Exponents in Archimedean Arthur packets
type: Item
instance of: scholarly article
title: Exponents in Archimedean Arthur packets
zbMATH Open document ID: 1276.22002
author: Nicolas Bergeron
author: Laurent Clozel
publication date: 2013-06-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: Let \(G\) be a connected real reductive group, \(B\) a Borel subgroup and \(T\) a maximal torus in \(B\). Let \(\theta\) be an automorphism of \(G\) of finite order preserving the splitting induced by the pair \((B,T)\). Consider the twisted space \(L=G\rtimes \theta\) and write \(T^\theta\) for the elements in \(T\) fixed by \(\theta\) and \(T^1\) for the connected component of \(T^\theta\). Assume \(T\) is maximally split over \({\mathbb R}\) and is contained in a minimal \(\theta\)-stable parabolic subgroup \(P\) of \(G\) with Langlands decomposition \(P=MAN\), where \(A\) is the split part of \(T\) and \(N\) is the unipotent radical. Denote by \({\mathfrak n}\) the complexified Lie algebra of \(N\). Let \((V_M,\tau)\) be an irreducible \(\theta\)-stable representation of \(M\) (i.e., \(\tau\) and \(\tau\circ\theta\) are equivalent) and \(\mu\) a \(\theta\)-stable character of \(A\) (i.e., \(\mu\circ\theta=\mu\)). Then an intertwining operator \(a_\theta\) between \(\tau\) and \(\tau\circ\theta\) defines an operator \(A_\theta\) on the unitarily induced representation \(\operatorname{Ind}_P^G(V_M\otimes{\mathbb C}_\mu)\). Let \(V\) be the Harish-Chandra module of an admissible irreducible \(\theta\)-stable representation \(\pi\) of \(G\) and let \(\Theta_{\pi,\theta}\) be the twisted character of \(\pi\). It is known (see [\textit{W. Casselman} and \textit{M. S. Osborne}, Compos. Math. 31, 219--227 (1975; Zbl 0343.17006)]) that the homology group \(H_q({\mathfrak n},V)\) is finite-dimensional for all \(q\geq 0\). Since \(T\) and \(\theta\) both act on \(H_q({\mathfrak n},V)\) via the operator \(A_\theta\), one can define for all \(t\in T\): \(\Theta_q^\theta(t,V)=\text{trace}(t\theta\mid H_q({\mathfrak n},V))\). In the paper under review, the authors prove that \[ \Theta_{\pi,\theta}(t)=\frac{\sum_q(-1)^q\Theta_q^\theta(t,V)}{D_{\mathfrak n}^\theta(t)} \] for all elements \(t\in T^1\) such that the absolute values of the complex eigenvalues of \(t\theta\) in \({\…
reviewed by: Salah Mehdi
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 6177078
zbMATH Keywords: unitary representations
zbMATH Keywords: exponents
zbMATH Keywords: Osborne conjecture
zbMATH Keywords: Arthur packet
MaRDI profile type: Publication
OpenAlex ID: W2318966419
cites work: Q3409176
cites work: A proof of Kirillov's conjecture
cites work: Q3328755
cites work: Sur les caractères des groupes de Lie réductifs non connexes. (On the characters of non-connected reductive Lie groups)
cites work: Q4112093
cites work: Corps de nombres peu ramifiés et formes automorphes autoduales
cites work: Q3019003
cites work: Q4075116
cites work: A property of the characters of irreducible representations of real semisimple Lie groups
cites work: Invariant Eigendistributions on a Semisimple Lie Group
cites work: Characters, asymptotic and n-homology of Harish-Chandra modules
cites work: The characters of some induced representations of semi-simple Lie groups
cites work: Automorphic forms on GL (2)
cites work: Representation theory of semisimple groups. An overview based on examples. With a new preface by the author.
cites work: Q4840147
cites work: Q4243095
cites work: STABLE TWISTED TRACE FORMULA: ELLIPTIC TERMS
cites work: On the definition of transfer factors
cites work: Comparaison des param\`etres de Langlands et des exposants \`a l'int\'erieur d'un paquet d'Arthur
cites work: The unitary dual of \(\mathrm{GL}(n)\) over an archimedean field
cites work: The twisted group \(\mathrm {GL}_N\) over a \(p\)-adic field. I
cites work: Transfer factors for the classical groups: a formulary
DOI: 10.5802/AIF.2757
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 6177078