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

2012/03/31 by Patrick Iglesias-Zemmour, François Ziegler, Francois Ziegler · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Barycentric coordinate system #Geometry #Hamiltonian (control theory) #Homotopy and Cohomology in Algebraic Topology #Mathematics #Orbit (dynamics) #Pure mathematics #Representation theory #math.RT #math.SG #msc:22D10 #msc:53D20 #msc:57T10

paper · pdf · doi:10.4310/jsg.2015.v13.n1.a3

published in Journal of Symplectic Geometry 13(1), 51-76 · 23 pages, 1 figure. Final preprint version, to appear in Journal of Symplectic Geometry

arxiv created 2013/11/07 · openalex publication_date 2015/01/01 · arxiv updated 2015/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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 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.

Citations

• Geometric quantization and multiplicities of group representations
• Polarization and unitary representations of solvable Lie groups
• Reduction of symplectic manifolds with symmetry