# On convexity and the Iwasawa decomposition of split real and complex Kac-
Moody groups

Paul Zellhofer  Christian-Albrechts-Universität zu Kiel  Ralf Köhl  Christian-
Albrechts-Universität zu Kiel

(December 5, 2023)

###### Abstract

We prove an analogue of Kostant’s convexity theorem for split real and complex
Kac-Moody groups associated to free and cofree root data. The result can be
seen as a first step towards describing the multiplication map in a Kac-Moody
group in terms of Iwasawa coordinates. Our method involves a detailed analysis
of the geometry of Weyl group orbits in the Cartan subalgebra of a real Kac-
Moody algebra. It provides an alternative proof of Kostant convexity for
semisimple Lie groups and also generalizes a linear analogue of Kostant’s
theorem for Kac-Moody algebras that has been established by Kac and Peterson
in 1984.

## Introduction

Kostant convexity is a fundamental property of semisimple Lie groups that has
been established by B. Kostant in [Kos73]. The origins of Kostant’s theorem
are related to the following problem. Let $G$ be a connected semisimple real
Lie group with Lie algebra $\mathfrak{g}$ and
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ a Cartan decomposition of
$\mathfrak{g}$, where $\mathfrak{k}$ is the fixed-point set of a Cartan
involution $\theta:\mathfrak{g}\to\mathfrak{g}$ and $\mathfrak{p}$ is the
$(-1)$-eigenspace of $\theta$. If $\mathfrak{a}$ is a maximal abelian subspace
of $\mathfrak{p}$, then the adjoint action of $\mathfrak{a}$ on $\mathfrak{g}$
is diagonalizable and the set $\Phi\subset\mathfrak{a}^{\ast}$ of non-zero
eigenvalues are the roots of $(\mathfrak{g},\mathfrak{a})$. It is an abstract
root system in $\mathfrak{a}^{\ast}$ whose Weyl group $W$ is finite. Denote by
$\mathfrak{u}_{+}$ the subalgebra of $\mathfrak{g}$ generated by all positive
root spaces, then
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{u}_{+}$ as vector
spaces. This is known as the Iwasawa decomposition of $\mathfrak{g}$. It is a
fundamental result in the theory of Lie groups that the Cartan involution
$\theta$ lifts to an involutive automorphism $\Theta:G\to G$ with differential
$\theta$. Its fixed-point set $K$ is a maximal compact subgroup of $G$ (if the
center of $G$ is finite) with Lie algebra $\mathfrak{k}$ which contains all
topological information about $G$ in the sense that the quotient space $G/K$
is contractible. In fact, $G/K$ is a simply connected Riemannian symmetric
space of non-compact type. The exponential map $\exp:\mathfrak{g}\to G$ is
injective on $\mathfrak{a}$ and $\mathfrak{u}_{+}$ and their exponential
images $A$ and $U_{+}$ are contractible subgroups of $G$. Here, $A$ is abelian
and $U_{+}$ is unipotent in the sense that its Lie algebra $\mathfrak{u}_{+}$
is nilpotent. The multiplication map $K\times A\times U_{+}\to G$ is a
diffeomorphism which is called the Iwasawa decomposition of $G$.
However, this map is not a homomorphism of groups which raises the question
whether it is possible to describe the multiplication in $G$ in Iwasawa
coordinates. More precisely, given two elements $g_{1}=k_{1}a_{1}u_{1}$ and
$g_{2}=k_{2}a_{2}u_{2}$ of $G$, is it possible to compute the Iwasawa
coordinates of the product $g_{1}g_{2}$ from the ones of $g_{1}$ and $g_{2}$?
The major difficulty in this problem is the fact that $K$ does not normalize
$A$ and $U_{+}$. A simpler question is to focus on one of the components and
restrict $g_{1}$ and $g_{2}$ to certain subgroups of $G$. In applications, one
is often interested in the $A$-component and we denote by $\Pi:G\to
A,\,g=kau\mapsto a$ the corresponding projection. It is clear that
$\Pi(kg)=\Pi(g)$ holds for all $k\in K$ and $g\in G$, but there is no obvious
way how to compute $\Pi(gk)$. Using the Cartan decomposition $G=KAK$, this
problem can be reduced to the case $g=a\in A$. Kostant’s convexity theorem
asserts that if $a\in A$ and $k\in K$ are arbitrary, then $\Pi(ak)$ is
contained in the (multiplicative) convex hull of the Weyl group orbit of $a$
and, as $k$ runs through all of $K$, every point of this convex hull is
attained in this way. More precisely, [Kos73, Theorem 4.1] states that

$\\{\Pi(ak)\mid k\in K\\}=\exp(\operatorname{conv}(W\cdot\log(a))).$ (0.1)

Kac-Moody groups are a natural generalization of semisimple Lie groups and
inherit many of their properties, often by construction. It has thus been
conjectured that a suitable analogue of Kostant’s theorem might be true in the
Kac-Moody setting as well. This question is also related to the problem of
determining whether the causal pre-order on Kac-Moody symmetric spaces is a
partial order or trivial (cf. [FHHK20, Proposition 7.46]). In general,
however, Kac-Moody groups carry no differentiable structure and an exponential
map between a Kac-Moody algebra and group cannot be globally defined, so it is
far more difficult to “linearize” problems about Kac-Moody groups.
In this article we prove an analogue of Kostant’s convexity theorem for
certain classes of split real and complex Kac-Moody groups. The result is
formulated as follows. Let $\mathbb{A}$ be a symmetrizable generalized Cartan
matrix with Weyl group $W$ and let $\mathcal{D}$ be a free and cofree Kac-
Moody root datum associated to $\mathbb{A}$. If
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$, the split minimal Kac-Moody group
$G$ over $\mathbb{K}$ of type $\mathcal{D}$ has an Iwasawa decomposition
$G=KAU_{+}$ generalizing the classical decomposition of a semisimple Lie
group. Here, $K$ is the fixed-point set of an involutive automorphism of $G$,
$A$ is the exponential image of the Cartan subalgebra $\mathfrak{a}$ of the
real Kac-Moody algebra of type $\mathcal{D}$ and $U_{+}$ is the subgroup of
$G$ generated by all positive root groups. The Weyl group $W$ acts on
$\mathfrak{a}$ and $A$ and the exponential map $\exp:\mathfrak{a}\to A$ is a
$W$-equivariant group isomorphism. The subset of $\mathfrak{a}$ where all
simple roots are non-negative is called the fundamental chamber and the union
$X$ of all its $W$-translates, called Weyl chambers, is the Tits cone of
$\mathfrak{a}$. In Theorem 4.1 we prove the following result.

######  Main Theorem.

Let $G$ be a free and cofree, split, minimal, real or complex Kac-Moody group,
$G=KAU_{+}$ an Iwasawa decomposition of $G$ and $\Pi:G\to A$ the associated
projection. Let $W$ denote the Weyl group of $\mathbb{A}$. If $a\in A$ is
contained in the exponential image of an open chamber of the Tits cone
$X\subset\mathfrak{a}$, then

$\\{\Pi(ak)\mid k\in K\\}=\exp(\operatorname{conv}(W\cdot\log(a))).$

If $\mathbb{A}$ is the Cartan matrix of a finite-dimensional complex
semisimple Lie algebra, then $G$ is a reductive Lie group and the result is
very similar to Kostant’s original theorem. In this case, the Weyl group $W$
is finite and $X=\mathfrak{a}$. In the general case, however, $W$ is typically
infinite and $X$ is a proper subset of $\mathfrak{a}$. Thus, it is a priori
not clear whether the set
$\operatorname{conv}(W\cdot\log(a))\subset\mathfrak{a}$ is well-behaved in any
sense. A major part of the proof of Theorem 4.1 consists of understanding the
structure of this set. Our approach is inspired by the methods of [Kos73]. To
obtain equality between the sets in (0.1), Kostant uses tools from the
representation theory of $G$ and an inductive argument. We proceed similarly,
but our proof avoids tools that are particular to Lie theory and instead lays
more emphasis on combinatorial properties of the Weyl group $W$, which is a
(generally infinite) Coxeter group with respect to a finite generating set
$S$. In particular, we relate the facial structure of
$\operatorname{conv}(W\cdot\log(a))$ to the Coxeter complex of $(W,S)$ and
present an inductive argument that allows to compute the Iwasawa coordinates
of an element of the form $ak_{1}k_{2}$ with $k_{1},k_{2}\in K$ by iteration.
In the Lie group case, our methods can even be extended in a suitable way to
provide a new proof of Kostant’s original result. However, Kac-Moody groups
usually do not admit a Cartan decomposition $G=KAK$, so our result does not
yield a description of $\\{\Pi(gk)\mid k\in K\\}$ for an arbitrary $g\in G$.
Furthermore, if $\mathbb{A}$ is of affine type, then $G$ is an example of a
group with an affine $BN$-pair. Such groups are always endowed with a
different variant of the Iwasawa decomposition and act on an affine building
which can be viewed as a discrete analogue of a symmetric space. Kostant’s
convexity theorem has been transferred to the setting of affine buildings by
P. Schwer111née Hitzelberger in [Hit10] and generalized in [Hit11]. Hence,
real and complex Kac-Moody groups of affine type can be endowed with two
different Iwasawa decompositions that both satisfy an analogue of Kostant’s
convexity theorem. The result of Schwer extends far beyond this setting as it
is formulated entirely in the language of buildings and does not require any
group action to begin with. In addition, it also applies to groups with an
affine $BN$-pair outside the Kac-Moody realm such as semisimple algebraic
groups over non-Archimedean local fields.
The text is organized as follows. We begin in Chapter 1 by reviewing some
basic concepts from the theory of Kac-Moody algebras and Coxeter groups. We
fix the notation that will be used throughout our discussion and introduce
some standing assumptions that will be necessary in the proof of Theorem 4.1.
In Chapter 2 we study the geometry of convex hulls of Weyl group orbits in
$\mathfrak{a}$ and obtain results that lie at the heart of the proof of our
main theorem. For this, we require some notions from convex geometry that we
collect in Section 2.1. Chapter 3 explains the fundamental properties of Kac-
Moody groups that are used in Section 4.1 where we prove Theorem 4.1.
Afterwards, we formulate a “linear” analogue of our main result in Section 4.2
which is closely related to a convexity theorem for Kac-Moody algebras that
was established by V.G. Kac and D.H. Peterson in [KP84]. We close our
discussion in Section 4.3 by comparing Theorem 4.1 to Kostant’s original
result and mentioning some possible generalizations and open problems.
The present article is part of the first author’s PhD-project which is
supported financially by the Studienstiftung des deutschen Volkes. He would
also like to thank the research group Algebra of CAU Kiel for many valuable
contributions both inside and outside of mathematics. Both authors thank
Timothée Marquis for various remarks on a preliminary version of this article.

##  1 Kac-Moody algebras

We begin by collecting the most important properties of Kac-Moody algebras and
fix the setting in which we are going to work in the following chapters. We
assume that the reader is familiar with the basic theory of these algebras and
standard terminology in the subject. More details, definitions and proofs of
all assertions in this chapter can be found in [Kac90, Chapter 1-3, 5, 9, 11]
and [Mar18, Chapter 3, 4, 6, 7]. Our notation and conventions mostly follow
[Mar18]. We also freely use standard terminology from the theory of Coxeter
groups (cf. [AB08, Chapter 2-3] or [BB05, Chapter 1-2]). In addition, we
employ the convention that we do not consider $0$ to be a natural number, i.e.
$\mathbb{N}=\\{1,2,3,\ldots\\}$, and write $\mathbb{N}_{0}$ for the set
$\\{0,1,2,3,\ldots\\}$ if that should be necessary. If $n\in\mathbb{N}$ is a
natural number that is fixed by the context, we set $I:=\\{1,2,\ldots,n\\}$.
A generalized Cartan matrix (“GCM”) is an integer matrix
$\mathbb{A}=(a_{ij})_{i,j\in I}\in\mathbb{Z}^{n\times n}$ which satisfies the
following three conditions for all $i,j\in I$: (i) $a_{ii}=2$, (ii)
$a_{ij}\leq 0$ for $i\neq j$, (iii) $a_{ij}=0$ if and only if $a_{ji}=0$. It
is said to be symmetrizable if there exist a diagonal matrix $D$ and a
symmetric matrix $B$ of size $n\times n$ such that $\mathbb{A}=DB$.
A Kac-Moody root datum is a quintuple
$\mathcal{D}=(I,\mathbb{A},\Lambda,(c_{i})_{i\in I},(h_{i})_{i\in I})$, where
$I$ is a set indexing a GCM $\mathbb{A}$ of size $n=|I|$, $\Lambda$ is a free
$\mathbb{Z}$-module of rank $d\in\mathbb{N}$, $(c_{i})_{i\in I}$ is a family
of elements of $\Lambda$ and $(h_{i})_{i\in I}$ is a family in the dual
$\mathbb{Z}$-module $\Lambda^{\vee}$ such that the natural pairing between
$\Lambda$ and $\Lambda^{\vee}$ satisfies $\langle c_{j},h_{i}\rangle=a_{ij}$
for all $i,j\in I$. We say that $\mathcal{D}$ is free (resp. cofree) if
$(c_{i})_{i\in I}\subset\Lambda$ (resp. $(h_{i})_{i\in
I}\subset\Lambda^{\vee}$) is linearly independent. Let $\mathbb{K}$ be a
field. The Kac-Moody algebra of type $\mathcal{D}$ over $\mathbb{K}$ is the
free Lie algebra $\mathfrak{g}$ over $\mathbb{K}$ generated by
$\mathfrak{h}:=\Lambda^{\vee}\otimes_{\mathbb{Z}}\mathbb{K}$ and $2n$ symbols
$\\{e_{i},f_{i}\mid i\in I\\}$ subject to the following relations:

$\displaystyle[h,h^{\prime}]$ $\displaystyle=0\text{ for
$h,h^{\prime}\in\mathfrak{h}$}$ (1.1) $\displaystyle[h,e_{i}]$
$\displaystyle=\langle c_{i},h\rangle e_{i}\text{ and }[h,f_{i}]=-\langle
c_{i},h\rangle f_{i}\text{ for $h\in\mathfrak{h}$ and $i\in I$}$
$\displaystyle[e_{i},f_{j}]$ $\displaystyle=-\delta_{ij}h_{i}\text{ for
$i,j\in I$}$ $\displaystyle\operatorname{ad}(e_{i})^{1-a_{ij}}e_{j}$
$\displaystyle=\operatorname{ad}(f_{i})^{1-a_{ij}}f_{j}=0\text{ for $i,j\in I$
with $i\neq j$}$

We call $\mathfrak{h}$ the Cartan subalgebra and $\\{e_{i},f_{i}\mid i\in
I\\}$ the Chevalley generators of $\mathfrak{g}$. The Chevalley involution of
$\mathfrak{g}$ is the involutive $\mathbb{K}$-linear Lie algebra automorphism
$\omega:\mathfrak{g}\to\mathfrak{g}$ defined by $\omega(e_{i}):=f_{i}$ and
$\omega(f_{i}):=e_{i}$ for all $i\in I$ and $\omega(h):=-h$ for all
$h\in\mathfrak{h}$. Let $\Phi_{0}:=\\{\alpha_{i}\mid i\in I\\}$ and
$\Phi_{0}^{\vee}:=\\{\alpha_{i}^{\vee}\mid i\in I\\}$ be two abstract sets and
denote by

$Q:=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}\qquad\text{and}\qquad
Q^{\vee}:=\bigoplus_{i\in I}\mathbb{Z}\alpha_{i}^{\vee}$

the free abelian groups generated by them. Then $\mathfrak{g}$ has a natural
$Q$-gradation

$\mathfrak{g}=\bigoplus_{\alpha\in Q}\mathfrak{g}_{\alpha}$

induced by the assignment
$\operatorname{deg}(e_{i})=-\operatorname{deg}(f_{i}):=\alpha_{i}$ for $i\in
I$ and $\operatorname{deg}(h):=0$ for $h\in\mathfrak{h}$. The elements of
$\Phi:=\\{\alpha\in Q\setminus\\{0\\}:\mathfrak{g}_{\alpha}\neq\\{0\\}\\}$ are
the roots of $\mathfrak{g}$, $\Phi_{0}\subset\Phi$ is the set of simple roots
and $\Phi_{0}^{\vee}$ is the set of simple coroots. Every root is a
$\mathbb{Z}$-linear combination of simple roots with coefficients of the same
sign, which defines two sets $\Phi_{+},\Phi_{-}\subset\Phi$, where

$\Phi_{\pm}:=\Phi\cap Q_{\pm}\qquad\text{and}\qquad
Q_{\pm}:=\pm\bigoplus_{i\in I}\mathbb{N}_{0}\alpha_{i},$

that satisfy $\Phi_{+}=-\Phi_{-}$ and $\Phi=\Phi_{+}\sqcup\Phi_{-}$. We
introduce a partial order on $Q$ by declaring $\alpha\leq\beta$ if and only if
$\beta-\alpha\in Q_{+}$. For $\alpha=\sum_{i\in I}k_{i}\alpha_{i}\in Q$ with
$k_{1},\ldots,k_{n}\in\mathbb{Z}$, the support and height of $\alpha$ are
defined by

$\operatorname{supp}(\alpha):=\\{i\in I:k_{i}\neq
0\\}\qquad\text{and}\qquad\operatorname{ht}(\alpha):=\sum_{i\in I}k_{i}$

and one defines the same notions for elements of $Q^{\vee}$ in the obvious
way. There exist canonical $\mathbb{Z}$-linear maps $c:Q\to\Lambda$ and
$h:Q^{\vee}\to\Lambda^{\vee}$ that satisfy $c(\alpha_{i})=c_{i}$ and
$h(\alpha_{i}^{\vee})=h_{i}$ for all $i\in I$. If $\mathcal{D}$ is free (resp.
cofree), then $c$ (resp. $h$) is injective. The Cartan subalgebra
$\mathfrak{h}=\mathfrak{g}_{0}\subset\mathfrak{g}$ acts diagonally on
$\mathfrak{g}$ via the adjoint action and we have
$\mathfrak{g}_{\alpha}\subset\\{x\in\mathfrak{g}:[h,x]=\langle
c(\alpha),h\rangle x\,\,\forall h\in\mathfrak{h}\\}$ for all $\alpha\in\Phi$,
but this inclusion is strict if $\mathcal{D}$ is not free. Hence, the
eigenspace decomposition for the adjoint action of $\mathfrak{h}$ is usually
coarser than the $Q$-gradation of $\mathfrak{g}$.

######  Remark 1.1.

It is immediate from the property $\langle c_{j},h_{i}\rangle=a_{ij}$ for all
$i,j\in I$ in the definition of a Kac-Moody root datum that if $\mathcal{D}$
is not cofree (resp. free), then the rows (resp. columns) of the GCM
$\mathbb{A}$ are linearly dependent over $\mathbb{Z}$. Conversely, every
$\mathbb{Z}$-linear dependence between the rows (resp. columns) of
$\mathbb{A}$ can be used to construct a root datum of rank $n$ that is not
cofree (resp. free). Hence, a GCM is invertible if and only if all associated
root data are free and cofree. In the non-invertible case, there always exist
Kac-Moody root data of rank $d<n$ which are therefore neither free nor cofree.
If $\mathbb{A}$ is arbitrary, there exists a Kac-Moody root datum
$\mathcal{D}^{\mathbb{A}}_{\text{Kac}}$ that is free and cofree such that
$d=2n-\operatorname{rank}(\mathbb{A})$ (cf. [Mar18, Example 7.10]). It is
minimal with these properties in the sense that every free and cofree root
datum has to satisfy $d\geq 2n-\operatorname{rank}(\mathbb{A})$. The Kac-Moody
algebra of type $\mathcal{D}^{\mathbb{A}}_{\text{Kac}}$ over $\mathbb{C}$
coincides with the algebra $\mathfrak{g}(\mathbb{A})$ defined in [Mar18,
Definition 3.17]. Similarly, there exists a cofree Kac-Moody root datum
$\mathcal{D}^{\mathbb{A}}_{\text{sc}}$ of rank $d=n$ such that the associated
complex Kac-Moody algebra is the derived algebra of $\mathfrak{g}(\mathbb{A})$
(cf. [Mar18, Example 7.11]). If $\mathbb{A}$ is invertible, these root data
are the same. If $\mathbb{A}$ is symmetrizable, then by the Gabber-Kac theorem
(cf. [Kac90, Theorem 9.11]), these two definitions also coincide (up to some
minor conventions; see [Mar18, Remark 3.19]) with the “classical” Kac-Moody
algebra (resp. its derived algebra) that was first introduced by V.G. Kac and
studied in [Kac90].

The Weyl group associated to $\mathbb{A}$ is the group $W$ generated by the
set $S:=\\{s_{i}\mid i\in I\\}$ subject to the relations
$(s_{i}s_{j})^{m_{ij}}=1$ for all $i,j\in I$, where $m_{ij}$ is equal to $1$
if $i=j$ and determined by the following table if $i\neq j$:

$\begin{array}[]{c|c|c|c|c|c}a_{ij}a_{ji}&0&1&2&3&\geq 4\\\ \hline\cr
m_{ij}&2&3&4&6&\infty\end{array}$ (1.2)

By definition, $(W,S)$ is a Coxeter system with Coxeter matrix
$M=(m_{ij})_{i,j\in I}$ and we denote by $l:W\to\mathbb{N}_{0}$ the length
function of $W$ with respect to the generating set $S$. The Weyl group
operates on $Q$ and $\Lambda$, where the action of the generator $s_{i}$
$(i\in I)$ is given by the following maps:

$\displaystyle Q$ $\displaystyle\to Q$ $\displaystyle\qquad\Lambda$
$\displaystyle\to\Lambda$ $\displaystyle\beta$
$\displaystyle\mapsto\beta-\langle\beta,\alpha_{i}^{\vee}\rangle\alpha_{i}$
$\displaystyle\qquad\lambda$
$\displaystyle\mapsto\lambda-\langle\lambda,h_{i}\rangle c_{i}$

By duality, $W$ also acts on $Q^{\vee}$ and $\Lambda^{\vee}$ such that the
natural pairings between $Q$ and $Q^{\vee}$ (resp. $\Lambda$ and
$\Lambda^{\vee})$ are $W$-invariant. In particular, these actions induce
$\mathbb{K}$-linear representations of $W$ on $\mathfrak{h}$ and
$\mathfrak{h}^{\ast}$ by extension of scalars. By definition, the canonical
maps $c:Q\to\Lambda$ and $h:Q^{\vee}\to\Lambda^{\vee}$ are $W$-equivariant.
The $W$-actions on $Q$ and $Q^{\vee}$ are faithful while the actions on
$\Lambda$ and $\Lambda^{\vee}$ (and hence on $\mathfrak{h}$ on
$\mathfrak{h}^{\ast}$) are not necessarily faithful if $\mathcal{D}$ is
neither free nor cofree. Since the actions of $W$ on $\Lambda$ and
$\Lambda^{\vee}$ are dual, it suffices that one of the maps $c$ or $h$ is
injective, i.e. that $\mathcal{D}$ is either free or cofree, in order to
obtain a faithful action. The set of roots $\Phi\subset Q$ is $W$-invariant
and the subset $\Phi^{\operatorname{re}}:=W\cdot\Phi_{0}\subset\Phi$ is called
the set of real roots. Its complement
$\Phi^{\operatorname{im}}:=\Phi\setminus\Phi^{\operatorname{re}}$, called the
set of imaginary roots, will not play a major role in our discussion. If
$\alpha\in\Phi^{\operatorname{re}}$, i.e. $\alpha=w(\alpha_{i})$ for some
$w\in W$ and $i\in I$, the element $\alpha^{\vee}:=w(\alpha_{i}^{\vee})\in
Q^{\vee}$ is independent of the representation of $\alpha$ as a $W$-translate
of a simple root and the associated reflection

$s_{\alpha}:Q\to
Q,\qquad\beta\mapsto\beta-\langle\beta,\alpha^{\vee}\rangle\alpha$

satisfies $s_{\alpha}=ws_{i}w^{-1}\in W$. For every $w\in W$ and $i\in I$, we
have

$w(\alpha_{i})\in\Phi_{+}\Longleftrightarrow l(w)<l(ws_{i}).$ (1.3)

We use the following standard terminology from the theory of Coxeter groups.
Given an element $w\in W$, there is a unique set $S(w)\subset S$ such that
every reduced decomposition of $w$ contains precisely the reflections in
$S(w)$. We call $\operatorname{supp}(w):=\\{i\in I:s_{i}\in S(w)\\}$ the
support of $w$. For any subset $J\subset I$, the subgroup $W_{J}$ of $W$
generated by $S_{J}:=\\{s_{j}\mid j\in J\\}$ is called a standard subgroup of
$W$. A left coset $wW_{J}$ with $w\in W$ is called a standard coset. If
$J=I\setminus\\{i\\}$ for some $i\in I$, we write $S^{(i)}$ and $W^{(i)}$
instead of $S_{I\setminus\\{i\\}}$ and $W_{I\setminus\\{i\\}}$. The Coxeter
complex of $(W,S)$ is the partially-ordered set (“poset”) of all standard
cosets of $(W,S)$ ordered by reversed inclusion and denoted by

$\Sigma(W,S)=\\{wW_{J}\mid w\in W,\,J\subset I\\}.$

We also write $\Sigma^{\ast}(W,S)$ for the poset with the same underlying set
but ordered by non-reversed inclusion. Every standard coset $wW_{J}\subset W$
has a unique element $w^{\prime}$ of minimal length which is characterized by
$l(w^{\prime}w_{J})=l(w^{\prime})+l(w_{J})$ for all $w_{J}\in W_{J}$.
Next, we introduce some notions from the representation theory of
$\mathfrak{g}$ (cf. [Mar18, Section 4.1] or [Kac90, Chapter 3 and 9]). A
representation $\rho$ of $\mathfrak{g}$ on a $\mathbb{K}$-vector space $V$ is
said to be integrable if $\mathfrak{h}$ acts diagonally on $V$ and for every
$i\in I$ and $v\in V$ there exists some $N\in\mathbb{N}$ such that
$\rho(e_{i})^{N}v=\rho(f_{i})^{N}v=0$. A functional
$\lambda\in\mathfrak{h}^{\ast}$ is called a dominant integral weight if it
satisfies $\langle\lambda,h_{i}\rangle\in\mathbb{N}_{0}$ for all $i\in I$. For
every such weight, there exists a unique irreducible integrable
$\mathbb{K}$-linear representation
$\rho_{\lambda}:\mathfrak{g}\to\mathfrak{gl}(L(\lambda))$ with highest weight
$\lambda$. This means that there exists a non-zero vector $v_{\lambda}\in
L(\lambda)$, which is unique up to scalars, such that
$\rho_{\lambda}(h)v_{\lambda}=\langle\lambda,h\rangle v_{\lambda}$ for all
$h\in\mathfrak{h}$, the subalgebra
$\mathfrak{u}_{+}:=\bigoplus_{\alpha\in\Phi_{+}}\mathfrak{g}_{\alpha}\subset\mathfrak{g}$
annihilates $v_{\lambda}$ and $L(\lambda)$ is generated by $v_{\lambda}$ as a
representation of the universal enveloping algebra of $\mathfrak{g}$. The
Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ acts diagonally on
$L(\lambda)$, its joint eigenvalues
$\operatorname{wt}(L(\lambda))\subset\mathfrak{h}^{\ast}$ are the weights of
$L(\lambda)$ and the joint eigenspace decomposition

$L(\lambda)=\bigoplus_{\mu\in\operatorname{wt}(L(\lambda))}L(\lambda)_{\mu}$

is the weight space decomposition of $L(\lambda)$. Every weight is of the form
$\lambda-c(\alpha)$ for some $\alpha\in Q_{+}$ and the set of weights is
$W$-invariant. If $\mu$ is any weight and $x\in\mathfrak{g}_{\alpha}$ for some
$\alpha\in\Phi$, then it is immediate from the defining relations (1.1) of
$\mathfrak{g}$ that

$\rho_{\lambda}(x)(L(\lambda)_{\mu})\subset L(\lambda)_{\mu+c(\alpha)}.$ (1.4)

Depending on the Kac-Moody root datum $\mathcal{D}$, there does not
necessarily exist a non-zero dominant integral weight. If $\mathcal{D}$ is
cofree, however, then $\\{h_{i}\mid i\in I\\}$ can be extended to a
$\mathbb{K}$-basis of $\mathfrak{h}$ and the corresponding dual basis consists
of dominant integral weights. If $\langle\lambda,h_{i}\rangle=0$ for all $i\in
I$, then the defining relations of $\mathfrak{g}$ imply that we may define a
representation $\rho_{\lambda}$ of $\mathfrak{g}$ on the one-dimensional
vector space $\mathbb{K}$ by setting
$\rho_{\lambda}(e_{i})=\rho_{\lambda}(f_{i}):=0$ for all $i\in I$ and
$\rho_{\lambda}(h)1:=\langle\lambda,h\rangle$ for all $h\in\mathfrak{h}$. This
clearly defines an irreducible representation of $\mathfrak{g}$ with highest
weight $\lambda$. Since $L(\lambda)$ is unique up to isomorphism, it follows
that $L(\lambda)$ is one-dimensional and isomorphic to $\mathbb{K}$ under the
map $v_{\lambda}\mapsto 1$.

######  Assumption 1.2.

From now on, we will always make the following assumptions which will play a
central role in the proof of our main theorem:

  * (i)

The GCM $\mathbb{A}$ is symmetrizable.

  * (ii)

The Kac-Moody root datum $\mathcal{D}$ is free and cofree.

  * (iii)

The field $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$.

The first assumption occurs frequently in the theory of Kac-Moody algebras and
will be used explicitly in the proof of Theorem 4.1. The necessity of the
third will already be apparent in the formulation of that result as it
involves an “exponential function” on the Cartan subalgebra of $\mathfrak{g}$
whose definition is based on the usual real or complex exponential map. In our
discussion, it will usually be possible to treat real and complex Kac-Moody
algebras at the same time, so we do not want to distinguish them in notation.
Hence, for the rest of this article, we denote by $\mathfrak{g}$ either the
real or the complex Kac-Moody algebra of type $\mathcal{D}$. If we need to
emphasize the field, we write $\mathfrak{g}_{\mathbb{R}}$ or
$\mathfrak{g}_{\mathbb{C}}$. It will, however, be very important to
distinguish the real Cartan subalgebra from the complex one, so we will write
$\mathfrak{a}:=\Lambda^{\vee}\otimes_{\mathbb{Z}}\mathbb{R}$ and
$\mathfrak{h}:=\Lambda^{\vee}\otimes_{\mathbb{Z}}\mathbb{C}$. We explicitly do
not denote these algebras by $\mathfrak{h}_{\mathbb{R}}$ and
$\mathfrak{h}_{\mathbb{C}}$ since we often want to view $\mathfrak{a}$ as a
subset of $\mathfrak{g}_{\mathbb{C}}$ and a notation like
$\mathfrak{h}_{\mathbb{R}}\subset\mathfrak{g}_{\mathbb{C}}$ might be
misleading. We denote by $\theta:\mathfrak{g}\to\mathfrak{g}$ the involutive
automorphism of $\mathfrak{g}$ obtained from pre-composing the Chevalley
involution of $\mathfrak{g}$ with complex conjugation. More precisely, if
$\mathbb{K}=\mathbb{R}$, then $\theta$ coincides with the Chevalley involution
$\omega_{\mathbb{R}}$ of $\mathfrak{g}_{\mathbb{R}}$, and if
$\mathbb{K}=\mathbb{C}$, then $\theta$ is the unique extension of
$\omega_{\mathbb{R}}$ to a $\mathbb{C}$-antilinear automorphism of
$\mathfrak{g}_{\mathbb{C}}$. In both cases, we call $\theta$ the compact
involution of $\mathfrak{g}$ and denote its fixed-point set by $\mathfrak{k}$.
Assumption (ii) is a simplification that requires some justification. It
allows us to identify $Q$ and $Q^{\vee}$ with subsets of $\Lambda$ and
$\Lambda^{\vee}$. For every $i\in I$, we will therefore write $\alpha_{i}$ for
$c_{i}$ and $\alpha_{i}^{\vee}$ for $h_{i}$ and suppress the inclusions $c$
and $h$ from the notation. With this convention, we may extend
$\Phi_{0}^{\vee}\subset\Lambda^{\vee}$ to an $\mathbb{R}$-basis
$\mathcal{B}:=\\{\alpha_{1}^{\vee},\ldots,\alpha_{n}^{\vee},h_{n+1},\ldots,h_{d}\\}$
of $\mathfrak{a}$, which also yields a $\mathbb{C}$-basis of $\mathfrak{h}$,
and we denote by
$\mathcal{B}^{\ast}:=\\{\omega_{1},\ldots,\omega_{n},\phi_{n+1},\ldots,\phi_{d}\\}\subset\mathfrak{a}^{\ast}$
the corresponding dual basis. Recall that the assumption that $\mathcal{D}$ is
free and cofree implies $d\geq 2n-\operatorname{rank}(\mathbb{A})\geq n$. The
complex Kac-Moody algebra $\mathfrak{g}_{\mathbb{C}}$ is then simply a direct
product of the Kac-Moody algebra $\mathfrak{g}(\mathbb{A})$ studied in
[Mar18], which coincides with the definition in [Kac90] by assumption (i) and
the Gabber-Kac theorem. More precisely, we have
$\mathfrak{g}_{\mathbb{C}}\cong\mathfrak{g}(\mathbb{A})\oplus\mathfrak{h}^{\prime}$
as a Lie algebra, where $\mathfrak{h}^{\prime}$ is a subspace of
$\mathfrak{h}$ of dimension $d-2n+\operatorname{rank}(\mathbb{A})$ on which
all roots vanish. The obvious analogue of this assertion holds in the real
case. This allows us to use most results of [Kac90] and [Mar18] in our
setting. Simply put, we study Kac-Moody algebras that behave exactly like the
classical ones, except that we allow the Cartan subalgebra to have dimension
larger than $2n-\operatorname{rank}(\mathbb{A})$. As discussed in Remark 1.1,
condition (ii) is always satisfied if $\mathbb{A}$ is invertible.
Moreover, the assumption that $\mathcal{D}$ is free guarantees that the
$\mathbb{R}$-linear representation $W\to\operatorname{GL}(\mathfrak{a})$ is
faithful and the set

$C:=\\{h\in\mathfrak{a}:\langle\alpha_{i},h\rangle>0\,\,\forall i\in I\\}$

is not empty. It is called the open fundamental chamber and its closure
(w.r.t. the standard Euclidean topology on $\mathfrak{a}$)

$\overline{C}:=\\{h\in\mathfrak{a}:\langle\alpha_{i},h\rangle\geq 0\,\,\forall
i\in I\\}$

is the closed fundamental chamber. It can equivalently be described as

$\overline{C}=\bigg{\\{}h\in\mathfrak{a}:\forall w\in
W:h-w(h)=\sum_{i=1}^{n}c_{i}\alpha_{i}^{\vee}\text{ for some
$c_{1},\ldots,c_{n}\in\mathbb{R}_{\geq 0}$}\bigg{\\}}.$ (1.5)

The $W$-translates of $\overline{C}$ are called Weyl chambers. Every such set
$w(\overline{C})\subset\mathfrak{a}$ with $w\in W$ is a simplicial cone which
is bounded by the walls $\\{w(V_{\alpha_{i}})\mid i\in I\\}$, where
$V_{\alpha_{i}}:=\operatorname{ker}(\alpha_{i})$. The union of all Weyl
chambers

$X:=\bigcup_{w\in W}w(\overline{C})\subset\mathfrak{a}$

is the Tits cone of $\mathfrak{a}$. The closed fundamental chamber is a strict
fundamental domain for the $W$-action on $X\subset\mathfrak{a}$, i.e. every
point of $X$ lies in the $W$-orbit of a unique point in $\overline{C}$. If
$h\in\overline{C}$ and $J:=\\{i\in I:\langle\alpha_{i},h\rangle=0\\}$, then
the stabilizer of $h$ is the standard subgroup $W_{J}\subset W$ generated by
the reflections $S_{J}=\\{s_{j}\mid j\in J\\}$.
Finally, Assumption 1.2 also has one important consequence for the
representation theory of $\mathfrak{g}$. If
$\lambda\in\mathfrak{a}^{\ast}\subset\mathfrak{h}^{\ast}$ is a dominant
integral weight, then by [Kac90, Proposition 9.4 and Lemma 11.5] there exists
a non-degenerate Hermitian form $H:L(\lambda)_{\mathbb{C}}\times
L(\lambda)_{\mathbb{C}}\to\mathbb{C}$ which is contravariant in the sense that

$H(x\cdot v,w)=-H(v,\theta(x)\cdot w)\qquad\text{for all
$x\in\mathfrak{g}_{\mathbb{C}}$ and $v,w\in L(\lambda)_{\mathbb{C}}$.}$ (1.6)

It is unique up to scalars and different weight spaces are $H$-orthogonal.
Since $\lambda|_{\mathfrak{a}}$ is real valued, it restricts to a non-
degenerate symmetric bilinear form on $L(\lambda)_{\mathbb{R}}$. If the GCM
$\mathbb{A}$ is symmetrizable, then $H$ is positive definite by [Kac90,
Theorem 11.7]. In particular, if $\mathcal{D}$ is free and cofree, the dual
basis $\mathcal{B}^{\ast}\subset\mathfrak{a}^{\ast}$ consists of dominant
integral weights. For ease of reference, we summarize the most important
consequences of Assumption 1.2 in the following proposition.

######  Proposition 1.3.

Suppose that Assumption 1.2 is in effect, then $d\geq
2n-\operatorname{rank}(\mathbb{A})$ and the following three properties hold:

  * (i)

The dual basis
$\mathcal{B}^{\ast}=\\{\omega_{1},\ldots,\omega_{n},\phi_{n+1},\ldots,\phi_{d}\\}\subset\mathfrak{a}^{\ast}$
consists of dominant integral weights. Each of the real or complex highest-
weight modules
$L(\omega_{1}),\ldots,L(\omega_{n}),L(\phi_{n+1}),\ldots,L(\phi_{d})$ can be
endowed with a positive definite, contravariant, symmetric bilinear or
Hermitian form $H$ such that weight spaces belonging to different weights are
$H$-orthogonal.

  * (ii)

For every $i\in\\{n+1,\ldots,d\\}$, the highest-weight module $L(\phi_{i})$ is
one-dimensional. The subalgebras $\mathfrak{u}_{+}$ and $\mathfrak{u}_{-}$ of
$\mathfrak{g}$ act trivially on $L(\phi_{i})$ and an element
$h\in\mathfrak{h}$ acts on $L(\phi_{i})$ by multiplication with
$\langle\phi_{i},h\rangle$.

  * (iii)

The open fundamental chamber $C\subset\mathfrak{a}$ is not empty.

##  2 Convex hulls of Weyl group orbits

The goal of this chapter is to study the geometry of $W$-orbits in
$\mathfrak{a}$ and relate it to the combinatorial structure of the Coxeter
system $(W,S)$. More precisely, we will be interested in the convex hull of
the $W$-orbit of an element $h\in\mathfrak{a}$:

$\Xi_{h}:=\operatorname{conv}(W\cdot h)$

The structure of this set depends heavily on the location of $h$ with respect
to the double Tits cone $X\cup-X\subset\mathfrak{a}$. One can distinguish the
following three possibilities. We say that $h$ is

  * •

regular if $h$ or $-h$ is contained in an open chamber of $X$.

  * •

singular if $h$ or $-h$ is contained in a wall of $X$.

  * •

external if neither $h$ nor $-h$ is contained in $X$.

In the present article, we will restrict our attention to regular elements,
which exist since $\mathcal{D}$ is assumed to be free. In this case, we shall
see that $\Xi_{h}$ has a simple combinatorial description in terms of the
Coxeter complex $\Sigma(W,S)$. Some of our results do indeed generalize to the
singular case and we briefly discuss this in Section 4.3. We emphasize that
Assumption 1.2 and the notational conventions following it are in effect for
this entire chapter.

######  Remark 2.1.

If the Weyl group $W$ is finite, then $\Xi_{h}$ is a convex polytope (see
Section 2.1 for the definition) and most results of Section 2.2 can be deduced
from the general theory of these sets. However, this is the case if and only
if $\mathbb{A}$ is the Cartan matrix of a finite-dimensional semisimple Lie
algebra which is a very small subclass of Kac-Moody algebras. If $W$ is
infinite, it is not even a priori clear whether $\Xi_{h}$ is a closed subset
of $\mathfrak{a}$.

Convention: From this point onward, the set of imaginary roots
$\Phi^{\operatorname{im}}$ will not appear in our discussion. Thus, we
simplify the notation and denote the set of real roots by $\Phi$ instead of
$\Phi^{\operatorname{re}}$.

###  2.1 Some results from convex geometry

To prepare for the study of $\Xi_{h}$, we begin by collecting some basic
properties of convex sets in finite-dimensional real vector spaces. We follow
the terminology in [Roc70]. More details and proofs of the assertions about
general convex sets can be found in [Roc70, §1, 6, 11, 17, 18] or [Gru07,
Chapter 3-5]. A thorough discussion of convex polytopes that goes way beyond
what is needed for this article is provided by [Gru07, Chapter 14-20].
Let $V$ be a real vector space of dimension $d<\infty$ equipped with its
standard Euclidean topology. For two points $x,y\in V$, we denote by $[x,y]$
the closed line segment connecting $x$ to $y$ and by $(x,y)$, $[x,y)$ and
$(x,y]$ the corresponding open and half-open line segments. Let $P\subset V$
be a convex subset, not necessarily closed. A face of $P$ is a non-empty
convex subset $F\subset P$ such that whenever $p_{1},p_{2}\in P$ satisfy
$(p_{1},p_{2})\cap F\neq\emptyset$, then we already have $[p_{1},p_{2}]\subset
F$. The dimension of a face is the dimension of its affine span. We write
$\mathcal{F}(P)$ for the partially-ordered set of faces of $P$ ordered by
inclusion. Observe that $P$ is trivially a face of itself, but we have
excluded the empty set from being a face. This is merely a convention that
will simplify the terminology in Section 2.2. Hence, $\mathcal{F}(P)$ has a
maximal element, but not necessarily a minimal one. The intersection of any
two faces of $P$ is either empty or again a face of $P$ and for any face $F$
of $P$, the faces of $F$ are precisely the faces of $P$ contained in $F$. If
$A\subset P$ is a convex subset and $F$ a face of $P$ which contains a point
from the relative interior of $A$, then it follows immediately from the
definition of a face that $A\subset F$. In particular, every proper face
$F\subsetneq P$ is contained in the topological boundary of $P$ and the
relative interiors of different faces are disjoint. For two points
$p_{1},p_{2}\in P$, we therefore have $[p_{1},p_{2}]\subset\partial P$ if and
only if $p_{1}$ and $p_{2}$ are contained in a common proper face. Note that
non-trivial faces do not exist if $P\cap\partial P=\emptyset$.
We will mostly be interested in a less general notion. Suppose that $E\subset
V$ is an affine hyperplane222By an affine hyperplane in $V$, we always mean an
affine subspace of codimension $1$. such that $P$ lies in one of the two
closed half-spaces defined by $E$ and $F:=P\cap E\neq\emptyset$. Then $F$ is a
face of $P$ and faces of this type are said to be exposed. The plane $E$ is
called a supporting hyperplane of $P$ and the half-space containing $P$ is a
supporting half-space. Exposed faces of dimension $0$ and $1$ are called
vertices and edges, those of dimension $\dim(P)-1$ are called facets and their
supporting hyperplanes are referred to as defining hyperplanes. If
$\dim(P)=d$, then defining hyperplanes are uniquely determined and, more
generally, two defining hyperplanes give rise to the same face if and only if
their intersections with the affine span of $P$ coincide. The trivial face $P$
is clearly exposed if $\dim(P)<d$ since every affine hyperplane $E$ containing
$P$ is supporting, but it is trivial in the sense that $P$ is contained in
both closed half-spaces defined by $E$. We also consider $P$ to be exposed if
$\dim(P)=d$. But $P$ may have non-exposed proper faces which are then
necessarily contained in $\partial P$. However, it follows from a separation
argument (cf. [Roc70, Theorem 11.6]) that every point of $P\cap\partial P$ is
contained in a non-trivial supporting hyperplane of $P$. Consequently, every
proper face of $P$ is contained in a proper exposed face. In particular, if
$P$ is closed, then every point of $P$ is either contained in the relative
interior of $P$ or lies on a non-trivial supporting hyperplane, which implies
that $P$ can be reconstructed as the intersection of all supporting half-
spaces containing it. In algebraic terms, this means that a point is contained
in $P$ if and only if it satisfies a family of linear (in-)equalities. If
$\dim(P)=d$, then every supporting hyperplane is non-trivial and none of these
inequalities are equalities. By replacing $V$ with the affine span of $P$, it
is usually no restriction to assume that $\dim(P)=d$, but for our purposes, it
will be more convenient to avoid this assumption.
As mentioned above, the faces of a face $F$ of $P$ are exactly the faces of
$P$ contained in $F$. In particular, this holds for exposed faces. More
generally, suppose that $E\subset V$ is an arbitrary hyperplane, not
necessarily supporting. If $P^{\prime}:=P\cap E$ is non-empty, then for every
face $F$ of $P$, the intersection $F\cap E$ is either empty or a face of
$P^{\prime}$. Conversely, it is not difficult to show that for every face
$F^{\prime}$ of $P^{\prime}$ there exists a face $F$ of $P$ such that
$F^{\prime}=F\cap E$. The intersection of all such faces of $P$ also has this
property, so if $F$ is assumed to be minimal, it is uniquely determined and
its dimension is given by

$\operatorname{dim}(F)=\begin{cases}\operatorname{dim}(F^{\prime})&\text{if
$F^{\prime}$ is a face of $P$ contained in $E$}\\\
\operatorname{dim}(F^{\prime})+1&\text{otherwise.}\end{cases}$ (2.1)

For a set $Y\subset V$, the convex hull of $Y$ is the smallest convex subset
of $V$ containing $Y$. It consists precisely of all finite convex combinations
$\sum_{y\in Y}\lambda_{y}y$, where $\lambda_{y}\in[0,1]$ for all $y\in Y$,
$\lambda_{y}>0$ only for finitely many $y\in Y$ and $\sum_{y\in
Y}\lambda_{y}=1$. It is denoted by $\operatorname{conv}(Y)$. If $Y$ is finite,
then $P:=\operatorname{conv}(Y)$ is compact and called a convex polytope. A
convex polytope $P$ has only finitely many faces which are all exposed. Every
face $F$ of $P$ is of the form $F=\operatorname{conv}(Y^{\prime})$ for some
subset $Y^{\prime}\subset Y$ and thus again a convex polytope. For every
$0\leq k\leq\dim(P)$, there exists a $k$-dimensional face of $P$ and all
maximal chains in $\mathcal{F}(P)$ have the same length. In particular, every
non-trivial face of $P$ is contained in a facet, i.e. a maximal proper face.
If $Y$ is infinite, however, it is not necessarily true that
$P:=\operatorname{conv}(Y)$ is closed, not even if $Y$ is uniformly discrete.
It might even be open in which case it cannot have any non-trivial faces.
Hence, the theory of convex polytopes cannot be applied in this situation and
it is not clear whether $\mathcal{F}(P)$ has any interesting properties.
However, it is still true that if $F\subset P$ is an exposed face, then there
exists a subset $Y^{\prime}\subset Y$ such that
$F=\operatorname{conv}(Y^{\prime})$.

###  2.2 Orbits of regular elements

We now turn to the study of $\Xi_{h}=\operatorname{conv}(W\cdot h)$ where
$h\in\mathfrak{a}$ is regular. After possibly replacing $h$ by $-h$, it is no
restriction to assume that $h$ is contained in an open chamber of $X$. In this
case, the $W$-translates $\\{w(h)\mid w\in W\\}$ are all distinct. Since
$\Xi_{h}$ is $W$-invariant, we may w.l.o.g. assume that $h\in C$, i.e.
$\langle\alpha_{i},h\rangle>0$ for all $i\in I$. Our goal is to show that
$\Xi_{h}$ is a closed convex set whose face poset $\mathcal{F}(\Xi_{h})$ is
isomorphic to $\Sigma^{\ast}(W,S)$, i.e. the poset of all standard cosets of
$(W,S)$ ordered by non-reversed inclusion. Our approach will consist of two
steps: First, we explicitly construct exposed faces of $\Xi_{h}$. Afterwards,
we show that $\Xi_{h}$ is closed and that every face is exposed.
First of all, note that the assumption that $\mathcal{D}$ is cofree implies
$\dim(\Xi_{h})=n$ since it is contained in the $n$-dimensional affine subspace
$E_{h}:=h+\mathfrak{a}^{\prime}$, where
$\mathfrak{a}^{\prime}:=\bigoplus_{i=1}^{n}\mathbb{R}\alpha_{i}^{\vee}$, and
the $n+1$ points $\\{h,s_{1}(h),\ldots,s_{n}(h)\\}\subset\Xi_{h}$ are affinely
independent. Next, $W$-invariance of $\Xi_{h}$ also implies that the set of
faces of $\Xi_{h}$ is $W$-invariant. Hence, every face of $\Xi_{h}$ is
$W$-equivalent to a face that meets $\overline{C}$. The same is also true for
the subset of exposed faces. Every exposed face $F$ of $\Xi_{h}$ is of the
form $F=\operatorname{conv}(W^{\prime}\cdot h)$ where $W^{\prime}$ is a subset
of $W$, so by $W$-invariance, it is also no restriction to assume that $1\in
W^{\prime}$ and hence $h\in F$. Thus, it is sufficient to construct exposed
faces containing $h$. To this end, we define for every $i\in I$ the hyperplane

$H^{(i)}:=\\{h^{\prime}\in
E_{h}:\langle\omega_{i},h^{\prime}\rangle=\langle\omega_{i},h\rangle\\}\subset
E_{h}$

In order to show that these hyperplanes give rise to faces of $\Xi_{h}$, we
need the following lemma from the general theory of Coxeter groups, which is a
refinement of (1.5).

######  Lemma 2.2.

If $h\in C$, then $h-w(h)=\sum_{i\in I}c_{i}\alpha_{i}^{\vee}$ with $c_{i}>0$
for all $i\in\operatorname{supp}(w)$ and $c_{i}=0$ for all
$i\notin\operatorname{supp}(w)$.

###### Proof.

By (1.5) and the fact that $c_{i}=0$ if $i\notin\operatorname{supp}(w)$, it
only remains to show that $c_{i}>0$ for all $i\in\operatorname{supp}(w)$.
Suppose that this was not the case, then we would have
$\langle\omega_{i},w(h)\rangle=\langle\omega_{i},h\rangle$ for some
$i\in\operatorname{supp}(w)$. But then every element in the left coset
$W^{(i)}w$ satisfies the same equation, so we may assume that $w$ is the
unique element of minimal length in that coset. In this case, it has the form
$w=s_{i}w^{\prime}$ for some $w^{\prime}\in W$ with $l(w^{\prime})=l(w)-1$ and
we have

$\langle\omega_{i},h\rangle=\langle\omega_{i},w(h)\rangle=\langle
s_{i}(\omega_{i}),w^{\prime}(h)\rangle=\langle\omega_{i},w^{\prime}(h)\rangle-\langle\alpha_{i},w^{\prime}(h)\rangle$

with $\langle\omega_{i},w^{\prime}(h)\rangle\leq\langle\omega_{i},h\rangle$ by
(1.5) and $\langle\alpha_{i},w^{\prime}(h)\rangle=\langle
w^{\prime-1}(\alpha_{i}),h\rangle>0$ by (1.3) because
$l(w^{\prime-1})=l(w^{\prime})<l(w)=l(w^{-1})=l(w^{\prime-1}s_{i})$. Hence, we
arrive at a contradiction. ∎

######  Proposition 2.3.

For every $i\in I$, the plane $H^{(i)}$ is a defining hyperplane of $\Xi_{h}$
and the corresponding facet of $\Xi_{h}$ is given by

$F^{(i)}:=\operatorname{conv}(W^{(i)}\cdot h),$

where $W^{(i)}$ is the subgroup of $W$ generated by
$S^{(i)}=S\setminus\\{s_{i}\\}$.

###### Proof.

If $h^{\prime}\in\Xi_{h}$, we can write it as $h^{\prime}=\sum_{w\in
W}\lambda_{w}w(h)$ with $\lambda_{w}\in[0,1]$ for all $w\in W$ and $\sum_{w\in
W}\lambda_{w}=1$, where $\lambda_{w}>0$ only for finitely many $w\in W$.
Moreover, it follows from (1.5) that $\langle\omega_{i},h-w(h)\rangle\geq 0$
for every $w\in W$, which implies that

$\langle\omega_{i},h^{\prime}\rangle=\sum_{w\in
W}\lambda_{w}\langle\omega_{i},w(h)\rangle\leq\sum_{w\in
W}\lambda_{w}\langle\omega_{i},h\rangle=\langle\omega_{i},h\rangle.$

Hence, $\Xi_{h}$ lies on one side of $H^{(i)}$ and $\Xi_{h}\cap H^{(i)}$ is an
exposed face. Moreover, $H^{(i)}$ contains $h$ and $s_{j}(h)$ whenever $j\neq
i$ since

$\langle\omega_{i},s_{j}(h)\rangle=\langle\omega_{i},h\rangle-\langle\alpha_{j},h\rangle\langle\omega_{i},\alpha_{j}^{\vee}\rangle=\langle\omega_{i},h\rangle.$

As these $n$ points are affinely independent, it follows that $\Xi_{h}\cap
H^{(i)}$ has dimension $n-1$ and is therefore a facet of $\Xi_{h}$. Iterating
the computation above shows that $H^{(i)}$ contains $w(h)$ for every $w\in
W^{(i)}$ and hence $F^{(i)}\subset\Xi_{h}\cap H^{(i)}$. Conversely, if
$w\notin W^{(i)}$, then every reduced decomposition of $w$ contains $s_{i}$,
i.e. $i\in\operatorname{supp}(w)$. By Lemma 2.2, this implies that $h-w(h)$ is
a linear combination of simple coroots in which $\alpha_{i}^{\vee}$ occurs
with a positive coefficient, so that
$\langle\omega_{i},w(h)\rangle<\langle\omega_{i},h\rangle$. Let now
$h^{\prime}\in\Xi_{h}\cap H^{(i)}$, then $h^{\prime}$ has to be a convex
combination of the points in $W\cdot h\cap H^{(i)}$ which we have just shown
to coincide with $W^{(i)}\cdot h$. Thus, it follows that $\Xi_{h}\cap
H^{(i)}\subset F^{(i)}$ and the proposition is proven. ∎

Observe that we consider the hyperplanes $H^{(1)},\ldots,H^{(n)}$ as subsets
of $E_{h}$, which is the affine span of $\Xi_{h}$, rather than the full vector
space $\mathfrak{a}$. If $d>n$, then for every $i\in I$ there are many
hyperplanes in $\mathfrak{a}$ that intersect $E_{h}$ in $H^{(i)}$, but this
extension process is irrelevant to our discussion since we are only interested
in intersections with $\Xi_{h}\subset E_{h}$. As a subset of $E_{h}$,
$H^{(i)}$ is uniquely determined by $F^{(i)}$. Since the set of exposed faces
is $W$-invariant and closed under intersections, the following corollary
follows immediately from the previous proposition.

######  Corollary 2.4.

For every subset $J\subset I$, the set $F_{J}:=\operatorname{conv}(W_{J}\cdot
h)$ is an exposed face of $\Xi_{h}$ of dimension $|J|$. More generally, for
every $w\in W$, the set $w(F_{J})$ is an exposed face of dimension $|J|$ and
coincides with $\operatorname{conv}(wW_{J}\cdot h)$. In particular, every
$W$-translate of $h$ is a vertex of $\Xi_{h}$.

If $J,J^{\prime}\subset I$ are two distinct subsets of $I$, then the affine
spans of the Weyl group orbits $W_{J}\cdot h$ and $W_{J^{\prime}}\cdot h$ do
not coincide, which implies that the faces $F_{J}$ and $F_{J^{\prime}}$ are
also distinct. Hence, $h$ is contained in at least $2^{n}$ faces of $\Xi_{h}$.
More generally, we have $w(F_{J})=w^{\prime}(F_{J^{\prime}})$ if and only if
$J=J^{\prime}$ and $w^{-1}w^{\prime}\in W_{J}$, so the map $wW_{J}\mapsto
w(F_{J})$ defines an order-preserving injection
$\Sigma^{\ast}(W,S)\to\mathcal{F}(\Xi_{h})$. In order to show that this map is
also surjective, we need more information on the topology of $\Xi_{h}$.

######  Lemma 2.5.

Let $h\in C$ and $h^{\prime}\in\Xi_{h}$ be such that
$v:=h-h^{\prime}=\sum_{i\in I}c_{i}\alpha_{i}^{\vee}$ satisfies $c_{i}>0$ for
all $i\in I$. Then the open line segment $(h,h^{\prime})$ is contained in the
relative interior of $\Xi_{h}\subset E_{h}$.

###### Proof.

Choose any $\lambda\in(0,1)$ and set $x:=h-\lambda v\in\Xi_{h}$. First, we
show that for every $i\in I$ there exists some $\mu_{i}>0$ such that
$x-\mu_{i}\alpha_{i}^{\vee}\in\Xi_{h}$ by writing this point as a convex
combination of $\\{h,h^{\prime},s_{i}(h)\\}$. To this end, choose $\mu_{i}>0$
such that $c:=\frac{\mu_{i}}{\langle\alpha_{i},h\rangle}<1-\lambda$, set
$b:=\lambda$ and $a:=1-b-c$. Then we have $a,b,c\in(0,1)$ and $a+b+c=1$.
Moreover, the following computation shows that the corresponding convex
combination is the desired point:

$\displaystyle ah+bh^{\prime}+cs_{i}(h)$
$\displaystyle=ah+bh+b(h^{\prime}-h)+c(h-\langle\alpha_{i},h\rangle\alpha_{i}^{\vee})$
$\displaystyle=(a+b+c)h-bv-c\langle\alpha_{i},h\rangle\alpha_{i}^{\vee}$
$\displaystyle=h-\lambda
v-\mu_{i}\alpha_{i}^{\vee}=x-\mu_{i}\alpha_{i}^{\vee}$

On the other hand, $\Xi_{h}$ also contains $x+\mu v$ for all
$0\leq\mu\leq\lambda$. Since $\lambda>0$ and $c_{i}>0$ for all $i\in I$, it
follows that for every $i\in I$, $\Xi_{h}$ also contains
$x+\mu_{i}^{\prime}\alpha_{i}^{\vee}$ for sufficiently small
$\mu_{i}^{\prime}>0$. Altogether, we obtain that $\Xi_{h}$ contains the cube

$x+\prod_{i\in I}[-\mu_{i},\mu_{i}^{\prime}]\alpha_{i}^{\vee}\subset E_{h}.$

Since the simple roots form a basis of $\mathfrak{a}^{\prime}$, this set has
non-empty relative interior in $E_{h}$, so $x$ lies in the relative interior
of $\Xi_{h}\subset E_{h}$. ∎

######  Proposition 2.6.

If $h\in C$, then $\Xi_{h}$ is closed.

###### Proof.

If $W$ is finite, then $\Xi_{h}$ is a convex polytope and hence compact. Thus,
for the rest of the proof, we may assume that $|W|=\infty$. For every
$k\in\mathbb{N}_{0}$, we set

$U_{k}:=\\{w\in W:l(w)\leq k\\}\qquad\text{and}\qquad
X_{k}:=\operatorname{conv}(U_{k}\cdot h).$

Since $U_{k}$ is finite, $X_{k}$ is a convex polytope and hence compact.
Moreover, these polytopes yield a filtration of $\Xi_{h}$ in the sense that
$X_{k}\subset X_{k+1}$ for all $k\in\mathbb{N}_{0}$ and
$\Xi_{h}=\bigcup_{k=0}^{\infty}X_{k}$. Let $(y_{m})_{m=1}^{\infty}$ be a
sequence in $\Xi_{h}$ converging to some $y\in E_{h}$. We are going to show
that there exists some $k^{\prime}\in\mathbb{N}_{0}$ such that $y_{m}\in
X_{k^{\prime}}$ for all $m\in\mathbb{N}$. Once this is established, we may
conclude that the sequence $(y_{m})_{m=1}^{\infty}$ is contained in the
compact set $X_{k^{\prime}}$, so that $y\in X_{k^{\prime}}\subset\Xi_{h}$. The
proof of this assertion will be reduced to a sequence of claims. Before we
formulate them, we need to introduce some notation. Throughout this proof, we
will identify $E_{h}$ with $\mathbb{R}^{n}$ by moving $h$ to $0$ and mapping
$\Phi_{0}^{\vee}$ to the standard basis of $\mathbb{R}^{n}$ in the natural
order. For every $i\in I$, $\omega_{i}-\langle\omega_{i},h\rangle$ corresponds
to the $i$-th coordinate functional under this identification. Thus, if we set
$\omega:=\sum_{i\in I}\omega_{i}$, then

$|\langle\omega,p\rangle-\langle\omega,q\rangle|=\bigg{|}\sum_{i=1}^{n}(p_{i}-q_{i})\bigg{|}\leq\sum_{i=1}^{n}|p_{i}-q_{i}|=||p-q||_{1},\qquad
p,q\in E_{h}\cong\mathbb{R}^{n},$ (2.2)

where $||\cdot||_{1}$ denotes the $\ell^{1}$-norm on $\mathbb{R}^{n}$. Since
all norms on a finite-dimensional vector space are equivalent, the
$\ell^{1}$-topology on $\mathbb{R}^{n}$ coincides with the standard Euclidean
topology. Next, set $U_{-1}:=\emptyset$ and define for every
$k\in\mathbb{N}_{0}$

$\displaystyle V_{k}$ $\displaystyle:=U_{k}\setminus U_{k-1}=\\{w\in
W:l(w)=k\\},$ $\displaystyle m_{+}(k)$ $\displaystyle:=\max_{w\in
V_{k}}\langle\omega,w(h)\rangle,$ $\displaystyle m_{-}(k)$
$\displaystyle:=\min_{w\in V_{k}}\langle\omega,w(h)\rangle,$ $\displaystyle
d(k)$ $\displaystyle:=m_{+}(k)-m_{-}(k).$

Since we assume that $W$ is infinite, we have $V_{k}\neq\emptyset$ for all
$k\in\mathbb{N}_{0}$. We set $c:=\min_{i\in I}\langle\alpha_{i},h\rangle>0$.
Claim 1: For every $w\in W$ and $i\in I$ such that $l(w)<l(ws_{i})$, we have
$\langle\omega,ws_{i}(h)\rangle\leq\langle\omega,w(h)\rangle-c$.
Proof of Claim 1: We have

$\langle\omega,ws_{i}(h)\rangle=\langle\omega,w(h)\rangle-\langle\alpha_{i},h\rangle\langle\omega,w(\alpha_{i}^{\vee})\rangle,$

where, by (1.3), we subtract a positive term if and only if $l(w)<l(ws_{i})$.
In that case, we have $\langle\omega,w(\alpha_{i}^{\vee})\rangle\geq 1$ which
proves the desired inequality.
Claim 2: For every $k\in\mathbb{N}_{0}$, there exists some
$k^{\prime}\in\mathbb{N}$ such that $m_{+}(k^{\prime})<m_{-}(k)$.
Proof of Claim 2: Let $k^{\prime}\geq k$ and $w\in V_{k^{\prime}}$ be
arbitrary and write $w=w_{1}w_{2}$ with $l(w_{1})=k$ and
$l(w_{2})=k^{\prime}-k$. Iterating the computation in Claim 1 shows that

$\langle\omega,w(h)\rangle\leq\langle\omega,w_{1}(h)\rangle-(k^{\prime}-k)c\leq
m_{+}(k)-(k^{\prime}-k)c=m_{-}(k)+d(k)-(k^{\prime}-k)c.$

Since the right-hand side does not depend on $w$, we conclude that

$m_{+}(k^{\prime})\leq m_{-}(k)+d(k)-(k^{\prime}-k)c.$

If we choose $k^{\prime}$ so large that $(k^{\prime}-k)c>d(k)$, the claim is
satisfied.
Claim 3: For every $k\in\mathbb{N}_{0}$, every proper face $F$ of $X_{k}$ is
either contained in a proper face of $\Xi_{h}$ or it is of the form
$F=\operatorname{conv}(U\cdot h)$ for a subset $U\subset V_{k}$.
Proof of Claim 3: The statement is trivial if $k=0$ since $X_{0}=\\{h\\}$ is a
vertex of $\Xi_{h}$. Thus, we may assume that $k\geq 1$. Since
$X_{k}=\operatorname{conv}(U_{k}\cdot h)$ is a convex polytope, every face is
exposed. Hence, $F$ is of the form $F=\operatorname{conv}(U\cdot h)$ for some
proper subset $U\subsetneq U_{k}$. If $U$ is contained in a non-trivial
standard coset of $W$, i.e. $U\subset wW_{J}$ for some $w\in W$ and
$J\subsetneq I$, then $F$ is contained in the proper face
$w(F_{J})=\operatorname{conv}(wW_{J}\cdot h)$ of $\Xi_{h}$. Otherwise, let
$u\in U$ be arbitrary, then for every $i\in I$, there exists some $u_{i}\in U$
such that $u_{i}\notin uW^{(i)}$. Equivalently, $u^{-1}u_{i}\notin W^{(i)}$
and hence $s_{i}\in S(u^{-1}u_{i})$. Now Lemma 2.2 implies that
$h-u^{-1}u_{i}(h)$ is a sum of simple coroots in which all coefficients are
non-negative and the coefficient of $\alpha_{i}^{\vee}$ is non-zero. Now
consider the point

$h^{\prime}:=\sum_{i=1}^{n}\frac{1}{n}u^{-1}u_{i}(h)=u^{-1}\bigg{(}\sum_{i=1}^{n}\frac{1}{n}u_{i}(h)\bigg{)}\in
u^{-1}(F).$

Then we have

$h-h^{\prime}=\sum_{i=1}^{n}\frac{1}{n}(h-u^{-1}u_{i}(h)),$

so that $h-h^{\prime}$ is a sum of simple coroots in which all coefficients
are positive. By Lemma 2.5, the open line segment $(h,h^{\prime})$ is
contained in the relative interior of $\Xi_{h}$. More precisely, if $l(u)<k$,
the proof of Lemma 2.5 actually shows that this line runs through the relative
interior of $u^{-1}(X_{k})$. Indeed, observe that in that proof we have only
used $h$, $h^{\prime}$ and the simple reflections to create open
neighbourhoods and $s_{i}(h)=u^{-1}us_{i}(h)\in u^{-1}(X_{k})$ for every $i\in
I$ since $l(us_{i})\leq l(u)+1\leq k$. But then the line segment
$(u(h),u(h^{\prime}))\subset F$ is contained in the relative interior of
$X_{k}$, contradicting the assumption that $F$ is a proper face of $X_{k}$.
Since $u\in U$ was arbitrary, $F$ can only be a face if $l(u)=k$ for all $u\in
U$, i.e. $U\subset V_{k}$.
Claim 4: If $q\in\Xi_{h}\setminus X_{k}$ for some $k\in\mathbb{N}_{0}$, then
$\langle\omega,q\rangle<m_{+}(k)$.
Proof of Claim 4: The claim is clear if $k=0$ since $X_{0}=\\{h\\}$ and every
point $q\in\Xi_{h}\setminus\\{h\\}$ satisfies
$\langle\omega,q\rangle<\langle\omega,h\rangle=m_{+}(0)$, so we may assume
that $k\geq 1$. Consider the line segment $L:=[h,q]\subset\Xi_{h}$ connecting
$h$ and $q$. Since $h\in X_{0}\subset X_{k}$ and $q\notin X_{k}$, there exists
a supporting hyperplane $E$ for $X_{k}$ such that $q$ and $h$ are not
contained in the same closed half-space defined by $E$ and $L$ intersects the
corresponding face $F:=X_{k}\cap E$. If $F$ was contained in a face of
$\Xi_{h}$, then we could choose $E$ to be a supporting hyperplane for
$\Xi_{h}$, but then $q\in\Xi_{h}$ would imply that $q$ and $h$ were both
contained in the same closed half-space. Thus, $F$ is of the form
$F=\operatorname{conv}(U\cdot h)$ for some $U\subset V_{k}$ by Claim 3. As
$k\geq 1$ and $q\notin F$, this face neither contains $h$ nor $q$, so $h$ and
$q$ lie in different open half-spaces defined by $E$ and the line $L$
intersects $F$ in an interior point $x\in(h,q)$. Since
$\langle\omega,u(h)\rangle\leq m_{+}(k)$ for all $u\in U$ and $x$ is a convex
combination of these vertices, we have $\langle\omega,x\rangle\leq m_{+}(k)$.
In addition, since $k\geq 1$, we also have $m_{+}(k)<\langle\omega,h\rangle$.
Hence, $\omega$ is strictly monotonically decreasing along $L$. As $x$ lies
strictly between $h$ and $q$, this yields that
$\langle\omega,q\rangle<\langle\omega,x\rangle\leq m_{+}(k)$.
Claim 5: Let $k\in\mathbb{N}_{0}$ and choose $k^{\prime}>k$ such that
$\delta:=m_{-}(k)-m_{+}(k^{\prime})>0$. Then $||p-q||_{1}>\delta$ for all
$p\in X_{k}$ and $q\in\Xi_{h}\setminus X_{k^{\prime}}$.
Proof of Claim 5: On the one hand, $p\in X_{k}$ is a convex combination of
$W$-translates of $h$ of length at most $k$. The value of $\omega$ on each of
these vertices is at least $m_{-}(k)$ since $m_{-}$ is monotonically
decreasing by Claim 1. This implies $\langle\omega,p\rangle\geq m_{-}(k)$. On
the other hand, we have $\langle\omega,q\rangle<m_{+}(k^{\prime})$ by Claim 4,
so (2.2) implies that

$||p-q||_{1}\geq|\langle\omega,p\rangle-\langle\omega,q\rangle|>m_{-}(k)-m_{+}(k^{\prime})=\delta.$

Conclusion of the proof: Since a convergent sequence is bounded, we find $R>0$
such that $(y_{m})_{m=1}^{\infty}$ is contained in a closed ball of radius $R$
(w.r.t. the $\ell^{1}$-norm) around $y_{1}$. In addition, since for every
$m\in\mathbb{N}$, $y_{m}$ is a finite convex combination of $W$-translates of
$h$, there exists some $k(m)\in\mathbb{N}_{0}$ such that $y_{m}\in X_{k(m)}$.
Set $k:=k(1)$ and choose $k^{\prime}>k$ such that
$\delta:=m_{-}(k)-m_{+}(k^{\prime})\geq R$. Since $||y_{1}-y_{m}||_{1}\leq R$
for all $m\in\mathbb{N}$, Claim 5 implies that the sequence
$(y_{m})_{m=1}^{\infty}$ must be contained in $X_{k^{\prime}}$, as required. ∎

With these topological results at hand, we are now in a position to show that
the exposed faces of $\Xi_{h}$ that we constructed so far actually make up the
entire face poset.

######  Proposition 2.7.

If $h\in C$ and $F$ is a face of $\Xi_{h}$, then $F=w(F_{J})$ for some $w\in
W$ and a unique subset $J\subset I$. In particular, every face of $\Xi_{h}$ is
exposed.

###### Proof.

We have already discussed uniqueness of $J$ after Corollary 2.4, so it remains
to show existence. If $F=\Xi_{h}$, the claim is satisfied if and only if $J=I$
and $w\in W$ is arbitrary. Thus, we may assume that $F$ is a proper face of
$\Xi_{h}$. We will argue by induction on $n=|I|$. If $n=1$, then $\Xi_{h}$ is
the line segment connecting the two distinct points $h$ and $s_{1}(h)$. The
only proper faces of $\Xi_{h}$ are those two points which have the required
description for $J=\emptyset$.
Let now $n>1$ be arbitrary and let $x$ be a point in the relative interior of
$F$. As $F$ is a proper face, the point $x$ lies on the boundary of $\Xi_{h}$
and since $\Xi_{h}$ is closed, it is contained in a non-trivial supporting
hyperplane $E$ of $\Xi_{h}$. Let $G:=\Xi_{h}\cap E$ be the corresponding
exposed face, then $G$ is also proper and $F\subset G$ because $G$ contains a
point from the relative interior of $F$. This exposed face is of the form
$G=\operatorname{conv}(W^{\prime}\cdot h)$ for some proper subset
$W^{\prime}\subset W$ and upon replacing $F$ by $w^{-1}(F)$ and $G$ by
$w^{-1}(G)$ for a suitable $w\in W$, we may assume that $1\in W^{\prime}$ and
$h\in G$. We will show that there exists some $i\in I$ such that
$W^{\prime}\subset W^{(i)}$.
Assume for a contradiction that for every $i\in I$, there exists some
$w_{i}\in W^{\prime}$ with $i\in\operatorname{supp}(w_{i})$. Then it follows
from Lemma 2.2 that $h-w_{i}(h)$ is a linear combination of simple coroots in
which $\alpha_{i}^{\vee}$ occurs with a positive coefficient. Since all other
coefficients are non-negative, we may form a suitable convex combination of
these points to deduce that there exists a point $y\in G$ such that
$h-y=\sum_{i\in I}c_{i}\alpha_{i}^{\vee}$ with $c_{i}>0$ for all $i\in I$. Now
Lemma 2.5 implies that the line connecting $h$ and $y$, which is contained in
$G$, runs through the interior of $\Xi_{h}$, a contradiction.
Hence, we must have $W^{\prime}\subset W^{(i)}$ for some $i\in I$, which
implies that $G$ is contained in the facet $F^{(i)}$. The same is then true
for $F$, so $F$ is a face of $F^{(i)}$. The induction hypothesis now implies
that there exists a unique (not necessarily proper) subset $J\subset
I\setminus\\{i\\}$ and some $w\in W^{(i)}$ such that
$F=w(F_{J})=\operatorname{conv}(wW_{J}\cdot h)$. Since we have already shown
that every face of this form is exposed, the proof is finished. ∎

The following theorem is now an immediate consequence of Corollary 2.4 and
Proposition 2.7 and provides a completely combinatorial description of the
face poset of $\Xi_{h}$.

######  Theorem 2.8.

If $h\in C$, then the faces of $\Xi_{h}$ correspond bijectively to the
standard cosets of $(W,S)$. More precisely, for every standard coset
$wW_{J}\subset W$ with $w\in W$ and $J\subset I$, the set
$w(F_{J})=\operatorname{conv}(wW_{J}\cdot h)$ is a face of $\Xi_{h}$ of
dimension $|J|$ and, conversely, every face is of this form for a unique
standard coset of $(W,S)$. In particular, the poset $\mathcal{F}(\Xi_{h})$ is
isomorphic to $\Sigma^{*}(W,S)$.

######  Remark 2.9.

This theorem shows that every vertex of $\Xi_{h}$, i.e. every $W$-translate of
$h$, is contained in precisely $2^{n}$ faces of $\Xi_{h}$. In particular,
every face is only contained in finitely many other faces. As every face is an
intersection of sufficiently many facets, this means that every point of
$\Xi_{h}$ is only contained in finitely many defining hyperplanes, so the
hyperplane arrangement

$\\{w(H^{(i)})\mid w\in W,\,i\in I\\},$

whose intersection pattern induces the facial structure on $\Xi_{h}$, is
locally finite. Note that these hyperplanes are not linear. In the language of
Coxeter groups, this is reflected in the fact that for every standard coset
$wW_{J}$, there exist only finitely many standard cosets of $(W,S)$ containing
it. In contrast, a face may well contain infinitely many other faces. In fact,
for every $J\subset I$, the face poset $\mathcal{F}(F_{J})$ is isomorphic to
$\Sigma^{\ast}(W_{J},S_{J})$, where $S_{J}=\\{s_{j}\mid j\in J\\}$, which is
finite if and only if the standard subgroup $W_{J}\subset W$ is finite. Since
the generating set $S$ is finite, this is the case if and only if the length
function of $W$ is bounded on $W_{J}$. By inspecting the proof of Proposition
2.6, this implies that the face $F_{J}$ is bounded (and hence, compact) if and
only if $W_{J}$ is finite, so the bounded faces of $\Xi_{h}$ correspond
precisely to the finite parabolic subgroups of $W$. Trivially, vertices and
edges contain only finitely many faces, so unbounded faces must be at least
two-dimensional. In combinatorial terms, this reflects the fact that there is
only one Coxeter group of rank $1$, namely the finite group $\mathbb{Z}_{2}$.
The group $W$ is said to be k-spherical for some $2\leq k\leq n$ if every
standard subgroup $W_{J}$ with $|J|\leq k$ is finite. In this case, every face
of dimension at most $k$ has only finitely many faces or, equivalently, is
bounded. The only infinite Coxeter group of rank $2$ is the infinite dihedral
group

$D_{\infty}=\langle s,t\mid s^{2}=t^{2}=1\rangle,$

so $W$ is 2-spherical if and only if the order of $s_{i}s_{j}\in W$ is finite
for all $i\neq j\in I$. By (1.2), this is the case if and only if the GCM
$\mathbb{A}$ satisfies $a_{ij}a_{ji}\leq 3$ for all $i\neq j\in I$. If
$\mathbb{A}$ is of affine type (cf. [Kac90, §4.3]), then $W$ is infinite but
every proper standard subgroup of $W$ is finite. In this case, $W$ is
$(n-1)$-spherical, so every proper face of $\Xi_{h}$ is bounded, but $\Xi_{h}$
itself is not.
This behaviour is in a sense “dual” to the cellular structure of the Tits
cone. A cell of $X$ is any set of the form $w(D_{J})\subset X$ for some $w\in
W$ and $J\subset I$, where

$D_{J}:=\\{h^{\prime}\in\mathfrak{a}:\langle\alpha_{i},h^{\prime}\rangle=0\,\,\forall
i\in J,\,\,\langle\alpha_{i},h^{\prime}\rangle>0\,\,\forall i\in I\setminus
J\\}.$

The sets $\\{D_{J}\mid J\subset I\\}$ are precisely the faces of the closed
fundamental chamber $\overline{C}$ and for $h^{\prime}\in w(D_{J})$, the
stabilizer of $h^{\prime}$ is the parabolic subgroup $wW_{J}w^{-1}\subset W$.
In particular, it depends only on the cell $w(D_{J})$, so the cell complex of
$X$ is isomorphic to the “usual” Coxeter complex $\Sigma(W,S)$ where standard
cosets are ordered by reversed inclusion. The Tits cone thus provides a
geometric realization of $\Sigma(W,S)$. Every cell of $X$ contains only
finitely many cells, but may well be contained in infinitely many different
ones. Observe that the cellular structure on $X$ is induced by the
intersection pattern of the hyperplane arrangement

$\\{w(\operatorname{ker}(\alpha_{i}))\mid w\in W,\,i\in I\\}.$

In contrast to the situation of $\Xi_{h}$, all these hyperplanes are linear,
which implies that this hyperplane arrangement is locally finite if and only
if it is finite or, equivalently, $W$ is finite.

Another useful result, which is a direct consequence of Theorem 2.8, is the
following. We use standard terminology of Coxeter complexes as described in
[AB08, Chapter 3].

######  Corollary 2.10.

If $h\in C$, the underlying graph of $\Xi_{h}$, i.e. the set of all edges and
vertices of $\Xi_{h}$, is isomorphic to the chamber system of the Coxeter
complex $\Sigma(W,S)$. In other words, two vertices $w_{1}(h)$ and $w_{2}(h)$
of $\Xi_{h}$ are joined by an edge if and only if the chambers $\\{w_{1}\\}$
and $\\{w_{2}\\}$ of $\Sigma(W,S)$ are adjacent, i.e. $w_{2}=w_{1}s_{i}$ for
some $i\in I$. Under this isomorphism, the graph distance in $\Xi_{h}$
corresponds to the chamber distance in $\Sigma(W,S)$ and minimal paths between
vertices of $\Xi_{h}$ are in bijective correspondence with minimal galleries
between the corresponding chambers. Equivalently, the underlying graph of
$\Xi_{h}$ is also isomorphic to the Cayley graph of $(W,S)$.

Finally, recall that a closed convex set can be written as the intersection of
all supporting half-spaces. As every proper face of $\Xi_{h}$ is contained in
a facet, $\Xi_{h}$ can also be reconstructed as the intersection of all
defining half-spaces. This observation yields the next corollary which will
play a crucial role in the proof of Theorem 4.1.

######  Corollary 2.11.

If $h\in C$, then a point $h^{\prime}\in\mathfrak{a}$ is contained in
$\Xi_{h}$ if and only if it satisfies the following (in-)equalities:

$\begin{cases}\langle\omega_{i},w(h^{\prime})\rangle\leq\langle\omega_{i},h\rangle&\text{for
all $w\in W$ and $i\in I$}\\\
\langle\phi_{i},h^{\prime}\rangle=\langle\phi_{i},h\rangle&\text{for all
$i\in\\{n+1,\ldots,d\\}$}\end{cases}$

###### Proof.

Since the set of defining hyperplanes of $\Xi_{h}$ is $\\{w^{-1}(H^{(i)})\mid
w\in W,\,i\in I\\}$ and $\Xi_{h}$ is the intersection of all defining half-
spaces, which we view as subsets of $E_{h}$, $h^{\prime}$ is contained in
$\Xi_{h}$ if and only if it lies in $E_{h}$ and satisfies the corresponding
inequalities
$\langle\omega_{i},w(h^{\prime})\rangle\leq\langle\omega_{i},h\rangle$ for all
$w\in W$ and $i\in I$. Moreover, the condition $h^{\prime}\in E_{h}$ is
equivalent to $h-h^{\prime}\in\mathfrak{a}^{\prime}$ and hence to
$\langle\phi_{i},h\rangle=\langle\phi_{i},h^{\prime}\rangle$ for all
$i\in\\{n+1,\ldots,d\\}$. ∎

It is also not difficult to show that the inequalities in Corollary 2.11
directly imply that $h^{\prime}$ lies in the Tits cone $X\subset\mathfrak{a}$.
Once this is established, it turns out that most inequalities are in a certain
sense redundant. In fact, if $h^{\prime}$ is already contained in
$\overline{C}$, then
$\langle\omega_{i},w(h^{\prime})\rangle\leq\langle\omega_{i},h^{\prime}\rangle$
for all $w\in W$ and $i\in I$ by (1.5), so that the inequalities with $w\neq
1$ are implied by those with $w=1$. If $h^{\prime}\in X$ does not necessarily
lie in $\overline{C}$, then $h^{\prime}$ is contained in $\Xi_{h}$ if and only
if its $W$-translate $h^{\prime\prime}\in(W\cdot h^{\prime})\cap\overline{C}$
has this property. As we have just seen, this is the case if and only if it
lies in $E_{h}$ and satisfies
$\langle\omega_{i},h^{\prime\prime}\rangle\leq\langle\omega_{i},h\rangle$ for
all $i\in I$.
We have now fully determined the face poset of $\Xi_{h}$ and found necessary
and sufficient conditions that decide whether a point in $\mathfrak{a}$ is
contained in $\Xi_{h}$ or not. Next, we want to study the intersections of
$\Xi_{h}$ with parallel translates of the defining hyperplanes. As we shall
see, these intersections are invariant under subgroups of $W$ of rank $n-1$
which will allow us to apply inductive arguments in the proof of Theorem 4.1.
For the rest of this section, we fix an arbitrary $i\in I$ and consider
parallel translates of the defining hyperplane $H^{(i)}$. Given any point
$y\in E_{h}$, we set $y_{i}:=\langle\omega_{i},y\rangle$. For every
$t\in\mathbb{R}$, we define

$H^{(i)}(t):=\\{y\in E_{h}:y_{i}=t\\},$

so that $H^{(i)}(t)$ is parallel to $H^{(i)}=H^{(i)}(h_{i})$. We have already
seen that every point $y\in\Xi_{h}$ satisfies $y_{i}\leq h_{i}$ with equality
if and only if $y\in F^{(i)}$. Thus, the intersection

$P^{(i)}(t):=\Xi_{h}\cap H^{(i)}(t)$

can only be non-empty if $t\leq h_{i}$. The existence of a lower bound depends
on whether the index of $W^{(i)}$ in $W$ is finite. This is obviously the case
if $W$ is finite. If $W$ is infinite, however, this is only possible in a
rather trivial situation.

######  Theorem 2.12.

Let $(W,S)$ be a Coxeter system, let $I_{1},\ldots,I_{m}\subset I$ be the
connected components of the Coxeter diagram of $(W,S)$ and set

$I_{\infty}:=\bigcup\\{I_{k}\mid\text{$1\leq k\leq m,\,W_{I_{k}}$ is
infinite}\\}.$

Then $W_{I_{\infty}}$ is the smallest finite-index standard subgroup of $W$.
If $W_{J}$ is a finite-index standard subgroup of $W$, then $I_{\infty}\subset
J$ and $W_{J}\cong W_{I_{\infty}}\times W_{J\setminus I_{\infty}}$. In
particular, if $(W,S)$ is infinite and irreducible, then $W$ does not contain
a non-trivial standard subgroup of finite index.

###### Proof.

[Hos02, Corollary 3.3 and 3.4] ∎

In the formulation of the following lemma, we use the fact that in every
Coxeter system $(W,S)$ with $|W|<\infty$, there exists a unique element
$w_{+}\in W$ such that $l(w_{+})$ is maximal.333In the literature on Coxeter
groups, it is customary to denote this element by $w_{0}$ rather than $w_{+}$.
We do not use that notation on purpose since it will sometimes be convenient
to be able to list elements of $W$ as $w_{0},w_{1},w_{2},\ldots$, where the
first element should not be confused with $w_{+}$ if $W$ is finite. It is of
order two and conjugation by $w_{+}=w_{+}^{-1}$ preserves the generating set
$S$. Hence, it induces an automorphism of the Coxeter diagram of $(W,S)$. It
is called the longest word of $W$.

######  Lemma 2.13.

  * (i)

If $[W:W^{(i)}]$ is finite, let $w_{+}$ be the longest word of $W_{I\setminus
I_{\infty}}$ and let $j\in I$ be the unique index such that
$s_{j}=w_{+}s_{i}w_{+}$. Then $P^{(i)}(t)$ is non-empty if and only if
$t\in[-h_{j},h_{i}]$.

  * (ii)

If $[W:W^{(i)}]$ is infinite, then $P^{(i)}(t)$ is non-empty for all
$t\in(-\infty,h_{i}]$.

###### Proof.

  * (i)

Since $W=W_{I_{\infty}}\times W_{I\setminus I_{\infty}}$ is the direct product
of an infinite and a finite Coxeter group, conjugation by the longest word of
$W_{I\setminus I_{\infty}}$ defines an automorphism $\sigma$ of the Coxeter
diagram of $(W,S)$ which fixes all vertices in $I_{\infty}$. Hence, we have
$w_{+}s_{k}w_{+}=s_{\sigma(k)}$ for all $k\in I$, where $\sigma(k)=k$ for all
$k\in I_{\infty}$. This implies that
$w_{+}(\alpha_{k}^{\vee})=\pm\alpha_{\sigma(k)}^{\vee}$ for all $k\in I$,
where the sign is positive if and only if $k\in I_{\infty}$ by (1.3) because
$l(w_{+})<l(w_{+}s_{k})$ if and only if $k\in I_{\infty}$. Thus, the action of
$w_{+}$ on $\mathfrak{a}^{\ast}$ fixes $\omega_{k}$ for every $k\in
I_{\infty}$ and for $k\in I\setminus I_{\infty}$, it maps $\omega_{k}$ to
$-\omega_{\sigma(k)}$. If $[W:W^{(i)}]$ is finite, then $i\notin I_{\infty}$
and $w_{+}(\omega_{i})=-\omega_{j}$ for $j=\sigma(i)\in I\setminus
I_{\infty}$. Since $i\in\operatorname{supp}(w_{+})$, we obtain from Lemma 2.2
that

$0<\langle\omega_{i},h-w_{+}(h)\rangle=h_{i}-\langle
w_{+}(\omega_{i}),h\rangle=h_{i}+h_{j},$

so the interval $[-h_{j},h_{i}]$ is well-defined. If $h^{\prime}\in\Xi_{h}$,
then by $W$-invariance of $\Xi_{h}$, we also have
$w_{+}(h^{\prime})\in\Xi_{h}$. Proposition 2.3 now implies that
$\langle\omega_{k},w_{+}(h^{\prime})\rangle\leq h_{k}$ for all $k\in I$ and
hence

$h_{i}^{\prime}=\langle\omega_{i},h^{\prime}\rangle=\langle
w_{+}(\omega_{i}),w_{+}(h^{\prime})\rangle=\langle-\omega_{j},w_{+}(h^{\prime})\rangle\geq-
h_{j}.$

Thus, in this case $H^{(i)}(t)$ can only intersect $\Xi_{h}$ if $t\geq-h_{j}$.
On the other hand, we know that $P^{(i)}(h_{i})=F^{(i)}$ is a face of
$\Xi_{h}$ and using the same reasoning, one verifies that

$P^{(i)}(-h_{j})=\operatorname{conv}(W^{(i)}w_{+}\cdot
h)=\operatorname{conv}(w_{+}W^{(j)}\cdot h)=w_{+}(F^{(j)})$

is also a face of $\Xi_{h}$ and thus non-empty. Since $\Xi_{h}$ is convex, it
follows that $P^{(i)}(t)$ is non-empty for all $t\in[-h_{j},h_{i}]$.

  * (ii)

Consider the orbit of $\omega_{i}$ under the $W$-action on
$\mathfrak{a}^{\ast}$. Since
$\langle\omega_{i},\alpha_{j}^{\vee}\rangle=\delta_{ij}$ for all $i,j\in I$,
it follows easily by induction on the word length (cf. the proof of [Kac90,
Proposition 3.12 a)]) that the stabilizer of $\omega_{i}$ is the standard
subgroup $W^{(i)}$. Since $[W:W^{(i)}]=\infty$, the $W$-orbit of $\omega_{i}$
is infinite. In addition, it is contained in $\omega_{i}-Q_{+}$ which implies
that the orbit contains elements of the form $\omega_{i}-\alpha$ with
$\alpha\in Q_{+}$ and $\operatorname{ht}(\alpha)$ unbounded. For every
$t\in\mathbb{R}$, we therefore find $w\in W$ such that
$t_{0}:=\langle\omega_{i},w(h)\rangle\leq t$. Since $P^{(i)}(t_{0})$ contains
$w(h)$, it is non-empty. By convexity, it follows that $P^{(i)}(t)$ is non-
empty for all $t\in[t_{0},h_{i}]$. As $t$ can be chosen arbitrarily small and
$t_{0}\leq t$, the assertion follows.

∎

Using the notation of the preceding lemma, we define an interval
$I_{h}\subset\mathbb{R}$ by

$I_{h}:=\begin{cases}[-h_{j},h_{i}]&\text{if $[W:W^{(i)}]<\infty$,}\\\
(-\infty,h_{i}]&\text{if $[W:W^{(i)}]=\infty$.}\end{cases}$

so that we may express the previous result result more briefly as

$\Xi_{h}=\bigcup_{t\in I_{h}}P^{(i)}(t).$ (2.3)

Our next goal is to determine the face poset of $P^{(i)}(t)$ for $t\in I_{h}$.
By the discussion in Section 2.1, for every face $F$ of $P^{(i)}(t)$ there
exists a unique minimal face $F^{\prime}$ of $\Xi_{h}$ such that
$F=F^{\prime}\cap H^{(i)}(t)$ and its dimension is determined by (2.1). In
particular, every face of $P^{(i)}(t)$ is exposed and every vertex of $F$ is
either a vertex of $F^{\prime}$ that happens to be contained in $H^{(i)}(t)$,
or lies in the interior of a unique edge of $F^{\prime}$. The first case is
only possible if $t=\langle\omega_{i},w(h)\rangle$ for some $w\in W$. Thus, in
general $F$ need not contain any vertices of $\Xi_{h}$, but it still inherits
the following property from $F^{\prime}$.

######  Lemma 2.14.

Every face of $P^{(i)}(t)$ is the convex hull of its vertices.

If $P^{(i)}(t)$ is a convex polytope, this is of course a well-known property.
For arbitrary closed convex sets, however, this is not necessarily true
without further requirements on the set. In our situation, the assertion can
be easily reduced to the compact case by a cut-off argument.

###### Proof.

Let $F$ be any face of $P^{(i)}(t)$ and choose $F^{\prime}$ as the smallest
face of $\Xi_{h}$ such that $F=F^{\prime}\cap H^{(i)}(t)$. By the preceding
discussion, the set $Y$ of vertices of $F$ is discrete and not empty and
$\operatorname{conv}(Y)\subset F$ holds by convexity. To show the converse, we
claim that for every $x\in F$, there exists a finite subset $Y^{\prime}\subset
Y$ such that $x\in\operatorname{conv}(Y^{\prime})$. Set $\omega:=\sum_{i\in
I}\omega_{i}\in\mathfrak{a}^{\ast}$, let $x\in F$ be arbitrary and consider
the closed half-space

$E_{+}:=\\{y\in E_{h}:\langle\omega,x\rangle\leq\langle\omega,y\rangle\\}.$

Then $F_{+}:=F\cap E_{+}$ is compact since $F=F^{\prime}\cap P^{(i)}(t)$ is
closed and each of the linearly independent functionals
$\omega_{1},\ldots,\omega_{n}\in\mathfrak{a}^{\ast}$ is bounded on $F_{+}$.
Every face of $F_{+}$ is exposed, so by [Roc70, Corollary 18.5.1], $F_{+}$ is
the convex hull of its vertices. Now every vertex of $F_{+}$ is either a
vertex of $F$ contained in $E_{+}$, or it lies in the interior of a unique
edge of $F$. Let $Y_{1}^{\prime}\subset Y$ be the set of vertices of $F$
contained in $E_{+}$ and let $Y_{2}^{\prime}\subset Y$ be the set of all
endpoints of edges of $F$ which start in $E_{+}$ and end outside of it. Since
$Y$ is discrete and $F_{+}$ is compact, the set
$Y^{\prime}:=Y_{1}^{\prime}\cup Y_{2}^{\prime}$ is finite and every vertex of
$F_{+}$ is either contained in $Y^{\prime}$ or it is a convex combination of
two points in $Y^{\prime}$. Hence, it follows that $x\in
F_{+}\subset\operatorname{conv}(Y^{\prime})\subset\operatorname{conv}(Y)$, as
claimed. As $x\in F$ was arbitrary, we conclude that
$F\subset\operatorname{conv}(Y)$. ∎

Moreover, $P^{(i)}(t)$ is invariant under the action of the rank-$(n-1)$
subgroup $W^{(i)}\subset W$ which is the Weyl group of the submatrix
$\mathbb{A}^{(i)}$ of $\mathbb{A}$ with the $i$-th row and column removed. The
set $\Phi^{(i)}$ of real roots of $\mathbb{A}^{(i)}$ can be identified with
the subset of $\Phi$ that lies in the $\mathbb{Z}$-span of
$\Phi_{0}\setminus\\{\alpha_{i}\\}$. The Kac-Moody root datum $\mathcal{D}$
with GCM $\mathbb{A}$ can be restricted to a free and cofree root datum with
GCM $\mathbb{A}^{(i)}$ without changing $\Lambda$ which allows us to define an
open fundamental chamber $C^{(i)}$ and a Tits cone $X^{(i)}$ for $W^{(i)}$ in
$\mathfrak{a}$ by

$C^{(i)}:=\\{h^{\prime}\in\mathfrak{a}:\langle\alpha_{k},h^{\prime}\rangle>0\,\,\forall
k\in I\setminus\\{i\\}\\}\qquad\text{and}\qquad X^{(i)}:=\bigcup_{w\in
W^{(i)}}w(\overline{C^{(i)}}).$

The closure of $C^{(i)}$ is a strict fundamental domain for the
$W^{(i)}$-action on $X^{(i)}\subset\mathfrak{a}$, so every face of
$P^{(i)}(t)\subset X^{(i)}$ is $W^{(i)}$-equivalent to a face that intersects
the closed fundamental chamber

$\overline{C^{(i)}}=\\{h^{\prime}\in\mathfrak{a}:\langle\alpha_{k},h^{\prime}\rangle\geq
0\,\,\forall k\in I\setminus\\{i\\}\\}.$

In particular, every vertex is $W^{(i)}$-equivalent to a vertex in that set.
More precisely, the next proposition asserts that there are only finitely many
such vertices and all of them are already contained in the open fundamental
chamber $C^{(i)}$. To distinguish between the different notions of chambers,
we refer to the $W$-translates of $\overline{C}$ as $W$-chambers and to the
$W^{(i)}$-translates of $\overline{C^{(i)}}$ as $W^{(i)}$-chambers and
likewise for the open chambers.

######  Proposition 2.15.

For every $t\in I_{h}$, the set $P^{(i)}(t)$ is the convex hull of a finite
union of $W^{(i)}$-orbits. More precisely, there exist finitely many vertices
$\\{x_{1},\ldots,x_{m}\\}$ of $P^{(i)}(t)$ that are contained in $C^{(i)}$ and

$P^{(i)}(t)=\operatorname{conv}\bigg{(}\bigcup_{r=1}^{m}W^{(i)}\cdot
x_{r}\bigg{)}.$

###### Proof.

Applying Lemma 2.14 to the trivial face $F=P^{(i)}(t)$ yields that
$P^{(i)}(t)$ is the convex hull of its vertices. Every such vertex is
$W^{(i)}$-equivalent to a vertex in $\overline{C^{(i)}}$, so $P^{(i)}(t)$ is
the convex hull of the union of all $W^{(i)}$-orbits of its vertices in
$\overline{C^{(i)}}$. We first show that there are only finitely many such
vertices. Recall that every vertex is of one of the following two types: It is
either a vertex of $\Xi_{h}$ contained in $H^{(i)}(t)$, i.e. equal to $w(h)$
for some $w\in W$, or it lies in the interior of a unique edge of $\Xi_{h}$.
First, there can only be finitely many vertices of the first type. If
$t\neq\langle\omega_{i},w(h)\rangle$ for all $w\in W$, there cannot be any
such vertices at all, so assume that $t=\langle\omega_{i},w(h)\rangle$ for
some $w\in W$ and let $\alpha\in Q_{+}$ be such that
$w^{-1}(\omega_{i})=\omega_{i}-\alpha$. For any other $w^{\prime}\in W$ with
$w^{\prime-1}(\omega_{i})=\omega_{i}-\beta$, we have
$\langle\omega_{i},w^{\prime}(h)\rangle=t$ if and only if
$\langle\alpha,h\rangle=\langle\beta,h\rangle$. If $\alpha=\beta$, then
$w^{-1}(\omega_{i})=w^{\prime-1}(\omega_{i})$, so that $w^{\prime}w^{-1}\in
W^{(i)}$ and $w(h)$ is $W^{(i)}$-equivalent to $w^{\prime}(h)$. Otherwise,
since $\langle\beta,h\rangle>0$ for all $\beta\in Q_{+}$, the equality
$\langle\alpha,h\rangle=\langle\beta,h\rangle$ is only possible for finitely
many $\beta\in Q_{+}$.
Second, this argument also shows that there can only be finitely many vertices
of the second type. Let $x$ be a vertex of $P^{(i)}(t)$ that is contained in
the interior of an edge of $\Xi_{h}$. The endpoints of that edge are $w(h)$
and $ws_{k}(h)$ for some $w\in W$ and $k\in I$ with
$\langle\omega_{i},w(h)\rangle>t$ and $\langle\omega_{i},ws_{k}(h)\rangle<t$.
By what we have just shown, there are only finitely many
$W^{(i)}$-inequivalent elements $w\in W$ with
$\langle\omega_{i},w(h)\rangle>t$ and from every such vertex $w(h)$, there are
only $n$ edges emanating from it, so there are only finitely many
$W^{(i)}$-inequivalent edges of $\Xi_{h}$ that cross $H^{(i)}(t)$.
Hence, there exist finitely many vertices $\\{x_{1},\ldots,x_{m}\\}$ of
$P^{(i)}(t)$ that lie in $\overline{C^{(i)}}$. We now show that they are
actually contained in $C^{(i)}$. To this end, it suffices to show that none of
these vertices is contained in a wall of $\overline{C^{(i)}}$. If $x_{r}$ is
of the first type, then $x_{r}$ is $W$-equivalent to $h$ which is assumed to
be contained in the open $W$-chamber $C$. This is a subset of $C^{(i)}$, so
$x_{r}$ is not contained in any wall of $\overline{C^{(i)}}$. If $x_{r}$ is of
the second type, it is contained in the interior of a unique edge $L$ of
$\Xi_{h}$ whose endpoints are $w_{1}(h)$ and $w_{2}(h)$ for some
$w_{1},w_{2}\in W$. By Corollary 2.10, this edge crosses precisely one wall,
so $\langle\alpha_{k},x_{r}\rangle=0$ for some $k\neq i$ is only possible if
$w_{2}=s_{k}w_{1}$. But then $\omega_{i}$ takes the same value at $w_{1}(h)$
and $w_{2}(h)$ which implies that $\omega_{i}$ is constant on $L$. Since
$x_{r}\in L\cap P^{(i)}(t)$, it follows that $L\subset P^{(i)}(t)$ is an edge
of $P^{(i)}(t)$. However, $x_{r}$ is an interior point of $L$ and a vertex of
$P^{(i)}(t)$, a contradiction. ∎

In particular, it follows that every vertex of $P^{(i)}(t)$ lies in an open
$W^{(i)}$-chamber $w(C^{(i)})$ for some $w\in W^{(i)}$. Moreover, every face
of $P^{(i)}(t)$ contains a vertex, so we also conclude that every face of
$P^{(i)}(t)$ is $W^{(i)}$-equivalent to a face which contains at least one of
the vertices $\\{x_{1},\ldots,x_{m}\\}$. We denote by
$P^{(i)}_{\operatorname{ess}}(t)$ the union of all faces of $P^{(i)}(t)$ which
are entirely contained in $C^{(i)}$. Equivalently, since $C^{(i)}$ is convex,
a face lies in $P^{(i)}_{\operatorname{ess}}(t)$ if and only if all its
vertices are contained in $\\{x_{1},\ldots,x_{m}\\}$. We call
$P^{(i)}_{\operatorname{ess}}(t)$ the essential part of $P^{(i)}(t)$. This
terminology is motivated by Proposition 2.17 below. To prepare for the proof,
we need some more information on the faces of $P^{(i)}(t)$. Observe that if
$m=1$, then $P^{(i)}(t)$ is the convex hull of a single $W^{(i)}$-orbit and
the arguments of this section prove that its face poset is isomorphic to
$\Sigma^{\ast}(W^{(i)},S^{(i)})$. If $m>1$, this is no longer true. Since
$W^{(i)}$ acts simply transitively on $W^{(i)}$-chambers, a face in the
essential part cannot be stabilized by any non-trivial element of $W^{(i)}$.
Conversely, the following result shows that if a face of $P^{(i)}(t)$ is not
contained in the essential part, it must have a certain degree of symmetry.

######  Proposition 2.16.

Every face of $P^{(i)}(t)$ is invariant under a parabolic subgroup of
$W^{(i)}$. More precisely, for every face $F$ of $P^{(i)}(t)$, there exist
$1\leq t\leq m$, $r_{1},\ldots,r_{t}\in\\{1,\ldots,m\\}$, $w\in W^{(i)}$ and a
standard subgroup $W_{J}\subset W^{(i)}$ such that

$F=\operatorname{conv}\bigg{(}\bigcup_{s=1}^{t}wW_{J}\cdot x_{r_{s}}\bigg{)}.$

The second statement is indeed a refinement of the first because if $F$ has
the asserted shape, then $F$ is invariant under the parabolic subgroup
$wW_{J}w^{-1}\subset W^{(i)}$.

###### Proof.

Since $F$ is the convex hull of its vertices, it suffices to show that the
vertices of $F$ form a union of orbits of a standard coset of
$(W^{(i)},S^{(i)})$. After replacing $F$ by $w^{-1}(F)$ for a suitable $w\in
W^{(i)}$, we may assume that at least one vertex of $F$ is contained in
$C^{(i)}$. We are going to show that there exists a unique maximal subset
$J\subset I\setminus\\{i\\}$ such that $F$ is invariant under the standard
subgroup $W_{J}\subset W^{(i)}$, so that $w(F)$ will be invariant under the
parabolic subgroup $wW_{J}w^{-1}$. If all vertices of $F$ are already
contained in the set $\\{x_{1},\ldots,x_{m}\\}$, the claim is satisfied only
by $J=\emptyset$. Otherwise, there exists an edge $L\subset F$ which connects
a vertex $x_{r}\in C^{(i)}$ to a vertex $y\in w(C^{(i)})$ for some $w\in
W^{(i)}\setminus\\{1\\}$. We claim that $w=s_{k}$ for some $k\neq i$ and
$y=s_{k}(x_{r})$.
Let $w=s_{i_{1}}\ldots s_{i_{l}}$ be a reduced decomposition of $w$ and set
$k:=i_{1}\in I\setminus\\{i\\}$. Then we have
$\langle\alpha_{k},x_{r}\rangle>0$ and $\langle\alpha_{k},y\rangle=\langle
w^{-1}(\alpha_{k}),w^{-1}(y)\rangle<0$ because
$w^{-1}(\alpha_{k})\in\Phi^{(i)}_{-}$ by (1.3) and $w^{-1}(y)\in C^{(i)}$.
Hence, $x_{r}$ and $y$ lie in open $W^{(i)}$-chambers on different sides of
the hyperplane $V_{\alpha_{k}}:=\operatorname{ker}(\alpha_{k})$, so the edge
$L$ crosses this plane at an interior point. Since the faces of $P^{(i)}(t)$
are $W^{(i)}$-invariant, $L^{\prime}:=s_{k}(L)$ is also an edge of
$P^{(i)}(t)$. As $s_{k}$ fixes an interior point of $L$, the two edges
$L^{\prime}$ and $L$ intersect at a point that lies in the relative interior
of both of them. This is only possible if $L^{\prime}=L$, which forces
$w=s_{k}$ and $y=s_{k}(x_{r})$. In particular, it shows that $L$ is invariant
under the rank-one standard subgroup of $W^{(i)}$ generated by $s_{k}$.
Next, we show that the entire face $F$ is $s_{k}$-invariant. To this end, let
$E\subset\mathfrak{a}$ be a supporting hyperplane for $F$ and denote by
$E^{\prime}:=E-x_{r}\subset\mathfrak{a}$ the corresponding parallel translate
containing the origin, i.e. the linear subspace corresponding to $E$. Observe
that $E$ contains the infinite line $L_{0}$ through the two distinct points
$x_{r}$ and $y=s_{k}(x_{r})$ which crosses the wall $V_{\alpha_{k}}$, so
$E^{\prime}$ contains the one-dimensional $s_{k}$-invariant subspace
$L_{0}^{\prime}:=L_{0}-x_{r}$ which is not contained in $V_{\alpha_{k}}$.
Therefore, $E^{\prime}$ and $V_{\alpha_{k}}$ intersect in a
$(d-2)$-dimensional subspace and every point $v\in E^{\prime}$ can be uniquely
written as $v=v_{1}+v_{2}$ with $v_{1}\in L_{0}^{\prime}$ and $v_{2}\in
E^{\prime}\cap V_{\alpha_{k}}$. This yields that
$s_{k}(v)=s_{k}(v_{1})+s_{k}(v_{2})=s_{k}(v_{1})+v_{2}\in E^{\prime}$, so
$E^{\prime}$, and hence $F$, is $s_{k}$-invariant. Continuing in this way for
all $W^{(i)}$-chambers adjacent to $C^{(i)}$ that $F$ meets, we conclude that
$F$ is invariant under a unique maximal standard subgroup $W_{J}$ of
$W^{(i)}$.
Hence, if $\\{x_{r_{1}},\ldots,x_{r_{t}}\\}$ are the vertices of $F$ contained
in $C^{(i)}$, then the set $W_{J}\cdot\\{x_{r_{1}},\ldots,x_{r_{t}}\\}$ also
consists of vertices of $F$. We claim that these are all the vertices of $F$.
Assume that this is not the case, then there exists a vertex $y_{1}$ in a
chamber $w_{1}(C^{(i)})$ for some $w_{1}\in W^{(i)}\setminus W_{J}$. Since the
underlying graph of $P^{(i)}(t)$ is connected, we may choose $y_{1}$ in such a
way that it is connected by an edge $L$ to a vertex of the form
$y_{2}=w_{2}(x_{r_{s}})$, where $s\in\\{1,\ldots,t\\}$ and $w_{2}\in W_{J}$.
As $F$ is $W_{J}$-invariant, $w_{2}^{-1}(L)$ is an edge of $F$ connecting
$x_{r_{s}}$ to a vertex $y^{\prime}:=w_{2}^{-1}(y_{1})\in
w_{2}^{-1}w_{1}(C^{(i)})$. Exactly as in the first part of the proof, we
conclude that $w_{2}^{-1}w_{1}=s_{l}$ for some $l\neq i$, but $s_{l}\notin
W_{J}$. This contradicts the maximality of $W_{J}$ and shows that the set of
vertices of $F$ is the union of the $W_{J}$-orbits of its vertices in
$C^{(i)}$. The proof is therefore complete. ∎

In order to construct the set $P^{(i)}(t)$ from its vertices, one first has to
form the union of a finite number of $W^{(i)}$-orbits and then take the convex
hull of the resulting set. However, with the previous proposition at hand, we
can now show that the essential part allows us to exchange these operations
and instead express $P^{(i)}(t)$ as the union of convex hulls of
$W^{(i)}$-orbits. Observe that if $m>1$, then the essential part contains
uncountably many points.

######  Proposition 2.17.

Every point in $P^{(i)}(t)$ lies in the convex hull of the $W^{(i)}$-orbit of
a point in $P^{(i)}_{\operatorname{ess}}(t)$, i.e.

$P^{(i)}(t)=\bigcup_{y\in
P^{(i)}_{\operatorname{ess}}(t)}\operatorname{conv}(W^{(i)}\cdot y).$

###### Proof.

Let $x\in P^{(i)}(t)$ be arbitrary and let $F$ be the smallest face of
$P^{(i)}(t)$ containing $x$. We prove the claim by induction on the dimension
of $F$. Since $P^{(i)}(t)$ is $W^{(i)}$-invariant, we may w.l.o.g. assume that
$x\in\overline{C^{(i)}}$, which implies that one of the vertices of $F$ is
contained in $C^{(i)}$. Let $x_{r}$ for some $r\in\\{1,\ldots,m\\}$ be one
such vertex.
If $\operatorname{dim}(F)=0$, then $x=x_{r}\in
P^{(i)}_{\operatorname{ess}}(t)$ and there is nothing to show. Assume now that
$\operatorname{dim}(F)>0$. If all vertices of $F$ are in
$\\{x_{1},\ldots,x_{m}\\}$, then $F\subset P^{(i)}_{\operatorname{ess}}(t)$
and we are done. Otherwise, Proposition 2.16 shows that $F$ is invariant under
a non-trivial standard subgroup $W_{J}\subset W^{(i)}$. Assume first that $x$
is not a global fixed-point of $W_{J}$ and choose some $k\in J$ such that
$s_{k}(x)\neq x$. Then $\omega_{k}$ is not bounded from above on the infinite
line $L_{0}$ spanned by $x$ and $s_{k}(x)$, so $L_{0}$ intersects the boundary
of $F$ in at least one point $x_{+}$ which is contained in a proper face of
$F$. By the induction hypothesis, there exists a point $y\in
P^{(i)}_{\operatorname{ess}}(t)$ such that
$x_{+}\in\operatorname{conv}(W^{(i)}\cdot y)$. Since $x$ and $s_{k}(x)$ are
contained in the relative interior of $F$ and $x_{+}$ in the boundary, $x_{+}$
cannot lie between $x$ and $s_{k}(x)$ on $L_{0}$. This implies that $x$ is a
convex combination of $x_{+}$ and $s_{k}(x_{+})$ and we conclude that
$x\in\operatorname{conv}(W^{(i)}\cdot y)$.
If $s_{k}(x)=x$ for all $k\in J$, then the line $L_{0}$ is not well-defined,
but this issue is easily dealt with. By minimality of $F$, $x$ has to lie in
the relative interior of $F$. Let $E$ be a supporting hyperplane for $F$ and
consider its parallel translate $E^{\prime}:=E-x$ containing the origin. Let
$k\in J$ be arbitrary, then $s_{k}\in W_{J}$ acts on $\mathfrak{a}$ as a non-
trivial involution, so one of its eigenvalues is $-1$. Moreover, it preserves
$E^{\prime}$ and acts non-trivially on it. Hence, there exists a non-zero
vector $v\in E^{\prime}$ such that $s_{k}(v)=-v$. Choosing $v\neq 0$
sufficiently small, we have $x^{\prime}:=x+v\in F$ and $x$ can be written as
the non-trivial convex combination

$x=\frac{1}{2}(x+v+x-v)=\frac{1}{2}(x^{\prime}+s_{k}(x^{\prime})).$

The same argument as above applied to the line spanned by $x^{\prime}$ and
$s_{k}(x^{\prime})$ yields that $x^{\prime}$, and hence $x$, lies in the
convex hull of $W^{(i)}\cdot y$ for some $y\in
P^{(i)}_{\operatorname{ess}}(t)$. ∎

######  Example 2.18.

We illustrate the content of Proposition 2.17 in Figure 1. Both pictures show
the union of the convex hulls of three $W^{(i)}$-orbits, where $W^{(i)}$ is
the symmetric group $S_{3}$ in (a) and $W^{(i)}=(\mathbb{Z}_{2})^{3}$ in (b).
The set $P^{(i)}(t)$ is the convex hull of these three orbits, which is not
depicted in the figure, and the essential part is highlighted in red. In both
examples, it is easy to imagine that as a point moves over the red region, the
corresponding $W^{(i)}$-orbits trace out the entire set $P^{(i)}(t)$. In
Figure (a), we have also drawn the reflecting hyperplanes which bound the Weyl
chambers. In Figure (b), these would be the coordinate hyperplanes and the
closed fundamental chamber is the octant of $\mathbb{R}^{3}$ where all
coordinates are non-negative.

(a) $W^{(i)}=S_{3}$

(b) $W^{(i)}=(\mathbb{Z}_{2})^{3}$

Figure 1: Two configurations of $P^{(i)}_{\operatorname{ess}}(t)$.

Proposition 2.17 will play an important role in the proof of Theorem 4.1. We
close this section by briefly outlining how the correspondence between minimal
edge paths in $\Xi_{h}$ and minimal galleries in the Coxeter complex
$\Sigma(W,S)$ (cf. Corollary 2.10) provides a method to determine the number
$m=m(t)$ of vertices of $P^{(i)}_{\operatorname{ess}}(t)$ for a varying $t\in
I_{h}$, which is especially interesting in the case of finite index. We use
the same notation as in Lemma 2.13. Suppose that $[W:W^{(i)}]<\infty$, let
$w_{+}$ be the longest word of $W_{I\setminus I_{\infty}}$ and let $j\in I$ be
the unique index such that $s_{j}=w_{+}s_{i}w_{+}$. Denote by $w_{-}$ the
unique element of minimal length in the coset $W^{(i)}w_{+}=w_{+}W^{(j)}$. The
faces $F^{(i)}$ and $w_{+}(F^{(j)})=\operatorname{conv}(W^{(i)}\cdot
w_{+}(h))$ of $\Xi_{h}$ are “opposite” in the sense that they have the same
shape, their defining hyperplanes are parallel and $\Xi_{h}$ lies between
them. The point $w_{-}(h)$ is the unique vertex of $w_{+}(F^{(j)})$ whose
distance to $h$ in the underlying graph of $\Xi_{h}$ is minimal. As this graph
is isomorphic to the underlying chamber system of $\Sigma(W,S)$, minimal edge
paths from $h$ to $w_{-}(h)$ are in bijective correspondence with reduced
decompositions of $w_{-}$. In addition, every such path is contained in the
open $W^{(i)}$-chamber $C^{(i)}$ and every vertex in $C^{(i)}$ lies on such a
path. With this information, it is not difficult to show that every minimal
edge path from $h$ to $w_{-}(h)$ intersects $H^{(i)}(t)$ in one of the
vertices $\\{x_{1},\ldots,x_{m}\\}$ of $P^{(i)}_{\operatorname{ess}}(t)$ and,
conversely, every vertex lies on such a minimal path. In particular, if
$w_{-}$ has a unique reduced decomposition, then there exists only one minimal
edge path from $h$ to $w_{-}(h)$ and we have $m(t)=1$ for all $t\in I_{h}$. In
this case, $P^{(i)}(t)$ is the convex hull of a single $W^{(i)}$-orbit. This
happens, for example, if $W$ is the symmetric group $S_{n}$ and
$i\in\\{1,n-1\\}$, but it is not the case if $n\geq 4$ and
$i\in\\{2,\ldots,n-2\\}$. We point out explicitly, however, that $m(t)$ is
usually not equal to the number of reduced decompositions of $w_{-}$ because
distinct minimal edge paths from $h$ to $w_{-}(h)$ may well intersect
$H^{(i)}(t)$ in the same point. For example, every such path begins with the
edge $[h,s_{i}(h)]$ and ends with $[w_{-}s_{j}(h),w_{-}(h)]$, so that $m(t)=1$
for all
$t\in[-\langle\omega_{j},h\rangle,-\langle\omega_{j},s_{j}(h)\rangle]\cup[\langle\omega_{i},s_{i}(h)\rangle,\langle\omega_{i},h\rangle]$.
In between these values, minimal edge paths typically split and merge. Since
the vertices of $P^{(i)}(t)$ are simply the intersection points of
$H^{(i)}(t)$ with edges of $\Xi_{h}$, varying $t$ can only change $m(t)$ if
this causes the hyperplane $H^{(i)}(t)$ to begin or to stop intersecting an
edge of $\Xi_{h}$. Hence, $m(t)$ is constant on the connected components of
the complement of the discrete set $\\{\langle\omega_{i},w(h)\rangle\mid w\in
W\\}$ in $I_{h}$.
If $[W:W^{(i)}]=\infty$, the element $w_{-}\in W$ is not defined which mirrors
the fact that the length of a left $W^{(i)}$-reduced element in $W$ is
unbounded. In this case, there is no canonical vertex of $\Xi_{h}$ that all
minimal edge paths in $C^{(i)}$ starting at $h$ can be extended to.
Nevertheless, the set of all $w\in W$ such that $w(h)$ is the first vertex in
a minimal edge path in $C^{(i)}$ starting at $h$ with
$\langle\omega_{i},w(h)\rangle\leq t$ is still finite and $m(t)$ can be
computed as in the finite-index case by counting reduced decompositions of
these elements and investigating whether the associated minimal paths
intersect $H^{(i)}(t)$ in distinct points.

##  3 Kac-Moody groups

Having only dealt with Kac-Moody algebras and their Weyl groups so far, we now
introduce the class of groups that will be relevant to our discussion. Kac-
Moody groups can be defined in many different ways and varying levels of
generality. A common approach that was introduced by J. Tits, following the
scheme-theoretic construction of the classical Chevalley groups, is to define
a Kac-Moody group over a field $\mathbb{K}$ as the group
$\mathcal{G}(\mathbb{K})$ of $\mathbb{K}$-points of a group functor
$\mathcal{G}:\text{$\mathbb{Z}$-alg}\to\text{Grp}$ satisfying a list of axioms
(cf. [Tit87, Section 2]). However, Tits also shows that over fields, this
functor is uniquely determined (up to a certain non-degeneracy assumption) and
the group $\mathcal{G}(\mathbb{K})$ can be given an explicit presentation by
generators and relations (cf. [Tit87, Section 3.6 and 3.9, Theorem 1]). For
our purposes, it will suffice to consider Kac-Moody groups defined via this
presentation which is called the constructive Tits functor in [Rém02, Section
8.3]. We explicitly point out that we continue to denote the set of real roots
by $\Phi$ instead of $\Phi^{\operatorname{re}}$. The set of imaginary roots
does not occur in this definition of Kac-Moody groups. In the first part of
this chapter, however, we do not require Assumption 1.2. For any field
$\mathbb{K}$, we write $\mathbb{K}^{\ast}:=\mathbb{K}\setminus\\{0\\}$ for its
group of invertible elements.
Let $\mathcal{D}=(I,\mathbb{A},\Lambda,(c_{i})_{i\in I},(h_{i})_{i\in I})$ be
an arbitrary Kac-Moody root datum of rank $d$, not necessarily free or cofree
here. For any field $\mathbb{K}$, we set
$T(\mathbb{K}):=\text{Hom}_{\text{Grp}}(\Lambda,\mathbb{K}^{\ast})\cong\Lambda^{\vee}\otimes_{\mathbb{Z}}\mathbb{K}^{\ast}$.
Hence, $T(\mathbb{K})$ is a free abelian group of rank $d$ isomorphic to
$(\mathbb{K}^{\ast})^{d}$. For $r\in\mathbb{K}^{\ast}$ and
$h\in\Lambda^{\vee}$, we denote by $r^{h}\in T(\mathbb{K})$ the homomorphism
$\lambda\mapsto r^{h}(\lambda):=r^{\langle\lambda,h\rangle}$. If
$\\{v_{1},\ldots,v_{d}\\}$ is a $\mathbb{Z}$-basis of $\Lambda^{\vee}$, then
$T(\mathbb{K})$ is generated by $\\{r^{v_{i}}\mid 1\leq i\leq
d,\,r\in\mathbb{K}^{\ast}\\}$. The canonical action of the Weyl group on
$\Lambda$ induces an action of $W$ on $T(\mathbb{K})$ by $w(t):=t\circ w^{-1}$
for $w\in W$ and $t\in T(\mathbb{K})$. If $t=r^{h}$ for some
$r\in\mathbb{K}^{\ast}$ and $h\in\Lambda^{\vee}$, then $w(r^{h})=r^{w(h)}$.
For any Kac-Moody root datum $\mathcal{D}$, the (split, minimal) Kac-Moody
group of type $\mathcal{D}$ over $\mathbb{K}$ is the group $G(\mathbb{K})$
generated by $T(\mathbb{K})$ and a family of subgroups
$(U_{\alpha}(\mathbb{K}))_{\alpha\in\Phi}$ indexed by the real roots, subject
to the relations in [Tit87, Section 3.6].444We do not give the exact
presentation here since its formulation requires concepts that we have not
discussed. Also, their precise form is largely irrelevant to our discussion.
What matters are the algebraic and representation-theoretic properties listed
below that follow from the relations. Here, $U_{\alpha}(\mathbb{K})$ is an
abelian subgroup of $G(\mathbb{K})$ isomorphic to the additive group
$(\mathbb{K},+)$. We choose an isomorphism $x_{\alpha}:\mathbb{K}\to
U_{\alpha}(\mathbb{K})$ and write $U_{\alpha}(\mathbb{K})=\\{x_{\alpha}(r)\mid
r\in\mathbb{K}\\}$. We call $T(\mathbb{K})$ the (standard $\mathbb{K}$-split)
torus and $(U_{\alpha}(\mathbb{K}))_{\alpha\in\Phi}$ the root groups of
$G(\mathbb{K})$. Every root group is normalized by the torus. In fact, for
every $\alpha\in\Phi,\,r\in\mathbb{K}$ and $t\in T(\mathbb{K})$, we have
$tx_{\alpha}(r)t^{-1}=x_{\alpha}(t(c(\alpha))r)$. We point out that we have
suppressed the root datum $\mathcal{D}$ from the notation, but the torus
$T(\mathbb{K})$ clearly depends on $\mathcal{D}$ and so do some of the
defining relations that we have not mentioned.

######  Remark 3.1.

Although we will not use this theory explicitly, we mention for completeness
that the tuple
$(G(\mathbb{K}),(U_{\alpha}(\mathbb{K}))_{\alpha\in\Phi},T(\mathbb{K}))$ is an
RGD-system of type $(W,S)$ (see [AB08, Section 8.6]). This is a simple
consequence of the fact that the real roots $\Phi$ are in bijective
correspondence with the roots of the Coxeter complex $\Sigma(W,S)$ and the
commutator relations between the root groups that are built into the
presentation of $G(\mathbb{K})$ (cf. [Mar18, Theorem 7.69]). In particular,
$G(\mathbb{K})$ acts strongly transitively on an associated Moufang twin
building which is a valuable tool to investigate its algebraic structure. We
refer to [AB08, Chapter 8] for more information.

To simplify the notation in the following discussion, we fix a field
$\mathbb{K}$ and write $G,T,U_{\alpha},\ldots$ instead of
$G(\mathbb{K}),T(\mathbb{K}),U_{\alpha}(\mathbb{K}),\ldots$ and only indicate
the field if there is danger of confusion. For every $i\in I$, set
$\tilde{s}_{i}:=x_{\alpha_{i}}(1)x_{-\alpha_{i}}(1)x_{\alpha_{i}}(1)$ and let
$\tilde{W}$ be the subgroup of $G$ generated by $\\{\tilde{s}_{i}\mid i\in
I\\}$. It is called the extended Weyl group. It induces the natural $W$-action
on $T$ in the sense that for every $w\in W$ there exists some
$\tilde{w}\in\tilde{W}$ such that $\tilde{w}t\tilde{w}^{-1}=w(t)$ for all
$t\in T$. Moreover, it also permutes the root groups by the rule
$\tilde{w}x_{\alpha}(r)\tilde{w}^{-1}=x_{w(\alpha)}(\pm r)$ for all
$\alpha\in\Phi$ and $r\in\mathbb{K}$, where the ambiguous sign need not
concern us here.
For any subset $J\subset I$, we set $\Phi(J):=\Phi\cap\bigoplus_{j\in
J}\mathbb{Z}\alpha_{j}$, $\Phi_{+}(J):=\Phi_{+}\cap\Phi(J)$ and
$\Phi^{\perp}_{+}(J):=\Phi_{+}\setminus\Phi_{+}(J)$ and define the following
three subgroups of $G$:

$\displaystyle G_{J}$ $\displaystyle:=\langle
T,U_{\alpha}\mid\alpha\in\Phi(J)\rangle$ $\displaystyle U_{+}^{\perp}(J)$
$\displaystyle:=\langle U_{\alpha}\mid\alpha\in\Phi_{+}^{\perp}(J)\rangle$
$\displaystyle P_{J}$ $\displaystyle:=\langle
T,U_{\alpha}\mid\alpha\in\Phi_{+}\cup\Phi(J)\rangle$

The group $P_{J}$ is called the standard parabolic subgroup of type $J$ and it
has a Levi decomposition

$P_{J}\cong G_{J}\ltimes U_{+}^{\perp}(J).$ (3.1)

The Levi factor $G_{J}$ is again a Kac-Moody group associated to the Kac-Moody
root datum $\mathcal{D}(J):=(J,\mathbb{A}_{J},\Lambda,(c_{j})_{j\in
J},(h_{j})_{j\in J})$, where $\mathbb{A}_{J}$ is the submatrix of $\mathbb{A}$
with rows and columns indexed by $J$. Note that if $\mathcal{D}$ is free
(resp. cofree), then $\mathcal{D}(J)$ has the same property. If $J=\emptyset$,
we set $B_{+}:=P_{\emptyset}$ and $U_{+}:=U_{+}^{\perp}(\emptyset)$ and obtain
$B_{+}\cong T\ltimes U_{+}$. The group $B_{+}$ is called the positive Borel
subgroup of $G$ and $U_{+}$ is the subgroup of $G$ generated by all
$U_{\alpha}$ with $\alpha\in\Phi_{+}$.
Next, we transfer some concepts from the representation theory of Kac-Moody
algebras (cf. Chapter 1) to the group level. Let $\mathfrak{g}$ be the Kac-
Moody algebra of type $\mathcal{D}$ over $\mathbb{K}$ with Cartan subalgebra
$\mathfrak{h}\subset\mathfrak{g}$ and let $\lambda\in\mathfrak{h}^{\ast}$ be a
dominant integral weight. Since the highest-weight module
$\rho_{\lambda}:\mathfrak{g}\to\mathfrak{gl}(L(\lambda))$ is integrable, it
can be lifted to a $\mathbb{K}$-linear representation
$\hat{\rho}_{\lambda}:G\to\operatorname{GL}(L(\lambda))$ by

$\displaystyle\hat{\rho}_{\lambda}(x_{\alpha}(r))$
$\displaystyle:=\exp(\rho_{\lambda}(re_{\alpha})),$
$\displaystyle\alpha\in\Phi,\,r\in\mathbb{K},$ (3.2)
$\displaystyle\hat{\rho}_{\lambda}(t)v_{\mu}$ $\displaystyle:=t(\mu)v_{\mu},$
$\displaystyle t\in T,\,\mu\in\operatorname{wt}(L(\lambda)),\,v_{\mu}\in
L(\lambda)_{\mu},$

where $e_{\alpha}\in\mathfrak{g}_{\alpha}$ is a suitably chosen non-zero
vector (cf. [Mar18, Remark 7.6]). In particular, it follows from (1.4) and
(3.2) that for $\alpha\in\Phi$, $r\in\mathbb{K}$ and $v_{\mu}\in
L(\lambda)_{\mu}$, we have $\rho_{\lambda}(re_{\alpha})^{N+1}v_{\mu}=0$ for
some $N\in\mathbb{N}_{0}$ and

$\hat{\rho}_{\lambda}(x_{\alpha}(r))v_{\mu}=v_{\mu}+\sum_{k=1}^{N}\text{(an
element of $L(\lambda)_{\mu+kc(\alpha)}$).}$ (3.3)

Let us now specialize to the case that the field $\mathbb{K}$ is either
$\mathbb{R}$ or $\mathbb{C}$. Let $\\{v_{1},\ldots,v_{d}\\}$ be a
$\mathbb{Z}$-basis of $\Lambda^{\vee}$, denote by $\mathfrak{a}$ the real
Cartan subalgebra of $\mathfrak{g}_{\mathbb{R}}$ and by $\mathfrak{h}$ the
complex Cartan subalgebra of $\mathfrak{g}_{\mathbb{C}}$. There exists a well-
defined surjective exponential map

$\exp:\mathfrak{h}\to
T(\mathbb{C}),\qquad\sum_{i=1}^{d}r_{i}v_{i}\mapsto\prod_{i=1}^{d}(e^{r_{i}})^{v_{i}},\qquad
r_{1},\ldots,r_{d}\in\mathbb{C},$

which corresponds to the coordinate-wise exponential map
$\mathbb{C}\to\mathbb{C}^{\ast}$ under the isomorphisms
$\mathfrak{h}\cong\mathbb{C}^{d}$ and
$T(\mathbb{C})\cong(\mathbb{C}^{\ast})^{d}$. It restricts to an injective map
$\exp:\mathfrak{a}\to T(\mathbb{R})$ whose image is isomorphic to
$(\mathbb{R}_{>0})^{d}$ and will be denoted by $A$. It can be viewed as a
subgroup of $G(\mathbb{R})$ and of $G(\mathbb{C})$. The polar decomposition in
$\mathbb{C}$ and its trivial restriction to $\mathbb{R}$ imply that
$T(\mathbb{K})\cong M\times A$, where $M$ is isomorphic to $(S^{1})^{d}$ if
$\mathbb{K}=\mathbb{C}$ and to $(\mathbb{Z}_{2})^{d}$ if
$\mathbb{K}=\mathbb{R}$. The extended Weyl group $\tilde{W}\subset G$
normalizes $A$ and $\exp:\mathfrak{a}\to A$ is $W$-equivariant.
Moreover, the compact involution $\theta:\mathfrak{g}\to\mathfrak{g}$
possesses an analogue on the group level. More precisely, there exists a
unique involutive automorphism $\Theta:G\to G$ which satisfies
$\Theta(x_{\alpha}(r))=x_{-\alpha}(\overline{r})$ for all $\alpha\in\Phi$ and
$r\in\mathbb{K}$ and $\Theta(r^{v_{i}})=\overline{r}^{-v_{i}}$ for all
$i\in\\{1,\ldots,d\\}$ and $r\in\mathbb{K}^{\ast}$. Here, the bars denote
complex conjugation which can be omitted if $\mathbb{K}=\mathbb{R}$. It is
called the compact involution of $G$. Its fixed-point set is denoted by $K$
and called the compact form of $G$. By definition of the extended Weyl group,
$\tilde{W}$ is contained in $K$.
Using the fact that all panels in the twin building associated to $G$ (cf.
Remark 3.1) are isomorphic to the projective line over $\mathbb{K}$, it is not
difficult to show that the action of $K$ on this building is chamber-
transitive and hence $G=KB_{+}$. Moreover, we have $B_{+}=T\ltimes U_{+}$ and
one easily verifies that $K\cap U_{+}=\\{1\\}$ and $K\cap T=M$. Since $T\cong
M\times A$, this shows that $G$ has an Iwasawa decomposition $G=KAU_{+}$ in
the sense that

the multiplication map $K\times A\times U_{+}\to G$ is bijective. (3.4)

Such a decomposition for certain types of Kac-Moody groups has been
established, for example, in [KP85, Proposition 5.1] or [FHHK20, Theorem
3.31]. The argument that we have indicated is inspired by [DMGH09, Section 3].
Note that the group $A$ is the same in the real and complex case, only $K$ and
$U_{+}$ depend on the field. The corresponding projection onto the
$A$-component will be denoted by

$\Pi:G\to A,\qquad g=kau\mapsto a.$

We study a particular property of this map in our main theorem. A key
ingredient of the proof will be the following facts about highest-weight
representations of $G$. As discussed at the end of Chapter 1, if
$\lambda\in\mathfrak{a}^{\ast}\subset\mathfrak{h}^{\ast}$ is a dominant
integral weight, there exists a contravariant symmetric bilinear or Hermitian
form $H$ on the irreducible real or complex highest-weight module
$L(\lambda)$. Using that $G$ is generated by the torus $T$ and all root groups
$U_{\alpha}$ with $\alpha\in\Phi$, it is straightforward to deduce from (3.2)
and (1.6) that the corresponding representation of $G$ on $L(\lambda)$
satisfies

$H(g\cdot v,w)=H(v,\Theta(g^{-1})\cdot w)\qquad\text{for all $g\in G$ and
$v,w\in L(\lambda)$.}$ (3.5)

Therefore, $H$ is $K$-invariant. If the GCM $\mathbb{A}$ is symmetrizable,
then $H$ is positive definite, so that $K$ acts on $L(\lambda)$ by unitary
operators. Moreover, $A$ acts diagonally on $L(\lambda)$ with positive
eigenvalues. In fact, if $a=\exp(h)$ with
$h=\sum_{i=1}^{d}r_{i}v_{i}\in\mathfrak{a}$, then a simple computation using
(3.2) shows that $a(\mu)=e^{\langle\mu,h\rangle}$ for every weight $\mu$ of
$L(\lambda)$, which is positive since $\langle\mu,h\rangle\in\mathbb{R}$. The
action of $U_{+}$ is determined by (3.3). In particular, if Assumption 1.2 is
satisfied, this discussion applies to the functionals in the dual basis
$\mathcal{B}^{\ast}=\\{\omega_{1},\ldots,\omega_{n},\phi_{n+1},\ldots,\phi_{d}\\}\subset\mathfrak{a}^{\ast}$
of Proposition 1.3.

##  4 The convexity theorem

We now come to our main theorem. Throughout this chapter, we suppose that
Assumption 1.2 is in effect. Let $G$ be the split Kac-Moody group over
$\mathbb{K}\in\\{\mathbb{R},\mathbb{C}\\}$ associated to a free and cofree
Kac-Moody root datum $\mathcal{D}$ of rank $d\geq
2n-\operatorname{rank}(\mathbb{A})$ and denote by $\mathfrak{a}$ the Cartan
subalgebra of $\mathfrak{g}_{\mathbb{R}}$ with basis
$\mathcal{B}=\\{\alpha_{1}^{\vee},\ldots,\alpha_{n}^{\vee},h_{n+1},\ldots,h_{d}\\}$.
Let $G=KAU_{+}$ be an Iwasawa decomposition of $G$ and define

$\Pi:G\to A,\qquad g=kau\mapsto a$

as the corresponding projection onto the $A$-component. It is clear that
$\Pi(kg)=\Pi(g)$ for all $k\in K$ and $g\in G$, but there is no obvious way
how to compute $\Pi(gk)$. The following theorem will answer this question in
the special case that $g=a\in A$ is contained in the exponential of an open
chamber of the Tits cone $X\subset\mathfrak{a}$. Recall that the exponential
map restricts to a $W$-equivariant isomorphism $\mathfrak{a}\to A$ whose
inverse will be denoted by $\log:A\to\mathfrak{a}$.

######  Theorem 4.1.

If $a\in A$ is contained in the exponential image of an open chamber of the
Tits cone $X\subset\mathfrak{a}$, then

$\\{\Pi(ak)\mid k\in K\\}=\exp(\operatorname{conv}(W\cdot\log(a))).$

In order to prove Theorem 4.1, let us fix some notation. Assume that $a\in A$
is fixed and set $h:=\log(a)\in\mathfrak{a}$. By assumption, $h$ is contained
in an open chamber of $X$. We write $\Xi_{h}:=\operatorname{conv}(W\cdot h)$
as in Chapter 2 and define

$\Pi_{a}:=\\{\Pi(ak)\mid k\in K\\}=\Pi(aK).$

Thus, we need to show that $\Pi_{a}=\exp(\Xi_{h})$. The proof will consist of
two steps: First, we show that $\Pi_{a}\subset\exp(\Xi_{h})$ by using tools
from the representation theory of $G$. Afterwards, we prove that
$\exp(\Xi_{h})\subset\Pi_{a}$ by induction on the size $n$ of the GCM
$\mathbb{A}$. Note that both sets do not depend on the $W$-translate of $a$
(resp. $h$). This is obvious for $\Xi_{h}$ and for $\Pi_{a}$, we may argue as
follows: If $w\in W$ is arbitrary, then we may choose a representative
$\tilde{w}\in\tilde{W}\subset K$ such that conjugation by $\tilde{w}$ induces
the action of $w$ on $A$ and we have

$\Pi_{w(a)}=\Pi(w(a)K)=\Pi(\tilde{w}a\tilde{w}^{-1}K)=\Pi(a\tilde{w}^{-1}K)=\Pi(aK)=\Pi_{a}.$

Thus, we may w.l.o.g. assume that $h\in C$ and $a\in\exp(C)$.

###  4.1 Proof of the main theorem

####  4.1.1 Step 1: Highest-weight representations

We begin by establishing that $\Pi_{a}\subset\exp(\Xi_{h})$. Let $k\in K$ be
arbitrary and write $ak\in G$ according to its Iwasawa decomposition as

$ak=k^{\prime}a^{\prime}u^{\prime}\qquad\text{with $k^{\prime}\in
K,\,a^{\prime}\in A,\,u^{\prime}\in U_{+}$.}$

In order to prove that $h^{\prime}:=\log(a^{\prime})$ is contained in
$\Xi_{h}$, we show that $h^{\prime}$ satisfies the inequalities

$\begin{cases}\langle\omega_{i},w(h^{\prime})\rangle\leq\langle\omega_{i},h\rangle&\text{for
all $w\in W$ and $i\in I$}\\\
\langle\phi_{i},h^{\prime}\rangle=\langle\phi_{i},h\rangle&\text{for all
$i\in\\{n+1,\ldots,d\\}$}\end{cases}$

of Corollary 2.11 by studying the action of $G=KAU_{+}$ in suitable highest-
weight representations. By Proposition 1.3 and the discussion in Chapter 3,
the dual basis
$\mathcal{B}^{\ast}=\\{\omega_{1},\ldots,\omega_{n},\phi_{n+1},\ldots,\phi_{d}\\}$
of $\mathfrak{a}^{\ast}$ consists of dominant integral weights and the group
$G$ acts on the corresponding real or complex highest-weight modules
$L(\omega_{1}),\ldots,L(\omega_{n}),L(\phi_{n+1}),\ldots,L(\phi_{d})$. Each of
these modules carries a positive definite, contravariant, symmetric bilinear
or Hermitian form $H$ such that weight spaces belonging to different weights
are orthogonal and by (3.5), the action of the subgroup $K\subset G$ is
unitary. The subgroup $A\subset G$ acts diagonally on these modules with
positive real eigenvalues and the action of $U_{+}$ is determined by (3.3).
Moreover, the modules $L(\phi_{n+1}),\ldots,L(\phi_{d})$ are one-dimensional
and the action of $U_{+}$ is trivial. We write $\hat{\rho}_{i}$ for the
corresponding homomorphism $G\to\operatorname{GL}(L(\omega_{i}))$ if $i\in I$
and $G\to\operatorname{GL}(L(\phi_{i}))$ if $i\in\\{n+1,\ldots,d\\}$ and
denote the associated representations of the Kac-Moody algebra $\mathfrak{g}$
by $\rho_{1},\ldots,\rho_{d}$.
First, we fix an arbitrary $i\in\\{n+1,\ldots,d\\}$ and show that
$\langle\phi_{i},h\rangle=\langle\phi_{i},h^{\prime}\rangle$. Since
$L(\phi_{i})$ is one-dimensional and the $K$-action is unitary, we have
$\hat{\rho}_{i}(K)\subset\\{\pm\operatorname{id}\\}$ and thus

$\hat{\rho}_{i}(ak)v=\pm\hat{\rho}_{i}(a)v=\pm a(\phi_{i})v$

for any vector $v\in L(\phi_{i})$. Moreover, $U_{+}$ acts trivially on
$L(\phi_{i})$ which implies that

$\hat{\rho}_{i}(k^{\prime}a^{\prime}u^{\prime})v=\hat{\rho}_{i}(k^{\prime}a^{\prime})v=a^{\prime}(\phi_{i})\hat{\rho}_{i}(k^{\prime})v=\pm
a^{\prime}(\phi_{i})v.$

Since $ak=k^{\prime}a^{\prime}u^{\prime}$, these two vectors must coincide.
Now $a(\phi_{i})$ and $a^{\prime}(\phi_{i})$ are both positive real numbers,
so if we choose $v\neq 0$, then equality is only possible if
$a(\phi_{i})=a^{\prime}(\phi_{i})$ or, equivalently,
$e^{\langle\phi_{i},h\rangle}=e^{\langle\phi_{i},h^{\prime}\rangle}$. As the
exponents are real numbers, this is equivalent to
$\langle\phi_{i},h\rangle=\langle\phi_{i},h^{\prime}\rangle$.
Thus, it only remains to show that if $w\in W$ and $i\in I$ are arbitrary,
then $\langle\omega_{i},w(h^{\prime})\rangle\leq\langle\omega_{i},h\rangle$.
Since the set of weights of $L(\omega_{i})$ is $W$-invariant,
$\mu:=w^{-1}(\omega_{i})$ is a weight of $L(\omega_{i})$ and we may choose a
vector $v\in L(\omega_{i})_{\mu}$ with $||v||:=H(v,v)=1$. As we assume
$\mathcal{D}$ to be free, every weight of $L(\omega_{i})$ is of the form
$\omega_{i}-\alpha$ for some $\alpha\in Q_{+}$. The eigenvalues of the
diagonal map $\hat{\rho}_{i}(a)$ are precisely the exponentials of the
eigenvalues of $\rho_{i}(h)$. Since $h\in C$, we have
$\langle\alpha,h\rangle>0$ for all $\alpha\in Q_{+}$, so the largest
eigenvalue is $a(\omega_{i})=e^{\langle\omega_{i},h\rangle}$. Moreover,
$u^{\prime}\in U_{+}$ is a product of elements in positive root groups, so
(3.3) implies that for every $\nu\in\operatorname{wt}(L(\omega_{i}))$ with
$\mu<\nu\leq\omega_{i}$, there exists a suitable weight-$\nu$ vector
$v_{\nu}\in L(\omega_{i})_{\nu}$ such that

$\hat{\rho}_{i}(u^{\prime})v=v+\sum_{\mu<\nu\leq\omega_{i}}v_{\nu}.$

On the one hand, this allows us to make the following computation:

$\displaystyle||\hat{\rho}_{i}(ak)v||$
$\displaystyle=||\hat{\rho}_{i}(k^{\prime}a^{\prime}u^{\prime})v||=\bigg{|}\bigg{|}\hat{\rho}_{i}(k^{\prime}a^{\prime})\bigg{(}v+\sum_{\mu<\nu\leq\omega_{i}}v_{\nu}\bigg{)}\bigg{|}\bigg{|}$
(4.1)
$\displaystyle=\bigg{|}\bigg{|}\hat{\rho}_{i}(k^{\prime})\bigg{(}a^{\prime}(\mu)v+\sum_{\mu<\nu\leq\omega_{i}}a^{\prime}(\nu)v_{\nu}\bigg{)}\bigg{|}\bigg{|}$
$\displaystyle=\bigg{|}\bigg{|}a^{\prime}(\mu)v+\sum_{\mu<\nu\leq\omega_{i}}a^{\prime}(\nu)v_{\nu}\bigg{|}\bigg{|}$
$\displaystyle\geq||a^{\prime}(\mu)v||=a^{\prime}(\mu)=a^{\prime}(w^{-1}(\omega_{i}))=w(a^{\prime})(\omega_{i})$

To obtain the inequality (4.1), we have used that the $K$-action is unitary,
that weight vectors in different weight spaces are orthogonal, $||v||=1$ and
that $a^{\prime}(\mu)=e^{\langle\mu,h^{\prime}\rangle}>0$. On the other hand,
the vector $\hat{\rho}_{i}(ak)v$ is contained in the image of the unit sphere
under the diagonal map $\hat{\rho}_{i}(a)$. More precisely, since
$L(\omega_{i})$ has a basis consisting of weight vectors and
$\hat{\rho}_{i}(k)v$ is a finite linear combination of basis vectors, there
exists some $m\in\mathbb{N}$ such that

$\hat{\rho}_{i}(k)v\in\bigoplus_{\begin{subarray}{c}\alpha\in Q_{+},\\\
\operatorname{ht}(\alpha)\leq
m\end{subarray}}L(\omega_{i})_{\omega_{i}-\alpha}=:V_{0},$

which is a finite-dimensional vector space, and $\hat{\rho}_{i}(k)v$ is
contained in the unit sphere of $V_{0}$. Since $\hat{\rho}_{i}(a)$ acts
diagonally on $L(\omega_{i})$, it preserves $V_{0}$ and the restriction of
$\hat{\rho}_{i}(a)$ to $V_{0}$ is represented by a finite diagonal matrix
whose diagonal entries are the values $a(\omega_{i}-\alpha)$, where $\alpha$
runs over all elements of $Q_{+}$ such that $\operatorname{ht}(\alpha)\leq m$
and $\omega_{i}-\alpha\in\operatorname{wt}(L(\omega_{i}))$. Hence,
$\hat{\rho}_{i}(ak)v$ is contained in the image of the unit sphere of $V_{0}$
under the diagonal map $\hat{\rho}_{i}(a)|_{V_{0}}$. This set is an ellipsoid
whose half-axes have lengths given by the absolute values of the eigenvalues
of $\hat{\rho}_{i}(a)|_{V_{0}}$. Since these are all positive, the absolute
values can be omitted. In particular, the longest half-axis is the maximal
eigenvalue $a(\omega_{i})$ and we conclude that

$w(a^{\prime})(\omega_{i})\leq||\hat{\rho}_{i}(ak)v||\leq a(\omega_{i}).$

This is equivalent to $e^{\langle\omega_{i},w(h^{\prime})\rangle}\leq
e^{\langle\omega_{i},h\rangle}$ and hence to
$\langle\omega_{i},w(h^{\prime})\rangle\leq\langle\omega_{i},h\rangle$ since
the exponents are real numbers. The proof of the inclusion
$\Pi_{a}\subset\exp(\Xi_{h})$ is therefore finished.

####  4.1.2 Step 2: Induction

We now proceed to show the converse inclusion $\exp(\Xi_{h})\subset\Pi_{a}$ by
induction on the size $n=|I|$ of the GCM $\mathbb{A}$. If $n=1$, then
$\mathbb{A}=(2)$ is the Cartan matrix of the finite-dimensional simple Lie
algebra $\mathfrak{sl}_{2}(\mathbb{K})$ and the associated Weyl group is
isomorphic to $\mathbb{Z}_{2}$. By [Mar18, Exercise 7.33], the Kac-Moody group
associated to the minimal free and cofree root datum
$\mathcal{D}^{\mathbb{A}}_{\text{Kac}}$ (which coincides with
$\mathcal{D}^{\mathbb{A}}_{\text{sc}}$ since $\mathbb{A}$ is invertible) is
$\operatorname{SL}_{2}(\mathbb{K})$ and $G$ is a semidirect extension of that,
i.e. $G\cong\operatorname{SL}_{2}(\mathbb{K})\rtimes T^{\prime}$ where
$T^{\prime}$ is a subtorus of $T$ of dimension $d-1$. In contrast to the Lie
algebra case (cf. the discussion after Assumption 1.2), this is not
necessarily a direct product.
Before we treat the case of an arbitrary free and cofree root datum, let us
assume that $\mathcal{D}=\mathcal{D}^{\mathbb{A}}_{\text{Kac}}$. In this case,
$G=\operatorname{SL}_{2}(\mathbb{K})$ and the proof can be completely reduced
to $(2\times 2)$-matrix computations. If we identify the positive (resp.
negative) root group of $G$ with the group of upper (resp. lower)
unitriangular matrices, then the compact involution of $G$ is the map
$g\mapsto(\overline{g}^{-1})^{T}$. It induces the usual Iwasawa decomposition
$G=KAU_{+}$, where $K=\operatorname{SO}(2)$ if $\mathbb{K}=\mathbb{R}$ and
$K=\operatorname{SU}(2)$ if $\mathbb{K}=\mathbb{C}$, $A$ is the group of
diagonal matrices with determinant $1$ and positive real entries (both in the
real and complex case) and $U_{+}$ is the upper unitriangular group. Hence, we
have

$h=\begin{pmatrix}r&0\\\ 0&-r\end{pmatrix}\text{and
}a=\exp(h)=\begin{pmatrix}e^{r}&0\\\ 0&e^{-r}\end{pmatrix}\text{for some
$r\in\mathbb{R}$.}$

Since the positive root of $\mathfrak{g}=\mathfrak{sl}_{2}(\mathbb{K})$ is the
map $\begin{pmatrix}r&0\\\ 0&-r\end{pmatrix}\mapsto 2r$ and we assume that
$h\in C$, we have $r>0$. The unique non-trivial element $s_{1}\in
W\cong\mathbb{Z}_{2}$ acts on the one-dimensional space $\mathfrak{a}$ as
$-\operatorname{id}$. We parametrize the group $\operatorname{SO}(2)\subset K$
as

$\operatorname{SO}(2)=\bigg{\\{}R_{\gamma}:=\begin{pmatrix}\cos(\gamma)&-\sin(\gamma)\\\
\sin(\gamma)&\cos(\gamma)\end{pmatrix}\,\bigg{|}\,\gamma\in[0,2\pi)\bigg{\\}}.$

######  Proposition 4.2.

For every $\gamma\in[0,2\pi)$, we have

$\Pi(aR_{\gamma})=\begin{pmatrix}\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}&0\\\
0&\frac{1}{\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}}\end{pmatrix}.$

###### Proof.

Throughout this proof, we denote by $(\cdot,\cdot)$ the standard symmetric
bilinear or Hermitian form on $\mathbb{K}^{2}$ and by $||\cdot||$ the induced
norm. To compute the Iwasawa decomposition of $aR_{\gamma}$, we need to apply
the Gram-Schmidt procedure to the column vectors of the matrix

$aR_{\gamma}=\begin{pmatrix}e^{r}\cos(\gamma)&-e^{r}\sin(\gamma)\\\
e^{-r}\sin(\gamma)&e^{-r}\cos(\gamma)\end{pmatrix}.$

Let $\\{v_{1},v_{2}\\}$ denote the two column vectors of this matrix, then the
Gram-Schmidt process turns $\\{v_{1},v_{2}\\}$ into an orthonormal basis
$\\{e_{1},e_{2}\\}$ of $\mathbb{K}^{2}$ as follows: First, set
$\tilde{e}_{1}:=v_{1}$ and

$\displaystyle\tilde{e}_{2}$
$\displaystyle:=v_{2}-\frac{(v_{1},v_{2})}{||v_{1}||^{2}}v_{1}=\begin{pmatrix}-e^{r}\sin(\gamma)\\\
e^{-r}\cos(\gamma)\end{pmatrix}-\frac{(e^{-2r}-e^{2r})\cos(\gamma)\sin(\gamma)}{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}\begin{pmatrix}e^{r}\cos(\gamma)\\\
e^{-r}\sin(\gamma)\end{pmatrix}$
$\displaystyle=\frac{1}{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}\bigg{[}\begin{pmatrix}-e^{r}\sin(\gamma)[e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)]\\\
e^{-r}\cos(\gamma)[e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)]\end{pmatrix}$
$\displaystyle\quad-\begin{pmatrix}(e^{-2r}-e^{2r})\cos(\gamma)\sin(\gamma)\cdot
e^{r}\cos(\gamma)\\\ (e^{-2r}-e^{2r})\cos(\gamma)\sin(\gamma)\cdot
e^{-r}\sin(\gamma)\end{pmatrix}\bigg{]}=\frac{1}{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}\begin{pmatrix}-e^{-r}\sin(\gamma)\\\
e^{r}\cos(\gamma)\end{pmatrix}.$

Then $e_{1}$ and $e_{2}$ are obtained by normalizing $\tilde{e}_{1}$ and
$\tilde{e}_{2}$ and we have

$||\tilde{e}_{1}||=\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}\qquad\text{and}\qquad||\tilde{e}_{2}||=\frac{1}{\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}}.$

Since column operations are realized by multiplication with elementary
matrices from the right, it follows that
$k^{\prime}=gu^{\prime-1}a^{\prime-1}$ where:

$\displaystyle k^{\prime}$
$\displaystyle=\frac{1}{\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}}\begin{pmatrix}e^{r}\cos(\gamma)&-e^{-r}\sin(\gamma)\\\
e^{-r}\sin(\gamma)&e^{r}\cos(\gamma)\end{pmatrix}\in\operatorname{SO}(2)\subset
K$ (4.2) $\displaystyle a^{\prime-1}$
$\displaystyle=\begin{pmatrix}\frac{1}{\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}}&0\\\
0&\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}\end{pmatrix}\in A$
$\displaystyle u^{\prime-1}$
$\displaystyle=\begin{pmatrix}1&-\frac{(e^{-2r}-e^{2r})\cos(\gamma)\sin(\gamma)}{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}\\\
0&1\end{pmatrix}\in U_{+}$

If we rearrange this expression as $g=k^{\prime}a^{\prime}u^{\prime}$, we
conclude that

$\Pi(aR_{\gamma})=a^{\prime}=\begin{pmatrix}\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}&0\\\
0&\frac{1}{\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}}\end{pmatrix}.$

∎

As we let $\gamma$ run through the interval $[0,2\pi)$, the expression
$\sqrt{e^{2r}\cos^{2}(\gamma)+e^{-2r}\sin^{2}(\gamma)}$ attains precisely the
values in the interval $[e^{-r},e^{r}]\subset\mathbb{R}_{>0}$, which is in
bijective correspondence with $[-r,r]\subset\mathbb{R}$ under the exponential
map. Therefore, $\Pi(aR_{\gamma})$ takes precisely the values in the
exponential of the line segment connecting

$h=\begin{pmatrix}r&0\\\ 0&-r\end{pmatrix}\qquad\text{with}\qquad
s_{1}(h)=\begin{pmatrix}-r&0\\\ 0&r\end{pmatrix},$

which is the convex hull of the $W$-orbit of $h$. This shows that
$\exp(\Xi_{h})\subset\Pi(a\operatorname{SO}(2))\subset\Pi(aK)=\Pi_{a}$. These
inclusions are automatically equalities if $\mathbb{K}=\mathbb{R}$, but they
are also equalities if $\mathbb{K}=\mathbb{C}$ by Step 1 of the proof. In
particular, we see that
$\Pi(a\operatorname{SO}(2))=\Pi(a\operatorname{SU(2)})$ in the complex case.
Let now $\mathcal{D}$ be an arbitrary free and cofree Kac-Moody root datum
associated to $\mathbb{A}=(2)$ and let $G$ be the corresponding Kac-Moody
group of type $\mathcal{D}$ over $\mathbb{K}$. Then the subgroup $G_{1}$ of
$G$ generated by $U_{\pm\alpha_{1}}$ and $T_{1}:=\\{r^{\alpha_{1}^{\vee}}\mid
r\in\mathbb{K}^{\ast}\\}$ is isomorphic to $\operatorname{SL}_{2}(\mathbb{K})$
under the map $\varphi:\operatorname{SL}_{2}(\mathbb{K})\to G$ defined by
$\varphi\begin{pmatrix}1&r\\\ 0&1\end{pmatrix}:=x_{\alpha_{1}}(r)$,
$\varphi\begin{pmatrix}1&0\\\ r&1\end{pmatrix}:=x_{-\alpha_{1}}(-r)$ for
$r\in\mathbb{K}$ and $\varphi\begin{pmatrix}r&0\\\
0&r^{-1}\end{pmatrix}:=r^{\alpha_{1}^{\vee}}$ for $r\in\mathbb{K}^{\ast}$. If
$T_{2}$ is any complementary subgroup to $T_{1}$ in $T$, then $G\cong
G_{1}\rtimes T_{2}$, where the conjugation action of $T_{2}$ is trivial on
$T_{1}$ and satisfies
$tx_{\pm\alpha_{1}}(r)t^{-1}=x_{\pm\alpha_{1}}(t(\pm\alpha_{1})r)$ for all
$t\in T_{2}$ and $r\in\mathbb{K}$. In particular, if $t(\alpha_{1})=1$, then
$t$ commutes with $G_{1}$ and thus lies in the center of $G$. The compact