buildings and classical groups - p. garrett

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1、BuildingsandClassicalGroupsc1995,PaulGarrett,garrett@math.umn.eduTypesetwithAMS-LaTeX1Garrett:`Introduction'iIntroductionThisbookdescribesthestructureoftheclassicalgroups,meaninggenerallineargroups,symplecticgroups,andorthogonalgroups,bothovergeneraleldsandinnerdetailoverp-adicelds.Tothise

2、nd,halfofthetextisasystematicdevelopmentofthetheoryofbuildingsandBN-pairs,bothsphericalandane,whiletheotherhalfisillustrationbyandapplicationtotheclassicalgroups.Theviewpointisthatbuildingsarethefundamentalobjects,usedtostudygroupswhichactuponthem.Thus,tostudyagroup,onediscoversorcon-structs

3、abuildingnaturallyassociatedtoit,onwhichthegroupactsnicely.Thisdiscussionisintendedtobeintelligibleaftercompletionofabasicgraduatecourseinalgebra,sothereareaccountsofthenecessaryfactsaboutgeometricalgebra,re ectiongroups,p-adicnumbers(andotherdiscreteval-uationrings),andsimplicialcomplexesand

4、theirgeometricrealizations.Itisworthnotingthatitisthebuilding-theoreticaspect,notthealgebraicgroupaspect,whichdeterminesthenatureofthebasicrepresentationtheoryofp-adicreductivegroups.OneimportantsourceofinformationforthisandrelatedmaterialisthemonumentaltreatiseofBruhat-Tits,whichappearedinse

5、veralparts,widelyseparatedintime.Thistreatiseconcernedmostlyapplicationofthetheoryofanebuildingstop-adicgroupsofthetheoryofanebuildings.Oneofthebasicpointsmade,andanideapervasiveinthework,isthatbuildingscanbeattachedinanintrinsicmannertoallp-adicreductivegroups.Butthispointisdiculttoapprec

6、iate,makingthissourcenotcongenialtobeginners,pre-sumingasitdoesthatthereaderknowsagreatdealaboutalgebraicgroups,andhasarmgraspofrootsystemsandre ectiongroups,havingpresumablyworkedalltheexercisesinBourbaki'sLietheorychaptersIV,V,VI.Incontrast,itisthisauthor'sopinionthat,ratherthanbeingcoroll

7、ariesofthetheoryofalgebraicgroups,themechanismbywhichasuitableactionofagroupuponabuildingilluminatesthegroupstructureisafundamentalthingitself.Still,muchofthematerialofthepresentmonographcanbefoundin,orinferredfrom,thefollowingitems:F.Bruhata