Coxeter orbits and Brauer trees III

Dudas, Olivier¹ ; Rouquier, Raphaël²

¹ Université Denis Diderot – Paris 7, UFR de Mathématiques, Institut de Mathématiques de Jussieu, Case 7012, 75205 Paris Cedex 13, France
² Department of Mathematics, UCLA, Box 951555, Los Angeles, California 90095-1555

Journal of the American Mathematical Society, Tome 27 (2014) no. 4, pp. 1117-1145

Résumé

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

Détail

DOI : 10.1090/S0894-0347-2014-00791-8

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