On the vanishing ranges for the cohomology of finite groups of Lie type

TL;DR

This paper advances understanding of the cohomology of finite groups of Lie type by establishing initial vanishing ranges and identifying the first non-trivial cohomology classes for certain types, using geometric and combinatorial methods.

Contribution

It determines initial vanishing ranges for the cohomology of finite Chevalley groups, improving previous results, and identifies the first non-trivial classes for types A_n and C_n when p exceeds the Coxeter number.

Findings

Established initial vanishing ranges for cohomology.

Identified first non-trivial cohomology classes for specific types.

Used geometric and combinatorial techniques involving line bundle cohomology.

Abstract

Let $G({\mathbb F}_{q})$ be a finite Chevalley group defined over the field of $q=p^{r}$ elements, and $k$ be an algebraically closed field of characteristic $p>0$. A fundamental open and elusive problem has been the computation of the cohomology ring $\opH^{\bullet}(G({\mathbb F}_{q}),k)$. In this paper we determine initial vanishing ranges which improves upon known results. For root systems of type $A_n$ and $C_n$, the first non-trivial cohomology classes are determined when $p$ is larger than the Coxeter number (larger than twice the Coxeter number for type $A_n$ with $n>1$ and $r >1$). In the process we make use of techniques involving line bundle cohomology for the flag variety $G/B$ and its relation to combinatorial data from Kostant Partition Functions.