C^∞-vectors of irreducible representations of exponential solvable Lie groups

Abstract

Let G be an exponential solvable Lie group, and π be an irreducible unitary representation of G. Then by induction from a unitary character of a connected subgroup, π is realized in an L²-space of functions on a homogeneous space. We are concerned with C^∞vectors of π from a viewpoint of rapidly decreasing properties. We show that the subspace \\mathcal{S}\\mathcal{E} consisting of vectors with a certain property of rapidly decreasing at infinity can be embedded as the space of the C^∞vectors in an extension of π to an exponential group including G. Using the space \\mathcal{S}\\mathcal{E}, we also give a description of the space \\mathcal{A}\\mathcal{S}\\mathcal{E} related to Fourier transforms of L¹-functions on G. We next obtain an explicit description of C^∞vectors for a special case. Furthermore, we consider a space of functions on G with a similar rapidly decreasing property and show that it is the space of the C^∞vectors of an irreducible representation of a certain exponential solvable Lie group acting on L²(G).