The Poisson- Plancherel formula for a real semialgebraic group. I: Fourier transform of orbital integrals
This paper is the first part of the authors' work to derive the Poisson- Plancherel formula for semialgebraic groups \(G\), a formula conjectured by \textit{M. Vergne} [A Plancherel formula without group representations, O. A. G. R. Conference INCREST, Bucarest, Roumania (1980)]. Let \(\mathfrak g\) be the Lie algebra of \(G\). Basing on works of \textit{M. Duflo} [see, for instance, Acta Math. 149, 153-213 (1982; Zbl 0529.22011)], they associate to any linear form on \(\mathfrak g\) a subgroup, called canonical acceptable subgroup, of \(G\) and study the conjugacy classes of the reductive factors of these subgroups. They then define orbital integrals on \(\mathfrak g\), generalizing those introduced by \textit{Harish-Chandra} [Ann. Math. 80, 551-593 (1964; Zbl 0152.134)] and \textit{M. Duflo} [Représentations unitaires des groupes de Lie et méthode des orbites. Actualités mathématiques, Actes 6ième Congr. Group. Math. Exp. Latine, Luxembourg 1981, 125-138 (1982; Zbl 0492.22008)]. The main result of this paper is an expression of the Fourier transforms of these orbital integrals. We consider a connected central subgroup \(Z\) of \(G\) contained in the unipotent radical of \(G\). Let \(\mathfrak z\) be the Lie algebra of \(Z\) and let \(\chi\) be a unitary character of \(Z\). To any elliptic element \(T\) of \(\mathfrak g\), they associate a \(\chi\)-tempered distribution \(M_{G, T, \chi}\), called \(\chi\)-orbital integral, whose Fourier transform is denoted by \(\Theta_{G, T, \chi}\). If we put \({\mathfrak g}_\chi^* = \{\xi \in {\mathfrak g}^*;\) \(\xi \mid {\mathfrak z} = (d \chi /\sqrt {-1}) |{\mathfrak z}\}\), \(\Theta_{G, T, \chi}\) is a tempered generalized function on \({\mathfrak g}^*_\chi\). When \(Z\) is trivial, we write simply \(M_{G, T}\) and \(\Theta_{G, T}\). In this case \(M_{G, T}\) is a tempered distribution on \(\mathfrak g\), supported by the \(G\)-orbit of \(T\). When \(G\) is reductive, \(M_{G, T}\) is, up to a scalar multiplication, the Harish-Chandra's orbital integral associated to \(T\), and \(\Theta_{G, T}\) is an essentially bounded Lebesgue measurable function on \({\mathfrak g}^*\), which is analytic on the Zariski open set \({\mathfrak g}^*_r\) of strongly regular elements of \({\mathfrak g}^*\). The main result expresses \(\Theta_{G, T, \chi}\) in terms of the Fourier transforms of Harish-Chandra's orbital integrals of some reductive factors of the generic canonical acceptable subgroups of \(G\). In particular, \(\Theta_{G, T, \chi}\) is an essentially bounded Lebesgue measurable function on \({\mathfrak g}^*_\chi\), which is analytic on a non-empty Zariski open subset of the trace \({\mathfrak g}^*_{\chi, r}\), of \(({\mathfrak g}/\text{ker} (d\chi/ \sqrt {-1}))^*\) on \({\mathfrak g}^*_\chi\).
scientific article; zbMATH DE number 515262