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.
Categories: math.RA math.AG