On the standard \(L\)-function for \(G_ 2\)
type: Item
instance of: scholarly article
title: On the standard \(L\)-function for \(G_ 2\)
zbMATH Open document ID: 0777.11016
DOI: 10.1215/S0012-7094-93-06915-3
author: David Ginzburg
published in: Duke Mathematical Journal
publication date: 1993-12-08T00:00:00Z
review text: The problem of determining the analytic continuation and functional equation of general \(L\)-functions associated to automorphic representations of reductive Lie groups and representations of the \(L\)- groups has been pursued by two different methods. The first is the interpretation of the \(L\)-function as arising in the constant term of Eisenstein series. The second is the generalized Rankin-Selberg integral representation of the \(L\)-function. In dealing with the case \(G_ 2\), \textit{I. Piatetski-Shapiro}, \textit{S. Rallis} and \textit{G. Schiffman} [ Am. J. Math. 114, 1269-1315 (1993; reviewed below)] have shown that the latter method gives new examples of \(L\)-functions not possible by the first method. In this paper another Rankin-Selberg integral which represents the standard \(L\)-function of \(G_ 2\) is constructed. It involves an Eisenstein series on the double cover of \(SL_ 2\). Since the poles of this Eisenstein series are known it can be shown that the partial standard \(L\)-function for \(G_ 2\) can have at most one simple pole.
reviewed by: Joachim Schwermer
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 179194
zbMATH Keywords: automorphic representations
zbMATH Keywords: Rankin-Selberg integral
zbMATH Keywords: standard \(L\)- function of \(G_ 2\)
zbMATH Keywords: Eisenstein series
zbMATH Keywords: simple pole
MaRDI profile type: Publication
cites work: Symmetric square \(L\)-functions on \(GL(r)\)
cites work: Q3926803
cites work: The conjugate classes of Chevalley groups of type \((G_2)\)
cites work: Q4173774
cites work: L-Functions for SOn x GLk.
cites work: A Rankin-Selberg integral for the adjoint representation of \(GL_ 3\)
cites work: On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series
cites work: Q3497288
cites work: Q3470542
cites work: Rankin-Selberg Convolutions
cites work: Automorphic forms on \(GL(3)\). I, II
cites work: Howe correspondences on a \(p\)-adic field
cites work: The symmetric-square L-function attached to a cuspidal automorphic representation of \(GL_ 3\)
cites work: Rankin-Selberg Integrals for the Group G 2
cites work: Theta Correspondence Associated to G 2
cites work: Rankin-Selberg convolutions for 𝑆𝑂_{2𝑙+1}×𝐺𝐿_{𝑛}: local theory
full work available at URL: s0012-7094-93-06915-3
OpenAlex ID: W1527251327
description: scientific article; zbMATH DE number 179194