On the quantization of spherical nilpotent orbits

Yang, Liang

doi:10.1090/S0002-9947-2013-05925-9

On the quantization of spherical nilpotent orbits
HTML articles powered by AMS MathViewer

by Liang Yang PDF

Trans. Amer. Math. Soc. 365 (2013), 6499-6515 Request permission

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

Jeffrey Adams, Jing-Song Huang, and David A. Vogan Jr., Functions on the model orbit in $E_8$, Represent. Theory 2 (1998), 224–263. MR 1628031, DOI 10.1090/S1088-4165-98-00048-X
Birne Binegar, On a class of multiplicity-free nilpotent $K_{\Bbb C}$-orbits, J. Math. Kyoto Univ. 47 (2007), no. 4, 735–766. MR 2413063, DOI 10.1215/kjm/1250692287
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
Roger Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), no. 3, 535–552. MR 985172, DOI 10.1090/S0894-0347-1989-0985172-6
Roger Howe and Chen-Bo Zhu, Eigendistributions for orthogonal groups and representations of symplectic groups, J. Reine Angew. Math. 545 (2002), 121–166. MR 1896100, DOI 10.1515/crll.2002.031
Jing-Song Huang and Jian-Shu Li, Unipotent representations attached to spherical nilpotent orbits, Amer. J. Math. 121 (1999), no. 3, 497–517. MR 1738410
M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47. MR 463359, DOI 10.1007/BF01389900
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
Anthony W. Knapp and David A. Vogan Jr., Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR 1330919, DOI 10.1515/9781400883936
Donald R. King, Classification of spherical nilpotent orbits in complex symmetric space, J. Lie Theory 14 (2004), no. 2, 339–370. MR 2066860
Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
Stephen S. Kudla and Stephen Rallis, Degenerate principal series and invariant distributions, Israel J. Math. 69 (1990), no. 1, 25–45. MR 1046171, DOI 10.1007/BF02764727
Soo Teck Lee and Chen-Bo Zhu, Degenerate principal series and local theta correspondence, Trans. Amer. Math. Soc. 350 (1998), no. 12, 5017–5046. MR 1443883, DOI 10.1090/S0002-9947-98-02036-4
Jian-Shu Li, Singular unitary representations of classical groups, Invent. Math. 97 (1989), no. 2, 237–255. MR 1001840, DOI 10.1007/BF01389041
Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi, Hiroshi Yamashita, and Shohei Kato, Nilpotent orbits, associated cycles and Whittaker models for highest weight representations, Société Mathématique de France, Paris, 2001. Astérisque No. 273 (2001) (2001). MR 1845713
Kyo Nishiyama and Chen-Bo Zhu, Theta lifting of holomorphic discrete series: the case of $\textrm {U}(n,n)\times \textrm {U}(p,q)$, Trans. Amer. Math. Soc. 353 (2001), no. 8, 3327–3345. MR 1828608, DOI 10.1090/S0002-9947-01-02830-6
Kyo Nishiyama and Chen-Bo Zhu, Theta lifting of unitary lowest weight modules and their associated cycles, Duke Math. J. 125 (2004), no. 3, 415–465. MR 2166751, DOI 10.1215/S0012-7094-04-12531-X
Tomasz Przebinda, Characters, dual pairs, and unitary representations, Duke Math. J. 69 (1993), no. 3, 547–592. MR 1208811, DOI 10.1215/S0012-7094-93-06923-2
Wilfried Schmid and Kari 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, DOI 10.2307/121129
David A. Vogan Jr., Associated varieties and unipotent representations, Harmonic analysis on reductive groups (Brunswick, ME, 1989) Progr. Math., vol. 101, Birkhäuser Boston, Boston, MA, 1991, pp. 315–388. MR 1168491
David A. Vogan Jr., Irreducibility of discrete series representations for semisimple symmetric spaces, Representations of Lie groups, Kyoto, Hiroshima, 1986, Adv. Stud. Pure Math., vol. 14, Academic Press, Boston, MA, 1988, pp. 191–221. MR 1039838, DOI 10.2969/aspm/01410191
David A. Vogan Jr., Representations of reductive Lie groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 245–266. MR 934226
David A. Vogan Jr., The method of coadjoint orbits for real reductive groups, Representation theory of Lie groups (Park City, UT, 1998) IAS/Park City Math. Ser., vol. 8, Amer. Math. Soc., Providence, RI, 2000, pp. 179–238. MR 1737729, DOI 10.1090/pcms/008/05
Chen-bo Zhu, Invariant distributions of classical groups, Duke Math. J. 65 (1992), no. 1, 85–119. MR 1148986, DOI 10.1215/S0012-7094-92-06504-5
Chen-Bo Zhu and Jing-Song Huang, On certain small representations of indefinite orthogonal groups, Represent. Theory 1 (1997), 190–206. MR 1457244, DOI 10.1090/S1088-4165-97-00031-9