Weighted projective spaces and minimal nilpotent orbits

TL;DR

This paper explores the structure of differential operator rings on resolutions of minimal nilpotent orbit closures in symplectic Lie algebras, revealing connections to weighted projective spaces and primitive ideals in the universal enveloping algebra.

Contribution

It introduces a novel link between differential operators on singularity resolutions, weighted projective spaces, and primitive ideals in the universal enveloping algebra of symplectic Lie algebras.

Findings

Identification of homomorphic images of subalgebras of the UEA of sp_{2n}

Construction of rings of differential operators on weighted projective spaces

Discovery of new primitive ideals related to 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 $\bar O_{\mathrm{min}}$ 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.