On the Cohomology of Chevalley Groups by Eric M. Friedlander and Brian Parshall

Let G be a simple, simply connected algebraic group defined and split over the prime field F p > 3. Let F be a finite dimensional rational G-module. As shown in [1] , for d suitably large, the Eilenberg-Mac Lane cohomology groups H*(G(Fpd), V) achieve a stable value H*en(G, V)9 the so-called generic cohomology of V. This generic cohomology can in turn be determined from the rational cohomology groups H^x(Gt V^), r > 1, where V^ stands for the module V with the action of G twisted by the rth power of the Frobenius morphism a: G —• G. In this paper we announce an explicit determination of the cohomology groups #* t(G, F< >), and hence of the generic cohomology groups H*(G, V), in a range of cohomology degrees (restricted by the prime p) for an arbitrary irreducible rational module V whose high weight lies in the bottom p-alcove. In particular, we obtain stability in H^en(G, V) for large p, answering a question raised by Scott [6] . Our methods involve a determination in a range of degrees of the cohomology of the restricted enveloping algebra of the Lie algebra of G. The general Theorem 4 below is motivated by the "experimental evidence" provided by Theorem 1. This result concerning the general linear group is particularly strong in terms of its range of degrees, its applicability to small fields, and its description of the fc-algebra structure. Here k is an algebraically closed field of characteristic p.