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?



reductive group orbits in P(V)?
Yes, the projectivized orbit of a highest weight vector is well known to be the only closed one. It's also an orbit under the compact real form $G\_u$, and as such is the unique $G\_u$-orbit that is a complex (hence Kähler) submanifold (work of Borel, Weil, Tits, Hirzebruch). In [this paper](http://www.ams.org/mathscinet-getitem?mr=661285) Kostant and Sternberg work out the conditions under which $G\_u\cdot[v]$ is a symplectic submanifold. For an exposition see e.g. Guillemin and Sternberg's *Symplectic Techniques in Physics*.


(Orbits under other real forms are thoroughly studied in J. A. Wolf, [The action of a real semisimple group on a complex flag manifold](http://www.ams.org/mathscinet-getitem?mr=251246).)


Your problem, however, is that of studying the orbits in the flag manifold $\Bbb P(V)=GL(V)/GL(V)\_{[w]}$ under any **complex** (irreducibly represented, reductive) $G\hookrightarrow GL(V)$. In this generality I'm not sure there is much one can say. For example, a theorem of Chevalley (J. Humphreys, *Linear algebraic groups*, 11.2) asserts that every $G/H$, $H$ Zariski closed in $G$, occurs as such an orbit for some (not necessarily irreducible) representation.


