Translation complete subgroups of affine Weyl-Heisenberg groups and their generalized wavelet systems

The $n$-dimensional affine Weyl-Heisenberg group is a Lie group typically parameterized as $G_{aWH} = \mathbb{T} \times \mathbb{R}^n \times \widehat{\mathbb{R}^n} \times \mathrm{GL}(n, \mathbb{R})$, generated by all translation, dilation, and modulation operators acting on $L^2(G)$. It was introduced by Torr\'esani and his coauthors as a common framework to discuss both wavelet and time-frequency analysis, as well as possible intermediate constructions. In this paper, we focus on a particular class of subgroups of $G_{aWH}$, namely those of the form $G = \mathbb{T} \times \mathbb{R}^n \times V \times H$, where $V$ is a subspace of $\mathbb{R}^n$ and $H$ is a closed subgroup of $\mathrm{GL}(n, \mathbb{R})$. The main goal is to identify pairs $(V, H)$ that ensure the existence of an associated inversion formula, through the notion of square-integrable representations. We derive an admissibility criterion that is largely analogous to the well-known Calder\'on condition for the fully affine case, corresponding to $V = \{ 0 \}$. %The criteria for such a characterization can be formulated and proved in a way that is in many respects analogous to the affine case. We then identify $G_{aWH}$ as a subgroup of the semidirect product of the $n$-dimensional Heisenberg group and the symplectic group $Sp(n,\mathbb{R})$, which acts via the extended metaplectic representation, and compare our admissibility conditions to existing criteria based on Wigner functions. Finally, we present a list of novel examples in dimensions two and three which illustrate the potential of our approach, and present some foundational results regarding the systematic construction, classification, and conjugacy of these groups.