# Complex Dirac structures with constant real index on flag manifolds

Cristian Ortiz and Carlos Varea  Cristian Ortiz
Instituto de Matemática e Estatística
Universidade de São Paulo,
Rua do Matão, 1010 - Cidade Universitária,
05508-090, São Paulo - São Paulo - Brasil.
<EMAIL_ADDRESS>Carlos Varea
Departamento de Matemática
Universidade Tecnológica Federal do Paraná,
Av. Alberto Carazzai, 1640 - Centro,
86300-000, Cornélio Procópio - Paraná - Brasil.
<EMAIL_ADDRESS>
###### Abstract.

In this paper we describe all invariant complex Dirac structures with constant
real index on a maximal flag manifold in terms of the roots of the Lie algebra
which defines the flag manifold. We also completely classify these structures
under the action of $B$-transformations.

###### 2010 Mathematics Subject Classification:

53D18; 22F30; 14M15

###### Contents

  1. 1 Introduction
  2. 2 Dirac structures and generalized complex structures
  3. 3 Flag manifolds
  4. 4 Complex Dirac structures on flag manifolds
    1. 4.1 Maximal isotropic subspaces of $(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}^{\ast}_{\alpha})\otimes\mathbb{C}$
    2. 4.2 Invariance
    3. 4.3 Involutivity
    4. 4.4 The real index
    5. 4.5 Real index zero
    6. 4.6 Real index two
    7. 4.7 Real index four
    8. 4.8 Real index six
  5. 5 Classification up to $B$-transformation
    1. 5.1 Classification on flag manifolds
  6. 6 Examples

##  1\. Introduction

A Dirac structure [8] on a manifold $M$ is a subbundle $L\subset TM\oplus
T^{\ast}M$ which is both maximal isotropic with respect to the natural
symmetric pairing defined on $TM\oplus T^{\ast}M$ and involutive with respect
to the Courant bracket. The natural pairing and the Courant bracket on
$TM\oplus T^{\ast}M$ extend naturally to the complexification $(TM\oplus
T^{\ast}M)\otimes\mathbb{C}$, allowing us to define complex Dirac structures.
If $L\subset(TM\oplus T^{\ast}M)\otimes\mathbb{C}$ is a complex Dirac
structure and $x\in M$, the real index of $L$ at $x$ is defined as
$\dim(L_{x}\cap\overline{L_{x}})$. A generalized complex structure [12, 10] on
$M$ can be seen as a complex Dirac structure whose real index at every point
is zero. Hence, generalized complex structures are particular instances of
complex Dirac structures with constant real index. These were introduced in
[1, 2] including not only generalized complex structures but also
presymplectic, transverse holomorphic and CR structures.

A diffeomorphism $\phi:M\to M$ lifts naturally to a vector bundle automorphism
$\tilde{\phi}:(TM\oplus T^{\ast}M)\otimes\mathbb{C}\to(TM\oplus
T^{\ast}M)\otimes\mathbb{C}$. One observes that if
$\tilde{\phi}(L_{x})=L_{\phi(x)}$, then real index of $L$ at $x\in M$
coincides with the real index at $\phi(x)$. In particular, if $G$ is a Lie
group acting on a manifold $M$ via $\phi_{g}:M\to M$ and $L\subseteq(TM\oplus
T^{\ast}M)\otimes\mathbb{C}$ is an invariant complex Dirac structure, i.e.
$\tilde{\phi_{g}}(L_{x})=L_{\phi_{g}(x)}$ for every $x\in M$ and $g\in G$,
then the real index of $L$ is constant along the orbits of the action. As a
consequence, every invariant complex Dirac structure on a homogeneous space
has constant real index.

The study of geometric structures which are invariant by the action of a Lie
group plays a central role in differential geometry, including: symplectic
actions and moment maps, isometric actions, holomorphic actions, among others.
Directly related to this work is the study of invariant geometric structures
on homogeneous spaces such as Kähler structures [3] and generalized complex
structures on both nilmanifolds [7] and flag manifolds [14, 15].

In this paper we are concerned with invariant complex Dirac structures on
maximal flag manifolds of complex semisimple Lie groups. Let $\mathfrak{g}$ be
a complex semisimple Lie algebra and let $G$ be a connected Lie group with Lie
algebra $\mathfrak{g}$. Then its maximal flag manifold is the homogeneous
space $\mathbb{F}=G/P$ where $P$ is a Borel subgroup of $G$ (minimal parabolic
subgroup). If $U$ is a compact real form of $G$, then $U$ acts transitively on
$\mathbb{F}$ so that we also have the homogeneous space $\mathbb{F}=U/T$ where
$T=P\cap U$ is a maximal torus of $U$. We are concerned with $U$-invariant
structures on $\mathbb{F}$. In general, invariant structures on a flag
manifold $\mathbb{F}$ are expected to be described in terms of roots of the
Lie algebra which defines the flag manifold, see for instance [4, 5, 13].

In [14], the second author studied invariant generalized complex structures on
$\mathbb{F}$. In this work, we show that the techniques of [14] naturally
extend to the setting of complex Dirac structures. The idea is to reduce the
problem at the origin $b_{0}$ of $\mathbb{F}$, then an invariant complex Dirac
structure on $\mathbb{F}$ will be completely described by a complex Dirac
structure $L$ on the vector space $T_{b_{0}}\mathbb{F}\oplus
T_{b_{0}}^{\ast}\mathbb{F}$ which is invariant under the adjoint
representation. We can decompose the Lie algebra of $U$ as
$\mathfrak{u}=\mathfrak{t}\oplus\mathfrak{m}$ where $\mathfrak{t}$ is the Lie
algebra of $T$ and $\mathfrak{m}=\sum_{\alpha}\mathfrak{u}_{\alpha}$ is the
sum of root spaces in $\mathfrak{u}$, that is,
$\mathfrak{u}_{\alpha}=(\mathfrak{g}_{\alpha}+\mathfrak{g}_{-\alpha})\cap\mathfrak{u}$
and $\mathfrak{g}_{\alpha}$ is the root space in the complex Lie algebra
$\mathfrak{g}$. Thus we can identify $T_{b_{0}}\mathbb{F}\oplus
T_{b_{0}}^{\ast}\mathbb{F}$ with two copies of $\mathfrak{m}$, namely
$T_{b_{0}}\mathbb{F}\oplus
T_{b_{0}}^{\ast}\mathbb{F}\approx\mathfrak{m}\oplus\mathfrak{m}^{\ast}$ where
$\mathfrak{m}^{\ast}=\sum_{\alpha}\mathfrak{u}_{\alpha}^{\ast}$. In the same
spirit of [14] we can decompose an invariant complex Dirac structure on
$\mathfrak{m}$ as a direct sum $L=\bigoplus_{\alpha}L_{\alpha}$ where each
$L_{\alpha}$ is an invariant complex Dirac structure on
$\mathfrak{u}_{\alpha}$.

Once we have an algebraic description of a complex Dirac structure, we proceed
to analyze the involutivity with respect to the Courant bracket. We obtain
algebraic conditions for a complex Dirac structure to be involutive depending
on a triple of positive roots $(\alpha,\beta,\alpha+\beta)$. Thus, in order to
simplify the computations we separate the cases according to the real index of
the corresponding triple of subspaces
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$.

As observed before, invariant complex Dirac structures on a homogeneous space
always have constant real index. Our main results are the following.

######  Theorem.

Let $L$ be an invariant complex Dirac structure on a flag manifold
$\mathbb{F}$. Then the real index of $L$ is constant and equal to $2k$ for
some $0\leq k\leq l$ where $l$ is the number of positive roots of the
associated Lie algebra $\mathfrak{g}$.

Conversely, if $l$ is the number of positive roots of the Lie algebra
$\mathfrak{g}$ which define the flag manifold $\mathbb{F}$, then there exists
an invariant complex Dirac structure on $\mathbb{F}$ with constant real index
equal to $2k$, with $k\in\\{0,1,\cdots,l\\}$.

######  Theorem.

Let $\mathfrak{g}$ be a semisimple Lie algebra and consider $\mathbb{F}$ the
maximal flag manifold associated to $\mathfrak{g}$. If $\mathfrak{g}$ has $l$
positive roots, then there exists an invariant Dirac structure on $\mathbb{F}$
with constant real index equal to $2k$, where $0\leq k\leq l$.

It is well-known that the vector bundle $(TM\oplus
T^{\ast}M)\otimes\mathbb{C}$ has additional symmetries which do not come from
a diffeomorphism of the base $M$, namely symmetries given by $B$-fields.
Bringing $B$-field invariance into the picture allows us to fully describe all
invariant complex Dirac structures up to $B$-fields.

######  Proposition.

Let $L_{\alpha}$ be an invariant complex Dirac structure on
$\mathfrak{u}_{\alpha}$. Then, up to $B$-transformations, we have only four
possibilities for $L_{\alpha}$.

  * (a)

$L_{\alpha}=(\mathfrak{u}_{\alpha})_{\mathbb{C}}$, that is, $L_{\alpha}$ is
the root space associated to the root $\alpha$;

  * (b)

$L_{\alpha}=(\mathfrak{u}^{\ast}_{\alpha})_{\mathbb{C}}$, that is,
$L_{\alpha}$ is the dual of the root space associated to the root $\alpha$;

  * (c)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}+\varepsilon_{\alpha}iS_{\alpha},A_{\alpha}^{\ast}+\varepsilon_{\alpha}iS^{\ast}_{\alpha}\\}$
with $\varepsilon_{\alpha}=\pm 1$, that is, $L_{\alpha}$ is the $i$-eigenspace
of an invariant generalized complex structure on $\mathfrak{u}_{\alpha}$ of
complex type.

  * (d)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}+\frac{i}{x_{\alpha}}A_{\alpha}^{\ast},S_{\alpha}+\frac{i}{x_{\alpha}}S^{\ast}_{\alpha}\\}$
with $x_{\alpha},a_{\alpha}\in\mathbb{R}$ and $x_{\alpha}\neq 0$, that is,
$L_{\alpha}$ is the $i$-eigenspace of an invariant generalized complex
structure on $\mathfrak{u}_{\alpha}$ of symplectic type.

In other words, if we denote by
$\textnormal{Dir}_{\mathbb{C}}(\mathfrak{u}_{\alpha})$ the set of all
invariant complex Dirac structures on $\mathfrak{u}_{\alpha}$, then

$\frac{\textnormal{Dir}_{\mathbb{C}}(\mathfrak{u}_{\alpha})}{B\textnormal{-fields}}=\\{(a),(b),(c),(d)\\}.$

The paper is organized as follows. In Section 2 we present a brief
introduction to Dirac structures and, as a particular case, we define
generalized complex structures. These are the basic concepts used throughout
the paper.

In Section 3 we introduce the concept of flag manifolds and fix notation that
will be useful in the paper.

In Section 4 we recall the classification of invariant generalized complex
structures on a maximal flag manifold as in [14]. Since generalized complex
structures are a special case of complex Dirac structures, we use this
classification as a motivation for the study of general invariant complex
Dirac structures on a maximal flag. Thus we give a description of the
invariant complex Dirac structures on $\mathbb{F}$. To obtain such a
description we first describe the maximal isotropic subspaces of
$\mathfrak{u}_{\alpha}$, for a fixed root $\alpha$. Then we analyze the
invariance and the real index of them and, finally, we study the involutivity
by considering different cases according to the real index.

In Section 5 we study the action of $B$-transformations on the space of
invariant complex Dirac structures found in Section 4, presenting a
classification of the possible invariant complex Dirac structures on flag
manifolds up to $B$-transformations.

Finally, Section 6 is devoted to give explicit examples of invariant complex
Dirac structures for the maximal flag manifolds associated to the Lie algebras
$\mathfrak{sl}(2,\mathbb{C})$ and $\mathfrak{sl}(3,\mathbb{C})$.

Acknowledgments: We would like to thank Elizabeth Gasparim and Fabricio
Valencia for several comments which have improved this work. C. Ortiz was
supported by Grant 2016/23746-6 Sao Paulo Research Foundation - FAPESP and by
the National Council of Research and Development CNPq-Brazil, Bolsa de
Produtividade em Pesquisa Grant 315502/2020-7. C. Varea was supported by the
Sao Paulo Research Foundation - FAPESP Grant 2020/12018-5.

##  2\. Dirac structures and generalized complex structures

In this section we introduce the basics on Dirac structures and generalized
complex structures. For more details see [6], [8] and [11].

Let $M$ be a smooth $n$-dimensional manifold. The vector bundle $TM\oplus
T^{\ast}M$ is endowed with a natural symmetric bilinear form of signature
$(n,n)$ defined by

$\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)),$

where $X+\xi,Y+\eta\in\Gamma(TM\oplus T^{\ast}M)$. The Courant bracket is the
skew-symmetric bracket defined on $\Gamma(TM\oplus T^{\ast}M)$ by

$[X+\xi,Y+\eta]=[X,Y]+\left(\mathcal{L}_{X}\eta-\mathcal{L}_{Y}\xi-\frac{1}{2}d\left(i_{X}\eta-
i_{Y}\xi\right)\right).$

######  Definition 2.1.

([8]) A Dirac structure on $M$ is a maximal isotropic subbundle $L\subseteq
TM\oplus T^{\ast}M$, whose space of sections is closed under the Courant
bracket.

######  Example 2.2.

Examples of Dirac structures are:

  * a)

Presymplectic structures: let $\omega\in\Omega(M)$ be a nondegenerate closed
$2$-form. Then

$L_{\omega}=\\{X+\iota_{X}\omega\ |\ X\in TM\\}$

is a Dirac structure.

  * b)

Poisson structures: let $\pi$ be a Poisson structure. Then

$\textnormal{Graph}(\pi)=\\{\iota_{\xi}\pi+\xi\ |\ \xi\in T^{\ast}M\\}$

is a Dirac structure.

It is important to notice that these notions can be extended to the
complexification $(TM\oplus T^{\ast}M)\otimes\mathbb{C}$ which allows us to
define a complex Dirac structure on $M$.

######  Definition 2.3.

A complex Dirac structure on $M$ is an involutive maximal isotropic subbundle
$L\subseteq(TM\oplus T^{\ast}M)\otimes\mathbb{C}$.

Given $L$ a complex Dirac structure on $M$ and $p\in M$, we note that $L|_{p}$
is a complex Dirac structure on $T_{p}M$.

######  Definition 2.4.

The real index of a complex Dirac structure $L|_{p}\subset(T_{p}M\oplus
T_{p}^{\ast}M)\otimes\mathbb{C}$ is given by
$\dim_{\mathbb{C}}(L|_{p}\cap\overline{L|}_{p})$.

A very special class of complex Dirac structures is given by those with real
index zero, called generalized complex structures. The condition
$L\cap\overline{L}=\\{0\\}$ allows us to describe a generalized complex
structure as an endomorphism $\mathcal{J}\colon TM\oplus T^{\ast}M\rightarrow
TM\oplus T^{\ast}M$ satisfying $\mathcal{J}^{2}=-1$ and
$\mathcal{J}^{\ast}=-\mathcal{J}$.

The basic examples of generalized complex structures on a manifold $M$ are:

$\mathcal{J}_{J}=\left(\begin{array}[]{cc}-J&0\\\
0&J^{\ast}\end{array}\right)\ \mathrm{and}\
\mathcal{J}_{\omega}=\left(\begin{array}[]{cc}0&-\omega^{-1}\\\
\omega&0\end{array}\right)$

where $J:TM\to TM$ is a complex structure and $\omega\in\Omega^{2}(M)$ is a
symplectic form.

Note that, given $L$ a maximal isotropic subbundle of $TM\oplus T^{\ast}M$ (or
its complexification) then $L$ is Courant involutive if and only if
$\operatorname{Nij}|_{L}=0$, where

$\operatorname{Nij}(A,B,C)=\frac{1}{3}\left(\langle[A,B],C\rangle+\langle[B,C],A\rangle+\langle[C,A],B\rangle\right),$

is the Nijenhuis operator.

##  3\. Flag manifolds

The aim of this section is to briefly review the concept of flag manifold. Let
$\mathfrak{g}$ be a semisimple Lie algebra and $G$ be a connected Lie group
with Lie algebra $\mathfrak{g}$. Let us consider a Cartan subalgebra
$\mathfrak{h}\subseteq\mathfrak{g}$ and $\Pi$ be a root system of
$\mathfrak{g}$ relative to $\mathfrak{h}$. For each root $\alpha\in\Pi$,
denote by $\mathfrak{g}_{\alpha}=\\{X\in\mathfrak{g}\ |\
\operatorname{ad}_{\mathfrak{h}}(X)=\alpha(\mathfrak{h})\cdot X\\}$ the root
space associated to $\alpha$. The Cartan–Killing form of $\mathfrak{g}$ is
defined by $\langle
X,Y\rangle=\operatorname{tr}(\operatorname{ad}_{X}\cdot\operatorname{ad}_{Y})$
and it restricts to a nondegenerate bilinear form on $\mathfrak{h}$. Given
$\alpha\in\mathfrak{h}^{\ast}$ we denote by $H_{\alpha}\in\mathfrak{h}$ the
element which is defined by $\alpha(\cdot)=\langle H_{\alpha},\cdot\rangle$,
and denote by $\mathfrak{h}_{\mathbb{R}}\subseteq\mathfrak{h}$ the real
subspace generated by $H_{\alpha}$, $\alpha\in\Pi$.

Let us choose a Weyl basis of $\mathfrak{g}$. Such a basis is given by root
vectors $X_{\alpha}\in\mathfrak{g}_{\alpha}$ satisfying $\langle
X_{\alpha},X_{-\alpha}\rangle=1$ and

$[X_{\alpha},X_{\beta}]=\left\\{\begin{array}[]{l}m_{\alpha,\beta}X_{\alpha+\beta},\
\textnormal{ if }\alpha+\beta\ \textnormal{ is a root}\\\ 0,\ \textnormal{
otherwise}\end{array}\right.$

with $m_{\alpha,\beta}\in\mathbb{R}$ obeying
$m_{-\alpha,-\beta}=-m_{\alpha,\beta}$.

Let $\Pi^{+}\subset\Pi$ be a choice of positive roots and denote by $\Sigma$
the corresponding simple root system. We define the Borel subalgebra
$\mathfrak{b}$ of $\mathfrak{g}$ by

$\mathfrak{b}=\mathfrak{h}\oplus\sum_{\alpha\in\Pi^{+}}\mathfrak{g}_{\alpha}.$

A subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ is parabolic if
$\mathfrak{b}\subseteq\mathfrak{p}$. In particular, when
$\mathfrak{p}=\mathfrak{b}$ we say that the parabolic subalgebra is minimal.

######  Definition 3.1.

Let $\mathfrak{p}$ be a minimal parabolic subalgebra. The maximal flag
manifold $\mathbb{F}$ associated to $\mathfrak{p}$ is the homogeneous space
$\mathbb{F}=G/P$, where $P\subseteq G$ is the parabolic subgroup generated by
$\mathfrak{p}$.

We also have a realization of a flag manifold using a compact real form of
$G$. Indeed, let $\mathfrak{u}$ be a compact real form of $\mathfrak{g}$, that
is, the real subalgebra

$\mathfrak{u}=\textnormal{span}_{\mathbb{R}}\\{i\mathfrak{h}_{\mathbb{R}},A_{\alpha},S_{\alpha}\
:\ \alpha\in\Pi^{+}\\},$

where $A_{\alpha}=X_{\alpha}-X_{-\alpha}$ and
$S_{\alpha}=i(X_{\alpha}+X_{-\alpha})$. Denote by $U=\exp\mathfrak{u}$ the
corresponding compact real form of $G$. Then the real representation
$\mathbb{F}=U/T$ is obtained by the transitive action of $U$ on $G/P$, where
the closed connected subgroup $T=P\cap U$ can be identified with a maximal
torus on $U$. The Lie algebra of $T$ is
$\mathfrak{t}=\mathfrak{p}\cap\mathfrak{u}$. Let $\mathfrak{m}$ be the
orthogonal complement (with respect to the Killing form) of $\mathfrak{t}$ in
$\mathfrak{u}$. Since $\mathbb{F}$ is a reductive homogeneous space, its
tangent space at the origin can be identified with $\mathfrak{m}$, that is,
$T_{b_{0}}\mathbb{F}\simeq\mathfrak{m}$ where $b_{0}=e\cdot T$ is the origin
of the flag manifold. Throughout the text we will consider only maximal flag
manifolds, so when we consider $\mathbb{F}$ a flag manifold it is understood
that $\mathbb{F}$ is a maximal flag manifold.

##  4\. Complex Dirac structures on flag manifolds

Our goal in this section is to describe the invariant complex Dirac structures
on a maximal flag manifold $\mathbb{F}=U/T$.

######  Definition 4.1.

Let $L$ be a complex Dirac structure on $\mathbb{F}$. We say that $L$ is
invariant if $L$ is invariant under action of the adjoint representation. That
is,

$(\operatorname{Ad}(g)\oplus\operatorname{Ad}^{\ast}(g))\cdot L\subseteq L,$

for every $g\in T$.

######  Example 4.2 (Invariant generalized complex structures on
$\mathbb{F}$).

In [14] was proved that an invariant generalized almost complex structure
$\mathcal{J}$ on $\mathbb{F}$ can be decomposed as
$\mathcal{J}=\bigoplus_{\alpha}\mathcal{J}_{\alpha}$ where
$\mathcal{J}_{\alpha}$ is the restriction of $\mathcal{J}$ to the subspace
$\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast}$, where
$\mathfrak{u}_{\alpha}=\textnormal{span}_{\mathbb{R}}\\{A_{\alpha},S_{\alpha}\\}$,
and each $\mathcal{J}_{\alpha}$ can assume only two forms:

  * a)

Complex type

$\mathcal{J}_{\alpha}=\pm\mathcal{J}_{0}=\pm\left(\begin{array}[]{cccc}0&-1&0&0\\\
1&0&0&0\\\ 0&0&0&-1\\\ 0&0&1&0\end{array}\right).$

  * b)

Symplectic type (up to $B$-transformation)

$\mathcal{J}_{\alpha}=\left(\begin{array}[]{cccc}a_{\alpha}&0&0&-x_{\alpha}\\\
0&a_{\alpha}&x_{\alpha}&0\\\ 0&-y_{\alpha}&-a_{\alpha}&0\\\
y_{\alpha}&0&0&-a_{\alpha}\end{array}\right)$

with $a_{\alpha},x_{\alpha},y_{\alpha}\in\mathbb{R}$ such that
$a_{\alpha}^{2}=x_{\alpha}y_{\alpha}-1$.

The integrability of the invariant generalized almost complex structures
described above depends on analyzing what happens with triples of the form
$(\mathcal{J}_{\alpha},\mathcal{J}_{\beta},\mathcal{J}_{\alpha+\beta})$
associated to triples of positive roots $(\alpha,\beta,\alpha+\beta)$.
Accordingly, we obtain that $\mathcal{J}$ is integrable if and only if for
each triple of positive roots $(\alpha,\beta,\alpha+\beta)$ we have that
$(\mathcal{J}_{\alpha},\mathcal{J}_{\beta},\mathcal{J}_{\alpha+\beta})$
corresponds to one of the rows of the following table

$\mathcal{J}_{\alpha}$ | $\mathcal{J}_{\beta}$ | $\mathcal{J}_{\alpha+\beta}$
---|---|---
complex | complex | complex
symplectic | complex ($\pm\mathcal{J}_{0}$) | complex ($\pm\mathcal{J}_{0})$
complex ($\pm\mathcal{J}_{0}$) | symplectic | complex ($\pm\mathcal{J}_{0}$)
complex ($\pm\mathcal{J}_{0})$ | complex ($\mp\mathcal{J}_{0})$ | symplectic
symplectic | symplectic | symplectic
Table 1. Integrability conditions.

where, in the first row, we can have one of the following combination of signs
$(\pm\mathcal{J}_{0},\pm\mathcal{J}_{0},\pm\mathcal{J}_{0})$,
$(\pm\mathcal{J}_{0},\mp\mathcal{J}_{0},\pm\mathcal{J}_{0})$ and
$(\pm\mathcal{J}_{0},\mp\mathcal{J}_{0},\mp\mathcal{J}_{0})$. Moreover, in the
last row, the following extra condition is required for integrability:

$\left\\{\begin{array}[]{cc}a_{\alpha+\beta}x_{\alpha}x_{\beta}-a_{\beta}x_{\alpha}x_{\alpha+\beta}-a_{\alpha}x_{\beta}x_{\alpha+\beta}=0\\\
x_{\alpha}x_{\beta}-x_{\alpha}x_{\alpha+\beta}-x_{\beta}x_{\alpha+\beta}=0.\end{array}\right.$
(1)

For more details see [14].

Our main goal is to generalize the results from invariant generalized complex
structures [14] to the setting of complex Dirac structures with constant real
index. To this end, we start by recalling the following result from [14].

######  Proposition 4.3.

Let $L$ be a subspace of $\mathfrak{m}\oplus\mathfrak{m}^{\ast}$. There is a
natural decomposition

$L=\sum_{\alpha}L\cap(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast}).$

Moreover, $L$ is a Dirac structure on $\mathfrak{m}$ if and only if
$L_{\alpha}:=L\cap(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast})$
is a Dirac structure on $\mathfrak{u}_{\alpha}$ for each positive root
$\alpha$.

Note that Proposition 4.3 can be extended to complex Dirac structures. So, for
our purposes it suffices to describe the invariant complex Dirac structures
$L_{\alpha}\subseteq\mathfrak{u}_{\alpha}\oplus\mathfrak{u}^{*}_{\alpha}$ for
each positive root $\alpha$. For that, first we will find all maximal
isotropic subspaces of
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}^{\ast}_{\alpha})\otimes\mathbb{C}$,
then we proceed to analyze the invariance, involutivity and real index of such
subspaces separately.

###  4.1. Maximal isotropic subspaces of
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}^{\ast}_{\alpha})\otimes\mathbb{C}$

Here we describe all maximal isotropic subspaces of
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}^{\ast}_{\alpha})\otimes\mathbb{C}$.

######  Proposition 4.4.

Let $L=\sum_{\alpha}L_{\alpha}$ be a maximal isotropic subspace of
$(\mathfrak{m}\oplus\mathfrak{m}^{\ast})\otimes\mathbb{C}$. Then, for each
root $\alpha\in\Pi$,
$L_{\alpha}\subseteq\mathfrak{u}_{\alpha}\oplus\mathfrak{u}^{*}_{\alpha}$ must
be one of the following:

  * (a)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}=(\mathfrak{u}_{\alpha})_{\mathbb{C}}$;

  * (b)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{-S_{\alpha}^{\ast},A_{\alpha}^{\ast}\\}=(\mathfrak{u}_{\alpha}^{\ast})_{\mathbb{C}}$;

  * (c)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},A_{\alpha}^{\ast}\\}$;

  * (d)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{S_{\alpha},-S_{\alpha}^{\ast}\\}$;

  * (e)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha}A_{\alpha}+b_{\alpha}S_{\alpha},a_{\alpha}A_{\alpha}^{\ast}+b_{\alpha}S_{\alpha}^{\ast}\\}$
with $a_{\alpha},b_{\alpha}\neq 0$;

  * (f)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha}A_{\alpha}+b_{\alpha}A_{\alpha}^{\ast},a_{\alpha}S_{\alpha}+b_{\alpha}S_{\alpha}^{\ast}\\}$
with $a_{\alpha},b_{\alpha}\neq 0$.

###### Proof.

Using that $\langle A_{\alpha},S_{\alpha}^{\ast}\rangle=-\langle
S_{\alpha},A^{\ast}_{\alpha}\rangle=1$ and $\langle
A_{\alpha},A_{\alpha}^{\ast}\rangle=\langle
A_{\alpha},S_{\alpha}\rangle=\langle
A_{\alpha},A_{\alpha}^{\ast}\rangle=\langle
S_{\alpha},S_{\alpha}\rangle=\langle
S_{\alpha},S_{\alpha}^{\ast}\rangle=\langle
A_{\alpha}^{\ast},A_{\alpha}^{\ast}\rangle=\langle
A_{\alpha}^{\ast},S_{\alpha}^{\ast}\rangle=\langle
S_{\alpha}^{\ast},S_{\alpha}^{\ast}\rangle=0$ we can check directly that the
subspaces $(a)$ – $(f)$ are in fact maximal isotropic subspaces on
$(\mathfrak{m}\oplus\mathfrak{m}^{\ast})\otimes\mathbb{C}$.

Now let us prove that these are the only possibilities for a maximal isotropic
subspace on
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast})\otimes\mathbb{C}$.
In fact, let $L_{\alpha}$ be a maximal isotropic subspace on
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast})\otimes\mathbb{C}$.
Since
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast})\otimes\mathbb{C}$
is 4-dimensional, it follows that $L_{\alpha}$ is 2-dimensional. Indeed,
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{X,Y\\}$ where
$X=a_{1}A_{\alpha}+b_{1}S_{\alpha}+c_{1}(-s_{\alpha}^{\ast})+d_{1}A_{\alpha}^{\ast}$
and
$Y=a_{2}A_{\alpha}+b_{2}S_{\alpha}+c_{2}(-s_{\alpha}^{\ast})+d_{2}A_{\alpha}^{\ast}$,
with $a_{i},b_{i},c_{i},d_{i}\in\mathbb{C}$. Therefore, we have $\langle
X,X\rangle=2(a_{1}c_{1}+b_{1}d_{1})$, $\langle
X,Y\rangle=a_{1}c_{2}+b_{1}d_{2}+c_{1}a_{2}+d_{1}b_{2}$ and $\langle
Y,Y\rangle=2(a_{2}c_{2}+b_{2}d_{2})$. Now, using these expressions and the
fact that $L_{\alpha}$ is isotropic, we can prove that $L_{\alpha}$ must be
one of those subspaces $(a)$ – $(f)$. The idea is to make some of the scalars
$a_{1},b_{1},c_{1},d_{1}$ equal to zero and analyze each case until the
possibilities are exhausted. Thus we have 14 possibilities to analyze. We will
do the first case and the other cases are similar.

Consider $b_{1}=c_{1}=d_{1}=0$, that is, $X=a_{1}\alpha$ with $a_{1}\neq 0$.
Then we have that $\langle X,Y\rangle=0$ if and only if $a_{1}c_{2}=0$,
therefore $c_{2}=0$. Then we can write
$Y=a_{2}A_{\alpha}+b_{2}S_{\alpha}+d_{2}A^{\ast}_{\alpha}$. Moreover, since
$\langle Y,Y\rangle=0$ we obtain $b_{2}d_{2}=0$. So, we get two subcases:

  * (i)

If $c_{2}=b_{2}=0$, we have $Y=a_{2}A_{\alpha}+d_{2}A^{\ast}_{\alpha}$. Thus,

$\displaystyle L_{\alpha}$ $\displaystyle=$
$\displaystyle\textnormal{span}_{\mathbb{C}}\\{a_{1}A_{\alpha},a_{2}A_{\alpha}+d_{2}A^{\ast}_{\alpha}\\}$
$\displaystyle=$
$\displaystyle\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},A^{\ast}_{\alpha}\\}.$

  * (ii)

If $c_{2}=d_{2}=0$, then $Y=a_{2}A_{\alpha}+b_{2}S_{\alpha}$. Hence,

$\displaystyle L_{\alpha}$ $\displaystyle=$
$\displaystyle\textnormal{span}_{\mathbb{C}}\\{a_{1}A_{\alpha},a_{2}A_{\alpha}+b_{2}S_{\alpha}\\}$
$\displaystyle=$
$\displaystyle\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}.$

Therefore, we conclude that subcases $(i)$ and $(ii)$ are exactly the cases
$(c)$ and $(a)$, respectively. ∎

###  4.2. Invariance

Remember that we are interested in the subspaces which are invariant with
respect to the isotropy representation. In our case, the isotropy
representation is equivalent to the adjoint representation which can be
identified with the torus action as we saw in [14]. So these subspaces
$L_{\alpha}$ listed in Proposition 4.4 are invariant if

$\left(\begin{array}[]{cccc}\cos\theta&-\sin\theta&0&0\\\
\sin\theta&\cos\theta&0&0\\\ 0&0&\cos\theta&-\sin\theta\\\
0&0&\sin\theta&\cos\theta\end{array}\right)\cdot L_{\alpha}\subset
L_{\alpha}.$

It is simple to see that the subspaces in $(a)$ and $(b)$ in Proposition 4.4
are invariant. Having into account that we are considering the ordered basis
$\mathcal{B}=\\{A_{\alpha},S_{\alpha},-S_{\alpha}^{\ast},A_{\alpha}^{\ast}\\}$,
let us check the other cases:

  * (c)

We have that $A_{\alpha}\in L_{\alpha}$, then

$\left(\begin{array}[]{cccc}\cos\theta&-\sin\theta&0&0\\\
\sin\theta&\cos\theta&0&0\\\ 0&0&\cos\theta&-\sin\theta\\\
0&0&\sin\theta&\cos\theta\end{array}\right)\cdot\left(\begin{array}[]{c}1\\\
0\\\ 0\\\ 0\end{array}\right)=\left(\begin{array}[]{c}\cos\theta\\\
\sin\theta\\\ 0\\\ 0\end{array}\right).$

But observe that $\cos\theta A_{\alpha}+\sin\theta S_{\alpha}\notin
L_{\alpha}$ when $\sin\theta\not=0$. Therefore, we conclude that $L_{\alpha}$
is not invariant.

  * (d)

The reasoning is analogous to case $(c)$ using the fact that $S_{\alpha}\in
L_{\alpha}$, concluding that $L_{\alpha}$ is not invariant.

  * (e)

Take $a_{1}A_{\alpha}+b_{1}S_{\alpha}\in L_{\alpha}$, then

$\left(\begin{array}[]{cccc}\cos\theta&-\sin\theta&0&0\\\
\sin\theta&\cos\theta&0&0\\\ 0&0&\cos\theta&-\sin\theta\\\
0&0&\sin\theta&\cos\theta\end{array}\right)\cdot\left(\begin{array}[]{c}a_{1}\\\
b_{1}\\\ 0\\\ 0\end{array}\right)=\left(\begin{array}[]{c}a_{1}\cos\theta-
b_{1}\sin\theta\\\ a_{1}\sin\theta+b_{1}\cos\theta\\\ 0\\\
0\end{array}\right).$

But $(a_{1}\cos\theta-
b_{1}\sin\theta)A_{\alpha}+(a_{1}\sin\theta+b_{1}\cos\theta)S_{\alpha}\in
L_{\alpha}$ if and only if there exists a scalar $k_{1}$ such that
$(a_{1}\cos\theta-
b_{1}\sin\theta)A_{\alpha}+(a_{1}\sin\theta+b_{1}\cos\theta)S_{\alpha}=k_{1}(a_{1}A_{\alpha}+b_{1}S_{\alpha})$,
in other words, the scalar $k$ satisfies

$\displaystyle\left\\{\begin{array}[]{l}a_{1}\cos\theta-
b_{1}\sin\theta=k_{1}a_{1}\\\
a_{1}\sin\theta+b_{1}\cos\theta=k_{1}b_{1}.\end{array}\right.$ (4)

Observe that, for $\theta=\pi/2$ we have

$\left\\{\begin{array}[]{l}-b_{1}=k_{1}a_{1}\\\
a_{1}=k_{1}b_{1}\end{array}\right.$

and, then we obtain $k_{1}^{2}=-1$ which implies that $k_{1}=\pm i$. Using the
fact that $a_{1}A_{\alpha}^{\ast}+b_{1}S_{\alpha}^{\ast}\in L_{\alpha}$ we
obtain an expression equivalent to Equation (4). Therefore, we have
$L_{\alpha}$ invariant when

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}\pm
iS_{\alpha},A_{\alpha}^{\ast}\pm iS_{\alpha}^{\ast}\\}.$

  * (f)

Using $a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha}\in L_{\alpha}$, we obtain

$\left(\begin{array}[]{cccc}\cos\theta&-\sin\theta&0&0\\\
\sin\theta&\cos\theta&0&0\\\ 0&0&\cos\theta&-\sin\theta\\\
0&0&\sin\theta&\cos\theta\end{array}\right)\cdot\left(\begin{array}[]{c}a_{1}\\\
0\\\ 0\\\ b_{1}\end{array}\right)=\left(\begin{array}[]{c}a_{1}\cos\theta\\\
a_{1}\sin\theta\\\ -b_{1}\sin\theta\\\ a_{1}\cos\theta\end{array}\right).$

and then $a_{1}\cos\theta A_{\alpha}+a_{1}\sin\theta
S_{\alpha}-b_{1}\sin\theta(-S_{\alpha}^{\ast})+a_{1}\cos\theta
A_{\alpha}^{\ast}\in L_{\alpha}$ if and only if there exist scalars
$k_{1},k_{2}$ such that $a_{1}\cos\theta A_{\alpha}+a_{1}\sin\theta
S_{\alpha}-b_{1}\sin\theta(-S_{\alpha}^{\ast})+a_{1}\cos\theta
A_{\alpha}^{\ast}=k_{1}(a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha})+k_{2}(a_{1}S_{\alpha}+b_{1}S_{\alpha}^{\ast})$,
or equivalently

$\displaystyle\left\\{\begin{array}[]{l}a_{1}\cos\theta=k_{1}a_{1}\\\
a_{1}\sin\theta=k_{2}a_{1}\\\ b_{1}\sin\theta=k_{2}b_{1}\\\
b_{1}\cos\theta=k_{1}b_{1}\end{array}\right.$ (9)

Since $a_{1},b_{1}\neq 0$ we have $k_{1}=\cos\theta$ and $k_{2}=\sin\theta$.
Since $a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha}\in L_{\alpha}$ we obtain an
expression equivalent to Equation (9). Therefore, $L_{\alpha}$ is invariant.

Thus, we have the following result.

######  Proposition 4.5.

The invariant maximal isotropic subspaces of
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast})\otimes\mathbb{C}$
are:

  1. (1)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}$;

  2. (2)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{-S^{\ast}_{\alpha},A^{\ast}_{\alpha}\\}$;

  3. (3)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}+i\varepsilon_{\alpha}S_{\alpha},A^{\ast}_{\alpha}+i\varepsilon_{\alpha}S_{\alpha}^{\ast}\\}$
with $\varepsilon_{\alpha}=\pm 1$;

  4. (4)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha},a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha}\\}$
with $a_{1},b_{1}\neq 0$.

###  4.3. Involutivity

Now we study the Courant involutivity of a maximal isotropic subspace of
$(\mathfrak{m}\oplus\mathfrak{m}^{\ast})\otimes\mathbb{C}$ . In [14] was
proved that the involutivity is given by analyzing the maximal isotropic
subspaces associated to triples of positive roots of the type
$(\alpha,\beta,\alpha+\beta)$, because for triples $(\alpha,\beta,\gamma)$,
where $\gamma\neq\alpha+\beta$, the Nijenhuis operator is immediately zero.
Since a maximal isotropic subspace of
$(\mathfrak{m}\oplus\mathfrak{m}^{\ast})\otimes\mathbb{C}$ can be written as
$L=\sum_{\alpha}L_{\alpha}$, we just need to analyze the possible combinations
for triples $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$, where $L_{\delta}$ can
be one of the four types listed on Proposition 4.5, for
$\delta\in\\{\alpha,\beta,\alpha+\beta\\}$.

Let $\alpha,\beta$ be roots such that $\alpha+\beta$ is a root as well. After
some computations, we have

$\displaystyle\operatorname{Nij}(A_{\alpha},S_{\beta},A_{\alpha+\beta}^{\ast})$
$\displaystyle=$
$\displaystyle-\operatorname{Nij}(A_{\alpha},A_{\beta},S_{\alpha+\beta}^{\ast})=\operatorname{Nij}(S_{\alpha},S_{\beta},S_{\alpha+\beta}^{\ast})=\operatorname{Nij}(S_{\alpha},A_{\beta},A_{\alpha+\beta}^{\ast})$
$\displaystyle=$
$\displaystyle-\operatorname{Nij}(A_{\alpha},S^{\ast}_{\beta},A_{\alpha+\beta})=\operatorname{Nij}(A_{\alpha},A^{\ast}_{\beta},S_{\alpha+\beta})=-\operatorname{Nij}(S_{\alpha},S^{\ast}_{\beta},S_{\alpha+\beta})$
$\displaystyle=$
$\displaystyle=-\operatorname{Nij}(S_{\alpha},A^{\ast}_{\beta},A_{\alpha+\beta})=-\operatorname{Nij}(A_{\alpha}^{\ast},S_{\beta},A_{\alpha+\beta})=\operatorname{Nij}(A_{\alpha}^{\ast},A_{\beta},S_{\alpha+\beta})$
$\displaystyle=$
$\displaystyle-\operatorname{Nij}(S_{\alpha}^{\ast},S_{\beta},S_{\alpha+\beta})=-\operatorname{Nij}(S_{\alpha}^{\ast},A_{\beta},A_{\alpha+\beta})\neq
0$

and the other cases are all zero. For more details see [14].

From the last we can conclude the following.

######  Proposition 4.6.

Let $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ be a triple where each of its
components is described by case (1) of Proposition 4.5, then the Nijenhuis
operator restricted to $L_{\alpha}\cup L_{\beta}\cup L_{\alpha+\beta}$
vanishes.

######  Proposition 4.7.

Let $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ such that at least two of them
are described by case (2) of Proposition 4.5, then the Nijenhuis operator
restricted to $L_{\alpha}\cup L_{\beta}\cup L_{\alpha+\beta}$ vanishes.

In what follows we will use the following notation:

######  Notation 4.8.

We will refer to $L_{\alpha}$ by its respective case number according to
Proposition 4.5. For example, let $(\alpha,\beta,\alpha+\beta)$ be a triple of
roots and consider $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ the complex Dirac
structures associated to these roots such that
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{A_{\beta}+i\varepsilon_{\beta}S_{\beta},A^{\ast}_{\beta}+i\varepsilon_{\beta}S_{\beta}^{\ast}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{a_{1}A_{\alpha+\beta}+b_{1}A^{\ast}_{\alpha+\beta},a_{1}S_{\alpha+\beta}+b_{1}S^{\ast}_{\alpha+\beta}\\}$.
Then we will refer to the triple $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ as
$(1,3,4)$, because $L_{\alpha}$ is associated to case (1), $L_{\beta}$ to case
(3) and $L_{\alpha+\beta}$ to case (4).

We will list below the other cases where the Nijenhuis operator is zero.
Fixing $L=L_{\alpha}\cup L_{\beta}\cup L_{\alpha+\beta}$, we have

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$ | $\operatorname{Nij}|_{L}=0$
---|---|---|---
1 | 3 | 4 | $\varepsilon_{\beta}=\varepsilon_{\alpha+\beta}$
1 | 4 | 4 | $\frac{a_{1}}{b_{1}}=\frac{a_{2}}{b_{2}}$
2 | 3 | 4 | $\varepsilon_{\beta}=\varepsilon_{\alpha+\beta}$
3 | 3 | 1 | $\varepsilon_{\alpha}=-\varepsilon_{\beta}$
3 | 3 | 2 | $\varepsilon_{\alpha}=-\varepsilon_{\beta}$
3 | 3 | 3 | $\varepsilon_{\alpha+\beta}-\varepsilon_{\alpha}-\varepsilon_{\beta}+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\alpha+\beta}=0$
3 | 3 | 4 | $\varepsilon_{\alpha}=-\varepsilon_{\beta}$
3 | 4 | 3 | $\varepsilon_{\alpha}=\varepsilon_{\alpha+\beta}$
4 | 4 | 1 | $\frac{a_{1}}{b_{1}}=-\frac{a_{2}}{b_{2}}$
4 | 4 | 4 | $a_{1}a_{2}b_{3}-a_{1}b_{2}a_{3}-b_{1}a_{2}a_{3}=0$
Table 2. Involutivity conditions

###  4.4. The real index

The next step is to determine the real index (see Definition 2.4) of each
Lagrangian subspace of
$(\mathfrak{u}_{\alpha}\oplus\mathfrak{u}_{\alpha}^{\ast})\otimes\mathbb{C}$.
For the first two cases of Proposition 4.5 it is simple to check that
$L_{\alpha}=\overline{L_{\alpha}}$ and therefore
$\dim(L_{\alpha}\cap\overline{L_{\alpha}})=2$. Moreover, note that the
subspaces of type (3) in Proposition 4.5 are the Lagrangian subspaces
described in [14] associated to generalized complex structures of complex
type, therefore $\dim(L_{\alpha}\cap\overline{L_{\alpha}})=0$. Then it remains
case (4) of Proposition 4.5 to be analyzed. We will see that in this case the
real index of $L_{\alpha}$ can be either zero or two.

Indeed, consider
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{1}A_{\alpha}+b_{1}A_{\alpha}^{\ast},a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha}\\}$
with $a_{1},b_{1}\neq 0$ and suppose $X\in
L_{\alpha}\cap\overline{L_{\alpha}}$. Then
$X=x_{1}(a_{1}A_{\alpha}+b_{1}A_{\alpha}^{\ast})+x_{2}(a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha})$
and also
$X=y_{1}(\overline{a_{1}}A_{\alpha}+\overline{b_{1}}A_{\alpha}^{\ast})+y_{2}(\overline{a_{1}}S_{\alpha}+\overline{b_{1}}S^{\ast}_{\alpha})$.
We need to analyze some cases:

  * 1.

If $x_{1}=0$ and $x_{2}\neq 0$, we have that $y_{1}=0$ and $y_{2}\neq 0$. Thus
$x_{2}(a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha})=y_{2}(\overline{a_{1}}S_{\alpha}+\overline{b_{1}}S^{\ast}_{\alpha})$,
that is,

$\left\\{\begin{array}[]{l}x_{2}a_{1}=y_{2}\overline{a_{1}}\\\
x_{2}b_{1}=y_{2}\overline{b_{1}}.\end{array}\right.$

From the system above we obtain
$\overline{\left(\frac{b_{1}}{a_{1}}\right)}=\frac{b_{1}}{a_{1}}$, therefore
$\frac{b_{1}}{a_{1}}\in\mathbb{R}$. But, if $\frac{b_{1}}{a_{1}}\in\mathbb{R}$
we also have that $a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha}\in
L_{\alpha}\cap\overline{L_{\alpha}}$. In fact, if
$\frac{b_{1}}{a_{1}}\in\mathbb{R}$ then
$\overline{\left(\frac{b_{1}}{a_{1}}\right)}=\frac{b_{1}}{a_{1}}$ which
implies that there is $k\in\mathbb{C}$ such that
$\left\\{\begin{array}[]{l}b_{1}=k\overline{b_{1}}\\\
a_{1}=k\overline{a_{1}}\end{array}\right.$. Therefore we can write
$a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha}=k(\overline{a_{1}}A_{\alpha}+\overline{b_{1}}A^{\ast}_{\alpha})\in\overline{L_{\alpha}}$.
So, we can conclude that $\dim(L_{\alpha}\cap\overline{L_{\alpha}})=2$ when
$\frac{b_{1}}{a_{1}}\in\mathbb{R}$ and
$\dim(L_{\alpha}\cap\overline{L_{\alpha}})=0$ otherwise.

  * 2.

If $x_{1}\neq 0$ and $x_{2}=0$, we have $y_{1}\neq 0$ and $y_{2}=0$ and it is
analogous to the previous item.

  * 3.

Now, let us suppose that $x_{i},y_{i}\neq 0$ for $i=1,2$. So
$x_{1}(a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha})+x_{2}(a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha})=y_{1}(\overline{a_{1}}A_{\alpha}+\overline{b_{1}}A^{\ast}_{\alpha})+y_{2}(\overline{a_{1}}S_{\alpha}+\overline{b_{1}}S^{\ast}_{\alpha})$
and then we have

$\left\\{\begin{array}[]{l}x_{1}b_{1}=y_{1}\overline{b_{1}}\\\
x_{1}a_{1}=y_{1}\overline{a_{1}}\end{array}\right.\textnormal{ and
}\left\\{\begin{array}[]{l}x_{2}b_{1}=y_{2}\overline{b_{1}}\\\
x_{2}a_{1}=y_{2}\overline{a_{1}}.\end{array}\right.$

Therefore, we can conclude that
$\frac{b_{1}}{a_{1}}=\overline{\left(\frac{b_{1}}{a_{1}}\right)}$ which
implies that $\frac{b_{1}}{a_{1}}\in\mathbb{R}$. From the first case we
conclude that $\dim(L_{\alpha}\cap\overline{L_{\alpha}})=2$.

Finally, observe that when $\dim(L_{\alpha}\cap\overline{L_{\alpha}})=0$ then
$L_{\alpha}$ defines a generalized complex structure, in this case we have a
generalized complex structure of symplectic type (up to $B$-transformation).
Due to [14] we know that $L_{\alpha}$ reduces to

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha}A_{\alpha}+(a_{\alpha}-i)A_{\alpha}^{\ast},x_{\alpha}S_{\alpha}+(a_{\alpha}-i)S_{\alpha}^{\ast}\\},$

with $a_{\alpha},x_{\alpha}\in\mathbb{R}$ and $x_{\alpha}\neq 0$.

As a direct consequence of Proposition 4.3 and the fact that, for each
positive root $\alpha$, the subspace $L_{\alpha}$ has real index zero or two
we have the following:

######  Theorem 4.9.

Let $L$ be an invariant complex Dirac structure on a flag manifold
$\mathbb{F}$. Then the real index of $L$ is constant and equal to $2k$ for
some $0\leq k\leq l$ where $l$ is the number of positive roots of the
associated Lie algebra $\mathfrak{g}$.

######  Remark 4.10.

Let $\mathbb{F}$ be a flag manifold with $\dim\mathbb{F}=2l$. So, by Theorem
4.9, if we want an invariant complex Dirac structure of real index $k=2r$,
where $0\leq r\leq l$, we need to join $r$ Lagrangian subspaces with real
index two and $l-r$ Lagrangian subspaces with real index zero, respecting the
fact that $\operatorname{Nij}|_{L}=0$.

######  Remark 4.11.

When $r=0$ in Remark 4.10 we recover the invariant generalized complex
structures described in [14].

In light of Remark 4.10, we will describe the triples of invariant complex
Dirac structures $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ separating them
according to the real index. The possible invariant complex Dirac structures
on the subspace $\mathfrak{u}_{\alpha}$ are described by Proposition 4.5, but
the case (4) can be of real index zero or two depending on $a_{1}$ and
$b_{1}$. So, to simplify the notation we will split these cases. From now on
we have:

$\displaystyle(1)\ $ $\displaystyle
L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}=(\mathfrak{u}_{\alpha})_{\mathbb{C}};$
$\displaystyle(2)\ $ $\displaystyle
L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{-S^{\ast}_{\alpha},A^{\ast}_{\alpha}\\}=(\mathfrak{u}_{\alpha}^{\ast})_{\mathbb{C}};$
$\displaystyle(3)\ $ $\displaystyle
L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}+i\varepsilon_{\alpha}S_{\alpha},A^{\ast}_{\alpha}+i\varepsilon_{\alpha}S_{\alpha}^{\ast}\\}\textnormal{
with }\varepsilon_{\alpha}=\pm 1$ $\displaystyle\textnormal{(generalized
complex structure of complex type)};$ (10) $\displaystyle(4.1)\ $
$\displaystyle
L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha},a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha}\\}\textnormal{
with }a_{1},b_{1}\neq 0\textnormal{ and }\frac{b_{1}}{a_{1}}\in\mathbb{R}$
$\displaystyle(4.2)\ $ $\displaystyle
L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha}A_{\alpha}+(a_{\alpha}-i)A^{\ast}_{\alpha},x_{\alpha}S_{\alpha}+(a_{\alpha}-i)S^{\ast}_{\alpha}\\}\textnormal{
with }x_{\alpha},a_{\alpha}\in\mathbb{R}\textnormal{ and }x_{\alpha}\neq 0.$
(generalized complex structure of symplectic type)

Therefore, now we get that cases (1), (2) and (4.1) have real index two and
cases (3) and (4.2) have real index zero.

From now on, we will use Notation 4.8 based on the cases appearing in (4.4)
instead of the cases from Proposition 4.5. For example, let
$(\alpha,\beta,\alpha+\beta)$ be a triple of roots and consider
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ the complex Dirac structures
associated to these roots such that
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{A_{\beta}+i\varepsilon_{\beta}S_{\beta},A^{\ast}_{\beta}+i\varepsilon_{\beta}S_{\beta}^{\ast}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha+\beta}A_{\alpha+\beta}+b_{\alpha+\beta}A^{\ast}_{\alpha+\beta},a_{\alpha+\beta}S_{\alpha+\beta}+b_{\alpha+\beta}S^{\ast}_{\alpha+\beta}\\}$
with $a_{\alpha+\beta},b_{\alpha+\beta}\neq 0$ and
$\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}\in\mathbb{R}$. Then we will refer
to the triple $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ as $(1,3,4.1)$,
because $L_{\alpha}$ is associated to case (1), $L_{\beta}$ to case (3) and
$L_{\alpha+\beta}$ to case (4.1).

###  4.5. Real index zero

The triple $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ has real index zero when
$L_{\alpha}$ ,$L_{\beta}$ and $L_{\alpha+\beta}$ have simultaneously real
index zero, that is, they are associated with a generalized complex structure.
Then the possible combinations are:

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$ | $\operatorname{Nij}=0$
---|---|---|---
3 | 3 | 3 | $\varepsilon_{\alpha+\beta}-\varepsilon_{\alpha}-\varepsilon_{\beta}+\varepsilon_{\alpha}\varepsilon_{\beta}\varepsilon_{\alpha+\beta}=0$
3 | 3 | 4.2 | $\varepsilon_{\alpha}=-\varepsilon_{\beta}$
3 | 4.2 | 3 | $\varepsilon_{\alpha}=\varepsilon_{\beta}$
4.2 | 4.2 | 4.2 | Equation 1

These are exactly the cases of invariant complex Dirac structures associated
with invariant generalized complex structures described in [14].

###  4.6. Real index two

We are going to analyze triples of subspaces
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ such that for two roots we have
$\dim(L_{\gamma}\cap\overline{L_{\gamma}})=0$ and for the other root we have
$\dim(L_{\gamma}\cap\overline{L_{\gamma}})=2$ where
$\gamma\in\\{\alpha,\beta,\alpha+\beta\\}$. Observe that the subspaces
$L_{\gamma}$ such that $\dim(L_{\gamma}\cap\overline{L_{\gamma}})=0$ are
associated to generalized complex structures. Thus we have the following
cases:

  1. (1)

Two of complex type;

  2. (2)

Two of symplectic type (up to $B$-transformation);

  3. (3)

One of complex type and one of symplectic type (up to $B$-transformation).

First of all, note that item (3) cannot happen, because it is not a case
presented in Table 2.

We start by studying item (1), that is, the cases where appear two subspaces
associated to invariant generalized complex structures of complex type. Given
the triple $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ then the subspaces
associated with the complex type can be $(L_{\alpha},L_{\beta})$ and
$(L_{\beta},L_{\alpha+\beta})$. The case $(L_{\alpha},L_{\alpha+\beta})$
coincides with $(L_{\beta},L_{\alpha+\beta})$ since
$\alpha+\beta=\beta+\alpha$.

######  Proposition 4.12.

Let $(\alpha,\beta,\alpha+\beta)$ be a triple of roots and consider
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ the associated triple of invariant
complex Dirac structures such that $L_{\alpha}$ and $L_{\beta}$ come from
invariant generalized complex structures of complex type, that is,

$L_{\gamma}=\textnormal{span}_{\mathbb{C}}\\{A_{\gamma}+\varepsilon_{\gamma}iS_{\gamma},A^{\ast}_{\gamma}+\varepsilon_{\gamma}iS^{\ast}_{\gamma}\\},$

with $\varepsilon_{\gamma}=\pm 1$ where $\gamma\in\\{\alpha,\beta\\}$. Then
$\operatorname{Nij}|_{L}=0$ if and only if
$\varepsilon_{\alpha}=-\varepsilon_{\beta}$ for every maximal isotropic
subspace $L_{\alpha+\beta}$ such that
$\dim_{\mathbb{C}}(L_{\alpha+\beta}\cap\overline{L_{\alpha+\beta}})=2$, where
$L=L_{\alpha}\cup L_{\beta}\cup L_{\alpha+\beta}$.

###### Proof.

Suppose that $L_{\alpha}$ and $L_{\beta}$ are invariant complex Dirac
structures associated to invariant generalized complex structures of complex
type. Thus,
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}+i\varepsilon_{\alpha}S_{\alpha},A^{\ast}_{\alpha}+i\varepsilon_{\alpha}S^{\ast}_{\alpha}\\}$
and
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{A_{\beta}+i\varepsilon_{\beta}S_{\beta},A^{\ast}_{\beta}+i\varepsilon_{\beta}S^{\ast}_{\beta}\\}$
where $\varepsilon_{\alpha},\varepsilon_{\beta}=\pm 1$. Meanwhile
$L_{\alpha+\beta}$ can be any of the three types of structure with real index
two, respecting certain conditions so that $\operatorname{Nij}|_{L}=0$. Such
conditions will tell us the possible combinations of signals
$\varepsilon_{\alpha}$ and $\varepsilon_{\beta}$. Let us analyze case by case:

  * a.

If
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha+\beta},S_{\alpha+\beta}\\}$,
then $\operatorname{Nij}|_{L}=0$ if and only if
$\varepsilon_{\alpha}=-\varepsilon_{\beta}$.

  * b.

If
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{-S^{\ast}_{\alpha+\beta},A^{\ast}_{\alpha+\beta}\\}$,
then $\operatorname{Nij}|_{L}=0$ if and only if
$\varepsilon_{\alpha}=-\varepsilon_{\beta}$.

  * c.

If
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha+\beta}A_{\alpha+\beta}+b_{\alpha+\beta}A^{\ast}_{\alpha+\beta},a_{\alpha+\beta}S_{\alpha+\beta}+b_{\alpha+\beta}S^{\ast}_{\alpha+\beta}\\}$
with $\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}\in\mathbb{R}$, then
$\operatorname{Nij}|_{L}=0$ if and only if

$\displaystyle\left\\{\begin{array}[]{l}\varepsilon_{\beta}+\varepsilon_{\alpha}=0\\\
1+\varepsilon_{\alpha}\varepsilon_{\beta}=0.\end{array}\right.$ (13)

Note that (13) is satisfied if and only if
$\varepsilon_{\alpha}=-\varepsilon_{\beta}$.

∎

######  Proposition 4.13.

Let $(\alpha,\beta,\alpha+\beta)$ be a triple of roots and consider
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ the associated triple of invariant
complex Dirac structures such that $L_{\beta}$ and $L_{\alpha+\beta}$ come
from invariant generalized complex structures of complex type, that is,

$L_{\gamma}=\textnormal{span}_{\mathbb{C}}\\{A_{\gamma}+\varepsilon_{\gamma}iS_{\gamma},A^{\ast}_{\gamma}+\varepsilon_{\gamma}iS^{\ast}_{\gamma}\\},$

with $\varepsilon_{\gamma}=\pm 1$ where $\gamma\in\\{\beta,\alpha+\beta\\}$.
Then $\operatorname{Nij}|_{L}=0$ if and only if
$\varepsilon_{\beta}=\varepsilon_{\alpha+\beta}$ for every maximal isotropic
subspace $L_{\alpha}$ such that
$\dim_{\mathbb{C}}(L_{\alpha}\cap\overline{L_{\alpha}})=2$, where
$L=L_{\alpha}\cup L_{\beta}\cup L_{\alpha+\beta}$.

###### Proof.

Suppose that $L_{\beta}$ and $L_{\alpha+\beta}$ are invariant complex Dirac
structures associated to invariant generalized complex structures of complex
type. Thus,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{A_{\beta}+i\varepsilon_{\beta}S_{\beta},A^{\ast}_{\beta}+i\varepsilon_{\beta}S^{\ast}_{\beta}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha+\beta}+i\varepsilon_{\alpha+\beta}S_{\alpha+\beta},A^{\ast}_{\alpha+\beta}+i\varepsilon_{\alpha+\beta}S^{\ast}_{\alpha+\beta}\\}$
where $\varepsilon_{\beta},\varepsilon_{\alpha+\beta}=\pm 1$. Meanwhile
$L_{\alpha}$ can be any of the three types of structure with real index two,
respecting certain conditions so that $\operatorname{Nij}|_{L}=0$. Let us
analyze case by case:

  * a.

If $L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}$,
then $\operatorname{Nij}|_{L}=0$ if and only if
$\varepsilon_{\beta}=\varepsilon_{\alpha+\beta}$.

  * b.

If
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{-S^{\ast}_{\alpha},A^{\ast}_{\alpha}\\}$,
then $\operatorname{Nij}|_{L}=0$ if and only if
$\varepsilon_{\beta}=\varepsilon_{\alpha+\beta}$.

  * c.

If
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{aA_{\alpha}+bA^{\ast}_{\alpha},aS_{\alpha}+bS^{\ast}_{\alpha}\\}$,
then $\operatorname{Nij}|_{L}=0$ if and only if

$\displaystyle\left\\{\begin{array}[]{l}\varepsilon_{\alpha+\beta}-\varepsilon_{\beta}=0\\\
1-\varepsilon_{\beta}\varepsilon_{\alpha+\beta}=0.\end{array}\right.$ (16)

Note that (16) is satisfied if and only if
$\varepsilon_{\beta}=\varepsilon_{\alpha+\beta}$.

∎

Finally we will study case (2), that is, the cases where appear two invariant
complex Dirac structures associated to invariant generalized complex
structures of symplectic type (up to $B$-transformation). We have four cases
to analyze, namely:

  * (2.1)

Suppose
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{x_{\beta}A_{\beta}+(a_{\beta}-i)A^{\ast}_{\beta},x_{\beta}S_{\beta}+(a_{\beta}-i)S^{\ast}_{\beta}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha+\beta}A_{\alpha+\beta}+(a_{\alpha+\beta}-i)A^{\ast}_{\alpha+\beta},x_{\alpha+\beta}S_{\alpha+\beta}+(a_{\alpha+\beta}-i)S^{\ast}_{\alpha+\beta}\\}$.
We have $\operatorname{Nij}|_{L}=0$ if and only if
$\frac{x_{\beta}}{a_{\beta}-i}=\frac{x_{\alpha+\beta}}{a_{\alpha+\beta}-i}$
which happens if and only if $x_{\beta}=x_{\alpha+\beta}$ and
$a_{\beta}=a_{\alpha+\beta}$.

  * (2.2)

Suppose
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha}A_{\alpha}+(a_{\alpha}-i)A^{\ast}_{\alpha},x_{\alpha}S_{\alpha}+(a_{\alpha}-i)S^{\ast}_{\alpha}\\}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{x_{\beta}A_{\beta}+(a_{\beta}-i)A^{\ast}_{\beta},x_{\beta}S_{\beta}+(a_{\beta}-i)S^{\ast}_{\beta}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha+\beta},S_{\alpha+\beta}\\}$.
We have $\operatorname{Nij}|_{L}=0$ if and only if
$\frac{x_{\alpha}}{a_{\alpha}-i}=-\frac{x_{\beta}}{a_{\beta}-i}$ which happens
if and only if $x_{\alpha}=-x_{\beta}$ and $a_{\alpha}=a_{\beta}$.

  * (2.3)

Suppose
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha}A_{\alpha}+b_{\alpha}A^{\ast}_{\alpha},a_{\alpha}S_{\alpha}+b_{\alpha}S^{\ast}_{\alpha}\\}$
with $\frac{b_{\alpha}}{a_{\alpha}}\in\mathbb{R}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{x_{\beta}A_{\beta}+(a_{\beta}-i)A^{\ast}_{\beta},x_{\beta}S_{\beta}+(a_{\beta}-i)S^{\ast}_{\beta}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha+\beta}A_{\alpha+\beta}+(a_{\alpha+\beta}-i)A^{\ast}_{\alpha+\beta},x_{\alpha+\beta}S_{\alpha+\beta}+(a_{\alpha+\beta}-i)S^{\ast}_{\alpha+\beta}\\}$.
We have $\operatorname{Nij}|_{L}=0$ if and only if

$a_{\alpha}x_{\beta}(a_{\alpha+\beta}-i)-a_{\alpha}(a_{\beta}-i)x_{\alpha+\beta}-b_{\alpha}x_{\beta}x_{\alpha+\beta}=0,$

which is equivalent to

$\displaystyle
x_{\beta}(a_{\alpha+\beta}-i)-(a_{\beta}-i)x_{\alpha+\beta}-\frac{b_{\alpha}}{a_{\alpha}}x_{\beta}x_{\alpha+\beta}=0.$
(17)

Separating the real and complex part, we have that (17) is satisfied when

$\displaystyle\left\\{\begin{array}[]{ll}x_{\beta}-x_{\alpha+\beta}=0\\\
x_{\beta}a_{\alpha+\beta}-a_{\beta}x_{\alpha+\beta}-\frac{b_{\alpha}}{a_{\alpha}}x_{\beta}x_{\alpha+\beta}=0.\end{array}\right.$
(20)

From the second expression of (20) we obtain

$\frac{b_{\alpha}}{a_{\alpha}}=\frac{x_{\beta}a_{\alpha+\beta}-a_{\beta}x_{\alpha+\beta}}{x_{\beta}x_{\alpha+\beta}}=\frac{a_{\alpha+\beta}}{x_{\alpha+\beta}}-\frac{a_{\beta}}{x_{\beta}}.$

  * (2.4)

Suppose
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha}A_{\alpha}+(a_{\alpha}-i)A^{\ast}_{\alpha},x_{\alpha}S_{\alpha}+(a_{\alpha}-i)S^{\ast}_{\alpha}\\}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{x_{\beta}A_{\beta}+(a_{\beta}-i)A^{\ast}_{\beta},x_{\beta}S_{\beta}+(a_{\beta}-i)S^{\ast}_{\beta}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha+\beta}A_{\alpha+\beta}+b_{\alpha+\beta}A^{\ast}_{\alpha+\beta},a_{\alpha+\beta}S_{\alpha+\beta}+b_{\alpha+\beta}S^{\ast}_{\alpha+\beta}\\}$.
We have $\operatorname{Nij}|_{L}=0$ if and only if

$x_{\alpha}x_{\beta}b_{\alpha+\beta}-x_{\alpha}(a_{\beta}-i)a_{\alpha+\beta}-(a_{\alpha}-i)x_{\beta}a_{\alpha+\beta}=0.$

With an argument analogous to the previous item we obtain

$\displaystyle x_{\beta}+x_{\alpha+\beta}=0\ \textnormal{ and }\
\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}=\frac{a_{\beta}}{x_{\beta}}+\frac{a_{\alpha}}{x_{\alpha}}.$

Putting all this information together we have:

######  Proposition 4.14.

Let $(\alpha,\beta,\alpha+\beta)$ be a triple of roots. The associated triple
of invariant complex Dirac structures
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ of real index two is involutive if
it coincides with one of the possibilities that appears in the following table

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$ | $\operatorname{Nij}=0$
---|---|---|---
3 | 3 |  1 $\vee$ 2 $\vee$ 4.1 |  Proposition 4.12
1 $\vee$ 2 $\vee$ 4.1 | 3 | 3 |  Proposition 4.13
1 | 4.2 | 4.2 |  $x_{\beta}=x_{\alpha+\beta}$ and $a_{\beta}=a_{\alpha+\beta}$
4.2 | 4.2 | 1 |  $x_{\alpha}=-x_{\beta}$ and $a_{\alpha}=a_{\beta}$
4.1 | 4.2 | 4.2 |  $x_{\beta}=x_{\alpha+\beta}$ and $\frac{b_{\alpha}}{a_{\alpha}}=\frac{a_{\alpha+\beta}}{x_{\alpha+\beta}}-\frac{a_{\beta}}{x_{\beta}}$
4.2 | 4.2 | 4.1 |  $x_{\beta}=-x_{\alpha+\beta}$ and $\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}=\frac{a_{\beta}}{x_{\beta}}+\frac{a_{\alpha}}{x_{\alpha}}$

######  Notation 4.15.

The symbol $\vee$ means, for example, in the first row that we can have the
triples $(3,3,1)$, $(3,3,2)$ or $(3,3,4.1)$.

###  4.7. Real index four

Here we are considering triples of subspaces
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ such that for two roots we have
$\dim(L_{\gamma}\cap\overline{L_{\gamma}})=2$ and for the other root we have
$\dim(L_{\gamma}\cap\overline{L_{\gamma}})=0$ where
$\gamma\in\\{\alpha,\beta,\alpha+\beta\\}$. That means we have one root
associated with an invariant complex Dirac structure that comes from an
invariant generalized complex structure.

Due to Proposition 4.7 one has that triples
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ such that two subspaces are given by
$L_{\gamma}=\textnormal{span}_{\mathbb{C}}\\{-S_{\gamma}^{\ast},A_{\gamma}^{\ast}\\}$,
with $\gamma\in\\{\alpha,\beta,\alpha+\beta\\}$, and the subspace associated
to the other root is of real index zero, then
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ is involutive of real index four.

Note that, according to Table 2, triples
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ of real index four are necessarily
described by the last row. Since we have an extra algebraic condition to
analyze, we have the following two cases:

  1. (1)

Suppose that $L_{\alpha}$ is of symplectic type (up to $B$-transformation),
that is,
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha}A_{\alpha}+(a_{\alpha}-i)A^{\ast}_{\alpha},x_{\alpha}S_{\alpha}+(a_{\alpha}-i)S^{\ast}_{\alpha}\\}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{a_{\beta}A_{\beta}+b_{\beta}A^{\ast}_{\beta},a_{\beta}S_{\beta}+b_{\beta}S^{\ast}_{\beta}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha+\beta}A_{\alpha+\beta}+b_{\alpha+\beta}A^{\ast}_{\alpha+\beta},a_{\alpha+\beta}S_{\alpha+\beta}+b_{\alpha+\beta}S^{\ast}_{\alpha+\beta}\\}$
with
$\frac{b_{\beta}}{a_{\beta}},\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}\in\mathbb{R}$.
In this case, we must have

$x_{\alpha}a_{\beta}b_{\alpha+\beta}-x_{\alpha}b_{\beta}a_{\alpha+\beta}-(a_{\alpha}-i)a_{\beta}a_{\alpha+\beta}=0.$

Since $a_{\beta},a_{\alpha+\beta}\neq 0$ the last expression can be rewritten
as

$x_{\alpha}\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}-x_{\alpha}\frac{b_{\beta}}{a_{\beta}}-(a_{\alpha}-i)=0,$

which is an absurd because
$x_{\alpha},a_{\alpha},\frac{b_{\beta}}{a_{\beta}},\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}\in\mathbb{R}$.
Therefore this case is not involutive.

  2. (2)

Suppose $L_{\alpha+\beta}$ of symplectic type (up to $B$-transformation), that
is,
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha}A_{\alpha}+b_{\alpha}A^{\ast}_{\alpha},a_{\alpha}S_{\alpha}+b_{\alpha}S^{\ast}_{\alpha}\\}$,
$L_{\beta}=\textnormal{span}_{\mathbb{C}}\\{a_{\beta}A_{\beta}+b_{\beta}A^{\ast}_{\beta},a_{\beta}S_{\beta}+b_{\beta}S^{\ast}_{\beta}\\}$
and
$L_{\alpha+\beta}=\textnormal{span}_{\mathbb{C}}\\{x_{\alpha+\beta}A_{\alpha+\beta}+(a_{\alpha+\beta}-i)A^{\ast}_{\alpha+\beta},x_{\alpha+\beta}S_{\alpha+\beta}+(a_{\alpha+\beta}-i)S^{\ast}_{\alpha+\beta}\\}$
with $\frac{b_{\alpha}}{a_{\alpha}},\frac{b_{\beta}}{a_{\beta}}\in\mathbb{R}$.
In this case, we must have

$a_{\alpha}a_{\beta}(a_{\alpha+\beta}-i)-a_{\alpha}b_{\beta}x_{\alpha+\beta}-b_{\alpha}a_{\beta}x_{\alpha+\beta}=0.$

Since $a_{\alpha},a_{\beta}\neq 0$ the last expression can be rewritten as

$(a_{\alpha+\beta}-i)-\frac{b_{\beta}}{a_{\beta}}x_{\alpha+\beta}-\frac{b_{\alpha}}{a_{\alpha}}x_{\alpha+\beta}=0,$

which cannot happen again because
$\frac{b_{\alpha}}{a_{\alpha}},\frac{b_{\beta}}{a_{\beta}},x_{\alpha+\beta},a_{\alpha+\beta}\in\mathbb{R}$.

Summarizing we obtain the following:

######  Proposition 4.16.

Let $(\alpha,\beta,\alpha+\beta)$ be a triple of roots. The associated triple
of invariant complex Dirac structures
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ of real index four is involutive if
it appears in the following table

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$ | $\operatorname{Nij}=0$
---|---|---|---
2 | 2 |  3 $\vee$ 4.2 |  Proposition 4.7
2 |  3 $\vee$ 4.2 | 2 |  Proposition 4.7

###  4.8. Real index six

This is the last case, here we consider the triples
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ where none of the subspaces is of
real index zero.

Let $(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ be a triple with
$L_{\gamma}=\textnormal{span}_{\mathbb{C}}\\{a_{\gamma}A_{\gamma}+b_{\gamma}A^{\ast}_{\gamma},a_{\gamma}S_{\gamma}+b_{\gamma}S^{\ast}_{\gamma}\\}$
for all $\gamma\in\\{\alpha,\beta,\alpha+\beta\\}$. In this case we must have

$a_{\alpha}a_{\beta}b_{\alpha+\beta}-a_{\alpha}b_{\beta}a_{\alpha+\beta}-b_{\alpha}a_{\beta}a_{\alpha+\beta}=0.$

But, since $a_{\alpha},a_{\beta},a_{\alpha+\beta}\neq 0$ we obtain that

$\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}-\frac{b_{\beta}}{a_{\beta}}-\frac{b_{\alpha}}{a_{\alpha}}=0,$

or equivalently,

$\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}=\frac{b_{\beta}}{a_{\beta}}+\frac{b_{\alpha}}{a_{\alpha}}.$

Therefore the cases of triples with real index six are:

######  Proposition 4.17.

Let $(\alpha,\beta,\alpha+\beta)$ be a triple of roots. The associated triple
of invariant complex Dirac structures
$(L_{\alpha},L_{\beta},L_{\alpha+\beta})$ of real index six is involutive if
it is one of the possibilities in the following table

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$ | $\operatorname{Nij}=0$
---|---|---|---
1 | 1 | 1 |  Proposition 4.6
2 | 1 | 2 |  Proposition 4.7
2 | 2 |  1 $\vee$ 2 $\vee$ 4.1 |  Proposition 4.7
2 | 4.1 | 2 |  Proposition 4.7
4.1 | 4.1 | 4.1 | $\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}=\frac{b_{\beta}}{a_{\beta}}+\frac{b_{\alpha}}{a_{\alpha}}$

Thus we exhausted all possibilities for triples of invariant complex Dirac
structures $(L_{\alpha},L_{\beta},L_{\beta})$.

At this point we are able to prove a converse of Theorem 4.9, that is, we can
prove that there exists an invariant complex Dirac structure on $\mathbb{F}$
with constant real index equal to $2k$, with $k\in\\{0,1,\cdots,l\\}$ where
$l$ is the number of positive roots of the Lie algebra $\mathfrak{g}$ which
defines the flag manifold $\mathbb{F}$.

######  Theorem 4.18.

Let $\mathfrak{g}$ be a semisimple Lie algebra and consider $\mathbb{F}$ the
maximal flag manifold associated to $\mathfrak{g}$. If $\mathfrak{g}$ has $l$
positive roots, then there exists an invariant Dirac structure on $\mathbb{F}$
with constant real index equal to $2k$, where $0\leq k\leq l$.

###### Proof.

Let us denote by $d_{i}$ the number of positive roots of height $i$, where
$d_{1}$ is the number of simple roots, $m$ is the highest height and
$d_{1}+d_{2}+\cdots+d_{m}=l$. We want to construct an invariant complex Dirac
structure $L$ with real index $2k$, with $0\leq k\leq l$.

If $k=0$ we have that $L$ comes from a generalized complex structure, whose
existence follows from [14].

In what follows we look at Dirac structures of type 1 and 2 according to
Proposition 4.5. If $0<k\leq d_{1}$, then for $k$ simple roots we consider
$L_{\alpha_{1}},\cdots,L_{\alpha_{k}}$ of type $2$, and $L_{\beta}$ of type 1
with same sign for all other positive root $\beta$.

If $d_{1}<k\leq d_{2}$, then for all simple root $\alpha$ we consider
$L_{\alpha}$ of type $2$, for $k-d_{1}$ roots of height 2 we also consider
$L_{\alpha}$ of type two and, for the orther roots, we consider $L_{\beta}$ of
type 1 with same sign for all other positive root $\beta$.

Proceeding iteratively, if $d_{i}<k\leq d_{i+1}$ then we consider $L_{\alpha}$
of type $2$ for all root of height $1,2,\cdots,d_{i}$, we also consider
$L_{\alpha}$ of type two for $k-(d_{1}+d_{2}+\cdots+d_{i})$ and, for the other
roots, we consider $L_{\beta}$ of type 1 with same sign for all other positive
root $\beta$.

Now, let us observe that $L$ constructed like above is involutive. In fact,
given a triple of roots $(\alpha,\beta,\alpha+\beta)$ then the possible
associated triple of subspaces are $(1,1,1)$, $(2,1,1)$, $(1,2,1)$, $(2,2,1)$
and $(2,2,2)$ which are always involutive thanks to the computations done in
the previous subsections. In fact, the construction of $L$ does not allow us
to have triples of subspaces $(1,1,2)$, $(2,1,2)$ or $(1,2,2)$, because the
root $\alpha+\beta$ is higher than $\alpha$ and $\beta$. ∎

##  5\. Classification up to $B$-transformation

In [2] was presented a classification up to $B$-transformations of complex
Dirac structures in terms of its real index, order and type. Before defining
order and type, we need to fix some notation.

Given $L\subset(V\oplus V^{\ast})_{\mathbb{C}}$ a complex Dirac structure on
$V$, we obtain the following subspaces

$\displaystyle K=\textnormal{Re}(L\cap\overline{L})\subseteq V\oplus
V^{\ast},\ \qquad E=pr_{V_{\mathbb{C}}}L\subseteq V_{\mathbb{C}}$
$\displaystyle\Delta=\textnormal{Re}(E\cap\overline{E})\subseteq V,\ \qquad
D=\textnormal{Re}(E+\overline{E})\subseteq V.$ (21)

Note that $E$ is the range and the other subspaces are real defined from $L$.
Moreover, recall that there exists a complex $2$-form
$\varepsilon\in\wedge^{2}E^{\ast}$ such that

$L=L(E,\varepsilon)=\\{X+\xi\in E\oplus V^{\ast}\ |\
\xi|_{E}=\iota_{X}\varepsilon\\}.$

Consider the real $2$-form

$\omega_{\Delta}\colon=\textnormal{Im}(\varepsilon)|_{\Delta}\in\wedge^{2}\Delta^{\ast},$

where $\textnormal{Im}(\varepsilon)$ means the imaginary part of
$\varepsilon$.

Now, we are able to define the type and order:

######  Definition 5.1.

Let $L$ be a linear complex Dirac structure. The order of $L$ is defined as

$\textnormal{order}(L)=\textnormal{codim}D.$

The order is always less than or equal to the real index of the linear complex
Dirac structure. In particular, in a generalized complex structure the order
is always zero.

######  Definition 5.2.

The type of a complex Dirac structure $L$ is

$\textnormal{type}(L)=\textnormal{dim}(E+\overline{E})-\textnormal{dim}E.$

Once we have fixed the notation we can present the classification done by
Agüero and Rubio in [2]. This classification has as key ingredients the
associated presymplectic subspace $(\Delta,\omega_{\Delta})$ and a certain
complex structure on $D/\Delta$, and states that a complex Dirac structure is
(up to $B$-transformations) the product of $(\Delta,\omega_{\Delta})$ with the
complex structure $D/\Delta$.

######  Proposition 5.3 ([2]).

Let $L$ be a complex Dirac structure with real index $r$ and order $s$. Then
$L$ is isomorphic to a $B$-transformation of the product of a complex Dirac
structure defined by a presymplectic structure with $(r-s)$-dimensional kernel
with a complex Dirac structure defined by a codimension-$s$ CR structure.

###  5.1. Classification on flag manifolds

We have proved that an invariant complex Dirac structure $L$ on a flag
manifold can be decomposed in terms of the root spaces, that is,
$L=\oplus_{\alpha}L_{\alpha}$ where $L_{\alpha}$ is a complex structure on
$\mathfrak{u}_{\alpha}$. Moreover, we described every possibility for these
subspaces $L_{\alpha}$ in (4.4). Therefore, using the real index, order and
type of each invariant complex Dirac structure presented in (4.4), we can
classify these structures up $B$-transformation.

######  Proposition 5.4.

Let $L_{\alpha}$ be an invariant complex Dirac structure on
$\mathfrak{u}_{\alpha}$. Then, up to $B$-transformations, we have only four
possibilities for $L_{\alpha}$.

  * (a)

$L_{\alpha}=(\mathfrak{u}_{\alpha})_{\mathbb{C}}$, that is, $L_{\alpha}$ is
the root space associated to the root $\alpha$;

  * (b)

$L_{\alpha}=(\mathfrak{u}^{\ast}_{\alpha})_{\mathbb{C}}$, that is,
$L_{\alpha}$ is the dual of the root space associated to the root $\alpha$;

  * (c)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}+\varepsilon_{\alpha}iS_{\alpha},A_{\alpha}^{\ast}+\varepsilon_{\alpha}iS^{\ast}_{\alpha}\\}$
with $\varepsilon_{\alpha}=\pm 1$, that is, $L_{\alpha}$ is the $i$-eigenspace
of an invariant generalized complex structure on $\mathfrak{u}_{\alpha}$ of
complex type.

  * (d)

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}+\frac{i}{x_{\alpha}}A_{\alpha}^{\ast},S_{\alpha}+\frac{i}{x_{\alpha}}S^{\ast}_{\alpha}\\}$
with $x_{\alpha},a_{\alpha}\in\mathbb{R}$ and $x_{\alpha}\neq 0$, that is,
$L_{\alpha}$ is the $i$-eigenspace of an invariant generalized complex
structure on $\mathfrak{u}_{\alpha}$ of symplectic type.

In other words, if we denote by
$\textnormal{Dir}_{\mathbb{C}}(\mathfrak{u}_{\alpha})$ the set of all
invariant complex Dirac structures on $\mathfrak{u}_{\alpha}$, then

$\frac{\textnormal{Dir}_{\mathbb{C}}(\mathfrak{u}_{\alpha})}{B\textnormal{-fields}}=\\{(a),(b),(c),(d)\\}.$

###### Proof.

Note that the subspaces described by Case 1 and Case 4.1 in (4.4) have the
same real index, order and type. In fact, if
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha},S_{\alpha}\\}=(\mathfrak{u}_{\alpha})_{\mathbb{C}}$,
then the subspaces described in (5) are given by
$K=\Delta=D=\mathfrak{u}_{\alpha}$ and
$E=(\mathfrak{u}_{\alpha})_{\mathbb{C}}$. Therefore, we have a complex Dirac
structure with real index $2$, order $0$ and type $0$. In the same way, if
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{\alpha}A_{\alpha}+b_{\alpha}A_{\alpha}^{\ast},a_{\alpha}S_{\alpha}+b_{\alpha}S^{\ast}_{\alpha}\\}$
with $\frac{b_{\alpha}}{a_{\alpha}}\in\mathbb{R}$, we have that
$K=L_{\alpha}$, $\Delta=D=\mathfrak{u}_{\alpha}$ and
$E=(\mathfrak{u}_{\alpha})_{\mathbb{C}}$. Thus, we have a complex Dirac
structure with real index $2$, order $0$ and type $0$.

Since these cases have the same real index, type and order, they are the same
subspaces up to $B$-transformation. In fact, we have that Case 1 is given by
$L((\mathfrak{u}_{\alpha})_{\mathbb{C}},0)$ and Case 4.1 is given by
$L((\mathfrak{u}_{\alpha})_{\mathbb{C}},\varepsilon)$ where
$\varepsilon=\frac{b_{\alpha}}{a_{\alpha}}(-S^{\ast}_{\alpha})\wedge
A^{\ast}_{\alpha}$ with $\frac{b_{\alpha}}{a_{\alpha}}\in\mathbb{R}$. Then,
using the fact that

$e^{B}\cdot L(E,\varepsilon)=L(E,\varepsilon+\iota^{\ast}B),$

consider $B\in\wedge^{2}(\mathfrak{u}_{\alpha})_{\mathbb{C}}^{\ast}$ such that
$\iota^{\ast}B=\varepsilon$ and we can conclude that

$e^{B}\cdot
L((\mathfrak{u}_{\alpha})_{\mathbb{C}},0)=L((\mathfrak{u}_{\alpha})_{\mathbb{C}},0+\iota^{\ast}B)=L((\mathfrak{u}_{\alpha})_{\mathbb{C}},\varepsilon).$

Note that, for Case 2 in (4.4) given by
$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{A_{\alpha}^{\ast},S_{\alpha}^{\ast}\\}=(\mathfrak{u}_{\alpha}^{\ast})_{\mathbb{C}}$,
we have $K=\mathfrak{u}_{\alpha}^{\ast}$ and $E=\Delta=D=\\{0\\}$, obtaining a
complex Dirac structure with real index $2$, order $2$ and type $0$.

Finally, we already know from [9] how Case 3 and Case 4.2 in (4.4) behave
under the action of $B$-transformation. Case 3 are fixed points of
$B$-transformations and Case 4.2 is a $B$-transformation of the $i$-eigenspace
of a symplectic structure. ∎

##  6\. Examples

In this section we will provide some examples of complex Dirac structures for
some flag manifolds. Remember that the possible invariant complex Dirac
structures on $\mathfrak{u}_{\alpha}$, where $\alpha$ is a positive root, were
described in (4.4). In the next examples we consider the flag manifolds
associated with the Lie algebras of $\mathfrak{sl}(2,\mathbb{C})$ and
$\mathfrak{sl}(3,\mathbb{C})$ due to the low number of roots, which allows us
to completely describe the invariant complex Dirac structures in these cases.

######  Example 6.1.

Let $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$, then we have
$\Sigma=\Pi^{+}=\\{\alpha\\}$ and the associated maximal flag manifold is
$\mathbb{F}=\mathbb{P}^{1}$. Since we have only one positive root for
$\mathfrak{g}$, then we may have only invariant complex Dirac structures of
real index zero or two.

Real index zero. If $L_{\alpha}$ is an invariant complex Dirac structure on
$\mathbb{F}$ with real index zero, then $L_{\alpha}$ is given by an invariant
generalized complex structure defined either by a complex structure or by a
$B$-transformation of a symplectic form.

Real index two. Now if $L_{\alpha}$ is an invariant Dirac structure on
$\mathbb{F}$ with real index two, then we have three possibilities:

  * •

The tangent space at the origin, that is,
$L_{\alpha}=(\mathfrak{u}_{\alpha})_{\mathbb{C}}$;

  * •

The cotangent space at the origin, that is,
$L_{\alpha}=(\mathfrak{u}_{\alpha}^{\ast})_{\mathbb{C}}$;

  * •

$L_{\alpha}=\textnormal{span}_{\mathbb{C}}\\{a_{1}A_{\alpha}+b_{1}A^{\ast}_{\alpha},a_{1}S_{\alpha}+b_{1}S^{\ast}_{\alpha}\\}$
with $a_{1},b_{1}\neq 0$ and $\frac{b_{1}}{a_{1}}\in\mathbb{R}$

We saw in Section 5 that, up to $B$-transformation, in this case we have only
the tangent and cotangent spaces at the origin as invariant complex Dirac
structures with real index two.

######  Example 6.2.

Consider the Lie algebra $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$, then we
have $\Sigma=\\{\alpha,\beta\\}$ is a simple root system and the corresponding
set of positive roots is $\Pi^{+}=\\{\alpha,\beta,\alpha+\beta\\}$. The
maximal flag manifold is given by
$\mathbb{F}=\mathbb{F}(1,2)\subset\mathbb{P}^{2}\times\mathbb{P}^{2}$ where

$\mathbb{F}(1,2)=\\{([x_{1},x_{2},x_{3}],[y_{1},y_{2},y_{3}])\in\mathbb{P}^{2}\times\mathbb{P}^{2}\
:\ x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}=0\\}.$

We can have invariant Dirac structures on $\mathbb{F}$ with real index zero,
two, four or six. Let us present the possibilities according to the real
index.

Real index zero. These are exactly the invariant complex Dirac structures on
$\mathbb{F}$ that come from invariant generalized complex structures.

Real index two. We can see on Section 4.6 the possible invariant complex Dirac
structures on $\mathbb{F}$ with real index two where each case has an
involutivity condition involving the signals for structures of type $3$, the
constants $a,b$ for type $4.1$ or the constants $x,a$ for type $4.2$. Then the
possible invariant complex Dirac structures are

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$
---|---|---
1 $\vee$ 2 $\vee$ 4.1 | 3 | 3
1 | 4.2 | 4.2
4.1 | 4.2 | 4.2

where $\varepsilon_{\beta}=\varepsilon_{\alpha+\beta}$ for the first row. For
$(1,4.2,4.2)$ we must have $x_{\beta}=x_{\alpha+\beta}$ and
$a_{\beta}=a_{\alpha+\beta}$ and, finally, for $(4.1,4.2,4.2)$ we must have
$x_{\beta}=x_{\alpha+\beta}$ and
$\frac{b_{\alpha}}{a_{\alpha}}=\frac{a_{\alpha+\beta}}{x_{\alpha+\beta}}-\frac{a_{\beta}}{x_{\beta}}$.

Real index four. The invariant complex Dirac structures on $\mathbb{F}$ with
real index four was described on Section 4.7. Different from the previous
cases, here we do not have algebraic condition for the involutivity, since the
cases always contain two invariant complex Dirac structure of type $2$ and
these cases were described on Proposition 4.7. More specifically

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$
---|---|---
2 | 2 | 3 $\vee$ 4.2
2 | 3 $\vee$ 4.2 | 2

Real index six. In this case the components $L_{\delta}$ have real index two,
which means that none of them come from a generalized complex structure. The
possible combinations for $L$ to be involutive are:

$L_{\alpha}$ | $L_{\beta}$ | $L_{\alpha+\beta}$
---|---|---
1 | 1 | 1
2 | 1 | 2
2 | 2 | 1
2 | 2 | 2
2 | 4.1 | 2
2 | 2 | 4.1
4.1 | 4.1 | 4.1

where the first and fourth cases are the tangent and cotangent spaces at the
origin of $\mathbb{F}$, respectively. The cases $(2,1,2)$, $(2,2,1)$,
$(2,4.1,2)$ and $(2,2,4.1)$ are described in Proposition 4.7. The case
$(4.1,4.1,4.1)$ must satisfy
$\frac{b_{\alpha+\beta}}{a_{\alpha+\beta}}=\frac{b_{\beta}}{a_{\beta}}+\frac{b_{\alpha}}{a_{\alpha}}$.

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