Totally Disconnected and Locally Compact Heisenberg-Weyl Groups

Harmonic analysis on ℤ(pℓ) and the corresponding representation of the Heisenberg-Weyl group HW[ℤ(pℓ),ℤ(pℓ),ℤ(pℓ)], is studied. It is shown that the HW[ℤ(pℓ),ℤ(pℓ),ℤ(pℓ)] with a homomorphism between them, form an inverse system which has as inverse limit the profinite representation of the Heisenberg-Weyl group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathfrak {HW}[{\mathbb{Z}}_{p},{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}]$\end{document}. Harmonic analysis on ℤp is also studied. The corresponding representation of the Heisenberg-Weyl group HW[(ℚp/ℤp),ℤp,(ℚp/ℤp)] is a totally disconnected and locally compact topological group.