On a correspondence between cuspidal representations of 𝐺𝐿_{2𝑛} and 𝑆𝑝̃_{2𝑛}

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