Journal of Symplectic Geometry

Volume 13 (2015)

Number 1

Primary spaces, Mackey’s obstruction, and the generalized barycentric decomposition

Pages: 51 – 76



Patrick Iglesias-Zemmour (Laboratoire d’Analyse, Topologie et Probabilités/CNRS, Aix-Marseille Université, Marseille, France)

François Ziegler (Department of Mathematical Sciences, Georgia Southern University, Statesboro, Ga., U.S.A.)


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) $\times$ (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 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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