Differential operators and reflection group of 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.