Complex structure on flag manifolds - MathOverflow

Complex structure on flag manifolds

2011-09-07T21:05:21Z

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 by Johannes Ebert for Complex structure on flag manifolds

2011-09-07T23:20:10Z

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.

Answer by Faisal for Complex structure on flag manifolds

2011-09-08T02:39:37Z

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 by Joseph Wolf for Complex structure on flag manifolds

2011-10-16T19:56:52Z

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.