What is the Explicit Relationship between Coadjoint Orbits and Flag Manifolds? 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.
 A: Let $G$ be a complex semi-simple Lie group. Then $$\mathcal O_\lambda\cong G/B\cong G/G_{\lambda}\cong G^{\mathbb C}/P$$ where $G^{\mathbb C}$ is the complexification of Lie group $G$ and in fact, every coadjoint orbit is projective variety with Kodiaira dimension $-\infty$. 
By following decomposition 
$$G^{\mathbb C}\cong G\times \mathfrak g^{*}\cong T^*G$$,
where, $G^{\mathbb C}:=\exp \{\mathfrak g+i\mathfrak g\}$.
If we take 
$\mu:T^*G\to \mathfrak g^*$, then by previous decomposition, $\mu^{-1}(\lambda)=G$, So, $$\mu^{-1}(\lambda)/G_\lambda\cong G/G_\lambda\cong \mathcal O_\lambda$$
Coadjoint orbits are symplectic quotient of cotangent bundle of Lie groups.
Two geometric property of coadjoint orbits.
They are Symplectic varieties and also Kaehler varieties. 
In fact if $M$ be a compact Kaehler manifold, then its symplectic quotient is also kaehler manifold. So, because $T^*G$ is Kahler manifold, so coadjoint orbit is also Kaehler.
Allen Knutson, says that coadjoint orbits are birationally equivalent to its open Bruhat cells.
If we complexify our coadjoint orbites, i.e, $$G^{\mathbb C}/G_\lambda^{\mathbb C}\cong \mathcal O_\lambda^{\mathbb C}$$, then the complexified of coadjoint orbits have hyper kahler structure and are Stein manifolds
Hirzebruch computed the first chern class of flag varieties and coadjoint orbits
He showed $c_1( \mathcal O_\lambda)=2\rho$, where $\rho=\frac{1}{2}\sum_{\alpha>0} \alpha$(half sum of positive roots of Lie group $G$)
In final, computing some coadjoint orbits
A: 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.
