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A result on the Gelfand-Kirillov dimension
of representations of classical groups


Author: Chen-bo Zhu
Journal: Proc. Amer. Math. Soc. 126 (1998), 3125-3130
MSC (1991): Primary 22E46, 22E47
DOI: https://doi.org/10.1090/S0002-9939-98-04418-9
MathSciNet review: 1452837
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $(G,G')$ be the reductive dual pair $(O(p,q),Sp(2n,\mathbb{R}))$. We show that if $\pi $ is a representation of $Sp(2n,\mathbb{R})$ (respectively $O(p,q)$) obtained from duality correspondence with some representation of $O(p,q)$ (respectively $Sp(2n,\mathbb{R})$), then its Gelfand-Kirillov dimension is less than or equal to
$(p+q)(2n-\frac{p+q-1}{2})$ (respectively $2n(p+q-\frac{2n+1}{2})$).


References [Enhancements On Off] (What's this?)

  • [A] J. Adams, The theta correspondence over $\mathbb{R}$, Preprint, Workshop at the University of Maryland (1994).
  • [BK] W. Borho and H. Kraft, Uber die Gelfand-Kirillov-dimension, Math. Ann. 220 (1976), 1-24. MR 54:367
  • [H1] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570. MR 90h:22015a
  • [H2] -, $\theta $-series and invariant theory, Proceedings of Symposia in Pure Mathematics 33 (1979), 275-285. MR 81f:22034
  • [H3] -, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), 535-552. MR 90k:22016
  • [H4] -, Dual pairs in physics: Harmonic oscillators, photons, electrons, and singletons, Lectures in Appl. Math., Vol. 21, Amer. Math. Soc., Providence, R.I. (1985), 179- 206. MR 86:22036
  • [H5] -, $L^{2}$ duality for stable reductive dual pairs, preprint.
  • [KL] G.R. Krause and T.H. Lenegan, Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics, V 116, Pitman Publishing Inc., 1985. MR 86g:16001
  • [L1] J.S. Li, On the singular rank of a representation, Proc. Amer. Math. Soc. 106 (2) (1989), 567-571. MR 89k:22029
  • [L2] -, Singular unitary representations of classical groups, Invent. Math. 97 (1989), 237-255. MR 90h:22021
  • [LZ1] S.T. Lee and C.B. Zhu, Degenerate principal series and local theta correspondence, to appear in the Trans. Amer. Math. Soc. CMP 97:11
  • [LZ2] -, Degenerate principal series and local theta correspondence II, Israel J. Math. 100 (1997), 29-59. CMP 97:11
  • [TZ] E.C. Tan and C.B. Zhu, Poincare series of holomorphic representations, Indag. Mathem. N.S. 7 (1) (1996), 111-126.
  • [V] D. Vogan, Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98. MR 58:22205
  • [W] H. Weyl, The classical groups, Princeton University Press, Princeton, New Jersey, 1946.
  • [ZH] C.B. Zhu and J.S. Huang, On certain small representations of indefinite orthogonal groups, Representation Theory 1 (1997), 190-206. CMP 97:14

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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9939-98-04418-9
Keywords: Classical groups, duality correspondence, Gelfand-Kirillov dimension
Received by editor(s): November 4, 1996
Received by editor(s) in revised form: February 27, 1997
Communicated by: Roe Goodman
Article copyright: © Copyright 1998 American Mathematical Society

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