Zak Transform for Semidirect Product of Locally Compact Groups

TL;DR

This paper introduces a Zak transform for semidirect product groups formed from a locally compact group and an LCA group, demonstrating its properties and applications to specific groups like SL(2,Z) and Weyl-Heisenberg groups.

Contribution

It defines a Zak transform on semidirect product groups with respect to a $ au$-invariant lattice and proves its Plancherel formula, extending harmonic analysis tools to these groups.

Findings

Zak transform satisfies the Plancherel formula.

Application to semidirect product groups like SL(2,Z) and Weyl-Heisenberg groups.

Provides a framework for harmonic analysis on these groups.

Abstract

Let $H$ be a locally compact group and $K$ be an LCA group also let $\tau:H\to Aut(K)$ be a continuous homomorphism and $G_\tau=H\ltimes_\tau K$ be the semidirect product of $H$ and $K$ with respect to $\tau$. In this article we define the Zak transform $\mathcal{Z}_L$ on $L^2(G_\tau)$ with respect to a $\tau$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\mathrm{SL}(2,\mathbb{Z})\ltimes_\tau\mathbb{R}^2$ and also the Weyl-Heisenberg groups.