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 complexified representations.