First cohomology for finite groups of Lie type: simple modules with small dominant weights

TL;DR

This paper studies the first cohomology groups of simple modules with small dominant weights for finite groups of Lie type, providing a complete description under mild conditions and extending previous results.

Contribution

It offers a comprehensive analysis of $H^1(G( ext{F}_q),L( ext{λ}))$ for small weights, with new proofs and broader applicability than prior work.

Findings

First cohomology groups have dimension at most one.

Complete description of $H^1$ for weights up to fundamental dominant weights.

Extends and simplifies earlier results on minimal nonzero dominant weights.

Abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(\lambda))$ where $L(\lambda)$ is the simple $G$-module of highest weight $\lambda$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $\lambda$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $\lambda$ is a minimal nonzero dominant weight.