Calogero-Moser versus Kazhdan-Lusztig cells

TL;DR

This paper introduces a new approach to defining cells in complex reflection groups using Calogero-Moser spaces and conjectures their equivalence to Kazhdan-Lusztig cells in real groups, supported by initial results.

Contribution

It proposes a novel definition of cells for complex reflection groups based on Calogero-Moser spaces and conjectures their equivalence to classical Kazhdan-Lusztig cells for real groups.

Findings

Initial evidence supports the conjecture of equivalence.

Provides a version of left cell representations for complex groups.

Sets the stage for detailed future study of Calogero-Moser cells.

Abstract

In 1979, Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter groups. They defined in particular partitions of the group in left and two-sided cells. In 1983, Lusztig generalized this theory to Hecke algebras of Coxeter groups with unequal parameters. We propose a definition of left cells and two-sided cells for complex reflection groups, based on ramification theory for Calogero-Moser spaces. These spaces have been defined via rational Cherednik algebras by Etingof and Ginzburg. We conjecture that these coincide with Kazhdan-Lusztig cells, for real reflection groups. Counterparts of families of irreducible characters have been studied by Gordon and Martino, and we provide here a version of left cell representations. The Calogero-Moser cells will be studied in details in a forthcoming paper, providing thus several results supporting our conjecture.