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Langlands' Philosophy and Koszul Duality

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Langlands' Philosophy and Koszul Duality LANGLANDS PHILOSOPHY AND KOSZUL DUALITY WOLFGANG SOERGEL Contents Introduction The basic conjecture The easiest example The geometry of AdamsBarbaschVogan Thanks Notations and conventions Groups and varieties Grothendieck groups Equivariant derived categories Geometric extension algebras Examples Tori Generic central character The case G SL Complex groups Compact groups Tempered representations The center and geometry Further conjectures Generalities on Koszul duality Application to our situation Motivation References Introduction In this article I will formulate conjectures relating representation theory to geometry and prove them in some cases I am trying to extend work of AdamsBarbaschVogan ABV and some joint work of myself with Beilinson and Ginzburg BGS All this should b e regarded as a contribution to the lo cal case of Langlands philosophy SOERGEL The lo calization of BeilinsonBernstein also establishes a relation b etween representation theory and geometry This is however very dierent from the relation to b e investigated in this article Whereas lo calization leads to geometry on the group itself or its ag manifold the results in ABV to b e extended in this article lead to geometry on the Langlands dual group The basic conjecture Let G b e only for a short moment a real reductive Zariskiconnected algebraic group We want to study admissible representations of the asso ciated Lie group GR of real p oints of G More precisely we want to study the category MGR of smo oth admissible representations of GR Remark that by a theorem of CasselmanWallach this is nothing else but the category of Harish Chandra mo dules for GR mo dulo the choice of a maximal compact subgroup K GR Let g b e the complexication of the Lie algebra of GR U mo d U U mo d U g the enveloping algebra of g and Z U mo d U the center of U mo d U Then MGR decomp oses according to innitesimal character into M MGR MGR where runs over the set Max Z of maximal ideals of Z There are several classications of the simple ob jects in MGR The most p opular ones are by Langlands Lan VoganZuckerman Vog and BeilinsonBernstein BB In this article I will exploit a version of the rst classication scheme due to Langlands and Shelstad and more precisely the form given to it by AdamsBarbaschVogan ABV Let me roughly explain its structure First of all we have to forget ab out the real algebraic group G and just remember the inner class of real forms for G C given by G In other words and other notations R we start out with a complex connected reductive algebraic group G endowed with an inner class of real forms To simplify the exp osition let us furthermore choose a nite central subgroup F of G To these data one asso ciates in a more or less unique way and to detail this more or less takes most of the time a nite family S of antiholomorphic involutions G G all F from the xed inner class a nite connected algebraic covering G of the dual group G ie F of the complex connected reductive algebraic group G whose ro ot system is dual to the ro ot system of G a collection fX g of complex algebraic G varieties MaxZ LANGLANDS PHILOSOPHY AND KOSZUL DUALITY I will give details in section For every antiholomorphic involution G G let GR denote the xed p oint set of the induced map GC GC and for any MaxZ let GR denote the set of isomorphism classes of simple ob jects in MGR Then AdamsBarbaschVogan ABV establish a bijection dep ending on additional choices b etween S a the set GR and S F b the set of isomorphism classes of pairs Y where Y is a G orbit on X and an irreducible G equivariant lo cal system F on Y Strictly sp eaking this classies the representations of a whole collec tion of real forms simultaneously rather then representations of a single group at a time To explain which geometric parameters Y b elong to which real form S wouldnt make go o d reading for this intro F duction compare Vog Let me mention however that any real form in the given inner class is equivalent to GR with S F if we choose F big enough hence the ab ove classication captures all real forms One p oint of setting up the ab ove classication scheme is the very b eautiful and miraculous fact that the KLdata on the representation theoretic side are mirrored by the ICdata on the geometric side This is the essence of Vogans character duality Vog I conjecture that this duality lifts to the level of categories To give more details I have to introduce more notation Let H b e a complex algebraic group and X a complex algebraic H variety Supp ose H decomp oses X into nitely many orbits Then we can form a graded C algebra Ext X as follows Take the equi H variant derived category D X as dened in BL This is a tri H angulated C category containing the category P X of equivariant H p erverse sheaves as a full ab elian sub category For F G D X let H us denote by Hom F G the space of homomorphisms in D X and H H n F G Hom F G n Since H was supp osed to act with put Hom H H nitely many orbits there are up to isomorphism only nitely many simple ob jects in P X Let L P X denote the direct sum of H H all simple ob jects ie take one from each isomorphism class Then put n Ext X Hom L L End L H n H H There is our graded C algebra We will refer to it as the geometric extension algebra For any ring R let R M od b e the category of all left R mo dules For n a graded C algebra A A dene the full sub category A N il n SOERGEL A M od by n A N il fM A M od j dim M A M for n g C So ob jects in A N il are not assumed to b e graded but only conti nous along the graduation Recall now the classication scheme from ABV The main ob jective of this article is to formulate motivate and discuss the following Basic conjecture There exists an equivalence of categories M Ext X N il MGR G F S F Remarks Take again our algebraic group H acting on our variety X with nitely many orbits Then the simple ob jects of Ext X N il H are in an obvious bijection with the simple ob jects of P X and H these in turn are parametrized by pairs Y with Y an H orbit on X and an irreducible H equivariant lo cal system on Y cf BL Thus we see that the simple ob jects on the left and on the right of our conjectured equivalence are parametrized by the same set Certainly our equivalence should identify simple ob jects which have he same parameter In this article I will check the conjecture in the following cases if G is a torus for arbitrary G and almost all central characters under the additional assumption F for G SL for complex groups more precisely when G G G for c c a semisimple simply connected complex algebraic group G and c F and the inner class of real forms is such that S consists just F of one and GR G C under the additional assumption c that the central character is integral regular I will explain in the last section how I exp ect this conjecture to b e related to Vogans character duality formulas Vog ABV It seems reasonable to make analogous conjectures for nonar chimedean elds as well In these cases X should b e replaced by spaces of quasiadmissible morphisms of the WeilDeligne group to the Langlands dual group I exp ect that one may identify in sp ecial situations the geometric extension algebras with Lusztigs graded ane Hecke algebras The easiest example To give the reader a more concrete idea of what is going on let me treat the easiest example Let us take G C endowed with the inner class of real forms consisting of the split form so that GR R and choose F LANGLANDS PHILOSOPHY AND KOSZUL DUALITY Then S will consist of the single element the covering G G F F will b e the identity C C and X will consist just of two p oints for every Max Z We nd that Ext X Ext two p oints G C F Ext p oint Ext p oint C C H B C C H B C C H P C C H P C C C t C t where B C P C is the classifying space of C and deg t On the other hand MR is just the category of continuous nite dimensional representations of R I leave it to the reader to check for every Max Z the equivalence C t C t N il MR This proves our conjecture in this sp ecial case The geometry of AdamsBarbaschVogan I promised at the b eginning that given a complex connected reductive algebraic group G endowed with an inner class of real forms and a nite central subgroup F I would construct a multiset S of real forms a nite con F nected covering G of the dual group and a collection fX g MaxZ F of G varieties To fullll this promise I have to make some choices In ABV the authors investigate in detail how these choices aect the nal outcome Let us choose an antiholomorphic involution G G in the given inner class that stabilizes a Borel subgroup Then form j j the real Lie group G G GC if we wouldnt abuse notation where Gal C R f g acts on G via this an tiholomorphic involution For any G G with Z G the conjugation int G G is an antiholomorphic involution inner to Let S G G b e a system of representatives for F the Gconjugacy classes of G G with F As explained we may consider S as a multiset of antiholomorphic involutions F G of G For example S H I warn the reader that for a complex algebraic group I quite often write just G for the complex p oints of G to b e denoted GC in more careful notation I hop e this do es not lead to confusion Let us choose in addition T B G a stable maximal torus and
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