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

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.