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.

Inhalt

I Preliminaries.- 1. Lie Groups and Lie Algebras.- 1.1. Basic Structure.- 1.2. Representations of Lie Groups.- 1.3. Representations of Lie Algebras.- 2. Theory of Fourier Transform.- 2.1. Distributions.- 2.2. 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.- 1.1. Indecomposable Modules.- 1.2. Standard Modules.- 1.3. Structure Theorem.- 2. Tensor Products.- 2.1. Tensor Product of Two Lowest Weight Modules.- 2.2. Formal Vectors.- 2.3. Tensor Product V? ? V??.- 3. Formal Eigenvectors.- 3.1. Action of Other Bases of sl(2).- 3.2. Formal e+-Null Vectors in (V? ? V??)~.- 3.3. Formal h Eigenvectors in U(v+, v-)~.- 3.4. Some Modules in U(v+, v-)~.- Exercises.- III Unitary Representations of the Universal Cover of SL(2, ?).- 1. Infinitesimal Classification.- 1.1. Unitarizability of Standard Modules.- 1.2. A Realization of U(v+, v-).- 1.3. Unitary Dual of SL(2, ?).- 2. Oscillator Representation.- 2.1. Theory of Hermite Functions.- 2.2. The Contragredient (?n, S(?n)).- 2.3. Tensor Product ?p ? ?q*.- 2.4. Case q = 0: Theory of Spherical Harmonics.- Exercises.- IV Applications to Analysis.- 1. Bochner's Periodicity Relations.- 1.1. Fourier Transform as an Element of $$ \mathop{{SL}}\limits^{ \sim } $$(2, ?).- 1.2. Bochner's Periodicity Relations.- 2. Harish-Chandra's Restriction Formula.- 2.1. Motivation: Case of O(3, ?).- 2.2. Harish-Chandra's Restriction Formula for U(n).- 2.3. Some Consequences.- 3. Fundamental Solution of the Laplacian.- 3.1. Fundamental Solution of the Definite Laplacian.- 3.2. Fundamental Solution of the Indefinite Laplacian.- 3.3. Structure of O(p,q)-Invariant Distributions Supported on the Light Cone.- 4. Huygens' Principle.- 4.1. The Propagator.- 4.2. Symmetries of the Propagator.- 4.3. Representation Theoretic Considerations.- 4.4. O+(n, 1)-Invariant Distributions.- 5. Harish-Chandra's Regularity Theorem for SL(2, ?), and the Rossman-Harish-Chandra-Kirillov Character Formula.- 5.1. Regularity of Invariant Eigendistributions.- 5.2. Tempered Distributions and the Character Formula.- Exercises.- V Asymptotics of Matrix Coefficients.- 1. Generalities.- 1.1. Various Decompositions.- 1.2. Matrix Coefficients.- 2. Vanishing of Matrix Coefficients at Infinity for SL(n, ?).- 3. Quantitative Estimates.- 3.1. Decay of Matrix Coefficients of Irreducible Unitary Representations of SL(2, ?).- 3.2. Decay of Matrix Coefficients of the Regular Representation of SL(2, ?).- 3.3. Quantitative Estimates for SL(n, ?).- 4. Some Consequences.- 4.1. Kazhdan's Property T.- 4.2. Ergodic Theory.- Exercises.- References.