base contenant les elements figurant dans La table 1 sont donnes par:
G. Malle
316
Table 1: Table des caracteres pour rt(B~e)) (1:81 1:82 )2m+1
TOl (To I T02 )2m
Tf,(ToJ 02)2m
Xij
(XiZj )2m+1
x2m+lz2m
x ,2m J m + p
XJk
0
,
(Xl
i
J
+ X2)( -XIX2 Zj Zk)m
pour Ies 2e camcteres Iineaires
pour Ies e(e - 1)/2 camcteres
Xij,
XJk
(zf
+ zf)( -XIX2 Zj Zk)m
et
de degre 2.
Notons que, apres changement de variables, les formules coincident avec celles de [10, (2.19)]. Alors Ie theoreme demontre la conjecture 2.20(2) de loco cit. dans Ie cas de dimension n = 2 pour les bases quasi-symetriques qui contiennent les representants indiques ci-dessus. Demonstration. Nous verifions que les fonctions rationnelles figurant dans Ie tMoreme satisfont au systeme lineaire (2.5). Soit B une base quasi-symetrique de rt(B~e)) telle que l'image de B sous la specialisation sur l'algebre de groupe de W = G(e, 1, 2) est formee des elements de W. Alors il est clair qu'un ensemble d'equations lineaires de rang maximal pour les degres relatifs peut s'obtenir en evaluant (2.5) sur un sousensemble C de B, tel que, apres specialisation, l'image de C contienne des representants de toutes les classes de conjugaison de W. Par exemple, cette observation s'applique a l'ensemble defini dans (3.3). Dans la preuve, l'identite ~
L...J j=l
n
y'j'-l
ITI=1 (Yj l#-i
- Yl)
__ (_l)n-Ioo,m
ITl:: I Yl
pour 0::; m ::; n - 1,
(3.5)
avec des indeterminees YI, ... ,Yn, nouS sera utile. Elle se demontre aisement en calculant la premiere colonne de l'inverse de la matrice de Vandermonde en YI, ... , Yn. Posons
pour i = 1,2. Cette somme peut etre interpretee comme sous-somme de (3.5) avec n = 2e, Yj := Zj, Ye+j := -XiZj/X3_i pour 1 :::; j :::; e. Puis, un calcul simple montre que pour 1 :::; m :::;
2e.
(3.6)
Regardons d'abord les valeurs sur les elements de la forme (TS1 TS2 )2m+1. Comme les caracteres de degre 2 prennent la valeur 0 sur ces elements, il faut alors demontrer que F 1,2m+1 + F2 ,2m+l = 0 pour 0 :::; m :::; e - 1. Cela resulte immediatement de (3.6). Pour les elements de la forme ~ (TS1 TS2 )2m la contribution par les caracteres de degre 2 est egale it
tt
j=l
k~l
X1X2 Zj Z k(zf 2(X1Zj
+
X2 Z k)(X1Zk
+ zf)( -X1X2 Zj Zk)m TII;ij,k Z[ + X2 Z j) TII;ij,k(Zj - Zl) TII;ij,k(Zk -
Zl)
k,<j
La somme interieure s'interprete alors comme sous-somme de (3.5), avec n = e + 1, Yi := Zi pour i i= I, Yl := -X1Zj/X2 et Ye+1 := -X2Zj/X1' (Notons que l'on a toujours m + p + 1 :::; e.) Apres avoir fait Ie remplacement correspondant, on obtient
en utilisant (3.6), ce qui est egal a la contribution des caracteres lineaires a 8o,m8o,p pres, demontrant la formule desiree dans ce cas. Enfin, pour les elements Ts 1 (Ts 1 TS2 ) 2m un calcul similaire donne Ie resultat. D
3B. Degres Relatifs de D~e) Pour obtenir les degres relatifs des algebres cyclotomiques correspondant a des groupes G(e,p, 2), p pair, nous devons aussi traiter Ie cas des groupes G(2e, 2, 2), de diagramme
D~e):
C
(3.7)
G. Malle
318
L'interet de l'algebre cyclotomique H(D~e), (Xl, X2; YI, Y2; Zl, ... ,Ze)) vient du resultat suivant, qui se demontre facilement a partir de [1, Prop. 1.6] :
Proposition 3.8 Pour tout pie il existe une specialisation des parametres u ....... ii de I'algebre H(D~e), u) telle que H(G(2e, 2p, 2), v) est une sousalgebre de H(D~e), ii). Les representations irreductibles de H(D~e)) sont donnees par
pour i,j E {1,2}, 1 ~ k ~ e, et
pour 1 ~ k < I ~ e, ou v = Vkl := w~ - X2Y2Zk - X2Y2 ZI, w 2 = XIX2YIY2ZkZI, et K = Q(u, w). L'element Z := SlS2S3 est central dans H(D~e)). L'ensemble
{SIZm, S2Zm I 0
~ m ~
e - I} U {~zm I m
+ 2p ~ 2e -I}
forme un systeme de representants des classes de conjugaison de G(2e, 2, 2). Ainsi nous obtenons la table des caracteres suivant : Table 2: Table des caracteres pour H(D~e))
Xijk
xii
TS,zm
TS2 zm
TI3 Z m
x'('+1 yjzf (Xl + X2)W m
x'('yj+1 zf
x'('Yjz;;'+P (z~ + zf)w m
(YI + Y2)W m
X~l (Xl +X2)(-W)m (YI
+ Y2)( _W)m
(Z~
+ zf)(-W)m
Theoreme 3.9 Les degres relatifs de H(D~e),(XI,X2;YI'Y2;ZI, ... ,Ze)) par rapport Ii chaque base contenant les elements figurant dans la table 2 sont donnes par:
pour les 4e caracteres lineaires
avec X i2 = degre 2.
Xi, y2 j
=
Yj, Z2 n
et
Xijk,
=
Zn,
' pour l es e 2 - e carac t eres
1,2 XkL
de
Demonstration. Nous utiIisons la meme methode que dans la demonstration du theorerne 3.4. Soit d'abord F-.
'=
',l,m'
L e
e
e
(
X3-iY3-j XiYjZk
k=l (X3-i -
Xi)(Y3-j -
Yj) OL,ek(Zk -
)m nL,ek
2
Zl
Zl) OL,ek(XiYjZk -
X3-iY3-jZL)
pour i,j E {I, 2}. En utilisant (3.5) on montre que
F...
',lom -
D
r3
. 3' -lorn
-'0
XiYj - X3-iY3-j c Vo (X3-i _ Xi) (Y3-j _ Yj) ,m
pour 0 S; m S; 2e - l.
(3.10) La contribution des caracteres xii de degre 2 sur les elements de la forme TI3 z m ala somme de (2.5) est
ou
0, Zn
designent des racines carrees de Xi, Yj, Zn telles que w Les caracteres X~l' conjugues des xiI> contribuent pour une somme similaire, et on voit qu'encore une fois l'identite (3.5) s'applique. Alors, un calcul simple donne Xi,
X1X2YIY2ZjZk.
1 2 (1 8°om 800P - 2 i,j=l XtYl)P
1) L ( . . + (X3-.Y3-1 . .)P F.',l,m+p
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comme contribution des caracteres de degre 2. Pour les caracteres lineaires il s'ajoute 2 1 ~ LJ ( . F .)P .',J,m +p, i,J=1
x'YJ
et (3.10) permet de conclure. Pour les elements de type T S1
zm et T zm Ie calcul est similaire. S2
0
4. Les groupes de reflexions primitifs de dimension deux 4A. Structure des groupes primitifs de dimension deux Soit w'---+ GL 2 (C) un groupe de reflexions irreductible de dimension 2, et W son image canonique dans PGL 2 (C). Si West primitif, on voit aisement que West isomorphe a I'un des groupes A 4 ,54 ou A 5 . Plus precisement, avec la notation de Shephard et Todd [Ill, I'image des groupes G4 , ... , G8 est A 4 , I'image de Gg , ... , G I5 est 54, et I'image de G 16 , ... ,G22 est A 5 . De plus, dans chacun de ces trois cas il existe un groupe de reflexions W maximal, tel que tous les W qui ont la meme image dans PGL 2 (C) peuvent etre realises comme sous-groupes (distingues) de W. II s'agit des groupes G7 ,GU,G I9 ; les diagrammes cyclotomiques associes sont les suivants : (4.1) avec n = 3,4,5 selon les cas (voir [2, 3D]), ou Ie cercle symbolise la relation rst = str = trs entre les trois generateurs r, s, t qui correspondent aux trois nceuds. On a Ie resultat suivant pour les algebres cyclotomiques correspondantes :
Proposition 4.2 50it W un groupe de reflexions de dimension 2 primitif 5upposons que les algebres cyclotomiques H(W ' , v) pour tous les W' tels que W = W' satisfont Ii la conjecture 2.3. Alors il existe une specialisation des parametres u f-> U de l'algebre H(W,u) telle que H(W,v) est une sous-algebre de H(W, u). Demonstration. Cela se demontre cas par cas. Nous donnons les details seulement pour Ie groupe W = G 13 , de presentation
avec trois generateurs Sl, S2, S3 (voir [5]). Alors W = G n et selon la definition 2.2 et (4.1) I'algebre H(W, u) est engendree par T I , T2 , T3
Degres relatifs
321
soumis aux relations T 1T2T3 2
II (T1 -
= T2T3T 1 = T3T 1T2 et
3
Ul,j) =
j=1
II (T2 -
4
U2,j) =
j=1
II (T3 -
U3,j) =
O.
j=1
Apres la specialisation (U3,1, U3,2, U3,3, U3,4)
les elements
U 1 :=
Ti,
U 2 :=
f--+
T 1 et
(~, -~,,;v:;., U 3 :=
-,;v:;.)
T2 verifient les relations 2
U 2 U 3 U 1 = U 1U 2 U 3 ,
UIU2U3U2U3 = U 3 U 1U 2 U 3 U 2 ,
II(U 1 - Vi) = i=1
0,
dans I'algebre obtenue par specialisation. De plus, la sous-algebre engendree par U 1 , U 2 et U 3 est d'indice 3 dans H(W), et comme nous supposons la validite de la conjecture 2.3, cela montre que (U1 , U 2 , U 3 ) est isomorphe a H(G I5 ) (voir 15] pour Ie diagramme cyclotomique de G I5 ). Specialisant encore U2,j f--+ avec (3 = exp(21fi/3) les elements VI := U 1 , V 2 := U 2 et V 3 := U;tU2 U 3 verifient
(1
(notons que maintenant Ui = Ti = 1), et comme precedemment on en deduit que VI, V 2 , V 3 engendrent une sous-algebre d'indice 2 de la specialisation de H( G15 ) isomorphe aH(Gd. Cela demontre que H(G I3 , (VI, V2; Ul,l, Ul,2)) est isomorphe a
(VI, \12, V3) ::; H(G u , ii) avec ii = (Ul,l, Ul,2; (3, (t, 1;~, -~,,;v:;., -y'v2). Les specialisations dans les autres cas se trouvent dans les tables 4, 6cl8.
0
Remarquons que la proposition precedente permet de ramener la determination des degres relatifs pour les groupes W consideres a la determination de ceux des trois groupes W. En effet, W etant une extension centrale du sous-groupe distingue W, les restrictions de tous les caracteres irreductibles de W a W sont irreductibles. Alors, Ie lien entre les degres relatifs est explique par I'observation evidente suivante :
Lemme 4.3 Soient W ::; W comme ci-dessus et H' ::; H l'inclusion de leurs algebres cyclotomiques (obtenues par specialisation) d'apres la proposition 4.2. Soient t' la restriction Ii H' d'une forme symetrisante t sur H et X' := xht, la restriction (irreductible) d'un caractere irreductible X E Irr(H). Alors les degres relatifs verifient 8X ,(t' ) = (W : W) 8x (t).
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Dans les sections suivantes, nous allons traiter les trois possibilites pour iV, en supposant que la conjecture 2.3 est satisfaite pour les groupes W concernes. 4B. Degnls Relatifs de D~,3
Le groupe de reflexions G 7 est un produit central G 7 = C l2 0C2 8£2(3) de 8£2(3) avec Ie groupe cyclique d'ordre 12. Le diagrarnme D = D(G7 ) est donne par (4.1) avec n = 3. Soit
H(D~,3, (Xl, X2; Yl, Y2, Y3; Zl, Z2, Z3)) I'algebre cyclotomique associee par la definition 2.2. Notons Sl, S2, S3 les trois generateurs de W = G 7 = W(D~,3) dans la presentation decrite par Ie diagramme cyclotomique (4.1) avec
s~ = s~ = s~ = 1, et T j ,T2,T3 Ies generateurs de H:= H(D~,3,(Xl,X2;Yl'Y2'Y3;ZI,Z2,Z3)) correspondants. Alors I'element Z := SlS2S3 est central dans W, et de meme Z := T l T2T3 est central dans H. On remarque que I'ensemble {Zi, SIZj, S2Z\ S3Zi I 0 ~ i ~ 11, 0 ~ j ~ 5}
est un systeme de representants des classes de conjugaison de W. Nous choisissons alors
comme sous-ensemble de la base quasi-symetrique cherchee. Theoreme 4.4 Les degres relatifs de H(D~,3, (Xl, X2; Yl, Y2, Y3; Zl, Z2, Z3)) par rapport Ii chaque base quasi-symetrique contenant C comme sousensemble sont donnes par: (X2 - Xl) (X2Y2Y3Z2Z3 - XIY?Z?) TIi=2 ((Yi - Yl)(Zi - Zl)) TIi,j=2(X2Yi Zj - XlYlZl) pour les 18 carocteres lineaires, -x~ x~ Yl Y4 y~ z? z~ z~ 2 Il;=2 ((Y3 - Yi)(Z3 - Zi)) m=l ((XiY3 Z3 - U)(XiY2Z2 - U)(XiY2Zl - u)) avec u 2 = XlX2YlY2ZlZ2 pour les 18 caraeteres de degre 2, et
-xI x~ Y{ y~ y~ Z{ z~ zj 3 (Xl - X2) Il;,j=I(XlYiZj - v)
avec v 3 = X?X2YlY2Y3ZIZ2Z3 pour les 6 caracteres de degre 3.
Degres relatifs
323
a l'aide du lemme 2.4. La validite de 2.3 implique en particulier qu'il existe une bijection '" : Irr(1-l) --=::... Irr(W),
Demonstration. Nous allons calculer les degn§s relatifs
qui preserve les degres, et telle que "'(X) est Ie caractere obtenu a partir de X par la specialisation Xi f--+ (_l)i, Yi, Zi f--+ (~. Ii s'ensuit que l'on peut determiner les valeurs X(Ti ) pour X E Irr(1-l) si on connait les valeurs "'(X)(Si)' Comme Z E 1-l est central, il agit par un scalaire dans to utes les representations R (absolument) irreductibles de 1-l. Pour une telle representation R de caractere X on a
(X(Z)/X(l))X(I) = det(R(Z)) = det(R(Tl )) det(R(T2)) det(R(T3)). Comme la valeur du membre de droite est connue, cela donne X( Z)X(I). En comparant avec la specialisation sur W on obtient X(Z). Comme Z agit par une matrice scalaire, les valeurs x(TiZ) verifient 1
x(TiZ) = X(l)x(Z)x(Ti)' Le raisonnement precedent montre qu'on peut evaluer tout caractere irreductible de 1-l sur tout element de C en connaissant seulement la table des caraeteres de W. Celle-ci peut etre calculee par exemple avec Ie logiciel Cayley. Explicitons alors les valeurs des caraeteres irreduetibles de 1-l sur les elements de C : Table 3: Table des caracteres pour 1-l(D~,3) degre 1
2 3
Zi x~Y~ z~ 2ui 3v i
TlZi x{+iYizi (Xl + X2)U i (2Xl + X2)V i
T2Z i xiy~+l z~
(Yl + Y2)U i (Yl + Y2 + Y3)V i
T3Z i xiyi Z~+l (Zl + Z2)Ui (Zl + Z2 + Z3)V i
Cela permet de verifier l'assertion du theoreme (au moins en principe, avec un ordinateur assez puissant). 0 Donnons maintenant quelques indications sur la maniere de resoudre Ie systeme d'equations lineaires obtenu ci-dessus. Meme avec les logiciels modernes il ne semble pas possible d'appliquer l'algorithme de Gauss et
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d'obtenir une solution dans un temps acceptable. Dans Ie cas de D~,3 la methode suivante mime au succes. On specialise sept des huit indeterminees Xi, Yi, Zi en des petits nombres premiers distincts. On peut alors resoudre Ie systeme (2.5) avec Ie logiciel Maple par exemple. Ensuite on devine les vrais degres a partir de leurs specialisations (vu les resultats, il n'est pas trop difficile de deviner ...), et enfin on verifie par une multiplication simple. Table 4: Specialisations des parametres pour ?t(D~·3)
[It] G7 Gs I G6 I G4
'D
indice 1 B~,3 2 G~3) 3 3) 6
D~,3
A
TI XI,X2 1, -1 XI,X2 1, -1
T2 YI, Y2, Y3 YI, Y2, Y3 1, (3, (j 1, (3, (j
T3 ZI, Z2, Z3 ZI, Z2, Z3 ZI, Z2, Z3 ZI, Z2, Z3
Comme on I'a deja remarque, les degres relatifs des autres groupes de reflexions W avec W = A 4 peuvent etre obtenus a partir des resultats ci-dessus par specialisation en utilisant Ie lemme 4.3. Dans la table 4 nous donnons les specialisations des parametres (XI,X2;YI,Y2;ZI,Z2,Z3) de ?t(D~·3) necessaires pour obtenir les sous-algebres cyclotomiques avec les diagrammes indiques. lei, comme dans les tables 6 et 8, (n, pour n E N, designe la racine primitive n-ieme de l'unite (n = exp(21fi/n). Pour les cas de G 4 et G s on retrouve ainsi les resultats de [2, Bern. 5.5 et 5.7] a ceci pres que dans Ie premier degre pour G s donne dans loc. cit. il est intervenu une faute de frappe: Ie premier facteur du denominateur doit etre change en (uvxy + W2Z2).
4C. Degres relatifs de D~,4 Soient maintenant W = Gil, un produit central C24 0C2 GL 2(3), de diagramme D~,4 dans (4.1) ou n = 4, et ?t(D~,4) I'algebre cyclotomique correspondante. Soient encore Sl, S2, S3 les trois generateurs de W = Gil avec Si d'ordre i + 1, 1 ~ i ~ 3, et Ti les generateurs de ?t = ?t(D~,4) correspondants. L'element Z = SIS2S3 est central dans W, et on verifie que {zi, SIZ i , S2 zi , S3 zi , s~zi I 0 ~ i ~ 23, 0 ~ j ~ Il} donne un systeme de representants des 96 classes de conjugaison de W. Notons C I'ensemble d'elements de ?t correspondants.
Theoreme 4.5 Les degres relatifs de ?t(D~,4, (XI, X2; YI, Y2, Y3; ZI, Z2, Z3, Z4)) par rapport Ii chaque base quasi-symetrique contenant C sont donnes par:
Degres relatifs
3
(X2- XI)
325
434
II (Yi-Yd II (Zi-ZJ) II II (X2Yi Zj- XIYIZI) (X2Y2Y3 Z2Z3- Xlyr Zr)
i=2 i=2 i=2j=2 2 2 2ZI3) X(X2Y2Y3Z2Z4 - XIYI2Zl2)( X2Y2Y3Z3Z4 - XIYI2Zl2)( X2Y2Y3Z2Z3Z4 - XIYI pour les
24
caracter-es lineaires,
Demonstration. Les valeurs des caracteres irreductibles sur les elements de C sont obtenues comme dans la demonstration du theoreme 4.4 (voir la table 5). 0
Table 5: Table des caracteres pour ?t(D~,4)
TIZj T1Zj T3Z i T 2Z i ;:r!,YizJ+~ x{+IYiZ{ x;yi+lzi xiyi zi+ 1 xiyizi I I I j 2r i (z? + z~)rj (Zl + Z2)r i (Xl + X2)r (YI + Y2)r i j 3si (2xI + X2)Sj (YI + Y2 + Y3)Si (ZI + z2 + Z3)Si (z? + Z~ + zns i i 4t (2X I + 2X2)ti (2YI + Y2 + Y3)t i (Zl + z2 + Z3 + Z4)t (zr + z~ + Z5 + Z~)tj Zi
(lei,
0 ~ i ~ 23, 0 ~ j ~ 11, et r 2
t 4 = xrX~YrY2Y3ZIZ2Z3Z4')
= XIX2YIY2ZIZ2, S3 = X?X2YIY2Y3ZIZ2Z3.
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Dans ce cas, comme pour Ie cas de D~,5 (voir plus loin), la simplification pour la solution du systeme d'equations decrit pour Ie cas de D~·3 n'est pas suffisante. lei, nous specialisons les 9 (resp. 10 dans Ie cas de D~,5) indeterminees en des nombres premiers, puis reduisons Ie systeme de 96 (resp. 270) equations obtenu modulo un autre nombre premier (par exemple Ie nombre 1009 convient). Avec Maple il est alors possible de verifier que Ie rang du systeme est encore maximal et de calculer les solutions mod 1009. Un processus de relevement de Hensel permet d'obtenir des approximations des solutions dans I'anneau 1009-adique ZlO09. (II se trouve qu'une approximation a 130 chiffres pres suffit.) Comme les coefficients des equations lineaires sont contenus dans Q, les solutions sont algebriques sur Q. Par I'algorithme LLL, on trouve les equations minimales sur Q satisfaites par ces solutions. On peut alors deviner les fonctions en 9 (resp. 10) variables ayant comme specialisations les nombres rationnels qui interviennent comme coefficients des polyn6mes minimaux. II reste a verifier les solutions. Dans la table 6 nous donnons Table 6: Specialisations des parametres pour 1-l(D~·4)
[11] G ll G lO G 15 I G9 G14 G8 G 13 G 12
I
D D~,4
B~,3
G~4)
I~3) (8) A~4)
K2
indice 1 2 2 3 4 6 6 12
Tl
T2
T3
Xl, X2
Yl, Y2, Y3
21, Z2, Z3, 24
1, -1
Yl,Y2,Y3
Xl, X2
Yl,Y2,Y3
y'Ul, JU2, -y'Ul, -JU2
Xl, X2
1, (3, (j
Xl, X2
Yl,Y2,Y3
1, -1
1, (3, (j 1, (3, (j 1, (3, (j
Zl, 22, Z3, 24 1, (4, -1, -(4 21,22,23,24
Xl, X2 Xl, X2
Zl, Z2,
23, 24
y'Ul, JU2, -y'Ul,-JU2 1,(4,-1,-(4
les specialisations des parametres necessaires pour obtenir les degres relatifs des autres algebres cyclotomiques 1-l(W) avec W = 8 4, Ainsi on retrouve les degres relatifs des algebres cyclotomiques correspondantes aux groupes G 8 et G 12 deja traitees dans [2, Bem. 5.10 et 5.13].
4D. Degres Relatifs de D~,5 Enfin, soient W = G 19 , un produit central C60 oc, 8L 2 (5), avec diagramme D = D~,5 dans (4.1) ou n = 5, et 1-l1'algebre correspondante. Soient Sl, S2, S3 les trois generateurs de W = G 19 avec Sl d'ordre 2, S2
Degres relatifs
327
d'ordre 3 et S3 d'ordre 5, et 1'; les generateurs de 'H = 'H(D~'s) correspondants. Notons encore z = SlS2S3 et Z := T IT2T3 les elements centraux. Alors {Zi, slzi, S2Z\ S3Zi, S~Zi I 0 ~ i ~ 59, 0 ~ j ~ 29} constitue un systeme de representants des classes de conjugaison de W. Notons C l'ensemble des elements de 'H correspondants figurant dans la table 7. Table 7: Table des caracteres pour 'H(D~'s)
Zi xiyi zi 2ri 3s i 4t i 5u i 6v i
T;Zi TIZi T2Z i x{+IYizi xiyi zi+ k xiyi+1 zi i (Xl + X2)r i (z~ + z5)r i (YI + Y2)r (2XI + X2)si (z~ + z~ + Z~)Si (YI + Y2 + Y3)Si (2XI + 2X2)ti (z~ + z~ + z~ + z:W (2YI + Y2 + Y3W (3XI + 2X2)Ui (2YI + 2Y2 + Y3)U i (z~ + z~ + z~ + z~ + Z~)Ui (3XI + 3X2)Vi (2YI + 2Y2 + 2Y3)V i (2z~ + z~ + z~ + z: + Z~)Vi
(lei, 0 ~ i ~ 59, 0 ~ j ~ 29, 1 ~ k ~ 2, et r 2 = XIX2YIY2ZIZ2, S3 = X~X2YIY2Y3ZIZ2Z3' t 4 = x~X~Y~Y2Y3ZIZ2Z3Z4' US = xix~Y~Y~Y3ZIZ2Z3Z4ZS, v 6 = xix~y~y?y~z~ Z2Z3Z4ZS.) Afin de rendre les formules dans Ie prochain enonce pas trop monstrueuses, nous avons adopte la convention suivante. Les facteurs qui interviennent dans les degres ont beaucoup de symetries; dans chaque cas, nous donnons seulement une partie des facteurs, les autres peuvent s'en deduire par application d'un groupe de symetries sur des sousensembles des indeterminees. Plus precisement, pour chaque facteur dans Ie numerateur ou Ie denominateur, on doit ajouter toutes les images differentes qui peuvent s'obtenir par l'action du groupe symetrique sur les ensembles de variables donnes dans la ligne suivante, en negligeant les changements de signe. Par exemple,
(Xl - X2)(XIYI - X2Y2) symetrise selon ({Xl, X2}1 {Y2' Y3}) est l'abreviation de
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EVidemment, ce1a donne Ie resultat seulement au signe pres. Le signe correct se deduit du fait que les degres deviennent des nombres rationnels positifs par specialisation sur Ie groupe de reflexions W .
Theoreme 4.6 Les degres relatifs de 1{(D~'s, (Xl,X2; Yl, Y2, Y3; Zl, ... , zs)) par rapport a. chaque base quasi-symetrique contenant C sont donnes par: x~o Y~o zF
(X2 - Xl)(Y2 - Yl)(Z2 - ZI)(X2Y2 Z2 - XlYlZl) (X2Y2Y3 Z2Z3 - xlyizi) X (X~Y2Y3Z2Z3Z4 -xiyi Z{)(X~Y~Y3Z2Z3Z4ZS-xiy{z{) (x~Y?Y~Z2Z3Z4ZS-X{Y{ Z{) (symetrise selon ({Y2, Y3} I{ Z2, Z3, Z4, zs})) pour les 30 caracteres lineaires, -x~ Y{ Y2 Y~o
zi zF
2(Y3 - Yl)(Y3 - Y2)(Z3 - ZI)(YlZlZ2 - Y3Z3Z4)(Y2ZIZ2 - Y3Z3Z4) X (YlY2ZIZ~ - Y~Z3Z4ZS)(XlYlZl - r)(x2Y3 Z3 - r)(xlY3Z3Z4ZS - rZIZ2) (symetrise selon ({Xl,X2}I{ZI,Z2}I{Z3,Z4,ZS})) avec r 2 = XlX2YlY2ZlZ2 pour les 60 caracteres de degre 2,
xI X~S Y~ Z{ zF 3(X2 - Xl)(Z4 - ZI)(X2Z4ZS - XlZlZ2)(X2YlZ4 - S)(XlYlZl - S)(XlX2YlY2Z4ZS - S2) (symetrise selon ({Yl, Y2, Y3} I{Zl, Z2, Z3} I{ Z4, zs})) avec S3 = X~X2YlY2Y3ZIZ2Z3 pour les 60 caracteres de degre 3,
x1 6 yp yJ3 Y~o z~ Z~2 4(Y2 - Yl)(Y3 - yJ)(zs - ZI)(XlYlZl - t)(XlY2 ZS - t)(XlY3ZS - t)x (XlX2YlY2ZIZ2 - t 2)(XlX2Y2Y3 ZIZS - t 2) (symetrise selon ({Xl, x2}I{ZI, Z2, Z3, Z4})) avec t 4 pour les 60 caracteres de degre 4, x1 8 X~8 yj3 y~l zI
= xix~yiY2Y3ZIZ2Z3Z4
5(X2 - XJ)(Y3 - Yl)(XlYlZl - U)(X2Y3Zl - U)(XlX2YlY2ZIZ2 - u 2) (symetrise selon ({Yl, Y2} I{ Zl, Z2, Z3, Z4, zs})) avec US = X{X~yiY~Y3ZIZ2Z3Z4ZS pour les 30 caracteres de degre 5, xi 4 X~l yl S zF z~ v 3 6(Z2 - ZJ)(XlYlZl - V)(X2YlZl - V)(XlX2YlY2ZIZ2 - v2)(xix2YlY2Y3ZIZ2Z3 - v 3) (symetrise selon ({Yl,Y2,Y3}I{Z2,Z3,Z4,ZS})) avec v = x{x~yiy~y~zi Z2 Z3Z4ZS 6
pour les 30 caracteres de degre 6.
Degres relatifs
329
Demonstration. Comme precedemment nous pouvons evaluer les caracteres irreductibles de 1t sur les elements de C. Les methodes dlkrites apres la demonstration du theoreme 4.5 fournissent alors la solution cherchee. D La table 8 contient les specialisations donnant les autres algebres cyclotomiques 1t(W) avec W = As. Table 8: Specialisations des parametres pour 1t(D~'s)
[11]
'D
indice
Gig GIS
D~,b
B~,3 G~S)
IG l7 I G 21 G I6 G 20
Gn
I~3) (10) A(S) 2 Ii,3(5)
M2
TI
T2
1 2
XI,X2
3
XI,X2
YI, Y2, Y3 YI, Y2, Y3 1, (3, (j
5 6 10 15
XI,X2
1, -1
1, -1 1, -1 XI,X2
YI, Y2, Y3 1, (3, (j YI, Y2, Y3 1, (3, (j
ZI, Z2, Z3,
T3 Z4, Zs
ZI,z2,z3,Z4,ZS
ZI, Z2, Z3, Z4, Zs 1, (s, (~, a ZI, Z2, Z3, Z4, Zs 1,(s,(l,a,a 1, (s, (~, (~, a
a,
5. Theories d-Howlett-Lehrer-Lusztig pour des blocs unipotents D'apres les resultats de [3], en particulier Ie Fundamental Theorem 3.2, on sait que pour chaque polynome cyclotomique X-y entre les caracteres unipotents 'Y E £ (G, (L, A)) et les caracteres irreductibles de l'algebre cyclotomique 1t(Wc(L, A)) associee au groupe de Weyl cyclotomique Wc(L, A). Si cette coincidence numerique s'explique par les conjectures enoncees dans [2, §1], il devrait alors exister une specialisation de l'algebre 1t(Wc (L, A)) telle que les degres des caracteres unipotents 'Y E £(G, (L, A)) soient donnes par la formule suivante : (5.1)
OU Dx"Y designe la specialisation du degre relatif de X-y. NOllS dirons qu'un
G. Malle
330
dans [2, §5] pour quelques cas exceptionnels et dans [10] pour les groupes classiques. Avec les formules obtenues ici il est possible de la verifier dans tous les cas ou Ie groupe de Weyl cyclotomique Wc(L, A) est isomorphe a un groupe de reflexions de dimension 2 :
Proposition 5.2 Soient G un groupe reductif fini simple de type exceptionnel, et £(G, (L, A)) un
Demonstration. Grace a [2, Foig. 5.6, 5.8, 5.11 et 5.14] il reste seulement les cas de la table 9 a considerer. Les specialisations des parametres se deduisent des cas de dimension 1, c'est a dire des cas OU Wc(L, A) est cyclique, qui figurent dans [2, 2B]. On fait alors les calculs avec les for0 mules donnees par les theoremes precedents pour verifier (5.1).
Table 9:
G
d
L
IEs
5 8 12
H4
3
4
A 1 1 1 1 1
parametres
Wc(L, A)
A5)
G~4)
B 4,3 2
I~'~(5)
M2
1,q,q2,q3,q4 1, q2 , q4, q6; 1, l 1,q3, -q3,l; 1, -q2,l 1,q,q2 1, q2
6. Degres unipotents Soient W Ie groupe de Weyl d'un groupe reductif fini G, et H(W) I'algebre de Iwahori-Hecke correspondante. L'interpretation de H(W) comme algebre d'endomorphismes du caractere de permutation de G sur Ie sous-groupe de Borel B montre que les degres relatifs, multiplies par I'indice de B dans G, donnent les degres des caracteres unipotents de
Degres relatifs
331
C qui interviennent dans 1~. II a ete remarque dans [9] que dans Ie cas plus general ou West un groupe de Coxeter fini, les degres relatifs de
1-l(W) (vus comme fonctions rationnelles du param€Me q) multiplies par Ie polyn6me de Poincare de W donnent des polyn6mes en q. De plus, I'ensemble de ces polyn6mes peut etre complete en un ensemble £(W) de polyn6mes reels en q, appele degres unipotents de W, ayant beaucoup de proprietes combinatoires en commun avec I'ensemble des degres des caracteres unipotents d'un groupe reductif. Dans [10] nous avons construit de tels ensembles £(W) pour deux families infinies de groupes de reflexions complexes imprimitifs W. Ayant calcule les degres relatifs des groupes de reflexions W primitifs de dimensions 2, il est alors naturel de se demander dans quels cas existent aussi des ensembles £(W) de degres unipotents, satisfaisant aux proprietes enumerees dans [10, Satz 1.1]. Proposition 6.1 Soit W un groupe de reflexions primitif Alors W possede un ensemble £(W) de degres unipotents satisfaisant aux conditions de [10, Satz 1.1} si et seulement si soit WE {A(3) C(3) A(4) 2, 2, 2,
1(3)(8)} 2
soit West de dimension au moins egale Ii 3 et est different de L 4 . Demonstration. Pour les cas de dimension au moins egale a 3, les ensembles £(W) sont donnes dans [4]. Si West de dimension 2, on specialise les parametres de I'algebre cyclotomique 1-l(W) par us.} >-->
d.
si j < ds ,
Us,d. >-->
q.
Puis, on verifie a I'aide des resultats ci-dessus que les degres relatifs multiplies par Ie polyn6me de Poincare donnent des polyn6mes seulement dans les cas enumeres dans la proposition. II reste alors a construire les ensembles £(W); les resultats seront publies dans [4]. 0 References
[:I.] S. Ariki, Representation theory of a Heeke algebra of C(r,p, n), J. Algebra 177 (1995), 164-185 [2] M. Broue et G. Malle, Zyklotomische Heckealgebren, Asterisque 212 (1993),119-189 [3] M. Broue, G. Malle et J. Michel, Generic blocks of finite reductive groups, Asterisque 212 (1993), 7-92
G. Malle
332
[4] M. Broue, G. Malle et J. Michel, "Reflection data" and their unipotent degrees, en preparation [5] M. Broue, G. Malle et R. Rouquier, On complex reflection groups and their associated braid groups, Representations of Groups, Canadian Mathematical Society, Conference Proceedings 16, Amer. Math. Soc. 1995, 1-13 [6] F. Digne et J. Michel, Cohomologie des varietes de Deligne-Lusztig et de certaines de leurs generalisations, prepublication (1994) 17] H. Kapp, Irreduzible Darstellungen und generische Grade von zyklotomischen Aigebren zu exzeptionellen komplexen Spiegelungsgruppen, Diplomarbeit Universitat Heidelberg (1994) 18] G. Lusztig, Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976), 101-159 [9] G. Lusztig, Coxeter groups and Asterisque 212 (1993), 191-203
unipotent
representations,
[HI] G. Malle, Unipotente Grade imprimitiver komplexer Spiegelungsgruppen, J. Algebra 177 (1995), 768-826 [11] G. C. Shephard et J. A. Todd, Finite unitary reflection groups, Ganad. J. Math. 6 (1954), 274-304 Universitat Heidelberg I.W.R., 1m Neuenheimer Feld 368 D-69120 Heidelberg, Germany [email protected] Received December 1994
Character Values of Iwahori-Hecke Algebras of Type BI Goiz Pfeiffer The concept of the character table of a generic Iwahori-Hecke algebra is introduced in [8] as a square matrix which maps under specialization to the character table of the corresponding Weyl group. The character tables for the series of Iwahori-Hecke algebras of type An are determined by a recursion formula which was originally proved in Ram's article [15] and by a different approach in [13]. The character tables of the IwahoriHecke algebras of exceptional type have been computed by Geck in [5] and [4] and by Geck and Michel in [7]. In this article we determine the character table of the generic IwahoriHecke algebra of type B n . The main result is given in Theorem (12.1). This follows from two deformations (10.3) and (11. 7) of the MurnaghanNakayama formula for the character values of symmetric groups. These two results are derived in the more general context of cyclotomic algebras of type B which have recently been defined by Broue and Malle [2] and by Ariki and Koike [1]. The central result which allows us to derive character formulas of this kind is the decomposition of certain character values in Theorem (9.3). The definition of the character table of an Iwahori-Hecke algebra is given in Section 1. In Section 3 the cyclotomic algebra of type B is introduced; Section 8 describes the representing matrices for its irreducible representations. Some facts about wreath products and their characters are collected in sections 5 and 6. In Section 7 we generalize the Littlewood-Richardson rule for an application to Weyl groups of type B n . The proof of the main results covers sections 9 to 12. Section 13 contains some concluding remarks. Computer programs for the explicit calculation of the character tables of generic Iwahori-Hecke algebras of type An, B n , D n and their specializations have been implemented in the GAP language [16] and are part of CHEVIE [6].
lThis article is part of the author's Ph.D. thesis [14] under the direction of Prof. H. Pahlings. It is a contribution to the DFG research project "Algorithmic Number Theory and Algebra." The author gratefully acknowledges financial support by the DFG and the Studienstiftung des deutschen Volkes.
334
G. Pfeiffer
1. The Character Table of an Iwahori-Hecke Algebra
We briefly recall the definition of the character table of an Iwahori-Heeke algebra from [8]. Let (W, S) be a Weyl group with generating set of simple reflections SeW. Let A be the ring of polynomials over Z in indeterminates q., s E S, such that qs = qs' whenever sand s' are conjugate in W. The generic Iwahori-Hecke algebra H associated to W is an associative A-algebra with basis Tw , w E W, and multiplication defined by Tww' qsT1 + (qs - l)Ts
if l(ww') = l(w) for s E S,
+ l(w'),
where l(w) is the usual length function on W. Denote by K the algebraic closure of the field of fractions of A. Then the K-algebra H K = K 0A H is semisimple and split over K. Let C be a conjugacy class of W and denote by Cmin the set of elements of minimal length in C. Choose an element We E Cmin for each class C of W. Then we have the following consequence of Theorem (1.1) in [8]. Theorem 1.1 Let X be a character of H K . (a) X is constant on {Tw I w E Cmin} for each conjugacy class C ofW. (b) For each w E W there exist polynomials fw,e E Z[qs that X(Tw)
Is
E S] such
= 'Lfw,e X(Twc ) e
where the sum is over all conjugacy classes C of w.
This enables the following natural definition of the character table of H K which is in fact independent of the actual choice of the We. Definition 1.2 The character table of H K is the square matrix
with rows labeled by the irreducible characters ¢i of H K and columns labeled by the conjugacy classes C of w.
Iwahori-Hecke Algebras of Type B
335
2. Basic Notations We shortly summarize the most important definitions for this article and cite two important results from the representation theory of symmetric groups. Let n be a positive integer. A partition a = [a1,"" aT] of n is a (finite) sequence of integers a1 ::::: .. , ::::: aT > 0, i = 1, ... , r, with L ai = n. We write lal = n and denote by l(a) = r the length of the partition a. The partition a is represented by a diagram, which consists of l(a) rows of boxes with ai boxes in the i-th row. This diagram also is denoted by a. A tableau 1I' assigns a positive integer to each box of the diagram a. 1I' is called a semistandard tableau, if these numbers are increasing along the rows and strictly increasing along the columns. If {3i is the number of entries i in 1I', then {3 = [{31, {32, ...] is called content of 1I' and a is the shape of 1I'. A semistandard tableau with content [In] = [1,1,1, ...] is called a standard tableau. In a standard tableau each of the numbers 1, ... , n occurs exactly once. The following picture shows from left to right the diagram of the partition a = [4,2, 1] of 7, the canonical standard tableau of shape a, a semistandard tableau of shape a with content {3 = [3,2,1,1] and a tableau of shape a, which is not semistandard.
~ ~ 234 5 6 7
112
2 4
3
Let 1I' be a standard tableau of shape a. Each number m E {I, ... ,n} has a unique position in 1I'. If m has position (i,j) in 1I', i.e., m lies in the i-th row and in the j-th column, then we denote by c(1I' : m) = j - i its content. This way the content measures the distance of m to the diagonal of the tableau 1I'. Furthermore the axial distance r(m1' m2) between m1 and m2 in 1I' is defined as r(m1, m2) = c(1I' : m2) - c(1I' : md. The content c(a) of the partition a is defined by c(a) = 2:;:'=1 c(1I' : m) and it is independent of the chosen standard tableau of shape a. In the above canonical tableau of shape a = [4,2,1] the numbers 1 and 6 have content 0, the number 5 has content -1 and the number 2 has content 1. The axial distance r(2, 5) therefore is -1 - 1 = -2. The content of the partition a is c( a) = 3. For partitions a and 'Y with l(a) ~ lb) and ai ~ 'Yi for i = 1, ... , l(a) we write a ~ 'Y.
336
G. Pfeiffer
Let 0: and 'Y be partitions with 0: ~ 'Y. Then the set theoretic difference 'Y \ 0: of the diagrams 'Y and 0: forms a skew diagram. The diagram 'Y \ 0: is a strip, if it does not contain any 2 x 2-block of boxes. If 'Y \ 0: is furthermore connected then it is called a hook. The hook length of a hook 'Y \ 0: is denoted by l~ and is defined as the number of rows which are occupied by the hook in the diagram 'Y, minus one. The connected components of a strip 'Y \ 0: are hooks and we denote by c~ their number and by l~ the sum of the corresponding hook lengths. The conjugacy classes and the irreducible characters of the symmetric group §n are parameterized by the partitions of n. Its character table is determined by the Murnaghan-Nakayama formula, which can now be stated. For this let Xl ] = l.
Theorem 2.1 (Murnaghan-Nakayama formula) Let'Y and W be partitions of n and let k
=
Wi
[WI'···' Wi-I, Wi+I,.··, WL(1l")].
x'Y(W)
for some i ~ l(w). Then =
L
Furthermore let p
=
(-I)I;(X u (p),
1'Y\ul=k
where the sum is taken over all partitions a hook.
0:
of n - k such that 'Y \
Proof. [10] (2.4.7), [11] (5.6).
0:
is
o
Recall that a lattice permutation is a sequence of positive integers where, for all i, j, the number of integers i among the first j elements is greater than or equal to the number of integers i + 1 among these. Let 'Y be a partition of n, 0: a partition of n - k, (3 a partition of k and denote by g~(3 the multiplicity of the character XU x x(3 in the restriction of the character X'Y of §n to §n-k x §k.
Theorem 2.2 (Littlewood-Richardson rule) For partitions 0: ofnk, (3 of k and 'Y of n the multiplicity g~(3 equals the number of skew semistandard tableaux of shape 'Y \ 0: with content (3, which yield lattice permutations if the entries are read row by row from right to left. In particular g~(3 = 0, whenever 0: ~ 'Y.
Proof. [10], (2.8.13).
o
3. The Cyclotomic Algebra of Type B}:") This section introduces the algebras we are dealing with. Let r be a positive integer. Following [1] and [2] we define a cyclotomic algebra
Iwahori-Hecke Algebras of Type B
337
associated to the complex reflection group C r I §n (s. sections 5 and 6 for some facts about groups of this type). For this let q, Qo, ... ,Qr-l be indeterminates over Q. Furthermore let A = Z[q, Qo, ... , Qr-d and let K be the field of fractions of A.
Definition 3.1 The cyclotomic algebra H~r) of type B~r) is generated as K -algebra by elements T, Sl, ... ,Sn-l satisfying the relations
(i) (T - Qo) ... (T - Qr-d = 0, (ii) S;
= q + (q - I)Si
for i
= 1, ... , n - 1,
(iii) TS1TS 1 = SlTS1T,
(iv) TSi
= SiT
for i
(v) SiSj = SjSi (vi) SiSi+lSi
=
> 1,
for all i,j with
Si+lSiSi+l
for i
Ii - jl 2: 2, =
and
1, ... , n - 2.
The relations (iii) - (vi) are called braid relations.
Proposition 3.2 H~r) is a semisimple algebra of dimension n! r n over K and it splits over K. Proof. [1] (3.10).
D
The elements S1, ... ,Sn-l generate a subalgebra of type A n- 1 which will be denoted by H n for short. We define elements T; E H¥) as To = T and T; = SiT;-lSi for i = 1, ... , n - 1. Then ([1], (3.4))
KT.oko '"
Tkn-1'T'
n-l.Lw
as a K-vector space. By specializing q f---> 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
n:::6
r-l _ _
II
c=o
of its inertia group
c Tensoring this character with the irreducible character n~:6 Xo of the inertia factor To/C~ ~ §!oOI x ... x §loT-ll
and inducing to X n yields the character XO of Xnirreducible character of X n has this form
Conversely each
Proposition 6.1 The characters XO form a complete system of pairwise different absolutely irreducible characters of X n if a runs through all rtuples of partitions of n.
o
Proof. [10] (4.4.3).
The character table of Cr I §n is determined by the following generalization of the Murnaghan-Nakayama formula (2.1).
Proposition 6.2 Let ( be a primitive r-th root of unity, let"( and 1r be r-tuples of partitions of n, and let k = 1r; for at < r, i :::; l( 1r t ). Moreover let p be the r-tuple of partitions of n - k, which is obtained from 1r by deleting the part 1r; = k. The value of the irreducible character X'"l on an element x E C r I Sn with cycle structure 1r is determined by
x'"l(1r)
=
L
(st( -l)I~Xo(p),
h\ol=k
where the sum ranges over all r-tuples of partitions a of n - k such that
"( \ a is a hook and where s = r("( \ a).
342
G. Pfeiffer
Proof. This follows from a more general formula for wreath products of finite groups with symmetric groups (s. [12] (4.4) and [18] (4.3)) by the fact that ((st) is the character table of the cyclic group Cr. For an explicit proof see [:I.] (2.2). [J 7. Two Littlewood-Richardson Rules In this section we derive two generalizations of the Littlewood-Richardson rule for Weyl groups W n of type B n . For this let a, 13 and 1 be double partitions of n - k, k and nand let g:{3 be the multiplicity of the character X'" x X{3 in the restriction of the character X"I of Wn to Wn - k X Wk. Then the computation of this multiplicity can be reduced to the classical Littlewood-Richardson rule
(2.2). Proposition 7.1 For double partitions a = (ao, ( 1 ) of n - k, "I "10 "11 (13 0 , 13 1 ) af k and 1 = (0 1 ,1 1) af n we have g"'{3 = g"'0{30g",I/31.
13
Proof. By F'robenius reciprocity g:{3 also is the multiplicity of X"I in the induced character (X'" x X(3)Wn , hence
(X'" x X(3)wn =
L g:{3X"I. "I
By construction the character X"I associated to a double partition 1 = (,,(°,1 1 ) of n is induced from an irreducible character ¢ "I of an inertia group T"I' which is isomorphic to the direct product Who l x Wh11. More precisely where the characters X"I', i = 0,1, of the symmetric groups §hil here are regarded as characters of Whil and where E is the linear character of Wh11 which maps the sign change on -1 and the other generators on 1. Similar statements hold for X'" and X{3. Since induction of characters commutes with the outer tensor product (s. [3], (43.2)) we have
X'" x X{3
= (¢",)wn -
k
X
(4J3)wk
= (¢'"
x 4J3)Wn - k XWk •
Moreover ¢'"
X
¢{3
= X'" ° X
EX
",I
X
{30
X
X
EX
{31
= (X'"
°x
{30
X )
X
E(X'"
1
X
{31
X ),
343
Iwahori-Hecke Algebras of Type B
Define ni
= 100i\ + \f3i\ for i = 0,1. Then
By the Littlewood-Richardson rule (2.2) we have
and
where in each case the sum is taken over all partitions ,../ of ni. So we finally have
[J
We nOw investigate the restriction of X"l to a subgroup of the form W n- k
X
§k.
Lemma 7.2 Let k :::; n. For the restriction of the character x"l of W n on Wn-k X §k the following holds.
where the sum is taken over all double partitions all partitions 8 of k.
0:
of n - k, f3 of k and
Proof. By construction X13 I§k = (x 13° X x131 )§k = 2:0 g~0131 XO, where the sum is taken over all partitions 8 of k and the coefficients g~0131 are given by the Littlewood-Richardson rule (2.2). The assertion then follows from the above Proposition (7.1). 0
344
G. Pfeiffer
If the components of the skew diagram I \ 0: are in a suitable way drawn one on top of the other the resulting diagram looks like a skew diagram of ordinary partitions. So it is possible to talk about tableaux of the form I \ 0: with content (3 for an ordinary partition (3 of n.
Proposition 7.3 For double partitions 0: of n - k, I of n and a partition (3 = [k-r, IT] of k the multiplicity g:{3 equals the number of skew standard tableaux of shape I \ 0: and of content (3, which yield lattice permutations if the entries are read row by row from right to left. In particular g:{3 = if 0: Cli'
°
Proof. By Lemma (7.2) we have
where the sum is taken over all double partitions 8 of k. We examine the nonzero summands. For this let g~Ob' > 0. Then 8i ~ (3 for i = 0,1, and since (3 = [k - r, IT] is a hook, the 8i also must be hooks. Therefore Ii \ o:i is a strip for i = 0, 1. Denote by C;~1I"3 the set of all standard tableaux of shape 7[1 \ 7[2, which are filled with content 7[3 according to the Littlewood-Richardson rule. Under these conditions there is a bijection a between the set
U{(To, T
1,
T2) I Ti E C::{3" i = 0,1, and T2 E C~Ob'}
b
and the set of all tableaux (T~, T{) of shape 1\0: and of content (3. For (T~, T{) we have here T~ = To, and T{ arises from T1 by replacing the entries in a suitable way, such that (T~, T{) has content (3.
a(To,T1 , T2) =
o Remark. Presumably a suitable generalization of Proposition (7.1) holds in arbitrary wreath products CT I §n' Moreover Proposition (7.3) certainly is valid for arbitrary partitions (3 and arbitrary r-tuples of partitions I and 0:. For this article, however, it is sufficient to have both propositions in the proven form.
8. Representing Matrices
Ariki and Koike ([1] (3.6)) give a method to construct matrices for all irreducible representations of H~T). This is a generalization of Hoefsmit's method (see [9] (2.2.6)) for the ordinary Iwahori-Hecke algebra of type
Bn-
345
IwahoT'i-Hecke Algebras of Type B
rr
For this let [ be an r-tuple of partitions of n, furthermore let = X"l(l) be the degree of the corresponding character of Cr I §n and let {1l'p I p = 1, be the set of all standard tableaux of shape [. These tableaux form the basis of the H~rLmodule V"I defined below. For k E Z and y E K let 0
0
•
,rr}
and let
M(k
)_
1
,y - ~(k, y)
[
q- 1 q~(k - 1, y)
~(k + 1, y) ]
_qky(q - 1)
be a 2 x 2-matrix over K. The action of T on V"I is diagonal and given by 1l'pT = Qr(ll'p:l)1l'P for all p = 1, ... ,
rr,
where the entry 1 lies in the component 1l';(ll'p:l) of 1l'p, The action of Si on V"I is given by 1l'pSi = q1l'p, if i and i
+ 1 lie in the same row of a component
of 1l'p, and by
if i and i + 1 lie in the same column of a component of 1l'po Otherwise 1l'q := 1l'p(i, i + 1) is again a standard tableau and Si acts on the subspace with basis (1l'p, 1l'q) as the matrix
Here r(i+ 1, i) is the axial distance between i+ 1 and i in 1l'p, and i (resp. + 1) lies in the component 1l';(ll'p:i) (resp. 1l';(ll'p:i+l)) of 1l'p, The axial distance between i + 1 and i in 1l'q equals -r(i + 1, i), and
i
M( -r(i + 1, i), Qr(ll'p:i+l)/Qr(ll'p:i))
=
[? ~] M(r(i + 1, i), Qr(ll'p:ij/Qr(ll'p:i+l)) [? ~]
So the action of Si on V"I is uniquely defined.
Proposition 8.1 The V"I form a complete system of absolutely irreducible, pairwise non-equivalent representations of H~r), if [ runs through all r-tuples of partitions of n.
Proof. [l] (3.7), (3.10).
o
G. Pfeiffer
346
By the above definition To = T acts in each of the given representations as a diagonal matrix. It turns out that all T;, i = 0, ... ,n - 1, act in such a way.
Proposition 8.2 Let i
~
n - 1 and let 11' be a standard tableau of shape
--y. Then
11'7:-, -- Qr(ll':i+l) qc(ll':i+l)+ill' , where c(lI' : i) denotes the content of i in the tableau 11'.
o
Proof. [1] (3.16), d. [9] (3.3.3).
For each h E HAT) the matrix of the action of h on V"I with respect to the basis of standard tableaux of shape "( is denoted by M"I(h). The character of h E HAT) on V"I is the trace of M"I(h) and is denoted by
XJ(h). 9. Subalgebras The following notation will be used throughout the rest of this article. Let k ~ n and set
H(T)
n-k
(T,Sl",.Sn-k-l) ,
H'k
(Sn-k+l,"" Sn-l) ,
H*k
(Tn-k, Sn-k+l, ... ,Sn-l) .
So H~T~k is a subalgebra of type B~2k' H~ ~ Hk is a subalgebra of type A k - 1 , and HZ corresponds to a subalgebra of type B k (d. Section 13). Let 0: be an r-tuple of partitions of n - k and define V"I\<> to be the K -vector space with the set of all standard tableaux of the form "( \ 0: as its basis. By Lemma (4.1) V"I has as a vector space the structure of a direct sum of tensor products, <>
We write 1r = 1r1 011'2 for the decomposition of a standard tableau 1r of shape "( into a standard tableau 11'1 of shape 0: and a standard tableau 1r2 of shape "( \ 0: according to Lemma (4.1). We may assume that the set of all standard tableaux of shape "( is ordered as follows. For this purpose we assume a linear ordering on the set of all r-tuples of partitions of n - k and for each r-tuple of partitions 0: of n - k a linear ordering on the set of all standard tableaux of shape 0: and a linear ordering on the set of all standard tableaux of shape "( \ 0:. Let 1r and 1r' be standard tableaux of shape "( with 1r = 1r101r2 for a 1r1 of shape 0: and 1r' = 1r~ 01r; for a 1r~ of shape 0:'. Then define 1r < 1r', if
347
IwahoT'i-Hecke Algebras of Type B
1. a < a', or 2. a
= a' and 11'1 <
11'~,
or
3. a = ex' and 11'1 = 11"1 and 11'2 < 11'2' Now certain matrices have the structure of block diagonal matrices, where each block is the Kronecker product of two smaller matrices. More precisely we have the two following lemmas.
Lemma 9.1 Let h E H~~k' Then M'Y(h) =
EB MO(h) 0
Id,
° where a runs over all r-tuples of partitions of n - k with a ~ I and Id denotes the r\o x r\o -identity matrix.
Proof. For 11' = 11'1 0 11'2 we have by the definition of the action in Section 8 1I'T = Qr(ll':I) (11'1 011'2) = 1I' I T 0 11'2, since 1 lies in the same component of 11'1 as in 11', whence r(1I' ; 1) = r(1I'1 ; 1). For all i S n - k we have r(1I' ; i) = r(1I'1 ; i) and c(1I' ; i) = C(1I'1 ; i). Therefore the axial distance r(i + 1, i) is the same in 11' and 11'1. Moreover 1I'(i, i + 1) = 1I'1(i, i + 1) 0 11'2 for all i < n - k. Consequently
Thus h E H;:~k acts on each subspace yo 011'2 of Y'Y as the matrix MO(h) according to Section 8. Due to the ordering on the basis of V'Y the matrix M'Y (h) is a block diagonal matrix. 0
Lemma 9.2 Let h E H k. Then for each r-tuple of partitions a of n - k there is a matrix M'Y\O(h), such that
° where a runs over all r-tuple of partitions of n - k with a ~ I and Id denotes the r x r-identity matrix. The map h t---+ M'Y\o (h) defines an action of H k on V'Y\O.
348
G. Pfeiffer
Proof. Let 'lr = 'lr l 0'lr2 be a standard tableau of the form f. By (8.2) we have 10\ 'lr ) Qr(]':n-k+l) q c(][':n-k+I)+n-k('lr I '
For i ~ n - k we have r('lr : i) = r('lr2 : i) and c('lr : i) = c('lr 2 : i). Therefore the axial distance r( i + 1, i) between i + 1 and i is the same in'lr and in 'lr2. Moreover 'lr(i,i + 1) = 'lr 1 0'lr2(i,i + 1). Thus for each r-tuple of partitions 0: of n - k and each 'lr l of shape 0: there is a matrix M'Y\o.(h), such that h E HZ acts on the subspace 'lr I 0 V'Y\o. as M'Y\o. (h). Due to the ordering on the basis of V'Y the matrix M'Y(h) is a block diagonal matrix. Since M'Y is a matrix representation of H;;) also M'Y\o. is a matrix 0 representation of HZ. We denote by XJ\o. the character of HZ on V'Y\o.. The above two lemmas can then be combined to the following crucial result on a decomposition of the character XJ. Theorem 9.3 Let hI E H~~k' h 2 E HZ and let [ be an r-tuple of partitions of n. Then
XJ(h lh2 ) =
L h\o.l=k
x~(hdxJ\0.(h2)'
where the sum ranges over all r-tuples of partitions 0: of n - k with 0:
~ [.
Proof. The matrix M'Y(h l h 2) is a block diagonal matrix since by (9.1) and (9.2)
M'Y(hd M 'Y(h2) E9(Mo.(hd 0 Id)(Id 0 M'Y\0.(h 2)) 0.
E9 Mo.(hd 0 M'Y\0.(h 2), where each block belongs to an r-tuple of partitions 0: of n - k and is a Kronecker product Mo. (hd 0 M'Y\0.(h 2). Therefore the character value is
XJ(h l h2) = trace(M'Y(h l h 2))
=
LX~(hl)XJ\0.(h2)' 0.
0
Iwahori-Hecke Algebras of Type B
349
10. Restriction to Type B x A From now on let T = 2. Then H~2) is the Iwahori-Hecke algebra of type B n . Remember that here QI = -1 and Qo = Q. Let k ::; n,
/'\, = Sn-k+1 ... Sn-I
E Wn
and so
TI<
= Sn-k+1 ... Sn-I
E H.
In this section we investigate the character values of elements of the form hTI< and in the next section the character values of elements of the form hTn_kTI< for some h E H~~k' We recall the following result about character values on Coxeter elements in the Iwahori-Hecke algebra Hn of type An-I. Proposition 10.1 Let I be a partition of n. Then
o
Proof. [13] (2.2)
Lemma 10.2 Let I be a double partition of and let 0: be a double partition of n - k. Then '"1\0
Xq
TI< = {(q_l)C~-I(_I)I~qk-I~-c~, 0
()
ifl\O:
is a strip, otherwise.
Proof. XJ\o is defined as a character of HZ in Section 9. By restriction to H~ we obtain XJ\o = L/3 g:/3X~, where the sum is over all partitions {3 of k and the coefficients 9:/3 are given by the Littlewood-Richardson rule (7.3). We do not need, however, all partitions (3 in order to determine the value of XJ\O(TI<)' By (10.1) we have X~(TI<) = 0, unless {3 = [k - T, IT] for some T ::; k - 1. In that case
Therefore
k-I
(T.I< ) -- ""' X'"1\0 L.J 9'"I0 ,[k-T,lrj (I)T qk-T-I . q T=O
For {3 = [k - T, IT] the multiplicity 9:/3 is by (7.3) equal to the number of skew semistandard tableaux of shape 1\0: and content {3, which yield lattice permutations if the entries are read row by row from right to left.
G. Pfeiffer
350
This means to place k - r symbols 1 and the symbols 2, ... r + 1 in a diagram of shape 1\0:. This is impossible if the diagram 1\0: contains a 2 x 2-block. Hence g:,[k-r,lrj i- 0 only if I \ 0: is a strip. So let I \ 0: be a strip. Then we write k-l
G("VJ, 0:)
'= '"' g"l •
L...J
r=l
o,[k-r,lr]
(_I)rqk-r-l = '"'(_I)r(][')qk-r(ll')-l L...J '
11'
where the sum ranges over all admissible tableaux 11' according to the Littlewood-Richardson rule and r(11') = r if 11' has content (3 = [k - r, I r ]. Since I \ 0: is a strip, its connected components are hooks. The first of these has the following form.
If I \ 0: is connected, then there is only one admissible tableau 11' of that shape with r(11') = l~ = s - 1. Hence
which proves the assertion in that case. If I \ 0: has more than one connected component, then each of the remaining connected components has one of the following two forms.
(1)
(2)
Iwahori-Hecke Algebras of Type B
351
Consider the diagram [\a.' which consists of all but the last connected components of [ \ 0: for a suitable double partition 0:' and assume by induction on the number c~ of connected components that
G(o:', [)
=
2] _lr(]["lqk-r(ll"l-1 =
(q - 1t:,-1 (_l)I:'qk'-c: / -I: ,
ll"
where k' = Ii \ 0:' I and the sum ranges over all admissible tableaux '][" for [\ a.' according to the Littlewood-Richardson rule. Note that C~, = c~-1. Observe that to each admissible tableau '][" in the above summation there correspond exactly two admissible tableaux ']['1 and ']['2 for [\ 0: with the following property. Both ']['1 and ']['2 consist of '][" and an additional connected component which for ']['1 has form (1) and for ']['2 has form (2) and where r(']['2) = r(']['d + 1. Their contribution to the sum G([, 0:) is (_lr(ll'd qk- r (ll'd- 1 + (_lr(1l'2l qk-r(1l'2l- 1 = (q _1)(_lr(ll'd qk- r (ll'd- 2 = (_lr(1l"l qkl -r(ll"l-1. (q -1). (_lr(ll'd-r(1l'/lqk-kl-r(ll'd+r(1l'/l-1.
Finally r(']['d - r('][") = l~ - l~, is independent of '][" and therefore summation over all admissible '][" for [ \ 0:' yields
G( [,0:') (q - 1) (_1)1~-1:, qk-k'-I~+I:,-1
G([,O:)
(q _ 1)ci.-1( _l)l~qk-I~-ci.
o
which completes the proof of the lemma. The recursion formula for this case can now be stated as follows.
Proposition 10.3 Let h E Hi~k. Then XJ(hTK ) =
L
(q -lt~-1(-1)I~qk-I~-C~X;(h),
h\ol=k
where the sum is taken over all double partitions is a strip.
0: ~ [
such that [ \
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(TK ) in (10.2). 0 11. Restriction to Type B x B In contrast to the Weyl group there is in general no subalgebra of type B n - k x B k in the Iwahori-Hecke algebra of type B n . Yet it is possible to pursue a similar strategy as in the preceding section.
352
G. Pfeiffer
Let
/'i,
and TI< be as in Section 10 and let
By specialization one obtains the following formula for wreath products from (9.3).
Lemma 11.1 Let WI E Wn- k and W2 E W;' Then X'"l(WIW2) =
L Xo (wdx'"l\o (W2)' 0<;;'"1
Corollary 11.2 Let I be a double partition of n, let 0: be a double partition of n - k and let i < 2. Then
X'"l\O(t~_k/'i,) = { ~-I)I~(_1)ij, if I
\ 0: is a hook, otherwise,
where j
=
r([ \ 0:).
Proof. This follows from the Murnaghan-Nakayama formula (6.2) for wreath products Cr/§n of cyclic with symmetric groups and the regularity 0 of the character table (xO(wd) of Wn- k in Lemma (11.1). We define
n-I
Tfj. =
II
T;
i=n-k and
k+l
'"I_c([)-c(o:) eo k for double partitions on V'"I\o.
0:
+n- -2with Ii \ 0:1 = k. Then Tfj. acts as a scalar
~ I
Lemma 11.3 Let'll' be a standard tableau of shape I \ 0:. Then n-I
'll'Tfj.
=
qke~
II
Qr(ll':i+l) 'll'.
i=n-k
Proof. By definition of Tfj. and Proposition (8.2) we have n-I
'll'Tfj. = 'll'
II
i=n-k
n-I '7', -
.1, -
II
(qC(ll':i+I)+iQ, r(t+ I ) ) 'll' .
i=n-k
Here I:~:Lk c('ll' : i + 1) = c([) - c(o:), which is the content of the skew diagram 1\0:, and I:~:Lk i = kn - k(k + 1)/2, whence the sum of the exponents of q equals ke~. 0
Iwahori-Hecke Algebras of Type B
Proof. For k
>
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<;"CP) fulfilling the following equalities
x(su) =
L cpE1rrk(CG (u),Bru(b))
d~u,cp)cp(s)
(1.3.1)
362
L. Puig
where X runs over Irr,dG, b), u runs over the set of p-elements of G, s runs over the set of p'-elements of Cc (u) and, for any
Zch(OGb) =
L
OXo
(1.5.1)
xElrr.c(C,b)
We are able to state the announced relationship between DC,b and Z(OGb) endowed with Zch(OGb). Theorem 1.6. There is an isomorphism of O-algebms
(1.6.1)
mapping Zch(OGb) onto Dn(K:) n Mn(O)Dc,b' Remark 1.7. If ME Mn(O) is invertible in Mn(K) then, since every K-algebra homomorphism Dn(K:) ---> K maps Dn(K) nMn(O)M onto a finitely generated O-submodule of K and since 0 is integrally closed, we have
(1.7.1)
and the intersection Dn(K) nMn(O)M is an ideal of Dn(O) contained in Dn(K:) nMn(O)M since
Remark 1.8. If M E Mn(O)* and N E Dn(O)* then we have
Dn(O) n Mn(O)MDG,bN = Dn(O) n Mn(O)DG,b Dn(O) n Mn(O)MDc,~ = Dn(O) n Mn(O)Dc,b'
(1.8.1)
The Center of a Block
363
In particular, when determining Z(VGb) and Zch(VGb) , we can replace Irrk(Cc(u), Br,.(b)) by any basic set for the pair (Cc(u), Br,.(b)), where u runs over a set of representatives for the G-conjugacy classes of all the p-elements of G. Corollary 1.9. Assume that b = 1 and denote by Tc a n x n-matrix
determined by the so-called ordinary character table of G. There is an V-algebra isomorphism (1.9.1)
mapping Zch(VG) onto Dn(K) n Mn(V)Tc. Proof. According to Remark 1.8, it suffices to replace Irrk(CC(U), 1) by the set of characteristic functions of the Cc(u )-conjugacy classes of p'-elements of Cc(u), where u runs over the set of p-elements of G. • Remark 1.10. Since neither 1 nor Tc depend on the choice of the prime p, it is not difficult to see that Corollary 1.9 remains true if we replace V by a suitable ring of algebraic integers. 2. The Center of a Symmetric V-Algebra 2.1. We will prove an analog of Theorem 1.6 in the more general frame of symmetric V-algebras having a split semisimple extension to K, since most of our arguments need only the symmetry of VGb. So, let A be a symmetric V-algebra, V-free of finite V-rank, and, denoting by [A, A] the V-submodule of A generated by the elements [a, a'l = aa' - a'a where a and a' run over A, let us set
Zo(A) = A/[A, A]
and
ZO(A) = Homo(Zo(A), V);
(2.1.1)
notice that, whereas ZO(A) has no V-torsion, Zo(A) is not necessarily V-free; moreover, the canonical map A ----+ Zo(A) induces an injective V-linear map
ZO(A)
----+
Homo(A, V)
(2.1.2)
and an V-linear form J.l E Homo(A, V) belongs to the image of ZO(A) if and only if it is symmetric (Le., J.l(aa') = J.l(a'a) for any a, a' E A); in that case we denote by J.l0 the corresponding element of ZO(A). On the other hand, it is clear that
Z(a) . [A, A] = [A, A]
(2.1.3)
364
L. Puig
and therefore Zo(A) and ZO(A) hold natural Z(A)-module structures; the key observation is the following statement (d. Lemma 1.7 in [6]). Proposition 2.2. The Z(A)-module ZO(A) is free of rank one and J.l 0 E ZO(A) is a generator if and only if the corresponding symmetric form J.l over A is nonsingular. Proof. Since A is symmetric, there is a symmetric nonsingular form J.l over A and in particular the V-linear map A
-----+
Homo(A, V)
(2.2.1 )
mapping a E A on a . J.l is bijective; moreover, since the group A* of invertible elements of A generates A as V-module, the V-linear form a· J.l is symmetric if and only if it is A*-stable and therefore, since J.l is already A *-stable, a . J.l is symmetric if and only if a belongs to Z (A). Consequently, the V-linear maps 2.1.2 and 2.2.1 induces a bijective Vlinear map between Z(A) and ZO(A) showing both that ZO(A) is a free Z(A)-module and that J.l 0 is a generator. Finally, if z E Z(A) then z . J.l 0 is another generator of ZO(A) if and only if z is a generator of Z(A), so if and only if z is invertible in Z(A); similarly, the V-linear form z . J.l over A is nonsingular if and only if z . J.l is a generator of the A-module Homo(A, V), so if and only if z is invertible in A. • Remark 2.3. Notice that Proposition 2.2 induces a unique bijection between the set of ideals of Z(A) and the set of Z(A)-submodules of ZO(A). 2.4. Let us consider now the extensions to K. The canonical map A -----+ IC 00 A is injective and it is clear that it induces
IC 00 Z(A) ~ Z(IC 00 A)
IC 00 [A, A] ~ [IC 00 A, IC 00 A]; (2.4.1) consequently, we get IC 00 Z(A)-module isomorphisms and
moreover, since Z(A) is V-free, the canonical map Z(A) -----+ IC00Z(A) is injective and the next corollary allows us to recognize its image. Corollary 2.5. If z E IC 00 Z(A) then z E 10 Z(A) if and only if we have z· (1 0 ZO(A)) C 10 ZO(A) in IC 00 ZO(A).
The Center of a Block
365
Proof. If /1-0 is a generator of the Z(A)-module ZO(A) and z . (10/1-°) belongs to 10 ZO(A) then there is z' E Z(A) such that z· (1 0/1-°) = 10 z' '/1-0 which forces z = 10 z'. • 2.6. Let us denote by £dA) the Grothendieck group of K 00 A ("e as "length"); since every K 00 A-module is isomorphic to K 00 M for a suitable A-module M, the trace over the K 00 A-modules induces a group homomorphism from £dA) to Homo(A, 0) and, since the trace is a symmetric linear form, we get from 2.1.2 an O-linear map (2.6.1 ) and we denote by Z~h (A) its image; that is to say, denoting by IrrK:( A) the set of characters over A of all the simple K 00 A-modules, we have
Z~h(A) =
L
Oxo.
(2.6.2)
XElrrdA)
Proposition 2.7. The O-linear map ch A is injective and if K 00 A is a split semisimple K-algebra then, for any X E IrrK:(A), OXo is a Z(A)-submodule of ZO(A) and we have (2.7.1)
Proof. If l:XElrrd A ) AxXO = 0 where Ax E 0 for any X E IrrK:(A),it suffices to evaluate the sum l:XE1rrd A ) AxX, extended to K 00 A, over every primitive idempotent of K 00 A to obtain Ax = 0 for any X E Irrx; (A). On the other hand, if K00 A is isomorphic to a direct product of matrix algebras over K, we have rankz(£K:(A))
= IIrrK:(A)I = dimK:(Z(K 00 A)) = dimK:(Zo(K 00 A))
and moreover, for any X E IrrK:(A), any z E Z(A) and any a E A, denoting by ao the image of a in Zo(A), we get (z· xO)(ao)
= x(za) = (X(z)/X(I))x(a) = (X(z)/X(I))xO(ao)
since z induces a homothetie over any simple K 00 A-module.
•
2.8. Asume now that K 00 A is a split semisimple K-algebra. In that case, by Proposition 2.7, Z~h(A) is a Z(A)-submodule of ZO(A)
L. Puig
366
and therefore, by Remark 2.3, it determines a unique ideal ZCh(A) of Z(A); moreover, since V 0z .cd A) ~ Z~h(A), the Z(A)-module structure of Z~h(A) and the V-linear map ch A determine an V-algebra homomorphism ml A : Z(A)
---+
Endo(V 0z .cdA)).
(2.8.1 )
Notice that, again by Proposition 2.7, the V-linear map ch A induces K 00 Endo(V 0z .cd A )) ~ EnddK 00 ZO(A)) ~ K 00 Endo(Zo(A)) ~ HomdK 00 ZO(A), K 0z .cdA))
(2.8.2)
~ K 00 Homo(Zo(A), V 0z .cdA))
and we denote by DdA) the set of elements of K00Endo(V0 z .cdA)) having a diagonal matrix in the canonical basis of K 0z .cdA); thus, the same proposition implies (2.8.3)
We are ready to give a description of Z(A) which will lead us to the announced proof of Theorem 1.6. Theorem 2.9. Assume that K 00 A is a split semisimple K-algebra. Then the map ml A is injective and we have 10 mlA(Z(A))
= DdA) n ((K 0 chA)-1
0
(1 0 Endo(Zo(A)))
0
(K 0 ch A ))
10 mlA(Zch(A)) = DdA)
n ((10 Homo(Zo(A), V
0z .cdA)))
0
(K 0 ch A )). (2.9.1)
Remark 2.10. Setting n = ranko(Z(A)) and denoting by D the matrix of the V-linear map ch A in the canonical V-basis of V 0z .cdA) and some V-basis of ZO(A), we get from 2.9.1 an V-algebra homomorphism (2.10.1 )
mapping ZCh(A) onto Dn(K)nMn(V)D. Conversely, if we know the Valgebra Z(A), together with its ideal Z(A), it is clear from Proposition 2.7 that ZCh(A) has a unique decomposition as a direct sum of ideals
367
The Center of a Block
of Z(A) of V-rank one and it suffices to express an V-basis of ZCh(A) coming from this decomposition in terms of an V-basis of Z(A) to obtain a square matrix V' such that V'=MVN
(2.10.2)
for some ME Mn(V)* and some N E Vn(V)*. Proof. The injectivity of ml A is an immediate consequence of Proposition 2.2 and then, since
dim,dK: 00 Z(A))
= IIrr,dA)I = rankz(£,dA)) = dim,dV,dA)), (2.9.2)
we get from 2.8.3 the equality (K: 0 mlA)(K: 00 Z(A)) = V,dA).
(2.9.3)
But, if z E K: 00 Z(A) then (K: 0 mlA)(z) belongs to (K: 0 chA)-l
0
(1 0 Endo(ZO(A)))
0
(K: 0 ch A)
(2.9.4)
if and only if z . (10 ZO(A)) C 10 ZO(A) which, by Corollary 2.5, is equivalent to the fact that z belongs to 1 0 Z(A), so that (K: 0 mlA)(z) belongs to 1 0 mlA(Z(A)); this proves the first equality in 2.9.1. Moreover, in that case, z belongs to 10 Zch(A) if and only if we have z· (10Z0(A)) C Z~h(A), that is to say, if and only if (K:0mlA)(z) belongs to (2.9.5) but, since Z~h(A) is the image of ch A which is injective, we have
and therefore the K:-vector space 2.9.5 becomes (2.9.7)
Remark 2.11. Although Proposition 2.2 could suggest that Z(A) and ZO(A) are "almost" the same object, notice that, whereas the last behaves functorially, the first does not. Explicitly, if B is another
L. Puig
368
symmetric V-algebra, any V-algebra homomorphism f: A [A, A] to [B, B] and therefore it induces V-linear maps
Zo(f): Zo(A)
----->
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)
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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
°
3.3. If U E is a pointed element over OGb then the element ui, where i runs over c, have the same image in Zo(OGb) that we denote by zo(u E); indeed, if a E (( OGb) (u)) * then we have
(3.3.1) moreover, it is clear that zo((uE)X) = zo(u E) for any x E G. We are ready to exhibit an O-basis of Zo(OGb). Theorem 3.4. The set {zo(uE)}u€E[£o(C,b) is an O-basis of Zo(OGb). Proof. Since k is algebraically closed, we can replace 1C by any finite extension without changing the set {zo(uE)}u€E[£o(C,b); hence, we may assume that 1C is a splitting field for G. Now, by 3.1.1, it suffices to
370
L. Puig
prove the theorem when b = 1. Moreover, we have already proved above that
1££o(C, 1)1 =
L
IIrrk(Gc(u), 1)1
(3.4.1)
when u runs over a set U of representatives for the conjugacy classes of p-elements of C; but, since k is algebraically closed, for any u E U, the cardinal of Irrk(Gc(u), 1) is the number of conjugacy classes of p'elements of Gc(u) and therefore their sum when u runs over U is equal to the number of conjugacy classes of elements of C, which coincides with the O-rank of Z(OC). In conclusion, we get (cf. Proposition 2.2)
1££o(C, 1)1 = ranko(Z(OC)) = ranko(Zo(OC) 1
(3.4.2)
and therefore it suffices to prove that the set {zo(uE)}..,E[£o(C,I) generates the O-module Zo(OC). Let x be an element of C and consider its decomposition x = us as a product of a p-element u and a p'-element s which centralize each other; in particular, s generates a commutative O-semisimple subalgebra T of (OC) (u) (actually T is isomorphic to the group algebra O( (s) )). Let S be a maximal commutative O-semisimple subalgebra of (OC) (u) containing T and denote by I the set of primitive idempotents of S; consequently, I is both a primitive decomposition of the unity in (OC)(u) and an O-basis of S; hence, there are Ai E 0, where i runs over I, such that x = L:iEI AiUi, and any i E I belongs to some point of u over OC; thus, setting AE = L:iEEnI Ai for any point c of u over OC and denoting by zo(x) the image of x in Zo(OC), we get (3.4.3) x
where c runs over the set of points of u over OC; now, the theorem follows from the next lemma. • Lemma 3.5. For any p-element u of C and any non-local point c of u over OC we have zo(u E ) = O.
Proof. Let i be an element of c; since Bru(i) = 0, we have u =I- 1 P and i = Tr~~~)(a) for a suitable a E (OC)(u ) (cf. 1.3 and [5]) and therefore it follows from Theorem 1 in [4] that there is a (u)-stable set P J of cardinal p of pairwise orthogonal idempotents of (OC)(u ) such
The Center of a Block
371
that i = L.jEJ j; hence, we have ui juj = ujUj = 0 for any j E J.
=
L.jEJ uj
=
L.jEJ[uj, j] since •
3.6. We are ready to complete the proof of Theorem 1.6. Denoting by {zO(uE)}"cE££o(C,b) the V-basis of ZO(VGb) which is the dual of the V-basis of Zo(VGb) provided by Theorem 3.4, we claim that 3.6.1. The matrix {d~c}XElrrdC,b)'''cE££o(C,b) of the V-linear map ch ocb in the canonical V-basis of V 0z .cdG, b) and the V-basis {zO(uE)}"cE££o(C,b) of ZO(VGb) coincides with DC,b throughout the bijection mapping any fiE E .c£o(G, b) on (u, CPE)'
Indeed, for any X E IrrdG, b), we have now
L
XO =
d~c zO(u E)
(3.6.2)
"cE.c£o(C,b) and therefore, for any fiE E .c£o(G, b), we get (3.6.3)
where i E c. Hence, it follows from Corollary 4.4 in [5] applied to A = VGb, that, for any p-element u of G and any p'-element s of Cc(u), we have (3.6.4) E
where c runs over the set of local points of u over VGb, and therefore, by 1.3.1, we get d~c
= d~u,cpc).
3.7. A last remark. Assume that b is primitive in Z(VG) and denote by d the defect of this block. In [2] Brauer has proved the existence of a finite set M(d) of square matrices, depending only on d, containing MDc,b for a suitable M E M n (V) *. Consequently, from Theorem 1.6, it is possible to determine, up to isomorphism, all the centers of blocks of defect d, endowed with their ideal of characters.
References
[1] R. Brauer, Some applications of the theory of blocks of characters of finite groups I, J. of Alg. 1 (1964), 152-167.
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L. Puig
[2] R. Brauer, Types of blocks of representations of finite groups, Proc. Symp. Pure Math. 21 (1971),7-11. [3] M. Broue, Radical, hauteurs, p-sections et blocs, Ann. of Math. 107 (1978), 89-107. [4] L.Puig, Sur un theoreme de Green, Math. Z. 166 (1979), 117-129. [5] L. Puig, Pointed groups and construction of characters, Math. Z. 176 (1981), 265-292. [6] L. Puig, Y. Usami, Perfect isometries for blocks with abelian defect groups and Klein four inertial quotients, J. of Alg. 160 (1993), 192-225. [7] W. Reynolds, Sections and ideals of centers of group elements, J. of Alg. 20 (1972) 176-181. CNRS, Institut de Mathematiques de Jussieu 6 Avenue Bizet 94340 Joinville Le Pont, France Received January 1995
Unipotent Characters of Finite Classical Groups Toshiaki Shoji
1. Introduction Let G be a connected reductive group defined over a finite field F q with Frobenius map F. One of the main problems in the representation theory of finite groups is to determine all the irreducible characters (on C or Ql) of G F and complete their character tables. Lusztig classified all the irreducible characters of G F and determined their degrees ([L2]). But as far as the character values are concerned, it is still open. In order to attack this problem, Lusztig proposed a general strategy for it and gave a conjecture. It is explained roughly as follows. Let V be the Ql-vector space of class functions of G F • As is well-known the set of irreducible characters of GF forms an orthonormal basis of V under the natural inner product. He constructed two other types of orthonormal basis of V, i.e., the one consisting of almost characters of G F , and the other consisting of characteristic functions of F-stable character sheaves on G. The significance of these two bases is as follows. The almost characters are closely related to Deligne-Lusztig's virtual characters and the decomposition of those to irreducible characters is explicitly known. So the determination of irreducible characters of G F is equivalent to that of almost characters. In turn, characteristic functions of character sheaves are more related to the geometry of conjugacy classes of G, and there exists a general algorithm for computing those characteristic functions (under a mild condition on the characteristic p of F q, i.e., p is almost good. See 2.2 for the definition). Under this situation, he conjectured that these two bases coincide with each other up to scalar constants. In the case where the center of G is connected, this conjecture was verified by the author ([S2, I, II]). So the remaining step towards the determination of irreducible characters is to determine those scalars involved in Lusztig's conjecture. This paper is an attempt at this last step. In this paper, we show that the scalars are actually determined in the special case where G is a classical group of split type with connected center provided that p is odd, and if almost characters are concerned with unipotent characters,
(Le., a linear combination of unipotent characters). This implies, in particular, that all the unipotent characters of such CF are computable. We note that in [L5], Lusztig determined the scalars concerning with almost characters which have a non-zero support on the set of unipotent elements in CF, in the case where G is adjoint simple and F is of split type under some congruence condition on p. Our result was inspired by his work. 2. Preliminaries on character sheaves 2.1. Let k be an algebraic closure of a finite field F q of q elements. We denote by C a connected reductive algebraic group defined over k. Let T be a maximal torus of C and W = Nc(T)jT a Weyl group of C with respect to T. Following Lusztig [L4], we shall review a definition of character sheaves on C. Let DC be the bounded derived category of constructible Ql-sheaves on C, and let MC be the full subcategory of DC consisting of perverse sheaves. Here l is a prime number invertible in k. For a torus 5 over k, S(5) will denote the set of local systems £. on 5 such that rank£. = 1 and that £.0 n c::: Ql for some integer n ~ 1, invertible in k. Take a local system£. in S(T) such that w* £. c::: £. for some w E W. Then one can construct a complex Kfj E DC as the set of in [L4, III, 12.1]. For each£. E S(T), we denote by isomorphism classes of irreducible perverse sheaves A on C such that A is a constituent of i-th cohomology perverse sheaf P Hi (Kfj) of Kfj for some wand some i E Z. The set of character sheaves of C is defined as the union of where£. runs over all the elements in S(T).
a.c
a.c,
a
2.2. Let p be the characteristic of k. We say that p is almost good for C if P is a good prime for each factor of C of exceptional type, and no conditions for factors of classical type. We assume, from now on, that p is almost good for C. Let E be the inverse image of a conjugacy class C in CjZO(C) under the natural map C -> 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.
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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
(2.4.1.) Then Qr,C,F,CPl is a C F -invariant function on the set of unipotent elements on C F , and it is independent of the choice of £ and E, v 1---+ x- 1svx, the pair (ZO(Lx)Cx , 1 ~:Fx) is a cuspidal pair on Lx, (1 ~:Fx is the inverse image of :Fx under the map ZO(Lx)Cx -> Cx). Let E. Under this notation, Lusztig has proved that
Theorem 2.6 (Character formula, L4, II, Th. 8.5])
XK,cp(SU) = IZg(s)FI- 1 I LF I-
1
L
IL;IQi~,~~,Fz,cpz (u).
xEC F
x-lsxEL'l
2.7. Let us fix an F-stable maximal torus T and an F-stable Borel subgroup B containing T. Let L be an F-stable Levi subgroup of an
Unipotent Characters
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F-stable parabolic subgroup P of G. We assume that P contains B and L contains T. Let A o be an F-stable cuspidal character sheaf on L corresponding to a cuspidal pair (E, £) as in 2.3. Let K = ind ~Ao, and let A = EndMcK be the endomorphism algebra of K in MG. It is known by [L3, 3.4] that A is isomorphic to the group algebra QdW,d twisted by a 2-cocycle, where WE
= {n E Nc(L)
I nEn- l
= E, ad(n)*£ ~ £}jL.
Assume now A o is F-stable, and let 'Po : F*£ ~£ be the mixed structure of £. We choose a specific isomorphism 'Po so that (2.7.1) for any 9 E EF, the linear map 'Pg : £g -> £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<», where <, > is the cyclic group generated by,. Since a A : VA -> VA is A-semilinear and bijective, W-module VA can be extended to an irreducible W<»-module. Once we choose an extension VA of VA to W<»-module, the action of a A on VA coincides with the action of, on VA up to a scalar multiple. We fix an extension VA of VA for each F -stable A E G appearing in K, and choose 'PA, by replacing by a scalar multiple if necessary, so that a A exactly coincides with, on VA. This gives the mixed structure 'PA : F* A~A.
T. Shoji
378
2.8. We now consider the special case where (17,£) is given in the following way; 17 = ZO(L)C with a unipotent class C in L, and £ = 1181 F where F is an irreducible local system on C. Then it is known by [L3, 3.4] that the assumption (2.7.2) is satisfied. Let A be an Fstable character sheaf on G appearing in the decomposition of K. By the generalized Springer correspondence ([L3, Th. 6.5]), the restriction of A to the unipotent variety G uni is isomorphic to IC(G',£')[d] with d = dimC' + dimZO(L), extended by 0 on Guni - G', where C' is a unipotent class in G and £' is an irreducible G-equivariant local system on C'. Since A is F -stable, (C', £') is also F-stable. The natural mixed structure on £' is inherited from 'PAin terms of the isomorphism H- d A Ie' ~ £'. In this way, for each F-stable unipotent class C' and F -stable G-equivariant irreducible local system £' on C', the natural mixed structure is defined. 2.9. We return to the situation in 2.7, and let Ao = IC(E,£)[dimE] be the cuspidal character sheaf on L. Let £0 = Ql E S(T) be the constant sheaf on T, and assume that A o E £.c o ' Then A satisfies the the former part of the condition (2.7.2) by Lemma 5.9 in [S2, I]. Furthermore, in view of [L2, (8.5.13), (8.5.3]] and [S2, I, (5.16.1)], we see that W£ = W = Nc(L)/ L. Now, for each w E W, choose a representative w E Nc(L), and take a E G such that a-I F(a) = F(W). We set L w = aLa-I, E w = aEa- 1 and £w = ad(a- 1 )*£, a local system on E w. For a given isomorphism 'Po : F* £ .c:::.£, one can construct an isomorphism ('Po)w : F*£w.c:::.£w as in [L4, II, 10.6], which induces 'Pw : F* Kw.c:::.Kw, where K w is a complex on G induced from the pair (Ew,£w) on L w. Note that L w is different from L W defined 1 in [loco cit.], i.e., we have L w = LF(w- ). Let W~ be the set of (isomorphism classes) of irreducible representations of W extendable to W<'Y>. For each E E W~, we choose an extension E of E so that E is defined over Q. Let A = AE be the character sheaf in G.co corresponding to E E W~. The mixed structure 'PA : F* A.c:::.A is determined by the requirement VA = E. We denote by XA the characteristic function of A with respect to 'PA. Then we have the following formula. (2.9.1)
XKw,'Pw =
L
Tr (,w, E)XAE
(w E W),
EEW~
by [L4, II, 10.4, 10.6]. (See also [S2, I, 5.17] for the inconsistency of the notation of Lusztig. Also note that K w is written as K W in [loco cit.,
I]).
Unipotent Characters
379
3. Split elements of classical groups 3.1. Let (C',£') be as in 2.8. The mixed structure on £' introduced in 2.8 was made explicit in [L5, 3.4] in the case where G is adjoint simple and F is of split type. Here we consider a more general situation such as G is the product of classical groups. For this, we need to choose some particular extension VA for each irreducible W-module VA in 2.7. So, let W be a Weyl group ~nd u be the Weyl group automorphism of order c. We denote by W the semidirect product W, where is the cyclic group of order c generated by u. Let E E W~. By modifying the preferred extension of Weyl groups given in [L4, IV, 17.2], we define a specific extension E of E, which we call a "good" extension, as follows. First assume that W is the Weyl group of type Cn or D n . If c = 1, the extension is trivial. We assume that W is of type D n and c = 2. Then irreducible representations of W, stable by u, are parametrized by unordered pair (a, (3) of partitions such that a =I- {3. By allowing 0 in the entries, we can express it as a : 0 :::; al :::; .. , :::; am, {3 : 0 :::; {31 :::; '" :::; {3m for some integer m ~ 1 such that E ai + E {3j = n. On the other hand, W is isomorphic to the Weyl group of type C n , and irreducible representations are parametrized by ordered pairs of partitions. For a given irreducible representation Eo<,/3 corresponding to (a,{3), we choose an extension Eo<,/3 E WII such that a > (3 with respect to the (downwards) lexicographic order of partitions (e.g., a > {3 if am > (3m). In a description of Eo<,/3 in terms of symbols (see 5.1 for the notation), this is equivalent to the following; let A = (~) E <J)~, lSI> ITI, be the symbol corresponding to Eo<,/3. Then the largest entry which appears only in one row appears in S. Note that our extension is different from the preferred extension in [L4, IV, 17.2], which is defined by the property that the smallest entry which appears in only one row appears in T. Next consider the case where W = WI X W2, with Wi a Weyl group of type C or D. The irreducible representation E of W is given by the external tensor product E I ~ E 2 , where E i is an irreducible W i module. If u stabilizes each Wi, the good extension E of E is defined as E = E I ~ E 2 , where E i are good extensions of E i . Assume that u permutes WI and W 2. We set WI = W 2. Then u : WI ---t WI is given by U(WI,W2) = (¢(W2),¢(WI)), where ¢ : WI ---t WI is an automorphism of the Weyl group. Since E = E I ~ E 2 is u-stable, we see that there exists a linear isomorphism hI : E 1 ---t E 2, h 2 : E 2 ---t E I such that h l h2 : E 2 ---t E 2 give the good extension E2 of W. We define a good extension E of E by
u(el Q9 e2) = h 2(e2) Q9 h1(el),
(ei
E E i ),
380
T. Shoji
(d. [L4, IV, 17.2, (e)]). 3.2. We consider the group G of the following type. (3.2.1) G is a quotient of G 1 x G 2 by its central subgroup, where G i is isomorphic to SPN or SON.
We shall choose a specific unipotent element u E C F , which is called a split element, for each F-stable unipotent class C in G. In general, if G is adjoint simple and F is of split type, a split element u is defined as the element such that F stabilizes each irreducible component of the variety of Borel subgroups of G containing u. Such an element exists, unless G is of type E s and q == -1 (mod 3), and is unique up to GF-conjugacy, (d. [BS], [SI]). In the case where G is a classical group, split elements are explicitly determined in [SI, 3.3], where they are called distinguished elements. Here we recall them and define split elements for SPN or SON. First we prepare some notation. For each even integer N ~ 2, let X N be the set of partitions>' = (F 1 2r2 ••• ) of N such that ri is even for odd i. Then the set of unipotent classes of SPN is in bijection with XN through Jordan's normal forms. Next, for any integer N ~ 1, let X N be the set of partitions>. = (F 1 2r2 •.• ) of N such that ri is even for even i. Then the set of unipotent classes of ON is in bijection with X N. Note that each unipotent class in ON gives rise to a unipotent class in SON except the case of the partition (2 r2 , 4r4 , ••• ), where the unipotent class in ON splits into two unipotent classes in SON' Let G = SPN. Then for a unipotent element u = u>. E G corresponding to >. E X N , Ac(u) = Zc(u)/zg(u) is isomorphic to (Z/2Z)t, where t is the number of even i such that ri =I- O. For such i, we denote by ai the corresponding generator of Ac(u). On the other hand, in the case where G = SON, we set G = ON. Then Ac(u) is isomorphic to (Z/2Z)t, where t is the number of odd i such that ri =I- O. Ac(u) is a subgroup of index 2 consisting of elements a = IT a~i such that Lei == 0 (mod 2), where ai is the generator of Ac(u) defined similarly to the case of SPN. For a given>. E XN (resp. >. E X N), we define the set T>. of sequences {3 = ({3i), where (3i = ±1 is attached to each generator ai E Ac(u>.) (resp. ai E Ac(u>.)), respectively. Let G = SPN or SOn, and take an F-stable unipotent class C in G. Then as is discussed in [SI, 3.1], the GF-conjugacy classes in C are uniquely determined by giving {3 E T>. for u>. E C. (For a given u' E C F , we put {3i = +1 (resp. -1) if the quadratic form associated to ZC(u')F corresponding to the row of length i is split (resp. nonsplit)). We call {3 the type of u' in CF. Now we arrange the set {aj I rj: odd} as {ajlla12""} so that j1 ~ 12 ~ "', and define an
Unipotent Characters
381
integer Cjk = [jk/2] + k. The following formula determines the split elements of adjoint groups of split type. (3.2.2) ([81, 3.3]) Assume that F is of split type. Then {3 = ((3j) corresponds to a split element in G / Z (G) if the following condition is satisfied: {3j = 1 if r j is even, and {3jk = C (resp. (3jk = c( -1 )jk) if r jk is odd and q == 1 (mod 4), (resp. q == -1 (mod 4)), respectively. Here C = ± 1 is a constant independent of j. We now define a split element u E C F by the condition in (3.2.2) in the case where G = SPN or SON and F is of split type. Hence for a given C, there exists two classes of split elements according to the case c = 1 or -1. We call u a positive (resp. negative) split element if c = 1 (resp. C = -1). We fix a split element u = u>. E C F . Then the component group Ac(u) (resp. Ac(u)) acts on the set T>. in such a way that Oi . {3j = {3j, (j =I- i), ai' {3i = -{3i for {3 = ({3i) E T>.. 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
m' ~ 2 dim Zu, we have H;:" (ZU, F) ~ H;:'~ (ZUll Fd Q9 H;:'; (ZU2' F2 ), where m~ and Fi are the corresponding objects forZui • Then it is easy to see that the Frobenius action on H;:" (ZU, F) induces qm' /2 times a A on a submodule isomorphic to VA' The case where G is a quotient of G 1 x G 2 by a central subgroup is easily deduced from the above case. 4. The evaluation of certain class functions 4.1. In this section, G will denote an adjoint simple group of type B n , C n or D n . Here we include the group of non-split type in our consideration, though it is not used later. We consider the complex K defined as in 2.9 with a fixed F q-structure
4.2. Let T, B, Land P be as in 2.7. We assume that L has the Dynkin diagram of the same type as G. Let W = Nc(T)/T be the Weyl group of G, and let W L be the Weyl subgroup of W corresponding to L. We fix a semisimple element sET such that s is isolated in G. Let W s be the reflection subgroup of the stabilizer of s in W which is a WeyI group of zg (s), and let Z s be a coset of W s in the set Z~ = {w E W I Fw(s) = s}. So, Zs is expressed as Zs = w1Ws for some WI E W. Here one can choose WI so that FWI : W s - W s leaves invariant the set of simple roots of zg (s) determined naturally from
384 the pair (T, B). Such a choice of WI is unique. Let Ws,L We note that
T. Shoji
= W s n W L.
(4.2.1) if wE W normalizes Ws,L, then W normalizes W L . In fact, let
Unipotent Characters
385
subgroups conjugate to L s are in 1 - 1 correspondence with F'-twisted conjugacy classes of W s '
4.4. Let us choose an element go E E, and write it as go = souo = uoso, where So is semisimple and uo is unipotent in L. Then we have the following. (4.4.1) For x E G, the condition that x-lsx E E l is equivalent to the condition that Lx = xLx- 1 n H is a Levi subgroup of H which is conjugate to ZZ(so) under G. In fact, assume that x-lsx EEl. Since s E Lx, Lx is a Levi subgroup o'f H. Moreover, since x-lsx EEl, x-lsx = zy-ISOY for some z E ZO(L), y E L. Hence
Conversely, assume that Lx satisfies the latter condition in (4.4.1). Since Lx is a Levi subgroup of H, s is contained in Lx, i.e., x-lsx E L. On the other hand, ZZ(x-lsx) has the same semisimple rank as ZZ(so). It follows that x-lsx is an isolated semisimple element in L. Now it is easily verified for a Levi subgroup L as in 4.2 that, two isolated semisimple elements having isomorphic centralizers are conjugate in its adjoint group. This implies that x-lsx == y-ISOY (mod ZO(L)) for some y E L, and so we have x-Isx EEl' We now prove the following lemma.
Lemma 4.5 Let 9 = su = us be the Jordan decomposition of 9 E GF', and let g' = s'u' = u's' E GF with u' = {3urrl, s' = (3s{3-I. Then for each wE W, we have the following. If s' is GF -conjugate to some element in E l , then we have
yEW
y-1wlTyEW.F(willT
where z = y-lwF(ywil) E W s in the sum, and (Cs,z,Fs,z) is a cuspidal pair on a Levi subgroup Ls,z of H, associated to z E W s with some mixed structure 'Pz : (F')* Fs,z.':.::4Fs,z. If s' is not conjugate to any element in E I , then XKw,'P,js'u') = O.
Proof. The second statement is clear from Theorem 3.6. So, we assume that s' is G F -conjugate to some element in E l . We denote by X the
left hand side of the formula in the lemma. It follows from Theorem 2.6, we have
xEC F
x-ls'xE(Ewh where Lw,x = xLwx- 1 n zg(s'). Put s~ = {3so{3-1 for So as in 4.4. Then by (4.4.1), we see that the condition in the sum is equivalent to the condition that Lw,x is a Levi subgroup of H' conjugate un(s~). We note, for a given L', that the mixed strucder G to ture (Cx,Fx,'Px) is mutually isomorphic for any x E GF such that xLwx- 1 = L'. In fact, assume xLwx- 1 = yLwy-1 with x, y E GF. Then y = xn for some n E Nc(Lw)F. It is clear that C x = Cy since the unipotent class carrying a cuspidal pair is unique. Moreover, since (adn)*£w~£w for n E Nc(L w), we see that F x c::: F y • Now, for v E C!" Tr ('Px, (Fx)v) = Tr ('Pw, (£w)x-lsvx)' But in our case, £w is written as p* F w for a local system F w on p( E w), (p is the natural map from E w to Ew/ZO(L w)), and 'Pw is induced from ('Pdw : F* Fw~Fw. On the other hand, the conjugation action n of n on L w = Lw/ZO(L w) induces an isomorphism (adn)* Fw~Fw, which is compatible with ('Pdw : F* Fw~Fw. It follows from this that the traces of ('PI)W of F w at the stalks p(x-Isvx) and p(y-I svy ) coincides with each other. Thus the mixed structure depends only on L'. We denote the mixed structure corresponding to L' by (C~, F~, 'P~)' Then we can write X as (4.5.1) X =1 H,F I-II L~ I-II Nc(Lw)F I I L': I Qf~,C~,F~,'PJU'),
zt
L L'
where the sum is taken over all L' such that L' is GF' -conjugate to L w and that L~ = L' n H' is G-conjugate to Z2 (s~). We now change the setting from (s', H', F) to (s, H, F') by making use of ad{3-1 : H' ~H. Then L w is transferred to a F'-stable Levi subgroup of G obtained from L by twisting W~IW, since
We denote this F'-stable Levi subgroup by L W -lw. Then one can l rewrite (4.5.1) as (4.5.2) F' I-II Nc (L w1 l w)F' ILJ " I L s/IF'I QL';,C~"F~"'P';(U H ), X = I H F'I-II L w1lw £"
387
Unipotent Chamcters
where the sum is taken over all F'-stable Levi subgroups L" of C which is C F' -conjugate to L W -1 w and that L~ = L" n H is C-conjugate to 1 -1 (so). Moreover, C~ = (3-1C;{3, F;' = (ad{3)* F; and 'P~ =
z2
W
1
w
(ad{3) * ('P~). Note that the map L" -. L" n H, from the set of Levi subgroups of C containing s and conjugate to L, to the set of Levi subgroups of H conjugate to L s , is a bijection. In fact, if L" = gLg- 1, then sand g-1 sg are both isolated elements in L. Hence there exists h E L such that h- 1g- 1sgh = zs for some z E Z(L). By replacing 9 by gh, we may assume that g-l sg = zs. Then L"nH = g(LnH)g-l. But in this case, it is easily checked that if two Levi subgroups in H are conjugate in C, then they are already conjugate in H. This gives a map of one direction. The map of the other direction is naturally constructed. Hence we get the bijection. As discussed in 4.2, the set of HF' -classes of P'stable Levi subgroups of H conjugate to L s is in bijection with the set Wsl ~F' of P'-twisted classes of W s . Similarly, the set of C F' -classes of pi-stable Levi subgroups of C conjugate to L is in bijection with the set WI ~F'. By 4.2, we have a natural map Wsl ~F'-' WI ~F'. This map is compatible with the map L" -. L" n H, i.e., the map Ls,w' f--t L w' (w' E W s ) gives the inverse of L" -. L" n H. Since w',w" E Ware F'-twisted conjugate if and only if W'F(Wl)U and w"P(Wl)U are W-conjugate, (4.5.2) implies that
X
=1
LF~1 WI
W
1-1
F'
~
I Ls.w' II
L.J
X
INc(L WI-1 w)F'1 x NH(Ls,w')
F'
-1
I
H
QLs.w"Cs,w"Fs,w"'Pw'(u),
w'EWs/~F'
w' F(wllO'~wO' where w'P(Wl)U ~ wu in the sum means the conjugation by W on Wu. Here we used the notation such as C s.w" Fs,w', etc. to denote C~,F;', etc. corresponding to L~ = Ls,w" Then we have, X
=1 LF~1 w 1-11 WI
Nc(L W -1 W )F' l
~ L..J
I
I L~,~, I X
w'EW8
w' F(W1 )O'~wO' x
I NH(Ls,w,)F' 1-11
ZW s(W'P(Wl)U)
II W 1-1 Qfs,w,,cs,w,,Fs,w"'Pw'(U)' S
where Zw s(w' P(Wl)U) is the stabilizer of w' P(Wl)U under the action of W s on the coset WsF(wdu. But since NH(Ls,w' )F' I L:~, -::::= Zw s (W'P(Wl)U), the last formula is equal to
I L~~1w 1-11
Nc (L w, 1w)F'
II W 1-1 S
L w'EWs
w' F(wllO'~wO'
Qfs,w,,cs,w,,F.,,w"'Pw' (u).
388
T. Shoji
Now using Nc(L WI-1 W l' /LF~, c::: Zw(wu), we have WI W
YEW
y- 1 W<7YEW.F(w,)<7
where z = y-IwF(yw~l) E W s . This proves the lemma.
o
4.6. In order to describe the value XKw,'Pw(S'u'), we need to make the isomorphisms 'Pz involved in Lemma 4.5 more explicit. But before doing so, we prepare some general facts on the splitting of conjugacy classes. In this subsection, we consider G as a connected algebraic group with Frobenius map F, and let 91 = SIUI = UISI E G F be the Jordan decomposition as before. We set H = Zg(SI) and H = ZC(SI)' Let A c (91) = ZC(91)/Zg(91) and AH(UI) = ZH(UI)/Z~(UI)' Since Zc(91) = Zjj(uI) and Zg(91) = Z~(UI), the group AH(UI) may be regarded as as subgroup of A c (91)' We consider the coset decomposition A c (91) = IJi AH(uI)ai. One can choose a representative ai in Zjj(uI) for each ai E A c (91). For a given a = ai, choose {3 E G such that a = (3-IF({3). Let 9' = (39{3-1 E G F , and define s',u',H' in a similar way as 9' by taking the conjugation of SI,UI,H, respectively. Then H,F -conjugacy classes contained in the H'-conjugacy class H' . u' of u' are in bijection with the F -twisted classes of AH' (u'). However by the map ad{3, the latter set is transferred to the set of aF-twisted classes of A H ( UI)' Hence the set of G F -conjugacy classes contained in the G-conjugacy class G . 91 of 91 is parametrized by the union of various H"'i F -conjugacy classes contained in the H -conjugacy class H . UI of UI' Under this correspondence, the H"'i F -conjugacy class in H . U corresponding to z E AH (UI) is attached to the G F -conjugacy class in G . 9 corresponding to zai E A c (91 ). 4.1. Returning to the setting in Lemma 4.5, we shall consider the isomorphism 'Pz : (F'y Fs,z.'::.::4Fs,z given there. Let eeL be the F-stable conjugacy class as given in 4.3. We choose a specific representative 91 E C F as follows; let 91 = SI UI be the Jordan decomposition of 91, where SI is semisimple and UI is unipotent. Then ZL(sd is a central product of two classical groups. We choose SI so that F leaves each factor of Z L (s I) invariant and acts as a split Frobenius map on it. (If G is of non-split type, we only assume that F leaves each factor of Z~ (s I) invariant.) This determines S I up to L F -conjugacy, in the case G F is of split type. Then we choose U I E (s Il a split element as defined in 3.2 rv 3.4, (note that Z~(SI) is the same kind of groups
Zl
Unipotent Chamcters
389
considered in section 3). We fix an isomorphism 'PI : F* F ~F by the requirement that (4.7.1) the linear map 'PI : F g1 - F g1 on the stalk F g1 of F at gI E C F induced from 'PI : F* F ~F coincides with q(dimL-dimC)/2 times identity. Note that 'PI induces an isomorphism 'Po : F* £ ~£ satisfying the assumption (2.7.1). Let PI be an irreducible character of AL (gd corresponding to the local system F on C. Since AL(gd is abelian, and F acts trivially on AL(gl), the trace of 'PIon F is described as follows. (For each a E AL(91), we denote by ga the element in C F obtained by twisting gl by a, i.e., ga = (3g(3-I for (3 E L such that iJ, = (3-1 F((3)).
(4.7.2) to gao
Tr('PI,Fx ) = PI (a)q(dimL-dimC)/2
if x is LF_conjugate
Next we consider the F'-stable subgroup L s , and the cuspidal pair (C S , F s ) in L s , where C s is a unipotent class in L s . We choose a split element VI E Cr and define a standard mixed structure 'ljJs : (F')*Fs~Fs by the condition that (4.7.3) the linear map 'ljJs : (FS )V1 - (FS )V1 on the stalk at VI E Cr induced from 'ljJs : (F ' )* Fs~Fs coincides with q(dim L.-dim C. -dim
ZO(L.))/2
times identity. For each Z E W s , we consider the cuspidal pair (Cs,z, Fs,z) on Ls,z as before. Then we can define the standard mixed structure 'ljJz : (F ' )* Fs,z~Fs,z inherited from 'ljJs : (F ' )* Fs~Fs as in [L4, II, 10.6]. We now assume that s' belongs to a class of Sl corresponding to a E AL(gl)' Then we may assume that s' = Sl and that F ' = iJ,F for such a. Therefore, by the parametrization in 4.6, one can attach to the element s' v~ = (3SVI (3-1 (or more precisely its image in L) an element ao E AL(91). Under this setting, we have the following lemma. Lemma 4.8 Let 'Pz : (F' )* Fs,z~Fs,z be the isomorphism induced from 'PI : F* F ~F as in Lemma 4.5. Then we have
T. Shoji
390
,-1
Proof. We choose, E H such that F'b) = F'(z), where z E NH(L s ) is a representative of z E W s . Then Cs,z = ,Cs,-1 and take v = ,vn- 1 E C{~. From the construction of the map 'ljJz, we have
On the other hand, it follows from the construction of 'Pz, we have
But the last number is equal to implies the lemma since
Pl(aO)q(dimL-dimC)/2
by (4.7.2). This
dimL - dimC = dimZL(s'vD = dimL s - dimCs - dimZo(L s ). 0 4.9. We consider the complex K induced from F on C c L as before. The mixed structure 'PI : F* F ~F defined in (4.7.1) induces the mixed structure 'Pw : F* K w ~Kw for each w E W. For each E E W"', we have a character sheaf AE which is a direct summand of K. The mixed structure 'PAE : F* AE~AE is determined uniquely from 'P : F* K ~K by the choice of an extension £ of E to W (d. 2.9). Here we determine 'P AE by choosing the good extension £ of E in the sense of 3.1. We denote by XAE the characteristic function of AE with respect to 'PAE' Then by (2.9.1), we have
(4.9.2)
XKw,'Pw =
L
Tr (uw, £)XAE
(w E W).
EEW.;x
Next, we consider the complex K s E MH induced from the cuspidal pair (Cs,Fs ) of L s . The endomorphism algebra EndMHKs is isomorphic to the group algebra Ql [Ws ] (d. 2.8) and the character sheaves of H appearing in K s are written as AE 1 , (E 1 E W;'). Now the mixed structure 'ljJs : (F't Fs~Fs determined in (4.7.3) induces the mixed structure 'ljJs : (F't Ks~Ks, and induces 'ljJz : (F't Ks,z~Ks,z, where Ks,z is the F'-stable complex on H obtained from K s by twisting by z E W s , in a similar way as K w . Since the action of F ' = FWI on W s is the conjugation action of UWl, the automorphism, = (F ' )-1 on W s (d. 2.7) is the conjugation by (UWl)-1 on W s' Thus we may identify the group W s<,> in 2.7 with the subgroup Ws of W generated by W s and UWl. For each E 1 E (Ws)~, we choose the good extension £1' This determines the mixed structure of A E1 , and we
Unipotent Chamcters
391
denote by XAE 1 the characteristic function with respect to this mixed structure. Then it follows from (2.9.1) that, XK.,.,1/Jz
L
=
Tr hz, EI )XAE 1.
EIE(W.)~
Now, the restriction of XK.,.,1/J% on the set of unipotent elements in H F' is nothing but the generalized Green function Qf.,%,C.,.,F.,.,1/J%' and the relationship between two maps 'Pz and 'ljJz are given in Lemma 4.8. Hence we have
<,
We now define, for each W s >-module VI, V2 , an inner product < VI, V2 >, by the following formula, < VI, V2
>,= IWsl- I
L
Trhw , VI) Tr(hw)-I, V2).
wEW.
Then the following corollary is easily deduced from Lemma 4.5, (4.9.2) and (4.9.3). Corollary 4.10 XAE(S'U')=PI(ao)
L
<EI,w.,EI>,XAE1(U),
EIE(W.)~x
5. Combinatorics on classical groups 5.1. In [L1], unipotent characters of classical groups are parametrized in terms of symbols. So, first we recall the definition of symbols. A symbol is an (unordered) pair (~) of finite subsets ofN = {O, 1,2,"'}, modulo the equivalence relation generated by the shift operation (~) rv (~:), where S' = {O}U(S+ 1), T' = {O}U(T+ 1). The rank of a symbol A = (~) is defined by
where for any real number z, [z] means the largest integer n such that n :S z. The defect of A is defined by d(A) = the absolute value of ISI-
T. Shoji
392
ITI. The rank and the defect are independent of the shift operation. We denote by ~ the set of symbols of rank n and defect d ~ 1. In the case of symbols of defect 0, a symbol A = (~) is said to be degenerate if S = T, and is said to be non-degenerate if S =I- T. We denote by ~ the set of symbols of rank n and defect 0, where degenerate symbols are counted twice. Let n be the set of symbols of rank n and odd defect. We define also ;t, ~ as follows. if.+ _.i.o
'*'n - '*'n
Il
if.d
'*'n'
~ =
Il
~.
d=2 (4)
d=O (4)
Then by [LI], unipotent characters of finite classical groups C F of type B n or en (resp. D n ) are parametrized by n, (resp. ;t or ~ according to the case where F is of split type or non-split type). Let W n be the Weyl group of type B n , and W n be the Weyl group of type D n which is a subgroup of W n . Then we have natural bijections
(5.1.1) as follows: Each irreducible representation E of W n is parametrized by an ordered pair of partitions (a, (J) of n. Here by allowing 0 on the entries of a, {3, we may assume that the partitions a, {3 have the following form. a : 0 :S: ao :S: a 1 :S: ... :S: a p , {3 : 0 :S: {31 :S: {32 :S: ... :S: {3p for some positive integer p such that L: ai + L: {3j = n. Then we define S = p,o, >'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 <
< O:p + 2p} < {3p + 2p - 2}
Then (~) E 1JI~~+I' and this gives a bijection 8'. We need also a bijection 8' : IJI~~ which is defined in a similar way as above, but first choose (0:, (3) such that 0 ::; 0:1 ::; 0:2 ::; ... ::; O:p,O ::; {31 ::; {32 ::; ... ::; {3p, and set A = {O:i + 2i - 2}, B = {{3i + 2i - 2}. Now, in view of (5.4.5), the generalized Springer correspondence is given by .T,' 1 d Wf'I .T,'O Hr h t he b IJectlOn Wf'I (N-d2)/2 ----; '.i' N-d2+1 an n ----; '.i'2n' He ave
W; ----;
oo
•
(5.7.4) ([L3, Th. 13.3]) The generalized Springer correspondence between WDV-d 2 )/2 and 1JI~_d2+1 is given by 8'(and by 8' for W; ----; '0 ) . 1J1 2n
Remark 5.8. The statement of (5.7.3) is not exactly the same as the one given in Theorem 12.3 in [L3]. Actually, Theorem 12.3 contains some error and it should be corrected in the form as above. In fact, by the argument given in the proof of Theorem 12.3, the determination of the generalized Springer correspondence is reduced to the case where m = 2. Let (Co'£o) be the cuspidal pair on L. We assume that LjZO(L) is isogeneous to Spj(j+l), and assume that Co corresponds to >. = (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,n or ;t, for which the constant (A such that RA = (AXAII,
400
T. Shoji
Remark 6.3. The similar results also hold for the case where G is a non-split group of type D n , which will be discussed elsewhere. 6.4. First we note that the proof of Theorem 6.2 is reduced to the case where AA is a cuspidal character sheaf. In fact, assume that A E GCo appears in the decomposition of K = ind ~Ao, where A o E Lc o is cuspidal. Then as discussed in 2.9, A = AA with A = A(E) for E E WI\. The mixed structure
6.6. The proof of Theorem 6.2 is easily reduced to the case where G is an adjoint simple group. 80, we assume that G = PSP2n, S02n+l or PS02n, with odd p. Moreover, in view of Lemma 6.5, we may assume that G contains a cuspidal character sheaf A o E Gco. Hence n (resp. ;t) contains the cuspidal symbol Ao, and so we have n = d2 + d (resp. n = 4d2) in the case where G is of type B n or C n (resp. D n ), respectively. Now cuspidal character sheaves on G are listed in [L4, V, 23.2]. We choose a specific cuspidal pair (C,£) for each G as follows. (a) G = PSP2n with n = d2 + d. C is a conjugacy class of G containing 9 = su = us, where s is an isolated semisimple element in G such that H = Z~(s) ~ (Spd2+d x SPd2+d)/{±1}, and u is a unipotent element in H such that the unipotent class C 1 containing u gives the cuspidal pair (C 1,£1) on H. Note that (C1,£1) is the unique cuspidal pair having unipotent supports, and it is obtained from the cuspidal pair (Co, £0) on H o = SPd2+d described in 5.6 as follows; C 1 = Co X Co, and if we choose u = (uo, uo) E Co x Co, and Po E AHo(uo)1\ corresponding to £0 on Co, then Por8Jpo E (AHa (UO) X AHa (uo))1\ factors through AH(u) and defines an irreducible character PI E AH(u)l\. £1 is defined as the one corresponding to Pl. Now Ac(s) ~ Z/2Z, and we see that AH(u) is a subgroup of Ac(g) of index 2 (d. 4.6), and so we have Ind~~~~)Pl = p+ p' for p,p' E Ac(g)l\. If we write £,£' the irreducible local systems on C corresponding to p, p' respectively, the
Unipotent Characters
401
pairs (C, £), (C, £/) are both cuspidal pairs of G. We choose p E Ac (g)1\ by the condition that p( a) = 1 for a E Ac (g) - A H(u) such that its representative a E G induces a graph automorphism on H permuting two components. We shall fix a mixed structure on (C,£). We choose s E G F such that F leaves two components of H invariant. Note that this property characterizes s up to GF-conjugacy. Next we choose u E H F a split element in the sense of section 3, and put 9 = su E F . We fix
c
(b) G = S02n+l with n = d2 + d. C is a conjugacy class of G containing 9 = su = us, where s is an isolated semisimple element in G such that H = Z~(s) ~ SO(d+l)2 X SOd2, and u is a unipotent element in H such that the unipotent class C 1 containing u gives the cuspidal pair (C 1, £1) on H. Then (C 1, £1) is the unique cuspidal pair on H having unipotent supports, and it is of the form C 1 = Co X Co, £1 = £0r8J£0, where (Co, £0) and (Co, £0) are cuspidal pairs of SO(d+l)2 and SOd2 described in 5.6. The irreducible character PI E AH(U) corresponding to £1 is determined as the external tensor product of two characters corresponding to £0, £0 from 5.6. Since Ac(s) ~ Z/2Z, AH(U) is an index two subgroup of Ac(g). Hence we have Ind~~i~)Pl = p + pi for p, pi E Ac (g)l\. If we write £, £1 the irreducible local systems on C corresponding to p,p', respectively, the pairs (C,£), (C,£/) give the cuspidal pairs of G. We shall choose p E Ac(g)1\ as follows. If we write H = Ho x H where H o is of type D and H is of type B, AH(U) is isomorphic to AHo(UO) x A H6(un), where U = (uo, un) E H. Then Ac(g) may be identified with Ajf)uo) x AH6 (un), where fIo is an orthogonal group containing H o as a subgroup of index 2. We now choose a E Ac(g) - AH(U) so that a coincides with the element ao E AHo(uo) - AHo(UO) given in 3.3. We define p E Ac(g)1\ by the condition that p( a) = 1. We fix a mixed structure of (C, £). We choose s E G F such that F acts as a standard F'robenius map on H, and take a split element U E HF. Then put 9 = su E C F . We fix <po : F*£~£ by the condition that (<po)g : £g ----; £g coincides with q(dimC-dimC)/2 times identity.
o,
o
(c) G = P S02n with n = 4d2. C is a conjugacy class of G containing 9 = su = us, where s is an isolated semisimple element in G such that H = Z~(s) ~ (S04d2 x S04d2)/{±1}, and U is a unipotent element in H such that the unipotent class C 1 containing U gives the cuspidal pair (C1,£I) on H. Then (C 1,£I) is the unique cuspidal pair on H having unipotent supports, and it is obtained from the cuspidal
T. Shoji
402
pair (Co, £0) on H o = S04d2 described in 5.6 as follows; C I = Co X Co, and we choose u = (uo, uo) E Co x Co. We choose Po E AHo(UO)'" corresponding to £0 on Co. Then Po I:8l Po E (A Ho (uo) X AHo(uo))'" factors through AH(U) and define a character PI E AH(u)"'. £1 is defined as the one corresponding to Pl. In this case, Ac(s) ~ Z/2Z x Z/2Z, and Ac(g)/AH(U) ~ Z/2Z x Z/2Z. Hence Ind~~~~)PI decomposes into 4 irreducible characters of Ac(u), all of which give the cuspidal pairs on C. We choose a specific pair (C, £) as follows; AH(u) is isomorphic to the quotient of AHo(uo) X A Ho (uo) by a subgroup < c > of order 2, and we may regard AH(U) as a subgroup of (AHo(uo) x AHo(uo))/<.c>, where fIo = 04d2. Then Ac(g)..::.AH(U) x , where b permutes two factors of A H(u), and a = (ao, ao) mod <.c> in A Ho (uo) x A Ho (uo) with ao E AHo(uo) -AHo(UO) as in 3.3. Then we choose P E Ac(g)'" by the condition that p(a) = p(b) = 1. Let (C, £) be the cuspidal pair corresponding to p. We fix a mixed structure of (C, f). We choose s E G F such that F acts on H as a standard Frobenius map, and choose a split element u E HF and put 9 = su E C F . We fix
(i) Ao = IC(C,£)[dimC], where (C,£) is the cuspidal pair given in 6.6. (ii) Under the mixed structure
RAo(x) = {
p(a)q(dimC-dimC)/2
if x = ga E C F ,
o
if x ~ C F ,
where ga is a representative of G F -conjugacy classes of C F corresponding to a E A c (g), (here 9 E C F is a canonical representative given in 6.6). 6.9. The remainder of this section is devoted to the proof of Proposition 6.7. Let G be as in the proposition. By induction on the rank of G, we may assume that Proposition 6.7 holds for smaller rank cases. Let L be
Unipotent Characters
403
an F-stable Levi subgroup contained in an F-stable parabolic subgroup P of G such that LcD contains a cuspidal character sheaf A~. We fix a mixed structure ~ Ai> : F* A~ .:;A~ as in (ii) of Proposition 6.7 (applying for L). Then for each AE appearing in ind ~A~ corresponding to E E W!\ , the isomorphism ~AE : F* A E .:;A E is determined. We denote by XAE the characteristic function of A E with respect to ~AE' Let Fa be the cuspidal family in n or ~ as in 5.1. Then Fa may be expressed explicitly as follows; G : type B n or Cn
Fa = {A = (:)
I 5uT = {O, 1, ... ,2d}, 5nT = 0, 151-ITI =
odd, 2: I}
G: type D n
Fa = {A = (:)
I 5uT = {O, 1, ... ,4d-1}, 5nT = 0, 151-ITI = even,
2: O}
We now define a subset V of Fa according to the case B n , Cn or Dn as follows; if G is of type B n , we define V as the set of symbols A = (~) in Fa such that 2i and 2i + 1 are simultaneously contained in 5 or T for any i, (0 ~ i ~ d - 1). If G is of type Cn, we define V as the set of symbols A = (~) in Fa such that 2i - 1 and 2i are simultaneously contained in 5 or T for any i, (1 ~ i ~ d). If G is of type D n , we define V as the set of symbols A = (~) in Fa such that 2i and 2i + 1 are simultaneously contained in 5 or T for any i, (0 ~ i ~ 2d - 1). For example,
It is easy to see that
(6.9.1 )
G : type B n or Cn G: type D n .
6.10 We now consider g' = s'u' E C F as in Lemma 4.5. Assume that we are in a situation as in 4.3, Le., we consider the cuspidal character
sheaf A~ E Leo, and K = ind ~A~ on G. We assume that L is a proper subgroup of G containing s, and let AA be the character sheaf in Ceo appearing in the decomposition of K. The following lemma is crucial for the proof of Proposition 6.7. Lemma 6.11 Let A E Fo - {A o}. Then
if A E V if A E Fo - V, where Pl(ao) is as in Corollary 4.10 and rnA = dimH - dimC l dimZO(L s ), (here H = Z~(s) and C 1 is the unipotent class of H containing u. L s = H n L). 6.12. We shall prove the lemma by making use of Corollary 4.10. First we consider the case wher G = PSP2n. Let A E Fo n ;'t+l for some integer t, (0 ~ t < d). Then AA = A E for some E E WI\. Here W = Nc(L)/L, and L is a Levi subgroup of type Ct 2+t. Hence W ~ W n - Ct 2+t). Moreover, L s is isogeneous to SPt2+t x SPt2+t modulo center, and W s ~ W Cn - t 2-t)/2 X WCn-tLt)/2' Each E 1 E W~ is expressed as E 1 = Ei r8JEi' for Ei, Ei' E WC~-t2-t)/2' For each E 1 E W~, we denote by C(E 1 ) the unipotent class in H corresponding to E 1 under the generalized Springer correspondence. We shall prove the following. (6.12.1) (i) If A E F o - V and E 1 appears in the decomposition of Elws ' then C(E 1 ) 1J C 1 • (ii) If A E V, there exists a unique E 1 appearing in the decomposition of Elws such that C(Ed ~ C 1 • In this case C(Ed = C 1 • Moreover, if we write E = E o ,{3, 0: : 0 ~ 0:0 ~ 0:1 ~ .,. ~ 0:2m,13: 0 ~ 131 ~ 132 ~ '" ~ 132m for some rn, then we have 0:2i = 0:2i-l,132j = 132j-l, (0 < i,j ~ rn), and E 1 = Ei r8J Ei' is given by Ei = Ei' = E O ',{3" where 0:' : 0 ~ 0:0 ~ 0:2 ~ .. . ~ 0:2m, 13' : 0 ~ 132 ~ 134 ~ .. . ~ 132m
We show (6.12.1) by induction on d. The case d = 1 is easily verified. We assume that (6.12.1) holds for d-1. Now A E Fon;'t+l is written as (6.12.2)
A = (S) = T
(AO < Al < /l1 < /l2 <
< Ad+t) . < /ld-t
Then E = E o ,{3 E WI\ such that A == A(E) is determined by the correspondence (5.1.1) and (5.1.2). In particular, if we write 0: : 0 ~ 0:0 ~ 0:1 ~ ... O:d+t, 13 : 0 ~ 131 ~ ... ~ 13d+t, we have O:d+t = Ad+t (d + t), 13d+t = /ld-t + t + d + 1. Since Ai, /lj E {a, 1, ... , 2d} and all of them are distinct, we have
405
Unipotent Chamcters
(6.12.3) O:d+t ~ d - t, (3d+t ~ d + t + 1 and one of the two inequalities is a strict inequality. Now take E 1 E W: such that C(E 1 ) ~ C 1 . C(E 1 ) is written as C(E 1 ) = Cb x C~, where Cb, C~ are unipotent classes in SPd2+d such that Cb ~ Co, C~ ~ Co. We write the partition of d 2 + d corresponding to Cb by Cb ...... (Fl, 2T2 , •.. , k Tk ) E X d 2+d' Hence, by (5.5.1) Cb determines a similarity class R' in IJ1 d2+d' By the generalized Springer correspondence given in (5.7.3), E 1 corresponds to ()' E R' n 1J1~2+d where j' = t + 1 if t is even and j' = -t if t is odd. We write E' = E O ',{3" where (0:', (3') is a pair of partitions of (n - t 2 - t) /2. We express them as 0:' : O:T ~ O:T -1 ~ ... ,{3' : {3T ~ {3T -1 ~ .... Then the explicit computation of the generalized Springer correspondence (5.4.3), (5.5.1) and (5.7.3) implies the following. (6.12.4) Either O:T
= [~] -
t or {3~
= [~J + t + 1 holds.
We now assume that E 1 appears in the restriction of E to W s . This implies that O:T ~ O:d+t and {3~ ~ {3d+t. Then we have (6.12.5) In the expression Cb ...... (I Tl , 2T2 , •.• ,kTk ) E X d 2+d, we have k = 2d, r2d = 1. The similar statement holds also for C~.
In fact, if k > 2d + 1, (6.12.4) contradicts (6.12.3). If k = 2d + 1, then rk ~ 2. In this case, (and also in the case where k = 2d, rk ~ 2), the chase of the generalized Springer correspondence implies that both of the equalities hold in (6.12.4), i.e., we have O:T = d-t,{3~ = d+t+1. This contradicts (6.12.3). Since Cb ~ Co, we have r2d = 1. Now let ()b E R' be the distinguished element corresponding to Cb. Then ()b can be expressed as
()0' _- (>'0 < >'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: <---+ (ITl, 2T2 , ••• , (2d-1 2d - 1 ), and similarly for C~. The following fact is easily verified.
r
(6.12.7) Under the generalized Springer correspondence, respond to the unipotent classes C:, C~, respectively.
E:,E~
cor-
We now return to A E ;'t+l and consider E E W:-Ct2+tl such that A(E) as before. It follows from (6.12.6) that O:d+t = O:d+t-l = d-t or {3d+t = {3d+t-l = d + t + 1 according as the case j > 0 or j < O. This implies that A = (~) in (6.12.2) has the form that {2d - 1,2d} c S (resp. {2d - 1,2d} c T) in the case where j > 0 (resp. j < 0). We define A* = (~:) E ~~/+l by S' = S - {2d - 1, 2d}, T' = T (resp. S' = S, T' = T - {2d - 1, 2d}) if j > 0 (resp. j < 0), where n' = d 2-d = (d-1)2+(d-1), and t' = t-2 or t+2 according asj > 0 or j < O. Then A* E (.1"*)0 n ~~'+l, where (.1"*)0 is the cuspidal family in n' consisting of (~:) such that S' U T' = {O, 1, ... ,2d - 2}. Moreover, the irreducible representation E* E W n '-Ct'2+t'l corresponding to A* is obtained from E by removing O:d+t and O:d+t-l (resp. {3d+t and {3d+t-d if j > 0 (resp. j < 0). We need the following combinatorial result, which is easily reduced to the case of symmetric groups, and is proved by making use of Littlewood-Richardson rule for symmetric groups (d. [M, I, 9]). A
=
(6.12.8) Let E>. E W~, EJ.LrFJEI/ E (Wm x W m , )'" where n = m+m', and>. = (0:, (3), J.L = Cr, 0), v = (1],~) are pairs of partitions of n, m, and m', respectively. We write them as 0: :
O:p
I : Ip
~
~
O:p-l
Ip-l
~
~
. .. ,
{3 : {3p
. .. ,
o : op
~
~
{3p-l Op-l
~
~
, .
Assume that O:p = Ip = a or {3p = op = a. We define pairs of partitions X and J.L' by removing O:p, Ip or {3p, op from>., J.L as follows. If O:p = Ip,
Unipotent Chamcters
407
we set >.' = (0:'; (3) with 0:' : O:p-l ~ O:p-2 ~ ... and J.L' = (,'; 0) with I' ; IP-l ~ IP-2 ~ .... If {3p = op, we set>.' = (0:, (3') with {3' : {3p-l ~ (3p-2 ~ ... , and J.L' = (,,0') with 0' ; Op-l ~ Op-2 ~ .... Then we have
Now one can apply the induction hypothesis to A. by making use of (6.12.8). (i) of (6.12.1) follows easily from this. For (ii), if A E V the uniqueness of E 1 and the fact that C(E 1 ) = C 1 is also clear from the above discussion. Thus (6.12.1) is proved. We shall now prove Lemma 6.11. We can apply Corollary 4.10. In view of (6.12.1), the only non-zero contribution for the right hand side of 4.10 is the term E 1 as in (6.12.1) in the case where A E V. In this case, it is easy to see that < Elw.,E 1 >w.= 1. Now, since u is a split element, we have XAEl (u) = (_I)c qmA /2 by Lemma 3.6, where c = dimC I + dimZO(L s ) (d. 2.8). But dimC I is even and since Ls/ZO(L s ) is isogeneous to SPd2+d x SPd2+d, we see that c == r(H) = n (mod 2). This implies the lemma. 6.13. The proof of Lemma 6.11 for the case of S02n+l or PS0 2n is done in a similar way. We just give the outline of the proof. First consider the case where G = S02n+l. We take A E Fon4>~t+l such that A = A(E) for E E W!\. Here W ~ W n - Ct 2+t) and H ~ SOCd+l)2 X SOd2. We consider the unipotent class C 1 = Co x Cb as in 6.6 (b). Take E 1 E W: such that C(E 1 ) :::> C 1 . Then C(E 1 ) = q x C~ where q :::> Co and C~ :::> Cb. We assume that E 1 appears in the restriction of Eon W s • Then by the similar but more precise argument as in 6.12, one can show that C 2 ...... (ITl, 2T2 , ... , (2d-ly2d-l, 2d+ 1) E X Cd + 1 )2. Then we consider the unipotent class C~ ...... (ITl, 2T2 , ... , (2d - lY2d-l- 1 ) E X Cd- 1)2, i.e., the one obtained from C 2 by removing2d-l,2d+1. Now, C~ x C~ E SOCd-l)2 X SOd2, and one can apply the induction argument as in 6.12. Next consider the case PS0 2n . In this case, the unipotent class C 1 cHis written as C 1 = Co X Co, where Co E S04d2 as in 6.6 (c). We take El E W: such that C(E 1 ) :::> C 1 . Then C(E 1 ) = q X C~ and we have C 2 :::> Co, C~ :::> Co. If we assume that E 1 appears in the restriction of E E W!\ on W s . Then similarly as in the case of S02n+l, one can show that C 2 ...... (ITl, 2T2 , ... , (4d- 3Y4d-3, 4d-l) E X 4d 2, and similarly for C~. Then we consider C~ ...... (Ft, 2T2 , ... (4d - 3Y4d-3 -1 ) and similarly for C;, i.e., C:, C; are obtained from q, C~ by removing 4d - 3, 4d - 1, respectively. Then C~ xC; E S04Cd-l)2 x S04Cd-l)2, and the induction can be applied to this situation. We omit the details.
408
T. Shoji
For a given A = A(E) E V, the unique element E 1 E W s such that C(Ed :::> C 1 and that E 1 appears in the restriction of Eon W s is given in the following way. Write E = E o ,{3 with 0: : 0 ::; 0:1 ::; ... ::; O:p+ 1, (3 : 0 ::; (31 ::; ..• ::; (3p (resp. 0: : 0 ::; 0: 1 ::; .•. ::; O:p, (3 : 0 ::; (31 ::; ... (3p) for some p 2: 1, if W is of type C (resp. type D). We choose 0:' : 0 ::; 0:1 ::; 0:3 ::; ••. ,(3' : 0 ::; (31 ::; (33 ::; ... ,0:" : 0 ::; 0:2 ::; 0:4 ::; ... , (3" : 0 ::; (32 ::; (34 ::; ... , and set E~ = Eo' ,{3', Ei' = Eo" ,{3'" Then E 1 = E~ 181 E~' gives the required representation. 6.14. We shall proceed to the proof of the proposition. We consider the equation (5.2.1). If Ai=- Ao , RA is expressed as R A = (_l)dimE(bxA", where (b is a constant associated to the cuspidal character sheaf in L.c o . But since L i=- e, we know, by induction, that (b = 1. Moreover, since L.c o contains a cuspidal character sheaf, semisimple rank of L is always is of type En, C n or D n ). It even, (it is t 2 + t or 4t 2 according as follows that dimE == n == 0 (mod 2). Hence we have RA = XA". For the cuspidal symbol A o , we write R Ao = (oXAo' By using Lemma 6.11, we evaluate the formula (5.2.1) at g' = s'u'. Then
e
PE(g') = 21k {(oXAo(g')
(6.14.1)
+
L
1JMm ,,/2},
AEV
A;iA o
where 1JA = P1 (ao) is determined from A and 8' u' is as in Lemma 6.11, and m A = dim H - dim C 1 - dim ZO (L s). Put bo = (dimH -dimC 1 -r(H))/2. Then bo = dimB~, where B~ is the variety of Borel subgroups of H containing the unipotent element u E C 1. Hence bo E Z. Moreover, if L is of type E t 2+t, C t 2+t or D 4t 2 according as is of type En, C n or D n , then the semisimple rank of L s is equal to t 2 + t, t 2 + t, 4t 2 accordingly, since 8 is isolated in L. Hence we have
e
e is of type En or C n A if e is of type D n . Let vt = V n ~t+1 for each t, (0 ::; t ::; d) if e is of type En or Cn, and let Vi' = V n ;t,4t for each t, (0 ::; t ::; d) if e is of type D n , (6.14.2)
m
2
if
+t +t 2bo + 4t 2
= { 2b o
respectively. Since 1JA is constant for all A E vt or Then we have the following formula.
e:
Vi',
we set 1JA = 1Jt.
type En or C n
(6.14.3)
PE(g') = 21d {(oXAo(g')
+
L O:::;t
2 1Jtlvtllo+(t +t)/2}.
Unipotent Characters
409
G: type D n (6.14.4)
PE(g') =
22~-1 {(oXAo(g') +
1]tl~'llo+2t2}.
L O':;t
We now show that (6.14.5) the support of A o coincides with C.
In fact, assume not. Then XAo (g') = 0 for any g' E C F . Since E E W!\, we have PE(g') E Z. Now we choose g' E C F so that s' = s. Then ao = 1 and we have p(ao) = 1 in Lemma 6.11. It follows that 1]t = 1 for any t. First consider the case where G is of type En or Cn. Then by (6.14.3) we have
L
2 IVillo+(t +t)/2
== 0 (mod
2).
O':;t
But since q is odd, and IVdl = 1, this contradicts to the fact that IVI = Lo
XAo(g') = P'(a)q(dimC-dimC)/2. Note that dim G - dim C = dim H - dim C 1 = 2bo + n. We take g' = g. Then a = 1 and so p'(a) = 1. Since 1]t = 1 for any t, by applying (6.14.3) and (6.14.4), we see that (0 E Q. Since (0 has an absolute value one, we have (0 = ±1. From now on, we assume that d 2: 2. The case d = 1 will be discussed in 6.16. We shall show that (6.14.6) Assume that d 2: 2. Then (0 = 1. First assume that G is of type En or Cn. We consider the equation (6.14.3). Since q is odd, PE(g') is divisible by qbo. Moreover, since d 2: 2, 2d == 0 (mod 4). Then it follows from (6.14.3), by taking g' = g, that we have (6.14.7)
2
(Oq(d +d)/2
+
L O':;t
2
IVilq(t +t)/2
== 0 (mod 4).
410
T. Shoji
If q == 1 (mod 4), we have (0 == 1 (mod 4) since IVI == 0 (mod 4), (d. (6.9.1)). This implies that (0 = 1. To prove (0 = 1 in the case where q == -1 (mod 4), it is enough to show the following formula.
L
(6.14.8)
2
(_I)(t +t)/2Ilitl = O.
O=:;t=:;d
We show (6.14.8). It is easy to see that
tl
2
Set = [t/2]. Then (_I)(t +t)/2 (resp. t = 2h + 1). Then
L
t 2t I
= (_I)t 1 (resp. (_I)t 1+1) if =
2
(_I)(t +t)/2Ilitl
O=:;t=:;d
~
(-I)
0=:;2tl =:;d = (_I)[d/2](1
t1
Cd/2f_tJ+ ~
(-I)tt+
I
Cd/2]:tl+l)
0=:;2tt+I =:;d
+ (_I))d =
O.
Hence (6.14.8) is verified, and so (6.14.6) follows. The case where G is of type D n is done in a similar way as in the case of q == 1 (mod 4) 2 since q2t == 1 (mod 4) for any t. Thus (6.14.6) is proved. Finally we show that £' coincides with £ given in the proposition. For this, we apply Corollary 4.10 for various s'. Assume that G is of type En or en' We choose s', u' E G F such that s'u' corresponds to a E Ac(g) given in 6.6. Then u is a split element in HF' with F' = iLF. Moreover, SVI E HF' given in 4.7 determines an element ao E AL(gl), which is the element of similar type as a E Ac(g). It follows, by induction, that we have PI (ao) = 1. In particular, we see that TIt = 1 for any t, (0 ~ t < d). This implies, by the similar argument as before (for the case ao = 1), that p(a) = 1. It follows that £' coincides with £ in the proposition. The similar argument works also for the case of type D n , where one can show that p(a) = p(b) = 1 for the generators a,b given in 6.6 (c). Thus the proposition is proved in the case where d? 2. In order to treat the case d = 1 we need an easy lemma, (d. [Se, 18.1 (v)]).
Lemma 6.15 Let r be a finite group and let su = us E r, where u is an element of order a power of p, and s is an element of order prime
411
Unipotent Characters
to p. We assume that sr = 1 for a prime number r such that r #- p. Let X be a character of r such that X(su),X(u) E Z. Then we have x(su) == X(u)
(mod r).
Proof. Let ( be a primitive r-th root of unity. Then X(su) may be written as r-l
X(su)
=
L
O:i(i,
i=O
where O:i are algebraic integers contained in a cyclotomic field K of degree a power of p. Then we have r-2
X(su)
= L(O:i -
O:r_l)(i.
i=O
= 1, (, ... , (r-2 are linearly independent on K((), we see that O:r-l E Z and 0:1 = 0:2 = ... = O:r-l' Since X(u) = L O:i
Since (0 0:0 0:0
+ (r -
l)O:r-l E
x(su)
= 0:0 -
Z, we have O:r-l
0:0, O:r-l E
== ao + (r -
Z. Hence
l)O:r-l
= X(u) (mod r).D
6.16. We shall prove the proposition in the case where d = 1. We have the following.
If G
= PSP4, then
(6.16.1) and if G (6.16.2)
, _ { 1 (mod 2) PE(g ) = 0 (mod 2)
if q == 1 (mod 4) if q == -1 (mod 4),
= PSO s , then PE(g') == 1 (mod 2).
(6.16.1) is verified directly using the character table of SP4 due to Srinivasan [Sr], or it can be verified by the similar method for the case of PSOs, which we explain now. So, we assume that G = PSO s . One can apply Lemma 6.15 with r = 2 to PE and we have PE(S'U') == PE(U') (mod 2). But for such a group, the values at unipotent elements of unipotent characters are computable by using Green functions of GF. (Note that, the argument in 6.14 implies that the almost character R Ao
412
T. Shoji
of G F has no supports on the set of unipotent elements, and all other RA are uniform functions. Green functions of G F are already identified with the ones associated to character sheaves by [S2, ID. Then using the table of Green functions for D 4 ([LS]), we obtain PE(U ' ) = q3. (6.16.2) follows from this. Now similar argument as before can be applied to this case also. For example, in the case of G = PSP4, we have, instead of (6.14.7), a formula (Oq
(d 2 +d)/2
+
IVt 1= { a -
2 (mod 4) 0 (mod 4)
== 1 (mod 4) ifq == -1 (mod 4). if q
This implies (0 = 1. The case where G = PSOs is dealt similarly. The determination of the local system £1 on C is also done in a similar way. This shows the proposition in the case where d = 1, and so completes the proof of it. Remark 6.17. Lemma 6.11 is a counterpart of 4.16 in [L5], where the similar property is proved for a certain subset X of a family V x V' C
[BS] W.M. Beynon and N. Spaltenstein, Green functions of finite Chevalley groups of type En, J. of Algebra 88 (1984), 584 -614. [LB] L. Lambe and B. Srinivasan, A computation of Green functions for some classical groups, Comm. in Algebra, 18 (1990), 35073545. [Ll] G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125-175. [L2] G. Lusztig, "Characters of reductive groups over a finite field", Ann. of Math. Studies, Vol. 107, Princeton Univ Press, Princeton, 1984. [L3] G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), 205-272. [L4] G. Lusztig, Character sheaves, I, Adv. in Math. 56 (1985), 193237, II, Adv. in Math. 57 (1985), 226-265, III, Adv. in Math. 57 (1985),266-315, IV, Adv. in Math. 59 (1986), 1-63, V, Adv. in Math. 61 (1986), 103-155.
Unipotent Characters
413
[L5] G. Lusztig, On the character values of finite Chevalley groups at unipotent elements, J. of Algebra 104 (1986), 146-194. [M] I.G. Macdonald, "Symmetric functions and Hall polynomials", Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 1979. [Se] J.P. Serre, "Linear representations of finite groups", GTM. 42, Springer-Verlag, New-York, 1977. [SI] T. Shoji, On the Green polynomials of classical groups, Invent. Math. 74 (1983), 237-267. [S2] T. Shoji, Character sheaves and almost characters of reductive groups, I, Adv. in Math. 111 (1995), 244 - 313, II, Adv. in Math. 111 (1995), 314 - 354. [Sr] B. Srinivasan, The characters of the finite symplectic group Sp(4,q), Trans. Amer. Math. Soc. 131 (1986),488-525. Department of Mathematics Science University of Tokyo Noda, Chiba 278, Japan [email protected] Received February 1995
A propos d'une conjecture de Langlands modulaire Marie-France Vigneras
Introduction
Dans cet article, nous allons etudier Ie comportement des conjectures locales de Deligne-Langlands par reduction modulo un nombre premier l, dans quelques cas modestes. On note R un corps commutatif algebriquement clos de caracteristique quelconque, F un corps commutatif local non archimedien de corps residueI F q de caracteristique p, F s une cloture algebrique separable de F, de corps residuel F q , WF Ie groupe de Weil de Fsi F. Admettant les conjectures locales lorsque R est de caracteristique nulle, nous aimerions les etendre au cas ou. R est de caracteristique l > 0. Les trois points fondamentaux de ces conjectures lorsque R est de caracteristique nulle, sont: 1) l'existence d'une bijection (respectant les fonctions L et E: de paires) entre les representations irreduetibles lisses de W F sur R de dimension n et les representations irreducibles supereuspidales lisses de GL(n, F) sur R, 2) la classification des modules simples de l'algebre de Heeke affine de GL(n, F) sur R, 3) la classification de toutes les representations irreductibles lisses de GL(n, F) sur R it partir de 1) et de 2). Note: 2) et 3) ont ete demontres par Zelevinski [Zl], 1) n'est pas connu en general quoique de tres nombreux cas Ie soient; pour F de caracteristique p > 0, Laumon, Rapoport, et Stuhler [LRS]. Pour F de caracteristique 0, si n = 2 (Kutzko), n = 3 [He], et plus generalement si n est un nombre premier it p [Moy], ou si n = p [KM], des bijections naturelles comme en 1) existent, il n'est cependant pas connu (sauf si n = 2) qu'elles conservent les fonctions L et E: de toutes les paires souhaitees. 2) est equivalent it la classification des representations
416
M.-F. Vigneras
imlductibles lisses de G L( n, F) sur R ayant un vecteur non nul invariant par Ie groupe d'Iwahori. Supposons que R est de caracteristique 1 > 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
M. -F. Vigneras
422
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<;ant par les deux cas extremes. a) a(x) est irreductible si et seulement si X est regulier sur F q. b ) a(l) est de longueur n, de sous-quotients les caracteres l/~ ---. R*, pour 0 ~ i ~ n - 1, ou x E R* engendre Ie groupe des racines n-emes de l'unite dans R*. L'ordre c(x, R*) de x dans R* est n si R est de caracteristique o. Si R est de caracteristique l, alors c(x, R*) = c(x, Fj) est la partie de n premiere it l.
WP
F;n_l ---. F;d
c) On ecrit X = f-LN ou N : est la norme, n = md, et f-L : Fd ---. R* un caractere regulier sur F q. Le nombre d est unique, egal au nombre de conjugues de X sur F q; Ie caractere f-L est unique car la norme est surjective. Du cote groupe de Weil, Ie caractere correspondant de WPn se prolonge it WPd. Par b), indwFd,wFn X est de longueur m, de sous-quotients les caracteres f-Ll/~, : W Pd ---. R*, pour o ~ i ~ m - 1, et pour x' un generateur du groupe des racines m-emes de I'unite dans R*. Si la caracteristique de Rest 0, c( x', R*) = m; si la caracteristique de Rest l, c(x' , Fj) est egal it la partie de m premiere it l. On a vu en 1.2 que Ie caractere non ramifie l/~, = l/x,; de WPd est la restriction d'un caractere l/x de Wp avec x d = X'i. Par a), a(f-Ll/~/) est irreductible egale it a(f-L)l/~. Le determinant de a(x) s'identifie au caractere moderement ramifie
XIF·. q
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424
1.16 Reduction modulo l . Soit l un nombre premier, et X : F;n ----t un caractere. La reduction modulo l commute avec l'induction, donc la semi-simplifiee de a(TiX) est egale it rl(a(x)). Pour decrire rlX, on ecrit X comme un produit
Q;
X=f-LN. Xl
Q;
OU f-L : F;d ----t est un caractere regulier s~r F q, d'ordre premier it l, N : F;n ----t F*d est la norme, et Xl un caractere d'ordre l. Le groupqe cyclique F;n est d'ordre qn - 1, donc si l ne divise pas qn _ 1 alors Xl est trivial. Si l divise qn - 1, alors Ie caractere Ti Xl est trivial et Tif-L : F;d ----t reste un caractere regulier sur F q. La representation a(Tif-LN) est decrite en (1.15).
If;
On s'interesse au cas ou a(x) est irreductible, alors X est regulier sur F q. Si l ne divise pas qn - 1, alors X = f-L et rla(x) est irreductible. C'est Ie cas par exemple pour l = p. Si l divise qn - 1, on sait qu'alors [livre III.2.1, DJI 2.3 page 268]
n = dE(qd, FiW pour un entier y 2:: O. Posons m = E( qd, FnlY; alors qd engendre Ie groupe des racines de l'unite d'ordre m dans On deduit de (1.15):
If;.
1.17 Proposition. La reduction modulo l de a(x) est egale Ii
et a(Tif-L) est irreductible. Dans le cas particulier ou l = p, rla(x) est irreductible. 1.18 Corollaire. La reduction modulo l d'une representation moderement ramifiee a E IrrQ1 Wp de dimension n et l-entiere, est de la forme
pour une representation T E IrrF1 Wp de dimension d, m = 1 ou m = E( qd ,FnlY pour un entier y 2:: 0, et md = n. G. Henniart a vu que (1.18) est general (l'hypothese "moderement ramifiee" est superflue). Ceci utilise Ie resultat suivant [Zink prop. 4.6.1].
425
Conjecture de Langlands modulaire
1.19 Lemme. Une representation p E Irr R P p normalisee par W p se prolonge d W p.
Preuve. La dimension d'une representation irreductible p de Pp est une puissance de p, car P p est un pro-p-groupe. Le groupe W pest un produit semi-direct de Pp par H = WpjPp (1.1). L'obstruction de Mackey pour prolonger p it W p est representee par un 2-coycle Q : H x H ---. R* dont les valeurs sont des racines de I'unite d'ordre divisant la dimension de p [KZ sect. 2.2]. Le p-groupe de Sylow H p de H est cyclique, et H2(Hp , R*) = HO(Hp , R*) est trivial. La restriction a Hp induit une injection de la partie p-primaire de H2(H, R*) dans H2(Hp , R*), donc Q est trivial. La reduction modulo l -I p d'une representation p E IrrR Pp est irreductible. Le lemme (1.19) et la theorie de Clifford permettent de decomposer une representation irreductible a E Irr R W p en deux morceaux: Ie sauvage et Ie modere. Soient p E Irr R Pp de groupe d'isotropie WE dans Wp, pi un prolongement de p a WE, et a mr E IrrR WE, de dimension n, moderement ramifiee et l-entiere. Alors la representation
est l-entiere, et irreductible.
1.20 Reduction modulo l. La reduction modulo l
-I p
de a est de
la forme
pour la representation irreductible
avec T mr de dimension d un entier y :::: o.
=
njm, et m
=
1 ou m
=
€(q~,FnlY pour
La reduction modulo p d'une representation irreductible moderement ramifiee est irreductible (1.18). La reduction modulo p d'une representation p E IrrQ ! Pp est un multiple de la representation triviale. Donc la reduction modulo p de a comme ci-dessus est moderement ramifiee, et de longueur dimp.
M.-F. Vigneras
426
2. Modules simples d'une algebre de Heeke affine modulaire.
2.1 La eombinatoire.
€-type. Soit € E N U 00; Ie groupe cyclique < x > d'ordre €, de generateur x est visualise comme un €-cercle oriente (€ points sur un cercle oriente si € < 00, ou les points entiers sur une droite orientee si € = 00). Pour tout entier m > 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
428
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-
434
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<;ant chaque poids X' qui intervient dans Mred par E w X'w- I lorsque parcourt les w E Sn comme ci-dessus. 2.14 Lemme.Soit X = (Xj) : A -> 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<;on it. retrouver ces signes. On a donc 2n - 1 possibilites. Dans Ie cas non generique, ou Ie support est Ie cercle, toutes les suites de e signes sont permises, sauf deux: celles ou tous les signes sont egaux. On a donc 2n - 2 possibilites. Comme on l'a vu en (2.6), les e-types l:(Yi' mi) de support C regulier sont classes par l'ensemble des Yi. Si on met Ie signe - sur chaque Yi, et Ie signe + sur les autres points de C, on obtient une bijection entre les bons e-types de support C et les H -modules simples de support C. Cette bijection est compatible avec la bijection (2.16) dans Ie cas regulier et generique. La conjecture (2.4) est vraie dans Ie cas regulier. Exemples. Dans Ie cas generique regulier n < e, Ie caractere trivial de poids (l,q, ... ,qn-l) correspond it. 1](1) = -1 et tous Ies autres signes 1]( qi), i =I- 1, egaux it. +1, Ie caractere signe de poids (qn-l, ... , 1) correspond it. tous les signes egaux it. -1, donc Ie e- type est
1 + q + ...
+ qn-l .
Dans Ie cas non generique regulier e = n, -Ies caracteres "triviaux" de poids qi X correspondent it. 1](qi) = -1, et tous Ies autres signes egaux it. +1, et aux e-types qi(1 ---+ ••• ---+ qe-l), - Ies caracteres "signes" qi(qe-l, ... , 1), correspondent it 1](qi) = +1, et tous Ies autres signes egaux it. -1, et aux e-types (qi-l,qi)
+
L ji-i,i-l
qj.
442
M.-F. Vigneras
2.18 Types associes aux caracteres de HR(n, q). Le caractere trivial X de poids (1, ... ,qn-1) correspond au c-type simple (1, m) := (1 --+ q ... --+ qm-1), et Ie caractere signe de poids (qn-1, ... ,1) qui est l'image du caractere trivial par l'involution de qi. Zelevinski, correspond au c-type
2:7:11
2.19 La conjecture (2.4) est vraie pour n
= 2.
Ceci resulte des cas precedents, puisque pour n = 2, on est dans Ie cas regulier ou generique (2.6), (2.11). De fa<;on explicite, on assode - au module simple 1(1 2 ) de caractere-poids 2(1,1), Ie type 1 + 1, - au caractere de poids (1, q), Ie type 1 --+ q, - au caractere de poids (q, 1), Ie type q --+ 1 si c = 2, ou Ie type 1 + q si c > 2.
2.20 Classification des HR(3, q)-modules simples. On suppose c > 1. a) La representation 1(1,1, q) est sans multiplicite de longueur 2, de sous-quotients simples de caracteres-poids
2(1, 1,q)
+ (l,q, 1),
2(q, 1, 1)
+ (l,q, 1)
si c(q) > 2 (cas gemkique). Si c(q) = 2, elle est de longueur 4 avec 3 sous-quotients simples non isomorphes, le caractere (1, q, 1) de multiplicite 2, et deux modules simples de dimension 2, de caractere-poids 2(1,1,q),2(q,1,1). b) La representation 1(1, q, q2) est sans multiplicite de longueur 4, de so us-quotients de caracteres-poids
si c(q) > 3 (cas regulier generique). Si c(q) = 3 (cas regulier non generique), elle est sanS multiplicite de longueur 6, ses sous-quotients sont les caracteres
Preuve. Le seul cas nouveau est la decomposition de 1(1 2 , q), c = 2. La reduction du cas generique regulier 1( 1, q, q2) modulo q2 = 1 montre que 1(1 2 , q) a 4 sous-quotients, Ie caractere (1, q, 1) de multiplicite 2, et deux modules M, M' de caractere-poids respectifs 2(1,1,q), 2(q, 1, 1). On applique (2.12), comme 1(1,1) est irreductible
443
Conjecture de Langlands modulaire
puisque q =I- 1, la multiplicite de (1,1, q) ou de (q, 1, 1) dans un module est nulle ou ~ 2. Les deux modules M, M' sont donc im§ductibles. 2.21 La conjecture (2.4) est vmie dans le cas n
= 3..
a) Si c > 1, on associe it. l'unique module simple 1(1 3 ) de caracterepoids 6(1,1,1), l'unique c-type 1 + 1 + 1 de support 13 . b) Si c > 2, on associe aux modules simples generiques de caracterepoids respectifs 2( 1, 1, q) + (1, q, 1) et 2(q, 1, 1) + (1, q, 1), les c-types bien ordonnes (1 ---+ q) + 1 et q + 1 + 1 (2.16 Exemple). Si c = 2, on associe au caractere (1, q, 1) Ie c-type (1 ---+ q ---+ 1). Le seul cas nouveau se presente avec les modules simples de caracterepoids respectifs 2(1, 1, q) et 2(q, 1,1), les bons c-types associes (1, q) + 1 et (q,l) + 1 correspondent aux decompositions en c-types simples de l'unique poids du module. II n'y a pas d'autre choix possible. c) Si c > 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.
452
M.-F. Vigneras
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