Harmonic Analysis on Semi-Simple Lie Groups II

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.