viii CONTENT S

3.6. The Group Algebra 48

3.7. The Measure Algebra 50

3.8. Adele Groups 51

Part 2.

REPRESENTATION THEORY AND COMPACT GROUPS

Chapter 4. Basic Representation Theory 55

4.1. Definitions and Examples 56

4.2. Subrepresentations and Quotient Representations 59

4.3. Operations on Representations 64

4.3A. Dual Space 64

4.3B. Direct Sum 64

4.3C. Tensor Product of Spaces 65

4.3D. Horn 67

4.3E. Bilinear Forms 67

4.3F. Tensor Products of Algebras 68

4.3G. Relation with the Commuting Algebra 69

4.4. Multiplicities and the Commuting Algebra 70

4.5. Completely Continuous Representations 72

4.6. Continuous Direct Sums of Representations 75

4.7. Induced Representations 77

4.8. Vector Bundle Interpretation 81

4.9. Mackey's Little-Group Theorem 82

4.9A. The Normal Subgroup Case 82

4.9B. Cohomology and Projective Representations 84

4.9C. Cocycle Representations and Extensions 85

4.10. Mackey Theory and the Heisenberg Group 87

93

93

96

97

99

101

104

107

107

107

110

111

111

112

113

115

Chapter

5.1.

5.2.

5.3.

5.4.

5.5.

5.6.

5.7.

5.8.

5.9.

5. Representations of Compact Groups

Finite Dimensionality

Orthogonality Relations

Characters and Projections

The Peter-Weyl Theorem

The Plancherel Formula

Decomposition into Irreducibles

Some Basic Examples

5.7A. The Group 17(1)

5.7B. The Group SU(2)

5.7C. The Group SO(3)

5.7D. The Group 50(4)

5.7E. The Sphere

S2

5.7F. The Sphere S3

Real, Complex and Quaternion Representations

The Frobenius Reciprocity Theorem