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Let $V$ be a real representation of a torus $T$ and assume that $V^T=0$. Then $V$ is a sum of $2$-dimensional representations. Assume that all isotypical subspaces have real dimension $2$; the total dimension being $2n$. Then, by Schur's Lemma, the endomorphism algebra $End_T (V)$ is $\mathbb{C} \oplus \ldots \mathbb{C}$ ($n$ times). There are $2^n$ unital homomorphisms $\mathbb{C} \to End_T V$, in other words, $2^n$ different invariant complex structures on $V$ (they differ by conjugation in each isotypical summand).

The adjoint representation of $T$ in $\mathfrak{g}/\mathfrak{t}$ satisfies these assumptions and the tangent bundle of $G/T$ is the bundle $G \times_T \frac{\mathfrak{g}}{\mathfrak{t}}$. So there is a $G$-invariant complex structure $T G/T$ (meaning, it is a complex vector bundle), in other words, $G/T$ has an invariant almost complex structure. The question is whether this almost complex structure is integrable.

This is discussed in Borel,Hirzebruch: Homogeneous spaces and characteristic classes I. To show the integrability of the almost structure, they use Newlander-Nirenberg's theorem. All the data are real-analytic by general Lie group theory. For real analytic data, the Newlander-Nirenberg theorem is pretty easy, using little more than Frobenius's theorem (the general theorem is a hard PDE result). The proof that the integrability conditions hold is Lie-algebraic.

It is relatively easy to the see that $G/T$ is Kähler by Lie theory: Pick an invariant hermitian metric. Its imaginary part is a nondegenerate $2$-form $\omega$. Next, $G/T$ is a symmetric space and on symmetric spaces, each invariant form as $\omega$ is closed. Thus the metric satisfies the Kaehler condition.

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Let $V$ be a real representation of a torus $T$ and assume that $V^T=0$. Then $V$ is a sum of $2$-dimensional representations. Assume that all isotypical subspaces have real dimension $2$; the total dimension being $2n$. Then, by Schur's Lemma, the endomorphism algebra $End_T (V)$ is $\mathbb{C} \oplus \ldots \mathbb{C}$ ($n$ times). There are $2^n$ unital homomorphisms $\mathbb{C} \to End_T V$, in other words, $2^n$ different invariant complex structures on $V$ (they differ by conjugation in each isotypical summand).

The adjoint representation of $T$ in $\mathfrak{g}/\mathfrak{t}$ satisfies these assumptions and the tangent bundle of $G/T$ is the bundle $G \times_T \frac{\mathfrak{g}}{\mathfrak{t}}$. So there is a $G$-invariant complex structure $T G/T$ (meaning, it is a complex vector bundle), in other words, $G/T$ has an invariant almost complex structure. The question is whether this almost complex structure is integrable.

This is discussed in Borel,Hirzebruch: Homogeneous spaces and characteristic classes I. To show the integrability of the almost structure, they use Newlander-Nirenberg's theorem. All the data are real-analytic by general Lie group theory. For real analytic data, the Newlander-Nirenberg theorem is pretty easy, using little more than Frobenius's theorem (the general theorem is a hard PDE result). The proof that the integrability conditions hold is Lie-algebraic.

It is relatively easy to the that $G/T$ by Lie theory: Pick an invariant hermitian metric. Its imaginary part is a nondegenerate $2$-form $\omega$. Next, $G/T$ is a symmetric space and on symmetric spaces, each invariant form as $\omega$ is closed. Thus the metric satisfies the Kaehler condition.