Second cohomology for finite groups of Lie type

TL;DR

This paper investigates the second cohomology groups of finite groups of Lie type, establishing conditions for isomorphisms with algebraic group cohomology and computing explicit examples.

Contribution

It provides new criteria for when the restriction map in second cohomology is an isomorphism and computes these cohomology groups in many cases, revealing new nonzero examples.

Findings

Restriction map is an isomorphism under mild conditions for certain weights.

Explicit computations of second cohomology for various finite groups of Lie type.

Identification of new instances of nonzero second cohomology groups.

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 $H^2(G,L(\lambda)) \rightarrow H^2(G(\mathbb{F}_q),L(\lambda))$ 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.