Nonvanishing of a certain sesquilinear form in the theta correspondence

He, Hongyu

doi:10.1090/S1088-4165-01-00140-6

Nonvanishing of a certain sesquilinear form in the theta correspondence
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by Hongyu He

Represent. Theory 5 (2001), 437-454

DOI: https://doi.org/10.1090/S1088-4165-01-00140-6

Published electronically: October 30, 2001

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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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