Calogero-Moser spaces vs unipotent representations

TL;DR

This paper explores the deep connections between Lusztig's classification of unipotent representations of finite reductive groups and the Poisson geometry of Calogero-Moser spaces, proposing conjectures and supporting evidence for their correspondence.

Contribution

It consolidates observations and formulates conjectures linking combinatorial representation theory with Calogero-Moser space geometry.

Findings

Families correspond to ${ m C}^ imes$-fixed points

Harish-Chandra series relate to symplectic leaves

Blocks are associated with symplectic leaves in fixed point subvarieties

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.