Complex structure on flag manifolds

Question

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.

Answer 1

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).

Answer 2

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.

Answer 3

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 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.