Harmonic Analysis on Semi-Simple Lie Groups II

6 Spherical Functions — The General Theory.- 6.1 Fundamentals.- 6.1.1 Spherical Functions — Functional Properties.- 6.1.2 Spherical Functions — Differential Properties.- 6.2 Examples.- 6.2.1 Spherical Functions on Motion Groups.- 6.2.2 Spherical Functions on Semi-Simple Lie Groups.- 7 Topology on the Dual Plancherel Measure Introduction.- 7.1 Topology on the Dual.- 7.1.1 Generalities.- 7.1.2 Applications to Semi-Simple Lie Groups.- 7.2 Plancherel Measure.- 7.2.1 Generalities.- 7.2.2 The Plancherel Theorem for Complex Connected Semi-Simple Lie Groups.- 8 Analysis on a Semi-Simple Lie Group.- 8.1 Preliminaries.- 8.1.1 Acceptable Groups.- 8.1.2 Normalization of Invariant Measures.- 8.1.3 Integration Formulas.- 8.1.4 A Theorem of Compacity.- 8.1.5 The Standard Semi-Norm on a Semi-Simple Lie Group.- 8.1.6 Completely Invariant Sets.- 8.2 Differential Operators on Reductive Lie Groups and Algebras.- 8.2.1 Radial Components of Differential Operators on a Manifold.- 8.2.2 Radial Components of Polynomial Differential Operators on a Reductive Lie Algebra.- 8.2.3 Radial Components of Left Invariant Differential Operators on a Reductive Lie Group.- 8.2.4 The Connection between Differential Operators in the Algebra and on the Group.- 8.3 Central Eigendistributions on Reductive Lie Algebras and Groups.- 8.3.1 The Main Theorem in the Algebra.- 8.3.2 Properties of FT-I.- 8.3.3 The Main Theorem on the Group.- 8.3.4 Properties of FT- II.- 8.3.5 Rapidly Decreasing Functions on a Euclidean Space.- 8.3.6 Tempered Distributions on a Reductive Lie Algebra.- 8.3.7 Rapidly Decreasing Functions on a Reductive Lie Group.- 8.3.8 Tempered Distributions on a Reductive Lie Group.- 8.3.9 Tools for Harmonic Analysis on G.- 8.4 The Invariant Integral on a Reductive Lie Algebra.- 8.4.1 The Invariant Integral — Definition and Properties.- 8.4.2 Computations in sl(2, R).- 8.4.3 Continuity of the Map f ? ? f.- 8.4.4 Extension Problems.- 8.4.5 The Main Theorem.- 8.5 The Invariant Integral on a Reductive Lie Group.- 8.5.1 The Invariant Integral — Definition and Properties.- 8.5.2 The Inequalities of Descent.- 8.5.3 The Transformations of Descent.- 8.5.4 The Invariant Integral and the Transformations of Descent.- 8.5.5 Estimation of ?f and its Derivatives.- 8.5.6 An Important Inequality.- 8.5.7 Convergence of Certain Integrals.- 8.5.8 Continuity of the Map f? ?f.- 9 Spherical Functions on a Semi-Simple Lie Group.- 9.1 Asymptotic Behavior of ?-Spherical Functions on a Semi-Simple Lie Group.- 9.1.1 The Main Results.- 9.1.2 Analysis in the Universal Enveloping Algebra.- 9.1.3 The Space S(?,?).- 9.1.4 The Rational Functions ??.- 9.1.5 The Expansion of ?-Spherical Functions.- 9.1.6 Investigation of the c-Function.- 9.1.7 Applications to Zonal Spherical Functions.- 9.2 Zonal Spherical Functions on a Semi-Simple Lie Group.- 9.2.1 Statement of Results — Immediate Applications.- 9.2.2 The Plancherel Theorem for I2(G).- 9.2.3 The Paley-Wiener Theorem for I2(G).- 9.2.4 Harmonic Analysis in I1(G).- 9.3 Spherical Functions and Differential Equations.- 9.3.1 The Weak Inequality and Some of its Implications.- 9.3.2 Existence and Uniqueness of the Indices I.- 9.3.3 Existence and Uniqueness of the Indices II.- 10 The Discrete Series for a Semi-Simple Lie Group — Existence and Exhaustion.- 10.1 The Role of the Distributions ?? in the Harmonic Analysis on G.- 10.1.1 Existence and Uniqueness of the ??.- 10.1.2 Expansion of Z-Finite Functions in C-(G).- 10.2 Theory of the Discrete Series.- 10.2.1 Existence of the Discrete Series.- 10.2.2 The Characters of the Discrete Series I — Implication of the Orthogonality Relations.- 10.2.3 The Characters of the Discrete Series II — Application of the Differential Equations.- 10.2.4 The Theorem of Harish-Chandra.- Epilogue.- Append.- 3 Some Results on Differential Equations.- 3.1 The Main Theorems.- 3.2 Lemmas from Analysis.- 3.3 Analytic Continuation of Solutions.- 3.4 Decent Convergence.- 3.5 Normal Sequences of is-Polynomials.- General Notational Conventions.- List of Notations.- Guide to the Literature.- Subject Index to Volumes I and II.