Generalized Weyl–Heisenberg (GWH) groups

Abstract

Let $$H$$ be a locally compact group, $$K$$ be a locally compact Abelian (LCA) group, $$\theta :H\rightarrow Aut(K)$$ be a continuous homomorphism, and let $$G_\theta =H\ltimes _\theta K$$ be the semi-direct product of $$H$$ and $$K$$ with respect to the continuous homomorphism $$\theta $$ . In this article, we introduce the Generalized Weyl–Heisenberg (GWH) group $${\mathbb {H}}(G_\theta )$$ associate with the semi-direct product group $$G_\theta $$ . We will study basic properties of $${\mathbb {H}}(G_\theta )$$ from harmonic analysis aspects. Finally, we will illustrate applications of these methods in the case of some well-known semi-direct product groups.