# Symmetric spaces as adjoint orbits and their geometries

Leonardo F. Cavenaghi† Instituto de Matemática, Estatística e Computação
Científica – Unicamp, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas,
SP, Brazil  ,  Carolina Garcia∗ Instituto de Matemática, Estatística e
Computação Científica – Unicamp, Rua Sérgio Buarque de Holanda, 651,
13083-859, Campinas, SP, Brazil  ,  Lino Grama⋆ Instituto de Matemática,
Estatística e Computação Científica – Unicamp, Rua Sérgio Buarque de Holanda,
651, 13083-859, Campinas, SP, Brazil  and  Luiz San Martin∙ Instituto de
Matemática, Estatística e Computação Científica – Unicamp, Rua Sérgio Buarque
de Holanda, 651, 13083-859, Campinas, SP, Brazil

###### Abstract.

We realize specific classical symmetric spaces, such as the semi-Kähler
symmetric spaces due to Berger, as the cotangent bundle of symmetric flag
manifolds. In particular, such realizations allow us to describe these
cotangent bundles’ geodesics and Lagrangian submanifolds. As a final
application, we provide the first examples of vector bundles over simply
connected nonnegatively curved manifolds that do not admit (connection)
metrics of nonnegative sectional curvature, even though their associated unit
sphere bundles do admit such metrics. Our examples come from explicit bundle
constructions over symmetric flag spaces.

##  1\. Introduction

Several authors, such as Berger, Kobayashi, Nomizu, and Helgason, have well-
studied symmetric spaces. Following [KN69], a symmetric space is a triple
$(G,H,\sigma)$ consisting of a connected Lie group $G$, a closed subgroup $H$
and an involutive automorphism $\sigma$ of $G$ such that
$G_{0}^{\sigma}\subset H\subset G^{\sigma}$, where $G^{\sigma}=\\{g\in
G:\sigma(g)=g\\}$ and $G_{0}^{\sigma}$ is the identity (connected) component
of $G^{\sigma}$. If $(G,H,\sigma)$ is a symmetric space, then $G/H$ is said to
be an affine symmetric manifold.

Let $\mathfrak{g}$ and $\mathfrak{h}$ be the Lie algebras of $G$ and $H$,
respectively, then the symmetric Lie algebra
$(\mathfrak{g},\mathfrak{h},\sigma)$ has the canonical decomposition
$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ satisfying
$[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m}$ and
$[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}$. If
$\mathrm{ad}_{\mathfrak{g}}(\mathfrak{h})$ is compact, then
$(\mathfrak{g},\mathfrak{h},\sigma)$ is called an _orthogonal symmetric Lie
algebra_. These are compact or non-compact types, and four classes with their
geometric properties obstructed. See [KN69, Theorem 8.6, p. 256]

######  Theorem 1.1.

Let $(G,H,\sigma)$ be a symmetric space with $\mathrm{Ad}_{G}(\mathfrak{h})$
compact and let $(\mathfrak{g},\mathfrak{h},\sigma)$ be its orthogonal
symmetric Lie algebra. Take any $G$-invariant Riemannian metric on $G/H$.
Then,

  * (1)

if $(\mathfrak{g},\mathfrak{h},\sigma)$ is of compact type, then $G/H$ is a
compact Riemannian symmetric space with nonnegative sectional curvature and
positive definite Ricci tensor

  * (2)

if $(\mathfrak{g},\mathfrak{h},\sigma)$ is of non-compact type, then $G/H$ is
a simply connected non-compact Riemannian symmetric space with non-positive
sectional curvature and negative-definite Ricci tensor and is diffeomorphic to
a Euclidean space.

In [Hel62], Helgason provides a table with Riemannian symmetric spaces. Some
of which are: the Grassmannian
$\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ of complex
subspaces of $\mathbb{C}^{n}$, the space $\mathrm{Sp}(n)/\mathrm{U}(n)$ of
complex structures on $\mathbb{H}^{n}$ compatible with its standard inner
product, and the space $\mathrm{SO}(2n)/\mathrm{U}(n)$ of orthogonal complex
structures on $\mathbb{R}^{2n}$. All these examples have orthogonal symmetric
Lie algebras.

On the one hand, the adjoint orbit of a complex semisimple Lie group, $G$, has
several realizations, as indicated in [GGSM17]. One of those realizations is
as the cotangent bundle of a complex flag manifold:

######  Theorem 1.2.

[GGSM17, Theorem 2.1] The adjoint orbit
$\mathcal{O}(H_{\Theta})=\mathrm{Ad}(G)\cdot H_{\Theta}\approx G/Z_{\Theta}$
of the characteristic element $H_{\Theta}$ is a $C^{\infty}$ vector bundle
over $\mathbb{F}_{\Theta}$ that is isomorphic to the cotangent bundle
$T^{*}\mathbb{F}_{\Theta}$. Moreover, we can write down a diffeomorphism
$\iota:\mathrm{Ad}(G)\cdot H_{\Theta}\rightarrow T^{*}\mathbb{F}_{\Theta}$
such that

  1. $(a)$

$\iota$ is equivariant with respect to the actions of $U$, that is, for all
$u\in U$,

$\iota\circ\mathrm{Ad}(u)=\tilde{u}\circ\iota,$

where $\tilde{u}$ is the lifting to $T^{*}\mathbb{F}_{\Theta}$ (via the
differential) of the action of $u$ on $\mathbb{F}_{\Theta}$; and

  2. $(b)$

the pullback of the canonical simplectic form on $T^{*}\mathbb{F}_{\Theta}$ by
$\iota$ is the (real) Kirillov-Kostant-Souriau form on the orbit.

Recall that every complex flag manifold is a homogeneous space $U/K$ where $U$
is a compact real form of a complex Lie group and $K\subset U$ is a
centralizer of a torus in $U$. Combining this with Theorem 1.2, an adjoint
orbit $\mathrm{Ad}(G)X$ for a characteristic element $X$ in $\mathfrak{g}$ is
a vector bundle over a complex flag manifold $U/K$, where $U$ is the compact
(connected) real form of $G$. Since the adjoint orbit of a complex simple Lie
group $G$ is contained in the Lie algebra $\mathfrak{g}$, this allows
$\mathrm{Ad}(G)X$ to inherit some required structures from the Lie algebra
$\mathfrak{g}$.

More precisely, consider a symmetric Lie algebra
$(\mathfrak{u},\mathfrak{k},\sigma)$. Let
$\mathfrak{u}=\mathfrak{k}+\mathfrak{m}$ be its canonical decomposition. If we
denote by $\mathfrak{g}$ and $\mathfrak{h}$ the complexifications of
$\mathfrak{u}$ and $\mathfrak{k}$, respectively, and by $\sigma^{c}$ the
involutive automorphism of $\mathfrak{g}$ induced by $\sigma$, one gets that

(1) $\mathfrak{u}^{*}:=\mathfrak{k}+\sqrt{-1}\mathfrak{m}.$

with $\sigma^{*}=\sigma^{c}\mid\mathfrak{u}^{*}$, is a symmetric sub-algebra
$(\mathfrak{u}^{*},\mathfrak{k},\sigma^{*})$ of
$(\mathfrak{g},\mathfrak{h},\sigma^{c})$, which is called the _dual of_
$(\mathfrak{u},\mathfrak{k},\sigma)$. In this work, the flag manifolds $U/K$
are affine symmetric manifolds. The adjoint orbits of $G$ contain the affine
symmetric manifold $U^{*}/K$, where $U^{*}$ is the connected Lie subgroup of
$G$ with Lie algebra $\mathfrak{u}^{*}$. Again, we can inherit the required
structures from the Lie algebra $\mathfrak{g}$ to $U^{*}/K$ seen as a
submanifold of the adjoint orbit.

The first part of this paper has the following content description

  * •

we prove that if a complex flag manifold $U/K$ is a symmetric space, then its
cotangent bundle is a symmetric space (Theorem A)

  * •

we determine an automorphism $\sigma$ of $G$ such that $(U,K,\sigma)$ and
$(G,G^{\sigma},\sigma)$ are symmetric spaces with $G/G^{\sigma}$ being a
realization of the cotangent bundle of $U/K$ (Theorem A)

  * •

we find Lagrangian submanifolds of the adjoint orbit $G/G^{\sigma}$ and obtain
nonrealization results (Theorem D)

######  Theorem A.

Let $\mathbb{F}_{\Theta}$ be a symmetric flag space identified with the
homogeneous space

(2) $\mathbb{F}_{\Theta}=U/U_{\Theta},$

as in Proposition 2.3 . Then there exists an element $H_{\Theta}$ in a fixed
Cartan sub-algebra of the compact Lie algebra $\mathfrak{u}$ such that for
$\sigma:=C_{\exp H_{\Theta}}$, we have

(3) $U_{\Theta}=\\{u\in U:C_{\exp H_{\Theta}}(u)=u\\}.$

Moreover, let $G$ be the complex Lie group and $U$ be its compact real form.
Then the cotangent bundle of $\mathbb{F}_{\Theta}$ has a realization as a
symmetric space. If we use also the notation $\sigma$ for
$\mathrm{Ad}_{\mathfrak{u}}(\exp H_{\Theta})$ and
$\mathrm{Ad}_{\mathfrak{g}}(\exp H_{\Theta})$, then
$(\mathfrak{u},\mathfrak{u}_{\Theta},\sigma)$ and
$(\mathfrak{g},\mathfrak{z}_{\mathfrak{g}}(H_{\Theta}),\sigma)$ are symmetric
Lie algebras.

A detailed explicit description of $\mathbb{F}_{\Theta}$ as $U/U_{\Theta}$ is
presented in Section 6. Related to that, we obtain

######  Theorem B.

The following flag manifolds $\mathbb{F}_{\Theta}$ are symmetric spaces with
corresponding symmetric cotangent bundle $T^{*}\mathbb{F}_{\Theta}$:

$\mathbb{F}_{\Theta}$ | $T^{*}\mathbb{F}_{\Theta}$
---|---
$\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ | $\mathrm{Sl}(p+q,\mathbb{C})/\mathrm{S}(\mathrm{Gl}(p,\mathbb{C})\times\mathrm{Gl}(q,\mathbb{C}))$
$\mathrm{Sp}(l)/\mathrm{U}(l)$ | $\mathrm{Sp}(l,\mathbb{C})/\mathrm{Gl}(l,\mathbb{C})$
$\mathrm{SO}(2l)/\mathrm{U}(l)$ |  $\mathrm{SO}_{\mathbb{C}}(2l)/\mathrm{Gl}(l,\mathbb{C})$.

At this point, we recall that in [Ber57], an affine symmetric space $G/H$
(resp. symmetric Lie algebra) is said to be $\mathbb{C}$-symmetric if there
exists some complex structure of the vector space $\mathfrak{m}$, invariant by
the linear representation ($\mathrm{Ad}(H),\mathfrak{m}$) [resp.
(ad($\mathfrak{h}$), $\mathfrak{m}$)]. On the other hand, we say that the
affine symmetric space $G/H$ (resp. the symmetric Lie algebra) is semi-Kähler
if it is $\mathbb{C}$-symmetric and for a fixed complex structure of
$\mathfrak{m}$, the representation ($\mathrm{Ad}(H),\mathfrak{m}$) [resp.
(ad($\mathfrak{h}$),$\mathfrak{m}$)] leaves invariant a non-degenerate
Hermitian form of $\mathfrak{m}$.

######  Proposition 1.3.

[Ber57] Let $(\mathfrak{g},\mathfrak{h},\sigma)$ be a symmetric Lie algebra
with $\mathfrak{g}$ simple. The symmetric Lie algebra is semi-Kähler if, and
only if, the isotropy sub-algebra $\mathfrak{h}$ contains the Lie algebra
$\mathbb{R}$.

If $(\mathfrak{g},\mathfrak{h},\sigma)$ is a symmetric Lie algebra, we name
$(\mathfrak{g},\mathfrak{h})$ as a symmetric Lie pair. Berger gives a table of
all symmetric Lie pairs indicating which are semi-Kähler. Some symmetric semi-
Kähler Lie pairs are:
$(\mathfrak{sl}(p+q,\mathbb{C}),\mathfrak{sl}(p,\mathbb{C})\oplus\mathfrak{sl}(q,\mathbb{C})\oplus\mathbb{C}^{*})$,
$(\mathfrak{sp}(n,\mathbb{C}),\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*})$
and
$(\mathfrak{so}(2n,\mathbb{C}),\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*})$.
Note that the isotropy sub-algebras contain the complex vector space
$\mathbb{C}$ in those examples. They contain $\mathbb{R}$. Theorem A verifies
that each of the semi-Kähler symmetric spaces above described is cotangent
spaces of some flag manifolds endowed with a symmetric space structure.

The symmetric Lie algebra of a flag symmetric space $\mathbb{F}_{\Theta}$ is
$(\mathfrak{u},\mathfrak{u}_{\Theta},\sigma)$, where we use also the notation
$\sigma$ for the automorphism $\mathrm{Ad}(\exp H_{\Theta})$ of
$\mathfrak{g}$. This symmetric Lie algebra has the canonical decomposition
$\mathfrak{u}=\mathfrak{u}_{\Theta}+\mathfrak{m}$. Then, we denote by
$\mathcal{S}$ the subset $\mathrm{Ad}(\exp(\sqrt{-1}\mathfrak{m})H_{\Theta}$
of the adjoint orbit $\mathrm{Ad}(G)H_{\Theta}$. One of our main results
states that $\mathcal{S}$ is a submanifold of the adjoint orbit
$\mathrm{Ad}(G)H_{\Theta}$. It admits a realization as the dual symmetric
space of $\mathbb{F}_{\Theta}$. Furthermore, $\mathcal{S}$ is a Riemannian
manifold diffeomorphic to a vector space. Also, if a fiber in the cotangent
bundle $T^{*}\mathbb{F}_{\Theta}$ intersects $\mathcal{S}$, then such fiber
and $\mathcal{S}$ are transverse.

In Sections 3.2 and 3.3, we present the examples 1 and 3. There we discuss the
real flag manifold $\mathrm{SO}(2)/\\{\pm I\\}\cong\mathrm{S}^{1}$, which is a
symmetric space, such that its cotangent bundle has a realization as the one-
sheeted hyperboloid $Q=\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}-z^{2}=1\\}$,
that is also a symmetric space. The flag manifold $\mathrm{S}^{1}$ is the
intersection of $Q$ with the plane $z=0$. We study the orbit of the
exponential of symmetric zero-trace matrices in the matrix

$\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right).$

Such an orbit has a realization as the curve
$\mathcal{C}=\\{(0,y,z)\in\mathbb{R}^{3}:y^{2}-z^{2}=1,\;y>0\\}$, which is
intuitively symmetric with respect to the axis $Y$. In the complex case, the
generalization of this orbit is the submanifold $\mathcal{S}$. Hence, these
examples work as “toy models” to the general description given by Theorem A.

Remarkable as well, in general, the intersection of the submanifold
$\mathcal{S}$ with the fibers of the bundle
$\mathrm{Ad}(G)H_{\Theta}\rightarrow\mathbb{F}_{\Theta}$ is either null or a
single point. Then, the projection of the cotangent bundle restricted to the
submanifold $\mathcal{S}$ is an injective map since each fiber intersecting
$\mathcal{S}$ does so in a single element. Then, $\mathcal{S}$ is continuously
deformed into a part of the sphere $\mathbb{F}_{\Theta}$:

######  Theorem C.

Let $\mathbb{F}_{\Theta}=U/U_{\Theta}$ be a flag symmetric space with
symmetric Lie algebra

$\mathfrak{u}=\mathfrak{u}_{\Theta}+\mathfrak{m}.$

Then,

  1. $(a)$

$\mathcal{S}$ is a submanifold of the adjoint orbit $G\cdot H_{\Theta}$, with
$G$ being the complex Lie group with compact real form $U$. Furthermore,

$\mathcal{S}\cong U^{*}/U_{\Theta},$

where $U^{*}/U_{\Theta}$ is the dual symmetric space of $\mathbb{F}_{\Theta}$

  2. $(b)$

the submanifold $\mathcal{S}$ is diffeomorphic to a vector space

  3. $(c)$

$\mathcal{S}$ is a Riemannian homogeneous space with a Riemannian structure
defined by the Cartan–Killing form

  4. $(d)$

any fiber in the cotangent bundle of $\mathbb{F}_{\Theta}$ intersecting the
submanifold $\mathcal{S}$ intersects it _transversally_.

Once this paper also deals with cotangent bundles, it is natural to define s
symplectic form

$\omega(X,Y)=\mathrm{Im}\mathcal{K}(X,\sigma Y),\qquad X,Y\in\mathfrak{g},$

where $\mathcal{K}(\cdot,\cdot)$ is the Cartan–Killing form and $\sigma$ is
the conjugation in $\mathfrak{g}$ concerning the compact real form
$\mathfrak{u}$. Since the adjoint orbit $\mathrm{Ad}(G)H_{\Theta}$ is
contained in the Lie algebra $\mathfrak{g}$, the symplectic form $\omega$
induces a symplectic form in the adjoint orbit through the pullback of
$\omega$. The submanifold $\mathcal{S}$ and the flag symmetric space
$\mathbb{F}_{\Theta}$ are Lagrangian submanifold of $\mathrm{Ad}(G)H_{\Theta}$
with respect to $\omega$. However, the submanifold $\mathcal{S}$ is not a
Lagrangian submanifold for the KKS form (Kirillov-Kostant-Souriau):

######  Theorem D.

The symmetric flag space $\mathbb{F}_{\Theta}=U/U_{\Theta}$ and its dual
symmetric space $\mathcal{S}$ are Lagrangian submanifolds of the adjoint orbit
$G\cdot H_{\Theta}$ with the symplectic form $\omega$, although $\mathcal{S}$
is not Lagrangian for the KKS form.

We evidence that, if $g\in G$, $X\in\mathfrak{u}$ and $g\in G\mapsto u\in U$,
then the projection of the geodesics $\\{g\exp(tX)\cdot
H_{\Theta}\\}_{t\in\mathbb{R}}$ of the adjoint orbit
$\mathrm{Ad}(G)H_{\Theta}$ over $\mathbb{F}_{\Theta}$ are also geodesics
$\\{u\exp(tX)\cdot H_{\Theta}\\}_{t\in\mathbb{R}}$ in the flag symmetric space
$\mathbb{F}_{\Theta}$. Furthermore, for $g\in G$,
$Y\in\mathfrak{n}_{\Theta}^{+}$ and $g\in G\mapsto u\in U$, once more the
projection of the geodesics $\\{g\exp(tY)\cdot H_{\Theta}\\}_{t\in\mathbb{R}}$
of the adjoint orbit $\mathrm{Ad}(G)H_{\Theta}$ over $\mathbb{F}$ are
geodesics $\\{u\cdot H_{\Theta}\\}$. Here, $\mathfrak{n}_{\Theta}^{+}$ is a
sub-algebra of $\mathfrak{g}$ isomorphic to the fiber of the cotangent bundle
$T^{*}(\mathbb{F}_{\Theta})$. Considering this, in Section 5.1, we describe
some geodesics employing our realizations, both in $\mathbb{F}_{\Theta}$ and
its cotangent bundle.

As a last contribution, in Section 5.2, we obtained the following application
regarding the existence of some vector bundles over simply connected manifolds
that do not admit metrics of nonnegative sectional curvature. However, which
corresponding sphere bundles do admit, answering in negative Problem 9 in
[Zil] for connection metrics:

######  Theorem E.

For any symmetric flag space $U/U_{\Theta}$, there is a vector bundle
$U^{*}/U_{\Theta}\rightarrow\widetilde{M}\rightarrow U/U_{\Theta}$ which does
not admit a connection metric of nonnegative sectional curvature whose base is
considered with the $-\mathcal{K}$ metric, that is, the
$\mathrm{Ad}(U_{\Theta})$ metric induced by the negative of the Cartan
Killing–form on $\mathfrak{u}$. Moreover, any sphere bundle obtained
considering spheres on the fiber $U/U_{\Theta}$ has nonnegative sectional
curvature.

The exciting aspect of Theorem E is due a kind of converse to
Cheeger–Gromoll’s Soul Theorem: _any complete non-compact manifold $M$ with a
metric of nonnegative sectional curvature is diffeomorphic to the normal
bundle of some compact totally geodesic manifold_ $\Sigma\subset M$, named as
_Soul_ of $M$. That is, _which vector bundles over compact, nonnegatively
curved base spaces can admit complete metrics of nonnegative curvature?_ Until
now was not known an example of a vector bundle over a simply connected
Riemannian manifold whose associated sphere bundle admits a metric of
nonnegative sectional curvature, although the corresponding vector bundle does
not. The description here of $\mathcal{S}$ as the dual of a symmetric space of
compact type allows us to construct the first example. However, as in [GÁ17],
or [GÁZ21], up some connecting sums, these vector bundles admit metrics of
nonnegative sectional curvature.

##  2\. Lie theoretical description of flag manifolds and symmetric spaces

###  2.1. The basics of root systems and parabolic Lie algebras

Most of this work is deeply based on Lie algebra theory instead of working
directly with Lie groups. Hence, this section provides a method to obtain
parabolic sub-algebras descriptions via root systems. We are particularly
interested in the algebraic descriptions of symmetric spaces and flag
manifolds. We will review some fundamental facts about Lie Theory, including
root systems. For further details, please see [GGSM17] and references. Our
quick recall of root systems aims to be self-contained.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathfrak{h}$ a
Cartan sub-algebra with _roots set_ denoted by $\Pi$. We recall that since
$\mathbb{C}$ is algebraically closed and of zero characteristic, once
$\mathfrak{g}$ is finite-dimensional, it follows that $\mathfrak{h}$ is
Abelian. Moreover, the adjoint representation
$\mathrm{ad}:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$ enjoys the
property that the image $\mathrm{ad}(\mathfrak{h})$ consists of _semisimple
operators_. Hence, all such operators are simultaneously diagonalizable.
Considering this, we thus decompose $\mathfrak{g}$ via appropriate invariant
subspaces

(4)
$\mathfrak{g}=\bigoplus_{\lambda\in\mathfrak{h}^{*}}\mathfrak{g}_{\lambda},$

where

(5)
$\mathfrak{g}_{\lambda}:=\ker\left(\mathrm{ad}(h)-\lambda(h)1:\mathfrak{g}\rightarrow\mathfrak{g}\right),$

for $h\in\mathfrak{h}$. Hence, the roots set is nothing but
$\Pi:=\left\\{\alpha\in\mathfrak{h}^{*}\setminus\\{0\\}:\mathfrak{g}_{\alpha}\not\equiv
0\right\\}$. We reinforce that it can also be proved that for each
$\alpha\in\Pi$, it holds that
$\mathrm{dim}_{\mathbb{C}}\mathfrak{g}_{\alpha}=1$.

We can extract from $\Pi$ a subset $\Pi^{+}$ completely characterized by both:

  1. (i)

for each root $\alpha\in\Pi$ only one of $\pm\alpha$ belongs to $\Pi^{+}$

  2. (ii)

for each $\alpha,\beta\in\Pi^{+}$ necessarily $\alpha+\beta\in\Pi^{+}$ if
$\alpha+\beta\in\Pi$.

To the set $\Pi^{+}$, we name _subset of positive roots_. We say that the
subset $\Sigma\subset\Pi^{+}$ consists of the _simple roots system_ if it
collects the positive roots, which can not be written as a combination of two
elements in $\Pi^{+}$.

Since $\mathfrak{h}$ is an abelian Lie sub-algebra, we can pick a basis
$\\{H_{\alpha}:\alpha\in\Sigma\\}$ to $\mathfrak{h}$ and complete it with
$\\{X_{\alpha}\in\mathfrak{g}_{\alpha}:\alpha\in\Pi\\}$, generating what is
called a Weyl basis of $\mathfrak{g}$: for any $\alpha,\beta\in\Pi$

  1. (1)

$\mathcal{K}(X_{\alpha},X_{-\alpha}):=\mathrm{tr}~{}(\mathrm{ad}(X_{\alpha}\circ\mathrm{ad}(X_{-\alpha}))=1$

  2. (2)

$[X_{\alpha},X_{\beta}]=m_{\alpha,\beta}X_{\alpha+\beta},~{}m_{\alpha,\beta}\in\mathbb{R}$,

where $\mathcal{K}$ is the _Cartan–Killing_ form of $\mathfrak{g}$, see [SM21,
p. 214].

Then $\mathfrak{g}$ decomposes into root subspaces:

$\mathfrak{g}=\mathfrak{h}+\sum_{\alpha\in\Pi}\mathbb{C}X_{\alpha}.$

It is straightforward from [SM21, Theorem 11.13, p. 224] that such a
decomposition implies that the compact real form of $\mathfrak{g}$ is given by

(6)
$\mathfrak{u}=\sqrt{-1}\mathfrak{h}_{\mathbb{R}}+\sum_{\alpha\in\Pi^{+}}\mathrm{Span}_{\mathbb{R}}(X_{\alpha}+X_{-\alpha},\sqrt{-1}(X_{\alpha}-X_{-\alpha}))$

Given $\Theta\subset\Sigma$, an element $H_{\Theta}\in\mathfrak{h}$ is said to
be characteristic for $\Theta$ if one recognizes
$\Theta=\\{\alpha\in\Sigma:\alpha(H_{\Theta})=0\\}$. In particular, such a
subset $\Theta$ defines a _parabolic sub-algebra_ $\mathfrak{p}_{\Theta}$,
that is, it contains a maximal solvable sub-algebra of $\mathfrak{g}$, i.e., a
_Borel sub-algebra_ , with parabolic subgroup $P_{\Theta}$. It also defines a
_flag manifold_ $\mathbb{F}_{\Theta}=G/P_{\Theta}$. It can be checked that the
parabolic sub-algebra is $\mathfrak{p}_{\Theta}=\oplus_{\lambda\geq
0}\mathfrak{g}_{\lambda}$, where $\lambda$ runs through the non-negative
eigenvalues of $\mathrm{Ad}(H_{\Theta})$. Conversely, starting with
$H_{0}\in\mathfrak{h}$ we define
$\Theta_{H_{0}}=\\{\alpha\in\Sigma:\alpha(H_{0})=0\\}$, leading to the
corresponding flag manifold $\mathbb{F}_{H_{0}}:=\mathbb{F}_{\Theta_{H_{0}}}$.

Now recall that for any subset $Q\subset\Pi$, we have the decomposition
$Q=Q^{s}\cup Q^{a}$, where $Q^{s}:=Q\cap(-Q)$, and $Q^{a}:=Q\setminus Q^{s}$.
In what follows we will always denote by $\mathfrak{g}(Q)$ the algebra
generated by $Q$

(7) $\mathfrak{g}(Q):=\langle[X_{\alpha},X_{-\alpha}]=H_{\alpha},\alpha\in
Q^{s}\rangle+\sum_{\alpha\in Q}\mathbb{C}X_{\alpha}.$

Once we have recalled several aspects of the roots system, let us explain how
a system of roots, via a characteristic element, can provide a helpful
description of parabolic sub-algebras.

Fix a subset $\Theta=\\{\alpha_{1},\cdots,\alpha_{m}\\}\subset\Sigma$ of
simple roots. We shall denote by $\langle\Theta\rangle$ the set of roots
generated by $\Theta$. Also, we denote the intersection
$\langle\Theta\rangle\cap\Pi^{\pm}$ by $\langle\Theta\rangle^{\pm}$. For each
subset $\Theta$ we get the decomposition of $\mathfrak{g}$:

(8)
$\mathfrak{g}=\mathfrak{n}^{-}_{\Theta}\oplus\mathfrak{z}_{\Theta}\oplus\mathfrak{n}_{\Theta}^{+},$

where

(9)
$\mathfrak{n}^{-}_{\Theta}=\mathfrak{g}(\Pi^{-}\setminus\langle\Theta\rangle^{-})$

and

(10)
$\mathfrak{n}^{+}_{\Theta}=\mathfrak{g}(\Pi^{+}\setminus\langle\Theta\rangle^{+})$

are nilpotent sub-algebras of $\mathfrak{g}$, and

$\mathfrak{z}_{\Theta}=\mathfrak{h}+\mathfrak{g}(\Theta)$

is a reductive sub-algebra, with

$\mathfrak{g}(\Theta):=\mathfrak{g}(\langle\Theta\rangle)=\langle
H_{\alpha}\rangle_{\alpha\in\langle\Theta\rangle}+\sum_{\alpha\in\langle\Theta\rangle}\mathbb{C}X_{\alpha}.$

This manner, one gets,

(11) $\mathfrak{z}_{\Theta}=\mathfrak{z}_{\mathfrak{g}}(H_{\Theta})$

Now one can construct a parabolic sub-algebra from a subset
$\Theta\subset\Sigma$ of simple roots of $\mathfrak{g}$:

(12)
$\mathfrak{p}_{\Theta}=\mathfrak{h}+\mathfrak{g}(\langle\Theta\rangle^{-}\cup\Pi^{+})=\mathfrak{z}_{\Theta}+\mathfrak{n}_{\Theta}^{+}=\sum_{\alpha\in\langle\Theta\rangle^{-}}\mathfrak{g}_{\alpha}+\mathfrak{b},$

where $\mathfrak{b}$ is a Borel sub-algebra.

Inspired by such a description, one defines flag manifolds associated with any
_connected complex semisimple Lie group_ $G$: since a parabolic subgroup
$P_{\Theta}$, with Lie algebra $\mathfrak{p}_{\theta}$, is the normalizer of
$\mathfrak{p}_{\Theta}$ in $G$, i.e., $P_{\Theta}=\\{g\in
G:\mathrm{Ad}(g)\mathfrak{p}_{\Theta}=\mathfrak{p}_{\Theta}\\}$, one has

######  Definition 1.

A _flag manifold_ of a complex semisimple Lie group $G$ is the quotient
$M=G/P_{\Theta}$ of $G$ by a parabolic subgroup $P_{\Theta}$.

###  2.2. Flag Symmetric Spaces and their algebraic aspects

This section will explore the Lie algebraic description of symmetric spaces,
such as given in [KN69, Chapter XI], jointly with their algebraic descriptions
of parabolic sub-algebras, to study symmetric spaces that are also flag
manifolds. More precisely, we call _symmetric flag spaces_ to symmetric
spaces, which are also flag manifolds.

We recall that a symmetric space is a triple $(G,H,\sigma)$ consisting of a
connected Lie group $G$, a closed subgroup $H$ of $G$, and an involutive
automorphism $\sigma$ of $G$ such that $H$ lies between $G_{\sigma}$ and the
identity component of $G_{\sigma}$, where $G_{\sigma}$ denotes the closed
subgroup of $G$ consisting of all elements left fixed by $\sigma$. An
infinitesimal version of a symmetric space, named as a symmetric Lie algebra,
is a triple $(\mathfrak{g},\mathfrak{h},\sigma)$ consisting of a Lie algebra
$\mathfrak{g}$, a sub-algebra $\mathfrak{h}$ of $\mathfrak{g}$, and an
involutive automorphism $\sigma$ of $\mathfrak{g}$ such that $\mathfrak{h}$
consists of all elements of $\mathfrak{g}$ which are left fixed by $\sigma$.

Every symmetric space $(G,H,\sigma)$ gives rise to a symmetric Lie algebra
$(\mathfrak{g},\mathfrak{h},\sigma)$ in a natural manner; $\mathfrak{g}$ and
$\mathfrak{h}$ are the Lie algebras of $G$ and $H$, respectively, and the
automorphism $\sigma$ of $\mathfrak{g}$ is the one induced by the automorphism
$\sigma$ of $G$. Conversely, if $(\mathfrak{g},\mathfrak{h},\sigma)$ is a
symmetric Lie algebra and $G$ is a simply connected Lie group with Lie algebra
$\mathfrak{g}$, the automorphism $\sigma$ of $\mathfrak{g}$ induces an
automorphism $\sigma$ of $G$. Moreover, for any subgroup $H$ lying between
$G_{\sigma}$ and the identity component of $G_{\sigma}$, the triple is a
symmetric space. The pair $(G,H)$ is called a symmetric pair.

Let $(\mathfrak{g},\mathfrak{h},\sigma)$ be a symmetric Lie algebra. Since
$\sigma$ is involutive, seeing it as a linear operator, one gets that its
eigenvalues are $1$ and $-1$. One then recoveries $\mathfrak{h}$ as the
eigenspace associated with $1$. Let $\mathfrak{m}$ be the eigenspace
associated with $-1$. The decomposition

(13) $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$

is called the _canonical decomposition_ of
$(\mathfrak{g},\mathfrak{h},\sigma)$.

Since the following shall play some role, it is worth recalling that

######  Proposition 2.1.

[KN69, Proposition 2.1, p. 226] If $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is
the canonical decomposition of a symmetric Lie algebra
$(\mathfrak{g},\mathfrak{h},\sigma)$, then

(14)
$[\mathfrak{h},\mathfrak{h}]\subset\mathfrak{h},\quad[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m},\quad[\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}.$

######  Proposition 2.2.

[KN69, Proposition 2.2, p. 227] Let $(G,H,\sigma)$ be a symmetric space and
$(\mathfrak{g},\mathfrak{h},\sigma)$ its symmetric Lie algebra. If
$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ is the canonical decomposition of
$(\mathfrak{g},\mathfrak{h},\sigma)$, then

(15) $\mathrm{Ad}(H)\mathfrak{m}\subset\mathfrak{m}.$

Let $\mathbb{F}_{\Theta}$ be a flag manifold, just as defined in Definition 1.
It is remarkable, considering the discussions on both sections 2.1 and 2.2,
that for flag-symmetric spaces, there exists an involutive inner automorphism
$\sigma$ over $\mathfrak{g}$ (unless explicitly said, we use the same notation
for the automorphism over $G$, $\mathfrak{u}$ and $U$) such that the
$\sigma$-fixed point set in $\mathfrak{g}$ is

(16)
$\mathfrak{g}^{\sigma}=\mathfrak{z}_{\Theta}=\mathfrak{z}_{\mathfrak{g}}(H_{\Theta}),$

where we do recall that $\mathfrak{z}_{\mathfrak{g}}(H_{\theta})$ is obtained
by the description given by equation (7). Then the tangent space of the
homogeneous space $G/G^{\sigma}$ at the origin can be identified with

$\mathfrak{m}_{G}\cong\mathfrak{n}_{\Theta}^{-}\oplus\mathfrak{n}_{\Theta}^{+},$

recall the definitions of both $\mathfrak{n}_{\Theta}^{\pm}$ on equations (9)
and (10), respectively.

Hence, in this context, the canonical decomposition (respecting Proposition
2.1 of the complex Lie algebra is given by

(17) $\mathfrak{g}=\mathfrak{z}_{\Theta}\oplus\mathfrak{m}_{G}.$

We remark that the real complex form $U$ of $G$ also defines an action of
$\mathbb{F}_{\Theta}$ via conjugation. If $U$ corresponds to the compact real
form of $G$ then it can be shown that $\mathbb{F}_{\Theta}=U/U_{\Theta}$ where
$U_{\Theta}=U\cap P_{\Theta}$.

######  Proposition 2.3.

Let $G$ be a Lie group with compact real form $U$ and let $P_{\Theta}$ be a
parabolic subgroup of $G$. Any flag manifold $M=G/P_{\Theta}$ can be
identified with the quotient $U/Z_{\Theta}^{\sigma}:=U/U_{\Theta}$ of a
semisimple compact Lie group $U\subset G$ modulo the centralizer
$Z_{\Theta}^{\sigma}$, associated to the adjoint representation
$\mathrm{Ad}(U)$, for some characteristic element $H_{\theta}$. Hence, it is
recovered by an adjoint orbit.

For a better understanding of such identification, and since this shall be
needed for our results, we present a sketch of the proof of Proposition 2.3.

###### Sketch of the proof of Proposition 2.3.

The decomposition given by equation (8) induces a decomposition of some open
subset of $G$ (connected complex semisimple Lie group with Lie algebra
$\mathfrak{g}$) into the product of correspondent subgroups:

$G_{\mathrm{reg}}=N_{\Theta}^{-}\cdot Z_{\Theta}\cdot N^{+}_{\Theta}$

with $Z_{\Theta}\cap N^{\pm}_{\Theta}=N_{\Theta}^{+}\cap
N_{\Theta}^{-}=\\{e\\}$.

The parabolic subgroup is

$P_{\Theta}=Z_{\Theta}\cdot N^{+}_{\Theta}$

This decomposition shows that the nilpotent subgroup $N_{\Theta}^{-}$ acts
transitively on the open dense subset
$M_{\mathrm{reg}}=G_{\mathrm{reg}}/P_{\Theta}$ of the flag manifold
$M=G/P_{\Theta}$. Hence, any complex coordinates on $N_{\Theta}^{-}$ define
local complex coordinates on $M$.

Let $\tau$ be the involution on $\mathfrak{g}$ for the real compact form
$\mathfrak{u}$ (the same notation is used for the Lie groups). A maximal
compact subgroup $G^{\tau}=:U$ of $G$ acts on $M$ transitively. For a fixed
$x\in M$, the isotropy subgroup in $U$ will be denoted by $U_{x}$. Denoting by
$\mathfrak{u}_{x}$ its corresponding Lie algebra, one has

$\mathfrak{u}_{x}=\mathfrak{u}\cap\mathfrak{p}_{\Theta}=\mathfrak{p}^{\tau}_{\Theta}=\\{X\in\mathfrak{p}_{\Theta}:\tau
X=X\\}$

Since the sub-algebra
$\mathfrak{z}_{\Theta}=\mathfrak{h}+\mathfrak{g}(\Theta)$ is invariant under
the involution $\tau$; and $\tau(X_{\alpha})=-X_{-\alpha}$ for all
$\alpha\in\Pi$, one gets

$\tau(\mathfrak{n}_{\Theta}^{+})=\mathfrak{n}_{\Theta}^{-},$

and that

(18)
$\mathfrak{u}_{\Theta}:=\mathfrak{p}^{\tau}_{\Theta}=\mathfrak{z}_{\Theta}^{\tau}=\mathfrak{h}^{\tau}+(\mathfrak{g}(\Theta))^{\tau}$

is a real compact form of the sub-algebra $\mathfrak{z}_{\Theta}$. Therefore,
$\mathfrak{u}_{\Theta}=\mathfrak{z}_{\Theta}\cap\mathfrak{u}$ is the
centralizer of a comutative sub-algebra
$\mathfrak{h}_{\Theta}:=\mathfrak{h}\cap\mathfrak{u}_{\Theta}$.

On the other hand, $\mathfrak{h}_{\Theta}$ generates a torus $T$ in the
compact connected semisimple Lie group $U$. Moreover, the centralizer of a
torus in a connected compact semisimple Lie group is connected, and there
exists an element $t_{0}$ in $\sqrt{-1}\mathfrak{h}_{\mathbb{R}}$ such that,
recalling the notation defined by equation (7), one gets

(19) $U_{\Theta}=Z_{U}(\mathfrak{h}_{\Theta})=Z_{U}(T)=Z_{U}(t_{0}).\qed$

##  3\. The Cotangent bundle of a symmetric flag space

As mentioned in the Introduction, in [GGSM17], the adjoint orbit $G\cdot
H_{\Theta}:=\mathrm{Ad}(G)H_{\Theta}$ realizes the cotangent bundle of the
complex flag manifold $\mathbb{F}_{\Theta}$.

######  Theorem 3.1.

[GGSM17, Theorem 2.1] The adjoint orbit
$\mathcal{O}(H_{\Theta})=\mathrm{Ad}(G)\cdot H_{\Theta}\approx G/Z_{\Theta}$
of the characteristic element $H_{\Theta}$ is a $C^{\infty}$ vector bundle
over $\mathbb{F}_{\Theta}$ that is isomorphic to the cotangent bundle
$T^{*}\mathbb{F}_{\Theta}$. Moreover, we can write down a diffeomorphism
$\iota:\mathrm{Ad}(G)\cdot H_{\Theta}\rightarrow T^{*}\mathbb{F}_{\Theta}$
such that

  1. $(a)$

$\iota$ is equivariant with respect to the actions of $U$, that is, for all
$u\in U$,

$\iota\circ\mathrm{Ad}(u)=\tilde{u}\circ\iota,$

where $\tilde{u}$ is the lifting to $T^{*}\mathbb{F}_{\Theta}$ (via the
differential) of the action of $u$ on $\mathbb{F}_{\Theta}$; and

  2. $(b)$

the pullback of the canonical simplectic form on $T^{*}\mathbb{F}_{\Theta}$ by
$\iota$ is the (real) Kirillov-Kostant-Souriau form on the orbit.

On the one hand, the realization of the cotangent bundle is generally with the
coadjoint orbit. However, we claim that such an orbit can be identified with
the adjoint orbit, also realized by the homogeneous space $G/Z_{\Theta}$,
where $Z_{\Theta}$ is the centralizer of $H_{\Theta}$ in $G$ by the adjoint
representation. The precise reason for this _exchangeability_ on the possible
representations is because _for semisimple Lie algebras, the adjoint
representation behaves like the coadjoint representation_.

Indeed, we recall that the _coadjoint representation_ $\mathrm{Ad}^{*}$ of $G$
on the dual $\mathfrak{g}^{*}$ of the Lie algebra $\mathfrak{g}$ is defined by
$\mathrm{Ad}^{*}(g)\alpha=\alpha\circ\mathrm{Ad}(g^{-1}),~{}g\in G$ and
$\alpha\in\mathfrak{g}^{*}$. Moreover, its infinitesimal representation
$\mathrm{ad}^{*}:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$ is given
by $\mathrm{ad}^{*}(X)\alpha=-\alpha\circ\mathrm{Ad}(X)$. Hence, $G$ acts on
$\mathfrak{g}^{*}$ by the representation $\mathrm{Ad}^{*}$, being the induced
action vector fields $\tilde{X}$, $X\in\mathfrak{g}$, given by
$\tilde{X}=\mathrm{ad}^{*}(X)$.

For $\alpha\in\mathfrak{g}^{*}$, the coadjoint orbit
$\mathrm{Ad}^{*}(G)\alpha$ is identified with the homogeneous space
$G/Z_{\alpha}$, where $Z_{\alpha}$ is the closed subgroup

$Z_{\alpha}=\\{g\in G:\alpha\circ\mathrm{Ad}(g^{-1})=\alpha\\}.$

The Lie algebra $\mathfrak{z}_{\alpha}$ of $Z_{\alpha}$, is given by
$\mathfrak{z}_{\alpha}=\\{X\in\mathfrak{g}:\alpha\circ\mathrm{ad}(X)=0\\}$.
Moreover, the tangent space of $\mathrm{Ad}^{*}(G)\alpha$ at $\alpha$ is given
by

$T_{\alpha}(\mathrm{Ad}^{*}(G)\alpha)=\\{\mathrm{ad}^{*}(X)\alpha:X\in\mathfrak{g}\\}.$

Following [SM21, p. 324], the Cartan–Killing form $\mathcal{K}(\cdot,\cdot)$
of a Lie algebra $\mathfrak{g}$ of the semisimple group $G$ is non-degenerate
and define an isomorphism $W:\mathfrak{g}\rightarrow\mathfrak{g}^{*}$ by
$W(X)(\cdot)=\mathcal{K}(X,\cdot)$. This isomorphism exchanges the adjoint and
coadjoint representations, i.e.,

$W\mathrm{Ad}(g)=\mathrm{Ad}^{*}(g)W$

for every $g\in G$, since $\mathcal{K}$ is $\mathrm{Ad}$-invariant. Hence, $W$
applies diffeomorphically the adjoint orbits in the coadjoint orbits.

Concerning Theorem 3.1, the projection $\pi:G\cdot
H_{\Theta}\rightarrow\mathbb{F}_{\Theta}$ is obtained via the action of $G$:
the canonical fibration $gZ_{\Theta}\in G/Z_{\Theta}\mapsto
gP_{\Theta}\in\mathbb{F}_{\Theta}$ has as fiber $P_{\Theta}/Z_{\Theta}$. In
this manner, in terms of the adjoint representation, the fiber is
$\mathrm{Ad}(P_{\Theta})\cdot H_{\Theta}$, which coincides with the affine
subspace $H_{\Theta}+\mathfrak{n}_{\Theta}^{+}$ (see [GGSM17, p. 364]).

Moreover, in [GGSM17, Section 2.2, p. 365] it is also described the
isomorphism of the adjoint orbit $G\cdot H_{\Theta}$ with the cotangent bundle
$T^{*}\mathbb{F}_{\Theta}$. The tangent space
$T_{b_{\Theta}}\mathbb{F}_{\Theta}$ of the flag manifold $\mathbb{F}_{\Theta}$
at the origin $b_{\Theta}$ can be identified with $\mathfrak{n}_{\Theta}^{-}$
and the isotropy representation
$U_{\Theta}\rightarrow$Gl$(T_{b_{\Theta}}\mathbb{F}_{\Theta})$ becomes the
restriction of the adjoint representation. Hence, the subspace
$\mathfrak{n}^{+}_{\Theta}$ is isomorphic to the dual
$(\mathfrak{n}_{\Theta}^{-})^{*}$ of $\mathfrak{n}_{\Theta}^{-}$ via the
Cartan–Killing form $\mathcal{K}(\cdot,\cdot)$ of $\mathfrak{g}$, being the
map
$X\in\mathfrak{n}_{\Theta}^{+}\mapsto\mathcal{K}(X,\cdot)\in(\mathfrak{n}_{\Theta}^{-})^{*}$
an isomorphism.

###  3.1. Characterization of some symmetric spaces and flag manifolds

We begin recalling that, according to [KN69, Chapter XI.7], the _orthogonal
symmetric Lie algebras_ are of compact and non-compact type. Moreover, they
are of four classes (these former concepts shall be approached in further
detail in Section 5.2). The types of symmetric spaces are related to their
geometric properties as follows

######  Theorem 3.2.

[KN69, Theorem 8.6, p. 256] Let $(G,H,\sigma)$ be a symmetric space with
$\mathrm{ad}_{\mathfrak{g}}(\mathfrak{h})$ compact and let
$(\mathfrak{g},\mathfrak{h},\sigma)$ be its orthogonal symmetric Lie algebra.
Take any $G$-invariant Riemannian metric on $G/H$. Then,

  * (1)

if $(\mathfrak{g},\mathfrak{h},\sigma)$ is of compact type, then $G/H$ is a
compact Riemannian symmetric space with non-negative sectional curvature and
positive definite Ricci tensor

  * (2)

if $(\mathfrak{g},\mathfrak{h},\sigma)$ is of non-compact type, then $G/H$ is
a simply connected non-compact Riemannian symmetric space with non-positive
sectional curvature and negative-definite Ricci tensor and is diffeomorphic to
a Euclidean space.

On the one hand, on [Hel62, Table V, p.518] we can find the following table of
irreducible Riemannian globally symmetric classical spaces of type I and III,
with $p+q=n$:

|  Compact symmetric space |  Noncompact symmetric space
---|---|---
AI |  $\mathrm{SU}(n)/\mathrm{SO}(n)$ |  $\mathrm{Sl}(n,\mathbb{R})/\mathrm{SO}(n)$
AII |  $\mathrm{SU}(2n)/\mathrm{Sp}(n)$ |  $\mathrm{SU}^{*}(2n)/\mathrm{Sp}(n)$
AIII |  $\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ |  $\mathrm{SU}(p,q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$
BDI |  $\mathrm{SO}(p+q)/\mathrm{SO}(p)\times\mathrm{SO}(q)$ |  $\mathrm{SO}_{o}(p,q)/\mathrm{SO}(p)\times\mathrm{SO}(q)$
DIII |  $\mathrm{SO}(2n)/\mathrm{U}(n)$ |  $\mathrm{SO}^{*}(2n)/\mathrm{U}(n)$
CI |  $\mathrm{Sp}(n)/\mathrm{U}(n)$ |  $\mathrm{Sp}(n,\mathbb{R})/\mathrm{U}(n)$
CII |  $\mathrm{Sp}(p+q)/\mathrm{Sp}(p)\times\mathrm{Sp}(q)$ |  $\mathrm{Sp}(p,q)/\mathrm{Sp}(p)\times\mathrm{Sp}(q)$

On the other hand, an essential class of symmetric spaces generalizing the
Riemannian symmetric spaces is _pseudo-Riemannian symmetric spaces_ , in which
an indefinite Riemannian metric replaces the Riemannian metric, which leads,
for instance, to Lorentzian symmetric spaces. Generally, symmetric and locally
symmetric spaces can be seen as affine symmetric spaces. As shall be further
approached in Section 5.2, if $G/H$ is a symmetric space, we can provide some
$G$-invariant natural connection associated with $G/H$. Conversely, a manifold
with such a connection is locally symmetric. It is a symmetric space up to
passing to its universal covering.

Such manifolds can also be described as those affine manifolds whose _geodesic
symmetries_ are all globally defined affine diffeomorphisms, generalizing the
Riemannian and pseudo-Riemannian case. In [Ber57, Tableau I, p. 114] we find
the following table of symmetric spaces for each _classical_ complex simple
Lie group:

G |  |  |
---|---|---|---
$\mathrm{Sl}(n,\mathbb{C})$ |  $G/\mathrm{SO}(n,\mathbb{C})$ |  $G/\left(\mathrm{S}(\mathrm{Gl}(k,\mathbb{C})\times\mathrm{Gl}(l,\mathbb{C}))\right)$, $k+l=n$ |  $G/\mathrm{Sp}(n,\mathbb{C})$
$\mathrm{SO}(n,\mathbb{C})$ |  $G/\left(\mathrm{SO}(k,\mathbb{C})\times\mathrm{SO}(l,\mathbb{C})\right)$, $k+l=n$ |  $G/\mathrm{Gl}(n/2,\mathbb{C})$, n even |
$\mathrm{Sp}(n,\mathbb{C})$ |  $G/\left(\mathrm{Sp}(k,\mathbb{C})\times\mathrm{Sp}(l,\mathbb{C})\right)$, $k+l=n$ |  $G/\mathrm{Gl}(n,\mathbb{C})$ |

The next table shows the flag manifolds of the classical Lie groups. Here,
consider $n=n_{1}+\cdots+n_{s}$, $n_{1}\geq\cdots\geq n_{s}\geq 1$ and
$l=l_{1}+\cdots+l_{k}+m$, $l_{1}\geq\cdots\geq l_{k}\geq 1$, $k,m\geq 0$.

|  Flag manifold
---|---
A |  $\mathrm{SU}(n)/\mathrm{S}(\mathrm{U}(n_{1})\times\cdots\times\mathrm{U}(n_{s}))$
B |  $\mathrm{SO}(2l+1)/\mathrm{U}(l_{1})\times\cdots\times\mathrm{U}(l_{k})\times\mathrm{SO}(2m+1)$
C |  $\mathrm{Sp}(l)/\mathrm{U}(l_{1})\times\cdots\times\mathrm{U}(l_{k})\times\mathrm{Sp}(m)$
D |  $\mathrm{SO}(2l)/\mathrm{U}(l_{1})\times\cdots\times\mathrm{U}(l_{k})\times\mathrm{SO}(2m)$

Comparing the tables above, we can see that

######  Lemma 1.

The symmetric spaces as in Theorem 3.1 that are also flag manifolds are

$\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q)),~{}~{}~{}\mathrm{Sp}(l)/\mathrm{U}(l)~{}~{}~{}\text{and}~{}~{}~{}\mathrm{SO}(2l)/\mathrm{U}(l),$

of types _A,C_ and _D_ , respectively.

######  Remark 1.

It is worth mentioning that we are frequently identifying the space
$\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}$ with
$\mathfrak{gl}(n,\mathbb{C})$. Precisely, following [Ber57], we make such
identification as follows: let $A\in\mathfrak{gl}(n,\mathbb{C})$. Consider the
injective $\mathcal{L}$ map between $\mathfrak{gl}(n,\mathbb{C})$ and
$\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}$ given by:

For $A\in\mathfrak{sl}(n,\mathbb{C})$ we make

$\mathcal{L}:A=\left[A+\left(\begin{array}[]{cc}-1&0\\\
0&\frac{1}{n-1}I_{n-1}\end{array}\right)\right]+\left(\begin{array}[]{cc}1&0\\\
0&-\frac{1}{n-1}I_{n-1}\end{array}\right)\mapsto
A+\frac{1}{1-n}\in\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}.$

For $A\in\mathfrak{gl}(n,\mathbb{C})\ominus\mathfrak{sl}(n,\mathbb{C})$ we let

$\mathcal{L}:A=\left[A-\frac{\mathrm{tr}~{}A}{n}I_{n}\right]+\frac{\mathrm{tr}~{}A}{n}I_{n}\mapsto\left[A-\frac{\mathrm{tr}~{}A}{n}I_{n}\right]+\frac{\mathrm{tr}~{}A}{n}\in\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}.$

The map $\mathcal{L}$ assures that
$\mathfrak{gl}(n,\mathbb{C})\subset\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}$
and the other containment is straightforward.

###  3.2. Symmetric spaces on the adjoint orbit

The flag manifolds given by Lemma 1 are Riemannian symmetric manifolds. If we
denote by $(\mathfrak{u},\mathfrak{k},\sigma)$ their orthogonal Lie algebra of
compact type, one gets that these are of class (I). Next, we shall see these
spaces as homogeneous spaces with canonical connections, which is also
invariant; see [KN69, Theorem 3.1, p. 230].

In Section 6, we shall prove Theorem 3.3 below, i.e., that the cotangent
bundle of a symmetric flag space is a symmetric space.

######  Theorem 3.3.

Let $\mathbb{F}_{\Theta}$ be a symmetric flag space identified with the
homogeneous space

(20) $\mathbb{F}_{\Theta}=U/U_{\Theta},$

as in Proposition 2.3. Then, there exists an element $H_{\Theta}$ in a fixed
Cartan sub-algebra of the compact Lie algebra $\mathfrak{u}$ such that for
$\sigma:=C_{\exp H_{\Theta}}$,

(21) $U_{\Theta}=\\{u\in U:C_{\exp H_{\Theta}}(u)=u\\}.$

Moreover, let $G$ be the complex Lie group such that $U$ is its compact real
form. Then, the cotangent bundle of $\mathbb{F}_{\Theta}$ has a realization as
a symmetric space.

If we also use the notation $\sigma$ for $\mathrm{Ad}_{\mathfrak{u}}(\exp
H_{\Theta})$ and $\mathrm{Ad}_{\mathfrak{g}}(\exp H_{\Theta})$, then
$(\mathfrak{u},\mathfrak{u}_{\Theta},\sigma)$ and
$(\mathfrak{g},\mathfrak{z}_{\mathfrak{g}}(H_{\Theta}),\sigma)$ are symmetric
Lie algebras.

According to Theorem 3.3, it is possible to find an element $H_{\Theta}$ in a
Cartan sub-algebra of $\mathfrak{g}$, determining the automorphism $\sigma$
for two symmetric spaces, and then, for two symmetric Lie algebras.

Since Lie groups here considered are all semisimple, the Cartan–Killing form
$\mathcal{K}(\cdot,\cdot)$ is a non-degenerate bilinear form, which is also
invariant by automorphisms. Then, it is $\mathrm{Ad}(H)$ invariant, where $H$
stands for the isotropy subgroup for symmetric flag spaces and adjoint orbits.
In the case of symmetric flag spaces, the Cartan Killing form is negative
definite. Hence, $-\mathcal{K}(\cdot,\cdot)$ generates a $U$-invariant
Riemannian metric on $\mathbb{F}_{\Theta}=U/U_{\Theta}$. However, for the
cotangent bundle $T^{*}\mathbb{F}_{\Theta}\cong G/Z_{\Theta}$, there exists no
$\mathrm{Ad}(Z_{\Theta})$-invariant (definite) metric, because $Z_{\Theta}$ is
never a group of isometries.

The proof of Theorem 3.3 is nothing but a generalization of the examples 1–3
below. We first present a real unidimensional flag to know
$\mathrm{SO}(2)/\\{\pm\mathrm{Id}\\}$, which is a symmetric space such that
its dual symmetric space is contained in the bi-dimensional adjoint orbit
diffeomorphic to the cotangent bundle of
$\mathrm{SO}(2)/\\{\pm\mathrm{Id}\\}$.

######  Example 1.

Let us consider the following basis of $\mathfrak{sl}(2,\mathbb{R})$:

(22) $A=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),\quad
B=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right),\quad
C=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right).$

The compact Lie group

$\mathrm{SO}(2)=\left\\{\left(\begin{array}[]{cc}\cos s&-\sin s\\\ \sin s&\cos
s\end{array}\right):s\in\mathbb{R}\right\\}$

determines the maximal flag manifold

(25) $\displaystyle\mathrm{SO}(2)\cdot B=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}\cos s&-\sin s\\\ \sin s&\cos
s\end{array}\right)\cdot B=\sin(2s)A+\cos(2s)B+0C:s\in\mathbb{R}\right\\}$
(26) $\displaystyle=$ $\displaystyle\mathrm{S^{1}}$

contained in the real vector space
$\mathfrak{sl}(2,\mathbb{R})\ominus\mathfrak{so}(2)=\mathrm{span}_{\mathbb{R}}\\{A,B\\}$.
This flag manifold is symmetric since it is the $1$-sphere $\mathrm{S^{1}}$.

The adjoint orbit $\mathrm{Sl}(2,\mathbb{R})\cdot
B:=\mathrm{Ad}(\mathrm{Sl}(2,\mathbb{R}))B$ consists in the set of matrices
$xA+yB+zC$ for $x,y,z\in\mathbb{R}$, with eigenvalues $\pm 1$, i.e.,

(27) $\mathrm{det}\left(\begin{array}[]{cc}y&x+z\\\
x-z&-y\end{array}\right)=\mathrm{det}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$

Then, the adjoint orbit $\mathrm{Sl}(2,\mathbb{R})\cdot B$ is the one-sheeted
hyperboloid

(28) $x^{2}+y^{2}-z^{2}=1.$

This manifold has a realization as the homogeneous space

$\mathrm{Sl}(2,\mathbb{R})/\mathrm{S}(\mathrm{Gl}(1,\mathbb{R})\times\mathrm{Gl}(1,\mathbb{R}))\newline
\cong\mathrm{Sl}(2,\mathbb{R})/\mathbb{R}^{*},$

where $\mathbb{R}^{*}=\mathbb{R}\setminus\\{0\\}$.

The cotangent bundle of $\mathrm{S^{1}}$ is also the union of fibers
$T^{*}\mathrm{S^{1}}=\bigcup_{k\in\mathrm{SO}(2)}k\cdot(B+\mathfrak{n}^{+})$,
with $\mathfrak{n}^{+}$ being the Lie sub-algebra of
$\mathfrak{sl}(2,\mathbb{R})$ defined by the upper triangular matrices. For
each element

(29) $k=\left(\begin{array}[]{cc}\cos t&\sin t\\\ -\sin t&\cos
t\end{array}\right)\in\mathrm{SO}(2),\quad t\in\mathbb{R},$

the fiber $k\cdot(B+\mathfrak{n}^{+})$ is the one-dimensional affine vector
space

(34) $\displaystyle\mathrm{Ad}(k)(B+\mathfrak{n}^{+}):=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}\cos t&\sin t\\\ -\sin t&\cos
t\end{array}\right)\cdot\left[B+\left(\begin{array}[]{cc}0&r\\\
0&0\end{array}\right)\right]:r\in\mathbb{R}\right\\}$ (37) $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}\cos(2t)-\frac{r}{2}\sin(2t)&-\sin(2t)+r(\cos
t)^{2}\\\ -\sin(2t)-r(\sin
t)^{2}&-\cos(2t)-\frac{r}{2}\sin(2t)\end{array}\right):r\in\mathbb{R}\right\\}$
(38) $\displaystyle=$
$\displaystyle\left\\{(-\sin(2t)+\frac{r}{2}\cos(2t))A+(\cos(2t)+\frac{r}{2}\sin(2t))B+\frac{r}{2}C:r\in\mathbb{R}\right\\}.$

A vector equation defining each fiber above is

(39) $\left(\begin{array}[]{c}x\\\ y\\\
z\end{array}\right)=\left(\begin{array}[]{c}-\sin(2t)\\\ \cos(2t)\\\
0\end{array}\right)+\frac{r}{2}\left(\begin{array}[]{c}\cos(2t)\\\ \sin(2t)\\\
1\end{array}\right)\quad r\in\mathbb{R}.$

######  Example 2.

Here we present an example of a low-dimension symmetric space, to be further
studied in Section 6. Consider the Lie group $G=\mathrm{Sl}(2,\mathbb{C})$,
with compact real form $U=\mathrm{SU}(2)$ and take the element in the Lie
algebra $\mathfrak{u}=\mathfrak{su}(2)$

(40) $H_{\Theta}=\frac{\sqrt{-1}\pi}{2}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right).$

This element defines the maximal flag manifold
$\mathbb{F}_{\Theta}=U/U_{\Theta}$ for $\Theta=\emptyset$, where $U_{\Theta}$
is the centralizer of $H_{\Theta}$ in $U$. The Lie group

(41) $\mathrm{SU}(2)=\left\\{\left(\begin{array}[]{cc}\alpha&\beta\\\
-\bar{\beta}&\bar{\alpha}\end{array}\right):\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}=1,\alpha,\beta\in\mathbb{C}\right\\}$

is diffeomorphic to the $3$-sphere $\mathrm{S^{3}}$ and $Z_{U}(H_{\Theta})$ is
the subgroup

(44) $\displaystyle\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(1))=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}\alpha&0\\\
0&\bar{\alpha}\end{array}\right):\lvert\alpha\rvert^{2}=1,\alpha\in\mathbb{C}\right\\}$
(47) $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}e^{i\theta}&0\\\
0&e^{-i\theta}\end{array}\right):\theta\in[0,2\pi]\right\\}$

diffeomorphic to the unidimensional sphere $\mathrm{S^{1}}$.

Then, the flag manifold $\mathbb{F}_{\Theta}$ is a symmetric space and is the
Riemann sphere

$\mathrm{SU}(2)/\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(1))=\mathrm{S^{3}}/\mathrm{S^{1}}=\mathbb{C}\mathrm{P}^{1}\cong
S^{2}.$

The cotangent bundle of
$\mathrm{SU}(2)/\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(1))$ has a
realization as the homogeneous space

$G/Z_{G}(H_{\Theta})=\mathrm{Sl}(2,\mathbb{C})/\mathrm{S}(\mathrm{Gl}(1,\mathbb{C})\times\mathrm{Gl}(1,\mathbb{C}))$

with

$\mathrm{S}(\mathrm{Gl}(1,\mathbb{C})\times\mathrm{Gl}(1,\mathbb{C}))=\left\\{\left(\begin{array}[]{cc}z&0\\\
0&z^{-1}\end{array}\right):z\in\mathbb{C}^{*}\right\\}\cong\mathbb{C}^{*}:=\mathbb{C}\setminus\\{0\\}$.

In Section 6, we prove that the cotangent bundle of the $2$-sphere is a
symmetric space since the element $H_{\Theta}$ defined in (40) is a particular
case of the matrix in the equation (75). This guarantees that the homogeneous
space $T^{*}(S^{2})=\mathrm{Sl}(2,\mathbb{C})/\mathbb{C}^{*}$ is a symmetric
space.

Furthermore, the following is another realization of the cotangent bundle of
$S^{2}$ as the adjoint orbit of $G=\mathrm{Sl}(2,\mathbb{C})$ in $H_{\Theta}$

$\displaystyle G\cdot H_{\Theta}=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}z_{1}&z_{2}\\\
z_{3}&z_{4}\end{array}\right)H_{\Theta}\left(\begin{array}[]{cc}z_{4}&-z_{2}\\\
-z_{3}&z_{1}\end{array}\right):z_{1}z_{4}-z_{2}z_{3}=1,z_{1},z_{2},z_{3}\in\mathbb{C}\right\\}$
$\displaystyle=$
$\displaystyle\left\\{\dfrac{\sqrt{-1}\pi}{2}\left(\begin{array}[]{cc}1+2z_{2}z_{3}&-2z_{1}z_{2}\\\
2z_{3}z_{4}&-1-2z_{3}z_{2}\end{array}\right):z_{1}z_{4}-z_{2}z_{3}=1,z_{1},z_{2},z_{3}\in\mathbb{C}\right\\}.$

######  Remark 2.

In all the cases of symmetric flag spaces to be considered here, we have
symmetric Lie algebras $(\mathfrak{g},\mathfrak{z}_{\Theta},\sigma)$ in the
cotangent bundle satisfying

  1. $(i)$

$\mathfrak{g}$ is simple and is a vector direct sum
$\mathfrak{n}_{\Theta}^{-}+\mathfrak{z}_{\Theta}+\mathfrak{n}_{\Theta}^{+}$
with the relations

(48)
$\begin{array}[]{ccc}\left[\mathfrak{z}_{\Theta},\mathfrak{z}_{\Theta}\right]\subset\mathfrak{z}_{\Theta},&\left[\mathfrak{z}_{\Theta},\mathfrak{n}_{\Theta}^{-}\right]\subset\mathfrak{n}_{\Theta}^{-},&\left[\mathfrak{z}_{\Theta},\mathfrak{n}_{\Theta}^{+}\right]\subset\mathfrak{n}_{\Theta}^{+},\\\
\left[\mathfrak{n}_{\Theta}^{-},\mathfrak{n}_{\Theta}^{+}\right]\subset\mathfrak{z}_{\Theta},&\left[\mathfrak{n}_{\Theta}^{-},\mathfrak{n}_{\Theta}^{+}\right]=0,&\left[\mathfrak{n}_{\Theta}^{+},\mathfrak{n}_{\Theta}^{+}\right]=0;\end{array}$

  2. $(ii)$

the canonical decomposition
$\mathfrak{g}=\mathfrak{z}_{\Theta}\oplus\mathfrak{m}_{G}$ is given by the
direct sum

$\mathfrak{m}_{G}=\mathfrak{n}_{\Theta}^{-}+\mathfrak{n}^{+}_{\Theta};$

  3. $(iii)$

with respect to the Cartan–Killing form $\mathcal{K}$ of $\mathfrak{g}$, the
subspaces $\mathfrak{n}_{\Theta}^{\pm}$ are dual to each other and, moreover,

$\mathcal{K}(\mathfrak{n}_{\Theta}^{-},\mathfrak{n}_{\Theta}^{+})=0\quad\mathrm{and}\quad\mathcal{K}(\mathfrak{n}_{\Theta}^{+},\mathfrak{n}_{\Theta}^{+})=0$

###  3.3. The dual symmetric space to a symmetric flag space

So far, the symmetric spaces considered are Riemannian, considering the
Cartan–Killing form as a metric. On the adjoint orbit of $G$, this metric is
indefinite and invariant for the symmetries. However, in the flag manifold
$\mathbb{F}_{\Theta}$, it is the opposite of a (definite) Riemannian
structure. Moreover, in the _symmetric dual space_ , it is an invariant metric
by a non-compact Lie group of symmetries.

In this section, we study the _dual symmetric spaces of the symmetric flag
spaces_ $(\mathfrak{u},\mathfrak{u}_{\Theta},\sigma)$. Recall that the dual-
symmetric Lie algebra is the direct sum (see [KN69, Section XI.8] or [Hel62,
Chapter V.2])

(49) $\mathfrak{u}^{*}=\mathfrak{u}_{\Theta}+\sqrt{-1}\mathfrak{m},$

with $\sigma^{*}=\mathrm{Ad}(e^{H_{\Theta}})\mid_{\mathfrak{u}^{*}}$. From the
expressions (106), (193) and (6.3), it is going to be clear that
$\mathfrak{u}^{*}$ is isomorphic to a classical real Lie algebra, being the
above decomposition, the Cartan decomposition of $\mathfrak{u}^{*}$ with
Cartan involution given by $\sigma^{*}$.

The Example 3, to be introduced, is a continuation of Example 1: where we deal
with $\mathrm{SO}(2)/\\{\pm I\\}\cong\mathrm{S^{1}}$, which is a symmetric
space. As we shall see, the cotangent bundle of $\mathrm{S^{1}}$ has a
realization as the one-sheeted hyperboloid

$Q=\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+y^{2}-z^{2}=1\\},$

being also a symmetric space. Moreover, the flag manifold $\mathrm{S^{1}}$
consists of the intersection of $Q$ with the plane $z=0$. Also, in Example 3,
the orbit of the exponential of symmetric zero-trace matrices in the matrix

$\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$

coincides with the curve

$\mathcal{C}=\\{(0,y,z)\in\mathbb{R}^{3}:y^{2}-z^{2}=1,\;y>0\\},$

which is, intuitively, symmetric for the axis $Y$, as we can see in Figure 1.

In the complex case, to generalize the previously described orbits and
correspondent curves, we consider the submanifold $\mathcal{S}$:

$\mathcal{S}:=\mathrm{Ad}(\mathrm{e}^{\sqrt{-1}\mathfrak{m}})H_{\Theta}\subset
G\cdot H_{\Theta}:=\mathrm{Ad}(G)H_{\Theta}.$

Observe that, according to Theorem 3.1, the cotangent bundle to any flag
manifold $\mathbb{F}_{\Theta}$ can always be realized as the adjoint orbit
$\mathrm{Ad}(G)H_{\Theta}$ for some characteristic element $\Theta$. Hence,
the submanifolds $\mathcal{S}$ can be conjectured to at least be a homogeneous
space.

On the other hand, Proposition 3.5 below shows that, in general, the
intersection of the submanifold $\mathcal{S}$ with the fibers of the bundle
$T^{*}\mathbb{F}_{\Theta}\cong\mathrm{Ad}(G)H_{\Theta}\rightarrow\mathbb{F}_{\Theta}$
is either null or a single point. Hence, the projection of the cotangent
bundle restricted to the submanifold $\mathcal{S}$ is an injective map since
each fiber intersecting $\mathcal{S}$ does so in a single element. Therefore,
$\mathcal{S}$ continuously deforms into a part of the sphere
$\mathbb{F}_{\Theta}$.

######  Theorem 3.4.

Let $\mathbb{F}_{\Theta}=U/U_{\Theta}$ be a flag symmetric space with
symmetric Lie algebra

$\mathfrak{u}=\mathfrak{u}_{\Theta}+\mathfrak{m}.$

Then,

  1. $(a)$

$\mathcal{S}$ is a submanifold of the adjoint orbit $G\cdot H_{\Theta}$, with
$G$ being the complex Lie group with compact real form $U$. Furthermore,

$\mathcal{S}\cong U^{*}/U_{\Theta},$

where $U^{*}/U_{\Theta}$ is the dual symmetric space of $\mathbb{F}_{\Theta}$

  2. $(b)$

the submanifold $\mathcal{S}$ is diffeomorphic to a vector space

  3. $(c)$

$\mathcal{S}$ is a Riemannian homogeneous space with a Riemannian structure
defined by the Cartan–Killing form

  4. $(d)$

let $u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ be a fiber in the cotangent
bundle of $\mathbb{F}_{\Theta}$ intersecting the submanifold $\mathcal{S}$,
then both spaces are transverse.

###### Proof.

  1. $(a)$

The quotient space $U^{*}/U_{\Theta}$ is an affine symmetric space by the
definition of dual symmetric space. Since $U^{*}$ is a subgroup of $G$ and
$U_{\Theta}=U\cap Z_{\Theta}$, then $(U^{*},U_{\Theta},\sigma^{*})$ is a
symmetric subspace of $(G,Z_{\Theta},\sigma)$, with
$\sigma^{*}=\sigma\mid_{U^{*}}$. According to [KN69, Theorem 4.1, p. 234],
$U^{*}/U_{\Theta}$ is a totally geodesic submanifold of $G/Z_{\Theta}$
(concerning the canonical connection of $G/Z_{\Theta}$) whose canonical
connection is the restriction of the canonical connection of $G/Z_{\Theta}$ to
$U^{*}/U_{\Theta}$.

We now apply Theorem 3.1 to the non-compact connected Lie group $U^{*}$,
obtaining that $U^{*}/U_{\Theta}$ is diffeomorphic to
$\exp(\sqrt{-1}\mathfrak{m})$, which is simply connected:

(50)
$\mathrm{Ad}(U^{*})H_{\Theta}=\mathrm{Ad}(e^{\sqrt{-1}\mathfrak{m}})\mathrm{Ad}(U_{\Theta})H_{\Theta}=\mathrm{Ad}(e^{\sqrt{-1}\mathfrak{m}})H_{\Theta}=:\mathcal{S}.$

  2. $(b)$

According to the previous item, $\mathcal{S}$ is diffeomorphic to
$\exp(\sqrt{-1}\mathfrak{m})$, thus, it is diffeomorphic to the vector space
$\sqrt{-1}\mathfrak{m}$.

  3. $(c)$

The tangent space of the dual symmetric space $U^{*}/U_{\Theta}$ at the origin
is isomorphic to $\sqrt{-1}\mathfrak{m}$ and the Cartan–Killing form in
$\sqrt{-1}\mathfrak{m}$ is
$\mathcal{K}(\sqrt{-1}X,\sqrt{-1}Y)=-\mathcal{K}(X,Y)=:\langle X,Y\rangle$ for
$X,Y\in\mathfrak{m}$. Since $\mathfrak{m}\subset\mathfrak{u}$, then the
Cartan–Killing form in $\sqrt{-1}\mathfrak{m}$ is positive defined. The
homogeneous space $U^{*}/U_{\Theta}$ admits invariant metrics since
$U_{\Theta}$ is compact. A natural metric in $U^{*}/U_{\Theta}$ is that given
by the restriction of the Cartan–Killing form to $\sqrt{-1}\mathfrak{m}$.
Since $\mathfrak{u}^{*}$ is simple, then the adjoint representation in
$\sqrt{-1}\mathfrak{m}$ is irreducible, and the invariant metric is
essentially unique, defined by the Cartan–Killing form.

  4. $(d)$

On the one hand, every fiber has the form
$u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})=u\cdot(N_{\Theta}^{+}\cdot
H_{\Theta})$, for $u\in U$. To say that $\mathcal{S}$ is transverse to
$H_{\Theta}+\mathfrak{n}_{\Theta}^{+}$ is equivalent to proving that for every
$x\in\mathcal{S}\cap u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ it must
holds

$T_{x}(\mathcal{S})+T_{x}(u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+}))=T_{x}(G\cdot
H_{\Theta}),$

for each $u\in U$ such that $\mathcal{S}\cap
u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\neq\emptyset$.

On the other hand, the Lie algebra $\mathfrak{g}$ can be seen as the set of
left-invariant fields. Equation

$\dfrac{d}{dt}\left(\mathrm{Ad}(e^{tX}(Y))\right)\mid_{t=0}=\left[X,Y\right],\qquad
X,Y\in\mathfrak{g},$

ensures that the tangent space of $\mathcal{S}$ at $e^{A}\cdot
H_{\Theta}=u\cdot(H_{\Theta}+X)=ue^{Z}\cdot H_{\Theta}$, with
$A\in\sqrt{-1}\mathfrak{m},X,Z\in\mathfrak{n}_{\Theta}^{+}$, $u\in U$ and
$e^{Z}\cdot H_{\Theta}=H_{\Theta}+X$, is precisely $[e^{A}\cdot
H_{\Theta},\sqrt{-1}\mathfrak{m}]$. Moreover, the tangent space to the fiber
$u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ at $e^{A}\cdot H_{\Theta}$
coincides with $[e^{A}\cdot H_{\Theta},\mathfrak{n}_{\Theta}^{+}]$. In this
manner, the sum of the former two tangent spaces is

(51) $[e^{A}\cdot H_{\Theta},\sqrt{-1}\mathfrak{m}]+[e^{A}\cdot
H_{\Theta},\mathfrak{n}_{\Theta}^{+}]=[e^{A}\cdot
H_{\Theta},\sqrt{-1}\mathfrak{m}+\mathfrak{n}_{\Theta}^{+}]=[e^{A}\cdot
H_{\Theta},\mathfrak{m}_{G}],$

i.e., it is the tangent space to the adjoint orbit $G\cdot H_{\Theta}$ at
$e^{A}\cdot H_{\Theta}$. ∎

######  Proposition 3.5.

The intersection of $\mathcal{S}$ with the fiber
$H_{\Theta}+\mathfrak{n}_{\Theta}^{+}$ is the point set $\\{H_{\Theta}\\}$.

###### Proof.

In Section 6, we shall compute the intersection of the submanifold
$\mathcal{S}$ with the fiber $H_{\Theta}+\mathfrak{n}_{\Theta}^{+}$ in the
three cases of symmetric flag spaces. The results will be evidenced in (112),
(200), and (255). ∎

The symmetry $\tilde{\sigma}$ in the submanifold $\mathcal{S}$ is given by

(52)
$\begin{array}[]{clcl}\tilde{\sigma}:&\mathcal{S}&\rightarrow&\mathcal{S}\\\
&e^{A}\cdot H_{\Theta}&\mapsto&\mathrm{C}_{e^{H_{\Theta}}}(e^{A})\cdot
H_{\Theta},\end{array}$

for $A\in\sqrt{-1}\mathfrak{m}$. Then, $\tilde{\sigma}(e^{A}\cdot
H_{\Theta})=\exp(\mathrm{Ad}(e^{H_{\Theta}}A)\cdot
H_{\Theta})=\exp(\sigma(A))\cdot H_{\Theta}=\exp(-A)\cdot H_{\Theta}$, since
$(\mathfrak{u}^{*},\mathfrak{u}_{\Theta},\sigma)$,
$\sigma=$Ad($e^{H_{\Theta}})$ is a symmetric Lie algebra. Once more, we
reinforce using the same notation $\sigma$ for the automorphism in the Lie
group and algebra. We prove:

######  Theorem 3.6.

Suppose that, for some $u\in U$, the fiber
$u\cdot(H_{\Theta}+\mathfrak{n}^{+}_{\Theta})$ intersects the submanifold
$\mathcal{S}$. Then, the fiber
$\sigma(u)\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ intersects
$\mathcal{S}$.

###### Proof.

Each element $e^{A}\cdot H_{\Theta}$ of $\mathcal{S}$ is an element of some
fiber of the cotangent bundle and has the form
$u\cdot(H_{\Theta}+X)=ue^{Z}\cdot H_{\Theta}$, for some $u\in U$ and
$Z,X\in\mathfrak{n}_{\Theta}^{+}$, since $\mathcal{S}\subset G\cdot
H_{\Theta}$. Hence,

$\displaystyle\tilde{\sigma}(e^{A}\cdot H_{\Theta})=$
$\displaystyle\tilde{\sigma}(ue^{Z}\cdot H_{\Theta})$ $\displaystyle
e^{-A}\cdot H_{\Theta}=$ $\displaystyle\sigma(u)\sigma(e^{Z})\cdot H_{\Theta}$
$\displaystyle\qquad=$ $\displaystyle\sigma(u)e^{-Z}\cdot H_{\Theta},\quad
Z\in\mathfrak{m}_{G}$ $\displaystyle\qquad\in$
$\displaystyle\sigma(u)\cdot(H_{\Theta}+\mathfrak{n}^{+}_{\Theta}).$

∎

However, not every fiber intersects $\mathcal{S}$. The following theorem
indicates some fibers which do not intersect $\mathcal{S}$.

######  Theorem 3.7.

The fiber $u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ does not intersect
$\mathcal{S}$ if $u\cdot\mathfrak{n}_{\Theta}^{+}\subset\mathfrak{u}^{*}$ and
$u\cdot H_{\Theta}\notin\mathfrak{u}_{\Theta}$.

###### Proof.

Let $u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ a fiber such that
$u\cdot\mathfrak{n}_{\Theta}^{+}\subset\mathfrak{u}^{*}$ and $u\cdot
H_{\Theta}\notin\mathfrak{u}_{\Theta}$. Suppose that
$u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathcal{S}\neq\varnothing.$
The fiber is contained in $u\cdot H_{\Theta}+\mathfrak{u}^{*}$. If
$u\cdot(H_{\Theta}+X)\in\mathcal{S}\subset\mathfrak{u}^{*}$, with
$X\in\mathfrak{n}_{\Theta}^{+}$, then $u\cdot
H_{\Theta}\in\mathfrak{u}^{*}\cap\mathfrak{u}=\mathfrak{u}_{\Theta}$ and this
is a contradiction. Hence

$(u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+}))\cap\mathcal{S}=\varnothing.\qed$

######  Theorem 3.8.

Let $u\cdot(H_{\Theta}+X)\in\mathcal{S}\subset T^{*}\mathbb{F}_{\Theta}$ be an
element that satisfies
$u\cdot\mathfrak{n}_{\Theta}^{+}\cap\mathfrak{u}^{*}=\\{0\\}$. Then, the
intersection of the submanifold $\mathcal{S}$ with the fiber
$u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ is
$\\{u\cdot(H_{\Theta}+X)\\}$.

###### Proof.

Consider an element $u\cdot(H_{\Theta}+Y)$ in the intersection
$\mathcal{S}\cap u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$. Then
$u\cdot(H_{\Theta}+Y)-u\cdot(H_{\Theta}+X)=u\cdot(Y-X)\in
u\cdot\mathfrak{n}_{\Theta}^{+}\cap\mathfrak{u}^{*}=\\{0\\}$, since
$\mathcal{S}\subset\mathfrak{u}^{*}$. Hence $X=Y$. ∎

######  Example 3 (Continuation of Example 1).

The symmetric part of the Cartan decomposition of
$\mathfrak{sl}(2,\mathbb{R})$ is generated by the matrices $A$ and $B$ (see
equation (22)). Thus, the submanifold $\mathcal{S}$ is represented by

$\displaystyle\mathrm{Ad}(e^{tA})B=:$ $\displaystyle\exp(tA)\cdot B$ (55)
$\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}\cosh t&\sinh
t\\\ \sinh t&\cosh t\end{array}\right)\cdot
B=0A+\cosh(2t)B-\sinh(2t)C:t\in\mathbb{R}\right\\}$ (56) $\displaystyle=$
$\displaystyle\\{(0,y,z)\in\mathbb{R}^{3}:y^{2}-z^{2}=1,\;y>0\\}$

and it is the intersection of the hyperboloid $x^{2}+y^{2}-z^{2}=1$ with the
semi-plane $\\{x=0,y>0\\}$ contained in the vector space
$\mathrm{span}_{\mathbb{R}}\\{B,C\\}$. Here, the submanifold $\mathcal{S}$ is
symmetric for the abelian vector subspace of $\mathfrak{sl}(2,\mathbb{R})$
generated by $B$. Hence, we have three symmetric spaces: The hyperboloid,
given by equation (28), the $1$-sphere, and a branch of the hyperbola (56),
contained in the plane $YZ$.

Figure 1. Real adjoint orbit.

The intersection of $\mathcal{S}$ with the fiber $k\cdot(B+\mathfrak{n})$ is
the set of the vectors $xA+yB+zC$ in (38), such that $x=0$ and $y>0$. These
conditions are equivalent to the system

(57) $\left\\{\begin{array}[]{c}-2\sin(2t)+r\cos(2t)=0\\\
2\cos(2t)+r\sin(2t)>0\end{array}\right.,t\in[0,\pi].$

This system has solution only for $t\in(-\frac{\pi}{4},\frac{\pi}{4})$. Thus,
the only fibers intersecting $\mathcal{S}$ are $k\cdot(B+\mathfrak{n})$ with
$k$ as in (29) and $t\in(-\frac{\pi}{4},\frac{\pi}{4})$. The intersection is
the only vector satisfying the equation (39) with $r=2\tan(2t)$. Furthermore,
note that when $k$ is equal to

(58) $\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
-1&1\end{array}\right)\;\mathrm{or}\;\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&-1\\\
1&1\end{array}\right),$

then $k\cdot B$ is equal to $-A$ and $A$, respectively. Hence, the fibers
passing by $\pm A$ do not intersect the submanifold $\mathcal{S}$.

We can see these results in Figure 2. All the lines in this figure are fibers,
and the sky blue lines are the fibers passing by $\pm A$. We can see how the
fibers passing by the points in the (green) semicircle of radius one contained
in $\\{(x,y,0):x\in\mathbb{R},y>0\\}$ intersect the submanifold $\mathcal{S}$
represented by the red curve.

Figure 2. Fibers of $T^{*}(\mathrm{S^{1}}$ and the submanifold $\mathcal{S}$.

######  Example 4 (Continuation of Example 2).

The manifold $\mathcal{S}:=(\exp\mathfrak{s})\cdot H_{\Theta}$, with

$\mathfrak{s}=\sqrt{-1}\mathfrak{m}=\left\\{\left(\begin{array}[]{cc}0&b\\\
\bar{b}&0\end{array}\right):b\in\mathbb{C}\right\\},$

is

$\displaystyle\mathcal{S}=e^{\sqrt{-1}\mathfrak{m}}\cdot H_{\Theta}=$
$\displaystyle\left\\{\dfrac{\sqrt{-1}\pi}{2}\left(\begin{array}[]{cc}\cosh(2\left|b\right|)&-\frac{b}{\left|b\right|}\sinh(2\left|b\right|)\\\
\frac{\bar{b}}{\left|b\right|}\sinh(2\left|b\right|)&-\cosh(2\left|b\right|)\end{array}\right):b\in\mathbb{C}\right\\}$

The diagonal entries of the matrices in $\mathcal{S}$ are complex numbers with
real parts equal to zero.

The fibers of the cotangent bundle are $u\cdot(H_{\Theta}+\mathfrak{n}^{+})$,
with $u=\left(\begin{array}[]{cc}\alpha&\beta\\\
-\bar{\beta}&\bar{\alpha}\end{array}\right)\in\mathrm{SU}(2)$ and each fiber
is the set

$\left\\{\left(\begin{array}[]{cc}\frac{\sqrt{-1}\pi}{2}(2\left|\alpha\right|^{2}-1)+x\alpha\bar{\beta}&-\sqrt{-1}\pi\alpha\beta+x\alpha^{2}\\\
-\sqrt{-1}\pi\bar{\alpha}\bar{\beta}-x\bar{\beta}^{2}&-\frac{\sqrt{-1}\pi}{2}(2\left|\alpha\right|^{2}-1)-x\alpha\bar{\beta}\end{array}\right):x\in\mathbb{C}\right\\}.$

To find out which fibers intersect $\mathcal{S}$, we are looking for elements
in $u\cdot(H_{\Theta}+\mathfrak{n}^{+})$ such that for each $\alpha$ and
$\beta$ as above, we must find if there exists $x\in\mathbb{C}$ satisfying the
following conditions:

(60) $\displaystyle\bar{x}\bar{\alpha}\beta=-x\alpha\bar{\beta}$ (61)
$\displaystyle\sqrt{-1}\pi\bar{\alpha}\bar{\beta}+\bar{x}\bar{\alpha}^{2}=-\sqrt{-1}\pi\bar{\alpha}\bar{\beta}-x\bar{\beta}^{2}$
(62)
$\displaystyle\frac{\pi}{2}(2\left|\alpha\right|^{2}-1)+\mathrm{Im}(x\alpha\bar{\beta})\geq\frac{\pi}{2}$

The first and second conditions come from the form of matrices in
$\mathfrak{su}(1,1)$ and the third condition is because, from (4), the
imaginary part of the first element in the diagonal of matrices in
$\mathcal{S}$ is greater than $\frac{\pi}{2}$.

If $\beta=0$, then $u\in\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(1))$ and
$\alpha\neq 0$, thus the fiber is $(H_{\Theta}+\mathfrak{n}^{+})$. The unique
$x\in\mathbb{C}$ that satisfies the three conditions above is $x=0$. Hence,
the intersection $(H_{\Theta}+\mathfrak{n}^{+})\cap\mathcal{S}$ is
$\\{H_{\Theta}\\}$.

If $\alpha=0$, then $\left|\beta\right|=1$ and the fiber is
$(H_{\Theta}+\mathfrak{n}^{-})$. The equation (61) tells us that $x=0$.
Putting $x=0$ in (62), we have a contradiction. Hence, the intersection
$(H_{\Theta}+\mathfrak{n}^{-})\cap\mathcal{S}$ is empty.

If $\alpha\neq 0\neq\beta$, from equations (60) and (61), we have the new
condition

$2\sqrt{-1}\pi\bar{\alpha}\beta=x(2\left|\alpha\right|^{2}-1).$

Here, we have two cases:

  * a)

If $2\left|\alpha\right|^{2}-1=0$, then $2\sqrt{-1}\pi\bar{\alpha}\beta=0$.
This is a contradiction, since $\bar{\alpha}\beta\neq 0$. Hence, the fibers
$u\cdot(H_{\Theta}+\mathfrak{n}^{+})$ with

$u\in\left\\{\left(\begin{array}[]{cc}\alpha&\beta\\\
-\bar{\beta}&\bar{\alpha}\end{array}\right):\lvert\alpha\rvert^{2}=\lvert\beta\rvert^{2}=\frac{1}{2},\alpha,\beta\in\mathbb{C}\right\\},$

has empty intersection with the manifold $\mathcal{S}$.

  * b)

If $2\left|\alpha\right|^{2}-1\neq 0$, from equations (60) and (61), we find
the complex number

$x=\frac{2\pi\sqrt{-1}}{2\left|\alpha\right|^{2}-1}\bar{\alpha}\beta.$

Condition (62), with $x$ as above, leads to the following condition

$\frac{1}{2}<\left|\alpha\right|^{2}<1.$

Hence, $(u\cdot(H_{\Theta}+\mathfrak{n}^{+}))\cap\mathcal{S}$ is non-empty if

(63) $u\in\left\\{\left(\begin{array}[]{cc}\alpha&\beta\\\
-\bar{\beta}&\bar{\alpha}\end{array}\right):\frac{1}{2}<\lvert\alpha\rvert^{2}\leq
1,\lvert\alpha\rvert^{2}+\lvert\beta\rvert^{2}=1,\alpha,\beta\in\mathbb{C}\right\\}.$

Furthermore,

$(u\cdot(H_{\Theta}+\mathfrak{n}^{+}))\cap\mathcal{S}=\left\\{\frac{\sqrt{-1}\pi}{2\left|\alpha\right|^{2}-1}\left(\begin{array}[]{cc}1&\alpha\beta\\\
-\bar{\alpha}\bar{\beta}&-1\end{array}\right)\right\\}.$

Considering the following basis for $\mathfrak{u}$:

$e_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\;e_{2}=\left(\begin{array}[]{cc}\sqrt{-1}&0\\\
0&-\sqrt{-1}\end{array}\right),\;e_{3}=\left(\begin{array}[]{cc}0&\sqrt{-1}\\\
\sqrt{-1}&0\end{array}\right),$

the matrices in $u\cdot H_{\Theta}$, for $u$ as in equation (63), have the
decomposition

$\pi\mathrm{Im}(\alpha\beta)e_{1}+\frac{\pi}{2}(2\left|\alpha\right|^{2}-1)e_{2}-\pi\mathrm{Re}(\alpha\beta)e_{3},$

with the second component greater than zero and

$\pi^{2}[\mathrm{Im}(\alpha\beta)]^{2}+\frac{\pi^{2}}{4}(2\left|\alpha\right|^{2}-1)^{2}+\pi^{2}[\mathrm{Re}(\alpha\beta)]^{2}=\frac{\pi^{2}}{4}.$

Hence, $u\cdot H_{\Theta}$, with $u$ as in (63), is the semi-sphere
$\\{(a,b,c)\in\mathfrak{su}(2):a^{2}+b^{2}+c^{2}=\frac{\pi^{2}}{4},\;b>0\\}$.
We then conclude that $\mathcal{S}$ is projected into the flag as the semi-
sphere through the fibers.

Since the manifold $\mathcal{S}$ is contained in

$\mathfrak{su}(1,1)=\left\\{\left(\begin{array}[]{cc}\alpha&\beta\\\
\bar{\beta}&-\alpha\end{array}\right):\beta\in\mathbb{C},\bar{\alpha}=-\alpha\right\\},$

we can express each of its elements as a combination of the matrices

$A=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),B=\left(\begin{array}[]{cc}\sqrt{-1}&0\\\
0&-\sqrt{-1}\end{array}\right),C=\left(\begin{array}[]{cc}0&\sqrt{-1}\\\
-\sqrt{-1}&0\end{array}\right).$

In that base, the manifold $\mathcal{S}$ is the part of the two-sheeted
hyperboloid $x^{2}+z^{2}-y^{2}=-\frac{\pi^{2}}{4}$ with positive
$Y$-component:

$\mathcal{S}=\left\\{\frac{\pi}{2}\sin t\sinh(2r)\cdot
A+\frac{\pi}{2}\cosh(2r)\cdot B-\frac{\pi}{2}\cos t\sinh(2r)\cdot
C:t\in[0,2\pi],r\geq 0\right\\}.$

Observe that the set of matrices $xA+yB+zC\in\mathfrak{su}(1,1)$ with
determinant equal to $\mathrm{det}(H_{\Theta})=\frac{\pi^{2}}{4}$ is the
hyperboloid $x^{2}+z^{2}-y^{2}=-\frac{\pi^{2}}{4}$ and has
$\left|\mathcal{W}\right|$ connected components where $\mathcal{W}$ is the
Weyl group of $\mathfrak{sl}(2,\mathbb{R})$. Also, the flag is maximal in this
example, and $H_{\Theta}$ is a regular element.

Let $w_{0}\in\mathcal{W}$ be the principal involution, then $w_{0}\cdot
H_{\Theta}=-H_{\Theta}$, and the adjoint orbit
$\mathrm{Ad}(\mathrm{SU}(1,1))(-H_{\Theta})=-\mathrm{Ad}(\mathrm{SU}(1,1))H_{\Theta}$
is the sheet of the hyperboloid $x^{2}+z^{2}-y^{2}=-\frac{\pi^{2}}{4}$ such
that $y<0$. Thus,

$\displaystyle\\{M\in\mathfrak{sl}(2,\mathbb{R}):\mathrm{det}(M)=\frac{\pi^{2}}{4}\\}=$
$\displaystyle\\{(x,y,z)\in\mathbb{R}^{3}:x^{2}+z^{2}-y^{2}=-\frac{\pi^{2}}{4}\\}$
$\displaystyle=$
$\displaystyle\mathrm{Ad}(\mathrm{SU}(1,1))H_{\Theta}\cup\mathrm{Ad}(\mathrm{SU}(1,1))w_{0}H_{\Theta}.$

##  4\. Symplectic forms and non-realizations of Lagrangian submanifolds

Any cotangent bundle to a Riemannian manifold is naturally a symplectic
manifold: let $(M,g)$ be a Riemannian manifold and denote by
$\pi:T^{*}M\rightarrow(M,g)$ its corresponding cotangent bundle. Choosing any
point $p\in T^{*}M$ is the same as picking a point $x\in M$ and a $1$-form
$\omega$ computed at $x\in M$. We then _tautologically_ introduce a $1$-form
$\theta$ on $T^{*}M$ by the relation

$\theta_{(x,\omega)}:=\pi^{*}\omega.$

That is, given a local smooth vector field $V$ in $TT^{*}M$, the tangent
bundle of the cotangent bundle, the application of the tautological $1$-form
$\theta$ to $V$ at $(x,\omega)$ is computed by projecting $V$ into the tangent
space of $M$ at $x$ using $d\pi:TT^{*}M\rightarrow TM$, and applying $\omega$
to this projection. The exterior derivative of $\theta$ converts
$(T^{*}M,d\theta)$ into a symplectic manifold. We refer to $d\theta$ as the
_canonical symplectic structure_ on $T^{*}M$.

As Theorem 3.4 has proved, the cotangent bundle to a symmetric flag space also
happens to be symmetric. Moreover, once we are dealing with semisimple Lie
algebras, let us define a $2$-form in $\mathfrak{g}$ by

(64) $\mathcal{H}_{\sigma}(X,Y)=-\mathcal{K}(X,\sigma Y)\qquad
X,Y\in\mathfrak{g}.$

Equation (64) defines a hermitian form in $\mathfrak{g}$, thus defining a
symplectic form in $\mathfrak{g}$:

(65) $\omega(X,Y):=-\mathrm{Im}\mathcal{H}_{\sigma}(X,Y)\qquad
X,Y\in\mathfrak{g}$

Since Theorem 3.1 ensures that symmetric flag space $\mathbb{F}_{\Theta}$ can
be considered as the adjoint orbit $\mathrm{Ad}(U)\cdot H_{\Theta}$ of the
compact Lie group $U$, contained in $\mathfrak{u}$, the conjugation $\sigma$
restricts to $\mathfrak{u}$ as the identity function. Then, the symplectic
form above in $\mathrm{Ad}(U)\cdot H_{\Theta}$ is

(66) $\omega(X,Y)=-\mathrm{Im}\mathcal{K}(X,Y)\qquad X,Y\in\mathfrak{u}.$

Moreover, item (b) of Theorem 3.1 ensures that this $2$-form induces a
symplectic form in the adjoint orbit through the pullback of $\omega$ of the
inclusion $G\cdot H_{\Theta}\hookrightarrow\mathfrak{g}$ and with $T_{g\cdot
H_{\Theta}}G\cdot H_{\Theta}=[\mathfrak{g},g\cdot H_{\Theta}]$, for each $g\in
G$. Such a $2$-form is what is called _the real Kirillov-Kostant-Souriau_
$\Omega$ (KKS) in the coadjoint orbit $\mathrm{Ad}^{*}(G)\alpha$, given by

(67) $\Omega_{\alpha}(\tilde{X}(\alpha),\tilde{Y}(\alpha))=\alpha[X,Y]\qquad
X,Y\in\mathfrak{g}.$

As we already mentioned, whenever $\mathfrak{g}$ is semisimple, there is a
natural correspondence induced by the Cartan–Killing form between the
coadjoint and the adjoint orbit, which justifies $\Omega$ being a symplectic
form in the adjoint orbit.

We prove:

######  Theorem 4.1.

The symmetric flag space $\mathbb{F}_{\Theta}=U/U_{\Theta}$ and its dual
symmetric space $\mathcal{S}$ are Lagrangian submanifolds of the adjoint orbit
$G\cdot H_{\Theta}$ with the symplectic form defined in (65).

###### Proof.

Note that the Cartan–Killing form is a symmetric $2$-form while $\omega$ is
anti-symmetric. Moreover, since the Lie algebra $\mathfrak{u}$ is real, the
symplectic form $\omega$ restricted to the submanifold $\mathbb{F}_{\Theta}$
is zero.

Now, let us see which is the restriction of the symplectic form above to the
submanifold $\exp(\sqrt{-1}\mathfrak{m})\cdot H_{\Theta}=U^{*}\cdot
H_{\Theta}$ of $G\cdot H_{\Theta}$. This submanifold is contained in
$\mathfrak{u}^{*}$ and each tangent space $T_{v\cdot H_{\Theta}}U^{*}\cdot
H_{\Theta}=[\mathfrak{u}^{*},v\cdot H_{\Theta}]$, for each
$v\in\exp(\sqrt{-1}\mathfrak{m})$, is also contained in $\mathfrak{u}^{*}$.
The real Lie algebra $\mathfrak{u}^{*}$ is invariant by the conjugation
$\sigma$. The Cartan–Killing form restricted to $\mathfrak{u}^{*}$ coincides
with the Cartan–Killing form of $\mathfrak{u}^{*}$ since it is a real form.
Then, the symplectic form restricted to $\mathfrak{u}^{*}$ is zero. ∎

On the other hand, the Kirillov-Kostant-Souriau symplectic form (KKS)
restricted to $\mathcal{S}$ is

(68) $\Omega_{\xi}([X,\xi],[Y,\xi])=\mathcal{K}(\xi,[X,Y]),\qquad
X,Y\in\mathfrak{u}^{*},\xi\in\mathcal{S}.$

Hence, if $\xi=H_{\Theta}$ and $X=\sqrt{-1}A_{\alpha},Y=\sqrt{-1}Z_{\alpha}$,
for some $\alpha\in\Pi^{+}\setminus\langle\Theta\rangle^{+}$, then

$\displaystyle\Omega_{H_{\Theta}}([\sqrt{-1}A_{\alpha},H_{\Theta}],[\sqrt{-1}Z_{\alpha},H_{\Theta}])=$
$\displaystyle\mathcal{K}(H_{\Theta},[\sqrt{-1}A_{\alpha},\sqrt{-1}Z_{\alpha}])$
$\displaystyle=$
$\displaystyle\mathcal{K}(H_{\Theta},-[A_{\alpha},Z_{\alpha}])$
$\displaystyle=$ $\displaystyle\mathcal{K}(H_{\Theta},2\sqrt{-1}H_{\alpha})$
$\displaystyle=$ $\displaystyle 2\sqrt{-1}\alpha(H_{\Theta})$
$\displaystyle\neq$ $\displaystyle 0.$

Therefore, the submanifold $\mathcal{S}$ is non-Lagrangian for the KKS form.
We thus have proved:

######  Corollary 4.2.

The diffeomorphism $\iota:(\mathrm{Ad}(G)\cdot
H_{\Theta},\Omega)\rightarrow(T^{*}\mathbb{F}_{\Theta},\omega)$ is not a
symplectomorphism.

##  5\. On the geometry of symmetric spaces and associated bundles: geodesics
and curvature

On the one hand, according to [KN69, Chapter 3, p.230-p.234], we know that to
each symmetric space $G/H$, one can guarantee the existence of an invariant
affine connection which recovers a principal bundle over $G/H$. Moreover,
there is a one-to-one correspondence between the set of $G$-invariant
connections in $G$ (connection in the bundle $G(M,H)$ which is invariant by
the left translations of $G$) and the set of linear mappings
$\Lambda_{\mathfrak{m}}:\mathfrak{m}\rightarrow\mathfrak{h}$ such that

$\Lambda_{\mathfrak{m}}(\mathrm{Ad}(h)X)=\mathrm{Ad}(h)(\Lambda_{\mathfrak{m}}(X)),\;X\in\mathfrak{m},h\in
H.$

If $\Lambda:\mathfrak{g}\rightarrow\mathfrak{g}$ is defined by

$\Lambda(X)=\left\\{\begin{array}[]{cc}X,&\;X\in\mathfrak{h},\\\
\Lambda_{\mathfrak{m}}(X),&\;X\in\mathfrak{m}\end{array}\right.$

one indetifies any $G$-invariant connection in $G$ with a connection form
$\omega$ as

$\Lambda(X)=\omega_{u_{0}}(\tilde{X})\qquad X\in\mathfrak{g},$

where $\tilde{X}$ denotes the natural lift to $G$ of a vector field
$X\in\mathfrak{g}$ of $M=G/H$ and $u_{0}\in G$ fixed. The invariant connection
corresponding to $\Lambda_{\mathfrak{m}}=0$ is called _canonical connection_
of $(G,H,\sigma)$ or $G/H$.

On the other hand, we recall that a homogeneous space $M=G/H$ with a
$G$-invariant indefinite Riemannian metric is said to be _naturally reductive_
if it admits an $\mathrm{Ad}(H)$-invariant decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ satisfying the condition

(69) $B(X,[Z,Y]_{\mathfrak{m}})+B([Z,X]_{\mathfrak{m}},Y)=0,\qquad\forall
X,Y,Z\in\mathfrak{m}$

where $B(\cdot,\cdot)$ is an $\mathrm{Ad}(H)$-invariant non-degenerate
symmetric bilinear form on $\mathfrak{m}$. We observe that _every symmetric
space is naturally reductive_ since $[X,Y]_{\mathfrak{m}}=0$ for all
$X,Y\in\mathfrak{m}$.

Once every reductive homogeneous space $M=G/H$ admits a _unique_ torsion-free
$G$-invariant affine connection ([KN69]), having the same geodesics as the
canonical connection, which is obtained for

(70) $\Lambda_{\mathfrak{m}}(X)(Y)=\dfrac{1}{2}[X,Y]_{\mathfrak{m}},\quad
X,Y\in\mathfrak{m},$

we say that such an invariant connection is the _natural torsion-free
connection_ on $G/H$ (with respect to the decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$).

Recall that if $(G,H,\sigma)$ is a symmetric space with $G$ semisimple, the
restriction of the Cartan–Killing form $\mathcal{K}(\cdot,\cdot)$ of
$\mathfrak{g}$ to $\mathfrak{m}$ defines a $G$-invariant (indefinite)
Riemannian metric on $G/H$, which induces the canonical connection on $G/H$.
Moreover, for the canonical connection of a symmetric space $(G,H,\sigma)$,
the homogeneous space $M=G/H$ with origin $o$ is a (complete) affine symmetric
space with symmetries $s_{x}$ such that, for each $X\in\mathfrak{m}$, the
curve $f_{t}=\exp tX$ in $G$ implies that the induced curve $x_{t}=f_{t}o$ in
$M=G/H$ is a geodesic. Conversely, every geodesic in $M$ starting from $o$ is
$f_{t}\cdot o=\exp tX\cdot o$ for some $X\in\mathfrak{m}$.

###  5.1. Geodesics on symmetric flag spaces and their cotangent spaces

Considering the essential aspects of the former section, let us describe some
geodesics on every studied flag symmetric space appearing in this paper. To do
so, recall that the tangent vector space to the adjoint orbit at the origin
can be expressed in the following ways (direct sums):

(71)
$\mathfrak{m}_{G}=\mathfrak{m}+\sqrt{-1}\mathfrak{m}=\mathfrak{m}+\mathfrak{n}_{\Theta}^{+}.$

Hence, every $X\in\mathfrak{m}_{G}$ admits the decompositions

(72) $\displaystyle X=X_{m}+X_{s},$ $\displaystyle\qquad
X_{m}\in\mathfrak{m},\;X_{s}\in\sqrt{-1}\mathfrak{m}$ (73) $\displaystyle
X=X_{m}+X_{f},$ $\displaystyle\qquad
X_{m}\in\mathfrak{m},\;X_{f}\in\mathfrak{n}_{\Theta}^{+}.$

With a canonical connection given by the Cartan–Killing form over
$G/Z_{\Theta}$, the maximal geodesics are the curves

$\gamma(t)=\pi_{G}(g\mathrm{exp}(tX)),\quad g\in G,\quad X\in\mathfrak{m}_{G}$

where $\pi_{G}:G\rightarrow G/Z_{\Theta}$ is the canonical projection defined
by $g\in G\mapsto g\cdot o$, with $o=1\cdot Z_{\Theta}$. The maximal geodesics
in $G/P_{\Theta}=U/U_{\Theta}$ are the curves

$\gamma_{U}(t)=\pi_{U}(u\,\mathrm{exp}(tA)),\quad u\in U,\quad
A\in\mathfrak{m}\cong T_{b_{\Theta}}\mathbb{F}_{\Theta}$

where $\pi_{U}:U\rightarrow U/U_{\Theta}$ is the canonical projection defined
by $u\in U\mapsto u\cdot b_{\Theta}$.

Furthermore, we can define another projection

$\begin{array}[]{rccccl}\pi:&G/Z_{\Theta}&\rightarrow&G/P_{\Theta}&\rightarrow&U/U_{\Theta}\\\
&gZ_{\Theta}&\mapsto&gP_{\Theta}&\mapsto&uU_{\Theta}\end{array}$

The following question arises: _is the projection of a geodesic in
$G/Z_{\Theta}$ a geodesic in $U/U_{\Theta}$?_ Here, we analyze a case where
the answer is yes. We will focus on studying geodesics that pass through the
origin.

From the decomposition in (73), if $[X_{\mathfrak{m}},X_{f}]=0$, then

$\mathrm{exp}(tX)=\mathrm{exp}(tX_{m})\,\mathrm{exp}(tX_{f})$

and

$\pi(\mathrm{exp}(tX)\cdot o)=\mathrm{exp}(tX_{\mathfrak{m}}\cdot
b_{\Theta}),$

since $X_{m}\in\mathfrak{m}$ and $X_{f}\in\mathfrak{n}_{\Theta}^{+}$.

When $X_{f}=0$, we get that the geodesic
$\gamma(t)=g\exp(tX_{\mathfrak{m}}\cdot o)$ is a horizontal curve in the
vector bundle $G/Z_{\Theta}\rightarrow U/U_{\Theta}$, and the projection in
$U/U_{\Theta}$ is the geodesic curve
$\gamma_{U}(t)=u\exp(tX_{\mathfrak{m}}\cdot b_{\Theta})$, where
$\pi_{GU}(g)=u$ is the projection of $G$ in $U$. We also have the case when
$X_{\mathfrak{m}}=0$: the geodesic is in the fiber $\mathfrak{n}_{\Theta}^{+}$
and the projection of this curve on $U/U_{\Theta}$ is
$\gamma_{U}(t)=b_{\Theta}$.

From the realization $T^{*}\mathbb{F}_{\Theta}=G/Z_{\Theta}$, the elements in
$g\cdot o\in G/Z_{\Theta}$ can be viewed as $uv\cdot
H_{\Theta}=u\cdot(H_{\Theta}+X)\in T^{*}\mathbb{F}_{\Theta}$, with $u\in
U,v\in N_{\Theta}^{+}$ such that $g=uv$ and $v\cdot H_{\Theta}=H_{\Theta}+X$.
Thus, the projection is given by $u\cdot(H_{\Theta}+X)\mapsto u\cdot
H_{\Theta}$. Hence, for $\\{u_{t}\\}\subset U$ and
$\\{H_{\Theta}+X_{t}=e^{W_{t}}\cdot H_{\Theta}\\}\subset N_{\Theta}^{+}\cdot
H_{\Theta}$, we get that

$\gamma(t)=e^{tZ}\cdot H_{\Theta}=u_{t}e^{W_{t}}\cdot
H_{\Theta}=u_{t}\cdot(H_{\Theta}+X_{t}).$

Then, the projection of the geodesic is $\pi(\gamma(t))=u_{t}\cdot
H_{\Theta}$.

When $Z\in\mathfrak{m}$, the geodesic $\gamma(t)=ge^{tZ}\cdot o$ is a
horizontal curve in the vector bundle for $g\in G$. Its projection on the flag
is the geodesic $\gamma_{U}(t)=ue^{tZ}\cdot b_{\Theta}$, with
$\pi_{GU}(g)=u\in U$. Moreover, when $Z\in\mathfrak{n}_{\Theta}^{+}$, then

$\begin{array}[]{ccc}\gamma(t)=g\exp(tZ)\cdot
o&\xrightarrow{\pi}&\gamma_{U}(t)=u\cdot b_{\Theta}.\end{array}$

Such a geodesic lies in the fiber
$u\cdot(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$ of the cotangent bundle, hence
being vertical.

###  5.2. Orthogonal symmetric Lie algebras and conditions to nonnegative
curvature on bundles

Let $(\mathfrak{g},\mathfrak{h},\sigma)$ be a symmetric Lie algebra. Consider
the Lie algebra $\mathrm{ad}_{\mathfrak{g}}(\mathfrak{h})$ of linear
endomorphisms of $\mathfrak{g}$ consisting of $\mathrm{ad}X$ where
$X\in\mathfrak{h}$. If the connected Lie group of linear transformations of
$\mathfrak{g}$ generated by $\mathrm{ad}_{\mathfrak{g}}(\mathfrak{h})$ is
compact, then $(\mathfrak{g},\mathfrak{h},\sigma)$ is called an _orthogonal
symmetric Lie algebra_.

Let $(G,H,\sigma)$ be a symmetric space such that $H$ has a finite number of
connected components, and suppose that $(\mathfrak{g},\mathfrak{h},\sigma)$ is
its symmetric Lie algebra. In that case, it can be checked that
$\mathrm{Ad}_{G}(\mathfrak{h})$ is compact if, and only if,
$(\mathfrak{g},\mathfrak{h},\sigma)$ is an orthogonal symmetric Lie algebra.
Furthermore, it holds that

######  Proposition 5.1.

Let $(\mathfrak{g},\mathfrak{h},\sigma)$ be an orthogonal symmetric Lie
algebra with $\mathfrak{g}$ simple. Let
$\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$ be the canonical decomposition. Then,

  1. (1)

$\mathrm{ad}\mathfrak{h}$ is irreducible on $\mathfrak{m}$.

  2. (2)

The Cartan–Killing form $\mathcal{K}$ of $\mathfrak{g}$ is (negative or
positive) defined on $\mathfrak{m}$.

A symmetric Lie algebra $(\mathfrak{g},\mathfrak{h},\sigma)$ is said to be
effective if $\mathfrak{h}$ contains no non-zero ideal of $\mathfrak{g}$.

An effective symmetric Lie algebra $(\mathfrak{g},\mathfrak{h},\sigma)$ is
_irreducible_ if $\mathrm{Ad}([\mathfrak{m},\mathfrak{m}])$ is a irreducible
representation on $\mathfrak{m}$. If $\mathfrak{g}$ is simple, then the
symmetric Lie algebra is irreducible. In general, an orthogonal symmetric Lie
algebra $(\mathfrak{g},\mathfrak{h},\sigma)$ with $\mathfrak{g}$ semisimple is
said to be of _compact type_ or _noncompact type_ according as the Cartan-
Kiling form $\mathcal{K}$ of $\mathfrak{g}$ is negative-defined or positive
definite on $\mathfrak{m}$. In what follows, we must consider Riemannian
symmetric spaces, which, accordingly to the former proposition, are related to
orthogonal symmetric Lie algebras for semisimple lie algebras. Our main goal
consists of taking advantage of the symmetric space realization of the
cotangent bundle to understand the geometry of some vector bundles. More
precisely, to understand the existence or not of metrics of nonnegative
sectional curvature on some bundles. See for instance [SW89, CDR92, BK01,
Tap03].

Since Cheeger–Gromoll’s Soul Theorem: _any complete noncompact manifold $M$
with a metric of nonnegative sectional curvature is diffeomorphic to the
normal bundle of some compact totally geodesic manifold_ $\Sigma\subset M$,
named as _Soul_ of $M$, it emerges the question, seen as a converse to this
result: _which vector bundles over compact, nonnegatively curved base spaces
can admit complete metrics of nonnegative curvature?_

On the one hand, there are vector bundles that are known not to admit
nonnegative curvature, but, at least to the best of the author’s knowledge, in
all such examples, the base space has an infinite fundamental group, see
[Tap03].

On the other hand, trivial positive results include all homogeneous vector
bundles over homogeneous spaces: let $K{<\,}H{<\,}U$ be compact Lie groups and
consider the submersion $\pi\colon(U/K,\mathcal{H})\to(U/H,b)$ defined by
$\pi(gK)=gH$, where $\mathcal{H}$ encodes the horizontal distribution defining
the submersion. Suppose that $(U/H,b)$ is normal homogeneous, where $b$ is the
normal homogeneous Riemannian metric on $U/H$, and
$g\cdot\mathcal{H}_{p}=\mathcal{H}_{gp}$ for all $g\in G$. Then, $U/K$ admits
a Riemannian submersion metric $\tilde{\textsl{g}}$ of nonnegative sectional
curvature with totally geodesic fibers.

Let $G$ be a complex simple Lie group, and let $U$ be its compact real form.
Considering the previous discussion, we also take $K=\\{e\\}$ and let $H$ be
the intersection $U\cap P_{\Theta}$ where $P_{\Theta}$ is a parabolic subgroup
of $G$. Then $U/H$ is a flag manifold. According to Theorem B, we have that
each pair $(U,H)$ there presented is a flag manifold that is also symmetric
and which its cotangent bundle $T^{*}(U/H)$ is a symmetric space. Moreover,
the submanifold $\mathcal{S}\subset T^{*}(U/H)$ is a symmetric space dual to
$U/H$. We take take the vector bundle

$\bar{\pi}:\mathcal{S}\hookrightarrow\widetilde{M}:=\left(U\times\mathcal{S}/\star\right)\rightarrow
U/H$

Moreover, observe that its fibers coincide with $\mathcal{S}$ while it defines
a vector bundle over $U/H$. Here, $\widetilde{M}$ is the orbit space of the
following $H$ action on $U\times\mathcal{S}$

$(H\ni h)\star\left((u,v)\in
U\times\mathcal{S}\right):=(uh,\mathrm{Ad}(h^{-1})v)\in U\times\mathcal{S}.$

The projection $\bar{\pi}([u,v]):=\pi(u)$ where $\pi:U\rightarrow U/H$ is the
orbit map $u\mapsto uH$.

Fixed the origin $o=eH\in U/H$ we denote $T_{o}(U/H):=\mathfrak{m}$ where
$\mathfrak{u}=\mathfrak{m}\oplus\mathfrak{u}_{\Theta}$ corresponds to the
appropriate Cartan decomposition of $\mathfrak{u}$. Since
$\mathcal{S}=\exp(\sqrt{-1}\mathfrak{m})$ one can define a Riemannian
submersion metric on $\bar{\pi}$ in the following manner:

$\textsl{g}(X+\sqrt{-1}Z,Y+\sqrt{-1}W):=-\mathcal{K}(X,Y)+\mathcal{K}(\sqrt{-1}Z,\sqrt{-1}W),~{}X,Y,Z,W\in\mathfrak{m}$

where $\mathcal{K}$ is the Cartan–Killing form of $\mathfrak{u}$. Note that
$X,Z\in\mathfrak{m}$ while $\sqrt{-1}Z,\sqrt{-1}W\in\sqrt{-1}\mathfrak{m}\cong
T_{o}(U^{*}/H)$, where $U^{*}/H\cong S$ is the dual symmetric space to $U/H$.
We also observe that any sphere bundle obtained out $\bar{\pi}$ admits a
submersion metric with nonnegative curvature since it constitutes a
homogeneous submersion: recall that $\mathcal{S}$ can be injectively projected
in $U/H$ (Theorem C). However, let us show that g does not have nonnegative
sectional curvature.

Observe that $U/H$ contains the topology of the bundle $\bar{\pi}$ since its
fibers are contractible. We then consider $U/H$ as a _Soul_ to $\bar{\pi}$. We
make use of the following result of Tapp–Strake–Walschap (see Theorem A and
Corollary 2.5 in [Tap03]): in order to
$\mathcal{S}\hookrightarrow(M,\textsl{g})\rightarrow(U/H,-\mathcal{K})$ to
have non-negative sectional curvature it must holds that

$R_{\textsl{g}}(X,Y,\sqrt{-1}Z,\sqrt{-1}W)^{2}\leq
4\mathrm{sec}_{U/H}(X,Y)\mathrm{sec}_{U^{*}/H}(\sqrt{-1}Z,\sqrt{-1}W),$

where
$R_{\textsl{g}}(X,Y,\sqrt{-1}Z,\sqrt{-1}W):=\textsl{g}(R_{\textsl{g}}(X,Y)\sqrt{-1}Z,\sqrt{-1}W)$
and $R_{\textsl{g}}$ is the Riemannian tensor associated to g. However, since
$U/H$ and $U^{*}/H$ are symmetric spaces dual to each other, there exists
$X,Y,Z,W\in\mathfrak{m}$ in a way that the former inequality can not hold,
finishing the proof Theorem E.

We have then presented the first examples of vector bundles over simply
connected manifolds that can not admit metrics of nonnegative sectional
curvature, with the base manifold of nonnegative sectional curvature and
finite fundamental group, but vector bundles for which the corresponding
sphere bundles can admit metrics of nonnegative sectional curvature. We solve,
in particular, Problem 9 in [Zil].

##  6\. Explicit descriptions of symmetric flag spaces

In this section we find the automorphisms $\sigma$ of the symmetric spaces
$(U,U_{\Theta},\sigma)$, its dual symmetric space $(U^{*},(U^{*})^{\sigma})$
and $(G,G^{\sigma})$. We search for an element $H_{\Theta}$ such that
$g_{\Theta}:=\exp(H_{\Theta})$ and $H_{\Theta}$ has the same centralizer in
$G$ and $U$, i.e.

(74) $Z_{G}(H_{\Theta})=Z_{G}(g_{\Theta})\qquad
Z_{U}(H_{\Theta})=Z_{U}(g_{\Theta}).$

The automorphism $\sigma$ is given by $C_{g_{\Theta}}$ for symmetric spaces
and symmetric Lie algebras; we use the same notation $\sigma$ to the
automorphism Ad$(g_{\Theta})$.

Assuming that $p+q=l+1$ and $p\leq q$, we get

$\begin{array}[]{|c|c|c|c|}\hline\cr
G&\mathrm{Sl}(l+1,\mathbb{C})&\mathrm{Sp}(l,\mathbb{C})&\mathrm{SO}(2l,\mathbb{C})\\\
\hline\cr\Theta&\Sigma\setminus\\{\alpha_{p}\\}&\Sigma\setminus\\{\alpha_{l}\\}&\Sigma\setminus\\{\alpha_{l}\\}\\\
\hline\cr\mathbb{F}_{\Theta}&\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))&\mathrm{Sp}(l)/\mathrm{U}(l)&\mathrm{SO}(2l)/\mathrm{U}(l)\\\
\hline\cr
T^{*}\mathbb{F}_{\Theta}&G/\mathrm{S}(\mathrm{Gl}(p,\mathbb{C})\times\mathrm{Gl}(q,\mathbb{C}))&G/\mathrm{Gl}(l,\mathbb{C})&G/\mathrm{Gl}(l,\mathbb{C})\\\
\hline\cr\end{array}$

The corresponding $H_{\Theta}$ satisfying (74) for $A_{l},C_{l}$ and $D_{l}$
cases are

$\dfrac{\sqrt{-1}\pi}{l+1}\left(\begin{array}[]{cc}qI_{p}&0\\\
0&-pI_{q}\end{array}\right),\qquad\dfrac{\sqrt{-1}\pi}{2}\left(\begin{array}[]{cc}\mathrm{Id}_{l}&0\\\
0&-\mathrm{Id}_{l}\end{array}\right)$
$\dfrac{\sqrt{-1}\pi}{2}\left(\begin{array}[]{cc}\mathrm{Id}_{l}&0\\\
0&-\mathrm{Id}_{l}\end{array}\right),$

respectively.

###  6.1. Case $A_{l}$

Let $G$ be the Lie group $\mathrm{Sl}(n,\mathbb{C})$ and let $U$ be the
compact Lie group $\mathrm{SU}(n)$, for $n=l+1$, with Lie algebras
$\mathfrak{g}$ and $\mathfrak{u}$, respectively. The corresponding flag
symmetric space is the set of Grassmannian of complex $p$-dimensional
subspaces of $\mathbb{C}^{p+q}$

$\mathbb{F}_{\Theta}=\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q)),\qquad
n=p+q,\;p\leq q,$

with

$\displaystyle\Theta=\Sigma\setminus\\{\alpha_{p}\\}=$
$\displaystyle\\{\alpha_{1}=\lambda_{1}-\lambda_{2},\cdots,\alpha_{p-1}=\lambda_{p-1}-\lambda_{p},\alpha_{p+1}=\lambda_{p+1}-\lambda_{p+2},\cdots,\alpha_{l}=\lambda_{l}-\lambda_{l+1}\\}$

where each $\lambda_{i}$ is given by

$\lambda_{i}:\mathrm{diag}\\{a_{1},\cdots,a_{l+1}\\}\mapsto a_{i}.$

The compact real form of $\mathfrak{g}$ is $\mathfrak{u}=\mathfrak{su}(n)$ and
an element in $\mathfrak{u}$ that defines the flag $\mathbb{F}_{\Theta}$ is
the matrix

(75) $H_{\Theta}:=\dfrac{i\pi}{n}\left(\begin{array}[]{cc}qI_{p}&0\\\
0&-pI_{q}\end{array}\right)\in\mathfrak{su}(n)\subset\mathfrak{sl}(n,\mathbb{C}).$

It belongs to the Cartan subalgebra of $\mathfrak{su}(n)$ and its exponential
is the matrix

(76) $g_{\Theta}:=\exp(H_{\Theta})=\left(\begin{array}[]{cc}e^{i\frac{\pi
q}{n}}I_{p}&0\\\ 0&e^{-i\frac{p\pi}{n}}I_{q}\end{array}\right).$

Since $p+q=n$, then $e^{i\frac{\pi}{n}(p+q)}=-1$. So, we have
$e^{i\pi\frac{q}{n}}\cdot e^{i\pi\frac{p}{n}}=-1$. Therefore,

$e^{i\pi\frac{q}{n}}=-e^{-i\pi\frac{p}{n}}$

and then

$\displaystyle g_{\Theta}=$
$\displaystyle\left(\begin{array}[]{cc}e^{i\pi\frac{q}{n}}I_{p}&0\\\
0&-e^{i\pi\frac{q}{n}}I_{q}\end{array}\right)$ $\displaystyle=$ $\displaystyle
e^{i\pi\frac{q}{n}}\left(\begin{array}[]{cc}I_{p}&0\\\
0&-I_{q}\end{array}\right)$ $\displaystyle:=$ $\displaystyle
e^{i\pi\frac{q}{n}}I_{p,q}\in\mathrm{SU}(n)$

where $I_{p,q}:=\begin{matrix}\left(\begin{array}[]{cc}I_{p}&0\\\
0&-I_{q}\end{array}\right)\end{matrix}$.

Furthermore, $(g_{\Theta})^{2}=(e^{i\pi\frac{q}{n}})^{2}\cdot
I_{p,q}^{2}=e^{2i\pi\frac{q}{n}}\cdot I_{n}$. This verifies that
$(g_{\Theta}^{2})^{n}=e^{2\pi qi}\cdot I_{n}=I_{n}$, i.e.,
$(g_{\Theta})^{2}\in Z(\mathrm{SU}(n))$, since $g_{\Theta}^{2}$ is a nth root
of unit.

We shall use the same notation $\sigma$ for the symmetric automorphism for Lie
groups and algebras. Seeing that $I_{p,q}^{-1}=I_{p,q}$, the inner
automorphism

$\begin{array}[]{rccl}\sigma:&\mathrm{SU}(n)&\rightarrow&\mathrm{SU}(n)\\\
&h&\mapsto&C_{g_{\Theta}}(h)=g_{\Theta}hg_{\Theta}^{-1}=e^{i\pi\frac{q}{n}}I_{p,q}he^{-i\pi\frac{q}{n}}I_{p,q}^{-1}=I_{p,q}hI_{p,q}\end{array}$

is an involution, since $C_{g_{\Theta}}^{2}(h)=I_{p,q}^{2}hI_{p,q}^{2}=h$, for
all $h\in\mathrm{SU}(n)$. The fixed point set of $C_{g_{\Theta}}$ in
$\mathrm{SU}(n)$ is:

$\displaystyle\\{h\in\mathrm{SU}(n):I_{p,q}hI_{p,q}=h\\}$ $\displaystyle=$
$\displaystyle\\{h\in\mathrm{SU}(n):I_{p,q}=h^{-1}I_{p,q}h\\}=Z_{\mathrm{SU}(n)}(I_{p,q}$
$\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}a_{p\times
p}&b_{p\times q}\\\ c_{q\times p}&d_{q\times
q}\end{array}\right)\in\mathrm{SU}(n):\left(\begin{array}[]{cc}a&-b\\\
-c&d\end{array}\right)=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}a_{p\times p}&0\\\ 0&d_{q\times
q}\end{array}\right)\in\mathrm{SU}(n)\right\\}$ $\displaystyle=$
$\displaystyle\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q)).$

Hence, the flag manifold
$\mathrm{SU}(n)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ is a symmetric
space. The tangent space to the symmetric flag at the origin is isomorphic to

(79) $\mathfrak{m}=\left\\{\left(\begin{array}[]{cc}0&Z\\\
-\bar{Z}^{T}&0\end{array}\right):Z_{p\times
q}\;\mathrm{complex}\,\mathrm{matrix}\right\\}$

and the symmetric pair is $(\mathfrak{su}(n),\mathfrak{u}_{\Theta})$, where
the subalgebra $\mathfrak{u}_{\Theta}$ consists of all elements in
$\mathfrak{su}(n)$ which are fixed by $C_{g_{\Theta}}$. This is also the Lie
algebra of $\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$

(80) $\mathfrak{u}_{\Theta}=\left\\{\left(\begin{array}[]{cc}A_{p\times
p}&0\\\ 0&B_{q\times
q}\end{array}\right):\mathrm{tr}(A)+\mathrm{tr}(B)=0,A\in\mathfrak{u}(p),\;B\in\mathfrak{u}(q)\right\\}.$

In order to study the symmetry of the cotangent bundle of the symmetric flag
space $\mathbb{F}_{\Theta}$, we study the centralizer of $H_{\Theta}$ in the
Lie group $G$,

$\displaystyle Z_{\mathrm{Sl}(n,\mathbb{C})}(H_{\Theta})=$
$\displaystyle\\{h\in\mathrm{Sl}(n,\mathbb{C}):\mathrm{Ad}(h)H_{\Theta}=H_{\Theta}\\}$
(87) $\displaystyle=$
$\displaystyle\left\\{h=\left(\begin{array}[]{cc}a_{p\times p}&b\\\
c&d_{q\times
q}\end{array}\right)\in\mathrm{Sl}(n,\mathbb{C}):h\left(\begin{array}[]{cc}qI_{p}&0\\\
0&-pI_{q}\end{array}\right)=\left(\begin{array}[]{cc}qI_{p}&0\\\
0&-pI_{q}\end{array}\right)h\right\\}$ (92) $\displaystyle=$
$\displaystyle\left\\{h\in\mathrm{Sl}(n,\mathbb{C}):\left(\begin{array}[]{cc}qa&-pb\\\
qc&-pd\end{array}\right)=\left(\begin{array}[]{cc}qa&qb\\\
-pc&-pd\end{array}\right)\right\\}$ (95) $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&d\end{array}\right)\in\mathrm{Sl}(n,\mathbb{C}):a_{p\times p},b_{q\times
q}\right\\}$ (96) $\displaystyle=$
$\displaystyle\mathrm{S}(\mathrm{Gl}(p,\mathbb{C})\times\mathrm{Gl}(q,\mathbb{C}))$

and note that the centralizer of the matrix $I_{p,q}$ in the group
$\mathrm{Sl}(n,\mathbb{C})$ is

$\displaystyle Z_{\mathrm{Sl}(n,\mathbb{C})}(I_{p,q})=$
$\displaystyle\left\\{h=\left(\begin{array}[]{cc}a_{p\times p}&b\\\
c&d_{q\times
q}\end{array}\right)\in\mathrm{Sl}(n,\mathbb{C}):h\left(\begin{array}[]{cc}I_{p}&0\\\
0&-I_{q}\end{array}\right)=\left(\begin{array}[]{cc}I_{p}&0\\\
0&-I_{q}\end{array}\right)h\right\\}$ $\displaystyle=$
$\displaystyle\left\\{h\in\mathrm{Sl}(n,\mathbb{C}):\left(\begin{array}[]{cc}a&-b\\\
c&-d\end{array}\right)=\left(\begin{array}[]{cc}a&b\\\
-c&-d\end{array}\right)\right\\}$ $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&d\end{array}\right)\in\mathrm{Sl}(n,\mathbb{C}):a_{p\times p},b_{q\times
q}\right\\}$ $\displaystyle=$
$\displaystyle\mathrm{S}(\mathrm{Gl}(p,\mathbb{C})\times\mathrm{Gl}(q,\mathbb{C}))$
$\displaystyle=$ $\displaystyle Z_{\mathrm{Sl}(n,\mathbb{C})}(H_{\Theta})$

We can extend the inner automorphism $\sigma=C_{g_{\Theta}}$ to the complex
Lie group $\mathrm{Sl}(n,\mathbb{C})$:

(97)
$\begin{array}[]{rccl}C_{g_{\Theta}}:&\mathrm{Sl}(n,\mathbb{C})&\rightarrow&\mathrm{Sl}(n,\mathbb{C})\\\
&h&\mapsto&ghg^{-1}=I_{p,q}hI_{p,q}.\end{array}$

Since we saw that it is an involution and its fixed point set is
$Z_{\mathrm{Sl}(n,\mathbb{C})}(I_{p,q})=Z_{\mathrm{Sl}(n,\mathbb{C})}(H_{\Theta})=\mathrm{S}(\mathrm{Gl}(p,\mathbb{C})\times\mathrm{Gl}(q,\mathbb{C}))$,
one gets that the cotangent bundle of $\mathbb{F}_{\Theta}$, as homogeneous
space,

$\mathrm{Sl}(n,\mathbb{C})/\mathrm{S}(\mathrm{Gl}(p,\mathbb{C})\times\mathrm{Gl}(q,\mathbb{C}))$

is a symmetric space associated to the symmetric pair
$(\mathfrak{sl}(n,\mathbb{C}),\mathfrak{sl}(p,\mathbb{C})\oplus\mathfrak{sl}(q,\mathbb{C})\oplus\mathbb{C}^{*})$.
In fact, let
$A\in\mathfrak{s}(\mathfrak{gl}(p,\mathbb{C})\times\mathfrak{gl}(q,\mathbb{C}))$,
then

$A=\left(\begin{array}[]{cc}D_{p\times p}&0\\\ 0&B_{q\times
q}\end{array}\right).$

  * (i)

If $A\in\mathfrak{sl}(p,\mathbb{C})\oplus\mathfrak{sl}(q,\mathbb{C})$, then

$A=\left[A+\left(\begin{array}[]{cc}-I_{p}&0\\\
0&\frac{p}{q}I_{q}\end{array}\right)\right]+\left(\begin{array}[]{cc}I_{p}&0\\\
0&-\frac{p}{q}I_{q}\end{array}\right)\mapsto A^{\prime}+\frac{p}{q}.$

  * (ii)

If $A\notin\mathfrak{sl}(p,\mathbb{C})\oplus\mathfrak{sl}(q,\mathbb{C})$, then

$A=\left[A+\mathrm{tr}(B)\left(\begin{array}[]{cc}I_{p}&0\\\
0&-I_{q}\end{array}\right)\right]+\mathrm{tr}(B)\left(\begin{array}[]{cc}-I_{p}&0\\\
0&I_{q}\end{array}\right)\mapsto A^{\prime}+\mathrm{tr}(B).$

In both of cases, $A^{\prime}$ belongs to
$\mathfrak{sl}(p,\mathbb{C})\oplus\mathfrak{sl}(q,\mathbb{C})\oplus\mathbb{C}^{*}$.
Hence
$\mathfrak{s}(\mathfrak{gl}(p,\mathbb{C})\times\mathfrak{gl}(q,\mathbb{C}))\subset\mathfrak{sl}(p,\mathbb{C})\oplus\mathfrak{sl}(q,\mathbb{C})\oplus\mathbb{C}^{*}$.
The other containment is straightforward. Thus, this symmetric Lie algebra is
in Table 3.1. Furthermore, the tangent space of that symmetric space at the
origin is isomorphic to

$\displaystyle\mathfrak{m}_{G}=$
$\displaystyle\mathfrak{m}+\sqrt{-1}\mathfrak{m}$ $\displaystyle\cong$
$\displaystyle\mathfrak{n}_{\Theta}^{-}+\mathfrak{n}_{\Theta}^{+}$ (100)
$\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}0&A\\\
B&0\end{array}\right):A_{p\times q},\;B_{q\times
p}\;\mathrm{complex}\;\mathrm{matrices}\right\\}$

where the elements in $\mathfrak{n}_{\Theta}^{-}$ are the matrices in 100 with
$A=0$, $\mathfrak{n}_{\Theta}^{+}$ is the set (100) with $B=0$ and
$\mathfrak{m}$ is the set in (79).

Directly applying Theorem 3.1, we get that

$G\cdot H_{\Theta}=G/Z_{\Theta}=T^{*}(U/U\cap
Z_{\Theta})=T^{*}(U/U_{\Theta}).$

Then,

$\mathrm{Sl}(n,\mathbb{C})/\mathrm{S}(\mathrm{Gl}(p,\mathbb{C})\times\mathrm{Gl}(q,\mathbb{C}))=T^{*}(\mathrm{SU}(n)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))).$

Thus, the cotangent bundle of the flag symmetric manifold
$\mathrm{SU}(n)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ is also a
symmetric space with the same involutive automorphism $\sigma=C_{g_{\Theta}}$.

Note that the inner automorphism can be lifted to the Lie algebra
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, obtaining the function

(101)
$\begin{array}[]{rccl}\sigma=\mathrm{Ad}(g_{\Theta}):&\mathfrak{sl}(n,\mathbb{C})&\rightarrow&\mathfrak{sl}(n,\mathbb{C})\\\
&Z&\mapsto&g_{\Theta}Zg_{\Theta}^{-1}=I_{p,q}ZI_{p,q}.\end{array}$

Denote by $\mathfrak{g}_{0}$ the non compact real form $\mathfrak{su}(p,q)$ of
$\mathfrak{g}$, which elements are the $n\times n$ complex matrices satisfying

(102) $I_{p,q}X+\bar{X}^{T}I_{p,q}=0,\qquad
I_{p,q}=\left(\begin{array}[]{cc}1_{p}&0\\\ 0&-1_{q}\end{array}\right).$

This Lie algebra is

(103) $\mathfrak{su}(p,q)=\left\\{\left(\begin{array}[]{cc}A_{p\times
p}&B_{p\times q}\\\ \bar{B}_{q\times p}^{T}&C_{q\times
q}\end{array}\right):A\in\mathfrak{u}(p),\;C\in\mathfrak{u}(q),\;\mathrm{tr}(A+C)=0\right\\},$

and has the Cartan decomposition
$\mathfrak{su}(p,q)=\mathfrak{k}+\mathfrak{s}$ with
$\mathfrak{k}=\mathfrak{u}_{\Theta}$, as in (80), and

(104) $\mathfrak{s}=\left\\{\left(\begin{array}[]{cc}0&B\\\
\bar{B}^{T}&0\end{array}\right):B_{p\times
q}\;\mathrm{complex}\;\mathrm{matrix}\right\\}.$

The involution Ad(exp$(H_{\Theta})$) in $\mathfrak{sl}(n,\mathbb{C})$,

(105) $\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\mapsto\left(\begin{array}[]{cc}A&-B\\\
-C&D\end{array}\right)$

restricted to $\mathfrak{su}(p,q)$ is also an inner automorphism since
$H_{\Theta}\in\mathfrak{u}_{\Theta}=\mathfrak{k}$. It also coincides with the
Cartan involution
$Z\in\mathfrak{su}(p,q)\mapsto-\bar{Z}^{T}\in\mathfrak{su}(p,q)$.

From the definition of $\mathfrak{m}$ in (79), we have that the vector space
$\sqrt{-1}\mathfrak{m}$ coincides with $\mathfrak{s}$ and therefore, the dual
symmetric algebra of $(\mathfrak{su}(n),\mathfrak{k},\sigma)$ is

(106) $\mathfrak{su}(n)^{*}=\mathfrak{su}(p,q)=\mathfrak{k}+\mathfrak{s}.$

In this way, as studied in Section 3.3, the dual symmetric
space$\mathrm{SU}(p,q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ of
$\mathrm{SU}(p+q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ is a
submanifold of the adjoint orbit $\mathrm{Sl}(n,\mathbb{C})\cdot H_{\Theta}$
(this $H_{\Theta}$ is from the $A_{l}$ case).

The dual symmetric space
$\mathrm{SU}(p,q)/\mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(q))$ is
diffeomorphic to $\mathcal{S}=\exp\mathfrak{s}\cdot H_{\Theta}$. To prove
Proposition 3.5, let us calculate the intersection
$\mathcal{S}\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$. Initially, note that

(107)
$H_{\Theta}\in(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathrm{Ad}(S)\cdot
H_{\Theta}\subset\mathfrak{su}(p,q)\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+}),$

where $S$ is the exponential of the vector space $\mathfrak{s}$. To calculate
the intersection
$\mathfrak{su}(p,q)\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$, we describe
the elements of the fiber passing through the origin $H_{\Theta}$:

(108) $\left(\begin{array}[]{cc}\dfrac{\sqrt{-1}\pi}{n}qI_{p}&Z_{p\times q}\\\
0&-\dfrac{\sqrt{-1}\pi}{n}pI_{q}\end{array}\right),$

with $Z\in\mathfrak{n}_{\Theta}^{+}$. These matrices belong to
$\mathfrak{su}(p,q)$ if, and only if, they satisfy the expression

$\left(\begin{array}[]{cc}I_{p}&0\\\
0&-I_{q}\end{array}\right)\left(\begin{array}[]{cc}\frac{\sqrt{-1}\pi}{n}qI_{p}&Z\\\
0&-\frac{\sqrt{-1}\pi}{n}pI_{q}\end{array}\right)+\left(\begin{array}[]{cc}-\frac{\sqrt{-1}\pi}{n}qI_{p}&0\\\
\bar{Z}^{T}&\frac{\sqrt{-1}\pi}{n}qI_{p}\end{array}\right)\left(\begin{array}[]{cc}I_{p}&0\\\
0&-I_{q}\end{array}\right)=\mathbf{0}$

which is equivalent to

(109) $\left(\begin{array}[]{cc}0&Z\\\
\bar{Z}^{T}&0\end{array}\right)=\mathbf{0}\Longleftrightarrow Z=\mathbf{0}.$

It means that the intersection
$\mathfrak{su}(p,q)\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\subset\\{H_{\Theta}\\}$.
Once from (196), we conclude that

(110)
$\mathfrak{su}(p,q)\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})=\\{H_{\Theta}\\}.$

it holds that

(111)
$H_{\Theta}\in((H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathcal{S}\subset(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathfrak{su}(p,q)^{*}(l)=\\{H_{\Theta}\\}.$

Hence,

(112)
$(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathcal{S}=\\{H_{\Theta}\\}.$

However, the fiber $-H_{\Theta}+\mathfrak{n}_{\Theta}^{-}$ is

(113) $\left(\begin{array}[]{cc}\dfrac{-\sqrt{-1}\pi}{n}qI_{p}&0\\\ Z_{p\times
q}&\dfrac{\sqrt{-1}\pi}{n}pI_{q}\end{array}\right)$

while the element $H_{\Theta}$ is the linear combination

$\displaystyle H_{\Theta}=2\sqrt{-1}\pi
q[H_{\alpha_{1}}+2H_{\alpha_{2}}+\cdots+pH_{\alpha_{p}}]$
$\displaystyle+2\sqrt{-1}\pi
p[(q-1)H_{\alpha_{p+1}}+(q-2)H_{\alpha_{p+2}}+\cdots+2H_{\alpha_{p+q-2}}+H_{\alpha_{l}}].$

Also,

$\mathcal{K}(\sqrt{-1}H_{\alpha_{p}},H_{\Theta})=-\pi.$

###  6.2. Case $C_{l}$

Let $G$ be the Lie group $\mathrm{Sp}(l,\mathbb{C})$ and $U$ be the compact
Lie group $\mathrm{Sp}(l)=\mathrm{Sp}(l,\mathbb{C})\cap\mathrm{SU}(2l)$, with
Lie algebras $\mathfrak{g}$ and $\mathfrak{u}$, respectively. The
corresponding flag symmetric manifold is the space of complex structures on
$\mathbb{H}^{l}$ compatible with its standard inner product, that is,

$\mathbb{F}_{\Theta}=\mathrm{Sp}(l)/\mathrm{U}(l)$

with $\Theta=\\{\lambda_{1}-\lambda_{2},\cdots,\lambda_{l-1}-\lambda_{l}\\}$
and each $\lambda_{i}$ given by

(114) $\lambda_{i}:\mathrm{diag}\\{a_{1},\cdots,a_{l}\\}\mapsto a_{i}.$

The Lie algebra
$\mathfrak{u}:=\mathfrak{sp}(l)=\mathfrak{sl}(l,\mathbb{C})\cap\mathfrak{su}(2l)$
is the set

(115) $\mathfrak{sp}(l)=\left\\{\left(\begin{array}[]{cc}a_{l\times l}&b\\\
-\bar{b}&\bar{a}\end{array}\right)\mid
a\in\mathfrak{u}(l),b:\,\mathrm{symmetric}\,\mathrm{complex}\,\mathrm{matrix}\right\\}$

which is a compact real form of the Lie algebra of $G$, denoted by
$\mathfrak{g}:=\mathfrak{sp}(l,\mathbb{C})=\\{A\in
M(2l,\mathbb{C}:AJ+JA^{T}=0\\}$, where
$J=\left(\begin{array}[]{cc}0&-Id_{l}\\\ Id_{l}&0\end{array}\right)$ with
$J^{2}=-Id_{2l}$. The elements of $\mathfrak{g}$ are $2l\times 2l$ complex
matrices which can be represented in blocks as

(116) $\left(\begin{array}[]{cc}a_{l\times l}&b\\\
c&-a^{T}\end{array}\right),\qquad b,c:\;\mathrm{symmetric}.$

Since $a,b,c$ are complex matrices, the matrix above is quaternionic.
Furthermore, the Lie group $G$ is the set

(119) $\displaystyle\mathrm{Sp}(l,\mathbb{C})=$ $\displaystyle\left\\{M\in
M_{2l\times 2l}(\mathbb{C}):M^{T}\Omega
M=\Omega\right\\},\quad\Omega=\left(\begin{array}[]{cc}0&1_{l}\\\
-1_{l}&0\end{array}\right)$ (122) $\displaystyle=$
$\displaystyle\left\\{M=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right):A,B,C,D\in M_{l\times
l}(\mathbb{C}),\mathrm{det}(M)=1\right\\}$

and

$\displaystyle-C^{T}A+A^{T}C$ $\displaystyle=0,$ $\displaystyle-C^{T}B+A^{T}D$
$\displaystyle=I_{l},$ (123) $\displaystyle-D^{T}B+B^{T}D$ $\displaystyle=0.$

An element in the Cartan subalgebra $\sqrt{-1}\mathfrak{h}_{\mathbb{R}}$ of
the compact real form $\mathfrak{u}$ that defines the symmetric flag manifold
is

(124) $H_{\Theta}:=\dfrac{\pi}{2}\left(\begin{array}[]{cc}\sqrt{-1}Id_{l}&0\\\
0&-\sqrt{-1}Id_{l}\end{array}\right)\in\mathfrak{sp}(l)\subset\mathfrak{sp}(l,\mathbb{C})$

and its exponential is the matrix

(125)
$g_{\Theta}:=e^{H_{\Theta}}=\left(\begin{array}[]{cc}\sqrt{-1}Id_{l}&0\\\
0&-\sqrt{-1}Id_{l}\end{array}\right)$

Since $H_{\Theta}=\frac{\pi}{2}g_{\Theta}$ and the centralizer
$Z_{\Theta}=Z_{G}(g_{\Theta})$ of $H_{\Theta}$ in $G$ satisfies

(126) $g_{\Theta}^{2}=-Id_{2l}\in Z(U)\subset Z(G),$

the inner automorphism

(127)
$\begin{array}[]{cccc}\sigma=C_{g_{\Theta}}:&\mathrm{Sp}(l,\mathbb{C})&\rightarrow&\mathrm{Sp}(l,\mathbb{C})\\\
&h&\mapsto&C_{g_{\Theta}}(h)=g_{\Theta}hg_{\Theta}^{-1}\end{array}$

is an involution (
$\sigma^{2}(h)=g_{\Theta}^{2}hg_{\Theta}^{-2}=hg_{\Theta}^{2}g_{\Theta}^{-2}=h$
). Its fixed point set is:

(130) $\displaystyle Z_{G}(g_{\Theta})=Z_{\Theta}=$
$\displaystyle\left\\{h=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{Sp}(l,\mathbb{C}):a,b,c,d\in
M(l,\mathbb{C}),g_{\Theta}h=hg_{\Theta}\right\\}$ (133) $\displaystyle=$
$\displaystyle\left\\{h=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{Sp}(l,\mathbb{C}):b=c=0\right\\}$ (136)
$\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&d\end{array}\right)\in\mathrm{Sp}(l,\mathbb{C})\right\\}$

But also,

(139) $\displaystyle\left(\begin{array}[]{cc}a&0\\\
0&d\end{array}\right)\in\mathrm{Sp}(l,\mathbb{C})$ (148)
$\displaystyle\Leftrightarrow$ $\displaystyle\left(\begin{array}[]{cc}a&0\\\
0&d\end{array}\right)^{T}\left(\begin{array}[]{cc}0&Id_{l}\\\
-Id_{l}&0\end{array}\right)\left(\begin{array}[]{cc}a&0\\\
0&d\end{array}\right)=\left(\begin{array}[]{cc}0&Id_{l}\\\
-Id_{l}&0\end{array}\right)$ (153) $\displaystyle\Leftrightarrow$
$\displaystyle\left(\begin{array}[]{cc}0&a^{T}d\\\
-d^{T}a&d\end{array}\right)=\left(\begin{array}[]{cc}0&Id_{l}\\\
-Id_{l}&0\end{array}\right)$ (154) $\displaystyle\Leftrightarrow$
$\displaystyle d=(a^{T})^{-1}.$

Then

(157) $\displaystyle Z_{\Theta}=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&(a^{T})^{-1}\end{array}\right)\in\mathrm{Sp}(l,\mathbb{C})\right\\}$ (158)
$\displaystyle\cong$ $\displaystyle\mathrm{Gl}(l,\mathbb{C})$

Thus,

(159)
$G/Z_{G}(g_{\Theta})=\mathrm{Sp}(l,\mathbb{C})/\mathrm{Gl}(l,\mathbb{C})$

is a symmetric space with symmetric pair
$(\mathfrak{sp}(l,\mathbb{C}),\mathfrak{sl}(l,\mathbb{C})\oplus\mathbb{C}^{*})$.
In fact, let $A\in\mathfrak{gl}(n,\mathbb{C})$.

  * (i)

If $A\in\mathfrak{sl}(n,\mathbb{C})$, then

$A=\left[A+\left(\begin{array}[]{cc}-1&0\\\
0&\frac{1}{n-1}I_{n-1}\end{array}\right)\right]+\left(\begin{array}[]{cc}1&0\\\
0&-\frac{1}{n-1}I_{n-1}\end{array}\right)\mapsto
A^{\prime}+\frac{1}{1-n}\in\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}.$

  * (ii)

If $A\notin\mathfrak{sl}(n,\mathbb{C})$, then
$A=\left[A-\frac{tr(A)}{n}I_{n}\right]+\frac{tr(A)}{n}I_{n}\mapsto\left[A-\frac{tr(A)}{n}I_{n}\right]+\frac{tr(A)}{n}\in\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}$.

Hence
$\mathfrak{gl}(n,\mathbb{C}))\subset\mathfrak{sl}(n,\mathbb{C})\oplus\mathbb{C}^{*}$.
The other containment is straightforward. Hence, this symmetric Lie algebra is
in Table 3.1. Once more, Theorem 3.1 implies that the cotangent bundle

(160) $G/Z_{\Theta}=G\cdot H_{\Theta}=T^{*}(U/U_{\Theta})$

is a symmetric space. Since $U_{\Theta}=Z_{\Theta}\cap\mathrm{Sp}(l)$ and
using the fact that
$\mathrm{Sp}(l)=\mathrm{Sp}(l,\mathbb{C})\cap\mathrm{SU}(2l)$, we have that

(163) $\displaystyle U_{\Theta}=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&(a^{T})^{-1}\end{array}\right)\in\mathrm{SU}(2l):a_{l\times
l}\;\mathrm{complex}\right\\}$ (168) $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&(a^{T})^{-1}\end{array}\right)\cdot\left(\begin{array}[]{cc}\bar{a}^{T}&0\\\
0&\bar{a}^{-1}\end{array}\right)=Id_{2l}\right\\}\subset\mathrm{Sp}(l,\mathbb{C})$
(171) $\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&(a^{T})^{-1}\end{array}\right)\in\mathrm{SU}(2l):a_{l\times
l}\in\mathrm{U}(l)\;\right\\}$ (172) $\displaystyle=$
$\displaystyle\mathrm{U}(l).$

Hence, the cotangent bundle of $\mathrm{Sp}(l)/\mathrm{U}(l)$:

(173)
$T^{*}(\mathrm{Sp}(l)/\mathrm{U}(l))=\mathrm{Sp}(l,\mathbb{C})/\mathrm{Gl}(l,\mathbb{C})$

is a symmetric space.

The Lie algebra $\mathfrak{g}=\mathfrak{sp}(l,\mathbb{C})$ has the real normal
form $\mathfrak{g}_{0}=\mathfrak{sp}(l,\mathbb{R})$

(174) $\mathfrak{g}_{0}=\left\\{\left(\begin{array}[]{cc}a_{l\times l}&b\\\
c&-a^{T}\end{array}\right)\mid b,c\;\mathrm{symmetrics}\right\\}$

which has the Cartan decomposition

$\mathfrak{g}_{0}=\mathfrak{k}+\mathfrak{s}$

where

(175) $\mathfrak{k}=\left\\{\left(\begin{array}[]{cc}a_{l\times l}&-b\\\
b&a\end{array}\right)\mid
a\in\mathfrak{o}(l),b:\mathrm{symmetric}\right\\}\cong\mathfrak{u}(l)$ (176)
$\mathfrak{s}=\left\\{\left(\begin{array}[]{cc}a_{l\times l}&b\\\
b&-a\end{array}\right):a,b\,\mathrm{symmetrics}\right\\}.$

The lift of the automorphism in (127) to the Lie algebra restricted to
$\mathfrak{g}_{0}$ is

(177)
$\begin{array}[]{cccc}\sigma=\mathrm{Ad}(g_{\Theta}):&\mathfrak{sp}(l,\mathbb{R})&\rightarrow&\mathfrak{sp}(l,\mathbb{R})\\\
&X&\mapsto&\mathrm{Ad}(g_{\Theta})X=g_{\Theta}Xg_{\Theta}^{-1}.\end{array}$

It does not coincide with the Cartan involution $\Theta(X)=-X^{T}$. However,
we have the symmetric pairs $(\mathfrak{sp}(l),\mathfrak{u}(l))$ and
$(\mathfrak{sp}(l,\mathbb{C}),\mathfrak{sl}(l,\mathbb{C})\oplus\mathbb{C})$,
since the Lie algebra of $Z_{\Theta}$ is
$\mathfrak{gl}(l,\mathbb{C})\cong\mathfrak{sl}(l,\mathbb{C})\oplus\mathbb{C}$.

The tangent bundle of the cotangent bundle of $\mathbb{F}_{\Theta}$, as a
homogeneous space, at the origin, is isomorphic to

(178) $\displaystyle\mathfrak{m}_{G}=$
$\displaystyle\mathfrak{n}_{\Theta}^{-}+\mathfrak{n}_{\Theta}^{+}$ (181)
$\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}0&Y_{l\times
l}\\\ X_{l\times
l}&0\end{array}\right):X,Y\,\mathrm{symmetric}\;\mathrm{complex}\;\mathrm{matrices}\right\\}$
(182) $\displaystyle=$ $\displaystyle\mathfrak{m}+\sqrt{-1}\mathfrak{m}$

where $\mathfrak{n}_{\Theta}^{\pm}$ is the set of matrices in (181) with $X=0$
and $Y=0$, respectively. The tangent bundle of the symmetric flag manifold at
the origin is isomorphic to

(183) $\displaystyle\mathfrak{m}\cong$
$\displaystyle\sum_{\alpha\in\Pi^{+}\setminus\left\langle\Theta\right\rangle^{+}}\mathfrak{u}_{\alpha}$
(186) $\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}0&B\\\
-\bar{B}&0\end{array}\right):B_{l\times
l}\,\mathrm{symmetric}\;\mathrm{complex}\right\\}$

where the subspace
$\mathfrak{u}_{\alpha}=$span${}_{\mathbb{R}}\\{A_{\alpha},Z_{\alpha}\\}$ is
generated by $A_{\alpha}=\left(\begin{array}[]{cc}0&E_{ij}+E_{ji}\\\
-E_{ij}-E{ji}&0\end{array}\right)$ and
$Z_{\alpha}=\left(\begin{array}[]{cc}0&\sqrt{-1}(E_{ij}+E_{ji})\\\
\sqrt{-1}(E_{ij}+E_{ji})&0\end{array}\right)$ with
$\alpha\in\Pi^{+}\setminus\langle\Theta\rangle^{+}=\\{\lambda_{i}+\lambda_{j},1\leq
i,j\leq l\\}$, for $E_{i,j}=\delta_{i}^{j}Id$.

The involution $\sigma$ for the symmetric Lie algebra
$(\mathfrak{sp}(l,\mathbb{C}),\mathfrak{gl}(l,\mathbb{C}),\sigma))$ is the
following automorphism, with $H_{\Theta}\in\mathfrak{sp}(l)$ as in (124),

(187) $\mathrm{Ad}(e^{H_{\Theta}}):\left(\begin{array}[]{cc}A_{l\times l}&B\\\
C&-A^{T}\end{array}\right)\in\mathfrak{sp}(l,\mathbb{C})\mapsto\left(\begin{array}[]{cc}A&-B\\\
-C&-A^{T}\end{array}\right).$

The involution above is also an automorphism of $\mathfrak{sp}(l)$, and this
symmetric Lie algebra has the canonical decomposition

(188) $\mathfrak{sp}(l)=\mathfrak{u}_{\Theta}+\mathfrak{m},$

with $\mathfrak{m}$ as in (186) and $\mathfrak{u}_{\Theta}$ being the
subalgebra such that $\sigma(W)=W$, for all $W\in\mathfrak{u}_{\Theta}$

(189) $\mathfrak{u}_{\Theta}=\left\\{\left(\begin{array}[]{cc}A&0\\\
0&\bar{A}\end{array}\right):A\in\mathfrak{u}(l)\right\\}\cong\mathfrak{u}(l).$

Then, the dual symmetric Lie algebra of
$(\mathfrak{sp}(l),\mathfrak{u}(l),\sigma)$ must have the canonical
decomposition

(190) $\mathfrak{sp}(l)^{*}=\mathfrak{u}_{\Theta}+\sqrt{-1}\mathfrak{m}.$

The set $\sqrt{-1}\mathfrak{m}$ is the vector space satisfying $\sigma(W)=-W$,
for all $W\in\sqrt{-1}\mathfrak{m}$, it is

(191) $\left\\{\left(\begin{array}[]{cc}0&B_{l\times l}\\\
\bar{B}&0\end{array}\right):A\;\mathrm{symmetric}\;\mathrm{complex}\;\mathrm{matrix}\right\\}$

and then

(192) $\mathfrak{sp}(l)^{*}=\left\\{\left(\begin{array}[]{cc}A&B\\\
\bar{B}&\bar{A}\end{array}\right):A\in\mathfrak{u}(l),B\;\mathrm{symmetric}\;\mathrm{complex}\;\mathrm{matrix}\right\\}.$

Let us define the following function

(193) $\left(\begin{array}[]{cc}A&B\\\
\bar{B}&\bar{A}\end{array}\right)\in\mathfrak{sp}(l)^{*}\mapsto\left(\begin{array}[]{cc}\mathrm{Re}(A)+\mathrm{Re}(B)&\mathrm{Im}(B)-\mathrm{Im}(A)\\\
\mathrm{Im}(A)+\mathrm{Im}(B)&\mathrm{Re}(A)-\mathrm{Re}(B)\end{array}\right)\in\mathfrak{sp}(l,\mathbb{R}).$

It is an isomorphism of Lie algebras since $A\in\mathfrak{u}(l)$ and B are
symmetric. The function above is known as the Cayley transform. Furthermore,

(194)
$\mathfrak{u}_{\Theta}\cong\mathfrak{k}\cong\mathfrak{u}(l),\qquad\sqrt{-1}\mathfrak{m}\cong\mathfrak{s},$

where $\mathfrak{sp}(l)=\mathfrak{k}+\mathfrak{s}$ is the Cartan decomposition
indicated in (175) and (176). Hence, the dual symmetric Lie algebra is
isomorphic to $(\mathfrak{sp}(l,\mathbb{R}),\mathfrak{k},\theta)$, with
$\theta$ being the Cartan involution $\theta(X)=\Omega X\Omega^{-1}$, where
$\Omega$ is as in (119).

The function defined in (193) transforms $H_{\Theta}\in\mathfrak{sp}(l)$ in

(195)
$\hat{H}_{\Theta}:=\dfrac{\sqrt{-1}\pi}{2}\left(\begin{array}[]{cc}0&-I_{l\times
l}\\\ I_{l\times l}&0\end{array}\right)\in\mathfrak{sp}(l,\mathbb{R}).$

The dual symmetric space
$\mathrm{Sp}(l)^{*}/U_{\Theta}\cong\mathrm{Sp}(l,\mathbb{R})/\mathrm{U}(l)$ is
diffeomorphic to $\mathcal{S}=\exp\sqrt{-1}\mathfrak{m}\cdot
H_{\Theta}\cong\exp\mathfrak{s}\cdot\hat{H}_{\Theta}$. To prove Proposition
3.5, let us calculate the intersection
$\mathcal{S}\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$. Initially, note that

(196)
$H_{\Theta}\in(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathcal{S}\subset\mathfrak{sp}(l)^{*}\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+}).$

To calculate the intersection
$\mathfrak{sp}(l)^{*}\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})$, we describe
the elements of the fiber passing through the origin $H_{\Theta}$:

(197) $\left\\{\left(\begin{array}[]{cc}\frac{\sqrt{-1}\pi}{2}I_{l\times
l}&Z\\\ 0&-\frac{\sqrt{-1}\pi}{2}I_{l\times l}\end{array}\right):Z\in
M_{l\times l}(\mathbb{C})\right\\}.$

with $Z\in\mathfrak{n}_{\Theta}^{+}$. These matrices belong to
$\mathfrak{sp}(l)^{*}$ if, and only if, they belong to the set in (192), i.e.,
$Z$ is a complex symmetric matrix and $-\bar{Z}=0$. It means that the
intersection
$\mathfrak{sp}(l)^{*}\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\subset\\{H_{\Theta}\\}$.
According to (196) we conclude that

(198)
$\mathfrak{sp}(l)^{*}\cap(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})=\\{H_{\Theta}\\}.$

Thus,

(199)
$H_{\Theta}\in((H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathcal{S}\subset(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathfrak{sp}(l)^{*}=\\{H_{\Theta}\\}.$

Hence,

(200)
$(H_{\Theta}+\mathfrak{n}_{\Theta}^{+})\cap\mathcal{S}=\\{H_{\Theta}\\}.$

Finally, the element $H_{\Theta}$ is the linear combination

$\displaystyle
2\pi(l+1)\sqrt{-1}[H_{\lambda_{1}-\lambda_{2}}+2H_{\lambda_{2}-\lambda_{3}}+\cdots+(l-1)H_{\lambda_{l-1}-\lambda_{l}}]$
$\displaystyle+l(l+1)\pi\sqrt{-1}H_{2\lambda_{l}}$

and

$\mathcal{K}(\sqrt{-1}H_{2\lambda_{l}},H_{\Theta})=-4\pi(l+1).$

###  6.3. Case $D_{l}$

Let $G$ be the Lie group $\mathrm{SO}(2l,\mathbb{C})$ and $U$ be the compact
Lie subgroup $\mathrm{SO}(2l)$, with Lie algebras

$\mathfrak{g}:=\mathfrak{so}(2l,\mathbb{C})=\left\\{\left(\begin{array}[]{cc}a&b\\\
-b^{T}&d\end{array}\right)\in\mathfrak{gl}(2l,\mathbb{C}):a_{l\times
l},d_{l\times l}\,\text{skew-symmetric}\right\\}$

and $\mathfrak{u}:=\mathfrak{so}(2l)$ respectively. The corresponding
symmetric flag manifold is the space of orthogonal complex structures on
$\mathbb{R}^{2l}$

$\mathbb{F}_{\Theta}=\mathrm{SO}(2l)/\mathrm{U}(l)$

with $\Theta=\\{\lambda_{1}-\lambda_{2},\cdots,\lambda_{l-1}-\lambda_{l}\\}$
and $\Sigma=\Theta\cup\\{\lambda_{l-1}+\lambda_{l}\\}$ where each
$\lambda_{i}$ was defined in (114).

As $\mathfrak{so}(2l,\mathbb{C})$ is defined over an algebraically closed
field, if two not degenerated quadratic forms are equivalents, then the
algebras of anti-symmetric matrices for the quadratic forms are both
isomorphic.

Let

(201) $F=\left(\begin{array}[]{cc}0&I_{l}\\\ I_{l}&0\end{array}\right),\qquad
f=\dfrac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\sqrt{-1}I_{l}&I_{l}\\\
I_{l}&\sqrt{-1}I_{l}\end{array}\right).$

Then,

$F=f^{T}I_{2l}f.$

Hence, the Lie algebra is isomorphic to

(202) $\mathfrak{g}_{1}=\\{A\in\mathfrak{sl}(2l,\mathbb{C}):AF+FA^{T}=0\\}$

Thus, $A\in\mathfrak{so}(2l,\mathbb{C})$ if, and only if,
$fAf^{-1}\in\mathfrak{g}_{1}$. In other words
$f\mathfrak{so}(2l,\mathbb{C})f^{-1}=\mathfrak{g}_{1}$ and the algebras
$\mathfrak{so}(2l,\mathbb{C})$ and $\mathfrak{g}_{1}$ are isomorphic. To make
easier the computations, we will use the Lie algebra $\mathfrak{g}_{1}$
instead of $\mathfrak{so}(2l)$, denoting by $G_{1}$ the connected Lie group
with Lie algebra $\mathfrak{g}_{1}$.

A Cartan subalgebra of $\mathfrak{g}_{1}$ is given by

(203) $\mathfrak{h}=\left\\{\left(\begin{array}[]{cc}\Lambda&0\\\
0&-\Lambda\end{array}\right):\Lambda_{l\times
l}\;\mathrm{diagonal}\;\mathrm{matrix}\right\\}.$

An element in the Cartan subalgebra $\sqrt{-1}\mathfrak{h}_{\mathbb{R}}$ of
the compact real form $f\mathfrak{u}f^{-1}=f\mathfrak{so}(2l)f^{-1}$ that
defines the symmetric flag manifold $\mathbb{F}_{\Theta}$ is

(204)
$\hat{H}_{\Theta}:=\dfrac{\pi}{2}\sqrt{-1}\left(\begin{array}[]{cc}I_{l\times
l}&0\\\ 0&-I_{l\times
l}\end{array}\right)\in\sqrt{-1}\mathfrak{h}_{\mathbb{R}}\subset\mathfrak{g}_{1}$

and

(205)
$g_{\Theta}:=\exp(\hat{H}_{\Theta})=\sqrt{-1}\left(\begin{array}[]{cc}I_{l\times
l}&0\\\ 0&-I_{l\times l}\end{array}\right).$

Since

(206) $g_{\Theta}^{2}=-Id_{2l}\in Z(G_{1})$

the inner automorphism

$\begin{array}[]{cccc}\sigma=C_{g_{\Theta}}:&G_{1}&\rightarrow&G_{1}\\\
&A&\mapsto&C_{g_{\Theta}}(A)=g_{\Theta}Ag_{\Theta}^{-1}\end{array}$

is an involution, by (206), because of
$(C_{g_{\Theta}}^{2}(A)=g_{\Theta}^{2}A(g_{\Theta}^{-1})^{2}=A$. Let us
calculate the fixed point set, which is the centralizer of $g_{\Theta}$ in
$G_{1}$. Since $H_{\Theta}=\frac{\pi}{2}g_{\Theta}$, then
$Z_{G_{1}}(H_{\Theta})=Z_{G_{1}}(g_{\Theta})$. Indeed,

$\displaystyle Z_{G_{1}}(g_{\Theta})=$ $\displaystyle\\{A\in
G_{1}:C_{g_{\Theta}}(A)=A\\}$ $\displaystyle=$ $\displaystyle\\{A\in
G_{1}:g_{\Theta}Ag_{\Theta}^{-1}=A\\}$ $\displaystyle=$ $\displaystyle\\{A\in
G_{1}:H_{\Theta}A=\dfrac{\pi}{2}Ag_{\Theta}\\}$ (211) $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}x_{l\times l}&y\\\ z&w_{l\times
l}\end{array}\right)\in
G_{1}H_{\Theta}A=\dfrac{\pi}{2}\sqrt{-1}\left(\begin{array}[]{cc}x&-y\\\
z&-w\end{array}\right)\right\\}$ (216) $\displaystyle=$
$\displaystyle\left\\{A=\left(\begin{array}[]{cc}x_{l\times l}&y\\\
z&w_{l\times l}\end{array}\right)\in
G_{1}:H_{\Theta}A=\left(\begin{array}[]{cc}\dfrac{\sqrt{-1}\pi}{2}x&-\dfrac{\sqrt{-1}\pi}{2}y\\\
\dfrac{\sqrt{-1}\pi}{2}z&-\dfrac{\sqrt{-1}\pi}{2}w\end{array}\right)\right\\}$
(219) $\displaystyle=$
$\displaystyle\left\\{A=\left(\begin{array}[]{cc}x_{l\times l}&y\\\
z&w_{l\times l}\end{array}\right)\in G_{1}:H_{\Theta}A=AH_{\Theta}\right\\}$
(220) $\displaystyle=$ $\displaystyle Z_{G_{1}}(H_{\Theta}).$

On the other hand, the set of fixed points of $\sigma=$Ad$(g_{\Theta})$ (which
coincides with 187 in $\mathfrak{g}_{1}$ is

(223) $\displaystyle\left\\{A=\left(\begin{array}[]{cc}a&b\\\
c&-a^{T}\end{array}\right)\in\mathfrak{g}_{1}:a,b,c,d\in
M(l,\mathbb{C}),g_{\Theta}Ag_{\Theta}^{-1}=A\right\\}$ (226) $\displaystyle=$
$\displaystyle\left\\{A=\left(\begin{array}[]{cc}a&b\\\
c&-a^{T}\end{array}\right)\in\mathfrak{g}_{1}:b=0=c\right\\}$ (229)
$\displaystyle=$ $\displaystyle\left\\{\left(\begin{array}[]{cc}a&0\\\
0&-a^{T}\end{array}\right)\in\mathfrak{sl}(2l,\mathbb{C})\right\\}$ (230)
$\displaystyle\cong$ $\displaystyle\mathfrak{gl}(l,\mathbb{C}).$

Using (220) and (230), we can see that

$G/Z_{\Theta}\cong\mathrm{SO}(2l,\mathbb{C})/\mathrm{Gl}(l,\mathbb{C})$

is a symmetric space. Its corresponding symmetric pair is
$(\mathfrak{so}(2l,\mathbb{C}),\mathfrak{sl}(l,\mathbb{C})\oplus\mathbb{C}^{*})$.
Hence, this symmetric Lie algebra is in Table 3.1. Theorem 3.1 implies that
the cotangent bundle

$G/Z_{\Theta}=G\cdot H_{\Theta}=T^{*}(U/U_{\Theta})$

is a symmetric space. In order to find out the set $U_{\Theta}$, since
$U=\mathrm{SO}(2l)$, let us use the matrix
$H_{\Theta}=f^{-1}\hat{H}_{\Theta}f\in\mathfrak{so}(2l)$. This matrix is

(231) $H_{\Theta}:=\dfrac{\pi}{2}\left(\begin{array}[]{cc}0&Id_{l}\\\
-Id_{l}&0\end{array}\right)\in\mathfrak{so}(2l,\mathbb{R})\subset\mathfrak{so}(2l,\mathbb{C}).$

Then,

(234) $\displaystyle U_{\Theta}=$
$\displaystyle\left\\{u=\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{SO}(2l):a,b\in M_{l\times
l}(\mathbb{R}),uH_{\Theta}=H_{\Theta}u\right\\}$ (237) $\displaystyle=$
$\displaystyle\left\\{\left(\begin{array}[]{cc}a_{l\times l}&-b\\\
b&a\end{array}\right)\in\mathrm{Sl}(2l,\mathbb{R}):aa^{T}+bb^{T}=I_{l\times
l},ba^{T}=ab^{T}\right\\}$ $\displaystyle\cong$
$\displaystyle\\{a+ib\in\mathrm{Gl}(l,\mathbb{C}):(a+ib)(a+ib)^{*}=Id\\}$
(238) $\displaystyle=$ $\displaystyle\mathrm{U}(l)$

Hence, the cotangent bundle of $\mathrm{SO}(2l)/\mathrm{U}(l)$:

(239)
$T^{*}(\mathrm{SO}(2l)/\mathrm{U}(l))\cong\mathrm{SO}_{\mathbb{C}}(2l)/\mathrm{Gl}(l,\mathbb{C})$