Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 22E46, 11F27

Retrieve articles in all journals with MSC (1991): 22E46, 11F27


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
Email: matzhucb@leonis.nus.edu.sg

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02036-4
PII: S 0002-9947(98)02036-4
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