Decomposition of modules over invariant differential operators

Let $G$ be a finite subgroup of the linear group of a finite-dimensional complex vector $V$, $B=\mathrm{S}(V)$ be the symmetric algebra, $\mathcal{D}={\mathcal{D}}_B^G$ the ring of $G$-invariant differential operators, and ${\mathcal{D}}^{-}$ its subring of negative degree operators. We prove that $M\mapsto M^{ann}={\mathrm{A}nn}_{{\mathcal{D}}^{-}}(M)$ defines an isomorphism between the category of $\mathcal{D}$-submodules of $B$ and a category of modules formed as lowest weight spaces. This is applied to a construction of simple $\mathcal{D}$-submodules of $B$ when $G$ is a generalized symmetric group, to show that $B^{ann}$ is a so-called Gelfand model. Using differential algebra and lowest weight methods we also prove branching rules, entailing the main results in the representation theory of the symmetric group, such as a differential construction of the Young basis.