1. Parabolic Deligne-Lusztig varieties
- Author
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François Digne, Jean Michel, Laboratoire Amiénois de Mathématique Fondamentale et Appliquée (LAMFA), Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Hecke algebra ,General Mathematics ,Deligne-Lusztig variety ,Group Theory (math.GR) ,0102 computer and information sciences ,Garside families ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,20G40 ,20F36 ,Mathematics::Group Theory ,Springer regular elements ,Monoid (category theory) ,Mathematics::Category Theory ,FOS: Mathematics ,Braid ,Order (group theory) ,Finite Chevalley group ,0101 mathematics ,Abelian group ,Mathematics::Representation Theory ,Broue conjectures ,Mathematics ,Conjecture ,010102 general mathematics ,Cohomology ,010201 computation theory & mathematics ,Ribbon category ,Mathematics - Group Theory - 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.
- Published
- 2014
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