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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)

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