Classification of degenerate Verma modules for E(5,10)

Given a Lie superalgebra $\frak g$ with a subalgebra $\frak g_{\geq 0}$, and a finite-dimensional irreducible $\frak g_{\geq 0}$-module $F$, the induced $\frak g$-module $M(F)=U({\frak g}) \otimes_{U(\frak g_{\geq 0})} F $ is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra ${\frak g}=E(5,10)$ with the subalgebra $\frak g_{\geq 0}$ of minimal codimension. This is done via classification of all singular vectors in the modules $M(F)$. Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as 'exceptional' de Rham complexes for $E(5,10)$.


Introduction
Recall that a linearly compact Lie (super)algebra g is defined by the property that, viewed as a vector space, g is linearly compact. According to E. Cartan's classification, the list of infinite-dimensional simple linearly compact Lie algebras consists of four Lie-Cartan series: W n , S n , H n , and K n .
Let F be an irreducible finite-dimensional g 0 -module, extend it to g ≥0 by letting all g j with j > 0 act by 0, and consider the finite Verma g-module where M(F ) is viewed as a vector space with discrete topology. These modules are especially interesting since their topological duals are linearly compact.
The first problem of representation theory of linearly compact Lie superalgebras is to classify their degenerate (i.e., non-irreducible) finite Verma modules and morphisms between them. This is equivalent to classification of singular vectors in these modules, i.e., those which are annihilated by g j with j ≥ 1. This problem was solved for Lie algebras W n , S n and H n by Rudakov [11], [12]. In particular, he showed that the degenerate finite W n -modules form the de Rham complex in a formal neighborhood of 0 in C n (rather its topological dual).
In a series of papers [7], [8], [10] this problem was solved for the exceptional linearly compact Lie superalgebra E (3,6). It turned out that all the morphisms between the degenerate finite Verma modules over E (3,6) can be arranged in an infinite number of complexes, and cohomology of these complexes was computed in [8] as well. The most difficult technical part of this work is [10], where all singular vectors have been classified.
In the subsequent paper [9] a solution to this problem was announced for E (3,8), and a conjecture on classification of degenerate finite Verma modules for E(5, 10) was posed, motivated by the singular vectors of degree 1 constructed there (the degree on M(F ) = U(g <0 ) ⊗ F is induced by the degree on g <0 = ⊕ j<0 g j ). In a more recent paper [13] it was proved that these are all singular vectors of degree 1, and also some singular vectors of degree 2,3,4 and 5 have been constructed. In the subsequent paper [2] it was shown that the singular vectors of degree less than or equal to 3 constructed by Rudakov are all singular vectors of degree less than or equal to 3. Actually, the morphisms of degrees 2, 3 and 5 corresponding to singular vectors constructed in [13] are composition of morphisms of degree 1 and 4, and the morphisms of degree 1 and 4 can be arranged in an infinite number of infinite complexes [13]. However, in Figure 2 of [13] there are two notable gaps in the complexes.
The key discovery of the present paper is the existence of morphisms of degree 7 and 11, which fill these gaps (see Figure 4). Moreover, we show that there are no further singular vectors (Theorem 10.1), thereby proving the conjecture from [9] on classification of degenerate finite Verma modules over E (5,10).
The proof of Theorem 10.1 goes as follows. First, using a result from [12] on S n -modules for n = 5, which is the even part of E(5, 10), we show that there are no singular vectors of degree greater than 14. Next we find that for degrees between 11 and 14 there is only one singular vector, it has degree 11 and defines a morphism from M(C 5 ) to M(C 5 * ), where C 5 is the standard sl 5 -module and C 5 * its dual. After that, using the techniques of [2], we show that in degrees between 6 and 10 the only singular vector has degree 7 and it defines a morphism from M(S 2 C 5 ) to M(S 2 C 5 * ). These are precisely the two morphisms, missing in Figure 2 of [13]. Finally, we show that in degrees less than or equal to 6 there are no other singular vectors as compared to [13]. The calculations involve solution of large systems of linear equations, which are performed with the aid of computer. Note also that the construction of morphisms is facilitated by the duality, constructed in [3], such that the morphism M(F ) → M(F 1 ) induces the morphism M(F 1 ) * → M(F ) * and for E (5,10) has the property that M(F ) * = M(F * ).
We have learned recently that Daniele Brilli obtained in [1] the upper bound 12 on the degrees of singular vectors for finite Verma modules over E (5,10), using the techniques of representation theory of Lie pseudoalgebras.
We consider the simple, linearly compact Lie superalgebra of exceptional type g = E(5, 10) whose even and odd parts are as follows: g0 consists of zero-divergence vector fields in five (even) indeterminates x 1 , . . . , x 5 , i.e., where ∂ i = ∂ x i , and g1 = Ω 2 cl consists of closed two-forms in the five indeterminates x 1 , . . . , x 5 . The bracket between a vector field and a two-form is given by the Lie derivative and for f, g ∈ C[[x 1 , . . . , where, for i, j, k, l ∈ [5], ε ijkl and t ijkl are defined as follows: if |{i, j, k, l}| = 4 we let t ijkl ∈ [5] be such that |{i, j, k, l, t ijkl }| = 5 and ε ijkl be the sign of the permutation (i, j, k, l, t ijkl ). If |{i, j, k, l}| < 4 then ε ijkl = 0.
From now on we shall denote dx i ∧ dx j simply by d ij .
The Lie superalgebra g has a consistent, irreducible, transitive Z-grading of depth 2 where, for k ∈ N, Note that g 0 ∼ = sl 5 , g −2 ∼ = (C 5 ) * , g −1 ∼ = 2 C 5 as g 0 -modules (where C 5 denotes the standard sl 5 -module). We set g − = g −2 ⊕ g −1 , g + = ⊕ j>0 g j and g ≥0 = g 0 ⊕ g + . We denote by U (resp. U − ) the universal enveloping algebra of g (resp. g − ). Note that U − is a g 0 -module with respect to the adjoint action: for x ∈ g 0 and u ∈ U − , We also point out that the Z-grading of g induces a Z-grading on the enveloping algebra U − . It is customary, though, to invert the sign of the degrees hence getting a grading over N. Note that the homogeneous component (U − ) d of degree d of U − under this grading is a g 0 -submodule.
If λ ∈ Λ is a weight, we use the following convention: for all 1 ≤ i < j ≤ 5 we let If V is a sl 5 -module and v ∈ V is a weight vector we denote by λ(v) the weight of v and by λ ij (v) = (λ(v)) ij . If λ = (a, b, c, d) ∈ Λ is a dominant weight, i.e. a, b, c, d ≥ 0, let us denote by F (λ) = F (a, b, c, d) the irreducible sl 5 -module of highest weight λ. In this paper we always think of F (a, b, c, d) as the irreducible submodule of . , x 5 } denotes the standard basis of C 5 , x ij = x i ∧ x j , and x * i and x * ij are the corresponding dual basis elements. Besides, for a weight λ = (a, b, c, d) we let λ * = (d, c, b, a), so that F (λ) * ∼ = F (λ * ).
Notice that, as a g 0 -module, g 1 ∼ = F (1, 1, 0, 0) and that x 5 d 45 is a lowest weight vector in g 1 . Moreover, for j ≥ 1, we have g j = g j 1 .

Generalized Verma modules and morphisms
We recall the definition and some properties of (generalized) Verma modules over E(5, 10), most of which hold in the generality of arbitrary Z-graded Lie superalgebras (for some detailed proofs see [2]).
Given a g 0 -module V , we extend it to a g ≥0 -module by letting g + act trivially, and define Note that M(V ) has a g-module structure by multiplication on the left, and is called the (generalized) Verma module associated to V . We also observe that M(V ) ∼ = U − ⊗ C V as g 0 -modules. If the g 0 -module V is finite-dimensional and irreducible, then we call M(V ) a finite Verma module (it is finitely-generated as a U − -module). We denote by M(λ) the finite Verma module M(F (λ)). A finite Verma module is said to be non-degenerate if it is irreducible and degenerate otherwise. (i) x i ∂ i+1 w = 0 for every i = 1, . . . , 4; (ii) zw = 0 for every z ∈ g 1 ; (iii) w does not lie in V .
We observe that the homogeneous components of positive degree of a singular vector are singular vectors. The same holds for its weight components. From now on we will thus assume that a singular vector is a homogeneous weight vector unless otherwise specified. Notice that if condition (i) is satisfied then condition (ii) holds if x 5 d 45 w = 0 since x 5 d 45 is a lowest weight vector in g 1 .
We recall that a minimal Verma module M(V ) is degenerate if and only if it contains a singular vector [2,Proposition 3.3].
Degenerate Verma modules can be described in terms of morphisms. A morphism ϕ : M(V ) → M(W ) can always be associated to an element Φ ∈ U − ⊗ Hom(V, W ) as follows: . We will say that ϕ (or Φ) is a morphism of degree d if u i ∈ (U − ) d for every i.
The following proposition characterizes morphisms between Verma modules. Proposition 3.3. [9,13] Let ϕ : M(V ) → M(W ) be the linear map associated with the element Φ ∈ U − ⊗ Hom(V, W ). Then ϕ is a morphism of g-modules if and only if the following conditions hold: We observe that, if M(V ) is a finite Verma module and condition (a) holds, it is enough to verify condition (b) for an element X generating g 1 as a g 0 -module and for v a highest weight vector in V .
We recall that a finite Verma module M(µ) contains a singular vector if and only if there exist a finite Verma module M(λ) and a morphism ϕ : M(λ) → M(µ) of positive degree [2,Proposition 3.5].
We recall the following duality on Verma modules which is established in [3] in a much wider generality.
is any basis of (U − ) d and {u * i , i ∈ I} is the corresponding dual basis. Definition 3.6. Let M(µ) be a finite Verma module and let π : M(µ) → U − ⊗ F (µ) µ be the natural projection, F (µ) µ being the weight space of F (µ) of weight µ. Given a singular vector w ∈ M(µ), we call π(w) the leading term of w.
It is shown in [2] that the leading term of a singular vector is non-zero, and therefore a singular vector is uniquely determined by its leading term.
The action of E(5, 10) on a module M restricts to an action of its even part on M. It is therefore natural to take into account the structure of M as an S 5 -module also. In order to do this we consider the grading on S 5 given by deg x i = 2 and deg(∂ i ) = −2 to be consistent with the embedding of S 5 in E(5, 10). The definition of a Verma module for S 5 is analogous to the one for E(5, 10). Rudakov classified all singular vectors for the infinite-dimensional Lie algebra S n in [12] and we recall here his results in the special case of S 5 .
Theorem 3.7. [12] The following is a complete list (up to multiplication by a scalar) of singular vectors w in Verma modules M(λ) for S 5 .
Theorem 3.7 provides the diagram of all non-zero morphisms between finite Verma modules for S 5 shown in Figure 1. ∈ Ω with i < j, then we let d p = d ij = dx i ∧ dx j . In order to avoid cumbersome notation, when no confusion may arise we will denote in this section the subset {i, j} simply as ij.
Let V be a finite dimensional g 0 -module. For all k ≥ 0 we let Note that M k (V ) is not an E(5, 10)-submodule of M(V ). Nevertheless the following result holds. Proof. It is enough to show that for all X ∈ S 5 , 1 ≤ j ≤ k, p 1 , . . . , p j ∈ Ω, and v ∈ V . We also show that and we prove that (1) and (2) hold simultaneously by a double induction on j and deg X. If j = 1 then (2) is trivial and (1) follows from (2). If deg X = −2 then (1) and (2) are both trivial, so we assume that j ≥ 2 and deg X ≥ 0. We have The latter summand clearly lies in M j (V ) by induction on j and so (1) will follow from (2). We have The former summand lies in M j (V ) by induction on j and the latter by induction on deg X: the result follows.
By Proposition 4.1 we have a filtration Proof. For all p 1 , . . . , p k ∈ Ω and every permutation σ of the indices {1, . . . , k} we have and so N k (V ) is generated as C[∂]-module by the elements d p 1 · · · d p k ⊗ v for all p 1 ≺ · · · ≺ p k and all v ∈ V . The result follows by Poincaré-Birkhoff-Witt theorem for U(g − ).
Next we observe that the subspace also has a special structure: Proof. The subspace F k (V ) of N k (V ) is an sl 5 -module since g −1 is a g 0 -module, and by the definition of N k (V ). The fact that F k (V ) is annihilated by (S 5 ) >0 follows easily by degree reasons. The second part follows from Proposition 4.2 and the first part.
This result together with Theorem 3.7 allows us to determine a first bound on the degree of singular vectors for E(5, 10). Proof. Let k be minimal such that w ∈ M k (V ). Then w is a fortiori either a highest weight vector in F k (V ) or a singular vector in the S 5 -Verma module N k (V ), and as such it has degree at most 4. It follows that w has degree at most k + 4, where k ≤ 10.

By construction we have
. . , 10 and in particular Every non-zero vector w in M(V ) can be expressed uniquely in the following form: We say in this case that w has height h and we call w ≺ h the highest term of w. Note that the height of an element does not depend on the order ≺, while its highest term does.
If w is homogeneous of degree d, then the term w ≺ j has the following form Note that if w is homogeneous of height h, then w ≺ j = 0 for all j ≡ h mod 2. Observe that, by construction, if w has height h, then and, in particular, w and w ≺ h lie in the same class in N h (V ). Theorem 3.7 provides us the following description of possible singular vectors for E(5, 10).
Corollary 5.1. Let w ∈ M(V ) be a singular vector of degree d and height h, and let w ≺ h be its highest term. Let w ≺ j be as in (4). Then one of the following applies: (ii) d = h + 2 and there exists i ∈ [5] such that is a highest weight vector for sl 5 in N h (V ) and and λ is a highest weight vector for sl 5 in N h (V ) and with λ(w) = (0, 0, 0, 1) and λ I: |I|=h d I v e 1 +e 5 ,I = (1, 0, 0, 0). Proof. This is a straightforward consequence of Theorem 3.7. We know that N h (V ) is a Verma module for S 5 and w ≺ h is annihilated by (S 5 ) >0 and by x i ∂ j for all i < j. In particular, if d = h, we have that the class of w ≺ h in N h (V ) is a genuine singular vector for S 5 : the classification of singular vectors in Theorem 3.7 then completes the proof. Note that if d = h, then the class of w ≺ h in N h (V ) is actually a highest weight vector in F h (V ), i.e. the sl 5 -module we are inducing from.
In this section we classify all possible singular vectors with degree strictly bigger than height, i.e. we treat the cases d = h + 2 and d = h + 4 in Corollary 5.1, and, in particular, we find all singular vectors of degree greater than 10. We fix the lexicographic order on The following inclusions are immediate from the definition of the action of g 0 and g 1 on M(V ): Due to (5), for X ∈ g 0 and w ∈ L h (V ), we adopt the following notation: Similarly, due to (6), for X ∈ g 1 and w ∈ L h (V ) we write: The following simple observation will be crucial in the sequel.
Remark 5.3. Let w ∈ M(V ) be a singular vector of height h. Then for all X ∈ g 1 we have Moreover, for all i = 1, 2, 3, 4 and It will be convenient to rephrase (9) in the following equivalent way: for all X ∈ g 1 we have (13) Xw Proof. By Corollary 5.1 we have By applying (13) with X = x k d kj and all k = j, we deduce that if v i,I = 0 then I must contain all pairs containing i and all pairs containing 5, and, in particular, w has height at least 7 since v 1,I = 0 for some I. If we apply (13) with X = x 1 d 23 + x 2 d 13 , we obtain −∂ 5 d 23 We deduce that, if v 1,I = 0 and I does not contain 23, then it necessarily contains 24, since all terms in the second summand do, and one can similarly show that I must contain 34 Permuting the roles of 2, 3, 4, this argument shows that I must contain at least two of the three pairs 23, 24, 34 and hence w has height at least 9. A singular vector of height 9 and degree 13 produces a morphism ϕ : is also a morphism of degree 13 and so we necessarily have λ = (1, 0, 0, 0). Therefore, if v 1,I = 0 then the weight of d I must be (0, 0, 0, 0), but one can easily check that there are no I with |I| = 9 such that λ(d I ) = (0, 0, 0, 0) (see [2, §6] for an easy way to compute the weight of the d I 's). If w has height 10 and degree 14, then by Corollary 5.1, and an argument analogous to the previous one shows that w ∈ M(1, 0, 0, 0). Now we can rule out the only left case with d = h + 4.
for some α 1 , . . . , α 5 ∈ C with α 1 = 0, and that w 8 has the following form: for some α M,I,k ∈ C. If we expand by (10) we obtain the relation Similarly, if we expand by (10) we obtain te relation and hence α 1 = 0, a contradiction.
Next target is to deal with the case d = h + 2 in Corollary 5.1. Proof. By (13) we have for all k = l This implies that, if v k,I = 0, then {k, l} ∈ I for all l = k. In particular, we immediately deduce that h ≥ 4 and , 5}} of cardinality h − 4, and similarly for I 2 , . . . , I 5 , where the vectors v j,I j have been reindexed. Now let k, l, m be distinct integers in [5] and use again (13) with the element X = x k d lm + x l d km . We obtain: Again, by (3), this implies that, if v k,I k = 0, then I k contains {l, m} (in which case the corresponding summand is zero), or it contains both pairs {l, r} and {l, s}, where {k, l, m, r, s} = [5]. It follows that I k must contain at least two pairs containing l (since if it does not contain one such pair it must contain the other two). This implies that I k contains at least four pairs, and this completes the proof that h ≥ 8.
If h = 8, by the previous argument, the two missing pairs in I k must contain the four elements distinct from k exactly once, and so the weight of ∂ k d 1k · · ·d kk · · · d 5k d I k is (0, 0, 0, 0).
We can now tackle the case of singular vectors of height 8 and degree 10.
where the sum is over all distinct i, j, k, l, m ∈ [5] such that j < k, l, and l < m (so we have exactly 15 summands). We also adopt the convention v i,lm,jk = v i,jk,lm for notational convenience. By construction, we immediately have (x i d ik ) 1 w 8 = 0 for all i = k. We will therefore consider elements in g 1 of the form which will allow us to deduce that w 8 = 0. To perform this computation efficiently we need the following notation. Let The reason for introducing this function is the following: let d(ij, kl) be the distance between the pairs ij and kl in the lexicographic order (i.e. in the graph represented in Figure 2; then one can easily check that for all i < j < k we have Let i, j, k, l, m be distinct such that i < j < k and l < m. We have By (3) and (14) we have Similarly, applying (3) and (15) we have By repeated application of Eq. (16) we obtain • v 3,12,45 = v 5,12,34 . All these equations together imply that all v i,jk,lm vanish.
We now consider the case of a singular vector w of height 9 and degree 11. In this case, as in the proof of Proposition 5.6, we can immediately deduce that w 9 must have the following form where the sum is over all distinct i, j, k with j < k (a total of 30 summands) and d ∨ jk = d Ω\{j,k} . As in the case of height 8, we can now proceed by applying all elements in g 1 of the form Lemma 5.8. If w is a singular vector of height 9 and degree 11 with highest term w 9 as in (18), then • v 1,23 = v 2,13 = v 3,12 ; • v 1,24 = v 2,14 = −v 4,12 ; • v 1,25 = v 2,15 = v 5,12 ; • v 1,34 = v 3,14 = v 4,13 ; Proof. All equalities are obtained using (3) and (13) applying elements 3,25 . All other equalities can be obtained similarly.
Thanks to Lemma 5.8 the highest term of the singular vector assumes the following form: Lemma 5.9. Let E i = x i ∂ i+1 ∈ g 0 and w be a singular vector of height 9 and degree 11 with w 9 as in (19) above. Then Proof. Recall the definition of E 0 1 from (7). By (11) we have . The result for E 1 follows. The other statements are obtained similarly. Lemma 5.9 is depicted in Figure 3, where an arrow from u i to u k labelled E j means E j .u i = u k and the absence of an arrow labelled E j coming out from u i means E j .u i = 0.
Finally, if u 1 = · · · = u 9 = 0 then u 10 = 0 is a highest weight vector, λ(w) = (0, 1, 0, 0) by Corollary 5.1 and so λ(u 10 ) = (0, 0, 0, 0). Now we show that cases (1) and (3)  Proof. In this case we have: where u is a generator of the trivial g 0 -module. By construction w 9 satisfies (9) for all X ∈ g 1 and (11) for all i. We will therefore take into account also (10) and (12) showing that there exists no w 7 which satisfies these equations. We start computing We have (20) We deduce in particular that Case (2) in Proposition 5.10 leads us to the following surprising discovery.
Theorem 5.12. The following vector is a (unique up to multiplication by a scalar) singular vector in M(0, 0, 0, 1) of degree 11, height 9, and weight (1, 0, 0, 0): Proof. We prefer to omit the long but elementary computations that show that this is indeed a singular vector. Its uniqueness follows from the fact that the term w 9 is determined up to a scalar by Lemma 5.9 and Proposition 5.10. So if w ′ is another singular vector with w 9 = w ′ 9 then w − w ′ would be a singular vector of degree 11 with height at most 7 and this would contradict Proposition 5.5.
The last possible case with d > h is ruled out by the following result.

Properties of ω I
In order to study morphisms between generalized Verma modules and to better understand their structure as g 0 -modules, a particular basis of U − has been introduced in [2]. The main goal of this section is to show that this basis is also extremely useful when considering the action of the whole g on a Verma module. We recall some technical notation needed to give an explicit definition of such a basis. We refer the reader to [2, §5] for further details.
We recall that (U − ) d denotes the homogeneous component of U − of degree d. We let I d = {I = (I 1 , . . . , I d ) : I l = (i l , j l ) with 1 ≤ i l , j l ≤ 5 for every l = 1, . . . , d}.
If I = (I 1 , . . . , I d ) ∈ I d we let d I = d I 1 · · · d I d ∈ (U − ) d , with d I l = d i l j l . Note that this notation is slightly different from the one adopted in Sections 4 and 5.
We let S d be the set of subsets of [d] of cardinality 2, so that |S d | = d 2 . Note that elements in I d are ordered tuples of ordered pairs, while elements in S d are unordered tuples of unordered pairs.
If {k, l} ∈ S d and I ∈ I d we let t I k ,I l = t i k ,j k ,i l ,j l and ε I k ,I l = ε i k ,j k ,i l ,j l . We also let D {k,l} (I) = 1 2 (−1) l+k ε I k ,I l ∂ t I k ,I l ∈ (U − ) 2 . if r ≥ 2 and D ∅ (I) = 1 (note that the order of multiplication is irrelevant as the elements D S j (I) commute among themselves). Definition 6.4. For I = (I 1 , . . . , I d ) ∈ I d and S = {S 1 , . . . , S r } ∈ SIF d we let C S (I) ∈ I d−2r be obtained from I by removing all I j such that j ∈ S k for some k ∈ [r]. Definition 6.5. For all I ∈ I d we let ). The main properties of the elements ω I have been obtained in [2, Proposition 5.6 and Theorem 5.8] and can be summarized in the following result. We will also need the following very useful notation. Let I ∈ I d and J ∈ I c , with c ≤ d. If there exists K ∈ I d−c such that x I = x J ∧ x K = 0 we let ω I\J = ω K , and we let ω I\J = 0 if such K does not exist. Note that this notation is well-defined also thanks to Proposition 6.6.
Instead of the explicit definition of the elements ω I given in Definition 6.5, we will need some (equivalent) recursive properties that they satisfy. Lemma 6.7. Let I = (I 1 , . . . , I d ). Then for all k > 1 we have Proof. We prove the first statement for all I ∈ I d by induction on k. If k = 2 we have D {1,2} (I) = − 1 2 ε I 1 ,I 2 ∂ t I 1 ,I 2 and so, letting J = (J 1 , . . . , J d−2 ) = (I 3 , . . . , I d ) we have If k > 2 we let J = (I 1 , . . . , I k−2 , I k , I k−1 , I k+1 , . . . , I d ) be obtained from I by swapping I k and I k−1 . We also observe that swapping k with k − 1 provides a bijection S → S ′ between elements in SIF d containing {1, k} and elements in SIF d containing {1, k − 1}; we also observe that by this bijection we have d C Equation (21) now follows from the first part observing that the first summand in the right-hand side of (21) is just The following result is probably the easiest way to handle and compute the elements ω I in a recursive way. Then Proof. By (21) and Proposition 6.6 we have since, clearly, ω I\(I j ,I k ) = −ω I\(I k ,I j ) for all k = j.
The following is an immediate consequence which is not needed in the sequel but sheds more light on the symmetric nature of the elements ω I 's. Corollary 6.9. We have Proof. We proceed by induction on d, the result being trivial for d = 1.
We have We now reformulate (21) in a way which is more suitable for our next arguments.
Lemma 6.10. Let I ∈ I d and let {i, j, r, s, t} = {1, 2, 3, 4, 5} be such that ε ij,rs = ε ij,st = ε ij,tr = 1. Then Next target is to study the commutator between an element in g 1 of the form x p d pq and a generic element ω I . In order to simplify the reading of the arguments we prefer to show the proof explicitly in the special case x 5 d 45 .
Proof. We first notice that in the left-hand side we have nonzero contributions only for those k such that I k = 12, 23, 31 (up to order). We compute the contribution of I k = 12, the others will be similar. We have ∂ α ω I\(I j ,αβ,βγ,γ4) .
Using Proposition 6.8 and the inductive hypothesis we have: We split this formula into three parts (according to the last three lines above): the first part is The third part is In order to compute the second part we notice, using Lemma 6.11, that the following holds: , ω I\I k + 3∂ 4 ω I\(12,23,31) − 3 2 (α,β,γ)∈S 3 ∂ α ω I\(αβ,βγ,γ4) .
Therefore the whole second part is The sum of the three parts gives the result.
One can analogously prove the following result. Then where S(a, b, c) denotes the set of permutations of {a, b, c}.

The fundamental equations
We are now going to use Theorem 6.13 to study possible morphisms of finite Verma modules ϕ : M(V ) → M(W ). Let ∼ be the equivalence relation on I d such that I ∼ I ′ if and only if ω I = ±ω I ′ , i.e. if I ′ can be obtained from I by permuting the pairs in I and the elements in each pair. By Proposition 3.3, Remark 3.5 and Proposition 6.6 we know that if a morphism has degree d then it can be expressed in the following way where the θ r 1 ,...,r l I : V → W are such that the map given by is a (well-defined) morphism of g 0 -modules. This fact permits us to easily compute the action of g 0 on the morphisms θ r 1 ,...,r l I 's. For example we have In order to prove the second equation we proceed in a similar way.
We can therefore compute where we have used Lemma 7.1.  (22). Then Proof. By Lemma 6.10, Lemma 7.1 and (24) we have The following result is fundamental in our study.
If p < γ we have  Note that all Equations appearing in Corollary 7.6 do not depend on the weights λ and µ: this observation will be the keypoint of our final classification.

Singular vectors of degree between 5 and 10
If w ∈ M(µ) is a singular vector of degree d at most 10 we know that it also has height d by the results in Section 4. In particular we can express it as where s is a highest weight vector in F (λ). Proof. Since w has height d, we know that there exists I ∈ I d such that θ I (s) = 0. Among all such I's choose I 0 such that θ I 0 (s) has maximal weight. Applying x i ∂ i+1 to w we obtain a term ω I 0 ⊗ x i ∂ i+1 .θ I 0 (s), which cannot be simplified by any other term. Therefore x i ∂ i+1 .θ I 0 (s) = 0.
If we fix any possible I 0 as in Lemma 8.1, we can consider all equations in Corollary 7.6 with weight µ and we observe that these equations do not depend on µ. For example, if we choose I 0 = (12, 24, 34, 45), we can consider Equation (33)  Proof. The proof is based on Corollary 7.6. The set of Equations (33)-(36) of weight λ(θ I 0 (s)) provides a system of homogeneous linear equations among all θ I (s), θ r J (s) and θ r 1 ,r 2 K (s) (with I ∈ I d , J ∈ I d−2 and K ∈ I d−4 ) such that θ I , θ r J and θ r 1 ,r 2 K have the same weight of θ I 0 , and which do not depend on (the weight of) s. This system can be solved with the help of a computer in all possible cases and one can check that it implies θ I 0 (s) = 0 in all cases, but in the three exceptions stated above.
We add a few words to explain what happens in the most complicated case, i.e., d = 10 and I 0 = (12,13,14,15,23,24,25,34,35,45). In this case 86 variables are involved: θ I 0 (s), 15 vectors of the form θ r J (s), and 70 vectors of the form θ r 1 ,r 2 K (s) with r 1 = r 2 . Equations (33) and (35) do not provide any condition. In Equation (34) we can choose (a, b, c, p, q) to be any permutation in S 5 and K = (ac, ap, aq, bp, bq, cp, pq), getting 120 linear equations among our 86 variables, and in Equation (36) we can choose (a, b, c, p, q) to be any permutation in S 5 (with a < b < c to avoid repeated equations) and K = (ap, aq, bp, bq, cp, cq, pq) getting 20 more equations. This system of 140 equations implies that all 86 variables involved vanish. Now we study the exceptions given by Theorem 8.2. The case of degree 7 leads to another new singular vector.

Conclusions
As a result of discussions in the previous Section we are now in a position to state a complete classification of singular vectors in finite Verma modules for E(5, 10).  Proof. In [9] singular vectors of degree 1 were constructed, and in [13] it was shown that there are no other ones. In [13] singular vectors of degree 2,3,4 and 5 were constructed, and it was shown in [2] that there are no other ones of degree 2 and 3. In the paper we show that there are no other singular vectors of degree 4 and 5. More precisely, singular vectors of degree 4 are classified in Section 9, singular vectors with degree equal to height greater than 5 are classified in Section 8, and singular vectors of degree greater than height are classified in Section 5.
This theorem gives an affirmative answer to the conjecture posed in [9].  Since a singular vector of weight µ in a finite Verma module with highest weight λ corresponds to a non-zero morphism M(µ) → M(λ), we can construct infinite sequences of morphisms as in Figure 4. All of these sequences are complexes (i.e. all compositions of consecutive morphisms vanish), except for the one through the origin; the latter becomes a complex if we replace the sequence M(0, 1, 0, 0) → M(0, 0, 0, 0) → M(1, 0, 0, 0) of morphisms of degree 1 by their composition of degree 2. This claim, when morphisms of degree 7 and 11 are not involved, follows from [13]; if they are involved, it follows since there are no morphisms of degree 8 and 12. Figure 5 represents all morphisms between finite degenerate Verma modules of degree 2 and 3; the corresponding bilateral infinite sequences shown in this picture are complexes, since any possible composition of two of these morphisms does not appear in Theorem 10.1.