Spherical varieties and integral representations of L-functions
type: Item
instance of: scholarly article
title: Spherical varieties and integral representations of L-functions
zbMATH Open document ID: 1253.11059
author: Yiannis Sakellaridis
published in: Algebra \& Number Theory
publication date: 2012-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.4245
full work available at URL: p01.xhtml
reviewed by: Zhengyu Mao
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 6064724
zbMATH Keywords: automorphic L-functions
zbMATH Keywords: spherical varieties
zbMATH Keywords: Rankin-Selberg integrals
zbMATH Keywords: periods of automorphic forms
MaRDI profile type: Publication
OpenAlex ID: W2144420366
arXiv ID: 0905.4245
DOI: 10.2140/ANT.2012.6.611
review text: A very important method in studying automorphic \(L\)-functions is the construction of integral representations of such \(L\)-functions. Previously many important works were done in the construction of such integral representations, however there was not a general theory and the constructions are often considered as ``accidents''.NEWLINENEWLINEIn this paper, the author gives a uniform interpretation of a large number of the previous constructions. The theory of spherical varieties play an essential role in this interpretation.NEWLINENEWLINEGiven a spherical variety \(X\) for a split reductive group \(G\), the paper defines a space of Schwartz functions on \(X\), and constructs the space of theta functions and \(X\)-Eisenstein series associated to the Schwartz functions. It is conjectured that these \(X\)-Eisenstein series admit meromorphic continuations.NEWLINENEWLINEThe \(X\)-Eisenstein series will be generating functions for certain automorphic \(L\)-functions, at least when \(X\) has a structure of a ``preflag bundle''. The conjectured meromorphic continuation of \(X\)-Eisenstein series will give the continuation of the \(L\)-functions immediately.NEWLINENEWLINEUnder this setting, the problem of constructing integral representations of automorphic \(L\)-functions becomes a problem of classifying spherical varieties with the preflag bundle structure. In the case when \(X\) is a smooth affine spherical variety, the preflag bundle structure is there automatically. Knop--Van Steirteghem classified the smooth affine spherical varieties up to coverings, central tori and \(G_m\)-fibrations. The paper goes through the list of smooth affine varieties and recovers most known constructions of \(L-\)functions, as well as some new ones (for \(L(\pi_1\otimes\pi_2,s_2) L(\pi_2,s_1-\frac12(n-1))\) when \(\pi_1,\pi_2\) are cuspidal representations of \(\text{GL}_n\), and \(L(\pi_1\otimes\pi_2,s_2+\frac12)L(\pi_2,s_1-\frac12 n)\) where \(\pi_1\) is a cuspidal representation of \(\…
description: scientific article; zbMATH DE number 6064724