Calogero–Moser versus Kazhdan–Lusztig cells

Cédric Bonnafé and Raphaël Rouquier

Vol. 261 (2013), No. 1, 45–51

DOI: 10.2140/pjm.2013.261.45

Abstract

In 1979, Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter groups, defining 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.

Keywords

Hecke algebra, reflection group, Cherednik algebra, Kazhdan–Lusztig theory

Mathematical Subject Classification 2010

Primary: 20C08

Milestones

Received: 6 March 2012

Revised: 18 May 2012

Accepted: 6 July 2012

Published: 28 February 2013