Irreducible bounded representations of exponential solvable Lie groups (Q2751522)

This is a survey article on the theory of irreducible bounded representations of exponential solvable Lie groups. The author first explains the algebraic structure of an exponential solvable Lie group \(G\) and the orbit method created by \textit{A. A. Kirillov} [Uspekhi Mat. Nauk 17, 53-104 (1962; Zbl 0106.25001)], refined by \textit{Bernat, Pukanszky} and \textit{Vergne} [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 (Paris 1972; Zbl 0248.22012)] to determine the unitary dual \(\widehat G\) of \(G\). He then sketches a brief history of the theory of simple \(L^1(G)\)-modules developed by Dixmier, Leptin, Poguntke, Jenkins and himself. This development finally led \textit{D. Poguntke} to give a complete description of these modules [Duke math. J. 50, 1077-1106 (1983; Zbl 0555.43005)]. Based on Poguntke's results, the author analyses in detail algebraically and topologically irreducible \(L^1(G)\)-modules in order to obtain the space of the equivalence classes of simple \(L^1(G)\)-modules. The paper contains no proofs and main references would be the following two preprints: \textit{J. Ludwig} et \textit{C. Molitor-Braun}, Représentations irréductibles bornées des groupes de Lie exponentiels; \textit{J. Ludwig}, \textit{S. Mint Elhacen} and \textit{C. Molitor-Braun}, Characterization of the simple \(L^1(G)\)-modules for exponential Lie groups.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00042].