We know this important fact from A.A.Kirillov that :


Every homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$ or a [central extension](http://en.wikipedia.org/wiki/Group_extension) of it.


But when can we precisely say that a homogeneous symplectic $G$-manifold is locally isomorphic to an orbit in the coadjoint representation of the group $G$


or 


A homogeneous symplectic $G$-manifold is central extension of it?


PS:Definition of coadjoint orbit;


Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra,and also $\mathfrak{g^\*}$ be the dual of Lie algebra, the coadjoint orbit is as follows


$\mathfrak{G}=\{Ad^\*(g)F, g\in G\}$ where $F\in\mathfrak{g^\*}$.



Coadjoint orbits and homogeneous symplectic $G$-manifolds
Certainly, if $G$ has a central extension, $G$ will act on the coadjoint orbits of the central extension. So I think your question might be "When can we be sure that our group has no central extensions?"


The central extensions of the Lie algebra are given by elements of $H^2({\mathfrak g})$. So you want that to vanish, which it does for e.g. compact semisimple groups.


The two cautionary examples to keep in mind are ${\mathbb R}^{2n}$ acting on $T^\* {\mathbb R}^n$ by translation, which leads one to the Heisenberg group central extension of ${\mathbb R}^{2n}$, and the free loop group $LK$ acting on the based loop space $\Omega K \cong LK/K$, which leads one to the affine Lie group central extension of $LK$. There, $H^2(L{\mathfrak K}) = H^2(LK) = \pi\_2(LK)^\* = \pi\_3(K)^\* = H^3(K)$ which is one-dimensional for $K$ simple and simply-connected.


There is an (in my opinion wrongheaded) definition of moment map that includes the composite ${\mathcal O} \hookrightarrow \widetilde{\mathfrak g}^\* \twoheadrightarrow {\mathfrak g}^\*$ where the first map is the inclusion of the coadjoint orbit, and the latter uses a noncanonical splitting of the central extension $\tilde{\mathfrak g}$. You can recognize this in the literature when people speak of "non-equivariant moment maps". Horrible. If a central extension is implicit I'd rather it be explicit.


