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Weighted projective spaces and minimal nilpotent orbits


Author: Carlo A. Rossi
Journal: Represent. Theory 12 (2008), 208-224
MSC (2000): Primary 13N10
DOI: https://doi.org/10.1090/S1088-4165-08-00328-2
Published electronically: April 17, 2008
MathSciNet review: 2403559
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Abstract: We investigate (twisted) rings of differential operators on the resolution of singularities of an irreducible component $ \overline X$ of $ \overline O_{\mathrm{min}}\cap\mathfrak{n}_+$ (where $ \overline O_{\mathrm{min}}$ is the (Zariski) closure of the minimal nilpotent orbit of $ \mathfrak{sp}_{2n}$ and $ \mathfrak{n}_+$ is the Borel subalgebra of $ \mathfrak{sp}_{2n}$) using toric geometry, and show that they are homomorphic images of a certain family of associative subalgebras of $ U(\mathfrak{sp}_{2n})$, which contains the maximal parabolic subalgebra $ \mathfrak{p}$ determining $ \overline O_{\min}$. Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same family of subalgebras. Finally, we investigate this family of subalgebras from the representation-theoretical point of view and, among other things, rediscover in a different framework irreducible highest weight modules for the UEA of $ \mathfrak{sp}_{2n}$.


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Additional Information

Carlo A. Rossi
Affiliation: Department of mathematics, ETH Zürich, 8092 Zürich, Switzerland
Email: crossi@math.ethz.ch

DOI: https://doi.org/10.1090/S1088-4165-08-00328-2
Received by editor(s): August 17, 2007
Received by editor(s) in revised form: November 8, 2007
Published electronically: April 17, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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