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Theta lifting of holomorphic discrete series: The case of $ U(n, n) \times U(p, q) $


Authors: Kyo Nishiyama and Chen-bo Zhu
Journal: Trans. Amer. Math. Soc. 353 (2001), 3327-3345
MSC (2000): Primary 22E46, 11F27
DOI: https://doi.org/10.1090/S0002-9947-01-02830-6
Published electronically: April 9, 2001
MathSciNet review: 1828608
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Abstract:

Let $ ( G, G' ) = ( U( n, n ), U( p, q ) ) \; ( p + q \leq n ) $ be a reductive dual pair in the stable range. We investigate theta lifts to $ G$ of unitary characters and holomorphic discrete series representations of $ G' $, in relation to the geometry of nilpotent orbits. We give explicit formulas for their $K$-type decompositions. In particular, for the theta lifts of unitary characters, or holomorphic discrete series with a scalar extreme $ K' $-type, we show that the $ K $ structure of the resulting representations of $G$is almost identical to the $K_{\mathbb{C} } $-module structure of the regular function rings on the closure of the associated nilpotent $K_{\mathbb{C} }$-orbits in $\mathfrak{s} $, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{s} $ is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.


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  • 1. J. Adams, $L$-functoriality for dual pairs. Orbites unipotentes et représentations, II. Astérisque No. 171-172, (1989), 85-129. MR 91e:22020
  • 2. R. Howe, $\theta$-series and invariant theory, in Automorphic forms, representations and $L$-functions, Proc. Sympos. Pure Math. 33 Amer. Math. Soc. (1979), pp. 275-286. MR 81f:22034
  • 3. R. Howe, Reciprocity laws in the theory of dual pairs, in Representation Theory of Reductive Groups (ed. P.C. Trombi), Progress in Mathematics 40 Birkhäuser (1983), pp. 159-175. MR 85k:22033
  • 4. R. Howe, Remarks on classical invariant theory. Trans. Amer. Math. Soc. 313 (1989), no. 2, 539-570. Erratum to ``Remarks on classical invariant theory". Trans. Amer. Math. Soc. 318 (1990), no. 2, 823. MR 90h:22015a; MR 90h:22015b
  • 5. R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535-552. MR 90k:22016
  • 6. J.-S. Huang and J.-S. Li, Unipotent representations attached to spherical nilpotent orbits, Amer. Jour. Math. 121 (1999), no. 3, 497-517. MR 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 57:3311
  • 8. B. Kostant, Lie group representations on polynomial rings, Amer. Jour. Math. 85 (1963), 327-404. MR 28:1252
  • 9. S. Kudla, Seesaw dual reductive pairs, ``Automorphic forms of several variables'', Progress in Mathematics 46 (1983), 244-268. MR 86b:22032
  • 10. S. T. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence, Trans. Amer. Math. Soc. 350 (1998), no. 12, 5017-5046. MR 99c:22021
  • 11. S. T. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence II, Israel J. Math., 100 (1997), 29-59. MR 99c:22022
  • 12. J.-S. Li, Singular unitary representations of classical groups, Invent. Math., 97 (1989), 237-255. MR 90h:22021
  • 13. J.-S. Li, Theta lifting for unitary representations with nonzero cohomology, Duke Math. J., 61 (1990), 913-937. MR 92f:22024
  • 14. Kyo Nishiyama, Multiplicity-free actions and the geometry of nilpotent orbits, Math. Ann. 318 (2000), 777-793. CMP2001:06
  • 15. Kyo Nishiyama, Theta lifting of two-step nilpotent orbits for the pair $ O(p, q) \times Sp(2n, {\mathbb{R} }) $. In Proceedings of the Symposium on ``Infinite Dimensional Harmonic Analysis'' (Kyoto, September 1999), pp. 278-289.
  • 16. Kyo Nishiyama, Hiroyuki Ochiai and Kenji Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules (Hermitian symmetric case), to appear in Astérisque.
  • 17. T. Przebinda, Characters, dual pairs, and unipotent representations. J. Funct. Anal. 98 (1991), no. 1, 59-96. MR 92d:22021
  • 18. T. Przebinda, Characters, dual pairs, and unitary representations. Duke Math. J. 69 (1993), no. 3, 547-592. MR 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), 1071-1118. CMP 2000:17
  • 20. E.-C. Tan and C.-B. Zhu, On certain distinguished unitary representations supported on null cones, Amer. J. Math. 120 (1998), no. 5, 1059-1076. MR 99h:22019
  • 21. P. Trapa, Annihilators, associated varieties, and the theta correspondence, preprint, November 1999.
  • 22. D. Vogan, Gelfand-Kirillov dimensions for Harish-Chandra modules, Inventiones Math. 48 (1978), 75-98. MR 58:22205
  • 23. D. A. Vogan, Jr., Associated varieties and unipotent representations, in Harmonic analysis on reductive groups (Brunswick, ME, 1989), 315-388, Progr. Math. 101, Birkhäuser Boston, Boston, MA, 1991. MR 93k:22012
  • 24. H. Weyl, The classical groups, Princeton University Press, 1946.
  • 25. C.-B. Zhu and J.-S. Huang, On certain small representations of indefinite orthogonal groups, Represent. Theory 1 (1997), 190-206 (electronic). MR 98j:22026

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Additional Information

Kyo Nishiyama
Affiliation: Faculty of Integrated Human Studies, Kyoto University, Sakyo, Kyoto 606-8501, Japan
Email: kyo@math.h.kyoto-u.ac.jp

Chen-bo Zhu
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Email: matzhucb@nus.edu.sg

DOI: https://doi.org/10.1090/S0002-9947-01-02830-6
Keywords: Reductive dual pair, theta lifting, holomorphic discrete series, nilpotent orbits, associated cycles
Received by editor(s): August 11, 2000
Received by editor(s) in revised form: November 8, 2000
Published electronically: April 9, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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