Diss. ETH No. 30394

Semialgebraic groups and
generalized affine buildings

A thesis submitted to attain the degree
of Doctor of Sciences
(Dr. sc. ETH Zürich)

presented by

Raphael Appenzeller

MSc Mathematics at ETH Zürich
born on 04.02.94

accepted on the recommendation of
Prof. Dr. Marc Burger, examiner
Prof. Dr. Petra Schwer, co-examiner

2024

## Abstract

We develop the theory of algebraic groups over real closed fields and apply
the results to construct a geometric object $\mathcal{B}$ and to prove that
$\mathcal{B}$ is an affine $\Lambda$-building. Real closed fields are ordered
fields that have particularly nice properties in the context of _semialgebraic
geometry_ , where objects are defined by polynomial equalities and
inequalities. Over the real numbers, algebraic groups are Lie groups and have
been studied extensively in the last century. Many results can be generalized
to algebraic groups over other real closed fields. We use a model theoretic
transfer principle to prove generalizations of statements about semisimple Lie
groups. In this direction we give proofs for the Iwasawa-decomposition $KAU$,
the Cartan-decomposition $KAK$ and the Bruhat-decomposition $BWB$. For
unipotent subgroups we prove the Baker-Campbell-Hausdorff formula and use it
to analyse root groups. We give a proof of the Jacobson-Morozov Lemma about
subgroups whose Lie algebra is isomorphic to $\mathfrak{sl}_{2}$ and we
describe other rank 1 subgroups which are the semisimple parts of Levi-
subgroups. We prove a semialgebraic version of Kostant’s convexity.

Over the reals, semisimple Lie groups are closely related to the symmetry
groups of symmetric spaces of non-compact type. Following an idea from [KT02],
these symmetric spaces can be described semialgebraically, which allows us to
consider their semialgebraic extension over any real closed field. Starting
from these non-standard symmetric spaces we use a valuation (with image some
non-discrete ordered abelian group $\Lambda$) on the fields to define a
$\Lambda$-pseudometric. Identifying points of distance zero results in a
$\Lambda$-metric space $\mathcal{B}$. Assuming that the root system of the
associated Lie group is reduced, we prove that $\mathcal{B}$ is an affine
$\Lambda$-building. The proof relies on a thorough analysis of stabilizers.

We further consider the special case of the group $\operatorname{SL}_{2}$,
where $\mathcal{B}$ is a $\Lambda$-tree. When $\Lambda<\mathbb{R}$ is a proper
subgroup, $\mathcal{B}$ is an incomplete metric space. We prove that even
after completing all the apartments, the metric space may still not be
complete. We also discuss the independence of axioms of $\Lambda$-trees and
$\Lambda$-buildings. We formalize the notions of $\Lambda$-metric spaces and
$\Lambda$-trees and produce computer verified proofs for some of our results
in the proof assistant $\operatorname{\mathsf{Lean}}$.

## Zusammenfassung

Wir entwickeln die Theorie der algebraischen Gruppen über reell
abgeschlossenen Körpern und verwenden die Resultate, um ein geometrisches
Objekt $\mathcal{B}$ zu konstruieren, und um zu zeigen, dass $\mathcal{B}$ ein
affines $\Lambda$-Gebäude ist. Reell abgeschlossene Körper sind geordnete
Körper, die besonders gute Eigenschaften in der _semialgebraischen Geometrie_
haben, wo Objekte mit polynomiellen Gleichungen und Ungleichungen definiert
werden. Algebraische Gruppen über den reellen Zahlen sind Lie-Gruppen und
wurden im letzten Jahrhundert ausgiebig studiert. Viele Resultate können für
algebraische Gruppen über reell abgeschlossenen Körpern verallgemeinert
werden. Wir verwenden ein Transfertheorem aus der Modelltheorie, um
Verallgemeinerungen von Aussagen über halbeinfache Lie-Gruppen zu zeigen. In
dieser Richtung geben wir Beweise für die Iwasawa-Zerlegung $KAU$, die Cartan-
Zerlegung $KAK$ und die Bruhat-Zerlegung $BWB$. Für unipotente Untergruppen
zeigen wir die Baker-Campbell-Hausdorff-Formel und verwenden sie, um
Wurzelgruppen zu untersuchen. Wir geben einen Beweis für das Jacobson-Morozov-
Lemma über Untergruppen mit Lie-Algebren, die isomorph zu $\mathfrak{sl}_{2}$
sind, und wir beschreiben andere Untergruppen mit Rang 1, die den
halbeinfachen Teilen von Levi-Untergruppen entsprechen. Wir beweisen eine
semialgebraische Version von Kostant’s Konvexität.

Über den reellen Zahlen sind halbeinfache Lie-Gruppen eng verwandt mit den
Symmetriegruppen von symmetrischen Räumen von nicht-kompaktem Typ. Nach einer
Idee von [KT02], können diese symmetrischen Räume semialgebraisch beschrieben
werden, was ihre semialgebraische Erweiterung zu einem beliebigen reell
abgeschlossenen Körper erlaubt. Mit diesen nicht-standard symmetrischen Räumen
können Bewertungen (mit Bild in einer nicht-diskreten geordneten kommutativen
Gruppe $\Lambda$) der Körper verwendet werden, um eine $\Lambda$-Pseudometrik
zu definieren. Werden dann Punkte der Distanz Null identifiziert, entsteht ein
$\Lambda$-metrischer Raum $\mathcal{B}$. Unter der Annahme, dass das
Wurzelsystem der entsprechenden Lie-Gruppe reduziert ist, zeigen wir, dass
$\mathcal{B}$ ein affines $\Lambda$-Gebäude ist. Der Beweis hängt von einer
gründlichen Untersuchung der Stabilisatoren ab.

Weiter untersuchen wir den Spezialfall der Gruppe $\operatorname{SL}_{2}$, wo
$\mathcal{B}$ ein $\Lambda$-Baum ist. Wenn $\Lambda<\mathbb{R}$ eine echte
Untergruppe ist, dann ist $\mathcal{B}$ ein unvollständiger metrsicher Raum.
Wir zeigen, dass $\mathcal{B}$ auch nachdem alle Segmente vervollständigt
wurden nicht zwingend vollständig sein muss. Weiter diskutieren wir die
Unabhängigkeit der Axiome von $\Lambda$-Bäumen und $\Lambda$-Gebäuden. Wir
formalisieren die Konzepte von $\Lambda$-metrischen Räumen und
$\Lambda$-Bäumen und produzieren Computer-überprüfte Beweise für einige der
Resultate mittels Beweisassistent $\operatorname{\mathsf{Lean}}$.

## Acknowledgements

I am very grateful for the support of many people during the last five years.
I have learned so much from my supervisor, Marc Burger, starting all the way
back in my first semester at university, when he introduced me to the
hyperbolic plane. I appreciate your continued support and also the freedom you
gave me to work on projects that interested me. I am thankful to Petra Schwer
for showing interest in my work, inviting me to Magdeburg and Heidelberg, and
for many long discussions about how to prove axiom (A2). I am looking forward
to working together with you in the future. For ideas and help with the
mathematics in this thesis, I would like to thank the experienced
mathematicians Linus Kramer, Anne Parreau, Gabriel Pallier and Auguste Hébert.

I spent a lot of time with my academic siblings, Luca De Rosa, Xenia Flamm and
Victor Jaeck. I learned so much from our weekly meetings and joint projects.
Luca’s relaxed approach to life, Xenia’s energetic ideas and the humor of
Victor are the perfect ingredients for a sustainably motivating team. Numerous
people enriched the time at ETH, among them fellow PhD students Jannick
Krifka, Merlin Incerti-Medici, Davide Spriano, Tommaso Goldhirsch, Lisa Ricci,
Paula Truöl, Francesco Fournier-Facio, Lauro Silini, Martina Jørgensen, Hjalti
Isleifsson, Alessio Cela, Aitor Iribaz-Lopez, Mireille Soergel, Fernando
Camacho Cadena, Merik Niemeyer, Kevin Klinge, Konstantin Andritsch and Segev
Gonen Cohen and also more senior people such as Peter Feller, Tom Ilmanen, Urs
Lang, Emmanuel Kowalski, Sebastian Baader, Matthew Cordes, Benjamin Brück,
Johannes Schmitt and Samir Canning. It was a pleasure to set up reading
courses with you, have après-lunch-meetings and arxiv-seminars, discuss
mathematics, go 3D-printing and do outreach projects, but also to spend some
quality free time together. Special thanks to Victor Jaeck and Segev Gonen
Cohen for reading parts of this thesis and providing valuable feedback before
I handed it in.

I would also like to thank my family and friends, foremost Ramona Ackermann,
for their support and for keeping up with my constant excitement about
mathematics.

###### Contents

  1. Abstract
  2. Acknowledgements
  3. 1 Introduction and main results
    1. 1.1 Semialgebraic groups and decompositions
    2. 1.2 The nonstandard symmetric space $\mathcal{X}_{\mathbb{F}}$
    3. 1.3 The building $\mathcal{B}$
    4. 1.4 Further results on $\Lambda$-trees
    5. 1.5 Formalization in the proof assistant $\operatorname{\mathsf{Lean}}$
  4. 2 Real closed fields
    1. 2.1 The transfer principle
    2. 2.2 Valuations
    3. 2.3 Examples
  5. 3 Affine $\Lambda$-buildings
    1. 3.1 $\Lambda$-metric spaces
    2. 3.2 $\Lambda$-trees
      1. 3.2.1 Example, Brumfiel’s tree
    3. 3.3 Root systems
    4. 3.4 Apartments
    5. 3.5 Affine $\Lambda$-buildings
    6. 3.6 Independence of axiom (A6)
  6. 4 Linear algebraic groups
    1. 4.1 Definitions
      1. 4.1.1 Examples
    2. 4.2 Morphisms and tori
      1. 4.2.1 Examples
    3. 4.3 The Lie algebra and the adjoint representation
    4. 4.4 Root systems and the spherical Weyl group
    5. 4.5 Borel subgroups, parabolic subgroups
      1. 4.5.1 Examples
  7. 5 Results about Lie algebras and split tori
    1. 5.1 Cartan involutions and Cartan decompositions
      1. 5.1.1 Examples
    2. 5.2 Real Cartan subalgebras and $\mathbb{R}$-split subalgebras
    3. 5.3 Lie algebras over real closed fields
    4. 5.4 Split tori of algebraic groups over real closed fields
  8. 6 Semialgebraic groups
    1. 6.1 Examples
    2. 6.2 Compatibility of the root systems
    3. 6.3 Iwasawa decomposition $G=KAU$
    4. 6.4 Cartan decomposition $G=KAK$
    5. 6.5 Bruhat decomposition $G=BWB$
    6. 6.6 Baker-Campbell-Hausdorff formula
    7. 6.7 The unipotent group $U$
    8. 6.8 Jacobson-Morozov Lemma for algebraic groups
    9. 6.9 Rank 1 subgroups
    10. 6.10 Kostant convexity
  9. 7 Definition of the building $\mathcal{B}$
    1. 7.1 Non-standard symmetric spaces
    2. 7.2 The pseudo-distance
    3. 7.3 The apartment
    4. 7.4 The building
  10. 8 Verification of the axioms for $\mathcal{B}$
    1. 8.1 Axioms (A1), (A3) and (TI)
    2. 8.2 Root groups and some Bruhat-Tits theory
      1. 8.2.1 Root group valuations
      2. 8.2.2 $W_{a}$-convexity for $U_{\mathbb{F}}$
      3. 8.2.3 Stabilizers of apartment, half-apartments and Weyl-chambers
      4. 8.2.4 Bruhat-Tits theory for root groups
      5. 8.2.5 Bruhat-Tits theory in rank 1
      6. 8.2.6 Bruhat-Tits theory in higher rank
    3. 8.3 Axiom (A2)
    4. 8.4 Axiom (A4)
    5. 8.5 Axiom (EC)
    6. 8.6 Beyond reduced root systems
  11. A Appendix: The building for $\operatorname{SL}(n,\mathbb{F})$

> _C’est avec la logique que nous prouvons et avec l’intuition que nous
> trouvons._
>
> Henry Poincaré

##  1 Introduction and main results

The rich interplay of group theory and geometry first popularized by Klein’s
Erlangen program enabled enormous progress in both geometry as well as group
theory in the last century. A connection between Riemannian symmetric spaces
and Lie groups was developed by Cartan in 1926, leading to the classification
of symmetric spaces [Car27]. While Lie groups rely on the real numbers,
developments in algebraic geometry allowed the study of algebraic groups over
arbitrary fields. In 1962 Jacques Tits introduced axioms for buildings as a
means to connect algebraic groups with incidence geometries [Tit62]. Buildings
had and continue to have a wide impact from the classification of finite
simple groups to rigidity theorems by Margulis and Mostow and to the study of
character varieties.

Symmetric spaces are Riemannian manifolds that at every point admit an
isometry given by reversing the direction of geodesics through the point
[Hel78, Ebe96]. Examples include the model spaces of constant curvature
(Euclidean space $\mathbb{E}^{n}$, spheres $\mathbb{S}^{n}$ and hyperbolic
space $\mathbb{H}^{n}$) but also manifolds whose curvature properties may
depend on the direction. Every symmetric space decomposes as a direct product
into symmetric spaces of compact type, Euclidean type and non-compact type.
The isometry group of a symmetric space is a Lie group and when the space is
of non-compact type, the Lie group is semisimple.

simplicial buildingsspherical buildings$\operatorname{rank}/\dim$discrete
affine buildings Euclidean buildings [KL97]$\mathbb{R}$-buildings [BT72,
Par00]$\Lambda$-buildings [Ben94, HIL24]
[KT02]trees$\mathbb{R}$-trees$\Lambda$-trees$1$$0$affine
buildings$1$$2$$2$$3$$3$$4$$\vdots$$\vdots$ Figure 1: Buildings were first
considered as simplicial objects, but for affine buildings the successively
more general notions of Euclidean buildings, $\mathbb{R}$-buildings and
$\Lambda$-buildings were introduced. All spherical buildings of rank at least
3 and all discrete affine buildings of rank at least 4 are associated to an
algebraic group and some field. In this thesis we consider the most general
notion, that of affine $\Lambda$-buildings.  Figure 2:  Two 3D printed models
of parts of a discrete affine building. The 3D models are available on
www.thingiverse.com/thing:6301564.

Buildings were first considered as simplicial complexes [Ron89, Bro89]. They
consist of apartments glued together according to some rules of symmetry. The
topological type of the apartments leads to a rough division of buildings, see
Figure 1. Buildings with simplicial divisions of spheres as apartments are
called spherical buildings and when the apartment is homeomorphic to
$\mathbb{R}^{n}$, the building is called affine, for some 3D printed models of
affine buildings, see Figure 2. There is a large collection of buildings whose
apartments may have various shapes. Spherical and affine buildings with
apartments of large enough dimension are classified [Wei03, Wei09]: they are
all associated to semisimple algebraic groups over certain fields. In the case
of affine buildings, the fields come with a discrete valuation.

Starting with [BT72], algebraic groups over fields with not necessarily
discrete valuations were considered, which lead to the introduction of non-
discrete affine buildings, or $\mathbb{R}$-buildings. These are defined as a
set together with an atlas of apartments satisfying five axioms, though the
exact choice of axioms varied by time and author. In this setting, apartments
are Euclidean affine spaces equipped with the action of an affine Weyl group.
Examples of $\mathbb{R}$-buildings were constructed and used in [BT72, KL97,
Par00].

In dimension 1, discrete affine buildings correspond exactly to simplicial
trees without leaves and $\mathbb{R}$-buildings correspond to
$\mathbb{R}$-trees, also called real trees, without endpoints. Real trees
appear in various mathematical areas, such as geometric group theory [Sha87]
and probability [Ald91]. Real trees can also be characterized as those
geodesically complete metric spaces that are $\delta$-hyperbolic for
$\delta=0$, in the sense of Gromov [Gro87]. In 1984, Morgan and Shalen [MS84]
introduced $\Lambda$-trees for any linearly ordered abelian group $\Lambda$,
see [Chi01]. This concept generalizes simplicial trees as $\mathbb{Z}$-trees
and real trees as $\mathbb{R}$-trees. When $\Lambda<\mathbb{R}$,
$\Lambda$-trees offer a new way of describing $\mathbb{R}$-trees with special
properties, for example a $\mathbb{Q}$-tree can be viewed as an
$\mathbb{R}$-tree that only branches at rational distances from some base
point. But there are also new phenomena when $\Lambda\not<\mathbb{R}$, for
example when $\Lambda=\mathbb{Z}\times\mathbb{Z}$ with the lexicographical
ordering, $\Lambda$-trees can be interpreted as trees of trees.

The concepts of non-discrete buildings and $\Lambda$-trees were unified by
Bennett [Ben94], leading to the concept of affine $\Lambda$-buildings or
generalized affine buildings. Similar to how the step from $\mathbb{R}$-trees
to $\Lambda$-trees required the addition of an axiom, Bennett increased the
number of axioms by one to six axioms (A1) - (A6) for affine
$\Lambda$-buildings. Various alternative sets of axioms were given in [BSS14].
In Chapter 3.2 and 3.6 we will investigate the independence of some of the
axioms for $\Lambda$-trees and affine $\Lambda$-buildings further. Bennett
used homothety classes of lattices to construct an affine $\Lambda$-building
associated to $\operatorname{SL}(n,\mathbb{F})$ where $\mathbb{F}$ is any
field with a valuation taking values in $\Lambda$. More recently [HIL24]
follow the approach in [BT72] to give examples in reductive groups. In [KT02],
Kramer and Tent suggest a construction of an affine $\Lambda$-building
$\mathcal{B}$ based on a semisimple linear algebraic group defined over a non-
Archimedean real closed field $\mathbb{F}$. A large part of this thesis is
concerned with the construction and analysis of $\mathcal{B}$. We will
construct $\mathcal{B}$ in Section 7 and prove that $\mathcal{B}$ satisfies
the axioms (A1) - (A6) for affine $\Lambda$-buildings in Section 8.

There are multiple links between symmetric spaces and buildings. In 1973,
Mostow [Mos73] proved a rigidity result on lattices in higher rank Lie groups,
now called Mostow-rigidity. A crucial step is to associate to a symmetric
space of non-compact type its building at infinity, a spherical building. In
1997, Kleiner and Leeb [KL97] showed that the asymptotic cone of a symmetric
space of non-compact type is a non-discrete building. They used this result to
show rigidity of quasi-isometries. The asymptotic cone is a special case of
the building $\mathcal{B}$, when the field is a Robinson field [KT02, BIPP21,
BIPP23].

A recent direction of applications lies in the theory of character varieties.
Character varieties are spaces consisting of representations up to
postconjugation of a finitely generated group $\Gamma$ into a Lie group $G$. A
prototypical example of a component of the character variety of a surface
group $\pi_{1}(S)$ in $\operatorname{SL}(2,\mathbb{R})$ is given by the space
of marked hyperbolic structures $\mathcal{T}(S)$ on a surface $S$111The space
of marked hyperbolic structures $\mathcal{T}(S)$ is often called Teichmüller
space.. Thurston [Thu88] introduced a compactification of $\mathcal{T}(S)$
which lead to the classification of the elements of the mapping class group.
The boundary points in Thurston’s compactification correspond to actions of
$\Gamma$ on certain $\mathbb{R}$-trees. Brumfiel [Bru88a] introduced the real
spectrum compactification of $\mathcal{T}(S)$ and showed that it surjects onto
Thurston’s compactification. For general character varieties, Thurston’s
compactification was generalized to the Weyl chamber length spectrum
compactification by Parreau [Par12]. Generalizing Brumfiel’s real spectrum
compactification, Burger, Iozzi, Parreau and Pozetti [BIPP21, BIPP23]
introduced the real spectrum compactification of character varieties, and
showed that it projects to the Weyl chamber length spectrum compactification.
Moreover the real spectrum compactification preserves components, and
therefore promises to be a useful tool in studying components. The points in
the boundary of the real spectrum compactification can be identified with
equivalence classes of representations $\Gamma\to G(\mathbb{F})$ where
$G(\mathbb{F})$ is the group of $\mathbb{F}$-points of a semisimple algebraic
group for a non-Archimedean real closed field $\mathbb{F}$. The group
$G(\mathbb{F})$ acts on the building $\mathcal{B}$ studied in this thesis, and
hence every point in the boundary of the real spectrum compactification
defines an action of $\Gamma$ on $\mathcal{B}$.

Apart from applications to the study of symmetric spaces and character
varieties, $\mathcal{B}$ is also an interesting affine $\Lambda$-building on
its own. In [Ben94], an affine $\Lambda$-building of type
$\tilde{\operatorname{A}}_{n}$ was constructed for every $\Lambda$ arising as
the valuation group of a valued field. In [HIL24] a building for quasi-split
reductive groups was constructed. Apart from abstractly turning any affine
$\mathbb{R}$-building into an affine $\Lambda$-building using the
functoriality result of [SS12], the building $\mathcal{B}$ is the first
example for general semisimple groups. Moreover in contrast to [BT72] and
[HIL24], the construction of $\mathcal{B}$ is relatively straightforward as it
does not rely on a large number of group-theoretic results. However the proof
that $\mathcal{B}$ is an affine $\Lambda$-building does rely on such results.

Central tools used in this thesis are real closed fields, semialgebraic
geometry and the transfer principle, for an introduction see [BCR98]. The
field of real numbers $\mathbb{R}$ is an example of a real closed field and
all real closed fields are similar to $\mathbb{R}$ in a strong model-theoretic
sense, namely the following transfer principle holds [Tar51]: a first-order
logic sentence holds over $\mathbb{R}$ if and only if it holds over any real
closed field, or expressed in terms of model theory, the theory of real closed
fields is complete. The transfer principle was used by Artin [Art27] to solve
in the affirmative Hilbert’s 17th problem on whether every positive real
polynomial can be represented as a square of rational functions.

###  1.1 Semialgebraic groups and decompositions

For a detailed introduction to affine $\Lambda$-buildings, real closed fields
and algebraic groups, see Sections 3, 2 and 4. Let $\mathbb{F}$ and
$\mathbb{K}$ be real closed fields such that
$\mathbb{K}\subseteq\mathbb{F}\cap\mathbb{R}$. Let $G$ be a Zariski-connected
semisimple, self-adjoint (if $g\in G$ then $g{{}^{\mathsf{T}}\\!}\in G$)
linear algebraic $\mathbb{K}$-group222The whole theory could also be
formulated for a group $\tilde{G}$ that is defined as a semialgebraic set, and
that lies between the semialgebraic connected component of the identity
$G_{\mathbb{F}}^{\circ}$ of $G_{\mathbb{F}}$ and $G_{\mathbb{F}}$ itself, so
$G_{\mathbb{F}}^{\circ}<\tilde{G}<G_{\mathbb{F}}$, see [BIPP23]. that is a
subgroup of $\operatorname{SL}(n)$. The $\mathbb{K}$-points of $G$ form a
semialgebraic subset $G_{\mathbb{K}}\subseteq\mathbb{K}^{n\times n}$. The
$\mathbb{R}$-extension $G_{\mathbb{R}}$ is a semisimple Lie group. We denote
by $G_{\mathbb{F}}$ the $\mathbb{F}$-extension. We will call the study of
$G_{\mathbb{F}}$ the _semialgebraic setting_ , as opposed to the algebraic
setting (studying $G$ as an algebraic group) or the Lie group setting
(studying $G_{\mathbb{R}}$). In this terminology a large part of the thesis is
dedicated to the development of the semialgebraic setting.

Many decompositions of $G_{\mathbb{F}}$ rely on the choice of a torus. The
following first result shows that all real closed fields define the same split
tori.

######  Theorem 5.17.

A torus $S<G$ is maximal $\mathbb{K}$-split if and only if it is maximal
$\mathbb{F}$-split. Moreover there is a self-adjoint maximal
$\mathbb{F}$-split torus.

To state decomposition theorems, we define certain subgroups of
$G_{\mathbb{F}}$, for details see Section 6. Let $S_{\mathbb{K}}$ be the
$\mathbb{K}$-points of a self-adjoint maximal $\mathbb{K}$-split torus. Let
$A_{\mathbb{K}}$ be the semialgebraic connected component of $S_{\mathbb{K}}$
and $A_{\mathbb{F}}$ its semialgebraic extension. Let
$K=G\cap\operatorname{SO}_{n}$, $K_{\mathbb{K}}$ the $\mathbb{K}$-points and
$K_{\mathbb{F}}$ the semialgebraic extension, which coincides with the
$\mathbb{F}$-points as $K$ is an algebraic group. Over the reals,
$K_{\mathbb{R}}$ is a maximal compact subgroup. We also extend the
semialgebraic groups
$N_{\mathbb{K}}=\operatorname{Nor}_{K_{\mathbb{K}}}(A_{\mathbb{K}})$ and
$M_{\mathbb{K}}=\operatorname{Cen}_{K_{\mathbb{K}}}(A_{\mathbb{K}})$ to
$N_{\mathbb{F}}$ and $M_{\mathbb{F}}$. An order on the root system $\Sigma$
associated to $A_{\mathbb{F}}$ allows us to define $U_{\mathbb{K}}$ and
$U_{\mathbb{F}}$ as the exponential of the sum of root spaces corresponding to
positive roots. There are various definitions of root systems and Weyl groups
in the literature. In Section 6.2 we use Theorem 5.17 to verify how these
objects defined via the theory of algebraic groups, real Lie groups, Lie
algebras and semialgebraic setting all coincide. This allows us to later use
these settings interchangeably.

######  Proposition 6.4.

The algebraic root system $\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ is isomorphic
to the root system $\Sigma$ from the real setting. The spherical Weyl groups
$\vphantom{W}{}_{\mathbb{K}}W,\vphantom{W}_{\mathbb{F}}W,N_{\mathbb{R}}/M_{\mathbb{R}},N_{\mathbb{F}}/M_{\mathbb{F}}$
and the group generated by reflections in roots of the root system $\Sigma$
are all isomorphic.

Decompositions play a central role in the theory of Lie groups, the theory of
algebraic groups and in this thesis. We prove the following versions of the
Iwasawa (KAU), Cartan (KAK) and Bruhat (BWB) decomposition for
$G_{\mathbb{F}}$.

######  Theorem 6.6 ($G=KAU$).

For every $g\in G_{\mathbb{F}}$, there are $k\in K_{\mathbb{F}},\,a\in
A_{\mathbb{F}},\,u\in U_{\mathbb{F}}$ such that $g=kau$. This decomposition is
unique.

######  Theorem 6.8 ($G=KAK$).

For every $g\in G_{\mathbb{F}}$, there are $k_{1},k_{2}\in
K_{\mathbb{F}},\,a\in A_{\mathbb{F}}$ such that $g=k_{1}ak_{2}$. In this
decomposition $a$ is uniquely determined up to a conjugation by an element of
$N_{\mathbb{F}}/M_{\mathbb{F}}$.

######  Theorem 6.10 ($G=BWB$).

For every $g\in G_{\mathbb{F}}$, there are $b_{1},b_{2}\in
B_{\mathbb{F}}:=M_{\mathbb{F}}A_{\mathbb{F}}U_{\mathbb{F}}$ and $n\in
N_{\mathbb{F}}$ such that $g=b_{1}nb_{2}$. In this decomposition $n$ is unique
up to multiplying by an element in $M_{\mathbb{F}}$. For the spherical Weyl
group $W_{s}:=N_{\mathbb{F}}/M_{\mathbb{F}}$, we have a disjoint union of
double cosets

$G_{\mathbb{F}}=\prod_{[n]\in W_{s}}B_{\mathbb{F}}nB_{\mathbb{F}}.$

In Sections 6.6, 6.7 we note that while in contrast to the setting of Lie
groups the exponential map is not defined for $G_{\mathbb{F}}$, it is still
defined for $U_{\mathbb{F}}$ as the elements of $U_{\mathbb{F}}$ are
nilpotent. As a consequence, the Baker-Campbell-Hausdorff formula holds for
elements in $U_{\mathbb{F}}$ which is useful in proofs using induction over
the root system.

In Section 6.8 we give a proof of the folklore result that the Jacobson-
Morozov Lemma holds for algebraic groups.

######  Proposition 6.17.

Let $g\in G$ be any unipotent element in a semisimple linear algebraic group
$G$ over an algebraically closed field $\mathbb{D}$ of characteristic $0$.
Then there is an algebraic subgroup $\operatorname{SL}_{g}<G$ with Lie algebra
$\operatorname{Lie}(\operatorname{SL}_{g})\cong\mathfrak{sl}_{2}$ and
$g\in\operatorname{SL}_{g}$. The element $\log(g)\in\operatorname{Lie}(G)$
corresponds to

$\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}\in\mathfrak{sl}_{2}.$

Moreover, if $g\in G_{\mathbb{F}}$ for a field
$\mathbb{F}\subseteq\mathbb{D}$, then $\operatorname{SL}_{g}$ can be assumed
to be defined over $\mathbb{F}$.

For us, the following more precise version of the Jacobson-Morozov Lemma, now
in the semialgebraic setting, will be useful. Note that the assumption on
$\alpha$ in Proposition 6.18 means that it can only be applied to reduced root
systems (if $\alpha\in\Sigma$, then $2\alpha\notin\Sigma$).

######  Proposition 6.18.

Let $\mathbb{F}$ be a real closed field with an order compatible valuation
$v\colon\mathbb{F}\to\Lambda\cup\\{\infty\\}$, $\alpha\in\Sigma$ and assume
$\mathfrak{g}_{2\alpha}=0$. Let $u\in(U_{\alpha})_{\mathbb{F}}$. Then there
are $X\in(\mathfrak{g}_{\alpha})_{\mathbb{F}}$ and $t\in\mathbb{F}$ such that
$u=\exp(tX)$ and $\max_{ij}\left\\{(-v)(X_{ij})\right\\}=0$. Let $(X,Y,H)$ be
the $\mathfrak{sl}_{2}$-triplet of the Jacobson-Morozov Lemma in the Lie
algebra setting, see Lemma 5.9. Then there is a morphism of algebraic groups
$\varphi\colon\operatorname{SL}(2,\mathbb{D})\to G$ defined over $\mathbb{F}$
such that $\varphi$ has finite kernel and

$\displaystyle\varphi\begin{pmatrix}1&t\\\
0&1\end{pmatrix}=u=\exp(tX)\quad\text{and}\quad\varphi\begin{pmatrix}1&0\\\
t&1\end{pmatrix}=\exp(tY).$

If $\varphi$ is not injective, then $\ker(\varphi)\cong\mathbb{Z}/2\mathbb{Z}$
and $\varphi$ factors through the isomorphism

$\operatorname{PGL}(2,\mathbb{D}):=\operatorname{SL(2,\mathbb{D})}/\ker(\varphi)\xrightarrow{\sim}\varphi(\operatorname{SL}(2,\mathbb{D}))$

which is also defined over $\mathbb{F}$. Moreover
$\varphi(g{{}^{\mathsf{T}}\\!})=\varphi(g){{}^{\mathsf{T}}\\!}$, for any
$g\in\operatorname{SL}(2,\mathbb{F})$.

The Jacobson-Morozov Lemma produces subgroups with Lie algebras isomorphic to
$\mathfrak{sl}_{2}$. The following theorem produces potentially larger rank
one subgroups $L_{\pm\alpha}$ associated to a root $\alpha\in\Sigma$ in the
algebraic setting. These subgroups are the semisimple parts of Levi subgroups.

######  Theorem 6.21.

Let $\alpha\in\Sigma$. Then there is a connected semisimple self-adjoint
linear algebraic group $L_{\pm\alpha}$ defined over $\mathbb{K}$ such that

  1. (i)

$\operatorname{Lie}(L_{\pm\alpha})=(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})\oplus(\mathfrak{g}_{-\alpha}\oplus\mathfrak{g}_{-2\alpha})\oplus([\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]+[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}])$.

  2. (ii)

$\operatorname{Rank}_{\mathbb{R}}(L_{\pm\alpha})=\operatorname{Rank}_{\mathbb{F}}(L_{\pm\alpha})=1$.

###  1.2 The nonstandard symmetric space $\mathcal{X}_{\mathbb{F}}$

Now $\mathbb{F}$ is a non-Archimedean real closed field with order compatible
valuation $v\colon\mathbb{F}\to\Lambda\cup\\{\infty\\}$ and the groups are
defined as in the previous subsection. The semialgebraic set

$P_{\mathbb{R}}:=\left\\{x\in\mathbb{R}^{n\times n}\colon
x=x{{}^{\mathsf{T}}\\!},\det(x)=1,x\text{ is positive definite }\right\\}$

is a model for the symmetric space
$\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n,\mathbb{R})$. For the Lie
group $G_{\mathbb{R}}<\operatorname{SL}(n,\mathbb{R})$, the symmetric space
$G_{\mathbb{R}}/K_{\mathbb{R}}$ can be realized as a totally geodesic
submanifold of $P_{\mathbb{R}}$ given by the semialgebraic orbit
$\mathcal{X}_{\mathbb{R}}:=G_{\mathbb{R}}.\operatorname{Id}\subseteq
P_{\mathbb{R}}$. The semialgebraic extension $\mathcal{X}_{\mathbb{F}}$ is
called the _non-standard symmetric space_.

Over the reals, the symmetric space $\mathcal{X}_{\mathbb{R}}$ comes equipped
with a Riemannian distance, such that $G_{\mathbb{R}}$ acts by isometries: the
maximal flats are isometric to a Euclidean vector space and for any two points
there is a maximal flat containing the two, so the distance is given as the
norm of the difference of the two. We mimic this construction in the
semialgebraic setting. First we project two arbitrary points to the _non-
standard maximal flat_
$A_{\mathbb{F}}.\operatorname{Id}\subseteq\mathcal{X}_{\mathbb{F}}$, which
contains the fundamental Weyl chamber $A_{\mathbb{F}}^{+}.\operatorname{Id}$
for the semialgebraic subset

$A_{\mathbb{F}}^{+}:=\left\\{a\in A_{\mathbb{F}}\colon\chi_{\alpha}(a)\geq
1\text{ for all }\alpha\in\Sigma_{>0}\right\\}.$

The Cartan decomposition KAK can be used to define a Weyl chamber valued
distance on $\mathcal{X}_{\mathbb{R}}$, the semialgebraic version of which is
the following Lemma.

######  Lemma 7.2.

The Cartan decomposition defines a Cartan-projection

$\displaystyle\delta_{\mathbb{F}}\colon\mathcal{X}_{\mathbb{F}}\times\mathcal{X}_{\mathbb{F}}$
$\displaystyle\to A_{\mathbb{F}}^{+}$ $\displaystyle(x,y)$
$\displaystyle\mapsto a$

which is invariant under the action of $G_{\mathbb{F}}$.

We then use the semialgebraic norm

$\displaystyle N_{\mathbb{F}}\colon A_{\mathbb{F}}$
$\displaystyle\to\mathbb{F}_{\geq 1}$ $\displaystyle a$
$\displaystyle\mapsto\prod_{\alpha\in\Sigma}\max\left\\{\chi_{\alpha}(a),\chi_{\alpha}(a)^{-1}\right\\},$

Over the reals, $d=\log\circ N_{\mathbb{R}}\circ\delta_{\mathbb{R}}$ defines a
metric on $\mathcal{X}_{\mathbb{R}}$. The logarithm is not a semialgebraic
function, but we can replace $\log$ by the valuation $-v$, to obtain a
symmetric non-negative function

$\displaystyle d\colon\mathcal{X}_{\mathbb{F}}\times\mathcal{X}_{\mathbb{F}}$
$\displaystyle\to\Lambda$ $\displaystyle(x,y)$
$\displaystyle\mapsto(-v)(N_{\mathbb{F}}(\delta_{\mathbb{F}}(x,y))).$

The function $d$ is not quite a $\Lambda$-metric, since there are points
$x\neq y\in\mathcal{X}_{\mathbb{F}}$ with $d(x,y)=0$, but we have the
following.

######  Theorem 7.5.

The function
$d\colon\mathcal{X}_{\mathbb{F}}\times\mathcal{X}_{\mathbb{F}}\to\Lambda$ is a
pseudo-distance on $\mathcal{X}_{\mathbb{F}}$.

To prove the triangle inequality in Theorem 7.5, we use the following two
results of independent interest. First, the Iwasawa decomposition KAU can be
used to define the semialgebraic retraction $a_{\mathbb{F}}\colon
G_{\mathbb{F}}\to A_{\mathbb{F}}$ by $g=uak\mapsto a=:a_{\mathbb{F}}(g)$. For
some $b\in A_{\mathbb{R}}$, Kostant’s convexity theorem then describes the set
$a_{\mathbb{R}}(K_{\mathbb{R}}b)$ as a polytope in $A_{\mathbb{R}}$ given by
the convex hull of the Weyl group orbit of $b$. In the semialgebraic setting,
we prove the following variation for the elements $b\in A_{\mathbb{F}}^{+}$
for certain characters $\chi_{i}\colon A_{\mathbb{F}}\to\mathbb{F}^{\times}$,
$i\in\\{1,2,\ldots,\dim(A_{\mathbb{R}})\\}$.

######  Theorem 6.28.

For all $b\in A_{\mathbb{F}}^{+}$, we have

$\left\\{a\in A_{\mathbb{F}}^{+}\colon\exists k\in
K_{\mathbb{F}},a=a_{\mathbb{F}}(kb)\right\\}=\left\\{a\in
A_{\mathbb{F}}^{+}\colon\chi_{i}(a)\leq\chi_{i}(b)\text{ for all }i\right\\}.$

Theorem 6.28 can then be used to show that the Iwasawa retraction
$\rho\colon\mathcal{X}_{\mathbb{F}}\to A_{\mathbb{F}}.\operatorname{Id}$
defined by $g.\operatorname{Id}\mapsto a_{\mathbb{F}}(g).\operatorname{Id}$
decreases distances.

######  Theorem 7.4.

The map $\rho\colon\mathcal{X}_{\mathbb{F}}\to
A_{\mathbb{F}}.\operatorname{Id}$ is a $d$-diminishing retraction to
$A_{\mathbb{F}}.\operatorname{Id}$.

###  1.3 The building $\mathcal{B}$

In view of Theorem 7.5, we can define the $\Lambda$-metric space
$\mathcal{B}:=\mathcal{X}_{\mathbb{F}}/\\!\\!\sim$ where $x\sim y$ whenever
$d(x,y)=0\in\Lambda$. We note that $G_{\mathbb{F}}$ acts transitively by
isometries on $\mathcal{B}$. We denote the basepoint by
$o:=[\operatorname{Id}]\in\mathcal{B}$. In Section 7.3, we verify that
$\mathbb{A}:=A_{\mathbb{F}}.\operatorname{o}\subseteq\mathcal{B}$ is an
apartment in the sense of affine $\Lambda$-buildings, and that
$\operatorname{Nor}_{G_{\mathbb{F}}}(A_{\mathbb{F}})/\operatorname{Cen}_{G_{\mathbb{F}}}(A_{\mathbb{F}})$
acts as the affine Weyl group $\mathbb{A}\rtimes W_{s}$, where $W_{s}$ is the
spherical Weyl group of the root system $\Sigma$. The inclusion
$f_{0}\colon\mathbb{A}\to\mathcal{B}$ can then be used to define an atlas
$\mathscr{F}=\left\\{g.f_{0}\colon\mathbb{A}\to\mathcal{B}\colon g\in
G_{\mathbb{F}}\right\\}.$ The main result of this thesis is that the
$\Lambda$-metric space $\mathcal{B}$ together with the atlas $\mathscr{F}$ is
a generalized affine building. Theorem 8.1 was announced without proof as
Theorem 5.7 in [KT02] and as Theorem 4.3 in [KT04].

######  Theorem 8.1.

If the root system $\Sigma$ is reduced, then the pair
$(\mathcal{B},\mathscr{F})$ is an affine $\Lambda$-building.

######  Remark 1.1.

The technical assumption that the root system $\Sigma$ is reduced is an
artifact of our use of the Jacobson-Morozov Lemma, Proposition 6.18. As a
consequence of the classification of real Lie groups, see for example the
appendix of [OV90], only Hermitian groups of non-tube type have non-reduced
root systems; and they are of type $\operatorname{BC}_{n}$ [GW24]. In the
study of character varieties, components with discrete and faithful
representations are only known to exist when the group admits a
$\Theta$-positive structure. All groups admitting a $\Theta$-positive
structure have reduced root systems and our theorem can be applied to points
in the boundary of the real spectrum compactification in this case.

To be more precise, we verify that $(\mathcal{B},\mathscr{F})$ satisfies the
following six axioms, which are equivalent to the original axioms for affine
$\Lambda$-buildings by the work of [BSS14].

  1. (A1)

For all $f\in\mathscr{F}$, $w\in W_{a}$, $f\circ w\in\mathscr{F}$.

  2. (A2)

For all $f,f^{\prime}\in\mathscr{F}$, the set $B=f^{-1}\left(f(\mathbb{A})\cap
f^{\prime}(\mathbb{A})\right)$ is $W_{a}$-convex and there is a $w\in W_{a}$
such that $f|_{B}=f^{\prime}\circ w|_{B}.$

  3. (A3)

For all $x,y\in X$, there is a $f\in\mathscr{F}$ such that $x,y\in
f(\mathbb{A})$.

  4. (A4)

For any sectors $s_{1},s_{2}\subseteq X$ there are subsectors
$s_{1}^{\prime}\subseteq s_{1},\,s_{2}^{\prime}\subseteq s_{2}$ such that
there is an $f\in\mathscr{F}$ with $s_{1}^{\prime},s_{2}^{\prime}\subseteq
f(\mathbb{A})$.

  5. (TI)

The function $d\colon\mathcal{B}\times\mathcal{B}\to\Lambda$ induced from the
distance on apartments (via axioms (A1), (A2) and (A3)) satisfies the triangle
inequality.

  6. (EC)

For $f_{1},f_{2}\in\mathscr{F}$, if $f_{1}(\mathbb{A})\cap f_{2}(\mathbb{A})$
is a half-apartment, then there exists $f_{3}\in\mathscr{F}$ such that
$f_{i}(\mathbb{A})\cap f_{3}(\mathbb{A})$ are half-apartments for
$i\in\\{1,2\\}$. Moreover $f_{3}(\mathbb{A})$ is the symmetric difference of
$f_{1}(\mathbb{A})$ and $f_{2}(\mathbb{A})$ together with the boundary wall of
$f_{1}(\mathbb{A})\cap f_{2}(\mathbb{A})$.

The proof of Theorem 8.1 uses all the theory we have developed for
$G_{\mathbb{F}}$ up to this point and relies on further analysis of
$\mathcal{B}$, developed in Section 8.2. The proof of (A2) is the most
involved among the axioms. We adapt some of the methods in [BT72] to our
setting to be able to prove (A2). For the special case of
$G_{\mathbb{F}}=\operatorname{SL}(n,\mathbb{F})$, a direct proof of axiom (A2)
is given in Appendix A. We will point out some results that may be of
independent interest. Many of the following results are about stabilizers of
certain subsets.

The stabilizer of $o\in\mathcal{B}$ was calculated by [Tho02] for the special
case when $\mathbb{F}$ is a Robinson field. For general fields, it has been
suggested by [KT02] and proven by [BIPP23]. Let
$O=\\{a\in\mathbb{F}\colon(-v)(a)\leq 0\\}$ be the valuation ring associated
to the valuation $v$ and let
$E_{\mathbb{F}}(O):=E_{\mathbb{F}}\cap\operatorname{SL}(n,O)$ for any
semialgebraic subset $E_{\mathbb{F}}\subseteq G_{\mathbb{F}}$.

######  Theorem 7.11.

The stabilizer of $o\in\mathcal{B}$ in $G_{\mathbb{F}}$ is
$G_{\mathbb{F}}(O)$.

As a consequence of the Iwasawa retraction Theorem 7.4, we give an Iwasawa
decomposition of the stabilizer of $o$.

######  Corollary 7.12.

There is an Iwasawa decomposition
$G_{\mathbb{F}}(O)=U_{\mathbb{F}}(O)A_{\mathbb{F}}(O)K_{\mathbb{F}}$, meaning
that for every $g\in G_{\mathbb{F}}(O)$ there are unique $u\in
U_{\mathbb{F}}(O),\,a\in A_{\mathbb{F}}(O),\,k\in
K_{\mathbb{F}}=K_{\mathbb{F}}(O)$ with $g=uak$.

In Section 8.2 we implement some ideas from [BT72] in the semialgebraic
setting. For the root groups
$(U_{\alpha})_{\mathbb{F}}=\exp((\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})_{\mathbb{F}})$,
we introduce root group valuations $\varphi_{\alpha}\colon
U_{\alpha}\to\Lambda\cup\\{-\infty\\}$ in Subsection 8.2.1. Taking a closer
look at the unipotent group, we notice that if $u\in U_{\mathbb{F}}$ sends
some point of the apartment to some other point of the apartment, then the two
points have to be the same.

######  Proposition 8.7.

For all $u\in U_{\mathbb{F}}$ and $a\in A_{\mathbb{F}}$,
$ua.o\in\mathbb{A}\iff ua.o=a.o.$

In fact, the fixed point set of elements in $U_{\mathbb{F}}$ is a half-
apartment.

######  Proposition 8.8.

Let $\alpha\in\Sigma$. For $u\in(U_{\alpha})_{\mathbb{F}}$ we have

$\left\\{p\in\mathbb{A}\colon
u.p\in\mathbb{A}\right\\}=\left\\{a.o\in\mathbb{A}\colon\varphi_{\alpha}(u)\leq(-v)\left(\chi_{\alpha}\left(a\right)\right)\right\\}$

and therefore this set is a half-apartment, when $u\neq\operatorname{Id}$.

This can be upgraded to a finite intersection of half-apartments for elements
in $U_{\mathbb{F}}$.

######  Proposition 8.10.

For $u\in U_{\mathbb{F}}$ there are $k_{\alpha}\in\Lambda\cup\\{-\infty\\}$
for $\alpha\in\Sigma_{>0}$ such that

$\\{p\in\mathbb{A}\colon u.p\in\mathbb{A}\\}=\left\\{a.o\in\mathbb{A}\colon
k_{\alpha}\leq(-v)\left(\chi_{\alpha}\left(a\right)\right)\text{ for all
}\alpha\in\Sigma_{>0}\right\\}$

and therefore the set of fixed points is a finite intersection of half-
apartments. If $u$ fixes all of $\mathbb{A}$, then $u=\operatorname{Id}$.

We proceed to describe the pointwise stabilizers of the whole apartment
$\mathbb{A}$, the fundamental Weyl chamber

$C_{0}=\\{a.o\in\mathbb{A}\colon(-v)(\chi_{\alpha}(a))\geq 0\\},$

and the half-apartments

$H_{\alpha}^{+}=\\{a.o\in\mathbb{A}\colon(-v)(\chi_{\alpha}(a))\geq 0\\}.$

######  Theorem 8.17.

The pointwise stabilizer of $C_{0}$ in $G_{\mathbb{F}}$ is
$B_{\mathbb{F}}(O)=U_{\mathbb{F}}(O)A_{\mathbb{F}}(O)M_{\mathbb{F}}$.

######  Theorem 8.19.

The pointwise stabilizer of $\mathbb{A}$ in $G_{\mathbb{F}}$ is
$A_{\mathbb{F}}(O)M_{\mathbb{F}}$.

######  Theorem 8.20.

Let $\alpha\in\Sigma$. The pointwise stabilizer of $H_{\alpha}^{+}$ in
$G_{\mathbb{F}}$ is
$(U_{\alpha})_{\mathbb{F}}(O)A_{\mathbb{F}}(O)M_{\mathbb{F}}$.

In Subsection 8.2.5, we take a closer look at the rank 1 subgroup
$(L_{\pm\alpha})_{\mathbb{F}}<G_{\mathbb{F}}$. We now restrict to groups
$G_{\mathbb{F}}$ with reduced root systems, to be able to use the Jacobson-
Morozov Lemma to find elements
$m(u)\in\operatorname{Nor}_{G_{\mathbb{F}}}(A_{\mathbb{F}})$ representing a
reflection along some hyperplane
$M_{\alpha,\ell}=\\{a.o\in\mathbb{A}\colon(-v)(\chi_{\alpha}(a))=\ell\\}$ for
$\alpha\in\Sigma$ and $\ell\in\Lambda$. A careful analysis of the rank one
subgroup
$L=\langle(U_{\alpha})_{\mathbb{F}},(U_{-\alpha})_{\mathbb{F}}\rangle<(L_{\pm\alpha})_{\mathbb{F}}$
results in the following decomposition of its stabilizer.

######  Proposition 8.39.

Assume $\Sigma$ is reduced. Let $\alpha\in\Sigma$, $\Omega\subseteq\mathbb{A}$
a non-empty finite subset. The pointwise stabilizers

$\displaystyle L_{\Omega}$
$\displaystyle:=L\cap\operatorname{Stab}_{G_{\mathbb{F}}}(\Omega),$
$\displaystyle U_{\alpha,\Omega}$
$\displaystyle:=(U_{\alpha})_{\mathbb{F}}\cap\operatorname{Stab}_{G_{\mathbb{F}}}(\Omega),$
$\displaystyle U_{-\alpha,\Omega}$
$\displaystyle:=(U_{-\alpha})_{\mathbb{F}}\cap\operatorname{Stab}_{G_{\mathbb{F}}}(\Omega),$
$\displaystyle N_{\Omega}$
$\displaystyle:=\operatorname{Nor}_{G_{\mathbb{F}}}(A_{\mathbb{F}})\cap\operatorname{Stab}_{G_{\mathbb{F}}}(\Omega)$

satisfy $L_{\Omega}=\langle
U_{\alpha,\Omega},U_{-\alpha,\Omega},(N_{\Omega}\cap
L)\rangle=U_{\alpha,\Omega}U_{-\alpha,\Omega}(N_{\Omega}\cap L).$

In Section 8.2.6, we upgrade the rank 1 result to $G_{\mathbb{F}}$. For a
subset $\Omega\subseteq\mathbb{A}$, let

$\displaystyle\hat{P}_{\Omega}$ $\displaystyle:=\langle
N_{\Omega},U_{\alpha,\Omega}\colon\alpha\in\Sigma\rangle$ $\displaystyle
U_{\Omega}^{+}$ $\displaystyle:=\langle
U_{\alpha,\Omega}\colon\alpha\in\Sigma_{>0}\rangle$ $\displaystyle
U_{\Omega}^{-}$ $\displaystyle:=\langle
U_{\alpha,\Omega}\colon\alpha\in\Sigma_{<0}\rangle.$

######  Theorem 8.49.

Assume $\Sigma$ is reduced. The pointwise stabilizer of an arbitrary subset
$\Omega\subseteq\mathbb{A}$ satisfies
$\operatorname{Stab}_{G_{\mathbb{F}}}(\Omega)=\hat{P}_{\Omega}=U_{\Omega}^{+}U_{\Omega}^{-}N_{\Omega}.$

The proof of Theorem 8.49 is tightly intertwined with the proof of axiom (A2).
It first relies on the following _mixed Iwasawa_ decomposition for
$G_{\mathbb{F}}$. For $o\in\mathcal{B}$, let $\hat{P}_{o}:=\hat{P}_{\\{o\\}}$.

######  Theorem 8.42.

Assume $\Sigma$ is reduced. Then
$G_{\mathbb{F}}=U_{\mathbb{F}}\cdot\operatorname{Nor}_{G_{\mathbb{F}}}(A_{\mathbb{F}})\cdot\hat{P}_{o}$.

With the help of the mixed Iwasawa decomposition we obtain Theorem 8.49 in the
case of finite subsets $\Omega\subseteq\mathbb{A}$. This statement for finite
subsets is then enough to prove the second part of axiom (A2). The full
Theorem 8.49 then follows from this part of (A2) and the rest of axiom (A2)
follows from the full Theorem 8.49.

###  1.4 Further results on $\Lambda$-trees

To build up intuition in the study of affine $\Lambda$-buildings it often
makes sense to first consider the rank one case of $\Lambda$-trees. We sketch
some results about $\Lambda$-trees obtained as part of the thesis, details can
be found in Section 3.2 and [App24, ARFJ24]. In the axiomatic
characterization, $\Lambda$-trees are defined as $\Lambda$-metric spaces
satisfying three axioms (1), (2) and (3). The following theorem gives a purely
algebraic condition on $\Lambda$ for when the geometric property of axiom (3)
follows from (1) and (2).

######  Theorem 3.2.

([App24, Theorem 1]) Let $\Lambda$ be an ordered abelian group. The following
are equivalent:

  * (a)

For every positive $\lambda_{0}\in\Lambda$, the set $\\{t\in\Lambda\colon
0\leq 2t\leq\lambda_{0}\\}$ has a maximum.

  * (b)

Every $\Lambda$-metric space that satisfies axioms (1) and (2) also satisfies
(3).

For the case of $G=\operatorname{SL}(2)$, Brumfiel [Bru88b] constructs a non-
standard hyperbolic plane with a pseudometric and proves that the quotient
$\mathcal{B}$ is a $\Lambda$-tree. Starting for simplicity with the field
$\mathbb{F}$ of Puiseux-series, see definition in Section 2.3, $\mathcal{B}$
is a $\mathbb{Q}$-tree $\mathcal{T}$. Completing all the segments of
$\mathcal{T}$ results in a $\mathbb{R}$-tree
$\overline{\mathcal{T}}^{\operatorname{sc}}$. The segment completion is a
special case of a general base change functor for affine $\Lambda$-buildings
investigated in [SS12]. The following theorem answers the question of whether
the segment-completion $\overline{\mathcal{T}}^{\operatorname{sc}}$ is
complete as a metric space.

######  Theorem 3.6.

([ARFJ24, Theorem 1]) Let $\mathcal{T}$ be the $\mathbb{Q}$-tree constructed
from the field of Puiseux-series in [Bru88b]. There is a Cauchy sequence in
the segment completion $\overline{\mathcal{T}}^{\operatorname{sc}}$ that does
not converge and hence $\overline{\mathcal{T}}^{\operatorname{sc}}$ is not
complete.

###  1.5 Formalization in the proof assistant $\operatorname{\mathsf{Lean}}$

Mathematics thrives on the intuition surrounding its objects but with its
rigor also promises to be a pathway to absolute truth. In practice,
mathematicians are human and mistakes happen. With the advent of computers, it
has become possible to verify proofs by machine and to considerably improve
the reliability of mathematical results. The $\operatorname{\mathsf{Lean}}$
theorem prover is a proof assistant developed mainly by Leonardo de Moura at
Microsoft Research [dMKA+15]. There is an extensive community-built
mathematical library mathlib [The19], which by now contains a large part of
undergraduate mathematics. Recently some more advanced projects such as a
formalization of perfectoid spaces [BCM20] have been completed and a project
to formalize Andrew Wiles’ proof of Fermat’s last theorem has been announced.

As a proof of concept, some parts of this thesis were formalized in
$\operatorname{\mathsf{Lean}}$. We built on the definition of ordered abelian
groups already present in mathlib. We first formalized the notions of
$\Lambda$-metric spaces and $\Lambda$-trees. We then showed some basic
properties of $\Lambda$-trees before successfully formalizing that (a) implies
(b) in Theorem 3.2. This project showed that it is possible to verify new
research results with a computer, however the time investment is substantial
and we would therefore currently recommend not to formalize most new results.
The $\operatorname{\mathsf{Lean}}$-files can be found in [App23].

##  2 Real closed fields

An _ordered field_ is a field together with a total order such that the sum
and the product of positive elements are positive. A field $\mathbb{F}$ is
called _real closed_ if it satisfies one of the following equivalent
conditions.

  1. (1)

There is a total order on $\mathbb{F}$ turning $\mathbb{F}$ into an ordered
field such that every positive element has a square root and every polynomial
of odd degree has a solution.

  2. (2)

There is an order on $\mathbb{F}$ that does not extend to any proper algebraic
field extension of $\mathbb{F}$.

  3. (3)

$\mathbb{F}$ is not algebraically closed but every finite extension is
algebraically closed.

  4. (4)

$\mathbb{F}$ is not algebraically closed but $\mathbb{F}[\sqrt{-1}]$ is
algebraically closed.

An ordered field is called _Archimedean_ if every element is bounded by a
natural number. The real numbers $\mathbb{R}$ and the subset of real algebraic
numbers are examples of Archimedean real closed fields. A major tool when
working with real closed fields is the following transfer principle from model
theory.

###  2.1 The transfer principle

Recall that a _first-order formula of ordered fields with parameters in
$\mathbb{F}$_ is a formula that contains a finite number of conjunctions
$\wedge$, disjunctions $\lor$, negations $\lnot$, and universal $\forall$ or
existential $\exists$ quantifiers on variables, starting from atomic formulas
which are formulas of the kind $f(x_{1},\ldots,x_{n})=0$ or
$g(x_{1},\ldots,x_{n})\leq 0$, where $f$ and $g$ are polynomials with
coefficients in $\mathbb{F}$. A first-order formula without free variables is
called a _sentence_.

######  Theorem 2.1.

(Transfer principle, [BCR98]) Let $\mathbb{F}$ and $\mathbb{F}^{\prime}$ be
real closed fields. Let $\varphi$ be a sentence with parameters in
$\mathbb{F}\cap\mathbb{F}^{\prime}$. Then $\varphi$ is true for $\mathbb{F}$
if and only if $\varphi$ is true for $\mathbb{F}^{\prime}$, formally
$\mathbb{F}\models\varphi\iff\mathbb{F}^{\prime}\models\varphi$.

Let $\varphi$ be a first-order formula with parameters in some field
$\mathbb{K}\subseteq\mathbb{F}\cap\mathbb{R}$ with $n$ free variables. Let $X$
be a subset of $\mathbb{K}^{n}$ which can be described as
$X=\\{x\in\mathbb{K}^{n}\colon\mathbb{F}\models\varphi(x)\\}$. It follows from
the transfer principle, that the _semialgebraic extension
$X_{\mathbb{F}}=\\{x\in\mathbb{F}^{n}\colon\mathbb{F}\models\varphi(x)\\}$ of
$X$_ depends only on $X$ and not on $\varphi$ and is thus well defined. Sets
of the form $\\{x\in k\colon k\models\varphi(x)\\}$ for any ordered field
$k\supseteq\mathbb{K}$ are called _semialgebraic_ sets.

###  2.2 Valuations

For an overview of valued fields, we recommend [BCS18]. A subring
$O\subseteq\mathbb{F}$ of an ordered field $\mathbb{F}$ is an _order convex
subring_ , if for all $a,b\in O,\ c\in\mathbb{F},\ a\leq c\leq b$ implies
$c\in O$. Note that every order convex subring is in particular a valuation
ring: for all $a\in\mathbb{F}$, we have $a\in O$ or $a^{-1}\in O$. Let
$(\Lambda,+)$ be an ordered abelian group. A _valuation_ on an ordered field
$\mathbb{F}$ is a map $v\colon\mathbb{F}\to\Lambda\cup\\{\infty\\}$ which
satisfies for all $a,b\in\mathbb{F}$

  * (1)

$v(a)=\infty$ if and only if $a=0$.

  * (2)

$v(ab)=v(a)+v(b)$.

  * (3)

$v(a+b)\geq\min\\{v(a),v(b)\\}$.

We say that the valuation is _order compatible_ , if $(-v)(a)\geq(-v)(b)$
whenever $a\geq b\geq 0$. We will often use the same letter for $v$ and
$v|_{\mathbb{F}_{>0}}\colon\mathbb{F}_{>0}\to\Lambda$. We will often be more
interested in $(-v)$ than in $v$, as $(-v)$ is order preserving.

There is a correspondence between order convex valuation rings and order
compatible valuations, which follows from a theorem by Krull [Kru39].

######  Theorem 2.2.

Every order convex valuation ring $O\subseteq\mathbb{F}$ gives rise to an
order compatible valuation

$\displaystyle v\colon\mathbb{F}$ $\displaystyle\to\Lambda\cup\\{\infty\\},$

where $\Lambda\cong\mathbb{F}^{\times}/O^{\times}$. On the other hand, every
order compatible valuation gives rise to a order convex valuation ring

$O=\\{a\in\mathbb{F}\colon v(a)>0\\}.$

Real closed fields may admit order compatible valuation rings. The ordered
abelian group $\Lambda$ of a valuation of a real closed field is always a
$\mathbb{Q}$-vector space.

###  2.3 Examples

The field of _Puiseux series_ over the real algebraic numbers
$\overline{\mathbb{Q}}^{\operatorname{rc}}$

$\mathbb{F}:=\left\\{\sum_{k=-\infty}^{k_{0}}c_{k}X^{\frac{k}{m}}\,\colon\,k_{0},m\in\mathbb{Z},\,m>0,\,c_{k}\in\overline{\mathbb{Q}}^{\operatorname{rc}},\,c_{k_{0}}\neq
0\right\\},$

is a non-Archimedean real closed field, where the usual order on
$\overline{\mathbb{Q}}^{\operatorname{rc}}$ is extended by $X>r$ for all
$r\in\overline{\mathbb{Q}}^{\operatorname{rc}}$ [BCR98]. An order compatible
valuation
$v\colon\mathbb{F}\to\Lambda\cup\\{\infty\\}=\mathbb{Q}\cup\\{\infty\\}$ is
given by the degree

$\displaystyle
v\left(\sum_{k=-\infty}^{k_{0}}c_{k}X^{k/m}\right)=-\frac{k_{0}}{m}.$

A _non-principal ultrafilter on $\mathbb{Z}$_ is a function
$\omega\colon\mathcal{P}(\mathbb{Z})\to\\{0,1\\}$ that satisfies

  1. (1)

$\omega(\emptyset)=0$, $\omega(\mathbb{Z})=1$

  2. (2)

If $A,B\subseteq\mathbb{Z}$ satisfy $A\cap B=\emptyset$, then $\omega(A\cup
B)=\omega(A)+\omega(B)$.

  3. (3)

All finite subsets $A\subseteq\mathbb{Z}$ satisfy $\omega(A)=0$.

Ultrafilters can be thought of as finitely-additive probability measures that
only take values in $0$ and $1$. The existence of non-principal ultrafilters
is equivalent to the axiom of choice, [Hal66]. For a given ultrafilter
$\omega$, we define the _hyperreal numbers_ $\mathbb{R}_{\omega}$ to be the
equivalence classes of infinite sequences
$\mathbb{R}_{\omega}=\mathbb{R}^{\mathbb{N}}/\\!\\!\sim$, where
$x=(x_{i})_{i\in\mathbb{N}}\sim y=(y_{i})_{i\in\mathbb{N}}$ if
$\omega(\\{i\in\mathbb{N}\colon x_{i}\neq y_{i}\\})=0$ or
$\omega(\\{i\in\mathbb{N}\colon x_{i}=y_{i}\\})=1$. We define addition and
multiplication componentwise, the multiplicative inverse is obtained by taking
the inverses of all non-zero entries, turning $\mathbb{F}_{\omega}$ into a
field. Considering constant sequences, the real numbers are a subfield of
$\mathbb{R}_{\omega}$. The hyperreals are an ordered field with respect to the
order defined by $[(x_{i})_{i\in\mathbb{N}}]\leq[(y_{i})_{i\in\mathbb{N}}]$ if
and only if $\omega(\\{i\in\mathbb{N}\colon x_{i}\leq y_{i}\\})=1$. The
hyperreals are real closed, since $\mathbb{R}$ is. One can check that the
hyperreals do not admit a valuation to a subgroup of $\mathbb{R}$, but they do
admit a valuation to $(\Lambda,+):=((\mathbb{R}_{\omega})_{\neq 0},\cdot)$,
namely the identity. The hyperreals are non-Archimedean, since the equivalence
class containing $(1,2,3,\ldots)$ is an _infinite_ element, meaning it is
larger than any natural number.

Let $b\in\mathbb{R}_{\omega}$ be an infinite element. Then

$O_{b}:=\\{x\in\mathbb{R}_{\omega}\colon|x|<b^{m}\text{ for some
}m\in\mathbb{Z}\\}$

is an order convex subring of $\mathbb{R}_{\omega}$ with maximal ideal

$J_{b}:=\\{x\in\mathbb{R}_{\omega}\colon|x|<b^{m}\text{ for all
}m\in\mathbb{Z}\\}.$

The _Robinson field_ associated to the non-principal ultrafilter $\omega$ and
the infinite element $b$ is the quotient $\mathbb{R}_{\omega,b}:=O_{b}/J_{b}$
[Rob96]. The Robinson field is a non-Archimedean real closed field. Note that
$[b]\in\mathbb{R}_{\omega,b}$ is a _big_ element, meaning that for all
$a\in\mathbb{R}_{\omega,b}$ there is an $n\in\mathbb{N}$ such that $a<b^{n}$.

Non-Archimedean ordered fields $\mathbb{F}$ with big elements admit an order
compatible valuation $v\colon\mathbb{R}\to\mathbb{R}\cup\\{\infty\\}$ by
letting $v(a)$ be the real number defined by the Dedekind cut

$\displaystyle A_{a}$ $\displaystyle:=\left\\{\frac{p}{q}\in\mathbb{Q}\colon
b^{p}\leq a^{q},q\in\mathbb{Z}_{>0},p\in\mathbb{Z}\right\\}$ $\displaystyle
B_{a}$ $\displaystyle:=\left\\{\frac{p}{q}\in\mathbb{Q}\colon b^{p}\geq
a^{q},q\in\mathbb{Z}_{>0},p\in\mathbb{Z}\right\\}$

for $a\in\mathbb{F}$. Note that when $\mathbb{F}$ is Archimedean, every
element $b>1$ is big and we can still define as above $v(a)=-\log_{b}|a|$,
which then is the usual logarithm with base $b$. However $v$ is not a
valuation in our sense, since it does not satisfy the strong triangle
inequality, condition (3) in the definition.

##  3 Affine $\Lambda$-buildings

###  3.1 $\Lambda$-metric spaces

An abelian group $(\Lambda,+)$ with a linear order such that $x,y\geq 0$
implies $x+y\geq 0$ is called an _ordered abelian group_. If an ordered
abelian group $\Lambda$ is isomorphic (as an ordered group) to a subgroup of
$(\mathbb{R},+)$, it is called _Archimedean_. Hahn’s embedding theorem
classifies all ordered abelian groups.

######  Theorem 3.1.

([Hah07]) For every ordered abelian group $(\Lambda,+)$ there is an ordered
set $\Omega$ such that $\Lambda<\mathbb{R}^{\Omega}$ as an ordered subgroup,
where

$\mathbb{R}^{\Omega}=\\{f\colon\Omega\to\mathbb{R}\colon\operatorname{supp}(f)\text{
is contained in a well ordered set }\\}$

is equipped with the lexicographical ordering.

Let $(\Lambda,+)$ be a non-trivial ordered abelian group, In particular,
$\Lambda$ has no torsion.

We will use the following generalization of metric spaces. If $X$ is a set and
$d\colon X\times X\to\Lambda$ a function, we call $(X,d)$ a _$\Lambda$
-pseudometric space_ if for all $x,y,z\in X$

  1. (1)

$d(x,x)=0$, $d(x,y)\geq 0$

  2. (2)

$d(x,y)=d(y,x)$

  3. (3)

$d(x,y)\leq d(x,z)+d(z,y)$.

If in addition, $d(x,y)=0$ implies $x=y$, then $(X,d)$ is called a _$\Lambda$
-metric space_. The axioms are direct generalizations of the notions of
(pseudo)metric spaces when $\Lambda=\mathbb{R}$.

###  3.2 $\Lambda$-trees

We define the notion of a $\Lambda$-tree following Chiswell’s book [Chi01],
where more details can be found. Let $(\Lambda,+)$ be an ordered abelian group
and $(X,d)$ a $\Lambda$-metric space. A _closed $\Lambda$-interval_ is a set
of the form

$[a,b]:=\\{\lambda\in\Lambda\colon a\leq\lambda\leq b\\}$

for $a\leq b\in\Lambda$. Closed $\Lambda$-intervals with
$d_{\Lambda}(t,t^{\prime})=|t^{\prime}-t|$ for $t,t^{\prime}\in[a,b]$ are
$\Lambda$-metric spaces. Let $X$ be any $\Lambda$-metric space. An isometric
embedding $\varphi\colon[a,b]\to X$ is called a _parametrization_ of its image
$s=\varphi([a,b])\subseteq X$, which is called a _segment_. Note that the set
of _endpoints_ $\\{\varphi(a),\varphi(b)\\}\subseteq X$ is independent of the
parametrization $\varphi$ of the segment $s$. A $\Lambda$-metric space is
_geodesic_ if for any two points $p,q\in X$ there exists a segment that has
$p$ and $q$ as endpoints. If there is only one such segment, then $X$ is
called _uniquely geodesic_ (or _geodesically linear_ in [Chi01]). In a
uniquely geodesic $\Lambda$-metric space, we denote a segment $s$ with
endpoints $x,y\in X$ by $s=[x,y]$.

A $\Lambda$-metric space $(X,d)$ is a _$\Lambda$ -tree_ if it satisfies the
following three axioms, two of which are illustrated in Figure 3.

  * (1)

$(X,d)$ is geodesic.

  * (2)

If two segments $s,s^{\prime}\subseteq X$ intersect in a single point $s\cap
s^{\prime}=\\{p\\}$, which is an endpoint of $s$ and $s^{\prime}$, then their
union $s\cup s^{\prime}$ is a segment.

  * (3)

If two segments $s,s^{\prime}\subseteq X$ have an endpoint $p$ in common, then
their intersection $s\cap s^{\prime}$ is a segment.

Figure 3: Illustration of axiom (2) (left) and axiom (3) (right) of
$\Lambda$-trees. In axiom (2) the segments $s,s^{\prime}$ are only allowed to
intersect in one point.

The notion of a $\Lambda$-tree generalizes trees from graph theory as
$\mathbb{Z}$-trees and real trees as $\mathbb{R}$-trees. It is known that for
$\Lambda=\mathbb{Z}$ and $\Lambda=\mathbb{R}$, axiom (3) follows from (1) and
(2), see [Chi01, Lemma I.2.3, Lemma I.3.6]. In [App24], we gave the following
complete characterization of the groups $\Lambda$ for which axiom (3) follows
from (1) and (2), answering for which $\Lambda$ condition (3) has to be
included.

######  Theorem 3.2.

Let $\Lambda$ be an ordered abelian group. The following are equivalent:

  * (a)

For every positive $\lambda_{0}\in\Lambda$, the set $\\{t\in\Lambda\colon
0\leq 2t\leq\lambda_{0}\\}$ has a maximum.

  * (b)

Every $\Lambda$-metric space that satisfies axioms (1) and (2) also satisfies
(3).

The ordered abelian group $\Lambda=\mathbb{Z}[1/3]<\mathbb{R}$ is an example
of a group for which axiom (3) is independent of (1) and (2), so axiom (3) can
not be omitted from the definition for general $\Lambda$. In [App24] we leave
open two questions.

######  Question 3.3.

Let $\Lambda$ be a non-trivial ordered abelian group. Is axiom (2) independent
of the axioms (1) and (3)?

######  Question 3.4.

Let $\Lambda$ be a non-trivial ordered abelian group. Is there a uniquely
geodesic $\Lambda$-metric space that does not satisfy axiom (2)?

In applications, the algebraic condition $\Lambda=2\Lambda$ is often
satisfied. Elsewhere in the thesis, $\Lambda$ will be the image of a valuation
$v\colon\mathbb{F}_{>0}\to\Lambda$ of a real closed field $\mathbb{F}$. Since
square roots exist for positive elements in $\mathbb{F}$, $\Lambda$ is
divisible by 2.

####  3.2.1 Example, Brumfiel’s tree

Let $\mathbb{F}$ be a non-Archimedean real closed field with valuation
$v\colon\mathbb{F}\to\Lambda\cup\\{\infty\\}$ where
$\Lambda\subseteq\mathbb{R}$. We follow [Bru88b] to construct a $\Lambda$-tree
associated to the hyperbolic plane and $\mathbb{F}$. We consider the upper
half-plane model $\mathbb{H}_{\mathbb{R}}:=\\{(x,y)\in\mathbb{R}^{2}\colon
y>0\\}$ of the hyperbolic plane and define the _non-standard hyperbolic plane_
as its semialgebraic extension $\mathbb{H}_{\mathbb{F}}$. The cross ratio can
be used to define the distance on $\mathbb{H}_{\mathbb{R}}$: for
$p,q\in\mathbb{H}_{\mathbb{R}}$ there is a unique hyperbolic line through $p$
and $q$ given by a half circle or a ray perpendicular to the $x$-axis. Let
$a,b\in\mathbb{R}\cup\\{\infty\\}$ be the two points at infinity of the line,
$a$ closer to $p$ and $b$ closer to $q$. The cross-ratio

$\operatorname{cr}_{\mathbb{R}}(p,q):=\operatorname{CR}(p,q;a,b):=\frac{|a-q|\cdot|b-p|}{|a-p|\cdot|b-q|}\in\mathbb{R}_{\geq
1}$

can then be used to describe the hyperbolic distance
$d_{\mathbb{R}}(p,q)=\log\operatorname{cr}_{\mathbb{R}}(p,q)$. We note that
$\operatorname{cr}_{\mathbb{R}}$ is a semialgebraic function, and
$d_{\mathbb{F}}(p,q):=(-v)\operatorname{cr}_{\mathbb{F}}(p,q)$ defines a
$\Lambda$-pseudo-distance on $\mathbb{H}_{\mathbb{F}}$. Then
$\mathcal{T}:=\mathbb{H}_{\mathbb{F}}/\\!\\!\sim$, where $p\sim q$ if
$d_{\mathbb{F}}(p,q)=0$ defines a $\Lambda$-metric space.

######  Theorem 3.5.

([Bru88b, Theorem (28)]) The $\Lambda$-metric space $\mathcal{T}$ is a
$\Lambda$-tree.

This construction is an easier special case of the construction of the
building $\mathcal{B}$ in Section 7 and follows along the same ideas: we first
find a semialgebraic model for the hyperbolic plane, define the non-standard
hyperbolic plane and a pseudometric on it and finally show that the quotient
defines a $\Lambda$-tree. Theorem 3.5 is a special case of our main theorem,
Theorem 8.1 for $\operatorname{SL}(2)$. Our theorem is more general in two
ways: we allow for higher rank symmetric spaces and thus obtain affine
$\Lambda$-buildings, and we allow $\Lambda$ to be any ordered abelian group,
not necessarily a subgroup of $\mathbb{R}$.

In joint work with De Rosa, Flamm and Jaeck [ARFJ24] we investigated the
$\mathbb{Q}$-tree $\mathcal{T}$ for $\mathbb{F}$ the Puiseux series. Our main
result states that completing all the segments of $\mathcal{T}$ does not
result in a complete metric space.

######  Theorem 3.6.

Let $\mathcal{T}$ be the $\mathbb{Q}$-tree constructed from the field of
Puiseux-series in [Bru88b]. There is a Cauchy sequence in the segment
completion $\overline{\mathcal{T}}^{\operatorname{sc}}$ that does not converge
and hence $\overline{\mathcal{T}}^{\operatorname{sc}}$ is not complete.

###  3.3 Root systems

A detailed treatment of root systems can be found in [Bou08a]. Let $V$ be a
finite dimensional Euclidean vector space with scalar product
$\langle\cdot,\cdot\rangle$. For $\alpha\in V$, the reflection along the
hyperplane $M_{\alpha}=\\{\beta\in V\colon\langle\alpha,\beta\rangle=0\\}$ is
given by

$r_{\alpha}(\beta)=\beta-2\frac{\langle\alpha,\beta\rangle}{\langle\alpha,\alpha\rangle}\alpha.$

A pair $(\Phi,V)$ where $\Phi\subseteq V$ is called a _root system_ if

  1. (R1)

$\Phi$ is finite, symmetric ($\Phi=-\Phi$), spans $V$ and does not contain
$0$.

  2. (R2)

For every $\alpha\in\Phi$ the reflection $r_{\alpha}\colon V\to V$ preserves
$\Phi$.

When $V$ can be determined from the context, the root system may be denoted by
just $\Phi$. A root system is called _crystallographic_ if it satisfies the
integrality condition

  1. (R3)

If $\alpha,\beta\in\Phi$, then
$2\langle\alpha,\beta\rangle/\langle\alpha,\alpha\rangle\in\mathbb{Z}$.

For crystallographic root systems, $\operatorname{Span}_{\mathbb{Z}}(\Phi)$ is
a lattice in $V$. A root system is called called _reduced_ if

  1. (R4)

For every $\alpha\in\Phi$, $\mathbb{R}\cdot\alpha\cap\Phi=\\{\pm\alpha\\}$.

Figure 4:  Two examples of crystallographic root systems. Type
$\operatorname{A}_{2}$ on the left, Type $\operatorname{BC}_{2}$ on the right.
While $\operatorname{A}_{2}$ is reduced, $\operatorname{BC}_{2}$ is not.

Figure 4 gives examples of two crystallographic root systems, one of them
reduced the other not. The _spherical Weyl group $W_{s}$_ of a root system is
the subgroup of isometries of $V$ generated by reflections along the
hyperplanes $M_{\alpha}$ for $\alpha\in\Phi$. A subset $\Delta\subseteq\Phi$
is a _basis of $\Phi$_, if it is a vector space basis of $V$ such that all
roots $\beta\in\Phi$ can be written as
$\beta=\sum_{\delta\in\Delta}\lambda_{\delta}\delta$ with
$\lambda_{\delta}\in\mathbb{Z}$ for $\delta\in\Delta$ and such that all
$\lambda_{\delta}$ have the same sign (all are non-negative or non-positive).
Every root system has a basis and the spherical Weyl group acts transitively
on the set of bases of $\Phi$. The cardinality of a basis $\Delta$ is called
the _rank_ of the root system $\Phi$ and coincides with $\dim(V)$.

The connected components of $V\setminus\cup_{\alpha\in\Phi}M_{\alpha}$ are
called _(open) chambers_. The spherical Weyl group acts simply transitively on
the set of chambers. Given a basis $\Delta$, the _fundamental Weyl chamber of
$\Delta$_ is

$WC(\Delta)=\left\\{v\in V\colon\langle v,\alpha\rangle>0\
\forall\alpha\in\Delta\right\\}.$

Conversely, every chamber $C$ determines a basis $B(C)$ formed by those
$\alpha\in\Phi$, that satisfy $\langle\alpha,v\rangle>0$ for all $v\in C$ and
that cannot be written as a sum of other such elements of $\Phi$, see [Bou08a,
VI§1.5]. A basis $\Delta$ determines the set of _positive roots_
$\Sigma_{>0}\subseteq\Phi$ given by positive integer combinations of elements
in the basis. Thus a basis determines a partial order on $\Sigma$ by
$\alpha<\beta$ if $\beta-\alpha\in\Sigma_{>0}$. A total order on $\Sigma$ with
the same positive elements can be obtained by choosing an order on the basis
and extending it lexicographically to
$\Sigma\subseteq\operatorname{Span}_{\mathbb{Z}}(\Sigma)$.

Let now $(\Phi,V)$ be a crystallographic root system. For every
$\alpha\in\Phi$ the _coroots_ 333The coroots $\alpha^{\vee}$ are not to be
confused with the _dual roots_ in $V^{\star}$, which are sometimes denoted by
the same symbol.

$\alpha^{\vee}:=\frac{2\alpha}{\langle\alpha,\alpha\rangle}\in V$

have the property that for all $\alpha,\beta\in\Sigma$

$\langle\alpha,\beta^{\vee}\rangle=\frac{2\langle\alpha,\beta\rangle}{\langle\beta,\beta\rangle}\in\mathbb{Z}$

by the crystallographic condition (R3).

###  3.4 Apartments

We introduce the model apartment $\mathbb{A}$. A more detailed introduction
can be found in [Ben94], and including the non-crystallographic case in
[Sch09]. The root systems in our setting will arise from algebraic groups, and
are therefore crystallographic, but possibly not reduced.

Let $\Lambda$ be a non-trivial ordered abelian group. Let $(\Phi,V)$ be a
crystallographic root system and
$L=\operatorname{Span}_{\mathbb{Z}}(\Phi)\subseteq V$ its associated lattice.
Since both $L$ and $\Lambda$ are free $\mathbb{Z}$-modules, we can define the
_model apartment_

$\mathbb{A}:=L\otimes_{\mathbb{Z}}\Lambda.$

For a basis $\Delta=\\{\alpha_{1},\ldots,\alpha_{r}\\}$, the
$\mathbb{Z}$-bilinear map

$\displaystyle L\times\Lambda$ $\displaystyle\to\Lambda^{r}$
$\displaystyle\left(\sum_{i=1}^{r}b_{i}\alpha_{i},\lambda\right)$
$\displaystyle\mapsto(b_{1}\lambda,\ldots,b_{r}\lambda)$

extends to a $\mathbb{Z}$-module isomorphism $\mathbb{A}\cong\Lambda^{r}$,
giving rise to the notation

$\mathbb{A}\cong\left\\{\sum_{\delta\in\Delta}\lambda_{\delta}\delta\colon\lambda_{\delta}\in\Lambda\right\\}.$

Note that if $\Lambda$ is a $\mathbb{Q}$-vector space, then we also have

$\mathbb{A}=\operatorname{Span}_{\mathbb{Q}}(\Phi)\otimes_{\mathbb{Q}}\Lambda.$

Since $W_{s}$ acts on $\Phi$, it also acts on $\mathbb{A}$, fixing
$0\in\mathbb{A}$. The apartment $\mathbb{A}$ itself acts by translation on
$\mathbb{A}$. Let $T\subseteq\mathbb{A}$ be a subgroup of the translation
group normalized by $W_{s}$. Then $W_{a}=W\ltimes T$ is called the _affine
Weyl group_ and acts on $\mathbb{A}$. If $T=\mathbb{A}$, $W_{a}$ is called the
_full affine Weyl group_. We denote the data of a model apartment together
with an affine Weyl group as $\mathbb{A}=\mathbb{A}(\Phi,\Lambda,T)$ since the
root system $\Phi$, the ordered abelian group $\Lambda$ and the subgroup of
translations $T$ fully determine $\mathbb{A}$ and the affine Weyl group.

For any root $\alpha\in\Phi$, the reflection $r_{\alpha}\colon\Phi\to\Phi$
extends to a _reflection_

$\displaystyle r_{\alpha}\colon\mathbb{A}$ $\displaystyle\to\mathbb{A}$
$\displaystyle\sum_{\delta\in\Delta}\lambda_{\delta}\delta$
$\displaystyle\mapsto\sum_{\delta\in\Delta}\lambda_{\delta}r_{\alpha}(\delta)$

and for any $t\in T$, maps of the form $t\circ r_{\alpha}\in W_{a}$ are called
_affine reflections_. The scalar product on $\Phi$ extends naturally to the
bilinear pairing

$\displaystyle\langle\cdot,\cdot\rangle\colon\mathbb{A}\times L$
$\displaystyle\to\Lambda$
$\displaystyle\left(\sum_{\delta\in\Delta}\lambda_{\delta}\delta,\sum_{\delta^{\prime}\in\Delta}\mu_{\delta^{\prime}}\delta^{\prime}\right)$
$\displaystyle\mapsto\sum_{\delta\in\Delta}\sum_{\delta^{\prime}\in\Delta}\mu_{\delta^{\prime}}\lambda_{\delta}\langle\delta,\delta^{\prime}\rangle$

but we notice that the scalar product can not be defined on all of
$\mathbb{A}\times\mathbb{A}$, since $\Lambda$ may not have a multiplication.
For $\alpha\in\Phi$ and $\lambda\in\Lambda$ we define the _affine wall_

$M_{\alpha,\lambda}:=\left\\{x\in\mathbb{A}\colon\langle
x,\alpha\rangle=\lambda\right\\}$

as well as the two _affine half-spaces_

$\displaystyle H_{\alpha,\lambda}^{+}$
$\displaystyle=\left\\{x\in\mathbb{A}\colon\langle
x,\alpha\rangle\geq\lambda\right\\}$ $\displaystyle H_{\alpha,\lambda}^{-}$
$\displaystyle=\left\\{x\in\mathbb{A}\colon\langle
x,\alpha\rangle\leq\lambda\right\\}$

defined by $M_{\alpha,\lambda}$. The intersection of the positive half-spaces
for all $\delta\in\Delta$ is called the _fundamental Weyl chamber_

$C_{0}:=\left\\{x\in\mathbb{A}\colon\langle x,\alpha\rangle\geq 0\
\forall\delta\in\Delta\right\\}=\bigcap_{\delta\in\Delta}H_{\delta,0}^{+}.$

The images of the fundamental Weyl chamber under the action of the affine Weyl
group are called _chambers_.

Note that for every $\alpha\in\Phi$,

$\alpha^{\vee}:=\frac{2\alpha}{\langle\alpha,\alpha\rangle}=\sum_{\delta\in\Delta}\langle\alpha^{\vee},\delta\rangle\delta=\sum_{\delta\in\Delta}2\frac{\langle\alpha,\delta\rangle}{\langle\alpha,\alpha\rangle}\delta\in
L$

by the crystallographic condition. The model apartment $\mathbb{A}$ can be
endowed with a $\Lambda$-metric. There are multiple ways to do this and we
will describe one. Since $\mathbb{A}$ may not admit a $\Lambda$-valued scalar
product, we instead use the $\Lambda$-valued norm
$N\colon\mathbb{A}\to\Lambda$ defined by

$N(x)=\sum_{\alpha\in\Phi_{>0}}\left|\left\langle
x,\alpha^{\vee}\right\rangle\right|,$

where $|\lambda|=\max\\{\lambda,-\lambda\\}$ is the absolute value. Then

$d(x,y)=N(x-y)$

turns $\mathbb{A}$ into a $\Lambda$-metric space. Note that when $x-y\in
C_{0}$,

$d(x,y)=\left\langle x-y,\sum_{\alpha\in\Phi_{>0}}\alpha^{\vee}\right\rangle.$

Note that the $\Lambda$-metric space $\mathbb{A}$ may not be uniquely
geodesic: there may be two or more distinct isometric embeddings of a
$\Lambda$-interval with coinciding endpoints. Since $\Lambda$ is just a group,
the classical notion of convexity using linear combinations does not make
sense. Instead we say that a subset $B\subseteq\mathbb{A}$ is _$W_{a}$
-convex_, if it is a finite intersection of affine half-spaces. Figure 5
illustrates that $\mathbb{A}$ may not be uniquely geodesic and that the union
of all geodesics is a $W_{a}$-convex set.

Figure 5:  Given two points $p,q\in\mathbb{A}$, the interval
$\\{r\in\mathbb{A}\colon d(p,r)+d(r,q)=d(p,q)\\}$ is a $W_{a}$-convex set
defined by the intersection of four half-planes parallel to the walls in this
example of type $\operatorname{A}_{2}$.

###  3.5 Affine $\Lambda$-buildings

Let $\mathbb{A}$ be an apartment of type
$\mathbb{A}=\mathbb{A}(\Phi,\Lambda,T)$ with affine Weyl group $W_{a}$ as in
the previous section. Let $X$ be a set and $\mathscr{F}$ a set of injective
maps $\mathbb{A}\to X$. The set $\mathscr{F}$ is called the _atlas_ or
_apartment system_ and its elements are called the _charts_. The images
$f(\mathbb{A})$ for $f\in\mathscr{F}$ are called _apartments_. The images of
walls, half-spaces and chambers are called _walls, half-apartments_ and
_sectors_.

The pair $(X,\mathscr{F})$ is called a _generalized affine building_ or a
_affine $\Lambda$-building_ of type $\mathbb{A}=\mathbb{A}(\Phi,\Lambda,T)$ if
the following six axioms are satisfied. Axioms (A4) and (A6) are illustrated
in Figures 6 and 7.

  1. (A1)

For all $f\in\mathscr{F}$, $w\in W_{a}$, $f\circ w\in\mathscr{F}$.

  2. (A2)

For all $f,f^{\prime}\in\mathscr{F}$, the set $B=f^{-1}\left(f(\mathbb{A})\cap
f^{\prime}(\mathbb{A})\right)$ is $W_{a}$-convex and there is a $w\in W_{a}$
such that $f|_{B}=f^{\prime}\circ w|_{B}.$

  3. (A3)

For all $x,y\in X$, there is a $f\in\mathscr{F}$ such that $x,y\in
f(\mathbb{A})$.

  4. (A4)

For any sectors $s_{1},s_{2}\subseteq X$ there are subsectors
$s_{1}^{\prime}\subseteq s_{1},s_{2}^{\prime}\subseteq s_{2}$ such that there
is an $f\in\mathscr{F}$ with $s_{1}^{\prime},s_{2}^{\prime}\subseteq
f(\mathbb{A})$.

  5. (A5)

For every $x\in X$ and $f\in\mathscr{F}$ with $x\in f(\mathbb{A})$ there is a
distance diminishing retraction $r_{x,f}\colon X\to f(\mathbb{A})$ such that
$f^{-1}(\\{x\\})=\\{x\\}$.

  6. (A6)

For $f_{1},f_{2},f_{3}\in\mathscr{F}$, if $f_{i}(\mathbb{A})\cap
f_{j}(\mathbb{A})$ are half-apartments for $i\neq j$, then
$f_{1}(\mathbb{A})\cap f_{2}(\mathbb{A})\cap f_{3}(\mathbb{A})\neq\emptyset$.

Figure 6: Axiom (A4) states that while arbitrary sectors $s_{1},s_{2}$ may not
lie in a common flat, they contain subsectors $s_{1}^{\prime},s_{2}^{\prime}$
contained in a common flat $f(\mathbb{A})$.  Figure 7: Axiom (A6) states that
whenever three apartments intersect pairwise in a half-apartment, then there
is at least one point in the common intersection of all three.

The distance in axiom (A5) is the function $d\colon X\times X\to\Lambda$
induced from the distance on the model apartment $\mathbb{A}$, whose existence
follows from axioms (A1), (A2) and (A3): for any two points $x,y\in X$ we use
axiom (A3) to find $f\in\mathscr{F}$ such that $x,y\in f(\mathbb{A})$ and
define $d(x,y)=d(f^{-1}(x),f^{-1}(y))\in\Lambda$. This is well defined, by
axioms (A1) and (A2). A $d$-diminishing retraction is then a function $r\colon
X\to f(\mathbb{A})$ satisfying $r(y)=y$ for all $y\in f(\mathbb{A})$ and

$d(r(x),r(y))\leq d(x,y)$

for all $x,y\in X$. Note that while axioms (A1) - (A3) can be used to define a
symmetric positive definite function $d$, axiom (A5) can be used to show that
$d$ satisfies the triangle inequality. As shown in [BSS14], in the presence of
the other five axioms, axiom (A5) is equivalent to the triangle inequality. In
fact, [BSS14] contains a number of collections of axioms that characterize
affine $\Lambda$-buildings. We will use the following characterization.

######  Theorem 3.7 (Theorem 3.1, (5) $\implies$ (1) [BSS14]).

Let $(X,\mathbb{F})$ be a set with an atlas such that the axioms (A1), (A2),
(A3) and (A4), as well as

  1. (TI)

The function $d$ induced from the distance in apartments satisfies the
triangle inequality.

and the exchange condition

  1. (EC)

For $f_{1},f_{2}\in\mathscr{F}$, if $f_{1}(\mathbb{A})\cap f_{2}(\mathbb{A})$
is a half-apartment, then there exists $f_{3}\in\mathscr{F}$ such that
$f_{i}(\mathbb{A})\cap f_{3}(\mathbb{A})$ are half-apartments for
$i\in\\{1,2\\}$. Moreover $f_{3}(\mathbb{A})$ is the symmetric difference of
$f_{1}(\mathbb{A})$ and $f_{2}(\mathbb{A})$ together with the boundary wall of
$f_{1}(\mathbb{A})\cap f_{2}(\mathbb{A})$.

are satisfied. Then $(X,\mathbb{F})$ is an affine $\Lambda$-building.

###  3.6 Independence of axiom (A6)

The independence of some of the axioms of affine $\Lambda$-buildings were
studied in the appendix of [BSS14]. For $\Lambda=\mathbb{R}$, Parreau [Par00]
showed that axiom (A6) follows from (A1) - (A5). When Bennett introduced
affine $\Lambda$-buildings he claimed to give an example of a rank 1
$\mathbb{Q}$-building that satisfies axioms (A1) - (A5), but not (A6) [Ben94,
Remark 3.3]. The example looked like it had the claimed properties, but did
not actually admit a valid apartment system. Translated in terms of
$\Lambda$-trees, the example would have been a $\mathbb{Q}$-tree where axioms
(1) and (2) would have held, but not axiom (3). In Theorem 3.2 we showed that
this is not possible. The next example shows that for
$\Lambda=\mathbb{Z}[1/3]$, there is a $\Lambda$-metric space $X$ that admits
an apartment system such that axioms (A1) - (A5) are satisfied, but (A6) is
not. The example can be generalized to any ordered abelian group $\Lambda$
with $\Lambda\neq 2\Lambda$ and is illustrated in Figure 8.

Figure 8:  For $\Lambda=\mathbb{Z}[1/3]$, there is a $\Lambda$-metric space
$X$ that satisfies axioms (A1) - (A5), but not (A6). It is important that
$1/2\notin\Lambda$, so that there is no point opposite of $[0]\in
S_{\Lambda}^{1}$, illustrated by a circle with white interior.

Let $\Lambda=\mathbb{Z}[1/3]$. We start with a $\Lambda$-circle
$S_{\Lambda}^{1}:=\Lambda/\mathbb{Z}$ of length 1. At every point $p\in
S_{\Lambda}^{1}$ we glue a $\Lambda$-ray
$r_{p}=[0,\infty)_{\Lambda}\subseteq\Lambda$ to obtain a $\Lambda$-metric
space

$X=\left(S_{\Lambda}^{1}\amalg\coprod_{p\in
S_{\Lambda}^{1}}r_{p}\right)/\\!\sim$

where $p\in S_{\Lambda}^{1}$ is identified with $0\in r_{p}$. For any two
points $p,q\in S_{\Lambda}^{1}$, a unique apartment is given by the union of
$r_{p}$, $r_{q}$ and the shorter segment in $S_{\Lambda}^{1}$ connecting $p$
to $q$. Since $1/2\notin\Lambda$, $p$ and $q$ can not lie opposite each other
on $S_{\Lambda}^{1}$. Axioms (A1), (A2), (A3) and (A4) are readily checked,
(A5) requires some thought and (A6) clearly does not hold. The uniqueness of
the apartment is important for the validity of (A2) and is the reason why this
example does not work for $\Lambda=\mathbb{Q}$.

In analogy to Theorem 3.2, we would like to ask for which $\Lambda$ axiom (A6)
is independent. Note that even for rank 1, independence of (3) does not
directly translate to independence of (A6).

######  Question 3.8.

Characterize the ordered abelian groups for which axiom (A6) is independent of
the other axioms. Is there even any ordered abelian group $\Lambda$ for which
axiom (A6) is independent?

##  4 Linear algebraic groups

###  4.1 Definitions

Let $\mathbb{K}\subseteq\mathbb{R}$ be a subfield (usually
$\mathbb{K}=\mathbb{Q}$ or $\mathbb{K}=\mathbb{R}$), $\mathbb{F}$ a real
closed field containing $\mathbb{K}$ and $\mathbb{D}$ an algebraically closed
field that contains both $\mathbb{R}$ and $\mathbb{F}$. We follow a naive
approach to algebraic groups, viewing them as matrix groups. Since we are
working with fields of characteristic 0, the algebraic geometry can be kept at
a minimum. For an extensive introduction to algebraic groups, we refer to
[Bor66], [Bor91], [Hum75] or [Zim84].

The general linear group $\operatorname{GL}_{n}(\mathbb{D})$ can be realized
as an affine algebraic variety

$\operatorname{GL}_{n}(\mathbb{D})\cong\left\\{\begin{pmatrix}A&\\\
&t\end{pmatrix}\in\mathbb{D}^{(n+1)\times(n+1)}\colon\det(A)\cdot
t=1\right\\}.$

A subgroup $G<\operatorname{GL}_{n}(\mathbb{D})$ is called a _linear algebraic
group defined over $\mathbb{K}$_, if it is a set of common zeros for a set of
polynomials in the coordinate ring
$\mathbb{K}[\operatorname{GL}_{n}(\mathbb{D})]:=\mathbb{K}[(x_{ij}),\det(x_{ij})^{-1}]$
of $\operatorname{GL}_{n}(\mathbb{D})$. We will not consider more general
algebraic groups and hence also call $G$ an _algebraic group_ or a
$\mathbb{K}$-_group_. For any commutative $\mathbb{K}$-subalgebra
$\mathbb{B}\subseteq\mathbb{D}$ containing $\mathbb{K}$, we let

$\operatorname{GL}_{n}(\mathbb{B})=\left\\{(a_{ij})\in\operatorname{GL}_{n}(\mathbb{D})\colon
a_{ij}\in\mathbb{B}\text{ and }\det(a_{ij})^{-1}\in\mathbb{B}\right\\}.$

The _group of $\mathbb{B}$-points of a linear algebraic group $G$_ is
$G_{\mathbb{B}}:=G\cap\operatorname{GL}_{n}(\mathbb{B})$. The
$\mathbb{R}$-points of a linear algebraic group $G$ form a real Lie group
[Mil13, Theorem 2.1].

Viewing $G\subseteq\operatorname{GL}_{n}(\mathbb{D})$ as an algebraic subset
of $\mathbb{D}^{n+1\times n+1}$, we endow $G$ with the Zariski-topology, whose
closed sets are given by common zeros of sets of polynomials in
$\mathbb{K}[\operatorname{GL}_{n}(\mathbb{D})]$. A connected linear algebraic
group $G$ is _semisimple_ if every closed connected normal abelian subgroup is
trivial.

####  4.1.1 Examples

The multiplicative group $\mathbb{G}_{m}=\operatorname{GL}_{1}(\mathbb{D})$ is
a linear algebraic group defined over $\mathbb{Q}$. Note that $\mathbb{G}_{m}$
is connected in the Zariski-topology. Its $\mathbb{C}$-points
$(\mathbb{G}_{m})_{\mathbb{C}}=\mathbb{C}\setminus\\{0\\}$ are connected in
the Euclidean topology but its $\mathbb{R}$-points
$(\mathbb{G}_{m})_{\mathbb{R}}=\mathbb{R}\setminus\\{0\\}$ are not.

If $V$ is a $\mathbb{D}$-vector space, then the group $\operatorname{GL}(V)$
of automorphisms of $V$ is an algebraic group. As $\operatorname{GL}(V)$
contains the closed connected normal abelian subgroup of scalar
multiplication, $\operatorname{GL}(V)$ is not semisimple.

The real Lie groups
$\operatorname{SL}_{n}(\mathbb{R}),\operatorname{SO}_{n}(\mathbb{R})$ and
$\operatorname{Sp}_{2n}(\mathbb{R})$ are groups of $\mathbb{R}$-points of the
linear algebraic groups
$\operatorname{SL}_{n}(\mathbb{D}),\operatorname{SO}_{n}(\mathbb{D})$ and
$\operatorname{Sp}_{2n}(\mathbb{D})$ defined over $\mathbb{Q}$.

###  4.2 Morphisms and tori

For a linear algebraic group $G$ we consider the ideal

$I(G)=\left\\{p\in\mathbb{D}\left[\left(x_{ij}\right),z\right]\colon
p\left(g,\det\left(g\right)^{-1}\right)=0\text{ for all }g\in G\right\\},$

which is finitely generated as a consequence of Hilbert’s Basis theorem. The
_coordinate ring_ of $G$ is

$\mathbb{D}[G]=\mathbb{D}\left[\left(x_{ij}\right),d\right]/I(G)$

and its elements are called _regular functions_ on $G$. Given a map
$\varphi\colon G\to H$ between algebraic groups $G$ and $H$, we can define the
_transposed map_ $\varphi^{\circ}\colon\mathbb{D}[H]\to\mathbb{D}[G]$ by
$\varphi^{\circ}(f)=f\circ\varphi$. A _morphism_ of linear algebraic groups
$G$ and $H$ is a group homomorphism $\varphi\colon G\to H$ whose transposed
map $\varphi^{\circ}$ is a ring homomorphism. If $G$ and $H$ are defined over
$\mathbb{K}$ and $\varphi^{\circ}$ maps $\mathbb{K}[H]$ into $\mathbb{K}[G]$,
then $\varphi$ is _defined over_ $\mathbb{K}$. An important example of a
morphism $G\to G$ is the conjugation $\operatorname{Int}(g)\colon h\mapsto
ghg^{-1}$ by an element $g\in G$, and if $g\in G_{\mathbb{K}}$, then
$\operatorname{Int}(g)$ is defined over $\mathbb{K}$.

######  Lemma 4.1.

Let $\varphi\colon G\to H$ be a morphism of algebraic $\mathbb{K}$-groups
$G<\operatorname{GL}(n,\mathbb{D}),H<\operatorname{GL}(m,\mathbb{D})$ defined
over $\mathbb{K}$. If we write componentwise

$\displaystyle\varphi_{\mathbb{F}}\colon G_{\mathbb{F}}$ $\displaystyle\to
H_{\mathbb{F}}$ $\displaystyle\begin{pmatrix}x_{11}&\ldots&\\\
\vdots&\ddots&\\\ &&x_{nn}\end{pmatrix}$
$\displaystyle\mapsto\begin{pmatrix}\varphi_{11}(x_{11},\ldots,x_{nn})&\ldots&\\\
\vdots&\ddots&\\\ &&\varphi_{mm}(x_{11},\ldots,x_{nn})\end{pmatrix}$

then all $\varphi_{ij}\colon G_{\mathbb{F}}\to\mathbb{F}$ are polynomials.

###### Proof.

Consider the polynomial representing $X_{ij}\in\mathbb{K}[H]$. Then

$\displaystyle\varphi^{\circ}(X_{ij})(x_{11},\ldots,x_{nn})$
$\displaystyle=X_{ij}(\varphi(x_{11},\ldots,x_{mm}))$
$\displaystyle=X_{ij}(\varphi_{11}(x_{11},\ldots,x_{nn}),\ldots,\varphi_{mm}(x_{11},\ldots,x_{nn}))$
$\displaystyle=\varphi_{ij}(x_{11},\ldots,x_{nn})$

and thus $\varphi_{ij}\in\mathbb{K}[G]$ is a polynomial. ∎

######  Proposition 4.2.

In characteristic $0$, a map $G\to H$ between algebraic groups $G,H$ is a
morphism if and only if it is a group homomorphism whose graph is a set of
common zeros of polynomials. The morphism is defined over $\mathbb{K}$ if and
only if all the polynomials can be chosen to have coefficients in
$\mathbb{K}$.

A linear algebraic group $T$ that is isomorphic to $(\mathbb{G}_{m})^{d}$ is
called a _$d$ -dimensional torus_. An element
$g\in\operatorname{GL}_{n}(\mathbb{D})$ is _semisimple_ if $\mathbb{D}^{n}$ is
spanned by eigenvectors of $g$, or equivalently, if $g$ is diagonalizable.

######  Theorem 4.3 (8.5 in [Bor91]).

For a Zariski-connected algebraic group $T$, the following conditions are
equivalent.

  1. 1.

$T$ is a torus.

  2. 2.

$T$ consists only of semisimple elements.

  3. 3.

The whole group $T$ is simultaneously diagonalizable.

Since Zariski-connectedness of algebraic groups and semisimplicity of elements
is preserved under morphisms [Bor91, 4.4(4)], the image of a torus under a
morphism is a torus.

A morphism $\chi\colon G\mapsto\mathbb{G}_{m}$ from an algebraic group $G$ to
$\mathbb{G}_{m}$ is called a _character_. The set of characters of $G$ is an
abelian group, denoted by $\hat{G}$. If $T\cong(\mathbb{G}_{m})^{d}$ is a
$d$-dimensional torus and an element $x\in T$ corresponds to
$(x_{1},\ldots,x_{d})\in(\mathbb{G}_{m})^{d}$, then every character $\chi$ of
$T$ is of the form

$\chi(x)=x_{1}^{n_{1}}\cdots x_{d}^{n_{d}}$

for some $n_{i}\in\mathbb{Z}$ and hence $\hat{T}=\mathbb{Z}^{d}$. If there is
an isomorphism $T\to(\mathbb{G}_{m})^{d}$ defined over $\mathbb{K}$, $T$ is
said to be _$\mathbb{K}$ -split_.

######  Theorem 4.4 (8.4 in [Bor91].).

For a torus $T$ defined over $\mathbb{K}$, the following conditions are
equivalent.

  1. 1.

$T$ is $\mathbb{K}$-split.

  2. 2.

All characters of $T$ are defined over $\mathbb{K}$,

A _maximal torus_ $T\subseteq G$ is a subgroup that is a torus and is not
properly contained in any other torus in $G$.

######  Theorem 4.5 (11.3 and 20.9 in [Bor91] ).

In a connected algebraic group $G$, all maximal tori of $G$ are conjugate. In
a connected semisimple $\mathbb{K}$-group $G$, the maximal $\mathbb{K}$-split
tori of $G$ are conjugate under $G_{\mathbb{K}}$.

The _rank_ of $G$ is the common dimension of the maximal tori and if $G$ is
semisimple and defined over $\mathbb{K}$, the $\mathbb{K}$-_rank_ of $G$ is
the common dimension of the maximal $\mathbb{K}$-split tori of $G$.

A morphism $\mathbb{G}_{m}\to G$ is a _multiplicative one-parameter subgroup
of $G$_. For a torus $T$ the one-parameter subgroups of $T$ are denoted by
$X_{\star}(T)$. For $\chi\in\hat{T}$ and $\lambda\in X_{\star}(T)$, the
composition $(\chi\circ\lambda)\colon\mathbb{G}_{m}\to\mathbb{G}_{m}$ is a
character of the torus $\mathbb{G}_{m}$ and hence sends $x\in\mathbb{G}_{m}$
to $x^{m}$ for some $m\in\mathbb{Z}$. This defines a map $b\colon\hat{T}\times
X_{\star}(T)\to\mathbb{Z},(\chi,\lambda)\mapsto m$.

######  Proposition 4.6.

[Bor91, Proposition 8.7] For any torus $T$, the map

$b\colon\hat{T}\times X_{\star}(T)\to\mathbb{Z}$

is a nondegenerate bilinear form.

####  4.2.1 Examples

The $\mathbb{R}$-group
$\operatorname{SO}(2)=\\{g\in\operatorname{SL}_{2}(\mathbb{C})\colon
gg{{}^{\mathsf{T}}\\!}=\operatorname{Id}\\}$ is a torus. An isomorphism with
$\mathbb{G}_{m}$ is given by

$\displaystyle\mathbb{G}_{m}$ $\displaystyle\to\operatorname{SO}(2)$
$\displaystyle a$
$\displaystyle\mapsto\frac{1}{2}\begin{pmatrix}a+a^{-1}&i(a^{-1}-a)\\\
i(a-a^{-1})&a+a^{-1}\end{pmatrix}$ $\displaystyle a+ib$
$\displaystyle\mapsfrom\phantom{\frac{1}{2}}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$

and the appearance of $i\in\mathbb{C}$ indicates that $\operatorname{SO}(2)$
is not $\mathbb{R}$-split. On the other hand, the $\mathbb{R}$-group
$A=\\{g\in\operatorname{SL}_{2}(\mathbb{C})\colon g\text{ is diagonal}\\}$ is
an $\mathbb{R}$-split torus. Since $\mathbb{C}$ is algebraically closed, both
tori are $\mathbb{D}$-split. Both $\operatorname{SO}(2)$ and $A$ are maximal
tori of $\operatorname{SL}_{2}$ and they are conjugate under
$\operatorname{SL}_{2}$ and $\operatorname{SL}_{2}(\mathbb{C})$, but not under
$\operatorname{SL}_{2}(\mathbb{R})$. The rank and the $\mathbb{R}$-rank of
$\operatorname{SL}_{2}$ is $1$.

###  4.3 The Lie algebra and the adjoint representation

Let $e\in G$ be the neutral element of an algebraic $\mathbb{K}$-group $G$.
There are multiple ways to define the Lie algebra of $G$, a few of which can
be found for instance in Chapters 5 and 9 of [Hum75]. A first explicit
definition uses the description of $G$ as a matrix group

$G=\left\\{(a_{ij})\in\operatorname{GL}_{n}(\mathbb{D})\colon
f(a_{ij})=0\text{ for all }f\in I\right\\}$

where $I\subset\mathbb{K}[(x_{ij}),\det(x_{ij})^{-1}]$ is a finite subset. For
the following definition we restrict ourselves to subgroups of
$\operatorname{SL}_{n}(\mathbb{D})$, so that we can ignore the dependence on
$\operatorname{det}(x_{ij})^{-1}$. For $f\in I$ we define the _differential of
$f$ at $e:=\operatorname{Id}$_ to be the linear polynomial

$\operatorname{d}_{e}\\!f=\sum_{ij}\frac{\partial f}{\partial x_{ij}}(e)\cdot
x_{ij}\in\mathbb{K}[(x_{ij})].$

and the _Zariski tangent space_

$T_{e}G:=\left\\{(a_{ij})\in\mathbb{D}^{n\times
n}\colon\operatorname{d}_{e}\\!f(a_{ij})=0\text{ for all }f\in I\right\\}.$

We now give another useful description of the tangent space in terms of
derivations.

A _derivation_ of the coordinate ring $\mathbb{D}[G]$ (viewed as a
$\mathbb{D}$-algebra) of a linear algebraic group $G$ is a linear map

$X\colon\mathbb{D}[G]\to\mathbb{D}[G]$

that satisfies the Leibnitz rule
$X(ff^{\prime})=X(f)f^{\prime}+fX(f^{\prime})$ for
$f,f^{\prime}\in\mathbb{D}[G]$. Algebraic groups act by left (and right)
translation on their coordinate rings $\mathbb{D}[G]$. For $g\in G$, the left
translation is given by $\lambda_{g}(f)(h)=f(g^{-1}h)$ for
$f\in\mathbb{D}[G],h\in G$ (and the right translation by
$\rho_{g}(f)(h)=f(hg)$). The Zariski tangent space $T_{e}G$ of $G$ can be
identified with the set of derivations $\mathfrak{g}$ of $\mathbb{D}[G]$ which
commute with right translations

$\displaystyle\mathfrak{g}$
$\displaystyle:=\left\\{X\colon\mathbb{D}[G]\to\mathbb{D}[G]\colon\begin{matrix}X\text{
is a derivation of }\mathbb{D}[G]\text{ with }\\\ X\circ\rho_{g}=\rho_{g}\circ
X\text{ for all }g\in G\end{matrix}\right\\}.$

Endowed with the Lie algebra structure defined by the bracket operation on
derivations, $\mathfrak{g}$ is called the _Lie algebra of_ $G$. For a
commutative $\mathbb{K}$-subalgebra $\mathbb{B}\subseteq\mathbb{D}$ containing
$\mathbb{K}$, we can use the identification $\mathfrak{g}=T_{e}G$, to define
the _$\mathbb{B}$ -points of the Lie algebra of $G$_

$\mathfrak{g}_{\mathbb{B}}=T_{e}G\cap\mathbb{B}^{n\times n}.$

We note that the real Lie algebra $\mathfrak{g}_{\mathbb{R}}$ is the Lie
algebra of the real Lie group $G_{\mathbb{R}}$ [Mil13, Theorem 2.1].

Let $\varphi\colon G\to H$ be a morphism of algebraic groups $G,H$. The
transposed map $\varphi^{\circ}\colon\mathbb{D}[H]\to\mathbb{D}[G]$ gives rise
to a linear map

$\displaystyle\operatorname{d}_{e}\\!\varphi\colon\mathfrak{g}$
$\displaystyle\to\mathfrak{h}$ $\displaystyle X$ $\displaystyle\mapsto
X\circ\varphi^{\circ}$

which is called the _differential_ $\operatorname{d}_{e}\\!\varphi$ of
$\varphi$ at $e\in G$.

The group $G$ acts on itself by conjugation $\operatorname{Int}(g)\colon G\to
G,h\mapsto ghg^{-1}$. The differential of $\operatorname{Int}(g)$ at $e$ is
denoted $\operatorname{Ad}(g)$. The morphism $\operatorname{Ad}\colon
G\to\operatorname{GL}(\mathfrak{g})$ is called the _adjoint representation of_
$G$ and is given by

$\operatorname{Ad}(g)(X)=gXg^{-1}\in T_{e}G$

when $X\in\mathfrak{g}$ is viewed as an element of $T_{e}G$. The differential
of $\operatorname{Ad}$ is called the _adjoint representation of
$\mathfrak{g}$_ and denoted by
$\operatorname{ad}\colon\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$.

###  4.4 Root systems and the spherical Weyl group

Let $G$ now be a semisimple algebraic group and $S$ a torus of $G$. Since
$\operatorname{Ad}(S)$ is also a torus, its elements are simultaneously
diagonalizable and the Lie algebra $\mathfrak{g}$ decomposes into eigenspaces

$\mathfrak{g}=\mathfrak{g}_{0}^{(S)}\oplus\bigoplus_{\alpha\neq
0}\mathfrak{g}_{\alpha}^{(S)}$

where

$\mathfrak{g}_{\alpha}^{(S)}=\\{X\in\mathfrak{g}\colon\operatorname{Ad}(s)(X)=\alpha(s)\cdot
X\ \text{ for all }s\in S\\}$

for some $\alpha\in\hat{S}$. Elements $\alpha\neq 0$ with
$\mathfrak{g}_{\alpha}^{(S)}\neq 0$, are called the _roots (relative to $S$)_
and $\mathfrak{g}_{\alpha}^{(S)}$ the _root spaces_. The set of all roots is
denoted by $\Phi(G,S)$. If $G$ is defined over $\mathbb{K}$ and $S$ is a
maximal $\mathbb{K}$-split torus of $G$, then
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi:=\Phi(G,S)$ is called the set of
$\mathbb{K}$-_roots_ of $G$. Since all maximal $\mathbb{K}$-split tori of $G$
are conjugate over $\mathbb{K}$ [Bor91, Theorem 20.9(ii)],
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$ only depends on $\mathbb{K}$ and not on
the choice of maximal $\mathbb{K}$-split torus $S$.

######  Theorem 4.7 (21.6 [Bor91]).

Let $G$ be a semisimple connected $\mathbb{K}$-group and $S$ a maximal
$\mathbb{K}$-split torus of $G$. Recall that $\hat{S}\cong\mathbb{Z}^{d}$. Let
$\Phi=\Phi(G,S)$. Then there is an admissible scalar product on the
$\mathbb{R}$-vector space $V=\hat{S}\otimes_{\mathbb{Z}}\mathbb{R}$ such that
($V$,$\Phi$) is a crystallographic root system.

If $S$ is a maximal $\mathbb{K}$-split torus, the _spherical Weyl group
relative to $\mathbb{K}$_ is

$\vphantom{W}{}_{\mathbb{K}}W=\vphantom{W}_{\mathbb{K}}W(S,G)=\operatorname{Nor}_{G}(S)/\operatorname{Cen}_{G}(S)$

and acts faithfully on $S$, $\hat{S}$ and
$\vphantom{\Phi}{}_{\mathbb{K}}\Phi$.

###  4.5 Borel subgroups, parabolic subgroups

Let $G$ be a connected algebraic group. A subgroup is _solvable_ if its
derived series consisting of iterated commutator groups terminates. A subgroup
$B<G$ is a _Borel subgroup_ of $G$ if it is maximal among the connected
solvable subgroups. A closed subgroup $P<G$ is _parabolic_ if and only if it
contains a Borel subgroup. The minimal parabolic subgroups are exactly the
Borel subgroups. In general, a Borel subgroup may not be defined over
$\mathbb{K}$ and in this case the minimal parabolic $\mathbb{K}$-subgroups may
not be Borel subgroups.

######  Theorem 4.8.

(Bruhat decomposition, [Bor91, Theorem 21.15]) Let $S$ be a maximal
$\mathbb{K}$-split torus and $P$ a minimal parabolic $\mathbb{K}$-subgroup
containing $S$. Denote by
$\pi\colon\operatorname{Nor}_{G}(S)\to\vphantom{W}_{\mathbb{K}}W$ the Weyl
group projection. Then
$G_{\mathbb{K}}=P_{\mathbb{K}}\operatorname{Nor}_{G_{\mathbb{K}}}(S_{\mathbb{K}})P_{\mathbb{K}}$,
in fact there is a disjoint union of double classes

$G_{\mathbb{K}}=\bigsqcup_{w\in\vphantom{W}_{\mathbb{K}}W}P_{\mathbb{K}}\pi^{-1}(w)P_{\mathbb{K}}.$

After choosing a minimal parabolic $\mathbb{K}$-subgroup $P$ containing a
maximal $\mathbb{K}$-split torus, any parabolic $\mathbb{K}$-subgroup
containing $P$ is called _standard parabolic_.

####  4.5.1 Examples

The group of diagonal matrices in
$\operatorname{SL}_{3}=\operatorname{SL}_{3}(\mathbb{D})$ is a maximal
$\mathbb{Q}$-split torus $T$. There are six minimal parabolic
$\mathbb{Q}$-subgroups (which in this case are Borel subgroups) containing
$T$. They are given by

$\displaystyle\left\\{\begin{pmatrix}\star&\star&\star\\\ 0&\star&\star\\\
0&0&\star\\\
\end{pmatrix}\in\operatorname{SL}_{3}\right\\},\quad\left\\{\begin{pmatrix}\star&0&\star\\\
\star&\star&\star\\\ 0&0&\star\\\
\end{pmatrix}\in\operatorname{SL}_{3}\right\\},\quad\left\\{\begin{pmatrix}\star&\star&\star\\\
0&\star&0\\\ 0&\star&\star\\\ \end{pmatrix}\in\operatorname{SL}_{3}\right\\},$
$\displaystyle\left\\{\begin{pmatrix}\star&0&0\\\ \star&\star&0\\\
\star&\star&\star\\\
\end{pmatrix}\in\operatorname{SL}_{3}\right\\},\quad\left\\{\begin{pmatrix}\star&0&0\\\
\star&\star&\star\\\ \star&0&\star\\\
\end{pmatrix}\in\operatorname{SL}_{3}\right\\},\quad\left\\{\begin{pmatrix}\star&\star&0\\\
0&\star&0\\\ \star&\star&\star\\\
\end{pmatrix}\in\operatorname{SL}_{3}\right\\}$

and correspond to the Weyl chambers on which the spherical Weyl group acts
transitively by conjugation. A corresponding set of representatives of
$\vphantom{W}{}_{\mathbb{K}}W$ is given by

$\displaystyle\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\end{pmatrix},\quad\begin{pmatrix}0&-1&0\\\ 1&\phantom{-}0&0\\\
0&\phantom{-}0&1\\\ \end{pmatrix},\quad\begin{pmatrix}1&0&\phantom{-}0\\\
0&0&-1\\\ 0&1&\phantom{-}0\\\ \end{pmatrix},$
$\displaystyle\begin{pmatrix}0&0&-1\\\ 0&1&\phantom{-}0\\\ 1&0&\phantom{-}0\\\
\end{pmatrix},\quad\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\\\
\end{pmatrix},\quad\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 1&0&0\\\ \end{pmatrix}.$

##  5 Results about Lie algebras and split tori

Let $G$ be a connected semisimple algebraic group defined over
$\mathbb{K}\subseteq\mathbb{R}$, which is invariant under transposition. The
Lie algebra $\mathfrak{g}=T_{e}G$ is defined by finitely many linear equations
with coefficients in $\mathbb{K}$ and we can therefore consider its
$\mathbb{K}$-points $\mathfrak{g}_{\mathbb{K}}\subseteq\mathbb{K}^{n\times
n}$, which is a $\mathbb{K}$-vector space and an algebraic (hence
semialgebraic) set. Let $\mathfrak{g}_{\mathbb{F}}$ be the semialgebraic
extension of $\mathfrak{g}_{\mathbb{K}}$ for any real closed field
$\mathbb{F}\supseteq\mathbb{K}$.

In this section, we recall facts about the real Lie group $G_{\mathbb{R}}$ and
its Lie algebra $\mathfrak{g}_{\mathbb{R}}$. We sometimes reference chapter 3
of S. Helgason’s book [Hel78] and chapter 6 of A. Knapp’s book [Kna96], a
compact account of which can be found in [Wis01].

###  5.1 Cartan involutions and Cartan decompositions

Recall that the _adjoint representation_ of the Lie algebra
$\mathfrak{g}_{\mathbb{R}}$

$\operatorname{ad}_{\mathfrak{g}_{\mathbb{R}}}:=\operatorname{d}_{e}\\!\operatorname{Ad}_{G_{\mathbb{R}}}\colon\mathfrak{g}_{\mathbb{R}}\to\mathfrak{gl}(\mathfrak{g}_{\mathbb{R}})$

is given by $X\mapsto[X,\cdot\,]$. The _Killing form_ $B$ is the bilinear form
on $\mathfrak{g}_{\mathbb{R}}$ defined by

$B(X,Y):=\operatorname{tr}\left(\operatorname{ad}(X)\circ\operatorname{ad}(Y)\right)$

for $X,Y\in\mathfrak{g}_{\mathbb{R}}$. A Lie algebra is called _simple_ if it
is non-abelian and does not contain any non-zero proper ideals. A Lie algebra
is called _semisimple_ if it is a direct product of simple Lie algebras. By
Cartan’s criterion, being semisimple is equivalent to having a non-degenerate
Killing form. Since $G_{\mathbb{R}}$ is semisimple, so is
$\mathfrak{g}_{\mathbb{R}}$. A Lie algebra automorphism
$\theta\colon\mathfrak{g}_{\mathbb{R}}\to\mathfrak{g}_{\mathbb{R}}$ is called
an _involution_ if $\theta^{2}=\operatorname{Id}$. For any involution
$\theta\colon\mathfrak{g}_{\mathbb{R}}\to\mathfrak{g}_{\mathbb{R}}$ we can
define a bilinear form $B_{\theta}$ by

$B_{\theta}(X,Y):=-B(X,\theta Y)$

for $X,Y\in\mathfrak{g}_{\mathbb{R}}$. If $B_{\theta}$ is positive-definite,
then $\theta$ is called a _Cartan involution_. The following is a technical
result on Cartan involutions which is stated as an exercise in [Hel78] and
proven in [Kna96, Theorem 6.16].

######  Lemma 5.1.

Let $\mathfrak{g}_{\mathbb{R}}$ be a real semisimple Lie algebra, $\theta$ a
Cartan involution and $\sigma$ any involution on $\mathfrak{g}_{\mathbb{R}}$.
Then there exists an automorphism
$\varphi\in\operatorname{Ad}(G_{\mathbb{R}})$ such that
$\varphi\theta\varphi^{-1}$ commutes with $\sigma$.

An application of the preceding Lemma is that all Cartan involutions are
conjugated, see [Kna96, Corollary 6.19].

######  Theorem 5.2.

Let $\mathfrak{g}_{\mathbb{R}}$ be a real semisimple Lie algebra. Any two
Cartan involutions of $\mathfrak{g}_{\mathbb{R}}$ are conjugate via an element
of $\operatorname{Ad}(G_{\mathbb{R}})$.

###### Proof.

Let $\theta$ and $\theta^{\prime}$ be two Cartan involutions of
$\mathfrak{g}_{\mathbb{R}}$. If $\theta$ and $\theta^{\prime}$ commute, they
have the same eigenspaces. We claim $E_{1}(\theta)=E_{1}(\theta^{\prime})$ and
$E_{-1}(\theta)=E_{-1}(\theta^{\prime})$, since otherwise if for instance
$\theta(X)=X$ and $\theta^{\prime}(X)=-X$, then

$B_{\theta^{\prime}}(X,X)=-B(X,\theta^{\prime}(X))=-B(X,-X)=B(X,\theta(X))=-B_{\theta}(X,X),$

but both $B_{\theta}(X,X)$ and $B_{\theta^{\prime}}(X,X)$ should be positive.
We conclude that if $\theta$ and $\theta^{\prime}$ commute, then
$\theta=\theta^{\prime}$, since $\theta$ and $\theta^{\prime}$ take the same
values on their eigenspaces. If $\theta$ and $\theta^{\prime}$ do not commute,
then we can apply Lemma 5.1 to find
$\varphi\in\operatorname{Ad}(G_{\mathbb{R}})$ such that
$\varphi\theta\varphi^{-1}$ commutes with $\varphi^{\prime}$, and therefore
$\varphi\theta\varphi^{-1}=\theta^{\prime}$ by the previous argument. ∎

A decomposition of $\mathfrak{g}_{\mathbb{R}}$ as a direct sum
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{p}.$ is called a
_Cartan decomposition_ if

$[\mathfrak{k},\mathfrak{k}]\subseteq\mathfrak{k},\quad[\mathfrak{k},\mathfrak{p}]\subseteq\mathfrak{p},\quad[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}$

and the Killing form $B$ is negative definite on $\mathfrak{k}$ and positive
definite on $\mathfrak{p}$. There is a correspondence between Cartan
involutions and Cartan decompositions. A Cartan involution $\theta$ defines a
decomposition into eigenspaces $\mathfrak{k}=E_{1}(\theta)$,
$\mathfrak{p}=E_{-1}(\theta)$ of $\mathfrak{g}_{\mathbb{R}}$. The bracket
relations can be checked using that $\theta$ commutes with the bracket
operation. Since then
$\operatorname{ad}(\mathfrak{k})\operatorname{ad}(\mathfrak{p})(\mathfrak{k})\subseteq\mathfrak{p}$
and
$\operatorname{ad}(\mathfrak{k})\operatorname{ad}(\mathfrak{p})(\mathfrak{p})\subseteq\mathfrak{k}$,
we have

$B(\mathfrak{k},\mathfrak{p})=\operatorname{Tr}(\operatorname{ad}(\mathfrak{k})\operatorname{ad}(\mathfrak{p}))=0$

and the decomposition is orthogonal. Since $\theta$ is a Cartan involution,
$B_{\theta}$ is positive definite, and thus $B$ is negative definite on
$\mathfrak{k}$ and positive definite on $\mathfrak{p}$. Starting from a Cartan
decomposition $\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{p}$, we
can define the involution

$\theta\colon X\mapsto\left\\{\begin{matrix}X&\text{ if }X\in\mathfrak{k}\\\
-X&\text{ if }X\in\mathfrak{p}\end{matrix}\right.$

which is compatible with the bracket operation and for which $B_{\theta}$ is
positive definite. To refine the decomposition further, we need the following
Lemma.

######  Lemma 5.3.

Let $\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{p}$ be a Cartan
decomposition with Cartan involution $\theta$ and let $X\in\mathfrak{p}$. The
map
$\operatorname{ad}(X)\colon\mathfrak{g}_{\mathbb{R}}\to\mathfrak{g}_{\mathbb{R}}$
is symmetric with respect to the scalar product $B_{\theta}$.

###### Proof.

We have

$\displaystyle B_{\theta}(\operatorname{ad}(X)(Y),Z)$
$\displaystyle=-B([X,Y],\theta(Z))=B(Y,[X,Z])=-B(Y,[\theta(X),\theta(Z)])$
$\displaystyle=-B(Y,\theta([X,Z]))=B_{\theta}(Y,\operatorname{ad}(X)Z)$

for all $X\in\mathfrak{p}$ and $Y,Z\in\mathfrak{g}_{\mathbb{R}}$. ∎

Let now $\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{p}$ be a fixed
Cartan decomposition with associated Cartan involution $\theta$. Let
$\mathfrak{a}\subseteq\mathfrak{p}$ be a maximal abelian subspace. We can
define the _real rank of $G_{\mathbb{R}}$_,
$\operatorname{rank}_{\mathbb{R}}(G_{\mathbb{R}}):=\operatorname{dim}(\mathfrak{a})$,
which is independent of $\mathfrak{p}$ as a consequence of Theorem 5.2 and
independent of $\mathfrak{a}$ since any two maximal abelian subspaces are
conjugate to each other [Kna96, Theorem 6.51]. The set
$\\{\operatorname{ad}(H)\colon H\in\mathfrak{a}\\}$ consists of symmetric,
hence diagonalizable linear maps. Since they all commute, they are
simultaneously diagonalizable. This results in the decomposition

$\mathfrak{g}_{\mathbb{R}}=\mathfrak{g}_{0}\oplus\bigoplus_{\alpha\in\Sigma}\mathfrak{g}_{\alpha}$

where for every $\alpha\in\mathfrak{a}^{\star}$ in the dual space
$\mathfrak{a}^{\star}$ of $\mathfrak{a}$

$\mathfrak{g}_{\alpha}=\left\\{X\in\mathfrak{g}_{\mathbb{R}}\colon[H,X]=\alpha(H)\cdot
X\text{ for all }H\in\mathfrak{a}\right\\}$

and where $\Sigma=\\{\alpha\in\mathfrak{a}^{\star}\colon\alpha\neq 0\text{ and
}\mathfrak{g}_{\alpha}\neq 0\\}$. The elements of $\Sigma$ are called
_restricted roots_ and $\mathfrak{g}_{\alpha}$ their associated _restricted
root spaces_. Note that in contrast to the decomposition of complex Lie
algebras, the root spaces may not be one-dimensional444In the real semisimple
Lie algebra $\mathfrak{sl}_{2}(\mathbb{C})$ we have
$\operatorname{dim}(\mathfrak{g}_{\alpha})=2$.. In Section 4.4 we defined the
root system of an algebraic group with a maximal split torus. In Section 6.2
we will show that these two approaches give the same root spaces and root
systems.

######  Proposition 5.4.

([Kna96, Proposition 6.40]) The _restricted root space decomposition_ is an
orthogonal direct sum with respect to $B_{\theta}$ and satisfies for all
$\alpha,\beta\in\Sigma$

  * (i)

$[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]\subseteq\mathfrak{g}_{\alpha+\beta}$,

  * (ii)

$\theta(\mathfrak{g}_{\alpha})=\mathfrak{g}_{-\alpha}$ and

  * (iii)

$\mathfrak{g}_{0}=\mathfrak{a}\oplus\mathfrak{z}_{\mathfrak{k}}(\mathfrak{a})$,
where
$\mathfrak{z}_{\mathfrak{k}}(\mathfrak{a})=\left\\{X\in\mathfrak{k}\colon[H,X]=0\text{
for all }H\in\mathfrak{a}\right\\}$.

The inner product $B_{\theta}$ may be restricted to $\mathfrak{a}$ and used to
set up an isomorphism $\mathfrak{a}^{\star}\cong\mathfrak{a}$ which turns
$\mathfrak{a}^{\star}$ into a Euclidean vector space.

######  Theorem 5.5.

([Kna96, Corollary 6.53]) The set of roots $(\Sigma,\mathfrak{a}^{\star})$ is
a crystallographic root system555$\Sigma$ may not be reduced, for example when
$G=\operatorname{SU}(2,1)$. .

By choosing an ordered basis of the root system $\Sigma$, we can define the
set of positive roots $\Sigma_{>0}$. Then

$\mathfrak{n}=\bigoplus_{\alpha\in\Sigma_{>0}}\mathfrak{g}_{\alpha}$

is a nilpotent subalgebra of $\mathfrak{g}_{\mathbb{R}}$. We have the
following properties.

######  Lemma 5.6.

([Kna96, Lemma 6.45]) Let $\mathfrak{g}_{\mathbb{R}}$ be a real semisimple Lie
algebra. There exists a basis of $\mathfrak{g}_{\mathbb{R}}$ such that the
matrices representing $\operatorname{ad}(\mathfrak{g}_{\mathbb{R}})$ have the
following properties

  1. 1.

The matrices of $\operatorname{ad}(\mathfrak{k})$ are skew-symmetric.

  2. 2.

The matrices of $\operatorname{ad}(\mathfrak{a})$ are diagonal.

  3. 3.

The matrices of $\operatorname{ad}(\mathfrak{n})$ are upper triangular with
$0$ on the diagonal.

We also have the Iwasawa decomposition on the level of Lie algebras.

######  Theorem 5.7.

([Kna96, Proposition 6.43]) Let $\mathfrak{g}_{\mathbb{R}}$ be a semisimple
Lie algebra. Then
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n}$
is a direct sum.

The following lemma about the Lie subalgebra

$\mathfrak{l}:=(\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha})\oplus(\mathfrak{g}_{-\alpha}\oplus\mathfrak{g}_{-2\alpha})\oplus([\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]+[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}])$

for some $\alpha\in\Sigma$ will be useful when we consider certain rank 1
subgroups in Section 6.9. Note that by Cartan’s criterion, $\mathfrak{l}$ is
semisimple, as the Killing form is the restriction of the definite Killing
form of $\mathfrak{g}$. Then
$\mathfrak{k}\cap\mathfrak{l}\oplus\mathfrak{p}\cap\mathfrak{l}$ is a Cartan
decomposition of $\mathfrak{l}$.

######  Lemma 5.8.

Let $\alpha\in\Sigma$ and $X\in\mathfrak{g}_{\alpha}\setminus\\{0\\}$. Then
the real rank $\operatorname{rank}_{\mathbb{R}}$ of $\mathfrak{l}$ is one. A
maximal abelian subspace of the symmetric part of $\mathfrak{l}$ is given by
$\mathfrak{a}\cap\mathfrak{l}=\langle[X,\theta(X)]\rangle$.

###### Proof.

We use the fact that $B_{\theta}$ gives us a scalar product on $\mathfrak{a}$
allowing us to identify $\mathfrak{a}\cong\mathfrak{a}^{\star}$, sending
$\alpha\in\Sigma$ to $H_{\alpha}$ defined by
$\alpha(H)=B_{\theta}(H_{\alpha},H)$ for all $H\in\mathfrak{a}$. Let
$X\in\mathfrak{g}_{\alpha},Y\in\mathfrak{g}_{-\alpha}$ and $H\in\mathfrak{a}$.
Then

$\displaystyle B_{\theta}([X,Y],H)$
$\displaystyle=-B([X,Y],\theta(H))=B([X,Y],H)=-B(Y,[X,H])$
$\displaystyle=B(Y,[H,X])=B(Y,\alpha(H)X)=\alpha(H)B(Y,X)$
$\displaystyle=B_{\theta}(H_{\alpha},H)B(X,Y)=B_{\theta}(B(X,Y)\cdot
H_{\alpha},H),$

where we used that $B$ is $\operatorname{ad}(X)$-invariant. The element

$W=[X,Y]-B(X,Y)\cdot H_{\alpha}\in\mathfrak{g}_{0}$

satisfies $B_{\theta}(W,H)=0$ for all $H\in\mathfrak{a}$, hence lies in
$\mathfrak{g}_{0}$ perpendicular to $\mathfrak{a}$, hence
$W\in\mathfrak{z}_{\mathfrak{k}}(\mathfrak{a})\subseteq\mathfrak{k}$ by
Proposition 5.4. Similarly, one can show that for any
$X^{\prime}\in\mathfrak{g}_{2\alpha}$ and
$Y^{\prime}\in\mathfrak{g}_{-2\alpha}$,
$[X^{\prime},Y^{\prime}]=W^{\prime}+2B(X^{\prime},Y^{\prime})\cdot H_{\alpha}$
for some $W^{\prime}\in\mathfrak{z}_{\mathfrak{k}}(\mathfrak{a})$. Any element
$Z\in\mathfrak{a}\cap\mathfrak{l}$ lies in
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]\oplus[\mathfrak{g}_{2\alpha},\mathfrak{g}_{-2\alpha}]$
since $\mathfrak{a}\subseteq\mathfrak{g}_{0}$ is orthogonal to
$\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha}\oplus\mathfrak{g}_{-\alpha}\oplus\mathfrak{g}_{-2\alpha}$.
Therefore, $Z$ can be written as
$Z=W+W^{\prime}+(B(X,Y)+2B(X^{\prime},Y^{\prime}))\cdot
H_{\alpha}\in\mathfrak{z}_{\mathfrak{k}}(\mathfrak{a})\oplus\mathfrak{a}$,
hence $W+W^{\prime}=0$ and $Z\in\langle H_{\alpha}\rangle$.

If $Y=\theta(X)$, then

$\theta(W)=\theta([X,\theta(X)]-B(X,Y)\cdot
H_{\alpha})=[\theta(X),X]-\theta(H_{\alpha})B(X,Y)=-W$

which implies $W\in\mathfrak{p}\cap\mathfrak{k}$, hence $W=0$. We have shown
that $[X,\theta(X)]\in\mathfrak{a}\cap\mathfrak{l}\subseteq\langle
H_{\alpha}\rangle$. Since $B$ is definite, $[X,\theta(X)]\neq 0$ and
$\mathfrak{a}\cap\mathfrak{l}=\langle[X,\theta(X)]\rangle$.

The map $\theta|_{\mathfrak{l}}$ is a Cartan-involution with Cartan
decomposition
$\mathfrak{l}=\mathfrak{k}\cap\mathfrak{l}\oplus\mathfrak{p}\cap\mathfrak{l}$.
There is a maximal abelian subspace of $\mathfrak{p}\cap\mathfrak{l}$
contained in $\mathfrak{a}$, which is equal to
$\mathfrak{a}\cap\mathfrak{l}=\langle[X,\theta(X)]\rangle$ by the above. The
dimension of a maximal abelian subspace of $\mathfrak{p}\cap\mathfrak{l}$ is
exactly the rank of $\mathfrak{l}$ and it is one. ∎

The following is a special case of the Jacobson-Morozov Lemma in the
literature. Note that this Lemma does not work for general
$X\in\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{2\alpha}$.

######  Lemma 5.9.

Let $\alpha\in\Sigma$ and $X\in\mathfrak{g}_{\alpha}\setminus\\{0\\}$. Then
there is a $Y\in\mathfrak{g}_{-\alpha}\setminus\\{0\\}$ and $H\in\mathfrak{a}$
such that $(X,Y,H)$ forms a $\mathfrak{sl}(2)$-triplet, i.e.

$[X,Y]=H,\quad[H,X]=2X,\quad[H,Y]=2Y.$

###### Proof.

Let $H_{\alpha}\in\mathfrak{a}$ be as in the proof of Lemma 5.8 defined by
$\alpha(H)=B_{\theta}(H_{\alpha},H)$ for all $H\in\mathfrak{a}$. Given
$X\in\mathfrak{g}_{\alpha}$, let

$Y:=\frac{-2}{B_{\theta}(X,X)\cdot\alpha(H_{\alpha})}\theta(X)\in\mathfrak{g}_{-\alpha}$

and $H:=[X,Y]$. In the proof of Lemma 5.8 we saw that

$[X,\theta(X)]=B(X,\theta(X))\cdot H_{\alpha}=-B_{\theta}(X,X)\cdot
H_{\alpha},$

whence $H\in\mathfrak{a}$ and moreover

$\displaystyle[H,X]$
$\displaystyle=\left[\left[X,\frac{-2}{B_{\theta}(X,X)\alpha(H_{\alpha})}\theta(X)\right],X\right]$
$\displaystyle=\frac{2}{\alpha(H_{\alpha})}\left[H_{\alpha},X\right]=\frac{2}{\alpha(H_{\alpha})}\alpha(H_{\alpha})X=2X$

and similarly $[H,Y]=-2Y$. ∎

####  5.1.1 Examples

Since $G_{\mathbb{R}}$ is invariant under transposition, $\sigma\colon
g\mapsto(g^{-1}){{}^{\mathsf{T}}\\!}$ is an involution of the Lie group whose
differential $\theta:=\operatorname{d}_{e}\\!\sigma\colon
X\mapsto-X{{}^{\mathsf{T}}\\!}$ is an involution on the Lie algebra
$\mathfrak{g}_{\mathbb{R}}$ which decomposes
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{p}\oplus\mathfrak{k}$ into a symmetric
part $\mathfrak{p}$ and an skew-symmetric part $\mathfrak{k}$. One can check
that

$[\mathfrak{k},\mathfrak{k}]\subseteq\mathfrak{k},\quad[\mathfrak{k},\mathfrak{p}]\subseteq\mathfrak{p}\quad\text{
and }\quad[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}$

and that the Killing form is negative definite on $\mathfrak{k}$ and positive
definite on $\mathfrak{p}$. Thus
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{k}\oplus\mathfrak{p}$ is a Cartan
decomposition and $\theta$ is a Cartan involution.

###  5.2 Real Cartan subalgebras and $\mathbb{R}$-split subalgebras

In this section we discuss Cartan subalgebras of a finite dimensional
semisimple real Lie algebra $\mathfrak{g}$. These results apply in particular
to the Lie algebra $\mathfrak{g}_{\mathbb{R}}$ of the real Lie group
$G_{\mathbb{R}}$.

An abelian subalgebra $\mathfrak{h}\subseteq\mathfrak{g}$ is called a _Cartan
subalgebra_ 666More generally, a Cartan subalgebra is defined to be a
nilpotent self-normalizing subalgebra. In our context, we restrict to
semisimple Lie algebras over fields of characteristic $0$, in which case
Cartan subalgebras are abelian.  if

$\mathfrak{h}=\operatorname{Nor}_{\mathfrak{g}}(\mathfrak{h}):=\left\\{X\in\mathfrak{g}\colon[X,Y]\in\mathfrak{h}\text{
for all }Y\in\mathfrak{h}\right\\}.$

An abelian subalgebra $\mathfrak{a}\subseteq\mathfrak{g}$ is called
_$\mathbb{R}$ -split_ if
$\operatorname{ad}(X)\colon\mathfrak{g}\to\mathfrak{g}$ is diagonalizable over
$\mathbb{R}$ for every $X\in\mathfrak{a}$. Let $r_{\mathbb{R}}(\mathfrak{g})$
be the maximal dimension of such an $\mathbb{R}$-split abelian subalgebra and
denote

$V(\mathfrak{g}):=\\{\mathfrak{a}\subseteq\mathfrak{g}\colon\text{$\mathfrak{a}$
is an $\mathbb{R}$-split abelian subalgebra with
$\dim(\mathfrak{a})=r_{\mathbb{R}}(\mathfrak{g})$}\\}.$

A _maximally $\mathbb{R}$-split Cartan subalgebra_ is a Cartan subalgebra
containing an element of $V(\mathfrak{g}_{\mathbb{R}})$ as a subset. Let

$\mathcal{C}(\mathfrak{g})=\left\\{\mathfrak{h}\subseteq\mathfrak{g}\colon\mathfrak{h}\text{
is a maximally $\mathbb{R}$-split Cartan subalgebra }\right\\}.$

Knapp [Kna96] uses slightly different definitions and names for these objects.
We will show here that they coincide. We start by showing that our notion of
Cartan subalgebra coincides with the notion in [Kna96].

######  Lemma 5.10.

Let $\mathfrak{h}$ be a subalgebra of a finite dimensional real semisimple Lie
algebra $\mathfrak{g}$. Then $\mathfrak{h}$ is a Cartan subalgebra if and only
if it satisfies one of the following conditions

  1. (i)

$\mathfrak{h}$ is abelian and
$\operatorname{Nor}_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}$.

  2. (ii)

The complexification $\mathfrak{h}_{\mathbb{C}}:=\mathfrak{h}\oplus
i\mathfrak{h}$ is abelian and
$\operatorname{Nor}_{\mathfrak{g}_{\mathbb{C}}}(\mathfrak{h}_{\mathbb{C}})=\mathfrak{h}_{\mathbb{C}}$.

  3. (iii)

The complexification $\mathfrak{h}_{\mathbb{C}}$ is nilpotent and
$\operatorname{Nor}_{\mathfrak{g}_{\mathbb{C}}}(\mathfrak{h}_{\mathbb{C}})=\mathfrak{h}_{\mathbb{C}}$.

  4. (iv)

The complexification $\mathfrak{h}_{\mathbb{C}}$ is nilpotent and

$\mathfrak{h}_{\mathbb{C}}=\left\\{X\in\mathfrak{g}_{\mathbb{C}}\colon\exists
n\in\mathbb{N},\ \forall H\in\mathfrak{h}_{\mathbb{C}},\
\operatorname{ad}(H)^{n}X=0\right\\}.$

  5. (v)

The complexification $\mathfrak{h}_{\mathbb{C}}$ is maximal abelian and the
subset
$\operatorname{ad}(\mathfrak{h}_{\mathbb{C}})\subseteq\mathfrak{gl}(\mathfrak{g}_{\mathbb{C}})$
is simultaneously diagonalizable.

###### Proof.

Notion (i) is our definition of Cartan subalgebra. For any
$X=X_{1}+iX_{2}\in\mathfrak{g}_{\mathbb{C}}$ and
$Y=Y_{1}+iY_{2}\in\mathfrak{g}_{\mathbb{C}}$, we have

$\operatorname{ad}(Y)X=[Y_{1}+iY_{2},X_{1}+iX_{2}]=[Y_{1},X_{1}]-[Y_{2},X_{2}]+i([Y_{1},X_{2}]+[Y_{2},X_{1}]).$

From this formula we can deduce that a subalgebra of a real Lie algebra is
abelian if and only if its complexification is abelian. It also follows that
for any real subalgebra $\mathfrak{h}\subseteq\mathfrak{g}$ we have

$\operatorname{Nor}_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}\iff\operatorname{Nor}_{\mathfrak{g}_{\mathbb{C}}}(\mathfrak{h}_{\mathbb{C}})=\mathfrak{h}_{\mathbb{C}}.$

It follows that notions (i) and (ii) coincide. For (ii) implies (iii), it
suffices to notice that every abelian subalgebra is nilpotent, the converse
uses semisimplicity and is given in [Kna96, Proposition 2.10]. Condition (iv)
is the definition used in [Kna96] and the equivalence of (iii) and (iv) is
given by [Kna96, Proposition 2.7]. The equivalence of (iv) and (v) is given by
[Kna96, Corollary 2.13]. ∎

A subset $\mathfrak{h}\subseteq\mathfrak{g}$ is called _$\theta$ -stable_ if
there is a Cartan involution $\theta$ such that
$\theta(\mathfrak{h})=\mathfrak{h}$. If $\theta$ is given by the context,
$\theta$-stable refers to that particular Cartan involution.

######  Proposition 5.11.

([Kna96, Prop 6.59]) Let $\mathfrak{g}$ be the Lie algebra of a real
semisimple Lie group $G$ and $\operatorname{Ad}(G)^{\circ}$ the connected
component of $\operatorname{Ad}(G)\subseteq\operatorname{GL}(\mathfrak{g})$ in
the Lie group topology. Any Cartan subalgebra $\mathfrak{h}$ is conjugate via
$\operatorname{Ad}(G)^{\circ}$ to a $\theta$-stable Cartan subalgebra.

If $\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{k}$ is the Cartan decomposition
associated to a Cartan involution $\theta$, and $\mathfrak{h}$ is a
$\theta$-stable Cartan subalgebra, then
$\mathfrak{a}:=\mathfrak{h}\cap\mathfrak{p}$ and
$\mathfrak{t}:=\mathfrak{h}\cap\mathfrak{k}$ are $\theta$-stable and
$\mathfrak{h}=\mathfrak{a}\oplus\mathfrak{t}$. A Cartan subalgebra is called
_maximally noncompact_ if $\dim(\mathfrak{a})$ is maximal. While all complex
Cartan subalgebras are conjugated [Kna96, Theorem 2.15], for real Cartan
subalgebras, the dimensions of the spaces $\mathfrak{a}$ and $\mathfrak{t}$
are preserved.

######  Proposition 5.12.

([Kna96, Prop 6.61]) Let $\mathfrak{g}$ be the Lie algebra of a semisimple
real Lie group $G$ and let $K$ be a subgroup with Lie algebra $\mathfrak{k}$
where $\mathfrak{g}=\mathfrak{p}\oplus\mathfrak{k}$ is the Cartan
decomposition associated to a Cartan involution $\theta$.

All maximally noncompact $\theta$-stable Cartan subalgebras are conjugate
under $\operatorname{Ad}(K)^{\circ}$.

Together, Propositions 5.11 and 5.12 allow us to extend our definition of
maximally noncompact $\theta$-stable Cartan subalgebras to all Cartan
subalgebras, by stipulating that a Cartan subalgebra is called _maximally
noncompact_ if it is conjugated to a $\theta$-stable maximally noncompact
Cartan subalgebra. We will now show that the set of maximally
$\mathbb{R}$-split Cartan subalgebras $\mathcal{C}(\mathfrak{g})$ coincides
with the set of maximally noncompact Cartan subalgebras.

######  Proposition 5.13.

For any maximally $\mathbb{R}$-split subalgebra
$\mathfrak{a}\subseteq\mathfrak{g}$ of a finite dimensional real semisimple
Lie algebra $\mathfrak{g}$, there is a maximally noncompact Cartan subalgebra
$\mathfrak{h}\supseteq\mathfrak{a}$. Every maximally noncompact Cartan
subalgebra contains a maximally $\mathbb{R}$-split subalgebra $\mathfrak{a}$
and if $\mathfrak{h}$ is $\theta$-stable, then
$\mathfrak{a}=\mathfrak{h}\cap\mathfrak{p}$ and
$\mathfrak{h}=\mathfrak{a}\oplus\mathfrak{t}$ in the decomposition above and
$r_{\mathbb{R}}(\mathfrak{g})=\dim(\mathfrak{a})$.

###### Proof.

We introduce a few concepts. If $k$ is a field and $\mathfrak{g}$ is a
semisimple $k$-Lie algebra, then an abelian subalgebra $\mathfrak{h}$ is
called _toral_ if
$\operatorname{ad}(\mathfrak{h})\subseteq\mathfrak{gl}(\mathfrak{g})$ consists
of linear maps that are diagonalizable over the algebraic closure of $k$. If
all the elements of $\operatorname{ad}(\mathfrak{h})$ are diagonalizable over
$k$ itself, then $\mathfrak{h}$ is called _$k$ -split toral_. If moreover
$\mathfrak{h}$ is maximal among all $k$-split toral subalgebras,
$\mathfrak{h}$ is called _maximal $k$-split toral_.

Let now $\mathfrak{g}$ be a finite dimensional real semisimple Lie algebra.
Let $\mathfrak{a}\in V(\mathfrak{g})$, in our terminology $\mathfrak{a}$ is
maximal $\mathbb{R}$-split toral. Let $\mathfrak{h}$ be maximal toral
containing $\mathfrak{a}$, which exists since $\mathfrak{g}$ is finite
dimensional. We want to show that $\mathfrak{h}$ is a Cartan subalgebra and
use characterization (v) of Lemma 5.10. Since $\mathfrak{h}$ is abelian, so is
its complexification $\mathfrak{h}_{\mathbb{C}}$. Since $\mathfrak{h}$ is
$\mathbb{C}$-split, all elements of
$\operatorname{ad}(\mathfrak{h}_{\mathbb{C}})=\operatorname{ad}(\mathfrak{h})\oplus
i\operatorname{ad}(\mathfrak{h})$ are diagonalizable over $\mathbb{C}$. This
means that $\mathfrak{h}_{\mathbb{C}}$ is a toral subalgebra of
$\mathfrak{g}_{\mathbb{C}}$ and by a general linear algebra fact,
$\operatorname{ad}(\mathfrak{h}_{\mathbb{C}})$ is simultaneously
diagonalizable. The complexification $\mathfrak{h}_{\mathbb{C}}$ is also
maximal abelian, since for all $X=X_{1}+iX_{2}\in\mathfrak{g}_{\mathbb{C}}$,
if $[X,\mathfrak{h}_{\mathbb{C}}]=0$, then we have in particular for
$H\in\mathfrak{h}$

$0=[X,H]=[X_{1}+iX_{2},H]=[X_{1},H]+i[X_{2},H],$

hence $X_{1},X_{2}\in\mathfrak{h}$, so $X\in\mathfrak{h}_{\mathbb{C}}$. This
shows that $\mathfrak{h}$ is a Cartan subalgebra by characterization (v).

By Proposition 5.11, there is a Cartan involution $\theta$ and a $g\in
G_{\mathbb{R}}$ such that $\operatorname{Ad}(g)(\mathfrak{h})$ is a
$\theta$-invariant Cartan subalgebra with decomposition
$\operatorname{Ad}(g)(\mathfrak{h})=\mathfrak{a}^{\prime}\oplus\mathfrak{t}^{\prime}$
as described before. Since $\mathfrak{a}^{\prime}\subseteq\mathfrak{p}$, it is
$\mathbb{R}$-split. In fact, for
$X=X_{\mathfrak{a}^{\prime}}+X_{\mathfrak{t}^{\prime}}\in\mathfrak{a}^{\prime}\oplus\mathfrak{t}^{\prime}$,
$\operatorname{ad}(X)$ is only diagonalizable over $\mathbb{R}$ if
$X_{\mathfrak{t}^{\prime}}=0$. Since diagonalizability is preserved under
conjugation, $\operatorname{Ad}(\mathfrak{a})\subseteq\mathfrak{a}^{\prime}$
and by maximality of $\mathfrak{a}$, we have
$\operatorname{Ad}(\mathfrak{a})=\mathfrak{a}^{\prime}$. Thus $\mathfrak{h}$
is maximally noncompact.

If we instead start with a maximally noncompact Cartan subalgebra
$\mathfrak{h}\subseteq\mathfrak{g}$, we similarly obtain (possibly using a
conjugation to a $\theta$-stable one) an $\mathbb{R}$-split subalgebra
$\mathfrak{a}$ with $\mathfrak{h}=\mathfrak{a}\oplus\mathfrak{t}$. A priori
$\mathfrak{a}$ is maximal $\mathbb{R}$-split as a subalgebra of
$\mathfrak{h}$, but if $\mathfrak{a}$ was contained in a larger subalgebra
$\mathfrak{a}^{\prime}$ maximal $\mathbb{R}$-split in $\mathfrak{g}$, then the
above construction would result in a Cartan subalgebra
$\mathfrak{h}^{\prime}=\mathfrak{a}^{\prime}\oplus\mathfrak{t}^{\prime}$ with
$\dim(\mathfrak{a}^{\prime})>\dim(\mathfrak{a})$, which is impossible, since
we assumed $\mathfrak{h}$ to be maximally noncompact, which means
$\dim(\mathfrak{a})$ is maximal. By definition, $r_{\mathbb{R}}(\mathfrak{g})$
is the maximal dimension of an $\mathbb{R}$-split abelian subalgebra, so
$r_{\mathbb{R}}(\mathfrak{g})=\dim(\mathfrak{a})$. ∎

We now turn back to the Lie algebra $\mathfrak{g}_{\mathbb{R}}$ of the
$\mathbb{R}$-points $G_{\mathbb{R}}$ of the self-adjoint semisimple algebraic
group $G$. We now collect a few properties of $V(\mathfrak{g}_{\mathbb{R}})$
and $\mathcal{C}(\mathfrak{g}_{\mathbb{R}})$.

######  Lemma 5.14.

Recall that for the algebraic group $G$, $\operatorname{rank}_{\mathbb{R}}(G)$
is the maximal dimension of any abelian subspace of
$\mathfrak{p}\subseteq\mathfrak{g}_{\mathbb{R}}$, where
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{p}\oplus\mathfrak{k}$ is the Cartan
decomposition associated to the Cartan involution
$X\mapsto-X{{}^{\mathsf{T}}\\!}$. We have

  * (1)

$r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})=\operatorname{rank}_{\mathbb{R}}(G)$.

  * (2)

$G_{\mathbb{R}}$ acts transitively on $\mathcal{C}(\mathfrak{g}_{\mathbb{R}})$
and $V(\mathfrak{g}_{\mathbb{R}})$.

  * (3)

$\mathcal{C}(\mathfrak{g}_{\mathbb{R}})$ contains a $\theta$-stable element.

###### Proof.

A maximally $\mathbb{R}$-split Cartan subalgebra, or by Proposition 5.13 a
maximally noncompact Cartan subalgebra can be obtained by starting with a
Cartan involution $\theta$, taking $\mathfrak{a}\subseteq\mathfrak{p}$ a
maximal abelian subspace of $\mathfrak{p}$ and taking
$\mathfrak{t}\subseteq\operatorname{Cen}_{\mathfrak{k}}(\mathfrak{a})$ a
maximal abelian subspace of $\operatorname{Cen}_{\mathfrak{k}}(\mathfrak{a})$.
Then $\mathfrak{h}=\mathfrak{a}\oplus\mathfrak{t}$ is a $\theta$-stable
maximally noncompact Cartan subalgebra, see [Kna96, Proposition 6.47]. This
shows (3).

By Proposition 5.13,
$r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})=\dim(\mathfrak{a})$ and since
$\mathfrak{a}$ is a maximal abelian subspace of $\mathfrak{p}$,
$r_{\mathbb{R}}(\mathfrak{g}_{\mathbb{R}})=\operatorname{rank}_{\mathbb{R}}(G)$,
showing (1).

By Propositions 5.11 and 5.12, $\operatorname{Ad}(G_{\mathbb{R}})$ and thus
$G_{\mathbb{R}}$ act transitively on $\mathcal{C}(\mathfrak{g}_{\mathbb{R}})$.
Next, let $\mathfrak{a},\mathfrak{a}^{\prime}\in
V(\mathfrak{g}_{\mathbb{R}})$. By Proposition 5.13, there are maximally
noncompact Cartan subalgebras $\mathfrak{h},\mathfrak{h}^{\prime}$ such that
$\mathfrak{a}\subseteq\mathfrak{h}$ and
$\mathfrak{a}^{\prime}\subseteq\mathfrak{h}^{\prime}$. By Proposition 5.12,
both $\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ are conjugated to a
$\theta$-stable maximally noncompact Cartan subalgebra
$\mathfrak{h}^{\prime\prime}$. By Proposition 5.13, the corresponding
conjugates of $\mathfrak{a}$ and $\mathfrak{a}^{\prime}$ coincide with the
intersection $\mathfrak{h}^{\prime\prime}\cap\mathfrak{p}$, hence with each
other. This shows that $\operatorname{Ad}(G_{\mathbb{R}})$ acts transitively
on $V(\mathfrak{g}_{\mathbb{R}})$ and concludes the proof of (2). ∎