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

TL;DR

This paper extends previous work on the cohomology of finite groups of Lie type, identifying the first non-trivial cohomology classes for various types when the prime exceeds the Coxeter number.

Contribution

It generalizes earlier results by locating the initial non-trivial cohomology classes for all remaining Lie types beyond A and C, under certain prime conditions.

Findings

Determined the first non-trivial cohomology classes for types B, D, E, F, G when prime > Coxeter number.

Extended the induction functor approach to new Lie types.

Provided a unified framework for cohomology computations across multiple Lie types.

Abstract

The computation of the cohomology for finite groups of Lie type in the describing characteristic is a challenging and difficult problem. In earlier work, the authors constructed an induction functor which takes modules over the finite group of Lie type to modules for the ambient algebraic group G. In particular this functor when applied to the trivial module yields a natural G-filtration. This filtration was utilized in the earlier work to determine the first non-trivial cohomology class when the underlying root system is of type A_{n} or C_{n}. In this paper the authors extend these results toward locating the first non-trivial cohomology classes for the remaining finite groups of Lie type (i.e., the underlying root system is of type B_{n}, C_{n}, D_{n}, E_{6}, E_{7}, E_{8}, F_{4}, and G_{2}) when the prime is larger than the Coxeter number.