Coxeter orbits and Brauer trees III

Olivier Dudas (1) , Raphaël Rouquier (2)

1 MI - Mathematical Institute [Oxford]
2 Oxford Mathematical Institute, UCLA Department of Mathematics

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 $\ell$-block of $E_7(q)$ when $\ell \mid \Phi_{18}(q)$ and $E_8(q)$ when $\ell \mid \Phi_{18}(q)$ or $\ell \mid \Phi_{30}(q)$.

Mots clés: Finite reductive groups; Brauer trees; Deligne-Lusztig varieties; Derived categories; Broue's conjecture

Domaines: Théorie des représentations [math.RT]

Olivier Dudas, Raphaël Rouquier. Coxeter orbits and Brauer trees III. 2012. ⟨hal-00686064⟩