## Regular functions of symplectic spherical nilpotent orbits and their quantizations

HTML articles powered by AMS MathViewer

- by Kayue Daniel Wong
- Represent. Theory
**19**(2015), 333-346 - DOI: https://doi.org/10.1090/ert/474
- Published electronically: December 17, 2015
- PDF | Request permission

## 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

## Bibliographic Information

**Kayue Daniel Wong**- Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
- MR Author ID: 1140325
- Email: makywong@ust.hk
- Received by editor(s): June 13, 2015
- Received by editor(s) in revised form: November 30, 2015
- Published electronically: December 17, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Represent. Theory
**19**(2015), 333-346 - MSC (2010): Primary 17B08, 22E10
- DOI: https://doi.org/10.1090/ert/474
- MathSciNet review: 3434893