Regular automorphisms and Calogero-Moser families

TL;DR

This paper investigates fixed points of automorphisms in Calogero-Moser spaces related to complex reflection groups, linking geometric fixed points to character tables, and exploring their connection to unipotent representations.

Contribution

It identifies fixed points of automorphisms in Calogero-Moser spaces and relates them to the character theory of complex reflection groups, proposing a conjecture for all such fixed points.

Findings

Determined some fixed points of automorphisms in Calogero-Moser spaces.

Linked fixed points to the character table of the associated reflection group.

Proposed a conjecture for all fixed points in the maximal component.

Abstract

We study the subvariety of fixed points of an automorphism of a Calogero-Moser space induced by a regular element of finite order of the normalizer of the associated complex reflection group $W$. We determine some of (and conjecturally all) the ${\mathbb{C}}^\times$-fixed points of its unique irreducible component of maximal dimension in terms of the character table of $W$. This 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.