Cuspidal Calogero-Moser and Lusztig families for Coxeter groups

TL;DR

This paper classifies cuspidal Calogero-Moser families for all infinite Coxeter groups by analyzing symplectic leaves and rigid modules, revealing they coincide with Lusztig families.

Contribution

It provides a comprehensive computation of cuspidal Calogero-Moser families across all infinite Coxeter groups and establishes their equivalence with Lusztig families.

Findings

Cuspidal Calogero-Moser families are classified for all infinite Coxeter groups.

These families are shown to be equivalent to Lusztig families.

The classification is based on symplectic leaf analysis and rigid modules.

Abstract

The goal of this paper is to compute the cuspidal Calogero-Moser families for all infinite families of finite Coxeter groups, at all parameters. We do this by first computing the symplectic leaves of the associated Calogero-Moser space and then by classifying certain "rigid" modules. Numerical evidence suggests that there is a very close relationship between Calogero-Moser families and Lusztig families. Our classification shows that, additionally, the cuspidal Calogero-Moser families equal cuspidal Lusztig families for the infinite families of Coxeter groups.