Second cohomology for finite groups of Lie type

Let G be a simple, simply-connected algebraic group defined over Fp. Given a power q=pr of p, let G(Fq)⊂G be the subgroup of Fq-rational points. Let L(λ) be the simple rational G-module of highest weight λ. In this paper we establish sufficient criteria for the restriction map in second cohomology H2(G,L(λ))→H2(G(Fq),L(λ)) to be an isomorphism. In particular, the restriction map is an isomorphism under very mild conditions on p and q provided λ 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 H2(G(Fq),L(λ)) in terms of rational cohomology for G. We apply our techniques to compute H2(G(Fq),L(λ)) in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.