Spherical Inversion on SLn(R)

Spherical Inversion on SLn(R) by Jay Jorgenson, Serge Lang.

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...

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Bibliographic Details
Main Authors: Jorgenson, Jay (Author), Lang, Serge (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 2001.
Edition:1st ed. 2001.
Series:Springer Monographs in Mathematics,
Springer eBook Collection.
Subjects:
Topological groups.
Lie groups.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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
  • §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 Formulas for Spherical Functions
  • §5. The c-Function and the First Spherical Asymptotics
  • §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
  • §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 and the 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
  • §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.

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