Non-Abelian Harmonic Analysis Applications of SL (2,?)

This book mainly discusses the representation theory of the special linear group 8L(2, 1R), and some applications of this theory. In fact the emphasis is on the applications; the working title of the book while it was being writ­ ten was "Some Things You Can Do with 8L(2). " Some of the applications are outside representation theory, and some are to representation theory it­ self. The topics outside representation theory are mostly ones of substantial classical importance (Fourier analysis, Laplace equation, Huyghens' prin­ ciple, Ergodic theory), while the ones inside representation theory mostly concern themes that have been central to Harish-Chandra's development of harmonic analysis on semisimple groups (his restriction theorem, regularity theorem, character formulas, and asymptotic decay of matrix coefficients and temperedness). We hope this mix of topics appeals to nonspecialists in representation theory by illustrating (without an interminable prolegom­ ena) how representation theory can offer new perspectives on familiar topics and by offering some insight into some important themes in representation theory itself. Especially, we hope this book popularizes Harish-Chandra's restriction formula, which, besides being basic to his work, is simply a beautiful example of Fourier analysis on Euclidean space. We also hope representation theorists will enjoy seeing examples of how their subject can be used and will be stimulated by some of the viewpoints offered on representation-theoretic issues.

I Preliminaries.- 1. Lie Groups and Lie Algebras.- 2. Theory of Fourier Transform.- 3. Spectral Analysis for Representations of ?n.- Exercises.- II Representations of the Lie Algebra of SL(2, ?).- 1. Standard Modules and the Structure of sl(2) Modules.- 2. Tensor Products.- 3. Formal Eigenvectors.- Exercises.- III Unitary Representations of the Universal Cover of SL(2, ?).- 1. Infinitesimal Classification.- 2. Oscillator Representation.- Exercises.- IV Applications to Analysis.- 1. Bochner’s Periodicity Relations.- 2. Harish-Chandra’s Restriction Formula.- 3. Fundamental Solution of the Laplacian.- 4. Huygens’ Principle.- 5. Harish-Chandra’s Regularity Theorem for SL(2, ?), and the Rossman-Harish-Chandra-Kirillov Character Formula.- Exercises.- V Asymptotics of Matrix Coefficients.- 1. Generalities.- 2. Vanishing of Matrix Coefficients at Infinity for SL(n, ?).- 3. Quantitative Estimates.- 4. Some Consequences.- Exercises.- References.