Computation of the homology of the complexes of finite Verma modules for $K'_4$

We compute the homology of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal A(K'_{4})$, associated with the conformal superalgebra $K'_{4}$, obtained in \cite{K4}. We use the computation of the homology in order to provide an explicit realization of all the irreducible quotients of finite Verma modules over $\mathcal A(K'_{4})$.

modules over A(K ′ 4 ) occur among cokernels, kernel and images of complexes in Figure 1. As an application of this result, we compute the size of all the irreducible quotients of finite Verma modules, that is defined following [17]. The paper is organized as follows. In section 2 we recall some notions on conformal superalgebras. In section 3 we recall the definition of the conformal superalgebra K ′ 4 and the classification of singular vectors obtained in [1]. In section 4 we find an explicit expression for the morphisms represented in Figure 1. In section 5 we recall the preliminaries on spectral sequences that we need. In section 6 we compute the homology of the complexes in Figure 1. Finally in section 7 we compute the size of all the irreducible quotients of finite Verma modules.

Preliminaries on conformal superalgebras
We recall some notions on conformal superalgebras. For further details see [15,Chapter 2], [13], [4], [2]. Let g be a Lie superalgebra; a formal distribution with coefficients in g, or equivalently a g−valued formal distribution, in the indeterminate z is an expression of the following form: a(z) = n∈Z a n z −n−1 , with a n ∈ g for every n ∈ Z. We denote the vector space of formal distributions with coefficients in g in the indeterminate z by g[[z, z −1 ]]. We denote by Res(a(z)) = a 0 the coefficient of z −1 of a(z). The vector space g[[z, z −1 ]] has a natural structure of C[∂ z ]−module. We define for all a(z) ∈ g[[z, z −1 ]] its derivative: ∂ z a(z) = n∈Z (−n − 1)a n z −n−2 .
A formal distribution with coefficients in g in the indeterminates z and w is an expression of the following form: where the coefficients (a(w) (j) b(w)) := Res z (z − w) j [a(z), b(w)] are formal distributions in the indeterminate w. Definition 2.3 (Formal Distribution Superalgebra). Let g be a Lie superalgebra and F a family of mutually local g−valued formal distributions in the indeterminate z. The pair (g, F) is called a formal distribution superalgebra if the coefficients of all formal distributions in F span g.
We define the λ−bracket between two formal distributions a(z), b(z) ∈ g[[z, z −1 ]] as the generating series of the (a(z) (j) b(z))'s: (ii) skew − symmetry : (iii) Jacobi identity : where p(a) denotes the parity of the element a ∈ R and p(∂a) = p(a) for all a ∈ R.
Due to this relation and (iii) in Definition 2.5, the map ∂ : R → R, a → ∂a is a derivation with respect to the n−products.
We say that a conformal superalgebra R is finite if it is finitely generated as a C[∂]−module. An ideal I of R is a C[∂]−submodule of R such that (a (n) b) ∈ I for every a ∈ R, b ∈ I, n ≥ 0. A conformal superalgebra R is simple if it has no non-trivial ideals and the λ−bracket is not identically zero. We denote by R ′ the derived subalgebra of R, i.e. the C−span of all n−products.
Definition 2.7. A module M over a conformal superalgebra R is a left Z 2 −graded C[∂]−module endowed with C−linear maps R → End C M , a → a (n) , defined for every n ≥ 0, that satisfy the following properties for all a, b ∈ R, v ∈ M , m, n ≥ 0: For an R−module M , we define for all a ∈ R and v ∈ M : A module M is called finite if it is a finitely generated C[∂]−module.
We can construct a conformal superalgebra starting from a formal distribution superalgebra (g, F). Let F be the closure of F under all the n−products, ∂ z and linear combinations. By Dong's Lemma, F is still a family of mutually local distributions (see [15]). It turns out that F is a conformal superalgebra. We will refer to it as the conformal superalgebra associated with (g, F). Let us recall the construction of the annihilation superalgebra associated with a conformal superalgebra R. Let R = R[y, y −1 ], set p(y) = 0 and ∂ = ∂ + ∂ y . We define the following n−products on R, for all a, b ∈ R, f, g ∈ C[y, y −1 ], n ≥ 0: In particular if f = y m and g = y k we have for all n ≥ 0: We observe that ∂ R is a two sided ideal of R with respect to the 0−product. The quotient Lie R := R/ ∂ R has a structure of Lie superalgebra with the bracket induced by the 0−product, i.e. for all a, b ∈ R, f, g ∈ C[y, y −1 ]: Definition 2.8. The annihilation superalgebra A(R) of a conformal superalgebra R is the subalgebra of Lie R spanned by all elements ay n with n ≥ 0 and a ∈ R. The extended annihilation superalgebra A(R) e of a conformal superalgebra R is the Lie superalgebra C∂ ⋉ A(R). The semidirect sum C∂ ⋉ A(R) is the vector space C∂ ⊕ A(R) endowed with the structure of Lie superalgebra determined by the bracket: For all a, b ∈ R, we have: [a λ , b µ ] = [a λ b] λ+µ and (∂a) λ = −λa λ (for a proof see [6]). Proposition 2.9 ( [9]). Let R be a conformal superalgebra. If M is an R-module then M has a natural structure of A(R) e -module, where the action of ay n on M is uniquely determined by a λ v = n≥0 λ n n! ay n .v for all v ∈ M . Viceversa if M is a A(R) e -module such that for all a ∈ R, v ∈ M we have ay n .v = 0 for n ≫ 0, then M is also an R-module by letting a λ v = n λ n n! ay n .v. Proposition 2.9 reduces the study of modules over a conformal superalgebra R to the study of a class of modules over its (extended) annihilation superalgebra. The following proposition states that, under certain hypotheses, it is sufficient to consider the annihilation superalgebra. We recall that, given a Z−graded Lie superalgebra g = ⊕ i∈Z g i , we say that g has finite depth d ≥ 0 if g −d = 0 and g i = 0 for all i < −d.
Proposition 2.10 ( [2], [12]). Let g be the annihilation superalgebra of a conformal superalgebra R. Assume that g satisfies the following conditions: L1: g is Z−graded with finite depth d; L2: There exists an element whose centralizer in g is contained in g 0 ; L3: There exists an element Θ ∈ g −d such that g i−d = [Θ, g i ], for all i ≥ 0. Finite modules over R are the same as modules V over g, called finite conformal, that satisfy the following properties: (1) for every v ∈ V , there exists j 0 ∈ Z, j 0 ≥ −d, such that g j .v = 0 when j ≥ j 0 ; (2) V is finitely generated as a C[Θ]−module.
Remark 2.11. We point out that condition L2 is automatically satisfied when g contains a grading element, i.e. an element t ∈ g such that [t, b] = deg(b)b for all b ∈ g.
Let g = ⊕ i∈Z g i be a Z−graded Lie superalgebra. We will use the notation g >0 = ⊕ i>0 g i , g <0 = ⊕ i<0 g i and g ≥0 = ⊕ i≥0 g i . We denote by U (g) the universal enveloping algebra of g.
Definition 2.12. Let F be a g ≥0 −module. The generalized Verma module associated with F is the g−module Ind(F ) defined by: Ind(F ) := Ind g g ≥0 (F ) = U (g) ⊗ U (g ≥0 ) F. If F is a finite−dimensional irreducible g ≥0 −module we will say that Ind(F ) is a finite Verma module. We will identify Ind(F ) with U (g <0 ) ⊗ F as vector spaces via the Poincaré−Birkhoff−Witt Theorem. The Z−grading of g induces a Z−grading on U (g <0 ) and Ind(F ). We will invert the sign of the degree, so that we have a Z ≥0 −grading on U (g <0 ) and Ind(F ). We will say that an element v ∈ U (g <0 ) k is homogeneous of degree k. Analogously an element m ∈ U (g <0 ) k ⊗ F is homogeneous of degree k. For a proof of the following proposition see [1]. Proposition 2.13. Let g = ⊕ i∈Z g i be a Z−graded Lie superalgebra. If F is an irreducible finite− dimensional g ≥0 −module, then Ind(F ) has a unique maximal submodule. We denote by I(F ) the quotient of Ind(F ) by the unique maximal submodule.
Definition 2.14. Given a g−module V , we call singular vectors the elements of: Homogeneous components of singular vectors are still singular vectors so we often assume that singular vectors are homogeneous without loss of generality. In the case V = Ind(F ), for a g ≥0 −module F , we will call trivial singular vectors the elements of Sing(V ) of degree 0 and nontrivial singular vectors the nonzero elements of Sing(V ) of positive degree. Theorem 2.15 ([18], [12]). Let g be a Lie superalgebra that satisfies L1, L2, L3, then: (i) if F is an irreducible finite−dimensional g ≥0 −module, then g >0 acts trivially on it; (ii) the map F → I(F ) is a bijective map between irreducible finite−dimensional g 0 −modules and irreducible finite conformal g−modules; (iii) the g−module Ind(F ) is irreducible if and only if the g 0 −module F is irreducible and Ind(F ) has no nontrivial singular vectors.
We recall the notion of duality for conformal modules (see for further details [4], [6]). Let R be a conformal superalgebra and M a conformal module over R.
Definition 2.16. The conformal dual M * of M is defined by: Definition 2.17. Let T : M → N be a morphism of R−modules, i.e. a linear map such that for all a ∈ R and m ∈ M : i: We denote by F the functor that maps a conformal module M over a conformal superalgebra R to its conformal dual M * and maps a morphism between conformal modules T : M → N to its dual T * : N * → M * . Proof. Let us consider an exact short sequence of conformal modules: Therefore we know that d 2 • d 1 = 0, d 1 is injective, d 2 is surjective and Ker d 2 = Im d 1 . We consider the dual of this sequence: By Theorem 2.18 and Remark 3.11 in [6], we know that d * 1 is surjective and d * 2 is injective. We have to show that Ker d * ). Since Ker d 2 = Im d 1 , this condition tells that β vanishes on Ker d 2 . We also know that for every p ∈ P , p = d 2 (n p ), for some n p ∈ N . We define α ∈ P * as follows, for all p ∈ P : Let us show that α actually lies in P * . For every p ∈ P : α λ (∂p) = α λ (∂d 2 (n p )) = α λ (d 2 (∂n p )) = −β λ (∂n p ).

4
In this section we recall some notions and properties about the conformal superalgebra K ′ 4 (for further details see [1], [2], [14]). We first recall the notion of the contact Lie superalgebra. Let (N ) be the Grassmann superalgebra in the N odd indeterminates ξ 1 , ..., ξ N . Let t be an even indeterminate and (1, N ) = C[t, t −1 ] ⊗ (N ). We consider the Lie superalgebra of derivations of (1, N ): One can define on (1, N ) a Lie superalgebra structure as follows: for all f, g ∈ (1, N ) we let We recall that K(1, N ) ∼ = (1, N ) as Lie superalgebras via the following map (see [11]): From now on we will always identify elements of K(1, N ) with elements of (1, N ) and we will omit the symbol ∧ between the ξ i 's. We adopt the following notation: we denote by I the set of finite sequences of elements in {1, . . . , N }; we will write I = i 1 · · · i r instead of I = (i 1 , . . . , i r ). Given I = i 1 · · · i r and J = j 1 · · · j s , we will denote i 1 · · · i r j 1 · · · j s by IJ; if I = i 1 · · · i r ∈ I we let ξ I = ξ i 1 · · · ξ ir and |ξ I | = |I| = r. We denote by I = the subset of I of sequences with distinct entries. We consider on K(1, N ) the standard grading, i.e. for every m ∈ Z and I ∈ I, deg(t m ξ I ) = 2m + |I| − 2. Now we want to recall the definition of the conformal superalgebra K N . In order to do this, we construct a formal distribution superalgebra using the following family of formal distributions: Note that the set of all the coefficients of formal distributions in F spans (1, N ) and the distributions are mutually local (for a proof see [1,Proposition 3.1]). The conformal superalgebra associated . For all I, J ∈ I the λ−bracket is given by for a proof see [1,Proposition 3.1]. K N is simple except for the case N = 4. If N = 4, K 4 = K ′ 4 ⊕ Cξ 1234 , where K ′ 4 is the derived subalgebra, i.e. the C−span of the n−products. K ′ 4 is a simple conformal superalgebra (see [14]).
, is a central extension of K(1, 4) + by a one−dimensional center CC: The extension is given by the 2−cocycle ψ ∈ Z 2 (K(1, 4) + , C) which computed on basis elements returns non−zero values in the following cases only (up to skew-symmetry of ψ): We denote with g := A(K ′ 4 ) = K(1, 4) + ⊕ CC. We recall from [1] the description of g. The grading on g is the standard grading of K(1, 4) + and C has degree 0. We have: The annihilation superalgebra g satisfies L1, L2, L3: L1 is straightforward; L2 follows by Remark 2.11 since t is a grading element for g; L3 follows from the choice Θ := −1/2 ∈ g −2 . We recall that g 0 = {C, t, ξ ij : 1 ≤ i < j ≤ 4} ∼ = so(4) ⊕ Ct ⊕ CC, where so(4) is the Lie algebra of 4 × 4 skew−symmetric matrices. In the above isomorphism the element ξ ij corresponds to the skewsymmetric matrix −E i,j + E j,i ∈ so(4). We consider the following basis of a Cartan subalgebra h: Let α x , α y ∈ h * be such that α x (h x ) = α y (h y ) = 2 and α x (h y ) = α y (h x ) = 0. The set of roots is ∆ = {α x , −α x , α y , −α y } and we have the following root decomposition: We will use the following notation: The set {e 1 , e 2 } is a basis of the nilpotent subalgebra g αx ⊕ g αy . We denote by g ss 0 the semisimple part of g 0 .
Remark 3.2. The sets {e x , f x , h x } and {e y , f y , h y } span two copies of sl 2 and we think of g ss 0 in the standard way as a Lie algebra of derivations. We have that: By direct computations, we obtain the following results.
By Lemma 3.3 to check whether a vector m in a g-module is a highest weight singular vector it is sufficient to show that it is annihilated by e 1 , e 2 , t(ξ 1 + iξ 2 ) and (ξ 1 + iξ 2 )ξ 3 ξ 4 .
Lemma 3.4 ([1]). As g ss 0 −modules: The isomorphism is given by: From now on it is always assumed that F is a finite−dimensional irreducible g ≥0 −module.
Given I = i 1 · · · i k ∈ I = , we will use the notation η I to denote the element η i 1 · · · η i k ∈ U (g <0 ) and we will denote |η I | = |I| = k.
Remark 3.6. Since C is central, by Schur's lemma, C acts as a scalar on F .
Remark 3.7 is used in [1] to construct the maps in Figure 1 of all possible morphisms between finite Verma modules in the case of K ′ 4 . The maps will be described explicitly in section 4. We now Remark 3.8. By the main result in [6], the conformal dual of a Verma module M (m, n, µ t , µ C ) is M (m, n, −µ t + a, −µ C + b), with a = str(ad(t) |g <0 ) = 2 and b = str(ad(C) |g <0 ) = 0, where g = A(K ′ 4 ), 'str' denotes supertrace, and 'ad' denotes the adjoint representation. In particular in Figure 1 the duality is obtained with the rotation by 180 degrees of the whole picture.
We introduce the following g 0 −modules: The subscripts [i, j] mean that t acts on V X , for X = A, B, C, D, as − 1 2 (x 1 ∂ x 1 +x 2 ∂ x 2 +y 1 ∂ y 1 +y 2 ∂ y 2 ) plus i Id and C acts on V X , for X = A, B, C, D, as 1 2 ( is assumed to be [0, 0] when it is omitted, i.e. for X = A. The elements of g ss 0 act on V X , for X = A, B, C, D, in the standard way: We introduce the following bigrading: The V m,n X 's are irreducible g 0 −modules. We point out that for m, n ∈ Z ≥0 : Hence for m, n ∈ Z ≥0 , V m,n A is the irreducible g 0 −module determined by coordinates (m, n) in quadrant A of Figure 1 Figure 1. We now recall the classification of highest weight singular vectors found in [1], using the notation of the V X 's for X = A, B, C, D and (11). Theorem 3.9 ([1]). Let F be an irreducible finite−dimensional g 0 −module, with highest weight µ. A vector m ∈ Ind(F ) is a non trivial highest weight singular vector of degree 1 if and only if m is (up to a scalar) one of the following vectors:   Remark 3.13. We point out that the highest weight singular vectors of Theorems 3.9, 3.10 and 3.9 are written differently from [1]. Indeed in [1] the irreducible g ss 0 −module of highest weight (m, n) with respect to h x , h y is identified with the space of bihomogeneous polynomials in the four variables x 1 , x 2 , y 1 , y 2 of degree m in the variables x 1 , x 2 , and of degree n in the variables y 1 , y 2 . We use instead the notation of the V X 's and (11) because it is convenient for the explicit description of the morphisms in Figure 1.
From Theorems 3.9, 3.10, 3.11 and 3.12 it follows that the module M (0, 0, 2, 0) does not contain non trivial singular vectors, hence it is irreducible due to Theorem 2.15.

The morphisms
In this section we find an explicit form for the morphisms that occur in Figure 1. We follow the notation in [17] and define, for every u ∈ U (g <0 ) and φ ∈ Hom for every u ′ ⊗ v ∈ U (g <0 ) ⊗ V X . From this definition it is clear that the map u ⊗ φ commutes with the action of g <0 . The following is straightforward.
Lemma 4.1. Let u ⊗ φ be a map as in (12). Let us suppose that u ⊗ φ = i u i ⊗ φ i where {u i } i and {φ i } i are bases of dual g 0 −modules and u i is the dual of φ i for all i. Then u ⊗ φ commutes with g 0 .
On the other hand we have: Let us suppose that Φ commutes with g ≤0 and that Φ(v) is a singular vector for every v highest weight vector in V m,n X and for all m, n ∈ Z. Then Φ is a morphism of g−modules.
Proof. Due to Lemma 2.3 in [17], it is sufficient to show that g >0 Φ(w) = 0 for every w ∈ V X , in order to prove that Φ commutes with g >0 . We know that g >0 Φ(v) = 0 for every v highest weight vector in V m,n X for all m, n ∈ Z. Let v be the highest weight vector in V m,n X , f one among f x , f y and g + ∈ g >0 . We have that: This can be iterated and we obtain that We consider, for j = 1, 2, the map ∂ x j : V X −→ V X that is the derivation by x j for X = A, D and the multiplication by ∂ x j for X = B, C. We define analogously, for j = 1, 2, the map ∂ y j : V X −→ V X , that is the derivation by y j for X = A, B and the multiplication by ∂ y j for X = C, D. We will often write, by abuse of notation, We point out that ∇ |M m,n X : M m,n X −→ M m−1,n−1 X for X = A, B, C, D; by abuse of notation we will write ∇ instead of ∇ |M m,n X . Remark 4.4. By (8) and (9) it is straightforward that ( Proposition 4.5. The map ∇ is the explicit expression of the g−morphisms of degree 1 in Figure  1 and ∇ 2 = 0.  (12). By Lemmas 4.1, 4.3 it follows that ∇ is a morphism of g−modules. The property ∇ 2 = 0 follows from the fact that ∇ is a map between Verma modules that contain only highest weight singular vectors of degree 1, by Theorems 3.9, 3.10, 3.11.  Figure 1 in quadrant D. We introduce the following notation:

Proof. It is a straightforward verification that
We denote M X ′ = U (g <0 ) ⊗ V X ′ . We point out that M X ′ is the direct sum of Verma modules of Figure 1 in quadrant X that lie on the axis n = 0. We consider the map τ 1 : M A ′ −→ M D ′ that is the identity. We have that: We call By abuse of notation, we also call ∇ 2 : for every m ≥ 0. By abuse of notation we will also write Proposition 4.6. The map ∇ 2 is the explicit expression of the morphisms of degree 2 in Figure 1 from the quadrant A to the quadrant D and from the quadrant B to the quadrant C; ∇ 2 ∇ = ∇∇ 2 = 0.
Proof. We first point out that: The map ∇ 2 commutes with g <0 by (12). By Lemmas 4.1, 4.3 it follows that ∇ 2 is a morphism of g−modules. Finally, ∇ 2 ∇ = ∇∇ 2 = 0 follows from the fact that due to Theorem 3.11, there are no highest weight singular vectors of degree 3 in the codomain of ∇ 2 ∇ and ∇∇ 2 .
By Proposition 4.6, it follows that, for every m ≥ 2, the maps are the morphisms represented in Figure 1 from the quadrant A to the quadrant D and, for every m ≥ 0, the maps We now define the map τ 3 : V 0,0 A −→ V 0,0 C that is the identity. We have that: We define the map ∇ 3 : as follows, using definition (12), for every m ∈ M 0,1 A : is the explicit form for the morphism of g−modules of degree 3 from quadrant A to quadrant C and ∇ 3 ∇ = ∇∇ 3 = 0.
A , is the highest weight singular vector of degree 3 in M −1,0 C , classified in Theorem 3.11. Indeed, the highest weight vector in V 0,1 A is y 1 and, by direct computation, ∇ 3 (y 1 ) = − m 3a , that is the highest weight singular vector of M (1, 0, 5 2 , − 1 2 ) found in Theorem 3.11. ∇ 3 commutes with g <0 due to (12). By a straightforward computation and (17) A . Therefore it is a morphism of g−modules due to Lemmas 4.2 and 4.3. Finally ∇ 3 ∇ = ∇∇ 3 = 0 since there are no singular vectors of degree 4 due to Theorem 3.12.
Let us define the maps ∆ + : M X −→ M X and ∆ − : M X −→ M X as follows: We point out that the morphism ∇, defined in (14), can be expressed also by Remark 4.8. By (8) and (9) it is straightforward that ( ∆ + ) 2 = 0, ( ∆ − ) 2 = 0 and ∆ + ∆ − + ∆ − ∆ + = 0. We introduce the following notation: We point out that M X ′′ is the direct sum of Verma modules of Figure 1 in quadrant X that lie on the axis m = 0. We consider the map τ 1 : M A ′′ −→ M B ′′ that is the identity. We have that: We call By abuse of notation, we also call ∇ 2 : for every n ≥ 0. By abuse of notation we will also write Proposition 4.9. The map ∇ 2 is the explicit expression of the morphisms of g−modules of degree 2 in Figure 1 from the quadrant A to the quadrant B and from the quadrant D to the quadrant C and ∇ 2 ∇ = ∇ ∇ 2 = 0.
Proof. We first point out that: i: for n ≥ 2, the map for v highest weight vector in V n A ′′ , is the highest weight singular vector of degree 2 in M n−2 B ′′ , classified in Theorem 3.10. Indeed, the highest weight vector in V n A ′′ is y n 1 and, by direct computation, , classified in Theorem 3.10. Indeed, the highest weight vector in V −n D ′′ is ∂ n y 2 and, by direct computation, The map ∇ 2 commutes with g <0 by (12). By Lemmas 4.1, 4.3 it follows that ∇ 2 is a morphism of g−modules. Finally, ∇ 2 ∇ = ∇ ∇ 2 = 0 follows from the fact that due to Theorem 3.11, there are no highest weight singular vectors of degree 3 in the codomain of ∇ 2 ∇ and ∇ ∇ 2 .
By Proposition 4.9, it follows that, for every for every n ≥ 2, the maps ∇ 2 : M n A ′′ −→ M n−2 B ′′ are the morphisms represented in Figure 1 from the quadrant A to the quadrant B and, for every n ≥ 0, the maps ∇ 2 |M −n are the morphisms from the quadrant D to the quadrant C.
We define the map ∇ 3 : as follows, using definition (12), for every m ∈ M 1,0 A : A , is the highest weight singular vector of degree 3 in M 0,−1 C , classified in Theorem 3.11. Indeed, the highest weight vector in V 1,0 A is x 1 and, by direct computation, ∇ 3 (x 1 ) = − m 3b , that is the highest weight singular vector of M (0, 1, 5 2 , 1 2 ) found in Theorem 3.11. ∇ 3 commutes with g <0 due to (12). By a straightforward computation and (17) A . Therefore it is a morphism of g−modules by Lemmas 4.2 and 4.3. Finally ∇ 3 ∇ = ∇ ∇ 3 = 0 since there are no singular vectors of degree 4 due to Theorem 3.12.
The following sections are dedicated to the computation of the homology of complexes in Figure  1. We will use techniques of spectral sequences, that we briefly recall for the reader's convenience in the next section.

Preliminaries on spectral sequences
In this section we recall some notions about spectral sequences; for further details see [17,Appendix] and [21, Chapter XI]. We follow the notation used in [17]. Let A be a module with a filtration: for fixed s and all p in Z. The classical case studied in [21, Chapter XI, Section 3] corresponds to s = 1. We will need the case s = 0. The filtration (21) induces a filtration on the module H(A) of the homology spaces of A: indeed, for every p ∈ Z, We denote by H(E) = H(E, d) the homology of E under the differential d that is the family } r∈Z is a sequence of families of modules with differential (E r , d r ) as in definition 5.1, such that, for all r, d r has degree −r and: Proposition 5.3. Let A be a module with a filtration as in (21) and differential as in (22). Therefore it is uniquely determined a spectral sequence, as in definition 5.2, E = {(E r , d r )} r∈Z such that: Proof. For the proof see [17,Appendix].
We point out that (27) states that E s is isomorphic to the homology of the module Gr A with respect to the differential induced by d.
Thus, by iteration we obtain the following inclusions: Let A be a module with a filtration as in (21) and differential as in (22). Let {(E r , d r )} r∈Z be the spectral sequence determined by Proposition 5.3. We define E ∞ p as: Let B be a module with a filtration as in (21). We say that the spectral sequence converges to B if, for all p: Proposition 5.6. Let A be a module with a filtration as in (21) and differential as in (22). Let us suppose that the filtration is convergent above and, for some N , F −N A = 0. Then the spectral sequence converges to the homology of A, that is: Proof. For the proof see [17,Appendix].
Remark 5.7. Let A be a module with a filtration as in (21) and differential as in (22). We moreover suppose that A = ⊕ n∈Z A n is a Z−graded module and d : A n −→ A n−1 for all n ∈ Z. Therefore the filtration (21) induces a filtration on each A n . The family {F p A n } p,n∈Z is indexed by (p, n). It is customary to write the indices as (p, q), where p is the degree of the filtration and q = n − p is the complementary degree. The filtration is called bounded below if, for all n ∈ Z, there exists a s = s(n) such that F s A n = 0. In this case the spectral sequence E = {(E r , d r )} r∈Z , determined as in Proposition 5.3, is a family of modules E r = E r p,q p,q∈Z indexed by (p, q), where E r p = p,q∈Z E r p,q , with the differential d r = d r p,q : E p,q −→ E p−r,q+r−1 p,q∈Z of bidegree (−r, r − 1) such that d p,q • d p+r,q−r+1 = 0 for all p, q ∈ Z. Equations (23), (24) ,(25), (26) and (27) can be written so that the role of q is explicit. For instance, Equation (23) can be written as: for all p, q ∈ Z. Equation (27) can be written as E s We now recall some results on spectral sequences of bicomplexes; for further details see [17] and [21, Chapter XI, Section 6].
Definition 5.8 (Bicomplex). A bicomplex K is a family {K p,q } p,q∈Z of modules endowed with two families of differentials, defined for all integers p, q, d ′ and d ′′ such that We can also think K as a Z−bigraded module where K = p,q∈Z K p,q . A bicomplex K as in Definition 5.8 can be represented by the following commutative diagram: (28) Definition 5.9 (Second homology). Let K be a bicomplex. The second homology of K is the homology computed with respect to d ′′ , i.e.: The second homology of K is a bigraded complex with differential d ′ : H ′′ p,q (K) −→ H ′′ p−1,q (K) induced by the original d ′ . Its homology is defined as: , and it is a bigraded module.
Definition 5.10 (First homology). Let K be a bicomplex. The first homology of K is the homology computed with respect to d ′ , i.e.: The first homology of K is a bigraded complex with differential d ′′ : H ′ p,q (K) −→ H ′ p,q−1 (K) induced by the original d ′′ . Its homology is defined as: , and it is a bigraded module.
Definition 5.11 (Total complex). A bicomplex K defines a single complex T = T ot(K): From the properties of d ′ and d ′′ , it follows that d 2 = 0.
We point out that T n is the sum of the modules of the secondary diagonal in diagram (28). We have that: The first filtration F ′ of T = T ot(K) is defined as: The associated spectral sequence E ′ is called first spectral sequence. Analogously we can define the second filtration and the second spectral sequence.
Proposition 5.12. Let (K, d ′ , d ′′ ) be a bicomplex with total differential d. The first spectral sequence E ′ = {(E ′r , d r )}, E ′r = p,q E ′r p,q has the property: p,q has the property: . If the first filtration is bounded below and convergent above, then the first spectral sequence converges to the homology of T with respect to the total differential d.
If the second filtration is bounded below and convergent above, then the second spectral sequence converges to the homology of T with respect to the total differential d.

Homology
In this section we compute the homology of the complexes in Figure 1. The main result is the following theorem.  Figure  1 and constructed as in Remark 3.7. Then Im ∇ is an irreducible g−submodule of M ( µ 1 , µ 2 , µ 3 , µ 4 ).
The aim of this section is to prove Theorem 6.1. Following [17], we consider the following filtration on U (g <0 ): for all i ≥ 0, F i U (g <0 ) is defined as the subspace of U (g <0 ) spanned by elements with at most i terms of g <0 . Namely: (14), it follows that ∇F i M X ⊂ F i+1 M X . Therefore M X is a filtered complex with the bigrading induced by (10) and differential ∇; moreover M X = ∪ i F i M X and F −1 M X = 0. Hence we can apply Propositions 5.3 and 5.6 to our complex (M X , ∇) and obtain a spectral sequence ( Thus we start by studying the homology of Gr M X . Remark 6.4. We observe that g contains a copy of W (1, 0) = p(t)∂ t via the injective Lie superalgebras morphism: In particular, we point out that g −2 is contained in this copy of W (1, 0).
We consider the standard filtration on Proof. We point out that L W j ⊆ k≥j g 2k , since p(t)∂ t ∈ L W deg(p(t))−1 corresponds to p(t) 2 ∈ g and deg( p(t) 2 ) = 2 deg(p(t)) − 2. Let us fix j and show the thesis by induction on i. It is clear that We now suppose that the thesis holds for i. Let w j ∈ L W j and u 1 u 2 ...u r ⊗v ∈ F i+1 M X , with r ≤ i+1 and u 1 , u 2 , ..., u r ∈ g <0 . We moreover suppose that, for some N , u s = Θ for all s ≤ N and u s ∈ g −1 for all s > N . We obtain that: ..u r ) ≥ 2j −1−r +1 and, by our assumption, u 2 , ..., u r ∈ g −1 . Therefore [w j , By (29) and the fact that W (1, 0) ∼ = Gr W (1, 0), we have that the action of W (1, 0) on M X descends on Gr M X .
By the Poincaré−Birkhoff−Witt Theorem, Gr U (g <0 ) ∼ = S(g −2 )⊗ (g −1 ); indeed, in U (g <0 ), η 2 i = Θ for all i ∈ {1, 2, 3, 4}. Therefore: We define: (29), it follows that L W 1 = L W 1 annihilates G X := (g −1 ) ⊗ V X . Therefore, as W−modules: We observe that Gr M X is a complex with the morphism induced by ∇, that we still call ∇. Indeed ∇F i M X ⊂ F i+1 M X for all i and therefore it is well defined the induced morphism that has the same expression as ∇ defined in (14), except that the multiplication by the w's must be seen as multiplication in Gr U (g <0 ) instead of U (g <0 ). Thus (G X , ∇) is a subcomplex of (Gr M X , ∇): indeed it is sufficient to restrict ∇ to G X ; the complex (Gr M X , ∇) is obtained from (G X , ∇) extending the coefficients to S(g −2 ). We point out that also the homology spaces H m,n (G X ) are annihilated by L W 1 . Therefore, as W−modules: By (30) and Propositions 5.3, 5.6, we obtain the following result. 6.1. Homology of the complexes G X . Motivated by Proposition 6.6, in this subsection we study the homology of the complexes G X 's. We will use arguments similar to the one used in [17] for E (3,6). We point out that, in order to compute the homology of the M X 's, we will compute the homology of the G X 's for X = A, C, D and we will use arguments of conformal duality, in particular Remark 3.8 and Proposition 2.19, for the case X = B. We denote by G X ′ := (g −1 ) ⊗ V X ′ . Let us consider the evaluation maps from V X to V X ′ that map y 1 , y 2 , ∂ y 1 , ∂ y 2 to zero and are the identity on all the other elements. We consider the corresponding evaluation maps from G X to G X ′ . We can compose these maps with ∇ 2 when X = A, B and obtain new maps, that we still call ∇ 2 , from G A to G D ′ and from G B to G C ′ respectively. We consider also the map from G A ′ to G D (resp. from G B ′ to G C ) that is the composition of we will call also this composition ∇ 2 . We let: Remark 6.7. The map ∇ is still defined on G X • since ∇∇ 2 = ∇ 2 ∇ = 0. The bigrading (10) induces a bigrading also on the G X • 's. We point out that G m,n A = G m,n A • for n > 0, G m,n D = G m,n D • for n < 0, G m,n B = G m,n B • for n > 0 and G m,n C = G m,n C • for n < 0. The complexes (G X • , ∇) start or end at the axes of Figure 1; thus for m, n ∈ Z: Motivated by Remark 6.8 and Proposition 6.6, we study the homology of the complexes G X • 's. We therefore introduce an additional bigrading as follows: The definition can be extended also to G X • . We point out that this new bigrading is related to the bigrading (10) by the equation p + q = n. We have that d ′ := ∆ + ∂ y 1 : We point out that i + and i − are isomorphic as x 1 ∂ x 1 − x 2 ∂ x 2 , x 1 ∂ x 2 , x 2 ∂ x 1 −modules; therefore, in the following, we will often write i when we are speaking of the x 1 ∂ x 1 − x 2 ∂ x 2 , x 1 ∂ x 2 , x 2 ∂ x 1 − module isomorphic to i + and i − . We introduce the following notation, for all a, b ∈ Z and p, q ≥ 0: , and for all a, b ∈ Z, p, q ≤ 0: . From now on we will use the notation i ). We point out that, as We observe that ∇ : G X (a, b) → G X (a, b) (resp. ∇ : G X • (a, b) → G X • (a, b)) and therefore G X (a, b) (resp. G X • (a, b)) is a subcomplex of G X (resp. G X • ); the G X (a, b)'s and G X • (a, b)'s are bicomplexes, with the bigrading induced by (31) and differentials d ′ = ∆ + ∂ y 1 and d ′′ = ∆ − ∂ y 2 . The computation of homology spaces of G X and G X • can be reduced to the computation for G X (a, b) and G X • (a, b). In the following lemmas we compute the homology of the G X • (a, b)'s. We start with the homology of the G X • (a, b)'s when either a or b do not lie in {0, 1, 2}. To prove the following results we will use Proposition 5.12. Lemma 6.9. Let us suppose that a > 2 or b > 2. Let k = max(a, b).
Then as Let us suppose that a < 0 or b < 0. Let k = min(a, b). Then, as Proof. We first observe that if a > 2 or b > 2 (resp. a < 0 or b < 0), then G X • (a, b) = G X (a, b) for X = A (resp. X = C, D); indeed they differ only when p + q = 0, that does not occur in this case. We prove the thesis in the case b > 2 for X = A and b < 0 for X = C, D; the case a > 2 for X = A and a < 0 for X = C, D can be proved analogously using the second spectral sequence instead of the first one.
This complex is the tensor product of a−p + y p 1 and the following complex, since a−p + y p 1 is not involved by d ′′ : We now show that this complex is exact except for the right end, in which the homology space is is zero if and only if p is constant. Hence the kernel is Cy b 2 . ii: Let us consider the map ∆ − ∂ y 2 : Hence an element of the kernel is such that: Thus at this point the sequence is exact.
Since the original complex was the tensor product with a−p + y p 1 , we have that the non zero homology space is a−p + y p 1 y b 2 and E ′ 1 • (a, b)) survives only for q = b. Now we should compute its homology with respect to d ′ , but the E We observe that the first filtration (F ′ p (G A (a, b))) n = h≤p (G A (a, b)) [h,n−h] is bounded below, since F ′ −1 = 0, and it is convergent above. Therefore by Proposition 5.12: Since there are no x 1 's and x 2 's involved, this means that H m,n (G A (a, b)) = 0 if m = 0 and H 0,n (G A (a, b)) = a+b−n Case D) In the case of G D (a, b), using the same argument, when b < 0 we obtain: Therefore: Since there are no x 1 's and x 2 's involved, this means that H m,n (G D (a, b)) = 0 if m = 0 and H 0,n (G D (a, b)) = a+b−n Case C) In the case of G C (a, b) when b < 0 we have the following complex with the differential This complex is the tensor product of a−p + ∂ −p y 1 and the following complex, since a−p + ∂ −p y 1 is not involved by d ′′ : We now show that this complex is exact except for the left end, in which the homology space is . Indeed: . We compute the kernel.
. We compute the kernel. Let Then (in particular p 2 has positive degree in ∂ x 2 ). Therefore an element of the kernel is such that: Thus at this point the sequence is exact.
iii: Let us consider the map If p has positive degree in ∂ x 1 , then: If p has positive degree in ∂ x 2 , then: If p is constant, it does not belong to the image of ∆ − ∂ y 2 . Therefore the homology space is isomorphic to Since the original complex was the tensor product with a−p + ∂ −p y 1 , then the non zero homology space is a−p p,q because the map d ′ is 0 on the E ′ 1 p,q 's (the image of the map d ′ always involves elements of positive degree in ∂ x 1 or ∂ x 2 that are 0 in E ′ 1 p,q for the previous computations). Since we have a one row spectral sequence, then E ′ 2 = ... = E ′ ∞ . Therefore: We observe that the first filtration (F ′ p (G C (a, b))) n = h≤p (G C (a, b)) [h,n−h] is bounded below, since F ′ n−1 = 0, and it is convergent above. Therefore by Proposition 5.12: Since there are no ∂ x 1 's and ∂ x 2 's involved, this means that H m,n (G C (a, b)) = 0 if m = 0 and H 0,n (G C (a, b)) = a+b−n−2 In Lemma 6.9 we computed the homology of the G X • (a, b)'s in the case that either a or b do not belong to {0, 1, 2}. In order to compute the homology of the G X • (a, b)'s in the case that both a and b belong to {0, 1, 2}, we need the following remark and lemmas. Remark 6.10. We introduce some notation that will be used in the following lemmas. Let 0 < b ≤ 2. Let us define: We have an isomorphism of bicomplexes γ : q] which is the valuating map that values y 1 and y 2 in 1 and is the identity on all the other elements. We consider on G A (a, b) the differentials d ′ = ∆ + and d ′′ = ∆ − induced by ∆ + ∂ y 1 and ∆ − ∂ y 2 for G A (a, b). We also define: The following is a commutative diagram: a, b). a, b)) is isomorphic, as a bicomplex, to G A • (a, b). Its diagram is the same of G A (a, b) except for p = q = 0. The diagram of G A (a, b) is the following, respectively for a = 0, a = 1, a ≥ 2: where the horizontal maps are d ′ and the vertical maps are d ′′ . The diagram of G A • (a, b) is analogous to this, except for p = q = 0, where a , that we shortly call Ker(∆ − ∆ + ) in the next diagram. The E ′1 spectral sequence of G A • (a, b), i.e. the homology with respect to ∆ − , is the following, respectively for a = 0, a = 1, a ≥ 2, b = 1 and a = 0, a = 1, a ≥ 2, b = 2 (the computation is analogous to Lemma 6.9): We have that, in the diagram of the E ′1 spectral sequence, only the rows for q = 0 and q = b are different from 0. The previous diagram will be the first step in Lemma 6.13 for the computation of the homology of the G A • (a, b)'s when a, b ∈ {0, 1, 2}. Analogously we define, for 0 ≤ b < 2: We have an isomorphism of bicomplexes γ : q] which is the valuating map that values ∂ y 1 and ∂ y 2 in 1 and is the identity on all the other elements. We consider on G C (a, b) the differentials d ′ = ∆ + and d ′′ = ∆ − induced by ∆ + ∂ y 1 and ∆ − ∂ y 2 for G C (a, b). We also define: We have the following commutative diagram: a, b). a, b)) is isomorphic, as a bicomplex, to G C • (a, b). Its diagram is the same of G C (a, b) except for p = q = 0. In the following diagram we shortly write CoKer(∆ − ∆ + ) for: The diagram of the bicomplex G C • (a, b) is the following, respectively for a = 2, a = 1 and a ≤ 0: · · · · · · · · · 2 + 2 where the horizontal maps are d ′ and the vertical maps are d ′′ .
The E ′1 spectral sequence of G C • (a, b) is the following, respectively for a = 2, a = 1, a ≤ 0, b = 1 and a = 2, a = 1, a ≤ 0, b = 0 (the computation is analogous to Lemma 6.9): We have that only the rows q = 0 and q = b − 2 are different from 0. We point out that, since b < 2: The isomorphism holds because b < 2 and we know, by Lemma 6.9, that is exact except for the right end. The previous diagram will be the first step in Lemma 6.13 for the computation of the homology of the G C • (a, b)'s when a, b ∈ {0, 1, 2}.
Proof. We first focus on 0 < b ≤ 2. In order to make the proof more clear, we show the statement for b = 1 that is more significant; the proof for b = 2 is analogous. We observe that, due to the definition of S(a, 1), H i (S(a, 1)) = H i (S(a + 1, 1)) for 0 ≤ i ≤ a. Hence it is sufficient to compute it for large a. We take a > 2; for sake of simplicity, we choose a = 3. The complex S(3, 1) reduces to: ). In this case the thesis reduces to show that: (S(3, 1)) ∼ = 0, H 2 (S(3, 1)) ∼ = 0, H 1 (S(3, 1)) ∼ = 0, H 0 (S(3, 1)) ∼ = 2 + . We point out that the complex S(3, 1) is isomorphic, via ∆ − , to the complex: that is exactly the row for q = 0 in the diagram of the E ′1 spectral sequence of G A • (3, 1) in Remark 6.10. In particular, since a = 3, this is the row for q = 0 and values of p respectively 0, 1, 2 and 3 from the left to the right. The isomorphism of the two complexes follows from b = 1 > 0 and the fact that, by Lemma 6.9, we know that is exact except for the left end.
Since E ′2 (G A • (3, 1)) has two nonzero rows for q = 0 and q = 1 (see the diagram in Remark 6.10), then the differentials d r p,q are all zero except for r = b + 1 = 2, q = 0, 1 < p ≤ 3. Indeed 1 < p ≤ 3 follows from the fact that: From the fact that the homology spaces of G A • (3, 1) and G A • (3, 1) are isomorphic and from Lemma 6.9, it follows that: By (32) we obtain that d 2 p,0 , for 1 < p ≤ 3, must be an isomorphism. Indeed, let us first show that d 2 p,0 , for 1 < p ≤ 3, is surjective. We point out that: using an argument similar to Lemma 6.9. By (32), we know that for n = p − 1 < 3: Moreover d r = 0 for r > 2 and d 2 p−2,1 = 0. Therefore d 2 p,0 must be surjective. Let us see that d 2 p,0 is injective. If p < 3, then E ′∞ p,0 ( G A • (3, 1)) = 0 since it appears in the sum by (32). Moreover is identically 0. Hence Ker(d 2 p,0 ) = 0. If p = 3, we know, by (32), that and E ′∞ p,0 ( G A • (3, 1)) appears in this sum. Moreover we know that since d r = 0, when r > 2, d 2 4,0 = d 2 2,1 = 0 and E ′2 2,1 ( G A • (3, 1)) ∼ = 1 + due to an argument similar to Lemma 6.9. The space E ′∞ 2,1 ( G A • (3, 1)) also appears in the sum (35) and therefore we conclude that E ′∞ p,0 ( G A • (3, 1)) = 0. Since d 2 p+2,−1 , given by (34), is identically 0, therefore Ker(d 2 p,0 ) = 0. Thus, by the fact that d 2 p,0 is an isomorphism, we obtain that E ′2 p,0 ( G A • (3, 1)) ∼ = 5−p + . Hence: . We now prove the statement in the case b = 0. Due to the definition of S(a, 0), H i (S(a, 0)) = H i (S(a + 1, 0)) for 0 ≤ i ≤ a. Hence it is sufficient to compute it for a = 2. The complex S(2, 0) reduces to: In this case the thesis reduces to show that: We compute the homology spaces by direct computations. Let us compute H 0 (S(2, 0)). We take ) has the following form: Hence: Therefore P lies in the kernel if and only if ∂ 2 Let us now compute H 1 (S(2, 0)). We take w 11 p(x 1 , x 2 ) + w 21 q(x 1 , ) has the following form: Hence: Therefore P lies in the kernel if and only if: . We obtain that: where Q 1 (x 1 ) (resp. Q 2 (x 2 )) is a polynomial expression costant in x 2 (resp. costant in x 1 ). Therefore, if P lies in the kernel then ∂ x 1 q = ∂ x 2 p + α, with α ∈ C. Let us consider an element of the kernel, we obtain that: We point out that w 21 w 12 ⊗ α does not lie in the image of the map ∆ − ( 0 Finally, let us compute H 2 (S(2, 0)). We take w 11 w 21 p(x 1 , ) has the following form: We point out that: Therefore every element of ∆ − ( 2 . Thus H 0 (S(2, 0)) ∼ = 0. Lemma 6.12. Let 0 ≤ b < 2. Let us consider the complex T (a, b) defined as follows: The homology spaces of the complex T (a, b), from left to right, are respectively isomorphic to: Proof. We first point out that the statement is obvious for b = 0 since in this case the complex is trivial and the homology spaces are obviously trivial. We now focus on b = 1. The complex T (a, b), due to its construction, has the property that H i (T (a, b)) = H i (T (a − 1, b)) for a ≤ i ≤ 2; then we can compute the homology for small a. Let us take a < 0. For sake of simplicity we focus on a = −1. The complex T (−1, 1) reduces to: The thesis reduces to show that: In order to prove the thesis, we use that the complex T (−1, 1) is isomorphic, via ∆ − , to the row for q = 0 in the diagram of the E ′1 spectral sequence of G C • (−1, 1) in Remark 6.10, that is: where we shortly write Ker(∆ − ) i,j for: We point out that in this case, the spaces Ker(∆ − ) 2,1 , Ker(∆ − ) 1,1 and Ker(∆ − ) 0,1 correspond respectively to the valus of p = −3, −2, −1 and q = 0 in the diagram of the E ′1 spectral sequence of G C • (−1, 1) (see Remark 6.10). The isomorphism between the two complexes follows from b = 1 < 2 and the fact that, by Lemma 6.9, we know that is exact except for the right end. In this case the complex E ′1 of G C • (−1, 1) has two nonzero rows, for q = 0 and q = b − 2 = −1, and therefore the differentials d r p,q are all zero except for r = 2, q = −1 and −2 < p ≤ 0. Indeed: We know, by Lemma 6.9, that: By (38) we obtain that d r p,q for r = 2, q = −1 and −2 < p ≤ 0 must be an isomorphism. Indeed, let us first show that d 2 p,q for q = −1 and −2 < p ≤ 0 is injective. We point out that: using an argument similar to Lemma 6.9. We know, by (38), that for n = p − 1 > −3: Hence E ′∞ p,−1 ( G C • (−1, 1)) = 0. Moreover d r = 0 for r > 2 and d 2 p+2,−2 = 0 since its domain is 0. Therefore d 2 p,−1 must be injective. Let us show that d 2 p,−1 is surjective. If p − 2 > −3, then E ′ ∞ p−2,0 ( G C • (−1, 1)) appears in the sum is identically 0 because the codomain is 0. Hence d 2 p,−1 must be surjective.
by (38). We know that E 1)) ∼ = 1 + due to an argument similar to Lemma 6.9. Since E ′ ∞ −2,−1 ( G C • (−1, 1)) also appears in the sum (41), we conclude that E ′ ∞ p−2,0 ( G C • (−1, 1)) = 0. Since d 2 p−2,0 , given by (40), is identically 0, therefore d 2 p,−1 must be surjective. Hence, by the fact that d 2 p,−1 is an isomorphism, we obtain that E and we obtain that: Now using Remark 6.10 and Lemmas 6.11, 6.12, we are able to compute the homology of the G X • (a, b)'s when a, b ∈ {0, 1, 2}. otherwise; Proof. We prove the statement in the case 0 ≤ a ≤ b ≤ 2 for X = A, D and 0 ≤ b ≤ a ≤ 2 for X = C using the theory of spectral sequences for bicomplexes; the case 0 ≤ b ≤ a ≤ 2 for X = A, D and 0 ≤ a ≤ b ≤ 2 for X = C can be proved analogously using the second spectral sequence instead of the first.
Case A) Let us first consider G A • (0, 0) = Ker(∇ 2 : 0 In this case the statement is straightforward. Indeed by a = b = 0 we deduce that p = q = 0. Therefore G m,n A • (0, 0) = 0 when n = 0, G 1,0 We therefore assume b > 0. As in Remark 6.10 we consider: We consider on this space the differentials d ′ = ∆ + and d ′′ = ∆ − induced by ∆ + ∂ y 1 and ∆ − ∂ y 2 for G A (a, b). As in Remark 6.10, the E ′1 spectral sequence of G A • (a, b), i.e. the homology with respect to ∆ − , is represented in following diagram: .

· · ·
We have that only the rows for q = 0 and q = b are different from 0. We observe that d ′ is 0 on the row q = b. Moreover d r p,q is 0 for r ≥ 2 because either the domain or the codomain of these maps are 0, since a ≤ b. Therefore E ′2 = ... = E ′∞ . We need to compute E ′2 for the row q = 0; for this computation we apply Lemma 6.11. We point out that the isomorphism in (33) of Lemma 6.11 was induced by ∇, that decreases the degree in is formed by elements with representatives of degree 1 in Indeed in this sum there is not the possibility (p, q) = (p, 0) with p ≤ a < b. We have that H 0,n ( G A • (a, b)) ∼ = a+b−n + , if n ≥ b > a. For n = a = b the result follows similarly. Case D) We define: We have an isomorphism of bicomplexes γ : G D (a, b) [p,q] −→ G D (a, b) [p,q] which is the valuating map that values ∂ y 1 and ∂ y 2 in 1 and is the identity on all the other elements. We consider on G D (a, b) the differentials d ′ = ∆ + and d ′′ = ∆ − induced by ∆ + ∂ y 1 and ∆ − ∂ y 2 for G D (a, b). We also define: We have the following commutative diagram: a, b). a, b)) is isomorphic, as a bicomplex, to G D • . Its diagram is the same of G D except for p = q = 0 (upper right point in the following diagram), where instead of a . Moreover we observe that G A ′ (0, 0) = 0, then G D • (0, 0) = G D (0, 0) and we can use the same argument of Lemma 6.9. We now assume b > 0. The diagram of G D (a, b) is the following: where the horizontal maps are d ′ and the vertical maps are d ′′ . In the following diagram we shortly write Ker(∆ − ) Im(∆ − ∆ + ) for the space: , and Ker(∆ − ) i,j for: The E ′1 spectral sequence of G D • (a, b) is (the computation is analogous to Lemma 6.9): · · · 0 0 0. 0 We observe that, since b > 0: The non zero row of the previous diagram is isomorphic, via ∆ − , to the following complex: The fact that the two complexes are isomorphic follows from b > 0 and that, by Lemma 6.9, the sequence is exact except for the left end. We observe that we can compute the homology of the complex (43) using the homology of S(2, b − 1) given by Lemma 6.11. Indeed (43) is different from S(2, b − 1) only at the right end, because the left end of (43) is ∆ − ( 2 ) that is the left end of S(2, b − 1). The homology at the right end of (43) is: (S(2, b − 1)).
We focus on the coefficient of ∂ x 1 ∂ y 1 in ∇v and z, that should be the same. We get a 1 Θw 11 = iw 11 w 21 w 12 that is impossible. ⊗ H −1,−1 (G C ) is the first step of the spectral sequence; this holds a contradiction using (4). By Proposition 6.14, we know that H −1,−1 (G C ) is one−dimensional. By the previous properties 0 = [z] ∈ H −1,−1 (G C ); hence [z] is a basis for the g 0 −module H −1,−1 (G C ).  Proof. It is a straightforward computation that x 1 ∂ x 2 .k = y 1 ∂ y 2 .k = 0. We point out that ∇k is a cycle in Gr M C since k ∈ F 4 M C and ∇k ∈ F 4 M C . Indeed in M C , by direct computations, ∇k = Θz. Moreover [k] lies in H 0,0 (G C ) since the terms of k that include Θ are in F 3 M , the other is in F 4 M C . By Proposition 6.14, we know that H 0,0 (G C ) is one−dimensional. By the previous computations 0 = [k] ∈ H 0,0 (G C ); hence [k] is a basis for the g 0 −module H 0,0 (G C ). We have also that ∇[k] = Θ[z].
Proof of Proposition 6.17. By (30) and Lemmas 6.18, 6.20 we know that as W−modules Finally we focus on the two remaining cases for M A . A / Im ∇ ∼ = C. Indeed Im ∇ is the g−module generated by the singular vector w 11 ⊗ 1 and we have that: x 2 ∂ x 1 .(w 11 ⊗ 1) = w 21 ⊗ 1, y 2 ∂ y 1 .(w 11 ⊗ 1) = w 12 ⊗ 1, In order to prove Proposition 6.21 we need the following lemma.
Proof of Proposition 6.21. By (30), Remark 6.22 and Lemma 6.23, we know that as W−modules Remark 6.24. We point out that for C = 0, the study of finite irreducible modules over K ′ 4 reduces to the study of finite irreducible modules over K 4 , already studied in [2]. In particular, for C = 0, the diagram of maps between finite Verma modules reduces to the diagonal m = n in the quadrants A and C of Figure 1. For K 4 the homology had been already computed in [2, Propositions 6.2, 6.4] using de Rham complexes. Propositions 6.17 and 6.21 are coherent with the results of [2, Propositions 6.2, 6.4] for K 4 .
Proof of Theorem 6.1. The proof follows combining the results of Propositions 6.15, 6.16, 6.17 and 6.21.

Size
The aim of this section is to compute the size of the irreducible quotients I(m, n, µ t , µ C ), where (m, n, µ t , µ C ) occurs among the weights in Theorems 3.9, 3.10, 3.11. This computation is an application of Theorem 6.1. For a S(g −2 )−module V , we define its size as (see [17]): size(V ) = 1 4 rk S(g −2 ) V.
Remark 7.2. We point out that it is sufficient to compute the size for modules I(m, n, − m+n 2 , m−n 2 ) of type A and I(m, n, 1 + n−m 2 , 1 + m+n 2 ) of type D and use conformal duality, since conformal dual modules have the same size. Let us show that the module I(m, n, m+n 2 + 2, n−m 2 ) of type C is the conformal dual of I(m + 1, n + 1, − m+n+2 2 , m−n 2 ) of type A, , when (m, n) = (0, 0). Indeed, by Remarks 3.7 and 3.8, we have the following dual maps, for m, n ≥ 0: , m−n 2 ). Using the same argument, it is possible to show that the module I(m, n, 1 + m−n 2 , −1 − m+n 2 ) of type B is the conformal dual of I(m + 1, n − 1, 1 + n−m−2 2 , 1 + m+n 2 ) of type D. 7.1. The character. We now introduce the notion of character, that will be used for the computation of the size. Let s be an indeterminate. We define the character of a g−module V , following [17], as: The character is a Laurent series in the indeterminate s; the coefficient of s k is the dimension of the eigenspace of V of eigenvalue k with respect to the action of −t ∈ g 0 .