Parabolic Deligne-Lusztig varieties

François Digne (1) , Jean Michel (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 IMJ - Institut de Mathématiques de Jussieu

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 is a {\em Garside category}, which plays an important role in our proof, so we devote the first part of our paper to the necessary background on Garside categories.

Mots clés: Deligne-Lusztig variety; Garside category; Springer regular elements

Domaines: Théorie des groupes [math.GR]

François Digne, Jean Michel. Parabolic Deligne-Lusztig varieties. 2011. ⟨hal-00634644v1⟩