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

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 Harish-Chandra Schwartz space had not yet made it into a book exposition. Previous expositions have dealt with a general, wide class of Lie groups. This has made access to the subject difficult for outsiders, who may wish to connect some aspects with several if not all other parts of mathematics, and do so in specific cases of intrinsic interest. The essential features of Harish-Chandra theory are exhibited on SLn(R), but hundreds of pages of background can be replaced by short direct verifications. The material becomes accessible to graduate students with especially no background in Lie groups and representation theory. Spherical inversion is sufficient to deal with the heat kernel, which is at the center of the authors' current research. The book will serve as a self-contained background for parts of this research

I Iwasawa Decomposition and Positivity -- ยง1. The Iwasawa Decomposition -- ยง2. Haar Measure and Iwasawa Decomposition -- ยง3. The Cartan Lie Decomposition, Polynomial Algebra and Chevalleyโ{128}{153}s 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 Ankerโ{128}{153}s 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