# 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.

• Every subset of the set of nodes of the Dynkin diagram corresponds to a coadjoint orbit (aka generalized flag variety). Was that your question? – André Henriques Nov 3 '14 at 15:06
• So flag varieties and coadjoint orbits are just two names for the same thing? – Dontok Bartalez Nov 3 '14 at 15:09
• @user49349: The question needs to be formulated much more carefully. Most important, there are infinitely many coadjoint orbits for a (complex?) semisimple Lie or algebraic group $G$ in the dual of its Lie algebra. Maybe you are only interested in "nilpotent" ones? (Also, $G/P$ is usually referred to as a "partial" or "generalized" flag variety when $P$ is larger than a Borel subgroup, as in Andre's comment.) What do you mean by "particular type"? – Jim Humphreys Nov 3 '14 at 15:25
• @Jim I've re-written the question to make it more precise, I hope this helps. – Dontok Bartalez Nov 3 '14 at 15:34

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.

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