Segal–Bargmann Transform and Paley–Wiener Theorems on Motion Groups

Abstract

We study the Segal–Bargmann transform on a motion group ℝ_n_ ⋉ K, where K is a compact subgroup of SO(n). A characterization of the Poisson integrals associated to the Laplacian on ℝ_n_ ⋉ K is given. We also establish a Paley–Wiener type theorem using complexified representations.