Graduate Texts in Mathematics 225 Editorial Board S Axler F.W Gehring K.A Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics TAKEUTU~G.Introductionto 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFER Topological Vector Spaces 2nded HILTON/STAMMBACH A Course in Homological Algebra 2nd ed MAC LANE Categories for the Working Mathematician 2nd ed HUGHESlPiPER Projective Planes J.-P SERRE A Course in Arithmetic TAKEUTU~G Axiomatic Set Theory HUMPHREYS Introduction to Ue Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSONIFuLi.ER Rings and Categories of Modules 2nd ed GoLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMos Measure Theory HALMos A Hilbert Space Problem Book 2nded HusEMOLLER Fibre Bundles 3rd ed HUMPHREYS Linear Algebraic Groups BARNESIMACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and Its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEy General Topology lAIusKIlSAMUEL Commutative Algebra Vol.I lAIusKIlSAMUEL Commutative Algebra VoI.II JACOBSON Lectures in Abstract Algebra I Basic Concepts JACOBSON Lectures in Abstract Algebra II Unear Algebra JACOBSON Lectures in Abstract Algebra ill Theory of Fields and Galois Theory HiRsCH Differential Topology 34 SPITZER Principles of Random Walk 2nded 35 ALExANDERlWERMER Several Complex Variables and Banach Algebras 3rd ed 36 KELLEy/NAMIOKA et a! Unear Topological Spaces 37 MoNK Mathematical Logic 38 GRAUERTIFRiTZSCHE Several Complex Variables 39 ARVESON An Invitation to c*-Algebras 40 KEMENY/SNEw'KNAPP Denumerable Markov Chains 2nd ed 41 APoSTOL Modular Functions and Dirichlet Series in Number Theory 2nded 42 J.-P SERRE Unear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LoSVE Probability Theory I 4th ed 46 LoSVE Probability Theory II 4th ed 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENEERG/WEIR Unear Geometry 2nded 50 EDWARDS Fermat's Last Theorem 51 KLiNGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRAVERIWATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BRoWN!PEARcy Introduction to Operator Theory I: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CRoWELLlFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 2nd ed 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 2nd ed 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOVIMERLZIAKOV Fundamentals of the Theory of Groups 63 BOLLOBAs Graph Theory (continued after index) Daniel Bump Lie Groups , Springer Daniel Bump Department of Mathematics Stanford University 450 Serra Mall, Bldg 380 Stanford, CA 94305-2125 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California, Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 20xx, 22xx Library of Congress Cataloging in Publication Data Bump, Daniel Lie groups / Daniel Bump p cm (Graduate texts in mathematics: 225) Includes bibliographical references and index ISBN 978-1-4419-1937-3 ISBN 978-1-4757-4094-3 (eBook) DOI 10.1007/978-1-4757-4094-3 [on file] ISBN 978-1-4419-1937-3 Printed on acid-free paper ©2004 Springer Science+Business Media New York Originally published by Springer-Verlag New York, LLC in 2004 Softcover reprint of the hardcover 1st edition 2004 All rights resetVed This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC , except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights springeronline com SPIN 10951439 Preface This book aims to be a course in Lie groups that can be covered in one year with a group of good graduate students I have attempted to address a problem that anyone teaching this subject must have, which is that the amount of essential material is too much to cover One approach to this problem is to emphasize the beautiful representation theory of compact groups, and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III But I did not want to omit important topics such as the Bruhat decomposition and the theory of symmetric spaces For these subjects, compact groups are not sufficient Part I covers standard general properties of representations of compact groups (including Lie groups and other compact groups, such as finite or padic ones) These include Schur orthogonality, properties of matrix coefficients and the Peter-Weyl Theorem Part II covers the fundamentals of Lie groups, by which I mean those subjects that I think are most urgent for the student to learn These include the following topics for compact groups: the fundamental group, the conjugacy of maximal tori (two proofs), and the Weyl character formula For noncompact groups, we start with complex analytic groups that are obtained by complexification of compact Lie groups, obtaining the Iwasawa and Bruhat decompositions These are the reductive complex groups They are of course a special case, but a good place to start in the noncompact world More general noncompact Lie groups with a Cartan decomposition are studied in the last few chapters of Part II Chapter 31, on symmetric spaces, alternates examples with theory, discussing the embedding of a noncompact symmetric space in its compact dual, the boundary components and Bergman-Shilov boundary of a symmetric tube domain, and Cartan's classification Chapter 32 constructs the relative root system, explains Satake diagrams and gives examples illustrating the various phenomena that can occur, and reproves the Iwasawa decomposition, formerly obtained for complex analytic groups, in this more general context Finally, Chapter 33 surveys the different ways Lie groups can be embedded in one another vi Preface Part III returns to representation theory The major unifying theme of Part III is Frobenius-Schur duality This is the correspondence, originating in Schur's 1901 dissertation and emphasized by Weyl, between the irreducible representations of the symmetric group and the general linear groups The correspondence comes from decomposing tensor spaces over both groups simultaneously It gives a dictionary by which problems can be transferred from one group to the other For example, Diaconis and Shahshahani studied the distribution of traces of random unitary matrices by transferring the problem of their distribution to the symmetric group The plan of Part III is to first use the correspondence to simultaneously construct the irreducible representations of both groups and then give a series of applications to illustrate the power of this technique These applications include random matrix theory, minors of Toeplitz matrices, branching formulae for the symmetric and unitary groups, the Cauchy identity, and decompositions of some symmetric and exterior algebras Other thematically related topics topics discussed in Part III are the cohomology of Grassmannians, and the representation theory of the finite general linear groups This plan of giving thematic unity to the "topics" portion of the book with Frobenius-Schur the unifying theme has the effect of somewhat overemphasizing the unitary groups at the expense of other Lie groups, but for this book the advantages outweigh this disadvantage, in my opinion The importance of Frobenius-Schur duality cannot be overstated In Chapters 48 and 49, we turn to the analogies between the representation theories of symmetric groups and the finite general linear groups, and between the representation theory of the finite general linear groups and the theory of automorphic forms The representation theory of GL( n, IFq) is developed to the extent that we can construct the cuspidal characters and explain HarishChandra's "Philosophy of Cusp Forms" as an analogy between this theory and the theory of automorphic forms It is a habit of workers in automorphic forms (which many of us learned from Piatetski-Shapiro) to use analogies with the finite field case systematically The three parts have been written to be somewhat independent One may thus start with Part II or Part III and it will be quite a while before earlier material is needed In particular, either Part II or Part III could be used as the basis of a shorter course Regarding the independence of Part III, the Weyl character formula for the unitary groups is obtained independently of the derivation in Part II Eventually, we need the Bruhat decomposition but not before Chapter 47 At this point, the reader may want to go back to Part II to fill this gap Prerequisites include the Inverse Function Theorem, the standard theorem on the existence of solutions to first order systems of differential equations and a belief in the existence of Haar measures, whose properties are reviewed in Chapter Chapters 17 and 50 assume some algebraic topology, but these chapters can be skipped Occasionally algebraic varieties and algebraic groups Preface vii are mentioned, but algebraic geometry is not a prerequisite For affine algebraic varieties, only the definition is really needed The notation is mostly standard In GL(n), I or In denotes the n x n identity matrix and if is any matrix, t denotes its transpose Omitted entries in a matrix are zero The identity element of a group is usually denoted but also as I, if the group is GL(n) (or a subgroup), and occasionally as e when it seemed the other notations could be confusing The notations c and ~ are synonymous, but we mostly use X C Y if X and Y are known to be unequal, although we make no guarantee that we are completely consistent in this If X is a finite set, IXI denotes its cardinality One point where we differ with some of the literature is that the root system lives in R ® X* (T) rather than in the dual space of the Lie algebra of the maximal torus T as in much of the literature This is of course the right convention if one takes the point of view of algebraic groups, and it is also arguably the right point of view in general since the real significance of the roots has to with the fact that they are characters of the torus, not that they can be interpreted as linear functionals on its Lie algebra To keep the book to a reasonable length, many standard topics have been omitted, and the reader may want to study some other books at the same time Cited works are usually recommended ones Acknowledgments The proofs in Chapter 36 on the Jacobi-Trudi identity were worked out years ago with Karl Rumelhart when he was still an undergraduate at Stanford Very obviously, Chapters 40 and 41 owe a great deal to Persi Diaconis, and Chapter 43 on Cauchy's identity was suggested by a conversation with Steve Rallis I would like to thank my students in Math 263 for staying with me while I lectured on much of this material This book was written using 'lEXmacs, with further editing of the exported Jb.'JEjX file The utilities patch and diff were used to maintain the differences between the automatically generated and the hand-edited 'lEX files The figures were made with MetaPost The weight diagrams in Chapter 24 were created using programs I wrote many years ago in Mathematica based on the Freudenthal multiplicity formula This work was supported in part by NSF grant DMS-9970841 Daniel Bump Contents Preface v Part I: Compact Groups Haar Measure Schur Orthogonality Compact Operators 17 The Peter-Weyl Theorem 21 Part II: Lie Group Fundamentals Lie Subgroups of GL(n, q Vector Fields 36 Left-Invariant Vector Fields 41 The Exponential Map 46 Tensors and Universal Properties 50 29 10 The Universal Enveloping Algebra 54 11 Extension of Scalars 58 12 Representations of 6l(2, q 62 13 The Universal Cover 69 x Contents 14 The Local Frobenius Theorem 79 15 Tori 86 16 Geodesics and Maximal Tori 94 17 Topological Proof of Cartan's Theorem 107 18 The Weyl Integration Formula 112 19 The Root System 117 20 Examples of Root Systems 127 21 Abstract Weyl Groups 136 22 The Fundamental Group 146 23 Semisimple Compact Groups 150 24 Highest-Weight Vectors 157 25 The Weyl Character Formula 162 26 Spin 175 27 Complexification 182 28 Coxeter Groups 189 29 The Iwasawa Decomposition 197 30 The Bruhat Decomposition 205 31 Symmetric Spaces 212 32 Relative Root Systems 236 33 Embeddings of Lie Groups 257 Part III: Topics 34 Mackey Theory 275 35 Characters of GL( n, q 284 36 Duality between Sk and GL(n, q 289 440 References 35 A Dold Lectures on Algebmic Topology Springer-Verlag, New York, 1972 Die Grundlehren der mathematischen Wissenschaften, Band 200 36 E Dynkin Maximal subgroups of semi-simple Lie groups and the classification of primitive groups of transformations Doklady Akad Nauk SSSR (N.S.), 75:333-336, 1950 37 E Dynkin Maximal subgroups of the classical groups Trudy Moskov Mat ObBe., 1:39-166, 1952 38 E Dynkin Semisimple subalgebras of semisimple Lie algebras Mat Sbornik N.S., 30(72):349-462, 1952 39 F Dyson Statistical theory of the energy levels of complex systems, I, II, III J Mathematical Phys., 3:140-156, 157-165, 166-175, 1962 40 S Eilenberg and N Steenrod Foundations of Algebmic Topology Princeton University Press, Princeton, New Jersey, 1952 41 H Freudenthal Lie groups in the foundations of geometry Advances in Math., 1:145-190 (1964), 1964 42 G Frobenius Uber die charakterisischen Einheiten der symmetrischen Gruppe S'ber Akad Wiss Berlin, 504-537, 1903 43 G Frobenius and I Schur Uber die rellen Darstellungen der endlichen Gruppen S'ber Akad Wiss Berlin, 186-208, 1906 44 W Fulton Young Tableaux, with applications to representation theory and geometry, volume 35 of London Mathematical Society Student Texts Cambridge University Press, Cambridge, 1997 45 W Fulton Intersection Theory, volume of Ergebnisse der Mathematik und ihrer Grenzgebiete Springer-Verlag, Berlin, second edition, 1998 46 I Gelfand, M Graev, and I Piatetski-Shapiro Representation Theory and Automorphic Functions Academic Press Inc., 1990 Translated from the Russian by K A Hirsch, Reprint of the 1969 edition 47 R.o Goodman and N Wallach Representations and Invariants of the Classical Groups, volume 68 of Encyclopedia of Mathematics and its Applications Cambridge University Press, Cambridge, 1998 48 R Gow Properties of the characters of the finite general linear group related to the transpose-inverse involution Proc London Math Soc (3),47(3):493-506, 1983 49 J Green The characters of the finite general linear groups Trans Amer Math Soc., 80:402-447, 1955 50 B Gross Some applications of Gelfand pairs to number theory Bull Amer Math Soc (N.S.), 24:277-301, 1991 51 P Halmos Measure Theory D Van Nostrand Company, Inc., New York, N Y.,1950 52 Harish-Chandra Eisenstein series over finite fields In Functional analysis and related fields (Proc Conf M Stone, Univ Chicago, Chicago, nt., 1968), pages 76-88 Springer, New York, 1970 53 M Harris and R Taylor The Geometry and Cohomology of Some Simple Shimum Varieties, volume 151 of Annals of Mathematics Studies Princeton University Press, Princeton, NJ, 2001 With an appendix by Vladimir G Berkovich 54 R Hartshorne Algebmic Geometry Springer-Verlag, New York, 1977 Graduate Texts in Mathematics, No 52 55 E Heeke Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklungen, I and II Math Ann., 114:1-28, 316-351, 1937 References 441 56 S Helgason Differential Geometry, Lie Groups, and Symmetric Spaces, volume 80 of Pure and Applied Mathematics Academic Press Inc [Harcourt Brace Jovanovich Publishers], New York, 1978 57 E Hewitt and K Ross Abstract Harmonic Analysis Vol I, Structure of topological groups, integration theory, group representations, volume 115 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences) Springer-Verlag, Berlin, second edition, 1979 58 H Hiller Geometry of Coxeter Groups, volume 54 of Research Notes in Mathematics Pitman (Advanced Publishing Program), Boston, Mass., 1982 59 W Hodge and D Pedoe Methods of Algebraic Geometry Vol II Cambridge Mathematical Library Cambridge University Press, Cambridge, 1994 Book III: General theory of algebraic varieties in projective space, Book IV: Quadrics and Grassmann varieties, Reprint of the 1952 original 60 R Howe 8-series and invariant theory In Automorphic Forms, Representations and L-Functions (Proc Sympos Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc Sympos Pure Math., XXXIII, pages 275-285 Amer Math Soc., Providence, R.I., 1979 61 R Howe Harish-Chandra Homomorphisms for p-adic Groups, volume 59 of CBMS Regional Conference Series in Mathematics Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1985 With the collaboration of Allen Moy 62 R Howe Heeke algebras and p-adic G Ln In Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993), volume 177 of Contemp Math., pages 65-100 Amer Math Soc., Providence, RI, 1994 63 R Howe Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond In The Schur lectures (1992) (Tel Aviv), volume of Israel Math Conf Proc., pages 1-182 Bar-Ilan Univ., Ramat Gan, 1995 64 R Howe and E.-C Tan Nonabelian Harmonic Analysis Universitext Springer-Verlag, New York, 1992 Applications of SL(2, R) 65 R Howlett and G Lehrer Induced cuspidal representations and generalised Heeke rings Invent Math., 58(1):37-64, 1980 66 E.Ince Ordinary Differential Equations Dover Publications, New York, 1944 67 N Inglis, R Richardson, and J Saxl An explicit model for the complex representations of Sn Arch Math (Basel), 54:258-259, 1990 68 I M Isaacs Character Theory of Finite Groups Dover Publications Inc., New York, 1994 Corrected reprint of the 1976 original [Academic Press, New York; MR 57 #417] 69 N.Iwahori On the structure of a Heeke ring of a Chevalley group over a finite field J Fac Sci Univ Tokyo Sect I, 10:215-236, 1964 70 N Iwahori Generalized Tits system (Bruhat decompostition) on p-adic semisimple groups In Algebraic Groups and Discontinuous Subgroups (Proc Sympos Pure Math., Boulder, Colo., 1965), pages 71-83 Amer Math Soc., Providence, R.I., 1966 71 N Iwahori and H Matsumoto On some Bruhat decomposition and the structure of the Heeke rings of p-adic Chevalley groups Inst Hautes Etudes Sci Publ Math., 25:5-48, 1965 72 N Jacobson Exceptional Lie Algebras, volume of Lecture Notes in Pure and Applied Mathematics Marcel Dekker Inc., New York, 1971 73 M Jimbo A q-analogue of U(gl(N + 1)), Heeke algebra, and the Yang-Baxter equation Lett Math Phys., 11(3):247-252, 1986 442 References 74 V Jones Hecke algebra representations of braid groups and link polynomials Ann of Math (2), 126:335-388, 1987 75 N Katz and P Sarnak Zeroes of zeta functions and symmetry Bull Amer Math Soc (N.S.), 36(1):1-26, 1999 76 N Kawanaka and H Matsuyama A twisted version of the Frobenius-Schur indicator and multiplicity-free permutation representations Hokkaido Math J., 19(3):495-508, 1990 77 J Keating and N Snaith Random matrix theory and ((1/2 + it) Comm Math Phys., 214(1):57-89, 2000 78 R King Branching rules for classical Lie groups using tensor and spinor methods J Phys A, 8:429-449, 1975 79 S Kleiman Problem 15: rigorous foundation of Schubert's enumerative calculus In Mathematical Developments Arising from Hilbert Problems (Proc Sympos Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974), pages 445482 Proc Sympos Pure Math., Vol XXVIII Amer Math Soc., Providence, R I., 1976 80 A Klyachko Models for complex representations of groups GL(n, q) Mat Sb (N.S.), 120(162)(3):371-386, 1983 81 A Knapp Representation Theory of Semisimple Groups, an overview based on examples, volume 36 of Princeton Mathematical Series Princeton University Press, Princeton, NJ, 1986 82 A Knapp Lie groups, Lie algebms, and Chomology, volume 34 of Mathematical Notes Princeton University Press, Princeton, NJ, 1988 83 A Knapp Lie Groups Beyond an Introduction, volume 140 of Progress in Mathematics Birkhauser Boston Inc., Boston, MA, second edition, 2002 84 M.-A Knus, A Merkurjev, M Rost, and J.-P Tignol The Book of Involutions, volume 44 of American Mathematical Society Colloquium Publications American Mathematical Society, Providence, RI, 1998 With a preface in French by J Tits 85 D Knuth The Art of Computer Progmmming Volume 3, Sorting and Searching Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973 Addison-Wesley Series in Computer Science and Information Processing 86 S Kobayashi and K Nomizu Foundations of Differential Geometry Vol Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963 87 A Koranyi and J Wolf Generalized Cayley transformations of bounded symmetric domains Amer J Math., 87:899-939, 1965 88 A Koranyi and J Wolf Realization of hermitian symmetric spaces as generalized half-planes Ann of Math (2), 81:265-288, 1965 89 J Landsberg and L Manivel The projective geometry of Freudenthal's magic square J Algebm, 239(2):477-512, 2001 90 S Lang Algebm, volume 211 of Graduate Texts in Mathematics SpringerVerlag, New York, third edition, 2002 91 R Langlands Euler Products Yale University Press, New Haven, Conn., 1971 A James K Whittemore Lecture in Mathematics given at Yale University, 1967, Yale Mathematical Monographs, 92 H B Lawson and M.-L Michelsohn Spin Geometry, volume 38 of Princeton Mathematical Series Princeton University Press, Princeton, NJ, 1989 93 D Littlewood The Theory of Group Characters and Matrix Representations of Groups Oxford University Press, New York, 1940 References 443 94 L Loomis An Introduction to Abstract Harmonic Analysis D Van Nostrand Company, Inc., Toronto-New York-London, 1953 95 I Macdonald Symmetric Functions and Hall Polynomials Oxford Mathematical Monographs The Clarendon Press Oxford University Press, New York, second edition, 1995 With contributions by A Zelevinsky, Oxford Science Publications 96 L Manivel Symmetric Functions, Schubert Polynomials and Degeneracy Loci, volume of SMFjAMS Texts and Monographs American Mathematical Society, Providence, RI, 2001 Translated from the 1998 French original by John R Swallow, Cours Specialises [Specialized Courses], 97 M Mehta Random Matrices Academic Press Inc., Boston, MA, second edition, 1991 98 J Milnor and J Stasheff Characteristic Classes Princeton University Press, Princeton, N J., 1974 Annals of Mathematics Studies, No 76 99 C Moeglin Representations of GL(n) over the real field In Representation theory and automorphic forms (Edinburgh, 1996), volume 61 of Proc Sympos Pure Math., pages 157-166 Amer Math Soc., Providence, RI, 1997 100 C Mceglin and J.-L Waldspurger Spectral Decomposition and Eisenstein Series, Une paraphrase de l'Ecriture fA paraphrase of Scripture}, volume 113 of Cambridge Tracts in Mathematics Cambridge University Press, Cambridge, 1995 101 I Pyateskii-Shapiro Automorphic Functions and the Geometry of Classical Domains Translated from the Russian Mathematics and Its Applications, Vol Gordon and Breach Science Publishers, New York, 1969 102 J Rogawski On modules over the Hecke algebra of a p-adic group Invent Math., 79:443-465, 1985 103 H Rubenthaler Les paires duales dans les algebres de Lie reductives Asterisque, 219, 1994 104 W Rudin Fourier Analysis on Groups Interscience Tracts in Pure and Applied Mathematics, No 12 Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962 105 B Sagan The Symmetric Group, representations, combinatorial algorithms, and symmetric junctions, volume 203 of Graduate Texts in Mathematics Springer-Verlag, New York, second edition, 2001 106 I Satake On representations and compactifications of symmetric Riemannian spaces Ann of Math (2), 71:77-110, 1960 107 I Satake Theory of spherical functions on reductive algebraic groups over p-adic fields Inst Hautes Etudes Sci Publ Math., 18:5-69, 1963 108 I Satake Classification Theory of Semi-simple Algebraic Groups Marcel Dekker Inc., New York, 1971 With an appendix by M Sugiura, Notes prepared by Doris Schattschneider, Lecture Notes in Pure and Applied Mathematics, 109 I Satake Algebraic Structures of Symmetric Domains, volume of Kano Memorial Lectures Iwanami Shoten and Princeton University Press, Tokyo, 1980 110 R Schafer An Introduction to Nonassociative Algebras Pure and Applied Mathematics, Vol 22 Academic Press, New York, 1966 111 J.-P Serre Galois Cohomology Springer-Verlag, Berlin, 1997 Translated from the French by Patrick Ion and revised by the author 112 E Spanier Algebraic Topology McGraw-Hill Book Co., New York, 1966 444 References 113 T Springer Galois cohomology of linear algebraic groups In Algebraic Groups and Discontinuous Subgroups (Proc Sympos Pure Math., Boulder, Colo., 1965), pages 149 158 Am~r Math Soc., Providence, R.I., 1966 114 T Springer Cusp Forms for Finite Groups In Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69), Lecture Notes in Mathematics, Vol 131, pages 97-120 Springer, Berlin, 1970 115 R Stanley Enumerative Combinatorics Vol 2, volume 62 of Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge, 1999 With a foreword by Gian-Carlo Rota and appendix by Sergey Fomin 116 G Szego On certain Hermitian forms associated with the Fourier series of a positive function Comm Sem Math Univ Lund [Medd Lunds Univ Mat Sem.), 1952(Tome Supplementaire):228-238, 1952 117 T Tamagawa On the (-functions of a division algebra Ann of Math (2), 77:387-405, 1963 118 J Tate Number theoretic background In Automorphic forms, representations and L-functions (Proc Sympos Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc Sympos Pure Math., XXXIII, pages 3-26 Amer Math Soc., Providence, R.I., 1979 119 J Tits Classification of algebraic semisimple groups In Algebraic Groups and Discontinuous Subgroups (Proc Sympos Pure Math., Boulder, Colo., 1965), pages 33-62, Providence, R.I., 1966, 1966 Amer Math Soc 120 V Varadarajan An Introduction to Harmonic Analysis on Semisimple Lie Groups, volume 16 of Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge, 1989 121 E Vinberg, editor Lie Groups and Lie Algebras, III, volume 41 of Encyclopaedia of Mathematical Sciences Springer-Verlag, Berlin, 1994 Structure of Lie groups and Lie algebras, A translation of Current problems in mathematics Fundamental directions Vol 41 (Russian), Akad Nauk SSSR, Vsesoyuz Inst Nauchn i Tekhn Inform., Moscow, 1990 [MR 91b:2200l], Translation by V Minachin [V V Minakhin], Translation edited by A L Onishchik and E B Vinberg 122 D Vogan Unitary Representations of Reductive Lie Groups, volume 118 of Annals of Mathematics Studies Princeton University Press, Princeton, NJ, 1987 123 N Wallach Real Reductive Groups I, volume 132 of Pure and Applied Mathematics Academic Press Inc., Boston, MA, 1988 124 A Weil L'integration dans les Groupes Topologiques et ses Applications Actual Sci Ind., no 869 Hermann et Cie., Paris, 1940 [This book has been republished by the author at Princeton, N J., 1941.] 125 A Weil Numbers of solutions of equations in finite fields Bull Amer Math Soc., 55:497-508, 1949 126 A Weil Algebras with involutions and the classical groups J Indian Math Soc (N.S.), 24:589-623 (1961), 1960 127 A Weil Sur certains groupes d'operateurs unitaires Acta Math., 111:143-211, 1964 128 A Weil Sur la formule de Siegel dans la tMorie des groupes classiques Acta Math., 113:1 87, 1965 References 445 129 H Weyl Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, i, ii and iii Math Zeitschri/t, 23:271309,24:328-395, 1925, 1926 130 J Wolf Complex homogeneous contact manifolds and quaternionic symmetric spaces J Math Mech., 14:1033-1047, 1965 131 J Wolf Spaces of Constant Curvature McGraw-Hill Book Co., New York, 1967 132 A Zelevinsky Induced representations of reductive p-adic groups II On irreducible representations of GL(n) Ann Sci Ecole Norm Sup (4), 13(2):165210, 1980 133 A Zelevinsky Representations of Finite Classical Groups, A Hopf algebra approach, volume 869 of Lecture Notes in Mathematics Springer-Verlag, Berlin, 1981 Index Abelian subspace, 238 absolute root system, 236, 237 Adams operations, 288 adjoint representation, 48, 49 admissible path, 95 affine Hecke algebra, 384 affine ring, 350 algebraic complexification, 185 algebraic cycle, 429 algebraic representation, 186 alternating map, 52, 290 analytic complexification, 182 anisotropic kernel, 220, 239 anisotropic torus, 423 antipodal map, 40 arclength, 95 Ascoli-Arzela Lemma, 19, 20, 22 atlas, 36 augmentation map, 392 automorphic cuspidal representation, 401 automorphic form, 399, 400 automorphic representation, 401 balanced map, 281 base point, 69 Bergman-Shilov boundary, 229 Bezout's Theorem, 430 bimodule, 281 Borel subgroup, 202 standard, 202 boundary Bergman-Shilov, 229 boundary component, 227, 229 boundary of a symmetric space, 224 bounded operator, 17 bracket Lie, 30 braid group, 189 braid relation, 189 branching rule, 339 Brauer's method of decomposing tensor products, 171 Bruhat decomposition, 256 Cartan decomposition, 76 Cartan involution, 212 Casimir element, 56, 400 Cauchy Identity, 347 Cauchy identity dual,355 Cayley numbers, 267 Cayley transform, 223, 224 center of a Lie algebra, 150 central character, 401 central orthogonal idempotents, 216 character, 6, 284 algebraic, 284 linear, 88 rational, 88 unipotent, 389 character, generalized, 15 character, virtual, 15 characteristic function of a measure, 325 Chow's Lemma, 429 Christoffel symbol, 97 circle Index Noneuclidean, 106 Circular Orthogonal Ensemble (COE), 329 Circular Symplectic Ensemble (CSE), 329 Circular Unitary Ensemble (CUE), 329 class function, 16 classical root systems, 130 Clifford algebra, 175 closed Lie subgroup, 29, 41 coalgebra, 306 commutator subgroup, 155 compact Lie algebra, 217 compact operator, 17 complementary minors, 297 complete reducibility, 65 complete symmetric polynomial, 284 complex analytic group, 86 complex and real representations, 58 complex Lie group, 86 complex manifold, 86 complexification, 60, 88 algebraic, 185 analytic, 182 torus, 88 concatenation of paths, 69 cone homogeneous, 230 self-dual, 230 conformal map, 105 conjugacy class indicator, 316 conjugate partition, 293 constant term, 401 contractible space, 69 contragredient representation, 9, 361 convolution, 16, 21 coordinate functions, 36 coordinate neighborhood, 36 coordinate ring, 350 coroot, 119 correlation, 329 covering map, 71 local triviality of, 71 pointed,71 trivial, 71 universal, 72 covering space morphism of, 71 Coxeter group, 189 447 cusp form, 399, 401 cuspidal representation, 397, 403, 404 CW-complex, 428 cycle type, 315 defined over a field, 364 derivation, 31 derived group, 155 diffeomorphism, 29, 36 differential of a Lie algebra homomorphism,45 discrete series, 397, 423 dominant weight, 144 dual Cauchy identity, 355 dual group, dual reductive pair, 269 dual symmetric spaces, 213 Dynkin diagram, 193 eigenspace, 17 eigenvalue, 17 eigenvector, 17 Einstein summation convention, 95 Eisenstein series, 397, 401 elementary symmetric polynomial, 284 ensemble, 327 equicontinuity, 19, 20 equivariant map, 10 Euclidean space, 117 evaluation map, 37 even partition, 365 exceptional group, 133 exceptional Jordan algebra, 231 exponential map, 31 extended Dynkin diagram, 258, 261, 262 extension of scalars, 58 exterior algebra, 352 faithful representation, 24 Ferrers' diagram, 293 fixed point, 107 isolated, 107 flag manifold, 92 folding, 265 Fourier inversion formula, Frobenius-Schur duality, 289, 385 Frobenius-Schur number, 362, 367 twisted, 367 fundamental dominant weight, 127, 144 448 Index fundamental group, 72 G-module, 275 Galois cohomology, 187 Gaussian binomial coefficient, 420, 436 Gaussian Orthogonal Ensemble (GOE), 328 Gaussian Symplectic Ensemble (GSE), 328 Gaussian Unitary Ensemble (GUE), 328 Gelfand pair, 376, 379 Gelfand subgroup, 376, 379 Gelfand-Graev representation, 382 Gelfand-Tsetlin pattern, 359 general linear group, 30 generalized character, 15 generator topological, 89 generic representation, 382 geodesic, 96, 98 geodesic coordinates, 99 geodesically complete, 102 germ, 36 graded algebra, 306 graded module, 306 Grassmannian, 432 Haar measure, left, right, half-integral weight, 179 Hamiltonian, 328 Hecke algebra, 376 affine, 384 Iwahori, 384 spherical, 384 Heine-Szego identity, 333 Hermitian form, Hermitian manifold, 221 Hermitian matrix, 76 positive definite, 76 Hermitian symmetric space, 221 highest-weight vector, 158, 169 highest-weight vectors, 157 Hilbert-Schmidt operator, 20 homogeneous space, 78 homomorphism Lie algebra, 44 homomorphism of G-modules, 10 homomorphism of Lie algebras, 54 homotopic, 69 homotopy, 69 path, 69 hook, 342 hook length formula, 342 Hopf algebra, 306, 307 horizontal strip, 344 hyperbolic space, 248 idempotents orthogonal central, 216 induced representation, 276 initial object, 50 inner form, 218 inner product, 7, 94 equivariant, invariant, integral curve, 46 integral manifold, 79 integral weight, 179 interlace, 357 intersection multiplicity, 429 intertwining integral, 403 intertwining operator, 10 support of, 279 invariant bilinear form, 55 invariants of a representation, 13 Inverse Function Theorem, 29 involution, 364, 376 Cartan,212 involution model, 365 involutory family, 79 irreducible character, irreducible representation, 6, 56 isolated fixed point, 107 isometric map, 105 Iwahori Hecke algebra, 384 Iwahori subgroup, 384 Iwasawa decomposition, 198 Jacobi identity, 30 Jacquet functor, 403 Jordan algebra, 230 Kawanaka and Matsuyama theorem, 367 Killing form, 55 Kronecker's Theorem, 89 Index Langlands correspondence, 422, 424 Laplace-Beltrami operator, 400 Lefschetz fixed-point formula, 107,437 Lefschetz number, 107 left invariant vector field, 41 length of a partition, 293 Levi subgroup, 386 Lie algebra, 30 center, 150 compact, 217 Lie algebra homomorphism, 44, 54 differential of, 45 Lie algebra representation, 48 Lie bracket, 30 Lie group, 41 reductive, 257 Lie subgroup, 29, 41 closed, 29, 41 Lie's theorem on solvable Lie algebras, 200 linear character, 88 linear equivalence of cycles, 429 Littlewood-Richardson rule, 342, 343 local coordinates, 36 local derivation, 38 local field, 397 local homomorphism, 74 local Langlands correspondence, 422 local subgroup, 82 local triviality, 71 locally closed subspace, 29 loop, 69 lowering operator, 365 magic square of Freudenthal, 232 manifold Hermitian, 221 Riemannian, 94 smooth,36 matrix coefficient, 348 matrix coefficient of a representation, 8, Metropolis algorithm, 385 model,365 model of a representation, 378, 381 module, 10 module of invariants, 67 monatomic representation, 410 monomial matrix, 382 449 morphism of covering maps, 71 multinomial coefficient, 345 multiplicity weight, 163 multiplicity-free representation, 340, 375 negative root, 136 nilpotent Lie algebra, 198 no small subgroups, 24 normalized induction, 402 observable, 328 octonions, 231, 267 one-parameter subgroup, 31 open Weyl chamber, 142 operator norm, 17 ordered partition, 404 orientation, 92 orthogonal group, 30 orthogonal representation, 361, 362 outer form, 218 parabolic induction, 397, 398 parabolic subgroup, 207, 209, 225, 258, 386 standard, 385, 386 partial order on root space, 129 partition, 293 conjugate, 293 even, 365 length,293 path,69 arclength, 95 concatenation of, 69 reparametrization, 69 reversal of, 70 trivial, 69 well-paced, 95 path of shortest length, 96 path of stationary length, 96 path-connected space, 69 path-homotopy, 69 path-lifting property, 71 permutation matrix, 301 Peter-Weyl Theorem, 7, 23-25, 169 Pieri's Formula, 343 Pieri's formula, 342, 344 Plancherel formula" 6, 450 Index pointed covering map, 71 pointed topological space, 69 polarization, 310 polynomial character, 373 Pontriagin duality, positive root, 127, 136 positive Weyl chamber, 129, 142 power-sum symmetric polynomial, 287 preatlas, 36 probability measure, 322 quadratic space, 30, 34 quantum group, 385 quasicharacter, modular, unitary, quasisplit group, 250 quaternionic representation, 361, 362 raising operator, 365 random matrix theory, 327 rank real,220 rank of a Lie group, 117 rational character, 88, 240 rational equivalence of cycles, 429 real form, 186 real representation, 361, 362 reduced norm, 236 reducible root system, 133, 193 reductive group, 236, 257 reflection, 117 regular element, 146, 254, 396 regular embedding, 258 regular function, 313 regular semisimple element, 413 relative root system, 145, 236, 237 relative Weyl group, 236, 245 reparametrization of a path, 69 representation, algebraic, 186 contragredient, 9, 361 cuspidal, 397 discrete series, 397 Lie algebra, 48 orthogonal, 361, 362 quaternionic, 361, 362 real, 361, 362 symplectic, 361, 362 trivial, 13 unitary, 24 restricted root system, 236, 237 Riemann zeta function, 329 Riemannian manifold, 94 Riemannian structure, 94 root, 118, 207 positive, 127, 136 simple, 127, 145 simple positive, 136 root folding, 265 root lattice, 127 root system absolute, 236, 237 reducible, 133, 193 relative, 236, 237 Schubert cell, 432 Schubert polynomial, 436 Schur orthogonality, 11, 12, 15 Schur polynomial, 297, 308 Schur's Lemma, 10 self-adjoint, 17 semisimple element, 396, 413 semisimple Lie algebra, 150 semisimple Lie group, 150 Siegel domain Type 1,233 Type 11,233 Siegel parabolic subgroup, 225 Siegel space, 221 Siegel upper half-space, 221 simple positive root, 145 simple reflection, 136, 206 simple root, 127, 136 simply-connected, 45 topological space, 70 simply-laced Dynkin diagram, 194 singular element, 146, 254 skew partition, 344 smooth manifold, 36 smooth map, 29, 36 smooth premanifold, 36 solvable Lie algebra, 198 Lie's theorem, 200 special linear group, 30 special orthogonal group, 30 special unitary group, 30 spin group, 78 Index spin representation, 175, 179 split group, 249 standard Borel subgroup, 202 standard parabolic subgroup, 258, 385, 386, 404 standard representation, 158 standard tableau, 341 stationary length, 96 Steinberg character, 389 strip horizontal, 344 vertical, 344 subgroup commutator, 155 submanifold, 29 subpermutation matrix, 378 summation convention, 95 support, 165 support of a permutation, 315 support of an intertwining operator, 279 symmetric algebra, 53, 352, 353 symmetric power, 52 symmetric space, 212 boundary, 224 dual,213 Hermitian, 221 irreducible, 215 reducible, 215 type I, 217 type II, 216 type III, 217 type IV, 216 symplectic group, 30 symplectic representation, 361, 362 tableau, 341 standard, 341 tangent bundle, 79 tangent space, 38 tangent vector, 38 tensor product, 50 terminal object, 50 Tits' system, 205, 206 Toeplitz matrix, 331 topological generator, 89 torus, 87, 413 anisotropic, 423 compact, 87 complex, 88 totally disconnected group, 24 451 triality, 265 triangulable, 107 trivial path, 69 trivial representation, 13 tube domain, 222 twisted Frobenius-Schur number, 367 type I symmetric spaces, 217 type II symmetric spaces, 216 type III symmetric spaces, 217 type IV symmetric spaces, 216 type of conjugacy class, 415 unimodular group, unipotent character, 389 unipotent matrix, 197 unipotent radical, 269 unipotent subgroup, 269 unitary group, 30 unitary representation, 7, 24 universal cover, 72 universal property, 50, 51 vector field, 39 left invariant, 41 subordinate to a family, 79 vertical strip, 344 virtual character, 15 weak convergence of measures, 322 weight, 125, 143, 162 dominant, 144 fundamental dominant, 127 half-integral, 179 integral, 179 weight diagram, 158 weight lattice, 127 weight multiplity, 163 well-paced, 95 Weyl chamber, 142 positive, 129 Weyl character formula, 165 Weyl dimension formula, 169 Weyl group, 91 relative, 236, 245 Weyl integration formula, 112 Young diagram, 293 Young tableau, 341 Zariski topology, 350 Graduate Texts in Mathematics (CillftiIfUe4/", pqe ii) 64 EDWARDS Fourier Series Vol I 2nd ed 65 WELLS Differential Analysis on Complex Manifolds 2nd ed 66 WATERHOUSE Introduction to Affine Group Schemes 67 SERRE Local Fields 68 WEIDMANN Linear Operators in Hilbert Spaces 69 LANG Cyclotomic Fields II 70 MASSEY Singular Homology Theory 71 FARKASIKRA Riemann Surfaces 2nd ed 72 STlll WELL Classical Topology and Combinatorial Group Theory 2nd ed 73 HUNGERFORD Algebra 74 DAVENPORT Multiplicative Number Theory 3rd ed 75 HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras 76 IITAKA Algebraic Geometry 77 HEeKE Lectures on the Theory of Algebraic Numbers 78 BURRIslSANKAPPANAVAR A Course in Universal Algebra 79 WALTERS An Introduction to Ergodic Theory 80 ROBINSON A Course in the Theory of Groups 2nd ed 81 FORSTER Lectures on Riemann Surfaces 82 BOTTITu Differential Forms in Algebraic Topology 83 WASHINGTON Introduction to Cyclotomic Fields 2nd ed 84 IRELANDlRoSEN A Classical Introduction to Modem Number Theory 2nd ed 85 EDWARDS Fourier Series Vol II 2nd ed 86 VAN LINT Introduction to Coding Theory 2nded 87 BROWN Cohomology of Groups 88 PIERCE Associative Algebras 89 LANG Introduction to Algebraic and Abelian Functions 2nd ed 90 BR0NDSTED An Introduction to Convex Polytopes 91 BEARDON On the Geometry of Discrete Groups 92 DIESTEL Sequences and Series in Banach Spaces 93 DuBROV]NfFoMENKO~OVll{OV Modem Geometry-Methods and Applications Part I 2nd ed 94 WARNER Foundations of Differentiable Manifolds and Lie Groups 95 SHIRYAEV Probability 2nd ed 96 CONWAY A Course in Functional Analysis 2nded 97 KOBUTZ Introduction to Elliptic Curves and Modular Forms 2nd ed 98 BROCKERfI'OM DIECK Representations of Compact Lie Groups 99 GROVEIBENSON Finite Reflection Groups 2nded 100 BERGICHRISTENSEN/RESSEL Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions 101 EDWARDS Galois Theory 102 VARADARAJAN Lie Groups, Lie Algebras and Their Representations 103 LANG Complex Analysis 3rd ed 104 DuBRoV]NfFo~o~ovll{ov.Modem Geometry-Methods and Applications Part II 105 LANG SL2(R) 106 SILVERMAN The Arithmetic of Elliptic Curves 107 OLVER Applications of Lie Groups to Differential Equations 2nd ed 108 RANGE Holomorphic Functions and Integral Representations in Several Complex Variables 109 LEHTo Univalent Functions and Teichmilller Spaces 110 LANG Algebraic Number Theory III HusEMOLLER Elliptic Curves 2nd ed 112 LANG Elliptic Functions 113 KARATZASISHREVE Brownian Motion and Stochastic Calculus 2nd ed 114 KOBUTZ A Course in Number Theory and Cryptography 2nd ed 115 BERGERlGoSTIAUX Differential Geometry: Manifolds, Curves, and Surfaces 116 KELLEy/SRINIVASAN Measure and Integral Vol I 117 J.-P SERRE Algebraic Groups and Class Fields 118 PEDERSEN Analysis Now 119 ROTMAN An Introduction to Algebraic Topology 120 ZIEMER Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation 121 LANG Cyclotomic Fields I and II Combined 2nd ed 122 REMMERT Theory of Complex Functions Readings in Mathematics 123 EBBINGHAus/HERMES et al Numbers Readings in Mathematics 124 DUBROVINlFoMENKOfNOVIKOV Modern Geometry-Methods and Applications Part III 125 BERENSTEIN/GAY Complex Variables: An Introduction 126 BOREL Linear Algebraic Groups 2nd ed 127 MASSEY A Basic Course in Algebraic Topology 128 RAUCH Partial Differential Equations 129 FULTONiHARRIS Representation Theory: A First Course Readings in Mathematics 130 DODSONIPOSTON Tensor Geometry 131 LAM A First Course in Noncommutative Rings 2nd ed 132 BEARDON Iteration of Rational Functions 133 HARRIs Algebraic Geometry: A First Course 134 ROMAN Coding and Information Theory 135 ROMAN Advanced Linear Algebra 136 ADKINS/WEINTRAUB Algebra: An Approach via Module Theory 137 AxLERiBOURDON/RAMEy Harmonic Function Theory 2nd ed 138 COHEN A Course in Computational Algebraic Number Theory 139 BREDON Topology and Geometry 140 AUBIN Optima and Equilibria An Introduction to Nonlinear Analysis 141 BECKERIWEISPFENNINGIKREDEL Grobner Bases A Computational Approach to Commutative Algebra 142 LANG Real and Functional Analysis 3rd ed 143 DooB Measure Theory 144 DENNIs/FARB Noncommutative Algebra 145 VICK Homology Theory An Introduction to Algebraic Topology 2nd ed 146 BRIDGES Computability: A Mathematical Sketchbook 147 ROSENBERG Algebraic K- Theory and Its Applications 148 ROTMAN An Introduction to the Theory of Groups 4th ed 149 RATCLIFFE Foundations of Hyperbolic Manifolds 150 EISENBUD Commutative Algebra with a View Toward Algebraic Geometry 151 SILVERMAN Advanced Topics in the Arithmetic of Elliptic Curves 152 ZIEGLER Lectures on Polytopes 153 FULTON Algebraic Topology: A First Course 154 BROWNIPEARCY An Introduction to Analysis 155 KAsSEL Quantum Groups 156 KECHRIS Classical Descriptive Set Theory 157 MALLIAVIN Integration and Probability 158 ROMAN Field Theory 159 CONWAY Functions of One Complex Variable II 160 LANG Differential and Riemannian Manifolds 161 BORWEIN/ERDELYI Polynomials and Polynomial Inequalities 162 ALPERINIBELL.Groupsand Representations 163 DIXONiMORTIMER Permutation Groups 164 NATHANSON Additive Number Theory: The Classical Bases 165 NATHANSON Additive Number Theory: Inverse Problems and the Geometry of Sumsets 166 SHARPE Differential Geometry: Cartan's Generalization ofK.lein's Erlangen Program 167 MORANDI Field and Galois Theory 168 EWALD Combinatorial Convexity and Algebraic Geometry 169 BHATIA Matrix Analysis 170 BREDON Sheaf Theory 2nd ed 171 PETERSEN Riemannian Geometry 172 REMMERT Classical Topics in Complex Function Theory 173 DIESTEL Graph Theory 2nd ed 174 BRIDGES Foundations of Real and Abstract Analysis 175 LICKORISH An Introduction to Knot Theory 176 LEE Riemannian Manifolds 177 NEWMAN Analytic Number Theory 178 CLARKEILEDYAEVISTERN/WOLENSKI Nonsmooth Analysis and Control Theory 179 DOUGLAS Banach Algebra Techniques in Operator Theory 2nd ed 180 SRIVASTAVA A Course on Borel Sets 181 KREss Numerical Analysis 182 WALTER Ordinary Differential Equations 183 MEGGINSON An Introduction to Banach Space Theory 184 BOLLOBAS Modem Graph Theory 185 CoxILrrrLFlO'SHEA Using Algebraic Geometry 186 RAMAKRIsHNANNALENZA Fourier Analysis on Number Fields 187 HARRIslMORRISON Moduli of Curves 188 GoWBLATI Lectures on the Hyperreals: An Introduction to Nonstandard Analysis 189 LAM Lectures on Modules and Rings 190 EsMONOFlMURTY Problems in Algebraic Number Theory 191 LANG Fundamentals of Differential Geometry 192 HIRSCHILACOMBE Elements of Functional Analysis 193 COHEN Advanced Topics in Computational Number Theory 194 ENGEIiNAGEL One-Parameter Sernigroups for Linear Evolution Equations 195 NATHANSON Elementary Methods in Number Theory 196 OSBORNE Basic Homological Algebra 197 EISENBUOIHARRIS The Geometry of Schemes 198 ROBERT A Course in p-adic Analysis 199 HEoENMALM/KoRENBLUMlZHU Theory of Bergman Spaces 200 BAO/CHERNISHEN An Introduction to Riemann-Finsler Geometry 201 HINDRy/SILVERMAN Diophantine Geometry: An Introduction 202 LEE Introduction to Topological Manifolds 203 SAGAN The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions 204 EscOFIER Galois Theory 205 FELIxlHALPERINITHOMAS Rational Homotopy Theory 2nd ed 206 MURTY Problems in Analytic Number Theory Readings in Mathematics 207 GoosnJROYLE Algebraic Graph Theory 208 CHENEY Analysis for Applied Mathematics 209 ARVESON A Short Course on Spectral Theory 210 ROSEN Number Theory in Function Fields 211 LANG Algebra Revised 3rd ed 212 MATOU~EK Lectures on Discrete Geometry 213 FRiTzsCHElGRAUERT From Holomorphic Functions to Complex Manifolds 214 lOST Partial Differential Equations 215 GoLOSCHMIDT Algebraic Functions and Projective Curves 216 D SERRE Matrices: Theory and Applications 217 MARKER Model Theory: An Introduction 218 LEE Introduction to Smooth Manifolds 219 MACLACHLANIREID The Arithmetic of Hyperbolic 3-Manifolds 220 NESTRUEV Smooth Manifolds and Observables 221 GRONBAUM Convex Polytopes 2nd ed 222 HALL Lie Groups, Lie Algebras, and Representations: An Elementary Introduction 223 VRETBLAD Fourier Analysis and Its Applications 224 WALSCHAP Metric Structures in Differential Geometry 225 BUMP: Lie Groups ... that [X, Yj E lli~ Lie Subgroups of GL( n, q 33 We see that Lie( G) is a Lie subalgebra of g[( n, q Thus, we are able to associate a Lie algebra with a Lie group Example 5.8 The Lie algebra of GL(n,... Classification (2000): 20xx, 22xx Library of Congress Cataloging in Publication Data Bump, Daniel Lie groups / Daniel Bump p cm (Graduate texts in mathematics: 225) Includes bibliographical references... G is a closed Lie subgroup of GL(n, C), and if X, Y E Lie( G), then [X, Yj E Lie( G) Proof It is evident that Lie( G) is mapped to itself under conjugation by elements of G Thus, Lie( G) contains