Unitary equivalence of lowest dimensional reproducing formulae of type $\mathcal{E}_2 \subset Sp(2,{\mathbb R})$

Abstract

All two-dimensional reproducing formulae, i.e. of $L^2({\mathbb R}^2)$, resulting out of restrictions of the projective metaplectic representation to connected Lie subgroups of $Sp(2,{\mathbb R})$ and of type $\mathcal{E}_2$, were listed and classified up to conjugation within $Sp(2,{\mathbb R})$ in [2], [3]. A full classification, up to conjugation within ${\mathbb R}^2 \rtimes Sp(1,{\mathbb R})$, of one-dimensional reproducing formulae, i.e. of $L^2({\mathbb R})$, resulting out of restrictions of the extended projective metaplectic representation to connected Lie subgroups of ${\mathbb R}^2 \rtimes Sp(1,{\mathbb R})$ was obtained in [13], [14]. In dimension one, there are no reproducing formulae with one-dimensional parametrizations, yet in dimension two, there are reproducing formulae with two-dimensional parametrizations. Two-dimensional reproducing subgroups of $Sp(2,{\mathbb R})$ of type $\mathcal{E}_2$ are a novelty. They exhibit completely new phase space phenomena. We show, that they are all unitarily equivalent via natural choices of coordinate systems, and we derive the consequences of this equivalence.