Nonvanishing of a certain sesquilinear form in the theta correspondence

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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. © 2001 American Mathematical Society.

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Representation Theory

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