Whittaker-orthogonal models, functoriality, and the Rankin-Selberg method
type: Item
instance of: scholarly article
title: Whittaker-orthogonal models, functoriality, and the Rankin-Selberg method
zbMATH Open document ID: 0782.11015
DOI: 10.1007/BF01232019
author: Daniel Bump
author: Solomon Friedberg
author: David Ginzburg
published in: Inventiones Mathematicae
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/144013
review text: The purpose of this paper is to exhibit a phenomenon in the Rankin- Selberg method, in which a Rankin-Selberg integral of an automorphic form on one group \((SO(2n+2))\) is used to represent an \(L\)-function of an automorphic form on a different group \((Sp(2n)\times GL(m))\). The two automorphic forms involved are related by a lifting of the sort described by the Langlands' functoriality conjecture. The role played by Whittaker models in the Rankin-Selberg method is taken over by a modification of it which they call a Whittaker-Orthogonal model (WO-model). In section 2, they prove that an irreducible admissible representation of \(GO(2n+2,F)^ 0\) for a non-archimedean local field \(F\) admits up to constant multiple at most one WO-functional. In section 3, they show that a nonramified principal series representation of this group has a WO- model if and only if it is a local lifting from \(GSp(2n)\), in which case they give an explicit formula for the spherical vector in the model analogous to the Casselman-Shalika formula for the spherical vector in a Whittaker model. In section 4, they give analogous results for the group \(SO(2n+2)\). From these results they represent for a nonramified place \(v\) the local factor \[ L_ v((m+2)s-(m+1)/2, \quad \pi_ v'\otimes \tau_ v, \text{ ``standard''}) \] of an \(L\)-function on \(Sp(2n)\times GL(m)\) with \(n\geq 2\) and \(m\leq n-1\) by the corresponding local factor of the Rankin-Selberg integral \[ \int_{X_ F\setminus X_ A} \int_{W_ F\setminus W_ A} \int_{SO_{2m+3}(F)\setminus SO_{2m+3}(A)} \varphi(xwg)\psi(w) E(g,s,\tau,f_ s) dg dw dx \] on \(SO(2n+2)\) by regarding \(SO(2m+3)\) as a subgroup of \(SO(2n+2)\). They also discuss the case \(m\geq n\).
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 126371
zbMATH Keywords: Rankin-Selberg integrals
zbMATH Keywords: \(L\)-functions for automorphic forms
zbMATH Keywords: Whittaker- Orthogonal model
zbMATH Keywords: irreducible admissible representation
zbMATH Keywords: non-archimedean local field
zbMATH Keywords: nonramified principal series representation
zbMATH Keywords: Rankin-Selberg integral
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 126371