Spherical Inversion on SL n (R)
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1 Jay Jorgenson Serge Lang Spherical Inversion on SL n (R) Springer
2 Contents Acknowledgments Overview Table of the Decompositions ix xi xvii CHAPTER I Iwasawa Decomposition and Positivity 1 1. The Iwasawa Decomposition 1 2. Haar Measure and Iwasawa Decomposition 5 3. The Cartan Lie Decomposition, Polynomial Algebra and Chevalley's Theorem Positivity Convexity The Harish-Chandra [/-Polar Inequality; Connection with the Iwasawa and Polar Decompositions 28 CHAPTER II Invariant Differential Operators and the Iwasawa Direct Image Invariant Differential Operators on a Lie Group The Projection on a Homogeneous Space The Iwasawa Projection on A Use of the Cartan Lie Decomposition The Harish-Chandra Transforms The Transpose and Involution 66
3 VI CONTENTS CHAPTER II! Characters, Eigenfunctions, Spherical Kernel and W-lnvariance Characters The (a, n)-characters and the Iwasawa Character The Weyl Group Orbital Integral for the Harish Transform W-Invariance of the Harish and Spherical Transforms A"-Bi-Invariant Functions and Uniqueness of Spherical Functions Integration Formulas and the Map x i-» x~ l W-Harmonic Polynomials and Eigenfunctions of ^-Invariant Differential Operators on A 114 CHAPTER IV Convolutions, Spherical Functions and the Mellin Transform Weakly Symmetric Spaces Characters and Convolution Operators Example: The Gamma Function Invariance or Bi-Invariance and Eigenfunctions of Convolutions Convolution Sphericality The Spherical Transform as Multiplicative Homomorphism The Mellin Transform and the Paley-Wiener Space Behavior of the Support 167 CHAPTER V Gelfand-Naimark Decomposition and the Harish-Chandra c-function The Gelfand-Naimark Decomposition and the Harish-Chandra Mapping of U into M\K The Bruhat Decomposition Jacobian Formulas Integral Formulas for Spherical Functions The C-Function and the First Spherical Asymptotics The Bhanu-Murty Formula for the C-Function Invariant Formulation on o^ Corollaries on the Analytic Behavior of C Ha r 214 CHAPTER VI Polar Decomposition The Jacobian of the Polar Map From /if-bi-invariant Functions on G to ^-Invariant Functions on a Appendix. The Bernstein Calculus Lemma Pulling Back Characters and Spherical Functions to a Lemmas Using the Semisimple Lie Iwasawa Decomposition The Transpose Iwasawa Decomposition and Polar Direct Image W-Invariants 253
4 CONTENTS Vii CHAPTER Vll The Casimir Operator Bilinear Forms of Cartan Type The Casimir Differential Operator The A-Iwasawa and Harish-Chandra Direct Images The Polar Direct Image 270 CHAPTER VIII The Harish-Chandra Series and Spherical Inversion Linear Independence of Characters Revisited Eigenfunctions of Casimir The Harish-Chandra Series and Gangolli Estimate The C-Function and the W-Trace The Helgason and Anker Support Theorems An L 2 -Estimate and Limit Spherical Inversion 304 CHAPTER IX General Inversion Theorems The Rosenberg Arguments Helgason Inversion on Paley-Wiener and the L 2 -Isometry The Constant in the Inversion Formula 321 CHAPTER X The Harish-Chandra Schwartz Space (HCS) and Anker's Proof of Inversion More Harish-Chandra Convexity Inequalities More Harish-Chandra Inequalities for Spherical Functions The Harish-Chandra Schwartz Space Schwartz Continuity of the Spherical Transform Continuity of the Inverse Transform and Spherical Inversion on HCS(K\G/K) Extension of Formulas by HCS Continuity An Example: The Heat Kernel The Harish Transform 367 CHAPTER XI Tube Domains and the L 1 (Even L p ) HCS Spaces The Schwartz Space on Tubes The Filtration HCS (p) (K\G/K) with 0 < p ^ The Inverse Transform Bounded Spherical Functions Back to the Heat Kernel 385
5 viii CONTENTS CHAPTER XII SL n (C) A Formula of Exponential Polynomials Characters and Jacobians The Polar Direct Image Spherical Functions and Inversion The Heat Kernel The Flensted-Jensen Decomposition and Reduction 406 Bibliography 411 Table of Notation 419 Index 423
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