Parabolic Deligne-Lusztig varieties
Résumé
Motivated by the Broué conjecture on blocks with abelian defect groups for finite reductive groups, we study ''parabolic'' Deligne-Lusztig varieties and construct on those which occur in the Broué conjecture an action of a braid monoid, whose action on their $\ell$-adic cohomology will conjecturally factor trough a cyclotomic Hecke algebra. In order to construct this action, we need to enlarge the set of varieties we consider to varieties attached to a ''ribbon category''; this category has a {\em Garside family}, which plays an important role in our constructions, so we devote the first part of our paper to the necessary background on categories with Garside families.
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