For the most part the authors are concerned with SL_n(R) and with invariant differential operators, the invariance being with respect to various subgroups. To a large extent, this book carries out the general results of Harish-Chandra.

Accessible to graduate students

Inhalt
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.