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

TL;DR

This paper extends the Segal-Bargmann transform and Paley-Wiener theorems to Heisenberg motion groups, characterizing Poisson integrals and establishing a Plancherel theorem through representation theory and Gutzmer's formula.

Contribution

It introduces a new analysis framework for the Heisenberg motion groups, including explicit representation realizations and a Paley-Wiener theorem.

Findings

Established the Plancherel theorem for Heisenberg motion groups.

Characterized Poisson integrals via Gutzmer's formula.

Proved a Paley-Wiener type theorem using complexified representations.

Abstract

We study the Segal-Bargmann transform on the Heisenberg motion groups $\mathbb{H}^n \ltimes K,$ 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 $\mathbb{H}^n \ltimes K,$ we prove the Plancherel theorem. A Paley-Wiener type theorem is proved using complexified representations.