\(L\)-functions and the oscillator representation
type: Item
instance of: scholarly article
title: \(L\)-functions and the oscillator representation
zbMATH Open document ID: 0605.10016
author: Steve Rallis
published in: Lecture Notes in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The theory of dual reductive pairs and the oscillator (Weil) representation yields 'correspondences' between the representations of these pairs over both local and global fields. The local theory is now fairly well understood but the global theory turns out to be more subtle. Around 1980 J.-L. Waldspurger demonstrated that the image of an automorphic representation under the Shimura correspondence is zero or not depends on, apart from local conditions, whether a certain \(L\)-function vanishes at a special point or not. In this book the author proposes a much broader framework in which to understand Waldspurger's theorem, and which leads to a very wide generalization. He also proposes novel methods of proof. Here this programme is described and carried through for a restricted class of dual reductive pairs; this class contains the case considered by Waldspurper but is much wider. The technique used here involves computing the \(L^2\)-norm of elements of the image under the correspondence and relating this to the Siegel theorem for a dual reductive pair related to the original one.
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 22-02
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 3977089
zbMATH Keywords: L-functions
zbMATH Keywords: oscillator representation
zbMATH Keywords: dual reductive pairs
zbMATH Keywords: automorphic representation
zbMATH Keywords: Shimura correspondence
zbMATH Keywords: \(L^2\)-norm
zbMATH Keywords: Rankin integral representations of standard L-functions
zbMATH Keywords: see-saw dual reductive pairs
zbMATH Keywords: Siegel-Weil formula
reviewed by: Samuel James Patterson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0077894
OpenAlex ID: W4213163565
DOI: 10.1007/BFB0077894
description: scientific article; zbMATH DE number 3977089