2021/12/23 by Bonnafé, Cédric
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2112.13684
openalex publication_date 2021/12/23 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Lusztig's classification of unipotent representations of finite reductive groups depends only on the associated Weyl group W (endowed with its Frobenius automorphism). All the structural questions (families, Harish-Chandra series, partition into blocks...) have an answer in a combinatorics that can be entirely built directly from W. Over the years, we have noticed that the same combinatorics seems to be encoded in the Poisson geometry of a Calogero-Moser space associated with W (roughly speaking, families correspond to ℂ^×-fixed points, Harish-Chandra series correspond to symplectic leaves, blocks correspond to symplectic leaves in the fixed point subvariety under the action of a root of unity). The aim of this survey is to gather all these observations, state precise conjectures and provide general facts and examples supporting these conjectures.