Given a complex semi-simple Lie group $G$, it acts smoothly on the dual $\frak{g}^\*$ of its Lie algebra $\frak{g}$ by the coadjoint action. The orbits of that action are called coadjoint orbits.


A maximal Zariski closed and connected solvable subgroup of $G$ is called a Borel subgroup; and $P$ is a parabolic subgroup of $G$ if it contains a Borel subgroup. We call the quotient $G/P$ a flag manifold. 


I know that these two families of spaces are related, but cannot find an exact statement of the relationship.



What is the Explicit Relationship between Coadjoint Orbits and Flag Manifolds?
Let $K$ be a maximal compact subgroup of $G$. Then $K$ acts transitively on each $G/P$, and up to $K$-isomorphism, the $K$-spaces obtained exactly match those occurring as coadjoint orbits of $K$ (acting on $\mathfrak k^\*$).


Basic example: $G=SO\_3(\mathbb C)$, with $G/P = \mathbb{CP}^1, pt$. Then $K=SO\_3(\mathbb R)$, acting on $\mathbb R^3$ with orbits = the concentric spheres and the origin.


