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Graded limits of minimal affinizations and Beyond: the multiplicty free case for type E6

We obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type E6. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit of the corresponding minimal affinizations of the associated quantum group. We prove that this is the case under further restrictions on the highest weight. Under another set of conditions on the highest weight, Chari and Greenstein have recently proved that they are projective objects of a full subcategory of the category of graded modules for the current algebra. Our formula applies to all of these projective modules.


Introduction
The problem of determining the structure of the minimal affinizations of quantum groups is one of the most studied problems in the finite-dimensional representation theory of quantum affine algebras in recent years (see [6] for a recent survey with a comprehensive list of references). In particular, determining the character of such representations when regarded as modules for the quantum group U q (g) over the underlying semisimple Lie algebra g is of special interest. Determining the character is theoretically equivalent to determining the multiplicity of the irreducible constituents of these representations when regarded as U q (g)-modules. In practice, computing the multiplicities out of a given character is a laborious task which can be performed algorithmically.
One of the methods which have been used to approach this problem is that of considering the classical limit of the given module and regard it as a representation for the current algebra g[t] = g ⊗ C[t]. This approach was first considered in [2,7] and it was then further developed in [9,10,21]. In this paper, we apply this method for g of type E 6 and obtain a formula for the multiplicities of the irreducible constituents of the graded pieces of these modules assuming certain conditions on the highest weight. Our formula actually holds for a larger class of g[t]-modules. Namely, given a dominant integral weight λ of g, the first author defined in [21] a g[t]-module denoted by M (λ). The definition is by generator and relations which naturally generalize the relations of the classical limits of Kirillov-Reshetikhin modules obtained in [2]. It was conjectured in [21] that M (λ) is isomorphic to the classical limit of the minimal affinizations of the irreducible U q (g)-module of highest-weight λ provided that there exists a unique equivalence class of minimal affinizations associated to λ. Our main results are a formula for the multiplicities of the irreducible constituents of the graded pieces of the modules M (λ) and the proof of the conjecture of [21] assuming certain conditions on λ. To explain these conditions, let us label the nodes of the Dynkin diagram of g as follows. Let I = {1, 2, . . . , 6} and identify it with the set of nodes of the Dynkin diagram of g following the above labeling. For an integral weight µ, the support of µ is the subset of I consisting of labels such that the value of µ on the corresponding co-root is nonzero. The connected closure of the support is the minimal connected subdiagram of the Dynkin diagram of g containing the nodes in the support of µ.
We mostly focus our study on the modules M (λ) with λ not supported in the trivalent node and prove that the character formula (3.12) below holds for all λ with support contained in one of the following subsets of I: {1, 2, 5, 6}, {1, 4, 5, 6}, {2, 4, 6}. Following the conjecture of [21], we conjecture that (3.12) holds for all λ not supported in the trivalent node and prove in such generality that (3.12) gives an upper bound for the multiplicities of the g-irreducible constituents of the graded pieces of M (λ) (see (3.10)). In particular, it follows from (3.9) that all irreducible constituents are multiplicity free (even if the grading is not taken into account). As a byproduct of the proof of (3.12), we obtain a realization of M (λ) as a submodule of the tensor product of the classical limits of certain Kirillov-Reshetikhin modules (Theorem 3.14(a)), thus establishing part of the conjecture of [21] for such λ.
Keeping the above conditions on λ and further assuming that the connected closure of the support of λ is of type A, we prove that M (λ) is isomorphic to the classical limit of the corresponding minimal affinizations when regarded as g[t]-modules (Theorem 3.14(b)). This establishes the other part of the conjecture of [21] for these values of λ. In particular, (3.12) gives the multiplicities of the irreducible constituents of the minimal affinizations when the support of λ is contained in one of the following subsets of I: {1, 2, 5}, {1, 4, 5}, {1, 2, 6}, {4, 5, 6}, {2, 4}. Moreover, we also prove that, if (3.12) indeed holds for any λ not supported in the trivalent node as conjectured, then we can include {1, 2, 4, 5} in this list. Dropping all the assumptions on λ except that the connected closure of its support is of type A, we prove that the classical limit of the corresponding minimal affinizations are quotients of M (λ) (Proposition 3. 15). This is a further step towards the proof of the conjecture of [21] in general. However, the graded character formula for the Kirillov-Reshetikhin modules associated to the trivalent node given in [16] implies that, if λ is supported on that node, then these modules are not multiplicity free. We remark that, in [22], Nakajima developed an algorithm for computing the t-analogue of the qcharacter of any finite-dimensional irreducible representation of the quantum affine algebra associated to any simply laced simple Lie algebra g. In particular, without any assumption on λ, the graded character of the classical limits of the minimal affinizations associated to λ can be computed using this algorithm. Theoretically, one can then compute the multiplicities from the character as mentioned in the first paragraph of this introduction. On the other hand, with the above assumptions on λ, formula (3.12) gives these multiplicities directly.
Let us explain the reasons behind the several aforementioned restrictions on λ. First we recall that, for simply laced g, there exists a unique equivalence class of minimal affinizations associated to λ if and only if the connected closure of its support is of type A. Let θ be the highest root of g and, given i ∈ I, let ǫ i (θ) be its coordinate in the basis of simple roots. Given a positive integer r, let g[t : r] be the quotient of g[t] by the ideal g ⊗ t r C [t]. It turns out that M (λ) factors to a module for g[t : r] where r is the maximum of ǫ i (θ) for i running on the support of λ. In particular, if g is of type E 6 , M (λ) can be regarded as a module for g[t : 3]. Moreover, if λ is not supported on the trivalent node, then M (λ) factors to a module for g[t : 2]. The category G 2 of graded g[t : 2]-modules with finite-dimensional graded pieces has been recently studied in [4,5] by exploring its interplay with the theory of Koszul algebras and quiver representations. The literature on the representation theory of g[t : r] for r > 2 is more limited and results such as the ones from [4,5] are yet to be established. Thus, we focus on the case that M (λ) factors to a g[t : 2]-module which, for type E 6 , is equivalent to assuming that λ is not supported on the trivalent node (as mentioned above, this is also the necessary and sufficient condition for the modules M (λ) to be multiplicity free). It follows from [5, Theorem 1] that, if λ satisfies certain conditions, then M (λ) is a projective object of a full subcategory of G 2 naturally attached to λ. Moreover, [5, Theorem 2] gives a graded character formula for M (λ) provided λ satisfies the conditions of [5, Theorem 1]. We remark that [5, Theorem 2] expresses the graded character of M (λ) in terms of an alternating sum of the graded characters of M (µ) with µ strictly smaller than λ with respect to the usual partial order on the weight lattice of g. Hence, the formula of [5, Theorem 2] is of recursive nature. For g of type E 6 , we prove that the conditions on λ required on [5, Theorem 1] is equivalent to requiring that the support of λ be contained in one of the following subsets of I: {1, 2, 5, 6}, {1, 4, 5, 6}. Therefore, (3.12) holds beyond the cases covered by [5, Theorem 2]. This latter list of subsets of I also hints that it should be expected that when the support of λ contains {2, 4} the situation should be more complicated than otherwise. Indeed, the proof of (3.12) for this case is significantly more technically involved than for the others.
The paper is organized as follows. In Section 1, we review the basic notation on simply laced simple Lie algebras and the associated loop algebras, current algebras, quantum groups, and quantum affine algebras. In Section 2, we review the relevant facts on the finite-dimensional representation theory of these algebras. After reviewing the classification of minimal affinizations in Subsection 3.1, the main results (Theorem 3.14, Proposition 3.15, the multiplicity free property (3.10), and the character formula (3.12)) are stated in Subsection 3.2. The relation of our results with those of [5] is explained in Subsection 3.3. The proofs are given in Section 4.

Acknowledgements:
The work of the first author was partially supported by CNPq. The M.Sc. studies of the second author, during which part of this work was done, were supported by FAPESP.

Quantum and classical loop algebras
Throughout the paper, let C, R, Z, Z ≥m denote the sets of complex numbers, reals, integers, and integers bigger or equal m, respectively. Given a ring A, the underlying multiplicative group of units is denoted by A × . The dual of a vector space V is denoted by V * . The symbol ∼ = means "isomorphic to". The cardinality of a set S will be denoted by |S|.
1.1. Classical algebras. Let I = {1, . . . , n} be the set of vertices of a finite-type simply laced Dynkin diagram and let g be the associated semisimple Lie algebra over C with a fixed Cartan subalgebra h. Fix a set of positive roots R + and let The simple roots will be denoted by α i and the fundamental weights by ω i , i ∈ I. Q, P, Q + , P + will denote the root and weight lattices with corresponding positive cones, respectively. Let also h i ∈ h, be the co-root associated to α i , i ∈ I. We equip h * with the partial order λ ≤ µ iff µ − λ ∈ Q + . Let C = (c ij ) i,j∈I be the Cartan matrix of g, i.e., c ij = α j (h i ). The Weyl group is denoted by W.
The subalgebras g ±α , α ∈ R + , are one-dimensional and [g ±α , g ±β ] = g ±α±β for every α, β ∈ R + . We denote by x ± α any generator of g ±α and, in case α = α i for some i ∈ I, we may also use the notation For each subset J of I let g J be the Lie subalgebra of g generated by x ± α j , j ∈ J, and define n ± J , h J in the obvious way. Let also Q J be the subgroup of Q generated by α j , j ∈ J, and R + J = R + ∩ Q J . Given λ ∈ P , let λ J be the restriction of λ to h * J and λ J ∈ P be such that λ J (h j ) = λ(h j ) if j ∈ J and λ J (h j ) = 0 otherwise. By abuse of language, we will refer to any subset J of I as a subdiagram of the Dynkin diagram of g. The support of µ ∈ P is defined to be the subdiagram supp(µ) ⊆ I given by supp(µ) = {i ∈ I : µ(h i ) = 0}. Let also supp(µ) be the minimal connected subdiagram of I containing supp(µ).
If a is a Lie algebra over C, define its loop algebra to beã = a ⊗ C C[t, t −1 ] with bracket given by [x ⊗ t r , y ⊗ t s ] = [x, y] ⊗ t r+s . Clearly a ⊗ 1 is a subalgebra ofã isomorphic to a and, by abuse of notation, we will continue denoting its elements by x instead of x ⊗ 1. We also consider the current algebra a[t] which is the subalgebra ofã given by a[t] = a ⊗ C[t]. Theng =ñ − ⊕h ⊕ñ + andh is an abelian subalgebra and similarly for g [t]. The elements x ± α ⊗ t r , x ± i ⊗ t r , and h i ⊗ t r will be denoted by x ± α,r , x ± i,r , and h i,r , respectively. Also, Diagram subalgebrasg J are defined in the obvious way.
Let U (a) denote the universal enveloping algebra of a Lie algebra a. Then U (a) is a subalgebra of U (ã). Given a ∈ C, let τ a be the Lie algebra automorphism of a[t] defined by τ a (x⊗f (t)) = x⊗f (t−a) for every x ∈ a and every f (t) ∈ C[t]. If a = 0, let ev a :ã → a be the evaluation map x⊗ f (t) → f (a)x. We also denote by τ a and ev a the induced maps U (a[t]) → U (a[t]) and U (ã) → U (a), respectively. Given a nonzero x ∈ a we shall denote by U (x) the universal enveloping algebra of the one-dimensional subalgebra generated by x regarded as a subalgebra of U (a).
For each i ∈ I and r ∈ Z, define elements Λ i,r ∈ U (h) by the following equality of formal power series in the variable u: 1.2. Quantum algebras. Let C(q) be the ring of rational functions on an indeterminate q and , h i,r (i ∈ I, r ∈ Z\{0}) and the following defining relations: for all sequences of integers r 1 , . . . , r m , where m = 1 − c ij , S m is the symmetric group on m letters, and the ψ ± i,r are determined by equating powers of u in the formal power series Denote by U q (ñ ± ), U q (h) the subalgebras of U q (g) generated by {x ± i,r }, {k ±1 i , h i,s }, respectively. Let U q (g) be the subalgebra generated by x ± i := x ± i,0 , k ±1 i , i ∈ I, and define U q (n ± ), U q (h) in the obvious way. U q (g) is a subalgebra of U q (g) and multiplication establishes isomorphisms of C(q)-vectors spaces: and Let J ⊆ I and consider the subalgebra U q (g J ) generated by k ±1 j , h j,r , x ± j,s for all j ∈ J, r, s ∈ Z, r = 0.
Although we are denoting the elements x ± i,r , h i,r , and Λ i,r above by the same symbol as their classical counterparts, this will not create confusion as it will be clear from the context.
Let U A (g) be the A-subalgebra of U q (g) generated by the elements (x ± i,r ) (k) , k ±1 i for i ∈ I, r ∈ Z, and k ∈ Z ≥0 . Define U A (g) similarly and notice that U A (g) = U A (g) ∩ U q (g). Henceforth a will denote a Lie algebra of the following set: g, n ± , h,g,ñ ± ,h. For the proof of the next proposition see [2, Lemma 2.1] and the locally cited references.
Regard C as an A-module by letting q act as 1 and set Denote byx the image of x ∈ U A (g) in U q (g). For a proof of the next proposition see [11,Proposition 9.2.3] and the locally cited references.
Proposition 1.2. U (g) is isomorphic to the quotient of U q (g) by the ideal generated by k i − 1. In particular, the category of U q (g)-modules on which k i act as the identity operator for all i ∈ I is equivalent to the category of allg-modules.
The algebra U q (g) is a Hopf algebra and induces a Hopf algebra structure (over A) on U A (g). Moreover, the induced Hopf algebra structure on U (g) coincides with the usual one (see [11,20]). On U q (g) we have for all i ∈ I. The next lemma is easily established (cf. [21, Lemma 1.5]).
. Then x ∈ U A (n − ) and An expression for the comultiplication ∆ of U q (g) in terms of the generators x ± i,r , h i,r , k ±1 i is not known. The following partial information will suffice for our purposes (see [21,Lemma 1.6] and the locally cited references).
1.3. The ℓ-weight lattice. Given a field F consider the multiplicative group P F of n-tuples of rational functions µ = (µ 1 (u), · · · , µ n (u)) with values in F such that µ i (0) = 1 for all i ∈ I. We shall often think of µ i (u) as a formal power series in u with coefficients in F. Given a ∈ F × and i ∈ I, let ω i,a be defined by Clearly, if F is algebraically closed, P F is the free abelian group generated by these elements which are called fundamental ℓ-weights. It is also convenient to introduce elements ω λ,a , λ ∈ P, a ∈ F, defined by If F is algebraically closed, introduce the group homomorphism (weight map) wt : P F → P by setting wt(ω i,a ) = ω i . Otherwise, let K be an algebraically closed extension of F so that P F can be regarded as a subgroup of P K and define the weight map on P F by restricting the one on P K .
Define the ℓ-weight lattice of U q (g) to be P q := P C(q) . The submonoid P + q of P q consisting of n-tuples of polynomials is called the set of dominant ℓ-weights of U q (g). Given λ ∈ P + q with λ i (u) = j (1 − a i,j u), where a i,j belongs to some algebraic closure of C(q), let λ − ∈ P + q be defined by λ − i (u) = j (1 − a −1 i,j u). We will also use the notation λ + = λ. Given ν ∈ P q , say ν = λµ −1 with λ, µ ∈ P + q , define a C(q)-algebra homomorphism Ψ ν : One easily checks that the map Ψ : P q → (U q (h)) * given by ν → Ψ ν is injective. Define the ℓ-weight lattice P ofg to be the subgroup of P q generated by ω i,a for all i ∈ I and all a ∈ C × or, equivalently, P = P C . Set also P + = P ∩ P + q . From now on we will identify P q with its image in (U q (h)) * under Ψ. Similarly, P will be identified with a subset of U (h) * via the homomorphism Ψ ν : U (h) → C determined by (1.6) and Ψ ν (h i ) = wt(ν)(h i ).
It will be convenient to introduce the following notation. Given i ∈ I, a ∈ C(q) × , r ∈ Z ≥0 , define If J ⊆ I and λ ∈ P q , let λ J be the associated J-tuple of rational functions. Notice that, if λ j (u) ∈ C(q j )(u) for all j ∈ J, λ J can be regarded as an element of the ℓ-weight lattice of U q (g J ). Let also λ J ∈ P q be such that (λ J ) j (u) = λ j (u) for every j ∈ J and (λ J ) j (u) = 1 otherwise.
Given i ∈ I and a ∈ C(q) × , define the simple ℓ-root α i,a by The subgroup of P q generated by the simple ℓ-roots is called the ℓ-root lattice of U q (g) and will be denoted by Q q . Let also Q + q be the submonoid generated by the simple ℓ-roots. Quite clearly wt(α i,a ) = α i . Define a partial order on P q by Remark. The elements α i,a were first defined in [15] where they were denoted by A i,aq . The term simple ℓ-root was introduced in [8] where an alternate definition in terms of an action of the braid group of g on P q was given. For more details on the ℓ-weight lattice see [19, Section 3] and the references therein.
2. Finite-dimensional representations 2.1. Simple Lie algebras. For the sake of fixing notation, we now review some basic facts about the representation theory of g and U q (g). For the details see [18] and [11] for instance.
Given a U q (g)-module V and µ ∈ P , let A nonzero vector v ∈ V µ is called a weight vector of weight µ. If v is a weight vector such that x + i v = 0 for all i ∈ I, then v is called a highest-weight vector. If V is generated by a highest-weight vector of weight λ, then V is said to be a highest-weight module of highest weight λ. A U q (g)-module V is said to be a weight module if V = µ ∈ P V µ . Denote by C q be the category of all finite-dimensional weight modules of U q (g). Analogous concepts for g-modules are defined similarly after setting Denote by C the category of finite-dimensional g-modules.
Let Z[P ] be the integral group ring over P and denote by e : P → Z[P ], λ → e λ , the inclusion of P in Z[P ] so that e λ e µ = e λ+µ . The character of an object V from C q or C is defined by The following theorem summarizes the basic facts about the categories C q and C.
Theorem 2.1. Let V be an object either of C q or of C. Then: (c) For each λ ∈ P + , the g-module V (λ) generated by a vector v satisfying , is irreducible and finite-dimensional. If V ∈ C is irreducible, then V is isomorphic to V (λ) for some λ ∈ P + . (d) For each λ ∈ P + the U q (g)-module V q (λ) generated by a vector v satisfying (e) For all λ ∈ P + , char(V q (λ)) = char(V (λ)).
If J ⊆ I we shall denote by V q (λ J ) the simple U q (g J )-module of highest weight λ J . Similarly V (λ J ) denotes the corresponding irreducible g J -module.
Assume g = g 1 ⊕ g 2 where g j are semisimple Lie algebras. Then P = P 1 × P 2 where P j is the weight lattice of g j for j = 1, 2, and so on. Given λ ∈ P + j , denote by V j (λ) the irreducible g j -module of highest-weight λ. If V 1 is a g 1 -module and V 2 is a g 2 -module, then V 1 ⊗ V 2 is naturally a g-module.
Lemma 2.4. Let V be a finite-dimensional g-module and suppose l ∈ Z ≥1 , ν k ∈ P, v k ∈ V ν k , for for some µ j ∈ P + , and let π j : V → V j be the associated projection for j = 1, . . . , m.
2.2. Loop algebras. Let V be a U q (g)-module. We say that a nonzero vector v ∈ V is an ℓ-weight vector if there exists λ ∈ P q and k ∈ Z >0 such that (η − Ψ λ (η)) k v = 0 for all η ∈ U q (h). In that case, λ is said to be the ℓ-weight of v. V is said to be an ℓ-weight module if every vector of V is a linear combination of ℓ-weight vectors. In that case, let V λ denote the subspace spanned by all ℓ-weight vectors of ℓ-weight λ. An ℓ-weight vector v is said to be a highest-ℓ-weight vector if ηv = Ψ λ (η)v for every η ∈ U q (h) and x + i,r v = 0 for all i ∈ I and all r ∈ Z. V is said to be a highest-ℓ-weight module if it is generated by a highest-ℓ-weight vector. Denote by C q the category of all finite-dimensional ℓ-weight modules of U q (g). Quite clearly C q is an abelian category.
Define the concepts of ℓ-weight vector, etc., forg in a similar way and denote by C the category of all finite-dimensionalg-modules. The next proposition is easily established using (2.3).
Proposition 2.5. If V is a highest-ℓ-weight module, then it has a unique proper submodule and, hence, a unique irreducible quotient.
Definition 2.6. Let λ ∈ P + q and λ = wt(λ). The Weyl module W q (λ) of highest ℓ-weight λ is the U q (g)-module defined by the quotient of U q (g) by the left ideal generated by the elements , and η − Ψ λ (η) for every i ∈ I, r ∈ Z, and η ∈ U q (h). Denote by V q (λ) the irreducible quotient of W q (λ). The Weyl module W (λ), λ ∈ P + , ofg is defined in a similar way. Its irreducible quotient will be denoted by V (λ).
We shall need the following lemma which is a consequence of the proof of Theorem 2.7.
Lemma 2.8. If V is a highest-ℓ-weight module ofg and v be a highest-ℓ-weight vector.
If J ⊆ I we shall denote by V q (λ J ) the U q (g J )-irreducible module of highest ℓ-weight λ J . Similarly V (λ J ) denotes the corresponding irreducibleg J -module. Similar notations for the Weyl modules are defined in the obvious way.
The next theorem was conjectured in [15] and proved in [14].
2.3. Classical limits. Denote by P + A the subset of P q consisting of n-tuples of polynomials with coefficients in A. Let also P × A be the subset of P + A consisting of n-tuples of polynomials whose leading terms are in Cq Z \{0} = A × . Given λ ∈ P + A , let λ be the element of P + obtained from λ by evaluating q at 1.

Recall that an
where C is regarded as an A-module by letting q act as 1. ThenL is ag-module by Proposition 1.2 and dim(L) = dim(V ). The next theorem is essentially a corollary of the proof of Theorem 2.7.
Theorem 2.11. Let V be a nontrivial quotient of W q (λ) for some λ ∈ P × A , v a highest-ℓ-weight vector of V , and L = U A (g)v. Then, L is a U A (g)-admissible lattice of V and char(L) = char(V ). In particular,L is a quotient of W (λ).
Definition 2.12. Let λ ∈ P × A , v be a highest-ℓ-weight vector of V q (λ) and L = U A (g)v. We denote by V q (λ) theg-moduleL.

Minimal affinizations and Beyond
3.1. Classification of minimal affinizations. We now review the notion of minimal affinizations of an irreducible U q (g)-module introduced in [1].
for some m µ (V ) ∈ Z ≥0 . Two affinizations of V q (λ) are said to be equivalent if they are isomorphic as U q (g)-modules. If λ ∈ P + q is such that wt(λ) = λ, then V q (λ) is quite clearly an affinization of V q (λ). The partial order on P + induces a natural partial order on the set of (equivalence classes of) affinizations of V q (λ). Namely, if V and W are affinizations of V q (λ), say that V ≤ W if one of the following conditions hold: A minimal element of this partial order is said to be a minimal affinization.
The next corollaries are easily established (recall from §1.1 that supp(λ) is the minimal connected subdiagram of I containing supp(λ)).
Corollary 3.2. Suppose λ ∈ P + is such that supp(λ) is of type A. Then, V q (λ) has a unique equivalence class of minimal affinizations.
Corollary 3.3. Given i ∈ I and m ∈ Z ≥0 , the modules V q (ω i,a,m ), a ∈ C(q) × , are the only minimal affinizations of V q (mω i ).
The modules V q (ω i,a,m ) are known as Kirillov-Reshetikhin modules.
We now state a few results which were used in the proof of Theorem 3.1 and will be useful for us as well. The proofs can be found in [12].
Definition 3.5. Suppose g is of type A. A connected subdiagram J ⊆ I is said to be an admissible subdiagram. If g is of type D or E, let i 0 ∈ I be the trivalent node. A connected subdiagram J ⊆ I is said to be admissible if J is of type A and J\{i 0 } is connected.
The next proposition was proved in [21, Proposition 3.7].

Graded characters.
Recall the definition of the maps τ a : g[t] → g[t] from subsection 1.1.
Let V be a Z ≥0 -graded vector space and denote its r-th graded piece by Let also m µ,r (V ) be the multiplicity of V (µ) as an irreducible constituent of V [r].
) by the left ideal generated by -module. Given λ ∈ P + one can consider the modules A(λ) defined in [21]. These are graded g[t]-modules which were proved to be finite-dimensional in [21,Proposition 3.15]. One can proceed similarly to prove that the modules M (mω i ) are finite-dimensional. Moreover, it was proved in [23, Proposition 5.2.5] that A(mω i ) ∼ = M (mω i ) (for a general simple Lie algebra g). We shall not need the modules A(λ) in this paper.
In particular, it follows that M (0) is the trivial representation and R + (i, 0, s) = R + for all i ∈ I, s ∈ Z ≥0 . Now, given λ ∈ P + and r ∈ Z ≥0 , set Notice R(mω i , r) = R(i, m, r) for all i ∈ I and m, r ∈ Z ≥0 .
) by the left ideal generated by generated by the top weight space.
Definitions 3.10 and 3.11 of M (mω i ) coincide since R(mω i , r) = R(i, m, r) for all i ∈ I, m, r ∈ Z ≥0 . The modules M (λ) are clearly Z ≥0 -graded. It follows from [21, Proposition 3.13] that M (λ) is a quotient of the module A(λ) of [21] and, hence, finite-dimensional. Moreover, one easily sees that T (λ) is a graded quotient of M (λ) for all λ ∈ P + (the details can be found in [23, Proposition 5.2.10]).
The following is the main conjecture of [21].
For the rest of the subsection assume that g is of type E 6 and that the nodes of the Dynkin diagram are labeled as in the introduction. We now state our main results.
Notice that part (a) of Theorem 3.14 and Proposition 3.12 together with the following proposition which will be proved in Subsection 4.4 imply part (b) of Theorem 3.14.
Proposition 3.15. Let λ ∈ P + be such that is of type A. Then, L(λ) is a quotient of M (λ).
As a byproduct of the proof of Theorem 3.14 we are able to compute char t (M (λ)) for λ as in the theorem. In particular, we compute char(V q (λ)) for all λ ∈ P + q such that wt(λ) satisfies the hypothesis of part (b) of the theorem. Let us now present these formulas and, along the way, explain the strategy of the proof of Theorem 3.14(a).
We have already seen that t µ,r ≤ m µ,r . Therefore, in order to prove the first isomorphism of Conjecture 3.13, it suffices to show that For r ∈ Z 6 , set and gr(r) = r 1 + r 2 + r 3 + r 4 + r 5 + r 6 . Let also A = {r ∈ Z 6 ≥0 : r 6 ≤ m 6 , r 3 ≤ m 5 , r 4 ≤ m 1 , r 1 + r 3 + r 5 ≤ m 2 , r 2 + r 4 + r 5 ≤ m 4 }, A µ = {r ∈ A : wt(r) = µ}, A r = {r ∈ A : gr(r) = r}, and A µ,r = A µ ∩ A r . The omission of the dependence of wt and A on λ in the notation will not create confusion. One easily checks that the function wt : Z 6 → P is injective and, if r ∈ A, then wt(r) ∈ P + . In particular, The basic idea for proving (3.8) is the same one used in [9, 10, 21]. Namely, in Subsection 4.5, we will use the defining relations of M (λ) to show that, Moreover, for λ as in Theorem 3.14, by performing some explicit computations in T (λ), we show in Subsection 4.7 that (3.11) t µ,r ≥ |A µ,r |.
Clearly (3.10) and (3.11) together imply (3.8). Moreover, (3.12) char t (M (λ)) = r∈A char(V (wt(r)))t gr(r) for all λ as in Theorem 3.14. In particular, for λ as in Theorem 3.14(b) and λ ∈ P + q such that V q (λ) is a minimal affinization of V q (λ), we have Remark. Similar results in the case that g is of classical type or G 2 were obtained in [9, 10, 21] (however, the definition of the modules T (mω i ) requires some extra care in the non simply laced case). Equation (3.12) (and similar ones for general g) was predicted in [16] in the case that λ = mω i for some i ∈ I, m ∈ Z ≥0 . However, the meaning of the gradation in [16] is related to the quantum context, whereas here it appears by computing the classical limit. It is not clear to us why these two gradations coincide. The formulas in [16] were obtained by assuming the Kirillov-Reshetikhin conjecture whose proof was later completed in [17]. Our results give an alternate proof of these formulas for g of type E 6 and i = 3. As mentioned in the introduction, M (mω 3 ) is not multiplicity free in general. Using the methods of this paper, we are able to prove that the isotypical components of M (mω 3 )[r] are exactly as given by [16]. However, so far we could only obtain an upper bound for m µ,r which is most often larger than the actual value of m µ,r .
We end this subsection by reviewing a construction used in [9, §2.6] which will be useful for us as well. Let V r , 0 ≤ r ≤ k, be g-modules such that (3.14) Hom where we assume that V k+1 = 0. Fix non-zero elements p r ∈ Hom g (g ⊗ V r , V r+1 ), 0 ≤ r ≤ k − 1, and set p k = 0. It is easily checked that the following formulas extend the canonical g-module structure to a graded 3.3. Projectivity. If supp(λ) is not of type A, then Proposition 3.15 is probably false. In fact, most likely, M (λ) is then a proper quotient of L(λ). We now explain the motivation for studying the modules M (λ) beyond the cases associated to minimal affinizations from the perspective of [5]. We begin with following straightforward lemma which has been implicitly used in [5].
Lemma 3.16. Let r ∈ Z >0 and V be a g[t]-module generated by a vector v satisfying (g⊗t r C[t])v = 0. Then, (g ⊗ t r C[t])V = 0.
We proceed by induction on m. If m = 0, we have (x ⊗ t s )w = 0 by hypothesis. Assume m > 0, let w ′ = (x 2 ⊗ t r 2 ) · · · (x m ⊗ t rm )v and assume, by induction hypothesis, that (y ⊗ t s )w ′ = 0 for all y ∈ g, s ≥ r. Then, given x ∈ g and s ≥ r, we have Both summands are zero by the induction hypothesis on m.
The next proposition follows immediately from the above lemma and the definition of M (λ).
If V is a g[t]-module as in Lemma 3.16, then the canonical projection g[t] → g[t : r] := g[t]/g ⊗ t r C[t] induces a g[t : r]-module structure on V . Chari and Greenstein in [4,5] initiated the study of the category G 2 of graded g[t : 2]-modules with finite-dimensional graded pieces (they do not assume g is simply laced). Given a subset Γ of P + × Z ≥ 0, they consider the full subcategories G 2 (Γ) of G 2 consisting of modules V such that V (µ) is an irreducible constituent of V [r] only if (µ, r) ∈ Γ. In particular, they consider subsets Γ of the following form. Given Ψ ⊆ R + and λ ∈ P , set Notice that (λ, 0) ∈ Γ(λ, Ψ) for any choice of Ψ and that Γ(λ, ∅) = {(λ, 0)}. If we regard V (λ) as a module for g[t : 2] by pulling back the canonical projection g[t : 2] → g[t : 1] = g, then V (λ) is an object of G 2 (Γ(λ, Ψ)). The full strength of the results of [5] is realized when Ψ is either empty or of the form Ψ ν for some ν ∈ P where Ψ ν = {α ∈ R + : (α, ν) = max{(β, ν) : β ∈ R + }} and (·, ·) is the bilinear form on P × P induced from the Killing form of g.
For λ as in Theorem 3.18, [5, Theorem 2] gives a formula for computing the graded character of M (λ) by induction on the cardinality of the set Γ(λ, Ψ λ ).
Let us return to the case that g is of type E 6 . It follows from the proof of Theorem 3.14 (see Lemma 4.11 below) that M (λ) is a module as in Lemma 3.16 with r = 3. Moreover, if λ(h 3 ) = 0, then we can take r = 2.
Proof. Recalling that (α i , ν) = 1 2 (α i , α i )ν(h i ) and using the characterizations of R(λ, 1) given by (4.7), one easily checks by inspection of Table 1 below that Clearly λ satisfies the hypothesis of the lemma iff it satisfies one of the conditions (a)-(d) above.
This immediately implies the following corollary of Theorem 3.18.
Similarly to the proof of Lemma 3.19, one easily checks that if {2, 4} ⊆ supp(λ), then Ψ λ = ∅ and Ψ λ = Ψ ν for all ν ∈ P . Therefore, λ satisfies the hypothesis of Theorem 3.18 iff it satisfies the hypothesis of Lemma 3. 19. It follows that every λ as in Theorem 3.18 satisfies the hypothesis of Theorem 3.14. On the other hand, if λ satisfies the hypothesis of Theorem 3.14 but not the one of Theorem 3.18, then {2, 4} ⊆ supp(λ) ⊆ {2, 4, 6}. In this case, we cannot conclude that M (λ) is a projective object in some subcategory of G 2 nor can we use [5, Theorem 2] to compute its graded character.
Remark. It is worth remarking that we will perform most of the proof of (3.11) using only the hypothesis λ(h 3 ) = 0. This provides some evidence that Conjecture 3.13 holds in complete generality. In particular, we conjecture that (3.12) is the graded character of M (λ) for all λ ∈ P + such that λ(h 3 ) = 0.

4.1.
On characters for type A 2 . We now record some lemmas about the characters of certain finitedimensional sl 3 -modules which will be needed in the proof of (3.11). To simplify some formulas, we introduce the notation of divided powers. If A is an associative algebra, x ∈ A, and r ∈ Z ≥0 , set x (r) = 1 r! x r . We will make use of the following result on representations of the 3-dimensional Heisenberg algebra which will also be used in the proof of (3.10). Thus, consider the three-dimensional Heisenberg Lie algebra H spanned by elements x, y, z where z is central and [x, y] = z. Part (a) of the following lemma is standard while a proof of part (b) can be found in [10, Lemma 1.5]. Recall that U (n − ) is Q + -graded and denoted by U (n − ) η the piece of degree η. For the remainder of this subsection we assume g = sl 3 and I = {1, 2}. Observe that the map n − → H given by Proof. Since dim(U (n − ) η ) = p(η) = min{k i , k j } + 1, it suffices to show that this set is linearly independent. Let us write x = x − i , y = x − j , and z = [x, y]. Then, by part (a) of Lemma 4.1 we have One now easily uses the PBW theorem to prove that these vectors, with 0 ≤ r ≤ min{k i , k j }, are linearly independent.
Conversely, assume that V ∼ = l s = 1 V (µ s ) and let V s , s = 1, . . . , l, be a submodule of V isomorphic to V (µ s ) and such that V = l s = 1 V s . Let also π s : V → V s be the associated projection. By Lemma 2.4 we can assume π s (v s ) is a highest weight vector of V s . Observe that the set ( is a basis for U (n − ) ηs by Lemma 4.2. In particular, the vectors ( Since we already know that dim((V s ) µ l ) = min{k s,1 , k s,2 } + 1, the claim follows. Let a r,s ∈ C be such that It follows that a r,t = 0 for all t = 1, . . . , l and 0 ≤ r ≤ min{k t,1 , k t,2 }.
Lemma 4.5. Let a, b, c, m ∈ Z ≥0 , i, j ∈ I, j = i, λ = mω i , and v ∈ V (λ) λ \{0}. Then, Proof. From the sl 2 representation theory we have ( and c ≤ m). Notice that this together with the second statement implies the first statement. We prove the second statement by induction on a ≥ 0. The case a = 0 is obvious. The induction step will however depend on the knowledge of the case a = 1. For convenience set x = x − j , y = x − i , and z = [x, y]. Using the well-known commutation relation in U (n − ) where, in the second equality, we used that xv = 0 and the last statement of Lemma 4.1. The case a = 1 follows. Then, for a > 1, using the induction hypothesis we get Since, by the case a = 1, we have c+s is a positive rational number.
Assume supp(λ) ⊆ {1, 2, 4, 5}. As in the previous case, one sees that x − β j ,1 v = 0 for all 20 ≤ j ≤ 23. If both m 2 and m 4 are nonzero, we are left to show that x − β j ,2 v = 0 for all 24 ≤ j ≤ 30. For 24 ≤ j ≤ 28, this is done as in the case m 3 = 0 while for j = 29, 30 this then follows from the relations

Quantized relations.
The goal of this subsection is to prove Proposition 3.15. We proceed as in the proof of [21,Proposition 3.22] where a similar statement for orthogonal Lie algebras was proved. First we record several previously proved results which will be used in the proof. Lemma 4.7 ([21,Lemma 4.18]). Suppose w is a highest-ℓ-weight vector of V q (ω i,a,m ) for some i ∈ I, a ∈ C(q) × , and m ∈ Z ≥0 . Then, The following proposition follows from the results of [3, Section 6]. . Let λ ∈ P + , a i ∈ C(q) × , i ∈ I, and λ = i∈I ω i,a i ,λ(h i ) . Then, there exists an ordering i 1 , . . . , i n of I such that V q (λ) is isomorphic to the U q (g)-submodule of V q (ω i 1 ,a i 1 ,λ(h i 1 ) ) ⊗ · · · ⊗ V q (ω in,a in ,λ(h in ) ) generated by the top weight space.

Proposition 4.10 ([21, Proposition 3.13]). Suppose λ ∈ P ×
A is such that V q (λ) is a minimal affinization and that J ⊆ I is an admissible subdiagram. Let v be a highest-ℓ-weight vector of V = V q (λ), λ = wt(λ), and a ∈ C × be such that λ = ω λ,a . Then If α ∈ R + J for some admissible diagram J, we shall refer to α as an admissible root.
Proof of Proposition 3.15. Let a ∈ C be such thatλ = ω λ,a . We fix a highest-ℓ-weight vector v of V = V q (λ) and a i ∈ A × , i ∈ I, such that λ = i∈I ω i,a i ,m i . Let alsov be the image of v in V and v ′ be the image ofv in L(λ). By Proposition 4.6, we need to show that x − α,1 v ′ = 0 for all α ∈ R + 1 . This is equivalent to showing that (4.2) x − α,1v = av for all α ∈ R + 1 . By Proposition 4.10, (4.2) holds if α is an admissible root. Therefore, it remains to show that (4.3) x − β j ,1v = av for all 7 < j < 20.
Assume first that supp(λ) ⊆ {1, 2, 3, 4, 5}. In this case α 6 ∈ R(λ, 0) and (4.3) with j ∈ {8, 9, 11, 12} follows from the following relations ] together with the fact that β 3 , β 4 , β 6 , and β 7 are admissible roots. Next, assume that we have proved (4.3) for j ∈ {10, 14, 15, 16}. Then, (4.3) for the remaining values of j follows from the following relations . In order to prove (4.3) for j ∈ {10, 14, 15, 16}, it suffices to find elements X j , X j,1 ∈ U A (ñ − ) such that , and X j,1 v = a j (q)X j v + x j v for some a j (q) ∈ A and x j ∈ U A (g) satisfying a j (1) = a and x j = 0. We prove the existence of such elements assuming The case a i+1 = a i q −(m i +m i+1 +1) , i < 5, is proved similarly using part (b) of Proposition 2.10 instead of part (a). Let i 0 = max{i ∈ I : m i = 0} (in the case a i+1 = a i q −(m i +m i+1 +1) , i < 5, we would use i 0 = min{i ∈ I : m i = 0}). The relations X j,1 v = a j (q)X j v + x j v of (4.4) are the quantized relations alluded to in the title of this subsection.
Let λ ′ be such that λ = λ ′ ω i 0 ,a i 0 ,m i 0 . Let also v 1 , v 2 be highest-ℓ-weight vectors of V q (λ ′ ) and V q (ω i 0 ,a i 0 ,m i 0 ), respectively. By (4.5), Proposition 4.8, and Corollary 4.9, the assignment 0 ,a i 0 ,m i 0 ). Henceforth, we identify v with v 1 ⊗ v 2 . We write down the proof of the existence of elements as in (4.4) for j = 16 assuming i 0 = 5 (the other cases are proved similarly and the computations are simpler). Set , and X 16,1 = [x − 5,1 , X 14 ]. Quite clearly, X 16 , X 16,1 ∈ U A (g) satisfy the first two identities in (4.4). By Lemmas 1.3 and 1.4, modulo an element of the form xv with x ∈ U A (g) ⊗ U A (g) such thatx = 0, we have . Using Lemma 4.7 we get Since a 16 (q) := a 5 q m 5 satisfies a 16 (1) = a, in order to prove that X 16 and X 16,1 satisfy the last identity of (4.4), it suffices to show that Notice that x + 5,r X 14 v 1 = 0 for all r ∈ Z and let W be the U q (g 5 )-submodule of V q (λ ′ ) generated by X 14 v 1 . Then, by Proposition 2.10(a), W is a highest-ℓ-weight module with highest ℓ-weight ω 5,a 4 q m 4 . It then follows from Lemma 4.7 that This and (4.5) imply (4.6).
Lemma 4.11. For every i ∈ I, m ∈ Z ≥0 , and α = j∈I a j α j ∈ R + we have α ∈ R(i, m, a i ). In particular:  Proof. Statements (a)-(e) follow from the first statement by inspection of Table 1. Conversely, clearly items (a)-(e) together imply the first statement. The proof is analogous to that of [2, Proposition 1.2] (see also [23,Lemma 5.2.8]). We omit the details.
Observe that the above Lemma together with (3.5) imply 0 for all i ∈ I, j < 20, k < 29, r ≥ 1, s ≥ 2 and R(λ, 3) = R + . Let R ′ (i, m, r) be the set on the righthand-side of the inclusion symbol of the appropriate item of Lemma 4.11. It will follow from Section 4.6 below that, if m > 0, then R(i, m, r) = R ′ (i, m, r).
So far we proved that the sum in (4.12) can be restricted to r ∈ Z 6 ≥0 such that r 4 ≤ m 1 , r 3 ≤ m 5 , r 1 + r 3 + r 5 ≤ m 2 , and r 2 + r 4 + r 5 ≤ m 4 . Now, observe that, for such r, wt(r) ∈ P + iff r ∈ A. Therefore, by Lemma 2.4, we must have a surjective homomorphism of g-modules for every r ∈ Z ≥0 and (3.10) follows.
We begin proving this in the case m 6 = 1 in which case we have T (λ) = M (λ) by definition. Observe that Hom g (g⊗V (ω 6 ), V (ω 6 )) = 0 which is true since V (ω 6 ) is isomorphic to the adjoint representation. Hence, we can apply the construction given by (3.15) with V 0 = V (ω 6 ) and V 1 = V (0). One easily checks that the highest weight vector w 0 of V 0 satisfies the relations satisfied by v and, hence, the module V constructed in this way is a quotient of M (ω 6 ). Since V [0] ∼ = V (ω 6 ) and V [1] ∼ = V (0), (4.14) follows. Moreover, we clearly have x − β 30 ,1 w 0 = 0 (otherwise the map p 0 would be zero) and, hence, In particular, (4.7) holds for i = 6.

Lower bounds.
We now complete the proof of (3.11) for λ as in Theorem 3.14. In fact, we will carry out most of the proof assuming only that λ(h 3 ) = 0. Recall the notation x r , r ∈ A, developed in Section 4.5. In addition, we shall use the following notation. Denote by v i,m Given s = (s i ) i∈I ∈ Z I ≥0 , let T s (λ) be the submodule of µ,r denote the multiplicity of V (µ) as an irreducible constituent of T s (λ) [r]. Observe that, since |A wt(r),gr(r) | = 1 for all r ∈ A, in order to prove (3.11), it suffices to prove that (4.18) for each r ∈ A there exists s ∈ Z I ≥0 such that t s wt(r),gr(r) ≥ 1.
Thus, r ∈ Z 6 [s 2 + s 4 + s 6 ] iff r = r j for some j ∈ Z 5 . For shortening some expressions, given j ∈ Z 5 , we may use the notation and let B(s) be the set of tuples j ∈ Z 5 ≥0 satisfying In Subsection 4.8 we will show that It follows from (4.21) that In particular, wt(r o ) is the unique maximal weight of T s (λ)[s 2 + s 4 + s 6 ] and, hence, Lemma 4.14. Let r ∈ A. Then, there exists s ∈ Z I ≥0 and j ∈ B(s) such that j 5 = 0 and r = r j . In particular, r ∈ A(s).
The above lemma shows that it suffices to show (4.18) in the case that r = r j for some s ∈ Z I ≥0 and j ∈ B(s) such that j 5 = 0. In this case, it follows from the proof of (4.21) (see the last line of Subsection 4.8) that x r j v s is a nonzero scalar multiple of . From now on we fix s ∈ Z I ≥0 , write B = B(s), and set B 0 = {j ∈ B(s) : j 5 = 0}.
Given k ∈ B, let It easily follows from (4.23) that By Lemma 2.4, (4.22), and the injectivity of wt : A → P , (4.18) holds for r = r k iff Equivalently, (4.18) holds for r = r k iff we have an isomorphism of g-modules Given j ∈ B 0 , define the height of j to be We prove (4.27) by induction on k = ht(k). Equation (4.24) implies that (4.27) holds for k = 0. Thus, assume k > 0 and, by induction hypothesis, that (4.27) holds for j ∈ B 0 such that ht(j) < k. It follows from the induction hypothesis and (4.25) that We are left to show that and similarly for J in place of J ± . Moreover, we have isomorphisms of vector spaces The first isomorphism above is clear and the second follows from Proposition 2.3. If j ∈ B k , it easily follows from (4.23) that Hence, we can use Lemma 4.3 to compute Plugging this in (4.28) we get (4.32) dim((V + k ) wt(r k ) ) = j∈B + k (min{k 2 −j 2 , k 2 −j 2 +j 4 −k 4 }+1)(min{k 1 −j 1 , k 1 −j 1 +j 3 −k 3 }+1).
Set p = (p 1 , p 5 ), x j,p = x p 5 ,p 4 (j),k 1 −j 1 −p 5 5,4,5 x p 1 ,p 2 (j),k 2 −j 2 −p 1 1,2,1 , and v j,p = x j,p v j . Thus, we want to show that the vectors v j,p are linearly independent for j and p as above. From now on, when now confusion arises, we simplify notation and write l − in place of l − (j), etc. Recall that To simplify the expression above, Also, using Lemma 4.12 we get up to nonzero scalar multiples. By applying the comultiplication one sees that v j,p is equal to χ x d 2 ,e 2 ,f 2 5,4,5 x a 2 ,b 2 ,c 2 1,2,1 where χ runs over the set of collections of nonnegative integers a l , b l , c l , d l , e l , f l satisfying Above we also used that x a,b,c 1,2,1 v 5,m 5 = x a,b,c 5,4,5 v 1,m 1 = x a,b,c 1,2,1 v 6 = x a,b,c 5,4,5 v 6 = 0 whenever a + b + c > 0. We will need to study the summands on the right-hand-side of (4.37).
Therefore, v j,p is a linear combination of vectors of the form and observe that the coefficient of v a,b,d,e in v j,p is nonzero iff j 1 ≤ k 1 − e, j 2 ≤ k 2 − b, a ≤ p 1 , d ≤ p 5 .
The above paragraph proves the induction step on n 2 . It remains to show that the induction on n 2 starts when n 1 > 0. Thus, assume n 1 > 0, n 2 = 0 and, by induction hypothesis on n 1 , that {v j,p : (j, p) ∈ S(n 1 − 1, 0)} is linearly independent. Let c j,p ∈ C be such that The proof of (4.47) is similar to that of (4.45) and we omit the details.