Let $N$ be a homogeneous space. Therefore we find a Liegroup $G$ and a isotropy-subgroup $K$ of $G$, such that we can identify $N = G/K$. Then we have a canonical action $l\colon G \times G/K \to G/K$ on $G/K$, given by left-multiplication.
We now identify the cotangentbundle $T^*N = G \times_K \mathfrak{k}^\circ$, where $\mathfrak{k}^\circ$ is the annihilator of $\mathfrak{k}$ in $\mathfrak{g}^*$ and we write the elements as $[g,\alpha] \in G \times_K \mathfrak{k}^°$
Then the lifted action $\lambda \colon G \times T^*N \to T^*N$ is $\lambda_g([h,\alpha]) =[gh, \alpha]$.
Under this identification, the momentum map for our $G$-action is:
$\Phi \colon T^* N \to \mathfrak{g}^*, \ [g,\alpha] \mapsto \operatorname{Ad}^*_{g^{-1}} \alpha$
Is it true, that $\Phi(T^*N)$ is a submanifold of $\mathfrak{g}^*$ and $\Phi \colon T^*N \to \Phi(T^*N)$ is a submersion?
1) If not, is it true, if we assume that $K$ acts on $\mathcal{O} \cap \mathfrak{k}^°$ transitively (for $\mathcal{O}$ an arbitrary $G$-orbit in $\mathfrak{g}^*$)?
2) is it true, if we assume 1) and $G$ acts properly on $T^*N$?
