Segal-Bargmann transform and Paley-Wiener theorems on Heisenberg motion groups (Q906163)

The author considers the Heisenbeg motion group \(\mathbb{H}\mathbb{M}=\mathbb{H}^n \rtimes K\), 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. He investigates the Segal-Bargmann transform on \(\mathbb{H}\mathbb{M}\) and characterizes the Poisson integrals associated to the Laplacian for \(\mathbb{H}\mathbb{M}\) using Gutzmer's formula. He also proves a Paley-Wiener type theorem involving complexified representations using explicit realisations of the irreducible unitary representations of \(\mathbb{H}\mathbb{M}\) which occur in the Plancherel identity.