Njvilenkin Special Functions and The

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10.1090/mmono/022 •.TRANSLATIONS 'O.'f. MATHiMATiCA W MONOGRAPH S VOLUME 22

N.J.Vilenkin

Special Functions and the Theory of Group Representations

If American Mathematical Society

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~~Translated from the Russian by V.N.Singh~~

Library of Congress Card No. 68—19438 Copyright © 196 8 by the American Mathematical Society Printed in the United States of America Reprinted wit h correction s 1988 . All Rights Reserved No portion of this f>ook may be reproduced without the written permission of the publisher

The pape r use d i n this book i s acid-free an d fall s within th e guideline s established t o ensure permanenc e an d durability - © Information o n Copying and Reprintin g can b e foun d a t th e bac k o f this volume . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 0 4 03 0 2 01 0 0 PREFACE

In this boo k w e stud y th e theor y o f specia l function s fro m th e group-theoreti c point o f view . A t firs t sight , th e theor y o f specia l function s appear s a s a chaoti c collection o f formulas : apar t fro m th e fac t tha t ther e exist s a n immens e aggregat e of th e specia l function s themselves , fo r eac h o f the m ther e are , a t present , al l sorts o f differentia l equations , integra l representations , recurrenc e formulas , com - position theorem s an d s o on . T o establish som e kin d o f orde r i n thi s chao s o f formulas seem s t o b e a completel y hopeles s task . However, th e developmen t o f th e theor y o f grou p representations ha s no w made i t possibl e t o comprehen d th e theor y o f th e mos t importan t classe s o f spe - cial function s fro m a singl e poin t o f view . W e note tha t th e appraisa l o f th e impor - tance o f individua l classe s o f specia l function s ha s change d greatl y durin g th e last hundre d years. I n th e middl e an d th e secon d hal f o f th e nineteent h century , elliptic an d relate d function s wer e regarde d a s th e mos t interesting . However , a s F. Klei n (se e ["^]) observe d i n on e o f hi s lectures , ther e exist s anothe r class o f specia l functions , a t leas t a s importan t i n vie w o f it s numerou s applications t o astronom y an d mathematica l physics , namel y thos e associate d with th e hypergeometri c function . Th e developmen t o f mathematic s ha s no w con - firmed Klein' s opinion ; th e hypergeometri c functio n an d it s variou s specia l an d degenerate cases , th e function s o f Besse l an d Legendre , th e orthogona l polyno - mials o f Jacobi , Cebysev , Laguerr e an d Hermit e etc . pla y a bi g rol e i n differen t branches o f mathematic s an d it s applications . Thi s clas s o f specia l function s is ofte n calle d th e specia l function s o f mathematica l physics . Thi s i s precisel y the class o f function s whic h lend s itsel f t o a group-theoretic treatment . The connectio n betwee n specia l function s an d grou p representations wa s first discovere d b y E . Carta n (se e [244]) . (However , a connection betwee n th e

111 IV PREFACE theory o f specia l function s an d th e theor y o f invariants , whic h i s on e o f th e aspects o f th e theor y o f grou p representations, wa s establishe d eve n earlier) . Th e application o f theor y o f representation s t o quantu m mechanic s playe d a significan t part i n th e investigatio n par t i n th e investigatio n o f thes e connections . Further wor k i n thi s fiel d wa s stimulate d b y th e researche s o f I . M . Ge l 'fan d and M . A . Nair n ark an d thei r student s an d collaborator s i n th e fiel d o f infinite - dimensional representation s o f groups . I n th e cours e o f thes e researches , a con - nection wa s establishe d betwee n th e theor y o f grou p representations an d automor - phic functions , th e theor y o f specia l function s ove r finit e field s wa s constructed , special function s wer e studie d i n homogeneous domains , an d so on . The ai m o f thi s boo k i s a systemati c accoun t o f th e theor y o f specia l function s from th e group-theoreti c poin t o f view . W e have restricte d ourselve s her e t o th e study o f classica l specia l function s an d thu s o f th e simples t groups . The boo k consists o f eleven chapters . I n th e firs t chapte r basi c concept s and fact s o f th e theor y o f transformatio n group s an d group representations ar e pre - sented. In th e secon d chapter , tw o mode l example s ar e analyzed , namel y th e additiv e group o f rea l number s an d th e grou p of rotation s o f th e circle . Thes e group s lead , respectively, t o th e exponentia l an d trigonometri c functions . I n this chapte r cer - tain fact s o f classica l harmoni c analysis , th e theor y o f Fourie r serie s an d integrals in rea l an d comple x domains , ar e also presented . The thir d chapter i s devote d t o representations o f th e grou p o f rotation s o f the three-dimensiona l euclidea n spac e an d th e locall y isomorphi c grou p SU(2) o f unitary unimodula r matrice s o f th e secon d order . Th e matri x element s o f irredu - cible unitar y representation s o f thes e group s ar e expresse d b y mean s o f Legendr e and Jacob i polynomials . Th e genera l propertie s o f th e matri x element s lea d t o various relation s fo r thes e polynomial s (orthogonalit y an d completenes s relations , recurrence formulas , compositio n theorem , etc.) . A t th e en d o f th e chapte r w e consider th e discrete analogu e o f Jacob i polynomials , namel y th e Clebsch - Gordon coefficients . In th e fourt h chapte r representation s o f th e grou p o f motion s o f th e euclidea n plane ar e studied . Th e matri x element s o f representation s o f thi s grou p ar e expressed b y mean s o f th e Bes s el functions , whic h allow s on e t o derive a numbe r of propertie s o f Besse l function s fro m group-theoreti c considerations . Furthe r properties o f Besse l function s an d th e closel y relate d Hanke l an d Macdonal d PREFACE v functions ar e studie d i n Chapte r V , devote d t o representations o f th e grou p o f motions o f th e pseudo-euclidea n plane . I n thi s chapte r w e deriv e a number o f integrals involvin g th e inde x o f cylindrica l functions . A t th e beginnin g o f th e chapter th e grou p o f linea r transformation s o f th e straigh t lin e an d th e theor y o f r-functions ar e considered . The sixt h an d sevent h chapter s ar e devote d t o representations o f th e grou p of rea l unimodula r matrice s o f th e secon d order . Representation s o f thi s grou p are connecte d wit h certai n classe s o f specia l functions . I n th e sixt h chapte r th e conical function s P^w + • \(ch 6) ar e studied . I n th e sevent h chapte r w e conside r the realizatio n o f representation s i n th e for m o f integra l operators , wit h th e hyper - geometric functio n a s th e kernel . Thi s lead s t o a numbe r o f propertie s o f th e hypergeometric function , som e o f the m well know n (th e Mellin-Barne s integra l representation), other s new . I n Chapte r VII I we similarl y stud y th e confluen t hypergeometric functio n and , i n greater detail , th e closel y relate d Whittake r func - tions. I t is show n tha t th e theor y o f Whittake r function s i s base d o n th e stud y o f representations o f th e grou p of thir d orde r triangula r matrices . Startin g fro m this , we deriv e variou s propertie s o f Whittake r functions , e.g . integra l representations , recurrence formulas , an d continual compositio n theorems . I n th e sam e chapte r Laguerre polynomial s ar e considered , an d a compositio n formul a i s derive d fo r them. All th e group s considere d i n Chapter s II—VII I have a ver y simpl e structure . More complicate d group s ar e considere d i n Chapter s IX—XI ; the y ar e th e group s of motion s o f th e n-dimensiona l sphere , o f th e rc-dimensional Lobacevski r spac e and o f th e n -dimensional euclidea n space . Here , however,w e d o no t conside r al l irreducible representation s o f thes e groups , bu t onl y th e so-calle d representation s of clas s 1 , an d w e d o not conside r al l matri x element s o f representations , bu t only th e matri x element s o f "nul l column*' . Thi s is foun d t o b e sufficien t fo r co n structing th e theor y o f Gegenbaue r polynomial s an d harmoni c polynomials , an d also fo r obtainin g th e decompositio n o f function s o n th e rc-dimensional spher e an d hyperboloid. I n Chapte r X I furthe r propertie s o f Besse l function s ar e derived . Unfortunately th e siz e o f th e boo k di d no t permi t goin g int o certai n question s connected wit h th e theor y o f representation s an d specia l functions . Thu s th e group o f motion s o f th e n -dimensional pseudo-euclidea n spac e ha s no t bee n con - sidered. Hardl y an y ligh t ha s bee n throw n upo n asymptoti c propertie s o f specia l functions. However , th e integra l representation s o f thes e functions , presente d i n VI PREFACE

the book , allo w on e t o obtain thei r asymptoti c expansion s b y th e usua l method s (see [6] , [62]) . The autho r expresses hi s profoun d gratitud e t o I . M . Gel fand , whos e advic e and instruction s wer e mad e us e o f throughou t th e wor k o n th e book , an d whos e part i n creating th e boo k i s difficul t t o overrate . Th e autho r als o thank s M . A . Naimark, M . I. Graev , I . I . Pjateckfi-Sapiro , F . A . Berezin , F . I . Karpelevic , S. G . Gindikin , an d D. P . Zelobenko , wit h who m questions o f th e theor y o f grou p representations an d th e theor y o f specia l function s wer e ofte n discussed . Numer - ous suggestion s o f M . I. Grae v influence d th e treatmen t o f som e question s an d th e final arrangemen t o f th e material . The conversation s wit h Ja . A. Smorodinsk n an d his students , o n th e appli - cations o f th e theor y o f grou p representations an d integra l transformation s t o physics, wer e highl y usefu l fo r th e author . V. V . Cukerma n too k upo n himsel f th e difficul t tas k o f verifyin g th e formulas . Great attentio n wa s pai d t o th e manuscript , a t al l stage s o f it s passage , b y A . Z . Ryvkin. S . A . Vilenkin a wen t throug h severa l variant s o f th e manuscript . T o them I express m y gratitude fo r thei r help, whic h considerabl y facilitate d prepara - tion o f th e manuscrip t fo r printing .

~~/V. J a. Vilenkin TABLE O F CONTENT S Page Introduction I~~

Chapter I . Grou p representations 7 §1. Basi c concept s o f th e theor y o f representation s 7 §2. Transformatio n group s an d thei r representation s 2 3 §3- Invarian t operator s an d th e theor y o f representation s 3 6 §4. Representation s o f compac t group s 4 2 Supplement t o Chapte r I . Som e fact s concernin g linea r space s 5 9 Chapter II . Th e additiv e grou p o f rea l number s an d th e exponentia l func - tion. Fourie r serie s an d integral s 7 0 §1. Th e exponentia l an d trigonometrica l function s 7 0 §2. Fourie r serie s 7 7 §3- Th e Fourie r integra l 8 0 §4. Fourie r transformatio n i n th e comple x domai n 8 9 Chapter III . Th e grou p o f secon d orde r unitar y matrice s an d th e polyno - mials o f Legendr e an d Jacob i 9 7 §1. Th e grou p SU(2) 9 7 §2. Irreducibl e unitar y representation s T.(u) 10 8 §3- Matri x element s o f th e representation s 7 ] (g). Th e polynomial s of Legendr e an d Jacob i 11 5 §4. Functiona l relation s fo r th e function s P l (z) 12 9 mn §5- Generatin g function s fo r P l (z) 14 7 §6. Expansio n o f function s o n th e grou p SU(2) 15 8 §7. Th e character s o f th e representation s T(u) 17 0 §8. Th e Clebsch-Gorda n coefficient s 17 4 Chapter IV . Representation s o f th e grou p o f motion s o f th e plan e an d Bessel function s 19 5 §1. Th e grou pA T (2) 19 5 §2. Irreducibl e unitar y representation s o f th e grou p M(2) 20 0 §3- Matri x element s o f representation s 71(g ) an d Besse l function s 20 5 §4. Functiona l relation s fo r Besse l function s 20 8

vii viii CONTENT S §5- Decomposition s o f th e representation s o f th e grou p Mil) an d th e Fourier-Bessel transformatio n 21 4 §6. Produc t o f representation s 22 4 §7. Besse l function s an d th e function s P l (x) 22 8 mn Chapter V . Representation s o f th e grou p o f motion s o f th e pseudo - euclidean plan e an d th e function s o f Besse l an d Macdonal d 23 1 §1. Representation s o f th e grou p o f linea r transformation s o f th e straight lin e an d T-function s 23 1 §2. Th e grou p IUH(2) o f motion s o f th e pseudo-euclidea n plan e 24 8 §3. Representation s o f MHU) 25 4 §4. Recurrenc e formula s an d th e differentia l equatio n fo r th e Macdonald an d Hanke l function s 26 3 §5- Functiona l relation s fo r th e Hanke l an d Macdonald function s 26 8 §6. Decompositio n o f th e quasi-regula r representatio n o f th e grou p MH(2) 28 1 Chapter VI . Representation s o f th e grou p QU (2) o f th e unimoduia r quasi - unitary matrice s o f th e secon d orde r an d th e function s o f Legendre an d Jacob i 28 8 §1. Th e grou p QU(2) 28 8 §2. Irreducibl e representation s o f QU (2) 29 5 §3. Matri x element s o f Ty(g) 30 7 §4. Functiona l relation s fo r W (chr) 32 1 mn §5- Decompositio n o f th e regula r representatio n o f th e grou p QU(2) 33 1 Chapter VII . Representation s o f th e grou p o f rea l unimoduia r matrice s and th e hypergeometri c functio n 34 4 §1. Th e hypergeometri c functio n 34 4

§2. Th e grou p SL(2 y R) o f rea l unimoduia r matrice s o f th e secon d order 35 0 §3- Irreducibl e representation s o f th e grou p SL(2 , R) 35 4 §4. Calculatio n o f th e kernel s o f Ryig) 36 4 §5- Recurrenc e formula s fo r th e hypergeometri c function . Th e hyper - geometric equatio n 37 3 §6. Integra l representation s an d th e compositio n formul a fo r th e hypergeometric functio n 38 1 §7. Representation s o f th e grou p o f rea l matrice s o f th e secon d orde r and Hanke l function s 39 5 CONTENTS i x Chapter VIII . Representation s o f th e grou p o f thir d orde r triangula r matrices an d th e Whittake r function s 40 0 §1. Whittake r function s an d th e confluen t hypergeometri c functio n 40 0 §2. Th e grou p o f thir d orde r triangula r matrice s an d it s representa - tions 40 2 §3» Functiona l relation s fo r Whittake r function s 41 1 §4. Integral s connecte d wit h th e Whittake r function s 41 9 §5. Th e Laguerr e polynomial s an d representations o f th e grou p o f complex thir d orde r triangula r matrice s 43 0 Chapter IX . Th e grou p o f rotation s o f /z-dimensiona l euclidea n spac e and Gegenbaue r function s 43 5 §1. Th e grou p SOU) 43 5 §2. Representation s o f clas s 1 of th e grou p S0(n) an d harmoni c polynomials 44 0 §3. Zona l spherica l function s o f representation s T nl(g) an d Gegen - bauer polynomial s 45 7 §4. Matri x element s o f th e nul l colum n 46 9 §5. Spherica l function s an d th e Laplac e operator . Polyspherica l functions 49 3 Chapter X . Representation s o f th e grou p o f hyperboli c rotation s o f ^-dimensional spac e an d Legendr e function s 50 3 §1. Pseudo-euclidea n spac e an d hyperboli c rotation s 50 3 §2. Representation s o f class 1 of th e grou p SH(n) 50 9 §3. Zona l an d associated spherica l function s o f th e representation s of clas s 1 of grou p SH(n) 51 9 §4. Decompositio n o f representation s o f th e grou p SH(n) an d th e Mehler-Fock transformatio n 53 0 §5- A Laplace operato r o n th e hyperboloid . Polyspherica l an d hori - spherical function s o n th e hyperboloi d 54 1 Chapter XI . Th e grou p o f motion s o f th e rc-dimensional euclidea n spac e and Besse l function s 54 7 §1. Th e grou p M(n) 54 7 §2. Irreducibl e representation s o f clas s 1 of th e grou p M{n) 54 9 §3- Zona l an d associated spherica l function s o f representation s o f class 1 o f th e grou p M(n) 55 1 §4. Limitin g proces s fo r th e spatia l dimension . Hermit e polynomial s 55 9 x CONTENT S

~~Remarks an d bibliographica l note s 56 9 Bibliography 57 5 Subject Inde x 60 1 Notation Inde x 61 1~~

~~REMARKS AND BIBLIOGRAPHICAL NOTES~~

~~Chapter I~~

The theor y o f representations o f linea r transformatio n group s wa s originate d by G . Frobeniu s [ 55]. The n Burnsid e an d I . Schu r foun d a substantiall y simple r approach b y bringin g th e representatio n matri x itself int o the foregroun d instead of it s trace, namel y th e Frobeniu s character . Th e infinitesima l approac h t o representa - tion theor y wa s develope d b y Is . Ca r tan [69], Th e infinitesima l theor y o f Li e groups i s expounde d i n th e book s o f Pontrjagi n [ 42], Helgaso n [ 56], Cebotare v [57], Chevalle y [ 6 0], Eisenhar t [ 6 i] an d Kowalewski [78] , The genera l theor y o f continuou s group s was , i n th e main , originate d b y L. S . Pontrjagin [ 42]. Invarian t integratio n o n Lie group s wa s introduce d b y Adol f Hurwitz i n 189 7 fo r proving theorem s o n invariants . I n 192 4 I . Schu r [290] applie d this proces s t o representation s o f compac t groups , i n particular , th e rea l ortho - gonal group . Th e existenc e o f invarian t measur e o n a n arbitrar y locally compac t group was prove d b y Haa r [252]. Induced representations o f group s wer e utilize d i n a serie s o f paper s o f I . M . Gel'fand an d M . A . Nair n ark [17]. Concernin g th e genera l theory , se e Macke y [277], Mautne r [ 280] an d Bruha t [ 24<>]. The theor y o f spherica l function s o n group s goe s bac k t o Ca r tan [2441; se e Weyl [ 299] also . Th e subsequen t developmen t o f th e theor y i s give n b y Gel'fand , Naimark an d Berezin [17] , [105], [129] , Vilenki n [n*], Godement [251] , Harish - Chandra [259], Mautne r [281], Yoshizaw a [3 0 3] an d man y othe r authors . The result s se t fort h her e have , i n th e main , bee n obtaine d b y I . Schur , F . Peter an d H . Weyl . Th e fundamenta l resul t o n completenes s o f th e syste m o f irreducible representation s o f a compac t Li e grou p was prove d b y Peter an d Wey l [284], fo r infinite-dimensiona l representations , se e Ann a Hurevitsch [ 264]. These result s wer e late r carrie d over , i n varyin g measure , t o locall y compac t l7 2 groups; se e Rudi n [90], Zelobenk o [ *]9 Macke y [197,198], Nalmar k [ <>4], Harish- Chandra [ 2 57-2 58] an d Mautne r [2 7 8]. Problem s simila r t o thos e involve d her e were studie d b y M . G . Krei n [192,1 9 3],

~~569 570 REMARKS AN D BIBLIOGRAPHICAL NOTE S~~

Supplement t o Chapte r I . Concernin g th e topic s touche d upo n here , se e th e books o f I . M . Gel'fand, M . I . Grae v an d th e autho r [14-15], an d th e pape r b y M . A . Naimark an d S . V . Fomi n [ 2°9].

~~Chapter I I~~

The theor y o f trigonometrica l serie s goe s bac k t o Bernoull i an d Euler . Thes e series wer e systernatricall y applie d b y Fourie r an d Cauch y i n th e 1820* s fo r solv - ing th e problem s o f mathematica l physics . Concernin g th e moder n theor y o f trigo - nometrical series , se e N . K. Bari , Trigonometrical Series, an d A . Zygmund , Trigo- nometrical Series, volume s 1 an d 2. Th e theor y o f Fourie r integral s i s expounde d in th e book s o f Bochne r [5 ] an d Titchmarsh [ 50]. Fourie r transformatio n i n th e complex domai n wa s firs t studie d i n detai l b y Wiene r an d Paley [ 12]. Th e proo f of th e inversio n formul a fo r Fourie r transformation , give n i n th e book , belong s t o Gel'fand [132] . From th e group-theoreti c poin t o f view , th e theor y o f trigonometrica l serie s and integrals i s a special cas e o f th e theor y o f character s o f commutativ e locall y compact groups , constructe d b y L. S . Pontrjagi n [ 42].

~~Chapter H I~~

The function s P (x) wer e introduce d almos t simultaneousl y b y Legendr e an d Laplace i n connectio n wit h th e stud y o f spheroida l attraction . The y wer e th e firs t a example o f orthogona l polynomials . Th e Jacob i polynomial s P^ k (x) ar e a gen - eralization o f the function s P (x). Concernin g th e genera l theor y o f orthogona l polynomials, se e Jackso n [ 22] an d Szegd [ 46]. Fo r a bibliograph y o n these poly - nomials, se e [ 92]. The classica l theor y o f Legendr e function s i s expounde d i n th e book s o f Hobson [19] , Whittake r an d Watso n [ 5 3], Lens e [79] , Robi n [89 ] an d Wangeri n [9 7,98]. Thes e function s ar e widel y applie d i n classica l an d quantu m physics . The stud y o f th e angula r momentu m operato r i n quantu m mechanic s le d t o a clari - fication o f th e connectio n betwee n th e polynomial s o f Legendr e an d Jacob i o n th e one hand , an d representation s o f th e grou p o f rotation s o f three-dimensiona l eu - clidean spac e o n th e other . Thes e matter s ar e explaine d i n th e book s o f Bhaga - vantam an d Venkatarayudu [ 2L Baue r [3], Beima n [ 4], Va n de r Waerde n [ 7], Wigner [ ll], Landa u an d LifSic [30] , Ljubarski i [33] , Heine [ 58], Hammermesc h [?2], Kaha n [?4], Racah [87 ] Wey l [99 ] an d others . REMARKS AN D BIBLIOGRAPHICA L NOTE S 571

A systemati c constructio n o f th e theor y o f Legendr e polynomial s o n th e basi s of th e theor y o f grou p representation s ha s bee n give n b y Gel'fan d an d Sapir o [ 155] see Gel ;fand, Minlo s an d Sapir o [ 16] also . I n these source s on e ca n fin d a n intro - duction to , an d a detaile d stud y of , th e function s P (x\ whic h ar e simila r t o Jacobi polynomials , ye t somewha t differen t fro m them . Th e result s o f §§5. 1 an d 5.2 ar e du e t o th e autho r [115,117], The Clebsch-Gorda n coefficient s ar e employe d fo r solvin g problem s i n spec - troscopy; se e Jucys , Levinsona s an d Vanaga s [ 63] an d di e book s mentione d above. Th e derivatio n o f propertie s o f thes e coefficients , give n here , is du e bas - ically t o th e author .

~~Chapter I V~~

The functio n } 0(x) wa s considere d i n 173 8 b y Danie l Bernoulli , whil e func - tions / (x) wit h integra l value s o f n wer e considere d i n 176 4 b y Euler , an d studied i n detai l b y Bessel i n 1824 . Concernin g th e classica l theor y o f Besse l functions, se e Watso n [ 8], Gra y an d Matthews [ 21], th e handboo k Higher Trans- cendental Functions [ 73] an d Nielsen [ 83]. Application s t o problem s i n mathe - matical physic s ar e given i n th e book s o f Korene v [ 27], Lebede v [31] , Mors e an d Feshbach [36 ] an d Ufljan d [ 5 4]. The autho r [ 1 !3] ha s constructe d certai n section s o f th e theor y o f Besse l functions o n th e basi s o f th e theor y o f representation s o f th e grou p of euclidea n plane motions . Se e Mille r [ 81L Konstantinov a an d Sokoli k [ 19°] an d Inon u an d Wigner [265]. Representation s o f th e universa l coverin g grou p fo r A/(2 ) hav e bee n studied b y Thom a [ 297]. Fo r a generalizatio n t o th e $-adi c case , se e Sait o [ 289].

~~Chapter V~~

Representations o f th e grou p o f linea r transformation s o f th e straigh t lin e have bee n studie d b y Gel'fan d an d Nair n ark [145]; se e Kohar i [ 267] also . Th e functions T{x) an d B (x, y) wer e introduce d b y Euler . Th e accoun t o f th e theor y of thes e functions , give n here , belong s t o th e author . Variou s generalization s o f the T-functio n hav e bee n considere d b y Godemen t [ 16°], Ge l fan d an d Grae v [ 143] and Gindiki n [ 158]. The classica l theor y o f th e Macdonal d an d Han k el function s i s presente d i n the treatis e o f Watso n [8], alread y mentioned . Th e autho r has studie d thes e func - tions o n th e basi s o f th e theor y o f representation s o f th e grou p o f pseudo-euclidea n 572 REMARKS AND BIBLIOGRAPHICA L NOTE S

plane motions . Som e of th e results (th e compositio n theorem , fo r example ) hav e been obtaine d independentl y b y V . S . Ryko . I t is possibl e tha t som e o f th e form - ulas give n her e ar e new .

~~Chapter V I~~

The grou p of rea l unitar y matrice s o f th e secon d orde r (i.e. th e grou p o f quasi - unitary matrice s o f th e secon d order ) wa s th e firs t noneompact , noncommutativ e group th e theor y o f whos e representation s wa s studie d i n detai l (Bargman n [ 2 36]). Interest i n th e representation s o f thi s grou p an d the closel y relate d grou p of uni - tary comple x matrice s o f th e secon d orde r wa s evoke d b y th e rol e o f thes e group s in physics (the y ar e isomorphi c t o th e three-dimensiona l an d four-dimensiona l Lorentz group s respectively) . Representation s o f th e grou p o f comple x secon d order matrice s wer e firs t studie d b y Gel'fand an d Nair n ark I 144]. Th e result s ob - tained her e wer e use d fo r a fundamenta l stud y o f representation s o f semisimpl e groups (Gel'fan d an d Naimar k [17^47L Gel'fan d an d Graev [135.136,137,138] , 176177 4 Berezin [103,104] , Xelobenko [ ]t Ehrenprei s an d Mautne r [2 9], Harish - Chandra [ 2 5 3-2 59] an d others). Man y o f th e result s o f harmoni c analysi s o n groups wer e firs t obtaine d o n group s o f th e secon d order ; moreover , al l o f thes e have no t as ye t bee n carrie d ove r t o a mor e general cas e (Bogaevsk n [ 109], Gel' - fand an d Fomi n [152] , Zelobenk o [172-174] , Naimar k [2 0 1,203,207], R omm [220 ,

221], Kunz e an d Stein [269,270 ] an d Pukanszk y [285.287] , The expositio n i n thi s chapte r basicall y follow s Bargmann . Man y of th e top - ics ar e presented i n th e boo k o f Ge l fand , Grae v an d the autho r [15], Chapte r 7 . Concerning othe r derivation s o f th e Planchere l formul a o n th e grou p SL(2 , R), se e Gel'fand an d Grae v [136 ] an d Pukanszk y [287] ,

~~Chapter VI I~~

The hypergeometri c functio n wa s introduce d b y Gauss. Concernin g th e class - ical theor y o f thi s function , se e Kratze r an d Fran z [28], Higher Transcendental Functions,volume 1 [73], an d Klein [ 77]. Th e accoun t o f th e theor y give n her e be - longs t o th e autho r [122,123], Severa l formula s o f thi s chapte r ar e new .

Chapter VII I Concerning th e theor y o f confluen t hypergeometri c function s an d th e closel y related Whittake r functions , se e Kratze r an d Fran z [ 2 8], Whittake r an d Watso n [53], Baile y [ 66], Buchhol z [ 68L Higher Transcendental Functions, volum e 1 [73], REMARKS AND BIBLIOGRAPHICAL NOTE S 57 3

and Tricom i [ 94]« Th e theor y o f representation s o f th e grou p o f triangula r thir d order matrices i s a special cas e o f th e genera l theor y o f representation s o f solv - able an d nilpoten t Li e group s (see Kirillo v [188], Dixmie r [167,248]) . The connectio n betwee n the theor y o f Whittake r function s an d th e theor y o f representations o f th e grou p o f triangula r matrice s wa s establishe d b y th e autho r [125]; se e Mille r [81 ] also . Severa l formula s o f thi s chapte r ar e new .

~~Chapter I X~~

Schur[290], Carta n [2 4 2,243], Wey l [300] an d Braue r [ 2 38] have give n th e classification o f representation s o f th e orthogona l grou p an d calculate d th e char - acters o f thes e representations . Concernin g thes e results , se e Murnagha n [37, 8 2] and Wey l [ 10] also . Gel'fan d an d Cetli n [ 154] hav e calculate d th e matri x elements o f irreducibl e infinitesima l representation s o f thi s group . Th e expressio n for th e matri x element s o f representation s i n term s o f Eule r parameter s ha s bee n 195 obtained b y th e autho r [114,121] a nd Lambin a [ ]. The theor y o f spherica l function s i n th e n-dimensiona l cas e i s expounde d i n the book s o f Appel l an d Kamp e d e Ferie t [ 65] an d Nielsen [ 85]. I n Higher Trans- cendental Functions, volum e 2 [ 73], thi s theor y i s presente d essentiall y o n th e basis o f representatio n theory . Th e realizatio n o f representation s give n i n ou r book belong s t o th e author , a s doe s th e theor y o f polyspherica l functions . Se e

~~Kirillov [186 ] a lSo. The connectio n betwee n th e theor y o f grou p representations an d Gegenbaue r polynomials wa s establishe d b y Carta n [244] ,~~

~~Chapter X~~

~~Representations o f th e grou p o f hyperboli c rotation s o f n-dimensiona l spac e and th e relate d specia l function s wer e obtained , independently , b y th e autho r~~

[114,119.121] an d Takahash i [292] ; se e Hira i [262 ] a lso. In th e four-dimensiona l cas e thes e representation s hav e bee n completel y described b y Gel'fan d an d Naimar k [144,16,39,201] , Thes e representations , thei r matrix element s an d relate d integra l transformation s hav e bee n studie d i n a number 162 of paper s (see , fo r example , Golode c [ ]? Dolginov , Toptygi n an d Moskale v [169-171], 2elobenk o [173.174] , L evinsonas an d Jucys [*96] , Popo v [218] , Rom m [220,221], Sapir o [226,226a] , feskin [ 2 2 8]. Jakahashi [2 9 1] andTamm[294]. The horispher e metho d fo r decomposin g representation s int o irreducibl e com - ponents ha s bee n develope d b y Gel'fan d an d Grae v [1 4 1,142]. Th e derivatio n o f 574 REMARK S AN D BIBLIOGRAPHICAL NOTE S

the Fock-Mehle r integra l transformatio n o n th e basi s o f th e horisphere metho d ha s been give n b y th e autho r [119.121], I t ca n b e show n tha t othe r integra l transform a tions (o f Kontorovich-Lebedev , Olevski i [ 214] an d others) ar e also obtaine d b y this method . Fo r th e four-dimensiona l cas e thi s ha s bee n don e b y th e autho r an d Smorodinskii [127]. Polyspherica l an d horispherica l function s o n th e hyperboloi d have bee n introduce d b y th e autho r [l24]. I n the three-dimensiona l cas e variou s orthogonal coordinat e systems , i n whic h th e Laplac e operato r admit s separatio n of variables , hav e bee n studie d b y Olevski i [ 215].

~~Chapter X I~~

The stud y o f propertie s o f Besse l function s i n connectio n wit h th e theor y o f representations o f th e grou p of n-dimensiona l euciidea n spac e motion s ha s bee n 28 266 carried ou t b y th e autho r [l 14.121 ] an d independentl y b y Orihar a [ ^]. It o [ ] has describe d al l irreducibl e unitar y representation s o f thi s group . Besse l func - tions o f matri x argumen t hav e bee n studie d b y Her z [261]- Concernin g th e theor y of Hermit e polynomials , se e Szegd[ 4 6], BIBLIOGRAPHY

~~Monographs an d Textbook s~~

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[286] , On the Kronecker product of irreducible unitary representations of the inhomogeneous Lorentz group, J . Math . Mech. 1 0 (1961), 475—491 . MR 25 #4034 . [287] , The Plancherel formula for the universal covering group of SL(R, 2) , Math. Ann . 15 6 (1964) , 96-143 - M R 3 0 #1215 . [288] A . Rubinowicz , Uber ein Additions-theorem fur Laguerreschen Polynomen, Zeeman Festschriften , 1935 , 143-147 . [289] M . Saito, Representations unitaires du groupe des deplacements du plan ^-adique, Proc . Japa n Acad . 3 9 (1963), 407-409 . M R 29 #3577 . [290] I . Schur , Neue Anwendungen der Integrairechunung auf Probleme der Invar- iantentheory, S.-B . Preuss . Akad . Wiss . Berlin , 1924 , 189-208 , 297-321 , 346-355. [290a] M . Sugiura , Representations of compact groups realized by spherical functions on symmetric spaces, Proc . Japa n Acad . 3 8 (1962) , 111—113 - MR 26 #258 . [29l] R . Takahashi , Sur les fonctions spheriques et la formule de Plancherel dans le groupe hyperbolique, Japa n J . Math . 1 3 (1961), 55-90 . M R 27 #2809 . [292] , Sur les representations unitaires des groupes de Lorentz general- ises, Bull . Soc . Math . Franc e 9 1 (1963) , 289-433 - M R 3 1 #3544 . [293] T . Takahashi , Generalized spherical harmonics as representation, matrix elements of rotation group, J . Phys . Soc . Japa n 7 (1952) , 307-312 . MR 14, 373. [294] J . Tamm , Die verallgemeinerten Kugelfunktionenund Wellenphysik, Z . Physik . 71, 141-150 . [295] N . Tatsuuma , Decomposition of representations of the three-dimensional Lorentz group, Proc . Japa n Acad . 3 8 (1962), 12-14 . M R 25 #4035 . [296] , Decomposition of Kronecker products of representations of the inhomogeneous Lorentz group, Proc . Japa n Acad . 3 8 (1962), 156—160 . MR 25 #4036 . [297] E . Thoma , Die unit'dren Darstellungen der universellen Uberlagerungsgruppe der B ewegungsgruppe des R 2, Math . Ann . 13 4 (1958) , 428-459 . MR 21 #821 . [298] L . H . Thomas , On unitary representations of the group of De Sitter space, Ann. o f Mat h (2 ) 4 2 (1941), 113-126 . M R 2, 216 . 600 BIBLIOGRAPHY

[299] H . Weyl , Harmonics on homogeneous manifolds, Ann . o f Math . 3 5 (1934) , 486-499. [300] , Theorie der Darstellung kontinuierlicher halbeinfacher Gruppen durch lineare Trans formation en, Math . Z . 2 3 (1925), 271-309 ; 2 4 (1925) , 228-395. [301] E . P . Wigner , On unitary representations of the inhomogeneous Lorentz group, Ann. o f Math . 4 0 (1939) , 149-204 . [302] S . Yamaguchi , On certain zonal spherical functions, Mem . Fac. Sci . Kyush u Univ. Ser . A 1 7 (1963) , 131-134 . M R 28 #5142 . [3O3] H . Yoshizawa , Group representations and spherical functions, Sugak u 12 (1960/61), 21-37 . (Japanese ) M R 26 #3817 . SUBJECT INDE X

Absolute o f Lobocevskii'space , 50 7 Canonical basis, 11 4 nl Adjoint, in § y 465 , 46 8 representation, 1 2 decomposition o f a polynomial , 44 6 space, 12 , 5 9 matrix, 46 9 Angles, Eule r — see Eule r angle s elements of , 469 , 47 1 Antilinear operator , 6 3 v v Cebysev polynomia l o f th e firs t kind , 13 4 Antiperiodic function , 8 3 Central function , 56 , 173—17 4 Associated Legendr e function — se e Parseval equality , 5 8 Legendre function , associate d series expansion , 57 , 17 4 Associated spherica l function—se e Character o f a representation , 1 9 spherical function , associate d Class o f transitivity , 2 4 Basis, 3 Clebsch-Gordan coefficients , 17 7 and th e function s P (z) , 18 8 biorthogonal, 1 2 mn canonical, 11 4 and Jacob i polynomials , 18 8 in § nl, 465 , 46 8 asymptotic formula , 229—23 0 17 computation, 179 , 18 0 in S^ 1 Sp2, 6 Bessel functions , 206 , 266 , 26 7 generating function , 19 3 and Hanke l functions , 266—26 7 recurrence formulas , 18 9 and Jacob i polynomials , 228—22 9 representation a s a sum , 181 , 18 2 and Legendr e polynomials , 228—22 9 special values , 183 , 18 4 and zona l spherica l functions , 552 — symmetry relations , 182—18 3 553 Clebsch-Gordan series , 18 6 composition theorem , 208 , 209 , 55 5 Coefficients — see correspondin g nam e differential equation , 211 , 21 2 Commutator, 2 2 generating function , 212 , 55 7 Conditionally periodi c function , 8 3 integral representation , 269 , 27 0 Confluent hypergeometri c function , 40 1 multiplication formulas , 210 , 227 , Conical functions , 31 5 556 Convergence, recurrence formulas , 211 , 21 3 in th e mean , 8 8 series expansion , 208 , 27 0 of matrices , 4 2 with opposit e indices , 20 7 Convolution o f functions , 5 5 Beta function , 24 3 Coordinates, expression b y th e gamm a function , bispherical, 48 9 243 Cartesian* an d pol y spherical, 49 7 Bispherical coordinates , 48 9 essentially followin g (preceding) , 495

~~601 602 SUBJECT INDE X~~

Coordinates, Expansion, horispherical, 54 4 by function s P l (x) , 163 , 16 5 hyperbolic, 50 $ by Gegenbaue r polynomials , 486—48 7 polar, i n pseudo-euclidea n space , by Jacob i functions , 33 6 248-249 of field s o n th e sphere , 16 9 polyspherical, 495 , 49 7 of functions , and Cartesia n coordinates , 49 7 on QU(2), 33 4 and th e Laplac e operator , 49 9 on 5(/(2) , 158 , 165-16 7 differential o f arcs , 497-49 8 on th e sphere , 16 7 on th e hyperboloid , 54 2 Exponential function , 7 spherical, 436 , 459 , 493-494 , 49 7 Factor-space, 1 5 subordinate, 495—49 6 Field o f quantitie s o n a sphere , 16 9 Cosine, hyperbolic , 7 5 Fourier coefficients , 7 8 Cross product , of a convolution , 5 5 of groups , 19 7 Fourier series , 77—7 8 representation of , 20 4 and Legendr e polynomials , 12 8 Cylindrical functions , 26 5 on compac t groups , 4 5 differential equation , 26 6 Fourier transform , 81 , 83 , 8 6 Decomposition, in comple x domains , 8 9 canonical, 44 6 inversion formula , 83—8 5 of function s o n homogeneou s spaces , of function s o f severa l variables , 52 88-89 of Kronecke r products , 174—17 5 inversion of , 8 9 Direct-sum decompositio n o f operators , 6 9 of square-integrabl e functions , 8 8 Direct su m o f Hilber t spaces , of x\ an d x*ty 24 5 continuous, 67—6 8 Fourier-Bessel transformation , 217 , 218 , orthogonal, 68—6 9 220 Dirichlet-Murphy integra l representations , analogue o f th e Planchere l formula , 155-156 218 Entire analyti c function s o f exponentia l inversion formula , 21 8 Functions —see als o correspondin g nam e type, 9 0 Functions, Equivalence o f representations , 1 1 antiperiodic, 8 3 Euclidean space , rotatio n of , 43 7 central — see centra l function s Euler angles , 9 8 conditionally periodic , 8 3 complex, 10 3 conical, 31 5 for hyperboli c rotations , 50 9 convolution of , 5 5 for product s o f matrices , 9 9 cylindrical, 26 5 for rotations , 105 , 106 , 43 9 exponential, 7 for SH(n), 50 8 finite, 89-9 0 Euler formulas , 7 2 generating, 14 7 SUBJECT INDE X 603

Functions, Gegenbauer polynomials , 45 8 homogeneous, 3 5 and associated Legendr e functions , horispherical, 54 6 484 and Hermit e polynomials , 559—56 1 infinitely differentiabie , 79 , 80 composition theorem , 471 , 472 on compac t groups , harmoni c analysi s differential equation , 45 9 of, 5 0 expansion by , 486—48 7 on th e circle, odd , even, 29 6 l generating function , 491—49 2 P mn(*\ 120 , 137 integral representation , 482—484 , and Clebsch-Gorda n coefficients , 487-488 188 multiplication formulas , 473—47 5 and Jacob i polynomials , 12 5 orthogonality relations , 46 1 circuit relations , 12 4 recurrence formulas , 459 , 460 composition theorem , 129-13 0 special cases , 46 1 differential equation , 13 7 Gel fand-Grae v integra l transformation , expansion by , 163—16 5 532-533 expansion o f b y associate d Group —see als o representatio n Legendre functions , 18 6 Group, expansion o f thei r product , 18 6 compact, Fourie r serie s on , 4 5 generating functions , 148 , 15 3 harmonic analysi s on , 5 0 integral representation , 121 , 153 continuous, 8 multiplication formula , 13 3 group rin g of , 5 6 orthogonality relations , 16 1 Lie, linear , 2 2 recurrence formulas , 135 , 140— locally isomorphic , 10 5 141, 145 , 147-153, 18 7 matrix, 21 , 42 series expansion s b y 163 , 16 5 compact, 42—4 3 special values , 12 1 — 122 locally compact , 4 3 symmetry relations , 122—12 3 A/(2), 195 , 200, 228 poly spherical, 50 2 Complexification, 20 0 rapidly decreasing , 8 2 Lie algebra , 198-20 0 spherical, 3 0 parametrization, 196 , 19 7 square-integrable, transformatio n of , Af(2, C) , 20 0 91 M(n), 54 7 trigonometrical, 7 2 parametrization, 54 8 xuJl an d x u~l , 245-24 6 MH(2), 24 9 Gamma function , 237 , 23 8 Lie algebra , 25 2 and th e bet a function , 24 3 parametrization, 25 1 complementation formula , 24 3 subgroup o f hyperboli c rotations , composition formula , 24 2 250 duplication formula , 24 4 of linea r transformation s o f th e properties, 238—24 1 straight line , 23 1 604 SUBJECT INDE X

~~Group, Group,~~

of motion s o f E n, 54 7 transformation, 2 3 of rea l numbers , additive, 7 0 effective, 2 4 of rea l unimodula r matrice s o f orde r 2 , transitive, 2 4 344-346 Hankel functions , 259 , 266 , 39 5 of triangula r matrice s o f order,3 , 400 , and Bessel functions , 266 , 26 7 402 composition theorems , 27 6 product o f (cross , o r semidirect) , 19 7 integral representation , 27 0 QV(2), 288 , 289 , 33 9 integral transformations , 280 , 284 - expansion o f function s on , 33 4 287 invariant integration , 29 4 multiplication theorems , 27 9 Laplace operator , 33 3 of th e firs t an d secon d types , 25 9 Lie algebra , 294 , 29 5 recurrence formulas , 26 5 mapping int o Stf(3) , 289,29 0 series expansion , 27 0 parametrization, 29 2 Harmonic, R (o f rea l numbers) , 7 0 analysis o f function s o n compac t Rn, 8 8 groups, 5 0 S//(2), 74 , 7 5 polynomial, 442 , 457 , 45 8 Sf/(3), 28 9 projection o f a homogeneous poly - SH(n), 503 , 505 , 50 6 nomial, 445 , 44 6 Euler angles, 50 8 Hermite polynomials , 56 0 SL(2, C) , 10 2 as limi t o f Gegenbaue r polynomial s SL(2, K) , 289-29 1 559-561 Lie algebra , 35 3 composition formulas , 562—56 3 parametrization, 350 , 35 1 differential equation , 561—56 2 subgroups SO(2) , SH(2), Z , 29 1 generating function , 56 2 S0(2), 71,.72 , 7 5 multiplication formulas , 562—56 3 coraplexification, 7 5 orthogonality relations , 56 4 integration on , 7 7 recurrence formulas , 56 2 SO (2, C) , 7 6 Hilbert-Schmidt operator , 6 1 SO(3), 104 , 22 8 Hilbert space—se e space , Hilber t SOW, 435 , 43 7 Homogeneous polynomial , Euler angles, 43 8 canonical expansion , 44 6 invariant integration , 439—44 0 harmonic projection , 445 , 44 6 invariant measure , 439—44 0 Horispheres an d integratio n i n Lobac'e v SU(2), 97 , 98 , 102-10 4 skn space , 531—53 2 complexifi cation, 10 2 Horispherical, expansion o f function s on , 158 , functions, 54 6 165, 16 7 rotation, 543—54 4 invariant measur e on , 158 , 15 9 Hyperbolic, Laplace operator , 14 4 cosine, 7 5 Lie algebra , 10 1 SUBJECT INDE X 605

Hyperbolic, Jacobi functions , rotation, 74 , 250 , 255-256 , 50 5 multiplication theorems , 321—32 5 Euler angles , 50 9 orthogonality relations , 34 3 recurrence formulas , 326 , 33 3 sine, 7 5 symmetry relations , 316—31 7 Hypergeometric differentia l equation , 38 0 Jacobi polynomials , 12 5 degenerate, 401—40 2 and Besse l functions , 228-22 9 Hypergeometric function , 344—346 , 38 0 and Clebsch-Gorda n coefficients , 18 8 and associate d Legendr e functions , and th e function s P (z)* 12 5 349 mn expression b y hypergeometri c func - and Jacob i functions , 34 9 tions, 34 8 and Jacob i polynomials , 34 8 orthogonality relations , 16 2 and Legendr e polynomials , 34 9 Jansen formula , 20 9 and zona l spherica l functions , 522 — 524 Kernel space , 6 5 composition theorems , 39 1 Kronecker (tensor ) product , confluent, 40 1 of linea r spaces , 5 9 integral representations , 381—38 4 of operators , 60 , 6 3 integral transformations , 37 0 of representations , 1 8 Mellin transformation , 38 6 decomposition of , 174—17 5

recurrence formulas , 39 1 TR(g)t 224-22 5 series expansion , 34 5 Kronecker symbol , 1 0 Hypergeometric series , 40 1 Laguerre polynomials , 430—43 1 confluent, 40 1 composition theorem , 43 3 Infinitesimal operator , 2 1 orthogonality relations , 43 2 Integral transformation , Laplace operator , 442 , 44 3 Dirichlet-Murphy, 155-15 6 in pol y spherical coordinates , 49 9 Gel'fand-Graev, 532-53 3 on a sphere , 49 3 of function s o n th e hyperboloid , 53 8 on th e hyperboloid , 54 2 Integral wit h respec t t o measure , invari - on th e uni t sphere , 145 , 49 3 ance of , 2 6 Legendre coefficients , 15 5 Jacobi functions , 31 0 Legendre functions , 31 4 composition theorems , 321—32 5 associated, 125-127 , 315 , 31 6 continual generatin g function , 330 — and Gegenbaue r polynomials , 331 484 and th e function s P _ (z) , 18 6 differential equation , 327 , 33 4 mn expansion by , 33 6 composition theorems , 32 6 expression b y hypergeometri c func - differential equation , 138 , 32 8 tions, 34 9 expression b y hypergeometri c generating function , 32 8 functions, 34 9 continual, 330-33 1 generating function , 155 , 52 9 integral representations , 310—31 4 multiplication theorems , 32 6 606 SUBJECT INDE X

Legendre functions , Macdonald function , associated, multiplication theorems , 27 9 orthogonality, 16 1 recurrence formulas , 26 5 recurrence formulas , 135 , 32 8 series expansion , 27 0 symmetry relations , 317—31 8 Massive subgroup , 3 0 composition theorems , 326 , 527—52 9 Matrix, continual generatin g function , 33 1 canonical, 46 9 differential equation , 32 8 elements of , 469 , 47 1 multiplication theorems , 326 , 527 — convergence of , 4 2 529 group, 21 , 4 2 recurrence formulas , 32 8 compact, 42—4 3 Legendre polynomials , 125—127 , 31 5 locally compact , 4 3 and Bessel functions , 228—22 9 Lie algebra , 2 2 and zona l spherica l functions , 128 , product, Eule r angles of , 9 9 129 quasi-unitary unimodula r o f orde r 2 , composition theorem , 13 2 288 differential equation , 13 8 tangent, 2 2 expression b y hype r geometric func - trace of , 1 9 tions, 34 9 unitary, u(4>, 0 , */0 , 98-9 9 Fourier serie s expansion , 12 8 Mean convergence , 8 8 generating function , 15 5 limit, 8 8 integral representations , 154 , 15 5 Measure, invariant , 26 , 29 4 orthogonality relations , 16 1 left (right) , 2 6 recurrence formulas , 156—15 8 Mehler-Fock transformation , 54 1 Lie group , linear , 2 2 Mellin transformation , 94-95 , 23 6 Lie matrix , algebra , 2 2 analogue o f th e Planchere l formula , Limit i n th e mean , 8 8 95 Linear space , inversion formula , 9 5 n-dimensional, 8 8 of th e functio n R\(g)(x) y 23 6 product of , 5 9 of th e hypergeometri c function , 38 6 unitary, 6 1 of th e Macdonal d function , 271 , 27 4 Lobacevskii space , 506 , 50 7 of Whittake r function s b y parameters , absolute of , 50 7 421 invariant integratio n b y horispheres , Mellin-Barnes representatio n o f Whittake r 531-532 functions, 41 9 Macdonald function , 25 9 Motions,

composition theorems , 27 6 of E v 19 5 differential equation , 26 6 of £„ , 54 7 integral representation , 26 9 of th e pseudo-euclidea n plane , 24 9 integral transformations , 284—28 7 reciprocal, 28 0 Neumann function , 267—26 8 Mellin transformatio n of , 271 , 27 4 series expansion , 27 1 SUBJECT INDE X 607

~~Operator—see correspondin g nam e Product, Operator, of representation s (Kronecke r o r ten - antilinear, 6 3 sor), 1 8 continuous direct-su m decomposition , decomposition of , 174—17 5~~

69 TR(g)7 224-22 5 \»- 14 2 Projection, harmonic , o f a homogeneou s polynomial, 445—44 6 Hermitian-ad joint, 1 3 Pseudo-euclidean plane , 24 8 infinitesimal, 2 1 analogue o f pola r coordinates , 248 — invariant fo r transformations , 4 1 249 Kronecker (tensor ) product , 60 , 6 3 distance betwee n points , 24 8 of Hilbert-Schmid t type , 6 1 motions of , 24 9 permutable wit h representations , 3 6 Pseudo-euclidean space , 50 3 wave, 3 6 Pseudo-sphere, 50 4 Orispheres — see horisphere s distance betwee n points , 50 6 Orthogonal complemen t o f a subspace , 1 7 Orthogonal direct-su m — see su m Quantity fiel d o n a sphere , 16 9 Parseval equality , 4 6 Representations, 8 for centra l functions , 5 8 adjoint, 1 2 Plancherel formula , 8 6 by shif t operators , 2 7 for th e Melli n transformation , 9 5 character of , 1 9 Polynomial — see correspondin g nam e completely reducible , 17 , 4 4 Polynomial, complete syste m of , 4 5 canonical decompositio n of , 44 6 direct su m of , 1 6 harmonic homogeneous , spac e of , direct-sum expansio n of , 1 6 442-443, 464-46 6 direct sum , orthogonal , 1 6 homogeneous, equivalent, 1 1 canonical decompositio n of , 44 6 faithful, 8 harmonic projectio n of , 445 , 44 6 finite-dimensional, 8 space of , 108 , 441 , 444 , 44 8 Hermitian-ad joint, 1 3 Polyspherical, induced, 31-32 , 34 1 coordinates — see coordinate s infinite-dimensional, 1 0 functions, 50 2 irreducible, 1 4 Power, symbolic , 18 8 Kronecker produc t of , 1 8 Product, decomposition of , 174—17 5

of Hilber t space s (tensor) , 6 2 TR(g)y 224-22 5 of linea r space s (Kronecke r o r tensor) , left (right ) regular , 2 8 59 matrix notatio n for , 8 , 1 9 of matrices , Eule r angle s of , 9 9 rt-dimensional, 9 of operator s (Kronecke r o r tensor) , of clas s 1 relative t o a subgroup , 2 9 60, 6 3 of som e group—se e nex t mai n entr y 608 SUBJECT INDE X

Representations, Representations of , operator-irreducible, 3 7 S/Y(2), 7 5 permutable wit h operators , 3 6 SH(n\ 50 9 product o f — see representations , adjoint, 51 1 Kronecker produc t of , complementary series , 51 8 reducible, 1 4 discrete series , 518-51 9 restriction of , 1 5 equivalent, 51 9 right regular , 2 8 fundamental series , 518—51 9 space of , 8 irreducible, 51 3 trivial, 8 quasi-regular, 533—53 4 unitary, 13 , 1 4 reducible, 51 5 completely reducible , 1 7 unitary, 516 , 51 8 weight of , 11 4 SL(2y C) , 11 0 with operato r factors , 3 2 SL(2, K) > 354 , 356 , 362 , 39 5 zonal spherica l function s fo r T* 1 (g) , SO (2), 72 , 76 , 7 8 457 S0(n), Representations of , infinitesimal operator s of , 11 0 compact groups , 4 4 in space s o f harmoni c an d homo - cross products , 20 4 geneous polynomials , 441 , 44 4 M<2), irreducible, 441 , 448 , 45 2 irreducible, 196 , 20 0 quasi-regular, 440 , 44 7 quasi-regular, 214 , 217 , 22 0 S(/(2), M{n), irreducible , 54 9 characters of , 170-17 2 MH (2), 25 0 infinitesimal operator s of , 11 0 irreducible, 254 , 25 5 invariant scala r product , 112—11 3 quasi-regular, 28 1 irreducible, 115-11 9 QU(2\ regular, 133 , 141 , 22 4 by infinitesima l operators , 29 8 the grou p o f irreducibl e triangula r induced, 34 1 matrices o f th e thir d order , 404 , irreducible, 295 , 299 , 300 , 306 - 405, 408 , 40 9 308 the grou p o f linea r transformation s o f partially equivalent , 30 2 the straigh t line , 232 , 233 , 24 6 quasi-regular, 34 2 Root o f a tree , 49 5 reducible, 299-30 0 Rotation, regular, 331 , 339-34 0 Euler angle s of , 105 , 106 , 43 9 unitary adjoint , 30 6 horispherical, 543—54 4 unitary, o f th e fundamenta l an d hyperbolic, 74 , 250 , 255-256 , 50 5 complementary series , 303 — Euler angle s of , 50 9 305 Of Ey 10 4 R, 7 0 of £ n, 43 7 n R y regular , 8 8 SUBJECT INDE X 609

~~Schur's Space, Lemma, 3 8 of a representation , 8 Corollaries, 3 9 295>29~~

Tangent matrix , 2 2 Whittaker, Tensor produc t —see Kronecke r product ; continual compositio n theorems , 42 5 product 429-430 Trace o f a matrix , 1 9 differential equation , 41 5 Transformation — see correspondin g nam e dual formulas , 42 8 Transitivity class , 2 4 functions, 400 , 40 1 Tree, 49 5 Mellin-Barnes representation , 41 9 root of , 49 5 Mellin transformatio n b y parameters , Trigonometrical functions , 7 2 421 Vector invarian t relativ e t o subgroups , 2 9 recurrence formulas , 411—41 4 series expansion , 40 1 Wave operator , 54 1 symmetry relations , 416—41 8 Weight o f a representation, 11 4 Wigner symbol , 18 3 Whittaker differentia l equation , 41 5 Zonal spherica l function s - se e spherica l functions, zona l NOTATION INDE X

~~Latin alphabe t g{~~

~~C(l, j) = C(l, t U, If j, k, m) - Clebsch - K(^j, S^j) —tenso r produc t o f Hilber t Gordan coefficients , 17 7 spaces, 6 2~~

~~C? (t) — Gegenbauer polynomials , 45 8 / (x ) — nth Besse l function , 20 6~~

£ —identit y operato r i n 2 , 8 /^(z)—Macdonald function , 25 9 E-i —three-dimensional euclidea n space , L(g),/?(g)- regular representations , 281 . 106 331, 44 0 E — n-dimensional euclidea n space , 43 5 (m « J )-Hgne r symbol , 18 3 n \tn m m I e—unit o f G, 8 1 2 3 l.i.m. —mea n limit , 8 8 F(a, j3 , y\ z ) — hypergeometric function , 344 — group o f motion s o f Eyt 19 5

M\2y C) — complexification o f W(2) , 20 0 f.(g) h , ; a -base s i n § , (g ) §2, 17 6 Af(rt) — group o f motion s o f E , 54 7 f\ * f? " convolution o f functions , 5 5 — groups o f motion s o f th e pseudo - G/W—homogeneous space , 2 5 euclidean plane , 24 9 g —transpos e o f th e matri x g, 4 3 M\ (z), V\ (z ) - Whittake r functions , g(a,, a 2, 0 ) —motio n o f th e pseudo - 400 euclidean plane , 249 , 25 0 /V^(z)—Neumann function , 267—26 8 g{a, b, a)-element s o f M(2), 19 6 PJ^^W-Jacobi polynomials , 12 5

~~611 612 NOTATION INDE X~~

Pi (2) —Legendre polynomials , 12 5 TiA —trac e o f a matrix , 1 9 t (&\ t itd, l x 0? ) — matrix element s Pi (2)—associate d Legendr e functions , OTI ^ ' mn ^ ' mn^> 125 of representations , 115 , 30 7

mn */)/)(&)— zonal spherica l function , 52 1 Qn(g)~representation o f MH{2), 25 5 Lj(nig) — associated spherica l function , 521 QU(2) — group o f quasi-unitar y unimodula r matrices o f orde r 2 , 28 8 u —matri x Hermitian-adjoin t t o u , 9 7

/? —additive grou p o f rea l numbers , 7 0 W\ (z),Mi (2)—Whittake r functions , Rn — n-dimensional linea r space , 8 8 400 Rig) — regular representatio n o f S0(2) , 7 8 [x, y] — bilinear for m i n pseudo-euclidea n K(g),L(g)-regular representations , 281 , space, 50 3 331, 44 0 Y,, i, 0) —associated spherica l func - /?» U)-representatio n o f SU(2), 14 1 tions, 12 9

Sn -uni t spher e i n E , 43 5 Z (2 ) —cylindrical function , 12 9 o —representatio n o f SH(n), 50 9 German alphabe t — group o f hyperboli c rotation s o f /l /z-dimensional space , 74 , 289 , 50 5 8 °'-509 SL{2, O-complexificatio n o f S(/(2) , 10 2 5) —space o f infinitel y differentiabl e S0(n)~ grou p o f rotation s o f E , 70 , 104 , functions o n th e circle , 20 0 435 5)x —spac e o f th e functio n iz), 295, 29 6 50(2, C) — complexification o f S0(2) , 7 6 £5/— invariant subspaces , i n whic h th e Sf/ (2) — group o f unitar y unimodula r matri - representations T,(li) ar e realized , ces o f orde r 2 , 9 7 175 signa — sign o f th e numbe r a , 3 5 J) 7 — space o f homogeneou s polynomial s of degre e 21, 10 8 T{g), T,(g), 7>), T R(g), T x{g), R xig)- $j>n —space o f homogeneou s harmoni c various representations , 110 , 200 , 201 , polynomials, 44 2 223, 254 , 295 , 404 , 408 , 54 9 Q —spac e adjoin t t o Q , 12 , 5 9 Tig) ® Qig) -Kronecke r produc t o f repre - sentations, 1 8 Q iS ) — space o f square-summabl e functions o n th e sphere , 44 0 T(g) — representation Hermitian-adjoin t t o su m r(«), 1 3 5, + Oj—direc t ° * spaces, 1 5 T ig) — representation adjoin t t o Tig), 1 2 3,0^2 —Kronecke r (tensor ) produc t o f spaces, 6 0 Tn ig) — irreducible representation s o f C , 2 . 3 —subspace s o f function s SOin), 44 1 n' m y m n r T (% ) — Cebysev polynomia l o f th e firs t on SU (2), 164-16 7 kind, 13 4 31 = G///—homogeneous space , 2 5 NOTATION INDE X 613

~~5p,(ch r)-Legendre functions , 31 4 15.rAx) — canonical basi s i n $j) n , 466 , 46 8~~

Sp?1 (z)— associated Legendr e functions , 0((X; y; z) — confluent hypergeometri c 315, 31 6 function, 40 1 Z 25 (c h r), SB * n (z)- Jacobi functions , <£>, 0, 0 —Eule r angles , 9 8 309, 31 0 0, r , ^ r — Euler angles , 29: > — space o f homogeneou s polynomials , X /M * Y f (g)-characters , 19,1/ 0 441 12 —subgroup o f hyperboli c rotations , 25 0 6-82, 22 5 Special symbol s Greek alphabe t &-Kronecker (tensor ) product , 18 , 60 , BU, y ) — beta function , 24 3 63, 17 6

~~r(z) — gamma function , 23 7 ^ ® ^^ o r ^ ^(g)-direc t orthogona l~~

A, A ft —Laplace operators , 143—14 5 sum of representations , 1 6 A; —restrictio n o f th e Laplac e operator , Q —the wav e operator , 54 1 142 DQ —Laplac e operato r o n a hyperboloid , S.—Kronecker symbol , 1 0 542 An (/? ) — (n - l)-dimensiona l Lobacevsk u space, 50 6 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given. Republication, systemati c copying, o r multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissionOams.org .