Dissertation Finiteness properties of S-arithmetic subgroups of Chevalley groups in characteristic 0 von Lovis Yannik Kirschner zur Erlangung des Grades eines Dr. rer. nat. Abgabe: 25.3.2025 Betreuung: Prof. Dr. Stefan Witzel Fachbereich Mathematik und Informatik, Physik, Geographie Justus-Liebig-Universität Gieÿen Abstract We consider in this thesis S-arithmetic subgroups of certain algebraic ma- trix groups de�ned over Q. The simplest example of such a group is Γ = SLn(Z[1/p]). Each of these groups is of type F∞ by a well-known result of Borel and Serre. On a formal level, this means that there is a K(Γ, 1) com- plex with �nite m-skeleton for every m ∈ N. A nice consequence is that Γ is �nitely presented. While the method of Borel and Serre is more algebraic, we give here a new, purely geometric, proof that uses Morse theory. Doing so, we �rst develop the terminology of a Morse function without crit- ical values greater than a constant r > 0, which is de�ned on the product of a Riemannian manifold and a metric space. After that, we deduce some properties from the reduction theory of S-arithmetic groups, which we trans- late into geometric terms to a space X, on which our group acts canonically. Finally, we construct a real-valued function on that space. We show that this is a Morse function in the sense above. From that we deduce the statement concerning the �niteness properties of the group. Zusammenfassung Wir betrachten in dieser Dissertation S-arithmetische Untergruppen von be- stimmten algebraischen Matrixgruppen über Q. Das einfachste Beispiel einer solchen ist Γ = SLn(Z[1/p]). Nach einem bekannten Resultat von Borel und Serre ist jede dieser Gruppen vom Typ F∞. Rein formal bedeutet das, dass es einen K(Γ, 1)-Komplex gibt, der endliches m-Skelett hat für jede natürliche Zahl m ∈ N. Eine schöne Folgerung daraus ist, dass Γ endlich präsentiert ist. Während die Methode von Borel und Serre algebraisch ist, geben wir hier einen neuen, rein geometrischen Beweis, der Morse-Theorie benutzt. Dazu entwickeln wir zunächst den Begri� einer Morse-Funktion ohne kri- tische Werte oberhalb einer Konstante r > 0, die auf dem Produkt einer Riemannschen Mannigfaltigkeit und eines metrischen Raumes de�niert ist. Danach folgern wir einige Eigenschaften aus der Reduktionstheorie S-arith- metischer Gruppen und übersetzen diese in geometrische Form auf einen Raum X, auf dem unsere Gruppe kanonisch wirkt. Schlieÿlich konstruieren wir eine reellwertige Funktion auf diesem Raum und zeigen, dass sie die Kri- terien einer Morse-Funktion wie oben erfüllt. Daraus leiten wir die Aussage über die Endlichkeitseigenschaften der Gruppe ab. 1 Contents 1 Introduction 4 I Fundamentals and background material 7 2 Proper maps and actions 7 3 Metric spaces of non-positive curvature 10 3.1 CAT(0) spaces and Busemann functions . . . . . . . . . . . . 10 3.2 Products of metric spaces . . . . . . . . . . . . . . . . . . . . 15 4 Finiteness properties 17 5 Real and complex manifolds 20 5.1 Smooth manifolds, vector �elds and �ows . . . . . . . . . . . 20 5.2 Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . 27 6 Lie algebras 31 7 Lie groups 32 7.1 Real Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 32 7.2 Complex Lie groups . . . . . . . . . . . . . . . . . . . . . . . 34 8 Riemannian manifolds 36 8.1 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . 36 8.2 Vector �elds de�ned on Riemannian manifolds . . . . . . . . . 39 9 Root systems 43 10 A�ne algebraic varieties 46 11 Algebraic matrix groups 49 11.1 Algebraic matrix groups as complex linear groups . . . . . . . 49 11.2 Algebraic matrix groups as functors . . . . . . . . . . . . . . . 58 11.3 The Iwasawa decomposition of a real algebraic matrix group . 60 2 12 Adeles, adelic groups and S-arithmetic groups 61 12.1 Absolute values . . . . . . . . . . . . . . . . . . . . . . . . . . 61 12.2 Adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 12.3 Adelic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 12.4 S-arithmetic groups . . . . . . . . . . . . . . . . . . . . . . . 64 13 Euclidean buildings 66 13.1 Buildings as chamber complexes . . . . . . . . . . . . . . . . . 66 13.2 Euclidean buildings as geometric objects . . . . . . . . . . . . 67 13.3 Algebraic groups and Euclidean buildings . . . . . . . . . . . 69 II Geometric proof of the Borel-Serre theorem 70 14 Morse theory on two factors 70 14.1 Morse theory in the non compact case . . . . . . . . . . . . . 71 14.2 Vector �elds and �ows . . . . . . . . . . . . . . . . . . . . . . 73 14.3 Morse theory on products . . . . . . . . . . . . . . . . . . . . 83 15 Conventions and notation 87 15.1 Conventions for the algebraic matrix group . . . . . . . . . . 87 15.2 Notation about the groups . . . . . . . . . . . . . . . . . . . . 90 15.3 Notation about the geometric space . . . . . . . . . . . . . . . 93 16 Reduction theory 95 16.1 Reduction theory of S-arithmetic groups . . . . . . . . . . . . 95 16.2 Geometric reduction theory . . . . . . . . . . . . . . . . . . . 100 17 Raghunathan's function on the space X 109 17.1 Construction of the function . . . . . . . . . . . . . . . . . . . 109 17.2 The function is Morse . . . . . . . . . . . . . . . . . . . . . . 113 18 Finiteness properties of the S-arithmetic group 136 3 1 Introduction In geometric group theory the structure of groups is examined via geometrical methods. The approach is often to construct a geometric space on which they act with good properties. The groups we will deal with in this thesis are S- arithmetic subgroups of algebraic matrix groups de�ned over Q. A typical example for this is Γ = SLn(Z[1/p]) for some prime p. Via the diagonal embedding one can view this as a discrete subgroup of SLn(R)× SLn(Qp). The latter acts with compact stabilizers on a contractible space X = X∞ × Xp where X∞ ∼= SLn(R)/ SO(n) is the symmetric space associated to SLn(R) and Xp is the Bruhat-Tits building associated to SLn(Qp), an example of a Euclidean building. Now consider a group as the fundamental group of some connected CW com- plex C: It is an easy consequence of algebraic topology that π1(C) is �nitely generated if C has �nite 1-skeleton, and �nitely presented if C has �nite 2- skeleton. This can be generalized to higher dimensions in the following way: The group π1(C) is of type Fn if C has �nite n-skeleton and the universal cover C̃ is contractible. Here we want to show that the group under consid- eration is of type F∞, that means of type Fn for all n. One way to do so is to construct a contractible CW complex on which our group acts with �nite stabilizers and with compact quotient. But in the setup described above the induced action of Γ fails to be cocompact, so one needs to modify the space X. Raghunathan [Rag68] considered arithmetic subgroups of semisimple alge- braic groups de�ned over Q. Here one can think of SLn(Z), which is a discrete subgroup of SLn(R). His method was to cut out some part of the symmetric space making the induced action of the arithmetic group cocom- pact. In fact, he constructed a smooth real valued function de�ned on the quotient and showed that it has no critical value greater than some constant r > 0. That the homotopy type does not change then follows from Morse theory as developed in [Mil63]. The way of Borel and Serre [BS73] to handle the problem of cocompactness was to attach a boundary to the space X∞, that is in some sense compatible with the action and does not change the homotopy type, but which makes the quotient compact. In [BS76] they extended their result to S-arithmetic groups: Theorem (Borel-Serre). Let G be a simply connected, Q-simple Chevalley group over Q. Then the S-arithmetic subgroup Γ = G(Z[ 1p1 , . . . , 1 ps ]) is of type F∞. The procedure is sketched in the book of Brown [Bro89, VII 1D and 2C]. 4 The aim of this thesis is to give a new proof of the Borel-Serre theorem about S-arithmetic groups but following the method of Raghunathan. To do so we will construct a Morse function with no critical values outside some compact interval, not only on the smooth factor X∞ but on the whole space X = X∞ ×Xp. Note that Morse theory in the classical sense like in [Mil63] only applies to smooth structure. So we need to extend it to a function de�ned on the product of a smooth and a non-smooth factor. Theorem. Let f : M × Z → R be a continuous function de�ned on the product of a smooth Riemannian manifold M and a �rst countable Hausdor� space Z. We assume that it ful�lls the following properties for some r > 0: a) The induced map fz : M → R de�ned by fz(p) := f(p, z) is smooth for every z ∈ Z. b) The norm of the gradient ∇fz is uniformly bounded from below on the set {p ∈M | fz(p) > r}. This bound is independent of z ∈ Z. c) The map Z → C1(M,R) de�ned by z 7→ fz is continuous with respect to the weak C1 topology. Then f−1(]−∞, R]) is homotopy equivalent to M × Z for all R > r. Such a map f we will call a Morse function with no critical values greater than r. The proof of the Borel-Serre theorem we will give here will be entirely ge- ometric. As an outlook, one possible application of the methods developed in this work could be to show a conjecture stated in the article of Hartnick and Witzel [HW22]: There the authors examine �niteness properties of so called approximate subgroups. These are de�ned as a symmetric subset Λ of a group G that contains the identity of the group, but is only closed under multiplication up to �niteness: This means ΛΛ ⊂ FΛ for a �nite set F ⊂ G. Examples of such approximate subgroups can be constructed in a similar way from algebraic groups like S-arithmetic subgroups. The authors inves- tigate �niteness properties for the case where S contains only �nite places. They conjecture that if S also has Archimedean places, the corresponding approximate subgroup is also of type F∞. It should be possible to extend our geometric proof to this, while the construction of Borel and Serre is too algebraic to apply here. The attentive reader will have noticed that the theorem Borel and Serre originally proved is much more general. This is due to technical assumptions that we will use in our argumentation. By making some adjustments, it should also be possible to prove the result in the full generality of [BS76]. 5 Let me outline the structure of this thesis, it is divided into two parts. Part I consists of Sections 2 to 13 and serves as an introductory part. It collects terminology and basic results that are needed later on. These include proper actions, CAT(0) spaces and Busemann functions, �niteness properties, (Rie- mannian) manifolds, Lie groups, root systems, algebraic matrix groups and Euclidean buildings. Those topics with which the reader is already familiar can be skipped. Speci�c notation or special results that are required will be clearly referenced to at the appropriate points. In Part II we will prove the theorems stated above. Section 14 begins with the theorem about Morse functions on two factors. In Section 15 we �x an algebraic matrix group G de�ned over Q. To work with it we will give a list of conventions that should apply to that group and introduce notation. We also de�ne the space X the S-arithmetic group acts on. Raghunathan lists several properties about arithmetic groups that are necessary for his proof, see [Rag68, (i)-(iv) on p.326]. In Section 16 we will deduce analogous statements from reduction theory of S-arithmetic groups using the classical results stated in [PR94] and [Bor63]. These we will translate to geometric properties of the action from Γ on the space X, and describe them in terms of Busemann functions. Here we use some constructions from [HW22]. Then in Section 17 we will de�ne a real valued function on the space X = X∞×Xp in a similar way Raghunathan does: It will be invariant under the action of the group Γ. Using the geometric results already mentioned we will show that it is a Morse function as described above. From this we will deduce the �niteness properties of S-arithmetic subgroups of G in Section 18. 6 Part I Fundamentals and background material In this part we give de�nitions for the terms needed later and collect some background material. Most of the time we give references for the statements needed. In some cases we need more speci�c results or easy consequences from the general theory, then we give short proofs. 2 Proper maps and actions Here we collect some basic facts about proper maps and group actions. A good source is the overview article by Kramer [Kra22]. De�nition 2.1. LetX,Y be Hausdor� spaces, where we additionally assume Y to be locally compact. A map f : X → Y is called proper if f−1(K) ⊂ X is compact for all compact subsets K ⊂ Y . Remark 2.2. There are various de�nitions for a map to be proper. But our de�nition is equivalent to the one given in [Kra22] since we assumed Y to be locally compact. We give a criterion for a map to be proper. Lemma 2.3. Let f : X → Y be a continuous map between topological Haus- dor� spaces, where we assume X to be a countable union of open, relatively compact subsets. Let further f ful�ll the following property: For every sequence (xn)n∈N in X with the property that the set {n ∈ N | xn ∈ C} is �nite for any compact subset C ⊂ X, the set {n ∈ N | f(xn) ∈ K} is also �nite for all compact subsets K ⊂ Y . Then f−1(K) is a compact subset of X for all compact K ⊂ Y . Proof. Let K ⊂ Y be compact, then the preimage f−1(K) ⊂ X is closed. Suppose it were not compact. By assumption X = ⋃ n∈N Un for some rela- tively compact open subsets Un of X. For every n ∈ N the preimage f−1(K) is not contained in ⋃n i=1 Ui, so we can pick some xn ∈ f−1(K)∖ ( ⋃n i=0 Ui). On the other hand, for all compact subsets C of X there is an m ∈ N such that C is a subset of the �nite union ⋃m i=0 Ui, which implies xn /∈ C for all n ≥ m. But this means that the set {n ∈ N | xn ∈ C} is �nite for any compact subset C of X, so {n ∈ N | f(xn) ∈ K} must be �nite, too. This is a contradiction, since every f(xn) is an element of K by de�nition. 7 De�nition 2.4. The continuous action of a topological Hausdor� group G on a locally compact space X is called proper if the map f : G×X → X ×X, (g, x) 7→ (g.x, x) is proper. The following is a useful characterization of proper group actions. Lemma 2.5. Let G be a topological Hausdor� group acting continuously on a locally compact space X. The following are equivalent: a) The action is proper. b) The set {g ∈ G | g.B ∩ C ̸= ∅} is compact for all compact subsets B,C ⊂ X. c) The set {g ∈ G | g.C ∩ C ̸= ∅} is compact for all compact subsets C ⊂ X. Proof. By [Kra22, Proposition 1.5] statements a) and b) are equivalent, and obviously b) implies c). So take B,C ⊂ X compact, then c) implies that {g ∈ G | g.(B ∪ C) ∩ (B ∪ C) ̸= ∅} is also compact. Using the fact that H := {g ∈ G | g.B ∩ C ̸= ∅} ⊂ {g ∈ G | g.(B ∪ C) ∩ (B ∪ C) ̸= ∅} we only have to show that H is closed in G. Take any net (gλ)λ∈Λ in H converging to some g ∈ G, we have to show that g ∈ H. We can choose nets (bλ)λ∈Λ ⊂ B and (cλ)λ∈Λ ⊂ C such that gλ.bλ = cλ for all λ ∈ Λ by de�nition of H. Since C is compact we get a subnet (cµ)µ converging to some c ∈ C. If we now consider the corresponding net (bµ)µ, then we can choose another subnet (bν)ν converging to some b ∈ B again by compactness. This leads to nets (gν)ν , (bν)ν and (cν)ν converging to g, b and c. By continuity of the action we get g.b = lim ν gν .bν = lim ν cν = c so in fact g ∈ H holds. Corollary 2.6. If G is a topological Hausdor� group acting properly on a locally compact space X, then any closed subgroup H ⊂ G acts also properly on X. Especially this holds for all discrete subgroups. Proof. Follows directly from the lemma above. Properness of group actions also passes over to products. 8 Lemma 2.7. Let (Gi)i∈I be a family of topological Hausdor� groups, each of them acting on a locally compact space Xi. Then the induced action of∏ i∈I Gi on ∏ i∈I Xi, is proper if and only if each of the actions Gi on Xi is proper. Proof. Follows from [Eng89, Theorem 3.7.9]. Just note that what we have called proper here is called perfect there. Finally we prove a criterion for a group action to be proper that we want to apply later. Since properness passes to closed subgroups by Corollary 2.6 this will also give a criterion for a discrete subgroup to act properly. Lemma 2.8. Let G be a locally compact group acting continuously and tran- sitively on a Hausdor� space X. Suppose there is a point x0 ∈ X such that: a) The stabilizer StabG(x0) is compact in G, b) the continuous map φ : G→ X, g 7→ g.x0 is open. Then for any compact subset C ⊂ X the set {g ∈ G | g.C ∩ C ̸= ∅} is compact. Consequently, if X is assumed to be locally compact, then the action is proper. Proof. It su�ces to �nd a compact subset A ⊂ G to a given C ⊂ X compact with A.x0 = C, because this implies that {g ∈ G | g.C ∩ C ̸= ∅} = AKA−1 is compact for K := StabG(x0). Under the given assumptions both sets indeed coincide: If ga.x0 = a′.x0 for a, a′ ∈ A and g ∈ G, then x0 = a−1ga′, so g ∈ aKa′−1 ⊂ AKA−1. On the other hand, if g = a′ka−1 ∈ AKA−1 then g.(a.x0) = a′.x0. So let A := {g ∈ G | g.x0 ∈ C} = φ−1(C) then A.x0 = C by transitivity. Thus we have to show that A is compact. For any a ∈ A choose a relatively compact open neighborhood Ua ⊂ G by local compactness of G. Since φ is open, the set ⋃ a∈A φ(Ua) is an open cover of the compact set C, so we can �nd U1, . . . , Un ∈ {Ua | a ∈ A} such that C ⊂ n⋃ i=1 φ(Ui) = φ ( n⋃ i=1 Ui ) = ( n⋃ i=1 Ui ) .x0. But this implies A = φ−1(C) ⊂ φ−1 (( n⋃ i=1 Ui ) .x0 ) = ( n⋃ i=1 Ui ) ·K since K = Stab(x0). Thus, A is a subset of a compact subset of G. Further- more, A is closed by de�nition, so it is indeed compact. 9 3 Metric spaces of non-positive curvature We give a short introduction into CAT(0) spaces. The goal of the �rst subsection is to de�ne the boundary at in�nity and Busemann functions and state some basic results about their interplay. In the second part we consider products of metric spaces. Main source for both parts is the book of Bridson and Hae�iger [BH99]. 3.1 CAT(0) spaces and Busemann functions Throughout this whole subsection let X be a metric space with metric d. De�nition 3.1. Let λ > 0. A map c : [a, b] → X with d(c(t), c(s)) = λ |t− s| for all s, t ∈ [a, b] ⊂ R is a linearly reparametrized geodesic. We call it a geodesic if λ = 1. Its image [c(a), c(b)] := im(c([a, b])) is a geodesic segment from c(a) to c(b). If for any two points x, y ∈ X there is a (unique) geodesic segment from x to y, then we call X a (unique) geodesic space. A geodesic ray is a map [0,∞[→ X such that the restriction to [a, b] is a geodesic for any [a, b] ⊂ [0,∞[. Similarly, we de�ne a geodesic line as a map R → X with the same property for all [a, b] ⊂ R. Equivalently, one could de�ne geodesics, geodesic rays and geodesic lines as isometric embeddings. Two geodesic rays c, c′ : [0,∞[→ X are called asymptotic if there is a constant k > 0 with d(c(t), c′(t)) < k for all t ≥ 0. This clearly de�nes an equivalence relation on the set of geodesic rays into X. Write c(∞) for the equivalence class of the ray c : [0,∞[→ X, and de�ne the boundary at in�nity of X as the set ∂X := {c(∞) | c : [0,∞[→ X is a geodesic ray}. In the following text we denote an element ξ ∈ ∂X as point at in�nity. The isometry group Isom(X) of a metric space X has a canonical action on ∂X since d(γ.c(t), γ.c′(t)) = d(c(t), c′(t)) for γ ∈ Isom(X). Thus, every group that acts via isometries on X also acts on its boundary. A geodesic triangle in X with vertices x, y, z ∈ X is the union ∆(x, y, z) := ([x, y] ∪ [y, z] ∪ [x, z]) ⊂ X of three geodesic segments. We call a triangle ∆(x, y, z) ⊂ R2 with d(x, y) = d(x, y), d(y, z) = d(y, z) and d(x, z) = d(x, z) a comparison triangle for ∆(x, y, z). A comparison point for p ∈ [x, y] ⊂ X is a point p ∈ [x, y] ⊂ R2 with d(p, x) = d(p, x). A geodesic triangle ∆(x, y, z) satis�es the CAT(0)- inequality, if for all p, q ∈ ∆(x, y, z) and comparison points p, q ∈ ∆(x, y, z) the inequality d(p, q) ≤ d(p, q) holds. 10 De�nition 3.2. A geodesic metric space X is a CAT(0) space, if for every triangle ∆(x, y, z) ⊂ X the CAT(0) inequality holds. Furthermore, we say a metric space X is locally CAT(0) or of non-positive curvature if for every x ∈ X there is an ε > such that Bε(x) = {y ∈ X | d(x, y) < ε} is a CAT(0) space when endowed with the canonical subspace metric. Every CAT(0) space X is in fact uniquely geodesic. Further it has the prop- erty that its geodesics vary continuously with their endpoints: If c, cn : [0, 1] → X are linearly reparametrized geodesics with the properties limn→∞ cn(0) = c(0) and limn→∞ cn(1) = c(1), then cn converges to c uniformly on the whole interval [0, 1]. For both statements see [BH99, Proposition II.1.4]. One can easily deduce: Lemma 3.3. Every CAT(0) space is contractible. Proof. See [BH99, Corollary II.1.5]. For any two geodesics c, c′ : [a, b] → X the function f : [a, b] → R; t 7→ d(c(t), c′(t)) is convex by [BH99, Proposition II.2.2]. Example 3.4. We give some examples of CAT(0) spaces here. a) Obviously, the Euclidean space Rn is a CAT(0) space. b) Every tree endowed with the canonical metric is a CAT(0) space, com- pare [BH99, Example II.1.15 (4)]. c) We construct the hyperboloid model Hn for the hyperbolic n-space: For x, y ∈ Rn+1 let x ◦ y := ∑n i=1 xiyi − xn+1yn+1. Then we de�ne Hn = {x ∈ Rn+1 | x ◦ x = −1 and xn+1 > 0} and set dHn(x, y) = arcosh(−x ◦ y). The induced map dHn de�nes a metric on Hn, see [Rat94, Theorem 3.2.2]. One can show that the corresponding metric space is uniquely geodesic, compare [BH99, Corollary I.2.8 (1)]. We de�ne a CAT(−1) space analogously to a CAT(0) space, but taking comparison triangles ∆ in H2 instead of R2. The space H2 canonically embeds as a subspace into Hn for all n ≥ 2. For all x, y, z ∈ Hn there is an isometry γ ∈ Isom(Hn) sending all the three points x, y, z into H2, this is proven in [Rat94, �3.2]. Thus, Hn is CAT(−1) so [BH99, Theorem II.1.12] implies that it is CAT(0). 11 d) A non-example is the sphere Sn := {x ∈ Rn+1 | |x| = 1} endowed with the angular metric d(u, v) := arccos(⟨u, v⟩) which is constructed in [BH99, Proposition I.2.1]. This fails to be CAT(0), because it is not contractible. More examples will be constructed later. Assume now the CAT(0) space X to be complete. For every ξ ∈ ∂X and every x ∈ X there exists a unique geodesic ray cx : [0,∞[→ X with cx(0) = x and cx(∞) = ξ. See [BH99, Proposition II.8.2]. We can endow the set X := X ∪ ∂X with the inverse limit topology of the system prr : Bs(x0) → Br(x0) for s ≥ r. Here prr is de�ned in the following way: Take for any y ∈ Bs(x0) the unique geodesic c : [0, b] → Bs(x0) with c(0) = x0 and c(b) = y. Then we set prr(y) = c(max{r, b}). A basis of this topology is given by the open sets of X together with the sets of the form U(c, r, ε) = {x ∈ X | d(x, c(0)) > r, d(prr(x), c(r)) < ε} where c : [0,∞[→ X is a geodesic ray and ε, r > 0. The inclusion X ↪→ X then becomes a homeomorphism onto its image. See [BH99, II.8.5 and 8.6] for details. Furthermore, any isometry γ ∈ Isom(X) extends naturally to a homeomorphism X → X by [BH99, Corollary II.8.9]. De�nition 3.5. Let c : [0,∞[→ X be a geodesic ray. a) The map βc : X → R; x 7→ limt→∞ t−d(x, c(t)) is called the Busemann function associated to c. This is well de�ned as the limit always exists by [BH99, Lemma II.8.18]. b) Let r ∈ R be any real number, the preimage β−1 c (]r,∞[) ⊂ X is called a horoball and β−1 c (r) is a horosphere. Remark 3.6. In some sources, for example [BH99], Busemann functions are de�ned as x 7→ limt→∞ d(x, c(t)) − t which is just −β in our de�nition. So we can apply their results by changing the sign if necessary. Further, a reparametrized Busemann function is a map of the form x 7→ aβ(x) for some a > 0 and a Busemann function β . Example 3.7. We describe easy examples of Busemann functions and the corresponding horoballs. a) First we consider a rather trivial example: Let c : R → R be the iden- tity, then the restriction to [0,∞[ will be a geodesic ray. The corre- sponding Busemann function is just the identity which can be seen by βc(x) = lim t→∞ t− d(c(t), x) = lim t→∞ t− t+ x = x so the horoballs are β−1 c (]r,∞[) =]r,∞[. 12 b) More generally, we consider the Euclidean space X = Rn, then any geodesic ray is of the form c(t) = b+ tu for some u ∈ Sn−1 and b ∈ Rn. The corresponding Busemann function is βc(x) = lim t→∞ t− ∥b+ tu− x∥ = lim t→∞ t− √ ⟨b− x, b− x⟩+ 2t ⟨b− x, u⟩+ t2 ⟨u, u⟩ = lim t→∞ t2 − (⟨b− x, b− x⟩+ 2t ⟨b− x, u⟩+ t2 ⟨u, u⟩) t+ √ ⟨b− x, b− x⟩+ 2 ⟨b− x, u⟩+ t2 ⟨u, u⟩ = lim t→∞ − ⟨b− x, b− x⟩+ 2t ⟨b− x, u⟩ t+ √ ⟨b− x, b− x⟩+ 2 ⟨b− x, u⟩+ t2 = −2 ⟨b− x, u⟩ 1 + √ 1 = ⟨x− b, u⟩ So the horoballs are the a�ne half-spaces β−1 c (]r,∞[) = {x ∈ Rn | ⟨x− b, u⟩ > r}. This is from [BH99, Example II.8.24 (1)]. More examples can be found in [BH99]. The next lemmas state useful prop- erties of Busemann functions. Lemma 3.8. Every Busemann function is distance decreasing and therefore continuous. Proof. Let X be a metric space, let c : [0,∞[→ X be a geodesic ray and let β = βc be the corresponding Busemann function. For any two points x, y ∈ X we have |β(y)− β(x)| = ∣∣∣ lim t→∞ t− d(y, c(t))− t+ d(x, c(t)) ∣∣∣ = lim t→∞ |d(x, c(t))− d(y, c(t))| ≤ d(x, y). Thus, limn→∞ β(xn) = β(x) if limn→∞ xn = x. Consider now a complete CAT(0) space X and let c, c′ : [0,∞[→ X be geodesic rays with c(∞) = c′(∞). Let further βc, βc′ : X → R be the associ- ated Busemann functions. Then βc − βc′ is constant as a function X → R by [BH99, Corollary II.8.20]. Therefore the following de�nition makes sense: We call a Busemann function β : X → R centered at ξ ∈ ∂X if β−βc = const for any geodesic ray c : [0,∞[→ X with c(∞) = ξ. We also call a horoball or a horosphere centered at ξ if the corresponding Busemann function is. 13 Lemma 3.9. Let X be a complete CAT(0) space and let β : X → R be a Busemann function centered at a point at in�nity ξ ∈ ∂X. For every geodesic ray c : [0,∞[→ X with c(∞) = ξ we have β(c(t)) = β(c(0)) + t for all t ∈ [0,∞[. If c extends to a geodesic line R → X this equation holds for all t ∈ R. The Busemann function associated to c ful�lls βc(c(0)) = 0. Proof. For t ≥ s we have d(c(t), c(s)) = t− s as c is an isometric embedding, so βc(c(s)) = lim t→∞ t− d(c(s), c(t)) = lim t→∞ t− t+ s = s holds for all s ∈ R, on which c is de�ned. Now choose a constant k ∈ R with β − βc = k, then the above implies β(c(t))− t = β(c(t))− βc(c(t)) = k. Therefore, β(c(0)) = k and thus β(c(t)) = β(c(0)) + t hold for all t ∈ R on which c is de�ned. Lemma 3.10. Let G be a group acting via isometries on X and let β : X → R be a Busemann function centered at a point ξ ∈ ∂X. For every g ∈ G the map x 7→ β(g.x) is a Busemann function centered at g−1.ξ. Proof. Let c be a geodesic ray with β = βc. Then βc(g.x) = lim t→∞ t− d(c(t), g.x) = lim t→∞ t− d(g−1.c(t), x) = βg−1.c(x) is a Busemann function associated to the geodesic ray g−1.c de�ned by t 7→ g−1.c(t). At this point we also de�ne the terminology of a proper metric space, that we will need later. De�nition 3.11. A metric space X is called proper if for all x ∈ X the closed balls Br(x) = {y ∈ X | d(x, y) ≤ r} are compact for any r ≥ 0. It follows directly from the de�nition that a metric space X is proper if and only if any bounded and closed subset A ⊂ X is compact. Remark 3.12. The terminology of a proper metric space comes from the following easy fact: A metric space is proper if and only if for every x0 ∈ X the real valued function dx0(x) := d(x0, x) is a proper map in the sense of De�nition 2.1. Properness gives a useful criterion to check if a metric space X is CAT(0). Proposition 3.13. Let X be a proper, uniquely geodesic metric space of non-positive curvature. Then X is a CAT(0) space. Proof. See [BH99, Corollary I.3.13 and Proposition II.4.9]. 14 3.2 Products of metric spaces In this subsection we will state some basic facts about products of CAT(0) spaces. Recall that for metric spaces (Xi, di)i=1,...,n the product metric d on X1 × . . . ×Xn is de�ned by d(x, y) = ( ∑n i=1 di(xi, yi) 2) 1 2 for two points x = (x1, . . . , xn), y = (y1, . . . , yn). In the following we will state the facts only for X = X1×X2 for the sake of brevity in the notation. They all extend easily via induction to the case with n factors. Lemma 3.14. For a product X = X1 × X2 of metric spaces the following statements hold: a) X is complete if and only if X1 and X2 are complete. b) X is a proper metric space if and only if X1 and X2 are proper. c) X is a geodesic space if and only if X1 and X2 are geodesic spaces. d) A map c : [0, a] → X, c(t) = (c1(t), c2(t)) is a linearly reparametrized geodesic if and only if c1 and c2 are linear reparametrized geodesics. e) A map c : [a, b] → X is a geodesic, if and only if c(t) = (c1(λ1t), c2(λ2t)) for some geodesics ci : [λia, λib] → Xi and λi ≥ 0 with λ21 + λ22 = 1. f) X is a CAT(0) space if and only if X1 and X2 are CAT(0). g) If Gi are groups acting via isometries on Xi for i = 1, 2 then G1 ×G2 acts via isometries on X. Proof. For a), c), d and g) see [BH99, Proposition I.5.3]. To b): The product B1 × B2 of two closed bounded subsets Bi ⊂ Xi is bounded and closed inX. Hence, ifX is proper, thenB1 andB2 are compact. On the other hand, if X1 and X2 are proper, then we can cover any bounded and closed subset B of X by two compact balls B1 and B2, so B is compact, too. To e): By c) the map c is a linearly reparametrized geodesic if and only if it has the form stated above for some λi ≥ 0. From d(c(s), c(t))2 = d1(c1(λ1s, λ1t)) 2 + d2(c1(λ2s, λ2t)) 2 = (λ21 + λ22) |s− t| follows the statement. To f): If X1 and X2 are CAT(0), then X is by [BH99, Example I.1.15 (3)], too. If on the other handX is a CAT(0) space, then both factors are geodesic by b). Furthermore the CAT(0) inequality must hold, because they embed as isometric subspaces. Note that e) extends to geodesic rays and lines by de�nition. 15 Corollary 3.15. Let X = X1 × X2 be a complete CAT(0) space. Assume c : [0,∞[→ X is a geodesic ray given by c(t) = (c1(λ1t), c2(λ2t)) like above and let x = (x1, x2) ∈ X be any point. The unique ray starting at x asymp- totic to c is given by cx(t) = (cx1(λ1t), cx2(λ2t)), where cxi : [0,∞[→ Xi are geodesic rays with cxi(0) = xi and cxi(∞) = ci(∞). Proof. Follows directly from d(c(t), cx(t)) 2 = d1(c1(λ1t), cx1(λ1t)) 2 + d2(c2(λ2t), cx2(λ2t)) 2 ≤ k21 + k22 for constants ki bounding di(ci(t), cxi(t)). If X = X1 ×X2 is a complete CAT(0) space, then the boundary at in�nity is the spherical join ∂X = ∂X1 ∗ ∂X2: This is the set of equivalence classes of ∂X1×∂X2× [0, π2 ] where two elements (ξ1, ξ2, θ) and (ξ′1, ξ ′ 2, θ ′) de�ne the same class if and only if (θ = 0 and ξ2 = ξ′2) or (θ = π 2 and ξ1 = ξ′1) holds. The ray c(t) = (c1(cos(θ)t), c2(sin(θ))) represents c(∞) = [c1(∞), c2(∞), θ] =: cos(θ)c1(∞) + sin(θ)c2(∞). Compare [BH99, Example II.8.11 (6)]. The notation �ts to the following observation about Busemann functions. Lemma 3.16. Let c : [0,∞[→ X = X1 × X2 be a geodesic ray given by c(t) = (c1(cos(θ)t), c2(sin(θ)t)). The corresponding Busemann function is βc((x1, x2)) = cos(θ)βc1(x1) + sin(θ)βc2(x2). Proof. See [BH99, Example II.8.24 (3)]. 16 4 Finiteness properties We begin this section by reminding the reader that a covering space projec- tion is a continuous map p : X̃ → X such that any point x ∈ X has an open neighborhood U ⊂ X so that p−1(U) = ∐ i∈I Ui is a disjoint union of open subsets Ui ⊂ X̃ and p induces homeomorphisms Ui → U . We call such a X̃ a covering space of X. If f : Y → X is a continuous map, then another map f̃ : Y → X̃ with the property p ◦ f̃ = f is a lift of f . Recall further that a topological space X is locally path connected if for any open subset U ⊂ X and any point x ∈ U there is a path connected neighborhood V ⊂ U of x. It is not hard to see that a covering space of a locally simply connected space is also locally simply connected and vice versa. Furthermore, we call X semilocally simply-connected if every point x ∈ X has a neighborhood U ⊂ X such that the canonical homomorphism π1(U, x) → π1(X,x) induced by the inclusion is trivial. These properties are not really harsh restrictions, for example manifolds and CW complexes are locally path connected and semilocally simply-connected: In fact both are locally contractible (for every U ⊂ X open and every x ∈ U there is a contractible neighborhood V ⊂ U of x), since manifolds are locally Euclidean, and for CW complexes see [Hat02, Proposition A.4]). One can show that for any path connected, locally path connected and semilocally simply-connected space X there is a covering space X̃ that is simply connected, see the construction in [Hat02, p.63-65]. To be semilo- cally simply-connected in fact is also a necessary condition for a space to have a simply connected covering space. We call such an X̃ the universal cover of X, because it ful�lls the following universal property: Proposition 4.1. Let X be path connected, locally path connected space and let p : X̃ → X be a covering projection, mapping a point x0 ∈ X to x̃0 ∈ X̃, where X̃ is assumed to be simply connected. Then for any other covering projection q : Z → X mapping z0 to x0 with Z path connected and locally path connected, there is a unique map p̃ : X̃ → Z with the properties q ◦ p̃ = p and p̃(x̃0) = z0. Proof. This follows directly from the lifting criterion and the unique lifting property, see [Hat02, Propositions 1.33 and 1.34]. This implies that the universal cover (if it exists) is unique up to homeomor- phism, see [Hat02, Proposition 1.37]. Further recall that a deck transformation of a covering space projection p : X̃ → X is a homeomorphism f : X̃ → X̃ such that p = p ◦ f . Obviously, the set of all deck transformations form a group with ◦ as group multipli- cation. The covering space projection is called normal if for every x ∈ X and every pair x̃, x̃′ ∈ p−1(x) there is a deck transformation mapping x̃ to 17 x̃′. This terminology comes from the fact that the covering space projection p is normal if and only if p∗(π1(X̃, x̃0)) ⊂ π1(X,x0) is a normal subgroup, where p∗ is the induced homomorphism between fundamental groups. See [Hat02, Proposition 1.39 a)]. De�nition 4.2. AK(Γ, 1) space is a path connected, locally path connected and semilocally simply-connected space X such that a) π1(X) ∼= Γ, b) the universal cover of X is contractible. If X is a CW complex, then we call it a K(Γ, 1) complex. De�nition 4.3. A path connected space X is called n-aspherical if any continuous map Sk → X has an extension Dk+1 → X for all 2 ≤ k ≤ n. It is called n-connected if the same holds for all 0 ≤ k ≤ n. Further we call it aspherical if it is n-aspherical for any n ∈ N. Remark 4.4. In [Geo08, p.162] a K(Γ, 1) complex is de�ned to be an as- pherical path connected CW complex X with π1(X,x0) = Γ. But since a path connected CW complex is aspherical if and only if its universal cover is contractible by [Geo08, Proposition 7.1.3] these two de�nitions coincide. De�nition 4.5. Let Γ be a group. a) We say Γ is of type Fn if there is a K(Γ, 1) complex X which has �nite n-skeleton. b) We call Γ of type F∞ if there is a K(Γ, 1) complex X which has �nite n-skeleton for every n ∈ N. Remark 4.6. a) One can show that Γ is of type F1 if and only if it is �nitely generated, and is of type F2 if and only if it is �nitely presented, see [Geo08, Proposition 7.2.1]. So being of type Fn is a topological generalization of of these two algebraic properties. b) The group Γ is of type F∞ if and only if it is of type Fn for all n ∈ N, see [Geo08, Proposition 7.2.2]. Example 4.7. Any �nite group is of type F∞. Proof. See [Geo08, Corollary 7.2.5]. This comes from the fact that a group is of type Fn if and only if some (and therefore any) of its �nite index subgroups is, see [Geo08, Corollary 7.2.4]. Thus, �niteness properties are in fact invariant under commensurability. Next we give a criterion for a group to be of type Fn. Recall that a pair (Λ,≤) is a directed set if ≤ is a preorder on the set Λ, such that for all λ, λ′ ∈ Λ there is µ with λ, λ′ ≤ µ. 18 De�nition 4.8. Let X be a topological space. a) A �ltration of X is a collection of subspaces Xλ indexed by a directed set (Λ,≤) such that Xλ is a subset of Xµ whenever λ ≤ µ and such that ⋃ λ∈ΛXλ = X. b) We call a �ltration (Xλ)λ∈Λ essentially n-connected if for every 0 ≤ i ≤ n the following holds: For all λ ∈ Λ there is a µ ∈ Λ, such that the group homomorphism πi(Xλ) → πi(Xµ) induced by the inclusion Xλ ⊂ Xµ, is trivial. c) Two �ltrations (Xλ)λ∈Λ and (Yµ)µ∈M of X are equivalent, if for all λ ∈ Λ there is µ ∈ M such that Xλ is a subset of Yµ, and if for all µ ∈M there is λ ∈ Λ with the converse subset relation. Lemma 4.9. Let (Xλ)λ∈Λ and (Yµ)µ∈M be two equivalent �ltrations of the same space X. One of them is essentially n-connected if and only if the other one is. Proof. See [HW22, Lemma 2.3]. Lemma 4.10. Let (X, d) be a CAT(0) space with a point o ∈ X and let G be a locally compact, second countable group acting on it properly and con- tinuously via isometries. A closed subgroup Γ of G is of type Fn if and only if the �ltration (Nr(Γ.o))r>0 is essentially (n−1)-connected. Here (Nr(Γ.o)) is de�ned to be the set of all x ∈ X such that d(γ.o, x) < r for some γ ∈ Γ. Proof. See [HW22, Proposition 2.23]. This will be our strategy to check the �niteness properties of a group Γ: Viewing it as a discrete subgroup of a locally compact, second countable group G, which acts naturally as described in the lemma above on a space X. We will construct a map f : X → R and show that forXt := f−1(]−∞, t]) the �ltration (Xt)t∈R is essentially n-connected and equivalent to (Nr(Γ.o))r>0. 19 5 Real and complex manifolds Here we give an introduction into the topic of manifolds. This section is divided into two subsections: In the �rst we deal with smooth manifolds, the second is about the holomorphic counterpart. 5.1 Smooth manifolds, vector �elds and �ows Main source here is the book of Tu [Tu11]. We also use some aspects of [Con01]. Recall that a topological manifold of dimension m ∈ N is a second countable topological Hausdor� space M , such that for any point p ∈ M there exists an open neighborhood U ⊂ M of p and a continuous map ϕ : U → Rm such that ϕ(U) ⊂ Rm is open and ϕ : U → ϕ(U) is a homeomorphism. We will refer to the pair (U, ϕ) as a chart around p and we will always assume that ϕ(p) = 0. Setting xj = prj ◦ϕ for j = 1, . . . ,m we get ϕ(q) = (x1(q), . . . , xm(q)) ∈ Rm for q ∈ U . We call (x1, . . . , xm) the coordinates of ϕ and refer to (U, x1, . . . , xm) as a local coordinate system around p. De�ne the chart map from a chart (U, ϕ) to another one (V, ψ) of M as ψ ◦ϕ−1 : ϕ(U ∩V ) → ψ(U ∩V ). We call it smooth or C∞ if it is smooth as a map of open subsets of Rm. An atlas of M is a set of charts U = {(Ui, ϕi) | i ∈ I} such that M = ⋃ i∈I Ui. It is called smooth if all of its chart maps are. It is called maximal if for any other atlas V = {(Vj , ϕj) | j ∈ J} with U ⊂ V we already have U = V. De�nition 5.1. A topological manifold M is called smooth if it has a max- imal smooth atlas U . The existence of a maximal smooth atlas is equivalent to the existence of some smooth atlas (not necessary maximal) since any smooth atlas is con- tained in a unique maximal one, see [Tu11, Proposition 5.10]. We make the following convention: If we talk of a manifold, we will always mean a smooth manifold. Example 5.2. Here are some examples of smooth manifolds. a) Obviously, Rm is a smooth manifold with one chart (Rm, id), compare [Tu11, Example 5.11]. b) Let M be a smooth manifold, then any open subset V ⊂ M is also a smooth manifold of the same dimension. An atlas on V is given by {(Ui ∩ V, ψi) | i ∈ I} where ψi = ϕi |Ui∩V for an atlas {(Ui, ϕi) | i ∈ I} for M . See [Tu11, Example 5.12]. 20 c) Let U ⊂ Rm be open and f : U → Rn be a smooth map, then ϕ : Γ(f) → U, (x, f(x)) 7→ x is a chart of the graph Γ(f) = {(x, y) ∈ U × Rn | f(x) = y}. So Γ(f) is also a smooth manifold, see [Tu11, Example 5.14]. This example shows that the hyperboloid Hn = {x ∈ Rn+1 | −x2n+1 + n∑ i=1 x2i = −1 and xn+1 > 0} is a smooth manifold of dimension n, because Hn = Γ(f) for the smooth map f : Rn → R, x 7→ √∑n i=1 x 2 i + 1. d) Using b) we see that GLn(R) = det−1(R ∖ {0}) is a smooth manifold of dimension n, see [Tu11, Example 5.15]. In fact this is also the prototype example of a Lie group, as multiplication and inversion are easily seen to be smooth. We will deal with that in the section about Lie groups. e) If M and N are smooth manifolds of dimensions m and n, then the productM×N is also a smooth manifold of dimensionm+n: An atlas is given by {Ui×Vj , (ϕi, ψj) | (i, j) ∈ I×J} for atlases {(Ui, ϕi) | i ∈ I} for M and {(VJ , ψj) | j ∈ J} for N . See [Tu11, Example 5.17 and Proposition 5.18]. Let M,N be smooth manifolds of dimension m and n. A continuous map f : M → N is called smooth in a point p ∈M if for some charts (U, ϕ) around p and (V, ψ) around f(p) the map (ψ ◦ f ◦ ϕ−1) : ϕ(U ∩ f−1(V )) → Rn is smooth as a real function. Further f is called smooth if it is smooth in any point p ∈M . A smooth map f : M → N is called a di�eomorphism if there is another smooth map g : N → M such that g ◦ f = id and f ◦ g = id. It is called a local di�eomorphism if for every point p ∈ M there is a neighborhood U ⊂M of p such that the restriction f |U : U → f(U) is a di�eomorphism. The de�nition of smoothness does not depend on the choice of the charts, see [Tu11, Proposition 6.7 and 6.8]. In the special case N = Rn with the atlas {(Rn, id)} from Example 5.2 one directly deduces: A continuous map f : M → Rn is smooth (in p) if and only if the map f ◦ ϕ−1 is smooth (in p) for any chart (U, ϕ) of M (around p). This shows that chart maps are smooth. Furthermore, one can show that the composition of two smooth maps is smooth again, see [Tu11, Proposition 6.9]. We denote by C∞(M) the set of smooth maps M → R. Taking a coordinate system (U, x1, . . . , xm) = (U, ϕ) around a point p ∈M we de�ne for f ∈ C∞ 21 the partial derivative with respect to xi as ∂ ∂xi |pf := ∂f ∂xi (p) := ∂(f ◦ ϕ−1) ∂xi (ϕ(p)), where the latter is the usual derivative at the point ϕ(p) ∈ Rm of the smooth function f ◦ ϕ−1 de�ned on the open subset ϕ(U) of Rm. In analogy to the above we write C∞(U) for the set of all smooth real valued functions de�ned on an open subset U ⊂ M . For a function f ∈ C∞(U) de�ned on an open neighborhood U of p we de�ne the germ of f as the equivalence class [f ]p given by the following relation: A function g ∈ C∞(V ) de�ned on the neighborhood V ⊂ M of p is equivalent to f if they coincide on an open set W ⊂ U ∩V containing the point p. We write C∞ p (M) for the set of all germs at p. One easily veri�es that addition [f ]p + [g]p = [f + g]p, multiplication [f ]p · [g]p = [f · g]p and scalar multiplication r · [fp] = [r · f ]p are well de�ned, such that C∞ p (M) becomes an R-algebra. A derivation at p or a point-derivation of C∞ p (M) is an R-linear map Xp : C ∞ p (M) → R such that Xp([fg]p) = Xp([f ]p)g(p) + f(p)Xp([g]p) holds. Since this de�nition does not depend on the choice of the representatives, we will write Xp(f) instead of Xp([f ]p). The set TpM of all derivations at p is a real vector subspace of the dual space C∞ p (M)∗, as one can verify easily. De�nition 5.3. The real vector space TpM is the tangent space of M at p. We will call an element Xp ∈ TpM a tangent vector at p. Remark 5.4. For an open neighborhood U ⊂M of a point p ∈M there is a canonical identi�cation TpU = TpM by de�nition, see [Tu11, Remark 8.2]. A direct calculation shows that the map ∂ ∂xi |p : C∞ p (M) → R, [f ]p 7→ ∂ ∂xi |pf is a derivation at p. In fact, the set { ∂ ∂x1 |p, . . . , ∂ ∂xm |p} is a basis for the tangent space TpM by [Tu11, Proposition 8.9]. Lemma 5.5. Let M be a smooth manifold of dimension m. Consider the local coordinate systems (U, x1, . . . , xm) and (V, y1, . . . , ym) around a point p ∈M . Then we have ∂ ∂xi = ∑m j=1 ∂yj ∂xi ∂ ∂yj on U ∩ V . Proof. See [Tu11, Proposition 8.10]. For a smooth map f : M → N we de�ne the di�erential of f at p ∈M as the map f∗,p : TpM → Tf(p)N, Xp 7→ f∗,p(Xp) where f∗,p(Xp)(g) := Xp(g ◦ f) 22 for any smooth map g : N → R. This is a linear map between the two vector spaces TpM and Tf(p)N because of f∗,p(Xp + rYp)(g) = Xp(g ◦ f) + rYp(g ◦ f) = f∗,p(Xp)(g) + rf∗,p(Yp)(g). As in the case of real maps Rm → Rn we have the chain rule (g ◦ f)∗,p = g∗,f(p) ◦ f∗,p at every point p ∈M for smooth maps f : M → N and g : N → P , see [Tu11, Theorem 8.5]. We get a more geometric intuition of the tangent space if we describe its vectors via curves: A curve into a smooth manifold M is a continuous map c : I → M where I ⊂ R is any interval. We call the curve smooth if there is an open interval J ⊂ R with containing I and a smooth map c̄ : J →M with c̄ |I= c. We call c a piecewise smooth curve if I = [a, b] for some a < b in R and if there are a = t0 < t1 < . . . < tk = b such that c|[ti,ti+1] is a smooth curve for all i = 0, . . . , k − 1. For t0 ∈ I we de�ne the velocity vector at c(t0) ∈ M of a smooth curve as the tangent vector c′(t0) := c∗( d dt |t0) ∈ Tc(t0)M . A direct calculation shows c′(t0)(f) = c∗ ( d dt |t0 ) (f) = d dt |t0(f ◦ c) = (f ◦ c)′(t0) for all f ∈ C∞ c(t0) (M). Note that the latter becomes the usual derivative at t0 of the real valued function f ◦ c de�ned on the interval I. If we pick a local coordinate system (U, x1, . . . , xm) around c(t) ∈ M , then describing the velocity vector via the induced basis we get c′(t) = ∑m i=1 ċ i(t) ∂ ∂xi |c(t) for ċi(t) := (xi ◦ c)′(t). See [Tu11, Proposition 8.15,]. On the other hand, let p ∈M be a point. For any tangent vector Xp ∈ TpM there is ε > 0 and a smooth curve c : ]− ε, ε[→M with c(0) = p and c′(0) = Xp. For this we have Xp(f) = (f ◦ c)′(0) ∈ R for all f ∈ C∞ p (M). More generally, if g : M → N is a smooth map, then g∗(Xp) = (g ◦c)′(0) ∈ Tg(p)N . See [Tu11, Propositions 8.16,8.17 and 8.18]. De�nition 5.6. The tangent bundle of a smooth manifold M is de�ned as the set TM := ⋃ p∈M{p} × TpM . De�ne a topology on TM in the following way: For any chart (U, ϕ) = (U, x1, . . . , xm) on M consider TU = ⋃ p∈U{p} × TpU = ⋃ p∈U{p} × TpM , see Remark 5.4. For all p ∈ U the set { ∂ ∂x1 |p, . . . , ∂ ∂xm |p} is a basis on TpM . Thus, picking any vector v ∈ TpM there are unique scalars a1(v), . . . , am(v) ∈ R such that v = ∑m i=1 a i(v) ∂ ∂xi |p. We de�ne a map ϕ̃ : TU → ϕ(U)× Rm, (p, v) 7→ (x1(p), . . . , xm(p), a1(v), . . . , am(v)). One can show that ϕ̃ is a bijection and therefore we get a topology on TU de�ning A ⊂ TU to be open if and only if ϕ̃(A) ⊂ ϕ(U) × Rm is open. A 23 basis for this topology on TM is given by B = ⋃ (U,ϕ) chart of M {A | A ⊂ TU open }. Using this topology TM becomes a smooth manifold with an atlas {(TUi, ϕ̃i)} for any atlas {(Ui, ϕi)} for M . See [Tu11, �12.1-12.2, p.129-133] for details. De�nition 5.7. A vector �eld on a smooth manifold M is a map X : M → TM, p 7→ (p,Xp). It is called smooth if it is a smooth map with respect to the manifold structure on TM de�ned above. We write X(M) for the set of smooth vector �elds on M . For any tangent vectorXp ∈ TpM there are scalars a1(p), . . . , am(p) ∈ R such that Xp = ∑m i=1 a i(p) ∂ ∂xi |p, where (U, x1, . . . , xm) is a coordinate system around p ∈ M . Thus, an arbitrary vector �eld X : M → TM induces maps ai : U → R and locally we can write X = ∑m i=1 a i ∂ ∂xi via the identi�cation X(p) = Xp. In fact, X de�nes a smooth vector �eld on U if and only if all the maps ai : U → R are smooth. See [Tu11, Lemma 14.1]. Lemma 5.8. Let X be any vector �eld on a smooth manifold M . The following are equivalent: a) X is a smooth vector �eld. b) X de�nes a smooth vector �eld on any chart (U, x1, . . . , xm). c) For any f ∈ C∞(M) the induced map Xf : M → R, p 7→ Xp(f) is smooth. Proof. See [Tu11, Propositions 14.2 and 14.3]. Remark 5.9. Point c) of the above lemma shows that we can see a smooth vector �eld X ∈ X(M) also as a derivation of the R-algebra C∞(M) since the identities X(f + rg)(p) = Xp(f + rg) = Xpf + rXpg = (Xf)(p) + r(Xg)(p) and X(fg)(p) = Xp(fg) = Xp(f)g(p)+f(p)Xp(g) = (X(f) ·g)(p)+(f ·X(g))(p) hold for any p ∈M, f, g ∈ C∞(M), r ∈ R. Consider a vector �eld X ∈ X(U) that is de�ned on a neighborhood U ⊂M of a point p. Then there is a smooth vector �eld X̃ ∈ X(M) that coincides with Y on a possibly smaller neighborhood V of p, see [Tu11, Proposition 14.4]. 24 Lemma 5.10. Let M be a smooth manifold and let p ∈ M be a point. For any tangent vector v ∈ TpM there is a smooth vector �eld X ∈ X(M) such that Xp = v. Proof. Choose a coordinate chart (U, x1, . . . , xm) around p and let a1, . . . , am be real numbers with v = ∑m i=1 a i ∂ ∂xi |p. We de�ne Xq = ∑m i=1 a i ∂ ∂xi |q for any q ∈ U . Obviously, the induced vector �eld X is smooth on U , so we can extend it smoothly to the whole manifold M . An integral curve or a trajectory of X ∈ X(M) is a smooth curve c : I → X for some open interval I ⊂ R containing 0, such that c′(t) = Xc(t) holds for all t ∈ I. We say the curve starts at the initial point c(0) ∈M . The integral curve is called maximal if for any other integral curve γ : J →M with I ⊂ J and γ |I= c we already have I = J . Observe that in a chart (U, ϕ) a smooth curve c : I → U is an integral curve of X = ∑m i=1 a i ∂ ∂xi ∈ X(U) if and only if ċi(t) = ai(c(t)) for all i = 1, . . . ,m, compare [Tu11, p.154]. Thus, the curve c is an integral curve for X with initial point p ∈ U if and only if the smooth curve (ϕ ◦ c) : I → ϕ(U) ⊂ Rm is a solution of the initial value problem γ′(t) = (a ◦ ϕ−1)(γ(t)), γ(0) = ϕ(p) ∈ ϕ(U) in Rm. Here a : U → Rm denotes the smooth function de�ned by q 7→ (a1(q), . . . , am(q)). Locally, such a curve always exists uniquely in the sense that any two of them agree on their common domain. See for example [Con01, Theorem 2.8.4 and Corollary 2.8.6.]. This extends globally to the whole manifold in the following way. Proposition 5.11. Let X ∈ X(M) be a vector �eld of a smooth manifold M . For any p ∈ M there is a unique maximal integral curve cp : Ip → M starting at p. Furthermore, the map F : Ω := ⋃ p∈M Ip × {p} →M, (t, p) 7→ cp(t) is smooth and Ω is an open subset of R×M . For any given p ∈M the interval Ip either contains [0,∞[ or cp(t) leaves any compact set as t→ sup(Ip). The same holds for ]−∞, 0] and t→ inf Ip. Proof. See [Pet, Theorem 2.2.4.]. We call the map F in the proposition above the �ow that is generated by the vector �eld X. In the case Ω = R×M we speak of a global �ow. De�nition 5.12. A vector �eld X ∈ X(M) is called complete if it generates a global �ow. We denote by Xc(M) the set of complete vector �elds on M . 25 The global �ow has the following useful property: Lemma 5.13. Let X ∈ Xc(M) be a complete vector �eld. Its global �ow F : R ×M → M ful�lls Ft ◦ Fs = Ft+s for the induced maps p 7→ Ft(p) = F (t, p) for all t, s ∈ R. Furthermore, F0(p) = p holds for every p ∈M . Each map Ft : M →M is invertible with F−1 t = F−t. Proof. Fix any s ∈ R. By de�nition Ft(Fs(p)) = Ft(cp(s)) = ccp(s)(t) so t 7→ Ft(Fs(p)) is the integral curve ofX with starting point cp(s). Consider the curve t 7→ c(t) := cp(t+ s) = Ft+s(p). For every f ∈ C∞(M) we have c′(t)(f) = (f ◦ c)′(t) = d dt |t(f ◦ c) = d dt |t+s(f ◦ cp) = c′p(t+ s)(f). So c′(t) = c′p(t + s) holds for all t ∈ R. Thus, c(0) = cp(0 + s) = cp(s) implies that c is also an integral curve of X with starting point cp(s). From uniqueness follows Ft+s = Ft ◦ Fs. By de�nition, F0(p) = cp(0) = p holds for all p ∈M . Consequently, (F−t ◦ Ft)(p) = F−t+t(p) = F0(p) = p implies F−t = F−1 t for all t,−t ∈ R. Recall that a smooth vector �eld X and a function f ∈ C∞(M) de�ne a smooth map Xf : M → R via p 7→ Xp(f). Thus, if Y ∈ X(M) is another vector �eld, the term Yp(Xf) makes sense for p ∈M . De�ne the Lie bracket [X,Y ] as follows: For any point p ∈ M and any smooth map f ∈ C∞(M) we set [X,Y ]p(f) := (XpY − YpX)(f). This de�nes again a smooth vector �eld on M , that means [X,Y ] ∈ X(M). See [Tu11, Proposition 14.10]. De�nition 5.14. Let M be a smooth manifold of dimension m. A sub- set S ⊂ M is called a regular submanifold of dimension k ≤ m, if for all p ∈ S there is a coordinate system (U, x1, . . . , xm) around p, such that∣∣{i ∈ {1, . . . ,m} | xi|U∩S ̸= 0} ∣∣ = k. We also say that S has codimension m− k. By permuting the coordinates if necessary, we can assume that ϕ |U∩S= (x1, . . . , xk, 0, . . . , 0). We call such a chart (U, ϕ) an adapted chart relative to S and set ϕS := (prRk ◦ϕ |U∩S) : U ∩ S → Rk. The pair (U ∩ S, ϕS) becomes a chart of S in the subspace topology. If {(Ui, ϕi) | i ∈ I} is a family of adapted charts on M such that all the chart maps are smooth and S ⊂ ⋃ i∈I Ui is a subset, then the family {(Ui ∩ 26 S, ϕiS) | i ∈ I} is a smooth atlas on S. Thus, S itself becomes a smooth manifold of dimension k using the notation of the de�nition above. See [Tu11, Proposition 9.4]. Consider now a smooth map f : M → N between smooth manifolds M,N of dimension dim(M) = m and dim(N) = n. We call f an immersion at p ∈M if the di�erential f∗,p : TpM → Tf(p)N is injective, and a submersion at p if it is surjective. Linearity of f∗ implies that m ≤ n (m ≥ n) if f is an immersion (a submersion) at p. Further f is an immersion (submersion) if it is an immersion (a submersion) at every point. A point p ∈ M is called a regular point and f(p) ∈ N is a regular value of the function if f∗,p : TpM → Tf(p)N is a submersion at p. Otherwise p is a critical point and f(p) is a critical value. We call the preimage f−1(c) of a point c ∈ N a level set of f . It is a regular if c ∈ N is a regular value of f . Example 5.15. Let M be a smooth manifold. a) If U ⊂M is an open subset, then the inclusion U ↪→M is a submersion and an immersion, see [Tu11, example on p.96]. This shows that a submersion does not need to be surjective in general. b) If S ⊂ M is a regular submanifold, then the inclusion S ↪→ M is an immersion by [Tu11, Theorem 11.14]. c) A point p ∈ M is a critical point of a smooth function f : M → R, if and only if ∂f ∂xi (p) = 0 for all i =, 1 . . . ,m for a chart (U, x1, . . . , xm) around p. See [Tu11, Proposition 8.23] for details. Using the fact that { ∂ ∂x1 |p, . . . ∂ ∂xm |p} is a basis for TpM this is equivalent to ∂f ∂xi (p) = 0 for every chart (U, x1, . . . , xm). The following is called regular level set theorem or implicit function theorem. Lemma 5.16. Let f : M → N be a smooth map between smooth manifolds M,N of dimensions m and n. If c ∈ N is a regular value of f and the preimage f−1(c) is not empty, then the level set f−1(c) ⊂ M is a regular submanifold of dimension m− n. In the case N = R the level set f−1(c) ⊂ M is a regular submanifold of dimension m− 1. Proof. See [Tu11, Theorems 9.8 and 9.9]. 5.2 Complex manifolds The de�nition and much of the terminology can be developed analogous to the real case replacing �Rm� by �Cm� and �smooth� by �holomorphic�. We 27 will give a short introduction and some basic terminology. A good reference is [FG02, Section IV.1]. We also use [Lee02, Section 1.1]. Let M be a topological Hausdor� space. A complex chart or complex coor- dinate system is a pair (U, ϕ) consisting of an open subset U ⊂ M and a homeomorphism ϕ : U → ϕ(U) for some open set ϕ(U) ⊂ Cm. We refer to m as the dimension of the complex chart. Two charts (U, ϕ) and (V, ψ) of the same dimension are complex compatible, if either U ∩ V = ∅ or the induced map ϕ◦ψ−1 : ψ(U ∩V ) → ϕ(U ∩V ) is holomorphic and has holomorphic in- verse (this is a map between open subsets of Cm.) A complex atlas U is a set of complex charts (U, ϕ) all of the same dimension such that the union of all U covers M and such that any two complex charts are complex compatible. An m-dimensional complex structure is the equivalence class of a complex atlas, where two atlases U and V are equivalent if all two coordinate systems (U, ϕ) ∈ U and (V, ψ) ∈ V are complex compatible. De�nition 5.17. A complex manifold of dimension m is a �rst countable topological Hausdor� space M endowed with an m-dimensional complex structure. To avoid confusion we make the following convention: If we talk of a (smooth) manifold, then we will always mean a smooth real manifold as de�ned in De�nition 5.1. Otherwise we will always use the phrase complex as above. Example 5.18. We give three easy examples. a) As a trivial example, the space Cm is a complex manifold of dimension m, its complex structure is given by (Cm, id). b) IfM is a complex manifold with complex structure U , then every open subset B ⊂ M is again a complex manifold of the same dimension. Its complex structure is given by the set of all (U ∩ B,ϕ|U∩B) for (U, ϕ) ∈ U . See [FG02, Examples 1. and 2. on p.155]. c) Let M and N be two complex manifolds of dimension m and n, then the product M × N carries the structure of a complex manifold of dimension n + m: If (U, ϕ) is a complex chart of M and (V, ψ) is a complex chart of N then (ϕ × ψ,U × V ) is a complex coordinate system of M × N , and the image ϕ(U) × ψ(V ) is an open subset of Cm × Cn. See the paragraph to product manifolds in [Lee02, Section 1.1]. 28 A continuous map f : M → N between two complex manifolds is holomorphic if for some (and therefore any [FG02, Proposition IV.1.5]) charts (U, ϕ) ofM and (V, ψ) of N containing f(U) the induced map (ψ◦f ◦ϕ−1) is holomorphic as a function between open subsets ϕ(U) ⊂ Cm and ψ(V ) ⊂ Cn. The holomorphic map f : M → N is biholomorphic or an isomorphism of complex manifolds if it has a holomorphic inverse f−1 : N →M . De�nition 5.19. A subset S of an m-dimensional complex manifold M is a complex submanifold of codimension k, if every point p ∈ S has an open neighborhood U ⊂ M and holomorphic functions f1, . . . , fk : U → C such that S ∩ U = {q ∈ U | f1(q) = . . . = fk(q) = 0} and rkp(f1, . . . , fk) = k. Here rk(f1, . . . , fk)p := rk (∂(fi◦ψ−1) ∂zj (ψ(p)) ) i=1,...,k j=1,...,m is the rank of the Jacobi matrix, which is independent of the choice of the chart (V, ψ) [FG02, p.161]. The submanifold S ⊂ M canonically becomes itself a complex manifold of dimension m− k: In fact there are charts (U, ϕ) of M covering S with ϕ = (z1, . . . , zm) such that U ∩S = {q ∈ U | zj(q) = 0 for j = m− k+1, . . . ,m} and the coordinate systems are then given by ((prCm−k ◦ϕ|U∩S), U ∩S). See [FG02, Proposition IV.1.7]. Next we de�ne the tangent space at a point p of a complex manifold M . A germ of holomorphic functions at p is an equivalence class of complex valued holomorphic functions de�ned on an open subset containing p, where two of them are equivalent if they coincide on a neighborhood of p that contains the intersection of their domains. The set of germs at p canonically has the structure of a C-algebra. A complex derivation at p is a complex valued C-linear map δ de�ned on the set of germs at p, that ful�lls the identity δ(fg) = δ(f)g(p)f(p)δ(g) (here we write shortly f and g for their corre- sponding equivalence classes). Analogously to the real case, they form a complex vector space, the tangent space TpM , and the derivations are there- fore also called tangent vectors. Choosing a complex chart (U, ϕ) and setting zj := prj ◦ϕ we obtain a basis for TpM by { ∂ ∂z1 |p, . . . , ∂ ∂zm |p} where ∂ ∂zj |p(f) is de�ned to be the usual complex derivative ∂f(ϕ(p)) ∂zj . The map Cm → TpM given by (c1, . . . , cm) 7→ ∑m j=1 cj ∂ ∂zj |p is therefore an isomorphism of complex vector spaces. See [FG02, Proposition IV.1.14]. Via the natural identi�cation R2 ∼= C every complex manifold M of dimen- sion m can be also viewed as a smooth real manifold of dimension 2m, since under this identi�cation every holomorphic function de�ned on open subsets of Cm induces canonically a smooth map on the corresponding open subset of R2m. We denote this smooth real manifold by MR. Rewriting the map ϕ of a chart (U, ϕ) of M as p 7→ (z1(p), . . . , zm(p)), every zj induces two real coordinate maps xj and yj , so the corresponding real chart is given by p 7→ (x1(p), y1(p), . . . , xm(p), ym(p)). We therefore get an isomorphism of 29 real vector spaces R2m → TpMR, (a1, b1, . . . , am, bm) 7→ m∑ j=1 aj ∂ ∂xj + bj m∑ j=1 ∂ ∂yj . But on the other hand, (a1, b1, . . . , am, bm) 7→ (a1 + ib1, . . . , am + ibm) is a canonical isomorphism R2m → Cm. As a complex vector space TpM is also one over R. Lemma 5.20. Let (U, ϕ) be a complex chart of M containing a point p. Let zj = prj ◦ϕ be the coordinates of the complex chart, and let xj , yj : M → R be the corresponding smooth maps with zj = xj + iyj. Then TpMR → TpM,  m∑ j=1 aj ∂ ∂xj |p + m∑ j=1 bj ∂ ∂yj |p  7→ m∑ j=1 (aj + ibj) ∂ ∂zj |p is an isomorphism of real vector spaces. Proof. See [Lee02, Proposition 1.1]. Thus we can calculate the underlying real tangent space of a complex man- ifold to determine complex tangent vectors TpM . Remark 5.21. In the sources we have given, the proof is done the other way around: There the basis for the corresponding real vector space TpMR is calculated �rst and from this the complex variant is deduced using the Cauchy-Riemann equations. See in [FG02, Section IV.1] or in [Lee02, Section 1.1] the corresponding paragraphs. You can also see directly: If S ⊂ M is a complex submanifold, then the isomorphism TpM → TpMR restricts to an isomorphism TpS → TpSR for p ∈ S. Let now f : M → N be a holomorphic map between two complex manifolds and let p ∈ M be a point. The di�erential of f at p is the C-linear map f∗,p : TpM → Tf(p)M de�ned like in the real case: For a tangent vector Xp ∈ TpM we de�ne (f∗,p(Xp))(g) := Xp(g ◦ f) for holomorphic functions g : V → C de�ned on an open set V ⊂ N containing f(p). 30 6 Lie algebras We collect a few facts and terminology about Lie algebras. A good source is [HN12, Section 5]. De�nition 6.1. A Lie algebra over a �eld k is a �nite dimensional k-vector space g together with a bilinear map [., .] : g× g → g, (X,Y ) 7→ [X,Y ] that has the following properties: 1) [X,X] = 0 for all X ∈ g, 2) [X, [Y,Z]] + [Z, [X,Y ]] + [Y, [Z,X]] = 0 for all X,Y, Z ∈ g. This map [., .] is called the Lie bracket of the Lie algebra and 2) is the Jacobi identity. Example 6.2. For every �eld k the vector space kn×n of n × n matrices together with [X,Y ] = XY − Y X for all X,Y ∈ kn×n is a Lie algebra. More general, for a k vector space V the set of endomorphisms End(V ) is a Lie algebra with Lie bracket [f, g] = (f ◦ g)− (g ◦ f). This is [HN12, Example 5.1.3]. See [HN12, Example 5.1.6] for more examples. A Lie subalgebra is a k-linear subspace h ⊂ g which is closed under the Lie bracket, that means [h, h] ⊂ h. A k-linear map φ : g → g′ is called a homomorphism of Lie algebras, if φ([X,Y ]) = [φ(X), φ(Y )]′ holds for all X,Y ∈ g. An ideal of g is a subalgebra h such that [g, h] lies in h. The Lie algebra is called solvable, if Dng = 0 holds for some n ∈ N, where we de�ne D0g := g and Dk+1g := [Dkg, Dkg] for all k ∈ N. By [HN12, Proposition 5.4.3] every Lie algebra has a maximal solvable ideal rad(g), the radical of g. De�nition 6.3. A Lie algebra g is called semisimple, if rad(g) = {0}. It is simple, if g and {0} are the only ideals. Every simple Lie algebra is semisimple [HN12, Lemma 5.5.2]. 31 7 Lie groups We give the de�nition of a Lie group (real and complex) and collect some basic properties. 7.1 Real Lie groups A good reference for Lie groups is the book of Hilgert and Neeb [HN12]. De�nition 7.1. A Lie group is a group G which also carries the structure of a smooth manifold such that the multiplication G×G→ G, (g, h) 7→ gh and inversion G→ G, g 7→ g−1 are smooth maps. The tangent space at the identity is the Lie algebra of the Lie group G, we denote it by g = T1G. This terminology comes from the fact that it carries in a canonical way the structure of a real Lie algebra as de�ned in the previous section, see [HN12, De�nition 9.1.7]. A homomorphism of Lie groups is a smooth map G → H that is also a group homomorphism. The exponential map of the Lie group is de�ned by expG : g → G, X 7→ γX(1), where γX : R → G is the unique homomorphism of Lie groups that ful�lls γX(0) = 1 and γ′X(0) = X. Note that [HN12, De�nition 9.2.2] coincides with ours by [HN12, Lemma 9.2.4]. Lemma 7.2 (One-parameter Group Theorem). For every X ∈ g the map R → G de�ned by t 7→ expG(tX) is a smooth group homomorphism. Con- versely, any continuous homomorphism R → G is of the form t 7→ expG(tX) for some X ∈ g. Proof. See [HN12, Theorem 9.2.15]. We call a homomorphism of Lie groups R → G a one-parameter group. The exponential map has the following useful property. Lemma 7.3. If φ : G→ H is a homomorphism of Lie groups, then we have expH ◦φ∗,1 = φ ◦ expG, where φ∗,1 : g → h is the di�erential at 1 ∈ G. Proof. See [HN12, Proposition 9.2.10]. For g ∈ G let cg : G → G be the conjugation map de�ned by cg(x) = gxg−1, this is a smooth group homomorphism. We de�ne Adg := (cg)∗,1 as the induced linear map of tangent spaces g → g. This implies Adg(X) = d dt |t=0 g expG(tX)g−1. The map Ad: g 7→ Adg is a homomorphism of groups G→ GL(g) because of cg ◦ ch = cgh, we call it the adjoint representation. Lemma 7.4. Let G0 be the connected component of 1 ∈ G. Then ker(Ad) is the center of G0. 32 Proof. See [HN12, Lemma 9.2.21]. We give an example. Example 7.5. The group G = GLn(R) of invertible real n × n matrices is a Lie group by [HN12, Example 9.1.4]. Its Lie algebra is g = Rn×n with the Lie bracket [X,Y ] = XY − Y X, and the exponential map is the usual matrix exponential exp(X) = ∑∞ k=0 Xk k! . [HN12, Example 9.2.3]. The adjoint representation is given by conjugation, this follows from Adg(X) = d dt |t=0 g exp(tX)g−1 = g ( d dt |t=0 exp(tX) ) g−1 = gXg−1. Proposition 7.6 (Closed Subgroup Theorem). Let G be a Lie group with Lie algebra g and let H be a closed subgroup of G. Then H is a submanifold of G and the restriction of the multiplication and inversion maps to H are smooth. Its tangent space at 1 is a Lie subalgebra h ⊂ g and its exponential map is expH = expG |h. Conversely, if a subgroup H of a Lie group G is again a Lie group with the induced topology, then it is closed. Proof. See [HN12, Theorem 9.3.7 and Proposition 9.3.9]. In view of this proposition we call a closed subgroup of a Lie group a Lie subgroup. Corollary 7.7. Let G be a closed subgroup of GLn(R). Then it is a Lie group, its Lie algebra g is a Lie subalgebra of Rn×n with Lie bracket given by [X,Y ] = XY − Y X. The exponential map is the usual matrix exponential restricted to g and the adjoint representation of G is given by conjugation. Proof. Follows directly from the above proposition, see also [HN12, Propo- sition 4.1.4]. We state some more facts that we will need later. Lemma 7.8. Let G be a Lie group with Lie algebra g. Then the subgroup generated by expG(g) is the connected component of G that contains the identity element. Proof. See [HN12, Lemma 9.2.9]. Let M be a smooth manifold. We call the action of a Lie group G on M smooth, if the induced map G×M →M is smooth. Lemma 7.9. Let G be a Lie group and H ⊂ G be a Lie subgroup. The quotient G/H of topological groups carries a canonical manifold structure such that the quotient map is a submersion of manifolds. Moreover, the action of G on G/H via left multiplication is smooth. Proof. See [HN12, Theorem 10.1.10]. 33 Lemma 7.10. Let G be a Lie group acting smoothly on a manifold M and let K ⊂ G be the stabilizier of a point p ∈M . Then K is a closed subgroup of G. If additionally the action is transitive, then the canonical map G/K →M is a di�eomorphism. Proof. See [HN12, Lemma 10.1.5 and Corollary 10.1.17]. An easy consequence of this is the following: If G → G′ is an isomorphism of Lie groups mapping a Lie subgroup K of G to a subgroup K ′ of G′, then the induced map G/K → G′/K ′ is a die�eomorphism. 7.2 Complex Lie groups Like in the case of smooth manifolds there is also a complex counterpart for Lie groups. We collect some facts about them. A good reference is [Lee02, Section 1.2]. De�nition 7.11. A complex Lie group is a group G which is also a complex manifold, such that the map G×G→ G, (g, h) 7→ g−1h is holomorphic. To avoid confusion we will always write complex Lie group when we mean such a structure as above. We will use the terminology Lie group only for the smooth real ones from De�nition 7.1. A homomorphism of complex Lie groups is a holomorphic map G → H between two complex Lie groups, that is also a group homomorphism. The Lie algebra g of a complex Lie group G is the tangent space T1G. This also has the canonical structure as a complex Lie algebra [Lee02, Theorem 1.7] and the di�erential of a homomorphism G → H is a homomorphism of the corresponding Lie algebras g → h [Lee02, Theorem 1.8]. The adjoint representation is the group homomorphism G → GL(g) given by g 7→ Adg, the latter is the di�erential of the conjugation automorphism cg(h) = ghg−1. Since every complex manifold is also a real manifold and holomorphic maps are smooth under the canonical identi�cation R2 ∼= C every complex Lie group G is also a (real) Lie group which we denote by GR. If g denotes the complex Lie algebra of G, then it also has the structure of a real vector space, and consequently can be viewed as a real Lie algebra. This is canonically isomorphic to the real Lie algebra gR that comes from the Lie group GR under the map from Lemma 5.20, see [Lee02, Proposition 1.9]. Further we de�ne the exponential map of G by expG(X) := expGR(X) for every X ∈ g using the identi�cation g = gR. Viewing this as a map exp: g → G the exponential is holomorphic, see [Lee02, Theorem 1.15] (the complex vector space g has a canonical structure as a complex manifold). 34 Example 7.12. The group G = GLn(C) is a complex manifold as it is an open subset of Cn×n given by det−1(C∖{0}). Multiplication and inversion are holomorphic and its Lie algebra is g = Cn×n, the corresponding Lie bracket is [X,Y ] = XY −Y X. The exponential map is the usual matrix exponential exp(X) = ∑∞ k=0 Xk k! and the adjoint action is given by conjugation Adg(X) = gXg−1 like in the real case. See the corresponding example in [Lee02, Section 1.3]. A complex Lie subgroup of a complex Lie group G is a subgroup H which is also a complex submanifold of G, and that is again a complex Lie group with this structure. Then HR is also a (real) Lie subgroup of GR and therefore is closed by the Closed Subgroup Theorem. Furthermore, the exponential map of H is expH = expG |h where h is the Lie algebra of H and Proposition 7.6 also implies that hR is a real Lie subalgebra of gR. Consequently, h must be a complex Lie subalgebra of g (the isomorphism gR → g of real vector spaces restricts to an isomorphism hR → h). Remark 7.13. In the real case to be a closed subgroup was already su�cient to be a submanifold which is again a Lie group with this structure. In the complex case, however, this is not true as the following simple example shows: The subgroup (R,+) of (C,+) is a closed and a real Lie subgroup, but surely R ⊂ C is not a complex submanifold. From Example 7.12 now directly follows: Corollary 7.14. Let G be a complex Lie subgroup of GLn(C). Then its Lie algebra g is a Lie subalgebra of Cn×n with the Lie bracket [X,Y ] = XY −Y X, the exponential map is given by the matrix exponential exp, and the adjoint action Adg is via conjugation. Example 7.15. Here are some examples of complex Lie subgroups ofGLn(C), they are taken from [Lee02, Section 1.3]. a) The group G = SLn(C) is a complex Lie subgroup of GLn(C) with Lie algebra g = {X ∈ Cn×n | tr(X) = 0}. Note that it is connected. b) The subgroup Bn(C) ≤ GLn(C) of upper triangular invertible matrices is a complex Lie subgroup. Its Lie algebra is the set of all upper triangular matrices in Cn×n. c) Furthermore, the group Un(C) of upper triangular matrices with 1 on the diagonal is also a complex Lie subgroup of GLn(C), which has the set of all upper triangular matrices with 0 on the diagonal as Lie algebra. Note that it is also a complex subgroup of Bn(C). 35 8 Riemannian manifolds Here we introduce some additional structure on a manifold, a so called Rie- mannian metric. This can be used to study manifolds under lots of geometric aspects. But it also allows us to de�ne more smooth structure like the gra- dient of a function. 8.1 Riemannian geometry A good introduction to this topic is the book of do Carmo [dC92], another one is [Pet16]. We will only introduce some very basic terminology that we need in this thesis, because we want to view a Riemannian manifold as a metric space and work with the metric directly. To learn about all the other important aspects like the exponential function, curvature and so on, see the references already mentioned. The interplay between the curvature in the sense of the Riemannian manifold and in the sense of the induced metric space is analyzed in [BH99, Appendix to Chapter II.1]. Throughout the whole subsection let M be a smooth manifold of dimension m. De�nition 8.1. A Riemannian metric or a Riemannian structure on M is a map ⟨., .⟩ : M → ⋃ p∈M {TpM × TpM → R}, p 7→ ⟨_,_⟩p such that ⟨_,_⟩p is a scalar product on the real vector space TpM and such that for all X,Y ∈ X(M) the induced map ⟨X,Y ⟩ : M → R, p 7→ ⟨Xp, Yp⟩p is smooth. We call a such a pair (M, ⟨., .⟩) a Riemannian manifold. Every smooth manifold can be endowed with a Riemannian structure, see [dC92, Proposition 1.2.10]. Using the Riemannian structure we can de�ne a lot of geometric terminology on the manifold M : For a point p ∈ M and a tangent vector v ∈ TpM de�ne the norm or length as ∥v∥p = √ ⟨v, v⟩p. Let c : [a, b] → M be a smooth curve, then the arc length of c is de�ned as ℓ(c) = ∫ b a ∥c ′(t)∥c(t)dt. Using this we can also de�ne the arc length of a piecewise smooth curve γ : [a, b] → M by ℓ(γ) = ∑k i=0 ℓ(γ|[ti,ti+1]) if γ is smooth on all intervals [ti, ti+1] for i = 0, . . . , k − 1. De�ne the arc length function of γ by s(t) = ℓ(γ|[a,t]) for t ∈ [a, b]. Furthermore, we de�ne the set Ωpg = {c : [0, 1] →M | c(0) = p, c(1) = q, c is piecewise smooth} and set d(p, q) = inf{ℓ(c) | c ∈ Ωpg} for any p, q ∈ M . This gives us a map d : M ×M → R. One can show that (M,d) is a metric space and that the topology on M induced by the metric d coincides with the original manifold topology. See [dC92, Propositions 7.2.5 and 7.2.6] for details. 36 Example 8.2. At this point we give some examples of Riemannian mani- folds. a) Let M = Rm together with the trivial chart (Rm, id). A Riemannian structure is given by 〈 ∂ ∂xi |p, ∂ ∂xj |p 〉 p = δij for all p ∈ Rm. So for any piecewise smooth curve c : [a, b] → Rm we have∥∥c′(t)∥∥2 c(t) = 〈 m∑ i=1 ċi(t) ∂ ∂xi |c(t), m∑ j=1 ċj(t) ∂ ∂xj |c(t) 〉 = m∑ i,j=1 ċi(t)ċj(t) 〈 ∂ ∂xi |c(t), ∂ ∂xj |c(t) 〉 = m∑ i=1 (ċi(t))2. Thus, the metric on Rm coming from the Riemannian structure coin- cides with the Euclidean metric. Compare [dC92, Example 1.2.4]. b) Let f : M → N be an immersion between smooth manifolds, where N is additionally assumed to be a Riemmannian manifold. Then ⟨u, v⟩p := ⟨f∗(u), f∗(v)⟩f(p) also de�nes a Riemannian structure on M , compare [dC92, Example 1.2.5]. An application of this is the following: Consider the smooth map h : Rm → R, x 7→ (∑m i=1 x 2 i ) − 1, then Sm−1 = h−1(0) is a regular submanifold by the implicit function theorem (Lemma 5.16). The in- clusion Sm−1 ↪→ Rm is an immersion by Example 5.15. So we can pull back the Riemannian structure on Rm via f like indicated above. The induced metric is the usual angular metric on Sm−1, see [BH99, Proposition I.6.17]. c) In Example 5.2 we have seen that Hn = Γ(f) ⊂ Rn × R is a smooth manifold for f : Rn → R, x 7→ √∑n i=1 x 2 i + 1 with an atlas consisting of one chart ϕ : Γ(f) → Rn, (x, f(x)) 7→ x. Thus, the graph Γ(f) ⊂ Rn × R is a regular submanifold of codimension 1 and the inclusion Γ(f) ↪→ Rn × R is an immersion by Example 5.15. We de�ne for p ∈ Rn × R = Rn+1 the symmetric bilinear form〈 ∂ ∂xi |p, ∂ ∂xj |p 〉 = { −1 if i = n+ 1 = j δij else. This surely is not positive-de�nite on Tp(Rn+1) but the restriction to TpHn is for any p ∈ Hn, see [BH99, p. 93] for details. The metric induced by this Riemannian structure on Hn coincides with the hyperbolic metric as de�ned in c) of Example 3.4, see [BH99, Proposition I.6.17]. 37 d) We also have seen that the product M × N of two smooth manifolds M,N is again a smooth manifold and one can show that the projection on each of the factors is a smooth map, see [Tu11, Example 6.12]. If both factors also have a Riemannian structure, then a Riemannian metric on the product is de�ned by ⟨u, v⟩(p,q) := ⟨ prM∗(u), prM∗(v)⟩p + ⟨ prN∗(u), prN∗(v)⟩q for (p, q) ∈ M × N and u, v ∈ T(p,q)(M × N). See [dC92, Example 1.2.7]. A di�eomorphism f : M → N between two Riemannian manifolds is called a Riemannian isometry if ⟨f∗(u), f∗(v)⟩f(p) = ⟨u, v⟩ holds for all u, v ∈ TpM . Further, f is a local Riemannian isometry if every point p ∈M has a neigh- borhood U ⊂ M such that the restriction f |U is a Riemannian isometry. It is not hard to see that any Riemannian isometry is also an isometry in the sense of metric spaces. But the converse is also true by a theorem of Myers-Steenrod: Proposition 8.3. Let M,N be Riemannian manifolds and let f : M → N be a bijection with d(f(p), f(q)) = d(p, q) for all p, q ∈ M . Then f is also a Riemannian isometry. Proof. See [Pet16, Theorem 5.6.15]. We call a Riemannian manifold complete if the induced metric space is com- plete. By the theorem of Hopf-Rinow a Riemannian manifold is complete if and only if the corresponding metric space is proper, see [dC92, Theorem 7.2.8]. The following provides a construction of a complete Riemannian manifold that we will work with later. Example 8.4. a) The set Sn(R) of symmetric n × n matrices with en- tries in R is a real vector space of dimension n(n+1) 2 , and therefore the open subset Pn(R) of all positive-de�nite matrices is a manifold of the same dimension. For every p ∈ Pn(R) the tangent space TpPn(R) is canonically given by Sn(R) and ⟨X,Y ⟩p = tr(p−1Xp−1Y ) for X,Y ∈ Sn(R) de�nes a Riemannian metric structure on Pn(R). The action of GLn(R) on Pn(R) given by g.p = gpg⊤ extends canonically to Sn(R) via g.X = gXg⊤. It is smooth and via Riemannian isometries. Fur- ther, it is transitive and the stabilizer of the unit matrix 1n is exactly O(n), the orthogonal group. Therefore, we have a natural identi�cation GLn(R)/O(n) ∼= Pn(R) via gO(n) 7→ gg⊤. See [BH99, II.10.31-34] for all of these statements. The subgroup O(n) of GLn(R) is maximal com- pact [BH99, Corollary II.10.41], and the Riemannian manifold Pn(R) is 38 a symmetric space by [BH99, Proposition II.10.34]. As a metric space it is proper and CAT(0) [BH99, Theorem II.10.39]. The geodesic lines c : R → Pn(R) with c(0) = p are exactly the maps t 7→ g exp(tX)g⊤ for g ∈ GLn(R) ful�lling gg⊤ = p and X ∈ Sn(R) with tr(X2) = 1. If ξ = c(∞) is the corresponding point in the boundary, then its stabilizer is Stab(ξ) = {g ∈ GLn(R) | lim t→∞ exp(−tX)g exp(tX) exists}. Consider the special case of the above X being a diagonal matrix, and let λ1 > . . . > λk be the di�erent entries on the diagonal, such that every λi shows up exactly ri times. Then g ∈ GLn(R) is an element of Stab(ξ) if and only if g =  a11 . . . . . . a1k 0 a22 . . . a2k ... . . . . . . ... 0 . . . 0 akk  where aij is an ri × rj matrix. See [BH99, Proposition II.10.64]. The corresponding Busemman functions are all smooth by [BH99, Propo- sition II.10.69]. b) More generally, let G ⊂ GLn(R) be a closed subgroup that is also closed under matrix transposition, that means g⊤ ∈ G holds for all g ∈ G. Then G is a Lie subgroup of GLn(R). Further we assume that the group to have the following property: ∀X ∈ Sn(R) : exp(X) ∈ G⇒ exp(RX) ⊂ G. (8.1) The space M := G ∩ Pn(R) is a totally geodesic Riemannian subman- ifold of Pn(R), which is therefore itself a complete CAT(0) symmetric space. The subgroup K := G ∩ O(n) is maximal compact and the maps G/K → M de�ned by gK 7→ gg⊤ and M × K → G de�ned by (m, k) 7→ mk are di�eomorphisms. Especially, M is the G orbit of 1n. See [BH99, Theorem II.10.58] for all of this. We further see that the action of G on M is proper by Lemma 2.8 as the quotient map G→ G/K is open. A criterion for the property in (8.1) is provided by [BH99, Lemma II.10.59]: If G = {g ∈ GLn(R) | ∀f ∈ F : f(g) = 0} for a �nite set of polynomials F with real coe�cients in n×n variables, then the desired property is ful�lled. 8.2 Vector �elds de�ned on Riemannian manifolds So far we only cared about metric aspects. But the Riemannian structure also has e�ects on smooth vector �elds and enables us to construct the gra- dient of a function. We deal with that in the following. 39 Lemma 8.5. Let M be a complete Riemannian manifold and let X ∈ X(M) be a smooth vector �eld. If the norm on X is uniformly bounded, then X is a complete vector �eld. Proof. Let C > 0 be a constant such that ∥X∥ ≤ C. Let further c : I → M be any integral curve of X starting at the point c(0) ∈ M . For every t ∈ I ∩ [0,∞[ we have d(c(0), c(t)) ≤ ∫ t 0 〈 c′(s), c′(s) 〉 c(s) ≤ C2 · t where d is the metric on M induced by the Riemannian structure. Thus, if T = sup(I) were a real number, the image c([0, T [) would be a subset of BC2T (c(0)), which is compact by completeness. This contradicts Proposi- tion 5.11. The same argument shows ]−∞, 0] ⊂ I. For every smooth function f ∈ C∞(M) on the Riemannian manifold M there is a unique vector �eld ∇f ∈ X(M) such that ⟨X,∇f⟩ = X(f) for every X ∈ X(M), see [Con01, Lemma 4.2.1.]. De�nition 8.6. This vector �eld ∇f is called the gradient or gradient vector �eld of f with respect to the Riemannian metric. Let us have a further look at the gradient vector �eld. When we write ∇f = ∑m j=1 d j ∂ ∂xj in a chart (U, ϕ), then for any other vector �eld X =∑m i=1 a i ∂ ∂xi ∈ X(U) we see ⟨X,∇f⟩ = m∑ i,j=1 aidjgij = m∑ i=1 ai  m∑ j=1 gijd j  for gij = 〈 ∂ ∂xi , ∂ ∂xj 〉 . On the other hand, X(f) = m∑ i=1 ai ∂f ∂xi . Since X ∈ X(M) was arbitrary, this leads to the system of linear equations ∂f ∂xi = m∑ j=1 gijd j i=1,. . . ,m so the coe�cients of the gradient are given byd1 ... dm  = G−1 ·  ∂f ∂x1 ... ∂f ∂xm  where G = (gij)ij=1,...,m. 40 Remark 8.7. This shows especially that the above de�nition of the gradient is a generalization of the standard de�nition of the gradient on Rm, because in this case G is the unit matrix. One can further deduce that the gradient of constant functions is equal to 0. This and the following observation will be useful later. Proposition 8.8. Let M be a Riemannian manifold and let fj ∈ C∞(M) and λj ∈ R for j = 1, . . . ,m. a) If f = ∑n j=1 λjfj, then ∇f = ∑n j=1 λj∇fj. b) If f = f1 · f2, then ∇f = ∇f1 · f2 + f1 · ∇f2. c) If f : M → R and g : R → R are smooth functions, then ∇(g ◦ f) = (g′ ◦ f) · ∇f . Proof. It su�ces to prove these statements in an arbitrary chart (U, ϕ). Let ∇f = ∑m i=1 d i ∂ ∂xi and ∇fj = ∑m i=1 d i j ∂ ∂xi in this chart. Furthermore, let G = (gij)ij=1,...,m for gij = 〈 ∂ ∂xi , ∂ ∂xj 〉 . If f = ∑n j=1 λjfj , then the fact ∂f ∂xi = ∑n j=1 λj ∂fj ∂xi showsd1 ... dm  = G−1 ·  ∂f ∂x1 ... ∂f ∂xm  = G−1 · n∑ j=1 λj  ∂fj ∂x1 ... ∂fj ∂xm  = n∑ j=1 λj d1j ... dmj  which proves a). But in a similar way assuming f = f1 · f2 the fact ∂f ∂xi = ∂f1 ∂xi · f2 + f1 · ∂f2∂xi shows d1 ... dm  = G−1 ·  ∂f ∂x1 ... ∂f ∂xm  = G−1 ·   ∂f1 ∂x1 ... ∂f1 ∂xm  · f2 + f1 ·  ∂f2 ∂x1 ... ∂f2 ∂xm   = G−1 ·  ∂f1 ∂x1 ... ∂f1 ∂xm   · f2 + f1 · G−1 ·  ∂f2 ∂x1 ... ∂f2 ∂xm   = d11 ... dm1  · f2 + f1 · d12 ... dm2  41 and this proves b). For c) we calculate G−1 ·  ∂(g◦f) ∂x1 ... ∂(g◦f) ∂xm  = G−1 · (g′ ◦ f) · ∂f ∂x1 ... (g′ ◦ f) ∂f ∂xm  = (g′ ◦ f) ·G−1 ·  ∂f ∂x1 ... ∂f ∂xm  which proves that the coe�cient vectors of ∇(g◦f) and (g′◦f) ·∇f coincide. Remark 8.9. It is possible to prove the proposition without choosing a coordinate system, but using the de�ning property of the gradient. Let f = ∑n i=1 λifi be a linear combination of smooth functions as in a) then ⟨X,∇f⟩ = X(f) = X ( n∑ i=1 λifi ) = n∑ i=1 λiX(fi) = n∑ i=1 λi ⟨X,∇fi⟩ = 〈 X, n∑ i=1 λi∇fi 〉 holds for all X ∈ X(M). We obtain ∇f = ∑n i=1 λi∇fi from uniqueness. The proof of part b is similar: Again, let X be any smooth vector �eld then ⟨X,∇(f1f2)⟩ = X(f1f2) = X(f1)f2 + f1X(f2) = ⟨X,∇f1⟩ · f2 + f1 · ⟨X,∇f2⟩ = ⟨X,∇f1 · f2⟩+ ⟨X, f1 · ∇f2⟩ = ⟨X,∇f1 · f2 + f1 · ∇f2⟩ implies ∇(f1f2) = ∇f1 · f2 + f1 · ∇f2. In c) we get the assertion from ⟨X,∇(g ◦ f)⟩ = X(g ◦ f) = (g ◦ f)∗(X) = (g′ ◦ f) · f∗(X) = (g′ ◦ f) ·X(f) = (g′ ◦ f) · ⟨X,∇f⟩ = 〈 X, (g′ ◦ f) · ∇f 〉 . Here we write f∗(X) for the induced map p 7→ f∗,p(Xp). 42 9 Root systems This paragraph collects some facts about root systems. We �x a real vector space V . An automorphism rα ∈ Aut(V ) with rα(α) = −α that �xes a hyperplane H in V pointwise is called a re�ection with respect to a vector α ̸= 0V . This property makes it unique. De�nition 9.1. An abstract root system is a �nite subset ϕ of V such that the following hold: a) The set ϕ generates V as a real vector space and does not contain 0V . b) For every α ∈ ϕ there is a re�ection rα that stabilizes ϕ as a set. c) For all α, β ∈ ϕ there is an integer nβ,α ∈ Z with rα(β) = β − nβ,αα. The elements of ϕ are called roots. We call the root system ϕ reduced if for all roots α and every a ∈ [−1, 1] from aα ∈ ϕ follows a ∈ {±1}. We make the convention every root system that we consider is reduced. The group W = W (ϕ) := ⟨rα | α ∈ ϕ⟩ is the Weyl group of ϕ. By b) of the de�nition above holds w.ϕ = ϕ for all w ∈W . Lemma 9.2. We can endow V with a scalar product ⟨., .⟩, such that the Weyl group acts via orthogonal transformations, that is ⟨w.x,w.y⟩ = ⟨x, y⟩ holds for all w ∈W and x, y ∈ V . Proof. See [Mil11, Proposition III.1.9]. We refer to such a ⟨., .⟩ as Weyl-invariant scalar product, and this makes (V, ⟨., .⟩) a Euclidean space. A subset ψ of ϕ is closed if α, β ∈ ψ ⇒ α+β ∈ ψ holds. A basis of ϕ is a subset ∆ ⊂ ϕ such that a) ∆ is a basis of V (in the sense of vector spaces), b) every β ∈ ϕ is a Z-linear combination β = ∑ α∈∆mα,βα such that all mα,β are either non-negative or non-positive. One can show that a basis always exists [Mil11, Proposition III.1.10]. Once it has been chosen we refer to the elements of ∆ as simple roots. If a root β = ∑ α∈∆mα,βα has all mα,β ≥ 0 then we call it a positive root. We write ϕ+ for the set of positive roots. Obviously we have ∆ ⊂ ϕ+ ⊂ ϕ. Lemma 9.3. With the above notation the following are equivalent for a subset ψ ⊂ ϕ: a) ψ is a closed subset and ψ ∪ −ψ is a partition of ϕ. 43 b) There is a vector x ∈ V such that ψ = {α ∈ ϕ | ⟨α, x⟩ > 0}. c) There is an order on V making it an ordered vector space such that each α ∈ ϕ is either positive or negative and ψ = V + ∩ ϕ for the set V + of positive elements. Proof. See [Bou05, Corollaries 1 and 2 in VI.1.7]. Such a subset ψ we the call system of positive roots in ϕ. This is also a set of positive roots in the previous sense, because the set of all α ∈ ψ with α ̸= β + γ for all β, γ ∈ ψ is then a basis of the root system ϕ making ψ its set of positive roots. Consequently, systems of positive roots and bases of the root system are in one to one relation. For all roots α, β ∈ ϕ we have ⟨α, β⟩ = ⟨rα(α), rα(β)⟩ = ⟨ − α, β − nβ,αα⟩ = −⟨α, β⟩+ nβ,α ⟨α, α⟩ and from this follows 2 ⟨α, β⟩ ⟨α, α⟩ = nβ,α ∈ Z. Thus rα(β) = β− 2⟨α,β⟩ ⟨α,α⟩ α. Furthermore, if α and β are di�erent simple roots, then ⟨α, β⟩ ≤ 0, because otherwise the root rα(β) = β−2 ⟨α,β⟩ ⟨α,α⟩α would be a linear combination of two simple roots with coe�cients of di�erent signs. Thus we have: Lemma 9.4. For all simple roots α, β ∈ ∆ the following hold: a) ⟨α, β⟩ ≤ 0 if α ̸= β, b) 2 ⟨α,β⟩ ⟨α,α⟩ is an integer. One could also start with a Euclidean vector space, and then de�ne a root system via re�ections with respect to the corresponding scalar product. See for example [AB08, Appendix B]. We give one more de�nition and state two consequences about root systems from [Rag68]. De�nition 9.5. For every simple root α we de�ne the corresponding fun- damental weight as the unique vector λα ∈ V such that ⟨λα, β⟩ = δα,β holds for all β ∈ ∆. To simplify the notation, we write ∆ = {α1, . . . , αr} for the set of simple roots. The fundamental weights are denoted by λj for j = 1, . . . , r with the property ⟨αi, λj⟩ = δij . 44 Lemma 9.6. Let I be any subset of {1, . . . , r}. Writing λj = ∑ i∈I aiαi +∑ i/∈I biλi then all the ai and bi are greater than or equal to 0. Proof. See [Rag68, Lemma 1.1]. Lemma 9.7. Let Λ = ∑r j=1mjλj be any linear combination of the funda- mental weights with all mj > 0. For every subset I of {1, . . . , r} we have Λ = ∑ j∈I mIjλj + ∑ i/∈I nIiαi with all mIj ≥ mj > 0 and all nIi ≥ 0. Setting ΛI = ∑ j∈I mIjλj we get that ΛI∪{i} − ΛI = ∑r k=1 cIikαk is always a non-negative linear combination of the simple roots. For i /∈ I we obtain cIii > 0. Proof. See [Rag68, Lemma 3.1 and the text preceding it]. 45 10 A�ne algebraic varieties We give a very short and naive introduction to the topic, as we will only need few basic results. Here we are guided by the description in [CSM95, Section III.1]. A standard reference is [Spr98, Sections 1.1-1.3]. We will only consider a�ne varieties over the �eld of complex numbers C. Let Am = C[T1, . . . , Tm] be the ring of complex polynomials in m variables T1, . . . , Tm. For a subset F ⊂ Am we de�ne V (F ) = {x ∈ Cm | ∀f ∈ F : f(x) = 0} and call this an algebraic subset of Cm. One easily deduces V (∅) = Cm and V (Am) = ∅. Furthermore, for a family (Fi)i∈I of subsets of Am we have ⋂ i∈I V (Fi) = V ( ⋃ i∈I Fi). For two subsets F,G ⊂ Am also holds V (F )∪V (G) = V (F ·G), see [CSM95, III.(1.1)]. Thus, the algebraic subsets of Cm ful�ll the axioms of a family of closed subsets of a topology on Cm. De�nition 10.1. We call the topology on Cm whose closed sets are exactly the algebraic subsets the Zariski topology. A Zariski closed subset V of Cm endowed with the corresponding relative topology is called an a�ne algebraic variety. It is immediate from the de�nition that the Zariski topology is coarser than the usual Euclidean topology on Cm, as every Zariski closed subset of Cm must also be closed in the standard topology. For any subset S ⊂ Cm we set I(S) = {f ∈ Am | ∀x ∈ S : f(x) = 0}. This is clearly an ideal in Am, which is �nitely generated by Hilbert's basis theorem. Lemma 10.2. The Zariski closure of a subset S ⊂ Cm is V (I(S)). Proof. We �rst prove that a set S is Zariski closed if and only if V (I(S)) = S. If V (I(S)) = S holds, then S clearly is closed in the Zariski topology. So let S = V (F ) be Zariski closed, we show that this implies S = V (I(S)). On the one hand, if x is an element of S, then f(x) = 0 for all f ∈ I(S) by de�nition, and therefore we have x ∈ V (I(S)). On the other hand, if y lies in V (I(S)), we obtain f(y) = 0 for all f ∈ F from F ⊂ I(S). So in this case we have y ∈ V (F ) = S. Obviously, if S is a subset of T we have I(T ) ⊂ I(S), and therefore V (I(S)) ⊂ V (I(T )) holds by [CSM95, (1.1)]. Thus, if T is the Zariski closure of S, then we get T ⊂ V (I(S)) ⊂ V (I(T )) = T , which implies T = V (I(S)). If X ⊂ Cm is an algebraic variety, then C[X] = Am/I(X) is the coordinate ring of X. 46 Remark 10.3. The usual de�nition of an a�ne algebraic variety goes as fol- lows, see [CSM95, p.142]: A map φ : X → C is called a regular function if there is a polynomial f ∈ Am with φ(x) = f(x) for all x ∈ X. Via pointwise addition and multiplication the set C[X] of regular functions on X forms a C-algebra. Every polynomial f ∈ Am canonically induces a regular function φf on X via φf (x) = f(x) and one can show that the map f 7→ φf is a homomorphism of C-algebras Am → C[X]. Its kernel is exactly I(X) so we get an isomorphism Am/I(X) ∼= C[X]. Under this identi�cation, evaluation at x ∈ X de�ned by evx : C[X] → C, evx(f) = f(x) de�nes a map from X to HomC(C[X],C) the set of all C-algebra morphisms, which in fact is a bijection. The closed subsets of X are then exactly VX(F ) = {x ∈ X | ∀f ∈ F ∈ : f(x) = 0} for F ⊂ C[X]. More generally than before one calls the pair (X,C[X]) an a�ne algebraic variety. But for our purposes, the previous naive de�nition is su�cient. We give some more terminology. A morphism between two a�ne varieties X ⊂ Cm and Y ⊂ Cn is a map X → Y given by x 7→ (f1(x), . . . , fn(x)) for polynomials f1, . . . , fn ∈ Am. The morphism is an isomorphism if it is bijective and its inverse map is also a morphism of varieties. The product X × Y is again an a�ne variety and its coordinate ring is C[X × Y ] ∼= C[X]⊗C C[Y ], see [CSM95, III.(1.9)]. Recall now that a topological space is irreducible if it is not the union of two proper non-empty closed subsets. Convention. For the rest of this section let X ⊂ Cm be an a�ne algebraic variety that is irreducible in the Zariski topology. Lemma 10.4. In the corresponding Euclidean topology (the relative topology coming from Cm endowed with the standard topology) X is connected. Proof. See [PR94, Theorem 3.5, p.118]. By [Spr98, Proposition 1.2.5] the ring C[X] is an integral domain. The dimension of X is the transcendence degree of the quotient �eld of C[X] over C. We write dim(X) for its dimension. Lemma 10.5. Let X,Y ⊂ Cm be irreducible varieties, such that Y is a proper subset of X. Then dim(Y ) < dim(X). Proof. See [Spr98, Proposition 1.8.2]. From this also follows that dim(X) ≤ m holds, because of dim(Cm) = m. Furthermore, if X0 ⊊ X1 ⊊ . . . ⊊ Xn = X is an ascending chain of closed irreducible varieties, then n ≤ dim(X). 47 A vector v ∈ Cm is a tangent vector at x ∈ X if for all f ∈ I(X) the polynomial h(T ) := f(x + Tv) is contained in the ideal of C[T ] generated by T 2. The set of all such v is the tangent space at x, written as TxX. By [Sha69, �II.1.2] a vector v ∈ Cm is a tangent vector at x if and only if ∀f ∈ I(X) : dxf(v) := m∑ i=1 vi ∂f ∂Ti |x = 0 where ∂f ∂Ti |x is the formal derivative of the polynomial f ∈ C[T1, . . . , Tm] evaluated at x. Since the map v 7→ dxf(v) is linear, this shows that the tangent space is indeed a complex vector space. In fact, by Hilbert's basis theorem we can write TxX = ⋂r i=1 ker(dxfi) for some polynomials f1, . . . , fr generating C[X]. A point x of X is called simple if dim(X) = dim(TxX). We give a criterion for that. Lemma 10.6. A point x ∈ X is simple if and only if there are polynomials f1, . . . , fr ∈ C[T1, . . . , Tm], where r = m − dim(X), and a Zariski open subset U ⊂ Cm with x ∈ {y ∈ U | ∀i = 1, . . . , r : fi(y) = 0} and such that the complex matrix a with entries aij = ( ∂fi∂Tj |x) for i = 1, . . . , r and j = 1, . . . ,m has rank r. Proof. See [PR94, Proposition 2.22, p.97]. The irreducible variety X is called smooth if every point x ∈ X is simple. In fact, as every Zariski open subset U of Cm is also open in the Euclidean topology, the above lemma shows that a smooth complex variety X is a complex submanifold of Cm of codimension r. 48 11 Algebraic matrix groups In this section we de�ne algebraic matrix groups, which are a special case of so-called linear algebraic groups. In the �rst subsection we de�ne them as a Zariski closed subgroup of some GLn(C). As such we can also view them as subvarieties of some Cm. Later we will see that this canonically induces a functor into the category of groups. 11.1 Algebraic matrix groups as complex linear groups Here we mainly follow the presentation in [PR94, Section 2.1], another good introduction is given by [CSM95, Chapter III]. Standard references for linear algebraic groups (although by far too general for our goals) are [Bor91] and [Spr98]. De�nition 11.1. We call a subgroupG ≤ GLn(C) an algebraic matrix group if there is a subset F of C[(Ti,j)i,j=1,...,n], the ring of polynomials in n × n variables, such that G = {g = (gij)i,j=1,...,n ∈ GLn(C) | ∀f ∈ F : f(g) = 0}. Equivalent to this de�nition is to say that G is (relatively) closed in the Zariski topology induced on GLn(C) ⊂ Cn×n. A trivial example is GLn = GLn(C) (here the set F can be chosen to be empty). We want to view G not only as a closed subgroup of GLn(C), but also as an a�ne variety. Therefore, we use the inclusion ι : GLn(C) ↪→ GLn+1(C), g 7→ ( g 0n×1 01×n det(g−1) ) whose inverse is clearly ρ : ι(GLn(C)) → GLn(C), x 7→ (xij)ij=1,...,n. Under this identi�cation GLn(C) is the intersection of the two Zariski closed subsets {x ∈ C(n+1)×(n+1) | ∀i, j = 1, . . . , n : xi,n+1 = 0 = xn+1,j} and {x ∈ C(n+1)×(n+1) | det((xij)i,j=1,...,n) · xn+1 − 1 = 0} and so ι(G) ⊂ GLn+1(C) must be Zariski closed, too. It follows directly that we can write ι(G) as the subset V ( F ∪ {xi,n+1 | i = 1, . . . , n} ∪ {xn+1,j | j = 1, . . . , n} ∪ {det((xi,j)i,j=1,...,n)xn+1,n+1 − 1} ) 49 of C(n+1)×(n+1) (here we view F as a set of polynomials in (n+1)×(n+1) vari- ables using the canonical inclusion C[T1,1, . . . , Tn,n] ⊂ C[T1,1, . . . , Tn+1,n+1]). Thus, ι : G → ι(G) is a homeomorphism with respect to the induced Zariski topologies, which is inverse to ρ : ι(G) → G. Remark 11.2. As closed subgroups of GLn(C) and GLn+1(C) the groups G and ι(G) are in fact Lie groups, and the identi�cation maps ι and ρ provide isomorphisms of Lie groups. The identi�cation therefore transfers not only algebraic but also topological structures. We shall see later that G is even a complex Lie group holomorphic to ρ(G). The coordinate ring of an algebraic matrix group G is I[G] = C[(Tij)i,j=1,...,n+1]/I(G) where I(G) = {f ∈ C[(Tij)i,j=1,...,n+1] | f(ι(G)) = 0}. A morphism (of algebraic groups) between two algebraic matrix groups G ⊂ GLn(C) and H ⊂ GLm(C) is a group homomorphism that is de�ned by polynomials, that is g 7→ (fi,j(g))i,j=1,...,m for some complex polynomials fi,j in n× n variables. A morphism is an isomorphism (of algebraic groups) if it is an isomorphism of groups and its inverse is also given by polynomials. Remark 11.3. This also means that any (iso-)morphism between algebraic matrix groups induces an (iso-)morphism of the corresponding a�ne vari- eties. As already mentioned before, our terminology of an algebraic matrix group is a special case of a so-called linear algebraic group. The most naive de�nition of that would be like above, but considering isomorphism classes of algebraic matrix groups. However, we do not want to do that here, because we will later work with a concrete subgroup of some GLn(C). Another way to de�ne linear algebraic groups is to start with a group whose underlying set is an a�ne variety and whose multiplication and inversion are morphisms of varieties. But all such groups embed into a closed subgroup of some GLn, see for example [Bor91, Proposition 1.10]. Let now k be a sub�eld of C (for example k = Q.) The algebraic matrix group G is de�ned over k or a k-group if the ideal I(G) is generat