Orthonormal Bases in the Orbit of Square-Integrable Representations of Nilpotent Lie Groups

Abstract

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2({\mathbf{R}}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$ there exist a discrete subset $\Gamma$ of $G/Z(G)$ and some (relatively) compact set $F\subseteq {\mathbf{R}}^d$ such that $$\{|F{|}^{-1/2}\pi (\gamma )1_F\mid \gamma \in \Gamma \}$$ forms an orthonormal basis of $L^2({\mathbf{R}}^d)$. This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis. The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set $\Gamma$ can be chosen to be a uniform subgroup of $G/Z$.