On the definition of transfer factors
type: Item
instance of: scholarly article
title: On the definition of transfer factors
zbMATH Open document ID: 0644.22005
DOI: 10.1007/BF01458070
author: D. Shelstad
author: Robert P. Langlands
published in: Mathematische Annalen
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164291
review text: ``Endoscopic groups'' in harmonic analysis over reductive groups \(G\) on local fields \(F\) are sophisticated objects and still elusive (an example is given by a 1-dimensional torus anisotropic over a local field \(F\) of characteristic 0 but split over the quadratic extension of \(F\), when \(G=\mathrm{SL}_2)\). Such groups were introduced with the object of understanding the inner structure of ``\(L\)-packets'' of irreducible representations of the group \(G(F)\) of \(F\)-valued points on \(G\) (if \(\overline F\) is the algebraic closure of F, conjugacy realized via elements of \(G(\overline F)\) intervenes during the application of harmonic analysis to number- theoretic problems like the study of \(L\)-functions; for irreducible representations of \(G(F)\), this conjugacy leads to a classification coarser than the usual equivalence and the coarse classes are called ``\(L\)-packets'' of irreducible representations). To a conjugacy class in the group \(H(F)\) of \(F\)-rational points on an endoscopic group \(H\) associated to \(G\), there correspond many conjugacy classes in \(G(F)\). ``Orbital integrals'' defined for pairs \(f\), \(f^H\) of functions respectively on \(G(F)\), \(H(F)\) are related by ``transfer factors''. The object of this important paper is, inter alia, to define explicitly ``transfer factors'', in the context of the correspondence of an \(f^H\) on \(H(F)\) to each \(f\) on \(G(F)\), with good functional behaviour: the definition, even for \(F=\mathbb R\), represents an improvement upon an earlier construction.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 4049901
zbMATH Keywords: Endoscopic groups
zbMATH Keywords: harmonic analysis over reductive groups
zbMATH Keywords: local fields
zbMATH Keywords: L-packets
zbMATH Keywords: irreducible representations
zbMATH Keywords: L-functions
zbMATH Keywords: orbital integrals
zbMATH Keywords: transfer factors
MaRDI profile type: Publication
cites work: Rational conjugacy classes in reductive groups
cites work: Stable trace formula: Cuspidal tempered terms
cites work: L-Indistinguishability For SL (2)
cites work: Stable Conjugacy: Definitions and Lemmas
cites work: Orbital Integrals on Forms of SL(3), I
cites work: Q3322323
cites work: L-indistinguishability for real groups
cites work: Linear algebraic groups
cites work: Conjugacy classes in algebraic groups. Notes by Vinay V. Deodhar
OpenAlex ID: W2011178942
description: scientific article; zbMATH DE number 4049901