Coxeter orbits and Brauer trees

TL;DR

This paper investigates the cohomology of Deligne-Lusztig varieties linked to Coxeter elements to understand the structure of blocks in finite reductive groups, confirming conjectures about Brauer trees and derived equivalences.

Contribution

It establishes the Brauer tree and derived equivalence for the principal l-block of finite reductive groups under specific cohomological assumptions, confirming key conjectures.

Findings

Determined the planar embedded Brauer tree of the block.

Confirmed Broué's conjecture in the geometric setting.

Connected cohomology properties to block structure.

Abstract

We study the cohomology with modular coefficients of Deligne-Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal l-block of a finite reductive group G(F_q) when the order of q modulo l is assumed to be the Coxeter number. These results include the determination of the planar embedded Brauer tree of the block (as conjectured by Hiss, L\"ubeck and Malle) and the derived equivalence predicted by the geometric version of Brou\'e's conjecture.