Koszul Duality for Parabolic and Singular Category O 1

Home , Category O

REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 3, Pages 139–152 (July 19, 1999) S 1088-4165(99)00055-2

KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY O

ERIK BACKELIN

Abstract. This paper deals with a generalization of the “Koszul duality theorem” for the Bernstein-Gelfand-Gelfand category over a complex semi- simple Lie-algebra, established by Beilinson, GinzburgO and Soergel in Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), 473– 527. In that paper it was proved that any “block” in , determined by an integral, but possibly singular weight, is Koszul (i.e. equivalentO to the category of ﬁnitely generated modules over some Koszul ring) and, moreover, that the “Koszul dual” of such a block is isomorphic to a “parabolic subcategory” of the trivial block in . We extend theseO results to prove that a parabolic subcategory of an integral and (possibly) singular block in is Koszul and we also calculate the Koszul dual of such a category. O

1. Introduction (Cf. the subsection about category in section 2.) Let g b h be a complex O ⊃ ⊃ semi-simple Lie-algebra, a Borel subalgebra and a Cartan subalgebra. Let φ, ψ h∗ ∈ be integral and dominant. Let φ be the block in the category deﬁned in [BGG1] corresponding to the weight φO.Letq(ψ) bbe the parabolicO subalgebra of g ψ ⊃ corresponding to ψ and denote by φ the subcategory of φ whose objects are O ψ O locally ﬁnite over q(ψ). Denote by Rφ the endomorphism ring of a (minimal) ψ projective generator of φ . The conclusion of this paper is the parabolic and singular-singular and parabolicO Koszul duality:

ψ ψ ! Theorem 1.1. The ring Rφ is Koszul. Its Koszul dual (Rφ ) is isomorphic to φ R w ψ. − 0 φ This theorem generalizes the main result of [BGS], Theorem 1.1.3, where R0 = 0 ! ∼ (Rφ) is proved. My proof depends heavily on their results but may have the ad- vantage that I am (once an isomorphism ( 0)! 0 is chosen) able to construct R0 ∼= R0 the isomorphism and not just prove its existence as in [BGS]. Actually, this isomorphism is induced by “translation onto the wall”. The methods used in this paper are elementary except for the proof of Proposi- tion 3.5 where some theory of perverse sheaves is needed. At the end of section 2 we consider as an example — in order to give the non- expert reader a feeling for these things — the sl(2, C)case.

Received by the editors August 24, 1998 and, in revised form, January 31, 1999. 1991 Mathematics Subject Classiﬁcation. Primary 17B10, 18G15, 17B20.

c 1999 American Mathematical Society

139 140 ERIK BACKELIN

Motivation. Let me start by mentioning some applications of the Koszul duality of [BGS]. First of all, the fact that 0 is Koszul and selfdual implies ([BGS]) that the Kazhdan-Lusztig conjecture, whichO is one of the deepest results in representation theory, holds for this category. However, there is no way to prove the Koszulity of 0 without using the Kazhdan-Lusztig conjectures. One could think of the KoszulityO property as a strengthening of the Kazhdan-Lusztig conjectures. The Koszul duality theorem for φ enables us to solve some questions working q(φO) with the Koszul dual category 0 . A famous theorem of Soergel ([Soe2]) states that the ext groups between a VermaO module and a simple module vanish in every second degree. From the standpoint of geometry, which applies to q(φ) (but not O0 to φ)—via -modules on the flag variety of G/Q,whereGis the Lie-group correspondingO toDg and Q is the subgroup with LieQ = q — it is not so hard q(φ) to show that this theorem actually holds for 0 . One then deduces Soergel’s theorem from the Koszul duality theorem. O A possible important application of the Koszul duality theorem in this paper is the following: There exists an equivalence between the category of finite- dimensional representations of a quantum group at a root of unity and a parabolic category of the corresponding affine Kac-Moody algebra; cf. [KL1], [KL2]. So in order to establishO a singular-parabolic duality for the category of finite-dimensional representations of the quantum group we need a parabolic and singular-singular and parabolic duality for of the Kac-Moody algebra. Theorem 1.1 indicates that such a duality should exist.O

2. Review of category and Koszul rings O Koszul rings. We refer to [BGS] for a detailed review of the theory of Koszul rings. Here we just recall

Definition 2.1. Let R = i 0 Ri be a positively graded ring with R0 semi-simple and put = . is≥ called Koszul if the right module admits R+ i>0 Ri R R0 ∼= R/R+ L i a graded projective resolution P • R0 such that P is generated by its component i L Pi in degree i. ! For a positively graded ring R put R := ExtR(R0,R0). This has the structure ! i of a graded ring, with (R )i =ExtR(R0,R0) and multiplication given by the cup- ! i product. If R is Koszul, we call R the Koszul dual of R.DenotebyextR(,) extensions in the category of graded (left) R-modules and for a graded R-module M andanintegerjdefine the graded R-module M j by M j i = Mi j .Letus h i h i − say that a graded R-module M is pure of weight i if M = M i. − Proposition 2.2. Let R be a positively graded ring with R0 semi-simple. Then i R is Koszul iff extR(R0,R0 j )=0whenever i = j.IfRis Koszul, then R is generated in degree 1 over itsh degreei 0 part and has6 relations only in degree 2.Then ! ! ! also R is Koszul and, if in addition R1 is right finitely generated over R0, (R ) ∼= R canonically. Remark 2.3. (i) Each ring R considered in this paper is finite-dimensional over C so the condition R1 is right finitely generated over R0 is automatically fulfilled. ! opp (ii) The standard definition of R is ExtR(R0,R0) . However, the category defined below admits a duality (e.g. [BGG1]) which restricts to a duality on eachO subcategory of we shall consider. This implies that the endomorphism ring of O KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY 141 O a projective generator in such a category is isomorphic to its own opposite ring and each ring R in this paper is either an endomorphism ring of this type or the opp ! ext-algebra of such a ring, hence R ∼= R. Thus, our definition of R coincides with the standard definition. Anyway, Proposition 2.2 holds with our definition of R! as well as with the standard definition. Aconvention.Each ring R under consideration in this paper will be indexed by φ a pair of weights in h∗,e.g. R=R. In order to not confuse the index 0 N 0 ∈ referring to grading degree with index 0 h∗ referring to weight we shall in the ∈ φ following always write (R)i when referring to grading degree i N,e.g.(R )0. ∈ 0 Category . Let g b h be a semi-simple Lie-algebra, a Borel subalgebra and a Cartan subalgebraO and⊃ let⊃ U(g) be the enveloping algebra of g.LetRbe the root- system, R+ the positive roots, B R+ the basis, R := R+ and let (W, S)bethe corresponding Coxeter-system and⊂ let w W be the− longest− element. Let ρ denote 0 ∈ the half-sum of the positive roots; for φ h∗,w W we put w φ := w(φ + ρ) ρ. Denote by H the coroot corresponding∈ to α R∈. · − α ∈ In this paper each weight φ h∗ is assumed to be integral, i.e. φ(Hα) Z, α ∈ ∈ ∀ ∈ B. φ is called dominant if φ(Hα) 1, α Band anti-dominant if φ(Hα) 1, α B.DenotebyL(φ) the≥− simple∀ module∈ with highest weight φ.(This≤ − ∀ ∈ terminology differs from that of [BGG1]; they denote by Lφ the simple module of highest weight φ ρ.) Let be the− category of all finitely generated U(g)-modules which are locally finite overO b and semi-simple over h. Each object in has finite length and has enough projectives, e.g. [BGG1]; denote by P (φ) a projectiveO cover of L(φ)inO . Let be the subcategory, so called block, of whose objects have compositionO Oφ O factors isomorphic to L(w φ),w W. The simple objects in φ are parametrized · ∈ O φ by the cosets W/Wφ,whereWφ := w W ; w φ = φ .DenotebyW the set of longest representatives of the elements{ ∈ in W/W· .Put}S =S W. We refer to φ φ ∩ φ (W, S, Sφ) as the Coxeter-system of (g, b,φ). A block in category depends only on this Coxeter-system: O Proposition 2.4 ([Soe1], Theorem 11). 1 Assume we are given a Coxeter-system (W, S, S ) where φ is any dominant weight. Let g0 b0 h0 be another semi- φ ⊃ ⊃ simple Lie-algebra, a Borel subalgebra and a Cartan subalgebra and let φ0 h0∗ ∈ be any dominant weight. Then any isomorphism (W, S, Sφ)˜(W0,S0,S0 ),w w0 → φ0 → induces an equivalence of categories (g, b)˜ (g0, b0) with L(w φ) L(w0 φ0). Oφ →Oφ0 · → · Let φ, ψ h∗ be dominant weights. ψ defines a set of singular simple roots ∈ Bψ := α B; ψ(Hα)= 1, and hence the parabolic subalgebra q(ψ) b, { ∈ −} α ψ ⊇ generated by b and the weight-spaces g− for α B .Let be the subcategory ∈ ψ Oφ of consisting of locally q(ψ)-finite objects. Oφ φ Translation functors. Recall the two translation functors T0 : 0 φ and T 0 : , translation onto and out of the wall, respectively,O e.g.→O [Jan]. By φ Oφ →O0 definition T φ = pr (E ), where E is an irreducible finite-dimensional g-module 0 φ ⊗C · 1This proposition actually holds for non-integral weights, with W replaced by the integral Weyl group of φ, replaced by... etc; so all results in this paper generalize to the non-integral Oφ case. 142 ERIK BACKELIN with extremal weight φ and pr is projection from onto the block ; T 0 = φ O Oφ φ pr0(F C ), where F is an irreducible finite-dimensional g-module with extremal weight⊗ φ,· pr is projection onto . They satisfy the following properties: − 0 O0 φ 0 Lemma 2.5. 1) T0 and Tφ are exact functors adjoint to each other. There is an φ 0 isomorphism of functors id φ ... id φ = T0 Tφ (card(Wφ) number of copies). O ⊕ ⊕ O ∼ ◦ 2) For each simple in there is exactly one simple in mapped to it by T φ. Oφ O0 0 More precisely, T φL(w 0) = L(w φ),ifw Wφ, else T φL(w 0) = 0. 0 · ∼ · ∈ 0 · 3) T 0 and T φ map projectives to projectives. Moreover, T 0P (w φ) = P (w 0). φ 0 φ · ∼ · 4) T φ and T 0 induce functors T φ : ψ ψ and T 0 : ψ ψ. Moreover, if 0 φ 0 O0 →Oφ φ Oφ →O0 T φL(w 0) =0(hence = L(w φ)) and belongs to ψ,thenL(w 0) ψ. 0 · 6 ∼ · Oφ · ∈O0 φ 0 Proof. 1) The exactness of T0 and Tφ follows from the fact that E and F are flat modules over C and that projecting to a block in is an exact functor. Next, note O that F ∼= HomC(E,C)withtheg-module structure given by (xφ)(e):= φ(xe)for x g,φ Hom (E,C),e E. Then, for M ,N ,wehave − ∈ ∈ C ∈ ∈O0 ∈Oφ φ Hom (T0 M,N)=Hom (prφ(E C M),N)=Hom (E C M,N) O O ⊗ O ⊗ 0 =Hom (M,F C N)=Hom (M,pr0(F C N)) = Hom (M,TφN). O ⊗ O ⊗ O φ 0 φ This means that T0 is left adjoint to Tφ . A similar argument shows that T0 is also 0 right adjoint to Tφ . The last statement follows from [Jan], 4.13 (2), and [BG], Theorem 3.3 2) is proved in [Jan], 4.12 (3). 0 φ 3) Let P φ be projective. We have Homg(Tφ P, )=Homg(P, T0 ( )). Since ∈O φ 0 P is projective and T0 is exact, we conclude that this functor is exact; hence Tφ P φ is projective. Analogously, T0 maps projectives to projectives. Next, from 2) we get 0 φ Homg(T P (w φ),L(x φ)) = Homg(P (w φ),T L(x φ)) = δ C φ · · · 0 · ∼ w,x for each x W .SinceweknowthatT0P(w φ) is projective, this formula implies ∈ φ · that T 0P (w φ) = P (w 0). φ · ∼ · For 4) we note that tensor product with a finite-dimensional module clearly preserves locally finiteness over q(ψ); hence we get the asserted induced functors. Assume 0 = T φL(w 0). Then 6 0 · 0 φ φ φ Hom (L(w 0),TφT0 L(w 0)) = Hom (T0 L(w 0),T0 L(w 0)) =0, O · · O · · 6 0 φ which implies that L(w 0) is isomorphic to a submodule of Tφ T0 L(w 0). Now, if φ · 0 φ · T0 L(w 0) is locally finite over q(ψ), then Tφ T0 L(w 0), and hence also L(w 0), have this· property. · · Parabolic inclusion and truncation. Let i := iψ : ψ be the inclusion. φ Oφ →Oφ We shall refer to this exact functor i as the parabolic inclusion functor. We define the parabolic truncation functor τ := τ ψ : ψ by τ (M ) := “the maximal φ Oφ →Oφ quotient of M which is locally finite over q(ψ)”. Then τ is right exact and left adjoint to i and it follows that τ maps projectives to projectives. Since a projective object is indecomposable if and only if it has a unique simple quotient, we see that τ maps indecomposable projectives to indecomposable projectives. (τ doesn’t KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY 143 O map arbitrary indecomposables to indecomposables; a counterexample can be given for sp(4, C).) It follows readily that τ maps a minimal projective generator to a minimal projective generator. It is also easy to verify that if P, Q are ∈Oφ ψ projective, then τ :Hom φ(P, Q) Hom (τP,τQ) is surjective. O → Oφ Lemma 2.6. a) i and τ commutes with translation functors. b) The functor Db( ψ) Db( ) induced by i is faithful. Oφ → Oφ Proof. a) Clearly i commutes with translations functor: we have for each M ∈Oφ and N ψ that ∈Oφ 0 0 ψ Hom (τTφM,N)=Hom φ(TφM,iN) Oφ O φ φ =Hom φ(M,T0 iN)=Hom φ(M,iT0 N) O O φ 0 =Hom ψ(τM,T0 N)=Hom ψ(TφτM,N) Oφ Oφ

0 0 φ φ so that τTφ = Tφτ. Analogously τT0 = T0 τ. b) The case when φ = 0 is proved by geometry in [BGS], Theorem 3.5.3. We have a commutative diagram of bounded derived categories:

i0 / b ψ / b

D ( 0 ) D ( 0)

O O O O 0 0 Tφ Tφ iψ

ψ φ / Db( ) / Db( ) Oφ Oφ

φ 0 Since T0 Tφ is isomorphic to card(Wφ) copies of the identity functor we see that 0 ◦ ψ Tφ is faithful on derived categories and we conclude that also iφ is faithful.

Endomorphism rings, ext-algebras and canonicity. The fact that projective covers are unique only up to non-unique isomorphism makes our constructions non-canonical and causes delicate problems with commutative diagrams. We can avoid these problems to some extent by choosing projective covers and maps between them commuting with various functors. Fix dominant weights φ and ψ.Fix projective covers P (w φ)ofL(w φ)in φ and put Pφ := P(w φ). Anal- · · O w W/Wφ · ogously, fix P (w ψ)andP and fix P (w 0) and P . According∈ to Lemma 2.5 3) ψ 0 P we fix for φ· isomorphisms 0 ( ·) ( 0). We get the (now) canonical w W Tφ P w φ ∼= P w ∈ 0 0 · 0 · isomorphism P0 = Tφ Pφ (Tφ Pφ)⊥ where (Tφ Pφ)⊥ = w W W φ P (w 0). ∼ ⊕ ∈ \ · It follows from the results in the previous paragraph that P ψ(w 0) := τ ψP (w 0) L · · is a projective cover of L(w 0) in ψ,ifL(w 0) ψ and τ ψP (w 0) = 0 else. Put · O0 · ∈O0 · P ψ = τ ψP which is a (minimal) projective generator of ψ. Define analogously 0 0 O0 P ψ(w φ):=τψP(w φ)andPψ := τ ψP . We get the canonical isomorphism φ · φ · φ φ φ ψ 0 ψ ( 0 ψ) where ( 0 ψ) = ψ( 0 ) .Put = ( 0) P0 ∼= Tφ Pφ Tφ Pφ ⊥ Tφ Pφ ⊥ τφ Tφ Pφ ⊥ L0 w WLw etc. ⊕ ∈ · L ψ ψ 0 Definition 2.7. Let Rφ := Endg(Pφ ) and write for simplicity Rφ := Rφ.Next, φ φ consider the ring Ext φ (Lψ,Lψ). This is a canonically graded ring with degree i Oψ 144 ERIK BACKELIN

i φ φ part Ext φ (Lψ,Lψ) and multiplication given by the cup-product. In [BGS], The- Oψ φ φ orem 1.1.3, it is proved that Ext φ (L ,L ) is Koszul and they also prove the exis- 0 0 0 O φ φ tence of an isomorphism between Ext φ (L ,L )andRφ. Under this isomorphism 0 0 0 φ φ O is Hom φ (L ,L ) mapped onto the subring of Rφ spanned by projections onto 0 0 0 indecomposableO projectives. We fix such an isomorphism. This defines a grading on Rφ making it a Koszul ring. The parabolic truncation functor gives in Proposi- ψ ψ ψ tion 3.2 a graded surjection Rφ Rφ defining a grading (Rφ )i on Rφ . With this grading we define a (canonically graded) algebra.

ψ ! ψ ψ Definition 2.8. (R ) := Ext ψ ((R )0, (R )0). φ Rφ φ φ Since each object in has finite length, it is known and easy to verify that ψ O ψ the functor Homg(Pφ , ) induces an equivalence of categories between φ and the · ψ ψ O category of finitely generated right Rφ -modules. Now clearly Lφ is mapped to the right ψ-module ( ψ) ψ ( ψ) under this equivalence; this gives the ring Rφ Rφ 0 ∼= Rφ / Rφ + isomorphism ψ ! ψ ψ ψ ψ (2.1) (Rφ ) =ExtRψ((Rφ )0, (Rφ )0) = Ext ψ (Lφ ,Lφ). φ ∼ Oφ We have the surjection ψ ψ (2.2) τφ : Rφ Rφ .

ψ 0 ψ 0 ψ 0 ψ The composition End ψ (Pφ ) End ψ (Tφ Pφ ) End ψ (Tφ Pφ Tφ Pφ )⊥)= Oφ → O0 → O ⊕ ψ End ψ (P0 ) gives the injection O0 (2.3) T 0 : Rψ , Rψ. φ φ → 0 Similary, we get a map (which by Lemma 2.6 is injective) (2.4) iψ :(Rψ)! , (R )!. φ φ → φ By Lemma 2.5 φ ψ ψ. Fix such an isomorphism and ﬁx an analogous T0 L0 ∼= Lφ isomorphism when ψ is replaced by 0 such that these two isomorphisms commute with parabolic inclusion. We get from (2.1) the ring homomorphism (2.5) T φ :(Rψ)! (Rψ)! 0 0 → φ which will be shown to be a surjection in Lemma 3.4 below.

The easiest example. Let g = sl(2,C), X ,X+ and H the standard Cheval- ley generators. has up to isomorphism two− simple objects: L(0) = C and O0 L( 2) = U(g)/(U(g)(H +2)+U(g)X+). Then P (0) := U(g)/(U(g)H + U(g)X+) − 2 and P ( 2) := U(g)/(U(g)(H +2)+U(g)X+) is a projective cover of L(0) and L( 2ρ)in− , respectively. Put P := P (0) P ( 2). This is a projective generator − O0 ⊕ − of 0.PutR=Endg(P). Then R is a five-dimensional algebra over C.Denoteby 1¯ theO image of 1 U(g)inP(0) and in P ( 2), respectively. A vector space basis of R is then given∈ by: π := id for i =0−, 2; α: P (0) P ( 2) : 1)¯ X 1;¯ i P (i) − → − → + β : P ( 2) P (0) : 1¯ X 1and¯ γ:P(0) P (0) : 1¯ X X+1.¯ − → → − → → − This makes R = R[0] R[1] R[2] a graded algebra with R[0] := π0,π 2 , R[1] = α, β and R[2] :=⊕ γ .⊕ This defines also an R-module structure{ on −R[0]} { } { } KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY 145 O by letting R[1] R[2] act trivially. The Koszul dual of the graded ring R is by ! ⊕ definition R := ExtR(R[0],R[0]). Put L := L(0) L( 2) and E := Ext (L, L). Let E[i]:=Exti (L, L), then E = E[0] E[1] ⊕E[2]− is a graded algebraO with multiplication givenO by the cup product. ⊕ ⊕ Now, E[0] has basis j0,j 2 corresponding to the identity on L(0),L( 2). E[1] − − has basis corresponding to the extension a: L( 2) , M(0) L(0), where M(0) = U(g)/(U(g)H+U(g)X ) is the Verma module of− highest→ weight zero, and b: L(0) , + → M(0)∗ L( 2), where M(0)∗ is the dual of M(0) ([BGG1]) E[2] has basis c := − ba Ext2 (L(0),L(0)). ∈ O ! As we have seen before E ∼= R . The Koszul duality theorem in our case then states that R ∼= E. Under this isomorphism, π0 j0,π 2 j 2,α a, β b and γ c. → − → − → → → 3. Parabolic and singular-singular and parabolic Koszul duality We have our fixed dominant weights φ and ψ. As mentioned before the set of isomorphism classes of simple objects in φ, which we denote by Irr φ, is naturally φ O O parametrized by W w L(w φ) Irr φ.Notethatw 0(Hα) 0, α φ 13 ↔ · ∈ O · ≥ ∀ ∈ Bφ w ( W ) − w 0 ; this means precisely that L(w 0) is locally finite over q(φ) ⇐⇒ ∈ φ 1 · φ 1 if and only if w (W )− w0. This gives the bijection W w L(w− w0 0) φ ∈ 3 ↔ · ∈ Irr 0 . O φ φ The projection Pφ P (w φ) defines an idempotent ew (Rφ)0, w W . φ · ∈ φ φ∈! Similary, the projection L0 L(w 0) defines an idempotent fw ((R0 ) )0 = φ φ 1 · ∈ φ − Hom φ (L0 ,L0), for w (W ) w0. TheO following theorem∈ was proved in [BGS], Theorem 1.1.3. (Their proof is partly reconstructed in our proof of Lemma 3.5.) φ Proposition 3.1. The rings Rφ and R0 are Koszul and there is an isomorphism φ ! φ φ φ Rφ ˜ (R0 ) such that ew f 1 , w W . → → w− w0 ∈ φ φ φ ! φ Clearly, the f 1 , w W formsabasisof((R ))0 over C. Hence the e , w− w0 0 w φ ∈ w W formsabasisof(Rφ)0, by the choice of grading on Rφ coming from the ∈ φ ! grading of (R0 ) and the above isomorphism. ψ Proposition 3.2. The ring Rφ is Koszul.

Proof. Put Ω := w W ; α Bψ : w φ(Hα) < 0 and let I Rφ be the ideal { φ ∈ ∃ ∈ · } ψ ⊂ generated by the ew for w Ω. We have (where mod-Rφ means ﬁnitely generated ψ ∈ right Rφ -modules) mod-Rψ = ψ = M :[M:L(w φ)] = 0, w Ω φ ∼ Oφ { ∈Oφ · ∀ ∈ } = M :Homg(P(w φ),M)=0, w Ω =mod-R /I. { ∈Oφ · ∀ ∈ }∼ φ ψ Since Rφ and Rφ/I are basic algebras, we now conclude that they are isomorphic. ψ By Proposition 3.1 Rφ is Koszul and we conclude that Rφ is a graded quotient ψ of Rφ with respect to the Koszul grading of Rφ. To show that Rφ is Koszul we i ψ ψ just have to prove (by Proposition 2.2) that ext ψ ((Rφ )0, (Rφ )0 j )=0when Rφ h i i=j. We have the morphism Db(mod-Rψ) Db(mod-R ) corresponding to 6 φ → φ 146 ERIK BACKELIN iψ : Db( ψ) Db( ), which is faithful by Lemma 2.6. Since this morphism is φ Oφ → Oφ also induced by a graded ring homomorphism R Rψ, we conclude that it gives φ → φ injections exti (( ψ) ( ψ) ) exti (( ) ( ) ). The Koszulity of ψ Rφ 0, Rφ 0 j , Rφ Rφ 0, Rφ 0 j Rφ Rφ h i → h i ensures that the latter expression vanishes for i = j. 6

The modular representation theoretic analogy of the following key lemma was proved in [AJS].

φ Lemma 3.3. Let L and L0 be two simples in 0.ThenT0 induces a surjection 1 1 φ φ O Ext (L, L0) Ext (T0 L, T0 L0). O0 → Oφ 0 φ Proof. (a)Putν=TφT0. Clearly the assertion of the lemma is equivalent to show- ing that the homomorphism Ext1 ( ) Ext1 ( ) given by the canonical 0 L, L0 φ L, νL0 O → O φ adjunction id ν is surjective. We assume without restriction that T L0 =0; → 0 6 then the morphism L0 νL0 gives an embedding which identifies L0 with soc(νL0) → as follows from the formula (for any simple L00 ) ∈O0 φ φ Homg(L00,νL0)=Homg(T L00,T L0)=δ C. 0 0 L00,L0 · We want to define a grading on ν and for this purpose we must pass to derived categories and modules over the coinvariant algebra. The discussion below should rather have been presented as a part of a comprehensive exposition of the theory. (b)PutS=S(h∗); we define a grading on S by putting h∗ in degree 1. Its graded quotient C := S/S SW is called the coinvariant algebra. Let Cφ := CWφ C + · ⊂ be the graded subalgebra of Wφ invariants. In [Soe1], Endomorphismensatz 7, an φ isomorphism C ˜ End (P (w0 φ)) of rings is constructed. → Oφ · Let Proj( φ) be the category of projective objects in φ. Define the functor O Oφ Vφ : Q Hom φ (P (w0 φ),Q) from Proj( φ)tomod-C := the category of → O · O right modules over Cφ (by means of the above ring isomorphism). It is proved in [Soe1], Struktursatz 9, that Vφ is fully faithful so that Vφ induces an equivalence of categories between Proj( φ) and its image category Im(Vφ). (c) Given a category ofO some modules over a graded ring R, we define its graded A version to be the category of those graded modules over R whose underlying R- A modules belong to . There is the grading-forgetting functor for : . Assume A A→A in additione we are given the data ( , , for). We say that a functor F : is B B A→B a grading of the functor F : if for F = F for. e Consider the grading of RA→Bgivene by Proposition◦ ◦ 3.1 and identify e ewithe the φ Oφ category of finitely generated right modulese over R ; we get graded categories φ Oφ φ φ and Proj^( φ) φ and also mod^ -C and Im^(Vφ) mod-^C . O ⊂O ⊂ f The eqivalence Vφ : Proj( φ)˜Im(Vφ) induces (by definition!) an equivalence O → f Proj^( φ)˜Im^(Vφ), which in turn induces an equivalence between homotopy cate- O → b b b gories K (Proj^( φ)) ˜ K (Im^(Vφ)). Remember that K (Proj^( φ)) is canonically O → O isomorphic to the bounded derived category Db( ); we get the equivalence Oφ b b f (3.1) Vφ : D ( φ)˜K (Im^(Vφ)). O → f KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY 147 O (d) There are the standard functors

φ resC :mod-C M MCφ mod-Cφ, C 3 → | ∈ C φ ind φ :mod-C M C φ M mod-C. C 3 → ⊗C ∈ φ ^Cφ ^C The fact that C is a graded subalgebra enables us to define resC and indCφ in ^Cφ such a way that they become adjoint to each other and resC preserves pure objects of any weight. φ Cφ It is proved in [Soe1], Theorem 10, that Vφ T0 = resC V0 : Proj( 0) 0 C ◦ ◦ O → Im(Vφ)andthatV0 Tφ =indCφ Vφ : φ Im(V0). ◦ ◦ O → 0 b Thus formula (3.1) gives us (abusing some notation) functors Tφ : D ( φ) b φ b b O → D ( 0)andT0 :D( 0) D( φ). O φO → O e e However, since T0 is clearly t exact (with respect to the standard t-structures eb e b e e φ on D ( 0)andD ( φ)), it follows that also T0 is t exact. This gives us the graded O O0 φ translation functor Tφ : φ 0 and analogously we get T0 : 0 φ.These O → O φ e O → O latter two functors are adjoint and T0 preserves pure objects of any weight. We also get the graded functore e ν :=eT 0 T φ on . e e e φ ◦ 0 O0 (e) A simple module in can bee given the (unique up to a shift) grading of O0 being pure. Denote by L ande L0 thesimpleobjectsine e e 0 which are pure of weight 0O φ φ 0 satisfying for(L)=Land for(L0)=L0.SinceTφand T0 are adjoint and T0 e e e preserves pure objects of weight 0, we see that the image of L0 νL0 lives in degree e e e e → e 0. Note also that it follows from (a) that the morphism id ν identifies L0 with → soc(νL0). e ee Let i be maximal such that (νL0) =0.Then(νL0) is semi-simplee (sincee it i 6 i is annihilatedee by the radical of R0 when we identify the category 0 with some O modules over R ), hence (νL0) soce (νL0), hence i =0.e 0 i ⊆e e Since soc(νL0)=L0 is simple and (νL0) soc(νL0) is non-zero we now get 0 ⊆ (νL0) = L0. We concludee thate νL0/L0 elivese only in degrees 1. 0 ≤− (f) Consideree the shorte exact sequenceee ee e e e e e 0 Le0 νL0 νL0/L0 0. → → → → It induces the exact sequence 1 1 1 (3.2) Ext (L, L0) Ext (L, νL0) Ext (L, νL0/L0). O0 → O0 → O0 Now let Ext1 ( , ) denote the first left derived functor of the (grading-preserving) O0 hom-functor in the graded category 0;weget g O Ext1 (for(M), for(N)) = Ext1 (M,N i ) O0 f 0 h i i O ∈Z Y g for each M,N . Thus (3.2) is given by exact sequences ∈ O0 1 1 1 Ext (L, L0 i ) Ext (L, νL0 i ) Ext (L, νL0/L0 i ) O0f h i → O0 h i → O0 h i and we have to prove that the first map is surjective for every . g g g i 1 e e 1e eφe φ e eφe e We have Ext (L, νL0 i )=Ext (T0L, T0 L0 i )andT0L0 is pure of weight O 0 h i Oφ h i 0. Koszulity of implies that T φL admits a graded projective resolution whose gφ 0 g O e ee e e e e e 1 jth component is generated in degree j. It follows easily that Ext (L, νL0 i )=0 e e O0 h i g e ee 148 ERIK BACKELIN

1 1 if i = 1. Hence it suﬃces to show that Ext (L, L0 1 ) Ext (L, νL0 1 )is 6 0 h i → 0 h i surjective. O O This is true, because by Koszulity again L admitsg e e a graded projectiveg e resolutionee whose jth component is generated in degree j and since νL0/L0 lives in degrees < 0, 1 we get Ext (L, νL0/L0 i ) = 0 unless i>1.e In particular this group vanishes for 0 h i i =1. O ee e g e e e φe ψ ! ψ ! Lemma 3.4. T0 induces a surjective algebra homomorphism (R0 ) (Rφ ) . ψ ψ Proof. Recall from (2.1) that this means that the homomorphism Ext ψ (L ,L ) 0 0 0 ψ ψ φ φ ψO ψ → Ext ψ (L ,L ) induced from T (and our ﬁxed isomorphism T L = L )issur- φ φ φ 0 0 0 ∼ φ jective.O Surjectivity in degree 0 is obvious. ψ ψ The category 0 (resp., φ ) is closed under extensions in 0 (resp., in φ)soif we interpret LemmaO 3.3 asO surjections between Yoneda extensions,O the surjectivityO in degree 1 is clear. Since the second algebra in the statement of the lemma is Koszul, it is generated in degree 1 over its degree 0 part and we are done.

! φ ! Proposition 3.5. One can choose isomorphisms R0 ˜ (R0) and Rφ ˜ (R0 ) satis-

fying the properties of Proposition 3.1 which make the→ diagram below→ commute: / / !

R0 ∼ (R0)

Tφ i

? / ∼ / φ ! Rφ (R0 ) Proof. (a) We keep the notations of the proof of Lemma 3.3. We get from the

discussion in (d) in the proof of Lemma 3.3 the commutative diagram / V /

R0 EndC (VP0)

T 0 indC

φ Cφ

? Vφ / Rφ / EndCφ (VφPφ) (b)LetXbe a smooth complex variety equipped with an algebraic B-action, such that the B-orbits form a stratification of X, consisting of finitely many linear affine spaces. We denote by (X) the bounded derived category of C-constructible sheaves on X, whose cohomologiesD are constant on B-orbits. Let (X) (X) be the subcategory of perverse sheaves. It follows from the Riemann-HilbertP ⊂D corre- spondence that (X) is canonically isomorphic to Db( (X)), (e.g. [BGS], Corol- D P lary 3.3.2). For (X), the hypercohomology complex H( ):=RΓ(X; ) F∈Db F F is an object in D (C), the derived category of vector spaces. Let CX denote the constant sheaf on X. The canonical isomorphism = C gives us a ring- F ∼ X ⊗F homomorphism End• (X)(CX ) EndD• b(C)(H ). This defines an action of the D → F cohomology ring H•(X)=End•(X)(CX)onH ; hence, hypercohomology gives a D F functor H: (X) H•(X)-Mod. D → A (minimal) simple generator of the category (X)isgivenby IC•(Y )where P Y runs over the set of B-orbits on X and IC•(Y ) denotes the intersection cohomol- L ogy sheaf on Y (recall that Y is simply connected). KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY 149 O (c)LetGbe the group of inner automorphisms of g, B Q the subgroups such that LieB = b and LieQ = q(φ). We put := (G/B)⊂, Q := (G/Q), := Q D D D D Pφ (G/B)and := (G/Q). We fix isomorphisms C = H•(G/B)andC = P P P φ ∼ ∼ H•(G/Q) such that the inclusion C , C corresponds to the pullback morphism ? → π :End•(CG/Q) End• (CG/B), where π : G/B G/Q is the projection, (cf. [BGG2]).D → D →

Let 0(w 0) := IC•(BwB/B) and 0 := w W 0(w 0); analogously, put φ L · Q ∈P φ L ∈ φ L · (w 0) := IC•(BwQ/Q) and := φ (w 0). 0 0 w LW 0 L · ! ∈P L φ ! ∈ Lφ φ· Recall (R0) = Ext 0 (L0,L0)and(R0) =Ext φ (L0 ,L0) canonically. We have ∼ O ∼L 0 φ φ φ φ O Ext φ (L0 ,L0)=Ext Q ( 0 , 0 ), by [BGS], Theorem 3.5.1. Also, O0 ∼ P L L φ φ φ φ φ φ End• Q ( 0 , 0 ) = EndD• b( Q)( 0 , 0 ) = Ext Q ( 0 , 0 ). D L L ∼ P L L ∼ P L L φ ! φ φ ! Thus, ( ) End Q ( ). In particular, ( ) End ( ). By [Soe1], R0 ∼= • 0 , 0 R0 ∼= • 0, 0 Erweiterungssatz 17,D theL hypercohomologyL induces ring isomorphismsD L L

φ φ φ End• ( 0, 0) EndC (H 0, H 0)& End•Q( 0, 0) EndCφ (H φ, H 0 ). D L L → L L D L L → L L Consider the diagram (where d =dim G/B dim G/Q):

C − C

o / o !

EndC (H 0, H 0) End• ( 0, 0) ∼ (R0)

O O L L ∼ D L L O indC ?

Cφ π [d] i

? ?

o / φ o φ φ ∼ φ ! EndCφ (H 0 , H φ) End• Q ( 0 , 0 ) (R0 ) L L ∼ D L L The second square commutes by [BGS], Theorem 3.5.3. The first square commutes C ? by definition if we replace indCφ by Hπ [d]. However, in [Soe1], Theorem 14, an isomorphism of functors π?[d] = indC : Q C-Mod is constructed, so the H ∼ Cφ H diagram commutes as asserted. ◦ D → (d) Isomorphisms V0P0(w 0) ∼= H 0(w 0), w W ,ofC-modules are constructed in [Soe1], Proposition 10 and· LemmaL 9. · ∈ That proof can be modified to the parabolic case (an ad hoc proof is also given φ in [BGS], Lemma 3.7.2) which gives the isomorphisms VφPφ(w φ) = H (w 0), · ∼ L0 · w W φ,ofCφ-modules; we fix these latter isomorphisms. ∈Now, according to (d) in the proof of Proposition 3.5 and [Soe1], Theorem 14, there are isomorphisms indC P (w φ) = P (w 0) and indC φ(w 0) = Cφ Vφ φ ∼ V0 0 Cφ H 0 ∼ φ · · L · H 0(w 0), w W . Fix such isomorphisms. L · ∈ φ Then we choose those isomorphisms V0P0(w 0) ∼= H 0(w 0) for w W which are compatible with all our fixed isomorphisms· above; finallyL · fix any isomorphisms∈ φ V0P0(w 0) ∼= H 0(w 0) for w W W . By construction this gives us a

commutative· diagram:L · ∈ \ /

EndC (V0P0, V0P0) ∼ / EndC (H 0, H 0)

O O O L L indC indC

Cφ Cφ

? ? / EndCφ (VφPφ, VφPφ) ∼ / EndCφ (H φ, H φ) L L Arranging the three commutative diagrams that occur in this proof into one big commutative diagram the theorem is proved. 150 ERIK BACKELIN

ψ ψ ψ ! ψ Recall the idempotents ew (Rψ)0 and f 1 ((R0 ) )0, w W , from the ∈ w− w0 ∈ ∈ ψ ! beginning of this section which correspond under the isomorphism Rψ ˜ (R0 ) given by Proposition 3.1. → Put ψ¯ := w ψ.(Note that there is no “ ” in this product.)Thenψ¯is a − 0 · dominant weight such that w0Sψ¯w0 = Sψ so that we have an isomorphism of Coxeter-systems (W, S, S ¯)˜(W, S, S ), w ww w. It follows from Proposition ψ → ψ 0 0 → 2.4 that this gives (after choosing a projective generator Pψ¯ and putting Rψ¯ := ψ¯ ψ ψ Endg(P ¯))aringisomorphismR¯˜Rψ such that e maps to e , w W . ψ ψ→ w0ww0 w ∈ Composing the surjection in Lemma 3.4 with the isomorphism R ˜ (Rψ)! and the ψ→ 0 isomorphism R ¯ ˜ R we get a surjection ψ→ ψ (3.3) : ( ψ)! with ( ψ¯ )= φ( ψ ) for ψ π Rψ¯ R , π ew0ww0 T0 f 1 , w W . φ w− w0 ∈ Put Ωφ := w w W ψw ; α B : w ψ¯(H ) < 0 and let Iφ be the ideal in ψ¯ { ∈ 0 0 ∃ ∈ φ · α } ψ¯ ψ¯ φ φ φ R ¯ generated by the e for w Ω .ThenR¯/I and R have — in analogy with ψ w ∈ ψ¯ ψ ψ¯ ψ¯ what we saw in the proof of Proposition 3.2 — isomorphic categories of ﬁnitely generated right modules (= φ ) and these algebras are basic so they must be iso- ∼ Oψ¯ morphic. We have the surjection φ : φ (after putting φ := End ( φ )) τψ¯ Rψ¯ Rψ¯ Rψ¯ g τψ¯Pψ¯ which clearly vanishes on φ so we get the induced surjection φ : φ φ Iψ¯ τψ¯ Rψ¯/Iψ¯ Rψ¯ which must be an isomorphism. Lemma 3.6. vanishes on φ;thus factors as π Iψ¯ π

φ τψ¯ φ ( ψ)! Rψ¯ Rψ¯ Rφ .

ψ¯ φ ψ ψ Proof. (a) We know that π(e )=T(f 1 ), for w W .PutΓ:= w0ww0 0 w− w0 ψ 1 ∈ w W;w−w0(α) R, α Bφ.Thusπ(ew0ww0 ) =0 w Γ, { ∈ ∈ − ∀ ∈ }φ 6 ⇐⇒ ∈ by Lemma 2.5 3) and the fact that w W w ( α ) R , α B φ .PutΛ:= ψ ¯ ∈ ⇐⇒ ∈ − ∀ ∈ w W ;w0ww0 ψ(Hα) 0 . If we can show that Γ Λ, the lemma is proved. {We∈ prove Γ = Λ. · ≥ } ⊆ (b)Λ Γ.Let w Λ. Then for each α Bφ we have (where ( , ) denotes the Killing-form):⊆ ∈ ∈ 0 w ww ψ¯(H )= w w ψ(H ) ≤ 0 0 · α − 0 · α 0 ( w w ψ,α)=( w w(ψ+ρ) ρ,α) ⇐⇒ ≤ − 0 · − 0 − 1 =( w w(ψ+ρ),α) 1= (ψ+ρ, w− w (α)) 1 − 0 − − 0 − 1 1 ( ψ + ρ, w− w0(α)) 1= w−w0(α) R, ⇐⇒ ≤− ⇒ ∈ − since ψ is dominant. Hence w Γ. (c) We conclude from Lemma∈ 2.5 that we have the bijection Γ w L(w φ) 3 ↔ · ∈ Irr ψ and we clearly have the bijection Λ w L(w ww ψ¯) Irr φ.Thus Oφ 3 ↔ 0 0 · ∈ Oψ¯ we have by (b) the inequality card(Irr φ) card(Irr ψ) and replacing ψ by φ Oψ¯ ≤ Oφ and φ by ψ¯ we get a reserve inequality which now gives Γ = Λ.

φ ψ ! Theorem 3.7. R w ψ is isomorphic to (Rφ ) . − 0 Remark 3.8. The obvious idempotents in (Rφ ) resp., in ((Rψ)!) correspond w0ψ 0 φ 0 under this isomorphism. − KOSZUL DUALITY FOR PARABOLIC AND SINGULAR CATEGORY 151 O Remark 3.9. One could probably combinatorially show that both rings in the theorem have the same dimension by constructing an inversion formula for the gener- alized Kazhdan-Lusztig polynomials associated to the highest weight category ψ Oφ (cf. [CPS]) which according to Lemma 3.6 would prove the theorem. Proof. Consider the diagram (with ψ¯ = w ψ) − 0 φ

/ / / / ! !

(3.4) R0 ∼ (R0) (Rφ)

T 0 ψ ψ

ψ¯ i0 iφ

φ ? ?

? T0 ψ

/ / / / ψ R ¯ ∼ ! ! ψ (R0 ) (Rφ ) The horizontal isomorphisms to the left can be chosen making the left square commutative. Indeed, if in the left lower corner we replace Rψ¯ by Rψ and its upgoing 0 0 arrow Tψ¯ by Tψ, this is the statement of Proposition 3.5. The reader may then T 0 0 φ verify that T : R ¯ , R equals the composition R ¯ ˜ R , R where the ﬁrst ψ¯ ψ → 0 ψ→ ψ → 0 map is induced from the isomorphism (W, S, S ¯) w w ww (W, S, S )of ψ 3 → 0 0 ∈ ψ Coxeter-systems.2 The right square clearly commutes and its horizontal maps are surjections by Lemma 3.4. Applying Lemma 3.6 to the composition of the maps in the bottom row of (3.4) (resp., in the top row with ψ = 0) we conclude — since parabolic truncation functors commute with translation functors — that the outer rectangle of (3.4)

factors through the commutative diagram

/ / φ / !

R0 (Rφ)

0 ψ

Tψ¯ iφ

φ ? ψ

/ / / ! Rψ¯ (Rφ )

φ The upper horizontal map must be an isomorphism since we know that R0 ! and (Rφ) (being isomorphic) have the same dimensions. Since the vertical maps are injective, we conclude that the lower horizontal map is injective; hence an isomorphism.

Acknowledgements I wish to thank Wolfgang Soergel for informing me about this problem and for generously sharing his deep insight in “graded representation theory” during many discussions. I also thank Steen Ryom-Hansen for useful discussions. This article was written at the Albert-Ludwigs-Universit¨at in Freiburg. The work was sup- ported by the TMR-project ERB FMRX-CT97-0100 from February to July 1998, for which I am very grateful. I also thank the Swedish Foundation for International Cooperation in Research and Higher Education (STINT) for ﬁnancing my work from August 1998 to present day.

2 0 0 Note that T : R ¯ , R0 is only well-deﬁned after choosing an isomorphism between T P ¯ ψ¯ ψ → ψ¯ ψ and a summand of P0 it is isomorphic to. What we really claim is that this isomorphism can be 0 chosen such that T : R ¯ , R0 equals the required composition. ψ¯ ψ → 152 ERIK BACKELIN

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Department of Mathematics, Albert-Ludwigs-Universitat, Eckerstr. 1, D-79104 Freiburg im Briesgau, Germany E-mail address: [email protected]