Conference Board of the Mathematical Sciences · Contents 1. Introduction 1 2. Representations of SU(2) 4 3. The general theory of linear representations of compact groups 9 4. Lie

Conference Boar d o f th e Mathematica l Science s

REGIONAL CONFERENCE SERIES IN MA THEMA TICS

supported b y th e

National Scienc e Foundatio n

Number 4 2

SPECIAL FUNCTIONS AND LINEAR REPRESENTATION S O F LIE GROUPS

by

JEAN DIEUDONNfe

Published fo r

Conference Boar d o f th e Mathematica l Science s

by th e

American Mathematica l Societ y

Providence, Rhod e Islan d

http://dx.doi.org/10.1090/cbms/042

Expository Lecture s

from th e CBM S Regional Conferenc e

held a t Eas t Carolin a Universit y

March 5-9 , 197 9

1980 Mathematics Subject Classification. Primary 22E45,43A90, 33A75; Secondary 20H10, 22C05, 22D10, 22D30, 22E60, 43A85, 46J05, 33A45.

Library o f Congres s Cataloging in Publication Dat a

Dieudonne, Jean Alexandre , 1906 -Special function s an d linea r representation s o f Li e groups . (Regional conferenc e serie s in mathematics ; no. 42) "Expository lecture s fro m th e CBM S regional conferenc e hel d a t Eas t Carolin a

University, Marc h 5-9 , 1979. " Bibliography: p . 1. Li e groups . 2 . Functions , Special . 3 . Representation s o f groups . I . Confer -

ence Boar d o f th e Mathematica l Sciences . II . Title . III . Series . QA1.R33 no . 42 [QA387 ] 510'.8 s [512'.55 ] ISBN 0-8218-1692-6 79-2218 0

Copyright © 198 0 b y th e America n Mathematica l Societ y

Printed i n th e Unite d State s o f Americ a

All rights reserved except those granted to the United States Government

This boo k ma y no t b e reproduce d i n an y for m withou t permissio n o f th e publishers .

Contents

1. Introductio n 1 2. Representation s of SU(2) 4 3. Th e general theory o f linear representations of compact groups 9 4. Li e theor y o f representation s o f compac t connecte d Li e groups. . 1 5 5. Induce d representations of compact groups 1 7 6. Spherica l functions o n compact groups 1 8 7. Examples ; spherical harmonics , 2 1 8. Th e general theory of spherical functions 2 5 9. Fourie r and Plancherel transforms 3 0 10. Extensio n o f th e Planchere l transfor m 3 4 11. Th e subtletie s o f harmonic analysi s 3 7

12. Differentia l propertie s o f spherica l function s o n Li e groups 4 0

13. Spherica l function s o n semisimpl e Li e groups 4 2

14. Mor e o n SL(2 , R) 4 5 15. Automorphi c function s 5 3 16. Group s o f isometrie s an d Besse l functions 5 5 17. Othe r specia l function s 5 8 References 5 9

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References

For th e proof s o f th e theorem s o n linea r representation s o f compac t groups , and thei r explicit determinatio n fo r compac t Li e groups , see

J. Dieudonne , Treatise on analysis, Vol. 5 , Academic Press , New York, 1977 , Chapter 21. The result s fro m spectra l theor y use d i n tha t chapte r ar e prove d i n Chapte r 15 . Th e

elementary notion s o n induce d representations , an d o n Gelfan d pair s an d spherica l functions , are describe d an d prove d i n Chapter 22 of th e sam e Treatis e (Vol . 6) , Academi c Press , 1978 ; the mai n result s o n commutativ e harmoni c analysi s ar e als o containe d i n tha t chapter .

A complete descriptio n o f th e irreducibl e representation s o f th e group s SU(2) , SL(2, R), M(2) (group o f motion s o f th e euclidea n plane) , together wit h a large numbe r o f formula s connecting the m wit h specia l functions , ar e t o b e foun d i n th e encyclopaedi c treatise :

N. J. Vilenkin , Special functions and the theory of group representations, Trans . Math. Monographs, Vol . 22, Amer. Math . Soc , Providence , RI , 1968 .

The mor e advance d theor y o f spherica l function s o n semisimpl e noncompac t Li e group s is partly treate d i n a n articl e o f R . Gangoll i i n Symmetric spaces, Short course s presente d a t Washington University , ed . by W . Boothby an d G . Weiss, Dekker, New York , 1972 .

Another articl e b y th e sam e autho r i n th e sam e boo k i s devoted t o th e relation s be-tween grou p representation s an d automorphi c functions , a s well as

I. Gelfand , M . Graev an d I . Pyatetskii-Shapiro, Representation theory and automorphic

functions, Saunders , Philadelphia, PA, 1969 .

For mor e informatio n o n sem i simple Li e groups , see also S. Helgason, Differential geometry and symmetric spaces, Academic Press , New York ,

1962. Finally, th e reader s interested i n th e relation s betwee n commutativ e harmoni c analysi s

and arithmeti c ma y consul t J. P . Kahane an d R . Salem , Ensembles parfaits et series trigonometriques, Hermann ,

Paris, 1963 . Y. Meyer, Algebraic numbers and harmonic analysis, North-Holland, Amsterdam , 1972 .

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