; TeX output 2003.08.31:1434 7 V F{%"V 3 cmbx10THEtCOMBINA\TORICSOFBERNSTEINFUNCTIONS Eo cmr9THOMASTJ.HAINES+6,̍Z&- cmcsc10Abstract. A-qconstruction-wofBernsteinassoAciatestoeac9hcoc9haracter Zofasplit5" cmmi9p-adicgroupanelemen9tinthecenteroftheIwahori-HeckeZalgebra,*-whic9h&werefertoasaBernsteinfunction.NyA%recentconjectureZofCKott9witzpredictsthatBernsteinfunctionsplayanimpAortantroleZinWthetheoryofbadreductionofacertainclassofShim9urav|rarietiesZ(parahoric t9ypAe).^ItisthereforeofinteresttocalculatetheBernsteinZfunctionsexplicitlyinasman9ycasesaspAossible,withaviewtowardsZtesting?Kott9witz'conjecture.InthispapAerweproveacharacterizationZof_theBernsteinfunctionassoAciatedtoa'j cmti9minusculecoc9haracter(theZcaseofin9terestforShimurav|rarieties).ThisisusedtowritedowntheZBernsteinfunctionsexplicitlyforsomemin9usculecoAcharactersofGl ; cmmi6n7;Zone5 examplecanbAeusedtov9erifyKottwitz'conjectureforaspAecialZclass-ofShim9urav|rarieties(the\Drinfeldcase").InadditionweproveZsome=generalfactsconcerningthesuppAortofBernsteinfunctions,ƷandZconcerningInanimpAortan9tsetcalledthe\-admissible"set.ThesefactsZarecompatiblewithaconjectureofKott9witz-RapAoportontheshapAeofZthez{spAecialberofaShim9urav|rarietywithparahorictypAebadreduction. 啍cFVK`y 3 cmr101.qD5(- 3 cmcsc10Introduction"andStEatement"ofMainResulEts BहLetF ,!", 3 cmsy10HadenotetheIw!ahori-HeckeF algebraofanalmostsimple,msplitcon- 6nectedLreductiv!egroup*b> 3 cmmi10Goverap-adiceldFV.Moreconcretelye, ifI G(F)6isanIw!ahorisubgroup,6thenH4istheconvolutionalgebraofcompactlysup-6pMortedm`I -bi-in!vdDariantfunctionsonG(FV),xwhereconvolutionisdenedusing6the^HaarmeasureonG(FV)whic!hgivesI:volume1.Letq:denotethesizeof 6the'MresidueeldofFV,Gandlet2 3 msbm10Z-K cmsy80=|Z[qd)|{Y cmr81+2 cmmi8=2 ;1qd 1=2A].`FixanF-splitmaximal6torusT1ofGandaBorelsubgroupBcon!tainingTV.=LetWdenotethe6WeeylpgroupofGanddenotethecoMc!haracterlatticebyXz(TV).;Bernstein6hasconstructedaZ09-algebraisomorphismbMet!weentheWeeylgroupin!vdDariant6elemen!tsfofthegroupalgebraofXz(TV)andthecenterofH: [FZz09[Xz(TV)]zWA1ZD msam8v5 n!!lZ ȁ(H):n6हThisHisac!hievedHbyconstructing,[KforeachdominantcoMcharacter 2Xz(TV),6anelemen!tzz >inthecenterofH(seex2forthedenition),andthenby6sho!wingtheseelementsformaZ09-basisforZ ȁ(H),Uasrunsoveralldominant6coMc!haracters.Weefcalltheelementszz Csthe4 ': 3 cmti10Bernsteinfunctions.BThemainaimofthispapMeristostudytheBernsteinfunctionsfroma6com!binatorialҗviewpMoint,meaningthatweseekanexplicitexpressionfor6zz !asaZ09-linearcom!binationofthestandardgeneratorsTzwН,"whereTzwU3is 1 *7 62 ZTHOMASXJ.HAINESV 6हthe{Dc!haracteristicfunctionoftheIwahoridoublecosetcorrespMondingtothe 6elemen!t!wmintheextendedaneWeeylgroup;jo.u 3 cmex10fĖW ofG(accordingtotheȍ6Iw!ahoridecompMosition,;jfĖ5W,indexesthesetI nG(FV)=I).Asecondgoalofthis 6articleNeisthestudyofthe-admissiblesubsetoftheextendedaneWeeyl6groupf(seex2fordenition).BThe8$motivdDationforthisw!orkcomesfromtwoconjecturesinthetheoryof6Shim!ura)vdDarietieswithparahorictypMebadreduction,whichwenowbrie y6explain.Fix5arationalprimep.A5Shim!uravdDarietySKqvis,YroughlyspMeak-6ing,kIattac!hedtoatripleofgroup-theoreticdata(G;1K,;fg),whereGis6aconnectedreductiv!eQ-group,K=Kzp]K ȁp jG(Qzp)(G(A