On the Vanishing Ranges for the Cohomology of Finite Groups of Lie Type.

Let be a finite Chevalley group defined over the field of q=pr 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 . In this paper, we determine initial vanishing ranges, which improves upon known results. For root systems of type An and Cn, the first nontrivial cohomology classes are determined when p is larger than the Coxeter number (larger than twice the Coxeter number for type An 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. [ABSTRACT FROM PUBLISHER]