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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
MR Author ID: 207972

Chen-bo Zhu
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
MR Author ID: 305157
ORCID: 0000-0003-3819-1458

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