A user's guide to the trace formula for covering groups - Wen
Contents
1. Jy fv depends on e the choice of Ky a Haar measure on the R v s a the ratio of the Haar measures on G F and G F For general S can be reduced to S v via Arthur s splitting formula e For general defined by a limiting process Trace formula for covers July 24 2012 19 39 Non invariant trace formula The refined geometric side Let us return to the refined geometric expansion We have we F ee gt J f 3 ute gt a 8 75 Ji 75 f MEL Mo O vec Fy X89 ws S depending on Supp f M F m s the set of M S equivalence classes defined by Arthur _ eee MS components outside S are in X and whose local component g in Mg is good using p M Fs x K5 Ms x K9 e The correspondence y yg described as above the classes y admitting a representative whose Trace formula for covers July 24 2012 20 39 CR esse One difficulty Arthur s proof can be adapted to prove the refined geometric expansion But it requires new ideas as well In the course of proof one has to show that a S 78 I ity 78 f behaves well with respect to the correspondence y Yg Unlike the case treated by Arthur the character o intervenes and i need a somewhat technical result of transport of structure for a S 7s and PAGE Trace formula for covers July 24 2012 21 39 Non invariant trace formula The refined geometric side Compression of coefficients2. Note the set of places S depends on the support of f Goal we want an expression depending only on a set V containing v vo and the ramified places and we use test functions of the form f fv fgv where fgv is the unit of the antigenuine spherical Hecke algebra w r t GY KV Theorem For V as above there are a set I L V of good conjugacy classes in Ly coefficients a 7 for T L V such that Trace formula for covers July 24 2012 22 39 Non invariant trace formula The refined geometric side The compressed coefficients al 5 are defined in terms of the global coefficients aM 5 for various M C L and S D V large enough the unramified weighted orbital integrals r i e the weighted orbital integrals of the unit in the genuine unramified spherical Hecke algebra GY w r t KY Trace formula for covers July 24 2012 23 39 Non invariant trace formula The refined spectral side The refined spectral side Suppose f to be left and right K finite from now on Then W If gt gt gt DT MEL Mo ren M1 LEL M sEWL M det s Lla s G tr Mz P EE f x n dr ular TI M the set of unitary irreps of M up to equivalence P P M arbitrary Mp p s 9 global intertwining operators M P an operator defined by a G L family arising from intertwining operators o fe unitary parabolic induction Trace formula for covers July 24
3. However the latter group is less manageable Trace formula for covers July 24 2012 16 39 Non invariant trace formula The refined geometric side Basic ideas for the refined geometric expansion Reduce to the study of JEZ D f unip Express Jae in terms of weighted unipotent orbital integrals twisted by o adapt Arthur s arguments Define weighted orbital integrals on G and deduce analogous descent formulas Compare the descent formulas to express 37 Jo f in terms of weighted orbital integrals on G Trace formula for covers July 24 2012 17 39 Non invariant trace formula The refined geometric side The result will be expressed in terms of good weighted orbital integrals Jus fs where S a sufficiently large finite set of places containing the archimedean and the ramified ones depending on Supp f o Ms p M Fs f fsfks where fgs is the unit in the antigenuine spherical Hecke algebra of GS wart KS Iles Ky g conjugacy class in Msg which is good i e Tys Ys iff zys ysx where p x When M G we get the usual orbital integrals Trace formula for covers July 24 2012 18 39 Non invariant trace formula The refined geometric side Regular semisimple weighted orbital integrals Let S v 7 M be semisimple regular and good in G M C G Levi Then JG fo D fo a4a um a de Le o vm M F G F K gt Rso is the volume for some convex polytope in a
4. absolutely convergent Arthur put some restriction on the infinitesimal characters of the oo part of 7 This seems to be avoidable by the work of Finis Lapid M ller Trace formula for covers July 24 2012 27 39 Non invariant trace formula The refined spectral side Local unitary weighted characters Let V a finite set of places such that either V contains some v oo or e the places in V have the same residual characteristic gt 0 P MU a parabolic subgroup with Levi component M D gt Mo For x II My unitary and genuine there is an operator Mu r P acting on the space of IC x such that Tiny T fv tr Mula PE a fv is well defined It does not depend on P Remark My r P is defined in terms of local intertwining operators and Harish Chandra s j functions they are canonical objects attached to M 7 When M G we get the usual characters Trace formula for covers July 24 2012 28 39 See Compression of coefficients Goal re index the fine spectral expansion by the local objects 7 My Theorem For any Levi L D Mo one can define a space Dati V of genuine representations of Ly endowed with a measure o coefficients a x for I_ L V such that for f fy fgv where fgv is the unit of the antigenuine spherical Hecke algebra w r t GV KV we have L 2 je 5 me al n J m 0 fy at IWE n_ div Here J 7 0 fv is the Fourier coefficie
5. ad hoc arguments e G GL n used in Mezo s work on metaplectic correspondence G SL 2 Hiraga lkeda eG Sp 2n the twofold cover of Sp 2n Weil G unitary group Trace formula for covers July 24 2012 37 39 Sinple rase ferrules Cuspidal test functions Let v be a place of F fy C Gy be antigenuine Definition We say fy is cuspidal if trace r f 0 for any x of the form T I 0 where P 4 G and is a tempered genuine irrep of My Let fv ey fo He we say fy is cuspidal at v if fy is cuspidal Recall the invariant distribution fy I fv has two expansions spectral and geometric Trace formula for covers July 24 2012 38 39 Invariant trace formula Simple trace formulas Theorem Let fv Levy fo He Put f fy fgv C G If fy is cuspidal at one place then the spectral expansion of I fy becomes 7 gt a nv Ja av fv my Cll disc G V or in global terms D a n Ja a f TEllaise G1 If f is cuspidal at two places then the geometric expansion of I f D JG P ser G1 V becomes Trace formula for covers July 24 2012 39 39
6. 2012 24 39 Non invariant trace formula The refined spectral side Local ingredients of the proof Mostly Harish Chandra s theory e local intertwining operators c functions p functions Plancherel formula normalization of local intertwining operators Archimedean case juggling with l functions e non archimedean case adapt Langlands proof e unramified case need some theory of unramified genuine principal series cf McNamara Global ingredients of the proof In view of Moeglin Waldspurger one can simply copy Arthur s arguments Trace formula for covers July 24 2012 25 39 Non invariant trace formula The refined spectral side Global weighted characters One can define e a set ITgisc M of genuine discrete parameters for each Levi M the discrete coefficients alt x for each 7 Mais M1 for i a the global weighted character Jalta f te Sarma PIS f where o my 7 expAm P is a parabolic with Levi component M e Ju my P is an intertwining operator coming from M m P e 1S is the normalized parabolic induction Warning Iais M is not necessarily contained in the discrete spectrum Trace formula for covers July 24 2012 26 39 Non invariant trace formula The refined spectral side Theorem We have wM n 2 e Se M L Mo relais 2 YOU aM rx rx f d Remark To make this integral
7. A user s guide to the trace formula for covering groups Wen Wei Li Pan Asian Number Theory Conference 2012 24 July 2012 Trace formula for covers July 24 2012 1 39 Introduction References Arthur s papers Moeglin and Waldspurger D composition spectrale et s ries d Eisenstein Progress in Math 113 1994 La formule des traces pour les rev tements de groupes r ductifs connexes I Le d veloppement g om trique fin arXiv 1004 4011 La formule des traces pour les rev tements de groupes r ductifs connexes II Analyse harmonique locale arXiv 1107 1865 La formule des traces pour les rev tements de groupes r ductifs connexes III Le d veloppement spectral fin arXiv 1107 2220 La formule des traces pour les rev tements de groupes r ductifs connexes IV Distributions invariantes in prepartation Trace formula for covers July 24 2012 2 39 Trace formula Arthur Selberg trace formula F number field A its ring of ad les G a connected reductive group over F G A Ker Ha where Hg G A gt ag is the Harish Chandra homomorphism R right regular representation of G on L G F G A f C G A e Kg the kernel of R f k x Kalz x for x G A The Arthur Selberg trace formula calculates a truncated integral of k x over G F G A geometric expansion J f spectral expansion Trace formula for covers July 24 2012 3 39 Trace formula Rou
8. central extension of G F related to algebraic K theory G GL metaplectic correspondence Flicker Kazhdan Patterson gt 1980 e G arbitrary Deligne and Brylinski 2001 classified their K extensions Trace formula for covers July 24 2012 6 39 A class of covers Genuine representations In harmonic analysis on may assume N um z C 2 1 for some m It suffices to study the representations 7 of G which are genuine i e m en Z for all Um e Test functions it suffices to consider m f with antigenuine f CR G ie f e f z Justification given a smooth irrep 7 of G let w N gt C be its central character on N Then it suffices to study the push forward of 1 N G G F gt 1 byw Trace formula for covers July 24 2012 10 2389 A class of covers Constraints on covers The class of such extensions under consideration should be stable under push forward by any homomorphism um Hm e stable under passage to Levi subgroups philosophy of cusp forms when F is global splitting G F G spectral decomposition see MW splittings over hyperspecial subgroups G o at almost all v such that IL Go G is continuous here we fix an integral model of G continued the corresponding antigenuine spherical Hecke algebra at v must be commutative Q decomposition of smooth irreps Ww Ww Existence of canonical splittings over un
9. ghly speaking e the geometric side distributions on G A indexed by rational conjugacy classes eg orbital integrals the spectral side distributions on G A indexed by automorphic representations eg characters Some of the applications Base change and Jacquet Langlands correspondence for GL n Endoscopic classification of representations of classical groups e Formula for the trace of Hecke operators e Various results in local harmonic analysis character identities etc e Applications in analytic number theory In each case it is crucial to have some refined versions of this trace formula Examples invariant trace formula stable trace formula Trace formula for covers July 24 2012 4 39 A class of covers Covers of connected reductive groups the local case Consider central extensions of locally compact groups as follows The local setting F local field G connected reductive F group 1 N G G F 1 where N is finite abelian The global setting F global field A its ring of ad les G connected reductive F group 1 N 6 G A 1 Trace formula for covers July 24 2012 5 39 A class of covers Examples G Sp 2n A Weil 1964 representation theoretic interpretation of Siegel modular forms of half integral weight G SL 2 or GL 2 Shimura 1973 Kubota G split simple and simply connected R Steinberg 1962 H Matsumoto 1969 constructed the universal
10. ion on Gy by C Gy 2 fu f fv fgv CPG For each Levi L choose a suitable space of test functions Hz together with a surjection Hz IH Find a good linear map He IH satisfying similar identities under conjugation as the weighted characters orbital integrals By induction define a distribution J IG so that J is invariant follows from the conditions above on e I factors through THe e we have the decomposition Dy z N eG Fema LEL Mo Wo Trace formula for covers July 24 2012 32 39 Invariant trace formula We shall apply a similar procedure to the distributions J in the refined expansions using Levi subgroups L gt M This will yield invariant distributions 1 Ce Iz T Invariant trace formula We ae gt WEI D AORN LEL Mo yer A V L gt Wa a n I 7 0 fv LEL Mo we W_ 1 V This is almost a formal consequence of the recipe above Remark For L G the distributions I Ja are the usual characters and orbital integrals Trace formula for covers July 24 2012 33 39 Choceoto Hi 1H Following Arthur we take H to be the space of Ky finite left and right C antigenuine functions on Lt IH to be a space of C valued functions on the tempered genuine dual of Le characterized by trace Paley Wiener theorems so that for any fy Hj T gt trace a fv lies in TH j
11. ipotent subgroups automatic gt notions of constant terms and Jacquet functors These conditions are satisfied by the K2 extensions of Brylinski Deligne Trace formula for covers July 24 2012 8 39 Non invariant trace formula Desiderata Goal establish the Arthur Selberg trace formula for a large class of covers Fix a minimal Levi Mo and set Mo Levi containing Mo The coarse trace formula Se 4 gt 4a o x Refined trace formula in terms of weighted characters and weighted orbital integrals wl oe ee 2 we 2 dif HA LEL Mo O yer Et V w IWF Ju _ L1 V Jz 7 0 f dr won Trace formula for covers July 24 2012 9 39 Non invariant trace formula The invariant trace formula W oe gt wey gt o MEG 1 f LEL Mo 0 yer Et v L z gt Wa a n I m 0 f dr LEL Mo WF Jn dav where the J are invariant distributions For L G we get the usual orbital integrals and characters e Simple trace formula for suitable choice of f only the terms with L G survive e Long term goal for some special G stabilization rewrite everything in terms of stable distributions on certain linear reductive groups Trace formula for covers July 24 2012 10 39 Non invariant trace formula Coarse trace formula Coarse trace formula Let p G G A be a cover Ker p Hm o G Ker Hgo p where Ha G A gt ag is the Harish Chandra homomorphis
12. m R right regular representation of G on L G F G f CX G antigenuine Kg the kernel of R f k x Ka for x G A Z p l x for any parabolic P MU Rp the right regular representation on L U A M F G and Kp its kernel Fix minimal Levi Mo and maximal compact K C G A in good relative position Set K p K ao amo Trace formula for covers July 24 2012 11 39 Non invariant trace formula Coarse trace formula Fix Py P Mo For T ao define the truncated kernel la Arthur kT x gt LP 42 4 PDPo Kea eet D 6 P F G F Theorem For T highly regular kT x is integrable over G F G There is an identity of absolutely convergent integrals Sarr 0 x Everything in sight is polynomial in T Trace formula for covers July 24 2012 12 39 Non invariant trace formula Coarse trace formula e Spectral side x ranges over cuspidal data M 0 where M D Mo is a Levi subgroup is a cuspidal automorphic representation of M inside L M F M e Geometric side o ranges over semisimple classes in G F The unipotent term di corresponds to 1 About the proof Combinatorics the same as the case of reductive groups Arthur e Spectral decomposition included in Moeglin Waldspurger e Geometric side the same as in the case of reductive groups we only look at conjugacy classes in G F Trace formula for covers July 24 2012 13 39 Coarse trace f
13. nt at 0 of J m fv Trace formula for covers July 24 2012 29 39 Non invariant trace formula The refined spectral side The compressed coefficients a x is defined using M the global discrete coefficients agi oE Iais M1 the normalizing factors r4 c for the intertwining operators of various genuine unramified representations c of GY w r t KY a for various M C L and In view of the works of Langlands and Shahidi r c is expected to be related to L functions of some linear group Cf McNamara Remark Note the formal resemblance between the refined geometric and spectral expansions with compressed coefficients Trace formula for covers July 24 2012 30 39 Invariant trace formula Invariant trace formula for covers Let V be a set of places containing the archimedean and ramified ones Motivation In harmonic analysis we usually choose test functions fy that are defined only through their o orbital integrals Je 7 fv or e characters Jg 7 fv eg the Euler Poincar functions These distributions fy J amp m fv resp Jay 7 fv are weakly dense in the space of invariant distributions on Gy Goal Express the distributions J in the refined trace formula in terms of invariant distributions on Levi subgroups Trace formula for covers July 24 2012 31 39 Arthur s idea Let J be the distribution of the refined trace formula viewed as a distribut
14. ormula Refinement Let To ao be the canonical element depending on K defined by Arthur Then The problem is to find explicit formulas for them Goal Express J f Jo f in terms of weighted orbital integrals and weighted characters local objects Isolate global and local information Trace formula for covers July 24 2012 14 39 Non invariant trace formula The refined geometric side Descend to the unipotent case Idea get rid of the cover on the geometric side For x y G A with liftings Z 7 G set 1 x y 2 gg Let o G F be semisimple Go Za o Then o defines a homomorphism Go A gt Hm Principle Let o be the G F orbit containing o Reduce JE f to Ja the unipotent term of the trace formula of Go twisted by the character More precisely some Levi subgroups of G may appear Remark we have Jordan decomposition on covers Trace formula for covers July 24 2012 15 39 UI died sea Example descent of orbital integrals The same formalism about o applies to the local case Let F be a local field p G G F a cover and f CX G antigenuine Theorem Sf C g F such that VS o exp X with X g F sufficiently close to 0 we have i a 1 ahe IDEI OF f DS x 2051p where DC and D are the Weyl discriminants on G and go respectively It s tempting to remove o and replace G F by Ker o a F
15. to be the map sending fy to a den 0 fy Recall J 7 0 fv is the 0 th Fourier coefficient of the weighted character Jz my fv Trace formula for covers July 24 2012 34 39 Invariant trace formula o 2 To show has image in IH a detailed analysis of weighted characters is needed We also need to assume the trace Paley Wiener theorem of Ky finite functions To show that the invariant distribution We ig I fv gt I ilf V do wI V factors through JH Arthur uses a global argument la Kazhdan via the trace formula For some technical reason this does not work for covers except for G GL n or G Sp 2n the twofold cover of Sp 2n We use a purely local argument via the invariant local trace formula for covering groups Trace formula for covers July 24 2012 35 ff 39 Invariant trace formula Trace Paley Wiener theorem Assumption For all Levi L C G the trace Paley Wiener theorem characterizing the image of fv r trace r fy for fy H holds where m is a genuine tempered irrep of Li This can be reduced to the case V v for genuine tempered irreps of Ly When v is nonarchimedean simply copy the arguments of Bernstein Deligne Kazhdan When v is archimedean some arguments of Clozel Delorme seem problematic for covers Trace formula for covers July 24 2012 36 39 Invariant trace formula For archimedean v some special cases can be check by
Download Pdf Manuals
Related Search