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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 [4, Rêves 1 et 2]. [ABSTRACT FROM AUTHOR]

We exhibit a simple representation of the finite symplectic group of Sp 8 ( q ) in characteristic ℓ | q 2 + 1 having two non-conjugate supercuspidal supports. This answers a question of M.-F. Vignéras. [ABSTRACT FROM AUTHOR]

Abstract: We determine the cohomology of Deligne–Lusztig varieties associated with some short-length regular elements for split groups of type and . As a byproduct, we obtain conjectural Brauer trees for the principal -block of and the principal -block of . [Copyright &y& Elsevier]

Abstract: Using virtual projective characters coming from the ℓ-adic cohomology of Deligne–Lusztig varieties, we determine new decomposition numbers for principal blocks of some groups of Lie type of small rank, including and Steinbergʼs triality groups . [Copyright &y& Elsevier]

Abstract: We study the quotient of parabolic Deligne–Lusztig varieties by a finite unipotent group where U is the unipotent radical of a rational parabolic subgroup . We show that in some particular cases the cohomology of this quotient can be expressed in terms of “smaller” parabolic Deligne–Lusztig varieties associated to the Levi subgroup L. [Copyright &y& Elsevier]

Abstract: We study the cohomology with modular coefficients of Deligne–Lusztig varieties associated to Coxeter elements. Under some torsion-free assumption on the cohomology we derive several results on the principal ℓ-block of a finite reductive group when the order of q modulo ℓ is assumed to be the Coxeter number. These results include the determination of the planar embedded Brauer tree of the block (as conjectured by Hiss, Lübeck and Malle (1995) in ) and the derived equivalence predicted by the geometric version of Brouéʼs conjecture (Broué and Malle, 1993, ). [Copyright &y& Elsevier]