QpQpdenoteageometricFrobeniuselementThenoneseekstoproveaformulaforthesemisimpleLefschetznumberLefss8rpShKp101trss8rpH15cShKp E Qp QX 0 14c 0 14O 1KpTO14 ID: 937609
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2T.Haineslet(G;h�1;KpKp)denoteShimuradatawherepisaxedrationalprimesuchthatthelevel-structuregroupfactorizesasKpKpG(Apf)G(Qp).Thisdatagivesrisetoaquasi-projectivevarietyShKp:=Sh(G;h�1;KpKp)overanumbereldEC.Letp2Gal( Qp=Qp)denoteageometricFrobeniuselement.Thenoneseekstoproveaformulaforthesemi-simpleLefschetznumberLefss(rp;ShKp)(1.0.1)trss(rp;Hc(ShKp E Qp; Q`))=X( 0; ;)c( 0; ;)O (1Kp)TO( r);(seex6:1formoredetails).Thetestfunction rappearinghereisthemostinterestingpartoftheformula.ExperiencehasshownthatwemayoftenndatestfunctionbelongingtothecenterZ(G(Qpr);Kpr)oftheHeckealgebraH(G(Qpr);Kpr),inawaythatisexplicitlydeterminedbytheE-rationalconjugacyclassfgof1-parametersubgroupsofGassociatedtotheShimuradata.InmostPELcaseswithgoodreduction,whereKpG(Qp)isahyperspecialmaximalcompactsubgroup,thiswasdonebyKottwitz(cf.e.g.[Ko92a]).WhenKpisaparahoricsubgroupofG(Qp)andwhenGQpisunramied,theKottwitzconjecturepredictsthatwecantake rtobeapowerofptimestheBernsteinfunctionzKp�;jarisingfromtheBernsteinisomorphismforthecenterZ(G(Qpr);Kpr)oftheparahoricHeckealgebraH(G(Qpr);Kpr)(seeConjecture7.1.2andx11).InfactKottwitzformulated(again,forunramiedgroupscomingfromShimuradata)acloselyrelatedconjectureconce
rningnearbycyclesonRapoport-ZinklocalmodelsofShimuravarieties,whichsubsequentlyplayedanimportantroleinthestudyoflocalmodels(Conjecture7.1.3).ItalsoinspiredimportantdevelopmentsinthegeometricLanglandsprogram,e.g.[Ga].BothversionsofKottwitz'conjectureswerelaterprovedinseveralparahoriccasesattachedtolinearorsymplecticgroups(see[HN02a,H05]).Inarecentbreakthrough,PappasandZhu[PZ]denedgroup-theoreticversionsofRapoport-Zinklocalmodelsforquitegeneralgroups,andprovedintheunramiedsituationstheanalogueofKottwitz'nearbycyclesconjectureforthem.Thesemattersarediscussedinmoredetailinx7andx8.Untilaround2009itwasstillnotclearhowonecoulddescribethetestfunctions rinalldeeperlevelsituations.Inthespringof2009theauthorandKottwitzformulatedaconjecturepredictingtestfunctions rforgenerallevelstructureKp.Thisisthetestfunctionconjecture,Conjecture6.1.1.Itpostulatesthatwemayexpress rintermsofadistributionZVEj0�;jintheBernsteincenterZ(G(Qpr))associatedtoacertainrepresentationVEj0�;j(denedin(6.1.2))oftheLanglandsL-groupL(GQpr).Letd=dim(ShKp).ThenConjecture6.1.1assertsthatwemaytake r=prd=2�ZVEj0�;j1Kpr2Z(G(Qpr);Kpr)theconvolutionofthedistributionZVEj0�;jwiththecharacteristicfunction1KprofthesubgroupKpr.Asshowninx7,thisspecializestotheKottwi
tzconjectureforparahoricsubgroupsinunramiedgroups.Conjecture6.1.1wassubsequentlyprovedforDrinfeldcaseShimuravarietieswith�1(p)-levelstructurebytheauthorandRapoport[HRa],andformodularcurvesandforDrinfeldcaseShimuravarietieswitharbitrarylevelstructurebyScholze[Sch1,Sch2].ThedistributionsinConjecture6.1.1arebestseenasexamplesofaconstructionV ZVwhichattachestoanyalgebraicrepresentationVoftheLanglandsdualgroupLG(forGanyconnectedreductivegroupoveranyp-adiceldF),anelementZVinthestableBernsteincenterofG=F. 4T.HainesweneedinourelaborationofthestableBernsteincenterinx5.Inparticularinx5:2wedescribetheenhancement(LLC+)whichplaysasignicantrolethroughoutthechapter,andexplainwhyitholdsforgenerallineargroupsinRemark5.2.3andCorollary5.2.6.ThedistributionsZVaredenedinx5:7,andareusedtoformulatethetestfunctionconjecture,Conjecture6.1.1,inx6:1.Intherestofx6,wedescribethenearbycyclesvariantConjecture6.1.2alongwithsomeoftheendoscopictransferconjecturesneededforthe\pseudostabilization",andassumingtheseconjecturesweproveinTheorem6.3.2theexpectedformofthesemi-simplelocalHasse-Weilzetafunctions,inthecompactandnon-endoscopiccases.Inx7wegiveaconcretereformulationofthetestfunctionconjectureinparahoriccases,recoveringtheKottwitzconjectureandgeneralizingittoallgroupsusingthematerialfromtheAp
pendix.Thepurposeofx8andx9istolistsomeoftheavailableevidenceforConjectures6.1.1and6.2.3.Inx10certaintestfunctionsaredescribedveryexplicitly.Finally,theAppendixgivesthetreatmentofBernsteinisomorphismsandthetransferhomomorphisms,alludedtoabove.Acknowledgments.IamverygratefultoGuyHenniartforsupplyingtheproofofProposition5.2.5andforallowingmetoincludehisproofinthischapter.IwarmlythankTimoRicharzforsendingmehisunpublishedarticle[Ri]andforlettingmequoteafewofhisresultsinLemma11.3.1.IamindebtedtoBrooksRobertsforprovingConjecture5.2.2forGSp(4)(seeRemark5.2.3).IthankmycolleaguesJe reyAdamsandNiranjanRamachandranforusefulconversations.IalsothankRobertKottwitzforhisin uenceontheideasinthischapterandforhiscommentsonapreliminaryversion.IthankMichaelRapoportformanystimulatingconversationsabouttestfunctionsovertheyears.Iamgratefultotherefereeforhelpfulsuggestionsandremarks.2.NotationIfGisaconnectedreductivegroupoverap-adiceldF,thenR(G)willdenotethecategoryofsmoothrepresentationsofG(F)onC-vectorspaces.Wewillwrite2R(G)irredor2(G=F)ifisanirreducibleobjectinR(G).IfGasabovecontainsanF-rationalparabolicsubgroupPwithF-LevifactorMandunipotentradicalN,denethemodulusfunctionP:M(F)!R�0byP(m)=jdet(Ad(m);Lie(N(F)))jFwherejjFisthenormalizedabsolutevalueonF.B
y1=2P(m)wemeanthepositivesquare-rootofthepositiverealnumberP(m).For2R(M),wefrequentlyconsiderthenormalizedinducedrepresentationiGP()=IndG(F)P(F)(1=2P):Welet1SdenotethecharacteristicfunctionofasubsetSofsomeambientspace.IfSG,letgS=gSg�1.IffisafunctiononS,denethefunctiongfongSbygf()=f(g�1g).ThroughoutthechapterweusetheWeilformofthelocalorglobalLanglandsL-groupLG.3.ReviewoftheBernsteincenterWeshallgiveabriefsynopsisof[BD]thatissuitableforourpurposes.Otherusefulreferencesare[Be92],[Ren],and[Roc].TheBernsteincenterZ(G)ofap-adicgroupGisdenedastheringofendomorphismsoftheidentityfunctoronthecategoryofsmoothrepresentationsR(G).Itcanalsoberealizedasanalgebra StableBernsteincenterandtestfunctions5ofcertaindistributions,astheprojectivelimitofthecentersofthenite-levelHeckealgebras,andastheringofregularfunctionsonacertainalgebraicvariety.Wedescribetheseinturn.3.1.Distributions.Westartbydeningtheconvolutionalgebraofdistributions.WewriteGfortherationalpointsofaconnectedreductivegroupoverap-adiceld.ThusGisatotallydisconnectedlocallycompactHausdor topologicalgroup.FurtherGisunimodular;xaHaarmeasuredx.LetC1c(G)denotethesetofC-valuedcompactlysupportedandlocallyconstantfunctionsonG.LetH(G;dx)=(C1c(G);dx),theconvolution
productdxbeingdenedusingtheHaarmeasuredx.AdistributionisaC-linearmapD:C1c(G)!C.Foreachf2C1(G)wedenef2C1(G)byf(x)=f(x�1)forx2G.WesetD(f):=D(f):WecanconvolveadistributionDwithafunctionf2C1c(G)andgetanewfunctionDf2C1(G),bysetting(Df)(g)=D(gf);where(gf)(x):=f(xg).ThefunctionDfdoesnotautomaticallyhavecompactsupport.WesayDisessentiallycompactprovidedthatDf2C1c(G)foreveryf2C1c(G).Wedenegfbygf(x):=f(g�1xg)forx;g2G.WesaythatDisG-invariantifD(gf)=D(f)forallg;f.ThesetD(G)GecofG-invariantessentiallycompactdistributionsonC1c(G)turnsouttohavethestructureofacommutativeC-algebra.Wedescribenexttheconvolutionproductanditsproperties.GivendistributionsD1;D2withD2essentiallycompact,wedeneanotherdistributionD1D2by(D1D2)(f)=D1(D2f):Lemma3.1.1.TheconvolutionproductsDfandD1D2havethefollowingproperties:(a)For 2C1c(G)letD dx(sometimesabbreviated dx)denotetheessentiallycompactdis-tributiongivenbyf7!RGf(x) (x)dx.ThenD dxf= dxf.(b)Iff2C1c(G),thenD(fdx)=(Df)dx.Inparticular,D 1dxD 2dx=D 1dx 2dx.(c)IfD2isessentiallycompact,then(D1D2)f=D1(D2f).IfD1andD2areeachessentiallycompact,soisD1D2.(d)IfD2andD3areessentiallycompact,then(D
1D2)D3=D1(D2D3).(e)AnessentiallycompactdistributionDisG-invariantifandonlyifD(1Ugdx)=(1Ugdx)DforallcompactopensubgroupsUGandg2G.Here1UgisthecharacteristicfunctionofthesetUg.(f)IfDisessentiallycompactandf1;f22C1c(G),thenD(f1dxf2)=(Df1)dxf2.Corollary3.1.2.Thepair(D(G)Gec;)isacommutativeandassociativeC-algebra.3.2.Theprojectivelimit.LetJGrangeoverthesetofallcompactopensubgroupsofG.LetH(G)denotetheconvolutionalgebraofcompactly-supportedmeasuresonG,andletHJ(G)H(G)denotetheringofJ-bi-invariantcompactly-supportedmeasures,withcenterZJ(G).TheringHJ(G)hasasuniteJ=1JdxJ,where1JisthecharacteristicfunctionofJanddxJistheHaarmeasurewithvoldxJ(J)=1.NotethatifJ0J,thendxJ0=[J:J0]dxJ. StableBernsteincenterandtestfunctions7WedeneH(L)1:=ker(H)H(F)1:=ker(H)\H(F):WealsodeneH(F)1H(F)1tobethekernelofthemapH(F)!X(Z(bH))I=torsderivedfromH.IfHisanisotropicmodulocenter,thenH(F)1istheuniquemaximalcompactsubgroupofH(F)andH(F)1istheuniqueparahoricsubgroupofH(F)(seee.g.[HRo]).Sometimesthetwosubgroupscoincide:forexampleifHisanyunramiedF-torus,thenH(F)1=H(F)1.WedeneX(H):=Homgrp(H(F)=H(F)1;C);thegroupofunramiedcharactersonH(F).ThisdenitionofX(H)agreeswiththeusualoneasin[BD].WedeneXw(H):=Ho
mgrp(H(F)=H(F)1;C)andcallitthegroupofweaklyunramiedcharactersonH(F).Wefollowthenotationof[BK]indiscussingsupercuspidalsupportsandinertialequivalenceclasses.Asindicatedearlierinx3:1,forconveniencewewillsometimeswriteGwhenwemeanthegroupG(F)ofF-pointsofanF-groupG.Acuspidalpair(M;)consistsofanF-LevisubgroupMGandasupercuspidalrepresentationonM.TheG-conjugacyclassofthecuspidalpair(M;)willbedenoted(M;)G.Wedenetheinertialequivalenceclasses:wewrite(M;)(L; )ifthereexistsg2GsuchthatgMg�1=Landg= forsome 2X(L).Let[M;]Gdenotetheequivalenceclassof(M;)G.If2R(G)irred,thenthesupercuspidalsupportofistheuniqueelement(M;)GsuchthatisasubquotientoftheinducedrepresentationiGP(),wherePisanyF-parabolicsubgrouphavingMasaLevisubgroup.LetXGdenotethesetofallsupercuspidalsupports(M;)G.Denotebythesymbols=[M;]Gatypicalinertialclass.Foraninertialclasss=[M;]G,denethesetXs=f(L; )Gj(L; )(M;)g.WehaveXG=asXs:WeshallseebelowthatXGhasanaturalstructureofanalgebraicvariety,andtheBernsteincomponentsXsformtheconnectedcomponentsofthatvariety.FirstweneedtorecallthevarietystructureonX(M).Asiswell-known,X(M)hasthestructureofacomplextorus.Todescribethis,werstconsidertheweaklyunrami
edcharactergroupXw(M).ThisisadiagonalizablegroupoverC.Infact,byKottwitzwehaveanisomorphismM(F)=M(F)1=X(Z(cM)I)=X((Z(cM)I)):ThismeansthatXw(M)(C)=Homgrp(M(F)=M(F)1;C)=Homalg(C[X((Z(cM)I))];C);inotherwords,(3.3.2)Xw(M)=(Z(cM)I): StableBernsteincenterandtestfunctions94.ThelocalLanglandscorrespondenceWeneedtorecallthegeneralformoftheconjecturallocalLanglandscorrespondence(LLC)foraconnectedreductivegroupGoverap-adiceldF.LetFdenoteanalgebraicclosureofF.LetWFGal(F=F)=:�FbetheWeilgroupofF.Ittsintoanexactsequenceoftopologicalgroups1// IF// WFval// Z// 1;whereIFistheinertiasubgroupof�Fandwhere,if2WFisageometricFrobeniuselement(theinverseofanarithmeticFrobeniuselement),thenval()=�1.HereIFhasitspronitetopologyandZhasthediscretetopology.SometimeswewriteIforIFinwhatfollows.RecalltheWeil-DelignegroupisW0F:=WFnC,wherewzw�1=jwjzforw2WFandz2C,withjwj:=qval(w)FforqF=#(OF=($F)),thecardinalityoftheresidueeldofF.ALanglandsparameterisanadmissiblehomomorphism':W0F!LG,whereLG:=bGoWF.Thismeans:'iscompatiblewiththeprojectionsW0F!WFand:LG!WF;'iscontinuousandrespectsJordandecompositionsofelementsinW0FandLG(cf.[Bo79,x8]forthedenitionofJordandecompositioninthegroupWFnCandwhatitmeanstorespectJordand
ecompositionshere);if'(W0F)iscontainedinaLevisubgroupofaparabolicsubgroupofLG,thenthatparabolicsubgroupisrelevantinthesenseof[Bo79,x3:3].(ThisconditionisautomaticifG=Fisquasi-split.)Let(G=F)denotethesetofbG-conjugacyclassesofadmissiblehomomorphisms':W0F!LGandlet(G=F)=R(G(F))irredthesetofirreduciblesmooth(oradmissible)representationsofG(F)uptoisomorphism.Conjecture4.0.1(LLC).Thereisanite-to-onesurjectivemap(G=F)!(G=F),whichsatisesthedesiderataof[Bo79,x10].Theber'over'2(G=F)iscalledtheL-packetfor'.WementionafewdesiderataoftheLLCthatwillcomeupinwhatfollows.First,LLCforGmisnothingotherthanLanglandsdualityforGm,whichwenormalizeasfollows:forTanysplittorustorusoverF,withdualtorusbT,Homconts(T(F);C)=Homconts(WF;bT)$'satises,forevery2X(T)=X(bT)andw2WF,(4.0.4)('(w))=((Art�1F(w))):HereArt�1F:WabF!FisthereciprocitymapoflocalclasseldtheorywhichsendsanygeometricFrobeniuselement2WFtoauniformizerinF.Next,wethinkofLanglandsparametersintwoways,eitherascontinuousL-homomorphisms':W0F!LGmodulobG-conjugation,orascontinuous1-cocycles'cocyc:W0F!bG StableBernsteincenterandtestfunctions11AssumeLLCholdsforG=F.LetbeaninnitesimalcharacterforG.Denetheinnitesimalclasstobethefollow
ingniteunionofL-packets:=a' ':Here'rangesoverbG-conjugacyclassesofadmissiblehomomorphismsW0F!LGsuchthat('jWF)bG=()bG,and'isthecorrespondingL-packetofsmoothirreduciblerepresentationsofG(F).5.2.LLC+.InordertorelatetheBernsteinvarietyXwiththevarietyYofinnitesimalcharacters,wewillassumetheLocalLanglandsCorrespondence(LLC)forGandallofitsF-Levisubgroups.WeassumeallthedesideratalistedbyBorelin[Bo79].TherearetwoadditionaldesiderataofLLCweneed.Denition5.2.1.WewilldeclarethatGsatisesLLC+iftheLLCholdsforGanditsF-Levisub-groups,andthesecorrespondencesarecompatiblewithnormalizedparabolicinductioninthesenseoftheConjecture5.2.2below,andinvariantundercertainisomorphismsinthesenseofConjecture5.2.7below.LetMGdenoteanF-Levisubgroup.ThentheinclusionM,!GinducesanembeddingLM,!LGwhichiswell-deneduptobG-conjugacy(cf.[Bo79,x3]).Conjecture5.2.2.(CompatibilityofLLCwithparabolicinduction)Let2(M=F)and2(G=F)andassumeisanirreduciblesubquotientofiGP(),whereP=MNisanyF-parabolicsubgroupofGwithF-LevifactorM.Thentheinnitesimalcharacters'jWF:WF!LGand'jWF:WF!LM,!LGarebG-conjugate.Remark5.2.3.(1)Theconjectureimpliesthattherestriction'jWFdependsonlyonthesupercus-pidalsupportof.Thislatterstatementisaformalconsequen
ceofVogan'sConjecture7.18in[Vo],buttheConjecture5.2.2isslightlymoreprecise.InProposition5.5.1wewillgiveaconstructionofthemapfinVogan'sConjecture7.18,bysendingasupercuspidalsupport(M;)G(a\classicalinnitesimalcharacter"in[Vo])totheinnitesimalcharacter('jWF)bG.Withthisformulation,theconditiononfimposedinVogan'sConjecture7.18isexactlythecompatibilityintheconjectureabove.(2)TheconjectureholdsforGLn,andisimplicitinthewaythelocalLanglandscorrespondenceforGLnisextendedfromsupercuspidalstoallrepresentations(seeRemark13.1.1of[HRa]).ItwasapointofdepartureinScholze'snewcharacterizationofLLCforGLn[Sch3],andthatpaperalsoprovidesanotherproofoftheconjectureinthatcase.(3)IwasinformedbyBrooksRoberts(privatecommunication),thattheconjectureholdsforGSp(4).(4)Givenaparameter':W0F!LG,thereexistsacertainP=MNandacertaintemperedparameter'M:W0F!LMandacertainreal-valuedunramiedcharacter MonM(F)whoseparameterisintheinterioroftheWeylchamberdeterminedbyP,suchthattheL-packet'consistsofLanglandsquotientsJ(M M),forMrangingoverthepacket'M.Theparameter'isthe 12T.Hainestwistof'Mbytheparameterassociatedtothecharacter M.Thisreducestheconjecturetothecaseoftemperedrepresentations.Onecanfurtherreducetothecaseofdiscreteseriesrepresentations.Thefollowingisaverynaturalkin
doffunctorialitywhichshouldbesatisedforallgroups.Conjecture5.2.4.(InvarianceofLLCunderisomorphisms)Suppose :(G;)f!(G0;0)isanisomorphismofconnectedreductiveF-groupstogetherwithirreduciblesmoothrepresentationsonthem.ThentheinducedisomorphismL :LG0f!LG(well-deneduptoaninnerautomorphismofbG),takesthebG0-conjugacyclassof'0:W0F!LG0tothebG-conjugacyclassof':W0F!LG.Proposition5.2.5.Conjecture5.2.4holdswhenG=GLn.Proof.(GuyHenniart).ItisenoughtoconsiderthecasewhereG0=GLnand isanF-automorphismofGLn.ThefunctorialpropertiesintheLanglandscorrespondenceforGLnare:(i)Compatibilitywithclasseldtheory,thatis,withthecasewheren=1.(ii)ThedeterminantoftheWeil-Delignegrouprepresentationcorrespondstothecentralchar-acter:thisisLanglandsfunctorialityforthehomomorphismdet:GLn(C)!GL1(C).(iii)Compatibilitywithtwistsbycharacters,i.e.,Langlandsfunctorialityfortheobvioushomo-morphismofdualgroupsGL1(C)GLn(C)!GLn(C).(iv)Compatibilitywithtakingcontragredients:thisisLanglandsfunctorialitywithrespecttotheautomorphismg7!tg�1(transposeinverse),sinceitisknownthatforGLn(F)thissendsanirreduciblerepresentationtoarepresentationisomorphictoitscontragredient.ThesepropertiesareenoughtoimplythedesiredfunctorialityforF-automorphismsofGLn.Whenn=1,thefunctorialityisobviousforanyF-e
ndomorphismofGL1.Whennisatleast2,anF-automorphismofGLninducesanautomorphismofSLnhenceanautomorphismoftheDynkindiagramwhichmustbetheidentityor,(whenn3)theoppositionautomorphism.HenceuptoconjugationbyGLn(F),theF-automorphismistheidentityonSLn,orpossibly(whenn3)transposeinverse.ConsequentlytheF-automorphismcanbereduced(bycomposingwithaninnerautomorphismorpossiblywithtransposeinverse)toonewhichistheidentityonSLn,henceisoftheformg7!gc(det(g))wherec2X(Z(GLn)).Butthisimpliesthatitistheidentityunlessn=2,inwhichcaseitcouldalsobeg7!gdet(g)�1.Inthatexceptionalcase,themapinducedonthedualgroupGL2(C)isalsog7!gdet(g)�1,andthedesiredresultholdsbyinvoking(ii)and(iii)above.Corollary5.2.6.LetM=GLn1GLnrGLnbeastandardLevisubgroup.Letg2GLn(F).ThenConjecture5.2.4holdsfortheisomorphismcg:Mf!gMgivenbyconjugationbyg.Proof.ItisenoughtoconsiderthecasewheregbelongstothenormalizerofMinGLn.LetTMbethestandarddiagonaltorusinGLn.Theng2NG(T)M.ThuscomposinggwithapermutationmatrixwhichnormalizesMwemayassumethatcgpreserveseachdiagonalfactorGLni.ThedesiredfunctorialityfollowsbyapplyingProposition5.2.5toeachGLni.ForthepurposesofcomparingtheBernsteincenterandthestableBernsteincenterasinPropo-sition5.5.1,weneedonlythisweakervariantofConjecture5.2.4.Conjec
ture5.2.7.(WeakinvarianceofLLC)LetMGbeanyF-Levisubgroupandletg2G(F).ThenConjecture5.2.4holdsfortheisomorphismcg:Mf!gM. 14T.HainesWecannowdenethenotionofinertialequivalence(1)bG(2)bGofinnitesimalcharacters.Denition5.3.3.Wesay(1)bGand(2)bGareinertiallyequivalentiffM1g=fM2g;thereexistsM2fM1g,and+12(1)bGand+22(2)bGwhoseimagesareminimallycontainedbyM,andanelementz2(Z(M)I),suchthat(z+1)M=(+2)M:Wewrite[]bGfortheinertialequivalenceclassof()bG.NotethatMautomaticallycontains(z+1)(WF)minimallyifitcontains+1(WF)minimally.Lemma5.3.4.Therelationisanequivalencerelationonthesetofinnitesimalcharacters.Proof.UseLemmas5.3.1and5.3.2.Remark5.3.5.Todene(1)bG(2)bGweusedthechoiceofbGbBbT(whichwasassumedtoformpartofa�F-invariantsplittingforbG)inordertodenethenotionofstandardLevisubgroupofLG.However,theequivalencerelationisindependentofthischoice,sinceasremarkedabove,anytwo�F-invariantsplittingsforbGareconjugateunderbG�F,by[Ko84a,Cor.1.7].Remark5.3.6.ThepropertyweneedofstandardLevisubgroupsMLGisthattheyaredecomposable,thatis,M:=M\bGisWF-stable,andM=MoWF.AnystandardLevisubgroupisdecomposable
.Inourdiscussion,wecouldhaveavoidedchoosinganotionofstandardLevi,byassociatingtoeach()bGauniqueclassofdecomposableLevisubgroupsfMg,allofwhicharebG-conjugate,suchthatfactorsminimallythroughsomeM2fMg.NowxastandardLevisubgroupMLG.WewritetMforaninertialequivalenceclassofadmissiblehomomorphismsWF!M.WewriteYtMforthesetofM-conjugacyclassescontainedinthisinertialclass.WewanttogivethissetthestructureofananealgebraicvarietyoverC.Denethetorus(5.3.1)Y(M):=(Z(M)I):ThenY(M)actstransitivelyonYtM.Fixarepresentative:WF!Mforthisinertialclass,sothattM=[]M.Lemma5.3.7.TheY(M)-stabilizerstab:=fz2Y(M)j(z)M=()Mgisnite.Proof.Thereexistsanintegerr1suchthatractstriviallyonM.ThegroupstabiscontainedinthepreimageofthenitegroupZ(M)�F\(M)derunderthenormhomomorphismNr:(Z(M)I)!Z(M)�F;z7!z(z)r�1(z)andthekernelofthishomomorphismisnite. 16T.HainesinplaceofF,itfollowsthat(z)jWE=zfjWE,wherezfisdenedasabove.Thusthemap(z)bG7!((z)jWE)bGliftstothemap(z)M7!(zfjWE)M7!(zfjWE)bG;andbeinginducedby(5.4.1),thelatterisanalgebraicmorphism.5.5.RelationbetweentheBern
steincenterandthestableBernsteincenter.ThevarietiesXandYaredenedunconditionally.Inordertorelatethem,weneedtoassumeLLC+holds.Proposition5.5.1.AssumeLLC+holdsforthegroupG.Thenthemap(M;)G7!('jWF)bGdenesaquasi-nitemorphismofanealgebraicvarietiesf:X!Y:ItissurjectiveifG=Fisquasi-split.ThereadershouldcomparethiswithConjecture7.18in[Vo].OurvarietystructureonthesetYisdi erentfromthatputforthbyVogan,andourfisgivenbyasimpleandexplicitrule.InviewofLLC+ourfautomaticallysatisestheconditionwhichVoganimposedonthemapinhisConjecture7.18:ifhassupercuspidalsupport(M;)G,thentheinnitesimalcharacterofisf((M;)G).Proof.Itiseasytoseethatthemap(M;)G7!('jWF)bGiswell-dened.Weneedtoshowthatanisomorphismcg:(M;)f!(gM;g)givenbyconjugationbyg2G(F)givesrisetoparameters':W0F!LM,!LGand'g:W0F!L(gM),!LGwhichdi erbyaninnerautomorphismofbG.InviewofConjecture5.2.7appliedtoM,theisomorphismL(gM)f!LMtakes'gtoancM-conjugateof'.OntheotherhandtheembeddingsLM,!LGandL(gM),!LGaredenedusingbasedrootsystemsinsuchawaythatitisobviousthattheyarebG-conjugate.Toexaminethelocalstructureofthismap,werstxaandastandardMthroughwhichfactorsminimally.Lett=[]bG.ThenoverYtthemapftakestheform(5.5.1)asM tX
sM!Yt:HeresMrangesovertheinertialclasses[M;]Gsuchthat('jWF)bGisinertiallyequivalentto()bG.Wenowxarepresentative(M;)forsM.Givensucha',itsrestriction'jWFfactorsthroughabG-conjugateofLM.But('jWF)bG()bGimpliesthat(uptoconjugationbybG)'jWFfactorsminimallythroughM.ThuswemayassumethatMLM.ThecorrespondinginclusionZ(cM),!Z(M)inducesamorphismofalgebraictoriY(cM)=(Z(cM)I)!(Z(M)I)=Y(M):Further,recallX(M)=Y(cM)bytheKottwitzisomorphism(ortheLanglandsdualityforquasi-characters),bytherule 7!zcocyc ().Taking(4.0.5)intoaccount,weseethat(5.5.1)onXsM,givenby(M; )G7!(' jWF)bGfor 2X(M)=stab,liftstothemap(5.5.2)X(M)=stab!Y(M)=stab; 18T.HaineswheneverdoesnotfactorthroughaproperLeviinLG,theinnitesimalclassconsistsofatmostoneL-packet.45.6.Aside:whendoesaninnitesimalclassconsistonlyofsupercuspidalrepresenta-tions?Proposition5.6.1.AssumeG=Fisquasi-splitandLLC+holdsforG.ThenconsistsentirelyofsupercuspidalrepresentationsifandonlyifdoesnotfactorthroughanyproperLevisubgroupLM(LG.Proof.Ifcontainsanonsupercuspidalrepresentationwithsupercuspidalsupport(M;)GforM(G,thenbyLLC+,wemayass
ume'jWF,andhence,factorsthroughtheproperLevisubgroupLM(LG.Conversely,iffactorsminimallythroughastandardLevisubgroupM(LG,thenwemustshowthatcontainsanonsupercuspidalrepresentationofG.SinceG=Fisquasi-split,wemayidentifyM=LMforanF-LevisubgroupM(G.Nowfort=[]bG,themap(5.5.1)issurjective.ForanyF-LevisubgroupM)M,acomponentoftheformX[M;]Ghasdimensiondim(Z(cM)I)dim(Z(dM)I)=dimYt.ThustheunionofthecomponentsoftheformX[M;]GwithM)McannotsurjectontoYt.ThustheremustbeacomponentoftheformX[M;]appearinginthelefthandsideof(5.5.1).Wemayassume'factorsthroughLMalongwith.Writing()cM=(zcocyc ')cMforsome 2X(M),itfollowsthatcontainsthenonsupercuspidalrepresentationswithsupercuspidalsupport(M; )G.5.7.ConstructionofthedistributionsZV.Let(r;V)beanite-dimensionalalgebraicrepre-sentationofLGonacomplexvectorspace.GivenageometricFrobeniuselement2WFandanadmissiblehomomorphism:WF!LG,wemaydenethesemi-simpletracetrss(();V)):=tr(r();Vr(IF)):Notethisisindependentofthechoiceof.ThisnotionwasintroducedbyRapoport[Ra90]inordertodenesemi-simplelocalL-functionsL(s;p;r),andisparall
eltothenotionfor`-adicGaloisrepresentationsusedin[Ra90]todenesemi-simplelocalzetafunctionsssp(X;s);seealso[HN02a,H05].Thefollowingresultisaneasyconsequenceofthematerialinx5:3.Proposition5.7.1.Themap7!trss(();V)denesaregularfunctiononthevarietyYhencedenesanelementZV2Zst(G)byZV(()bG)=trss(();V):WeusethesamesymbolZVtodenotethecorrespondingelementinZ(G)givenviaZst(G)!Z(G).Thelatterhastheproperty(5.7.1)ZV()=trss('();V)forevery2(G=F),whereZV()standsforZV((M;)G)if(M;)Gisthesupercuspidalsupportof. 4Noteaddedinproof(Feb.2014):Infactthisstatementholds:ifdoesnotfactorthroughaproperLevisubgroupofLG,thenthereisatmostonewaytoextendittoanadmissiblehomomorphism':W0F!LG. StableBernsteincenterandtestfunctions21FollowingRapoport'sstrategy(cf.[Ra90],[Ra05],[H05]),oneseekstondanaturalintegralmodelMKpoverOEforShKp,andthenrephrasetheaboveconjectureusingthemethodofnearbycyclesR :=R MKp( Q`).Conjecture6.1.2.ThereexistsanaturalintegralmodelMKp=OEforShKp,suchthat(6.1.3)Xx2MKp(kEj0)trss(rp;R x)=X( 0; ;)c( 0; ;)O(1Kp)TO( r);where r=prd=2ZVEj0�;j1KprasinConjecture6.1.1.Remark6.1.3.Implicitinthisconjectureisthatthemethodofnearbycyclescanbeusedforcompactly-suppo
rtedcohomology.Infactwecouldconjecturethereexistsasuitablynicecompact-icationofMKp=OEsothatthenaturalmapHic(MKp OE Fp;R ( Q`))!Hic(ShKp E Qp; Q`)isaGalois-equivariantisomorphism.ForG=GSp2gandwhereMKpisthenaturalintegralmodelforShKpforKpanIwahorisubgroup,thiswasprovedbyBenoitStroh.Ofcourse,oneisreallyinterestedinintersectioncohomologygroupsoftheBaily-Borelcompactication(seefootnote5),andinfactStroh[Str]computedthenearbycyclesandveriedtheanalogueoftheKottwitzconjectureonnearbycycles(seeConjecture7.1.3below)forthesecompactications.Remark6.1.4.SomeunconditionalversionsofConjectures6.1.1and6.1.2havebeenproved.Seex8.6.2.EndoscopictransferofthestableBernsteincenter.PartoftheLanglands-Kottwitzap-proachistoperforma\pseudostabilization"of(6.1.1),andinparticularprovethe\fundamentallem-mas"thatarerequiredforthis.Thestabilizationexpresses(6.1.1)intheformPHi(G;H)STe(h),thesumoverglobalQ-ellipticendoscopicgroupsHforGofthe(G;H)-regularQ-ellipticpartofthegeometricsideofthestabletraceformulafor(H;h)(cf.notationof[Ko90]),foracertaintransferfunctionh2C1c(H(A)).(Bycontrastin\pseudostabilization"whichisusedincertainsituations,oneinsteadwrites(6.1.1)intermsofthetraceformulaforGandnotitsquasi-splitinnerform,andthisissometimesenough,asine.g.Theorem6.3.2below.)Forstabilizationoneneeds
toproduceelementshp2C1c(H(Qp))whichareFrobenius-twistedendoscopictransfersof r.TheexistenceofsuchtransfershpisduemainlytotheworkofNg^o[Ngo]andWaldspurger[Wal97,Wal04,Wal08].Butwehopetohaveapriorispectralinformationaboutthetransferredfunctionshp.Aguidingprincipleisthatthenearbycyclesonanappropriate\localmodel"forShKpshouldnaturallyproduceacentralelementasatestfunction r,whichshouldcoincidewiththatgivenbythetestfunctionconjecture(cf.Conjecture6.1.2);thenitsspectralbehaviorisknownbyconstruction.Inthatcaseonecanformulateaconjecturalendoscopictransferhpof rwithknownspectralbehavior.GeneralFrobenius-twistedendoscopictransferhomomorphismsZst(GQpr)!Zst(HQp)willbedescribedelsewhere.Hereforsimplicitywecontentourselvestodescribetwospecialcases:standard(untwisted)endoscopictransferofthegeometricBernsteincenter,andthebasechangetransferforthestableBernsteincenter. StableBernsteincenterandtestfunctions23somemildrestrictionsontheresiduecharacteristicofF),andthusc(ZHVj1KHn)shouldbeanexplicittransferofZGV1KGn.AFrobenius-twistedanalogueofFerrari'stheoremtogetherwiththeFrobenius-twistedanalogueofConjecture6.2.2wouldgiveanexplicitFrobenius-twistedtransferofthetestfunction rfromConjecture6.1.1,ifKpisaprincipalcongruencesubgroup.6.2.2.BasechangeofthestableBernsteincenter.Weret
urntothesituationofProposition5.4.1,butwespecializeittocyclicGaloisextensionsofFandfurthermoreweassumeG=Fisquasi-split.LetE=FbeanynitecyclicGaloissubextensionof F=FwithGaloisgrouphi,andwithcorrespondinginclusionofWeilgroupsWE,!WF.If 2H(G(E))andf2H(G(F))arefunctionsinthecorrespondingHeckealgebrasoflocallyconstantcompactly-supportedfunctions,thenwesay ;fareassociated(orfisabase-changetrans-ferof ),ifthefollowingresultholdsforthestable(twisted)orbitalintegrals:foreverysemisimpleelement 2G(F),wehave(6.2.1)SO (f)=X( ;)SO( )wherethesumisoverstable-conjugacyclasses2G(E)withsemisimplenormN,andwhere( ;)=1ifN= and( ;)=0otherwise.Seee.g.[Ko86],[Ko88],[Cl90],or[H09]forfurtherdiscussion.Conjecture6.2.3.Intheabovesituation,considerZ2Zst(GE),andconsideritsimage,alsodenotedbyZ,inZ(GE).ConsiderbE=F(Z)2Zst(G)(cf.Def.5.4.2)andalsodenotebybE=F(Z)itsimageinZ(G).ThenbE=F(Z)isthebase-changetransferofZ2Z(GE),inthefollowingsense:wheneverafunctionf2C1c(G(F))isabase-changetransferof 2C1c(G(E)),thenbE=F(Z)fisabase-changetransferofZ .Proposition6.2.4.Conjecture6.2.3holdsforGLn.Proof.ThemostecientprooffollowsScholze'sproofofTheoremCin[Sch2]whichmakesessen-tialuseoftheexistenceofcyclicbasechangeliftsforGLn
.Let2(GLn=F)beatemperedirreduciblerepresentationwithbasechangelift2(GLn=E),atemperedrepresentationwhichischaracterizedbythecharacteridentity((g;))=(Ng)forallelementsg2GLn(E)withregularsemisimplenormNg([AC,Thm.6.2,p.51]).Here(g;)2GLn(E)oGal(E=F)andactsonbythenormalizedintertwinerI:!of[AC,p.11].Supposefisabase-changetransferof .UsingtheWeylintegrationformulaanditstwistedanalogue(cf.[AC,p.36]),weseethattr(( ;)j)=tr(fj):MultiplyingbytheconstantZ()=bE=F(Z)(),wegettr((Z ;)j)=tr(bE=F(Z)fj):(UseCorollary3.2.1anditstwistedanalogue.)Thereexistsabase-changetransferh2C1c(GLn(F))ofZ ([AC,Prop.3.1]).UsingthesameargumentasaboveforthepairZ andh,weconcludethattr(bE=F(Z)f�hj)=0foreverytemperedirreducible2(GLn=F).ByKazhdan'sdensitytheorem(Theorem1in[Kaz])theregularsemi-simpleorbitalintegralsofbE=F(Z)fandhagree.Thusthe(twisted)orbitalsintegralsofbE=F(Z)fand matchatallregularsemi-simpleelements,andhenceatallsemi-simpleelementsbyClozel'sShalikagermargument([Cl90,Prop.7.2]). 26T.HainesItfollowsthatLefss(jp;ShKp)=Xm()tr(fpf(j)pf1j)=XfX121m(f 1)tr(fpjpf)tr(f(j)pjp)tr(f1j
1)=Xfa(f)dim(Kf)pjfd=2trss('p(jp);VE0�);thelastequalityby(6.3.3).BydenitionwehavelogLss(s;p;rp)=1Xr=1trss('p(rp);rp)p�rs r:Now(6.3.6)followsbyinvoking(6.3.5).Remark6.3.3.UnconditionalversionsofTheorem6.3.2areavailableforsomeparahoricorpro-p-Iwahorilevelcases,orforcertaincompact\Drinfeldcase"Shimuravarietieswitharbitrarylevel;thesecasesarealludedtoinx8.6.4.RelationwithgeometricLanglands.Forsimplicity,assumeGissplitoverap-adicorlocalfunctioneldF.AssumeGsatisesLLC+.FromtheconstructionofZVinProposition5.7.1,wehaveamapK0RepC(bG)!Z(G;J)(6.4.1)V7!ZV1JforanycompactopensubgroupJG(F),whichgivesrisetoacommutativediagramZ(G;J)�J1I Z(G;I)�I1K K0Rep(bG)Sat// Bern55 88 H(G;K)wheneverJIKwhereIresp.KisanIwahoriresp.specialmaximalcompactsubgroup,andwherethebottomtwoarrowsaretheBernsteinresp.Satakeisomorphisms.WewarnthereaderthattheobliquearrowK0Rep(bG)!Z(G;J)isinjectivebutnotsurjectiveingeneral,andalsoitisadditivebutnotanalgebrahomomorphismingeneral.Gaitsgory[Ga]constructedthetwoarrowsSatandBerngeometricallywhenFisafunctioneld,usingnearbycyclesforadegenerationoftheaneGrassmannianGrGtotheane agvarietyFlforG.Onecanhopethat,asintheIwahoricase[Ga],onecanco
nstructthearrowK0Rep(bG)!Z(G;J)categoricallyusingnearbycyclesforasimilardegenerationofGrGtoa\partialane agvariety",namelyanfpqc-quotientLJ=L+JwhereJisasmoothconnectedgroupschemeover Fp[[t]]withgenericberJFp((t))=GFp((t))andJ(Fp[[t]])=J.HereLJ(resp.L+J)istheind-scheme(resp.scheme)overFprepresentingthesheafofgroupsforthefpqc-topologywhosesectionsforanFp-algebraRaregivenbyLJ(SpecR)=J(R[[t]][1 t])(resp.L+J(SpecR)=J(R[[t]])). 28T.HainesLemma7.1.1.Intheabovesituation,(7.1.2)ZVEj0�;j1Kpr=z�;j;wheretheBernsteinfunctionz�;j(cf.Denition11.10.2)istheuniqueelementofZ(G(F);KF)whichacts(ontheleft)onthenormalizedinducedrepresentationiGB( )KFbythescalarP2W(F)(�)( ),foranyunramiedcharacter :T(F)!C.Ofcoursetheadvantageofz�;jisthatunlikethelefthandsideof(7.1.2),itisdeneduncon-ditionally.Arelativelyself-contained,elementary,andecientapproachtoBernsteinfunctionsisgiveninx11.ThusConjecture6.1.1inthissituationisequivalenttotheKottwitzConjecture.Conjecture7.1.2.(Kottwitzconjecture)InthesituationwhereGQprisunramiedandKpisaparahoricsubgroup,thefunction rin(6.1.1)maybetakentobeprd=2z�;j.Conjecture7.1.2wasformulatedbyKottwitzin1998,about11yearsearlierthanConjecture6.1.1.Thereisacloselyrelate
dconjectureofKottwitzconcerningnearbycyclesonRapoport-ZinklocalmodelsMlocKpforShKp.Wereferto[RZ,Ra05]fordenitionsoflocalmodels,andto[H05,HN02a]forfurtherdetailsaboutthefollowingconjectureinvariousspecialcases.Conjecture7.1.3.(KottwitzConjectureforNearbyCycles)WriteGfortheBruhat-TitsparahoricgroupschemeoverZprwithgenericberGQprandwithG(Zpr)=Kpr.LetGtdenotetheanalogousparahoricgroupschemeoverFpr[[t]]withthe\same"specialberasG.ThenthereisanL+Gt;Fpr-equivariantembeddingofMlocKp;Fprintotheane agvarietyLGt;Fpr=L+Gt;Fpr,viawhichwecanidentifythesemisimpletraceofFrobeniusfunctionx7!trss(Frpr;R MlocKpx)onx2MlocKp(Fpr)withthefunctionpdr=2z�;j2Z(Gt(Fpr((t)));Gt(Fpr[[t]])).7.2.Thequasisplitcase.ThegroupbGIFisapossiblydisconnectedreductivegroup,withmaximaltorus(bTIF)(seetheproofofTheorem8.2of[St]).NowwemayrestricttherepresentationVE0�tothesubgroupbGIFoWFbGoWF.Let beaweaklyunramiedcharacterofT(F);by(3.3.2)wecanview 2(bTIF)F.TheonlybTIF-weightspacesof(VE0�)1oIFwhichcontributeto(7.0.4)areindexedbytheF-xedweights,i.e.bythoseinX(bTIF)F.(ItisimportanttonotethatitistheweightspacesforthediagonalizablegroupbTIF,andnotforthemaximaltorus(bTIF),whichcomeinhere.)ThisisconsistentwithTheorem11.10.1oftheAppendix,andmaybe
expressedasfollows.Proposition7.2.1.Inthegeneralquasisplitsituation,ZVEj0�;j1KpristheuniquefunctioninZ(G(Qpr);Kpr)whichactsontheleftoneachweaklyunramiedprincipalseriesrepresentationiGB( )KFbythescalar(7.0.4),andthusisacertainlinearcombinationofBernsteinfunctionsz�;jwhere�2X(bTIF)FrangesovertheW(G;A)-orbitsofF-xedbTIF-weightsinVE0�.ItisaninterestingexercisetowriteoutthelinearcombinationsofBernsteinfunctionsexplicitlyineachgivencase.Oncethisisdone,theresultcanbeusedtondexplicitdescriptionsoftestfunctionsforinnerformsofquasi-splitgroups.Thisisthesubjectofthenextsubsection.7.3.Passingfromquasisplittogeneralcasesviatransferhomomorphisms. StableBernsteincenterandtestfunctions31ThisiscompatiblewithcalculationsofRapoportofthetraceofFrobeniusonnearbycyclesofthelocalmodelsforsuchsituations,see[Ra90].Thusthenormalizedtransferhomomorphismgivesagroup-theoreticframeworkwithwhichwecouldmakefurtherpredictionsaboutnearbycyclesonthelocalmodelsattachedtonon-quasiplitgroupsG,assumingweknowexplicitlythecorrespondingtestfunctionforaquasisplitinnerformofG.Proof.BythenalsentenceofProposition11.12.6,wesimplyneedtointegratethefunctionprd=2z�2Z(GLn(Qpr);Kpr)overtheberoftheKottwitzhomomorphismvaldetover1m2C[Z].Thi
sisacombinatorialproblemwhichcouldbesolvedsinceweknowprd=2z�explicitly.However,itiseasiertousegeometry.Translating\integrationovertheberoftheKottwitzhomomorphism"intermsoflocalmodelsgivesustheequalityCq=Xx2Mloc(Fq)Tr(rp;R Mloc(Q`)x):(Here`isarationalprimewith`6=p.)ButthespecialberofMlocembedsintotheane agvarietyFlGLnforGLn=Fp,andundertheprojectionp:FlGLn!GrGLntotheaneGrassmannian,MlocmapsontoGr(m;n)andRp(R Mloc(Q`))=Q`,theconstant`-adicsheafonGr(m;n)indegree0.ThusweobtainCq=Xx2Gr(m;n)(Fq)Tr(rp;(Q`)x)=#Gr(m;n)(Fq)asdesired.(ThereadershouldnotethesimilaritywithProp.3.17in[Ra90],whichisjustiedinaslightlydi erentway.)8.Overviewofevidenceforthetestfunctionconjecture8.1.Goodreductioncases.IncaseGQpisunramiedandKpisahyperspecialmaximalcompactsubgroupofG(Qp),weexpectSh(G;h�1;KpKp)tohavegoodreductionoverOEp.InPELcasesthiswasprovedbyKottwitz[Ko92a].InthesamepaperforPELcasesoftypeAorC,itisprovedthatthefunction r=1Kpr(p�1)Kprsatises(6.1.1),whichcanbeviewedasverifyingConjecture6.1.1forthesecases.8.2.Parahoriccases.AssumeKpisaparahoricsubgroup.WewilldiscussonlyPELShimuravarieties.HeretheapproachisviatheRapoport-ZinklocalmodelMlocKpforasuitableintegralmodelMKpforShKpandthemainideasareduetoRapoport
.Werefertothesurveyarticles[Ra90],[Ra05],and[H05]formoreabouthowlocalmodelstinwiththeLanglands-Kottwitzapproach.Formuchmoreaboutthegeometryoflocalmodels,wereferthereadertothesurveyarticleofPappas-Rapoport-Smithing[PRS]andthereferencestherein.Usinglocalmodels,therststeptoprovingConjecture6.1.2istoproveConjecture7.1.3.Therstevidencewaspurelycomputational:in[H01],z�;jwascomputedexplicitlyintheDrinfeldcaseandtheresultwascomparedwithRapoport'scomputationofthenearbycyclesinthatsetting,provingConjecture7.1.3directly.ThisresultmotivatedKottwitz'moregeneralconjectureandalsoinspiredBeilinsonandGaitsgorytoconstructthecenterofananeHeckealgebraviaanearbycyclesconstruction,afeatcarriedoutin[Ga].Thenin[HN02a]Gaitsgory'smethodwasadaptedtoproveConjecture7.1.3forthesplitgroupsGLnandGSp2n.Thisinturnwasusedtodemonstrate StableBernsteincenterandtestfunctions33levelcases,havethefavorablepropertythattheirnearbycyclesnaturallygiverisetodistributionsintheBernsteincenter.Itremainsaninterestingproblemtondsuchintegralmodelsinmorecases,andtobetterunderstandtheroleoftheBernsteincenterinthestudyofShimuravarieties.9.EvidenceforconjecturesontransferoftheBernsteincenterHerewepresentsomeevidenceforthegeneralprinciplethatthe(stable/geometric)Bernsteincen-terisparticularlywell-behavedwi
threspectto(twisted)endoscopictransfer.TheprimaryevidencethusfarconsistsofsomeunconditionalanaloguesofConjecture6.2.3.LetG=Fbeanunramiedgroup,andletFr=FbethedegreerunramiedextensionofFinsomealgebraicclosureofF.In[H09,H12],theauthordenedbasechangehomomorphismsbr:Z(G(Fr);Jr)!Z(G(F);J);whereJG(F)iseitheraparahoricsubgrouporapro-pIwahorisubgroup,andwhereJristhecorrespondingsubgroupofG(Fr).Thenwehave\base-changefundamentallemmas"ofthefollowingform.8Theorem9.0.1.Forany r2Z(G(Fr);Jr),thefunctionbr( r)isabase-changetransferof rinthesenseof(6.2.1).ByKottwitz[Ko86],thefunction1Jisabase-changetransferof1Jr.HenceforanyVr2Rep(L(GFr)),Conjecture6.2.3predictsthatbFr=F(ZVr)1Jisabase-changetransferofZVr1Jr.ThisisaconsequenceofTheorem9.0.1,becauseofthefollowingcompatibilitybetweenthebase-changeoperationsin[H09,H12]andinthecontextofstableBernsteincenters(cf.Prop.5.4.1).Lemma9.0.2.Intheabovesituations,br(ZVr1Jr)=bFr=F(ZVr)1J.Proof.FirstassumeJisaparahoricsubgroup.Let beanyunramiedcharacterofT(F).ItisenoughtoshowthatthetwofunctionsactontheleftbythesamescalaroneveryunramiedprincipalseriesrepresentationiGB( )J.LetNr:T(Fr)!T(F)bethenormhomomorphism.Bythedenitionofbrin[H09],br(ZVr1Jr)actsbythescalarbywhichZVr1JractsoniGrBr( &
#14;Nr)Jr.ThisisthescalarbywhichZVractsoniGrBr( Nr),whichinviewofLLC+is(9.0.1)trss('Tr Nr(rF);Vr)=trss('T (rF);Vr):ButtherighthandsideisthescalarbywhichbFr=F(ZVr)1JactsoniGB( )J.Theequality'Tr Nr(rF)='T (rF)weusedin(9.0.1)followsfromthecommutativityofthediagramofLanglandsdualitiesfortoriHomconts(T(F);C)// Nr H1conts(WF;LT)Res Homconts(T(Fr);C)// H1conts(WFr;LTr)whichwasprovedin[KV,Lemma8.1.3]. 8Relatingtopro-pIwahorilevel,amuchstrongerresultisprovedin[H12]concerningthebasechangetransferofBernsteincentersofBernsteinblocksfordepth-zeroprincipalseriesrepresentations. 34T.HainesNowsupposeJ=I+isapro-pIwahorisubgroup.Thenthesameargumentworksgiventhefollowingfact:foranydepth-zerocharacter :T(F)1!Candanyextensionofittoacharacter~ onT(F),andanyzr2Z(G(Fr);I+r),thefunctionbr(zr)actsoniGB(~ )I+bythescalarbywhichzractsoniGrBr(~ Nr)I+r.ThisfollowsfromthedenitionofbrgiveninDenition10.0.3of[H12],using[H12,Lemma4.2.1].LetusalsomentionagainScholze'sTheoremCin[Sch2],whichessentiallyprovesConjecture6.2.3forGLn(seeProposition6.2.4).10.Explicitcomputationofthetestfunctions10.1.Parahoriccases.Conjecture6.1.1impliesthattestfunctionsarecompatiblewithchangeoflevel.Thereforeforthepurpo
sesofcomputingthemforparahoriclevel,thekeycaseiswhereKpisanIwahorisubgroup.Thus,fortherestofthissubsectionweconsideronlyIwahorilevelstructure.Sincetestfunctionsattachedtoquasisplitgroupsshoulddetermine,inacomputableway,thoseforinnerforms(byConjecture7.3.1andProposition11.12.6),itisenoughtounderstandquasisplitgroups.ViaProposition7.2.1thisboilsdowntogivingexplicitdescriptionsoftheBernsteinfunctionsz�;j,assumingwehavealreadyexpressedthetestfunctionexplicitlyintermofthese{thisisautomaticforunramiedgroupsusingtheKottwitzConjecture(Conjecture7.1.2).LetusthereforeconsidertheproblemofexplicitlycomputingBernsteinfunctionszattachedtoanygroupG=FandanIwahorisubgroupIG(F)(Fbeinganylocalnon-archimedeaneld).ForsimplicityconsiderthecasewhereG=Fisunramied,andregardasadominantcoweightinX(A).ThewhichariseinConjecture7.1.2areminuscule;however,weconsiderwhicharenotnecessarilyminusculehere.LetfWdenotetheextendedaneWeylgroupofGoverF(cf.x11).Attachedtoisanthe-admissiblesetAdm()fW,denedbyAdm()=fx2fWjxt;forsome2W(G;A)g;wheredenotestheBruhatorderonfWdeterminedbytheIwahorisubgroupIandwheretdenotesthetranslationelementinfWcorrespondingto2X(A).The-admissiblesethasbeenstudiedfori
tsrelationtothestraticationbyIwahori-orbitsinthelocalmodelMlocKp;formuchinformationsee[KR],[HN02b],[Ra05].ThestrongestcombinatorialresultsrelatinglocalmodelsandAdm()areduetoBrianSmithling,seee.g.[Sm1,Sm2,Sm3].Forourpurposes,thesetAdm()entersbecauseitisthesetindexingthedoublecosetsinthesupportofz.Proposition10.1.1.ThesupportofzistheunionSx2Adm()IxI.Proof.ThiswasprovedusingthetheoryofalcovewalksaselaboratedbyGortz[G],intheAppendixto[HRa].ItappliestoaneHeckealgebraswitharbitraryparameters,hencethecorrespondingresultholdsforarbitrarygroups,notjustunramiedgroups.Thefollowingexplicitformulawasprovedin[H01]andin[HP].LetTx=1IxIforx2fW.Intheformulashereandbelow,q=pristhecardinalityoftheresidueeldofF.Proposition10.1.2.Assumeisminuscule.AssumetheparametersfortheIwahoriHeckealgebraareallequal.Thenq`(t)=2z=(�1)`(t)Xx2Adm()(�1)`(x)Rx;t(x)(q)Tx; 36T.Hainesdenesadistributioninthesenseof(3.2.1).Thisdistributionisq1=2ZVwhereVisthestandardrepresentationC2oftheLanglandsdualgroupGL2(C).Inanunpublishedwork,Kottwitzgaveanotherproofofthispropositionandalsodescribedthesamedistributionintermsofafamily(n)noffunctionsn2Z(GL2(F);In)whereInrangesoverthe\barycentric"Moy-PrasadltrationinthestandardIwahorisubgrou
pIGL2(F).Byacompletelydi erenttechnique,in[Var]SandeepVarmaextendedboththeresultsofScholzeandKottwitzstatedabove,bydescribingthedistributionsattachedtoV=Symr(C2)whererisanyoddnaturalnumberlessthanp,theresidualcharacteristicofF.11.Appendix:Bernsteinisomorphismsviatypes11.1.StatementofPurpose.AlanRocheprovedthefollowingbeautifulresultin[Roc],Theorem1.10.3.1.Theorem11.1.1(Roche).LetebeanidempotentintheHeckealgebraH=H(G(F)).ViewHasasmoothG(F)-moduleviatheleftregularrepresentation,andwritee=Ps2SesaccordingtotheBernsteindecompositionH=Ls2SHs.LetSe=fs2Sjes6=0g,andconsiderthecategoryRSe(G(F))=Qs2SeRs(G(F))anditscategoricalcenterZSe=Qs2SeZs.LetZ(eHe)denotethecenterofthealgebraeHe.Thenthemapz7!z(e)denesanalgebraisomorphismZSef!Z(eHe).Roche'sproofisdecidedlynon-elementary:besidesthematerialdevelopedin[Roc],itreliesonsomedeepresultsofBernsteincitedthere,mostimportantlyBernstein'sSecondAdjointnessTheoremandtheconstructionofanexplicitprogeneratorforeachBernsteinblockRs(G(F)).InthischapterweuseonlytheveryspecialcaseofRoche'stheoremwheree=eJforaparahoricsubgroupJG(F).Wewillexplainamoreelementaryapproachtothisspecialcase.ItwillrelyonlyonthepartofBernstein'stheoryembodiedinProposition11.7.3below.Formally,theinputsneededare,rst,theexistenceofBernstein'scategoricaldecompositionR
(G)=QsRs(G),whichisprovedforinstancein[Roc,Thm.1.7.3.1]inanelementaryway,and,second,theinternalstructureoftheBernsteinblockRs(G)associatedtoacuspidalpairs=[(M(F);~ ]GwhereMisaminimalF-LevisubgroupofGand~ isacharacteronM(F)whichistrivialonitsuniqueparahoricsubgroup.Forsuchcomponents,progeneratorscanbeconstructedinanelementaryway,withoutusingBernstein'sSecondAdjointnessTheorem.InfactinwhatfollowswedescribethisinternalstructureusingafewstraightforwardelementsofthetheoryofBushnell-Kutzkotypes,allofwhicharecontainedin[BK].Fore=eJRoche'stheoremgivestheidenticationofthecenteroftheparahoricHeckealgebra,inotherwordsaBernsteinisomorphismforthemostgeneralcase,whereG=FisarbitraryandJG(F)isanarbitraryparahoricsubgroup.HoweverwewillprovideaproofonlyforthecrucialcaseofJ=I,anIwahorisubgroupofG(F).ThegeneralparahoriccaseshouldfollowformallyfromtheIwahoricase,followingthemethodofTheorem3.1.1of[H09],providedoneiswillingtorelyonsomebasicpropertiesofintertwinersforprincipalseriesrepresentations(apurelyalgebraictheoryofsuchintertwinerswasdetailedforsplitresp.unramiedgroupsin[HKP]resp.[H07],andtheextensiontoarbitrarygroupsshouldbesimilarto[H07]). 38T.HainesFurther,theBruhatorderandlengthfunction`onWa extendintheusualwaytofW,andwehaveforw2fWands2Wa representingasimplea&#
14;nere ection,theusualBN-pairrelations(11.3.2)IsIwI=8:IswI;ifwswIwI[IswI;ifsww:11.3.2.Iwahorifactorization.LetP=MNbeanF-rationalparabolicsubgroupwithLevifactorM,unipotentradicalNandoppositeunipotentradical N.LetIH=I\HforH=N; N;orM.Lemma11.3.3.Intheabovesituation,wehavetheIwahorifactorization(11.3.3)I=INIMI N:Proof.Weusethenotationof[BT2].By[BT2,5.2.4]with :=a,wehaveGa(O\)=U+\aU�\aN\a;whereN\a:=N\\Z(O\)U\a.SinceZ(O\)U\aGa(O\),wehaveN\a=N\\Ga(O\)=Z(O\):Thekeyinclusionhere,N\\Ga(O\)Z(O\);translatesinournotationtoNG(A)(F)\IM(F)1,whichcanbededucedfromLemma11.3.1(a).TranslatingagainbacktoournotationwegetI=INI NIMwhichisthedesiredequalitysinceIMnormalizesI N.11.3.3.OnM(F)1=M(F)1.Lemma11.3.4.Thefollowingbasicstructuretheoryresultshold:(a)Inthenotationof[HRo],wehaveM(F)1=M(F)1=~K=K,whichinjectsintoG(F)1=G(F)1.ThusM(F)1=M(F)1\G(F)1.(b)TheWeylgroupW(G;A)actstriviallyonM(F)1=M(F)1.(c)LetaB(G;L)denotethealcoveinvariantunderthegroupAut(L=F)hiwhichcorrespondstotheIwahoriIG(F).WeassumeIK.ThenthenaiveIwahori~I:=G(F)1\Fix(a)hasthefollowingpropertiesM(F)1=M(F)1=~I=I=~K=K.~I=IM(F)1.Proof.Part(a):inthenotationof[HRo],weknowthatM;tors=~K=K([HRo,Prop.11.1.4]).Applyi
ngthistoG=MwegetM;tors=M(F)1=M(F)1.SoM(F)1=M(F)1=~K=K.By(8.0.1)andLemma8.0.1in[HRo],thelatterinjectsintoG(F)1=G(F)1.Thenalstatementfollows.Part(b):By[HRo,Lemma5.0.1],W(G;A)hasrepresentativesinK\NG(A)(F).Thusitisenoughtoshowthatifn2K\NG(A)(F)andm2M(F)1,thennmn�1m�12M(F)1.Thisfollowsfrom(a),sinceweclearlyhavenmn�1m�12M(F)1\G(F)1.Part(c):FirstnotethatM(F)1~IandM(F)1I.Thusthereisacommutativediagram~I=I// ~K=KM(F)1=M(F)1OO 88 StableBernsteincenterandtestfunctions39Theobliquearrowisbijectiveby(a).Weclaimthehorizontalarrowisinjective,thatis,~I\K=I.But~I\K=G(F)1\Fix(a)\G(F)1\Fix(vF),wherevFisthespecialvertexinB(Gad;F)correspondingtoK(cf.[HRo,Lem.8.0.1]).Thus~I\K=G(F)1\Fix(a)=IbyRemark8.0.2of[HRo].Itnowfollowsthatallarrowsinthediagramarebijective.Thisimpliesbothstatementsin(c).Remark11.3.5.LetP=MNbeasabove.Wededucefrom(c)and(11.3.3)theIwahorifactor-izationfor~I(11.3.4)~I=INM(F)1I N;usingthefactthatM(F)1normalizesINandI N.11.3.4.Iwasawadecomposition.NextweneedtoestablishasuitableformoftheIwasawadecompo-sition.LetP=MNbeasabove.Lemma11.3.6.TheinclusionNG(A)(F),!G(F)inducesbijectionsfW:=NG(A)(F)=M(F)1f!M(F)1N(F)nG(F)=I(11.3.5)W(G;A)=NG(A)(F)=M(F)f!P(F)nG(F)=I:(11.3.6)Proof.InviewofthedecompositionfW= MoW(G;A)(cf.Lemma3.0.1(III)of[HRo]plus
Lemma11.3.1(b))andtheKottwitzisomorphism Mf!M(F)=M(F)1(cf.Lemma3.0.1of[HRo]),itsucestoprove(11.3.5).Forx2B(G;F),letPxG(F)denotethesubgroupxingx.By[Land],Proposition12.9,wehaveG(F)=N(F)NG(A)(F)Px:Forsucientlygenericpointsx2a,wehavePx=~I,whichisM(F)1IbyLemma11.3.4(c).SinceM(F)1NG(A)(F),wehaveG(F)=N(F)NG(A)(F)Iandthemap(11.3.5)issurjective.Toproveinjectivity,assumen1=um0n2jforu2N(F),m02M(F)1,n1;n22NG(A)(F),andj2I.Thereexistsz2Z(M)(F)suchthatzuz�12IN(cf.e.g.[BK,Lem.6.14]).Thenzn2=(zuz�1)m0zn2j2Izn2I;andsoby(11.3.1),zn2zn2moduloM(F)1.Lemma11.3.7.Ifx;y2fWandM(F)1N(F)xI\IyI6=;,thenxyintheBruhatorderonfWdeterminedbyI.Proof.ThisfollowsfromtheBN-pairrelations(11.3.2)asintheproofoftheClaiminLemma1.6.1of[HKP].11.3.5.TheuniversalunramiedprincipalseriesmoduleM.DeneM=Cc(M(F)1N(F)nG(F)=I):Thesubscript\c"meansweareconsideringfunctionssupportedonnitelymanydoublecosets.SomebasicfactsaboutMweregivenin[HKP]forthespecialcasewhereGissplit,andhereweneedtostatethosefactsinthecurrentgeneralsituation.AbbreviatebysettingH=H(G(F);I)andR=C[M].Thenf2HactsontheleftonMbyrightconvolutionsbyf,whichisdenedbyf(g)=f(g�1).Thesamegoesforthenormalized StableBernsteincenterandtestfunctions4111.5.Wh
y(I;1)isanSG-type.WedeneSG=f[t]Gj[t]M2SMg.Themap[M;~ ]M7![M;~ ]Gisinjective:if[M;~ 1]G=[M;~ 2]G,thenthereexistsn2NG(A)(F)suchthatn(~ 1)=~ 2forsomecharacteronM(F)=M(F)1.RestrictingtoM(F)1andusingn( 1)= 1(Lemma11.3.4(b)),wesee 1= 2.SoSM=SGvia[t]M7![t]G.Wesawabovethat(M(F)1;1)isanSM-type.Thefactthat(I;1)isanSG-typefollowsfrom[BK,Thm.8.3],oncewecheckthefollowingproposition.Proposition11.5.1.Thepair(I;1)isaG-coverfor(M(F)1;1)inthesenseof[BK,Denition8.1].Proof.Weneedtocheckthethreeconditions(i-iii)ofDenition8.1.First(i),thefactthat(I;1)isdecomposedwithrespectto(M;P)inthesenseof[BK,(6.1)],followsfromtheIwahorifactorizationI=INIMI NdiscussedinRemark11.3.5.TheequalityI\M(F)=M(F)1givescondition(ii).Finallywemustprove(iii):foreveryF-parabolicPwithLevifactorM,thereexistsaninvertibleelementofH(G(F);I)supportedonIzPI,wherezPbelongstoZ(M)(F)andisstrongly(P;I)-positive.TheexistenceofelementszP2Z(M)(F)whicharestrongly(P;I)-positiveisprovedin[BK,Lemma6.14].Anycorrespondingcharacteristicfunction1IzPIisinvertibleinH(G(F);I),asfollowsfromtheIwahori-MatsumotopresentationofH(G(F);I).(Thispresentationitselfiseasytoproveusing(11.3.2).)11.6.StructureoftheBernsteinvarieties.LetR(G)denotethecategoryofsmoothrepresenta-tionsofG(F),
andletR (G)denotethefullsubcategorycorrespondingtotheinertialclass[M;~ ]G.Thatis,arepresentation(;V)2R(G)isanobjectofR (G)ifandonlyifforeachirreduciblesubquotient0of,thereexistsanextension~ of suchthat0isasubquotientofIndGP(1=2P~ ).WereviewthestructureoftheBernsteinvarietiesXG andXM .Inthisdiscussion,foreach wexanextension~ of onceandforall{thestructureswedenewillbeindependentofthechoiceof~ ,i.e.uniquelydeterminedby(M; )uptoauniqueisomorphism.AsasetXG (resp.XM )consistsoftheelements(M;~ )G(resp.(M;~ )M)belongingtotheinertialequivalenceclass[M;~ ]G(resp.[M;~ ]M)asrangesoverthesetX(M)ofunramiedC-valuedcharactersonM(F)(unramiedmeansitfactorsthroughM(F)=M(F)1).ThemapX(M)!XM ,7!(M;~ )M,isabijection.SinceX(M)isacomplextorus,thisgivesXM thestructureofacomplextorus.Morecanonically,XM isjustthevarietyofallextensions~ of ,anditisatorsorunderthetorusX(M).Nowx~ again.ThereisasurjectivemapXM !XG (M;~ )M7!(M;~ )G:SinceW:=W(G;A)actstriviallyonM(F)1=M(F)1(Lemma11.3.4),onecanprovethatthebersofthismaparepreciselytheW-orbitsonXM .ThusassetsWnXM =XG ;andthisgivesXG thestructur
eofananevarietyoverC.HavingchosentheisomorphismX(M)=XM asabove,wecantransporttheW-actiononXM overtoanactiononX(M).Thisactiondependsonthechoiceof~ andisnottheusualactionunless~ isW-invariant.WeobtainXG =Wn~ X(M),wherethelatterdenotesthequotientwithrespecttothisunusualactiononX(M). StableBernsteincenterandtestfunctions43Concretely,themapZ(G(F);I)!Q C[XG ],z7!^z,ischaracterizedasfollows:forevery andevery(M;~ )G2XG ,z2Z(G(F);I)actsonIndGP(1=2P~ )Ibythescalar^z(~ ).LetussingleoutwhathappensinthespecialcaseofG=M.WecanidentifyH(M(F);M(F)1)=C[M].LeteM denotetheidempotentinH(M(F);M(F)1)analogoustoe ,forthecaseG=M.ByPropositions11.7.2and11.7.3forG=M,wehave(11.7.1)H(M(F);M(F)1)=Y eM H(M(F);M(F)1)eM =Y C[XM ];thelastequalityholdingsinceH(M(F);M(F)1)isalreadycommutative.Thus,theringeM H(MF);M(F)1)eM canberegardedastheringofregularfunctionsonthevarietyXM ofallextensions~ of .11.8.TheembeddingofC[M]WintoZ(G(F);I).WemakeuseofthefollowingspecialcaseofageneralconstructionofBushnell-Kutzko[BK]:foranyF-parabolicPwithLevifactorM,thereisaninjectivealgebrahomomorphismtP:H(M(F);M(F)1)!H(G(F);I)whichisuniquelycharacterizedbythepropertythatforeach(;V)2RI(G),v2VI,andh2H(M(F);M(F)1),w
ehavetheidentity(11.8.1)q(tP(h)v)=hq(v):Hereq:VIf!VM(F)1Nisanisomorphism,whichisinducedbythecanonicalprojectionV!VNtothe(unnormalized)Jacquetmodule.See[BK,Thm.7.9].Itturnsoutthatitisbettertoworkwithadi erentnormalization.WedeneanotherinjectivealgebrahomomorphismP:H(M(F);M(F)1)!H(G(F);I)h7!tP(�1=2Ph):Thenusing(11.8.1)Psatises(11.8.2)q(P(h)v)=(�1=2Ph)q(v):Weview~ asavaryingelementoftheZariski-densesubsetSofthevarietyofallcharactersonthenitely-generatedabeliangroupM(F)=M(F)1,consistingofthoseregularcharacters~ suchthatV(~ ):=iGP(~ ):=IndGP(1=2P~ )isirreducibleasanobjectofR(G).WeapplytheabovediscussiontotherepresentationsV:=V(~ )with~ 2S.ByaresultofCasselman[Cas],weknowthatasM(F)-modulesVN=Mw2W1=2P(w~ )andthatM(F)1actstriviallyonthismodule.Nowsupposeh2C[M]W.Then�1=2PhactsonVN=VM(F)1Nbythescalarh(~ )(viewinghasaregularfunctiononXM ).Itfollowsfrom(11.8.2)thatP(h)actsbythescalarh(~ )oniGP(~ )I,for~ 2S:Nowletf2Hbearbitrary,andset:=fP(h)�P(h)f2H.Weseethat(11.8.3)actsbyzerooniGP(~ )Iforevery~ 2S. 46T.Haineswhere()designatestheGaloisactiononG(forWeyl-groupequivariancesee[HRo,x
12:2]).Since MisatorsorforMadthishomomorphismdoesnotdependonthechoiceof 02 M.Further,itdependsonlyonthechoiceofAandA,andnotonthechoiceoftheparabolicsubgroupsPMandBTwemadeinconstructingit.Denition11.12.1.LetJG(F)andJG(F)beanyparahoricsubgroupsandchoosecompatiblemaximalF-splittoriAresp.Aasabove.Thenwedenethetransferhomomorphismt:Z(G(F);J)!Z(G(F);J)tobetheuniquehomomorphismmakingthefollowingdiagramcommuteZ(G(F);J)t// o Z(G(F);J)o C[X(cT)FIF]W(G;A)tA;A// C[X(Z(cM))FIF]W(G;A);wheretheverticalarrowsaretheBernsteinisomorphisms.By[BT2,4.6.28],anytwochoicesforA(resp.A)areJ-(resp.J-)conjugate.UsingCorollary11.11.1itfollowsthattisindependentofthechoiceofAandAandisacompletelycanonicalhomomorphism.Remark11.12.2.ThemaptA;A:X(cT)FIF!X(Z(cM))FIFissurjective.ViatheKottwitzhomomorphismwemayviewthisasthecomposition(11.12.1)T(F)=T(F)1// M(F)=M(F)1 �10// M(F)=M(F)1wheretherstarrowisinducedbytheinclusionT,!M.ItisenoughtoobservethatM(F)=T(F)M(F)1,whichinturnfollowsfromtheIwasawadecomposition(cf.(11.3.6))forM(F),whichstatesthatM(F)=T(F)UM(F)KMforan
F-rationalBorelsubgroupBM=TUMandaspecialmaximalparahoricsubgroupKMinM,andfromthevanishingoftheKottwitzhomomorphismonUM(F)KM.11.12.2.Normalizedtransferhomomorphism.Thetransferhomomorphismisslightlytoonaive,anditisnecessarytonormalizeitinordertogetahomomorphismwhichhastherequiredproperties.WeneedtodenenormalizedhomomorphismsetA;AonWeyl-groupinvariants,forwhichthefollowinglemmaisneeded.Lemma11.12.3.RecallthatT=CentG(A)isamaximaltorusinGdenedoverF;letSbetheFun-splitcomponentofT,amaximalFun-splittorusinGdenedoverFandcontainingA.WehaveT=CentG(A)=CentG(S).ChooseamaximalFun-splittorusSGwhichisdenedoverFandwhichcontainsA,andsetT=CentG(S).Choose 02 Msuchthat 0isdenedoverFunandsatises 0(S)=Sandhence 0(T)=T.ThenthediagramW(G;A) \0// o W(G;A)=W(M;A)o W(G;S)=W(M;S)F 0// W(G;S)=W(M;S)F 48T.HainesSettingt= 0(n0)�1tandusingat=at(whichfollowsfromW(G;A)-invariance),wewritetheaboveasXt7!mat�1=2B( 0(n0)t)1=2P(n0m):TheindexsetisstableundertheW(M;A)-actiononT(F)=T(F)1.IfwelookatthesumovertheW(M
;A)-orbitofasingleelementt0,withstabilizergroupStab(t0),weget(11.12.4)1 jStab(t0)jat0Xy�1=2BM( 0(n0)yt0);whereyrangesoverW(M;A).Nown2NG(S)(L)FNG(A)(F)=NG(M)(F).Hence 0(n)mn= 0(n0)normalizesMaswellasT,andthusconjugationby 0(n0)takesBMtoanotherF-rationalBorelsubgroupofMcontainingT.Usingthisitisclearthat(11.12.4)isunchangedifthesuperscript 0(n0)isomitted,andthisprovesourclaim.Forthesamereason(11.12.3)isindependentofthechoiceofPandB,andsimilarly~tA;AisindependentofthechoiceofPandB,andofthechoiceofFun-rational 02 M.NowwegiveanormalizedversionofDenition11.12.1.Denition11.12.5.Wedenethenormalizedtransferhomomorphism~t:Z(G(F);J)!Z(G(F);J)tobetheuniquehomomorphismmakingthefollowingdiagramcommuteZ(G(F);J)~t// o Z(G(F);J)o C[X(cT)FIF]W(G;A)~tA;A// C[X(Z(cM))FIF]W(G;A);wheretheverticalarrowsaretheBernsteinisomorphisms.Aswasthecasefort,thehomomorphism~tisindependentofthechoiceofAandA,anditisacompletelycanonicalhomomorphism.Thefollowingshowsitiscompatiblewithconstanttermhomomorphisms.11.12.3.Normalizedtransferhomomorphismsandconstantterms.Weusethenotationofx11:11.WriteL=CentG(AL)forsometorusA
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