isotropy of the cotangent lift of a group action

Question

I asked this question in stack exchange but have not received an answer, so I am posting it here.

Given a group action on a manifold (e.g. configuration space of coordinates), cotangent-lift it to the phase space (i.e. coordinates and momentum), which is the appropriate cotangent bundle of the base manifold.

For a point on the base, there is an isotropy that fixes the coordinate variable; I naively think that the isotropy of the lifted action for a point in the cotangent fiber should be a subgroup of the corresponding isotropy on the base because both coordinate and momentum variables would have to be fixed now.

It seems that the size of the isotropy for the lifted action would depend on the momentum component: for example, along the zero section in the cotangent bundle (i.e. zero momentum for any coordinate point), the isotropy remains the same; for a generic momentum value, however, it may become a proper subgroup.

In general, momentum may not be independent of coordinate for they usually have to satisfy constraints (e.g. the moment map). So if I wish to find out the isotropy for the cotangent-lifted action at a point in the phase space, would I have to first solve the constraints -- provided that it is possible -- to figure out which momentum point I am looking at and then compute the isotropy explicitly? Or are there general facts that relate the isotropy on the base and that for the cotangent lift?

Thank you!

Answer 1

Since the projection $\pi: T^* Q \to Q$ is equivariant, the stabilizer $G_p$ of a point $p \in T^*_q Q$ is indeed a subgroup of the stabilizer of the base point $G_q$. In fact, $G_p$ is also the stabilizer of $p$ under the action of $G_q$ on the fiber $T^*_q Q$. But without knowing what momentum you are looking at, its hard to say more. The zero-level set of the momentum map $J$ is however special: roughly speaking a subgroup is an orbit type of $J^{-1}(0)$ only if it occurs also as an orbit type of $Q$, see Theorem 5 in On the geometry of reduced cotangent bundles at zero momentum (the idea of proof might also give you information about orbit types for other level sets, but I'm not aware of a more general statement in the literature).

You might be interested in papers about symplectic reduction for cotangent bundles; they usually deal with issues like this. In this setting, you get a refinement of the classical stratification of the reduced phase space: the set of all points in $T^* Q$ that have a certain orbit type for the $G$-action on $T^* Q$ and may project onto different orbit types for the $G$-action on $Q$; and fixing the orbit type on $Q$ gives you then a decomposition of the orbit-momentum-stratum. What helped me to understand things better was to look at particular examples and work out the orbit type decomposition. The Harmonic oscillator is discussed from this perspective in detail in Example 2.31 of my paper Singular symplectic cotangent bundle reduction of gauge field theory.