%0 Journal Article
%T Parabolic Deligne-Lusztig varieties
%A Fran£żois Digne
%A Jean Michel
%J Mathematics
%D 2011
%I arXiv
%R 10.1016/j.aim.2014.02.023
%X Motivated by the Brou\'e 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 Brou\'e 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.
%U http://arxiv.org/abs/1110.4863v3