Coxeter orbits and Brauer trees III

TL;DR

This paper proves key conjectures relating to derived equivalences and Brauer trees in modular representation theory of finite groups of Lie type, specifically for Coxeter elements and certain blocks, advancing understanding of their structure.

Contribution

It establishes the Broué conjecture for Coxeter elements in good characteristic and confirms a conjecture on Brauer trees, determining decomposition matrices for specific blocks of E_7(q) and E_8(q).

Findings

Proved Broué's conjecture for Coxeter elements in good characteristic.

Confirmed a conjecture on Brauer trees of certain blocks.

Determined decomposition matrices for principal blocks of E_7(q) and E_8(q).

Abstract

This article is the final one of a series of articles on certain blocks of modular representations of finite groups of Lie type and the associated geometry. We prove the conjecture of Brou\'e on derived equivalences induced by the complex of cohomology of Deligne-Lusztig varieties in the case of Coxeter elements whenever the defining characteristic is good. We also prove a conjecture of Hi{\ss}, L\"ubeck and Malle on the Brauer trees of the corresponding blocks. As a consequence, we determine the Brauer trees (in particular, the decomposition matrix) of the principal $\ell$-block of $E_7(q)$ when $\ell \mid \Phi_{18}(q)$ and $E_8(q)$ when $\ell \mid \Phi_{18}(q)$ or $\ell \mid \Phi_{30}(q)$.