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arXiv · 0708.1714

Weighted projective spaces and minimal nilpotent orbits

C. A. Rossi
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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.

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C. A. Rossi. 2007-11-06. Weighted projective spaces and minimal nilpotent orbits. https://arxiv.org/abs/0708.1714

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