Zak transform for semidirect product of locally compact groups (Q390258)

The authors extend the notion of ZAK transform to semidirect product groups of the form \(G_\tau=H \ltimes_\tau K\), where \(K\) is a locally compact abelian group, \(H\) is a locally compact group and \(\tau\) is a continuous homomorphism from \(H\) to the group of automorphisms, \(\mathrm {Aut}(K)\), of \(K\). As a result, they prove that the ZAK transform for this type of semidirect product satisfies the Plancherel Formula. Finally, they apply their methods to some semidirect products such as the Weyl-Heisenberg group and \(\mathrm{SL}(2, \mathbb Z) \ltimes_\tau \mathbb R^2\).