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Bedenotebyd#2 cmmi8l!ȹ(G) thesmallestdegreeofanon-tri3vialprojectiverepresentationofGoveraeldofcharacteristic lC.InthisnoteweconsiderthoseGarisingfromtheexceptionalgroupsofLietype, i.e.,thenitesimplegroupsinthefollo3winglist: 8color push Black color pop>j2 |{Y cmr82B5B2O[(222m+1!D);UP22 G2Ź(322m+1);UPG2(qn9);23D4=8(q);22 F4\(222m+1!D);hQ>F4(qn9);UP22 E6s(q);UPE6(q);UPE7(q);UPE8(q);where mUR%!",
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msbm10Nandqn9isanarbitrarypo3werofaprimep.ForXjagroupGinthislistletxl~Gp4:=G(qn9)scubeacorrespondingnitegroupof5LietypearisingasgroupofxedpointsunderaFrobeniusmapofasimplesimply-connected*algebraicgroup.UptoanitenumberofexceptionsxV,~G istheuni3versalcoveringgroupofGandGPg=xkW~UG=XZ ܞ(x~G D).:So,inthiscase,thesmallestdegreeslofnon-tri3vialprojectiverepresentationsofGareequaltothesmallestnon-tri3vial degreesofrepresentationsofx~GD(whichisaperfectgroup).AslthemainresultofthisnotewedetermineinSectionhtml:#color push rgb 0.6179 0.0236 0.08942 html: color poptherstfe3wsmall-est?non-tri3vialdegreesofcomplexrepresentationsofthegroupsx쟹~GI,Otogetherwiththeir-:multiplicities.?W
BegettheseasanapplicationofDeligne-LusztigtheoryandLusztig'Ws classicationofirreduciblecharactersofnitegroupsofLietype.They*groupswithexceptionaluni3versalcoverings(aswellastheT|itsgroup22F4(2)2&K cmsy809)arelistedanddealtwithinSectionhtml:#color push rgb 0.6179 0.0236 0.08943 html: color pop.?Thiscompletesthedeterminationof d0(G)forallgroupsintheabove list.InSectionhtml:#color push rgb 0.6179 0.0236 0.08944 html: color popwecollectfortherst3vetypesofgroupssomevaluesdl!ȹ(G)forlG"aprimenotequaltop. Theinformationiscompleteintherstthreecases.Thisimproves