Exponents in Archimedean Arthur packets
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 \({\mathfrak n}\) are less than \(1\). Here \(D_{\mathfrak n}^\theta\) denotes the twisted Weyl denominator given by \(D_{\mathfrak n}^\theta(t)=\det((1-\theta t)_{\mid_{\mathfrak n}})\). This result is a twisted version of a proof by \textit{H. Hecht} and \textit{W. Schmid} of the Osborne conjecture [Acta Math. 151, 49--151 (1983; Zbl 0523.22013)]. As a byproduct, the authors, comparing the exponents (i.e., characters of the split components \(A\)) of representations in Arthur packets, obtain an Archimedean analog of a result due to \textit{C. Moeglin} [J. Lie Theory 19, No. 4, 797--840 (2009; Zbl 1189.22010)] which states that a local Arthur packet contains the corresponding local \(L\)-packet and representations which are more tempered. The latter statement follows a general principle formulated by \textit{L. Clozel} in [Clay Mathematics Proceedings 13, 99--106 (2011; Zbl 1243.22016)].
scientific article; zbMATH DE number 6177078
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