Weyl's Character Formula for Algebraic Groups

ID:40985021

大小:947.38 KB

页数:21页

时间:2019-08-12

Weyl's Character Formula for Algebraic Groups_第1页
Weyl's Character Formula for Algebraic Groups_第2页
Weyl's Character Formula for Algebraic Groups_第3页
Weyl's Character Formula for Algebraic Groups_第4页
Weyl's Character Formula for Algebraic Groups_第5页
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

当前文档最多预览五页,下载文档查看全文

此文档下载收益归作者所有

当前文档最多预览五页,下载文档查看全文
正文描述:

《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

显示全部收起
温馨提示:
1. 部分包含数学公式或PPT动画的文件,查看预览时可能会显示错乱或异常,文件下载后无此问题,请放心下载。
2. 本文档由用户上传,版权归属用户,天天文库负责整理代发布。如果您对本文档版权有争议请及时联系客服。
3. 下载前请仔细阅读文档内容,确认文档内容符合您的需求后进行下载,若出现内容与标题不符可向本站投诉处理。
4. 下载文档时可能由于网络波动等原因无法下载或下载错误,付费完成后未能成功下载的用户请联系客服处理。
关闭