Weighted projective spaces and minimal nilpotent orbits

We investigate (twisted) rings of differential operators on the resolution of singularities of an irreducible component $\overline X$ of $\overline O_{\mathrm{min}}\cap\mathfrak{n}_+$ (where $\overline O_{\mathrm{min}}$ is the (Zariski) closure of the minimal nilpotent orbit of $\mathfrak{sp}_{2n}$ and $\mathfrak{n}_+$ is the Borel subalgebra of $\mathfrak{sp}_{2n}$) using toric geometry, and show that they are homomorphic images of a certain family of associative subalgebras of $U(\mathfrak{sp}_{2n})$, which contains the maximal parabolic subalgebra $\mathfrak{p}$ determining $\overline O_{\min}$. Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same family of subalgebras. Finally, we investigate this family of subalgebras from the representation-theoretical point of view and, among other things, rediscover in a different framework irreducible highest weight modules for the UEA of $\mathfrak{sp}_{2n}$.