Representations of exponential solvable Lie groups induced from maximal subgroups
Let \(G\) be an exponential solvable Lie group. The author studies in detail the mechanism of inducing process up to \(G\) from a unitary representation of a maximal closed connected proper subgroup \(H\). Namely, let \(\nu\) be an irreducible unitary representation of \(H\) and let \(\pi = \text{Ind}^ G_ H \nu\) be the unitary representation of \(G\) induced from \(\nu\). The author analyses this situation in the framework of the orbit method. When \(\dim G/H = 1\), this analysis was previously given by himself [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 127-164 (1990; Zbl 0724.22006)]. This paper treats the case where \(\dim G/H = 2\) and derives seven mutually exclusive possibilities for the structure of the inducing up. Let \(\mathfrak g\) (resp. \(\mathfrak h\)) be the Lie algebra of \(G\) (resp. \(H\)) and let \(\Omega_ H(\nu) \subset {\mathfrak h}^*\) be the coadjoint \(H\)- orbit corresponding to \(\nu\). There is the unique 1-codimensional subalgebra \(\mathfrak k\) of \(\mathfrak h\) which is a 3-codimensional ideal in \(\mathfrak g\). One takes \(\psi\) in \(\Omega_ H(\nu)\) and considers the irreducible unitary representation \(\gamma\) of \(K = \text{exp }{\mathfrak k}\), which corresponds to \(\theta = \psi| {\mathfrak k}\). Then the key determining factor is the stability subgroup \(G(\gamma)\) of \(\gamma\) in \(G\).
scientific article; zbMATH DE number 474972