表示论基础教程（英文影印版）

　　本书是一部很受欢迎的教材，初版于1991年，至今已被springer出版社重印5次。全书分为四部分，26章，书中主要论述李群、李代数和经典群的有限维表示， 可作为大学高年级学生, 研究生及教师的教学用书。
　　 目次：（一）有限群：有限群表示；特征；实例；ed表示；ud 、gl2和 fq表示；外尔结构。（二）李群和李代数：李群；李代数和李群；李代数的初始分类；一维、二维和三维中的李代数；sl2 c表示；sl3 c表示。（三）经典李代数及其示；任意半单李代数的结构与表示； sl4 c和sln c;辛李代数；sp6c和sp2n c;正交李代数；so6 c、 so7 c和som c;so m c自旋表示。（四）李理论：复单李群的分类；g2和其它例外李代数；复李群；外尔特征公式；实李代数和李群。
　　 读者对象：数学及物理学专业的高年级本科生、研究生和教师。

preface
using this book
part i: finite groups
1. representations of finite groups
1.1: definitions
1.2: complete reducibility; schur's lemma
1.3: examples: abelian groups;
2. characters
2.1: characters
2.2: the first projection formula and its consequences
2.3: examples: 4 and 4
2.4: more projection formulas; more consequences
3. examples; induced representations; group algebras; real
representations
3.1: examples: 5 and 5
3.2: exterior powers of the standard representation of d
3.3: induced representations
3.4: the group algebra
3.5: real representations and representations over subfields of c
4. representations of gd: young diagrams and frobenius's

　　The primary goal of these lectures is to introduce a beginner to the finite- dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book.
　　Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is (e.g., a cohomology group, tangent space, etc.). As a consequence, many mathematicians other than specialists in the field (or even those who think they might want to be) come in contact with the subject in various ways. It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference.
　　This idea essentially determines the choice of material covered here. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific. For a start, what kind of group G are we dealing with--a finite group like the symmetric group or the general linear group over a finite field GLn(Fq), an infinite discrete group like SLN(Z), a Lie group like SL.C, or possibly a Lie group over a local field? Needless to say, each of these settings requires a substantially different approach to its representation theory. Likewise, what sort of vector space is G acting on: is it over C, R, Q, or possibly a field of finite characteristic? Is it finite dimensional or infinite dimensional, and if the latter, what additional structure (such as norm, or inner product) does it carry? Various combinations of answers to these questions lead to areas of intense research activity in representation theory, and it is natural for a text intended to prepare students for a career in the subject to lead up to one or more of these areas. As a corollary, such a book tends to get through the elementary material as quickly as possible: if one has a semester to get up to and through Harish-Chandra modules, there is little time to dawdle over the representations of and SL3C.
　　By contrast, the present book focuses exactly on the simplest cases: repre- sentations of finite groups and Lie groups on finite-dimensional real and complex vector spaces. This is in some sense the common ground of the subject, the area that is the object of most of the interest in representation theory coming from outside.
　　The intent of this book to serve nonspecialists likewise dictates to some degree our approach to the material we do cover. Probably the main feature of our presentation is that we concentrate on examples, developing the general theory sparingly, and then mainly as a useful and unifying language to describe phenomena already encountered in concrete cases. By the same token, we for the most part introduce theoretical notions when and where they are useful for analyzing concrete situations, postponing as long as possible those notions that are used mainly for proving general theorems.
　　Finally, our goal of making the book accessible to outsiders accounts in part for the style of the writing. These lectures have grown from courses of the second author in 1984 and 1987, and we have attempted to keep the informal style of these lectures. Thus there is almost no attempt at efficiency: where it seems to make sense from a didactic point of view, we work out many special cases of an idea by hand before proving the general case; and we cheerfully give several proofs of one fact if we think they are illuminating. Similarly, while it is common to develop the whole semisimple story from one point of view, say that of compact groups, or Lie algebras, or algebraic groups, we have avoided this, as efficient as it may be.
　　It is of course not a strikingly original notion that beginners can best learn about a subject by working through examples, with general machinery only introduced slowly and as the need arises, but it seems particularly appropriate here. In most subjects such an approach means one has a few out of an unknown infinity of examples which are useful to illuminate the general situation. When the subject is the representation theory of complex semisimple Lie groups and algebras, however, something special happens: once one has worked through all the examples readily at hand--the "classical" cases of the special linear, orthogonal, and symplectic groups--one has not just a few useful examples, one has all but five "exceptional" cases.
　　This is essentially what we do here. We start with a quick tour through representation theory of finite groups, with emphasis determined by what is useful for Lie groups. In this regard, we include more on the symmetric groups than is usual. Then we turn to Lie groups and Lie algebras. After some preliminaries and a look at low-dimensional examples, and one lecture with some general notions about semisimplicity, we get to the heart of the course: working out the finite-dimensional representations of the classical groups.
　　For each series of classical Lie algebras we prove the fundamental existence theorem for representations of given highest weight by explicit construction. Our object, however, is not just existence, but to see the representations in action, to see geometric implications of decompositions of naturally occurring representations, and to see the relations among them caused by coincidences between the Lie algebras.
　　The goal of the last six lectures is to make a bridge between the example- oriented approach of the earlier parts and the general theory. Here we make an attempt to interpret what has gone before in abstract terms, trying to make connections with modern terminology. We develop the general theory enough to see that we have studied all the simple complex Lie algebras with five exceptions. Since these are encountered less frequently than the classical series, it is probably not reasonable in a first course to work out their representations as explicitly, although we do carry this out for one of them. We also prove the general Weyl character formula, which can be used to verify and extend many of the results we worked out by hand earlier in the book.
　　Of course, the point we reach hardly touches the current state of affairs in Lie theory, but we hope it is enough to keep the reader's eyes from glazing over when confronted with a lecture that begins: "Let G be a semisimple Lie group, P a parabolic subgroup .... "We might also hope that working through this book would prepare some readers to appreciate the elegance (and efficiency) of the abstract approach.
　　In spirit this book is probably closer to Weyl's classic [Wel] than to others written today. Indeed, a secondary goal of our book is to present many of the results of Weyl and his predecessors in a form more accessible to modern readers. In particular, we include Weyl's constructions of the representations of the general and special linear groups by using Young's symmetrizers; and we invoke a little invariant theory to do the corresponding result for the orthogonal and symplectic groups. We also include Weyl's formulas for the characters of these representations in terms of the elementary characters of symmetric powers of the standard representations. (Interestingly, Weyl only gave the corresponding formulas in terms of the exterior powers for the general linear group. The corresponding formulas for the orthogonal and symplectic groups were only given recently by Koike and Terada. We include a simple new proof of these determinantal formulas.)
　　More about individual sections can be found in the introductions to other parts of the book.
　　Needless to say, a price is paid for the inefficiency and restricted focus of these notes. The most obvious is a lot of omitted material: for example, we include little on the basic topological, differentiable, or analytic properties of Lie groups, as this plays a small role in our story and is well covered in dozens of other sources, including many graduate texts on manifolds. Moreover, there are no infinite-dimensional representations, no Harish-Chandra or Verma modules, no Steifel diagrams, no Lie algebra cohomology, no analysis on symmetric spaces or groups, no arithmetic groups or automorphic forms, and nothing about representations in characteristic p ] 0. There is no consistent attempt to indicate which of our results on Lie groups apply more generally to algebraic groups over fields other than R or C (e.g., local fields). And there is only passing mention of other standard topics, such as universal enveloping algebras or Bruhat decompositions, which have become standard tools of representation theory. (Experts who saw drafts of this book agreed that some topic we omitted must not be left out of a modern book on representation theory--but no two experts suggested the same topic.)
　　We have not tried to trace the history of the subjects treated, or assign credit, or to attribute ideas to original sources--this is far beyond our knowl- edge. When we give references, we have simply tried to send the reader to sources that are as readable as possible for one knowing what is written here. A good systematic reference for the finite-group material, including proofs of the results we leave out, is Serre [Se2]. For Lie groups and Lie algebras, Serre [Se3], Adams [Ad], Humphreys [Hull, and Bourbaki [Bour] are recommended references, as are the classics Weyl [Wel] and Littlewood [Litl].
　　We would like to thank the many people who have contributed ideas and suggestions for this manuscript, among them J-F. Burnol, R. Bryant, J. Carrell, B. Conrad, P. Diaconis, D. Eisenbud, D. Goldstein, M. Green, P. Griffiths, B. Gross, M. Hildebrand, R. Howe, H. Kraft, A. Landman, B. Mazur, N. Chriss, D. Petersen, G. Schwartz, J. Towber, and L. Tu. In particular, we would like to thank David Mumford, from whom we learned much of what we know about the subject, and whose ideas are very much in evidence in this book.
　　Had this book been written 10 years ago, we would at this point thank the people who typed it. That being no longer applicable, perhaps we should thank instead the National Science Foundation, the University of Chicago, and Harvard University for generously providing the various Macintoshes on which this manuscript was produced. Finally, we thank Chan Fulton for making the drawings.
　　Bill Fulton and Joe Harris