Second cohomology for finite groups of Lie type

Abstract

Let $G$ be a simple, simply-connected algebraic group defined over ${\mathbb{F}}_p$. Given a power $q=p^r$ of $p$, let $G({\mathbb{F}}_q)\subset G$ be the subgroup of ${\mathbb{F}}_q$-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 $H^2(G,L(\lambda ))\to H^2(G({\mathbb{F}}_q),L(\lambda ))$ 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 $H^2(G({\mathbb{F}}_q),L(\lambda ))$ in terms of rational cohomology for $G$. We apply our techniques to compute $H^2(G({\mathbb{F}}_q),L(\lambda ))$ in a wide range of cases, and obtain new examples of nonzero second cohomology for finite groups of Lie type.

29 pages, GAP code included as an ancillary file. Rewritten to include the adjoint representation in types An, B2, and Cn. Corrections made to Theorem 3.1.3 and subsequent dependent results in Sections 3-4. Additional minor corrections and improvements also implemented