Nonvanishing of a certain sesquilinear form in the theta correspondence

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.