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.

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 Formulasfor 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 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.- §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.