Coxeter orbits and Brauer trees III

2012/04/07 by Dudas, Olivier, Rouquier, Raphaël
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1204.1606

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 whenever the defining characteristic is good. 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 trees (in particular, the decomposition matrix) of the principal ℓ-block of E₇(q) when ℓ | Φ₁₈(q) and E₈(q) when ℓ | Φ₁₈(q) or ℓ | Φ₃₀(q).

Related

• Coxeter orbits and Brauer trees II
• The Brauer trees of unipotent blocks
• Cohomology of Deligne-Lusztig varieties for short-length regular elements in exceptional groups
• On the Cohomology of Deligne-Lusztig Varieties
• Derived categories and Deligne-Lusztig varieties II