Nonvanishing of a Certain Sesquilinear Form in the Theta Correspondence

Suppose 2n + 1 ≥ p + q. In an earlier paper in 2000 we study a certain sesquilinear form (, )π introduced by Jian-Shu Li in 1989. For π in the semistable range of θ(MO(p, q) → MSp2n(R)), if (, )π does not vanish, then it induces a sesquilinear form on θ(π). In another work in 2000 we proved that (, )π is positive semidefinite under a mild growth condition on the matrix coefficients of π. In this paper, we show that either (, )π or (, )π⊗det is nonvanishing. These results combined with one result of Przebinda suggest the existence of certain unipotent representations of Mp2n(R) beyond unitary representations of low rank.