Analysis of some monomial representations of exponential solvable Lie groups

Let G be an exponential solvable Lie group and H an analytic subgroup of G. Let. chi be a unitary character of H and tau = Ind(H)(G)chi. We provide a necessary and a sufficient condition for the representation tau to split finitely on its isotopic components for adapted triplets. For the case in which H is maximal, we provide a concrete and smooth intertwining operator and its inverse for the disintegration of tau into irreducibles.