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Representation Theory

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

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Nonvanishing of a certain sesquilinear form in the theta correspondence
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by Hongyu He PDF
Represent. Theory 5 (2001), 437-454 Request permission


Suppose $2n+1 \geq p+q$. In an earlier paper in 2000 we study a certain sesquilinear form $(,)_{\pi }$ introduced by Jian-Shu Li in 1989. For $\pi$ in the semistable range of $\theta (MO(p,q) \rightarrow MSp_{2n}(\mathbb {R}))$, if $(,)_{\pi }$ does not vanish, then it induces a sesquilinear form on $\theta (\pi )$. In another work in 2000 we proved that $(,)_{\pi }$ is positive semidefinite under a mild growth condition on the matrix coefficients of $\pi$. In this paper, we show that either $(,)_{\pi }$ or $(,)_{\pi \otimes \det }$ is nonvanishing. These results combined with one result of Przebinda suggest the existence of certain unipotent representations of $Mp_{2n}(\mathbb {R})$ beyond unitary representations of low rank.
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Additional Information
  • Hongyu He
  • Affiliation: Department of Mathematics and Statistics, Georgia State University, Atlanta, Georgia 30303-3083
  • Email:
  • Received by editor(s): April 24, 2001
  • Received by editor(s) in revised form: July 30, 2001
  • Published electronically: October 30, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Represent. Theory 5 (2001), 437-454
  • MSC (2000): Primary 22E45
  • DOI:
  • MathSciNet review: 1870597