Regular functions of symplectic spherical nilpotent orbits and their quantizations

Wong, Kayue Daniel

doi:10.1090/ert/474

Regular functions of symplectic spherical nilpotent orbits and their quantizations
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by Kayue Daniel Wong

Represent. Theory 19 (2015), 333-346

DOI: https://doi.org/10.1090/ert/474

Published electronically: December 17, 2015

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Abstract:

We study the ring of regular functions of classical spherical orbits $R(\mathcal {O})$ for $G = Sp(2n,\mathbb {C})$. In particular, treating $G$ as a real Lie group with maximal compact subgroup $K$, we focus on a quantization model of $\mathcal {O}$ when $\mathcal {O}$ is the nilpotent orbit $(2^{2p}1^{2q})$. With this model, we verify a conjecture by McGovern and another conjecture by Achar and Sommers related to the character formula of such orbits. Assuming the results in a preprint of Barbasch, we will also verify the Achar-Sommers conjecture for a larger class of nilpotent orbits.

References

Pramod Narahari Achar, Equivariant coherent sheaves on the nilpotent cone for complex reductive Lie groups, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2717009
Pramod N. Achar and Eric N. Sommers, Local systems on nilpotent orbits and weighted Dynkin diagrams, Represent. Theory 6 (2002), 190–201. MR 1927953, DOI 10.1090/S1088-4165-02-00174-7
Jeffrey Adams and Dan Barbasch, Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 (1995), no. 1, 1–42. MR 1346217, DOI 10.1006/jfan.1995.1099
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
Dan Barbasch and David A. Vogan Jr., Unipotent representations of complex semisimple groups, Ann. of Math. (2) 121 (1985), no. 1, 41–110. MR 782556, DOI 10.2307/1971193
Dan Barbasch, The unitary dual for complex classical Lie groups, Invent. Math. 96 (1989), no. 1, 103–176. MR 981739, DOI 10.1007/BF01393972
D. Barbasch, Regular Functions on Covers of Nilpotent Coadjoint Orbits, http:// arxiv.org/abs/0810.0688v1, 2008
Dan Barbasch and Pavle Pandžić, Dirac cohomology and unipotent representations of complex groups, Noncommutative geometry and global analysis, Contemp. Math., vol. 546, Amer. Math. Soc., Providence, RI, 2011, pp. 1–22. MR 2815128, DOI 10.1090/conm/546/10782
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
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
Tanya Chmutova and Viktor Ostrik, Calculating canonical distinguished involutions in the affine Weyl groups, Experiment. Math. 11 (2002), no. 1, 99–117. MR 1960305, DOI 10.1080/10586458.2002.10504473
Mauro Costantini, On the coordinate ring of spherical conjugacy classes, Math. Z. 264 (2010), no. 2, 327–359. MR 2574980, DOI 10.1007/s00209-008-0468-5
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
Hanspeter Kraft and Claudio Procesi, On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982), no. 4, 539–602. MR 694606, DOI 10.1007/BF02565876
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
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
Soo Teck Lee and Chen-Bo Zhu, Degenerate principal series and local theta correspondence. II, Israel J. Math. 100 (1997), 29–59. MR 1469104, DOI 10.1007/BF02773634
Soo Teck Lee and Chen-Bo Zhu, Degenerate principal series and local theta correspondence. III. The case of complex groups, J. Algebra 319 (2008), no. 1, 336–359. MR 2378075, DOI 10.1016/j.jalgebra.2007.06.018
George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
G. Lusztig, Notes on unipotent classes, Asian J. Math. 1 (1997), no. 1, 194–207. MR 1480994, DOI 10.4310/AJM.1997.v1.n1.a7
William M. McGovern, Rings of regular functions on nilpotent orbits and their covers, Invent. Math. 97 (1989), no. 1, 209–217. MR 999319, DOI 10.1007/BF01850661
William M. McGovern, Completely prime maximal ideals and quantization, Mem. Amer. Math. Soc. 108 (1994), no. 519, viii+67. MR 1191608, DOI 10.1090/memo/0519
Dmitri I. Panyushev, Some amazing properties of spherical nilpotent orbits, Math. Z. 245 (2003), no. 3, 557–580. MR 2021571, DOI 10.1007/s00209-003-0555-6
Eric Sommers, A generalization of the Bala-Carter theorem for nilpotent orbits, Internat. Math. Res. Notices 11 (1998), 539–562. MR 1631769, DOI 10.1155/S107379289800035X
Eric Sommers, Lusztig’s canonical quotient and generalized duality, J. Algebra 243 (2001), no. 2, 790–812. MR 1850659, DOI 10.1006/jabr.2001.8868
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., 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
K. Wong, Regular Functions of Nilpotent Orbits and Normality of their Closures, http://arxiv.org/abs/1302.6627, 2013
Kayue Wong, Dixmier algebras on complex classical nilpotent orbits and their representation theories, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Cornell University. MR 3193183
Liang Yang, On the quantization of spherical nilpotent orbits, Trans. Amer. Math. Soc. 365 (2013), no. 12, 6499–6515. MR 3105760, DOI 10.1090/S0002-9947-2013-05925-9