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

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

2010-10-06

arxiv.org/abs/1010.1203v3

Mathematics

Group Theory

24 pages, 5 figures, 6 tables. Typos corrected and some proofs streamlined over previous version

Scientific paper

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.

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