Second cohomology for finite groups of Lie type

Let G be a simple, simply-connected algebraic group defined over F-p. Given a power q = p(r) of p, let G(F-q) subset of G be the subgroup of F-q-rational points. Let L(lambda) be the simple rational G-module of highest weight lambda. In this paper we establish sufficient criteria for the restriction map in second cohomology H-2(G, L(lambda)) -> H-2(G(F-q), L(lambda)) to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on p and q provided lambda is less than or equal to a fundamental dominant weight. Even when the restriction map is not an isomorphism, we are often able to describe H-2(G(F-q), L(lambda)) in terms of rational cohomology for G. We apply our techniques to compute H-2(G(F-q), L(lambda)) in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type. (c) 2012 Elsevier Inc. All rights reserved.