Group Representations and Special Functions

I.- 1. Groups and Homogeneous Spaces.- 1.1. Groups.- 1.2. Differentiate Manifolds.- 1.3. Lie Groups and Lie Algebras.- 1.4. Transformation Groups. Invariant Tensor Fields.- 1.5. Additional Structures on Manifolds.- 1.6. The Hurwitz Measure.- 1.7. Quasi-Invariant Measures.- 1.8. Elements of the Classification of Lie Groups and Algebras.- 2. Representations of Locally Compact Groups.- 2.1. Definition of a Representation. Examples.- 2.2. Basic Constructions. Induced Representations.- 2.3. Further Constructions of Representations.- 2.4. Intertwinning Operators. Unitary Equivalence of Representations.- 2.5. Positive Definite Measures and Cyclic Representations.- 2.6. Matrix Elements of Representations.- 2.7. Group Algebra Representations and Group Representations.- 2.8. The Universal Enveloping Algebra of a Lie Group Algebra. The Differential of a Representation.- 3. Decomposition Theory of Unitary Representations.- 3.1. Irreducible Representations. Schur’s Lemma.- 3.2. Classical Fourier Transformation.- 3.3. The Fourier Transforms of Functions in D (Rn).- 3.4. Analysis on the Multiplicative Group R+. The Mellin Transformation.- 3.1. The Circle Group and the Fourier Series.- 3.2. Fourier Analysis on a Commutative Locally Compact Group.- 4. Representations of Compact Groups.- 4.1. Operators of the Hilbert-Schmidt Type.- 4.2. The Tensor Product of Hilbert Spaces.- 4.3. The Frobenius Theorem.- 4.4. The Peter-Weyl Theory.- 4.5. The Orthogonality Relations of Matrix Elements.- 4.6. Characters of Finite-Dimensional Representations.- 4.7. Harmonic Analysis on Compact Groups and on Their Homogeneous Spaces.- 5. Theory of Spherical Functions.- 5.1. The Spherical Integral Equation.- 5.2. Spherical Functions and Spherical Representations.- 5.3. Existence of Spherical Functions. Gelfand Pairs.- 5.4. Differentiability of Spherical Functions on Lie Groups.- II.- 6. The Euler ?- and B-Functions.- 6.1. Definition of the ?-Function.- 6.2. The Fourier Transformation and the Mellin Transformation.- 6.3. The Reflection Formula for the ?-Function.- 6.4. The Riemann ?-Function.- 7. Bessel Functions.- 7.1. The Group of Rigid Motions of R2.- 7.2. Spherical Representations of the Group M(2).- 7.3. Properties of the Bessel Functions.- 7.4. Harmonic Analysis on the Symmetric Space of the Motion Group M(2). The Fourier-Bessel Transformation.- 8. Theory of Jacobi and Legendre Polynomials.- 8.1. Representations of the Group SL(2, C) on a Space of Polynomials.- 8.2. Properties of the Representations Tl and Their Consequences.- 8.3. Integral Equations for the Functions Pjkl.- 8.4. The Differential of the Representation Tl. Recurrence and Differential Equations for the Functions Pmnl.- 8.5. Characters of Irreducible Representations and New Integral Formulas for Legendre Functions.- 8.6. Harmonic Analysis on the Group SU(2) and the Sphere S2.- 8.7. Decomposition of the Tensor Product of Representations Tl. The Clebsch-Gordan Coefficients.- 9. Gegenbauer Polynomials.- 9.1. Information about the Group SO(n) and the Homogeneous Space Sn-1.- 9.2. Spherical Representations of the Group SO(n).- 9.3. Gegenbauer’s Equation and Basic Recurrences.- 9.4. Integral Formulas for the Gegenbauer Polynomials.- 9.5. A Mean Value Theorem for a Spherical Function.- 10. Jacobi and Legendre Functions.- 10.1. Structure of the Group SL(2, R) and Its Homogeneous Spaces.- 10.2. Induced Representations of the Group SL(2,R).- 10.3. Properties of the Representation U? and the Function Bmnl.- 10.4. Differentials of the Representations U?. Recurrence Relations. Irreducibility.- 10.5. Harmonic Analysis on the Disc SU(1, 1)/K.- Chapter11. Harmonic Analysis on the Lobatschevsky space.- 11.1. The Group SL(2, C). Induced Spherical Representations.- 11.2. On the Structure of the Lobatschevsky Space.- 11.3. The Spherical Fourier Transformation on ?.- 11.4. Decomposition into Plane Waves on ?.- 11.5. Differential Properties of Spherical Functions.- 11.6. The Gelfand-Graev Transformation.- 11.7. Irreducibility Problems of the Representations Ul.- 12. The Laguerre Polynomials.- 12.1. The Group, the Representation, Matrix Elements.- 12.2. Basic Properties of the Laguerre Polynomials.- 12.3. Differential Properties of the Laguerre Polynomials.- 12.4. One-Dimensional Harmonic Oscillator and the Hermite Polynomials.- 12.5. Connection between the Laguerre Polynomials and the Jacobi Functions.- 12.6. Orthogonality Relations for the Laguerre Polynomials.- Chapter13. The Hypergeometric Equation.- 13.1. The Second Order Homogeneous Linear Differential Equation on C.- 13.2. Solutions of the Hypergeometric Equation in the Form of Euler Integrals.- 13.3. The Hypergeometric Function for Some Special Values of the Parameters.- 13.4. The Confluent Hypergeometric Equation and the Confluent Hypergeometric Function.- III.- 14. Affine Transformations.- 14.1. Associated Vector Bundles.- 14.2. Operations on Differential Forms.- 14.3. Affine Connections.- 14.4. Parallel Translation. Geodesies. The Exponential Mapping.- 14.5. Covariant Differentiation.- 14.6. Affine Mappings.- 14.7. The Riemannian Connection. Sectional Curvature.- 15. Symmetric Spaces.- 15.1. Definitions and Examples.- 15.2. Affine Connection on a Symmetric Space.- 15.3. Structure of the Group of Displacements of a Symetric Space.- 15.4. Geometry of Symmetric Spaces.- 15.5. Riemannian Symmetric Spaces. Riemann Pairs.- 15.6. A Symmetric Pair is a Gelfand Pair.- 16. General Harmonic Analysis on a Symmetric Space.- 17. Semisimple Algebras. Semisimple Groups. Symmetric Spaces of the Non-Compact Type.- 17.1. Compact Lie Algebras.- 17.2. Structure of Semisimple Algebras.- 17.3. Iwasawa Decomposition of an Algebra and of a Group.- 17.4. The Weyl Group.- 17.5. Boundary of a Symmetric Space of the Non-Compact Type.- 17.6. Planes and Horocycles in a Symmetric Space.- 18. Harmonic Analysis on Symmetric Spaces of the Non-Compact Type.- 18.1. Plane Waves and Spherical Functions.- 18.2. The Fourier Transformation on a Symmetric Space.- 18.3. Properties of Spherical Functions.- 18.4. Asymptotic Behaviour of a Spherical Function. The Harish-Chandra c(•)-Function.- 18.5. Properties of the Harish-Chandra c(•)-Function.- 18.6. The Plancherel Formula for the Fourier transformation on a Symmetric Space.- 18.7. The Radon Transformation.- 18.8. The Paley-Wiener Theorem.- Table of Formulas.- References.- List of Symbols.- Author Index.