Edward Cline

Rational and Generic Cohomology

Let G be a semisimple algebraic group defined and split over k,=GF(p). For q=p”, let G(q) be the subgroup of GF(q)-rational points. The main objective of this paper is to relate the cohomology of the finite groups G(q) to the rational cohomology of the algebraic group G. Let I/ be a finite dimensional rational G-module, and, for a non-negative integer e, let V(e) be the G-module obtained by “twisting” the original G-action on V by the Frobenius endomorphism x++xtPel of G. Theorem (6.6) states that, for sufficiently large q and e (depending on I’ and n), there are isomorphisms H”(G, V(e))gH’(G(q), V(e))rH”(G(q), V) where the first map is restriction. In particular, the cohomology groups H”(G(q), V) have a stable or “generic” value H;,,(G, V). This phenomenon had been observed empirically many times (cf. [6, 203). The computation of generic cohomology reduces essentially to the computation of rational cohomology. One (surprising) consequence is that Hi,,(G, V) does not depend on the exact weight lattice for a group G of a given type cf. (6.10), though this considerably affects the structure of G(q). We also obtain that rational cohomology takes a stable value relative to twisting i.e., for sufficiently large E, we have semilinear isomorphisms H”(G, V(E)) % H”(G, V(e)) for all e 2 F. This paper contains many new results on rational cohomology beyond those required for the proof of the main theorem. We mention in particular the vanishing theorems (2.4) and (3.3), and especially the results (3.9) through (3.11) which relate H2(G, V) and Extk( K W) to the structure of Weyl modules. These results explain for example the generic values of H’ determined in [6], cf. (7.6). Also, it is shown in Theorem (3.12) that every finite dimensional rational G-module has a finite resolution by finite dimensional acyclic G-modules. A key ingredient in the proofs is an important theorem of G. Kempf [I93 on the vanishing of cohomology of certain homogeneous line bundles. This result is translated into the language of rational cohomology in (1.2), and is used in

Cohomology of finite groups of lie type, I

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Induced Modules and Extensions of Representations

Let G be a connected affine algebraic group over an algebraically closed field k, and let B be a Borel subgroup of G. If V is a rational B-module we can ask for conditions under which V extends to G, that is, under which the action of B extends to a rational action of G on V. Since G/B is a complete variety, we have MtnlG_--__ M for any rational G-module M, cf. (1.3); so if any extension of V exists, it must be unique. The main result of this paper is that V is extendible to G if and only if it is extendible to every minimal parabolic subgroup P>B. Here minimal means that P properly contains B and that no other closed subgroup of P does so; equivalently, P has semisimple rank one. We also show that if P contains a Levi complement L, then V extends to P iff VILnn extends to L; so, for reductive G, the extendibility of V reduces to an SL 2 question. We obtain the main result as a corollary to a general theorem on induced modules. For any parabolic subgroup P >B, and any rational B-module V, let VIe denote the corresponding module induced from B to P. If P~ . . . . . P, is any sequence of such Ps, we let VI e ..... e. denote the P.-module vlvlnlV2-I r obtained by successively inducing to P~_ 1, restricting to B, and inducing to P~. We prove the B-module structure of V[ e ...... e. depends only on the set-theoretic product P~ ... P., and that VI v ..... e . _ VIG as P.-modules in case this product is G. At the same time we prove a factorization theorem for the affine coodinate ring R(G) of G. Let Px . . . . . P. be parabolic subgroups containing B such that G = P1... P.. Then we have R(G)~-R(P,)@n R(P,_ O@B @B R(PO, where in general for a right B-module M and a left B-module N, we denote by M@BN the set of fixed points of B in M @ N relative to the action b. (m@ n)= m b -1@ b n. The above results are in turn derived from simple considerations involving the Demazure desingularization of Schubert varieties [9], and are inspired by Kempf [12] and by Andersens efforts [1] toward a reorganization of Kempfs work. A simplified version of Andersens main reduction is given in the appendix.