Smooth decomposition of finite multiplicity monomial representations for a class of completely solvable homogeneous spaces (Q1919875)

Let \(G=\exp{\mathfrak g}\) be a connected, simply connected completely solvable Lie group with Lie algebra \(\mathfrak g\). Suppose that \(G\) is the semidirect product \(G=NH\), where \(N\) is nilpotent and normal in \(G\), and \(H\) is abelian and acts semisimply on \(N\) with real eigenvalues. Let \(\chi\) be such a unitary character of \(H\) that the induced representation \(\tau=\text{ind}^G_H \chi\) is of finite multiplicities. In this elaborated paper, the author first studies the algebraic structure of \(\mathfrak g\). In particular, \(N\) turns out two step. Then he analyzes the monomial representation \(\tau\) in orbit picture to get Penney's distributions and \textit{R. Penney's} Plancherel formula [J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016)] (cf. also a paper of the reviewer [Pac. J. Math. 127, 329-352 (1987; Zbl 0588.22008)] and of \textit{R. L. Lipsman} [Pac. J. Math. 140, 117-147 (1989; Zbl 0645.43010)]). In various situations one finds a formal candidate for Penney's distribution, but it is often difficult to prove that this candidate gives a real generalized vector of \(\tau\). The author fully uses his detailed algebraic description of \(\mathfrak g\) to overcome this essential problem. It is also shown that \(\tau\) has uniform multiplicity \(2^u\) with \(2u=\dim(N/\text{cent}(N))\). Finally two illustrating examples are presented.