on the unipotent characters of the ree groups of type g_2

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1、THEU
TETCHARACTERSFTHEREEGRUSFTYEG2
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ERBRUATAbstra t.Thisnoteis on ernedwiththeunipotenthara tersoftheReegroupsoftypeG2.Wedeterminetherootsofunityasso iatedbyusztigand2G2(32n+1)Digne-i heltounipotenthara tersofandweprovethatthe2G2(32n+1)FouriermatrixofdenedbyGe kandallesatises

2、a onje tureofDigne-i hel.urmaintoolistheShintanides entofReegroupsoftypeG2.1.
ntrodu tionetGbea onne tedredu tivegroupdenedovertheniteeldFqwithqelementsofhara teristi p>0,andletFbethe orrespondingFrobeniusmap.etTbeamaximalrationaltorusofG, ontainedinarationalBorelsubgroupBofG.WedenotebyW=N

3、G(T)/TtheWeylgroupofG.Forw∈W,thereistheF orrespondingDeligne-usztighara terRwofthenitexed-pointsubgroupG.Wereferto[4,7.7℄forapre ise onstru tion.ThenwedenethesetofunipotentFhara tersofGbyU(GF)={χ∈Irr(GF)

4、hχ,RiwGF6=0forsomew∈W}.Fusztig[9℄andDigne-i hel[5℄asso iatedtoeveryχ∈U(G)arootofunityωχ

5、∈Qℓ(whereQℓistheℓ-adi eldforℓ6=p)asfollows.WedenotebyδtheorderoftheautomorphismofWindu edbyF.Forw∈W,wedenote−1bynw∈NG(T)anelementsu hthatnwT=w.WesetXw={xB

6、xF(x)∈BnwB}the orrespondingDeligne-usztigvariety.Foreveryintegerjwedenotej(X,Q)byHcwℓthej-thℓ-adi ohomologyspa ewith ompa tsupportoverQℓ,δF

7、asso iatedtoXw.Therefore,thegroupshFiandGa tonXw,whi hindu esjjFδarXiv:0807.3766v1[math.RT]23Jul2008linearoperationsonHc(Xw,Qℓ).Hen eHc(Xw,Qℓ)isaQℓG-module,andFδa tsonthisspa easalinearendomorphism.WealsoxaneigenvalueλofF,andδFwedenotebyFλ,jitsgeneralizedeigenspa e.Sin ethea tionsofhFiandGF ommu

8、te,thespa eFλ,jisaQℓG-module.oreover,theirredu ible onstituentsFofGo urringinthismoduleareunipotent.Conversely,foreveryunipotenthar-×Fa terχofG,therearew∈W,λ∈Qℓandj∈N,su hthatχo ursinthehara terasso iatedtoFλ,j.usztig[9℄andDigne-i hel[5℄haveshownthats/2×λ=qωχforsomenon-negativeintegersandaro

9、otofunityωχ∈Qℓwhi hdependsonlyonχ.Thesetofrootsasso iatedasabovetotheunipotenthara tershavebeen omputedbyusztigin[9℄forniteredu tivegroupsifFissplit.oreoverusztig omputedin[8℄thesetofrootsfortheunipotenthara tersap