Jack polynomials and the coinvariant ring of $G(r,p,n)$

Abstract

We study the coinvariant ring of the complex reflection group $G(r,p,n)$ as a module for the corresponding rational Cherednik algebra $\HH$ and its generalized graded affine Hecke subalgebra $\mathcal{H}$. We construct a basis consisting of non-symmetric Jack polynomials, and using this basis decompose the coinvariant ring into irreducible modules for $\mathcal{H}$. The basis consists of certain non-symmetric Jack polynomials, whose leading terms are the ``descent monomials'' for $G(r,p,n)$ recently studied by Adin, Brenti, and Roichman and Bagno and Biagoli. The irreducible $\mathcal{H}$-submodules of the coinvariant ring are their ``colored descent representations''.