# Zak transform for semidirect product of locally compact groups

Research paper by Ali Akbar Arefijamaal, Arash Ghaani Farashahi

Indexed on: 10 Apr '13Published on: 10 Apr '13Published in: Analysis and Mathematical Physics

#### Abstract

Let $$H$$ be a locally compact group and $$K$$ be an LCA group also let $$\tau :H\rightarrow Aut(K)$$ 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 analyze how these technique apply for the semidirect product group $$\mathrm SL (2,\mathbb{Z })\ltimes _\tau \mathbb{R }^2$$ and also the Weyl-Heisenberg groups.