Automorphisms and symplectic leaves of Calogero-Moser spaces

TL;DR

This paper investigates the structure of symplectic leaves in Calogero-Moser spaces fixed by automorphisms, providing a parametrization that extends previous work and hints at deep connections with representation theory.

Contribution

It introduces a new parametrization of symplectic leaves in fixed point subvarieties of Calogero-Moser spaces, generalizing earlier results and linking geometry with representation theory.

Findings

Parametrization of symplectic leaves via Harish-Chandra style methods

Extension of Bellamy and Losev's work to fixed point subvarieties

Connections to unipotent representations of finite reductive groups

Abstract

We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by an element of finite order of the normalizer of the associated complex reflection group $W$. We give a parametrization {\it \`a la Harish-Chandra} of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero-Moser spaces and unipotent representations of finite reductive groups, which will be the theme of a forthcoming paper.