Automorphic forms and applications 9781470439118, 1470439115

Citation preview

Automorphic Form s and Application s

This page intentionally left blank

https://doi.org/10.1090//pcms/012

IAS/PARK CIT Y MATHEMATICS SERIE S Volume 1 2

Automorphic Form s and Application s Peter Sarna k Freydoon Shahid i Editors

tBEM^

American Mathematica l Societ y Institute for Advanced Stud y

J o h n C . Polking , Serie s E d i t o r P e t e r Sarnak , Volum e E d i t o r Freydoon Shahidi , Volum e E d i t o r I A S / P a r k Cit y M a t h e m a t i c s I n s t i t u t e r u n s m a t h e m a t i c s e d u c a t i o n p r o g r a m s t h a t brin t o g e t h e r hig h schoo l m a t h e m a t i c s teachers , researcher s i n m a t h e m a t i c s a n d m a t h e m a t i c education, u n d e r g r a d u a t e m a t h e m a t i c s faculty , g r a d u a t e s t u d e n t s , a n d u n d e r g r a d u a t e s t p a r t i c i p a t e i n distinc t b u t overlappin g p r o g r a m s o f researc h a n d education . T h i s volum contains t h e lectur e note s fro m t h e G r a d u a t e S u m m e r Schoo l p r o g r a m . 2000 Mathematics Subject

g s o e

Classification. P r i m a r y 1 1 -06 , 1 1 F1 2 , 1 1 F66 , 1 1 F70 , 1 1 F72 , 11G40, 1 1 M36 , 1 1 T60 , 22E46 , 81 Q50 .

Library o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Automorphic form s an d application s / Pete r Sarnak , Freydoo n Shahidi , editors , p. cm . — (IAS/Par k Cit y mathematic s series , ISS N 1 079-563 4 ; v. 1 2 ) Includes bibliographica l references . ISBN 978-0-821 8-2873- 1 (alk . paper ) 1. Automorphi c forms . 2 . Automorphi c functions . 3 . Form s (Mathematics ) I ter. II . Shahidi , Freydoon .

. Sarnak , Pe -

QA353.A9A92 200 7 S I S ^ — d c 2 2 200604803

6

C o p y i n g an d reprinting . Materia l i n thi s boo k ma y b e reproduce d b y an y mean s fo r edu cational an d scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y services tha t collec t fee s fo r deliver y o f document s an d provide d tha t th e customar y acknowledg ment o f th e sourc e i s given . Thi s consen t doe s no t exten d t o othe r kind s o f copyin g fo r genera l distribution, fo r advertisin g o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercial us e o f materia l shoul d b e addresse d t o th e Acquisition s Department , America n Math ematical Society , 20 1 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n also b e mad e b y e-mai l t o [email protected] . Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e firs t pag e o f each article. ) © 200 7 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Copyright o f individua l article s ma y rever t t o th e publi c domai n 2 8 year s after publication . Contac t th e AM S fo r copyrigh t statu s o f individua l articles . 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 th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 21 1

2 1 1 1 0 09 08 0 7

Arman d Bore l 1923-200 3

This page intentionally left blank

Dedication t o Arman d Bore l This volum e o f th e IAS/Par k Cit y Mathematic s Institut e Lectur e Note s i s dedi cated t o Arman d Borel . Professo r Bore l die d o n Augus t 1 1 , 2003 , whil e h e wa s a Professo r Emeritu s a t th e Institut e fo r Advance d Study , wher e h e ha d bee n a member o f Facult y sinc e 1 957 . Professor Borel' s researc h covered man y fields , includin g pioneerin g work s i n al gebraic an d arithmeti c group s an d th e theor y o f automorphi c forms . H e wa s a principal lecture r i n th e IAS/PCM I Graduat e Summe r Schoo l durin g th e summe r of 2002 . Th e note s fo r hi s lecture s appea r i n thi s volume , startin g o n pag e 5 .

This page intentionally left blank

Contents

Preface xii

i

Introduction Armand Borel , Automorphi c Form s o n Reductiv e Group s 5 Introduction 7 1. Notatio n 7 2. Notio n o f automorphi c for m 8 3. Firs t propertie s o f automorphi c form s 9 4. Reductiv e group s (review )

2

5. Arithmeti1 c subgroups . Reductio n theor y

9

6. Constan t terms . Th e basi c estimat e 2

4

7. Finit e dimensionalit y o f A(T, J , £) 2

8

8. Convolutio n operator s o n cuspida l function s 2

9

9. Automorphi c form s an d th e regula r representatio n o n T\G 3

0

10. A

2

decompositio n o f th e spac e o f automorphi c forms . 3

11. Som e estimate s o f growt h function s 3

3

12. Eisenstei n serie s 3

5

Bibliography 3

9

L. Clozel , Spectra l Theor y o f Automorphi c Form s 4 1 Foreword 4

3

Lecture 1 . Mostl y SL(2 ) 4

7

x CONTENT

S

Lecture 2 . Th e spectra l decompositio n o f L Arthur's conjecture s 5

2

(G(Q)\G(A)): 7

Lecture 3 . Know n bound s fo r th e cuspida l spectru m an d th e Burger-Sarna k method 6

5

Lecture 4 . Applications : contro l o f th e spectru m 7

9

Appendix: Al l reductiv e adeli c group s ar e tam e 8

7

Bibliography 8

9

James W . Cogdell , L-function s an d Convers e Theorem s fo r GL n 9

5

Introduction 9

7

Lecture 1 . Fourier expansion 1 s an d multiplicit y on e 0 1 1 1 Lecture 2 . Eulerian integral s fo r GL n 1 Lecture 3 . Local L-function s 2

3

Lecture 4 . Globa l L-function s 3

7

Lecture 5 . Convers e theorem s 4

7

Lecture 6 . Convers 1 e theorem s an d functorialit y 6 1 Bibliography 7

3

Philippe Michel , Analyti c Numbe r Theor y an d Familie s o f Automorphic L-function s 7

9

Foreword 8 1 Lecture 1 . Analytic propertie s o f individua 1 l L-function s 8

7

Lecture 2 . A revie w o f classica 1 l automorphi c form s 2 1 Lecture 3 . Large siev e inequalitie s 22

3

Lecture 4 . Th e subconvexit y proble m 24 1 Lecture 5 . Som e application s o f subconvexit y 26

7

Bibliography 28

5

Freydoon Shahidi , Langlands-Shahid i Metho d 29

7

Foreword 29

9

Lecture 1 . Basic concept s 30 1 Lecture 2 . Eisenstein serie s an d L-function s 30

7

Lecture 3 . Functional equation 1 s an d multiplicativit y 3

5

CONTENTS x

i

Lecture 4 . Holomorphy an d boundedness ; application s 32 1 Bibliography 32

7

Audrey Terras , Arithmetica l Quantu m Chao s 33 1 Abstract 33

3

Lecture 1 . Finite model s 33

5

Lecture 2 . Three symmetri c space s 35

5

Bibliography 37 1 David A . Vogan , Jr , Isolate d Unitar y Representations . 37

7

Bibliography 39

7

Wen-Ching Winni e Li , Ramanuja n Graph s an d Ramanuja n Hypergraphs 39

9

Introduction 40 1 Lecture 1 . Ramanujan graph s an d connection s wit h numbe r theor y 40 1 Lecture 2 . Ramanujan hypergraph s 4 Bibliography 42

3 5 5

This page intentionally left blank

Preface The IAS/Par k Cit y Mathematic s Institut e (PCMI ) wa s founde d i n 1 99 1 as par t o f the "Regiona l Geometr y Institute " initiativ e o f th e Nationa l Scienc e Foundation . In mid 1 99 3 the progra m foun d a n institutiona l hom e at th e Institute fo r Advance d Study (IAS ) i n Princeton , Ne w Jersey . The IAS/Par k Cit y Mathematic s Institut e encourage s bot h researc h an d ed ucation i n mathematic s an d foster s interactio n betwee n th e two . Th e three-wee k summer institut e offer s program s fo r researcher s an d postdoctora l scholars , gradu ate students , undergraduat e students , hig h schoo l teachers , undergraduat e faculty , and researcher s i n mathematic s education . On e o f PCMF s mai n goal s i s t o mak e all o f the participant s awar e o f the tota l spectru m o f activitie s tha t occu r i n math ematics educatio n an d research : w e wis h t o involv e professiona l mathematician s in educatio n an d t o brin g moder n concept s i n mathematic s t o th e attentio n o f educators. T o tha t en d th e summe r institut e feature s genera l session s designe d to encourag e interactio n amon g th e variou s groups . In-yea r activitie s a t th e site s around th e countr y for m a n integra l par t o f th e Hig h Schoo l Teacher s Program . Each summe r a differen t topi c i s chose n a s th e focu s o f th e Researc h Progra m and Graduat e Summe r School . Activitie s i n th e Undergraduat e Summe r Schoo l deal wit h thi s topi c a s well . Lectur e note s fro m th e Graduat e Summe r Schoo l ar e being publishe d eac h yea r i n thi s series . Th e firs t twelv e volume s are : • Volum e 1 : Geometry and Quantum Field Theory (1 991 ) • Volum e 2 : Nonlinear Partial Differential Equations in Differential Geometry (1 992 ) • Volum e 3 Complex Algebraic Geometry (1 993 ) • Volum e 4 Gauge Theory and the Topology of Four-Manifolds (1 994 ) • Volum e 5 Hyperbolic Equations and Frequency Interactions (1 995 ) • Volum e 6 Probability Theory and Applications (1 996 ) • Volum e 7 Symplectic Geometry and Topology (1 997 ) • Volum e 8 Representation Theory of Lie Groups (1 998 ) • Volum e 9 Arithmetic Algebraic Geometry (1 999 ) • Volum e 1 0 : Computational Complexity Theory (2000 ) • Volum e 1 1 : Quantum Field Theory, Super symmetry,and Enumerative Geometry (2001 ) • Volum e 1 2 : Automorphic Forms and their Applications (2002 ) Volumes ar e i n preparatio n fo r th e subsequen t years . xii i

XIV

PREFACE

Some materia l fro m th e Undergraduat e Summe r Schoo l i s publishe d a s par t of the Studen t mathematica l Librar y serie s of the America n Mathematica l Society . We hope t o publis h materia l fro m othe r part s o f the IAS/PCM I i n the future . Thi s will include material fro m th e Hig h Schoo l Teachers Program an d publication s doc umenting th e interactiv e activities , whic h ar e a primar y focu s o f the PCMI . A t th e summer institut e lat e afternoon s ar e devote d t o seminar s o f commo n interes t t o all participants . Man y dea l wit h curren t issue s i n education : other s trea t mathe matical topic s a t a leve l tha t encourage s broa d participation . Th e PCM I ha s als o spawned interaction s betwee n universitie s an d hig h school s a t a loca l level . W e hope t o shar e thes e activitie s wit h a wide r audienc e i n futur e volumes .

https://doi.org/10.1090//pcms/012/01

Introduction The 200 2 IAS/Park Cit y Mathematic s Institut e o n "Automorphi c Form s an d thei r Applications" too k place in Park City , Utah fro m Jul y 1 to Jul y 20 . Approximatel y 57 from th e graduat e progra m plu s man y o f th e researc h progra m participant s at tended differen t portion s of the Graduate Summe r School . Th e school was narrowe d to cove r topic s relate d t o development s i n analyti c aspect s o f th e subjec t a s wel l as a numbe r o f introductor y course s o n relevan t topics . Th e lecture s wer e divide d into si x genera l areas : (1) Basic Theory of Eisenstein Series b y Armand Bore l and Josep h Bernstein , (2) Converse Theorems and Langlands-Shahidi Method b y James Cogdel l an d Preydoon Shahidi , (3) Ramanujan Conjectures and Applications, e.g., Ramanujan Graphs b y Laurent Clozel , Wen-Chin g Winni e L i an d Alain e Valette , (4) Analytic Theory of GL{2) Forms and L-functions b y Phillip e Michel , (5) Arithmetic Quantum Chaos b y Zee v Rudnic k an d Audre y Terras , (6) Unipotent Flows on T\G and Applications b y Ale x Eskin . Also, Pete r Sarna k gav e tw o introductor y lecture s outlinin g th e topic s covere d b y the Graduat e Summe r School . The presen t volum e contain s th e lecture s o f A . Borel , L . Clozel , J . Cogdell , W. Li , P . Michel , F . Shahid i an d A . Terras , togethe r wit h a manuscrip t b y Davi d Vogan, "Isolate d unitar y representations" , whic h i s related t o th e topi c covere d b y Clozel. Recent importan t instance s o f Langland s Functorialit y Conjecture , includin g the existenc e o f the thir d an d th e fourt h symmetri c power s of cusp form s o n GL(2) as automorphi c form s o n GL(4) an d GL(b) establishe d b y Ki m an d Shahidi , hav e relied o n applyin g Convers e Theorem s o f Cogdel l an d Piatetski - Shapiro t o ana lytic propertie s o f certai n automorphi c L-functions , obtaine d fro m th e Langlands Shahidi method . Thi s i s a metho d whic h relie s o n th e stud y o f th e constan t an d non-constant term s o f Eisenstei n serie s attache d t o generi c cuspida l automorphi c representations o f Lev i subgroup s o f maxima l paraboli c subgroup s o f quasi-spli t connected reductiv e group s ove r numbe r fields. I t thu s relie s o n a goo d under standing o f th e theor y o f Eisenstei n series . Lectures b y Bore l an d Bernstei n hav e addresse d thes e issue s b y coverin g bot h the genera l aspect s o f th e theor y o f automorphi c form s o n G(Ap), i.e. , tha t o f un derstanding L 2(ZG(AF)G(F)\G(AF)) o r classicall y L 2(T\G) fo r a reductive grou p G ove r a numbe r fiel d F. Thi s i s th e conten t o f BoreF s lectures . O n th e othe r l

2

INTRODUCTION

hand, Bernstein' s lectures , whic h unfortunatel y ar e no t include d i n thi s volume , addressed th e theor y o f Eisenstei n serie s an d thei r meromorphi c continuatio n vi a spectral theor y an d a principl e o f uniqueness . (For th e cas e that th e Eisenstei n se ries are induced fro m cuspida l dat a o n the Levi subgroup, a treatment alon g simila r such lines is due to Selberg and can be found i n S. Wong [Memoir s of the A.M.S. Vol. 83 (1 990)]. ) Lecture s o n Convers e Theorem s an d th e Langlands-Shahid i metho d were delivere d b y Cogdel l an d Shahid i an d thei r lectur e note s ar e include d here . One of the significant consequence s of the recent result s on functoriality ar e new bounds toward s th e Ramanuja n an d Selber g Conjecture s fo r GL(2) du e t o Kim , Sarnak an d Shahidi . Th e Conjecture s ca n b e formulate d quit e generall y fo r an y connected reductiv e algebrai c grou p ove r a numbe r field o r a functio n field. Th e basic assertio n i s tha t th e loca l component s o f cuspida l representation s o n adeli c points o f thes e group s mus t b e tempered , i.e. , the y mus t hav e matri x coefficient s which li e i n L 2 + e (ZG(F V)\G(FV)) fo r an y e > 0 , wher e F v i s a loca l completio n of F . Her e ZQ i s th e cente r o f G. Fo r GL(n) ove r functio n fields th e conjecture s are a consequenc e o f Lafforgue's proo f o f Langlands ' globa l reciprocit y Conjecture ; however, fo r a numbe r field F nothin g tha t stron g i s known . Wha t i s know n ar e quite shar p approximations . Fo r GL(n) ove r a number field thi s i s due t o Rudnick , Luo an d Sarna k wh o us e method s fro m th e analyti c theor y o f L-function s (specif ically Rankin-Selber g L-functions) . Man y o f th e ne w bound s obtaine d fo r cus p forms o n GL(2) an d fo r othe r reductiv e group s ar e a consequenc e o f th e GL(n) bounds an d transfe r principles . Ther e ar e certai n instance s fo r whic h th e con jecture i s full y establishe d suc h a s tha t b y Delign e fo r classica l holomorphi c cus p forms o n th e uppe r half-plan e an d certai n case s o f Hilbert modula r form s an d uni tary group s (Harris , Taylo r Blasius) . Thes e conjecture s wer e th e topi c o f ClozeF s Lectures whic h ar e include d here . Beside s th e case s mentione d abov e h e als o ad dresses th e connectio n wit h Arthur' s conjecture s o n th e non-tempere d spectru m in L 2(ZG(&F)G(F)\G(AF)) an d th e subgroup , induction/restriction , techniqu e o f Burger-Li-Sarnak givin g globa l informatio n fro m loca l calculations . In th e cas e o f definit e quaternio n algebra s th e no w prove n Ramanuja n Con jectures hav e application s t o combinatoric s an d compute r scienc e throug h th e con struction o f Ramanuja n Graph s (thes e ar e optimall y highl y connecte d bu t spars e graphs calle d "expanders") . Othe r application s o f thi s propert y o f larg e spectra l gaps ar e t o th e countin g an d equi-distributio n o f integra l point s o n homogeneou s spaces. Th e lecture s b y L i whic h ar e include d i n th e volum e cove r thes e topic s a s well as such highe r dimensiona l combinatoria l object s calle d Ramanuja n buildings . The lecture s b y Valett e als o discusse d th e closel y relate d notio n o f propert y T fo r groups. The techniques of studying averages over "familie s o f automorphic L-functions " have prove n t o b e ver y powerfu l i n applications , particularl y fo r establishin g sub convex estimate s fo r automorphi c L-function s o n thei r critica l lin e a s wel l a s fo r establishing non-vanishin g results . Fo r GL2/Q th e sub-convexit y wa s pioneere d b y Duke-Fieldlander an d Iwaniec and especially important i s their method o f amplifica tion. Ther e are plenty of applications through special value formulae fo r L-function s at th e centra l point . Thes e includ e a solutio n o f Hilbert's elevent h proble m o n rep resentations o f integer s b y quadrati c form s ove r numbe r fields (Cogdell-Piatetski Shapiro-Sarnak), rank s o f Jacobian s o f modula r curve s ... . al l thi s an d mor e ar e discussed i n th e detaile d note s b y Miche l whic h ar e included .

INTRODUCTION

3

A topic of current interes t i n mathematical physics is arithmetic quantum chaos. Quantum chao s i s concerned wit h th e stud y o f th e semiclassica l limi t o f th e quan tization o f a classicall y chaoti c Hamiltonian . Ther e i s ampl e numerica l experimen tation i n thi s directio n i n th e physic s literature . Thi s ha s le d t o man y observe d phenomena abou t whic h the usual methods of microlocal analysis can say very little. For a system suc h a s the geodesi c flo w o n th e uni t tangen t bundl e o f a n arithmeti c hyperbolic surface , numbe r theoreti c technique s ca n b e employe d t o resolv e som e of th e centra l problem s (thi s stud y i s no w terme d "arithmeti c quantu m chaos") . The inpu t include s th e proble m o f subconvexit y fo r highe r degre e L-function s a s well a s mor e traditiona l diophantin e analysis . Terras ' lectures , whic h ar e include d here, giv e a n introductio n t o thi s subjec t a s wel l a s a descriptio n an d analysi s o f some finite field analogues . Rudnick' s lectures , whic h ar e no t included , describe d some delicate applications o f the Selber g Trace Formula to study fluctuations o f th e smoothed remainde r ter m i n WeyP s la w fo r th e eigenvalu e coun t fo r th e modula r group. A very powerfu l too l that ha s emerge d i n recent year s in connection wit h prob lems o f equi-distributio n o f measure s i n T\G associate d wit h arithmeti c an d spec tral geometr y i s ergodi c theory . Specificall y Ratner' s Theore m give s a complet e and simpl e classificatio n o f suc h measure s whic h ar e invarian t unde r a unipoten t action. Thi s allow s one, in case s wher e suc h a n invarianc e ca n b e demonstrated , t o establish quit e genera l an d ofte n strikin g equi-distributio n results . I n hi s lecture s Eskin gav e a n accoun t o f Ratner's wor k a s wel l a s a numbe r o f applications. Thes e included quantitativ e version s o f Oppenheim' s Conjectur e du e t o Eskin , Marguli s and Moze s a s wel l a s arithmetica l application s du e t o Vatsa l an d Cornu t t o rank s of a n ellipti c curve , define d ove r Q , i n anticyclotomi c towers . Since th e tim e o f th e conferenc e ther e hav e bee n man y impressiv e advance s in th e abov e topics . Noteworth y amon g the m i s th e solutio n b y E . Lindenstraus s (Annals of Math., Vol .1 6 3 (2006) , 1 65-21 9 ) o f th e quantu m uniqu e ergodicit y conjecture fo r hyperboli c arithmeti c surfaces . I t combine s th e numbe r theoreti c methods wit h recen t advance s b y Einsiedler-Katok-Lindenstraus s o n classificatio n of measure s o n T\G whic h ar e invarian t unde r highe r ran k Carta n actions . The succes s o f th e 200 2 Graduat e Summe r Schoo l lie s o n th e contribution s from man y people . Ou r warmes t thank s ar e du e t o ou r lecturer s fo r thei r clea r and accessibl e presentations . W e ar e particularl y gratefu l t o thos e wh o provide d us wit h thei r lectur e notes . W e als o lik e t o than k ou r teachin g assistant s fo r thei r useful proble m session s a s wel l a s thei r hel p i n preparin g th e lectures . Finally thank s ar e du e t o Her b Clemen s an d th e entir e PCM I Steerin g Com mittee for askin g us to organize thi s Summe r School , an d particularl y Joh n Morga n and Joh n Polkin g fo r al l thei r help . Last , bu t no t least , w e lik e t o than k Cather ine Giesbrech t an d th e entir e IAS/PCM I staf f fo r helpin g mak e thi s progra m th e success i t became . Peter Sarna k an d Freydoo n Shahidi , Volume Editor s an d Graduate Summe r Schoo l Organizer s June, 200 6

This page intentionally left blank

https://doi.org/10.1090//pcms/012/02

Automorphic Form s o n Reductiv e Group s Armand Bore l

This page intentionally left blank

IAS/Park Cit y Mathematic s Serie s Volume 1 2 , 200 2

Automorphic Form s o n Reductiv e Group s Armand Bore l

Introduction The goa l o f thes e note s i s th e basi c theor y o f automorphi c form s o n reductiv e groups, u p t o an d includin g th e convergenc e o f Eisenstei n serie s fo r larg e value s of the parameters . The book [6 ] was written wit h th e general case in mind. Familiarit y wit h i t wil l be helpful . W e shal l als o refe r t o i t fo r proof s vali d wit h littl e o r n o modificatio n in th e genera l case .

1. Notatio n 1.1. Le t X b e a set an d f yg strictl y positiv e rea l function s o n X. W e write f 0 suc h tha t f(x) < cg(x) fo r al l # £ X; similarly , / y g i f g - < /, an d / x g i f / ~ < g an d g -< f. 1.2. Le t G b e a group . Th e lef t (resp . right ) translatio n b y g G G i s denote d l g (resp r g); thes e ac t o n function s vi a (1) l

g

• fix) = f{9~ lx), r 9

g

• f{x) = f(xg)

x

If A i s a subse t o f G , the n A = g.A.g~ an d MA = {g e G | 9 A = A], ZA={geG |

9

a = a,(aeA)}.

1.3. Le t G b e a Li e group an d g its Li e algebra. Th e latte r ma y b e viewe d a s th e tangent spac e T\ (G) a t th e identity , o r a s th e spac e o f left-invarian t vecto r fields on G. I f X\ G Ti(G), th e associate d vecto r field i s x i— • x • X\. Th e actio n o f X o n functions i s given b y

Xf=±f(xetx)\t=0 1

School of mathematics, Institut e fo r Advance d Study , Einstei n Dr. , Princeton , N J 08540 . ©2007 America n Mathematica l Societ y 7

8

ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS

The universa l envelopin g algebr a U(g) i s identified wit h th e algebr a of left-invarian t differential operators ; th e elemen t X1 X 2 . .. X n act s vi a XXX2 . . . X nf(x) =

^

A+

ati . . . at

/(xe

tlXl

e ^ . . . e ^)\U =

0.

n

Let Z(g) b e th e cente r o f U{g). I f G i s connected , i t i s identifie d wit h th e algebra o f lef t an d righ t invarian t differentia l operators . I f G i s connecte d an d reductive, i t i s a polynomia l algebr a o f ran k equa l t o th e ran k o f G. 1.4. Le t G b e unimodular . Th e convolutio n u *v o f two function s i s defined b y (2) u*v{x)=

\

u(xy)v{y~ x)dy =

/ u(y)v(y~

1

x)dy

JG JG

whenever th e integra l converges . I t i s a smoothin g operator : i f u i s continuous an d v G C™(G), the n (3) D(u*v) =u*Dv 1 DeU(g) and, i n particular , u*v G C°°(G). I t extend s t o distribution s an d i s associative. I f g is identifie d t o distribution s wit h suppor t {1 } , then Xf = / * (—X); se e sectio n 2.2 o f [6] .

2. Notio n o f automorphi c for m Let G be a subgrou p o f finite inde x i n the grou p o f real points o f a connected semi simple algebrai c grou p G define d ove r R . Le t K b e a maxima l compac t subgrou p of G. The n X = G/K i s th e Riemannia n symmetri c spac e o f noncompac t typ e o f G. Le t T C G b e a discret e subgroup . 2.1. A continuous functio n / G G(G, C) i s an automorphi c for m fo r T i f it satisfie s the followin g conditions : (Ai) / ( 7 i ) = / ( i ) ( 7 e r , i £ G ) . (A2) / i s i^-finit e o n th e right . (A3) / i s Z(fl)-finite . (A4) / i s o f moderat e growt h (o r slowl y increasing) . Explanation, f i s infinit e o n th e righ t righ t mean s tha t th e se t o f righ t translate s rkf, k G K i s containe d i n a finite dimensiona l space . / i s Z(g)-finit e mean s tha t Z(g)f i s finite dimensiona l or , equivalently , tha t ther e exist s a n idea l J o f finite co-dimension i n Z(g) whic h annihilate s / . I f / i s not C°° , thi s i s understood i n th e sense o f distributions , bu t i n an y cas e / wil l b e analytic , cf . below . B y definition , G C SLiv(M) , an d i s closed . Le t ||p| | b e th e Hilbert-Schmid t nor m o f g G S L ^ M ). Thus ||g|| 2 = t r ^g.g) = Y2i j 9ij- The n / i s of moderate growth o r slowly increasing if there exist s m G Z suc h tha t \f(x)\*\\x\\m,(x€G). Let u m b e th e semi-nor m o n C(G 1 C ) define d b y u m(f) = sup/(x).||x||~ m . The n / is slowly increasin g i f an d onl y i f v m(f) < oo for som e m. W e shal l sometime s cal l m a boun d fo r th e growth . W e not e som e elementar y propertie s o f | | ||: (nl) \\x.y\\ < ||x||.||2/|| , an d ther e i s a n m suc h tha t ||x _ 1 || ^ < ||#||m . (n2) I f C , C ar e compact subset s o f G , the n He.y.e' H x \\y\\ (c G G, d G C\y G G).

3. FIRS T PROPERTIE S O F AUTOMORPHI C FORM S

9

2.2. R e m a r k . Th e notio n o f moderat e growt h (bu t no t th e exponen t m) i s independent o f th e embedding . On e ca n als o defin e a canonica l Hilbert-Schmid t norm a s follows: O n g, let K(x, y) = t r (adxoady ) b e the Killing form, an d le t 8 be the Carta n involutio n o f G wit h respec t t o K. The n th e for m (x , y) = —K(6x, y) is positiv e definit e o n g , an d th e associate d Hilbert-Schmid t nor m o n th e adjoin t group i s ||p|| 2 = t r ( A d % - 1 . A d # ) (Exercise) . 2.3. Relatio n wit h classica l automorphi c form s o n th e uppe r hal f plane . Here G = SL 2 (R), K = S 0 2 , an d X = {z € C \ Sz > 0 }, th e actio n o f G bein g defined b y (a b\ az +b \c dj cz + dK } Let (cz + d) m = /j,(g, z). I t i s an automorph y factor , i.e . (4) V>(9-9\ z) = /i(p , 9'-z)v>(9\ z )K i s th e isotrop y grou p o f i € X. Equatio n (4 ) give s fo r k,k f € K, an d z = i , i) = /x(fc, i)/i(^/ , i),

(5) /i(fcfc'

i.e. k i-^ * /i(fc , i) i s a characte r Xm o f K. Let r b e a subgrou p o f finite inde x i n SL 2 (Z). A n automorphi c for m f on X of weigh t m i s a functio n satisfyin g

(Ai>) f( 7.z) = M(7,z)f(z) (

7er,2ei)

(A2') / i s holomorphi c (A3') / i s regular a t th e cusps . Let / b e th e functio n o n G define d b y

f(g) = »(9,ir

1

f(g.i).

Then (Al' ) fo r / implie s (Al ) an d (A2 ) fo r / b y a simpl e computatio n usin g (5) . Note i n particula r (6) f(g.k)

= ~f{g)x~m{k).

The conditio n (A2' ) implie s tha t / i s a n eigenfunctio n o f th e Casimi r operato r C. As C generate s 2(g), thi s yields (A3) . Conside r the cusp at oo . I n the inverse image of th e "Siege l set " \x\ < c an d y > t , wher e c an d t ar e positiv e constants , i t i s easily seen that \\g\\ x yz , henc e moderate growt h mean s - < y 771 fo r som e m. O n th e other hand , T contain s a translatio n x n a ; + p, fo r som e non-zer o intege r p. Sinc e / i s invarian t unde r thi s translatio n i n th e x direction , / admit s a developmen t i n a Lauren t serie s J2 n a n exp( 2 7 r m z ). Fo r / t o b e bounde d b y y m i n th e Siege l set , i t is necessar y an d sufficien t tha t a n = 0 fo r n < 0 . Thi s i s th e regularit y conditio n (A3'), (cf . [6] , 5.1 4 ) 3 . Firs t p r o p e r t i e s o f a u t o m o r p h i c form s In thi s section , G , K, X, an d T ar e a s i n 1 .1 , and / i s an automorphi c for m fo r I \ 3.1. / is analytic. Fo r thi s i t suffices , b y a regularit y theorem , t o sho w tha t i t i s annihilated b y a n analyti c ellipti c operator . Let g = t ® p b e th e Carta n decompositio n o f g , wher e p i s th e orthogona l complement o f t wit h respec t t o th e Killin g form . Th e latte r i s negativ e (resp . positive) definit e o n t (resp . p) . Le t {x^ (resp . {%} ) b e a n orthonorma l basi s o f

10

ARMAND BOREL, AUTOMORPHIC FORMS ON REDUCTIVE GROUPS

t (resp . p) . The n a Casimi r operato r ca n b e writte n C g — — ^xj + J2vh a n d Ct = ]T ^ xf i s a Casimi r operato r fo r t. Le t O = 2C$ + C g. I t i s an analyti c ellipti c operator. W e clai m tha t / i s annihilate d b y som e non-constan t polynomia l i n O , which will prove our assertion . Th e functio n / i s annihilated b y a n idea l J o f finite codimension in Z(g) (b y (A3 ) ) and, sinc e / i s K-finite o n the right, i t is annihilate d by a n idea l [ of finite codimensio n o f U{t). Therefor e / i s annihilate d b y a n ide a of finite codimensio n o f th e subalgebr a Z(g).U(t) oiU(g). Bu t the n ther e exist s a polynomial o f strictl y positiv e degre e P(£L) in O belonging t o tha t ideal . 3.2. A functio n a o n G i s sai d t o b e X-invarian t i f a(k.x) = a(x.k) fo r al l x G G, k G K. W e hav e th e followin g theorem : Theorem. Given a neighborhood U of 1 in G, there exists a K-invariant function a G C2°(U) such that f = / • a . This follow s fro m th e fac t tha t / i s Z-finit e an d X-finit e o n on e side , b y a theorem o f Harish-Chandra . Se e ([1 3] , theore m 1 ) o r ([1 ] , 3.1 ) , an d fo r SL2(R ) ([6], 2.1 3) . Als o se e Sectio n 9. 4 below . 3.3. A smooth functio n u on G is said to be of uniform moderate growth (bounde d by m) i f ther e exist s m G Z suc h tha t v m(Df) < o o fo r al l D €lt(g). An elementar y computatio n show s tha t i f ^ m (u) < oo , the n oo

vm(u*a)
0 such tha t W 0 such tha t xCy C Gciiaii.Hj/Hj s o that (9 ) implie s (1 0) . 3.5. W e shal l nee d th e followin g lemma : Lemma. Let a E C£° (G). There exists n £ N such that \u*a(x)\ -
0, SU

ca(x) x-^ £

)(x)\
.x)

However, th e derivativ e i s alon g e sXj viewe d a s a rig/i£-invarian t vecto r field, whereas w e conside r usuall y derivative s wit h respec t t o /e^-invarian t fields (se e Section 1 .2) . Se t y s = e sXjx; i t lie s in S ' . (72) jLf

1 j+

(e»x'x) =

±f j+l(erX'ya)\r=o =

^fi+iiVs-yT^^y.)^

Since th e exponential commute s wit h G , we have: y-\