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Koszul Duality Patterns in Representation Theory JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Number 2, April 1996 KOSZUL DUALITY PATTERNS IN REPRESENTATION THEORY ALEXANDER BEILINSON, VICTOR GINZBURG, AND WOLFGANG SOERGEL Contents 1. Introduction 473 2. Koszul rings 479 3. Parabolic-singular duality and Koszul duality 496 4. Koszul rings arising from ‘mixed’ geometry 513 References 525 1. Introduction 1.1. Parabolic-singular duality. Let g b h be a complex semisimple Lie algebra, a Borel subalgebra and a Cartan subalgebra⊃ ⊃ contained in it, respectively. Let U = U(g) be the universal enveloping algebra of g. Let be the category of all finitely generated U-modules which are locally finite overOb and semisimple over h, see for example [BG80]. For λ h∗ consider in the Verma module ∈ O M(λ)=U U(b) Cλ, its simple quotient L(λ) and the (indecomposable) projective cover P (λ)of⊗ L(λ)in . Let Z U be theO center. Let L be the direct sum of all simple highest weight modules⊂ L(λ) which have trivial∈O infinitesimal character, i.e. which have the same annihilator in Z as the trivial representation C = L(0). There are but finitely many of those, parametrized by the Weyl group. Let P be the direct sum of their respective projective covers. The first statement of∈O the following theorem is the main result of [Soe90]. Theorem 1.1.1. There exists an isomorphism of finite dimensional C-algebras End P = Ext• (L, L), O ∼ O i where Ext• stands for the direct sum of all Ext , made into a ring via the cup product. Furthermore, Ext• (L, L) is a Koszul ring. O We have to explain what a Koszul ring is. Received by the editors November 13, 1991 and, in revised form, February 16, 1995. 1991 Mathematics Subject Classification. Primary 17B10; Secondary 16A03. The first author was partially supported by an NSF grant. The second author thanks Harvard University and MIT, where part of this work was written. The third author thanks the MPI and DFG for financial support. c 1996 American Mathematical Society 473 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 474 A. BEILINSON, V. GINZBURG, AND W. SOERGEL Definition 1.1.2. A Koszul ring is a positively graded ring A = j 0 Aj such ≥ that (1) A0 is semisimple and (2) A0 considered as a graded left A-module admits a graded projective resolution L 2 1 0 P P P A0 ···→ → → i i i such that P is generated by its degree i component, i.e., P = APi . Morally a Koszul ring is a graded ring that is “as close to semisimple as a Z-graded ring possibly can be”. This statement will be made more precise by Proposition 2.1.3. We explain next the meaning of the ring E =End P. If is an abelian category whose objects have finite length and P a projectiveO A generator of (i.e. a projective object surjecting onto each simple∈A object in ), then there is aA canonical standard (see, e.g., [Bas68], the exercise following TheoremA 1.3 from Chapter II) equivalence of categories Hom (P, ): Mof-E, E =End P, A A→ A where Mof- stands for the category of finitely generated right modules. Now let + Z =AnnZC Zbe the augmentation ideal of the center. Apply the considera- ⊂ + n tions above to the category 0 = M (Z ) M=0forn 0 .Thusweget an equivalence of categoriesO { ∈O| } 0 = Mof-(End P ). O ∼ O So the theorem says in particular that the category 0 is governed by a Koszul ring, i.e. that it is in some sense “graded semisimple”.O We hope to convince our readers that this is a good way to formalize the beauty of this part of representation theory, and we believe that similar results should hold for all kinds of representation theories associated to root systems. As a first step we explain how to generalize the above to other infinitesimal characters. Let ρ h∗ be the half sum of positive roots, i.e. of the roots in b. Let ( , ) be the Coxeter∈ system of g b and let w be the longest element of the W S ⊃ ◦ Weyl group . For w ,λ h∗ we put w λ = w(λ + ρ) ρ. For any subset W ∈W ∈ · − ι —here ι is a variable index—let ι be the subgroup generated by ι, S ⊂S ι W ⊂W S wι ι its longest element and the set of longest representatives of the ∈W W ⊂W cosets / ι. W W For λ h∗ integral dominant but perhaps singular (in other words, λ is integral and λ + ∈ρ lies in the closure of the dominant Weyl chamber) define the category λ to consist of all objects of whose (generalized) infinitesimal character O ⊂O O coincides with the one of L(λ). Set λ = s s λ=λ .Then L(x λ) x λ S { ∈S| · } { · } ∈W represents the isomorphism classes of simple objects in λ. The P (x λ)arethe O · projective covers of L(x λ)in λ as well. · O q For q g a parabolic subalgebra containing b define the subcategory 0 ⊂ O ⊂O to consist of all q-locally finite objects. Let q be the simple reflections q Sq ⊂S 1 q q corresponding to q. For all x we set Lx = L(x− w 0) .The Lx ∈W q ◦ · q ∈O represent the isomorphism classes of simple objects in . Any Lx has a projective q q O cover Px in . We will proveO in section 3 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use KOSZUL DUALITY PATTERNS 475 Theorem 1.1.3. Let λ, q be as above and suppose λ = q. Then there are iso- S S morphisms of finite dimensional C-algebras q q End λ ( P (x λ)) = Ext• q ( Lx, Lx), O q · ∼ O q End ( Px ) = Ext• λ ( L(x λ), L(x λ)), O L ∼ O L L· · λ q where the sums go overL x = . Both algebrasL on theL right are Koszul rings and, moreover, these rings∈W are KoszulW dual to one another. Remarks. (1) We have to explain when two Koszul rings are dual to one another. For every positively graded ring A we may consider A0 as a left A-module A0 = A/A>0 and form the positively graded ring E(A)=ExtA•(A0,A0). For a Koszul ring A satisfying a suitable finiteness condition we prove as Theorem 2.10.2 that E(A) is Koszul as well and E(E(A)) = A canonically. Motivated by this equality, we call E(A) the dual of A. b (2) Remark that = 0. So the theorem tells us in particular that the Koszul ring from TheoremO O 1.1.1 is selfdual. q λ (3) All sums go over x = .Let 1x be the obvious pairwise orthogonal idempotents in our four∈W algebras.W The isomorphisms of the theorem can (and will) be chosen so that the 1x correspond to each other under the isomorphisms above. (4) From what we explained before it is clear that the endomorphism algebras q of projectives appearing in the theorem govern the categories λ and respectively. O O (5) The projective covers appearing in the theorem are unique only up to non- unique isomorphism, hence we cannot expect the isomorphisms predicted by the theorem to be canonical. We can however construct without choices a collection of possible P (x λ) along with a canonical isomorphism as in the theorem for this special collection.· q (6) Morally we should rather consider categories λ and for Langlands-dual O O Lie algebras. In fact one might then view λ as a category of Harish-Chandra modules for a complex group and q as aO category of perverse sheaves on the corresponding space of LanglandsO parameters, more precisely its modified version defined by Adams-Barbasch-Vogan [ABV92]. Our theorem can then be understood as a refinement of Langlands’ philosophy in the case of complex groups. More details on this point of view can be found in [Soe92]. This article is the result of our efforts to join the two partially wrong and quite incomplete preprints [Soe89b] and [BG86] into one readable article. First we approach the above results in as elementary a way as possible. The only facts relying on “mixed” geometry we really use are the computation of the intersection cohomology of Schubert varieties and Lemma 3.5.4. In the last section we adopt the more powerful technique of mixed geometry. 1.2. Koszul duality. The results described here and proved in section 2 are essen- tially not new but rather generalizations of results of Priddy [Pri70], L¨ofwall [L¨of86], Backelin [Bac82], Backelin-Fr¨oberg [BF85], Bernstein-Gelfand-Gelfand [BGG78] and specializations of results from [BGS88]. We start by repeating our definition of a Koszul ring. Definition 1.2.1. A Koszul ring is a positively graded ring A = j 0 Aj such that (1) A is semisimple and (2) A considered as a graded left A-module≥ admits 0 0 L License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 476 A. BEILINSON, V. GINZBURG, AND W. SOERGEL a graded projective resolution 2 1 0 P P P A0 ···→ → → i i i such that P is generated by its component of degree i, P = APi . Remarks. (1) A complex of graded A-modules 2 1 0 P P P A0 ···→ → → as in the definition is unique up to unique isomorphism and has a natural right A0-action.
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