Cohomology of Deligne-Lusztig varieties for short-length regular elements in exceptional groups

TL;DR

This paper computes the cohomology of specific Deligne-Lusztig varieties in exceptional groups and proposes conjectural Brauer trees for certain blocks in E7 and E8, advancing understanding in algebraic group theory.

Contribution

It provides explicit cohomology calculations for Deligne-Lusztig varieties in types F4, E6, E7, and E8, and introduces conjectural Brauer trees for key blocks.

Findings

Cohomology of Deligne-Lusztig varieties for certain elements in F4, E6, E7, E8.

Conjectural Brauer trees for principal blocks of E7 and E8.

Enhanced understanding of modular representation theory of exceptional groups.

Abstract

We determine the cohomology of Deligne-Lusztig varieties associated to some short-length regular elements for split groups of type F4 and En. As a byproduct, we obtain conjectural Brauer trees for the principal Phi_{14}-block of E7 and the principal Phi_{24}-block of E8.