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.
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