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Notes on Symmetric Spaces Notes on symmetric spaces Jonathan Holland Bogdan Ion arXiv:1211.4159v1 [math.DG] 17 Nov 2012 Contents Preface 5 Chapter1. Affine connections and transformations 7 1. Preliminaries 7 2. Affine connections 10 3. Affine diffeomorphisms 13 4. Connections invariant under parallelism 17 Chapter 2. Symmetric spaces 21 1. Locally symmetric spaces 21 2. Riemannian locally symmetric spaces 22 3. Completeness 23 4. Isometry groups 25 Chapter 3. Orthogonal symmetric Lie algebras 27 1. Structure of orthogonal symmetric Lie algebras 28 2. Rough classification of orthogonal symmetric Lie algebras 30 3. Cartan involutions 31 4. Flat subspaces 33 5. Symmetric spaces from orthogonal symmetric Lie algebras 35 6. Cartan immersion 38 Chapter 4. Examples 41 1. Flat examples 41 2. Spheres 41 3. Projective spaces 44 4. Negatively curved spaces 48 Chapter 5. Noncompact symmetric spaces 53 Chapter 6. Compact semisimple Lie groups 59 1. Introduction 59 2. Compact groups 59 3. Weyl’s theorem 61 4. Weight and root lattices 62 5. The Weyl group and affine Weyl group 63 6. Determining the center of G 65 7. Symmetric spaces of type II 67 8. Relative root systems 68 9. Fixed points of involutions 72 10. Symmetric spaces of type I 74 3 4 CONTENTS Chapter 7. Hermitian symmetric spaces 75 1. Hermitian symmetric Lie algebras 75 2. Hermitian symmetric spaces 78 3. Bounded symmetric domains 81 4. Structure of Hermitian symmetric Lie algebras 84 5. Embedding theorems 89 Chapter 8. Classification of real simple Lie algebras 93 1. Classical structures 93 2. Vogan diagrams 94 Bibliography 107 Index 109 Preface These are notes prepared from a graduate lecture course by the second author on symmetric spaces at the University of Pittsburgh during the Fall of 2010. We make no claim to originality, either in the nature of the results or in exposition. In both capacities, we owe a large debt to the notes of Borel [4], as well as to Kobayashi and Nomizu [11], [12] for the earlier chapters, and Helgason [5] for the later chapters. Several changes were made during the preparation of these notes from the material as it was presented in the course. First, many proofs have been corrected or streamlined. Whereever possible, results have been more carefully stated. Some exercises have been added to the text. In addition, we have tried to provide references to further reading. We have also devoted an entire chapter to a naive treatment of examples in an effort to illustrate some of the general features of symmetric spaces. Most of the material that we have drawn upon is quite standard, but some of it is difficult to find. The earlier chapters draw primarily from Borel [4] and Kobayashi and Nomizu [11]. Chapter 6 draws from Borel [4] and Humphreys [7], although there are many more detailed accounts of compact Lie groups; see Knapp [10] for a thorough treatment. The use of Vogan diagrams in Chapter 8 instead of the traditional Satake diagrams seems to be due to Knapp [10]. At present, these notes are a first draft—still very much a work in progress. As of this draft, there are a number of notable omissions: Certain portions of the text have not been optimally organized. For • instance, we have interspersed discussion of non-compact orthogonal symmetric Lie algebras in the chapter on compact orthogonal symmetric Lie algebras. At the Lie algebra level, the two notions are dual, but doing it this way avoided certain awkwardness in discussing the “maximally compact” (as opposed to “maximally split”) Cartan subalgebra. Furthermore, the lectures themselves scrupulously avoided the familiar • theorem–proof paradigm. We have attempted in these notes to organize things more around results, though. However, in some sections it was not clear exactly how to do this, and in these sections the text often becomes more obscure. We have not yet prepared suitable figures, although figures were pre- • sented in the lecture, and would shed a great deal of light on certain examples. We have omitted the discussion of Satake diagrams in favor of Vogan • diagrams, although Satake diagrams were presented in one of the lectures. 5 6 PREFACE We do not have plans to add this content, since it can be easily found in Helgason, and does not add any substantial results to those we have already obtained. The final chapter on analysis on symmetric spaces is not yet completed. • The lectures presented a very skeletal outline of the theory of spherical functions that we should like to include here. In the mean time, the expos- itory articles [19], [18] cover approximately the same content, although from a slightly different viewpoint. Jonathan Holland Bogdan Ion University of Pittsburgh December, 2010 CHAPTER 1 Affine connections and transformations 1. Preliminaries This section sets the notational conventions and basic facts of differential ge- ometry that will be essential to subsequent investigations. We shall omit proofs of, and even precise citations for, these facts. Two standard references for this material are Michor [16] and Kol´aˇr, Michor, Slov´ak [13]. 1.1. Tensors. Let M be a smooth manifold of dimension n. We denote by C ∞ the sheaf of smooth functions on M. A tangent vector at a point p M is an R-linear ∈ derivation vp : C ∞ R. The vector space of tangent vectors at p is denoted TMp, p → and the disjoint union over all p M, denoted TM, is the tangent bundle and carries ∈ a natural vector bundle structure. The dual bundle TM∗ is the cotangent vector. Tensors on M are formed by taking tensor products of copies of TM with copies of p TM∗. Denote by Tq M the bundle of tensors p T M = TM TM TM∗ TM∗ . q ⊗···⊗ ⊗ ⊗···⊗ p q | {z } | p {z } T p Tensor fields are sections of the tensor bundle Tq M, and here are denoted by q M. T 1 ff T 0 The space of vector fields is thus 0 M and the space of di erential forms is 1 M. T 1 It is worth remarking that these are all sheaves, and in fact 0 is the sheaf of derivations of C ∞ to itself. The resulting sheaf of C ∞-modules is locally free of rank n, and so this shows that TM is actually a vector bundle. 1.2. Lie derivatives. If φ : M N is a differentiable map, then for each p M, → = ∈ the pushforward φ ,p : TMp TNφ(p) is defined by (φ ,pXp)( f ) Xp( f φ) for ∗ → ∗ ◦ f C ∞ . The pushforward is also denoted by dφp. The pullback φ∗ : TN∗ TM∗ ∈ φ(p) φ(p) → p is the transpose of φ ,p. If φ is a diffeomorphism, then we defined φ on one-forms 1 ∗ 1 ∗ by φ = (φ− )∗ and φ∗ on vectors by φ∗ = (φ− ) . When φ is a diffeomorphism, these operators∗ extend in a unique manner to isomorphisms∗ of the full tensor algebra. T 1 ff Let X 0 M. An integral curve of X through a point p is a di erentiable ∈ C = d curve γ :( ǫ,ǫ) M such that, for all f ∞(M), (X f ) γ dt ( f γ). Taking f in turn to be− each of→n functionally independent∈ coordinates,◦ this defines◦ a first-order differential equation in the coordinates of γ, so that through each point p M, there exists an integral curve defined on a small enough interval. For each p ∈ M, there is an open neighborhood U M containing p and ǫ> 0 such that each point∈ ⊂ q U has an integral curve γq through it defined for all t ( ǫ,ǫ). The map ∈ ∈ − (q, t) γq(t) : U ( ǫ,ǫ) M is smooth, by smooth dependence of solutions on initial7→ conditions.× This− mapping→ is the local flow of the vector field X, and we shall denote the local flow by (q, t) exp(tX)(q). Shrinking U and choosing a smaller ǫ 7→ 7 8 1. AFFINE CONNECTIONS AND TRANSFORMATIONS if necessary, exp is a diffeomorphism of U onto its image for all t ( ǫ,ǫ). Indeed, 1 1 ∈ − exp− (tX) = exp( tX) is smooth on U. − T 1 The Lie derivative of a tensor field T along a vector field X 0 M is defined by ∈ exp(tX)∗T T LXT = lim − . t 0 t → For instance, it follows from the definition of the exponential map that LX f = X f for a function f C ∞(M). The Lie derivative along X is an R-linear derivation of ∈ C ∞-modules. In particular, LX(S T) = LXS T + S LXT. ⊗ ⊗ ⊗ Note also that the Lie derivative commutes with tensor contraction, since by defi- nition the pullback commutes with tensor contraction. If X, Y T 1M, then the Lie bracket of X and Y is the commutator of derivations: ∈ 0 [X, Y] f = XY f YX f. − T 1 With this operation, 0 M becomes a Lie algebra. The Lie bracket is related to the Lie derivative by LXY = [X, Y]. A proof of this is sketched below using differential forms. 1.3. Differential forms. Let TM∗ be the exterior algebra of the cotangent ∧ bundle, and let Ω denote the sheaf of sections of TM∗. This is a graded algebra ∧ whose graded components Ωk are the spaces of k-forms. Let Der(Ω) be the space of graded derivations on Ω, equipped with the supercommutator degD deg D′ [D, D′] = DD′ ( 1) D′D. − − Then Der(Ω) is a Lie superalgebra graded by degree. A Lie superalgebra is a Lie algebra in the category of super vector spaces. A 2 super vector space is a vector space with a Z grading V = V0 V1.
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