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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9947-01-02830-6
PII: S 0002-9947(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