Theta lifting of nilpotent orbits for symmetric pairs
HTML articles powered by AMS MathViewer
- by Kyo Nishiyama, Hiroyuki Ochiai and Chen-bo Zhu PDF
- Trans. Amer. Math. Soc. 358 (2006), 2713-2734 Request permission
Abstract:
We consider a reductive dual pair $(G, G’)$ in the stable range with $G’$ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent $K’_{\mathbb {C}}$-orbits, where $K’$ is a maximal compact subgroup of $G’$ and we describe the precise $K_{\mathbb {C}}$-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair $( G, K)$. As an application, we prove sphericality and normality of the closure of certain nilpotent $K_{\mathbb {C}}$-orbits obtained in this way. We also give integral formulas for their degrees.References
- Walter Borho and Hanspeter Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), no. 1, 61–104 (German, with English summary). MR 522032, DOI 10.1007/BF02566256
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Neil Chriss and Victor Ginzburg, Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997. MR 1433132
- 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
- Andrzej Daszkiewicz, Witold Kraśkiewicz, and Tomasz Przebinda, Nilpotent orbits and complex dual pairs, J. Algebra 190 (1997), no. 2, 518–539. MR 1441961, DOI 10.1006/jabr.1996.6910
- A. Daszkiewicz, W. Kraśkiewicz and T. Przebinda, Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements, preprint.
- Stephen S. Gelbart, A theory of Stiefel harmonics, Trans. Amer. Math. Soc. 192 (1974), 29–50. MR 425519, DOI 10.1090/S0002-9947-1974-0425519-8
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- R. Howe, $\theta$-series and invariant theory, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 275–285. MR 546602
- Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539–570. MR 986027, DOI 10.1090/S0002-9947-1989-0986027-X
- 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, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
- 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
- Shohei Kato and Hiroyuki Ochiai, The degrees of orbits of the multiplicity-free actions, Astérisque 273 (2001), 139–158 (English, with English and French summaries). Nilpotent orbits, associated cycles and Whittaker models for highest weight representations. MR 1845716
- 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 group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
- Shin-ichi Izumi, An abstract integral. II, Proc. Imp. Acad. Tokyo 16 (1940), 87–89. MR 1840
- Kyo Nishiyama, Multiplicity-free actions and the geometry of nilpotent orbits, Math. Ann. 318 (2000), no. 4, 777–793. MR 1802510, DOI 10.1007/s002080000141
- Kyo Nishiyama, Theta lifting of two-step nilpotent orbits for the pair $\mathrm O(p,q)\times \textrm {Sp}(2n,\Bbb R)$, Infinite dimensional harmonic analysis (Kyoto, 1999) Gräbner, Altendorf, 2000, pp. 278–289. MR 1851930
- Kyo Nishiyama, Hiroyuki Ochiai, and Kenji Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules—Hermitian symmetric case, Astérisque 273 (2001), 13–80 (English, with English and French summaries). Nilpotent orbits, associated cycles and Whittaker models for highest weight representations. MR 1845714
- 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
- Takuya Ohta, Nilpotent orbits of $Z_4$-graded Lie algebra and geometry of the moment maps associated to the dual pair $( U(p,q), U(r,s) )$, preprint.
- Takuya Ohta, Nilpotent orbits of $Z_4$-graded Lie algebra and geometry of the moment maps associated to the dual pairs $(\mathrm {O}(p,q), \mathrm {Sp}(2n, \mathbb {R}))$ and $(\mathrm {O}^*(2n), \mathrm {Sp}(p,q))$, in preparation.
- V. L. Popov and E. B. Vinberg, Invariant theory, in “Algebraic geometry. IV.”, Encyclopaedia of Mathematical Sciences, 55. Springer-Verlag, Berlin, 1994, pp. 123–284.
- 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
- Igor R. Shafarevich, Basic algebraic geometry. 1, 2nd ed., Springer-Verlag, Berlin, 1994. Varieties in projective space; Translated from the 1988 Russian edition and with notes by Miles Reid. MR 1328833
- È. B. Vinberg and V. L. Popov, A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 749–764 (Russian). MR 0313260
- 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
- H. Weyl, The classical groups, Princeton University Press, 1946.
Additional Information
- Kyo Nishiyama
- Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo, Kyoto 606-8502, Japan
- MR Author ID: 207972
- Email: kyo@math.kyoto-u.ac.jp
- Hiroyuki Ochiai
- Affiliation: Department of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
- Email: ochiai@math.nagoya-u.ac.jp
- Chen-bo Zhu
- Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
- MR Author ID: 305157
- ORCID: 0000-0003-3819-1458
- Email: matzhucb@nus.edu.sg
- Received by editor(s): December 18, 2003
- Received by editor(s) in revised form: August 14, 2004
- Published electronically: December 20, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 2713-2734
- MSC (2000): Primary 22E46, 11F27
- DOI: https://doi.org/10.1090/S0002-9947-05-03826-2
- MathSciNet review: 2204053
Dedicated: Dedicated to Roger Howe on his sixtieth birthday