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Representations of Finite Groups of Lie Type

Permanent Link: http://ufdc.ufl.edu/UFE0022445/00001

Material Information

Title: Representations of Finite Groups of Lie Type
Physical Description: 1 online resource (123 p.)
Language: english
Creator: Nguyen, Hung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: finite, groups, lie, representations, type
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Let G be a finite quasi-simple group of Lie type. One of the important problems in modular representation theory is to determine when the restriction of an absolutely irreducible representation of G to its proper subgroups is still irreducible. The solution for this problem is a key step towards classifying all maximal subgroups of finite classical groups. We solve this problem for the cases when $G$ is a Lie group of the following types: $G_2(q)$, $^2B_2(q)$, $^2G_2(q)$, and $^3D_4(q)$. One of the main tools to approach the above problem, and many others, is the classification of low-dimensional representations of finite groups of Lie type. Low-dimensional complex representations were first studied in Tiep-Zalesskii for finite classical groups and then in Lubeck for exceptional groups. We extend the results in Tiep-Zalesskii for symplectic and orthogonal groups to a larger bound. More explicitly, we classify irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and orthogonal groups $Spin_n^\pm(q)$ of degrees up to the bound $D$, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups, $D=q^{4n-8}$ for orthogonal groups in odd dimension, and $D=q^{4n-10}$ for orthogonal groups in even dimension.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hung Nguyen.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Tiep, Pham H.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022445:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022445/00001

Material Information

Title: Representations of Finite Groups of Lie Type
Physical Description: 1 online resource (123 p.)
Language: english
Creator: Nguyen, Hung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: finite, groups, lie, representations, type
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Let G be a finite quasi-simple group of Lie type. One of the important problems in modular representation theory is to determine when the restriction of an absolutely irreducible representation of G to its proper subgroups is still irreducible. The solution for this problem is a key step towards classifying all maximal subgroups of finite classical groups. We solve this problem for the cases when $G$ is a Lie group of the following types: $G_2(q)$, $^2B_2(q)$, $^2G_2(q)$, and $^3D_4(q)$. One of the main tools to approach the above problem, and many others, is the classification of low-dimensional representations of finite groups of Lie type. Low-dimensional complex representations were first studied in Tiep-Zalesskii for finite classical groups and then in Lubeck for exceptional groups. We extend the results in Tiep-Zalesskii for symplectic and orthogonal groups to a larger bound. More explicitly, we classify irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and orthogonal groups $Spin_n^\pm(q)$ of degrees up to the bound $D$, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups, $D=q^{4n-8}$ for orthogonal groups in odd dimension, and $D=q^{4n-10}$ for orthogonal groups in even dimension.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Hung Nguyen.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Tiep, Pham H.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022445:00001


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1

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Iamindebtedtomyadvisor,Prof.PhamHuuTiep,whohasgivenmeconstantencouragementanddevotedguidanceoverthelastfouryears.IwouldliketothankProf.AlexanderDranishnikov,Prof.PeterSin,Prof.MeeraSitharamandProf.AlexandreTurull,forservingonmysupervisorycommitteeandgivingmevaluablesuggestions.IamalsopleasedtothanktheDepartmentofMathematicsattheUniversityofFloridaandtheCollegeofLiberalArtsandSciencesforhonoringmewiththe4-yearAlumniFellowshipandtheCLASDissertationFellowship.Finally,Ithankmyparents,mysister,andmywifeforalltheirloveandsupport. 3

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page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 6 LISTOFSYMBOLS .................................... 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 1.1OverviewandMotivation ............................ 9 1.2Results ...................................... 11 2IRREDUCIBLERESTRICTIONSFORG2(q) ................... 15 2.1Preliminaries .................................. 16 2.2DegreesofIrreducibleBrauerCharactersofG2(q) .............. 19 2.3Proofs ...................................... 21 2.4SmallGroupsG2(3)andG2(4) ......................... 28 3IRREDUCIBLERESTRICTIONSFORSUZUKIANDREEGROUPS ..... 56 3.1SuzukiGroups .................................. 56 3.2ReeGroups ................................... 57 4IRREDUCIBLERESTRICTIONSFOR3D4(q) .................. 60 4.1BasicReduction ................................. 60 4.2RestrictionstoG2(q)andto3D4(p ..................... 62 4.3RestrictionstoMaximalParabolicSubgroups ................ 64 5LOW-DIMENSIONALCHARACTERSOFTHESYMPLECTICGROUPS .. 67 5.1Preliminaries .................................. 67 5.1.1StrategyoftheProofs .......................... 67 5.1.2CentralizersofSemi-simpleElements ................. 68 5.2UnipotentCharacters .............................. 75 5.2.1UnipotentCharactersofSO2n+1(q)andPCSp2n(q) ......... 75 5.2.2UnipotentCharactersofP(CO2n(q)0) ................. 83 5.2.3UnipotentCharactersofP(CO+2n(q)0) ................. 88 5.3Non-unipotentCharacters ........................... 93 6LOW-DIMENSIONALCHARACTERSOFTHEORTHOGONALGROUPS 106 6.1OddCharacteristicOrthogonalGroupsinOddDimension ......... 106 4

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......... 109 6.3OddCharacteristicOrthogonalGroupsinEvenDimension ......... 111 6.4GroupsSpin12(3) ................................ 115 REFERENCES ....................................... 118 BIOGRAPHICALSKETCH ................................ 123 5

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Table page 2-1DegreesofirreduciblecomplexcharactersofG2(q) ................. 48 2-2Degreesof2-BrauercharactersofG2(q),qodd ................... 49 2-3Degreesof3-BrauercharactersofG2(q),3-q 50 2-4Degreesof`-BrauercharactersofG2(q),`5and`-q 51 2-5Degreesof`-BrauercharactersofG2(q),`5and`-q 52 2-6FusionofconjugacyclassesofPinG2(3) ...................... 53 2-7FusionofconjugacyclassesofPinG2(4) ...................... 54 2-8FusionofconjugacyclassesofQinG2(4) ...................... 55 5-1Low-dimensionalunipotentcharactersofSp2n(q),n6,qodd,I ........ 103 5-2Low-dimensionalirreduciblecharactersofSp2n(q),n6,qodd,II ....... 104 5-3Low-dimensionalirreduciblecharactersofSp2n(q),n6,qodd,III ....... 105 6

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LetGbeanitegroup,`beaprimenumber,andFbeanalgebraicallyclosedeldofcharacteristic`.Wewrite forthecenterofG O`(G): forthemaximalnormal`-subgroupofG forthesetofirreduciblecomplexcharactersofG,orthesetofirreducibleCG-representation IBr`(G): forthesetofirreducible`-BrauercharactersofG,orthesetofabsolutelyirreducibleG-representationincharacteristic` forthesmallestdegreeofirreducibleFG-modulesofdimension>1 forthelargestdegreeofirreducibleFG-modules forthesecondsmallestdegreeofirreducibleFG-modulesofdimension>1 ford0(G),m0(G),d2;0(G),respectively fortherestrictionofacharacter2Irr(G)to`-regularelementsofG forGLn(q) forGUn(q) seethediscussionbeforeLemma 5.1.4 forthequotientofCSp2n(q)byitscenter seethediscussionbeforeLemma 5.1.5 forthequotientofCO2n(q)0byitscenter fortheSpingroup.Modulosomeexceptions,itistheuniversalcoverofthesimpleorthogonalgroupPn(q) 7

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LetGbeanitequasi-simplegroupofLietype.OneoftheimportantproblemsinmodularrepresentationtheoryistodeterminewhentherestrictionofanabsolutelyirreduciblerepresentationofGtoitspropersubgroupsisstillirreducible.Thesolutionofthisproblemisakeysteptowardsclassifyingallmaximalsubgroupsofniteclassicalgroups.WesolvethisproblemforthecaseswhereGisaLiegroupofthefollowingtypes:G2(q),2B2(q),2G2(q),and3D4(q). Oneofthemaintoolstoapproachtheaboveproblem,andmanyothers,istheclassicationoflow-dimensionalrepresentationsofnitegroupsofLietype.Low-dimensionalcomplexrepresentationswererststudiedin[ 70 ]forniteclassicalgroupsandthenin[ 50 ]forexceptionalgroups.Weextendtheresultsin[ 70 ]forsymplecticandorthogonalgroupstoalargerbound.Moreexplicitly,weclassifyirreduciblecomplexcharactersofthesymplecticgroupsSp2n(q)andorthogonalgroupsSpinn(q)ofdegreesuptotheboundD,whereD=(qn1)q4n10=2forsymplecticgroups,D=q4n8fororthogonalgroupsinodddimension,andD=q4n10fororthogonalgroupsinevendimension. 8

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IfGisaprimitivepermutationgroupwithapointstabilizerMthenMisamaximalsubgroupofG.ThankstoworksofAschbacher,O'Nan,Scott[ 3 ],andofLiebeck,Praeger,Saxl,andSeitz[ 47 ],[ 48 ],mostproblemsinvolvingsuchaGcanbereducedtothecasewhereGisaniteclassicalgroup.IfGisaniteclassicalgroup,afundamentaltheoremofAschbacher[ 2 ]statesthatanymaximalsubgroupMofGisamemberofeitheroneofeightfamiliesCi;1i8,ofnaturallydenedsubgroupsofGorthecollectionSofcertainquasi-simplesubgroupsofG.Conversely,ifM2[8i=1Ci,thenthemaximalityofMhasbeendeterminedbyKleidmanandLiebeckin[ 40 ].ItremainstodeterminewhichM2SareindeedmaximalsubgroupsofG.Thisquestionleadstoanumberofimportantproblemsconcerningmodularrepresentationsofnitequasi-simplegroups.Oneofthemistheirreduciblerestrictionproblem. ThesolutionforProblemAwhenG=Z(G)isasporadicgroupislargelycomputational.WhenGisacoverofasymmetricoralternatinggroup,ProblemAisquitecomplicatedandalmostdonein[ 6 ],[ 41 ],[ 42 ],and[ 57 ].OurmainfocusisonthecasewhereGisa 9

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43 ]whenGisoftypeA. OneoftheimportantingredientsneededtosolveProblemAistheclassicationoflow-dimensionalirreduciblerepresentationsofnitegroupsofLietype.Thisisthereasonwhywehavebeenworkingsimultaneouslyontheirreduciblerestrictionproblemandtheproblemofdetermininglow-dimensionalrepresentations. SupposethatG=G(q)isanitegroupofLietypedenedoveraeldofqelements,whereq=pnisaprimepower.LowerboundsforthedegreesofnontrivialirreduciblerepresentationsofGincrosscharacteristic`(i.e.`6=p)werefoundbyLandazuri,SeitzandZalesskiiin[ 45 ],[ 59 ]andimprovedlaterbymanypeople.Theseboundshaveprovedtobeveryusefulinvariousapplications.Weareinterestedinnotonlythesmallestrepresentation,butmoreimportantly,thelow-dimensionalrepresentations. Low-dimensionalcomplexrepresentationswererststudiedbyTiepandZalesskii[ 70 ]forniteclassicalgroupsandthenbyLubeck[ 50 ]forexceptionalgroups.Forrepresentationsovereldsofcrosscharacteristics,thisproblemhasbeenstudiedrecentlyin[ 5 ],[ 23 ]forSLn(q);[ 25 ],[ 32 ]forSUn(q);[ 25 ]forSp2n(q)withqodd;and[ 24 ]forSpn(q)withqeven.LetHbeoneofthesegroups,andletthesmallestdegreeofnontrivialirreduciblerepresentationsofHincrosscharacteristicbedenotedbyd(H).ThemainpurposeofthesepapersistoclassifytheirreduciblerepresentationsofHofdegreesclosetod(H),andtoprovethatthereisarelatively\big"gapbetweenthedegreesoftheserepresentationsandthenextdegree. Acommonapplicationoflow-dimensionalrepresentationsisasfollows.SupposewewanttoprovesomestatementPinvolvingrepresentations'ofnitegroupsG.First,onetriestoreducetothecasewhenGisquasi-simple.Then,withGbeingquasi-simple,oneshowsthatPholdsif'hasdegreegreaterthanacertainboundd.Atthisstage,resultsonlow-dimensionalrepresentationsshouldbeappliedtoidentifytherepresentation'ofdegreedandtoestablishPdirectlyfortheserepresentations.Wereferto 10

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68 ]forasurveyofrecentprogressandmoredetailedapplicationsoflow-dimensionalrepresentationsofnitegroupsofLietype. Thesmallestdegreeofnon-trivialirreduciblecharactersofG2(q)wasdeterminedbyHiss.BasedonresultsofHissandShamashaboutcharactertables,decompositionnumbers,andBrauertreesofG2(q),wefoundthesecondsmallestdegreeofirreduciblecharactersofG2(q)incrosscharacteristic.ThisseconddegreeplaysacrucialroleinsolvingProblemAwhenG=G2(q).AnotheringredientisthemodularrepresentationtheoryofvariouslargesubgroupsofG2(q),suchasSL3(q)[ 64 ]andSU3(q)[ 20 ].Thefollowingtheoremisthefocusofchapter2.

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4 ]onunionsof`-blocks,aswellasresultsofGeck[ 21 ]onbasicsetsofBrauercharactersin`-blocks.RecentresultsofHimstedt[ 26 ]oncharactertablesofparabolicsubgroupsof3D4(q)havealsoproveduseful.Itturnsoutthattherearenoirreduciblerestrictionsfor3D4(q)andthiswillbeshowninchapter4. tations,orequivalently,low-dimensionalcomplexcharacters.LetustemporarilydenotebyGeitherthesymplecticgroupsSp2n(q)ortheorthogonalgroupsSpinn(q).ThesmallestnontrivialcomplexcharacterofGwasdeterminedbyTiepandZalesskiiin[ 70 ].Furthermore,whenGistheoddcharacteristicsymplecticgroup,theyclassiedallirreduciblecomplexcharactersofdegreeslessthan(q2n1)=2(q+1).Itturnsoutthat,uptothisbound,Ghasfourirreduciblecharactersofdegrees(qn1)=2,whicharetheso-calledWeilcharacters,andthesmallestunipotentcharacterofdegree(qn1)(qnq)=2(q+1). Wewanttoextendtheseresultstoalargerbound.Moreprecisely,weclassifytheirreduciblecomplexcharactersofGofdegreesuptotheboundD,whereD=(qn1)q4n10=2forsymplecticgroups,D=q4n8fororthogonalgroupsinodddimension,and 12

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24 ]andthereforeitisnotinourconsideration. Ourmainresultsonthistopicarethefollowingtheorems,whichwillbeprovedinchapters5and6.Wenotethatcouldbeunderstoodas1andviceverse,dependingonthecontext. 5-1 5-2 5-3 (attheendofchapter5). Furthermore,when(n;)=(5;),GhasexactlyqcharactersofdegreeD(5;q;),1characterofdegreeq2(q4+1)(q5+1)=(q+1),andnomorecharactersofdegreesuptoq10.

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Furthermore,when(n;)=(5;),GhasexactlyqcharactersofdegreeD(5;q;),1characterofdegreeq2(q4+1)(q5+1)=(q+1),andnomorecharactersofdegreesuptoq10.

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LetGbeanitegroupofLietypeG2(q)denedoveraniteeldwithq=pnelements,wherepisaprimenumberandnisapositiveinteger.LetMbeamaximalsubgroupofGandlet'beanabsolutelyirreduciblerepresentationofGincrosscharacteristic`.Thepurposeofthischapteristondallpossibilitiesof'andMsuchthat'jMisalsoirreducible. Inordertodothat,weusetheresultsaboutmaximalsubgroups,charactertables,blocks,andBrauertreesofGobtainedbymanyauthors.ThelistofmaximalsubgroupsofGisdeterminedbyCoopersteinin[ 13 ]forp=2andKleidmanin[ 39 ]forpodd.ThecomplexcharactertableofGisdeterminedbyChangandRee([ 10 ])forp5,byEnomoto([ 16 ])forp=3,andbyEnomotoandYamada([ 17 ])forp=2.Inaseriesofpapers[ 29 ],[ 33 ],[ 34 ],[ 61 ],[ 62 ],and[ 63 ],HissandShamashhavedeterminedtheblocks,Brauertreesand(almostcompletely)thedecompositionnumbersforG. Inordertosolveourproblem,itturnsouttobeusefultoknowthedegreesoflow-dimensionalirreducibleBrauercharactersofG.Thesmallest(ortherst)degreeofnon-trivialcharactersofGwasdeterminedbyHissandwillbeusedhere.Forthesecondsmallestdegree,wecomparedirectlythedegreesofirreducibleBrauercharactersandgettheexactformulaforitwhichisgiveninTheorem 2.2.1 .UsingtherstdegreeofG,wecanexcludemanymaximalsubgroupsofGbyReductionTheorem 2.3.4 .Theremainingmaximalsubgroupsaretreatedindividuallybyvarioustools,includingtheseconddegreeandmodularrepresentationtheoryoflargesubgroupsofGsuchasSL3(q)[ 64 ]andSU3(q)[ 20 ]. Thischapterisorganizedasfollows.Inx1,westatesomelemmaswhichwillbeusedlater.Inx2,wecollectthedegreesofirreducibleBrauercharactersandgiveformulasfortherstandseconddegreesofG2(q).x3isdevotedtoproveTheoremB.SmallgroupsG2(3)andG2(4)andtheircoverswillbetreatedinx4. 15

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Proof. Assumeg0isanyelementinZG(V).ThenX(g0)isascalarmatrixandthereforeitcommuteswithX(g)foreveryg2G.Henceg0commuteswithgforeveryg2GsinceKer(X)=1.Inotherwords,ZG(V)Z(G),whichimpliesthelemma. Proof. 2.1.2 16

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Proof. (ii)Assumethecontrary:VjHisirreducible.Thenitiswell-knownthatO`(H)actstriviallyonV.ItfollowsthatCV(O`(H))=V.ThisandthehypothesisO`(H)6=1leadtoacontradictionby(i). Proof.

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Proof. Proof. Nextwesupposethatforeachg2G,1(g)\Hconsistsofexactlyoneelement.ThenjHisanisomorphism.Inotherwords,H'GandM:G=M:Hbecomesasplitextension.WithoutlosswemayidentifyHwithG.NoweveryprojectivecomplexirreduciblerepresentationofGliftstoalinearrepresentationofM:G.ThenalsoliftstothelinearrepresentationjGofG.ThustheSchurmultiplierMofGistrivial,acontradictionagain.Wehaveshownthat(H)isapropersubgroupofG.Then(H)iscontainedinamaximalsubgroupofG,sayK.WehaveH1(K).SinceHismaximal,H=1(K)andwearedone. 18 ]LetGbeanitegroupandHG.SupposejG:Hj=pisprimeand2IBr`(G).Theneither

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35 ,p.190]LetGbeanitegroupandHbeanormalsubgroupofG.Let2Irr(G)and2Irr(H)beaconstituentofjH.Then(1)=(1)dividesjG=Hj. 56 ]LetBbean`-blockofgroupG.Assumethatall2B\Irr(G)areofthesamedegree.ThenB\IBr`(G)=fgandb=forevery2B\Irr(G). 69 ,Theorem1.6]LetGbeantegroupofLietype,ofsimplyconnectedtype.AssumethatGisnotoftypeA1,2A2,2B2,2G2,andB2.IfZisalong-rootsubgroupandVisanontrivialirreduciblerepresentationofG,thenZmusthavenonzeroxedpointsonV. Inthissection,wecollectthedegreesofirreduciblecharactersofG2(q)(bothcomplexandBrauercharacters)obtainedbymanypeople.Thenwewillrecallthevalueofd`(G)anddeterminethevalueofd2;`(G)when`iscoprimetoq. ThedegreesofirreduciblecomplexcharactersofG2(q)canbereadofrom[ 10 ],[ 16 ],[ 17 ]andarelistedinTable 2-1 .Fromthistable,weget and 6q(q1)2(q2q+1);p=2,3orq=5,7;q4+q2+1;p5,q>7;(2.2) foreveryq5.Moreover,Irr(G)containsauniquecharacterofdegreelargerthan1butlessthand2;C(G)andthischaracterhasdegreedC(G). Thedegreesofirreducible`-BrauercharactersofG2(q)when`jjGjand`-qcanbereadofrom[ 29 ],[ 33 ],[ 34 ],[ 61 ],[ 62 ],and[ 63 ].TheyarelistedinTables 2-2 2-3 2-4 ,and 2-5 .Comparingthesedegreesdirectly,weeasilygetthevalueofd`(G)when`jjGj,`-q.Combiningthisvaluewithformula 2.1 ,wegetthatifq5and`-qthen 19

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When`-jGj,weknowthatIBr`(G)=Irr(G)andthevalueofd2;C(G)isgiveninformula( 2.2 ).Formula( 2.2 )anddirectcomparisonofthedegreesofirreducible`-BrauercharactersofGwhen`jjGjyieldthefollowingtheorem.Weomitthedetailsofthisdirectcomputation. 6q(q1)2(q2q+1);p=3orq=5,7;q4+q2;p5,q11; 6q(q1)2(q2q+1);q=5,7,orp=2,q1(mod3);=q4+q2+1;p5,q1(mod3)andq11;q4q3+q2;q13andq1(mod3); 6q(q1)2(q2q+1);p=2,3orq=5,7;q4+q2+1;p5,q11: 20

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17 ],[ 20 ],[ 30 ],and[ 64 ]. 3)q31. 3)q3+1. Proof. 64 ,p.487],SL3(q)hasthefollowingconjugacyclasses:C(0)1,C(0)2,C(0;0)3,C(k)4,C(k)5,C(k;l;m)6,C(k)7andC(k)8.Foranyx2G,wedenotebyK(x)theconjugacyclasscontainingx.ThenwehaveC(0)1K(1),C(0)2K(u1),C(0;0)3K(u6),C(k)4K(k2)[K(h1b),C(k)5K(k2;1)[K(h1b;1),C(k;l;m)6K(h1),C(k)7K(hb)[K(h2b)andC(k)8K(h3).TakingthevaluesofX32and(q21 3)q31ontheseclasses,weget 3)q31(C(0)1)=X32(1)=q31,(q21 3)q31(C(0)2)=X32(u1)=1,(q21 3)q31(C(0;0)3)=X32(u6)=1,(q21 3)q31(C(k)4)=X32(k2)=X32(h1b)=q1,(q21 3)q31(C(k)5)=X32(k2;1)=X32(h1b;1)=1,(q21 3)q31(C(k;l;m)6)=X32(h1)=0,(q21 3)q31(C(k)7)=X32(hb)=X32(h2b)=(kq21 3+qkq21 3)=(!k+wk),(q21 3)q31(C(k)8)=X32(h3)=0; 3.WehaveshownthatX32jSL3(q)=(q21 3)q31ifq1(mod3)andqodd.Thecaseqisevenisprovedsimilarly. (ii)NowweshowthatX32jSU3(q)isirreduciblewhenq1(mod3).Moreprecisely,X32jSU3(q)=(q21 3)q3+1.Letusconsiderthecaseqisodd.WeneedtondthefusionofconjugacyclassesofSU3(q)inG.From[ 20 ,p.565],SU3(q)hasthefollowingconjugacyclasses:C(0)1,C(0)2,C(0;0)3,C(k)4,C(k)5,C(k;l;m)6,C(k)7andC(k)8.WehaveC(0)1K(1),C(0)2K(u1),C(0;0)3K(u4),C(k)4K(k2)[K(h2a),C(k)5K(k2;1)[K(h2a;1), 21

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3)q3+1ontheseclasses,weget 3)q3+1(C(0)1)=X32(1)=q3+1,(q21 3)q3+1(C(0)2)=X32(u1)=1,(q21 3)q3+1(C(0;0)3)=X32(u4)=1,(q21 3)q3+1(C(k)4)=X32(k2)=X32(h2a)=q+1,(q21 3)q3+1(C(k)5)=X32(k2;1)=X32(h2a;1)=1,(q21 3)q3+1(C(k;l;m)6)=X32(h2)=0,(q21 3)q3+1(C(k)7)=X32(ha)=X32(h1a)=(kq21 3+qkq21 3)=(!k+wk),(q21 3)q3+1(C(k)8)=X32(h6)=0; 3.Wehaveshownthat,whenq1(mod3)andqisodd,X32jSU3(q)2Irr(SU3(q))andX32jSU3(q)=(q21 3)q3+1.Thecaseqevenisprovedsimilarly.Weomitthedetails. 3)q312IBr`(SL3(q))for`-q. Proof. 3)q311Zq1.WehavedimV=q31.Accordingto[ 23 ,x4],Vhastheform(SC(s;(1))SC(t;(1)))"Gwheres2Fqandthasdegree2overFq.UsingCorollary2.7of[ 23 ],since1and2arecoprime,weseethatVisirreducibleinanycrosscharacteristic.ThisimpliesthatVjSL3(q)isalsoirreducibleinanycrosscharacteristicandtherefore\(q21 3)q312IBr`(SL3(q))forevery`-q. 20 ]. 3)q3+12IBr2(SU3(q))when`=2. Proof. 3)q3+1byforshort.Sinceq1(mod3),GU3(q)=SU3(q)Z(GU3(q))withZ(GU3(q))'Zq+1.Hence,SU3(q)hasthesamedegrees 22

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72 ]thatevery'2IBr2(GU3(q))eitherliftstocharacteristic0or'(1)=q(q2q+1)1.ThisandthecharactertableofSU3(q)givesthepossiblevaluesfordegreesofirreducible2-BrauercharactersofSU3(q):1,q2q,q2q+1,q(q2q+1)1,q(q2q+1),(q1)(q2q+1),q3,q3+1,and(q+1)2(q1). Assumebisreduciblewhen`=2.Thenbisthesumofmorethanoneirreducible2-BrauercharactersofSU3(q).Inotherwords,(1)=q3+1isthesumofmorethanonevalueslistedabove.Thisimpliesthatbmustincludeacharacterofdegreeeither1,orq2q,orq2q+1.Onceagain,by[ 72 ],thischaracterliftstoacomplexcharacterthatwedenoteby.Clearly,andbelongtothesame2-blockofSU3(q).Wewillusecentralcharacterstoshowthatthiscannothappen. LetRbethefullringofalgebraicintegersinCandamaximalidealofRcontaining2R.Itisknownthatandareinthesame2-blockifandonlyif whereKisanyclasssumand!isthecentralcharacterassociatedwith.Thevalueof!onaclasssumisgivenbelow:!(K)=(g)jKj 3.4 )impliesthat whereg2GandjgGjdenotesthelengthoftheconjugacyclasscontainingg. Considertherstcasewhen(1)=1.Itmeansthatisthetrivialcharacter.In( 3.5 ),takegtobeanyelementintheconjugacyclassC(1)7(notationin[ 20 ]),wehave(g) 1q3(q3+1)=q3q3(q3+1)2.Notethat\Z=2Z.Sinceq3q3(q3+1)isanoddnumber,wegetacontradiction. 23

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3.5 ),takegtobeanyelementintheconjugacyclassC(1)7,wehave(g) Finally,if(1)=q2q+1then=(u)q2q+1forsome1uq.In( 3.5 ),takegtobeanyelementintheconjugacyclassC(q+1)7.Notethatsinceq1(mod3),wehave(g) Proof. 2.1.1 ,wehaved`(G)p 2.1 )and( 2.3 ),wehaved`(G)q31if3-qandd`(G)q4+q2if3jqforevery`-q.Therefore, Here,wewillonlygivetheproofforthecasep5.Theproofsforp=2andp=3aresimilar.Accordingto[ 39 ],ifMisamaximalsubgroupofG=G2(q),q=pn,p5,thenMisG-conjugatetooneofthefollowinggroups: 1. 2. (SL2(q)SL2(q))2,involutioncentralizer, 3. 23L3(2),onlywhenp=q, 4. 5. 24

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7. 8. 9. 10. Considerforinstancethecase5)with3.Thenp 3.6 ).Thecases2),3),6)-10)areexcludedsimilarly. (i) maximalparabolicsubgroupsPa;Pb, (ii) (iii) 2.1.11 ,ZmusthavenonzeroxedpointsonV.Inotherwords,CV(Z)=fv2Vja(v)=vforeverya2Zg6=0.ThereforeVjPaisreduciblebyLemma 2.1.4 .Equivalently,'jPaisreducible. 1 ],[ 16 ],and[ 17 ],wehavemC(Pb)=q(q1)(q21)forq5.Supposethat'jPbisirreducible,then'(1)q(q1)(q21)byLemma 2.1.1 .If3jqthend`(G)q4+q2becauseofformula( 2.3 ).Thenwehaved`(G)q4+q2>q(q1)(q21)'(1)andthiscannothappen.Soqmustbecoprimeto3. ItiseasytocheckthatmC(Pb)
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6q(q1)2(q2q+1).RecallthatjPbj=q6(q21)(q1),whichisnotdivisiblebyX18(1)forq=5or7.ItfollowsthatX18jPbisreducibleandsois'18jPb.Wehaveshownthattheuniquepossibilityfor'isthenontrivialcharacterofsmallestdegreewhen3-q.RecallthatX32(1)=q3+,whichdoesnotdividejPbjandthereforeX32jPbisreducible. If`=3andq1(mod3)then=dX32c1G.AssumethatjPb=dX32jPbc1Pbisirreducible.ThereducibilityofX32jPbandtheirreducibilityofdX32jPbc1PbimpliesthatX32jPb=+whereb=c1Pb,2Irr(Pb)andb2IBr3(Pb).Wethenhave(1)=X32(1)1=q3,whichisacontradictionsincePbhasnoirreduciblecomplexcharacterofdegreeq3. Itremainstoconsiderthecasewhen`6=3orqisnotcongruentto1modulo3.ThenjPb=dX32jPb,whichisreducibleasnotedabove. 64 ],weknowthatmC(SL3(q))=(q+1)(q2+q+1)foreveryq5.Therefore,mC(SL3(q):2)2(q+1)(q2+q+1).Since'jSL3(q):2isirreducible,'(1)2(q+1)(q2+q+1).Similarargumentsasaboveshowthatqisnotdivisibleby3. ByTheorem 2.2.1 ,theinequality'(1)2(q+1)(q2+q+1)canholdonlyif'isthenontrivialcharacterofsmallestdegreeor'18whenq=5.NotethatX18(1)=280andjSL3(q):2j=744;000whenq=5andthereforeX18(1)-jSL3(q):2j.Hence,X18jSL3(q):2isreducibleandsois'18jSL3(q):2whenq=5.Again,theuniquepossibilityfor'iswhen3-q. If`=3andq1(mod3)then=dX32c1G.AssumethatjSL3(q):2isirreducible.LetVbeanirreducibleFG-module,char(F)=3,aordingthecharacter.ThenVjSL3(q):2isanirreducibleF(SL3(q):2)-module.LetbeageneratorforthemultiplicativegroupFqandIbetheidentitymatrixinSL(3;Fq).ConsiderthematrixT=q1 3I.Wehave=Z(SL3(q))andhenceESL3(q):2.Sinceord(T)=3,6O3(SL3(q):2)andthereforeO3(SL3(q):2)isnontrivial.ItfollowsthatVjSL3(q):2isreduciblebyLemma 2.1.4 26

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Whenq1(mod3),wehave=dX32.ByLemmas 2.3.1 and 2.3.2 ,wegetthatjSL3(q)=dX32jSL3(q)=\(q21 3)q312IBr`(SL3(q))for`-q,asweclaimintheitem(i)ofTheoremB. 64 ],wehavemC(SU3(q))=(q+1)2(q1)foreveryq5andthereforemC(SU3(q):2)2(q+1)2(q1)forq5.Hence'(1)2(q+1)2(q1)bytheirreducibilityof'jSU3(q):2.Again,qmustbecoprimeto3. ByTheorem 2.2.1 ,theinequality'(1)2(q+1)2(q1)canholdonlyif'isthenontrivialcharacterofsmallestdegreeor'18ofdegree1 6q(q1)2(q2q+1)whenq=5.Notethat'18=dX18.By[ 20 ],thedegreesofirreduciblecomplexcharactersofSU3(5)are:1,20,125,21,105,84,126,144,28and48.Whenq=5,X18(1)=280.Therefore,X18jSU3(5)isthesumofatleast3irreduciblecharacters.SinceSU3(5)isanormalsubgroupofindex2ofSU3(5):2,byCliord'stheorem,X18jSU3(5):2isreduciblewhenq=5.Thisimpliesthat'18jSU3(5):2isalsoreduciblewhenq=5.Insummary,theuniquepossibilityfor'iswhen3-q. Ifq1(mod3)then=dX32ofdegreeq31.RecallthatjSU3(q):2j=2q3(q3+1)(q21),whichisnotdivisiblebyq31foreveryq5.HenceX32jSU3(q):2isreducibleandsoisjSU3(q):2.Nowitremainstoconsiderq1(mod3). Firstweassumethat`=2.Then=dX32.ByLemmas 2.3.1 and 2.3.3 ,wehavejSU3(q)=dX32jSU3(q)=\(q21 3)q3+12IBr2(SU3(q)).ThereforejSU3(q):2isirreduciblewhen`=2.Next,if`=3then=dX32c1G.ByLemma 2.3.1 ,wehavejSU3(q)=dX32jSU3(q)b1SU3(q)=\(q21 3)q3+1b1SU3(q)=cq3,whichisanirreducible3-BrauercharacterofSU3(q)by[ 20 ,p.573].Finally,if`6=2;3then=dX32.ByLemma 2.3.1 ,wehavedX32jSU3(q)=\(q21 3)q3+1whichisirreducibleagainby[ 20 ].Therefore,jSU3(q):2isalsoirreducible,asweclaimintheitem(ii)ofTheoremB. 27

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2.2.1 ,theuniquepossibilityfor'is.Sinceq=q20,q1(mod3). If`=3then=dX32c1G.Hence,jG2(q0)=dX32jG2(q0)b1G2(q0).RecallthatX32(1)=q3+1andjG2(q0)j=q60(q601)(q201)=q3(q31)(q1).Itiseasytoseethat(q3+1)-q3(q31)(q1)foreveryq5.SoX32jG2(q0)isreducible.AssumethatdX32jG2(q0)b1G2(q0)isirreducible.ThenX32jG2(q0)=+whereb=b1G2(q0),2Irr(G2(q0))andb2IBr3(G2(q0)).Wethenhave(1)=X32(1)1=q3=q60.SoistheSteinbergcharacter.From[ 33 ],weknowthatthereductionmodulo3oftheSteinbergcharacterisreducible,whichcontradictsb2IBr3(G2(q0)). If`6=3then=dX32.Therefore,jG2(q0)=dX32jG2(q0).FromthereducibilityofX32jG2(q0)asnotedabove,jG2(q0)isalsoreducible.2 Inthissection,wemainlyuseresultsandnotationof[ 12 ]and[ 36 ].WenoticethattheuniversalcoverofG2(3),whichis3G2(3),hastwopairsofcomplexconjugateirreduciblecharactersofdegree27.Thepurposeofthissectionistoprovethefollowingtheorem.

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2.4.1 holdsinthecaseG=G2(3),M=U3(3):2. Proof. 12 ,p.14],wehavemC(U3(3))=32andmC(U3(3):2)=64.Thus,if'jMisirreduciblethen'(1)64.InspectingthecharactertablesofG2(3)in[ 12 ,p.60]and[ 36 ,p.140,142,143],weseethat'(1)=14or64. NotethatG2(3)hasauniqueirreduciblecomplexcharacterofdegree14whichisdenotedby2andeveryreductionmodulo`6=3of2isstillirreducible.Nowwewillshowthat2jU3(3)=6,whichistheuniqueirreduciblecharacterofdegree14ofU3(3).Supposethat2jU3(3)6=6,then2jU3(3)isreducibleanditisthesumofmorethanone 29

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By[ 12 ,p.14],U3(3):2hasonecomplexcharactersofdegree64whichwedenoteby.Also,G2(3)hastwoirreduciblecomplexcharactersofdegree64thatare3and4asdenotedin[ 12 ,p.60].Wewillshowthat3;4jU3(3):2=.Notethatthetwoconjugacyclassesof(U3(3):2)-subgroupsofG2(3)arefusedunderanouterautomorphismofG2(3),whichstabilizeseachof3and4.HencewithoutlosswemayassumethatU3(3):2istheoneconsideredin[ 16 ,p.237].Checkingdirectly,itiseasytoseethatthevaluesof3and4coincidewiththoseofateveryconjugacyclassesexcepttheclassofelementsoforder3atwhichweneedtocheckmore.U3(3):2hastwoclassesofelementsoforder3,3Aand3B.Using[ 16 ,p.237]tondthefusionofconjugacyclassesofU3(3)inG2(3)andthevaluesof3and4(whichare12(k)in[ 16 ]),weseethattheclasses3A,3BofU3(3)arecontainedintheclasses3A,3EofG2(3),respectively.Wealsohave3(3A)=4(3A)=(3A)=8and3(3E)=4(3E)=(3B)=2.Thus,3jU3(3):2=4jU3(3):2=.By[ 36 ,p.140,142,143],anyreductionmodulo`6=3of3aswellas4isirreducible.Also,thereductionmodulo`ofisirreducibleforevery`6=3,7.Therefore,c3jU3(3):2andc4jU3(3):2arethesameandirreducibleforevery`6=3,7.When`=7,m7(U3(3))=28andm7(U3(3):2)=56.Sincec3(1)=c4(1)=64,c3jU3(3):2andc4jU3(3):2arereduciblewhen`=7. 2.4.1 holdsinthecaseG=G2(3),M=23L3(2).

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When`-jG2(3)j(`6=2,3,7,and13),from[ 44 ],weknowthatc2j23L3(2)isirreducible.When`=2,wehavem2(23L3(2))=m2(L3(2))p jIrr(E)nf1Egj=26:3.Thus,thereductionmodulo7ofisitselfandthereforeitisirreducible.Hence\2jEL3(2)isalsoirreduciblewhen`=7. 2.4.1 holdsinthecaseG=3G2(3),M=3:Por3:QwhereP,QaremaximalparabolicsubgroupsofG2(3). Proof. 16 ,p.217].If'j3:Pisirreduciblethen'(1)mC(3:P)p 12 ]and[ 36 ](notethatweonlyconsiderfaithfulcharacters),wehave'(1)=27and'isactuallythereductionmodulo`6=3ofoneoffourirreduciblecomplexcharactersofdegree27of3G2(3).Fromnowon,wedenotethesecharactersby24, 12 ,p.60])and25, 12 ,p.60]). Nowwewillshowthat24j3:Pisirreducible.Notethatifg1andg2arethepre-imagesofanelementg2G2(3)underthenaturalprojection:3G2(3)!G2(3),then24(g1)=!24(g2)where!isacubicrootofunity.Thereforewehave[24j3:P;24j3:P]3:P=1 3jPjPx23:P24(x) 12 ,p.60].Thefusion 31

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16 ,p.217].BycomparingtheordersofcentralizersofconjugacyclassesofG2(3)in[ 16 ,p.239]withthosein[ 12 ,p.60],wecanndacorrespondencebetweenconjugacyclassesofG2(3)inthesetwopapers.ThelengthofeachconjugacyclassofPcanbecomputedfrom[ 16 ,p.217,218].AlltheaboveinformationiscollectedinTable 2-6 Fromthistable,weseethatthevalueof24iszeroatanyelement By[ 16 ,p.217],theclassD1ofPiscontainedintheclassD11ofG2(3).Thisclassistheclass4Aor4Baccordingtothenotationin[ 12 ,p.60].FirstweassumeD11is4A.Thenbylookingatthevaluesofonecharacterofdegree273ofG2(3)bothin[ 16 ]and[ 12 ],itiseasytoseethatD2D12=12AandE2(i)E2=8A.Therefore,Equation( 4.7 )becomes Next,weassumeD11is4B.ThenD12=12BandE2=8BandEquation( 4.7 )becomes Soinanycase,wehave[24j3:P;24j3:P]3:P=1 Notethatc24isirreducibleforevery`6=3by[ 36 ,p.140,142,143].Wewillshowthatc24j3:Pisalsoirreducibleforevery`6=3.ThestructureofPis[35]:2S4.WedenotebyO3themaximalnormal3-subgroup(oforder35)ofP.Thentheorderofanyelement 32

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[24j3:O3;24j3:O3]3:O3=1 35272=3. ByCliord'stheory,24j3:O3=ePti=1iwheree=[24j3:O3;1]3:O3and1,2,...,tarethedistinctconjugatesof1in3:P.Soe2:t=3andthereforee=1,t=3.Thus,24j3:O3=1+2+3.When`6=3,biisclearlyirreducible.ByLemma 2.1.5 ,c24j3:Pisirreduciblewhen`6=3. Usingsimilararguments,wealsohavec25j3:Pisirreducibleforevery`6=3,andthereforec 2.1.6 2.4.1 holdsinthecaseG=3G2(3),M=3:(U3(3):2). Proof. When`=2or7,U3(3):2doesnothaveanyirreduciblerepresentationincharacteristic`ofdegree27.Therefore,`6=2;3and7.Thatmeans`-jMjandIBr`(M)=Irr(M).Soweonlyneedtoconsiderthecomplexcase`=0. Itisobviousthat'j3(U3(3):2))isirreducibleifandonlyif'j(U3(3):2))isirreducible.Moreover,since'(1)=27,'j(U3(3):2))isirreducibleifandonlyif'jU3(3)isirreducible.Nowwewillshowthateither24jU3(3)or25jU3(3)isirreducibleandtheotherisreducible. From[ 12 ,p.14],weknowthatU3(3)hastheuniqueirreduciblecharacterofdegree27,whichisdenotedby10.Thischaracterisextendedtotwocharactersofdegree27ofU3(3):2.Itiseasytoseethattheclasses4A,4BofU3(3)iscontainedinthesameclassofG2(3),whichis4Aor4B.Fordeniteness,wesupposethatthisclassis4A.Thenthe 33

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NotethatG2(3)hastwonon-conjugatemaximalsubgroupswhichareisomorphictoU3(3):2.WedenotethesegroupsbyM1andM2.Supposethat25jM1isirreducibleand24jM1isreducible.LetbeanautomorphismofG2(3)suchthat(M1)=M2.Then(M2)=M1,24=25and25=24.ByLemma 2.1.6 ,25jM2isreducibleand24jM2isirreducible. 2.4.1 holdsinthecaseG=3G2(3),M=3:(L3(3):2). Proof. 12 ,p.13].Therefore,if'jMisirreduciblethen'(1)=27and'isrestrictionmodulo`6=3of24,25, When`=2or13,L3(3):2doesnothaveanyirreducible`-Brauercharacterofdegree27.Therefore`6=2,3,13andwehaveIBr`(M)=Irr(M).Thusweonlyneedtoconsiderthecomplexcase`=0. Since'(1)=27,itisobviousthat'j3(L3(3):2))isirreducibleifandonlyif'jL3(3)isirreducible.Nowwewillshowthateither24jL3(3)or25jL3(3)isirreducibleandtheotherisreducible. From[ 12 ,p.13],weknowthatL3(3)hasauniqueirreduciblecharacterofdegree27,whichisdenotedby11.Thischaracterisextendedtotwocharactersofdegree27ofL3(3):2.Theclass4AofL3(3)iscontainedinaclassofelementsoforder4ofG2(3),whichis4Aor4B.Fordeniteness,wesupposethatthisclassis4A.InL3(3)wehave(8A)24A,(8B)24AandinG2(3),(8A)24A,(8B)24B.Sotheclasses8A, 34

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NotethatG2(3)hastwonon-conjugatemaximalsubgroupswhichareisomorphictoL3(3):2.WedenotethesegroupsbyM1andM2.Supposethat24jM1isirreducibleand25jM1isreducible.LetbeanautomorphismofG2(3)suchthat(M1)=M2.Then(M2)=M1,24=25and25=24.ByLemma 2.1.6 ,wehave24jM2isreducibleand25jM2isirreducible. 2.4.1 holdsinthecaseG=3G2(3),M=3:(L2(8):3). Proof. 12 ,p.6],theSchurmultiplierofL2(8)istrivial.So3:(L2(8))=3L2(8).Therefore,mC(M)3mC(3L2(8))=27.Henceif'jMisirreduciblethen'(1)=27and'mustbetherestrictionmodulo`6=3ofoneofcharacters24,25, When`=2,7,Mdoesnothaveanyirreducible`-Brauercharacterofdegree27.So`6=2;3;7andthereforeitisenoughtoconsiderthecomplexcase`=0. Inspectingcharactervaluesatelementsoforder7,itiseasytoseethat24jL2(8)=25jL2(8)isthesumofthreeirreduciblecharactersofdegree9whicharefusedinM.Therefore,24jMaswellas25jMareirreducible. 2.4.1 holdsinthecaseG=G2(4),M=U3(4):2. Proof. Nowitremainstoconsiderthecasewhen'isareductionmodulo`6=2ofthecharacter3(asdenotedin[ 12 ,p.98])ofdegree78.SinceU3(4):2hasnocomplex 35

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2.4.1 holdsinthecaseG=G2(4),M=J2. Proof. 12 ,p.98].First,weshowthat4jJ2isactuallyirreducible.Moreprecisely,4jJ2=20,where20istheuniqueirreduciblecomplexcharacterofJ2ofdegree300. Itiseasytoseethatthevaluesof4and20arethesameatconjugacyclassesofelementsoforder5,7,8,10,and15.Theuniqueclass12Aofelementsoforder12inJ2isrealandthereforeitiscontainedinarealclassofG2(4).Henceitiscontainedinclass12AofG2(4)andwehave4(12A)=20(12A)=1.Weseethat(12A)3=4AinbothG2(4)andJ2.Thereforetheclass4AofJ2iscontainedintheclass4AofG2(4)andwealsohave4(4A)=20(4A)=4.Nowwemovetoclassesofelementsoforder2,3and6.SinceJ2isasubgroupofG2(4),either2J2(theuniversalcoverofJ2)or2J2isasubgroupof2G2(4)(theuniversalcoverofG2(4)).NotethatdC(2G2(4))=12anddC(J2)=dC(2J2)=14.So2J2cannotbeasubgroupof2G2(4)andtherefore2J2isasubgroupofG2(4).From[ 12 ],theclass2AofG2(4)liftstotwoinvolutionclassesof2G2(4)andtheclass2BofG2(4)liftstoaclassofelementsoforder4of2G2(4).Inthesameway,theclass2AofJ2liftstotwoinvolutionclassesof2J2andtheclass2BofJ2liftstoaclassofelementsoforder4of2J2.Theseimplythattheclasses2Aand2BofJ2arecontainedintheclasses2Aand2BofG2(4),respectively.Again,wehave4(2A)=20(2A)=20and4(2B)=20(2B)=0.Usingsimilararguments,wealsocanshowthattheclasses6Aand6BofJ2arecontainedintheclasses6Aand6BofG2(4)respectivelyandwealsohave4(6A)=20(6A)=1,4(6B)=20(6B)=0.InbothG2(4)andJ2,wehave(6A)2=3Aand(6B)2=3B.Thatmeanstheclasses3Aand3BofJ2arecontainedintheclasses3Aand3BofG2(4),respectively.Onemoretime, 36

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Notethat5= 2.4.1 holdsinthecaseG=2G2(4),M=2:P. Proof. Notethatifg1andg2arepre-imagesofanelementg2G2(4)underthenaturalprojection:2G2(4)!G2(4),then(g1)=(g2).Therefore[j2:P;j2:P]2:P=1 2jPjPx22:P(x) 12 ,p.98].In[ 17 ,p.357],wehavethefusionofconjugacyclassesofPinG2(4).BycomparingtheordersofcentralizersofconjugacyclassesofG2(4)in[ 17 ,p.364]withthosein[ 12 ,p.98]andlookingatthevaluesofirreduciblecharactersofdegrees65,78,wecanndacorrespondencebetweenconjugacyclassesofG2(4)inthesetwopapers.ThelengthofeachconjugacyclassofPcanbecomputedfrom[ 17 ,p.357].AlltheaboveinformationiscollectedinTable 2-7 .Fromthistable,weget Therefore,[j2:P;j2:P]2:P=1 37

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1024whichisnotaninteger.Thereforetherstcasemusthappenandwehave[j2:O2;j2:O2]2:O2=1 2.1.5 ,wehavebj2:Pisirreducibleforevery`6=2. 38

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2.4.1 holdsinthecaseG=2G2(4),M=2:Q. Proof. 2.4.10 ,weonlyneedtoshowthatbj2:Qisirreducibleforevery`6=2,whereistheuniqueirreduciblecharacterofdegree12of2G2(4).First,letusprovethatj2:Qisirreducible. Wehave[j2:Q;j2:Q]2:Q=1 2jQjPx22:Q(x) 12 ,p.98].In[ 17 ,p.361],wehavethefusionofconjugacyclassesofQinG2(4)andthelengthofeachconjugacyclassofQ.ThiswillbecollectedinTable 2-8 .Fromthistable,weget Therefore,[j2:Q;j2:Q]2:Q=1 Next,weshowthatbj2:Qisalsoirreducible.SetO2=24+6tobethemaximalnormal2-subgroupofQ.Sincej2:Qisirreducible,byCliord'stheorem,j2:O2=ePti=1i,wheree=[j2:O2;1]2:O2and1,2,...,tarethedistinctconjugatesof1in2:Q.Wehave12=(1)=et1(1).NotethatO2isa2-groupofexponent4andso1(1)2f1;2;4g.Thereforeet2f3;6;12g.Setm=e2t=[j2:O2;j2:O2]2:O2=1 2-8 ,weseethat( 1024(122+n(4)2)wherenisthesumofthelengthsofconjugacyclassesinO2atwhichthevaluesofis4.Hencen=(1024m144)=16.Wealsohaven15+360+240+240=855and5jn.Thisimpliesthatm13.Sinceet2f3;6;12g,itfollowsthate2t2f3;6;9;12g.Ifm=3,resp.9,12,thenn=183,resp. 39

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Wehaveshownthat[j2:O2;j2:O2]2:O2=6.Soe2:t=6andthereforee=1,t=6.Thusj2:O2=P6i=1i.ByLemma 2.1.5 ,bj2:Qisirreducibleforevery`6=2. 2.4.1 holdsinthecaseG=2G2(4),M=2:(U3(4):2). Proof. 12 ,p.30].Hence,if'jMisirreduciblethen'(1)150.Fromthecharactertableof2G2(4),wehave'(1)=12when`6=2,'(1)=104when`6=2;5or'(1)=92when`=5. 36 ,p.72],weseethatMdoesnothaveanyirreducible5-Brauercharacterofdegree92. 12 ,p.98].First,weshowthat34jMisirreducible. FromLemma 2.4.8 ,weknowthattherestrictionoftheuniqueirreduciblecharacter2ofdegree65ofG2(4)toU3(4)isirreducibleandequaltotheuniquerationalirreduciblecharacterofdegree65ofU3(4).Bylookingatthevaluesofthesecharacters,weseethattheinvolutionclass2AofU3(4)iscontainedintheclass2AofG2(4)andtheclass5EofU3(4)iscontainedintheclass5Aor5BofG2(4).Fordeniteness,wesupposethatthisclassis5A.Nowassumethat34jU3(4)containstheuniqueirreduciblecharacterofdegree64ofU3(4).Since34(2A)=8,bylookingatthevaluesofirreduciblecharactersofU3(4)attheinvolutionclass2A,weseethattheotherirreducibleconstituentsof34jU3(4)are 40

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Wehaveshownthat34jU3(4)doesnotcontaintheirreduciblecharacterofdegree64ofU3(4).InspectingthecharactertableofU3(4)in[ 12 ,p.30],weseethatU3(4)hasfourirreduciblecharactersofdegree52,whicharedenotedby9,10,11and12.Notethat10= 2(1+p Itiseasytoseethat35=34and11+12=(9+10)wheretheoperatoristhealgebraicconjugation:r+sp When`6=2;5,thereductionsmodulo`of34and35arestillirreducible.Similarargumentsshowthatc34jMaswellasc35jMareirreducible. 2.4.1 holdsinthecaseG=2G2(4),M=2:(SL3(4):2). Proof. 12 ,p.24,25],weseethatdC(3L3(4))=15. 41

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Inspectingthecharactertableof6L3(4)in[ 12 ,p.25],wehavemC(M)2mC(6L3(4))=120.Thereforeif'jMisirreduciblethen'(1)120.Hence'(1)=12when`6=2,'(1)=104when`6=2;5or'(1)=92when`=5.Alsofromcharactertablesof6L3(4),weseethatMdoesnothaveanyfaithfulirreduciblecharacterofdegree92or104.Sotheuniquepossibilityfor'isareductionmodulo`6=2oftheuniqueirreduciblecharacterofdegree12of2G2(4). Wehave16=O3(Z6)M.SoO3(M)6=1andthereforebyLemma 2.1.4 ,bjMisreduciblewhen`=3.Nowwesuppose`6=2;3.From[ 12 ,p.23],Mcanbeeither6:L3(4):21or6:L3(4):22or6:L3(4):23.Notethat6:L3(4):21haselementsoforder24and2G2(4)doesnothaveanyelementoforder24.Therefore,Mis6:L3(4):22or6:L3(4):23.Ineithercase,by[ 12 ,p.25]and[ 36 ,p.56,58],wehaved`(M)=12forevery`6=2;3.HencebjMisirreducibleforevery`6=2;3. 2.4.1 (i)Accordingto[ 12 ,p.61],ifMisamaximalsubgroupofG2(3)thenMisG2(3)-conjugatetooneofthefollowinggroups: 1. 2. 3. 4. 5. 23L3(2), 6. 7. [25]:32:2. ByLemmas 2:4:2 ,weneedtoconsiderthefollowingcases: 1)M=PorQ.ThestructureofPaswellasQis35:2S4.Sotheyaresolvable.Itiswell-knownthateveryBrauercharacterofthemisliftabletocomplexcharacters. 42

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3)M=L3(3):2.From[ 12 ,p.13],wehavemC(L3(3):2)=52.Soif'jMisirreduciblethen'(1)52andtherefore'(1)=14.So'isareductionmodulo`6=3oftheuniquenontrivialirreduciblecharacterofdegree14,whichisdenotedby2in[ 12 ,p.60].Since14-jL3(3):2j=11;232,2jL3(3):2isreducibleandsoisc2jL3(3):2. 4)M=L2(8):3.From[ 12 ,p.6],wehavemC(L2(8):3)=27.Again,if'jMisirreduciblethen'(1)=14.Inspecting[ 12 ,p.6]and[ 36 ,p.6],weseethatL2(8):3doesnothaveanyirreducibleBrauercharacterofdegree14.Therefore,'jL2(8):3isreducibleforevery'2IBr`(G2(3))with`6=3. 6)M=L2(13).From[ 12 ,p.8],wehavemC(L2(13))=14.Soif'jMisirreduciblethen'(1)=14.Thatmeanstheuniquepossibilityfor'isthereductionmodulo`6=3of2.Nowwewillshowthat2jL2(13)isreducibleandthereforec2jL2(13)isalsoreducibleforevery`6=3.Suppose2jL2(13)isirreducible.ThenitisoneofthetwocharactersofL2(13)ofdegree14.Thevaluesofthesecharactersontheuniqueclassofelementsoforder3ofL2(13)is1.Ontheotherhand,by[ 12 ,p.60,61],thisclassiscontainedintheclass3DofG2(3)and2(3D)=2,acontradiction. 7)M=[25]:32:2.WehavemC([25]:32:2)p (ii)Inthispart,weonlyconsiderfaithfulirreduciblecharactersof3G2(3).TheyarecharacterswhicharenotinatedfromirreduciblecharactersofG2(3).ByLemma 2.1.7 ,amaximalsubgroupof3G2(3)isthepre-imageofamaximalsubgroupofG2(3)underthenaturalprojection:3G2(3)!G2(3).Wedenoteby3:Xthepre-imageofXunder.ByLemmas 2:4:4 ,weneedtoconsiderthefollowingcases: 43

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6)M=3:L2(13).From[ 12 ,p.8],theSchurmultiplierofL2(13)hasorder2.So3:L2(13)=3L2(13).ThereforewehavemC(M)=mC(L2(13))=14.Ontheotherhand,thedegreeofanyfaithfulirreducibleBrauercharacterof3G2(3)isatleast27.Sowedonothaveanyexampleinthiscase. 7)M=3:(25:32:2).WehavemC(M)p (iii)Accordingto[ 12 ,p.97],ifMisamaximalsubgroupofG2(4)thenMisG2(4)-conjugatetooneofthefollowinggroups: 1. 2. 3. 4. 5. 6. 7. ByLemmas 2:4:8 ,weneedtoconsiderthefollowingcases. 1)M=PorQ.From[ 17 ],itiseasytoseethatmC(P)aswellasmC(Q)arelessthan256.Soif'jMisirreduciblethen'(1)<256.InspectingthecomplexandBrauercharactertablesofG2(4),wehavetwocases: 12 ,p.98].Notethatboth65and78donotdividejPj=jQj=184;320.Sobothc2jP;Qandc3jP;Qarereducible. 44

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17 ,p.358],weseethatthereisnoirreduciblecharacterofPofdegree64andwegetacontradiction.TheargumentsforQisexactlythesame. 3)M=SL3(4):2.ThiscaseistreatedsimilarlyasCase3whenq1(mod3)intheproofofTheoremB. 4)M=U3(3):2.WehavemC(U3(3):2)=64.Sotheuniquepossibilityfor'isthecharacterofsmallestdegree64when`=3.Butm3(U3(3):2)=30andthereforethereisnoexampleinthiscase. 5)M=A5A5.WehavemC(A5)=5andthereforemC(A5A5)=25.Ontheotherhandd`(G2(4))64forevery`6=2.So'jA5A5isreducibleforevery'2IBr`(G2(4))with`6=2. 6)M=L2(13).WehavemC(L2(13))=14andd`(G2(4))64forevery`6=2.So'jL2(13)isreducibleforevery'2IBr`(G2(4))with`6=2. (iv)Inthispart,weonlyconsiderfaithfulirreduciblecharactersof2G2(4).TheyarecharacterswhicharenotinatedfromirreduciblecharactersofG2(4).ByLemma 2.1.7 ,amaximalsubgroupof2G2(4)isthepre-imageofamaximalsubgroupofG2(4)underthenaturalprojection:2G2(4)!G2(4).Wedenoteby2:Xthepre-imageofXunder.ByLemmas 2:4:10 ,weneedtoconsiderthefollowingcases: 4)M=2:(U3(3):2).SincetheSchurmultiplierofU3(3)istrivial,2:U3(3)=2U3(3)and2:(U3(3):2)=(2U3(3):2.Soif'jMisirreduciblethen'(1)2mC(U3(3))=64.Therefore'isthereductionmodulo`6=2of,theuniqueirreduciblecomplexcharacterofdegree12of2G2(4).AssumejMisirreducible.UsingthecharactertableofU3(3)in[ 12 ,p.14],itiseasytoseethatjU3(3)=22,where2istheuniqueirreduciblecharacterofdegree6ofU3(3).Thereforej2U3(3)=2(2),whereisthenontrivialirreducible 45

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2.1.8 implythatjMisreducible,acontradiction. 5)M=2:(A5A5).WedenotebyAandBthepre-imagesoftherstandsecondtermsA5(inA5A5),respectively,undertheprojection.Foreverya2A,b2B,wehave([a;b])=[(a);(b)]=1.Therefore,[a;b]2Z2whereZ2=Z(2G2(4))Z(M).Thisimpliesthat[[A;B];A]=[[B;A];A]=1.By3-subgrouplemma,wehave[[A;A];B]=1.SincetheSchurmultiplierofA5is2,Ais2A5or2:A5,theuniversalcoverofA5.IfA=2:A5then[A;A]=A.IfA=2A5then[A;A]=A5.So,inanycase,[A;B]=1orinotherwordsAcentralizesB.ThatmeansM=(AB)=Z2. WehavemC(M)p SupposeisanyirreduciblecharacterofMofdegree12.ThenwecanregardasanirreduciblecharacterofABwithZ2Ker.Assumethat=ABwhereA2Irr(A)andB2Irr(B).Therearetwopossibilities: 12 ,p.2],weseethatA5doesnothaveanyirreduciblecharacterofdegree2or6.SoA=2:A5.Thevalueofanyirreduciblecharacterofdegree2of2:A5atanyconjugacyclassofelementsoforder5is1 2(1p 12 ,p.2],weseethatthevalueofanyirreduciblecharacterofdegree3ofA5or2:A5atanyconjugacyclassofelementsoforder5is1 2(1p Wehaveshownthatanyirreduciblecharacterofdegree12ofMisnotrational.Ontheotherhand,isrational.SojMisreducibleandthereforebjMisalsoreducibleforevery`6=2. 46

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12 ,p.8],wehavemC(2:L2(13))=14.Soif'jMisirreduciblethen'isthereductionmodulo`6=2of.Thevalueofattheuniqueconjugacyclassofelementsoforder7ofG2(4)is2.InspectingthecharactertablesofL2(13)anditsuniversalcover[ 12 ,p.8],weseethatMmustbetheuniversalcoverofL2(13)andjMisasumoftwoirreduciblecharactersofdegree6.ThatmeansjMisreducibleandthereforethereisnoexampleinthiscase. 7)M=2:J2.WealreadyknowfromtheproofofLemma 2.4.9 thatMistheuniversalcoverofJ2.Inspectingthecharactertableof2:J2in[ 12 ,p.43]and[ 36 ,p.103,104,105],weseethatthedegreesoffaithfulirreduciblecharactersof2G2(4)andthoseof2:J2aredierentfromeachother.Therefore'j2:J2isreducibleforeveryfaithfulirreduciblecharacter'of2G2(4).2

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DegreesofirreduciblecomplexcharactersofG2(q) CharacterDegree 3q(q4+q2+1)X141 3q(q4+q2+1)X151 2q(q+1)2(q2q+1)X161 6q(q+1)2(q2+q+1)X171 2q(q1)2(q2+q+1)X181 6q(q1)2(q2q+1)X19(k)1 3q(q1)2(q+1)2X31q3(q3+)X32q3+X33q(q+)(q3+)X21q2(q4+q2+1)X22q4+q2+1X23q(q4+q2+1)X24q(q4+q2+1)X1aq(q+1)(q4+q2+1)X01a(q+1)(q4+q2+1)X1bq(q+1)(q4+q2+1)X01b(q+1)(q4+q2+1)X2aq(q1)(q4+q2+1)X02a(q1)(q4+q2+1)X2bq(q1)(q4+q2+1)X02b(q1)(q4+q2+1)X1(q+1)2(q4+q2+1)X2(q1)2(q4+q2+1)Xaq61Xbq61X3(q21)2(q2q+1)X6(q21)2(q2+q+1) where: (i) (ii) (iii) Thistableiscollectedfrom[ 10 ],[ 16 ],and[ 17 ]. 48

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Degreesof2-BrauercharactersofG2(q),qodd Character4j(q1)4j(q+1) 6(q1)2(6q4+(83)q3+(103+)q2+(83)q+6)1 6(q1)2(6q4+(83)q3+(103+)q2+(83)q+6) 3(q1)(q4+q3+2q2+2q+3)1 3(q1)(q4+q3+2q2+2q+3)'141 3(q1)(q4+q3+2q2+2q+3)1 3(q1)(q4+q3+2q2+2q+3)'15q4+q2q4+q2'171 2q(q1)2(q2+q+1)1 2q(q1)2(q2+q+1)'181 6q(q1)2(q2q+1)1 6q(q1)2(q2q+1)'19(k)1 3q(q1)2(q+1)21 3q(q1)2(q+1)2 where: (i) (ii) 0q1ifp6=3and02qifp=3, (iii) 01 3(q+2), (iv) 11 3(q+1). Therefore,ifp6=3then'12(1)1 3(q1)2(q+1)(q3+2q2+q+3)andifp=3then'12(1)1 3(q1)2(q3+2q2+4q+3).Moreover,when4j(q+1)andq1(mod3),wehave'31(1)1 3(q1)2(q2+q+1)(2q2+2q+3). Thistableiscollectedfrom[ 34 ]. 49

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Degreesof3-BrauercharactersofG2(q),3-q 2q '1111 6(q1)2(6q4+(92)q3+(9+4)q2+(92)q+6)1 6(q1)2(6q4+(1123)q3+(13+43)q2+(1123)q+6) 6q(q2q+1)((1)q2+(4+2)q+(1))1 6(q21)(q3+3q2q+6) 2(q5+q4+q2+q2)1 2q(q+1)2(q2q+1)'16q3q31'171 2q(q1)2(q2+q+1)1 2q(q1)2(q2+q+1)'181 6q(q1)2(q2q+1)1 6q(q1)2(q2q+1)'191 3q(q1)2(q+1)21 3q(q1)2(q+1)2'21q2(q4+q2+1)(q1)2(q4+q2+1)'22q4+q2+1q4+q2+1'23q(q4+q2+1)(q1)(q4+q2+1)'24q(q4+q2+1)(q1)(q4+q2+1)'1aq(q+1)(q4+q2+1)(q21)(q4+q2+1)'01a(q+1)(q4+q2+1)(q+1)(q4+q2+1)'1bq(q+1)(q4+q2+1)(q21)(q4+q2+1)'01b(q+1)(q4+q2+1)(q+1)(q4+q2+1)'2aq(q1)(q4+q2+1)(q1)2(q4+q2+1)'02a(q1)(q4+q2+1)(q1)(q4+q2+1)'2bq(q1)(q4+q2+1)(q1)2(q4+q2+1)'02b(q1)(q4+q2+1)(q1)(q4+q2+1)'1(q+1)2(q4+q2+1)(q+1)2(q4+q2+1)'2(q1)2(q4+q2+1)(q1)2(q4+q2+1)'aq61q61'bq61q61'3(q21)2(q2q+1)(q21)2(q2q+1)'6(q21)2(q2+q+1)(q21)2(q2+q+1) where: (i) (ii) when3j(q1),'14(1)2f1 6q(q2q+1)(q2+4q+1),q2(q2q+1)g 2(q1)2(q4+2q3+3q+2), (iii) when3j(q+1),'121 4(q1)2(q+2)2(q2+q+1). Thistableiscollectedfrom[ 33 ]. 50

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Degreesof`-BrauercharactersofG2(q),`5and`-q 2q11 6(q+2),21 3(q+1) 6(q1)2(6q4+(83)q3+(103+)q2+(83)q+6) 3q(q4+q2+1)1 3(q1)(q4+q3+2q2+2q+3)'141 3q(q4+q2+1)1 3(q1)(q4+q3+2q2+2q+3)'151 2q(q+1)2(q2q+1)1 2q(q+1)2(q2q+1)'161 6q(q+1)2(q2+q+1)1 6q(q+1)2(q2+q+1)'171 2q(q1)2(q2+q+1)1 2q(q1)2(q2+q+1)'181 6q(q1)2(q2q+1)1 6q(q1)2(q2q+1)'19(k)1 3q(q1)2(q+1)21 3q(q1)2(q+1)2 where: (i) (ii) (iii) when`j(q+1),'12(1)1 18(q1)2(13q44q3+26q2+46q+18), (iv) when`j(q+1)andq1(mod3),'31(1)1 3(q31)(2q3+q3). Thistableiscollectedfrom[ 29 ]and[ 62 ]. 51

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Degreesof`-BrauercharactersofG2(q),`5and`-q 6(6q67q53q4+8q33q27q+6)q61 6q(q+1)2(q2+q+1)+1 3q(q4+q2+1)1 3q(q4+q2+1)'141 3q(q4+q2+1)1 3q(q4+q2+1)'151 2(q5+q4+q2+q2)1 2q(q+1)2(q2q+1)'161 6q(q+1)2(q2+q+1)1 6q(q+1)2(q2+q+1)1'171 2q(q1)2(q2+q+1)1 2q(q1)2(q2+q+1)'181 6q(q1)2(q2q+1)1 6q(q1)2(q2q+1)'19(k)1 3q(q1)2(q+1)21 3q(q1)2(q+1)2 '33(q2+q1)(q3+1)ifq1(mod3)q(q1)(q31)ifq1(mod3)q(q+)(q3+) where: (i) (ii) Thistableiscollectedfrom[ 61 ]. 52

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FusionofconjugacyclassesofPinG2(3) Fusionin[ 16 ]Correspondingclassin[ 12 ]LengthValueof24 53

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FusionofconjugacyclassesofPinG2(4) Fusionin[ 17 ]Correspondingclassin[ 12 ]LengthValueof A0A01A112A1A12A34A2A12A604A3A22B2400A41A314A1204A42A324C3600A5A44B2400A61A22B9600A62A314A14404A63A324C14400A71A518A57600A72A528B57602B0B03A3206B1B16A9602[B2]B16A38402[B3]B16A38402B2(0)B2(0)12A38402B2(i)B2(i)(i=1;2)12B;12C38400C31(i)C21(twoclasses)3B51200C32(i)C22(twoclasses)6B153600C41(i)C21(twoclasses)3B10240C42(i)C22(twoclasses)6B153600D11(i)D11(i)(twoclasses)5C;5D30723D12(i)D12(i)(twoclasses)10A;10B92161E(i)E1(i)(fourclasses)15C;15D122880 54

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FusionofconjugacyclassesofQinG2(4) Fusionin[ 17 ]Correspondingclassin[ 12 ]LengthValueof A0A01A112A1A12A154A2A22B480A31A22B2400A32A314A3604A33A324C3600A41A12A2404A42(0)A314A2404A42(i)A4(i=1;2)4B2400A5(t)A2;A32;A4(fourclasses)2B;4C;4B7200A61A518A57600A62A528B57602B0(i)B0(i=1;2)3A646B1(i)B1(i=1;2)6A9602B2(i)B1(i=1;2)6A38402B3(i;0)B2(0)(i=1;2)12A38402B3(i;j)B2(j)(i=1;2;j=1;2)12B;12C38400C21C213B51200C22C226B153600C31(i)C21(twoclasses)3B51200C32(i)C22(twoclasses)6B153600D11(i)D21(i)(twoclasses)5A;5B30722D12(i)D22(i)(twoclasses)10C;10D92160E(i)E2(i)(fourclasses)15A;15B122881 55

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ProofofTheoremC.Accordingto[ 67 ],ifMisamaximalsubgroupofGthenMisG-conjugatetooneofthefollowinggroups: 1. 2. 3. 4. 5. ByLemma 2.1.1 andtheirreducibilityof'jM,wehavep 45 ].ItfollowsthatjMjq(q1)2=2andthereforeMcanonlybethemaximalparabolicsubgroupofG. FromthecomplexcharactertableofPgivenin[ 52 ,p.157],wehavemC(P)=(q1)p 7 ],weobtainthat'=c1orc2,where1and2aretwoirreduciblecomplexcharactersofSz(q)ofdegree(q1)p Firstweconsiderthecase`=0.Comparingdirectlythevaluesofcharactersonconjugacyclasses,weseethat1jP=2and2jP=3,where2and3aretwocomplexirreduciblecharactersofPofdegree(q1)p 52 ]. Next,wewillshowthatbi,i=2,3,arealsoirreduciblewhen`6=0;2.Assumethecontrarythatb2isreducible.Thenitisthesumofmorethanone`-BrauerirreduciblecharactersofP.Thesecharactershavedegreeslessthan(q1)p 52 ,p.157].Since`isoddandord(f)=4,fisan`-regularelement.InspectingthecharactertableofP, 56

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Inspectingthecharactertableof2:Sz(8),weseethatif'isafaithfulirreduciblecharacterof2:Sz(8),then'(1)8.Thereforeif'jMisirreduciblethenp 36 ,p.64].NotethatP=[26]:7and2:P=2:([26]:7)=[27]:7.Alsofrom[ 36 ,p.64],thevalueof'11atanynontrivial2-elementis0.Itfollowsthat['11j[27];'11j[27]][27]=82=26=1andhence'11j[27]isirreducible.So'11j2:Pisalsoirreducible. 36 ,p.65].Since16>p 2.1.8 SowhenG=2:Sz(8),supposethat'isfaithful,then'jMisirreducibleifandonlyifM=2:Pand'istheuniqueirreducible5-Brauercharacterofdegree8. ProofofTheoremD.Accordingto[ 37 ,p.181],ifMisamaximalsubgroupofG,MisG-conjugatetooneofthefollowinggroups: 1. 2. 2L2(q),involutioncentralizer, 57

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(22D(q+1)=2):3, 4. 5. 6. ByLemma 2.1.1 andtheirreducibilityof'jM,wehavep 45 ].Therefore,jMjq2(q1)2.ThisinequalityhappensifandonlyifMisthemaximalparabolicsubgroupP.Furthermore,fromthecomplexcharactertableofPisgivenin[ 46 ,p.88],wehavem`(P)mC(P)=q(q1). Assume`5,usingtheresultsaboutBrauertreesofGin[ 31 ],itiseasytocheckthatd`(G)=q2q+1.Therefore,thereisnoexampleinthiscasesinced`(G)>m`(P).Itremainstoconsiderthecase`=2.Nowwehaveq(q1)m2(P)'(1)d2(G)q(q1).So'(1)=q(q1).Wewillcheckall2-blocksofGwhicharestudiedin[ 46 ]and[ 74 ].Wealsousenotationinthesepapers. 1)9,10,iareof2-defect0.Theirdegreesarealllargerthanq(q1). 2)Thereisone2-blockofdefect1.Allcharactersinthisblockarerand0rofdegreeq3+1.ByLemma 2.1.10 ,thereisaunique2-Brauerirreduciblecharacterofdegreeq3+1inthisblock. 3)Thereareseveral2-blocksofdefect2.Everycharacterintheseblockshasdegree(q1)(q2q+1).ApplyingLemma 2.1.10 again,all2-Brauerirreduciblecharactersintheseblockshavedegree(q1)(q2q+1). 4)Theprincipalblockanditsdecompositionmatrixaredescribedin[ 46 ].Thedegreesofirreducible2-Brauercharactersinthisblockare:'1(1)=1,'2(1)=q(q1),'3(1)=(q1)(q2(q+1)(p Wehaveshownthattheuniquepossibilityfor'is'='2,thenontrivialconstituent(ofdegreeq2q)ofthereductionmodulo2oftheuniqueirreduciblecomplexcharacterofdegreeq2q+1intheprincipleblock. 58

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46 ,p.88])ispositive.Ontheotherhand,'2(X)=qwhichisnegative.Wegetacontradiction.2

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ThemainpurposeofthischapteristoproveTheoremE.Inotherwords,weshowthattherestrictionofanyirreduciblerepresentationof3D4(q)toanyitspropersubgroupisreducible. Theorem4.1.1(ReductionTheorem). Proof. 51 ,Theorem4.1],d`(3D4(q))q5q3+q1forevery`coprimetoq.Next,accordingto[ 38 ],ifMisamaximalsubgroupofG,butMisnotamaximalparabolicsubgroup,thenMisG-conjugatetooneofthefollowinggroups: 1. 2. 3. 4. 5. 6. ((Zq2+q+1)SL3(q)):f+:2,wheref+=(3;q2+q+1), 7. ((Zq2q+1)SU3(q)):f:2,wheref=(3;q2q+1), 8. (Zq2q+1)2:SL2(3), 9. Weneedtoconsiderthefollowingcases: 60

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2.1.1 2.1.3 2.1.9 ,mC(M)2:f+:mC((Zq2+q+1)SL3(q))2:f+:mC(SL3(q)).From[ 64 ],wehave Itiseasytocheckthat2:f+:mC(SL3(q))
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12 ]and[ 36 ],weseethatdeg(')=25for`=3ordeg(')=26for`6=3.Moreover,when`6=3,'isthereductionmodulo`oftheuniqueirreduciblecomplexrepresentationofdegree26.Since26-jMj=1296,Mdoesnothaveanyirreduciblecomplexrepresentationofdegree26,whencejMand'jMmustbereducible.When`=3,M=((Z3)SU3(2)):3:2'31+2:2S4.Som3(M)=m3(31+2:2S4)=m3(2S4)mC(2S4)p InthissectionwehandletwoofthemaximalsubgroupssingledoutinTheorem 4.1.1 Proof. 2.1.1 ,'(1)


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21 ]statesthatthe`-modulardecompositionmatrixofGhasalowerunitriangularshape.Inparticular,thisimpliesthat'isanintegrallinearcombinationof^,with2E(G;1),thesetofunipotentcharactersofG.Buteachsuchis-invariant,whence'is-invariant.Nowassumethatqiseven.Then`6=2,andsoitisagoodprimeforR,and`doesnotdividejZ(R)j,whereRisthesimple,simplyconnectedalgebraicgroupoftypeD4.Hence,bythemainresultof[ 22 ],f^j2E(G;1)gisabasicsetofBrauercharactersofE`(G;1).Itfollowsthat'isanintegrallinearcombinationof^,with2E(G;1),andsoitis-invariantasabove. Considerthesemidirectproduct~G=Ghi.ThenG~G,and~G=Giscyclic.Since'is~G-invariant,itextendsto~Gby[ 18 ,TheoremIII.2.14].ButC~G(M)36=1,hence'jMcannotbeirreduciblebyLemma 2.1.3 Proof. ItiswellknownthatVjZaordsallthenontriviallinearcharactersofthelong-rootsubgroupZ:=Z(P0)(whichiselementaryabelianoforderq2),andthecorrespondingeigenspacesVarepermutedregularlybythetorusZq21.LetU=q2+16denotetheunipotentradicalofPandconsideranysuch.ThenIBr`(U)containsauniquerepresentation(ofdegreeq8),onwhichZactsviathecharacter.Moreover,sinceP0=U'SL2(q6)hastrivialSchurmultiplierandisperfect,thisrepresentationofUextendstoauniquerepresentationofP0,whichwedenotebyE.ByCliordtheory,theP0-moduleVisisomorphictoEAforsomeA2IBr`(P0=U).SupposethatAcontainsanontrivialcompositionfactor,asaSL2(q6)-module.Thendim(A)(q61)=2.Itfollows 63

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dim(V)(q21)q8(q61)=2:(2.1) Ontheotherhand,theirreducibilityofVjHimpliesthatdim(V)


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Proof. 26 ]and[ 28 ]thatgisQ-conjugatetothelong-rootelementu=x3+2(1).Butthengisap-element,acontradiction.HenceOr(Q)=1. (ii)Letbeanirreducibleconstituentof'jU,andletIdenotetheinertiagroupofinQ.ByCliordtheory,'=IndQI()forsome2IBr`(I)whoserestrictiontoUcontains.Sincep6=`,wemayviewasanordinarycharacterofU.Byourassumption,6=1U.ThestructureofI=Uisdescribedin[ 26 ],[ 28 ].Inparticular,if2jq,thenI=Uisalwayssolvable.Ontheotherhand,ifqisodd,thenI=Uissolvable,exceptforoneorbit,thekernelofanycharacterinwhichhowevercontainsalong-rootelementx3+2(1)(inthenotationof[ 26 ]).Recallweareassumingthat'isfaithfulifqisodd.ItfollowsthatineithercaseI=Uissolvable,andsoIissolvable.BytheFong-SwanTheorem,liftstoacomplexcharacterofI.Hence'liftstothecomplexcharacter:=IndQI(). Nowassumethat'isfaithfulbutK:=Ker()isnon-trivial;inparticular,`6=0.IfKisnotan`-group,thenKcontainsanon-trivial`0-elementg.Since'(g)=(g)=(1)='(1),weseethat'isnotfaithful,acontradiction.HenceKisan`-group,andsoO`(Q)6=1,contradicting(i). Proof. 2.1.11 .SowemayassumethatM=Q,theothermaximalparabolicsubgroupofG.Alsoassumethecontrary:'jQisirreducible. 65

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26 ],[ 27 ],[ 28 ].Clearly,theyareconjugateinG,so'(u)='(v).ByLemma 4.3.1 ,'jQliftstoacomplexirreduciblecharacterofQwhichisalsofaithful.Sinceuandvare`0-elements,wehave'(u)=(u)and'(v)=(v).Itfollowsthat NotethatZ:=Z(Op(Q))=X3+2X3+3hasorderq2,andconsistsoftheq21Q-conjugatesofuand1.ThusQactstransitivelyonZnf1gandonIrr(Z)nf1Zg.SinceKer()=1,weconcludethat(u)=(1)=(q21). Firstconsiderthecaseqisodd.Thenu,resp.v,belongstotheQ-conjugacyclassc1;1,resp.c1;2,inthenotationof[ 26 ].Accordingto[ 26 ],thefaithfulcharactermustbeoneofj(k),16j20.Ifj=16or17,then(v)isexplicitlycomputedin[ 26 ],andoneseesthat( 3.3 )isviolated.Nowsupposethatj=18or19.Then(u)=q3(q31)=2.Ontheotherhand,accordingtoProposition2.1oftheAppendixof[ 55 ],(v)=mq(q31)withm(q21)=2.Itfollowsthat(v)>(u),violating( 3.3 ).Finally,supposethatj=20.Then(u)=q3(q31).Meanwhile,byProposition2.1oftheAppendixof[ 55 ],(v)=mq(q31)withm(q21).Itfollowsthat(v)>(u),againviolating( 3.3 ). Nextweconsiderthecaseqiseven.Thenu,resp.v,belongstotheQ-conjugacyclassc1;1,resp.c1;7,inthenotationof[ 28 ].Accordingto[ 28 ],thefaithfulcharactermustbeoneofj(k),14j16.Ifj=14or15,then(u)isexplicitlycomputedin[ 28 ],andoneseesthat( 3.3 )isviolated.Finally,supposethatj=16.Then(u)=q3(q31).Ontheotherhand,byProposition1.1oftheAppendixof[ 55 ],(v)=mq(q31)withm(q21).Itfollowsthat(v)>(u),againviolating( 3.3 ). 4.1.1 toMtogetfourpossibilities(i){(iv)forM.NoneofthemcannothoweveroccurbyTheorems 4.2.1 4.2.2 ,and 4.3.2 .2

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ThepurposeofthischapteristoclassifyirreduciblecomplexcharactersofthesymplecticgroupsSp2n(q),qodd,ofdegreesupto(qn1)q4n10=2. Thischapterisorganizedasfollows.Inx1,wedescribethestrategytodeterminethelow-dimensionalcomplexcharactersofniteclassicalgroupsandgivestructuresofcentralizersofsemi-simpleelementsinthesymplecticandorthogonalgroups.Unipotentcharactersofbothsymplecticandorthogonalgroupswillbedeterminedinx2.Wemustincludetheorthogonalgroupsinthischaptersincetheirunipotentcharacterswillbeusedtodeterminenon-unipotentcharactersofthesymplecticgroups.Inthelastsectionx3,low-dimensionalnon-unipotentcharactersofSp2n(q)willbehandled. 5.1.1StrategyoftheProofs LetGbeeitherthesymplecticgroupsSp2n(q)ortheorthogonalgroupsSpinn(q),whereqisapowerofaprimep.LetGbethealgebraicgroupandFtheFrobeniusendomorphismonGsuchthatG=GF.LetGbethedualgroupofGandFthedualFrobeniusendomorphismanddenoteG=GF.Lusztig'sclassication(seechapter13of[ 15 ])saysthatthesetofirreduciblecomplexcharactersofGispartitionedintoLusztigseriesE(G;(s))associatedtovariousgeometricconjugacyclasses(s)ofsemi-simpleelementsofG.Infact,E(G;(s))isthesetofirreducibleconstituentsofaDeligne-LusztigcharacterRGT(),where(T;)isofthegeometricconjugacyclassassociatedto(s).TheelementsofE(G;(1))arecalledunipotentcharactersofG.WhenGisaconnectedreductivegroup,foranysemi-simpleelements2G,thereisabijection7!fromE(G;(s))toE(CG(s);(1))suchthat 67

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SupposethatwearedeterminingirreduciblecomplexcharactersofGofdegreesuptoacertainboundD.UnipotentcharactersofGaswellasGareclassiedbyLusztig(seex13.8of[ 9 ])andwewillusethattondthelow-dimensionalunipotentcharactersofG.Whenacharacterisnotunipotent,i.e.(s)6=(1),basedonthestructureofCG(s),weestimate(G:CG(s))p0andcomeupwithsomecertaincaseswhenCG(s)islargeenough.Morespecically,fromformula 1.1 ,weseethatif(1)jGjp0=D.ThefollowingPropositionisusedfrequentlytodetermineunipotentcharactersofCG(s). 15 ]). 8 ],[ 19 ],and[ 71 ]).Sinceproofsofthefollowinglemmasaresimilar,weomitmostoftheirproofsexceptthelastone's.

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70 ].Whenqisodd,itisalsoshownin[ 70 ]thatCO2n(q)0=fg2CO2n(q)jdet(g)=(g)ngandCO2n(q)0isactuallyasubgroupofindex2ofCO2n(q)suchthatCO2n(q)0=SO2n(q)'CO2n(q)=GO2n(q)'Zq1.

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Proof. Weconsidertwofollowingcases: 70

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(x2)m)x=s(V);Vi:=(P(x) 1a)First,wewillshowthatCGO(W)(s)'GUm0(qk)andCCO(W)(s)'CGO(W)(s)Zq1,whereW=Vifori=1;:::;l,m0=mi,andk=deg(fi)=2.NotethatthecharacteristicpolynomialofsactingonWisfm0i(x).Let2Fq2kbeaneigenvalueoftheactionofsonW.ThenalleigenvaluesoftheactionofsonWare;q;:::;q2k1.Since1isalsoarootoffi(x)and1arenot,1=qk.ConsiderfW=WFqFq2k.Fixabasis(fi)inW,anddeneaFrobeniusendomorphism:ixifi7!ixqifionfW,wherexi2Fq2k.ThesimplicityofsimpliesthatfW=fW1fW2k,wherefWi=Ker(sqi1).WeseethatpermutesthefWi'scyclically:(fWi)=fWi+1,wherefW2k+1=fW1.Letg2CO(W)becommutingwithsjW.ThengpreserveseachfWi.Moreover,itiseasytoseethatgalsocommuteswith.ThisimpliesthattheactionofgonfWiscompletelydeterminedbyitsactiononfW1:g(iw)=i(gw)forw2fW1.SoCCO(W)(s),!GLm0(q2k).Ifu2fWiandv2fWjthen(u;v)=(su;sv)=qi1+qj1(u;v).Therefore, Chooseabasis(u1;:::;um0)infW1.Then(vi)isabasisforfW1+k,wherevi=k(ui).Weseethat(ui;vj)=(ui;vj)q.Hence,(ui;vj)qk=(kui;kvj)=(vi;uj)=(uj;vi).Inotherwords,tU=Uqk.Thus,togetherwith 1.2 ,Udeterminesanon-degenerateHermitianformonanm0-dimensionalF2k-space. IfgactsonfW1withmatrixA=(aij)(withrespecttothebasis(ui)),thengactsonfW1+kwithmatrixAqk=(aqkij)(withrespecttothebasis(vi)).From 1.2 ,g2CGO(W)(s)if 71

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1b)Next,wewillshowthatCGO(W)(s)'GLn0(qk)andCCO(W)(s)'GLn0(qk)Zq1,whereW=Ujforj=1;:::;l0,n0=nj,andk=deg(gj).NotethatthecharacteristicpolynomialofsactingonWisgn0j(x)bgn0j(x).Let2Fqkbearootofgj.Thenalltherootsofgjare;q;:::;qk1andalltherootsofbgjare1;:::;qk1.Let(fi)beabasisofW,anddeneaFrobeniusendomorphism:ixifi7!ixqifionfW:=WFqFqk,wherexi2Fqk.ThesimplicityofsimpliesthatfW=fW1fWkfW01fW0k,wherefWi=Ker(sqi1)andfW0j=Ker(sqj1).WeseethatpermutesthefWi'sandfW0j'scyclically:(fWi)=fWi+1,(fW0j)=fW0j+1,wherefW2k+1=fW1andfW02k+1=fW01.Letg2CO(W)commutingwithsjW.ThengpreserveseachfWi,fW0j.Again,theactionofgonfWiscompletelydeterminedbyitsactiononfW1andfW01.SoCCO(W)(s),!GL2n0(qk).Wealsohave Choosebasis(ui)and(vi)offW1andfW01,respectively.SupposethatgactsonfW1withmatrixA=(aij)(withrespecttothebasis(ui))andonfW01withmatrix(bij)(withrespecttothebasis(vi)).From 1.3

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1c)Lastly,wewillshowthatCGO(V)(s)'GOm(q2)andCCO(V)(s)'CGO(V)(s)Zq1.Again,wehaveeV=eV1eV2,whereeV=VFqFq2,eV1=Ker(s1),andeV2=Ker(s+1).Furthermore,eV1?eV2andthereforeeV1,eV2arenon-degenerate.Similarargumentsasin1a),weseethattheactionofanelementg2CCO(V)(s)oneViscompletelydeterminedbyitsactiononeV1.Hence,CGO(V)(s)'GO(eV1)=GOm(q2)andCCO(V)(s)'CGO(V)(s)Zq1,sincemiseven. Combiningwhatwehaveprovedin1a);1b);1c),wehavethatCGO(V)(s)'CGO(V)(s)YiCGO(Vi)(s)YjCGO(Uj)(s)''GOm(q2)YiGUmi(qdeg(fi)=2)YjGLnj(qdeg(gj)): 73

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Since1=n=det(s)=(1)2mk0,k0iseven.Setk0=2kandV0:=Ker(s1);V00:=Ker(s+1);Vi:=(P(x) 5.1.5 iswrongwhens=2CO2n(q)0.Hereisoneexample.Supposethatqisodd.LetV=F2q=f(x;y)jx;y2Fqgbethevectorspaceofdimension2.ThequadraticformQ(x;y)=xyisnon-degenerateofWittindex1.TheGrammatrixofQcorrespondingtothebasisf(1;0);(0;1)gisJ=0110.Bydenition,GO+2(q)isthegroupofmatricesA2GL2(q)suchthattAJA=J.Therefore,GO+2(q)=a00a1;0aa10ja2Fq: 74

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5.2.1UnipotentCharactersofSO2n+1(q)andPCSp2n(q) Proposition5:1in[ 70 ]showsthatSO2n+1(q)aswellasPCSp2n(q)haveauniqueunipotentcharacterofminimaldegree(qn1)(qnq)=2(q+1)andanyothernon-trivialunipotentcharacterhasdegreegreaterthanq2n=2(q+1).WemimicitsproofandgetthefollowingProposition,whichclassiesunipotentcharactersofdegreesuptoq6n15=2. 5-1 ,wheredegreeofeachcharacter(labeledbyasymbol)iscalculated. 9 ,p.466,467],weknowthattheunipotentcharactersofGareparametrizedbysymbolsoftheform=123:::a12:::b 22=n: 2a+b1 2q(a+b22)+(a+b42)+QiQik=1(q2k1)QjQjk=1(q2k1):

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(q2n41)(qn31)(qn61) andq6D(n)asrequired.If11,then=(k;l;n+1kl)with1kD(n) foreveryn6.Itremainstoconsider(k;l)6=(1;2).Thenk+l4andl3.Itfollowsthat3k+l1and3lnkl.Therefore,2(k+l1)(nkl)6(n4). 76

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2q2klq6n24q2k+l>1 2q6n14>D(n)andwearedone.Nowwemayassumea5. Supposethat11.Considertheunipotentcharacter0labeledbythesymbol0ofrankn1,where0=(11;2;:::;a).Wehave(1) (q2q11)(qaq11)q2n1 22=nandhencen1=2++aa1 22(a1)1+a21 410.Therefore,(1)=0(1)q20=2>q7.Itfollowsthat(1)q7D(n1)>D(n).Nowwemayassumethat1=0. Nextweassumethatii12forsomei2.Considertheunipotentcharacter0labeledbythesymbol0ofrankn1,where0=(1;:::;i1;i1;i+1;:::;a).Similarly,wehave(1)=0(1)>q2(ni)=2.Byinductionhypotheses,0(1)D(n1).Ifni4then(1)q7D(n1)>D(n).Nowsupposeni3.NotethatPii=a1 22+n.Ifia1thena1n3andan2.Thena1 22+n0+1++(a3)+(n3)+(n2)ora28a9+4n0.Thisisacontradictionbecausea5andn7.Thereforei=aandwehave0+1++(a2)+a=a1 22+n.Sincea(n3),a24a90andthereforea=5.Then=(0;1;2;3;n2)anditiseasytocheckthat(1)>D(n).Nowwemayassume=(0;1;2;:::;a2;a1).Consider0=0;0,where0=(0;1;2;:::;a4;a3),and0=.Notethata2=4n+1andtherankof0isna+1.Wehave(1) 2q(a22)Qa2k=1(q2k1)Qa1k=1(q2k1)==(q2(na+2)1):::(q2n1) (q21):::(q2(a1)1)qa1qa2

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3)Fromnowonweassumethatb1.Atthispointwesupposethat(1;1)6=(1;0)and11.Considerthecharacter0labeledbythesymbol00ofrankn1,where0=(11;2;:::;a)and0=.If002Ln1then212n2;3n20;1n22;2n21.ItiseasytocheckthatthedegreesofcharacterscorrespondingtothesesymbolsaregreaterthanD(n).Nowwecanassume00isnotinLn1.Wehave(1) 2(q211)>q2(n1) Similarly,for(1;1)6=(0;1)and11,weconsiderthecharacter0labeledbythesymbol00ofrankn1,where0=and0=(11;2;:::;b).If002Ln1then22n21;0n23;1n22;012n12;0122n1.Again,itiseasytocheckthatthedegreesofcharacterscorrespondingtothesesymbolsaregreaterthanD(n).Nowwecanassume00isnotinLn1.Wehave(1) 2(q211)>q2(n1) 78

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Itremainstoconsiderthecasewhere(1;1)=(0;1)or(1;0). 4)Herewesupposethat(1;1)=(1;0)but6=(1;2;:::;a).Thenthereexistsani2suchthatii1+2.Wechooseitobesmallestpossible.Ifa=2thenb=1and=1n02Ln.Hencea3.Considerthecharacter0labeledbythesymbol00ofrankn1,where0=(1;:::;i1;i1;i+1;:::;a)and0=.If002Ln1thencanonlybe13n101.ThedegreeofthecorrespondingcharacterisgreaterthanD(n).Nowwecanassumethat00isnotinLn1.Byinductionhypothesis,0(1)Dn1.SetT1=Yi0q7andtherefore(1)D(n)asdesired.Nowweassumeni2.Therearetwocases: qi1q,T2=1andT3qa+1 (qna1)(qna+1+1)>q2a2:

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Similarly,supposethat(1;1)=(0;1)but6=(1;2;:::;b);inparticular,b2.Thenthereexistsanindexj2suchthatjj1+2.Wechoosejtobesmallestpossible.Considerthecharacter0labeledbythesymbol00ofrankn1,where0=and0=(1;:::;j1;j1;j+1;:::;b).If002Ln1thencanonlybe01n113.ThedegreeofthecorrespondingcharacterisgreaterthanD(n)andhencewearedoneinthiscase.Nowwecanassume00isnotinLn1.SetU1=Yj0q7andtherefore(1)Dnasdesired.Nowweassumenj2.Therearetwocases: qj1q,T3qj+1 qj1q,V2=1andV3qa+1 (qnb1+1)(qnb1)>q2b1:

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5)Herewesupposethat1=0and=(1;2;:::;a).Considerthecharacter0labeledbythesymbol00=012:::a23:::b Similarly,supposethat1=0and=(1;2;:::;b)but6=(0;1;:::;a1).Thenthereexistsani2suchthatii1+2.Wechooseitobesmallestpossible.Considerthecharacter0labeledbythesymbol00=1:::i1i1i+1:::a1:::b 81

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2q4q2n1 2q2q2n1 6)Finally,weconsiderthecasewhere=012:::a112:::b.Considerthecharacter0labeledbythesymbol00=012:::a212:::b1ofrankn1.Thenwehave(1) (qb1)(qa1+1)>q2n1

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5.2.1 .Thecasen=5canbeveriedeasilybyusingTable26of[ 11 ]. Proposition7:1in[ 70 ]showsthattheprojectiveconformalorthogonalgroupoftype-,P(CO2n(q)0),hasauniqueunipotentcharacterofminimaldegree(qn+1)(qn1q)=(q21)andanyothernon-trivialunipotentcharacterhasdegreegreaterthanq2n2.WemimicitsproofandgetthefollowingProposition,whichclassiesunipotentcharactersofdegreesuptoq4n10. Proof. 9 ,p.475,476],weknowthattheunipotentcharactersofGareparametrizedbysymbolsoftheform=123:::a12:::b; 22#=n: 2a+b2 2q(a+b22)+(a+b42)+QiQik=1(q2k1)QjQjk=1(q2k1): 83

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11 ,Table31].Therestoftheproofestablishestheinductionstepforn6. 1)Atthispointwesupposethat(1;1)6=(1;0)and11(eventuallybmaybezero).Considertheunipotentcharacter0labeledbythesymbol00ofrankn1,where0=(11;2;:::;a)and0=.If002Ln1then=2n2andtherefore(1)=(q2n21)(qn+1)(qn2q2)=(q21)(q41)>q4n10.Sowecansupposethat00isnotinLn1.Byinductionhypothesis,0(1)>q4(n1)10.Observethatn=Pii+Pjj(a+b)22(a+b) 4a1+(ab)2 2(q211)(qn+1)(qn11) 2(q2n81)>q4; 2)Similarly,supposing(1;1)6=(0;1)and11,considertheunipotentcharacter0labeledbythesymbol00ofrankn1,where0=and0=(11;2;:::;b).If002Ln1then=01n12andtherefore(1)=(qn+1)(qn11)(qn1q)(qn1+q2)=2(q21)2>q4n10.Sowecansupposethat00isnotinLn1.Byinductionhypothesis,0(1)>q4(n1)10.SetT1=bYj=2qjq1 2(q211)>q4: 84

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Nextwesupposethat=(0;1;:::;a1).Then(1)=(qn+1)f(a)andn=a2=4,where 2a2 2q(a22)+(a42)+Qa1i=0Qik=1(q2k1)(2.4) foranyevena.Itiseasytoshowthatf(a)=f(a2)>qa(a1)(a3)=2=26>q3a2=4=q3nfora6.Furthermore,f(4)>1.Itfollowsthat(1)>q4n10andwearedone. Itremainstoconsiderthecasewhereb1and(1;1)=(0;1)or(1;0). 4)Herewesupposethat(1;1)=(1;0)but6=(1;2;:::;a).Thenthereexistsi2suchthatii1+2.Chooseitobesmallestpossible.Considertheunipotentcharacter0labeledby00ofrankn1,with0=(1;:::;i1;i1;i+1;:::;a)and0=.Bytheinductionhypothesis,0(1)>q4n14.SetU1=Yi0
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(qa1q)(qa1q2);U2=1;U3qa+1 (qa1q)(qa1q2)qa+1 5)Similarly,supposethat(1;1)=(0;1)but6=(1;2;:::;b).Thenthereexistsj2suchthatjj1+2.Considertheunipotentcharacter0labeledby00ofrankn1,with0=and0=(1;:::;j1;j1;j+1;:::;b).Bytheinductionhypothesis,0(1)>q4n14.SetV1=Yj0
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2(q2j1)>q4; Nextweconsiderthecasewhere=(0;1;:::;b1).Observethatn=a+(ab)2=4.Ifb=1thenn=(a+1)2=4and(1)>(qn+1)f(a+1)wherefisthefunctiondenedinformula( 2.4 ).Sincen6,itfollowsthata7.Wehavealreadyprovedthatf(a+1)>q3n10fora5.Therefore(1)>q4n10asrequired.Nowwecanassumeb2.Considertheunipotentcharacter0labeledby00ofrankn1,with0=(1;2;:::;a1)and0=(0;1;:::;b2).Since00isnotinLn1,againbytheinductionhypothesis,0(1)>q4n14.Furthermore,(1) (qa1)(qb1+1)>q2n3>q4: 7)Finally,wesupposethat1=0and=(1;2;:::;b).Firstweconsiderthecasewhere6=(0;1;:::;a1).Thenthereexistsi2suchthatii1+2.Chooseitobesmallestpossible.Considertheunipotentcharacter0labeledby00ofrankn1,with0=(1;:::;i1;i1;i+1;:::;a)and0=.If002Ln1then=02n11andbycheckingdirectlywehave(1)>q4n10.Thereforewecansupposethat00isnotinLn1.Inotherwords,bytheinductionhypothesis,0(1)>q4n14.Remarkthatni0.SetW1=Yi0
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(qi11)(qi1q)qi+q qi1+q>>(qn+1)(qn11) (qn1+1)(qn21)q2iq2 Thelastcongurationwehavetohandleisthat=(0;1;:::;a1)and=(1;2;:::;b).Considertheunipotentcharacter0labeledby00ofrankn1,with0=(0;1;:::;a2)and0=(1;2;:::;b1)(ifb=1,0isjustempty).Bytheinductionhypothesis,0(1)>q4n14.Furthermore,(1) (qa1)(qb1+1)>q2n3>q4: Proposition7:2in[ 70 ]showsthattheprojectiveconformalorthogonalgroupoftype+,P(CO+2n(q)0),hasauniqueunipotentcharacterofminimaldegree(qn1)(qn1+q)=(q21)andanyothernon-trivialunipotentcharacterhasdegreegreaterthanq2n2. 88

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Proof. 9 ,p.471,472],weknowthattheunipotentcharactersofGareparametrizedbysymbolsoftheform=123:::a12:::b; 22#=n: 2cq(a+b22)+(a+b42)+QiQik=1(q2k1)QjQjk=1(q2k1): Whenn=5,Propositioncanbeverieddirectlybyusing[ 11 ,Table30].DenoteLn=ff(n);(0)g;f(n1);(1)g;f(0;1);(1;n)gg.Wewillprovebyinductiononn5thatif=;andf;g=2Ln,then(1)>q4n10,provided(n;q)6=(5;2).Therestoftheproofestablishestheinductionstepforn6. 1)Atthispointwesupposethat(1;1)6=(1;0)and11(eventuallybmaybezero).Considertheunipotentcharacter0labeledbythesymbol00ofrankn1, 89

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4a1+(ab)2 2(q211)aYi=2qiq1 4(q211)(qn1)(qn1+1) 4(q2n81)>q4; Thecase(1;1)6=(0;1)and11canbereducedto1)byinterchangingand. 2)Nowweconsiderthecasewhereb=0and1=0.Thena0(mod4).Firstwesupposethat6=(0;1;:::;a1).Thenthereexistsi2suchthatii12.Wechooseitobesmallestpossible.Considerthecharacter0correspondingtothesymbol0,where0=(:::;i1;i1;i+1;:::).Notethatni(a2)2=4.Therefore,(1)=0(1)>(qn+1)(qn11)=2(q2i1)>q4ifa8.Itisobviousthat0=2Ln1.If(n;q)=(6;2)and0=(0;1;2;5)or(0;1;3;4),wecancheckdirectlythat(1)>q4n10.Otherwise,theinductionhypothesis0(1)>q4n14impliesthat(1)>q4n10.Ifa=4andni3,thenagain(1)=0(1)>q4,so(1)>q4n10.Ifa=4andni2,theni=aand=(0;1;2;n1).Directcomputationshowsthat(1)>q4n10andwearedone.Nextwesupposethat=(0;1;:::;a1).Then(1)=(qn1)f(a)andn=a2=4,wheref(a)isthefunctiondenedin( 2.4 ).Notethatn6,soa8.Wealreadyshowedthatf(a)>q3nfora6.Hence,(1)>q4n10asdesired. Again,thecasea=0and1=0canbereducedto2)byinterchangingand.Therefore,itremainstoconsiderthecasewherea;b1and(1;1)=(0;1)or(1;0).Byasimilarreasonasabove,itisenoughtosupposethat(1;1)=(1;0). 90

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qi1q1 2qi+1 (qn2q)(qn2q2);U2=1;U3qn1+1 (qn2q)(qn2q2)qn1+1 (qn1q)(qn1q2)(qn+1)(qn+q)(qn+q2) (qn1+1)(qn1+q)(qn1+q2)=(qnq)(qnq2) (qn1q)(qn1q2)(qn+q)(qn+q2) (qn1+q)(qn1+q2)>q4; 91

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(qn21)(qn2q)>q(qn1q) (qn2q);V3qn1+q qn2+q: (qn2q)qn1+q qn2+q>q4; Nextweconsiderthecasewhere=(0;1;:::;b1).Considertheunipotentcharacter0labeledby00ofrankn1,with0=(1;2;:::;a1)and0=(0;1;:::;b2)(ifa=1,0isjustempty;ifb=1,0isjustempty).Itisobviousthatf0;0gisnotinLn1andtherefore0(1)>q4n14bytheinductionhypothesis.Furthermore,(1) (qa1)(qb1+1)>q2n3>q4: 92

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5.2.1 uptodegree(q2n21)(q2n1)(qn21)(qn2q3)=2(q21)(q41)(q3+1),whichisgreaterthanq4n10(qn1)=2whenn6.Therefore,TheoremFwillbeprovedifwecanclassifynon-unipotentcharactersofGofdegreesuptoq4n10(qn1)=2.WeknowthatthedualgroupofGisG=SO2n+1(q)andeverynon-unipotentcharacter2Irr(G)isparametrizedbyapair((s);)where(s)isanontrivialgeometricconjugacyclassofasemi-simpleelements2GandisanirreducibleunipotentcharacterofthecentralizerC:=CG(s).Moreover,(1)=(G:C)p0(1). 66 ]and[ 75 ].Letdenoteageneratoroftheeldofq4elements,=q21,=q2+1,=q1,=q+1.LetT1=f1;:::;(q3)=2g,T2=f1;:::;(q1)=2gifqisoddandT1=f1;:::;(q2)=2g,T2=f1;:::;q=2gifqiseven.Furthermore,whenqisodd,letR1=fj2Z:1j
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Proof. 5.1.2 (q+1)(q21)(qnm(1)nm):=f(m;n): 2)Casem=n2.Firstwesupposethatk=n2.Then(1)=(q2n21)(q2n1) 70 ,Proposition5.1].Then(1)(q2n21)(q2n1)(qn21)(qn2q) 2(q+1)2(q21)>D(n): Next,weconsiderk=0then(1)(qn21)(q2n21)(q2n1) 2(q+1)(q21)>D(n):

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(q+1)(q21)Qn2i=k+1(q2i1)(qn2k1) 2Qn2ki=1(q2i1)>(q2n21)(q2n1) (q+1)(q21)1 2q2k(n2k)(qn2k1)>(q2n21)(q2n1) (q+1)(q21)1 2q2(n3)>D(n): 2(q2)(1),whereisaunipotentcharacterofC'GO12n2(q)GL21(q)=(SO12n2(q)2)GL21(q),1;2=1.NotethatGL21(q)'Zq2hasonlyoneunipotentcharacter,whichisthetrivialone.Supposethatisanirreducibleconstituentof(!)SO12n2(q)21GL21(q)where!isaunipotentcharacterofSO12n2(q).ByProposition 5.1.1 ,theunipotentcharactersofSO2n2(q)areoftheformf,whererunsovertheunipotentcharactersofP(CO2n2(q)0)andfisthecanonicalhomomorphismf:SO2n2!P(CO2n2(q)0).Inparticular,degreesoftheunipotentcharactersofSO2n2(q)arethesameasthoseofP(CO2n2(q)0).Therefore,byPropositions7.1,7.2of[ 70 ],if!isnontrivialthen!(1)(qn1+1)(qn2q)=(q21).Then(1)(qn1+1)(q2n1) 2(q2)(qn1+1)(qn2q) Assumek=n1.Then(1)=q2n1 5.2.2 ,!iseitheroneofcharactersofdegrees1,(qn11)(qn1q)=2(q+1),(qn1+1)(qn1+q)=2(q+1),(qn1+1)(qn1q)=2(q1), 96

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Assumek=1.Then(1)(G:C)p0(qn21)(q2n21)(q2n1) 2(q+1)(q21)>D(n): 2(q1)(q2)(1); 70 ].Then,(1)(q2n1)(q2n21) 2(q+1)2(qn21)(qn2q) 2(q+1)>D(n): 2(q21)(q2(n1k)1)(qn1k1)q2n1 2q2k(n1k)(qn1k1)q2n1 2q4(n3)(q21)>D(n): 2Qnki=1(q2i1)(1); 2q2k(nk)(qnk1). Assumek=n1.Then(1)=q2n1 2(q)(1).Wehave=!,where!isaunipotentcharactersofSO2n1(q)andisoneoftwoextensionsof1SO12(q)toGO12(q). 97

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5.2.2 ,!iseitheroneofcharactersofdegrees1,(qn11)(qn1q)=2(q+1),(qn1+1)(qn1+q)=2(q+1),(qn1+1)(qn1q)=2(q1),(qn11)(qn1+q)=2(q1)or!(1)>q4n12.Intheformercase,6)holds.Inthelattercase,(1)>q2n1 2(q)q4n12>D(n). Assumek=0.Then(1)=qn+ 5.2.4 ,iseitheroneofcharactersofdegrees1,(qn1)(qn1+q)=(q21),(q2nq2)=(q21),or(1)>q4n10.Inthelattercase,(1)=qn+1 2(1)qn+1 2!(1)>qn+1 2q4n10>D(n): 65 ],thecharacteraordedbytherank3permutationmoduleofGO+2n(q)onthesetofallsingular1-spacesofF2nqisthesumofthreeunipotentcharactersofdegrees1,(qn1)(qn1+q)=(q21),and(q2nq2)=(q21).Hence,7)holdsinthiscase. 5.2.3 ,iseitheroneofcharactersofdegrees1,(qn+1)(qn1q)=(q21),(q2nq2)=(q21),or(1)>q4n10.Inthelattercase,(1)=qn1 2(1)qn1 2!(1)>qn1 2q4n10=D(n): 65 ],thecharacteraordedbytherank3permutationmoduleofGO2n(q)onthesetofallsingular1-spacesofF2nqisthesumofthreeunipotentcharactersofdegrees1,(qn+1)(qn1q)=(q21),and(q2nq2)=(q21).Hence,8)holdsinthiscase. Assumek=1.Then(1)=(q2n1)(qn1+) 2(q21)(1),whereisaunipotentcharacterofC'SO3(q)GO2n2(q).Supposethatisanirreducibleconstituentof!'Cwhere!;'areunipotentcharactersofSO3(q);SO2n2(q),respectively.Letusconsider 98

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70 ],wehave'(1)(qn1+1)(qn2q)=(q21).Then(1)(q2n1)(qn11)(qn1+1)(qn2q) 2(q21)2>D(n): Assumek=2.Then(1)=(q2n21)(q2n1)(qn21) 2(q21)(q41)(1)>D(n): 2(q21)(q2)(1); 70 ],!(1)(qn21)(qn2q)=2(q+1)andtherefore(1)(q2n21)(q2n1)(qn21)(qn2q) 4(q+1)(q41)>D(n): 99

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ConsiderthetransformationT:V!VdenedbyT(M)=MT,thetransposeofM.ItisclearthatTpreservesQandthereforeT2GO+4(q).Moreoverdet(T)=1.SoGO+4(q)=SO+4(q)o.WehaveT1(A;B)T(M)=BTM(AT)1foreveryA;B2GL2(q).SoTxes12andmapsoneoff11L2;1L12gtotheother.Inotherwords,theunipotentcharacterofdegreeq2ofSO+4(q)hastwoextensionstoGO+4(q)andtheinductionsoftwounipotentcharactersofdegreeqofSO+4(q)toGO+4(q)areequalandirreducible.So10)holdsinthiscase. 100

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2Qnki=1(q2i1)>>1 2q2k(nk)(qnk1)1 2q6(n3)(q31)>D(n): 5.3.1 anditsproof,wehavenotshownhowtocountthenumberofsemi-simpleconjugacyclasses(s)inSO2n+1(q)foracertainC:=CG(s),whichwillimplythenumberofirreduciblecharactersofSp2n(q)ateachdegree.Actually,thewaytocountthemisprettysimilarinallthecasesfrom2)to10)inProposition 5.3.1 .First,fromthestructureofCG(s),weknowtheformofthecharacteristicpolynomialaswellastheeigenvaluesofs.Ingeneral,wehaveSpec(s)=f1;:::;1| {z }2k+1;1;:::;1| {z }2(mk);:::|{z}2(nm)g: 73 ,(2.6)],if1=2Spec(s),thereisexactlyoneGO2n+1(q)-conjugacyclassofsemi-simpleelements(s)foragivenSpec(s).ThisclassisalsoanSO2n+1(q)-conjugacyclasssince(GO2n+1(q):CGO2n+1(q)(s))=(SO2n+1(q):CSO2n+1(q)(s))(seeLemma 5.1.2 ).Thesituationisalittlebitdierentwhen12Spec(s).Inthatcase,againby[ 73 ,(2.6)],thereareexactlytwoGO2n+1(q)-conjugacyclassofsemi-simpleelements(s)foragivenSpec(s),inwhichCGO2n+1(q)(s)'GO2k+1(q)GO2(mk)(q)Qti=1GLiai(qki)(+foroneclassandfortheother).TheseclassesarealsoSO2n+1(q)-conjugacyclassbythesamereasonasbefore.So,inordertocountthenumberofsemi-simpleconjugacyclasses(s)withagivencentralizerC,weneedtocountthenumberofchoicesofSpec(s).ThisisdemonstratedinTables 5-2 5-3 WenishthissectionbytwofollowingCorollariesofTheoremF.

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Low-dimensionalunipotentcharactersofSp2n(q),n6,qodd,I CharactersSymbolsDegrees 1Gn1201n(qn1)(qnq) 2(q+1)301n(qn+1)(qn+q) 2(q+1)41n0(qn+1)(qnq) 2(q1)50n1(qn1)(qn+q) 2(q1)602n1(q2n1)(qn11)(qn1q2) 2(q41)702n1(q2n1)(qn1+1)(qn1+q2) 2(q41)82n10(q2n1)(qn1+1)(qn1q2) 2(q21)290n12(q2n1)(qn11)(qn1+q2) 2(q21)2101n11q(q2n1)(q2n2q2) (q21)211012n1(q2nq2)(qn1)(qnq2) 2(q41)120121n(q2nq2)(qn+1)(qn+q2) 2(q41)1312n01(q2nq2)(qn+1)(qnq2) 2(q21)21401n12(q2nq2)(qn1)(qn+q2) 2(q21)2

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Low-dimensionalirreduciblecharactersofSp2n(q),n6,qodd,II Spec(s)CG?(s)(1)(1)]ofcharacters (q1)(q21)(q3)=2 (q1)(q21)(q3)=2 (q+1)(q21)(q1)=2 (q+1)(q21)(q1)=2 (q1)2(q3)(q5)=8 (q+1)2(q1)(q3)=8 2(q1)2(q3) 2(q+1)2(q1)

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Low-dimensionalirreduciblecharactersofSp2n(q),n6,qodd,III Spec(s)CG?(s)(1)(1)]ofchars 2(q)(q1)2(q3) 2(q)(q+1)2(q1) 2(q)(1)4 (qn)(qn1+q) 2(q21)4 2(q21)4 2(q21)4 2(q21)4 2(q21)(q2)4 2(q21)(q2)4 (q21)21 Here,(?)is1,(qn11)(qn1q)=2(q+1),(qn1+1)(qn1+q)=2(q+1),(qn1+1)(qn1q)=2(q1),or(qn11)(qn1+q)=2(q1).

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5.2.2 .Sowecanassumeisnotunipotentsuchthat(1)q4n8.Supposethatisparametrizedby((s);),wheresisanon-trivialsemi-simpleelementinG.ConsideraninverseimagesofsineG.Then(s)isanon-trivialconjugacyclassofsemi-simpleelementsineG.SetC=CG(s)andC=CeG(s).LetgbeanyelementineGsuchthattheimageofginGbelongstoC.Thengsg1=(g)sforsome(g)2Zq1.Therefore,(s)=(gsg1)=((g)s)=(g)2(s).Itfollowsthat(g)is1or1.Hence(G:C)p0(eG:C)p0=2: 5.1.4 ,C'(Spm(q2)Qti=1GLiai(qki))Zq1,wherei=,ti=1kiai=nm.Therefore,(eG:C)p0(q21)(q2n1) (q41)(q2m1)(q+1)(q21)(qnm(1)nm)8>><>>:minf(q21)(q2n1) (q41)(q2n1);(q21)(q2n1) (q+1)(q21)(qn(1)n)gif2jnminf(q21)(q2n1) (q41)(q2n21)(q+1);(q21)(q2n1) (q+1)(q21)(qn(1)n)gif2-n=8><>:minf(q21)(q61)(q2n21);(q1)(qn+(1)n)gif2jnminf(q21)(q61)(q2n1)=(q+1);(q1)(qn+(1)n)gif2-n;

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5.1.4 ,C'(Sp2k(q)Sp2(mk)(q)Qti=1GLiai(qki))Zq1,wherei=,ti=1kiai=nm.ItiseasytoseethatjSp2k(q)Sp2(mk)(q)jp0Qmi=1(q2i1)andjQti=1GLiai(qki)jp0Qnmi=1(qi(1)i).Therefore,(eG:C)p0(q2(m+1)1)(q2n1) (q+1)(q21)(qnm(1)nm)=:f(m;n): 2)Whenm=n1.If1kn2then(eG:C)p0(q2n21)(q2n1)=(q+1)(q21).Again,(1)(eG:C)p0=2>q4n8.Sok=0orn1.ModuloZ(eG),wecanassumethatk=n1.Therearetwocases: 73 ,(2.6)].ThisconjugacyclassisalsotheclassofsinCSp2n(q).Therefore,thereisexactlyoneconjugacyclassofsemi-simpleelements(s)inGsuchthatSpec(s)=f1;:::;1,;1gforeach=k;k2T1. Intwoabovesituations,iff1;:::;1;;1g=f;:::;;;1gforsome2Fq,then=1sincen5.Therefore,CisthecompleteinverseimageofCineG.Inotherwords,C=C=Z(eG)andhence(G:C)p0=(eG:C)p0.Considerthecanonicalhomomorphismf:Sp2n2(q)GL1(q),!C!C,=,whosekerneliscontainedin 107

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5.1.1 ,theunipotentcharactersofSp2n2(q)GL1(q)areoftheformf,whererunsovertheunipotentcharactersofC.Inparticular,istrivialor(1)(qn11)(qn1q)=2(q+1)byProposition5.1of[ 70 ].Inthelattercase,(1)q2n1 2(q+1)>q4n8: 3)Whenm=n.Since(s)isnon-trivial,weassume1kn1. First,ifk=1orn1,moduloZ(eG),wemayassumeSpec(s)=f1;1;1;:::;1g.Again,C=C=Z(eG)and(G:C)p0=(eG:C)p0.Thereisauniquepossibilityfor(s)inthiscase.Wehave(1)=q2n1 5.1.1 againforthecanonicalhomomorphismf:Sp2(q)Sp2n2(q),!C!C,weseethateitheristrivial,ortheunipotentcharacterofdegreeq,or(1)(qn11)(qn1q)=2(q+1).Inthelastcase,(1)q2n1 2(q+1)>q4n8: Next,ifk=2orn2,thenoneagaincanshowthatC=C=Z(eG)sincen5.Thereforewehave(1)(eG:C)p0(q2n21)(q2n1) (q21)(q41)>q4n8: 2(q21)(q41)(q61)>q4n8:

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5.2.3 5.2.4 .Nowweassumeisnotunipotent.Supposethatisparametrizedby((s);),wheresisanon-trivialsemi-simpleelementinGandisaunipotentcharacterofC:=CG(s)suchthat(1)q4n10.SetC0=CGO2n(q)(s).SinceCisasubgroupofC0ofindex1or2,(G:C)20=(GO2n(q):C0)20. LetV=F2nqbeendowedwithanon-degeneratequadraticformQ(:).FixabasisofVandletJbetheGrammatrixofQcorrespondingtothisbasis.ThentsJs=J.HenceSpec(s)=Spec(ts)=Spec(Js1J1)=Spec(s1).DenotethecharacteristicpolynomialofsactingonVbyP(x)2Fq[x]anddecomposeP(x)intodistinctirreduciblepolynomialsoverFq:P(x)=(x1)2mlYi=1fmii(x)l0Yj=1gnjj(x)bgjnj(x); ByLemma 5.1.3 ,C0'GO2m(q)Qti=1GLiai(qki),wherei=,ti=1kiai=nm.Since(s)isnon-trivial,m
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(q+1)(q21)>q4n10: 2)Whenm=n1.ThenC'GO2n2(q)GL1(q)with=.SinceGL1(q)'2(q),C0'GO2n2(q)GL1(q).ItisobviousthatC=2n2(q)GL1(q).Therearetwocases: 73 ,(3.7)].Since(G:C)=(GO2n(q):C0),GO2n(q)-conjugacyclassofsisalsoaG-conjugacyofs.Therefore,thereisalsoexactlyoneconjugacyclassofsemi-simpleelements(s)inGsuchthatSpec(s)=f1;:::;1,;1gforeach=k;k2T1. Wehave(1)=(qn)(qn1+) 70 ],(1)(qn1)(qn2+q)

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3)Whenm=0.Ifn6,westillhavef(0;n)=Qni=1(qi+(1)i)=(qn+1)>q4n10.Nowweconsiderthecasen=5.Since(1)q4n10,C0=GU5(q).ThisforcesG=10(q).Then(G:C)20=(q1)(q2+1)(q31)(q4+1)andtherefore(1)=1.Thereareexactlyq=2conjugacyclasses(s)inGO10(q)ofsemi-simpleelementssuchthatC0'GU5(q),whichishappenedwhenSpec(s)=f;;;;;1;1;1;1;1g,=k;k2T2.NotethatGU5(q)10(q),soC=GU5(q)and(GO10(q):C0)=2(10(q):C).Inotherwords,jsGO10(q)j=2js10(q)j.Thereforethereareexactlyqconjugacyclasses(s)in10(q)suchthatC=GU5(q).Thisgivesqcharactersofdegree(q1)(q2+1)(q31)(q4+1)of10(q).2 5.2.3 5.2.4 .Sowecanassumethatisnotunipotent.WedenoteeG:=CO2n(q),eG0:=CO2n(q)0,andZ:=Z(eG)'Z(eG0)'Zq1.Supposethatsisanon-trivialsemi-simpleelementinG.ConsideraninverseimagesofsineG0.Then(s)isanon-trivialconjugacyclassofsemi-simpleelementsineG.SetC:=CG(s),eC:=CeG(s),andeC0:=CeG0(s).Supposethatisparametrizedby((s););2Irr(C),suchthat(1)q4n10.LetgbeanyelementineG0suchthattheimageofginGbelongstoC.Thengsg1=(g)sforsome(g)2Zq1.Therefore,(s)=(gsg1)=((g)s)=(g)2(s).Itfollowsthat(g)is1 111

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5.1.5 ,eC'(GOm(q2)Qti=1GLiai(qki))Zq1,wherei=,ti=1kiai=nm.DenotethegroupQti=1GLiai(qki)byAforshort.Nowwewillshowthat,whenn6,(eG:eC)p0>4q4n10,andtherefore(1)(G:C)p0(eG:eC)p0=4>q4n10.ItiseasytoseethatjAjp0Qnmi=1(qi(1)i).Hence,(eG:eC)p02(q21)(q2n1) (qn+1)(q+1)(q21)(qnm(1)nm)jGOm(q2j=:f(m;n): Nowletusconsiderthecase(n;)=(5;+).NotethatGU5(q)isnotasubgroupofGO+10(q).Therefore,ifA6=GL5(q),onecanshowthat(eG:eC)p0>4q10andhence(1)>q10.IfA=GL5(q),thenthecharacteristicpolynomialofshastheformP(x)=(x)5(x1)5with2Fqand:=(s).Nowwewillshowthat(G:C)p0=(eG0:eC0)p0andtherefore(1)(eG:eC)p0=2=(q+1)(q2+1)(q3+1)(q4+1)>q10.Supposethats1;s22eG0areeG0-conjugateandtheirimagesinGarethesame.Thens1=s2.LetV1:=Ker(s)andV2:=Ker(s1).NotethatV1andV2aretotallyisotropicsubspacesinVandV1\V2=f0g.Bydenition,anelementineG0cannotcarryV1toV2.Thisimpliesthats1=s2.Inotherwords,jsGj=jseG0jandhence(G:C)p0=(eG0:eC0)p0asdesired. Thenextcaseis(n;)=(5;).NotethatGL5(q)isnotasubgroupofGO10(q).Therefore,ifA6=GU5(q),oneagaincanshowthat(1)>q10.IfA=GU5(q),thenthecharacteristicpolynomialofshastheformP(x)=[(x)(x1)]5,where(x)(x1)isanirreduciblepolynomialoverFq.Notethatthereare(q+1)=2choicesforsuchapair(;1).Sothereareexactly(q+1)=2conjugacyclasses(s)ineGsuchthateC'GU5(q).Thatmeansthereareexactly(q+1)suchconjugacyclasses(s)in 112

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5.1.5 ,eC'(GO2k(q)GO2m2k(q)Qti=1GLiai(qki))Zq1,wherei=,ti=1kiai=nm. ItiseasytoseethatjGO2k(q)GO2m2k(q)jp02Qmi=1(q2i1)=(qm1)ifm1andjQti=1GLiai(qkijp0Qnmi=1(qi(1)i).Therefore,withconventionthatq01=2,wehave(eG:eC)p0(q2(m+1)1)(q2n1)(qm1) (qn+1)(q+1)(q21)(qnm(1)nm)=:g(m;n): 2)Whenm=0,theng(0;n)=2Qni=1(qi+(1)i)=(qn+1).Westillhaveg(0;n)>4q4n10ifn6.Thecase(n;)=(5;+)canbearguedsimilarlyaswhen(s)isnotasquareinFq.Soweonlyneedtoconsiderthecase(n;)=(5;).NotethatGL5(q)isnotasubgroupofGO10(q).Therefore,ifA6=GU5(q),oneagaincanshowthat(1)>q10.SowecanassumeA=GU5(q).ThenthecharacteristicpolynomialofshastheformP(x)=[(x)(x1)]5,where(x)(x1)isanirreduciblepolynomialoverFq.Notethatthereareexactly(q1)=2suchapair(;1).Repeatingargumentsaswhen(s)isnotasquareinFq,wegetexactly(q1)=2charactersofdegree(q1)(q2+1)(q31)(q4+1)inthiscase. 3)Whenm=n1.If1kn2then(eG:eC)p0(q2n21)(qn1)(qn21)=2(q+1)2>4q4n10.Hence,(1)(G:C)p0>q4n10.Sok=0orn1.Withnoloss,wecanassumek=n1.NotethatSpec(s)=f1;:::;1,;1ginthiscase.Iff1;:::;1;;1g=f;:::;;;1gforsome2Fq,then=1sincen5. 113

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73 ,(2.6)].ThisconjugacyclassisalsotheclassofsineG0.Therefore,thereisexactlyoneconjugacyclassofsemi-simpleelements(s)inGsuchthatSpec(s)=f1;:::;1,;1gforeach=k;k2T1. UsingProposition 5.1.1 forthecanonicalhomomorphismf:SO2n2(q)GL1(q),!eC0!C,=,weseethattheunipotentcharactersofSO2n2(q)GL1(q)areoftheformf,whererunsovertheunipotentcharactersofC.Inparticular,byPropositions7.1,7.2of[ 70 ],istrivialor(1)(qn1)(qn2+q)=(q21).Inthelattercase,(1)(qn)(qn1+) 4)Whenm=n.Since(s)isnon-trivial,weassume1kn1. Ifk=1orn1,moduloZ,wemayassumeSpec(s)=f1;1;1;:::;1g.TheneC=(GO2(q)GO2n2(q))Zq1andeC0'(SO2(q)SO2n2(q))2Zq1.Again,C=eC0=Zand(G:C)p0=(eG:eC)p0.Thereisauniquepossibilityfor(s)foreach 114

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2(q)(1)whereisaunipotentcharacterofC.Considerthecanonicalhomomorphismf:(SO2(q)SO2n2(q))2,!eC0!C,whosekerneliscontainedinthecenterof(SO2(q)SO2n2(q))2andimagecontainsthecommutatorgroupofCsince(SO2(q)SO2n2(q))2containsthecommutatorgroupofeC0.ByProposition 5.1.1 ,theunipotentcharactersof(SO2(q)SO2n2(q))2areoftheformf,whererunsovertheunipotentcharactersofC.Inparticular,isoneoftwolinearunipotentcharactersor(1)(qn1)(qn2+q)=(q21).Inthelattercase,(1)(qn)(qn1+) 2(q)(qn1)(qn2+q) 8(q21)(q41)>q4n10:2 Inthissection,weclassifytheirreduciblecomplexcharactersofG=Spin12(3),=,ofdegreesupto4315.Theargumentsinthissectionaresimilartothoseinx6:3andwewillkeepallnotationfromthere.TwofollowingLemmascanbecheckedbydirectcomputation.

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2Q5i=1(3i+1),providedthatG=Spin+12(3). Next,weconsiderthecaseeC'GU6(3)Z2.Notethattherearetwosemi-simpleconjugacyclasses(s)ineGsothateC'GL6(3)Z2,whichishappenedwhenthecharacteristicpolynomialofsis(x2+x1)6or(x2x1)6.Inthiscase,eC0=eC,(eG:eC)=2(eG0:eC0),andthereforethereare4suchconjugacyclassesofsemi-simpleelementsineG0.ModuloZ,wegetexactlytwosemi-simpleconjugacyclasses(s)inG.Furthermore,C'eC=Z.Therefore,wegettwocharactersofdegreeQ5i=1(3i+(1)i)ofG=Spin+12(3). 2Q5i=1(3i+(1)i),providedthatG=Spin+12(3). 116

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4(35+)(343) 8>4315: Nowwesupposem=6.First,Ifk=1or5,wehave=(1)(36)(35+)=2(3),where=andisaunipotentcharacterofC=eC0=Z,eC0'(SO2(3)SO10(3))2Z2.When=+,theinequality(1)<4315forcestobelinear.Therefore,isoneof2charactersofdegree(36)(35+)=4.When=,notethattheunipotentcharactersof(SO2(3)SO10(3))2areoftheformf,wherefisthecanonicalhomomorphismf:(SO2(3)SO10(3))2!C.SinceSO2(3)hasauniqueunipotentcharacterwhichistrivial,theunipotentcharactersof(SO2(3)SO10(3))2areactuallyunipotentcharactersofGO10(3).Therefore,isoneof2unipotentcharactersofdegree1;2unipotentcharactersofdegree(35+)(343)=8;or(1)(31032)=8.Hencethereare4charactersofdegreeslessthan4315inthiscase.Next,ifk=2or4,wehave(1)=(34+)(3101)(36) 2(321)(32)(1); 117

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Iwasbornin1980inThanhHoa,Vietnam.IgraduatedfromLamSonHighSchoolin1997.IearnedmyBSinmathematicsfromVietnamNationalUniversityin2001.IenteredGraduateSchoolattheUniversityofFloridain2003.IcompletedmyPhDinMathematicsattheUniversityofFloridain2008. 123