Weighted projective spaces and minimal nilpotent orbits

Abstract

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 ${\bar{O}}_{\min }$ 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.

15 pages; misprints corrected; some terminology mistakes have been corrected