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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]

In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author in [19]. As a consequence, the Brauer trees of principal ℓ-blocks of finite groups are known for ℓ > 71. [ABSTRACT FROM AUTHOR]

We use the progenerator constructed in a previous work to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees. [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]

We prove a long-standing conjecture of Geck which predicts that cuspidal unipotent characters remain irreducible after $$\ell $$ -reduction. To this end, we construct a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand-Graev representations. This is achieved by showing that cuspidal representations appear in the head of generalised Gelfand-Graev representations attached to cuspidal unipotent classes, as defined and studied in Geck and Malle (J Lond Math Soc 2(53):63-78, 1996). [ABSTRACT FROM AUTHOR]

Abstract: We formulate a strong positivity conjecture on characters afforded by the Alvis–Curtis dual of the intersection cohomology of Deligne–Lusztig varieties. This conjecture provides a powerful tool to determine decomposition numbers of unipotent ℓ-blocks of finite reductive 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]