Symplectic and Contact Imprimitivity

A famous theorem of Mackey characterizes those unitary G-modules V that are induced from a closed subgroup H ⊂ G by the presence of a system of imprimitivity based on G/H: that is, a G-invariant, commutative C∗ -subalgebra of End(V ) whose spectrum is, as a G-space, homogeneous and isomorphic to G/H. In this work, we similarly characterize those Hamiltonian G-spaces X that are induced from H (in the sense of Kazhdan-Kostant-Sternberg, 1978) by the presence of a (symplectic) system of imprimitivity based on G/H: that is, a G-invariant, Poisson commutative subalgebra f of C ∞(X), consisting of functions whose Hamiltonian flow is complete, and such that the image of the moment map X → f ∗ is homogeneous and isomorphic to G/H. Likewise, we characterize induced Kostant-Souriau bundles over Hamiltonian G-spaces by the presence of a (contact) system of imprimitivity. This result is a key ingredient in the Mackey ‘normal subgroup analysis’ of Hamiltonian and Kostant-Souriau G-spaces.