Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. One way to construct a complex structure on $G/T$ is to choose a nilpotent subalgebra $\mathfrak{n}^+$ corresponding to some choice of positive roots of the complexified Lie algebra of $G$. We let $J\_0$ be the corresponding almost complex structure on $\mathfrak{n}^+\oplus \mathfrak{n}^-$, which yields an invariant almost complex structure $J\_x$ at every $x\in G/T$ and show that it is integrable using the Newlander-Nirenberg theorem. Another way to show the existence of a complex structure on $G/T$ is to let $B$ be a Borel subgroup of the complexification $G^{\mathbb C}$ of $G$ and show that $G^{\mathbb C}/B$ is diffeomorphic to $G/T$.


If the positive roots of $B$ are the same as those we used to choose $\mathfrak{n}^+$ will the complex structures be the same? I.e. will $J\_x$ be equal to the usual $J'\_x$ on $G^\mathbb{C}/B$ after pulling it back to be a map on $T\_x(G^\mathbb{C}/B)$? Or are the two complex structures merely "equivalent?" And how are they equivalent?



Equivalence of complex structures on flag manifold
Under the natural identification $G/T\to G^c/B$ they coincide. This is Borel-Hirzebruch ([1958](//ams.org/mathscinet-getitem?mr=102800), §14.3) $=$ Bourbaki ([2005](//ams.org/mathscinet-getitem?mr=2109105), §IX.4, Exercise 8 *g*)).


