A mock metaplectic representation

TL;DR

This paper characterizes admissible vectors for a new class of unitary representations called mock metaplectic, which generalize certain symplectic group representations, with applications in signal analysis like shearlets.

Contribution

It introduces a new representation framework called mock metaplectic, providing necessary and sufficient conditions for admissibility, and decomposes it into irreducible components.

Findings

Provides criteria for admissible vectors in mock metaplectic representations

Decomposes the representation into irreducible components

Connects the theory to applications in signal analysis such as shearlets

Abstract

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 $\hat A$ and on $M$. The representation $U$ generalizes the restriction of the metaplectic representation to triangular subgroups of $Sp(d,\R)$, whence the name "mock metaplectic". For simplicity, we content ourselves with the case where $A=\R^n$ and $M=\R^d$. 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.