%0 Journal Article
%T Second cohomology for finite groups of Lie type
%A Brian D. Boe
%A Brian Bonsignore
%A Theresa Brons
%A Jon F. Carlson
%A Leonard Chastkofsky
%A Christopher M. Drupieski
%A Niles Johnson
%A Daniel K. Nakano
%A Wenjing Li
%A Phong Thanh Luu
%A Tiago Macedo
%A Nham Vo Ngo
%A Brandon L. Samples
%A Andrew J. Talian
%A Lisa Townsley
%A Benjamin J. Wyser
%J Mathematics
%D 2011
%I arXiv
%R 10.1016/j.jalgebra.2012.02.028
%X 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 $H^2(G,L(\lambda)) \rightarrow H^2(G(\mathbb{F}_q),L(\lambda))$ 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.
%U http://arxiv.org/abs/1110.0228v2