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

We consider the Heisenberg motion groups HM = H-n (sic) K, where H-n is the Heisenberg group and K is a compact subgroup of U(n) such that (K, H-n) is a Gelfand pair. We study the Segal-Bargmann transform on HM and characterise the Poisson integrals associated to the Laplacian for HM using Gutzmer's formula. We also prove a Paley-Wiener type theorem involving complexified representations using explicit realisations of some unitary irreducible representations of HM.