## On the quantization of spherical nilpotent orbits

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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 $Sp(2n,\mathbb {R})$.## References

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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
- 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
- © Copyright 2013
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**365**(2013), 6499-6515 - MSC (2010): Primary 20G15, 22E46
- DOI: https://doi.org/10.1090/S0002-9947-2013-05925-9
- MathSciNet review: 3105760