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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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