Some Macdonald-Mehta integrals by brute force

SOMEMACDONALD-MEHTAINTEGRALSBYBRUTEFORCEFrankG.Garvan*Abstract.BombieriandSelbergshowedhowMehta's[6;p.42]integralcouldbeevaluatedusingSelberg's[7]integral.Macdonald[5;xx5,6]conjecturedtwodierentgeneralizationsofMehta'sintegralformula.TherstgeneralizationisintermsofniteCoxetergroupsanddependsononeparameter.Thesecondgeneralizationisintermsofrootsystemsandthenumberofparametersinequaltothenumberofdierentrootlengths.InthecaseofWeylgroupsMacdonaldshowedhowtherstgeneralizationfollowsfromthesecond.WegiveaproofoftheI3caseoftherstgeneralizationandtheF4caseofthesecondgeneralization.AswellwegiveatwoparametergeneralizationforthedihedralgroupH2n2.Theparametersareconstantoneachofthetwoorbits.WenotethattheG2caseofthesecondgeneralizationfollowsfromourtwo-parameterversionforH62.OurproofsdrawonideasfromAomoto's[1]proofofSelberg'sintegralandZeilberger's[10]proofoftheG_2caseoftheMacdonaldMorris[5;Conj.3.3]constanttermrootsystemconjecture.Theproblemisreducedtosolvingasystemoflinearequations.TheseequationsweregeneratedandsolvedbythecomputeralgebrapackageMAPLE.1.Introduction.In1967Mehta[6;p.42]conjecturedthat(1.1)ZRnekxk2=2jD(x)j2kdx=(2)n=2nYj=1(jk+1)(j+1):Here,kisanycomplexnumberwithRe(k)0,dx=dx1:::dxnisLebesguemeasure,kxk2=x21+


+x2n,andD(x)=Qij(xixj).E.BombieriandSelbergshowedhow(1.1)followsfromSelberg'sintegralZ[0;1]nnYi=1xa1i(1xi)b1jD(x)j2cdx(1.2)=nYi=1(a+(ni)c)(b+(ni)c)(ic+1)(a+b+(2ni1)c)(c+1):See[5;p.1000].Macdonald[5;xx5,6]conjecturedtwodierentgeneralizationsof(1.1).TherstgeneralizationisintermsofCoxetergroups.LetGbeaniteCox-etergroup,i.e.anitegroupofisometriesofRngeneratedbyreectionsSinhyperplanesthroughtheorigin.Theequationsofthesehyperplanesareofthe*InstituteforMathematicsanditsApplications,UniversityofMinnesota,Minneapolis,Minnesota,55455.Currentaddress,SchoolofMathematics,MacquarieUniversity,Sydney,NewSouthWales2109,Australia.TypesetbyAMS-TEX12FRANKG.GARVAN*formhS(x)=Pni=1aixi=0.NormalizeeachhS(uptosign)byrequiringthatPa2i=2,andletP(x)=QShS(x)betheproductofthesenormalizedlinearforms,theproductbeingoverallreectionsSinG.LetdibethedegreesofthefundamentalpolynomialinvariantsofG.Macdonald[5;Conj.5.1]conjecturedMacdonald-MehtaConjectureI.IfkisanycomplexnumberwithRe(k)0,then(Mac-MehI)1(2)n=2ZRnekxk2=2jP(x)jkdx=nYj=1(k2dj+1)(k2+1):WhenGisthesymmetricgroupSn,actingonRnbypermutingthecoordinates(Mac-MehI)reducesto(1.1).A.RegevobservedthatwhenGisBnorDnthen(Mac-MehI)istrueforallk,againbySelberg'sintegral.Macdonaldshowedthat(Mac-MehI)istruefork=1andGaWeylgroup,andforarbitrarykwhenGisdihedral.AsnotedbyMacdonaldthedihedralcasecanbecomputedbytransformingtopolarcoordinates.Wenotethat(Mac-MehI)maybegeneralizedasfollows:forareectionS2GweletkSbeanycomplexnumberwithRe(kS)0suchthatkS1=kS2wheneverhS1;hS2belongtothesameorbitwhenGactsonthesetofhyperplanes.InthiscaseifjP(x)jkintheintegrandoftheleftsideof(Mac-MehI)isreplacedbyQSjhS(x)jkSthentheresultingintegralcanbeevaluatedasaniceproductofgammafunctions.IfGisaWeylgroupthenthisintegralreducestotheonegivenbelowin(Mac-MehII).Theonlyothernon-transitivenon-WeylirreducibleCoxetergroupsarethedihedralgroupsH2m2.Inthiscasetherearetwoorbits.Theintegralisgivenbelowin(1.4).Asintheequalparametercasetheevaluationfollowseasilybytransformingtopolarcoordinates.Thedetailsaregiveninx2.Theorem(1.3).Ifa;b2CwithRe(a);Re(b)0andm2Nthen22m(a+b)=2ZR2m1Yk=0jcoskmx1+sinkmx2jajcos(2k+1)2mx1+sin(2k+1)2mx2jb(1.4)
e(x21+x22)=2dx1dx2=(a+1)(b+1)(m(a+b)2+1)(a2+1)(b2+1)(a+b2+1):Macdonald'ssecondgeneralizationisintermsofrootsystems.LetSbea(notnecessarilyreduced)rootsystemconsistingoflinearformsonarealEuclideanspaceA.Wenormalizethelinearformsa2Ssothattheyhavenormp2.Letkbecomplex-numberswithrealpart0suchthatk=kifkk=kk,andletP(x)=Q2S+j(x)jkbetheproductofthesenormalizedlinearforms,weightedaccordingtothemultiplicityk,overthesetofpositiveroots.MacdonaldconjecturedMacdonald-MehtaConjectureII.(Mac-MehII)ZAekxk2=2P(x)d(x)=Y2S+(12k+14k=2+12(k;_)+1)(14k=2+12(k;_)+1);SOMEMACDONALD-MEHTAINTEGRALSBYBRUTEFORCE3where_isthecoroot2=kk2,k=2=0if12=2S,k=12P2S+k,andistheGaussianmeasureonA.Macdonaldhasshownthat(Mac-MehII)istrueinthefollowingthreecases:(a)Sisofclassicaltype(An;Bn;Cn;Dn;BCn)(bySelberg'sintegral).(b)SistherestrictedrootsystemofasymmetricspaceG=Kandthekarethemultiplicitiesmoftheroot.(c)S=G2andthekareallequal.Case(c)followsfromthefactthatwhenSisreducedandthekareallequal(Mac-MehII)reducesto(Mac-MehI)andfromthefactthattheWeylgroupofG2isH62.WenotethatthegeneralG2casefollowsfrom(1.4)withm=3.Inx3weintroducesomenotationandprovesomepreliminaryresults.Inx4wepresentacomputerapproachforhandling(Mac-MehI)foragivenCoxetergroup.Inx5wedescribesomemodicationsofthisapproachsoastohandle(Mac-MehII).WehavesucessfullyimplementedthisapproachonthecomputertoprovetheI3caseoftherstconjectureandtheF4caseofthesecondconjecture.ThedetailsoftheI3casearegiveninx6.SomedetailsoftheF4casearegiveninx7.OurcomputerprogramsarewritteninFORTRANorMAPLEandwererunonanAPOLLODN-5800attheI.M.A.,Univers