Invariant differential operators and the generalized symmetric group

In this paper we study the decomposition of the direct image of ${\pi }_{+}({\text{Oc}}_X)$ the polynomial ring ${\text{Oc}}_X$ as a $\text{D}$-module, under the map $\pi :\text{spec}{\text{Oc}}_X\to \text{spec}{\text{Oc}}_X^{G(r,n)}$, where ${\text{Oc}}_X^{G(r,n)}$ is the ring of invariant polynomial under the action of the wreath product $G(r,p):=\text{ZZ}\mathrm{/}r\text{ZZ}\wr {\text{Sc}}_n$. We first describe the generators of the simple components of ${\pi }_{+}({\text{Oc}}_X)$ and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a $\text{D}$-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.