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Intégrales invariantes et formules de caractères pour un groupe de Lie connexe à radical co-compact. (Invariant integrals and character formulas for a connected Lie group with cocompact radical). (French) Zbl 0786.22013
Let $$G$$ be a simply connected unimodular Lie group, with cocompact radical. Let $$\ell$$ be a linear form on the Lie algebra $${\mathfrak g}$$ of $$G$$; assume $$\ell$$ is admissible, $$\ell$$ has a solvable polarization satisfying Pukanszky’s condition, and the stabilizer algebra $${\mathfrak g}(\ell)$$ is reductive in $${\mathfrak g}$$. From $$\ell$$, one can build an equivalence class of irreducible unitary representations $$T$$ of $$G$$.
The main result of the paper is a character formula for $$T$$: $\text{tr } T(\varphi)= \int_{G(\ell)/Z} d\dot x\int_ Z F_{\ell\varphi}(xz) X_ \ell(xz) dz,$ (convergent integrals), where $$G(\ell)$$ is the stabilizer group of $$\ell$$, $$Z$$ is the center of $$G$$ and $$\varphi$$ is a smooth function on $$G$$ with arbitrary compact support. The function $$F_{\ell\varphi}$$ constructed here is the invariant integral of $$\varphi$$ with respect to $$\ell$$. It is smooth on an open dense subset of $$G(\ell)$$, compactly supported modulo $$Z$$, but needs not be summable on $$G(\ell)$$. The function $$X_ \ell$$ is the unitary character of $$G(\ell)$$ defined by $$i\ell$$.
This formula extends several previously known results.
Reviewer: F.Rouvière (Nice)
MSC:
 22E30 Analysis on real and complex Lie groups 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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References:
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