2.1

Throughout this paper, we denote by $\mathbb {C}$ the field of complex numbers and by $\mathbb {Z}$ (resp. $\mathbb {Z}_{+}$ , $\mathbb {N}$ ) the subset of integers (resp. nonnegative, positive integers). Given an indeterminate t, we let $\mathbb C[t]$ (resp. $\mathbb C[t,t^{-1}]$ , $\mathbb C(t)$ ) be the ring of polynomials (resp. Laurent polynomials, rational functions) in the variable t. For a Lie algebra $\mathfrak {a}$ , let $\mathbf {U}(\mathfrak {a})$ be the universal enveloping algebra of $\mathfrak {a}$ . Furthermore, let

$$ \begin{align*}\mathfrak{a}[t^{\pm 1}]:=\mathfrak{a}\otimes \mathbb{C}[t,t^{-1}]\ \ \ (\text{resp.} \ \mathfrak{a}[t]:=\mathfrak{a}\otimes \mathbb{C}[t])\end{align*} $$

be the loop algebra (resp. current algebra) of $\mathfrak {a}$ with Lie bracket

$$ \begin{align*}[x\otimes t^r, y\otimes t^s]=[x,y]\otimes t^{r+s},\ \ x,y\in\mathfrak{a},\ \ r,s\in \mathbb{Z}\ \ (\text{resp.}\ r,s\in\mathbb{Z}_+).\end{align*} $$

A finite-dimensional $\mathbb {Z}$ -graded $\mathfrak {a}[t]$ -module is a $\mathbb {Z}$ -graded vector space admitting a compatible graded action of $\mathfrak {a}[t]$ :

$$ \begin{align*}V=\bigoplus_{k\in \mathbb{Z}} V[k],\ \ (a\otimes t^r) V[k]\subseteq V[k+r],\ \ a\in \mathfrak{a}, \ r\in\mathbb{Z}_+.\end{align*} $$

Given a $\mathbb {Z}$ -graded space V, let $\tau ^{*}_p V$ be the graded space whose rth graded piece is $V[r-p]$ .

2.2

Let $\mathfrak {g}$ be the Lie algebra $\mathfrak {sl}_{n+1}$ with Cartan matrix $(c_{i,j})_{1\leq i,j\leq n}$ , and let $\mathfrak {h}$ be a Cartan subalgebra of $\mathfrak {g}$ . We denote by R the corresponding set of roots and $\{\alpha _1,\dots ,\alpha _n\}$ and $\{\varpi _1,\dots ,\varpi _n\}$ be a set of simple roots and fundamental weights, respectively. The set of positive roots is denoted by $R^+$ , the $\mathbb {Z}$ (resp. $\mathbb {Z}_+$ )-span of the simple roots by Q (resp. $Q^+$ ), and the $\mathbb {Z}$ (resp. $\mathbb {Z}_+$ )-span of the fundamental weights by P (resp. $P^+$ ). We define as usual a partial order on P by $\lambda \succeq \mu $ if $\lambda -\mu \in Q^+$ . Note that for two positive roots $\alpha ,\beta \in R^+$ with $\beta \succ \alpha $ , we either have $\beta -\alpha \in R^+$ or there exists $\gamma _1,\gamma _2\in R^+$ such that

(2.1) $$ \begin{align}\beta-\gamma_1\in R^+,\ \ \alpha=\beta-\gamma_1-\gamma_2.\end{align} $$

For $\gamma =\sum _{i=1}^nr_i\alpha _i\in Q^+$ , define its height by $|\gamma |=\sum _{i=1}^n r_i$ . Given a root $\alpha \in R^+$ , let $x_{\alpha }^{\pm }$ be the corresponding root vector of weight $\pm \alpha $ and $h_{\alpha }$ the corresponding coroot. We have a triangular decomposition

$$ \begin{align*}\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus \mathfrak{n}^{+},\ \ \mathfrak{n}^{\pm}=\bigoplus_{\alpha\in R^+} \mathbb{C} \cdot x_{\alpha}^{\pm}.\end{align*} $$

If $\alpha =\alpha _i+\cdots +\alpha _j$ and $1\leq i\leq j\leq n$ , we abbreviate in the rest of the paper

$$ \begin{align*}x^{\pm}_{\alpha}:=x^{\pm}_{i,j},\ x^{\pm}_{i,i}:=x^{\pm}_i,\ h_{\alpha}:=h_{i,j},\ h_{i,i}:=h_i.\end{align*} $$

For $\lambda \in P^+$ , let $V(\lambda )$ be the unique irreducible representation of $\mathfrak {g}$ of highest weight $\lambda $ and set

$$ \begin{align*}\max(\lambda)=\max\{i: &\lambda(h_i)>0\},\ \ \min (\lambda)=\min\{i: \lambda(h_i)>0\},\ \ \mathrm{supp}_1(\lambda)=\{i: \lambda(h_i)>0\},\\&\mathrm{supp}_2(\lambda)=\{(i,j): i<j,\ [i,j]\cap \mathrm{supp}_1(\lambda)=\{i,j\}\}.\end{align*} $$

Moreover, we set

$$ \begin{align*}P^+(1)=\{\lambda\in P^+: \lambda(h_i)\leq 1 \text{ for all } 1\leq i\leq n\}.\end{align*} $$

2.3

Let $\widehat {\mathfrak {g}}$ be the untwisted affine Lie algebra associated with $\mathfrak {g}$ which is realized as

$$ \begin{align*}\widehat{\mathfrak{g}}=\mathfrak{g}[t^{\pm 1}]\oplus \mathbb{C}K\oplus \mathbb{C}d,\end{align*} $$

where K is required to be central and the Lie bracket is defined as

$$ \begin{align*}[x\otimes t^{r},y\otimes t^s\kern-1.2pt]\kern1.4pt{=}\kern1.4pt[x,y]\otimes t^{r+s}\kern1.4pt{+}\kern1.4pt\mathrm{tr}(xy)K,\ \ [d,x\otimes t^r]\kern1.4pt{=}\kern1.4pt r(x\otimes t^r),\ \ \ x,y\kern1.4pt{\in}\kern1.4pt\mathfrak{g},\ \ r,s\kern1.4pt{\in}\kern1.4pt\mathbb{Z}.\end{align*} $$

The commutator subalgebra $[\widehat {\mathfrak {g}},\widehat {\mathfrak {g}}]$ modulo the center is the loop algebra $\mathfrak {g}[t^{\pm 1}]$ , and note that the element d defines a grading on the loop algebra. The $\mathbb {Z}_+$ -graded subalgebra is the current algebra $\mathfrak {g}[t]$ associated with $\mathfrak {g} $ . In the rest of the paper, we abbreviate

$$ \begin{align*}&\mathbf{U}=\mathbf{U}(\mathfrak{g}[t]),\ \ \mathbf{U}^{\pm}=\mathbf{U}(\mathfrak{n}^{\pm}[t]),\ \ \mathbf{U}^0=\mathbf{U}(\mathfrak{h}[t]),\\\mathbf{U}_{\ell}=\mathbf{U}(\mathfrak{g}[t^{\pm 1}]),\ \ &\mathbf{U}_{\ell}^{+}=\mathbf{U}(\mathfrak{n}^{+}[t^{\pm 1}]),\ \ \mathbf{U}_{\ell}^{-}=\mathbf{U}(\mathfrak{n}^{-}[t^{\pm 1}]),\ \ \mathbf{U}^0_{\ell}=\mathbf{U}(\mathfrak{h}[t^{\pm 1}]).\end{align*} $$

So as vector spaces,

$$ \begin{align*}\mathbf{U}\cong \mathbf{U}^-\otimes \mathbf{U}^{0} \otimes \mathbf{U}^+,\ \ \mathbf{U}_{\ell}\cong \mathbf{U}_{\ell}^-\otimes \mathbf{U}_{\ell}^{0} \otimes \mathbf{U}_{\ell}^+.\end{align*} $$

2.4

Recall that $\mathbb {C}(q)$ denotes the field of rational functions in an indeterminate q. We discuss in the rest of this section quantum loop algebras, their representations (of type 1), and graded limits. Set

$$ \begin{align*}[m]=\frac{q^{m}-q^{-m}}{q-q^{-1}},\ \ [m]!=[m][m-1]\cdots [1],\ \ \begin{bmatrix}m \\r \end{bmatrix}=\frac{[m]!}{[r]![m-r]!},\ \ \ r,m\in \mathbb{Z}_+,\ m\geq r.\end{align*} $$

The quantum loop algebra $\mathbf {U}_q(\mathfrak {g}[t^{\pm 1}])$ is the $\mathbb {C}(q)$ -algebra generated by elements

$$ \begin{align*}\tilde{x}_{i,r}^{{}\pm{}},\ \tilde{k}_i^{{}\pm 1},\ \tilde{h}_{i,s}\ \ (1\leq i\leq n, r\in \mathbb{Z}, s\in \mathbb{Z}\backslash\{0\})\end{align*} $$

subject to the following relations:

$$ \begin{align*}&\tilde{k}_i\tilde{k}_i^{-1} = \tilde{k}_i^{-1}\tilde{k}_i =1,\ \ [\tilde{k}_i,\tilde{k}_j]=[\tilde{k}_i,\tilde{h}_{j,r}] =[\tilde{h}_{i,r},\tilde{h}_{j,s}]=0,\ \ \tilde{k}_i\tilde{x}_{j,r}^{{}\pm{}}\tilde{k}_i^{-1} = q^{{}\pm c_{ij}}\tilde{x}_{j,r}^{{}\pm{}},\\&[\tilde{h}_{i,r} , \tilde{x}_{j,s}^{{}\pm{}}] = \pm\frac1r[rc_{ij}]\tilde{x}_{j,r+s}^{{}\pm{}},\\&\tilde{x}_{i,r+1}^{{}\pm{}}\tilde{x}_{j,s}^{{}\pm{}} -q^{{}\pm a_{ij}}\tilde{x}_{j,s}^{{}\pm{}}\tilde{x}_{i,r+1}^{{}\pm{}} =q^{{}\pm a_{ij}}\tilde{x}_{i,r}^{{}\pm{}}\tilde{x}_{j,s+1}^{{}\pm{}} -\tilde{x}_{j,s+1}^{{}\pm{}}\tilde{x}_{i,r}^{{}\pm{}},\\&[\tilde{x}_{i,r}^{{}\pm{}},\tilde{x}_{j,s}^{{}\pm{}}]=0,\ \text{if }c_{i,j}=0\\&\tilde{x}_{j,s}^{{}\pm{}} \tilde{x}_{i, r_{1}}^{{}\pm{}}\tilde{x}_{i,r_{2}}^{{}\pm{}}+\tilde{x}_{j,s}^{{}\pm{}} \tilde{x}_{i, r_{2}}^{{}\pm{}}\tilde{x}_{i,r_{1}}^{{}\pm{}}+x_{i, r_{1}}^{{}\pm{}}\tilde{x}_{i,r_{2}}^{{}\pm{}}\tilde{x}_{j,s}^{{}\pm{}}+x_{i, r_{2}}^{{}\pm{}}\tilde{x}_{i,r_{1}}^{{}\pm{}}\tilde{x}_{j,s}^{{}\pm{}}\\ &\quad=[2](\tilde{x}_{i, r_{1}}^{\pm} \tilde{x}_{j,s}^{{}\pm{}}\tilde{x}_{i, r_{2}}^{\pm}+\tilde{x}_{i, r_{2}}^{\pm} \tilde{x}_{j,s}^{{}\pm{}} \tilde{x}_{i, r_{1}}^{{}\pm{}}),\ i\neq j\\&[\tilde{x}_{i,r}^+ , \tilde{x}_{j,s}^-]=\delta_{ij} \frac{\phi_{i,r+s}^+ - \phi_{i,r+s}^-}{q - q^{-1}},\end{align*} $$

where $\phi _{i,r}^{\pm }$ is determined by equating coefficients of powers of u in

$$ \begin{align*}\Phi_i^{\pm}(u)=\sum_{r\in\mathbb{Z}}\phi_{i,\pm r}^{{}\pm{}}u^{{} r} = \tilde{k}_i^{{}\pm 1} \mathrm{exp}\left(\pm(q-q^{-1})\sum_{s=1}^{\infty}\tilde{h}_{i,\pm s} u^{{}s}\right).\end{align*} $$

Denote by $\mathbf {U}_q(\mathfrak {g}) \left (\text {resp. } \mathbf {U}_q(\mathfrak {h}[t^{\pm 1}])\right )$ the subalgebra generated by

$$ \begin{align*}\left\{\tilde{x}_{i,0}^\pm, \tilde{k}_i^{\pm 1}, 1\leq i\leq n\right\}\ \ \left(\text{resp. } \left\{\tilde{k}_i^{\pm1}, \tilde{h}_{i,s}, 1\leq i\leq n, s\in\mathbb{Z}\backslash\{0\}\right\}\right).\end{align*} $$

Then $\Lambda _{i,r}$ together with $\tilde {k}_i^{\pm 1}$ , $1\leq i\leq n$ , $r\in \mathbb {Z}$ , generate $\mathbf {U}_q(\mathfrak {h}[t^{\pm 1}])$ as an algebra where the elements $\Lambda _{i,r}$ are obtained by equating powers of u in the formal power series

$$ \begin{align*} \Lambda_i^\pm(u)=\sum_{r=0}^\infty \Lambda_{i,\pm r} u^{r}= \exp\left(-\sum_{s=1}^\infty\frac{\tilde{h}_{i,\pm s}}{[s]}u^s\right). \end{align*} $$

2.5

We consider the dominant $\ell $ -weight lattice of $\mathbf {U}_q(\mathfrak {g}[t^{\pm 1}])$ defined as the monoid $\mathcal P^+$ of n-tuples of polynomials $\boldsymbol {\pi } = (\boldsymbol {\pi }_1(u),\dots ,\boldsymbol {\pi }_n(u))$ with coefficients in $\mathbb {C}(q)[u]$ (the polynomial algebra in an indeterminate u with coefficients in the field $\mathbb {C}(q)$ ) such that $\boldsymbol {\pi }_i(0)=1$ for all $1\leq i\leq n$ . Given $a\in \mathbb {C}(q)^{\times }$ and $1\leq i\leq n$ , define the fundamental $\ell $ -weight $\boldsymbol {\varpi }_{i,a}\in \mathcal P^+$ by

$$ \begin{align*}(\boldsymbol{\varpi}_{i,a})_j(u) = (1-\delta_{i,j}au).\end{align*} $$

The $\ell $ -weight lattice $\mathcal P$ is the free abelian group generated by fundamental $\ell $ -weights. We consider the map $\boldsymbol {\Psi }:\mathcal {P}\rightarrow \mathbf {U}_q(\mathfrak {h}[t^{\pm 1}])^{*},\ \boldsymbol {\pi }\rightarrow \boldsymbol {\Psi }_{\boldsymbol {\pi }}$ (which turns out to be injective) by the following rule on the generators. For $\boldsymbol {\pi }=\boldsymbol {\pi }'\widetilde {\boldsymbol {\pi }}^{-1}$ with $\boldsymbol {\pi }',\widetilde {\boldsymbol {\pi }}\in \mathcal {P}^+$ , let $\boldsymbol {\Psi }_{\boldsymbol {\pi }}$ be the algebra homomorphism determined by

$$ \begin{align*}\boldsymbol{\Psi}_{\boldsymbol{\pi}}(\tilde{k}_i^{\pm1})=q^{\pm \mathrm{wt}(\boldsymbol{\pi})(h_i)},\ \ \boldsymbol{\Psi}_{\boldsymbol{\pi}}(\Lambda^{\pm}_{i}(u))=\frac{\boldsymbol{\pi}^{\prime\pm}_i(u)}{\widetilde{\boldsymbol{\pi}}^{\pm}_i(u)},\end{align*} $$

where the weight map is the group homomorphism $\mathrm {wt}:\mathcal P \to P$ , $\mathrm {wt}(\boldsymbol \varpi _{i,a})=\varpi _i$ and $\boldsymbol {\pi }^{+}_i(u)=\boldsymbol {\pi }^{}_i(u)$ whereas $\boldsymbol {\pi }^{-}_i(u)$ is the polynomial obtained from $\boldsymbol {\pi }_i(u)$ by replacing each $\boldsymbol {\varpi }_{i,a}$ by $\boldsymbol {\varpi }_{i,a^{-1}}$ .

2.6

A nonzero vector v of a $\mathbf {U}_q(\mathfrak {g}[t^{\pm 1}])$ -module V is called

• • an $\ell $ -weight vector of $\ell $ -weight $\boldsymbol {\pi }\in \mathcal P$ if there exists $k\in \mathbb {N}$ such that

  $$ \begin{align*}(H-\boldsymbol{\Psi}_{\boldsymbol{\pi}}(H))^kv=0 \ \ \text{ for all }\ \ H\in \mathbf{U}_q(\mathfrak{h}[t^{\pm 1}]);\end{align*} $$

• • a highest $\ell $ -weight vector if we have

  $$ \begin{align*}Hv=\boldsymbol{\Psi}_{\boldsymbol{\pi}}(H)v \ \ \text{for all}\ \ H\in \mathbf{U}_q(\mathfrak{h}[t^{\pm 1}]) \ \ \text{and}\ \ \tilde{x}_{i,r}^+v=0 \quad\text{for all}\quad 1\leq i\leq n,\ r\in\mathbb Z.\end{align*} $$

Moreover, a module V is called

• • an $\ell $ -weight module if every vector of V is a linear combination of $\ell $ -weight vectors;

• • a highest $\ell $ -weight module if it is generated by a highest $\ell $ -weight vector.

Let $\mathcal C_q$ be the category of all finite-dimensional $\ell $ -weight modules of $\mathbf {U}_q(\mathfrak {g}[t^{\pm 1}])$ . This category is abelian and stable under tensor product. Moreover, if V is an object of $\mathcal C_q$ , we have for $\mu \in P$

$$ \begin{align*}\left\{v\in V: \tilde{k}_iv=q^{\mu(h_i)}v \text{ for all }1\leq i\leq n\right\}=\bigoplus_{\boldsymbol{\pi} : \mathrm{wt}(\boldsymbol{\pi})=\mu} V_{\boldsymbol{\pi}},\end{align*} $$

where $V_{\boldsymbol {\pi }}\subseteq V$ denotes the subspace spanned by all $\ell $ -weight vectors of $\ell $ -weight $\boldsymbol {\pi }$ . In particular, V is a type 1 representation.

The irreducible objects in $\mathcal C_q$ were classified and are obtained as follows. Let $\boldsymbol {\pi }\in \mathcal {P}^+$ and $W_q(\boldsymbol {\pi })$ be the $\mathbf {U}_q(\mathfrak {g}[t^{\pm 1}])$ -module generated by an element $w_{\boldsymbol {\pi }}$ with defining relations

$$ \begin{align*}\tilde{x}_{i,r}^+w_{\boldsymbol{\pi}}=(\tilde{x}_{i,0}^-)^{\mathrm{wt}(\boldsymbol{\pi})(h_i)+1}w_{\boldsymbol{\pi}}=(H-\boldsymbol\Psi_{\boldsymbol{\pi}}(H))w_{\boldsymbol{\pi}}=0,\ 1\leq i\leq n,\ r\in\mathbb Z,\ H\in \mathbf{U}_q(\mathfrak{h}[t^{\pm 1}]).\end{align*} $$

Since $W_q(\boldsymbol {\pi })$ is a highest $\ell $ -weight module with $\dim (W_q(\boldsymbol {\pi })_{\boldsymbol {\pi }})=1$ , it has a unique irreducible quotient $V_q(\boldsymbol {\pi })$ . The next theorem can be derived from [Reference Chari and Pressley9, Reference Chari and Pressley10].

2.7

There are several ways to study the structure of these representations. One method would be to determine the classical decomposition with respect to the Hopf subalgebra $\mathbf {U}_q(\mathfrak {g})$ . If $V\in \mathcal {C}_q$ , then V can be viewed as a $\mathbf {U}_q(\mathfrak {g})$ -module and hence (the category of finite-dimensional $\mathbf {U}_q(\mathfrak {g})$ -modules is semisimple)

$$ \begin{align*}V\cong \bigoplus_{\mu\in P^+} V_q(\mu)^{c_{\mu}},\end{align*} $$

where $V_q(\mu )$ is the irreducible highest weight module of $\mathbf {U}_q(\mathfrak {g})$ of highest weight $\mu $ .

We describe now the graded limit approach to the irreducible objects in $\mathcal {C}_q$ . Let $\mathbf {A}=\mathbb {Z}[q,q^{-1}]$ and denote by $\mathbf {U}_{q,\mathbf {A}}(\mathfrak {g})$ and $\mathbf {U}_{q,\mathbf {A}}(\mathfrak {g}[t^{\pm 1}])$ the $\mathbf {A}$ -form of $\mathbf {U}_{q}(\mathfrak {g})$ and $\mathbf {U}_{q}(\mathfrak {g}[t^{\pm 1}])$ , respectively (for a precise definition, see [Reference Lusztig26]). These are free modules over the ring $\mathbf {A}$ such that

$$ \begin{align*}\mathbf{U}_{q}(\mathfrak{g})\cong \mathbf{U}_{q,\mathbf{A}}(\mathfrak{g})\otimes_{\mathbf{A}} \mathbb{C}(q),\ \ \mathbf{U}_{q}(\mathfrak{g} [t^{\pm 1}])\cong \mathbf{U}_{q,\mathbf{A}}(\mathfrak{g} [t^{\pm 1}])\otimes_{\mathbf{A}} \mathbb{C}(q).\end{align*} $$

We could try to mimic the same kind of construction for an arbitrary irreducible object in $\mathcal {P}^+$ , but the existence is not guaranteed in general. Assume that $V_q(\boldsymbol {\pi })$ admits an $\mathbf {A}$ -form, i.e., there is a representation $V_{q,\mathbf {A}}(\boldsymbol {\pi })$ of $\mathbf {U}_{q,\mathbf {A}}(\mathfrak {g}[t^{\pm 1}])$ such that

$$ \begin{align*}V_{q}(\boldsymbol{\pi})\cong V_{q,\mathbf{A}}(\boldsymbol{\pi})\otimes_{\mathbf{A}} \mathbb{C}(q).\end{align*} $$

If such an $\mathbf {A}$ -form exists, the classical limit is defined as

$$ \begin{align*}\overline{V_{q}(\boldsymbol{\pi})}:= V_{q,\mathbf{A}}(\boldsymbol{\pi})\otimes_{\mathbf{A}} \mathbb{C},\end{align*} $$

where $\mathbb {C}$ is viewed as an $\mathbf {A}$ -module by letting q act as $1$ . Since $\mathbf {U}_{q,\mathbf {A}}(\mathfrak {g}[t^{\pm 1}])\otimes _{\mathbf {A}} \mathbb {C}$ is a quotient of the universal enveloping algebra $\mathbf {U}_{\ell }$ , we obtain that $\overline {V_{q}(\boldsymbol {\pi })}$ is a module for $\mathbf {U}_{\ell }$ and hence for $\mathbf {U}$ by restriction. The graded limit $L(\boldsymbol {\pi })$ is obtained by pulling back the $\mathbf {U}$ -module $\overline {V_{q}(\boldsymbol {\pi })}$ via the automorphism

$$ \begin{align*}\mathfrak{g}[t]\rightarrow \mathfrak{g}[t],\ \ x\otimes t^r\mapsto x\otimes (t-1)^r.\end{align*} $$

So whenever $V_q(\boldsymbol {\pi })$ admits an $\mathbf {A}$ -form, we can associate a representation $L(\boldsymbol {\pi })$ of $\mathbf {U}$ with it.

2.8

Let $\mathcal {P}^+_{\mathbb {Z}}$ be the submonoid of $\mathcal {P}^+$ generated by the elements $\boldsymbol {\varpi }_{i,a}$ with $a\in q^{\mathbb {Z}}$ . Furthermore, we denote by $\mathcal {P}_{\mathbb {Z}}^+(1)$ the subset of $\mathcal {P}^+_{\mathbb {Z}}$ consisting of elements

$$ \begin{align*}\boldsymbol{\varpi}_{i_1,a_1}\cdots \boldsymbol{\varpi}_{i_k,a_k},\ \ 1\leq i_1<i_2\cdots<i_k\leq n,\ \ a_j\in q^{\mathbb{Z}}\end{align*} $$

such that

$$ \begin{align*}a_{j+1}=a_jq^{\pm(i_{j+1}-i_j+2)}\end{align*} $$

and

$$ \begin{align*}a_{j+1}=a_jq^{\pm(i_{j+1}-i_j+2)}\implies a_{j+2}=a_{j+1}q^{\mp(i_{j+2}-i_{j+1}+2)}.\end{align*} $$

Then $V_q(\boldsymbol {\pi })$ admits an $\mathbf {A}$ -form for all $\boldsymbol {\pi }\in \mathcal {P}^+_{\mathbb {Z}}(1)$ (see, for example, [Reference Chari and Pressley11]) and the corresponding graded limit $L(\boldsymbol {\pi })$ has been described in terms of generators and relations in [Reference Brito, Chari and Moura7]. Before we state their result, we introduce the following definition.

Remark The above class of representations has been studied in [Reference Biswal, Chari, Shereen and Wand4, Section 2.2] in the context of Macdonald polynomials and fusion products of level two and level one Demazure modules. In their notation, $N_{\boldsymbol {\zeta }}$ was denoted by $M(\lambda _2,2\lambda _0+\lambda _1)$ and the defining relations look more complicated. However, the conditions $\max (\lambda _0)<\min (\lambda _1)$ and $\max (\lambda _2)<\min (\lambda _1)$ ensure that the defining relations of $M(\lambda _2,2\lambda _0+\lambda _1)$ stated in [Reference Biswal, Chari, Shereen and Wand4, Section 2.2] can be simplified to the ones in the definition of $N_{\boldsymbol {\zeta }}$ . To see this, we simply have to write

$$ \begin{align*}\hspace{0,1cm} \left(x_{i,j}^{-}\otimes t^{(\lambda_0+\lambda_2)(h_{i,j})+\lceil\frac{m-r}{2}\rceil}\right)= \big[\cdots&\big[\big[\big[(x_{i,i_r-1}^{-}\otimes t^{(\lambda_0+\lambda_2)(h_{i,i_{r}-1})},(x_{i_r,i_{r+1}}^{-}\otimes t)\big]&\\&,(x_{i_{r+1}+1,i_{r+2}-1}^{-}\otimes 1)\big],(x_{i_{r+2},i_{r+3}}^{-}\otimes t)\big],\dots,(x_{i_{m-1}+1,j}^{-}\otimes 1)\big],\end{align*} $$

where $\lambda _1=\varpi _{i_1}+\cdots +\varpi _{i_k}, 1\leq i_1<\cdots <i_k\leq n$ and m and r are determined by $i_{r-1}<i\leq i_r$ and $i_{m-1}\leq j< i_m$ ; we understand $i_0=0$ and $i_{k+1}=n+1$ .

The first part of the next theorem has been proved in [Reference Brito, Chari and Moura7, Theorem 1], and the second part follows from [Reference Biswal, Chari, Shereen and Wand4, Proposition 2.5(i)] and Remark 2.8.

2.9

We describe now the importance of these representations and the relation to the HL category.

Definition We call $V_q(\boldsymbol {\pi })$ a prime irreducible representation of $\mathbf {U}_{q}(\mathfrak {g}[t^{\pm 1}])$ if

$$ \begin{align*}V_q(\boldsymbol{\pi})\cong V_q(\boldsymbol{\pi}^1)\otimes \cdots \otimes V_q(\boldsymbol{\pi}^s)\end{align*} $$

implies that $(s-1)$ factors are trivial representations.

It is clear that $V_q(\boldsymbol {\pi })$ is either prime or can be written as a tensor product of non-trivial prime representations; however, the uniqueness of such a decomposition is not known in general. The motivation of this paper is to determine the structure of the prime objects in the HL category by describing the graded characters of their graded limits. First, we recall the definition of the HL category.

Let $\kappa : \{1,\dots ,n\}\rightarrow \mathbb {Z}$ be a (height) function satisfying $|\kappa (i+1)-\kappa (i)|= 1$ for $1\leq i\leq n$ , and let $Q_{\kappa }$ be the corresponding quiver whose vertices are indexed by $\{1,\dots ,n\}$ and there is an edge $i\rightarrow i+1$ if $\kappa (i)<\kappa (i+1)$ and $i\leftarrow i+1$ otherwise. The HL category $\mathcal {C}_{q,\kappa }$ is the full subcategory of $\mathcal {C}_q$ whose objects have all its Jordan–Hölder components of the form

$$ \begin{align*}V_q(\boldsymbol{\pi}),\ \ \boldsymbol{\pi}\in \mathcal{P}_{\mathbb{Z}}^+(\kappa,1),\end{align*} $$

where $\mathcal {P}_{\mathbb {Z}}^+(\kappa ,1)$ is the submonoid of $\mathcal {P}^+$ generated by $\boldsymbol {\varpi }_{i,a},\ a\in \{q^{\kappa (i)},q^{\kappa (i)+2}\}, 1\leq i\leq n.$ The following results have been proved in [Reference Brito and Chari6, Reference Hernandez and Leclerc18, Reference Hernandez and Leclerc19].

Theorem 3 The category $\mathcal {C}_{q,\kappa }$ is closed under tensor products. Let $V_q(\boldsymbol {\pi })$ be a prime irreducible object in $\mathcal {C}_{q,\kappa }$ . Then $\boldsymbol {\pi }\in \{\boldsymbol {\varpi }_{i,q^{\kappa (i)}}\boldsymbol {\varpi }_{i,q^{\kappa (i)+2}},\boldsymbol {\varpi }_{i,q^{\kappa (i)+2}},\boldsymbol {\varpi }_{i,q^{\kappa (i)}}\}$ for some $i\in \{1,\dots ,n\}$ or there exists an interval $J\subseteq [1,n]$ such that

$$ \begin{align*}\boldsymbol{\pi}=\boldsymbol{\pi}_{\kappa,J}:=\prod_{i\in J_{\mathrm{sink}}} \boldsymbol{\varpi}_{i,q^{\kappa(i)}}\prod_{i\in J_{\mathrm{source}}} \boldsymbol{\varpi}_{i,q^{\kappa(i)+2}},\end{align*} $$

where $J_{\mathrm {sink}}$ (resp. $J_{\mathrm {source}}$ ) are the sinks (resp. sources) of $Q_{\kappa }$ contained in J. Conversely, all these representations are prime objects in $\mathcal {C}_{q,\kappa }$ .

In fact, we have

$$ \begin{align*}\mathcal{P}_{\mathbb{Z}}^+(1)=\{\boldsymbol{\pi}_{\kappa,J}: \kappa \text{ height function}, J\subseteq [1,n]\text{ interval}\}.\end{align*} $$

The only nontrivial direction is derived as follows. Let $\boldsymbol {\pi }=\boldsymbol {\varpi }_{i_1,a_1}\cdots \boldsymbol {\varpi }_{i_k,a_k}\in \mathcal {P}_{\mathbb {Z}}^+(1)$ such that $a_2=a_1q^{(i_2-i_1+2)}$ , then we choose $J=[i_1,i_k]$ and $\kappa $ to be the height function given by $a_{1}=q^{\kappa (i_1)}$ and $i_1,i_3,\dots $ are the sinks and $i_2,i_4,\dots $ are the sources. If $a_2=a_1q^{-(i_2-i_1+2)}$ , we simply change the role of sinks and sources. Then we have $\boldsymbol {\pi }=\boldsymbol {\pi }_{\kappa ,J}$ .

3.1

We shall view $V(\lambda )$ for $\lambda \in P^+$ as a $\mathbb {Z}$ -graded space by declaring that each element has degree zero. Let M be a finite-dimensional and graded $\mathbf {U}$ -module. Then there exists a decreasing filtration

(3.1) $$ \begin{align}M=L_0\supseteq L_1\supseteq\cdots \supseteq L_k\supseteq L_{k+1}=0\end{align} $$

of graded $\mathfrak {g}[t]$ -submodules such that

$$ \begin{align*}L_i/L_{i+1}\cong \tau^*_{p_i} V(\nu_i),\ \ p_i\in\mathbb{Z},\ \nu_i\in P^+,\ 0\leq i\leq k.\end{align*} $$

We denote by $[M: \tau _p^{*}V(\lambda )]$ the multiplicity of $\tau _p^{*}V(\lambda )$ in the filtration (3.1) of M; note that this number is independent of the choice of the filtration. The graded character of M is defined as follows:

$$ \begin{align*}\mathrm{ch}_{\mathrm{gr}}(M):=\sum_{\mu\in P^+}[M: V(\mu)]_q \ \mathrm{ch}_{\mathfrak{h}} V(\mu),\ \ [M: V(\mu)]_q:= \sum_{p=0}^{\infty}\ [M: \tau_p^{*}V(\mu)] \cdot q^p,\end{align*} $$

where $\mathrm {ch}_{\mathfrak {h}} V(\mu )$ denotes the usual $\mathfrak {h}$ -character of $V(\mu )$ which can be computed, for example, with the Weyl character formula (see [Reference Humphreys20, Section 24.3]) or the various existing combinatorial models for $V(\mu )$ (Young tableux, path model, Gelfand–Tsetlin pattern, etc.).

3.2

In order to state our results, we need some more notation. Given $\gamma =\sum _{i=1}^nr_i\alpha _i\in Q^+$ we write $\boldsymbol {\mu }\vdash \gamma $ if $\boldsymbol {\mu }$ is a multipartition

(3.2) $$ \begin{align}\boldsymbol{\mu}=(\mu_1,\dots,\mu_n),\ \ \mu_i\vdash r_i,\ \ \mu_i=(\mu_i^{1}\geq \mu_i^{2}\geq\dots\geq \mu_i ^{r_i}\geq 0).\end{align} $$

The tuple of empty partitions is denoted by $\boldsymbol {\emptyset }$ . We fix a dominant integral weight $\lambda \in P^+$ and a multipartition $\boldsymbol {\mu }$ as in (3.2). We naturally identify a partition with a Young diagram and define

• • $\mu _i(s):= \text {number of boxes in the first }s\text { columns of }\mu _i$ ,

• • $m_{i,r}:= \text { number of rows of length }r\text { in }\mu _i$ ,

• • $d(\mu _i):= \text {number of rows of }\mu _i$ .

We set

$$ \begin{align*}P^{\boldsymbol{\mu},\lambda}_{s,i}=\lambda(h_i)-2\mu_i(s)+\mu_{i-1}(s)+\mu_{i+1}(s),\ \ \ \ 1\leq i\leq n,\ 1\leq s\leq r_i\end{align*} $$

and

$$ \begin{align*}K^{\lambda}_{\boldsymbol{\mu}}=\sum_{i=1}^n\Big(\sum_{j=1}^{r_i}\big(2j\mu_i^{j}-\mu_{i+1}(\mu_i^{j})\big)-\lambda(h_i) \cdot d(\mu_i)\Big),\end{align*} $$

where we understand $\mu _0(\cdot )=\mu _{n+1}(\cdot )=0$ .

The following lemma gives certain relations which are needed later.

Lemma Let $\boldsymbol {\mu }$ be a multipartition and $i\leq j$ such that $m_{i,1},\dots ,m_{j,1}>0$ . We denote by $\boldsymbol {\tilde \mu }$ the multipartition obtained from $\boldsymbol {\mu }$ by removing a box in the last row of each partition $\mu _i,\dots ,\mu _j$ . For all $\lambda \in P^+$ with $\lambda -\alpha _{i,j}\in P^+$ and $\lambda (h_{i,j})=2$ , we have

$$ \begin{align*}K^{\lambda}_{\boldsymbol{\mu}}=K^{\lambda-\alpha_{i,j}}_{\boldsymbol{\tilde\mu}}+(j-i),\ \ P^{{\boldsymbol\mu},\lambda}_{r,k}= P^{{\boldsymbol{\tilde\mu}},\lambda-\alpha_{i,j}}_{r,k}\ \forall r,k.\end{align*} $$

Proof Let $\boldsymbol {\tilde \mu }=(\tilde {\mu }_1,\dots ,\tilde {\mu }_n)$ . The lemma follows from the following equations:

$$ \begin{align*}\lambda-\alpha_{i,j}=\lambda-\varpi_i-\varpi_j+\varpi_{i-1}+\varpi_{j+1},\ \ d(\mu_a)=\begin{cases}d(\tilde{\mu}_a)+1,& a\in\{i,\dots,j\}, \\ \\ d(\tilde\mu_a),& \text{otherwise}, \\\end{cases}\end{align*} $$

$$ \begin{align*}\mu_a^b=\begin{cases}\tilde\mu^b_a,& a\notin\{i,\dots,j\}\text{ or }b\neq d(\mu_a), \\ \\ \tilde\mu^b_a+1,& \text{otherwise}, \end{cases}\ \ \ \ \ \ \mu_a(r)=\begin{cases}\tilde\mu_a(r),& a\notin\{i,\dots,j\}, \\ \\ \tilde\mu_a(r)+1,& \text{otherwise}, \end{cases}\end{align*} $$

$$ \begin{align*}\sum_{b=1}^{r_a}\mu_{a+1}(\mu_a^b)=\begin{cases}\displaystyle\sum_{b=1}^{r_a}\tilde{\mu}_{a+1}(\tilde{\mu}_a^b),& a,a+1\notin\{i,\dots,j\} \\\displaystyle\sum_{b=1}^{r_a}\tilde{\mu}_{a+1}(\tilde{\mu}_a^b)+d(\tilde{\mu}_{a+1})+d(\tilde{\mu}_a)+1,& a,a+1\in\{i,\dots,j\}, \\ \displaystyle\sum_{b=1}^{r_a}\tilde{\mu}_{a+1}(\tilde{\mu}_a^b)+d(\tilde{\mu}_{a+1}),& a=j, \\ \displaystyle\sum_{b=1}^{r_a}\tilde{\mu}_{a+1}(\tilde{\mu}_a^b)+d(\tilde{\mu}_{a}),& a=i-1. \\\end{cases}\end{align*} $$

The next definition introduces the main combinatorial objects of this paper.

Definition Let $\gamma \in Q^+$ be a nonzero element, $\boldsymbol {\mu }\vdash \gamma $ be a multipartition, $p\in \mathbb {Z}_+$ , and $\boldsymbol {\zeta }=(\lambda ,\lambda _0,\lambda _1,\lambda _2)$ be an admissible quadrupel. We define $L^p_{\boldsymbol {\mu },\boldsymbol {\zeta }}$ to be the number of lattice points in the polytope $\mathcal {S}^p_{\boldsymbol {\mu },\boldsymbol {\zeta }}$ consisting of all points

$$ \begin{align*}\left(C_{d,r,i}\right)_{1\leq i\leq n,\hspace{0.2em} 1\leq r\leq r_i,\hspace{0.2em} 1\leq d\leq m_{i,r}}\in \mathbb{R}^{d(\mu_1)}\times \cdots \times \mathbb{R}^{d(\mu_n)}\end{align*} $$

satisfying the following inequalities:

(3.3) $$ \begin{align}C_{d,r,i}\geq 0 \ \ \forall \ d,r,i,\ \ \ \ \sum_{d=1}^{m_{i,r}}C_{d,r,i}\leq P^{\boldsymbol{\mu},\lambda}_{r,i} \ \ \forall\ r,i, \ \ \ \ \sum_{d,r,i}d\cdot C_{d,r,i}=|\gamma|-p-K^{\lambda}_{\boldsymbol{\mu}},\end{align} $$

(3.4) $$ \begin{align}\sum_{i=a}^{b}C_{m_{i,1},1,i}\geq 1,\ \ \forall (a,b)\in\mathrm{supp}_2(\lambda_1) ,\ \ \ \ C_{m_{i,1},1,i}\geq 1,\ \ \forall \ i\in \mathrm{supp}_1(\lambda_0).\end{align} $$

We extend our definition for $\gamma =0$ and set $L^p_{\boldsymbol {\emptyset },\boldsymbol {\zeta }}=\delta _{p,0}$ .

Remark We keep the notation of Definition 3.2. If $|\mathrm {supp}_1(\lambda _1)|\leq 1$ , then the first family of inequalities in (3.4) vanish and $\sum _{p\in \mathbb {Z}_+}L^p_{\boldsymbol {\mu },\boldsymbol {\zeta }}$ counts the number of lattice points of a polytope with inequalities

$$ \begin{align*}C_{d,r,i}\geq 0 \ \ \forall \ d,r,i,\ \ \ \ \sum_{d=1}^{m_{i,r}}C_{d,r,i}\leq Q^{\boldsymbol{\mu},\lambda}_{r,i} \ \ \forall\ r,i,\end{align*} $$

where $Q^{\boldsymbol {\mu },\lambda }_{r,i}= P^{\boldsymbol {\mu },\lambda }_{r,i}-1$ if $(r,i)\in \{1\}\times \mathrm {supp}_1(\lambda _0)$ and $Q^{\boldsymbol {\mu },\lambda }_{r,i}= P^{\boldsymbol {\mu },\lambda }_{r,i}$ otherwise. Hence,

(3.5) $$ \begin{align}\sum_{\boldsymbol{\mu}\vdash \gamma}\sum_{p\in\mathbb{Z}_+}L^p_{\boldsymbol{\mu},\boldsymbol{\zeta}}=\sum_{\boldsymbol{\mu}\vdash \gamma}\prod_{i=1}^n\prod_{r=1}^{r_i}\begin{bmatrix}{Q^{\boldsymbol{\mu},\lambda}_{r,i} +m_{i,r}}\\{m_{i,r}}\end{bmatrix}=[N_{\boldsymbol{\zeta}}: V(\lambda-\gamma)]_{q=1},\end{align} $$

where the last equation follows from the following two observations:

• • If $|\mathrm {supp}_1(\lambda _1)|\leq 1$ , then $N_{\boldsymbol {\zeta }}$ is simply a tensor product of finite-dimensional irreducible $\mathfrak {g}$ -modules

  (3.6) $$ \begin{align}N_{\boldsymbol{\zeta}}\cong_{\mathfrak{g}} \bigotimes_{i=1}^n\left( V(\varpi_i)^{\otimes \lambda_2(h_i)}\otimes V(2\varpi_i)^{\otimes \lambda_0(h_i)}\right)\otimes V(\lambda_1).\end{align} $$
  This can be derived from the generators and relations description [Reference Naoi28, Theorem B] of fusion products of KR modules and the fact that KR modules are classically irreducible in type A.

• • The numerical multiplicities in tensor products of finite-dimensional irreducible $\mathfrak {g}$ -modules corresponding to multiples of fundamental weights are given by sums of products of binomial coefficients (see, for example, [Reference Di Francesco and Kedem13]). In the case of (3.6), they are exactly given by the sums of products of binomials as in (3.5).

We will prove the following result in the rest of the paper.

Theorem 4 Let $p\in \mathbb {Z}_+$ , $\nu \in P^+$ , and $\boldsymbol {\pi }=\boldsymbol {\varpi }_{i_1,a_1}\cdots \boldsymbol {\varpi }_{i_k,a_k}\in \mathcal {P}_{\mathbb {Z}}^+(1)$ . Then

$$ \begin{align*}\left[L(\boldsymbol{\pi}): \tau_p^{*}V(\nu)\right]=\begin{cases}\ 0,& \text{if }\nu\notin \mathrm{wt}(\boldsymbol{\pi})-Q^+,\\ \\ \ \displaystyle \sum_{\boldsymbol{\mu}\vdash \gamma} L^p_{\boldsymbol{\mu},\left(\mathrm{wt}(\boldsymbol{\pi}),0,\mathrm{wt}(\boldsymbol{\pi}),0\right)},& \text{if }\nu=\mathrm{wt}(\boldsymbol{\pi})-\gamma.\end{cases}\end{align*} $$

The strategy of the proof is as follows; let $\boldsymbol {\zeta }=(\lambda ,\lambda _0,\lambda _1,\lambda _2)$ be an admissible quadrupel. We will use the dual functional realization of loop algebras to prove in Section 6.4 that

$$ \begin{align*}\left[N_{\boldsymbol{\zeta}}: \tau_p^{*}V(\lambda-\gamma)\right]\leq \sum_{\boldsymbol{\mu}\vdash \gamma}L_{\boldsymbol{\mu},\boldsymbol{\zeta}}^{p}.\end{align*} $$

Subsequently, we will show in Theorem 6 the reverse estimate under the restriction $\lambda _2\in P^+(1)$ . The key step for the reverse estimate is the following recursion satisfied by the numbers $L_{\boldsymbol {\mu },\boldsymbol {\zeta }}^{p}$ ; the main theorem is then obtained as a corollary by choosing the quadruple $(\mathrm {wt}(\boldsymbol {\pi }),0,\mathrm {wt}(\boldsymbol {\pi }),0)$ .

Proposition Let $\boldsymbol {\mu }$ be a multipartition, $p\in \mathbb {Z}_+$ , and $\boldsymbol {\zeta }=(\lambda ,\lambda _0,\lambda _1,\lambda _2)$ be an admissible quadrupel. Suppose that

$$ \begin{align*}\lambda_1=\varpi_{i_1}+\cdots+\varpi_{i_k},\ \ 1\leq i_1<\cdots<i_k\leq n,\ k\geq 2.\end{align*} $$

Then

$$ \begin{align*}L^p_{\boldsymbol{\mu},\boldsymbol{\zeta}}=\begin{cases} L^p_{\boldsymbol{\mu},\boldsymbol{\zeta}_1},&\text{ if }\exists \ i\in\{i_1,\dots,i_2\}\text{ with }m_{i,1}=0,\\ \\ L^p_{\boldsymbol{\mu},\boldsymbol{\zeta}_1}-(1-\delta_{p,0})L^{p-1}_{\boldsymbol{\tilde \mu},\boldsymbol{\zeta}_2},&\text{ otherwise}, \end{cases}\end{align*} $$

where $\boldsymbol {\tilde \mu }$ is the multipartition obtained from $\boldsymbol {\mu }$ by removing a box in the last row of each partition $\mu _{i_1},\dots ,\mu _{i_2}$ and $\boldsymbol {\zeta }_1,\boldsymbol {\zeta }_2$ are defined as in (2.3) and (2.4).

Proof If $m_{i,1}=0$ for some $i\in \{i_1,\dots ,i_2\}$ , the defining inequalities of $\mathcal {S}^p_{\boldsymbol {\mu },\boldsymbol {\zeta }}$ and $\mathcal {S}^p_{\boldsymbol {\mu },\boldsymbol {\zeta }_1}$ coincide and the statement is immediate. So suppose that $m_{i,1}\geq 1$ for all $i\in \{i_1,\dots ,i_2\}$ and set $\epsilon _i=1$ if $i\in \{i_1,\dots ,i_2\}$ and $\epsilon _i=0$ otherwise. In particular, $\boldsymbol {\mu }$ is not the tuple of empty partitions and we have $L^0_{\boldsymbol {\mu },\boldsymbol {\zeta }}=0=L^0_{\boldsymbol {\mu },\boldsymbol {\zeta }_1}$ . Hence, we can assume $p\in \mathbb {N}$ in the rest of the proof. The definition of $\boldsymbol {\tilde \mu }$ implies $d(\tilde {\mu }_i)=d(\mu _i)-\epsilon _i$ and we consider the projection and inclusion maps

$$ \begin{align*}\rho:\mathbb{R}^{d(\mu_1)}&\times \cdots \times \mathbb{R}^{d(\mu_n)}\rightarrow \mathbb{R}^{d(\tilde\mu_1)}\times \cdots \times \mathbb{R}^{d(\tilde\mu_n)},\\\iota:\mathbb{R}^{d(\tilde\mu_1)}&\times \cdots \times \mathbb{R}^{d(\tilde\mu_n)}\rightarrow \mathbb{R}^{d(\mu_1)}\times \cdots \times \mathbb{R}^{d(\mu_n)}.\end{align*} $$

To be more precise, the projection $\rho $ applied to an element $\left (C_{d,r,i}\right )\in \mathbb {R}^{d(\mu _1)}\times \cdots \times \mathbb {R}^{d(\mu _n)}$ drops the entries $C_{m_{i_1,1},1,i_1},\dots ,C_{m_{i_2,1},1,i_2}$ corresponding to the last rows of the partitions $\mu _{i_1},\dots ,\mu _{i_2}$ and the inclusion $\iota $ fills the aforementioned places with zeros. By comparing once more the defining inequalities of the polytopes, we observe that $\mathcal {S}^p_{\boldsymbol {\mu },\boldsymbol {\zeta }_1}$ is obtained from $\mathcal {S}^p_{\boldsymbol {\mu },\boldsymbol {\zeta }}$ by dropping the inequality $C_{m_{i_1,1},1,i_1}+\cdots +C_{m_{i_2,1},1,i_2}\geq 1.$ Hence, we have a disjoint union

$$ \begin{align*}\mathcal{S}^p_{\boldsymbol{\mu},\boldsymbol{\zeta}}\ \dot\cup \ \mathcal{S}=\mathcal{S}^p_{\boldsymbol{\mu},\boldsymbol{\zeta}_1},\end{align*} $$

where $\mathcal {S}$ consists of all points $\left (C_{d,r,i}\right )_{1\leq i\leq n,\hspace {0.2em} 1\leq r\leq r_i,\hspace {0.2em} 1\leq d\leq m_{i,r}}$ in $\mathcal {S}^p_{\boldsymbol {\mu },\boldsymbol {\zeta }_1}$ satisfying

$$ \begin{align*} C_{m_{i_1,1},1,i_1}=\cdots=C_{m_{i_2,1},1,i_2}=0.\end{align*} $$

We finish the proof by showing that $\rho (\mathcal {S})\subseteq \mathcal {S}^{p-1}_{\boldsymbol {\tilde \mu },\boldsymbol {\zeta }_2}$ and $\iota (\mathcal {S}^{p-1}_{\boldsymbol {\tilde \mu },\boldsymbol {\zeta }_2})\subseteq \mathcal {S}$ . Since both inclusions are proven similarly, we focus on the first one. From Lemma 3.2 (note that $\lambda (h_{i_1,i_2})=2$ ), we know that each point in $\rho (\mathcal {S})$ satisfies the inequalities (3.3) of $\mathcal {S}^{p-1}_{\boldsymbol {\tilde \mu },\boldsymbol {\zeta }_2}$ . The inequalities (3.4) of $\mathcal {S}^{p-1}_{\boldsymbol {\tilde \mu },\boldsymbol {\zeta }_2}$ read as follows:

$$ \begin{align*}\sum_{i=i_j+\delta_{j,2}}^{i_{j+1}}C_{m_{i,1},1,i}\geq 1,\ \ j\in J,\ \ \ \ C_{m_{i,1},1,i}\geq 1,\ \ \forall \ i\in \mathrm{supp}_1(\lambda_0),\end{align*} $$

where $J=\{2,\dots ,k-1\}$ if $i_{2}+1<i_3$ or $k=2$ and $J=\{2,4,\dots ,k-1\}$ otherwise. Since $\mathcal {S}\subseteq \mathcal {S}^p_{\boldsymbol {\mu },\boldsymbol {\zeta }_1}$ , they are obviously satisfied by all points in $\rho (\mathcal {S})$ and thus $\rho (\mathcal {S})\subseteq \mathcal {S}^{p-1}_{\boldsymbol {\tilde \mu },\boldsymbol {\zeta }_2}$ .

3.3

We end this section by discussing a few examples.

4.1

We introduce the family of truncated $\mathfrak {g}[t]$ -modules indexed by pairs $(\boldsymbol {\xi },\lambda )$ where $\boldsymbol {\xi }=(\xi _{\alpha })_{\alpha \in R^+}$ is a tuple of nonnegative integers indexed by the set of positive roots of $\mathfrak {g}$ and $\lambda $ is a dominant integral weight (abbreviate as usual $\xi _{\alpha _{i,j}}=\xi _{i,j}$ etc.). We define $M_{\boldsymbol {\xi },\lambda }:=\mathbf {U}/\mathcal {I}_{\boldsymbol {\xi },\lambda }$ , where $\mathcal {I}_{\boldsymbol {\xi },\lambda }\subseteq \mathbf {U}$ is the left ideal generated by the elements

$$ \begin{align*}\mathfrak{n}^+[t],\ h_i\otimes t^r,\ h_i-\lambda(h_i)\cdot 1\ (r\in\mathbb{N},\ 1\leq i\leq n),\end{align*} $$

$$ \begin{align*}(x_i^{-}\otimes 1)^{\lambda(h_i)+1}\ (1\leq i\leq n),\ (x^{-}_{\alpha}\otimes t^{\xi_{\alpha}}),\ \alpha\in R^+.\end{align*} $$

In other words, $M_{\boldsymbol {\xi },\lambda }$ is the cyclic quotient of the local Weyl module $W_{\mathrm {loc}}(\lambda )$ by the submodule generated by

$$ \begin{align*}\left\{(x^{-}_{\alpha}\otimes t^{\xi_{\alpha}})w_{\lambda} : \alpha \in R^+\right\},\end{align*} $$

where $w_{\lambda }$ is the highest weight generator of $W_{\mathrm {loc}}(\lambda )$ . We call $\boldsymbol {\xi }$ normalized if $\xi _{\alpha }=\min \{\xi _{\beta }: \beta \succeq \alpha \}$ for all $\alpha \in R^+$ . Given an arbitrary $\boldsymbol {\xi }$ , we obviously have $M_{\boldsymbol {\xi },\lambda }\cong M_{\boldsymbol {\xi }',\lambda }$ , where $\boldsymbol {\xi }'$ is defined by $\xi _{\alpha }':=\min \{\xi _{\beta }: \beta \succeq \alpha \}$ . Hence, we can replace $\boldsymbol {\xi }$ by the normalized tuple $\boldsymbol {\xi }'$ , and suppose in the rest of the paper without loss of generality that $\boldsymbol {\xi }$ is normalized unless otherwise stated.

4.2

The aim of this section is to describe the $\mathbf {U}^{-}$ structure of $M_{\boldsymbol {\xi },\lambda }$ . We set $x^{(r)}:=\frac {1}{r!} x^r$ for an element $x\in \mathfrak {g}[t]$ , $r\in \mathbb {Z}_+$ and

$$ \begin{align*}\mathbf{x}_i^{-}(r,s):=\sum_{} (x_i^{-}\otimes 1)^{(b_0)}\cdots (x_i^{-}\otimes t^s)^{(b_s)},\end{align*} $$

where the sum runs over all tuples $(b_0,\dots ,b_s)$ of nonnegative integers satisfying $b_0+\cdots +b_s=r$ and $b_1+2b_2+\cdots + sb:s=s$ . The following result is a slight modification of [Reference Biswal and Kus5, Lemma 20].

Lemma Let V be a $\mathbf {U}$ -representation and $v\in V$ a weight vector such that

$$ \begin{align*}(h\otimes t^{r+1})v=(x_i^+\otimes t^r)v=0,\ \text{for all } h\in \mathfrak{h},\ r\in \mathbb{Z}_+.\end{align*} $$

Then we have for all $r,s\in \mathbb {Z}_+$ with $r+s\geq 1+\lambda (h_i)$ ,

(4.1) $$ \begin{align}\mathbf{U}\cdot \mathbf{x}_i^{-}(r,s).v \subseteq \sum_{u+w\geq 1+\lambda(h_i)}\mathbf{U}^{-}\cdot \mathbf{x}_i^{-}(u,w)v.\end{align} $$

Proof From [Reference Biswal and Kus5, Proof of Lemma 20], we obtain up to a nonzero constant

$$ \begin{align*}(h\otimes t^{\ell})\cdot \mathbf{x}_i^{-}(r,s)v=r\cdot \mathbf{x}_i^{-}(r,s+\ell)v-\sum_{j=0}^{\ell-1}(x_i^{-}\otimes t^j)\cdot \mathbf{x}_i^{-}(r-1,s+\ell-j)v.\end{align*} $$

Hence, the above element lies in the $\mathbf {U}^-$ -span of elements of the form $\mathbf {x}_i^{-}(u,w)v$ with $u+w\geq 1+\lambda (h_i)$ . The fact that an arbitrary product of elements in $(\mathfrak {h}\otimes t\mathbb {C}[t])$ applied to $\mathbf {x}_i^{-}(r,s)v$ lies in the right-hand side of (4.1) follows from $[\mathfrak {h},\mathfrak {n}^{-}]\subseteq \mathfrak {n}^-$ and induction on the length. Now we consider the element $(x_i^+\otimes t^{a})\cdot \mathbf {x}_i^{-}(r,s)v$ . If $a=1$ , then clearly (up to a constant)

$$ \begin{align*}(x_i^+\otimes t)\cdot \mathbf{x}_i^{-}(r,s)v=\mathbf{x}_i^{-}(r-1,s+1)v.\end{align*} $$

Otherwise, we choose $h\in \mathfrak {h}$ with $\alpha _i(h)\neq 0$ and get

$$ \begin{align*}(x_i^+\otimes t^{a})\cdot \mathbf{x}_i^{-}(r,s)v=\big[h\otimes t,x_i^+\otimes t^{a-1}\big]\cdot \mathbf{x}_i^{-}(r,s)v\end{align*} $$

and the claim in this case (length one) follows by induction on a. Again, the general case follows by induction on the length and the Poincaré–Birkhoff–Witt (PBW) theorem.

By construction, we have an isomorphism of $\mathbf {U}^{-}$ -modules:

$$ \begin{align*}\mathbf{U}^{-}/\mathcal{J}_{\boldsymbol{\xi},\lambda} \xrightarrow{\sim} M_{\boldsymbol{\xi},\lambda},\ \ \mathcal{J}_{\boldsymbol{\xi},\lambda}:=(\mathcal{I}_{\boldsymbol{\xi},\lambda}\cap \mathbf{U}^{-}).\end{align*} $$

Proposition We have that $\mathcal {J}_{\boldsymbol {\xi },\lambda }$ is the left ideal in $\mathbf {U}^{-}$ generated by the elements

(4.2) $$ \begin{align} &\mathbf{x}_{i}^{-}(r,s),\ 1\leq i\leq n,\ r\in \mathbb{N},\ s\in\mathbb{Z}_+:\ r+s\geq 1+\lambda(h_i), \end{align} $$

(4.3) $$ \begin{align} &(x^{-}_{\alpha}\otimes t^{k}),\ \alpha\in R^+,\ k\geq \min\{\xi_{\beta}: \beta\succeq \alpha\}.\end{align} $$

Proof We first show that each element in (4.2) and (4.3) is contained in $\mathcal {J}_{\boldsymbol {\xi },\lambda }$ . Let $\beta \in R^+$ such that $\beta \succeq \alpha $ and $\xi _{\beta }$ is minimal with this property. Either $\alpha =\beta $ or $\beta -\alpha \in R^+$ or we can find $\gamma _1,\gamma _2\in R^+$ such that (2.1) holds. We assume that the latter holds, since all cases are treated similarly. We get

$$ \begin{align*}(x_{\alpha}^{-}\otimes t^{\xi_{\beta}+p})=(x_{\gamma_2}^{+}\otimes 1)(x_{\gamma_1}^{+}\otimes t^p)(x_{\beta}^{-}\otimes t^{\xi_{\beta}})\quad\mod \mathbf{U} \cdot \mathfrak{n}^+[t],\ \ \forall p\geq 0.\end{align*} $$

Since $\mathfrak {n}^+[t]\subseteq \mathcal {I}_{\boldsymbol {\xi },\lambda }$ and $(x_{\beta }^{-}\otimes t^{\xi _{\beta }})\in \mathcal {I}_{\boldsymbol {\xi },\lambda },$ we also have $(x_{\alpha }^{-}\otimes t^{\xi _{\beta }+p})\in \mathcal {I}_{\boldsymbol {\xi },\lambda }\cap \mathbf {U}^-$ and all elements in (4.3) are contained in $\mathcal {J}_{\boldsymbol {\xi },\lambda }$ . The containment of (4.2) follows from the Garland identities (see, for example, [Reference Chari and Pressley11, Lemma 1.3] for the current formulation)

$$ \begin{align*}(x_i^+\otimes t)^{(s)}(x_i^-\otimes 1)^{(s+r)}-(-1)^s\mathbf{x}_{i}^{-}(r,s)\in \mathbf{U}\cdot \mathfrak{n}^+[t]\oplus \mathbf{U}\cdot (\mathfrak{h}\otimes t\mathbb{C}[t])\subseteq \mathcal{I}_{\boldsymbol{\xi},\lambda}\end{align*} $$

and the fact that $(x_i^-\otimes 1)^{(s+r)}\in \mathcal {I}_{\boldsymbol {\xi },\lambda }$ for all $r+s\geq 1+\lambda (h_i)$ . In order to finish the proof, we do the following procedure. We write each element in $\mathcal {I}_{\boldsymbol {\xi },\lambda }$ in PBW order:

$$ \begin{align*}\mathbf{U}\cong \mathbf{U}\cdot\mathfrak{n}^+[t]\ \oplus \ \mathbf{U}^{-}\cdot \mathbf{U}^0\cdot (\mathfrak{h}\otimes t\mathbb{C}[t])\ \oplus \ \mathbf{U}^{-}\cdot\mathbf{U}(\mathfrak{h}).\end{align*} $$

After that, we take the projection of that element onto $\mathbf {U}^{-}\cdot \mathbf {U}(\mathfrak {h})$ and substitute $h=\lambda (h)$ for all $h\in \mathfrak {h}$ . This gives an element in $\mathcal {J}_{\boldsymbol {\xi },\lambda }$ and it is clear that each element in $\mathcal {J}_{\boldsymbol {\xi },\lambda }$ appears in this way. So the strategy of the proof is to show that by doing this procedure we only get elements in the left ideal in $\mathbf {U}^{-}$ generated by the elements (4.2) and (4.3). Note that it is enough to project all elements in

(4.4) $$ \begin{align}\sum_{i=1}^n \mathbf{U}\cdot (x_i^-\otimes 1)^{\lambda(h_i)+1}+\sum_{\alpha\in R^+}\mathbf{U}\cdot (x^-_{\alpha}\otimes t^{\xi_{\alpha}}).\end{align} $$

Using Lemma 4.2, we see that the projection of each element in (4.4) is contained in

$$ \begin{align*}\sum_{i=1}^n\sum_{\substack{r,s:\\ r+s>\lambda(h_i)}} \mathbf{U}^{-}\cdot\mathbf{x}^-_i(r,s)+\sum_{\alpha\in R^+}\sum_{k\geq \min\{\xi_{\beta}: \beta\succeq \alpha\}}\mathbf{U}^{-}\cdot (x^-_{\alpha}\otimes t^{k})\mathbf{U}(\mathfrak{h}).\end{align*} $$

This finishes the proof of the proposition.

4.3

Our next goal is to understand the left ideal $\mathcal {L}_{\boldsymbol {\xi },\lambda }:=\mathbf {U}_{\ell }^-\mathcal {J}_{\boldsymbol {\xi },\lambda }$ generated by $\mathcal {J}_{\boldsymbol {\xi },\lambda }$ in the loop algebra. The reason is that the same proof as in [Reference Naoi28, Lemma 4.2] shows that we have a linear isomorphism

(4.5) $$ \begin{align}\mathbf{U}_{\ell}^-/\left(\mathfrak{n}^-[t^{-1}]\mathbf{U}_{\ell}^{-}+\mathcal{L}_{\boldsymbol{\xi},\lambda})\right)\cong \mathbf{U}^-/(\mathfrak{n}^-\mathbf{U}^{-}+\mathcal{J}_{\boldsymbol{\xi},\lambda})\end{align} $$

preserving the $-Q^+$ -grading. This isomorphism is induced from the projection $\mathbf {U}_{\ell }^-\rightarrow \mathbf {U}^-$ with respect to the decomposition

$$ \begin{align*}\mathbf{U}_{\ell}^-\cong (\mathfrak{n}^-\otimes t^{-1}\mathbb{C}[t^{-1}])\mathbf{U}_{\ell}^-\oplus \mathbf{U}_{}^-\end{align*} $$

and its importance will be discussed later. The explicit generators of $\mathcal {L}_{\boldsymbol {\xi },\lambda }$ are given as follows. Let $N=\text {max}\{\lambda (h_i),\xi _{i}: 1\leq i\leq n\}$ and define

$$ \begin{align*}\widehat{X}_i^{-}(z)=\sum^N_{m=-\infty} (x_{i}^{-}\otimes t^m) z^{-m-1},\ 1\leq i\leq n.\end{align*} $$

We denote by $\widehat {\mathbf {x}}_{i}^{-}(r,s)$ the coefficient in front of $z^{-(r+s)}$ in the series $\left (\widehat {X}_i^{-}(z)\right )^r$ .

Lemma The $\mathbf {U}_{\ell }^-$ left ideal $\mathcal {L}_{\boldsymbol {\xi },\lambda }$ is generated by the elements

(4.6) $$ \begin{align}&\quad\notag \widehat{\mathbf{x}}_{i}^{-}(r,s),\ 1\leq i\leq n,\ r\in \mathbb{N},s\in\mathbb{Z}_+:\ r+s\geq 1+\lambda(h_i),\\&\quad (x^{-}_{\alpha}\otimes t^{k}),\ \alpha\in R^+,\ k\geq \min\{\xi_{\beta}: \beta\succeq \alpha\}.\end{align} $$

Proof The proof is similar to the proof of [Reference Naoi28, Lemma 4.2] using Proposition 4.2.

Now we come back to the importance of the graded version of (4.5). Recall that $M_{\boldsymbol {\xi },\lambda }[p]$ denotes the pth graded piece of $M_{\boldsymbol {\xi },\lambda }$ for $p\in \mathbb {Z}_+$ (similarly for $\mathbf {U}^{-}$ ). Setting

$$ \begin{align*}\mathbf{U}_{\ell}^-[p]:=(\mathfrak{n}^-\otimes t^{-1}\mathbb{C}[t^{-1}])\mathbf{U}_{\ell}^-\oplus \mathbf{U}_{}^-[p], \end{align*} $$

we obtain

(4.7) $$ \begin{align}[M_{\boldsymbol{\xi},\lambda}: \tau^{*}_p & V(\lambda-\gamma)]\notag=\text{dim} \left(M_{\boldsymbol{\xi},\lambda}[p]/\mathfrak{n}^-M_{\boldsymbol{\xi},\lambda}[p]\right)_{\lambda-\gamma}&\\&=\notag\text{dim}\left(\mathbf{U}^-[p]/\left(\mathfrak{n}^-\mathbf{U}^-[p]+\mathcal{J}_{\boldsymbol{\xi},\lambda}\cap \mathbf{U}^-[p]\right)\right)_{-\gamma}&\\&=\text{dim} \left(\mathbf{U}_{\ell}^-[p]/\left((\mathfrak{n}^-\otimes t^{-1} \mathbb{C}[t^{-1}])\mathbf{U}_{\ell}^-+\mathfrak{n}^-\mathbf{U}^-[p]+\mathcal{L}_{\boldsymbol{\xi},\lambda}\cap\mathbf{U}_{\ell}^-[p]\right)\right)_{-\gamma}.\end{align} $$

Hence, the graded character of $M_{\boldsymbol {\xi },\lambda }$ is determined by the numbers (4.7).

5.1

In the rest of this section, we fix an element $\gamma = \sum ^n_{i=1}r_i \alpha _i$ of $Q^+$ . Consider the generating current

$$ \begin{align*}x_\alpha^{-}(z)=\sum_{r\in\mathbb{Z}} (x_{\alpha}^{-}\otimes t^r) z^{-r-1},\ \alpha\in R^+.\end{align*} $$

The graded component $\left (\mathbf {U}_{\ell }^-\right )_{-\gamma }$ with respect to the $\mathfrak {h}$ -grading is generated by the coefficients of products of generating currents of the form

$$ \begin{align*}x^{-}_{\alpha_{i_1}}(z_1)\cdots x^{-}_{\alpha_{i_{k}}}(z_{k}),\ \gamma=\alpha_{i_1}+\alpha_{i_2}+\cdots+\alpha_{i_k}.\end{align*} $$

Note that we have two types of operator product expansion relations induced from the commutation relations and the Serre relations (see, for example, [Reference Ardonne and Kedem1, Section 4]). The commutation relation between currents implies the following: for $\alpha ,\beta \in R^+$ with $\alpha +\beta \in R^+$ , we have that

(5.1) $$ \begin{align}x^{-}_{\alpha}(z_1)x^{-}_{\beta}(z_2)=\frac{x^{-}_{\alpha+\beta}(z_2)}{(z_1-z_2)}+ \text{"terms with no pole at }z_1=z_2,\text{"}\end{align} $$

where the expansion of the denominator is taken in the region $|z_1|>|z_2|$ . As a result of the Serre relations in $\mathfrak {g}$ , we have

(5.2) $$ \begin{align}\left((z_{2,i}-z_{1,j})(z_{1,i}-z_{1,j})x_{\alpha_i}^{-}(z_{1,i})x_{\alpha_i}^{-}(z_{2,i})x_{\alpha_j}^{-}(z_{1,j})\right)\Big|_{z_{1,i}=z_{2,i}=z_{1,j}}=0,\ \ \text{if } |i-j|=1.\end{align} $$

We will see at the end of this subsection that a typical element in the dual space of $\left (\mathbf {U}^-_{\ell }\right )_{-\gamma }$ can be viewed as a rational function in the variables

$$ \begin{align*}\mathbf{w}_{\gamma}=\{ w_{i,r} : 1\leq i \leq n , 1 \leq r \leq r_i\}.\end{align*} $$

Due to the restrictions coming from the OPE relations (see (5.1) and (5.2)), we will get some restrictions on these rational functions. The first OPE relation (5.1) states that the functions will have at most a simple pole whenever $w_{i,r}=w_{i+1,s}$ . The second one (5.2) tells us that the evaluation of the functions at $w_{i,1}=w_{i,2}=w_{i\pm 1,1}$ must vanish. Moreover, since $[x_{\alpha _{i}}^{-}(z),x_{\alpha _{i}}^{-}(w)]=0$ , the functions have a symmetry under the exchange of variables $w_{i,r}\leftrightarrow w_{i,s}$ . This motivates the next definition.

Definition Let $\mathbb {U}_{\gamma }$ be the subspace of rational functions in the variables $\mathbf {w}_{\gamma }$ which are of the form

$$ \begin{align*}g(\mathbf{w}_{\gamma})=\frac{f(\mathbf{w}_{\gamma})}{\Delta_{\gamma}},\ \ \ \ \Delta_{\gamma}:=\prod_{\substack{1\leq r\leq r_i,\ 1\leq s\leq r_{i+1},\\ 1\leq i\leq n-1}}(w_{i,r}-w_{i+1,s}),\end{align*} $$

where $f(\mathbf {w}_{\gamma })$ is a Laurent polynomial in the variables $\mathbf{w}_{\gamma }$ , invariant under the action of the parabolic subgroup $S_{r_1}\times \cdots \times S_{r_n}$ ( $S_{r_i}$ permutes the variables $w_{i,r},\ 1\leq r\leq r_i$ ), and vanishing under the specialization $w_{i,1}=w_{i,2}=w_{i\pm 1,1}$ .

The residue of $g(\mathbf{w}_{\gamma })\in \mathbb {U}_{\gamma }$ viewed as a function in $w_{i,1}$ is defined as follows. We consider the Laurent series expansion of $g(\mathbf{w}_{\gamma })$ in a punctured disk $\{0<|w_{i,1}|<\epsilon \}$ by expanding all $(w_{i,1}-w_{i+1,s})^{-1}$ (resp. $(w_{i-1,r}-w_{i,1})^{-1}$ ) in $g(\mathbf{w}_{\gamma })$ by using

$$ \begin{align*}-\delta(w_{i+1,s},w_{i,1})\ \ (\text{resp. } \delta(w_{i-1,r},w_{i,1})),\end{align*} $$

where $\delta (z,w)=\sum _{r\geq 0}z^{-r-1}w^{r}$ . Then, $\text {Res}_{w_{i,1}}\left (g(\mathbf{x}_{\gamma })\right )$ is defined to be the coefficient of $\left (w_{i,1}\right )^{-1}$ in this series expansion. We set

$$ \begin{align*}R_{i,p}(g(\mathbf{w}_{\gamma})):=\text{Res}_{w_{i,1}}\left(\left(w_{i,1}\right)^{p}g(\mathbf{x}_{\gamma})\right)\end{align*} $$

and view it as a function in $\mathbb {U}_{\gamma -\alpha _i}$ by re-indexing the set $\{w_{i,2},\dots ,w_{i,r_i}\}$ to $\{w_{i,1},\dots ,w_{i,r_i-1}\}$ which is possible by the symmetry of these functions. The following theorem can be found in [Reference Ardonne, Kedem and Stone2, Theorem 3.3].

Proposition The space of functions $\mathbb {U}_{\gamma }$ is dual to the graded component $\left (\mathbf {U}^-_{\ell }\right )_{-\gamma }$ with pairing

$$ \begin{align*}\langle \cdot , \cdot \rangle: \left(\mathbf{U}^-_{\ell}\right)_{-\gamma} \times \mathbb{U}_{\gamma} \rightarrow \mathbb{C}\end{align*} $$

given by the rule

(5.3) $$ \begin{align}\left\langle (x^{-}_{i_1}\otimes t^{p_1})(x^{-}_{i_2}\otimes t^{p_2})\cdots (x^{-}_{i_k}\otimes t^{p_k}),g(\mathbf{w}_{\gamma})\right\rangle =R_{i_1,p_1}R_{i_2,p_2}\cdots R_{i_k,p_k}\left(g(\mathbf{w}_{\gamma})\right)\end{align} $$

for all $p_1,\dots ,p_k\in \mathbb {Z}$ and $i_1,\dots ,i_k\in \{1,\dots ,n\}$ with $\gamma =\alpha _{i_1}+\cdots +\alpha _{i_k}$ .

5.2

Recall that the dual space of a quotient $V/W$ is isomorphic to the space of functions f in $V^{*}$ satisfying $f(W)=0$ . Hence, using Proposition 5.1, we get that the dual space of

$$ \begin{align*}\left(\mathbf{U}_{\ell}^-[p]/\left((\mathfrak{n}^-\otimes t^{-1}\mathbb{C}[t^{-1}])\mathbf{U}_{\ell}^-+\mathfrak{n}^-\mathbf{U}^-[p]+\mathcal{L}_{ \boldsymbol{\xi},\lambda}\cap\mathbf{U}_{\ell}^-[p]\right)\right)_{-\gamma}\end{align*} $$

consists of all functions $g(\mathbf {w}_{\gamma })\in \mathbb {U}_{\gamma }$ satisfying

(5.4) $$ \begin{align} \left\langle (x_{i}^{-}\otimes t^k)\cdot \left(\mathbf{U}^-_{\ell}\right)_{-\gamma+\alpha_i}, g(\mathbf{w}_{\gamma})\right\rangle=0,\ \ \forall i\in \{1,\dots,n\},\ k\leq 0, \end{align} $$

(5.5) $$ \begin{align} \left\langle \left(\mathbf{U}^-[q]\right)_{-\gamma}, g(\mathbf{w}_{\gamma})\right\rangle=0,\ \ \forall q\neq p, \end{align} $$

and

(5.6) $$ \begin{align} \left\langle \mathcal{L}_{ \boldsymbol{\xi},\lambda}\cap \left(\mathbf{U}^-_{\ell}[p]\right)_{-\gamma}, g(\mathbf{w}_{\gamma})\right\rangle=0.\end{align} $$

We claim that the above three conditions are equivalent to the conditions (5.4) and (5.5) and

(5.7) $$ \begin{align} \left \langle \mathcal{L}_{ \boldsymbol{\xi},\lambda}\cap \left(\mathbf{U}^-_{\ell}\right)_{-\gamma}, g(\mathbf{w}_{\gamma})\right\rangle=0. \end{align} $$

To see this, we only have to show that a function satisfying (5.4)–(5.6) also satisfies (5.7). This claim is only nontrivial for elements in $\left (\mathbf {U}^-[q]\right )_{-\gamma +r\alpha _i}\widehat {\mathbf {x}}_i^{-}(r,s)$ considered as a subset of $\mathcal {L}_{ \boldsymbol {\xi },\lambda }\cap \left (\mathbf {U}^-_{\ell }\right )_{-\gamma }$ . Obviously,

$$ \begin{align*}\left(\mathbf{U}^-[q]\right)_{-\gamma+r\alpha_i}\widehat{\mathbf{x}}_i^{-}(r,s)\subseteq \left((\mathfrak{n}^-\otimes t^{-1}\mathbb{C}[t^{-1}])\mathbf{U}_{\ell}^-\oplus \mathbf{U}^-[q+s]\right)_{-\gamma}.\end{align*} $$

So, if $q+s=p$ , then the latter element is contained in $\mathcal {L}_{ \boldsymbol {\xi },\lambda }\cap \left (\mathbf {U}^-_{\ell }[p]\right )_{-\gamma }$ and the claim follows by our assumption. If $q+s\neq p$ , then the claim follows from (5.4) and (5.5).

So summarizing, in order to determine (4.7), we are interested in all rational functions $g(\mathbf{w}_{\gamma })\in \mathbb {U}_{\gamma }$ satisfying (5.4), (5.5), and (5.7). The characterization of functions satisfying (5.4) is a slight modification of the result [Reference Naoi28, Lemma 4.6] and is stated in the next lemma.

Lemma A function $g(\mathbf {w}_{\gamma })=\frac {f(\mathbf {w}_{\gamma })}{\Delta _{\gamma }}\in \mathbb {U}_{\gamma }$ satisfies (5.4) if and only if

$$ \begin{align*}\text{deg}_{w_{i,1}}f(\mathbf{w}_{\gamma}) \leq r_{i-1}+ r_{i+1}-2 \text{ for all } i \in \{1,\dots,n\},\end{align*} $$

where we understand $r_0=r_{n+1}=0$ .

5.3

In this subsection, we investigate the meaning of (5.5) and (5.7). We set $e_{\gamma }=\sum _{i=1}^{n-1} r_ir_{i+1}$ .

It remains to observe the restrictions coming from (5.7). Recall that $(x^{-}_{i}\otimes t^{k})\in \mathcal {L}_{ \boldsymbol {\xi },\lambda }$ for all $k\geq N$ . For $r\leq r_i$ , we have

$$ \begin{align*}\left\langle \widehat{X}_i^{-}(z)^r, g(\mathbf{w}_{r \alpha_i})\right\rangle=g(\mathbf{w}_{r \alpha_i})\Big|_{w_{i,1}=\cdots=w_{i,r}=z}\end{align*} $$

and hence the condition

$$ \begin{align*}\left\langle \left(\mathbf{U}_{\ell}^-\right)_{\gamma-r\alpha_i} \widehat{\mathbf{x}}_i^{-}(r,s), g(\mathbf{w}_{\gamma})\right\rangle=0,\ \ r+s\geq 1+\lambda(h_i),\end{align*} $$

means that the order of the pole of $g(\mathbf {w}_{\gamma })|_{w_{i,1}=\cdots =w_{i,r}=z}$ at $z=0$ can be at most $\lambda (h_i)$ . In particular, a function $g(\mathbf{w}_{\gamma })$ satisfying (5.7) has this property by Lemma 4.3. The interpretation of (4.6) is explained in the proof of Theorem 5. First, we define the main object of this section.

Now we state the main theorem of this section.

Theorem 5 Let $ \boldsymbol {\xi },\lambda ,\gamma ,p$ be as in Definition 5.3. We have an isomorphism of vector spaces

$$ \begin{align*}\mathcal{V}_{ \boldsymbol{\xi},\lambda,\gamma,p}\cong \left(\mathbf{U}^-[p]/(\mathfrak{n}^-\mathbf{U}^-[p]+\mathcal{J}_{ \boldsymbol{\xi},\lambda}\cap \mathbf{U}^-[p] )\right)_{-\gamma}^*\end{align*} $$

and hence the graded character of $M_{ \boldsymbol {\xi },\lambda }$ is given by

$$ \begin{align*}\mathrm{ch}_{\mathrm{gr}} (M_{ \boldsymbol{\xi},\lambda})=\sum_{\gamma\in Q^+} \left(\sum_{p\in\mathbb{Z}_+} \dim(\mathcal{V}_{ \boldsymbol{\xi},\lambda,\gamma,p}) \ q^p\right)\mathrm{ch}_{\mathfrak{h}} V(\lambda-\gamma).\end{align*} $$

Proof It is clear that the space of all functions in $\mathbb {U}_{\gamma }$ satisfying the properties (5.4)–(5.7) describes the aforementioned dual space as a vector space. Using Lemmas 5.2 and 5.3 and the discussion preceding Definition 5.3, it has only to be checked that Definition 5.3(3) is equivalent to the fact that

(5.8) $$ \begin{align}\left\langle \left(\mathbf{U}_{\ell}^{-}\right)_{\gamma-\alpha}(x^{-}_{\alpha}\otimes t^k),\frac{f(\mathbf{x}_\gamma)}{\Delta_{\gamma}}\right\rangle =0,\ \ \alpha\in R^+,\ \ k\geq \xi_{\alpha}.\end{align} $$

But if we write

$$ \begin{align*}(x^{-}_{i,j}\otimes t^k)=[(x^{-}_{j}\otimes t^{k_j}),[(x^{-}_{j-1}\otimes t^{k_{j-1}}),\dots,[(x^{-}_{i+1}\otimes t^{k_{i+1}}),(x^{-}_{i}\otimes t^{k_i})]\dots]]\end{align*} $$

for some $k_i,\dots ,k_{j}\in \mathbb Z$ with $k_1+\dots +k_{\ell }=k$ , this follows from

(5.9) $$ \begin{align} [R_{j,k_j},[R_{{j-1},k_{j-1}},&\dots,[R_{i+1,k_{i+1}},R_{i,k_i}]\dots]] \frac{f(\mathbf{x}_\gamma)}{\Delta_{\gamma}} &\\&\notag =\text{Res}_{w_{j,1}} \Big\{(w_{i,1}-w_{i+1,1})\cdots (w_{j-1,1}-w_{j,1}) \frac{f(\mathbf{x}_\gamma)}{\Delta_{\gamma}}\Big|_{w_{i,1}=\cdots=w_{j,1}=z}\cdot (w_{j,1})^k\Big\}, \end{align} $$

which has been proved in [Reference Naoi28, Section 4.2]. So (5.8) is equivalent to the statement that the order of the pole of

$$ \begin{align*}\left((w_{i,1}-w_{i+1,1})\cdots (w_{j-1,1}-w_{j,1})\frac{f(\mathbf{x}_\gamma)}{\Delta_{\gamma}}\right)|_{w_{i,1}=w_{i+1,1}=\cdots=w_{j,1}=z}\end{align*} $$

at $z=0$ is at most $\xi _{i,j}$ . The character equality follows from (4.7).

Example Let $n=2$ , $\xi _{\alpha _1}=\xi _{\alpha _2}=\xi _{\alpha _{1,2}}=2$ , $\gamma =2\alpha _1+\alpha _2$ , and $\lambda =7\varpi _1+5\varpi _2$ . We consider Laurent polynomials $f(\mathbf{w}_{\gamma })$ in the variables $\{w_{1,1}, w_{1,2}, w_{2,1}\}$ which are symmetric under the exchange of variables $w_{1,1} \leftrightarrow w_{1,2}$ and satisfying

$$ \begin{align*}w_{1,1}^{2} &f(\mathbf{w}_{\gamma})\in \mathbb{C}[w_{1,1}, w^{\pm 1}_{1,2},w^{\pm 1}_{2,1}],\ \ w_{2,1}^{2} f(\mathbf{w}_{\gamma})\in \mathbb{C}[x^{\pm 1}_{1,1}, w^{\pm 1}_{1,2},w_{2,1}],\ \ \text{deg}_{w_{2,1}}f(\mathbf{w}_{\gamma})\leq 0,\\&\text{deg}_{w_{1,1}}f(\mathbf{w}_{\gamma})\leq -1,\ \ z^{2} f(\mathbf{w}_{\gamma})\big|_{w_{1,1}=w_{2,1}=z}\in \mathbb{C}[w^{\pm 1}_{1,2}, z],\ \ f(\mathbf{w}_{\gamma})\big|_{w_{1,1}=w_{1,2}=w_{2,1}}=0.\end{align*} $$

This is a vector space of dimension 2 with basis

$$ \begin{align*}f_1(\mathbf{w}_{\gamma})&=\left(w_{1,1}\right)^{-2}\left(w_{1,2}\right)^{-2}\left(w_{2,1}\right)^{-2}\left(\left(w_{2,1}\right)^2+w_{1,1}w_{1,2}-w_{1,2}w_{2,1}-w_{1,1}w_{2,1}\right),\\f_2(\mathbf{w}_{\gamma})&=\left(w_{1,1}\right)^{-2}\left(w_{1,2}\right)^{-2}\left(w_{2,1}\right)^{-2}\left(w_{1,2}\left(w_{2,1}\right)^2+w_{1,1}\left(w_{2,1}\right)^2-2w_{1,1}w_{1,2}w_{2,1}\right).\end{align*} $$

So Theorem 5 implies $[M_{\boldsymbol {\xi },\lambda }: V(\lambda -\gamma )]_q=q^2+q^3$ .

6.1

We first record the following lemma.

Lemma Let $f(\mathbf {w}_{\gamma })$ be a polynomial in the variables $\mathbf {w}_{\gamma }$ such that the specialization at $w_{i,1}=w_{i+1,1}=\cdots =w_{j,1}=z$ is divisible by z for all pairs $(i,j)\in \{(i_{\ell },i_{\ell +1}):1\leq \ell \leq k-1\}$ . Then $f(\mathbf {w}_{\gamma })$ is contained in the monomial ideal

$$ \begin{align*}\mathcal{I}^{}_{i_1,\dots,i_k}:=\left\langle w_{i_1,1}^{c_{i_1}}w_{i_1+1,1}^{c_{i_1+1}}\cdots w_{i_{k},1}^{c_{i_{k}}}: c_{i_{\ell}}+c_{i_{\ell}+1}+\cdots+c_{i_{\ell+1}}= 1\ \forall 1\leq \ell\leq k-1\right\rangle.\end{align*} $$

Proof Suppose that we can write

$$ \begin{align*}f(\mathbf{w}_{\gamma})=A(\mathbf{w}_{\gamma})+B(\mathbf{w}_{\gamma}),\end{align*} $$

where $A(\mathbf {w}_{\gamma })\in \mathcal {I}_{i_1,\dots ,i_k}$ and $B(\mathbf {w}_{\gamma })$ is a nonzero polynomial with $B(\mathbf {w}_{\gamma })\notin \mathcal {I}_{i_1,\dots ,i_k}$ . Without loss of generality, we can assume that none of the monomials in $B(\mathbf {w}_{\gamma })$ is contained in $\mathcal {I}_{i_1,\dots ,i_k}$ . We write

$$ \begin{align*}B(\mathbf{w}_{\gamma})=a_1 m_1(\mathbf{w}_{\gamma})+\cdots+a_r m_r(\mathbf{w}_{\gamma})\end{align*} $$

for some $a_1,\dots ,a_r\in \mathbb {C}^{\times }$ and monomials $m_1(\mathbf {w}_{\gamma }),\dots ,m_r(\mathbf {w}_{\gamma })$ . Since $m_1(\mathbf {w}_{\gamma })\notin \mathcal {I}_{i_1,\dots ,i_k}$ , there exists $j\in \{1,\dots ,k-1\}$ with $m_1(\mathbf {w}_{\gamma })$ is independent of the variables $w_{i_j,1},w_{i_j+1,1},\dots ,w_{i_{j+1},1}$ . Now we collect all monomials which are independent of the aforementioned variables, say for simplicity $m_1(\mathbf {w}_{\gamma }),\dots ,m_p(\mathbf {w}_{\gamma })$ for some $p\in \{1,\dots ,r\}$ , and the monomials $m_{p+1}(\mathbf {w}_{\gamma }),\dots , m_r(\mathbf {w}_{\gamma })$ which depend on (at least) one of the aforementioned variables. This implies that

$$ \begin{align*}a_1m_1(\mathbf{w}_{\gamma})+\cdots+a_p m_p(\mathbf{w}_{\gamma})=f(\mathbf{w}_{\gamma})-A(\mathbf{w}_{\gamma})-\sum^r_{k=p+1}a_{k}m_{k}(\mathbf{w}_{\gamma}) \end{align*} $$

is divisible by z after specializing $w_{i_j,1}=w_{i_j+1,1}=\dots =w_{i_{j+1},1}=z$ . This is a contradiction and we must have $B(\mathbf {w}_{\gamma })=0$ .

It is not difficult to translate the dimension of $\mathcal {V}_{\boldsymbol {\xi },\lambda ,\gamma ,p}$ into the language of linear algebra by multiplying the Laurent polynomials in Definition 5.3 by the factor $\prod _{i,r}w_{i,r}^{\lambda (h_i)}$ and using Lemma 6.1. We consider a typical polynomial

$$ \begin{align*}\sum_{\mathbf{c}}a_{\mathbf{c}} \prod_{i=1}^n w_{i,1}^{c_{i,1}}\cdots w_{i,r_i}^{c_{i,r_i}},\ \ a_{\mathbf{c}} \in \mathbb{C}.\end{align*} $$

By the symmetry, we get $a_{\mathbf {c}}=a_{\sigma (\mathbf {c})}$ for each $\sigma \in S_{r_1}\times \cdots \times S_{r_n}$ . Hence, the coefficients are determined by complex numbers $\left (a_{\boldsymbol {\mu }}\right )_{\boldsymbol {\mu }}$ where $\boldsymbol {\mu }=(\mu _1,\dots ,\mu _n)$ is a tuple of partitions and each $\mu _i$ is a partition with at most $r_i$ rows and whose entries are bounded by $r_{i-1}+r_{i+1}-2+\lambda (h_i)$ by Definition 5.3(1). Continuing in this way gives a long list of constraints which is in general hard to calculate. In the next subsection, we use a different approach developed in [Reference Ardonne and Kedem1, Reference Ardonne, Kedem and Stone2].

6.2

We recall a filtration on the space of rational functions $\mathbb {U}_{\gamma }$ from [Reference Ardonne and Kedem1, Section 4.1]. Let $\boldsymbol \mu = (\mu _1,\dots ,\mu _n)$ be a multipartition such that $\mu _i\vdash r_i$ for all $1\leq i\leq n$ . Recall that $m_{i,r}$ denotes the number of rows of length r in the partition $\mu _i$ . Our aim is to define a specialization map

$$ \begin{align*}\varphi_{\boldsymbol \mu}: \mathbb{U}_{\gamma}\rightarrow \mathbb{H}_{\boldsymbol \mu},\end{align*} $$

where $\mathbb {H}_{\boldsymbol \mu }$ is the space of rational functions in the variables

$$ \begin{align*}\mathbf{y}_{\boldsymbol\mu}=\{y_{i,r,u}: 1\leq i\leq n, 1\leq r\leq r_i, 1\leq u \leq m_{i,r}\}.\end{align*} $$

It means that we have a variable for each row in the multipartition. The specialization map $\varphi _{\boldsymbol \mu }$ does the following: for each $i\in \{1,\dots ,n\}$ , we fill the boxes of $\mu _i$ with the variables $x_{i,r}$ , $1\leq r\leq r_i$ , and specialize all variables in the uth row of length r in $\mu _i$ to $y_{i,r,u}$ . By the symmetry, this definition is independent of the filling.

Example Let $n=2$ and $\gamma =2\alpha _1+\alpha _1$ . Consider the multipartition

If we apply the specialization map to the functions in Example 5.3 (ignoring the fact that they are not contained in $\mathbb {U}_{\gamma }$ ), we get

$$ \begin{align*}&\varphi_{\boldsymbol{\mu}}(f_1(\mathbf{x}_{\gamma}))=y_{1,2,1}^{-4}y_{2,1,1}^{-2}\left(y_{2,1,1}^2+y_{1,2,1}^2-2y_{2,1,1}y_{1,2,1}\right),\\&\varphi_{\boldsymbol{\mu}}(f_2(\mathbf{x}_{\gamma}))=2y_{1,2,1}^{-4}y_{2,1,1}^{-2}\left(y_{1,2,1}y_{2,1,1}^2-y_{1,2,1}^2y_{2,1,1}\right).\end{align*} $$

Now we recall the definition of a filtration on the space $\mathbb {U}_{\gamma }$ . First, we define a lexicographical ordering on the set of multipartitions. We say $\boldsymbol \nu \succ \boldsymbol \mu $ if there exists some $j\in \{1,\dots ,n\}$ such that $\nu _d=\mu _d$ for all $d<j$ and $\nu _j>\mu _j$ , where the latter is the usual lexicographical order on partitions which is a total order. Let

$$ \begin{align*} \Gamma_{\boldsymbol\mu} = \underset{\boldsymbol\nu\succ\boldsymbol\mu}{\bigcap} \ker\varphi_{\boldsymbol \nu},\quad \Gamma^{\prime}_{\boldsymbol\mu} = \underset{\boldsymbol \nu\succeq \boldsymbol\mu}{\bigcap} \ker\varphi_{\boldsymbol \nu}\subseteq \Gamma_{\boldsymbol\mu}. \end{align*} $$

Then we immediately get

$$ \begin{align*}\Gamma_{\boldsymbol\mu}\subseteq\Gamma_{\boldsymbol\nu},\ \ \text{if}\ \ \boldsymbol\nu \succ \boldsymbol \mu\end{align*} $$

and hence $(\Gamma _{\boldsymbol \mu })$ defines a filtration on $\mathbb {U}_{\gamma }$ :

(6.1) $$ \begin{align}\{0\} \subseteq \Gamma_{\boldsymbol\mu_1}\subseteq\cdots \subseteq \Gamma_{\boldsymbol\mu_t}= \mathbb{U}_{\gamma},\end{align} $$

where $\{\boldsymbol \mu _1\prec \cdots \prec \boldsymbol \mu _t\}$ is the set of multipartitions $\boldsymbol \mu = (\mu _1,\dots ,\mu _n)$ with $\mu _i\vdash r_i$ for all $1\leq i\leq n$ . We say

$$ \begin{align*} (r,u)<(s,v) \quad \text{ if }r>s\text{ or if }r=s\text{ and }u<v. \end{align*} $$

Let

$$ \begin{align*}\Omega_{\boldsymbol\mu}=\frac{\displaystyle \prod_{1\leq i \leq n}\prod_{(r,u)<(s,v)} (y_{i,r,u}-y_{i,s,v})^{2s} } {\displaystyle \prod_{1\leq i<n}\ \prod_{r,s,u,v} (y_{i,r,u} - y_{i+1,s,v})^{\min\{r, s\}}}.\end{align*} $$

We recall the following important result from [Reference Ardonne, Kedem and Stone2, Theorem 3.6].

Lemma The induced map $\kern1pt\overline {\varphi }_{\boldsymbol \mu }: \Gamma _{\boldsymbol \mu }/\Gamma ^{\prime }_{\boldsymbol \mu }\rightarrow \mathbb {H}_{\boldsymbol \mu }$ of the specialization map $\varphi _{\boldsymbol \mu }$ is an isomorphism of graded vector spaces onto its image. Moreover, the image is the space of rational functions of the form

$$ \begin{align*} \Omega_{\boldsymbol\mu}\cdot h(\mathbf{y}_{\boldsymbol\mu}), \end{align*} $$

where $h(\mathbf {y}_{\boldsymbol \mu })$ is an arbitrary Laurent polynomial in the variables $\mathbf {y}_{\boldsymbol \mu }$ , symmetric with respect to the exchange of variables $y_{i,r,u}\leftrightarrow y_{i,r,v}$ .

Now (6.1) induces a filtration

$$ \begin{align*}\{0\} \subseteq \mathcal{V}_{\boldsymbol \xi,\lambda,\gamma,p}\cap \Gamma_{\boldsymbol\mu_1}\subseteq\cdots \subseteq \mathcal{V}_{\boldsymbol\xi,\lambda,\gamma,p}\cap\Gamma_{\boldsymbol\mu_t}= \mathcal{V}_{\boldsymbol \xi,\lambda,\gamma,p}\end{align*} $$

and hence

$$ \begin{align*}\dim(\mathcal{V}_{\boldsymbol\xi,\lambda,\gamma,p})=\sum_{\boldsymbol\mu} \dim \left((\mathcal{V}_{\boldsymbol\xi,\lambda,\gamma,p}\cap \Gamma_{\boldsymbol\mu})/(\mathcal{V}_{\boldsymbol\xi,\lambda,\gamma,p}\cap \Gamma^{\prime}_{\boldsymbol\mu})\right)=\sum_{\boldsymbol\mu} \dim (\varphi_{\boldsymbol \mu}(\mathcal{V}_{\boldsymbol\xi,\lambda,\gamma,p}\cap \Gamma^{}_{\boldsymbol\mu})).\end{align*} $$

6.3

In what follows, we will describe the functions in $\varphi _{\boldsymbol \mu }(\mathcal {V}_{\boldsymbol \xi ,\lambda ,\gamma ,p}\cap \Gamma ^{}_{\boldsymbol \mu })$ . Recall the role of the space $\mathcal {V}_{\boldsymbol \xi ,\lambda ,\gamma ,p}$ from Theorem 5. By Definition 5.3, we have a list of restrictions to the function $h(\mathbf {y}_{\boldsymbol \mu })$ . The second property implies that the Laurent polynomial $h(\mathbf {y}_{\boldsymbol \mu })$ has a pole at $y_{i,r,u}$ of order at most $\lambda (h_i)$ . Hence, we can write the Laurent polynomials in $\varphi _{\boldsymbol \mu }(\mathcal {V}_{\boldsymbol \xi ,\lambda ,\gamma ,p}\cap \Gamma ^{}_{\boldsymbol \mu })$ as

(6.2) $$ \begin{align}\frac{\Omega_{\boldsymbol\mu}}{\displaystyle \prod_{1\leq i\leq n}\prod_{(r,u)}(y_{i,r,u})^{\lambda(h_i)}}\ h_1(\mathbf{y}_{\boldsymbol\mu}),\end{align} $$

where $h_1(\mathbf {y}_{\boldsymbol \mu })$ is an arbitrary polynomial symmetric under the exchange of variables $y_{i,r,u}\leftrightarrow y_{i,r,v}$ . The first condition in Definition 5.3 implies that

$$ \begin{align*}\text{deg}_{y_{i,r,u}}\varphi_{\boldsymbol\mu}(g(\mathbf{x}_{\gamma}))\leq -2r.\end{align*} $$

Since the degree of the first term in (6.2) with respect to the variable $y_{i,r,u}$ is given by $-P^{\boldsymbol {\mu },\lambda }_{r,i}-2r$ (recall the definition from Section 3), we obtain that

$$ \begin{align*}\text{deg}_{y_{i,r,u}}h_1(\mathbf{y}_{\boldsymbol\mu})\leq P^{\boldsymbol{\mu},\lambda}_{r,i}.\end{align*} $$

Since the specialization map preserves the homogeneous degree, we have also that $\varphi _{\boldsymbol \mu }(g(\mathbf {x}_{\gamma }))$ is homogeneous of degree $-p-|\gamma |$ and hence

$$ \begin{align*}\mathrm{deg}(\Omega_{\boldsymbol\mu})=-p-|\gamma|-\mathrm{deg}(h_1(\mathbf{y}_{\boldsymbol\mu}))+\sum_{i=1}^n \lambda(h_i) \cdot d(\mu_i).\end{align*} $$

Let $j_{i,r,u}$ be the row index of $\mu _i$ corresponding to the pair $(r,u)$ . Note that $\mu _i^{j_{i,r,u}} = r$ . It follows that the degree of the numerator of $\Omega _{\boldsymbol {\mu }}$ is given by

$$ \begin{align*} \sum_{i=1}^n \sum_{(r,u) < (s,v)} 2s & = \sum_{i=1}^n \sum_{(s,v)} 2 ( j_{i,s,v}-1) \mu_i^{j_{i,s,v}} = \sum_{i=1}^n \sum_{j = 1}^{r_i} 2 (j-1) \mu_i^j\\ &= - 2|\gamma| + \sum_{i=1}^n \sum_{j = 1}^{r_i} 2 j \mu_i^j, \end{align*} $$

where $(r,u) < (s,v)$ runs over all pairs of integer tuples satisfying $1 \le r,s \le r_i$ , $1 \le u \le m_{i,r}$ , and $1 \le v \le m_{i,s}$ . Similarly, the degree of the denominator is given by

$$ \begin{align*} \sum_{i = 1}^{n-1} \sum_{r,u,s,v} \min\{ r,s \} &= \sum_{i = 1}^{n -1}\sum_{j = 1}^{r_i} \sum_{l = 1}^{r_{i+1}} \min\{ \mu_i^{j},\mu_{i+1}^{l} \} = \sum_{i = 1}^{n-1} \sum_{j = 1}^{r_i} \mu_{i+1}(\mu_i^j), \end{align*} $$

where the second sum runs over all $1 \le r \le r_i$ , $1 \le u \le m_{i,r}$ , $1 \le s \le r_{i+1}$ , and $1 \le u \le m_{i+1,s}$ . Therefore,

$$ \begin{align*} \mathrm{deg}(\Omega_{\boldsymbol\mu})=-2 |\gamma|+K_{\boldsymbol{\mu}}^{\lambda}+\sum_{i=1}^n \lambda(h_i) \cdot d(\mu_i) \end{align*} $$

and thus we get that $h_1(\mathbf {y}_{\boldsymbol \mu })$ is homogeneous of degree $-p+|\gamma |-K^{\lambda }_{\boldsymbol \mu }$ . Now we investigate the meaning of the last condition of Definition 5.3. Consider the polynomial $\tilde {f}(\mathbf {w}_{\gamma })=f(\mathbf {w}_{\gamma }) \prod _{i,r}w_{i,r}^{\lambda (h_i)}$ and note that the specializations

$$ \begin{align*} \tilde{f}(\mathbf{w}_{\gamma}) \Big|_{w_{i_{j},1}=\cdots=w_{i_{j+1},1}=z}&=z^2\cdot \prod_{(i,r)\notin \{(i_j,1),\dots,(i_{j+1},1)\}}w_{i,r}^{\lambda(h_i)}\cdot f(\mathbf{w}_{\gamma})\Big|_{w_{i_{j},1}=\cdots=w_{i_{j+1},1}=z}&\\&=z^2\cdot \prod_{(i,r)\notin \{(i_j,1),\dots,(i_{j+1},1)\}}w_{i,r}^{\lambda(h_i)}\cdot (a z^{-1}+b+cz+\cdots),\end{align*} $$

$$ \begin{align*} \tilde{f}(\mathbf{w}_{\gamma}) \Big|_{w_{\ell,1}=z}&=z^{(2\lambda_0+\lambda_2)(h_{\ell})}\cdot \prod_{(i,r)\neq (\ell,1)}w_{i,r}^{\lambda(h_i)}\cdot f(\mathbf{w}_{\gamma})\Big|_{w_{\ell,1}=z}&\\&=z^{(2\lambda_0+\lambda_2)(h_{\ell})}\cdot \prod_{(i,r)\neq (\ell,1)}w_{i,r}^{\lambda(h_i)}\cdot (a z^{-(\lambda_0+\lambda_2)(h_{\ell})}+bz^{-(\lambda_0+\lambda_2)(h_{\ell})+1}+\cdots)\end{align*} $$

are divisible by z for all $1\leq j<k$ and $\ell \in \mathrm {supp}_1(\lambda _0)$ ; recall that $\lambda _0\in P^+(1)$ . Hence, we can apply Lemma 6.1 to $\tilde {f}(\mathbf {w}_{\gamma })$ and obtain that

(6.3) $$ \begin{align}\displaystyle \prod_{1\leq i \leq n}\prod_{(r,u)<(s,v)} (y_{i,r,u}-y_{i,s,v})^{2s} \displaystyle \prod_{1\leq i<n}\ \prod_{r,s,u,v} (y_{i,r,u} - y_{i+1,s,v})^{rs-\min\{r, s\}} \prod_{i,r,u}y_{i,r,u}^{(r-1)\lambda(h_i)}\ h_1(\mathbf{y}_{\boldsymbol\mu})\end{align} $$

is contained in the intersection of ideals

(6.4) $$ \begin{align}\mathcal{Y}:=\bigcap_{\substack{s_{i_1},\dots,s_{i_k}\\ v_{i_1},\dots,v_{i_k}}}\mathcal{Y}_{\substack{s_{i_1},\dots,s_{i_k}\\ v_{i_1},\dots,v_{i_k}}}\cap \bigcap_{\substack{\ell\in\mathrm{supp}(\lambda_0),\\ 1\leq r\leq r_{\ell},\\ 1\leq u\leq m_{\ell,r}}}\left \langle y_{\ell,r,u} \right\rangle, \end{align} $$

where

$$ \begin{align*}\mathcal{Y}_{\substack{s_{i_1},\dots,s_{i_k}\\ v_{i_1},\dots,v_{i_k}}}:=\Big\langle y_{i_1,s_{i_1},v_{i_1}}^{c_{i_1}}y_{i_1+1,s_{i_1+1},v_{i_1+1}}^{c_{i_1+1}}\cdots y_{i_{k},s_{i_k},v_{i_k}}^{c_{i_{k}}}:\ \ \sum_{p=i_j}^{i_{j+1}} c_{p} = 1,\ \forall j\in[1,k)\Big\rangle.\end{align*} $$

Clearly, a polynomial $f(\mathbf {y}_{\boldsymbol {\mu }})$ is contained in $\mathcal {Y}_{\substack {s_{i_1},\dots ,s_{i_k}\\ v_{i_1},\dots ,v_{i_k}}}$ if and only if

(6.5) $$ \begin{align} f(\mathbf{y}_{\boldsymbol{\mu}})|_{y_{i_j,s_{i_j},v_{i_j}}=\cdots =y_{i_{j+1},s_{i_{j+1}},v_{i_{j+1}}}=z} \ \text{ is divisible by }z\end{align} $$

for all $j\in [1,k).$ However, this property is already satisfied by the prefactor of $h_1(\mathbf {y}_{\boldsymbol {\mu }})$ in (6.3) unless there exists $j\in [1,k)$ with $s_{i_j}=\cdots =s_{i_{j+1}}=1$ . Moreover, if there exists such a j with $s_{i_j}=\cdots =s_{i_{j+1}}=1$ , then the prefactor is a polynomial which has a nonzero constant term after evaluating the variables $y_{i_j,s_{i_j},v_{i_j}}=\cdots =y_{i_{j+1},s_{i_{j+1}},v_{i_{j+1}}}=z$ (viewed as a polynomial in z). Hence, the discussion gives

• • The expression (6.3) is contained in $\mathcal {Y}_{\substack {s_{i_1},\dots ,s_{i_k}\\ v_{i_1},\dots ,v_{i_k}}} \iff h_1(\mathbf {y}_{\boldsymbol {\mu }})$ satisfies (6.5) for all $j\in [1,k)$ with $s_{i_j}=\cdots =s_{i_{j+1}}=1$ .

Similarly, the prefactor of $h_1(\mathbf {y}_{\boldsymbol {\mu }})$ in (6.3) is already contained in $\displaystyle \bigcap _{\substack {\ell \in \mathrm {supp}(\lambda _0),\\ 2\leq r\leq r_{\ell },\\ 1\leq u\leq m_{\ell ,r}}}\left \langle y_{\ell ,r,u} \right \rangle $ and thus

• • The expression (6.3) is contained in $\displaystyle \bigcap _{\substack {\ell \in \mathrm {supp}(\lambda _0),\\ 1\leq r\leq r_{\ell },\\ 1\leq u\leq m_{\ell ,r}}}\left \langle y_{\ell ,r,u} \right \rangle \iff h_1(\mathbf {y}_{\boldsymbol {\mu }})\in \displaystyle \bigcap _{\substack {\ell \in \mathrm {supp}(\lambda _0),\\ 1\leq u\leq m_{\ell ,1}}}\left \langle y_{\ell ,1,u} \right \rangle $ .

This implies the following lemma.

Lemma The containment of (6.3) in the ideal (6.4) is equivalent to the fact that $h_1(\mathbf {y}_{\boldsymbol {\mu }})$ is contained in the intersection

$$ \begin{align*}\bigcap_{\substack{s_{i_1},\dots,s_{i_k}\\ v_{i_1},\dots,v_{i_k}}}\widetilde{\mathcal{Y}}_{\substack{s_{i_1},\dots,s_{i_k}\\ v_{i_1},\dots,v_{i_k}}}\cap \bigcap_{\substack{\ell\in\mathrm{supp}(\lambda_0),\\ 1\leq u\leq m_{\ell,1}}}\left \langle y_{\ell,1,u} \right\rangle,\end{align*} $$

where $\widetilde {\mathcal {Y}}_{\substack {s_{i_1},\dots ,s_{i_k}\\ v_{i_1},\dots ,v_{i_k}}}$ is the ideal generated by

$$ \begin{align*}y_{i_1,s_{i_1},v_{i_1}}^{c_{i_1}}y_{i_1+1,s_{i_1+1},v_{i_1+1}}^{c_{i_1+1}}\cdots y_{i_{k},s_{i_k},v_{i_k}}^{c_{i_{k}}},\end{align*} $$

where for each $1\leq j\leq k-1$ we have

$$ \begin{align*}c_{i_j}+c_{i_j+1}+\cdots+c_{i_{j+1}}= \begin{cases}1,& \text{ if }s_{i_j}=\cdots=s_{i_{j+1}}=1,\\ 0,& \text{ otherwise. } \end{cases}\end{align*} $$

Example We consider $r_1=2$ and $r_2=1$ ( $\lambda _1=\varpi _1+\varpi _2$ , $\lambda _0=\lambda _2=0$ ) with the two obvious tuples of partitions. In the case $\boldsymbol {\mu }=(\mu _1,\mu _2)$ with , we get that

$$ \begin{align*}(y_{1,2,1}-y_{2,1,1}) y_{1,2,1} h_1(\mathbf{y}_{\mu})\in \left \langle y_{1,2,1},y_{2,1,1} \right\rangle,\end{align*} $$

and in the other case

$$ \begin{align*}(y_{1,1,1}-y_{1,1,2})^2h_1(\mathbf{y}_{\mu})\in \left \langle y_{1,1,1},y_{2,1,1} \right\rangle \cap \left \langle y_{1,1,2},y_{2,1,1} \right\rangle= \left \langle y_{1,1,1}y_{1,1,2},y_{2,1,1} \right\rangle.\end{align*} $$

In the first case, we have no restriction on $h_1(\mathbf {y}_{\mu }), $ and in the second case, we actually need

$$ \begin{align*}h_1(\mathbf{y}_{\mu})\in \left \langle y_{1,1,1}y_{1,1,2},y_{2,1,1} \right\rangle.\end{align*} $$

Proposition The dimension of $\varphi _{\boldsymbol {\mu }}(\mathcal {V}_{\boldsymbol {\xi },\lambda ,\gamma ,p}\cap \Gamma _{\boldsymbol {\mu }})$ is less or equal to the dimension of the space $\mathcal {W}$ consisting of all homogeneous polynomials $f(\mathbf {y}_{\boldsymbol \mu })$ of degree $-p+|\gamma |-K^{\lambda }_{\boldsymbol {\mu }}$ in the ideal

(6.6) $$ \begin{align}\bigcap_{\substack{s_{i_1},\dots,s_{i_k}\\ v_{i_1},\dots,v_{i_k}}}\widetilde{\mathcal{Y}}_{\substack{s_{i_1},\dots,s_{i_k}\\ v_{i_1},\dots,v_{i_k}}}\cap \bigcap_{\substack{\ell\in\mathrm{supp}(\lambda_0),\\ 1\leq u\leq m_{\ell,1}}}\left \langle y_{\ell,1,u} \right\rangle\end{align} $$

symmetric with respect to the exchange of variables $y_{i,r,u}\leftrightarrow y_{i,r,v}$ and satisfying

$$ \begin{align*}\text{deg}_{y_{i,r,u}}f(\mathbf{y}_{\boldsymbol\mu})\leq P^{\boldsymbol{\mu},\lambda}_{r,i},\ \forall i,r,u.\end{align*} $$

Proof The discussion in Section 6.3 shows that each element in $\varphi _{\boldsymbol {\mu }}(\mathcal {V}_{\boldsymbol {\xi },\lambda ,\gamma ,p}\cap \Gamma _{\boldsymbol {\mu }})$ is of the form (6.2) with $h_1(\mathbf {y}_{\mu })\in \mathcal {W}$ and hence the upper bound for the dimension is obtained.

6.4

If we denote by $e_d^{(i,r)}(\mathbf {y}_{\boldsymbol \mu })$ , $1\leq d\leq m_{i,r}$ the dth elementary symmetric polynomial in the variables $\{y_{i,r,1},\dots , y_{i,r,m_{i,r}}\}$ , we can rewrite the statement of Proposition 6.3 as follows. The dimension of $\varphi _{\boldsymbol \mu }(\mathcal {V}_{\boldsymbol {\xi },\lambda ,\gamma ,p}\cap \Gamma _{\boldsymbol \mu })$ is less or equal to the dimension of the space spanned by all monomials

$$ \begin{align*}\prod_{i=1}^n\prod_{r=1}^{r_i}\prod_{d=1}^{m_{i,r}} (e_d^{(i,r)}(\mathbf{y}_{\boldsymbol\mu}))^{C_{d,r,i}},\ \ C_{d,r,i}\in\mathbb{Z}_+\end{align*} $$

with

$$ \begin{align*}&\sum_{d=1}^{m_{i,r}}C_{d,r,i}\leq P^{{\boldsymbol\mu},\lambda}_{r,i},\ \forall r,i, \ \ \ \ \sum_{i,r,d}d\cdot C_{d,r,i}=-p+|\gamma|-K^{\lambda}_{\boldsymbol{\mu}},\\&\sum_{i=i_j}^{i_{j+1}}C_{m_{i,1},1,i}\geq 1, \ \ \forall j\in[1,k),\ \ \ \ C_{m_{i_1},1,i}\geq 1,\ \ \forall \ i\in\mathrm{supp}_1(\lambda_0).\end{align*} $$

Therefore,

(6.7) $$ \begin{align}[N_{\boldsymbol\zeta}:\tau_p^*V(\lambda-\gamma)]=\sum_{\boldsymbol\mu \vdash \gamma}\dim(\varphi_{\boldsymbol\mu}\left(\mathcal{V}_{\boldsymbol{\xi},\lambda,\gamma,p}\cap \Gamma_{\boldsymbol\mu})\right)\leq \sum_{\boldsymbol\mu \vdash \gamma} L_{\boldsymbol\mu,\boldsymbol\zeta}^p.\end{align} $$

Now Theorem 4 follows from the next theorem and Theorem 2(1); the fact $[L(\boldsymbol {\pi }):\tau _p^*V(\nu )]=0$ for all $\nu \notin \mathrm {wt}(\boldsymbol {\pi })-Q^+$ is obvious by weight reasons.

Theorem 6 Let $\gamma \in Q^+$ , $p\in \mathbb {Z}_+$ , and $\boldsymbol \zeta =(\lambda ,\lambda _0,\lambda _1,\lambda _2)$ be an admissible quadrupel such that $\lambda _2\in P^+(1)$ . Then

$$ \begin{align*}[N_{\boldsymbol{\zeta}}:\tau_p^*V(\lambda-\gamma)]= \sum_{\boldsymbol\mu \vdash \gamma} L_{\boldsymbol\mu,\boldsymbol{\zeta}}^p.\end{align*} $$

Proof The proof is by induction on $|\mathrm {supp}_1(\lambda _1)|$ . In the case of $|\mathrm {supp}_1(\lambda _1)|\leq 1$ , the theorem follows from Remark 3.2 (see in particular (3.5)) since the assumption $[N_{\boldsymbol {\zeta }}:\tau _p^*V(\lambda -\gamma )]< \sum _{\boldsymbol \mu \vdash \gamma } L_{\boldsymbol \mu ,\boldsymbol {\zeta }}^p$ would lead together with (6.7) to a contradiction:

$$ \begin{align*}[N_{\boldsymbol{\zeta}}:V(\lambda-\gamma)]_{q=1}=\sum_{k\in\mathbb{Z}_+}[N_{\boldsymbol{\zeta}}:\tau_k^*V(\lambda-\gamma)]< \sum_{k\in\mathbb{Z}_+}\sum_{\boldsymbol\mu \vdash \gamma} L_{\boldsymbol\mu,\boldsymbol{\zeta}}^k=[N_{\boldsymbol{\zeta}}:V(\lambda-\gamma)]_{q=1}.\end{align*} $$

Now let $|\mathrm {supp}_1(\lambda _1)|\geq 2$ and write $\lambda _1=\varpi _{i_1}+\cdots +\varpi _{i_k},\ 1\leq i_1<\cdots <i_k\leq n$ . Since $\lambda _2\in P^+(1)$ , we can use Theorem 2(2) and get

$$ \begin{align*}[N_{\boldsymbol{\zeta}}:\tau_p^*V(\lambda-\gamma)]=[N_{\boldsymbol{\zeta}_1}:\tau_p^*V(\lambda-\gamma)]-(1-\delta_{p,0}) [N_{\boldsymbol{\zeta}_2}:\tau_{p-1}^*V(\lambda-\gamma)].\end{align*} $$

Since $\boldsymbol {\zeta }_1$ is again admissible and $\lambda _2+\varpi _{i_1}\in P^+(1)$ (see the definition in (2.3)), we can apply our induction hypothesis to get

$$ \begin{align*}[N_{\boldsymbol{\zeta}_1}:\tau_p^*V(\lambda-\gamma)]=\sum_{\boldsymbol\mu \vdash \gamma} L_{\boldsymbol\mu,\boldsymbol{\zeta}_1}^p.\end{align*} $$

In general, we cannot apply the induction hypothesis to the second summand, but using (6.7) (which holds for all admissible quadruples), we can at least say

$$ \begin{align*}(1-\delta_{p,0})[N_{\boldsymbol{\zeta}_2}:\tau_{p-1}^*V(\lambda-\gamma)]\leq (1-\delta_{p,0}) \sum_{\boldsymbol{\tilde\mu}\vdash (\gamma-\alpha_{i_1,i_2})} L_{\boldsymbol{\tilde\mu},\boldsymbol{\zeta}_2}^{p-1}.\end{align*} $$

Hence,

$$ \begin{align*}[N_{\boldsymbol{\zeta}}:\tau_p^*V(\lambda-\gamma)]\geq \sum_{\boldsymbol\mu \vdash \gamma} L_{\boldsymbol\mu,\boldsymbol{\zeta}_1}^p-(1-\delta_{p,0})\sum_{\boldsymbol{\tilde\mu} \vdash (\gamma-\alpha_{i_1,i_2})} L_{\boldsymbol{\tilde\mu},\boldsymbol{\zeta}_2}^{p-1}=\sum_{\boldsymbol\mu \vdash \gamma} L_{\boldsymbol\mu,\boldsymbol{\zeta}}^p,\end{align*} $$

where the last equality follows from Proposition 3.2. So the theorem is proved by using once more (6.7).