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

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

2012-03-26

arxiv.org/abs/1203.5723v1

Mathematics

Symplectic Geometry

21 pages, 1 figure

Scientific paper

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. This provides the missing piece for a full "Mackey theory" of hamiltonian G-spaces, where G is an overgroup in which N is normal.

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