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.



Complex structure on flag manifolds
This is essentially a more "condensed" version of Johannes Ebert's answer.


From the root space decomposition
$$ \mathfrak g /\mathfrak t \otimes \mathbb C = \oplus\_{\alpha \in \Phi} R\_\alpha, $$
one can see that a choice $\Phi^+$ of positive roots gives rise to a $G$-invariant almost complex structure on $G/T$. Indeed, simply require that $\oplus\_{\alpha \in \Phi^+} R\_\alpha$ be the $(1,0)$ part of complexified tangent space of $G/T$ at $T$, and then translate. Conversely, and in the same way, a $G$-invariant almost complex structure on $G/T$ gives rise to a choice of positive roots.


The interesting part is that the integrability of such an almost complex structure boils down to having that
$$ [R\_\alpha, R\_\beta] \subset R\_{\alpha+\beta} $$
whenever $\alpha$, $\beta$, and $\alpha+\beta$ are in $\Phi$, which of course we have. Now apply Newlander--Nirenberg.


This kind of result and reasoning is valid for more general types of homogeneous spaces $G/H$. The Borel--Hirzebruch article mentioned by Johannes is good reading, as is the book by Yang on almost complex homogeneous spaces (see Chapter II in particular).


