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# Kac–Moody Symmetric Spaces: An Addendum

Ralf Köhl and Christian Vock

###### Abstract

Kac–Moody symmetric spaces have been introduced by Freyn, Hartnick, Horn and
the first-named author for centered Kac–Moody groups, that is, Kac–Moody
groups that are generated by their root subgroups. In the case of non-
invertible generalized Cartan matrices this leads to complications that –
within the approach proposed originally – cannot be repaired in the affine
case.
In the present article we propose an alternative approach to Kac–Moody
symmetric spaces which for invertible generalized Cartan matrices provides
exactly the same concept, which for the non-affine non-invertible case
provides alternative Kac–Moody symmetric spaces, and which finally provides
Kac–Moody symmetric spaces for affine Kac–Moody groups.
In a nutshell, the original intention by Freyn, Hartnick, Horn and first-named
author was to construct symmetric spaces that likely lead to primitive actions
of the Kac–Moody groups; this, of course, cannot work in the affine case as
affine Kac–Moody groups are far from simple.

##  1 Introduction

Freyn, Hartnick, Horn and the first author developed the theory of Kac–Moody
symmetric spaces for non-affine real split Kac–Moody groups in [FHHK20]. Note
that this is a generalization of Riemannian symmetric spaces of non-compact
type. The authors of [FHHK20] use Kac–Moody groups generated by their root
groups, called centered Kac–Moody groups. For these groups, the automorphism
group is well understood by [Cap09], which is a central part of their theory
(see Sections 6 and 7 in [FHHK20]).

In this paper, we develop a theory for symmetric spaces of real or complex
Kac–Moody groups of arbitrary type, based on the basic ideas of [FHHK20].
The main difference between [FHHK20] and the present article is that we need
to extend the Cartan subalgebra of the underlying Kac–Moody algebra, as
proposed in Kac’s book [Kac90], to ensure that the roots in the affine case
are linearly independent so that, among other things, we obtain a faithful
action of the Weyl group on the Cartan subalgebra. This leads to some subtle
problems throughout this paper.
But first we need to define a symmetric space for arbitrary Kac–Moody groups
over $\mathbb{C}$ or $\mathbb{R}$, which is done using Loos’ definition of an
abstract symmetric space, see [Loo69, Chapter 2].

######  Proposition 1.1 (3.3)

Let $G$ be a Kac–Moody group over $\mathbb{R}$ or $\mathbb{C}$ of arbitrary
type and $K$ the set of fixed points of $G$ with respect to the Chevalley
involution, i.e. $K=G^{\Theta}$. Define the map

$\displaystyle\begin{split}\mu\colon G/K\times G/K&\to G/K\\\
(gK,hK)&\mapsto\tau(g)\Theta(h)K.\end{split}$

The map $\mu$ is continuous, and if $\tau(G)\cap K=\\{e\\}$, then $(G/K,\mu)$
is a symmetric space.

Having defined the symmetric spaces for arbitrary Kac–Moody groups, we now
want to study the structure of these symmetric spaces. For this we need
_flats_. To define them, we need the _topological Iwasawa decomposition_ for
complex Kac–Moody groups. The real version is given for centered Kac–Moody
groups in [FHHK20, Theorem 3.31], and for the general case we can use the same
proof strategy, since in [HK23] J. Grüning and the first author proved a
theorem connecting Kac–Moody groups of arbitrary type and strong topological
buildings. Therefore, we can prove the following:

######  Theorem 1.2 (3.11)

Let $G$ be a non-spherical split Kac–Moody group over $\mathbb{R}$ or
$\mathbb{C}$. Then the Iwasawa decompositions

$\displaystyle m_{1}$ $\displaystyle\colon U_{\pm}\times A\times K\to G,$
$\displaystyle m_{2}$ $\displaystyle\colon K\times A\times U_{\pm}\to G$

are homeomorphisms.

Using this statement, we can define and characterize flats in real or complex
Kac–Moody symmetric spaces of arbitrary type, see 3.13.
Further, we will be able to translate most of the statements from [FHHK20,
Section 6] into the more general context. Here we need to pay attention to the
structure of our Kac–Moody group, because the basic statement on which the
first half of Section 6 in [FHHK20] is based comes from [Cap09], where the
author analyzes automorphisms of centered Kac–Moody groups. Thus, for our
Kac–Moody group that is enlarged by a torus, we need to restrict ourselves to
the centered subgroup in order to apply the statement [Cap09, Theorem 2.4].
For the second part, it is important that we have enlarged the Cartan
subalgebra at the beginning: This gives us a faithful representation, as used
in [FHHK20, Section 6.14], which allows us to describe local actions on flats.

Once we have developed the theory of Kac–Moody symmetric spaces over
$\mathbb{R}$ or $\mathbb{C}$ of arbitrary type sufficiently far, we can also
define the causal boundary, as in [FHHK20, Section 7]. Since we are taking a
similar approach, we can take the main definitions from that section and then
formulate the propositions in this general case. Overall, this leads to the
following theorem about the building at infinity of an affine Kac–Moody
symmetric space.

######  Theorem 1.3 (5.9)

Let $G$ be a Kac–Moody group over $\mathbb{R}$ or $\mathbb{C}$ of affine type,
$G^{\dagger}<G$ its centered subgroup and $X_{1}$ the corresponding Kac–Moody
symmetric space. Then the following statements are true:

  1. 1

$\mathcal{M}_{\parallel}$ has a unique polyhedral complex structure, so that
the map

$\varphi_{x}\colon\mathcal{M}_{x}\hookrightarrow\mathcal{M}\rightarrow\mathcal{M}_{\parallel}$

is a geometric isomorphism for every point $x\in X_{1}$. In particular the map
respects the weak topology and maps open cells homeomorphically to open cells.

  2. 2

The automorphism group of the symmetric space $X_{1}$ acts on the causal
boundary by geometric automorphisms.

  3. 3

The unique ideal polyhedral complex structure on $\mathcal{M}_{\parallel}$ is
geometrically isomorphic to the $\mathfrak{a}$-realization
$\left|\Delta\right|_{\mathfrak{a}}$ of the twin building $\Delta_{\pm}$.

The structure of this article is as follows:
In Section 2 we give a brief overview of the basics of Kac–Moody theory.
Section 3 begins by introducing the notion of symmetric space due to Loos,
which we use to define Kac–Moody symmetric spaces over
$k\in\\{\mathbb{C},\mathbb{R}\\}$ of arbitrary type.
After defining symmetric spaces related to Kac–Moody groups, we analyze them.
To do this, we must first formulate the topological Iwasawa decomposition for
split Kac–Moody groups over $k$ of arbitrary type, which we then use to define
flats. Thereafter, we characterize the flats.
Following the structure of the underlying article [FHHK20], we connect in
Section 4 the automorphism group of the symmetric space to the automorphism
group of the corresponding Kac–Moody group. Then we consider the local
actions, i.e., the actions on the flats of the symmetric space. Thus, we can
compare the Weyl group of a Kac–Moody group with the Weyl group of a symmetric
space. Towards the end of this section, we connect the local action to the
global action.
In the last section, Section 5, we develop the theory of a building at
infinity for Kac–Moody symmetric spaces over $k$ of arbitrary type.

Acknowledgment:
The authors thank the DFG SPP 2026 priority program “Geometry at infinity” for
partial financial support via KO 4323/14. The first half of the project, both
authors were affiliated with and funded by JLU Giessen, the second half
affiliated with and funded by CAU Kiel. The authors would also like to thank
Paul Zellhofer in particular for a wealth of very helpful discussions and
comments.

##  2 Basics

In this section, we give a brief overview of the basic notions as well as the
construction of a Kac–Moody group. Most of this can be found in [Kac90],
[Tit87], [Rém02] or [Mar18].

###### Definition 2.1

Denote with $\mathbb{A}=\left(a_{ij}\right)_{1\leq i,j\leq
n}\in\mathbb{Z}^{n\times n}$ a square matrix over the integers which fulfills
the following properties

  * •

$a_{ii}=2$ for all $i=1,\ldots,n$,

  * •

$a_{ij}\leq 0$ for all $i,j=1,\ldots,n$, $i\neq j$, and

  * •

$a_{ij}=0$ if and only if $a_{ji}=0$.

In the following, we will refer to $\mathbb{A}$ as a _generalized Cartan
matrix_.

At this point we define an affine generalized Cartan matrix, which will be of
interest in this article. Let $v\in\mathbb{R}^{n}$ be a vector, then we write
$v\geq 0$ if $v_{i}\geq 0$ for all $1\leq i\leq n$. A generalized Cartan
matrix $\mathbb{A}$ is said to be of the _affine type_ if the following
properties are satisfied:

  1. 1

The corank of $\mathbb{A}$ is $1$.

  2. 2

There exists a vector $u>0$ such that $\mathbb{A}u=0$ and $\mathbb{A}v\geq 0$
implies $\mathbb{A}v=0$.

For details on the classification of $\mathbb{A}$, see [Kac90, Theorem 4.3].
Additionally, we only consider _symmetrizable_ generalized Cartan matrices
$\mathbb{A}$, this means: There must be a symmetric matrix
$B\in\mathbb{R}^{n\times n}$ and an invertible diagonal matrix
$D\in(\mathbb{R}^{\times})^{n\times n}$ such that $\mathbb{A}=DB$, cf. [Kac90,
§2.1].

In [Kac90] the notion of a realization of $\mathbb{A}$ is introduced, which
assigns a triple to a generalized Cartan matrix, allowing the construction of
a Kac–Moody algebra. Here, we use a more general approach to Kac–Moody
algebras that allows us also define a Kac–Moody group associated to the
algebra in a rigorous way. This definition is due to [Rém02, Chapitre 8] or
[Mar18, Definition 7.9].

###### Definition 2.2

Let $\mathbb{A}=\left(a_{ij}\right)_{i,j\in I}$ be a generalized Cartan matrix
with index set $I$, $\Lambda$ a free $\mathbb{Z}$-module of finite rank and
$\Lambda^{\vee}$ the $\mathbb{Z}$-dual of $\Lambda$. The elements $c_{i}$ of
$\Lambda$ and the elements $h_{i}$ of $\Lambda^{\vee}$ need to satisfy
$c_{j}(h_{i})=a_{ij}$. The _Kac–Moody root datum_ is the quintuple
$\mathcal{D}\coloneqq\left(I,\mathbb{A},\Lambda,(c_{i})_{i\in I},(h_{i})_{i\in
I}\right).$

###### Remark 2.3

Consider a generalized Cartan matrix $\mathbb{A}$ of arbitrary type and a
corresponding Kac–Moody root datum $\mathcal{D}$. Define the _Cartan
subalgebra_
$\mathfrak{h}_{\mathcal{D}}\coloneqq\Lambda^{\vee}\otimes_{\mathbb{Z}}\mathbb{C}$
with elements, $\alpha_{i}^{\vee}\coloneqq h_{i}\otimes 1$ and define the dual
space
$\mathfrak{h}_{\mathcal{D}}^{*}\coloneqq\Lambda\otimes_{\mathbb{Z}}\mathbb{C}$
with elements $\alpha_{i}\coloneqq c_{i}\otimes 1$. Call the elements
$\Pi=\\{\alpha_{i}\mid i\in I\\}\subset\mathfrak{h}^{*}$ _simple roots_ ,
$\Pi^{\vee}=\\{\alpha_{i}^{\vee}\mid i\in I\\}\subset\mathfrak{h}$ _simple
coroots_ and denote with $\Delta^{re}$ the set of real roots (cf. [Kac90,
§5.1] or [Mar18, Chapter 6.1]).

According to the approach proposed in Kac’s book [Kac90, §1.1], one wants the
sets $\Pi$ and $\Pi^{\vee}$ to be linearly independent.To work with any
generalized Cartan matrix $\mathbb{A}$, we need to take care of the case where
$\mathbb{A}$ is not invertible.
In this case it follows from the relation
$\langle\alpha_{i},\alpha_{j}^{\vee}\rangle=a_{ij}\in\mathbb{A}$, that $\Pi$
is linearly dependent. Therefore, we extend the Cartan subalgebra
$\mathfrak{h}$ to solve this problem, as done in [Kac90, Proof of Proposition
1.1] and [Mar18, Chapter 3.5 and Example 7.10]: Let
$\mathbb{A}\in\mathbb{Z}^{n\times n}$ and $l$ be the rank of $\mathbb{A}$,
then the extended Cartan subalgebra must be
$\mathfrak{h}\cong\mathbb{C}^{2n-l}$. The dual space is also extended and
constructed so that the roots still satisfy the relation to $\mathbb{A}$. Let
us now denote by $\mathcal{D}_{Kac}^{\mathbb{A}}$ the root datum where the
elements $c_{i}$ and $h_{i}$ are linearly independent and $\Lambda$ has
minimal rank with respect to these properties, i.e. the rank of $\Lambda$ is
equal to $2n-l$. Moreover, denote by $\\{v_{i}\\}$ the $\mathbb{Z}$-basis of
the extended $\Lambda$, where $h_{i}=v_{i}$ for $1\leq i\leq n$. We follow
here the notation and idea of [Mar18, Example 7.10].

According to [Rém02, Chapitre 7] or Definition 7. 13 in [Mar18], one can
define a _Kac–Moody algebra $\mathfrak{g}_{\mathcal{D}}$ of type
$\mathcal{D}$_ as a Lie algebra with generators $\mathfrak{h}_{\mathcal{D}}$,
$\\{e_{i}\\}_{i\in I}$ and $\\{f_{i}\\}_{i\in I}$, where $e_{i}$ and $f_{i}$
are to be symbols satisfying the following relations.

$\displaystyle[\mathfrak{h}_{\mathcal{D}},\mathfrak{h}_{\mathcal{D}}]$
$\displaystyle=0$ $\displaystyle[h,e_{i}]$ $\displaystyle=c_{i}(h)e_{i}$
$\displaystyle[h,f_{i}]$ $\displaystyle=-c_{i}(h)f_{i}$
$\displaystyle[e_{i},f_{j}]$ $\displaystyle=-\delta^{i}_{j}h_{i}$
$\displaystyle(\operatorname{ad}e_{i})^{1-a_{ij}}e_{j}$
$\displaystyle=(\operatorname{ad}f_{i})^{1-a_{ij}}f_{j}=0\;(i\neq j)$

for $h\in\mathfrak{h}_{\mathcal{D}}$ and $i,j\in I$. In the following, we
denote with $\mathfrak{g}(\mathbb{A})$ the Kac–Moody algebra of type
$\mathcal{D}_{\text{Kac}}^{\mathbb{A}}$.

Now one can associate to a Kac–Moody root datum $\mathcal{D}$ a Kac–Moody
group $G_{\mathcal{D}}(\mathbb{C})$ using a basis of type $\mathcal{D}$ as in
[Tit87, Section 2] (and [Mar18, Definition 7.77]). Therefore, one need the
_split torus scheme_ which is a group functor
$\mathfrak{T}_{\Lambda}\colon\mathbb{Z}\text{-alg}\to\text{Grp}$, associated
to the root datum $\mathcal{D}$, given by
$\mathfrak{T}_{\Lambda}(R)\coloneqq\Lambda^{\vee}\otimes_{\mathbb{Z}}R^{\times}$
for any ring $R$, yielding that $\mathfrak{T}_{\Lambda}(R)$ is isomorphic to
$s$ copies of the multiplicative group functor $\mathbb{G}_{m}(R)\coloneqq
R^{\times}$, where $s$ is the rang of $\Lambda$. As an alternative, one can
think of the split torus scheme as
$\mathfrak{T}_{\Lambda}(R)=\operatorname{Hom}_{\mathbb{Z}\text{-alg}}(\mathbb{Z}[\Lambda],R)$.
Thus, a _basis of type $\mathcal{D}$_ is defined as a triple
$\mathcal{F}\coloneqq\left(\mathcal{G},(\varphi_{i})_{i\in I},\eta\right)$
where $\mathcal{G}\colon\mathbb{Z}\text{-alg}\to\text{Grp}$ is a group
functor, $(\varphi_{i})_{i\in I}$ is a assortment of morphisms of functors
$\varphi_{i}\colon\operatorname{SL}_{2}\to\mathcal{G}$ and a morphism of
functors $\eta\colon\mathfrak{T}_{\Lambda}\to\mathcal{G}$. One call the group
functor $\mathcal{G}$ _Tits functor of type $\mathcal{D}$_ if the following
axioms are satisfied

  1. KMG.1

If $k$ is a field, $\mathcal{G}(k)$ is generated by
$\varphi_{i}(\operatorname{SL}(2,k))$ and by
$\eta(\mathfrak{T}_{\Lambda}(k))$.

  2. KMG.2

For every ring $R$, the homomorphism
$\eta\colon\mathfrak{T}_{\Lambda}(R)\to\mathcal{G}(R)$ is injective.

  3. KMG.3

For $i\in I$ and $r\in R^{\times}$, one has

$\varphi_{i}\begin{pmatrix}r&0\\\
0&r^{-1}\end{pmatrix}=\eta\left(r^{h_{i}}\right),$

where $r^{h_{i}}$ denotes the element $\lambda\mapsto
r^{\langle\lambda,h_{i}\rangle}$ of the torus for each
$h_{i}\in\Lambda^{\vee}$ and $\lambda\in\Lambda$.

  4. KMG.4

If $\iota\colon R\to k$ is an injective morphism of a ring $R$ in a field $k$,
then $\mathcal{G}(\iota)\colon\mathcal{G}(R)\to\mathcal{G}(k)$ is injective.

  5. KMG.5

There is a homomorphism

$\operatorname{Ad}\colon\mathcal{G}(\mathbb{C})\to\operatorname{\text{Aut}}([\mathfrak{g}(\mathbb{A}),\mathfrak{g}(\mathbb{A})])$

whose kernel is contained in $\eta(\mathfrak{T}_{\Lambda}(\mathbb{C}))$, such
that for $c\in\mathbb{C}$ and $i\in I$,

$\operatorname{Ad}\left(\varphi_{i}\begin{pmatrix}1&c\\\
0&1\end{pmatrix}\right)=\exp(\operatorname{ad}ce_{i}),\;\operatorname{Ad}\left(\varphi_{i}\begin{pmatrix}1&0\\\
c&1\end{pmatrix}\right)=\exp(\operatorname{ad}-cf_{i}),$

and for $t\in\mathfrak{T}_{\Lambda}(\mathbb{C})$ and $i\in I$,

$\operatorname{Ad}(\eta(t))(e_{i})=t(c_{i})\cdot
e_{i},\;\operatorname{Ad}(\eta(t))(f_{i})=t(-c_{i})\cdot f_{i}.$

For the remainder of this article, let $k\in{\mathbb{R},\mathbb{C}}$ and
denote with $G$ the evaluation of the Tits functor $\mathcal{G}$ w.r.t. the
Kac–Moody root datum $\mathcal{D}_{\text{Kac}}^{\mathbb{A}}$ on the filed $k$.
Call $G$ then a _split (minimal) Kac–Moody group over $k$ (of type
$\mathbb{A}$)_. In the following, we will only refer to Kac–Moody groups over
$k$; if differences concerning the type of $\mathbb{A}$ are important, they
will be mentioned explicitly.

###### Remark 2.4

Let $G$ be a split Kac–Moody group over $k$ of type $\mathbb{A}$ with
underlying Kac–Moody algebra $\mathfrak{g}(\mathbb{A})$. Recall that every
Kac–Moody algebra $\mathfrak{g}(\mathbb{A})$ possess a _Weyl group_
$\mathcal{W}$ which is a subgroup of $\operatorname{GL}(\mathfrak{h})$
generated by a set $S=\\{r_{1}^{\vee},\ldots,r_{n}^{\vee}\\}$, where
$r_{i}^{\vee}$ is a reflection given by

$\displaystyle r_{i}^{\vee}\colon\mathfrak{h}$ $\displaystyle\to\mathfrak{h}$
$\displaystyle h$ $\displaystyle\mapsto r_{i}^{\vee}(h)\coloneqq
h-\alpha_{i}(h)\alpha_{i}^{\vee}.$

See [Mar18, Chapter 4.2], [Kac90, §3.7] or [FHHK20, A.18] for more details
about the Weyl group and its properties.
For any simple real root $\alpha_{i}\in\Delta^{re}$ define the following
subgroups of $G$

$U_{\alpha_{i}}\coloneqq\varphi_{i}\left(\left\\{\begin{pmatrix}1&r\\\
0&1\end{pmatrix}\middle|\;r\in k\right\\}\right)\;\text{ and
}\;\;U_{-\alpha_{i}}\coloneqq\varphi_{i}\left(\left\\{\begin{pmatrix}1&0\\\
-r&1\end{pmatrix}\middle|\;r\in k\right\\}\right).$

Now define the following group element for each simple root $\alpha_{i}$

$\tilde{s}_{i}\coloneqq\varphi_{i}\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$

The group generated by these elements,
$\widetilde{\mathcal{W}}\coloneqq\langle\tilde{s}_{i}\mid i\in I\rangle$, is
called the _extended Weyl group_. A detailed discussion of the extended Weyl
group and its properties can be found in [GHKW17, Section 18], [FHHK20,
Section 3.13] or [KP85, p. 172].
By the definition of a real root, each $\alpha\in\Delta^{re}$ can be written
as $\alpha=w.\alpha_{i}$ for $\alpha_{i}\in\Pi$ and $w\in\mathcal{W}$.
Moreover, using the extended Weyl group, for each real root
$\alpha\in\Delta^{re}$ we can define the corresponding _root group_
$U_{\alpha}$ as follows

$U_{\alpha}\coloneqq\tilde{w}U_{\alpha_{i}}\tilde{w}^{-1},$

where $\tilde{w}\in\widetilde{\mathcal{W}}$, see [FHHK20, Section 3.13].
Furthermore, denote with $T$ the evaluation of the split torus scheme with
respect to the root datum $\mathcal{D}_{\text{Kac}}^{\mathbb{A}}$ on $k$. Then
the triple $(G,(U_{\alpha})_{\alpha\in\Delta^{re}},T)$ by [Mar18, Theorem
7.69] forms a (linear) RGD system. This leads to a twin building which is
associated with the Kac–Moody group $G$; we denote it by $\Delta_{\pm}$. For
details on the theory of RGD systems, see [AB08, Chapter 8].

For a split Kac–Moody group $G$ over $k$ one defines the _Kac–Peterson
topology_ as the final group topology with respect to the maps
$\varphi_{i}\colon\operatorname{SL}(2,k)\to G$ and $\eta\colon T\to G$, where
$\operatorname{SL}(2,k)$ and $T$ are equipped with their natural Lie group
topologies. For details of construction, see [HKM13, Section 7.5].
This topology turns $G$ into a topological group with the following
properties:

  1. 1

$G$ is Hausdorff ([HKM13, Proposition 7.21] and [HK23, Definition and Remark
2.1]).

  2. 2

$G$ is a $k_{\omega}$ group ([HKM13, Proposition 7.10] and [HK23, Definition
and Remark 2.1]).

  3. 3

The Kac–Peterson topology induces the unique Lie group topology on $T$, in
particular $T$ is a closed subgroup ([HK23, Definition and Remark 2.1] and
[HKM13, Corollary 7.17]).

For more details, refer to [HKM13, Chapter 7], [HK23] and [Har20]. Also, many
of these properties were formulated by Kac and Peterson in section 4G of
[KP83] without proof.

##  3 Symmetric Space

###  3.1 Symmetric Spaces According Loos

Before we define a Kac–Moody symmetric space, we recall the definition of an
abstract symmetric space according to Ottmar Loos, cf. [Loo69, Chapter II],
and several important facts. Loos comes up with this abstract definition by
generalizing Riemannian symmetric spaces.

######  Definition 3.1 (Definition 1, Chapter II in [Loo69])

A topological space $X$ with a continuous map, which is called _reflection_ ,
$\mu\colon X\times X\to X$, $(x,y)\mapsto\mu(x,y)\coloneqq x.y$ is called a
_symmetric space_ $(X,\mu)$ if the map satisfies the following properties:

  1. S.1

$x.x=x$ for all $x\in X$,

  2. S.2

$x.(x.y)=y$ for all $x,y\in X$,

  3. S.3

$x.(y.z)=(x.y).(x.z)$ for all $x,y,z\in X$, and

  4. S.4

for every $x\in X$ there is a neighborhood $U$ such that $x.y=y$ implies $x=y$
for all $y\in U$.

If the space $X$ is of non-compact type, one can use a global condition
instead of the local condition S.4:

$\displaystyle\begin{matrix}\textbf{S.4}_{\text{global}}&x.y=y\Longrightarrow
x=y\;\;\forall x,y\in X\end{matrix}$

Since we are interested in generalizations of Riemannian symmetric spaces of
non-compact type leading to Kac–Moody symmetric spaces, each symmetric space
that we consider in this article will automatically satisfy
$\textbf{S.4}_{\text{global}}$ instead of simply S.4.
Call a continuous map $f\colon X_{1}\to X_{2}$ between two symmetric spaces
$(X_{1},\mu_{1})$ and $(X_{2},\mu_{2})$ a _morphism of symmetric spaces_ , if
$f(\mu_{1}(x,y))=\mu_{2}(f(x),f(y))$. A symmetric space is called _pointed_ if
it has a chosen base point $\beta$. A morphism between pointed symmetric
spaces is a morphism between their underlying symmetric spaces with the
additional property that the base point is preserved.
In the rest of the article, if the map is clear from the context, we will
write $X$ for the symmetric space pair $(X,\mu)$ for short.

###### Example 3.2

One of the first examples that comes to mind of a symmetric space is the
$n$-dimensional Euclidean space $\mathbb{E}^{n}$ for any $n\in\mathbb{N}$. In
particular, the symmetric space is given by the pair $\mathbb{R}^{n}$ together
with the point reflection on $x\in\mathbb{R}^{n}$, i.e. the symmetric space
map $\mu_{\text{Eucl}}$ is defined as $\mu_{\text{Eucl}}(x,y)\coloneqq 2x-y$.

Following [FHHK20, Section 2.15] one recall the necessary properties and
definitions of a symmetric space $(X,\mu)$.

  1. 1

A _midpoint_ of two points $x,y\in X$ is a point $m\in X$, such that
$\mu(m,x)=y$ and $\mu(m,y)=x$.

  2. 2

Call a subspace $U\subseteq X$

    1. 1

a _reflection subspace_ , if for all pair of points $x,y\in U$ the image
$\mu(x,y)$ is contained in $U$.

    2. 2

_midpoint convex_ , if for all $x,y\in U$ there is a midpoint of $x$ and $y$
in $U$.

  3. 3

Two points $x,y\in X$ _weakly commute_ , if for every point $p\in X$ applies
$x.(p.(y.p))=y.(p.(x.p))$.

  4. 4

Two points $x,y\in X$ _commute_ , if for all points $p,q\in X$ applies
$x.(p.(y.q))=y.(p.(x.q))$.

  5. 5

A reflection subspace $F\subseteq X$ is a _(weak) flat_ , if it is closed,
midpoint convex, contains at least two points and if all points $x,y\in F$
(weakly) commute.

Moreover a symmetric subspace $F\subseteq X$ is called _Euclidean flat of rank
$n$_ if it is isomorphic to the $n$-dimensional Euclidean space
$\mathbb{E}^{n}$ as a symmetric space and if $F$ is closed. In the case of
rank one, the flat is called a _geodesic_ (cp. [FHHK20, Definition 2.23]).

###  3.2 Kac–Moody Symmetric Space

As we have seen, one needs a map $\mu$ with several properties to obtain a
symmetric space. It turns out, that the Chevalley involution together with the
complex conjugation is essential to define such a map $\mu$.
At the level of a Kac–Moody algebra, the Chevalley involution composed with
the complex conjugation due to [Cap09, Section 8.2] leads to a continuous
involution $\Theta$ on a Kac–Moody group $G$ over $k$ with respect to the Kac-
Peterson topology. Note, that for $k=\mathbb{R}$ the $\Theta$ is just the
Chevalley involution, since the complex conjugation conjugation fixes the real
numbers elementwise. The fixed point set of $\Theta$ is denoted by $K$, i.e.

$K\coloneqq\\{g\in G\mid\Theta(g)=g\\}.$

Now we define another continuous map which proves to be very useful in the
definition and survey of symmetric spaces: The _twist map_ of $G$ (also called
the _Lang map_ with respect to the involution $\Theta$) is defined as follows

$\displaystyle\begin{split}\tau\colon G&\to G\\\ g&\mapsto\tau(g)\coloneqq
g\Theta(g^{-1}).\end{split}$ (1)

Since the map $\Theta$ is continuous and $G$ is a topological group, it
follows that $\tau$ is also continuous. Note also that $K$ is a closed
subgroup, since $\Theta$ is continuous and thus the quotient $G/K$ is a
topological Hausdorff space.

###### Proposition 3.3

Let $G$ be a Kac–Moody group over $k$. Define the map

$\displaystyle\begin{split}\mu\colon G/K\times G/K&\to G/K\\\
(gK,hK)&\mapsto\tau(g)\Theta(h)K.\end{split}$ (2)

The map $\mu$ is continuous, and if $\tau(G)\cap K=\\{e\\}$, then $(G/K,\mu)$
is a symmetric space with the natural action

$G\to\operatorname{Sym}(G/K)\,,\;g\mapsto(hK\mapsto ghK),$

which is given by automorphisms.

###### Remark 3.4

Before we prove this statement we need an auxiliary statement which allows us
to prove axiom $\textbf{S.4}_{\text{global}}$ in the context of Kac–Moody
groups over $k\in\\{\mathbb{R},\mathbb{C}\\}$ of arbitrary type. Call an
element $g$ of a Kac–Moody group $G$ with the property

$\Theta(g)=g^{-1}$

_symmetric_.

######  Lemma 3.5 (Lemma 4.2 in [Hor17])

Let $G$ be a Kac–Moody group of type $\mathbb{A}$ and denote with
$\Delta_{\pm}$ the associated twin building. For a symmetric element $g\in G$
the statements below are equivalent:

  1. 1

The element $g$ fixes a $\Theta$-stable twin apartment chamberwise.
$\Theta$-stable means that a twin apartment of $\Delta_{\pm}$ is invariant as
a set under the action of $\Theta$.

  2. 2

The element $g$ fixes a twin apartment chamberwise.

  3. 3

The element $g$ stabilizes a chamber.

  4. 4

For all chambers $D$ in the twin building $\Delta_{\pm}$, the length of any
minimal gallery from $D$ to a chamber in the set of $G$-orbits $\\{g^{n}.D\mid
n\in\mathbb{Z}\\}$, is bounded.

  5. 5

For some chamber $D$ in the twin building $\Delta_{\pm}$, the length of any
minimal gallery from $D$ to a chamber in the set of $G$-orbits $\\{g^{n}.D\mid
n\in\mathbb{Z}\\}$, is bounded.

  6. 6

The element $g$ stabilizes a spherical residue in either half of the twin
building.

###### Proof

Let $g$ be a symmetric element of $G$. For obvious reasons, in the
implications $\ref{itm:fir}\Rightarrow\ref{itm:sec}$,
$\ref{itm:sec}\Rightarrow\ref{itm:thr}$, and
$\ref{itm:for}\Rightarrow\ref{itm:fiv}$ is nothing to prove.
The idea now is to prove that $\ref{itm:thr}\Rightarrow\ref{itm:fir}$, thus
the first three points are equivalent. Then we show that

$\ref{itm:thr}\Rightarrow\ref{itm:for}\Rightarrow\ref{itm:fiv}\Rightarrow\ref{itm:six}\Rightarrow\ref{itm:sec},$

which gives us the desired statement.

  1. $\ref{itm:thr}\Rightarrow\ref{itm:fir}$

Let $D$ be a chamber in $\Delta_{\pm}$ which is stabilized by $g$, i.e.
$g.D=D$. If $g$ stabilizes $D$, then also $g^{-1}$ stabilizes $D$:

$D=(g^{-1}g).D=g^{-1}.(g.D)=g^{-1}.D.$

Hence one obtains by a straight forward calculation,

$\Theta(D)=\Theta(g.D)=g^{-1}.\Theta(D),$

that $g$ stabilizes the $\Theta$-stable twin apartment $\Sigma(D,\Theta(D))$,
since $\Theta(D)$ is opposite of $D$.

  2. $\ref{itm:thr}\Rightarrow\ref{itm:for}$

First, recall that the building $\Delta_{\pm}$ associated with the Kac–Moody
group $G$ is of type $(\mathcal{W},S)$.
Let $D^{\prime}\in\Delta_{+}$ be a chamber in the positive half of the twin
building which is stabilized by $g$, also let $D\in\Delta_{+}$ be an arbitrary
chamber and $n\in\mathbb{Z}$. Then we know that

$\delta_{+}(D^{\prime},D)=\delta_{+}(g^{n}.D^{\prime},g^{n}.D)=\delta_{+}(D^{\prime},g^{n}.D).$

Next, denote with $l\colon\mathcal{W}\to\mathbb{N}$ the length function of the
Weyl group $\mathcal{W}$ with respect to the generating set $S$. By [AB08,
Definition 5.1 (WD3)] one derives a triangle inequality, which is given for
the length function $l$, so that one can compute

$l(\delta_{+}(D,g^{n}.D))\leq
l(\delta_{+}(D,D^{\prime}))+l(\delta_{+}(D^{\prime},g^{n}.D))=2l(\delta_{+}(D,D^{\prime})).$

Hence the orbit $\\{g^{n}.D\mid n\in\mathbb{Z}\\}$ is bounded and by the
symmetry arguments of the to halves of a twin building, the claim follows.

  3. $\ref{itm:fiv}\Rightarrow\ref{itm:six}$

This statement is exactly the Bruhat-Tits fixed point theorem applied on the
CAT(0) realization of $\Delta_{\pm}$, see [AB08, Corollary 12.67].

  4. $\ref{itm:six}\Rightarrow\ref{itm:sec}$

As we have already seen, if $g$ stabilizes a chamber $D$, its inverse $g^{-1}$
also stabilizes $D$. The same calculation can be done to see, that if $g^{-1}$
stabilizes $D$, then also $g$ stabilizes $D$:

$D=(gg^{-1}).D=g.(g^{-1}.D)=g.D.$

Now let $R$ be a spherical residue stabilized by $g$ and by the computation

$\Theta(R)=\Theta(g.R)=g^{-1}.\Theta(R),$

and the previous comment, follows that $\Theta(R)$ is also stabilized by $g$.
Hence the symmetric element $g$ stabilizes the spherical residue in each half
of $\Delta_{\pm}$.
The idea is now, to show that $g$ fixes a chamber in $R$ and hence a twin
apartment in $\Delta_{\pm}$. In particular, if $g$ fixes a chamber $D$ in $R$,
$g$ fixes also $\Theta(D)$, which is an opposite chamber. This leads to the
fact, that $g$ fixes then the twin apartment $\Sigma(D,\Theta(D))$ point-wise,
i.e. $g$ stabilizes a twin apartment chamberwise.
By [AB08, Proposition 67.27], one knows that the stabilizer of a spherical
residue of type $J$ equals a (spherical) parabolic subgroup $P_{J}$ (compare
also discussion before [Mar18, Proposition 7.75]). Since $g$ stabilizes $R$
and an opposite residue $\Theta(R)$, it is an element of

$g\in P_{J}\cap\Theta(P_{J}).$

Using the fact, that a parabolic subgroup, in the context of a Kac–Moody
group, has a Levi decomposition (see [Rém02, 6.2.2 (Théorème)] or [Mar18,
Proposition 7.75]) one can analyze this intersection. Therefore, denote by

$P_{J}=G_{J}\ltimes U_{J}^{+}\,\text{ and }\Theta(P_{J})=G_{J}\ltimes
U_{J}^{-}$

the Levi decomposition of $P_{J}$ and note that $\Theta(P_{J})$ is obtained by
applying the Chevalley involution.
Here we used the notion from [Mar18, Chapter 7], in particular the group
$G_{J}$ stands for a subgroup of $G$, which is generated by the torus $T$ and
the root groups $U_{\alpha}$, where $\alpha$ are the roots contained in the
span of simple roots corresponding to $J$, see the discussion before [Mar18,
Proposition 7.75]. The second part of the Levi decomposition, $U_{J}^{\pm}$,
is the normal subgroup of $U_{\pm}$ generated by $U_{\alpha}$, where $\alpha$
is in the complement of the roots used to generate $G_{J}$.
Since $g$ is an element of the intersection, it follows that it can be written
in two ways:

$xu_{+}=g=yu_{-},$

where $x,y\in G_{J}$, $u_{+}\in U_{J}^{+}$ and $u_{-}\in U_{J}^{-}$. In fact,
one realizes that

$g^{-1}.R=(u_{-}^{-1}y^{-1}).R=u_{-}^{-1}.R=R,$

since $y^{-1}$ is an element of the stabilizer. It yields that $u_{-}\in
P_{J}\cap U_{J}^{-}=\\{1\\}$, and analogously one can show with
$\Theta(P_{J})$ that $u_{+}$ must also be trivial.
And so, the action of $g$ on the residue depends only on $G_{J}$, and since
$\mathcal{W}_{J}$ is finite, this is a Chevalley group, i.e. a linear
algebraic group. By [Bor91, Proposition 1.10] we can consider this group as a
closed subgroup of $\operatorname{GL}(n,k)$,
$k\in\\{\mathbb{R},\mathbb{C}\\}$. To clarify the notation, we write $M_{g}$
for the corresponding matrix in the subgroup of $\operatorname{GL}(n,k)$.
The Chevalley involution translates for these matrices to transpose-inverse,
composed with complex conjugation if $k=\mathbb{C}$. Taking a closer look on
the property that $g$ is symmetric turns $M_{g}$ it into a Hermitian matrix,
in particular

$\Theta(M_{g})=M_{g}^{-1}\Longleftrightarrow\left(\overline{M_{g}}\right)^{-T}=M_{g}^{-1}\Longleftrightarrow
M_{g}^{T}=\overline{M_{g}}.$

Thus, $M_{g}$ is diagonalizable and hence it is conjugated to an diagonal
matrix $D_{g}$. Translated back into the language of Kac–Moody groups, this
leads to the fact that $g$ is conjugate to an element of the standard torus
$T$ of the Kac–Moody group $G$. Since the standard torus fixes the pair of
fundamental chambers $(C,C^{\prime})$ in $\Delta_{\pm}$, a conjugated element
fixes a translated pair of the fundamental chambers. In detail if $T$ fixes
$C$ and $aga^{-1}\in T$ denotes the conjugated torus version of $g$, one
calculates

$T.C=C\Longrightarrow\left(aga^{-1}\right).C=C\Longrightarrow
g.\left(a^{-1}.C\right)=a^{-1}.C.$

By the argument from above, $g$ fixes then the twin apartment
$\Sigma(a^{-1}.C,a^{-1}.\Theta(C))$ chamberwise. Note that all of these
arguments apply to both halves of the twin building for reasons of symmetry.

∎

######  Proof (Proof of 3.3)

Since $G$ is a topological group and $\Theta$ and $\tau$ are continuous, the
map $\mu$ is continuous as well. The verification of axioms S.1, S.2, and S.3
is a straightforward calculation and can be found in the proof of Proposition
4.2 in [FHHK20]. Note that this is just a simple calculation, without assuming
any special type of $\mathbb{A}$ or using the underlying field of the group.
To prove that $K\cap\tau(G)$ is trivial, one may mimic the proof given for
[FHHK20, Proposition 3.38]:
Let $g$ be an element of $\tau(G)$, i.e. $g=h\Theta(h)^{-1}$, then one
calculates

$\Theta(g)=\Theta(h\Theta(h)^{-1})=\Theta(h)h^{-1}=g^{-1}.$

Thus, if $g\in K\cap\tau(G)$, one realizes that $g=\Theta(g)=g^{-1}$, from
which it follows that $g$ has order at most $2$, leading to the fact that the
orbits of $g$ are bounded as in point 4 of 3.5. Hence by the proof of 1 from
3.5 one knows, that $g$ fixes the $\Theta$-stable twin apartment
$\Sigma(C,\Theta(C))$ chamberwise. Thus $g$ is contained in a corresponding
torus $T^{\prime}$ of $G$. By [FHHK20, Theorem 3.31 (iii)], based on the
Iwasawa decomposition 3.11, which we will prove later, states that $K$ acts
transitively on both halves of the associated twin building. Hence there is a
$k\in K$ such that $kT^{\prime}k^{-1}=T$, and that $kg\Theta(k)^{-1}$ is an
element of $T$ and of $K\cap\tau(G)$. Since the proof of [FHHK20, Lemma 3.27
(i)] does not use assertions about the type or field of the Kac–Moody group we
can use the statement and conclude that $T\cap\tau(G)=A$. From [FHHK20, Lemma
3.26 (iii)], which again will not require any assertions about the type or
field of $G$, it immediately follows that

$g\in T\cap\tau(G)\cap K=A\cap K=\\{e\\}.$

∎

###### Definition 3.6

Let $G$ be a Kac–Moody group over $k\in\\{\mathbb{R},\mathbb{C}\\}$ of
arbitrary type $\mathbb{A}$. Then the pair $(G/K,\mu)$ is called a _complex
(or real) Kac–Moody symmetric space (of type $\mathbb{A}$)_.

###### Remark 3.7

  1. 1

Note that this definition of the reflection map $\mu$ also holds for any
topological group with an involution. Moreover, the first three properties in
3.1 can be proved without any assumption about the group, see the calculation
in the proof of [FHHK20, Proposition 4.2]. Thus any topological group with an
involution that satisfies the last property gives rise to a symmetric space.
Let’s take as an illustration 3.2: the group is given by $(\mathbb{R}^{n},+)$
and the involution is given by $\Theta\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$,
$\Theta(x)=-x$. Then one computes for $\tau(x)=x+\Theta(x)^{-1}=2x$ and hence
the reflection of the symmetric space is given by
$\mu(x,y)=\tau(x)+\Theta(y)+K=2x-y$ and $K$ is equal to
$\\{0\\}\subseteq\mathbb{R}^{n}$.

  2. 2

The definition given here of a Kac–Moody symmetric space of arbitrary type
over $\mathbb{R}$ or $\mathbb{C}$ is the same as in [FHHK20], since the
problems arise only in the further study of the symmetric space. To change the
underlying field from the real to the complex numbers requires a generalized
version of the topological Iwasawa decomposition, see 3.11. The step between
the non-affine and affine cases is somewhat more subtle:
For a non-affine generalized Cartan matrix, there is an approach for symmetric
spaces described in [FHHK20]. In the affine case one has to extend the Cartan
subalgebra and consequently the torus, see 2.3. This leads to several
problems, see Section 4 or 3.14. Therefore, the approach taken in this article
to allow affine generalized Cartan matrices leads to a larger symmetric space
than in [FHHK20], and this approach works effectively for all matrices in the
end. This means that for non-affine Kac–Moody groups with non-invertible
generalized Cartan matrix, there are two symmetric spaces to consider. For
details, see Section 6.
Moreover, in Section 4.2 (or [FHHK20, Section 6]), a faithful action of the
Weyl group is used to study local actions on the symmetric space. This action
is given by isometries of a non-degenerate symmetric bilinear form that
depends on the generalized Cartan matrix $\mathbb{A}$ and in particular on the
assumption that $\mathbb{A}$ is symmetrizable. For details on the bilinear
form, see [Kac90, §2.0] or [FHHK20, A.22]. To obtain such a faithful Weyl
group action, the authors in [FHHK20] use an _semisimple adjoint quotient_ of
a Kac–Moody group, which means that they consider the quotient of the
Kac–Moody group with its connected central subgroup. This central subgroup
corresponds to the radical of the bilinear form on
$\mathfrak{h}^{\prime}=\sum_{i=1}^{n}\mathbb{C}\alpha_{i}^{\vee}$ ([Kac90,
Lemma 2.1 a)]), so the bilinear form on this quotient is not degenerate.
By enlarging the Cartan subalgebra in the case of affine Kac–Moody algebras,
the bilinear form is well-defined and not degenerate on $\mathfrak{h}$
([Kac90, Lemma 2.1 b), Theorem 2.2 b)]). Thus one has again a faithful action
of the Weyl group which can be used. But then the group also enlarges, which
complicates further studies, for example the analysis of automorphisms, see
Section 4.

To analyze the structure of the Kac–Moody symmetric space of a Kac–Moody group
of type $\mathbb{A}$, we first have to define flats. For this purpose we can
use the fact that the Cartan subalgebra is
$\mathfrak{h}\cong\mathbb{C}^{2n-l}$, where $l$ is the rank of the generalized
Cartan matrix $\mathbb{A}\in\mathbb{Z}^{n\times n}$. The next step is to
define a subgroup $A\coloneqq\tau(T)$.
We are now interested in describing the structure of $A$ and for this recall
that the torus $T$ is isomorphic to $(k^{\times})^{2n-l}$,
$k\in\\{\mathbb{R},\mathbb{C}\\}$, by definition. By [Cap09, Section 8.2]
follows that $\Theta(t)=\overline{t}^{-1}$, hence one calculates

$\displaystyle\tau(t)$ $\displaystyle=t\cdot\theta(t)^{-1}$
$\displaystyle=t\cdot\bar{t}=|t|^{2}.$

Therefore, $A=\tau(T)\cong(\mathbb{R}_{>0})^{2n-l}$ is independent of the
choice of the underlying field $k\in\\{\mathbb{R},\mathbb{C}\\}$.

For $k=\mathbb{R}$ the torus has as a topological group the decomposition
$(\mathbb{R}^{\times})^{2n-l}\cong T\cong M\times A$, where $M$ is its
_torsion subgroup_ equipped with the discrete topology, i.e. $M$ is the unique
maximal finite subgroup of order $2^{2n-l}$.
In the complex case, one can use the polar decomposition and deduce that

$(\mathbb{C}^{\times})^{2n-l}\cong T\cong(\SS^{1})^{2n-l}\times A.$

Corresponding to the real case, one define the _maximal compact subgroup_
$M\cong(\SS^{1})^{2n-l}$ as the unique maximal compact subgroup.
Note that in the real case $M$ is called the torsion subgroup and it is the
only maximal finite subgroup (see [FHHK20, discussion after Proposition
3.11]). Further, every finite discrete subgroup is compact, which means the
name of $M$ in the complex is adequate.
Define a Borel subgroup of $G$ by $B_{\pm}\coloneqq T\ltimes U_{\pm}$, where
$U_{\pm}$ is the subgroup generated only by the positive (or negative) root
groups. It can be shown that

$\displaystyle T\cap K=B_{\pm}\cap K=M$ (3)

also holds for $k=\mathbb{C}$, as in the real case, see [FHHK20, Lemma 3.26
(iii)]. Note that this statement works in the general case, since the
calculation of the necessary fact about the elements of $A$ is the same. In
detail, this can be seen as follows: Let $g\in B_{\pm}\cap K$ and recall that
$\Theta(B_{+})=B_{-}$, see [FHHK20, Lemma 3.21], it should be noted that this
statement does not take into account the underlying field of the group nor the
type of the generalized Cartan matrix. Then $g$ is an element of $T=B_{+}\cap
B_{-}$, since $\Theta(g)=g$, which gives that $B_{\pm}\cap K=T\cap K$. For the
next equality, consider an element $t\in T$ and use the decomposition $T=MA$,
i.e., write $t=ma$ with $m\in M$ and $a\in A$. Then one calculates

$\tau(t)=|t|^{2}=|m^{2}|\cdot|a^{2}|=|a|^{2}=a^{2}.$

It follows that $\tau(t)=e$ if and only if $a=e$, since
$A\cong(\mathbb{R}_{>0})^{2n-l}$. Moreover, since we know from [FHHK20, Lemma
3.24 (iii)], which is a straightforward calculation without assumptions, that
$\tau^{-1}(e)=K$, it follows that $T\cap K=M$.

In the following we denote by $M$ both subgroups such that for any
$k\in\\{\mathbb{R},\mathbb{C}\\}$ there exists the decomposition $T\cong
M\times A$. If it is necessary to distinguish between the real and the complex
case, we will point it out.

The recently defined subgroup $A$ of the torus $T$ can be connected to a
subset of the Cartan subalgebra $\mathfrak{h}$. To see this, we first recall
that $T$ is defined by a split torus scheme $\mathfrak{T}_{\mathcal{D}}$,
where $\mathcal{D}$ denotes a root datum. In this article, we have fixed
$\mathcal{D}=\mathcal{D}_{\text{Kac}}^{\mathbb{A}}$. By this choice of a root
datum, we have determined a basis of $\mathfrak{h}$, in particular, the
elements $v_{i}\in\Lambda^{\vee}$ form a basis ([Mar18, Example 7.10]).
Together with the definition of the torus $T$, we know that $T$ is generated
by the following elements

$\\{\lambda^{v_{i}}\mid\lambda\in\mathbb{C}^{\times},\,1\leq i\leq 2n-l\\},$

see [Mar18, Example 7.25]. Following a similar approach to [Mar18, Exercise
7.26], one can define an exponential map between $\mathfrak{h}$ and $T$ in the
following sense

$\displaystyle\exp\colon\mathfrak{h}$ $\displaystyle\to T$ (4)
$\displaystyle\lambda v_{i}$ $\displaystyle\mapsto(e^{\lambda})^{v_{i}},$

where $\mathfrak{h}=\Lambda^{\vee}\otimes_{\mathbb{Z}}\mathbb{C}$, see 2.3.
Note, that $\lambda$ is an element from $\mathbb{C}$, so with
$\mathfrak{h}\cong\mathbb{C}^{2n-l}$ and
$T\cong\left(\mathbb{C}^{\times}\right)^{2n-l}$ this map is equivalent to the
usual exponential map. Now, to obtain a connection with $A$, we will define a
real form of the Cartan subalgebra

$\mathfrak{a}\coloneqq\operatorname{span}_{\mathbb{R}}\left(\alpha_{1}^{\vee},\ldots,\alpha_{2n-l}^{\vee}\right)\cong\mathbb{R}^{2n-l}.$

Restricting the exponential map to $\mathfrak{a}$, we obtain that

$\exp\mid_{\mathfrak{a}}\colon\mathfrak{a}\to A$

is bijective using the identification $A\cong(\mathbb{R}_{>0})^{2n-l}$, which
comes from the recent discussion of the structure of $A$.

###  3.3 The Iwasawa Decomposition

In the theory of symmetric spaces, it is necessary to define flats because
they are the most important principles for understanding actions and causal
boundaries. To do so, one needs the topological Iwasawa decomposition, i.e.
that the multiplication induces a homeomorphism

$m_{\pm}\colon U_{\pm}\times A\times K\to G.$

Here $U_{\pm}$ denotes the subgroups of $G$ which are generated by the root
groups $U_{\alpha}$ for all real positive or negative roots
$\alpha\in\Delta^{re}$.
The underlying component to prove the topological variant is to show the
algebraic decomposition. In general this is known for centered Kac–Moody
groups, see [KP85, Section 3], or for real centered Kac–Moody groups it can be
found in [FHHK20, Theorem 3.31 (ii)].

###### Proposition 3.8

Let $G$ be a Kac–Moody group over $k\in\\{\mathbb{R},\mathbb{C}\\}$ and of
arbitrary type $\mathbb{A}$. Then $G$ decomposes into

$G=U_{\pm}AK=KAU_{\pm}.$

Moreover, the maps,

$\displaystyle m_{1}$ $\displaystyle\colon U_{\pm}\times A\times K\to G$
$\displaystyle m_{2}$ $\displaystyle\colon K\times A\times U_{\pm}\to G,$

given by multiplication, are bijective.

###### Proof

By the axioms of an RGD system, one can decompose $G$ into $TG^{\dagger}$,
where $T$ is the torus and $G^{\dagger}$ is the group generated by the root
subgroups. In particular, $G^{\dagger}$ is the centered Kac–Moody group
associated with $G$ which is studied in [FHHK20], for details see later 3.15.
Since the underlying field $k$ plays no role in the algebraic decomposition,
and $G^{\dagger}$ is a centered Kac–Moody group, one can use [FHHK20, Theorem
3.31 (ii)]:

$G^{\dagger}=U_{\pm}A^{\dagger}K^{\dagger}=K^{\dagger}A^{\dagger}U_{\pm},$

where $K^{\dagger}\coloneqq K\cap G^{\dagger}$ and $A^{\dagger}\coloneqq A\cap
G^{\dagger}$.
Recall that $T=MA$, $M\subseteq K$ by Equation 3 and that $T$ normalizes
$U_{\pm}$. Now we put everything together and compute

$\displaystyle G$ $\displaystyle=TG^{\dagger}$
$\displaystyle=TU_{\pm}A^{\dagger}K^{\dagger}$
$\displaystyle=U_{\pm}TA^{\dagger}K^{\dagger}$
$\displaystyle=U_{\pm}A^{\dagger}(AM)K^{\dagger}$ $\displaystyle=U_{\pm}AK$

since the only part which distinguishes $K^{\dagger}$ from $K$ is contained in
the additional torus $T$, and by Equation 3 this is the subgroup $M$. The
other order of the decomposition results from the symmetry arguments.

To see that these maps $m_{\pm}$ are bijective, we need to check injectivity.
To do this, we assume that $u_{1}a_{1}k_{1}=u_{2}a_{2}k_{2}$. By reordering
follows,

$K\ni k_{1}k_{2}^{-1}=a_{1}^{-1}u_{1}^{-1}u_{2}a_{2}\in AU_{\pm}A\subseteq
B_{\pm}.$

Hence $a_{1}^{-1}u_{1}^{-1}u_{2}a_{2}\in K$ and $B_{\pm}\cap K=M\subseteq T$,
see Equation 3. It follows that

$u_{1}^{-1}u_{2}\in U_{\pm}\cap T=\\{1\\}\Longrightarrow u_{1}=u_{2},$

(cp. [AB08, Lemma 7.62]) and from this, that

$a_{1}^{-1}a_{2}\in M\cap A=\\{1\\}\Longrightarrow a_{1}=a_{2}.$

It follows that $k_{1}=k_{2}$ and thus $m_{\pm}$ is injective. ∎

Now for the topological version, the idea is to use the same strategy as in
the proof of the special case of the topological Iwasawa decomposition seen in
[FHHK20, Theorem 3.31 (ii)]. At this stage, the authors could only prove this
special case, since it was not yet known that every Kac–Moody group has a
strong topological building (see the results by J. Grüning and the first
author of this article from [HK23, Appendix B]). This property allows us to
use formula (9) from [FHHK20, page 39] in a more general context, and thus we
are able to prove [FHHK20, Proposition 3.36] for Kac–Moody groups of any type
over $k$.
For an introduction to topological buildings see [HKM13, Section 3.1] and for
strong topological buildings in particular see [HKM13, Definition 3.21].

In below we consider only Kac–Moody groups over $\mathbb{A}$, where
$\mathbb{A}$ is a symmetrizable Cartan matrix of arbitrary non-spherical type.
In the case where $\mathbb{A}$ is of spherical type, the topological Iwasawa
decomposition is a well-known fact, see for example [Kna02, Theorem 6.46].
Further, there are also proofs of the topological Iwasawa decomposition with
respect to other groups, e.g. for $\operatorname{GL}(n,\mathbb{R})$ ([PR94,
Proposition 3.12]) or in greater generality for reductive $\mathbb{R}$-groups
([PR94, Theorem 3.9]).

###### Lemma 3.9

Let $G$ be a Kac–Moody group over $k$. Then the fibration

$p_{\pm}\colon\nicefrac{{G}}{{AU_{\pm}}}\to\nicefrac{{G}}{{B_{\pm}}}$

is trivial.

###### Proof

First, note that $G$ leads to a strong topological twin building, according to
a result of J. Grüning and R. Köhl, see [HK23, Corollary B.8]. Furthermore,
note that $T\cong A\times M$ as topological groups and that $U_{\mp}\times
T\times U_{\pm}$ is open. In particular the map $U_{\mp}\times T\times
U_{\pm}\to B_{\mp}B_{\pm}$ is a homeomorphism (cf. [AB08, Section 8.8] and
[HK23, Corollary B.7]) and $B_{\mp}B_{\pm}$ is open (cf. [HKM13, Lemma 6.1]).
Hence

$1\times M\times 1\to U_{\mp}\times T\times U_{\pm}\to U_{\mp}\times A\times
U_{\pm}$

provides a local trivialization of the fibration

$M\to\nicefrac{{G}}{{AU_{\pm}}}\to\nicefrac{{G}}{{B_{\pm}}},$

in particular, every point in $G/AU_{\pm}$ has a neighborhood that looks like
an open set in $G/B_{\pm}$ times $M$. Again, by the property that the
associated topological twin building is strong, we know by [HKM13, Theorem
5.13] that the building is contractible. Hence the fibration $p_{\pm}$ is
trivial. ∎

A necessary condition for the proof of the topological Iwasawa decomposition
is the following lemma, which is a generalization of [FHHK20, Proposition
3.36]. The proof of this theorem uses the theory of $k_{\omega}$-pairs, for
details see [FT77] or [FHHK20, Definition 3.33 et seq.].

###### Lemma 3.10

Let $G$ be a Kac–Moody group over $k$ and define the maps

$\displaystyle\iota_{\pm}\colon K$ $\displaystyle\to G/AU_{\pm}$
$\displaystyle k$ $\displaystyle\mapsto kAU_{\pm}.$

These maps are homeomorphisms.

###### Proof

The goal is to use [FHHK20, Lemma 3.34], which is a purely topological
statement about $k_{\omega}$-spaces. According to the algebraic Iwasawa
decomposition 3.8 it follows that the maps are $\iota_{\pm}$ are continuous
bijections, since the multiplication is continuous.
Next define

$G_{l}^{\pm}\coloneqq\bigcup_{w\in\mathcal{W},\,l(w)\leq l}B_{\pm}wB_{\pm},$

and denote with $\widetilde{D}_{l,\pm}$ the image of $G_{l}^{\pm}$ in
$G/AU_{\pm}$ and with $D_{l,\pm}$ the image of $G_{l}^{\pm}$ in $G/B_{\pm}$.
By [HKM13, Corollary 3.13] follows that

$\left(\nicefrac{{G}}{{B_{\pm}}},\left(D_{l,\pm}\right)\right)$

is a $k_{\omega}$-pair. Note that the preimage of $D_{l,\pm}$ under the
fibration $p_{\pm}$ is exactly $\widetilde{D}_{l,\pm}$ and that the fibration
by 3.9 is trivial, so we can apply [FHHK20, Lemma 3.35], which works here,
since the statement is just a topological assertion about $k_{\omega}$ pairs,
so we observe that

$\left(\nicefrac{{G}}{{AU_{\pm}}},\left(\widetilde{D}_{l,\pm}\right)\right)$

is also a $k_{\omega}$-pair.
In order to apply Lemma 3.34 from [FHHK20], it remains to prove that
$(K,(K_{l}^{\pm}))$ is a $k_{\omega}$-pair, where $K_{l}^{\pm}\coloneqq K\cap
G_{l}^{\pm}$. Therefore by [HKM13, Corollary 7.11] one knows that

$G=\varinjlim G_{l}^{\pm}$

and in particular that

$K=\varinjlim K_{l}^{\pm}.$

To conclude that $(K,(K_{l}^{\pm}))$ is a $k_{\omega}$ pair, we must show that
the subsets $(K_{l}^{\pm})$ are compact. For this we recall that a Kac–Moody
group possesses a twin building $\Delta_{\pm}$ of type $(\mathcal{W},S)$,
which is constructed using the RGD system. Moreover, for a chamber $C$ in
$\Delta_{\pm}$ one can define a set

$E_{w}(C)=\\{D\in\mathcal{C}_{\pm}\mid\delta_{\pm}(C,D)=w\\},$

called _Schubert cell_ , for details see [HKM13, p.8]. In particular, these
cells are compact, see [HK23, Corollary 3.10].
Denote by $C_{\pm}$ the fundamental chambers of $\Delta_{\pm}$ and recall that
the fundamental chambers correspond to $B_{\pm}$. Therefore, we can rewrite
the $G_{l}^{\pm}$ as

$G_{l}^{\pm}=\bigcup_{w\in\mathcal{W},\,l(w)\leq l}E_{w^{-1}}(C_{\pm}).$

Since this is a finite union, we know that $G_{l}^{\pm}$ is compact, and the
intersection $G_{l}^{\pm}\cap K=K_{l}^{\pm}$ is compact since $K$ is a closed
set. Therefore, the pair $(K,(K_{l}^{\pm}))$ is a $k_{\omega}$ pair.
Furthermore,

$\iota_{\pm}(K_{l}^{\pm})=\widetilde{D}_{l,\pm}$

which allows us to apply the desired lemma and finish the proof. ∎

###### Theorem 3.11

Let $G$ be a non-spherical split Kac–Moody group over $k$. Then the Iwasawa
decompositions

$\displaystyle m_{1}$ $\displaystyle\colon U_{\pm}\times A\times K\to G,$
$\displaystyle m_{2}$ $\displaystyle\colon K\times A\times U_{\pm}\to G$

are homeomorphisms.

###### Proof

We prove here only the statement for $m_{1}$, since the arguments for $m_{2}$
are the same because of symmetry.
By the algebraic Iwasawa decomposition 3.8 we know that $m_{\pm}$ is a
bijection, so it follows from the continuity of multiplication in $G$ that
$m_{\pm}$ is continuous. Thus, all that remains is to construct an inverse
continuous map as in the proof of [FHHK20, Proposition 3.31].
Let $g\in G$ and define the two elements

$k(g)\coloneqq\iota_{\pm}^{-1}\left(gAU_{\pm}\right)\;\text{ and
}\;h(g)\coloneqq k(g)^{-1}g,$

where the maps $\iota_{\pm}$ are the same as in 3.10. Therefore we define the
map

$\displaystyle n_{\pm}\colon G$ $\displaystyle\to K\times AU_{\pm}$
$\displaystyle g=k(g)h(g)$ $\displaystyle\mapsto(k(g),h(g)),$

which is a continuous map, since $G$ is a topological group and $\iota_{\pm}$
is a homeomoprhism. Moreover, $n_{\pm}$ is inverse to $m_{\pm}$, which
finishes the proof. ∎

###  3.4 Flats

Having proved the topological Iwasawa decomposition for any split Kac–Moody
group over $k$, we can formulate the following definition of an important
object for Kac–Moody symmetric spaces.

###### Proposition 3.12

Let $G$ be a split Kac–Moody group over $k$ and
$\mathfrak{a}\coloneqq\operatorname{span}_{\mathbb{R}}\left(\alpha_{1}^{\vee},\ldots,\alpha_{2n-l}^{\vee}\right)$,
moreover equip $\mathfrak{a}$ with its Euclidean symmetric space structure as
in 3.2. Then the map

$\displaystyle\begin{split}\varphi_{g}\colon\mathfrak{a}&\to gAK\\\ X&\mapsto
g\exp(X)K\end{split}$ (5)

is an isomorphism of symmetric spaces for every $g\in G$. In particular, the
subset $gAK$ of the Kac–Moody symmetric space $G/K$ is an Euclidean flat of
dimension $\dim\mathfrak{a}=2n-l$.

This characterizes all maximal flats of a Kac–Moody symmetric space, as we
will see later.

###### Proof

By the definition of an Euclidean flat one need to check that $gAK\subset G/K$
is closed and that $\varphi_{g}$ is an isomorphism of symmetric spaces. 3.11
tells that the multiplication

$m\colon U_{\pm}\times A\times K\to G$

is a homeomorphism. This yields that the subspace $AK$ and $gAK$, for any
$g\in G$, are closed in $G/K$.
In the definition of a Kac–Moody symmetric space (cf. 3.3) we have given an
action of the group $G$ on $G/K$ by an automorphism, wherefore we can assume
the case $g=e$. Let $X,Y\in\mathfrak{a}$ and use that for all $t\in A$ follows
that $\Theta(t)=t^{-1}$, since they are diagonal matrices with real entries.
Then one computes

$\displaystyle\mu(\varphi_{e}(X),\varphi_{e}(Y))$
$\displaystyle=\tau(\exp(X))\Theta(\exp(Y))K$
$\displaystyle=\exp(X)\Theta(\exp(X))^{-1}\Theta(\exp(Y))K$
$\displaystyle=\exp(2X-Y)K=\varphi_{e}(\mu_{Eucl}(X,Y)),$

where it is used that $\exp$ is a group homomorphism and that $\mathfrak{a}$
is the preimage of $A=\tau(T)$, hence the restricted exponential function is
bijective. ∎

Call the subset $gAK$ for every $g\in G$ a _standard flat_ of the Kac–Moody
symmetric space.

######  Theorem 3.13 (cf. Theorem 5.17, [FHHK20])

Let $G$ be split Kac–Moody group over $k$. Then every weak flat in the
corresponding symmetric space $G/K$ is contained in a standard flat.
Especially

  1. 1

standard flats are exactly the maximal flats;

  2. 2

all weak flats are Euclidean;

  3. 3

all weak flats are flats;

  4. 4

$G$ acts transitively on maximal flats.

Before proving this theorem, we need to recall a technical detail. The proof
given in [FHHK20, Theorem 5.17] uses the _group model_ , which is another
version for presenting the symmetric space, see [FHHK20, Section 4.12] for
more details. To be precise, if we consider centered Kac–Moody groups, as done
in [FHHK20], one can show that $\langle\tau(G)\rangle=G$ (see [FHHK20,
Proposition 3.39]). But when we work with non-centered Kac–Moody groups, this
statement is no longer true; in addition, this statement is necessary to show
that the group model is isomorphic to the coset model we work with.

######  Remark 3.14 (The Group Model)

Let $G$ be a split Kac–Moody group over $k$ of arbitrary type. Recall that $G$
is generated by the torus $T$ and the group generated by all root subgroups
(see (KMG 1) in the definition of the Kac–Moody group in Section 2 or (RGD 4)
in [AB08, Definition 7.82]). This decomposition of $G$, and in particular this
subgroup generated by all root groups, are important for the development of
the theory of Kac–Moody symmetric spaces of arbitrary type, as they represent
just an essential difference from the theory in [FHHK20]. In [FHHK20] only
centered Kac–Moody groups of non-affine type are studied, i.e. Kac–Moody
groups generated only by their root subgroups. We give the following
definition to make this subgroup more accessible.

###### Definition 3.15

Let $G$ be a split Kac–Moody group over $k$. Denote with $G^{\dagger}<G$ the
subgroup of $G$ generated by the root subgroups of $G$, i.e.

$G^{\dagger}\coloneqq\langle U_{\alpha}\mid\alpha\in\Delta^{re}\rangle.$

The group is called a _centered subgroup_.

By definition we can write $G=G^{\dagger}T=G^{\dagger}AM$. Moreover the action
of $G$ on the symmetric space $G/K$ is given by the map

$\displaystyle G\times G/K$ $\displaystyle\to G/K$ $\displaystyle(g,hK)$
$\displaystyle\mapsto g\bullet hK\coloneqq ghK.$

Next, we want to compute the kernel of this action. Thus, if $g\in G$ acts
trivially on $G/K$, then $gK=K$ must be true. This implies that the kernel of
the action is a subset of $K$. Since [FHHK20, Lemma 3.2] is a simple
computation without including the type of $G$, we can use the fact that
$\tau^{-1}(e)=K$. From this results the following calculation

$\displaystyle g\bullet hK=ghK=hK$ $\displaystyle\Rightarrow h^{-1}gh\in K$
$\displaystyle\Rightarrow\tau\left(h^{-1}gh\right)=e$
$\displaystyle\Rightarrow\left(h^{-1}gh\right)\Theta\left(h^{-1}gh\right)^{-1}=e$
$\displaystyle\Rightarrow\left(h^{-1}gh\right)\Theta\left(h^{-1}\right)\Theta\left(g^{-1}\right)\Theta(h)=e$
$\displaystyle\Rightarrow\left(h^{-1}g\right)\tau(h)g^{-1}\Theta(h)=e$
$\displaystyle\Rightarrow g\tau(h)g^{-1}=\tau(h).$

Hence, if $g\in G$ acts trivially on $G/K$, then it centralizes $\tau(G)$, in
other words the kernel of the action is given by $C_{K}(\tau(G))$.

We define next the quotient

$\left(G^{\dagger}A\right)_{\operatorname{eff}}\coloneqq\nicefrac{{G^{\dagger}A}}{{C_{K}(G^{\dagger}A)}},$

which denotes the group that acts effectively on the symmetric space. Recall
that $\tau(A)=A$, which follows directly from the calculation in the
discussion after 3.7, and that $\langle\tau(G^{\dagger})\rangle=G^{\dagger}$,
which is the statement of [FHHK20, Proposition 3. 39] (which is usable since
only centered Kac–Moody groups are considered in the article, i.e. Kac–Moody
groups equal $G^{\dagger}$), one can see that this notion indeed is
reasonable. Altogether, it can be determined that

$\langle\tau(G^{\dagger}A)\rangle=G^{\dagger}A.$

Based on this fact, one observes that an element $g\in G$ acting trivially on
$G/K$ centralizes $G^{\dagger}A$.

As a next step we recall the _transvection group_ of a symmetric space. It is
the subgroup generated by two point reflections of the automorphism group of
the symmetric space. In specific

$\displaystyle\operatorname{Trans}(G/K)\coloneqq\langle s_{gK}\circ s_{hK}\mid
gK,hK\in G/K\rangle,$ (6)

where $s_{gK}\colon G/K\to G/K$, $hK\mapsto\mu(gK,hK)$ denotes the point
reflection on $gK\in G/K$.
The idea is to formulate [FHHK20, Proposition 4.11] for Kac–Moody groups of
arbitrary type, i.e. to associate the transvection group with
$\left(G^{\dagger}A\right)_{\operatorname{eff}}$. This means that we have to
take care of the torus, which is important in the affine case. For this
purpose we take a closer look at a point reflection $s_{gK}$:

$\displaystyle s_{gK}(hK)$ $\displaystyle=\mu(gK,hK)=\tau(g)\Theta(h)K$
$\displaystyle=g\Theta\left(g^{-1}h\right)K$
$\displaystyle=gC_{K}\left(G^{\dagger}A\right)\bullet\left(\Theta\left(g^{-1}h\right)K\right)$
$\displaystyle=gC_{K}\left(G^{\dagger}A\right)\bullet\left(s_{eK}\left(g^{-1}hK\right)\right)$
$\displaystyle=gC_{K}\left(G^{\dagger}A\right)\bullet\left(s_{eK}\left(g^{-1}C_{K}\left(G^{\dagger}A\right)\bullet(hK)\right)\right)$
$\displaystyle=\left(gC_{K}\left(G^{\dagger}A\right)\circ s_{eK}\circ
g^{-1}C_{K}\left(G^{\dagger}A\right)\right)(hK),$

where we slightly overload the notation in the last line. Now we can rewrite
the following

$\displaystyle\begin{split}s_{gK}\circ
s_{eK}&=gC_{K}\left(G^{\dagger}A\right)\circ s_{eK}\circ
g^{-1}C_{K}\left(G^{\dagger}A\right)\circ s_{eK}\\\
&=g\Theta\left(g^{-1}\right)C_{K}\left(G^{\dagger}A\right)=\tau(g)C_{K}\left(G^{\dagger}A\right)\end{split}$
(7)

and finally we observe for the generators of the transvection group the
following representation with $\tau$-elements

$\displaystyle s_{gK}\circ s_{hK}$ $\displaystyle=\left(s_{gK}\circ
s_{eK}\right)\circ\left(s_{hK}\circ s_{eK}\right)^{-1}$
$\displaystyle=\tau(g)\tau(h)^{-1}C_{K}\left(G^{\dagger}A\right).$

Note that these computations are performed in the proof of [FHHK20,
Proposition 4.11] for centered Kac–Moody groups, except that they must use a
different kernel of the action.

Using the decomposition $G=G^{\dagger}T=G^{\dagger}AM$ together with the fact
$M\subseteq K$ (see Equation 3) we deduce that
$\nicefrac{{G}}{{K}}=\nicefrac{{G^{\dagger}AM}}{{K}}=\nicefrac{{G^{\dagger}A}}{{K}}.$
This leads to the fact that

$\left(G^{\dagger}A\right)_{\operatorname{eff}}=\operatorname{Trans}(G/K).$

Once we have established the connection between the transvection group and the
quotient of $G$ acting faithfully on $G/K$, we use this to develop an
isomorphism between $G/K$ and $\tau(G^{\dagger}A)$ that allows us to equip
$\tau(G^{\dagger}A)$ with a symmetric space structure. To establish this
connection, we use the same strategy as in [FHHK20, Section 4.12].
Since [FHHK20, Section 2] only deals with a general symmetric spaces, we have
by [FHHK20, Remark 2.10] a bijective map

$\displaystyle q\colon\nicefrac{{\left(G^{\dagger}A\right)}}{{K}}$
$\displaystyle\to T(G/K)=\\{s_{gK}\circ s_{eK}\mid
gK\in\nicefrac{{\left(G^{\dagger}A\right)}}{{K}}\\}$ $\displaystyle gK$
$\displaystyle\mapsto s_{gK}\circ s_{eK}.$

This map induces a symmetric space structure on the set $T(G/K)$ where the
reflection map on $T(G/K)$ is given by $\mu_{T}(s,t)\coloneqq st^{-1}s$ (for
details see [FHHK20, Remark 2.10]). In particular it gives an isomorphism of
the symmetric spaces

$\left(G/K,\mu\right)\cong\left(T(G/K),\mu_{T}\right).$

To see this, we need to check if $\mu$ is preserved by the transport to
$T(G/K)$. Therefore, we calculate the image of the reflection map $\mu$ of
$G/K$ under $q$:

$\displaystyle q(\mu(gK,hK))=q(\tau(g)\Theta(h)K)=s_{\tau(g)\Theta(h)K}\circ
s_{eK}.$

And then the corresponding image of $q(gK)$ and $q(hK)$ under the reflection
map $\mu_{T}$:

$\displaystyle\mu_{T}(q(gK),q(hK))$ $\displaystyle=\mu_{T}(s_{gK}\circ
s_{eK},s_{hK}\circ{eK})$ $\displaystyle=(s_{gK}\circ s_{eK})\circ(s_{eK}\circ
s_{hK})\circ(s_{gK}\circ s_{eK})$ $\displaystyle=s_{gK}\circ s_{hK}\circ
s_{gK}\circ s_{eK}$ $\displaystyle=s_{s_{gK}(hK)}\circ s_{eK},$

where we used [FHHK20, Lemma 2.5] in the last step. If we now determine the
index of the first point reflection, we get the desired result:

$\displaystyle s_{gK}(hK)=\mu(gK,hK)=\tau(g)\Theta(h)K$
$\displaystyle\Rightarrow s_{s_{gK}(hK)}\circ
s_{eK}=s_{\tau(g)\Theta(h)K}\circ s_{eK}.$

Thus, as we saw in Equation 7, only $\tau(G)$ elements are needed to form
$T(G/K)$, i.e., there is a surjective map

$\widetilde{\pi}\colon\tau(G)\to T(G/K).$

Since $\tau(G)\cap K=\\{e\\}$ (see [FHHK20, Proposition 3.38], where no
arguments are used that take into account that the group is centered or non-
affine), it follows that $\ker(\widetilde{\pi})$ is trivial. Consequently,
$\widetilde{\pi}$ is a bijection.
By transport, as before, the symmetric space structure of $(T(G/K),\mu_{T})$
establishes an isomorphism of the symmetric spaces between $(T(G/K),\mu_{T})$
and $(\tau(G),\mu_{\tau})$, where $\mu_{\tau}(x,y)=xy^{-1}x$ is the
corresponding induced reflection map.
In the end we have an isomorphism of symmetric spaces between

$\left(G/K,\mu\right)\cong\left(\tau(G),\mu_{\tau}\right).$

We call $\left(\tau(G),\mu_{\tau}\right)$ the _group model_ and $(G/K,\mu)$
the _coset model_.

Having established the isomorphism between the group model and the coset
model, we can return to the proof 3.13. The proof is essentially the same as
the proof of [FHHK20, Theorem 5.17], since we can also use the group model and
most of the arguments do not take into account the type of the Kac–Moody
symmetric space.

######  Proof (Proof of 3.13)

The proof for $k=\mathbb{R}$ is the same as the proof of Theorem 5.17 in
[FHHK20]. The argumentation is not affected by the fact that the matrix is of
an arbitrary type, since by [Mar18, Theorem 7.69] any Kac–Moody group over a
field posses an RGD system. Hence for $k=\mathbb{C}$ one have to take a closer
look on claim $5$ and claim $8$.
For $k=\mathbb{R}$ claim 5 uses reductive split real Lie groups and that this
is a subgroup of $\operatorname{GL}(n+1,\mathbb{R})$. In the complex case, the
chain of reasoning is essentially the same, primarily the underlying field of
the considered reductive Lie group changes. Furthermore, we need [PR94,
Theorem 3.7 and Proposition 3.9] to keep the argument that $\Theta$ embeds
into the matrix group as conjugate, transpose invert. Everything else is
analogous to the real case, since a reductive split complex Lie group has
similar properties as a reductive split real Lie group.
Claim $8$ uses the assertion that if a flat $F$ is midpoint convex (cf.
[FHHK20, Corollary 5.11], which is applicable because the group model exists
in the general case (3.14) and the quoted theorem uses only one calculation,
which is independent of the structure of the Kac–Moody group) and contains the
neutral element, then every point $x\in F$ is diagonalizable. Here
diagonalizability is a property of the action of the group $G$ on the
associated building, see Section 3.14 and Definition 5.8 of [FHHK20]. Note
that this action is independent of the choice of Kac–Moody group, because each
Kac–Moody group has an associated twin building and an action induced by it.
This statement is still true in the complex case, since it uses only midpoint
convexity and general properties of the continuous map $\tau$. ∎

##  4 Automorphism of Kac–Moody Symmetric Spaces

As a next step in developing the theory of symmetric spaces for Kac–Moody
groups over $k$ of arbitrary type, one can analyzes the automorphism group of
the whole symmetric space and the local action on flats. For the non-affine
real case, in chapter 6 of [FHHK20], the authors observe several statements
for the global automorphism group on the symmetric space as well as for the
local automorphism group with respect to the flats.
The main goal in the following is to transfer the statements of [FHHK20,
Chapter 6]. A central assumption in [FHHK20], allowing to decompose an
automorphism, is that the RGD system is centered, i.e. the Kac–Moody group is
generated only by its root groups. Here, we want to establish similar results
for Kac–Moody groups $G$ of arbitrary type. Therefore, we have to take care of
the extended torus (which happens for Kac–Moody groups of type $\mathbb{A}$,
where $\mathbb{A}$ is not invertible), so the assumption that the RGD system
is centered cannot hold in this context, in particular $G$ is generated by the
torus and its root groups (cf. 2.3 or 2.4). In order to transfer the
statements from [FHHK20, Chapter 6] concerning the decomposition of
automorphisms, we need to work with the centered subgroup $G^{\dagger}$
(3.15). But before doing so, we need to check that $G^{\dagger}$ is a
characteristic subgroup, i.e. that $G^{\dagger}$ is mapped to itself by any
automorphism of $G$. Note that for centered Kac–Moody groups, this subgroup is
equal to the Kac–Moody group, thus in the real and non-affine case, we return
the group studied in [FHHK20].

###### Lemma 4.1

Let $G$ be a split Kac–Moody group over $k$. Then $G^{\dagger}$ is a
characteristic subgroup of $G$, i.e.

$f(G^{\dagger})\subseteq G^{\dagger}$

for all $f\in\operatorname{\text{Aut}}(G)$.

###### Proof

The idea is to prove that $G^{\dagger}$ is the derived subgroup of $G$, then
the result follows immediately, since derived subgroups are characteristic.
By the properties of an RGD system, note that $G=TG^{\dagger}$ (see [AB08,
Chapter 8]), and that $G^{\dagger}$ is a normal subgroup of $G$, since

$T=\bigcap_{\alpha\in\Delta^{re}}{N_{G}(U_{\alpha})},$

cp. [AB08, Corollary 8.79]. Moreover $G/G^{\dagger}$ is abelian, therefore
$[G,G]\subseteq G^{\dagger}$.
On the other hand, by definition $G^{\dagger}$ is generated by the root
subgroups, in particular $G^{\dagger}$ is generated by the standard rank one
subgroups

$G_{i}=\langle
U_{\alpha_{i}},U_{-\alpha_{i}}\rangle\cong\operatorname{SL}(2,k).$

Due to the fact that $\operatorname{SL}(2,k)$ is perfect for
$k\in\\{\mathbb{R},\mathbb{C}\\}$, one deduces that $G^{\dagger}$ is perfect.
Thus

$[G,G]\subseteq G^{\dagger}=[G^{\dagger},G^{\dagger}]\subseteq[G,G]$

which finishes the proof. ∎

###### Remark 4.2

  1.   2. 1

There is also an action of $\operatorname{\text{Aut}}(G)$ on $G^{\dagger}$
given by the homomorphism
$\operatorname{\text{Aut}}(G)\to\operatorname{\text{Aut}}(G^{\dagger})$ which
restricts any $f\in\operatorname{\text{Aut}}(G)$ on $G^{\dagger}$.

  3. 2

Since we are interested in statements about Kac–Moody symmetric spaces, we
also need to define a symmetric space with respect to the centered subgroup
$G^{\dagger}$. In particular, let $K^{\dagger}\coloneqq G^{\dagger}\cap K$,
and since $\Theta$ is an automorphism of $G$, it follows that
$(G^{\dagger})^{\Theta}=K^{\dagger}$. Now define the corresponding Kac–Moody
symmetric space

$\left(\nicefrac{{G^{\dagger}}}{{K^{\dagger}}},\mu\mid_{G^{\dagger}}\right)$

and denote it by $X_{1}$. Note that this is the unreduced Kac–Moody symmetric
space, which is also treated in part in the work on which this is based, see
[FHHK20, Definition 4.5 (i)].

###  4.1 Global Automorphism

Below let $G$ be a Kac–Moody group over $k$ and let $G^{\dagger}<G$ be the
centered subgroup. Since $G^{\dagger}$ is characteristic, one can take a
closer look on the automorphism group
$\operatorname{\text{Aut}}(G^{\dagger})$. By [CM05, Theorem 2.4] any
automorphism of $G^{\dagger}$ can be written as a product of an inner
automorphism, the Chevalley involution, a diagonal automorphism, a graph
automorphism and a field automorphism.
For $k=\mathbb{R}$, there are no non-trivial field automorphism and the
Chevalley involution is $\Theta$. In the case $k=\mathbb{C}$ there is only one
non-trivial (continuous) field automorphism, namely the complex conjugation.
With this concrete decomposition of an automorphism we can concretely describe
the automorphism group of $G^{\dagger}$ and relate it to the automorphism
group of the symmetric space $X_{1}$.

######  Proposition 4.3 (cf. Proposition 6.5, [FHHK20])

Let $G$ be a split Kac–Moody group over $k$ and $G^{\dagger}<G$ the centered
subgroup. Then one can decompose the automorphism group, i.e.

$\operatorname{\text{Aut}}(G^{\dagger})\cong\left(\operatorname{Ad}(G^{\dagger})\rtimes\left(D\times\langle\Theta\rangle\right)\right)\rtimes\operatorname{\text{Aut}}(\Gamma),$

where $\operatorname{Ad}(G^{\dagger})\coloneqq G^{\dagger}/Z(G^{\dagger})$,
$D$ is a group of diagonal automorphisms and
$\operatorname{\text{Aut}}(\Gamma)$ denotes the diagram automorphisms of the
corresponding Dynkin diagram with respect to $\mathbb{A}$.

Note that this explicit statement about the structure of the automorphism
group is mainly based on the result of [Cap09, Theorem 2.4].

###### Proof

The case $k=\mathbb{R}$ is proven in [FHHK20, Proposition 6.5], where it is
formulated for any generalized Cartan-Matrix $\mathbb{A}$.
For the complex case, we recall the decomposition of an automorphism of
$G^{\dagger}$ ([CM05, Theorem 2.4]: Denote with $c_{g}$ an inner automorphism,
with $\omega$ the Chevalley involution, with $d$ a diagonal automorphism, with
$\gamma$ a diagram automorphism and with $f$ a field automorphism. Then any
automorphism $\alpha\in G^{\dagger}$ can be written as

$\alpha=c_{g}\circ\omega\circ d\circ\gamma\circ f.$

In the complex case, there is only one non-trivial continuous field
automorphism $f$, the complex conjugation. In particular, $f$ commutes with
all other automorphism in this composition and the generated subgroups of
these homomorphisms intersect trivially. By the definition of the involution
$\Theta$ follows $\omega\circ f=\Theta$. ∎

###### Remark 4.4

Recall that an RGD system of $G$ is given by
$(G,(U_{\alpha})_{\alpha\in\Delta^{re}},T)$, and by definition $G^{\dagger}$
is a subgroup of $G$ generated by the root groups $U_{\alpha}$. Further, the
torus $T$ can be written as the intersection of the normalizers of all the
root groups with respect to the group $G$, see [AB08, Corolarry 8.79].
Therefore, one can obtain an RGD system of $G^{\dagger}$ from the RGD system
of $G$ by minimizing the torus so that it satisfies the latter property. Thus,
an RGD system of $G^{\dagger}$ is given by
$(G^{\dagger},(U_{\alpha})_{\alpha\in\Delta^{re}},T^{\prime})$, where
$T^{\prime}$ denotes the smaller torus defined by

$T^{\prime}=\bigcap_{\alpha\in\Delta^{re}}N_{G^{\dagger}}(U_{\alpha})=T\cap
G^{\dagger}.$

Since the torus does not play a crucial role in the construction of the twin
building from an RGD system of $G$, one notices that the twin buildings of
$G^{\dagger}$ and $G$ are the same.
Hence it follows from [FHHK20, Proposition 6.4] that

$\operatorname{\text{Aut}}(G^{\dagger})\hookrightarrow\operatorname{\text{Aut}}(\Delta)$

which allows us to interpret automorphisms of $G^{\dagger}$ as automorphisms
of the corresponding twin building $\Delta_{\pm}$ from the RGD system
associated with the Kac–Moody group $G$. By an automorphism of $\Delta$ we
mean here a map of the chamber set $\Delta=\Delta_{+}\cup\Delta_{-}$ into
itself, preserving the adjacency relation and the opposition relation.
Note that the assertion of Proposition 6.4 from [FHHK20] remains true since we
consider $G^{\dagger}$, which is a centered Kac–Moody group and hence
considered in [FHHK20].

We conclude this section by pointing out the following statement, which
describes the automorphism group of a Kac–Moody symmetric space coming from
the centered subgroup. Since $X_{1}$ corresponds to the unreduced Kac–Moody
symmetric space in [FHHK20], the proof of this theorem is the same as that of
the quoted theorem. Note that the underlying field plays no role in the proof
of this statement.

######  Theorem 4.5 (Theorem 6.12 (i) [FHHK20])

Let $G$ be a split Kac–Moody group over $k$ and $G^{\dagger}<G$ its centered
subgroup. Then

$\operatorname{\text{Aut}}\left(X_{1}\right)\cong\operatorname{\text{Aut}}_{\text{eff}}(G^{\dagger})\coloneqq\left(\nicefrac{{G^{\dagger}}}{{C_{K^{\dagger}}(G^{\dagger})}}\rtimes(D\times\langle\Theta\rangle)\right)\rtimes\operatorname{\text{Aut}}(\Gamma),$

where $C_{K^{\dagger}}(G^{\dagger})$ is the centralizer of $G^{\dagger}$ in
$K^{\dagger}$.

###  4.2 Local Action

For general symmetric spaces (cf. Section 3.1) one can define local
transformations of flats. Here we briefly give the definitions and transfer
them to Kac–Moody symmetric spaces. Since the local actions do not take into
account the structure of the automorphism group, we can study the local
actions for the entire symmetric space $X=G/K$ derived from a split Kac–Moody
group $G$ over $k$ of type $\mathbb{A}$.
In what follows, we will often refer to statements from [FHHK20] without
saying each time why one can apply the statement, since most statements we
will use do not take into account the underlying field or type of the
generalized Cartan matrix. If there are problems with the field or type, we
will point them out.

###### Definition 4.6

Let $X$ be a symmetric space according to Loos which contains a maximal
Euclidean flat $F$ and let $p\in F$.

  1. 1

Call the pair $(p,F)$ consisting of a point $p\in F$ and a maximal flat $F$ a
_pointed maximal flat_ (cp. [FHHK20, Definition 2.32]).

  2. 2

Call a point $q\in F$ _singular_ with respect to $p$ if there exists a second
maximal flat $E\subseteq X$ such that $p,q\in E$ and $E\neq F$. Otherwise call
$q$ _regular_. Denote with ${F^{\text{sing}}}(p)$ the subset of singular
points of $(p,F)$ and with ${F^{\text{reg}}}(p)$ the set of regular points
(cp. [FHHK20, Definition 2.35]). Note that
$F={F^{\text{reg}}}(p)\sqcup{F^{\text{sing}}}(p)$.

  3. 3

The isomorphism of symmetric spaces $\varphi\colon\mathbb{R}^{n}\to F$ is a
_chart of_ $F$ _centered at $p$_ if $p=\varphi(0)$ (cp. [FHHK20, Definition
2.32]).

  4. 4

A map $f\colon F\to F$ is called _linear at $p$_ if for some chart $\varphi$
centered at $p$ we have $\varphi^{-1}\circ
f\circ\varphi\in\operatorname{GL}(2n-l,\mathbb{R})$, in particular $f$ is
invertible (cp. [FHHK20, Definition 2.35]).

  5. 5

A _local transformation_ of a pointed flat $(p,F)$ is a map $f\colon F\to F$
which is linear at $p$ and preserves the decomposition of $F$ into regular and
singular points (cp. [FHHK20, Definition 2.35]). Denote with
$\operatorname{GL}(p,F,{F^{\text{sing}}}(p))$ the set of local
transformations. Note that $\operatorname{GL}(p,F,{F^{\text{sing}}}(p))$ forms
a group.

In order to apply these definitions to Kac–Moody symmetric spaces, one has to
keep in mind that singular points can be described in terms of real form
$\mathfrak{a}$. In the beginning of section $6.14$ in [FHHK20] the authors
give such a description for non-affine generalized Cartan matrices. The
difficulty of applying this description to the affine case is, that in
[FHHK20] they are only interested in the semisimple adjoint quotient of a
Kac–Moody group (cp. [FHHK20, Definition 3.8])and for Kac–Moody groups of
affine type the quotient prevents that a presentation is faithful and that the
corresponding bilinear form is non-degenerated (cp. [FHHK20, Section A.22]).
In the following, we give a short overview how to fix this problems in the
affine case for real or complex split Kac–Moody groups. Therefore we define
the Weyl group and the necessary representation.

Recall that every Kac–Moody algebra $\mathfrak{g}(\mathbb{A})$ admits a _Weyl
group_ $\mathcal{W}$, 2.4, which is a subgroup of
$\operatorname{GL}(\mathfrak{h})$. The action of $\mathcal{W}$ stabilizes
$\mathfrak{h}$, and since the reflections depend on the generalized Cartan
matrix $\mathbb{A}$, the action also stabilizes the real form $\mathfrak{a}$,
so we can define the _Kac–Moody representation_

$\displaystyle\begin{split}\rho_{KM}\colon\mathcal{W}\to\operatorname{GL}(\mathfrak{a}).\end{split}$
(8)

By [Kac90, Proposition 3.12] this representation is faithful. According to
[Kac90, §2.1] one can define an invariant symmetric bilinear form on
$\mathfrak{a}$ if the generalized Cartan matrix $\mathbb{A}$ is symmetrizable,
denoted by B. With this bilinear form B one can as well define the reflections
$r_{i}^{\vee}$, since it is not degenerate on $\mathfrak{h}$ (cf. [Kac90,
Lemma 2.1 b)], [Kac90, Theorem 2.2] and [FHHK20, Proposition A.23]). And from
the definition with B it follows that the reflections are orthogonal to B.

The elements of $\mathcal{W}$ acts as linear reflections under the
representation $\rho_{KM}$, hence we can define a _reflection hyperplane_
w.r.t. a real root $\alpha\in\Delta^{re}$ as the fixed points of the
associated reflection $r_{\alpha}^{\vee}\in\mathcal{W}$ under $\rho_{KM}$,
i.e.

$H_{\alpha}\coloneqq\operatorname{Fix}(\rho_{KM}(r_{\alpha}^{\vee}))\subseteq\mathfrak{a}.$

By the non-degeneracy of B on $\mathfrak{h}$, and in particular on
$\mathfrak{a}\subseteq\mathfrak{h}$, one observes that a reflection under the
representation is the unique B-orthogonal reflection at the hyperplane
$H_{\alpha}$. Moreover there is a one-to-one correspondence between the
positive real roots $\alpha\in\Delta^{re,+}$ and the hyperplanes $H_{\alpha}$.
With this background one can define singular points with respect to the real
form:

${\mathfrak{a}^{\text{sing}}}\coloneqq\bigcup_{\alpha\in\Delta^{re,+}}H_{\alpha}.$

On the other hand one can define singular points on $\mathfrak{a}$ by using
the description of singular points of a flat and the map $\varphi_{e}$ coming
from 3.12, i.e. let $(e,AK)$ be the standard flat, then define

$\varphi_{e}^{-1}\left({AK^{\text{sing}}}(e)\right).$

######  Proposition 4.7 (Proposition 6.16 in [FHHK20])

Let $G$ be a split Kac–Moody group over $k$ and $X=G/K$ be the corresponding
symmetric space. Then

$\bigcup_{\alpha\in\Delta^{re,+}}H_{\alpha}={\mathfrak{a}^{\text{sing}}}=\varphi_{e}^{-1}\left({AK^{\text{sing}}}(e)\right).$

###### Proof

The proof uses only arguments based on the RGD system of Kac–Moody groups and
by [Mar18, Theorem 7.69] any Kac–Moody group over $k$ of arbitrary type has an
RGD system. To be precise, no statements are used specifically for centered
RGD systems. ∎

Next define the group of linear maps which preserves the decomposition of a
flat into regular and singular points w.r.t. the real form:

$\operatorname{GL}(\mathfrak{a},{\mathfrak{a}^{\text{sing}}})\coloneqq\\{f\in\operatorname{GL}(\mathfrak{a})\mid
f({\mathfrak{a}^{\text{sing}}})={\mathfrak{a}^{\text{sing}}}\\}.$

Let $(p,F)$ be a pointed flat in the Kac–Moody symmetric space $X$ and
$f\in\operatorname{GL}(p,F,{F^{\text{sing}}})$. Call $f$ a _local
automorphism_ of the flat $(p,F)$ if $\varphi^{-1}\circ f\circ\varphi$ is
B-orthogonal for some chart $\varphi$ centered at $p$ and denote the subgroup
of local automorphism by
$\operatorname{\text{Aut}}(p,F)<\operatorname{GL}(p,F,{F^{\text{sing}}})$.
By [FHHK20, Corollary 6.19] one can write a local transformation of $(p,F)$ as
a product of a homothety and a B-orthogonal transformation. Note that this is
a corollary of Proposition 6.16 in [FHHK20], and hence it is a corollary of
4.7 since no further assumptions are made. Therefore, the statement is valid
in our more general context.
Since a local automorphism $f\in\operatorname{\text{Aut}}(p,F)$ is just a
B-orthogonal transformation without a non-trivial homothety, one can define an
action $\mathbb{R}_{>0}$ on the flat $F$ by homotheties and receive the
splitting

$\operatorname{GL}(p,F,{F^{\text{sing}}})\cong\mathbb{R}_{>0}\times\operatorname{\text{Aut}}(p,F).$

######  Proposition 4.8 (Corollary 6.20 in [FHHK20])

Let $\mathbb{A}$ be non-spherical and $G$ be a split Kac–Moody group over $k$.
Then for every pointed flat $(p,F)$ in the symmetric space $X=G/K$ one has

$\operatorname{\text{Aut}}(p,F)\cong(\mathcal{W}\rtimes\operatorname{\text{Aut}}(\mathcal{W},S))\times\mathbb{Z}/2\mathbb{Z}$

and hence

$\operatorname{GL}(p,F,{F^{\text{sing}}})\cong\mathbb{R}_{>0}\times(\mathcal{W}\rtimes\operatorname{\text{Aut}}(\mathcal{W},S))\times\mathbb{Z}/2\mathbb{Z}$

###### Proof

Recall that for the simplicial Coxeter complex $\Sigma$ of the Coxeter system
$(\mathcal{W},S)$, one denotes with $\operatorname{\text{Aut}}(\Sigma)$ the
group of simplicial automorphisms and with
$\operatorname{\text{Aut}}(\mathcal{W},S)$ the group of automorphisms of the
Coxeter graph, cf. [FHHK20, Section A.1]. Define the group
$\operatorname{O}(\mathfrak{a},{\mathfrak{a}^{\text{sing}}})\coloneqq\operatorname{O}(\mathfrak{a},\textbf{B})\cap\operatorname{GL}(\mathfrak{a},{\mathfrak{a}^{\text{sing}}})$,
where $\operatorname{O}(\mathfrak{a},\textbf{B})$ denotes the group of
orthogonal linear maps with respect to the bilinear form $B$. By Proposition
A.32 and Remark A.33 (ii) of [FHHK20] one deduce that
$\operatorname{\text{Aut}}(\Sigma)\times\mathbb{Z}/2\mathbb{Z}\cong\operatorname{O}(\mathfrak{a},{\mathfrak{a}^{\text{sing}}})$,
in particular it is
$\operatorname{\text{Aut}}(\Sigma)\times\mathbb{Z}/2\mathbb{Z}\cong\operatorname{\text{Aut}}(p,F)$
by revealing the definitions. The cited statements are working for affine
Kac–Moody groups, since they need a faithful presentation $\rho_{KM}$ and a
non-degenerated bilinear form B on $\mathfrak{a}$ which are both given by the
previous discussion of this section.
Together with [FHHK20, Lemma A.31] one obtains the first isomorphism. The
second follows immediately by the previous discussion. ∎

###  4.3 Comparing Weyl Groups

In the general setting of symmetric spaces according to Loos (cf. Section 3.1)
one can define a Weyl group of the symmetric space using an action on the
symmetric space. Therefore let $(X,\mu)$ be a symmetric space which contains a
maximal Euclidean flat. Moreover let $H$ be a subgroup of
$\operatorname{\text{Aut}}(X)$ which acts transitively on the maximal flats of
$X$. Define the stabilizer and fixator of a pointed flat $(p,F)$ in $X$ as the
following

$\operatorname{Stab}_{H}(p,F)\coloneqq\\{h\in H\mid h(F)=F\,,\;h(p)=p\\}$

and

$\operatorname{Fix}_{H}(p,F)\coloneqq\\{h\in H\mid\forall f\in F:h(f)=f\\}.$

Then the _geometric Weyl group_ of the action of $H$ on $X$,
$W(H\curvearrowright X)$, is given by

$W(H\curvearrowright
X)\coloneqq\nicefrac{{\operatorname{Stab}_{H}(p,F)}}{{\operatorname{Fix}_{H}(p,F)}}$

and it is independent of the choice of the pointed flat up to conjugation, cf.
[FHHK20, Proposition 2.36].
Recall that Section 2 of [FHHK20] only talks about general symmetric spaces in
the sense of Loos (cf. 3.1), so any statement formulated there can be used in
the context of this article.

The question arises whether this Weyl group of a symmetric space $G/K$ with
corresponding split Kac–Moody group $G$ is related to the Weyl group of $G$.
In order to answer this question, we need to review a couple of things.

  1. 1

Since $G$ possesses a twin building on which $G$ acts strongly transitively,
one knows that there is an isomorphism

$\nicefrac{{N_{G}(T)}}{{T}}\cong\mathcal{W},$

see [AB08, Section 8.6.2]. This section explains that both, the Weyl group and
$N_{G}(T)$, act on the root groups $\\{U_{\alpha}\mid\alpha\in\Delta^{re}\\}$
and that the torus $T$ acts trivial.
Recall from 4.4 that the RGD systems of $G^{\dagger}$ and $G$ lead to the same
twin building. Moreover, it follows from [AB08, Proposition 8.82], that the
kernel of the action of $G$ on the building is contained in $T$. Based on this
fact, we see that the quotient w.r.t. $G^{\dagger}$

$\displaystyle\nicefrac{{N_{G^{\dagger}}\left(T\cap
G^{\dagger}\right)}}{{(T\cap
G^{\dagger})}}=\nicefrac{{N_{G^{\dagger}}\left(T^{\prime}\right)}}{{T^{\prime}}}\cong\mathcal{W}$
(9)

is also isomorphic to the Weyl group of $G$.

  2. 2

Recall from 2.4, that the extended Weyl group is given by
$\widetilde{\mathcal{W}}\coloneqq\langle\tilde{s}_{i}\mid i\in I\rangle$.
According to [Mar18, Corollary 7.54] and [Mar18, Proposition 4.18 (2)] the
extended Weyl group surjects on $\mathcal{W}$ (alternatively see [GHKW17,
Observation 18.11 (c)]), in particular
$\operatorname{Ad}(\tilde{s}_{\alpha_{i}})\mid_{\mathfrak{h}}=r_{\alpha_{i}}^{\vee}$.
Since $\widetilde{\mathcal{W}}$ is a subgroup of $G^{\dagger}$, one can apply
[KP85, Proposition 2.1], which gives the isomorphism

$\nicefrac{{\widetilde{\mathcal{W}}}}{{(\widetilde{\mathcal{W}}\cap T)}}\cong
W.$

Note that $\tilde{s}_{i}^{2}$ is an element of the torus (see [KP85, (2.4)] or
[Mar18, Exercise 7.49]), i.e. they generate a subgroup of
$\widetilde{\mathcal{W}}\cap T$ of order $2^{n}$. By [KP85, Corollary 2.3
(a)], all elements of $\widetilde{\mathcal{W}}\cap T$ have order less than or
equal to $2$. In the real case, i.e. $k=\mathbb{R}$, we know from the
discussion in Section 3.2 that $M$ is the unique maximal finite subgroup of
elements of order $2^{n}$, so $\widetilde{\mathcal{W}}\cap T$ is contained in
$M$. In particular, the intersection $\widetilde{\mathcal{W}}\cap T$ is in
fact equal to $M$ (For the specific arguments, see [FHHK20, Section 3.13],
where only the arguments from the previous discussion here are used.). In
other words, for $k=\mathbb{R}$ we have the following isomorphism

$\nicefrac{{\widetilde{\mathcal{W}}}}{{M}}\cong W.$ (10)

######  Proposition 4.9 (Proposition 6.23 in [FHHK20])

Let $G$ be a split Kac–Moody group over $\mathbb{R}$ and $(p,F)$ a pointed
flat in the symmetric space $(X=G/K,\mu_{G})$. Then

  1. 1

$\operatorname{Stab}_{G}(p,F)\cong N_{K}(T)$ and
$\operatorname{Fix}_{G}(p,F)\cong M$, and

  2. 2

$W(G\curvearrowright X)\cong\mathcal{W}$.

###### Proof

By [FHHK20, Proposition 2.36(i)] one can choose for the pointed maximal flat
$(p,F)$ the standard flat $(e,AK)$. With respect to the action of $G$ on $X$,
the fixator of $e$ is given by $K$. Next, recall that $N_{G}(T)$ is the
stabilizer of the standard flat, for the computation of this fact see [FHHK20,
Remark 5.5]. Note that this is just a direct calculation without using the
underlying field or structure of $G$. It therefore follows that
$\operatorname{Stab}_{G}(e,AK)=N_{K}(T)$, and using [FHHK20, Corollary 3.28],
which works for Kac–Moody groups of arbitrary type $\mathbb{A}$ but only over
$\mathbb{R}$, yields the fact that $\widetilde{\mathcal{W}}\cong N_{K}(T)$.
The group $M$ is contained in $K$ (Equation 3) and centralizes $A$, i.e. it is
a subgroup of $\operatorname{Fix}_{G}(p,F)$. Thus, it follows from Equation 10
that

$\nicefrac{{\operatorname{Stab}_{G}(e,AK)}}{{M}}\cong\nicefrac{{N_{K}(T)}}{{M}}\cong\nicefrac{{\widetilde{\mathcal{W}}}}{{M}}\cong\mathcal{W}.$

As in the discussion above in the definition of $\widetilde{\mathcal{W}}$, one
can consider the adjoint action of the quotient $\widetilde{\mathcal{W}}/M$ on
the real form $\mathfrak{a}$ using the Kac–Moody representation $\rho_{KM}$.
In particular, $\rho_{KM}$ is faithful and therefore $M$ must be equal to
$\operatorname{Fix}_{G}(p,F)$. ∎

###  4.4 From Local to Global

To conclude this survey, we give a statement that relates the local action on
flats to the global action in the context of affine Kac–Moody groups over
$\mathbb{R}$. The non-affine version of this statement is given in [FHHK20,
Theorem 6.25]. The theorem shows, that the local action of
$W(\operatorname{Aut}(X_{1})\curvearrowright X_{1})$ takes values in the group
of local automorphism.

######  Theorem 4.10 (cp. [FHHK20], Theorem 6.25)

Let $G$ be a split Kac–Moody group over $\mathbb{R}$ of type $\mathbb{A}$,
where $\mathbb{A}$ is non-spherical, symmetrizable and affine. Furthermore let
$G^{\dagger}$ be the centered subgroup of $G$ and denote with $X_{1}$ the
corresponding Kac–Moody symmetric space and let $(p,F)$ be a pointed maximal
flat in $X_{1}$. Then the following diagram commutes

$W(\operatorname{Aut}(X_{1})\curvearrowright
X_{1})$$\operatorname{GL}\left(p,F,F^{\text{sing}}(p)\right)$$\left(\mathcal{W}\rtimes\operatorname{\text{Aut}}(\Gamma)\right)\times\mathbb{Z}/2\mathbb{Z}$$\mathbb{R}_{>0}\times\left(\mathcal{W}\rtimes\operatorname{\text{Aut}}(\mathcal{W},S)\right)\times\mathbb{Z}/2\mathbb{Z}$,$\pi$$\iota$$\cong$$\cong$

where $\pi$ represents the canonical inclusion and
$\operatorname{\text{Aut}}(\mathcal{W},S)$ the automorphism group of
$\mathcal{W}$ which preserves the set of generators $S$ as a set. In addition,
every local automorphism extends to a global automorphism if and only if
$\operatorname{\text{Aut}}(\Gamma)=\operatorname{\text{Aut}}(W,S)$.

###### Proof

First, note that the right isomorphism follows from 4.8 and the map $\iota$ is
straightforward, since the geometric Weyl group takes values in the group of
local transformations.
For the left isomorphism, recall the definition

$W(\operatorname{Aut}(X_{1})\curvearrowright
X_{1})\coloneqq\nicefrac{{\operatorname{Stab}_{\operatorname{\text{Aut}}(X_{1})}(p,F)}}{{\operatorname{Fix}_{\operatorname{\text{Aut}}(X_{1})}(p,F)}}$

and because of [FHHK20, Proposition 2.36] (Recall that this statement is made
about general symmetric spaces on Loos.), note that one can consider the
standard pointed flat $(e,AK)$ instead of $(p,F)$ without loss of generality.
The idea is to copy the proof of the non-affine version [FHHK20, Theorem
6.25]. To do this, one must use the decomposition of
$\operatorname{\text{Aut}}(G^{\dagger})$ and check if the quotient of the
stabilizer of the standard flat and the fixator of the standard flat is still
isomorphic to $\mathcal{W}$. The rest of the proof remains the same, since the
decomposition in the reduced case (which is used in the non-affine case) and
in the non-reduced case differs only in the inner automorphisms.
Therefore let $G^{\dagger}$ be the centered subgroup of $G$. Recall from 4.5
that there is the isomorphism

$\operatorname{\text{Aut}}(X_{1})\cong\left(\nicefrac{{G^{\dagger}}}{{C_{K^{\dagger}}(G^{\dagger})}}\rtimes(D\times\langle\Theta\rangle)\right)\rtimes\operatorname{\text{Aut}}(\Gamma).$

The Chevalley involution $\Theta$ acts on $(e,AK)$ by inversion, a diagonal
automorphism fixes $(e,AK)$ point-wise, a diagram automorphism stabilizes
$(e,AK)$ and moreover can be considered as a subgroup of
$\operatorname{\text{Aut}}(W,S)$ (see [FHHK20, Remark 6.3]).
Thus, a diagonal automorphism has no effect, the diagram automorphism remains,
and the Chevalley involution corresponds to the generator
$\mathbb{Z}/2\mathbb{Z}$.
Now one has to analyze the stabilizer and the fixator of $AK$ with respect to
$G^{\dagger}$. By 4.9 one knows that

$\operatorname{Stab}_{G^{\dagger}}(e,AK)\cong N_{K^{\dagger}}(T\cap
G^{\dagger})\quad\text{and}\quad\operatorname{Fix}_{G^{\dagger}}(e,AK)\cong
M,$

since the arguments do not include the structure of the surrounding group with
respect to the differences between $G^{\dagger}$ and $G$. Finally, one can
consider these subgroups of $G^{\dagger}$ in terms of the inner automorphisms
of the decomposition $\operatorname{\text{Aut}}(X_{1})$, i.e.

/StabG†(e,AK)/CK†(G†)FixG†(e,AK)/CK†(G†)
$\displaystyle\cong\nicefrac{{\operatorname{Stab}_{G^{\dagger}}(e,AK)}}{{\operatorname{Fix}_{G^{\dagger}}(e,AK)}}$
$\displaystyle\cong\mathcal{W},$

where the last step is given by 4.9. Thus we have shown the isomorphisms of
the vertical arrows, the commutativity of the diagram follows from the fact
that the horizontal arrows are inclusions. ∎

##  5 The Causal Boundary

Considering rays in a Kac–Moody symmetric space starting at a point and going
to infinity, one can ask for the set of all points at infinity, called the
boundary at infinity or the _causal boundary_. Here we describe this set of
rays and give it a combinatorial structure that follows from the structure of
the corresponding Kac–Moody group. To be precise: one uses the building
belonging to the Kac–Moody group to construct and connect different
realizations as polyhedral complexes on the set of rays. The results for the
non-affine case are presented in Section 7 from [FHHK20].
To establish the general statements for real or complex split Kac–Moody groups
of arbitrary type, we have to take care of the affine case again. Therefore,
compared to [FHHK20], we have a non-centered Kac–Moody group. As a
consequence, we do not have a strong statement connecting the automorphism
groups, see [FHHK20, Theorem 6.12]. We have only the weaker statement 4.5.
For this section, let $G$ be a split Kac–Moody group over $k$ of arbitrary
type $\mathbb{A}$, $G^{\dagger}<G$ its centered subgroup and denote with
$X_{1}$ the corresponding Kac–Moody symmetric space $G^{\dagger}/K^{\dagger}$.
Within this general context, the aim is to reproduce the main statements of
section 7 of [FHHK20]. Therefor two necessary terms of this section are the
_positive causal structure_ on $X_{1}$, denoted by
$\left(C_{x}^{+}\right)_{x\in X_{1}}$, and the _negative causal structure_ on
$X_{1}$, $\left(C_{x}^{-}\right)_{x\in X_{1}}$. These objects and their
properties are defined in [FHHK20, Definition 7.7].

In what follows, we will often refer to the article [FHHK20]. Since we have
focused on $G^{\dagger}$ in this section, it is clear that the main difference
between this article and [FHHK20] is the underlying field. Therefore, most of
the statements work in our setting, since the field has the largest impact on
the topology. However, if there are (subtle) differences, we will point them
out.

######  Definition 5.1 (cf. Definition 7.10 in [FHHK20])

A continuous map $r\colon[0,\infty)\to X_{1}$ is called a _causal ray_ if the
image is contained in a geodesic of $X_{1}$, i.e.
$\operatorname{im}(r)\subseteq F$, where $F$ is a Euclidean flat of rank $1$,
and for every $t\in[0,\infty)$ there exists an $\varepsilon>0$ such that

$r((t,t+\varepsilon))\subset C_{r(t)}^{+}.$

Otherwise, call $\gamma$ a _anti-causal ray_ if the image is contained in a
geodesic of $X_{1}$ and for every $t\in[0,\infty)$ there exists an
$\varepsilon>0$ such that

$r((t,t+\varepsilon))\subset C_{r(t)}^{-}.$

In the following we denote with $R(X_{1})$ the set of all geodesic rays
$r\colon[0,\infty)\to X_{1}$ and with $\mathcal{M}\subseteq R(X_{1})$ the
subset of all causal and anti-causal rays. This subset $\mathcal{M}$ is called
_municipality_ of $X_{1}$. Furthermore, denote all rays of $\mathcal{M}$
starting in $x\in X_{1}$ by $\mathcal{M}_{x}$.

###### Remark 5.2

Since the topic is very technical, we refer the reader to [FHHK20] for
detailed discussions. Here we will only give motivations, quote the most
necessary definitions, and show when necessary that the statement is true in
the affine case. Most definitions and statements apply to both affine and non-
affine cases, since they are based on the observations of Section 4 (cf.
[FHHK20, Section 6]). Therefore, to make all statements from this section
applicable, we have focused on $G^{\dagger}$ and thus on $X_{1}$.

###  5.1 Polyhedral Complex Structure and Realizations

An obligatory notion that we will need is that of the polyhedral complex. To
specify this, recall that a half-space in $\mathbb{R}^{n}$ is given as the
closed connected component of the complement of an affine hyperplane. We call
a non-empty intersection of finitely many half-spaces a _polyhedron_. A
_polyhedral complex_ is a set $A$ together with a family of injective maps of
polyhedra into $A$; the mappings must of course satisfy some additional
properties, see [FHHK20, Definition 7.17]. It follows from the definition that
polyhedra are always closed and convex, but not necessarily compact, which
means that the cells in the complex are not necessarily compact.
In following we are interested in ideal polyhedral complexes, i.e., if $A$ is
a polyhedral complex, then a subset $B\subset A$ is called an _ideal
polyhedral complex_ with completion $A$ if $B$ is a union of open cells and
contains all faces of codimension $0$ and $1$ (cf. [FHHK20, Definition 7.18]).
The finest topology that makes all inclusions continuous is the weak topology
on a polyhedral complex $A$. Equipped with a topology, one can define a
partial ordering on the set of open cells (the _polyhedral poset structure_
$\Sigma(A)$ of $A$), and so define that $A$ is a realization of a poset
$\Sigma$ if the polyhedral poset structure of $A$ is iosmorphic to $\Sigma$.
Finally, one can consider cells in the polyhedral complex $A$ which are not a
face of any other cell, called _chambers_. For a finite set $S$, we can define
an $S$-coloring of a polyhedral poset structure as a map from the comaximal
elements of $\Sigma(A)$ to $S$ which restricts to a bijection on the
codimension $1$ faces of each single chamber.
Therefore, we can distinguish between the following types of isomorphism of
polyhedral complexes with respect to the different structures. Let $A,B$ be
two ideal polyhedral complexes, then $A$ and $B$ are called …

  1. 1

_geometrically isomorphic_ , if there exists a homeomorphism with respect to
the weak topologies which maps open cells onto open cells.

  2. 2

_combinatorially isomorphic_ , if there exists an order-preserving map such
that the underlying posets $\Sigma(A)$ and $\Sigma(B)$ are isomorphic.

  3. 3

_chamber isomorphic_ , if $A$ and $B$ are both $S$-colored and there exists a
bijection between the chamber systems, such that the coloring is preserved.

Note that this order reflects the strength of the isomorphisms, i.e. the
geometric isomorphism is the strongest and the chamber isomorphism is the
weakest and preserves little information of the complex.
For more details about (ideal) polyhedral complexes see [FHHK20, Section
7.16].

Recall the definition of the Tits cone (cf. see [Mar18, Chapter 4.3]): The set

$C_{0}\coloneqq\\{h\in\mathfrak{a}\mid\alpha_{i}(h)\geq 0\;\forall i\in I\\},$

where $\alpha_{i}$ stands for the simple roots in the Kac–Moody algebra
$\mathfrak{g}(\mathbb{A})$, is called the _fundamental chamber_. The set

$C=\bigcup_{w\in\mathcal{W}}w(C_{0}),$

where $\mathcal{W}$ stands for the corresponding Weyl group of
$\mathfrak{g}(\mathbb{A})$, is called the _Tits cone_.

One can turn the fundamental chamber $C_{0}$ into a polyhedral cone by
considering $C_{0}\subset\mathfrak{a}$ as a set bounded by the root
hyperplanes $H_{\alpha_{i}}$ of the simple roots (see Section 4.2). For a
detailed definition of a polyhedral cone, see the discussion before [FHHK20,
Definition 7.18]. The polyhedral cells in this cone are given by the
intersections of theses hyperplanes. Furthermore, the generating set $S$ of
the Weyl group $W$ can be used to color the surfaces in a natural way. For
this purpose, label each reflection hyperplane with the corresponding simple
reflection $\rho_{KM}\left(r_{\alpha_{i}}^{\vee}\right)$.
Using this polyhedral complex structure one can define a realization of the
building $\Delta_{\pm}$ corresponding to a Kac–Moody group $G$. To do this,
consider the intersection of this cone with any sphere centered at $0$, in
particular, denote with
$\SS(\mathfrak{a})\coloneqq\left(\mathfrak{a}\setminus\\{0\\}\right)/\,\mathbb{R}^{>0}$
the space $\mathfrak{a}$ modulo homotheties considered as a real vector space.
Then there is a natural projection
$\SS\colon(\mathfrak{a}\setminus\\{0\\})\to\SS(\mathfrak{a})$, where we denote
by $\SS(C_{0})$ the image of the fundamental chamber with the polyhedral cone
structure under this projection.

Next consider the direct product $\Delta_{\pm}\times\SS(C_{0})$ and define a
relation $(\mathcal{C}_{1},x)\sim(\mathcal{C}_{2},x)$ if the chambers
$\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are $s$-adjacent for some $s\in S$ and
$x$ is contained in the closure of the face of $\SS(C_{0})$ with the label
$s$. The quotient of the direct product with respect to the relation leads
then to a colored ideal polyhedral complex
$\left|\Delta_{\pm}\right|_{\mathfrak{a}}$, for details see [FHHK20, Section
7.20]. Let

$\left|\Delta\right|_{\mathfrak{a}}\coloneqq\left|\Delta_{+}\right|_{\mathfrak{a}}\sqcup\left|\Delta_{-}\right|_{\mathfrak{a}}.$

###### Remark 5.3

  1.   2. 1

Note that via the embedding
$\operatorname{\text{Aut}}(X_{1})\hookrightarrow\operatorname{\text{Aut}}(\Delta)$
(cf. 4.4) the automorphism group of a Kac–Moody symmetric space acts on the
realization $\left|\Delta\right|_{\mathfrak{a}}$.

  3. 2

Since the construction of $\left|\Delta\right|_{\mathfrak{a}}$ works exactly
the same in the affine case as in the non-affine case, we can use all the
statements from the subsections 7.20 and 7.22 of [FHHK20]. In particular,
[FHHK20, Proposition 7.26]: There exists a unique ideal polyhedral structure
on $\mathcal{M}_{x}$, $x\in X_{1}$, so that for every pointed flat $F\ni x$
the subset of all rays contained in $F$ is an ideal polyhedral subcomplex.
Moreover, this structure implies that $\mathcal{M}_{x}$ is
$\operatorname{\text{Aut}}(X_{1})_{x}$-equivariantly geometrically isomorphic
to $\left|\Delta\right|_{\mathfrak{a}}$.

  4. 3

Recall that $\Sigma_{sph}(\Delta_{\pm})$ is the set consisting of all
parabolic subgroups of spherical type, which has a poset structure by reverse
inclusion (cp. [FHHK20, Definition 7.21]). Hence, by Proposition 7.26 of
[FHHK20] follows that $\mathcal{M}_{x}$ realizes $\Sigma_{sph}(\Delta_{\pm})$
as an ideal polyhedral complex.

Overall this justifies that we call $\mathcal{M}_{x}$ the _twin building at
the horizon of_ $x$. In the following we use automorphisms, so we consider
only the symmetric space $X_{1}$ of the centered subgroup $G^{\dagger}<G$ of a
Kac–Moody group $G$.

Next, we use this ideal polyhedral structure on $\mathcal{M}_{x}$ to obtain an
ideal polyhedral structure on the whole municipality
$\mathcal{M}=\bigsqcup_{x\in X_{1}}{\mathcal{M}_{x}}$.
Therefore fix a base point $e$ of the symmetric space $X_{1}$ and based on
[FHHK20, Proposition 7.26] an
$\operatorname{\text{Aut}}(X_{1})_{e}$-equivariant geometric isomorphism

$\left|\iota_{e}\right|\colon\left|\Delta\right|_{\mathfrak{a}}\to\mathcal{M}_{e}$

with underlying combinatorial isomorphism that preserves the coloring

$\iota_{e}\colon\Sigma_{sph}(\Delta_{\pm})\to\Sigma_{e},$

where $\Sigma_{e}$ is the underlying poset of $\mathcal{M}_{e}$ (see the
discussion after Proposition 7.26 in [FHHK20]).
Finally equip

$X_{1}\times\left|\Delta\right|_{\mathfrak{a}}=\bigsqcup_{x\in
X_{1}}\left|\Delta\right|_{\mathfrak{a}}$

with the ideal polyhedral structure coming from each
$\left|\Delta\right|_{\mathfrak{a}}$, turning it into an ideal polyhedral
complex.

######  Proposition 5.4 (cf. Proposition 7.29 in [FHHK20])

Let $G$ be a split Kac–Moody group over $k$, $G^{\dagger}<G$ its centered
subgroup and $X_{1}$ the corresponding symmetric space. The geometric
isomorphism $\left|\iota_{e}\right|$ admits a unique color preserving
$\operatorname{\text{Aut}}(X_{1})$-equivariant extension

$\displaystyle\left|\iota\right|\colon
X_{1}\times\left|\Delta\right|_{\mathfrak{a}}$ $\displaystyle\to\mathcal{M}$
$\displaystyle(x,\delta)$
$\displaystyle\mapsto\left|\iota_{x}\right|(\delta),$

such that
$\left|\iota_{x}\right|\colon\left|\Delta\right|_{\mathfrak{a}}\to\mathcal{M}_{x}$
is an geometric isomorphism for every $x\in X_{1}$.

Note that this extension means that at the base point $e$ the extension
matches $\left|\iota_{e}\right|$, i.e.

$\left|\iota_{e}\right|(\cdot)=\left|\iota\right|(e,\cdot).$

Moreover, this statement implies that the underlying combinatorial isomorphism
$\iota_{e}$ can be extended to a unique
$\operatorname{\text{Aut}}(X_{1})$-equivariant automorphism

$\displaystyle\iota\colon X_{1}\times\Sigma_{sph}(\Delta_{\pm})$
$\displaystyle\to\Sigma$ $\displaystyle(x,C)$
$\displaystyle\mapsto\iota_{x}(C),$

such that $\iota_{x}\colon\Sigma_{sph}(\Delta_{\pm})\to\Sigma_{x}$ is an
isomorphism of posets for any $x\in X_{1}$. Here $\Sigma=\bigsqcup_{x\in
X_{1}}\Sigma_{x}$ stands for the underlying poset structure of $\mathcal{M}$.

###### Proof

The non-affine case is shown in the proof of [FHHK20, Proposition 7.29]. For
the affine case, one argues the same, except for one detail: in the non-affine
case the authors are working with the reduced case, i.e. they prove this
statement for a smaller automorphism group.
Since the arguments for the underlying combinatorial isomorphism of posets are
given, which implies a geometric isomorphism, we work with
$\iota_{x}\colon\Sigma_{sph}(\Delta_{\pm})\to\Sigma_{x}$, as in the proof of
[FHHK20, Proposition 7.29].
For the adaptation of the automorphism group in the arguments of the proof,
one uses $\operatorname{\text{Aut}}(X_{1})$, in particular, the isomorphism
$\operatorname{\text{Aut}}(X_{1})\cong\operatorname{\text{Aut}}_{\text{eff}}(G^{\dagger})$
(4.5). Since we are working with $\iota_{x}$, $C$ is a parabolic subgroup in
$G^{\dagger}$. Let $f\in\operatorname{\text{Aut}}(X_{1})$ and
$f^{\prime}\in\operatorname{\text{Aut}}_{\text{eff}}(G^{\dagger})$ the
corresponding image under the isomorphism, then $f^{\prime}(C)$ is again a
parabolic subgroup.
The rest of the proof works now the same as the proof for Proposition 7.29 in
the article [FHHK20]. ∎

On the way to the definition of a causal boundary by means of asymptotic
curves, one needs a way to connect the polyhedral structure of two different
sets of rays $\mathcal{M}_{x}$ and $\mathcal{M}_{y}$ for any pair of points
$x,y\in X_{1}$. Since $\left|\iota_{x}\right|$ and $\left|\iota_{y}\right|$
are geometrical isomorphisms of the ideal polyhedral complex structures, one
can define a map

$\displaystyle\left|\iota_{x,y}\right|\coloneqq\left|\iota_{y}\right|\circ\left|\iota_{x}\right|^{-1}.$
(11)

This map is then a coloring-preserving geometrical isomorphism of ideal
polyhedral complexes and there is a commuting diagram:

$\mathcal{M}_{x}$$\mathcal{M}_{y}$$\left|\Delta\right|_{\mathfrak{a}}$$\left|\Delta\right|_{\mathfrak{a}}$id$\left|\iota_{x}\right|$$\left|\iota_{x,y}\right|$$\left|\iota_{y}\right|$

###  5.2 Asymptotic Causal Rays

Similar to the boundary term for hyperbolic groups, one wants to define the
term asymptotically to determine the boundary at infinity of a symmetric
space. Here the motivation of the term is similar to that in Riemannian
symmetric spaces. There, two geodesic rays are said to be asymptotic if they
lie within a bounded Hausdorff distance.

######  Definition 5.5 (cf. Definition 7.34 in [FHHK20])

Let $G$ be a Kac–Moody group over $k$ and $G^{\dagger}<G$ its centered
subgroup. Also, $X_{1}$ is the corresponding symmetric space and $x,y\in X$
are two points. Call two geodesic rays $c\in\mathcal{M}_{x}$ and
$\tilde{c}\in\mathcal{M}_{y}$ _asymptotic_ if
$\left|\iota_{x,y}\right|(c)=\tilde{c}$, denoted by $c\parallel\tilde{c}$.

###### Remark 5.6

The property that two geodesic rays are parallel is an equivalence relation:

  1. 1

reflexivity: Follows through $\left|\iota_{x,x}\right|=\textbf{id}$.

  2. 2

symmetry: Let $c\parallel\tilde{c}$, where $c\in\mathcal{M}_{x}$ and
$\tilde{c}\in\mathcal{M}_{y}$, i.e. $\left|\iota_{x,y}\right|(c)=\tilde{c}$.
Then it follows $c=\left|\iota_{x,y}\right|^{-1}(\tilde{c})$ since
$\left|\iota_{x,y}\right|$ is an isomorphism.

  3. 3

transitivity: Let $c\in\mathcal{M}_{x}$, $c^{\prime}\in\mathcal{M}_{y}$ and
$c^{\prime\prime}\in\mathcal{M}_{z}$ and $c\parallel
c^{\prime}(\Leftrightarrow\left|\iota_{x,y}\right|(c)=c^{\prime})$ and
$c^{\prime}\parallel
c^{\prime\prime}(\Leftrightarrow\left|\iota_{y,z}\right|(c^{\prime})=c^{\prime\prime})$.
Then

$\displaystyle\left|\iota_{x,z}\right|(c)$
$\displaystyle\overset{\ref{Eq:Iota-
xy}}{=}(\left|\iota_{z}\right|\circ\left|\iota_{x}\right|^{-1})(c)$
$\displaystyle=(\left|\iota_{z}\right|\circ(\left|\iota_{y}\right|^{-1}\circ\left|\iota_{y}\right|)\circ\left|\iota_{x}\right|^{-1})(c)$
$\displaystyle=(\left|\iota_{y,z}\right|\circ\left|\iota_{x,y}\right|)(c)$
$\displaystyle=\left|\iota_{y,z}\right|(c^{\prime})=c^{\prime\prime}.$

######  Definition 5.7 (cf. Definition 7.40 in [FHHK20])

The _causal boundary_ of $X_{1}$ is the set

$\mathcal{M}_{\parallel}\coloneqq\nicefrac{{\mathcal{M}}}{{\parallel}}$

of asymptoticity classes of causal and anti-causal rays in $X_{1}$.

The next statement relates the asymptotic property to the corresponding
properties in Euclidean and hyperbolic space.
Therefore, recall that the standard rank one subgroups $G_{i}$, $i\in I$, of a
Kac–Moody group $G$ over $k$ are isomorphic to $\operatorname{SL}(2,k)$. For
$k=\mathbb{R}$, the corresponding fixed points under the Chevalley innvolution
$\Theta$ are given by $K_{i}=G_{i}^{\Theta}=\operatorname{SO}(2)$. For the
case $k=\mathbb{C}$ one has for the fixed point set
$K_{i}=G_{i}^{\Theta}=\operatorname{SU}(2)$. It follows that for
$k=\mathbb{R}$ the hyperbolic $2$-space
$\mathbb{H}^{2}=\operatorname{SL}(2,\mathbb{R})/\operatorname{SO}(2)$ (cp.
[Voi21, 34.6.2]) and for $k=\mathbb{C}$ the hyperbolic $3$-space
$\mathbb{H}^{3}=\operatorname{SL}(2,\mathbb{C})/\operatorname{SU}(2)$ (cp.
[Voi21, 36.3.13]) is contained in $X_{1}$. To keep the notation simple, we
call a subset $\mathbb{H}_{(i)}^{n}$ for $(n=2,3)$ and $i\in I$ a _hyperbolic
$n$-space_ in $X_{1}$, where the subscript $i$ refers to the standard rank one
subgroup $G_{i}$.

######  Proposition 5.8 (cp. Proposition 7.36 in [FHHK20])

Let $G$ be a Kac–Moody group over $k$, $G^{\dagger}<G$ its centered subgroup
and $X_{1}$ the corresponding symmetric space. Further let $x,y$ two points in
$X_{1}$ and $c\in\mathcal{M}_{x}$, $\tilde{c}\in\mathcal{M}_{y}$ two geodesic
rays. Then the equivalence relation that rays are asymptotic satisfies the
following properties:

  1. 1

For every ray $c_{1}\in\mathcal{M}_{x}$ there exists a unique geodesic ray
$c_{2}\in\mathcal{M}_{y}$ such that they are asymptotic, i.e. $c_{1}\parallel
c_{2}$.

  2. 2

The equivalence relation is invariant under automorphisms of $X_{1}$, i.e. if

$c\parallel\tilde{c}\Rightarrow f(c)\parallel f(\tilde{c})$

for all $f\in\operatorname{\text{Aut}}(X_{1})$.

  3. 3

If $c,\tilde{c}$ are contained in a standard hyperbolic plane (or space), then
$c\parallel\tilde{c}$ if and only if they are asymptotic in the sense of
hyperbolic $n$-spaces ($n=2,3)$.

  4. 4

If $c,\tilde{c}$ are contained in a common maximal pointed flat $(p,F)$, then
$c\parallel\tilde{c}$ if and only if they are asymptotic in the sense of the
Euclidean space.

###### Proof

The proof from [FHHK20, Proposition 7.36] goes in both the complex and real
case for all statements except point III. In the complex case we have the
surrounding hyperbolic space $\mathbb{H}^{3}$, which is the problem of the
third point. As a first step to get around this problem, let us first look at
the definition of an asymptotic curve in hyperbolic $n$-spaces. Two geodesic
rays in $\mathbb{H}^{n}$ ($n=2,3)$ are asymptotic if they converge to the same
point in the boundary of the space, i.e. if they have the same endpoints in
the Poincarè half space model.
Let $c\in\mathcal{M}_{x}$ and $\tilde{c}\in\mathcal{M}_{y}$, by the assertion
of the second statement, one can assume that $c$ and $\tilde{c}$ are contained
in one $\mathbb{H}^{3}_{(i)}$ for some $i\in I$. To make the work with the
hyperbolic space simpler, one can identify $\mathbb{H}^{3}_{(i)}$ with the
upper half-space $\mathbb{C}\times\mathbb{R}_{>0}$ (cp. [Voi21, 36.1.1]),
where we identify the base point $e$ with the complex unit $i$. As well, the
image of the building at the horizon of $e$,

$\left|\iota_{e}\right|^{-1}\left(\mathbb{H}^{3}_{(i)}\right)\subseteq\left|\Delta\right|_{\mathfrak{a}},$

can be identified with the boundary of $\mathbb{H}^{3}_{(i)}$, i.e. with
$\mathbb{C}P^{1}=\mathbb{C}\cup\\{\infty\\}$ (cp. [Voi21, 36.1.4]). This
identification is so that geodesics in $\mathbb{H}^{3}_{(i)}$ starting from
$e$ are identified with the endpoint of the corresponding geodesic in
$\mathbb{H}^{3}$. Recall, that the geodesics in the upper half-space model of
$\mathbb{H}^{3}$ are the semicircles orthogonal to $\mathbb{C}$ and vertical
rays, see [Voi21, 36.1.5].
The rest of this prove is now to show, that $\left|\iota_{x,y}\right|$ only
translates the endpoints of the geodesics. The arguments of the proof given in
[FHHK20] for Proposition 7.36 (A3) are independent of the ambient hyperbolic
space, such that it can be copied. ∎

Now we formulate the main statement characterizing the causal boundary of
Kac–Moody symmetric spaces. The non-affine real version is given in [FHHK20,
Corollary 7.41] and we can generalize the first three points to the general
framework. The last point, i.e. $(iv)$ from [FHHK20, Corollary 7.41], is a bit
more difficult, since the automorphism group of the symmetric space is no
longer isomorphic to the automorphism group of $G$. Based on the discussion in
this section, the arguments in the proof of the following theorem are exactly
the same as in the proof of the corresponding statements in the non-affine
real version.

######  Theorem 5.9 (cf. Corollary 7.41 in [FHHK20])

Let $G$ be a Kac–Moody group over $k$ of affine type, $G^{\dagger}<G$ its
centered subgroup and $X_{1}$ the corresponding Kac–Moody symmetric space.
Then the following statements are true:

  1. 1

$\mathcal{M}_{\parallel}$ has a unique polyhedral complex structure, so that
the map

$\varphi_{x}\colon\mathcal{M}_{x}\hookrightarrow\mathcal{M}\rightarrow\mathcal{M}_{\parallel}$

is a geometric isomorphism for every point $x\in X_{1}$. In particular the map
respects the weak topology and maps open cells homeomorphically to open cells.

  2. 2

The automorphism group of the symmetric space $X_{1}$ acts on the causal
boundary by geometric automorphisms.

  3. 3

The unique ideal polyhedral complex structure on $\mathcal{M}_{\parallel}$ is
geometrically isomorphic to the $\mathfrak{a}$-realization
$\left|\Delta\right|_{\mathfrak{a}}$ of the twin building $\Delta_{\pm}$.

##  6 Comparing Symmetric Spaces

Having set up a symmetric space for Kac–Moody groups of arbitrary type over
the field $k\in\\{\mathbb{R},\mathbb{C}\\}$, one can compare it with the
symmetric space developed in [FHHK20], which works for non-affine Kac–Moody
groups over $\mathbb{R}$.
Since a Kac–Moody algebra requires a linearly independent set of simple roots
due to the definition of Kac used in [FHHK20], the present work uses the root
datum $\mathcal{D}_{Kac}^{\mathbb{A}}$. In contrast, the authors of [FHHK20]
have chosen to use the simply connected root term
$\mathcal{D}_{sc}^{\mathbb{A}}$, see [Mar18, Example 7.11], which does not
imply that the simple roots are linearly independent. Due to this, the authors
in [FHHK20] work with the derived Kac–Moody algebra (see [Mar18, Definition
7.13]), and use the Cartan subalgebra of this Kac–Moody algebra to construct
the standard flat $A$. This is one of the main differences between this
article and [FHHK20]: In the present work, we extend the Cartan subalgebra to
achieve linear independence of the simple roots, and thus also obtain a larger
standard flat. For details, see 2.3.

As explained in 2.3, one encounters a problem when the generalized Cartan
matrix is non-invertible, since then the simple roots are not linearly
independent. Therefore, for non-affine non-invertible generalized Cartan
matrices, there is a kernel of the bilinear form B in the Cartan subalgebra of
the derived Kac–Moody algebra, which is equal to the center ([Kac90, Lemma 2.1
a)]). To get around this problem, the authors of [FHHK20] use the simple
adjoint quotient of a Kac–Moody group: that is the quotient of the group by
the center of the bilinear form which is lifted to the group. This quotient of
the original group allows them to establish, for example, the useful
statements [FHHK20, Proposition 6.4] and [FHHK20, Theorem 6.12 (ii)].

In the context of the present article, these statements are no longer true,
since we have to extend the Kac–Moody group by a torus resulting from the
Cartan subaglebra. This larger torus is indeed a problem when analyzing the
automorphism group of the Kac–Moody group and connecting it to the
automorphism group of the associated symmetric space. The problem arises from
Caprace’s fundamental theorem, which is the basis for understanding the
automorphisms of a Kac–Moody group: the theorem controls only the
automorphisms of a Kac–Moody group generated by the root groups, see [Cap09,
Theorem 4.2]. On the extended torus, one is asking basically for the
automorphism group of $(\mathbb{R},+)$. This group is too large to classify,
so it is not possible to say anything about the automorphisms on the torus.
Nevertheless, in order to do an analysis of the automorphisms and the building
at infinity of the attached symmetric space, we must restrict ourselves to the
subgroup $G^{\dagger}$ of a Kac–Moody group generated only by its root groups,
since it allows us to use Caprace’s theorem. These results can be found in 4.5
and 5.9.
But this restriction means that in the general case, 5.9, we cannot set up a
version of [FHHK20, Corollary 7.41 (iv)], since there it is assumed that one
has this concrete isomorphism between the automorphism group of the Kac–Moody
group and the symmetric space. In the concrete case of this article, there is
only one isomorphism between a larger automorphism group and the symmetric
space, so we cannot achieve the unique determinability assumed in the
corollary [FHHK20, Corollary 7.41 (iv)].

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