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

Let C be an exponential solvable Lie group, and pi be an irreducible unitary representation of C. Then by induction from a unitary character of a connected subgroup, pi is realized in an L-2-space of functions oil a homogeneous space. We are concerned with C-vectors of pi from a viewpoint of rapidly decreasing properties. We show that the subspace f E consisting of vectors with a certain property of rapidly decreasing at infinity can be embedded as the space of the C-vectors in ail extension of T to an exponential group including G. Using the space f E, we also give a description of the space A f E related to Fourier transforms of L-1-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 L-2(G).