Orthonormal bases in the orbit of square-integrable representations of nilpotent Lie groups

Abstract

Let G be a connected, simply connected nilpotent group and π be an irreducible unitary representation of G that is square-integrable modulo its center Z(G) on L2(Rd). We prove that under reasonably weak conditions on G and π there exist a discrete subset Γ of G/Z(G) and some (relatively) compact set F⊆Rd such that{|F|−1/2π(γ)1F|γ∈Γ} forms an orthonormal basis of L2(Rd). 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 Γ can be chosen to be a uniform subgroup of G/Z.