`GOOD--ELEMENTS-OF-FINITE-COXETER-GROUPS-AND-REPRE

‘GOOD’ELEMENTSOFFINITECOXETERGROUPSANDREPRESENTATIONSOFIWAHORI{HECKEALGEBRASMeinolfGeckandJeanMichel1IntroductionandmainresultsLetW=W(M)beaCoxetergroup,thatis,agroupwithapresentationhs2Ijs2=1;sts:::|{z}mstfactors=tst:::|{z}mstfactorsfors;t2I;s6=tiwhereIisanitesetandM=(mst)s;t2Iisasymmetricmatrixofpositiveintegerswithmss=1andmst1ifs6=t.WeshallassumethroughoutthatMischosensothatWisanitegroup.TheArtin-TitsbraidgroupB=B(M)associatedtoWisthegroupdenedbythepresentationhs2Ijsts:::|{z}mstfactors=tst:::|{z}mstfactorsfors;t2I;s6=ti:Usually,wewillusethesamesymbolsforgeneratorsandelementsofWandofB,exceptedthatwewilluseboldlettersforelementsofB.ThisisjustiedbythefactthatthereisacanonicalsectiontothenaturalquotientmapB!Wgivenasfollows.Letw2Wandchooseaminimalexpressionw=s1:::sm.Thenitiswell-knownthatthecorrespondingproductw=s1:::sm2Bisindependentofthechosenminimalexpression.LetB+bethesubmonoidofBconsistingofallwordsinthegeneratorss2IofB(wordswithnoinverses);inthisway,wecanidentifyWwiththesubsetB+redofB+whichconsistsofelementsofthesamelengthastheirimageinW.NowletCbeaconjugacyclassinW,andletCminbethesetofelementsofminimallengthinC.Ithasbeenshownin[15]thattheelementsinCminhavere-markablepropertieswithrespecttoconjugationinWandintheassociatedgenericIwahori-HeckealgebraH.(ActuallythesepropertiesholdinBsincetheproofdoesnotusethequadraticrelationsinH.)Here,wewillshowthatthereexistelementsinCminwhich,whenconsideredaselementsofB,alsohaveremarkablepropertieswithrespecttotakingpowersofthem.Ourmainapplicationwillbeageneralformulafortheabsolutevaluesoftheeigenvaluesoftheirreduciblerepresenta-tionsofH;thiswillallowus,inparticular,todeterminethepreviouslyunknown1991MathematicsSubjectClassication20C20,20F3612Geck-MichelcharactertableoftheIwahori-HeckealgebraoftypeE8(theonlyIwahori-Heckeal-gebrawhosetablewasmissinguptonow,seetheremarksfollowingProposition1.3below).Letusnowexplaininmoredetailourmainresults.TheyaremotivatedbytheresultsofDeligne[10]andBrieskorn{Saito[4]aboutthesolutionofthewordproblemandanormalformforelementsinB+.FollowingBrieskorn-Saito,suchanormalformisgivenasfollows.Letw2B+,andletJ=J(w)bethesetofalls2IsuchthatsdivideswinB+fromtheleft.Thentheleastcommonmultipleofthesegeneratorsalsodividesw.NowthisleastcommonmultipleiswJ,the(imageinthebraidmonoidofthe)longestelementoftheparabolicsubgroupWJofWgeneratedbytheelementsJ.HencewecandividewbywJ,andproceedinthesamewaywiththequotient.Thus,wcanbewrittenintheformw=wJ1:::wJrforsuitablesubsetsJiIsuchthatJi=J(wJi:::wJr)fori=1;:::;r.Theorem1.1LetCbeaconjugacyclassinW,andletdbetheorderoftheelementsinC.Thenthereexistsanelementw2Cmin,astrictlydecreasingsequenceofsubsetsJ1J2:::JrofI,andpositiveintegersd1;d2;:::;drsuchthattherelationwd=wd1J1wd2J2:::wdrJrholdsinB+:Notethatthediarenecessarilyeven(fortherelationwd=1tohold).Suchanelementw2Cminwillbecalled‘good’.Moreover,ifdiseventhenagoodelementw2Cmincanbechosensothatwd=2isgivenbytheanalogousexpressionasabovewhereeachdiisreplacedbydi=2.Suchanelementwillthenbecalled‘verygood’.ThepointoftheTheoremisthepropertythatthesubsetsJiformadecreasingsequencefromwhichitfollowsthattheelementswdiJicommutewitheachother(seethedescriptionofthecentreofB+in[10],[4]).Theseresultscanbeseenasageneralizationofthewell-knownfactthat,ifCistheclasssuchthatCministhesetofCoxeterelementsandhistheCoxeternumber,thenwh=w2Iforw2Cmin,andifhiseventhenthereexistsanelementw2Cminsuchthatwh=2=wI.Theyalsogeneralizeresultsof[5]ontheclassofregularelementsoforderdwhenthelengthofelementsofCminis2N=d(whereNisthenumberofreectionsinW).TheseelementsarecharacterizedbythefactthattherelationinTheorem1.1holdswithr=1,andallelementsofCminaregood.(However,theyarenotnecessarilyallverygoodwhendiseven.)Itisreadilycheckedthat,inordertoproveTheorem1.1,itissucienttoconsideronlyirreducibleCoxetergroups.Theproofwillthenbegivencase{by{caseinxx2and3whereweuseareductionargument,asfollows.LetCbeaconjugacyclassinW.WesaythatCiscuspidalifC\WJ=;foreverypropersubsetJI.ItisthenclearthatTheorem1.1needonlybeprovedforcuspidalclassesofW.ForeachirreducibleCoxetergroupWwestartwithalistofconjugacyclassesforwhichitiseasytoverifythatitcontainsallcuspidalclassesofW.Wethenchecktheexistenceofgoodrespectivelyverygoodelementsintheclassesinthislist,whichisallweneedinordertoproveTheorem1.1.Inx2weconsiderCoxetergroupsofclassicaltype.FortypeAntheonlyclassinourlististheclassoftheCoxeterelementsandTheorem1.1iscoveredbytheabovementionedgeneralresultaboutCoxeterelements.FortypesBnandDnourlistconsistsofthoseclassesforwhichthe‘standard’representativesof[15]don’tGoodelementsandrepresentations3lieinproperparabolicsubgroups,andcheckthatthesestandardrepresentativesaregoodrespectivelyverygood.Fortheexceptionalandnon-crystallographictypes,weuseoncemoretheGAP[23]programsmentionedin[15](seealso[14])toproduceacompletelistofallclassesCsuchthatCmin\WJ=;forpropersubsetsJI.AnimplementationinGAPoftheDelignenormalformforelementsinB+writtenbythesecondnamedauthoristhenappliedtondallgoodrespectivelyverygoodelements.Theseresultsaregiveninx3inthefo