Group Representations and Special Functions (Mathematics and its Applications)

Instead of cover......Page 1
Series......Page 2
Title page......Page 3
Copright page......Page 4
Editor\'s Preface......Page 5
Table of Contents......Page 7
Preface......Page 13
1.1. Groups......Page 17
1.2. Differentiable Manifolds......Page 31
1.3. Lie Groups and Lie Algebras......Page 57
1.4. Transformation Groups. Invariant Tensor Fields......Page 69
1.5. Additional Structures on Manifolds......Page 78
1.6. The Hurwitz Measure......Page 85
1.7. Quasi-Invariant Measures......Page 101
1.8. Elements of the Classification of Lie Groups and Algebras......Page 105
CHAPTER 2. Representations of Locally Compact Groups......Page 112
2.1. Definition of a Representation. Examples......Page 113
2.2. Basic Constructions. Induced Representations......Page 119
2.3. Further Constructions of Representations......Page 127
2.4. Intertwinning Operators. Unitary Equivalence of Representations......Page 131
2.5. Positive Definite Measures and Cyclic Representations......Page 134
2.6. Matrix Elements of Representations......Page 140
2.7. Group Algebra Representations and Group Representations......Page 144
2.8. The Universal Enveloping Algebra of a Lie Group Algebra. The Differential of a Representation......Page 146
CHAPTER 3. Decomposition Theory of Unitary Representations......Page 157
3.1. Irreducible Representations. Schur\'s Lemma......Page 158
3.2. Classical Fourier Transformation......Page 168
3.3. The Fourier Transforms of Functions in $\\mathcal{D}(\\mathbb{R}^n)$......Page 172
3.4. Analysis on the Multiplicative Group $\\mathbb{R}_+$. The Mellin Transformation......Page 174
3.5. The Circle Group and the Fourier Series......Page 175
3.6. Fourier Analysis on a Commutative Locally Compact Group......Page 180
CHAPTER 4. Representations of Compact Groups......Page 189
4.1. Operators of the Hilbert-Schmidt Type......Page 191
4.2. The Tensor Product of Hubert Spaces......Page 194
4.3. The Frobenius Theorem......Page 197
4.4. The Peter-Weyl Theory......Page 201
4.5. The Orthogonality Relations of Matrix Elements......Page 204
4.6. Characters of Finite-Dimensional Representations......Page 208
4.7. Harmonic Analysis on Compact Groups and on Their Homogeneous Spaces......Page 210
CHAPTER 5. Theory of Spherical Functions......Page 223
5.1. The Spherical Integral Equation......Page 224
5.2. Spherical Functions and Spherical Representations......Page 234
5.3. Existence of Spherical Functions. Gelfand Pairs......Page 236
5.4. Differentiability of Spherical Functions on Lie Groups......Page 242
Part II......Page 251
6.1. Definition of the $\\Gamma$-Function......Page 259
6.2. The Fourier Transformation and the Mellin Transformation......Page 262
6.3. The Reflection Formula for the $\\Gamma$-Function......Page 263
6.4. The Riemann $\\zeta$-Function......Page 269
7.1. The Group of Rigid Motions of $\\mathbb{R}^2$......Page 274
7.2. Spherical Representations of the Group $M(2)$......Page 275
7.3. Properties of the Bessel Functions......Page 280
7.4. Harmonic Analysis on the Symmetric Space of the Motion Group $M(2)$. The Fourier-Bessel Transformation......Page 285
8.1. Representations of the Group SL(2,$\\mathbb{C}$) on a Space of Polynomials......Page 295
8.2. Properties of the Representations $T^l$ and Their Consequences......Page 300
8.3. Integral Equations for the Functions $P^l_{jk}$......Page 303
8.4. The Differential of the Representation $T^l$. Recurrence and Differential Equations for the Functions $P^l_{mn}$......Page 310
8.5. Characters of Irreducible Representations and New Integral Formulas for Legendre Functions......Page 314
8.6. Harmonic Analysis on the Group SU(2) and the Sphere $S^2$......Page 316
8.7. Decomposition of the Tensor Product of Representations $T^l$. The Clebsch-Gordan Coefficients......Page 320
9.1. Information about the Group SO($n$) and the Homogeneous Space $S^{n-1}$......Page 338
9.2. Spherical Representations of the Group SO($n$)......Page 339
9.3. Gegenbauer\'s Equation and Basic Recurrences......Page 346
9.4. Integral Formulas for the Gegenbauer Polynomials......Page 348
9.5. A Mean Value Theorem for a Spherical Function......Page 352
10.1. Structure of the Group SL(2,$\\mathbb{R}$) and Its Homogeneous Spaces......Page 360
10.2. Induced Representations of the Group SL(2,$\\mathbb{R}$)......Page 368
10.3. Properties of the Representation $\\mathcal{U}^\\sigma$ and the Function $\\mathfrak{P}^l_{mn}$......Page 374
10.4. Differentials of the Representations $\\mathcal{U}^\\sigma$. Recurrence Relations. Irreducibility......Page 379
10.5. Harmonic Analysis on the Disc SU(1,1)/$\\mathcal{K}$......Page 383
11.1. The Group SL(2,$\\mathbb{C}$). Induced Spherical Representations......Page 405
11.2. On the Structure of the Lobatschevsky Space......Page 409
11.3. The Spherical Fourier Transformation on $\\mathcal{H}$......Page 416
11.4. Decomposition into Plane Waves on $\\mathcal{H}$......Page 418
11.5. Differential Properties of Spherical Functions......Page 427
11.6. The Gelfand-Graev Transformation......Page 429
11.7. Irreducibility Problems of the Representations $\\mathcal{U}^l$......Page 434
12.1. The Group, the Representation, Matrix Elements......Page 440
12.2. Basic Properties of the Laguerre Polynomials......Page 445
12.3. Differential Properties of the Laguerre Polynomials......Page 446
12.4. One-Dimensional Harmonic Oscillator and the Hermite Polynomials......Page 450
12.5. Connection between the Laguerre Polynomials and the Jacobi Functions......Page 453
12.6. Orthogonality Relations for the Laguerre Polynomials......Page 455
13.1. The Second Order Homogeneous Linear Differential Equation on $\\mathbb{C}$......Page 461
13.2. Solutions of the Hypergeometric Equation in the Form of Euler Integrals v......Page 474
13.3. The Hypergeometric Function for Some Special Values of the Parameters......Page 479
13.4. The Confluent Hypergeometric Equation and the Confluent Hypergeometric Function......Page 481
Introduction......Page 490
14.1. Associated Vector Bundles......Page 494
14.2. Operations on Differential Forms......Page 496
14.3. Affine Connections......Page 499
14.4. Parallel Translation. Geodesies. The Exponential Mapping......Page 507
14.5. Covariant Differentiation......Page 511
14.6. Affine Mappings......Page 515
14.7. The Riemannian Connection. Sectional Curvature......Page 520
15.1. Definitions and Examples......Page 533
15.2. Affine Connection on a Symmetric Space......Page 536
15.3. Structure of the Group of Displacements of a Symétrie Space......Page 543
15.4. Geometry of Symmetric Spaces......Page 548
15.5. Riemannian Symmetric Spaces. Riemann Pairs......Page 553
15.6. A Symmetric Pair is a Gelfand Pair......Page 560
CHAPTER 16. General Harmonic Analysis on a Symmetric Space......Page 566
17.1. Compact Lie Algebras......Page 576
17.2. Structure of Semisimple Algebras......Page 580
17.3. Iwasawa Decomposition of an Algebra and of a Group......Page 592
17.4. The Weyl Group......Page 597
17.5. Boundary of a Symmetric Space of the Non-Compact Type......Page 602
17.6. Planes and Horocycles in a Symmetric Space......Page 606
18.1. Plane Waves and Spherical Functions......Page 621
18.2. The Fourier Transformation on a Symmetric Space......Page 631
18.3. Properties of Spherical Functions......Page 634
18.4. Asymptotic Behaviour of a Spherical Function. The Harish-Chandra $c(\\cdot)$-Function......Page 639
18.5. Properties of the Harish-Chandra $c(\\cdot)$-Function......Page 642
18.6. The Plancherel Formula for the Fourier Transformation on a Symmetric Space......Page 647
18.7. The Radon Transformation......Page 649
18.8. The Paley-Wiener Theorem......Page 658
Table of Formulas......Page 662
References......Page 688
List of Symbols......Page 696
Author Index......Page 698
Subject Index......Page 699