Holomorphically induced representations of exponential solvable semi-direct product groups \(\mathbb{R} \ltimes \mathbb{R}^{n}\) (Q2340118)

Let \(G\) be an exponential solvable Lie group with Lie algebra \(\mathfrak g\) defined by a semi-direct product of \(\mathbb R\) and \(\mathbb R^n\). The author studies a holomorphically induced representation \(\rho\) of \(G\) from a real linear form \(f\) of \(\mathfrak g\) and a 1-dimensional complex subalgebra \(\mathfrak h\) of \(\mathfrak g_{\mathbb C}\) such that the space \(\mathfrak h + \overline{\mathfrak h}\) generates \(\mathfrak g_{\mathbb C}\). Under some assumptions on the structure of the Lie algebra, the author obtains a sufficient condition for the non-triviality of \(\rho\) and decompose \(\rho\) into a multiplicity-free direct integral of irreducible representations.