Parabolic Deligne-Lusztig varieties

Abstract

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.

References in corpus (2)

• Minimal length elements of finite Coxeter groups
• On the Cohomology of Deligne-Lusztig Varieties

Cited by in corpus (9)

• Split Spetses for primitive reflection groups
• Geometric construction of Heisenberg-Weil representations for finite unitary groups and Howe correspondences
• Perverse Equivalences and Broué's Conjecture II: The Cyclic Case
• Diagrammatics for Coxeter groups and their braid groups
• Minimally dominant elements of finite Coxeter groups
• Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature $(1,n-1)$
• Mod $\ell$ cohomology of some Deligne-Lusztig varieties for ${\mathrm{GL}}_n(q)$
• Intersections of blocks of cyclotomic Hecke algebras
• Diagonal Orbits in a Type A Double Flag Variety of Complexity One