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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Degenerate principal series and
local theta correspondence

Authors: Soo Teck Lee and Chen-bo Zhu
Journal: Trans. Amer. Math. Soc. 350 (1998), 5017-5046
MSC (1991): Primary 22E46, 11F27
MathSciNet review: 1443883
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Abstract: In this paper we determine the structure of the natural $\widetilde{U}(n,n)$ module ${\Omega^{p,q}(l)}$ which is the Howe quotient corresponding to the determinant character $\det^l$ of $U(p,q)$. We first give a description of the tempered distributions on $M_{p+q,n}(\mathbb C)$ which transform according to the character $\det^{-l}$ under the linear action of $U(p,q)$. We then show that after tensoring with a character, ${\Omega^{p,q}(l)}$ can be embedded into one of the degenerate series representations of $U(n,n)$. This allows us to determine the module structure of ${\Omega^{p,q}(l)}$. Moreover we show that certain irreducible constituents in the degenerate series can be identified with some of these representations ${\Omega^{p,q}(l)}$ or their irreducible quotients. We also compute the Gelfand-Kirillov dimensions of the irreducible constituents of the degenerate series.

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Additional Information

Soo Teck Lee
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore

Chen-bo Zhu
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore

Keywords: degenerate principal series, local theta correspondence, Howe quotients, unitary representations, Gelfand-Kirillov dimension
Received by editor(s): May 16, 1995
Received by editor(s) in revised form: January 27, 1997
Article copyright: © Copyright 1998 American Mathematical Society

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