Graded Limits of Minimal Affinizations in Type D

We study the graded limits of minimal affinizations over a quantum loop algebra of type D in the regular case. We show that the graded limits are isomorphic to multiple generalizations of Demazure modules, and also give their defining relations. As a corollary we obtain a character formula for the minimal affinizations in terms of Demazure operators, and a multiplicity formula for a special class of the minimal affinizations.


Introduction
Let g be a complex simple Lie algebra, Lg = g⊗C[t, t −1 ] the associated loop algebra, and U q (Lg) the quantum loop algebra. In [1], Chari introduced an important class of finite-dimensional simple U q (Lg)-modules called minimal affinizations. For a simple U q (g)-module V , we say a simple U q (Lg)-module V is an affinization of V if the highest weight of V is equal to that of V . One can define a partial ordering on the equivalence classes (the isomorphism classes as a U q (g)-module) of affinizations of V , and modules belonging to minimal classes are called minimal affinizations (a precise definition is given in Section 2.6). For example, a Kirillov-Reshetikhin module is a minimal affinization whose highest weight is a multiple of a fundamental weight. Minimal affinizations have been the subjects of many articles in the recent years. See [7,10,12,18,19,21] for instance. For the original motivations of considering minimal affinizations, see [1,Introduction]. Given a minimal affinization, one can consider its classical limit. By restricting it to the current algebra g[t] = g ⊗ C[t] and taking a pull-back, a graded g[t]-module called graded limit is obtained. Graded limits are quite important for the study of minimal affinizations since the U q (g)-module structure of a minimal affinization is completely determined by the U (g)-module structure of its graded limit.
Graded limits of minimal affinizations were first studied in [2,5] in the case of Kirillov-Reshetikhin modules, and subsequently the general ones were studied in [18]. In that paper, Moura presented several conjectures for the graded limits of minimal affinizations in general types, and partially proved them. Graded limits of minimal affinizations in type ABC were further studied in [21]. In that paper the author proved that the graded limit of a minimal affinization in these types is isomorphic to a certain g[t]-module D(w • ξ 1 , . . . , w • ξ n ). Here w • is the longest element of the Weyl group of g, ξ j are certain weights of the affine Lie algebra g which are g-dominant, and D(w • ξ 1 , . . . , w • ξ n ) is a g[t]-submodule of a tensor product of simple highest weight g-modules, which is generated by the tensor product v w•ξ 1 ⊗ · · · ⊗ v w•ξn of the extremal weight vectors with weights w • ξ j . As a corollary of this, a character formula for minimal affinizations was given in terms of Demazure operators. In addition, the defining relations of graded limits conjectured in [18] were also proved. In type ABC a minimal affinization with a fixed highest weight is unique up to equivalence, and the graded limit of a minimal affinization depends only on the equivalence class. As a consequence, the module D(w • ξ 1 , . . . , w • ξ n ) can be determined from the highest weight only. (If the highest weight is λ = 1≤i≤n λ i i where i are fundamental weights, then ξ j are roughly equal to λ i ( i +a i Λ 0 ) where Λ 0 is the fundamental weight of g associated with the distinguished node 0, and a i = 1 if the simple root α i is long and a i = 1/2 otherwise. For the more precise statement, see [21].) In contrast to this, in type D there are nonequivalent minimal affinizations with the same highest weights. It was proved in [10], however, that even in type D if the given highest weight satisfies some mild condition (see Section 2.6), then there are at most 3 equivalence classes of minimal affinizations with the given highest weight. We say a minimal affinization is regular if its highest weight satisfies this condition. The purpose of this paper is to study the graded limits of regular minimal affinizations of type D using the methods in [21].
In the sequel we assume that g is of type D n . Let π be Drinfeld polynomials and assume that the simple U q (Lg)-module L q (π) associated with π is a regular minimal affinization. Then in a certain way we can associate with L q (π) a vertex s ∈ {1, n − 1, n} of the Dynkin diagram of g (see Section 2.6). In the case where the number of equivalence classes are exactly 3, this s parameterizes the equivalence class of L q (π). In this paper we show that there exists a sequence ξ (s) 1 , . . . , ξ (s) n of g-dominant g-weights such that the graded limit L(π) of L q (π) is isomorphic to j depends not only on the highest weight of L q (π) but also s, and the correspondence is less straightforward compared with the case of type ABC (see Section 3.1 for the precise statement). As a consequence, we give a character formula for L q (π) in terms of Demazure operators (Corollary 3.5). We also prove the defining relations of the graded limits L(π) conjectured in [18] (Theorem 3.2), which also depends not only on the highest weight but also s.
Recently Sam proved in [22] some combinatorial identity in type BCD, and gave a multiplicity formula for minimal affinizations in type BC using the identity and results in [4] and [21]. By applying the identity of type D to our results, we also obtain a similar multiplicity formula for a special class of minimal affinizations in type D, which gives multiplicities in terms of the simple Lie algebra of type C (Corollary 3.8).
The proofs of most results are similar to those in [21] and are in some respects even simpler since the type D is simply laced. For example we do not need the theory of q-characters, which was essentially needed in loc. cit.
The organization of the paper is as follows. In Section 2, we give preliminary definitions and basic results. In particular, we recall the definition of the modules D(ξ 1 , . . . , ξ p ), the classification of regular minimal affinizations of type D, and the definition of graded limits. In Section 3 we state Theorems 3.1 and 3.2, and discuss some of their corollaries. The proofs of Theorems 3.1 and 3.2 is given in Section 4.

Preliminaries
2.1 Simple Lie algebra of type D n Let I = {0, 1, . . . , n} and C = (c ij ) i,j∈ I be the Cartan matrix of type D (1) n whose Dynkin diagram is as follows: Let J ⊆ I be a subset. In this paper, by abuse of notation, we sometimes denote by J the subdiagram of (2.1) whose vertices are J.
Let I = I \ {0}, C = (c ij ) i,j∈I be the Cartan matrix of type D n , and g the complex simple Lie algebra associated with C. Let h be a Cartan subalgebra and b a Borel subalgebra containing h. Denote by ∆ the root system and by ∆ + the set of positive roots, and let θ ∈ ∆ + be the highest root. Let α i and i (i ∈ I) be the simple roots and fundamental weights respectively, and set 0 = 0 for convenience. Let P be the weight lattice and P + the set of dominant integral weights. Let W denote the Weyl group with simple reflections s i (i ∈ I), and w • ∈ W the longest element.
For each α ∈ ∆ denote by g α the corresponding root space, and fix nonzero elements e α ∈ g α , f α ∈ g −α and α ∨ ∈ h such that We also use the notation e i = e α i , f i = f α i for i ∈ I. Set n ± = α∈∆ + g ±α . For a subset J ⊆ I, denote by g J ⊆ g the semisimple Lie subalgebra corresponding to J, and let h J = i∈J Cα ∨ i ⊆ h.

Af f ine Lie algebra of type
⊕ CK ⊕ Cd be the affine Lie algebra with Cartan matrix C, where K is the canonical central element and d is the degree operator. Naturally g is regarded as a Lie subalgebra of g. Define a Cartan subalgebra h and a Borel subalgebra b as follows: Set n + = n + ⊕ g ⊗ tC[t]. We often consider h * as a subspace of h * by setting K, λ = d, λ = 0 for λ ∈ h * . Let ∆ be the root system of g, ∆ + the set of positive roots, ∆ re the set of real roots and ∆ re + = ∆ + ∩ ∆ re . Set α 0 = δ − θ, e 0 = f θ ⊗ t, f 0 = e θ ⊗ t −1 and α ∨ 0 = K − θ ∨ . Denote by Λ 0 ∈ h * the unique element satisfying K, Λ 0 = 1 and h, Λ 0 = d, Λ 0 = 0, and define P , P + ⊆ h * by P = P ⊕ ZΛ 0 ⊕ Cδ and P + = ξ ∈ P | α ∨ i , ξ ≥ 0 for all i ∈ I .
Let W denote the Weyl group of g with simple reflections s i (i ∈ I). We regard W naturally as a subgroup of W . Let : W → Z ≥0 be the length function. Let ( , ) be the unique non-degenerate W -invariant symmetric bilinear form on h * satisfying Let Σ be the group of Dynkin diagram automorphisms of g, which naturally acts on h * and g, and W the subgroup of GL( h * ) generated by W and Σ. Note that we have W = Σ W . The length function is extended on W by setting (τ w) = (w) for τ ∈ Σ, w ∈ W .
Denote by V (λ) for λ ∈ P + the simple g-module with highest weight λ, and by V (Λ) for Λ ∈ P + the simple highest weight g-module with highest weight Λ. For a finite-dimensional semisimple h-module (resp. h-module) M we denote by ch h M ∈ Z[h * ] (resp. ch h M ∈ Z[ h * ]) its character with respect to h (resp. h). We will omit the subscript h or h when it is obvious from the context.

Loop algebras and current algebras
Given a Lie algebra a, its loop algebra La is defined as the tensor product a ⊗ C[t, t −1 ] with the Lie algebra structure given by [x ⊗ f, y ⊗ g] = [x, y] ⊗ f g. Let a[t] and t k a[t] for k ∈ Z >0 denote the Lie subalgebras a ⊗ C[t] and a ⊗ t k C[t] respectively. The Lie algebra a[t] is called the current algebra associated with a.
For a ∈ C × , let ev a : Lg → g denote the evaluation map defined by ev a (x ⊗ f ) = f (a)x, and let V (λ, a) for λ ∈ P + be the evaluation module which is the simple Lg-module defined by the pull-back of V (λ) with respect to ev a . An evaluation module for g[t] is defined similarly and is denoted by V (λ, a) (λ ∈ P + , a ∈ C).

b-submodules
. , ξ p be a sequence of elements belonging to the Weyl group orbits W ( P + ) of dominant integral weights of g. We define a b-module D(ξ 1 , . . . , ξ p ) as follows. For each 1 ≤ j ≤ p let Λ j ∈ P + be the unique element satisfying ξ j ∈ W Λ j , and take a nonzero vector v ξ j in the If (α i , ξ j ) ≤ 0 for all i ∈ I and 1 ≤ j ≤ p, then D(ξ 1 , . . . , ξ p ) can be regarded as a g[t] ⊕ CK ⊕ Cdmodule and in particular a g[t]-module.
Some of D(ξ 1 , . . . , ξ p ) are realized in a different way. To introduce this, we prepare some notation. Assume that V is a g-module and D is a b-submodule of V . For τ ∈ Σ, we denote by F τ V the pull-back (τ −1 ) * V with respect to the Lie algebra automorphism τ −1 on g, and define a b-submodule F τ D ⊆ F τ V in the obvious way. It is easily proved that F τ D(ξ 1 , . . . , ξ p ) ∼ = D(τ ξ 1 , . . . , τ ξ p ).
For i ∈ I let p i denote the parabolic subalgebra b ⊕ Cf i ⊆ g, and set F i D = U ( p i )D ⊆ V to be the p i -submodule generated by D. Finally we define F w D for w ∈ W as follows: let τ ∈ Σ and w ∈ W be the elements such that w = τ w , and choose a reduced expression w = s i 1 · · · s i k . Then we set Proposition 2.1 ([21, Proposition 2.7]). Let Λ 1 , . . . , Λ p be a sequence of elements of P + , and w 1 , . . . , w p a sequence of elements of W . We write w [r,s] = w r w r+1 · · · w s for r ≤ s, and assume that (w [1,p] where e λ (λ ∈ P ) are the generators of Z[ P ]. For w ∈ W with a reduced expression w = s i 1 · · · s i k , we set D w = D i 1 · · · D i k . If w ∈ W and w = τ w (τ ∈ Σ, w ∈ W ), we set D w = τ D w . The operator D w is called a Demazure operator. The character of the right-hand side of (2.2) is expressed using Demazure operators by [16,Theorem 5], and as a consequence we have the following (see also [21, Corollary 2.8]).
Proposition 2.2. Let Λ j ∈ P + and w j ∈ W (1 ≤ j ≤ p) be as in Proposition 2.1. Then we have

Quantum loop algebras and their representations
The quantum loop algebra U q (Lg) is a C(q)-algebra generated by x ± i,r , k ±1 i and h i,m (i ∈ I, r ∈ Z, m ∈ Z \ {0}) subject to certain relations (see, e.g., [6,Section 12.2]). U q (Lg) has a Hopf algebra structure [6,17]. In particular if V and W are U q (Lg)-modules then V ⊗ W and V * are also U q (Lg)-modules, and we have ( which is isomorphic to the quantized enveloping algebra associated with g. For a subset J ⊆ I, let U q (Lg J ) denote the subalgebra generated by We recall basic results on finite-dimensional U q (g)-and U q (Lg)-modules. Note that in the present paper we assume that g is of type D, and when g is non-simply laced some of indeterminates q appearing below should be replaced by In this article we will only consider modules of type 1. For a finite-dimensional module V of type 1, we set ch V = λ∈P e λ dim V λ ∈ Z[P ]. For λ ∈ P + we denote by V q (λ) the finitedimensional simple U q (g)-module of type 1 with highest weight λ. The category of finitedimensional U q (g)-modules of type 1 is semisimple, and every simple object is isomorphic to V q (λ) for some λ ∈ P + . We say that a U q (Lg)-module V is highest -weight with highest -weight vector v and highest -weight ]] are defined as follows: for some polynomials π i (u) ∈ C(q)[u] whose constant terms are 1.
(ii) Conversely, for every I-tuple of polynomials π = π 1 (u), . . . , π n (u) such that π i (0) = 1, there exists a unique (up to isomorphism) finite-dimensional simple highest -weight The I-tuple of polynomials π = π 1 (u), . . . , π n (u) are called Drinfeld polynomials, and we will say by abuse of terminology that the highest -weight of V is π if the highest -weight 3). We denote by L q (π) the finite-dimensional simple U q (Lg)module of type 1 with highest -weight π, and by v π a highest -weight vector of L q (π).
Let i →ī be the bijection I → I determined by αī = −w • (α i ).
). For any Drinfeld polynomials π we have as U q (Lg)-modules, where π * = πī q −h ∨ u i∈I and h ∨ is the dual Coxeter number.

Minimal af f inizations
For an I-tuple of polynomials π = π i (u) i∈I , set wt(π) = i∈I i deg π i ∈ P + .
(ii) Affinizations V and W of V q (λ) are said to be equivalent if they are isomorphic as U q (g)modules. We denote by [V ] the equivalence class of V .
with some m µ (V ) ∈ Z ≥0 . Let V and W be affinizations of V q (λ), and define m µ (V ), m µ (W ) as above. We write [V ] ≤ [W ] if for all µ ∈ P + , either of the following holds: Then ≤ defines a partial ordering on the set of equivalence classes of affinizations of V q (λ) [1, Proposition 3.7].

Definition 2.6 ([1]). We say an affinization
is minimal in the set of equivalence classes of affinizations of V q (λ) with respect to this ordering.
Let us recall the classification of minimal affinizations in the regular case of type D, which was given in [10]. (Similar results also hold in type E. See [7] for type ABCF G, in which minimal affinizations are unique up to equivalence.) For that, we fix several notation. Set S = {1, n − 1, n} ⊆ I and define the subsets I s ⊆ I (s ∈ S) by I 1 = {1, 2, . . . , n − 3}, I n−1 = {n − 1}, I n = {n}. Note that I s is the connected component of the subdiagram I \ {n − 2} containing s, and I \ I s is the maximal type A subdiagram of I not containing s. For s ∈ S, ε ∈ {±}, λ = i∈I λ i i ∈ P + and a ∈ C(q) × , define Drinfeld polynomials π ε s (λ, a) as follows: • When s = 1, set π ε 1 (λ, a) = i∈I π (i) λ i ,a i (the product being defined component-wise) with a 1 = a and Remark 2.7. The Drinfeld polynomials π ε s (λ, a) are determined so that they satisfy the following property: if r ∈ S \ {s} and J (i) If supp(λ) ∩ I s = ∅ for some s ∈ S, then there exists a unique equivalence class of minimal affinizations of V q (λ), and the equivalence class is given by L q π ε r (λ, a) ε ∈ {±}, a ∈ C(q) × with r ∈ S \ {s} (here the choice of r is irrelevant since π ± r (λ, a) = π ± r (λ, a) holds for r, r ∈ S \ {s}).
We call λ ∈ P + regular if λ satisfies one of the assumptions of (i) or (ii) in Theorem 2.8. We call a minimal affinization is regular if its highest weight is regular. Remark 2.9 ([9]). In the remaining case when supp(λ) ∩ I s = ∅ for all s ∈ S and λ n−2 = 0, the number of equivalence classes of minimal affinizations increases unboundedly with λ, and the classification of minimal affinizations has not been given except for the type D 4 .

Classical limits and graded limits
Let A = C[q, q −1 ] be the ring of Laurent polynomials with complex coefficients, and denote by U A (Lg) the A-subalgebra of U q (Lg) generated by k ±1 i in a similar way. We define C-algebras U 1 (Lg) and U 1 (g) by where we identify C with A/ q − 1 . Then the following C-algebra isomorphisms are known to hold [17], [6, Proposition 9.3.10]: Let π = π 1 (u), . . . , π n (u) be Drinfeld polynomials, and assume that there exists b ∈ C × such that all the roots of π i (u)'s are contained in the set bq Z (it is known that in order to describe the category of finite-dimensional U q (Lg)-modules, it is essentially enough to consider representations attached to such families of Drinfeld polynomials. For example, see [13,Section 3.7]). Note that π ± s (λ, a) satisfies this assumptions when a ∈ C × q Z . Let L A (π) be the U A (Lg)-submodule of L q (π) generated by a highest -weight vector v π . Then by the isomorphism (2.4), becomes a finite-dimensional Lg-module, which is called the classical limit of L q (π).
Define a Lie algebra automorphism ϕ b : We consider L q (π) as a g[t]-module by restriction, and define a g[t]-module L(π) by the pullback ϕ * b L q (π) . We call L(π) the graded limit of L q (π). In fact, at least when π = π ± s (λ, a), it turns out later from our main theorems that L(π) is a graded g[t]-module, which justifies the name "graded limit". Setv π = 1 ⊗ A v π ∈ L(π), which generates L(π) as a g[t]-module. The following properties of graded limits are easily proved from the construction (see [2]). Lemma 2.10. Assume wt(π) = λ.
(ii) The vectorv π satisfies the relations iv π = 0 for i ∈ I.
(iv) For every µ ∈ P + we have where the left-and right-hand sides are the multiplicities as a U q (g)-module and g-module, respectively.

Main theorems
Denote by τ 0,1 ∈ Σ (resp. τ n−1,n ∈ Σ) the diagram automorphism interchanging the nodes 0 and 1 (resp. n − 1 and n). We will not use other elements of Σ in the sequel. For s ∈ S and 1 ≤ j ≤ n, define ξ (s) j = ξ (s) j (λ) ∈ P as follows: • When s = 1, let m, m be such that {m, m } = {n − 1, n}, λ m = max{λ n−1 , λ n } and λ m = min{λ n−1 , λ n }, and define • When s = n − 1, set ξ   Note that if α ∈ ∆ (s) + , then the coefficient of α i in α is 0 or 1 for all i ∈ I. Theorem 3.2. The graded limit L(π s ) is isomorphic to the cyclic g[t]-module generated by a nonzero vector v subject to the relations We prove Theorems 3.1 and 3.2 in Section 4. the author also conjectured that the graded limit L(π s ) is isomorphic to the g[t]-submodule of L π ε s (λ I s , a) ⊗ L π ε s (λ I\I s , a) generated by the tensor product of highest weight vectors. This is easily deduced from Theorem 3.1.
The proof of the following lemma is straightforward.
In particular, each Λ (s) j belongs to P + . We obtain the following chain of isomorphisms from Theorem 3.1, Lemma 3.4(ii), and Proposition 2.1: where the second isomorphism obviously holds by definition. Hence by Proposition 2.2 and Lemma 2.10(iii), the following holds.
Hence we also obtain the character in the case s = n − 1.
(i) It is possible to use other elements of W in the expression of ch L q (π s ). That is, if w j ∈ W For example w j = s j−1 s j−2 · · · s 1 τ 0,1 satisfy the above conditions when s = 1. Our choice is made so that the results are stated in a uniform way.
(ii) The right-hand side of the isomorphism (3.1) has a crystal analog, and using this we can express the multiplicities of L q (π s ) in terms of crystal bases. For the details, see [21,Corollary 4.11].
Our next result is a formula for multiplicities of simple finite-dimensional U q (g)-modules in L q (π 1 ) which can be deduced from our Theorem 3.2 and the results of [4] and [22]. For that, we prepare a lemma.
Lemma 3.7. Assume that V is a cyclic finite-dimensional g[t]-module generated by a h-weight vector v, and n + [t] ⊕ th[t] acts trivially on v. Let µ ∈ P + , and W be the g[t]-submodule of V ⊗ V (µ, 0) generated by v ⊗ v µ , where v µ denotes a highest weight vector. Then for every ν ∈ P + , we have where [ : ] denotes the multiplicity as a g-module.
Proof . Note that and W is a finite-dimensional g-module, we see that Hence the assertion follows from (3.2).

K. Naoi
Let sp 2n−2 be the simple Lie algebra of type C n−1 , and denote by P sp its weight lattice and by sp i (1 ≤ i ≤ n−1) its fundamental weights. We assume that sp i are labeled as [15,Section 4.8]. Define a map ι : Corollary 3.8. For every µ ∈ P + , we have otherwise.

Proofs of main theorems
Note that the theorems for s = n − 1 and s = n are equivalent because of the existence of diagram automorphism of g interchanging n − 1 and n. Therefore, throughout this section we assume that s = n − 1, and prove the theorems only for the case s = 1, n. Similarly as the previous section we fix λ = i∈I λ i i ∈ P + , ε ∈ {±} and a ∈ C × q Z , and write π s = π ε s (λ, a). Let M s (λ) denote the module defined in Theorem 3.2. We shall prove the existence of three surjective g[t]-module homomorphisms. More presicely, we prove M s (λ) L(π s ) in Section 4.1,

Proof for M s (λ) L(π s )
Though the proof is similar to that in [18], we will give it for completeness.
Let v πs be a highest -weight vector of L q (π s ), and setv πs = 1 ⊗ v πs ∈ L(π s ). In order to prove M s (λ) L(π s ), it is enough to show the relations since the other relations hold by Lemma 2.10(ii).
Let r ∈ S \ {s} and J = I \ I r . Then the subalgebra U q (Lg J ) ⊆ U q (Lg) is a quantum loop algebra of type A. By [10, Lemma 2.3], the U q (Lg J )-submodule of L q (π s ) generated by v πs is isomorphic to the simple U q (Lg J )-module with highest -weight (π s ) i (u) i∈J . Denote this U q (Lg J )-submodule by L q . Then we see from Remark 2.7 and [10, Theorem 3.1] that L q is also simple as a U q (g J )-module. From this and the construction of graded limits, it follows that the Lg J -submodule is simple as a g J -module. Hence the restriction of the surjective homomorphism L(π s ) V (λ, 0) in Lemma 2.10(i) to L is an isomorphism, which obviously implies (g J ⊗ t)v πs = 0. Now the relations (4.1) obviously follow from the definition of ∆
Proof . In both the cases s = 1 and s = n, it follows from Lemma 3.4(i) that holds for every 1 ≤ j ≤ k ≤ n − 1. Hence ρ n,j (α) > 0 implies β ∈ −∆ + and k ≥ 1, and if β = − i∈I t i α i then we have This immediately implies the assertion (i). Note that ρ i,j (α) = ρ i+1,j (s i α) holds for 1 ≤ i ≤ n−1 by the definition of ρ i,j . Hence ρ i,j (α) > 0 implies that we have either α = α i or α = s i γ for some γ ∈ ∆ re + such that ρ i+1,j (γ) > 0. From this, the assertion (ii) is easily proved by the descending induction on i.
For 0 ≤ i ≤ n and 1 ≤ j ≤ n − 1, set For α = β + kδ ∈ ∆ re with β ∈ ∆ and k ∈ Z, denote by x α ∈ g the vector e β ⊗ t k . For i ∈ I, define a Lie subalgebra n i of n + by We shall determine the generators of the annihilators Ann U ( n + ) v(i, j) inductively, along the lines of [21,Section 5.1]. For that, we need the following lemma which is proved in [14,Section 3] (see also [21,Lemma 5.3]).
Lemma 4.2. Let V be an integrable g-module, T a f inite-dimensional b-submodule of V , i ∈ I and ξ ∈ P such that (α i , ξ) ≥ 0. Assume that the following conditions hold: (i) T is generated by a h-weight vector v ∈ T ξ satisfying e i v = 0.
(ii) There is an ad(e i )-invariant left U ( n i )-ideal I such that Let where r i denotes the algebra automorphism of U ( g) corresponding to the reflection s i .

It remains to show that (
which is the weight of v(i, j).
In the sequel we write ρ = ρ 0,1 for brevity. Note that we have The following assertions are proved from the definition of ξ (s) j 's by a direct calculation.
(ii) Assume that s = n.
(b) We have Let v M ∈ M s (λ) denote the generator in the definition. The proof of the following lemma is elementary (see, e.g., [20,Lemma 4.5]).
Proof . Since th[t]v M = 0 and the h-weights of both v D and v M are λ, it suffices to show x ρ(α)+1 α v M = 0 for all α ∈ ∆ re + . First we consider the case s = 1. The assertion for α = −(α p,n + α q,n−1 ) + δ with p < q < n is proved by applying Lemma 4.4 with β = −α p,m +δ and γ = −α q,m . The assertion for remaining α ∈ ∆ re + is easily proved from the defining relations of M 1 (λ).

Proof for
We shall prove the assertion in the case ε = + (the case ε = − is similarly proved). For i ∈ I, set which is the distance between the nodes i and n − 2 in the Dynkin diagram. We need the following lemma that is proved from [3].
(i) Let i 1 , . . . , i p ∈ I, b 1 , . . . , b p ∈ C(q) × and l 1 , . . . , l p ∈ Z >0 , and assume that Then the submodule of L q π (i 1 ) lp,bp generated by the tensor product of highest -weight vectors is isomorphic to L q p k=1 π (i k ) l,bq k is simple.
Proof . (i) For r < s, it is directly checked that L q π (is) ls,bs ⊗ L q π (ir) lr,br satisfies the condition of [3, Corollary 6.2], which assures that the module is generated by the tensor product of highest -weight vectors. Hence is also generated by the tensor product of highest -weight vectors (see [3, sentences above Corollary 6.2]). Now the assertion (i) follows by dualizing the statement and applying Lemma 2.4 (the bijection i →ī in the lemma is τ n−1,n ). (ii) We see from the above argument that L q (π (i) l,b )⊗ L q (π (j) l,bq k ) is both cyclic and cocyclic, and hence simple. Lemma 4.7. Let i ∈ I \ {n − 2, n − 1}, b ∈ C × q Z and l ∈ Z ≥0 . The graded limit L π (i) l,b π (n−1) l,bq p i −1 is isomorphic to D lw • ( i + n−1 + Λ 0 ) as a g[t]-module.
Hence by Sections 4.1 and 4.2, there exists a surjective homomorphism l,bq p i −1 .

K. Naoi
Therefore it suffices to show the equality of the dimensions. By Lemma 4.6(ii), we have L q π (i) l,b π (n−1) l,bq p i −1 ∼ = L q π (i) l,b ⊗ L q π (n−1) l,bq p i −1 , which implies dim C L π (i) l,b π (n−1) l,bq p i −1 = dim C(q) L q π (i) l,b π (n−1) l,bq p i −1 = dim C(q) L q π (i) l,b · dim C(q) L q π (n−1) where the last equality follows from [5,Proposition 5.1.3]. On the other hand, we have by [11,Theorem 1]. Hence the assertion is proved. Now let us begin the proof of the assertion L(π s ) D w • ξ (s) 1 , . . . , w • ξ (s) n . First we prove this in the case s = n. Let (a i ) i∈I be the sequence in the definition of π n = π + n (λ, a), and define U q (Lg)-modules L q [j] for 1 ≤ j ≤ n − 2 and j = n by if j < , j = n − 2, n, λ j ,a j π (n−1) λ j ,a j q p j −1 , otherwise. (4.4) There exists an injective U q (Lg)-module homomorphism L q (π n ) → L q [1] ⊗ · · · ⊗ L q [n − 2] ⊗ L q [n] by Lemma 4.6, and this induces a U A (Lg)-module homomorphism where we set Applying C ⊗ A − and taking the pull-back with respect to the automorphism ϕā, we obtain a g[t]-module homomorphism L(π n ) → j L[j], where L[j] denotes the graded limit or the tensor product of the two graded limits. Note that, by construction, this homomorphism maps a highest weight vector of L(π n ) to the tensor product of highest weight vectors. By Lemma 4.7 and [5, Proposition 5.1.3], we have Hence in order to complete the proof, it suffices to show that D (λ −λ)w • ( + Λ 0 ),λw • ( + n−1 + Λ 0 ) ∼ = D w • ξ (n) , and The first isomorphism follows since we have D w • ξ (n) ∼ = F w•σ Λ (n) ∼ = F w•σ D (λ −λ)Λ 0 ⊗ D λ ( n−1 + Λ 0 ) ∼ = D (λ −λ)w • ( + Λ 0 ),λw • ( + n−1 + Λ 0 ) by Proposition 2.1, and the second one is also proved similarly. The proof for s = n is complete. The case s = 1 can be proved by a similar (and simpler) argument in which we replace the definition of L q [i] given in (4.4) by the following: λm−λ m ,a m q λm , j = n.