Author | Jorgenson, Jay. author |
---|---|
Title | Spherical Inversion on SLn(R) [electronic resource] / by Jay Jorgenson, Serge Lang |
Imprint | New York, NY : Springer New York, 2001 |
Connect to | http://dx.doi.org/10.1007/978-1-4684-9302-3 |
Descript | XX, 426 p. 1 illus. online resource |
I Iwasawa Decomposition and Positivity -- ยง1. The Iwasawa Decomposition -- ยง2. Haar Measure and Iwasawa Decomposition -- ยง3. The Cartan Lie Decomposition, Polynomial Algebra and Chevalleyโs Theorem -- ยง4. Positivity -- ยง5. Convexity -- ยง6. The Harish-Chandra U-Polar Inequality; Connection with the Iwasawa and Polar Decompositions -- II Invariant Differential Operators and the Iwasawa Direct Image -- ยง1. Invariant Differential Operators on a Lie Group -- ยง2. The Projection on a Homogeneous Space -- ยง3. The Iwasawa Projection on A -- ยง4. Use of the Cartan Lie Decomposition -- ยง5. The Harish-Chandra Transforms -- ยง6. The Transpose and Involution -- III Characters, Eigenfunctions, Spherical Kernel and W-Invariance -- ยง1. Characters -- ยง2. The (a, n)-Characters and the Iwasawa Character -- ยง3. The Weyl Group -- ยง4. Orbital Integral for the Harish Transform -- ยง5. W-Invariance of the Harish and Spherical Transforms -- ยง6. K-Bi-Invariant Functions and Uniqueness of Spherical Functions -- ยง7. Integration Formulas and the Map x ? x-1 -- ยง8. W-Harmonic Polynomials and Eigenfunctions of W-Invariant Differential Operators on A -- IV Convolutions, Spherical Functions and the Mellin Transform -- ยง1. Weakly Symmetric Spaces -- ยง2. Characters and Convolution Operators -- ยง3. Example: The Gamma Function -- ยง4. K-Invariance or Bi-Invariance and Eigenfunctions of Convolutions -- ยง5. Convolution Sphericality -- ยง6. The Spherical Transform as Multiplicative Homomorphism -- ยง7. The Mellin Transform and the Paley-Wiener Space -- ยง8. Behavior of the Support -- V Gelfand-Naimark Decomposition and the Harish-Chandra c-Function. -- ยง1. The Gelfand-Naimark Decomposition and the Harish-Chandra Mapping of U? into M\K -- ยง2. The Bruhat Decomposition -- ยง3. Jacobian Formulas -- ยง4. Integral Formulas for Spherical Functions -- ยง5. The c-Function and the First Spherical Asymptotics -- ยง6. The Bhanu-Murty Formula for the c-Function -- ยง7. Invariant Formulation on 1 -- ยง8. Corollaries on the Analytic Behavior of cHar -- VI Polar Decomposition -- ยง1. The Jacobian of the Polar Map -- ยง2. From K-Bi-Invariant Functions on G to W-Invariant Functions on a.. -- Appendix. The Bernstein Calculus Lemma -- ยง3. Pulling Back Characters and Spherical Functions to a -- ยง4. Lemmas Using the Semisimple Lie Iwasawa Decomposition -- ยง5. The Transpose Iwasawa Decomposition and Polar Direct Image -- ยง6. W-Invariants -- VII The Casimir Operator -- ยง1. Bilinear Forms of Cartan Type -- ยง2. The Casimir Differential Operator -- ยง3. The A-Iwasawa and Harish-Chandra Direct Images -- ยง4. The Polar Direct Image -- VIII The Harish-Chandra Series and Spherical Inversion -- ยง0. Linear Independence of Characters Revisited -- ยง1. Eigenfunctions of Casimir -- ยง2. The Harish-Chandra Series and Gangolli Estimate -- ยง3. The c-Function and the W-Trace -- ยง4. The Helgason and Anker Support Theorems -- ยง5. An L2-Estimate and Limit -- ยง6. Spherical Inversion -- IX General Inversion Theorems -- ยง1. The Rosenberg Arguments -- ยง2. Helgason Inversion on Paley-Wiener and the L2-Isometry -- ยง3. The Constant in the Inversion Formula -- X The Harish-Chandra Schwartz Space (HCS) and Ankerโs Proof of Inversion -- ยง1. More Harish-Chandra Convexity Inequalities -- ยง2. More Harish-Chandra Inequalities for Spherical Functions -- ยง3. The Harish-Chandra Schwartz Space -- ยง4. Schwartz Continuity of the Spherical Transform -- ยง5. Continuity of the Inverse Transform and Spherical Inversion on HCS(K\G/K) -- ยง6. Extension of Formulas by HCS Continuity -- ยง7. An Example: The Heat Kernel -- ยง8. The Harish Transform -- XI Tube Domains and the L1 (Even Lp) HCS Spaces -- ยง1. The Schwartz Space on Tubes -- ยง2. The Filtration HCS(p)(K\G/K) with 0 < p ? 2 -- ยง3. The Inverse Transform -- ยง4. Bounded Spherical Functions -- ยง5. Back to the Heat Kernel -- XII SL n (C) -- ยง1. A Formula of Exponential Polynomials -- ยง2. Characters and Jacobians -- ยง3. The Polar Direct Image -- ยง4. Spherical Functions and Inversion -- ยง5. The Heat Kernel -- ยง6. The Flensted-Jensen Decomposition and Reduction -- Table of Notation