Primary Spaces, Mackey's Obstruction, and the Generalized Barycentric Decomposition

Understand

We call a hamiltonian N-space \emph{primary} if its moment map is onto a single coadjoint orbit.

• The question has long been open whether such spaces always split as (homogeneous) x (trivial), as an analogy with representation theory might suggest.
• For instance, Souriau's \emph{barycentric decomposition theorem} asserts just this when N is a Heisenberg group.
• For general N, we give explicit examples which do not split, and show instead that primary spaces are always flat bundles over the coadjoint orbit.