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.