Basic Lie Theory Hossein Abbaspour Martin Moskowitz ii To Gerhard Hochschild Contents Preface and Acknowledgments ix Notations xiii 0 Lie Groups and Lie Algebras; Introduction 1 0.1 Topological Groups . 1 0.2 LieGroups ......................... 6 0.3 CoveringMapsandGroups . 10 0.4 Group Actions and Homogeneous Spaces . 15 0.5 LieAlgebras......................... 25 1 Lie Groups 31 1.1 Elementary Properties of a Lie Group . 31 1.2 Taylor’s Theorem and the Coefficients of expX expY . 39 1.3 Correspondence between Lie Subgroups and Subalgebras 45 1.4 The Functorial Relationship . 48 1.5 The Topology of Compact Classical Groups . 60 1.6 The Iwasawa Decompositions for GL(n, R) and GL(n, C) 67 1.7 The Baker-Campbell-Hausdorff Formula . 69 2 Haar Measure and its Applications 89 2.1 Haar Measure on a Locally Compact Group . 89 2.2 Properties of the Modular Function . 100 2.3 Invariant Measures on Homogeneous Spaces . 101 2.4 Compact or Finite Volume Quotients . 106 v vi 2.5 Applications......................... 112 2.6 Compact linear groups and Hilbert’s 14th problem . 121 3 Elements of the Theory of Lie Algebras 127 3.1 Basics of Lie Algebras . 127 3.1.1 Ideals and Related Concepts . 127 3.1.2 Semisimple Lie algebras . 138 3.1.3 Complete Lie Algebras . 139 3.1.4 Lie Algebra Representations . 140 3.1.5 The irreducible representations of sl(2, k) . 142 3.1.6 Invariant Forms . 145 3.1.7 Complex and Real Lie Algebras . 147 3.1.8 Rational Forms . 148 3.2 EngelandLie’sTheorems . 150 3.2.1 Engel’s Theorem . 150 3.2.2 Lie’s Theorem . 153 3.3 Cartan’s Criterion and Semisimple Lie algebras . 158 3.3.1 SomeAlgebra.. .. .. .. .. .. 158 3.3.2 Cartan’s Solvability Criterion . 162 3.3.3 Explicit Computations of Killing form . 166 3.3.4 Further Results on Jordan Decomposition . 170 3.4 Weyl’s Theorem on Complete Reducibility . 173 3.5 Levi-Malcev Decomposition . 180 3.6 Reductive Lie Algebras . 188 3.7 The Jacobson-Morozov Theorem . 193 3.8 Low Dimensional Lie Algebrasover R and C ....... 198 3.9 Real Lie Algebras of Compact Type . 202 4 The Structure of Compact Connected Lie Groups 207 4.1 Introduction......................... 207 4.2 Maximal Tori in Compact Lie Groups . 208 4.3 Maximal Tori in Compact Connected Lie Groups . 210 4.4 TheWeylGroup ...................... 217 4.5 What goes wrong if G is not compact . 221 vii 5 Representations of Compact Lie Groups 223 5.1 Introduction......................... 224 5.2 The Schur Orthogonality Relations . 226 5.3 Compact Integral Operators on a Hilbert Space . 228 5.4 The Peter-Weyl Theorem and its Consequences . 234 5.5 Characters and Central Functions . 243 5.6 Induced Representations . 250 5.7 Some Consequences of Frobenius Reciprocity . 255 6 Symmetric Spaces of Non-compact type 261 6.1 Introduction......................... 261 6.2 The Polar Decomposition . 264 6.3 The Cartan Decomposition . 267 6.4 The Case of Hyperbolic Space and the Lorentz Group . 274 6.5 The G-invariant Metric Geometry of P .......... 278 6.6 The Conjugacy of Maximal Compact Subgroups . 289 6.7 The Rank and Two-Point Homogeneous Spaces . 294 6.8 The Disk Model for Spaces of Rank 1 . 299 6.9 Exponentiality of Certain Rank 1 Groups . 304 7 Semisimple Lie Algebras and Lie Groups 313 7.1 Root and Weight Space Decompositions . 313 7.2 CartanSubalgebras. 316 7.3 Roots of Complex Semisimple Lie Algebras . 323 7.4 Real Forms of Complex Semisimple Lie Algebras . 337 7.5 The Iwasawa Decomposition . 343 8 Lattices in Lie Groups 355 8.1 Lattices in Euclidean Space . 355 8.2 GL(n, R)/ GL(n, Z) and SL(n, R)/ SL(n, Z) ....... 360 8.3 Lattices in more general groups . 371 8.4 Fundamental Domains . 374 9 Density results for cofinite Volume Subgroups 377 9.1 Introduction......................... 377 9.2 A Density Theorem for cofinite Volume Subgroups . 379 viii 9.3 Consequences and Extensions of the Density Theorem . 389 A Vector Fields 397 B The Kronecker Approximation Theorem 403 C Properly discontinuous actions 407 D The Analyticity of Smooth Lie Groups 411 References 413 Index 421 Preface and Acknowledgments In the view of the authors, and as we hope to convince the reader, Lie theory, broadly understood, lies at the center of modern mathematics. It is linked to algebra, analysis, algebraic and differential geometry, topol- ogy and even number theory, and applications of some of these other subjects are crucial to many of the arguments we shall present here. This also holds true in the opposite direction: Lie theory can be used to clarify or derive results in these other areas. In a philosophical sense Lie groups are pervasive within much of mathematics, for whenever one has some system, the “automorphisms” of it will frequently be a Lie group. This even occurs in the oldest deductive system in mathematics, namely Euclidean geometry. Here the key issue is congruent figures, particularly triangles. Two such planar triangles are congruent if and only if they differ by an element of the Lie group E(2) = O(2, R) ⋉R2, the group of rigid motions of the Euclidean plane. The reader will find in these many interrelations a vast panorama well worth studying. This book is the result of courses taught by one of the authors over many years on various aspects of Lie theory at the City University of New York Graduate Center. The primary reader to which it is ad- dressed is a graduate student in mathematics, or perhaps physics, or a researcher in one of these subjects who wants a comprehensive ref- erence work in Lie theory. However, by a judicious selection of topics, some of this material could also be used to give an introduction to the subject to well-grounded advanced undergraduate mathematics majors. ix x Preface and Acknowledgments For example, Chapters 3 and most of 7 could form a semester’s course in Lie algebras. Similarly, Chapters 0, 2 and 5 (respectively 0, 2 and 8) could be a semester’s course in integration in topological groups and their homogeneous spaces (respectively lattices and their applications). For the reader’s convenience we have included a diagram of the inter- dependence of the chapters. We have also tried to make the text as self-contained as possible even at the cost of increased length. We shall assume the reader has some knowledge of basic group theory, topology, and linear algebra, and a general acquaintance with the grammar of mathematics. While reading this book one may wish to consult some of the other books on the subject for clarification, or to see another view- point or treatment; especially useful books are listed in the bibliography. We have not attempted to detail the historical development of our sub- ject, nor to systematically give credit to the individual researchers who discovered these results. The book’s organization is as follows: Chapter 0 introduces the players; topological and Lie groups, cover- ings, group actions, homogeneous spaces, and Lie algebras. Chapter 1 deals with the correspondence between Lie groups and their Lie algebras, subalgebras and ideals, the functorial relationship determined by the exponential map, the topology of the classical groups, the Iwasawa decomposition in certain key cases, and the Baker Campbell Hausdorff theorem, and local Lie groups. Chapter 2 concerns Haar measure both on a group and on cocom- pact and finite volume homogeneous spaces together with a number of applications. Chapter 3 gives the elements of Lie algebra theory in some consid- erable detail (except for the detailed structure of complex semisimple Lie algebras, which we defer until Chapter 7). Chapter 4 deals with the structure of a compact connected Lie group in terms of a maximal torus and the Weyl group. Chapter 5 contains the representation theory of compact groups. Chapter 6 concerns symmetric spaces of non-compact type. Chapter 7 presents the detailed structure of complex semisimple Lie algebras. Preface and Acknowledgments xi Chapter 8 gives an introduction to lattices in Lie groups. Chapter 9 presents a “density theorem” for cofinite volume sub- groups of certain Lie groups. Although we have included a rather detailed and extensive index, it might be helpful to inform the reader of what we are not doing here. We do not deal extensively with the theory of algebraic groups, nor with transformation groups, although each of these makes some appearance. Similarly, we do not prove that any connected Lie group is, as a man- ifold, the direct product of a Euclidean space and a maximal compact subgroup, K; nor that any two such Ks are conjugate. But we do this in important special cases. We do not deal, except by example, with the theory of faithful representations. We have also omitted the Weyl character formula, the universal enveloping algebra, the classification of complex simple algebras and, with the exception of one example, branching theorems. We would like to thank Frederick Greenleaf, Adam Kor´anyi, Keivan Mallahi and Grigory Margulis for reading various chapters of our book and making a number of valuable suggestions. Of course any errors or misstatements are the sole responsibility of the authors. The authors would like to thank Richard Mosak for his help in the final preparation of this manuscript. We also thank Isabelle and Anita for their extraordinary patience during the several years that it took to bring this project to completion. Hossein Abbaspour Martin Moskowitz Max-Planck Institut f¨ur Mathematik The Graduate School Bonn The City University of New York New York xii Interdependency Chart Chapter 0 Chapter 1 Chapter 3 Chapter 2 Chapter 6 Chapter 4 Chapter 8 Chapter 5 Chapter 7 Chapter 9 Notations xiii Notations For a complex number z, z is the real part and z is the imaginary ℜ ℑ part.