Harmonic Analysis of Spherical Functions on Real Reductive Groups
Analysis on Symmetric spaces, or more generally, on homogeneous spaces of semisimple Lie groups, is a subject that has undergone a vigorous development in recent years, and has become a central part of contemporary mathematics. This is only to be expected, since homogeneous spaces and group represen...
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Format:  eBook 
Language:  English 
Published: 
Berlin, Heidelberg
Springer Berlin Heidelberg
1988, 1988

Edition:  1st ed. 1988 
Series:  Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics

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Online Access:  
Collection:  Springer Book Archives 2004  Collection details see MPG.ReNa 
Table of Contents:
 Notes on Chapter 6
 7. LpTheory of HarishChandra Transform. Fourier Analysis on the Spaces Cp(G//K)
 §7.1. Radial Components and Their Expansions
 § 7.2. The Differential Equations, Initial Estimates, and the Approximating Sequence
 § 7.3. Expressions for ?0 — ?n0, ?n0 — ?n10, and Estimates for ?0 — exp(?0)?n0
 § 7.4. Further Study of the ?n0. The Matrices ?q
 § 7.5. The Functions ?q
 § 7.6. Asymptotic Expansions for ??
 § 7.7. The Tube Domains ??, *?? and the Function Spaces ?(??), ?(??)
 § 7.8. The Spaces Cp(G//K)
 § 7.9. Study of the Functions ?q
 § 7.10. Wave Packets and the Transform Theory for Cp(G//K)
 Notes on Chapter 7
 § 3.2. Determination of All Elementary Spherical Functions. The Functional Equations
 § 3.3. The HarishChandra Transform
 § 3.4. Finite Dimensional Representation Theory of G and Its Consequences for the HFunction and the Elementary Spherical Functions
 § 3.5. Convexity Properties of the HFunction
 Notes on Chapter 3
 4. The HarishChandra Series for ?? and the cFunction
 §4.1. Radial Components of Spherical Differential Operators on A+
 § 4.2. The Radial Component of the Casimir Operator
 § 4.3. Construction of the Eigenfunctions on G+
 § 4.4. The HarishChandra Series for ?? and the cFunction
 § 4.5. Estimates for the HarishChandra Series When ? Becomes Unbounded
 § 4.6. Estimates for the Elementary Spherical Functions. The Functions ? and ?
 § 4.7. The cFunction
 Notes on Chapter 4
 5. Asymptotic Behaviour of Elementary Spherical Functions
 § 5.1. The Case When rk(G/K) =1
 1. The Concept of a Spherical Function
 § 1.1. Review of Some Basic Notions of Representation Theory
 § 1.2. Decomposition of a Representation with Respect to a Compact Subgroup K and Kfinite Representations
 § 1.3. Elementary Spherical Functions of Arbitrary Type
 § 1.4. Spherical Functions on Lie Groups
 § 1.5. Gel’fand Pairs (G, K)
 § 1.6. Plancherel Formula for G/K
 § 1.7. Eigenfunction Expansions in G/K
 Notes on Chapter 1
 2. Structure of Semisimple Lie Groups and Differential Operators on Them
 § 2.1. Groups of Class ?
 § 2.2. Iwasawa Decomposition. Roots. Weyl Group
 § 2.3. Parabolic Subalgebras and Parabolic Subgroups
 § 2.4. Integral Formulae
 § 2.5. Flag Manifolds, Bruhat Decomposition and Related Integral Formulae
 § 2.6. Differential Operators on G and G/K
 Notes on Chapter 2
 3. The Elementary Spherical Functions
 § 3.1. Principal Series Representations and Integral Representations for Their Matrix Coefficients
 § 5.2. The Basic Differential Equations Viewed as a Perturbation of a Linear System: The Regular Case
 § 5.3. Radial Components on M10? and M10+
 § 5.4. The Basic Differential Equations Viewed as a Perturbation of a Linear System: The General Case
 § 5.5. Spectral Theory of Representations of Polynomial Rings Associated to Finite Reflexion Groups
 § 5.6. The Initial Estimates
 § 5.7. Perturbations of Linear Systems (with A Priori Estimates)
 § 5.8. Asymptotics of ?0(?:·) on M10+. The Function ?
 § 5.9. Asymptotics of ?(?:·)
 § 5.10. Complements. Constant Term for Tempered ?Finite Functions
 Notes on Chapter 5
 6. The L2Theory. The HarishChandra Transform on the Schwartz Space of G//K
 § 6.1. The Schwartz Spaces C(G) and C(G//K)
 § 6.2. The HarishChandra Transform on C(G//K)
 § 6.3. Wave Packets in C(G//K)
 § 6.4. Statements of the Main Theorems
 § 6.5. The Method of HarishChandra
 § 6.6. The Method of GangolliHelgasonRosenberg