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Unitary dual of GL(n) at archimedean places and global Jacquet–Langlands correspondence

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  08 June 2010

A. I. Badulescu  and
D. Renard
A. I. Badulescu
Affiliation:
Université Montpellier 2, Case Courrier 051, Place Eugène Bataillon, 34095 Montpellier cedex, France
D. Renard
Affiliation:
Centre de mathématiques Laurent Schwartz, École Polytechnique, 91 128 Palaiseau cedex, France (email: renard@math.polytechnique.fr)
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Abstract

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In a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

Keywords

Jacquet–Langlands correspondenceautomorphic representationsunitary dualLanglands functoriality

MSC classification

Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E50: Representations of Lie and linear algebraic groups over local fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 5 , September 2010 , pp. 1115 - 1164
Copyright
Copyright © Foundation Compositio Mathematica 2010

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