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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the quantization of spherical nilpotent orbits


Author: Liang Yang
Journal: Trans. Amer. Math. Soc. 365 (2013), 6499-6515
MSC (2010): Primary 20G15, 22E46
Published electronically: April 25, 2013
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Abstract: Let $ G$ be the real symplectic group $ Sp(2n,\mathbb{R})$. This paper determines the global sections of certain line bundles over the spherical nilpotent $ K_{\mathbb{C}}$-orbit $ \mathcal {O}$. As a consequence, Vogan's conjecture for these orbits is verified. The conjecture holds that there exists a unique unitary $ (\mathfrak{g},K)$-module structure on the space of the algebraic global sections of the line bundle associated to the admissible datum, provided that the boundary $ \partial \overline {\mathcal {O}}$ has complex codimension at least $ 2$ in $ \overline {\mathcal {O}}$. Similar results are obtained for the metaplectic twofold cover $ Mp(2n,\mathbb{R})$ of $ Sp(2n,\mathbb{R})$.


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Additional Information

Liang Yang
Affiliation: Department of Mathematics, Sichuan University, Chengdu, 610064, People’s Republic of China
Email: malyang@scu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05925-9
PII: S 0002-9947(2013)05925-9
Keywords: Admissible data, spherical nilpotent orbits, Vogan's conjecture
Received by editor(s): November 9, 2011
Received by editor(s) in revised form: May 20, 2012
Published electronically: April 25, 2013
Additional Notes: Part of this work was included in the author’s Ph.D. thesis
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.