Translation by the full twist and Deligne-Lusztig varieties

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 Broue-Michel's conjecture on the disjointness of the cohomology for the variety associated to any good regular element. That conjecture was inspired by Broue's abelian defect group conjecture and the specific form Broue conjectured for finite groups of Lie type [4, Reves 1 et 2]. (C) 2019 Published by Elsevier Inc.