Admissible vectors for mock metaplectic representations

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 C1 intertwining map between the actions of H on the dual group 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=Rn and M=Rd. 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.