One-W-type modules for rational Cherednik algebra and cuspidal two-sided cells

TL;DR

This paper classifies certain simple modules of rational Cherednik algebras for finite Weyl groups, linking them to Lusztig's cuspidal two-sided cells, and explores their Dirac cohomology to reveal relations with Calogero-Moser cells.

Contribution

It provides a classification of modules that are irreducible on W and connects this classification to Lusztig's cuspidal two-sided cells, using Dirac cohomology techniques.

Findings

Classification of simple modules related to cuspidal two-sided cells

Computation of Dirac cohomology for these modules

Identification of relations between Calogero-Moser and two-sided cells

Abstract

We classify the simple modules for the rational Cherednik algebra that are irreducible when restricted to W, in the case when W is a finite Weyl group. The classification turns out to be closely related to the cuspidal two-sided cells in the sense of Lusztig. We compute the Dirac cohomology of these modules and use the tools of Dirac theory to find nontrivial relations between the cuspidal Calogero-Moser cells and the cuspidal two-sided cells.