Coxeter orbits and Brauer trees III

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 Broué on derived equivalences induced by the complex of cohomology of Deligne-Lusztig varieties in the case of Coxeter elements. We also prove a conjecture of Hiß, Lübeck, and Malle on the Brauer trees of the corresponding blocks. As a consequence, we determine the Brauer tree (in particular, the decomposition matrix) of the principal ℓ \ell -block of E 7 ( q ) E_7(q) when ℓ ∣ Φ 18 ( q ) \ell \mid \Phi _{18}(q) and E 8 ( q ) E_8(q) when ℓ ∣ Φ 18 ( q ) \ell \mid \Phi _{18}(q) or ℓ ∣ Φ 30 ( q ) \ell \mid \Phi _{30}(q) .