  We investigate (twisted) rings of differential operators on the resolution of
singularities of a particular irreducible component of the (Zarisky) closure of
the minimal orbit $\bar O_{\mathrm{min}}$ of $\mathfrak{sp}_{2n}$, intersected
with the Borel subalgebra $\mathfrak n_+$ of $\mathfrak{sp}_{2n}$, using toric
geometry and show that they are homomorphic images of a subalgebra of the
Universal Enveloping Algebra (UEA) of $\mathfrak{sp}_{2n}$, which contains the
maximal parabolic subalgebra $\mathfrak p$ determining the minimal nilpotent
orbit. Further, using Fourier transforms on Weyl algebras, we show that
(twisted) rings of well-suited weighted projective spaces are obtained from the
same subalgebra. Finally, investigating this subalgebra from the
representation-theoretical point of view, we find new primitive ideals and
rediscover old ones for the UEA of $\mathfrak{sp}_{2n}$ coming from the
aforementioned resolution of singularities.
