Other Title s in This Series 148 Vladimi r I. Piterbarg, Asymptotic methods i n the theory of Gaussia n processe s an d fields, 199 5 147 S . G . Gindiki n and L. R . Volevich , Mixe d problem s for partia l differentia l equation s with quasihomogeneous principal part, 199 5 146 L . Ya. Adrianora , Introduction to linea r systems of differential equations , 199 5 145 A . N. Andriano v an d V, G. Zhuravlev , Modular forms an d Hecke operators, 199 5 144 O . V . Troshkin, Nontraditional methods in mathematical hydrodynamics , 199 5 143 V . A. Malyshe v an d R. A. Minlos, Linea r infinite-particle operators , 199 5 142 N . V . Krylov, Introduction to th e theory of diffusio n processes , 199 5 141 A . A. Davydov , Qualitative theory of control systems , 199 4 140 Aizfl c I. Volpert , Vitaly A. Volpert , an d Vladimir A. Volpert , Traveling wave solutions of paraboli c systems, 199 4 139 I . V . Skrypnlk, Methods for analysis of nonlinear elliptic boundary value problems, 199 4 138 Yu . E Razmyslov , Identities of algebra s and thei r representations, 199 4 137 F . I. Karpelevich an d A. Ya. Kreinin , Heavy traffi c limits for multiphase queues, 199 4 136 Masayosh i Miyanishi, Algebraic geometry, 199 4 135 Masar u Takeuchi, Modern spherical functions , 199 4 134 V . V. Prasolov, Problems and theorem s in linear algebra, 199 4 133 P . I. Naumki n an d I. A. Shishmarev, Nonlinear nonlocal equations in the theory of waves, 199 4 132 Hajim e Urakawa, Calculus of variation s and harmonic maps, 199 3 131 V . V. Sharko, Functions on manifolds: Algebrai c and topological aspects , 199 3 130 V . V. Vershinin, Cobordisms and spectral sequences, 199 3 129 Mitsu o Morimoto, An introductio n to Sato's hyperfunctions, 199 3 125 V . P. Orevkov, Complexity of proofs and their transformations in axiomatic theories , 199 3 127 F . L. Zak, Tangents and secants of algebrai c varieties, 199 3 126 M . L. Agranovsku , Invariant functio n space s on homogeneou s manifolds of Li e group s and applications, 199 3 125 Masayosh i Nagata, Theory of commutative fields, 199 3 124 Masahis a Adachi, Embeddings and immersions, 199 3 123 M . A. Akivis and B. A. Rosenfeld , felie Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hoa, Theory of entire and meromorphic functions: Deficient an d asymptotic values and singular directions, 199 3 121 I . B. Fesenk o and S. V . Vostokov, Loca l fields and their extensions: A constructiv e approach , 199 3 120 Takeyuk i Hida an d Masuyuki Hitsoda, Gaussian processes, 199 3 119 M . V . Karasev and V . P. Maslov, Nonlinear Poisson brackets. Geometry and quantization, 199 3 118 Kenkkh i Iwasawa , Algebraic functions, 199 3 117 Bori s Zilber, Uncountably categorical theories , 199 3 116 G . M. Fel'dman , Arithmetic of probabOit y distributions, and characterization problems on abelia n groups, 199 3 115 Nikola i V . IV&DOY , Subgroups of Teichmuller modular groups, 199 2 114 Seiz d ltd, Diffusion equations , 199 2 113 Michai l Zhitomirskil , Typica l singularitie s of differentia l i-form s an d Pfaffia n equations , 199 2 112 S . A. Lomov , Introduction to th e general theory of singular perturbations, 199 2 111 Simo n Gindikin , Tube domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introductio n to complex analysi s Part II. Functions of severa l variables, 199 2 109 Isa o Miyadera, Nonlinear semigroups, 199 2 108 Take o Yokonmna, Tensor spaces and exterior algebra, 199 2 107 B . M. Makarov , M. G . Goluzina , A. A. Lodkin, an d A. N. Podkorytov , Selecte d problem s in rea l analysis, 199 2 (Continued in the back of this publication) This page intentionally left blank 10.1090/mmono/040 TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME 40 D. P. Zelobenko Compact Lie Groups and Their Representations ||f^l]| American Mathematical Society KOMF1AKTHME TPynribl JI M H M X nPE^CTABJIEHHfl &. n. )KEJIOEEHK O M3joiaTejibCTBO ^HayKa ^ nnaBHaa peaaKtm a <|)H3HKO-MaTeMaTHHecKOH JiMTepaTyp w MocKBa 197 0 Translated fro m th e Russia n b y Israel Progra m fo r Scientifi c Translation s 2000 Mathematics Subject Classification. Primar y 22-02 ; Secondary 17-02 . Library o f Congres s Cataloging-in-PubHcatio n Dat a Zhelobenko, Dmitri ! Petrovich. Compact Li e groups an d thei r representations . (Translations of mathematica l monographs , v . 40) Translation o f Kompaktny e grupp y L i i ikh predstavleniia . Bibliography: p . 1. Li e groups. 2 . Representation s o f groups . I . Title. II . Series. QA387.Z443I3 5I2'.5 5 73-1718 5 ISBN 0-8218-1590- 3 ISSN 0065-928 2 Copying an d reprinting . Individua l reader s of thi s publication , an d nonprofi t librarie s actin g fo r them , are permitted to make fai r use of th e material, such a s to copy a chapter for use in teaching or research. Permission i s granted t o quot e brie f passages fro m thi s publicatio n i n reviews , provide d th e customar y acknowledgment o f th e source is given. Republication, systemati c copying, or multiple reproductio n o f an y material i n this publication (in - cluding abstracts) i s permitted only unde r license from th e American Mathematica l Society . Request s fo r such permissio n shoul d b e addressed t o th e Assistant t o the Publisher , America n Mathematica l Society , P.O. Box 6248, Providence, Rhode Island 02940-6248. Request s can also be made by e-mail t o reprint- permission&aath•ams.org. © Copyrigh t 197 3 b y th e America n Mathematica l Society . A H rights reserved . Second printin g with corrections , 197 8 The America n Mathematica l Societ y retain s al l rights except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America. @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit the AMS home page at URL : http://www.aias.org/ 12 1 1 1 0 9 8 7 6 5 0 6 0 5 04 03 02 0 1 TABLE O F CONTENT S PREFACE 1 PART I. INTRODUCTION 5 CHAPTER I . TOPOLOGICA L GROUPS. LIE GROUPS 6 1. Definitio n o f a group 6 2. Topologica l group s 8 3. Parametri c groups and Li e groups 1 2 4. Li e theory 1 4 5. Locall y isomorphic Li e groups 1 8 6. Invarian t forms on Li e groups 2 1 7. Metrics . Haa r measure 2 3 CHAPTER II . LINEA R GROUPS 2 6 8. Genera l linear group. Exponential 2 6 9. Genera l linea r group. Fundamental decomposition s 2 7 10. Linea r groups associated wit h second-order form s 3 0 11. Quaternion s 3 3 12. Simpl y connected group s 3 6 13. Complexificatio n 3 9 14. Transformation s i n the class of tensors 4 0 CHAPTER III . FUNDAMENTA L PROBLEMS OF REPRESENTATIO N THEORY 4 4 15. Function s o n a homogeneous spac e 4 4 16. Terminolog y o f representatio n theory 4 6 17. Reductio n o f the fundamental proble m 5 0 18. Elementar y harmonic s 5 1 19. Algebra s and groups of an equation 5 4 20. Schur' s Lemm a 5 6 21. Burnside' s Theorem 5 9 22. Grou p algebras and thei r representations 6 2 23. Statemen t o f the fundamental problem s 6 3 v VI TABLE O F CONTENT S PART II. ELEMENTARY THEOR Y 6 7 CHAPTER IV . COMPAC T LIE GROUPS. GLOBA L THEORE M 6 8 24. Definitio n o f compact groups 6 8 25. Statemen t of the global theore m 7 0 26. The averaging technique 7 1 27. Orthogonalit y propert y 7 3 28. Approximatio n lemma for a linea r group G 7 4 29. Fourie r series on a linear group G 7 4 30. Completio n of the proof for a linear group G 7 6 3L Completio n of the proof in the general case 7 8 32. Harmoni c analysis on a homogeneous manifold 7 9 33. Character s 8 1 34. Representatio n theory of finite groups 8 2 35. Universalit y of the group U(n) 8 4 CHAPTER V . TH E INFINITESIMA L METHO D I N REPRESENTATIO N THEOR Y 8 7 36. Differentia l o f a representation 8 7 37. Irreducibl e representations of the group SU(2) 9 0 38. Matri x elements of the group SU(2) 9 6 39. Som e groups related to SU(2) 10 1 40. Problem s arising in applications of the infinitesimal metho d 10 4 CHAPTER VI. ANALYTI C CONTINUATIO N 10 9 41. Genera l principle of analytic continuation 10 9 42. Hypercompac t Li e groups. Weyl's "unitarian trick" 11 3 43. Bicomple x Li e groups and Lie algebras 11 5 44. Complexificatio n o f U(n). Weights and roots 11 8 45. Mode l of irreducible representations of SU(3) 12 1 CHAPTER VII.