Translation by the full twist and Deligne-Lusztig varieties

TL;DR

This paper proves several conjectures about the cohomology of Deligne-Lusztig varieties, including invariance, behavior under the full twist, and parity vanishing, with implications for Broué-Michel's conjecture in type A groups.

Contribution

It establishes new invariance and parity properties of Deligne-Lusztig cohomology and confirms Broué-Michel's conjecture for split groups of type A.

Findings

Proved invariance under conjugation in the braid group.

Established behavior of cohomology with respect to the full twist.

Demonstrated parity vanishing of the cohomology for the full twist variety.

Abstract

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 Brou\'e-Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Brou\'e's abelian defect group conjecture and the specific form Brou\'e conjectured for finite groups of Lie type.