Asymptotic Representations of Quantum Affine Superalgebras

We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that we established previously on tensor products of these modules, and realize their inductive limits as modules over its Borel subalgebra, the so-called $q$-Yangian. A new generic asymptotic limit of the same inductive systems is proposed, resulting in modules over the full quantum affine superalgebra. We derive generalized Baxter's relations in the sense of Frenkel-Hernandez for representations of the full quantum group.


Introduction
Let q be a non-zero complex number which is not a root of unity. Let g := gl(M, N ) be the general linear Lie superalgebra. Let U q ( g) be the associated quantum affine superalgebra. (We refer to §2.2.2 for the precise definition.) This is a Hopf superalgebra neither commutative nor co-commutative, and it can be seen as a deformation of the universal enveloping algebra of the following affine Lie superalgebra: Here the E ij for 1 ≤ i, j ≤ M + N are the elementary matrices in g.
In this paper, we are concerned with a distinguished family of finite-dimensional U q ( g)modules, the so-called Kirillov-Reshetikhin modules. We would like to construct inductive systems of these modules, and realize their inductive limits as modules over the upper Borel subalgebra of U q ( g), which is in our context the q-Yangian Y q (g). These limits may or may not carry an action of the full superalgebra U q ( g), and they may also be finite-dimensional.
1.1. Background. Our motivation for studying representations of quantum affine superalgebras comes, on the one hand, from the integrability structure of AdS/CFT correspondence where various quantum superalgebras related to sl(2, 2) show up (see for example [BGM12]), and on the other hand, from the generalization of Hernandez-Jimbo's construction of asymptotic representations related to Baxter's Q-operators to the super case. In is paper, we concentrate on the second point.
In the early seventies, towards the study of the eight-vertex model, Baxter [Ba72] introduced the concept of Q-operators and T-Q functional relations to determine the eigenvalues of transfer matrices. Ever since, various progress has been made towards understanding and generalizing Baxter's Q-operators and T-Q relations, notably the approach by using representation theory of quantum affine (super)algebras.
In a series of papers [BLZ96,BLZ97,BLZ99], Bazhanov-Lukyanov-Zamolodchikov (BLZ for short) generalized T, Q-operators from lattice models to integrable quantum field theory. The idea goes roughly as follows. Firstly, one gets a representation W of the lower Borel subalgebra B − of the quantum affine algebra U q ( sl 2 ), which is either a finite-dimensional evaluation module for the lattice model, or an infinite-dimensional vertex representation for the quantum field theory. W is referred to as a quantum space. Secondly, one constructs an L-operator, which is an element of the completed tensor product B + ⊗ EndW . (In both integrable structures, L is presumed to be the universal R-matrix R ∈ B + ⊗ B − of U q ( sl 2 ) with B − specialized to EndW . See [BLZ99,Conjecture] for the statement and [BHK02, §3.3] for a proof.) Thirdly, T, Q-operators, as elements of EndW , are defined as twisted traces of L over various representations of B + : finite-dimensional evaluation representations for T and oscillator representations for Q. Baxter's T-Q relations are then deduced from tensor product decompositions of representations of B + . The oscillator representations are constructed quite explicitly in [BLZ99] by adapting the so-called oscillator realization of B + . There are several generalizations on BLZ's construction of oscillator representations and T-Q relations when replacing U q ( sl 2 ) by higher rank quantum affine (super)algebras. In the super case, this has been partly done: by Bazhanov-Tsuboi [BT08] for the quantum affine superalgebra U q ( sl(2, 1)), and later generalized by Tsuboi [Ts12] for the quantum affine superalgebra U q ( gl(M, N )); by Kulish-Zeitlin [KZ05] and Ip-Zeitlin [IZ14] for the (twisted) quantum affine superalgebra C to construct finite-dimensional representations is already an interesting problem. Nevertheless, Drinfeld realizations corresponding to various Dynkin diagrams for this quantum affine superalgebra are known [HSTY08].
We remark also that in the super case, except for some small rank quantum affine superalgebras such as C (2) q (2) [IZ14] and U q ( gl(2, 2)) [Ga98], there is still no explicit formula of Damiani type [Da98] for the universal R-matrix of quantum affine superalgebras.
1.2. Asymptotic representations. Recently, Hernandez-Jimbo [HJ12] constructed the analogue of oscillator representations for an arbitrary (non-twisted) quantum affine algebra U q ( g ′′ ). Their construction is based on a well-studied family of finite-dimensional modules, the so-called Kirillov-Reshetikhin modules. They first constructed inductive systems of Kirillov-Reshetikhin modules, and then endowed their inductive limits with actions of the upper Borel subalgebra in an asymptotic way, resulting in oscillator representations for U q ( g ′′ ). The asymptotic construction eventually enables Frenkel-Hernandez [FH13] to interpret generalized T-Q relations in terms of representations and to prove a conjecture of Frenkel-Reshetikhin on the spectra of quantum integrable systems [FR99]. We refer to [He14] for a general review.
The advantage of Hernandez-Jimbo's construction is that the asymptotic modules have simpler representation structures compared to finite-dimensional modules and they give rise to generalized T-Q relation in [FH13] whose proof does not need universal R-matrix. (The complete proof of Frenkel-Reshetikhin conjecture however does.) In the present paper, we would like to apply Hernandez-Jimbo's asymptotic construction to the quantum affine superalgebra U q ( g). As we shall see, their inductive systems of Kirillov-Reshetikhin modules give rise to the oscillator representations of the upper Borel subalgebra Y q (g) in [BT08,Ts12]. We also propose a new application of these inductive systems by realizing their inductive limits as one-parameter families of representations of the full quantum affine superalgebra.
1.3. Main results. To state in a neat way the resulting asymptotic representations in this paper, let us introduce several notations (slightly different from the main text).
Let I 0 := {1, 2, · · · , M + N − 1} be the set of Dynkin vertices for the Lie superalgebra g. For r a Dynkin vertex and f (z) a rational function with f (0) ∈ C × , there is a simple Y q (g)-module, denoted by S r (f ), which is generated by a highest ℓ-weight vector v satisfying φ r (z)v = f (z)v and φ i (z)v = v whenever i = r. Here the φ i (z), as formal power series in Y q (g), are quantum affine analogues of diagonal matrices in g.
Notably, for r a Dynkin vertex, a a non-zero complex number (called spectral parameter), and k a positive integer (called lever of representation), the simple module S r (q k r 1−za 1−zaq 2k r ) is called a Kirillov-Reshetikhin module. It is a finite-dimensional evaluation module, and its Y q (g)-module structure can be extended to that of U q ( g)-module.
1.3.1. Inductive systems of Kirillov-Reshetikhin modules. From now on, fix the Dynkin vertex r and the spectral parameter a. Our first main task in this paper is to construct an inductive system of the Kirillov-Reshetikhin modules S r (q k r 1−za 1−zaq 2k r ) with respect to the level of representation (see §4.4-4.5) (1.1) F k,l : S r (q l r 1 − za 1 − zaq 2l r ) −→ S r (q k r 1 − za 1 − zaq 2k r ) for l < k.
The idea of construction is the same as that of Hernandez-Jimbo [HJ12]: by using the cyclicity property of some particular tensor products of Kirillov-Reshetikhin modules. In our case, the following guarantees the existence of inductive systems (Theorem 3.1): (A) Assume l 1 < l 2 < l 3 . The tensor product S r ( 1−zaq 2l 2 as a Y q (g)-module, is of highest ℓ-weight. The proof relies on a more general cyclicity result in our previous paper [Zh14] and on some duality argument. (See §3 for details.) One special feature of the above inductive system is that the structural maps F k,l do not respect the U q ( g)-module structures. That is to say, given a generator x of the algebra, the two maps F k,l x and xF k,l are in general different. Nevertheless, one can establish stability and asymptotic properties of the F k,l , which enable us to express asymptotically these maps xF k,l , with l, x fixed and with k varying; they turn out to be Laurent polynomials in q k of a particular form (see Propositions 4.6-4.7 for precise statements).
In the non-graded case, to prove these properties, Hernandez-Jimbo used some deep theory of q-characters of tensor products [He10, Proposition 3.2], which is by no means available to us as we do not even have the notion of q-character for representations of quantum affine superalgebras. However, when working with the RTT realization of U q ( g), we are able to prove these two properties in a straightforward manner.
1.3.2. Asymptotic construction of Hernandez-Jimbo. The argument of Hernandez-Jimbo, which provides the inductive limit V ∞ of the inductive system (1.1) with a module structure of the upper Borel subalgebra, can be adapted to our situation without difficulty. Take the highest ℓ-weight vectors v k in S r (q k r 1−za 1−zaq 2k r ) as an example. They give rise to the same vector v ∞ ∈ V ∞ . To get φ r (z)v ∞ , note first of all φ r (z)v k = q k r 1−za 1−zaq 2k r v k . By forgetting the term q k r , and then by taking the limit lim k→∞ q k r = 0, we obtain φ r (z)v ∞ = (1 − za)v ∞ . The stability and asymptotic properties of F k,l explains in a conceptual way the validity of this argument for all vectors in V ∞ . Eventually, we get a representation (ρ, V ∞ ) of Y q (g) having S r (1 − za) as a simple sub-quotient, which is the desired oscillator representation of Y q (g) within the framework of Bazhanov-Tsuboi [BT08,Ts12].
There are formal series , giving rise to the expression lim k→∞ q −2k r , which is nonsense. For this reason, S r (1 − za) does not carry an action of U q ( g).
1.3.3. Generic asymptotic construction. Starting from the same inductive system (1.1) and from the same stability and asymptotic properties, we propose in this paper a new limit process which endows the inductive limit V ∞ with U q ( g)-module structures. Again take the highest ℓ-weight vectors v k as an example. We have the asymptotic expressions φ r (z)v k = q k r 1−za 1−zaq 2k r v k . Now, fix a non-zero complex number b. By taking the limit lim k→∞ q k r = b, we obtain φ r (z)v ∞ = b 1−za 1−zab 2 v ∞ . Note that the expression lim k→∞ q −k r = b −1 makes perfect sense. Again, thanks to the stability and asymptotic properties, we get a representation (ρ b , V ∞ ) of the full quantum affine superalgebra U q ( g) on the inductive limit. In particular, it has a simple sub-quotient S r (b 1−za 1−zab 2 ). Informally, one can think of (ρ + , V ∞ ) as (ρ 0 , V ∞ ).
1.3.4. Category O and q-character for representations of Y q (g). Now, following [HJ12], we introduce a category of representations of Y q (g) including all the Kirillov-Reshetikhin modules, (ρ + , V ∞ ) and (ρ b , V ∞ ) constructed above and we define the notions of q-character χ q and normalized q-character χ q . Let us put together the main results obtained in this paper on category O and (normalized) q-character.
(B) As in the non-graded case [He05,HJ12,MY14], there is a classification of simple Y q (g)-modules, simple U q ( g)-modules, and finite-dimensional simple Y q (g)-modules in category O in terms of rational functions (Lemma 6.6). Contrary to the nongraded case, there are finite-dimensional simple Y q (g)-modules which cannot be U q ( g)-modules. (C) In the case g = gl(1, 1), the explicit formula χ q (S 1 (f )) for all rational function f such that f (0) = 1 is deduced. See Equation (6.39). (D) If S is an evaluation module of a polynomial representation, or of the dual of a polynomial representation of U q (g), then χ q (S) is multiplicity-free, and it is a polynomial in the [A i,x ] −1 where i ∈ I 0 and x ∈ C × (Corollaries 7.2,7.5). Here as in the non-graded case, the A i,x are the generalized simple roots, and U q (g) is the finite-type quantum superalgebra. (E) Fix a Dynkin vertex r ∈ I 0 and a spectral parameter a. Then for all b ∈ C × as formal power series in the [A i,x ] −1 with coefficients 0 or 1. (ρ + , V ∞ ) is a simple Y q (g)-module, and (ρ + , V ∞ ) is simple provided some generic condition on b.
(Corollaries 8.8,8.15) (F) Assume that b is generic. Then related to the following chains of subalgebras: an evaluation module of the dual of a polynomial representation, we do not have similar closed formula for χ q (S).
The asymptotic constructions is quite Lie theoretic. It should eventually be done for more general quantum affine superalgebras like U q ( D(2, 1; α)), once we know how to define analogues of Kirillov-Reshetikhin modules, their character formulas and some cyclicity result of particular tensor products. (This is the reason why we are restricted to U q ( g) in the present paper.) Moreover, such constructions should work for non-quantum algebras: semisimple Lie algebras, current algebras and affine algebras, as we have good candidates for Kirillov-Reshetikhin modules such as Demazure modules and Weyl modules. We hope to return to these issues in future works.
1.4. Outline. This paper is organized as follows. In §2, we recall the definitions concerning quantum (affine) superalgebras and Yangians. Then we study the Gelfand-Tsetlin bases for certain finite-dimensional simple representations for the quantum superalgebra U q (g). §3 proves a cyclicity result of tensor products of Kirillov-Reshetikhin modules to be needed in the construction of inductive systems of Kirillov-Reshetikhin modules. In §4-5 we carry out in detail the asymptotic constructions for the quantum affine superalgebra U q ( g), which are illustrated with explicit examples in §5.3. In §6 we introduce category O and q-character, discuss their general properties, and study in detail the case g = gl(1, 1) in §6.6. In §7 we compute the normalized q-character for some evaluation modules. Then in §8 we compute normalized q-character of asymptotic modules and establish Gelfand-Tsetlin bases for them.
About notations, in the introduction, for the sake of simplicity and for unifying the cases r ≤ M and r > M , we have used the nonstandard notation S r (f ) for the Kirillov-Reshetikhin modules. In the main text, we adopt the more classical notation W (r) k,a . Relevant results and proofs are usually divided into two parts depending on r. Vyjayanthi Chari for their interest in the present work and for valuable discussions. Part of it was done while he was visiting Centre de Recherches Mathématiques in Montréal. He would like to thank Masaki Kashiwara for remarks on cyclicity of tensor products, and thank Eugene Mukhin and Weiqiang Wang for remarks on Gelfand-Tsetlin basis and q-character.

Preliminaries
In this section we first collect basic facts about the RTT realizations of the quantum affine superalgebra U q ( g), the q-Yangian Y q (g), and the quantum superalgebra U q (g). Next, we review part of the Schur-Weyl duality theory for tensor powers of the natural representation of U q (g), following Benkart-Kang-Kashiwara [BKK00]. Then we show the existence of Gelfand-Tsetlin basis for certain simple representations of U q (g): polynomial representations and their duals.
2.1. Conventions. Throughout this paper, all the vector superspaces and superalgebras are defined over the base field C. We fix q ∈ C to be non-zero and not a root of unity. Fix M, N ∈ Z >0 . Let I := {1, 2, · · · , M + N }. Define the following maps: Set q i := q d i . Set P := ⊕ i∈I Zǫ i . Let (, ) : P × P −→ Z be the bilinear form defined by (ǫ i , ǫ j ) = δ ij d i . Let | · | : P −→ Z 2 be the morphism of abelian groups such that In the following, only three cases of |x| ∈ Z 2 are admitted: x ∈ I; x ∈ P; x is a Z 2homogeneous vector of a vector superspace. If a vector space V = α∈P (V ) α carries a grading by an abelian group P , then we write |x| P = α for α ∈ P and x ∈ (V ) α .
Let V := i∈I Cv i be the vector superspace with Z 2 -grading |v i | = |i|. Then EndV is naturally a superalgebra. Let E ij ∈ EndV be the linear endomorphism v k → δ jk v i . In particular, |E ij | = |i| + |j|. Let g be the general linear Lie superalgebra associated with V, which is, the vector superspace EndV endowed with the Lie bracket: for f, g ∈ EndV homogeneous. In the following, EndV will always be viewed as a superalgebra, and g as a Lie superalgebra, although they are the same as vector superspaces.
2.2. Quantum superalgebras. In this subsection, we review the RTT realizations of the quantum affine superalgebra U q ( g), the q-Yangian Y q (g), and the finite type quantum superalgebra U q (g), following [Zh14].
Remark that (2.4)-(2.6) are operator equations in ]. Let us express for example Equation (2.5) in terms of matrix coefficients: (2.9) (2.11) and antipode S : Here the RHS of the above formulas are well defined thanks to Relation (2.7).
2.2.3. q-Yangian. The subalgebra of U q ( g) generated by the s ii ) −1 is called the q-Yangian, denoted by Y q (g). As shown in [Zh14, Proposition 3.10], these generators together with Z 2 -grading (R2) in §2.2.2, Relation (2.8) with the t ij , and Relation (2.5) (or equivalently Relation (2.9)) give a full presentation for the superalgebra Y q (g).
Here, as usual, T = i≤j t ji ⊗ E ji , S = i≤j s ij ⊗ E ij ∈ U q (g) ⊗ EndV. U q (g) is endowed with a Hopf superalgebra structure with similar coproduct as in Formulas (2.10)-(2.11).
The relationship between U q ( g) and U q (g) is explained as follows.
We understand that s ji = t ij = 0 in the superalgebra U q (g) for 1 ≤ i < j ≤ M + N . The morphism ev is called an evaluation morphism. It is clear that ev • ι = Id Uq(g) .
2.2.5. Structures of quantum superalgebras. Let us gather together in this paragraph the main properties of U q ( g), Y q (g), U q (g) which will be used later on.
(a) For a ∈ C × , there is an automorphism of Hopf superalgebra ij . Φ a restricts to an automorphism of the q-Yangian still denoted by Φ a : Y q (g) −→ Y q (g). Let us define the evaluation morphism ev a := ev • Φ a . So ev 1 = ev. (b) The following relations hold in U q ( g) in view of Equations (2.4)-(2.8): Relation (2.15) gives rise to the weight grading on U q ( g): for α ∈ Q, In particular, for i, j ∈ I we have |s There exists a superalgebra automorphism These automorphisms behave well under coproduct in the following way: Moreover φ (f,g) restricts to a superalgebra automorphism of q-Yangian denoted by φ g : Y q (g) −→ Y q (g) as it does not depend on f . (d) The following defines an isomorphism of Hopf superalgebras ji . Here ε ij := (−1) |i|(|i|+|j|) for i, j ∈ I.
We remark that the automorphism Ψ and Relations (2.15)-(2.17) degenerate directly to the quantum superalgebra U q (g) thanks to Proposition 2.1. In particular, U q (g) is Q-graded in the obvious way.
2.3. Schur-Weyl duality. There is a natural representation ρ (1) of U q (g) on the vector superspace V defined by the following matrix equations [Zh14, §4.4]: We would like to understand the U q (g)-module structure of the tensor powers V ⊗s .
2.3.1. Highest weight representations. Let λ ∈ P. Up to isomorphism, there exists a unique simple U q (g)-module, denoted by L(λ), which is generated by a vector v λ satisfying: For example, as U q (g)-modules, V ∼ = L(ǫ 1 ) in view of the following equations: Characters. Usually a U q (g)-module V is P-graded via the action of the s ii : From the Q-grading on U q (g) we see that (U q (g)) α (V ) β ⊆ (V ) α+β for α ∈ Q and β ∈ P. Assume furthermore that all weight spaces (V ) α are finite-dimensional. Introduce characters Here Z P is the abelian group of functions P −→ Z and [α] = δ α,· . For example, is the group ring of P over Z.
2.3.3. Young combinatorics. Let us introduce several combinatorial objects before stating Schur-Weyl duality.
(1) Let P ⊆ P be the subset consisting of λ = i∈I λ i ǫ i such that For such λ ∈ P, define ht(λ) := i∈I λ i ∈ Z ≥0 .
(2) A g-Young diagram is a Young diagram Y such that (M + 1, N + 1) / ∈ Y . In other words, it is a finite subset Y ⊂ Z >0 × Z >0 satisfying Let YD be the set of g-Young diagrams.
(3) For Y ∈ YD, a g-Young tableau of shape Y is a map f : Let B(Y ) be the set of g-Young tableaux of shape Y .
(4) For λ = i λ i ǫ i ∈ P, define Y λ to be the g-Young diagram formed of such (i, j) that Let Y : P −→ YD, λ → Y λ be the bijective map thus obtained. On the other hand, we shall make the obvious inclusions: Theorem 2.4.
[BKK00] For all s ∈ Z ≥1 , the U q (g)-module V ⊗s is completely reducible. More precisely, we have a decomposition into simple sub-U q (g)-modules as follows: Here c λ ∈ Z ≥0 and c λ = 0 if and only if ht(λ) = s. Furthermore, for λ ∈ P (2.21) Such representations L(λ) with λ ∈ P are usually called polynomial representations, as they appear as simple submodules of tensor powers of the natural representation.
Let us end this paragraph with the following simple application of the character formula (2.21). This result, on the asymptotic behaviour of weight spaces, serves as a motivation for our limit construction carried out later.
Proof. Note that k̟ r ∈ P for all k ∈ Z >0 . Furthermore, In view of Formula (2.21), we only need to check the following: for all k > lr and for all f ∈ B Y k̟r such that we must have f (i, 1) = i for 1 ≤ i ≤ r. Let us fix such an f . The condition (L(l̟ r )) l̟r−β = 0 says the existence of g ∈ B(Y l̟r ) such that This says that In particular, c i ≥ k − l for 1 ≤ i ≤ r. We prove by induction on 1 ≤ i ≤ r that: f (i, j) = i for all 1 ≤ j ≤ k−il. For i = 1, this is obvious from the definition of Young tableau. Suppose the assertion for i − 1 true. If there exists 1 Then we have the following: (1) m ≤ i, from the definition of Young tableau; (2) if m = i, then 1 ≤ n < j 0 . In particular, n ≤ k − il − 1; (3) if m < i, then k − (i − 1)l < n ≤ k as f (i − 1, j) = i − 1 for 1 ≤ j ≤ k − (i − 1)l. Note that (m, n), (m ′ , n) ∈ f −1 (i) forces m = m ′ . By counting the number of n in f −1 (i) we get ♯f −1 (i) = c i ≤ k − il − 1 + (i − 1)l = k − l − 1, which contradicts with the fact that c i ≥ k − l. Thus, f (i, j) = i for all 1 ≤ j ≤ k − il and 1 ≤ i ≤ r, as desired.
For example, when r = 1, all the weight spaces of L(k̟ 1 ) are one-dimensional.
2.4. Gelfand-Tsetlin basis. Following Remark 2.3, let X be the set of pairs (a, b) ∈ Z 2 such that: either (a = M, 1 ≤ b ≤ N ); or (1 ≤ a ≤ M, b = 0). Let k → (M k , N k ) be the unique bijective map I −→ X such that M k + N k = k for k ∈ I. For k ∈ I, let U q (g k ) be the subalgebra of U q (g) generated by the s ij , t ji with 1 ≤ i ≤ j ≤ k. Then U q (g k ) is a sub-Hopf-superalgebra canonically isomorphic to U q (gl(M k , N k )), and we have the following sequence of inclusions of Hopf superalgebras: Moreover, the restriction of f to this subset gives us a g k−1 -Young tableau of corresponding shape.
As in the non-graded case, Gelfand-Tsetlin patterns are in one-to-one correspondence with Young tableaux.
For k ∈ I, λ ∈ P M k ,N k and s ∈ Z 2 , let L(λ; g k ) be the simple U q (g k )-module generated by a highest weight vector of highest weight λ and of Z 2 -degree |λ|; let C s be the onedimensional U q (g k )-module of zero weight and of Z 2 -degree s. The following is a direct consequence of Theorem 2.4. Corollary 2.9. Let 2 ≤ k ≤ M + N and let λ ∈ P M k ,N k . Then the U q (g k−1 )-module Res Uq(g k ) Uq(g k−1 ) L(λ; g k ) is completely reducible with the following decomposition: as U q (g k−1 )-modules. Now we get an explicit description of Gelfand-Tsetlin basis as follows.

Dual Gelfand-Tsetlin basis.
We shall also need Gelfand-Tsetlin bases for U q (g)modules L(λ) * with λ ∈ P. In general, if H is a Hopf superalgebra and if V is an H-module, then the vector superspace V * = hom(V, C) is endowed with an H-module structure by: ) be a Gelfand-Tsetlin basis for L(λ). Let (v * λ : λ ∈ GT(λ)) be its dual basis for L(λ) * . Since the embeddings U q (g k−1 ) −→ U q (g k ) respect the Hopf superalgebra structures, and since the restricted modules Res for λ ∈ GT(λ) and k ∈ I. For this reason, call (v * λ : λ ∈ GT(λ)) a dual Gelfand-Tsetlin basis. Let us compute the highest weight of the U q (g)-module L(λ) * when λ ∈ P. Let Y λ = Y(λ) be the associated g-Young diagram. For all i, j ∈ Z >0 , define . Then the highest and lowest weights λ, λ b of L(λ) can be written as: Moreover, as U q (g)-modules, we have the identification In the non-graded case, λ b = w 0 (λ) where w 0 is a longest element of the Weyl group associated with the Lie algebra. In particular, (λ + µ) b = λ b + µ b for λ, µ dominant. Such additive formula is no longer true in our case. For example, take (M, N ) = (2, 1). Then (ǫ 1 ) b = ǫ 3 and (2ǫ 1 ) b = ǫ 2 + ǫ 3 .
Remark 2.11. (1) For the quantum superalgebra U q (g), Palev-Stoilova-Van der Jeugt [PSV94] established the Gelfand-Tsetlin bases for the so-called essentially typical representations. In our situation, for λ ∈ P, L(λ) is essentially typical if and only if (M, N ) ∈ Y λ . When this is the case, a combinatorial description of the set GT(λ) similar to the example above has been given (cf. Equations (9)-(10) in loc. cit). Moreover, explicit actions of the generators e ± i with respect to the Gelfand-Tsetlin bases were given therein.
(2) For the Lie superalgebra g, there are analogs of such representations L 0 (λ) as λ ∈ P constructed as simple submodules of tensor powers of the natural representations of g on V, also called covariant representations. (One can view L(λ) as a deformation of L 0 (λ).) In [Mo10], Molev constructed Gelfand-Tsetlin bases for all the L 0 (λ) and deduced explicit formulas of the actions of the generators E ij with i, j ∈ I, |i − j| = 1 with respect to these bases. The main ingredients used by Molev are the Yangian Y (gl N ) and the Mickelsson-Zhelobenko algebra Z(g, gl M ).
(3) We shall give another characterization of Gelfand-Tsetlin basis for L(λ) and dual Gelfand-Tsetlin basis for L(λ) * within the framework of representations of the quantum affine superalgebra U q ( g). (See Propositions 7.1 and 7.4.)

Kirillov-Reshetikhin module and tensor products
We prove a cyclicity result on tensor products of Kirillov-Reshetikhin modules over U q ( g), based on our previous result [Zh14,Theorem 4.4] and on duality arguments.

Kirillov-Reshetikhin modules. Let us recall the definition of Kirillov-Reshetikhin modules
Similarly, the notions of lowest ℓ-weight vector and lowest ℓ-weight module are defined by For example, let v k̟r ∈ L(k̟ r ) be as in §2.3.1, and let v s ∈ C s be a non-zero vector with s = |k̟ r |.
k,a is a highest ℓ-weight vector and The rest of this section is devoted to proving the following cyclicity result.
Theorem 3.1. Let r ∈ I 0 , a ∈ C × and l 1 , l 2 , l 3 ∈ Z >0 such that l 1 < l 2 < l 3 . ( Let us recall another cyclicity result before turning to the proof.
k,a;J be the corresponding Kirillov-Reshetikhin modules over U q ( g ′ ). Then, as The first one is afforded by the Hopf superalgebra structure on U q ( g), the corresponding U q ( g)-module still denoted by V * as in §2.5. The second one is the pull back Ψ * V * , where Ψ is the superalgebra isomorphism given by (2.19). Let us write In this case, v is also called a cogenerator and we write U q ( g)v =: socle(V ).
Remark 3.3. Let V be a U q ( g)-module P-graded with respect to the action of the s (0) ii : The proof of Theorem 3.1 (1) will go as follows: first we compute the duals (W (r) k,a ) ∨ and rephrase (1) as a statement of cogenerators; next we realize the duals (W (r) k,a ) ∨ as simple submodules arising from cogenerators and apply Theorem 3.2 and Remark 3.4 to conclude.
We close this subsection with the following convention. Let S 1 , S 2 be U q ( g)-modules. We say that S 1 ≃ S 2 if there exists a one-dimensional U q ( g)-module D making the two U q ( g)-modules S 1 and S 2 ⊗ D isomorphic. Remark that ≃ does not change the property of being of highest ℓ-weight, of lowest ℓ-weight, or cogenerated.
3.2. Duals of Kirillov-Reshetikhin modules. Throughout this subsection, 1 ≤ r ≤ M . We shall determine (W In view of the P-grading on the simple module (W From Formulas (2.12)-(2.13) we get We are led to determine the inverses of X(z), Y (z). In view of Proposition 2.1, Therefore, it is enough to find the inverse of Let i, j ∈ I. When i = j, When i = j, Let (x ij (z)) i,j∈I be the inverse of M M,N (z). Then Remark 3.5. One can write down explicitly the inverse of M M,N (z): It follows from Equation (3.26), Theorem 3.2 (II) and Remark 3.3 that the tensor product r j=1 W (1) In particular, the submodule U q ( g)v is simple and it is exactly the socle of r j=1 W (1) 1,bq −2j . By comparing the lowest ℓ-weight vectors, we see that On the other hand, the U q ( g)-module ( r j=1 W (1) 1,bq −2j ) ∨ is of lowest ℓ-weight generated by v * by Remark 3.3, and its simple quotient is the twisted dual of U q ( g)v. Making use of Equation (3.26) we conclude that 1,bq −2j is of highest ℓ-weight. We argue in a similar way as in the preceding paragraph (replacing lowest by highest): Similar duality argument shows that 3.3. Proof of Theorem 3.1. According to §3.1.1, it is enough to prove (1). Fix 1 ≤ r ≤ M and a ∈ C × . Let l 1 , l 2 , l 3 ∈ Z >0 be such that l 1 < l 2 < l 3 . Set c := q 2(M −N −r+1) . In view of Remark 3.3 (1) is equivalent to the following assertion: l 3 −l 2 ,aq −2l 2 ) ∨ is cogenerated by the tensor product of highest ℓ-weight vectors. On the other hand, Equations (3.27) and (3.28) say that By Theorem 3.2, Remark 3.3 and Equation (3.27), the big tensor product is cogenerated by the tensor product of highest ℓ-weight vectors. Hence (2) follows from Remark 3.4.

Asymptotic construction of Hernandez-Jimbo
In this section, following the idea of Hernandez-Jimbo in [HJ12], we construct inductive systems of Kirillov-Reshetikhin modules and endow Y q (g)-module structures on their inductive limits with the help of asymptotic algebras.
To the end of this section, we will construct Y q (g)-modules L ± r,a where r ∈ I 0 and a ∈ C × such that L + r,a contains a highest ℓ-weight vector v + with 4.2. Asymptotic algebras. We propose in this subsection two versions of asymptotic algebras in the super case. Note that in the non-graded case, asymptotic algebras have been defined for all quantum affine algebras in [HJ12, §2.2].
Definition 4.1. DefineỸ q (g) and Y q (g) to be the two subalgebras of Y q (g) generated bys ij with i, j ∈ I, n ∈ Z ≥0 respectively. Herẽ CallỸ q (g) and Y q (g) asymptotic algebras.
Indeed, one can write out the full defining relations of asymptotic algebras in terms of the generatorss ij . Take Y q (g) for example. It is the superalgebra defined by (As1) generatorss Remark also thatỸ q (g) and Y q (g) are Q-homogeneous subalgebras as their generators are Q-homogeneous. EndowỸ q (g) and Y q (g) with the following C[P]-module structures: From the Ice Rule (2.3) and §2.2.3 comes the following: We adapt the notion of Q-graded modules over asymptotic algebras in [HJ12,§2.4].
The following corollary is an application of the lemma above on the semi-direct product constructions of q-Yangian. It is parallel to [HJ12, Proposition 2.4].
Corollary 4.4. Let V = α∈Q V (α) be a Q-gradedỸ q (g)-module (resp. Y q (g)-module) in the sense of Hernandez-Jimbo. Then the module structure over the asymptotic algebra can be extended to that over Y q (g) by setting Let v lk ∈ Z (r) (l < k, a) be a highest ℓ-weight vector.
The following lemma is a direct consequence of Theorem 3.1 (1).
In consequence, for l < k, there exists a unique morphism of U q ( g)-modules k,a are P-graded with respect to the action of the s (0) ii . Also, Theorem 2.4 gives a combinatorial description of dimensions of the weight spaces (W Proposition 4.6. The maps (F kl : l < k ∈ Z >0 ) verify the following properties: for all i, j ∈ I, n ∈ Z ≥0 and l, k ∈ Z >0 with t(l) < k. Proof.
(1) comes easily from the action of the s i,i+1 v lk = 0, we see that F k,l respects the action of thes (0) i,i+1 with i ∈ I 0 . We are left to verify the following: let β ∈ Q ≥0 \ {0} and x ∈ (W (r) l,a ) l̟r−β be such thats l,a = ev * a L(l̟ r ) with L(l̟ r ) the simple U q (g)-module of highest weight l̟ r . Let such x ∈ L(l̟ r ) be given (we use the quantum superalgebra U q (g) and its generators s ij , t ij ). Then s i,i+1 x = 0 for all i ∈ I 0 . It follows from Equation (2.17) and Proposition 2.1 that s ij x = 0 for all 1 ≤ i < j ≤ M + N . x is a highest weight vector for the U q (g)-module L(l̟ r ), which is impossible.
For (3), remark first that the U q ( g)-module S := W (r) l,a ⊗ Z (r) (l < k, a) ⊗ Z (r) (k < u, a) is of highest ℓ-weight by Theorem 3.1 (1). In consequence, F u,k • (F k,l ⊗ Id Z (r) (k<u,a) ) is the unique morphism of U q ( g)-modules F : S −→ W (r) u,a sending v l ⊗ v lk ⊗ v ku to v u . On the other hand, that Z (r) (l < k, a) ⊗ Z (r) (k < u, a) is of highest ℓ-weight gives us a surjective morphism of U q ( g)-modules It follows from the uniqueness of F that k<u,a) ).
Applying the above equation to W (4) comes from the asymptotic behaviour of dimensions of weight subspaces observed in Corollary 2.6.
We remark that in the non-graded case [HJ12], (4) is proved by using a deep property [He10, Proposition 3.2] of q-character concerning tensor product of two vectors (not necessarily two modules). In particular, it was shown [HJ12, Lemma 4.4] that t(l) = l + 1.
Let us fix t : Z >0 −→ Z >0 in (4). Thanks to (2) the following operators are well-defined for i, j ∈ I, n ∈ Z ≥0 and k, l ∈ Z >0 such that k > t(l). Also, (W (r) k,a , F kl ) is an inductive system of vector superspaces. Let (W ∞ , F k ) be its inductive limit with Proof. It is enough to establish the existence of these maps A ij ) = 0 in the following cases: either (n ≥ 2), or (n = 1, i < j), or (n = 0, i > j).
In other words, ρ k (s . Observe furthermore that s j ′ j (z)v lk = 0 if j > r and j ′ = j. Hence, when j > r, we have l,a . This says in particular that F −1 ij ). We are left to consider the case i > j ≤ r.
for i ′ ∈ I 0 . In order to establish Equation (4.30), it is enough to prove the following assertions AS(β) for all β ∈ Q ≥0 .
We argue by induction on ℓ(β). First suppose that β = 0. Then up to scalar x = v l . For k > t(l), define Here the second equality comes from Case B and the definition of F k,l . We have therefore As we see in the proof of Proposition 4.6, this says that The 2 × 2 matrix above is diagonalizable with eigenvalues q, q −1 . We can therefore find x ′ , y ′ ∈ (W (r) t(l),a ) t(l)̟r−αr which are linear combinations of w t(l)+1 , w t(l)+2 such that for all k > t(l). This proves the assertion AS(0). Suppose m > 0 and that the assertions For the last term, note that ρ l (s l,a ) l̟r−β+α i ′ . Hence the assertion AS(β−α i ′ ) applies. We can find The rest is parallel. Case C.3: i > r + 1 and j = r. This comes from Case C.2 and Equation (2.17): ir . Remark that (|r + 1| + |r|)(|i| + |r + 1|) = 0, so there is no sign on the left hand side.
Case C.4: j < r < i. This comes from Cases C.1-C.3 and the following relation in U q ( g): ij . This concludes the proof.
From the above proof, we see that: if i < j then A    ij (l) = 0 for n = 1. 4.3.3.Ỹ q (g)-module structure on W ∞ . Since t is strictly increasing, we get another inductive system of vector superspaces (W (r) t(k),a , F f (k),t(l) ) over the linearly ordered set Z >0 . Furthermore, the following defines a morphism of inductive systems which induces an isomorphism of inductive limits. Identify their inductive limits.
Let us show that (A a ) gives a morphism of inductive system of vector superspaces. In other words, Indeed, for all k > t(l + 1), we have The desired equations follow. We obtain therefore linear endomorphisms of W ∞ : ij is Z 2 -homogeneous of degree |i| + |j| in view of Equation (4.30). Now, as in [HJ12,§4], the assignments ij defines a representation of the super-algebraỸ q (g) on the vector superspace W ∞ . In other words, the A and by using Equation (4.30), we see that (As3) is indeed true for the A (n) ij .
4.3.4. Y q (g)-module structure on W ∞ . Proposition 4.6 (1) implies that W ∞ is endowed with a Q-grading: k,a ) α+k̟r . Now Equation (4.30) says that this is a Q-grading in the sense of Hernandez-Jimbo. In conclusion, W ∞ becomes a Y q (g)-module thanks to Corollary 4.4.
From the proof of Proposition 4.7, we see that for α ∈ Q, x ∈ (W ∞ ) (α) and j ∈ I, Moreover, W ∞ contains a highest ℓ-weight vector v ∞ := F k v k with the same action of the s ii (z) as that of the highest ℓ-weight vector v − ∈ L − r,a in §4.1. For this reason, set L − r,a to be the Y q (g)-module W ∞ thus obtained.

Asymptotic construction of L +
r,a with 1 ≤ r ≤ M . This is parallel to the construction of L − r,a . We start from an inductive system of Kirillov-Reshetikhin modules, establish stability (Proposition 4.6) and asymptotic (Proposition 4.7) properties for this inductive system, and endow Y q (g)-module structure on the inductive limit through an asymptotic algebra. As the proofs of these properties are identical to the case of L − r,a , we omit them in this subsection.
The Kirillov-Reshetikhin modules involved will be W (r) k,aq 2k for k ∈ Z >0 . For l < k, set k,aq 2k (resp. v kl ∈ Z kl ) be a highest ℓ-weight vector. For l < k, the U q ( g)-module Z kl ⊗ W (r) l,aq 2l is of highest ℓ-weight with simple quotient isomorphic to W (r) k,aq 2k . This affords a unique morphism of U q ( g)-modules k,aq 2k be the restriction: x → F k,l (v kl ⊗ x). Then the F k,l verify all the properties in Proposition 4.6 with W (r) k,a replaced by W (r) k,aq 2k everywhere. Furthermore, a detailed analysis as in the proof of Proposition 4.7 shows that: for all i, j ∈ I, n ∈ Z ≥0 and l ∈ Z >0 , there exist linear operators A ij (l) for all k > t(l).
As before, the A t(k),aq 2t(k) , F t(k),t(l) ). We obtain therefore a representation of the asymptotic algebra Y q (g) on the inductive limit: Also W + r,a is Q-graded in the sense of Hernandez-Jimbo: for α ∈ Q, x ∈ (W + r,a ) (α) if x ∈ F k (W (r) k,aq 2k ) α+k̟r for some k ∈ Z >0 . In this way, we get a representation of Y q (g) on W + r,a by Corollary 4.4. For α ∈ Q, x ∈ (W + r,a ) (α) and j ∈ I, The vector v ∞ := F k (v k ) ∈ W + r,a verifies the same conditions as v + ∈ L + r,a in §4.1. Let L + r,a be the Y q (g)-module W + r,a thus obtained.

4.5.
Construction of L ± r,a for M +1 ≤ r ≤ M +N −1. For this purpose, let g ′ = gl(N, M ) be as in §3.1.1. Recall that we have defined the quantum affine superalgebra U q ( g ′ ) by the RTT generators s  ij;J as in §3.1.1. Let Y q (g ′ ) be the subalgebra of U q ( g ′ ) generated by the s (n) ij;J . Then the isomorphism f J,I defined by Formula (3.24) restricts to an isomorphism of Hopf superalgebras which we write as f J,I ji . We construct as in §4.3-4.4 the Y q (g ′ )-modules L ± r,a;J as limits of corresponding Kirillov-Reshetikhin modules for 1 ≤ r ≤ N and a ∈ C × . Now the Y q (g)-modules L ± r,a for M + 1 ≤ r ≤ M + N − 1 and a ∈ C × are realized as: Then the tensor product Z kl ⊗ W (r) l,a is of highest ℓ-weight. This affords an inductive system (W k,a comes from the surjective map The rest is completely parallel to §4.4. Following [FH13], the Y q (g)-modules L ± r,a are called positive/negative prefundamental modules. We shall see that they are always simple. Contrary to the non-graded case where all prefundamental modules are infinite-dimensional, L ± r,a is finite-dimensional if and only if r = M . This says that there are "more" finite-dimensional simple Y q (g)-modules than finite-dimensional simple U q ( g)-modules, as we have seen in [Zh14, §5] on representation theory of Y q (gl(1, 1)). See also Proposition 6.7. 4.6. Relation with Tsuboi's work. The Y q (g)-modules L ± r,a have been constructed by Tsuboi in a different way. In [Ts12], Tsuboi proposed the notion of a contracted quantum superalgebra. This is the superalgebra defined by generatorsṡ ii ,ṡ −1 ii ,ṫ ii ,ṡ jk ,ṫ kj for i, j, k ∈ I, j < k with Z 2 -degrees |ṡ ii | = |ṡ −1 ii | = |ṫ ii | = 0, |ṡ jk | = |ṫ kj | = |j| + |k| subject to the following relations (takeṪ = jkṫ jk ⊗ E jk ,Ṡ = jkṡ jk ⊗ E jk ) iiṡ ii . LetU q (g) be the superalgebra obtained. Then the proof of Proposition 2.1 implies thaṫ ii defines a morphism of superalgebras for a ∈ C × . LetL + r be the simpleU q (g)-module generated by a highest weight vector v + : Similarly, letL − r be the simpleU q (g)-module generated by a highest weight vector v − : Then, as Y q (g)-modules, L ± r,a ∼ =ė v * aL ± r . Tsuboi constructed theL ± r via the q-oscillator realizations of the contracted quantum superalgebraU q (g), which is a generalization of the construction carried out in [BT08] for U q ( sl(2, 1)) to the case U q ( gl(M, N )).

Generic asymptotic construction
In this section, based on the inductive systems of Kirillov-Reshetikhin modules and their stability and asymptotic properties in he previous section, we propose a new asymptotic construction which realizes the inductive limits as modules over the full quantum affine superalgebra U q ( g) instead of the q-Yangian Y q (g).
Let us fix two parameters a, b ∈ C × and a Dynkin node r ∈ I 0 . To the end of this section, we shall construct a U q ( g)-module, written as L r,a (b), which has a non-zero vector v of Z 2 -degree 0 such that whereas in the case M + 1 ≤ r < M + N k,aq 2k and fix v k ∈ V k a vector of highest ℓ-weight for k ∈ Z >0 . Let (F k,l : V l −→ V k ) l<k be the inductive system of vector superspaces constructed in §4.4. Let (V ∞ , F l : Let S be the subset of U q ( g) consisting of the RTT generators s ij for i, j ∈ I, n ∈ Z ≥0 . Let ρ k be the representation of U q ( g) on V k for k ∈ Z >0 . Let t : Z >0 −→ Z >0 be a strictly increasing function such that (Proposition 4.6) It follows that ρ k (s)F k,l V l ⊆ F k,t(l) V t(l) for k > t(l) and s ∈ S. Since the F k,l are injective, the operators F −1 k,t(l) ρ k (s)F k,l : V l −→ V t(l) for s ∈ S, k > t(l) are well-defined.
Lemma 5.1. Let l ∈ Z >0 and s ∈ S. Then there exists uniquely a Hom(V l , V t(l) )-valued Laurent polynomial P l,s (z) ∈ Hom(V l , V t(l) )[z, z −1 ] such that Furthermore, the coefficients of z n in P l,s (z) are non-zero only if −2 ≤ n ≤ 3.
Proof. Assume without loss of generality s = s On the other hand, by definition of the F k,l Combining with Equation (4.31), we find a Laurent polynomial P l,s (z) with the desired property. Clearly such P l,s is unique.
Now we argue as in §4.3.3. By using the defining property of the P l,s we see that for all s ∈ S and l ∈ Z >0 : F t(l+1),t(l) P l,s (z) = P l+1,s (z)F l+1,l ∈ Hom(V l , V t(l+1) )[z, z −1 ].
Let P s [n] be the inductive limit of the above morphism. Then P s [n] ∈ End(V ∞ ) as both inductive systems give rise to the same inductive limit. As a result, the following assignment defines a representation of the quantum affine superalgebra U q ( g) on V ∞ . Let us compute the action of s In other words, P l,s (n) ij v ∞ = 0. Next, assume i = j, in view of the following equation In a similar way, we get s ii (z)v ∞ and t ii (z)v ∞ by regarding q k in the above equation as b.
The U q ( g)-module V ∞ is the desired L r,a (b), as v ∞ verifies Equation (5.32).

5.2.
Construction of L r,a (b) with r > M . In this case, by abuse of language, set V k := W (r) k,a . Following §4.5, let v k ∈ V k be a highest ℓ-weight vector and let F k,l : V l −→ V k be the structural maps of the inductive system of vector superspaces As before, choose a strictly increasing function t : Z >0 −→ Z >0 so that the following operators (ρ k denotes the representation of U q ( g) on V k ) with s ∈ S, k > t(l) are well-defined. The following lemma is proved in a similar way as Lemma 5.1.
Lemma 5.2. Let l ∈ Z >0 and s ∈ S. Then there exists uniquely a Hom(V l , V t(l) )-valued Laurent polynomial P l,s (z) ∈ Hom(V l , V t(l) )[z, z −1 ] such that F −1 k,t(l) ρ k (s)F k,l = P l,s (z)| z=q k for k > t(l).
Furthermore, the coefficients of z n in P l,s (z) are non-zero only if −2 ≤ n ≤ 1.
The rest is also parallel to the preceding subsection. We get a representation of U q ( g) on the vector superspace V ∞ . Moreover, v ∞ is killed by s ij (z), t ij (z) whenever 1 ≤ i < j ≤ n. Also by replacing q k and v l , v t(l) in the following equation with b and v ∞ , we conclude that v ∞ verifies Equation (5.33): The U q ( g)-module V ∞ is the desired L r,a (b).

5.3.
Examples. Let us give three examples to illustrate the general construction. In this subsection g = gl(2, 1). We construct L ± 2,a , L 2,a (b) explicitly. Fix a, b ∈ C × .
k,a be a lowest ℓ-weight vector. Define Then by Theorem 2.4, (v 1 , v 2 , v 3 , v 4 ) constitute a basis for the U q ( g)-module W (2) k,a . Moreover, from Relation 2.5, we deduce the explicit action of the s ij (z). Let ρ k be the representation corresponding to the U q ( g)-module W Here the E ij : v l → δ jl v i are the linear transformations on the underlying vector superspace By using these A ij (z), we get the Y q (g)-module L − 2,a = W defined by: Remark 5.3. It is straightforward to check that the representation (W, ρ − ) is simple. Following §2.4, let Y q (g 2 ) be the subalgebra of Y q (g) generated by the s ii ) −1 with n ∈ Z ≥0 and 1 ≤ i, j ≤ 2. Then as superalgebras Y q (g 2 ) ∼ = Y q (gl(2, 0)). Furthermore, is a Krull-Schmidt decomposition of the Y q (g 2 )-module Res Yq(g) Yq(g 2 ) W into indecomposable submodules. Note that the third factor is not simple. Hence the underlying Y q (g 2 )-module structure on W is not semi-simple. 5.3.2. Construction of L 2,a (b) and L + 2,a . For k ∈ Z >0 , let w 4 ∈ W (2) k,aq 2k be a lowest ℓ-weight vector. Define w 3 := s  (w 1 , w 2 , w 3 , w 4 ) is a basis for W (2) k,aq 2k . In the following, we identify the underlying vector superspaces of the W (r) k,aq (2k) for k ∈ Z >0 with the vector superspace W = W (2) 1,aq 2 by using this preferred basis. Let ρ k be the representation associated to the U q ( g)-module W Now by replacing ρ k , q k in the above formulas with ρ, b respectively, we get a representation ρ of U q ( g) W := 4 i=1 Cw i for W (2) 1,aq 2 . The corresponding U q ( g)-module is L 2,a (b). As before, for Similarly, we get the Y q (g)-module L + 2,a = W : Remark 5.4. The Y q (g)-module L + 2,a defined above is easily seen to be simple. Furthermore, contrary to Remark 5.3, it is semi-simple as a Y q (g 2 )-module.
We point out that L − 2,a and L + 2,a have been constructed by Bazhanov-Tsuboi [BT08, Appendix B.3] as representations of the upper Borel subalgebra of U q ( g) defined by Drinfeld-Jimbo generators, bearing the name W −+ 3 (x) and W +− 3 (x) respectively after scattering.

Category O and q-character
We have constructed in §4-5 the asymptotic modules: L ± r,a as modules over Y q (g) and L r,a (b) as modules over U q ( g), as certain limits of Kirillov-Reshetikhin modules W (r) k,a . In this section, we introduce a category O of Y q (g)-modules including these three kinds of modules following Hernandez-Jimbo [HJ12] and study the Frenkel-Reshetikhin q-character for this category.
6.1. Quantum Berezinian. The quantum affine superalgebra admits another system of generators, the so-called Drinfeld generators, X ± i,n , K ± j,±s where i ∈ I 0 , j ∈ I, n ∈ Z and s ∈ Z ≥0 , arising from Gauss decomposition: For example, K + 1 (z) = s 11 (z) and K − 1 (z) = t 11 (z). Now for i ∈ I 0 = I, we have We refer to [Zh14,§3.4] for more details on the relations and on the coproduct of these Drinfeld generators. Recall the definition of the d i , θ i in §2.1 and in Remark 3.5. For i ∈ I, define the quantum Berezinian Here ]. The following result comes from [Zh14, Theorem 3.2, Proposition 3.13].
Corollary 6.1. Let k ∈ I. (1) (2) For all i, j ∈ I such that i, j ≤ k, we have C k (z)s ij (w) = s ij (w)C k (z) and C k (z)t ij (w) = t ij (w)C k (z) as formal power series in U q ( g).
Let C q ( g) be the subalgebra of U q ( g) generated by the (s (0) ii ) −1 and by the coefficients of the C i (z). Then C q ( g) is indeed a commutative subalgebra of Y q (g). Let C i,0 ∈ Y q (g) be the constant term of C i (z). Then (6.35) 6.2. Weights and ℓ-weights. By using the commutative subalgebra C q ( g) of Y q (g), let us introduce the notion of a weight and an ℓ-weight. Denote P := (C × ) I . Endow it with an additive abelian group structure: Then the assignment ǫ i → ǫ ′ i extends uniquely to an injective homomorphism of abelian groups P −→ P. From now on, we view P as a free abelian subgroup of P and identify ǫ i = ǫ ′ i for i ∈ I. In this way, Q ⊂ P ⊂ P.
Similarly, denote P := (C[[z]] × ) I . Endow it with a multiplicative abelian group structure: Let ̟ : P −→ P and σ : P −→ P be two maps defined as follows: These are homomorphisms of abelian groups. Furthermore ̟ • σ = Id P . For the precise statements of results, we shall not identify P as a sub-abelian-group of P by σ. Let us introduce the analogues of ǫ i , α j ∈ P in P to be used later.
Definition 6.2. For i ∈ I and a ∈ C, define X i,a ∈ P by For i ∈ I 0 , define A i,a := X i,a X −1 i+1,a ∈ P. Call A i,a a generalized simple root if a = 0.
By definition ̟(X i,a ) = ǫ i and ̟(A i,a ) = α i . Let V be a Y q (g)-module. For a = (a i ) i∈I ∈ P, define the weight space Note that when a ∈ P, the definition of weight space (V ) a is the same as the one given in Remark 3.3, if we view U q ( g)-modules therein as Y q (g)-modules. Similarly, for f = (f i (z)) i∈I , define the ℓ-weight space ) n x = 0 for i ∈ I and n ≫ 0}.
Example 2. For λ ∈ P and a ∈ C × , the evaluation modules ev * a L(λ), viewed as Y q (g)modules, are in category O. Furthermore, the Y q (g)-modules L ± r,a for 1 ≤ r ≤ M + N − 1 are in category O. Indeed, the weights of these modules lie in −Q ≥0 , and for α ∈ Q ≥0 , where the limit at the RHS exists thanks to Corollary 2.6 in the case r ≤ M and thanks to §3.1.1 in the case r > M .
Example 3. Let r ∈ I 0 and a, b ∈ C × . Then L r,a (b), viewed as a Y q (g)-module, is in category O. Consider the case r ≤ M . Let us come back to the situation of §5.1. Let α ∈ Q ≥0 . For l ∈ Z >0 and x ∈ (V l ) l̟r−α , we have It follows that s In other words, F l (x) ∈ (L r,a (b)) λ r,b −α where λ r,b ∈ P is given by (λ r,b ) j = b min(r,j) . In consequence, (L r,a (b)) λ = 0 only if λ = λ r,b − α for some α ∈ Q ≥0 , in which case This says that L r,a (b) is in category O. When r > M , similarly one can find λ r,b ∈ P with (λ r,b ) j = b max(j−r,0) such that the above statements on weight spaces of L r,a (b) in the case r ≤ M remain true. 6.4. Character and q-character. To define classical character and q-character, let us first introduce the target rings. Let E ⊂ Z P be the set of maps c : P −→ Z satisfying c(α) = 0 for all α outside a finite union of sets of the form λ − Q ≥0 . Endow E with a ring structure: In particular, we see that E contains the group ring Z[P]. For V a Y q (g)-module in category O, define its classical character as in §2.3.2: Let E ℓ ⊂ Z P be the set of maps c : P −→ Z satisfying c(f ) = 0 for all f such that ̟(f ) is outside a finite union of sets of the form µ − Q ≥0 and such that for each α ∈ P, there are finitely many f verifying ̟(f ) = α and c(f ) = 0. Make E ℓ into a ring: Extend ̟ : P −→ P to a surjective ring morphism ̟ : E ℓ −→ E and σ : P −→ P to an injective ring morphism σ : E −→ E ℓ .
Let V be a Y q (g)-module in category O. For α ∈ P, the weight space (V ) α , being finitedimensional, is stable by the action of C q (g). It follows that (V ) α (and hence V ) admits an ℓ-weight space decomposition V = f ∈ P (V ) f with finite-dimensional ℓ-weight spaces. Define the q-character Note that for all f ∈ P, Hence In other words, χ q is a refinement of χ.
Remark 6.3. We shall also need the notion of a normalized q-character. Let V be in category O. Assume that there exists a λ ∈ P such that: Clearly, for such a Y q (g)-module V , the weight λ and the ℓ-weight f are uniquely determined. Define the normalized character χ and normalized q-character χ q as follows: In the following, whenever we write χ(V ) or χ q (V ), it should be understood implicitly that V verifies the above weight conditions.
Example 4. Let r ∈ I 0 and a, b ∈ C × . Then from Examples 2-3 we see that (6.37) χ(L ± r,a ) = χ(L r,a (b)) = lim Lemma 6.4. Let V, W be in category O. Then Proof. The proof is exactly the same as in the non-graded case (see [FR99, Remark 2.6]) in view of the coproduct formula in Corollary 6.1 (1), by using the partial order on P induced by Q ≥0 .
Remark 6.5. Let V, W be in category O. In general, it is not true that For example, if V, W verify such weight conditions that their normalized characters exist (Remark 6.3), then so does V ⊗ W and 6.5. Simple modules in category O. As in the non-graded case [HJ12, Theorem 3.11], we also have a classification of simple modules in category O in terms of highest ℓ-weight. Following [Zh14, §5.1.3], let R be the subset of (1 + zC[[z]]) I 0 consisting of I 0 -tuples of power series (f i (z) : i ∈ I 0 ) such that f i (z) ∈ C(z) for all i ∈ I 0 .
(1) For all f = (f i (z) : i ∈ I 0 ) ∈ R, there exists uniquely a simple Y q (g)module generated by a highest ℓ-weight vector v with Moreover, such a module is in category O. Let V (f ) be the Y q (g)-module thus obtained.
(2) All simple modules in category O can be factorized uniquely into V (f ) ⊗ C f with f ∈ R and C f one-dimensional.
Proof. The proof of Part (2) is standard as in the non-graded case. Let S be a simple module in category O. Then S must be a highest ℓ-weight Y q (g)-module. Let v ∈ S be a highest ℓ-weight vector with weight λ ∈ P. Then it is enough to show that This comes essentially from the fact that (S) λ−α i is finite-dimensional combined with the following relations and Q-grading on U q ( g): The proof of Part (1) is the same as that of [HJ12, Theorem 3.11] or [Zh14, Lemma 5.1] by realizing V (f ) as a simple sub-quotient of certain tensor product of the L ± r,a and one-dimensional Y q (g)-modules.
As in [Zh14, §5.1.1], one-dimensional Y q (g)-modules are factorized uniquely into the form Here C a is the usual sign module, C a = Cv is the Y q (g)-module with Example 5. For r ∈ I 0 and a ∈ C × , let ̟ ± r,a ∈ R be such that (̟ ± r,a ) j := (1 − zaδ jr ) ±1 ∈ 1 + zC [[z]]. Then the Y q (g)-module L ± r,a has a simple sub-quotient isomorphic to V (̟ ± r,a ) ⊗ C f ± r,a where f + r,a = 1 and f − r,a = 1 − za. Example 6. For r ∈ I 0 and a, b ∈ C × , the Y q (g)-module L r,a (b) has a simple sub-quotient isomorphic to V (f ) ⊗ D with D one dimensional and f ∈ R defined by We end this subsection with the following observations.
(2) V (f ) admits a U q ( g)-module structure extending that of Y q (g) up to tensor product by one-dimensional modules if and only if for all i ∈ I 0 , seen as a meromorphic function, The proof of this proposition is again standard as in the non-graded case, by using the Drinfeld generators of U q ( g). See [HJ12, §3.2] for Part (1), [He05,Lemma 4.11] and [MY14, Theorem 3.6] for Part (2). For example, As seen from their constructions in §4-5 the L r,a (b) are U q ( g)-modules, whereas the L ± r,a are not. Note that this proposition also gives a classification of finite-dimensional simple U q ( g)-modules in terms of highest ℓ-weight, as done in [Zh13] with a quite different approach. 6.6. Category O of the q-Yangian Y q (gl(1, 1)). In this subsection, we compute the normalized q-character for all simple modules V (f ) in category O of the q-Yangian Y q (gl(1, 1)). This serves as the first non-trivial example for the study of normalized q-character of the asymptotic modules. Note that in this case R is the set of rational functions of the form P (z) Q(z) where P (z), Q(z) ∈ 1 + zC[z] are co-prime. 6.6.1. Prime simple modules. Let a, b ∈ C be such that a = b. As we see in [Zh14, §5.2.2], the simple module V ( 1−za 1−zb ) is two-dimensional with basis v 1 , v 2 and with the action of the s ij (z) given as follows: . Let us compute the action of K + i (z). By the Gauss decomposition, K + 2 (z) = s 22 (z) + s 21 (z)s 11 (z) −1 s 12 (z) It follows that ( §6.1) By definition of normalized q-character in Remark 6.3, In particular, the normalized q-character of V ( 1−za 1−zb ) depends only on a ∈ C.
(1) Let 1 = f ∈ R. Then χ q (V (f )) is a polynomial in the [A 1,a ] −1 with a ∈ C × if and only if f (∞) = 0. Retain the above factorization for f ∈ R. Then V (f ) has multiplicity-free q-character if and only if a i = a j for all 1 ≤ i < j ≤ max(s, t).
(2) χ q (V ( 1 1−za )) = σ χ(V ( 1 1−za )). In other words, the normalized character and normalized q-character for the simple module V ( 1 1−za ) coincide. In the non-graded case, as proved by Hernandez-Jimbo [HJ12, Theorem 7.5] in the simply-laced case and later by Frenkel-Hernandez [FH13,Theorem 4.1] in full generality, this is true for the positive prefundamental modules within the framework of Hernandez-Jimbo, and it gives rise to Baxter's T-Q relation for an arbitrary non-twisted quantum affine algebra.
(3) As in [Zh14,§5], let F be the category of finite-dimensional representations of Y q (g). It is not true that F must be a subcategory of O. Nevertheless, the q-character χ q (V ) ∈ Z[ P] ⊂ E ℓ is still well-defined for V in F and it induces a ring homomorphism where K 0 (F) is the usual Grothendieck ring of the tensor category F. In order to have an injective q-character map as in the non-graded case, we need to take into account the Z 2 -grading. This is done by extending χ q to the super q-character map s.χ q : where s. dim W := dim W 0 + ε dim W 1 is the superdimension associated with a vector superspace [Se96,§1]. The resulting map s.χ q : K 0 (F) −→ Z[ P] ⊗ Z Z[ε]/(ε 2 ) is an injective ring homomorphism. In a similar way as in Remark 6.3, normalized super q-character s. χ q can be defined, and Equation (6.39) becomes (1 + ε[A 1,a i q ] −1 ).
(4) If we work directly with category O, then s.χ q (V ) is still well-defined and it induces an injective ring homomorphism s.χ q : is certain completion of the usual Grothendieck ring K 0 (O). Already in the non-graded case [HJ12, Proposition 3.12] concerning the injective q-character morphism, a completed version K c 0 (O) of Grothendieck ring is implicitly used. We are grateful to David Hernandez for this comment.

q-character of evaluation representations
In this section, we study the (normalized) q-character of the evaluation modules ev * a L(λ) and ev * a L(λ) * for λ ∈ P, following the idea of Frenkel-Mukhin [FM02,Lemma 4.7] relating ℓ-weight spaces to (dual) Gelfand-Tsetlin bases.
7.1. q-character and Gelfand-Tsetlin bases. For k ∈ I, let U q ( g k ) be the subalgebra of U q ( g) generated by the s ij with i, j ≤ k. Then as seen in Corollary 6.1, the coefficients of C k (z) are central elements in U q ( g k ). Moreover the following are isomorphic as superalgebras: U q ( g k ) ∼ = U q ( gl(M k , N k )) where M k , N k are defined at the beginning of §2.4. Also from Proposition 2.1 ev a U q ( g k ) ⊆ U q (g k ).
Proof. The idea is the same as that of [FM02,Lemma 4.7]. For completeness, let us explain briefly the main steps. Let (v λ : λ ∈ GT(λ)) be a basis of L(λ) verifying Corollary 2.10. Fix a Gelfand-Tsetlin pattern λ = (λ (k) ) k∈I ∈ GT(λ) and let f := GT −1 λ (λ) be the associated Young tableau. For k ∈ I, there exists a sign module C s such that v λ is contained in a simple sub-U q ( g k )-module S isomorphic to Here the second equation above comes from Gauss decomposition and from the fact that v + is a highest ℓ-weight vector of the U q ( g k )-module S. It is enough to show For example, let us assume k > M . Then That V (λ; a) has multiplicity-free q-character comes from Definition 2.2 and from the fact that the X i,b with i ∈ I, b ∈ C × are algebraically independent in E ℓ .
As an immediate corollary, we have Proof. Let λ = i∈I1 x i ǫ i ∈ P. Let Y λ be the associated g-Young diagram. Let f + ∈ B(Y λ ) be the g-Young tableau corresponding to the Gelfand-Tsetlin pattern Then for all f ∈ B(Y λ ) and (i, j) ∈ Y λ , we have f (i, j) ≥ f + (i, j). The rest is clear from Equation (7.41) and Definition 6.2.
be the set of g-Young tableaux of shape Y . In view of Proposition 7.1, it is enough to prove the following: (1) Let f, g, h ∈ B be such that in the group P: Remark that the X i,x with i ∈ I, x ∈ C × generate a free abelian subgroup of P. Let us compare the index x of the X i,x on both sides. For 1 ≤ i ≤ r by assumption Hence the X i,ab 2 q 2(i−k)−1 for 1 ≤ i ≤ r on the RHS must disappear. Furthermore, i ≤ h(i, k) and h(i, k) ≤ h(i ′ , k) for 1 ≤ i ≤ i ′ ≤ k. It follows that h(i, k) = i for 1 ≤ i ≤ r. By definition of a g-Young tableau, this says that h(i, j) = i for all (i, j) ∈ Y .
As we see in Example 7 (by taking g = gl(2, 1), r = 2, b = q −1 ), The Lemma above is false if we remove the condition on b. More generally, let 1 ≤ s < r ≤ M and define three g-Young tableaux f, g, h of shape Y as follows: Then (1) in the proof of the Lemma is satisfied by taking b = q −s . 7.2. q-character and dual Gelfand-Tsetlin bases. Our next task is to consider such U q ( g)-modules ev * a L(λ) * with λ ∈ P. Proposition 7.4. Let λ ∈ P. Let (v * λ : λ ∈ GT(λ)) be a dual Gelfand-Tsetlin basis for the U q (g)-module L(λ) * as in §2.5. Then there exist f i,λ (z) ∈ C[[z]] for i ∈ I and λ ∈ GT(λ) such that in the U q ( g)-module ev * a L(λ) Furthermore, if f i,λ (z) = f i,µ (z) for λ, µ ∈ GT(λ), then λ (i) = µ (i) . In particular, ev * a L(λ) * , viewed as a Y q (g)-module in category O, has multiplicity-free q-character.
Contrary to Proposition 7.1, we do not have a closed formula for χ q (ev * a L(λ) * ).
Proof. The existence of the f i,λ (z) comes from the defining properties of the dual Gelfand-Tsetlin basis in §2.5. We only need to show that In the following, we show that λ Since v * λ is contained in a sub-U q (g i )-module of L(λ) * isomorphic to L(−λ (i) b ; g i ) ⊗ C |λ−λ (i) | , we get an explicit formula for f i,λ : Note that s − 1 + a i+1−s < t − 1 + a i+1−t for 1 ≤ s < t ≤ i. Hence a s = a ′ s for 1 ≤ s ≤ i.
Then the x l , y j (similar for the x ′ l , y ′ j ) verify the following conditions: (A) x 1 ≥ x 2 ≥ · · · ≥ x M and y 1 ≥ y 2 ≥ · · · ≥ y k ; (B) if y j ≤ r, then x r+1 = 0 (with the convention x M +1 = x M +2 = · · · = 0). Our aim is to show that x l = x ′ l and y j = y ′ j for 1 ≤ l ≤ M and 1 ≤ j ≤ k. Note that and N ′ (z), D ′ (z) are similarly defined using x ′ l , y ′ j . We shall prove N (z) = N ′ (z). If one of the fractions N (z) D(z) , N ′ (z) D ′ (z) is in reduced form, then by counting the zeros and the poles we get N (z) = N ′ (z). Suppose therefore neither N (z) D(z) nor N ′ (z) D ′ (z) is reduced. Let P (resp. Z) the set of poles (resp. zeros) not including ∞ of the rational function N (z −1 ) D(z −1 ) . Let P 0 (resp. Z 0 ) be the set of zeros of D(z −1 ) (resp. N (z −1 )). Define similarly P ′ , Z ′ , P ′ 0 , Z ′ 0 . Then P = P ′ , Z = Z ′ , P P 0 , Z Z 0 . Moreover, there exists 1 ≤ j 1 < j 2 < · · · < j s ≤ k such that series f i,µ (z)f i,λ (z) −1 can be written as a product of q 1−zbq 1−zbq −1 with b ∈ C × . We consider the case i = M + k with 1 ≤ k ≤ N . (The case i ≤ M is similar and simpler.) Let g := GT −1 λ (λ). Then as in the proof of Corollary 7.2 we have g(x, y) ≤ f − (x, y) for (x, y) ∈ Y λ .
As before, for θ ∈ P M i ,N i , let θ b be the lowest weight of the U q (g i )-module L(θ; g k ). Write Then as in the proof of Proposition 7.1 we have It is therefore enough to show that x j ≤ x ′ j and y l ≤ y ′ l for 1 ≤ j ≤ M and 1 ≤ l ≤ k. Let us verify y l ≤ y ′ l for 1 ≤ l ≤ k. (The case for x is parallel.) For this, we use the description in §2.5 on the relationship between Young diagrams and lowest weight vectors. Namely, we have (Definition 2.7) This says that y l ≤ y ′ l , as desired.
Example 8. Let (M, N ) = (1, 2) and g = gl(1, 2). Consider the U q ( g)-module W (2) k,a . Let us compute its normalized q-character by using dual Gelfand-Tsetlin basis. The g-Young diagram in this case is Let v i , w j ∈ W (2) k,a be dual Gelfand-Tsetlin vectors associated with f i , g j . Then v 0 is a highest ℓ-weight vector. The action of C q (g) on these vectors becomes:

Gelfand-Tsetlin basis of asymptotic representations
In this section, we study the normalized q-character for the asymptotic modules L + r,a and L r,a (b) constructed in previous sections §4-5 and we establish Gelfand-Tsetlin basis for these modules (provided that b is generic).
8.1. Gelfand-Tsetlin basis. For k ∈ I, let Y q (g k ) be the subalgebra of Y q (g) generated by the s Then there exists canonical isomorphisms of superalgebras We have therefore a chain of sub-superalgebras of Y q (g): V is semi-simple as a Y q (g k )-module for all k ∈ I 0 and simple as a Y q (g)-module; (B2) for all j ∈ Λ there exists a sequence (S s ) s∈I such that: S s is a simple sub-Y q (g s )module; S s ⊂ S s+1 for 1 ≤ k < M + N ; v j ∈ S 1 ; (B3) for all 1 ≤ k ≤ M + N and for S a simple sub-Y q (g s )-module, the decomposition of Res

Yq(gs)
Yq(g s−1 ) S into simple sub-Y q (g s−1 )-modules is multiplicity-free. Now we can state the central result of this section.
Theorem 8.1. Let r ∈ I 0 , a, b ∈ C × be such that b / ∈ ±q Z . Then the Y q (g)-modules L + r,a and L r,a (b) admit Gelfand-Tsetlin bases.
The idea of proof goes as follows. Fix r ∈ I 0 . There exists an inductive system (V k , F k,l ) constructed in §4.4-5 where V k are properly defined Kirillov-Reshetikhin modules. The representations of Y q (g) on L + r,a and on L r,a (b) are built upon the same underlying space, namely the inductive limit (V ∞ , F k ). For all k ∈ Z >0 , the Y q (g)-module V k admits a Gelfand-Tsetlin basis. Furthermore the structural maps F k,l : V l −→ V k respect Gelfand-Tsetlin bases on both sides. From this we construct directly a basis of V ∞ which serves at the same time as a Gelfand-Tsetlin basis for L + r,a and for L r,a (b). The next two subsections are devoted to the detailed proof of this theorem in cases r ≤ M and r > M . Along the proof, we shall find the simple sub-Y q (g k )-modules in (B2) in a combinatorial way (by using semi-infinite Young tableaux).
Before the proof, let us make the following convention. For 1 ≤ s ≤ M + N , related to the q-Yangian Y q (g s ) ∼ = Y q (gl(M s , N s )) we can define in exactly the same way the category O s , the group P Ms,Ns , the generalized simple roots A i,a;s ∈ P Ms,Ns for 1 ≤ i < s, the ring (E ℓ ) s , the q-character χ s q and the normalized q-characterχ s q . We identify P Ms,Ns (resp. (E ℓ ) s ) with a subset of P (resp. E ℓ ) in the following natural way: Then under this identification, A i,a;s = A i,a for 1 ≤ i < k as seen from Definition 6.2.
8.2. Proof of Theorem 8.1 when r ≤ M . Fix a ∈ C × . Let us be in the situation of the first paragraph of §5.1, so that we have an inductive system of vector superspaces (V k , F k,l ) with inductive limit (V ∞ , F k ). Note that we have fixed a highest ℓ- k,aq 2k , L + r,a , L r,a (b). Let t : Z >0 −→ Z >0 be a strictly increasing function such that the F k,t(l) ρ k (s (n) ij )F k,l for k > t(l) are well-defined. (See Proposition 4.6.) 8.2.1. Compatibility of Gelfand-Tsetlin bases and structural maps. Fix l, k ∈ Z >0 such that l < k. Choose for all l > 0 a Gelfand-Tsetlin basis (v λ : λ ∈ GT(l̟ r )) of V l satisfying the properties of Corollary 2.10. (Note that V k ∼ = L(k̟ r ) as U q (g)-modules.) Lemma 8.2. In the above situation, we have: for all λ = (λ (i) ) ∈ GT(l̟ r ), Proof. This comes from Remark 6.5 combined with (the proof of) Proposition 7.1 on relationship between Gelfand-Tsetlin vectors and ℓ-weight spaces and on multiplicity-free property of q-character, in view of the definition of F k,l in §4.4.

8.2.2.
Semi-infinite Young tableau and Gelfand-Tsetlin bases. Based on Lemmas 8.2 and 2.8, we introduce another index set of Gelfand-Tsetlin basis of V k . This will be convenient for the statement of results. Let Y (r) be the subset of Z 2 consisting of (i, j) such that 1 ≤ i ≤ r and j < 0. Let B (r) be the set of functions f : Y (r) −→ I satisfying (T1)-(T3) in Definition 2.2 and: for all 1 ≤ i ≤ r, there exists j < 0 such that f (i, j) = i. Such an f is also called a semi-infinite Young tableau. For k > 0, let B (r) k be the subset of B (r) consisting of such f that f (i, j) = i for j < −k. We have therefore a chain of subsets of B (r) : There is a canonical bijective map π k : B(Y k̟r ) −→ B (r) k sending g, a g-Young tableau of shape Y k̟r , to a semi-infinite Young tableau f where f (i, j) = i (j < −k), g(i, j + k + 1) (j ≥ −k).
Let f 0 ∈ B (r) be such that f 0 (i, j) = i. By using Lemma 2.8, Corollary 2.10 and Lemma 8.2, one can construct by induction on k > 0, a Gelfand-Tsetlin basis {v[k, f ]|f ∈ B (r) k } of V k verifying the following properties.
For f ∈ B (r) and s ∈ I, let S s (f ) be the sub-vector-superspace of V ∞ spanned by the v[g] where g ∈ B (r) satisfies g(i, j) = f (i, j) whenever f (i, j) > s or g(i, j) > s. Clearly For 1 ≤ i ≤ min(r, s), define l i := ♯{j < 0 | f (i, j) > s}. Set λ s (l, f ) := min(r,s) i=1 (l − l i )ǫ i . Let f ↓s ∈ B (r) be the semi-infinite Young tableau obtained from f by replacing any f (i, j) ≤ s with i. At last, let Φ l , Φ + , Φ b ∈ P such that The precise formulas for Φ l , Φ + , Φ b will be given in the proof of Lemma 8.7. Lemma 8.3. Let f ∈ B (r) l and s ∈ I. The sub-vector-superspace F −1 l S s (f ) ⊆ V l is a simple sub-Y q (g s )-module isomorphic to ev * aq 2l L(λ s (l, f ); g s ) ⊗ D where D is a sign module. Moreover, v[l, f ↓s ] is a highest ℓ-weight vector of the Y q (g s )-module F −1 l S s (f ). Proof. Indeed, F −1 l S s (f ) is the sub-vector-superspace of V l spanned by the v[l, g] where g ∈ B (r) l and g(i, j) = f (i, j) whenever g(i, j) > s or f (i, j) > s. In view of (c), the map π l , and the correspondence between Young tableaux and Gelfand-Tsetlin patterns in Lemma 2.8, the lemma is evident.
Lemma 8.4. For s ∈ I and f ∈ B (r) , the sub-vector-superspace S s (f ) of V ∞ is stable by ρ x (Y q (g s )) whenever x ∈ {+, b}.
Proof. By definition, S s (f ) is stable by the ρ x (s (0) ii ). (See also Lemma 8.7 below for more general statements.) It is enough to show that S s (f ) is stable by the ρ x ( s (n) ij ) where n ∈ Z ≥0 and 1 ≤ i, j ≤ s. Let us assume f ∈ B (r) v . For k > v, the preceding lemma says that F −1 k S s (f ) ⊆ V k is stable by the ρ k ( s (n) ij ) with 1 ≤ i, j ≤ s. It follows that (v < k < t(k) < u) On the other hand, we know from §4.4 the following asymptotic formula ij )F u,k = A k + q 2u B k for u > t(k) where A k , B k : V k −→ V t(k) are linear operators depending only on i, j, n and inductively on k. As u can be arbitrarily large, we must have for all k > v. But this says exactly that ρ + ( s (n) ij ) (resp. ρ b ( s (n) ij )), being defined as inductive limit of the A k (resp. A k + b 2 B k ), stabilizes S s (f ), as desired.
Based on the proof of the above lemma, the following lemma is clear.
Remark 8.6. Lemmas 8.3-8.5 can be rephrased as follows. The original inductive system (V k , F k,l ) admits a sub-inductive-system of vector superspaces of the form: The two asymptotic constructions in §4.4 and in §5.1 can be restricted to this sub-inductivesystem, resulting in asymptotic modules (ρ + , S s (f )) and (ρ b , S s (f )) over Y q (g s ).
Lemma 8.7. For f ∈ B (r) , there exists uniquely Φ f ∈ P such that whenever f ∈ B (r) l : Moreover, Φ f is a product of the A −1 i,x where i ∈ I 0 and x ∈ aq 2Z+1 . Proof. For such l and f , v[l, f ] is a Gelfand-Tsetlin vector. In view of Proposition 7.1, Cv[l, f ] is an ℓ-weight space of (ρ l , V l ). Hence by Corollary 7.2 there exists a unique Φ f as a product of the A −1 i,x with i ∈ I 0 , x ∈ aq 2Z+1 such that )v[f ], remark that for all k > l, by assumption F k,l v[l, f ] = v[k, f ]. Hence for all k > t(l) , f ] On the other hand, in view of the following explicit formulas for Φ k , Φ + , Φ b : (1 − zaθ −1 j ) (i > r), we see from the construction of ρ b in §5.1 that (by replacing q k with b everywhere) Finally, for ρ + (C i (z))v[f ], from the above formulas of Φ k , Φ + we deduce that there exists a formal power series h i (w) ∈ 1 + wC[[w]] for i ∈ I such that for all k > t(l), From Gauss decomposition observe that C i (z)C i (0) −1 ∈ Y q (g) [[z]]. Now from the construction of ρ + in §4.4 we obtain (by taking q k = 0) It is therefore enough to compute ρ + (C i (0))v[f ]. For this, note that in (ρ l , V l ), Hence in (ρ + , V ∞ ), the Q-degree of v[f ] is given by the second term of the RHS. From the proof of Proposition 7.1 we get an explicit expression of the (Φ f ) i (0): In consequence, ρ + (C i (z))v[f ] = (Φ + Φ f ) i (z)v[f ], as desired.
We arrive at the following important consequence of the preceding lemmas: simplicity of S M +N (f ) for ρ + and ρ b .
Corollary 8.8. Let 1 ≤ r ≤ M and a, b ∈ C × . Then χ q (L + r,a ) = χ q (L r,a (b)) = lim k→∞ χ q (W (r) k,aq 2k ) as formal power series in the [A i,x ] −1 where i ∈ I 0 and x ∈ aq 2Z+1 . Furthermore, L + r,a is a simple Y q (g)-module. L r,a (b) is simple if b / ∈ ±q Z >−r .
Proof. Let us explain firstly that the limit of normal q-characters above makes sense. For V a module in category O with well-defined normalized character (Remark 6.3), set and from the normalized q-character of L r,a (b) we deduce that there exists k > l such that k,aq 2k ) = 1 = d(Φ 2 , W (r) k,ab 2 q 2k ).
Lemma 8.9. Let W be a Y q (g)-module in category O admitting normalized character. Let (k n ) n∈Z >0 be a strictly increasing sequence of integers. For n ∈ Z >0 take W n := Φ * q 2kn W . If Φ ∈ P is such that d(Φ, W n ) > 0 for all n, then Φ ∈ σP.
Remark 8.10. Corollary 8.8 says that L r,a (b) is simple if b ∈ ±q Z >−r . In the case r = M , it is possible to determine the b ∈ ±q Z −r making L M,a (b) non-simple as L M,a (b) is of dimension 2 M N . In the case r < M , if b ∈ ±q k with k ≥ 0, then L r,a (b) has a finite-dimensional simple sub-quotient isomorphic to W (r) k,aq 2k (up to tensor product by one-dimensional modules, with the convention that W (r) 0,a is the trivial module). Since L r,a (b) is infinite-dimensional, it is not simple. It remains to find integers 1 ≤ s < r such that L r,a (±q −s ) is not simple. Unfortunately, limited by Lemma 7.3, we still do not know the answer.
The proof of Corollary 8.8 applied perfectly if we replace S M +N (f ) with more general S s (f ), we obtain the following corollary, whose proof is omitted, on normalized q-character formulas and on simplicity of asymptotic modules. (1) The Y q (g s )-modules (ρ + , S s (f )) and (ρ b , S s (f )) are in category O s and χ s q (ρ + , S s (f )) = χ s q (ρ b , S s (f )) = lim k→∞ χ s q (ev * aq 2k L(λ s (k, f ); g s )) as formal power series in the [A i,x ] −1 with 1 ≤ i < s and x ∈ aq 2Z+1 . (2) The Y q (g s )-module (ρ + , S s (f )) is simple, while the Y q (g s )-module (ρ b , S s (f )) is simple if b / ∈ ±q Z . 8.2.3. End of proof of Theorem 8.1 when r ≤ M . Assume b / ∈ ±q Z . Let x ∈ {+, b}. Consider the Y q (g)-module (ρ x , V ∞ ). According to Corollary 8.11, its basis {v[f ] | f ∈ B (r) } together with the chains (one for each f ∈ B (r) ) v[f ] ∈ S 1 (f ) ⊆ S 2 (f ) ⊆ S 3 (f ) ⊆ · · · ⊆ S M +N −1 (f ) ⊆ S M +N (f ) = V ∞ Let r ∈ I 0 be such that M < r < M + N . Consider the U q ( g)-module L r,a (b). By pulling it back with respect to f J,I : U q ( g ′ ) −→ U q ( g), we get a U q ( g ′ )-module f * J,I L r,a (b) such that where v ∈ L r,a (b) is a highest ℓ-weight vector of L r,a (b). By comparing the normalized characters and highest ℓ-weights of f * J,I L r,a (b) and L M +N −r,ab −2 ;J (b) we conclude that Corollary 8.15. Let M < r < M + N and a, b ∈ C × . Assume that b / ∈ ±q Z >−M −N+r . Then the U q ( g)-module L r,a (b) is simple.
In other words, when b is generic, χ q (V (θ i,a,b )) is independent of b. Note that a similar phenomenon happens for more general rational functions than the 1−za 1−zb , as we have observed in Equation (6.39) concerning the q-Yangian Y q (gl(1, 1)).