How fat is a fat bundle?

Let P → M be a principal G-bundle with connection 1-form θ and curvature Θ. For a subset S of g^* the given connection is S-fat (Weinstein, [5]) if for every μ in S the form μ ° Θ is nondegenerate on each horizontal subspace in TP. Let K be a compact group and K/H be its coadjoint orbit. The orthogonal projection t → h defines a connection on the principal H-bundle K → K/H. We show that this connection is fat off certain walls of Weyl chambers in h^*. We then apply the result to the construction of symplectic fiber bundles over K/H. As an example, we show how higher-dimensional coadjoint orbits fiber symplectically over lower-dimensional orbits.