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

Abstract

Letkkbe an algebraically closed field of characteristicp>0p > 0, and letGGbe a simple, simply connected algebraic group defined overFp\mathbb {F}_p. Givenr≥1r \geq 1, setq=prq=p^r, and letG(Fq)G(\mathbb {F}_q)be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology groupH1⁡(G(Fq),L(λ))\operatorname {H}^1(G(\mathbb {F}_q),L(\lambda )), whereL(λ)L(\lambda )is the simpleGG-module of highest weightλ\lambda. Under certain very mild conditions onppandqq, we are able to completely describe the first cohomology group whenλ\lambdais 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 whenλ\lambdais a minimal non-zero dominant weight.