Diploma-Thesis

Basics on Hermitian Symmetric Spaces

Tobias Strubel

Advisor: Prof. Dr. Marc Burger

ETH Zurich, Wintersemester 2006/2007

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Contents

0 Introduction 3

1 Basics 51.1 Basics in Riemannian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Riemannian Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Involutive Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Hermitian Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Some Topics on Symmetric Spaces 312.1 Complexification and Cartan Decomposition . . . . . . . . . . . . . . . . . . . 312.2 Totally Geodesic Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Bounded Symmetric Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.5 Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5.1 The Borel Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.5.2 The Harish-Chandra Embedding . . . . . . . . . . . . . . . . . . . . . 41

2.6 Filtrations, Gradations and Hodge Structures . . . . . . . . . . . . . . . . . . 462.7 Symmetric Cones and Jordan Algebras . . . . . . . . . . . . . . . . . . . . . . 50

2.7.1 Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.7.2 Jordan Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.7.3 Correspondence between Cones and Jordan Algebras . . . . . . . . . . 52

A Appendix 54A.1 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56A.3 Algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57A.4 Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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0 Introduction

Consider a space on which we have a notion of length and curvature. When should one call itsymmetric? Heuristically a straight line is more symmetric than the graph of x5−x4− 9x3−3x2− 2x+7, a circle is more symmetric than an ellipse and a ball is more symmetric than anegg. Why? Assume we have a ball and an egg, both with blue surface without pattern. If youclose your eyes and I rotate the ball, you can’t reconstruct the rotation. But if I rotate theegg, you can reconstruct at least the rotations along two axes. The reason is the curvature.The curvature of the ball is the same in each point (i.e. constant), the curvature of the eggis not. Should we call a space symmetric if its curvature is constant? No, because there areto less. Every manifold of constant curvature can be obtained by one of the following: froma n-dimensional sphere if the curvature is positive, from a n-dimensional euclidian space ifthe curvature is zero and from a n-dimensional hyperbolic space if the curvature is negative.So we go a step back and consider spaces on which a “derivative” of the curvature vanishes.This gives an interesting class of spaces, called locally symmetric spaces. They are defined asRiemannian manifold, where the covariant derivative of the curvature tensor vanishes. We’llshow that this happens if and only if there exists for every point x a local isometry whichfixes x and acts by multiplication by −1 on the tangent spaces at x. A Riemannian manifoldis called symmetric if this local symmetry extends to a global isometry. Surprisingly its groupof automorphisms acts transitively on the symmetric space and it is homogeneous. The studyand classification can be reduced to the study of Lie algebras equipped with an involutiveautomorphism. We treat this relation explicitly in the first three sections. Elie Cartan usedthis fact in the early 20th century to classify them. In the 60th Koecher studied them via arelation between symmetric cones and Jordan algebras. We will sketch this relation in Section2.7.2.

This text should be an introduction to symmetric spaces readable for a third year student.We presuppose only the knowledge of an introduction to differential geometry and basicknowledge in the theory of Lie groups and Lie algebras.

The first chapter gives an introduction to the notion and the basic theory of symmetricspaces. The first section is a short introduction to Riemannian geometry. We present there inshort the theory needed later in the text. In the second section we define symmetric spaces asRiemannian manifold whose curvature tensor is invariant under parallel transport. Furtherwe give some examples and deduce from the definition that a symmetric space has the formG/K for a Lie group G (the automorphism group) and a compact subgroup K. The Liealgebra g of G decomposes in a natural way into k + p. In the third section we discuss thedecomposition of Lie algebras, which can be uses to decompose symmetric spaces. The fourthand last section treats Hermitian symmetric spaces. They are symmetric spaces which havea complex structure. There can be characterized as the spaces G/K where the center of K isnon-trivial.

In the second chapter we treat some topics on symmetric spaces. The first section explainsthe notion of a Cartan decomposition of a Lie algebra, which is generalization of decompositionof g = k+p. The second and the third section introduce the rank. The next two chapters showthat Hermitian symmetric spaces of the non-compact type are exactly the bounded domainsin Cn. In the sixth section gives an application of Hermitian symmetric spaces as the modulispace of variations of Hodge structures on a vector space V . The last chapter explains aone-to-one correspondence between algebraic objects (Jordan algebras) and geometric ones(symmetric cones).

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A very good introduction to Riemannian geometry is Lees Book [Le97]. THE standardbook for symmetric spaces is Helgason [He78]. It is complete and can be used as an intro-duction to Riemannian geometry too, but it is technical and its structure is not very good.A short and precise treatment of symmetric spaces can be found in [Bo98]. Further goodintroductions are the book of Wolf [Wo67] and the text of Koranyi [Ko00]. The latter is avery nice introduction for the short reading. The texts [Ma06] and [Pa06] should be readtogether. The first gives an introduction from the differential geometers point of view, thelatter from the algebraists point of view. Delignes course notes [De73] are very interesting,since his way to symmetric spaces is different from the usual ways. But the text is not soeasy to read, since the proof are discontinuous and he omitted big parts without telling whathe omits.

Zurich, March 2007 Tobias Strubel. tstrubel@math.ethz.ch

1 Basics

1.1 Basics in Riemannian Geometry

In this section we will introduce notions and some general facts from Riemannian geometry.We will need them for the definition and the study of symmetric spaces. A very well readableand detailed introduction is [Le97].

All manifolds and vector fields are assumed to be C∞.

Definition 1.1.1. Let M be a manifold and X(M) the vector space of vector fields on M .An affine connection ∇ assigns to each X ∈ X(M) a linear mapping ∇X of X(M) into itself,satisfying the following two conditions:

(i) ∇fX+gY = f∇X + g∇Y ;

(ii) ∇X(fY ) = f∇XY + (Xf)Y .

for f, g ∈ C∞(M) and X,Y ∈ X(M). The second condition is sometimes called Leibniz rule.The operator ∇X is called covariant derivative along X.

Let ∇ be a affine connection on M and ϕ a diffeomorphism of M . Put

∇′XY := dϕ−1∇dϕXdϕY.

One can easily check that this defines a connection. The affine connection∇ is called invariantunder ϕ, if ∇′ = ∇. In this case ϕ is called an affine transformation of M .A tensor of type (r, s) over a vector space V is an element of (V ∗)⊗r ⊗V ⊗s

. A tensor of type(r, s) over a manifold is a section in the bundle (TM∗)⊗r ⊗ (TM)⊗s

. By abuse of notationwe call them tensor, too. The tensors T of type (2, 1) and R of type (3, 1) are defined by

T (X, Y ) := ∇XY −∇Y X − [X, Y ]; R(X,Y ) := ∇X∇Y −∇Y∇X −∇[X,Y ].

T is called torsion, R is the curvature.

Let (U, x) be a local chart on M . We write ∂i for ∂∂xi

and with this notation we can write

∇∂i∂j =∑

k

Γkij∂k,

with Γkij ∈ C∞(U). The Γk

ij are called Christoffel symbols.

Remark 1.1.2. The vector (∇XY )p depends only on the values of X and Y on a neighbor-hood of p. We show this for Y , for X the proof works similarly. Let Y and Y be vectorfields which coincide on a neighborhood U of p. We want to prove that ∇XYp = ∇xYp. Bylinearity this is true if and only if Y − Y = 0 on U implies ∇X(Y − Y )p = 0. Now let Y bea vector field which vanishes on a neighborhood of p. We show that ∇XYp = 0. To do this,choose a bump function ϕ with support in U and with ϕ(p) = 1. Note that ϕY ≡ 0. Withthe linearity we have ∇X(ϕY ) = 0. And by Definition 1.1.1 (ii) we get

0 = ∇X(ϕY ) = (Y · ϕ)X + ϕ∇XY.

The first term of the right hand side is zero, since Y vanishes on the support of ϕ. Therefore(∇XY )p = 0.

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Now we use this result to show, that (∇XY )p depends only on the values of Y in aneighborhood of p and on X(p). We choose local coordinates and we can write X =

∑Xi∂i.

Again by linearity we can assume that the Xi(p) = 0. Since we proved that (∇XY )p dependsonly on the values of X and Y in a neighborhood of p we get with Definition 1.1.1 (ii)

(∇XY )p = (∇PXi∂iY )p =

∑Xi(p)(∇∂i

Y )p = 0.

Definition 1.1.3. Let γ be a curve in M and X a vector field on M with X(γ(t)) = γ(t).Such a vector field exists always, because on can construct X with the help of local charts ina neighborhood of γ and a partition of unity. A vector field Y is called parallel (along γ) if(∇XY )γ(t) = 0 for all t.A geodesic is a curve for which every vector field X with X(γ(t)) = γ(t) is parallel along γ,i.e.

∇γ(t)γ(t) = 0 ∀t.Let γ be a curve. A vector field along γ is a smooth map Y : I ⊂ R→ TM with Y (t) ∈ Tγ(t)Mfor all t.

Remark 1.1.4. The definition of parallelity makes sense, since by Remark 1.1.2 the vector(∇XY )p depends only on Y and the value of X in p. Therefore the definition is independentof the extension of X on M .

Let γ be a curve and X and Y a vector fields, where γ(t)i = Xi(γ(t)). In local coordinatesX and Y can be written as

∑Xi∂i respectively

∑Y i∂i and we have

∇XY =∑

k

∑

i

Xi(∂iYk)∂k +

∑

i,j

XiY jΓkij∂k

.

Since on γ the γi(t) = Xi(γ(t)), for the vector field Y being parallel is equivalent to

dY k

dt+

∑

i,j

Γki,j

dxi

dtY j = 0. (1)

for all k.

Proposition 1.1.5. Let p ∈ M and X 6= 0 in TpM . There exists a unique maximal geodesicγ in M such that

γ(0) = 0 and γ(0) = X.

We will denote this geodesic by γX .

One can find straight from the definitions of Γkij and the geodesic a system of differential

equation for a geodesic (the geodesic equation):

d2xk

dt2+

∑

i,j

Γkij

dxi

dt

dxj

dt= 0.

By the existence and uniqueness theorem for linear differential equations it has a uniquesolution. See [He78, Ch. I.6] for details.

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Remark 1.1.6. Let X ∈ TpM and γ be a geodesic with γ(0) = p. Then there exists a vectorfield along γ with X(γ(0)), which is parallel along γ. Indeed let (x1, . . . , xm) be a local chartwith (x1(p), . . . , xm(p)) = 0. By the existence and uniqueness theorem for ordinary differentialequations and Equation (1) we know that there exists, for every vector X = (X1, . . . , Xm) inTpM , functions

Y i(t) = ϕi(t,X1, . . . , Xm)

with Y i(0) = ϕi(0, X1, . . . , Xm)) = Xi. The vector field (Y 1, . . . , Y m) is therefore parallelwith respect to γ. The mapping Y (0) 7→ Y (t) is linear, since ϕi(0, X1, . . . , Xm) is linearin X1, . . . , Xm and ϕi is unique. From the uniqueness also follows that Y (0) 7→ Y (t) is inisomorphism from Tγ(0)M to Tγ(t)M . We call this map parallel transport (see [He78, Ch.I.5]).Let X be a vector field and denote by τs,p the parallel transport from TpM to Tγ(s)M alongthe geodesic γX(p). Let A be a tensor of type (l, k). Since τs,p is an isomorphism betweenTpM and Tγ(s)M we can transport the tensor Ap on TpM to Tγ(0)M via

(τs,pAp)(a1, . . . , al, b1, . . . , bk) := Ap(τ−1

s,p a1, . . . , τ−1s,p al, τ

∗s,pb1, . . . , τ

∗s,pb

k),

for a1, . . . , al ∈ Tγ(s)M and b1, . . . , bk ∈ T ∗γ(s)M . Define

(∇XA)p := lims→0

1s((τ−1

s,p Aγ(s))−Ap).

This is the covariant derivative for tensor fields. A tensor is called invariant under paralleltransport, if τsA = A for all s. This is the case if and only if ∇XA = 0 for all vector fields X.

Definition 1.1.7. Let p be a point in M . For every X ∈ TpM there exists a geodesic γX .This defines a function from TpM to M

expp : X 7→ γX(1)

if γX(1) is defined. It is called the exponential map. Its differential exists and it can be seenas a map TpM → TpM and a direct calculation shows that it is the identity. By the localinversion theorem expp is a local diffeomorphism between non-empty open sets U ⊂ TpM andV ⊂ M . Its inverse is noted logp.A neighborhood N0 of 0 ∈ TpM is called normal if expp is a diffeomorphism of N0 onto anopen neighborhood Np of p and for all X ∈ N0 and 0 ≤ t ≤ 1 then tX ∈ N0. Np is also callednormal.

For a more precise treatment of the exponential map see [He78, Ch. I.6] or [Le97, Ch. 5].

Let X be a vector field. Define the 1-forms

ωi(Xj) := δij ωi

j =∑

k

Γikjω

k.

The following theorem shows that these two forms are determined by the torsion and cur-vature tensor. Since they define the connection ∇, the torsion and curvature tensor determinethe connection.

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Theorem 1.1.8. Let T be the torsion tensor and R the curvature tensor. Then

dωi = −∑

p

ωip ∧ ωp +

12

∑

j,k

T ijkω

j ∧ ωk

dωil = −

∑p

ωip ∧ ωp

l +12

∑

j,k

Riljkω

j ∧ ωk.

These equations are called the Cartan structure equations.Let Y1, . . . , Ym be a basis for the tangent space TpM and let N0 be a normal neighborhood ofthe origin in TpM and Np := expN0. Let Y ∗

1 , . . . , Y ∗m be the vector fields on Np adapted to

Y1, . . . , Ym. Let Φ : R× Rn → M be given by

Φ : (t, a1, . . . , am) → exp(ta1Y1 + . . . + tamYm).

Then the dual forms Φ∗ωi and Φ∗ωji are given by

Φ∗ωi = aidt + ωi, Φ∗ωil = ωi

l ,

where ωi and ωij are 1-forms in da1, . . . , dam. They are given by

∂ωi

∂t= dai +

∑

k

akωik +

∑

j,k

T ijkajω

k. ωit=0(t : aj; dak) = 0 : (2)

∂ωil

∂t=

∑

j,k

Riljkajω

k ωil(t; aj ; dak)t=0 = 0 (3)

For a proof see [He78, I.8].

Theorem 1.1.9. Let M and M ′ be differentiable manifolds with affine connections ∇ and ∇′.Assume that the torsion tensors T and T ′ and the curvature tensors R and R′ are invariantunder parallel transport, i.e.:

∇T = 0 = ∇′T ′, ∇R = 0 = ∇′R′.

Let x ∈ M and x′ ∈ M ′ and A : TxM → Tx′M′ be an isomorphism of the tangent spaces

at x and x′ sending T and R onto T ′ and R′. Then there exists a local isomorphism1 a :(M,∇, x) → (M ′,∇′, x′) whose differential in x is A.

Proof. Definea(y) := expx′(A logx y).

It seems to be a good candidate for our local isomorphism. It is in fact a local diffeomorphism.Now we have to check, that it is an affine transformation. Before we show that, we take acloser look on what it means to be an affine transformation. On M respectively on M ′ wehave

∇XiXj =∑

ΓkijXk ∇′X′

iX ′

j =∑

Γ′kijX′k,

1An isomorphism is a diffeomorphism which is an affine transformation in both directions.

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if Xi and X ′i are local basis’ for vector fields in M respectively M ′. A map a : M → M ′ is an

affine transformation if and only if

∇′XY = da∇da−1Xda−1Y.

If X ′i = daXi and a is affine, we have

∑

k

ΓkijXk = ∇XiXj = ∇da−1X′

ida−1X ′

j = da−1(∑

Γ′kijX′k

)

= da−1(∇′X′

iXj

)=

∑(Γ′kij ◦ a)Xk.

hence Γ′kij ◦a = Γkij is a sufficient and necessary condition for a to be an affine transformation.

Lets check this for our a.Choose normal neighborhoods N0 and N ′

0 of the origins in TxM and Tx′M′ and a basis

Y1, . . . , Ym of TxM . Put Nx := exp(N0) and N ′x′ := expN ′

0. Then AY1, . . . , AYn is a basisof Tx′M

′ and the coefficients of T and T ′ with respect to these basis’ are constant, since allderivatives vanishes by ∇T = 0. Furthermore they are the same for T and T ′ since A mapsT to T ′. This also holds for the coefficients of R and R′. Put

Φ(t, a1, . . . , am) = expx(ta1Y1 + · · · tamYm) (4)Φ′(t, a1, . . . , am) = expx′(ta1Y1 + · · · tamYm). (5)

We have by construction Φ′ = a ◦ Φ. The Cartan structure equations say

Φ∗ωi = aidt + ωi, (Φ′)∗ω′i = aidt + ω′i

Φ∗ωij = ωi

j , (Φ′)∗ω′ij = ω′ij

By equations (2) and (3) in Theorem 1.1.8 ωi = ω′i and ω′ij = ωij , since both sides are solutions

of the same differential equation because the coefficients of T and T ′ respectively R and R′

are the same. With (4), (5) and Φ′∗ = Φ∗ ◦ a∗ we get

Φ∗ωi = aidt + ωi = aidt + ω′i = (Φ′)∗ω′i = Φ∗ ◦ a∗(ω′i)

and with a similar calculationΦ∗ωi

j = Φ∗ ◦ a∗ω′ij .

Putting t = 1 we obtain

ωi = a∗(ω′i), ωij = a∗(ω′ij).

Finally we have∑

k

Γikjω

k = ωij = a∗(ω′ij) =

∑

k

(Γ′ikj ◦ a)a∗(ω′k) =∑

k

(Γ′ikj ◦ a)ωk,

so Γikj = Γ′ikj ◦ a.

The theorem above is Lemma IV.1.2 in [He78].

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Proposition 1.1.10. Under the hypotheses of Theorem 1.1.9 assume that M and M ′ aresimply connected and geodesically complete. Then a can be continued to a unique globalisomorphism.

Proof. Let a := exp′p(A logp) and x be a point in M . We are searching for a point y in M ′,such that y = a(x). Since M is geodesically complete, we can join p and x by a geodesic γ(see Theorem A.1.1), such that p = γ(0) and x = γ(1). A transports the initial vector inTpM to a vector in Tp′M which generates a geodesic γ in M ′. Put a(x) := γ(1). Since M andM ′ are simply connected, this is welldefined and unique. See [He78, Thm. IV.5.6] or [Wo67,Ch. 1.8] for details.

Corollary 1.1.11. Let (M,∇) be such that ∇T = 0 = ∇R and M simply connected andgeodesically complete. Let γ(t) be a geodesic (i.e. ∇γ(t)γ(t) = 0 for every t). There exists aunique one-parameter group of automorphisms τ(γ, u) of M that induces translation γ(t) 7→γ(t + u) on γ and displaces the tangent bundle on γ by parallel transport.

Proof. Fix t ∈ R and denote by ϑ the parallel transport from Tγ(t)M to Tγ(t+u). Since ϑ isan isomorphism we have by Theorem 1.1.9 and Proposition 1.1.10 a unique automorphismτ(γ, u) which maps γ(t) to γ(t+u) and dτ(γ, u)γ(t) = ϑ. Being an automorphism τ(γ, u) mapsgeodesics on geodesics. Since it acts as parallel transport from Tγ(t)M to Tγ(t)M if maps γ(t)to γ(t + u), hence γ on itself. Using the geodesic equation one can show that τ(γ, u) mapsγ(s) to γ(s + u) for all s. By the uniqueness of the solution of the Equation (1) τ(γ, u) actsby parallel transport on the tangent bundle of γ.

Remark 1.1.12. Conversely if such a one-parameter group of automorphisms exists, M isgeodesically complete. If a geodesic γ exists on [−1, 1], then we have γ(n + a) = τ(γ, n)γ(a),if n ∈ Z and a ∈ [0, 1).

Corollary 1.1.13. Let M a manifold with an affine connection ∇. The following conditionsare equivalent:

(i) T = 0 = ∇R;

(ii) for all x ∈ M there exist sx, a local automorphism of (M,∇) inducing t 7→ −t on TxM .

Proof.(i)⇒ii) Use Theorem 1.1.9 with A = −1.

(ii)⇒i) Fix x ∈ M . s := sx is a local automorphism of (M,∇), i.e. we have

(∇XY )y = (ds−1(∇dsXdsY ))s(y)

for all vectorfields X and Y and y in a neighborhood of x. Using this fact andthe definition of R and T we see that ∇ZR(X, Y ) = ds−1∇ds−1ZR(dsX, dsY )ds andT (X,Y ) = ds−1T (dsX, dsY ). dsx = −1, hence

T (X,Y )x = −T (X, Y )x ∇ZR(X,Y )x = −∇ZR(X,Y )x,

since ∇ is invariant under s. But x was arbitrary, so ∇R ≡ 0 ≡ T .

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Definition 1.1.14. Let M be a manifold and g a (2, 0) tensor field on M such that gp ispositive definite for all p ∈ M , i.e. gp is a scalar product on TpM . Such a g is a Riemannianmetric. A connection ∇ on a manifold equipped with a Riemannian metric is called Levi-Civita-Connection2 if

(i) ∇XY −∇Y X = [X, Y ] (torsion free)

(ii) ∇Xg ≡ 0 for all X (invariant under parallel transport).

By a Riemannian manifold we mean a triple (M, g,∇) where M is a connected manifold, ga riemannian metric and ∇ a Levi-Civita-Connection. Usually we simply write M .

Remark 1.1.15. The condition (ii) for the Levi-Civita Connection is equivalent to X ·g(Y, Z) = g(∇XY, Z) + g(Y,∇XZ).The Levi-Civita Connection exists and it is unique. From the definition one can get theformula

2g(X,∇ZY ) = Z ·g(X, Y )+g(Z, [X,Y ])+Y ·g(X, Y )+g(Y, [X,Y ])−X ·g(Y, Z)−g(X, [Y,Z]).

This formula and the fact that g is non-degenerate can be used to prove uniqueness andexistence (see [Le97, Ch.5]).The tensor g is called metric, because one can use it to measure the length of a path γ by

L(γ) :=∫

γ

√gγ(t)(γ(t), γ(t))dt.

For p, q ∈ M we define

d(p, q) := inf{L(γ)| γ a path joining p and q}.

This defines a distance function on M .

Definition 1.1.16. Let M be a Riemannian manifold and p ∈ M a point. Let S be atwo-dimensional subspace of TpM . The sectional curvature is defined by

K(S) = −gp(Rp(Y,Z)Y,Z)|Y ∧ Z|2 ,

(c.f. [He78, Thm. I.12.2]) where Y and Z are linearly independent vectors in S and |Y ∧ Z|denotes the area of the parallelogram spanned by Y and Z.

For details see [He78, I.12]).For the proof of the next proposition we need a technical lemma, it is Lemma I.12.4 from

[He78].

Lemma 1.1.17. Let E be a vector space over F (field of characteristic 0). Suppose B :E × E × E ×E → F is quadrilinear and satisfies the identities

(i) B(X,Y, Z, T ) = −B(Y, X,Z, T );

(ii) B(X,Y, Z, T ) = −B(X, Y, T, Z);2Sometimes it is called Riemannian connection.

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(iii) B(X,Y, Z, T ) + B(Y,Z, X, T ) + B(Z, X, Y, T ) = 0;

(iv) B(X,Y, X, Y ) = 0;

then B ≡ 0.

Proof. The proof is a pure calculation without tricks.

Proposition 1.1.18. Let M be a Riemannian manifold. The following conditions are equiv-alent:

(i) ∇R = 0;

(ii) the sectional curvature is invariant under parallel transport;

(iii) for all x ∈ M , y 7→ expx(− logx(y)) is a local isometry.

Proof. (i) ⇒ (ii) Let p ∈ M and S a two dimensional subvectorspace of TpM . The sectionalcurvature K in p is defined as

K(S) = −gp(Rp(Y,Z)Y,Z)|Y ∧ Z|2 .

The metric g and the curvature R are invariant under parallel transport (since ∇R ≡ 0) andthe same holds for the length of the two vectors and the angle between them, since lengthand angle are measured with the help of the invariant g. Hence K is invariant under paralleltransport.ii) ⇒ iii) Denote by τ the parallel transport from p to q along a geodesic. By the invarianceof K we have

gp(Rp(X, Y )X, Y ) = gq(Rq(τX, τY )τX, τY ) (6)

and by the invariance of g

gp(Rp(X,Y )X,Y ) = gq(τRp(X,Y )X, τY ) (7)

holds for every X, Y ∈ TpM . Let

B(X, Y, Z, T ) := gq(Rq(τX, τY )τZ, τT )− gq(τRp(X, Y )Z, τT )

If we show that B ≡ 0, then τRp = Rq, hence∇R = 0, since g is non-degenerate. To show B ≡0 we use Lemma 1.1.17. Lets check the assumptions. (i) is true, since R is skew-symmetric.Checking (ii) needs a straightforward calculation which uses the definition of R and the factthat g is invariant under parallel transport and hence Z ·g(X, Y ) = g(∇ZX, Y )+g(X,∇ZY ).(iii) is exactly the Bianchi-identity. (iv) is clear since the difference of the equations (6) and(7) is zero. By Corollary 1.1.13 ϕ(y) := expq(− logq(y)) is a local automorphism of (M,∇).It remains to show, that it is an isometry. Multiplication with −1(= dϕq) is an isometry onTqM . Then

gp(X,Y ) = gq(τX, τY ) = gϕ(q)(dϕqτX, dϕqτY ).

This last equals gϕ(p)(dϕpX, dϕqY ) because ϕ is an automorphism of (U,∇), i.e. the paralleltransport τ commutes with dϕ.(iii) ⇒ (i) follows from Corollary 1.1.13.

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1.2 Riemannian Symmetric Spaces

Definition 1.2.1. If a Riemannian manifold M satisfies the equivalent conditions of Propo-sition 1.1.18, it is called locally symmetric. We call the map y 7→ expx(− logx y) the geodesicsymmetry in x. If M is connected and for all x ∈ M there exists an isometric involution sx

with x as an isolated fixed point, M is called (Riemannian) symmetric.

Remark 1.2.2. Since x is an isolated fixed point, dsx|x acts as multiplication with −1 onthe tangent space in x.A symmetric space is complete. Indeed note first that for every point x on a geodesic γ thesymmetry sx maps the geodesic on itself because sx is an isometry. Therefore every geodesicis either closed or can be continued to infinity. By the Hopf-Rinow-Theorem (Theorem A.1.1)two arbitrary points can be joined by a geodesic arc and M is complete. Conversely if M islocally symmetric, complete and simply connected it is symmetric, because by Hopf-Rinow-Theorem the completeness implies that M is geodesically closed and therefore by Proposition1.1.10 such a local symmetry extends to a global one.Let M be a complete locally symmetric space. Its universal cover is locally symmetric too.Since it is complete an simply connected, it is globally symmetric. Therefore complete, locallyconnected spaces are quotients of global ones.

Example 1.2.3. The euclidian spaces En, the spheres Sn and the hyperbolic spaces Hn seenas Riemannian manifolds are symmetric spaces. There curvatures are constant, hence they arelocally symmetric by Proposition 1.1.18. They are simply connected and complete, thereforesymmetric. For n = 2 it is easy to see how the geodesic symmetry acts on the space. Since itacts as multiplication with −1 on the tangent space, sp is the rotation by π around p in E2.In S2 the symmetry is rotation by π around the axis through p and −p. If one looks at theupper half plane model for H2, it is the inversion at the halfcircle through p perpendicular tothe x-axis with radius =p, composed with inversion at the geodesic line parallel to the y-axisthrough p.

Example 1.2.4. Let P := P (n,R), the set of the symmetric positive definite matrices inSL(n,R). It is clearly an open subset of the vector space of the symmetric matrices. Itsdimension is n(n + 1)/2. First we equip it with a Riemannian metric. On TpP (which is thevector space of symmetric matrices) we define a scalar product by

< X, Y >p:= tr(p−1Xp−1Y ).

This is a Riemannian metric.SL(n,R) acts on P via

g · p := gpg>,

where g ∈ SL(n,R) and p ∈ P . The differential is X 7→ gXg>. The metric given above isinvariant under this action since

< gXg>, gY g> >g·p = tr((gpg>)−1gXg>(gpg>)−1gY g>) = tr(g>p−1Xp−1Y (g>)−1)

= tr(p−1Xp−1Y ) =< X,Y >p .

This explains why we defined the metric in this way.Now we want for each point p ∈ P a symmetry sp. Put sp(q) := p(q>)−1p. It is clearly aninvolution which fixes p. We need to show that sp is an automorphism. It is a composition

14

of the maps σ : q 7→ q−1 and q 7→ p · q. We showed above, that the latter is an isometry. Toshow that σ is an isometry we calculate its derivative. Let X ∈ TpP be a tangent vector. Itis the initial vector of a curve p + tX. Let p−1 + tX be it inverse curve. Then

1 = (p + tX)(p−1 + tX) = 1 + t(Xp−1 + pX) +O(t2),

hence dσp(X) = X = −p−1Xp−1. Therefore

< X, Y >p−1= tr(pXpY ) = tr(Xp−1Y p−1) = tr(p−1Xp−1Y ) =< X,Y >p,

where we have the general fact tr(AB) = tr(BA). The differential of the map q 7→ p · q is themap itself and the composition of the two differentials is multiplication with −1. Hence P isa symmetric space.

Proposition 1.2.5. Let M be a Riemannian symmetric space γ be a geodesic. Denote by sx

the symmetry in x ∈ M . We write st for sγ(t). Then:

i) st(γ(t + u)) = γ(t− u);

ii)τ(γ, u) = su/2 ◦ s0

andst ◦ τ(γ, u) ◦ s−1

t = τ(γ,−u).

Proof. i) The geodesics γ(t+u) and γ(t−u) are generated by γ(t) respectively by −γ(t) =dstγ(t). By uniqueness of this geodesics we have st(γ(t + u)) = γ(t− u).

ii) With i): su(γ(t)) = su(γ(u+(t−u))) = γ(u−(t−u)) = γ(2u−t). Hence su/2◦s0(γ(t)) =su/2(γ(−t)) = γ(u+ t) = τ(γ, u)(t). And st ◦τ(γ, u), ◦st(γ(r)) = st ◦τ(γ, u)(γ(2t−r)) =st(γ(2t− r + u) = γ(r − u) = τ(γ,−u)γ(r).

Let M be a symmetric Riemannian space and G its group of isometries. We endow G withthe compact-open topology It is generated by sets W (C,U) := {f ∈ G|f(C) ⊂ U}, whereC ⊂ G is compact and U ⊂ G is open.It is the coarsest topology such that the evaluationmap is constant. It turns G into a topological group which acts continuously on M . It is afinite dimensional Lie group (see Theorem A.1.3). For details on this topology see [Ha02, p.529f] and [He78, Ch. IV.2].

It acts transitively on M . Indeed take two points p and q and join them by a geodesic γso that p = γ(0) and q = γ(1). The symmetry in γ(1/2) maps one to the other, since it is anisometry. Further τ(γ, 1) maps p to q. Since τ is a one parameter subgroup of G, (write τu

for τ(γ, u)) for u, v ∈ R τv+u = τv ◦ τu.Now we want to show that the identity component of G, denoted by G0, acts transitively

on M . To do that let V =⋂l

i=1 W (Ci, Ui) be an open subset of the identity component whichcontains the identity. Since e ∈ V we have Ci ⊂ Ui. The metric d(x, τu(x)) measures how fara point x is moved by τu. It depends continuously of x, because d and τu are continuous inx .It has for a fixed u a maximum on Ci, since Ci is compact. This maximum is continuousas a function of u. We call this function m and we have m(0) = 0. Furthermore we can coverCi by a finite number of open balls which are contained in Ui and such that the open balls

15

with half radius still cover Ci. Denote by r the minimal radius of these balls. If one choosesui such that m(ui) < r/2, one has τui(Ci) ⊂ Ui. For all 0 < u < ui for all i the map τu iscontained in V . Since τ is a one-parameter group, we have τu+v = τu ◦ τv. Therefore G0 actstransitively on M .

Let K be the stabilizer of a point x ∈ M . It is compact (Theorem A.1.2). HenceM = G/K (see Theorem A.1.4). On G there exists a natural involution σ : g 7→ sxgsx. Fork ∈ K we have d(sxksx) = (−1)dk(−1) = dk and since K injects into O(TxM) (LemmaA.1.5) we have sxksx = k and all elements of K are left fixed by σ. Therefore dσ is theidentity on k := Lie(K). Let p ⊂ Lie(G) be the set of infinitesimal generators of the oneparameter subgroups τ(γ, u), where γ is a geodesic through x. The involution σ acts on p bymultiplication with −1 because of Proposition 1.2.5 (ii). The vector space p can be identifiedwith TxM , because for two geodesics γ and γ with γ(0) = γ(0) and γ(0) = s ˙γ(0) with s > 0we have γ(t) = γ(st) and therefore τ(σ, u) = τ(γ, u/s). Hence τ(γ, u) is uniquely determinedby a vector in TxM , its direction gives γ and its length u and vice versa. By dimension reasonswe have g = k⊕ p where p is the eigenspace of σ to the eigenvalue −1 and k is the eigenspaceto the eigenvalue 1. If not otherwise stated, in the following G denotes the isometry group ofthe Riemannian symmetric space M and K the stabilizer of a point p ∈ M . The Lie algebrag of G admits always the decomposition k⊕ p with respect to the involution σ.

This shows

Theorem 1.2.6. Let M be a symmetric space and G its group of isometries. Then M = G/Kwhere K is the stabilizer of a point x ∈ M .

Example 1.2.7. Lets look how our examples are as homogeneous spaces. The isometrygroup of Rn is Rn n O(n) and we have O(n + 1) for Sn. For Hn we have O(n, 1)+. It is thesubgroup of O(n, 1) of index 2, which preserves the upper sheet Hn. The group of isometriesof P (n,R) is SL(n,R). The point-stabilizer in the first three cases is O(n), in the last case itis SO(n). Therefore we have:

Rn = Rn nO(n)/O(n)Sn = O(n + 1)/O(n)Hn = O(n, 1)+/O(n)P (n,R) = SL(n,R)/SO(n)

For details for the calculation of the isometry groups see [Br99, Ch. I].

Remark 1.2.8. Now we want to see how we can build a symmetric space from a given Liegroup G and a compact subgroup K. We assume that

(i) M = G/K is connected and

(ii) there exists an involution σ of G such that Lie(G)dσe = Lie(K) and Kσ = K.

We want to put a structure of a Riemannian symmetric space on M . First we choose Q a G-invariant Riemannian structure. Such a structure exists always. We can define a K-invariantinner product on TeM via

< a, b >=∫

K(dk · a, dk · b)dµ(k),

16

where (·, ·) is an arbitrary positive definite bilinear form on TeM and dµ(k) is the Haar-measure3 on K. The integral exists since K is compact and acts continuously on M . Now

hgK(X, Y ) :=< dg−1X, dg−1Y >

defines an invariant riemannian structure on M . It is well-defined since < ·, · > is K-invariant,hence it does not depend on the representant of gK.Now we are searching for a suitable point-symmetry for our M . Our σ is an involution,therefore g decomposes into the direct sum k + p, where k is the eigenspace to the eigenvalue1 (and the Lie algebra of K) and p is the eigenspace to the eigenvalue −1. Therefore ithas eK as an isolated fixed point and it is a symmetry in eK. Since M is homogeneousthis gives a symmetry for every gK ∈ M , putting spK(gK) := pσ(p−1q)K. It is clearly aninvolution and it is an automorphism of M , since σ and multiplication with an element of Gare automorphisms.

In Proposition 1.3.10 we will see that if G is semi-simple, connected and acts faithfully onM , it is the identity component of the group of isometries of M .

Example 1.2.9. Let G = SO(2, n) (with n > 3) the group of matrices in GL(2+n,R) whichleave the bilinear form of type (2, n) invariant, which is given by

(x|y) := x1y1 + x2y2 − x3y3 − . . .− xn+2yn+2

for x = (x1, . . . , xn+2) and y = (y1, . . . , yn+2). This form is defined with respect to a basis(e1, . . . , e2+n). We can use this basis to define the standard scalar product and therefore anorm on R2 and Rn. We denote the scalar product by 〈·, ·〉 and the norm by ‖ · ‖.The Lie algebra of G consists of matrices of the form

(X1 X2

X>2 X3

)

where X1 ∈ M(2, 2,R) skew-symmetric, X2 ∈ M(2, n,R) arbitrary and X3 ∈ M(n, n,R)skew-symmetric. One sees directly, that the subalgebra k consisting of matrices

(X1 00 X3

)

is compact. The corresponding Lie group K is S(O(2) × O(n)). Let σ : G → G the mapwhich maps g to (g>)−1. It is clearly an involution. The fixed points of σ are exactely thematrices in K. Its differential acts as X 7→ −X> on the Lie algebra g of G. The fixed pointsare exactly the matrices in k. Therefore G/K is a symmetric space.

A model of this symmetric space is the space G of the two dimensional subspaces of R2⊕Rn

on which (·|·) is positive definite. The group G acts on it linearly. We show now that theaction is transitive. We have clearly V0 := R2⊕{0} ∈ G. If we can show, that every point in Gcan be mapped to V0, we are done. Let V ∈ G. By V ⊥ we denote the orthogonal complementof V with respect to (·|·). Since this bilinear form is non-degenerated on V we can decomposeR2 ⊕ Rn = V ⊕ V ⊥. If we show, that our bilinear form is negative definite on V ⊥, we canfind a orthogonal basis {w1, . . . , wn} of V > and a orthogonal basis {v1, v2} of V . The matrix(v1, v2, w1, . . . , wn) is by orthogonality in SO(2, n) and it maps V0 to V . Hence the action is

3The Haar-measure exists since K is compact.

17

transitive. It remains to show that, (·|·) is negative definite on V >. Lets do this: let {v1, v2}be a orthogonal basis of V . Lets decompose them as v1 = ξ1 + η1 and v2 = ξ2 + η2 withξi ∈ R2 and ηi ∈ Rn and we have ‖ξi‖ > ‖ηi‖. A vector ξ + η ∈ V > with ξ ∈ R2 and η ∈ Rn

satisfies (u1 + u2|αv1 + βv2) = 0, hence 〈ξ, αξ1 + βη2〉 = 〈η, αη1 + βη2〉. Since {ξ1, ξ2} form abasis of R2 we have

‖ξ‖ = sup‖αξ1+βξ2‖

|〈ξ, αξ1 + βξ2〉| = sup‖αξ1+βξ2‖

|〈η, αη1 + βη2|

≤ sup‖αξ1+βξ2‖

‖η‖ · ‖αη1 + βη2‖ < sup‖αξ1+βξ2‖

‖η‖ · ‖αξ1 + βξ2‖ = ‖η‖.

We used in the first step a characterization of the norm, the third step is the Cauchy-Schwarzinequality and the fourth step uses αv1 + βv2 ∈ V . It is important that we have there “<”and not “≤”!

The space G can be realized as a subset of M(2, n,R). Let v = v1 + v2 ∈ V withv1 ∈ R2 and v2 ∈ Rn. The projection on R2 is injective since v1 = 0 if and only if v2 = 0.By dimension reasons it is also surjective and we can choose a basis {ξ1, ξ2} of V withξ1 = {1, 0, Z11, . . . , Z1n} and ξ2 = {0, 1, Z21, . . . , Z2n}. By definition of G we have

∑j Z2

ij < 1for j = 1, 2. Conversely every 2× n matrix defines an element of G via

V := {(v, Zv)|v ∈ R2}.

Therefore we have a one-to-one correspondence between G and the set of matrices M := {Z =(Z1, Z2) ∈ M2,n(R)|Z1 · Z1 < 1, Z2 · Z2 < 1}.

Example 1.2.10. Let G := SL(n,R), K := SO(n,R) and σ : G → G with σ : g 7→ (g>)−1.M := G/K is connected, since G and K are. σ is an involution and its fix-point set is bydefinition exactly K. If the fixed point set of dσe is Lie(K) we know, that M is a symmetricspace. Let X ∈ Lie(G). It generates a path e + tX in G. The path σ(e + tX) = (e + tX>)−1

can be written in the form e + tdσeX and is satisfies therefore

e = (e + tX>)(e + tdσeX) = e + t(X> + dσeX) +O(t2).

This means dσeX = −X>. The fixed point set of this involution is the vector space of skew-symmetric matrices, which is equal to the Lie algebra of SO(n,R). Therefore M = G/K is asymmetric space.

In Example 1.2.4 we constructed a symmetric space P as the set of symmetric positivedefinite matrices in SL(n,R). The identity matrix I is clearly in P . The involution on theisometry group of P given by σ : g → sIgsI can be calculated explicitly. Let q ∈ P :

sI(g · (sI(q))) = sI(g · (q>)−1) = sI(gq>)−1g>) = (g−1)>qg−1 = (g>)−1 · q.

Hence σ maps g to (g>)−1. This is the same action as above. Hence G/K = P .

Proposition 1.2.11. Let R denote the curvature tensor of the space Riemannian symmetricspace G/K with G and K as usual and let Q be the Riemannian structure on G/K. Then atthe point o ∈ G/K

Ro(X, Y )Z = −[[X,Y ], Z], for X, Y, Z ∈ p.

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This proposition is Theorem IV.4.2 in [He78]. In the following section we will use thefact that a symmetry sx in a symmetric space M gives in a natural way an involution onits group of automorphisms. We will use its differential, which is an involutive Lie algebraautomorphism, to study the structure of symmetric spaces.

Proposition 1.2.12. Let G and K be as above. The following conditions are equivalent

(i) G acts faithfully on M = G/K;

(ii) K acts faithfully on p = Lie(G)/Lie(K)=tangent space to M at the origin.

Proof. (i)⇒ (ii). K < G acts faithfully on M and K is the stabilizer of the point eK ∈ M .If K acts not faithfully on p there exists k1, k2 ∈ K with k1 6= k2 and dk1 = dk2 in TeKM .But this is by Lemma A.1.5 a contradiction.(ii) ⇒ (i). Take h ∈ G with h(gK) = gK for all g ∈ G. This holds in particular for g = eand have h ∈ K. From h = id on M follows that dh = id on p. K acts faithfully on p andtherefore h = e. Hence G acts transitively on M .

If the two conditions hold, σ is the identity on K, because it acts as identity on Lie(K).

Now we need some notions and theory of Lie algebras.

Definition 1.2.13. Let g be a Lie algebra. By ad we denote the adjoint representation of g

in itself by

ad(X) :g → End(g),X 7→ {ad(X) : Y 7→ [X, Y ]}

with X, Y ∈ g.The Killing form of g is defined as

B(X, Y ) := tr(ad(X)ad(Y )).

We denote the Killing form of a Lie algebra always by B or by Bg if there is danger ofconfusion.

Proposition 1.2.14. The Killing form has the following properties:

(i) It is a symmetric bilinear form, i.e. B(X, Y ) = B(Y, X) for all X,Y ∈ g.

(ii) It is infinitesimally invariant under ad, i.e. for all X,Y, Z ∈ g we have

B(ad(Z)X,Y ) = −B(X, ad(Z)Y ).

This is equivalent to B([X, Y ], Z) = B(X, [Y, Z]).

(iii) Let i ⊂ g be an ideal. Then the Killing form on i equals the Killing form on g restrictedto i× i.

Proof. (i) The bilinearity comes from the bilinearity of tr(XY ) and the linearity of ad.The symmetry is a consequence of the general fact that for matrices A and B we havetr(AB) =

∑AijBji =

∑BijAji = tr(BA).

19

(ii) With the Jacobi-Identity we get:

ad([Z, Y ])(A) =[A, [Y, Z]] = −[Y, [Z, A]]− [Z, [A, Y ]]=− ad(Y )ad(Z)(A) + ad(Z)ad(Y )(A)

Therefore

B(ad(Z)X, Y ) = tr(ad([Z, X])ad(Y )) = tr(ad(Z)ad(X)ad(Y ))− tr(ad(X)ad(Z)ad(Y ))= tr(ad(X)ad(Y )ad(Z))− tr(ad(X)ad(Z)ad(Y )) = tr(ad(X)ad([Y, Z]))= −B(X, ad(Z)Y )

(iii) Since i is a subvectorspace of g, one can complete a basis of i to a basis of g. Since it isan ideal in g (i.e. ad(x)g ⊂ i) for x ∈ i the matrix of ad(x) in g has the form

(A ∗

0

)

where A is the matrix of ad(x) in i.

Definition 1.2.15. The adjoint representation Ad of G in g is given by Ad(g) := dcg|e wherecg : h → ghg−1 is the conjugation by g.

The differential of Ad in e is ad. The Killing form is therefore Ad-invariant, since byProposition 1.2.14 (ii) B(ad(Z)X,Y ) = B(X, ad(Z)Y ) = 0.

We have for k ∈ K and a geodesic through e

kτ(γ, u)k−1 = τ(kγ, u).

Since γ is generated by a vector X = γ(0) ∈ p and K acts faithfully on p via k 7→ dk, we seethat kγ is generated by dkX. Hence the adjoint representation of K on p is faithful.

Further we have that ad(X) is a skew-symmetric matrix (Appendix Lemma A.1.6), ifX ∈ k. Therefore it is diagonalisable with eigenvalues in iR (Appendix Lemma A.1.7) andhence B(X,X) = tr(ad(X)2) = −∑

λ2j ≤ 0, where λj ∈ R and iλj are the eigenvalues of

ad(x). If tr(ad(X)2) = 0 then the only eigenvalue of ad(X) is zero and therefore ad(X) = 0.But then

Ad ◦ exp(X) = exp ◦ad(X) = 0,

and since the action of K on p is faithful, X is zero. This shows that B|k×k is negative definite.

Definition 1.2.16. A Lie algebra is semi-simple if the Killing form is non-degenerate.

1.3 Involutive Lie Algebras

We have seen, that one can introduce on the group of automorphims G of a symmetric spaceM an involution in a natural way. We fixed a point x ∈ M and set σ : g 7→ sxgsx. This isclearly an involution, since sx is. This involution gives an involution on g (the Lie algebraof G) by taking the differential at the identity dσ|e. We have also seen, that we can write g

as k ⊕ p where k and p are the eigenspaces to the eigenvalues 1 respectively −1 of dσe. Wewill always write p and k for these eigenspaces. By abuse of notation we write σ instead ofdσe. First we define two properties and show that they are fulfilled for Lie algebras that comefrom symmetric spaces.

20

Definition 1.3.1. An involutive Lie algebra (g, σ) is a Lie algebra together with an involutiveLie algebra automorphism4 σ.

Let (g, σ) be an involutive Lie algebra. By Remark A.1.8 σ is diagonalisable with eigen-values 1 and −1. Therefore we can decompose g into k⊕ p where k and p are the eigenspacesto the eigenvalues 1 respectively −1 of σ. We have

[k, k] ⊂ k, [k, p] ⊂ p, [p, p] ⊂ k. (8)

because k and p are defined as eigenspaces of σ. k is a subalgebra of g. Since σ is an Liealgebra automorphism it commutes with the bracket and we have ad(σX) = σad(X)σ−1. Asa general fact we know that the trace is invariant under conjugation. Therefore we have fork ∈ k and p ∈ p

B(k, p) =tr(ad(k)ad(p)) = tr(σad(k)σσ−1ad(p)σ−1) = tr(ad(σk)ad(σp)) = −tr(ad(k)ad(p))=−B(k, p),

hence B(p, k) = 0. Lets start our discussion of (g, σ).Our main reference here is [Bo98].

Definition 1.3.2. An involutive Lie algebra (g, σ) is said to be reduced, if k contains nonon-zero ideal of g.

Let i ⊂ k be an ideal of g. Then by definition of an ideal we have [p, i] ⊂ i ⊂ k and by from(8) we deduce [p, i] ⊂ p. Since p ∩ k = {0} the ideal i is contained in the centralizer of p in k.

Proposition 1.3.3. An involutive Lie algebra (g, σ) is reduced if and only if the representationof k in p given by k 7→ adg(k)|p is faithful.

Proof. The kernel of this representation is an ideal contained in k and it is by definition of therepresentation equal to the centralizer of p in k, denoted by z(p)k. To show that the kernel isin fact an ideal, we need to show, that for r in the kernel and g = k + p ∈ k⊕ p we have that[g, r] is in the kernel, i.e. [[g, r], p] = 0 for each p ∈ p. We use the Jacobi-Identity:

[[k + p, r], p] = [[k, r], p] = [k, [r, p]] + [r, [p, k]].

The right hand side is zero because p and [p, k] are in p.If (g, σ) is reduced the kernel of the adjoint representation is zero, since it is an ideal in k andvice versa.

From this proposition we see that an involutive Lie algebra which comes from a symmetricspace is reduced, since by Proposition 1.2.12 the representation k 7→ adg(k)|p is faithful.Since z(p)k is the greatest ideal of g contained in k, it is invariant under σ and σ induces aninvolution on g/z(p)k, which becomes a reduced involutive Lie algebra, denoted the reducedinvolutive Lie algebra associated to (g, σ).

Definition 1.3.4. An orthogonal involutive Lie algebra is an involutive Lie algebra (g, σ) onwhich there exists a positive non-degenerate quadratic form which is invariant under σ andinfinitesimally invariant5 under adgk.

4A vector space isomorphism with σ([X, Y ]) = [σ(X), σ(Y )] for all X, Y ∈ g.5i.e. Q(ad(Z)X, Y ) = −Q(X, ad(Z)Y ).

21

Proposition 1.3.5. An involutive Lie algebra is orthogonal if and only if k is compact andad(k)|p leaves a positive non-degenerate quadratic form infinitesimally invariant.

Proof. If (g, σ) is orthogonal, there exists Q a positive non-degenerate quadratic form invari-ant under σ. By definition of k and p as eigenspaces of σ we get Q(k, p) = 0 and Q = Q1 +Q2

where Q1 = Q|k×k and Q2 = Q|p×p. Since ad(k) leaves the positive non-degenerate quadraticform Q1 invariant, k is compact. Q2 is invariant under ad(k), too.Conversely if k is compact, it leaves a positive non-degenerate quadratic form Q1 infinitesi-mally invariant. Denote by Q2 the positive non-degenerate form on p invariant under ad(k).Q := Q1 + Q2 is by construction invariant under σ and infinitesimally invariant under ad(k),hence (g, σ) is orthogonal.

Let (g, σ) be the involutive Lie algebra that comes from a symmetric space. We know fromour discussion above, that k is compact, hence ad(k) contains only skew-symmetric matrices.Therefore we can construct on p a positive non-degenerate quadratic form Q invariant underadgk by fixing a basis of p and take Q as the usual quadratic form (x, y) := x> · y. We haveshown, that ad(k) is skew-symmetric for each k ∈ k, therefore this bilinear form has all desiredproperties and (g, σ) is orthogonal.

Definition 1.3.6. A Lie algebra is said to be flat if its Lie bracket is constantly zero.

Theorem 1.3.7. Let (g, σ) be a reduced orthogonal involutive Lie algebra. Then (g, σ) is thedirect product of a reduced orthogonal involutive flat Lie algebra and of reduced semi-simpleirreducible orthogonal involutive Lie algebras (gi, σi),(i = 1, . . . , a). This decomposition isunique up to the order of the factors and z(p) = p0.

Proof. Let Q be a positive non-degenerate quadratic form on p, infinitesimally invariantunder adgk. Q exists since (g, σ) is orthogonal. Let A be the linear map of p into itself,defined by Q(Ax, y) = B(x, y) for every x, y ∈ p. If Q and B are the matrices of thecorresponding bilinear forms A = Q−1B, where Q is invertible since it is non-degenerate. SinceQ and B are both symmetric forms, we have Q(Ax, y) = B(x, y) = B(y, x) = Q(Ay, x) =Q(x,Ay). Therefore A is symmetric, hence diagonalisable. We have a decomposition of p

into eigenspaces

p =r⊕

i=0

qi,

such thatA|qi = ci · id (ci ∈ R, c0 = 0, ci 6= cj for j 6= i).

q0 is the kernel of the Killing form, i.e. it contains every x ∈ g with B(x, y) = 0 for all y ∈ q.It is contained in p since B is non-degenerate on k. The qi are subvectorspaces of p. Forx ∈ qi and y ∈ qj , we have B(x, y) = ciQ(x, y) = cjQ(x, y) = 0 if i 6= j, hence

B(qi, qj) = Q(qi, qj) = 0, (i 6= j), B(q0, p) = 0

andB|qi = ciQ|qi .

Since B and Q are non-degenerate and infinitesimally invariant under adgk, A commuteselementwise with adgk. Therefore ad(k) leaves the eigenspaces of A fixed and hence we have

22

[k, qi] ⊂ qi. Let U ⊂ p be a subvectorspace invariant under ad(k) and U⊥ its orthogonalcomplement in p with respect to Q. Then we have for all u ∈ U, v ∈ U⊥, k ∈ k:

Q(u, ad(k)v) = −Q(ad(k)u, v) = 0.

Hence ad(k)v is contained in U⊥, because it is orthogonal to u. Therefore we can decomposeqi in subvectorspaces invariant minimal under adk and there exists a direct sum decompositionof

⊕ri=1 qi in

⊕ai=1 pi, where each pi is invariant minimal under adk. Each pi is contained in

a qj and pi and pl are orthogonal with respect to Q if l 6= i.Furthermore we have [pi, pj ] = 0 for i 6= j, since for k ∈ k:

B(k, [pi, pj ]) = B([k, pi], pj) ⊂ B(pi, pj) = 0,

since pi and pj are orthogonal with respect to Q. B is non-degenerate on k, therefore [pi, pj ] =0. The same argumentation shows that [qi, qj ] = 0, if i 6= j. We will need this fact later.Let now

gi := [pi, pi] + pi, (i = 1, . . . , a).

[pi, pi] is contained in k because σ acts as the identity on it, thus the sum is direct. Furthermorewe have

[gi, gj ] = 0, (i 6= j), [k, gi] ⊂ gi (1 ≤ i, j ≤ a)

and[p0, gi] = 0, [p, gi] = [pi, gi] ⊂ gi.

gi is an ideal of g which follows from the Jacobi identity and the equations above. Thereforethe Killing form on gi is the Killing form on g restricted to gi × gi. It is non-degenerateon the pi, since it is a multiple of Q and it is non-degenerate on [pi, pi] ⊂ k. Hence it isnon-degenerate on gi and therefore gi is semi-simple. We have, again by the above formulaeand the Jacobi-identity for i 6= j

B(gi, gj) = B(pi, pj) + B(pi, [pj , pj ]) + B(pj , [pi, pi]) + B([pi, pi], [pj , pj ]) = 0.

Therefore the gi are linearly independent. Let m be their sum. It is a semi-simple ideal of g

invariant under σ, therefore g = g0 × m, with g0 = z(g). Denoting by σi the restriction of σto gi we have:

(g, σ) = (g0, σ0)× (g1, σ1)× . . . (ga, σa),

with (gi, σi) semi-simple, irreducible for i ≥ 1 and (gi, σi) reduced for all i’s. (g0, σ0) isflat.

This proof shows that for i 6= 0, B|pi is either negative definite or positive definite, sinceit is a multiple of the positive definite bilinear form Q. Therefore either gi is compact (if B|pi

is negative definite) or gi is non-compact and [pi, pi] is a maximal compact subalgebra. Inthe first case we say that it is of compact type, in the second it is of non-compact type. Thisproperty is directly related to the sectional curvature.

Corollary 1.3.8. Let (g, σ) an orthogonal involutive Lie algebra, G a Lie group with Liealgebra g and K a subgroup corresponding to k. Assume that K is connected and closed. LetQ be an arbitrary K-invariant Riemannian structure on G/K. Then

i) If G/K is of Euclidean type, then G/K has sectional curvature everywhere = 0.

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ii) If G/K is of compact type, then G/K has sectional curvature everywhere ≥ 0.

iii) If G/K is of non-compact type, then G/K has sectional curvature everywhere ≤ 0.

Proof. The tangent space at o can be identified with p. Let S be two-dimensional subspace ofTpM and X and Y two orthonormal vectors in S. Proposition 1.2.11 tells us that R(X,Y )X =−[[X, Y ], X] and by the definition of the sectional curvature (Definition 1.1.16, we get

K(S) = −Q0(R(X, Y )X, Y ) = Q0([[X,Y ], X], Y )

since the latter is zero in the Euclidean case, we have proved i).The other two cases are not difficult. We assume that g is semi-simple and use the resultsof Theorem 1.3.7. We decomposed p into subspaces qi as eigenspaces to the eigenvalues ci ofthe matrix A defined by Q0(AX, Y ) = B(X, Y ) and for Xi, Yi ∈ qi we have

B(Xi, Yi) = ciQ0(Xi, Yi).

And [qi, qj ] = 0 holds if i 6= j. With X =∑

Xi and Y =∑

Yi (Xi, Yi ∈ qi we get[X, Y ] =

∑[Xi, Yi] and [[Xi, Yi], X] = [[Xi, Yi], Xi]. Hence

K(S) = Q0([[X, Y ], X], Y ) =∑

i

Q0([[Xi, Yi], Xi], Yi) =∑ 1

ciB([Xi, Yi], [Xi, Yi]). (9)

If G/K is of compact type, by definition the ci are all negative and the sectional curvature isnegative. If G/K is of non-compact type, the ci are all positive and the sectional curvatureis positive.

Proposition 1.3.9. For an involutive semi-simple reduced Lie algebra (g, σ)

g = p⊕ [p, p]

holds.

Proof. First note, that the Killing-form B on g is non-degenerate, since g is semi-simple.Therefore we can decompose g in a direct sum of p + [p, p] and c := (p + [p, p])>. Since[p, p] ⊂ k, the first sum is also direct. c, being orthogonal to p, is contained in k. If we show,that c is zero, we are done, because B is non-degenerate. To do that we will show that it isan ideal of g contained in k and must therefore be zero.Fix c ∈ c and let p, p ∈ p and g0 = p0 + k0 ∈ g = p⊕ k.

B(g0, [p, c]) = B([g0, p], c) = B([p0, p]︸ ︷︷ ︸∈[p,p]

, c) + B([k0, p]︸ ︷︷ ︸p

, c) = 0,

hence [p, c] = 0, since g0 was arbitrary and B is non-degenerate. With the Jacobi-identity weget immediately that [[p, p], c] = 0. We are not far from our result, since

B([g0, c], p) = B(g, [c, p]) = 0B([g0, c], [p, p]) = B(g, [c, [p, p]]) = 0.

This shows that c is an ideal of g contained in k which must be zero, since g is reduced.Therefore

g = p⊕ [p, p]

holds.

24

This property characterizes the isometry group of a symmetric space. In Remark 1.2.8 weannounced the following proposition:

Proposition 1.3.10. Let G be a connected semi-simple Lie group, K a compact subgroupand σ an involution on G. Assume that

i) M = G/K is connected,

ii) there exists an involution σ of G such that Lie(G)dσe = Lie(K) and Kσ = K and

iii) G acts faithfully on M .

Then G is the identity component of the group of isometries on M .

Proof. Let G′ denote the group of isometries of M . M can be written in the form G′/K ′

where K ′ is a compact subgroup of G′. G acts by definition of M isometrically on M andsince it is connected, it can be identified with a subgroup of the identity component of G′.g′, the Lie algebra of G′, is reduced since, by Proposition 1.2.12 K ′ and therefore by LemmaA.1.5 k act faithfully on p ' TpM . Furthermore g′ is semi-simple, since G′ has no abelianfactor. Hence g′ = p⊕ [p, p].The Lie algebra g of G is a subalgebra of g′ and it decomposes therefore into eigenspaces p⊕ kwith respect to σ. Since p ' TpM ' p and g is by assumption reduced and semi-simple, wehave

g = p⊕ [p, p] = g′

and therefore G is the identity component of G′.

Remark 1.3.11. Helgason classifies in [He78] irreducible orthogonal involutive Lie algebras,where a Lie algebra is irreducible, if adg(k) acts irreducibly and faithful on p. By assumptionin our case here, (g, σ) is reduced, hence adg(k) acts faithful on p. Let i ⊂ ki be an ideal of gi,i.e. [i, gi] ⊂ i. Then i is an ideal of g =

⊕ai=0 gi since [gi, gj ] = 0 if (i 6= j). We have assumed

(g, σ) to be reduced, therefore i is trivial and hence gi is reduced. By construction of pi theaction of adgk is irreducible, hence the (gi, σi) are irreducible in the sense of Helgason.

By Theorem 1.3.7 we can assume that (g, σ) is either flat or a semi-simple reduced or-thogonal involutive Lie algebra. If (g, σ) is flat, it corresponds to an euclidean space, sincewe have a vector space with vanishing Lie bracket.A compact semi-simple Lie algebra g (ignoring the involution) decomposes in a product ofsimple ideals

g =n⊕

i=1

hi.

σ(hi) is simple, hence σ(hi) = hj . We have two cases:

i) i = j. In this case (hi, σ|hi) is a simple reduced orthogonal involutive Lie algebra.hi = ki ⊕ pi where ki and pi are the eigenspaces with respect to the eigenvalues 1 and−1.

ii) i 6= j. Then hj and hi are isomorphic via σ and (hi ⊕ σ(hi), σ|hi⊕σ(hi)) is a reducedorthogonal involutive Lie algebra. σ acts on it via (x, y) 7→ (y, x).

25

They are both invariant under σ. This gives us a new decomposition of g

g =⊕

i

li,

where li is an ideal in g, invariant under σ. It can therefore be decomposed into eigenspacesli = ki + pi. The li are ideals, hence [li, lj ] ⊂ li for all i and j. Therefore [li, lj ] ⊂ li ∩ lj .The last is trivial if i 6= j, therefore [ki, pj ] = [ki, kj ] = 0 if i 6= j. Since k acts by assumptionirreducibly on p and ad(ki)(pj) = 0 for i 6= j we have p = pi for some i. We can assume i = 1without loss of generality. For i 6= 1 we have [kj , p1] = 0, hence kj is an ideal of g in k. Butad(k) acts faithfully on p, hence k = k1. This shows that g = l1.An irreducible Lie algebra g of compact type has the form

I. (g, σ) is a compact simple Lie algebra and σ is any involution on it.

II. g is the direct sum l1 ⊕ l2 and σ acts on it via (x, y) 7→ (y, x).

If (g, σ) is a non-compact irreducible orthogonal involutive Lie algebra, we can construct adual compact Lie algebra by putting

g = k⊕ p 7→ g := k⊕ ip ⊂ gC.

g is also irreducible, since taking this dual does not change the action of k on p respectivelyip.Let g be in case I. Assume g to be not simple. Then there exists a decomposition in non-zeroideals n1 and n2. They can be decomposed into k1 +p1 and k2 +p2, hence (k1 + ip1)+(k2 + ip2)is a decomposition of g into non-zero ideals. Since this is impossible, g is simple. A similarargumentation shows that the complexification gC of g is simple, too.If g is in case II., its dual admits a complex structure (see [He78, Thm. V.2.4]) and it issimple. Therefore we have the following two possibilities for the non-compact case.

III. g is a simple, noncompact Lie algebra over R, the complexification gC is a simple Liealgebra over R and σ is an involutive automorphism of g such that the fixed points forma compactly imbedded subalgebra.

IV. g is a simple Lie algebra over C. Here σ is the conjugation of g with respect to amaximal compactly imbedded subalgebra.

Theorem 1.3.12. Let M be a simply connected symmetric space of compact or non-compacttype. Then M is product

M = M1 × . . .×Mn,

where the factors Mi are irreducible.

Proof. We proofed that the Lie algebra g of its group of isometries decomposes into semi-simple factors (Theorem 1.3.7)

g = g1 × . . .× gn

with gi = pi ⊕ ki. Hence G = G1 × . . . × Gn where gi is the Lie algebra of Gi. FurtherK = K1 × . . .×Kn and ki is the Lie algebra of Ki. Therefore M = M1 × . . .×Mn.

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1.4 Hermitian Symmetric Spaces

Definition 1.4.1. A Hermitian manifold is a differentiable Riemannian manifold with theRiemannian connection ∇ whose tangent bundle is endowed with a Hermitian structure, i.e.a (1, 1)-tensor J with J2 = −1, which leaves the metric invariant (g(JX, JY ) = g(X, Y )).

Like in the study of riemannian symmetric spaces we start with a local property.

Proposition 1.4.2. Let M be a Hermitian manifold. The following conditions are equivalent:

i) ∇R = 0 = ∇J ;

ii) for all x ∈ M , sx : y 7→ expx(− logx y) is a local automorphism at x of M , i.e. sx leaves∇ and J invariant.

Proof. i)⇒ii). From Corollary 1.1.13 we know that sx is a local diffeomorphism which leaves∇ invariant, because ∇R = 0. J is invariant under parallel transport, since ∇J = 0. Sincedsx acts by multiplication with −1 on TxM , it commutes with Jx and hence J and sx arecompatible.ii)⇒i). If ii) is satisfied every canonical tensor of odd degree is zero, because it is invariant bysx. Therefore ∇J and ∇R are both zero. (Same argumentation as in Corollary 1.1.13.)

Definition 1.4.3. A Hermitian manifold M is locally symmetric if the equivalent conditionsof Proposition 1.4.2 are satisfied. It is called Hermitian symmetric if it is connected and forall x ∈ M there exits an involutive automorphism sx of M with x as isolated fixed point.

Everything we said about Riemannian symmetric spaces applies to Hermitian symmetricspaces, too.

Corollary 1.4.4. A locally symmetric Hermitian manifold is Kahler. If g is the metric onM , then

ω(X, Y ) := g(X,JY )

is a Kahler form, i.e. ω is a symplectic, closed 2-form.

The following fact is a central fact in our study of hermitian symmetric spaces.

Corollary 1.4.5. Let M be a simply connected symmetric Hermitian space and let G bethe identity component of its group of automorphisms. Let x be a point of M and let K itsstabilizer. There exists a homomorphism ux : U1 → K such that ux(z) induces multiplication6

by z on TxM .

Proof. We use Theorem 1.1.9 and Proposition 1.1.10 to construct ux. For a given z ∈ U1,we have in the notation of Theorem 1.1.9 M = M ′, x = x′, ∇ = ∇′ and A is multiplicationwith z. By Proposition 1.4.2 and the definition of the Riemannian connection we have ∇T =0 = ∇R. If we show that R is invariant under the automorphism v 7→ zv on TpM (i.e.zR(zX, zY )zZ = R(X,Y )Z), then we are done. Note that

g(R(X,Y )Z, T ) = g(R(Z, T )X,Y ) (10)

6U1 := {z ∈ C : |z| = 1} and for v ∈ TxM and z = a + ib ∈ U1we define z · v by av + bJv.

27

holds. A proof of this fact can be found7 in [He78, Lem. I.12.5].By assumption J is invariant under parallel transport. Hence it commutes with the con-travariant derivative ∇X for all X. Therefore

R(X, Y )(JZ) = ∇X∇Y (JZ)−∇Y∇X(JZ)−∇[X,Y ](JZ) = JR(X,Y )Z.

Now we can show what we wanted:

g(zR(zX, zY )zZ, T ) = g(R(zX, zY )zzZ, T ) = g(R(Z, T )zX, zY ) = g(zR(Z, T )zX, Y )= g(R(Z, T )zzX, Y ) = g(R(X, Y )Z, T ).

Remember the maps τ(γ, u) introduced in Corollary 1.1.11. They are composites of sym-metries and are therefore automorphisms of M . As before, we have

M = G/K and Lie(G) = k⊕ p

where G is the identity component of the Lie group of automorphisms of M and K thestabilizer of a x ∈ M . σ denotes the automorphism of G induced by sx.

Before we continue, we have a look at an example.

Example 1.4.6. Let V be a n-dimensional complex vector space endowed with a non-degenerate Hermitian form h of signature (p, q), i.e. p is the maximal dimension of a subspaceL ⊂ V such that h|L is positive definite. We can choose a basis (e1, . . . , ep, ep+1, . . . , ep+q) ofV such that h is of the form

h(z, w) =p∑

i=1

ziwi −q∑

j=1

zj+pwj+p.

Denote W+ the space spanned by e1, . . . , ep. Its complement with respect to h is spannedby ep+1, . . . , ep+q. Denote by X the space of p-dimensional subspaces W ⊂ V with h|W ispositive definite. X is an open subset of the Grassmannian Grp(V ), hence a manifold. FixW ∈ X. Since h|W is non-degenerated, we can decompose V into the direct sum of W andW⊥, the orthogonal complement with respect to h. An element x of V can be decomposedinto w + w⊥. Denote sW the map which maps w + w⊥ to w − w⊥. It is the reflection withrespect to W . For two orthogonal vectors x, y ∈ V (i.e. h(x, y) = 0) we have

h(x + y, x + y) =h(x, x) + h(y, y) + h(x, y) + h(y, x) = h(x, x) + h(y, y)− h(y, x)− h(x, y)=h(x− y, x− y).

Therefore the reflection sW lies in SU(p, q), the subgroup of SL(n,C) which contains allmatrices which leaves h invariant and sW induces a map from X into X, also denoted by sW .It is involutive with W as an isolated fixed point. This makes X into a symmetric space.

Now we want to see how X can be realized as a homogeneous space G/K. Let G :=SU(p, q) and fix W+ ∈ X as a base point. G acts transitively on X, since one can choose

7Or one can proof it oneself by using the definition of R and the invariance of g under paralleltransportationin the form X · g(Y, Z) = g(∇XY, Z) + g(Y,∇XZ) to show that g(R(X, Y )Z, T ) = −g(R(X, Y )T, Z) and usethis, the Bianchi identity and Lemma 1.1.17 to prove Equation 10.

28

a basis of W and one of W⊥ and write down a matrix for h with respect to this basis.This matrix is hermitian, hence diagonalizable and by scaling it can be brought to the formdiag(1, . . . , 1,−1, . . . ,−1) with p-times 1 and q-times −1. Therefore W can be mapped ontoW+ by an element of G and G acts transitively on X.

Let g be in G. Write g in the form(

A BC D

)with A ∈ M(p, p,C), B ∈ M(p, q,C), C ∈

M(q, q,C) and D ∈ M(q, p,C). As a direct calculation shows the following conditions musthold:

A∗A− C∗C = idp

B∗B −D∗D = −idq

B∗A−D∗C = 0det(g) = 1.

Such a matrix g maps W+ into itself if and only if C = 0. From the third equation follows thatin this case B = 0 because by the first equation A is invertible. The first two equations tellus that A and D are hermitian, since they leave a positive-definite Hermitian form invariant.Therefore the stabilizer of W is equal to S(U(p) × U(q)) the matrices of U(p) × U(q) withdet(up) det(uq) = 1 for up ∈ U(p) and uq ∈ U(q). We can write X as the quotient G/K =SU(p, q)/S(U(p)× U(q)). A discussion of SU(1, 1) and the associated symmetric space, thehyperbolic plane, can be found in Example 2.5.3 and Proposition A.4.7.

Remark 1.4.7. The complex structure on M induces on p(= TxM) a complex structure J ,which is invariant under K, i.e. J commutes with dk|e for k ∈ K. Conversely, if M = G/Kis Riemannian and symmetric, every complex K-invariant structure on p endows M with ahermitian symmetric structure, since the structure can transported along the geodesics toevery point and this is well-defined since the hermitian structure on p is K-invariant.

Remark 1.4.8. If Je is a k-invariant complex structure on p, i.e. J([k, p]) = [k, Jp] then thepi from theorem 1.3.7 are complex subspaces, because they are disjoint representations of k

and hence [k, Jpi] ⊂ pi. By Theorem 1.3.7 the Lie algebra of the group of automorphismsdecomposes into a direct sum of a flat Lie algebra and semi-simple involutive Lie algebras gi

which can be decomposed as ki + pi. There exists subgroups Ki corresponding to ki. HenceM decomposes into Hermitian factors

M = M0 ×∏

Mi,

where M0 is flat.

Remark 1.4.9. Let G be the identity component of the group of isometries of M . Assumethat it is semi-simple. Then it has the same Lie algebra [p, p]⊕p as the analogous group for theuniversal covering M of M (Proposition 1.3.9). Therefore we have by Corollary 1.4.5 a mapux : U1 → K where K is the stabilizer of a point x ∈ M . Composing ux with the canonicalprojection M → M gives a map from U1 to K the stabilizer of a point in M . Since ux acts asmultiplication with z on p and since the canonical projection is a local diffeomorphism, thecomposition acts as multiplication with z on p.

Proposition 1.4.10. i) The subgroup K is the centralizer of ux(U1) in G and K is con-nected.

29

ii) The center of G is trivial.

iii) The Hermitian symmetric space M is simply connected.

iv) If (g, σ) is indecomposable, then ux(U1) is the center of K.

Proof. i) Let K ′ be the centralizer of ux(U1). First we show K ′ ⊂ K using Lie(K ′) ⊂Lie(K). Remark that the Lie algebra of K is the fixed point set of the action of dσe

in g. As σ is multiplication with −1 on p, we have σ = Adux(−1). By definitionAdux(−1)(= σ) is the differential of g 7→ ux(−1)gux(−1)−1, which is the identity onK ′. Hence dσe leaves Lie(K ′) fixed and therefore Lie(K ′) ⊂ Lie(K). K ′ is the centralizerof a torus, thus it is connected (see Appendix, Proposition A.4.1). Therefore K ′ is asubgroup of K.

Conversely, note that the representation of k on p and the complex structure J onp commute and and that the representation of k on p is faithful. Therefore we haveK ⊂ K ′, hence K = K ′ and K is connected.

ii) We have seen above that kτ(γ, u)k−1 = τ(kγ, u) for all k ∈ K, where γ is a geodesicstarting in x. By i) the center Z of G must be contained in K. If k is in Z it mustcommute with τ(γ, u) and u ∈ R and hence τ(γ, u) = τ(kγ, u) for all k ∈ Z. But sinceK acts faithful on p this is only possible for k = e, i.e. Z is trivial.

iii) Let π : G → G be the universal covering of G. The Lie algebras of G and G are bothequal to p ⊕ [p, p]. On G exists an involution σ with dσ = dσ. Denote by K the fixedpoint group of G, it is connected. We have π(K) = K, since the Lie algebras of Kand K are equal and they are both connected. From Remark 1.2.8 we know that G/Kis a symmetric space. We can apply i) and K contains the center of G. Thereforeπ−1(K) = K. Hence

G/K = G/K = M,

is simply connected, since G is and K is connected.

iv) Since g is indecomposable, the representation of K in p is irreducible. The commutatorof K in End(p) the same as the commutator of K in EndC(p), since the complex structureon J commutes with the K-action. By Schur’s lemma the center of K consists only ofmultiples of the identity matrix. Since K is compact, the center is, as a closed subgroup,compact to. Hence the center is ux(U1).

Proposition 1.4.11. Let M be a symmetric Riemannian space and let G be the identitycomponent of its group of automorphisms, x ∈ M , K its stabilizer in G and σ the involutionon G generated by conjugation with sx. Assume that G is semi-simple, K is connected and(g, σ) is indecomposable. The following conditions are equivalent:

i) the representation p of k is not absolutely simple (i.e. pC is not simple);

ii) the center of k is non-zero;

iii) M admits the structure of a symmetric Hermitian space (compatible with its Riemannianstructure).

30

Proof. i)⇒ iii). By assumption pC decomposes into a direct sum of two non-trivial complexsubrepresentations p− and p+. We can embed p into pC = p ⊕ ip and project pC onto p±.Composing these maps we get two maps

p → pC → p±.

Note that p ∈ p ⊂ pC can be written uniquely as p+ + p− with p− ∈ p− and p+ ∈ p+.Since p ∩ p− = {0} = p ∩ p+ this maps are injective. Therefore p+ and p− must have thesame dimension, hence the maps are isomorphisms. Therefore we can transport the complexstructure of p+ to p. Since p can be identified with the tangent space of M in some point thistangent space can be endowed with a complex structure too. By Remark 1.4.7 this induces acomplex structure on M .iii) ⇒ i). Let J denote the complex structure on p. Since J2 is symmetric it can be diago-nalized over C with eigenvalues i and −i. The eigenspace to i is an ad(k) invariant subspaceof the complexification of p, therefore the representation is not absolutely simple.iii) ⇒ ii) follows from Proposition 1.4.10.ii) ⇒ iii) By Schur’s Lemma (Lemma A.3.11) the center of K acts on pC as multiplicationwith complex diagonal matrices. Since K is compact, it contains only multiples of the identitymatrix with U1. Since the representation is faithful, we get a complex structure on p andtherefore one on M .

Example 1.4.12. This shows that P (n,R) is a Hermitian symmetric space if and only ifn = 2. We saw that P (n,R) = SL(n,R)/SO(n,R) and the stabilizer of a point equalsSO(n,R), which has non-trivial center if and only if n = 2 (see [Hi91, p. 328]).

Example 1.4.13. Remember Example 1.2.9. We saw there that the symmetric space

SO(2, n)/S(O(2)×O(n))

can be realized as the space G of two dimensional subspaces of R2⊕Rn on which the bilinearform of signature (2, n) is positive definite. Since we are dealing with two-dimensional realsubspaces of Rn+2 it seems possible, that we can put a complex structure on G. And thisis in fact true: since the center of S(O(2) × O(n)) contains at least O(2), it is non-trivialand by Proposition 1.4.11, there exists a complex structure on G. There exists a holomorphicdiffeomorphism of

D := {z ∈ Cn|1− 2〈z, z〉+ |zz>| > 0, |z| < 1}to M .

Remark 1.4.14. Let M = G/K be a compact simple symmetric hermitian space. For x ∈ Mlet ux : U1 → G be as before. In the representation Adux on Lie(G)C it acts only throughthe characters 1 (on kC) and z and z (on pC).

Conversely, if G is a compact adjoint simple group and u : U1 → G acts like aboveon Lie(G)C. Its centralizer K is connected since it is the centralizer of a torus. It has forLie algebra the subspace fixed by the involution Adu(−1) of Lie(G) and G/K is Hermitiansymmetric.

The classification problem of irreducible Hermitian symmetric spaces is therefore reducedto the classification of compact adjoint simple groups with a morphism u : U1 → G such thatits representation on Lie(G)C acts only through the characters 1 (on kC) and z and z (on pC).

31

2 Some Topics on Symmetric Spaces

2.1 Complexification and Cartan Decomposition

Let V be a finite dimensional vector space over R. A complex structure of V is a R-linearendomorphism J with J2 = −1. This endomorphism is something like the multiplication withi in a complex vector space. A vector space V with a complex structure can be turned intoa vector space V over C by putting

(a + ib)X := aX + bJX,

for a, b ∈ R and X ∈ V . The dimension of V over C is 12 dimR V .

A Lie algebra v over R is said to have a complex structure, if it has a complex structureJ as a vector space and

[X,JY ] = J [X,Y ] (11)

holds for X,Y ∈ v. Remark that the Lie bracket on a complex Lie algebra is C-linear, henceit commutes with i. If (11) would not hold, it would not be possible to turn it into a complexLie algebra like we did it with the vector space V . Bur if (11) hold, it is possible as one caneasily check.

Now let W be a finite dimensional vector space over R (without further structure). If onewants to extend W to a complex vector space WC of the same dimension8, i.e. make a scalarextension, one has several possibilities (which all yield the same result). The most obviousway may be to take a basis {e1, . . . , en} of W and define

WC :=n∑

i=1

Cei.

This works but one has to check that WC does not depend on the chosen basis. Anotherpossibility uses the observation that for a, b ∈ R we have i(a+ib) = −b+ia, i.e. multiplicationwith i maps the vector (a, b) to (−b, a). Defining J : W × W → W × W by (X,Y ) 7→(−Y, X) for X, Y ∈ W gives us a complex structure on W ×W which we also denote by WC.Furthermore one can write WC := W + iW as a formal sum and define multiplication with iin the obvious way. Last but not least one can use some theory and put WC := W ⊗R C. Allfour possibilities are isomorphic. The vector space WC is called complexification of W .

The complexification works for Lie algebra too. Let g be a Lie algebra and gC its com-plexification (as a vector space). It consists of all symbols X + iY with X,Y ∈ g. ForX + iY, Z + iT ∈ gC we define

[X + iY, Z + iT ] := [X, Z]− [Y, T ] + i([Y, Z] + [Z, T ]).

This makes gC to a complex Lie algebra.Let g be a Lie algebra over C. A real form of g is a subalgebra g0 of g considered as a

real Lie algebra, s.t.g = g0 ⊕ Jg0

as a R Lie algebra.In this case each Z ∈ g can be written as X + iY with unique X,Y ∈ g0. The map

σ : X + iY 7→ X − iY is called conjugation with respect to g0.An important theorem is the following:

8dimRW = dimCWC

32

Theorem 2.1.1. Every semi-simple Lie algebra over C has a real form which is compact.

See [He78, Thm. III.6.3] for a proof. It uses a root decomposition to write down explicitlythe compact real form.

Definition 2.1.2. Let g0 be a semi-simple Lie algebra over R. Denote by g its complexifica-tion an by σ the conjugation in g with respect to g0. A decomposition of g0

g0 = k0 + p0

where k0 is a subalgebra of g0 and p0 is a subvectorspace is a Cartan decomposition if thereexists a compact real form gk of g such that

σ(gk) ⊂ gk, k0 = g0 ∩ gk, p0 = g0 ∩ (igk).

The following theorem and Theorem 2.1.3 show that each semi-simple Lie algebra admitsa Cartan decomposition.

Theorem 2.1.3. Let g0 be a semi-simple Lie algebra over R, g ist complexification and u

any compact real from of g. Let σ denote the conjugation of g with respect to g0. Then thereexists an automorphism ϕ of g such that the compact real form ϕ(u) is invariant under σ.

One can show that a decomposition of a semi-simple Lie algebra g0 into a subalgebra k0and a vector space p0 is a Cartan decomposition if an only if the bilinear form

Bτ (X, Y ) := −B(X, τY ),

where τ : k + p 7→ k − p, is positive definite. An involution τ on a semi-simple Lie algebra iscalled Cartan involution, if Bτ (X, Y ) := −B(X, τY ) is positive definite. There exists a one-to-one correspondence between Cartan involutions and Cartan decompositions. Therefore thedecomposition of g in the direct sum of k and p in the context of the non-compact symmetricspaces is a Cartan decomposition of g.

2.2 Totally Geodesic Submanifolds

Definition 2.2.1. Let M be a Riemannian manifold. A submanifold S is called geodesicin a point p ∈ S, if each geodesic in M , which is tangent to S in p, is a curve in S (i.e. adifferentiable map from I ⊂ R to S). The submanifold S is called totally geodesic if it isgeodesic in each point p ∈ S.

One can show (see [He78, Lem I.14.2]) that geodesics in M which are contained in S aregeodesics in S and vice versa.

Note that a totally geodesic submanifold of a locally symmetric space is locally symmetric,since the restriction of the geodesic symmetry is again a geodesic symmetry.For symmetric spaces there exists a nice description of totally geodesic submanifold in termsof the Lie algebra of the group of automorphisms. For the central theorem we need somefacts, which we don’t want to prove it here, since it would be just copying from Helgason.We will only sketch the proofs. All details can be found in [He78], Chapters I.14 and V.6.

Theorem 2.2.2. Let M be a Riemannian manifold an S a connected, complete submanifoldof M . Then S is totally geodesic if and only if parallel transport in M along curves in Stransports tangents to S into tangents to S.

33

The proof uses the differential equations for parallelity which we introduced for Remark1.1.6 (Equation (1)).

Theorem 2.2.3. Let M = G/K be a symmetric space. For X ∈ p we put TX := (ad(X))2.We fix a point p ∈ M and, as usual, we identify TpM with p. The latter can be consideredas a manifold whose tangent space at each point is identified with itself. The exponential mapexp : p 7→ G/K has in X ∈ p the differential d expX : p 7→ p, which is given by

d expX = dLexp X |p ◦∞∑

n=0

(TX)n

(2n + 1)!,

where Lg : M → M denotes the left-action of G on M .

This is [He78, Thm. IV.4.1]. The next theorem is the main theorem on totally geodesicsubmanifolds in Riemannian symmetric spaces. It relates totally geodesic submanifolds tosubspaces of p, called Lie triple systems. A subspaces s of p is called Lie triple system, if itsatisfies [X, [Y, Z]] ∈ s for all X, Y, Z ∈ s.

Theorem 2.2.4. Let M be a Riemannian symmetric space and s be a Lie triple systemcontained in p. Then S = exp(s) has a natural differentiable structure in which it is a totallygeodesic submanifold of M with TpS = s.Conversely, if S is a totally geodesic submanifold of M with p ∈ S, then TpM = s.

Proof. Let s be a Lie triple system. Since s ⊂ p, we have [s, s] ⊂ [p, p] ⊂ k. From the Jacobiidentity we get

[[X, Y ], [U, V ]] + [U, [V, [X,Y ]]] + [V, [[X, Y ], U ] = 0.

Combined with the fact, that s is a Lie triple system we get that [s, s] is a subalgebra of k.Therefore the g′ := s + [s, s] is a subalgebra of g. Denote by G′ the corresponding subgroupof G and by K ′ the stabilizer of p in G′. Define S := G′ · p. By definition of K ′ we have abijection between G′/K ′ and S via g · p → gK ′. Therefore we can pull the topology and thedifferential structure from G′/K ′ to S. The tangent space of S in p is s. The geodesics in Mthrough p have the form exp(tX) · p with X ∈ p. Such a geodesic is tangent to S if and onlyif X ∈ s. Hence S is geodesic in p. Since G acts transitively on M , it is everywhere geodesic,hence totally geodesic and S = exp(s).

Conversely: let S be a totally geodesic submanifold of M . Denote by s the tangent spaceat a point p. For t ∈ R the vector A := d exptY (X) is tangent to S at the point exp(tY ),since exptX maps s to S because the exponential function on S is the restriction of the oneon M . As above we have dLexp(−tY )A is parallel along exp(tY ), hence dLexp(−tY )A ∈ s. FromTheorem 2.2.3 we know that this is equal to

∞∑

n=0

TY

(2n + 1)!(X),

therefore this is in s to. This implies that TX(Y ) ∈ s for all X,Y ∈ s. We have by definitionof TX

TY +Z − TY − TZ = adY adZ + adZadY.

The left hand side is in s, hence the same holds for the right hand side. Applying it to X ∈ s

we get with the Jacobi identity

2[Y, [Z, X]] + [X, [Y,Z]] ∈ s.

34

Interchanging X and Y gives

4[X, [Z, Y ]] + 2[Y, [X, Z]] ∈ s.

Added to the equation above this shows that [X, [Y,Z]] ∈ s.

This is [He78, Thm. IV.7.2].

Remark 2.2.5. A totally geodesic submanifold S of a symmetric space M is a symmetricspace. We saw in the proof of Theorem 2.2.4, that g′ := s + [s, s] is a subalgebra of g. Theautomorphism σ of g which maps g ∈ G to spgsp can be restricted to a map g′ → g′, thereforeS is a symmetric space and can be written as G′/K ′.

2.3 Rank

Definition 2.3.1. A Riemannian manifold is flat if the curvature vanishes identically. Therank of a symmetric space M is the maximal dimension of a flat, totally geodesic submanifoldof M .

The rank is always bigger or equal 1, since a one-dimensional subspace is always flat.One can describe maximal flat totally geodesic submanifolds in terms of the Lie algebra, usingTheorem 2.2.4.

Theorem 2.3.2. Let M be a symmetric space of the compact or the non-compact type. Let pbe any point in M and we identify the tangent space in p with p. Let s be a Lie triple systemcontained in p. Then the totally geodesic submanifold S := exp(s) is flat if and only if s isabelian.

Proof. Remember Corollary 1.3.8 where we studied the connection between (non-)compact-ness and the scalar curvature of symmetric spaces. Recall Equation (9)

K(S) = Q0([[X, Y ], X], Y ) =∑

i

Q0([[Xi, Yi], Xi], Yi) =∑ 1

ciB([Xi, Yi], [Xi, Yi]),

where in our case the ci are all positive or all negative since our M , since we assumed M tobe of compact or non-compact type. If we want to apply this formula, we have to proof thatS is symmetric and can be written as G′/K ′, since we proved Corollary 1.3.8 for spaces ofthis form. But this is true by Remark 2.2.5.

Theorem 2.3.3. Let G be a Lie group with Lie algebra g and g = k+p a Cartan decompositionand (g, σ) and (gU , σ) the corresponding (dual) orthogonal Lie algebras and K be the subgroupwith Lie algebra k. Let a ⊂ p be any maximal abelian subspace and A ⊂ G respectivelyAU ⊂ GU the corresponding subgroups to a respectively aU . Then

G = KAK, GU = KAUK.

For the proof we need a Lemma

Lemma 2.3.4. Let a and a′ be two maximal abelian subgroups of p. Then

i) There exists an element H ∈ a whose centralizer in p is a.

35

ii) There exists an element k ∈ K, such that Ad(k)a = a′.

iii) p =⋃

k∈K Ad(k)a.

This is [He78, Lem. V.6.3]. A short and nice proof can be found there.Now we want to prove Theorem 2.3.3. If K0 is the identity component of K, we have bycompleteness (Hopf-Rinow) that G ·K0 = exp(p). With the Lemma above, we can find forg ∈ G elements H ∈ p and k ∈ K such that gK0 = exp(Ad(k)H). Since exp(Ad(k)H) =k exp(H)k−1, therefore G = k exp(H)k−1k0 for some k0 ∈ K0.

2.4 Bounded Symmetric Domains

A domain is a connected open set in Cn. A domain D is called bounded symmetric domain,if D is bounded and for every z ∈ D there exists a biholomorphic map sz : D → D which isinvolutive and has z as an isolated fixed point.

Example 2.4.1. Let D be the unit disc in C. It is clearly a bounded domain. Lets make itsymmetric. The map s0 : z 7→ −z is a biholomorphic map from D to D. and it has 0 as anisolated fixed point. If we know, that the group SU(1, 1) acts transitive holomorphically onD, we can choose for z ∈ D a g ∈ SU(1, 1), which maps g to 0 and put sz := g−1s0g. This isfor every z ∈ D a point symmetric.Lets discuss the action of SU(1, 1) on D. The definition and of SU(1, 1) can be found inExample 1.4.6. It acts via Mobius transformations

(a bb a

)· Z =

az + b

bz + a

on D. The point 0 is mapped to b/a. For an arbitrary z ∈ D we have 1 − |z|2 > 0. Put

a := 1/(1 − |z|2) and b := z/(1 − |z|2). The matrix(

a bb a

)is in SU(1, 1) and it maps 0 to

z. Therefore the action is transitive.

The natural question at this point is: Given a bounded symmetric domain. Is there a(natural?) metric on D, such that D is a Riemannian/Hermitian symmetric space? Theanswer is yes. The metric is called Bergman metric. It is discussed shortly in [Ko00] andmore explicitly in [He78].

Let D be a domain. By H2(D) we denote the space of holomorphic functions that arequare integrable with respect to the Lebesgue measure. If D is bounded the polynomialsare in H2(D), hence it is infinite dimensional. Furthermore it is a complete Hilbert space.We denote the scalar product by (·|·). Consider for w ∈ D the linear functional on H2(D)given by f 7→ f(w). By the Riesz representation theorem there exists Kw ∈ H2(D) withf(w) = (f |Kw) for all f ∈ H2(D). Sometimes we write K(z, w) for Kw(z). The function Kis defined on D ×D and it has the property

K(z, w) = Kw(z) = (Kw|Kz) = (Kz|Kw) = Kz(w) = K(w, z).

Hence K(z, z) = |Kz‖2 ≥ 0. The function K is called Bergman kernel.For any complete orthogonal system {ϕk} we have

K(z, w) =∑

k

ϕk(z)ϕk(w).

36

Since Kw ∈ H2(D), the Bergman kernel is holomorphic in the first and antiholomorphic inthe second variable.

If F : D → D′ is an isomorphism9 between two domains with Bergman kernels K respec-tively K ′, we have

K(z, w) = K ′(F (z), F (w))jF (z)jF (w),

where jF is the (complex) Jacobi determinant of F . A proof of this fact can be found in theAppendix (Proposition A.4.5).

Let D be a bounded domain and K its Bergman kernel. Let for z ∈ D

gjk(z) :=∂2

∂zj∂zklog K(z, z).

This defines an invariant Hermitian metric on D. Hermitian means, that for z ∈ D thematrix gjk(z) defines a Hermitian form on the tangent space TpD. To see that, note thatgjk(z) = gkj(z). It remains to show that it is positive definite on TzD. This can be done bya direct calculation using a orthonormal system {ϕn} of H2(D). A direct calculation showsalso that F is an isometry between D and D′ equipped with their Bergman metrics.

2.5 Embedding

Our reference in this section is [Ko00] and [Wo72].

2.5.1 The Borel Embedding

In this section we will introduce the Borel embedding. It embeds a Hermitian symmetricspace of non-compact type into its compact dual. In the text one will find numbers (n). Theyrefer to the corresponding number in Example 2.5.3.

Let g = k⊕ p be a non-compact, orthogonal, involutive Lie algebra. Since we are lookingat Hermitian symmetric spaces we assume that the center of k is non-trivial. An equiva-lent assumption is, that k is not semi-simple. In this case we have a n ∈ k, such that theBk(n, k) = 0. Since n is in a compact Lie algebra the matrix of ad(n) is skew-symmetric,hence diagonalisable with eigenvalues of the form it with t ∈ R. Therefore B(n, n) being thesum of squares is zero if and only if every summand is zero. Hence ad(n) ≡ 0 and n is in thecenter of k. This shows that a non-semi simple Lie algebra has a non-trivial center. However,in the following the center of k is not trivial. (1)

Further we assume that k acts faithful on p. exp k =: K acts via orthogonal linear transfor-mations (via k 7→ dk) faithful on p and therefore on the complexification pC too. The centerof K (which is by the above discussion at least one-dimensional) consists of automorphismsof the representation of K in p. By Schur’s lemma (Lemma A.3.11) it can only contain di-agonal matrices. Since K is compact and the center is closed, is must be isomorphic to U1.Since the representation is faithful there exists a complex structure J = ad(z) on p, withz ∈ U1. J commutes with the ad(u) and it is diagonalisable. Therefore pC decomposes intotwo eigenspaces p+ and p− with respect to the eigenvalues i and −i. Let p1, p2 ∈ p+ andJ = ad(z) be the complex structure of p (2). With the Jacobi-Identity we get

J [p1, p2] = ad(z)[p1, p2] = [z, [p1, p2]] = [[z, p1], p2] + [p1, [z, p2]] = 2i[p1, p2],

9Biholomorphic map.

37

hence [p1, p2] = 0, since J has only eigenvalues i and −i. This shows that p+ is abelian. SinceJ is in the center of k the spaces p± are invariant under ad(k).

Let now GC be the simply connected group with Lie algebra gC. Denote by KC, K, P+, P−,G and GU the subgroups corresponding to the Lie algebras kC, k, p+, p−, g andgU (where gU :=k+ip) (4). On GC there exists an involution σ which is induced by the map g1+ig2 7→ g1−ig2

on gC since this map induces a local involution on GC, which can be continued by Proposition1.1.10. In the same way we can find involutions τ and θ, which leave gU respectively kC fixed.G, KC and GU are closed since they are the connected components of fixed point subgroupsof the involutions σ, τ and θ. K is closed, since it is the connected component of G ∩KC.Lets consider the case P±. The subalgebra ad(p+) of ad(gC) consists of nilpotent elements.This can be seen by applying ad(p) a few times to a p ∈ pC and a k ∈ k. For example

ad(p)p ∈ kC ⇒ ad2(p)p ∈ p+ ⇒ ad3(p)p = 0,

since p+ is abelian. From [He78, Cor VI.4.4] we know, that exp : ad(p+) → Ad(P+) isregular10 and surjectiv. P+ is the pre-image under this covering homomorphism of thissimply connected group. This shows that exp : p+ → P+ is a homeomorphism and P+ isclosed.Lets have a look at KCP+. It is a subgroup of GC with Lie algebra kC + p−. It is closedbecause kC is the normalizer of p− in gC and hence KCP− must be the identity componentof the normalizer of p− (5).Let M∗ = GC/KCP−. Since it is the quotient of a complex group by a complex subgroup,M∗ is a complex manifold.

With these notations we have:

Theorem 2.5.1. The space GU/K is a compact Hermitian symmetric. The map i : GU/K →M∗, defined by gK 7→ gKCP− is a holomorphic diffeomorphism (see Example 2.5.3 (6)).

Proof. The tangent space of M∗ at the point o := eKCP− is p+. The map i is induced bythe map from GU into GC. It is well-defined since K ⊂ KCP−. Its differential embeds gU

into gC. Since gU ∩ (kC + p−) = k, the differential of i in eK is the identity. Hence i is alocal injection and it is holomorphic. The image of i is the orbit of o of the action of GU onM∗ via multiplication. It is open, since i is a local injection. But it is also compact, since itis the image of a compact set (GU/K) under the continuous function i. Hence it is closed.This shows that the image of i is M∗ and i is surjective and GU/K is the universal coveringof M∗. But M∗ is simply connected since GC is and KCP− is connected. Therefore i is adiffeomorphism.

Now we show that one can embed M into M∗.

Theorem 2.5.2. M = G/K is a Hermitian symmetric space. The map j : M → M∗ definedby j(gK) := gKCP− is a G-equivariant holomorphic diffeomorphism onto an open subset ofM∗ (see Example 2.5.3 (7)).

Proof. We have to show first, that the map is injective. To do that, remark that two pointsgo and ho are equal in M∗ if and only if h−1g ∈ KCP−. Since M equals G/K, two points gKand hK are equal if and only if h−1g ∈ K. Therefore it suffices to show G∩KCP− = K. Since

10A map is regular if its differential is bijectiv.

38

G = KP (with P = exp(p)) it suffices to show that P ∩KCP− = {e}. Suppose p = kp− ∈ Pwith k ∈ KC and p− ∈ P−. We have

p−1 = στp = στ(kp−) = k(p−)−1.

since a direct calculation, using only the definitions of σ and τ gives us

dσdτ : k1 + p1 + ik2 + ip2 = k1 − p1 − ip2 + ik2.

We get p = kp− = p−k−1 and therefore p2 = (p−)2. Applying τ at both sides gives τ(p2) =p2 = (p−)2 = τ(p−)2. By definition of τ this is in P+. Since the AdGC is faithful we havep2 = e, because P+ and P− are represented by upper respectively lower triangular matrices.Therefore p2 = e implies p = e.The proof, that j is holomorphic diffeomorphism works like in the proof of the theoremabove.

Example 2.5.3. This example should make clear everything that happens in the sectionabove. In the appendix (Details), one finds a list of the groups appearing here.

(1) Let G be SU(1, 1) the group of 2× 2-matrices of determinant 1 which leave the bilinearform of signature (1, 1) invariant. Its elements are of the form

(a bb a

)

where a, b ∈ C and |a|2−|b|2 = 1 (see Proposition A.4.7. Its Lie algebra su(1, 1) consistsof matrices of the form (

ix y + izy − iz −ix

)

with x, y, z ∈ R. A proof of this fact can be found in [He78, p. 444]. su(1, 1) hastherefore the matrices

e1 :=(

0 1/21/2 0

), e2 :=

(0 i/2

−i/2 0

), e3 :=

(i/2 00 −i/2

)

as a basis and the Lie bracket relations are:

[e1, e2] = −e3, [e1, e3] = −e2, [e2, e3] = e1.

This follows from a simple computation. Denote by k the space spanned by e3 and byp the space spanned by e1 and e2. k generates a compact subgroup K of G, namely thesubgroup consisting of matrices of the form

(z−1 00 z

),

with z ∈ U1. The killing form of k is constantly zero since k is one-dimensional andhence abelian.

39

(2) The complexification of p (denoted by pC) is given by p⊗ C = p + ip. The Lie algebrak acts via the adjoint representation on p and the complex structure J = ad(e3). Itis diagonalisable over C with eigenvalues i and −i and the eigenspaces p+ and p− arespanned by {e1− ie2, e2 + ie1} respectively {e1 + ie2,−e2 + ie1}. One sees immediatelythat they have complex structures and that they are conjugate.

(3) The vector space p+ has the matrices b1 := e1−ie2 =(

0 10 0

)and b2 := ib1 = e2+ie1 =

(0 i0 0

)as a basis. Calculating exp(zb1) =

(1 z0 1

)with z ∈ C one sees that P+ is

equal to C, since it is R2 as a vector space and it is complex. The same holds for P−.

(4) The Lie algebra gU is k + ip. It has the matrices

ie1 =(

0 i/2i/2 0

), ie2 =

(0 −1/2

1/2 0

), e3 =

(i/2 0

0,−i/2

)

as a basis. Therefore gU consists of matrices of the form(

ix iy − ziy + z −ix

)

with x, y, z ∈ R. Consulting [He78, Ch. X.2] one finds that this is the Lie algebra ofSU∗(2), the group of matrices which commute with the map

ψ :{

C2 → C2

(u, v) 7→ (v,−u)

Direct calculation shows that

SU∗(2) ={(

a b−b a

): a, b ∈ C, |a|2 + |b|2 = 1

}.

The complexification gC of g has the matrices e1, e2 and e3 as a complex basis. Anotherbasis is

a1 := e1 − ie2, a2 := e1 + ie2, a3 := −2ie3.

Calculation its brackets

[a1, a2] = −a3, [a1, a3] = −2a2, [a2, a3] = 2a2

and comparing with [He78, Ch. X.2], we see, that gC = sl(2,C). Therefore GC =SL(2,C).P+ and P− are isomorphic to C as we have seen in (3). KC is isomorphic to C∗ via

z 7→(

1/z 00 z

).

(5) What is GU/K? Let(

a b−b a

)in GU . If a = a1+ia2 and b = b1+ib2, with a1, a2, b1, b2 ∈

R, the condition |a|2 + |b|2 = 1 rewrites as

a21 + a2

2 + b21 + b2

2 = 1.

40

Therefore GU can be identified with S3, the 3-sphere in C2 ' R4. The circle-groupS1 ' U1 acts on it via right-multiplication

(a b−b a

)(z 00 z

)=

(za zb−zb za

).

The action is faithful, since a or b is non-zero. There is an important and well-knownaction of S1 on S3, namely the Hopf-Filtration. It has S2 as the base space, S3 as thetotal space and S1 as the fiber. For details to this fibration see [Ha02]. The map

(a b−b a

)7→ b

a∈ C ∪ {∞}

is a surjective map from GU to S2. If b/a = b/a, then a/a = b/b =: z and we have

|a|2 + |b|2 = 1 = |a|2 + |b|2 = |z|2(|a|2 + |b|2),

hence z ∈ U1 = K. Therefore GU/K = S2.

Now we want to determine G/K. If one multiplies an arbitrary(

a bb a

)∈ G with an

arbitrary(

z 00 z

)∈ K we get a matrix

(za zbzb za

).

An element in M is uniquely determined by a and b. Since z ∈ U1 we can assume thata ∈ R. If b = b1 + ib2 with a ∈ R our condition |a|2− |b|2 = 1 becomes a2− b2

1− b22 = 1.

But this is exactly the condition for a point (a, b1, b2) ∈ R to be in the upper sheet ofthe hyperboloid defined by the equation x2 − y2 − z2 = 1.

In Example 1.4.6 we realized G/K as the space X of one-dimensional subspaces W of C2

on which the quadratic form h(x, y) = x1y1− x2y2 is positive definite. Let (z1, z2) spana W . Since |z1|2 − |z2|2 > 0, z1 6= 0 and we can assume without loss of generality thatz1 = 1 (because 1

z1(z1, z2) ∈ W ). Therefore W depends only of z2 and since 1−|z2|2 > 0

we have 1 > |z2|2 and X can be identified with the unit-disk.

What is GC/KCP−? Let(

a bc d

)be in GC and

(x 0y z

)∈ KCP−. We have

(a bc d

)·(

x 0y z

)=

(xa + yb b/xxc + yd d/x

)

Putting x = d and y = −c we get (with ad− bc = 1)(

1 b/d0 1

),

with b/d = ∞ if d = 0. Therefore we have in every equivalence class an element of thisform, hence we can identity the set of this equivalence classes with S2.

41

(6) The map i : GU/K → GC/KCP− maps gK to gKCP−. It is a map from S2 to S2. Letx be in S2 = C ∪ {∞}. If x = ∞ put a := 0 and b := 1. If x 6= 0 we can write x as

b/a with |a|2 + |b|2 = 1. A representant of x is in both cases(

a b−b a

). This matrix

is contained in GC and represent therefore an equivalence classes of GC/KCP−. With

the last part of (5), every class has one and only one element of the form(

1 b/a0 1

).

Therefore i is the identity on the Riemann sphere.

(7) The same calculation shows that j maps a point (a, b1, b2) ∈ G/K to b/a ∈ GC/KCP−,with b := b1 + ib2. Since a2 − b2

1 − b22 = 1 we have |a|2 − |b|2 = 1 and therefore

|a|2 = |b|2 + 1 > |b|2. This shows that j(G/K) is the lower hemisphere in GC/KCP−.

2.5.2 The Harish-Chandra Embedding

In this section we will use the same notation as in the section on the Borel embedding.We saw in the preceding section that one can embed a Hermitian symmetric space M ofnon-compact type into its compact dual M∗. In this section we will show that there exista map ξ : p+ → M∗ which is an embedding. In our example 2.5.3 p+ ' C and M∗ = S2.Later we will see that the image of M in M∗ is contained in the image of p+ under ξ. Itis a diffeomorphism onto its image and we can use ξ−1 to send i(M) ⊂ M∗ onto a boundeddomain in the vector space p+.Let now g be a Lie algebra as in the preceding chapter. Let h ⊂ k a maximal abeliansubalgebra. This h contains the center and its complexification hC is maximal abelian in gC.Remember the map τ : gC → gC defined by p1 + ip2 + k1 + ik2 7→ −p1 + ip2 + k1− ik2. Definea bilinear form Bτ on g by Bτ (x, y) := −B(x, τy). A straightforward calculation11 showsthat Bτ is positive definite Hermitian. The map ad(ih) : gC → gC leaves the Killing form ong infinitesimally invariant and one can easily check by direct calculation that ad(ih) ◦ τ =−τ ◦ ad(ih) for all h ∈ h. These two facts combined tell us, that ad(ih) is Hermitian. Hencead(ih) is semi-simple (i.e. every invariant subspace of gC has an invariant complement).Therefore hC is a Cartan subalgebra of gC. We can decompose gC into root spaces gα withrespect to hC. Denote by ∆ the set of roots. We define a positive root system by: α > 0 if andonly if −iα(z) > 0, for the z ∈ hC that induces the complex structure on p+ via J = ad(z).The set of positive roots is denoted by ∆+ and we put ∆− := −∆+. Since for x ∈ gα we have

−x = J2x = ad(z)2x = α(z)2x,

we have α(z) = ±i and we defined a root to be positive if α(z) = +i and negative if α(z) = −i.By Theorem A.2.4 there exists for every root α a unique Hα ∈ hC such that

α(H) = 2B(H, Hα)B(Hα,Hα)

∀H ∈ hC.

Further we can find eα ∈ gα with τeα = −e−α and [eα, e−α] = Hα. To see that, take for everyα ∈ ∆+ a nonzero vector eα ∈ gα and put e−α := −τ eα. Note that we have for H ∈ hC

B(H, [eα, e−α]) = B([H, eα], e−α) = α(H)B(eα, e−α).11This calculation needs only the definition of τ and the positivity respectively the negativity of B on k and

p.

42

Hence Hα is a multiple of [eα, e−α] and we can find xα ∈ C such that for eα := xαeα we have[eα, e−α] = Hα.

The root space gα is either in kC of in pC, since the decomposition of gC into kC+ p+ + p−is the decomposition with respect to eigenspaces of ad(z) with z ∈ U1.

If gα ⊂ kC we say that α is a compact root, if gα ⊂ pC, α is non-compact and we denote theset of positive non-compact roots by Φ. We denote the set of compact roots by ∆K and thenon-compact roots by ∆P and we put ∆+

K := ∆K ∩∆+ and so on. The vector space pC is thedirect sum of the gα with α non-compact. If α is a positive root, −α is a root too (TheoremA.2.4) and it is negative- Therefore the number of positive and negative non-compact rootsare equal. For a positive root α we have

Jeα = ad(z)eα = α(z)eα = ieα.

Therefore⊕

α∈Φ gα ⊂ p+ and⊕

−α∈Φ gα ⊂ p− and by dimension reasons we have

p± =⊕

α∈±Φ

gα. (12)

Definition 2.5.4. Two roots α, β ∈ ∆ are strongly orthogonal, denoted α ° β, if neitherα± β is a root.

If α ° β then [gα, g±β] ⊂ gα±β = 0 and by general theory of root they are orthogonal.One can construct inductively a maximal set of strongly orthogonal non-compact roots

Ψ := {ψ1, . . . , ψr}

where ψj+1 is a minimal root, strongly orthogonal to ψ1, . . . , ψj .Since eα = −τe−α the vectors

xα,0 := eα + e−α and yα,0 := i(eα − e−α), α ∈ ∆+P

form a basis of p, since they are in p by definition of τ and they are linearly independent.Therefore

xα := ixα,o = i(eα + e−α) and yα = iyα,0 := −eα + e−α, α ∈ ∆+P

are a basis of ip.We define subspaces of pC by

a :=∑

ψ∈Ψ

Rxψ,0 ⊂ p, aU :=∑

ψ∈Ψ

Rxψ ⊂ ip.

They are abelian subalgebras of pC since the elements of Ψ are pairwise strongly orthogonal.Further they are maximal abelian since Ψ is maximal.There are groups A ⊂ G and AU ⊂ GU with Lie algebras a respectively aU .For α ∈ ∆ put

gC[α] := CHα + gα + g−α.

By construction this is a 3-dimensional simple subalgebra of gC and it has real forms

g[α] = g ∩ gC[α] and gU [α] = gU ∩ gC[α].

43

That defines simple groups GC[α], G[α] and GU [α] with Lie algebras gC[α], g[α] and gU [α].If Γ ⊂ Ψ we have the direct sums

gC[Γ] :=∑

α∈Γ

gC[α], g[Γ] :=∑

α∈Γ

g[α], gU [Γ] :=∑

α∈Γ

gU [α]

with corresponding groups GC[Γ], G[Γ] and GU [Γ]. By definition of Hα and eα, we have ingC[α] the relations:

[eα, e−α] = Hα, [eα,Hα] = −2eα, [e−α,Hα] = 2e−α.

Therefore gC[α] ' sl(2,C). See Example 2.5.3 for details.

Theorem 2.5.5 (Polydisc Theorem). Let Γ ⊂ Ψ and x0 ∈ M a point fixed by K. ThenGC[Γ] · x0 = GU [Γ] · x0 is a holomorphically embedded submanifold of M , that is a product of|Γ| Riemann spheres and G[Γ] · x0 is a holomorphically embedded submanifold of M∗ that isproduct of |Γ| hemispheres of GU [Γ] ·x0. Further we have M∗ = K ·GU [Ψ] and M = K ·G[Ψ].

Proof. Example 2.5.3 tells us that GU [α] · x0 is the Riemann sphere and we can considerG[α] · x0 as a hemisphere in GU [α] · x0. We know that G = KAK and GU = KAUK, hence

M = G · x0 = KAK · x0 = KA · x0 =⊂ K ·G[Ψ] · x0

andM∗ = GU · x0 = KAUK · x0 = KAU · x0 =⊂ K ·GU [Ψ] · x0

With the Polydisc Theorem, we can prove the following theorem, which tells us, that wecan realize a hermitian symmetric space of non-compact type as a bounded domain.

Theorem 2.5.6. The map φ : P+ × KC × P− → GC given by (p+, k, p−) 7→ m+kp− is acomplex diffeomorphism onto a dense open subset of GC that contains G. In particular

ξ :

{p+ → M = G/KCP−

p 7→ exp(p)KCP−

is a complex diffeomorphism of p+ onto a dense open subset of M∗ that contains M . Fur-thermore ξ−1(M) is a bounded domain in p+.

Proof. First we prove that φ is one-to-one. Assume p+1 k1p

−1 = p+

2 k2p−2 . Then (p+

1 )−1p+2 =

k1p−1 (k2p

+2 )−1 ∈ P+ ∩KCP−. We show in Lemma 2.5.7 that P+ ∩KCP− = {e}, therefore

p+1 = p+

2 . One can similarly prove that p−1 = p−2 and this gives us the injectivity of φ.Since dφ : p++kC+p− → p++kC+p− is an isomorphism, the map φ is a local diffeomorphism,hence its image is open.To show, that the image is dense and that ξ−1(M) is a bounded domain in p+, we restrictus to the case where GC = GC[α]. In the end we will generalize the result using the PolydiscTheorem.Let GC = GC[α]. Then GC = SL(2,C)/(±I) and we realize

eα :=(

0 10 0

), e−α :=

(0 01 0

), Hα :=

(0 1−1 0

).

44

In particular

xα,0 =(

0 11 0

), xα =

(0 ii 0

).

A direct calculation shows that

exp(txα,0) =(

cosh t sinh tsinh t cosh t

)=

(1 tanh t0 1

)(1

cosh t 00 cosh t

)(1 0

tanh t 0

)

= exp({tanh t}eα) exp({− log cosh t}Hα) exp({tanh t}e−α)

and

exp(txα) =(

cosh(it) sinh(it)sinh(it) cosh(it)

)=

(cos t i sin ti sin t cos t

)

=(

1 i tan t0 1

)(1

cos t 00 cos t

) (1 0

i tan t 0

).

Note that

ξ(p+) = exp(Ceα) ·KCP− ={(

1 z0 1

)·KCP−∣∣z ∈ C

}

and GU = KAUK, where AU = exp(Rxα) is the subgroup generated by a = Reα, since α isthe only positive non-compact root. Therefore

ξ(p+) = {k · exp(txα)KCP−|k ∈ K, t ∈ R and cos t 6= 0},and this shows that ξ(p+) is dense in M∗. Furthermore we have M ' G/KCP−. Note that

for(

a bb a

)∈ G there exists a unique

(x y0 z

)∈ KCP− with

(a bb a

) (x 0y z

)=

(1 b/a0 1

).

Since |a|2 − |b|2 = 1, we can write a = w cosh t and b = w sinh t with w, w ∈ U1 and t ∈ R.Hence we have:

M ={(

1 z0 1

)KCP−∣∣ |z| = tanh t for some real t

}

={(

1 z0 1

)KCP−∣∣|z| < 1

}= ξ({zeα| |z| < 1}),

hence ξ−1(M) is a bounded domain in p+ and we proved the theorem for GC = GC[α]. Thegeneral case follows from the Polydisc Theorem via

M∗ = {k exp(∑

ψ∈Ψ

tψxψ)KCP+|k ∈ K, tψ ∈ R}

ξ(p+) = {k exp(∑

ψ∈Ψ

tψxψ)KCP+|k ∈ K, tψ ∈ R and cos tψ 6= 0}

M = {kξ(∑

ψ∈Ψ

zψeψ)|k ∈ K and |zψ| < 1}.

45

Lemma 2.5.7. With the notion from above we have P+ ∩KCP− = {e}.

Proof. Let e 6= g ∈ P+ ∩KCP−. We know from our discussion above that exp : p+ → P+ isa diffeomorphism. Since g ∈ P+, there exists a unique x ∈ p+ with expx = g. We can writex =

∑α∈Φ cαeα, because of equation (12). Let β be a root with α−β > 0 if α ∈ Φ and α−β

is a root. Such a β exists. Denoting n+ =∑

α>0Ceα, we have

[x, e−β] = cβHβ +∑

α∈Φ,α6=β

cβ[eα, e−β].

The sum is in n+, since [eα, e−β] ∈ gα−β and by our choice of β, either the space gα−β is zeroor α− β is a positive root. Therefore

ad(x)e−β = [x, e−β] ≡ cβhβ mod n+. (13)

We know by general theory that Ad(g) = Ad(expx) = exp(ad(x)). By the definition of expas the power series exp(A) =

∑∞i=0

Ai

i! we get

Ad(g) = exp(ad(x))e−β ≡ e−β + ad(x)e−β +12ad(x)2e−β + . . .

For positive roots α and α we have [eα, eα] ⊂ gα+α ⊂ n+ and since Hβ ∈ hC, we have[Hβ, eα] = α(Hα)eα ∈ gα ⊂ n+, hence ad(x)ne−β ∈ n+ for n ≥ 2. Since by equation (13)ad(x)e−β ≡ cβHβ mod n+, we get

Ad(g)e−β ≡ e−β︸︷︷︸∈p−

+ cβHβ︸ ︷︷ ︸∈kC

mod n+.

This follows from g ∈ P+.But g is also in KCP−, which normalizes p−. Hence Ad(g)e−β ∈ p− which is a contradiction.Therefore our assumption that there is e 6= g ∈ P+ ∩ KCP− is false and P+ ∩ KCP− istrivial.

Example 2.5.8. We use Example 2.5.3 to explain the Harrish-Chandra embedding. We

saw that p+ is C. The map exp : p+ → P+ which maps(

0 a0 0

)to

(1 a0 1

)is clearly a

diffeomorphism. Since (1 a0 1

)·(

x 0y z

)=

(x + ya za

y z

),

every equivalence class in the image of p+ contains one and only one element of the form(1 a0 1

)with a ∈ C. The image is therefore the sphere without the north-pole. We saw

in Example 2.5.3, that one can embed G/K into the 2-sphere as the lower hemisphere. Itpreimage is the unit disc.

46

2.6 Filtrations, Gradations and Hodge Structures

In the following part we well need some notions and theory of algebraic groups. See theAppendix for a short overview or [Mi06] for a more detailed introduction.

Let k be a field and V a finite dimensional vector space over k. A Z-gradations of V is adecomposition

V =⊕

i∈ZV i

where the V i are subvectorspaces of V . Given such a gradation we can construct a represen-tation (V, w) of Gm, the multiplicative group of k, via

w(λ)v := λiv, for v ∈ V.

Conversely, every representation of (Gm)k decomposes into representations by characters(maps from (Gm)k into Gm). This hold especially for representations of Gm (see [Mi06,Ch.9]). Since the maps from Gm to Gm are all of the form z 7→ zm, every representation ofGm defines a gradation as above. Therefore the category of representations of Gm and thecategory of graded vector spaces are isomorphic.Every gradation defines an increasing (resp. decreasing) filtration by

W i(V ) =⊕

j≤i

V j

(resp.Wi(V ) =⊕

j≥i

V j).

Let G be a reductive group over a field k of characteristic zero and let w : Gm → G be amorphism. For any representation % : G → GL(V ) of G, % ◦ w : Gm → GL(V ) defines anincreasing filtration W of V .The map w : Gm → G defines a functor from the category of G representations to the categoryof Z-graded vector spaces. This functor is compatible with tensorproduct and with takingthe dual. Conversely: a functor with these properties is induced by some w. This followsfrom the Tannaka duality since such a functor is a functor between the representations ofG and the representations of Gm which is by the Tannaka-duality induced by a morphismw : Gm → G. For more informations to the Tannaka duality see [JS91] or [Mi06, Ch. 24].

Proposition 2.6.1. Let n ∈ Z and H a real finite dimensional vector space. Given one ofthe following, one can construct the other two.

i) A bigradation of the complexification12 HC:

HC =⊕

p+q=n

Hp,q,

such that Hq,p is the complex conjugate of Hp,q.

ii) A finite decreasing filtration F on HC such that

HC = F p ⊕ F q

for p + q = n + 1.12HC := V ⊗ C

47

iii) An action h of the real algebraic group C∗ on H s.t. x ∈ R∗ ⊂ C∗ acts as multiplicationby x−n.

Proof. a) Given i), we construct ii). Define F p :=⊕

i≥p H i,j . This is clearly a decreasingfiltration. We have to show that for given p and q with p+q = n+1 the complexificationHC equals F p ⊕ F q. Remark, that i + j = n and H l,k is the complex conjugate of Hk,l.

F p⊕ F q =

⊕

i≥p

H i,j

⊕

⊕

l≥q

H l,k

=

⊕

i ≥ pl ≥ n + 1− p

H i,j⊕Hk,l =⊕

p+q=n

Hp,q = HC.

b) Given ii), we construct i). Define Hp,q := F p ∩ F q. Note that Hq,p = F p ∩ F q is thecomplex conjugate of Hp,q. Since F is a filtration we have

HC = . . . ⊃ F p ⊃ F p+1 ⊃ . . . .

In additional we have HC = F p+1 ⊕ F q if p + q = n. Therefore we have Hp,q =F p ∩ F q = F p/F p+1. Since the dimension of HC is finite and F is a filtration we haveHC =

⊕p+q=n F p/F p+1 =

⊕p+q=n Hp,q.

iii) Given i) we construct iii) viah(z)v = z−pz−qv

for z ∈ C∗ and v ∈ Hp,q.

iv) Given iii). Every representation of C∗ in an algebraic group is multiplication withz−pz−q.

Definition 2.6.2. Let H be a vector space. Given i), ii) or iii) of Proposition 2.6.1 is a Hodgestructure on H. The filtration in ii) is called Hodge filtration.Let H be a vector space equipped with a Hodge structure. A polarization ψ on H is a bilinearform invariant under h(U1), such that ψ(x, h(−i)y) is symmetric and positive definite.

Example 2.6.3. Let H be a real vector with a complex structure J . The complex structuregives an action of C∗ on H, via (a+ib) ·v := (a+bJ)v for a+ib ∈ C∗ and v ∈ H. Denote by Hthe set that equals H as a vector space equipped with the action of C∗ via (a+ib)·v := (a−bJ)b.By Proposition 2.6.1 this gives a Hogde structure on H with H−1,0 := H and H0,−1 := H.With the notations above we have F 0 = H0,−1 and F−1 = H0,−1⊕H−1,0. H → HC/F 0(HC)is a C-linear isomorphism.In this case a polarization is nothing but the imaginary part of a positive definite hermitianform. Hermitian means here, that φ(λx, µy) = λφ(x, y)µ, for x, y ∈ H and λ, µ ∈ C. Since itis non-degenerate, we can find a basis {e1, . . . , em} of H which is orthonormal with respectto φ. By assumption φ(ei, ei) is bigger than zero, hence real. If x =

∑xiei and y =

∑yiei

we haveφ(x, y) =

∑xiyiφ(ei, ei) =

∑yixiφ(ei, ei) = φ(y, x) = φ(y, x).

Denote by ϑ the real part of φ and by ψ its imaginary part. We write zx instead of h(z)x.For z ∈ U1 we have

φ(zx, zy) = zzφ(x, y) = φ(x, y),

48

that means φ and therefore ψ are invariant under h(U1). We have

φ(x,−iy) = φ(iy, x) = φ(y,−ix),

hence φ(x,−iy) is symmetric. The last thing to check is the positive definiteness. Since φ ispositive definite (and therefore real) we have φ(x, x) = ϑ(x, x) > 0 and

ϑ(x,−iy) + iψ(x,−iy) = φ(x,−iy) = iφ(x, y) = iϑ(x, y)− ψ(x, y),

comparison of real and imaginary part of both sides shows that ψ(x,−iy) = ϑ(x, y) andtherefore it is positive definite.For the converse on can define φ(x, y) := ψ(x,−iy) + iψ(x, y).

Remark 2.6.4. Let w : Gm → G be a morphism. It defines for every representation (V, %)of G a filtration. We look at the case where this representation is the adjoint representationof G on g = Lie(G). We use the following notations: w(Gm) is abelian, hence contained in amaximal torus T . We denote by w the Lie algebra of w(Gm) and by t the Lie algebra of T .For i ∈ Z let

gi := {x ∈ g|Ad(w(λ))x = λix, ∀λ ∈ Gm}and for α : t → R linear

gα := {x ∈ g|ad(h)x = α(h)x, ∀h ∈ t}.

One sees immediately, that the gi and the gα are similar. They are both eigenspaces to somecommon eigenvalues. The non-empty gα are rootspaces of g. We define a positive root system.w is one-dimensional, hence the value of every root α on w is determined by an arbitrary non-zero vector a ∈ w. Roots α which have the property that the restriction to the one-dimensionalsubspace w is multiplication with a negative scalar, are now called positive. Let α be a positiveroot and i := α(a). Since exp(α(a)) = exp(ad(a)) = Ad(exp(a)) with exp(a) ∈ w(Gm) thereexists a λ ∈ Gm with w(λ) = exp(a). Since exp(α(a)) = Ad(exp(a)) = λi we have i ∈ Z−and gα ⊂ gi for some i ∈ Z−. Therefore gα ⊂ gi for i ≤ 0 and therefore gα ⊂ W0(g). Thisshows that W0(g) is a parabolic Lie algebra. It generates a parabolic subgroup W0(G) of G,which respects the filtration W on every representation of G.

Remark 2.6.5. Let M be a Hermitian symmetric space and G the identity component ofits group of automorphisms. For x ∈ M we have by Remark 1.4.9 a morphism ux : U1 → Gsending z to the automorphism of M that fixes and acts as multiplication by z on TxM . Forg ∈ G the maps ugx and guxg−1 act both as multiplication by z on TgxM and leave gx fixed.Therefore they are equal by Lemma A.1.5. We saw in Proposition 1.4.10 that the centralizerof ux is equal to the stabilizer of x. Therefore the mapping x 7→ ux identifies M with the setof conjugates of ux.

Definition 2.6.6. Let V be a n-dimensional vector space. The set of d-dimensional subvectorspaces of V , denoted by Gd(V ) is called Grassmann variety.

Let d = (d1, . . . , dr) with 0 < d1 < . . . < dr < n. A flag F of type d is a sequence ofdecreasing vector spaces V i

F : V ⊂ V 1 ⊂ . . . ⊂ V r ⊂ 0

with dimV i = di. The set of flags of type d is called flag variety.

49

The group GL(V ) acts transitively on the set of bases of V , hence it acts transitively onGd(V ). Let W be in Gd(V ) and denote by P (W ) the closed subset of GL(V ) which stabilizesW . Since the action of GL(V ) is transitive, we have

GL(V )/P (W ) ' Gd(W ).

Hence Gd(V ) is a manifold. The tangent space to Gd(V ) at W is

TW (Gd(V )) ' Hom(W,V/W ).

See [Mi04, p. 13f.].The map F 7→ (V i) from Gd(V ) to

∏i Gdi(V ) realizes Gd(V ) as a closed subset of∏

i Gdi(V ). The proof that Gd(V ) is a manifold works like the proof for Gd(V ). The tangentspace of Gd(V ) at a point F consists of homomorphisms ϕi : V i → V/V i with ϕi|V i+1 ≡ ϕi+1

mod V i+1.

Definition 2.6.7. Let V p,q be a real Hodge decomposition for the real vector space V . AHodge tensor is a multilinear map (r-tensor)

t : V × . . .× V︸ ︷︷ ︸r−times

→ R

which is left invariant by the action of h.

Now we will discuss Variations of Hodge structures and we will see that they are para-metrized by Hermitian symmetric spaces.

Fix a real vector space V and let S be a connected complex manifold. A family of Hodgestructures (V p,q

s ) parametrized by s ∈ S is said to be continuous if, for each (p, q) the V p,qs

vary continuously with s, i.e. d(p, q) = dimV p,q is constant and s 7→ V p,qs ∈ Gd(p,q)(V ) is

continuous. Let d = (. . . , d(p), . . .) with d(p) =∑

r≥p d(r, s). A continuous family of Hodgestructures (V p,q

s ) is holomorphic if the Hodge filtration Fs vary holomorphically, i.e.

ψ : s 7→ Fs ∈ Gd(V )

is holomorphic. The differential of ψ at s is C-linear:

dψs : TsS → TFs(Gd(V )) ⊂ ⊕p hom(F ps , V/F p

s ).

If the image of dφs is contained in

⊕p hom(F ps , F p−1

s /F ps )

for all s, then the holomorphic family is called a variation of Hodge structures on S.Let V be a real vector space and let T = {ti} be a finite family of tensors with t0 a

non-degenerate bilinear form on V . Let d : Z× Z→ N be such that

• d(p, q) = 0 for almost all p, q;

• d(q, p) = d(p, q);

• d(p, q) = 0 unless p + q = n.

50

Let S(d, T ) be the set of all Hodge structure (V p,q) on V such that

• dimV p,q = d(p, q) for all p, q;

• each t ∈ T is a Hodge tensor of V p,q;

• t0 is a polarization of V p,q.

The set S(d, T ) is a subspace of∏

d(p,q)6=0 Fd(p,q)(V ). Let S(d, T )+ be a connected componentof S(d, T ).

The next theorem is the reason why we introduced Hodge structures in the context ofsymmetric spaces.

Theorem 2.6.8. (a) If nonempty, S(d, T )+ has a unique structure of a complex manifoldfor which (V, V p,q

S ) is a holomorphic family of Hodge structures.

(b) With this complex structure, S(d, T )+ is a hermitian symmetric domain if (V, V p,qs ) is

a variation of Hodge structures.

(c) Every irreducible hermitian symmetric domain is of the form S(d, T ) for a suitable V ,d and T .

Proof. To show (a), first of all one shows that S+ can be written as a homogeneous space,namely as a quotient of the smallest subgroup of GL(V ) which contains h(S) for all h ∈ S+

and a point stabilizer. The complex structure on S+ comes from an embedding into Gd(V ).For part (b) see [De79]. The proof uses Remark 1.4.14.Given an irreducible Hermitian symmetric domain. We are searching for a vector space V ,

a suitable set of tensors T and a map d. The vector space V arises as a self dual representationG → GL(V ) where G the algebraic subgroup of GL(h) for h = Lie(Hol(M)+) with G(R)+ =Hol(M)+. One can show that there exists a non-degenerate bilinear form t0 on V fixed byG. Furthermore one can find a set of tensors such that G is the subgroup of GL(V ) whichfixes T . Let x ∈ D and let hx : S→ U1 → ux be the map which maps z to ux(z/z). This hx

defines a Hodge structure for which the t ∈ T are Hodge tensors and t0 is a polarisation.See [De79, 1.3] and [Mi04, Ch. 2] for details

2.7 Symmetric Cones and Jordan Algebras

In this chapter we will give a short introduction to symmetric cones and Jordan algebras andtheir correspondence. A good reference is [FK94].

2.7.1 Cones

Let V be a finite dimensional real Euclidean vector space with scalar product (·|·). A subsetC ⊂ V is called cone if it is non-empty and if for all λ > 0 and c ∈ C the vector λv ∈ C. Acone is convex if and only if λ, µ > 0 and x, y ∈ C imply xλx + µy ∈ C. A cone C is properif C ∩ (−C) = {0}. The dual open cone of a convex cone C is defined by

C∗ :={y ∈ Y |(x|y) > 0 ∀x ∈ C \ {0}}.

An open convex cone C is said to be self dual if C = C∗. Such a cone is proper since for anon-zero x ∈ C ∩ (−C) we have −x ∈ C and from (x| − x) < 0 we deduce see that C = C∗ isimpossible.

51

The automorphism group G(C) of an open convex cone C is defined by

G(C) = {g ∈ GL(V )|gC = C}.

It is a closed subgroup of GL(V ) hence a Lie group. The open cone C is said to be homogeneousif G(C) acts transitively on it. In this case we can apply Theorem A.1.4. A cone is said tobe symmetric if it is homogeneous and self dual.

Example 2.7.1. In Example 1.2.4 we discussed P = P (n,R), the set of positive definitematrices in Sym(n,R). It is clearly a cone in Rn×n. The closure P of P is the set of positivematrices. We want to prove that P is a symmetric cone. We know from the discussion inExample 1.2.4 that the group GL(n,R) acts via

g · x := gxg>

linearly on P and that this action is transitive. It remains to show that P is self-dual. Toshow that we need a criterion for y ∈ Sym(n,R) to be positive definite. An inner product onSym(n,R) is given by

(x|y) = tr(xy) =∑

i,j

xijyij .

The quadratic form Q associated to the symmetric matrix x is give by

Q(ξ) =∑

i,j

xi,jξiξj

for ξ ∈ Rn. Therefore we haveQ(ξ) = (x|ξξ>).

Back to our problem: let y ∈ Q∗. For a non-zero ξ ∈ Rn the matrix x = ξξ> is positive andsymmetric hence x ∈ P \ {0}. By the definition of P ∗ we get

k∑

j=1

yijξiξj = (y|x) > 0.

The vector ξ was arbitrary, hence y ∈ P .Any element x ∈ p \ {0} can be written as

x =k∑

j=1

αjα>j

where the αj are independent vectors in Rn and k ≥ 1. Therefore, if y ∈ P :

(y|x) =k∑

j=1

(y|αjα>j ) > 0

since all (y|αjα>j ) = Q(αj) > 0. Hence P ⊂ P ∗. This shows P = P ∗, therefore P is a

symmetric cone.

52

One can turn a symmetric cone C into a Riemannian symmetric space like we did it withbounded domains in Cn. First we introduce a metric on C. This works like in the case of thebounded domains. Define

ϕ(x) :=∫

C∗e−(x|y)dy.

This integral exists, it is uniformly convergent for x in any compact subset and it is C∞. Forall g ∈ G(C) we have

ϕ(gx) = | det g|−1ϕ(x).

The map x 7→ x∗ := ∇ log ϕ(x) is a bijection of C onto C∗. It has a unique fixed point.We define for x ∈ C and u, v ∈ V

Gx(u, v) :=∂2

∂u∂vlog ϕ(x).

Where One can show (see [FK94, Ch. I.4]) that is in fact a metric, which is invariant underG(C). The map x 7→ x∗ becomes an isometry, hence C is a Riemannian symmetric space.

2.7.2 Jordan Algebras

A Jordan algebra is a vector space V with a bilinear (not necessarily associative!) productV × V → V, (x, y) 7→ xy with

yx = xy

x(x2y) = x2(xy).

Define for L(x) the map L(x) : V → V which maps y to xy. The second property says thatL(x) and L(x2) commute.

Example 2.7.2. Let V be the algebra of n× n matrices over a field F . For x, y ∈ V put

x ◦ y :=12(xy + yx).

This defines a non-associative Jordan algebra as one can easily check by a direct calculation.

A Jordan algebra is said to Euclidian if there exists a positive definite bilinear form (·|·)such that (L(x)u|v) = (u|L(x)|v) for all x, u, v ∈ V .

2.7.3 Correspondence between Cones and Jordan Algebras

Let V be an Euclidian Jordan algebra. Let

Q := {x2|x ∈ V }.

Clearly, Q is a cone. The interior C of Q is a symmetric cone. This is not obvious and theproof is very technical. It can be found in [FK94, p. 46ff]. The cone C is the connectedcomponent of e in the set of invertible elements.

Let now C be a symmetric cone in a Euclidian space V . We denote by G(C) its automor-phism group, by G the identity component of G(C) and by K the subgroup G ∩O(V ). Onecan show that there exists a point e ∈ C whose stabilizer is K. We write g for the Lie algebra

53

of G and k for the Lie algebra of K. An element X of g belongs to k if an only if X · e = 0.We have G · e = C and one can show that g · e = V . Therefore we have a subspace p of g

which is isomorphic to V via X 7→ X · e. We denote by L its inverse: for x ∈ V , L(x) is theunique element in p such that L(x) · e = x.

Definingxy := L(x)y

gives V the structure of a Jordan algebra with identity element e. We have xy − yx =[L(x), L(y)] · e = 0, since [p, p] ⊂ k and k · e = 0. The proof of the second property of a Jordanalgebra is again very technical and therefore omitted. It can be found in [FK94, p. 49f].

A Appendix

A.1 Riemannian geometry

Theorem A.1.1 (Hopf-Rinow). Let M be a Riemannian manifold. The following threeconditions are equivalent

(i) M is geodesically complete;

(ii) M is complete;

(iii) Each bounded subset U ⊂ M is relatively compact.

These three conditions imply

(iv) Two points in M can be joined by a geodesic.

Proof. This proof is due to de Rham in [dRh52].“(iii) implies (ii)” is trivial, “(ii) implies (i)” is easy, since every non-closed geodesic whichcan not be continued to infinity contains a non-convergent Cauchy-sequence. Therefore itsuffices to show that (i) implies (iii) and (iv).Fix a ∈ V . An initial vector of a geodesic arc γ from a to x is a tangent vector of γ with thesame length and direction. By assumption i) for every vector in TaM there exits a geodesicarc starting in a in the direction of this vector and with the same length.For r ≥ 0 let Sr be the set of points x ∈ M with d(x, a) ≤ r. Denote by Er the subset of Sr

which contains the points that can be joined with a by a geodesic. Er is compact. To see thistake a sequence xk in Er. Every xk can be joined with a by a geodesic γk, parameterized suchthat γk(0) = a and γk(1) = xk. Therefore γk is contained in the ball with radius r and center0 in TaM , which is compact. Hence γk has an accumulation point X. X generates a geodesicγX and γX(1) is an accumulation point of xk. Therefore Er is compact. We will show thatEr = Sr for all r ≥ 0. This relation is true for r = 0. If it is true for r = R > 0, then it isalso true for r < R. Conversely if it holds for every r < R, it also holds for r = R, since ER

is closed. We will show that if ER = SR, then there exists s > 0 such that ER+s = SR+s.First note, that the metric d(x, y) is defined as the infimum of the length’ of pathes joining xand y. Hence for all n ∈ N we can find for all y ∈ M a path (not necessarily a geodesic) froma to y such that its length is smaller than d(a, y) + h−1. Denote by xn the last point of thispath which is contained in ER = SR. Then we have d(a, xn) = R and

d(xn, y) ≤ d(a, y)−R + n−1.

From the triangle inequality we get

d(a, y)−R = d(a, y)− d(a, xn) ≤ d(xn, y).

Since (xn) ⊂ ER it has an accumulation point x and by the inequalities above we get d(a, y) =d(x, y) + R.On M there exists a continuous function s : M → R, such that if d(x, y) ≤ s(x), thenx and y can be joined by a unique geodesic of length d(x, y). Every point in M has anormal neighborhood, hence s can be chosen to be positive. Since ER is compact, s has aminimum on it, denote it by s(> 0). s depends continuously of R. Let y ∈ M such thatR < d(a, y) ≤ R+s. There exists x ∈ ER such that d(a, x) = R and d(x, y) = d(a, y)−R ≤ s.

55

Hence x and y respectively x and a can be joined by geodesics. The path between a and y haslength d(a, y) and is therefore a geodesic. Hence y ∈ ER+s. This shows that if ER = SR thenER+s(R) = SR+s(R) hence ER+s(R)+s(R+s(R)) = SR+s(R)+s(R+s(R)) and so on. If the sequenceRn+1 := Rn + s(Rn) converges to R ∈ R, then

R = limn→∞Rn = lim

n→∞(Rn + s(Rn)) = R + s(R),

therefore s(R) = 0. Since s is positive, Rn can not converge. Hence SR = ER for all R.

Theorem A.1.2. The stabilizer K of a point p ∈ M in the group of isometries G is compact.

Proof. G is equipped with the compact open topology, which is generated by sets of the formW (C,U) := {f ∈ G|f(C) ⊂ U}, where C,U ⊂ G with C compact and U open. K is a closedsubset of W ({p}, U). The last has compact closure, hence K is compact. A more detailedproof can be found in [He78, I.3].

Theorem A.1.3. The automorphisms of a symmetric space form a finite dimensional Liegroup G.

Proof. If G is a Lie group, it has a Lie algebra g. The stabilizer of a point p is a compactsubgroup, which can be embedded in O(TpM) via k 7→ dk, hence it is a Lie (sub-)group withLie algebra k. Therefore g decomposes into k ⊕ p. If we can determine p we can generate Gin a neighborhood of e via exp and with some technical stuff one can in fact show, that Gis a Lie group. The transformations τ(γ, u), defined in Corollary 1.1.11 with γ(0) = p areelements of G. They are generated by elements of TpM . In fact one can show (see [He78,I.3]) that they do the job.

Theorem A.1.4. Let G be a locally compact group with a countable base. Suppose G is atransitive topological transformation group of a locally compact Hausdorff space M . Let p beany point in M and K the subgroup of G which leaves p fixed. Then K is closed and themapping gH 7→ g · p is a homeomorphism of G/H onto M .

Proof. Since K is the stabilizer of p it is clear that the map gH 7→ g · p is welldefined andbijectiv. See [He78, Thm. IV.3.3] for details.

Lemma A.1.5. Let M be a Riemannian manifold, ϕ and ψ two isometries of M onto itself.Suppose there exists a point x ∈ M for which ϕ(x) = ψ(x) and dϕx = dψx. Then ϕ = ψ.

Proof. We may assume ϕ(x) = x and dϕx is the identity. In an arbitrary normal neighborhoodof x all points are left fixed by ϕ because ϕ is an isometry. Since M is connected each pointy ∈ M can be connected to x by a chain of overlapping normal neighborhoods and henceϕ(y) = y.

Lemma A.1.6. If K is a compact Lie subgroup of a Lie group G, then the matrices of ad(k)are all skew-symmetric.

Proof. Ad(K) leaves (on g) the positive non-degenerate bilinear form

< a, b >=∫

K(Ad(k) · a,Ad(k) · b)dk,

56

invariant. (·, ·) was an arbitrary positive definite bilinear form on g. Therefore Ad(K) is aclosed subgroup of the orthogonal group that leaves < ·, · > fixed, which is compact. Hencethe matrices in Lie(Ad(K)) are skew-symmetric. Since for k ∈ k we have exp(k) ∈ K and

exp ◦ad(k) = Ad ◦ exp(k) ∈ Ad(K).

Therefore ad(k) ∈ Lie(Ad(K)) and hence ad(k) consists only of skew-symmetric matrices.

Lemma A.1.7. An skew-symmetric real matrix A is diagonalisable and all eigenvalues areof the form it with t ∈ R.

Proof. A skew-symmetric matrix over R can be seen as a hermitian matrix over C. If it isdiagonalisable, its eigenvalues must have the form it with t ∈ R.We know from linear algebra, that a symmetric matrix is diagonalisable. A2 is symmetric andhence diagonalisable. Let λ ∈ R be an eigenvalue of A2 and e 6= 0 an eigenvector with respectto λ. The space X spanned by e and Ae is invariant under A because Ae and A2e = λe arecontained in X. Now we have two possibilities. The dimension of X is either 1 or 2. In thefirst case Ae = λe with λ2 = λ. Therefore A can be written as

(A 00 λ

)

with A a (n− 1)× (n− 1)-matrix and since A is skew-symmetric, λ and λ are zero.If λ = 0 and Ae = λe, we get 0 = A2e = λ2e and since e 6= 0 by assumption we get λ = 0and therefore dimX = 1. Therefore if dimX = 2 we have λ 6= 0. The matrix of A restrictedto X with respect to the basis {e, Ae} is

(0 λ1 0

)

Since λ 6= 0, this matrix is diagonalisable, because it has two distinct eigenvalues ±√

λ.

Remark A.1.8. We can use the same argumentation to show that a matrix A is diagonaliz-able if its square is diagonalizable with non-zero entries on the diagonal. To see that, take aneigenvector e and look at the space X = {e,Ae}. If it is one-dimensional we have Ae = λe.If the dimension is 2, A acts as matrix

(0 λ1 0

)

which is diagonalizable since λ is non-zero. This works especially for involutions and complexstructures on vector spaces.

A.2 Roots

A good introduction to roots is [Kn96]. But everything we need can also be found in [He78,Ch.III.3].

Definition A.2.1. Let g be a Lie algebra. A subalgebra h ⊂ g is called Cartan subalgebraif for all h ∈ h the map ad(h) is semi-simple, i.e. every invariant subspace has an invariantcomplement.

57

Theorem A.2.2. Every semi-simple Lie algebra over C has a Cartan subalgebra.

Definition A.2.3. Let α : h → C be a linear functional. We put

gα := {X ∈ g|ad(H)X = α(H)X∀H ∈ h}.

If gα 6= 0, the functional α is called root and gα is the root space.

Theorem A.2.4. Let g be a Lie algebra and h a Cartan subalgebra of g. We denote by ∆the set of non-zero roots of g. Then

(i) g = h⊕⊕gα;

(ii) dimC gα = 1;

(iii) if α and β are roots with α + β 6= 0, then gα and gβ are orthogonal with respect to Bg;

(iv) B|h×h is non-degenerate and for every linear functional α on h exists a unique Hα ∈ h

with

α(H) = 2B(H,Hα)B(Hα,Hα)

∀H ∈ h

and α(Hα) 6= 0;

(v) if α ∈ ∆ then −α ∈ ∆ and [gα, g−α] = CHα;

(vi) if α + β 6= 0 then [gα, gβ] = gα+β.

A.3 Algebraic groups

We follow [Mi06].Let k be a field. An algebraic group is a group defined by polynomials. Examples areSL(V ), it is defined as {A ∈ Mn×n(k)|det(A) − 1 = 0} or GL(V ), which is defined as{(A, t)|A ∈ Mn×n(k), t ∈ k : det(A)t − 1 = 0}, where Mn×n(k) denotes the vector space ofn× n-matrices over k. Since a polynomial over k is also a polynomial over a k-algebra A weneed a concretization of the definition:

Definition A.3.1. Let G be a functor of k-algebras into groups. G is an affine algebraicgroup if there exists a finitely generated k-algebra A such that

G(R) = Homk−algebra(A,R)

is functorially in R.

To see why this definition makes sense, G be a group defined by the sets of zeros ofpolynomials f1, . . . , fn and A be the ideal in k[X1, . . . , Xn] generated by the fi. Then G(R) =Hom(k[X1, . . . , Xn]/A,R).By [Mi06, Ch. 3] one can embed every algebraic group in the general linear group over avector space V .

Example A.3.2. (i) Ga is defined by the set of zeros of the zero-polynomial. Ga(k) = kas an additive group. In the above notation it is Homk(k[X], R).

58

(ii) Gm is defined as the set of zeros of the polynomial XY − 1. Gm(k) = k∗ is themultiplicative group of k. We have Gm(R) = Homk(k[X, X−1], R).

Definition A.3.3. (i) An algebraic group is a split torus if it is isomorphic to a productof copies of Gm, and it is a torus if it becomes a split torus over the algebraic closureof k.

(ii) Let R be a ring with 1. x ∈ R is called unipotent, if 1− x is nilpotent13

(iii) A group G is solvable if there exists a chain of subgroups

G = G1 ⊃ G2 ⊃ . . . ⊃ Gn = 1,

such that Gi is normal in Gi+1 and Gi/Gi+1 is abelian.

(iv) The radical of an algebraic group G is the largest14 normal solvable subgroup. It isdenoted by RG.

(v) The unipotent radical is the intersection of RG and the set of unipotent elements of G.It is denoted by RuG.

(vi) A smooth connected algebraic group G 6= 1 is semisimple if it has no smooth connectednormal commutative subgroup other than the identity, and it is reductive if the onlysuch subgroups are tori.

Proposition A.3.4. Let k be a field of characteristic zero. Every algebraic group over k hasa radical and a unipotent radical.

Proof. Let H and N be normal algebraic solvable subgroups of G. Then HN is normal andsolvable. (See [Mi06, p. 94])

Remark A.3.5. A group is reductive if and only if its unipotent radical is zero. A reductivegroup has the property that every representation decomposes in irreducible representations.

Definition A.3.6. The derived group Gder is the intersection of the normal algebraic sub-groups N of G such that G/N is commutative.

Theorem A.3.7. If G is reductive, then the derived group Gder of G is semisimple, theconnected center Z◦ of G is a torus, and Z◦ ∩ Gder is the (finite) center of Gder; moreover,Z◦ ·Gder = G.

Definition A.3.8. A Levi-subgroup H has the property G = RuGoH.

Definition A.3.9. (i) A parabolic subgroup P is a subgroup, such that G/P is projective.

(ii) A Borel subgroup is a maximal connected solvable algebraic subgroup.

Proposition A.3.10. A subgroup P is parabolic if and only if it is contained in a Borelsubgroup. Every parabolic subgroup can be identified with a stabilizer of a (not necessarilycomplete) flag.

Lemma A.3.11. (Schur’s lemma). Let G be a group and (V, %) a irreducible complex repre-sentation of G, then the only automorphism of this representation, i.e. vectorspace-morphismswhich commute with %(G) are complex multiples of the identity matrix.

13i.e. there exists a n ∈ N with (1− x)n = 0.14Largest means that it contains every other normal solvable subgroup.

59

A.4 Details

Proposition A.4.1. The centralizer of a torus in a connected compact Lie group is connected.

This proposition is Corollary 4.51 in [Kn96]. The proof needs two lemmas:

Lemma A.4.2. Let A be a compact abelian Lie group. Denote by A0 its identity component.Assume that A/A0 is cyclic. Then A has an element whose powers are dense in A.

Proof. Since A0 is connected, closed and abelian it is a compact torus, hence isomorphic toS1 × . . .× S1. Therefore there exists a0 ∈ A0 such that the powers are dense in A0. Since Ais compact, the factor group A/A0 is finite (say of order N) and we can choose by assumtiona representative b of a generating coset. Clearly bN ∈ A0. Since A0 is a torus we can findc ∈ A0 such that bNcN = a0. The closure of the powers of bc contains A0 and a representativeof each coset in A/A0, hence it contains A.

Lemma A.4.3. Let G be a compact connected Lie group and let S be a torus of G. If g ∈ Gcentralizes S, then there exists a torus S′ in G containing both S and g.

Proof. Let A be the closure of⋃∞

n=−∞ gnS. This is a subgroup of G, since for gns, gmt ∈ Awe have gns(gmt)−1 = gm−nst−1 ∈ A. The identity component of A0 is closed, abelian andconnected, hence a torus. Since it is also open, the set

⋃∞n=−∞ gnA0 is open in A and it

contains⋃∞

n=−∞ gnS. Therefore⋃n

n=−∞ gnA0 = A. By compactness of A0 some nonzeropower of g is in A0. Therefore A/A0 is cyclic and by Lemma [??] we can find a ∈ A whosepowers are dense in A. Since A is compact, we can find an element X of the Lie algebra a

of A such that a = exp X. The closure of {exp tX|t ∈ R} is a torus S′ containing A, andtherefore S and g.

Proof of Proposition A.4.1: The centralizer of S is by the lemma above the union of tori,hence connected.

Proposition A.4.4. Let D and D′ be domains, F : D → D′ an isomorphism and denote byjF the Jacobi determinant of F . The map σ : H2(D′) → H2(D) given by f 7→ (f ◦ F )jF is aHilbert space isomorphism.

Proof. The linearity is obvious. The inverse of σ is give by f 7→ (f ◦ F−1)j−1F . It exists since

F is an isomorphism, which implies the existence of F−1 and j−1F . The last thing to check is

that σ leaves the scalar product invariant. Let f, g ∈ H2(D):

((f ◦ F )jF |(g ◦ F )jF ) =∫

Df(F (z))g(F (z))jF (z)jF (z)dµ(z) =

∫

D′f(z′)g(z′)dµ(z) = (f |g).

In the last step we used the fact that |jF |2 is the Jacobi determinant of F regarded as amap R2n → R2n. This can be seen by writing ∂

∂z = ∂∂x + i ∂

∂y and using the Cauchy-Riemanndifferential equations.

Proposition A.4.5. For the Bergman kernels K of D and K ′ of D′ we have

K ′(F (z), F (w))jF (z)jF (w) = K(z, w).

60

Proof. Let f : D′ → C in H2(D) and w ∈ D. We have

f(F (w))jF (w) = (f |KF (w))jF (w) = ((f ◦ F )jF |(K ′F (w) ◦ F )jF )jF (w)

= ((f ◦ F )jF |(K ′F (w) ◦ F )jF jF (w)),

where we applied in the first step the defrinitio of K ′ on f : D′ → C. In the second step we usedProposition A.4.4. One the other hand, if one applies the definition of K to (f ◦F )jF : D → C,we get

f(F (w))jF (w) = ((f ◦ F )jF (w)|Kw).

Example A.4.6.

G = SU(1, 1) ={(

a bb a

) ∣∣a, b ∈ C, |a|2 − |b|2 = 1}

;

g = su(1, 1) ={(

ix y + izy − iz −ix

) ∣∣x, y, z ∈ R}

;

k ={(

ix 00 −ix

) ∣∣x ∈ R}

;

p ={(

0 y + izy − iz 0

) ∣∣x, y ∈ R}

;

K ={(

a 00 a

) ∣∣a ∈ C, |a|2 = 1}

;

gU ={(

ix iy − ziy + z −ix

) ∣∣x, y, z ∈ R}

;

GU ={(

a −bb a

) ∣∣a, b ∈ C, |a|2 + |b|2 = 1}

;

gC ={(

a bc −a

) ∣∣a, b, c ∈ C}

;

GC ={(

a bc d

) ∣∣a, b, c, d ∈ C, ad− bc = 1}

;

kC ={(

a 00 −a

) ∣∣a ∈ C}

;

KCP− ={(

a 0c d

) ∣∣a, c, d ∈ C, ad = 1}

;

Proposition A.4.7. The matrices in SU(1, 1) are exactly those of the form

(a bb a

)

with a, b ∈ C and |a|2 − |b|2 = 1.

61

Proof. Let(

a bc d

)be a matrix in SU(1, 1). By definition we have

(1 00 −1

)=

(a cb d

) (1 00 −1

)·(

a bc d

)=

(aa− cc ab− cdab− cd bb− dd

),

hence

aa− cc = 1 (14)dd− bb = 1 (15)ab = cd (16)ad− bc = 1. (17)

We have to show that c = b and d = a. We multiply (14) and (15), their product is equal to1 and we get:

(aa− cc)(dd− bb) = aadd− ccdd− aabb + ccbb = aadd− 2ccdd + ccbb

=(aa− cc)dd− cc(dd− bb) = dd− cc = 1.

We used (16) in the second step and (14) and (15) in the last step. Again with (14) and (15)we see with dd − cc = 1 that bb = cc and dd = aa. Therefore the absolute values of a andd respectively b and c are equal and there exist two complex numbers s1 and s2 in the unitcircle such that b = s1c and d = s2a. With (14) and (17) we have

ad− bc = as2a− cs1c = 1 = aa− cc

and thereforea(s2a− a) = c(s1c− c).

With (16) we get 0 = ab− cd = acs1 − cas2. Adding ac− ac gives

c(as2 − a) = a(cs1 − c).

(14) can be written as |a|2 − |c|2 = 1, hence a 6= 0. Now assume cs2 − c 6= 0. Since a isnon-zero the right-hand side of the above equation is non-zere, hence the left-hand side andwe can divide by all factors of the equations. This yields

a

c=

s1c− c

s2a− a=

c

a,

hence aa = cc with is a contradiction to (14). Therefore b = s1c = c. If c is non-zero, d = afollows from (16), if c is zero, it follows from 17 and we are done.

62

References

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