garrett p. buildings and classical groups (web draft, 1995)(340s)_matg_

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1、BuildingsandClassicalGroupsc1995,PaulGarrett,garrett@math.umn.eduTypesetwithAMS-LaTeX1Garrett:`Introduction'iIntroductionThisbookdescribesthestructureoftheclassicalgroups,meaninggenerallineargroups,symplecticgroups,andorthogonalgroups,bothovergeneraleldsandi

2、nnerdetailoverp-adicelds.Tothisend,halfofthetextisasystematicdevelopmentofthetheoryofbuildingsandBN-pairs,bothsphericalandane,whiletheotherhalfisillustrationbyandapplicationtotheclassicalgroups.Theviewpointisthatbuildingsarethefundamentalobjects,usedtostud

3、ygroupswhichactuponthem.Thus,tostudyagroup,onediscoversorcon-structsabuildingnaturallyassociatedtoit,onwhichthegroupactsnicely.Thisdiscussionisintendedtobeintelligibleaftercompletionofabasicgraduatecourseinalgebra,sothereareaccountsofthenecessaryfactsaboutgeo

4、metricalgebra,re ectiongroups,p-adicnumbers(andotherdiscreteval-uationrings),andsimplicialcomplexesandtheirgeometricrealizations.Itisworthnotingthatitisthebuilding-theoreticaspect,notthealgebraicgroupaspect,whichdeterminesthenatureofthebasicrepresentationtheo

5、ryofp-adicreductivegroups.OneimportantsourceofinformationforthisandrelatedmaterialisthemonumentaltreatiseofBruhat-Tits,whichappearedinseveralparts,widelyseparatedintime.Thistreatiseconcernedmostlyapplicationofthetheoryofanebuildingstop-adicgroupsofthetheoryo

6、fanebuildings.Oneofthebasicpointsmade,andanideapervasiveinthework,isthatbuildingscanbeattachedinanintrinsicmannertoallp-adicreductivegroups.Butthispointisdiculttoappreciate,makingthissourcenotcongenialtobeginners,pre-sumingasitdoesthatthereaderknowsagreatde

7、alaboutalgebraicgroups,andhasarmgraspofrootsystemsandre ectiongroups,havingpresumablyworkedalltheexercisesinBourbaki'sLietheorychaptersIV,V,VI.Incontrast,itisthisauthor'sopinionthat,ratherthanbeingcorollariesofthetheoryofalgebraicgroups,themechanismbywhichas

8、uitableactionofagroupuponabuildingilluminatesthegroupstructureisafundamentalthingitself.Still,muchofthematerialofthepresentmonographcanbefoundin,orinferredfrom,thefollowingitems:F.Bruhata