Triangular Subgroups of $Sp(d,{\mathbb R})$ and Reproducing Formulae

Abstract

We consider the (extended) metaplectic representation of the semidirect product $\mathcal{G}={\mathbb H}^d\rtimes Sp(d,{\mathbb R})$ between the Heisenberg group and the symplectic group. Subgroups $H=\Sigma \rtimes D$, with $\Sigma$ being a $d\times d$ symmetric matrix and $D$ a closed subgroup of $GL(d,{\mathbb R})$, are our main concern. We shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in [5]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schr\"odinger representation of ${\mathbb R}^{2d}$ or the wavelet representation of ${\mathbb R}^d\rtimes D$, with $D$ closed subgroup of $GL(d,{\mathbb R})$. Finally, we shall provide new examples of reproducing groups of the type $H=\Sigma\rtimes D$, in dimension $d=2$.