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# Weighted projective spaces and minimal nilpotent orbits

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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.

### Most cited references2

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### The minimal nilpotent orbit, the Joseph ideal, and differential operators

(1988)
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### Actions of tori on weyl algebras

(1988)
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### Author and article information

###### Journal
2007-08-13
2007-11-06
0708.1714