Journal of Lie Theory Volume 14 (2004) 151–163 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Koszul Duality of Translation—and Zuckerman Functors

Steen Ryom-Hansen

Communicated by B. Ørsted

Abstract. We review Koszul duality in representation theory of category O, especially we give a new presentation of the Koszul duality functor. Combining this with work of Backelin, we show that the translation and Zuckerman functors are Koszul dual to each other, thus verifying a conjecture of Bernstein, Frenkel and Khovanov. Finally we use Koszul duality to give a short proof of the Enright-Shelton equivalence.

1. Introduction.

The definite exposition of Koszul duality in representation theory is the paper of Beilinson, Ginzburg and Soergel [5]. The main theme of that paper is that the category O of Bernstein, Gelfand and Gelfand is the module category of a Koszul ring, from which many properties of O can be deduced very elegantly. They also consider singular as well as parabolic versions of O and show that they are are equivalent to each other under the Koszul duality functor. Two of the main tools to obtain these results were the Zuckerman functors and the translation functors.

In a recent paper by Bernstein, Frenkel and Khovanov [6] these functors are studied on certain simultaneously singular and parabolic subcategories of O, in type A. They show, partially conjecture, that each one of them – in combination with an equivalence of certain categories due to Enright-Shelton – can be used to categorify the Temperley-Lieb algebra and also conjecture that these two pictures should be Koszul dual to each other.

The purpose of this note is to show that the translation– and Zuckerman functors are indeed Koszul dual to each other. By this we mean that both admit graded versions and that these correspond under the Koszul duality functor. Although this of course is one of the main philosophical points of [5], a rigorous proof was never given.

We furthermore use the Koszul duality theory to give a simple proof of the Enright-Shelton equivalence.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

∗It is a pleasure to thank Wolfgang Soergel for many useful discussions. Most of this work was done while I was a research assistant at the University of Freiburg, Germany.

Added: Recently C. Stroppel [11] has obtained a proof of the full Bernstein-Frenkel-Khovanov conjectures, in part using results from the present paper.

2. Preliminaries

In this section we will mostly recall some of the basic definitions and concepts from the theory of Koszul duality, see also [5].

Let 𝔤 be a complex semisimple Lie algebra containing a Borel algebra 𝔟 with Cartan part 𝔥 and let O be the category of 𝔤-modules associated to it by Bernstein, Gelfand and Gelfand. For λ ∈ 𝔥^{∗} integral but possibly singular, denote by O_{λ} the subcategory of O consisting of modules of generalized central character χ_{λ} – we refer to it as the singular category O. Let S be the set of simple reflections corresponding to our data, and let S_{λ} be the subset consisting of those reflections that fix λ under the dot operation. Then S_{λ} defines a parabolic subalgebra 𝔮(λ). We define the parabolic category O^{λ} to consist of the 𝔮(λ)-finite objects in O. For λ = 0 we omit the index, i.e. we write O_{0} = O, this should not cause confusion.

Let P denote the sum of all indecomposable projectives in O, hence P is a projective generator of O. Analogously, we can construct projective generators P_{λ} (resp. P^{λ}) of O_{λ} (resp. O^{λ}). Let A = End_{O} P (resp. A_{λ} = End_{O_{λ}} P_{λ}, A^{λ} = End_{O^{λ}} P^{λ}). By general principles, we can then identify O with Mod−A, the category of finitely generated A-right modules (and analogously for A_{λ}, A^{λ}).

Let T_{0}^{λ} : O → O_{λ} , T_{λ}^{0} : O_{λ} → O be the Jantzen translation functors onto and out of the wall. Passing to Mod−A, T_{0}^{λ} corresponds to the functor

Mod−A → Mod−A_{λ} : M ↦ M ⊗_{A} X

where X is the (A, A_{λ}) bimodule X = Hom_{O}( P_{λ}, T_{0}^{λ} P ). There is a similar description of T_{λ}^{0} : O_{λ} → O.

Let τ_{λ} : O → O^{λ} be the parabolic truncation functor, by definition it takes M ∈ O to its largest 𝔮(λ)-finite quotient. It is right exact and left adjoint to the inclusion functor ι_{λ} : O^{λ} → O, which is exact, and it thus takes projectives to projectives. It even takes indecomposable projectives to indecomposable projectives. Its top degree left derived functor is the Zuckerman functor that takes a module M ∈ O to its largest 𝔮(λ)-finite submodule.

On the Mod−A level, τ_{λ} is given by the tensor product with the (A, A^{λ})- bimodule Y = Hom_{O^{λ}}( P^{λ}, τ_{λ}P ). However, from the above considerations we deduce that Y = A^{λ}, the left A-structure coming from τ_{λ} and the right A^{λ}- structure coming from the multiplication in A^{λ}.

Let us finally recall the definition of a Koszul ring.

∗ Supported by EPSRC grant M22536 and by the TMR-network algebraic Lie Theory ERB-FMRX-CT97-0100.

Definition 2.1. Let R = ⨁_{i≥0} R_{i} be a positively graded ring with R_{0} semisimple and put R_{+} = ⨁_{i>0} R_{i}. R is called Koszul if the right module R_{0} ≅ R/R_{+} admits a graded projective resolution P^{•} ↠ R_{0} such that P^{i} is generated by its component P_{i}^{i} in degree i. The Koszul dual ring of R is defined as R^{!} := Ext_{R}^{•}(R_{0}, R_{0}).

The main point of [5] is now that all the rings appearing above can be given a Koszul grading.

3. Koszul duality.

The main purpose of this section is to give a new construction of the Koszul duality functor.

Koszul duality is an equivalence at the level of derived categories between the graded module categories of a Koszul ring and its dual. It exists under some mild finiteness conditions. The concept appeared for the first time in the paper [3] where it is shown that for any finite dimensional vector space V the derived graded module categories of the symmetric algebra SV and the exterior algebra ⋀ V^{∗} are equivalent. The argument used there carries over to general Koszul rings and is the one used in [5].

We shall here present another approach to the Koszul duality functor, using the language of differential graded algebras (DG-algebras). Although also the papers [4] and [9] link DG-algebras with Koszul duality, we were rather inspired by the construction of the localization functor in the book of Bernstein and Lunts [7].

Let us start out by repeating the basic definitions.

Definition 3.1. A DG-algebra A = (A, d) is a graded associative algebra A = ⨁_{i=−∞}^{∞} A^{i} with a unit 1_{A} ∈ A^{0} and an additive endomorphism d of degree 1, s.t.

d^{2} = 0
d(a · b) = da · b + (−1)^{deg(a)}a · db
and d(1_{A}) = 0.

Definition 3.2. A left module M = (M, d_{M}) over a DG-algebra A = (A, d) is a graded unitary right A-module M = ⨁_{i=−∞}^{∞} M^{i} with an additive endomorphism d_{M} : M → M of degree 1, s.t. d_{M}^{2} = 0 and

d_{M}(am) = da · m + (−1)^{deg(a)}a · d_{M}m

Definition 3.3. A right module M = (M, d_{M}) over a DG-algebra A = (A, d) is a graded unitary right A-module M = ⨁_{i=−∞}^{∞} M^{i} with an additive endomorphism d_{M} : M → M of degree 1, s.t. d_{M}^{2} = 0 and

d_{M}(ma) = d_{M}m · a + (−1)^{deg(m)}m · da

Unless otherwise stated, A will from now on be the ring End_{O}(P) from the former section.

By the selfduality theorem of Soergel [10] we know that A = Ext_{O}^{•}(L, L), where L is the sum of all simples in O. This provides A with a grading which by [10] or [5] is a Koszul grading.

We regard A as a DG-algebra A with A = A^{0}. At this stage we neglect the grading on A, hence A just has one grading, the DG-algebra grading.

Let K^{•} be the Koszul complex of A, see e.g. [5]. It is a projective resolution of k (which corresponds to the sum of all simples in O). In [5] K^{•} is a resolution of left A-modules of k; we however prefer to modify it to a resolution of right A-modules. Furthermore we change the sign of the indices so that the differential has degree +1.

Let k_{w} denote the simple A-module corresponding to L_{w} ∈ O and let K_{w}^{•} be a projective resolution of k_{w}. The Koszul complex K^{•} is then quasiisomorphic to ⨁_{λ} K_{w}^{•}.

We can consider K^{•} as a DG right module of the DG-algebra A, and as such it is K-projective in the sense of [7]: the homotopy category of DG-modules of A is simply the homotopy category of complexes of A-modules, hence any quasiisomorphism from K^{•} to an DG-module M is a homotopy isomorphism.

Given two DG-modules M, N of the (arbitrary) DG-algebra A, recall the construction of the complex Hom_{A}^{•}(M, N):

Hom_{A}^{n}(M, N) := Hom_{A} (M, N[n])
df := d_{N}f − (−1)^{n}fd_{M}, f ∈ Hom_{A}^{n}(M, N).

If we perform this construction on K^{•}, we obtain a DG-algebra

End_{A}^{•}(K^{•}) = Hom_{A}^{•}(K^{•}, K^{•}),

the multiplication being given by composition of maps (notice that this construction does not involve the Koszul grading on A).

The signs match up to give K^{•} the structure of an End_{A}^{•}(K^{•}) left DGmodule.

Let for any DG-algebra A, D^{b}(Mod−A) denote the bounded derived category of right A-modules, see for example Bernstein-Lunts [7] for an exposition of this theory.

Let D^{b}(Mof −A) denote the bounded derived category of finitely generated A-modules, and let D^{b}(Mof −A) be the corresponding category of A-modules. Let D^{b}(Mof − End_{A}^{•}(K^{•})) be the subcategory of D^{b}(Mof − End_{A}^{•}(K^{•})) consisting of finitely generated End_{A}^{•}(K^{•})-modules in the algebra sense. Then we have

Lemma 3.4. D^{b}(Mof − End_{A}^{•}(K^{•})) is generated, in the sense of triangulated categories, by the Hom_{A}^{•}(K^{•}, K_{w}^{•})’s.

Proof. Since our ring Ais Koszul, the DG-algebras

End_{A}^{•} (K^{•}) and (Ext_{A}^{•}(k, k), d = 0)

are quasiisomorphic [5, Theorem 2.10.1] and hence their bounded derived categories are equivalent. Under this equivalence the End_{A}^{•} (K^{•})-module Hom_{A}^{•}(K^{•}, K_{w}^{•}) becomes the (Ext_{A}^{•} (k, k) , d = 0)-module (Ext_{A}^{•} (k, k_{w}) , d = 0).

Now a certain polynomial DG-algebra B with d = 0 is studied in chapter 11 of [7]. Corollary 11.1.5 there is the statement that D^{b}(Mof −B) is generated by B itself. The proof of this Corollary 11.1.5 relies on the differential being zero for B and of the fact that finitely generated projective modules over a polynomial algebra are free. We do not have this last property in our situation and should therefore adjust the statement of the Lemma accordingly. Let us give a few details on the modifications:

Let thus (M, d_{M}) be an object of D^{b}(Mof −(Ext_{A}^{•}(k, k), d = 0)). Then we have a triangle in that category of the form

Ker d_{M} → M → M/ Ker d_{M} →

But the first and the third terms have zero differential so we may assume that d_{M} = 0.

Let now

0 → P^{−n} → P^{−n+1} → … →^{ϵ} M → 0

be a projective graded resolution of M considered as Ext_{A}^{•}(k, k)-module. Defining P ∈ D^{b}(Mof −(Ext_{A}^{•}(k, k), d = 0)) as having k’th DG-part ⊕_{i}P_{k−i}^{i} and differential from the resolution, we get a quasiisomorphism P ≅ M induced by ϵ.

On the other hand the finitely generated, graded projectives over Ext_{A}^{•} (k, k) are the direct sums of the Ext_{A}^{•} (k, k_{w})’s and we are done.

Now since K^{•} is K-projective, it can be used in itself to calculate RHom_{A}(K^{•}, M) for an A-module M. But K^{•} is an End_{A}^{•}(K^{•})- module, so we get a functor:

RHom_{A}(K^{•}, −) : D^{b}(Mod−A) → D^{b}(Mod− End_{A}^{•}(K^{•}))

This might require a little consideration: one should check that RHom_{A}(K^{•}, −) takes homotopies to homotopies, even when considered as a functor to the category of right End_{A}^{•}(K^{•})-modules, and likewise for quasiisomorphisms. For homotopies, one checks this by hand (after having consulted e.g. Bernstein-Lunts [7] for the definition of the homotopy category of DG-modules). For quasiisomorphisms one uses the standard argument: if f : M → N is a quasiisomorphism, then Hom_{A}^{•}(K^{•}, C(f)) is acyclic (C(f) denoting the cone of f) and so on: the action of End_{A}^{•}(K^{•}) is irrelevant for this argument.

Now we also have a functor in the other direction:

− ⊗^{L}_{End_{A}^{•}(K^{•})} K^{•} : D^{b}(Mod−A) ← D^{b}(Mod− End_{A}^{•}(K^{•}))

Recall that this construction involves the replacement of

N ∈ D^{b}(Mod− End_{A}^{•}(K^{•}))

by a K-flat object (actually a K-projective object will do), to which it is quasi-

isomorphic. The action of A on K^{•} then provides N ⊗^{L}_{End_{A}^{•}(K^{•})} K^{•} with the structure of DG-module over A:

d(n ⊗ ka) = d(n) ⊗ ka + (−1)^{deg(n)}n ⊗ d(ka)
= d(n) ⊗ ka + (−1)^{deg(n)}n ⊗ d(k)a
= d(n ⊗ k)a

since the differential of A is zero. Once again, one should here check that the functor makes sense as a functor into the category of A-modules.

We can then prove the following Theorem:

Theorem 3.5. The functor RHom_{A}(K^{•}, −) establishes an equivalence of the triangulated categories D^{b}(Mof −A) and D^{b}(Mof − End_{A}^{•} (K^{•})). The inverse func-

tor is − ⊗^{L}_{End_{A}^{•}(K^{•})} K^{•}.

Proof. Note first that D^{b}(Mof −A) is generated by the K_{w}^{•}, since they are resolutions of the simple A-modules and short exact sequences O-modules induce triangles in D^{b}(Mod−A).

But then RHom_{A}(K^{•}, M) belongs to D^{b}(Mof − End_{A}^{•}(K^{•})) for any M ∈ D^{b}(Mof − End_{A}^{•}(K^{•})), since it clearly does so for each of the generators K_{w}^{•}.

We furthermore see that RHom_{A}(K^{•}, M) is a K-projective End_{A}^{•}(K^{•})- module for any M in D^{b}(Mof −A) since it clearly holds for K_{w}^{•} and the Kprojectives form a triangulated subcategory of K(Mod− End_{A}^{•}(K^{•})), the homotopy category of End_{A}^{•}(K^{•})-modules.

We then obtain a morphism ψ_{M} in D^{b}(Mod − A) in the following way.

ψ_{M} : RHom_{A}(K^{•}, M) ⊗^{L}_{End_{A}^{•}(K^{•})} K^{•} → M : f ⊗ k ↦ f(k)

One checks that ψ_{M} defines a natural transformation from the functor

RHom_{A}(K^{•}, −) ⊗^{L}_{End_{A}^{•}(K^{•})} K^{•} to the identity functor Id. It is a quasiisomorphism for M = K_{w}^{•}, and thus for all M.

On the other hand we have for a K-projective End_{A}^{•}(K^{•})-module N the canonical morphism ϕ_{N} given as follows:

\begin{matrix}ϕ_{N} : N & → & RHom_{A} (K^{•}, N ⊗^{L}_{End_{A}^{•}(K^{•})} K^{•}) \\ n_{i} ∈ N^{i} & ↦ & ( k_{j} ∈ K^{j} ↦ n_{i} ⊗ k_{j}) \end{matrix}

One readily sees that ϕ_{N} is a quasiisomorphism for N = Hom_{A}^{•}(K^{•}, K_{w}^{•}). But these objects generate D^{b}(Mof − End_{A}^{•}(K^{•})) and we can argue as above.

Let us finally comment on the shifts [1]. It should here be noticed that since we are working with right DG-modules, the appropriate definition of M[1] for a DG-module M over a DG-algebra A is the following:

(M[1])^{i} = M^{i+1}, d_{M[1]} = −d_{M} and m ∘ a = ma

where m ∘ a is the multiplication in M[1] while ma is the multiplication in M. So unlike the left module situation, the A structure on M is here left unchanged. One now checks that the functors commute with the shifts [1].

Now A has a Koszul grading so we may consider the category mof −A of finitely generated, graded A-modules. Let us use a lower index to denote the graded parts with respect to this ℤ-grading. This passes to the derived category of graded DG-modules which we denote by D^{b}(mof − A). It may be identified with the derived category of mof −A and carries a twist ⟨1⟩ which we arrange the following way:

M⟨1⟩_{i} = M_{i−1}

Now the grading on K^{•} induces a grading on End_{A}^{•}(K^{•}) as well; it satisfies the rule

End_{A}^{•}(K^{•})_{i} := end_{A}^{•}(K^{•}, K^{•}⟨−i⟩)

With this grading on End_{A}^{•}(K^{•}), we may consider its graded module category, which we denote D^{b}(mof − End_{A}^{•} (K^{•})). Let us moreover denote the subcategory of this generated, with twists, by the graded summands of the DG-module End_{A}^{•}(K^{•}) itself by

D^{b}(⟨End_{A}^{•} (K^{•})⟩)

(It would have been more consistent with the earlier notation to write D^{b}(mof − End_{A}^{•} (K^{•})) for this category; we however prefer the above notation in order to save space).

It is now straightforward to prove the following strengthening of the former theorem:

Theorem 3.6. The functor RHom_{A}(K^{•}, −) establishes an equivalence of the triangulated categories D^{b}(mof − A) and D^{b}(⟨End_{A}^{•} (K^{•})⟩). It commutes with the twists [1] and ⟨1⟩.

Our next task will be to study the DG-algebra (Ext_{A}^{•}(K^{•}), d = 0). More precisely, we are going to compare D^{b} (⟨Ext_{A}^{•}(K^{•}), d = 0⟩) with the standard derived category of the category of finitely generated, graded modules over Ext_{A}^{•}(k, k), i.e. D^{b} (mof − Ext_{A}^{•}(k, k)). Here, we define the ℤ-grading on (Ext_{A}^{•}(K^{•}), d = 0) to be the negative of the DG-grading.

An object of D^{b} ⟨(mof − Ext_{A}^{•}(k, k)) is a bounded complex
0 → P^{−n} → P^{−n+1} → … → P_{0} → … → P^{m−1} → P^{m} → 0

of projective graded Ext_{A}^{•}(k, k)-modules (so the differential has degree 0). Using this, one constructs a DG-module P of the DG-algebra (Ext_{A}^{•}(k, k), d = 0) in the following way:

0 → P^{−n}⟨−n⟩ → P^{−n+1}⟨−n + 1⟩ → … → P^{0}⟨0⟩ → ⋯ P^{m}⟨m⟩ → 0.

In other words, P is the module P = ⨁_{n} P^{n} whose k’th DG-piece is P^{k} = ⨁_{i} P_{k−i}^{i}. We make P into a graded DG-module by the rule P_{j} = ⨁_{i} P_{−j}^{i}. This construction defines a functor

F : D^{b} (mof − Ext_{A}^{•}(k, k)) → D^{b} (mod−(Ext_{A}^{•}(k, k), d = 0))

since a graded homotopy between two morphisms f, g : P^{•} → Q^{•} of modules will be mapped to a homotopy in the DG-category.

I will now argue that actually F is a functor into the category D^{b} (⟨Ext_{A}^{•}(k, k), d = 0⟩). To see this note first that the P_{i}⟨i⟩ all lie in this category since they are summands of (shifts of) (Ext_{A}^{•}(k, k), d = 0). On the other hand F can be viewed as an iterated graded mapping cone construction in the DGsense, starting with the morphism P^{m−1}⟨m⟩ → P^{m}⟨m⟩, which is already a DGmorphism, and the claim follows. The graded mapping cone of a graded DGmodule morphism M →^{u} N is given by C(u) = N ⊕ M[1] as DG-module, and reversing the degrees with respect to the ℤ-grading.

It is now clear that F commutes with the twists [1]. On the other hand F takes a sequence of Ext_{A}^{•}(k, k)-modules isomorphic to a standard triangle

M →^{u} N → C(u) → M[1]

to a sequence of graded DG-modules isomorphic to

F (M) →^{F (u)} F (N) → F (C(u)) → F (M[1]).

But this is easily seen to be a standard triangle of DG-modules; in other words F is a triangulated functor.

One furthermore checks that F is full and faithful and it is thus an equivalence of triangulated categories once we have shown that the generators (Ext_{A}^{•}(k, k_{w}), d = 0) lie in the image of F. But this is clear.

Since

M_{1} → M_{2} → M_{3} →

is a triangle if and only if

F (M_{1}) → F (M_{2}) → F (M_{3}) →

is a triangle we deduce that the inverse functor G is triangulated as well.

Our next task will be to analyze the behavior of the twists ⟨1⟩ with respect to F. Now apart from the change of the sign of the differentials, we obtain the relation:

F (M⟨1⟩) = F (M)[−1]⟨−1⟩.

On the other hand, let for a complex P^{•} of graded Ext_{A}^{•}(k, k)-modules P = ⨁ P_{n}^{k} be the total module. We may then consider the map σ on P defined as follows:

p ∈ P_{n}^{k} ↦ (−1)^{n}p.

One now checks that σ defines an isomorphism between the functors F (−⟨1⟩) and F (−)[−1]⟨−1⟩.

Recall once again that since Ais Koszul, the DG-algebras

End_{A}^{•} (K^{•}) and (Ext_{A}^{•}(k, k), d = 0)

are quasiisomorphic so there is an equivalence

ϕ : D^{b}(Mod− End_{A}^{•} (K^{•})) → D^{b} (Mod−(Ext_{A}^{•}(k, k), d = 0)) .

A closer look at the above quoted proof reveals that the quasiisomorphism is even a graded one so we obtain an equivalence

ϕ : D^{b}(⟨End_{A}^{•} (K^{•})⟩) → D^{b} (⟨Ext_{A}^{•}(k, k), d = 0⟩)

Let us gather the results in one theorem.

Theorem 3.7. The composition of the above functors

G ∘ ϕ ∘ RHom_{A}(K^{•}, −) : D^{b}(mof −A) → D^{b} (mof − Ext_{A}^{•}(k, k))

is an equivalence of triangulated categories: the Koszul duality functor. It commutes with [1] and satisfies the rule

F (M⟨1⟩) = F (M)[−1]⟨−1⟩.

4. Translation and Zuckerman functors.

Let us return to the translation functors to and from the wall T_{0}^{λ} : O → O_{λ}, T_{λ}^{0} : O → O_{λ}. We saw in the first section, that they can be viewed as functors between the categories Mod −A and Mod −A_{λ}. Now, in [2] it is shown that they can be lifted to graded functors

T_{0}^{λ} : mod −A → mod −A_{λ}
T_{λ}^{0} : mod −A_{λ} → mod −A

that still are adjoint and exact and such that T_{0}^{λ} takes pure objects of weight n to pure objects of the same weight; this construction is based on Soergel’s theory of modules over the coinvariants of the Weyl group. See also [1], where graded translation functors are constructed in the setting of modular representation theory.

The graded translation functor T_{0}^{λ} induces a homomorphism of graded DGalgebras End_{A}^{•}(K^{•}) → End_{A_{λ}}^{•}(T_{0}^{λ}K^{•}). But T_{0}^{λ} and T_{λ}^{0} are exact and K^{•} is a projective graded resolution of the pure module k, so T_{0}^{λ}K^{•} is a projective graded resolution of k_{λ}. Hence T_{0}^{λ}K^{•} is graded homotopic to the Koszul complex K_{λ}^{•} of A_{λ} and this implies that End_{A_{λ}}^{•}(T_{0}^{λ}K^{•}) and End_{A_{λ}}^{•}(K_{λ}^{•}) are quasiisomorphic graded DG-algebras.

We already saw that A being a Koszul ring implies that

End_{A}^{•}(K^{•}) ≅ (Ext_{A}^{•}(k, k), d = 0)

and by the above and since A_{λ} is Koszul as well we have that

End_{A_{λ}}^{•}(T_{0}^{λ}K^{•}) ≅ (Ext_{A_{λ}}^{•}(T_{0}^{λ}k, T_{0}^{λ}k), d = 0) ≅ (Ext_{A_{λ}}^{•}(k_{λ}, k_{λ}), d = 0)

We now obtain the following commutative diagram of functors:

D^{b}(mod−A) ←^{− ⊗ K^{•}}  D^{b}(⟨End_{A}^{•}(K^{•})⟩) ←  D^{b}(⟨Ext_{A}^{•}(k, k), d = 0⟩)
↓ T^{λ}_{0}  ↓  ↓ T^{λ}_{0}
D^{b}(mod−A_{λ}) ←^{− ⊗ T^{λ}_{0}K^{•}} D^{b}(⟨End_{A_{λ}}^{•}(T_{0}^{λ}K^{•})⟩) ← D^{b}(⟨Ext_{A_{λ}}^{•}(T_{0}^{λ}k, T_{0}^{λ}k), d = 0⟩)

where the middle vertical arrow is the graded tensor product functor

− ⊗_{End_{A}^{•}(K^{•})} End_{A}^{•}(T_{0}^{λ}K^{•})

while the first and third vertical arrows are the graded translation functors. The commutativity of the first square is here obvious, whereas the quasiisomorphism of (Ext_{A}^{•}(k, k), d = 0)) × End_{A_{λ}}^{•}(T_{0}^{λ}K^{•})-bimodules

End_{A}^{•}(T_{0}^{λ}K^{•}) ≅ Ext_{A_{λ}}^{•}(T_{0}^{λ}k, T_{0}^{λ}k)

gives the natural transformation that makes the second square commutative.

By the same reasoning as in the last section, the two lower arrows define equivalences of triangulated categories.

Now Backelin [2] shows that one can choose the isomorphisms Ext_{A}^{•}(k, k) ≃ A and Ext_{A_{λ}}^{•}(T_{0}^{λ}k, T_{0}^{λ}k) ≃ A^{λ} to obtain the following commutative diagram:

Ext_{A}^{•}(k, k)  ⟶^{∼} A
T^{λ}_{0} ↓  ↓ τ
Ext_{A_{λ}}^{•}(T_{0}^{λ}k, T_{0}^{λ}k) ⟶^{∼} A^{λ}

The key point is here that both vertical maps are surjections: for τ this is clear, while for T_{0}^{λ} an argument involving the Koszul property of Ais required (one can here give a simple alternate argument along the lines of the Cline, Parshall, Scott approach to Kashdan-Luzstig theory [8]). It then follows that the kernels of the two vertical maps are the ideals generated by corresponding idempotents (thus in the degree 0 part).

Although the diagram involves non-graded maps, it can be used to give τ a grading – and then of course it is a commutative diagram of graded homomorphisms.

This diagram, on the other hand, gives rise to the following commutative diagram of functors:

D^{b}(⟨Ext_{A}^{•}(k, k), d = 0⟩)  ←^{F} D^{b}(mod−A)
T^{λ}_{0} ↓  ↓ τ
D^{b}(⟨Ext_{A_{λ}}^{•}(T_{0}^{λ}k, T_{0}^{λ}k), d = 0⟩) ←^{F} D^{b}(mod−A^{λ})

We now join the two diagrams of functors to obtain a diagram, in which the upper arrows compose to the Koszul duality functor of O while the composition of the lower arrows is isomorphic to the Koszul duality functor of O_{λ}. Let us formulate this as a Theorem Theorem 4.1. The translation– and Zuckerman functors are Kozsul to each other, in other words there is a commutative diagram of functors:

D^{b}(mod−A) ←^{D} D^{b}(mod−A)
T^{λ}_{0} ↓  ↓ τ
D^{b}(mod−A_{λ}) ←^{D_{λ}} D^{b}(mod−A^{λ})

where D (resp. D_{λ}) is the Koszul duality functor as described above. This is the Theorem announced in the introduction of the paper.

5. The Enright-Shelton Equivalence

We now consider the categorification of the Temperley-Lieb algebra. Let thus 𝔤 := gl_{n} and O := O(𝔤l_{n}). Let ϵ_{k}, k ∈ {1, 2, …, n} be the standard basis of the weight lattice and define for k ∈ {1, 2, …, n}, λ_{k} := ϵ_{1} + ϵ_{2} + … ϵ_{k}. We then let O_{k,n−k} be the singular block of O consisting of modules with central character θ(λ_{k}). So the Verma module with highest weight λ_{k} − ρ lies in O_{k,n−k}. Let 𝔤_{i}, 1 ≤ i ≤ i − 1 be the subalgebra of 𝔤 consisting of the matrices whose entries are nonzero only on the intersection of the i-th and (i + 1)-th rows and columns. We then denote by O_{k,n−k}^{i} the parabolic subcategory of O_{k,n−k} whose modules are the locally 𝔤_{i}-finite ones in O_{k,n−k}.

We shall also consider the following dual picture: let 𝔭_{k} be the parabolic subalgebra of 𝔤, whose Levi part is 𝔤_{k}⊕𝔤_{n−k} and such that 𝔫_{+} ⊆ 𝔭_{k} and let O^{k,n−k} be the the full subcategory of O consisting of the locally 𝔭_{k} -finite modules. Choose an integral dominant regular weight μ and integral dominant subregular weights μ_{i} on the i-th wall, i = 1, 2, …, n − 1 (so the coordinates of μ_{i} in the ϵ_{i} basis are strictly decreasing, except for the i-th and the (i + 1)-th that are equal). Let finally O_{i}^{k,n−k} be the subcategory of O^{k,n−k} with central character θ(μ_{i}).

All of this is the setup of [6]

Let R_{k,n−k}^{i}(resp.R_{i}^{k,n−k}) be the endomorphism ring of the minimal projective generator of O_{k,n−k}^{i} (resp. O_{i}^{k,n−k}). The following theorem is a direct consequence of Backelin’s work [2]:

Theorem 5.1. (R_{k,n−k}^{i})^{!} = R_{i}^{k,n−k}

Proof. The main result of [2] is that

(R_{ϕ}^{ψ})^{!} = R_{−w_{0}ψ}^{ϕ}

with the notation as in [5].

Now one should first observe that c Id ∈ 𝔤l_{n} acts on O_{λ} through multiplication with c ∑ λ_{i}; hence O_{λ} is equivalent to the category O\overline{λ} of 𝔰l_{n}-modules, where \overline{λ} denotes the image of λ under the projection of the weight lattice with kernel ℤ ∑ ϵ_{i}. We can thus restrict ourselves to the semisimple situation and may indeed use the results quoted.

Now μ_{i} and 𝔰l_{i} ⊆ 𝔰l_{n} are both given by the simple root α_{i} = ϵ_{i} − ϵ_{i+1} and also 𝔭_{k,n−k} and λ_{k} are given by the same simple roots (= Δ \ {α_{i}}). Finally, we have in type A that w_{0}λ = −λ and the theorem follows.

As a corollary we obtain a simple proof of the following equivalence of categories first proven by Enright and Shelton.

Corollary 5.2. O_{1}^{k,n−k} ≅ O^{k−1,n−k−1}

Proof. There is first of all a standard equivalence of categories:

O_{k,n−k}^{1} ≅ O_{k−1,n−k−1}

The functor ν_{n} from left to right takes the sum of all weight spaces of weight ϵ_{1} + x_{3}ϵ_{3} + … x_{n}ϵ_{n} − ρ_{n} with x_{i} ∈ ℤ. This is a 𝔤l_{n−2}-module and ν_{n} is then the tensor product of it with the module defined by ϵ_{1} + ϵ_{2} + … ϵ_{n−2}. The inverse functor comes from an induction procedure, see [6] for the details. Now this equivalence gives us a ring isomorphism

(R_{k,n−k}^{1})^{!} ≅ (R_{k−1,n−k−1})^{!}

which combined with the theorem yields a ring isomorphism

ξ : R_{1}^{k,n−k} ≅ R^{k−1,n−k−1}

But then the module categories of R_{1}^{k,n−k} and R^{k−1,n−k−1} are equivalent, by the restriction and extension of scalars along ξ. The corollary is proved.

This might be useful in proving the full conjectures of [6].

References

[1] Andersen, H. H. , J. C. Jantzen, and W. Soergel, Representations of Quantum Groups at a p-th root of unity and of semisimple Groups in Characteristic p: Independence of p, Astérisque 220, 1994, 1–320.

[2] Backelin, E., Koszul Duality for Parabolic and Singular Category O, Representation Theory, 3, (1999), 139–152.

[3] Bernstein, J., I. Gelfand, and S. Gelfand, Algebraic bundles over ℙ^{n} and problems of linear algebra, Funcional Analysis and its Appl. 12 (1978), 66–67.

[4] Beilinson, A., J. Ginzburg, and V. Schechtmann, Koszul Duality, J. of Geometry and Physics 5 (1988), 317–350.

[5] Beilinson, A., V. Ginzburg, and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9, (1996), 473–527.

[6] Bernstein, J., I. Frenkel, and M. Khovanov, A categorification of the Temperley-Lieb Algebra and Schur quotients of U(𝔰l_{2}) via projective and Zuckerman functors, Selecta Math. (N.S.), 5 (1999), 199–241.

[7] Bernstein, J., and V. Lunts, “Equivariant Sheaves and Functors,” Lecture Notes in Mathematics (Springer-Verlag) 1578, (1994).

[8] Cline, E., B. Parshall, and L. Scott, Abstract Kashdan-Luzstig Theories, Tôhoko Math. J. 45 (1993), 511–534.

[9] B. Keller, E., Deriving DG-categories, Ann. Sci. École Norm. Sup. 27 (1994), 63–102.

[10] Soergel, W., Kategorie O, Perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421–445.

[11] Stroppel, C., Categorifications of the Temperley 2-category, tangles and cobordisms via projective functors , Preprint, C.Stroppel@mcs.le.ac.uk.

Steen Ryom-Hansen Matematisk Afdeling Universitetsparken 5 DK-2100 København Ø Danmark steen@math.ku.dk

Received October 16, 2002 and in final form October 13, 2003