Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

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

Author(s): Chen-bo Zhu
Journal: Proc. Amer. Math. Soc. 126 (1998), 3125-3130.
MSC (1991): Primary 22E46, 22E47
MathSciNet review: 1452837
Retrieve article in: PDF
This article is available free of charge

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:

[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

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 22E46, 22E47

Retrieve articles in all Journals with MSC (1991): 22E46, 22E47

Additional Information:

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

DOI: 10.1090/S0002-9939-98-04418-9
PII: S 0002-9939(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
Copyright of article: Copyright 1998, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia