Second cohomology for finite groups of Lie type

Brian D. Boe, Brian Bonsignore, Theresa Brons, Jon F. Carlson, Leonard Chastkofsky, Christopher M. Drupieski, Niles Johnson, Daniel K. Nakano, Wenjing Li, Phong Thanh Luu, Tiago Macedo, Nham Vo Ngo, Brandon L. Samples, Andrew J. Talian, Lisa Townsley, Benjamin J. Wyser

Published 2011-10-02, updated 2011-12-14Version 2

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.