Primary Spaces, Mackey’s Obstruction etc.
March 27, 2012, update November 8, 2013
Joint work  with Francois Ziegler.
We call a hamiltonian N-space primary if its moment map is onto a single coadjoint orbit. The question has long been open whether such spaces always split as (homogeneous) × (trivial), as an analogy with representation theory might suggest. For instance, Souriau’s 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

To be published in “Journal of Symplectic Geometry”. Volume 13, Number 1, 55–80, 2015.
that primary spaces are always flat bundles over the coadjoint orbit. This provides the missing piece for a full “Mackey theory” of hamiltonian G-spaces, where G is an overgroup in which N is normal.