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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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Degenerate principal series and local theta correspondence
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by Soo Teck Lee and Chen-bo Zhu PDF
Trans. Amer. Math. Soc. 350 (1998), 5017-5046 Request permission

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
  • Email: matleest@leonis.nus.edu.sg
  • Chen-bo Zhu
  • Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore
  • MR Author ID: 305157
  • ORCID: 0000-0003-3819-1458
  • Email: matzhucb@leonis.nus.edu.sg
  • Received by editor(s): May 16, 1995
  • Received by editor(s) in revised form: January 27, 1997
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 5017-5046
  • MSC (1991): Primary 22E46, 11F27
  • DOI: https://doi.org/10.1090/S0002-9947-98-02036-4
  • MathSciNet review: 1443883