On lifting from classical groups to \(\text{GL}_N\)
type: Item
instance of: scholarly article
title: On lifting from classical groups to \(\text{GL}_N\)
zbMATH Open document ID: 1028.11029
author: Henry H. Kim
author: Freydoon Shahidi
author: James Cogdell
author: I. I. Pyatetskii-Shapiro
published in: Publications Mathématiques
publication date: 2003-03-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104177
review text: Langlands' functorial lift is one of the central questions in the theory of automorphic forms. In this paper, the four authors unify their methods into a powerful machinery and prove the existence of a weak functorial lift of a generic cuspidal representation of \(\text{SO}_{2n+1}\) to \(\text{GL}_{2n}\). In the follow-u paper ``Functoriality for the classical groups'', the authors establish functoriality from the split classical groups \(\text{SO}_{2n+1}\), \(\text{SO}_{2n}\) and \(\text{Sp}_{2n}\) to \(\text{GL}_{2n}\), for globally generic cuspidal representations. Their approach is based on the Converse Theorem of Cogdell and Piatetski-Shapiro and the Langlands-Shahidi method, which studies automorphic \(L\)-functions via the constant and non-constant terms of the Eisenstein series. Functoriality is a part of the Langlands program. It is a manifestation of the (conjectural) Langlands correspondence. The principle of functoriality states that any \(L\)-homomorphism of \(L\)-groups \( ^L H \to ^L G\) should determine a lifting of admissible or automorphic representations of \(H\) to admissible or automorphic representations of \(G\). If \(H=\text{SO}_{2n+1}\) and \(G=\text{GL}_{2n}\), the connected components of the \(L\)-groups are \(^L H^0=\text{Sp}_{2n}(\mathbb{C})\) and \(^L G^0=\text{GL}_{2n}(\mathbb{C})\), and there is a natural embedding \(^L H^0 \hookrightarrow ^L G^0\). The philosophy of Langlands then says that associated to this map of dual groups there should exist a lifting of automorphic forms on \(\text{SO}_{2n+1}(\mathbb{A})\) to \(\text{GL}_{2n}(\mathbb{A})\). Let \(\pi = \otimes' \pi_v\) be an irreducible automorphic representation of \(\text{SO}_{2n+1}(\mathbb{A})\). A weak lift of \(\pi\) is an automorphic representation \(\Pi = \otimes' \Pi_v\) of \(\text{GL}_{2n}(\mathbb{A})\) such that, at almost all places, \(\Pi_v\) is a local lift of \(\pi_v\). More precisely, for every Archimedean place \(v\) and for almost all non-Archimedean places \(v…
reviewed by: Dubravka Ban
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E55
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11R39
zbMATH DE Number: 1874405
zbMATH Keywords: Langlands functorial lift
zbMATH Keywords: \(\text{SO}_{2n+1}\)
zbMATH Keywords: \(\text{GL}_{2n}\)
zbMATH Keywords: generic cuspidal automorphic representations
zbMATH Keywords: weak functorial lift
MaRDI profile type: Publication
OpenAlex ID: W2046099957
DOI: 10.1007/S10240-001-8187-Z
description: scientific article; zbMATH DE number 1874405