On standard \(L\)-functions for unitary groups
type: Item
instance of: scholarly article
title: On standard \(L\)-functions for unitary groups
zbMATH Open document ID: 0888.11019
author: Keiji Takano
published in: Proceedings of the Japan Academy. Series A
publication date: 1998-05-27T00:00:00Z
review text: Let \(K/k\) be a quadratic extension of number fields, and \(G_n\) be the associated quasi-split unitary group of \(2n\times 2n\) matrices. Then the \(L\)-group of \(G_n\) is a semidirect product \(^L G=GL_{2n}({\mathbb{C}})\rtimes \text{Gal}(K/k)\), and admits a \(4n\)-dimensional representation \(\rho\), the so-called standard representation. Let \({\mathcal A}\) denote the adeles of \(k\) and let \(\pi=\otimes \pi_v\) be a cuspidal automorphic representation of \(G_n({\mathcal A})\). Let \(S\) denote the set of places which are either archimedean or else finite but with the data ramified. In this paper the author shows that the partial \(L\)-function \(L_S(s,\pi,\rho)=\prod_{v\not\in S}L(s,\pi_v,\rho),\) when multiplied by a suitable normalizing factor (a product of Dedekind zeta functions), has meromorphic continuation to the entire complex plane with at most a finite number of poles. A similar result for the standard \(L\)-function of the split orthogonal group \(G=O_{n.n}\) was proved by \textit{S. Gelbart, I. I. Piatetski-Shapiro} and \textit{S. Rallis} [Explicit constructions of automorphic \(L\)-functions, Lect. Notes Math. 1254. Springer-Verlag (1987; Zbl 0612.10022), Chapter 1], using their doubling method to write a suitable Rankin-Selberg integral. A similar Rankin-Selberg integral is given here, and the author notes that the unfolding into local integrals and the unramified computation are similar to that work. The main new ingredient is the study of the local integrals at the remaining `bad' places. The author rewrites these integrals in terms of Godement-Jacquet integrals, and obtains their continuations. At a finite ramified place, he moreover shows that the local integral may be made constant by means of a suitable choice of test function.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F55
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 1022282
zbMATH Keywords: automorphic form
zbMATH Keywords: standard \(L\)-function
zbMATH Keywords: unitary group
zbMATH Keywords: doubling method
zbMATH Keywords: automorphic representation
zbMATH Keywords: partial \(L\)-function
zbMATH Keywords: meromorphic continuation
zbMATH Keywords: Rankin-Selberg integral
zbMATH Keywords: local integrals
zbMATH Keywords: Godement-Jacquet integrals
reviewed by: Solomon Friedberg
MaRDI profile type: Publication
full work available at URL: pjaa.73.5
OpenAlex ID: W2051919097
cites work: Q4200278
cites work: Q3926802
cites work: Zeta functions of simple algebras
cites work: ε factor of representations of classical groups
DOI: 10.3792/PJAA.73.5
description: scientific article; zbMATH DE number 1022282