Invariant differential operators and the generalized symmetric group

TL;DR

This paper investigates the decomposition of polynomial rings as D-modules under invariant maps related to generalized symmetric groups, providing explicit generators, multiplicities, and the structure of differential operators on Specht polynomials.

Contribution

It introduces a detailed decomposition of polynomial rings as D-modules under the action of wreath product groups, including generators, multiplicities, and the structure of invariants and differential operators.

Findings

Explicit generators of simple components identified

Multiplicity formulas for components derived

Decomposition of localized polynomial rings achieved

Abstract

In this paper we study the decomposition of the direct image of $\pi_+(\Oc_{X})$ the polynomial ring $\Oc_X$ as a $\D$-module, under the map $\pi: \spec \Oc_{X} \to \spec \Oc_{X}^{G(r,n)}$, where $\Oc_{X}^{G(r,n)}$ is the ring of invariant polynomial under the action of the wreath product $G(r,p):= \ZZ / r \ZZ \wr \Sc_n $. We first describe the generators of the simple components of $\pi_+(\Oc_X)$ and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a $\D$-module decomposition of the polynomial ring localized at the discriminant of $\pi$. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials.