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Transactions of the American Mathematical Society

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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
MathSciNet review: 3105760
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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

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.