Explicit orbital parameters and the Plancherel measure for exponential Lie groups
Let \(G\) be an exponential solvable Lie group with Lie algebra \({\mathfrak g}\) and \({\mathfrak g}^*\) the dual vector space of \({\mathfrak g}\). Through the celebrated orbit method [cf. \textit{P. Bernat}, \textit{N. Conze}, \textit{M. Duflo}, \textit{M. Levy-Nahas}, \textit{M. Rais}, \textit{P. Renouard} and \textit{M. Vergne}, Représentations des groupes de Lie résolubles (Dunod, Paris) (1972; Zbl 0248.22012)], the unitary dual \(\widehat G\), namely the set of equivalence classes of irreducible unitary representations of \(G\), is realized by the space \({\mathfrak g}^*/G\) of coadjoint orbits of \(G\). The classical Plancherel formula for nonunimodular locally compact topological groups [cf. \textit{M. Duflo} and \textit{C. C. Moore}, J. Funct. Anal. 21, 209--243 (1976; Zbl 0317.43013)] is combined with the orbit method to yield its orbital picture [\textit{M. Duflo} and \textit{M. Raïs}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 9, 107--144 (1976; Zbl 0324.43011)]. Based upon his previous work on the structure of the orbit space \({\mathfrak g}^*/G\) [Trans. Am. Math. Soc. 332, No. 1, 241--269 (1992; Zbl 0763.22008)], the author elaborates a detailed analysis of orbital parameters and decomposes the Lebesgue measure on \({\mathfrak g}^*\) into semi-invariant orbital measures and a natural measure on an explicit cross-section for generic coadjoint orbits. This decomposition leads him to a precise and explicit description of the Plancherel measure.
scientific article; zbMATH DE number 5011507