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Zbl 0748.22007
Bouaziz, Abderrazak
Inversion formula for twisted orbital integrals. (Formule d'inversion d'intégrales orbitales tordues.) (French)
[J] Compos. Math. 81, No.3, 261-290 (1992). ISSN 0010-437X; ISSN 1570-5846/e

The aim of the paper is to prove a formula of the following type: $$\int\sb{G/G\sp \sigma}\varphi(g\sigma g\sp{-1})d\dot g=\int({1\over 2}\sum\sb \tau q(f,\tau) \text{tr }\pi\sb{f,\tau}(\varphi))dm(G.f).$$ Here $G$ is a semisimple algebraic group defined over $\bbfR$, identified with the group of its complex points, $G\sp \sigma$ is a real form of $G$, and $\varphi$ is a test function; the left-hand side is an orbital integral of $\varphi$. The right-hand side provides a Fourier inversion formula for this orbital integral, i.e. a decomposition by means of characters $\text{tr} \pi\sb{f,\tau}$ of irreducible unitary representations of $G$. Here the integral, with respect to the measure $dm$, runs over the set of all $\sigma$-stable admissible coadjoint orbits $(G,f)$. The representation $\pi\sb{f,\tau}$ comes out from Duflo's construction, where $\tau$ is a parameter running over a two- element set. The coefficients $q(f,\tau)$ are specific $G$-invariant functions of $f$ introduced in the text, but not explicitly computed. The proof of the formula is inspired by Duflo-Vergne's method for the Plancherel formula.
[F.Rouvière (Nice)]

MSC 2000:: *22E46 Semi-simple Lie groups and their representations

Keywords: semisimple algebraic group; orbital integral; Fourier inversion; irreducible unitary representations

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