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Inversion formula of orbital integrals on reductive Lie groups. (Formule d’inversion des intégrales orbitales sur les groupes de Lie réductifs.) (French) Zbl 0842.22013
Let \(G\) be a reductive Lie group in Harish-Chandra’s class, with Lie algebra \(\mathfrak g\). The orbital integral of a compactly supported smooth function \(f\) on \(G\) is defined as \[ Jf(x) = |\text{det}(1 - \text{Ad } x^{-1})_{{\mathfrak g}/{\mathfrak h}}|^{1\over 2} \int_{G/H} f(gxg^{-1}) d\dot g, \] where \(x\) is a regular semisimple element of \(G\), \(H\) is the Cartan subgroup of \(G\) containing \(x\), and \(\mathfrak h\) is the Lie algebra of \(H\) (assuming here for simplicity that all Cartan subgroups are abelian). The paper is entirely devoted to the proof of an inversion formula of the following form: \(Jf(x) = \Sigma \int_{\widehat{H}} \psi_{\widehat{h}} \theta_{\widehat{h}} (f) d\widehat{h},\) where \(\Sigma\) runs over the (finite) set of conjugacy classes of Cartan subgroups \(H\) and \(\widehat{H}\) is the character group of \(H\). The distribution \(\theta_{\widehat{h}}\) is the invariant eigendistribution constructed by Harish-Chandra, generically the trace of some tempered irreducible representation of \(G\). One of the main tasks of the present paper is to give an inductive construction of the orbital functions \(\psi_{\widehat{h}}\), by means of \(\theta\)-stable Levi subgroups.
Reviewer: F.Rouvière (Nice)

22E46 Semisimple Lie groups and their representations
43A80 Analysis on other specific Lie groups
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