Say $G$ is a reductive group over $\mathbb{C}$.  We can take 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?