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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
DOI: https://doi.org/10.1090/S0002-9947-2013-05925-9
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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  • 1. J. Adams, J.-S. Huang and D.A. Vogan, Functions on the model orbit in $ E\sb 8$, Represent. Theory 2 (1998), 224-263. MR 1628031 (99g:20077)
  • 2. B. Binegar, On a class of multiplicity-free nilpotent $ K\sb \mathbb{C}$-orbits, J. Math. Kyoto Univ. 47 (2007), no. 4, 735-766. MR 2413063 (2010b:22017)
  • 3. D. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993. MR 1251060 (94j:17001)
  • 4. R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535-552. MR 985172 (90k:22016)
  • 5. R. Howe and C.-B. Zhu, Eigendistributions for orthogonal groups and representations of symplectic groups. J. Reine Angew. Math. 545 (2002), 121-166. MR 1896100 (2003g:20077)
  • 6. J.-S. Huang and J.-S. Li, Unipotent representations attached to spherical nilpotent orbits. Amer. J. Math. 121 (1999), no. 3, 497-517. MR 1738410 (2000m:22018)
  • 7. M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials. Invent. Math. 44 (1978), no. 1, 1-47. MR 0463359 (57:3311)
  • 8. A.W. Knapp, Representation theory of semisimple groups. An overview based on examples. Princeton Mathematical Series, 36. Princeton University Press, Princeton, NJ, 1986. MR 855239 (87j:22022)
  • 9. A.W. Knapp and D.A. Vogan, Cohomological induction and unitary representations. Princeton Mathematical Series, 45. Princeton University Press, Princeton, NJ, 1995. MR 1330919 (96c:22023)
  • 10. D.R. King, Classification of spherical nilpotent orbits in complex symmetric space, J. Lie Theory 14 (2004), no. 2, 339-370. MR 2066860 (2005e:22012)
  • 11. B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387. MR 0142696 (26:265)
  • 12. S. Kudla and S. Rallis, Degenerate principal series and invariant distribution, Israel J. Math. 69 (1990), 25-45. MR 1046171
  • 13. S.L. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence. Trans. Amer. Math. Soc. 350 (1998), no. 12, 5017-5046. MR 1443883 (99c:22021)
  • 14. J.-S. Li, Singular unitary representations of classical groups. Invent. Math. 97 (1989), no. 2, 237-255. MR 1001840 (90h:22021)
  • 15. Kyo Nishiyama, H. Ochiai, K. Taniguchi and H. Yamashita, Nilpotent orbits, associated cycles and Whittaker models for highest weight representations. Astérisque No. 273 (2001). Société Mathématique de France, Paris, 2001. pp. i-vi and 1-163. MR 1845713 (2002b:22025)
  • 16. Kyo Nishiyama and C.-B. Zhu, Theta lifting of holomorphic discrete series: the case of $ U(n,n)\times U(p,q)$. Trans. Amer. Math. Soc. 353 (2001), no. 8, 3327-3345. MR 1828608 (2002e:22017)
  • 17. Kyo Nishiyama and C.-B. Zhu, Theta lifting of unitary lowest weight modules and their associated cycles. Duke Math. J. 125 (2004), no. 3, 415-465. MR 2166751 (2006f:22007)
  • 18. T. Przebinda, Characters, dual pairs, and unitary representations. Duke Math. J. 69 (1993), no. 3, 547-592. MR 1208811 (94i:22036)
  • 19. W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive Lie groups. Ann. of Math. (2) 151 (2000), no. 3, 1071-1118. MR 1779564 (2001j:22017)
  • 20. D.A. Vogan, Associated varieties and unipotent representations, Harmonic analysis on reductive groups (W. Barker and P. Sally, eds.), 315-388, Birkhäuser, Boston-Basel-Berlin, 1991. MR 1168491 (93k:22012)
  • 21. D.A. Vogan, Irreducibility of discrete series representations for semisimple symmetric spaces. Representations of Lie groups, Kyoto, Hiroshima, 1986, 191-221, Adv. Stud. Pure Math., 14, Academic Press, Boston, MA, 1988. MR 1039838 (91b:22023)
  • 22. D.A. Vogan, Representations of reductive Lie groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 245-266, Amer. Math. Soc., Providence, RI, 1987. MR 934226 (89h:22034)
  • 23. D.A. Vogan, The method of coadjoint orbits for real reductive groups. Representation theory of Lie groups (Park City, UT, 1998), 179-238, IAS/Park City Math. Ser., 8, Amer. Math. Soc., Providence, RI, 2000. MR 1737729 (2001k:22027)
  • 24. C.-B. Zhu, Invariant distributions of classical groups. Duke Math. J. 65 (1992), no. 1, 85-119. MR 1148986 (92k:22022)
  • 25. C.-B. Zhu, J.-S. Huang, On certain small representations of indefinite orthogonal groups. Represent. Theory 1 (1997), 190-206. MR 1457244 (98j:22026)

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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: https://doi.org/10.1090/S0002-9947-2013-05925-9
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.

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