When people study representations, they pay a lot of attention to the irreducible ones. This is justified by the fact that, roughly speaking, every unitary representation decomposes into a direct integral of irreducible representations—so, if you understand irreducible representations, you understand everything.

To make this statement precise, you need to put conditions on the group and the representation. I think it's enough to say that the group is separable and locally compact, the vector space being acted on is a separable complex Hilbert space, and the representation is strongly continuous. See Theorems 2.8 and 2.9 in Mackey's Theory of Unitary Group Representations for details.

When people study symplectic and Poisson manifolds that carry a group action, they pay a lot of attention to the homogeneous ones—the ones where the group action is transitive. These usually appear in the guise of coadjoint orbits (see §4 of Kirillov's Lectures on the Orbit Method). Is this justified by a corresponding decomposition theorem? If so, what are the necessary conditions on the group, the space, and the action?

I can think of two ways to atttempt a decomposition of a symmetric Poisson manifold.

  1. Just write the manifold as the union of the orbits of the group action. The orbits are, of course, homogeneous, but I can't see any reason why they should be Poisson submanifolds. In fact, when the action is Hamiltonian, I'm pretty sure the only way for an orbit to be a Poisson submanifold is for it to be a component of a symplectic leaf. This would mean, in particular, that no decomposition into Poisson submanifolds is possible if the original manifold is symplectic.

  2. Express the manifold as the colimit of its open group-invariant submanifolds. This fixes the problem with the first attempt, because an open submanifold is automatically Poisson. When an open invariant submanifold is small, I imagine it looks like a thickened version of an orbit, so you can think of this as a less naive version of the orbit decomposition. This attempt has its own problem, though: I can't see any reason why an open invariant submanifold, even a very thin one, should look like a homogeneous Poisson manifold.


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