Springer.Quantum.Groups.The.Loop.Grassmannian.And.The.Springer.Resolution

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1、QuantumGroups,theloopGrassmannian,andtheSpringerresolution.SergeyArkhipov,RomanBezrukavnikov,andVictorGinzburgAbstractWeestablishequivalencesofthefollowingthreetriangulatedcategories:G(Ne)←→DDquantum(g)←→Dcoherentperverse(Gr).Here,Dquantum(g)istheder

2、ivedcategoryoftheprincipalblockoffinitedimensionalrepresentationsofthequantizedenvelopingalgebra(atanoddrootofunity)ofacomplexsemisimpleLiealgebrag;thecategoryDG(Ne)isdefinedintermsofcoherentsheavesonthecotangentbundleonthecoherent(finitedimensional)flag

3、manifoldforG(=semisimplegroupwithLiealgebrag),andthecategoryDperverse(Gr)isthederivedcategoryofperversesheavesontheGrassmannianGrassociatedwiththeloopgroupLG∨,whereG∨istheLanglandsdualgroup,smoothalongtheSchubertstratification.TheequivalencebetweenDqu

4、antum(g)andDG(Ne)isan‘enhancement’oftheknownex-coherentpression(duetoGinzburg-Kumar)forquantumgroupcohomologyintermsofnilpotentvariety.TheequivalencebetweenDperverse(Gr)andDG(Ne)canbeviewedasa‘categorification’oftheiso-coherentmorphismbetweentwocomple

5、telydifferentgeometricrealizationsofthe(fundamentalpolynomialrepresentationofthe)affineHeckealgebrathathasplayedakeyroleintheproofoftheDeligne-Langlands-Lusztigconjecture.Onerealizationisintermsoflocallyconstantfunctionsontheflagmanifoldofap-adicreductiv

6、egroup,whiletheotherisintermsofequivariantK-theoryofacomplex(Steinberg)varietyforthedualgroup.Thecompositeofthetwoequivalencesaboveyieldsanequivalencebetweenabeliancategoriesofquantumgrouprepresentationsandperversesheaves.Asimilarequivalenceatanevenr

7、ootofunitycanbededuced,followingLusztigprogram,fromearlierdeepresultsofKazhdan-LusztigandKashiwara-Tanisaki.Ourapproachisindependentoftheseresultsandistotallydifferent(itdoesnotrelyonrepresentationtheoryofKac-Moodyalgebras).ItalsogiveswaytoprovingHump

8、hreys’conjecturesontiltingUq(g)-modules,aswillbeexplainedinaseparatepaper.TableofContents1.IntroductionI.Algebraicpart2.Variousquantumalgebras3.Algebraiccategoryequivalences4.ProofofInductiontheoremarXiv:math.RT/0304173v321Apr20045.ProofofQuantumgrou