Invariant almost complex structures on real flag manifolds

Abstract

In this work, we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures. We check which invariant almost complex structures are integrable and prove that only some flag manifolds of the Lie algebra $C_{l}$ admit complex structures.

1 Introduction

A flag manifold of a non-compact semisimple Lie algebra $\mathfrak {g}$ is a quotient space $\mathbb { F}_{\varTheta }=G/P_{\varTheta }$, where G is a connected group with Lie algebra $\mathfrak {g}$ and $P_{\varTheta }$ is a parabolic subgroup. If $K\subset G$ is a maximal compact subgroup and $K_{\varTheta }=K\cap P_{\varTheta }$, then the flag $\mathbb {F}_{\varTheta }$ can be written in the form $\mathbb {F}_{\varTheta }=K/K_{\varTheta }$.

In this work, we study the existence and integrability of invariant almost complex structures on real flag manifolds $\mathbb {F}_{\varTheta }$ in the case that $\mathfrak {g}$ is a split real form of a complex simple Lie algebra. Our goal is to make an exhaustive investigation of the real flag manifolds $\mathbb {F}_{\varTheta }$ that admit K-invariant almost complex structures and to verify their integrability, that is, when they are indeed complex structures.

The invariant geometry of complex flag manifolds has been extensively studied. Regarding invariant geometry of complex flag manifolds, the literature is exhaustive and goes back to Borel [2] and Wolf-Gray [21, 22]. Recent works are [1, 3,4,5, 7, 9, 13, 14, 19, 20] and [1].

For real flag manifolds, the literature is much more sparse. There is no systematic treatment of the invariant geometric structures on these flag manifolds. An attempt to fill this gap was made recently by Patrão and San Martin [15] who provide a detailed analysis of the isotropy representations for the flag manifolds of the split real forms of the complex simple Lie algebras.

In this paper, we rely on the results of [15] to build (or to prove the non-existence of) K-invariant almost complex structures on the real flag manifolds. The conclusion is that only a few flag manifolds (associated to split real forms) admit K-invariant almost complex structures. In this sense, we obtain the following result.

Theorem 1

A real flag manifold $\mathbb {F}_\varTheta =K/K_\varTheta$ admits a K-invariant almost complex structure if and only if it is a maximal flag of type $A_3$, $B_2$, $G_2$, $C_l$ for l even or $D_l$ for $l\ge 4$, or if it is one of the following intermediate flags:

• of type $B_3$ and $\varTheta =\{\lambda _1-\lambda _2, \lambda _2-\lambda _3\}$;

• of type $C_l$ with $\varTheta =\{\lambda _{d}-\lambda _{d+1},\ldots ,\lambda _{l-1}-\lambda _l,2\lambda _l\}$ for $d>1$, d odd.

• of type $D_l$ with $l=4$ and $\varTheta$ being one of: $\{ \lambda _{1}-\lambda _{2},\lambda _{3}-\lambda _{4}\}$, $\{ \lambda _{1}-\lambda _{2},\lambda _{3}+\lambda _{4}\}$, $\{ \lambda _{3}-\lambda _{4},\lambda _{3}+\lambda _{4}\}$.

The next step is to check which of the existing almost complex structures are integrable. By making computations with the Nijenhuis tensor, we arrive at the following result.

Theorem 2

A real flag manifold $\mathbb {F}_\varTheta =K/K_\varTheta$ admits K-invariant complex structures if and only if it is of type $C_l$ and $\varTheta =\{\lambda _d-\lambda _{d+1},\ldots ,\lambda _{l-1}-\lambda _l,2\lambda _l\}$ with $d>1$, d odd.

These complex flag manifolds are realized as manifolds of flags $\left( V_{1}\subset \cdots \subset V_{k}\right)$ of subspaces of $\mathbb {R}^{2l}$ that are isotropic with respect to the standard symplectic form of $\mathbb {R}^{2l}$. Moreover, $\mathbb {F}_\varTheta$ is finitely covered by $U(l)/U(l-d)$ and the complex structures on $\mathbb {F}_\varTheta$ can be lifted to this covering space.

To prove the results above, we mainly use the isotropy decomposition of $T_{b_\varTheta }\mathbb {F}_\varTheta$, the tangent space of the flag a the origin $b_\varTheta$. In [15] there are described the $K_\varTheta$-invariant and irreducible components of this representation obtaining a decomposition

$$\begin{aligned} T_{b_{\varTheta }}\mathbb {F}_{\varTheta }=V_{1}\oplus \ldots \oplus V_{k}. \end{aligned}$$

This decomposition is essential to find K-invariant geometries on $\mathbb {F}_{\varTheta }$. It is well known that the compact isotropy group is a product $K_\varTheta =M(K_\varTheta )_0$ where M is the isotropy of the maximal flag and $(K_\varTheta )_0$ the connected component of the identity. An almost complex structure commutes with the isotropy representation of $K_\varTheta$ if and only if it commutes with the M and $(K_\varTheta )_0$ representations on the tangent space. This allows us to split the proofs in two stages: study M-invariance on the one hand, and the condition of commutativity with ${{\mathrm{ad}}}_X$ for all $X\in \mathfrak {k}_\varTheta = \mathrm {Lie}(K_\varTheta )$, on the other hand.

A necessary and sufficient condition for a real flag to admit M-invariant almost complex structures is that every M-equivalence class on $\varPi ^{+}{\setminus } \langle \varTheta \rangle ^+$ has an even amount of elements. Two roots $\alpha$ and $\beta$ lie in the same M-equivalence class if the representations of M on $\mathfrak {g}_{\alpha }$ and $\mathfrak {g}_{\beta }$ are equivalent. This condition is necessary for $\mathbb {F}_\varTheta$ to admit $K_\varTheta$ invariant almost complex structures, so by inspection of these equivalence classes we discard many flags manifolds. For the remaining cases, we focus on the $\mathfrak {k}_\varTheta$ representation on $T_{b_\varTheta }\mathbb {F}_\varTheta$. We should remark that in all cases we give the almost complex structures explicitly, in a constructive way. Integrability is proved by computing the Nijenhuis tensor.

It is worth stressing a main difference in the isotropy representation of $K_{\varTheta }$ between the real case and the complex case. In the real flag, there are cases where two $K_{\varTheta }$-invariant and irreducible components are equivalent. In the complex case, this fact does not occur. Consequently, on the complex case, the $K_{\varTheta }$-invariant and irreducible components, in the isotropy representation of $\mathbb {F}_{\varTheta }$, are invariant by almost complex structures. On the real flag, there are cases where $JV_i=V_j$, for $V_i$ and $V_j$ equivalent $K_{\varTheta }$-invariant and irreducible components.

This work is organized in the following manner. In Sect. 2 we fix notations and present the first results on existence of M-invariant complex structures. We give necessary and sufficient conditions for a flag manifold to admit such structure. In the case of a maximal flag, that is $\varTheta =\emptyset$, this is all we need to pursue our study since $K_\varTheta =M$. Section 3 focuses in this case. Section 4 deals with intermediate flags, that is $\varTheta \ne \emptyset$. We only consider those intermediate flags verifying the necessary condition of Sect. 2. The full comprehension of the isotropy representation of $K_\varTheta$ is needed, so we fully describe it for the cases under study. The propositions in Sects. 3, 4 account to Theorems 1, 2 above.

2 Notation and preliminary results

We refer to [11, 17] for further developments of the concepts in this section. We assume throughout the paper that $\mathfrak {g}$ is the split real form of a complex simple Lie algebra $\mathfrak {g}_{\mathbb {C}}$. If $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {a} \oplus \mathfrak {n}$ is an Iwasawa decomposition then $\mathfrak {a}$ is a Cartan subalgebra. Denote $\varPi$ the set of roots of $\mathfrak {g}$ associated to $\mathfrak {a}$. If $\alpha \in \mathfrak {a}^{*}$ is a root then we write

$$\begin{aligned} \mathfrak {g}_{\alpha }=\{X\in \mathfrak {g}:\mathrm {ad}\left( H\right) X=\alpha \left( H\right) X,~H\in \mathfrak {a}\} \end{aligned}$$

for the corresponding root space, which is one-dimensional since $\mathfrak {g}$ is split. Let $\varPi ^+$ be a set of positive roots and $\varSigma$ the corresponding positive simple roots.

The set of parabolic Lie subalgebras of $\mathfrak {g}$ is parametrized by the subsets of simple roots $\varSigma$. Given $\varTheta \subset \varSigma$, the corresponding parabolic subalgebra is given by

$$\begin{aligned} \mathfrak {p}_\varTheta =\mathfrak {a}\oplus \sum _{\alpha \in \varPi ^+} \mathfrak {g}_\alpha \oplus \sum _{\alpha \in \left\langle \varTheta \right\rangle ^-}\mathfrak {g}_\alpha =\mathfrak {a}\oplus \sum _{\alpha \in \left\langle \varTheta \right\rangle ^+\cup \left\langle \varTheta \right\rangle ^-} \mathfrak {g}_\alpha \oplus \sum _{\alpha \in \varPi ^+\backslash \left\langle \varTheta \right\rangle ^+}\mathfrak {g}_\alpha \end{aligned}$$

where $\left\langle \varTheta \right\rangle ^\pm$ is the set of positive/negative roots generated by $\varTheta$.

Denote by G the group of inner automorphisms of $\mathfrak {g}$, which is connected and generated by $\exp {{\mathrm{ad}}}(\mathfrak {g})$ inside $\mathrm{GL}(\mathfrak {g})$. Let K be the maximal compact subgroup of G, then K is generated by ${{\mathrm{ad}}}(\mathfrak {k})$. The standard parabolic subgroup $P_\varTheta$ of G is the normalizer of $\mathfrak {p}_\varTheta$ in G. The associated flag manifold is defined by $\mathbb {F}_\varTheta =G/P_\varTheta$. The compact subgroup K acts transitively on $\mathbb {F}_\varTheta$ so we obtain $\mathbb {F}_\varTheta =K/K_\varTheta$ where $K_\varTheta =K\cap P_\varTheta$. Fixing an origin $b_\varTheta$ in $\mathbb {F}_\varTheta$, we identify the tangent space $T_{b_\varTheta }\mathbb {F}_\varTheta$ with the nilpotent Lie algebra

$$\begin{aligned} \mathfrak {n}_\varTheta ^-= \sum _{\alpha \in \varPi ^-\backslash \left\langle \varTheta \right\rangle ^-}\mathfrak {g}_\alpha . \end{aligned}$$

In $\mathfrak {n}^-$, the isotropy representation of $K_\varTheta$ on $T_{b_\varTheta }\mathbb {F}_\varTheta$ is just the adjoint representation, since $\mathfrak {n}^-_\varTheta$ is normalized by $K_\varTheta$. The Lie algebra $\mathfrak {k}_\varTheta$ of $K_\varTheta$ is

$$\begin{aligned} \mathfrak {k}_\varTheta =\sum _{\alpha \in \left\langle \varTheta \right\rangle ^+\cup \left\langle \varTheta \right\rangle ^-} (\mathfrak {g}_\alpha \oplus \mathfrak {g}_{-\alpha })\cap \mathfrak {k}. \end{aligned}$$

Compactness of K implies that $\mathfrak {k}_\varTheta$ admits a reductive complement $\mathfrak {m}_\varTheta$ so that $\mathfrak {k}=\mathfrak {k}_\varTheta \oplus \mathfrak {m}_\varTheta$ and $T_{b_\varTheta }\mathbb {F}_\varTheta$ is identified also with $\mathfrak {m}_\varTheta$. The map $X_\alpha \longrightarrow X_{\alpha }-X_{-\alpha }$ for $\alpha \in \varPi ^-\backslash \left\langle \varTheta \right\rangle ^-$ is a $K_\varTheta$ invariant map from $\mathfrak {n}_\varTheta ^-$ to $\mathfrak {m}_\varTheta$. Along the paper, we will call isotropy representation either the representation of $K_\varTheta$ on $\mathfrak {n}_\varTheta ^-$ or on $\mathfrak {m}_\varTheta$, without making any difference or special mention. In some cases, we will even use $\mathfrak {n}_\varTheta ^+$ instead of $\mathfrak {n}_\varTheta ^-$.

Let M be the centralizer of $\mathfrak {a}$ in K. Then $K_\varTheta =M\cdot (K_\varTheta )_0$ where $(K_\varTheta )_0$ is the connected component of the identity of $K_\varTheta$. Thus M acts on $T_{b_{\varTheta }}\mathbb {F}_{\varTheta }$ by restricting the isotropy representation of $K_\varTheta$. The group M is finite and acts on $\mathfrak {n}_\varTheta ^-$ leaving each root space $\mathfrak {g}_\alpha$ invariant. Moreover, if $m\in M$ and $X\in \mathfrak {g}_\alpha$ then $\mathrm {Ad}(m)X=\pm X$. Two roots $\alpha$ and $\beta$ are called M-equivalent, which we will denote by $\alpha \sim _M \beta$, if the representations of M on the root spaces $\mathfrak {g}_\alpha$ and $\mathfrak {g}_\beta$ are equivalent. The M-equivalence classes were described in [15].

When $\varTheta =\emptyset$, we drop all the sub-indexes $\varTheta$. The associated flag manifold is the maximal flag $\mathbb {F}=K/M$ and the tangent space at the origin b will be identified with $\mathfrak {n}^-$.

Let U be a group of linear maps of the vector space V. A subspace $W\subset U$ is U-invariant if $ux\in W$ for all $x\in W$ and for all $u\in U$. A complex structure on V is endomorphism $J:V\longrightarrow V$ such that $J^2=-1$ and it is said to be U-invariant if $uJ=Ju$ for all $u\in U$. We shall prove two technical results.

Lemma 1

Let $W\subset V$ be a U-invariant space. Then the following statements are true:

• 1. JW is U-invariant as well.

• 2. W is irreducible if and only if JW is irreducible.

• 3. The representations of U on W and JW are equivalent.

• 4. If W is irreducible then either $W\cap JW=\{0\}$ or $JW=W$.

• 5. If $\dim W=1$ then $W\cap JW=\{0\}$.

Proof

Take $u\in U$ and $x\in W$. Then, $uJx=Jux\in JW$ showing that JW is U -invariant.

Suppose that W is irreducible and let $A\subset JW$ be a U-invariant subspace. Then $J^{-1}A=JA\subset W$ is also U-invariant. Hence, $JA=W$ or $JA=\{0\}$, which implies that $A=JW$ or $A=\{0\}$. Thus JW is irreducible.

As J commutes with the elements of U, the map $J:W\rightarrow JW$ intertwines the representations on W and JW so that they are equivalent. Since $W\cap JW\subset W$ is U-invariant and W is irreducible we get item 4. Finally $W\cap JW=\{0\}$ if $\dim W=1$ because the eigenvalues of J are $\pm \,i$ hence W is not invariant by J. $\square$

Lemma 2

Let $W_i$, $i=1,2$ be U-invariant and irreducible subspaces of V such that $W_1\cap W_2=0$ and the representation of U on $W_1$ is not equivalent to that on $W_2$. If $V=W_1\oplus W_2\oplus W$ for some complementary subspace W and J is a U-invariant complex structure, then $Jw_1\in W_1\oplus W$ for all $w_1\in W_1$.

Proof

Consider $P:V\longrightarrow W_2$ the projection map with respect to the decomposition above. The map $P\circ J:W_1\longrightarrow W_2$ is U-invariant and bijective if nonzero, since its domain and target spaces are irreducible. Thus it is an equivalence between the representations of U, if nonzero. Therefore, $P\circ J\equiv 0$ and the result follows. $\square$

Under the hypothesis of the lemma above, in the particular case of $V=W_1\oplus W_2$ we have $JW_i=W_i$, $i=1,2$.

From the general theory of invariant tensors on homogeneous manifolds, we know that K-invariant almost complex structures on the flag manifold $\mathbb {F}_\varTheta =K/K_\varTheta$ are in one-to-one correspondence with $K_\varTheta$-invariant complex structures $J:T_{b_{\varTheta }}\mathbb {F}_{\varTheta }\rightarrow T_{b_{\varTheta }}\mathbb {F}_{\varTheta }$. Recall that $T_{b_{\varTheta }}\mathbb {F}_{\varTheta }$ identifies with $\mathfrak {n}_\varTheta ^-$ (or $\mathfrak {m}_\varTheta$) and this identification preserves the $K_\varTheta$ representation. So K-invariant almost complex structures on $\mathbb {F}_\varTheta$ also correspond to $K_\varTheta$-invariant complex structures on $\mathfrak {n}_\varTheta ^-$.

Let $J:\mathfrak {n}_\varTheta ^-\longrightarrow \mathfrak {n}_\varTheta ^-$ be a complex structure and assume it is only M-invariant. Since $K_\varTheta =M(K_\varTheta )_0$ we have that J is also $K_\varTheta$-invariant if and only if J commutes with the elements in $(K_\varTheta )_0$, or equivalently, ${{\mathrm{ad}}}_XJ=J{{\mathrm{ad}}}_X$ for all $X\in \mathfrak {k}_\varTheta$ (because of connectedness).

Proposition 1

Let $\mathbb {F}_{\varTheta }$ be a real flag manifold associated to a split real form. Then a necessary and sufficient condition for the existence of a M-invariant complex structure $J:T_{b_{\varTheta }}\mathbb {F}_{\varTheta }\rightarrow T_{b_{\varTheta }}\mathbb {F}_{\varTheta }$ is that the amount of elements in each M-equivalence class $\left[ \alpha \right]$ in $\varPi ^{-}{\setminus } \langle \varTheta \rangle ^-$ is even.

In this case, the M-invariant complex structures are given by direct sums of invariant structures on the subspaces $V_{\left[ \alpha \right] }=\sum _{\beta \sim _{M}\alpha }\mathfrak {g}_{\beta }\subset \mathfrak {n} _{\varTheta }^{-}$. In a subspace $V_{\left[ \alpha \right] }$ the set of M -invariant structures is parametrized by $\mathrm {Gl}(d,\mathbb {R})/\mathrm { Gl}(d/2,\mathbb {C})$, where $d=\dim V_{\left[ \alpha \right] }$.

Proof

If $\alpha \in \varPi ^{-}\backslash \langle \varTheta \rangle ^-$ then $\mathfrak {g} _{\alpha }\in \mathfrak {n}_{\varTheta }^{-}$ and $\dim \mathfrak {g}_{\alpha }=1$ (because $\mathfrak {g}$ is a split real form). The subspace $J\mathfrak {g} _{\alpha }\subset \mathfrak {n}_{\varTheta }^{-}$ is different of $\mathfrak {g} _{\alpha }$ by 5 in Lemma 1 and the representation of M in $J \mathfrak {g}_{\alpha }$ is equivalent to the representation on $\mathfrak {g} _{\alpha }$. Lemma 2 implies that $J\mathfrak {g}_{\alpha }$ is contained in the subspace $V_{\left[ \alpha \right] }=\sum _{\beta \sim _{M}\alpha }\mathfrak {g }_{\beta }$. Applying the same argument to the roots $\beta$ that are M -equivalent to $\alpha$, we obtain $JV_{\alpha }=V_{\alpha }$. As $J^{2}=-1$, it follows that $\dim V_{\alpha }$ is even and, hence, the amount of roots M-equivalent to $\alpha$ is even. This proves that the condition is necessary.

To see the sufficiency take a M-equivalent class $[\alpha ]$ so that by assumption the subspace $V_{[\alpha ]}=\sum _{\beta \sim _{M}\alpha } \mathfrak {g}_{\beta }$ is even dimensional. Given $m\in M$ we have $\mathrm {Ad}\left( m\right) X=\pm X$ if X belongs to a root space $X\in \mathfrak {g}_{\beta }$. In this equality, the sign does not change when $\beta$ runs through a M -equivalence class. It follows that $\mathrm {Ad}\left( m\right) =\pm 1$ on $V_{[\alpha ]}$. Hence any complex structure on $V_{[\alpha ]}$ is M-invariant. Taking direct sums of complex structures on the several $V_{ \left[ \alpha \right] }$ we get M-invariant complex structures on $T_{b_{\varTheta }}\mathbb {F}_{\varTheta }\simeq \mathfrak {n}_{\varTheta }^{-}$.

Finally the set of complex structures in a d-dimensional real space (d even) is parametrized by $\mathrm {Gl}(d,\mathbb {R})/\mathrm {Gl}(d/2,\mathbb {C })$. $\square$

We use the results in [15] to present in Table 1 all possible subsets $\varTheta \subset \varSigma$ for which the M-equivalence classes in $\varPi ^{-}{\setminus } \langle \varTheta \rangle ^-$ have an even amount of elements. Even though we do not give the explicit computations to construct this table, we present the M-equivalence classes for some cases in the followings sections.

Table 1 M-equivalence classes in $\varPi ^{-}{\setminus } \langle \varTheta \rangle ^-$ with even elements

Complex structures on $\mathbb {F}_\varTheta$ which are invariant under K are induced by $K_\varTheta$-invariant complex structures on the tangent space and, in particular, are M-invariant. Hence Proposition 1 and a simple inspection of Table 1 give the following result.

Proposition 2

Let $\mathbb {F}_{\varTheta }$ be a real flag manifold associated to a split real form. If $\mathbb {F}_{\varTheta }$ admits a K -invariant almost complex structure, then $\varTheta$ is in Table 1.

An invariant complex structure $J:\mathfrak {n}_\varTheta ^-\longrightarrow \mathfrak {n}_\varTheta ^-$ induced is integrable if the Nijenhuis tensor vanishes, that is if

$$\begin{aligned} N_J(X,Y):=[JX,JY]-[X,Y]-J[JX,Y]-J[X,JY]=0,\; \text{ for } \text{ all } X,Y \in \mathfrak {n}_\varTheta ^-. \end{aligned}$$

3 K-invariant complex structures on maximal flags

For a maximal flag manifold, the isotropy subgroup $\mathbb {K}_\varTheta$ is the centralizer of $\mathfrak {a}$ inside K, that is, $\mathbb {K}_\varTheta =M$. Hence Proposition 1 solves the question of existence of almost complex structures, remaining only integrability to be solved. The main result of this section is the following.

Proposition 3

The maximal real flag $\mathbb {F}$ associated to a split real form admits a K-invariant almost complex structure if and only if $\mathbb {F}$ is of type $A_3$, $B_2$, $G_2$, $C_l$ for even l and $D_l$ for $l\ge 4$. None of these structures is integrable.

Proof

By Proposition 1, a maximal flag $\mathbb {F}$ admits an M-invariant almost complex structure if and only if it appears in Table 1.

Recall that a M-invariant almost complex structure in F is given by an endomorphism $J:\mathfrak {n}^-\longrightarrow \mathfrak {n}^-$ which is a sum of almost complex structures $J_{[\alpha ]}:V_{[\alpha ]}\longrightarrow V_{[\alpha ]}$, for $\alpha \in \varPi ^-$. We address integrability of these structures by fixing one of these $J:\mathfrak {n}^-\longrightarrow \mathfrak {n}^-$ and we study case by case.

Notice that if $V_{[\alpha ]}$ is two dimensional with basis $\mathcal B$, then the matrix of $J_{[\alpha ]}$ in $\mathcal B$ is

$$\begin{aligned} \left( \begin{matrix} a&{}\frac{-(1+a^2)}{c}\\ c&{}-a \end{matrix}\right) , \text{ with } \, a,c\in \mathbb {R}, \,c\ne 0. \end{aligned}$$

(1)

• $Case A_3$ The M-equivalence classes of negative roots are:

  $$\begin{aligned} \{\lambda _2-\lambda _1,\lambda _4-\lambda _3\},\ \{\lambda _3-\lambda _1,\lambda _4-\lambda _2\}\ \text{ e } \{\lambda _4-\lambda _1,\lambda _3-\lambda _2\}. \end{aligned}$$

  Thus for $i=2,3,4$, $\dim V_{[\lambda _{i}-\lambda _1]}=2$ and it is spanned by $\{E_{i1},E_{st}\}$ with $s>t$, $\{s,t\}\cap \{i,1\}=\emptyset$ and $\{s,t\}\cup \{i,1\}=\{1,\ldots ,4\}$; here $E_{jk}$ is the $4\times 4$ matrix with 1 in the jk entry and zero elsewhere. For $i=2,3,4$, let $a_i,c_i\in \mathbb {R}$ such that $J|_{V_{[\lambda _{i}-\lambda _1]}}$ in this basis has the following form

  $$\begin{aligned} \left( \begin{matrix} a_i&{}\frac{-(1+a_i^2)}{c_i}\\ c_i&{}-a_i \end{matrix}\right) ,\quad c_i\ne 0. \end{aligned}$$

  Explicit computations give

  $$\begin{aligned} N_J(E_{21},E_{31})= & {} (c_3-c_2)c_4E_{32}+(c_2a_3-a_2c_3+a_4(c_3-c_2))E_{41},\\ N_J(E_{21},E_{41})= & {} c_4(a_3-a_2)E_{31}+c_4(c_2+c_3) E_{42}. \end{aligned}$$

  These two equations cannot be zero simultaneously since $c_i\ne 0$. Thus the Nijenhuis tensor does not vanish and J is not integrable.

• $Case B_2$ The M-equivalence classes of negative roots are

  $$\begin{aligned} \{\lambda _2-\lambda _1,-\lambda _2-\lambda _1 \}\ \text{ e } \{-\lambda _1,-\lambda _2\}. \end{aligned}$$

  Let $X_{21}$, $Y_{21}$, $X_{1}$ and $X_{2}$ be elements of a Weyl basis generating $\mathfrak {g}_{\lambda _2-\lambda _1}$, $\mathfrak {g} _{-\lambda _2-\lambda _1}$, $\mathfrak {g}_{-\lambda _1}$ and $\mathfrak {g} _{-\lambda _2}$, respectively. Thus J verifies

  $$\begin{aligned} \begin{array}{ll} JX_{21} =a_{21}X_{21}+c_{21}Y_{21}, &{} JX_{1} =a_1X_1+c_{1}X_{2}, \\ JY_{21}=-(1+a_{21}^2)X_{21}/c_{21}-a_{21}Y_{21}, &{}JX_{2}= -(1+a_1^2)X_{1}/c_{1}-a_1X_2, \end{array} \end{aligned}$$

  with $c_1,c_{21}\ne 0$.

  Let $m=m_{\lambda _2-\lambda _1,-\lambda _2}\ne 0$ be the corresponding coefficient in the Weyl basis, that is, $[X_{21},X_2]=m X_{1}$. Then

  $$\begin{aligned} \begin{array}{ccl} N_J(X_{21},X_{1}) &{} = &{} [JX_{21},JX_{1}]- [X_{21},X_{1}]-J[X_{21},JX_{1}] -J[JX_{21},X_{1}] \\ &{} = &{} -m c_{1}^2X_{2} +mc_1(a_{21}-a_1)X_1 \end{array} \end{aligned}$$

  which is never zero since $mc_1^2\ne 0$. Therefore, J is not integrable.

• $Case C_4$ The M-equivalence classes are:

  $$\begin{aligned}&\{\pm \,\lambda _2-\lambda _1,\pm \,\lambda _4-\lambda _3\}, \{\pm \,\lambda _3-\lambda _1,\pm \,\lambda _4-\lambda _2\}, \{\pm \,\lambda _4-\lambda _1,\pm \,\lambda _3-\lambda _2\},&\\&\{-2\lambda _i:\ i=1,\ldots , 4\}.&\end{aligned}$$

  Notice that $\dim V_{[2\lambda _1]}=\dim V_{[\lambda _{i}-\lambda _1]}=4$ for $i=2,3,4$. Let $(a_{ij})_{ij}$, $(b_{ij})_{ij}$, $(c_{ij})_{ij}$ be the matrices corresponding to $J|_{V_{[\lambda _2-\lambda _1]}}$, $J|_{V_{[\lambda _3-\lambda _1]}}$, $J|_{V_{[\lambda _4-\lambda _1]}}$, respectively, in a Weyl basis of $\mathfrak {n}^-$.

  Then $N_J(X_{-\lambda _2-\lambda _1},X_{-2\lambda _2})=0$ and $N_J(X_{-\lambda _4-\lambda _3},X_{-2\lambda _4})=0$ imply $a_{12}=a_{34}=0$ and moreover $a_{14}^2+a_{24}^2\ne 0$ because otherwise $X_{-\lambda _4-\lambda _3}$ would be an eigenvector of J. Analogously we obtain $b_{12}=b_{34}=c_{12}=c_{34}=0$ and $b_{14}^2+b_{24}^2\ne 0$, $c_{14}^2+c_{24}^2\ne 0$.

  With these conditions, $N_J(X_{-\lambda _2-\lambda _1},X_{-2\lambda _4})=0$ imply $a_{32}=0$ and $a_{42}\ne 0$. Similar computations give $b_{32}=c_{32}=0$ and $b_{42}\ne 0$, $c_{42}\ne 0$. Now $J^2=-1$ imply $a_{14}=b_{14}=c_{14}=0$.

  All this account to $N_J(X_{\lambda _2-\lambda _1},X_{-\lambda _3-\lambda _1})=0$ and $N_J(X_{\lambda _2-\lambda _1},X_{-\lambda _4-\lambda _1})=0$ only if, respectively, $a_{31}=c_{42}$ and $a_{31}=-c_{42}$. This clearly cannot hold since $c_{42}\ne 0$.

• $Case C_l$, l even and $l\ge 6$. The M-equivalence classes are

  $$\begin{aligned} \{\pm \lambda _s- \lambda _i\}, \;1\le i<s\le l,\ \text{ and } \ \{2\lambda _1, \ldots ,2\lambda _l\}. \end{aligned}$$

  Let $X_{si}$, $Y_{si}$ and $X_{j}$ be the generators of the roots spaces $\mathfrak {g}_{\lambda _s-\lambda _i}$, $\mathfrak {g }_{-\lambda _s-\lambda _i}$ and $\mathfrak {g}_{-2\lambda _j}$, respectively, corresponding to a Weyl basis. In this case, we have $\dim V_{[\lambda _{s}-\lambda _i]}=2$ while $\dim V_{[2\lambda _1]}=l$, even. Thus $JX_{1} =\sum _{j=1}^l b_j X_{j}$ and for $s=1, \ldots ,l$ we have

  $$\begin{aligned} JX_{s1} =a_{s1}X_{s1}+c_{s1}Y_{s1}, \quad JY_{s1}=-\frac{(1+a_{s1}^2)}{c_{s1}}X_{s1}-a_{s1}Y_{s1}, \quad c_{s1}\ne 0 . \end{aligned}$$

  We compute the Nijenhuis tensor on the vectors $X_1$ and $X_{s1}$, for $s=2,\ldots ,l$. Denote $m=m_{\lambda _s-\lambda _1,-2\lambda _s}\ne 0$, then we get

  $$\begin{aligned} N_J(X_{s1},X_{1})= & {} [JX_{s1},JX_{1}]- [X_{s1},X_{1}]-J[X_{s1},JX_{1}] -J[JX_{s1},X_{1}] \\= & {} [a_{s1}X_{s1}+c_{s1}Y_{s1},\sum _{j=1}^l b_j X_{j}]-b_s m JY_{s1}\\= & {} a_{s1}b_s m Y_{s1}-b_s m (-\frac{(1+a_{s1}^2)}{c_{s1}}X_{s1}-a_{s1}Y_{s1})\\= & {} b_s m \frac{(1+a_{s1}^2)}{c_{s1}} X_{s1}+a_{s1}(b_sm+1) Y_{s1}. \end{aligned}$$

  Hence $N_J(X_{s1},X_{1})=0$ if and only if $b_{s}m=0$. Thus J integrable implies $b_s=0$ for $s=2,\ldots ,l$. and therefore $JX_1=b_1X_1$, which contradicts the fact that $J^2=-1$. Thus J is not integrable.

• $Case D_4$. The M-equivalence classes are

  $$\begin{aligned} \{\pm \lambda _2-\lambda _1,\pm \lambda _4-\lambda _3\}, \{\pm \lambda _3-\lambda _1,\pm \lambda _4-\lambda _1\}, \{\pm \lambda _4-\lambda _1,\pm \lambda _3-\lambda _2\}. \end{aligned}$$

  Clearly, $\dim V_{[\lambda _{i}-\lambda _1]}=4$ for $i=2,3,4$. We proceed as in the $C_4$ case. Let $(a_{ij})_{ij}$, $(b_{ij})_{ij}$, $(c_{ij})_{ij}$ be the matrices corresponding to $J|_{V_{[\lambda _2-\lambda _1]}}$, $J|_{V_{[\lambda _3-\lambda _1]}}$, $J|_{V_{[\lambda _4-\lambda _1]}}$, respectively, in a Weyl basis of $\mathfrak {n}^-$.

  By imposing $N_J(X_\gamma ,X_\delta )=0$ for $\gamma \in [\lambda _3-\lambda _1]$ and $\delta \in [\lambda _4-\lambda _1]$ we obtain that the matrix of $J|_{V_{[\lambda _4-\lambda _1]}}$ in the Weyl basis is

  $$\begin{aligned} \left( \begin{matrix} -b_{44} &{}\quad -b_{34} &{}\quad b_{24} &{}\quad b_{14}\\ -b_{43} &{}\quad -b_{33} &{}\quad b_{23} &{}\quad b_{13}\\ b_{42} &{}\quad b_{32} &{}\quad -b_{22} &{}\quad -b_{12}\\ b_{41} &{}\quad b_{31} &{}\quad -b_{21} &{}\quad -b_{11} \end{matrix}\right) .\end{aligned}$$

  With this, $N_J(X_{\lambda _2-\lambda _1},X_{-\lambda _4-\lambda _1})=0$, $N_J(X_{-\lambda _4-\lambda _3},X_{-\lambda _3-\lambda _1})=0$ and $N_J(X_{\lambda _4-\lambda _3},X_{-\lambda _3-\lambda _1})=0$ imply $b_{12}b_{32}=0$, $b_{12}b_{42}= 0$ and $b_{32}b_{42}= 0$. But we know that $a_{12}^2+a_{32}^2+a_{42}^2\ne 0$ since $X_{-\lambda _2-\lambda _1}$ is not an eigenvector. So we conclude that only one of $b_{12},b_{32},b_{42}$ is not zero. In each of the three cases, we obtain $a_{12}=a_{32}=a_{42}=0$ if $N_J$ vanishes, which cannot happen since $X_{-\lambda _2-\lambda _1}$ is not an eigenvector of J.

• $Case D_l$, $l\ge 5$. The M-equivalence classes are:

  $$\begin{aligned} \{\pm \lambda _j-\lambda _i\}, \quad 1\le i<j\le l. \end{aligned}$$

  For $1\le i<j\le l$, we have $\dim V_{[\lambda _{j}-\lambda _i]}=2$; let $X_{ij}$ be a generator of $\mathfrak {g}_{\lambda _i-\lambda _j}$ and let $Y_{ij}$ be a generator of $\mathfrak {g}_{\lambda _i+\lambda _j}$. Thus $V_{[\lambda _j-\lambda _i]}$ is spanned by $\{X_{ij},Y_{ij}\}$ and J in this basis has a matrix of the form

  $$\begin{aligned} \left( \begin{matrix} a_{ij}&{}\frac{-(1+a_{ij}^2)}{c_{ij}}\\ c_{ij}&{}-a_{ij} \end{matrix}\right) , \text{ where } c_{ij}\ne 0. \end{aligned}$$

  Conditions $N_J(X_{13},X_{23})=0$ and $N_J(X_{12},X_{23})=0$ imply

  $$\begin{aligned} \frac{m_{\lambda _1-\lambda _2,\lambda _2+\lambda _3}}{m_{\lambda _1-\lambda _2,\lambda _2-\lambda _3}}=\frac{c_{13}}{c_{23}}=-\frac{m_{\lambda _1-\lambda _3,\lambda _2+\lambda _3}}{m_{\lambda _1+\lambda _3,\lambda _2-\lambda _3}}. \end{aligned}$$

  (2)

  Now using Jacobi identity, we have

  $$\begin{aligned} \begin{array}{lll} 0 &{} = &{} [Y_{23},[X_{12},X_{23}]]-[[Y_{23},X_{12}],X_{23}]-[X_{12},[Y_{23},X_{23}]] \\ &{} = &{} m_{\lambda _1-\lambda _2,\lambda _2-\lambda _3}[Y_{23},X_{13}]+m_{ \lambda _2 +\lambda _3,\lambda _1-\lambda _2}[X_{23},Y_{13}] \\ &{} = &{} \left( m_{\lambda _1-\lambda _2,\lambda _2-\lambda _3}m_{\lambda _2+\lambda _3,\lambda _1- \lambda _3}+m_{\lambda _2+\lambda _3,\lambda _1-\lambda _2}m_{\lambda _2- \lambda _3,\lambda _1+\lambda _3}\right) Y_{12}. \end{array} \end{aligned}$$

  Thus

  $$\begin{aligned} m_{\lambda _1-\lambda _2,\lambda _2-\lambda _3}m_{\lambda _2+\lambda _3,\lambda _1- \lambda _3}= & {} -m_{\lambda _2+\lambda _3,\lambda _1-\lambda _2}m_{\lambda _2- \lambda _3,\lambda _1+\lambda _3}\\= & {} -m_{\lambda _1-\lambda _2,\lambda _2+\lambda _3}m_{\lambda _1+\lambda _3, \lambda _2-\lambda _3}, \end{aligned}$$

  and therefore

  $$\begin{aligned} \frac{m_{\lambda _1-\lambda _2,\lambda _2+\lambda _3}}{m_{\lambda _1-\lambda _2,\lambda _2-\lambda _3}}=\frac{m_{\lambda _1-\lambda _3,\lambda _2+\lambda _3}}{m_{\lambda _1+\lambda _3,\lambda _2-\lambda _3}}. \end{aligned}$$

  (3)

  This equation clearly contradicts (2) and hence J is not integrable.

• $Case G_2$ The M-equivalence classes are

  $$\begin{aligned} \{-\lambda _1, -2\lambda _2-\lambda _1\},\ \{- \lambda _2-\lambda _1,-3\lambda _2-\lambda _1\},\ \{ -\lambda _2,-3\lambda _2-2\lambda _1\}. \end{aligned}$$

  For $(i,j)\in \{(1,0),(0,1),(1,1)\}$, $\dim V_{[-i\lambda _1-j\lambda _2]}=2$. In a Weyl basis of $\mathfrak {n}^-$, we have that the matrix of $J|_{V_{[-i\lambda _1-j\lambda _2]}}$ has the form

  $$\begin{aligned} \left( \begin{matrix} a_{ij}&{}\frac{-(1+a_{ij}^2)}{c_{ij}}\\ c_{ij}&{}-a_{ij} \end{matrix}\right) , \text{ where } c_{ij}\ne 0. \end{aligned}$$

  Denote $m=m_{-(\lambda _1+\lambda _2),-\lambda _2}$ then

  $$\begin{aligned} N_J(X_{-\lambda _1-\lambda _2},X_{-\lambda _2})= & {} m(a_{11}a_{01}-1)X_{-\lambda _1-2\lambda _2}-m(a_{11}+a_{01})JX_{-\lambda _1-2\lambda _2}\\= & {} m\left( (a_{11}a_{01}-1)+a_{10}(a_{11}+a_{01})\right) X_{-\lambda _1-2\lambda _2}\\&\qquad +\,m(a_{11}+a_{01})\frac{1+a_{10}^2}{c_{10}}X_{-\lambda _1}. \end{aligned}$$

  Thus

  $$\begin{aligned} N_J(X_{-\lambda _1-\lambda _2},X_{-\lambda _2})=0 \;\Leftrightarrow \; a_{01}=-a_{11}\, \text{ and } \, a_{11}a_{01}=1, \end{aligned}$$

  and J is not integrable.

$\square$

4 K-invariant complex structures on intermediate flags

In this section, we study existence of invariant almost complex structures on intermediate flags $\mathbb {F}_\varTheta$, and their integrability. We obtain the classification of the flags admitting K-invariant complex structures, only some of type $C_l$ do, and also we describe the complex structures explicitly.

Proposition 2 states that if $\mathbb {F}_\varTheta =K/K_\varTheta$ with $\varTheta \ne \emptyset$ admits a K-invariant almost complex structure, then $\mathbb {F}_\varTheta$ is one of the following:

• of type $B_3$ and $\varTheta =\{\lambda _1-\lambda _2, \lambda _2-\lambda _3\}$;

• of type $C_l$ with $l=4$ and $\varTheta =\{\lambda _1-\lambda _2,\lambda _3-\lambda _4\}$ or $\varTheta =\{\lambda _3-\lambda _4,2\lambda _4\}$; or $l\ne 4$ and $\varTheta =\{\lambda _{d}-\lambda _{d+1},\ldots ,\lambda _{l-1}-\lambda _l,2\lambda _l\}$ for $d>1$, d odd.

• of type $D_l$ with $l=4$ and $\varTheta$ being one of: $\{ \lambda _{1}-\lambda _{2},\lambda _{3}-\lambda _{4}\}$, $\{ \lambda _{1}-\lambda _{2},\lambda _{3}+\lambda _{4}\}$, $\{ \lambda _{3}-\lambda _{4},\lambda _{3}+\lambda _{4}\}$, $\{\lambda _1-\lambda _{2}, \lambda _{2}-\lambda _{3},\lambda _{3}-\lambda _{4}\}$, $\{\lambda _1-\lambda _{2}, \lambda _{2}-\lambda _{3},\lambda _{3}+\lambda _{4}\}$, $\{\lambda _2-\lambda _{3}, \lambda _{3}-\lambda _{4},\lambda _{3}+\lambda _{4}\}$; or $l\ge 5$ and $\varTheta =\{\lambda _d-\lambda _{d+1},\cdots , \lambda _{l-1}-\lambda _{l},\lambda _{l-1}+\lambda _{l}\}$ for $1<d\le l-1$.

We analyse the cases B, C and D separately in the next subsections. We need to treat them separately since the isotropy representations differ significantly. In the three cases, we start by imposing necessary conditions for the flag to admit an invariant complex structure, which we shall describe in the next paragraph. We obtain that only in few cases one can obtain that type of structure.

Recall that K-invariant almost complex structures on $\mathbb {F}_\varTheta$ are in one-to-one correspondence with $K_\varTheta$-invariant maps $J:\mathfrak {n}_\varTheta ^-\longrightarrow \mathfrak {n}_\varTheta ^-$ such that $J^2=-1$.

Assume $J:\mathfrak {n}_\varTheta ^-\longrightarrow \mathfrak {n}_\varTheta ^-$ is $K_\varTheta$-invariant and $J^2=-1$. Then J is necessarily M-invariant since $M\subset K_\varTheta =M(K_\varTheta )_0$, hence by Proposition 1 we have

$$\begin{aligned} JV_{[\alpha ]}=V_{[\alpha ]} \text{ for } \text{ each } \alpha \in \varPi ^-\backslash \left\langle \varTheta \right\rangle ^-. \end{aligned}$$

(4)

In addition, J is also $(K_\varTheta )_0$ invariant and therefore

$$\begin{aligned}&{{\mathrm{ad}}}_{X}\, J=J\,{{\mathrm{ad}}}_X \text{ for } \text{ all } X\in \mathfrak {k}_\varTheta .&\end{aligned}$$

(5)

Assume $\mathfrak {n}_\varTheta ^-=W_1\oplus \cdots \oplus W_s$ is a decomposition on $K_\varTheta$-invariant and irreducible subspaces. If the representation on $W_i$ is not equivalent to the representation on any other $W_j$, $j\ne i$ then $JW_i=W_i$ because of Lemma 2. Notice that if this is the case $W_i$ is even dimensional. To the contrary, if $JW_i=W_j$ for some $i\ne j$, then the $K_\varTheta$ representation on these subspaces are equivalent, and J gives such an equivalence.

To address the non-existence of almost complex structures, we prove that some of the necessary conditions above cannot hold simultaneously. For the cases where an invariant almost complex structure does exist, we use these necessary conditions to build them explicitly. Notice that, for instance, if $J:\mathfrak {n}_\varTheta ^-\longrightarrow \mathfrak {n}_\varTheta ^-$ with $J^2=-1$ satisfying (4) and (5) is $K_\varTheta$ invariant.

We remark that the conditions related to the $K_\varTheta$ and $\mathfrak {k}_\varTheta$ representation on $\mathfrak {n}_\varTheta ^-$ are dealt with through a description of $\mathfrak {g}$ as a matrix Lie algebra. Integrability of the almost complex structure is established by computing the Nijenhuis tensor, as in the maximal flag case.

4.1 Flags of $B_3=\mathfrak {so}(3,4)$.

The set of simple roots is $\varSigma =\{\lambda _1-\lambda _2,\lambda _2-\lambda _3,\lambda _3\}$, and we take $\varTheta =\{\lambda _1-\lambda _2, \lambda _2-\lambda _3\}$ obtaining $\langle \varTheta \rangle =\pm \{\lambda _1-\lambda _2, \lambda _2-\lambda _3, \lambda _1-\lambda _3\}$. Notice that the flag is a six dimensional manifold. The M-equivalence classes outside of $\varTheta$ are: $\{\lambda _1+\lambda _2, \lambda _3\}$, $\{\lambda _1+\lambda _3, \lambda _2\}$ and $\{\lambda _2+\lambda _3, \lambda _1\}$. The compact subgroup $\left( K_{\varTheta }\right) _0$ is isomorphic to $\text{ SO }\text{( }3)$.

We consider the realization of $B_{3}=\mathfrak {so}(3,4)$ in real matrices of the type

$$\begin{aligned} \left( \begin{array}{ccc} 0 &{}\quad \beta &{}\quad \gamma \\ -\gamma ^{T} &{}\quad A &{}\quad B \\ -\beta ^{T} &{}\quad C &{}\quad -A^{T} \end{array} \right) , \end{aligned}$$

with A, B, C are $3\times 3$ matrices, $\beta ,\gamma$ $1\times 3$ matrices and $B+B^{T}=C+C^{T}=0$. Then, $\left( K_{\varTheta }\right) _{0}$ (respectively M ) is given by matrices of the form

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad g &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad g \end{array} \right) , \end{aligned}$$

with $g\in \text{ SO }\text{( }3)$ (respectively, g diagonal with entries $\pm 1$ and an even amount of $-1$ entries). The root space corresponding to the short root $\lambda _{1}$ is given by matrices where the components A, B, C and $\beta$ vanish and $\gamma$ is a multiple of $e_{1}=(1,0,0)$. The same holds for the roots $\lambda _{2}$ and $\lambda _{3}$ with $e_{2}=(0,1,0)$ and $e_{3}=(0,0,1)$, respectively. The root spaces corresponding to $\lambda _{i}+\lambda _{j}$ have B as unique non-vanishing component and it has the following form, depending on the long root:

$$\begin{aligned}&\lambda _{1}+\lambda _{2}:B=\left( \begin{array}{ccc} 0 &{}\quad -1 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{array} \right) \ \ \ \lambda _{1}+\lambda _{3}:B=\left( \begin{array}{ccc} 0 &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 0 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 \end{array} \right)&\\&\lambda _{2}+\lambda _{3}:B=\left( \begin{array}{ccc} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -1 \\ 0 &{}\quad 1 &{}\quad 0 \end{array} \right) .&\end{aligned}$$

The subspaces $V_{c}=\sum _{i}\mathfrak {g}_{\lambda _{i}}$ and $V_{l}=\sum _{i,j} \mathfrak {g}_{\lambda _{i}+\lambda _{j}}$ are both invariant subspaces under the adjoint representation of $K_{\varTheta }=M\cdot \text{ SO }\text{( }3)$. The representation of the $\mathrm {SO}\left( 3\right)$ on $V_{c}$ is isomorphic to canonical representation on $\mathbb {R}^{3}$, while the representation on $V_{l}$ is the adjoint representation. These two representations of $\text{ SO }\text{( }3)$ are isomorphic. In fact, an isomorphism is constructed via the identification of $\mathbb {R}^{3}$ with the imaginary quaternions $\mathbb {H}$: if $p,q\in \mathbb {H}$ then ${{\mathrm{ad}}}(q)p=[q,p]\in \mathrm {Im}\ \mathbb {H}$ and ${{\mathrm{ad}}}(q)\in \mathfrak {so}(3)$ that commutes with the representations of the $\text{ SO }\text{( }3)$. This isomorphism also commutes with the representations of M. Indeed, considering the basis $\{e_{1},e_{2},e_{3}\}=\{i,j,k\}\in \mathbb {R}^{3}=\mathrm {Im}\ \mathbb {H}$, we have

$$\begin{aligned}&\mathrm {ad}(i)=\left( \begin{array}{ccc} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad -2 \\ 0 &{}\quad 2 &{}\quad 0 \end{array} \right) , \; \mathrm {ad}(j)=\left( \begin{array}{ccc} 0 &{}\quad 0 &{}\quad 2 \\ 0 &{}\quad 0 &{}\quad 0 \\ -2 &{}\quad 0 &{}\quad 0 \end{array} \right)&\\&\text{ and }\;\; \mathrm {ad}(k)=\left( \begin{array}{ccc} 0 &{}\quad -2 &{}\quad 0 \\ 2 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{array} \right) .&\end{aligned}$$

The isomorphism $P: V_c\rightarrow V_l$ takes the root spaces $\mathfrak {g}_{\lambda _{1}}$, $\mathfrak {g}_{\lambda _{2}}$ and $\mathfrak {g}_{\lambda _{3}}$ to the root spaces $\mathfrak {g}_{\lambda _{2}+\lambda _{3}}$, $\mathfrak {g} _{\lambda _{1}+\lambda _{3}}$ and $\mathfrak {g}_{\lambda _{1}+\lambda _{2}}$ , respectively. In addition, it commutes with the representation of $(K_{\varTheta })_0$ and with the representations of M. Therefore, $P: V_c\rightarrow V_l$ commutes with the representation of $K_{\varTheta }$.

Proposition 4

The flag manifold $\mathbb {F}_\varTheta$ of $B_3$ with $\varTheta =\{\lambda _1-\lambda _2,\lambda _2-\lambda _3\}$ admits K-invariant almost complex structures and each of them is given by $J_a$ for some $a \ne 0$ where $J_a:\mathfrak {n}_\varTheta ^+\longrightarrow \mathfrak {n}_\varTheta ^+$ is defined by

$$\begin{aligned} J_a(X)=aP(X) \text{ if } X \in V_c,\quad J_a(X)=-a^{-1}P^{-1}(X) \text{ if } X\in V_l. \end{aligned}$$

These structures are not integrable.

Proof

We have $\mathfrak {n}_{\varTheta }^+=V_c\oplus V_l$ as $K_\varTheta$-invariant irreducible subspaces and because of the reasoning above, $J_a$ is indeed invariant by $K_{\varTheta }$. Thus, there is a one-parameter family of invariant almost complex structures on $\mathbb {F}_{\varTheta }$.

Furthermore, a $K_\varTheta$-invariant complex structure J on $\mathfrak {n}_\varTheta ^+$ is of this form. In fact, any $K_\varTheta$-invariant complex structure $J:\mathfrak {n}_\varTheta ^+\longrightarrow \mathfrak {n}_\varTheta ^+$ interchanges $V_c$ with $V_l$ by 4. in Lemma 1, since these are irreducible odd dimensional subspaces. Moreover, the subspaces $\mathfrak {g}_{\lambda _1+\lambda _2}\oplus \mathfrak {g}_{\lambda _3}$, $\mathfrak {g}_{\lambda _1+\lambda _3}\oplus \mathfrak {g}_{\lambda _2}$, $\mathfrak {g}_{\lambda _2+\lambda _3}\oplus \mathfrak {g}_{\lambda _1}$ are J-invariant because of (4). The fact that ${{\mathrm{ad}}}_XJ=J{{\mathrm{ad}}}_X$ for all $X\in \mathfrak {k}_\varTheta$ implies that J is actually a multiple of P.

These structures are never integrable. In fact, $[V_c,V_c]= V_l$ and $[V_l,\mathfrak {n}_\varTheta ^+]=0$. Thus, for $X,Y\in V_c$ we have $J_aX, J_aY \in V_l$ and therefore $N_{J_a}(X,Y)=-[X,Y]$. Hence $N_J$ never vanishes. $\square$

Remark 1

This flag $\mathbb {F}_\varTheta$ of type $B_3$ and $\varTheta =\{\lambda _1-\lambda _2, \lambda _2-\lambda _3\}$ is the Grassmannian of three-dimensional isotropic subspaces of $\mathbb {R}^{7}$, that is, three-dimensional subspaces in which the quadratic form matrix

$$\begin{aligned} \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1_{3\times 3} \\ 0 &{} 1_{3\times 3} &{} 0 \end{array} \right) \end{aligned}$$

vanishes. The proposition above gives a family of K-invariant almost complex structures on this flag which is parametrized by $\mathbb {R}\backslash \{0\}$.

4.2 Flags of $C_l=\mathfrak {sp}(l,\mathbb {R})$

The set of simple roots is $\varSigma =\{\lambda _1-\lambda _2,\ldots ,\lambda _{l-1}-\lambda _l,2\lambda _l\}$. For the analysis of these flags, we separate the case $l=4$ where the M-equivalence classes are different from the general case.

4.2.1 Case $C_l$, $l\ne 4$

Assume $l\ne 4$ and let $\varTheta =\{\lambda _{d+1}-\lambda _{d+2},\ldots ,\lambda _{l-1}-\lambda _{l},2\lambda _l\}$ with $d\in \{0,\cdots ,l\}$ and d even. Notice that $\varTheta$ gives a Dynkin sub-diagram $C_p$ of $C_l$ with $p=l-d$, thus $\mathfrak {k}_{\varTheta }$ is the maximal compact subalgebra of $\mathfrak {sp}(p,\mathbb {R})$, that is, $\mathfrak {k} _{\varTheta }\simeq \mathfrak {u}(p)$.

The M-equivalence classes in $\varPi ^+\backslash \left\langle \varTheta \right\rangle ^+$ are

$$\begin{aligned} \{\lambda _i-\lambda _{j},\lambda _i+\lambda _{j}\},\, 1\le i\le d,\ i<j\le l,\; \text{ and } \;\{2\lambda _{1}, \ldots , 2\lambda _{d}\}. \end{aligned}$$

For each positive root $\alpha$ denote $\mathfrak {t}_\alpha =(\mathfrak {g}_{\alpha }\oplus \mathfrak {g}_{-\alpha })\cap \mathfrak {k}$. Then $\mathfrak {k}=\mathfrak {k}_\varTheta \oplus \mathfrak {m}_\varTheta$ where $\mathfrak {k}_\varTheta$ is the vector space sum of $\mathfrak {t}_\alpha$ where $\alpha$ runs in $\left\langle \varTheta \right\rangle ^+$ and

$$\begin{aligned} \mathfrak {m}_\varTheta =\sum _{1\le i\le d, i<j\le l}\mathfrak {t}_{\lambda _i-\lambda _j} \oplus \mathfrak {t}_{2\lambda _1}\oplus \cdots \oplus \mathfrak {t}_{2\lambda _d} \end{aligned}$$

is a reductive complement of $\mathfrak {k}_\varTheta$.

The invariant and irreducible subspaces of $\mathfrak {m}_\varTheta$ by the $K_{\varTheta }$ action were described in [15, Section 5.3] and we present them below. Define

$$\begin{aligned} R= & {} \{\lambda _i\pm \lambda _j: 1\le i<j\le d\}\cup \{2\lambda _i: 1\le i\le d \}.\\ \varPi _i= & {} \{\lambda _i\pm \lambda _j: d+1\le j\le l\},\quad i=1\ldots ,d, \end{aligned}$$

and let $W_R=\sum _{\alpha \in R}\mathfrak {k}_{\alpha }$ and $W_i=\sum _{\alpha \in \varPi _i}\mathfrak {k}_{\alpha }$, $i=1,\ldots ,d$. We have

$$\begin{aligned} \mathfrak {m}_{\varTheta }=W_R\oplus \sum _{i=1}^{d}W_i \end{aligned}$$

(6)

and the subspaces above are M-invariant.

If $\alpha \in R$ and $\beta \in \varTheta$, then $\pm \alpha \pm \beta$ is never a root so $[Y,X]=0$ for any $Y\in \mathfrak {k} _{\varTheta }$ and $X\in W_R$. Thus $\text{ Ad }\text{( }g)X=X$ for any $g \in (K_{\varTheta })_0$, since $(K_{\varTheta })_0$ is connected, and therefore $W_R$ is invariant by $\text{ Ad }\text{( } K_{\varTheta })$.

Each subspace $W_i$ is $K_\varTheta$ invariant and is irreducible subspace and the respective representations are not equivalent if $i\ne j$ (see [15, Lemma 5.11]). We make use of the following isomorphism between the compact algebra $\mathfrak {k}$ and $\mathfrak {u}(l)$ given by

$$\begin{aligned} \left( \begin{array}{cc} A &{} -B \\ B &{} A \end{array} \right) \longmapsto A+iB, \quad A+A^T=B-B^T=0. \end{aligned}$$

The isomorphism takes $\mathfrak {k}_{\varTheta }$ in the algebra of anti-Hermitian matrices of the form

$$\begin{aligned} \mathfrak {k}_{\varTheta }: \left( \begin{array}{cc} 0 &{}\quad 0 \\ 0 &{}\quad X \end{array} \right) , \end{aligned}$$

(7)

being X a $p \times p$ matrix. Moreover, $W_R$ corresponds to the matrices of the form

$$\begin{aligned} W_R: \left( \begin{array}{cc} * &{}\quad 0 \\ 0 &{}\quad 0 \end{array} \right) , \end{aligned}$$

with $d \times d$ upper left block, while the subspace $W=\sum _{i=1}^{d}W_i$ corresponds to

$$\begin{aligned} W: \left( \begin{array}{cc} 0 &{}\quad -\overline{C}^T \\ C &{}\quad 0 \end{array} \right) , \end{aligned}$$

(8)

where C is $d\times p$. A subspace $W_j$ is given by those matrices C having non-vanishing entries in column j. The representation of $\mathfrak {k}_{\varTheta }$ in W is given by the adjoint action:

$$\begin{aligned} \left[ \left( \begin{array}{cc} 0 &{}\quad 0 \\ 0 &{}\quad X \end{array} \right) , \left( \begin{array}{cc} 0 &{}\quad -\overline{C}^T \\ C &{}\quad 0 \end{array} \right) \right] = \left( \begin{array}{cc} 0 &{}\quad \overline{C}^TX \\ XC &{}\quad 0 \end{array} \right) . \end{aligned}$$

Thus C having non-vanishing entries on column j implies that the same occurs for XC. So the subspaces $W_j$ are, in fact, invariant.

The image of $\mathfrak {k}_{\lambda _j-\lambda _k}$ in $\mathfrak {u}(l)$ through the isomorphism is generated by the real anti-symmetric matrix $A_{jk}=E_{jk}-E_{kj}$, while the image of $\mathfrak {k}_{\lambda _j+\lambda _k}$ is generated by the imaginary symmetric matrix $S_{jk}=i(E_{jk}+E_{kj})$.

Lemma 3

• 1. An almost complex structure $J: \mathfrak {m}_\varTheta \longrightarrow \mathfrak {m}_\varTheta$ is M-invariant if and only if J leaves invariant each subspace $\mathfrak {k}_{\lambda _i-\lambda _j}\oplus \mathfrak {k}_{\lambda _i+\lambda _j}$ and $\mathfrak {k}_{2\lambda _1}\oplus \cdots \oplus \mathfrak {k}_{2\lambda _{d}}$.

• 2. An M-invariant almost complex structure J is $K_{\varTheta }$-invariant if and only if for each $j=1,\ldots ,d$ there is some $\varepsilon _{j}=\pm 1$ such that $JA_{kj}=\varepsilon _{j}S_{kj}$ and $JS_{kj}=-\varepsilon _{j}A_{kj}$ for all $d<k\le l$.

Proof

Let $J: \mathfrak {m}_\varTheta \longrightarrow \mathfrak {m}_\varTheta$ be an isomorphism such that $J^2=-1$. From Proposition 1 and taking into account the M-equivalence classes given above we have that J is M-invariant if and only if it preserves each $\mathfrak {k}_{\lambda _i-\lambda _j}\oplus \mathfrak {k}_{\lambda _i+\lambda _j}$ and $\mathfrak {k}_{2\lambda _1}\oplus \cdots \oplus \mathfrak {k}_{2\lambda _{d}}$.

Now assume J is M-invariant, then J is $K_\varTheta$-invariant if and only if ${{\mathrm{ad}}}_YJ=J{{\mathrm{ad}}}_Y$ for all $Y\in \mathfrak {k}_\varTheta$.

Notice that J preserves each $W_i$ and $W_R$ in (6). Since $[X,Y]=0$ for all $Y\in \mathfrak {k}_\varTheta$, $X\in W_R$ we see that $J|_{W_R}$ is $K_\varTheta$-invariant. Recall that $W_i$ is spanned by $A_{ji}$, $S_{ji}$ with $d+1\le j\le l$.

Let $Y\in \mathfrak {k}_\varTheta$ be as in (7) with X imaginary diagonal matrix, i.e. $X=\mathrm {diag}(ia_1,\ldots ,ia_m)$. We have $\text{ ad }\text{( }Y)A_{kj}=a_jS_{kj}$ and $\text{ ad }\text{( } Y)S_{kj}=-a_jA_{kj}$ for some $a_j\in \mathbb {R}$. That is, $\mathfrak {k}_{\lambda _j-\lambda _k}\oplus \mathfrak {k}_{\lambda _j+\lambda _k}$ is invariant by $\text{ ad }\text{( }Y)$ and the matrix of $\text{ ad }\text{( }Y)$ in the basis $\{A_{kj},S_{kj}\}$ is

$$\begin{aligned} \left( \begin{array}{cc} 0 &{} -a_j \\ a_j &{} 0 \end{array} \right) . \end{aligned}$$

(9)

If we denote $J_{kj}$ the restriction of J to $\mathfrak {k}_{\lambda _j-\lambda _k}\oplus \mathfrak {k}_{\lambda _j+\lambda _k}$, for $k>j$ we see that $J_{kj}$ commutes with ${{\mathrm{ad}}}(Y)$ only when its matrix in the basis $\{A_{kj},S_{kj}\}$ is

$$\begin{aligned} J_{kj}=\varepsilon _{kj}\left( \begin{array}{cc} 0 &{}\quad -1 \\ 1 &{}\quad 0 \end{array} \right) \quad \text{ with } \varepsilon _{kj}=\pm 1. \end{aligned}$$

(10)

Fix $j\in \{1,\ldots ,d\}$ and let $l\ge s,t\ge d+1$, consider Z be as in Eq. (7) with $X=E_{ts}-E_{st}$ and let D be as in Eq. (8) with $C=E_{sj}$. Then

$$\begin{aligned} \text{ ad }\text{( }Z)D=\left( \begin{array}{cc} 0 &{}\quad -\overline{XC}^T \\ XC &{}\quad 0 \end{array} \right) , \qquad \text{ with } XC=E_{tj}. \end{aligned}$$

This implies that $\text{ ad }\text{( }Z)A_{sj}=A_{tj}$ and $\text{ ad }\text{( }Z)S_{sj}=S_{tj}$. Recall that J in the basis restricted to $\mathfrak {k}_{\lambda _j-\lambda _k}\oplus \mathfrak {k}_{\lambda _j+\lambda _k}$ has a matrix of the form in Eq. (10) in the appropriate basis. In order J to commute with ${{\mathrm{ad}}}(Z)$ above, we need

$$\begin{aligned} \varepsilon _{tj}S_{tj}=JA_{tj}=J {{\mathrm{ad}}}(Z) A_{sj}= {{\mathrm{ad}}}(Z)J A_{sj}={{\mathrm{ad}}}(Z)\varepsilon _{sj}S_{sj}=\varepsilon _{sj}S_{tj}. \end{aligned}$$

Thus $\varepsilon _{sj}=\varepsilon _{tj}$ for all $l\ge s,t\ge d+1$, and we define $\varepsilon _j$ this value. We have then $JA_{kj}=\varepsilon _{j}S_{kj}$ and $JS_{kj}=-\varepsilon _{j}A_{kj}$ for all $d<k\le l$.

Next we prove that this condition is sufficient for J to commute with the adjoint of elements in $\mathfrak {k}_\varTheta$. Indeed, for j, s, t as above, we only have left to verify that J commutes with matrices Z as in Eq. (7) with $X=i(E_{ts}+E_{st})$. We consider D as in Eq. (8) with $C=E_{sj}$, then $XC=iE_{tj}$ and we obtain $\text{ ad }\text{( }Z)A_{sj}=S_{tj}$. Likewise, if $C=iE_{sj}$, then $XC=-E_{tj}$ and thus $\text{ ad }\text{( }Z)S_{sj}=-A_{tj}$. Therefore,

$$\begin{aligned} \text{ ad }\text{( }Z)JA_{sj}=\varepsilon _{j}\text{ ad }\text{( }Z)S_{sj}=- \varepsilon _{j}A_{tj}=JS_{tj}=J\text{ ad }\text{( }Z)A_{sj} \end{aligned}$$

and

$$\begin{aligned} \text{ ad }\text{( }Z)JS_{sj}=-\varepsilon _{j}\text{ ad }\text{( }Z)A_{sj}=- \varepsilon _{j}S_{tj}=-JA_{tj}=J\text{ ad }\text{( }Z)S_{sj}. \end{aligned}$$

$\square$

Remark 2

The set of K invariant almost complex structures on the flags $\mathbb {F}_\varTheta$ in Lemma 3 is parametrized by $\text{ Gl }\text{( }d-1,\mathbb {R})/ \text{ Gl }\text{( }d-1/2,\mathbb {C})\times ( \mathbb {R}^2\cup \mathbb {R}^2 )^{d(d-1)}\times \mathbb {Z}_2^{d}.$

The component $\text{ Gl }\text{( }d-1,\mathbb {R})/ \text{ Gl }\text{( }d-1/2,\mathbb {C })$ corresponds to the complex structures on the space generated by long roots outside $\langle \varTheta \rangle ^+$. The component $(\mathbb {R}^2\cup \mathbb {R}^2 )^{d(d-1)}$ corresponds to the structures on the spaces generated by the roots $\{\lambda _j-\lambda _k,\lambda _j+\lambda _k\}$. The set $\mathbb {R}^2\cup \mathbb {R}^2$ is the disjoint union of the two copies of $\mathbb {R}^2$, that is $\text{ Gl }\text{( }2,\mathbb {R})/ \text{ Gl }\text{( }1, \mathbb {C})$. Finally, $\mathbb {Z}_2^{(d-1)}$ parametrizes the signs $\varepsilon _{j}$.

We introduce two technical lemmas which will lead to the determination of the integrable structures.

Lemma 4

Let J be a $K_\varTheta$-invariant almost complex structure. If J is integrable then for each $i,j\in \{1,\ldots ,d\}$, $j>i$, we have $JA_{ji}=c_{ji}S_{ji}$ and $JS_{ji}=-c_{ji}A_{ji}$, with $c_{ji}=\pm 1$.

Proof

Take $1\le i<j\le d$ then by M-invariance $JS_{ii}=\sum _k b_{ki}S_{kk}$ and

$$\begin{aligned} J|_{\{A_{ji},S_{ji}\}}= \left( \begin{matrix} a_{ji}&{}-\frac{1+a_{ji}^2}{c_{ji}}\\ c_{ji}&{}-a_{ji} \end{matrix}\right) \text{ where } c_{ji}\ne 0. \end{aligned}$$

We have

$$\begin{aligned}&N_J(S_{ii},A_{ji}) =A_{ji}\left( 2c_{ji}(b_{ii}-b_{ji})+2(b_{ji}-b_{ii})\frac{(1+a_{ji}^2)}{c_{ji}}-2c_{ji}a_{ji}-2a_{ji}\frac{(1+a_{ji}^2)}{c_{ji}}\right)&\\&\qquad \qquad +S_{ji}\left( 2a_{ji}(b_{ji}-b_{ii})+2+2a_{ji}(b_{ji}-b_{ii})-2(a_{ji}^2+c_{ji} ^2)\right)&\end{aligned}$$

Therefore,

$$\begin{aligned} N_J(S_{ii},A_{ji})=0\Leftrightarrow & {} \left\{ \begin{array}{l}(b_{ii}-b_{ji})(c_{ji}^2-1-a_{ji}^2)-a_{ji}(c_{ji}^2+1+a_{ji}^2)=0\\ c_{ji}^2=2a_{ji}(b_{ji}-b_{ii})+1-a_{ji}^2\end{array}\right. \\\Leftrightarrow & {} \left\{ \begin{array}{l} c_{ji}^2=2a_{ji}(b_{ji}-b_{ii})+1-a_{ji}^2\\ ((b_{ii}-b_{ji})^2+1)a_{ji}=0\end{array}\right. \\\Leftrightarrow & {} \left\{ \begin{array}{l} a_{ji}=0\\ c_{ji}=\pm 1\end{array}\right. . \end{aligned}$$

$\square$

Up to this moment, we have proved that if J is $K_\varTheta$-invariant and integrable then for each $j=1,\ldots ,d$:

• $JA_{kj}=c_{kj}S_{kj}$ and $JS_{kj}=-c_{kj}A_{kj}$ for $k=1,\ldots ,d$, $k\ne j$ and

• $JA_{kj}=\varepsilon _{j}S_{kj}$ and $JS_{kj}=-\varepsilon _{j}A_{kj}$ for all $k=d+1,\ldots , l$.

where $\varepsilon _j,c_{kj}\in \{\pm 1\}$. To simplify notation in the following lemma we write

$$\begin{aligned} JA_{kj}=\mu _{kj}S_{kj}, \quad JS_{kj}=-\mu _{kj}A_{kj} \text{ for } \text{ all } j=1,\ldots ,d, \,j< k\ne l. \end{aligned}$$

(11)

Lemma 5

Let J be a $K_\varTheta$-invariant (integrable) complex structure. Then for any triple $k>j>s$ such that $j,s\in \{1,\ldots ,d\}$ the possible values for $(\mu _{ks},\mu _{kj},\mu _{js})$ are:

$$\begin{aligned} (\mu _{ks},\mu _{ks},\mu _{ks}), \;(\mu _{ks},-\mu _{ks},\mu _{ks})\, \text{ and } (\mu _{ks},\mu _{ks},-\mu _{ks})., \quad \mu _{ks}=\pm 1. \end{aligned}$$

In particular, if $\varepsilon _j=-\varepsilon _s$ then $c_{js}=\varepsilon _s$.

Proof

By Eq. (11) we obtain

$$\begin{aligned} 0\,=\,N_J(A_{kj},A_{ks})= & {} \left( 1+\mu _{kj}\mu _{js}- \mu _{kj}\mu _{ks}-\mu _{ks}\mu _{js}\right) A_{js}\\= & {} \left( (\mu _{kj}-\mu _{ks})\mu _{js}+( \mu _{ks}-\mu _{kj})\mu _{ks}\right) A_{js}\\= & {} \left( (\mu _{js}-\mu _{ks})\mu _{kj}+( \mu _{js}-\mu _{ks})\mu _{js}\right) A_{js}. \end{aligned}$$

From the second row of this equation, we see that $\mu _{kj}=-\mu _{ks}$ implies $\mu _{js}=\mu _{ks}$; while the third row implies $\mu _{kj}=-\mu _{js}=\mu _{ks}$ if $\mu _{js}=-\mu _{ks}$. We conclude then that the possible values for the triple $(\mu _{ks},\mu _{kj},\mu _{js})$ are: $(\mu _{ks},\mu _{ks},\mu _{ks})$, $(\mu _{ks},-\mu _{ks},\mu _{ks})$ and $(\mu _{ks},\mu _{ks},-\mu _{ks})$. $\square$

Proposition 5

Let $J:\mathfrak {m}_\varTheta \longrightarrow \mathfrak {m}_\varTheta$ be such that $J^2=-1$ and moreover it preserves $\mathfrak {k}_{2\lambda _1}\oplus \cdots \oplus \mathfrak {k}_{2l_d}$ and $JA_{kj}=\mu _{kj}S_{kj}$, $JS_{kj}=-\mu _{kj}A_{kj}$ for all $j=1,\ldots ,d$, $j<k\le d$, with $\mu _{kj}=\pm 1$.

Then J is $K_\varTheta$-invariant and integrable if and only if the following hold:

• for each $j=1,\ldots ,d$, $\mu _{kj}=\varepsilon _j$ for all $k=d+1,\ldots ,l$.

• for each triple $k>j>s$ such that $j,s\in \{1,\ldots ,d\}$ the coefficients $(\mu _{ks},\mu _{kj},\mu _{js})$ are one of the following:

  $$\begin{aligned} (\mu _{ks},\mu _{ks},\mu _{ks}), \;(\mu _{ks},-\mu _{ks},\mu _{ks})\, \text{ and } (\mu _{ks},\mu _{ks},-\mu _{ks}). \end{aligned}$$

  Conversely, any K-invariant complex structure on $\mathbb {F}_\varTheta$ is induced by J as above.

Proof

It is necessary for J to be M-invariant to preserve $\mathfrak {k}_{2\lambda _1}\oplus \cdots \oplus \mathfrak {k}_{2l_d}$ and $\mathfrak {k}_{\lambda _j-\lambda _k}\oplus \mathfrak {k}_{\lambda _j+\lambda _k}$. The conditions above are necessary as proved in Lemma 3 in order J to be $K_\varTheta$-invariant and Lemmas 4, 5 to be integrable. As seen there, such J verifies $N_J(S_{kk},A_{kj})=0$ $j=1,\ldots ,d$, $j<k\le l$ and $N_J(A_{kj},A_{ks})=0$ for each triple in the second item. To show that these conditions are sufficient, we have to show that i) $N_J(S_{kk},S_{kj})=0$, ii) $N_J(S_{kj},S_{ks})=0$, iii) $N_J(S_{kj},A_{ks})=0$ and iv) $N_J(S_{jj},S_{ss})=0$ for all $j>s\in \{1,\ldots ,d\}$ and $k>j>s$.

Clearly (iv) holds since these matrices are diagonal. Moreover, $N_J(A_{kj},A_{ks})=N_J(S_{kj},S_{ks})$ so (ii) also holds. Similar computations as in the proof of Lemma 4 give (i). Finally $N_J(S_{kj},A_{ks})=\left( -1-\mu _{kj}\mu _{js}+ \mu _{kj}\mu _{ks}+\mu _{ks}\mu _{js}\right) S_{js}$ so reasoning as in Lemma 5 one obtains (iii). $\square$

Example 1

We consider the flag $\mathbb {F}_{\varTheta }$ of $C_3$, with $\varTheta =\{2\lambda _3\}$. The component $W_R$ of tangent space at the origin of flag is given by sum of $\mathfrak {k}_{\alpha }$, $\alpha \in R$, and has the following form: $R=\{\lambda _1\pm \lambda _2\} \cup \{2\lambda _1,2\lambda _2\}$ . The components $W_j$ are determined by the sets of roots

$$\begin{aligned} \varPi _1 = \{\lambda _1\pm \lambda _3\},\quad \varPi _2 = \{\lambda _2\pm \lambda _3\}. \end{aligned}$$

Fix $\varepsilon _j=\pm 1$ $j=1,2$ $\nu =\pm 1$ such that

$$\begin{aligned} (\varepsilon _1,\varepsilon _2,\nu )\in \{(1,1,1),(-1,-1,-1),(1,-1,1),(-1,1,-1),(1,1,-1),(-1,-1,1)\}, \end{aligned}$$

and let $a_{11},c_{11}\in \mathbb {R}$ s.t. $c_{11}\ne 0$. The following table gives all $K_\varTheta$-invariant integrable complex structures J in $\mathbb {F}_\varTheta$.

Components	$K_{\varTheta }$-invariant complex structures
$W_1$	$JA_{31}=\varepsilon _{1}S_{31}$, $JS_{1}=-\varepsilon _{1}A_{31}$,
$W_2$	$JA_{32}=\varepsilon _{2}S_{32}$, $JS_{32}=-\varepsilon _{2}A_{32}$
$W_R$	$\begin{array}{l} JA_{21}=\nu S_{21}, JS_{21}=-\nu A_{21}, \\ JS_{11}=a_{11}S_{11}+c_{11}S_{22},\\ JS_{22}=-\frac{1+a_{11}^2}{c_{11}} S_{11}-a_{11}S_{22} \end{array}$

4.2.2 Case $C_4$

The M-equivalence classes of positive roots are

$$\begin{aligned} \{\lambda _1\pm \lambda _2,\lambda _3\pm \lambda _4\}, \;\{\lambda _1\pm \lambda _3,\lambda _2\pm \lambda _4\} \{\lambda _1\pm \lambda _4,\lambda _2\pm \lambda _3\},\;\{2\lambda _1, 2\lambda _2, 2\lambda _3, 2\lambda _4\}. \end{aligned}$$

Proposition 6

The real flag $\mathbb {F}_\varTheta$ of $C_4$ with $\varTheta = \{\lambda _1-\lambda _2, \lambda _3-\lambda _4\}$ does not admit K-invariant almost complex structures.

Proof

According to [15, Section 5.3], the $K_\varTheta$ irreducible components of $\mathfrak {n}_\varTheta ^-$ are given by

$$\begin{aligned}&\begin{array}{rclrcl} V_1 &{} = &{} \langle X_{2\lambda _1}-X_{2\lambda _2}, X_{-\lambda _2-\lambda _1}\rangle &{}V_4&{}=&{} \langle X_{2\lambda _3}+X_{2\lambda _4} \rangle ,\\ V_2&{}=&{}\langle X_{2\lambda _1}+X_{2\lambda _2} \rangle , &{} V_3 &{} = &{} \langle X_{2\lambda _3}-X_{2\lambda _4},X_{-\lambda _4-\lambda _3}\rangle \end{array}&\\&\begin{array}{rcl} V_5 &{} = &{} \langle X_{\lambda _3-\lambda _1}+X_{\lambda _4-\lambda _2},X_{\lambda _3-\lambda _2}-X_{\lambda _4-\lambda _1} \rangle \\ V_6&{}= &{} \langle X_{\lambda _3-\lambda _2}+X_{\lambda _4-\lambda _1}, X_{\lambda _4-\lambda _2}-X_{\lambda _3-\lambda _1} \rangle , \\ V_7 &{} = &{} \langle X_{-\lambda _3-\lambda _1}+X_{-\lambda _4-\lambda _2},X_{-\lambda _3-\lambda _2}-X_{-\lambda _4-\lambda _1} \rangle \\ V_8&{}= &{} \langle X_{-\lambda _3-\lambda _2}+X_{-\lambda _4-\lambda _1}, X_{-\lambda _4-\lambda _2}-X_{-\lambda _3-\lambda _1} \rangle , \end{array} \end{aligned}$$

where $X_{\alpha }$ is a generator of root space $\mathfrak {g}_{\alpha }$.

The components $V_2$, $V_5$ and $V_6$ are equivalent to the components $V_4$, $V_7$ and $V_8$, respectively. The subspaces $V_1$ and $V_3$ are neither equivalent between them nor to any other representation subspace.

Assume J is a $K_\varTheta$-invariant complex structure J on $\mathfrak {n}_\varTheta ^-$. Then $JV_1=V_1$ since it is irreducible and non-equivalent to any other representation subspace. Moreover, $V_{[-\lambda _2-\lambda _1]}=\mathfrak {g}_{-\lambda _2-\lambda _1}\oplus \mathfrak {g}_{-\lambda _4-\lambda _3}$ and J preserves this subspaces too because of its M-invariance. Therefore, $V_1\cap V_{[-\lambda _2-\lambda _1]}=\left\langle X_{-\lambda _2-\lambda _1}\right\rangle$ is an invariant subspace of J, which is a contradiction. So we conclude that no K-invariant complex structure exists in this case. $\square$

Fix $\varTheta =\{\lambda _3-\lambda _4,2\lambda _4\}$ for $C_4$. The $K_{\varTheta }$-irreducible components of $\mathfrak {m}_\varTheta$ are [15, Section 5.3]:

$$\begin{aligned}&\begin{array}{rclrcl} V_1 &{} = &{} \mathfrak {g}_{-2\lambda _1},&{}\quad V_3 &{} = &{} \mathfrak {g}_{\lambda _2-\lambda _1},\\ V_2 &{} = &{} \mathfrak {g}_{-2\lambda _2}, &{} \quad V_4 &{} = &{} \mathfrak {g}_{-\lambda _2-\lambda _1},\end{array}&\nonumber \\&\begin{array}{rcl} V_5 &{} = &{} \mathfrak {g}_{\lambda _3-\lambda _1} \oplus \mathfrak {g}_{-\lambda _3-\lambda _1}\oplus \mathfrak {g} _{\lambda _4-\lambda _1}\oplus \mathfrak {g}_{-\lambda _4-\lambda _1}, \\ V_6 &{} = &{} \mathfrak {g}_{\lambda _3-\lambda _2} \oplus \mathfrak {g}_{-\lambda _3-\lambda _2}\oplus \mathfrak {g} _{\lambda _4-\lambda _2}\oplus \mathfrak {g}_{-\lambda _4-\lambda _2}. \end{array} \end{aligned}$$

(12)

The components $V_1$ and $V_3$ are equivalent to, respectively, the components $V_2$ and $V_4$. The components $V_5$ and $V_6$ are not equivalent.

As in the previous section, we consider the isomorphism between $\mathfrak {k}$ and $\mathfrak {u}(4)$. Under this map, $\mathfrak {k}_\varTheta =\left\langle \{A_{43},S_{43},S_{33},S_{44}\}\right\rangle$ and

$$\begin{aligned} \mathfrak {m}_\varTheta =W_R\oplus \bigoplus _{ {\begin{matrix}j=1,2\\ k=3,4\end{matrix}}} W_{kj} \end{aligned}$$

where $W_R=W_R^1\oplus W_{21}$ with $W_R^1=\left\langle \{S_{11},S_{22}\}\right\rangle$ and $W_{kj}=\left\langle \{A_{kj},S_{kj}\}\right\rangle$.

Proposition 7

The flag manifold $\mathbb {F}_\varTheta$ of $C_4$ with $\varTheta =\{\lambda _3-\lambda _4,2\lambda _4\}$ admits K-invariant almost complex structures and each of them is induced by a map $J:\mathfrak {m}_\varTheta \longrightarrow \mathfrak {m}_\varTheta$ verifying

$$\begin{aligned} \begin{array}{rclrcll} JS_{11}&{}=&{}\nu _1 S_{22}, &{}JS_{22}&{}=&{}-\nu _1^{-1}S_{11}&{} \text{ with } \nu _1\ne 0,\\ JA_{21}&{}=&{}\nu _2 S_{21}, &{} JS_{21}&{}=&{}-\nu _2^{-1} A_{21} &{} \text{ with } \nu _2\ne 0,\\ JA_{kj}&{}=&{}\varepsilon _{j} S_{kj}, &{}JS_{kj}&{}=&{}-\varepsilon _{j}A_{kj} &{} \text{ for } k\in \{3,4\},\,j\in \{1,2\},\\ &{}&{}&{}&{}&{}&{} \text{ with } \varepsilon _{j}=\pm 1. \end{array} \end{aligned}$$

Such structure is integrable if and only if $\nu _2=\pm 1$ and $\nu _2=\varepsilon _1$ if $\varepsilon _2=-\varepsilon _1$.

Proof

We already know that $\mathbb {F}_\varTheta$ admits M-invariant almost complex structures and such J is the direct sum of almost complex structures in each $V_{[\alpha ]}$, $\alpha \in \varPi ^+\backslash \left\langle \varTheta \right\rangle ^+$. In this case, the M-equivalence classes are

$$\begin{aligned} \{\lambda _1-\lambda _2,\lambda _1+\lambda _2\},\; \{\lambda _1\pm \lambda _3,\lambda _2\pm \lambda _4\},\;\{\lambda _1\pm \lambda _4,\lambda _2\pm \lambda _3\},\;\{2\lambda _1, 2\lambda _2\}. \end{aligned}$$

So, in particular, $W_R^1$, $W_{21}$, $W_{31}\oplus W_{42}$ and $W_{32}\oplus W_{41}$ are J-invariant.

Moreover, since $V_5=W_{31}\oplus W_{41}$ and $V_6=W_{32}\oplus W_{42}$ in (12) are irreducible and non-equivalent, we have $JV_5=V_5$ and $JV_6=V_6$. Therefore, each $W_{kj}$, $k=3,4$, $j=1,2$ is invariant, since it can be described as an intersection of $V_{[\alpha ]}$ and $V_t$ for suitable root and index.

We proceed as in the general case $C_l$, $l \ne 4$ to show that J has the form given in the statement of the proposition.

For any $Y\in \mathfrak {k}_\varTheta$ and $Z\in W_R$, we have $[Y,Z]=0$ so J restricted to this subspace is also $\mathfrak {k}_\varTheta$-invariant. Let $Y=a_3 S_{33} +a_4 S_{44}\in \mathfrak {k}_\varTheta$, then ${{\mathrm{ad}}}_Y J=J{{\mathrm{ad}}}_Y$ implies that for $k=3,4$, $j=1,2$ the matrix of $J|_{W_{kj}}$ in the basis $\{A_{kj},S_{kj}\}$ is

$$\begin{aligned} \mu _{kj}\left( \begin{array}{cc} 0&{}\quad -1\\ 1&{}\quad 0 \end{array}\right) ,\, \mu _{kj}=\pm 1. \end{aligned}$$

Now let $Y=a_3 A_{43} +a_4 S_{43}\in \mathfrak {k}_\varTheta$ and let $Z\in W_{kj}$ with $k=3,4$, then ${{\mathrm{ad}}}_Y JZ =J{{\mathrm{ad}}}_Y Z$ holds if and only if $\varepsilon _{4j}=\varepsilon _{3j}$ for $j=1,2$. It is not hard to see that these conditions are also sufficient for J to be $K_\varTheta$-invariant.

To address integrability, notice that, as in the general case, we have

$$\begin{aligned} N_J(S_{11},A_{21})= & {} -2\left( \nu _1( \nu _2-\nu _2^{-1})A_{21}+(-1+\nu _2^{2})S_{21}\right) \\ N_J(A_{41},A_{42})= & {} \left( \varepsilon _1\varepsilon _2-1+(\varepsilon _1-\varepsilon _2)\nu _2^{-1}\right) A_{21} \end{aligned}$$

Therefore, J is integrable if $\nu _2=\pm 1$ and $\nu _2=\varepsilon _1$ in the case that $\varepsilon _1=\varepsilon _2$. One can check that these conditions are sufficient for J to be integrable. $\square$

4.3 Flags of $D_l=\mathfrak {so}(l,l)$

A root system is given by $\pm \lambda _i\pm \lambda _j$, $i\ne j$, and the corresponding set of simple roots is given by $\varSigma =\{\lambda _1-\lambda _2,\ldots , \lambda _{l-1}-\lambda _l,\lambda _{l-1}+\lambda _l\}$, $1\le i<j\le l$. The maximal compact subalgebra of $\mathfrak {so}(l,l)$ is $\mathfrak {k}\simeq \mathfrak {so}\left( l\right) \oplus \mathfrak {so}\left( l \right)$.

As in the $C_l$ case, we deal first with the case $D_l$ with $l\ge 5$ and later we address the case of $l=4$ because of the difference between the M-equivalence classes.

4.3.1 Case $D_l$, $l\ge 5$

We consider $\varTheta =\{\lambda _{d}-\lambda _{d+1},\ldots ,\lambda _{l-1}-\lambda _{l},\lambda _{l-1}+\lambda _{l}\}$, this gives a sub-diagram $D_p$ of $D_l$ with $p=l-d+1$, thus $\mathfrak {k}_\varTheta \simeq \mathfrak {so}\left( p\right) _{1}\oplus \mathfrak {so}\left( p\right) _{2}$. The set $\langle \varTheta \rangle$ of roots generated by $\varTheta$ is given by

$$\begin{aligned} \langle \varTheta \rangle =\{\pm \left( \lambda _{i}\pm \lambda _{j}\right) :d\le i<j\le l\}. \end{aligned}$$

The roots in $\varPi ^+\backslash \left\langle \varTheta \right\rangle ^+$ are

$$\begin{aligned} \lambda _i\pm \lambda _j \text{ with } 1\le i<j\le d, \quad \text{ and } \lambda _i\pm \lambda _j \text{ with } i=1,\ldots ,d-1,\,j=d, \ldots l. \end{aligned}$$

and the M-equivalence classes are $\{\lambda _i-\lambda _j,\lambda _i+\lambda _j\}$. Consider the subsets of roots in $\varPi ^+\backslash \left\langle \varTheta \right\rangle ^+$:

$$\begin{aligned} R= & {} \{\lambda _i\pm \lambda _j: 1\le i<j\le d\}\\ \varPi _i= & {} \{\lambda _i\pm \lambda _j:d\le j \le l \}, \quad i=1, \ldots ,d-1 \end{aligned}$$

and let $W_R=\sum _{\alpha \in R}\mathfrak {g}_\alpha$ and $W_i=\sum _{\alpha \in \varPi _i} \mathfrak {g}_{\alpha }$. Clearly we obtain

$$\begin{aligned} \mathfrak {n}_\varTheta ^+=W_R\oplus \sum _{i=1}^{d-1}W_i. \end{aligned}$$

(13)

The subspace $W_R$ is $K_\varTheta$ invariant and irreducible. Moreover, each $W_i$ decomposes as $W_i=V^1_i\oplus V_i^2$, where $V_i^j$ is irreducible $K_\varTheta$-invariant and the representations are not equivalent [15]. We present an explicit description of these subspaces.

A split real form of $D_{l}$ is $\mathfrak {so}\left( l,l\right)$ and it is represented by real matrices of the form

$$\begin{aligned} \left( \begin{array}{cc} A &{} B \\ C &{} -A^{T} \end{array} \right) , \ \text{ where } \ B+B^{T}=C+C^{T}=0. \end{aligned}$$

(14)

The algebra $\mathfrak {g}\left( \varTheta \right)$ generated by $\mathfrak {g} _{\alpha }$, $\alpha \in \langle \varTheta \rangle$ is given by matrices in Eq. (14) such that A, B and C have the form

$$\begin{aligned} \left( \begin{array}{cc} 0 &{}\quad 0 \\ 0 &{}\quad *\end{array} \right) , \end{aligned}$$

where the nonzero part is squared of size $p=l-d+1$. The Lie algebra $\mathfrak {g}\left( \varTheta \right)$ is of type $D_{p}$, isomorphic to $\mathfrak {so}\left( p,p\right)$.

The compact part $\mathfrak {k}$ inside $\mathfrak {so}(l,l)$ is given by the subset matrices in (14) having the form

$$\begin{aligned} \left( \begin{array}{cc} A &{} B \\ B &{} A \end{array} \right) , \ \text{ where } \ A+A^{T}=B+B^{T}=0. \end{aligned}$$

It is well known that $\mathfrak {k}$ decomposes as a sum of two ideals, both isomorphic to $\mathfrak {so}\left( l\right)$. The compact Lie algebra $\mathfrak {k}_\varTheta$ lies inside $\mathfrak {k}$ and also inside $\mathfrak {g}(\varTheta )$ and consists of matrices of the form

$$\begin{aligned} \left( \begin{array}{cc} A &{} B \\ B &{} A \end{array} \right) , \ \text{ with } A, B \in \left\langle \{E_{st}-E_{ts}: d\le s<t\le l\}\right\rangle . \end{aligned}$$

(15)

The Lie algebra $\mathfrak {k}_\varTheta$ also decomposes as a sum of two ideals, both isomorphic to $\mathfrak {so}\left( p\right)$, which are

$$\begin{aligned} \mathfrak {so}\left( p \right) _{1}= & {} \left\{ \left( \begin{array}{cc} A &{} A \\ A &{} A \end{array} \right) :A \in \left\langle \{E_{st}-E_{ts}: d\le s<t\le l\}\right\rangle \right\} ,\\ \mathfrak {so}\left( p\right) _{2}= & {} \left\{ \left( \begin{array}{cc} A &{} -A \\ -A &{} A \end{array} \right) :A \in \left\langle \{E_{st}-E_{ts}: d\le s<t\le l\}\right\rangle \right\} . \end{aligned}$$

Fix $i\in \{1,\ldots , d-1\}$ and denote $S_{i}=\{X=(a_{st})\in \mathfrak {gl}(l,\mathbb {R}):\, a_{st}=0 \text{ for } \text{ all } (st)\notin \{(ij): \,j=d,\ldots ,l\}\}$. For any $j=d, \ldots , l$ the root space $\mathfrak {g}_{\lambda _i-\lambda _j}$ is represented by matrices (14) where $A=E_{ij}$, $C=B=0$; meanwhile, $\mathfrak {g}_{\lambda _i+\lambda _j}$ is represented by the matrices of the above form where $B=E_{ij}-E_{ji}$, $A=C=0$. Thus $W_i$ is given by

$$\begin{aligned} \left( \begin{array}{cc} X &{} Y-Y^t \\ 0 &{} -X^{T} \end{array} \right) ,\quad \text{ where } X, Y \in S_i. \end{aligned}$$

(16)

For $Z\in S_i$ denote

$$\begin{aligned} X_Z=\left( \begin{array}{cc} Z &{} Z-Z^{T} \\ 0 &{} -Z^{T} \end{array} \right) ,\quad Y_Z=\left( \begin{array}{cc} Z&{} -Z+Z^{T} \\ 0 &{} -Z^{T} \end{array} \right) \end{aligned}$$

(17)

and define $V_i^1=\{X_Z:\, Z\in S_i\}$, $V_i^2=\{Y_Z:\,Z\in S_i\}$. Clearly, $V_i^1,V_i^2\subset W_i$. Moreover, a matrix as in (16) can be written as the sum of two matrices in (17) by taking $Z=(X+Y)/2$, $Z'=(X-Y)/2$. Thus we obtain $W_i=V_i^1\oplus V_i ^2$ and $\dim V_i^1=\dim V_i^2=l-d+1=p=|\varTheta |$.

We compute the $\mathfrak {k}_\varTheta$ action on $V_i^1$ and $V_i^2$: let $N\in \mathfrak {k}_\varTheta$ as in (15) and let $X_Z\in V_i^1$, $Y_Z\in V_i^2$ then $AZ=0=BZ$, $Z^TA=0=Z^TB$ so

$$\begin{aligned}{}[N,X_Z] = X_{-Z(A+B)}, \; \text{ and } \; [N,Y_Z] =Y_{-Z(A-B)}. \end{aligned}$$

This implies that $\mathfrak {so}(p)_2$ acts trivially on $V_i^1$ while for $N\in \mathfrak {so}(p)_1$ the action is $[N,X_Z]=X_{-2ZA}$. Similarly, $\mathfrak {so}(p)_1$ acts trivially on $V_i^2$ while for $N\in \mathfrak {so}(p)_2$ the action is $[N,X_Z]=X_{-2ZA}$. We conclude that the $\mathfrak {k}_\varTheta$ representation on $V_i^1$ is equivalent to the $\mathfrak {so}(p)\oplus \mathfrak {so}(p)$ representation on $\mathbb {R}^p$ where the action of $\mathfrak {so}(p)_1$ is the canonical and the action of $\mathfrak {so}(p)_2$ is trivial. Similarly, the $\mathfrak {k}_\varTheta$ representation on $V_i^2$ is equivalent to the $\mathfrak {so}(p)\oplus \mathfrak {so}(p)$ representation on $\mathbb {R}^p$ where the action of $\mathfrak {so}(p)_1$ is by zero and the action of $\mathfrak {so}(p)_2$ is the canonical one.

We keep $i=1, \ldots , d-1$ fixed. Let $s,t\in \{d\ldots l\}$, $s\ne t$ and consider $N_{st}^1$ being as in (15) with $A=E_{st}-E_{ts}$ and $B=A$ (i.e. $N\in \mathfrak {so}(p)_1$). Then

$$\begin{aligned} \left[ N_{st}^1,X_{E_{is}}\right] =-2X_{E_{it}} \text{ and } [N_{st}^1,X_{E_{it}}]=2X_{E_{is}}, \text{ while } [N_{st}^1,Y_{E_{ij}}]=0 \text{ for } \text{ all } j. \end{aligned}$$

(18)

Similarly, denote $N_{st}^2\in \mathfrak {so}(p)_2$ being the matrix in $\mathfrak {k}_\varTheta$ associated to $A=E_{st}-E_{ts}$ and $B=-A$, then

$$\begin{aligned} \left[ N_{st}^2,Y_{E_{is}}\right] =-2Y_{E_{it}} \text{ and } [N_{st}^2,Y_{E_{it}}]=2Y_{E_{is}}, \text{ while } [N_{st}^2,X_{E_{ij}}]=0 \text{ for } \text{ all } j. \end{aligned}$$

(19)

Having described the $\mathfrak {k}_\varTheta$ representation on $\mathfrak {n}_\varTheta ^+$ we can state:

Proposition 8

The real flags $\mathbb {F}_\varTheta$ of $D_l$ with $l \ge 5$ and $\varTheta \ne \emptyset$ do not admit $K_\varTheta$-invariant complex structures.

Proof

Assume $J:\mathfrak {n}_\varTheta ^+\longrightarrow \mathfrak {n}_\varTheta ^+$ is a $K_\varTheta$-invariant almost complex structure. As it is M-invariant and each subspace in (13) is sum of M-equivalence classes, we have that $JW_R=W_R$ and $JW_i=W_i$ for all $i=1, \ldots ,d-1$.

Recall that $W_i$ is not irreducible, for $i=1, \ldots , d-1$. Instead $W_i=V_i^1\oplus V_i^2$ where each of these subspace is invariant and irreducible by the $K_\varTheta$ action, and the induced representations are not equivalent [15]. By Lemma 2, we conclude that $V_i^1$, $V_i^2$ are J-invariant. In particular, $V_i^1$ and $V_i^2$ are even dimensional and thus p is even.

Fix $i=1,\ldots ,d-1$ and let $j\in \{d,\ldots ,l\}$. In the notation (17) one can see that $\mathfrak {g}_{\lambda _i-\lambda _j}\oplus \mathfrak {g}_{\lambda _i+\lambda _j}=\left\langle \{X_{E_{ij}},Y_{E_{ij}}\}\right\rangle$, which is a J-invariant subspace of $W_i$ because of the M-invariance of J. Thus $JX_{E_{ij}}=a_{ij}X_{E_{ij}}+c_{ij}Y_{E_{ij}}$ with $c_{ij}\ne 0$. For any $s\in \{d,\ldots ,l\}$, $s\ne j$ we apply (18) and obtain

$$\begin{aligned} {{\mathrm{ad}}}_{N_{sj}^1}\, JX_{E_{ij}}= & {} {{\mathrm{ad}}}_{N_{sj}^1}(a_{ij}X_{E_{ij}}+c_{ij} Y_{E_{ij}})=-2a_{ij}X_{E_{is}},\quad \text{ while } \\ J{{\mathrm{ad}}}_{N_{sj}^1}\, X_{E_{ij}}= & {} J(-2X_{E_{is}})=-2(a_{is} X_{E_{is}}+c_{is}Y_{E_{is}}), \end{aligned}$$

but $c_{is}\ne 0$, contradicting the $K_\varTheta$-invariance of J. $\square$

4.3.2 Case $D_4$

Now we proceed to the study of flags of $D_4$ with $\varTheta$ as in Table 1. The M-equivalence classes of positive roots in $D_4$ are:

$$\begin{aligned}&\{\lambda _{1}-\lambda _{2},\lambda _{1}+\lambda _{2},\lambda _{3}-\lambda _{4}, \lambda _{3}+\lambda _{4}\},\ \{\lambda _{1}-\lambda _{3},\lambda _{1}+\lambda _{3},\lambda _{2}-\lambda _{4}, \lambda _{2}+\lambda _{4}\}&\\&\{\lambda _{1}-\lambda _{4},\lambda _{1}+\lambda _{4},\lambda _{2}-\lambda _{3}, \lambda _{2}+\lambda _{3}\}.&\end{aligned}$$

As in the general case, we work with the split form $\mathfrak {so}(4,4)$. In what follows we denote by $X_{ij}=E_{i,j}-E_{l+j,l+i}$ a generator of $\mathfrak {g}_{\lambda _i-\lambda _j}$ and by $Y_{ij}=E_{i,l+j}-E_{j,l+i}$ a generator $\mathfrak {g}_{\lambda _i+\lambda _j}$, where $E_{i,j}$ is the $8\times 8$ matrix with 1 in the position ij and zeroes elsewhere.

The group M consists of $8\times 8$ diagonal matrices ${{\mathrm{diag}}}(\epsilon _1,\epsilon _2, \epsilon _3,\epsilon _4,\epsilon _1,\epsilon _2, \epsilon _3,\epsilon _4)$ where $\epsilon _i=\pm 1$ and $\epsilon _1\epsilon _2 \epsilon _3\epsilon _4 =1$, that is, there is an even amount of $-1$’s in the diagonal of matrices of M.

Proposition 9

The real flag manifold $\mathbb {F}_\varTheta$ of type $D_4$ with $\varTheta =\{\lambda _1-\lambda _2,\lambda _3-\lambda _4\}$ admits $K_\varTheta$ invariant almost complex structures. These structures are not integrable.

Proof

The following is the decomposition of $\mathfrak {n}_\varTheta ^+$ in $K_\varTheta$ invariant and irreducible subspaces

$$\begin{aligned} \mathfrak {n}_\varTheta ^+=\mathfrak {g}_{\lambda _1+\lambda _2}\oplus \mathfrak {g}_{\lambda _3+\lambda _4}\oplus \sum _{i=1}^4 V_i, \end{aligned}$$

where

$$\begin{aligned} V_1= & {} \langle X_{13}-X_{24},X_{14}+X_{23} \rangle \\ V_2= & {} \langle X_{13}+X_{24},X_{14}-X_{23}\rangle \\ V_3= & {} \langle Y_{13}-Y_{24},Y_{14}+Y_{23} \rangle \\ V_4= & {} \langle Y_{13}+Y_{24},Y_{14}-Y_{23}\rangle \end{aligned}$$

The map $T_{13}:V_1\longrightarrow V_3$ defined by $T_{13}(X_{13}-X_{24})=Y_{13}-Y_{24}$ and $T_{13}(X_{14}+X_{23})=Y_{14}+Y_{23}$ commutes with ${{\mathrm{ad}}}_{\mathfrak {k}_\varTheta }$. Moreover, the linear map $T_{24}:V_2\longrightarrow V_4$, verifying $T_{24}(X_{13}+X_{24})=Y_{23}+Y_{24}$ and $T_{24}(X_{14}-X_{23})=Y_{14}-Y_{23}$ commutes with the adjoints of $\mathfrak {k}_{\varTheta }$. Therefore, the $(K_\varTheta )_0$ representations on $V_1$ and $V_3$ and the representations on $V_2$ and $V_4$ are equivalent. One can see that these two different representations are not equivalent.

Assume $J:\mathfrak {n}_\varTheta ^+\longrightarrow \mathfrak {n}_\varTheta ^+$ is a $K_\varTheta$-invariant almost complex structure. The M-invariance implies that $V_{[\alpha ]}$ is M-invariant. For instance, $V_{[\lambda _1-\lambda _3]}=\left\langle \{ X_{13},Y_{13},X_{24},Y_{24}\}\right\rangle$ is invariant under J. Because of the $\mathfrak {k}_\varTheta$ representations described above, we have that $JV_1=V_1$ or $JV_1=V_3$. In the first case, we may have $X_{13}-X_{24}$ as an eigenvalue of J, which is not possible, so we obtain $JV_1=V_3$ and $J (X_{13}-X_{24})=c_1(Y_{13}-Y_{24})$ for some $c_1\ne 0$. By analogous reasoning, we obtain that J is as follows:

$$\begin{aligned} \begin{array}{rcl} JY_{12} &{} = &{} aY_{12}+cY_{34}, \\ JY_{34} &{} = &{} (1+a^2)Y_{12}/c-aY_{34}, \\ J(X_{13}-X_{24}) &{} = &{} c_1(Y_{13}-Y_{24}),\\ J(X_{14}+X_{23}) &{} = &{}c_2(Y_{14}+Y_{23}), \\ J(X_{13}+X_{24}) &{} = &{} c_3(Y_{13}+Y_{24}),\\ J(X_{14}-X_{23}) &{} = &{} c_4(Y_{14}-Y_{23}), \end{array} \end{aligned}$$

where $c_i, c\ne 0$. But $J{{\mathrm{ad}}}_X={{\mathrm{ad}}}_XJ$ for $X\in \mathfrak {k}_\varTheta$ implies $c_1=c_4$ and $c_2=c_3$. Direct computations show that this is M-invariant and $J{{\mathrm{ad}}}_X={{\mathrm{ad}}}_X J$ for all $X\in \mathfrak {k}_\varTheta$, therefore, a $K_\varTheta$-invariant almost complex structure.

Regarding integrability, it suffices to remark that, for instance, $N_J(Y_{12},X_{13}-X_{24})$ is never zero. $\square$

Proposition 10

The real flag manifold $\mathbb {F}_\varTheta$ of type $D_4$ with $\varTheta =\{\lambda _1-\lambda _2,\lambda _3+\lambda _4\}$ admits $K_\varTheta$ invariant almost complex structures. These structures are not integrable.

Proof

We proceed as in the previous proof. The following is a decomposition into $K_\varTheta$ invariant and irreducible subspaces

$$\begin{aligned} \mathfrak {n}_\varTheta ^+=\mathfrak {g}_{\lambda _1+\lambda _2}\oplus \mathfrak {g}_{\lambda _3-\lambda _4}\oplus \sum _{i=1}^4 V_i, \end{aligned}$$

where

$$\begin{aligned} V_1= & {} \langle X_{13}-Y_{24},Y_{14}+X_{23} \rangle \\ V_2= & {} \langle X_{13}+Y_{24},Y_{14}-X_{23}\rangle \\ V_3= & {} \langle Y_{13}-X_{24},X_{14}+Y_{23} \rangle \\ V_4= & {} \langle Y_{13}+X_{24},X_{14}-Y_{23}\rangle \end{aligned}$$

The subspace $V_1$ is $\mathfrak {k}_{\varTheta }$-equivalent to the subspace $V_3$ and the subspace $V_2$ is $\mathfrak {k}_{\varTheta }$-equivalent to the subspace $V_4$ through the following linear transformations $T_{13}:V_1\longrightarrow V_3$ and $T_{24}:V_2\longrightarrow V_4$, given by $T_{13}(X_{13}-Y_{24})=Y_{13}-X_{24}$, $T_{13}(Y_{14}+X_{23})=X_{14}+Y_{23}$, $T_{24}(X_{13}+Y_{24})=Y_{13}+X_{24}$ and $T_{24}(Y_{14}-X_{23})=X_{14}-Y_{23}$. The other representations are not $\mathfrak {k}_\varTheta$ equivalent.

Assume J is a $K_\varTheta$-invariant almost complex structure. As before, $JV_1=V_3$ and $JV_2=V_4$ and J verifies

$$\begin{aligned} \begin{array}{rcl} JY_{12} &{} = &{} aX_{34}+cY_{12},\\ JX_{34} &{} = &{} (1+a^2)X_{34}/c-aY_{12}, \\ J(X_{13}-Y_{24}) &{}=&{}c_1(Y_{13}-X_{24}), \\ J(Y_{14}+X_{23}) &{} = &{} c_2(X_{14}+Y_{23}),\\ J(X_{13}+Y_{24}) &{} = &{} c_3(Y_{13}+X_{24}),\\ J(Y_{14}-X_{23}) &{} =&{} c_4(X_{14}-Y_{23}), \end{array} \end{aligned}$$

J commuting with ${{\mathrm{ad}}}_X$, for $X\in \mathfrak {k}_\varTheta$ implies $c_1=c_2$ and $c_3=c_4$, and any such J commutes with all ${{\mathrm{ad}}}_X\in \mathfrak {k}_\varTheta$, so it is $(K_\varTheta )_0$ invariant. One can verify that J is also M-invariant.

Again, it is possible to see that $N_J(Y_{12},X_{13}-Y_{24})$ never vanishes. $\square$

Proposition 11

The real flag manifold $\mathbb {F}_\varTheta$ of type $D_4$ with $\varTheta =\{\lambda _3-\lambda _4,\lambda _3+\lambda _4\}$ admits $K_\varTheta$ invariant almost complex structures. These structures are not integrable.

Proof

The following is a decomposition into $(K_\varTheta )_0$ invariant and irreducible subspaces

$$\begin{aligned} \mathfrak {n}_\varTheta ^+=\mathfrak {g}_{\lambda _1-\lambda _2}\oplus \mathfrak {g}_{\lambda _1+\lambda _2}\oplus \sum _{i=1}^4 V_i, \end{aligned}$$

where

$$\begin{aligned} V_1= & {} \langle X_{13}+Y_{13},X_{14}+Y_{14} \rangle \\ V_2= & {} \langle X_{13}-Y_{13},X_{14}-Y_{14}\rangle \\ V_3= & {} \langle X_{23}+Y_{23},X_{24}+Y_{24} \rangle \\ V_4= & {} \langle X_{23}-Y_{23},X_{24}-Y_{24}\rangle \end{aligned}$$

The subspace $V_1$ is $\mathfrak {k}_{\varTheta }$-equivalent to the subspace $V_3$ and the subspace $V_2$ is $\mathfrak {k}_{\varTheta }$-equivalent to the subspace $V_4$. Indeed, we consider the linear transformations $T_{13}:V_1\longrightarrow V_3$ given by $T_{13}(X_{13}+Y_{13})=X_{24}+Y_{24}$ and $T_{13}(X_{14}+Y_{14})=-(X_{23}+Y_{23})$ and $T_{24}:V_2\longrightarrow V_4$ given by $T_{24}(X_{13}-Y_{13})=X_{24}-Y_{24}$ and $T_{24}(X_{14}-Y_{14})=-(X_{23}-Y_{23})$.

Any $(K_{\varTheta })_0$-invariant complex structure J is of form

$$\begin{aligned} \begin{array}{rcl} JX_{12} &{} = &{} aX_{12}+cY_{12},\\ JY_{12} &{} = &{} (1+a^2)X_{12}/c-aY_{12}, \\ J(X_{13}+Y_{13}) &{} = &{} c_1(X_{24}+Y_{24}), \\ J(X_{14}+Y_{14}) &{} = &{} -c_1(X_{23}+Y_{23}),\\ J(X_{13}-Y_{13}) &{} = &{} c_2(X_{24}-Y_{24}),\\ J(X_{14}-Y_{14}) &{} = &{} -c_2(X_{23}-Y_{23}), \end{array} \end{aligned}$$

Direct computations show that this is also M-invariant and therefore $K_\varTheta$-invariant. For such structure, $N_J(X_{12},X_{13}+Y_{13})$ never vanishes. $\square$

Proposition 12

The real flag manifolds $\mathbb {F}_\varTheta$ of type $D_4$ where $\varTheta$ is one of the following sets:

• $\varTheta _1=\{\lambda _1-\lambda _2,\lambda _2-\lambda _3,\lambda _3-\lambda _4\}$,

• $\varTheta _2=\{\lambda _1-\lambda _2,\lambda _2-\lambda _3,\lambda _3+\lambda _4\}$,

• $\varTheta _3=\{\lambda _2-\lambda _3,\lambda _3-\lambda _4,\lambda _3+\lambda _4\}$,

do not admit $K_\varTheta$-invariant almost complex structures.

Proof

Below we give the respective decompositions of $\mathfrak {n}_{\varTheta _i}^+$ in $K_\varTheta$ invariant and irreducible subspaces.

$$\begin{aligned} \mathfrak {n}_{\varTheta _1}^{-}= & {} \langle Y_{12}+Y_{34}, Y_{13}-Y_{24}, Y_{14}+Y_{23}\rangle \oplus \langle Y_{12}-Y_{34}, Y_{13}+Y_{24}, Y_{14}-Y_{23} \rangle .\\ \mathfrak {n}_{\varTheta _2}^{-}= & {} \langle Y_{12}+X_{34}, Y_{13}-X_{24}, X_{14}+Y_{23}\rangle \oplus \langle Y_{12}-X_{34}, Y_{13}+X_{24}, X_{14}-Y_{23} \rangle .\\ \mathfrak {n}_{\varTheta _3}^-= & {} \langle X_{12}+Y_{12}, X_{13}+Y_{13},X_{14}+Y_{14}\rangle \oplus \langle X_{12}-Y_{12}, X_{13}-Y_{13},X_{14}-Y_{14}\rangle . \end{aligned}$$

We see that each of them decomposes as a sum of two irreducible subspaces $V_1$ and $V_2$ which induce non-equivalent representations and such that $\dim V_1=\dim V_2=3$. Lemma 2 implies that any $K_\varTheta$-invariant complex structure preserves each of these irreducible components, which is not possible since these are odd dimensional. Therefore, $\mathbb {F}_{\varTheta _i}$ does not admit K-invariant almost complex structures for $i=1,2,3$. $\square$