First cohomology groups of Chevalley groups in cross characteristic

Abstract

Let $G$ be a simple Chevalley group defined over $\mathbb{F}_{q}$. We show that if $r$ does not divide $q$ and $k$ is an algebraically closed field of characteristic $r$, then very few irreducible $kG$-modules have nonzero $H^1(G,V)$. We also give an explicit upper bound for $\dim H^1(G,V)$ for $V$ an irreducible $kG$-module that does not depend on $q$, but only on the rank of the group. Cline, Parshall and Scott showed that such a bound exists when $r|q$. We obtain extremely strong bounds in the case that a Borel subgroup has no fixed points on $V$.