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We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of Lie groups and stuff, and it's also related to quantization. However, is it true that given a classical mechanics system, you could always realize this as a coadjoint orbit of some Lie group? And would finding its corresponding representation be be equivalent with finding its corresponding quantization?

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There is a general result by Patrick Iglesias-Zemmour that every symplectic manifold is a coadjoint orbit of its symplectomorphism group.

However, from the quantization point of view, this result is not constructive because the symplectomorphism group is infinite dimensional; and in addition, not much is known about the geometry and topology of symplectomorphism groups in general, let alone its representation theory. Please see the following review by McDuff.

In the case of standard coadjoint orbits of some compact group $G$, then $G$ itself is a subgroup of the group of symplectomorphisms. In this case the geometric quantization program can always be carried out to quantize the coadjoint orbit. However, even in the case of non-compact semisimple Lie groups and solvable Lie groups, there are some trouble in the method of orbits, please see Kirillov.

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