$C^{\infty}$-vectors of irreducible representations of exponential solvable Lie groups

Abstract

Let $G$ be an exponential solvable Lie group, and $\pi$ be an irreducible unitary representation of $G$. Then by induction from a unitary character of a connected subgroup, $\pi$ is realized in an $L^2$-space of functions on a homogeneous space. We are concerned with $C^\infty$vectors of $\pi$ from a viewpoint of rapidly decreasing properties. We show that the subspace $\mathscr{PE}$ consisting of vectors with a certain property of rapidly decreasing at infinity can be embedded as the space of the $C^\infty$vectors in an extension of $\pi$ to an exponential group including $G$. Using the space $\mathscr{PE}$, we also give a description of the space $\mathscr{APE}$ related to Fourier transforms of $L^1$-functions on $G$. We next obtain an explicit description of $C^\infty$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 $C^\infty$vectors of an irreducible representation of a certain exponential solvable Lie group acting on $L^2(G)$.