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

Let a Lie group G act on a symplectic manifold X in Hamiltonian fashion, i.e., the action preserves the 2 form of X and we have an equivariant momentum map X > Lie(G)*. If N is a normal subgroup of G, "Symplectic Mackey Theory" reduces the study of such actions to that of i) coadjoint orbits of N and ii) symplectic actions of subgroups of G/N.

Just as its cousin in representation theory, this analysis has 3 steps of which the last concerns the "primary" situation where X > Lie(G)* > Lie(N)* is onto a single coadjoint orbit U of N. So far this step had only been elucidated in the case where X splits as a product U x Z.

In this talk I will describe joint work with P. Iglesias showing that (1) X does not always split in this way; (2) X is always a flat bundle over U. This enables us to complete the Mackey analysis in the general case.