On the Calogero-Moser space associated with dihedral groups

TL;DR

This paper investigates conjectures relating Calogero-Moser space structures to Hecke algebra notions specifically for dihedral groups, advancing understanding of their algebraic and geometric connections.

Contribution

It provides an analysis of Calogero-Moser and Kazhdan-Lusztig notions for dihedral groups, testing conjectures in this specific case.

Findings

Calogero-Moser and Kazhdan-Lusztig notions coincide for dihedral groups

Verification of conjectures in the dihedral group case

Enhanced understanding of the geometric-algebraic relationship in this context

Abstract

Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author have attached to any finite complex reflection group $W$ several notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular characters), completing the notion of Calogero-Moser families defined by Gordon. If moreover $W$ is a Coxeter group, they conjectured that these notions coincide with the analogous notions defined using the Hecke algebra by Kazhdan and Lusztig (or Lusztig in the unequal parameters case). In the present paper, we aim to investigate these conjectures whenever $W$ is a dihedral group.