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Symmetric Spaces of the Non-Compact Type : Lie Groups

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Symmetric Spaces of the Non-Compact Type : Lie Groups SYMMETRIC SPACES OF THE NON-COMPACT TYPE : LIE GROUPS by Paul-Emile PARADAN Abstract. — In these notes, we give first a brief account to the theory of Lie groups. Then we consider the case of a smooth manifold with a Lie group of symmetries. When the Lie group acts transitively (e.g. the manifold is homogeneous), we study the (affine) invariant connections on it. We end up with the particuler case of homogeneous spaces which are the symmetric spaces of the non-compact type. R´esum´e (Espaces sym´etriques de type non-compact : groupes de Lie) Dans ces notes, nous introduisons dans un premier les notions fon- damentales sur les groupes de Lie. Nous abordons ensuite le cas d’une vari´et´ediff´erentiable munie d’un groupe de Lie de sym´etries. Lorsque le groupe de Lie agit transitivement (i.e. la vari´et´eest homog`ene) nous ´etudions les connexions (affines) invariantes par ce groupe. Finalement, nous traitons le cas particulier des espaves sym´etriques de type non- compact. Contents 1. Introduction..................................... .... 2 2. Lie groups and Lie algebras: an overview............. 2 3. Semi-simple Lie groups............................. .. 23 4. Invariant connections............................. ... 33 5. Invariant connections on homogeneous spaces. 36 References......................................... ..... 46 2000 Mathematics Subject Classification. — 22E15, 43A85, 57S20. Key words and phrases. — Lie group, connection, curvature, symmetric space. 2 PAUL-EMILE PARADAN 1. Introduction This note is meant to give an introduction to the subjects of Lie groups and of equivariant connections on homogeneous spaces. The final goal is the study of the Levi-Civita connection on a symmetric space of the non-compact type. An introduction to the subject of “symmetric spaces” from the point of view of differential geometry is given in the course by J. Maubon [5]. 2. Lie groups and Lie algebras: an overview In this section, we review the basic notions concerning the Lie groups and the Lie algebras. For a more complete exposition, the reader is invited to consult standard textbooks, for example [1], [3] and [6]. Definition 2.1. — A Lie group G is a differentiable manifold(1)which is also endowed with a group structure such that the mappings G × G −→ G, (x, y) 7−→ xy multiplication G −→ G, x 7−→ x−1 inversion are smooth. We can define in the same way the notion of a topological group: it is a topological space(2) which is also endowed with a group structure such that the ‘multiplication’ and ‘inversion’ mappings are continuous. The most basic examples of Lie groups are (R, +), (C −{0}, ×), and the general linear group GL(V ) of a finite dimensional (real or complex) (1)All manifolds are assumed second countable in this text. (2)Here “topological space” means Hausdorff and locally compact. SYMMETRIC SPACES : LIE GROUPS 3 vector space V . The classical groups like SL(n, R) = {g ∈ GL(Rn), det(g)=1}, n t O(n, R) = {g ∈ GL(R ), gg = Idn}, n t U(n) = {g ∈ GL(C ), gg = Idn}, p+q t Idp 0 O(p, q) = {g ∈ GL(R ), gIp,qg = Ip,q}, where Ip,q = 0 −Idq 0 −Id Sp(R2n) = {g ∈ GL(R2n), tgJg = J}, where J = n Idn 0 are all Lie groups. It can be proved by hand, or one can use an old Theorem of E. Cartan. Theorem 2.2. — Let G be a closed subgroup of GL(V ). Then G is an embedded submanifold of GL(V ), and equipped with this differential structure it is a Lie group. The identity element of any group G will be denoted by e. We write the tangent spaces of the Lie groups G,H,K at the identity element e respectively as: g = TeG, h = TeH, k = TeK. Example : The tangent spaces at the identity element of the Lie groups GL(Rn), SL(n, R), O(n, R) are respectively gl(Rn) = {endomorphisms of Rn}, sl(n, R) = {X ∈ gl(Rn), Tr(X)=0}, o(n, R) = {X ∈ gl(Rn), tX + X =0}, n t o(p, q) = {X ∈ gl(R ), XIdp,q + Idp,qX =0}, where p + q = n. 2.1. Group action. — A morphism φ : G → H of groups is by defi- nition a map that preserves the product : φ(g1g2)= φ(g1)φ(g2). Exercise 2.3. — Show that φ(e)= e and φ(g−1)= φ(g)−1. Definition 2.4. — A (left)actionofagroup G on a set M is a mapping (2.1) α : G × M −→ M such that α(e, m) = m, ∀m ∈ M, and α(g, α(h, m)) = α(gh, m) for all m ∈ M and g, h ∈ G. 4 PAUL-EMILE PARADAN Let Bij(M) be the group of all bijective maps from M onto M. The conditions on α are equivalent to saying that the map G → Bij(M),g 7→ αg defined by αg(m)= α(g, m) is a group morphism. If G is a Lie (resp. topological) group and M is a manifold (resp. topological space), the action of G on M is said to be smooth (resp. continuous) if the map (2.1) is smooth (resp. continuous). When the notations are understood we will write g · m, or simply gm, for α(g, m). A representation of a group G on a real (resp. complex) vector space V is a group morphism φ : G → GL(V ) : the group G acts on V through linear endomorphisms. Notation : If φ : M → N is a smooth map between differentiable manifolds, we denote by Tmφ : TmM → Tφ(m)N the differential of φ at m ∈ M. 2.2. Adjoint representation. — Let G be a Lie group and let g be the tangent space of G at e. We consider the conjugation action of G on itself, defined by −1 cg(h)= ghg , g,h ∈ G. The mappings cg : G → G are smooth and cg(e) = e for all g ∈ G, so one can consider the differential of cg at e Ad(g)= Tecg : g → g. Since cgh = cg ◦ch we have Ad(gh) = Ad(g)◦Ad(h). That is, the mapping (2.2) Ad : G −→ GL(g) is a smooth group morphism which is called the adjoint representation of G. The next step is to consider the differential of the map Ad at e: (2.3) ad= TeAd : g −→ gl(g). This is the adjoint representation of g. In (2.3), the vector space gl(g) denotes the vector space of all linear endomorphisms of g, and is equal to the tangent space of GL(g) at the identity. Lemma 2.5. — We have the fundamental relations • ad(Ad(g)X) = Ad(g) ◦ ad(X) ◦ Ad(g)−1 for g ∈ G,X ∈ g. SYMMETRIC SPACES : LIE GROUPS 5 • ad(ad(Y )X) = ad(Y ) ◦ ad(X) − ad(X) ◦ ad(Y ) for X,Y ∈ g. • ad(X)Y = −ad(Y )X for X,Y ∈ g. Proof. — Since Ad is a group morphism we have Ad(ghg−1) = Ad(g) ◦ Ad(h) ◦ Ad(g)−1. If we differentiate this relation at h = e we get the first point, and if we differentiate it at g = e we get the second one. For the last point consider two smooth curves a(t), b(s) on G d d with a(0) = b(0) = e, dt [a(t)]t=0 = X, and dt [b(t)]t=0 = Y . ∂2f We will now compute the second derivative ∂t∂s (0, 0) of the map f(t, s) = a(t)b(s)a(t)−1b(s)−1. Since f(t, 0) = f(0,s) = e, the term ∂2f ∂t∂s (0, 0) is defined in an intrinsic manner as an element of g. For ∂f the first partial derivatives we get ∂t (0,s) = X − Ad(b(s))X and ∂f ∂2f ∂s (t, 0) = Ad(a(t))Y − Y . So ∂t∂s (0, 0) = ad(X)Y = −ad(Y )X. Definition 2.6. — If G is a Lie group, one defines a bilinear map, [−, −]g : g × g → g by [X,Y ]g = ad(X)Y . It is the Lie bracket of g. The vector space g equipped with [−, −]g is called the Lie algebra of G. We have the fundamental relations • anti-symmetry : [X,Y ]g = −[Y,X]g • Jacobi identity : ad([Y,X]g) = ad(Y ) ◦ ad(X) − ad(X) ◦ ad(Y ). On gl(g), a direct computation shows that [X,Y ]gl(g) = XY − YX. So the Jacobi identity can be rewritten as ad([X,Y ]g) = [ad(X), ad(Y )]gl(g) or equivalently as (2.4) [X, [Y,Z]g]g + [Y, [Z,X]g]g + [Z, [X,Y ]g]g =0 forall X,Y,Z ∈ g. Definition 2.7. — • A Lie algebra g is a real vector space equipped with the antisymmetric bilinear map [−, −]g : g × g → g satisfying the Jacobi identity. • A linear map φ : g → h between two Lie algebras is a morphism of Lie algebras if (2.5) φ([X,Y ]g) = [φ(X),φ(Y )]h. Remark 2.8. — We have defined the notion of a real Lie algebra. How- ever, the definition goes through on any field k, in particular when k = C we shall speak of complex Lie algebras. For example, if g is a real Lie 6 PAUL-EMILE PARADAN algebra, the complexified vector space gC := g ⊗ C inherits a canonical structure of complex Lie algebra. The map ad : g → gl(g) is the typical example of a morphism of Lie algebras. This example generalizes as follows. Lemma 2.9. — Consider a smooth morphism Φ : G → H between two Lie groups.
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