On multiplicity-free weight modules over quantum affine algebras

TL;DR

This paper constructs and analyzes multiplicity-free weight modules over quantum affine algebras, introducing shiftability conditions and computing their highest $\

Contribution

It introduces the shiftability condition and provides a new framework for understanding infinite-dimensional multiplicity-free modules.

Findings

Defined shiftability condition for quantum affine algebras

Constructed infinite-dimensional multiplicity-free weight modules

Computed highest $\

Abstract

In this note, our goal is to construct and study the multiplicity-free weight modules of quantum affine algebras. For this, we introduce the notion of shiftability condition with respect to a symmetrizable generalized Cartan matrix, and investigate its applications on the study of quantum affine algebra structures and the realizations of the infinite-dimensional multiplicity-free weight modules. We also compute the highest $\ell$-weights of the infinite-dimensional multiplicity-free weight modules as highest $\ell$-weight modules.

Equations

$$\tilde{d}_{i}=\left\{\begin{array}[]{ll}1,&\text{ if }r=1\text{ or }X_{N}^{(r)}=A_{2n}^{(2)},\\ d_{i},&\text{otherwise}.\end{array}\right.$$

$$K_i{K}_i^{-1}=K_i^{-1}{K}_i=1,K_i{K}_j=K_j{K}_i,$$

$$K_i{X}_j^{\pm }{K}_i^{-1}=q_i^{\pm a_{ij}}{X}_j^{\pm },$$

$$X_i^{+}{X}_j^{-}-X_j^{-}{X}_i^{+}={\delta }_{ij}\frac{K_i-K_i^{-1}}{q_i-q_i^{-1}},$$

$$\sum _{k=0}^{1-a_{ij}}(-1{)}^k{\left[\genfrac{}{}{0ex}{}{1-a_{ij}}{k}\right]}_{q_i}(X_i^{\pm }{)}^k{X}_j^{\pm }(X_i^{\pm }{)}^{1-a_{ij}-k}=0,\text{ for }i\ne j,$$

$$[m{]}_q=\frac{q^m-q^{-m}}{q-q^{-1}},[m{]}_q^{!}=[m{]}_q[m-1{]}_q\cdots [1{]}_q,{\left[\genfrac{}{}{0ex}{}{m}{r}\right]}_q=\frac{[m{]}_q^{!}}{[r{]}_q^{!}[m-r{]}_q^{!}}.$$

$$\Delta (K_i)=K_i\otimes K_i,\Delta (X_i^{+})=X_i^{+}\otimes 1+K_i\otimes X_i^{+},$$

$$\Delta (X_i^{-})=X_i^{-}\otimes K_i^{-1}+1\otimes X_i^{-},$$

$$S(X_i^{+})=-K_i^{-1}{X}_i^{+},S(X_i^{-})=-X_i^{-}{K}_i,S(K_i)=K_i^{-1},$$

$$ϵ(X_i^{+})=0=ϵ(X_i^{-}),ϵ(K_i)=1.$$

$$x_{\sigma (i),k}^{\pm }={\omega }^k{x}_{i,k}^{\pm },h_{\sigma (i),k}^{\pm }={\omega }^k{h}_{i,k}^{\pm },k_{\sigma (i)}^{\pm 1}=k_i^{\pm 1},$$

$$k_i{k}_i^{-1}=k_i^{-1}{k}_i=1,k_i{k}_j=k_j{k}_i,k_i{h}_{j,l}=h_{j,l}{k}_i,$$

$$k_i{x}_{j,k}^{\pm }={q_{\bar{i}}}^{\pm a_{\bar{i}\bar{j}}}{x}_{j,k}^{\pm }{k}_i,$$

$$[h_{i,k},h_{j,l}]={\delta }_{k,-l}\frac{1}{k}(\sum _{s=1}^r[\frac{k({\bar{\alpha }}_i,{\bar{\alpha }}_{{\sigma }^s(j)})}{d_{\bar{i}}}{]}_{\bar{i}}{\omega }^{ks})\frac{C^k-C^{-k}}{q_{\bar{i}}-{q_{\bar{i}}}^{-1}},$$

$$[h_{i,k},x_{j,l}^{\pm }]=\pm \frac{1}{k}(\sum _{s=1}^r[\frac{k({\bar{\alpha }}_i,{\bar{\alpha }}_{{\sigma }^s(j)})}{d_{\bar{i}}}{]}_{\bar{i}}{\omega }^{ks})C^{\mp |k|/2}{x}_{j,k+l}^{\pm },$$

$$[x_{i,k}^{+},x_{j,l}^{-}]=(\sum _{s=1}^r\frac{{\delta }_{{\sigma }^s(i)j}{\omega }^{sl}}{{\tilde{d}}_{\bar{i}}})\frac{C^{(k-l)/2}{\psi }_{i,k+l}^{+}-C^{-(k-l)/2}{\psi }_{i,k+l}^{-}}{q_{\bar{i}}-{q_{\bar{i}}}^{-1}},$$

$$\sum _{k=0}^{\infty }{\psi }_{i,\pm k}^{\pm }{z}^{\pm k}=k_i^{\pm 1}\exp (\pm (q_{\bar{i}}-{q_{\bar{i}}}^{-1})\sum _{l=1}^{\infty }{h}_{i,\pm l}{z}^{\pm l}),$$

$$U\cong U(⩽)\otimes U(0)\otimes U(⩾)$$

$$(\lambda +\mu )(u)=(\lambda \otimes \mu )\circ \Delta (u),(-\lambda )(u)=\lambda \circ S(u)$$

$$V_{\lambda }=\{v\in V|u.v=\lambda (u)v,\forall u\in U^0\}.$$

$$V_{\lambda }=\underset{\mathbf{\gamma }:\mathrm{wt}(\mathbf{\gamma })=\lambda }{⨁}{V}_{\mathbf{\gamma }},$$

$$V_{\mathbf{\gamma }}=\{v\in V_{\lambda }|\forall 1\le i\le N,k\ge 0,\exists m\in {\mathbb{Z}}_{>0},({\psi }_{i,\pm k}^{\pm }-{\gamma }_{i,\pm k}^{\pm }{)}^m.v=0\},$$

$$\sum _{k=0}^{\infty }{f}_{i,{\tilde{d}}_{i}k}^{+}{z}^k=f_i(z)=\sum _{k=0}^{\infty }{f}_{i,-{\tilde{d}}_{i}k}^{-}{z}^{-k}$$

$$f_i(z)=\{\begin{array}{ll}{q}_n^{2\mathrm{deg}{P}_n}\frac{P_n(q_n^{-4}z)}{P_n(z)}& \text{if}(X_N^{(r)},i)=(A_{2n}^{(2)},n),\\ q_i^{\mathrm{deg}{P}_i}\frac{P_i(q_i^{-2}z)}{P_i(z)}& \text{otherwise}.\end{array}$$

$$fg={\zeta }_j^{-1}(f){\zeta }_i^{-1}(g).$$

$$y_i^{\pm 1}=\prod _{j\in I}{x}_j^{\pm a_{ji}}.$$

$$\left\{\begin{array}[]{l}\zeta_{i}(\phi_{i})-\phi_{i}=\{y_{i}\}_{i},\\[6.0pt] \phi_{i}\phi_{j}=\zeta_{j}^{-1}(\phi_{i})\zeta_{i}^{-1}(\phi_{j}),\end{array}\quad i,j\in I,i\neq j.\right.$$

$$A_n^{(1)}(n\ge 1):$$

$$C_n^{(1)}(n\ge 2):$$

$$\displaystyle\hskip 71.13188pt\cdots,\{q_{n-1}b_{C}z_{n-1}\}_{n-1}\{b_{C}z_{n}\}_{n-1},\{q_{n}b_{C}z_{n}\}_{n}\{b_{C}z_{n}^{-1}\}_{n}\Big{)}$$

$$A_{2n}^{(2)}(n\ge 1):$$

$$\displaystyle\hskip 71.13188pt\cdots,\{\imath q^{\frac{1}{2}}z_{n-1}\}_{n-1}\{\imath q^{-\frac{1}{2}}z_{n}\}_{n-1},\frac{\imath}{q_{n}-q_{n}^{-1}}\{\imath q^{\frac{1}{2}}z_{n}\}_{n}\Big{)}$$

$$D_{n+1}^{(2)}(n\ge 2):$$

$$\displaystyle\hskip 99.58464pt\cdots,\{\imath qz_{n-1}\}_{n-1}\{\imath q^{-1}z_{n}\}_{n-1},\frac{\imath}{q_{n}-q_{n}^{-1}}\{\imath qz_{n}\}_{n}\Big{)}$$

$$z_i=x_{i-1}^{-1}{x}_i,z_0=(z_1\cdots z_n{)}^{-1},y_i=z_i{z}_{i+1}^{-1},b_{A_n^{(1)}}\in {\mathbb{C}}^{\times }\text{in }{A}_n^{(1)};$$

$$z_i=x_{i-1}^{-1}{x}_i,y_0=z_1^{-2},y_i=z_i{z}_{i+1}^{-1},y_n=z_n^2,b_{C_n^{(1)}}=q^{-1/4}\text{ or }ıq^{-1/4}\text{in }{C}_n^{(1)};$$

$$z_i=x_{i-1}^{-1}{x}_i,z_n=x_{n-1}^{-1}{x}_n^2,y_0=z_1^{-2},y_i=z_i{z}_{i+1}^{-1},y_n=z_n,b_{A_{2n}^{(2)}}=ıq^{-\frac{1}{2}}\text{in }{A}_{2n}^{(2)};$$

$$z_1=x_0^{-2}{x}_1,z_i=x_{i-1}^{-1}{x}_i,z_n=x_{n-1}^{-1}{x}_n^2,y_i=z_i{z}_{i+1}^{-1},y_n=z_n,y_0=z_1^{-1},b_{D_{n+1}^{(2)}}=ıq^{-1}\text{in }{D}_{n+1}^{(2)}.$$

$$[\mathbf{a},{\mathbf{a}}^{+}{]}_{\nu }=\mathbf{k},$$

Full text

1. Introduction

Let $U_{q}(\mathfrak{g})$ be the quantum affine algebra (without derivation) associated to an affine Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ in which $q$ is not a root of unity. In this note, we are concerned with infinite-dimensional multiplicity-free weight representations, i.e., all of their weight subspaces are one-dimensional, over $U_{q}(\mathfrak{g})$. As we shall see, these representations are the basic representations towards to the infinite dimensional modules of quantum affine algebras.

In the classical cases, the multiplicity-free weight representations over finite-dimensional simple Lie algebras, or more general, the bounded weight representations have been extensively studied in [3, 6, 17, 18]. These representations play a crucial role in the classification of simple weight modules of finite dimensional simple Lie algebras (cf. [32]). For the quantum groups of finite type, Futorny-Hartwig-Wilson [16] gave a classification of all infinite-dimensional irreducible multiplicity-free weight representations of type $A_{n}$. Recently, the infinite-dimensional multiplicity-free weight representations of the quantum groups of types $A_{n},B_{n}$ and $C_{n}$ were constructed in [10].

As an important class of multiplicity-free weight modules, the $q$-oscillator representations over $U_{q}(\mathfrak{g})$ of types $A_{n}^{(1)}$, $C_{n}^{(1)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$ have been obtained in the works of T. Hayashi, A. Kuniba, M. Okado [19, 25, 26, 27]. Our goal is to construct infinite-dimensional multiplicity-free weight representations of $U_{q}(\mathfrak{g})$ in a general way. For this, associated to each symmetrizable generalized Cartan matrix we introduce a system of equations in a Laurent polynomial ring $\mathcal{A}$ (essentially, the Cartan part of $U_{q}(\mathfrak{g})$) by the shift operators. Call the corresponding generalized Cartan matrix satisfies the shiftability condition if the system of equations has solutions (see Subsection 4.1). One result of this note is that an affine Cartan matrix satisfies the shiftability condition if and only if the relevant Dynkin diagram is one of the types mentioned above (see Theorem 4.2). The solutions allow us to define $U_{q}(\mathfrak{g})$-module structures on $\mathcal{A}$, and to relate the quantum affine algebra structures with the $n$-fold of quantized oscillator algebra. Our method for the construction is parallel with the earlier work concerning $U^{0}$-free modules [10]. Namely we can get the multiplicity-free weight modules of $U_{q}(\mathfrak{g})$ by applying the “weighting” procedure to the above modules on $\mathcal{A}$. In particular, the $q$-oscillator representations can also be reconstructed.

For the study of weight representations of quantum affine algebras, the concepts of $\ell$-weights and $\ell$-weight vectors were proved especially useful, which allow one to refine the spectral data properly in weight representations. For example, we have the classification of irreducible finite-dimensional representations (cf. [7, 9]) and infinite-dimensional weight representation of quantum affine algebras in [20, 33] by highest $\ell$-weights (Note that their highest $\ell$-weights are determined by Drinfeld polynomials and rational functions, respectively). In this note, we shall compute explicitly the highest $\ell$-weight of the $q$-oscillator representations. For the type $A_{n}^{(1)}$, the highest $\ell$-weights of $q$-oscillator representations also were discussed in [4, 5, 29].

This note is organized as follows. In Section 2, we give some necessary notations, and review two presentations of quantum affine algebras. In Section 3 we recall the definition of highest $\ell$-weight representations. Then we obtain the classification of highest $\ell$-modules with finite weight multiplicities in general. In Section 4, we introduce the notion of shiftability condition, and present the solutions to the corresponding system of equations, which allow us to study the compatible structures of quantum affine algebras with the $n$-fold of quantized oscillator algebra. In Section 5 the infinite-dimensional multiplicity-free weight modules are constructed. In Section 6, we compute the highest $\ell$-weight of the $q$-oscillator representations.

Conventions. Let $\mathbb{Z},\mathbb{R}$, and $\mathbb{C}$ be the sets of integers, real numbers and complex numbers respectively, denote $\mathbb{C}\setminus\{0\}$ by $\mathbb{C}^{\times}$, the set of nonnegative integers by $\mathbb{Z}_{\geq 0}$, and the notation $\delta_{ij}$ stands for the Kronecker symbol in this note.

2. Preliminaries and notations

First, let us recall some necessary notations and two presentations of quantum affine algebras based on [2, 15, 24].

2.1. Affine Kac-Moody algebras

Let $\mathfrak{g}=\mathfrak{g}(X_{N}^{(r)})$ be an affine Kac-Moody algebra with respect to the generalized Cartan matrix $A=(a_{ij})_{i,j\in I}$ of type $X_{N}^{(r)}$ where $I=\{0,1,\cdots,n\}$ is an indexed set and $X_{N}^{(r)}$ is a Dynkin diagram from Table Aff $r$ of [24], except in the case of $X_{N}^{(r)}=A_{2n}^{(2)}(n\geq 1)$, where we reverse the numbering of the simple roots.

Let $\{\alpha_{i}\}_{i\in I}\subset\mathfrak{h}^{*}$ (resp. $\{\alpha_{i}^{\vee}\}_{i\in I}\subset\mathfrak{h}$) denote the set of simple roots (resp. simple coroots) such that $\langle{\alpha_{j},\alpha_{i}^{\vee}}\rangle=a_{ij}$. Let $Q=\oplus_{i\in I}\mathbb{Z}\alpha_{i}$ and $Q_{+}=\oplus_{i\in I}\mathbb{Z}_{\geq 0}\alpha_{i}$ be the root lattice and the set of positive roots of $\mathfrak{g}$, respectively. Assume that $\delta=\sum a_{i}\alpha_{i}$ and $c=\sum a_{i}^{\vee}\alpha_{i}^{\vee}$ are the smallest positive imaginary root and a center element of $\mathfrak{g}$, where $a_{i}$ and $a_{i}^{\vee}$ are the numerical labels of the Dynkin diagrams of $X_{N}^{(r)}$ and its dual, respectively. Let $\{\omega_{i}\}_{i\in I}$ denote the fundamental weights of $\mathfrak{g}$, i.e., $\langle{\omega_{i},\alpha_{j}^{\vee}}\rangle=\delta_{ij}$ for $i,j\in I$.

Let $W$ be the affine Weyl group of $\mathfrak{g}$ (which is a subgroup of the general linear group of $\mathfrak{h}^{*}$) generated by the simple reflections $s_{i}(\lambda)=\lambda-\langle{\lambda,\alpha_{i}^{\vee}}\rangle\alpha_{i},\lambda\in\mathfrak{h}^{*},i\in I$. Note that $w(\delta)=\delta$ for all $w\in W$. Set $I_{0}=I\setminus\{0\}$. Denote by $\mathring{W}$ the subgroup of $W$ generated by the simple reflections $s_{i}$ for $i\in I_{0}$. It is a finite group.

Take the nondegenerate symmetric bilinear form $(,)$ on $\mathfrak{h}^{*}$ invariant under the action of $W$, which is normalized uniquely by $(\lambda,\delta)=\langle\lambda,c\rangle$ for $\lambda\in\mathfrak{h}^{*}$. Define $D$ as the diagonal matrix $\mathrm{diag}(d_{0},\cdots,d_{n})$ with $d_{i}=a_{i}^{-1}a_{i}^{\vee}$. Then $(\alpha_{i},\alpha_{j})=d_{i}a_{ij}$ for all $i,j\in I$. Let $\triangle$ be the root system of $\mathfrak{g}$, $\triangle^{\pm}=\triangle\cap(\pm Q_{+})$ and let $\triangle^{\mathrm{re}}=\triangle\setminus\mathbb{Z}\delta$ be the set of real roots. For each $\alpha\in\triangle^{\mathrm{re}}$ we set $\tilde{d}_{\alpha}=\mathrm{max}(1,(\alpha,\alpha)/2)$. In particular, write $\tilde{d}_{i}$ simply for $\tilde{d}_{\alpha_{i}}$. Then

[TABLE]

Denote by $\mathring{A}=(a_{ij})_{i,j\in I_{0}}$ the Cartan matrix of finite type, and let $\mathring{\mathfrak{g}}$ be the associated simple finite-dimensional Lie algebra. Then $\{\alpha_{i}\}_{i\in I_{0}}$ is a set of simple roots for $\mathring{\mathfrak{g}}$. Let $\mathring{Q}=\oplus_{i\in I_{0}}\mathbb{Z}\alpha_{i}$ be the root lattice for $\mathring{\mathfrak{g}}$, $\widetilde{P}$ the weight lattice of the euclidean space $\mathbb{R}\otimes_{\mathbb{Z}}\mathring{Q}\subset\mathfrak{h}^{*}$ defined as $\widetilde{P}=\oplus_{i\in I_{0}}\mathbb{Z}\tilde{\omega}_{i}$, where $(\tilde{\omega}_{i},\alpha_{j})=\delta_{ij}\tilde{d}_{i}$. Then $\mathring{Q}$ can be naturally embedded into $\widetilde{P}$, which provides a $W$-invariant action on $\mathfrak{h}^{*}$ by $x(\lambda)=\lambda-(x,\lambda)\delta$ for $x\in\widetilde{P},\lambda\in\mathfrak{h}^{*}$.

Define the extended Weyl group by $\widetilde{W}=\mathring{W}\ltimes\widetilde{P}$. We also have $\widetilde{W}=W\ltimes\mathscr{T}$, where $\mathscr{T}=\{w\in\widetilde{W}|w(\triangle^{+})\subset\triangle^{+}\}$, which is a subgroup of the group of diagram automorphisms. An expression for $w\in\widetilde{W}$ is called reduced if $w=\tau s_{i_{1}}\cdots s_{i_{l}}$, where $\tau\in\mathscr{T}$ and $l$ is minimal. We call the minimal integer $l$ the length of $w$, and denote it by $l(w)$.

2.2. Quantum affine algebras

The quantum affine algebra $U_{q}(\mathfrak{g})$ in the Drinfeld-Jimbo realization [14, 22] is the unital associative algebra over $\mathbb{C}$ generated by $X_{i}^{+}$, $X_{i}^{-}$, $K_{i}^{\pm 1}$, $i\in I$ with the following relations:

[TABLE]

where $q\in\mathbb{C}^{\times}$ is not a root of unity and $q_{i}=q^{d_{i}}$. Here we have used the standard notations:

[TABLE]

In particular, denote $[m]_{q_{i}}$ by $[m]_{i}$ for simplicity.

Let $U^{0}$ be the commutative subalgebra of $U:=U_{q}(\mathfrak{g})$ generated by $K_{i},K_{i}^{-1},i\in I$. It is clear that each element in $U^{0}$ is a linear combination of the monomials $K_{\beta}:=K_{0}^{b_{0}}K_{1}^{b_{1}}\cdots K_{n}^{b_{n}}$ for $\beta=\sum_{i\in I}b_{i}\alpha_{i}\in Q.$ In particular, $K_{\delta}$ is a central element in $U$. Let $U^{+}$ (resp. $U^{-}$) denote the span of monomials in $X^{+}_{i}$ (resp. $X_{i}^{-}$). Recall $U$ has a canonical triangular decomposition $U\cong U^{-}\otimes U^{0}\otimes U^{+}$. For later use, we note that $U^{+}$ is graded by $Q_{+}$ in the usually way: $U^{+}=\oplus_{\beta\in Q_{+}}U^{+}_{\beta}$.

Let us recall the Hopf algebra structure of $U$ with the coproduct $\Delta$, the antipode $S$, the counit $\epsilon$ defined as follows:

[TABLE]

There exists another presentation of $U$ due to Drinfeld [15]. Just like the realizations of the affine Kac-Moody algebras $\mathfrak{g}$ as (twisted) loop algebras, this presentation of $U$ is generated by the Drinfeld’s “loop-like” generators.

Consider the root datum $(X_{N},\sigma)$ with $\sigma$ a diagram automorphism of $X_{N}$ of order $r$. Let $\bar{A}=(\bar{a}_{ij})_{1\leq i,j\leq N}$ be the Cartan matrix of the type $X_{N}$, and let $\omega$ be a fixed primitive $r$-th root of unity. Note that if $r=1$ (i.e., $\sigma$ is an identity) we have $N=n$, $\bar{A}=\mathring{A}$; if $r>1$, then $X_{N}$ is one of the simply laced types: $A_{N}(N\geq 2)$, $D_{n+1}(n\geq 2)$, $E_{6}$. We use $\bar{i}\in I_{0}$ to stand for one representative of the $\sigma$-orbit of $i$ on $\{1,2,\cdots,N\}$ such that $\bar{i}\leq\sigma^{s}(i)$ for any $s$. Take the set of simple roots $\{\bar{\alpha}_{i}\}_{1\leq i\leq N}$ and the normalized bilinear form $(,)$ (by abuse of notation) such that $(\bar{\alpha}_{i},\bar{\alpha}_{j})=d_{i}a_{ij}$ if $r=1$, otherwise $(\bar{\alpha}_{i},\bar{\alpha}_{j})=\bar{a}_{ij}$ for $1\leq i,j\leq N$.

The quantum affine algebra $U$ (add the central elements $K_{\delta}^{\pm 1/2}$) is isomorphic to the algebra generated by $x_{i,k}^{\pm}(1\leq i\leq N,k\in\mathbb{Z})$, $h_{i,k}(1\leq i\leq N,k\in\mathbb{Z}\setminus\{0\})$, $k_{i}^{\pm 1}(1\leq i\leq N)$ and the central elements $C^{\pm 1/2}$, subject to the following relations:

[TABLE]

where $\psi_{i,k}^{\pm}$’s are the elements determined by the following identity of the formal power series in $z$:

[TABLE]

together with the quantum Serre-Drinfeld relations, whose explicit forms will be not used in this note. One can refer to [15] for more details and to [1, 23] and [11, 12, 13]111The author used the notations ${\widetilde{H}}^{\pm}_{i,l}$, $H_{i,l}$, which are related with $\psi_{i,l}^{\pm}$, $h_{i,l}$ defined in this note by ${\widetilde{H}}^{\pm}_{i,l}=C^{l/2}k_{i}^{\mp 1}\psi_{i,l}^{\pm}$ and $H_{i,l}=C^{l/2}h_{i,l}$. for the proof.

Under the isomorphism, we have $X_{i}^{\pm}=x_{i,0}^{\pm}$, $K_{i}^{\pm 1}=k_{i}^{\pm 1}$ for $i\in I_{0}$, and $K_{\delta}=C$. Note that $\psi_{i,-k}^{+}=\psi_{i,k}^{-}=0$ for any positive integers $k$, and $\psi_{i,0}^{\pm}=k_{i}^{\pm 1}$ from the identity (2.6).

From the relations in Drinfeld presentation, $U$ is essentially generated by the generators $x_{i,\tilde{d}_{i}k}^{\pm}(i\in I_{0},k\in\mathbb{Z})$, $h_{i,\tilde{d}_{i}k}(i\in I_{0},k\in\mathbb{Z}\setminus\{0\})$, $k_{i}^{\pm 1}(i\in I_{0})$ and the central elements $C^{\pm 1/2}$ (see [12, Proposition 4.25]). Moreover, the quantum affine algebra $U$ has a triangular decomposition [8, 9]:

[TABLE]

where $U(\geqslant)$ (resp. $U(\leqslant)$) is the subalgebra generated by $x_{i,\tilde{d}_{i}k}^{+}$ (resp. $x_{i,\tilde{d}_{i}k}^{-}$), $i\in I_{0}$, $k\in\mathbb{Z}$, and $U(0)$ is the subalgebra generated by $C^{\pm 1/2}$, $k_{i}^{\pm}$, $h_{i,k},i\in I_{0},k\in\mathbb{Z}\setminus\{0\}$.

3. Highest $\ell$-weight representations with finite weight multiplicities

In this section, we recall basis notations of representations over quantum affine algebras: weight modules, $\ell$-weights, and highest $\ell$-weight modules. Most of the definitions and results in this section are well-known, one can refer to [7, 33].

3.1. Highest $\ell$-weight modules

We begin with the notion of highest $\ell$-weight modules. Thanks to the Hopf algebra structure of $U^{0}$ (inherits from $U$), the set of all algebra characters of $U^{0}$, i.e., all algebra homomorphisms from $U^{0}$ to $\mathbb{C}$, has an abelian group structure, the addition and the inverse are given by

[TABLE]

for any algebra characters $\lambda,\mu$, and $u\in U^{0}$. Denote this group simply by $(\mathscr{X},+)$. Any $\beta\in\mathfrak{h}^{*}$ induces a character in $\mathscr{X}$ by assigning $K_{i}$ to $q^{(\beta,\alpha_{i})}$ for $i\in I$, which is unique up to scalars of $\delta$, so we still denote it by $\beta\in\mathscr{X}$.

For a $U$-module $V$ and $\lambda\in\mathscr{X}$, define

[TABLE]

By the defining relations (2.2) $X_{i}^{\pm}.V_{\lambda}\subset V_{\lambda\pm\alpha_{i}}$. If $V_{\lambda}$ is nonzero, then we say $\lambda$ is a weight of $V$, and $V_{\lambda}$ is a weight space of weight $\lambda$, a nonzero vector $v\in V_{\lambda}$ is called a weight vector of weight $\lambda$. If the weight space $V_{\lambda}$ is finite-dimensional, then $\dim V_{\lambda}$ is called the multiplicity of the weight $\lambda$. Call $V$ a weight module if $V=\oplus_{\lambda}V_{\lambda}$. Moreover, a weight module $V$ is said to be multiplicity-free if $\dim V_{\lambda}\leq 1$ for all $\lambda\in\mathscr{X}$.

Throughout this note, we assume that the central element $C$ acts trivially on a $U$-module. So any weight $\lambda$ of a $U$-module is level-one, that is, $\lambda(K_{\delta})=1$.

Note that the actions of $\psi_{i,k}^{\pm}$’s on a $U$-module commute each other by (2.5) and (2.6). For a weight $\lambda$ of $V$ with finite multiplicity, we may refine the weight space $V_{\lambda}$ as

[TABLE]

[TABLE]

where $\boldsymbol{\gamma}=(\gamma_{i,\pm k}^{\pm})_{1\leq i\leq N,k\in\mathbb{Z}_{\geq 0}}$ is any $N$-tuple of sequences of complex numbers satisfying that $\gamma_{i,0}^{+}\gamma_{i,0}^{-}=1$ and $\gamma_{\sigma(i),\pm k}^{\pm}=\omega^{\pm k}\gamma_{i,\pm k}^{\pm}$ for all $1\leq i\leq N$, and we associate $\boldsymbol{\gamma}$ with a level-one weight $\mathrm{wt}(\boldsymbol{\gamma})\in\mathscr{X}$ by setting $\mathrm{wt}(\boldsymbol{\gamma})(K_{i})=\gamma_{i,0}^{+}$ for any $i\in I_{0}$. Call such a sequence $\boldsymbol{\gamma}$ an $\ell$-weight, $V_{\boldsymbol{\gamma}}$ the $\ell$-weight space of $\boldsymbol{\gamma}$ if $V_{\boldsymbol{\gamma}}$ is not zero.

Given an $\ell$-weight $\boldsymbol{\gamma}$. The defining relations in the Drinfeld presentation imply that $\boldsymbol{\gamma}$ is completely determined by the tuple of complex numbers $(\gamma_{i,\pm\tilde{d}_{i}k}^{\pm})_{i\in I_{0},k\in\mathbb{Z}_{\geq 0}}$. Note that $\gamma_{i,k}^{\pm}$’s for $\tilde{d}_{i}\nmid k$ are zero. Hence we may write $\boldsymbol{\gamma}\equiv(\gamma_{i,\pm\tilde{d}_{i}k}^{\pm})_{i\in I_{0},k\in\mathbb{Z}_{\geq 0}}$ directly without any ambiguity.

Now we can define the highest $\ell$-weight modules.

Definition 3.1**.**

We say $V$ is a highest $\ell$-weight modules of highest $\ell$-weight $\boldsymbol{\gamma}$ if $V=U.v$ for some non-zero vector $v\in V$ such that $x_{i,k}^{+}.v=0$ for $1\leq i\leq N,k\in\mathbb{Z}$, and $\psi_{i,\pm k}^{\pm}.v=\gamma_{i,\pm k}^{\pm}v$ for $1\leq i\leq N,k\in\mathbb{Z}_{\geq 0}$. By $(\ref{Triang})$ $\dim V_{\boldsymbol{\gamma}}=1$, so $v$ is unique up to a scalar; we call it the highest $\ell$-weight vector of $V$.

3.2. The classification theorem: rationality

In this subsection we give the classification of simple highest $\ell$-weight modules with finite weight multiplicity, which appeared in [33] for untwisted cases.

We say an $\ell$-weight $\boldsymbol{f}=(f_{i,\pm\tilde{d}_{i}k}^{\pm})_{i\in I_{0},k\in\mathbb{Z}_{\geq 0}}$ is rational if there is a tuple of complex-valued rational functions $(f_{i}(z))_{i\in I_{0}}$ in a formal variable $z$ such that for each $i\in I_{0}$, $f_{i}(z)$ is regular at [math] and $\infty$, $f_{i}(0)f_{i}(\infty)=1$ and

[TABLE]

in the sense that the left and right hand sides are the Laurent expansions of $f_{i}(z)$ at [math] and $\infty$, respectively.

Let $\mathcal{R}$ be the set of all rational $\ell$-weights. Then $\mathcal{R}$ forms an abelian group with the group operation $(\boldsymbol{f},\boldsymbol{g})\mapsto\boldsymbol{f}\boldsymbol{g}$ being given by component-wise multiplication of the corresponding tuples of rational functions. In what follows, we do not always distinguish between a rational $\ell$-weight $\boldsymbol{f}$ and the corresponding tuple $(f_{i}(z))_{i\in I_{0}}$ of rational functions.

Recall from [7, 9] that simple finite-dimensional modules of $U$ are highest $\ell$-weight modules, and their highest $\ell$-weights $\boldsymbol{f}$ are parametrized by the tuples of the Drinfeld polynomials. More precisely, there exists a tuple of polynomials $(P_{i}(z))_{i\in I_{0}}$ with all $P_{i}(z)$ having constant coefficient $1$ such that $\boldsymbol{f}$ satisfies that for $i\in I_{0}$,

[TABLE]

Therefore, the highest $\ell$-weight of any simple finite-dimensional module is rational.

In general, we have the following theorem.

Theorem 3.2**.**

Let $V$ be an irreducible highest $\ell$-weight module. Then all weight spaces of $V$ are finite-dimensional if and only if its highest $\ell$-weight $\boldsymbol{f}$ belongs to $\mathcal{R}$.

Proof.

For the non-twisted cases, one can refer to [33, Theorem 3.7] and the references therein. The proof of the twisted cases is essentially parallel to that of the untwisted cases thanks to the triangular decomposition (2.7) of the Drinfeld realization. ∎

4. Shiftability conditions and algebra homomorphisms

In this section, the notion of shiftability condition with respect to a generalized Cartan matrix will be introduced, and the compatible structures of the quantum affine algebras with the $n$-fold of $q$-oscillator algebras are given from the $q$-shiftability condition.

4.1. Shiftability conditions

Given any symmetrizable generalized Cartan matrix $A=(a_{ij})_{i,j\in I}$. Let $\mathcal{A}$ be the Laurent polynomial ring over $\mathbb{C}$ in the variables $x_{i},i\in I$, i.e., $\mathcal{A}=\mathbb{C}[x_{i}^{\pm 1},i\in I].$ For each $i\in I$, consider the algebra automorphism $\zeta_{i}:\mathcal{A}\rightarrow\mathcal{A}$ given by $\zeta_{i}(x_{j})=q_{i}^{-\delta_{ij}}x_{j}$ for $j\in I$. For any distinct $i,j\in I$, we say a pair of Laurent polynomials $(f,g)$ in $\mathcal{A}$ is $(i,j)$-shiftable if $f,g$ satisfy the equation

[TABLE]

Set $\{x\}_{i}:=\frac{x-x^{-1}}{q_{i}-q_{i}^{-1}}$ for any unit $x$ in $\mathcal{A}$, and write $\{x\}=\frac{x-x^{-1}}{q-q^{-1}}$ for simplicity. Define the elements $y_{i},y_{i}^{-1}\in\mathcal{A}$ as follows:

[TABLE]

Consider the following system of equations with respect to the variables $\phi_{i},i\in I$ in $\mathcal{A}$:

[TABLE]

In general, this system of equations does not always have solutions. It depends on the choice of the generalized Cartan matrix $A$. Therefore, we can say $A$ admits the $q$-shiftability condition when the corresponding system of equations (4.1) has a solution.

By a quick computation, we obtain a family of solutions to (4.1) for $A$ of types $A_{2}$ and $A_{1}^{(1)}$.

Example 4.1**.**

$\mathrm{(i)}$* For the type $A_{2}$, a pair of Laurent polynomials $(\phi_{1},\phi_{2})$, where $\phi_{1}=\{qbx_{1}\}\{bx_{1}^{-1}x_{2}\}$ and $\phi_{2}=\{qbx_{1}^{-1}x_{2}\}\{bx_{2}^{-1}\}$ for each scalar $b\in\mathbb{C}^{\times}$ is a solution;*

$\mathrm{(ii)}$* For the type $A_{1}^{(1)}$, consider the Laurent polynomials $\phi_{0}=\{qbx_{0}x_{1}^{-1}\}\{bx_{0}^{-1}x_{1}\}$ and $\phi_{1}=\{qbx_{0}^{-1}x_{1}\}\{bx_{0}x_{1}^{-1}\}$ for any scalar $b\in\mathbb{C}^{\times}$. It is easy to check that $(\phi_{0},\phi_{1})$ is a solution.*

In what follows, the $q$-shiftability condition for the generalized Cartan matrices of affine types will be investigated. Now assume that $A$ is an affine Cartan matrix as in Section 2. Then we have the first main result in this section.

Theorem 4.2**.**

There exists an $(n+1)$-tuple of Laurent polynomials in $\mathcal{A}$ satisfying the system of equations $(\ref{mainequ})$ if and only if $A$ is the type $A_{n}^{(1)}(n\geq 1),C_{n}^{(1)}(n\geq 2),A_{2n}^{(2)}(n\geq 1)$ or $D_{n+1}^{(2)}(n\geq 2)$.

The proof of Theorem 4.2 will be given in Appendix A. Here we list all tuples of Laurent polynomials $(\phi_{i})_{i\in I}$ satisfying (4.1) for each affine Cartan matrix $A$ in the theorem.

[TABLE]

where $\imath=\sqrt{-1}$. The elements $z_{i}\in\mathcal{A}$, and the relations in our notations are given as follows for each type:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

By our convention, the Dynkin diagrams of the above four types and the corresponding $q_{i}=q^{d_{i}}$ are the following:

[math]$q$ 1$$q$$2$$q$$n-1$$q$$n$$q… $(A^{(1)}_{n})$

[math]q$$1$$q^{\frac{1}{2}}$${n-1}$$q^{\frac{1}{2}}$$n$$q$$...$$(C_{n}^{(1)})

[math]q^{2}$$1$$q$${n-1}$$q$$n$$q^{\frac{1}{2}}$$...$$(A_{2n}^{(2)})

[math]q$$1$$q^{2}$${n-1}$$q^{2}$$n$$q$$...$$(D_{n+1}^{(2)})

Remark 4.3**.**

One can also consider the shiftability condition for a generalized Cartan matrix $A$ in the classical sense. More precisely, consider the polynomial ring $\mathcal{A}^{+}=\mathbb{C}[x_{i},i\in I]$, and the algebra automorphisms ${\zeta}_{i}:\mathcal{A}^{+}\rightarrow\mathcal{A}^{+}$ defined by ${\zeta}_{i}(x_{j})=x_{j}-\delta_{ij}$ for all $j\in I$. Denote ${y}_{i}=\sum_{j\in I}a_{ij}x_{j}$. Then a similar system of equations in $\mathcal{A}^{+}$ (replace $\{y_{i}\}_{i}$ in $(\ref{mainequ})$ by $y_{i}$) can be obtained.

4.2. Quantized oscillator algebra and algebra homomorphisms

One interesting application of $q$-shiftability condition is to study the compatible structures of quantum affine algebras of types $X_{N}^{(r)}$ with the $n$-fold of quantized oscillator algebra.

Fix $\nu\in\mathbb{C}^{\times}$. The (symmetric) quantized oscillator algebra $\mathscr{B}_{\nu}$ is the unital associative algebra over $\mathbb{C}$ generated by four elements $\mathbf{a}^{+}$, $\mathbf{a}$, $\mathbf{k}^{\pm 1}$ subject to the relations

[TABLE]

where $[x,y]_{\nu}:=xy-\nu^{-1}yx$. Then we have $\mathbf{a}^{+}\mathbf{a}=\{\mathbf{k}\}$, $\mathbf{a}\mathbf{a}^{+}=\{\nu\mathbf{k}\}$ and $\{\mathbf{k}\}\mathbf{a}^{+}=\mathbf{a}^{+}\{\nu\mathbf{k}\}$, $\mathbf{a}\{\mathbf{k}\}=\{\nu\mathbf{k}\}\mathbf{a}$ in $\mathscr{B}_{\nu}$.

One can easily check the following results.

Lemma 4.4**.**

$\mathrm{(i)}$* There exists a unique $\mathbb{C}$-algebra automorphism (an involution) $\vartheta:\mathscr{B}_{\nu}\rightarrow\mathscr{B}_{\nu}$ such that $\vartheta(\mathbf{a}^{+})=-\mathbf{a}$, $\vartheta(\mathbf{a})=\mathbf{a}^{+}$ and $\vartheta(\mathbf{k})=\nu^{-1}\mathbf{k}^{-1}$.

$\mathrm{(ii)}$ For any $b\in\mathbb{C}^{\times}$ and $m\in\mathbb{Z}$, there exists a family of $\mathbb{C}$-algebra automorphisms $\theta_{b,m}:\mathscr{B}_{\nu}\rightarrow\mathscr{B}_{\nu}$ such that $\theta_{b,m}(\mathbf{a})=b\mathbf{k}^{m}\mathbf{a}$, $\theta_{b,m}(\mathbf{a}^{+})=b^{-1}\mathbf{a}^{+}\mathbf{k}^{-m}$ and $\theta_{b,m}(\mathbf{k}^{\pm 1})=\mathbf{k}^{\pm 1}$.*

Consider the algebra $\mathscr{B}_{\nu}^{\otimes n}$ of the $n$-fold tensor product of $\mathscr{B}_{\nu}$. Denote the generators of its $i$-th component by $\mathbf{a}^{+}_{i}$, $\mathbf{a}_{i}$ and $\mathbf{k}_{i}^{\pm 1}$, which satisfy the above relations. Let $U_{q}(X_{N}^{(r)})$ be the quantum affine algebra $U$ of the type $X_{N}^{(r)}$ in Theorem 4.2. For convenience, if $X$ is the type $A$ then we shall deal with $A_{n-1}^{(1)}(n\geq 2)$ instead of $A_{n}^{(1)}(n\geq 1)$ from now on.

Fix a solution $(\phi_{i})_{i\in I}$ in Subsection 4.1. We define the algebra homomorphism $\pi_{X_{N}^{(r)}}:U_{q}(X_{N}^{(r)})\rightarrow\mathscr{B}_{\nu}^{\otimes n}$ in the following way: regard $\phi_{i}$ and $\zeta_{i}(\phi_{i})$ as the images of $X_{i}^{-}X_{i}^{+}$ and $X_{i}^{+}X_{i}^{-}$ respectively under $\pi_{X_{N}^{(r)}}$ by setting $\mathbf{k}_{i}=q^{-1}b_{A}^{-1}z_{i}^{-1}$ for the type $A_{n-1}^{(1)}$, and $\mathbf{k}_{i}=\imath\nu^{-1/2}z_{i}^{-1}$ otherwise (Here we consider the solution with $b_{C}=\imath q^{-1/4}$ for the type $C_{n}^{(1)}$), where $\nu$ is defined as in the following proposition for each type. Then the relations (2.3) for $i=j$ holds under $\pi_{X_{N}^{(r)}}$ since $\phi_{i}$ satisfies $\zeta_{i}(\phi_{i})-\phi_{i}=\{y_{i}\}_{i}$. In this sense, $\mathcal{A}_{0}:=\mathbb{C}[\mathbf{k}_{1}^{\pm 1},\cdots,\mathbf{k}_{n}^{\pm 1}]$ is a subalgebra of $\mathcal{A}$, and $\pi_{X_{N}^{(r)}}(K_{i})=y_{i}$ for $i\in I$. On the other hand, we choose $\pi_{X_{N}^{(r)}}(X_{i}^{\pm})\in\mathscr{B}_{\nu}^{\otimes n}$ satisfying that

[TABLE]

for any $f\in\mathcal{A}_{0}$. The above choice yields the relations (2.1)-(2.4) hold. Then we get the following algebra homomorphisms, which were obtained in [19, 28].

Proposition 4.5** ([19, 28]).**

For a parameter $z$, there exist algebra homomorphisms from $U_{q}(X_{N}^{(r)})$ to $\mathscr{B}_{\nu}^{\otimes n}[z,z^{-1}]$ defined as follows:

($A_{n-1}^{(1)},\nu=q$)**

$\pi_{A_{n-1}^{(1)},z}:U_{q}(A_{n-1}^{(1)})\rightarrow\mathscr{B}_{\nu}^{\otimes n}[z,z^{-1}]$

[TABLE]

In this type, we always read the index $i$ as $i$ modulo $n$.

($C_{n}^{(1)},\nu=q^{\frac{1}{2}}$)**

$\pi_{C_{n}^{(1)},z}:U_{q}(C_{n}^{(1)})\rightarrow\mathscr{B}_{\nu}^{\otimes n}[z,z^{-1}]$

[TABLE]

($A_{2n}^{(2)},\nu=q$)**

$\pi_{A_{2n}^{(2)},z}:U_{q}(A_{2n}^{(2)})\rightarrow\mathscr{B}_{\nu}^{\otimes n}[z,z^{-1}]$

[TABLE]

($D_{n+1}^{(2)},\nu=q^{2}$)**

$\pi_{D_{n+1}^{(2)},z}:U_{q}(D_{n+1}^{(2)})\rightarrow\mathscr{B}_{\nu}^{\otimes n}[z,z^{-1}]$

[TABLE]

where $\tau_{\nu}={(\nu+1)}/{(\nu-1)}$. ∎

5. Multiplicity-free weight modules

In this section, we construct the multiplicity-free weight representations over $U$ from the solutions and the algebra homomorphisms in the previous section. Throughout this section, we assume that $U$ is the quantum affine algebra of type $X_{N}^{(r)}$ in Proposition 4.5.

5.1. Module structures on $\mathcal{A}$

In order to construct the multiplicity-free weight representations, we first consider the auxiliary $U$-module structures on $\mathcal{A}$.

Let us fix some notations here. Note that $\alpha_{0}$ and $\alpha_{n}$ are long roots in the type $C_{n}^{(1)}$, while both of them are short roots in $D_{n+1}^{(2)}$. In addition, by our assumption, $\alpha_{0}$ is long, $\alpha_{n}$ is short in $A_{2n}^{(2)}$. We define a pair $\kappa:=(\kappa_{1},\kappa_{2})$ of signs such that $\kappa_{1}$, $\kappa_{2}$ are equal to [math] or $1$, which depends on the length of the roots $\alpha_{0}$ and $\alpha_{n}$ for each type, that is,

[TABLE]

Fix a solution $(\phi_{i})_{i\in I}$, and recall the units $z_{i}$’s for each type, and the shift operators $\zeta_{i}$ defined in Subsection 4.1. Put $b=b_{X_{N}^{(r)}}$. Then we have

Theorem 5.1**.**

Let $z$ be a parameter valued in $\mathbb{C}^{\times}$. For an $n$-tuple $\mathbf{f}=(f_{i})_{1\leq i\leq n}$ satisfying that $f_{i}$ is $1$ or $bz_{i}-b^{-1}z_{i}^{-1}$, ${1\leq i\leq n}$, there exists a $U$-module structure on the algebra $\mathcal{A}$ for each type defined in the following:

For the type $A_{n-1}^{(1)}$,

[TABLE]

and $K_{i}^{\pm 1}.u=y_{i}^{\pm 1}u$, for any $u\in\mathcal{A}$.

For other types,

[TABLE]

and $K_{i}^{\pm 1}.u=y_{i}^{\pm 1}u$, for any $u\in\mathcal{A}$.

Proof.

It is easy to check that $\phi_{i}=\zeta_{i}^{-1}(X_{i}^{+}.1)(X_{i}^{-}.1)=(X_{i}^{-}X_{i}^{+}).1$ for each $i$, where $1$ is the unit element in $\mathcal{A}$. It suffices to show that the actions $X_{i}^{\pm}.1,i\in I$ do hold under the defining relations (2.1)-(2.4). It is clear for the relations (2.1)-(2.2), and the cases that $i=j$ and $|i-j|>1$ in the relations (2.3). To check the rest part of (2.3), we first assume $a_{ij}a_{ji}=1$. Then we have to show

[TABLE]

If $j=i+1$ then $\zeta_{i}(f_{j+1})=f_{j+1}$, $\zeta_{j}^{-1}(f_{i})=f_{i}$, and $\zeta_{i}(f_{j})=\zeta_{j}^{-1}(f_{j})$; while $i=j+1$ we have $\zeta_{j}^{-1}(f_{i+1})=f_{i+1}$, $\zeta_{i}(f_{j})=f_{j}$, and $\zeta_{i}(f_{i})=\zeta_{j}^{-1}(f_{i})$. Both cases imply the above equality holds. When $X_{N}^{(r)}\neq A_{n-1}^{(1)}$, a direct computation yields the following equalities:

[TABLE]

Then similar arguments for the case that $a_{ij}a_{ji}=2$ are true. Any tuple $(\phi_{i})_{i}$ satisfying (4.1) and the choice of $(f_{i})_{i}$ also guarantee that these actions hold under the quantum Serre relations (2.4). ∎

Denote the above $U$-module by $S_{z}(\mathbf{f})$. Note that $S_{z}(\mathbf{f})$ is finitely $U^{0}$-generated instead of $U^{0}$-diagonalizable when restricted as a $U^{0}$-module. Recall $\pi_{X_{N}^{(r)}}(K_{i})=y_{i}$ in the construction in Subsection 4.2. Then we have

[TABLE]

In particular, $K_{\delta}$ acts trivially on $S_{z}(\mathbf{f})$.

Now let us explain the “weighting” procedure mentioned in the introduction. That is, for any $U$-module $S_{z}(\mathbf{f})$, we associate a weight module $\mathcal{W}(S_{z}(\mathbf{f}))$ in the following way. Consider the algebra embedding from $U^{0}$ to $\mathcal{A}$ by assigning $K_{i}$ to $y_{i}$, $i\in I$, which induces a natural group surjection from the group of characters of $\mathcal{A}$ to $\mathscr{X}$. For any character $\varphi$ of $\mathcal{A}$, denote by $\mathfrak{m}_{\varphi}(:=\ker\varphi)$ the corresponding maximal ideal of $\mathcal{A}$. Extend $\alpha_{j}\in\mathscr{X}$ to a character of $\mathcal{A}$ by setting $\alpha_{j}(x_{i})=q_{i}^{\delta_{ij}}$, $i\in I$ (so $\alpha_{j}(y_{i})=q_{j}^{a_{ji}}$, we still denote it by $\alpha_{j}$), then we have

[TABLE]

Define

[TABLE]

where $\varphi$ is over all characters of $\mathcal{A}$.

Corollary 5.2**.**

For any $U$-module $S_{z}(\mathbf{f})$, we have $\mathcal{W}(S_{z}(\mathbf{f}))$ is a weight module, and all its simple subquotients are multiplicity-free.

Proof.

It is clear that $S_{z}(\mathbf{f})/\mathfrak{m}_{\varphi}S_{z}(\mathbf{f})$ is $1$-dimensional and $K_{i}$ acts it diagonally. In particular, $K_{\delta}$ acts by $1$. The first assertion follows from the previous statements, and the $\lambda$-weight space $\mathcal{W}(S_{z}(\mathbf{f}))_{\lambda}=\oplus_{\bar{\varphi}=\lambda}S_{z}(\mathbf{f})/\mathfrak{m}_{\varphi}S_{z}(\mathbf{f})$, where $\bar{\varphi}$ means the image of $\varphi$ in $\mathscr{X}$. Since we have

[TABLE]

the second assertion follows. ∎

Remark 5.3**.**

In fact, the $U$-module $\mathcal{W}(S_{z}(\mathbf{f}))$ is a $q$-analog of the coherent family in the sense of [32]. This “weighting” procedure was first suggested by O. Mathieu in the paper [34].

Now let us study the possible highest weights of $\mathcal{W}(S_{z}(\mathbf{f}))$ when restricted as a $U_{q}(\mathring{\mathfrak{g}})$-module. Assume that the weight vector $1+\mathfrak{m}_{\varphi}S_{z}(\mathbf{f})$ of $\mathcal{W}(S_{z}(\mathbf{f}))$ is a highest weight vector for some $\varphi$. Then we have $X_{i}^{+}.(1+\mathfrak{m}_{\varphi}S_{z}(\mathbf{f}))=0$ for $i\in I_{0}$, which implies that

[TABLE]

The weight $\lambda=\bar{\varphi}$ is level-one, which is determined uniquely by the values $\lambda(K_{i}),i\in I_{0}$. Therefore, all level-one weights can be seen automatically as weights over $U_{q}(\mathring{\mathfrak{g}})$. As a result, we can obtain the following result.

Corollary 5.4**.**

Let $\lambda\in\mathscr{X}$ be a weight of $\mathcal{W}(S_{z}(\mathbf{f}))$ for some $\mathbf{f}$. If $\lambda$ is a highest $U_{q}(\mathring{\mathfrak{g}})$-weight, then up to twistings by the automorphisms of $U_{q}(\mathring{\mathfrak{g}})$, we have

1. (1)

for the type $A_{n-1}^{(1)}$, the weight $\lambda$ is of the form $\omega_{0}+a\omega_{s}-(a+1)\omega_{s+1}$ for some $a\in\mathbb{C}$ and $s\in I$ up to a scalar of $\delta$; 2. (2)

for the type $C_{n}^{(1)}$, the weight $\lambda$ has the form $\omega_{0}/2+\omega_{n-1}-3\omega_{n}/2$ or $(\omega_{0}-\omega_{n})/2$ up to scalars of $\delta$; 3. (3)

for the type $A_{2n}^{(2)}$ (resp. $D_{n+1}^{(2)}$), the weight $\lambda=(\lambda(K_{0}),\cdots,\lambda(K_{n}))$ is defined as

[TABLE]

Proof.

The result can be deduced directly from (5.2). For example, in the type $A_{n-1}^{(1)}$, these equations (5.2) become

[TABLE]

where we denote $\varphi(z_{i})$ by $m_{i}$ for $i\in I$. Let $\lambda=\bar{\varphi}$. Then $\lambda(K_{i})=m_{i}m_{i+1}^{-1}$. To solve the equations (5.3), we divide it into two cases: if $\{qbm_{1}\}$ is not zero, then $\{bm_{i}\}=0$ for $i\geq 2$; if $\{qbm_{1}\}$ is zero, then we assume that $s$ is the maximal index such that $\{qbm_{s}\}$ is zero, then $\{qbm_{i}\}=0$ for $i\leq s$ and $\{bm_{j}\}=0$ for $j\geq s+2$. In the first case, the possible solutions are $m_{1}=\pm b^{n-1}$, $m_{i}=\pm b^{-1}$ for $i\neq 1$. Then up to twistings by sign automorphisms of $U_{q}(\mathring{\mathfrak{g}})$, we have the weight $\lambda$ is given by

[TABLE]

which is of the form $(a+1)\omega_{0}-(a+1)\omega_{1}$ for some $a\in\mathbb{C}$. In the second case, $m_{i}=\pm q^{-1}b^{-1}$ for $0\leq i\leq s$, $m_{s+1}=\pm q^{s+1}b^{n-1}$, and $m_{j}=\pm b^{-1}$ otherwise. Then up to signs, we have the weight $\lambda$ is the following:

[TABLE]

which is exactly of the form in (1). So the assertion (1) follows. ∎

All simple subquotients obtained in Corollary 5.4 can be realized as $q$-oscillator representations by using the Fock space representations of $\mathscr{B}_{\nu}$ and the algebra homomorphisms in Proposition 4.5 (cf. e.g. [25]). In the following subsection, we shall recall the $q$-oscillator representations.

5.2. Realization of multiplicity-free weight modules

Let $F=\oplus_{m\in\mathbb{Z}_{\geq 0}}\mathbb{C}|{m}\rangle$ be the Fock space representation of $\mathscr{B}_{\nu}$ on which the generators $\mathbf{a}^{+}$ and $\mathbf{a}$ act as the creation and annihilation operators respectively, and the element $\mathbf{a}^{+}\mathbf{a}=\{\mathbf{k}\}_{\nu}$ corresponds to the number operator, more precisely, for any $m\in\mathbb{Z}_{\geq 0}$,

[TABLE]

In particular, $\mathbf{a}.|{0}\rangle=0$.

Denote this representation by $\rho:\mathscr{B}_{\nu}\rightarrow\mathrm{End}_{\mathbb{C}}(F)$. For any $b\in\mathbb{C}^{\times}$ and $\varepsilon\in\{0,1\}$, we denote by $\rho_{\varepsilon,b}$ the composition $\rho\circ\vartheta^{\varepsilon}\circ\theta_{b,0}$ (cf. Lemma 4.4). Then $F$ has a new $\mathscr{B}_{\nu}$-module structure via $\rho_{\varepsilon,b}$.

Definition 5.5**.**

Let $z$ be a parameter valued in $\mathbb{C}^{\times}$. We define the representation $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ of $U$ on the space $F^{\otimes n}$ via the composition of the algebra homomorphisms $\pi_{z}:=\pi_{X_{N}^{(r)},z}$ defined in Proposition 4.5 and

[TABLE]

where $\boldsymbol{\varepsilon}=(\varepsilon_{i})_{i}\in\{0,1\}^{n}$, and $\mathbf{b}=(b_{i})_{i}\in(\mathbb{C}^{\times})^{n}$.

For any $n$-tuple $(m_{i})_{i}\in(\mathbb{Z}_{\geq 0})^{n}$, we use $|{\mathbf{m}}\rangle:=|{m_{1}}\rangle\otimes\cdots\otimes|{m_{n}}\rangle$ for the basis vector of $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$. Let $\mathbf{e}_{j}$ be the $j$-th standard vector in $\mathbb{Z}^{n}$ with $1$ at the $j$-th term and [math] otherwise for $1\leq j\leq n$. Moreover, set $\bf 0$ for $(0,\cdots,0)\in\mathbb{Z}^{n}$.

For $n\geq 2$, note that $U$-module actions of $X^{\pm 1}_{i},K_{i}$ for $1\leq i\leq n-1$ on $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$, by Definition 5.5, can be written down uniformly as follows:

[TABLE]

for $\mathbf{m}\in(\mathbb{Z}_{\geq 0})^{n}$, where $\nu$ is defined in Proposition 4.5 for each type, and $|{\mathbf{m}}\rangle$ for $\mathbf{m}\notin(\mathbb{Z}_{\geq 0})^{n}$ can be read as [math]. Here we remark that the $U_{q}(A_{n-1}^{(1)})$-module actions of $X^{\pm}_{0},K_{0}$ on $|{\mathbf{m}}\rangle$ also have the above forms where we understand the indices $i,i+1$ as $n,1$ (mod $n$) respectively.

Regard $U_{q}(A_{n-1})$ as the subalgebra of $U_{q}(\mathfrak{g})(n\geq 2)$ via forgetting the actions of the Drinfeld-Jimbo generators indexed by [math] and $n$. One can check that $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ as a $U_{q}(A_{n-1})$-module is a multiplicity-free weight module. In fact, $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ has the following direct sum decomposition:

[TABLE]

For any $\mathbf{m}\in\mathbb{Z}^{n}$, denote $|\mathbf{m}|_{\boldsymbol{\varepsilon}}=\sum_{i}(-1)^{\varepsilon_{i}}m_{i}$. Each $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,(l)}$ is an irreducible, multiplicity-free weight $U_{q}(A_{n-1})$-module by the formulae (5.4)-(5.6) as $q$ (or $\nu$) is not a root of unity.

Fix $1\leq i\leq n$. Define the algebra homomorphism $\delta_{i}:\mathcal{A}_{0}\rightarrow\mathbb{C}$ by $\mathbf{k}_{j}\mapsto\nu^{\delta_{ij}}$ for $1\leq j\leq n$. Then it induces an algebra character $\tilde{\delta}_{i}\in\mathscr{X}$ by

[TABLE]

Then we have

Proposition 5.6**.**

For $\boldsymbol{\varepsilon}\in\{0,1\}^{n}$, $\mathbf{b}\in(\mathbb{C}^{\times})^{n}$, we have

1. (i)

$\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$* is a weight module with $\dim(\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}})_{\lambda}\leq 1$ for any $\lambda\in\mathscr{X}$.* 2. (ii)

If $\dim(\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}})_{\lambda}=1$, then there exists $\mathbf{m}\in(\mathbb{Z}_{\geq 0})^{n}$ such that $(\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}})_{\lambda}=\mathbb{C}|{\mathbf{m}}\rangle$ with

[TABLE]

Proof.

It is clear that $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ is a weight module. By (5.6) and the actions of $K_{0}$ and $K_{n}$, which are defined for $X_{N}^{(r)}\neq A_{n-1}^{(1)}$ as follows:

[TABLE]

where $\kappa_{1}$ and $\kappa_{2}$ are defined in Subsection 5.1, the relative weight of $|{\mathbf{m}}\rangle$ is given by the right hand side of the equality (5.7). By the above statement, $\dim(\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}})_{\lambda}\leq 1$ for any $\lambda\in\mathscr{X}$. ∎

Consider the following decomposition of $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$:

[TABLE]

For $0\leq s\leq n$, let $\boldsymbol{\varepsilon}_{>s}\in\{0,1\}^{n}$ satisfy

[TABLE]

For example, $\boldsymbol{\varepsilon}_{>0}=(1,\cdots,1)$ and $\boldsymbol{\varepsilon}_{>n}=(0,\cdots,0)$.

Then we have

Proposition 5.7**.**

For any $\boldsymbol{\varepsilon}\in\{0,1\}^{n}$, $\mathbf{b}\in(\mathbb{C}^{\times})^{n}$, we have

1. (i)

As a $U_{q}(A_{n-1}^{(1)})$-module, $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,(l)}$ is irreducible for any admissible $l\in\mathbb{Z}$; it is a highest $\ell$-weight module with a highest $\ell$-weight vector $v_{l,s}$ if and only if $\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{>s}$ for some $0\leq s\leq n$, where

[TABLE] 2. (ii)

As $U_{q}(C_{n}^{(1)})$-modules, $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,+}$ and $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,-}$ are irreducible; they are highest $\ell$-weight modules with highest $\ell$-weight vector $v^{+}=|{\bf 0}\rangle$ and $v^{-}=|{\mathbf{e}_{n}}\rangle$ respectively whenever $\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{>n}$. 3. (iii)

As a $U_{q}(A_{2n}^{(2)})$ or $U_{q}(D_{n+1}^{(2)})$-module, $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ is irreducible; it is a highest $\ell$-weight module with a highest $\ell$-weight vector $v=|{\bf 0}\rangle$ whenever $\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{>n}$.

Proof.

Note that $K_{\delta}$ acts trivially on $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ by (5.1). The defining relations (2.5) imply that the actions of $h_{i,k}$, $1\leq i\leq N$, $k\in\mathbb{Z}\setminus\{0\}$ on $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ commute pairwise. Hence $\mathcal{F}^{z}_{\boldsymbol{\varepsilon},\mathbf{b}}$ is an $\ell$-weight $U$-module. It is clear that $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,(l)}$ is closed under the action of $U_{q}(A_{n-1}^{(1)})$. The irreducibility of $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,(l)}$ can be checked by the actions (5.4)-(5.6). Note that for $\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{>0}$ (resp. $\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{>n}$), $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,(l)}$ is finite dimensional for $l<0$ (resp. $l\geq 0$). As a $U_{q}(A_{n-1})$-module (ignore the actions of $X_{0}^{\pm},K_{0}$) $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{z,(l)}$ is a highest weight module iff $\varepsilon_{1}\leq\cdots\leq\varepsilon_{n}$, and the corresponding highest weight vector can be chosen as (5.10), which is also a highest $\ell$-weight vector by weight consideration.

For $X_{N}^{(r)}\neq A_{n-1}^{(1)}$, the actions of $X_{0}^{+}$ and $X_{n}^{+}$ are given by

[TABLE]

and $X_{n}^{+}.|{\mathbf{m}}\rangle=x|{\mathbf{m}-(-1)^{\varepsilon_{n}}(\kappa_{2}+1)\mathbf{e}_{n}}\rangle$, where $a\in\mathbb{C}^{\times}$ is defined as follows:

[TABLE]

and $\tau_{\nu,\kappa_{2}}=(\nu-\kappa_{2}+1)/(\nu+\kappa_{2}-1)$. Similarly, we can obtain the actions of $X_{0}^{-}$ and $X_{n}^{-}$. Therefore, the assertions (ii) and (iii) can be deduced directly from the above actions. ∎

6. Highest $\ell$-weights

In this section, we focus on the irreducible highest $\ell$-weight representations constructed in the previous section, and compute their highest $\ell$-weights explicitly.

6.1. Multiplicity-free highest $\ell$-weight modules

Fix $0<s<n$. Let $\mathcal{W}_{s}:=F^{\otimes n}$ be the $U_{q}(A_{n-1}^{(1)})$-module defined as follows (see also [29]):

[TABLE]

and

[TABLE]

where $1\leq i<s<j\leq n-1$ and $\mathbf{m}\in(\mathbb{Z}_{\geq 0})^{n}$. From the actions (5.4)-(5.6), $\mathcal{W}_{s}$ is just the twisting of the module $\mathcal{F}^{1}_{\boldsymbol{\varepsilon},\mathbf{b}}$ where $\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{>s}$, $\mathbf{b}=(1,\cdots,1)$ by the automorphism of $U_{q}(A_{n-1}^{(1)})$ sending $X_{k}^{\pm}$ to $-X_{k}^{\pm}$, $s\leq k\leq n$ along with other Drinfeld-Jimbo generators fixed. Denote $\mathcal{W}_{s}^{(l)}$ as the $l$-th irreducible component of $\mathcal{W}_{s}$, i.e., $\mathcal{W}_{s}^{(l)}=\oplus_{|\mathbf{m}|_{\boldsymbol{\varepsilon}}=l}\mathbb{C}|{\mathbf{m}}\rangle$.

Let $(X_{N}^{(r)},\nu)$ be one of the types in Proposition 4.5 except $(A_{n-1}^{(1)},q)$. Let $\mathcal{W}=F^{\otimes n}$ be the $U_{q}(X_{N}^{(r)})$-module defined as (see [27]):

[TABLE]

where $\mathbf{m}\in(\mathbb{Z}_{\geq 0})^{n}$, $|\kappa|=\kappa_{1}+\kappa_{2}$ and $\tau_{\nu,\kappa_{i}}=(\nu-\kappa_{i}+1)/(\nu+\kappa_{i}-1)$. Here $\kappa_{i}$’s are defined in Subsection 5.1. This module can be obtained from $\mathcal{F}^{1}_{\boldsymbol{\varepsilon},\mathbf{b}}$ with $\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{>n}$ and $\mathbf{b}=(1,\cdots,1)$ by the automorphism of $U_{q}(X_{N}^{(r)})$ defined as

[TABLE]

and other generators are fixed. For the type $C_{n}^{(1)}$, denote the irreducible components $\mathcal{F}_{\boldsymbol{\varepsilon},\mathbf{b}}^{1,\pm}$ of $U_{q}(C_{n}^{(1)})$-module $\mathcal{W}$ by $\mathcal{W}^{\pm}$, for convenience.

Lemma 6.1**.**

Let $L(\boldsymbol{f})$ be an irreducible highest $\ell$-weight $U$-module with $\boldsymbol{f}=(f_{i}(z))_{i\in I_{0}}\in\mathcal{R}$. If $\dim L(\boldsymbol{f})_{\mathrm{wt}(\boldsymbol{f})-\alpha_{i}}=1$ for some $i\in I_{0}$ then $f_{i}(z)$ satisfies that

[TABLE]

where $a,b\in\mathbb{C}$ satisfy that $f_{i,2\tilde{d}_{i}}^{+}=af_{i,\tilde{d}_{i}}^{+}$ and $f_{i,\tilde{d}_{i}}^{+}=bf_{i,0}^{+}$.

Proof.

Suppose that $v\in L(\boldsymbol{f})$ is a nonzero $\ell$-weight vector of $\boldsymbol{f}$. Note that $\{x_{i,k}^{-}.v,k\in\mathbb{Z}\}$ spans the weight space $L(\boldsymbol{f})_{\mathrm{wt}(\boldsymbol{f})-\alpha_{i}}$. If $\dim(L(\boldsymbol{f})_{\mathrm{wt}(\boldsymbol{f})-\alpha_{i}})=1$, then there exist $j\in\mathbb{Z}$ such that $x_{i,\tilde{d}_{i}j}^{-}.v$ is nonzero, and $a\in\mathbb{C}$ such that

[TABLE]

Consider the actions of $x_{i,\tilde{d}_{i}k}^{+}$, $k\in\mathbb{Z}$ on (6.2). The defining relations (2.5) imply that

[TABLE]

for any $k\in\mathbb{Z}$. Since $f_{i,k}^{-}=f_{i,-k}^{+}=0$ for $k>0$, we have $f_{i,\tilde{d}_{i}(k+1)}^{+}=af_{i,\tilde{d}_{i}k}^{+}$ for any $k>0$. Take the series $f_{i}(z)=\sum_{k=0}^{\infty}z^{k}f_{i,\tilde{d}_{i}k}^{+}$, we have

[TABLE]

Hence $f_{i}(z)$ has the rational form (6.1). ∎

6.2. Highest $\ell$-weights

Let us first study some properties of the Weyl group and the description of the root vectors of quantum affine algebras, which will enable us to compute the highest $\ell$-weight explicitly.

Lemma 6.2**.**

Let $i,j\in I$, and $i\neq j$.

1. (1)

If $a_{ij}a_{ji}=1$, then $s_{j}s_{i}\alpha_{j}=\alpha_{i}$. 2. (2)

If $a_{ij}a_{ji}=2$, then $s_{i}s_{j}s_{i}\alpha_{j}=\alpha_{j}$.

Proof.

Both (1) and (2) are easy facts deduced from $s_{j}s_{i}\alpha_{j}=(a_{ij}a_{ji}-1)\alpha_{j}-a_{ij}\alpha_{i}$ and

[TABLE]

respectively. ∎

Recall the braid group operators associated to $\widetilde{W}$ introduced by Lusztig [30]. For each simple reflection $s_{i}$, there is an algebra automorphism $T_{i}=T_{s_{i}}$ of $U$ defined by

[TABLE]

where $\beta\in Q$ and $(X_{i}^{\pm})^{(k)}=(X_{i}^{\pm})^{k}/[k]^{!}_{i}$. Then $\Phi T_{i}=T_{i}^{-1}\Phi$, where $\Phi$ is the $\mathbb{C}$-linear anti-automorphism of $U$ sending $X_{i}^{\pm}$ to $X_{i}^{\pm}$, $K_{i}$ to $K_{i}^{-1}$ for $i\in I$. For any $\tau\in\mathscr{T}$, define $T_{\tau}$ by $T_{\tau}(X_{i}^{\pm})=X_{\tau(i)}^{\pm}$ and $T_{\tau}(K_{i})=K_{\tau(i)}$.

For later use, we list some well-known properties of braid group operators (cf. [31, 1]). Choose one element $w\in\widetilde{W}$. If $\tau s_{i_{1}}s_{i_{2}}\cdots s_{i_{m}}$ is a reduced expression of $w$, then the automorphism $T_{w}=T_{\tau}T_{i_{1}}T_{i_{2}}\cdots T_{i_{m}}$ of $U$ is independent on the choice of the reduced expression of $w$. In particular, one reduced expression can be transformed to another by a finite sequence of braid relations. If $w(\alpha_{i})=\alpha_{j}$ then $T_{w}(X_{i}^{+})=X^{+}_{j}$. Moreover, if $w=s_{i_{1}}s_{i_{2}}\cdots s_{i_{m}}$ is a reduced expression and $l(ws_{i})=l(w)+1$, then we have $T_{w}X^{+}_{i}\in U^{+}$.

Remark 6.3**.**

For $i\leq j$, put $\mathbf{s}_{(i,j)}=s_{i}s_{i+1}\cdots s_{j}.$

$(\rm i)$* In the type $A_{n-1}^{(1)}$, a reduced expression of $\tilde{\omega}_{i}$, $1\leq i\leq n-1$ can be chosen as:*

[TABLE]

where $\tau$ is the diagram automorphism of $A_{n-1}^{(1)}$ sending $j$ to $j+1(\mathrm{mod}\;n)$ for $j\in I$ (cf. **[21, Subsection 3.3]**).

$(\rm ii)$* In the type $A_{2n}^{(2)}$, the reduced expression of $\tilde{\omega}_{n}$ can be chosen (cf. [11, Corollary 4.2.4]) as:*

[TABLE]

$(\rm iii)$* In the type $C_{n}^{(1)}$ (resp. $D_{n+1}^{(2)}$), the reduced expressions of $\tilde{\omega}_{n-1}$ and $\tilde{\omega}_{n}$ can be chosen as:*

[TABLE]

respectively, where $\tau$ is the diagram automorphism of $C_{n}^{(1)}$ (resp. $D_{n+1}^{(2)}$) sending $i$ to $n-i$ for $i\in I$.

Now, let us define the root vectors in $U$. We refer the reader to [2] for the construction of root vectors $X^{+}_{\beta},\beta\in\triangle$ (i.e., $E_{\beta}$’s defined therein). In particular, the real root vectors $X^{+}_{k\tilde{d}_{i}\delta\pm\alpha_{i}}$ are described explicitly by

[TABLE]

Then $X^{+}_{k\tilde{d}_{i}\delta\pm\alpha_{i}}\in U^{+}$. The imaginary root vectors are defined by

[TABLE]

and define the elements $X^{+}_{i,k\tilde{d}_{i}\delta}$ by the following formal series in $z$:

[TABLE]

Under the isomorphism of two presentations of $U_{q}(\mathfrak{g})$, the generators $\psi_{i,k\tilde{d}_{i}}^{+}$ and the imaginary root vectors $\tilde{\psi}_{i,k\tilde{d}_{i}}$ are related (cf. [1, 12]), more precisely, for $k>0$ and $i\in I_{0}$, we have

[TABLE]

where $o:I_{0}\rightarrow\{\pm 1\}$ is a map such that $o(i)=-o(j)$ whenever

1. i)

$a_{ij}\leq 0$ implies that $o(i)o(j)=-1$, 2. ii)

in the twisted cases different from $A_{2n}^{(2)}$, if $a_{ij}=-2$ then $o(i)=1$.

Note that $o(n)=1$ in the type $D_{n+1}^{(2)}$ as $a_{n,n-1}=-2$. Thus, we can deduce that the map $o:I_{0}\rightarrow\{\pm 1\}$ is uniquely determined in the type $D_{n+1}^{(2)}$.

In Lemma 6.1, the scalars $a$ and $b$ can be described by the root vectors according to the above relations, which will become more computable in our case. Let $v\in L(\boldsymbol{f})$ be a nonzero $\ell$-weight vector of $\boldsymbol{f}$. Since $C^{1/2}$ acts trivially on $L(\boldsymbol{f})$ and $k_{i}$ commutes with $\tilde{\psi}_{i,k\tilde{d}_{i}}$, it implies that

[TABLE]

Lemma 6.4**.**

For any $i\in I_{0}$, $k\in\mathbb{Z}_{>0}$, we have the root vectors $X^{+}_{k\tilde{d}_{i}\delta-\alpha_{i}}$ in $U_{q}(X_{N}^{(r)})$ have the following relations:

[TABLE]

Proof.

We may use the following relations [11, Proposition 2.2.4, Corollary 3.2.4] (cf. [1]): for $k\in\mathbb{Z}_{>0}$,

[TABLE]

Note that $X^{+}_{i,\tilde{d}_{i}\delta}\in U$ is defined by the formal series (6.4). Since $\tilde{\psi}_{i,k\tilde{d}_{i}}=[X^{+}_{k\tilde{d}_{i}\delta-\alpha_{i}},X^{+}_{i}]_{q_{i}^{2}}$ in $(\ref{Equ:psi})$, by comparing the coefficients of $z$ in (6.4), we can get

[TABLE]

which implies the lemma by (6.7) and (6.8). ∎

Let $\alpha\in Q_{+}$. We introduce the height $\mathrm{ht}\alpha$ of $\alpha$ as $\mathrm{ht}\alpha=\sum_{i\in I}m_{i}$ if $\alpha=\sum_{i\in I}m_{i}\alpha_{i}$. Define a subset $Q_{+}(\alpha)$ of $Q_{+}$ as follows:

[TABLE]

Let $U^{+}(\alpha)$ be the subspace of $U^{+}_{\alpha}$ defined as $U^{+}(\alpha)=\sum_{\beta\in Q_{+}(\alpha)}U^{+}_{\beta}X^{+}_{\alpha-\beta}$.

Lemma 6.5**.**

$(\rm 1)$* For $i\in I_{0}$, the root vector $X^{+}_{\delta-\alpha_{i}}$ in $U_{q}(A_{n-1}^{(1)})$ has the following form:*

[TABLE]

$(\rm 2)$* In $U_{q}(C_{n}^{(1)})$,*

[TABLE]

$(\rm 3)$* In $U_{q}(A_{2n}^{(2)})$,*

[TABLE]

$(\rm 4)$* In $U_{q}(D_{n+1}^{(2)})$,*

[TABLE]

Proof.

Thanks to the reduced expressions of $\tilde{\omega}_{i}$ in Remark 6.3, the lemma can be deduced directly by the definition. One can refer to [21, Lemma 4.7] for the assertion (1). To see the remaining assertions we define the operators $\mathcal{D}^{(1)}_{i}$ and $\mathcal{D}^{(2)}_{i}$ of $U$ for $i\in I_{0}$ by $\mathcal{D}^{(1)}_{i}(X)=[X,X^{+}_{i}]_{q_{i}}$ and $\mathcal{D}^{(2)}_{i}(X)=[[X,X^{+}_{i}],X^{+}_{i}]_{q}$ for any $X\in U$, respectively. In the type $C_{n}^{(1)}$, we note that $T_{\tau}\mathcal{D}^{(s)}_{i}=\mathcal{D}^{(s)}_{n-i}T_{\tau}$ and $T_{j}\mathcal{D}^{(s)}_{i}=\mathcal{D}^{(s)}_{i}T_{j}$ for $|i-j|>1$ and $s=1,2$. Denote $T_{\mathbf{s}_{(i,j)}}=T_{(i,j)}$ for simplicity. Then we have

[TABLE]

due to $\mathbf{s}_{(0,n)}s_{n-1}\alpha_{n}=\alpha_{n}$. Moreover, for any $0<i<n-1$, we have $(\mathbf{s}_{(0,n)}s_{n-1})^{i-1}\alpha_{1}=\alpha_{i}$ and $\mathbf{s}_{(i+1,n-1)}\mathbf{s}_{(i,n-2)}\alpha_{n-2}=\alpha_{i}$ by using Lemma 6.2, thus we get

[TABLE]

and

[TABLE]

Finally, the definition of the root vectors and Remark 6.3 imply that

[TABLE]

where $\mathcal{D}_{i,q}^{(1)}(X)=[X,X^{+}_{i}]_{q}$ for $X\in U$, and

[TABLE]

which deduce the assertion (2). Similarly, we can prove that

[TABLE]

which imply (3) and (4). ∎

Remark 6.6**.**

In order to simplify computations in the following theorem for the type $A_{2n}^{(2)}(n\geq 2)$, we actually only need the two terms of $X^{+}_{\delta-\alpha_{n}}$:

[TABLE]

which can be deduced directly by the formula (6.9).

Now we compute the highest $\ell$-weights of the $q$-oscillator representations defined in Subsection 6.1.

Theorem 6.7**.**

$(\rm 1)$* Fix $0<s<n$ and $l\in\mathbb{Z}$. The $U_{q}(A_{n-1}^{(1)})$-module $\mathcal{W}_{s}^{(l)}$ has the highest $\ell$-weight $\boldsymbol{f}=(f_{i}(z))_{i\in I_{0}}$ given as follows:*

[TABLE]

for $1\leq i\leq n-1$, where $u=o(s)(-q^{-1})^{n}z$,

$(\rm 2)$* The highest $\ell$-weight of the $U_{q}(C_{n}^{(1)})$-module $\mathcal{W}^{+}$ (resp. $\mathcal{W}^{-}$) is given as follows:*

[TABLE]

where $u=o(n)q^{-n-1}z$.

$(\rm 3)$* The highest $\ell$-weight of the $U_{q}(A_{2n}^{(2)})$ (resp. $U_{q}(D_{n+1}^{(2)})$)-module $\mathcal{W}$ is given by*

[TABLE]

where $u=o(n)\imath\tau_{q}q^{-2n-1}z$ (resp. $u=q^{-2n}z$).

Proof.

The proof of the first assertion can be found in [29, Theorem 4.10]. For (2), we have verified in Proposition 5.7 that $v^{+}=|{{\bf 0}}\rangle$ and $v^{-}=|{\mathbf{e}_{n}}\rangle$ are highest $\ell$-weight vectors of $U_{q}(C_{n}^{(1)})$-modules $\mathcal{W}^{+}$ and $\mathcal{W}^{-}$ respectively. Therefore, it follows from Lemma 6.1 and the formulae (6.6) that we only need to compute the actions of $X^{+}_{2\tilde{d}_{i}\delta-\alpha_{i}}$ and $\tilde{\psi}_{i,\tilde{d}_{i}}$ on $v^{\pm}$.

Note that $X^{+}_{j}.v^{\pm}=0$ for all $j\in I_{0}$. By using Lemma 6.5 we have

[TABLE]

and then

[TABLE]

Therefore, we have

[TABLE]

and

[TABLE]

On the other hand,

[TABLE]

and then

[TABLE]

In other cases, we can check that $X^{+}_{\delta-\alpha_{i}}.v=0$, then $\tilde{\psi}_{i,1}.v=0$ and $X^{+}_{2\delta-\alpha_{i}}.v=0$. Thus, we get (2) as desired.

To get (3), let $v:=|{0}\rangle$. We first focus on the type $D_{n+1}^{(2)}$. By Lemma 6.5(4) we have

[TABLE]

and then

[TABLE]

For $i\neq n$, we have $X^{+}_{\tilde{d}_{i}\delta-\alpha_{i}}.v=0$, then $\tilde{\psi}_{i,\tilde{d}_{i}}.v=0$ and $X^{+}_{2\tilde{d}_{i}\delta-\alpha_{i}}.v=0$. Thus, the assertion (3) for the type $D_{n+1}^{(2)}$ is proved. In the type $A_{2n}^{(2)}$, Lemma 6.5(3) yields

[TABLE]

Note that all terms in the expression of $X^{+}_{\delta-\alpha_{n}}$ vanish on the vector $|{\mathbf{e}_{n}}\rangle$ except for the two terms in (6.11). We can compute the following action by using (6.11):

[TABLE]

Therefore,

[TABLE]

Then $a=o(n)\tau_{q}q^{-2n-3/2}$ and $b=o(n)\tau_{q}q^{-2n}(q^{-1/2}+q^{-3/2})$. The assertion (3) for the type $A_{2n}^{(2)}$ follows from Lemma 6.1 and Corollary 5.4(3). ∎

Appendix A Proof of Theorem 4.2

A.1. Proof of Theorem 4.2

For generalized Cartan matrices of finite types, the corresponding system of equations $(\ref{mainequ})$ has been solved in [10], the prescription used in this appendix is parallel with the one in there.

Given an affine Cartan matrix $A$. Fix one $i\in I$, denote $\mathcal{A}_{x_{i}}$ the subalgebra of $\mathcal{A}$ generated by all $x_{j}^{\pm 1}$ for $j\neq i$. Then there is a natural isomorphism $\mathcal{A}\cong\mathcal{A}_{x_{i}}[x_{i}^{\pm 1}].$ Suppose that $(\phi_{i})_{i\in I}$ is any solution to the system of equations (4.1).

We have the following two crucial lemmas.

Lemma A.1**.**

Any $\phi\in\mathcal{A}$ satisfying $\zeta_{i}(\phi)-\phi=\{y_{i}\}_{i}$ has the form $\phi=\beta^{+}_{i}y_{i}+\phi_{0}+\beta^{-}_{i}y_{i}^{-1}$, where $\phi_{0}\in\mathcal{A}_{x_{i}}$, $\beta^{+}_{i}=-q_{i}(q_{i}-q_{i}^{-1})^{-2}$ and $\beta^{-}_{i}=-q_{i}^{-1}(q_{i}-q_{i}^{-1})^{-2}$.

Proof.

Let $\phi=\sum_{k}\phi_{k}x_{i}^{k}$ with $\phi_{k}\in\mathcal{A}_{x_{i}}$. Then $\zeta_{i}(\phi)-\phi=\{y_{i}\}_{i}$ implies that

[TABLE]

Hence $\phi_{k}$ is zero unless $k=0,\pm 2$, and

[TABLE]

So the lemma is proved. ∎

Therefore, we may always assume that $\phi_{i}$ in the system of equations (4.1) satisfies $\phi_{i}=\beta^{+}_{i}y_{i}+\phi_{i,0}+\beta^{-}_{i}y_{i}^{-1}$ where $\phi_{i,0}\in\mathcal{A}_{x_{i}}$.

Note that any pair $(\phi_{i},\phi_{j})$ is $(i,j)$-shiftable. This condition can further restrict the choices of $\phi_{i,0}$ and $\phi_{j,0}$ when the nodes $i$ and $j$ are not connected in the Dynkin diagram of $A$, namely, $a_{ij}=0$. More precisely, we have

Lemma A.2**.**

If $a_{ij}=0$, then both $\phi_{i,0}$ and $\phi_{j,0}$ lie in $\mathcal{A}_{x_{i}}\cap\mathcal{A}_{x_{j}}$.

Proof.

Since $a_{ij}=0$, then $a_{ji}=0$, we have $y_{i}\in\mathcal{A}_{x_{j}}$ and $y_{j}\in\mathcal{A}_{x_{i}}$. If $\phi_{i,0}=0$, there is nothing to do. Assume that $\phi_{i,0}$ is not zero. We rewrite $\phi_{i,0}$ uniquely as the Laurent polynomial in $x_{j}$, i.e., a unique form in $\mathcal{A}_{x_{j}}[x_{j}^{\pm 1}]$. Take the nonzero term in this form of $\phi_{i,0}$ such that $x_{j}$ has the highest (resp. lowest) power, denoted by $\phi_{i,max}$ (resp. $\phi_{i,min}$), then the shiftability of $(\phi_{i},\phi_{j})$ implies that

[TABLE]

where $m=\mathrm{deg}_{x_{j}}\phi_{i,max}$ and $l=\mathrm{deg}_{x_{j}}\phi_{i,max}$. Hence $m=0=l$. Then we can conclude that $\phi_{i,0}\in\mathcal{A}_{x_{i}}\cap\mathcal{A}_{x_{j}}$ as desired. Similarly, we have $\phi_{j,0}\in\mathcal{A}_{x_{i}}\cap\mathcal{A}_{x_{j}}$. ∎

Let us first focus on the rank-two cases. Fix $i\neq j$ in $I$ and $J=\{i,j\}$. Due to Lemma A.2 we may assume that the nodes $i$ and $j$ are connected. Without loss of generality, we set $\lambda=a_{ij},\mu=a_{ji}$ and $|\lambda|\geq|\mu|$, then

[TABLE]

Then $y_{i}=x_{i}^{2}x_{j}^{\mu}$ and $y_{j}=x_{i}^{\lambda}x_{j}^{2}$ in this case. Assume that $\varphi_{i}$ and $\varphi_{j}$ have the forms as in Lemma A.1, i.e.,

[TABLE]

where $\varphi_{l,0}\in\mathcal{A}_{x_{l}},l\in J$, and let $\varphi_{i}$ and $\varphi_{j}$ satisfy the equality

[TABLE]

Record the $(x_{i},x_{j})$-degrees of a monomial $u$ in $\mathcal{A}$ by a degree vector

[TABLE]

Then a Laurent polynomial $f$ corresponds to a matrix with each column vector representing for the $(x_{i},x_{j})$-degrees of certain term of $f$. Moreover, if $\varphi_{i,0}$ is not zero (resp. $\varphi_{j,0}$ is not zero), then we use one vector with a parameter $s$ (resp. $t$)

[TABLE]

to stand for the possible $(x_{i},x_{j})$-degrees of $\varphi_{i,0}$ (resp. $\varphi_{j,0}$). For example, by Lemma A.2, if $a_{ij}=0$, then $s$ and $t$ always equal [math]. Therefore, we obtain the following matrix with possible $(x_{i},x_{j})$-degrees of terms of $\varphi_{i}\varphi_{j}$:

[TABLE]

where the first row of the above matrix is the corresponding shifted coefficients in $\zeta_{i}^{-1}(\varphi_{j})\zeta_{j}^{-1}(\varphi_{i})$. The terms with shifted coefficient $1$ can be cancelled on the left and right hand sides of the equality $(\ast)$, then we may omit such terms. Therefore, we have the following matrix

[TABLE]

One useful statement is that if a shifted coefficient is not $1$, then the corresponding degree vector has to be equal to another one in the matrix $(\rm{M1})$ by the equality $(\ast)$. Therefore we can determine all possible $(x_{i},x_{j})$-degrees of $\varphi_{i,0}$ and $\varphi_{j,0}$ as follows:

[TABLE]

Note that there is no $(x_{i},x_{j})$-degree vector for the type $G_{2}$ satisfying the above statement, so does for types $G_{2}^{(1)}$ and $D_{4}^{(3)}$. We have obtained all solutions for the type $A_{1}^{(1)}$ in Example 4.1. In the type $A_{2}^{(2)}$, we substitute the reduced forms of $\varphi_{i,0}$ and $\varphi_{j,0}$, i.e., $\varphi_{i,0}=0,\varphi_{j,0}\in\mathbb{C}^{\times}$, into (4.1), and then get

[TABLE]

By our assumption in Section 2, we have $i=1,j=0$ and $\phi_{0}=\varphi_{j},\phi_{1}=\varphi_{i}$ for the type $A_{2}^{(2)}$.

Let us turn to the higher rank cases. The next result tells us how to “glue” the rank-two cases together.

Lemma A.3**.**

Let $j\in I$ be a node which connects to the other two distinct nodes $i$ and $l$ in the Dynkin diagram. Assume that $\phi_{j,0}\neq 0$ and the pair of integers $(m,t)$ is the $(x_{i},x_{l})$-degree of any nonzero (monomial) term of $\phi_{j,0}$. Then we have $mt\leq 0$.

Proof.

Otherwise, assume that $mt>0$ and the corresponding nonzero term of $\phi_{j,0}$ is $\phi_{j,0}^{(1)}$. Without loss of generality, we may let $m>0$ and $t>0$. Consider the term $\beta^{-}_{i}y_{i}^{-1}\phi_{j,0}^{(1)}$ of $\phi_{i}\phi_{j}$ which has the factor $x_{i}^{m-2}x_{j}^{-a_{ji}}x_{l}^{t-a_{li}}$. So we have the shifted coefficient $q_{i}^{m}q_{j}^{-a_{ji}}=q_{i}^{m-a_{ij}}$ in $\zeta_{j}^{-1}\phi_{i}\zeta_{i}^{-1}\phi_{j}$ is not $1$. However, there is no other term in $\phi_{i}\phi_{j}$ whose $(x_{i},x_{j},x_{l})$-degree vector equals $(m-2,-a_{ji},t-a_{li})$. It is a contradiction. Hence $mt\leq 0$. ∎

The Lemma A.3 implies that there is no solution to the system of equations (4.1) for $A$ whose Dynkin diagram contains $D_{4}$ or $F_{4}$ as a subdiagram.

So far, we have ruled out all affine Cartan matrices except that of types $A_{n}^{(1)}(n\geq 1),C_{n}^{(1)}(n\geq 2),A_{2n}^{(2)}(n\geq 1)$ or $D_{n+1}^{(2)}(n\geq 2)$. Now we can substitute the reduced forms of $\phi_{i,0}$’s into the system of equations (4.1) to determine the coefficients of the possible terms. Then we obtain all solutions as listed below Theorem 4.2. Therefore, Theorem 4.2 is proved as desired.