Grundlehren der 335 mathematischen Wissenschaften ASeriesofComprehensiveStudies inMathematics Serieseditors M.Berger B.Eckmann P.delaHarpe F.Hirzebruch N.Hitchin L.Hörmander M.-A.Knus A.Kupiainen G. Lebeau M.Ratner D.Serre Ya.G.Sinai N.J.A.Sloane B.Totaro A.Vershik M.Waldschmidt Editor-in-Chief A.Chenciner J.Coates S.R.S.Varadhan . Colin J. Bushnell Guy Henniart The Local Langlands Conjecture for GL(2) ABC Colin J. Bushnell King's College London Department of Mathematics Strand, London WC2R 2LS UK e-mail: [email protected] Guy Henniart Université de Paris-Sud et umr 8628 du CNRS Département de Mathématiques Bâtiment 425 91405 Orsay France e-mail: [email protected] LibraryofCongressControlNumber:2006924564 MathematicsSubjectClassification(2000): Primary:i1F70,22E50,20G05 Secondary:i1D88,11F27,22C08 ISSN0072-7830 ISBN-10 3-540-31486-5SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-31486-8SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorand SPiusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11604440 41/SPI 543210 For Elisabeth and Lesley Foreword This book gives a complete and self-contained proof of Langlands’ conjecture concerning the representations of GL(2) of a non-Archimedean local field. It has been written to be accessible to a doctoral student with a standard grounding in pure mathematics and some extra facility with local fields and representations of finite groups. It had its origins in a lecture course given by the authors at the first Beijing-Zhejiang International Summer School on p-adic methods, held at Zhejiang University Hangzhou in 2004. We hope this isfoundafittingresponsetotheeffortsoftheorganizersandtheenthusiastic contribution of the student participants. King’s College London and Universit´e de Paris-Sud. Contents Introduction................................................... 1 Notation.................................................... 3 Notes for the reader .......................................... 4 1 Smooth Representations ................................... 7 1. Locally Profinite Groups ................................. 8 2. Smooth Representations of Locally Profinite Groups ......... 13 3. Measures and Duality.................................... 25 4. The Hecke Algebra ...................................... 33 2 Finite Fields............................................... 43 5. Linear Groups .......................................... 43 6. Representations of Finite Linear Groups.................... 45 3 Induced Representations of Linear Groups................. 49 7. Linear Groups over Local Fields........................... 50 8. Representations of the Mirabolic Group .................... 56 9. Jacquet Modules and Induced Representations .............. 61 10. Cuspidal Representations and Coefficients .................. 69 10a. Appendix: Projectivity Theorem .......................... 73 11. Intertwining, Compact Induction and Cuspidal Representations ......................................... 76 4 Cuspidal Representations.................................. 85 12. Chain Orders and Fundamental Strata ..................... 86 13. Classification of Fundamental Strata ....................... 95 14. Strata and the Principal Series ............................100 15. Classification of Cuspidal Representations ..................105 16. Intertwining of Simple Strata .............................111 17. Representations with Iwahori-Fixed Vector .................115 X Contents 5 Parametrization of Tame Cuspidals ........................123 18. Admissible Pairs ........................................123 19. Construction of Representations...........................125 20. The Parametrization Theorem ............................129 21. Tame Intertwining Properties .............................131 22. A Certain Group Extension...............................134 6 Functional Equation .......................................137 23. Functional Equation for GL(1) ............................138 24. Functional Equation for GL(2) ............................147 25. Cuspidal Local Constants ................................155 26. Functional Equation for Non-Cuspidal Representations.......162 27. Converse Theorem.......................................170 7 Representations of Weil Groups ...........................179 28. Weil Groups and Representations..........................180 29. Local Class Field Theory .................................186 30. Existence of the Local Constant ...........................190 31. Deligne Representations ..................................200 32. Relation with (cid:1)-adic Representations.......................201 8 The Langlands Correspondence............................211 33. The Langlands Correspondence ...........................212 34. The Tame Correspondence................................214 35. The (cid:1)-adic Correspondence ...............................221 9 The Weil Representation ..................................225 36. Whittaker and Kirillov Models ............................226 37. Manifestation of the Local Constant .......................230 38. A Metaplectic Representation .............................236 39. The Weil Representation .................................245 40. A Partial Correspondence ................................249 10 Arithmetic of Dyadic Fields ...............................251 41. Imprimitive Representations ..............................251 42. Primitive Representations ................................257 43. A Converse Theorem.....................................262 11 Ordinary Representations .................................267 44. Ordinary Representations and Strata ......................267 45. Exceptional Representations and Strata ....................279 Contents XI 12 The Dyadic Langlands Correspondence ....................285 46. Tame Lifting............................................286 47. Interior Actions .........................................295 48. The Langlands-Deligne Local Constant modulo Roots of Unity................................................297 49. The Godement-Jacquet Local Constant and Lifting ..........304 50. The Existence Theorem ..................................307 51. Some Special Cases ......................................313 52. Octahedral Representations...............................316 13 The Jacquet-Langlands Correspondence ...................325 53. Division Algebras........................................326 54. Representations .........................................328 55. Functional Equation .....................................331 56. Jacquet-Langlands Correspondence ........................334 References.....................................................339 Index..........................................................345 Some Common Symbols ......................................349 Some Common Abbreviations .................................351 Introduction We work with a non-Archimedean local field F which, we always assume, has finite residue field of characteristic p. ThusF is either a finite extension of the field Q of p-adic numbers or a field F ((t)) of formal Laurent series, in p pr one variable, over a finite field. The arithmetic of F is encapsulated in the Weil group W of F: this is a topological group, closely related to the Galois F group of a separable algebraic closure of F, but with rather more sensitive properties. One investigates the arithmetic via the study of continuous (in theappropriatesense)representationsofW overvariousalgebraicallyclosed F fieldsofcharacteristiczero,suchasthecomplexfieldCorthealgebraicclosure Q of an (cid:1)-adic number field. (cid:1) Sticking to the complex case, the one-dimensional representations of W F are the same as the characters (i.e., continuous homomorphisms) F× → C×: this is the essence of local class field theory. The n-dimensional analogue of a character of F× = GL (F) is an irreducible smooth representation of the 1 group GL (F) of invertible n×n matrices over F. As a specific instance of a n widespeculativeprogramme,Langlands[55]proposed,inapreciseconjecture, that such representations should parametrize the n-dimensional representa- tions of W in a manner generalizing local class field theory and compatible F with parallel global considerations. The excitement provoked by the local Langlands conjecture, as it came to beknown,stimulated aperiodofintenseandwidespreadactivity, reflectedin the pages of [8]. The first case, where n = 2 and F has characteristic zero, was started in Jacquet-Langlands [46]; many hands contributed but Kutzko, bringing two new ideas to the subject, completed the proof in [52], [53]. Sub- sequently, the conjecture has been proved in all dimensions, first in positive characteristic by Laumon, Rapoport and Stuhler [58], then in characteristic zerobyHarrisandTaylor [38], alsobyHenniart[43]onthebasisofanearlier paper of Harris [37]. 2 Introduction Throughout the period of this development, the subject has largely re- mained confined to the research literature. Our aim in this book is to provide a navigable route into the area with a complete and self-contained account of thecasen=2,inatolerablenumberofpages,relyingonlyonmaterialreadily available in standard courses and texts. Apart from a couple of unavoidable caveats concerning Chapter VII, we assume only the standard representation theory for finite groups, the beginnings of the theory of local fields and some very basic notions from topology. Inconsequence,ourmethodsareentirelylocalandelementary.Apartfrom ChapterI(whichcouldequallyserveasthestartofatreatiseontherepresen- tation theory of p-adic reductive groups) and some introductory material in Chapter VII, we eschew all generality. Whenever possible, we exploit special features of GL(2) to abbreviate or simplify the arguments. Thedesiretobebothcompactandcompleteremovestheoptionofappeal- ing to results derived from harmonic analysis on ad`ele groups (“base change” [57],[1])whichoriginallyplayedadeterminingroˆle.Thisparticularconstraint hasforcedustogivethefirstproofoftheconjecturethatcanclaimtobecom- pletely local in method. Thereisanassociatedloss,however.ThelocalLanglandsConjectureisjust aspecificinstanceofawideprogramme,encompassinglocalandglobalissues and all connected reductive algebraic groups in one mighty sweep. Beyond the minimal gesture of Chapter XIII, we can give the reader no idea of this. Nor have we mentioned any of the geometric methods currently necessary to prove results in higher dimensions. Fortunately, the published literature contains many fine surveys, from Gelbart’s book [32], which still conveys the breadth and excitement of the ideas, to the new directions described in [4]. The approach we take is guided by [46] and [50–53], but we have re- arranged matters considerably. We have separated the classification of repre- sentationsfromthefunctionalequation.WehaveimportedideasofBernstein and Zelevinsky into the discussion of non-cuspidal representations. While the treatmentofcuspidalrepresentationsisessentiallythatofKutzko,itisheavily informed by hindsight. We have given precedence to the Godement-Jacquet version of the functional equation and so had to treat the Converse Theorem inanovelmanner,owingsomethingtoideasofG´erardinandLi.Thereisalso somedegreeofnoveltyinourtreatmentoftheKirillovmodelandtherelation between the functional equation it gives and that of Godement and Jacquet. We have given a quick and explicit proof of the existence of the Langlands correspondence, in the case p(cid:2)=2, at an early stage. The case p = 2 has many pages to itself. The method is essentially that of Kutzko, but we have had to bring a new idea to the closing pages (the treatment of the so-called octahedral representations) to avoid an appeal to