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.
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190
COHOhlOLOGY OF FIXITE GROUPS OF LIE TYPE. I 19’
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Manuscrit rep le 20 juillet 1973
Rhist le 15 dhembre 1973.
191