Given a Lie group G and a homogeneous space (in particular a manifold) X/G, what does it mean for a differential form on X/G to be G-invariant?
The link here: https://en.wikipedia.org/wiki/Coadjoint_representation defines a symplectic structure on coadjoint orbits and says the differential form is G-invariant, but does not give any indication about what G-invariant means in this case.
Let $M$ be any smooth manifold so that $G$ acts on it smoothly. Then a differential $k$-forms $\omega$ on $M$ is $G$-invariant if
$$ g^* \omega = \omega , \ \ \ \forall g\in G.$$
So in order to talk about $G$-invariant differential forms, one needs an action on the manifolds.
It is not clear what $G$ actions do we have on $X/G$. Indeed for any manifolds $N$, the manifold $X = N\times G$ has an obvious $G$ action so that $X/G$ is diffeomorphic to $N$.
In the case of coadjoint orbit, it is indeed $G/stab(F)$, so it has a natural $G$ action and thus makes sense to talk about $G$-invariant two forms.