Calogero-Moser spaces vs unipotent representations

Abstract

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 ${\mathbb{C}}^\times$-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.