Lifting cusp forms on \(\text{GL}_{2n}\) to \(\widetilde{\text{Sp}}_{2n}\): The unramified correspondence
type: Item
instance of: scholarly article
title: Lifting cusp forms on \(\text{GL}_{2n}\) to \(\widetilde{\text{Sp}}_{2n}\): The unramified correspondence
zbMATH Open document ID: 0991.11024
DOI: 10.1215/S0012-7094-99-10009-3
author: David Ginzburg
author: Steve Rallis
author: David Soudry
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 1425258
zbMATH Keywords: Jacquet module
zbMATH Keywords: Fourier-Jacobi coefficients
zbMATH Keywords: theta function
zbMATH Keywords: multiplicity one theorem
zbMATH Keywords: self-dual cuspidal automorphic representations
zbMATH Keywords: generalization of the classical Shimura correspondence
zbMATH Keywords: lift
zbMATH Keywords: metaplectic cover
zbMATH Keywords: Eisenstein series
zbMATH Keywords: \(L\)-function
zbMATH Keywords: Jacobi module
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
cites work: On explicit lifts of cusp forms from \(\text{GL}_m\) to classical groups
cites work: On a correspondence between cuspidal representations of 𝐺𝐿_{2𝑛} and 𝑆𝑝̃_{2𝑛}
cites work: $L$-functions for symplectic groups
cites work: On Euler Products and the Classification of Automorphic Representations I
cites work: On Euler Products and the Classification of Automorphic Forms II
cites work: Q3470542
full work available at URL: s0012-7094-99-10009-3
OpenAlex ID: W2043195836
description: scientific article; zbMATH DE number 1425258