A mock metaplectic representation

We obtain necessary and sufficient conditions for the admissible vectors of a new unitary non irreducible representation $U$. The group $G$ is an arbitrary semidirect product whose normal factor $A$ is abelian and whose homogeneous factor $H$ is a locally compact second countable group acting on a Riemannian manifold $M$. The key ingredient in the construction of $U$ is a $C^1$ intertwining map between the actions of $H$ on the dual group $\widehat{A}$ and on $M$. The representation $U$ generalizes the restriction of the metaplectic representation to triangular subgroups of $Sp(d,\mathbb{R})$, whence the name “mock metaplectic”. For simplicity, we content ourselves with the case where $A={\mathbb{R}}^n$ and $M={\mathbb{R}}^d$. The main technical point is the decomposition of $U$ as direct integral of its irreducible components. This theory is motivated by some recent developments in signal analysis, notably shearlets. Many related examples are discussed.