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

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

2009-05-29

arxiv.org/abs/0906.0026v2

Mathematics

Group Theory

Scientific paper

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.

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