TITLE: Complex structure on flag manifolds
QUESTION [13 upvotes]: Let $G$ be a compact Lie group and $T$ a maximal torus of $G$.  Then the flag manifold $G/T$ is a complex manifold and a symplectic manifold.  One way to see the symplectic structure is to view $G/T$ as the co-adjoint orbit of a generic element $F_0 \in Lie(T)^*$.  Then the symplectic structure is given by
$$
\omega_F(X^+_F, Y^+_F) = F([X,Y]), 
$$
where $X^+,Y^+$ are the fundamental vector fields corresponding to $X,Y \in Lie(G)$ and $F \in Orbit(F_0)$.
All the references I've seen get the complex structure on $G/T$ by showing it is isomorphic to $G^{\mathbb C}/B$ where $G^{\mathbb C}$ is the complexification of $G$ and $B$ is a Borel subalgebra.
My question is if there is a way to get the complex structure explicitly in terms of the Lie algebras of $G$ and $T$, in a similar vein to how I defined the symplectic structure.

REPLY [6 votes]: Let $G_C$ denote the complexification of the compact connected Lie group $G$ and let $H$ be the centralizer of any toral subgroup (not necessarily maximal) in $G$.  Then there is a (complex) parabolic subgroup $Q \subset G_C$ with $G$ (as a subgroup of $G_C$) transitive on the complex manifold $Z := G_C/Q$ and $H = G \cap Q$.  So $G/H$ is realized as the complex flag manifold $Z$.  Here $H_C$ is the reductive component of $Q$, and the choice of $Q$ with given reductive part $H_C$ gives the complex structure: the antiholomorphic tangent space is the Lie algebra of the unipotent radical of $Q$.  These choices are parameterized by the quotient $W_G/W_H$ of the Weyl groups of $G$ and $H$.
Your question is about the special case where $H = T$, a maximal torus in $G$.  
The flag manifolds have many beautiful properties.  They were first described by Jacques Tits in his thesis published by the Belgian Academy of Sciences in 1954.