# A lightcone embedding of the twin building of a
hyperbolic Kac–Moody group

Lisa Carbone  Department of Mathematics, Rutgers University, New Brunswick,
New Jersey, USA<EMAIL_ADDRESS>,  Alex Feingold  Dept. of Math. Sci.,
The State University of New York, Binghamton, New York 13902-6000
<EMAIL_ADDRESS>and  Walter Freyn  Formerly at Fachbereich
Mathematik, Technical University of Darmstadt
Darmstadt, Germany<EMAIL_ADDRESS>
###### Abstract.

The twin building of a Kac–Moody group $G$ encodes the parabolic subgroup
structure of $G$ and admits a natural $G$–action. When $G$ is a complex
Kac–Moody group of hyperbolic type, we construct an embedding of the twin
building of $G$ into the lightcone of the compact real form of the
corresponding Kac–Moody algebra. When $G$ has rank $2$, we construct an
embedding of the spherical building at infinity into the set of rays on the
boundary of the lightcone.

This material is based upon work supported by the National Science Foundation
under Grant No. 1002477. The first author was supported in part by NSF grant
number DMS–1101282. All authors wish to thank the IHÉS for support during
various visits in May 2013, 2014 and 2015. The second author wishes to thank
the Max-Planck Institute for Gravitational Physics (Albert Einstein
Institute), Potsdam, Germany, for support during visits in June 2013, 2014 and
2015.

###### Contents

  1. 1 Introduction
  2. 2 Kac–Moody algebras and Kac–Moody groups
    1. 2.1 Kac–Moody algebras
    2. 2.2 Kac–Moody groups
    3. 2.3 $BN$–pair and Tits building of a minimal Kac–Moody group
    4. 2.4 Compact real forms of Kac–Moody algebras and groups
  3. 3 Tits cone and lightcone of hyperbolic Kac–Moody algebras of compact type
  4. 4 Group actions of the compact real form $K$
    1. 4.1 The Adjoint action of $K$ on $\mathfrak{k}$
    2. 4.2 The local structure of the Adjoint action
    3. 4.3 The action of the compact real form on the twin building
  5. 5 Simplicial structure on $\mathfrak{H}_{r}$
  6. 6 The main embedding theorem
  7. 7 Cartan involutions and the embedding of the twin building
  8. 8 Understanding of the Tits building in rank 2
  9. 9 Embedding the spherical building at infinity in rank $2$
  10. 10 Conclusion and further directions

##  1\. Introduction

Buildings were introduced by J. Tits in the 1950’s to provide a ‘geometric’
interpretation of finite dimensional simple algebraic groups and finite groups
of Lie type. Following ideas originally developed by F. Klein in his Erlanger
program, Tits aimed at understanding a large class of groups, containing
simple algebraic groups and finite groups of Lie type, as the automorphism
groups of carefully constructed geometric objects called ‘buildings’. It turns
out that the buildings of Tits correspond to groups which admit an additional
structure, called a _$BN$ –pair_ or equivalently a _Tits system_. This
structure also gives rise to a Bruhat decomposition for the corresponding
group. Tits’s approach was the geometric counterpart to the ‘algebraic’
construction of these groups by C. Chevalley using automorphisms of Lie
algebras [Che55].

In general, the building ${\mathbb{B}}$ of a group $G$ with a $BN$–pair is an
abstract simplicial complex constructed from group theoretical data. Namely,
simplices in $\mathcal{B}$ are in bijection with the union of all cosets
$G/P$, where $P$ runs through a set of representatives of the conjugacy
classes of parabolic subgroups. These simplices satisfy boundary relations
which can be phrased in terms of inclusions of these cosets.

For Kac–Moody groups, the closest infinite dimensional analogue of simple
algebraic groups, the structure of buildings and $BN$–pairs is richer. As
Kac–Moody groups have two conjugacy classes of Borel subgroups, they admit the
definition of two ‘opposite’ $BN$–pairs which together form a ‘twin
$BN$–pair’. Consequently the geometry associated to a Kac–Moody group $G$
naturally consists of two related components. This object is called a ‘twin
building’ $\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ associated to a
twin $BN$–pair, ($B^{+},B^{-},N$), where the subgroups $B^{+}$ and $B^{-}$ are
the standard Borel subgroups constructed from the positive and negative roots
of the Kac–Moody algebra respectively. Thus by construction the building is
related to the combinatorial structure of its Kac-Moody groups.

Kac-Moody algebras and groups fall naturally into three types, finite type,
affine type and indefinite type. While Kac-Moody groups of finite type (simple
Lie groups) and of affine type are well-understood, there are far fewer
results known for the indefinite type. The most important subclass of
indefinite type is the hyperbolic type, studied since the 1980s by various
authors [LM79, Fei80, FF83, KMW88, KM95] but certainly of interest to
physicists [Jul85, DHN02, DH09, Wes01].

While the algebraic properties of the hyperbolic Kac-Moody groups and algebras
attracted some attention, there have been only a few mathematical results
concerning their geometry, for example, the study of homogeneous or symmetric
spaces associated to them, and the study of ‘polar actions’ or isoparametric
submanifolds. These classes of objects are very well understood for finite
dimensional Lie groups and for affine Kac-Moody groups. It is hoped that the
understanding of this geometry, called Kac-Moody geometry, will shed new light
on algebraic and structural properties of these groups. As a first step
towards the goal of understanding the geometry of hyperbolic Kac-Moody
algebras and groups, we show in this work that for hyperbolic Kac–Moody groups
$G$ over ${\mathbb{C}}$, the associated Tits building is not only an abstract
simplicial complex admitting an action of $G$, but that it admits a natural
embedding inside the compact real form $\mathfrak{k}$ of the Kac–Moody algebra
$\mathfrak{g}$. As a consequence, the structure of the Tits building and of
the compact real form of the Kac–Moody algebra are closely related.

Our results generalize work of Quillen and Mitchell in the finite dimensional
case, and Kramer and Freyn in the affine case. Mitchell, in a paper based on
ideas of Quillen, used embeddings of spherical buildings associated to simple
(real or complex) Lie groups into the tangent space of the associated
noncompact real symmetric space [Mit88]. In particular, the topological
building of a real noncompact Lie group $G$ with maximal compact subgroup $K$
is canonically identified with a space homeomorphic to the unit sphere in the
tangent space of the noncompact real symmetric space $G/K$. For example, the
topological building of type $A_{1}$, isomorphic to $S^{1}$, can be embedded
into the unit circle in the tangent space of
$\mathcal{H}^{2}=SL_{2}(\mathbb{R})/SO(2)$ which is $\mathbb{R}^{2}$.

Kramer gave a topological construction of the complex twin building of type
$A_{n}^{(1)}$ and an equivariant embedding of this building into the
associated affine Kac–Moody algebra $\mathfrak{g}$ [Kra02].

Freyn gave a 2–parameter family $\varphi_{\ell,r}$ of equivariant embeddings
of affine ‘twin cities’ $\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ into
the ‘$s$-representations’ of affine Kac–Moody symmetric spaces [Fre09, Fre11,
Fre12b, Fre13c, Fre13b]. A ‘twin city’ is the natural completion of a twin
building; affine twin cities correspond to completions of affine Kac–Moody
groups in a similar way as twin buildings correspond to minimal Kac–Moody
groups. Twin cities carry a natural topology that is derived from the topology
on the corresponding Kac–Moody group.

Denoting by $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ the Cartan
decomposition, and restricting Freyn’s result to non-completed affine Kac-
Moody algebras, yields an identification of the twin building with the
intersection $\mathfrak{p}_{\ell,r}$ of the sphere of squared length
$\ell\in\mathbb{R}$ with horospheres parametrized by $r_{d}=\pm r\not=0$,
where $r_{d}$ is the real coefficient of the derivation $d$. The positive and
negative components of the twin building, $\mathcal{B}^{+}$ and
$\mathcal{B}^{-}$, are immersed into the two sheets of $\mathfrak{p}_{\ell,r}$
described by $r_{d}<0$ respectively $r_{d}>0$.

For affine Kac-Moody groups of the compact type, which are symmetric spaces
called of _‘type II’_ following Helgason’s classification (see [Hel01,
Fre13a]), this result includes embeddings of complex buildings into the
compact forms of affine Kac–Moody algebras; this case is the affine
counterpart to the embeddings constructed in this paper for symmetrizable
hyperbolic Kac–Moody algebras. We refer the reader to [Fre09, Fre12a, Fre11,
Fre12b] for additional details.

For $\mathfrak{g}$ a hyperbolic Kac–Moody algebra we are motivated in part by
the appearance of $\mathfrak{g}/\mathfrak{k}$ and $G/K$ in coset models of
certain supergravity theories, where coset spaces of the split real forms
occur as parameter spaces for the scalar fields of the theory [Jul85, DHN02,
Wes01].

The results in this paper can be extended to include a wider class of
Kac–Moody algebras, namely Lorentzian algebras. Nevertheless as soon as one
leaves the realm of hyperbolic Kac–Moody algebras, the equivalence between the
lightcone and the Tits cone breaks down. Consequently, the geometry of the
buildings and their lightcone embeddings becomes more complicated. On the
other hand interesting new phenomena occur, for example there are infinite
families of rank 3 Lorentzian, non–hyperbolic Kac–Moody algebras whose
buildings are isomorphic. Up to isomorphism, the same hyperbolic building will
embed into each algebra in this infinite family, but in different ways. An
extensive collection of examples is being worked out by A. Tichai [Tic14].

The authors wish to thank Peter Abramenko for his helpful comments on an
earlier draft of the manuscript. They would also like to thank Victor Kac for
helpful comments in May 2015 at IHÉS. AF would like to thank Kai-Uwe Bux, Max
Horn, Tobias Hartnick, Ralf Köhl and Peter Abramenko for helpful discussions
at the June 2015 conference on “Generalizations of Symmetric Spaces” in
Israel.

##  2\. Kac–Moody algebras and Kac–Moody groups

###  2.1. Kac–Moody algebras

A Kac–Moody algebra $\mathfrak{g}_{{\mathbb{F}}}(A)$ over a field
${\mathbb{F}}$ may be constructed by generators and relations using a
collection of data which includes a matrix $A=(a_{ij})_{i,j\in I}$ called a
generalized Cartan matrix satisfying the following conditions for all $i,j\in
I=\\{1,\dots,\ell\\}$:

$a_{ij}\in{\mathbb{Z}},\qquad a_{ii}=2,\qquad a_{ij}\leq 0\hbox{ if }i\neq
j,\quad\hbox{ and }\quad a_{ij}=0\iff a_{ji}=0.$

A generalized Cartan matrix $A$ is indecomposable if there is no partition of
the set $I=I_{1}\cup I_{2}$ into non-empty subsets so that $a_{ij}=0$ for
$i\in I_{1}$ and $j\in I_{2}$. The matrix $A$ is called symmetrizable if there
is an invertible diagonal matrix $D=diag(d_{1},\cdots,d_{\ell})$ such that
$DA=(d_{i}a_{ij})$ is symmetric. One distinguishes various types of
generalized Cartan matrices:

  * -

Finite type: $A$ is positive-definite. In this case $A$ is the Cartan matrix
of a finite dimensional semisimple Lie algebra and $det(A)>0$.

  * -

Affine type: $A$ is positive-semidefinite, but not positive-definite, and all
minors are positive definite. In this case $det(A)=0$.

  * -

Hyperbolic type: $A$ is neither of finite nor affine type, but every proper,
indecomposable submatrix is either of finite or of affine type. In this case
$det(A)<0$.

A complex Kac–Moody algebra $\mathfrak{g}_{\mathbb{C}}(A)$ has at least two
real forms, that is, real Lie algebras $\mathfrak{g}_{\mathbb{R}}$ such that
$\mathfrak{g}_{\mathbb{C}}(A)={\mathbb{C}}\otimes\mathfrak{g}_{\mathbb{R}}$.
The split real form of $\mathfrak{g}_{\mathbb{C}}(A)$ is
$\mathfrak{g}_{\mathbb{R}}(A)$ (see [BVBPBMR95]). From this point on, all
generalized Cartan matrices in this paper are assumed to be indecomposable,
symmetrizable and of hyperbolic type.

Given field ${\mathbb{F}}={\mathbb{C}}$ or ${\mathbb{F}}=\mathbb{R}$:

  * -

a hyperbolic generalized Cartan matrix $A=(a_{ij})_{i,j\in I}$, and

  * -

a vector space $\mathfrak{h}$ over ${\mathbb{F}}$ (which will play the role of
a Cartan subalgebra) with $dim_{\mathbb{F}}(\mathfrak{h})=\ell$, and basis
$\\{h_{i}\ |\ i\in I\\}$,

then there is a set of simple roots $\Pi=\\{\alpha_{j}\ |\ j\in
I\\}\subseteq\mathfrak{h}^{*}$ such that the pairing
$\langle\cdot,\cdot\rangle:\mathfrak{h}^{*}\times\mathfrak{h}\to{\mathbb{F}}$
given by $\langle\alpha,h\rangle=\alpha(h)$ satisfies
$\alpha_{i}(h_{j})=a_{ij}$ for all $i,j\in I$, and the hyperbolic Kac–Moody
Lie algebra $\mathfrak{g}=\mathfrak{g}_{\mathbb{F}}(A)$ is generated by the
elements $\\{e_{i},f_{i},h_{i}\ |\ i\in I\\}$, subject to the relations
([Kac90, GK81]):

  * -

$[h_{i},h_{j}]=0$,

  * -

$[h,e_{i}]=\langle\alpha_{i},h\rangle e_{i}$,
$h\in\mathfrak{h}\qquad\qquad\textrm{and}\qquad[h,f_{i}]=-\langle\alpha_{i},h\rangle
f_{i}$, $h\in\mathfrak{h}$,

  * -

$[e_{i},f_{j}]=\delta_{ij}h_{i}$,

  * -

$(ad_{e_{i}})^{1-a_{ij}}(e_{j})=0,\ i\neq
j\qquad\textrm{and}\qquad(ad_{f_{i}})^{1-a_{ij}}(f_{j})=0,\ i\neq j$,

where $ad_{x}(y)=[x,y]$. For each $i\in I$, let $\mathfrak{sl}_{2}^{i}$ be the
Lie subalgebra (isomorphic to $\mathfrak{sl}_{2}({\mathbb{F}})$) with basis
$\\{e_{i},f_{i},h_{i}\\}$, so that $\mathfrak{g}$ is generated by these
subalgebras. The abelian Lie subalgebra $\mathfrak{h}$ with basis $\\{h_{i}\
|\ i\in I\\}$ is called the standard Cartan subalgebra of $\mathfrak{g}$.

The algebra $\mathfrak{g}=\mathfrak{g}(A)$ is infinite dimensional since $A$
is not positive definite, and it admits an invariant symmetric bilinear form
$(\hskip 2.0pt,\hskip 2.0pt)$ which is unique up to a global scaling factor
([Kac90], section II), and which extends the form on $\mathfrak{h}$ given by
$2(h_{i},h_{j})/(h_{j},h_{j})=\alpha_{i}(h_{j})=a_{ij}$. The nondegeneracy of
the pairing $\langle\cdot,\cdot\rangle$ between $\mathfrak{h}^{*}$ and
$\mathfrak{h}$ determines a corresponding form on $\mathfrak{h}^{*}$. This
means that $(\alpha_{i},\alpha_{j})=(h_{i},h_{j})$ and

$a_{ji}(\alpha_{i},\alpha_{i})/2=(\alpha_{j},\alpha_{i})=(\alpha_{i},\alpha_{j})=a_{ij}(\alpha_{j},\alpha_{j})/2$

so that we may take the diagonal matrix $D=diag(d_{1},\cdots,d_{\ell})$ with
$d_{i}=2/(\alpha_{i},\alpha_{i})$ and the symmetric matrix
$DA=(d_{i}a_{ij})=(2a_{ij}/(\alpha_{i},\alpha_{i}))$. The standard way to
choose the global scaling factor is so that the longest square length of any
simple root is $2$. The adjoint action of $\mathfrak{h}$ on $\mathfrak{g}$ is
diagonalizable, and the simultaneous nonzero eigenspaces for that action,

$\mathfrak{g}_{\alpha}\ =\ \\{x\in\mathfrak{g}\mid[h,x]=\alpha(h)x,\
h\in\mathfrak{h}\\}$

for $\alpha\neq 0$ are called root spaces.

The root system of $\mathfrak{g}$ is the set
${\Phi}=\\{\alpha\in\mathfrak{h}^{*}\ |\ \alpha\neq
0,\mathfrak{g}_{\alpha}\neq 0\\}$, and the $\mathbb{Z}$-span of ${\Phi}$,
called the root lattice of $\mathfrak{g}$, is denoted by $Q$. From the
relations defining $\mathfrak{g}$ we see that for each $i\in I$,
$\mathfrak{g}_{\alpha_{i}}={\mathbb{C}}e_{i}$ and
$\mathfrak{g}_{-\alpha_{i}}={\mathbb{C}}f_{i}$, so that
$\pm\alpha_{i}\in{\Phi}$. In fact, we have
$Q=\mathbb{Z}\alpha_{1}\oplus\dots\oplus\mathbb{Z}\alpha_{\ell}$ is a free
$\mathbb{Z}$-module. Every $\alpha\in{\Phi}$ can be written uniquely as
$\alpha=\sum_{i=1}^{\ell}k_{i}\alpha_{i}$ where either all $k_{i}\geq 0$, in
which case $\alpha$ is called positive, or all $k_{i}\leq 0$, in which case
$\alpha$ is called negative. The set of all positive roots is denoted
${\Phi}^{+}$, and the set of all negative roots is denoted by ${\Phi}^{-}$.
Any root is either positive or negative.

For each simple root $\alpha_{i}$, $i\in I=\\{1,\dots,\ell\\}$, we define the
simple root reflection

(1) $w_{i}(\alpha_{j}):=\alpha_{j}-\alpha_{j}(h_{i})\alpha_{i}\,.$

The $w_{i}$ generate a group $W=W(A)$ of orthogonal transformations of
$\mathfrak{h}^{*}$, called the Weyl group of $A$. The non-degenerate pairing
between $\mathfrak{h}$ and $\mathfrak{h}^{*}$ gives a corresponding action of
$W$ as orthogonal transformations on $\mathfrak{h}$. A root $\alpha\in{\Phi}$
is called a real root if there exists $w\in W$ such that $w\alpha$ is a simple
root. A root $\alpha$ which is not real is called imaginary. We denote by
${\Phi}^{re}$ the set of all real roots and ${\Phi}^{im}$ the set of all
imaginary roots.

The multibracket,
$[e_{i_{1}},e_{i_{2}},\cdots,e_{i_{n}}]=ad_{e_{i_{1}}}ad_{e_{i_{2}}}\cdots
ad_{e_{i_{n-1}}}e_{i_{n}}$ is in the root space $\mathfrak{g}_{\alpha}$ for
$\alpha=\sum_{j=1}^{n}\alpha_{i_{j}}\in{\Phi}^{+}$, while a similar
multibracket with each $e_{i_{j}}$ replaced by $f_{i_{j}}$ is in
$\mathfrak{g}_{-\alpha}$. Therefore, $\mathfrak{g}$ has a root space
decomposition ([Kac90], Theorem 1.2)

$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in{\Phi}}\mathfrak{g}_{\alpha}=\mathfrak{h}\
\oplus\ \mathfrak{g}^{+}\oplus\ \mathfrak{g}^{-}$

where

$\mathfrak{g}^{+}=\bigoplus_{\alpha\in{\Phi}^{{}^{+}}}\mathfrak{g}_{\alpha},\qquad\qquad\mathfrak{g}^{-}=\bigoplus_{\alpha\in{\Phi}^{{}^{-}}}\mathfrak{g}_{\alpha}.$

The standard positive Borel subalgebra $\mathfrak{b}\equiv\mathfrak{b}^{+}$ is
defined by $\mathfrak{b}^{+}=\mathfrak{h}\oplus\mathfrak{g}^{+}$ and the
standard negative Borel subalgebra by
$\mathfrak{b}^{-}=\mathfrak{h}\oplus\mathfrak{g}^{-}$.

###  2.2. Kac–Moody groups

There are various ways to define abstract Kac-Moody groups (see for example
[KP85, Tit87, Rém02]). The main point of the abstract approach is to give a
flexible definition of Kac-Moody groups, allowing the construction of groups
whose Adjoint action is not faithful. There are indeed important examples of
that kind, the smallest one being the finite type Kac-Moody group
$SL(2,{\mathbb{C}})$, where the two matrices $\pm Id$ both act as the identity
operator in the Adjoint representation. Hence the Adjoint representation of
this Kac-Moody group is actually the group $PSL(2,{\mathbb{C}})$, and that is
what we get by using the definition of the adjoint Kac-Moody group given in
equation (2) for the Cartan matrix $A=[2]$ of type $A_{1}$.

By the definition of the abstract Kac-Moody group, there is a surjective group
homomorphism: $Ad:G\longrightarrow G^{ad}$ from an abstract Kac-Moody group
onto the Adjoint Kac-Moody group whose kernel is exactly the center of G (see
[Rém02, 9.6.2]). Since, as we will see in subsection 2.3, for the subgroups
$B^{\pm}$ and $N$ defined there, we have $B^{\pm}\cap N$ is abelian, and the
center of $G$ is the kernel of the action of $G$ on the twin building (see
[Cap09, Lemma 1.7.]). Hence, without loss of generality, to understand the
action on twin buildings we can work with the Adjoint Kac-Moody group. Our
references for this section are [Kum02, KP85, MP95].

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra over ${\mathbb{C}}$,
$L$ be a complex vector space and let $\pi:\mathfrak{g}\longrightarrow End(L)$
be any integrable representation, so that all $\pi(e_{i})$ and $\pi(f_{i})$
are locally nilpotent on $L$ and the linear operators

$\chi^{\pi}_{{\alpha}_{i}}(t)=exp(\pi(te_{i}))\qquad\hbox{and}\qquad\chi^{\pi}_{{-\alpha}_{i}}(t)=exp(\pi(tf_{i})),\
\hbox{ for }t\in{\mathbb{C}},$

are well-defined in $GL(L)$. Let the minimal Kac–Moody group associated to
such a representation $\pi$ be the group generated by these operators,

$G^{\pi}=G^{\pi}({\mathbb{C}})=\langle\chi^{\pi}_{{\alpha}_{i}}(t),\chi^{\pi}_{{-\alpha}_{i}}(t)\mid
i\in I,\ t\in{\mathbb{C}}\rangle\leq GL(L)\,.$

In fact, for any $x\in\mathfrak{g}_{\alpha}$, $\alpha\in{\Phi}^{re}$ the
operator $\pi(x)$ is locally nilpotent on $L$, so $\chi^{\pi}_{x}=exp(\pi(x))$
is well-defined and these give the real root groups $U_{\alpha}^{\pi}$. Now
apply the above to the adjoint representation

$ad:\mathfrak{g}\longrightarrow End(\mathfrak{g}),$

which is integrable. This gives the minimal adjoint Kac–Moody group

(2)
$G^{ad}=G^{ad}({\mathbb{C}})=\langle\chi^{ad}_{{\alpha}_{i}}(t),\chi^{ad}_{{-\alpha}_{i}}(t)\mid
i\in I,\ t\in{\mathbb{C}}\rangle\leq GL(\mathfrak{g})$

and for all $x\in\mathfrak{g}_{\alpha}$, $\alpha\in{\Phi}^{re}$ we have well-
defined operators $\chi^{ad}_{x}=exp(ad_{x})\in GL(\mathfrak{g})$ giving the
real root groups $U_{\alpha}^{ad}$. These generators act on $\mathfrak{g}$ as
Lie algebra automorphisms, so we have $G^{ad}\leq Aut(\mathfrak{g})$.

Since $ad_{e_{i}}$ and $ad_{f_{i}}$ are locally nilpotent, $\mathfrak{g}$ is
the direct sum of finite dimensional $\mathfrak{sl}_{2}({\mathbb{C}})$-modules
for each of the subalgebras $\mathfrak{sl}_{2}^{i}$. For fixed $i\in I$, on
each such summand the exponentials above generate the group $SL_{2}^{i}$
isomorphic to $SL_{2}({\mathbb{C}})$, so $G$ is also generated by the
subgroups $SL_{2}^{i}$, $i\in I$.

The operators $\pi(h)\in End(L)$ for $h\in\mathfrak{h}$ are semisimple so they
can also be exponentiated to give a commutative group of operators
$T_{\mathbb{C}}^{\pi}=T_{\mathbb{C}}^{\pi}(G)=\\{\chi^{\pi}_{h}(t)=exp(\pi(th))\
|\ h\in\mathfrak{h},t\in{\mathbb{C}}\\}\leq G^{\pi}$ which is called the
standard maximal torus of $G^{\pi}$. We also define the standard Borel
subgroups $(B^{\pi})^{\pm}=T_{\mathbb{C}}^{\pi}\ \langle U^{\pi}_{\alpha}\ |\
\alpha\in(\Phi^{re})^{\pm}\rangle$ and the normalizer of
$T_{\mathbb{C}}^{\pi}$ denoted by $N_{\mathbb{C}}^{\pi}$.

It is well known that the operators

$\widetilde{w}^{ad}_{i}=exp(ad_{e_{i}})exp(ad_{-f_{i}})exp(ad_{e_{i}})=exp(ad_{-f_{i}})exp(ad_{e_{i}})exp(ad_{-f_{i}})\
,1\leq i\leq\ell,$

in $G^{ad}$, generate a subgroup $\widetilde{W}^{ad}$ in $G^{ad}$ such that
the restriction of $\widetilde{w}^{ad}_{i}$ to the standard Cartan subalgebra
$\mathfrak{h}$ equals the simple Weyl group reflection $w_{i}$ and
$\widetilde{w}^{ad}_{i}(e_{i})=-f_{i}$. It means that $W$ is a homomorphic
image of $\widetilde{W}^{ad}$. Note that $\widetilde{W}^{ad}$ is a subgroup of
$N_{\mathbb{C}}^{ad}$ and that $N_{\mathbb{C}}^{ad}/T_{\mathbb{C}}^{ad}\cong
W$.

###  2.3. $BN$–pair and Tits building of a minimal Kac–Moody group

Our references for this section are [AB08, Ron89]. A Kac–Moody group $G$ over
${\mathbb{C}}$ has subgroups $B^{\pm}\subseteq G$ and $N\subseteq G$, with
twin $BN$–pair $(B^{+},B^{-},N)$ satisfying the Tits axioms:

  * T 1:

$G=\langle B^{\pm},N\rangle$, $H=B^{\pm}\cap N\lhd N$, $W=N/H$.

  * T 2:

Define $S$ by the condition $W=\langle S\rangle$ and for $1\leq i\leq\ell$,
$w_{i}\in S$ we have $w_{i}^{2}=1$.

  * T 3:

For $w_{i}\in S$ and $w\in W$ we have

$w_{i}B^{\pm}w\subseteq B^{\pm}w_{i}wB^{\pm}\cup B^{\pm}wB^{\pm}\,.$

  * T 4:

For $w_{i}\in S$ we have

$w_{i}B^{\pm}w_{i}^{-1}\not\subseteq B^{\pm}\,.$

For a hyperbolic Kac-Moody group $G_{\mathbb{C}}(A)$ we have
$T_{\mathbb{C}}=H$. Thus the group $T(G)=N\cap B^{\pm}$ is a normal subgroup
of $N$. The group $W=N_{G}(T(G))/T(G)$ is isomorphic to our earlier definition
in section 2.1 of the Weyl group $W$ as a group of orthogonal transformations
of $\mathfrak{h}^{*}$ given by formula (1).

We identify $W$ (non-canonically) with a subset (not a subgroup) of $N$ which
contains exactly one representative for every element of $W$. By abuse of
notation, this set of representatives will also be called $W$. We have the
(positive and negative) standard Borel subgroups, corresponding to the
standard Borel subalgebras, $B^{\pm}=T(G)U^{\pm}$ where $U^{+}$ is generated
by all positive real root groups and $U^{-}$ is generated by all negative real
root groups. The $BN$–pairs $(B^{+},N)$ and $(B^{-},N)$ have Birkhoff and
Bruhat decompositions:

$G\quad=\quad\coprod_{w\in W}B^{\pm}wB^{\mp}\quad=\quad\coprod_{w\in
W}B^{\pm}wB^{\pm}\,.$

A proper subgroup $P^{\pm}$ of $G$ is called parabolic when it contains a
conjugate of a Borel subgroup $B^{\pm}$, and it is called _positive_ or
_negative_ , depending on the sign. For each subset $J\subset I$ define the
subgroup $W_{J}=\langle w_{j}\ |\ j\in J\rangle$ of $W$ and the corresponding
subgroups of $G$,

$P_{J}^{\pm}\quad=\quad\coprod_{w\in W_{J}}B^{\pm}wB^{\pm}\,.$

Note that $P_{I}^{\pm}=G$ is not parabolic, $P_{\emptyset}^{\pm}=B^{\pm}$, and
we write $P_{i}^{\pm}=P_{\\{i\\}}^{\pm}$. For $J\subsetneq I$ we call
$P_{J}^{\pm}$ a standard parabolic subgroup, and these form a complete set of
representatives of the conjugacy classes of parabolic subgroups, so there are
$2\cdot(2^{\ell}-1)$ conjugacy classes of parabolic subgroups. A parabolic
subgroup $P^{\pm}$ is called _maximal_ if there is no parabolic subgroup
$P^{\prime\pm}$ such that $P^{\pm}\subsetneq P^{\prime\pm}$. For each $i\in
I$, $P_{[i]}^{\pm}=P_{I\backslash\\{i\\}}^{\pm}$ is a maximal standard
parabolic subgroup, so there are $2\ell$ conjugacy classes of maximal
parabolic subgroups, $\ell$ positive and $\ell$ negative.

A Tits building consists of a simplicial complex $\mathcal{B}$ together with a
collection of subcomplexes $\mathfrak{A}$, each of which is called an
apartment, such that

  1. (1)

each apartment is a Coxeter complex,

  2. (2)

each pair of chambers, i.e. simplices of maximal dimension in $\mathcal{B}$,
is contained in a common apartment,

  3. (3)

for two apartments $A$ and $A^{\prime}$ there is an isomorphism
$\varphi:A\rightarrow A^{\prime}$, fixing the intersection $A\cap A^{\prime}$.

A twin building $\mathcal{B}$ consists of a disjoint union
$\mathcal{B}^{+}\cup\mathcal{B}^{-}$ of buildings, which are ‘twinned’. This
twinning can be described in various ways, for example via twin apartments. We
will not go into details here, but direct the interested reader to [AR98].

From now on let $\mathcal{B}$ denote the twin building associated to a Kac-
Moody group $G$. The simplices of $\mathcal{B}$ are in bijection with
parabolic subgroups in such a way that simplices in $\mathcal{B}^{+}$
correspond to positive parabolic subgroups and simplices in $\mathcal{B}^{-}$
correspond to negative parabolic subgroups. The vertices ($0$-simplices) of
the twin building $\mathcal{B}$ are in bijection with maximal parabolic
subgroups in $G$. Chambers are in bijection with positive and negative Borel
subgroups. We will denote these simplices by their corresponding parabolic
subgroups. Since parabolic subgroups are self-normalizing, the simplicies in
the building can be equivalently indexed by the coset spaces $G/P_{J}^{\pm}$,
where $\\{P_{J}^{\pm}\ |\ J\subsetneq I\\}$ is a complete set of
representatives of the conjugacy classes of parabolic subgroups.

The incidence relation on the set of vertices is given by intersections of
parabolic subgroups as follows. For $0\leq r\leq\ell-1$ the $r+1$ vertices
$P_{[i_{1}]}^{\pm},\dots,P_{[i_{r+1}]}^{\pm}$ span an $r$-simplex if and only
if the intersection $P_{[i_{1}]}^{\pm}\cap\dots\cap
P_{[i_{r+1}]}^{\pm}=P_{[i_{1},\dots,i_{r+1}]}^{\pm}$ is a parabolic subgroup.
Hence $\mathcal{B}$ is a simplicial complex of dimension
$dim(\mathcal{B})=\ell-1$. We note that for Kac–Moody groups, the intersection
of a positive parabolic subgroup with a negative parabolic subgroup never
contains a Borel subgroup; hence the associated geometry consists of two
buildings $\mathcal{B}^{+}$ and $\mathcal{B}^{-}$, corresponding to the two
opposite $BN$–pairs $(B^{+},N)$ and $(B^{-},N)$, yielding a twin building
$\mathcal{B}=\mathcal{B}^{+}\sqcup\mathcal{B}^{-}$. In $\mathcal{B}^{\pm}$ the
$(\ell-1)$-simplex
$P_{[1,\dots,{\ell}]}^{\pm}=P_{[I]}^{\pm}=P_{\emptyset}^{\pm}=B^{\pm}$ is
called the fundamental chamber of $\mathcal{B}^{\pm}$. Each fundamental
chamber has boundary consisting of the simplices
$\Delta_{J}^{\pm}=P_{J}^{\pm}$, and has closure

$\Delta^{\pm}=\bigcup_{J\subsetneq I}P_{J}^{\pm}\,.$

Using the property that the simplices in each building are in bijection with
the union of the coset spaces $\bigcup_{J\subsetneq I}G/P_{J}^{\pm}$, we
describe the buildings $\mathcal{B}^{+}$ and $\mathcal{B}^{-}$ associated to a
Kac–Moody group $G$ as follows:

(3) $\mathcal{B}^{\pm}:=\left(G/B^{\pm}\times\Delta^{\pm}\right)/\sim$

The equivalence relation $\sim$ is defined by
$(fB^{\pm},\Delta_{J}^{\pm})\sim(gB^{\pm},\Delta_{J^{\prime}}^{\pm})$ in
$(G/B^{\pm},\Delta^{\pm})$ if and only if
$\Delta_{J}^{\pm}=\Delta_{J^{\prime}}^{\pm}$ (so $J=J^{\prime}$) and
$f^{-1}gP_{J}^{\pm}\subset P_{J}^{\pm}$.

Hence on the chambers $\Delta_{\emptyset}^{\pm}$, the equivalence relation
$\sim$ is trivial, while on simplices in the boundary it is nontrivial.

Let $\phi:G\rightarrow G$ be an involution centralizing the Weyl group and
such that $\phi(B^{\pm})=B^{\mp}$. Then $\phi$ induces a twin building
involution as follows (for details see [DMGH09, Hor09])

(4) $\phi(gB^{\pm},\Delta_{J}^{\pm})=(\phi(g)B^{\mp},\Delta_{J}^{\mp})\,.$

One can give a geometric realization of a building as follows. Let
$\\{e_{1},\cdots,e_{\ell}\\}$ denote the standard orthonormal basis of
$\mathbb{R}^{\ell}$. Each $r$-simplex is identified with a copy of the
standard simplex

$\Delta^{r}:=\left\\{x=\sum_{i=1}^{r+1}a_{i}e_{i}\in\mathbb{R}^{r+1}\ \big{|}\
0\leq a_{i}\leq 1,\sum_{i=1}^{r+1}a_{i}=1\right\\}$

inherits the topology from $\mathbb{R}^{r+1}$. Appropriate identifications
must be made among the copies of these standard simplices in order to reflect
the incidence structure among the simplices in the building. For details see
[AB08].

For the buildings associated with the hyperbolic Kac-Moody groups we wish to
study, the geometric realization of apartments in the buildings
$\mathcal{B}^{+}$ and $\mathcal{B}^{-}$ can be chosen to be isometric to
hyperbolic spaces tesselated by the action of the hyperbolic Weyl group $W$.

###  2.4. Compact real forms of Kac–Moody algebras and groups

Let $\mathfrak{g}=\mathfrak{g}_{{\mathbb{C}}}(A)$ be a complex Kac–Moody
algebra and let $\mathfrak{h}$ be the standard Cartan subalgebra. The Cartan
involution

${\omega}_{0}:\mathfrak{g}\longrightarrow\mathfrak{g}$

is the automorphism of $\mathfrak{g}$ determined by
${\omega}_{0}(e_{i})=-f_{i}$, $\omega_{0}(f_{i})=-e_{i}$ and
${\omega}_{0}(h_{i})=-h_{i}$. Composing ${\omega}_{0}$ with complex
conjugation, we obtain a conjugate linear involution $\omega$, called the
Cartan-Chevalley involution. Then
$\mathfrak{k}=\\{x\in\mathfrak{g}\mid\omega(x)=x\\}$ is a Lie algebra over
$\mathbb{R}$ called the compact real form of $\mathfrak{g}$ ([Kum02] p. 243).

Note that $\omega_{0}$ and $\omega$ both centralize the Weyl group, so they
each induce twin building involutions via formula (4).

We may give generators for the compact real form $\mathfrak{k}$ as follows.
For each $j=1,\cdots,\ell$, let $\mathfrak{g}_{j}=\mathfrak{sl}_{2}^{j}$ be
the Lie subalgebra of $\mathfrak{g}$ isomorphic to
$\mathfrak{sl}_{2}({\mathbb{C}})$ with basis $\\{e_{j},f_{j},h_{j}\\}$, so
that $\mathfrak{g}$ is generated by the subalgebras $\mathfrak{g}_{j}$ and
$\omega(\mathfrak{g}_{j})=\mathfrak{g}_{j}$. Then the real Lie algebra of
fixed points of $\omega$ on $\mathfrak{g}_{j}$,
$\mathfrak{k}_{j}=\mathfrak{su}_{2}^{j}$ has basis

$x_{j}=\frac{1}{2}(e_{j}-f_{j}),\quad y_{j}=\frac{{\bf
i}}{2}(e_{j}+f_{j}),\quad z_{j}=\frac{{\bf i}}{2}(h_{j})$

with brackets $[x_{j},y_{j}]=z_{j}$, $[y_{j},z_{j}]=x_{j}$,
$[z_{j},x_{j}]=y_{j}$, and the compact real form $\mathfrak{k}$ is generated
by all of the subalgebras $\mathfrak{k}_{j}$, $j=1,\cdots,\ell$ (see
Proposition 1 in [Ber85]). A Cartan subalgebra in the compact real form
$\mathfrak{k}$ is an abelian subalgebra whose complexification is a Cartan
subalgebra in $\mathfrak{g}$. The standard Cartan subalgebra
$\mathfrak{t}=\mathfrak{h}\cap\mathfrak{k}$ in $\mathfrak{k}$ has real basis
$\\{z_{j}\ |\ 1\leq j\leq\ell\\}$.

Let $G=G_{\mathbb{C}}(A)$ be the complex adjoint Kac–Moody group associated to
$\mathfrak{g}$. The involution $\omega$ of $\mathfrak{g}$ lifts to a unique
involution of $G$, also denoted by $\omega$, exchanging positive and negative
real root groups since $\omega(\mathfrak{g}_{\alpha})=\mathfrak{g}_{-\alpha}$.
We have the following more general lemma about the action of any Lie algebra
automorphism, which we will apply to $\omega$ as well as to
$\widetilde{w}^{ad}_{i}\in\widetilde{W}^{ad}$.

######  Lemma 2.1.

For any $\alpha\in{\Phi}^{re}$, $e_{\alpha}\in\mathfrak{g}_{\alpha}$ and any
$\phi\in Aut(\mathfrak{g})$, we have $\phi\circ
exp(ad_{e_{\alpha}})\circ\phi^{-1}=exp(ad_{\phi(e_{\alpha})})$ so that, in
particular, $\omega U^{ad}_{\alpha}\omega^{-1}=U^{ad}_{-\alpha}$ and
$\widetilde{w}^{ad}_{i}U^{ad}_{\alpha_{i}}(\widetilde{w}^{ad}_{i})^{-1}=U^{ad}_{-\alpha_{i}}$
for $1\leq i\leq\ell$.

###### Proof.

For any $x\in\mathfrak{g}$, since $\phi$ is a Lie algebra automorphism, we
have

$\phi(exp(ad_{e_{\alpha}})x)=\phi\left(\sum_{k\geq
0}\frac{1}{k!}(ad_{e_{\alpha}})^{k}(x)\right)=\sum_{k\geq
0}\frac{1}{k!}(ad_{\phi(e_{\alpha})})^{k}(\phi(x))=exp(ad_{\phi(e_{\alpha})})(\phi(x))\,.$

The formula with $\phi=\omega$ gives $\omega
U^{ad}_{\alpha}\omega^{-1}=U^{ad}_{-\alpha}$ since
$\omega(\mathfrak{g}_{\alpha})=\mathfrak{g}_{-\alpha}$, and with
$\phi=\widetilde{w}^{ad}_{i}$ gives
$\widetilde{w}^{ad}_{i}U^{ad}_{\alpha_{i}}(\widetilde{w}^{ad}_{i})^{-1}=U^{ad}_{-\alpha_{i}}$
since $\widetilde{w}^{ad}_{i}(e_{i})=-f_{i}$. ∎

We set $K=Fix_{G}(\omega)$. Then $K$ is called the unitary form or compact
real form of $G$. We will use the latter by analogy with the finite
dimensional case, even though $K$ is not compact. The group $K$ is generated
by subgroups $K_{j}$ such that $\mathfrak{k}_{j}=Lie(K_{j})$ for each
$j=1,\dots,\ell$ ([Cap09], [Tit86], [KP85]).

For each $v=a+b{\bf i}\in{\mathbb{C}}$ and $1\leq j\leq\ell$ we write a
generator of $T_{\mathbb{C}}^{ad}$ as

$exp(ad_{vh_{j}})=exp(ad_{ah_{j}})exp(ad_{b{\bf
i}h_{j}})=exp(ad_{ah_{j}})exp(ad_{b2z_{j}})\,.$

This gives the decomposition $T_{\mathbb{C}}^{ad}=T_{\mathbb{R}}^{ad}\,T$
where

(5) $T_{\mathbb{R}}^{ad}=\langle exp(ad_{ah_{j}})\ |\ a\in\mathbb{R},1\leq
j\leq\ell\rangle$

is the split real torus and $T=\langle exp(ad_{bz_{j}})\ |\
b\in\mathbb{R},1\leq j\leq\ell\rangle$ is the compact real torus.

It is clear from the two expressions for $\widetilde{w}^{ad}_{i}$ that
$\widetilde{W}^{ad}\leq K$, but it is not so obvious that these operators can
be expressed as a single exponential
$\widetilde{w}^{ad}_{i}=exp(ad_{\pi\,x_{i}})$. This is a special case of the
formula $\widetilde{w}^{\pi}_{i}=exp(\pi(\pi\,x_{i}))$ for any integrable
representation $\pi:\mathfrak{g}\to End(V)$, where the inner $\pi$ is in
$\mathbb{R}$, proven in [FF16] and first found in certain important cases by
[DH09].

######  Proposition 2.2.

All Cartan subalgebras of $\mathfrak{k}$ are conjugate under the action of
$K$.

This result follows from [Cap09], Proposition 8.1 (iii). See also [KW92],
Corollary 5.33 and [KP87], Proposition 3.5.

##  3\. Tits cone and lightcone of hyperbolic Kac–Moody algebras of compact
type

Let $\mathfrak{g}=\mathfrak{g}_{\mathbb{C}}(A)$ be a Kac–Moody algebra and let
$\mathfrak{h}$ be its standard Cartan subalgebra. The split real form
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{g}_{\mathbb{R}}(A)$ of $\mathfrak{g}$
contains the $\mathbb{R}$–subalgebra
$\mathfrak{h}_{\mathbb{R}}=\mathfrak{h}\cap\mathfrak{g}_{\mathbb{R}}$ which is
just the $\mathbb{R}$–span of $\\{h_{i}\ \mid\ i\in I\\}$. We call
$\mathfrak{h}_{\mathbb{R}}$ the standard Cartan subalgebra of
$\mathfrak{g}_{\mathbb{R}}$.

The Weyl group $W$ has been defined above as the group of orthogonal
transformations of $\mathfrak{h}^{*}$ generated by the simple reflections. The
isomorphism between $\mathfrak{h}^{*}$ and $\mathfrak{h}$ gives the action of
$W$ on $\mathfrak{h}$, where the formula for simple reflections is just
$w_{i}(h_{j})=h_{j}-\alpha_{j}(h_{i})h_{i}$. This same formula restricted to
$\mathfrak{h}_{\mathbb{R}}$ gives the action of $W$ and it also determines the
action of $W$ on $\mathfrak{t}$ by
$w_{i}(z_{j})=z_{j}-\alpha_{j}(h_{i})z_{i}$. These operators are orthogonal
with respect to the restriction of the bilinear form $(\cdot,\cdot)$ to
$\mathfrak{h}_{\mathbb{R}}$ and to $\mathfrak{t}$, and therefore $W$ preserves
each surface of constant square length,
$(\mathfrak{h}_{\mathbb{R}})_{r}=\\{x\in\mathfrak{h}_{\mathbb{R}}\ |\
(x,x)=r\\}$ and $\mathfrak{t}_{r}=\\{x\in\mathfrak{t}\ |\ (x,x)=r\\}$.

Since we are assuming that the Cartan matrix $A$ is hyperbolic, the form
$(\cdot,\cdot)$ is Lorentzian on $\mathfrak{h}_{\mathbb{R}}$ and on
$\mathfrak{t}$. The surface where $r=0$ is called the nullcone, each surface
where $r<0$ is called timelike, and each surface where $r>0$ is called
spacelike. The set of all timelike points has two connected components, one
called forward and denoted $TL^{+}$ and the other called backward and denoted
$TL^{-}$. We have $TL^{-}=-TL^{+}$.

Each of these components is preserved by the linear action of $W$, which acts
consistently on rays, $Ray_{x}=\\{rx\ |\ 0<r\in\mathbb{R}\\}$ since
$w(rx)=rw(x)$. A fundamental domain for the action of $W$ on each of the
timelike components is defined by

$\mathcal{C}^{\pm}=\\{x\in TL^{\pm}\ |\ \alpha_{i}(x)\geq 0,1\leq
i\leq\ell\\}.$

The union

$X^{\pm}=\bigcup_{w\in W}w(\mathcal{C}^{\pm})$

is called the positive (respectively negative) Tits cone and $X=X^{+}\cup
X^{-}$ is called the Tits cone.

Clearly, we have $X^{-}=-X^{+}$. For $\mathfrak{g}$ hyperbolic,
$X^{\pm}\supseteq TL^{\pm}$, since it is possible that $\mathcal{C}^{\pm}$,
and therefore $X^{\pm}$, contains rays on the nullcone. This happens for the
rank $3$ hyperbolic Cartan matrices

$\begin{bmatrix}2&-2&0\\\ -2&2&-1\\\
0&-1&2\end{bmatrix}\qquad\hbox{and}\qquad\begin{bmatrix}2&-2&-2\\\ -2&2&-2\\\
-2&-2&2\end{bmatrix}$

corresponding to the hyperbolic Kac–Moody algebra $\mathcal{F}$ [FF83] whose
Weyl group is the hyperbolic triangle group $T(2,3,\infty)$, and the “ideal”
hyperbolic Kac–Moody algebra $\mathcal{I}$ whose Weyl group is the hyperbolic
triangle group $T(\infty,\infty,\infty)$, respectively. For $A$ strictly
hyperbolic, that is, whose principal minors are of finite type, we have
$X^{\pm}=TL^{\pm}$. Such an example is $E_{10}$.

We have the following description of the closure of the Tits cone (see
[Kac90], 5.10.2)

$\overline{X}=\overline{X^{+}}\cup\overline{X^{-}}=\\{h\in\mathfrak{h}_{\mathbb{R}}\mid(h,h)\leq
0\\}.$

We introduce the notations

$\mathcal{L}_{\mathfrak{h}_{\mathbb{R}}}=\\{h\in\mathfrak{h}_{\mathbb{R}}\mid(h,h)\leq
0\\},$
$\mathcal{L}_{\mathfrak{h}_{\mathbb{R}}}^{0}=\\{h\in\mathfrak{h}_{\mathbb{R}}\mid(h,h)<0\\}\
\hbox{ and }$
$\partial\mathcal{L}_{\mathfrak{h}_{\mathbb{R}}}=\\{h\in\mathfrak{h}_{\mathbb{R}}\mid(h,h)=0\\}.$

######  Proposition 3.1.

Let $\mathfrak{g}$ be a complex Kac–Moody algebra, $\mathfrak{k}$ its compact
real form and $\mathfrak{g}_{\mathbb{R}}$ its split real form. Let
$\mathfrak{h}$, $\mathfrak{h}_{\mathbb{R}}$, $\mathfrak{t}$ be the standard
Cartan subalgebras of $\mathfrak{g}$, $\mathfrak{g}_{\mathbb{R}}$,
$\mathfrak{k}$, respectively. Then
$\mathfrak{h}_{\mathbb{R}}\cong\mathbb{R}^{\ell-1,1}$ has signature
$(\ell-1,1)$ and

$\mathfrak{t}={{\bf i}}\mathfrak{h}_{\mathbb{R}},$

where ${{\bf i}}^{2}=-1$ and ${{\bf i}}\mathfrak{h}_{\mathbb{R}}=\\{{{\bf
i}}x\mid x\in\mathfrak{h}_{\mathbb{R}}\\}$. The signature of
$\mathfrak{t}={{\bf i}}\mathfrak{h}_{\mathbb{R}}\cong\mathbb{R}^{1,\ell-1}$ is
$(1,\ell-1)$.

We use a sign convention on $\mathfrak{k}$ adopted from the theory of finite
dimensional Riemannian symmetric spaces [Hel01] and set

(6) $(\cdot,\cdot)_{\mathfrak{k}}=-(\cdot,\cdot)|_{\mathfrak{k}}.$

Note that in the affine case $(\cdot,\cdot)_{\mathfrak{k}}$, this sign
convention naturally occurs in the loop group realizations [PS86, Fre09]. With
respect to $(\cdot,\cdot)_{\mathfrak{k}}$, the Cartan subalgebra
$\mathfrak{t}$ has Lorentzian signature $(\ell-1,1)$.

######  Lemma 3.2.

The invariant symmetric bilinear form $(\cdot,\cdot)_{\mathfrak{k}}$ on
$\mathfrak{k}$ is Lorentzian with signature $(\infty,1)$.

###### Proof.

As a vector space, a complex hyperbolic Kac–Moody algebra $\mathfrak{g}$ has a
basis consisting of the Cartan subalgebra generators, $h_{i},i=1,\dots,\ell$,
and certain multibrackets $[e_{i_{1}},e_{i_{2}},\cdots,e_{i_{n}}]$ and
$[f_{i_{1}},f_{i_{2}},\cdots,f_{i_{n}}]$. The action of $\omega$ on such
multibrackets is simply
$\omega([e_{i_{1}},e_{i_{2}},\cdots,e_{i_{n}}])=(-1)^{n}[f_{i_{1}},f_{i_{2}},\cdots,f_{i_{n}}]$
and $\omega(h_{i})=-h_{i}$ so an $\mathbb{R}$-basis for the compact real form
$\mathfrak{k}$ consists of the compact Cartan basis elements $z_{i}=\frac{{\bf
i}}{2}h_{i},i=1,\dots,\ell$, and the elements obtained from basis
multibrackets above

$\frac{1}{2}([e_{i_{1}},e_{i_{2}},\cdots,e_{i_{n}}]+[f_{i_{1}},f_{i_{2}},\cdots,f_{i_{n}}])\hbox{
and }\frac{{\bf
i}}{2}([e_{i_{1}},e_{i_{2}},\cdots,e_{i_{n}}]-[f_{i_{1}},f_{i_{2}},\cdots,f_{i_{n}}])\
\hbox{ for }n\hbox{ even},$
$\frac{1}{2}([e_{i_{1}},e_{i_{2}},\cdots,e_{i_{n}}]-[f_{i_{1}},f_{i_{2}},\cdots,f_{i_{n}}])\hbox{
and }\frac{{\bf
i}}{2}([e_{i_{1}},e_{i_{2}},\cdots,e_{i_{n}}]+[f_{i_{1}},f_{i_{2}},\cdots,f_{i_{n}}])\
\hbox{ for }n\hbox{ odd}.$

In particular, for $n=1$ these include the elements
$x_{i}=\frac{1}{2}(e_{i}-f_{i})$ and $y_{i}=\frac{{\bf i}}{2}(e_{i}+f_{i})$.

Following our sign convention, the Cartan subalgebra $\mathfrak{t}$ spanned by
the elements $z_{i},i=1,\dots,\ell$, has signature $(\ell-1,1)$. Using our
sign convention and the characterization of the $ad$–invariant scalar product
in [Kac90], Equation 2.2.1, applied to the basis vectors $y_{i}$ and $z_{i}$,
we find that the bilinear form on each subalgebra
$\mathfrak{k}_{i}=\mathfrak{su}_{2}^{i}$ is positive definite.

Furthermore the $ad$–invariant bilinear form has the following properties on
root spaces (see [Kac90], Sections 2.1 and 2.2). The root spaces
$\mathfrak{g}_{\alpha}$ and $\mathfrak{g}_{-\alpha}$, which are interchanged
by $\omega$, have dual bases

$\\{e_{\alpha}^{j}\ |\ 1\leq j\leq\textrm{dim}(\mathfrak{g}_{\alpha})\\}\hbox{
and }\\{f_{\alpha}^{j}=e_{-\alpha}^{j}\ |\ 1\leq
j\leq\textrm{dim}(\mathfrak{g}_{\alpha})\\},\hbox{ respectively}$

such that

$(f_{\alpha}^{j},e_{\beta}^{m})=\delta_{j,m}\delta_{\alpha,\beta}.$

As a consequence, for positive roots $\alpha$, the basis elements of
$\mathfrak{k}$, $x_{\alpha}^{j}=\frac{1}{2}(e_{\alpha}^{j}-f_{\alpha}^{j})$
and $y_{\alpha}^{j}=\frac{{{\bf i}}}{2}(e_{\alpha}^{j}+f_{\alpha}^{j})$
satisfy

$(x_{\alpha}^{j},x_{\beta}^{m})=-\frac{1}{2}\delta_{j,m}\delta_{\alpha,\beta}=(y_{\alpha}^{j},y_{\beta}^{m})\hbox{
and }(x_{\alpha}^{j},y_{\beta}^{m})=0.$

Hence it follows from (6) that

$(x_{\alpha}^{j},x_{\beta}^{m})_{\mathfrak{k}}=\frac{1}{2}\delta_{j,m}\delta_{\alpha,\beta}=(y_{\alpha}^{j},y_{\beta}^{m})_{\mathfrak{k}}\hbox{
and }(x_{\alpha}^{j},y_{\beta}^{m})_{\mathfrak{k}}=0.$

The Cartan subalgebra has signature $(\ell-1,1)$ and for each positive root
$\alpha$ the subspaces

$\langle x_{\alpha}^{j}\ |\ 1\leq
j\leq\textrm{dim}(\mathfrak{g}_{\alpha})\rangle\hbox{ and }\langle
y_{\alpha}^{j}\ |\ 1\leq j\leq\textrm{dim}(\mathfrak{g}_{\alpha})\rangle$

in $\mathfrak{k}$ each have positive signature and are orthogonal to each
other. For different $\alpha$ these spaces are orthogonal to each other and to
the Cartan subalgebra $\mathfrak{t}$. Thus $\mathfrak{k}$ has Lorentzian
signature $(\infty,1)$. ∎

This could also be deduced from [RC89], Section 2.1.

######  Remark 3.3.

Note that the bilinear form $(\cdot,\cdot)$ on $\mathfrak{g}_{\mathbb{R}}$ is
indefinite with signature $(\infty,\infty)$ because, while the split Cartan
$\mathfrak{h}_{\mathbb{R}}$ has the signature $(\ell-1,1)$, each pair of dual
root vectors $\\{e_{\alpha}^{j},f_{\alpha}^{j}\\}$ forms a hyperbolic plane,
and for different positive $\alpha$ and distinct $j$ these planes are
orthogonal.

We recall that the $ad$–invariant bilinear form on $\mathfrak{h}_{\mathbb{R}}$
extends ${\mathbb{C}}$–bilinearly to the complexification
$\mathfrak{h}_{{\mathbb{C}}}$. In particular, for
$x\in\mathfrak{h}_{\mathbb{R}}$ we have

(7) $({{\bf i}}x,{{\bf i}}x)=-(x,x)=(x,x)_{\mathfrak{k}}$

so the map $\varphi:\mathfrak{h}_{\mathbb{R}}\longrightarrow\mathfrak{t}$
given by $\varphi(x)={{\bf i}}x$ is an isometry. In addition, for $w\in W$ we
have $\varphi(wx)=w\varphi(x)$, hence $\varphi$ is $W$-equivariant.

We introduce notations for certain subsets in $\mathfrak{k}$:

$\mathcal{L}_{\mathfrak{k}}=\\{x\in\mathfrak{k}\mid(x,x)_{\mathfrak{k}}\leq
0\\},\quad\mathcal{L}_{\mathfrak{t}}=\mathcal{L}_{\mathfrak{k}}\cap\mathfrak{t}=\varphi(\mathcal{L}_{\mathfrak{h}_{\mathbb{R}}}),$
$\mathcal{L}_{\mathfrak{k}}^{0}=\\{x\in\mathfrak{k}\mid(x,x)_{\mathfrak{k}}<0\\},\quad\mathcal{L}_{\mathfrak{t}}^{0}=\mathcal{L}_{\mathfrak{k}}^{0}\cap\mathfrak{t}=\varphi(\mathcal{L}_{\mathfrak{h}_{\mathbb{R}}}^{0}),$
$\partial\mathcal{L}_{\mathfrak{k}}=\\{x\in\mathfrak{k}\mid(x,x)_{\mathfrak{k}}=0\\},\quad\partial\mathcal{L}_{\mathfrak{t}}=\partial\mathcal{L}_{\mathfrak{k}}\cap\mathfrak{t}=\varphi(\partial\mathcal{L}_{\mathfrak{h}_{\mathbb{R}}}).$

The ones inside $\mathfrak{t}$ are related by the $W$-invariant isometry
$\varphi$ to the corresponding subsets defined earlier in
$\mathfrak{h}_{\mathbb{R}}$. Furthermore, we have

$\varphi(TL^{\pm})=TL_{\mathfrak{t}}^{\pm},\quad\varphi(\mathcal{C}^{\pm})=\mathcal{C}_{\mathfrak{t}}^{\pm},\quad\varphi(X^{\pm})=X_{\mathfrak{t}}^{\pm},\quad\varphi(\overline{X}^{\pm})=\overline{X}_{\mathfrak{t}}^{\pm},\quad\varphi(\overline{X})=\overline{X}_{\mathfrak{t}}$

which correspond to those subsets defined earlier in
$\mathfrak{h}_{\mathbb{R}}$. We call $X_{\mathfrak{t}}$ the Tits cone of
$\mathfrak{t}$ and note that
$\overline{X}_{\mathfrak{t}}=\mathcal{L}_{\mathfrak{t}}$.

######  Remark 3.4.

We will refer to $\mathcal{L}_{\mathfrak{k}}$ as the lightcone of
$\mathfrak{k}$ and $\mathcal{L}_{\mathfrak{t}}$ as the lightcone of
$\mathfrak{t}$.

##  4\. Group actions of the compact real form $K$

###  4.1. The Adjoint action of $K$ on $\mathfrak{k}$

Recall that $K$ denotes the compact form of the complex adjoint Kac–Moody
group $G$. We set

$\mathfrak{H}=\bigcup_{k\in
K}k\mathfrak{t}k^{-1}=\\{kxk^{-1}\in\mathfrak{k}\mid k\in
K,x\in\mathfrak{t}\\}\,.$

By Proposition 2.2 all Cartan subalgebras of $\mathfrak{k}$ are $K$–conjugate,
thus the definition of $\mathfrak{H}$ is independent of the choice of
$\mathfrak{t}$.

######  Proposition 4.1.

Let $\mathfrak{t}$ be any Cartan subalgebra of $\mathfrak{k}$ and let $z={\bf
i}h\in\mathfrak{t}$ for $h\in\mathfrak{h}_{\mathbb{R}}$ satisfying
$\alpha(h)\neq 0$ for all $\alpha\in\Phi^{+}$. Then the subspace
$[\mathfrak{k},z]$ of $\mathfrak{k}$ has basis

$\bigcup_{\alpha\in\Phi^{+}}\\{x_{\alpha}^{j}\ |\ 1\leq
j\leq\textrm{dim}(\mathfrak{g}_{\alpha})\\}\cup\\{y_{\alpha}^{j}\ |\ 1\leq
j\leq\textrm{dim}(\mathfrak{g}_{\alpha})\\}\,.$

###### Proof.

For $\alpha\in\Phi^{+}$ and $1\leq j\leq\textrm{dim}(\mathfrak{g}_{\alpha})$
we have

$[x_{\alpha}^{j},z]=-[{\bf
i}h,\textrm{\scriptsize$\frac{1}{2}$\normalsize}(e_{\alpha}^{j}-f_{\alpha}^{j})]=-\textrm{\scriptsize$\frac{{\bf
i}}{2}$\normalsize}[h,(e_{\alpha}^{j}-f_{\alpha}^{j}))]=-\textrm{\scriptsize$\frac{{\bf
i}}{2}$\normalsize}\alpha(h)(e_{\alpha}^{j}+f_{\alpha}^{j})=-\alpha(h)y_{\alpha}^{j}\,,$
$[y_{\alpha}^{j},z]=-[{\bf i}h,\textrm{\scriptsize$\frac{{\bf
i}}{2}$\normalsize}(e_{\alpha}^{j}+f_{\alpha}^{j})]=\textrm{\scriptsize$\frac{1}{2}$\normalsize}[h,(e_{\alpha}^{j}+f_{\alpha}^{j}))]=\textrm{\scriptsize$\frac{1}{2}$\normalsize}\alpha(h)(e_{\alpha}^{j}-f_{\alpha}^{j})=\alpha(h)x_{\alpha}^{j}$

and $[\mathfrak{t},z]=0$, so no basis vectors of $\mathfrak{t}$ are in
$[\mathfrak{k},z]$. ∎

By analogy with the finite dimensional and affine cases, the following
proposition shows that in the hyperbolic case, the $K$–orbits, $K\cdot
z=\\{kzk^{-1}\in\mathfrak{k}\mid k\in K\\}$ for each $z\in\mathfrak{t}$,
intersect each Cartan subalgebra orthogonally. For $z\in\mathfrak{t}$ let
$T_{z}(K\cdot z)$ be the tangent space of the submanifold $K\cdot z$ at the
point $z$.

######  Proposition 4.2.

Let $\mathfrak{t}$ be any Cartan subalgebra of $\mathfrak{k}$. The orbits
$K\cdot z$ for $z\in\mathfrak{t}$ are orthogonal to $\mathfrak{t}$ with
respect to the $ad$–invariant bilinear form $(\cdot,\cdot)_{\mathfrak{k}}$,
that is, $(T_{z}(K\cdot z),\mathfrak{t})_{\mathfrak{k}}=0$.

###### Proof.

For $z\in\mathfrak{t}$ we have

$T_{z}(K\cdot z)=[\mathfrak{k},z]\,.$

By definition, for $w\in T_{z}(K\cdot z)$, there is some $y\in\mathfrak{k}$
such that $w=[y,z]$. Since the form $(\cdot,\cdot)_{\mathfrak{k}}$ is
$ad$–invariant, for $z^{\prime}\in\mathfrak{t}$ we obtain

$(w,z^{\prime})_{\mathfrak{k}}=([y,z],z^{\prime})_{\mathfrak{k}}=(y,[z,z^{\prime}])_{\mathfrak{k}}=(y,0)_{\mathfrak{k}}=0.$

∎

Let $X\subset V$ be a subset of a real vector space $V$ equipped with a
bilinear form $(\cdot,\cdot)$. For any real number $r$ we define a ‘sphere’ of
radius $r$ in $X$ by

$X_{r}:=\\{x\in X\ |\ (x,x)=r\\}\,.$

If $(\cdot,\cdot)$ is Lorentzian and $r<0$ then $X_{r}$ is a two-sheeted
hyperboloid of constant curvature $\kappa=-\frac{1}{r^{2}}$ whose connected
components we call $X_{r}^{+}$ and $X_{r}^{-}$. In particular we will use
$\mathfrak{t}_{r}\subset\mathcal{L}_{\mathfrak{t}}^{0}\subset\mathfrak{t}$,
$\mathfrak{k}_{r}\subset\mathcal{L}_{\mathfrak{k}}^{0}\subset\mathfrak{k}$ and
$\mathfrak{H}_{r}\subset\mathfrak{H}$ and note that
$\mathfrak{k}_{0}=\partial\mathcal{L}_{\mathfrak{k}}$ and
$\mathfrak{t}_{0}=\partial\mathcal{L}_{\mathfrak{t}}$. For $\ell>2$ we find
$\mathfrak{t}_{-1}^{+}\cong\mathfrak{t}_{-1}^{-}\cong\mathbb{H}^{\ell-1}$,
$(\ell-1)$–dimensional hyperbolic space.

######  Definition 4.3 (Lightlike closure).

Let $V$ be a possibly infinite dimensional real vector space equipped with a
Lorentzian form $(\cdot,\cdot)$ and let $\partial\mathcal{L}_{V}=\\{x\in V\ |\
(x,x)=0\\}$ denote its nullcone. The boundary at infinity
$B^{\infty}(\partial\mathcal{L}_{V})$ of the nullcone consists of all rays
$Ray_{x}$ for $0\neq x\in\partial\mathcal{L}_{V}$. If $dim(V)=2$ it consists
of four points. If $dim(V)>2$ it consists of two components, corresponding to
the future timelike boundary $B^{\infty}(\partial\mathcal{L}_{V})^{+}$ and the
past timelike boundary $B^{\infty}(\partial\mathcal{L}_{V})^{-}$, each of
which can be identified with a sphere of dimension $\ell-2$ We define the
lightlike closure of $V$ by

$\widehat{V}=V\cup B^{\infty}(\partial\mathcal{L}_{V})\,.$

For each $r<0$, define $B^{\infty}(V_{r}^{\pm})$, the boundary at infinity of
$V_{r}^{\pm}$, to be equivalence classes of geodesic rays, where two rays are
equivalent if the distance between them is finite at all points. For details,
see [Ebe96]. Note that for each $r<0$, $B^{\infty}(V_{r}^{\pm})$ can be
identified with $B^{\infty}(\partial\mathcal{L}_{V})^{\pm}$. Using this
observation we define the lightlike closure of $V_{r}^{\pm}$ to be

(8) $\widehat{V}_{r}^{\pm}=V_{r}^{\pm}\cup
B^{\infty}(\partial\mathcal{L}_{V})^{\pm}\,.$

Since the bilinear form $(\cdot\ ,\ \cdot)_{\mathfrak{k}}$ is $ad$–invariant,
the Adjoint action of $K$ on $\mathfrak{k}$ preserves the surfaces
$\mathfrak{H}_{r}\subset\mathfrak{H}$. For $k\in K$ we have
$k\cdot\mathfrak{t}_{r}=(k\cdot\mathfrak{t})_{r}$. There is an induced action
of $K$ on $B^{\infty}(\partial\mathcal{L}_{\mathfrak{t}})$ and for each $r<0$
on $B^{\infty}(\partial\mathcal{L}_{\mathfrak{t}_{r}})^{\pm}$, as well as on
${\widehat{\mathfrak{t}}}$ and on ${\widehat{\mathfrak{t}}}_{r}^{\pm}$.

Hence, for each $r<0$, the Adjoint action of $K$ on $\mathfrak{k}$ induces a
well–defined action on the surface:

$K:\mathfrak{H}_{r}\longrightarrow\mathfrak{H}_{r},\qquad k\cdot
x=Ad_{k}(x)=kxk^{-1}$

as well as on the lightlike closure $\widehat{\mathfrak{H}}_{r}$.

Since all Cartan subalgebras are conjugate, we can define the following
surjective map

$\psi:K\times\mathfrak{t}_{r}\longrightarrow\mathfrak{H}_{r},\qquad\psi(k,x)=kxk^{-1}\,.$

Note that the choice of the standard Cartan subalgebra $\mathfrak{t}$ in the
definition of $\psi$ does not restrict the generality.

Let $T=exp(\mathfrak{t})=T(G^{ad})\cap K$ be the torus associated to
$\mathfrak{t}$ and let

$N=N_{K}(T)=N_{G^{ad}}(T(G^{ad}))\cap K$

be the normalizer of $T$ in $K$. For $k\in K$, $t\in T$ and $u\in\mathfrak{t}$
we have:

$Ad_{kt}(u)=ktut^{-1}k^{-1}=kuk^{-1}=Ad_{k}(u).$

Thus the Adjoint action on $\mathfrak{t}$ is $T$–invariant and factors to the
quotient space $K/T$ yielding a surjective map

$\psi:K/T\times\mathfrak{t}_{r}\longrightarrow\mathfrak{H}_{r}\,,\qquad\psi(kT,u)=kuk^{-1}\,.$

For any Cartan subalgebra $\mathfrak{t}^{\prime}=k\mathfrak{t}k^{-1}$ of
$\mathfrak{k}$, $\mathcal{L}_{\mathfrak{t}^{\prime}}$ coincides with the
closure of the Tits cone of $\mathfrak{t}^{\prime}$,
${\overline{X}}_{\mathfrak{t}^{\prime}}$. Hence
${\overline{X}}_{\mathfrak{t}^{\prime}}$ is the closure of the cone
$\\{su\in\mathfrak{t}^{\prime}\ |\ s>0,u\in\mathfrak{t}^{\prime}_{r}\\}$, for
any fixed $r<0$, which includes the boundary
$\partial\mathcal{L}_{\mathfrak{t}^{\prime}}$. We distinguish two cases:

  1. (1)

When every proper Cartan submatrix of $A$ is of finite type (i.e. $A$ is
$(\ell-1)$-spherical or strictly hyperbolic) then we have
$X_{\mathfrak{t}^{\prime}}=\mathcal{L}_{\mathfrak{t}^{\prime}}^{0}$ and there
is a bijection between each surface $(\mathfrak{t}^{\prime}_{r})^{\pm}$ for
$r<0$ and the set of rays in $\mathcal{L}_{\mathfrak{t}^{\prime}}^{0}$. In
this case we do not need to consider the lightlike closure
$\widehat{\mathfrak{H}}_{r}$ in order to embed the building in it.

  2. (2)

When $A$ contains an affine Cartan submatrix the fundamental chambers
$\mathcal{C}^{\pm}$ contain rays that accumulate at rays on the lightcone
$\partial\mathcal{L}_{\mathfrak{t}^{\prime}}$. So the corresponding points on
the surface $(\mathfrak{t}^{\prime}_{-1})^{\pm}$ accumulate at points on the
boundary of the $(\ell-1)$–dimensional hyperbolic space $\mathbb{H}^{\ell-1}$.
Therefore, in these cases we do have to consider the lightlike closure in
order to embed the building. In the example of $\mathcal{F}$, the surfaces
$(\mathfrak{t}^{\prime}_{-1})^{\pm}\cong\mathbb{H}^{2}$ are isometric to the
Poincaré disk, and the tesselation by the hyperbolic Weyl group
$T(2,3,\infty)$ includes chambers which have an ideal vertex on the boundary.
In such an example, the building would have $0$-simplices corresponding to
those ideal vertices, so to achieve our goal of embedding the twin building of
a hyperbolic algebra inside the lightcone of $\mathfrak{k}$, we must use
$\widehat{\mathfrak{H}}_{r}$.

Since $W=N/T\leq K/T$, the restriction to $\mathfrak{t}_{r}$ of the Adjoint
action of $N$ on $\mathfrak{k}_{r}$ coincides with the Weyl group action of
$W$ on $\mathfrak{t}_{r}$.

Restriction of the second coordinate of the domain of $\psi$ to either
fundamental domain $\mathcal{C}_{r}^{\pm}$ for the action of $W$ on
$\mathfrak{t}_{r}$ gives the surjective maps

(9)
$\psi^{\pm}:K/T\times\mathcal{C}_{r}^{\pm}\longrightarrow\mathfrak{H}^{\pm}_{r},\qquad\psi^{\pm}(kT,u)=kuk^{-1}\,.$

###  4.2. The local structure of the Adjoint action

In this section we describe the geometry ‘close’ to a fixed Cartan subalgebra
in the compact real form $\mathfrak{k}$. We show that for
$i\in\\{1,\dots,{\ell}\\}$, the Adjoint action of the fundamental
$SU(2)_{i}$–subgroup of the compact real form $K$, is a rotation of the
standard Cartan subalgebra around the hyperplane

$L_{\mathfrak{t},i}:=\\{z\in\mathfrak{t}\ |\ \alpha_{i}(z)=0\\}$

fixed by the generator $w_{i}$ of $W$.

The standard Cartan subalgebra is given by
$\mathfrak{t}=\mathbb{R}z_{1}+\dots+\mathbb{R}z_{\ell}$. For $1\leq i\leq\ell$
and $s,t\in\mathbb{R}$, we have $exp(sx_{i}+ty_{i})\in SU(2)_{i}\leq K$ and
for $z\in\mathfrak{t}$ we have $[sx_{i}+ty_{i},z]=-{\bf
i}\alpha_{i}(z)(tx_{i}-sy_{i})=0$ when $\alpha_{i}(z)=0$ which means
$exp(sx_{i}+ty_{i})(z)=z$ when $\alpha_{i}(z)=0$. So $L_{\mathfrak{t},i}$ is
the fixed point set in $\mathfrak{t}$ of $exp(ad_{sx_{i}+ty_{i}})$. For each
$s,t\in\mathbb{R}$,
$exp(ad_{sx_{i}+ty_{i}})\mathfrak{t}=\mathfrak{t}^{i}(s,t)$ is either another
Cartan subalgebra such that
$\mathfrak{t}\cap\mathfrak{t}^{i}(s,t)=L_{\mathfrak{t},i}$ or else
$\mathfrak{t}=\mathfrak{t}^{i}(s,t)$. Note that with
$v=(\frac{s}{2}+\frac{t{\bf i}}{2})\in{\mathbb{C}}$ we have
$sx_{i}+ty_{i}=ve_{i}-\overline{v}f_{i}$, so

$exp(ad_{sx_{i}+ty_{i}})=exp(ad_{ve_{i}-\overline{v}f_{i}})\in(\mathfrak{su}_{2})_{i}$

gives another parameterization of $\\{\mathfrak{t}^{i}(s,t)\ |\
s,t\in\mathbb{R}\\}=\\{exp(ad_{ve_{i}-\overline{v}f_{i}})\mathfrak{t}\ |\
v\in{\mathbb{C}}\\}$. For each $1\leq i\leq\ell$ the family of distinct Cartan
subalgebras obtained this way can be parameterized by a 2–sphere with
antipodes identified, as we will show using the first part of the following
theorem.

######  Theorem 4.4.

( [FF16]) For any $s,t\in\mathbb{R}$ such that $0<r^{2}=s^{2}+t^{2}$ and for
any $z\in\mathfrak{t}$, we have

  1. (1)

$\displaystyle exp(ad_{sx_{i}+ty_{i}})z=z-{{\bf
i}}\alpha_{i}(z)(cos(r)-1)z_{i}-{{\bf
i}}\alpha_{i}(z)\frac{sin(r)}{r}(tx_{i}-sy_{i})$,

  2. (2)

$\displaystyle exp(ad_{sx_{i}+ty_{i}})x_{i}=x_{i}-\frac{t\
sin(r)}{r}z_{i}-\frac{t}{r^{2}}(cos(r)-1)(tx_{i}-sy_{i})$,

  3. (3)

$\displaystyle exp(ad_{sx_{i}+ty_{i}})y_{i}=y_{i}+\frac{s\
sin(r)}{r}z_{i}-\frac{s}{r^{2}}(cos(r)-1)(tx_{i}-sy_{i})$.

###### Proof.

We prove only the first relation since the others are analogous. We have

$\begin{array}[]{lllll}(ad_{sx_{i}+ty_{i}})^{1}z&=&[sx_{i}+ty_{i},z]&=&-{\bf
i}\alpha_{i}(z)(tx_{i}-sy_{i})\,,\\\
(ad_{sx_{i}+ty_{i}})^{2}z&=&[sx_{i}+ty_{i},-{\bf
i}\alpha_{i}(z)(tx_{i}-sy_{i})]&=&{\bf i}\alpha_{i}(z)(s^{2}+t^{2})z_{i}\,,\\\
(ad_{sx_{i}+ty_{i}})^{3}z&=&[sx_{i}+ty_{i},\alpha_{i}(z)(s^{2}+t^{2})z_{i}]&=&{\bf
i}\alpha_{i}(z)(s^{2}+t^{2})(tx_{i}-sy_{i})\,,\\\
(ad_{sx_{i}+ty_{i}})^{4}z&=&[sx_{i}+ty_{i},{\bf
i}\alpha_{i}(z)(s^{2}+t^{2})(tx_{i}-sy_{i})]&=&-{\bf
i}\alpha_{i}(z)(s^{2}+t^{2})^{2}z_{i}\,.\end{array}$

Using $r^{2}=s^{2}+t^{2}\neq 0$, it is clear that for $n\geq 1$ we have

$(ad_{sx_{i}+ty_{i}})^{2n}z=-{\bf i}\alpha_{i}(z)(-1)^{n}(r^{2})^{n}z_{i}$

and for $n\geq 0$ we have

$(ad_{sx_{i}+ty_{i}})^{2n+1}z=-{\bf
i}\alpha_{i}(z)(-1)^{n}(r^{2})^{n}(tx_{i}-sy_{i})$

so we get

$exp(ad_{sx_{i}+ty_{i}})z=z-{\bf
i}\alpha_{i}(z)\sum_{n=1}^{\infty}\frac{(-1)^{n}r^{2n}}{(2n)!}z_{i}-{\bf
i}\alpha_{i}(z)\sum_{n=0}^{\infty}\frac{(-1)^{n}r^{2n}}{(2n+1)!}(tx_{i}-sy_{i})$
$=z-{\bf i}\alpha_{i}(z)(cos(r)-1)z_{i}-{\bf
i}\alpha_{i}(z)\frac{sin(r)}{r}(tx_{i}-sy_{i}).$

∎

######  Corollary 4.5.

( [FF16]) For each $1\leq i\leq\ell$, the family of distinct Cartan
subalgebras $\\{\mathfrak{t}^{i}(s,t)\ |\ s,t\in\mathbb{R}\\}$, including
$\mathfrak{t}$, is parametrized by the unit hemisphere

$\\{(sin(r)sin(\psi),-sin(r)cos(\psi),cos(r))\in\mathbb{R}^{3}\ |\ 0\leq
r<\pi,0\leq\psi<\pi\\}$

where $r=\sqrt{s^{2}+t^{2}}\geq 0$ and $\psi$ is defined when $r>0$ by

$sin(\psi)=\frac{t}{r}\quad\hbox{and}\quad cos(\psi)=\frac{s}{r}\,.$

###### Proof.

The Cartan subalgebra $\mathfrak{t}^{i}(s,t)$ is spanned by
$L_{\mathfrak{t},i}$ and

$exp(ad_{sx_{i}+ty_{i}})z_{i}=sin(r)(sin(\psi)x_{i}-cos(\psi)y_{i})+cos(r)z_{i}$

from theorem 4.4 (1) and the fact that $\alpha_{i}(z_{i})\neq 0$. Since for
$0\leq r<\pi$ and $0\leq\psi<\pi$ no two vectors of the above form are
colinear, all of the corresponding subspaces $\mathfrak{t}^{i}(s,t)$ are
distinct. But for antipodal points on the unit sphere, $(r,\psi)$ and
$(\pi-r,\psi+\pi)$, the corresponding subspaces are the same. ∎

###  4.3. The action of the compact real form on the twin building

To describe the action of $K$ on the twin building, we use the Iwasawa
decomposition [DMGH09] which yields

$G=KAU^{\pm},$

where $G$ denotes a complex Kac–Moody group, $K$ denotes the compact real form
of $G$, $A\cong\mathbb{R}^{rank(G)}$ is an abelian subgroup and $U^{\pm}$ is
the group generated by all positive (respectively negative) real root groups.
Recall that $\mathfrak{t}$ denotes a Cartan subalgebra in $\mathfrak{k}$, and
T$=exp(\mathfrak{t})$ its torus. Using $T_{\mathbb{C}}(G)=TA$, the
decomposition $B^{\pm}=T_{\mathbb{C}}(G)U^{\pm}$ and the Iwasawa
decomposition, we have

(10) $G/B^{\pm}\cong K/T\,.$

Recalling the description of the building from equation (3) in section 2.3

(11) $\mathcal{B}^{\pm}=\left(G/B^{\pm}\times\Delta^{\pm}\right)/\sim\,,$

we obtain an equivalent new description by equation (10):

(12) $\mathcal{B}^{\pm}=\left(K/T\times\Delta^{\pm}\right)/\sim\,.$

In the description of the twin building
$\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ given by equation (11), the
natural action of $G$ via left multiplication is apparent, while in the
description of $\mathcal{B}$ as in equation (12) the $G$-symmetry is broken to
$K$–symmetry. Note that in equation (11) the cosets of the two opposite
buildings $\mathcal{B}^{+}$ and $\mathcal{B}^{-}$ are defined with respect to
different subgroups $B^{+}$ and $B^{-}$ respectively. Hence there are
subgroups of $G$ which act differently on $\mathcal{B}^{+}$ and
$\mathcal{B}^{-}$. For example, for any $g\in G$ the coset $gB^{+}\in G/B^{+}$
is fixed by all elements in the subgroup $B^{+}_{g}=gB^{+}g^{-1}$, but that
subgroup acts transitively on $\\{fB^{-}\in G/B^{-}\ |\ g^{-1}f\in
B^{+}B^{-}\\}$ which certainly contains $gB^{-}$. But the description of
$\mathcal{B}^{\pm}$ in equation (12) shows that the action of $K$ is the same
in both. In particular this means that the group $K$ does not act transitively
on any apartment system. More precisely, for $A\in\mathcal{B}^{\pm}$ an
apartment, define the orbit $\mathcal{A}_{K}(A):=K\cdot A$. Then for each
chamber $c\in\mathcal{B}^{\pm}$ there is exactly one apartment
$A_{c}\in\mathcal{A}_{K}(A)$ such that $c\in A_{c}$.

An apartment $A$ is called $\omega$–stable iff $\omega(A)=A$. If $A$ is
$\omega$–stable, the set $\mathcal{A}_{K}(A)$ contains all $\omega$–stable
apartments.

Let $c=(fB^{\pm},\Delta_{\emptyset}^{\pm})$ be any chamber in
$\mathcal{B}^{\pm}=\left(G/B^{\pm}\times\Delta^{\pm}\right)/\sim$, and let
$\mathcal{C}ham(\mathcal{B}^{\pm})$ denote the set of all chambers of the
building $\mathcal{B}^{\pm}$. For $i\in I$, the $i$-wall of $c$ is
$(fB^{\pm},\Delta_{i}^{\pm})$ and the $i$–residue of $c$ is defined to be
$R_{i}(c)=\\{d=(gB^{\pm},\Delta_{\emptyset}^{\pm})\in\mathcal{C}ham(\mathcal{B}^{\pm})\
|\ (fB^{\pm},\Delta_{i}^{\pm})\sim(gB^{\pm},\Delta_{i}^{\pm})\\}$. Then we
have $R_{i}(c)\cong{\mathbb{P}}^{1}({\mathbb{C}})$. Identifying
$\mathbb{P}^{1}({\mathbb{C}})$ with the Riemann sphere
$\widehat{{\mathbb{C}}}={\mathbb{C}}\cup\infty$, the action of the subgroup
$SU(2)_{i}$ on $R_{i}(c)$ can be identified with the action of $SU(2)$ on
$\widehat{{\mathbb{C}}}$ by Möbius transformations. Additional details may be
found in [AB08], Chapter 6.

##  5\. Simplicial structure on $\mathfrak{H}_{r}$

In this section we define for each $r<0$ a simplicial structure on the set
$\mathfrak{H}_{r}\subset\mathfrak{k}_{r}$. We begin in the standard Cartan
subalgebra with $\mathfrak{t}_{r}\subset\mathfrak{t}$. Recall, that
$\mathfrak{t}_{r}=\mathfrak{t}_{r}^{+}\cup\mathfrak{t}_{r}^{-}$ is (up to
rescaling) isometric to a pair of hyperbolic spaces (for $\ell>2$) which are
both tesselated by the action of the Weyl group $W$. We call an element
$x\in\mathfrak{t}_{r}$ singular if it is fixed by a Weyl group element $w\neq
1$ and call $x$ regular otherwise. Let $\mathfrak{t}^{sin}_{r}$ be the set of
all singular elements in $\mathfrak{t}_{r}$ and let $\mathfrak{t}^{reg}_{r}$
be the set of all regular elements, and similarly we have sets
$(\mathfrak{t}^{sin}_{r})^{\pm}$ and $(\mathfrak{t}^{reg}_{r})^{\pm}$. Then we
have the decomposition

$\mathfrak{t}_{r}=\mathfrak{t}^{sin}_{r}\cup\mathfrak{t}^{reg}_{r}\,.$

We denote by $Comp_{r}^{\pm}$ the set of connected components of
$\mathfrak{t}^{reg}_{r}$. In each sheet one connected component coincides with
the fundamental domain $\mathcal{C}^{\pm}$. For each sign $\pm$, the Weyl
group acts simply transitively on that set $Comp_{r}^{\pm}$. We use the Weyl
group action to index the elements of $Comp_{r}^{\pm}$ as follows: Let $1\in
W$ denote the identity element. In each sheet $\mathfrak{t}_{r}^{\pm}$ we
index the fundamental chamber $\mathcal{C}^{\pm}$ with $1$. Then we index the
connected component $c^{\pm}=w\mathcal{C}^{\pm}\in Comp_{r}^{\pm}$ by $w$
yielding

$Comp_{r}^{\pm}=\\{(\mathfrak{t}^{reg}_{r})_{w}^{\pm}\mid w\in
W\\}\subset(\mathfrak{t}^{reg}_{r})^{\pm}\,.$

Let $\overline{(\mathfrak{t}^{reg}_{r})}_{w}^{\pm}$ denote the closure of the
component $(\mathfrak{t}^{reg}_{r})_{w}^{\pm}$ and let $\mathcal{U}^{\pm}$
denote the union

$\mathcal{U}^{\pm}=\bigcup_{w\in
W}\overline{(\mathfrak{t}^{reg}_{r})}_{w}^{\pm}$

so $\mathcal{U}^{\pm}$ covers $\mathfrak{t}_{r}^{\pm}$.

For any minimal subset $J\in W$ such that the intersection

$S_{J}^{\pm}=\bigcap_{w\in
J}\overline{(\mathfrak{t}^{reg}_{r})}_{w}^{\pm}\subset\mathfrak{t}_{r}^{\pm}$

is nonempty, we identify the interior of $S_{J}^{\pm}$ with a simplex of
dimension $\ell-|J|$.

For example, each connected component in $Comp_{r}^{\pm}$ is an open subset of
$\mathfrak{t}_{r}^{\pm}$, hence of dimension $\ell-1$. It follows that it is
identified with a simplex of dimension $\ell-1$.

It is straightforward to check that $\mathcal{S}^{\pm}=\\{S_{J}^{\pm}\mid
J\subset W,S_{J}^{\pm}\neq\emptyset\\}$ is a Coxeter complex [AB08] for the
Weyl group $W$. That is, $\mathcal{S}^{\pm}$ admits a $W$–action which is
simply transitively on simplices of maximal dimension. Thus for any two
simplices of maximal dimension, there is a chain of maximal dimensional
simplices such that two consecutive ones share a common boundary. Since our
compact real Kac–Moody algebra $\mathfrak{k}$ has rank $\ell$, all of its
Cartan subalgebras, $k\mathfrak{t}k^{-1}$, for $k\in K$, have dimension $\ell$
and each surface $(k\mathfrak{t}k^{-1})_{r}$ of radius $r<0$ has dimension
$\ell-1$. Thus simplices of maximal dimension have dimension $\ell-1$ and
their boundaries have dimension $\ell-2$.

We may now extend this simplicial structure to all of $\mathfrak{H}_{r}$ as
follows. A similar simplicial structure may be defined on any Cartan
subalgebra in $\mathfrak{H}_{r}$. We must check that the Weyl group
tesselations on different Cartan subalgebras fit together in a well–defined
way. Cartan subalgebras intersect exactly in hyperplanes fixed by Weyl group
elements (as in section 4.2), hence they intersect in singular elements. Thus
each simplex of maximal dimension lies in exactly one Cartan subalgebra. Thus
the simplicial structures on different Cartan subalgebras fit together,
leading to a tesselation of $\mathfrak{H}_{r}$ into simplices.

Each simplex of codimension $1$ lies in the intersection of a fixed point
hyperplane $L_{k\mathfrak{t}k^{-1},i}$ with the surface
$(k\mathfrak{t}k^{-1})_{r}$ of radius $r<0$ as calculated in Section 4.2,
while simplices of codimension $n$ lie in the intersection of $n$ such
hyperplanes with the surface. Restriction of the Adjoint action to a single
Cartan subalgebra coincides with the Weyl group action. This follows easily
from the fact that the Weyl group $W=N/T$ is a quotient of the normalizer of
$T$. Hence simplicies of lower dimension are fixed by nontrivial subgroups of
$W$.

##  6\. The main embedding theorem

An ‘embedding’ of the twin building
$\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ of the Kac-Moody group $G$
with identity element $e$ into the compact real form $\mathfrak{k}$ is a
bijective simplicial map from $\mathcal{B}$ onto the simplicial structure
defined on $\widehat{\mathfrak{H}}_{r}$ for each $r<0$ defined in section 5.

######  Theorem 6.1.

Let $A$ be a symmetrizable hyperbolic generalized Cartan matrix,
$\mathfrak{g}$ its complex Kac–Moody algebra and $G$ its complex Kac–Moody
group and let $r<0$. Let $K$ be the compact real form of $G$ and let
$\mathfrak{k}$ be its Lie algebra. Let
$\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ be the geometric realization
(section 2.3) of the twin building of $G$ over ${\mathbb{C}}$.

  1. (1)

There is a $K$–equivariant embedding
$\Psi_{r}:\mathcal{B}\hookrightarrow\widehat{\mathfrak{H}}_{r}\subset\widehat{\mathfrak{k}}$,
that is, the following diagram commutes:

$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{K}$$\scriptstyle{\Psi_{r}}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{r}}$$\textstyle{\widehat{\mathfrak{H}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textrm{Ad}_{K}}$$\textstyle{\widehat{\mathfrak{H}}_{r}}$

where $\textrm{Ad}_{K}$ denotes the Adjoint action and $K\subset G$ acts on
$\mathcal{B}$ by left multiplication.

  2. (2)

When $A$ is strictly hyperbolic the $K$–equivariant restrictions
$\Psi_{r}^{\pm}:\mathcal{B}^{\pm}\hookrightarrow\mathfrak{H}_{r}^{\pm}$ are
bijective, otherwise,
$\Psi_{r}^{\pm}:\mathcal{B}^{\pm}\hookrightarrow\widehat{\mathfrak{H}}_{r}^{\pm}$
are injective.

  3. (3)

There is a $W$–equivariant embedding of the fundamental apartment of
$\mathcal{B}^{\pm}$ into
$\widehat{\mathfrak{t}}^{\pm}_{r}\subset\widehat{\mathfrak{t}}_{r}$.

  4. (4)

The embedding of the twin building is a simplicial complex isomorphism with
respect to the simplicial structure on $\widehat{\mathfrak{H}}_{r}$ discussed
in section 5.

######  Remark 6.2.

The image ${\rm Im\,}(\Psi_{r})$ is contained in the interior of the lightcone
if $A$ is strictly hyperbolic. Examples of this type are the rank $2$
hyperbolic Kac-Moody algebras (see section 8). Otherwise it contains points on
the boundary of the lightcone. Examples of this type are the algebras
$\mathcal{F}$ and $\mathcal{I}$.

###### Proof.

To construct the embedding of the twin building we use the description given
in equation 12

$\mathcal{B}^{\pm}=\left(K/T\times\Delta^{\pm}\right)/\sim\,.$

Our construction is in three steps. We first define the embedding of the
fundamental chamber. Then we make use of the $K$-action on the building and
the Adjoint action of $K$ on the Lie algebra. In a third step we establish the
properties claimed in the theorem.

Part 1: We choose the fundamental chambers in the buildings
$\mathcal{B}^{\pm}$ to be
$c^{\pm}=(1T,\Delta_{\emptyset}^{\pm})\in\mathcal{B}^{\pm}$. The $\ell$ walls
of the fundamental chamber correspond to the simplices $(1T,\Delta_{i}^{\pm})$
for $1\leq i\leq\ell$, its vertices correspond to the simplices
$(1T,\Delta_{[i]}^{\pm})$. Recall that $[i]=I\backslash\\{i\\}$. Two pairs
$(fT,\Delta_{i}^{\pm})$ and $(gT,\Delta_{i}^{\pm})$ describe the same simplex
if any only if $fK_{i}=gK_{i}$ where $K_{i}=K\cap P_{i}=K\cap(B\sqcup
Bw_{i}B)$; similarly for any subset $J\subsetneq I$ two $J$-cells
$(fT,\Delta_{J}^{\pm})$ and $(fT,\Delta_{J}^{\pm})$ are equivalent if and only
$fK_{J}=gK_{J}$ for $K_{J}=K\cap P_{J}$.

For $r<0$ we choose a connected component $(\mathfrak{t}_{r}^{reg})^{+}_{1}$
where $1\in W$ denotes the identity element and we define

(13) $(\mathfrak{t}_{r}^{reg})_{1}^{-}:=-(\mathfrak{t}_{r}^{reg})_{1}^{+}\,.$

If $A$ is strictly hyperbolic the fundamental domain is contained in the
interior of $\mathfrak{t}_{r}$; hence its closure is contained in
$\mathfrak{t}_{r}$. On the other hand if $A$ is not strictly hyperbolic then
it is unbounded and its closure is contained in $\widehat{\mathfrak{t}}_{r}$
but not in $\mathfrak{t}_{r}$. We identify the boundary hyperplanes
$L_{\mathfrak{t},i}$ of the fundamental chamber with the generators $w_{i}$,
$1\leq i\leq\ell$, of the Weyl group $W$. We also identify intersections of
hyperplanes $L_{\mathfrak{t},i_{1}}\cap\dots\cap L_{\mathfrak{t},i_{n}}$ for
distinct indices $\\{i_{1},\dots,i_{n}\\}\subset I$ with the subset of
generators $\\{w_{i_{1}},\dots,w_{i_{n}}\\}$. Vertices correspond to the
intersection of $(\ell-1)$ hyperplanes and hence to subsets
$[i]=I\backslash\\{i\\}$. For a subset $J\subsetneq I$ we define the boundary
components

$\mathcal{C}_{J}^{\pm}=\mathcal{C}^{\pm}\cap\bigcap_{j\in
J}L_{\mathfrak{t},j}\,.$

We define $\Psi_{r}(1T,\Delta_{[i]}^{\pm})=\mathcal{C}_{[i]}^{\pm}$ and extend
this map to $\Delta^{\pm}$ by mapping a point $x$ in the geometric realization
of the simplex $(1T,\Delta_{J}^{\pm})$ with normalized barycentric coordinates
$x=[\lambda_{1}:\dots:\lambda_{\ell}]$ to the point with the same normalized
hyperbolic barycentric coordinates.

Recall the definition of normalized hyperbolic barycentric coordinates: Let
$\Delta$ be a simplex in $n$-dimensional hyperbolic space with vertices
$(v_{0},\dots,v_{n})$. Some vertices may be on the boundary of hyperbolic
space. We denote by $V=V(\Delta)$ the volume of $\Delta$. For any point
$p\in\Delta$ we can define $n+1$ simplices $\Delta_{[i]}$ for $0\leq i\leq n$,
spanned by the $(n+1)$-tuple of vertices $(v_{0},\dots,p,\dots,v_{n})$, where
the vertex $v_{i}$ has been replaced by $p$. Let $V_{i}=V(\Delta_{[i]})$
denote the volume of $\Delta_{[i]}$. Then the normalized hyperbolic
barycentric coordinates of $p$ are given by

$\left[\frac{V_{0}}{V}:\dots:\frac{V_{n}}{V}\right]\,.$

This yields a simplicial map
$\Psi_{r}:(1T,\Delta^{\pm})\longrightarrow\mathcal{C}^{\pm}$.

Part 2: Let $x^{\pm}\in\Delta^{\pm}$. We extend the map defined in Part $1$ to
a map

$\Psi_{r}:\mathcal{B}\hookrightarrow\widehat{\mathfrak{H}}_{r}$

by defining

$\Psi_{r}(kT,x^{\pm})=Ad_{k}\,\left[\Psi_{r}(1T,x^{\pm})\right]\,.$

We have to check that $\Psi_{r}$ is well-defined. Assume we have two
equivalent elements $(k_{1}T,x_{1}^{\pm})\sim(k_{2}T,x_{2}^{\pm})$, hence
$x_{1}^{\pm}=x_{2}^{\pm}$ and assuming $x_{1}^{\pm}\in\Delta^{\pm}_{J}$ for
some subset $J\subsetneq I$, we have $k_{1}K_{J}=k_{2}K_{J}$. Then there is
some $l\in K_{J}$ such that $k_{1}=k_{2}l$ and we have

$\Psi_{r}(k_{1}T,x_{1}^{\pm})=Ad_{k_{1}}\,\left[\Psi_{r}(1T,x_{1}^{\pm})\right]=Ad_{k_{2}l}\,\left[\Psi_{r}(1T,x_{1}^{\pm})\right]\,.$

Since $x_{1}^{\pm}\in\Delta_{J}^{\pm}$ we have

$\Psi_{r}(1T,x_{1}^{\pm})\in\bigcap_{j\in J}L_{\mathfrak{t},j}\,.$

Hence $Ad_{l}\,\Psi_{r}(1T,x_{1}^{\pm})=\Psi_{r}(1T,x_{1}^{\pm})$. Thus, from
$Ad_{k_{2}l}=Ad_{k_{2}}Ad_{l}$ we get

$Ad_{k_{2}l}\,\left[\Psi_{r}(1T,x_{1}^{\pm})\right]=Ad_{k_{2}}\,\left[\Psi_{r}(1T,x_{2}^{\pm})\right]=\Psi_{r}(k_{2}T,x_{2}^{\pm})\,.$

Part 3: We have to check that $\Psi_{r}$ satisfies the properties stated in
the theorem.

Proof of (4): Remark that $\Psi_{r}$ maps the unique simplex
$(kT,\Delta_{J}^{\pm})$ for $J\subsetneq I$ spanned by vertices
$(kT,\Delta^{\pm}_{[i]})$, $i\in J$, onto the simplex spanned by
$\Psi_{r}(kT,\Delta^{\pm}_{[i]})$, $i\in J$. If two simplices
$(k_{1}T,\Delta^{\pm}_{J})$ and $(k_{2}T,\Delta^{\pm}_{J})$ share a common
boundary $(lT,\Delta^{\pm}_{L})$, for $J\subsetneq L$ in $\mathcal{B}$ then by
definition we have $k_{1}T\subset Ad_{l}\,K_{L}$ and $k_{2}T\subset
Ad_{l}\,K_{L}$. But then
$\Psi_{r}(k_{1}T,\Delta^{\pm}_{L})=\Psi_{r}(lT,\Delta^{\pm}_{L})=\Psi_{r}(k_{2}T,\Delta^{\pm}_{L})$
is the commonly shared boundary of the simplices
$\Psi_{r}(k_{1}T,\Delta^{\pm}_{J})$ and $\Psi_{r}(k_{2}T,\Delta^{\pm}_{J})$.
Hence $\Psi_{r}$ preserves the simplicial structure of $\mathcal{B}$ and is
thus a simplicial complex isomorphism, establishing (4).

Proof of (1): Recall the left $K$–action on the building $\mathcal{B}^{\pm}$:

$K:\mathcal{B}^{\pm}\longrightarrow\mathcal{B}^{\pm},\quad
k\cdot(fT,\Delta^{\pm})\mapsto(kfT,\Delta^{\pm})\,.$

We need to verify that for $k_{1},k_{2}\in K$ and $J\subsetneq I$

$\Psi_{r}(k_{1}\cdot(k_{2}T,\Delta^{\pm}_{J}))=Ad_{k_{1}}\,\Psi_{r}(k_{2}T,\Delta^{\pm}_{J})\,.$

We have

$\displaystyle\Psi_{r}(k_{1}\cdot(k_{2}T,\Delta^{\pm}_{J}))$
$\displaystyle=\Psi_{r}(k_{1}k_{2}T,\Delta^{\pm}_{J})=Ad_{k_{1}k_{2}}\,\Psi_{r}(1T,\Delta^{\pm}_{J})=$
$\displaystyle=Ad_{k_{1}}\,(Ad_{k_{2}}\,\Psi_{r}(1T,\Delta^{\pm}_{J}))=Ad_{k_{1}}\,(\Psi_{r}(k_{2}T,\Delta^{\pm}_{J}))\,.$

Proof of (2): Assume two simplices $(k_{1}T,\Delta_{J}^{\pm})$ and
$(k_{2}T,\Delta_{J}^{\pm})$ satisfy

$\Psi_{r}(k_{1}T,\Delta_{J}^{\pm})=\Psi_{r}(k_{2}T,\Delta_{J}^{\pm})\,.$

Then we have

$Ad_{k_{1}}\,\Psi_{r}(1T,\Delta_{J}^{\pm})=Ad_{k_{2}}\,\Psi_{r}(1T,\Delta_{J}^{\pm})$

which implies
$Ad_{k_{1}^{-1}k_{2}}\Psi_{r}(1T,\Delta_{J}^{\pm})=\Psi_{r}(1T,\Delta_{J}^{\pm})$.

As $\Psi_{r}(1T,\Delta_{J}^{\pm})=\mathcal{C}^{\pm}_{J}$ its stabilizer in $K$
is $K_{J}$. Hence $k_{1}^{-1}k_{2}\in K_{J}$. Thus
$(k_{1}T,\Delta_{J}^{\pm})\sim(k_{2}T,\Delta_{J}^{\pm})$ which proves
injectivity.

Suppose we have an arbitrary element $x^{\pm}\in\mathfrak{H}_{r}^{\pm}$. Then
there is some group element $k\in K$ such that
$Ad_{k}\,x^{\pm}\in\mathfrak{t}^{\pm}_{r}$ so it is in the closure of some
chamber, $(\mathfrak{t}^{reg}_{r})^{\pm}_{w}$, uniquely labeled by an element
$w\in W$. The action of the corresponding element
$\widetilde{w}\in\widetilde{W}^{ad}\leq K$ matches the action of $w$ on
$\mathfrak{t}$. So we have
$Ad_{\widetilde{w}^{-1}}\,Ad_{k}\,x^{\pm}\in\mathcal{C}^{\pm}_{r}$. Therefore
$x^{\pm}\in\,Ad_{k^{-1}}\,Ad_{\widetilde{w}}\,\Psi_{r}(1T,\Delta^{\pm})=\Psi_{r}(k^{-1}\widetilde{w}T,\Delta^{\pm})\,.$
This shows that in the case when $A$ is strictly hyperbolic,
$\Psi_{r}:\mathcal{B}^{\pm}\to\mathfrak{H}_{r}^{\pm}$ is surjective. Otherwise
there can be elements $x^{\pm}\in B^{\infty}(\mathfrak{H}_{r}^{\pm})$ which
are in Im$(\Psi_{r})$, but some may not. It would be interesting to understand
precisely which points in $B^{\infty}(\mathfrak{H}_{r}^{\pm})$ are in
Im$(\Psi_{r})$. The proof of (3) is now clear. ∎

######  Remark 6.3.

Based on our understanding of the examples $\mathcal{F}$ and $E_{10}$, we
believe that the intersection of the nullcone with Im$(\Psi_{r})$ can be
characterized as those rays on the nullcone which contain roots of
$\mathfrak{g}$. Each such ray of null vectors corresponds to a copy of an
affine KM subalgebra inside $\mathfrak{g}$. Each such ray should be conjugate
under the action of the Weyl group $W$ to one ray in the closure of the
fundamental chamber, so the number of $W$ orbits is the number of ideal points
in that fundamental domain. Classifying those rays then becomes a significant
problem involving the arithmetic properties of $W$, essentially understanding
its cusps as a modular group. Such an analysis was carried out for $E_{10}$ in
Lemma 5.2 of [KPN12].

Thus for each $r<0$ we have established an embedding $\Psi_{r}$ of the twin
building $\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ into
$\mathfrak{H}_{r}^{\pm}$ which is inside the lightcone

$\mathcal{L}_{\mathfrak{k}}=\\{x\in\mathfrak{k}\mid(x,x)\leq 0\\}$

of the compact real form $\mathfrak{k}$ of a strictly hyperbolic Kac–Moody
algebra $\mathfrak{g}$, and into its lightlike closure,
$\widehat{\mathfrak{H}}_{r}^{\pm}$, otherwise.

######  Remark 6.4.

Recall that $\mathcal{B}^{\pm}$ is contractible since each apartment is
contractible (see [Ron89]). Since $\Psi_{r}$ for $r\leq 0$ is a simplicial
complex isomorphism onto its image, the image $\Psi_{r}(\mathcal{B}^{\pm})$ is
also contractible.

This observation yields the corollary

######  Corollary 6.5.

The spaces $\widehat{\mathfrak{H}}_{r}^{\pm}$ are contractible.

######  Remark 6.6.

The analogs of theorem 6.1 for finite dimensional compact Lie groups and
affine Kac–Moody groups are well-known (see [Ebe96, Mit88, Hei06, Fre12a]). It
can be generalized in these cases to s-representations and relates in this way
the (twin) building to the isotropy representations of finite dimensional
Riemannian symmetric spaces and affine Kac-Moody symmetric spaces
respectively. The existence of hyperbolic Kac–Moody symmetric spaces is
conjectural at this stage. We hope to take this up elsewhere.

##  7\. Cartan involutions and the embedding of the twin building

Recall from section 2.4 that the Cartan involution is the automorphism
${\omega}_{0}:\mathfrak{g}\longrightarrow\mathfrak{g}$ determined by
${\omega}_{0}(e_{i})=-f_{i}$, $\omega_{0}(f_{i})=-e_{i}$ and
${\omega}_{0}(h_{i})=-h_{i}$, and $\omega$ is the conjugate linear
automorphism defined as the composition of $\omega_{0}$ with complex
conjugation. This gives

$\omega_{0}(\mathfrak{t}^{\pm})=\mathfrak{t}^{\mp}\,\qquad\textrm{and}\qquad-\omega(\mathfrak{t}^{\pm})=\mathfrak{t}^{\mp}\,$

as well as

$\omega_{0}(\mathfrak{t}^{\pm}_{r})=\mathfrak{t}_{r}^{\mp}\,\qquad\textrm{and}\qquad-\omega(\mathfrak{t}^{\pm}_{r})=\mathfrak{t}^{\mp}_{r}\,$

for each $r\in\mathbb{R}$, so that

$\omega_{0}(\widehat{\mathfrak{t}}^{\pm}_{r})=\widehat{\mathfrak{t}}_{r}^{\mp}\,\qquad\textrm{and}\qquad-\omega(\widehat{\mathfrak{t}}^{\pm}_{r})=\widehat{\mathfrak{t}}^{\mp}_{r}\,.$

######  Proposition 7.1.

The following diagram is commutative:

$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\omega}$$\scriptstyle{\Psi_{r}}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{r}}$$\textstyle{\mathfrak{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-\omega}$$\textstyle{\mathfrak{k}}$

###### Proof.

By definition $-\omega|_{\mathfrak{k}}=-Id$. As before, we set
$\mathcal{B}^{\pm}=(K/T\times\Delta^{\pm})/\sim$. The action of the Cartan
involution $\omega$ on $\mathcal{B}$ is defined by

$\omega(kT,\Delta^{\pm}_{J})=(kT,\Delta^{\mp}_{J})\,.$

Let $(kT,\Delta^{\pm}_{J})\in\mathcal{B}^{\pm}$. Then we have

$-\omega(\Psi_{r}(kT,\Delta^{\pm}_{J}))=-\omega(Ad_{k}\,\Psi_{r}(1T,\Delta^{\pm}_{J}))=-Ad_{\omega(k)}\,\omega(\Psi_{r}(1T,\Delta^{\pm}_{J}))=-Ad_{k}\Psi_{r}((1T,\Delta^{\pm}_{J}))\,.$

On the other hand

$\Psi_{r}(\omega(kT,\Delta^{\pm}_{J}))=\Psi_{r}(kT,\Delta^{\mp}_{J})=Ad_{k}\Psi_{r}((1T,\Delta^{\mp}_{J}))=-Ad_{k}\Psi_{r}((1T,\Delta^{\pm}_{J}))$

where the last equality comes from Equation (13), finishing the proof. ∎

######  Remark 7.2.

The choice of $-\omega$ on $\mathfrak{k}$ comes from the identification of
$\mathfrak{k}$ with the $\mathfrak{p}$-component of the Cartan decomposition
of $\mathfrak{g}$ via the relation $\mathfrak{p}=i\mathfrak{k}$.

We note that while $\Psi_{r}$ is a simplicial complex isomorphism, the
apartment system of $\mathcal{B}$ has no direct geometric interpretation in
$\Psi_{r}(\mathcal{B})$. Our embedding identifies $\omega$–stable twin
apartments in $\mathcal{B}$ with Cartan subalgebras in $\mathfrak{k}$. The
other apartments are hidden because $\Psi_{r}$ is only $K$–equivariant, but
not $G$–equivariant.

##  8\. Understanding of the Tits building in rank 2

In this section assume that $\mathfrak{g}=\mathfrak{g}(A)$ is a rank $2$
hyperbolic Kac-Moody Lie algebra, so that

$A=\begin{bmatrix}2&-a\\\ -b&2\end{bmatrix},\ ab>4.$

We impose the additional conditions that $a>1$ and $b>1$ because otherwise the
real root groups $U_{i}$ defined below may not be abelian (see [Mor88,
CMS15]).

Rank $2$ hyperbolic Kac-Moody algebras were studied intensively by Lepowsky
and Moody [LM79], by Feingold [Fei80] for the “Fibonacci hyperbolic”
($a=b=3$), and by Kang and Melville [KM95]. In these rank 2 cases the
hyperboloids $\mathfrak{t}_{r}$ for $r\neq 0$ are hyperbolas, and there is no
topological distinction between “one-sheeted” for $r>0$ and “two-sheeted” for
$r<0$, as there is in higher rank. In this section we will try to provide a
detailed construction of the twin building,
$\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$, whose simplicial structure
is a pair of trees related by a twin building involution, and give the
apartment system and the action of the complex group $G_{\mathbb{C}}$ on each
tree. We will also give an explicit description of the smaller apartment
system and the action of the compact real form $K$, which will provide more
details of the embedding of the building into the compact real form in this
rank 2 case. In the next section we show how this allows the embedding of a
spherical building at infinity.

The eigenvalues of $A$ are $\lambda_{\pm}=2\pm\sqrt{ab}$ so $\lambda_{+}>0$
and $\lambda_{-}<0$. This means that the signature of the bilinear form
determined by $A$ on the split real Cartan subalgebra
$\mathfrak{h}_{\mathbb{R}}$ is $(1,1)$. By Equation (6), the bilinear form on
the compact real Cartan subalgebra $\mathfrak{t}=i\mathfrak{h}_{\mathbb{R}}$
also has signature $(1,1)$.

The Weyl group of $\mathfrak{g}$ is the infinite dihedral group

$W=\langle w_{1},w_{2}\mid
w_{1}^{2}=1=w_{2}^{2}\rangle\cong{\mathbb{Z}}/2{\mathbb{Z}}\ast{\mathbb{Z}}/2{\mathbb{Z}}\cong{\mathbb{Z}}\rtimes\\{\pm
1\\}$

which acts on $\mathfrak{h}\subset\mathfrak{g}$ as well as on
$\mathfrak{h}_{\mathbb{R}}\subset\mathfrak{g}_{\mathbb{R}}$ and on
$\mathfrak{t}\subset\mathfrak{k}$. Denote the infinite cyclic subgroup of $W$
by

$W^{even}=\\{(w_{2}w_{1})^{m}\ |\ m\in\mathbb{Z}\\}\cong\mathbb{Z},$

which has index $2$ in $W$, so that for $i=1,2$,

$W^{odd}=\\{(w_{2}w_{1})^{m}w_{i}\ |\ m\in\mathbb{Z}\\}$

is the other coset, and we have the relations

$w_{i}(w_{2}w_{1})^{m}w_{i}^{-1}=(w_{2}w_{1})^{-m}\quad\hbox{ for
}m\in\mathbb{Z},i=1,2.$

With $t=w_{2}w_{1}$ and $r=w_{i}$ this gives the presentation of $W$ as the
infinite dihedral group $W=\langle t,r\ |\ r^{2}=1,rtr^{-1}=t^{-1}\rangle$. It
can be useful to index the elements of $W$ by the integers so that $W^{even}$
and $W^{odd}$ correspond to even and odd integers, respectively (after making
a definite choice for $i$, say $i=2$:

$w(n)=\begin{cases}(w_{2}w_{1})^{m}\ \hbox{ if }n=2m,\\\
(w_{2}w_{1})^{m}w_{2}\ \hbox{ if }n=2m+1.\end{cases}$

Then we have $w_{1}w(n)=w(-1-n)$ and $w_{2}w(n)=w(1-n)$ for $n\in\mathbb{Z}$,
and

$w(n)^{-1}=\begin{cases}(w_{2}w_{1})^{-m}\ \hbox{ if }n=2m,\\\
w_{2}(w_{2}w_{1})^{-m}\ \hbox{ if }n=2m+1\end{cases}=\begin{cases}w(-n)\
\hbox{ if }n=2m,\\\ w(n)\ \hbox{ if }n=2m+1.\end{cases}$

It is straightforward to check that for any $n,k\in\mathbb{Z}$, we have

$w(n)w(k)=\begin{cases}w(n+k)\ \hbox{ if }n=2m,\\\ w(n-k)\ \hbox{ if
}n=2m+1.\end{cases}$

We will also find it useful to similarly label certain elements of
$\widetilde{W}$ by the integers in the same way:

$\widetilde{w}(n)=\begin{cases}(\widetilde{w}_{2}\widetilde{w}_{1})^{m}\
\hbox{ if }n=2m,\\\ (\widetilde{w}_{2}\widetilde{w}_{1})^{m}\widetilde{w}_{2}\
\hbox{ if }n=2m+1.\end{cases}$

We define the non-standard partition of the real roots of $\mathfrak{g}$,
${\Phi}^{re}={\Phi}_{1}\cup{\Phi}_{2}$ where

${\Phi}_{1}=W^{even}\alpha_{1}\cup
W^{odd}\,\alpha_{2}=W^{even}\\{\alpha_{1},-\alpha_{2}\\}$

and

${\Phi}_{2}=W^{even}\alpha_{2}\cup
W^{odd}\,\alpha_{1}=W^{even}\\{-\alpha_{1},\alpha_{2}\\}\,.$

See Figure 1 as an illustration for the “Fibonacci” rank 2 hyperbolic root
system, and see Figure 2 for the rank 2 hyperbolic root system coming from the
Cartan matrix $A$ given at the beginning of Section 8, with $a=2$ and $b=3$,
which has simple roots of different lengths. The real roots are on the red
hyperbolas, and the non-standard partition is according to whether a root is
on a left branch or a right branch.

Figure 1: The Fibonacci root system and non-standard partition of real roots

${\Phi}^{re}={\Phi}_{1}\cup{\Phi}_{2}$ where
${\Phi}_{1}=W^{even}\\{\alpha_{1},-\alpha_{2}\\}$ and
${\Phi}_{2}=W^{even}\\{-\alpha_{1},\alpha_{2}\\}$.

For $i=1,2$ recall that

$L_{\mathfrak{t},i}=\\{x\in\mathfrak{t}\ |\ w_{i}(x)=x\\}=\\{x\in\mathfrak{t}\
|\ \alpha_{i}(x)=0\\}$

are the lines in $\mathfrak{t}$ fixed by the simple reflections $w_{i}$,
respectively. Then

$L_{\mathfrak{t},1}=\\{t(az_{1}+2z_{2})\in\mathfrak{t}\ |\
t\in\mathbb{R}\\}\quad\hbox{and}\quad
L_{\mathfrak{t},2}=\\{t(2z_{1}+bz_{2})\in\mathfrak{t}\ |\ t\in\mathbb{R}\\}.$

For $i=1,2$ the line $L_{\mathfrak{t},i}$ is the intersection of the family of
Cartan subalgebras,

$\\{exp(ad_{sx_{i}+ty_{i}})\mathfrak{t}=\mathfrak{t}^{i}(s,t)\ |\
s,t\in\mathbb{R}\\}$

parametrized by the $2$-sphere with antipodes identified, the complex
projective space ${\mathbb{P}}^{1}({\mathbb{C}})$ (Theorem 4.4).

Figure 2: The root system and non-standard partition of real roots for a rank
2 hyperbolic

with unequal root lengths, $a=2$, $b=3$.

We also have the corresponding statements for the split real form. For $i=1,2$
recall that in the split real Cartan, $\mathfrak{h}_{\mathbb{R}}$, we have

$L_{\mathfrak{h}_{\mathbb{R}},i}=\\{x\in\mathfrak{h}_{\mathbb{R}}\ |\
w_{i}(x)=x\\}=\\{x\in\mathfrak{h}_{\mathbb{R}}\ |\ \alpha_{i}(x)=0\\}$

are the lines in $\mathfrak{h}_{\mathbb{R}}$ fixed by the simple reflections
$w_{i}$, respectively. Then

$L_{\mathfrak{h}_{\mathbb{R}},1}=\\{t(ah_{1}+2h_{2})\in\mathfrak{h}_{\mathbb{R}}\
|\ t\in\mathbb{R}\\}\quad\hbox{and}\quad
L_{\mathfrak{h}_{\mathbb{R}},2}=\\{t(2h_{1}+bh_{2})\in\mathfrak{h}_{\mathbb{R}}\
|\ t\in\mathbb{R}\\}.$

In Figures 1 and 2 those fixed lines are the inner green lines.

For $i=1,2$ the line $L_{\mathfrak{h}_{\mathbb{R}},i}$ is the intersection of
the family of Cartan subalgebras,

$\\{exp(ad_{se_{i}+tf_{i}})\mathfrak{h}_{\mathbb{R}}=\mathfrak{h}_{\mathbb{R}}^{i}(s,t)\
|\ s,t\in\mathbb{R}\\}$

parametrized by pairs of antipodal points on a $1$-sheeted hyperboloid.

Let $(G,B^{\pm},N)$ be a $BN$–pair for the complex Kac–Moody group
$G=G_{\mathbb{C}}(A)$. The standard parabolic subgroups $P^{\pm}_{J}$ for
$J\subsetneq\\{1,2\\}$ are

$P^{\pm}_{\emptyset}=B^{\pm},\quad P^{\pm}_{1}=B^{\pm}\sqcup
B^{\pm}w_{1}B^{\pm},\quad\hbox{and}\quad P^{\pm}_{2}=B^{\pm}\sqcup
B^{\pm}w_{2}B^{\pm}.$

As a simplicial complex, the twin building
$\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ of $(G,B^{\pm},N)$ is a pair
of homogenous ${\mathbb{P}}^{1}({\mathbb{C}})$-trees. The vertices of
$\mathcal{B}$ are in bijection with the conjugates of $P^{\pm}_{1}$ and
$P^{\pm}_{2}$ in $G$, while the set of edges (chambers) are in bijection with
conjugates of $B^{\pm}$. We identify the sets of vertices with the disjoint
union of cosets

$V(\mathcal{B}^{\pm})\quad\cong\quad G/P^{\pm}_{1}\sqcup G/P^{\pm}_{2}.$

The set of edges is given by

$E(\mathcal{B}^{\pm})\quad\cong\quad G/B^{\pm}.$

The group $G$ acts by left multiplication on cosets. There are natural
projections on cosets induced by the inclusion of $B^{\pm}$ in $P^{\pm}_{1}$
and $P^{\pm}_{2}$:

$\pi_{i}:G/B^{\pm}\longrightarrow G/P^{\pm}_{i},\ i=1,2\,.$

If $v_{i}\in G/P^{\pm}_{i}$ is a vertex, and
$St^{\mathcal{B}}(v_{i})=\pi_{i}^{-1}(v_{i})$ is the set of edges with origin
$v_{i}$, then we may index $St^{\mathcal{B}}(v_{i})$ by
$P^{\pm}_{i}/B^{\pm}\subseteq G/B^{\pm}$, $i=1,2$.

Apartments in $\mathcal{B}^{\pm}$ are infinite lines, but not every infinite
line in the tree is an apartment. The fundamental apartment $A^{\pm}$ in
$\mathcal{B}^{\pm}$ is a union of chambers which are Weyl group translates of
the fundamental chamber $C^{\pm}=(1B^{\pm},\Delta_{\emptyset}^{\pm})$, which
is fixed under the action of $B^{\pm}$, which contains the root group
$U^{\pm}$. The boundary vertices of that fundamental chamber are denoted by
$v_{i}^{\pm}=(1B^{\pm},\Delta_{i}^{\pm})$ for $i=1,2$, corresponding to the
maximal parabolic subgroups $P_{i}^{\pm}$. By definition, the vertex
$v_{i}^{\pm}=(1B^{\pm},\Delta_{i}^{\pm})$ is stabilized by the generator
$w_{i}$ of the Weyl group $W$. The apartment system $\mathcal{A}^{\pm}$ in
$\mathcal{B}^{\pm}$ is the set of apartments obtained from fundamental
apartment $A^{\pm}$ by the action of $G$. The question is how precisely can we
describe that apartment system.

In $\mathfrak{t}$, for each $r<0$ the hyperbola $\mathfrak{t}_{r}$ has two
connected components, $\mathfrak{t}_{r}^{\pm}$, which are each tessellated by
the action of $W$ into intervals separated by vertices which are the
intersections of $\mathfrak{t}_{r}$ with the $W$-images of the two lines
$L_{\mathfrak{t},i}$ fixed by $w_{i}$, $i=1,2$. The fundamental domain for the
$W$ action on $\mathfrak{t}_{r}^{\pm}$ is $\mathcal{C}^{\pm}$, which
corresponds to the fundamental chamber of $A^{\pm}$, and the other chambers of
$A^{\pm}$ correspond to:

$\\{\dots,\ w_{1}w_{2}w_{1}\mathcal{C}^{\pm},\ w_{1}w_{2}\mathcal{C}^{\pm},\
w_{1}\mathcal{C}^{\pm},\ \mathcal{C}^{\pm},\ w_{2}\mathcal{C}^{\pm},\
w_{2}w_{1}\mathcal{C}^{\pm},\ w_{2}w_{1}w_{2}\mathcal{C}^{\pm},\ \dots\\}\,$

so that adjacent chambers in this list are adjacent in the apartment, sharing
a common vertex. For example, the common vertex
$v_{1}^{\pm}=w_{1}\mathcal{C}^{\pm}\cap\mathcal{C}^{\pm}$ corresponds to the
maximal parabolic subgroup $P_{1}^{\pm}$, and
$v_{2}^{\pm}=\mathcal{C}^{\pm}\cap w_{2}\mathcal{C}^{\pm}$ corresponds to the
maximal parabolic subgroup $P_{2}^{\pm}$. It means that this apartment is a
line in the tree, infinite in both directions. Note that according to the
geometric pictures shown in Figures 1 and 2, this left-to-right ordering is
natural in $A^{+}$, but would be reversed in $A^{-}$ because the reflecting
mirrors of $w_{1}$ and $w_{2}$ are crossed in the lower lightcone. The order
would not be reversed if we think of viewing the lower lightcone from the
origin, facing south. We use the integers $n\in\mathbb{Z}$ to label each
chamber in $A^{\pm}$, as we did above to label elements of $W$,

$\mathcal{C}^{\pm}(n)=w(n)\mathcal{C}^{\pm}=\begin{cases}(w_{2}w_{1})^{m}\mathcal{C}^{\pm}\
\hbox{ if }n=2m,\\\ (w_{2}w_{1})^{m}w_{2}\mathcal{C}^{\pm}\ \hbox{ if
}n=2m+1\end{cases}$

and its boundary vertices

$v^{\pm}_{i}(n)=w(n)v^{\pm}_{i}=\begin{cases}(w_{2}w_{1})^{m}v^{\pm}_{i}\
\hbox{ if }n=2m,\\\ (w_{2}w_{1})^{m}w_{2}v^{\pm}_{i}\ \hbox{ if
}n=2m+1\end{cases}$

but notice that each vertex has two labels:
$v^{\pm}_{1}(2m)=v^{\pm}_{1}(2m-1)$ and $v^{\pm}_{2}(2m)=v^{\pm}_{2}(2m+1)$.
The chambers with boundary vertices may be written as

$[v^{\pm}_{1}(2m),\mathcal{C}^{\pm}(2m),v^{\pm}_{2}(2m)]\quad\hbox{and}\quad[v^{\pm}_{2}(2m+1),\mathcal{C}^{\pm}(2m+1),v^{\pm}_{1}(2m+1)]$

where we read the order of vertex, chamber, vertex, left to right. When
$w(2k+1)$ acts on one of these, it will reverse the positions of the vertices.
This can be seen using the formula $w(2k+1)w(n)=w(2k+1-n)$. We use this
notation to describe a (left or right) “ray” of chambers starting with
$\mathcal{C}^{\pm}(n)$ and including all boundary vertices:

$L^{\pm}_{ray}(n)=\\{\mathcal{C}^{\pm}(m)\ |\ m\leq n\\}\quad\hbox{and}\quad
R^{\pm}_{ray}(n)=\\{\mathcal{C}^{\pm}(m)\ |\ m\geq n\\}.$

Note that the action of the Weyl group generators on the chambers of $A^{\pm}$
is given by the simple formulas

$w_{1}\mathcal{C}^{\pm}(n)=\mathcal{C}^{\pm}(-1-n)\quad\hbox{and}\quad
w_{2}\mathcal{C}^{\pm}(n)=\mathcal{C}^{\pm}(1-n)$

for any $n\in\mathbb{Z}$, so that
$(w_{2}w_{1})^{m}\mathcal{C}^{\pm}(n)=\mathcal{C}^{\pm}(n+2m)$ for any
$m,n\in\mathbb{Z}$, and the action on rays is

$w_{1}L^{\pm}_{ray}(n)=R^{\pm}_{ray}(-1-n)\quad\hbox{and}\quad
w_{2}L^{\pm}_{ray}(n)=R^{\pm}_{ray}(1-n)$

so for each $m,n\in\mathbb{Z}$ we have

$(w_{2}w_{1})^{m}L^{\pm}_{ray}(n)=L^{\pm}_{ray}(n+2m)\quad\hbox{and}\quad(w_{2}w_{1})^{m}R^{\pm}_{ray}(n)=R^{\pm}_{ray}(n+2m).$

We can describe the action of the real root groups
$U_{\pm(w_{1}w_{2})^{m}\alpha_{i}}$, $i=1,2$, $m\in\mathbb{Z}$, on the
fundamental apartments $A^{\pm}$.

Using the notation ${\hat{i}}=3-i$, we label the roots in ${\Phi}_{i}$ by the
integers as follows:

${\Phi}_{i}(n)=\begin{cases}w(n)\alpha_{i}\quad\hbox{if }n=2m\\\
w(n)\alpha_{\hat{i}}\quad\hbox{if
}n=2m+1\end{cases}=\begin{cases}(w_{2}w_{1})^{m}\alpha_{i}\quad\hbox{if
}n=2m\\\ (w_{2}w_{1})^{m}w_{2}\alpha_{\hat{i}}\quad\hbox{if
}n=2m+1\end{cases}$

so that the labels of roots in both branches are the integers in order,
negative to positive going in ${\Phi}_{1}$ from top to bottom, but going in
${\Phi}_{2}$ from bottom to top:

${\Phi}_{i}(n)\in{\Phi}^{+}\hbox{ for }\begin{cases}n\leq 0\quad\hbox{if
}i=1\\\ n\geq 0\quad\hbox{if }i=2.\end{cases}$

We also have $w_{1}{\Phi}_{1}(n)={\Phi}_{2}(1-n)$ and
$w_{2}{\Phi}_{1}(n)={\Phi}_{2}(-1-n)$.

Here is another useful application of the labeling of the real roots by
${\Phi}_{i}(n)$, $n\in\mathbb{Z}$, as shown in the paragraph above. A
consistent choice of real root vectors given by

$e_{{\Phi}_{i}(n)}=\begin{cases}e_{w(n)\alpha_{i}}\quad\hbox{if }n=2m\\\
e_{w(n)\alpha_{\hat{i}}}\quad\hbox{if
}n=2m+1\end{cases}=\begin{cases}\widetilde{w}(n)e_{\alpha_{i}}\quad\hbox{if
}n=2m\\\ \widetilde{w}(n)e_{\alpha_{\hat{i}}}\quad\hbox{if
}n=2m+1\end{cases},$

and then we would have $\widetilde{w}_{j}e_{\alpha}=e_{w_{j}\alpha}$ for all
real roots $\alpha$ and $j=1,2$. $L^{\pm}_{ray}(n)$ is fixed by
$U_{\pm{\Phi}_{2}(n)}$ and $R^{\pm}_{ray}(n)$ is fixed by
$U_{\pm{\Phi}_{1}(n)}$ for all $n\in\mathbb{Z}$, but
$U_{\pm{\Phi}_{2}(n)}R^{\pm}_{ray}(n-1)$ is a distinct family of rays in
$\mathcal{B}^{\pm}$ indexed by ${\mathbb{C}}$ whose intersection with
$L^{\pm}_{ray}(n)$ is only the vertex $v_{2}^{\pm}(n)$ if $n$ is even,
$v_{1}^{\pm}(n)$ if $n$ is odd, and $U_{\pm{\Phi}_{1}(n)}L^{\pm}_{ray}(n+1)$
is a distinct family of rays in $\mathcal{B}^{\pm}$ indexed by ${\mathbb{C}}$
whose intersection with $R^{\pm}_{ray}(n)$ is only the vertex $v_{1}^{\pm}(n)$
if $n$ is even, $v_{2}^{\pm}(n)$ if $n$ is odd.

######  Proposition 8.1.

We have the following results about the action of real root groups
$U_{\pm{\Phi}_{i}(k)}$, $k\in\mathbb{Z}$, on the chambers
$\mathcal{C}^{\pm}(n)$, in the fundamental apartment $A^{\pm}$.

  1. (1)

The chambers of $L^{\pm}_{ray}(k)=\\{\mathcal{C}^{\pm}(n)\ |\ n\leq k\\}$ are
each fixed by $U_{\pm{\Phi}_{2}(k)}$ but
$U_{\pm{\Phi}_{2}(k)}R^{\pm}_{ray}(k+1)$ is a family of distinct rays in
$\mathcal{B}^{\pm}$ indexed by ${\mathbb{C}}$ whose intersection with
$L^{\pm}_{ray}(k)$ is only the vertex $v^{\pm}_{2}(k)$.

  2. (2)

The chambers of $R^{\pm}_{ray}(k)=\\{\mathcal{C}^{\pm}(n)\ |\ n\geq k\\}$ are
each fixed by $U_{\pm{\Phi}_{1}(k)}$ but
$U_{\pm{\Phi}_{1}(k)}L^{\pm}_{ray}(k-1)$ is a family of distinct rays in
$\mathcal{B}^{\pm}$ indexed by ${\mathbb{C}}$ whose intersection with
$R^{\pm}_{ray}(k)$ is only the vertex $v^{\pm}_{1}(k)$.

###### Proof.

(1) We know that $\mathcal{C}^{\pm}(n)=w(n)\mathcal{C}^{\pm}$ and that for
real $\alpha\in{\Phi}$, $U_{\alpha}\mathcal{C}^{\pm}=\mathcal{C}^{\pm}$ when
$\alpha\in{\Phi}^{\pm}$. So
$U_{\alpha}\mathcal{C}^{\pm}(n)=\mathcal{C}^{\pm}(n)$ when
$w(n)^{-1}U_{\alpha}w(n)\mathcal{C}^{\pm}=\mathcal{C}^{\pm}$, that is, when
$U_{w(n)^{-1}\alpha}\mathcal{C}^{\pm}=\mathcal{C}^{\pm}$. But that happens
when $w(n)^{-1}\alpha\in{\Phi}^{\pm}$. Recall that

$w(n)^{-1}=\begin{cases}w(-n)\ \hbox{ if }n=2r,\\\ w(n)\ \hbox{ if
}n=2r+1\end{cases}\qquad\hbox{and}\qquad w(n)w(k)=\begin{cases}w(n+k)\ \hbox{
if }n=2r,\\\ w(n-k)\ \hbox{ if }n=2r+1\end{cases}$

so for fixed $k\in\mathbb{Z}$, any $n\leq k$, and

$\alpha=\pm{\Phi}_{2}(k)=\begin{cases}\pm w(k)\alpha_{2}\quad\hbox{if }k=2m\\\
\pm w(k)\alpha_{1}\quad\hbox{if }k=2m+1\end{cases}$

we have

$w(n)^{-1}\alpha=\pm
w(n)^{-1}{\Phi}_{2}(k)=\pm\begin{cases}w(n)^{-1}w(k)\alpha_{2}\quad\hbox{if
}k=2m\\\ w(n)^{-1}w(k)\alpha_{1}\quad\hbox{if }k=2m+1\end{cases}$
$=\pm\begin{cases}w(-n)w(k)\alpha_{2}\quad\hbox{if }n=2r,\ k=2m\\\
w(-n)w(k)\alpha_{1}\quad\hbox{if }n=2r,\ k=2m+1\\\
w(n)w(k)\alpha_{2}\quad\hbox{if }n=2r+1,\ k=2m\\\
w(n)w(k)\alpha_{1}\quad\hbox{if }n=2r+1,\
k=2m+1\end{cases}=\pm\begin{cases}w(-n+k)\alpha_{2}\quad\hbox{if }n=2r,\
k=2m\\\ w(-n+k)\alpha_{1}\quad\hbox{if }n=2r,\ k=2m+1\\\
w(n-k)\alpha_{2}\quad\hbox{if }n=2r+1,\ k=2m\\\ w(n-k)\alpha_{1}\quad\hbox{if
}n=2r+1,\ k=2m+1\end{cases}$ $=\pm\begin{cases}{\Phi}_{2}(k-n)\quad\hbox{if
}n=2r,\ k=2m\\\ {\Phi}_{2}(k-n)\quad\hbox{if }n=2r,\ k=2m+1\\\
{\Phi}_{1}(n-k)\quad\hbox{if }n=2r+1,\ k=2m\\\ {\Phi}_{1}(n-k)\quad\hbox{if
}n=2r+1,\ k=2m+1\end{cases}\in\pm{\Phi}^{+}={\Phi}^{\pm}$

since

${\Phi}_{i}(s)\in{\Phi}^{+}\hbox{ for }\begin{cases}s\leq 0\quad\hbox{if
}i=1\\\ s\geq 0\quad\hbox{if }i=2.\end{cases}$

For $\alpha=\pm{\Phi}_{2}(k)$ and $t\in{\mathbb{C}}$ let
$g(t)=exp(ad_{te_{\alpha}})\in U_{\alpha}$ and consider the family of rays

$U_{\alpha}R^{\pm}_{ray}(k+1)=\\{g(t)R^{\pm}_{ray}(k+1)\ |\
t\in{\mathbb{C}}\\}$

with chambers $\\{g(t)\mathcal{C}^{\pm}(n)\ |\ n\geq k+1,t\in{\mathbb{C}}\\}$.
Two such rays are certainly distinct if their first chambers are distinct, so
suppose that $g(t_{1})\mathcal{C}^{\pm}(k+1)=g(t_{2})\mathcal{C}^{\pm}(k+1)$,
that is, $g(t_{1})w(k+1)\mathcal{C}^{\pm}=g(t_{2})w(k+1)\mathcal{C}^{\pm}$.
Let $w=w(k+1)$ and $g^{w}=w^{-1}gw$, so we have
$g(t_{1})^{w}\mathcal{C}^{\pm}=g(t_{2})^{w}\mathcal{C}^{\pm}$ which gives
$\mathcal{C}^{\pm}=(g(t_{1})^{w})^{-1}g(t_{2})^{w}\mathcal{C}^{\pm}=g(-t_{1})^{w}g(t_{2})^{w}\mathcal{C}^{\pm}=g(-t_{1}+t_{2})^{w}\mathcal{C}^{\pm}$.
Therefore, $g(-t_{1}+t_{2})^{w}\in B^{\pm}$. But for any $t\in{\mathbb{C}}$,
we have $g(t)^{w}\in U_{w^{-1}\alpha}$. Since

$w^{-1}=w(k+1)^{-1}=\begin{cases}w(-k-1)\quad\hbox{if }k\hbox{ is odd}\\\
w(k+1)\quad\hbox{if }k\hbox{ is even}\end{cases}$

we get

$w^{-1}\alpha=\pm\begin{cases}w(-k-1)w(k)\alpha_{1}\quad\hbox{if }k\hbox{ is
odd}\\\ w(k+1)w(k)\alpha_{2}\quad\hbox{if }k\hbox{ is
even}\end{cases}=\pm\begin{cases}w(-1)\alpha_{1}\quad\hbox{if }k\hbox{ is
odd}\\\ w(1)\alpha_{2}\quad\hbox{if }k\hbox{ is even}\end{cases}$
$=\pm\begin{cases}w_{1}\alpha_{1}\quad\hbox{if }k\hbox{ is odd}\\\
w_{2}\alpha_{2}\quad\hbox{if }k\hbox{ is
even}\end{cases}=\pm\begin{cases}-\alpha_{1}\quad\hbox{if }k\hbox{ is odd}\\\
-\alpha_{2}\quad\hbox{if }k\hbox{ is even}\end{cases}$

so $w^{-1}\alpha\in-{\Phi}^{\pm}$ which means $U_{w^{-1}\alpha}\leq U^{\mp}$.
Since $g(-t_{1}+t_{2})^{w}\in B^{\pm}\cap U^{\mp}=\\{1\\}$ we get
$g(-t_{1}+t_{2})^{w}=1$ so $g(-t_{1}+t_{2})=1$ so $t_{1}=t_{2}$.We have shown
that this family consists of distinct rays indexed by ${\mathbb{C}}$.

The proof of (2) is similar. ∎

##  9\. Embedding the spherical building at infinity in rank $2$

Let $G=G_{\mathbb{C}}(A)$ be a rank 2 hyperbolic Kac–Moody group with compact
real form $K$. Let $\mathcal{B}=\mathcal{B}^{+}\cup\mathcal{B}^{-}$ denote its
twin building, whose simplicial structure must be a pair of trees. We define
the twin spherical building at infinity,
$\mathcal{B}_{\infty}=\mathcal{B}_{\infty}^{+}\cup\mathcal{B}_{\infty}^{-}$,
by doing so for each tree $\mathcal{B}^{\pm}$ as follows. A ray in a tree is a
sequence of connected vertices and edges $(v_{1},e_{1},v_{2},e_{2},\cdots)$
with an initial vertex, $v_{1}$, infinite in only one direction. Two rays are
called equivalent if their intersection is infinite, that is, they have a
common tail. An equivalence class of rays is called an end of the tree. A line
in a tree is a sequence of connected vertices and edges infinite in both
directions, so it can be expressed as the union of two rays having a finite
nonempty intersection, which can always be taken to be exactly one vertex. So
each line has two ends, the classes of those two rays. For each apartment,
$X=X_{ray1}\cup X_{ray2}$ in $\mathcal{B}^{\pm}$ there an apartment
$[X]=[X_{ray1}]\cup[X_{ray2}]$ in $\mathcal{B}_{\infty}^{\pm}$ whose two
chambers $[X_{ray1}]$ and $[X_{ray2}]$ uniquely determine the line $X$ as
follows.

Without loss of generality, we may assume that $X_{ray1}\cap
X_{ray2}=\\{X_{0}\\}$ is exactly one vertex. Suppose another apartment
$Y=Y_{ray1}\cup Y_{ray2}$ with $Y_{ray1}\cap Y_{ray2}=\\{Y_{0}\\}$ also
exactly one vertex, has $[X_{ray1}]=[Y_{ray1}]$ and $[X_{ray2}]=[Y_{ray2}]$,
so the intersections $I_{1}=X_{ray1}\cap Y_{ray1}$ and $I_{2}=X_{ray2}\cap
Y_{ray2}$ are both infinite rays. The intersection $I_{1}\cap I_{2}\subseteq
X_{ray1}\cap X_{ray2}$ is either one vertex $\\{X_{0}\\}$ or empty, and
$I_{1}\cap I_{2}\subseteq Y_{ray1}\cap Y_{ray2}$ is either one vertex
$\\{Y_{0}\\}$ or empty. So if $I_{1}\cap I_{2}$ is non-empty, then it is one
vertex and that vertex is $X_{0}=Y_{0}$, so $X=Y$. Assume now that $I_{1}\cap
I_{2}$ is empty and write $I_{1}=(v_{1},e_{1},v_{2},e_{2},\cdots)$ and
$I_{2}=(w_{1},f_{1},w_{2},f_{2},\cdots)$ so $v_{1}$ is a vertex in
$X_{ray1}\cap Y_{ray1}$ and $w_{1}$ is a vertex in $X_{ray2}\cap Y_{ray2}$ but
the connected path in $X$ from $v_{1}$ to $w_{1}$ goes through $X_{0}$ while
the connected path in $Y$ from $v_{1}$ to $w_{1}$ goes through $Y_{0}$. If
those two paths were distinct, there would be a loop in the tree, so they must
be identical, giving $X=Y$.

For any apartment $X$ in $\mathcal{B}^{\pm}$ and for any vertex $v$ in $X$, we
may write $X=X_{1}(v)\cup X_{2}(v)$ where $X_{i}(v)$, $i=1,2$, are the rays
starting from $v$ and going towards the two ends of $X$, which have been
labeled $1$ and $2$. Let us suppress the $\pm$ for the moment to simplify the
notation. Vertex $v$ determines a partition of the tree into two “rooted”
trees, $Tree_{i}(v)$, $i=1,2$, such that

$Tree_{1}(v)\cap Tree_{2}(v)=\\{v\\},\quad X_{i}(v)\subset Tree_{i}(v),i=1,2.$

There can be no other vertices or edges in common between these two trees,
since otherwise there would be a loop passing through that intersection and
$v$. Sometimes the phrase “Nile Delta” is used to describe $X_{1}(v)\cup
Tree_{2}(v)$ and $X_{2}(v)\cup Tree_{1}(v)$, since they look like a long river
splitting into tributaries at the point $v$. We may use this terminology later
to help understand the action of the group $G$ on the tree.

Roughly speaking, a spherical building at infinity of a tree
$\mathcal{B}^{\pm}$ consists of a ‘sufficiently large’ set of ends. The
minimal spherical building at infinity may be constructed as follows. For each
sign $\pm$ let $A^{\pm}$ be the fundamental apartment of $\mathcal{B}^{\pm}$
with fundamental chamber $\mathcal{C}^{\pm}$, and let
$\mathcal{A}^{\pm}=\\{g\cdot A^{\pm}\ |\ g\in G\\}$ be the set of all
$G$–translates of $A^{\pm}$. Then $\mathcal{A}^{\pm}$ is an apartment system
(see [AB08]). A chamber in $\mathcal{B}_{\infty}^{\pm}$ is an end of an
apartment in $\mathcal{A}^{\pm}$, and $Cham_{\infty}^{\pm}$ denotes the set of
all chambers of $\mathcal{B}_{\infty}^{\pm}$. Of course, this building is
highly degenerate. As a chamber complex it consists of a set of points, and
its apartments are exactly subsets of two distinct points corresponding to the
two ends of a line. It is straightforward to show that $G$ acts doubly
transitively on $Cham_{\infty}^{+}$. (See Lemma 9.1.) So in this rank 2 case,
using this doubly transitive action, the existence of a spherical $BN$–pair
with Weyl group $W_{\infty}=\\{\pm 1\\}\cong\mathbb{Z}/2\mathbb{Z}$ follows,
thus relating the spherical building at infinity to the structure of $G$. We
denote all objects associated to the spherical building at infinity, i.e.
apartments, chambers or the Weyl group, with the subscript ‘$\infty$’. Our
construction of the spherical BN-pair follows the approach used in [CG03] to
define spherical BN-pairs for rank 2 hyperbolic Kac-Moody groups over finite
fields, which are then used to construct lattices in trees.

######  Lemma 9.1.

The action of $G$ on $\mathcal{B}_{\infty}^{\pm}$ is doubly transitive.

###### Proof.

For $1\leq i\leq 2$, let $c_{i}^{\pm},d_{i}^{\pm}\in Cham_{\infty}^{\pm}$ we
want to show that there is $g\in G$ such that $g\cdot c_{i}^{\pm}=d_{i}^{\pm}$
for $i=1,2$. By construction of $\mathcal{B}_{\infty}^{\pm}$ there exist
apartments $A_{c}^{\pm}$ and $A_{d}^{\pm}$, such that $c_{i}^{\pm}$ are ends
of $A_{c}^{\pm}$ and $d_{i}^{\pm}$ are ends of $A_{d}^{\pm}$. As the action of
$G$ is transitive on the apartment system of $\mathcal{B}^{\pm}$, there is a
group element $g_{0}\in G$ such that $g_{0}\cdot A_{c}^{\pm}=A_{d}^{\pm}$. If
$g_{0}\cdot c_{i}^{\pm}=d_{i}^{\pm}$ we are done. Otherwise, $g_{0}\cdot
c_{1}^{\pm}=d_{2}^{\pm}$ and $g_{0}\cdot c_{2}^{\pm}=d_{1}^{\pm}$ so we define
$g_{0}^{\prime}=w_{1}g_{0}$ and see that $g_{0}^{\prime}\cdot
c_{1}^{\pm}=w_{1}\cdot d_{2}^{\pm}=d_{1}^{\pm}$ and $g_{0}^{\prime}\cdot
c_{2}^{\pm}=w_{1}\cdot d_{1}^{\pm}=d_{2}^{\pm}$ so
$g=g_{0}^{\prime}=w_{1}g_{0}$ is the desired element of $G$. ∎

In this section, for $G$ of rank 2, we extend the embedding of the twin
building $\mathcal{B}^{\pm}$, given in Theorem 6.1 to the spherical building
at infinity.

In the rank 2 case, the split real form of the Cartan subalgebra,
$\mathfrak{h}_{\mathbb{R}}$, and the compact real form of the Cartan
subalgebra, $\mathfrak{t}$, have a bilinear form with signature $(1,1)$. So
for each $0\neq r\in\mathbb{R}$, $\mathfrak{t}_{r}$ is a hyperbola with two
connected components (branches) $\mathfrak{t}_{r}^{\pm}$, and
$\mathfrak{t}_{0}=\partial\mathcal{L}_{\mathfrak{t}}$ is a pair of lines (the
asymptotes). We have $\mathcal{L}_{\mathfrak{t}}=\cup_{r\leq
0}\mathfrak{t}_{r}$ and the real roots of $\mathfrak{g}$ are on the hyperbolas
$\mathfrak{t}_{(\alpha_{i},\alpha_{i})}$ for $i=1,2$. For symmetric Cartan
matrices, $A$, all real roots have the same length, so this is just one
hyperbola. All imaginary roots of $\mathfrak{g}$ are in
$\mathcal{L}_{\mathfrak{t}}^{0}=\\{x\in\mathfrak{t}\ |\ (x,x)<0\\}$, which is
the Tits cone. We define an equivalence relation on the nonzero vectors in
$\mathfrak{t}_{0}$ by saying that two nonzero points, $x$ and $y$, are
equivalent when $x=ry$ for some $0<r\in\mathbb{R}$. There are exactly four
equivalence classes of such points, corresponding to the four rays in
$\mathfrak{t}_{0}$, which we denote by $\\{x_{i}^{+},x_{i}^{-}\ |\ i=1,2\\}$.
This corresponds to the “lightlike closure”
$B^{\infty}(\partial\mathcal{L}_{\mathfrak{t}})$ in definition 4.3. To be more
precise, $x_{i}^{\pm}$ denotes the ray in $\mathfrak{t}_{0}^{\pm}$ such that
$x_{2}^{-}=-x_{1}^{+}$, $x_{1}^{-}=-x_{2}^{+}$ and

$(x_{1}^{+},\alpha_{1})<0,\quad(x_{1}^{+},\alpha_{2})>0,\quad(x_{2}^{+},\alpha_{1})>0,\quad(x_{2}^{+},\alpha_{2})<0.$

We will show that $\\{x_{1}^{+},x_{2}^{+}\\}$ corresponds to the fundamental
apartment $A^{+}$ and $\\{x_{1}^{-},x_{2}^{-}\\}$ corresponds to the
fundamental apartment $A^{-}$. Define the halos, positive and negative, of
$\mathfrak{g}$ to be the union of all $K$ conjugates of $\\{x_{i}^{\pm}\ |\
i=1,2\\}$

$\Xi_{\infty}^{\pm}=\\{kx_{i}^{\pm}k^{-1}\ |\ k\in K,i=1,2\\}$

and let the twin halo of $\mathfrak{g}$ be
$\Xi_{\infty}=\Xi_{\infty}^{+}\cup\Xi_{\infty}^{-}$. This will be where we
embed the twin spherical building at infinity $\mathcal{B}_{\infty}$ in the
next theorem.

######  Theorem 9.2.

There is a $K$–equivariant bijective map
$\Psi_{\infty}:\mathcal{B}_{\infty}\to\Xi_{\infty}$ such that the following
diagram commutes:

$\textstyle{\mathcal{B}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g\in
K}$$\scriptstyle{\Psi_{\infty}}$$\textstyle{\mathcal{B}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\infty}}$$\textstyle{\Xi_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Ad_{g}}$$\textstyle{\Xi_{\infty}}$

######  Remark 9.3.

Such a spherical building at infinity exists only in the highly degenerate
case of rank 2. The analogous construction for higher rank hyperbolic Kac-
Moody groups would have to involve a new kind of structure beyond the theory
of buildings, perhaps replacing Coxeter groups with some other class of
groups.

Our strategy for the proof of this result is to follow the proof of Theorem
6.1, but since chambers consist only of points, there are some
simplifications. On the other hand, we need a more precise characterization of
the Borel subgroups stabilizing chambers in $Cham_{\infty}^{\pm}$. Thus we
first study the existence of a spherical $BN$–pair with Weyl group
$W_{\infty}$.

Recall that $A^{\pm}$ denotes the fundamental apartment in $\mathcal{B}^{\pm}$
so it is a line in that tree with two ends. Let $e_{i,\infty}^{\pm}$, $i=1,2$,
denote the two ends of that line, that is, the equivalence classes of certain
rays defined as follows. As shown in Proposition 8.1, for each $i=1,2$,
$m\in\mathbb{Z}$, the subgroup $U_{(w_{1}w_{2})^{m}\alpha_{i}}$ fixes the end
$e_{i,\infty}^{\pm}$ if we define

$e_{1,\infty}^{+}=[R_{ray}^{+}(m)],\quad e_{2,\infty}^{+}=[L_{ray}^{+}(m)],$
$e_{1,\infty}^{-}=[L_{ray}^{-}(m)],\quad e_{2,\infty}^{-}=[R_{ray}^{-}(m)].$

Furthermore, we have shown that $U_{-(w_{1}w_{2})^{m}\alpha_{1}}$ fixes
$e_{2,\infty}^{\pm}$ and $U_{-(w_{1}w_{2})^{m}\alpha_{2}}$ fixes
$e_{1,\infty}^{\pm}$. This means that
$\omega(e_{1,\infty}^{+})=e_{2,\infty}^{-}$ and
$\omega(e_{2,\infty}^{+})=e_{1,\infty}^{-}$. We now define the stabilizer of
$e_{i,\infty}^{\pm}$ and its complement

$B_{i,\infty}^{\pm}=\\{g\in G\mid g\cdot
e_{i,\infty}^{\pm}=e_{i,\infty}^{\pm}\\}\qquad\hbox{ and }\qquad
Q_{i,\infty}^{\pm}=\\{g\in G\mid g\cdot e_{i,\infty}^{\pm}\neq
e_{i,\infty}^{\pm}\\}\,.$

Then for any $m\in\mathbb{Z}$, we have established that
$U_{(w_{1}w_{2})^{m}\alpha_{1}}$ and $U_{-(w_{1}w_{2})^{m}\alpha_{2}}$ are in
$B_{1,\infty}^{\pm}$, and that $U_{(w_{1}w_{2})^{m}\alpha_{2}}$ and
$U_{-(w_{1}w_{2})^{m}\alpha_{1}}$ are in $B_{2,\infty}^{\pm}$.

Recall the non-standard partition of the real roots of $\mathfrak{g}$,
${\Phi}^{re}={\Phi}_{1}\cup{\Phi}_{2}$ where

${\Phi}_{1}=W^{even}\\{\alpha_{1},-\alpha_{2}\\}\qquad\hbox{and}\qquad{\Phi}_{2}=W^{even}\\{-\alpha_{1},\alpha_{2}\\}\,.$

######  Definition 9.4.

For $i=1,2$, we set $U_{i}=\langle U_{\alpha}\mid\alpha\in{\Phi}_{i}\rangle$.

We have just shown the following result.

######  Proposition 9.5.

For $i=1,2$ we have $U_{i}\leq B_{i,\infty}^{\pm}$.

######  Lemma 9.6.

We have the decomposition
$Q_{i,\infty}^{\pm}=B_{i,\infty}^{\pm}(-1)B_{i,\infty}^{\pm}$ or equivalently
a Bruhat decomposition

(14) $G=B_{i,\infty}^{\pm}\sqcup B_{i,\infty}^{\pm}(-1)B_{i,\infty}^{\pm}\,.$

###### Proof.

The action of $G$ on $\mathcal{B}_{i,\infty}^{\pm}$ is doubly transitive by
lemma 9.1, and by definition $B_{i,\infty}^{\pm}$ is the stabilizer of
$e_{i,\infty}^{\pm}$. We continue the proof for the case $i=1$, since the
other case is similar.

We will show that for any pair of chambers $c_{1}^{\pm},c_{2}^{\pm}\in
Cham_{\infty}^{\pm}$ there is a $g\in Q_{1,\infty}^{\pm}$ mapping
$c_{1}^{\pm}$ onto the end $e_{1,\infty}^{\pm}$ of the fundamental apartment
$A^{\pm}$ and $c_{2}^{\pm}$ onto the other end $e_{2,\infty}^{\pm}$ of
$A^{\pm}$. To do this, we chose a (uniquely defined) apartment
$A_{c}^{\pm}\in\mathcal{B}^{\pm}$, whose ends are $c_{1}^{\pm}$ and
$c_{2}^{\pm}$. We chose first $b_{1}\in B_{1,\infty}^{\pm}$ such that
$b_{1}c_{1}^{\pm}=e_{2,\infty}^{\pm}$. Then we apply the Weyl group element
$-1\in W_{\infty}$, so $(-1)b_{1}c_{1}^{\pm}=e_{1,\infty}^{\pm}$. Transitivity
of $B_{1,\infty}^{\pm}$ on ends other than $e_{1,\infty}^{\pm}$ assures the
existence of an element $b_{2}\in B_{1,\infty}^{\pm}$ such that
$b_{2}(-1)b_{1}c_{2}^{\pm}=e_{2,\infty}^{\pm}$. As $b_{2}\in
B_{1,\infty}^{\pm}$, we have
$e_{1,\infty}^{\pm}=b_{2}e_{1,\infty}^{\pm}=b_{2}(-1)b_{1}c_{1}^{\pm}$. We now
see that $g=b_{2}(-1)b_{1}\in Q_{1,\infty}^{\pm}$ because
$b_{2}(-1)b_{1}e_{1,\infty}^{\pm}=b_{2}e_{2,\infty}^{\pm}\neq
e_{1,\infty}^{\pm}$ since the action of group elements is bijective. See
Figure 3. ∎

Figure 3: Apartments and Ends in $\mathcal{B}_{\infty}^{\pm}$ with superscript
$\pm$ suppressed.

######  Theorem 9.7.

For each $i=1,2$, a rank 2 hyperbolic Kac–Moody group $G=G_{\mathbb{C}}(A)$
has a spherical $BN$–pair at infinity with $B=B_{i,\infty}^{\pm}$,
$N=N_{G}(T_{\mathbb{C}})$ and Weyl group $W_{\infty}=\mathbb{Z}/2\mathbb{Z}$.

###### Proof.

Note that $H=N\cap B_{i,\infty}^{\pm}=W^{even}T_{\mathbb{C}}$ giving

(15) $W_{\infty}=N/H=\mathbb{Z}/2\mathbb{Z}\,.$

The theorem follows from Lemma 9.6. (See Theorem 11.96 and Corollary 11.98 in
[AB08].) ∎

Recall our description of the simplicial complex $\mathcal{B}^{\pm}$ in
Section 2.3, equation (3) in the general case. Applying that description to
the spherical $BN$-pair at infinity in Theorem 9.7, we have

(16)
$\mathcal{B}_{\infty}^{\pm}=(G/B_{i,\infty}^{\pm}\times\Delta_{\infty}^{\pm})/\sim$

where $\Delta_{\infty}^{\pm}$ is the fundamental chamber in the fundamental
apartment and $\sim$ is the equivalence relation which would only be non-
trivial on simplices in the boundary of $\Delta_{\infty}^{\pm}$, so that
relation is trivial. The fundamental apartment of $\mathcal{B}_{\infty}^{\pm}$
is the two-element set
$A_{\infty}^{\pm}=\\{e_{1,\infty}^{\pm},e_{2,\infty}^{\pm}\\}$, corresponding
to the ends of the fundamental apartment $A^{\pm}$ in $\mathcal{B}^{\pm}$.
Either of these two points could be chosen as the fundamental chamber. The
apartment system of $\mathcal{B}_{\infty}^{\pm}$ is the collection of two-
element sets consisting of the ends of apartments in the apartment system
$\mathcal{A}^{\pm}=\\{gA^{\pm}\ |\ g\in G\\}$ of $\mathcal{B}^{\pm}$. The
action of $-1\in W_{\infty}$ on each apartment
$A^{\pm}_{c}=\\{c_{1}^{\pm},c_{2}^{\pm}\\}$ interchanges those two chambers,
$(-1)\cdot c_{1}^{\pm}=c_{2}^{\pm}$. So the set of all chambers of
$\mathcal{B}_{\infty}^{\pm}$ is the $G$ orbit of any one chamber,
$\\{ge_{1,\infty}^{\pm}\ |\ g\in G\\}$. Since $G=B_{i,\infty}^{\pm}\sqcup
B_{i,\infty}^{\pm}(-1)B_{i,\infty}^{\pm}$, for each $i=1,2$, the chambers of
$\mathcal{B}_{\infty}^{\pm}$ are

$B_{1,\infty}^{\pm}e_{1,\infty}^{\pm}\cup
B_{1,\infty}^{\pm}(-1)B_{1,\infty}^{\pm}e_{1,\infty}^{\pm}=\\{e_{1,\infty}^{\pm}\\}\cup
B_{1,\infty}^{\pm}e_{2,\infty}^{\pm}.$

Another interpretation of this result is that the set of ends of apartments in
$\mathcal{B}^{\pm}$ consists of one end of fundamental apartment $A^{\pm}$
(the class of any right ray in $A^{+}$, any left ray in $A^{-}$), and the
images of the opposite end of $A^{\pm}$ under the stabilizer of the first end.
To get all apartments in the apartment system of $\mathcal{B}_{\infty}^{\pm}$
we apply the same reasoning to pairs, starting with the fundamental apartment
$A_{\infty}^{\pm}=[e_{1,\infty}^{\pm},e_{2,\infty}^{\pm}]$. Its orbit under
$G$ consists of two parts,

$B_{1,\infty}^{\pm}[e_{1,\infty}^{\pm},e_{2,\infty}^{\pm}]=[B_{1,\infty}^{\pm}e_{1,\infty}^{\pm},B_{1,\infty}^{\pm}e_{2,\infty}^{\pm}]=[e_{1,\infty}^{\pm},B_{1,\infty}^{\pm}e_{2,\infty}^{\pm}]$

and

$B_{1,\infty}^{\pm}(-1)[e_{1,\infty}^{\pm},B_{1,\infty}^{\pm}e_{2,\infty}^{\pm}]=B_{1,\infty}^{\pm}[e_{2,\infty}^{\pm},B_{2,\infty}^{\pm}e_{1,\infty}^{\pm}]=[B_{1,\infty}^{\pm}e_{2,\infty}^{\pm},B_{1,\infty}^{\pm}B_{2,\infty}^{\pm}e_{1,\infty}^{\pm}].$

NOTE: These statements are still under construction.

Recall the _even_ subgroup of the Weyl group $W$ has been defined to be

$W^{even}=\\{t^{m}=(w_{1}w_{2})^{m}\mid m\in\mathbb{Z}\\}\cong\mathbb{Z}$

so that $W=W^{even}\cup W^{odd}$ where $W^{odd}=w_{1}\,W^{even}$. The action
of $W^{even}$ on an apartment in $\mathcal{B}^{\pm}$, $\\{C(n)\ |\
n\in\mathbb{Z}\\}$, with chambers $C(n)$, (a line in the tree) is by
translation of chambers, $t^{m}C(n)=C(n+2m)$, leaving the two ends fixed. In
particular we have $W^{even}\subset B_{i,\infty}^{\pm}$ for $i=1,2$.

We define

$B_{W}^{\pm}=\bigcap_{w\in W}wB^{\pm}w^{-1}.$

As $B^{\pm}$ consists of all elements in $G$ fixing the fundamental chamber
$\mathcal{C}^{\pm}\subset A^{\pm}$, its conjugate $wB^{\pm}w^{-1}$ fixes the
chamber $w\mathcal{C}^{\pm}\in A^{\pm}$. Hence $B_{W}^{\pm}$ can be
characterized as follows:

$B^{\pm}_{W}=\\{g\in G\mid g|_{A^{\pm}}=Id\\}.$

In particular we have $B_{W}^{\pm}\subset B_{i,\infty}^{\pm}$ for $i=1,2$.

Recall the definition of $U^{\pm}$ from section 4.3. As we defined
$B_{W}^{\pm}$, we similarly define the group

$U_{W}^{\pm}=\bigcap_{w\in W}wU^{\pm}w^{-1}$

and since $B^{\pm}=T_{\mathbb{C}}U^{\pm}$, we have the relation

(17) $B_{W}^{\pm}=T_{\mathbb{C}}U_{W}^{\pm}\,.$

######  Remark 9.8.

Each element $w\in W$ has a length, the minimum number of simple reflections
in an expression for $w$. But that number is also the number of positive real
roots sent by $w$ to $\Phi^{re-}$. Since
$\widetilde{w}U_{\alpha}\widetilde{w}^{-1}=U_{w\alpha}$, as longer elements
are applied, more positive root generators in $U^{+}$ are eliminated from the
intersection (they are not in $U^{+}$), and in the complete intersection all
positive root generators are eliminated. But that means that what is left must
correspond to the imaginary root spaces, which can only be the result of
complicated commutators of real root generators. Hidden inside this
intersection may be the structures giving the imaginary root multiplicities.

######  Lemma 9.9.

Each group $U_{i}$, $i=1,2$, is abelian and is normalized by $W^{even}$, so
$U_{i}\,W^{even}=W^{even}\,U_{i}$. Furthermore,
$U_{i}\,W^{odd}=W^{odd}\,U_{\hat{i}}$.

###### Proof.

Under our assumptions about the Cartan matrix of the rank $2$ hyperbolic Kac-
Moody algebra $\mathfrak{g}$, it can be seen that for any
$\alpha,\beta\in{\Phi}_{i}$ we have $\alpha+\beta\notin{\Phi}\cup\\{0\\}$ (see
[Mor88, CMS15]). Then we have the Lie algebra commutator
$[x_{\alpha},x_{\beta}]=0$ for any real root vectors
$x_{\alpha}\in\mathfrak{g}_{\alpha}$ and $x_{\beta}\in\mathfrak{g}_{\beta}$,
so $[ad_{x_{\alpha}},ad_{x_{\beta}}]=0$. Since
$U_{\alpha}=\\{exp(ad_{sx_{\alpha}})\ |\ s\in{\mathbb{C}}\\}$ and
$U_{\beta}=\\{exp(ad_{tx_{\beta}})\ |\ t\in{\mathbb{C}}\\}$, each element of
$U_{\alpha}$ commutes with each element of $U_{\beta}$.

To see that $W^{even}$ normalizes each group $U_{i}$, note that for any $w\in
W$ and for any $\alpha\in{\Phi}_{i}$, we have
$w\,exp(ad_{sx_{\alpha}})\,w^{-1}=exp(ad_{sw(x_{\alpha})})\in
U_{w(\alpha)}\leq U_{i}$ since $W^{even}{\Phi}_{i}={\Phi}_{i}$. On the other
hand, $W^{odd}{\Phi}_{i}={\Phi}_{\hat{i}}$, so we get the final statement. ∎

######  Lemma 9.10.

$W$ normalizes $B_{W}^{\pm}$ and $U_{W}^{\pm}$, so

$W\,B_{W}^{\pm}=B_{W}^{\pm}\,W\quad\textrm{and}\quad
W\,U_{W}^{\pm}=U_{W}^{\pm}\,W\,.$

###### Proof.

Obvious. ∎

######  Corollary 9.11.

We have $W^{even}$ normalizes $B_{W}^{\pm}$ and $U_{W}^{\pm}$, so

$W^{even}\,B_{W}^{\pm}=B_{W}^{\pm}\,W^{even}=B_{1,\infty}^{\pm}\cap
B_{2,\infty}^{\pm}\quad\textrm{and}\quad
W^{even}\,U_{W}^{\pm}=U_{W}^{\pm}\,W^{even}\,.$

###### Proof.

By definition $B_{W}^{\pm}$ is the pointwise stabilizer of the standard
apartment $A^{\pm}$, while $W^{even}$ contains the translations of the
standard apartment. Hence the group $B_{W}^{\pm}W^{even}$ contains all
elements in $G$ which preserve both ends of the standard apartments $A^{\pm}$
in $\mathcal{B}^{\pm}$. Hence we have

$W^{even}B_{W}^{\pm}=B_{W}^{\pm}W^{even}=B_{1,\infty}^{\pm}\cap
B_{2,\infty}^{\pm}\,.$

∎

######  Proposition 9.12.

We have $U_{2}\cong\\{f:\mathbb{Z}\to{\mathbb{C}}\ |\ supp(f)<\infty\\}$ under
pointwise addition.

###### Proof.

We assume a choice of real root vectors
$\\{e_{\alpha}\in\mathfrak{g}_{\alpha}\ |\ \alpha\in{\Phi}^{re}\\}$ has been
made as follows. We begin by letting $e_{\alpha_{i}}=e_{i}$ be the simple root
generators of $\mathfrak{g}$, and then set
$e_{-\alpha_{i}}=\widetilde{w}_{i}(e_{i})=-f_{i}$. Application of additional
generators $\widetilde{w}_{j}$ fills out the rest of the real root vectors,
and in this way we have arranged it so that we have
$\widetilde{w}_{i}(e_{\alpha})=e_{w_{i}\alpha}$ for $i=1,2$ for all
$\alpha\in{\Phi}^{re}$. Each element of $u\in U_{2}$ is a finite product of
elements $exp(ad_{z_{\alpha}e_{\alpha}})$ for $\alpha\in{\Phi}_{2}$,
$z_{\alpha}\in{\mathbb{C}}$, but we have a labeling by the integers of
${\Phi}_{2}$, so $u$ can be written as

$u=\prod_{n\in\mathbb{Z}}exp(ad_{z_{n}e_{{\Phi}_{2}(n)}})$

where only a finite number of the $z_{n}$ are non-zero, and the order of the
factors can be decreasing since $U_{2}$ is abelian. We associate $u$ with the
function $f_{u}:\mathbb{Z}\to{\mathbb{C}}$ such that $f_{u}(n)=z_{n}$, so
$supp(f_{u})<\infty$. Conversely, each function $f:\mathbb{Z}\to{\mathbb{C}}$
with $supp(f)<\infty$ defines an element

$u_{f}=\prod_{n\in\mathbb{Z}}exp(ad_{f(n)e_{{\Phi}_{2}(n)}})\in U_{2}.$

The pointwise addition of two such functions corresponds to the product in
$U_{2}$, that is, for any $u,v\in U_{2}$, $f_{uv}=f_{u}+f_{v}$. ∎

From now on we use the functional notation $f_{u}$ for $u\in U_{2}$, and
introduce some new notations for $supp(f_{u})$ in order to express its action
on $A^{\pm}$. Define $Max(f_{u})=Max\\{n\in\mathbb{Z}\ |\ f_{u}(n)\neq 0\\}$
and $Min(f_{u})=Min\\{n\in\mathbb{Z}\ |\ f_{u}(n)\neq 0\\}$ so that

$supp(f_{u})=\\{Max(f_{u})=m_{1}>m_{2}>\cdots>m_{r}=Min(f_{u})\\}.$

We call these indices the hinges of $u$ and for each hinge $m_{j}$, we call
$f_{u}(m_{j})$ the direction of that hinge, so that this information
determines the apartment $uA^{\pm}$ with end $e^{\pm}_{2,\infty}$ as a
deformation of $A^{\pm}$. For $f_{u}$ for $u\in U_{2}$, note that

$L^{\pm}_{ray}(m_{1})\subset L^{\pm}_{ray}(m_{2})\subset\cdots\subset
L^{\pm}_{ray}(m_{r})$

and $U_{\pm{\Phi}_{2}(m_{j})}$ fixes $L^{\pm}_{ray}(m_{j})$, so $u$ fixes
$L^{\pm}_{ray}(m_{1})$ and moves the segments of chambers $(m_{1},m_{2}]$,
$(m_{2},m_{3}]$, $\cdots$, $(m_{r-1},m_{r}]$, $(m_{r},e_{1})$, whose union is
the ray $R^{\pm}_{ray}(m_{1}+1)$. So

$uA^{\pm}=L^{\pm}_{ray}(m_{1})\cup u(m_{1},m_{2}]\cup
u(m_{2},m_{3}]\cup\cdots\cup u(m_{r-1},m_{r}]\cup uR^{\pm}_{ray}(m_{1}+1)$

and at each hinge $m_{j}$ we have a direction
$f_{u}(m_{j})=z_{j}\in{\mathbb{C}}$. Thus, $uA^{\pm}$ is the apartment with
ends $e_{2}$ and $ue_{1}$. We have used the abbreviations $(m_{j},m_{j+1}]$
for a sequence of connected chambers with labels corresponding to that
interval of labels, and $e_{i}$ for $e^{\pm}_{i,\infty}$. See Figure 4 for an
illustration.

Figure 4: Apartments $A=A^{\pm}$ and $uA$ for $u\in U_{2}$

with $supp(f_{u})=\\{m_{1}>m_{2}>\cdots>m_{r}\\}$ and $f_{u}(m_{j})=z_{j}$.

We could have associated a function $f:\mathbb{Z}\to{\mathbb{C}}$ with finite
support with an element of $U_{1}$ or $U_{2}$, but that would have caused some
confusion about the choice. We know that $w_{i}$, $i=1,2$, each switch the
ends of $A^{\pm}$, their action on the chambers is by

$w_{1}\mathcal{C}^{\pm}(n)=\mathcal{C}^{\pm}(1-n)\quad\hbox{and}\quad
w_{2}\mathcal{C}^{\pm}(n)=\mathcal{C}^{\pm}(-1-n),$

and for any $n\in\mathbb{Z}$ we have

$\widetilde{w}_{1}exp(ad_{f(n)e_{{\Phi}_{2}(n)}})\widetilde{w}_{1}^{-1}=exp(ad_{f(n)\widetilde{w}_{1}e_{{\Phi}_{2}(n)}})=exp(ad_{f(n)e_{w_{1}{\Phi}_{2}(n)}})=exp(ad_{f(n)e_{{\Phi}_{1}(1-n)}})$

so

$\widetilde{w}_{1}u_{f}\widetilde{w}_{1}^{-1}=\prod_{n\in\mathbb{Z}}exp(ad_{f(n)e_{{\Phi}_{1}(1-n)}})\in
U_{1}$

allows us to write elements of $U_{1}$ as
$\widetilde{w}_{1}u_{f}\widetilde{w}_{1}^{-1}$. There is a similar formula
using $\widetilde{w}_{2}$:

$\widetilde{w}_{2}u_{f}\widetilde{w}_{2}^{-1}=\prod_{n\in\mathbb{Z}}exp(ad_{f(n)e_{{\Phi}_{1}(-1-n)}})\in
U_{1}.$

We wish to use these descriptions of $U_{1}$ and $U_{2}$, and their actions on
the tree, to understand what are all possible ends of apartments in
$\mathcal{B}^{\pm}$. The problem is that $G=\langle U_{1},U_{2}\rangle$ so
that in principle, we have to consider all finite sequences of products
$U_{1}U_{2}U_{1}U_{2}\cdots U_{1}e_{i}$ to get all possible ends. These divide
into those coming from $U_{1}e_{2}$ and those coming from $U_{2}e_{1}$. The
situation is symmetric so it would be enough to understand whether
$U_{1}U_{2}e_{1}$ reaches more ends than $U_{2}e_{1}$. It seems unlikely that
$U_{1}U_{2}e_{1}=U_{2}e_{1}$, that is, the ends that can be reached by the
action of $U_{2}$ on $e_{1}$ are all the ends of the building other than
$e_{2}$. The ends $\\{ue_{1}\ |\ u\in U_{2}\\}$ must all be distinct, since
otherwise there would be a loop in the tree, and none of them can be $e_{2}$
since $ue_{2}=e_{2}=ue_{1}$ would give $e_{1}=e_{2}$. Similarly, the ends
$\\{ue_{2}\ |\ u\in U_{1}\\}$ must all be distinct, and none of them can be
$e_{1}$. Thus it appears that the set of all ends of the building
$\mathcal{B}^{\pm}$, denoted by $Ends(\mathcal{B}^{\pm})$, includes at least

$\\{e_{2,\infty}^{\pm}\\}\cup\\{u_{2}e_{1,\infty}^{\pm}\ |\ u_{2}\in
U_{2}\\}\quad\hbox{and}\quad\\{e_{1,\infty}^{\pm}\\}\cup\\{u_{1}e_{2,\infty}^{\pm}\
|\ u_{1}\in U_{1}\\}.$

######  Conjecture 9.13.

Let $A_{g}^{\pm}=g\cdot A^{\pm}$ for some $g\in B_{i,\infty}^{\pm}$. Then