On a correspondence between cuspidal representations of $\operatorname{GL}_{2n}$ and $\widetilde {\operatorname{Sp}}_{2n}$

Let $\eta$ be an irreducible, automorphic, self-dual, cuspidal representation of $\operatorname{GL}_{2n}(\mathbb A)$, where $\mathbb A$ is the adele ring of a number field $K$. Assume that $L^S(\eta,\Lambda^2,s)$ has a pole at $s=1$ and that $L(\eta, \frac 12)\neq 0$. Given a nontrivial character $\psi$ of $K\backslash\mathbb A$, we construct a nontrivial space of genuine and globally $\psi^{-1}$-generic cusp forms $V_{\sigma _{\psi}(\eta)}$ on $\widetilde{\operatorname{Sp}}_{2n}(\mathbb A)$-the metaplectic cover of ${\operatorname{Sp}}_{2n}(\mathbb A)$. $V_{\sigma _{\psi}(\eta)}$ is invariant under right translations, and it contains all irreducible, automorphic, cuspidal (genuine) and $\psi^{-1}$-generic representations of $\widetilde{\operatorname{Sp}}_{2n}(\mathbb A)$, which lift (``functorially, with respect to $\psi$") to $\eta$. We also present a local counterpart. Let $\tau$ be an irreducible, self-dual, supercuspidal representation of $\operatorname{GL}_{2n}(F)$, where $F$ is a $p$-adic field. Assume that $L(\tau,\Lambda^2,s)$ has a pole at $s=0$. Given a nontrivial character $\psi$ of $F$, we construct an irreducible, supercuspidal (genuine) $\psi^{-1}$-generic representation $\sigma _\psi(\tau)$ of $\widetilde{\operatorname{Sp}}_{2n}(F)$, such that $\gamma(\sigma _\psi(\tau)\otimes\tau,s,\psi)$ has a pole at $s=1$, and we prove that $\sigma _\psi(\tau)$ is the unique representation of $\widetilde{\operatorname{Sp}}_{2n}(F)$ satisfying these properties.