This book is based on several courses given by the authors since 1966. It introduces the reader to the representation theory of compact Lie groups. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and establish the theory and to indicate relations to other branches of mathematics. Lie algebras, though mentioned occasionally, are not used in an essential way. The material as well as its presentation are classical; one might say that the foundations were known to Hermann Weyl at least 50 years ago.

Contents

CHAPTER I Lie Groups and Lie Algebras

1. The Concept ofa Lie Group and the Classical Examples

2. Left-Invariant Vector Fields and One-Parameter Groups

3. The Exponential Map

4. Homogeneous Spaces and Quotient Groups

5. Invariant Integration

6. Clifford Algebras and Spinor Groups

CHAPTER II Elementary Representation Theory

1. Representations

2. Semisimple Modules

3. Linear Algebra and Representations

4. Characters and Orthogonality Relations

5. Representations of SU(2), SO(3), U(2), and O(3)

6. Real and Quaternionic Representations

7. The Character Ring and the Representation Ring

8. Representations of Abelian Groups

9. Representations of Lie Algebras

10. The Lie Algebra s1(2,0

CHAPTER III Representative Functions

1. Algebras of Representative Functions

2. Some Analysis on Compact Oroups

3. The Theorem of Peter and Weyl

4. Applications of the Theorem of Peter and Weyl

5. Generalizations of the Theorem of Peter and Weyl

6. Induced Representations

7. Tannaka-KreIn Duality

8. The Complexification ofCompact Lie Groups

CHAPTER IV The Maximal Torus of a Compact Lie Group

1. Maximal Tori

2. Consequences of the Conjugation Theorem

3. The Maximal Tori and Weyl Groups of the Classical Groups

4. Cartan Subgroups of Nonconnected Compact Groups

CHAPTER V Root Systems

1. The Adjoint Representation and Groups of Rank 1

2. Roots and Weyl Chambers

3. Root Systems

4. Bases and Weyl Chambers

5. Dynkin Diagrams

6. The Roots of the Classical Groups

7. The Fundamental Group, the Center and the Stiefel Diagram

8. The Structure of the Compact Groups

CHAPTER VI Irreducible Characters and Weights

1. The Weyl Character Formula

2. The Dominant Weight and the Structure of the Representation Ring

3. The Multiplicities of the Weights of an Irreducible Representation

4. Representations of Real or Quaternionic Type

5. Representations of the Classical Groups

6. Representations of the Spinor Groups

7. Representations of the Orthogonal Groups

Bibliography

Symbol Index

Subject Index

CHAPTER 1

Lie Groups and Lie Algebras

In this chapter we explain what a Lie group is and quickly review the basic

concepts ofthe theory ofdifferentiable manifol]ds. The first section illustrates

the notion of a Lie group with classical examples of matrix groups from

linear algebra. The spinor groups are treated in a separate section, ?, but

the presentation of the general theory of representations in this book pre-

supposes no knowledge of spinor groups. They only appear as examples

which, although important, may be skipped. In Ё2, 3, and 4 we construct the

exponential map and exploit it to obtain elementary information about the

structure of subgroups and quotients, and in ? we explain how to construct

an invariant integral using differential forms. We quote Stokes' theorem to

get a result about mapping degrees which we shall use in Chapter IV.

1. The Concept of a Lie Group and the

Classical Examples

The concept of a Lie group arises naturally by merging the algebraic notion

ofa group with the geometric notion of a differentiable manifold. However,

the classical examples, as well as the methods of investigation, show the

theory of Lie groups to be a significant geometric extension of linear algebra

and analytic geometry.

(1.1) Definition. A Lie group is a differentiable manifold G which is also a

group such that the group multiplication

(and the map sending g to g ) is a differentiable map. A homomorphism

of Lie groups is a differentiable group homomorphism between Lie groups.

For us the word differentiable means infinitely often differentiable.

Throughout this book we use the words differentiable, smooth, and C as

synonymous.

The identity map on a Lie group is a homomorphism, and composing

homomorphisms yields a homomorphism-Lie groups and homomor-

phisms form a category. One may define the usual categorical notions: in

particular, an isomorphism (denoted by ) is an invertible homomorphism.

We will use e or 1 to denote the identity element of G, although we will

sometimes use E when considering a matrix group and 0 when considering

an additive abelian group.

The reader should know what a group is, and the concept of a differen-

tiable manifold should not be new. Nonetheless, we review a few facts about

manifolds.

(1.2) Definition. An n-dimensional(differentiable) manifoldM is a Hausdorff

topological space with a countable (topological) basis, together with a

maximal differentiable atlas. This atlas consists of a family of charts

h: U U R, where the domains of the charts, {U}, form an open

cover of M, the U are open in R, the charts (local coordinates) h are

homeomorphisms, and every change of coordinates h = hy o h is differ-

entiable on its domain ofdefinition hV ).

The atlas is maximal in the sense that it cannot be enlarged to another

differentiable atlas by adding more charts, so any chart which could be added

to the atlas in a consistent fashion is already in the atlas.

A continuous map f: M -> N of differentiable manifolds is called

differentiable if, after locally composing with the charts ofM and N, it induces

a differentiable map of open subsets of Euclidean spaces.