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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Theta lifting of holomorphic discrete series: The case of $U(n, n) \times U(p, q)$
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by Kyo Nishiyama and Chen-bo Zhu PDF
Trans. Amer. Math. Soc. 353 (2001), 3327-3345 Request permission

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
  • 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
  • MR Author ID: 305157
  • ORCID: 0000-0003-3819-1458
  • Email: matzhucb@nus.edu.sg
  • Received by editor(s): August 11, 2000
  • Received by editor(s) in revised form: November 8, 2000
  • Published electronically: April 9, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3327-3345
  • MSC (2000): Primary 22E46, 11F27
  • DOI: https://doi.org/10.1090/S0002-9947-01-02830-6
  • MathSciNet review: 1828608