Dualities and derived equivalences for category O

We determine the Ringel duals for all blocks in the parabolic versions of the BGG category O associated to a reductive finite dimensional Lie algebra. In particular we find that, contrary to the original category O and the specific previously known cases in the parabolic setting, the blocks are not necessarily Ringel self-dual. However, the parabolic category O as a whole is still Ringel self-dual. Furthermore, we use generalisations of the Ringel duality functor to obtain large classes of derived equivalences between blocks in parabolic and original category O. It seems that only some special cases were known thus far, where the proof relied purely on a geometric approach to category O. We subsequently classify all derived equivalence classes of blocks of category O in type A which preserve the Koszul grading.


Introduction and main results
For a reductive finite dimensional complex Lie algebra g with a fixed triangular decomposition, consider the Bernstein-Gelfand-Gelfand category O from [BGG] (see also [Hu]) and its parabolic generalisation by Rocha-Caridi in [RC]. It is well-known that blocks in parabolic category O are described by quasi-hereditary algebras which possess a Koszul grading, see [Ba1,BGS,So1,Ma2]. Moreover, graded lifts of blocks of category O are Koszul dual to graded lifts of blocks in the parabolic versions of O. It was proved in [So2] that parabolic category O, as a whole, is Ringel self-dual. It is well-known that for full category O, even every individual block is Ringel selfdual, as follows from [So1,Struktursatz 9]. The same holds for the principal block in parabolic category O, see [MS3]. One of our observations which motivated the present paper is that arbitrary blocks in parabolic category O are, surprisingly, not Ringel self-dual in general. This is our first main result.
Theorem A. Blocks in parabolic category O are, in general, not Ringel self-dual. However, every block is Ringel dual to another block in the same parabolic category. The Ringel dual of a block is also equivalent to a block with the same central character in a different parabolic version of O.
Ringel duality between two blocks always implies an equivalence, as triangulated categories, of the bounded derived categories of these blocks given by the derived Ringel duality functor, see [Rin,Ha,Ric1]. From our proof of Theorem A, it follows easily that, for parabolic category O, this equivalence lifts to the graded setting for the Koszul grading. On the other hand, the Koszul duality functor, see [BGS, MOS], also induces an equivalence between the derived categories of a graded lift of a block 1 and its Koszul dual. However, this equivalence does not correspond to a derived equivalence in the ungraded sense. Moreover, we show explicitly that two Koszul dual algebras need not be derived equivalent in the sense of [Ric1], by considering an example for category O.
We are interested in derived equivalences in the sense of [Ric1] which lift to the graded setting, see explicit Definition 3.2. We refer to this as a gradable derived equivalence or being gradable derived equivalent. Hence, for parabolic category O, the Ringel duality functor induces a gradable derived equivalence, whereas the Koszul duality functor does not.
In the current paper we start the systematic study of derived equivalences for (parabolic) category O. Some interesting derived equivalences for g = sl(n) have been obtained by Chuang and Rouquier in [CR] and by Khovanov in [Kh]. By construction, the ones in [CR] are gradable. We take an approach independent from the previous results but recover the derived equivalences in [CR, Kh]. Our results are rather conclusive for category O for type A, whereas there remain open questions for the parabolic versions of O over other reductive Lie algebras.
Consider a reductive finite dimensional complex Lie algebra g with a fixed Cartan subalgebra h and a fixed Borel subalgebra b containing h. Consider O = O(g, b) and let W = W (g : h) be the Weyl group and Λ int the set of integral weights. For a coset Λ ∈ h * /Λ int , we denote the corresponding integral Weyl group by W Λ . For a dominant λ ∈ Λ, the stabiliser of λ in W Λ under the dot action is denoted by W Λ,λ and the block in the category O(g, b) containing the simple highest weight module with highest weight λ by O λ (g, b). Our second main result is an analogue of [So1,Theorem 11] for derived categories, restricted to type A.
To present the remainder of the main results, we need more notation. We consider again an arbitrary complex reductive Lie algebras g and henceforth only consider integral weights, which is justified by the results in [So1]. Hence Λ = Λ int and we leave out the reference to Λ. For every integral dominant weight λ, the block O λ now consists of all modules in O with the same generalised central character as the simple module with highest weight λ. To any integral dominant weight µ, we associate a parabolic subalgebra q µ of g, uniquely defined by the fact that the Weyl group of its Levi factor is W µ . The full subcategory of O consisting of all modules which are locally q µ -finite is denoted by O µ . The integral part of this category decomposes naturally into subcategories O µ λ . Graded lifts of the latter categories, with respect to the Koszul grading, are denoted by Z O µ λ . Theorem C. Consider the algebra g = sl(n) and four integral dominant weights λ, λ ′ , µ, µ ′ . If we have isomorphisms of groups then there is a gradable derived equivalence between O µ λ and O µ ′ λ ′ . In particular, we have equivalences of triangulated categories A more complicated formulation of this result for arbitrary g can be found in Theorem 6.1. Theorem C generalises [Kh,Proposition 7], which corresponds precisely to the case λ = λ ′ = 0 in the ungraded setting. Furthermore, our approach gives an explicit form of the functor and the tilting complex, which describe the equivalence.
It also provides a purely algebraic proof, whereas the proof in [Kh] depends on the geometric description of O. Theorem C, for µ = µ ′ , gives the derived equivalences which can be constructed from [CR,Theorem 6.4 and Section 7.4]. It seems that in this case even the functors describing the derived equivalence are isomorphic, as can be checked by hand for small cases. In [CR] these functors have the elegant property that they are defined directly to act between the two relevant categories. On the other hand, they are defined in terms of total complexes for a complex of functors which becomes arbitrarily big. The functors in the current paper have the drawback that they are defined implicitly by using an auxiliary regular block in category O, see Theorem D. The advantage is that the functor on the latter category inducing the equivalence is a well-understood functor with elegant properties.
Our results are formulated in terms of four types of functors on category O, viz. projective, Zuckerman, twisting and shuffling functors, see [MS2] or the preliminaries for an overview. Twisting functors commute with projective functors, see [AS], and shuffling functors commute with Zuckerman functors, see [MS2]. The non-trivial commutation relations between twisting and Zuckerman functors, and between shuffling and projective functors, lie at the origin of the failure of blocks to be Ringel self-dual. It is also these commutation relation that we exploit to obtain the derived equivalences. As an extra result, we extend the Koszul duality between projective and Zuckerman functors of [Ry] to the full parabolic and singular setting.
For any x ∈ W , the derived twisting functor LT x and shuffling functor LC x are auto-equivalences of the category D b (O 0 ). For an integral dominant ν, we introduce the notation W † ν for the subgroup of W generated by all simple reflections which are orthogonal to all reflections in W ν . Let w ν 0 stand for the longest element of W ν . Theorem C can be obtained by an iterative application of the following theorem.
Theorem D. Consider g a finite dimensional complex reductive Lie algebra and an integral dominant ν, λ, µ ∈ h * such that W ν is of type A.
restricts to an equivalence of triangulated categories between two subcategories, equivalent to, respectively, restricts to an equivalence of triangulated categories between two subcategories, equivalent to, respectively, Note that the two parts can be interpreted as Koszul duals of one another using [MOS,Section 6.5]. When w ν 0 = w 0 and µ = 0 in Theorem D(ii), the categories O λ and O λ ′ are equivalent by [So1,Theorem 11]. However, our equivalence of the derived categories is not induced by that equivalence, but is rather given by the derived Ringel duality functor.
The paper is organised as follows. In Section 2 we recall some results on category O and Koszul and Ringel duality. In Section 3 we give an explicit example of Koszul dual algebras which are not derived equivalent and introduce the notion of gradable derived equivalence. We use this to study shuffling in the parabolic setting. In Section 4 we obtain several results on the graded lifts of translation functors. In particular, we extend the Koszul duality of [Ry] between translation functors and parabolic Zuckerman functors to the generality we will need it. In Section 5 we study the commutation relations between shuffling and projective functors. In Section 6 we construct the derived equivalences between blocks in parabolic category O, proving Theorems C and D. This is used in Section 7 to classify the blocks in category O for Lie algebras of type A up to gradable derived equivalence, proving Theorem B. In Section 8 we determine the Ringel duals of all blocks in parabolic category O, proving Theorem A, and study the Koszul-Ringel duality functor.

Preliminaries
We work over C. Unless explicitly stated otherwise, commuting diagrams of functors commute only up to a natural isomorphism. Graded always refers to Z-graded.
2.1. Category O and its parabolic generalisations. We consider the BGG category O, associated to a triangular decomposition of a finite dimensional complex reductive Lie algebra g = n − ⊕ h ⊕ n + , see [BGG, Hu]. For any weight ν ∈ h * , we denote the corresponding simple highest weight module by L(ν). We also introduce an involution on h * by setting ν = −w 0 (ν), with w 0 the longest element of the Weyl group W = W (g : h). We denote by ·, · a W -invariant inner product on h * and the set of integral, not necessarily regular, dominant weights by Λ + int . For any λ ∈ Λ + int , the indecomposable block in category O containing L(λ) is denoted by O λ .
For B the set of simple positive roots and µ ∈ Λ + int , set B µ = {α ∈ B | µ+ρ, α = 0}. Let u − µ be the subalgebra of g generated by the root spaces corresponding to the roots in −B µ . Then we have the parabolic subalgebra q µ of g, given by The full subcategory of O λ with objects given by the modules in O λ which are U (q µ )locally finite is denoted by O µ λ . These are subcategories of parabolic category O as introduced in [RC]. By construction, O µ λ is a Serre subcategory of O λ . We denote the corresponding exact full embedding of categories by ı µ : O µ ֒→ O. The left adjoint of ı µ is the corresponding Zuckerman functor, denoted by Z µ , it is given by taking the largest quotient inside O µ . The categories O µ 0 and O λ are indecomposable. The category O µ λ may decompose, e.g. in the case g = B(2) = so(5), for w λ 0 = s and w µ 0 = t, where s and t are the two different simple reflections. We define the set X λ as the set of longest representatives in W of cosets in W/W λ . The non-isomorphic simple objects in the category O λ are indexed by X λ : The set of such shortest representatives x ∈ X λ is denoted by X µ λ . We denote by d the usual duality on O, which restricts to O µ and to each block in these categories, see [Hu,Section 3.2]. For x ∈ X µ λ , consider the following structural modules in O µ λ : the standard module (or generalised Verma module) Consider a minimal projective generator of O µ λ given by (1) where P µ (x · λ) is the indecomposable projective cover of L(x · λ) in O µ λ , and the algebra A µ λ := End g (P µ λ ). Then we have the usual equivalence of categories O µ λ→ mod-A µ λ ; M → Hom g (P µ λ , M ).
We consider the Bruhat order ≤ on W , with the convention that e is the smallest element. It restricts to the Bruhat order on X µ λ . The order on the weights is defined by x · λ ≤ y · λ if and only if y ≤ x. From the BGG Theorem on the structure of Verma modules, see e.g. [Hu,Section 5.1], it follows that the algebras A µ λ are quasi-hereditary with respect to the poset of weights X µ λ · λ. The standard modules coincide with the ones above.
Remark 2.1. We will use the term (generalised) tilting module for a module of a finite dimensional algebra satisfying properties (i)-(iii) in [Ha,Section III.3]. When we refer to the modules in (parabolic) category O that simultaneously admit a standard and a costandard filtration, see e.g. [Hu,Chapter 11], we will use the term quasi-hereditary tilting module, or q.h. tilting module. Then a q.h. tilting module is a (special case of) a partial generalised tilting module, whereas the characteristic q.h. tilting module is a generalised tilting module, see e.g. [Rin,Theorem 5].
When µ is regular, meaning that the corresponding parabolic category O µ is the usual category O, we leave out the reference to µ. Similarly, we will leave out λ (as in L(x) := L(x.λ)), or replace it by 0, whenever it is regular. By application of [So1,Theorem 11], all categories O µ λ , with λ arbitrary integral regular dominant and µ fixed, are equivalent, justifying this convention.
Consider the translation functor θ on λ : O 0 → O λ to the λ-wall and also its adjoint θ out λ : O λ → O 0 , which is the translation out of the λ-wall, see [Hu,Chapter 7]. For x ∈ W , denote by θ x the unique projective functor on O 0 which maps P (e) to P (x), see [BGe]. Note that, in particular, θ out λ • θ on λ = θ w λ 0 . By [Hu,Theorem 7.9] or [Ja], for any x ∈ W , we have for shift in position in the complex and degree in the modules. This corresponds to the conventions in [BGS], but differs slightly from [MOS]. A module M , regarded as an object in the derived category put in position zero, is denoted by M • .
For any Koszul algebra B, [BGS,Theorem 2.12.6] introduces the Koszul duality functor K B , a covariant equivalence of triangulated categories As in [MOS,Section 3] or [MSa, Section 2], we introduce the full subcategory of D b (B-gmod) of linear complexes of projective modules in B-gmod, which we denote by LP B . Then [MSa,Theorem 2.4] (or, more generally, [MOS,Theorem 12]) establishes, for any quadratic algebra B, an equivalence From [MOS,Chapter 5] it follows that ǫ B is isomorphic to the restriction of K B in case B is Koszul.
As proved in [Ba1], A µ λ has a Koszul grading, where the Koszul dual algebra is E(A µ λ ) ∼ = A λ µ , see also [BGS,So1,Ma2]. We also have (A µ λ ) opp ∼ = A µ λ (as graded algebras) as an immediate consequence of the duality functor d. The algebras A µ λ are even standard Koszul in the sense of [ADL], see [Ma2].
The graded module categories are denoted by Z O µ λ := A µ λ -gmod. We will sometimes replace the notation HomZ O by hom O .
It is more convenient to work with the composition of the usual Koszul duality functor with the duality d to obtain a contravariant functor where we also identify the graded module categories corresponding to the isomorphism E(A µ λ ) ∼ = A λ µ . This functor satisfies [BGS,Theorem 3.11.1]. Similarly, we define ǫ µ λ = dǫ A µ λ , as a contravariant equivalence of categories We conclude this subsection with the introduction of the graded lifts to Z O of the translation functors on O, as studied in [St1]. We denote them by the same symbols as on O and use the grading convention of [St1]. This means that the graded version of equation (2) is ), see also [MOS,Lemma 38]. This implies that 2.3. Twisting and shuffling functors. We will use the twisting functor T s , which is an endofunctor on each integral block O λ corresponding to a simple reflection s, see [AS,MS2]. For any w ∈ W with reduced expression w = s 1 s 2 · · · s m , we can define the functor T w = T s 1 T s 2 · · · T sm , where the resulting functor does not depend on the choice of a reduced expression, see [KM,Corollary 11]. The functor T w is right exact and its derived functor LT w is an auto-equivalence of D b (O 0 ), see [AS,Corollary 4.2]. This property extends to a singular block of category O, see e.g. [CM1,Proposition 5.11], so we have an auto-equivalence Twisting functors admit graded lifts, see [MO, Appendix] or [KM,Theorem 1.1]. By [AS,Theorem 3.2], the following diagram commutes: The shuffling functor C s corresponding to a simple reflection s, see [Ca,MS2], is the endofunctor of O 0 defined as the cokernel of the adjunction morphism from the identity functor to the projective functor θ s . For any w ∈ W with reduced expression w = s 1 s 2 · · · s m , we can define the functor where the resulting functor does not depend on the choice of a reduced expression, see [MS1,Lemma 5.10] or [KM,Theorem 2] and [MOS,Section 6.5]. The functor C w is right exact and LC w is an auto-equivalence, with inverse dLC w d, of D b (O 0 ), see [MS1,Theorem 5.7], so The two basic types of derived auto-equivalences in this sections will be exploited to obtain more complicated derived equivalences in the remainder of the paper. We note that they also appear in more abstract generality in [We,Theorem 6.15  We also consider the following covariant right exact functor with * being the canonical duality functor from mod-R(B) to R(B)-mod. We call R B the Ringel duality functor. From [Rin,Theorem 6] and [MS3, Proposition 2.2], it follows that R B restricts to an equivalence between the additive subcategories of projective modules of B and tilting modules of R(B) and between the additive subcategories of tilting modules of B and injective modules of R(B). It also restricts to an equivalence between the category of B-modules with standard flag and the category of R(B)-modules with costandard flag. Finally, its left derived functor induces an equivalence As O λ and O µ 0 are Ringel self-dual, it is natural to compose the Ringel duality functor with a fixed equivalence which realises the self-duality. In case λ = 0, we can choose the Ringel duality functor as [MS3,Proposition 4.4]. In particular, the restriction of L l(w µ 0 ) C w 0 to the category O µ 0 is a right exact functor. In case µ = 0, the Ringel duality functor can be interpreted as 2.5. The centre and coinvariants. For an integral dominant λ, consider B λ and the corresponding semisimple Lie algebra g λ , generated by the root spaces of g corresponding to elements in ±B λ . The Weyl group of this algebra is isomorphic to W λ . Then h λ := g λ ∩ h is a Cartan subalgebra in g λ . Now we can consider the algebra of coinvariants for W λ given by The algebra C(W λ ) inherits a positive grading from S(h λ ) which is defined by giving constants degree 0 and elements of h degree 2.
For the particular case W λ = W , we set C := C(W ). For w ∈ W , let w C denote the C-C bimodule obtained from C C C by twisting the left action of C by w (note that W acts on C by automorphisms). For X a subgroup of W , let C X denote the algebra of X-invariants in C. Similarly, for a simple reflection s, we denote by C s the algebra of s-invariants in C.

Consider again an integral dominant λ and set
for any integral dominant λ. All these statement admit canonical graded lifts, as the grading on O can be introduced via V λ , see e.g. [St1]. [Br,Lemma 6.2] or [MS3, Theorem 5.2(2)], it follows that C λ is isomorphic to the centre of A λ . More generally, we denote the centre of A µ λ by C µ λ . In the remainder of this subsection we consider g = sl(n). Then W ∼ = S n and C µ λ has been calculated in [Br,St3]. For an integral dominant λ, it follows from [So1,Theorem 11] that O λ is uniquely determined, up to equivalence, by a composition p(λ) = (p 1 , · · · , p k ) of n. This composition is defined by demanding that W λ , as a subgroup of S n , is naturally given by S p 1 × S p 2 × · · · × S p k . It is well-known, by a result of Borel, that C λ is, as a graded algebra, isomorphic to the cohomology ring of a partial flag variety. The Hilbert-Poincaré polynomial for C λ , with p(λ) as above, is hence well-known, see e.g. [Ch], and is given by 2.6. Extension quivers in parabolic category O. We demonstrate the basic property that the Ext 1 -quiver of a parabolic singular block can be read off immediately from the Ext 1 -quiver of the principal block O 0 .
). Now, the Koszul duality of [BGS] implies an isomorphism Applying the same procedure again, but now to the right-hand side above, concludes the proof.

The elements of Ext
, L(y · λ)) correspond to morphisms from P µ (y · λ) to P µ (x · λ) such that the top of P µ (y · λ) maps to the top of the radical of P µ (x · λ). As the (Koszul) grading on A µ λ is such that the degree 0 part is semisimple and the algebra is generated by the degree 0 and 1 parts, this can also be expressed as Proposition 2.2 and equation (9) then imply the following corollary.

Graded versus non-graded derived equivalences
3.1. The example of the Koszul duality functor. The Koszul duality functor in equation (3) gives an equivalence between bounded derived categories of graded modules. We demonstrate, however, that Koszul dual algebras are, in general, not derived equivalent as ungraded algebras.
Proposition 3.1. Despite the existence of an equivalence of triangulated categories are not derived equivalent, if their centres are not isomorphic, see [Ric1,Proposition 9.2]. Consider the socle of the left regular module for the two centres. For A λ this socle is simple, by Subsection 2.5. It follows easily from [Br,Lemma 6.2] that the number of simple modules in that socle for A λ must be at least the number of non-isomorphic indecomposable projective-injective modules. This is the cardinality of the right Kazhdan-Lusztig cell of w λ 0 w 0 .
3.2. Gradable derived equivalences. In this subsection we define a special case of the concept of a derived equivalence (see [Ric1,Definition 6.5]), which we will investigate for category O further in the paper.
Definition 3.2. Two finite dimensional graded algebras B and D are said to be gradable derived equivalent if one of the following equivalent properties is satisfied.
, with inverse G, such that F and G admit graded lifts.
Remark 3.3. It is clear that the failure of the Koszul duality functor to satisfy the requirement in Definition 3.2(i), see equation (5), is precisely what prevents it from being the graded lift of an ungraded equivalence.
Now we state and prove two simple propositions which demonstrate equivalence of the two properties in Definition 3.2.
Proposition 3.4. Consider two finite dimensional graded algebras B and D such that there is an equivalence If F commutes with 1 , thenF is a graded lift of an equivalence Moreover, if G is inverse to F , then G is the graded lift of an inverse G to F .
Proof. Take a minimal projective generator P D for D-mod and the graded lift P D . Denote by T • the object in D b (B-mod) obtained by forgetting the grading on F −1 (P • D ). One deduces straightforwardly that T • is a tilting complex according to [Ric1,Definition 6.5 The first assertion of the proposition then follows from [Ric1,Theorem 6.4]. Now we consider the inverse G. From F G ∼ = Id ∼ = G F and F 1 ∼ = 1 F it follows that G 1 ∼ = 1 G. Hence we can use the first part to obtain that G is the graded lift of a triangulated functor G. The construction, moreover, implies that F G and GF acts as identity functors restricted to the subcategories of projective modules. As they are triangulated functors it follows that F and G are mutually inverse.
Proposition 3.5. Let B and D be finite dimensional graded algebras and an equivalence with inverse G. Assume that F and G admit graded lifts F and G, respectively. Then F gives an equivalence of triangulated categories Proof. First we prove that F is essentially surjective (dense). Consider the forgetful functor f D : [BGS,Lemma 2.5.3], the proof of which extends easily to the derived category, we then find that there is j ∈ N such that X The density of F hence follows.
Here F , f B and f D act by isomorphisms and hence so does F . As F respects it then follows easily that F is full and faithful. Hence F is an equivalence of categories. By construction it is also an equivalence of triangulated categories.
Finally, we note the following immediate extension of [Ric1, Proposition 9.2].
Lemma 3.6. Consider two finite dimensional graded algebras B, D which are gradable derived equivalent. Then there is an isomorphism of graded algebras Z(B) ∼ = Z(D), with canonically inherited grading on the centres.

3.
3. An application: derived shuffling in the parabolic setting. We use the results of the previous subsection to extend (12) to the form of (10) and (11). First we point out two subtleties (a) and (b). Consider the exact inclusion ı µ : O µ 0 → O 0 leading to a faithful (see e.g. [Ba1, Lemma 2.6]) triangulated functor 0 is a full Serre subcategory of O 0 and C s restricts to a right exact endofunctor of O µ 0 , which we denote by C µ s . However, its left derived functor LC µ s need not be isomorphic to the restriction of LC s to D b (O µ 0 ), viewed as a subcategory as above. A trivial example is g = sl (2), as then C µ s ∼ = 0 for µ singular. However, a restriction of LC s is never zero as LC s is an equivalence.
However, this subcategory is neither full, nor isomorphism closed. It is not full as, for instance, there can be higher extensions The functor LC s maps, by the definition of a derived functor, objects in is hence a non-trivial problem. Proposition 3.7. For any w ∈ W , there is an endofunctor LC w of D b (O µ 0 ), which yields an auto-equivalence and admits a commuting diagram The same holds in the graded setting.
Proof. An inverse of LT w is given by by dLT w d, see [AS,Section 4]. Proposition 3.5 thus implies that the graded lift of LT w induces an auto-equivalence of D b ( Z O λ ). So the diagram (11) admits a graded lift with equivalences on the horizontal arrows. Then we apply [MOS,Sections 6.4 and 6.5], see also [Ry] or Proposition 4.7. This implies a commutative diagram, where the horizontal arrows are equivalences As LT w is a triangulated functor which commutes with 1 , equation (5) implies that F commutes with 1 , so Proposition 3.4 implies that F is the lift of an ungraded equivalence and hence a gradable derived equivalence.
Remark 3.8. Consider the complex 0 → Id O 0 → θ s → 0 of exact functors, where the non-zero map is given by the adjunction morphism, cf. [Ric2] and [MS1,Remark 5.8]. Applying this to a complex in D b (O 0 ) and taking the total complex defines a triangulated endofunctor of D b (O 0 ), isomorphic to LC s . This endofunctor, by construction, preserves the image of D b (O µ 0 ) under ı µ , since both the identity and θ s do. This gives an alternative construction of the equivalence in Proposition 3.7.

Graded translation functors
4.1. Translation through the principal block. Proposition 4.1. As graded functors, we have In particular, ±l(w λ 0 ) are exactly the extremal degrees in which Id appears.
Proof. The ungraded statement follows easily from [BGe,Theorem 3.3] and [Ja,Formula 4.13 (1)]. It thus suffices to prove that Using equation (15) and ignoring an overall grading shift, we find So it suffices to prove that This extends to a graded morphism S(h) ։ S(h λ ) which we compose with the canonical surjection S(h λ ) ։ C(W λ ) to find ξ : S(h) ։ C(W λ ). By construction S(h) W + , and hence the corresponding ideal, is in the kernel of ξ. This implies a morphism of graded algebras η : C ։ C(W λ ).
As (C λ ) + = C W λ + is in the kernel of η, this proves the desired inequalities.
As an application, we prove the following proposition, which we need later.
Then θ out λ induces isomorphisms Proof. Consider i ∈ Z, then equation (8) and Proposition 4.1 imply that Now, if i = 0, the result follows by the assumptions as c 0 = 1. If i = 1, the result follows from the assumptions and c 1 = 0 and c 0 = 1.
This proposition generalises Corollary 2.3 and hence provides an alternative proof.

4.2.
Translating standard modules. In this subsection we completely describe the graded translation of standard modules to and from the wall. These results generalise [St1,Theorem 8.1(3) and Theorem 8.2(2)] to arbitrary walls and the parabolic setting. Consider the bijection which is inverse to multiplication and denote by b 1 λ and b 2 λ the composition of b λ with the projection on the X λ -component and the W λ -component, respectively. otherwise.
In particular, for any y ∈ X λ and u ∈ W λ , we have moreover, there are no other standard modules appearing in the filtration.
In the remainder of this subsection we prove these two theorems.
Lemma 4.5. For any x ∈ X λ , u ∈ W λ and j ∈ N, we have Proof. We prove the Koszul dual statement, which is dim Ext j O (∆(vy), L(y)) = δ j,l(v) , for any y ∈ X λ , v ∈ W λ and i ∈ N. That the left-hand side is zero, for j > l(v), follows immediately from [Hu,Theorem 6.11]. For any simple reflection s ∈ W λ and v ∈ W λ such that sv > v, the procedure in the proof of [Ma1, Proposition 3] shows dim Ext k O (∆(svy), L(y)) = dim Ext k−1 O (∆(vy), L(y)), which proves the claim inductively.
Proof of Theorem 4.3. It suffices to prove the non-parabolic case, as the remainder follows via an application of the Zuckerman functor, which commutes with translation functors. The non-parabolic result follows from equation (7), Lemma 4.5 and [Ja,Formula 4.12(2)].
Proof of Theorem 4.4. For any y ∈ X µ , [Hu,Theorem 3.3(d)] and equation (8) yield ). Theorem 4.3 then implies that the only y which can appear non-trivially are those for which we have b 1 λ (y) = x, which is precisely the set xW λ ⊂ X µ . Remark 4.6. By using Proposition 4.7, an alternative proof of Theorem 4.4 would be to determine a resolution of standard modules in O µ by Verma modules. This can be obtained by applying parabolic induction to g of the BGG resolutions for the Levi subalgebra of q µ , see e.g. [Hu,Section 6].
Proposition 4.7. There are commutative diagrams of functors as follows: Before proving this we list some consequences. Recall the c j 's from Proposition 4.1.
For any M in Z O λ which is locally U (q µ )-finite and for any k ∈ N, we have Proof. This is a direct application of Proposition 4.7 to Proposition 4.1. In particular, the maps in the complex LZ µ • ı µ (N • ) are trivial for any N ∈ Z O µ λ , which yields the result about the cohomology functors.
Remark 4.9. Note that Corollary 4.8 also implies the isomorphism as all projective modules and homomorphism spaces between them are gradable.
Corollary 4.10. Consider two objects X • and Y Then ı µ induces an isomorphism Proof. As ı µ is faithful, the morphism is always injective. So it suffices to prove that the dimensions of the spaces of homomorphisms agree. This follows from given by adjunction, and Remark 4.9.
Corollary 4.11. Consider two objects X • and Y Proof. The property in the graded setting follows immediately from Proposition 4.1 and Corollary 4.8. The ungraded version now follows from Remark 4.9.
Now we turn to the proof of Proposition 4.7.
Lemma 4.12. The translation functor θ out λ : restricts to a full and faithful functor θ out λ : LP µ λ → LP µ , the image of which is the full subcategory of LP µ given by linear complexes of projective modules in add(θ out λ P µ λ ).
Proof. Recall that the categories of linear complexes of projective modules are full subcategories of the derived categories. By equation (9), θ out λ restricts to a functor from LP µ λ to LP µ . To prove that this is full and faithful, consider two linear complexes P • and Q • of projective modules in Z O µ λ . Proposition 4.2 now implies The description of the image is a consequence of Corollary 2.3.
Corollary 4.13. The following is a commutative diagram of functors: Proof. Using the standard properties of K λ µ , see [BGS,Theorem 3.11.1], and equation (9) implies that, for any x ∈ X µ λ , Corollary 2.3 can then be applied to show that the complexes corresponding to θ out µ (ǫ λ µ ) −1 M and (ǫ λ ) −1 ı µ M , for any M in Z O µ λ , are isomorphic. As, by construction, this isomorphism is natural, this concludes the proof.
Proof of Proposition 4.7. The first diagram follows from [MOS,Theorem 30 and Proposition 21], in combination with Corollary 4.13. The second diagram is the adjoint reformulation of the first.

4.4.
A generalisation of Bott's theorem. Take ρ to be the half of the sum of positive roots. Then Z −ρ is the Zuckerman functor to the category of finite dimensional modules. Bott's extension of the Borel-Weil theorem then reads L k Z −ρ (∆(x)) = δ k,l(x) L(e) and L k Z −ρ (∆(y · λ)) = 0, for all k ∈ N, for all x ∈ W and all y ∈ X λ with λ singular. For an overview of the approach with Zuckerman functors, see e.g. [Co, dS]. We can now generalise this. Let b µ : W → W µ × X µ denote the bijection defined as the inverse of multiplication. Define x 1 and x 1 by b µ (x) = (x 1 , x 1 ) as in [Le,Proposition 3.4].
For any x ∈ X λ and any integral dominant µ, we have In particular, for any y ∈ X µ and u ∈ W µ , we have L k Z µ (∆(uy)) = δ k,l(u) ∆ µ (y).
Setting µ = −ρ (so x = x 1 and x 1 = e) yields the original result of Bott.
Proof. This follows from a direct application of the Koszul duality functor to the statement in Theorem 4.3 by using Proposition 4.7.
Remark 4.15. This could also be proved directly by using parabolic induction to reduce to the Bott's theorem for the Levi subalgebra. This would yield an alternative proof of Theorem 4.3 by applying Proposition 4.7.
4.5. Translation through the wall. In this subsection we gather some technical results on translation through the wall which will be needed later and which can be formulated most generally by ignoring grading.

Lemma 4.16. Consider an indecomposable
Applying θ out λ to the above isomorphism implies that there must be some p ∈ N for which θ out λ Y • ∼ = (X • ) ⊕p . Applying θ on λ , while using the ungraded version of Proposition 4.1, to the latter isomorphism then yields This implies the claim with k = |W λ |/p.
Lemma 4.18. Consider a module M ∈ Ob(O 0 ) such that θ w λ 0 M ∼ = M ⊕|W λ | and for which V yields an algebra isomorphism V : End O 0 (M )→ End C (VM ). Then V λ also induces an algebra isomorphism V λ : Proof. Equation (15) and the faithfulness of θ out λ and induction imply that we have a commuting diagram of algebra homomorphisms: As the upper horizontal arrow is an isomorphism by assumption, so is V λ .
Proof. Applying θ on λ to θ out λ X • ∼ = ı µ Y • and using the commutation of inclusion and translation, implies Applying θ out λ to the latter yields an isomorphism ı µ Y • ∼ = ı µ θ out λ Z • , so the conclusion follows from Corollary 4.11.

Shuffling and projective functors
Unlike twisting functors, shuffling functors do not commute with projective functors in general. Here we investigate their intertwining relations. We consider the principle block O 0 for an arbitrary reductive Lie algebra g.
Theorem 5.1. For two simple reflections s, t ∈ W , we have The above statement is formulated in [MS2,Section 11], however, the proof of [MS2, Section 11] is not complete (it proves only a special case of the statement). Below, we follow the alternative argument outlined in [MS2,Remarks 11.2 and 11.4].
Set Q (w) := add(C w P ), with w ∈ W , and P a projective generator for O 0 .
(i) For ν ∈ Λ + int and a simple reflection s ∈ W ν ×W † ν , we have C w ν 0 θ s ∼ = θ w ν if and only if the category Q (w ν 0 ) is stable under action of θ w ν 0 sw ν 0 . (ii) For any y ∈ W , we have C w 0 θ y ∼ = θ w 0 yw 0 C w 0 .
Remark 5.4. Note that the stability of Q (w) is not an obvious property. Already for g = sl(3), the category Q (s) is not stable under projective functors. Now we start the proofs. Define the graded lift of C t via the exact sequence (17) Id 1 adj −→ θ t → C t → 0.
Lemma 5.5. For a simple reflection t, define the functor F t as the kernel of the adjunction morphism in (17).
Proof. As Id is exact, F t is left exact. Then θ t T t is also left exact and so is T t θ t , as θ t maps injective modules to injective modules. By construction, we have This implies immediately that θ t F t ∼ = 0.
To prove that F t θ t acts trivially on all simple modules, it suffices to consider a simple module L with θ t L = 0. But then θ t L has simple socle L and the latter is not annihilated by θ t . This means that the adjunction morphism is injective on both L and θ t L, which implies F t θ t L = 0. The claim follows.
Proof of Theorem 5.1. Assume first that s and t commute but are distinct, in particular we have θ s θ t ∼ = θ ts ∼ = θ t θ s . We can compose (17) with θ s on both sides. This yields exact sequences From [BGe,Theorem 3.5] and Kazhdan-Lusztig combinatorics it follows that the morphism θ s 1 → θ st is unique, up to a non-zero scalar, implying C t θ s ∼ = θ s C t . Now consider the case s = t. Using Lemma 5.5 and the relation θ 2 s ∼ = θ s 1 ⊕ θ s −1 we obtain two short exact sequences The only possible way to have such an injection θ s 1 ֒→ θ s 1 ⊕ θ s −1 corresponds to θ s 1 → θ s 1 . This implies indeed θ s C s ∼ = θ s −1 ∼ = C s θ s .
Finally, assume that the simple reflections s and t together generate a group isomorphic to S 3 . Using the definitions of C s and C t via (17), we construct a commutative diagram with exact rows and columns as follows: Now we compose every functor in the diagram with the exact functor θ s and use the above result, for the case s = t, in the first row of the following diagram: The kernel of this map can be described as the direct sum of the image of β 1 with the preimage under β 2 of the image of h 1 . Because the top row is a split exact sequence, the latter corresponds to the image of θ s under g 1 . This implies the exact sequence From Kazhdan-Lusztig combinatorics we know that θ s θ t θ s ∼ = θ sts ⊕θ s . As there is no graded morphism θ s → θ sts and the only graded morphism θ s → θ s is the identity, up to a scalar, we finally obtain an exact sequence Analogously we can compose θ t with every functor in the first diagram, eventually yielding an exact sequence From Kazhdan-Lusztig combinatorics it, moreover, follows that there is only one graded morphism θ s θ t 1 → θ sts , which concludes the proof.
Proof of Corollary 5.2. If s ∈ W † ν , the result follows immediately from Theorem 5.1. The case s ∈ W ν can be reduced to W ν ∼ = S n generated by s 1 , · · · , s n and s = s i , for some 1 ≤ i ≤ n. In case i ≤ n/2, we take the reduced form w ν 0 = s n s n−1 · · · s 2 s 1 s n s n−1 · · · s 3 s 2 · · · s n s n−1 s n−2 s n s n−1 s n ; if i ≥ n/2, we take w ν 0 = s 1 s 2 · · · s n−1 s n s 1 s 2 · · · s n−2 s n−1 · · · s 1 s 2 s 3 s 1 s 2 s 1 . In both cases, apply Theorem 5.1 repeatedly, giving the result with s ′ = s n−i+1 .
Lemma 5.6. For any simple reflection t ∈ W , we have VC t ∼ = t C ⊗ C V.
Proof. Since both sides are right exact, it is enough to check the isomorphism on the category of projective modules in O 0 . [Ba2,Theorem 4.9] implies that there is a unique injective bimodule homomorphism C ֒→ C ⊗ C t C. We claim that the cokernel of this maps is isomorphic to t C. Indeed, this cokernel is isomorphic to C both as a left and as a right C-module. As a C t -C t -bimodule, we obviously get a decomposition of the cokernel into a direct sum of two copies of C t . To determine the action of the remaining generator, it is enough to compute the rank two case. In that case the statement is easily checked by a direct computation.
Corollary 5.7. For ν ∈ Λ + int and a simple reflection Proof. By Subsection 2.5, we have By induction and the choice of a reduced expression for w ν 0 , Lemma 5.6 implies of C-C bimodules. Combining the three isomorphisms yields the claim.
Lemma 5.8. Soergel's combinatorial functor V is full and faithful when restricted to the category Q (w) , for any w ∈ W .
Proof. As LC w is an auto-equivalence of D b (O 0 ), see equation (12), C w provides an isomorphism from Hom O 0 (P, P ′ ) to Hom O 0 (C w P, C w P ′ ) for any two projective modules P, P ′ in O 0 . The claim hence reduces to the statement that the functor VC w is full and faithful on the category of projective modules. By Lemma 5.6 this reduces to [So1,Struktursatz 9].
Proof of Proposition 5.3. We prove claim (i) first. It is clear that if the proposed equivalence of functors holds, then the fact that the category Q (w ν 0 ) is stable under θ w ν 0 sw ν 0 follows from the fact that θ s preserves the category of projective modules. Hence we focus on the other direction of the claim.
The two composed functors in the proposition are right exact, as shuffling functors are right exact and projective functors are exact functors mapping projective modules to projective modules. Hence it suffices to prove the isomorphism as functors restricted to Q (e) , the full subcategory of projective modules in O 0 .
By assumption both functors restrict to functors between Q (e) and Q (w ν 0 ) . Hence, the combination of Lemma 5.8 and Corollary 5.7 concludes the proof of claim (i). Now we consider claim (i) for w ν 0 = w 0 . As the category Q (w 0 ) is the category of q.h. tilting modules, see e.g. [MS3,Section 4.2], it is stable under the action of all projective functors. Therefore claim (ii) holds for y = s a simple reflection. The full statement then follows by induction on the length of y and [Ma1, Equation (1)].
We have the following application of Proposition 5.3(2).
]. Moreover, the same equation holds considering ∆ µ (x · λ) • as an object of D b (O µ λ ) and LC w 0 the endofunctor on D b (O µ 0 ) defined in Proposition 3.7.
Proof. By Theorem 4.3 and Proposition 5.3(2), we have Applying then [MS3,Proposition 4.4(1)], gives The result now follows by applying Theorem 4.3. The reformulation inside D b (O µ 0 ) follows from the commuting diagram in Proposition 3.7 and Corollary 4.11.
Finally, we derive useful expression for regular parabolic q.h. tilting modules.
. This module is the translation through the wall of a simple module, which has simple top and hence is indecomposable.
Then A µ λ and A µ ′ λ ′ are gradable derived equivalent, so, in particular, we have equivalences of triangulated categories For sl(n), the formulation of the theorem simplifies substantially. In this case, without loss of generality, we can take ν 1 = ν 2 = −ρ, so G 2 = G ′ 2 = H 2 = H ′ 2 = {e}. Then we obtain precisely Theorem C.
The remainder of this section is devoted to the proof of Theorem 6.1.
. Hence (18) and Lemma 4.18 imply . The algebra can then be computed by using equation (15) and Lemma 5.6, Under the isomorphism C λ ∼ = C λ ′ , we can identify the modules . This formula implies that θ on λ ′ S x decomposes into |W λ ′ | indecomposable direct summands. Equation (18) and Corollary 4.17 thus imply that there is an indecomposable

Now equation (19) can be rewritten as
The same calculation for P λ , rather than P (x · λ), implies As LC w ν 0 is an auto-equivalence of D b (O 0 ), the module S does not have self-extensions. As θ out For any projective module P in O 0 , we claim that θ on λ ′ C w ν 0 P is a module in add(T ). Indeed, by Corollary 5.2 we obtain where C w ν 0 θ w λ 0 P is clearly a module in add(S) and θ on λ ′ S ∼ = T ⊕|W λ | .
As LC w ν 0 is an auto-equivalence of D b (O 0 ), [Ric1,Theorem 6.4] implies that C w ν 0 P 0 is a generalised tilting module. Hence, P 0 has a coresolution by modules in add(C w ν 0 P 0 ). By the above paragraph, applying the exact functor θ on λ ′ leads to a coresolution of θ on λ ′ P 0 by modules in add(T ). As T has no self-extensions by equation (21), this immediately implies that every indecomposable summand in θ on λ ′ P 0 has such a coresolution. Thus, in particular, there is some r ∈ N for which there is an exact sequence where T i ∈ add(T ).
Equations (21) and (22) imply that T is a generalised tilting module, or a tilting complex (contained in one position) in the sense of [Ric1]. The equivalence then follows from equation (20) and [Ha,Theorem III.2.10] or [Ric1,Theorem 6.4].
Finally, we prove the existence of the commutative diagram. It suffices to prove that LC w ν 0 • θ out λ ∼ = θ out λ ′ • F λλ ′ , restricted as functors on the category of projective modules in O λ , as both are triangulated functors. By construction, the functor F λλ ′ acts on the category of projective modules by mapping P (x · λ) to T x and its action on morphisms corresponds to the algebra isomorphism determined in equation (20). The equivalence of the functors LC w ν 0 • θ out λ and θ out λ ′ • F λλ ′ acting between add(P λ ) and add(S), thus follows from construction of the isomorphism (20). Lemma 6.3. In the setup of Lemma 6.2, any object X Proof. According to Corollary 4.11, it is sufficient to prove that That this is true follows from the fact that Zuckerman and inclusion functors commute with translation functors (and hence also with shuffling functors), see e.g. [Ba1, Lemma 2.6(a)] and the diagram in Lemma 6.2.
Lemma 6.4. Consider λ, ν as in Lemma 6.2 and an arbitrary µ ∈ Λ + int . There is a an equivalence F µ λλ ′ : of triangulated categories which admits a commutative diagram of functors Proof. Consider the minimal projective generator P µ λ of O µ λ in (1). By Lemma 6.2, there is a complex where the latter isomorphism follows from Corollary 4.10. Lemma 6.2 and Proposition 3.7 yield θ out λ ′ S • ∼ = ı µ LC w ν 0 θ out λ P µ λ . Lemma 4.19 thus implies the existence of The second property in (24) and the faithfulness of θ out λ ′ give an injection The right-hand side is zero, if k = 0, by Proposition 3.7. So we find The first property in equation (24) and equation (25) allow us to apply Corollary 4.10 and deduce, using equation (23), that By equations (25) and (26), the result would follow from [Ric1,Theorem 6.4], if we could show that add(T • ) generates D b (O µ λ ) as a triangulated category. Now we focus on the latter statement. Remark 4.9 and Lemma 6.3 yield As, for every x ∈ X λ , as a triangulated category (which is an immediate consequence of Lemma 6.2) hence implies that add(T • ) generates D b (O µ λ ′ ) as a triangulated category. The existence of the commuting diagram then follows by the same arguments as in the proof of Lemma 6.2.
Lemma 6.5. The functor F µ λλ ′ from Lemma 6.4 admits a graded lift, yielding an equivalence of triangulated categories Proof. As translation, shuffling and inclusion functors admit graded lifts, it follows easily that so does F µ λ,λ ′ . We can switch the roles of λ and λ ′ and construct a functor F µ λ ′ λ . We define the (gradable) functor G µ λ ′ λ = dF λ ′ λ d. Then G µ λ ′ λ admits commutative diagrams with dLC w ν 0 d, analogous to the ones in Lemmata 6.2 and 6.4 and is also gradable. By construction, F µ λλ ′ G µ λ ′ λ and, respectively, G µ λ ′ λ F µ λλ ′ , are isomorphic to the respective identity functors when restricted to the categories of projective modules in O µ λ ′ and O µ λ , respectively. Hence G µ λ ′ λ is an inverse to F µ λ ′ λ . The result thus follows from Proposition 3.5.
Then A µ λ and A µ λ ′ are gradable derived equivalent. In particular, we have equivalences of triangulated categories Proof. There are k ∈ N and p i ∈ N, for 1 ≤ i ≤ k, such that p 1 + · · · + p k = n and G 1 = S p 1 × S p 2 × · · · × S p k as a subgroup of S n . Then G 2 , as a subgroup of S n , is equal to for some σ ∈ S k . It suffices to prove the lemma for σ such that it only interchanges two neighbors (j, j + 1) for some 1 ≤ j < k. Then we choose an integral dominant ν such that W ν is equal to the subgroup of W ν 1 ⊂ W defined as For the case we consider, this leads to w λ ′ 0 = w ν 0 w λ 0 w ν 0 and the equivalences follow from Lemma 6.4, Lemma 6.5 and Definition 3.2.
Proof of Theorem 6.1. To prove the theorem it suffices to restrict to either λ = λ ′ or µ = µ ′ as the full statement follows by composition of the two. The case µ = µ ′ is precisely Lemma 6.6.
The graded equivalence for λ = λ ′ follows immediately from composing Lemma 6.6 with the equivalence in equation (4). The fact that the equivalence then descends to the non-graded categories follows from Proposition 3.4 and equation (5).
Finally, we note how the results in this section lead to a proof of Theorem D. Part (ii) follows immediately from Lemmata 6.2, 6.4 and 6.5. Part (i) then follows from Koszul duality as in the proof of Theorem 6.1. The link with derived twisting functors follows from [MOS,Theorem 39].

The classification for type A
In this section we prove the following theorem, which implies Theorem B.
Note that, as we only consider type A in this section, we do not need to distinguish between isomorphisms of Coxeter groups and isomorphisms of Coxeter systems. Before proving Theorem 7.1 we need the following preparatory lemma.
Lemma 7.2. Consider Coxeter groups W and W ′ of type A with respective Young subgroups X and X ′ . There is a graded algebra isomorphism C(W ) X ∼ = C(W ′ ) X ′ if and only if, for some decompositions into products of irreducible Weyl groups, we have k = m and there is a permutation ϕ on {1, 2, . . . , k} such that X ∩X i ∼ = X ′ ∩X ′ ϕ(i) and X i ∼ = X ′ ϕ(i) for all i = 1, 2, . . . , k.
Proof. First we assume that W and W ′ are simple. Without loss of generality, we assume that W ∼ = S n and W ′ ∼ = S m with m ≥ n. Denote the compositions of n and m, corresponding to X and X ′ , by (p 1 , · · · , p k ) and (q 1 , · · · , q l ), respectively. A graded algebra isomorphism implies, in particular, a graded vector space isomorphism, so equation (16) . If m > n, then the irreducible factorisation of the righthand side contains a term (1 − exp( 2πi m )z), with i 2 = −1, which never occurs in the factorisation of the left-hand side. Hence m = n, or W ∼ = W ′ . The same argument can now be used if the maximum of {p j , 1 ≤ j ≤ k} were larger than the maximum of {q j , 1 ≤ j ≤ l}. By similar reasoning and induction it then follows that the compositions are the same up to ordering and hence we find X ∼ = X ′ . Now consider an arbitrary Coxeter group W of type A. It follows easily from the definitions that, if we consider the decomposition W ∼ = W 1 × W 2 × · · · × W k , for some k, such that each W i is irreducible and set X i = X ∩ W i , then we have The general result now follows.
Proof of Theorem 7.1. By [So1,Theorem 11], we can take reductive Lie algebrasg andg ′ for which there are Borel subalgebrasb andb ′ , Weyl groups W andW , and integral dominant weightsλ andλ ′ such that λ ′ under the above isomorphisms. Moreover, by [Mat,Proposition A.4], we can choseg andg ′ to be of type A. By (a), the properties in claim (I) and claim (II) are true if and only if they are true for Oλ (g,b) and Oλ ′ (g ′ ,b ′ ). Furthermore (b) and (c) imply that also the property in claim (III) is true if and only if it holds for W andW . Therefore it suffices to prove the equivalences between the three statements restricted to the integral setting. So, henceforth we can assume Λ = Λ int and Λ ′ = Λ ′ int .
Claim (I) implies claim (II) by Lemma 3.6. Now assume that claim (II) holds. The results repeated in Subsection 2.5 imply that there must be a graded isomorphism between C λ (W ) and C λ ′ (W ′ ). Application of Lemma 7.2 then proves that claim (II) implies claim (III). The fact that claim (III) implies claim (I) is an immediate consequence of Lemma 6.6.

Ringel duality
In this section we study Ringel duality for arbitrary blocks in parabolic category O for arbitrary reductive Lie algebras. We will obtain two version of the Ringel duality functor, which generalise the ones in equations (13) and (14).
8.1. Ringel duality for parabolic category O. The main result here is: Theorem 8.1. Consider a reductive Lie algebra g and λ, µ ∈ Λ + int . (i) There are isomorphisms of (graded) algebras: Before proving Theorem 8.1, we establish some preparatory lemmata.
Proof. Using [MOS,Section 6.5] and Proposition 4.7, one can consider the Koszul dual of (the graded version of) Corollary 5.9. Corollary 4.11 and equation (11) then allow us to interpret this as the proposed isomorphism. We also provide the sketch of a proof without the use of Koszul duality, following the proof of [MS3,Proposition 4.4(1)]. Applying parabolic induction to the classical BGG resolution (see [Hu,Chapter 6] or [Le]) for the Levi subalgebra of q µ , yields an exact complex As Verma modules are acyclic for T w 0 , see [AS,Theorem 2.2], we can compute The complex D • = LT w 0 C • = T w 0 C • is hence of the form We have w 0 w µ 0 x · λ = y · λ with y := w 0 w µ 0 xw λ 0 ∈ X µ λ . As the maps in the complex satisfy a dual version of the ones in the complex C • , we find , which concludes the proof.
Lemma 8.4. The restriction of the endofunctor L l(w µ 0 ) T w 0 of O λ to the full subcategory O µ λ is right exact.
Proof. Lemma 8.3 implies that L i T w 0 N = 0, if i = l(w µ 0 ), for any N ∈ Ob(O µ λ ) with standard flag. So, in particular, for projective objects of O µ λ . The fact that L i T w 0 M = 0 for i < l(w µ 0 ) and arbitrary M ∈ O µ λ can then be derived by induction on the projective dimension of M in O µ λ , by considering short exact sequences.
Proof of Theorem 8.1. First we prove part (ii). By Lemmata 8.3 and 8.4, the endofunctor L l(w µ 0 ) T w 0 of O λ restricts to a right exact functor R µ λ : O µ λ → O λ which maps standard modules to modules contained in the subcategory O µ λ . Subsequently, all simple modules, being the tops of standard modules, are mapped to modules in this subcategory. As O µ λ is a full Serre subcategory, this leads to the conclusion that we actually obtain a right exact functor R µ λ : O µ λ → O µ λ . By Lemma 8.3, this functor restricted to the full subcategory of modules in O µ λ with a standard flag, yields an exact functor from this category to the category of modules in O µ λ with costandard flag. This is an equivalence as it has as inverse, namely, the restriction of dL l(w µ 0 ) T w 0 d, by Lemma 8.3, equation (10)  In particular, this implies R(A µ λ ) ∼ = A µ λ . By [So1,Theorem 11] there is an isomorphism A λ ∼ = A λ . Under this isomorphism, the set of idempotents corresponding to X µ λ are mapped to the ones corresponding to X µ λ , implying that A µ λ ∼ = A µ λ . This proves the ungraded isomorphisms of algebras in (i).
By the previous conclusions in this proof, [MS3,Proposition 2.2(2)] implies that the functor L l(w µ 0 ) T w 0 maps a minimal projective generator of O µ λ to a characteristic q.h. tilting module of O µ λ ′ . As the derived functor is an equivalence between the derived categories and is gradable, it follows that R(A µ λ ) ∼ = A µ λ holds as graded algebras. Remark 8.5. Moreover, [MS3, Proposition 2.2] also implies that When g is not of type A, this is outside the scope of Theorem 6.1. However it can be proved similarly as will be done in the next subsection.
Remark 8.6. An anonymous referee brought to our attention an insightful different method for proving Theorem 8.1(i). We set q µ = l⊕w + , with w + the radical and l the Levi subalgebra of the parabolic subalgebra q µ , leading to parabolic decomposition g = w − ⊕ l ⊕ w + . Consider A = U (g), B = U (w + ), H = U (l) and B = U (w − ). This gives a triangular decomposition in the sense of [GGOR, Section 2.1], when taking into account footnote 1.
For a complex Lie algebra a such that dim(a) = d < ∞, the extension groups Ext • U (a) (C, U (a)) can be calculated from the Chevalley-Eilenberg complex This can be re-interpreted as the Koszul resolution of the trivial a-module C, shifted over d positions; Hence B is Gorenstein and one can apply [GGOR,Proposition 4.3], which implies that the Ringel dual of O µ λ is the category of right A-modules, which are finitely generated, H-semisimple (and locally finite), B-locally nilpotent and which admit generalised central character χ λ . To interpret these as left A = U (g)-modules we can apply an anti-automorphism of U (g). The principal automorphism corresponding to X → −X for X ∈ g shows that the Ringel dual is equivalent to O µ λ . The Chevalley anti-automorphism of U (g) gives an equivalence with O µ λ .
8.2. An alternative approach. The following result is a reformulation of the results in the previous subsection, but we provide an alternative proof, stressing the link with our construction of derived equivalences.
The isomorphism of the endomorphism algebras then follows from the fact that both modules correspond to complexes contained in one position exchanged by an equivalence of triangulated categories.
The proof of Proposition 8.7 then follows easily from this corollary. 8.3. Parabolic and singular Koszul-Ringel duality. In this subsection we compose the Koszul duality (4) with the Ringel duality. We also study the link with the category of linear complexes of q.h. tilting modules LT µ λ , a full subcategory of D b ( Z O µ λ ), see [MO]. By [MO,Corollary 6], the category LT µ λ is equivalent to gmod-R(A µ λ ) ! . Hence Theorem 8.1 and equation (6) implies that there are equivalences of categories We now give explicit descriptions of these equivalences, which will be useful for practical application in e.g. [CM2].
(ii) The functor LR µ λ induces an equivalence of categories , which restricts to an equivalence of categories Proof. The equivalences of derived categories (in the ungraded sense) follow from [MS3, Proposition 2.2(2)] and Theorem 8.1. The graded version then follows from application of Proposition 3.5. Alternatively, the (graded) equivalence can be proved as in Section 6 by using Proposition 5.3(ii) rather than Corollary 5.2. It is a general feature that the Ringel duality functor maps indecomposable projective modules to indecomposable q.h. tilting modules, see e.g. [MS3,Proposition 2.1(2)]. The explicit description then follows from the action on standard modules in Theorem 8.1.
We compose functors to obtain the contravariant Koszul-Ringel duality functor yielding an equivalence of triangulated categories.
The following is a standard property for categories of linear complexes.
Proof. The property is clearly true for complexes contained in one position as, by construction, The full statement then follows by induction, using distinguished triangles in D b ( Z O µ λ ) such that the cone belongs to LT µ λ .
8.4. A block which is not Ringel self-dual. Consider g = sl(4), with W = S 4 with simple reflections s 1 , s 2 , s 3 and choose λ, µ ∈ Λ + int such that w λ 0 = s 3 and w µ 0 = s 1 . The Ext 1 -quiver of O µ λ and O µ λ can be obtained from Proposition 2.2 and [St2, Appendix A]. As we have a duality preserving isomorphism classes of simple modules, we neglect the directions on the edges. We order the simple modules according to the Bruhat order. The Ext 1 -quiver of O µ λ is given by L(s 3 · λ) L(s 2 s 3 · λ) ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ P P P P P P P P P P P P L(s 2 s 1 s 3 · λ) P P P P P P P P P P P P L(s 3 s 2 s 3 · λ) ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ L(s 3 s 2 s 1 s 3 · λ) We have w µ 0 = s 3 and similarly we obtain the Ext 1 -quiver of O µ λ , ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ L(s 1 s 2 s 1 s 3 · λ) L(s 2 s 1 s 3 s 2 s 3 · λ) This implies that O µ λ contains precisely one simple module which has a first extension with only one other simple module. Its projective cover is a non-simple standard module, so is not injective. Also O µ λ has precisely one simple module which has a first extension with only one other simple module. This module is a simple standard module, which thus has an injective projective cover by [Ir]. This implies that O µ λ is not equivalent to O µ λ .