Unitary representations of cyclotomic rational Cherednik algebras

TL;DR

This paper classifies all irreducible unitary modules in category O for rational Cherednik algebras of type G(r,1,n), providing explicit combinatorial formulas and algorithms for their graded characters and unitarity conditions.

Contribution

It introduces a combinatorial algorithm to determine unitarity regions for modules of rational Cherednik algebras, extending previous results to classical types.

Findings

Explicit combinatorial formulas for graded characters.

An algorithm for unitarity parameter regions.

Closed-form solutions for symmetric and classical types.

Abstract

We classify the irreducible unitary modules in category O for the rational Cherednik algebras of type G(r,1,n) and give explicit combinatorial formulas for their graded characters. More precisely, we produce a combinatorial algorithm determining, for each r-partition of n, the closed semi-linear set of parameters for which the contravariant form on the irreducible representation with the given r-partition as lowest weight is positive definite. We use this algorithm to give a closed form answer for the Cherednik algebra of the symmetric group (recovering a result of Etingof-Stoica and the author) and the Weyl groups of classical type.