Parabolic Deligne-Lusztig varieties

TL;DR

This paper investigates parabolic Deligne-Lusztig varieties related to the Broué conjecture, constructing a braid monoid action on their cohomology and introducing categorical tools involving Garside families.

Contribution

It introduces a new framework for parabolic Deligne-Lusztig varieties using ribbon categories with Garside families, linking braid monoid actions to cohomology in the context of Broué's conjecture.

Findings

Constructed a braid monoid action on the cohomology of certain varieties.

Extended the class of varieties considered to those attached to ribbon categories.

Provided foundational background on categories with Garside families.

Abstract

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.