Now in paperback, this graduate-level textbook is an introduction to the representation theory of semi-simple Lie groups. As such, it will be suitable for research students in algebra and analysis, and for research mathematicians requiring a readable account of the topic. The author emphasizes the development of the central themes of the sunject in the context of special examples, without losing sight of its general flow and structure. The book concludes with appendices sketching some basic topics with a comprehensive guide to further reading.
The representation theory of locally compact groups has been vig orously developed in the past twenty-five years or so; of the various branches of this theory, one of the most attractive (and formidable) is the representation theory of semi-simple Lie groups which, to a great extent, is the creation of a single man: Harish-Chandra. The chief objective of the present volume and its immediate successor is to provide a reasonably self-contained introduction to Harish-Chandra's theory. Granting cer tain basic prerequisites (cf. infra), we have made an effort to give full details and complete proofs of the theorems on which the theory rests. The structure of this volume and its successor is as follows. Each book is divided into chapters; each chapter is divided into sections; each section into numbers. We then use the decimal system of reference; for example, 1. 3. 2 refers to the second number in the third section of the first chapter. Theorems, Propositions, Lemmas, and Corollaries are listed consecutively throughout any given number. Numbers which are set in fine print may be omitted at a first reading. There are a variety of Exam ples scattered throughout the text; the reader, if he is so inclined, can view them as exercises ad libitum. The Appendices to the text collect certain ancillary results which will be used on and off in the systematic exposi tion; a reference of the form A2.
Analysis on Symmetric spaces, or more generally, on homogeneous spaces of semisimple Lie groups, is a subject that has undergone a vigorous development in recent years, and has become a central part of contemporary mathematics. This is only to be expected, since homogeneous spaces and group representations arise naturally in diverse contexts ranging from Number theory and Geometry to Particle Physics and Polymer Chemistry. Its explosive growth sometimes makes it difficult to realize that it is actually relatively young as mathematical theories go. The early ideas in the subject (as is the case with many others) go back to Elie Cart an and Hermann Weyl who studied the compact symmetric spaces in the 1930's. However its full development did not begin until the 1950's when Gel'fand and Harish Chandra dared to dream of a theory of representations that included all semisimple Lie groups. Harish-Chandra's theory of spherical functions was essentially complete in the late 1950's, and was to prove to be the forerunner of his monumental work on harmonic analysis on reductive groups that has inspired a whole generation of mathematicians. It is the harmonic analysis of spherical functions on symmetric spaces, that is at the focus of this book. The fundamental questions of harmonic analysis on symmetric spaces involve an interplay of the geometric, analytical, and algebraic aspects of these spaces. They have therefore attracted a great deal of attention, and there have been many excellent expositions of the themes that are characteristic of this subject.
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.
This (post) graduate text gives a broad introduction to Lie groups and algebras with an emphasis on differential geometrical methods. It analyzes the structure of compact Lie groups in terms of the action of the group on itself by conjugation, culminating in the classification of the representations of compact Lie groups and their realization as sections of holomorphic line bundles over flag manifolds. Appendices provide background reviews.
This book presents an account of recent results on the theory of representations and the harmonic analysis of free groups. It emphasizes the analogy with the theory of representations of noncompact semisimple Lie groups and restricts the focus to a class of irreducible unitary representations.
* First of three independent, self-contained volumes under the general title, "Lie Theory," featuring original results and survey work from renowned mathematicians. * Contains J. C. Jantzen's "Nilpotent Orbits in Representation Theory," and K.-H. Neeb's "Infinite Dimensional Groups and their Representations." * Comprehensive treatments of the relevant geometry of orbits in Lie algebras, or their duals, and the correspondence to representations. * Should benefit graduate students and researchers in mathematics and mathematical physics.
This book presents a number of important contributions focusing on harmonic analysis and representation theory of Lie groups. All were originally presented at the 5th Tunisian–Japanese conference “Geometric and Harmonic Analysis on Homogeneous Spaces and Applications”, which was held at Mahdia in Tunisia from 17 to 21 December 2017 and was dedicated to the memory of the brilliant Tunisian mathematician Majdi Ben Halima. The peer-reviewed contributions selected for publication have been modified and are, without exception, of a standard equivalent to that in leading mathematical periodicals. Highlighting the close links between group representation theory and harmonic analysis on homogeneous spaces and numerous mathematical areas, such as number theory, algebraic geometry, differential geometry, operator algebra, partial differential equations and mathematical physics, the book is intended for researchers and students working in the area of commutative and non-commutative harmonic analysis as well as group representations.
This book is suitable for advanced undergraduate and graduate students in mathematics with a strong background in linear algebra and advanced calculus. Early chapters develop representation theory of compact Lie groups with applications to topology, geometry, and analysis, including the Peter-Weyl theorem, the theorem of the highest weight, the character theory, invariant differential operators on homogeneous vector bundles, and Bott's index theorem for such operators. Later chapters study the structure of representation theory and analysis of non-compact semi-simple Lie groups, including the principal series, intertwining operators, asymptotics of matrix coefficients, and an important special case of the Plancherel theorem. Teachers will find this volume useful as either a main text or a supplement to standard one-year courses in Lie groups and Lie algebras. The treatment advances from fairly simple topics to more complex subjects, and exercises appear at the end of each chapter. Eight helpful Appendixes develop aspects of differential geometry, Lie theory, and functional analysis employed in the main text.
This book presents the text of the lectures which were given at the NATO Advanced Study Institute on Representations of Lie groups and Harmonic Analysis which was held in Liege from September 5 to September 17, 1977. The general aim of this Summer School was to give a coordinated intro duction to the theory of representations of semisimple Lie groups and to non-commutative harmonic analysis on these groups, together with some glance at physical applications and at the related subject of random walks. As will appear to the reader, the order of the papers - which follows relatively closely the order of the lectures which were actually give- follows a logical pattern. The two first papers are introductory: the one by R. Blattner describes in a very progressive way a path going from standard Fourier analysis on IR" to non-commutative harmonic analysis on a locally compact group; the paper by J. Wolf describes the structure of semisimple Lie groups, the finite-dimensional representations of these groups and introduces basic facts about infinite-dimensional unitary representations. Two of the editors want to thank particularly these two lecturers who were very careful to pave the way for the later lectures. Both these chapters give also very useful guidelines to the relevant literature.
