Wreath Macdonald polynomials at q=t as characters of rational Cherednik algebras

TL;DR

This paper extends the connection between Macdonald polynomials and rational Cherednik algebra characters from symmetric groups to wreath product groups, showing that their characters correspond to specialized wreath Macdonald polynomials.

Contribution

It generalizes Gordon's result to wreath product groups, establishing that their rational Cherednik algebra characters are given by wreath Macdonald polynomials at q=t.

Findings

Characters are given by specialized wreath Macdonald polynomials.

Generalization from symmetric groups to wreath product groups.

Provides a new link between algebraic characters and symmetric functions.

Abstract

Using the theory of Macdonald, Gordon showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg associated to the symmetric group $\mathfrak{S}_n$ are given by plethystically transformed Macdonald polynomials specialized at q=t. We generalize this to restricted rational Cherednik algebras of wreath product groups $C_\ell \wr \mathfrak{S}_n$ and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman.