Say $G$ is a reductive group over $\mathbb{C}$. We can take an a dominant highest weight $\lambda$ and look at the action of $G$ on $X = \mathbb{P} V(\lambda)$. The stabilizer of the class of the highest weight vector is a parabolic subgroup so the orbit is isomorphic to $G/P$. What about the other orbits in $X$? If $[v] \in X - G/P$ then is there a good description of $G.[v]$ or its closure? If this is hard for general $[v] \in X - G/P$ are there any conditions you can put on $[v]$ that make it easier? Is $G/P$ the only closed orbit?
Does anyone know of references that address these questions?
