Affine category O, Koszul duality and Zuckerman functors
aa r X i v : . [ m a t h . R T ] J u l Affine category O, Koszul dualityand Zuckerman functors
Ruslan MaksimauInstitut Montpelliérain Alexander Grothendieck (CNRS: UMR 5149),Université Montpellier 2,Case Courrier 051,Place Eugène Bataillon,34095 MONTPELLIER Cedex,[email protected], [email protected].
Abstract
The parabolic category O for affine gl N at level − N − e admits astructure of a categorical representation of e sl e with respect to some end-ofunctors E and F . This category contains a smaller category A thatcategorifies the higher level Fock space. We prove that the functors E and F in the category A are Koszul dual to Zuckerman functors.The key point of the proof is to show that the functor F for the cate-gory A at level − N − e can be decomposed in terms of the components ofthe functor F for the category A at level − N − e − . To prove this, weuse the following fact from [9]: a category with an action of e s l e +1 containsa (canonically defined) subcategory with an action of e s l e .We also prove a general statement that says that in some generalsituation a functor that satisfies a list of axioms is automatically Koszuldual to some sort of Zuckerman functor. Contents O b gl N . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Affine Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.8 Categorical representations . . . . . . . . . . . . . . . . . . . . . 132.9 The category O . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.10 The choice of F . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.11 The standard representation of e sl e . . . . . . . . . . . . . . . . . 152.12 Categorical representation in the category O . . . . . . . . . . . . 162.13 The commutativity in the Grothendieck groups . . . . . . . . . . 182.14 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.15 The category A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.16 The change of level for A . . . . . . . . . . . . . . . . . . . . . . 202.17 The category A . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.18 The categorical representation in the category O over the field K T d,R , T d,R . . . . . . . . . . . . . . . . . . . . . . . 242.20 The proof of invertibility . . . . . . . . . . . . . . . . . . . . . . . 252.21 Rational Cherednik algebras . . . . . . . . . . . . . . . . . . . . . 272.22 Cyclotomic rational Cherednik algebras . . . . . . . . . . . . . . 292.23 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . 31 O . . . . . . . . . . . . . . . . . . . . . . 353.3 Linkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.4 Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 The action on standard and projetive modules . . . . . . . . . . 413.6 The functor V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.7 The cohomology of Schubert varieties . . . . . . . . . . . . . . . 433.8 Graded lifts of the functors . . . . . . . . . . . . . . . . . . . . . 463.9 The case W µ ′ ⊂ W µ . . . . . . . . . . . . . . . . . . . . . . . . . 49 O . . . . . . . . . . . . 564.8 The restriction to the category A . . . . . . . . . . . . . . . . . . 624.9 Zuckerman functors for the category A + . . . . . . . . . . . . . . 654.10 The Koszul dual functors in the category A . . . . . . . . . . . . 664.11 The case k = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 Introduction
Let O ν − e be the parabolic category O with parabolic type ν of the affine versionof the Lie algebra gl N at level − N − e . In [15], a categorical representation ofthe affine Kac-Moody algebra e sl e in O ν − e is considered. In particular, this meansthat there are exact biadjoint functors E i , F i : O ν − e → O ν − e for i ∈ [0 , e − whichinduce a representation of the Lie algebra e sl e on the Grothendieck group [ O ν − e ] of O ν − e . The precise definition of a categorical representation is given in Section2.8. The category O ν − e admits a decomposition O ν − e = M µ ∈ Z e O νµ that lifts the decomposition of the e sl e -module [ O ν − e ] in a direct sum of weightspaces.The category O νµ is Koszul by [17]. Its Koszul dual category is the category O µν, + defined similarly to O µν at a positive level. In particular, the Koszul du-ality exchanges the parameter ν ( the parabolic type ) with the parameter µ ( thesingular type ). The Koszul duality yields an equivalence of bounded derivedcategories D b ( O νµ ) ≃ D b ( O µν, + ) . More details about the Koszul duality can befound in [3].Let α , · · · , α e − be the simple roots of e sl e . We have E i ( O νµ ) ⊂ O νµ + α i , F i ( O νµ ) ⊂ O νµ − α i . The aim of this paper is to prove that Koszul dual functors D b ( O µν, + ) → D b ( O µ + α i ν, + ) , D b ( O µν, + ) → D b ( O µ − α i ν, + ) to the functors E i : D b ( O νµ ) → D b ( O νµ + α i ) , F i : D b ( O νµ ) → D b ( O νµ − α i ) . are the Zuckerman functors.Unfortunately, we cannot solve this problem for the full category O . But weare able to do this for a subcategory A of O .By definition, the Zuckerman functor is a composition of a parabolic inclu-sion functor with a parabolic truncation functor. Thus it is natural to try todecompose the functors E i and F i in two "smaller" functors. More precisely,we want to show that the functor F i : O νµ → O νµ ′ (and similarly for E i ) betweenthe blocks O νµ and O νµ can be decomposed as O νµ → O νµ → O νµ , where theblock O νµ is "more regular" than O νµ and than O νµ . The additional difficultyis that there is no good candidate for such a block at the level − N − e , butthere is one at another level: − N − ( e + 1) . We abuse the terminology using the word "block" here. We don’t claim that O νµ isindecomposable. The word "block" is used here by analogy with the non-parabolic finite typecategory O , where similar subcategories are indeed blocks. O νµ , E i , F i be defined in the same way as O νµ , E i , F i with e replaced by e + 1 . Let α , · · · , α e be the simple roots of e sl e +1 . Fix k ∈ [0 , e − . For an e -tuple µ = ( µ , · · · , µ e ) we set µ = (cid:26) ( µ , · · · , µ k , , µ k +1 , · · · , µ e ) if k = 0 , (0 , µ , · · · , µ e ) if k = 0 . Note that we have ( µ − α k ) = µ − α k − α k +1 .By [5], there is an equivalence of categories θ : O νµ → O νµ . The direct sum ofsuch equivalences identifies the category O ν − e with a direct factor of the category O ν − ( e +1) . We want to compare the e sl e -action on O ν − e with the e sl e +1 -action on O ν − ( e +1) . More precisely, we want to prove the following conjecture. Conjecture 1.1.
The following diagram of functors is commutative. O νµ O νµ − α k O νµ − α k − α k +1 O νµ O νµ − α k ✲ F k ✲ F k +1 ❄ θ − ✻ θ ✲ F k (1)Now, if the conjecture is true, then it implies a decomposition that we ex-pected. After that we could use an argument similar to [13] to show that thefunctor F k is Koszul dual to the parabolic inclusion functor and the functor F k +1 is Koszul dual to the parabolic truncation functor. Then we can deducethat F k is Koszul dual to the Zuckerman functor (which is the composition ofthe parabolic inclusion functor and the parabolic truncation functor). Thus theproblem is reduced to the proof of this conjecture.It is not hard to see that the diagram from Conjecture 1.1 is commutative atthe level of Grothendieck groups. In the case of the category O of gl N (insteadof affine gl N ) this is already enough to prove the analogue of Conjecture 1.1,using the theory of projective functors. Indeed, [1, Thm. 3.4] implies that twoprojective functors are isomorphic if their actions on the Grothendieck groupcoincide. Unfortunately, there is no satisfactory theory of projective functorsfor the affine case (an attempt to develop such a theory was given in [6]).We choose another strategy to prove this conjecture. It is based on the mainresult of [9], relating the notion of a categorical representaion of e sl e with thenotion of a categorical representation of e sl e +1 . Let us fix the following inclusionof Lie algebras e sl e ⊂ e sl e +1 e r e r if r ∈ [0 , k − , [ e k , e k +1 ] if r = k,e r +1 if r ∈ [ k + 1 , e − , r f r if r ∈ [0 , k − , [ f k +1 , f k ] if r = k,f r +1 if r ∈ [ k + 1 , e − . The Lie algebra e sl e has a categorical representation in the category O ν − e while the Lie algebra e sl e +1 has a categorical representation in the category O ν − ( e +1) . By [9, Thm. 3.5], each category C with a categorical action of e sl e +1 contains (under some assumptions) a (canonically defined) subcategory C ⊂ C that inherits a categorical action of e sl e from the categorical action of e sl e +1 on C . In particular, if we take C = O ν − ( e +1) , then the subcategory C can be easilyidentified with O ν − e using equivalences θ like in (1).We get two categorical representations of e sl e in O ν − e : • the original one, • the e sl e -categorical representation structure induced from the e sl e +1 -categoricalrepresentation structure in O ν − ( e +1) .To prove Conjecture 1.1, it is enough to prove that these two categorical repre-sentation structures are the same.Unfortunately we cannot apply the uniqueness theorem for categorical rep-resentations because the e sl e -module categorified by O ν − e is not simple. However,we can obtain a weaker version of Conjecture 1.1. The category O ν − e containssubcategories A ν [ α ] , parameterized by α ∈ Q + e , where Q + e is the positive partof the root lattice of e sl e . The direct sum of such categories categorifies the Fockspace. In this case we can use the technique similar to one used in [15]. Thistechnique allows to prove in some cases that two categorical representations thatcategorify the Fock space are the same. We get the following.For i ∈ [0 , e − we have F i ( A ν [ α ]) ⊂ A ν [ α + α i ] . Let A ν [ α ] be defined inthe same way as A ν [ α ] with respect to the parameter e + 1 instead of e . Let | α | be the height of α . For each α ∈ Q + e , we construct α = φ ( α ) ∈ Q + e +1 , where themap φ : Q e → Q e +1 is defined in Section 2.2 (see also Section 2.3).The main result of Section 2 is the following theorem. Theorem 1.2.
Assume that e > and ν = ( ν , · · · , ν l ) satisfies ν r > | α | foreach r ∈ [1 , l ] . There exists β ∈ Q + e +1 such that for each α ∈ Q + e as above thereare equivalences of categories θ ′ α : A ν [ α ] → A ν [ β + α ] and θ ′ α + α k : A ν [ α + α k ] → A ν [ β + α + α k + α k +1 ] such that the following diagram is commutative A ν [ β + α ] F k +1 F k −−−−−→ A ν [ β + α + α k + α k +1 ] θ ′ α x θ ′ α + αk x A ν [ α ] F k −−−−→ A ν [ β + α + α k ] . To get a categorification of the Fock space, we sum by α , but the values of N and ν arenot fixed. For each α we need N and ν such that ν r > | α | . See [15, Sec. 7.4] for more details. F in the category A (Theorem 1.2). We do this in the following way. We con-sider a category A that is equivalent to A as a category, but the functors E i and F i on A are defined with respect to another categorical action that comes fromlevel − N − ( e + 1) . We want to find a (probably different) equivalence between A and A that identifies the functors. The paper [15] compares A (together with E i and F i ) with the category O for the rational Cherednik algebra. We use asimilar argument to compare A with the same category O . As a consequence,we manage to compare A with A (together with the functors E i and F i ).In Section 3 we prove that in some cases the functors E and F for thecategory O admit graded lifts. For this we use Soergel’s functor V .In Section 4 we prove that the functors E and F for the category A areKoszul dual to Zuckerman functors. In fact, in Section 4.5 we prove a moregeneral and more abstract statement that says that in some general situation afunctor that satisfies a list of axioms is automatically Koszul dual to some sortof Zuckerman functor. The proof of this statement uses the approach of [13].The technique of restriction of categorical representations in [9] is developedfor solving the problem in the present paper. However, this technique has anindependent interest: another application is given in [14, Sec. 7].It is important to emphasize the relation between the present paper andthe preprint [10]. The preprint [10] is expected to be published as two differentpapers: the first of them is [9], the second one is the present paper. The first partcontains general results about KLR algebras and categorical representations.The second part is an application of the first part. The present paper is rewritten(compared to [10]) in a way that we never use KLR algebras explicitly. Thismakes the paper more independent from [9]. O For a noetherian ring A we denote by mod( A ) the abelian category of left finitelygenerated A -modules. We denote by N the set of non-negative integers. By acommutative diagram of functors we always mean a diagram that commutes upto an isomorphism of functors. Let
Γ = (
I, H ) be a quiver without -loops with the set of vertices I and the setof arrows H . For i, j ∈ I let h i,j be the number of arrows from i to j and setalso a i,j = 2 δ i,j − h i,j − h j,i . Let g I be the Kac-Moody algebra over C associatedwith the matrix ( a i,j ) . More precisely, the algebra g I is generated by elements6 i , f i , h i for i ∈ I satisfying the following relations: [ e i , f j ] = δ ij h i , [ h i , e j ] = a i,j e j , [ h i , f j ] = − a i,j f j , [ h i , h j ] = 0 , ad( e i ) − a ij ( e j ) = 0 if i = j, ad( f i ) − a ij ( f j ) = 0 if i = j. For each i ∈ I let α i be the simple root corresponding to e i . Set Q I = M i ∈ I Z α i , Q + I = M i ∈ I N α i . Let X I be the free abelian group with basis { ε i ; i ∈ I } . Set also X + I = M i ∈ I N ε i . For α ∈ Q + I denote by | α | its height, i.e., for α = P i ∈ I d i α i , d i ∈ N , we have | α | = P i ∈ I d i . Set I α = { i = ( i , · · · , i | α | ) ∈ I | α | ; P | α | r =1 α i r = α } .Let Γ ∞ = ( I ∞ , H ∞ ) be the quiver with the set of vertices I ∞ = Z and theset of arrows H ∞ = { i → i + 1; i ∈ I ∞ } . In this case we will simply write sl ∞ for the Lie algebra g I ∞ . We stress that here we get a "two-sided" sl ∞ , thegenerators of sl ∞ are parameterized by Z and not by N .Assume that e > is an integer. Let Γ e = ( I e , H e ) be the quiver with theset of vertices I e = Z /e Z and the set of arrows H e = { i → i + 1; i ∈ I e } . Then g I e is the Lie algebra e sl e = sl e ⊗ C [ t, t − ] ⊕ C .Assume that Γ = (
I, H ) is a quiver whose connected components are of theform Γ e , with e ∈ N , e > or e = ∞ . For i ∈ I denote by i + 1 and i − the(unique) vertices in I such that there are arrows i → i + 1 and i − → i .Let us also consider the following additive map ι : Q I → X I , α i ε i − ε i +1 . Fix a formal variable χ and set X χI = X I ⊕ Z χ . We can lift the Z -linear map ι to a Z -linear map ι χ : Q I → X χI , α i ε i − ε i +1 − χ. Note that the map ι χ : Q I → X χI is injective (while ι is not injective). Wemay omit the symbols ι , ι χ and write α instead of ι ( α ) or ι χ ( α ) .We will also abbreviate Q e = Q I e , X e = X I e , X χe = X χI e . .2 Doubled quiver Now we recall the notion of a doubled quiver introduced in [9, Sec. 2B].Let
Γ = (
I, H ) be a quiver without -loops. Fix a decomposition I = I ⊔ I such that there are no arrows between the vertices in I . In this section wedefine a doubled quiver Γ = (
I, H ) associated with (Γ , I , I ) . The motivationof this definition is that there is a relation between categorical representationsof g I and of g I , see [9] for more details. Later, we will apply this relation to thecategorical representation in the category O .The idea of how we get Γ from Γ is to "double" each vertex in the set I (we do not touch the vertices from I ). We replace each vertex i ∈ I by acouple of vertices i and i with an arrow i → i . Each arrow entering to i should be replaced by an arrow entering to i , each arrow coming from i shouldbe replaced by an arrow coming from i . See [9, Sec. 2B] for a more formaldefinition of the quiver Γ . This construction will be used in the present paperfor two special types of quivers mentioned in Section 2.3.Set I ∞ = ` d ∈ N I d , I ∞ = ` d ∈ N I d , where I d , I d are the cartesian products.The concatenation yields a monoid structure on I ∞ and I ∞ . Let φ : I ∞ → I ∞ be the unique morphism of monoids such that for i ∈ I ⊂ I ∞ we have φ ( i ) = (cid:26) i if i ∈ I , ( i , i ) if i ∈ I . There is a unique Z -linear map φ : Q I → Q I such that φ ( I α ) ⊂ I φ ( α ) for each α ∈ Q + I . It is given by φ ( α i ) = (cid:26) α i if i ∈ I ,α i + α i if i ∈ I . Let φ denote also the unique additive embedding φ : X I → X I , ε i ε i ′ , (2)where i ′ = (cid:26) i if i ∈ I ,i if i ∈ I . (3) We will use the construction from the previous section only for two special typesof quivers.First, consider the quiver
Γ = Γ e , for e > . We have I = I e = Z /e Z .Fix k ∈ [0 , e − and set I = { k } , I = I \{ k } . In this case the quiver Γ isisomorphic to Γ e +1 . The isomorphism Γ e ≃ Γ e +1 at the level of vertices is i i if i ∈ [0; k − ,k k,k k + 1 ,i i + 1 if i ∈ [ k + 1 , e − .
8o avoid confusion, for i ∈ I = I e +1 we will write α i and ε i instead of α i and ε i respectively.Let Υ: Z → Z be the map given for a ∈ Z , b ∈ [0 , e − by Υ( ae + b ) = (cid:26) a ( e + 1) + b if b ∈ [0 , k ] ,a ( e + 1) + b + 1 if b ∈ [ k + 1 , e − . (4)Note that we have the following commutative diagram Z Υ −−−−→ Z y y I e −−−−→ I e +1 , where the bottom map is i i ′ , see (3).Now we describe the second quiver that we are interested in. Let e Γ = ( e I, e H ) be the disjoint union of l copies of Γ ∞ .Write e α i and e ε i instead of α i and ε i respectively for each i ∈ e I . We identifyan element of e I with an element ( a, b ) ∈ I ∞ × [1 , l ] in the obvious way. Considerthe decomposition e I = e I ⊔ e I such that ( a, b ) ∈ e I if and only if a ≡ k mod e .In this case the quiver e Γ is isomorphic to e Γ . More precisely, in this case we have ( a, b ) = (Υ( a ) , b ) , ( a, b ) = (Υ( a ) , b ) , ( a, b ) = (Υ( a ) + 1 , b ) . To avoid confusion, we will always write e φ for any of the maps e φ : e I ∞ → e I ∞ , Q e I → Q e I , X e I → X e I in Section 2.2.Consider the quiver homomorphism π e : e Γ → Γ e such that π e : e I → I, ( a, b ) a mod e. Similarly, we have a quiver homomorphism π e +1 : e Γ → Γ e +1 . They yield Z -linearmaps π e : Q e I → Q e , π e : X e I → X e , π e +1 : Q e I → Q e +1 , π e +1 : X e I → X e +1 . The following diagrams are commutative for α ∈ Q + e , e α ∈ Q + e I such that π e ( e α ) = α , Q e I e φ −−−−→ Q e Iπ e y π e +1 y Q e φ −−−−→ Q e +1 X e I e φ −−−−→ X e Iπ e y π e +1 y X e φ −−−−→ X e +1 e I e α e φ −−−−→ e I e φ ( e α ) π e y π e +1 y I αe φ −−−−→ I φ ( α ) e +1 .4 Deformation rings In this section we introduce some general definitions from [15] for a later use.Let us fix an integer e > .We call deformation ring ( R, κ, τ , · · · , τ l ) a regular commutative noetherian C -algebra R with equipped with a homomorphism C [ κ ± , τ , · · · , τ l ] → R . Let κ, τ , · · · , τ l denote also the images of κ, τ , · · · , τ l in R . A deformation ring is in general position if any two elements of the set { τ u − τ v + aκ + b, κ − c ; a, b ∈ Z , c ∈ Q , u = v } have no common non-trivial divisors. A local deformation ring is a deformationring which is a local ring and such that τ , · · · , τ l , κ − e belong to the maximalideal of R . Note that each C -algebra that is a field has a trivial local deformationring structure, i.e., such that τ = · · · = τ l = 0 and κ = e . We always consider C as a local deformation ring with a trivial deformation ring structure. A C -algebra R is called analytic if it is a localization of the ring of germs ofholomorphic functions on some compact polydisc D ⊂ C d for some d > .We will write κ = κ ( e + 1) /e , τ r = τ r ( e + 1) /e . We will abbreviate R for ( R, κ, τ , · · · , τ l ) and R for ( R, κ, τ , · · · , τ l ) .Let us give names to two special assumptions on R that will be often usedin the paper. Assumption 1. R is a local analytic deformation ring in general positionof dimension .We denote by k the residue field and by K the field of fractions of R . As-sumption 1 is necessary to be able to study categorical representations in theaffine category O , see [15].Now, we introduce a modified version of this assumption. This is necessaryto be able to state results that are true over R , over K and over k withoutspeaking about K and k separately, see for example Proposition 2.8. Assumption 2.
The ring R is either as in Assumption 1 or is the fractionfield or the residue field of a ring satisfying Assumption 1.If R satisfies the second part of Assumption 2, we simply mean k = K = R .Let R be as in Assumption 1. Consider the element q e = exp(2 π √− /κ ) in R . This element specializes to ζ e = exp(2 π √− /e ) in k . If R satisfiesAssumption 2, then it has a deformation ring structure. The element q e stillmakes sense in R . If R is a resudue field of a ring satisfying Assumption 1, thenwe mean that q e is ζ e . b gl N Fix positive integers N , l and e such that e > . Let R be a deformation ring,see Section 2.4. Set g R = gl N ( R ) , b g R = b gl N ( R ) = gl N ( R )[ t, t − ] ⊕ R ⊕ R∂. i, j ∈ [1 , N ] let e i,j ∈ g R denote the matrix unit. Let h R ⊂ g R be theCartan subalgebra generated by the e i,i ’s, and ǫ , · · · , ǫ N be the basis of h ∗ R dual to e , , · · · , e N,N . Let P = Z ǫ ⊕ · · · ⊕ Z ǫ N be the weight lattice of g R . Weidentify P with Z N . Let Π , b Π ⊂ h ∗ R be the sets of simple roots of g R and b g R .Let W = S N be the Weyl group of g R and f W = W ⋉ Z Π , c W = W ⋉ P be theaffine and the extended affine Weyl groups.Then we define the element wt e ( λ ) ∈ X e given by wt e ( λ ) = N X s =1 ε λ r , where we write ε λ r for ε ( λ r mod e ) .We will abbreviate P [ µ ] = { λ ∈ P ; wt e ( λ ) = µ } . (5)Similarly, we consider the weight wt χe ( λ ) = N X r =1 ε λ r + ( N X r =1 λ r ) χ ∈ X χe . Finally, let X e [ N ] ⊂ X e be the subset given by X e [ N ] = { µ = e X r =1 µ r ε r ∈ X e ; µ r > , e X r =1 µ r = N } . We may identify µ with the tuple ( µ , · · · , µ e ) if no confusion is possible.Now, consider the Cartan subalgebra b h R = h R ⊕ R ⊕ R∂ of b g R . Let α , α , · · · , α N − ⊂ b h ∗ R and ˇ α , ˇ α , · · · , ˇ α N − ⊂ b h R be the simple roots andcoroots of b g R respectively. Let Λ and δ be the elements of b h ∗ R defined by δ ( ∂ ) = Λ ( ) = 1 , δ ( h R ⊕ R ) = Λ ( h R ⊕ R∂ ) = 0 . Let ( • , • ): b h ∗ R × b h ∗ R → R be the bilinear form such that λ (ˇ α r ) = ( λ, α r ) , λ ( ∂ ) = ( λ, Λ ) , ∀ λ ∈ b h ∗ R . Set P R = P ⊗ Z R . Given an l -tuple of positive integers ν = ( ν , · · · , ν l ) suchthat P lr =1 ν r = N , we define ρ = (0 , − , · · · , − N + 1) ,ρ ν = ( ν , ν − · · · , , ν , · · · , , · · · , ν l , · · · , ,τ = ( τ ν , · · · , τ ν l l ) , where τ ν r r means ν r copies of τ r . Set also b ρ = ρ + N Λ , e λ = λ + τ + z λ δ − ( N + κ )Λ , (6)where z λ = ( λ, ρ + λ ) / κ . Denote by b p R,ν the parabolic subalgebra of b g R ofparabolic type ν . For a ν -dominant weight λ ∈ P let ∆( λ ) R be the parabolicVerma module with highest weight e λ and ∆ λR = ∆( λ − ρ ) R . We will also skipthe subscript R when R = C . 11 .6 Affine Weyl groups Assume that R = C . In this section we discuss some combinatorial aspects ofthe c W -action on b h ∗ .The group c W is generated by { π, s i ; i ∈ Z /N Z } modulo the relations s i = 1 ,s i s j = s j s i ∀ i = j ± ,s i s i +1 s i = s i +1 s i s i +1 ,πs i +1 = s i π. Let f W be the subgroup of c W generated by { s i ; i ∈ Z /N Z } . The group c W acts on P in the following way: • s r switches of the r th and ( r + 1) th components of λ if r = 0 , • s ( λ , · · · , λ N ) = ( λ N − e, λ , · · · , λ N − , λ + e ) , • π ( λ , · · · , λ N ) = ( λ , · · · , λ N , λ + e ) .We will call this action of c W on P the negative e - action . We will always consideronly negative actions of c W on P up to Section 4.7. So we can skip the word"negative". We may write P ( e ) = P to stress that we consider the e -action of c W on P . The map P ( e ) → b h ∗ , λ ] λ − ρ + b ρ is c W -invariant. This means that the weights λ , λ ∈ P are in the same c W -orbitif and only if the highest weights of the Verma modules ∆ λ and ∆ λ are linkedwith respect to the Weyl group c W , see [17, Sec. 3.2] and [5, Sec. 2.3] for moredetails about linkage. Note that P = ` µ ∈ X e [ N ] P [ µ ] is the decomposition of P into c W -orbits with respect to the e -action. An element λ ∈ P is e - anti-dominant if λ λ · · · λ N λ + e .Recall the map Υ: Z → Z from (4). Applying Υ coordinate by coordinate tothe elements of P we get a map Υ: P ( e ) → P ( e +1) . Lemma 2.1.
The map Υ: P ( e ) → P ( e +1) is c W -invariant and takes e -anti-dominant weights to ( e + 1) -anti-dominant weights. Let R be a commutative ring with . Fix an invertible element q ∈ R . Definition . The affine Hecke algebra H d,R ( q ) is the R -algebra generatedby T , · · · , T d − and the invertible elements X , · · · , X d modulo the followingdefining relations X r X s = X s X r , T r X r = X r T r if | r − s | > ,T r T s = T s T r if | r − s | > , T r T r +1 T r = T r +1 T r T r +1 ,T r X r +1 = X r T r + ( q − X r +1 , T r X r = X r +1 T r − ( q − X r +1 , ( T r − q )( T r + 1) = 0 . l -tuple Q = ( Q , · · · , Q l ) ∈ R l . Definition . The cyclotomic Hecke algebra H Qd,R ( q ) is the quotient of H d,R ( q ) by the two-sided ideal generated by ( X − Q ) · · · ( X − Q l ) . Let R be a C -algebra. Fix an invertible element q ∈ R , q = 1 . Let C be an exact R -linear category. Definition . A representation datum in C is a tuple ( E, F, X, T ) where ( E, F ) is a pair of exact functors C → C and X ∈ End( F ) op , T ∈ End( F ) op areendomorphisms of functors such that for each d ∈ N , there is an R -algebrahomomorphism ψ d : H d,R ( q ) → End( F d ) op given by X r F d − r XF r − ∀ r ∈ [1 , d ] ,T r F d − r − T F r − ∀ r ∈ [1 , d − . Now, assume that R = k is a field. Assume that C is a Hom -finite abeliancategory.
Remark . Assume that we have a representation datum in a k -linear category C such that the functors E and F are biadjoint. Then by adjointness we havean algebra isomorphism End( E d ) ≃ End( F d ) op . In particular we get an algebrahomomorphism H d, k ( q ) → End( E d ) .Let F be a subset of k × . We view F as the vertex set of a quiver Γ F withan arrow i → j if and only if j = qi . Definition . An g F -categorical representation in C is the datum of a rep-resentation datum ( E, F, X, T ) and a decomposition C = L µ ∈ X F C µ satisfyingthe conditions ( a ) and ( b ) below. For i ∈ F let E i , F i be endofunctors of C suchthat for each M ∈ C the objects E i ( M ) , F i ( M ) are the generalized i -eigenspacesof X acting on E ( M ) and F ( M ) respectively, see also Remark 2.5. We assumethat ( a ) F = L i ∈ F F i and E = L i ∈ F E i , ( b ) E i ( C µ ) ⊂ C µ + α i and F i ( C µ ) ⊂ C µ − α i .For exemple, in the case when the quiver Γ F is isomorphic to Γ e , thenwe have g F = e sl e . So we will say "an e sl e -categorical representation" insteadof "a g F -categorical representation". However, we should remember that thisdefinition depends on the choice of the set F . O Let R be a deformation ring. Fix an l -tuple of positive integers ν = ( ν , · · · , ν l ) such that P lr =1 ν r = N . First we define an R -deformed version of the parabolic13ategory O for b gl N . Recall that we identify the weight lattice P with Z N . Wesay that λ ∈ P is ν - dominant if λ r > λ r +1 for each r ∈ [1 , N − \{ ν , ν + ν , · · · , ν + · · · + ν l } . Let P ν be the set of ν -dominant weights of P . Setalso P ν [ µ ] = P ν ∩ P [ µ ] , where P [ µ ] is as in (5). Let O νR be the R -linear abeliancategory of finitely generated b g R -modules M which are weight b h R -modules, andsuch that the b p R,ν -action on M is locally finite over R , and the highest weightof any subquotient of M is of the form e λ with λ ∈ P ν , where e λ is defined in (6).Let O νµ,R be the Serre subcategory of O νR generated by the modules ∆ λR for all λ ∈ P ν [ µ ] . Let O ν, ∆ µ,R ⊂ O νµ,R be the full subcategory of ∆ -filtered modules.We will omit the upper index ν if ν = (1 , , · · · , . Assume λ ∈ P . In thecase if R = k is a field we denote by L ( λ ) k the simple quotient of ∆( λ ) k . Inthe case if R is local with residue fields k , the simple module L ( λ ) k ∈ O k has asimple lift L ( λ ) R ∈ O R such that L ( λ ) k = k ⊗ R L ( λ ) R (see [4, Sec. 2.2]). Setalso L λR = L ( λ − ρ ) R . F In this section we define some sets F and F k . We will see later that these setsare related with the categorical representations in the categories O ν − e,K and O ν − e, k .We assume that R is as in Assumption 1. As above, we fix an l -tuple ν = ( ν , · · · , ν l ) of positive integers. Put Q r = exp(2 π √− ν r + τ r ) /κ ) , r ∈ [1 , l ] . (7)The canonical homomorphism R → k maps q e to ζ e and Q r to ζ ν r e , where q e and ζ e are as in Section 2.4.Now, consider the subset F of R given by F = [ r ∈ Z ,t ∈ [1 ,l ] { q re Q t } . Denote by F k the image of F in k with respect to the surjection R → k . Moreprecisely, we have F k = { ζ re ; r ∈ Z } (i.e., the set F k is the set of e th roots ofunity in k ). Recall from Section 2.8 that we consider F (and F k ) as a vertexset of a quiver. The set F is a vertex set of a quiver that is a disjoint union of l infinite linear quivers. The set F k is a vertex set of a cyclic quiver of length e .We fix the following identifications I e ≃ F k , i ζ ie , e I ≃ F , ( a, b ) exp(2 π √− a + τ b ) /κ ) . In particular, we identify the quivers Γ e and e Γ with the quivers Γ F k and Γ F respectively. We have g F k = e sl e On the other hand, the Lie algebras g F isisomorphic to ( sl ∞ ) ⊕ l .Now we consider some special cyclotomic Hecke algebras. Set H νd,R ( q e ) = H Qd,R ( q e ) , where Q = ( Q , . . . , Q l ) is the l -tuple defined by (7). Similarly, we14efine the algebras H νd,K ( q e ) and H νd, k ( ζ e ) (in the last case, we replace Q by itsimage in k ).It is useful to think of the algebras H νd,K ( q e ) and H νd, k ( ζ e ) as cyclotomicKLR algebras defined with respect to the quivers Γ F and Γ F k respectively, see[10, Cor. 2.18]. However, we don’t use this point of view explicitly in the presentpaper. e sl e Let e i , f i , h i , be the generators of the complex Lie algebra e sl e = sl e ⊗ C [ t, t − ] ⊕ C , here i ∈ I e . Let V e be a C -vector spaces with canonical basis { v , · · · , v e } and set U e = V e ⊗ C [ z, z − ] . The vector space U e has a basis { u r ; r ∈ Z } where u a + eb = v a ⊗ z − b for a ∈ [1 , e ] , b ∈ Z . It has a structure of an e sl e -module suchthat f i ( u r ) = δ i ≡ r u r +1 , e i ( u r ) = δ i ≡ r − u r − . Let { v ′ , · · · , v ′ e +1 } , { u ′ r ; r ∈ Z } denote the bases of V e +1 and U e +1 .Fix an integer k < e . Consider the following inclusion of vector spaces V e ⊂ V e +1 , v r (cid:26) v ′ r if r k,v ′ r +1 if r > k. It yields an inclusion sl e ⊂ sl e +1 such that e r e r if r ∈ [1 , k − , [ e k , e k +1 ] if r = k,e r +1 if r ∈ [ k + 1 , e − ,f r f r if r ∈ [1 , k − , [ f k +1 , f k ] if r = k,f r +1 if r ∈ [ k + 1 , e − ,h r h r if r ∈ [1 , k − ,h k + h k +1 if r = k,h r +1 if r ∈ [ k + 1 , e − . This inclusion lifts uniquely to an inclusion e sl e ⊂ e sl e +1 such that e (cid:26) e if k = 0 , [ e , e ] else ,f (cid:26) f if k = 0 , [ f , f ] else ,h (cid:26) h if k = 0 ,h + h else . Consider the inclusion U e ⊂ U e +1 such that u r u ′ Υ( r ) .15 emma 2.7. The embeddings V e ⊂ V e +1 and U e ⊂ U e +1 are compatible withthe actions of sl e ⊂ sl e +1 and e sl e ⊂ e sl e +1 respectively. Set ∧ ν U e = ∧ ν U e ⊗ · · · ⊗ ∧ ν l U e . For each λ ∈ P ν define the followingelement in ∧ ν U e : ∧ νλ = ( u λ ∧ · · · ∧ u λ ν ) ⊗ · · · ⊗ ( u λ + ··· + λ l − +1 ∧ · · · ∧ u λ ν + ··· + λ νl ) . The obvious e sl e -action on U e yields an e sl e -action on ∧ ν U e . We identify theabelian group X e / Z ( ε + · · · + ε e ) with the weight lattice of sl e . In particulareach element µ ∈ X e yields a weight of sl e . For each µ ∈ X e [ N ] let ( ∧ ν U e ) µ bethe weight space in ∧ ν U e corresponding to µ . O Set O ν − e,R = L µ ∈ X e [ N ] O νµ,R and similarly for O ν, ∆ − e,R . Now we define a rep-resentation datum in the category O ν, ∆ − e,R . See [15, Sec. 5.4] for more details.In Sections 2.12-2.17 we assume that R is as in Assumption 2. For an exactcategory C denote by [ C ] its complexified Grothendieck group. The followingproposition holds, see [15]. Proposition 2.8.
There is a pair of exact endofunctors E , F of O ν, ∆ − e,R suchthat the following properties hold. ( a ) The functors E , F commute with the base changes K ⊗ R • , k ⊗ R • . ( b ) ( O ν, ∆ − e,R , E, F ) admits a representation datum structure (with respect to q = q e ). ( c ) The pair of functors ( E, F ) is biadjoint. It extends to a pair of biadjointfunctors O ν − e,R → O ν − e,R if R is a field. ( d ) There are decompositions E = L i ∈ I e E i , F = L i ∈ I e F i such that E i ( O ν, ∆ µ,R ) ⊂ O ν, ∆ µ + α i ,R , F i ( O ν, ∆ µ,R ) ⊂ O ν, ∆ µ − α i ,R . ( e ) There is a vector space isomorphism [ O ν, ∆ µ,R ] ≃ ( ∧ ν U e ) µ such that the func-tors F i , E i act on [ O ν, ∆ − e,R ] = L µ ∈ X e [ N ] [ O ν, ∆ µ,R ] as the standard generators e i , f i of e sl e . ( f ) If R = k with the trivial deformation ring structure, then E i , F i yield acategorical representation of e sl e in O ν − e, k (with respect to the set F as inSection 2.10). Fix k ∈ [0 , e − . Recall the map Υ: P → P from Section 2.6 and the map φ : X e → X e +1 from (2). Set µ ′ = µ − α k and µ = φ ( µ ) . Set, µ = µ − α k and16 ′ = µ − α k − α k +1 . Note that Υ( P [ µ ]) ⊂ P [ µ ] . For k = 0 we have µ = ( µ , · · · , µ k , µ k +1 , · · · , µ e ) ,µ ′ = ( µ , · · · , µ k − , µ k +1 + 1 , · · · , µ e ) ,µ = ( µ , · · · , µ k , , µ k +1 , · · · , µ e ) ,µ = ( µ , · · · , µ k − , , µ k +1 , · · · , µ e ) ,µ ′ = ( µ , · · · , µ k − , , µ k +1 + 1 , · · · , µ e ) . For an e -tuple a = ( a , · · · , a e ) of non-negative integers we set a = (1 a , · · · , e a e ) .Note that we have Υ(1 µ ) = 1 µ , Υ(1 µ ′ ) = 1 µ ′ . (8) Remark . The set P [ a ] is a c W -orbit in P ( e ) . It is a union of f W -orbitsand each of them contains a unique e -anti-dominant weight. By definition,the weight a ∈ P [ a ] is e -anti-dominant. However, there is no canonical wayto choose an anti-dominant element in P [ a ] . In the case k = 0 we need tochange our convention and to set a = (0 a e , a e , · · · , ( e − a e − ) . This changeis necessary to have (8).First, assume that l = N and ν = (1 , , · · · , . Lemma 2.10.
There is an equivalence of categories θ µµ : O µ,R → O µ,R such that θ µµ (∆ λR ) ≃ ∆ Υ( λ ) R . It restricts to an equivalence of categories θ µµ : O ∆ µ,R → O ∆ µ,R .Proof. For each n ∈ Z the weight π n (1 µ ) is e -anti-dominant. Let O π n (1 µ ) ,R ⊂ O µ,R be the Serre subcategory generated by the Verma modules of the form ∆ wπ n (1 µ ) R with w ∈ f W . We have O µ,R = M n ∈ Z O π n (1 µ ) ,R . (9)The weights π n (1 µ ) ∈ P ( e ) and π n (1 µ ) ∈ P ( e +1) have the same stabilizers in f W . Thus by [5, Thm. 11] (see also [15, Prop. 5.24]) we have an equivalence ofcategories O π n (1 µ ) ,R ≃ O π n (1 µ ) ,R , ∆ wπ n (1 µ ) R ∆ wπ n (1 µ ) R ∀ w ∈ f W .
Taking the sum by all n ∈ Z we get an equivalence of categories θ µµ : O µ,R ≃ O µ,R , ∆ w (1 µ ) R ∆ w (1 µ ) R ∀ w ∈ c W .
Recall that we have
Υ(1 µ ) = 1 µ . Thus by c W -invariance of Υ we get θ µµ (∆ λR ) ≃ ∆ Υ( λ ) R ∀ λ ∈ P [ µ ] . emark . Notice that [5] yields an equivalence of categories over a field. Itis explained in [15] how to get from it an equivalence of categories O ∆ R . First,comparing the endomorphisms of projective generators one gets an equivalenceof the abelian categories O R . Then, comparing the highest weight structure inboth sides, we deduce an equivalence of additive categories O ∆ R .The equivalence θ µµ restricts to equivalences O νµ,R ≃ O νµ,R and O ν, ∆ µ,R ≃ O ν, ∆ µ,R for each parabolic type ν , see [15, Sec. 5.7.2]. We will also call this equivalence θ µµ . We obtain equivalences of categories θ µ ′ µ ′ : O νµ ′ ,R ≃ O νµ ′ ,R and θ µ ′ µ ′ : O ν, ∆ µ ′ ,R ≃ O ν, ∆ µ ′ ,R in a similar way. Conjecture 2.12.
There are the following commutative diagrams O ν, ∆ µ,R O ν, ∆ µ ,R O ν, ∆ µ ′ ,R O ν, ∆ µ,R O ν, ∆ µ ′ ,R ✲ F k ✲ F k +1 ❄ θ µ ′ µ ′ ✻ θ µµ ✲ F k and O ν, ∆ µ,R O ν, ∆ µ ,R O ν, ∆ µ ′ ,R O ν, ∆ µ,R O ν, ∆ µ ′ ,R ❄ θ µµ ✛ E k ✛ E k +1 ✻ θ µ ′ µ ′ ✛ E k . We have the following commutative diagram of vector spaces [ O ν, ∆ − ( e +1) ,R ] −−−−→ ∧ ν U e +1 ⊕ µ θ µµ x x [ O ν, ∆ − e,R ] −−−−→ ∧ ν U e , where the horizontal maps are respectively the isomorphisms of e sl e -modulesand e sl e +1 -modules from Proposition 2.8 ( e ) , the right vertical map is given bythe injection U e → U e +1 in Section 2.11. Moreover, the right vertical mapis a morphism of e sl e -modules where ∧ ν U e +1 is viewed as an e sl e -module viathe inclusion e sl e ⊂ e sl e +1 introduced in Section 2.11. Thus ⊕ µ θ µµ : [ O ν, ∆ − e,R ] → [ O ν, ∆ − ( e +1) ,R ] is a morphism of e sl e -modules which intertwines18 [ E r ] with [ E r ] , [ F r ] with [ F r ] if r ∈ [1 , k − , • [ E k ] with [ E k E k +1 ] − [ E k +1 E k ] , [ F k ] with [ F k +1 F k ] − [ F k F k +1 ] , • [ E r ] with [ E r +1 ] , [ F r ] with [ F r +1 ] if r ∈ [ k + 1 , e − .In particular, we see that the diagrams from Conjecture 2.12 commuteat the level of Grothendieck groups. Since there is no good notion ofprojective functors in the affine category O , this is not enough to proveour conjecture. A partition of an integer n > is a tuple of positive integers ( λ , · · · , λ s ) suchthat λ > λ > · · · > λ s and P st =1 λ t = n . Denote by P n the set of all partitionsof n and set P = ` n ∈ N P n . For a partition λ = ( λ , · · · , λ s ) of n , we set | λ | = n and ℓ ( λ ) = s . An l - partition of an integer n > is an l -tuple λ = ( λ , · · · , λ l ) ofpartitions of integers n , · · · , n l > such that P lt =1 n t = n . Let P ln be the setof all l -partitions of n and set P l = ` n ∈ N P ln .A partition λ can be represented by a Young diagram Y ( λ ) and an l -partition λ = ( λ , · · · , λ l ) by an l -tuple of Young diagrams Y ( λ ) = ( Y ( λ ) , · · · , Y ( λ l )) .Let λ ∈ P l be an l -partition. For a box b ∈ Y ( λ ) situated in the i th row, j th column of the r th component we define its residue Res ν ( b ) ∈ I as ν r + j − i mod e and its deformed residue g Res ν ( b ) ∈ e I as ( ν r + j − i, r ) . Set Res ν ( λ ) = X b ∈ Y ( λ ) α Res ν ( b ) ∈ Q + e , g Res ν ( λ ) = X b ∈ Y ( λ ) e α g Res ν ( b ) ∈ Q + e I . Now for α ∈ Q + e and e α ∈ Q + e I set P lα = { λ ∈ P l ; Res ν ( λ ) = α } , P l e α = { λ ∈ P l ; g Res ν ( λ ) = e α } . This notation depends on ν . We may write P lα,ν and P l e α,ν to specify ν . We havedecompositions P ld = M α ∈ Q + e , | α | = d P lα , P lα = M e α ∈ Q + e +1 ,π e ( e α )= α P l e α . A Let P νd ⊂ P ld be the subset of the elements λ = ( λ , · · · , λ l ) such that ℓ ( λ r ) ν r for each r ∈ [1 , l ] . We can view each λ ∈ P νd as the weight in P given by ( λ , · · · , λ ℓ ( λ ) , ν − ℓ ( λ ) , λ , · · · , λ ℓ ( λ ) , ν − ℓ ( λ ) , · · · , λ l , · · · , λ lℓ ( λ l ) , ν l − ℓ ( λ l ) ) . We abbreviate ∆[ λ ] R = ∆ λ + ρ ν R . 19 efinition . Let A νR [ d ] ⊂ O ν − e,R be the Serre subcategory generated bythe modules ∆[ λ ] R with λ ∈ P νd , see Section 2.5. Denote by A ν, ∆ R [ d ] the fullsubcategory of ∆ -filtered modules in A νR [ d ] .The restriction of the functor F to the subcategory A ν, ∆ R [ d ] yields a functor F : A ν, ∆ R [ d ] → A ν, ∆ R [ d +1] . However, it is not true that E ( A ν, ∆ R [ d +1]) ⊂ A ν, ∆ R [ d ] .Nevertheless, we can define a functor E : A ν, ∆ R [ d + 1] → A ν, ∆ R [ d ] that is leftadjoint to F : A ν, ∆ R [ d ] → A ν, ∆ R [ d + 1] , see [15, Sec. 5.9]. This can be donein the following way. Let h be the inclusion functor from A ν, ∆ R [ d ] to O ν, ∆ − e,R .Abusing the notation, we will use the same symbol for the inclusion functorfrom A ν, ∆ R [ d + 1] to O ν, ∆ − e,R . Let h ∗ be the left adjoint functor to h . We definethe functor E for the category A ν, ∆ R as h ∗ Eh .There is a decomposition A νR [ d ] = L α ∈ Q + e , | α | = d A νR [ α ] , where A νR [ α ] is theSerre subcategory of A νR [ d ] generated by the Verma modules ∆[ λ ] R such that λ ∈ P lα . The functors E , F admit decompositions E = M i ∈ I e E i , F = M i ∈ I e F i such that for each α ∈ Q + e and i ∈ I e we have E i ( A ν, ∆ R [ α ]) ⊂ A ν, ∆ R [ α − α i ] , F i ( A ν, ∆ R [ α ]) ⊂ A ν, ∆ R [ α + α i ] . Note that the functor E i for the category A ν, ∆ R is the restriction of the functor E i for the category O ν, ∆ − e,R if i = 0 . Thus for i = 0 the pair of functors ( E i , F i ) for the category A ν, ∆ R is biadjoint. But we have only a one-side adjunction ( E , F ) . Note also that if R is a field, then we can define the functors as above(with the same adjunction properties) for the category A νR instead of A ν, ∆ R .Let us write ∅ for the empty l -partition. Note that ∆[ ∅ ] R = ∆ ρ ν R is theVerma module of highest weight ^ ρ ν − ρ . Since, ρ ν lies in P [ wt e ( ρ ν )] , we have ∆[ ∅ ] R ∈ O ν wt e ( ρ ν ) ,R . More generally, fix an element α = P i ∈ I e d i α i in Q + e . Put µ = wt e ( ρ ν ) − α ∈ X e . See [15, Sec. 2.3] for the definition of a highest weightcategory over a local ring. The following proposition holds, see [15, Sec. 5.5]. Proposition 2.14.
The category A νR [ α ] is a full subcategory of O νµ,R that is ahighest weight category. For λ ∈ P ld let P [ λ ] R , ∇ [ λ ] R and T [ λ ] R be the projective, costandard andthe tilting objects in A νR [ d ] with parameter λ , see [15, Prop. 2.1]. A For λ , λ ∈ P we write λ > λ if ( λ ) r > ( λ ) r for each r ∈ [1 , N ] . Here, ( λ i ) r is the r th entry of λ i for each r . We identify Q e with a sublattice of X χe via themap ι χ defined in Section 2.1. 20 emma 2.15. ( a ) For each λ , λ ∈ P we have wt χe ( λ ) − wt χe ( λ ) ∈ Q e . ( b ) If we also have λ λ , then wt χe ( λ ) − wt χe ( λ ) ∈ Q + e .Proof. It is enough to assume that we have λ = λ − ǫ r for some r ∈ [1 , N ] . Inthis case we have wt χe ( λ ) − wt χe ( λ ) = α i , where i ∈ I e is the residue of theinteger ( λ ) r modulo e .Let φ : Q e → Q e +1 and φ : X e → X e +1 be as in Section 2.2 (see also Section2.3) and Υ be as in (4). Set α = φ ( α ) ∈ Q e +1 , µ = φ ( µ ) ∈ X e +1 and β = wt χe +1 ( ρ ν ) − wt χe +1 (Υ( ρ ν )) . By Lemma 2.15, we have β ∈ Q + e +1 . Proposition 2.16.
The equivalence of categories θ µµ takes the subcategory A νR [ α ] of O νµ,R to the subcategory A νR [ β + α ] of O νµ,R . First, we prove the following lemma.
Lemma 2.17. If λ , λ ∈ P , then wt χe +1 (Υ( λ )) − wt χe +1 (Υ( λ )) = φ ( wt χe ( λ ) − wt χe ( λ )) . Proof of Lemma 2.17.
It is enough to prove the statement in the case wherewe have λ = λ − ǫ r for some r ∈ [1 , N ] . In this case we have wt χe ( λ ) − wt χe ( λ ) = α i , where i is the residue of ( λ ) r modulo e . If i = k then we have wt χe +1 (Υ( λ )) − wt χe +1 (Υ( λ )) = α i ′ = φ ( α i ) , where i ′ is as in (3). If i = k thenwe have wt χe +1 (Υ( λ )) − wt χe +1 (Υ( λ )) = α k + α k +1 = φ ( α k ) . Proof of Proposition 2.16.
By definition, A νR [ α ] ⊂ O νµ,R is the Serre subcate-gory of O νµ,R generated by ∆ λR such that the weight λ ∈ P ν satisfies λ > ρ ν and wt χe ( ρ ν ) − wt χe ( λ ) = α . Here > is the order defined before Lemma 2.15.As θ µµ (∆ λR ) is isomorphic to ∆ Υ( λ ) R , Lemma 2.17 implies that θ µµ ( A νR [ α ]) isthe Serre subcategory of O νµ,R generated by ∆ λR for λ ∈ P ν such that λ > Υ( ρ ν ) and wt χe +1 (Υ( ρ ν )) − wt χe +1 ( λ ) = α .Moreover, for each module ∆ λR ∈ O νµ,R , the weight λ has no coordinates thatare congruent to k + 1 modulo e + 1 . Then λ satisfies λ > ρ ν if and only if itsatisfies λ > Υ( ρ ν ) . We have wt χe +1 ( ρ ν ) − wt χe +1 (Υ( ρ ν )) = β . Thus θ µµ ( A νR [ α ]) is the Serre subcategory of O νµ,R generated by the modules ∆ λR where λ runsover the set of all λ ∈ P ν such that λ > ρ ν and wt χe +1 ( ρ ν ) − wt χe +1 ( λ ) = α + β .This implies θ µµ ( A νR [ α ]) = A νR [ α + β ] . A From now on, to avoid cumbersome notation we will use the following abbrevi-ations. First, for each α ∈ Q + e we set A νR [ α ] = A νR [ β + α ] , A νR [ d ] = M | α | = d A νR [ α ] , A νR = M d ∈ N A νR [ d ] . E , · · · , E e − , F , · · · , F e − of A ν, ∆ R (or of A νR is R is a field) by F = F (cid:12)(cid:12) A ν, ∆ R , · · · , F k − = F k − (cid:12)(cid:12) A ν, ∆ R , F k = F k +1 F k (cid:12)(cid:12) A ν, ∆ R ,F k +1 = F k +2 (cid:12)(cid:12) A ν, ∆ R , · · · , F e − = F e (cid:12)(cid:12) A ν, ∆ R , (10) E = E (cid:12)(cid:12) A ν, ∆ R , · · · , E k − = E k − (cid:12)(cid:12) A ν, ∆ R , E k = E k F k +1 (cid:12)(cid:12) A ν, ∆ R ,E k +1 = E k +2 (cid:12)(cid:12) A ν, ∆ R , · · · , E e − = E e (cid:12)(cid:12) A ν, ∆ R . We precise that here we use the functor E for the category A νR . This functoris not just the naive restriction of the functor E for the category O ν − ( e +1) ,R ,see Section 2.15.By definition, we have E i ( A ν, ∆ R [ α ]) ⊂ A ν, ∆ R [ α − α i ] and F i ( A ν, ∆ R [ α ]) ⊂A ν, ∆ R [ α + α i ] . Consider the endofunctors E = L i ∈ I e E i and F = L i ∈ I e F i of A ν, ∆ R . We have E ( A ν, ∆ R [ d ]) ⊂ A ν, ∆ R [ d − and F ( A ν, ∆ R [ d ]) ⊂ A ν, ∆ R [ d + 1] .Let θ α : A νR [ α ] → A νR [ α ] be the equivalence of categories in Proposition 2.16.Taking the sum over α ’s, we get an equivalence θ : A νR → A νR and an equivalence θ d : A νR [ d ] → A νR [ d ] . Moreover, we have the following commutative diagrams ofGrothendieck groups [ A ν, ∆ R ] F i −−−−→ [ A ν, ∆ R ] θ y θ y [ A ν, ∆ R ] F i −−−−→ [ A ν, ∆ R ] , [ A ν, ∆ R ] E i ←−−−− [ A ν, ∆ R ] θ y θ y [ A ν, ∆ R ] E i ←−−−− [ A ν, ∆ R ] , (11)see Section 2.13.For λ ∈ P νd we set ∆[ λ ] R = ∆ Υ( ρ ν + λ ) R ∈ A νR [ d ] . By construction we have θ d (∆[ λ ] R ) ≃ ∆[ λ ] R . Let P R [ λ ] , ∇ R [ λ ] and T R [ λ ] be the projective, the costan-dard and the tilting object with parameter λ in A νR [ d ] . O overthe field K Assume that R is as in Assumption 1. In this section we compare the categori-cal representation in O ν − e,K with the representation datum in O ν, ∆ − e,R introducedabove. Recall that the categorical representation in O ν − e,K is defined with re-spect to the set F in Section 2.10.First, for each λ ∈ P we define the following weight in X + e I f wt e ( λ ) = l X r =1 ν + ··· + ν r X t = ν + ··· + ν r − +1 e ε ( λ t ,r ) . e µ ∈ X + e I let O ν e µ,K be the Serre subcategory of O ν − e,K generated by theVerma modules ∆ λK such that f wt e ( λ ) = e µ . This decomposition is a refinementof the decomposition O ν − e,K = L µ ∈ X e O νµ,K introduced in Section 2.9. Moreprecisely, we have O νµ,K = M e µ ∈ X + e I ,π e ( e µ )= µ O ν e µ,K . Similarly, there are decompositions E = M j ∈ e I E j , F = M j ∈ e I F j such that E j and F j map O e µ,K to O e µ + e α j ,K and O e µ − e α j ,K respectively. For i ∈ I e , we set E i = M j ∈ e I,π e ( j )= i E j , F i = M j ∈ e I,π e ( j )= i F j . We have commutative diagrams O ν, ∆ − e,R E i −−−−→ O ν, ∆ − e,RK ⊗ R • y K ⊗ R • y O ν, ∆ − e,K E i −−−−→ O ν, ∆ − e,K , O ν, ∆ − e,R F i −−−−→ O ν, ∆ − e,RK ⊗ R • y K ⊗ R • y O ν, ∆ − e,K F i −−−−→ O ν, ∆ − e,K . For each element e α ∈ Q + e I let A νK [ e α ] be the Serre subcategory of A νK gen-erated by the Verma modules ∆[ λ ] K such that λ ∈ P l e α,ν . Similarly to Section2.15, we have E j ( A νK [ e α ]) ⊂ A νK [ e α − e α j ] , F j ( A νK [ e α ]) ⊂ A νK [ e α + e α j ] . See [15, Sec. 7.4] for details.Similarly, for j ∈ e I ≃ e I we can define the endofunctor E j , F j of the cate-gories O ν − ( e +1) ,K and A νK .The decomposition A νK [ α ] = L π e ( e α )= α A νK [ e α ] yields a decomposition A νK [ α ] = L π e ( e α )= α A νK [ e α ] . We also consider the endofunctors E j , F j of A νK such that for j = ( a, b ) ∈ e I we have the following analogue of (10): E j = ( E (Υ( a ) ,b ) (cid:12)(cid:12) A νK if π e ( j ) = k,E (Υ( a ) ,b ) E (Υ( a )+1 ,b ) (cid:12)(cid:12) A νK if π e ( j ) = k,F j = ( F (Υ( a ) ,b ) (cid:12)(cid:12) A νK if π e ( j ) = k,F (Υ( a )+1 ,b ) F (Υ( a ) ,b ) (cid:12)(cid:12) A νK if π e ( j ) = k. We have E j ( A νK [ e α ]) ⊂ A νK [ e α − e α j ] and F j ( A νK [ e α ]) ⊂ A νK [ e α + e α j ] .23 .19 The modules T d,R , T d,R Assume that R is as in Assumption 2.Consider the module T d,R = F d (∆[ ∅ ] R ) in A νR [ d ] and the module T d,R = F d (∆[ ∅ ] R ) in A νR [ d ] . The commutativity of the diagram (11) implies that wehave the following equality of classes [ θ d ( T d,R )] = [ T d,R ] in [ A νR [ d ]] . The mod-ules T d,R ∈ A νR [ d ] and T d,R ∈ A νR [ d ] are tilting because ∆[ ∅ ] R ∈ A νR [0] and ∆[ ∅ ] R ∈ A νR [0] are tilting and the functor F preserves tilting modules, see [15,Lem. 8.33, Lem. 5.16 (b)]. Since a tilting module is characterized by its class inthe Grothendieck group, we deduce that there is an isomorphism of modules θ d ( T d,R ) ≃ T d,R . (12)The representation datum in O ν, ∆ − e,R given in Proposition 2.8, we have ahomomorphism H d,R ( q e ) → End( T d,R ) op . Similarly, we have homomorphisms H d, k ( ζ e ) → End( T d, k ) op and H d,K ( q e ) → End( T d,K ) op . The given homomor-phisms commute with the base changes k ⊗ R • and K ⊗ R • , see [15, Prop. 8.30].Now we are going to construct a similar homomorphism H d,R ( q e ) → End( T d,R ) op (13)using the representation datum in O ν, ∆ − ( e +1) ,R . To avoid confusion, we stressthat we are not going to use (12) to define (13). We are going to construct (13)using the representation datum in O ν, ∆ − ( e +1) ,R and not the representation datumin O ν, ∆ − e,R .Now, assume that R is as in Assumption 1. A notion of a categorical repre-sentation of ( e sl e , sl ⊕ l ∞ ) is given in [9, Def. B.17]. The triple ( O ν, ∆ − e,R , O ν − e, k , O ν − e,K ) matchs this definition. (The only difference is that the categories that we con-sider here do not satisfy the Hom -finiteness condition that is assumed in [9].However, it is possible to truncate the categories to make them
Hom -finite.)Similarly, the triple ( O ν, ∆ − ( e +1) ,R , O ν − ( e +1) , k , O ν − ( e +1) ,K ) is a categorical represen-tation of ( e sl e +1 , sl ⊕ l ∞ ) . Apply [9, Thm. B.18], we get a categorical representationof ( e sl e , sl ⊕ l ∞ ) on some triple of subcategories. In particular this yields a homo-morphism (13).Similarly, we have homomorphisms H d, k ( ζ e ) → End( T d, k ) op and H d,K ( q e ) → End( T d,K ) op . The given homomorphisms commute with the base changes k ⊗ R • and K ⊗ R • . Lemma 2.18.
The homomorphism H d, k ( ζ e ) → End( T d, k ) factors through ahomomorphism ψ d, k : H νd, k ( ζ e ) → End( T d, k ) .Proof. The statement follows from the lemma below.Only in the lemma below, we assume that k is an arbitrary field and F ⊂ k × is an arbitrary subset. 24 emma 2.19. Let C = L µ ∈ X F C µ be a categorical representation of g F over k , see Definition 2.6. Let M ∈ C be an object such that there are non-negativeintegers t i for i ∈ F such that t i is non-zero only for finitely many i and • End( M ) ≃ k , • E i F i ( M ) ≃ M ⊕ t i , ∀ i ∈ F .Then the polynomial Q i ∈ F ( X − i ) t i is in the kernel of the homomorphism H d, k ( q ) → End( F d ( M )) op for each d ∈ N .Proof. It is enough to prove the statement for d = 1 . By adjointness we havethe following isomorphisms of vector spaces Hom( F i ( M ) , F i ( M )) ≃ Hom( E i F i ( M ) , M ) ≃ Hom(
M, M ) ⊕ t i ≃ k t i . Each F i ( M ) is killed by ( X − i ) t i . Then F ( M ) is killed by Q i ∈ F ( X − i ) t i .Now, we get statements similar to Lemma 2.18 for R and K . Lemma 2.20.
The homomorphism H d,K ( q e ) → End( T d,K ) factors through ahomomorphism ψ d,K : H νd,K ( q e ) → End( T d,K ) .Proof. This statement also follows from Lemma 2.19.
Lemma 2.21.
The homomorphism H d,R ( q e ) → End( T d,R ) factors through ahomomorphism ψ d,R : H νd,R ( q e ) → End( T d,R ) .Proof. This statement follows from Lemma 2.20 and from the commutativity ofthe following diagram H d,R ( q e ) −−−−→ End( T d,R ) op y y H d,K ( q e ) −−−−→ End( T d,K ) op . We still assume that R is as in Assumption 1. From now on, we assume ν r > d for each r ∈ [1 , l ] . The goal of this section is to prove that under this assumptionthe homomorphism ψ d,R in Lemma 2.21 is an isomorphism.Consider the functors Ψ νd : A νR [ d ] → mod( H νd,R ( q e )) , M Hom( T d,R , M ) , Ψ νd : A νR [ d ] → mod( H νd,R ( q e )) , M Hom( T d,R , M ) , Hom( T d,R , M ) and Hom( T d,R , M ) are considered as H νd,R ( q e ) -moduleswith respect to the homomorphisms ψ d,R and ψ d,R .Let us abbreviate Ψ = Ψ νd , Ψ = Ψ νd , T R = T d,R and T R = T d,R . We maywrite Ψ R , Ψ R to specify the ring R . For λ ∈ P ld denote by S [ λ ] R the Spechtmodule of H νd,R ( q e ) . We will use similar notation for k or K instead of R . Seealso [15, Sec. 2.4.3].Let us identify e I ≃ F ⊂ K as in Section 2.10. For each j = ( j , · · · , j n ) ∈ e I d ,the Hecke algebra H νd,K ( q e ) contains an idempotent e ( j ) such that for each fi-nite dimensional H νd,K ( q e ) -module M the subspace j M ⊂ M is the general-ized eigenspace of the commuting operators X , X , · · · , X n with eigenvalues j , j , · · · , j n respectively. The idempotent e ( j ) acts on T K = F d (∆[ ∅ ] K ) byprojection onto F j d F j d − · · · F j (∆[ ∅ ] K ) .Now, for each e α ∈ Q + e I , set e ( e α ) = P j ∈ e I e α e ( j ) and F e α = M j =( j , ··· ,j d ) ∈ e I e α F j F j · · · F j d . We obviously have F d = M e α ∈ Q + e I , | e α | = d F e α . The idempotent e ( e α ) acts on T K = F d (∆[ ∅ ] K ) by projection onto F e α (∆[ ∅ ] K ) . Lemma 2.22. ( a ) The homomorphism ψ d,K : H νd,K ( q e ) → End( T K ) op is anisomorphism. ( b ) For each λ ∈ P ld we have Ψ(∆[ λ ] K ) ≃ S [ λ ] K .Proof. First, we prove that the homomorphism ψ d,K is injective. The algebra H νd,K ( q e ) is finite dimensional and semisimple. Its center is spanned by theidempotents e ( e α ) such that e α ∈ Q + e I and | e α | = d .Since the idempotent e ( e α ) acts on T K by projection onto F e α (∆[ ∅ ] K ) , to provethe injectivity of ψ d,K we need to check that F e α (∆[ ∅ ] K ) is nonzero whenever e ( e α ) is nonzero. Similarly to the argument in Section 2.13, we see that the equivalence θ : A νK ≃ A νK yields an isomorphism of Grothendieck groups [ A νK ] ≃ [ A νK ] thatcommutes with functors F j . Thus the module F e α (∆[ ∅ ] K ) ∈ A νK is nonzero ifand only if the module F e α (∆[ ∅ ] K ) ∈ A νK is nonzero. By [15, Prop. 5.22 (d)],the module F e α (∆[ ∅ ] K ) ∈ A νK is nonzero whenever e ( e α ) is nonzero. Thus ψ d,K is injective.Thus it is also surjective because dim K H νd,K ( q e ) = dim K End( T K ) op = dim K End( T K ) op , where the first equality holds by [15, Prop. 5.22 (d)] and the second holds by(12). This implies part ( a ) .The discussion above implies that T K contains each ∆[ λ ] K , λ ∈ P ld as adirect factor. In particular T K is a projective generator of A νK [ d ] . Thus Ψ K is26n equivalence of categories. It must take ∆[ λ ] K to S [ λ ] K because S [ λ ] K is theunique simple module in the block mod( H ν e α,K ( q e )) of mod( H νd,K ( q e )) . Lemma 2.23. ( a ) The homomorphism ψ d,K : H νd,R ( q e ) → End( T R ) op is anisomorphism. ( b ) For each λ ∈ P ld we have Ψ(∆[ λ ] R ) ≃ S [ λ ] R .Proof. Consider the endomorphism u of H νd,R ( q e ) obtained from the followingchain of homomorphisms u : H νd,R ( q e ) ψ d,R → End A ν ( T R ) op θ − d → End A ν ( T R ) op ψ − d,R → H νd,R ( q e ) . The invertibility of ψ d,R is equivalent to the invertibility of u . By [15, Prop. 2.23]to prove that u is an isomorphism it is enough to show that its localization Ku : H νd,K ( q e ) → H νd,K ( q e ) is an isomorphism and that Ku induces the iden-tity map on Grothendieck groups [mod( H νd,K ( q e ))] → [mod( H νd,K ( q e ))] . Thebijectivity of Ku follows from Lemma 2.22 ( a ) .Now we check the condition on the Grothendieck group. We already knowfrom [15, Prop. 5.22 (c)] and the proof of Lemma 2.22 that Ψ K and Ψ K areequivalences of categories. Thus, by semisimplicity of the categories A νK [ d ] , A νK [ d ] and mod( H νd,K ( q e )) , we have an isomorphism of functors Ψ K ≃ Ψ K ◦ θ d because Ψ K ( M ) ≃ Ψ K ◦ θ d ( M ) for each M ∈ A νK [ d ] . This implies that Ku isthe identity on the Grothendieck group. This proves part ( a ) .Part ( b ) follows from Lemma 2.22 and the characterization of Specht mod-ules, see [15, Sec. 2.4.3]. Remark . There is no reason why the automorphism u : H νd,R ( q e ) → H νd,R ( q e ) in the proof of Lemma 2.23 should be identity. Because of this, the functor Ψ has no reason to coincide with Ψ ◦ θ d . However the automorphism u of H νd,R ( q e ) induces an autoequivalence u ∗ of mod( H νd,R ( q e )) such that we have Ψ = u ∗ ◦ Ψ ◦ θ d . (14)Now, specializing to k , we obtain the following. Corollary 2.25. ( a ) The homomorphism ψ d, k : H νd, k ( ζ e ) → End( T k ) op is anisomorphism. ( b ) For each λ ∈ P ld we have Ψ k (∆[ λ ] k ) ≃ S [ λ ] k . Let R be a local commutative C -algebra with residue field C . Let W be acomplex reflection group. Denote by S = S ( W ) and A the set of pseudo-reflections in W and the set of reflection hyperplanes respectively. Let h be the27eflection representation of W over R . Let c : S → R be a map which is constanton the W -conjugacy classes.Denote by h• , •i the canonical pairing between h ∗ and h . For each s ∈ S fixa generator α s ∈ h ∗ of Im( s (cid:12)(cid:12) h ∗ − and a generator ˇ α s ∈ h of Im( s (cid:12)(cid:12) h − suchthat h α s , ˇ α s i = 2 . Definition . The rational Cherednik algebra H c ( W, h ) R is the quotient ofthe smash product RW ⋉ T ( h ⊕ h ∗ ) by the relations [ x, x ′ ] = 0 , [ y, y ′ ] = 0 , [ y, x ] = h x, y i − X s ∈ S c s h α s , y ih x, ˇ α s i s, for each x, x ′ ∈ h ∗ , y, y ′ ∈ h . Here T ( • ) denotes the tensor algebra.Denote by O c ( W, h ) R the category O of H c ( W, h ) R , see [7, Sec. 3.2] and [15,Sec. 6.1.1]. Let E be an irreducible representation of C W . Definition . A Verma module associated with E is the following module in O c ( W, h ) R ∆ R ( E ) := Ind H c ( W, h ) R RW ⋉ R [ h ∗ ] ( RE ) . Here h ⊂ R [ h ∗ ] acts on RE by zero.The category O c ( W, h ) R is a highest weight category over R with standardmodules ∆ R ( E ) .We call a subgroup W ′ of W parabolic if it is a stabilizer of some point of b ∈ h . In this case W ′ is a complex reflection group with reflection represen-tation h / h W ′ , where h W ′ is the set of W ′ -stable points in h . Moreover, themap c : S ( W ) → R restricts to a map c : S ( W ′ ) → R . There are induction andrestriction functors O Ind WW ′ : O c ( W ′ , h / h W ′ ) R → O c ( W, h ) R , O Res WW ′ : O c ( W, h ) R → O c ( W ′ , h / h W ′ ) R , see [2]. The definitions of these functors depend on b but their isomorphismclasses are independent of the choice of b .The following lemma holds. Lemma 2.28.
Assume that W ′ and W ′′ are conjugated parabolic subgroups in W . Let P ∈ O c ( W, h ) R be a projective module. Then the following conditionsare equivalent • the module P is isomorphic to a direct factor of the module O Ind WW ′ ( P ′ ) for some projective module P ′ ∈ O c ( W ′ , h / h W ′ ) R , • the module P is isomorphic to a direct factor of a module O Ind WW ′′ ( P ′′ ) for some projective module P ′′ ∈ O c ( W ′′ , h / h W ′′ ) R .Proof. Let w be an element of W such that wW ′ w − = W ′′ . The conjugationby w yields an isomorphism W ′ ≃ W ′′ . Hence, the element w takes h W ′ to h W ′′ .Thus we get an algebra isomorphism H c ( W ′ , h / h W ′ ) R ≃ H c ( W ′′ , h / h W ′′ ) R and28n equivalence of categories O c ( W ′ , h / h W ′ ) R ≃ O c ( W ′′ , h / h W ′′ ) R . Moreover,the conjugation by w yields an automorphism t of H c ( W, h ) R such that for each x ∈ h ∗ , y ∈ h , u ∈ W we have t ( x ) = w ( x ) , t ( y ) = w ( y ) , t ( u ) = wuw − . The following diagram of functors is commutative up to equivalence of functors O c ( W, h ) R t ∗ ←−−−− O c ( W, h ) R O Ind WW ′ x O Ind WW ′′ x O c ( W ′ , h / h W ′ ) R ←−−−− O c ( W ′′ , h / h W ′′ ) R To conclude, we need only to prove that the pull-back t ∗ induces the identitymap on the Grothendieck group of O c ( W, h ) R (and thus it maps each projectivemodule to an isomorphic one). This is true because t ∗ maps each Verma module ∆( E ) R to an isomorphic one because the representation E of W does not changethe isomorphism class when we twist the W -action by an inner automorphism. From now on, we assume that R = C or that R is as in Assumption 1 withresidue field k = C .Let Γ ≃ Z /l Z be the group of complex l th roots of unity and set Γ d = S d ⋉ Γ d . For γ ∈ Γ , r ∈ [1 , l ] denote by γ r the element of Γ d having γ atthe position r and at other positions. Let s r,t be the transposition in S d exchanging r and t . For γ ∈ Γ , r, t ∈ [1 , l ] set s γr,t := s r,t γ r γ − t ∈ Γ d . Fromnow on we suppose that the group W is Γ d and h = R d is the obvious reflectionrepresentation of Γ d . Assume also that h, h , · · · , h l − are some elements of R and set h − = h l − . Let us chose the parameter c in the following way c ( s γr,t ) = − h for each r, t ∈ [1 , t ] , r = t , γ ∈ Γ ,c ( γ r ) = − l − X p =0 γ − p ( h p − h p − ) for each r ∈ [1 , l ] , γ ∈ Γ , γ = 1 . Let ν , · · · , ν l be as above. We set h = − /κ, h p = − ( ν p +1 + τ p +1 ) /κ − p/l, p ∈ [1 , l − . Let us abbreviate O νR [ d ] = O c (Γ d , R d ) R . Consider the KZ -functor KZ νd : O νR [ d ] → mod( H νd,R ( q e )) introduced in [15, Sec. 6]. Denote by ∗ O νR [ d ] the category definedin the same way as O νR [ d ] with replacement of ( ν , · · · , ν l ) by ( − ν l , · · · , − ν ) and ( τ , · · · , τ l ) by ( − τ l , · · · , − τ ) . Similarly, denote by ∗ H νd,R ( q e ) the affine Heckealgebra defined in the same way as H νd,R ( q e ) with the replacement of parametersas above. There is also a KZ -functor ∗ KZ νd : ∗ O νR [ d ] → mod( ∗ H νd,R ( q e )) .29he simple C Γ d -modules are labeled by the set P ld . We write E ( λ ) for thesimple module corresponding to λ . Set ∆[ λ ] R = ∆( E ( λ )) R . Similarly, write P [ λ ] R and T [ λ ] R for the projective and tilting object in O νR [ d ] with index λ .The category ∗ O νR [ d ] is the Ringel dual of the category O νR [ d ] , see [15,Sec. 6.2.4]. In particular we have an equivalence between the categories of stan-dardly filtered objects R : ∗ O νR [ d ] ∆ → ( O νR [ d ] ∆ ) op . Let proj( R ) be the categoryof projective finitely generated R -modules. There is an algebra isomorphism ι : H νd,R ( q e ) → ( ∗ H νd,R ( q e )) op , T r
7→ − q e T − r , X r X − r , see [15, Sec. 6.2.4]. It induces an equivalence R H = ι ∗ ( • ∨ ): mod( ∗ H νd,R ( q e )) ∩ proj( R ) → (mod( H νd,R ( q e )) ∩ proj( R )) op , where • ∨ is the dual as an R -module. By [15, (6.3)], the following diagram offunctors is commutative ∗ O νR [ d ] ∆ R −−−−→ ( O νR [ d ] ∆ ) op ∗ KZ νd y KZ νd y mod( ∗ H νd,R ( q e )) ∩ proj( R ) R H −−−−→ (mod( H νd,R ( q e )) ∩ proj( R )) op . (15) Lemma 2.29.
Assume that d = 1 . For each l -partition of λ we have anisomorphism of H ν ,R ( q e ) -modules KZ ν ( P [ λ ] R ) ≃ Ψ ν ( T [ λ ] R ) .Proof. The proof is similar to the proof of [15, Prop. 6.7]. The commutativityof the diagram (15) implies KZ ν ( P [ λ ] R ) = KZ ν ( R ( T [ λ ] R )) = R H ( ∗ KZ ν ( T [ λ ] R )) . To conclude, we just need to compare the highest weight covers R H ◦ ∗ KZ ν and Ψ ν of H ν ,R ( q e ) using Lemma 2.22 ( b ) and [15, Prop. 2.21].Let O + µ,R be the affine parabolic category O associated with the parabolictype consisting of the single block of size ν + · · · + ν l . We define the categories A + R [ d ] , A + R [ d ] and O + R [ d ] similarly. In this case we will also write the upperindex + in the notation of modules and functors (for example ∆ + [ λ ] R , T + d,R , KZ + d , etc.) Let also H + d,R ( q e ) be the cyclotomic Hecke algebra with l = 1 . It isisomorphic to the Hecke algebra of S d .We can prove the following result. Lemma 2.30.
For each λ ∈ P we have KZ +2 ( P + [ λ ] R ) ≃ Ψ +2 ( T + [ λ ] R ) .Proof. Similarly to the proof of Lemma 2.29 we compare the highest weightcovers R H ◦ ∗ KZ +2 and Ψ +2 of H +2 ,R ( q e ) using Lemma 2.22 ( b ) and [15, Prop. 2.21].30enote by Ind d,νd, + the induction functor with respect to the inclusion H + d,R ( q e ) ⊂ H νd,R ( q e ) . We will also need the following lemma. Lemma 2.31.
Assume ν r > for each r ∈ [1 , l ] . Assume also that e > . Foreach λ ∈ P there exists a tilting module T λ,R ∈ A νR [2] such that Ψ ν ( T λ,R ) ≃ Ind ,ν , + (Ψ +2 ( T + [ λ ] R )) .Proof. Set λ + = (2) , λ − = (1 , . We have ζ e = − because e > . In this casethe algebra H +2 , k ( ζ e ) is semisimple. The category A + k [2] is also semisimple. Thisimplies T +2 ,R ≃ ∆[ λ + ] R ⊕ ∆[ λ − ] R = T [ λ + ] R ⊕ T [ λ − ] R . By definition, we have Ψ +2 ( T +2 ,R ) ≃ H +2 ,R ( q e ) and Ψ ν ( T ,R ) ≃ H ν ,R ( q e ) . Thisimplies Ψ ν ( T ,R ) ≃ Ind ,ν , + (Ψ +2 ( T +2 ,R )) . By the proof of [15, Prop. 6.8], the functor Ψ ν takes indecomposable factorsof T ,R to indecomposable modules. Thus, by (14), the functor Ψ ν takes in-decomposable factors of T ,R to indecomposable modules. Thus there is a de-composition T ,R = T λ + ,R ⊕ T λ − ,R such that T λ + ,R , T λ − ,R satisfy the requiredproperties. In this section we finally give a proof of over main result.A priori there is no reason to have the following isomorphism of functors Ψ να ≃ Ψ να ◦ θ α . However, we can modify the equivalence θ α to make this true.For d < d we have an inclusion H νd ,R ( q e ) ⊂ H νd ,R ( q e ) . Let Ind d d : mod( H νd ,R ( q e )) → mod( H νd ,R ( q e )) be the induction with respect to this inclusion. The followinglemma can be proved similarly to [15, Lem. 5.41]. Lemma 2.32.
Assume that ν r > d for each r ∈ [1 , l ] . Then the followingdiagram of functors is commutative. A νR [ d ] F −−−−→ A νR [ d + 1] Ψ νd y Ψ νd +1 y mod( H νd,R ( q e )) Ind d +1 d −−−−→ mod( H νd +1 ,R ( q e )) For a partition λ denote by λ ∗ the transposed partition. For an l -partition λ = ( λ , · · · , λ l ) set λ ∗ = (( λ l ) ∗ , · · · , ( λ ) ∗ ) . There is an algebra isomorphism IM: H νd,R ( q e ) → ∗ H νd,R ( q e ) , T r
7→ − q e T − r , X r X − r , IM ∗ : mod( ∗ H νd,R ( q e )) → mod( H νd,R ( q e )) be the inducedequivalence of categories. We have IM ∗ ( S [ λ ∗ ] R ) ≃ S [ λ ] R . (16)The following proposition is proved in [15, Thm. 6.9]. Proposition 2.33.
Assume that ν r > d for each r ∈ [1 , l ] . Then there is anequivalence of categories γ d : ∗ O νR [ d ] ≃ A νR [ d ] taking ∆[ λ ∗ ] R to ∆[ λ ] R . Moreover,we have the following isomorphism of functors Ψ νd ◦ γ d ≃ IM ∗ ◦ ∗ KZ νd . Now, we prove a similar statement for A νR [ d ] . For each reflection hyperplane H of the complex reflection group Γ d let W H ⊂ Γ d be the pointwise stabilizerof H . Proposition 2.34.
Assume that ν r > d for each r ∈ [1 , l ] and e > . Thereis an equivalence of categories γ d : ∗ O νR [ d ] ≃ A νR [ d ] , taking ∆[ λ ∗ ] R to ∆[ λ ] R .Moreover, we have the following isomorphism of functors Ψ νd ◦ γ d ≃ IM ∗ ◦ ∗ KZ νd .Proof. The proof is similar to the proof of [15, Thm. 6.9]. We set C = ∗ O νR [ d ] , C ′ = A νR [ d ] . Consider the following functors Y : C → mod( H νd,R ( q e )) , Y = IM ∗ ◦ ∗ KZ νd ,Y ′ : C ′ → mod( H νd,R ( q e )) , Y ′ = Ψ νd . By [15, Prop. 2.20] it is enough to check the following four conditions.(1) We have Y (∆[ λ ∗ ] R ) ≃ Y ′ (∆[ λ ] R ) and the bijection ∆[ λ ∗ ] R ∆[ λ ] R be-tween the sets of standard objects in C and C ′ respects the highest weightorders.(2) The functor Y is fully faithful on C ∆ and C ∇ .(3) The functor Y ′ is fully faithful on C ′ ∆ and C ′∇ .(4) For each reflection hyperplane H of Γ d and each projective module P ∈O ( W H ) R we have KZ νd ( O Ind Γ d W H P ) ∈ F ′ ( C ′ tilt ) . It is explained in the proof of [15, Thm. 6.9] that condition (4) announcedhere implies the fourth condition in [15, Prop. 2.20].We have Y (∆[ λ ∗ ] R ) ≃ Y ′ (∆[ λ ] R ) by Lemma 2.23 ( b ) , [15, Lem. 6.6] and (16).The composition of the equivalence θ d : A νR [ d ] → A νR [ d ] with the equivalence γ d : ∗ O νR [ d ] ≃ A νR [ d ] yields an equivalence of highest weight categories C ≃ C ′ that takes ∆[ λ ∗ ] R to ∆[ λ ] R . This implies (1) .Condition (2) is already checked in [15, Sec. 6.3.2].The functor Ψ νd is fully faithful on A ν, ∆ R [ d ] and A ν, ∇ R [ d ] by [15, Thm. 5.37 (c)].Thus the functor Ψ νd is fully faithful on A ν, ∆ R [ d ] and A ν, ∇ R [ d ] by (14). This implies (3) .Let us check condition (4) . There are two possibilities for the hyperplane H .32 The hyperplane is
Ker( γ r − for r ∈ [1 , d ] . By Lemma 2.28, we canassume that H = Ker( γ − . By Lemma 2.29 there exists a tiltingmodule T ∈ A νR [1] such that KZ ν ( P ) ≃ Ψ ν ( T ) . We get KZ νd ( O Ind Γ d W H P ) ≃ Ind d (KZ ν ( P )) ≃ Ind d (Ψ ν ( T )) ≃ Ψ νd ( F d − ( T )) . Here the first isomorphism follows from [15, (6.1)], the third isomorphismfollows from Lemma 2.32. • The hyperplane is
Ker( s γr,t − for r, t ∈ [1 , d ] , γ ∈ Γ . By Lemma 2.28, wecan assume that H = Ker( s , ) . By Lemma 2.30 there is a tilting module T + ∈ A + R [2] such that KZ +2 ( P ) ≃ Ψ +2 ( T + ) . By Lemma 2.31 there is atilting module T ∈ A νR [2] such that Ind ,ν , + (Ψ +2 ( T + )) ≃ Ψ ν ( T ) . Thus weget Ind ,ν , + KZ +2 ( P ) ≃ Ψ ν ( T ) .We obtain KZ νd ( O Ind Γ d W H P ) ≃ Ind d,ν , + (KZ +2 ( P )) ≃ Ind d,ν ,ν (Ψ ν ( T )) ≃ Ψ νd ( F d − ( T )) . Here the first isomorphism follows from [15, (6.1)], the third isomorphism followsfrom Lemma 2.32.Now, composing the equivalences of categories in Propositions 2.33, 2.34 weobtain the following result.
Corollary 2.35.
Assume that ν r > d for each r ∈ [1 , l ] and e > . There isan equivalence of categories θ ′ d : A νR [ d ] ≃ A νR [ d ] such that we have the followingisomorphism of functors Ψ νd ◦ θ ′ d ≃ Ψ νd . For each α ∈ Q + e such that | α | = d let θ ′ α : A νR [ α ] ≃ A νR [ α ] be the restrictionof θ ′ d .From now on we work over the field C . The following lemma can be provedby the method used in [15, Sec. 5.9]. Lemma 2.36.
Assume that ν r > d for each r ∈ [1 , l ] . The following diagramsare commutative modulo an isomorphism of functors. A ν [ d ] F −−−−→ A ν [ d + 1] Ψ νd y Ψ νd +1 y mod( H νd ( ζ e )) Ind d +1 d −−−−→ mod( H νd +1 ( ζ e )) A ν [ d ] F −−−−→ A ν [ d + 1] Ψ νd y Ψ νd +1 y mod( H νd ( ζ e )) Ind d +1 d −−−−→ mod( H νd +1 ( ζ e )) Theorem 2.37.
Assume that ν r > d for each r ∈ [1 , l ] and e > . Then thefollowing diagram is commutative A ν [ d ] F k −−−−→ A ν [ d + 1] θ ′ d x θ ′ d +1 x A ν [ d ] F k −−−−→ A ν [ d + 1] . In particular, for each α ∈ Q + e such that | α | = d , the following diagram iscommutative A ν [ α ] F k −−−−→ A ν [ α + α k ] θ ′ α x θ ′ α + αk x A ν [ α ] F k −−−−→ A ν [ α + α k ] . Proof.
The result follows from Corollary 2.35, Lemma 2.36 and an argumentsimilar to [16, Lem. 2.4].
For any noetherian ring A , let mod( A ) be the category of finitely generated left A -modules. For any noetherian Z -graded ring A = L n ∈ Z A n , let grmod( A ) bethe category of Z -graded finitely generated left A -modules. The morphisms in grmod( A ) are the morphisms which are homogeneous of degree zero. For each M ∈ grmod( A ) and each r ∈ Z denote by M r the homogeneous component ofdegree r in M . For n ∈ Z let M h n i be the n th shift of grading on M , i.e., wehave ( M h n i ) r = M r − n . For each Z -graded finite dimensional C -vector space V ,let dim q V ∈ N [ q, q − ] be its graded dimension, i.e., dim q V = P r ∈ Z (dim V r ) q r .The following lemma is proved in [3, Lem. 2.5.3]. Lemma 3.1.
Assume that M is an indecomposable A -module of finite length.Then, if M admits a graded lift, this lift is unique up to grading shift andisomorphism.Definition . A Z - category (or a graded category ) is an additive category e C with a fixed auto-equivalence T : e C → e C . We call T the shift functor. For each X ∈ e C and n ∈ Z , we set X h n i = T n ( X ) . A functor of Z -categories is a functorcommuting with the shift functor.For a graded noetherian ring A the category grmod( A ) is a Z -category where T is the shift of grading, i.e., for M = ⊕ n ∈ Z M n ∈ grmod( A ) , k ∈ Z , we have T ( M ) k = M k − . 34 efinition . Let C be an abelian category. We say that an abelian Z -category e C is a graded version of C if there exists a functor F C : e C → C and a gradednoetherian ring A such that we have the following commutative diagram, wherethe horizontal arrows are equivalences of categories and the top horizontal arrowis a functor of Z -categories e C −−−−→ grmod( A ) F C y forget y C −−−−→ mod( A ) . In the setup of Definition 3.3, we say that an object e X ∈ e C is a gradedlift of an object X ∈ C if we have F C ( e X ) ≃ X . For objects X, Y ∈ C withfixed graded lifts e X, e Y the Z -module Hom C ( X, Y ) admits a Z -grading given by Hom C ( X, Y ) n = Hom e C ( e X h n i , e Y ) . In the sequel we will often denote the object X and its graded lift e X by the same symbol. Definition . For two abelian categories C , C with graded versions e C , e C we say that the functor of Z -categories e Φ: e C → e C is a graded lift of a functor Φ: C → C if F C ◦ e Φ = Φ ◦ F C . O We can extend the Bruhat order on f W to an order on c W in the followingway. For each w , w ∈ c W we have w w if and only if there exists n ∈ Z such that w π n , w π n ∈ f W and we have w π n w π n in f W . Note that theorder on c W is defined in such a way that for w , w ∈ c W we can have w w only if f W w = f W w .Fix µ = ( µ , · · · , µ e ) ∈ X e [ N ] . Let W µ be the stabilizer of the weight µ ∈ P in f W (or equivalently in c W ). Let J µ (resp. J µ, + ) be the set of shortest(resp. longest) representatives of the cosets c W /W µ in c W . For each v ∈ c W put v J µ = { w ∈ J µ ; w v } and v J µ, + = { w ∈ J µ, + ; w v } . They are finiteposets.Assume that R is a local deformation ring. Let v O µ,R be the Serre sub-category of O µ,R generated by the modules ∆ w (1 µ ) R with w ∈ v J µ . This is ahighest weight category, see [17, Lem. 3.7]. Note that the definition of the cat-egory v O µ,R does not change if we replace v by the minimal length element in vW µ (i.e., by an element of J µ ). However, in some situations it will be moreconvenient to assume that v is maximal in vW µ (and not minimal).Recall the decomposition O µ,R = M n ∈ Z O π n (1 µ ) ,R in (9). Note that the definition of the order on c W implies that the category v O µ,R lies in O π n (1 µ ) ,R , where n ∈ Z is such that v ∈ f W π n .35 .3 Linkage We still consider the non-parabolic category O . In particular we have l = N .Let k be a deformation ring that is a field. Recall that the affine Weyl group f W is generated by reflections s α , where α is a real affine root. Now we considerthe following equivalence relation ∼ k on P . We define it as the equivalencerelation generated by λ ∼ k λ when e λ + b ρ = s α ( e λ + b ρ ) for some real affineroot α . The definition of ∼ k depends on k because the definitions of e λ and b ρ depend on the elements τ r , κ ∈ k .Now, let R be a deformation ring that is a local ring with residue field k .Then for λ , λ ∈ P we write λ ∼ R λ if and only if we have λ ∼ k λ .Note that the definition of the equivalence relation above is motivated by [4,Thm. 3.2].In the particular case when R is a local deformation ring, the equivalencerelation ∼ R coincides with the equivalence relation ∼ C because we have τ r = 0 and κ = e in the residue field of R . The relation ∼ C can be easily described interms of the e -action of c W on P , introduced in Section 2.6. We have λ ∼ C λ if and only the elements λ + ρ and λ + ρ of P ( e ) are in the same f W -orbit. Remark . Let k be as above. ( a ) Assume that for each r, t ∈ [1 , l ] such that r = t we have τ r − τ t Z . Inthis case the equivalence relation ∼ k is the equality. ( b ) Assume that we have τ r − τ t ∈ Z for a unique couple ( r, t ) as above.In this case each equivalence class with respect to ∼ k contains at most twoelements. ( c ) Let R be as local deformation ring in general position with the field offractions K . By ( a ) , the equivalence relation ∼ K is just the equality. Now, let p be a prime ideal of height in R . In this case, each equivalence class with respectto ∼ R p contains at most two elements (this follows from [15, Prop. 5.22 ( a ) ], ( a ) and ( b ) ).The relation ∼ R yields a decomposition of the category O − e,R in a direct sumof subcategories, see [4, Prop. 2.8]. More precisely, let Λ be an ∼ R -equivalenceclass in P . Let O Λ ,R be the Serre subcategory of O − e,R generated by ∆( λ ) for λ ∈ Λ . Then we have O µ,R = M Λ ⊂ P [ µ ] − ρ O Λ ,R . (17)For example, if R is a local deformation ring, then this decomposition coincideswith (9). The following lemma explains what happens after the base change,see [4, Lem. 2.9, Cor. 2.10]. Lemma 3.6.
The R and T be deformation rings that are local and let R → T be a ring homomorphism. ( a ) The equivalence relation ∼ T is finer than the relation ∼ R . ( b ) Let Λ be an equivalence class with respect to ∼ R . Then T ⊗ R O Λ ,R isequal to L Λ ′ O Λ ′ ,T , where the sum is taken by all ∼ T -equivalence classes Λ ′ in Λ . efinition . We say that the category O Λ ,R is generic if Λ contains a uniqueelement and subgeneric if it contains exactly two elements.More details about the structure of generic and subgeneric categories can befound in [5, Sec. 3.1]. We assume that R is a deformation ring that is a local ring with the residue field k and the field of fractions K . Recall that we have l = N because we considerthe non-parabolic category O .Let Λ be an equivalence class in P with respect to ∼ R . Consider the category O Λ ,R as in (17). There is a partial order on Λ such that λ λ when e λ − e λ is a sum of simple roots. There exists an element λ ∈ Λ such that Λ is minimalin Λ with respect to this order. Assume that Λ is finite. Definition . The antidominant projective module in O Λ ,R is the projectivecover in O Λ ,R of the simple module L R ( λ ) , where λ is the minimal element in Λ . (The existence of the protective cover as above is explained in [4, Thm. 2.7].)This notion has no sense if Λ is infinite. However we can consider the trun-cated version. Fix v ∈ c W . We have a truncation of the decomposition (17): v O µ,R = M Λ v O Λ ,R , (18)where we put v O Λ ,R = O Λ ,R ∩ v O µ,R .By [4, Thm. 2.7] each simple module in v O µ,R has a projective cover. Asabove, we denote by λ the element of Λ that is minimal in Λ with respect tothe order . Definition . The antidominant projective module in v O Λ ,R is the projectivecover in v O Λ ,R of the simple module L R ( λ ) .From now on we assume that R is a local deformation ring in general position,see Section 2.4. Let k and K be the residue field and the field of fractions of R respectively. We set h = τ l − τ − κ + e and h r = τ r +1 − τ r for r ∈ [1 , l − . Wehave h r = 0 for each r ∈ [0 , l − because the ring is assumed to be in generalposition. Under the assumption on R , the decomposition (18) contains only oneterm. Let v P µR be the antidominant projective module in v O µ,R , i.e., v P µR is theprojective cover of L π n (1 µ ) R , where n is such that we have π n v . Lemma 3.10. ( a ) The module v P µR has a ∆ -filtration such that each Vermamodule in the category v O µ,R appears exactly ones as a subquotient in this ∆ -filtration. ( b ) For each base change R ′ ⊗ R • , where R ′ is a deformation ring that islocal, the module R ′ ⊗ R v P µR splits into a direct sum of anti-dominant projectivemodules in the blocks of the category v O µ,R ′ . roof. The first part in ( a ) holds by [5, Thm. 2 (2)] and the second part in ( a ) holds by the proof of [5, Lem. 4]. Finally, ( b ) follows from [5, Rem. 5].We will need the following lemma. Lemma 3.11.
Let A be a ring. Then the center Z (mod( A )) of the category mod( A ) is isomorphic to the center Z ( A ) of the ring A .Proof. There is an obvious injective homomorphism Z ( A ) → Z (mod( A )) . Weneed only to check that it is also surjective.Let z be an element of the center of mod( A ) . By definition, z consists of anendomorphism z M of M for each M ∈ mod( A ) such that these endomorphismscommute with all morphisms between the modules in mod( A ) . Then z A is anendomorphism of the A -module A that commutes with each other endomor-phism of the A -module A . Thus z A is a multiplication by an element a in thecenter of A .Now we claim that for each module M ∈ mod( A ) the endomorphism z M is the multiplication by a . Fix m ∈ M . Let φ : A → M be the morphism of A -modules sending to m . We have φ ◦ z A = z M ◦ φ . Then z M ( m ) = z M ◦ φ (1) = φ ◦ z A (1) = φ ( a ) = am. This completes the proof.Let Z µ,R (resp. v Z µ,R ) be the center of the category O µ,R (resp. v O µ,R ). Proposition 3.12.
The evaluation homomorphism v Z µ,R → End( v P µR ) is anisomorphism.Proof. The statement is proved in [5, Lem. 5]. There are however some subtlepoints that we explain.Firstly, the statement of [5, Lem. 5] announces the result for the non-truncated category O . But in fact, the main point of the proof of [5, Lem. 5] isto show the statement first in the truncated case.Secondly, [5, Lem. 5] assumes that the deformation ring R is the localizationof the symmetric algebra S ( b h ) at the maximal ideal generated by b h . Let us sketchthe argument of [5, Lem. 5] to show that it works well for our assumption on R .Denote by ev R : v Z µ,R → End( v P µR ) the homomorphism in the statement.Let I ( R ) be the set of prime ideals of height in R . We claim that we have v Z µ,R = \ p ∈ I ( R ) v Z µ,R p , (19)where the intersection is taken inside of v Z µ,K . Really, let v A µ,R be the en-domorphism algebra of the minimal projective generator of v O µ,R . We havean equivalence of categories v O µ,R ≃ mod( v A µ,R ) . By Lemma 3.11 we have analgebra isomorphism v Z µ,R ≃ Z ( v A µ,R ) . The same is true if we replace R by R p or K . By [4, Prop. 2.4] we have v A µ,R p ≃ R p ⊗ R v A µ,R , v A µ,K ≃ K ⊗ R v A µ,R .38he algebra v A µ,R is free over R as an endomorphism algebra of a projectivemodule in v O µ,R . Thus we have v A µ,R = T p ∈ I ( R ) v A µ,R p , where the intersec-tion is taken in v A µ,K . If we intersect each term in the previous formula with v Z µ,K = Z ( v A µ,K ) , we get (19).Similarly, we have End( v P µR ) = \ p ∈ I ( R ) End( R p ⊗ R v P µR ) incide of End( K ⊗ R v P µR ) .To conclude, we only need to show that the evaluation homomorphisms ev R p : v Z µ,R p → End( R p ⊗ R v P µR ) , ev K : v Z µ,K → End( K ⊗ R v P µR ) are isomorphisms for each p ∈ I ( R ) and that the following diagram is commu-tative End( R p ⊗ R v P µR ) −−−−→ End( K ⊗ R v P µR ) ev R p x ev K x v Z µ,R p −−−−→ v Z µ,K The commutativity of the diagram is obvious. Since R is in general position,the category v O µ,K is semisimple, see Remark 3.5. Moreover, for each p ∈ I ( R ) ,the category v O µ,R p is a direct sum of blocks with at most two Verma modules ineach, see Remark 3.5. Similarly, by Lemma 3.10 ( b ) the localisation R p ⊗ R v P µR of the antidominant projective module splits into a direct sum of antidominantprojective modules. Now, the invertibility of ev R p and ev K follows from [5,Lem. 2].We will need the following lemma. Lemma 3.13.
Assume that C is an abelian category and C is an abeliansubcategory of C . Let i : C → C be the inclusion functor. For each object M ∈ C we assume that M has a maximal quotient that is in C and we denotethis quotient by τ ( M ) . Then we have the following. ( a ) The functor τ : C → C is left adjoint to i . ( b ) Let L be a simple object in C . Assume that L has a projective cover P in C . Then τ ( P ) is a projective cover of L in C .Proof. Take M ∈ C and N ∈ C . For each homomorphism f : M → N we have M/ Ker f ≃ Im f ∈ C . Thus Ker f must contain the kernel of M → τ ( M ) . Thisimplies that each homomorphism f : M → N factors through τ ( M ) . This proves ( a ) .Now, we prove ( b ) . We have a projective cover p : P → L in C . First, weclam that the object τ ( P ) is projective in C . Really, by ( a ) the functors from C to the category of Z -modules Hom C ( τ ( P ) , • ) and Hom C ( P, • ) are isomorphic.Thus the first of them should be exact because the second one is exact by39he projectivity of P . This shows that τ ( P ) is projective in C . Denote by p the surjection p : τ ( P ) → L induced by p : P → L . Let t be the surjection t : P → τ ( P ) . We have p = p ◦ t . Now we must prove that each proper submodule K ⊂ τ ( P ) is in ker p . Really, if this is not true for some K , then p ( t − ( K )) is nonzero. Then we have p ( t − ( K )) = L because the module L is simple.Then by the definition of a projective cover we must have t − ( K ) = P . This isimpossible because t is surjective and K is a proper submodule of τ ( P ) . Remark . Let A be a graded noetherian ring. Let I ⊂ A be a homogeneousideal. Put C = mod( A ) , C = mod( A/I ) , e C = grmod( A ) , e C = grmod( A/I ) .There is an obvious inclusion of categories i : C → C and it has an obviousgraded lift e i : e C → e C . The left adjoint functor τ to i is defined by τ ( M ) = M/IM and the left adjoint functor e τ to e i is also defined by e τ ( M ) = M/IM .This implies that the functor e τ is a graded lift of τ .Recall that we denote by s , · · · , s N − the simple reflections in f W . Proposition 3.15.
We have an isomorphism Z µ,R ≃ ( z w ) ∈ Y w ∈ J µ R ; z w ≡ z s r w mod h r ∀ r ∈ [0 , N − , w ∈ J µ ∩ s r J µ which maps an element z ∈ Z µ,R to the tuple ( z w ) w ∈ J µ such that z acts on theVerma module ∆ w (1 µ ) R by z w .Proof. The statement is proved in [4, Thm. 3.6] in the case where R is thelocalization of the symmetric algebra S ( b h ) at the maximal ideal generated by b h . Similarly to Proposition 3.12, the same proof works under our assumptionon the ring R .Proposition 3.15 has the following truncated version that can be proved inthe same way using the approach of localizations at the prime ideals of height1. (See, for example, the proof of Proposition 3.12). For each such localizationthe result becomes clear by [4, Cor. 3.5] and Remark 3.5. Proposition 3.16.
We have an isomorphism v Z µ,R ≃ ( z w ) ∈ Y w ∈ v J µ R ; z w ≡ z s r w mod h r ∀ r ∈ [0 , N − , w ∈ v J µ ∩ s rv J µ (20) which maps an element z ∈ v Z µ,R to the tuple ( z w ) w ∈ v J µ such that z acts onthe Verma module ∆ w (1 µ ) R by z w . v ∈ c W , set v J = { w ∈ c W ; w v } and v Z R ≃ ( ( z w ) ∈ Y w ∈ v J R ; z w ≡ z s r w mod h r ∀ r ∈ [0 , N − , w ∈ v J ∩ s rv J ) . If v is in J µ, + , then the group W µ acts on v Z R by w ( z ) = z ′ where z ′ x = z xw − for each x ∈ v J . Note that the algebra v Z W µ R of W µ -invariant elements in v Z R is obviously isomorphic to the right hand side in (20). Thus Proposition 3.16identifies the center v Z µ,R of v O µ,R with v Z W µ R . As above, we fix k ∈ [0 , e − and set µ ′ = µ − α k . From now on, we assumethat R is as in Assumption 1 with residue field k = C .From now on we also always assume that we have W µ ⊂ W µ ′ . This happensif and only if we have µ k = 1 . In this case we have J µ ′ ⊂ J µ and J µ ′ , + ⊂ J µ, + . From now on we always assume that the element v is in J µ ′ , + (thus v is also in J µ, + ). We have an inclusion of algebras v Z µ ′ ,R ⊂ v Z µ,R because v Z µ ′ ,R ≃ v Z W µ ′ R and v Z µ,R ≃ v Z W µ R . Let Res: mod( v Z µ,R ) → mod( v Z µ ′ ,R ) and Ind: mod( v Z µ ′ ,R ) → mod( v Z µ,R ) be the restriction and the induction functors.We may write Res µ ′ µ and Ind µµ ′ to specify the parameters.It is easy to see on Verma modules using two lemmas below that the functors E k and F k restrict to functors of truncated categories F k : v O ∆ µ,R → v O ∆ µ ′ ,R , E k : v O ∆ µ ′ ,R → v O ∆ µ,R . Lemma 3.17.
For each w ∈ c W , we have F k (∆ w (1 µ ) R ) ≃ ∆ w (1 µ ′ ) R .Proof. Since µ k = 1 , the weight w (1 µ ) ∈ P has a unique coordinate containingan element congruent to k modulo e . Let r ∈ [1 , N ] be the position number ofthis coordinate. By Proposition 2.8 ( e ) , we have [ F k (∆ w (1 µ ) R )] = [∆ w (1 µ )+ ǫ r R ] .The equality of classes in the Grothendieck group implies that we have an iso-morphism of modules F k (∆ w (1 µ ) R ) ≃ ∆ w (1 µ )+ ǫ r R . Finally, since w (1 µ ) + ǫ r = w (1 µ ′ ) , we get F k (∆ w (1 µ ) R ) ≃ ∆ w (1 µ ′ ) R . Lemma 3.18.
For each w ∈ c W , we have [ E k (∆ w (1 µ ′ ) R )] = P z ∈ W µ ′ /W µ [∆ wz (1 µ ) R ] .Proof. By Proposition 2.8 ( e ) , we have [ E k (∆ w (1 µ ′ ) R )] = X r [∆ w (1 µ ′ ) − ǫ r R ] , (21)where the sum in taken by all indices r ∈ [1 , N ] such that the r th coordinate of w (1 µ ) is congruent to k + 1 modulo e . For each such r we have w (1 µ ′ ) − ǫ r = z (1 µ ) for a unique element z ∈ W µ ′ /W µ . Moreover, the obtained map r z is a bijection from the set of possible indices r to W µ ′ /W µ . Thus (21) can berewritten as [ E k (∆ w (1 µ ′ ) R )] = X z ∈ W µ ′ /W µ [∆ wz (1 µ ) R ] . Lemma 3.19.
We have E k ( v P µ ′ R ) ≃ v P µR .Proof. By Lemma 3.10 ( a ) , the class [ v P µR ] of v P µR in the Grothendieck groupof v O ∆ µ,R is the sum of all classes of Verma modules of the category v O ∆ µ,R and similarly for [ v P µ ′ R ] . Taking the sum in the equality in Lemma 3.18 overall w ∈ v J µ ′ , we get [ E k ( v P µ ′ R )] = [ v P µR ] . Finally, this yields an isomorphism E k ( v P µ ′ R ) ≃ v P µR because the modules E k ( v P µ ′ R ) and v P µR are projective.Fix an isomorphism E k ( v P µ ′ R ) ≃ v P µR as above. Then by functoriality ityields an algebra homomorphism End( v P µ ′ R ) → End( v P µR ) . Lemma 3.20.
The following diagram of algebra homomorphisms is commuta-tive
End( v P µ ′ R ) −−−−→ End( v P µR ) x x v Z µ ′ ,R −−−−→ v Z µ,R , where the top horizontal map is as above, the bottom horizontal map is theinclusion and the vertical maps are the isomorphisms from Proposition 3.12.Proof. Note that each element in
End( v P µR ) is induced by the center v Z µ,R .In partilucar, each endomorphism of v P µR preserves each submodule of v P µR .Moreover, by Lemma 3.10 ( a ) , each Verma module in v O µ,R is isomorphic to asubquotient of v P µR . Thus, by Proposition 3.12 and Proposition 3.16, an elementof End( v P µR ) is determined by its action on the subquotients of a ∆ -filtration of v P µR .Fix an element z = ( z w ) in v Z µ ′ ,R , see Proposition 3.16. Fix also a ∆ -filtration of v P µ ′ R . The element z acts on v P µ ′ R in such a way that it preserveseach component of the ∆ -filtration and the induced action on the subquotient ∆ w (1 µ ′ ) R of v P µ ′ R is the multiplication by z w .For each w ∈ c W , the module E k (∆ w (1 µ ′ ) R ) is ∆ -filtered. The subquotientsin this ∆ -filtration can be described by Lemma 3.18. Since the functor E k isexact, the ∆ -filtration of v P µ ′ R induces a ∆ -filtration of v P µR ≃ E k ( v P µ ′ R ) . Thusthe image of z by v Z µ ′ ,R → End( v P µ ′ R ) → End( v P µR ) ∆ -filtration of v P µR in the following way: it actson the subquotient ∆ w (1 µ ) R of v P µR by the multiplication by z w . On the otherhand, the image of z by v Z µ ′ ,R → v Z µ,R → End( v P µR ) acts on the subquotients in the same way. This proves the statement becausean element of End( v P µR ) is determined by its action on the subquotients of a ∆ -filtration of v P µR . V Now, we assume that v is an arbitrary elements of c W . We have a functor V µ,R : v O µ,R → mod( v Z µ,R ) , M Hom( v P µR , M ) . Set v Z µ = C ⊗ R v Z µ,R and v Z = C ⊗ R v Z R . By [4, Prop. 2.6] we have C ⊗ R v P µR = v P µ . Next, [4, Prop. 2.7] yields an algebra isomorphism v Z µ ≃ End( v P µ ) . Now, consider the functor V µ : v O µ → mod( v Z µ ) , M Hom( v P µ , M ) . A Koszul grading on the category v O µ is constructed in [17]. Let us denoteby v e O µ the graded version of this category.The functor V above has following properties. Proposition 3.21. ( a ) The functor V µ,R is fully faithful on v O ∆ µ,R . ( b ) The functor V µ is fully faithful on projective objects in O µ . ( c ) The functor V µ admits a graded lift e V µ : v e O µ → grmod( v Z µ ) .Proof. Part ( a ) is [5, Prop. 2] (1). Part ( b ) is [17, Prop. 4.50] ( b ) . Part ( c ) isgiven in the proof of [17, Lem. 5.10]. All cohomology groups in this section have coefficients in C .Set G = GL N . Let B ⊂ G ( C (( t ))) be the standard Borel subgroup. Let P µ ⊂ G ( C (( t ))) be the parabolic subgroup with Lie algebra b p µ . Let X µ be thepartial affine flag ind-scheme G ( C (( t ))) /P µ . The affine Bruhat cells in X µ areindexed by J µ . For w ∈ J µ we denote by X µ,w (resp. X µ,w ) the correspondingfinite dimensional affine Bruhat cell (resp. Schubert variety). Note that we have X µ,w ≃ C ℓ ( w ) . The following statement is proved in [17, Prop. 4.43 (a)]. Lemma 3.22.
Assume v ∈ J µ ∩ f W . There is an isomorphism of graded algebrasbetween v Z µ and the cohomology H ∗ ( X µ,v ) . X µ,w and X µ,w to an arbitrary w ∈ J µ in order to get an extended version of the previous lemma.Let π be the cyclic shift the of Dynkin diagram of type A (1) N − that takes theroot α i to the root α i − for i ∈ Z /N Z . It yields an automorphism π : G → G .Then for each n ∈ Z we have a parabolic subgroup π n ( P µ ) ⊂ G ( C (( t ))) . Recallthat the symbol π also denotes an element of c W , see Section 2.6. Let X nµ bethe partial affine flag ind-scheme defined in the same way as X µ with respectto the parabolic subgroup π n ( P µ ) ⊂ G ( C (( t ))) instead of P µ . In particular, wehave X µ = X µ . Let us use the abbreviation π n ( W µ ) for the subgroup π n W µ π − n of f W . Note that the group π n ( W µ ) is the Weyl group of the Levi of π n ( P µ ) .The Bruhat cells and the Schubert varieties in X nµ are indexes by the shortestrepresentatives of the cosets in f W /π n ( W µ ) . For such a representative w let X nµ,w (resp. X nµ,w ) be the Bruhat cell (resp. Schubert variety) in X nµ .Assume that v ∈ J µ . Then v has a unique decomposition of the form v = wπ n , such that w is minimal in wπ n ( W µ ) . Then we set X µ,v = X nµ,w and X µ,v = X nµ,w . Note that for v ∈ J µ we have X µ,v ≃ C ℓ ( v ) . We get the followinggeneralization of the lemma above. Lemma 3.23.
Assume v ∈ J µ . There is an isomorphism of graded algebrasbetween v Z µ and the cohomology H ∗ ( X µ,v ) .Proof. Consider the decomposition v = wπ n as above. By definition, the trun-cated category v O µ is a Serre subcategory of O π n (1 µ ) . It is generated by modules L xπ n (1 µ ) , where x ∈ f W is such that x w . Note also that the stabilizer of theweight π n (1 µ ) in f W is π n ( W µ ) . Then, by [17, Prop. 4.43 (a)], we have an iso-morphism of graded algebras v Z µ = H ∗ ( X nµ,w ) . On the other hand the variety X µ,v is defined as X nµ,w .Now, assume that v ∈ J µ ′ , + . Recall that in this case we have an inclusionof algebras v Z µ ′ ,R ⊂ v Z µ,R because of the assumption W µ ⊂ W µ ′ . We want toshow that after the base change we get an inclusion of algebras v Z µ ′ ⊂ v Z µ .However, this is not obvious because the functor C ⊗ R • is not left exact. Butthis fact can be justified using geometry. The injectivity of the homomorphism v Z µ ′ → v Z µ is a consequence of Lemma 3.24 below.Denote by w µ the longest elements in W µ . The shortest elements in vW µ and vW µ ′ are respectively vw µ and vw µ ′ . By Lemma 3.23, we have algebraisomorphisms v Z µ ≃ H ∗ ( X µ,vw µ ) and v Z µ ′ ≃ H ∗ ( X µ ′ ,vw µ ′ ) .The group f W is a Coxeter group. In particular we have a length function ℓ : f W → N . We can extend it to c W be setting ℓ ( wπ n ) = ℓ ( w ) for each n ∈ Z and w ∈ f W . Now we are ready to prove the following result. Lemma 3.24.
There is the following isomorphism of graded v Z µ ′ -modules v Z µ ≃ µ k +1 M r =0 v Z µ ′ h r i . roof. Let J µµ ′ be the set of shortest representatives of classes in W µ ′ /W µ . Wehave the following decomposition into affine cells X µ,vw µ = a w ∈ v J µ X µ,w = a w ∈ v J µ ′ a x ∈ J µµ ′ X µ,wx . This yields v Z µ ≃ H ∗ ( X µ,vw µ ) ≃ L w ∈ v J µ H ∗ ( X µ,w ) h ℓ ( vw µ ) − ℓ ( w ) i≃ L w ∈ v J µ ′ L x ∈ J µµ ′ H ∗ ( X µ ′ ,wx ) h ℓ ( vw µ ) − ℓ ( w ) − ℓ ( x ) i . We also have X µ,v = ` w ∈ v J µ X µ,w . This implies v Z µ ′ ≃ H ∗ ( X µ ′ ,vw µ ′ ) ≃ L w ∈ v J µ ′ H ∗ ( X µ ′ ,w ) h ℓ ( vw µ ′ ) − ℓ ( w ) i Note that we have ℓ ( w µ ′ ) − ℓ ( w µ ) = µ k +1 . Moreover, for each w ∈ v J µ ′ and x ∈ J µµ ′ the variety X µ,wx is an affine fibration over X µ ′ ,w . This implies v Z µ ≃ M x ∈ J µµ ′ v Z µ ′ h ℓ ( vw µ ) − ℓ ( vw µ ′ ) − ℓ ( x ) i = µ k +1 M r =0 v Z µ ′ h r i . We will write
Ind and
Res for the induction and restriction functors
Ind: mod( v Z µ ′ ) → mod( v Z µ ) , Res: mod( v Z µ ) → mod( v Z µ ′ ) . We fix the graded lifts of g Res and g Ind of the functors
Res and
Ind in the followingway g Res( M ) = M h− µ k +1 i , g Ind( M ) = v Z µ ⊗ v Z µ ′ M. Now, Lemma 3.24 implies the following.
Corollary 3.25. ( a ) The pair of functors (Res , Ind) is biadjoint. ( b ) The pairs of functors ( g Ind , g Res h µ k +1 i ) and ( g Res , g Ind h− µ k +1 i ) are ad-joint. ( c ) g Res ◦ g Ind = Id ⊕ [ µ k +1 +1] q := µ k +1 M r =0 Id h r − µ k +1 i , where Id is the identity endofunctor of the category grmod( Z µ ′ ) . .8 Graded lifts of the functors As above we assume W µ ⊂ W µ ′ and that v ∈ J µ ′ , + . Lemma 3.26.
The following diagram of functors is commutative v O ∆ µ,R F k −−−−→ v O ∆ µ ′ ,R V µ,R y V µ ′ ,R y mod( v Z µ,R ) Res −−−−→ mod( v Z µ ′ ,R ) . Proof.
Let M be an object in v O ∆ µ,R . We have the following chain of isomor-phisms of v Z µ ′ ,R -modules. V µ ′ ,R ◦ F k ( M ) ≃ Hom( v P µ ′ R , F k ( M )) ≃ Hom( E k ( v P µ ′ R ) , M ) ≃ Hom( v P µR , M ) ≃ V µ,R ( M ) Here, the v Z µ,R -modules in the last two lines are considered as v Z µ ′ ,R -moduleswith respect to the inclusion v Z µ ′ ,R ⊂ v Z µ,R . The third isomorphism in thechain is an isomorphism of v Z µ ′ ,R -modules by Lemma 3.20. Lemma 3.27.
The following diagram of functors is commutative v O ∆ µ,R E k ←−−−− v O ∆ µ ′ ,R V µ,R y V µ ′ ,R y mod( v Z µ,R ) Ind ←−−−− mod( v Z µ ′ ,R ) . Proof.
Let M be an object in v O ∆ µ ′ ,R . We have the following chain of isomor-phisms of v Z µ,R -modules. V µ,R ◦ E k ( M ) ≃ Hom( v P µR , E k ( M )) ≃ Hom( F k ( v P µR ) , M ) ≃ Hom( V µ ′ ,R ◦ F k ( v P µR ) , V µ ′ ,R ( M )) ≃ Hom(Res ◦ V µ,R ( v P µR ) , V µ ′ ,R ( M )) ≃ Hom( v Z µ,R , V µ ′ ,R ( M )) ≃ Ind ◦ V µ ′ ,R ( M ) Here the third isomorphism holds by Proposition 3.21 ( a ) , the fourth isomor-phism holds by Lemma 3.26. The last isomorphism holds because, by Corollary3.25 ( a ) , the functor Hom v Z µ ′ ,R ( v Z µ,R , • ) , which is obviously right adjoint to Res , is isomorphic to
Ind .Now, Lemmas 3.26-3.27 imply the following.46 orollary 3.28.
The following diagrams of functors are commutative v O µ F k −−−−→ v O µ ′ V µ y V µ ′ y mod( v Z µ ) Res −−−−→ mod( v Z µ ′ ) , v O µ E k ←−−−− v O µ ′ V µ y V µ ′ y mod( v Z µ ) Ind ←−−−− mod( v Z µ ′ ) . Proof.
Passage to the residue field in Lemma 3.27 implies that the diagramsin the statement are commutative on ∆ -filtered objects. A standard argument(see for example the proof of Lemma 3.29) shows that the commutativity on ∆ -filtered objects implies the commutativity.Let v O proj µ and v e O proj µ be the full subcategories of projective modules in v O µ and v e O µ respectively. The fully faithfulness of the functor V µ on projective mod-ules implies the fully faithfulness of the functor e V µ on projective modules. Thesefunctors identify v O proj µ, and v e O proj µ with some full subcategories in mod( v Z µ ) and grmod( v Z µ ) that we denote mod( v Z µ ) proj and grmod( v Z µ ) proj respectively.Since the functor F k takes projective modules to projective modules, the com-mutativity of the diagram in Corollary 3.28 implies that the functor Res takesthe category mod( v Z µ ) proj to mod( v Z µ ′ ) proj . This implies that its graded lift g Res takes grmod( v Z µ ) proj to grmod( v Z µ ′ ) proj . Similar statements hold for Ind and g Ind . Lemma 3.29. ( a ) The functors E k and F k admit graded lifts e E k : v e O µ ′ → v e O µ and e F k : v e O µ → v e O µ ′ . They can be chosen in such a way that the condition belowholds. ( b ) The following pairs of functors are adjoint ( e F k , e E k h− µ k +1 i ) , ( e E k , e F k h µ k +1 i ) . Proof.
Let us prove ( a ) . We give the prove only for the functor F k . The prooffor E k is similar. The proof below is similar to the proof of [17, Lem. 5.10].As explained above, the functor g Res restricts to a functor grmod( v Z µ ) proj → grmod( v Z µ ′ ) proj . Together with the equivalences of categories v e O proj µ ≃ grmod( v Z µ ) proj and v e O proj µ ′ ≃ grmod( v Z µ ′ ) proj obtained by restricting e V µ and e V µ ′ this yields afunctor e F k : v e O proj µ → v e O proj µ ′ . Next, we obtain a functor of homotopy categories e F k : K b ( v e O proj µ ) → K b ( v e O proj µ ′ ) . Since the categories v e O µ and v e O µ ′ have finiteglobal dimensions, we have equivalences of categories K b ( v e O proj µ ) ≃ D b ( v e O µ ) K b ( v e O proj µ ′ ) ≃ D b ( v e O µ ′ ) . Thus we get a functor of triangulated categories e F k : D b ( v e O µ ) → D b ( v e O µ ′ ) . If we repeat the same construction for non-gradedcategories, we obtain a functor F k : D b ( v O µ ) → D b ( v O µ ′ ) that is the same as thefunctor between the bounded derived categories induced by the exact functor F k : v O µ → v O µ ′ , see Corollary 3.28. This implies that the following diagram iscommutative D b ( v e O µ ) e F k −−−−→ D b ( v e O µ ′ ) forget y forget y D b ( v O µ ) F k −−−−→ D b ( v O µ ′ ) Since the bottom functor takes v O µ to v O µ ′ , the top functor takes v e O µ to v e O µ ′ .This completes the proof of ( a ) .Now we prove ( b ) . The functors e E k and e F k are constructed as unique functorssuch that we have the following commutative diagrams v O µ e F k −−−−→ v O µ ′ e V µ y e V µ ′ y mod( v Z µ ′ ) g Res −−−−→ mod( v Z µ ′ ) , v O µ e E k ←−−−− v O µ ′ e V µ y e V µ ′ y mod( v Z µ ) g Ind ←−−−− mod( v Z µ ′ ) . (22)By Corollary 3.25 ( b ) and Proposition 3.21 ( b ) , the restrictions of the pairs ( e F k , e E k h− µ k +1 i ) and ( e E k , e F k h µ k +1 i ) to the subcategories of projective objectsare biadjoint. We can conclude using the lemma below. Lemma 3.30.
Let C , C be abelian categories of finite global dimension and let C ′ , C ′ be the full subcategories of projective objects. Assume that E : C → C , F : C → C are exact functors. Assume that E and F send projective objects toprojective objects and denote E ′ : C ′ → C ′ , F ′ : C ′ → C ′ the restrictions of E and F . Assume that the pair ( E ′ , F ′ ) is adjoint. Then the pair ( E, F ) is adjoint.Proof. Let ε ′ : E ′ F ′ → Id , η ′ : Id → F ′ E ′ be the counit and the unit of the adjoint pair ( E ′ , F ′ ) .We can extend the functors E ′ and F ′ to functors E ′ : K b ( C ′ ) → K b ( C ′ ) and F ′ : K b ( C ′ ) → K b ( C ′ ) of the homotopy categories of bounded complexes.The counit ε ′ and the unit η ′ extend to natural transformations of functors ofhomotopy categories. These extended natural transformations still satisfy theproperties of the counit and the unit of an adjunction. Thus the extended pair ( E ′ , F ′ ) is adjoint.Since the categories C and C have finite global dimensions, we have equiv-alences of categories K b ( C ′ ) ≃ D b ( C ) , K b ( C ′ ) ≃ D b ( C ) . (23)48y construction, the functors E : D b ( C ) → D b ( C ) , F : D b ( C ) → D b ( C ) (24)obtained from functors E ′ and F ′ via the equivalences (23) coincide with thefunctors induced from E : C → C and F : C → C . The pair of functors ( E, F ) in(24) is adjoint with a counit ε and a unit η , obtained from ε ′ and η ′ . These counitand unit restrict to natural transformations of functors of abelian categories E : C → C and F : C → C . This proves the statement.We need the following lemma later. Lemma 3.31.
We have the following isomorphism of functors e F k e E k ≃ Id ⊕ [ µ k +1 +1] q := µ k +1 M r =0 Id h r − µ k +1 i , where Id is the identity endofunctor of the category v e O µ ′ .Proof. By Corollary 3.25 ( c ) we have g Res ◦ g Ind ≃ Id ⊕ [ µ k +1 +1] q . Then the dia-grams (22) and Proposition 3.21 ( b ) yield an isomorphism e F k e E k ≃ Id ⊕ [ µ k +1 +1] q on projective modules in v e O µ ′ . This isomorphism can be extended to the cate-gory v e O µ ′ in the same way as in the proof of Lemma 3.30. W µ ′ ⊂ W µ In the sections above we assumed W µ ⊂ W µ ′ (or equivalently µ k = 1 ). Inthis section we announce similar results in the case W µ ′ ⊂ W µ (or equivalently µ k +1 = 0 ). All the proofs are the same as in the previous case but the roles of E k and F k should be exchanged.Here we always assume that v is in J µ, + (thus also in J µ ′ , + ). In contrastwith the situation above, we have v Z µ ′ ⊂ v Z µ . Consider the induction andthe restriction functors Ind: mod( v Z µ ′ ) → mod( v Z µ ) and Res: mod( v Z µ ) → mod( v Z µ ′ ) .Similarly to Corollary 3.28 we can prove the following statement. Lemma 3.32.
The following diagrams of functors are commutative v O µ F k −−−−→ v O µ ′ V µ y V µ ′ y mod( v Z µ ) Ind −−−−→ mod( v Z µ ′ ) , v O µ E k ←−−−− v O µ ′ V µ y V µ ′ y mod( v Z µ ) Res ←−−−− mod( v Z µ ′ ) . Lemma 3.33. ( a ) The functors E k and F k admit graded lifts e E k : v e O µ ′ → v e O µ .They can be chosen in such a way that the conditions below hold. ( b ) The following pairs of functors are adjoint ( e F k , e E k h µ k − i ) , ( e E k , e F k h− µ k + 1 i ) . ( c ) We have the following isomorphism of functors e E k e F k ≃ Id ⊕ [ µ k ] q := µ k − M r =0 Id h r − µ k + 1 i , where Id is the identity endofunctor of the category v e O µ . Let B be a C -algebra isomorphic to a finite direct sum of copies of C . We have B = L λ ∈ Λ C e λ for some idempotents e λ . Definition . Let bmod( B ) be the category of finite dimensional ( B, B ) -bimodules.A bimodule M ∈ bmod( B ) can be viewed just as a collection of finitedimensional C -vector spaces e λ M e µ for λ, µ ∈ Λ . To each bimodule M ∈ bmod( B ) we can associate a bimodule M ⋆ ∈ bmod( M ) as follows M ⋆ =Hom bmod( B ) ( M, B ⊗ C B ) . The bimodule structure on M ⋆ is defined in thefollowing way. For f ∈ M ⋆ , m ∈ M , b , b ∈ B we have b f b ( m ) = f ( b mb ) . Lemma 4.2.
Assume that
M, N ∈ bmod( B ) , X, Y ∈ mod( B ) , Z ∈ mod( B ) op .Then we have the following isomorphisms: (a) Hom bmod( B ) ( M, N ) ≃ L λ,µ ∈ Λ Hom C ( e λ M e µ , e λ M e µ ) , (b) Hom B ( X, Y ) ≃ L λ ∈ Λ Hom C ( e λ M, e λ M ) , (c) X ⊗ B Z = L λ ∈ Λ Xe λ ⊗ C e λ Z , (d) e λ M ⋆ e µ ≃ ( e µ M e λ ) ∗ , where • ∗ is the usual duality for C -vector spaces, (e) Hom B ( M ⋆ ⊗ B X, Y ) ≃ Hom B ( X, M ⊗ B Y ) , (f) ( M ⊗ B N ) ⋆ ≃ N ⋆ ⊗ B M ⋆ .Proof. Parts (a), (b), (c) are obvious. Part (d) follows from (a). Part (e) followsfrom (b), (c), (d). Part (f) follows from (c), (d).50 .2 Quadratic dualities
Let A = ⊕ n ∈ N A n be a finite dimensional N -graded algebra over C . Assume that A is semisimple and basic. Let T A ( A ) = L n ∈ N A ⊗ n be the tensor algebraof A over A , here A ⊗ n means A ⊗ A A ⊗ A · · · ⊗ A A with n components A . The algebra A is said to be quadratic if it is generated by elements ofdegree and with relations in degree , i.e., the kernel of the obvious map T A ( A ) → A is generated by elements in A ⊗ A A . Definition . Consider the ( A , A )-bimodule morphism φ : A ⊗ A A → A given by the product in A . Let φ ⋆ : A ⋆ → A ⋆ ⊗ A A ⋆ be the dual morphism to φ , see Lemma 4.2, here • ⋆ is as in Section 4.1 with respect to B = A . The quadratic dual algebra to A is the quadratic algebra A ! = T A ( A ⋆ ) / (Im φ ⋆ ) . Remark . In the previous definition we do not assume that the algebra A isquadratic itself. However, if it is true, we have a graded C -algebra isomorphism ( A ! ) ! ≃ A .Let C be an abelian category such that its objects are graded modules.Denote by Com ↓ ( C ) the category of complexes X • in C such that the j th gradedcomponent of X i is zero when i >> or i + j << . Similarly, let Com ↑ ( C ) the category of complexes X • in C such that the j th graded component of X i iszero when i << or i + j >> . Denote by D ↓ ( C ) and D ↑ ( C ) the correspondingderived categories of such complexes. We will use the following abbreviations D ↓ ( A ) = D ↓ (grmod( A )) , D ↑ ( A ) = D ↑ (grmod( A )) , D b ( A ) = D b (grmod( A )) . In the situation above we have the following functors K : D ↓ ( A ) → D ↑ ( A ! ) and K ′ : D ↑ ( A ! ) → D ↓ ( A ) called quadratic duality functors . See [13, Sec. 5] formore details. Let A = L n ∈ N A n be a finite dimensional N -graded C -algebra such that A issemisimple. We identify A with the left graded A -module A ≃ A/ ⊕ n> A n . Definition . The graded algebra A is Koszul if the left graded A -module A admits a projective resolution · · · → P → P → P → A such that P r isgenerated by its degree r component.If A is Koszul, we consider the graded C -algebra A ! = Ext ∗ A ( A , A ) op andwe call it the Koszul dual algebra to A . The following is well-known, see [3]. Proposition 4.6.
Let A be a Koszul C -algebra. Assume that A and A ! arefinite dimensional. Then, the following holds. ( a ) The algebra A is quadratic. The Koszul dual algebra A ! coincides withthe quadratic dual algebra. ( b ) The algebra A ! is also Koszul and there is a graded algebra isomorphism ( A ! ) ! ≃ A . c ) There is an equivalence of categories K : D b ( A ) → D b ( A ! ) , M RHom A ( A , M ) . If A is Koszul, then the functors K and K ′ from the previous section aremutually inverse. Moreover, the equivalence K of bounded derived categories inProposition 4.6 ( c ) is the restriction of the functor K from the previous section. Definition . Let A and B be Koszul algebras. We say that the functor Φ: D b ( A ) → D b ( B ) is Koszul dual to the functor Ψ: D b ( A ! ) → D b ( B ! ) if thefollowing diagram of functor is commutative D b ( A ) Ψ −−−−→ D b ( B ) K y K y D b ( A ! ) Φ −−−−→ D b ( B ! ) . In this section we recall some results from [13] about linear complexes. Let A be as in Section 4.2. Definition . Let LC ( A ) be the category of complexes · · · → X k − → X k →X k +1 → · · · of projective modules in grmod( A ) such that for each k ∈ Z eachindecomposable direct factor P of X k is a direct factor of A h k i . Proposition 4.9.
There is an equivalence of categories ǫ A : LC ( A ) ≃ grmod( A ! ) . Let us describe the construction of ǫ − A . Let M = ⊕ n ∈ Z M n be in grmod( A ! ) .The graded A ! -module structure yields morphisms of A -modules f ′ n : A !1 ⊗ M n → M n +1 for each n ∈ Z . We have Hom A ( A !1 ⊗ A M n , M n +1 ) = Hom A ( M n , ( A !1 ) ⋆ ⊗ A M n +1 )= Hom A ( M n , A ⊗ A M n +1 ) . Let f n : Hom A ( M n , A ⊗ A M n +1 ) be the image of f ′ n by the chain of iso-morphisms above.We have ǫ − A ( M ) = · · · ∂ k − → X k − ∂ k − → X k ∂ k → X k +1 ∂ k +1 → · · · with X k = A h k i ⊗ A M k and ∂ k : A h k i ⊗ A M k → A h k + 1 i ⊗ A M k +1 , a ⊗ m ( a ⊗ Id)( f k ( m )) . The quadratic duality functor discussed in the previous section can be char-acterized as follows, see [13, Prop. 21].52 emma 4.10.
Up to isomorphism of functors, the following diagram is com-mutative: D ↑ ( LC ( A )) D ↓ ( A ) D ↑ ( A ! ) ✑✑✑✑✰ Tot ✛ K ′ ◗◗◗◗❦ ǫ − A , where Tot is the functor taking the total complex.
Let { e λ ; λ ∈ Λ } be the set of indecomposable idempotents of A , i.e., we have A = L λ ∈ Λ C e λ . Denote by e ! λ the corresponding idempotent of A !0 via theidentification A ≃ A !0 . For each subset Λ ′ ⊂ Λ set e Λ ′ = P λ ∈ Λ ′ e λ . Considerthe graded algebras A Λ ′ = e Λ ′ Ae Λ ′ , Λ ′ A = A/ ( e Λ \ Λ ′ ) . Similarly, we can define A !Λ ′ and Λ ′ A ! .We have a functor F : grmod( A Λ ′ ) → grmod( A ) , M Ae Λ ′ ⊗ A Λ ′ M . Notealso that the category grmod( Λ ′ A ! ) can be viewed as a subcategory of grmod( A ! ) containing modules that are killed by e Λ \ Λ ′ . Let ι : grmod( Λ ′ A ! ) → grmod( A ! ) be the inclusion. The following proposition is proved in [13, Thm. 28]. Proposition 4.11. (a)
The quadratic dual algebra to A Λ ′ is isomorphic to Λ ′ A ! . (b) The following diagram commutes up to isomorphism of functors. D ↓ ( A ) K ′ ←−−−− D ↑ ( A ! ) F x ι x D ↓ ( A Λ ′ ) K ′ ←−−−− D ↑ ( Λ ′ A ! ) Idea of proof of (b).
By Lemma 4.10 it is enough to proof the commutativity ofthe following diagram. LC ( A ) ǫ − A ←−−−− grmod( A ! ) F x ι x LC ( A Λ ′ ) ǫ − A Λ ′ ←−−−− grmod( Λ ′ A ! ) We can generalize this result as follows.53 emma 4.12.
Let A ′ be a finite dimensional N -graded C -algebra. Assume thatfor some subset Λ ′ ⊂ Λ there is a graded (unitary) homomorphism ψ : A ′ → A Λ ′ such that (a) ψ is an isomorphism in degrees and , (b) ψ induces an isomorphism between the kernel of A ′ ⊗ A ′ A ′ → A ′ and thekernel of ( A Λ ′ ) ⊗ ( A Λ ′ ) ( A Λ ′ ) → ( A Λ ′ ) .Then the quadratic dual of A ′ is isomorphic to Λ ′ A .Consider the graded ( A, A ′ ) -bimodule Ae Λ ′ , where the right A ′ -module struc-ture is obtained from the right A Λ ′ -module structure using ψ . Consider the func-tor T : grmod( A ′ ) → grmod( A ) , M Ae Λ ′ ⊗ A ′ M . Then the following diagramcommutes up to an isomorphism of functors. D ↓ ( A ) K ′ ←−−−− D ↑ ( A ! ) T x ι x D ↓ ( A ′ ) K ′ ←−−−− D ↑ ( Λ ′ A ! ) Proof.
By definition, the quadratic dual of A ′ depends only on the algebra A ′ ,the ( A ′ , A ′ ) -bimodule A ′ and the kernel of A ′ ⊗ A ′ A ′ → A ′ . Thus the quadraticdual algebras of A ′ and A Λ ′ are isomorphic. Finally, Proposition 4.11 (a) impliesthat the quadratic dual of A ′ is isomorphic to Λ ′ A .Now, by Lemma 4.10 is enough to prove the commutativity of the followingdiagram up to an isomorphism of functors. LC ( A ) ǫ − A ←−−−− grmod( A ! ) T x ι x LC ( A ′ ) ǫ − A ′ ←−−−− grmod( Λ ′ A ! ) By analogy with the definition of the functor T , consider the functor Φ: grmod( A ′ ) → grmod( A Λ ′ ) , M A Λ ′ ⊗ A ′ M . For each λ ∈ Λ ′ let e ′ λ be the idempotent in A ′ such that ψ ( e ′ λ ) = e λ . We have Φ( A ′ e ′ λ ) = A Λ ′ e λ for each λ ∈ Λ ′ . Inparticular Φ induces a bijection between the indecomposable direct factors of A ′ and A Λ ′ . Thus Φ induces a functor Φ: LC ( A ′ ) → LC ( A Λ ′ ) . Note that bydefinition the boundary maps in the complexes of the category LC ( • ) are of de-gree . Thus, by (a) and (b) the functor Φ induces an equivalence of categories Φ: LC ( A ′ ) → LC ( A Λ ′ ) .Consider the following diagram, where the functor F is as before Proposition4.11. LC ( A ) Id ←−−−− LC ( A ) ǫ − A ←−−−− grmod( A ! ) T x F x ι x LC ( A ′ ) Φ − ←−−−− LC ( A Λ ′ ) ǫ − A Λ ′ ←−−−− grmod( Λ ′ A ! ) ǫ − A ′ = Φ − ◦ ǫ − A Λ ′ .Let us check that Φ ◦ ǫ − A ′ = ǫ − A Λ ′ . This is clear on objects because ǫ − A Λ ′ ( M ) = · · · ∂ ′ k − → A Λ ′ h k i ⊗ ( A Λ ′ ) M k ∂ ′ k → A Λ ′ h k + 1 i ⊗ ( A Λ ′ ) M k +1 ∂ ′ k +1 → · · · ,ǫ − A ′ ( M ) = · · · ∂ ′′ k − → A ′ h k i ⊗ A ′ M k ∂ ′′ k → A ′ h k + 1 i ⊗ A ′ M k +1 ∂ ′′ k +1 → · · · . The boundary maps are defined as follows ∂ ′ k : A Λ ′ h k i ⊗ ( A Λ ′ ) M k → A Λ ′ h k + 1 i ⊗ ( A Λ ′ ) M k +1 , a ⊗ m ( a ⊗ Id)( f n ( m )) ,∂ ′′ k : A ′ h k i ⊗ A ′ M k → A ′ h k + 1 i ⊗ A ′ M k +1 , a ⊗ m ( a ⊗ Id)( f n ( m )) , where f n : M n → ( A Λ ′ ) ⊗ ( A Λ ′ ) M n +1 and f n : M n → A ′ ⊗ A ′ M n +1 are definedin the same way as f n in the definition of ǫ − . Thus it is also clear that Φ commutes with the boundary maps. Remark . Condition (b) is necessary only to deduce that ( A ′ ) ! ≃ ( A Λ ′ ) ! .Without this condition we know only that the algebra ( A ′ ) ! is isomorphic toa quotient of ( A Λ ′ ) ! . Thus condition (b) can be replaced by the requirement dim( A ′ ) ! = dim Λ ′ A ! .We can reformulate Lemma 4.12 in the following way. Corollary 4.14.
Let A ′ be an N -graded finite dimensional C -algebra with basic A ′ such that the indecomposable idempotents of A ′ are parameterized by a subset Λ ′ of Λ , i.e., we have A ′ = L λ ∈ Λ ′ C e ′ λ . Assume that dim( A ′ ) ! = dim Λ ′ A ! .Assume also that there is an exact functor T : grmod( A ′ ) → grmod( A ) such that (a) T ( A ′ e ′ λ ) = Ae λ ∀ λ ∈ Λ ′ , (b) the functor T yields an isomorphism Hom A ′ ( A ′ e ′ λ h i , A ′ e ′ µ ) ≃ Hom A ( Ae λ h i , Ae µ ) .Then the quadratic dual for A ′ is Λ ′ A ! and the following diagram commutes upto isomorphism of functors. D ↓ ( A ) K ′ ←−−−− D ↑ ( A ! ) T x ι x D ↓ ( A ′ ) K ′ ←−−−− D ↑ ( Λ ′ A ! ) Proof.
Condition (a) implies that the functor T yields a homomorphism ofgraded algebras ψ : A ′ → A Λ ′ . Moreover, condition (b) implies that ψ satis-fies condition (a) of Lemma 4.12. Finally, the assumption dim( A ′ ) ! = dim Λ ′ A implies that ψ satisfies condition (b) of Lemma 4.12, see Remark 4.13. The func-tor T hare can be identified with the functor T = Ae Λ ′ ⊗ A ′ • in the statementof Lemma 4.12, see [18, Lem. 3.4]. Thus the statement follows from Lemma4.12. 55 .6 Zuckerman functors Fix v ∈ c W . Let ν and ν be two different parabolic types such that W ν ⊂ W ν .By definition of the parabolic category O , there is an inclusion of categories v O ν µ ⊂ v O ν µ . We denote by inc the inclusion functor. We may write inc = inc ν ν to specify the parameters. The functor inc admits a left adjoint functor tr . For M ∈ v O ν µ , the object tr( M ) is the maximal quotient of M that is in v O ν µ , seeLemma 3.13 ( a ) . We call the functor tr the parabolic truncation functor. Wemay write tr ν ν to specify the parameters.Now, we assume that ν and ν are two arbitrary parabolic types. Then thereis a parabolic type ν such that we have W ν = W ν ∩ W ν . The Zuckermanfunctor
Zuc ν ν (or simply Zuc ) is the composition
Zuc ν ν = tr ν ν ◦ inc ν ν .The parabolic inclusion functor is exact. The parabolic truncation functoris only right exact. This implies that the Zuckerman functor is right exact.Now, we are going to grade Zuckerman functors. Let v A νµ be the endomor-phism algebra of the minimal projective generator of v O νµ (or simply v A µ in thenon-parabolic case). We have v O νµ ≃ mod( v A νµ ) . The Koszul grading on v A νµ isconstructed in [17]. The graded version v e O νµ of v O νµ is the category grmod( v A νµ ) .Moreover, the algebra v A νµ is the quotient of v A µ by a homogeneous ideal I ν . Byconstruction, the grading on v A νµ is induced from the grading on v A µ . Assumethat ν and ν are such that W ν ⊂ W ν . Then we have I ν ⊂ I ν . This impliesthat the graded algebra v A ν µ is isomorphic to the quotient of the graded alge-bra v A ν µ by the homogeneous ideal I ν /I ν . This yields an inclusion of gradedcategories v e O ν µ ⊂ v e O ν µ . Let us denote by f inc ν ν (or simply f inc ) the inclusionfunctor. It is a graded lift of the functor inc . Similarly, its left adjoint functor e tr ν ν is a graded lift of the functor tr , see Remark 3.14. Thus we get graded lifts g Zuc ν ν of the Zuckerman functor Zuc ν ν for arbitrary parabolic types ν and ν .Similarly, we can define the parabolic inclusion functor, the parabolic trun-cation functor, the Zuckerman functor and their graded versions for the affinecategory O at a positive level. O As above, we assume W µ ⊂ W µ ′ . Set J νµ = { w ∈ J µ ; w (1 µ ) ∈ P ν } . Notethat the inclusion J µ ′ ⊂ J µ induces an inclusion J νµ ′ ⊂ J νµ . For v ∈ c W we set v J νµ = { w ∈ J νµ ; w v } .As in Section 3 we assume that we have W µ ⊂ W µ ′ . Fix a parabolic type ν = ( ν , · · · , ν l ) ∈ X l [ N ] .Assume v ∈ J νµ ′ w µ ′ . The functors F k : v O µ → v O µ ′ , E k : v O µ ′ → v O µ restrict to functors of parabolic categories F k : v O νµ → v O νµ ′ , E k : v O νµ ′ → v O νµ . w ∈ v J νµ . Let v P w (1 µ ) be the projective cover of L w (1 µ ) in v O νµ . (Note that we do not indicate the parabolic type ν in our notations formodules to simplify the notations.) We fix the grading on L w (1 µ ) such that itis concentrated in degree zero when we consider L w (1 µ ) as an v A νµ -module (seeSection 4.6 for the definition of v A νµ ). A standard argument shows that themodules v P w (1 µ ) and ∆ w (1 µ ) admit graded lifts. (The graded lift of v P w (1 µ ) canbe constructed as the projective cover of the graded lift of L w (1 µ ) in v e O νµ . Theexistence of graded lifts of projective modules implies the existence of gradedlifts of Verma modules, see [12, Cor. 4].) We fix the graded lifts of v P w (1 µ ) and ∆ w (1 µ ) such that the surjections v P w (1 µ ) → L w (1 µ ) and ∆ w (1 µ ) → L w (1 µ ) arehomogeneous of degree zero, see also Lemma 3.1.The following lemma is stated in the parabolic category O . Lemma 4.15. ( a ) For each w ∈ v J νµ ′ , we have E k ( v P w (1 µ ′ ) ) = v P w (1 µ ) . ( b ) For each w ∈ v J νµ , we have F k ( L w (1 µ ) ) = (cid:26) L w (1 µ ′ ) if w ∈ v J νµ ′ , else . Proof.
First, we prove ( a ) in the non-parabolic situation (i.e., for ν = (1 , , · · · , ).The modules E k ( v P w (1 µ ′ ) ) and v P w (1 µ ) are both projective. Thus it is enoughto show that their classes in the Grothendieck group are the same. To showthis, we compare the multiplicities of Verma modules in the ∆ -filtrations of E k ( v P w (1 µ ′ ) ) and v P w (1 µ ) .We need to show that for each x ∈ v J µ ′ we have [ E k ( v P w (1 µ ′ ) ) , ∆ x (1 µ ′ ) ] = [ v P w (1 µ ) , ∆ x (1 µ ) ] . By Lemma 3.18, for each x ∈ v J µ , the multiplicity [ E k ( v P w (1 µ ′ ) ) , ∆ x (1 µ ) ] isequal to the multiplicity [ v P w (1 µ ′ ) , ∆ x (1 µ ′ ) ] . So, we need to prove the equality [ v P w (1 µ ′ ) , ∆ x (1 µ ′ ) ] = [ v P w (1 µ ) , ∆ x (1 µ ) ] . The last equality is obvious because both of these multiplicities are given by thesame parabolic Kazhdan-Lusztig polynomial. See, for example, [11, App. A] formore details about multiplicities in the parabolic category O for b gl N .Now, we prove ( b ) . Since the set of simple modules in the parabolic category O is a subset of the set of simple modules of the non-parabolic category O , it isenough to prove ( b ) in the non-parabolic case.For each w ∈ v J µ and x ∈ v J µ ′ , we have Hom( v P x (1 µ ′ ) , F k ( L w (1 µ ) )) ≃ Hom( E k ( v P x (1 µ ′ ) ) , L w (1 µ ) ) ≃ Hom( v P x (1 µ ) , L w (1 µ ) ) . dim Hom( v P x (1 µ ′ ) , F k ( L w (1 µ ) )) = δ x,w . Since dim Hom( v P x (1 µ ′ ) , M ) counts the multiplicity of the simple module L x (1 µ ′ ) in the module M (this factcan be proved in the same way as [8, Thm. 3.9 (c)]), this proves ( b ) .Finally, we prove ( a ) in the parabolic situation. For each w ∈ v J νµ ′ and each x ∈ v J νµ we have Hom( E k ( v P w (1 µ ′ ) ) , L x (1 µ ) ) ≃ Hom( v P w (1 µ ′ ) , F k ( L x (1 µ ) )) ≃ (cid:26) Hom( v P w (1 µ ′ ) , L x (1 µ ′ ) ) if x ∈ v J νµ ′ else , where the second isomorphism follows from ( b ) . This implies that we have dim Hom( E k ( v P w (1 µ ′ ) ) , L x (1 µ ) ) = δ w,x . Thus we have E k ( v P w (1 µ ′ ) ) ≃ v P w (1 µ ) .The definitions of the graded lifts e E k and e F k in Lemma 3.29 depend on thechoice of the graded lift e V µ of V µ . Note that we have the following isomorphismof v Z µ -modules V µ ( v P µ ) ≃ v Z µ for all µ ∈ X e [ N ] . By Lemma 3.1, for eachchoice of the graded lift e V µ , we have e V µ ( v P µ ) ≃ v Z µ h r i for some r ∈ Z . Fromnow on, we always assume that the graded lift e V µ is chosen in such a way thatwe have an isomorphism of graded v Z µ -modules e V µ ( v P µ ) ≃ v Z µ (without anyshift r ).In the following statement we consider the non-parabolic situation. Lemma 4.16.
For each w ∈ v J µ ′ , the graded module e E k (∆ w (1 µ ′ ) ) has a graded ∆ -filtration with constituents ∆ wz (1 µ ) h ℓ ( z ) i for z ∈ J µµ ′ .Proof. First, we prove that e E k takes the graded anti-dominant projective mod-ule to the graded anti-dominant projective module, i.e., that we have e E k ( v P µ ′ ) ≃ v P µ .By Lemma 3.1, the graded lift of v P µ is unique up to graded shift. Thus,by Lemma 4.15, we have e E k ( v P µ ′ ) = v P µ h r i for some r ∈ Z . We need to provethat r = 0 .Recall that the graded lift e E k of E k is constructed in the proof of Lemma3.29 in such a way that the following diagram is commutative v O µ e E k ←−−−− v O µ ′ e V µ y e V µ ′ y mod( v Z µ ) g Ind ←−−−− mod( v Z µ ′ ) . Moreover, by definition, we have the following isomorphisms of graded modules e V µ ( v P w (1 µ ) ) ≃ v Z µ , e V µ ′ ( v P w (1 µ ′ ) ) ≃ v Z µ ′ , g Ind( v Z µ ′ ) = v Z µ . This implies that we have r = 0 . 58ow we prove the statement of the lemma. The module e E k (∆ w (1 µ ′ ) ) has agraded ∆ -filtartion because it has a ∆ -filtration as an ungraded module, see [11,Rem. 2.13]. The constituents (up to graded shifts) are ∆ wz (1 µ ) , z ∈ W µ ′ /W µ by Lemma 3.18. We need only to identify the shifts. The graded multiplicitiesof Verma modules in projective modules are given in terms of Kazhdan-Lusztigpolynomials in [11, App. A]. In particular, [11, Lem. A.4 (d)] implies that,for each w ∈ v J µ , the module ∆ w (1 µ ) appears as a constituent in a graded ∆ -filtration of v P µ once with the graded shift by ℓ ( w ) . Similarly, for each w ∈ v J µ ′ , the module ∆ w (1 µ ′ ) appears as a constituent in a graded ∆ -filtrationof v P µ ′ once with the graded shift by ℓ ( w ) . Now, since e E k ( v P µ ′ ) ≃ v P µ , we seethat, for each w ∈ v J µ ′ and each z ∈ J µµ ′ , the module ∆ wz (1 µ ) appears in the ∆ -filtration of e E k (∆ w (1 µ ′ ) ) with the graded shift by ℓ ( z ) .In the following lemma me consider the general (i.e., parabolic) situation. Lemma 4.17.
For each w ∈ v J νµ ′ , we have e E k ( v P w (1 µ ′ ) ) = v P w (1 µ ) .Proof. By Lemmas 3.1 and 4.15, we have e E k ( v P w (1 µ ′ ) ) = v P w (1 µ ) [ r ] for someinteger r . We must show that the shift r is zero.First, we prove this in the non-parabolic case. The module ∆ w (1 ′ µ ) (resp. ∆ w (1 µ ) ) is contained in each ∆ -filtration of v P w (1 µ ′ ) (resp. v P w (1 µ ) ) only onceand without a graded shift. Moreover, by Lemma 4.16 the module ∆ w (1 µ ) iscontained in each ∆ -filtration of e E k (∆ w (1 ′ µ ) ) only once and without a gradedshift. This implies that the graded shift r is zero.The parabolic case follows from the non-parabolic case. Really, the pro-jective covers of simple modules in the parabolic category O are quotients ofprotective covers in the non-parabolic category O (see Lemma 3.13 ( b ) ). Thusthe shift r should be zero in the parabolic case because it is zero in the non-parabolic case.Let us check that the functor e E k : v e O νµ → v e O νµ ′ satisfies the hypotheses ofCorollary 4.14. Condition ( a ) follows from Lemma 4.17.Let P and Q be projective covers of simple modules in v e O µ graded as above.To check ( b ) , we have to show that we have an isomorphism Hom( e E k ( P ) h i , e E k ( Q )) ≃ Hom( P h i , Q ) . We have
Hom( e E k ( P ) h i , e E k ( Q )) ≃ Hom( P, e F k +1 e E k +1 ( Q ) h µ k +1 − i ) ≃ Hom( P, [ µ k +1 + 1] q ( Q ) h µ k +1 − i ) ≃ Hom(
P, Q h− i ) L ⊕ µ k +1 r =1 Hom(
P, Q h r − i ) ≃ Hom( P h i , Q ) . Here the first isomorphism follows from Lemma 3.29 ( b ) , the second isomorphismfollows from Lemma 3.31. The last isomorphism holds because Hom(
P, Q h r i ) is59ero for r > because the Z -graded algebra End( M w ∈ v J νµ v P w (1 µ ′ ) ) has zero negative homogeneous components (as it is Koszul).For each µ = ( µ , · · · , µ e ) we set µ op = ( µ e , · · · , µ ) . We can define thepositive level version O νµ, + of the category O νµ in the following way. For each λ ∈ P we set e λ + = λ + z λ δ + ( e − N )Λ , where z λ = ( λ, ρ + λ ) / e . For each λ ∈ P ν denote by + ∆( λ ) the Verma module with highest weight e λ + and denoteby + L ( λ ) its simple quotient. We will also abbreviate + ∆ λ = + ∆( λ − ρ ) and + L λ = + L ( λ − ρ ) . Let O νµ, + be the Serre subcategory of O ν generated by thesimple modules + L λ for λ ∈ P ν [ µ op ] . Similarly to the negative e -action of c W on P described in Section 2.6 we can consider the positive e -action on P . Wedefine the positive e -action in the following way: the element w ∈ c W sends λ to − w ( − λ ) (where w ( − λ ) corresponds to the negative e -action). The notion ofthe positive e -action of c W on P is motivated by the fact that the map P → b h ∗ , λ ] λ − ρ + + b ρ is c W -invariant. We say that an element λ ∈ P is e - dominant if we have λ > λ > · · · > λ N > λ − e . Fix an e -dominant element + µ ∈ P [ µ op ] . (We cantake for example + µ = ( e µ , · · · , µ e ) ). Note that the stabilizer of + µ in c W withrespect to the positive e -action is W µ . From now on, each time when we write w (1 + µ ) we mean the positive e -action on P and each time when we write w (1 µ ) we mean the negative e -action.Recall that J µ, + is the subset of c W containing all w such that w is maximalin wW µ . Set J νµ, + = { w ∈ J µ, + ; w (1 + µ ) ∈ P ν } . Note that the inclusion J µ ′ ⊂ J µ induces an inclusion J νµ ′ ⊂ J νµ . For v ∈ c W we set v J νµ = { w ∈ J νµ ; w v } and v J νµ, + = { w ∈ J νµ, + ; w v } .We have the following lemma. Lemma 4.18. ( a ) There is a bijection J νµ → J µν, + given by w w − . ( b ) For each v ∈ J νµ , there is a bijection v J νµ → v − J µν, + given by w w − .Proof. Part ( a ) follows from [17, Cor. 3.3]. Part ( b ) follows from part ( a ) .Similarly to the truncated version v O νµ of O νµ , we can define the truncatedversion v O νµ, + of O νµ, + . We define v O νµ, + as the Serre quotient of O νµ, + , wherewe kill the simple module + L w (1 + µ ) for each w ∈ J νµ, + − v J νµ, + .By [17, Thm. 3.12], for v ∈ J νµ , the category v e O νµ is Koszul dual to thecategory v − e O µν, + . The bijection between the simple modules in v e O νµ and theindecomposable projective modules in v − e O µν, + given by the Koszul functor K is such that for each w ∈ v J νµ the module L w (1 µ ) corresponds to the projectivecover of + L w − (1 + ν ) . 60e should make a remark about our notation. Usually, we denote by e thenumber of components in µ and we denote by l the number of components in ν .So, when we exchange the roles of µ and ν and we consider the category O µν, + ,we mean that this category is defined with respect to the level l − N (and not e − N ).Now, assume again that v is in J νµ ′ w µ ′ . Then we have vw µ ∈ J νµ and vw µ ′ ∈ J νµ ′ . In this case the Koszul dual categories to v O νµ and v O νµ ′ are w µ v − O µν, + and w µ ′ v − O µ ′ ν, + . Lemma 4.19. ( a ) We have w µ ′ v − J µ ′ ν, + = w µ v − J µν, + ∩ J µ ′ ν, + . ( b ) We have w µ ′ v − J µ ′ ν, + = w µ v − J µ ′ ν, + . Proof.
Let us prove ( a ) . By Lemma 4.18 the statement is equivalent to vw µ ′ J νµ ′ = vw µ J νµ ∩ J νµ ′ . Moreover, by definition, we have vw µ ′ J νµ ′ = v J νµ ′ and vw µ J νµ = v J νµ . Thus, thestatement is equivalent to v J νµ ′ = v J νµ ∩ J νµ ′ . The last equality is obvious.Part ( b ) follows from part ( a ) .Now, put u = w µ v − . The discussion above together with Lemma 4.19shows that the Koszul dual categories to to v O νµ and v O νµ ′ are u O µν, + and u O µ ′ ν, + .We get the following result. Theorem 4.20.
Assume that we have W µ ⊂ W µ ′ . ( a ) The functor e F k : D b ( v e O νµ ) → D b ( v e O νµ ′ ) is Koszul dual to the shiftedparabolic truncation functor e tr h µ k +1 i : D b ( u e O µν, + ) → D b ( u e O µ ′ ν, + ) . ( b ) The functor e E k : D b ( v e O νµ ′ ) → D b ( v e O νµ ) is Koszul dual to the parabolicinclusion functor f inc: D b ( u e O µ ′ ν, + ) → D b ( u e O µν, + ) .Proof. We have checked above that the functor e E k : v e O νµ → v e O νµ ′ satisfies thehypotheses of Corollary 4.14. Thus Corollary 4.14 implies part ( b ) . Part ( a ) follows from part ( b ) by adjointness.Similarly to the situation W µ ⊂ W µ ′ , we can do the same in the situation W µ ⊂ W µ ′ (see also Section 3.9). In this case we should take v ∈ J νµ w µ and put u = w µ ′ v − . We get the following theorem. Theorem 4.21.
Assume that we have W µ ′ ⊂ W µ . ( a ) The functor e F k : D b ( v e O νµ ) → D b ( v e O νµ ′ ) is Koszul dual to the parabolicinclusion functor f inc: D b ( u e O µν, + ) → D b ( u e O µ ′ ν, + ) . ( b ) The functor e E k : D b ( v e O νµ ′ ) → D b ( v e O νµ ) is Koszul dual to the shiftedparabolic truncation functor e tr h µ k − i : D b ( u e O µ ′ ν, + ) → D b ( u e O µν, + ) . .8 The restriction to the category A The goal of this section is to restrict the results of the previous section tocategory A .We have seen that we can grade the functor E k and F k for category O when we have W µ ⊂ W µ ′ or W µ ′ ⊂ W µ . Let us show that in this cases wecan also grade similar functors for the category A . We have A ν [ α ] ⊂ v O νµ and A ν [ α + α k ] ⊂ v O νµ ′ . Denote by h the inclusion functor from A ν [ α ] to v O νµ .Abusing the notation, we will use the same symbol for the inclusion functorfrom A ν [ α + α k ] to v O νµ ′ . Let h ∗ and h ! be the left and right adjoint functors to h . The functor F k for the category A is defined as the restriction of the functor F k for the category O . This restriction can be written as h ! F k h . The functor E k for the category O does not preserve the category A in general. The functor E k for the category A is defined in [15, Sec. 5.9] as h ∗ E k h . It is easy to seethat we can grade the functor h and its adjoint functors in the same way aswe graded Zuckerman functors. Thus we obtain graded lifts e E k and e F k of thefunctors E k and F k for the category A . Moreover, we still have the adjunctions ( e E k , e F k h µ k +1 i ) (when W µ ⊂ W µ ′ ) and ( e E k , e F k h − µ k i ) (when W µ ′ ⊂ W µ ) inthe category A .We do not have adjunctions in other direction in general. However, if addi-tionally we have ν r > | α | for each r ∈ [1 , l ] , then the functors E k and F k for thecategory A are biadjoint by [15, Lem. 7.6]. This means that there is no differencebetween h ∗ E k h and h ! E k h . Thus we also get the adjunctions ( e F k , e E k h− µ k +1 i ) (when W µ ⊂ W µ ′ ) and ( e F k , e E k h µ k − i ) (when W µ ′ ⊂ W µ ) in the category A .(In fact, we always have the adjunctions in both directions if k = 0 because inthis case the functor E k for the category A is just the restriction of the functor E k for the category O and similarly for e E k .)We start from a general lemma. Let A be a finite dimensional Koszul algebraover C . Let { e λ ; λ ∈ Λ } be the set of indecomposable idempotents in A . Fix asubset Λ ′ ⊂ Λ . Assume that the algebra Λ ′ A (see Section 4.5 for the notations) isalso Koszul. Then we have an algebra isomorphism ( Λ ′ A ) ! ≃ ( A ! ) Λ ′ . The gradedalgebra Λ ′ A is a quotient of the graded algebra A by a homogeneous ideal.In particular we have an inclusion of categories ι : grmod( Λ ′ A ) → grmod( A ) .Moreover, there is a functor τ : grmod( A ! ) → grmod(( A ! ) Λ ′ ) , M e !Λ ′ M. The functors ι and τ are both exact. They yield functors between derivedcategories ι : D b ( Λ ′ A ) → D b ( A ) and τ : D b ( A ! ) → D b (( A ! ) Λ ′ ) .Since the algebra A is Koszul, there is a functor K : D b ( A ) → D b ( A ! ) definedby K = RHom( A , • ) , see Section 4.3. We will sometimes write K A to specifythe algebra A .In the following lemma we identify ( Λ ′ A ) ! = ( A ! ) Λ ′ . Lemma 4.22.
We have the following isomorphism of functors D b ( Λ ′ A ) → D b (( A ! ) Λ ′ ) K Λ ′ A ≃ τ ◦ K A ◦ ι. roof. For a complex M ∈ D b ( Λ ′ A ) , we have τ ◦ K A ◦ i ( M ) ≃ τ (RHom A ( A , M )) ≃ RHom A ( e Λ ′ A , M ) ≃ RHom Λ ′ A (( Λ ′ A ) , M ) ≃ K Λ ′ A ( M ) . Fix α ∈ Q + e . Consider the category A ν [ α ] as in Section 2.15. Let µ be suchthat A ν [ α ] is a subcategory of O νµ . (Then A ν [ α + α k ] is a subcategory of O νµ ′ .)Assume that we have W µ ⊂ W µ ′ . Assume that v ∈ J νµ ′ w µ ′ is such that A ν [ α ] isa subcategory of v O νµ and A ν [ α + α k ] is a subcategory of v O νµ ′ . Put u = w µ v − .The category A ν [ α ] is also Koszul. Denote by e A ν [ α ] its graded version. TheKoszul dual category to A ν [ α ] is a Serre quotient of the category u O µν, + (see[11, Rem. 3.15]). Let us denote this quotient and its graded version by A µ + [ α ] and e A µ + [ α ] respectively. (We will also use similar notations for A ν [ α + α k ] .)First, we prove the following lemma. Lemma 4.23.
Assume that we have W µ ⊂ W µ ′ and k = 0 . ( a ) The inclusion of categories u O µ ′ ν, + ⊂ u O µν, + yields an inclusion of cate-gories A µ ′ + [ α + α k ] ⊂ A µ + [ α ] . ( b ) The inclusion of categories u e O µ ′ ν, + ⊂ u e O µν, + yields an inclusion of cate-gories e A µ ′ + [ α + α k ] ⊂ e A µ + [ α ] .Assume that we have W µ ⊃ W µ ′ and k = 0 . ( c ) The inclusion of categories u O µν, + ⊂ u O µ ′ ν, + yields an inclusion of cate-gories A µ + [ α ] ⊂ A µ ′ + [ α + α k ] . ( d ) The inclusion of categories u e O µν, + ⊂ u e O µ ′ ν, + yields an inclusion of cate-gories e A µ ′ + [ α ] ⊂ e A µ ′ + [ α + α k ] .Proof. Denote by p and p respectively the quotient functors p : u O µ ′ ν, + → A µ ′ + [ α + α k ] , p : u O µν, + → A µ + [ α ] . To prove ( a ) and ( b ) , it is enough to prove that each simple module in u O µ ′ ν, + is killed by the functor p if and only if it is killed by the functor p . We canget the combinatorial description of the simple modules killed by p and p respectively using [11, Rem. 2.18].For each w ∈ v J νµ ′ (resp. w ∈ v J νµ ), the simple module + L w − (1 + ν ) is killed by p (resp. p ) if and only if the simple module L w (1 µ ′ ) ∈ v O νµ ′ is not in A ν [ α + α k ] (resp. the simple module L w (1 µ ) ∈ v O νµ is not in A ν [ α ] ). So, we need to showthat for each w ∈ v J νµ ′ the module L w (1 µ ′ ) ∈ v O νµ ′ is in A ν [ α + α k ] if and onlyif the module L w (1 µ ) ∈ v O νµ is in A ν [ α ] . Finally, we have to show that for each63 ∈ v J νµ ′ we have w (1 µ ′ ) > ρ ν if and only if we have w (1 µ ) > ρ ν . (Here theorder is as in Section 2.16.)It is obvious that w (1 µ ) > ρ ν implies w (1 µ ′ ) > ρ ν because we have w (1 µ ′ ) > w (1 µ ) . Now, let us show the inverse statement. Note that we have w (1 µ ′ ) = w (1 µ ) + ǫ r , where r ∈ [1 , N ] is the unique index such that w (1 µ ) r ≡ k mod e . Assume that we have w (1 µ ′ ) > ρ ν but not w (1 µ ) > ρ ν . Then we have w (1 µ ′ ) r = ( ρ ν ) r . Assume first that ( ρ ν ) r = 1 . In particular this implies r < N .Since the weight w (1 µ ) is in P ν , we have w (1 µ ′ ) r +1 = w (1 µ ) r +1 < w (1 µ ) r = ( ρ ν ) r − ρ ν ) r +1 . This contradicts to w (1 µ ′ ) > ρ ν . Now, assume that we have ( ρ ν ) r = 1 . Sincewe have ( ρ ν ) r ≡ w (1 µ ′ ) r ≡ k + 1 mod e , this implies k = 0 . This contradictswith the assumption k = 0 . This proves the statement.The proof of ( c ) , ( d ) is similar to the proof of ( a ) and ( b ) .In the case W µ ⊂ W µ ′ , k = 0 , the lemma above allows us to define theparabolic inclusion functor inc: A µ ′ + [ α + α k ] → A µ + [ α ] and the parabolic trun-cation functor tr: A µ + [ α ] → A µ ′ + [ α + α k ] and their graded versions f inc and e tr .Similarly, in the case W µ ⊃ W µ ′ , k = 0 , the lemma above allows us to definethe parabolic inclusion functor inc: A µ + [ α ] → A µ ′ + [ α + α k ] and the parabolictruncation functor tr: A µ ′ + [ α + α k ] → A µ + [ α ] and their graded versions f inc and e tr . Theorem 4.24.
Assume that we have W µ ⊂ W µ ′ . ( a ) The functor e F k : D b ( e A ν [ α ]) → D b ( e A ν [ α + α k ]) is Koszul dual to theshifted parabolic truncation functor e tr h µ k +1 i : D b ( e A µ + [ α ]) → D b ( e A µ ′ + [ α + α k ]) . ( b ) The functor e E k : D b ( e A ν [ α + α k ]) → D b ( e A ν [ α ]) is Koszul dual to theparabolic inclusion functor f inc: D b ( e A µ ′ + [ α + α k ]) → D b ( e A µ + [ α ]) .Now, assume that we have W µ ′ ⊂ W µ . ( c ) The functor e F k : D b ( e A ν [ α ]) → D b ( e A ν [ α + α k ]) is Koszul dual to theparabolic inclusion functor f inc: D b ( e A µ + [ α ]) → D b ( e A µ ′ + [ α + α k ]) . ( d ) The functor e E k : D b ( e A ν [ α + α k ]) → D b ( e A ν [ α ]) is Koszul dual to theshifted parabolic truncation functor e tr h µ k − i : D b ( e A µ ′ + [ α + α k ]) → D b ( e A µ + [ α ]) .Proof. Let us prove ( b ) .Let v ∈ J νµ ′ w µ ′ be such that A ν [ α ] is a subcategory of v O νµ ′ and A ν [ α + α k ] is a subcategory of v O νµ ′ . Then the same is true for graded versions. Denoteby i the inclusion functor from e A ν [ α ] to v e O νµ . Let τ : u e O µν, + → e A µ + [ α ] be thenatural quotient functor. 64onsider the following diagram D b ( e A µ + [ α ]) f inc ←−−−− D b ( e A µ ′ + [ α + α k ]) τ x τ x D b ( v e O µν, + ) f inc ←−−−− D b ( v e O µ ′ ν, + ) K x K x D b ( v e O νµ ) E k ←−−−− D b ( v e O νµ ′ ) i x i x D b ( e A ν [ α ]) E k ←−−−− D b ( e A ν [ α + α k ]) . The commutativity of the top and bottom rectangles is obvious. The commuta-tivity of the middle rectangle follows from Theorem 4.20 ( b ) . Now, by Lemma4.22, the big rectangle in the diagram above yields the following commutativediagram D b ( e A µ + [ α ]) e inc ←−−−− D b ( e A µ ′ + [ α + α k ]) K x K x D b ( e A ν [ α ]) F k ←−−−− D b ( e A ν [ α + α k ]) . This proves ( b ) .Part ( a ) follows from ( b ) by adjointness. We can prove ( c ) in the same wayas ( b ) , using Theorem 4.21 ( a ) . Part ( d ) follows from ( c ) by adjointness. A + Fix u ∈ c W . The Zuckerman functor Zuc: u O µν, + → u O µ ′ ν, + (see Section 4.6) is acomposition of a parabolic inclusion functor and a parabolic truncation functor u O µν, + inc → u O µ ′′ ν, + tr → u O µ ′ ν, + , where the parabolic type µ ′′ is chosen such that W µ ′′ = W µ ∩ W µ ′ (in fact, we can take µ ′′ = µ ). Now we are going to givean analogue of the Zuckerman functor for the category A + , i.e., we want todefine a functor Zuc + k : A µ + [ α ] → A µ ′ + [ α + α k ] . (Recall that the categories A µ + [ α ] and A µ ′ + [ α + α k ] are Serre quotients of u O µν, + and u O µ ′ ν, + respectively for u bigenough.) The main difficulty to give such a definition is that we have no obviouscandidate to replace the category u O µ ′′ ν, + .Let us write A instead of A to indicate that the category is defined withrespect to e +1 instead of e . Let us identify A µ + [ α ] ≃ A µ + [ β + α ] and A µ ′ + [ α + α k ] ≃ A µ ′ + [ β + α + α k + α k +1 ] (see Proposition 2.16). Assume that we have k = 0 .Then by Lemma 4.23 we have the following inclusion of categories A µ + [ β + α ] ⊂ A µ + [ β + α + α k ] ⊃ A µ ′ + [ β + α + α k + α k +1 ] . Zuc + k : A µ + [ α ] → A µ ′ + [ α + α k ] as thecomposition of the parabolic inclusion functor with the parabolic truncationfunctor A µ + [ α ] inc → A µ + [ β + α + α k ] tr → A µ ′ + [ α + α k ] . We define the Zuckerman functor
Zuc − k : A µ ′ + [ α + α k ] → A µ + [ α ] in a similar way.We can also define the graded version g Zuc ± k of the Zuckerman functors by replac-ing the functors inc and tr by their graded versions f inc and e tr . Unfortunatelythis approach does not allow to define the Zuckerman functors for k = 0 be-cause of the assumption k = 0 in Lemma 4.23. The definition of the Zuckermanfunctors for k = 0 will be given in Section 4.11. A Theorem 4.25.
Assume that we have ν r > | α | for each r ∈ [1 , l ] , e > and k = 0 . ( a ) The functor F k : A ν [ α ] → A ν [ α + α k ] has a graded lift e F k such that thefunctor e F k : D b ( e A ν [ α ]) → D b ( e A ν [ α + α k ]) is Koszul dual to the shifted Zucker-man functor g Zuc + k h µ k +1 i : D b ( e A µ + [ α ]) → D b ( e A µ ′ + [ α + α k ]) . ( b ) The functor E k : A ν [ α + α k ] → A ν [ α ] has a graded lift such that the func-tor e E k : D b ( e A ν [ α + α k ]) → D b ( e A ν [ α ]) is Koszul dual to the shifted Zuckermanfunctor g Zuc − k h µ k − i : D b ( e A µ + [ α ]) → D b ( e A µ ′ + [ α + α k ]) .Proof. By Theorem 2.37 we have the following commutative diagram A ν [ β + α ] A ν [ β + α + α k ] A ν [ β + α + α k + α k +1 ] A ν [ α ] A ν [ α + α k ] ✲ F k ✲ F k +1 ❄✻ ✲ F k Here the vertical maps are some equivalences of categories. By unicity ofKoszul grading (see [3, Cor. 2.5.2]) there exist unique graded lifts of verticalmaps such that they are equivalences of graded categories and they respect thechosen grading of simple modules (i.e., concentrated in degree ). Moreover,the top horizontal maps have graded lifts because for a suitable v we have A ν [ β + α ] ⊂ v O νµ , A ν [ β + α + α k ] ⊂ v O νµ , A ν [ β + α + α k + α k +1 ] ⊂ v O νµ ′ and W µ ⊃ W µ ⊂ W µ ′ . This implies that there is a graded version e F k ofthe functor F k such that it makes the graded version of the diagram abovecommutative.Since the categories A ν [ α ] and A ν [ β + α ] are equivalent, their Koszul dualcategories are also equivalent. We can chose the equivalences ( A ν [ α ]) ! ≃ A µ + [ α ] ( A ν [ β + α ]) ! ≃ A µ + [ α ] in such a way that the vertical map in the diagramis Koszul dual to the identity functor. We can do the same with the categoriesin the right part of the diagram above.By Theorem 4.24, the left top functor in the graded version of the dia-gram above is Koszul dual to the parabolic inclusion functor f inc and the topright functor in the diagram is Koszul dual to the graded shift e tr h µ k +1 i of theparabolic truncation functor. By definition (see Section 4.6), the Zuckermanfunctor is the composition of the parabolic inclusion and the parabolic trunca-tion functors. This implies that the functor e F k : D b ( e A ν [ α ]) → D b ( e A ν [ α + α k ]) is Koszul dual to the shifted Zuckerman functor g Zuc + k h µ k +1 i . This proves ( a ) .We can prove ( b ) in the same way. By adjointness, the diagram above yieldsa similar diagram for the functor E . This diagram allows to grade the functor E k . Then we deduce the Koszul dual functor to E k in the same way as in ( a ) . k = 0 Now, we are going to get an analogue of Theorem 4.25 in the case k = 0 . Themain difficulty in this case is that we cannot define Zuckerman functors for thecategory A + in the same was as in Section 4.9 because Lemma 4.23 fails. Tofix this problem we replace the category A by a smaller category A .Assume that we have k = 0 and W µ ⊃ W µ ′ . In particular this implies µ = 0 .Let A ν [ α + α ] be the Serre subcategory of A ν [ α + α ] generated by simplemodules L λ such that the weight λ ∈ P has no coordinates equal to . It is ahighest weight subcategory. Remark . ( a ) The category A ν [ α + α ] inherits the Koszul grading fromthe category A ν [ α + α ] in the following way. We know that there is a Koszulalgebra A such that A ν [ α + α ] ≃ mod( A ) . Let { e λ ; λ ∈ Λ } be the set ofindecomposable idempotents of A . Then by [11, Lem. 2.17] there is a subset Λ ′ ⊂ Λ such that we have A ν [ α + α ] ≃ mod( Λ ′ A ) (see Section 4.5 for thenotations). Moreover, the Koszul dual algebra to Λ ′ A is A !Λ ′ .Since, we have mod( A ! ) ≃ A µ ′ + [ α + α ] , the Koszul dual category A µ ′ + [ α + α ] to A ν [ α + α ] is a Serre quotient of A µ ′ + [ α + α ] . The quotient functor a : A µ ′ + [ α + α ] → A µ ′ + [ α + α ] can be seen as the functor a : mod( A ! ) → mod( A !Λ ′ ) , M e !Λ ′ M. ( b ) The left adjoint functor b : A µ ′ + [ α + α ] → A µ ′ + [ α + α ] to a can be seen as b : mod( A !Λ ′ ) → mod( A ! ) , M A ! e !Λ ′ ⊗ A !Λ ′ M. a and b have obvious graded lifts e a : e A µ ′ + [ α + α ] → e A µ ′ + [ α + α ] , e b : e A µ ′ + [ α + α ] → e A µ ′ + [ α + α ] . By Proposition 4.11, the functor e b is Koszul dual to the inclusion functor e A ν [ α + α ] → e A ν [ α + α ] . Then, by adjointness, the functor e a is Koszul dualto the right adjoint functor to the inclusion functor above.It is easy to see from the action of F on Verma modules (see Proposition2.8 ( e ) ) that the image of the functor F : A ν [ α ] → A ν [ α + α ] is in A ν [ α + α ] .Moreover, recall from Section 2.15 that the functor E : O νµ ′ → O νµ does not take A ν [ α + α ] to A ν [ α ] . (The reader should pay attention to the fact that thefunctor E for the category A is not defined as the restriction of the functor E for the category O .) However, it is easy to see from the action of E onVerma modules (see Proposition 2.8 ( e ) ) that the functor E for the category O takes A ν [ α + α ] to A ν [ α ] . Thus we get a functor E : A ν [ α + α ] → A ν [ α ] . Thisfunctor also coincides with the restriction of the functor E : A ν [ α + α ] → A ν [ α ] to the category A ν [ α + α ] .The following statement can be proved in the same way as Lemma 4.23. Lemma 4.27.
Assume that we have W µ ⊃ W µ ′ . ( a ) The inclusion of categories u O µν, + ⊂ u O µ ′ ν, + yields an inclusion of cate-gories A µ + [ α ] ⊂ A µ ′ + [ α + α ] . ( b ) The inclusion of categories u e O µ ′ ν, + ⊂ u e O µν, + yields an inclusion of cate-gories e A µ ′ + [ α + α ] ⊂ e A µ + [ α ] . The lemma above allows us to define the inclusion and the truncation func-tors inc: A µ + [ α ] → A µ ′ + [ α + α ] , tr: A µ ′ + [ α + α ] → A µ + [ α ] and their graded versions f inc , e tr .We still assume k = 0 but we do not assume W µ ⊃ W µ ′ any more. We definethe Zuckerman functors Zuc ± for this case. Let us identify A µ + [ α ] ≃ A µ + [ β + α ] and A µ ′ + [ α + α k ] ≃ A µ ′ + [ β + α + α k + α k +1 ] . By Lemmas 4.23, 4.27 we have thefollowing inclusions of categories A µ + [ β + α ] ⊂ A µ + [ β + α + α ] , A µ + [ β + α + α ] ⊃ A µ ′ + [ β + α + α + α ] . We define the Zuckerman functor
Zuc +0 : A µ + [ α ] → A µ ′ + [ α + α ] as the composition A µ + [ α ] inc → A µ + [ β + α + α ] b → A µ + [ β + α + α ] tr → A µ ′ + [ α + α ] . Similarly, we define the Zuckerman functor
Zuc − : A µ ′ + [ α + α ] → A µ + [ α ] as thecomposition A µ ′ + [ α + α ] inc → A µ + [ β + α + α ] a → A µ + [ β + α + α ] tr → A µ + [ α ] . inc , tr , a , b by their graded versions f inc , e tr , e a , e b yieldsgraded versions g Zuc +0 and g Zuc − of the Zuckerman functors.Now, similarly to Theorem 4.24 we can prove the following. Theorem 4.28.
Assume that we have k = 0 and W µ ⊃ W µ ′ . ( a ) The functor e F : D b ( e A ν [ α ]) → D b ( e A ν [ α + α ]) is Koszul dual to theparabolic inclusion functor f inc: D b ( e A µ + [ α ]) → D b ( e A µ ′ + [ α + α ]) . ( b ) The functor e E : D b ( e A ν [ α + α ]) → D b ( e A ν [ α ]) is Koszul dual to the shiftedparabolic truncation functor e tr h µ − i : D b ( e A µ ′ + [ α + α ]) → D b ( e A µ + [ α ]) . Finally, we get an analogue of Theorem 4.25 in the case k = 0 . Theorem 4.29.
Assume that we have ν r > | α | for each r ∈ [1 , l ] and e > . ( a ) The functor F : A ν [ α ] → A ν [ α + α ] has a graded lift e F such that thefunctor e F : D b ( e A ν [ α ]) → D b ( e A ν [ α + α ]) is Koszul dual to the shifted Zucker-man functor g Zuc +0 h µ i : D b ( e A µ + [ α ]) → D b ( e A µ ′ + [ α + α ]) . ( b ) The functor E : A ν [ α + α ] → A ν [ α ] has a graded lift e E such that thefunctor e E : D b ( e A ν [ α + α ]) → D b ( e A ν [ α ]) is Koszul dual to the shifted Zucker-man functor g Zuc − h µ − i : D b ( e A µ + [ α ]) → D b ( e A µ ′ + [ α + α ]) .Proof. The proof is similar to the proof of Theorem 4.25. To prove ( a ) we shouldconsider the diagram as in the proof of Theorem 4.25 with an additional term. A ν [ β + α ] A ν [ β + α + α ] A ν [ β + α + α ] A ν [ β + α + α + α ] A ν [ α ] A ν [ α + α ] ✲ F ✲ ✲ F ❄✻ ✲ F We prove ( b ) in the same way by considering the diagram obtained from thediagram above by adjointness. Note that in this case we have the adjunction ( F , E ) (and not only ( E , F ) ) because of the assumption on ν . Acknowledgements
I am grateful for the hospitality of the Max-Planck-Institut für Mathematik inBonn, where a big part of this work is done. I would like to thank Éric Vasserotfor his guidance and helpful discussions during my work on this paper. I wouldalso like to thank Cédric Bonnafé for his comments on an earlier version of thispaper. 69 eferences [1] J. N. Bernstein, S. I. Gelfand,
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