Multiplicity one theorems for Fourier-Jacobi models (Q2851609)

Let \(k\) be a non-Archimedean local field of characteristic zero. Let \(G\) denote the group \(\mathrm{GL}(n)\), \(U(n)\), or \(\mathrm{Sp}(2n)\), defined over \(k\), and regarded as a subgroup of the symplectic group \(\mathrm{Sp}(2n)\). Let \(\widetilde{G}\) be the double cover of \(G\) induced by the metaplectic cover \(\widetilde{\mathrm{Sp}}(2n)\) of \(\mathrm{Sp}(2n)\). Denote by \(\omega_\psi\) the smooth oscillator representation of \(\widetilde{\mathrm{Sp}}(2n)\) corresponding to a nontrivial character \(\psi\) of \(k\). The author proves that for every irreducible admissible smooth representation \(\pi\) of \(G\), and any genuine irreducible admissible smooth representation \(\pi'\) of \(\widetilde{G}\), one has NEWLINE\[NEWLINE \dim \text{Hom}_G (\pi' \otimes \omega_\psi \otimes \pi ,\mathbb C)\leq 1. NEWLINE\]NEWLINE For earlier ``multiplicity one theorems'' see [\textit{A. Aizenbud} et al., Ann. Math. (2) 172, No. 2, 1407--1434 (2010; Zbl 1202.22012)].