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