If $G$ is a complex semi simple Lie group and $P$ is a parabolic subgroup of it, the quotient $G/P$ is endowed with a Kahler structure and the action of $G$ on $G/P$ is holomorphic with respect to this structure. For $A\in H_2(G/P, \mathbb{Z})$ we define $\mathcal{M}_A$ as the moduli space of homolomorphic curves $\mu:\mathbb{CP}^1\to G/P$ whose degree is $A.$

There is a natural action of $G$ on $\mathcal{M}_A/PSL(2, \mathbb{C})$. under which assumptions on $G, P$ and $A$ is this a transitive action?

For example, when $G/P=G(k, n)$, $G=Sl(n, \mathbb{C}),$ and $P$ is certain group of block upper triangular matrices and $G(k, n)$ denotes the grassmannian manifold of $k$-complex dimensional subspaces of $\mathbb{C}^n,$ the second homological group $H_2(G(k ,n), \mathbb{Z})$ is cyclic, so we may consider the moduli space of holomorphic curves $\mathcal{M}_A$ where $A$ is a generator of the second homological group. In this case the moduli space of unparametrized holomorphic curves of degree one is the flag manifold $Fl(k-1, k+1; n)$ (see for example lemma 3.2 of http://arxiv.org/PS_cache/alg-geom/pdf/9403/9403010v1.pdf) and the action of $Sl(n, \mathbb{C})$ on $Fl(k-1, k+1; n)$ is transitive.