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Nonvanishing of a certain sesquilinear form in the theta correspondence

Author: Hongyu He
Journal: Represent. Theory 5 (2001), 437-454
MSC (2000): Primary 22E45
Published electronically: October 30, 2001
MathSciNet review: 1870597
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Abstract: 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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Hongyu He
Affiliation: Department of Mathematics and Statistics, Georgia State University, Atlanta, Georgia 30303-3083

Received by editor(s): April 24, 2001
Received by editor(s) in revised form: July 30, 2001
Published electronically: October 30, 2001
Article copyright: © Copyright 2001 American Mathematical Society