Parabolic Deligne–Lusztig varieties

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 ℓ-adic cohomology will conjecturally factor through 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 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.