On comparing the cohomology of algebraic groups, nite Chevalley groups and Frobenius kernels

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1 Journal of Pure and Applied Algebra 163 (2001) On comparing the cohomology of algebraic groups, nite Chevalley groups and Frobenius kernels Christopher P. Bendel a, Daniel K. Nakano b; ;1, Cornelius Pillen c a Department of Mathematics, Statistics and Computer Science, University of Wisconsin, Stout, Menomonie, WI 54751, USA b Department of Mathematics and Statistics, Utah State University, Lund Hall, Logan, UT , USA c Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA Received 28 April 2000 Communicated by E.M. Friedlander Abstract Let G be a semisimple simply connected algebraic group dened and split over the eld F p with p elements, G(F q) be the nite Chevalley group consisting of the F q-rational points of G where q = p r, and G r be the rth Frobenius kernel of G. This paper investigates relationships between the extension theories of G, G(F q), and G r over the algebraic closure of F p. First, some qualitative results relating extensions over G(F q) and G r are presented. Then certain extensions over G(F q) and G r are explicitly identied in terms of extensions over G. c 2001 Elsevier Science B.V. All rights reserved. MSC: Primary 20C; 20G; secondary 20J06; 20G10 1. Introduction 1.1. Let G be a semisimple simply connected algebraic group dened and split over the eld F p with p elements. Let k denote the algebraic closure of F p. Let F : G G (1) be the Frobenius map. For a xed r 1, let G r be the kernel of the rth iteration of the Frobenius map. Furthermore, let G(F q ) where q = p r be the nite Chevalley Corresponding author. Tel.: ; fax: addresses: bendelc@uwstout.edu (C.P. Bendel), nakano@sunfs.math.usu.edu (D.K. Nakano), pillen@mathstat.usouthal.edu (C. Pillen). 1 Research was supported in part by NSF grant DMS /01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S (01)00024-X

2 120 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) group consisting of the F q -rational points of G. For over 30 years, there has been a considerable amount of eort in relating the representation theory of reductive algebraic groups, Frobenius kernels and nite groups of Lie type in the dening characteristic, for example [5]. In this paper we will be primarily interested in the following three questions: (1.1.1) Can all extensions between G r -modules be found via an extension theory for G? (1.1.2) Can all extensions between G(F q )-modules be found via an extension theory for G? (1.1.3) Given two rational G-modules M and N, is there a relationship between Ext G r (M; N ) and Ext G(F q)(m; N )? Humphreys [9] and Andersen [1, p. 388] rst posed question (1:1:2) for Ext 1 between simple G(F q )-modules. Andersen was able to answer this question armatively for simple G(F q )-modules in several generic cases (see [1, Theorem 3:2]). The methods employed in this paper will be functorial and dierent from the ideas used in the past. In order to answer questions (1:1:1) and (1:1:2) we compare the category of G(F q )-modules and G r -modules with truncated categories of G-modules which contain enough projective objects. This will be accomplished by constructing two Grothendieck spectral sequences. These truncated categories of G-modules are highest weight categories, and are thus equivalent to module categories for nite-dimensional quasi-hereditary algebras. Our procedure will also demonstrate why it is easier to provide an answer to (1:1:1) as opposed to (1:1:2) for simple modules (see the Corollary given in 6:2). The idea of comparing the cohomology of nite groups with extensions of modules over certain nite-dimensional algebras was also used earlier in work of Doty, Erdmann and the second author [7]. In that paper, it was shown that one can construct a spectral sequence involving extensions for the Schur algebra S(n; d) (n d) to extensions for the symmetric group on d-letters d. Results comparing the cohomology in these two categories involving specic classes of modules are given in [14]. Applications involving computations of cohomology are provided in [8,17]. Question (1:1:3) is the most intractable question of the three. We will show that ideas from [15] can be used to provide a vanishing criterion of cohomology for nite Chevalley groups in terms of vanishing of Lie algebra cohomology. These results only work in the case when r = 1, but have the advantage that there are no strong restrictions on the size of the prime Now we will describe the contents of the paper in greater detail. The results in Sections 2 and 3 provide qualitative results involving the relationship between the cohomology of G(F q ) and G r. In Section 2, we give a criterion for projectivity of modules over G(F q ). A criterion for G r was given earlier in [3, Theorem(4:2:1)]. This criterion involves induced modules, Weyl modules for G and principal series modules for G(F q ). In the following section, we investigate sucient conditions to insure the vanishing of Ext n G(F p)(m; N ) for M; N mod(g). This is given in terms of the vanishing

3 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) of certain weight spaces in the cohomology for U 1 where U is the unipotent radical of the Borel subgroup B of G. The remainder of the paper is devoted to providing answers to questions (1:1:1) and (1:1:2). In Section 4, we rst dene a saturated set of weights s in X (T) + and let C s be the full subcategory of G-modules whose composition factors all have highest weight in s. We proceed by constructing two functors G s ( ) and H s ( ) fromthe category of G(F q )-modules and the category of G r -modules, respectively, into C s. For M C s and N 1 mod(g(f q )) and N 2 mod(g r ) two spectral sequences are obtained: E i;j 2 = Exti G(M; R j G s (N 1 )) Ext i+j G(F (M; N q) 1); (1.1) E i;j 2 = Exti G(M; R j H s (N 2 )) Ext i+j G r (M; N 2 ): (1.2) The higher right derived functors R j G s and R j H s are of particular interest since these provide us with vital information about the connection between these categories. The composition factors of these functors when applied to a module can be found by calculating extensions (in G(F q )org r ) between the module and the appropriate projective cover in C s. The constructions in Section 4 allow us to prove stability results in Section 5. In particular, the Theoremof 5:2, the Corollary of 5:3, and the Theoremof 5:5 by and large provide an answer to questions (1:1:1) and (1:1:2) given in Section 1. In Section 6 we look at the image of H s on a simple module and show in the Theoremof 6:1 that this is a semisimple G-module for large enough primes p. As a byproduct one obtains a formula in the Corollary of 6:2 which allows one to compute Ext 1 between two simple modules in mod(g r ) by knowing Ext 1 between simples in mod(g). We demonstrate in Section 7 that the situation for nite Chevalley groups is not as straightforward as in the Frobenius kernel case by looking at G s on simple modules. Our analysis shows that one can recover the isomorphisms given in [2] (see the Corollary given in 7:1) in a more functorial manner via our approach. The Theoremgiven in 7:6 generalizes Andersen s results for the generic case [2, Theorem3:2] allowing for a larger class of modules and extensions of higher degree. Furthermore, in the Theoremgiven in 7:5 we are able to give a complete description of the cohomology of the nite group G(F q ) and all its simple modules in terms of cohomology for the algebraic group with simple modules in the truncated category C s, provided the prime is large enough Notation. Unless otherwise stated G will always denote a semisimple simply connected algebraic group dened and split over the nite eld F p with p elements for a prime p. Here k denotes the algebraic closure of F p. Let be a root systemassociated to G with respect to a maximal split torus T. Moreover, let + (resp. ) be positive (resp. negative) roots and be a base consisting of simple roots. Let B be a Borel subgroup containing T corresponding to the positive roots and U be the unipotent radical of B. The Euclidean space associated with will be denoted by E and the inner product on E will be denoted by ;. The Weyl group W is the group generated by reections

4 122 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) associated to the root system and the ane Weyl group W p is the group generated by W and translations by elements in pz. Let X (T) be the integral weight lattice obtained from. The set X (T ) has a partial ordering dened as follows. If ; X (T) then if and only if N. If =2= ; is the coroot corresponding to, then the set of dominant integral weights is dened by X (T) + = { X (T ): 0 6 ; for all }: Furthermore, the set of p r -restricted weights is X r (T )={ X (T ): 0 6 ; p r for all }: The ane Weyl group W p acts on X (T ) via the dot action given by w =w(+) where w W (w W p ), X (T ), and is the half sumof positive roots. Let H be an ane algebraic group scheme over k and let H r =ker F r. Here F : H H (1) is the Frobenius map and F r is the rth iteration of the Frobenius map. We note that there is a categorical equivalence between restricted Lie(H)-modules and H 1 -modules. For any group scheme H, let mod(h) be the category of nite-dimensional rational H-modules. If r 1 and q = p r, let H(F q ) be the nite group obtained by taking the F q rational points of H. For a reductive algebraic group G the simple modules will be denoted by L() and the induced modules by H 0 () = ind G B, where X (T) +. For the innitesimal group scheme G r, the simple modules will be denoted by L r (). If X r (T), then L r () = L(). The induced and coinduced modules are given by Z r() = ind Gr B r and Z r () = coind Gr B r for X r (T ). If X r (T), the injective hull of L r () inmod(g r ) is Q r (). This is also the projective cover of L r () asag r -module [11, II. 11.5(4)]. For the nite group G(F q ), the simple modules are L() for X r (T). Moreover, for each X (T (F q )), the induced module is given by M r () = ind G(Fq) B(F. q) 2. Projectivity results 2.1. It is not known how to construct the simple G r -modules nor how one can compute the dimensions of these modules. So, for practical purposes it does not make sense to test a module for projectivity by showing that extensions between the module and the simple modules are zero. There are other canonical modules like Weyl modules, induced modules and Verma modules whose construction is well-known and whose character can be determined. In [3, Theorem (4:2:1)], it was shown that one can test projectivity for a module over G r by examining the extensions between the module and a family of canonical modules. The result is presented below. Theorem. Let G be a reductive algebraic k-group scheme and be a set of representatives for X (T )=p r X (T ). Moreover; let M mod(g). The following statements are equivalent: (a) M is projective in mod(g r );

5 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) (b) Ext i G r (M; L r())=0 for all i 0; (c) Ext i G r (M; H 0 ())=0 for all i 0; (d) (i) Ext i G r (M; Z r())=0 for all i 0 (ii) R i ind G B (Hom Br (M; )( r) )=0 for all i 0; (e) Ext i G r ( L r();m)=0 for all i 0; (f ) Ext i G r ( V ();M)=0 for all i 0; (g) (i) Ext i G r ( Z r();m)=0 for all i 0 (ii) R i ind G ( B HomB + ( r ; M)( r)) =0 for all i 0. Moreover; statements (a) (c), (e); and (f ) are equivalent for M mod(g r ) In general, the simple G(F q )-modules have also not been determined. The following result shows that projectivity over G(F q ) can also be checked by using principal series modules, Weyl modules or induced modules. Theorem. Let be a set of representatives for X (T)=p r X (T); and = X (T(F q )). Moreover; let M mod(g(f q )). The following statements are equivalent: (a) M is projective in mod(g(f q )); (b) Ext i G(F q)(m; L r())=0 for all i 0; (c) Ext i G(F q)(m; H 0 ())=0 for all i 0; (d) Ext i G(F q)( V ();M)=0 for all i 0; (e) Ext i G(F q)(m; M r())=0 for all i 0. Proof. If M mod(g(f q )) then (a) (b). Furthermore, (a) implies (c) (e). In order to nish the proof we show that (e) (a), (c) (b) and (d) (b). (e) (a): Fromour assumption and Frobenius reciprocity we have ( 0 = Ext i G(F q) M; ) ( r () = Ext i B(Fq) M; M ) : Therefore, M is projective in mod(b(f q )), and thus projective in mod(u(f q )). Since U(F q )isthep-sylow subgroup in G(F q ), it follows that M is projective in mod(g(f q )). (c) (b) and (d) (b): Dene the following order relation on X (T). Let 6 Q on X (T) if and only if is a non-negative rational linear combination of simple roots. Moreover, let C Z = { X (T ): 0 + ; 6 p 1 for all + }: We will prove the statement by using induction on the order relation. Assume rst that condition (c) holds. If C Z then H 0 ()=L r (), so Ext i G(F q)(m; L r ())=0 for all i 0 for all C Z. Now for an arbitrary assume that Ext i G(F q)(m; L r ())=0 for all i 0 and Q with. There exists a short exact sequence of G-modules given by 0 L r () H 0 () N 0: All composition factors L() of N satisfy. If then by induction Ext i G(F q)(m; L r ())=0 for all i 0. On the other hand, if then write = 0 +p r 1

6 124 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) where 0. By using Steinberg s tensor product theorem, it follows that Ext i G(F q)(m; L r ()) = Ext i G(F q)(m; L r ( 0 ) L r ( 1 )): But Q so all the composition factors of L r ( 0 ) L r ( 1 ) have high weights which are less than. Consequently, Ext i G(F q)(m; L r ( 0 ) L r ( 1 )) = 0, and Ext i G(F q)(m; N)=0 for all i 0. Consider the long exact sequence in cohomology: 0 Hom G(Fq)(M; L r ()) Hom G(Fq)(M; H 0 ()) Hom G(Fq)(M; N) Ext 1 G(F q)(m; L r ()) Ext 1 G(F q)(m; H 0 ()) Ext 1 G(F q)(m; N) Ext 2 G(F q)(m; L r ()) : We have Ext i G(F q)(m; L r ()) = 0 for all i 2. Therefore, by Bendel and Nakano [3, Proposition 2.4.2(c)] one has Ext i G(F q)(m; L r ()) = 0 for all i 0. To show that (d) (b), one can use an analogous inductive argument. 3. Vanishing of cohomology for G(F p ) and G Let M be a rational G-module. In [15] it was shown that if M is projective over G 1 then M is projective over G(F p ). The converse to this statement also holds if one assumes that the composition factors of M have high weights which are not too large (i.e. M is in the p-bounded category). A natural generalization to this question is the following. (3:1:1) Let M and N be rational G-modules. Under what conditions does Ext i G 1 (M; N) = 0 imply that Ext i G(F p)(m; N ) = 0 for all i 0? Observe if N = L 1() or H 0 () then this holds by Lin and Nakano [15, Corollary 3:5], Bendel and Nakano [3, Theorem(4:2:1)] and the Theoremgiven in 2:2. In order to demonstrate the subtlety of this problem, we present the following example. Let G =SL 2 (k) with p 7. There are three blocks for k SL 2 (F p ) and (p+1)=2 blocks for G 1. Let L 1 () be in the principal block for SL 2 (F p ), but not in the principal block for G 1. Then for some i 0, H i (SL 2 (F p );L 1 ()) 0, but H i (G 1 ;L 1 ()) = 0 for all i Let M and N be B-modules. According to [15, Theorem 3:2], there exists a spectral sequence E i;j 1 = Exti+j U 1 (M; N ) i Ext i+j U(F p) (M; N ): (3.1) The following result demonstrates that the T(F p ) invariants on the U 1 -cohomology are directly related to the calculation of the B(F p ) and the G(F p ) cohomology. Theorem. Let n 0. (a) If M; N mod(b) and Ext n U 1 (M; N ) T (Fp) =0 then Ext n B(F p)(m; N)=0. (b) If M; N mod(g) and Ext n U 1 (M; N ) T (Fp) =0 then Ext n G(F p)(m; N)=0.

7 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Proof. (a) First observe that B(F p )=T (F p ) n U(F p ) and modules over kt(f p ) are semisimple. Therefore, the Lyndon Hochschild Serre spectral sequence yields for n 0: Ext n B(F p)(m; N ) = Ext n U(F p)(m; N) T (Fp) : (3.2) The spectral sequence (3.1) for M; N mod(b) has dierentials which are T(F p )- homomorphisms. The xed point functor ( ) T (Fp) is exact. This means that we can obtain another spectral sequence from(3.1) E i;j 1 = Exti+j U 1 (M; N ) T (Fp) i Ext i+j B(F p) (M; N): (3.3) Next, observe that Ext n U 1 (M; N ) T(Fp) = Ext i+j U 1 (M; N) T(Fp) i : (3.4) i60 j=n i Consequently, if Ext n U 1 (M; N ) T (Fp) = 0 then Ext i+j U 1 (M; N) T(Fp) i = 0 for all i; j such that i + j = n, thus Ext n B(F p)(m; N )=0. (b) Since U(F p ) is the p-sylow subgroup of G(F p ) the restriction map res: Ext n G(F p)(m; N ) Ext n B(F p)(m; N ) is injective for all n 0. The result now follows frompart (a). 4. Spectral sequences 4.1. Let G be an algebraic k-group scheme and H be a closed subgroup scheme of G. Moreover, let M mod(g) and N mod(h). There exists a rst quadrant spectral sequence [11, I. 4.5 Proposition]: E i;j 2 = Exti G(M; R j ind G H (N )) Ext i+j H (M; N): (4.1) When G is as dened in 1.3 and H = G(F q ) the induction functor ind G G(F q) is exact because H is a nite algebraic k-group. In this case the spectral sequence (4.1) collapses and yields the following isomorphism: (4:1:1) For i 0, Ext i G(F q)(m; N ) = Ext i G(M; ind G G(F q) N). Similarly, if G is reductive and H = G r then G=G r is ane. Therefore, the induction functor ind G G r is exact. The spectral sequence (4.1) collapses in this case and yields the following isomorphism: (4:1:2) For i 0, Ext i G r (M; N ) = Ext i G(M; ind G G r N). Isomorphisms (4:1:1) and (4:1:2) provide armative answers to questions (1:1:1) and (1:1:2). Unfortunately, for practical purposes this answer is not satisfactory because the modules ind G G(F q) N and ind G G r N are, in general, innite-dimensional and dicult to compute. We will need to modify this construction to provide a better solution to computing cohomology for G(F q ) and G r via cohomology for G.

8 126 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Let us review some general properties about coalgebras and comodules. The conventions and results will follow those proved in [6, Section 1]. Let C be any coalgebra, and Mod(C) be the category of right comodules for C. Now suppose that B is a subcoalgebra of C and let Mod B (C) be the full subcategory of Mod(C) whose objects belong to B. IfM Mod(C), then let M B be the unique maximal C-subcomodule of M belonging to B. The functor M M B is a functor frommod(c) tomod B (C). Since the image of the structure map M B M B C is contained in M B B, one can regard M B as a B-comodule. The functor F B : M M B, frommod(c) to Mod(B), is left exact and takes injectives to injectives. Let be a set of simple C-comodules and M Mod(C). Let O (M) be the unique maximal C-subcomodule such that all the composition factors of which lie in. Set C()=O (C). Then C() is a subcoalgebra of C. Furthermore, O (M) is the same as F C() (M) if we regard F C() (M) in Mod(C) by ination. Let G be an algebraic k-group scheme and H be a closed subgroup scheme of G. Set C =k[g] and let be a set of simple C-comodules (or equivalently simple rational G-modules). Consider the functor T=F C() ind G H. This is a functor frommod(k[h]) to Mod(C()). We can now construct the following spectral sequence. Theorem. Let G be an algebraic k-group scheme with H a closed subgroup scheme of G. Let M Mod(C()) and N Mod(H). Then there exists a rst quadrant spectral sequence E i;j 2 = Exti C()(M; R j T(N )) Ext i+j H (M; N): Proof. The functor S = Hom C() (M; ) is left exact. Moreover, the functor T : Mod(H) Mod(C()) takes injectives to injectives because F C() and ind G H have this property. Furthermore, note that since M Mod(C()) we have S T( ) = Hom C() (M; T( )) = Hom C (M; ind G H ( )) = Hom H (M; ): The result now follows by Jantzen [11, I. Proposition 4:1] Let G be as dened in 1.3 and let be a subset of X (T) +. In the context of the previous section, we will abuse notation and also let denote the set of simple rational G-modules having highest weight in. Since O is equivalent to F C(),it follows that Mod(C()) is equivalent to the full subcategory of G-modules whose composition factors have highest weight in. Now assume that is saturated. That is, if and X (T ) + such that 6 then. In this case, Donkin proved the following isomorphism [6, (2.1f) Theorem]: (4:3:1) For M; N Mod(C()), Ext n G(M; N) = Ext n C()(M; N) for n 0. For a xed r 1, let q = p r and for all 0 s6 p 1 let s = { X (T ) + : + ; 0 2sp r ; 0 };

9 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) where 0 denotes the highest short root. Observe that if 6 then = c where c 0. Then ( + ) ( + );0 = c ; 0 0: Therefore, + ; 0 + ; 0. This computation shows that s is saturated. Furthermore, let C s be the full subcategory of G-modules all of whose composition factors have highest weights in s. For s=1, the category C 1 essentially coincides with the p r -bounded category as dened in [11, p. 360]. Since the category C s is equivalent to the category Mod(C( s )) (as noted more generally above), the isomorphism in (4:3:1) holds for modules in C s Let G s = F C(s) ind G G(F q) and H s = F C(s) ind G G r. The Theoremgiven in 4:2 (with H = G(F q )orh = G r and = s ) can be combined with the isomorphism in (4:3:1) to construct the following spectral sequences. Theorem. Let M C s. (a) For N mod(g(f q )) there exists a spectral sequence E i;j 2 = Exti G(M; R j G s (N )) Ext i+j G(F q) (M; N): (b) For N mod(g r ) there exists a spectral sequence E i;j 2 = Exti G(M; R j H s (N )) Ext i+j (M; N): G r 4.5. Given s with corresponding simple G-module L(), let P() denote the projective cover of L() in the category C s. Projective covers exist in the category C s since it is equivalent to the category Mod(C( s )) and C( s ) is a nite-dimensional coalgebra (see the discussion in 4:2). The following result describes the composition factors of the higher right derived functors of G s and H s. Theorem. For j 0: (a) If M mod(g(f q )) then [R j G s (M): L()] = dim k Ext j G(F q) (P();M): (b) If N mod(g r ) then [R j H s (N ): L()] = dim k Ext j G r (P();N): Proof. We will prove part (a). The proof for part (b) is completely analogous. Since R j G s (M) is a module in C s, by denition of P(), we have [R j G s (M): L()] = dim k Hom Cs (P();R j G s (M)): (4.2) Fromthe Theoremgiven in 4:4 there is a spectral sequence E i;j 2 = Exti C s (P();R j G s (M)) Ext i+j G(F q) (P();M):

10 128 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Since P() is projective in C s, the spectral sequence collapses to E 0;j 2 = Hom Cs (P();R j G s (M)) = Ext j G(F (P();M); q) fromwhich the result follows For a p r -restricted weight X r (T), if p 2(h 1), the G r structure on Q r () lifts to a G-structure [12]. Moreover, when considered as a G-module, Q r () is p r -bounded and can be identied as the injective hull (and projective cover) of L() in the p r -bounded category C 1 [11, p. 360]. In other words, for a restricted weight, the module P() may be identied with Q r (). The following proposition, which is an adaptation of Lemma 2:2 of [2], gives us information about P() for general s. For convenience, let 0 = X r (T). Note that any weight s may be expressed as = 0 + p r 1 with 0 X r (T), that is with 0 being p r -restricted. Further, for s 1, 1 must satisfy 1 ; 2s(h 1) (or 1 + ; (2s + 1)(h 1)) for short and X (T) +. Hence if p (2s + 1)(h 1), the weight 1 lies in the bottomalcove C Z. Proposition. Assume s 1; M C s ; and one of the following two conditions holds: (i) = 0 + p r 1 s and p (2s + 1)(h 1); (ii) = 0 + p r 1 s 1 and p 2s(h 1). Then the following hold: (a) Ext 1 G(Q r ( 0 ) L( 1 ) (r) ;M)=0. (b) The projective module P() in C s is a quotient of Q r ( 0 ) L( 1 ) (r). Proof. Let S =Q r ( 0 ) L( 1 ) (r). For part (a), it suces to show that Ext 1 G(S; L())=0 for all composition factors L() ofm. Since M is in C s, any such are in s, and so may be expressed as = 0 + p r 1 with 0 X r (T). The short exact sequence of group schemes 1 G r G G=G r 1 gives rise to the Lyndon Hochschild Serre spectral sequence E i;j 2 = Exti G=G r (k; Ext j G r (S; L())) Ext i+j G (S; L()): Since Q r ( 0 ) is projective over G r, S is also. Hence, the spectral sequence collapses and gives an isomorphism Ext i G(S; L()) = Ext i G=G r (k; Hom G (S; L()) for all i. In particular, we have Ext 1 G(S; L()) = Ext 1 G=G r (k; Hom Gr (S; L())) = Ext 1 G=G r (k; Hom Gr (Q r ( 0 ) L( 1 ) (r) ;L( 0 ) L( 1 ) (r) )) = Ext 1 G=G r (k; Hom Gr (Q r ( 0 );L( 0 )) (L( 1 ) (r) ) L( 1 ) (r) ) { Ext 1 = G=Gr (k; k (L( 1 ) (r) ) L( 1 ) (r) ) if 0 = 0 0 otherwise

11 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) { Ext 1 = G=Gr (L( 1 ) (r) ;L( 1 ) (r) ) if 0 = 0 0 otherwise =0 by the Linkage Principle (cf. [11, II. 6.17]). More precisely, under condition (i), as both and lie in s and p (2s + 1)(h 1), both 1 and 1 lie in the bottom alcove C Z. And so the claimfollows fromthe Linkage Principle. On the other hand, under condition (ii), lies in s 1 while lies in s. Now, the condition on p implies that 1 + ; (2(s 1) + 1)(h 1) 6 p (h 1) and 1 + ; (2s + 1)(h 1) 6 p +(h 1) for short X (T) +. In other words, 1 lies in the bottom alcove and moreover, more than h 1 beneath the upper wall, while either lies in the bottomalcove or no more than h 1 above the upper wall. Hence, the desired vanishing again follows fromthe Linkage Principle. For part (b), by denition of P(), there exists a surjection P() L() of G-modules. On the other hand, since there is a surjection Q r ( 0 ) L( 0 ), there is also a surjection (of G-modules) S = Q r ( 0 ) L( 1 ) (r) L( 0 ) L( 1 ) (r) = L(). To see that the map S L() lifts to S P(), consider the short exact sequence 0 N P() L() 0 and the corresponding long exact sequence of Ext-groups: 0 Hom G (S; N ) Hom G (S; P()) Hom G (S; L()) Ext 1 G(S; N) : By part (a), Ext 1 G(S; N ) = 0 and so the map Hom G (S; P()) Hom G (S; L()) is a surjection, giving the desired lifting The preceding proposition allows us to obtain some information about the higher right derived functors of G s and H s. Corollary. Let s 1 and p 2s(h 1) with M mod(g(f q )) and N mod(g r ). If j 0 then (a) R j G s (M) contains no composition factors with high weight in s 1 ; (b) R j H s (N ) contains no composition factors with high weight in s 1. Proof. Let = 0 + p r 1 be a weight in s 1. By the Proposition given in 4:6(b), the projective cover P() inc s is a quotient of Q r ( 0 ) L( 1 ) (r). But, the highest weight of Q r ( 0 ) L( 1 ) (r) is 2(p r 1) + w p r 1. This weight is in s and so Q r ( 0 ) L( 1 ) (r) lies in C s. By the projectivity of P() inc s, P() must be a G-summand of Q r ( 0 ) L( 1 ) (r). The latter module is projective as a G r -module and G(F q )-module. The assertion now follows by the Theorem given in 4:5. For s 1 and s 1, the proof in fact shows that the module P() inc s may be identied with Q r ( 0 ) L( 1 ) (r) since the G-head of each module is L().

12 130 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Stability of extensions 5.1. The rst result of this section is a vanishing result involving the extensions of certain G-modules with the higher right derived functors of G s and H s. Proposition. Let s 1; M mod(g(f q )) and N mod(g r ) and let L C s be such that L has only p r -restricted composition factors in its head. If p 2s(h 1) then (a) Ext i G(L; R j G s (M))=0; (b) Ext i G(L; R j H s (M))=0 for 0 6 i 6 s 1 and j 0. Proof. (a) Let P 0 be the projective cover of L in C s and R 1 be the kernel of the map P 0 L. Inductively, set P i to be the projective cover of R i and R i+1 to be the kernel of P i R i. With this procedure we have constructed a projective resolution of L in C s : P 2 P 1 P 0 L 0: Since the head of L contains only p r -restricted composition factors, it follows by the Proposition given in 4:6(b) that the composition factors of P 0 (and R 1 ) are in 1. An inductive argument shows that the composition factors of P i are in i+1 and the composition factors in the head of P i are in i for 0 6 i 6 s 1. Therefore, by the Corollary given in 4:7, Hom G (P i ;R j G s (M)) = 0 for j 0 and 0 6 i 6 s 1. Consequently, Ext i G(L; R j G s (M)) = 0 for j 0 and 0 6 i 6 s 1. Part (b) follows by a similar argument With the preceding results, one can provide positive answers to questions (1:1:1) and (1:1:2) by identifying certain extensions over G(F q )org r with extensions between nite-dimensional G-modules. Theorem. Let s 1; M mod(g(f q )) and N mod(g r ) and let L C s be such that L has only p r -restricted composition factors in its head (as G-module). If p 2s(h 1) then there are the following isomorphisms and embeddings: (a) Ext i G(F q)(l; M) = Ext i G(L; G s (M)) for all 0 6 i 6 s; (b) Ext s+1 G (L; G s(m)), Ext s+1 G(F q) (L; M); (c) Ext i G r (L; N ) = Ext i G(L; H s (N )) for all 0 6 i 6 s; (d) Ext s+1 G (L; H s(n )), Ext s+1 G r (L; N ). Proof. The proofs of the two cases are analogous and we prove parts (a) and (b). Fromthe Theoremgiven in 4:4 there is a spectral sequence E i;j 2 = Exti G(L; R j G s (M)) Ext i+j G(F q) (L; M): By the Propostion given in 5:1, we have Ext i G(L; R j G s (M)) = 0 for j 0 and 0 6 i 6 s 1. The spectral sequence has dierential with bidegree (r; 1 r). Consequently, Ext i G(F q)(l; M) = Ext i G(L; G s (M)) for 0 6 i 6 s. Furthermore, since E i;j 2 = 0 for j 0

13 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) and 0 6 i 6 s 1, the image of the dierentials are zero in E s+1;0 2, thus E s+1;0 2, (L; M). Ext s+1 G(F q) 5.3. The succeeding corollaries are immediate applications of the theorem to determining extensions between simple modules and Weyl modules. Corollary. Let p 2(h 1) and ; X r (T). Then there exists the following isomorphisms and embeddings: (i) Ext 1 G(F q)(l();l()) = Ext 1 G(L(); G 1 (L())); (ii) Ext 2 G(L(); G 1 (L())), Ext 2 G(F q)(l();l()); (iii) Ext 1 G(F q)(v ();L()) = Ext 1 G(V (); G 1 (L())); (iv) Ext 2 G(V (); G 1 (L())), Ext 2 G(F q)(v ();L()); (v) Ext 1 G r (L();L()) = Ext 1 G(L(); H 1 (L())); (vi) Ext 2 G(L(); H 1 (L())), Ext 2 G r (L();L()); (vii) Ext 1 G r (V ();L()) = Ext 1 G(V (); H 1 (L())); (viii) Ext 2 G(V (); H 1 (L())), Ext 2 G r (V ();L()) The following examples demonstrate that in order to study higher extensions between simple modules it is not sucient to truncate the induction functor from the nite group (or the rst Frobenius kernel) to the algebraic group at the level of twice the Steinberg weight. The methods for computing the functors G 1 ( ) and H 1 ( ) will be described in Sections 6 and 7. Example. Let G = SL 2 (k) and G(F q )=SL 2 (F p ) with p 7. By a direct computation we have G 1 (L(p 2)) = L(p 2) L(2p 3): The simple module L(p 4) is not in the same G-block as L(p 2) or L(2p 3), thus Ext 2 G(L(p 4); G 1 (L(p 2))) = 0. On the other hand, the structure of the projective indecomposable modules are known for G(F q )=SL 2 (F p ). One can construct the minimal projective resolution of the simple module L(p 4) to show that Ext 2 G(F p)(l(p 4);L(p 2)) 0. Consequently, Ext 2 G(F p)(l(p 4);L(p 2)) Ext 2 G(L(p 4); G 1 (L(p 2))). Example. Let G = SL 2 (k) and G 1 =(SL 2 ) 1 with p 7. We have H 1 (L(p 2)) = L(p 2) L(2p 2) L(2p 2): We have Ext j G 1 (L(p 2);L(p 2))=0 for j odd and Ext 2 G 1 (L(p 2);L(p 2)) = L(2) (1) as a G-module. There exists a spectral sequence E i;j 2 = Exti G=G 1 (k; Ext j G 1 (L(p 2);L(2p 2)) Ext i+j G (L(p 2);L(2p 2)): Observe by the Steinberg tensor product theoremthat Ext j G 1 (L(p 2);L(2p 2)) = Ext j G 1 (L(p 2);L(p 2)) L(1) (1) :

14 132 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Therefore, E 1;1 2 =0, E2;0 2 =Ext2 G=G 1 (k; L(1) (1) )=0 and E 0;2 2 =Hom G=G 1 (k; L(2) (1) L(1) (1) )= 0. This shows that Ext 2 G(L(p 2);L(2p 2))=0. A similar spectral sequence argument can be used to show that Ext 2 G(L(p 2);L(p 2)) = 0. It follows that Ext 2 G(L(p 2); H 1 (L(p 2)))=0, and Ext 2 G 1 (L(p 2);L(p 2)) Ext 2 G(L(p 2); H 1 (L(p 2))) Small primes. The results in 5:3 rely on the fact that the prime p is not too small. In this section we obtain a slightly weaker version of the Corollary given in 5:3 for arbitrary primes. Theorem. Let ; X r (T ) with. Then (a) Ext 1 G(F q)(l();l()) = Ext 1 G(L(); G 1 (L())); (b) Ext 1 G r (L();L()) = Ext 1 G(L(); H 1 (L())). Proof. (a) From [19, Lemma 1:4] one obtains the following statement. Let X r (T) and let M be a G(F q )-module that contains only simple composition factors whose p r -restricted highest weights satisfy 6 (p r 1)+w 0 where w 0 is the long element in the Weyl group. Then the G(F q )-head of St r M contains only simple modules L() whose p r -restricted highest weights satisfy. In particular it follows that for Hom G (St r L((p r 1) + w 0 );L()) Hom G(Fq)(St r L((p r 1) + w 0 );L())=0: On the other hand, Hom G (St r L((p r 1) + w 0 );L()) = HomG(Fq)(St r L((p r 1) + w 0 );L()) = k: Therefore, we have the following short exact sequence 0 R St r L((p r 1) + w 0 ) L() 0: (5.1) Note that this is a short exact sequence in the category C 1 because St r L((p r 1)+ w 0 ) C 1. By using the fact that the Steinberg module is projective over G(F q )we obtain the following exact sequence: 0 Hom G(Fq)(L();L()) Hom G(Fq)(St r L((p r 1) + w 0 );L()) Hom G(Fq)(R; L()) Ext 1 G(F q)(l();l()) 0: If then by our previous observations we have Hom G(Fq)(L();L()) = Hom G(Fq)(St r L((p r 1) + w 0 );L()): Consequently, Hom G(Fq)(R; L()) = Ext 1 G(F q)(l();l()): (5.2)

15 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Fromthe short exact sequence (5.1) one obtains the following long exact sequence: 0 Hom G (L(); G 1 (L())) Hom G (St r L((p r 1) + w 0 ); G 1 (L())) Hom G (R; G 1 (L())) Ext 1 G(L(); G 1 (L())) Ext 1 G(St r L((p r 1) + w 0 ); G 1 (L())) : Observe that by adjointness we have for 0 = Hom G(Fq)(L();L()) = Hom G (L(); G 1 (L())); 0 = Hom G(Fq)(St r L((p r 1) + w 0 );L()) = HomG (St r L((p r 1) + w 0 ); G 1 (L())): Similarly, for = it follows that Hom G(Fq)(L();L()) = Hom G (L(); G 1 (L())) = k; Hom G(Fq)(St r L((p r 1) + w 0 );L()) = HomG (St r L((p r 1) + w 0 ); G 1 (L())) = k: The long exact sequence and these isomorphisms now dene an injective map Hom G (R; G 1 (L())) Ext 1 G(L(); G 1 (L())): But, R C 1 so by adjointness we have Hom G(Fq)(R; L()) = Hom G (R; G 1 (L())): It follows by (5.2) that there exists an injective map Ext 1 G(F q)(l();l()) Ext 1 G(L(); G 1 (L())): The ve term exact sequence of the spectral sequence in 4.4 makes the base map E 1;0 E 1 an injective map in the other direction. Therefore, one obtains the desired isomorphism Ext 1 G(F q)(l();l()) = Ext 1 G(L(); G 1 (L())): A similar argument can be used to prove part (b). 6. Cohomology for Frobenius kernels 6.1. In the previous section, extensions of simple modules over G(F q ) and G r were identied with certain extensions over G. These G extensions involve the induction functors G s and H s. In this and the following section, we study G s (L()) and H s (L()) for X r (T ), in order to improve this identication to involve only simple modules. We begin by showing that the module H s (L()) is semisimple for X r (T).

16 134 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Theorem. Let p (2s + 1)(h 1) and X r (T). Then H s (L()) is semisimple. Moreover, H s (L()) = s Hom G (L(); H s (L())) L() = s L() dim k Hom Gr (L();L()) : Proof. First, consider the socle of H s (L()) over G (or equivalently in the category C s ): soc G H s (L()) = s Hom G (L(); H s (L())) L() = s L() dim k Hom Gr (L();L()) by adjointness. So, if H s (L()) is semisimple, then it has the claimed form. The Theoremgiven in 4:5 can be used to obtain information about the composition factors of H s (L()). Specically, we have [H s (L()) : L()] = dim k Hom Gr (P();L()) 6 dim k Hom Gr (Q r ( 0 ) L( 1 ) (r) ;L()) (by the Proposition of 4:6(b)): As a G r -module, L( 1 ) (r) is trivial and so Q r ( 0 ) L( 1 ) (r) = Q r ( 0 ). Further, since 0 and are both p r -restricted weights, Hom Gr (Q r ( 0 );L()) = Hom Gr (L( 0 );L()), and this will be non-zero if and only if 0 =. Hence, the only possible composition factors of H s (L()) are those L() with = + p r. As noted in 4:6, since is in s and is p r -restricted, must lie in the bottom alcove. To show that H s (L()) is semisimple, it now suces to show that Ext 1 G(L( 1 );L( 2 )) = 0 for any such weights 1 = + p r 1 and 2 = + p r 2. The argument is similar to that of the Proposition given in 4:6. The short exact sequence of group schemes 1 G r G G=G r 1 gives rise to the Lyndon Hochschild Serre spectral sequence E i;j 2 = Exti G=G r (k; Ext j G r (L( 1 );L( 2 ))) Ext i+j G (L( 1);L( 2 )): The beginning of the 5-termexact sequence is However, 0 E 1;0 E 1 E 0;1 = Hom G=Gr (k; Ext 1 G r (L( 1 );L( 2 ))) ::: : Ext 1 G r (L( 1 );L( 2 )) = Ext 1 G r (L() L( 1 ) (r) ;L() L( 2 ) (r) ) = Ext 1 G r ( L(); L())=0; since there are no self-extensions over G r (cf. [11, II. 12.9]). Hence there is an isomorphism Ext 1 G(L( 1 );L( 2 )) = E 1 = E 1;0 = Ext 1 G=G r (k; Hom Gr (L( 1 );L( 2 ))):

17 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Finally, we have Ext 1 G=G r (k; Hom Gr (L( 1 );L( 2 ))) =Ext 1 G=G r (k; Hom Gr (L() L( 1 ) (r) ;L() L( 2 ) (r) )) = Ext 1 G=G r (k; Hom Gr (L();L()) (L( 1 ) (r) ) L( 2 ) (r) ) = Ext 1 G=G r (k; k (L( 1 ) (r) ) L( 2 ) (r) ) = Ext 1 G=G r (L( 1 ) (r) ;L( 2 ) (r) )=0 by the Linkage Principle since 1 and 2 lie in the same alcove The following result shows that one can completely determine the extensions of simple modules in mod(g r ) by knowing the extension theory of simple modules in mod(g). Corollary. Let p (2s + 1)(h 1) and ; X r (T). Moreover, let L C s be such that L has only p r -restricted composition factors in its head. Then (a) For 0 6 i 6 s there exists the following isomorphism of k-vector spaces: Ext i G r (L; L()) = Hom Gr (L();L()) Ext i G(L; L()): s (b) In particular for 0 6 i 6 s, we have Ext i G r (L();L()) = Hom Gr (L();L()) Ext i G(L();L()): s (c) For p 3(h 1), we have Ext 1 G r (L();L()) = Hom Gr (L();L()) Ext 1 G(L();L()); 1 where 1 is the set of all p r -bounded weights. Proof. The result follows by combining the isomorphism Ext i G r (L; L()) = Ext i G(L; H s (L())) for 0 6 i 6 s of the Theoremgiven in 5.2 with the preceding identication of H s (L()) given in the Theoremof 6.1. Observe that the restriction on the prime indicates that the direct sums appearing here may be taken over those s with = + p r and C Z. 7. Cohomology for nite Chevalley groups 7.1. In this section, we use the module G s (L()) to recover and extend the results of [2] on extensions of G(F q )-modules. The hypotheses of these results depend on the

18 136 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) type of G. This is because when G is not of type A 1, ; for all simple roots, while ; 0 = 2 for the simple root = 0 in type A 1. First, consider the embedding L() G s (L()) of G-modules, which corresponds to the identity map under the isomorphism Hom G (L(); G s (L()) = Hom G(Fq)(L();L()). Proposition. Let s 1 and p (2s + 1)(h 1). Suppose ; X r (T) satisfy + ; 0 pr p r 1 1 where =2 if G is of type A 1 and =1 in all other cases. Then Ext 1 G(L(); G s (L())=L())=0: Proof. Fromthe Theoremgiven in 4:5, L() is the only restricted composition factor of G s (L()) and moreover appears only once. Hence, the composition factors of G s (L())=L() are not restricted. Let L() be a non-restricted composition factor of G s (L()) and write = 0 + p r 1 as usual. Since 0 is restricted and is not, 1 is necessarily non-zero. According to the Theoremof 4:5 and the Proposition of 4:6(b), we have [G s (L()) : L()] = dim k Hom G(Fq)(P();L()) 6 dim k Hom G(Fq)(Q r ( 0 ) L( 1 );L()) = dim k Hom G(Fq)(Q r ( 0 );L() L( w 0 1 )): So for L() to be a composition factor, we must have Hom G(Fq)(Q r ( 0 );L(!)) 0 for some composition factor L(!) ofl() L( w 0 1 ). By Jantzen [13] or Chastkofsky [4] (see also [2, 1(2)]), Q r ( 0 ) decomposes as a direct sum Q r ( 0 ) = U r() (of projective indecomposables for G(F q )) where each satises ; 0 pr 1+ 0 ;0. Hence, there is at least one! with! = 0 or!; 0 pr 1+ 0 ;0. On the other hand, by Lemma 2.5 of [2], if Ext 1 G(L();L()) 0, then + 0 ;0 p r 1 ;0 pr 1. Hence, for such and any weight! of L() L( w 0 1 ), we have!; ;0 p r p r ; ;0 (by original assumption) 6 (p r 1) 1 ;0 p r 1 +; ;0 = p r 1 ;0 p r 1 ; ;0 (by Lemma 2:5 of [2]): Hence, Ext 1 G(L();L()) = 0 for all composition factors L() ofg s (L())=L(). Corollary. Assume p 3(h 1). Let ; X r (T) satisfy +; 0 pr p r 1 1 where =2 if G is of type A 1 and =1 in all other cases. Then (a) (Theorem 2:8 of [2]) Ext 1 G(F q)(l();l()) = Ext 1 G(L();L()). (b) Moreover, there is an embedding Ext 2 G(L();L()), Ext 2 G(F q)(l();l()). Proof. For (a), consider the the short exact sequence 0 L() G 1 (L()) G 1 (L())=L() 0:

19 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) This short exact sequence induces a long exact sequence Hom G (L(); G 1 (L())=L()) Ext 1 G(L();L()) Ext 1 G(L(); G 1 (L())) Ext 1 G(L(); G 1 (L())=L()) : The rst termis zero since G 1 (L())=L() has no restricted composition factors, and the last termis zero by the Proposition given in 7:1. Hence, there is an isomorphism Ext 1 G(L();L()) = Ext 1 G(L(); G 1 (L())) and the claimed isomorphism follows from the Corollary given in 5:3. For (b), continuing the above long exact sequence, we have Ext 1 G(L(); G 1 (L())=L()) Ext 2 G(L();L()) Ext 2 G(L(); G 1 (L()) : Again, since the rst termis zero by the Proposition, we have an embedding Ext 2 G(L();L()), Ext 2 G(L(); G 1 (L())): The result follows by combining this with the embedding Ext 2 G(L(); G 1 (L())), Ext 2 G(F q)(l();l()) of the Corollary given in 5: More generally, semisimplicity of G s (L()) can be used to identify G(F q )- extensions with G-extensions. Theorem. Let s 1 and X r (T ). Then G s (L()) is semisimple if and only if G s (L()) = Hom G(Fq)(L();L()) L(): s Proof. Consider the socle of G s (L()) over G (or equivalently in the category C s ): soc G G s (L()) = s Hom G (L(); G s (L())) L() = s Hom G(Fq)(L();L()) L(): So G s (L()) is semisimple if and only if it has the claimed form. The previous result together with the Theoremof 5:2 yields the following corollary. Corollary. Let s 1; p 2s(h 1); X r (T), and let L C s be such that L has only p r -restricted composition factors in its head (as G-module). Assume that G s (L()) is semisimple. Then for 0 6 i 6 s there exists the following isomorphism of vector spaces: Ext i G(F q)(l; L()) = Hom G(Fq)(L();L()) Ext i G(L; L()): s

20 138 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) In particular, for p 2(h 1), we have Ext 1 G(F q)(l; L()) = 1 Hom G(Fq)(L();L()) Ext 1 G(L; L()); where 1 is the set of all p r -bounded weights We will now rene the result given in the Corollary of 7:2 by identifying homomorphisms over G(F q ) with homomorphisms over G. Lemma. Let p 2(h 1), and ; ; X r (T) with ; 0 pr 1. Then Hom G (L();L() L()) = Hom G(Fq)(L();L() L()): Proof. First, it is shown that the G-socle of L() L() contains only simple modules with p r -restricted highest weights. Assume that 0 Hom G (L( 0 ) L( 1 ) (r) ;L() L()); where 0 X r (T ). Without loss of generality, we may assume that 0 ; 0 ; 0 otherwise replace 0 by w 0 and by w 0 0. For the simple module L( 0 + p r 1 ) to appear as a composition factor in L() L() it is necessary that 0 + p r 1 ; ; ; 0 ; which implies that p r 1 ;0 6 ; 0 pr 1 and forces 1 =0. The above argument shows that soc G L() L() = i L( i) where all i are p r -restricted and satisfy ; 0 6 i;0 6 + ; 0. Moreover, the G-socle of L() L() is contained in the G(F q )-socle. Using the injectivity of Q r () inthe p r -bounded category one can embed L() L() in i Q r( i ). The G(F q )-socle of L() L() is therefore contained in the G(F q )-socle of i Q r( i ). Now assume that dim k Hom G (L();L() L()) dim k Hom G(Fq) (L();L() L()). Without loss of generality, we may assume this time that ; 0 ; 0. It follows that dim k Hom G (L(); i Q r( i )) dim k Hom G(Fq)(L(); i Q r( i )): That implies that, for at least one i, dim k Hom G (L();Q r ( i )) dim k Hom G(Fq)(L();Q r ( i )): The module Q r ( i ) are also injective as G(F q )-modules. A formula by Jantzen [13] and Chastkofsky [4] tells us how the Q r ( i ) split into indecomposable injective G(F q )-modules, denoted here by U r (). The module Q r ( i ) contains U r ( i ) exactly once, while any U r () with i that appears as a summand forces i ;0 +pr 1 6 ; 0 (see also [2]). Clearly i. Therefore we obtain the following sequence of inequalities: ; 0 ; 0 + p r 1= ; 0 + p r 1 6 i ;0 + p r 1 6 ; 0 6 ; 0 : Now p r 1 6 ; 0 pr 1 gives us the desired contradiction. For large enough primes one obtains the following result:

21 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Theorem. Let s 1; p r 2s(h 1), and X r (T) and assume that G s (L()) is semisimple. For any weight s we set = 0 + p r 1 with 0 X r (T). Then G s (L()) = s Hom G (L( 0 ) L( 1 );L()) L(): Proof. According to the Theoremgiven in 7:2 we have G s (L()) = s Hom G(Fq)(L();L()) L(): For any weight s we set = 0 +p r 1 with 0 X r (T). Then L() = L( 0 ) L( 1 ) as G(F q )-modules and so G s (L()) = Hom G(Fq)(L( 0 ) L( 1 );L()) L(): s Moreover, p r 1 ;0 6 ; 0 2spr (h 1) implies that 1 ;0 2s(h 1) 6 p r 1. Therefore, the theorem now follows by applying the preceding Lemma to Hom G(Fq)(L( 0 );L() L( w 0 1 )). Corollary. Let s 1; p 2s(h 1); X r (T), and let L C s be such that L has only p r -restricted composition factors in its head (as G-module). We assume that G s (L()) is semisimple. For any weight s we set = 0 + p r 1 with 0 X r (T). Then for 0 6 i 6 s, Ext i G(F q)(l; L()) = Hom G (L( 0 ) L( 1 );L()) Ext i G(L; L()): s In particular, for p 2(h 1), we have Ext 1 G(F q)(l; L()) = Hom G (L( 0 ) L( 1 );L()) Ext 1 G(L; L()); 1 where 1 is the set of all p r -bounded weights We now identify some for which G s (L()) is indeed semisimple. Let X [s]={ X (T) + ; 0 2s(h 1)}. Theorem. Let s 1; p (2s + 1)(h 1). Then G s (k) is semisimple. Moreover, G s (k) = L((p r w 0 )): X [s] Proof. The Theoremof 4:5 can be used to obtain information about the composition factors of G s (k). Specically, we have [G s (k):l()] = dim k Hom G(Fq)(P();k) 6dim k Hom G(Fq)(Q r ( 0 ) L( 1 ) (r) ;k) (by the Proposition of 4:6(b)): As a G(F q )-module, L( 1 ) (r) is isomorphic to L( 1 ) and so [G s (k):l()] 6 dim k Hom G(Fq)(Q r ( 0 );L( w 0 1 )):

22 140 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) Let us assume that dim k Hom G(Fq)(Q r ( 0 );L( w 0 1 )) = 0 unless 0 = w 0 1 : Then, for any s, we have [G s (k):l()] 6 dim k Hom G(Fq)(Q r ( 0 );L( w 0 1 )) = dim k Hom G(Fq)(L( 0 );L( w 0 1 )) = dim k Hom G(Fq)(L();k) = dim k Hom G (L(); G s (k)): Under our assumption it then follows fromthe Theoremgiven in 7:2 that G s (k) is semisimple. Moreover, all composition factors have highest weight of the form = 0 + p r 1 = w p r 1 and appear with multiplicity one. An easy weight computation shows that = w p r 1 s if and only if 1 ;0 2s(h 1): The module Q r ( 0 ) is projective over G(F q ) and dim k Hom G(Fq)(Q r ( 0 );L( w 0 1 )) equals the multiplicity of the indecomposable projective G(F q )-module U r ( w 0 1 )asa summand of Q r ( 0 ). It remains to be shown that this multiplicity [Q r ( 0 );U r ( w 0 1 )]= 0 unless 0 = w 0 1. As pointed out in the proof of the Lemma given in 7:3, [Q r ( 0 ); U r ( w 0 1 )] 0 and 0 w 0 1 imply that p r ;0 +pr 1 6 w 0 1 ;0 6 1 ;0 : But this is impossible because 1;0 2s(h 1). We denote by T the translation functor as dened in [11, II. 7]. Corollary. Let s 1; p (2s + 1)(h 1), and let X r (T) with 0 6 i 6 s. (a) If is p-regular then H i (G(F q );L()) = H i (G; T 0 L( + p r )): (b) If is p-singular then H i (G(F q );L()) = 0: X [s] Proof. Fromthe preceding theoremone can see that all the composition factors of G s (k) have p-regular highest weight. Therefore, there can be no non-zero cohomology for simple modules with p-singular highest weight. We may assume that is p-regular. H i (G(F q );L()) = Ext i G(Fq)(k; L()) = Ext i G(Fq)(L( w 0 );k) = Ext i G (L( w 0 ); G s (k)) (by the Theoremgiven in 5:2) = Ext i G(L( w 0 );L((p r w 0 ))) (by the preceding Theorem) X [s] = Ext i G(L();L( + p r )) ([11; I: 4:4]) X [s] = Ext i G(T0 (k);l( + p r )) X [s]

23 C.P. Bendel et al. / Journal of Pure and Applied Algebra 163 (2001) = Ext i G(k; T 0 L( + p r )) ([11; II: 7:6]) X [s] = H i (G; T 0 L( + p r )): X [s] For i = 1 we can do slightly better. It follows from [2, Lemma 2.5] that 0 H 1 (G; T 0 L( + p r )) forces ; 0 6 h 1. Therefore for p 3(h 1) one obtains H 1 (G(F q );L()) = { X (T ) + ; 0 6h 1} H 1 (G; T 0 L( + p r )): 7.5. For large primes we can now prove the existence of a one-to-one correspondence between the ith cohomology for certain simple G-modules and the ith cohomology for the simple G(F q )-modules. Following [10] we dene W p to be the ane group generated by the ordinary Weyl group and the translations by elements of px (T). For large p any p-regular weight has trivial stabilizer in W p [10, Lemma 5]. Theorem. Let s 1; p (4s + 1)(h 1); X r (T) with ; 0 2s(h 1), and u W p such that u is p r -restricted. Then for 0 6 i 6 s, H i (G(F q );L(u )) = H i (G; L(u 0+p r )): Moreover, all the non-zero ith cohomology for simple modules of the nite group can be obtained in this fashion. Proof. Assume that 0 H i (G(F q );L()). By the previous corollary there exists a X (T) + with ; 0 2s(h 1) and 0 Hi (G; T 0 L( + p r )): Let be the highest weight of the simple module T 0 L( + p r ). By the Linkage Principle there exists a unique u W p such that = u 0+p r. 0 and are inside the same alcove. Therefore, T0 T 0 L( + p r )=L( + p r ). Now clearly = u. It remains to be shown that there exists no other weight X (T ) + with ; 0 2s(h 1) and 0 Hi (G; T 0 L(+pr )) such that = v for some v W p. Suppose that u = = v. Then is contained in the W p -orbit of. Both and are p-regular and inside the lowest alcove. By Jantzen [10, Lemma 1] there exists a unique element w of the Weyl group and a unique weight w that is the sumof some distinct fundamental weights such that = w + p w.if w 0, then there exists a simple root such that ; w( + ); +1=p. But the left-hand side is less than or equal to 2s(h 1) 1+(2s + 1)(h 1) 1+1=(4s + 1)(h 1) 1 p. Therefore, w =0. Both and are p-regular dominant weights. Thus, w is the identity. Hence, u = v and =. For groups whose root lattice is identical to the weight lattice the condition on p can be weakened to p (2s + 1)(h 1). The following example illustrates how one can use the preceding Theoremto compute the G(F q )-cohomology via the G-cohomology.