On the restricted Verma modules at the critical level

We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with special emphasis on their Jordan-H"older multiplicities. The Feigin-Frenkel conjecture gives a formula for these multiplicities that involves the periodic Kazhdan-Lusztig polynomials. We prove this conjecture for all subgeneric blocks and for the case of anti-dominant simple subquotients.


Introduction
The representation theory of affine Kac-Moody algebras at the critical level is one of the essential ingredients in the approach towards the geometric Langlands conjectures proposed by Beilinson and Drinfeld (cf. [BD96]). In particular, the correspondence between the center of the (completed) universal enveloping algebra of an affine Kac-Moody algebra at the critical level and the geometry of the space of opers associated with the Langlands dual datum (cf. [FF92]) is one of the main tools used in the construction of a part of the global correspondence in [BD96].
1.1. The local geometric Langlands conjectures. In [FG06] Frenkel and Gaitsgory formulated the local geometric Langlands conjectures, which relate the critical level representation theory of an affine Kac-Moody algebra to the geometry of an affine flag manifold. In a series of subsequent papers, the authors proved parts of these conjectures. In particular, in the paper [FG07] a derived equivalence between a certain category of D-modules on the affine flag variety and a derived version of the affine category O at the critical level was constructed using a localization functor of Beilinson-Bernstein type. It seems, however, hard to control the action of this equivalence on the respective hearts of the triangulated categories, and hence it is not yet possible to deduce information on the simple critical characters of the category O from the Frenkel-Gaitsgory result.
1.2. The Andersen-Jantzen-Soergel approach. In this paper we study the critical representation theory by a very different method that is inspired by results of Jantzen (cf. [Jan79]) and Soergel (cf. [Soe90]) in the case of finite dimensional complex Lie algebras and of Andersen, Jantzen and Soergel in the case of modular Lie algebras and quantum groups (cf. [AJS94]). The analogous results for the non-critical blocks of O for a symmetrizable Kac-Moody algebra can be found in [Fie06]. The main idea of this approach is to first describe the generic and subgeneric blocks of the respective representation theory in as much detail as possible, and then to deform an arbitrary block in such a way that it can be viewed as an intersection of generic and subgeneric blocks. This intersection procedure should then be described using only some underlying combinatorial datum (like for example the associated integral Weyl group).
As a result we hope to be able to construct equivalences between various categories or links to categories defined in topological terms in the framework of the geometric Langlands program, and to deduce information on the respective simple characters. More specifically, we hope to find a correspondence between certain intersection cohomology sheaves on the Langlands dual affine flag variety and critical representations.
This paper provides the first steps in the approach described above. Its main result is the calculation of the simple characters in the subgeneric critical blocks. In order to explain our result, let us consider the respective categories in some more detail.
1.3. Critical representations of affine Kac-Moody algebras. Let us denote by g an affine Kac-Moody algebra and by h ⊂ g its Cartan subalgebra (for the specialists it should be noted here that we consider the central extension of a loop algebra together with the grading operator). The one-dimensional center of g acts semisimply on each module in the category O. Accordingly, each block of the category O, i.e. each of its indecomposable direct summands, determines a central character. There is one such character, called the critical character, which is distinguished in more than one respect.
In this paper we focus on the following feature of the critical blocks. Let h ⋆ be the dual space of the Cartan subalgebra and denote by δ ∈ h ⋆ the smallest positive imaginary root. Then the corresponding simple highest weight module L(δ) is of dimension one, and the tensor product ·⊗ C L(δ) defines a shift functor T on O that is an equivalence. Now the critical blocks are exactly those that are preserved by the functor T . This allows us to consider, for each critical block, the corresponding graded center A (see Section 4.3).
The graded center is huge and intimately related to the center of the (completed) universal enveloping algebra at the critical level, which was determined in the fundamental work of Feigin and Frenkel (cf. [FF92]) (conjecturally, A is a quotient of the latter center). We use the results of Feigin-Frenkel to describe the action of A on the Verma modules contained in the critical blocks.
1.4. The restricted Verma modules. Let ∆(λ) be the Verma module with highest weight λ ∈ h ⋆ . We define the restricted Verma module ∆(λ) with highest weight λ as the quotient of ∆(λ) by the ideal of A generated by the homogeneous constituents A n with n = 0. For our approach, the restricted Verma modules, not the ordinary ones, should be considered as the "standard objects" in the critical blocks.
We denote the irreducible quotient of ∆(λ) by L(λ). Results of Frenkel and Feigin-Frenkel yield the characters of the restricted Verma modules, and the knowledge of the character of L(λ) for all λ is equivalent to the knowledge of the Jordan-Hölder multiplicities [∆(λ) : L(µ)] for all pairs λ, µ of critical weights.
1.5. The Feigin-Frenkel conjecture. Let us choose a critical indecomposable block O Λ of O and let us identify the index Λ with the subset of highest weights of the simple modules in O Λ . Since Λ is critical we have λ + δ ∈ Λ if and only if λ ∈ Λ.
Let W be the affine Weyl group associated with our data, and W ⊂ W the finite Weyl group. The integral Weyl group W(Λ) corresponding to Λ is generated by the reflections with respect to those real roots that satisfy a certain integrality condition with respect to Λ. In the critical case, W(Λ) is the affinization of the corresponding finite integral Weyl group W(Λ) ⊂ W.
We now define λ ∈ Λ to be dominant resp. anti-dominant if it is dominant resp. anti-dominant with respect to the action of W(Λ), i.e. if it is the highest resp. smallest element in its W(Λ)-orbit. We say that λ is regular if it is regular with respect to W(Λ) (note that here we only refer to the finite integral Weyl group).
As mentioned before, W(Λ) is the affinization of W(Λ). In [Lus80] Lusztig associated with a pair w, x ∈ W(Λ) the periodic polynomial p x,w ∈ Z[v] (in Lusztig's paper these polynomials were indexed not by affine Weyl group elements, but by alcoves, see Section 4.5 for more details). The Feigin-Frenkel conjecture is the following.
Conjecture 1.1. Let Λ be a critical equivalence class.
(2) The restricted Verma multiplicities: Suppose that λ ∈ Λ is regular and dominant. Under some further regularity conditions on Λ (cf. Section 4.8), we have The Feigin-Frenkel conjecture fits very well into a broader picture that was anticipated by Lusztig in his ICM address in 1990 in Kyoto (cf. [Lus91]). There, Lusztig conjecturally linked the representation theory of modular Lie algebras, of quantum groups and of critical level representations of an affine Kac-Moody algebra to the topology of semi-infinite flag manifolds.
In [AF09] we use the results of the present article in order to prove part (1) of the Feigin-Frenkel conjecture, the restricted linkage principle.
(2) If Λ is subgeneric, then we have for all γ, ν ∈ Λ The above theorem confirms the Feigin-Frenkel conjecture in the respective cases. In [AF09] we use it in order to describe the structure of the restricted category O Λ completely for subgeneric Λ. As for generic Λ the structure of O Λ is easy to determine using the results of Feigin and Frenkel, we completed the first part of the Andersen-Jantzen-Soergel approach towards the description of the category O at the critical level. 1.7. Acknowledgments. Both authors wish to thank the Emmy Noether Center in Erlangen, where parts of the research for this paper were done, for its hospitality and support. The second author wishes to thank Nara Women's University for its hospitality and the Landesstiftung Baden-Württemberg for supporting the project.

Affine Kac-Moody algebras
In this section we recall the fundamentals of the theory of affine Kac-Moody algebras. The main references are the textbooks [Kac90] and [MP95]. Our basic data is a finite dimensional simple complex Lie algebra g. We denote by k : g × g → C its Killing form.
2.1. The construction of g. From g we construct the (untwisted) affine Kac-Moody algebra g as follows. We first consider the loop algebra g ⊗ C C[t, t −1 ] for which the commutator is the C[t, t −1 ]-linear extension of the commutator of g. That means that we have [x ⊗ t n , y ⊗ t m ] = [x, y] ⊗ t m+n for x, y ∈ g and m, n ∈ Z. The loop algebra has an up to isomorphism unique non-split central extension g of rank one. As a vector space we have g = g ⊗ C C[t, t −1 ] ⊕ CK, and the Lie bracket is given by for x, y ∈ g, n, m ∈ Z (here δ a,b denotes the Kronecker delta). In the last step of the construction we add the outer derivation operator [D, ·] = t ∂ ∂t and get the affine Kac-Moody algebra g := g ⊕ CD with the Lie bracket Let us fix a Borel subalgebra b ⊂ g and a Cartan subalgebra h ⊂ b. Then the corresponding Borel subalgebra b of g is given by 2.2. Affine roots. We denote by V ⋆ the dual of a vector space V and we write ·, · : V ⋆ × V → C for the canonical pairing. Let R ⊂ h ⋆ be the set of roots of g with respect to h. The projection h → h along the decomposition h = h ⊕ CK ⊕ CD allows us to embed h ⋆ inside h ⋆ . In particular, we can view any α ∈ R as an element in h ⋆ . Let us define δ, For α ∈ R let us denote by g α ⊂ g the corresponding root space. The root spaces of g with respect to h are g α+nδ = g α ⊗ t n for α ∈ R, n ∈ Z, g nδ = h ⊗ t n for n ∈ Z, n = 0.
The subsets R re := {α + nδ | α ∈ R, n ∈ Z}, are called the sets of real roots and of imaginary roots, resp.
Let R + ⊂ R be the set of roots of b with respect to h. Then the set R + of roots of b with respect to h is Let Π ⊂ R + be the set of simple roots and denote by γ ∈ R + the highest root. Then the set of simple affine roots is For each real root α ∈ R re the corresponding root space g α is one-dimensional, and so is the commutator [ g α , g −α ] ⊂ h. The (affine) coroot α ∨ associated with α is the unique element in [ g α , g −α ] on which α takes the value 2. Note that α ∨ is contained in h ⊕ CK, so we have δ, α ∨ = 0.
2.3. The Weyl groups. For α ∈ R re we define the reflection s α : h ⋆ → h ⋆ by s α (λ) := λ − λ, α ∨ α. We denote by W ⊂ GL( h ⋆ ) the affine Weyl group, i.e. the subgroup generated by the reflections s α for α ∈ R + . The subgroup W ⊂ W generated by the reflections s α with α ∈ R leaves the subset h ⋆ ⊂ h ⋆ stable and can be identified with the Weyl group of g.
Let ρ ∈ h ⋆ be an element with the property ρ, α ∨ = 1 for each simple affine root α. Note that ρ is only defined up to the addition of a multiple of δ (the span of the affine coroots is h ⊕ CK, and the simple coroots form a basis in this space). Yet all constructions in the following that use ρ do not depend on this choice. So let us fix such an element ρ once and for all.
The dot-action W × h ⋆ → h ⋆ , (w, λ) → w.λ, of the affine Weyl group on h ⋆ is obtained by shifting the linear action in such a way that −ρ becomes a fixed point, i.e. it is given by w.λ := w(λ + ρ) − ρ for w ∈ W and λ ∈ h ⋆ . Note that since δ, α ∨ = 0 we have s α (δ) = δ for all α ∈ R re . Hence w(δ) = δ for all w ∈ W (so the dot-action is independent of the choice of ρ, as we claimed above).
2.4. The invariant bilinear form. Denote by (·, ·) : g × g → C a nondegenerate and symmetric bilinear form that is invariant, i.e. that satisfies ([x, y], z) = (x, [y, z]) for all x, y, z ∈ g. Such a form is determined up to multiplication with a non-zero constant. We will use the form given by (x ⊗ t n , y ⊗ t m ) = δ n,−m k(x, y), (K, D) = 1 for x, y ∈ g, m, n ∈ Z. Then (·, ·) induces a non-degenerate bilinear form on the Cartan subalgebra h and hence yields an isomorphism h ∼ → h ⋆ , which is the direct sum of the isomorphism h → h ⋆ given by the Killing form k and the isomorphism CK ⊕ CD → CΛ 0 ⊕Cδ that maps K to δ and D to Λ 0 . We get an induced symmetric non-degenerate form on the dual h ⋆ that is given explicitly by (α, β) = k(α, β), for α, β ∈ h ⋆ (here we denote by k also the form on h ⋆ that is induced by the Killing form). It is invariant under the linear action of the affine Weyl group, i.e. we have (w(λ), w(µ)) = (λ, µ)

The category O for an affine Kac-Moody algebra
Having recalled the fundamental structural results for an affine Kac-Moody algebra we now turn to its representation theory. We restrict ourselves to representations in the affine category O.
3.1. The category O. Let M be a g-module. Its weight space corresponding to λ ∈ h ⋆ is Any non-zero element m ∈ M λ is said to be of weight λ. We say that M is a weight module if M = λ∈ h ⋆ M λ . We say that M is locally b-finite if all finitely generated b-submodules of M are finite dimensional. The affine category O is defined as the full subcategory of the category of g-modules that consists of all locally b-finite weight modules.
3.2. Highest weight modules. Our choice of positive roots defines a partial order on h ⋆ : we set ν ν ′ if and only if ν ′ − ν ∈ Z ≥0 R + . A highest weight module of highest weight λ ∈ h ⋆ is a g-module M that contains a generator v = 0 of weight λ such that g α v = 0 for all α ∈ R + . Then λ is indeed the highest weight of M, i.e. M µ = 0 implies µ λ. Each highest weight module is contained in O.
For λ ∈ h ⋆ denote by C λ the one-dimensional h-module corresponding to λ. We extend the h-action to a b-action using the homomorphism b → h of Lie algebras that is left inverse to the inclusion h ⊂ b. That means that g α acts trivially on C λ for all α ∈ R + . The induced module is called the Verma module corresponding to λ. It contains a unique simple quotient L(λ), and both ∆(λ) and L(λ) are highest weight modules of highest weight λ.
Moreover, the modules L(λ) for λ ∈ h ⋆ form a full set of representatives of the simple isomorphism classes of O, i.e. each simple object in O is isomorphic to L(λ) for a unique λ ∈ h ⋆ .
3.3. Characters. Let Z[ h ⋆ ] = λ∈ h ⋆ Ze λ be the group algebra of the additive group h ⋆ . Let Z[ h ⋆ ] ⊂ λ∈ h ⋆ Ze λ be the subgroup of elements (c λ ) that have the property that there exists a finite set {µ 1 , . . . , µ n } ⊂ h ⋆ such that c λ = 0 implies λ ≤ µ i for at least one i.
Let O f ⊂ O be the full subcategory of modules M that have the property that the weight spaces M λ are finite dimensional and such that there exist µ 1 , . . . , µ n ∈ h ⋆ such that M λ = 0 implies λ ≤ µ i for at least one i. For each object M of O f we can then define its character as The character of a Verma module is easy to calculate. For each λ ∈ h ⋆ we have Fie06]). The principal aim of our research project is to calculate ch L(λ) for the critical highest weights λ. [DGK82]). Note that the sum on the right hand side is, in general, an infinite sum. We define the multiplicity of L(ν) in M as

Multiplicities. Suppose again that M is an object in
The matrix [∆(λ) : L(µ)] is invertible, so the problem of calculating ch L(µ) for all µ ∈ h ⋆ is equivalent to the calculation of the multiplicities [∆(λ) : L(µ)] for all λ, µ ∈ h ⋆ .
3.5. Block decomposition. Denote by "∼" the equivalence relation on h ⋆ that is generated by λ ∼ µ if [∆(λ) : L(µ)] = 0. For an equivalence class Λ ∈ h ⋆ / ∼ we define the full subcategory O Λ of O that consists of all objects M with the property that [M : L(λ)] = 0 implies λ ∈ Λ. We have the following decomposition result.
is an equivalence of categories.
In particular, the equivalence relation "∼" is generated by the partial order relation " ".
In this case, the structure of the block O Λ can be completely described in terms of the group W(Λ) (which turns out to be a Coxeter group) and the singularity of its orbit Λ, cf. [Fie06].

3.7.
A duality on O f . We will later need the following duality functor. For convenience we only define it on the full subcategory O f of O that we defined earlier. All the modules that we encounter in this article belong to O f . For i.e. the involution induced on g by the root system automorphism that sends α ∈ R to −α ∈ R. Then M ⋆ ∈ O f and we indeed get a duality functor on O f . It is exact and maps irreducible modules to irreducible modules. A quick look at characters shows the following.
For each λ ∈ h ⋆ we denote by ∇(λ) := ∆(λ) ⋆ the dual of the Verma module with highest weight λ. By the above lemma and the exactness of the duality, ∇(λ) and ∆(λ) have the same Jordan-Hölder multiplicities, and ∇(λ) has a simple socle which is isomorphic to L(λ).

The critical hyperplane
In this section we recall the notion of a critical weight for the affine Kac-Moody algebra g. We introduce a shift functor T on each of the critical blocks and study the corresponding graded center A = n∈Z A n . For a critical weight λ ∈ h ⋆ we define the restricted Verma module ∆(λ) as the quotient of ∆(λ) by the ideal of A generated by n =0 A n . We state some fundamental properties of these modules in Theorem 4.7. The proof of this theorem is due to Feigin and Frenkel. Then we recall the Feigin-Frenkel conjecture on the Jordan-Hölder multiplicities for restricted Verma modules and, finally, state the main result of this article in Theorem 4.9. 4.1. A shift functor. The defining relations of g show that the derived Lie algebra of g coincides with the central extension g of the loop algebra, i.e.
Hence [ g, g] is of codimension one in g, so the quotient g/[ g, g] is a onedimensional Lie algebra. Each character of g/[ g, g] gives rise to a one dimensional module of g. In this way we get the simple modules L(ζδ) for ζ ∈ C.
Let us define the shift functor The action of g on the tensor product is the usual one: X.(m ⊗ l) = X.m ⊗ l + m ⊗ X.l for X ∈ g, m ∈ M and l ∈ L(δ). The functor T is exact and preserves the categories O f and O. Clearly, it is an equivalence on these categories with inverse It is given by the tensor product with the one-dimensional module L(nδ).
The following lemma is easy to prove (for part (3) use the facts that Now we come to the definition of critical equivalence classes, critical weights and critical blocks.
Lemma 4.2. For an equivalence class Λ ∈ h ⋆ / ∼ the following are equivalent.
If the above conditions on Λ are satisfied, we say that Λ is a critical equivalence class. In this case we call each element λ of Λ a critical weight, or of critical level. We let h ⋆ crit be the set of weights of critical level, i.e. the union of the critical equivalence classes. Condition (4) above shows that h ⋆ crit is an affine hyperplane in h ⋆ . It is called the critical hyperplane.
Proof. From the definition of the blocks, the exactness of T and Lemma 4.1 we deduce the equivalence of (1) and (2).

4.2.
The action of W in the critical hyperplane. Let us fix for the next sections a critical equivalence class Moreover, we have s α+nδ ∈ W(Λ) if and only if s α ∈ W(Λ). We set W(Λ) = W(Λ) ∩ W. Then W(Λ) is the affinization of W(Λ).
For a subset Λ of h ⋆ we denote by Λ its image in h ⋆ /Cδ.
Definition 4.4. We say that Note that part (3) of the above definition refers to the action of the finite Weyl group only. Set R(Λ) + := R(Λ) ∩ R + .
Definition 4.5. Let ν ∈ Λ. We say that 4.3. The graded center of a critical block. Since Λ is supposed to be critical, we can consider the functor T as an auto-equivalence on the block O Λ .
Let n ∈ Z and let z be a natural transformation from the functor T n on O Λ to the identity functor id on O Λ . Note that z associates with any M ∈ O Λ a homomorphism z M : T n M → M in such a way that for any homomorphism Denote by A n = A n (Λ) the complex vector space of all natural transformations z as above such that ) that makes A := n∈Z A n into a graded C-algebra. It is associative, commutative and unital.
4.4. Restricted Verma modules. Let λ ∈ Λ and n ∈ Z. Each z ∈ A n defines a homomorphism Each such homomorphism is zero if n > 0. Define ∆(λ) − ⊂ ∆(λ) as the submodule generated by the images of all the homomorphisms z ∆(λ) for z ∈ A n and n < 0. Consider the formal power series (1 + q j + q 2j + . . . ) rank g , and let us define for n ∈ Z the number p(n) ∈ N as the coefficient of q n in the above series.
We denote by n + := α∈ R + g α the subalgebra of g corresponding to the positive affine roots. For a g-module M we denote by M n + the set of n +invariant vectors. It is a h-submodule of M, hence we can also define its weight spaces M n + ν for ν ∈ h ⋆ . The following theorem lists the most important properties of the restricted Verma modules. The proofs of the following statements (2), (3) and (4), as well as the main step in the proof of (1), are due to Feigin and Frenkel. We recall the main arguments in Section 5.
(4) We have ∆(λ) n + λ−nδ = {0} for n = 0. 4.5. A conjecture. The character of L(λ) for a critical highest weight λ is not yet known in general. But we have a formula for the characters of the restricted Verma modules and the simple characters can be calculated once the Jordan-Hölder multiplicites [∆(λ) : L(µ)] for critical weights λ and µ are determined. In the following we state a conjecture that gives a formula for these multiplicities in terms of periodic Kazhdan-Lusztig polynomials. It is due to Feigin and Frenkel. Let us denote by A the set of alcoves for the affine action of W on h ⋆ and fix a base alcove A = A e . The set A is acted upon by W and we denote by A w := w(A e ) the image of the base alcove under the transformation given by w ∈ W. For alcoves A and B there is the periodic polynomial p A,B ∈ Z[v], defined by Lusztig (we refer to [Fie07] for details). Denote by w 0 ∈ W the longest element in the finite Weyl group.  for all x ∈ W(Λ).
This conjecture is closely related to an anticipated relation between representations of a small quantum group, the topology of semi-infinite flag manifolds and the restricted critical level representations of an affine Kac-Moody algebra, cf. [Lus91]. We prove part (1) of the above conjecture in [AF09]. 4.6. The main result. In the following theorem we summarize the main results of this article.
Theorem 4.9. Suppose that Λ ⊂ h ⋆ is a critical equivalence class.
(2) If Λ is subgeneric and ν ∈ Λ is dominant, then we have for all n ≥ 0 where α is the unique positive root in R(Λ).
Let us restate the above results in terms of restricted Verma modules.
Corollary 4.10. Let Λ ⊂ h ⋆ be a critical equivalence class.
(1) Suppose that ν ∈ Λ is anti-dominant. Then for any γ ∈ Λ we have In particular, Conjecture 4.8 holds for the anti-dominant multiplicities.
(2) If Λ is subgeneric, then we have for all γ, ν ∈ Λ where α is the unique positive root in R(Λ). In particular, Conjecture 4.8 holds in the subgeneric cases.
In the following section we recall the results of Feigin and Frenkel on the center at the critical level and deduce Theorem 4.7. In Section 6 we study the structure of projective objects in a critical block O Λ . In particular, we provide some results on the action of the graded center A on a projective object. In Section 7 we introduce the BRST-cohomology functor and recall the main Theorem of [Ara07]. In Section 8 we use the results of Sections 5, 6 and 7 to give a proof of Theorem 4.9.

The Feigin-Frenkel center
In this section we recall the fundamental results on the Feigin-Frenkel center [FF92] at the critical level. The main references are the textbooks [FBZ04] and [Fre07].
5.1. The universal affine vertex algebra at the critical level. Set where C crit is the one-dimensional representation of g⊗ C C[t] ⊕ CK ⊕ CD on which g⊗ C C[t] ⊕ CD acts trivially and K acts as multiplication by the critical value −ρ, K . The space V crit (g) has a natural structure of a vertex algebra, and is called the universal affine vertex algebra associated with g at the critical level (see e.g., [Kac98, §4.9]). Let a → a(z) = n∈Z a (n) z −n−1 be the state-field correspondence. The map Y (?, z) is uniquely determined by the condition where 1 is the vacuum vector 1⊗1.
The vertex algebra V crit (g) is graded by the Hamiltonian −D. If a ∈ V crit (g) is an eigenvector of −D, its eigenvalue is called the conformal weight and is denoted by ∆ a . We denote by ∂ the translation operator. It satisfies , z).

The Feigin-Frenkel center.
Let z( g) be the center of the vertex algebra V crit (g): One has where G is the adjoint group of g and G This defines a filtration of a vertex algebra. Let gr V crit (g) be the associated graded vertex algebra: gr V crit (g) = p F p V crit (g)/F p−1 V crit (g). It is a commutative vertex algebra and one has as differential rings 1 and G[[t]]-modules, where g ∞ is the infinite jet scheme of g (cf. [EF01]) and g is identified with g ⋆ . Below we shall identify gr V crit (g) with C[g ∞ ]. The natural projection g ∞ → g gives the embedding C Let {F p z( g)} be the induced filtration of z( g), gr z( g) the associated graded vertex algebra. Certainly, the image of the natural embedding gr z( g) ֒→ Letp According to [BD96] (see also [EF01]), one has FF92], see also [Fre07]). The embedding is an isomorphism.

The action of the Feigin-Frenkel center on objects with critical level.
For k ∈ C we denote by O k the category O at level k, i.e. the direct summand of the category O on which K acts as multiplication with k. In particular, we denote by O crit the category O at critical level. Let M be an object of O crit . Then M is naturally a graded module over the vertex algebra V crit (g), and hence, it is a graded module over its center z( g). Thus M can be viewed as a graded module over the polynomial ring in an obvious manner. Here Z n is the subspace of Z spanned by elements p nr with n 1 + · · · + n r = n. Set FG06], see also [Fre07, Theorem 9.5.3]). For any λ ∈ h ⋆ crit , ∆(λ) is free over Z − . Moreover, the natural map Z − n → Hom(∆(λ+nδ), ∆(λ)) is a bijection for all n ≤ 0.
We now construct a natural map Recall that L(δ) is one-dimensional. So we can choose a generator l of L(δ). This gives us, for any g-module M, a map M ⊗ L(δ) → M, m ⊗ l → m. Since L(δ) is trivial as a g-module, this map is a g-module homomorphism. By iteration we get g-module homomorphisms T n M = M ⊗ L(δ) ⊗n → M for all n ≥ 0. Using the element l ⋆ ∈ L(δ) ⋆ that is dual to l we analogously get g-module homomorphisms T −n M = M ⊗ (L(δ) ⋆ ) ⊗n → M. Now suppose that M is contained in a critical block of O. Let n ∈ Z and z ∈ Z n . Then the composition of the map T n M → M constructed above and the action map z : M → M yields now a g-module homomorphism T n M → M, as it now also commutes with the action of D. This gives us a natural transformation T n → id and we get an element in A n that we associate with z. Hence we constructed a map Z n → A n .

Projective objects
For a general equivalence class Λ ∈ h ⋆ / ∼ the block O Λ does not contain enough projective objects (this includes, for example, all critical equivalence classes). However, there is a way to overcome this problem by restricting the set of possible weights for the modules under consideration. This means that we have to consider the truncated subcategories of O.
6.1. The truncated categories. Let us fix a (not necessarily critical) equivalence class Λ ∈ h ⋆ / ∼ . We use the following notation: We write {≤ ν} for the set {ν ′ ∈ Λ | ν ′ ≤ ν} and use the similar notations {< ν}, {≥ ν}, etc. for the analogously defined sets. We consider the topology on Λ that is generated by the basic open sets {≤ ν} for ν ∈ Λ. Hence a subset J of Λ is open in this topology if and only if for all ν, ν ′ ∈ Λ with ν ′ ≤ ν, ν ∈ J implies ν ′ ∈ J . In addition to the submodules and quotient modules defined above we will also need the following subquotient modules that are associated with locally closed subsets. For any subset K of Λ define For each µ ∈ h ⋆ the multiplicity [M : ∆(µ)] := #{i | µ = µ i } is independent of the chosen filtration. It is a well-known fact that Ext 1 (∆(µ), ∆(λ)) = 0 implies that λ > µ. Hence we can find a filtration for M as in the definition such that µ i > µ j implies i < j. From this one easily gets the following lemma. 6.4. Projective objects. We say that J ⊂ Λ is bounded if for any λ ∈ J the set of µ ∈ J with µ ≥ λ is finite.
(1) There exists an (up to isomorphism unique) projective cover P J (λ) of (2) If J ′ ⊂ J is an open subset, then we have an isomorphism (P J (λ)) J ′ ∼ = P J ′ (λ). (3) The object P J (λ) admits a Verma flag and for each γ ∈ Λ we have the BGG-reciprocity formula By Lemma 6.5 and the theorem above, if ν, λ ∈ Λ are such that ν ≥ λ, then the module P J (λ) [ν] does not depend on the open set J as long as ν ∈ J . We denote this object by P (λ) [ν] .
Let J ⊂ Λ be open and bounded and λ ∈ J . Suppose that ν ∈ J is a maximal element. Then we can consider P (λ) [ν] as a subspace in P J (λ). We will need the following result later.
The last equation holds since P (λ) [ν] is isomorphic to a direct sum of copies of ∆(ν) and dim Hom(∆(ν), ∇(ν)) = 1. Since the map referred to in the lemma is injective and since the dimensions of its source and its image coincide, it is bijective.
6.5. The Casimir operator. We call a g-module M smooth if for all m ∈ M we have g α .m = 0 for all but a finite number of positive affine roots α ∈ R.
In particular, each locally b-finite g-module is smooth. Recall that on the full subcategory g-mod sm of smooth representations there is an endomorphism of the identity functor, the Casimir operator C : id → id (its construction can be found, for example, in [Kac90, Section 2.5]). We will only need the following property of C.
The construction of C depends on (·, ·), hence there is no ambiguity in the statement of the proposition.
Let Λ ⊂ h ⋆ be a (not necessarily critical) equivalence class. For λ, µ ∈ Λ we have c λ = c µ , so Λ defines a unique scalar c Λ ∈ C such that C acts by multiplication with c Λ on each Verma module in O Λ . Now suppose that λ, µ ∈ Λ are such that they form an atom in the partially ordered set Λ, i.e. suppose that λ < µ and that there is no ν ∈ Λ with λ < ν < µ. Then the object P µ (λ) is an extension of the Verma modules ∆(λ) and ∆(µ), each occurring once (this is, by the BGG-reciprocity, equivalent to [∆(λ) : L(λ)] = [∆(µ) : L(λ)] = 1, which can be deduced easily from the analog of Jantzen's sum formula in the Kac-Moody case, cf. [KK79]). Hence we have a short exact sequence Lemma 6.9. The endomorphism C − c Λ id : P µ (λ) → P µ (λ) is non-zero.
Proof. For the proof we use deformation theory, cf. [Fie03] and [AF09]. Denote by A = C[[t]] the completed polynomial ring in one variable and by Q = Quot A its quotient field. Let us fix γ ∈ h ⋆ with the property that (γ+ρ, λ) = (γ+ρ, µ). Let S = S( h) be the symmetric algebra over h and consider the algebra map τ : S → A that is determined by τ (h) = γ(h)t for all h ∈ h. This makes A, and hence Q, into a local S-algebra.
In [Fie03] we constructed the deformed categories O A and O Q as full subcategories of g⊗ C A-mod and g⊗ C Q-mod. We showed that the functors ⊗ A C and The categories O A and O Q contain deformed Verma modules ∆ A (ν) and ∆ Q (ν), resp. On their highest weight spaces the Cartan algebra h acts by the character ν + τ , which is considered as a linear map from h to A and Q, resp. In particular, the Casimir operator C acts on ∆ Q (ν) as multiplication with c ν+γt = (ν + γt + ρ, ν + γt + ρ) − (ρ, ρ).
The analogous definition as in the non-deformed case gives truncated cat- ). The category O Q is semi-simple, i.e. each object is isomorphic to a direct sum of Verma modules ∆ Q (λ). Now each P ν A (ν ′ ) ⊗ A Q admits a Q-deformed Verma flag, hence it splits into a direct sum of Q-deformed Verma modules. The Verma multiplicities of P ν A (ν ′ ), of P ν (ν ′ ) and of P ν A (ν ′ ) ⊗ A Q coincide. Now let λ, µ ∈ Λ be as in the statement of the lemma. Note that c λ+γt ≡ c µ+γt mod t, but our choice of γ implies c λ+γt = c µ+γt . Let us suppose that the action of C − c Λ on P µ (λ) was zero. From the above, our assumptions on λ and µ and the Jantzen sum formula we get an inclusion . On each of the modules above the Casimir operator acts. Our assumptions imply that the image of the action of C − c Λ on the module on the left is contained in tP µ A (λ), hence t −1 (C − c Λ ) is a well-defined operator on P µ A (λ). On the module on the right this operator acts diagonally with eigenvalues in C[[t]] which are distinct modulo t. Hence P µ A (λ) decomposes according to the inclusion above, which clearly cannot be the case.
Lemma 6.10. Suppose that λ, µ ∈ Λ are as above and that is a short exact sequence, where X is a module of highest weight λ. If C acts on M as a scalar, then there is a submodule Y of M with highest weight λ that maps surjectively onto X.
Proof. Since X is a module of highest weight λ and since the weights of M are smaller or equal to µ, there is a map f : P µ (λ) → M such that the composition 0 with exact rows. In order to prove the lemma it is enough to show that ∆(µ) ⊂ P µ (λ) is in the kernel of f , i.e. that the left vertical map f ′ is zero. Since any endomorphism of a Verma module is either zero or injective, it suffices to show that f ′ is not injective.
By assumption, the Casimir element C acts on M by a scalar, which has to be c By the previous lemma, (C − c Λ )P µ (λ) is non-zero, so the kernel of f ′ is not trivial, hence f ′ is not injective, hence it must be zero, which is what we wanted to show. 6.6. The action of A on projective objects. Let us fix now a critical equivalence class Λ ⊂ h ⋆ . Let λ ∈ Λ and n ≥ 0. In this section we study the action of A n on P λ (λ − nδ), i.e. we want to study the map A n → Hom(T n P λ (λ − nδ), P λ (λ − nδ)).

Then the action map
is surjective.
Proof. Note that P (λ) λ ∼ = ∆(λ) and that P (λ − nδ) [λ] is isomorphic to a direct sum of (P (λ − nδ) [λ] : ∆(λ))-copies of ∆(λ). By our assumption and the BGG-reciprocity, this number is p(n). Hence the spaces on the right hand side of our map are of dimension p(n). Now the following Lemma 6.13 shows that the image of the action map A n → Hom(T n P λ (λ − nδ), P λ (λ − nδ)) is of dimension p(n). From this we deduce our claim. 6.7. A duality on A. Let Λ ⊂ h ⋆ again be a critical equivalence class. In this section we define an algebra involution Fix n ∈ Z and choose z ∈ A n . We define Dz ∈ A −n as follows. Let One immediately checks that we get a natural transformation Dz : As O Λ is filtered by the truncated categories, and as each indecomposable projective object in a truncated category is also contained in O f Λ , this induces a natural transformation Dz : T n → id between the functors on the whole block O Λ , hence an element in A −n . Now we prove the statement that remained open in the proof of Proposition 6.11. Fix λ ∈ Λ and n ≥ 0.
Proof. By the definition of the duality we have For each homomorphism g : The strategy of the proof is the following. Suppose that z P λ (λ−nδ) = 0. We show that there is a map g such that the top right composition in the diagram above is non-zero. From this we deduce that z ∇(λ) = 0. We show that z ∇(λ) = 0 implies z P λ (λ−nδ) = 0 in a similar way.
The last statement of the lemma follows from the previous result and Theorem 4.7.
6.8. A variant for the subgeneric cases. We will also need the following variant of Proposition 6.11 in the case that Λ is critical and subgeneric. Suppose that α is the positive root in R(Λ). Fix λ ∈ Λ and n ≥ 0. We study the action of A n on the projective cover P α↑λ (λ − nδ), i.e. we now consider the map A n → Hom(T n P α↑λ (λ − nδ), P α↑λ (λ − nδ)).
Let us consider the composition where the last map is induced by the functor (·) [α↑λ] .
Proof. Consider the homomorphisms Hom(P (λ) [α↑λ] , P (λ − nδ) [α↑λ] ) By Proposition 6.11, the lower composition is surjective. But the kernel of the upper composition is contained in the kernel of the lower composition, as P (λ − nδ) [α↑λ,λ] is a direct sum of non-split extensions of ∆(α ↑ λ) and ∆(λ). Since the spaces on the top and the bottom share the same dimension, also the upper composition is surjective.

The BRST cohomology
To prove Theorem 4.9 we need a result from [Ara07], which we explain below.
7.1. The BRST cohomology associated with the quantized Drinfeld-Sokolov reduction. Denote by n − := α∈R + g −α the nilpotent subalgebra of g corresponding to the set of negative roots. Let Ψ be a non-degenerate character of n − in the sense of Kostant [Kos78], i.e.

The space
∞ 2 +• is graded by charge: ∞ 2 +• = i∈Z ∞ 2 +i , where the charges of 1, ψ α (n) and ψ −α (n) for α ∈ R + , n ∈ Z, are 0, 1, and −1, respectively. Also, we view Define an odd operator Q of charge 1 on C • (M) by where x α is a (fixed) root vector in g α for any α ∈ R − and [x α , x β ] = γ c γ α,β x γ . The operator Q is well-defined because M ∈ O. One has Q 2 = 0. (1) One has H i (M) = 0 for all i = 0 and M ∈ O crit . In particular, the functor F is exact.

j≥1
(1 − q j ) −rank g , Remark 7.2. In general, the correspondence M → H 0 (M) defines a functor from O k to the category of graded modules over the W -algebra W k (g) associated with g at level k, which coincides [FF92] with z( g) if the level k is critical. In [Ara07], it was proved that the functor H 0 (?) is exact and H 0 (L(λ)) is zero or irreducible for any λ at any level k.

The proof of the main Theorem
We have collected all the ingredients for the proof of our main Theorem 4.9. We start with claim (1). Let us state it again: Theorem 8.1. Let Λ ⊂ h ⋆ be a critical equivalence class and suppose that ν ∈ Λ is anti-dominant. Then for all w ∈ W(Λ) and n ≥ 0 we have Proof. By Theorem 7.1, (1), the functor F is exact. Hence we have Note that γ ∈ Λ is anti-dominant if and only if γ = ν + rδ for some r ∈ Z. Since w.ν, D = ν, D for all w ∈ W (as α, D = 0 for all finite roots α), the claim now follows directly from Theorem 7.1, (2).
It remains to prove part (2) of Theorem 4.9. Let us recall the statement: For the proof of the above statement we need the following result.
In particular, let us consider the chain of inclusions From now on we view each of these modules as a submodule of all the modules appearing on its right (note that there is more than one inclusion ∆(λ)֒→∆(α ↑ 2 λ), for example). By Theorem 5.2, there is an element z ∈ Z − , uniquely defined up to multiplication with a scalar in C, such that ∆(λ) = z∆(α ↑ 2 λ). It is easy to see that this implies ∆(α ↑ λ) = z∆(α ↑ 3 λ).