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Representation Theory

ISSN 1088-4165



Weighted projective spaces and minimal nilpotent orbits

Author: Carlo A. Rossi
Journal: Represent. Theory 12 (2008), 208-224
MSC (2000): Primary 13N10
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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Carlo A. Rossi
Affiliation: Department of mathematics, ETH Zürich, 8092 Zürich, Switzerland

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.