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

TL;DR

This paper investigates primary Hamiltonian N-spaces, revealing they are flat bundles over coadjoint orbits rather than always splitting as homogeneous times trivial, thus advancing the Mackey theory for Hamiltonian G-spaces.

Contribution

It provides explicit examples of primary spaces that do not split and characterizes them as flat bundles, filling a gap in the Mackey theory for Hamiltonian G-spaces.

Findings

Primary spaces are flat bundles over coadjoint orbits.

Counterexamples show primary spaces do not always split as homogeneous times trivial.

The results extend Mackey theory to Hamiltonian G-spaces with N normal.

Abstract

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.