Symplectic submanifolds of cotangent bundles of Lie groups

Question

So, my question specifically pertains to $T^*SO(3)$ but I guess adjusted it could be asked about Lie groups in general. The canonical symplectic form on the cotangent bundle is invariant under the cotangent lifted action of $SO(3)$ on $T^*SO(3)$. Symplectic reduction by this mapping then sends this form to the usual Lie Poisson structure on $\mathfrak{so}^*(3)$, which restricts to a symplectic form on coadjoint orbits. For some original reason which is now all but forgotten, I wanted to understand the structure of hypersurfaces of $T^*SO(3)$ which map to coadjoint orbits.

The coadjoint orbits of $\mathfrak{so}^*(3)$ are simply spheres centered at the origin which one can think of as constant angular momenta. The hypersurface in $T^*SO(3)$ which maps to a particular coadjoint orbit is then a sphere bundle over $SO(3)$. The symplectic form on $T^*SO(3)$ when restricted to such a hypersurface should then give a foliation of the hypersurface by symplectic submanifolds but I have a real problem seeing what these submanifolds should be. Frankly I'm feeling a bit tortured by the question, because I feel it should be so easy, but once I try to write anything down I get all tangled up. If anybody has an answer or a source it would be much appreciated.

Answer 1

The preimage of a coadjoint orbit under a moment map is, under a mild transversality assumption, a coisotropic submanifold; so its null foliation $\smash{\ker(\omega_{|\Phi^{-1}(\mathcal O)})}$ is not by symplectic but by isotropic leaves (symplectic orthogonals to coisotropic): see Guillemin & Sternberg (1984, Thm 26.2).

That theorem also describes the leaf through a point $x$ completely explicitly: it is the orbit of $x$ under the identity component, $G^{\mathrm o}_y$, of the coadjoint stabilizer of $y=\Phi(x)$.

In your case of $T^*G$, one also knows that the (symplectic) leaf space of this foliation is $\mathcal O\times\mathcal O$: see “symplectic Plancherel theorem” in Guillemin & Sternberg (1983, Thm 2.3).