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cmcsc10R.SkipGaribaldi,AnneQug*eguiner-MaUTthieu, xJean-PierreTignol$(w|{Y cmr8ReceivÎed: FJebruaryX2,2001 ~Revised: AprilX25,2001[nCommÎunicatedXbyUlfRehmann*㲍 Abstract.gF*or
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cmmi10Aacentralsimplealgebraofdegree2n,Athenth exteriorXpGoweralgebra^ 0er cmmi7nq~Aisendowedwithaninvolutionwhichpro- videsaninterestinginvqariantofA.ZDInthecasewhereAisisomorphic tot̵QM
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B=forsomequaternionalgebraQ,|wedescribGethisinvolution quiteexplicitlyintermsofthenormformforQandthecorrespGonding involutionUUforB q.qٍ 1991UUMathematicsSub 8jectClassication:q16K20(11E8120G05)ō Keywords"andPhrases:XT*raceforms,,involutions,central"simplealge- brasōTheclassicationofirreduciblerepresentationsofasplitsemisimplesimplyconnected_1algebraicgroupGover_1anarbitraryeldFiswell-known:~they_1areinone-to-onecorrespGondencewiththeconeofdominantweightsofG.>IF*urther-more,onecantellwhetherornotanirreduciblerepresentationisorthogonalorsymplectick(=suppGortsaG-invqariantkbilinearformwhichisrespectivelysym-metricorskew-symmetric)byinspGectingthecorrespondingdominantweight[html:11 html:
,M4x3.11].ī(ThroughoutthispapGer,weonlyconsidereldsofcharacteristic6=V2,cf.{html:1.8 html:.)F9ASG-invqariantbilinearformonanirreduciblerepresentationisnecessarilyUUuniqueuptoascalarmultiple.IftheassumptionthatGissplitisdroppGed,thentheGaloisgroup ofaseparable+closureFsrofFWover+ȵFactsontheconeofdominantweights(viathe?so-called\-action"),D?andthisactionmaybGenontrivial.jThoseirreduciblerepresentations correspGondingtodominantweightswhicharenotxedby arenot^0denedover^0Fc.XAlthoughanirreduciblerepresentationwhosedominantweight;isxedby maynotbGeFc-dened,)Athereisalwayssomecentralsimple ZDocumentuUa#Mathematica6(2001)99{120 d *e̍6XIhtml: html:100%vR.S.Garibaldi,A.Qug*eguiner-MaUTthieu,J.-P.Tignol+4e̍6XIFc-algebraAandamapG<7!S LٓR cmr71|s(A)denedoverF*whichisanappropriate 6XIdescent\of,see[html:14 html:
]or[html:12 html:,p.230,Prop.1]fordetails.ThealgebraAis6XIuniquelyO_determineduptoFc-isomorphism._Ifisorthogonalorsymplectic6XIover
FsF:,thenitiseasytoshowthatAsuppGortsauniqueG-invqariantinvolution6XI
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msbm10Qp6XIandF*=Rin[html:4 html: ]and[html:5 html:], Qbutoveranarbitraryeldtheproblemismuchmore6XIdicult2 html: ].6XIW*erestrictourattentiontosimplyconnectedgroupsoftypGe^1xA2nO! cmsy7 1e;thatis,՜to6XItheVcaseG=S L1|s(A)VforAacentralsimpleFc-algebraofdegree2n.UMoreover,6XIwewillfoGcusonthefundamentalirreduciblerepresentationcorrespGondingto6XItheAmiddlevertexoftheDynkindiagramofG,ټwhichsuppGortsaG-invqariant6XIinvolutionUU
8.6XIF*oranynonnegativeintegerkT2n,QthereisacentralsimpleFc-algebra^k됵A6XIattachedHtoAcalledthekP ':
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WqhasnotbGeenknownforanyalgebraAofindex8.CIfA6XIisPatensorproGductofquaternionalgebras,weprovide(inhtml:1.6 html:bGelow)aformula6XIthatUUgives
㍲intermsofthenormformsofthequaternionalgebras.M6XIDescribingUthisparticularinvolutionU
LisalsointerestingfromthepGointofview6XIofwgroupsoftypGe^1~BD2n m.w-SuchagroupisisogenoustoGXP=Spin(E ;[ٲ)wforE@a6XIcentralIsimplealgebraofdegree4nand"anorthogonalinvolutionwithtriv-6XIialZdiscriminant. If93ishypGerbolic,[thenZEpisisomorphictoM2|s(A)forsome6XIalgebraŵAofdegree2n.5Theanalogueofthedirectsumofthetwohalf-spin6XIrepresentationsAforSpin(M2|s(A);[ٲ)overFisthemapG!S L1|s(C (M2(A);[ٲ))6XIwhereǵC (M2|s(A);[ٲ)denotestheevenCliordalgebraof(M2(A);[ٲ).NThisalge- ZDocumentuUa#Mathematica6(2001)99{120 e e̍6XIhtml: html:({-InvolutionsandTraceFUormsonExteriorPowers{*101+4e̍6XIbra(\isendowed(\withacanonicalinvolution(\(\ fe cbwhich(\isG-invqariant;7Zit(\ismostly 6XIhypGerbolic~butcontainsanontrivialpiecewhichisisomorphicto(^nq~A;
8).6XIPleaseUUsee[html:3 html: ]foraprecisestatementand[html:10 html:
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\ cmmi5k_B"(x2|s):(1.2)This|#formalsohasanaturaldescriptionfromtherepresentation-theoreticview- pGoint:8}ThegroupS L1|s(B q)actsonthevectorspace^k됵B q,чandwhenB9!issplit^k됵BϲisP^isomorphictoatensorproGductofanirreduciblerepresentationwithits?dual,zseeSectionhtml:2 html:.1Consequently*,thereisacanonicalS L1|s(B q)-invqariantquadraticUUformon^k됵B q;itistk.W*e0lett^+፴m
g˲andt^ ፴mdenotetherestrictionsoftmtothesubspacesSym(^mB q;
m)andSkew[l(^mB q;
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m, sothattm
=Pt^+፴m ^Yõt^ ፴m.TheformsthusdenedarerelatedUUbythefollowingequation,proveninhtml:5.5 html::html: html:Theorem1.3.G7XIntheWittringofFc,thefol lowinge}'qualityholds:a7Qh2i8m 1Xk+B=0( 1)k됵tk=\(.
S t^ ፴m(
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^ifmiso}'dd. ZDocumentuUa#Mathematica6(2001)99{120 f #e̍6XIhtml: html:102%vR.S.Garibaldi,A.Qug*eguiner-MaUTthieu,J.-P.Tignol+4e̍6XIThesimilarityclassofqA Qisdeterminedbythefollowingtheorem,Eproveninhtml:5.7 html::6XIhtml: html:2cTheorem1.4.G7XIf*Gniseven,Oߵnׂ=2m,the*GsimilarityclassofqA
c}'ontainsthe quadr}'aticform:荍4lt+፴m { 8t ፴m+nQ `1ĵt ፴m+XG0k+B