Exercise 8, § 4

Let H be a connected closed subgroup of G of maximum rank, $\mathfrak{h}$ its Lie algebra, X the manifold $G/H$ and V the tangent space of X at the point corresponding to the class of H; identify V with the quotient $\mathfrak{g}/\mathfrak{h}$.

a) If $j$ is an almost complex structure (Differentiable and Analytic Manifolds, Results, 8.8.3) on X, denote by $V''(j)$ the subspace of the complex vector space $\mathbf{C}\otimes V$ consisting of the elements $u$ such that $j(u) =-iu$, and by $\mathfrak{q}(j)$ the subspace of $\mathfrak{g}_{\mathbf{C}}$ that is the inverse image of $V''(j)$, so that the canonical map $V\rightarrow \mathfrak{g}_{\mathbf{C}}/\mathfrak{q}(j)$ is a $\mathbf{C}$-linear isomorphism when V is given the $\mathbf{C}$-vector space structure induced by $j$. Prove that the map $j \rightarrow \mathfrak{q}(j)$ is a bijection from the set of almost complex structures on X invariant under G to the set of complex subspaces $\mathfrak{p}$ of $\mathfrak{g}_{\mathbf{C}}$ satisfying the following conditions:

b) There exists such a structure on X if and only if the commuting fields of the irreducible subrepresentations of the adjoint representation of H on V are all isomorphic to $\mathbf{C}$; in that case, there are $2^s$ such structures, where $s$ is the number of irreducible subrepresentations of V (use Exerc. 7).

c) Let $j$ be an almost complex structure on X, invariant under G, and $\mathfrak{p}=$ $\mathfrak{q}(j)$ the associated subspace. Then, $j$ is integrable (that is, associated to a complex-analytic manifold structure on X, cf. Differentiable and Analytic Manifolds, Results, 8.8.5 to 8.8.8) if and only if $\mathfrak{p}$ satisfies the condition

d) There exists a complex structure (that is, an integrable almost complex structure) on X invariant under G if and only if H is the centralizer of a torus in G (show that conditions (1) to (4) above imply that $\mathfrak{p}$ is a parabolic subalgebra of $\mathfrak{g}_{\mathbf{C}}$ (Chap. VIII, §3, no. 5)); in that case, these complex structures correspond bijectively to the parabolic subalgebras $\mathfrak{p}$ of $\mathfrak{g}_{\mathbf{C}}$ that are the direct sum of $\mathfrak{h}_{\mathbf{C}}$ and their unipotent radical (cf. Chap. VIII, §3, no. 4).

e) Prove that there exist exactly Card(W) complex structures on $G/T$ invariant under G; if $\sigma$ and $\sigma '$ are two such structures, there exists a unique element $w\in W$ such that the canonical operation of $w$ on $G/T$ (by inner automorphisms) transforms $\sigma$ into $\sigma '$. If $w$ is a non-identity element of W, the operation of $w$ on $G/T$ is not $\mathbf{C}$-analytic for any complex structure on $G/T$ invariant under G.

f) Determine the complex structures on $\mathbf{S}^2$ invariant under $\mathbf{S}\mathbf{O}(3)$.

$g)^*$ With the notations of Exerc. $8d)$, let $G_c$ be the complexification of G and P the complex Lie subgroup of $G_c$ with Lie algebra $\mathfrak{p}$. Show that the canonical map $G/H\rightarrow G_c/P$ is an isomorphism of complex-analytic manifolds.\*