Differential operators and reflection group of type $B_n$

TL;DR

This paper investigates the polynomial representation of the quantum Olshanetsky-Perelomov system for type $B_n$ reflection groups, focusing on invariant differential operators and their decomposition via group representation theory.

Contribution

It introduces a new module structure over the Weyl algebra for type $B_n$ reflection groups and explicitly decomposes the polynomial representation using higher Specht polynomials.

Findings

Explicit generators for simple components are provided.

Decomposition of polynomial representations is achieved.

Enhanced understanding of invariant differential operators for type $B_n$.

Abstract

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group $W$ of type $B_n$. We endow the polynomial ring ${\mathbb C} [x_1,\ldots\\\ldots, x_n]$ with a structure of module over the Weyl algebra associated with the ring ${\mathbb C} [x_1,\ldots,x_n]^{W}$ of invariant polynomials under a reflections group $W$ of type $B_n$. Then we study the polynomial representation of the ring of invariant differential operators under the reflections group $W$. We use the group representation theory namely the higher Specht polynomials associated with the reflection group $W$ and establish a decomposition of that structure by providing explicitly the generators of the simple components.