Unitary representations of Hecke algebras of complex reflection groups

TL;DR

This paper classifies all unitary representations of Hecke algebras associated with complex reflection groups, especially focusing on the symmetric group, and connects these to rational Cherednik algebra representations.

Contribution

It provides a complete classification of unitary representations for Hecke algebras of the symmetric group and links them to rational Cherednik algebra representations via the KZ functor.

Findings

Complete classification for symmetric group Hecke algebra

Connection established between Cherednik algebra and Hecke algebra representations

Identification of unitary representations in category O

Abstract

We begin the study of unitary representations of Hecke algebras of complex reflections groups. We obtain a complete classification for the Hecke algebra of the symmetric group $\mathfrak{S}_n$ over the complex numbers. Interestingly, the unitary representations in the category $\OO$ for the rational Cherednik algebra of type A studied in \cite{ESG} correspond to the unitary representations of the corresponding Hecke algebra via the KZ functor defined in \cite{GGOR}.