Segal-Bargmann transform and Paley-Wiener theorems on Heisenberg motion groups

We study the Segal-Bargmann transform on the Heisenberg motion groups ${\mathbb{H}}^n⋉K,$ where ${\mathbb{H}}^n$ is the Heisenberg group and $K$ is a compact subgroup of $U(n)$ such that $(K,{\mathbb{H}}^n)$ is a Gelfand pair. The Poisson integrals associated to the Laplacian for the Heisenberg motion group are also characterized using Gutzmer’s formulae. Explicitly realizing certain unitary irreducible representations of ${\mathbb{H}}^n⋉K,$ we prove the Plancherel theorem. A Paley-Wiener type theorem is proved using complexified representations.