Author | Gangolli, Ramesh. author |
---|---|
Title | Harmonic Analysis of Spherical Functions on Real Reductive Groups [electronic resource] / by Ramesh Gangolli, Veeravalli S. Varadarajan |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1988 |
Connect to | http://dx.doi.org/10.1007/978-3-642-72956-0 |
Descript | XIV, 365 p. online resource |
1. The Concept of a Spherical Function -- ยง 1.1. Review of Some Basic Notions of Representation Theory -- ยง 1.2. Decomposition of a Representation with Respect to a Compact Subgroup K and K-finite Representations -- ยง 1.3. Elementary Spherical Functions of Arbitrary Type -- ยง 1.4. Spherical Functions on Lie Groups -- ยง 1.5. Gelโfand Pairs (G, K) -- ยง 1.6. Plancherel Formula for G/K -- ยง 1.7. Eigenfunction Expansions in G/K -- Notes on Chapter 1 -- 2. Structure of Semisimple Lie Groups and Differential Operators on Them -- ยง 2.1. Groups of Class ? -- ยง 2.2. Iwasawa Decomposition. Roots. Weyl Group -- ยง 2.3. Parabolic Subalgebras and Parabolic Subgroups -- ยง 2.4. Integral Formulae -- ยง 2.5. Flag Manifolds, Bruhat Decomposition and Related Integral Formulae -- ยง 2.6. Differential Operators on G and G/K -- Notes on Chapter 2 -- 3. The Elementary Spherical Functions -- ยง 3.1. Principal Series Representations and Integral Representations for Their Matrix Coefficients -- ยง 3.2. Determination of All Elementary Spherical Functions. The Functional Equations -- ยง 3.3. The Harish-Chandra Transform -- ยง 3.4. Finite Dimensional Representation Theory of G and Its Consequences for the H-Function and the Elementary Spherical Functions -- ยง 3.5. Convexity Properties of the H-Function -- Notes on Chapter 3 -- 4. The Harish-Chandra Series for ?? and the c-Function -- ยง4.1. Radial Components of Spherical Differential Operators on A+ -- ยง 4.2. The Radial Component of the Casimir Operator -- ยง 4.3. Construction of the Eigenfunctions on G+ -- ยง 4.4. The Harish-Chandra Series for ?? and the c-Function -- ยง 4.5. Estimates for the Harish-Chandra Series When ? Becomes Unbounded -- ยง 4.6. Estimates for the Elementary Spherical Functions. The Functions ? and ? -- ยง 4.7. The c-Function -- Notes on Chapter 4 -- 5. Asymptotic Behaviour of Elementary Spherical Functions -- ยง 5.1. The Case When rk(G/K) =1 -- ยง 5.2. The Basic Differential Equations Viewed as a Perturbation of a Linear System: The Regular Case -- ยง 5.3. Radial Components on M10? and M10+ -- ยง 5.4. The Basic Differential Equations Viewed as a Perturbation of a Linear System: The General Case -- ยง 5.5. Spectral Theory of Representations of Polynomial Rings Associated to Finite Reflexion Groups -- ยง 5.6. The Initial Estimates -- ยง 5.7. Perturbations of Linear Systems (with A Priori Estimates) -- ยง 5.8. Asymptotics of ?0(?:ยท) on M10+. The Function ? -- ยง 5.9. Asymptotics of ?(?:ยท) -- ยง 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 -- ยง 6.1. The Schwartz Spaces C(G) and C(G//K) -- ยง 6.2. The Harish-Chandra Transform on C(G//K) -- ยง 6.3. Wave Packets in C(G//K) -- ยง 6.4. Statements of the Main Theorems -- ยง 6.5. The Method of Harish-Chandra -- ยง 6.6. The Method of Gangolli-Helgason-Rosenberg -- Notes on Chapter 6 -- 7. Lp-Theory of Harish-Chandra Transform. Fourier Analysis on the Spaces Cp(G//K) -- ยง7.1. Radial Components and Their Expansions -- ยง 7.2. The Differential Equations, Initial Estimates, and the Approximating Sequence -- ยง 7.3. Expressions for ?0 โ ?n0, ?n0 โ ?n-10, and Estimates for ?0 โ exp(?0)?n0 -- ยง 7.4. Further Study of the ?n0. The Matrices ?q -- ยง 7.5. The Functions ?q -- ยง 7.6. Asymptotic Expansions for ?? -- ยง 7.7. The Tube Domains ??, *?? and the Function Spaces ?(??), ?(??) -- ยง 7.8. The Spaces Cp(G//K) -- ยง 7.9. Study of the Functions ?q -- ยง 7.10. Wave Packets and the Transform Theory for Cp(G//K) -- Notes on Chapter 7