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

Suparna Sen 1
  • 1 Department of Mathematics, Indian Institute of Science, Bangalore 560012, India. Current address: Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India

Abstract

We consider the Heisenberg motion groups ℍ𝕄 = ℍnK, where ℍn is the Heisenberg group and K is a compact subgroup of U(n) such that (K,ℍn) 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.

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