Does every complex flag manifold have a natural Kähler structure? If so, what is it?



Kahler structure on flag manifolds
Every flag manifold $M=G^{\mathbb{C}}/P=G/C(S)$ where $P$ is a parabolic subgroup and $C(S)=P\cap G$ is the centralizer of a torus $S\subset G$, admits a finite number of
invariant Kähler structures. In particular the complex presentation $G^{\mathbb{C}}/P$
gives rise to an finite number of invariant complex structures (i.e. integrable almost complex structures commuting with the isotropy representation of $M$). Any such complex structure is determined by an invariant ordering $R\_{M}^{+}$ on the set of **complementary roots**
$R\_{M}=R\backslash R\_{K}$ of $M$ and explicitly is given by
$$
J\_{o}E\_{\pm \alpha}=\pm i E\_{\pm\alpha}, \quad a\in R\_{M}^{+}
$$
where $E\_{\alpha}$ are root vectors with respect a Weyl basis of $\frak{g}^{\mathbb{C}}$.


On the other hand, the real presentation $G/C(S)$ makes $M$ a (homogeneous) Kähler manifold,
as a (co)-adjoint orbit of an element $w\in\frak{g}$ in the Lie algebra $\frak{g}$ of the compact connected (semi)simple Lie group. **Flag manifolds exhaust all compact homogeneous Kähler manifolds corresponding to a compact connected semi-simple Lie group.** 


To be more specific, $M$ admits a finite number of Kähler structures which are parametrized by the well-known $\frak{t}$-chambers (connected components of the set of regular elements of $\frak{t}$) where 


$$
{\frak{t}} =( H\in{\frak{h}} : (H, \Pi\_{0})=0 )
 $$


is a real form of the center ${\frak{z}}$ of the isotropy subgroup $K=C(S)$.


Here $\frak{h}$ is the Cartan subalgebra corresponding to a maximal torus $T$ of $G$ which contains $S$, and $\Pi\_{0}\subset\Pi$ is the subgroup of simple roots which define (the semi-simple part of) the complexification $\frak{k}^{\mathbb{C}}$ (note that $K=C(S)=P\cap G$ is a reductive Lie group). We have


$$
{\frak{z}}^{\mathbb{C}}={\frak{t}}\oplus i {\frak{t}}, \ \ \ {\frak{k}}^{\mathbb{C}}={\frak{z}}^{\mathbb{C}}\oplus{\frak{k}}\_{ss}^{\mathbb{C}}
$$


where ${\frak{z}}^{\mathbb{C}}$ is the complexification of the center ${\frak{z}}$ and ${\frak{k}}\_{ss}^{\mathbb{C}}$ is the semi-simple part of the reductive complex Lie subalgebra ${\frak{k}}^{\mathbb{C}}$


In particular, there exists a natural 1-1 correspondence between elements from ${\frak{t}}$ and 
closed invariant 2-forms on $M$. Symplectic 2-forms (non-degenerate) correspond to regular elements $t$ of ${\frak{t}}$.


Note that the corresponding symplectic form corresponding to a regular element $t\_{0}$ is the Kirillov-Kostant-Souriau 2-form in the (co)-adjoint orbit $Ad(G)t\_{0}$, that is


$$
\omega\_{t\_{0}}(X, Y)=B(t\_{0}, [X, Y]), \ \ X, Y\in T\_{t\_{0}}M.
$$


For more details see: D. Alekseevsky: Flag manifolds (11. Yugoslav Geometrical seminar, Divcibare, 10-17 October 1993, 3-35.
This article is a very good review on the geometry of flag manifolds.


