Lifting cusp forms on \(\text{GL}_{2n}\) to \(\widetilde{\text{Sp}}_{2n}\): The unramified correspondence
This paper is one of a series where the authors construct what should be a generalization of the classical Shimura correspondence. The goal is a lift from self-dual, cuspidal automorphic representations of \(\text{GL}_{2n}(k_A)\) to the classical metaplectic cover of \(\text{Sp}_{2n}(k_A)\) over a number field \(k\). The construction is in two steps; the first is an induction to \(\text{Sp}_{4n}(k_A)\) by means of taking the residue of Eisenstein series. In the second step the authors form an integral against a theta function. They conjecture this space to be irreducible if the original representation was irreducible. In another paper they show that the representation space is nontrivial. In this paper they prove two theorems which show that at most all places the local component of the lift is uniquely determined through an \(L\)-function of the original representation. The main idea of the proof is the introduction of a suitable notion of a Fourier-Jacobi module; this is defined as a certain Jacobi module. It then transpires that for this module a ``multiplicity one'' theorem holds and that one can determine by calculation the appropriate \(L\)-function of the lift.
self-dual cuspidal automorphic representations
generalization of the classical Shimura correspondence
scientific article; zbMATH DE number 1425258