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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Main Authors: Gangolli, Ramesh, Varadarajan, Veeravalli S. (Author)
Corporate Author: SpringerLink (Online service)
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
Subjects:
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • Notes on Chapter 6
  • 7. Lp-Theory of Harish-Chandra 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 — ?n-10, 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 Harish-Chandra Transform
  • § 3.4. Finite Dimensional Representation Theory of G and Its Consequences for the H-Function and the Elementary Spherical Functions
  • § 3.5. Convexity Properties of the H-Function
  • Notes on Chapter 3
  • 4. The Harish-Chandra Series for ?? and the c-Function
  • §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 Harish-Chandra Series for ?? and the c-Function
  • § 4.5. Estimates for the Harish-Chandra Series When ? Becomes Unbounded
  • § 4.6. Estimates for the Elementary Spherical Functions. The Functions ? and ?
  • § 4.7. The c-Function
  • 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 K-finite 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 L2-Theory. The Harish-Chandra Transform on the Schwartz Space of G//K
  • § 6.1. The Schwartz Spaces C(G) and C(G//K)
  • § 6.2. The Harish-Chandra 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 Harish-Chandra
  • § 6.6. The Method of Gangolli-Helgason-Rosenberg