Spherical Inversion on SLn(R)
Harish-Chandra's general Plancherel inversion theorem admits a much shorter presentation for spherical functions. The authors have taken into account contributions by Helgason, Gangolli, Rosenberg, and Anker from the mid-1960s to 1990. Anker's simplification of spherical inversion on the H...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
New York, NY
Springer New York
2001, 2001
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| Edition: | 1st ed. 2001 |
| Series: | Springer Monographs in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- §6. The Bhanu-Murty Formula for the c-Function
- §7. Invariant Formulation on 1
- §8. Corollaries on the Analytic Behavior of cHar
- VI Polar Decomposition
- §1. The Jacobian of the Polar Map
- §2. From K-Bi-Invariant Functions on G to W-Invariant Functions on a..
- Appendix. The Bernstein Calculus Lemma
- §3. Pulling Back Characters and Spherical Functions to a
- §4. Lemmas Using the Semisimple Lie Iwasawa Decomposition
- §5. The Transpose Iwasawa Decomposition and Polar Direct Image
- §6. W-Invariants
- VII The Casimir Operator
- §1. Bilinear Forms of Cartan Type
- §2. The Casimir Differential Operator
- §3. The A-Iwasawa and Harish-Chandra Direct Images
- §4. The Polar Direct Image
- VIII The Harish-Chandra Series and Spherical Inversion
- §0. Linear Independence of Characters Revisited
- §1. Eigenfunctions of Casimir
- §2. The Harish-Chandra Series and Gangolli Estimate
- §3. The c-Function and the W-Trace
- §5. Back to the Heat Kernel
- XII SL n (C)
- §1. A Formula of Exponential Polynomials
- §2. Characters and Jacobians
- §3. The Polar Direct Image
- §4. Spherical Functions and Inversion
- §5. The Heat Kernel
- §6. The Flensted-Jensen Decomposition and Reduction
- Table of Notation
- I Iwasawa Decomposition and Positivity
- §1. The Iwasawa Decomposition
- §2. Haar Measure and Iwasawa Decomposition
- §3. The Cartan Lie Decomposition, Polynomial Algebra and Chevalley’s Theorem
- §4. Positivity
- §5. Convexity
- §6. The Harish-Chandra U-Polar Inequality; Connection with the Iwasawa and Polar Decompositions
- II Invariant Differential Operators and the Iwasawa Direct Image
- §1. Invariant Differential Operators on a Lie Group
- §2. The Projection on a Homogeneous Space
- §3. The Iwasawa Projection on A
- §4. Use of the Cartan Lie Decomposition
- §5. The Harish-Chandra Transforms
- §6. The Transpose and Involution
- III Characters, Eigenfunctions, Spherical Kernel and W-Invariance
- §1. Characters
- §2. The (a, n)-Characters and the Iwasawa Character
- §3. The Weyl Group
- §4. Orbital Integral for the Harish Transform
- §5. W-Invariance of the Harish and Spherical Transforms
- §4. The Helgason and Anker Support Theorems
- §5. An L2-Estimate and Limit
- §6. Spherical Inversion
- IX General Inversion Theorems
- §1. The Rosenberg Arguments
- §2. Helgason Inversion on Paley-Wiener and the L2-Isometry
- §3. The Constant in the Inversion Formula
- X The Harish-Chandra Schwartz Space (HCS) and Anker’s Proof of Inversion
- §1. More Harish-Chandra Convexity Inequalities
- §2. More Harish-Chandra Inequalities for Spherical Functions
- §3. The Harish-Chandra Schwartz Space
- §4. Schwartz Continuity of the Spherical Transform
- §5. Continuity of the Inverse Transform and Spherical Inversion on HCS(K\G/K)
- §6. Extension of Formulas by HCS Continuity
- §7. An Example: The Heat Kernel
- §8. The Harish Transform
- XI Tube Domains andthe L1 (Even Lp) HCS Spaces
- §1. The Schwartz Space on Tubes
- §2. The Filtration HCS(p)(K\G/K) with 0 < p ? 2
- §3. The Inverse Transform
- §4. Bounded Spherical Functions
- §6. K-Bi-Invariant Functions and Uniqueness of Spherical Functions
- §7. Integration Formulas and the Map x ? x-1
- §8. W-Harmonic Polynomials and Eigenfunctions of W-Invariant Differential Operators on A
- IV Convolutions, Spherical Functions and the Mellin Transform
- §1. Weakly Symmetric Spaces
- §2. Characters and Convolution Operators
- §3. Example: The Gamma Function
- §4. K-Invariance or Bi-Invariance and Eigenfunctions of Convolutions
- §5. Convolution Sphericality
- §6. The Spherical Transform as Multiplicative Homomorphism
- §7. The Mellin Transform and the Paley-Wiener Space
- §8. Behavior of the Support
- V Gelfand-Naimark Decomposition and the Harish-Chandra c-Function.
- §1. The Gelfand-Naimark Decomposition and the Harish-Chandra Mapping of U? into M\K
- §2. The Bruhat Decomposition
- §3. Jacobian Formulas
- §4. Integral Formulasfor Spherical Functions
- §5. The c-Function and the First Spherical Asymptotics