资源描述：
《Weyl's Character Formula for Algebraic Groups》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、Inventionesmath.5,85--105(1968)Weyl'sCharacterFormulaforAlgebraicGroupsT.A.SPmNGER(Utrecht)IntroductionH.WEYL'SformulaforthecharacterofirreduciblerepresentationsofcompactsemisimpleLiegroups,provedbyhimbytranscendentalmethods([13],p.358)isactuallyastatementofapurelyalgebraicnature.Moreoverthere
2、isanobviousvariantwhichmakessensefortherationalrepresentationsofconnectedsemisimplealgebraicgroupsoveranalgebraicallyclosedfieldofcharacteristic0,whichonederiveseasilyfromWEYL'Soriginalformula,forinstancebyinvokingthe"Lefschetzprinciple".Ofcourseaproofofanalgebraicstatementalongsuchlinesisnott
3、oosatisfactory.ThealgebraicproofofWEYL'SformulagivenbyFREUDENTHAL([5],seealso[8],Ch.VIII,whasthedisadvantagethatitoperateswiththeLiealgebra,sothattherestillisatransitiontobemadefromLiealgebratogroup,whichonewouldratheravoidindealingwithalgebraicgroups.Inthepresentnotea"global"proofofWEYL'Schar
4、acterformulawillbegiven,whichoperateswiththegroupitself.Theideasusedinthisproofarequitefamiliar,inoneformoranother.Themaintoolisanidentity(Proposition2.9)involvingtheCasimiroperatorwhichisthealgebraicanalogofoneduetoHARISH-CHANDRA(andwhichisalsoimplicitin[5]).Anadvantageofourmethodis,thatitdoe
5、snotcompletelybreakdownincharacteristicp>0,sothatonecanextractinformationaboutcertainirreduciblerepresentationsincharacteristicp>0(viz.thoseforwhichpis"largewithrespecttothehighestweight",inasensemademorepreciseby4.3).Theresultsare,however,farfromconclusiveanditdoesnotseemlikelythatthemethodso
6、fthisnotearesufficienttoobtainacompletesolutionoftheproblemoffindingageneralcharacterformulaincharacteristicp.TheauthorisindebtedtoJ.TITSforvarioususefulremarks.1.Preliminaries1.1.Letkbeanalgebraicallyclosedfieldofcharacteristicp.LetGbeaconnectedlinearalgebraicgroup,definedoverk.Wemayandshalli
7、dentifyitwithitsgroupofk-rationalpoints.Wereferto[4]fortherelevantfactsaboutalgebraicgroups.7Inventionesmath.,VoL586T.A.SPRINGER:TheringofregularfunctionsonGisdenotedbyk[G].ThegroupGactsonk[G]byleft(right)translations7(g)(resp.c~(g)),wh
