THE LOCAL LANGLANDS CONJECTURE FOR GL(2)
BASE CHANGE WE REGARD THIS CASE AS BEING PARTICULARLY IMPORTANT IT REMAINS THE ONE INSTANCE OF THE LOCAL CONJECTURE IN WHICH THE DETAIL IS SU?CIENTLY COMPLEX TO BE INTERESTING, YET SU?CIENTLY VISIBLE TO ILLUMINATE THE MIRACLE THAT IS THE LANGLANDS CORRESPONDENCE EVEN AFTER 25 YEARS, IT STANDS AS A STURDY CORRECTIVE TO OVER-OPTIMISTIC ATTITUDES TO MORE GENERAL PROBLEMS AS LIGHT RELIEF, WE HAVE BROADENED THE PICTURE WITH SOME DISCUSSION OF ?-ADIC REPRESENTATIONS, SINCE THESE PROVIDE A FORUM IN WHICH THE CORRESPONDENCE ?NDS MUCH OF ITS APPLICATION THE ?NAL CHAPTER XIII STANDS OUTSIDE THE MAIN SEQUENCE THERE, D IS THE QUATERNION DIVISION ALGEBRA OVER F THE IRREDUCIBLE REPRESENTATIONS OF D× = GL1(D) CAN BE CLASSI?ED BY A METHOD PARALLEL TO THAT USED FOR GL2(F) THE JACQUET-LANGLANDS CORRESPONDENCE PROVIDES A CANONICAL CONNECTION BETWEEN THE REPRESENTATION THEORIES OF D× AND GL2(F) WE INCLUDE IT AS AN INDICATION OF FURTHER DIMENSIONS IN THE SUBJECT GIVEN THE EXPERIENCE OF GL2(F), IT IS A FAIRLY STRAIGHTFORWARD MATTER WHICH WE HAVE LEFT AS A SEQUENCE OF EXERCISES ACKNOWLEDGEMENT THE ?NAL DRAFT WAS READ BY CORINNE BLONDEL, WHOSE ACUTE COMMENTS LED US TO REMOVE A LARGE NUMBER OF MINOR ERRORS AND OBSCURITIES, ALONG WITH A COUPLE OF MORE SIGNI?CANT LAPSES IT IS A PLEASURE TO RECORD OUR DEBT TO HER NOTATION WE LIST SOME STANDARD NOTATIONS WHICH WE USE REPEATEDLY, WITHOUT ALWAYS RECALLING THEIR MEANING F = A NON-ARCHIMEDEAN LOCAL ?ELD; O = THE DISCRETE VALUATION RING IN F; P = THE MAXIMAL IDEAL OF O; K = O/P; P = THE CHARACTERISTIC OF K; Q = |K|; UF = THE GROUP OF UNITS OF O; U N F = 1+PN, N ?1 (THUS THE CHARACTERISTIC OF F IS 0 OR P: WE NEVER NEED TO IMPOSE ANY FURTHER RESTRICTION ) IN ADDITION, ?F : F × ?Z IS THE NORMALIZED (SURJECTIVE) ADDITIVE VALUATION AND ?X?= Q??F (X) WE DENOTE BY µF THE GROUP OF ROOTS OF UNITY IN F OF ORDER PRIME TO P IF E/F IS A ?NITE ?ELD EXTENSION, WE USE THE ANALOGOUS NOTATIONS OE, PE, ETC THE NORM MAP E× ?F × IS DENOTED NE/F , AND THE TRACE E ?F IS TRE/F THE RAMI?CATION INDEX AND THE RESIDUE CLASS DEGREE ARE E(E|F), F(E|F) RESPECTIVELY THE DISCRIMINANT IS DE/F = PD+1, D = D(E|F) THE SYMBOL TR IS RESERVED FOR THE TRACE OF AN ENDOMORPHISM, SUCH AS A MATRIX OR A GROUP REPRESENTATION, AND DET IS INVARIABLY THE DETERMINANT IF R IS A RING WITH 1, R× IS ITS GROUP OF UNITS AND MN(R) IS THE RING OF N×N MATRICES OVER R WHEN R IS COMMUTATIVE, GLN(R) (RESP SLN(R)) IS THE THE LOCAL LANGLANDS CONJECTURE FOR GL(2) 0 ? , 1 ? 0 , A 0 A RESPECTIVELY UNLESS OTHERWISE SPECI?ED, A = M2(F) AND G = GL2(F) NOTES FOR THE READER PREREQUISITES WE ASSUME THE BEGINNINGS OF THE REPRESENTATION THEORY OF ?NITE GROUPS, INCLUDING MACKEY THEORY: THE ?RST 11 SECTIONS OF [77] COVER IT ALL, BAR A COUPLE OF RESULTS REQUIRING REFERENCE TO [26] OF NON-ARCHIMEDEAN LOCAL ?ELDS, WE NEED GENERAL STRUCTURE THEORY AS FAR AS THE DISCRIMINANT AND STRUCTURE OF TAME EXTENSIONS, PLUS BEHAVIOUR OF THE NORM IN TAME OR QUADRATIC EXTENSIONS PRACTICALLY EVERYTHING CAN BE FOUND IN [30] OR THE ?RST TWO PARTS OF [76], WHILE [87] IS THE SOURCE OF MANY OF THE IDEAS HERE FROM TOPOLOGY AND MEASURE THEORY, BEYOND THE MOST ELEMENTARY CONCEPTS, WE COVER PRACTICALLY EVERYTHING WE NEED ALL THIS MATERIAL IS COMMONLY AVAILABLE IN MANY BOOKS: WE MENTION ONLY PERSONAL FAVOURITES FROM CHAPTER VII ONWARDS, WE RELY ON LOCAL CLASS ?ELD THEORY NO DETAIL IS INVOLVED, SO WE HAVE BEEN ABLE TO TAKE AN AXIOMATIC APPROACH THE READER MIGHT CONSULT THE COMPACT [68] OR [74], [76] MORE SERIOUS IS THE TREATMENT IN §30 OF THE EXISTENCE OF THE LANGLANDS-DELIGNE LOCAL CONSTANT THIS DEPENDS ON AN INTERPLAY BETWEEN LOCAL AND GLOBAL ?ELDS USING SOME DEEP (BUT CLASSI- CAL) THEOREMS THE READER COULD AGAIN TAKE AN AXIOMATIC APPROACH WE HAVE INCLUDED A BRIEF ACCOUNT WHICH IS COMPLETE MODULO THE CLASSICAL BACKGROUND (THE REQUISITE MATERIAL IS IN [68] OR [54] ) NAVIGATION SECTIONS ARE NUMBERED CONSECUTIVELY THROUGHOUT THE BOOK EACH SECTION IS DIVIDED INTO (USUALLY) SHORT PARAGRAPHS, NUMBERED IN THE FORM Y Z A REFERENCE Y Z PROPOSITION MEANS THE (ONLY) PROPOSITION IN PARAGRAPH Y Z CHAPTER I STANDS ALONE, AND COULD SERVE AS AN INTRODUCTION TO MUCH WIDER AREAS CHAPTER II IS ELEMENTARY, AND COULD BE READ ?RST PARTS OF CHAPTERS VII AND X CAN BE READ INDEPENDENTLY CHAPTER XIII COULD BE READ DIRECTLY AFTER CHAPTER VI OTHERWISE, THE LOGICAL DEPENDENCE IS LINEAR AND FAIRLY RIGID PRINCIPAL SERIES (OR NON-CUSPIDAL) REPRESENTATIONS FORM A DISTINCT SUB- THEME AT A ?RST READING, THIS COULD BE EDITED OUT OR PURSUED EXCLUSIVELY, ACCORDING TO TASTE (FOR A DI?ERENT APPROACH, EMPHASIZING NON-CUSPIDAL REP- RESENTATIONS AND THEIR IMPORTANCE FOR L-FUNCTIONS, SEE BUMP’S BOOK [10] ) ANOTHER “SHORT COURSE” OPTION WOULD BE TO STOP AT THE END OF CHAPTER VIII, BY WHICH STAGE THE ARGUMENT IS COMPLETE FOR ALL BUT DYADIC ?ELDS F THE LOCAL LANGLANDS CONJECTURE FOR GL(2) EXERCISES A FEW EXERCISES ARE SCATTERED THROUGH THE TEXT THESE ARE INTENDED TO ILLUMINATE, ENTERTAIN, OR TO INDICATE DIRECTIONS WE DO NOT FOLLOW ONLY THE SIMPLER ONES EVER MAKE A SERIOUS CONTRIBUTION TO THE MAIN ARGUMENT NOTES WE HAVE APPENDED BRIEF NOTES OR COMMENTS TO SOME CHAPTERS, TO IN- DICATE FURTHER READING OR WIDER PERSPECTIVES THEY TEND TO PRE-SUPPOSE GREATER EXPERIENCE THAN THE MAIN TEXT HISTORY WE HAVE WRITTEN AN ACCOUNT OF THE SUBJECT, NOT ITS HISTORY: THAT WOULD BE A SEPARATE PROJECT OF COMPARABLE SCOPE WE HAVE MADE NO ATTEMPT AT A COMPLETE BIBLIOGRAPHY WE HAVE CITED SOURCES OF MAJOR IMPORTANCE, AND THOSE WE HAVE FOUND HELPFUL IN THE PREPARATION OF THIS VOLUME WE HAVE ALSO MENTIONED A NUMBER OF RECENT WORKS, ALONG WITH OLDER ONES THAT, IN OUR OPIN- ION, REMAIN VALUABLE TO ONE LEARNING THE SUBJECT THE LOCAL LANGLANDS CONJECTURE FOR GL(2) THE LOCAL LANGLANDS CONJECTURE FOR GL(2) SMOOTH REPRESENTATIONS 1 LOCALLY PROFINITE GROUPS 2 SMOOTH REPRESENTATIONS OF LOCALLY PROFINITE GROUPS 3 MEASURES AND DUALITY 4 THE HECKE ALGEBRA THIS CHAPTER IS INTRODUCTORY AND FOUNDATIONAL IN NATURE WE DE?NE A CLASS OF TOPOLOGICAL GROUPS, THE LOCALLY PRO?NITE GROUPS, AND STUDY THEIR SMOOTH REPRESENTATIONS ON COMPLEX VECTOR SPACES THESE REPRESENTATIONS ARE OFTEN IN?NITE-DIMENSIONAL, BUT SMOOTHNESS IMPOSES A DRASTIC CONTINUITY CONDITION NONTHELESS, THIS CLASS OF OBJECTS IS QUITE WIDE: IT INCLUDES, FOR EXAMPLE, ALL REPRESENTATIONS OF DISCRETE GROUPS WE START BY RECALLING SOME STANDARD FACTS WE THEN DEVELOP THE ELEMEN- TARY ASPECTS OF SMOOTH REPRESENTATION THEORY, VERY MUCH GUIDED BY THE OR- DINARY REPRESENTATION THEORY OF ?NITE GROUPS WE OCCASIONALLY TURN TO NON- ARCHIMEDEAN LOCAL ?ELDS AS A SOURCE OF EXAMPLES THE TOPIC OF HAAR MEASURE AND INTEGRATION ON TOPOLOGICAL GROUPS NECESSARILY ENTERS THE PICTURE SINCE WE HAVE ONLY TO DEAL WITH LOCALLY PRO?NITE GROUPS, THIS IS A STRAIGHTFORWARD MATTER OF WHICH WE GIVE JUST AS MUCH AS WE NEED WHILE WE WILL ULTIMATELY BE CONCERNED ONLY WITH NON-ARCHIMEDEAN LOCAL ?ELDS F AND ASSOCIATED GROUPS LIKE GL2(F), THERE IS NOTHING TO BE GAINED FROM SPECIALIZATION AT THIS STAGE LOOKING BEYOND THE CON?NES OF THE PRESENT BOOK, THERE IS MUCH TO BE LOST WE THEREFORE WORK, THROUGHOUT THIS CHAPTER, IN QUITE EXTREME GENERALITY