Zak Transform for Semidirect Product of Locally Compact Groups

Let $H$ be a locally compact group and $K$ be an LCA group also let $τ:H\to Aut(K)$ be a continuous homomorphism and $G_τ=H\ltimes_τK$ be the semidirect product of $H$ and $K$ with respect to $τ$. In this article we define the Zak transform $\mathcal{Z}_L$ on $L^2(G_τ)$ with respect to a $τ$-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 $\mathrm{SL}(2,\mathbb{Z})\ltimes_τ\mathbb{R}^2$ and also the Weyl-Heisenberg groups.

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