Springer.Some.Arithmetical.Results.On.Semisimple.Lie.Algebras

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1、PUBLICATIONSMATHÉMATIQUESDEL’I.H.É.S.TONNYA.SPRINGERSomearithmeticalresultsonsemi-simpleLiealgebrasPublicationsmathématiquesdel’I.H.É.S.,tome30(1966),p.115-141.©Publicationsmathématiquesdel’I.H.É.S.,1966,to

2、usdroitsréservés.L’accèsauxarchivesdelarevue«Publicationsmathématiquesdel’I.H.É.S.»(http://www.ihes.fr/IHES/Publications/Publications.html),impliquel’accordaveclesconditionsgénéralesd’utilisation(http://www.numdam.org/legal.php).Touteutilisationcommercialeouim

3、pressionsystématiqueestconstitutived’uneinfractionpénale.Toutecopieouimpressiondecefi-chierdoitcontenirlaprésentementiondecopyright.ArticlenumérisédanslecadreduprogrammeNumérisationdedocumentsanciensmathématiqueshttp://www.numdam.org/SOMEARITHMETICALRESULTSONSE

4、MI-SIMPLELIEALGEBRAST.A.SPRINGERIntroduction.Thepresentpaperhaditsorigininanattempttoprovetheexistenceofregularunipotentelementsinasemi-simplelinearalgebraicgroupGoveranalgebraicallyclosedfieldk(ofarbitrarycharacteristicp).Theattemptwasnotcompletelysuccessful,

5、itturnedouttogiveresultsonlyundersome(rathermild)restrictionsonp.Ourmethod,whichisgivenin§4ofthispaper,makesanessentialuseoftheexplicitformulasforthestructureoftheunipotentpartUofaBorelsubgroupBofG,whichareduetoChevalley([9]).Theapplicationoftheseformulastoour

6、problemleadsonetoinves-tigateanarithmeticalproblemaboutsemi-simpleLiealgebras.Thisisthefollowingproblem.LetQbeaLiealgebraovertheringofintegersZ,associatedwithacomplexsemi-simpleLiealgebra.Let{^}bethesetofc

7、fsimpleroots.Determinetheelementarydivisorsoftheendomorphismad(^z)of9.Thisarithmeticalproblemisdealtwithin§2ofthepaper(aftersomeintroductorymaterialin§oand§i).Theresultsforthecasethattherootsystemof9issimple,aregivenin(2.6).Theirproofsrelyheavilyontheexplicitk

8、nowledgeofthesimplerootsystems.Weneed,forexample,fortheexceptionalsimpletypesEg,Ey,Eg,F4,tablesgivingthepositiverootswhenexpressedinthesimpleones(thesetablesaregiveninanappendix).W