Totally Disconnected and Locally Compact Heisenberg-Weyl Groups

Abstract

Harmonic analysis on ℤ(p l ) and the corresponding representation of the Heisenberg-Weyl group HW[ℤ(p l ),ℤ(p l ),ℤ(p l )], is studied. It is shown that the HW[ℤ(p l ),ℤ(p l ),ℤ(p l )] with a homomorphism between them, form an inverse system which has as inverse limit the profinite representation of the Heisenberg-Weyl group \(\mathfrak {HW}[{\mathbb{Z}}_{p},{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}]\). Harmonic analysis on ℤ p is also studied. The corresponding representation of the Heisenberg-Weyl group HW[(ℚ p /ℤ p ),ℤ p ,(ℚ p /ℤ p )] is a totally disconnected and locally compact topological group.