INSTITUT DES HAUTES ETUDES

SCIENTIFIQUES

ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES (P. 5 a 99)

by Robin HARTSHORNE

RIEMANN-ROCH FOR SINGULAR VARIETIES (p. IOI i 146)

by Paul BAUM, William FULTON and Robert AlACPHERSO,N

RATIONAL EQUIVALENCE ON SINGULAR VARIETIES (P- 147 i 167)

by WiIIiam FULTON

COHOMOLOGY OF FINITE GROUPS OF LIE TYPE, I (p. 169 i 191)

by Edward CLIYVE, Brian PARSHALL, and Leonard SCOTT

MINIMAL INJECTIVE RESOLUTIONS WITH APPLICATIONS TO DUALIZING MODULES AND GORENSTEIN MODULES

(P. ‘93 A 2x5)

by Robert FOSSUAI, Hans-Bjorn FOXBY, PhiIIip GRIFFITH, Idun REITEN

1975

PUBLICATIONS MATHGMATIQUES, No 45 LE BOISMARIE - 91.BURES-SUR-YVETTE

COHOMOLOGY OF FINITE GROUPS OF LIE TYPE, I

by EDWARD CLINE, BRIAN PARSHALL, and LEONARD SCOTT

In this paper we determine Hl(G, V) for G a finite Chevalley group over k=GF(q),

q > 3, and V belonging to the class of “ minimal ” irreducible KG-modules. The modules

under consideration are described precisely in 4 I ; they include all the standard and

spin modules, as well as some adjoint modules and exterior products. Many occur

naturally as sections in Chevalley groups of larger rank (‘).

Our approach is thematically Lie-theoretic, and relatively free of explicit calculation.

All lower bounds on cohomology are determined by examining indecomposable modules

for G constructed from appropriate irreducible modules for the corresponding complex

Lie algebra (cf. (I. 2) and (4.2~)) ; in particular, we never have to explicitly exhibit

any cocycles. Upper bounds are obtained by studying interactions bet\veen the “ roots ”

and “ weights ” for G which arise from their analogues in the algebraically closed case

(cf. 0 2, (4.2~), and 4 5).

Many of the adjoint modules were treated by Hertzig [14], and certain of the

classical cases have been studied by D. Higman [IS], H. Pollatsek [22], and 0. Taussky

and H. Zassenhaus [27]. It should be noted that these papers contain results for more

general fields than we consider, especially the fields of 2 and 3 elements. Nevertheless

for K=GF(q), q>3, our results include all the above with the exception of the adjoint

module of type F, (“) .

1lre include (cf. 5 5) a proof of a result on Extl(U, V) = H1(G, U OV) for

G= SL(2, 2’9 stated by G. Higman in his notes on odd characterizations [IS] and

used there in the analysis of certain n-local subgroups.

One can also obtain lower bounds on cohomology by means of the Cartan-Eilenberg

stability theorem [5; p. 2591. An interesting by-product of our investigation is a new

interpretation of this theorem in terms of a pre\iously unnoticed action of Hecke algebras

on cohomology (cf. 4 6).

(I) Also, they include (essentially) the modules V for which (G, V) is a quadratic pair in the sense of Thompson [28].

(“) Since the writing of this paper, the fields of 2 and 3 elements have been treated by Wayne Jones in his thesis. By means of an elegant theorem describing the behavior of restriction to a Levi complement in a suitable maximal parabolic subgroup, he is able to reduce the question of upper bound to low rank cases.

169

22

ljcl ED\V.-\RD CLINE, BRI.\S PARSHALL, LEON;\RD SCOTT

I. Representations (“).

Let C be a finite root system in a Q-vector space E endowed with a positive definite

symmetric bilinear form ( , ) invariant under the \Veyl group JV of C. For cc, PEE,

(s) denotes the angle formed by x and p. Also, Cshort denotes the set of roots in S

of minimal length, while Clonn denotes the set of roots in S which are not short. It

will be convenient for later reference to fix the following notation for a fundamental

system A of the indecomposable root systems:

4

B f

c f

*f

Et?

Ei

E*

F4

G 0-4

/

Ql aa

(3) More details concerning the representation theory of algebraic and Chevalley groups can be found in [g], lzjl, 1% and 1291.

170

COHOblOLOGY OF FINITE GROUPS OF LIE TYPE, I 171

For I < i < !, A,; =A; denotes the fundamental dominant weight corresponding -- to the root cr,~A. Recall hire is defined by the condition that (k;, KY) = Sij, where

K: = ~cc~/(v.~, K~) is the coroot corresponding to ‘Jo. The dominant weights are the

non-zero elements of E of the form A = Cn,&, n, a non-negative integer. z

Minimal weights

\Z’e partially order the vector space E by the relation w 2 u iff w-u is a non-

negative integral combination of the “i. A dominant weight A is called minimal if it is

minimal relative to this partial order. 1Ve will write A=A(X) for the set of dominant

weights in E, and denote the set of minimal elements of A by A,,,.

Let 2 be the complex semisimple Lie algebra with root system C relative to a

fixed Cartan subalgebra 5j of 9.. Let AEA, and let 332=9J1(h) denote the irreducible

G-module of dominant weight A. Suppose o~Au(o} satisjes w 2 A. Choose a weight w’ n

of -5 in ‘9J32 minimal with respect to ~‘2 w, and icrite G = w’- w = 2 pi, where the pi i-l

are fundamental roots. If o is not a weight of 5 in YJJ?, then Z $ o, so

0<(3,Z)=(~, i F;), i=l

whence (G;, &)>o for some j. Since (w, $,) 20 for all k, (w’, pj) =(G+w, pj)>o.

This means [I 7; Theorem I, p. I IZ] that u’----$~ is a lveight of $5 in m, contradicting

the minimality of w’. Hence, w is in fact a weight of 3 in 9X. This slightly generalizes

-by essentially the same argument-a result of Freudenthal [12].

We can now determine the elements in A,,,. Let ?,‘EA,,,. First, consider the case

when A belongs to the root lattice ZS of E. From the previous paragraph, o is a weight

of $ in 9J1, whence, by irreducibility, ‘J is a \$.eight of Sj in 9JJz for some root v. Replacing v

by a suitable \V-conjugate, Ice can assume that \JEA. The minimality of ?\ implies

then that A==‘/. If C has one root length, ),=‘J is the maximal (relative to 2) root:

while if S has t\<o root lengths, A=V is the maximal short root. Conaersel_?l, this shows

the maximal short root belongs to A,, . \Ye enumerate these elements of A,,:

.A, , h + ?v ; B,, I.,; C,, j.f-,; *,, Q-,; E,, ‘h2; Ei, A,; Es, &; F,, A,; G,> he

Next, suppose j&En,,, , i.$ZC. Let C” be the dual root system to ?3 [2; p. 1441, and let yV be the maximal root in 5’ (relati1.e to the positive system defined by 4’).

Then [2; Ex. 24, p. 2261 i. is minimal iff (y’, A) = I. We can therefore enumerate these

additional elements of A,, as follows:

.4,, Ai, I 5 i%t; B,, A,; C,, h,; D,, hi, i= 1, 2, C; E,, ?t1, i\,; E;, hi (4).

(*) The weights in this list are the “minimal” dominant xrights of Chevalley [7; Esp. 211.

172 EDWARD CLINE, BRIAN PARSHALL, LEONARD SCOTT

For future reference we list here the m&ma1 roots which are not minimal dominant

weights :

B,, h-1; G, 5%; F,, h; G,, A,.

These roots together with the maximal short roots listed above are precisely the

elements of X n A.

Fix AEA,. Let U be the universal enveloping algebra of 9, and let

(X,, H,I ~cEC, PEA} be a Chevalley basis for 2 [26; p. 61. U, and U; denote the

Z-subalgebras of LI generated by the X:/n! (rn~Z+) for ~EC and NEC-, respectively.

Fk v + o in the A-weight space !Dmh of& in 9J1, and set 1I= vLI;, a I&-stable lattice in %.

Let G’ be the universal (or simply connected) Chevalley group over the algebraic

closure K of k defined by 2 (or C). Let T’ be the masimal K-split torus of G’ corre-

sponding to 5, and let X(T’) be the character module for T’ [I ; p. xgg] (j). We

identify C with the root system of T’ in G’, so X;cX’(T’) sX’(T’)OQ=E. For aEC,

V; denotes the corresponding one-dimensional root subgroup (normalized by T’), and

x, : K-+Uz is the isomorphism of [26; p. II].

Since U, stabilizes bf, G’ acts in a natural fashion on the K-space

S’=KOhl=KuG=KvU=-,

where u’- denotes the unipotent radical of the Bore1 subgroup I?- defined by T’ and

-A. s’ is an indecomposable G-module of dominant weight h, and if x’ is a maximal

submodule, S/X’ is the irreducible KG’-module of dominant weight A. If X’ CO,

let w be a maximal weight of T* in X. Then 05 A. Since the A-weight space S;

of T‘ in s’ is one-dimensional, x’=o and S’ is irreducible when A#ZC. When

AEZC, o =o and x’ is contained in the zero weight space S; of T’ in s’.

Assume AeR,n ZC, i.e. h is the maximal short root v in C. We claim x’ consists

of the set Y’ of vectors WEST fixed by the root subgroups UX, - ae A. Indeed, T’ and

the Va generate B’- (this follows by a standard argument from the formulas [26;

pp. 148, qr]), hence Y”=(ze,ES;,] wB*-= w>. Clearly, x’ g Y’, while if w is fixed by B’-,

it is fixed by G’ (the morphism G-ts’ defined by g+ wg factors through the complete

variety G/B’-, and hence is constant [20; p. 1041).

Write v=S/x’. Let C’={aEC[ u is a weight of T’ in V’>, and set A.‘= X’n A.

When C has only one root length, A’=A, while if C has two root lengths, A’ consists of

the set of short roots in A. The non-zero weights of T* in S’ are precisely the elements

of C’, and for CIEC’, the corresponding weight space SX is one-dimensional. Also,

(s) We recall briefly the interpretation of T’ in the notation of [26; p. 431. FVhen G’ is universal, T‘ is the direct product of the subgroups {h,(t) 1 ~EK”} for cz~A. S’(T’) identifies naturally with the lattice L,:

if pEL,, we view p as the character p : T’+K’ defined by ~(~fi~h,J$)) = ifilti”*a?.

172

COHOXIOLOGY OF FISITE GROUPS OF LIE TYPE, I ’ 73

dimS~=dim%Jnl,=]A’] [18; Eq. 50, p. 2611. Let C’=C(A’) denote the Cartan

matrix of A’. Let r(p) denote the rank of C’ modulo p, where p is the characteristic

of k. We can now state (“) :

Theorem (I. I ). - Let A = v, S’, x’, v’ be as above. Then

dim,= =dim,Sc--dim,V; =/ A’I-r(p).

Proof. -For MEA, fix O=I=V,EM so that Zv,=Mn’m,. Because S’=KvU-

and because the Uz, for XE-A, generate U*-, we have by [26; Lemma 72, p. 2og] that

S;=(v,X-,]u~h’). But dim Si=dim ‘331,=/A’], so {u,X-. ] aEA’) form a basis

for SG. For r, PEA’, write u,X-.X8=(x, p>vs. Then it is easy to see [21; proof

of Prop. 21 that the vr, for y~l’, can be adjusted so that the matrix - ({ tc, p)) is the

matrix C’. Thus: 2 c,u,X-.EY (for c,EK) iff for all PEA’ CI E A’

hence iff

for all PEA’. It follows that dim s’ = 1 A’] --T(P), as desired. Q .E.D.

For convenience we tabulate the number dim,X’ for those C and p where it is

nonzero :

c 4 B, C, % %+I E, Ei F, G,

P pip+1 2 Pie 2 2 3 2 3 2

dim, X I I I 2 I I I I I

For a k-subgroup W of G’ (e.g. G’, T’, U*-, B*-, etc.) we shall denote by H the

subgroup H; of k-rational points of H’. It should be noted here that since G’ is uni\:ersal,

G is just the corresponding universal Chevalley group over k [26; Cor. 3, p. 651.

If \v* is a K-vector space defined over k, Ire let W denote the k-subspace of

k-rational points 12;;. The G-module S above is endolved with a natural k-structure

S = k@XI, and this induces k-structures on x’ and V*. Note v* is an irreducible

k-rational G-module and remains irreducible upon restriction to G. This follows

from Steinberg’s theorem [IS; Th. 431 since here the dominant weight A= En,hi

satisfies 0 5 ni 5 q- I, except when G=SL(2, 2), h=ctl. \\Then we wish to emphasize

the dependence of S, X, and V= S/X on A, we denote them by S(A), X(A), and V(i<)

respectively.

(O) A similar result is stated in [3; p. 151.

‘74 ED\VARD CLINE, BRI.-\N P;\RSH;\LL. LEONARD SCOTT 1

We claim S is indecomposable when q > 2. Let N be a direct summand of S which

covers 1’. Recall hi=&;; hence for some c~EX, ZJ’=U+U~EN. Since v has weight :

and q>2, there is a JET with h(t)+ I, except when 1 is of type A, and q = 3. Witt’

this exception, N contains (~-h(t)) u, hvhence contains v, thus N=V; otherwise X=c

by (I . I) and S=V is irreducible, so indecomposable (;).

d lower bound for H’

Let q>2 and let S be the dual module to S. Let r’= j 1’ I- r(p), Since S has

a unique r’-dimensional submodule, 5 has a unique submodule I^v = X1 of codimension r’.

Then s/fi=T is isomorphic to the dual module of X, and hence is a trivial KG-module.

\Ye note also that o is the only fixed point of G in S, else S lvould contain a submodule

of codimension I, which is absurd. Hence (see (2. se)) from the follolving exact sequence

of G-modules

0+l^\*+~*2+0

\\;e get the exact sequence of cohomology groups:

\\‘hen ?.=v is the maximal short root, it is stable under the opposition involution I

of h (L=- w,,, where w,, is the unique element in 1v such that w,(A)=-A), so V

is self-dual as a G-module. Thus, V is self-dual as a G-module. Since \T is the dual

module to S/X =V, we obtain the following, using the fact that X = o if A+ZE:

Theorem (I .2 ). - Let q>2 and let A be a minirrul dominant weight. Then

dim,X(A)sdim,Hl(G, V(A)).

Recall that dim,X(h) = dim,=(A) is o unless A=.), in which case it is given by

Theorem (I. I) and the table which accompanies it.

2. Cohomology.

In this section we outline some basic homological results and apply these to the

cohomology of Chevalley groups.

Let A be a finite group, and V a kA-module. The first cohomology group H’(A, V)

is defined to be Z’(A, V)/B’(A, V) where

WA> V)={y : A-tVIy(xy)=y(xly+y(p)}, B’(A, V)=(.( : A-tVIy(x)=u-vx for some fixed UEV).

(‘) IVhen Q=Z, S may not be indecomposable, e.g. G =SL(n, 2), h=x,. This is the only case in which S is not indecomposable [30].

COHO1iOLOGY OF FIXITE GROVPS OF LIE TYPE, I ‘75

The elements of Z’=Z’(A, V) are called cocycles and those of B1 are called coboundaries.

Two cocycles y, y’ are said to be cohomologous (y-y’) if y- ~‘EB’, or equi\,alently,

if y and y’ determine the same cohomology class [y] = [y’] in H’.

There are other ways to think of H’. For instance we can associate with each

YEZ’ a complement A, to V in the split extension A. V by the formula

(2.X) 4, =txY(x) 1=4L

and conversely each complement determines a cocycle. Two complements A,,, A,

are conjugate in A.V iff y-y’; more precisely, A”, =A,, for an element VEV iff

y’(x) = y(x) + u- vx. Hence H’(A, V) may be regarded as the collection of conjugacy

classes of complements to V in the split extension A. V.

For some calculations it is more useful to think in terms of the homomorphism

f, : A-+%<A.V defined by

(2.2) f,w=xYw

In general if f : A-+A.V is any function such that f(x) =x (mod V) and y is defined

by (2.2), then y is a cocycle iff f is a homomorphism.

(2.3) \1’e list some standard properties of HI (“).

a) H’(A, V) is a k-vector space in a natural way. In fact, for y~Zl and cEk

we have cy~ Z r defined by (c)(x) =cy(x). Clearly B’ is a k-subspace of Z’.

b) More generally if f : V-+V’ is any homomorphism of kA-modules (“) we

can define yJeZ’(A, V’) by the formula (yf)(x) = y(x)5 The resulting map

Z’(A, V) --+ Z’(A, V’)

induces a map on H’.

c) If 9 : A’-+A is a homomorphism of groups, then ‘2 induces a natural kA’-module

.structure on V, and for ysZ’(A, V) \ve can define yy~Zl(A’, V) by (py)(x’) =Y(x’~),

x’ GA’. The resulting map Z’(A, V) -+ Z’(A’, V) induces a map on H’.

This map is called inflation \vhen ‘p is the natural projection of A’ onto a quotient

group A, and restriction (y iA’) when 9 is the natural inclusion from a subgroup A’ to A.

In case aEA, BLA, and ‘2 =a-l : B”-+B, then ‘2 does not induce a map

Z1(B, V) + Z’(B”, V) by the formula above, since it is required there that V be given

a “ new ” kB”-module structure (induced by y). However the map 8~ va is an

isomorphism from this “ new ” kBn-module back to the “ old “. Applying paragraph b),

we obtain a legitimate map a-‘ya from Z’(B, V) to Z1(B”, V), which induces a map on H’.

(s) A general reference for the cohomology of groups is [5; Ch. 121, though the reader can doubtless supply proofs here without difficulty.

(“) Similar statements apply iff is just a homomorphism of ZA-modules; in particular cohomology groups of Galois conjugate modules are isomorphic abelian groups.

175

176 EDWARD CLINE, BRIAN PARSHhLL, LEONARD SCOTT

\Ve shall denote a-‘ya by y’, and record here the formula

y”(x) = y(P-‘)u (xEB~).

Also we note that y’wy when DEB.

d) If kH’(A, V) and B< A then the class _ c = 8 lB E H’(B, V) has the property

of “ stability ” :

2 B1p=2 ! B;lB” for each n E A.

e) If 1Y is a kX-submodule of V we have a natural exact sequence (*“)

o+lv:L+v-i+ (V/W)” +H’(A, W)+H’(A, V)+H’(X, V/W).

Also, we have an exact sequence of cocycles

o+Z’(:l, y+-Z’(A, V)JZl(i\, V/\V).

f) If B .4 we have an exact “ inflation-restriction ” sequence

o-tH’(X/B, VB)+H1(i\, \‘)+H’(B, V)“B.

g) If B<A and the characteristic p of k does not divide the index [A : B], then

the restriction map H’(A, V) + H’(B, V) is injective. If in addition B .; A, then

H’(A, V) -+ H’(B, V)“!B is an isomorphism (see also $ 6).

h) If V is a projective ,&A-module, then H’(A, V) = o.

The following two propositions extend the results of $4 to central factors and direct

products; they play no further role in this paper.

Proposition (2.4). - Supp ose V-‘= o and B is a subgroup of the center of A. Then (11)

a) If VB=V, then H’(A, V) z H’(A/B, V),

b) If VB=o, then H’(A, V) = o (“).

Proof. - By (2.3f) we have an exact sequence

(*I o+H’(A/B, V”) +H’(A, V) +H’(B, V)“‘“.

If VB=V, then B1(B, V) =O and Z’(B, V) is just the collection of group homomorphisms

from B to V; H’(B, V)“lB may be identified with the .4-homomorphisms from B to V,

whence is o since V’=o and B is central. Thus, H1(A, V) = H’(A/B, V).

If VB= o, then certainly VB1=o where B=B, xB,, B, a p’-group and B, a

p-group. Applying the sequence (*) with B, in place of B gives H’(A, V) = o since

H’(B,, V) = o (‘“).

(I”) Here ( )^ denotes the fixed points of 2, e.g. VA = (u ~\.‘ua = L’ for all D 6.4). (‘l) If V is irreducible then of course \” =V or VB = o. (“) A number of special cases of the results in 5 4 can be obtained effortlessly from this fact. (Is) An alternate argument, well-known to finite group theorists, can be made from (2. I) and the fact

that C,. ,(q,) = C,(B,).

176

C:OHO~lOJ~OGY OF FINITE GROUPS OF LIE TYPE, I ‘57

Proposition (2.5). - Supp ose :\=;I, x A, and V =V,O V, where V, is a k&-module

and V, is a k&-module. Assume V$ = o. Then

H’(A,x4, V,OV,) zH’(A,, V,)OV+ (‘“).

Proof. - Again (2.3 f) gives an exact sequence

o+H’(A1, (V,OV,)“+H’(A,x4, V,@V,)+H’(AZ, VIOVJA1.

It is easily checked that (V,0VJA2=V1%V$, H1(A,, VIOV$) “H’(A,, V,)OV$,

and H’(X,, V,OV,)“l E (V,OH’(A,, %‘,))“I EV$@H’(A~, V,) = o. The result follows.

-liotation. - For the rest of this section, G denotes, as in $ I (see especially footnote 5),

the group of k-rational points of a universal Chevalley group G’ over K. Similar

conventions hold for B, T, U, U, (XE C). Thus, B is the semi-direct product T.U,

T={;~lh,i(~)Ir.iEkX, * L ‘<‘>, U is a Sylow p-subgroup of G, and U, =(x,(c)] EEk}.

Weights of T; Galois equivalence

Let X(T) denote the collection of homomorphisms (or “ weights “) o : T-+k”.

For VEX we let (as in 5 I) V, denote the associated “ weight space ”

{vEVI vt=o(t)v for all teT}.

We think of X(T) additi\.ely, so for example V, denotes the weight space associated

with the trivial weight w(t)=1 (JET).

A root aEC determines an element, still denoted a, of X(T) by the formula

x,(F$),=x,(a(t)t), E,Ek, tET. In $ 3 the extent to which this notation is ambiguous is

exactly determined, that is, when distinct elements of C determine the same element

of X(T).

1Ye shall say two weights wl: W*E X(T) are Galois equivalent (wlw WJ if WY = w2

for some automorphism G of k, or more precisely, if c+(t)” = c+(t) for all tET. The

importance of this concept lies in the follo\lTing result.

Proposition (2.6). - Let L be a kTU,-module which is I-dimeruional over k and on which

T acts with weight w. Then

dim,Z’(U,, L)‘= I

i

if w-a

0 otherwise.

Proof. - U, acts trivially on L of course, since L is I-dimensional. Thus Zl(U,, L)T

is the collection of k,T-homomorphisms from U, to L, where k,= GF(p). In all cases

U, is an irreducible k,T-module, so k@&U, is a sum of distinct Galois conjugates.

Thus Z’(U,, L)T=Homk,T(Ua, L) 2: Hom,,(kO,.U,, L) has dimension I or o o\‘er k,

depending on whether w-a or not.

(14) Suitably interpreted this result is jut a very special case of the Kiinneth formula [x9; p. 1661.

178 EDWARD CLINE, BRIAS PARSHALL. LEOSARD SCOTT

I -parameter cohomologv

The result just established readily yields information concerning the cohomology

of kTU,-modules L of arbitrary finite dimension.

Lemma (2.7). - Let L be a ,jnite-dimensional kTU,-module. Then

a) dim,Z’(U,, L)T<tiz2dimiiL,,

b) dim,B’(U,, L)T = dim,L,/L>,

c) dim,H’(U,, L)T< c dim,L,-dim,L,/Lp. o-*

Proof. - c) is of course a trivial consequence of a) and b), and the fact that T is

a $-group.

a) is trivial if L =o. So assume L =I= o and let ;\I be a maximal kTU,-submodule

of L. By (2.3e) \ve have an exact sequence

o-+ Z’(U,, 1f)T-t zyu,, L)T+ Z’(U,, L/M)?

11 has codimension I, since U, is a p-group and all irreducible KT-modules have

dimension I. Applying (2 .6) we obtain easily that

dim,Z’(U,, L/hI)T= 2 dim,L, - z dim,hI,, 0 - z (1, 5 1

from which a) follows by induction.

To prove 6) we observe that the map &-+B1(U,, L)T which sends P, in L, to

the coboundary u-+C,-!,u (z~cU,) is surjective (the result then follows since the kernel

of this map is obviously LF): Let u-e--!ZL be an element of B’(U,, L)T; thus

e- eu = (e-em-l) t

for each &T, UEU,, or C--et=(e--O)u. Hence ! is a fixed point of T modulo Lv=.

Since T is a p’-group there exists I,,ELT=L,, with P--ePOELCa, that is, the original

coboundary u&‘-&‘u is equal to the coboundary u+P,-!,u. Thus L,+B’(U,, L)’

is surjective.

Upper bounds for H’(G, V)

JVe can now prove the main theorem of this section.

Theorem (2.8). - Suppose +‘I Cf is a set of roots whose corresponding root groups

generate U, that is, U=(U,(CXE$J). Then

dim,H’(G, V)<~~,Jdim,Z’(U,, V)T-dimkVO+dim,VB (‘“).

Proof. - The defining formula y(xy) = y(x)y +Y(JJ) for elements y of Z’(U, V)

and the fact that U=(U, [ a+) insure that the natural product of restriction maps

(Is) Usually VB = o in this paper in view of (3.5) and [26, Theorem 46, p. 2391.

COHOXIOLOGY OF FISITE GROUPS OF LIE TYPE, I ‘3

Zl(U, 1’) + zv+Z1(Ua, V) is injective. Hence Z’(U, V)‘-+ n Zl(U,, V)T is injective, aE+

and dim,: Z’(U, V)Tlzz,Jdim, Z’(U,, V)T (‘“).

On the other hand, the map V,+ B’(U, V)’ which takes P,, in V,, to the coboundary

uti&--POu has kernel Vf=V’, hence dim, V,-- dim, VB< dim, B’(U, V)T. Thus

by (2.s) dim, H’(G, V) 5 dim, H’(B, V) = dim, H1(U, V)’

= dim, Z’(U, V)‘- dim, B1(U, V)’

5 c dim, Zl(U,, V)‘- dim, V, + dim, V’. as+

Q .E.D.

For n~Cu{o) set n,= c dim,V,. w-b

Corollary (2. g), - Under the hJ,pothesis of (2. S), ;f VB = o the72

dim, H’(G, V) 5 c n,--71, (“). UEcJ

Proof. - Apply (2.7a) and (2.8).

3. Restrictions of roots and Galois equivalences (l”).

As in 3 I, G’.denotes the universal Chevalley group over K defined by C; X’=X*(T’)

is the free abelian group generated by the fundamental dominant weights. Since T’

is k-split, T has exponent q-1, hence any homomorphism y- : T*+ KX maps T into k”)

and so there is a natural restriction homomorphism 2 : X’(Y) --f X(T) (‘9).

Proposition (3. I ) (‘“). - Assume C is indecomposable, and q>3. Let /3 + y be elements

of Cu{o}. Then one of the following holds:

a) P(P) *P(Y);

b) q=4> S is of ppe G,, p, -( are long, and (2;) = y;

(‘s) Similar considerations appear in Hertzig [x3]. (a’) In particular H’(G, V)= o if all n,‘s are o. \Ve mention here that T. A. Springer has obtained a

splitting criterion (for exact sequences of modules of a semisimple simply connected algebraic group over an algebraically closed field of characteristic p) which also is described in terms of weights in the modules invol\*ed [24; Proposition 4. j].

(Is) Parts of Propositions (3.1) and (3.3) in this section are contained in Chevalley [8; Lemma r I]. (I”) For cc ES c X’(T*), ;(a)(~) is x(t) as defined in 4 2 (see footnote 5 and [26; Lemma ~qt, p. 271). (*“) The fibres of p restricted to To= S U{O} partition To as To = n,~. uns, with o E no. When 4=2.

S=O, X=0,. \lyhen 9 = 3, the situation-now more complicated-entails the following alternatives: I) S is of f(f T-I)

type BC (Ci3), J=Y, n,={o}, q={Iz ?, 5 (~a,+ ~~)}a~ for some LOEN’, I <i< (!J - - 2” and nj = { & m} for

some a short, ($) <j<s; 2) S is of t>pe Cf (Iltj, 5=(0$-t, nO={o}USlon~~ ni={*u,, i(r,+cxs)}~ for some w~\Y, for i>t; 3) S isoftype Dp ((>3), 5=(i), no={o}, ni={tml, &xe}w for some WEM’, i>r; 4) C is of type Fp, short a, j > 3; 5) C

s=rj, no={o}, ni={+zl}(w,~,w,,)w forsome wEW, i=I, 2, 3, nj={+a} for some is of type G, , 5=3. no={o}, ni={fz1,f(x1$zz2)}w forsome WE\Y for i>5; 6)inthe

remaining cases J = /I; j/z and each ni is of the form { + a}. A proof of this statement can be based on the fact that the fibres of p on 2 form systems of imprimitivity for the action of \V. All such systems can be easily determined, since the stabilizer TV; of 1: = maximal long or maximal short root is generated by the fundamental reflections it contains.

r8o EDbVARD CLINE, BRI.-\N PARSHALL, LEOSARD SCOT?

.--. c) (?l) q=4, C is of ppe A,, (p, 7)=2x/3;

d) q=5, C is of type C,, B--y is long.

Proof. - Since ker(p) =(4--1)x’, we need only consider the map

; : x’--+ X’/(q-I)??.

\Vrite <EXI as a Z-linear combination of fundamental weights

then F(?J=o iff mc,ir~ (modq--r) for each i.

Let c~EC, and PEC u(o) be distinct elements such that $‘(x)=;(p). JVe may

assume 01 is a dominant weight (see the lists in $ I). Clearly F(u)+o, and if ?; is not of

type G, or A,, the congruences m,,i=m3,i (mod q-1) for all i, together with q>3,

imply p is a multiple of a, hence p = -a. Evidently this implies mcr,i> I for the non-

zero m,,i, so by inspection C is of type C,, CX=~A~=-~, q=5 and d) holds. If C is of

type G,, 6) follows by inspection. If C is of type .A!, I> I, then p = m,h, +m,A(

tvhere rn,r~ (modq-1) for i=r,t. This implies q =4 and m,-I = o or -3,

\vhence either p=-u or PE{>~~--~A(, AI--22A1). Clearly p=/= -a, and since ~,--22~~,

A(--2hl are roots only when e = 2, c) holds. Q .E.D.

The Galois equivalence relation on X(T) induces a similar relation on x’, viz.,

for ?., p E X(T’) we say h is Galois equivalent to p (A-F) provided ~(A)--(P) in the

sense of $ 2. If hm p and q=p”, then

(3.2) pjmh,i=mm,i (modp”-I) for some j, oI,j<n, all i.

iVe proceed to determine Galois equivalences among roots.

Proposition (3.3). - Assume C is indecomposable and q>3. If tt, ~EC are distinct

Galois equivalent roots, then one of the following occurs:

a) q=4, C is not of ppe G, or A,, a =-@;

b) q=4, C is of gbe G,, u = -9 OT x, p are both long (all long roots are equivalen

here) ;

c) q=4, ~~OftypeA,, all roots are Galois equivalent;

d) q=5, C is of ppe C,, a=--$ is long;

e) q=g, C is of ape CI (!>I), u= -p is long.

Proof. - If-q is prime, c-t iff p(c) =p(<), since k = GF(q) has no non-trivial

automorphism; hence d) holds if q=j. If q =4 and S is of type G, or A,, 6) and c)

hold by inspection. \Ve may assume henceforth q+ j and C is not of type G, or A,

when q=4.

(“I) The authors thank the referee for pointing out this case.

180

COHO1IOLOGY OF FISITE GROUPS OF LIE TYPE, I 181

Also we may assume ‘1. is a dominant iveight, and p- cx via a non-trivial auto-

morphism of GF( q) .

Suppose m,,i 51 for all i. Then (3.2) reads

mp,i G

1

P j if magi = 1 (modp”-I), for some o<j<n.

0 otherwise

Since 1 rnp,; I< 2 when C is not of type G,, $ must be a multiple of I; hence p = - TY.,

@+I-o (modp”-I), j=r, n=2, p=2, and u) holds.

Otherwise C is of type C, and SC= 2~~; then (3.2) reads

i

2pj if i=E mb,i G

0 if ids (modP”-1),

for some o<j<n. Since / m,,i / 5 2, /3 must be a multiple of a, so p = - r-, and either

a) or e) holds. Q .E.D.

Now we consider the Galois equi\,alences between roots and minimal dominant

Iv-eights. We may assume the minimal dominant weight ii is not in the root lattice.

Then A = A~ is fundamental and C is not of type G,. Thus for u.EC, 1 m,,jl 5 2. If

we assume q>3, the congruences m,,j ro (modq--r) for j=l=z’ imply cr.=mh for

m=k2. By the lists in $ I, C must be of type C, (!>I). Since pj-m=o (modp”--I)

for some osj<n, p=2. If m = - 2, necessarily n= 2. We summarize:

Proposition (3.4). - Assume C is indecomposable and q>3 (‘“). If ?,EA,, ?,+ZS, and ;f AwaGE, then C is of ape C, (&‘>I), A=>\,, and p=2. If q==+ a=hzh(, while

;f q>4, a=2AI.

(3.5) We remark here that, for q>3 and AEA,, 1,+0. It follows that

dim, Vi = dim, V0

\+vhere Vi is as in $ I and V,, is as in 4 2.

4. Examples.

With the bounds developed so far, it is an easy matter to compute H’(G, V) for

irreducible KG-modules V=V(A) lvhere i. is a minimal dominant weight. I2:e ha\c

the following possibilities for A:

(4. Ia) A is not Galois equi\.alent to a root;

(4.16) i, is not a root, but is Galois equivalent to a root;

(4. xc) A is the maximal short root.

(?*) N%cn k = GF(3) the Gal& equivalences al-c covered by footnote (*‘).

IS1

181 EDWARD CLINE, BRIAS PARSHALL, LEONARD SCOTT

If p=2, G’ is universal of type C, (C 2 I), and C* is universal of type B, ; let

: : T* , +G” be the isogeny defined by the special isomorphism Y(T*)OQ: r(?)@Q

given by p(xJ = 2Zl and T(c+)=& for i> I [7; Esp. 18, 231. Then P(A,)=~, if

P> I, while I = 2X1 =Zl if P = I. Also, the restriction of L to G is an isomorphism

from c onto G. 3-0~ by (3.4), (4.rb) occurs or+ zhen p = 2, C Zs of &be C, and A = A, ;

hence in order to treat case (4. I b) , it srlfJe.s to consider the cases zejhen G is of ppe B,(2’“) and

A=& for !>I, or of ppe A1(2”), h=2).,=a, for C=I. This reduces (q.Ib) to (4.1~).

Theorem (4.2 ). - Assume q> 3. If A satisjes

a) (4. Ia), then H’(G, V)=o;

b) (4. Ib), then H’(G, 1:) E X0.) where ?.=I.., is a fundamental weight of the dual

system of ppe B, (see the above paragraph);

C) (4. IL), then H’(G, 1’) -‘X(A), or q== 5 and G=X,(5) in which case H’(G, 17) is I -dimensional.

Proof. - Since A is minimal, e\-cry nonzero weight of T in V is It’-conjugate to A.

If A satisfies (4. IU) and BEX:, then np =o and (2 .g) implies H’(G, V)=o.

For the remaining cases, we assumejrst that q>4, and if q = 5 or g, G is not of ppe A,.

Since \ve ha\.e shown that (4.16) reduces to (4. IC), we assume A is the maximal short

root in C. BY (I.21

dim, X(A) 5 dim, H’(G, V).

If C has two root lengths and y~&~~, (3.3) implies ny = o, while if ~~~~~~~~~

ny=I by (3.3). Hence by (2. g) (23), if A’= A n CJhoti

dim, H’(G, V) L,F&n, -n,.

By (1.1) and (3.5), ]~‘/=n,fdim,X(h); hence

dim, X(A)~dim, H’(G, V)Ldim, X(h).

Now assume q =4 and G is not of type A,. \Ve show

(4.3) if

dim, Z’(U,, V)T< =%Ort,

- otherwise.

If- ~qooq, (4.3) follows from (3.3). Assume rl.~ X’ahort. Let G,=(U,,U-.) and

V,,,=XG,, then V,,, E\‘-.@V,,@V,, and every proper G-submodule of V,,, is

contained in (V,,,),; consequently V,,, has a unique maximal proper submodule

x E (V(x)) 0 * By (2.7a), Z’(U,, X)T = 0, hence (2.3e) yields an injection

0 + WJ,, y,,Y -+ WJ,, yJT

(23) Here we are using the fact that U=/Ujllz~lj when 4>3. It suffices to consider the rank 1 case. It is almost trivial for Ap and Cl,, while it follows for G, by (3.3) and the fact that T normalizes (U,la~h).

COHO\fOI.OGS OF FISITE GROCPS OF LIE TYPE, I ‘83

where v,,, =V,,,/X. By (2. ~a) again, Z’(U,, V/VIIJT= o, hence (2.3e) yields

WJ,, VT = WJ,, V(,JT,

so it suffices to show

(4.4)

Now vCa, is an irreducible kG,-module. Since the torus T,=Tn G, acts with

weight a on V,, it follows v(Z) is n-dimensional. Set ~*a=(~(b])*cc.

If Y~wJa, &)lT, the homomorphism u~~uy(u) of (2.2) yields (uY(u))~=I,

whence u fixes y(u). Writing y(u) = u=(u) + u-=(u) where ZJ* =(u) ET* ~, we see that

V-~(U) is fixed by u since u,(u) is automatically fixed by u. Since T acts irreducibly

on U,, u-, (u) is fixed by U,. But since vta, is z-dimensional and U, is non-trivial,

U, fixes a unique line in V,,,. Necessarily u-~(zJ)=o for all UEU,. By (2.3~7)

o-tzyu,, V~)T-+Z1(U~, v(,,)T:zl(U~, v(a)/,,,lV,)T is exact. We hav,e just shown YC is the zero map, so (4.4) follows from (2.6).

BY (2.8)

dim, H’(G, V)<aFA,dim, Z’(U,, V)T--n,.

Hence dim, H’(G, V) 5 11’ [ -n, = dim, X(A).

For the remainder of the proof, we consider the exceptional cases A,(q), A,(5)

and A,(g). Assume jirst that G=A,(4), h=&+h. Let y~2?(U, V)‘. The homo-

morphism (2.2) and the fact that U, has exponent 2 imply Y(x&E))EV% for each

~EC, (EA. This yields the following form for y:

Y(x~,(5))=v,,,,(5)+v,,,.~(E)+v,l,.1+.,(F)+v,,,-,,(F),

(4.5) Y(~~,(~))=v,*,,(~)~v,,,.1(~)+vcL*,2~i~1(~)+~~,,-~,(5)~

Y (xq + a, (~))=v?,t11,0(5)+v,,+.,,.,+.,(E)+v,l+.l,.1(5)+v,,+~,,.,(4),

where for each appropriate pair 9, VEC u(o)

%> y : k--f (v:)k

is an additive homomorphism lvhich, because of the T-stability of y, satisfies

(4.6) vp,,(?(05) =v(t)va,.(t) for teT.

Thus, us,0 =o for p>o.

Next consider the maps 0,, e : (Vi), --f (Vi), defined for p>o, F;EK by

O,, &w) = w- wxa(E) for we,.

If /3>0 and 6, yEkX, then

ker(eB, E)=ker($,J=i=o

while

183

184 E D LV A R D C I. I h- E:, B R I .A N P r\ K S H c\ I. I>. 1, E 0 N .-I R 1) S C: 0 T 1

It follows that 1i.e may adjust :’ by a T-stable I-coboundary to eliminate u,~, ?, and

V II, a,; thus

:/(x*,(~))=v,,,z,..,(~)+L’,,,-,,(~),

(4.5’) Y(%,(4 = u,,, Lx+ .*(-) + U?,, -&),

‘l+I, T .,W) =v,, LIZ, r,II,(?) A- V,&.,, J3-r) tz’x,+l~, &T),

for 5, I-EIZ.

If we assume uZ,, --x y *o, and compute the lTz,- component of :r( [x,,(5), x,%(7)])

using (2.2), (4.5’), (4.6) and [26, Lemma 72, p. 2og] ( or the fact that we know explicitly

the action of G on \’ here), \ve obtain a non-zero \.ector z.YE\?,, and constants A, B, C

with B+o such that

for all 5, YE-h. This is a contradiction, hence zj,>, --1 =o. X symmetric argument

s110ws v,,, -3 = 0. Since V c p, 1) - t, - a recompLitation of yi [x,,(E)> xs,(7)]) shob3 -i z 0 on U,, i- It)

thus

(4.5’!)

Now let [+H’(G, V), and choose 7 so its restriction *( to U is T-stable, hence,

after adjustment by a T-stable coboundary, so that %f satisfies (4.5”). BY (2.3d)> -lL -: 11; n cm- y/UnGm for every WE!V. Let LV==W,+W~,; then

UnU’“=U,, and p=(y”-y&.,,

is a T-stable coboundary, hence there exists a vector CE\’ such that

p(x,,(:)) = v-zx,,(;) for Ffck,

and vt- L’E\rbi,: for tET.

Since T is a 2’-group, we may choose v so that UE(V;I)~, whence

dJ,,) E K*)k *

However, direct computation of p shows P(U,~) c (\7y,,)k +(V& x,)k, hence p = o.

Now from (4.5”)

?“(U,,) =TW,,) s P’L,),n PC,+ a,)k= 0,

so $.a* = 0. By a symmetric argument, ylv,, = o, so it follows from (2.35) that

H’(G, V) = o completing the proof in this case.

Suppose next G=&(5), and A=u,. By (2.9) and (3.3), we have

dim,H’(G, V)<n=-n,,=~.

On the other hand, a computation shows that the module S(6A) formed as in 4 I is

indecomposable, and has a unique submodule .J of dimension 4 which contains V. J is

181

COHOMOLOGY OF FINITE GROUPS OF LIE TYPE, I ‘85

indecomposable and an application of the long exact sequence (2.3e) as in (I .2) yields

a lower bound of I on dim, H’(G, V) completing the proof in this case (‘4).

Finally suppose G=A,(g), and ?,=x,.,. Let y~Z’(u, V)T. Then by (2.35) it

suffices to show y-0. As in the case G-A,(q), write

Y(G,(W = Q(S) + VA0 + Ll(S)

where up : k+ (V;), is a function satisfying z+(~~(t)S)= F(t)up(E), for EEA, kT.

Since V/(v’,l +-Vi), is a trivial U-module, v-, is additke. If 0 is a primitive 4th root

of unity in GF(g) and if we write 5 = a+ b0 ’ (a, BcGF(3)), then

v+,(4) = (a + W’)x,,( I) = t3u-aL( I).

By computing the component of y(x,,(t +T)) in (VG), in t\vo r%vays for 5, TEGF(g),

we obtain

~,(4+~)=~,(s)+v,(~)+F;3~~~,,(I)x,,.

If VL1(I)SO, LI(I)X,,f 0, hence by the symmetry of ~(5 + 7) in < and 7, <“T = 73<

for all 5, I-ok, a contradiction. Hence vWa, = o. Also v,, is a Z,(T)-homomorphism,

hence z+,= o. For some <sky, we may choose v,-,~(V~), such that

~f(X,,(E))=v,,(F)=vo-vox~~(E).

By T-stability, -{(x,,(p)) = v~--z~,,x,~(~) for all p, so y-o. Q.E.D

We summarize our results in a table:

TABLE (4.5)

Type

Char k =p

9>3 Dominant \$‘eight dim, V dim, H’(G, V)

Al

2

2fq\ 9=5

i 4>5

Xl = 2Al(Al) 2 I(I)

El = 2% (4) 3(z) I(O)

‘% = 2~1(~1) 3(g) 40)

arbitrary Ai (‘?‘) 0

A* (P’I) (P+-I,P)=1 i*=A,+A( (l+I)2-I 0

(Pi-I,P)=P P (e-t 1)2-2 I

(“) .4n alternate proofin this case is as follows: If R denotes the permutation representation of &(5) on 5 letters over k=GF(5), then its restriction to a Sylow 5-subgroup is evidently the regular representation, hence R is indecomposable. It evidently contains unique submodules 31 and m of dimensions I and 4 respectively, and 51/rnE-V. Kow the exact sequence (2.2) applied to R/m yields a lower bound of I for the dimension of H1(G, V) as in (1.2).

24

TABLE (4.5) (suite)

Type

Char k =p

4>3 Dominant FVeight dim, V dim, H’(G, V)

b (!>2)

arbitrary

2

odd

> ‘1 2( 0

v = A I 21 I

v ne+r 0

2 A, 2P I

c, ([>I) odd

(e> P) =P

1.f 2P 0

V =A,-, (e-I)(ae+I)-I I

‘I 0

arbitrary

odd

A,; i=I, 2,t

i*=A,-1

re; i=e,

2(-l; i+,P

(nr-1)e

2!(4!--I)-2

(2!+ I) (@+ I)-1 I

arbitrary 4, A6 27 0

E6 3 i” = I.2 77 I

(3, PI=1 P 78 0

arbitrary 17 56 0

ET 2 CL =A1 132 I

(2, PI=1 CL 133 0

Et3 arbitrary CL=% 248 0

3 v I F4 =I., 25

(P> 3)=x V 26 0

2 v=A, 6 I G2

(A 2)=1 v 7 0

COHOMOLOGY OF FINITE GROUPS OF I.IE TYPE, I ‘87

5. A result on Ext for SL(2, 2”).

Let G=SL(2, 2”), n.21. Let CT be the automorphism of G induced by the field

automorphism t;,t’, t~k = GF(2”). Let p be the irreducible representation of G over k

defined by the dominant weight A=?,, (5 I), and set pi= pea’ for oli<n. Let hl,

be the KG-module corresponding to pi.

Theorem (5.x) [16; p. 291. - Ext’(l&, l,IJ=o, for all i,j.

Proof. - Since ExtO(V, V’) 2 Hom,(V, V’) N (VTOV)G N HO(G, VOV’) for all

finite dimensional kG-modules V, V’, we have Ext’(V, V’) = H’(G, <rOV’) by a standard

dimension shift. Since each of the modules Mj is self-dual

Ext’& r\$) N H’(G, h,f;O Mj).

Assume first that i+j, say i>j. The weights of T in 11, are i lib, hence the

weights of T in V=51,0Mj are p1=(pi+2j)h, p2=(2i-2j)h, p.3=-p2, pa=-pl.

This module V is the irreducible KG-module defined by the dominant weight (li + 19)~

([26; Thm. 431 or [25]). Since ;>j,

(5.=3) t*1 * a1 = 0: ;

(5.2b) p2-tl-p3 iff n=2, i=I, and j=o (“).

Since V is self-dual, it has [g; COI. I. 5g] unique KU,-submodules m =VW, and hl of

dimensions I and 3 respectively. If H1(G, V)+ o, the same is true of Hl(U,, V)’

by (2.39). Since T is a 2’-group, there exists a non-zero T-stable cocycle y : U,+W.

If CEK, (x,(c)~(x~(c)))~=I by (2.2), 1 ience y(x,(c)) lies in the centralizer C&(x,(c)).

(5.3) If c*o, C&(x,(c)) is not T-stable.

Otherwise, for tcT, C,(x~(c))‘=C,(x,(c)‘) =C,(x,(x(t)c)), hence

C&(X~((C)) =C,(U,) = m

for every c*o. Now y : U,-+m is in Z’(U,, m)T, contradicting (2.7~) and (5.2~).

\Ye leave the exceptional case n = 2, i = I, j = o of (5.2b) to the reader (%),

and assume henceforth p( * v. for some e = 2 or 3. By (5.3) (or by tensor product

considerations) V is not a uniserial kTU,-module, and so there is a a-dimensional

kTU,-submodule V, of 11 with (Jl/Vr)ll, = ?\I/V,. Sow y(x,(c)) lies in (&(x,(c)) C1I;

(a) If pi-a, then 2f(2i+2j)z~ (mod 2”-I) for some I, o<l<rl. If r, 5 are least residues of i $ I and j-r I respectively, then 2r+ 2s- I E o (mod 2” - I ). This implies r =s and ij (mod n), lvhence i =j, a contradiction.

Suppose ~~NGL~~~; then for some o<f, m<n, we have 2((2;-2j)E 1~ 2m(2i-li) (mod 2”--1). Setting e=i-j andf= IP--ml weobtain 2e-~~ 2U(mod2’1-~) for some o<u <n, and (21-+1)(2~-1)~0 (mod 2”-I). The first congruence implies 21-t 1 s o (mod 2”- I), lchence since oa<n, f =I and so n=2 as required.

(*s) Actually it is obvious that Hl(G, V) = o here, since V is the Steinberg module, which is projective [IO].

188 E D \\. A R D C L I X E, B R I -4 N P .I R S H A L L, L E 0 N .A R D S C 0 T 1’

and by (2.7a) the projection of y on 11/V, is o, whence ~c takes values in V,. Because

of (j.3), we must have V,nC,(x,(c))=m. But now y takes values in m, again contra-

dicting (2.7~) and (5.2~).

Assume i = j, and q>2 (we leave the case q =2 to the reader (z’)), then

V= Xi@>& is a Galois conjugate of ~loO~LIo = Xl09 11, zHom,(>l,, 11,). By foot-

note g (or (2.3~)) it suffices to treat the case V= Hom,(M,, 11,). Evidently V has

unique submodules m and hf of dimensions I and 3, and 11/m is a Galois conjugate

of MO. Applying (2.3e) to o+m+V+V/m+o gives o+Hi(G, V) -H’(G, V/AI)

exact. Applying (2.3e) again to the sequence o-+Sl,lm-~\r/m+Vj;\I-~o yields

dim,Ht(G, V/m) = dim,H’(G, 11/m) -I. By (4.2~) dimkH1(G, M/m) = I (‘“). Q.E.D.

6.. Action of Hecke algebras on cohomology.

Let A be a finite group and \’ a kX-module. 1l’hen B is a subgroup of A whose

index [A : B] is not divisible by the characteristic p of k, the restriction map

Hl(X, V) -+ H’(B, V) is injective and its image consists of stable classes, as mentioned

in (2.35, d). Denote here the collection of stable classes in H’(B, V) by H’(B, V)B\2”B.

Then the stability theorem of Cartan-Eilenberg [j; p. 2391 asserts

H’(A, V) z H’(B, V)B’a’B;

indeed Cartan-Eilenberg prove the analogous result for n-dimensional cohomology,

ncz.

Our notation suggests that the stable classes are the “ fixed points ” for an action

of the Hecke algebra on H’(B, V). In fact the Hecke algebra B\A/B does act naturally

on H”(B, V) for all integers n, all subgroups B of A, and all ZA-modules V. If, as in

the present case, V is a p-group and p r [A : B], then the stable classes may be interpreted

as “ fixed points ” of this action.

JVe sketch the details. The Z-linear combinations in the rational group algebra QA

of elements -!- BuB form a Z-algebra B\.4/B. PI __

Here BaB denotes the sum of all members

of the B, B double coset BuB of A. For ~EH”(B,V) define

where lB is corestriction (defined in dimension o to be a sum of multiplications by coset

representatives) (s”). It is easily checked that this defines an action of B\A/B on HO(B, V)

for all V, also l%O(R, V), hence on H”(B, V) by dimension shifting.

lVhen i’ is a p-group and pr [A : B] the stable classes may be interpreted as

“ fixed points ” for B\A/B in the following way: There is a natural homomorphism

(?‘) Again this case is immediate since hi, is the Steinberg module. (z8) It should he noted that, for q>.g, the upper bound in (4.2~) depends only on (2.8;. (?“) A similx definition appears in Shimura, Infroduction to the adhrnetic theory of aukmorphic fun&nc, Prin-

ceton University Press, 1 IYJ j 1.

COHOlIOLOGY OF FINITE GROUPS OF LIE TYPE, I ‘89

from the Hecke algebra B\A/B into Z which may be obtained from the augmentation

QA+ Q; the value of this homomorphism on _f_ BaB is [B : B”n B]. PI-

So if IZI is any

module for B\A/B it is reasonable to call rnE 11 a “ fixed point ” for B\A/B provided

m ( 1

L BaB = [B : B”nB]m for all UEA. IP-

Now observe that in case hf is the set of

B-fixed points of an A-module 1’ w-e have m(~&~)=mlAjB for any rnEM (in

the sum a ranges over a set of B, B double coset representatives). It follows easily that

multiplication by C -!- BaB is the same as IA 1 a on H”(B, V) for any 12. a IBI-

So if rnE H”(B, V)

is a “ fixed point ” for the action of B\A/B we have m IA 1s = T [B : Ban B]m = [A : B]m.

In the present case (pr [A : B]) th is implies that m is the restriction of an element of H”(A, V),

and so certainly a stable class. On the other hand all stable classes are obviously “ fixed

points ” for the action of B\A/B, since restriction followed by corestriction is multiplication

by the index. Thus we have shozon that, when V is a p-group and p7 [A : B], the stable classes

are precisely the “jxed points ” of the Hecke algebra B\A/B.

The homological “ explanation ” for this Hecke algebra action seems to be the

fact that Exti(TIA, V) z Exti(T, VIB). Here IA and 1s denote induction and restriction.

Ext;(TIA, V) is obviously a HomA(TIA, T]*) module. In case T= Z with trivial

B-action, Hom,(T iA, TIA) is well-known to be isomorphic to the Hecke algebra, while

Ext;(T, V]s) g H”(H, V]s).

Finally we mention that there seems to be no corresponding Hecke algebra action

for algebraic K-theory. It is of course possible to formally reproduce the definition

p IBaB =$]B”nR]B (IBI -1

but this simply does not define an action of the Hecke

algebra B\‘/B-not even on K,(CB) xvhere C = complex numbers, B = cyclic group of

order two, A =symmetric group on three letters.

We conclude this section Lvith an application of the Hecke algebra action. .4

somewhat less ob\.ious proof can be given directly.

Corollary (6. I ) (“O). - Let G be aJinite group with a BN-pair (see [2; Chapter IV]),

and let V be a kG-module. Let \Y=X/(B n N) be the W’ql group of G, and let {w,}, E 3 be

the associated set of fundamental reflections. Assume that the characteristic p of k does not

divide [G : B].

Then a necessary and sufjcient condition that a class

PE H’V, V

be stclble (pfc(B1’.nn=pjB~nB for all WE\Y) is that pW~IB~xnB=p.IB~*.nB for EELI.

iao) This result has been obtained independently by George Glauberman.

189

‘90 ED\V.iRD CLINE, BRI;\X PARSH;\LL, LEON.\RD SCOTT

Proof. - The hypothesis implies

p =[B :BnB%]p

for each a~&,. Hence TV- is a “ fixed point ” for the subalgebra generated by the various

LBw B. PIA

It is an easy exercise (“I) to show from the axioms for a BY-pair that this

subalgebra is in fact the full Hecke algebra. Q .E.D.

REFERENCES

[I] A. BOWL, Linear algebraic groups, New York, Benjamin (rg6g),

[2] ?;. BOURB.AM, G~oirpes et nI,odbres rle Lie, chap. I\‘, V, VI, Paris, Hermann (1968).

[3] S. BURGOYNE, hIodular representations of some finite groups, Proc. of Symposia in Pure Math., vol. -XXI,

Representation theory of Jinite groups and related topics, A.M.S. (1g71), 13-17.

[4] P. CAMERO?I and J. SEIDEL, Quadratic forms over GF(2), Indog. .\fath., 35 (tg73), x-17. [j] H. CAKTAN and S. EILENBERG, Homolo,oicnl nl,oebm, Princeton University Press (1956).

[6] R. CARTER, Simple grotrpr of Lie ~vpe, New York, John Wiley & Sons (1972). [7] C. CHEV.~LLEY, Cinssijicntion des groupes de Lie alg+!briques, vol. 2, Secretariat math., I I, rue P.-Curie,

Paris (tgj6-Igj8).

[8] -, SW certains groupes simples, To^hoku hfoth. Journal, 7 (Ig55), 14-66.

[g] C. IV. CURTIS, Irreducible representations of finite groups of Lie type, JournalJiir Math., 219 (rg65), I8o-Igg-

[IO] -, The Steinberg character of a finite group with a (B, N)-pair, J. Algebra, 4 (rg66), 433-441. [I I] E. B. D~~KIN, Semisimple subalgebras of semisimple Lie algebras, Am. Math. Sot. Tranrl. Ser., 2, 6 (x957),

I I r-245 (=Mat. Sbornik K. S., 30 (rg52), 3-19-462).

[12] H. FREUDENTHAL, The existence of a vector of weight o in irreducible Lie groups without centre, Proc. A.M.S.,

7 (1gj6), 175-176. [13] D. HERTZIG, Cohomology of algebraic groups, J. Algebra, 6 (rg67), 317-334. [t4] -, Cohomology of certain Steinberg groups, Bull. A.M.S., 75 (1g69), 35-36. [Is] D. G. HIGM.AX, Flag-transitive coliineation groups of finite projective spaces, Illinois J. of flfath., 6 (Ig62),

434-446. [16] G. Hrc-rr.~, Odd charncteritations ofpile simple groups, University of hfichigan notes (1968).

[17] J. HUYPHREYS, hlodular representations of classical Lie algebras and semisimple groups, 3. Algebra, 19

(197’)> 51-79. [x8] N. J.ACOBSOX, Lie algebras, iXew York, Interscience (1962).

[Ig] S. ?vl.<c LANE, Homology, New York, Academic Press (1963).

[20] D. .\IUMFORD, Introduction to algebraic geometry, Harvard notes. [21] B. PARSHALL, Regular elements in algebraic groups of prime characteristics, Proc. A.ltl.S., 39 (Ig73), 57-62.

[22] H. POLLATSEK, First cohomology of some linear groups over fields of characteristic two, Illinois 3. of Math.,

15 (197Ih 393-417. [23] D. QUILLEN, On the cohomology and K-theory of the general linear groups over a finite field, Annals oIf &fath.,

93 (Ig70), 5j2-j86.

(31) It is enough to show -!- BwB i Bru B = -!- Bww B whenever I(ww,) > I(w) and 1u, is a fundamental /B~---IJBI---I_

reflection. lB/----L

Since (BwB)(Bw,B)=Bwuw,B here, the above equation is certainly true up to a scalar multiple of

the right hand side. X formal calculation shows this multiple is If this number is

not I, then B’O n BIW Q B

h 1 BwrB A Bw-‘Blow, I.

T J so BwnBw,Bf a, which is impossible since (BwB)(Bw,B) ABWB = 0.

190

COHOhlOLOGY OF FIXITE GROUPS OF LIE TYPE. I 19’

f24] T. A. SPRINGER, Weyl’s character formula for algebraic groups, Inuent. .If&., 5 (Ig68), 85-105.

[25] R. STEINBERG, Representations of algebraic groups, Nagoya Math. J., 22 (cg63), 33-56.

1261 -, Lecfures on Chevalley groups (mimeographed notes), Yale University (1968).

[27] 0. TAUSSKY and H. ZASSENHAUS, On the r-cohomology of the general and special linear groups, ,4equntiones

math., 5 (IgTo), x29-201.

[28] J. THOMPSON, Quadratic pairs (unpublished).

[zg] J. TITS, Reprksentations 1inCaires irrkductibles d’un groupe rPductif sur un corps quelconque, Journalfir Math.,

247 (IgTr), xg6mo.

1303 W. J. WONG, Irreducible modular representations of finite Chevalley groups, 3. Algebra, 20 (tg72), 355-367.

Manuscrit rep le 20 juillet 1973

Rhist le 15 dhembre 1973.

191