Translation by the full twist and Deligne-Lusztig varieties

Cédric Bonnafé (1) , Olivier Dudas (2) , Raphaël Rouquier (3)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
3 Department of Mathematics [UCLA]

We prove several conjectures about the cohomology of Deligne-Lusztig varieties: invariance under conjugation in the braid group, behaviour with respect to translation by the full twist, parity vanishing of the cohomology for the variety associated with the full twist. In the case of split groups of type A, and using previous results of the second author, this implies Broué-Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broué's abelian defect group conjecture and the specific form Broué conjectured for finite groups of Lie type.

Domaines: Théorie des représentations [math.RT]

Cédric Bonnafé, Olivier Dudas, Raphaël Rouquier. Translation by the full twist and Deligne-Lusztig varieties. Journal of Algebra, 2020, 558, pp.129-145. ⟨10.1016/j.jalgebra.2019.10.010⟩. ⟨hal-01901594v2⟩