Cell modules and canonical basic sets for Hecke algebras from Cherednik algebras

TL;DR

This paper develops a method to label irreducible representations of specialized Hecke algebras of complex reflection groups using Cherednik algebra techniques, establishing connections with cell modules and canonical basic sets.

Contribution

It introduces a new approach combining category O, KZ functor, and combinatorics to construct canonical basic sets for non-semisimple Hecke algebra specializations.

Findings

Canonical basic sets constructed in many cases.

Standard modules' images match cell modules when Hecke algebra is cellular.

Provides a framework linking Cherednik algebras and Hecke algebra representations.

Abstract

In this note we are interested in labelling the irreducible representations of non-semisimple specialisations of Hecke algebras of complex reflection groups. We will use category O for the rational Cherednik algebra and the KZ functor together with elementary algebraic and combinatorial arguments to construct "canonical basic sets" in many cases. We will also show that the images of the standard modules through the KZ functor agree with the appropriate cell modules, whenever the Hecke algebra has a cellular structure.