Finite dimensional simple modules of deformed current Lie algebras

TL;DR

This paper classifies all finite dimensional simple modules of deformed current Lie algebras, advancing understanding of their representation theory and connections to cyclotomic q-Schur algebras.

Contribution

It provides a complete classification of finite dimensional simple modules for deformed current Lie algebras, a new development in their representation theory.

Findings

Complete classification of finite dimensional simple modules

Enhanced understanding of deformed current Lie algebra representations

Connections to cyclotomic q-Schur algebras at q=1

Abstract

The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.

Equations

$${\mathbb{B}}^{⟨Q_i⟩}=\{\begin{array}{ll}\{0\}& \text{ if }{Q}_i=0,\\ \mathbb{C}& \text{ if }{Q}_i\ne 0,\end{array}$$

$$\mathbb{C}[x{]}_{\mathrm{monic}}^{⟨Q⟩}=\{\begin{array}{ll}\mathbb{C}[x{]}_{\mathrm{monic}}& \text{ if }Q=0,\\ \{\phi \in \mathbb{C}[x{]}_{\mathrm{monic}}\mid Q^{-1}\text{ is not a root of }\phi \}& \text{ if }Q\ne 0.\end{array}$$

$$(\mathbf{\phi },\mathbf{\beta })=(({\phi }_i,{\beta }_i){)}_{1\le i\le m-1}\mapsto {\mathbf{u}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta })=({\mathbf{u}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta }{)}_{i,t}{)}_{1\le i\le m-1,t\ge 0},$$

$${\mathbf{u}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta }{)}_{i,t}=\{\begin{array}{ll}{\gamma }_{i,1}^t+{\gamma }_{i,2}^t+\cdots +{\gamma }_{i,n_i}^t& \text{ if }{Q}_i=0,\\ {\gamma }_{i,1}^t+{\gamma }_{i,2}^t+\cdots +{\gamma }_{i,n_i}^t+Q_i^{-t}{\beta }_i& \text{ if }{Q}_i\ne 0\end{array}$$

$$\displaystyle\big{(}\bigotimes_{j=1}^{m-1}\bigotimes_{k=1}^{n_{j}}L(\omega_{j})^{\mathbf{{ev}}_{\gamma_{j},k}}\big{)}\otimes\mathcal{L}^{\bm{\beta}},$$

$$(\mathbf{\phi },\mathbf{\beta },\mathbf{h})=(({\phi }_i,{\beta }_i{)}_{1\le i\le m-1},(h_t{)}_{t\ge 0})\mapsto {\tilde{\mathbf{u}}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta },\mathbf{h})=({\tilde{\mathbf{u}}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta },\mathbf{h}{)}_{j,t}{)}_{1\le j\le m,t\ge 0}$$

$${\tilde{\mathbf{u}}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta },\mathbf{h}{)}_{j,t}=\{\begin{array}{ll}\sum _{k=j}^{m-1}{\mathbf{u}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta }{)}_{k,t}+h_t& \text{ if }1\le j\le m-1\text{ and }t\ge 0,\\ h_t& \text{ if }j=m\text{ and }t\ge 0,\end{array}$$

$$\displaystyle\big{(}\bigotimes_{j=1}^{m-1}\bigotimes_{k=1}^{n_{j}}L(\widetilde{\omega}_{j})^{\widetilde{\mathbf{{ev}}}_{\gamma_{j},k}}\big{)}\otimes\widetilde{\mathcal{L}}^{\bm{\beta}}\otimes\widetilde{\mathcal{L}}^{\mathbf{h}}.$$

$$\mathcal{L}({\tilde{\mathbf{u}}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta },\mathbf{h}))\cong \mathcal{L}({\mathbf{u}}^{⟨\mathbf{Q}⟩}(\mathbf{\phi },\mathbf{\beta })),$$

$$L({\tilde{\omega }}_j{)}^{{\widetilde{\mathbf{ev}}}_{{\gamma }_j,k}}\cong L({\omega }_j{)}^{{\mathbf{ev}}_{{\gamma }_j,k}},{\tilde{\mathcal{L}}}^{\mathbf{\beta }}\cong {\mathcal{L}}^{\mathbf{\beta }}\text{ and }{\tilde{\mathcal{L}}}^{\mathbf{h}}\cong {\mathcal{L}}^{(0,\dots ,0)}$$

$$[{\mathcal{J}}_{i,s},{\mathcal{J}}_{j,t}]=0,$$

$$[{\mathcal{J}}_{j,s},{\mathcal{X}}_{i,t}^{\pm }]=\pm a_{ji}{\mathcal{X}}_{i,s+t}^{\pm },$$

$$[{\mathcal{X}}_{i,t}^{+},{\mathcal{X}}_{j,s}^{-}]={\delta }_{ij}({\mathcal{J}}_{i,s+t}-Q_i{\mathcal{J}}_{i,s+t+1}),$$

$$[{\mathcal{X}}_{i,t}^{\pm },{\mathcal{X}}_{j,s}^{\pm }]=0\text{ if }j\ne i\pm 1,$$

$$[{\mathcal{X}}_{i,t+1}^{+},{\mathcal{X}}_{i\pm 1,s}^{+}]=[{\mathcal{X}}_{i,t}^{+},{\mathcal{X}}_{i\pm 1,s+1}^{+}],[{\mathcal{X}}_{i,t+1}^{-},{\mathcal{X}}_{i\pm 1,s}^{-}]=[{\mathcal{X}}_{i,t}^{-},{\mathcal{X}}_{i\pm 1,s+1}^{-}],$$

$$[{\mathcal{X}}_{i,s}^{+},[{\mathcal{X}}_{i,t}^{+},{\mathcal{X}}_{i\pm 1,u}^{+}]]=[{\mathcal{X}}_{i,s}^{-},[{\mathcal{X}}_{i,t}^{-},{\mathcal{X}}_{i\pm 1,u}^{-}]]=0,$$

$$[{\mathcal{I}}_{i,s},{\mathcal{I}}_{j,t}]=0,$$

$$[{\mathcal{I}}_{j,s},{\mathcal{X}}_{i,t}^{\pm }]=\pm a_{ji}^{\prime }{\mathcal{X}}_{i,s+t}^{\pm },$$

$$[{\mathcal{X}}_{i,t}^{+},{\mathcal{X}}_{j,s}^{-}]={\delta }_{ij}({\mathcal{J}}_{i,s+t}-Q_i{\mathcal{J}}_{i,s+t+1}),\text{where we put }{\mathcal{J}}_{i,t}={\mathcal{I}}_{i,t}-{\mathcal{I}}_{i+1,t},$$

$${\mathcal{E}}_{i,j;t}=\{\begin{array}{ll}[{\mathcal{X}}_{i,0}^{+},[{\mathcal{X}}_{i+1,0}^{+},\dots ,[{\mathcal{X}}_{j-2,0}^{+},{\mathcal{X}}_{j-1,t}^{+}]\dots ]]& \text{ if }j>i,\\ [{\mathcal{X}}_{i-1,0}^{-},[{\mathcal{X}}_{i-2,0}^{-},\dots ,[{\mathcal{X}}_{j+1,0}^{-},{\mathcal{X}}_{j,t}^{-}]\dots ]]& \text{ if }j<i.\end{array}$$

$$\{{\mathcal{X}}_{i,t}^{+}|1\le i\le m-1,t\ge 0\}\text{ and }\{{\mathcal{X}}_{i,t}^{-}|1\le i\le m-1,t\ge 0\}$$

$$\{{\mathcal{J}}_{i,t}|1\le i\le m-1,t\ge 0\}(\text{resp. }\{{\mathcal{I}}_{i,t}|1\le i\le m,t\ge 0\}).$$

$${\rm Y}:{\mathfrak{sl}}_m^{⟨\mathbf{Q}⟩}[x]\to {\mathfrak{gl}}_m^{⟨\mathbf{Q}⟩}[x]\text{ such that }{\mathcal{X}}_{i,t}^{\pm }\mapsto {\mathcal{X}}_{i,t}^{\pm },\text{ and }{\mathcal{J}}_{i,t}\mapsto {\mathcal{I}}_{i,t}-{\mathcal{I}}_{i+1,t}.$$

$${\mathfrak{sl}}_m^{⟨\mathbf{Q}⟩}[x]={\mathbf{n}}^{-}\oplus {\mathbf{n}}^0\oplus {\mathbf{n}}^{+}\text{ and }{\mathfrak{gl}}_m^{⟨\mathbf{Q}⟩}[x]={\mathbf{n}}^{-}\oplus {\tilde{\mathbf{n}}}^0\oplus {\mathbf{n}}^{-}(\text{as vector spaces}).$$

$$\{{\mathcal{E}}_{i,j;t}|1\le i<j\le m,t\ge 0\}(\text{resp. }\{{\mathcal{E}}_{i,j;t}|1\le j<i\le m,t\ge 0\})$$

$$\{{\mathcal{J}}_{i,t}|1\le i\le m-1,t\ge 0\}(\text{resp. }\{{\mathcal{I}}_{j,t}|1\le j\le m,t\ge 0\})$$

$$[K_i,K_j]=0,[K_j,e_i]=a_{ji}^{\prime }{e}_i,[K_j,f_i]=-a_{ji}^{\prime }{f}_i,$$

$$[e_i,f_j]={\delta }_{ij}{H}_i,\text{ where }{H}_i=K_i-K_{i+1},$$

$$[e_i,e_j]=[f_i,f_j]=0\text{ if }j\ne i\pm 1,[e_i,[e_i,e_{i\pm 1}]]=[f_i,[f_i,f_{i\pm 1}]]=0.$$

$${\mathbf{ev}}_{\gamma }:U({\mathfrak{sl}}_m^{⟨\mathbf{Q}⟩}[x])\to U({\mathfrak{sl}}_m)\text{ by }{\mathcal{X}}_{i,t}^{+}\mapsto (1-Q_i\gamma ){\gamma }^t{e}_i,{\mathcal{X}}_{i,t}^{-}\mapsto {\gamma }^t{f}_i,{\mathcal{J}}_{i,t}\mapsto {\gamma }^t{H}_i$$

$${\widetilde{\mathbf{ev}}}_{\gamma }:U({\mathfrak{gl}}_m^{⟨\mathbf{Q}⟩}[x])\to U({\mathfrak{gl}}_m)\text{ by }{\mathcal{X}}_{i,t}^{+}\mapsto (1-Q_i\gamma ){\gamma }^t{e}_i,{\mathcal{X}}_{i,t}^{-}\mapsto {\gamma }^t{f}_i,{\mathcal{I}}_{j,t}\mapsto {\gamma }^t{K}_j.$$

$$\zeta :\Gamma (\mathbf{m})\to \{1,2,\dots ,m\}\text{ such that }(i,k)\mapsto \sum _{j=1}^{k-1}{m}_j+i.$$

$$[{\mathcal{I}}_{(i,k),s},{\mathcal{I}}_{(j,l),t}]=0,[{\mathcal{I}}_{(j,l),s},{\mathcal{X}}_{(i,k),t}^{\pm }]=\pm a_{(j,l)(i,k)}^{\prime }{\mathcal{X}}_{(i,k),s+t}^{\pm },$$

$$[{\mathcal{X}}_{(i,k),t}^{+},{\mathcal{X}}_{(j,l),s}^{-}]={\delta }_{(i,k)(j,l)}\{\begin{array}{ll}{\mathcal{J}}_{(i,k),s+t}& \text{ if }i\ne m_k,\\ -{\hat{Q}}_k{\mathcal{J}}_{(m_k,k),s+t}+{\mathcal{J}}_{(m_k,k),s+t+1}& \text{ if }i=m_k,\end{array}$$

$$[{\mathcal{X}}_{(i,k),t}^{\pm },{\mathcal{X}}_{(j,l),s}^{\pm }]=0\text{ if }(j,l)\ne (i\pm 1,k),$$

$$[{\mathcal{X}}_{(i,k),t+1}^{+},{\mathcal{X}}_{(i\pm 1,k),s}^{+}]=[{\mathcal{X}}_{(i,k),t}^{+},{\mathcal{X}}_{(i\pm 1,k),s+1}^{+}],[{\mathcal{X}}_{(i,k),t+1}^{-},{\mathcal{X}}_{(i\pm 1,k),s}^{-}]=[{\mathcal{X}}_{(i,k),t}^{-},{\mathcal{X}}_{(i\pm 1,k),s+1}^{-}],$$

$$[{\mathcal{X}}_{(i,k),s}^{+},[{\mathcal{X}}_{(i,k),t}^{+},{\mathcal{X}}_{(i\pm 1,k),u}^{+}]]=[{\mathcal{X}}_{(i,k),s}^{-},[{\mathcal{X}}_{(i,k),t}^{-},{\mathcal{X}}_{(i\pm 1,k),u}^{-}]]=0,$$

Full text

1. 0 Introduction
2. 1 Deformed current Lie algebras $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ and $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$
3. 2 Representations of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$
4. 3 Representations of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$
5. 4 Rank $1$ case ; some relations in $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$
6. 5 Rank $1$ case ; finite dimensional simple modules of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$
7. 6 Finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules
8. 7 Finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules
9. A Some combinatorics

0. Introduction

0.1. The deformed current Lie algebra $\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ was introduced in [W] to study the representation theory of cyclotomic $q$-Schur algebras at $q=1$. In this paper, we introduce the deformed current Lie algebra $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ and $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ over $\mathbb{C}$ associated with the special linear Lie algebra $\mathfrak{sl}_{m}$ and general linear Lie algebra $\mathfrak{gl}_{m}$ respectively. $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) is a deformation of the current Lie algebra $\mathfrak{sl}_{m}[x]=\mathfrak{sl}_{m}\otimes_{\mathbb{C}}\mathbb{C}[x]$ (resp. $\mathfrak{gl}_{m}[x]=\mathfrak{gl}_{m}\otimes_{\mathbb{C}}\mathbb{C}[x]$) with deformation parameters $\mathbf{Q}=(Q_{1},Q_{2},\dots,Q_{m-1})\in\mathbb{C}^{m-1}$. Note that $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) is coincide with $\mathfrak{sl}_{m}[x]$ (resp. $\mathfrak{gl}_{m}[x]$) if $Q_{i}=0$ for all $i=1,2,\dots,m-1$. The Lie algebra $\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ introduced in [W] is isomorphic to $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ under a suitable choice of deformation parameters $\mathbf{Q}$ (Lemma 1.7).

0.2. The differences of the representation theory of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ from one of $\mathfrak{sl}_{m}[x]$ appear in the following two points. The deformed current Lie algebra $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ has a family of $1$-dimensional representations $\{\mathcal{L}^{\bm{\beta}}\mid\bm{\beta}\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}\}$, where

[TABLE]

although the $1$-dimensional representation of $\mathfrak{sl}_{m}[x]$ is only the trivial representation (Lemma 6.2). (We remark that $\mathcal{L}^{(0,\dots,0)}$ is the trivial representation of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$.)

The second difference appears in the evaluation modules. For each $\gamma\in\mathbb{C}$, we can consider the evaluation homomorphism $\mathbf{{ev}}_{\gamma}:U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])\rightarrow U(\mathfrak{sl}_{m})$ which is a deformation of the evaluation homomorphism for $\mathfrak{sl}_{m}[x]$ (see the paragraph 1.4 for the definition). Then we can consider the evaluation modules by regarding $U(\mathfrak{sl}_{m})$-modules as $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules through the evaluation homomorphism $\mathbf{{ev}}_{\gamma}$. The evaluation homomorphism $\mathbf{{ev}}_{\gamma}$ is surjective if $\gamma\not=Q_{i}^{-1}$ for all $i=1,2,\dots,m-1$ such that $Q_{i}\not=0$. However, $\mathbf{{ev}}_{\gamma}$ is not surjective if $\gamma=Q_{i}^{-1}$ for some $i=1,2,\dots,m-1$. Moreover, in general, the evaluation module of a simple $U(\mathfrak{sl}_{m})$-module at $\gamma\in\mathbb{C}$ is not simple if $\gamma=Q_{i}^{-1}$ for some $i=1,2,\dots,m-1$ (see Remark 5.9).

0.3. It is a purpose of this paper to classify the finite dimensional simple modules of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ and $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$. A classification of the finite dimensional simple modules for the original current Lie algebra is well-known (e.g. [C], [CP]). The classification for $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) is an analogue of the original case.

Since $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ has the triangular decomposition (Proposition 1.4), we can develop the usual highest weight theory (see §2). In particular, any finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to a highest weight module $\mathcal{L}(\mathbf{u})$ of highest weight $\mathbf{u}\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$ (Proposition 2.6). Then it is enough to determine the highest weights such that the corresponding simple highest weight modules are finite dimensional. We obtain a classification of such highest weights as follows. Let $\mathbb{C}[x]_{\operatorname{monic}}$ be the set of monic polynomials over $\mathbb{C}$ with the indeterminate variable $x$. For each $Q\in\mathbb{C}$, put

[TABLE]

We define the map $\prod_{i=1}^{m-1}(\mathbb{C}[x]_{\operatorname{monic}}^{\langle Q_{i}\rangle}\times\mathbb{B}^{\langle Q_{i}\rangle})\rightarrow\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$,

[TABLE]

by

[TABLE]

when $\varphi_{i}=(x-\gamma_{i,1})(x-\gamma_{i,2})\dots(x-\gamma_{i,n_{i}})$ ($1\leq i\leq m-1$). Then we have the following classification of finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules (Theorem 6.4).

**Theorem: **

$\{\mathcal{L}(\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta}))\mid(\bm{\varphi},\bm{\beta})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]_{\operatorname{monic}}^{\langle Q_{i}\rangle}\times\mathbb{B}^{\langle Q_{i}\rangle})\}$ gives a complete set of isomorphism classes of finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules.

We remark that $\mathcal{L}(\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta}))$ is isomorphic to a subquotient of

[TABLE]

where $\{\omega_{j}\mid 1\leq j\leq m-1\}$ is the set of fundamental weights for $\mathfrak{sl}_{m}$, $L(\omega_{j})$ ($1\leq j\leq m-1$) is the simple highest weight $U(\mathfrak{sl}_{m})$-module of highest weight $\omega_{j}$ and $L(\omega_{j})^{\mathbf{{ev}}_{\gamma_{j},k}}$ is the evaluation module of $L(\omega_{j})$ at $\gamma_{j,k}$.

We also see that any finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to a highest weight module $\mathcal{L}(\widetilde{\mathbf{u}})$ of highest weight $\widetilde{\mathbf{u}}\in\prod_{j=1}^{m}\prod_{t\geq 0}\mathbb{C}$ (Proposition 3.3). Note that $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ is a Lie subalgebra of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (Proposition 1.4 (iii)). The difference of representations of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ from one of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ is given by the family of $1$-dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules $\{\widetilde{\mathcal{L}}^{\mathbf{h}}\mid\mathbf{h}\in\prod_{t\geq 0}\mathbb{C}\}$. We remark that $\widetilde{\mathcal{L}}^{\mathbf{h}}$ ($\mathbf{h}\in\prod_{t\geq 0}\mathbb{C}$) is isomorphic to the trivial representation $\mathcal{L}^{(0,\dots,0)}$ as a $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module when we restrict the action. We obtain the classification of finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules as follows. We define the map $\prod_{i=1}^{m-1}(\mathbb{C}[x]_{\operatorname{monic}}^{\langle Q_{i}\rangle}\times\mathbb{B}^{\langle Q_{i}\rangle})\times\prod_{t\geq 0}\mathbb{C}\rightarrow\prod_{j=1}^{m}\prod_{t\geq 0}\mathbb{C}$,

[TABLE]

by

[TABLE]

where $\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta})_{k,t}$ is determined by (0.3.1). Then we have the following classification of finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules (Theorem 7.4).

**Theorem: **

$\{\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))\mid(\bm{\varphi},\bm{\beta},\mathbf{h})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]_{\operatorname{monic}}^{\langle Q_{i}\rangle}\times\mathbb{B}^{\langle Q_{i}\rangle})\times\prod_{t\geq 0}\mathbb{C}\}$ gives a complete set of isomorphism classes of finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules.

We remark that $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ is isomorphic to a subquotient of

[TABLE]

(See §7 for definitions of $L(\widetilde{\omega}_{j})^{\widetilde{\mathbf{{ev}}}_{\gamma_{j},k}}$, $\widetilde{\mathcal{L}}^{\bm{\beta}}$ and $\widetilde{\mathcal{L}}^{\mathbf{h}}$.) We also remark that

[TABLE]

as $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules when we restrict the action.

Acknowledgements: This work was supported by JSPS KAKENHI Grant Number JP16K17565.

1. Deformed current Lie algebras $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ and $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$

In this section, we give a definition of deformed current Lie algebras $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ and $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$, and also give some basic facts. The definition of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ in this section is different from one of $\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ given in [W]. The relation between $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ and $\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ is given in Lemma 1.7.

Definition 1.1**.**

Put $\mathbf{Q}=(Q_{1},Q_{2},\dots,Q_{m-1})\in\mathbb{C}^{m-1}$. We define the Lie algebra $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ over $\mathbb{C}$ by the following generators and defining relations:

**Generators: **

$\mathcal{X}_{i,t}^{\pm}$, $\mathcal{J}_{i,t}$ ($1\leq i\leq m-1$, $t\geq 0$).

**Relations: **

[TABLE]

where we put $a_{ji}=\begin{cases}2&\text{ if }j=i,\\ -1&\text{ if }j=i\pm 1,\\ 0&\text{ otherwise}.\end{cases}$

We also define the Lie algebra $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ over $\mathbb{C}$ by the following generators and defining relations:

**Generators: **

$\mathcal{X}_{i,t}^{\pm}$* ($1\leq i\leq m-1$, $t\geq 0$), $\mathcal{I}_{j,t}$ ($1\leq j\leq m$, $t\geq 0$).*

**Relations: **

[TABLE]

together with the relations (L4)-(L6) in the above. In the relation (L’2), we put $a^{\prime}_{ji}=\begin{cases}1&\text{ if }j=i,\\ -1&\text{ if }j=i+1,\\ 0&\text{ otherwise}.\end{cases}$

1.2. We call $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) the deformed current Lie algebra associated with the special linear Lie algebra $\mathfrak{sl}_{m}$ (resp. the general linear Lie algebra $\mathfrak{gl}_{m}$). If $Q_{i}=0$ for all $i=1,2,\dots,m-1$, then $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) coincides with the current Lie algebra $\mathfrak{sl}_{m}[x]=\mathfrak{sl}_{m}\otimes_{\mathbb{C}}\mathbb{C}[x]$ (resp. $\mathfrak{gl}_{m}[x]=\mathfrak{gl}_{m}\otimes_{\mathbb{C}}\mathbb{C}[x]$) associated with $\mathfrak{sl}_{m}$ (resp. $\mathfrak{gl}_{m}$). We can also regard $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) as a filtered deformation of $\mathfrak{sl}_{m}[x]$ (resp. $\mathfrak{gl}_{m}[x]$) in a similar way as in [W, Proposition 2.13].

1.3. For $1\leq i\not=j\leq m$ and $t\geq 0$, we define an element $\mathcal{E}_{i,j;t}\in\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathcal{E}_{i,j;t}\in\mathfrak{gl}_{n}^{\langle\mathbf{Q}\rangle}[x]$) by

[TABLE]

In particular, we have $\mathcal{E}_{i,i+1;t}=\mathcal{X}_{i,t}^{+}$ and $\mathcal{E}_{i+1,i;t}=\mathcal{X}_{i,t}^{-}$.

Let $\mathbf{n}^{+}$ and $\mathbf{n}^{-}$ be the Lie subalgebra of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (also of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) generated by

[TABLE]

respectively. Let $\mathbf{n}^{0}$ (resp. $\widetilde{\mathbf{n}}^{0}$) be the Lie subalgebra of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ (resp. $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$) generated by

[TABLE]

By the relation (L1) (resp. (L’1)), we see that $\mathbf{n}^{0}$ (resp. $\widetilde{\mathbf{n}}^{0}$) is a commutative Lie subalgebra of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}}[x]$ (resp $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$).

Proposition 1.4**.**

1. (i)

$\{\mathcal{E}_{i,j;t}\,|\,1\leq i\not=j\leq m,\,t\geq 0\}\cup\{\mathcal{J}_{i,t}\,|\,1\leq i\leq m-1,\,t\geq 0\}$* gives a basis of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$.* 2. (ii)

$\{\mathcal{E}_{i,j;t}\,|\,1\leq i\not=j\leq m,\,t\geq 0\}\cup\{\mathcal{I}_{j,t}\,|\,1\leq j\leq m,\,t\geq 0\}$* gives a basis of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$.* 3. (iii)

There exists an injective homomorphism of Lie algebras

[TABLE] 4. (iv)

We have the triangular decomposition

[TABLE]

In particular,

[TABLE]

gives a basis of $\mathbf{n}^{+}$ (resp. $\mathbf{n}^{-}$), and

[TABLE]

gives a basis of $\mathbf{n}^{0}$ (resp. $\widetilde{\mathbf{n}}^{0}$).

Proof.

(i) and (ii) are proven in a similar way as in the proof of [W, Proposition 2.6]. By checking the defining relations, we see that $\Upsilon$ is well-defined. We also see that $\Upsilon$ is injective by investigating the basis given in (i) and (ii) under the homomorphism $\Upsilon$. Then we have (iii). (iv) folloes from (i) and (ii). ∎

1.5. Evaluation homomorphisms and evaluation modules.

The general linear Lie algebra $\mathfrak{gl}_{m}$ is a Lie algebra over $\mathbb{C}$ generated by $e_{i},f_{i}$ ($1\leq i\leq m-1$) and $K_{j}$ ($1\leq j\leq m$) together with the following defining relations:

[TABLE]

The special linear Lie algebra $\mathfrak{sl}_{m}$ is a Lie subalgebra of $\mathfrak{gl}_{m}$ generated by $e_{i},f_{i},H_{i}$ ($1\leq i\leq m-1$).

For each $\gamma\in\mathbb{C}$, by checking the defining relations, we have the homomorphisms of algebras (evaluation homomorphism)

[TABLE]

and

[TABLE]

Clearly, the homomorphism $\mathbf{{ev}}_{\gamma}$ (resp. $\widetilde{\mathbf{{ev}}}_{\gamma}$) is surjective if $\gamma\not=Q_{i}^{-1}$ for all $i=1,\dots,m-1$ such that $Q_{i}\not=0$.

For a $U(\mathfrak{sl}_{m})$-module $M$ (resp. a $U(\mathfrak{gl}_{m})$-module $M$), we can regard $M$ as a $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module (resp. a $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module) through the evaluation homomorphism $\mathbf{{ev}}_{\gamma}$ (resp. $\widetilde{\mathbf{{ev}}}_{\gamma}$). We call it the evaluation module, and denote it by $M^{\mathbf{{ev}}_{\gamma}}$ (resp. $M^{\widetilde{\mathbf{{ev}}}_{\gamma}}$).

1.6. In the rest of this section, we give a relation with the Lie algebra $\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ introduced in [W, Definition 2.2].

Let $\mathbf{m}=(m_{1},\dots,m_{r})$ be an $r$-tuple of positive integers such that $\sum_{k=1}^{r}m_{k}=m$. Put $\varGamma(\mathbf{m})=\{(i,k)\,|\,1\leq i\leq m_{k},\,1\leq k\leq r\}$ and $\varGamma^{\prime}(\mathbf{m})=\varGamma(\mathbf{m})\setminus\{(m_{r},r)\}$. Then we have the bijective map

[TABLE]

For $(i,k)\in\varGamma(\mathbf{m})$ and $j\in\mathbb{Z}$ such that $1\leq\zeta((i,k))+j\leq m$, put $(i+j,k)=\zeta^{-1}(\zeta((i,k))+j)$. For $(i,k)\in\varGamma^{\prime}(\mathbf{m})$ and $(j,l)\in\varGamma(\mathbf{m})$, put $a^{\prime}_{(j,l)(i,k)}=a^{\prime}_{\zeta((j,l))\zeta((i,k))}$. Take $\widehat{\mathbf{Q}}=(\widehat{Q}_{1},\dots,\widehat{Q}_{r-1})\in\mathbb{C}^{r-1}$. Then the Lie algebra $\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ in [W, Definition 2.2] is defined by the generators $\mathcal{X}_{(i,k),t}^{\pm}$, $\mathcal{I}_{(j,l),t}$ ($(i,k)\in\varGamma^{\prime}(\mathbf{m})$, $(j,l)\in\varGamma(\mathbf{m})$, $t\geq 0$) together with the following defining relations:

[TABLE]

where we put $\mathcal{J}_{(i,k),t}=\mathcal{I}_{(i,k),t}-\mathcal{I}_{(i+1,k),t}$. Then we have the following isomorphism between $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ and $\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ under the suitable choice of the deformation parameters $\mathbf{Q}$.

Lemma 1.7**.**

Assume that $\widehat{Q}_{i}\not=0$ for all $i=1,2,\dots,r-1$. We take $\mathbf{Q}=(Q_{1},Q_{2},\dots,Q_{m-1})\in\mathbb{C}^{m-1}$ as

[TABLE]

Then we have the isomorphism of Lie algebras $\Phi:\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]\rightarrow\mathfrak{g}_{\widehat{\mathbf{Q}}}(\mathbf{m})$ such taht

[TABLE]

Proof.

We see the well-definedness of $\Phi$ by checking the defining relations. The inverse homomorphism of $\Phi$ is given by

[TABLE]

∎

2. Representations of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$

In this section, we give some fundamental results for finite dimensional $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules by using the standard argument.

2.1. Put $\mathfrak{h}=\bigoplus_{i=1}^{m-1}\mathbb{C}\mathcal{J}_{i,0}\subset\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$, then $\mathfrak{h}$ is a commutative Lie subalgebra of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$. (Note that, if $Q_{i}=0$ for all $i=1,\dots,m-1$, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{sl}_{m}$.) Let $\mathfrak{h}^{\ast}$ be the dual space of $\mathfrak{h}$. For each $i=1,2,\dots,m-1$, we take $\alpha_{i}\in\mathfrak{h}^{\ast}$ as $\alpha_{i}(\mathcal{J}_{j,0})=a_{ji}$ for $j=1,\dots,m-1$. Put $Q^{+}=\bigoplus_{i=1}^{m-1}\mathbb{Z}_{\geq 0}\alpha_{i}\subset\mathfrak{h}^{\ast}$. We define the partial order on $\mathfrak{h}^{\ast}$ by $\lambda\geq\mu$ if $\lambda-\mu\in Q^{+}$ for $\lambda,\mu\in\mathfrak{h}$.

2.2. For $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-mdoule $M$, we consider the decomposition $M=\bigoplus_{\lambda\in\mathfrak{h}^{\ast}}\widetilde{M}_{\lambda}$, where $\widetilde{M}_{\lambda}=\{x\in M\,|\,(h-\lambda(h))^{N}\cdot x=0\text{ for }h\in\mathfrak{h}\text{ and }N\gg 0\}$, namely $M=\bigoplus_{\lambda\in\mathfrak{h}^{\ast}}\widetilde{M}_{\lambda}$ is the decomposition to the generalized simultaneous eigenspaces for the action of $\mathfrak{h}$. By the relation (L2), we have

[TABLE]

Thus, if $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $M\not=0$ is finite dimensional, there exists $\lambda\in\mathfrak{h}^{\ast}$ such that $\widetilde{M}_{\lambda}\not=0$ and $\mathcal{X}_{i,t}^{+}\cdot\widetilde{M}_{\lambda}=0$ for all $i=1,2,\dots,m-1$ and $t\geq 0$. On the other hand, $\widetilde{M}_{\lambda}$ ($\lambda\in\mathfrak{h}^{\ast}$) is closed under the action of $\mathbf{n}^{0}$ by the relation (L1). Thus, we can take a simultaneous eigenvector $v\in\widetilde{M}_{\lambda}$ for the action of $\mathbf{n}^{0}$. Then we have the following lemma.

Lemma 2.3**.**

For a finite dimensional $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $M\not=0$, there exists $v_{0}\in M$ ($v_{0}\not=0$) satisfying the following conditions:

1. (i)

$\mathcal{X}_{i,t}^{+}\cdot v_{0}=0$* for all $i=1,\dots,m-1$ and $t\geq 0$,* 2. (ii)

$\mathcal{J}_{i,t}\cdot v_{0}=u_{i,t}v_{0}$* ($u_{i,t}\in\mathbb{C}$) for each $i=1,\dots,m-1$ and $t\geq 0$.*

Moreover, if $M$ is simple, we have $M=U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])\cdot v_{0}$.

2.4. Highest weight modules. For $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $M$, we say that $M$ is a highest weight module if there exists $v_{0}\in M$ satisfying the following conditions:

1. (i)

$M$ is generated by $v_{0}$ as a $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module. 2. (ii)

$\mathcal{X}_{i,t}^{+}\cdot v_{0}=0$ for all $i=1,\dots,m-1$ and $t\geq 0$. 3. (iii)

$\mathcal{J}_{i,t}\cdot v_{0}=u_{i,t}v_{0}$ ($u_{i,t}\in\mathbb{C}$) for each $i=1,\dots,m-1$ and $t\geq 0$.

In this case, we say that $(u_{i,t})_{1\leq i\leq m-1,t\geq 0}\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$ is the highest weight of $M$, and that $v_{0}$ is a highest weight vector of $M$.

Let $M$ be a highest weight $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module with a highest weight $\mathbf{u}=(u_{i,t})_{1\leq i\leq m-1,t\geq 0}\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$ and a highest weight vector $v_{0}\in M$. Thanks to the triangular decomposition (Proposition 1.4 (iv)) together with the above conditions, we have $M=U(\mathbf{n}^{-})\cdot v_{0}$. Let $\lambda_{\mathbf{u}}\in\mathfrak{h}^{\ast}$ be as $\lambda_{\mathbf{u}}(\mathcal{J}_{i,0})=u_{i,0}$ for $i=1,\dots,m-1$. By $M=U(\mathbf{n}^{-})\cdot v_{0}$ and the relation (L2), we have the weight space decomposition

[TABLE]

and we also have $\dim_{\mathbb{C}}M_{\lambda_{\mathbf{u}}}=1$.

2.5. Verma modules. For $\mathbf{u}=(u_{i,t})\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$, let $\mathfrak{I}(\mathbf{u})$ be the left ideal of $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$ generated by $\mathcal{X}_{i,t}^{+}$ ($1\leq i\leq m-1$, $t\geq 0$) and $\mathcal{J}_{i,t}-u_{i,t}$ ($1\leq i\leq m-1$, $t\geq 0$). We define the Verma module $\mathcal{M}(\mathbf{u})=U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])/\mathfrak{I}(\mathbf{u})$. Then $\mathcal{M}(\mathbf{u})$ is a highest weight module of highest weight $\mathbf{u}$, and any highest weight module of highest weight $\mathbf{u}$ is realized as a quotient of the Verma module $\mathcal{M}(\mathbf{u})$. By the weight space decomposition (2.4.1), we see that $\mathcal{M}(\mathbf{u})$ has the unique maximal proper submodule $\operatorname{rad}\mathcal{M}(\mathbf{u})$. Put $\mathcal{L}(\mathbf{u})=\mathcal{M}(\mathbf{u})/\operatorname{rad}\mathcal{M}(\mathbf{u})$, then we have the following proposition.

Proposition 2.6**.**

For $\mathbf{u}=(u_{i,t})\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$, a highest weight simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module of highest weight $\mathbf{u}$ is isomorphic to $\mathcal{L}(\mathbf{u})$. Moreover, any finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to $\mathcal{L}(\mathbf{u})$ for some $\mathbf{u}=(u_{i,t})\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$.

Proof.

By Lemma 2.3, a finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is a highest weight module. Then we have the proposition by the above arguments. ∎

3. Representations of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$

For finite dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules, we can develop a similar argument as in the case of $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$ discussed in the previous section. In this section, we give only some notation for $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules.

3.1. Highest weight modules. For $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $M$, we say that $M$ is a highest weight module if there exists $v_{0}\in M$ satisfying the following conditions:

1. (i)

$M$ is generated by $v_{0}$ as a $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module. 2. (ii)

$\mathcal{X}_{i,t}^{+}\cdot v_{0}=0$ for all $i=1,\dots,m-1$ and $t\geq 0$. 3. (iii)

$\mathcal{I}_{j,t}\cdot v_{0}=\widetilde{u}_{j,t}v_{0}$ ($\widetilde{u}_{j,t}\in\mathbb{C}$) for each $j=1,\dots,m$ and $t\geq 0$.

In this case, we say that $(\widetilde{u}_{j,t})_{1\leq j\leq m,t\geq 0}\in\prod_{j=1}^{m}\prod_{t\geq 0}\mathbb{C}$ is the highest weight of $M$, and that $v_{0}$ is a highest weight vector of $M$.

3.2. Verma modules. For $\widetilde{\mathbf{u}}=(\widetilde{u}_{j,t})\in\prod_{j=1}^{m}\prod_{t\geq 0}\mathbb{C}$, let $\mathfrak{I}(\widetilde{\mathbf{u}})$ be the left ideal of $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$ generated by $\mathcal{X}_{i,t}^{+}$ ($1\leq i\leq m-1$, $t\geq 0$) and $\mathcal{I}_{j,t}-\widetilde{u}_{j,t}$ ($1\leq j\leq m$, $t\geq 0$). We define the Verma module $\mathcal{M}(\widetilde{\mathbf{u}})=U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])/\mathfrak{I}(\widetilde{\mathbf{u}})$. Then $\mathcal{M}(\widetilde{\mathbf{u}})$ is a highest weight module of highest weight $\widetilde{\mathbf{u}}$, and any highest weight module of highest weight $\widetilde{\mathbf{u}}$ is realized as a quotient of the Verma module $\mathcal{M}(\widetilde{\mathbf{u}})$. $\mathcal{M}(\widetilde{\mathbf{u}})$ has the unique maximal proper submodule $\operatorname{rad}\mathcal{M}(\widetilde{\mathbf{u}})$. Put $\mathcal{L}(\widetilde{\mathbf{u}})=\mathcal{M}(\widetilde{\mathbf{u}})/\operatorname{rad}\mathcal{M}(\widetilde{\mathbf{u}})$, then we have the following proposition.

Proposition 3.3**.**

For $\widetilde{\mathbf{u}}=(\widetilde{u}_{j,t})\in\prod_{j=1}^{m}\prod_{t\geq 0}\mathbb{C}$, a highest weight simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module of highest weight $\widetilde{\mathbf{u}}$ is isomorphic to $\mathcal{L}(\widetilde{\mathbf{u}})$. Moreover, any finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to $\mathcal{L}(\widetilde{\mathbf{u}})$ for some $\widetilde{\mathbf{u}}=(u_{j,t})\in\prod_{j=1}^{m}\prod_{t\geq 0}\mathbb{C}$.

4. Rank $1$ case ; some relations in $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$

4.1. Take $Q\in\mathbb{C}$, then $\mathfrak{sl}_{2}^{\langle Q\rangle}[x]$ is a Lie algebra over $\mathbb{C}$ generated by $\mathcal{X}_{t}^{\pm}$ and $\mathcal{J}_{t}$ ($t\in\mathbb{Z}_{\geq 0}$) together with the following defining relations:

[TABLE]

(In the rank $1$ case, we omit the first index of the generators since it is trivial.) By checking the defining relations, we see that there exists the algebra anti-automorphism ${\dagger}:U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])\rightarrow U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$ such that

[TABLE]

Clearly, ${\dagger}^{2}$ is the identity on $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$.

4.2. For $t,b\in\mathbb{Z}_{\geq 0}$, we define an element $\mathcal{X}_{t}^{+(b)}$ (resp. $\mathcal{X}_{t}^{-(b)}$) of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$ by

[TABLE]

For convenience, we put $\mathcal{X}_{t}^{\pm(b)}=0$ for $b\in\mathbb{Z}_{<0}$.

For $t,p,h\in\mathbb{Z}_{\geq 0}$, we define an element $\mathcal{X}_{t}^{+((p);h)}$ (resp. $\mathcal{X}_{t}^{-((p);h)}$) of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$ by

[TABLE]

Clearly, we have ${\dagger}(\mathcal{X}_{t}^{+((p);h)})=\mathcal{X}_{t}^{-((p);h)}$. For examples, we have

[TABLE]

For $s,p\in\mathbb{Z}_{\geq 0}$, we define an element $\mathcal{J}_{s}^{\langle p\rangle}$ of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x]))$ inductively on $p$ by

[TABLE]

For examples, we have

[TABLE]

Lemma 4.3**.**

For $s,t,p\in\mathbb{Z}_{\geq 0}$, we have the following relations in $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$.

1. (i)

$\displaystyle[\mathcal{J}_{s}^{\langle p\rangle},\mathcal{X}_{t}^{+}]=\sum_{z=1}^{p}(-1)^{z+1}(z+1)\mathcal{J}_{s}^{\langle p-z\rangle}\mathcal{X}_{t}^{+((z);s)}$. 2. (ii)

$\displaystyle[\mathcal{J}_{s}^{\langle p\rangle},\mathcal{X}_{t}^{-}]=-\sum_{z=1}^{p}(-1)^{z+1}(z+1)\mathcal{X}_{t}^{-((z);s)}\mathcal{J}_{s}^{\langle p-z\rangle}$.

Proof.

(ii) follows from (i) by applying the algebra anti-automorphism ${\dagger}$ defined in (4.1.1). Then, we prove only (i) by the induction on $p$.

If $p=0$, (i) is clear. If $p>0$, by the definition (4.2.2), we have

[TABLE]

By the assumption of the induction, we have

[TABLE]

Applying the assumption of the induction again, we have

[TABLE]

By the definition (4.2.2), we have

[TABLE]

By the definition (4.2.1), we have

[TABLE]

Put

[TABLE]

By the definition (4.2.1), we also have

[TABLE]

Put $z^{\prime}=z+k$ and $w^{\prime}=w+l$, we have

[TABLE]

By the induction on $k$, we can show that

[TABLE]

Then, we have

[TABLE]

and by the definition of (4.2.1), we have

[TABLE]

By the definition (4.2.2), we have

[TABLE]

Combining (4.3.1) with (4.3.2), (4.3.3), (4.3.5) and (4.3.6), we have

[TABLE]

∎

Lemma 4.4**.**

For $s,t,h\in\mathbb{Z}_{\geq 0}$ and $p\in\mathbb{Z}_{>0}$, we have

[TABLE]

Proof.

By the definitions (4.2.1), (4.2.2) and the defining relation (L3), we have

[TABLE]

∎

Lemma 4.5**.**

For $s,t,h\in\mathbb{Z}_{\geq 0}$ and $p\in\mathbb{Z}_{>0}$, we have the following relations.

1. (i)

$\displaystyle[\mathcal{J}_{s}^{\langle 1\rangle},\mathcal{X}_{t}^{+((p);h)}]=2\mathcal{X}_{s+t-h}^{+((p+1);h)}$. 2. (ii)

$\displaystyle[\mathcal{J}_{s}^{\langle 1\rangle},\mathcal{X}_{t}^{-((p);h)}]=-2\mathcal{X}_{s+t-h}^{-((p+1);h)}$.

Proof.

(ii) follows from (i) by applying the algebra anti-automorphism ${\dagger}$ defined in (4.1.1). Then, we prove (i).

By the definition (4.2.1), we have

[TABLE]

Applying Lemma 4.3 (i), we have

[TABLE]

Then, by the definition (4.2.1) again, we have

[TABLE]

On the other hand, we have

[TABLE]

Thus, we have (i). ∎

Lemma 4.6**.**

For $s,t,c\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Proof.

We prove the lemma by the induction on $c$. If $c=0$, it is clear. If $c=1$, it is the defining relation (L3). If $c>1$, by the assumption of the induction, we have

[TABLE]

Then, by the defining relations (L3), (L4) and Lemma 4.3 (ii), we have

[TABLE]

∎

4.7. A partition $\lambda=(\lambda_{1},\lambda_{2},\dots)$ is a non-increasing sequence of non-negative integers which has only finitely many non-zero terms. The size of a partition $\lambda$ is the sum of all terms of $\lambda$, and we denote it by $|\lambda|$. Namely, we have $|\lambda|=\sum_{i\geq 1}\lambda_{i}$. If $|\lambda|=n$, we say that $\lambda$ is a partition of $n$, and we denote it by $\lambda\vdash n$. The length of $\lambda$ is the maximal $i$ such that $\lambda_{i}\not=0$, and we denote the length of $\lambda$ by $\ell(\lambda)$. For a partition $\lambda=(\lambda_{1},\lambda_{2},\dots)$, let $m_{j}(\lambda)$ ($j\in\mathbb{Z}_{>0}$) be the multiplicity of $j$ in $\lambda$. Then, for a partition $\lambda$ and $t,h\in\mathbb{Z}_{\geq 0}$, we define an element $\mathcal{X}_{t}^{+(\lambda;h)}$ (resp. $\mathcal{X}_{t}^{-(\lambda;h)}$) of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$ by

[TABLE]

where we note the defining relation (L4). Clearly, we have ${\dagger}(\mathcal{X}_{t}^{+(\lambda;h)})=\mathcal{X}_{t}^{-(\lambda;h)}$. For examples, we have

[TABLE]

For $t,h,k,b,p\in\mathbb{Z}_{\geq 0}$, we define an element $\mathcal{X}_{t}^{+(b;p|k;h)}$ (resp. $\mathcal{X}_{t}^{-(b;p|k;h)}$) of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$ by

[TABLE]

Note the defining relation (L4), we see that ${\dagger}(\mathcal{X}_{t}^{+(b;p|k;h)})=\mathcal{X}_{t}^{-(b;p|k;h)}$. For examples, we have

[TABLE]

For the element $\mathcal{X}_{t}^{\pm(b;p|k;h)}\in U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$, we prepare the following technical formulas.

Lemma 4.8**.**

For $t,h,k,b,p\in\mathbb{Z}_{\geq 0}$, we have the following equations for the element $\mathcal{X}_{t}^{\pm(b;p|k;h)}$ of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$.

1. (i)

If $b-p<0$, we have $\mathcal{X}_{t}^{\pm(b;p|k;h)}=0$. 2. (ii)

*If $k=0$, we have $\mathcal{X}_{t}^{\pm(b;p|0;h)}=\mathcal{X}_{t}^{\pm(b-p)}$. *

If $k=1$, we have $\mathcal{X}_{t}^{\pm(b;p|1;h)}=\mathcal{X}_{t}^{\pm((1);h)}\mathcal{X}_{t}^{\pm(b-p-1)}$. 3. (iii)

If $p=b$, we have $\mathcal{X}_{t}^{\pm(b;b|k;h)}=\begin{cases}1&\text{ if }k=0,\\ 0&\text{ if }k\not=0.\end{cases}$ 4. (iv)

If $b,p>0$, we have $\mathcal{X}_{t}^{\pm(b;p|k;h)}=\mathcal{X}_{t}^{\pm(b-1;p-1|k;h)}$. 5. (v)

If $b,k>0$, we have

[TABLE] 6. (vi)

If $b>0$, we have

[TABLE]

Proof.

(i), (ii), (iii) and (iv) are clear from definitions.

We prove (v). Note that $\sum_{z\geq 1}zm_{z}(\lambda)=k$ for a partition $\lambda$ of $k$. Then, by the definition (4.7.2), we have

[TABLE]

On the other hand, by the definition (4.7.1), we have

[TABLE]

for each $z$ such that $m_{z}(\lambda)\not=0$. Thus, we have

[TABLE]

We prove (vi). By the definition (4.7.2), we have

[TABLE]

Note that $\ell(\lambda)=\sum_{z\geq 1}m_{z}(\lambda)$, $(b-p-\ell(\lambda))\mathcal{X}_{t}^{\pm(b-p-\ell(\lambda))}=\mathcal{X}_{t}^{\pm}\mathcal{X}_{t}^{\pm(b-p-\ell(\lambda)-1)}$ and the defining relation (L4), we have

[TABLE]

In a similar argument as in the proof of (v), we have

[TABLE]

Then, by (v) and the definition (4.7.2), we have

[TABLE]

∎

Lemma 4.9**.**

For $s,t,c,p,k\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Proof.

If $c=0$, the equation (4.9.1) follows from Lemma 4.8 (i) and (iii). Then, we prove (4.9.1) by the induction on $k$ in the case where $c>0$.

If $k=0$, we see that (4.9.1) is just the formula in Lemma 4.6 by Lemma 4.8 (ii).

If $k>0$, by Lemma 4.8 (v) and the defining relation (L4), we have

[TABLE]

Applying the assumption of the induction, we have

[TABLE]

Applying Lemma 4.4 and Lemma 4.5 (ii), we have

[TABLE]

Put

[TABLE]

then we have

[TABLE]

By Lemma 4.8 (v) together with (L4), we have

[TABLE]

Put $z^{\prime}=k-z$ in $(*2)$ and apply Lemma 4.8 (iv), we have

[TABLE]

Put $h=z+y$ in $(*3)$ , we have

[TABLE]

Applying Lemma 4.8 (v) together with (L4), we have

[TABLE]

By (4.9.4) and (4.9.5), we have

[TABLE]

We also have

[TABLE]

Put $h=k-y+1$, we have

[TABLE]

Put

[TABLE]

By (4.2.1), we have

[TABLE]

Put $y^{\prime}=w+y$, we have

[TABLE]

Note that $\sum_{w=\max\{0,y^{\prime}-(z+1)\}}^{\min\{h-(z+1),y^{\prime}\}}\begin{pmatrix}h-(z+1)\\ w\end{pmatrix}\begin{pmatrix}z+1\\ y^{\prime}-w\end{pmatrix}=\begin{pmatrix}h\\ y^{\prime}\end{pmatrix}$ by (4.3.4), we have

[TABLE]

(Use (4.2.1) again.) Then, we have

[TABLE]

Then we have

[TABLE]

where the last equation follows from Lemma 4.8 (v).

By (4.9.2), (4.9.3), (4.9.6) and (4.9.7), we have

[TABLE]

∎

Proposition 4.10**.**

For $s,t,b,c\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Proof.

We prove (4.10.1) by the induction on $b$. If $b=1$, (4.10.1) follows from Lemma 4.6 together with Lemma 4.8.

If $b>1$, we have

[TABLE]

by the assumption of the induction. Applying Lemma 4.6 and Lemma 4.9, we have

[TABLE]

On the other hand, by Lemma 4.3, we have

[TABLE]

Put

[TABLE]

then we have

[TABLE]

where we note that $\mathcal{X}_{t}^{+(b-1;p|l;s+t)}=0$ if $p=b$ by Lemma 4.8 (i).

By Lemma 4.8 (ii) together with (L4), we have

[TABLE]

Put $l^{\prime}=l+z$ in $(*3)$, we have

[TABLE]

Then we have

[TABLE]

where we note that $\sum_{z=1}^{0}\mathcal{X}_{t}^{+((z);s+t)}\mathcal{X}_{t}^{+(b-1;p|l-z;s+t)}=0$. Applying Lemma 4.8 (vi), we have

[TABLE]

Put $p^{\prime}=p+1$ in $(*4)$, we have

[TABLE]

where we note that $\mathcal{X}_{t}^{+(b-1;p^{\prime}-1|l;s+t)}=0$ if $p^{\prime}=b+1$, and $\mathcal{X}_{s}^{-(c;p^{\prime}|z;s+t)}=0$ if $p^{\prime}=c+1$ by Lemma 4.8 (i). Note that

[TABLE]

we have

[TABLE]

Put $k^{\prime}=k-z+1$, we have

[TABLE]

Since $\mathcal{J}_{k^{\prime}(s+t)+w}^{\langle 1\rangle}=\mathcal{J}_{k^{\prime}(s+t)+w}+(-Q)\mathcal{J}_{k^{\prime}(s+t)+w+1}$, we see that

[TABLE]

Thus we have

[TABLE]

where the last equation follows from (4.2.2). Then we have

[TABLE]

Applying Lemma 4.8 (iv), we have

[TABLE]

Put $p^{\prime}=p+1$ in $(*5)$, we have

[TABLE]

where we note that $\mathcal{X}_{t}^{+(b-1;p^{\prime}-1|l;s+t)}=0$ if $p^{\prime}=b+1$, and $\mathcal{X}_{s}^{-(c;p^{\prime}|k+1;s+t)}=0$ if $p^{\prime}=c+1$ by Lemma 4.8 (i). Put $k^{\prime}=k+1$, we have

[TABLE]

Applying Lemma 4.8 (iv), we have

[TABLE]

By (4.10.2), (4.10.3), (4.10.4), (4.10.5) and (4.10.6), we have

[TABLE]

∎

5. Rank $1$ case ; finite dimensional simple modules of $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$

In this section, we classify the finite dimensional simple $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-modules.

5.1. $1$-dimensional representations.

First, we consider $1$-dimensional representations of $\mathfrak{sl}_{2}^{\langle Q\rangle}[x]$. Let $L=\mathbb{C}v$ be a $1$-dimensional $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module with a basis $\{v\}$, then $\mathcal{J}_{t}$ ($t\geq 0$) acts on $v$ as a scalar multiplication. If $\mathcal{X}_{t}^{+}\cdot v\not=0$ (resp. $\mathcal{X}_{t}^{-}\cdot v\not=0$), then $\mathcal{X}_{t}^{+}\cdot v$ (resp. $\mathcal{X}_{t}^{-}\cdot v$) is an eigenvector for the action of $\mathcal{J}_{0}$ whose eigenvalue is different from one of $v$ by the defining relation (L2). This is a contradiction since $L$ is $1$-dimensional. Thus, we have $\mathcal{X}_{t}^{\pm}\cdot v=0$ for $t\geq 0$. Moreover, by the defining relation (L3), we have $(\mathcal{J}_{t}-Q\mathcal{J}_{t+1})\cdot v=(\mathcal{X}_{t}^{+}\mathcal{X}_{0}^{-}-\mathcal{X}_{0}^{-}\mathcal{X}_{t}^{+})\cdot v=0$. This implies that $\mathcal{J}_{t}\cdot v=0$ for $t\geq 0$ if $Q=0$, and that $\mathcal{J}_{t}\cdot v=Q^{-t}\mathcal{J}_{0}\cdot v$ for $t>0$ if $Q\not=0$.

We define the set $\mathbb{B}^{\langle Q\rangle}$ by

[TABLE]

For each $\beta\in\mathbb{B}^{\langle Q\rangle}$, we can define the $1$-dimensional $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module $\mathcal{L}^{\beta}=\mathbb{C}v_{0}$ such that

[TABLE]

by checking the defining relations of $\mathfrak{sl}_{2}^{\langle Q\rangle}[x]$. Note that $\mathcal{L}^{0}$ is the trivial representation. Now we obtain the following lemma.

Lemma 5.2**.**

Any $1$-dimensional $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module is isomorphic to $\mathcal{L}^{\beta}$ for some $\beta\in\mathbb{B}^{\langle Q\rangle}$.

5.3. Recall from §2, a finite dimensional simple $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module is isomorphic to a simple highest weight module $\mathcal{L}(\mathbf{u})$ for some highest weight $\mathbf{u}=(u_{t})\in\prod_{t\geq 0}\mathbb{C}$ (Proposition 2.6), where we omit the first index for the highest weight. Then, in order to classify the finite dimensional simple $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module, it is enough to classify the highest weight $\mathbf{u}$ such that $\mathcal{L}(\mathbf{u})$ is finite dimensional.

In order to obtain a necessary condition for $\mathbf{u}$ such that $\mathcal{L}(\mathbf{u})$ is finite dimensional, we prepare the following lemma.

Lemma 5.4**.**

Let $M$ be a finite dimensional $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module. Take an element $v\in M$ satisfying

[TABLE]

for some $u_{t}\in\mathbb{C}$ ($t\in\mathbb{Z}_{\geq 0}$) and $n\in\mathbb{Z}_{\geq 0}$. (In fact, a such element exists by Lemma 2.3.) Then, for $s,t\in\mathbb{Z}_{\geq 0}$, we have

[TABLE]

Proof.

By the assumption $\mathcal{X}_{0}^{-(n+1)}\cdot v=0$ and Proposition 4.10, we have

[TABLE]

By the definition, we have $\mathcal{X}_{s}^{+(n;p|l;s)}=\sum_{\lambda\vdash l}\mathcal{X}_{s}^{+(\lambda;s)}\mathcal{X}_{s}^{+(n-p-\ell(\lambda))}$. Thus, by the definition of $\mathcal{X}_{s}^{+(\lambda;s)}$ and the assumption $\mathcal{X}_{t}^{+}\cdot v=0$ ($t\geq 0$), we have

[TABLE]

Then (5.4.1) implies that

[TABLE]

By the definition, we have

[TABLE]

Thus, we have

[TABLE]

By multiplying $\mathcal{X}_{t}^{+}$ from left to this equation, we have

[TABLE]

where we use Lemma 4.4 and the fact $\mathcal{X}_{t}^{+}\mathcal{J}_{s}^{\langle n-k\rangle}\cdot v=0$. This implies the Lemma. ∎

This Lemma implies the following proposition which gives a necessary condition for $\mathbf{u}$ such that $\mathcal{L}(\mathbf{u})$ is finite dimensional.

Proposition 5.5**.**

Let $M$ be a finite dimensional $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module. Take an element $v\in M$ satisfying

[TABLE]

for some $u_{t}\in\mathbb{C}$ ($t\in\mathbb{Z}_{\geq 0}$) and $n\in\mathbb{Z}_{\geq 0}$.

1. (i)

If $Q=0$, we have $u_{0}=n$, and there exist $\gamma_{1},\gamma_{2},\dots,\gamma_{n}\in\mathbb{C}$ such that

[TABLE]

where $p_{t}(\gamma_{1},\dots,\gamma_{n})=\gamma_{1}^{t}+\gamma_{2}^{t}+\dots+\gamma_{n}^{t}$. 2. (ii)

If $Q\not=0$, there exist $\beta,\gamma_{1},\gamma_{2},\dots,\gamma_{n}\in\mathbb{C}$ such that

[TABLE]

where $p_{t}(\gamma_{1},\dots,\gamma_{n})=\gamma_{1}^{t}+\gamma_{2}^{t}+\dots+\gamma_{n}^{t}$.

Proof.

(i). Assume that $Q=0$. Then, $\mathfrak{sl}_{2}^{\langle 0\rangle}[x]$ coincides with the current Lie algebra $\mathfrak{sl}_{2}[x]$ of $\mathfrak{sl}_{2}$. Moreover, the Lie subalgebra of $\mathfrak{sl}_{2}[x]$ generated by $\mathcal{X}_{0}^{\pm}$ and $\mathcal{J}_{0}$ is isomorphic to $\mathfrak{sl}_{2}$. Thus, by the representation theory of $\mathfrak{sl}_{2}$, we have $u_{0}=n$.

For $u_{1},\dots,u_{n}$, there exist $\gamma_{1},\dots,\gamma_{n}\in\mathbb{C}$ such that

[TABLE]

by Lemma A.2.

By the definition, we have

[TABLE]

since we assume $Q=0$. By the induction on $k$ together with (5.5.1), (5.5.2) and (A.1.1), we can show that

[TABLE]

where $e_{k}(\gamma_{1},\dots,\gamma_{n})=\sum_{1\leq i_{1}<i_{2}<\dots<i_{k}\leq n}\gamma_{i_{1}}\gamma_{i_{2}}\dots\gamma_{i_{k}}$.

By the induction on $t$, we prove that

[TABLE]

If $t\leq n$, (5.5.4) follows from (5.5.1). If $t>n$, by Lemma 5.4 in the case where $s=1$, we have

[TABLE]

By the assumption of the induction together with (5.5.3) and (A.1.2), we have

[TABLE]

(ii). Assume that $Q\not=0$. For $u_{0},u_{1},\dots,u_{n}$, there exist $\beta,\gamma_{1},\dots,\gamma_{n}\in\mathbb{C}$ such that

[TABLE]

by Lemma A.2.

By the induction on $k$, we prove that

[TABLE]

where $\theta_{i}=1-Q\gamma_{i}$ ($1\leq i\leq n$) and $e_{k}(\theta_{1},\dots,\theta_{n})=\sum_{1\leq i_{1}<i_{2}<\dots<i_{k}\leq n}\theta_{i_{1}}\theta_{i_{2}}\dots\theta_{i_{k}}$.

In the case where $k=1$, we have $\mathcal{J}_{0}^{\langle 1\rangle}\cdot v=(\mathcal{J}_{0}+(-Q)\mathcal{J}_{1})\cdot v$. Then we have $\mathcal{J}_{0}^{\langle 1\rangle}\cdot v=e_{1}(\theta_{1},\dots,\theta_{n})v$ by (5.5.5).

In the case where $1<k\leq n$, by the definition, we have

[TABLE]

Applying the assumption of the induction to the right-hand side, we have

[TABLE]

where we note that $\mathcal{J}_{0}^{\langle 0\rangle}=e_{0}(\theta_{1},\dots,\theta_{n})=1$. On the other hand, by (5.5.5), we have

[TABLE]

where we note that $\sum_{w=0}^{z}\begin{pmatrix}z\\ w\end{pmatrix}(-1)^{w}=0$. By (5.5.7) and (5.5.8) together with (A.1.1), we have (5.5.6).

By the induction on $t$, we prove that

[TABLE]

If $t\leq n$, (5.5.9) follows from (5.5.5). If $t>n$, by Lemma 5.4 in the case where $s=0$, we have

[TABLE]

By (5.5.6) , we have

[TABLE]

On the other hand, for $k\geq 0$, we have

[TABLE]

since $\mathcal{J}_{(t-n-1)+w}^{\langle 1\rangle}=\mathcal{J}_{(t-n-1)+w}+(-Q)\mathcal{J}_{(t-n-1)+w+1}$. Then, by (5.5.11) and the assumption of the induction, we have

[TABLE]

and

[TABLE]

for $k=0,1,\dots,n-1$. Moreover, by the direct calculations, we have

[TABLE]

for $k\geq 0$, where $p_{k+1}^{(\bm{\gamma})}(\theta_{1},\dots,\theta_{n})=\gamma_{1}^{t-n-1}\theta_{1}^{k+1}+\gamma_{2}^{t-n-1}\theta_{2}^{k+1}+\dots+\gamma_{n}^{t-n-1}\theta_{n}^{k+1}$. Then, by (5.5.10), (5.5.12), (5.5.13) and (5.5.14), we have

[TABLE]

Applying (A.3.2) to the right-hand side, we have

[TABLE]

∎

5.6. By Lemma 5.2 and Proposition 5.5, we see that the highest weight $\mathbf{u}=(u_{t})_{t\geq 0}$ of a simple highest weight $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module $\mathcal{L}(\mathbf{u})$ has the form

[TABLE]

for some $n\in\mathbb{Z}_{\geq 0}$ and $\beta,\gamma_{1},\gamma_{2},\dots,\gamma_{n}\in\mathbb{C}$ if $\mathcal{L}(\mathbf{u})$ is finite dimensional.

Let $\mathbb{C}[x]$ be the polynomial ring over $\mathbb{C}$ with the indeterminate variable $x$, and let $\mathbb{C}[x]_{\operatorname{monic}}$ be the subset of $\mathbb{C}[x]$ consisting of monic polynomials. We define the set $\mathbb{C}[x]^{\langle Q\rangle}_{\operatorname{monic}}$ by

[TABLE]

Recall that

[TABLE]

We define the map

[TABLE]

by

[TABLE]

when $\varphi=(x-\gamma_{1})(x-\gamma_{2})\dots(x-\gamma_{n})$. We see that the map (5.6.2) is injective, and it gives a bijection between $\mathbb{C}[x]^{\langle Q\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q\rangle}$ and the set of highest weight $\mathbf{u}=(u_{t})_{\geq 0}$ satisfying (5.6.1), where we note that

[TABLE]

Then we have the following corollary of Lemma 5.2 and Proposition 5.5.

Corollary 5.7**.**

Any finite dimensional simple $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module is isomorphic to $\mathcal{L}(\mathbf{u}^{\langle Q\rangle}(\varphi,\beta))$ for some $(\varphi,\beta)\in\mathbb{C}[x]^{\langle Q\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q\rangle}$. Moreover, $\mathcal{L}(\mathbf{u}^{\langle Q\rangle}(\varphi,\beta))\not\cong\mathcal{L}(\mathbf{u}^{\langle Q\rangle}(\varphi^{\prime},\beta^{\prime}))$ if $(\varphi,\beta)\not=(\varphi^{\prime},\beta^{\prime})$.

5.8. Recall the evaluation modules from the paragraph 1.4. Let $L(2)$ be the two-dimensional simple $U(\mathfrak{sl}_{2})$-module, and $v_{0}\in L(2)$ be a highest weight vector. We consider the evaluation module $L(2)^{\mathbf{{ev}}_{\gamma}}$ at $\gamma\in\mathbb{C}$, then we see that

[TABLE]

in $L(2)^{\mathbf{{ev}}_{\gamma}}$.

For $(\varphi=(x-\gamma_{1})(x-\gamma_{2})\dots(x-\gamma_{n}),\beta)\in\mathbb{C}[x]^{\langle Q\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q\rangle}$, we consider the $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module

[TABLE]

where $\mathcal{L}^{\beta}$ is the $1$-dimensional $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module given in the paragraph 5. Let $v_{0}^{(k)}\in L(2)^{\mathbf{{ev}}_{\gamma_{k}}}$ ($1\leq k\leq n$) be a highest weight vector, and $\mathcal{L}^{\beta}=\mathbb{C}w_{0}$. Put $v_{(\varphi,\beta)}=v_{0}^{(1)}\otimes v_{0}^{(2)}\otimes\dots\otimes v_{0}^{(n)}\otimes w_{0}$. Then, for $t\geq 0$, we have

[TABLE]

and

[TABLE]

Let $\mathcal{N}^{\prime}_{(\varphi,\beta)}$ be the $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-submodule of $\mathcal{N}_{(\varphi,\beta)}$ generated by $v_{(\varphi,\beta)}$. Then (5.8.1) and (5.8.2) imply that $\mathcal{N}^{\prime}_{(\varphi,\beta)}$ is a highest weight module of highest weight $\mathbf{u}^{\langle Q\rangle}(\varphi,\beta)$, and $\mathcal{N}^{\prime}_{(\varphi,\beta)}/\operatorname{rad}\mathcal{N}^{\prime}_{(\varphi,\beta)}$ is isomorphic to the simple highest weight module $\mathcal{L}(\mathbf{u}^{\langle Q\rangle}(\varphi,\beta))$. From the construction, $\mathcal{L}(\mathbf{u}^{\langle Q\rangle}(\varphi,\beta))\cong\mathcal{N}^{\prime}_{(\varphi,\beta)}/\operatorname{rad}\mathcal{N}^{\prime}_{(\varphi,\beta)}$ is finite dimensional for each $(\varphi,\beta)\in\mathbb{C}[x]^{\langle Q\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q\rangle}$. Combining with Corollary 5.7, we have the following classification of finite dimensional simple $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-modules.

Theorem 5.9**.**

For $(\varphi,\beta)\in\mathbb{C}[x]^{\langle Q\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q\rangle}$, the highest weight simple $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-module $\mathcal{L}(\mathbf{u}^{\langle Q\rangle}(\varphi,\beta))$ of highest weight $\mathbf{u}^{\langle Q\rangle}(\varphi,\beta)$ is finite dimensional, and we have that

[TABLE]

for $(\varphi,\beta),(\varphi^{\prime},\beta^{\prime})\in\mathbb{C}[x]^{\langle Q\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q\rangle}$. Moreover,

[TABLE]

gives a complete set of isomorphism classes of finite dimensional simple $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-modules.

Remark 5.10. If $Q\not=0$, the evaluation module $L(2)^{\mathbf{{ev}}_{Q^{-1}}}$ at $Q^{-1}$ is not simple. Recall that $L(2)$ is the two dimensional simple $U(\mathfrak{sl}_{2})$-module with a highest weight vector $v_{0}$. Put $v_{1}=f\cdot v_{0}$. Then we see that $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])\cdot v_{1}=\mathbb{C}v_{1}$ is a proper $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-submodule of $L(2)^{\mathbf{{ev}}_{Q^{-1}}}$. Moreover, we have $L(2)^{\mathbf{{ev}}_{Q^{-1}}}/\mathbb{C}v_{1}\cong\mathcal{L}^{1}$ and $\mathbb{C}v_{1}\cong\mathcal{L}^{-1}$ as $U(\mathfrak{sl}_{2}^{\langle Q\rangle}[x])$-modules.

6. Finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules

In this section, we classify the finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules. By Proposition 2.6, any finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to the simple highest weight module $\mathcal{L}(\mathbf{u})$ of highest weight $\mathbf{u}=(u_{i,t})\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$. Thus, it is enough to classify the highest weight $\mathbf{u}$ such that $\mathcal{L}(\mathbf{u})$ is finite dimensional.

6.1. $1$-dimensional representations. First, we consider the $1$-dimensional representations of $\mathfrak{sl}_{m}^{\langle Q\rangle}[x]$. For each $i=1,2,\dots,m-1$, by checking the defining relations, we have the homomorphism of algebras

[TABLE]

Let $L=\mathbb{C}v$ be a $1$-dimensional $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module. For each $i=1,2,\dots,m-1$, when we regard $L$ as a $U(\mathfrak{sl}_{2}^{\langle Q_{i}\rangle}[x])$-module through the homomorphism $\iota_{i}$, we see that $L$ is isomorphic to $\mathcal{L}^{\beta_{i}}$ for some $\beta_{i}\in\mathbb{B}^{\langle Q_{i}\rangle}$ by Lemma 5.2. Thus, we have

[TABLE]

for some $\bm{\beta}=(\beta_{i})_{1\leq i\leq m-1}\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}$.

On the other hand, by checking the defining relations, we can define the $1$-dimensional $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $\mathcal{L}^{\bm{\beta}}=\mathbb{C}v$ by (6.1.2) for each $\bm{\beta}=(\beta_{i})\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}$. Now we proved the following lemma.

Lemma 6.2**.**

Any $1$-dimensional $U(\mathfrak{sl}_{m}^{\langle Q\rangle})$-module is isomorphic to $\mathcal{L}^{\bm{\beta}}$ for some $\bm{\beta}\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}$.

6.3. For $\mathbf{u}=(u_{i,t})\in\prod_{i=1}^{m-1}\prod_{t\geq 0}\mathbb{C}$, let $v_{0}$ be a highest weight vector of the simple highest weight $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $\mathcal{L}(\mathbf{u})$. When we regard $\mathcal{L}(\mathbf{u})$ as a $U(\mathfrak{sl}_{2}^{\langle Q_{i}\rangle}[x])$-module through the homomorphism $\iota_{i}$ in (6.1.1) for each $i=1,\dots,m-1$, we see that the $U(\mathfrak{sl}_{2}^{\langle Q_{i}\rangle}[x])$-submodule of $\mathcal{L}(\mathbf{u})$ generated by $v_{0}$ is a highest weight $U(\mathfrak{sl}_{2}^{\langle Q_{i}\rangle}[x])$-module of highest weight $\mathbf{u}_{i}=(u_{i,t})_{t\geq 0}\in\prod_{t\geq 0}\mathbb{C}$ with the highest weight vector $v_{0}$. Then, if $\mathcal{L}(\mathbf{u})$ is finite dimensional, we see that $\mathbf{u}_{i}=\mathbf{u}^{\langle Q_{i}\rangle}(\varphi_{i},\beta_{i})$ for some $(\varphi_{i},\beta_{i})\in\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}$ by Theorem 5.9 (or Corollary 5.7).

For (\bm{\varphi},\bm{\beta})=((\varphi_{i},\beta_{i}))_{1\leq i\leq m-1}\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}, we define

[TABLE]

by

[TABLE]

when $\varphi_{i}=(x-\gamma_{i,1})(x-\gamma_{i,2})\dots(x-\gamma_{i,n_{i}})$ ($1\leq i\leq m-1$). Then we have that

[TABLE]

for each $i=1,2,\dots,m-1$. From the definition, we see that

[TABLE]

for (\bm{\varphi},\bm{\beta}),(\bm{\varphi}^{\prime},\bm{\beta}^{\prime})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}. By the above argument, any finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to $\mathcal{L}(\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta}))$ for some (\bm{\varphi},\bm{\beta})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}.

On the other hand, for each (\bm{\varphi},\bm{\beta})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}, we can construct a finite dimensional highest weight $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module of highest weight $\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta})$ as follows.

Let $\omega_{j}$ $(1\leq j\leq m-1)$ be the fundamental weight of $\mathfrak{sl}_{m}$, and $L(\omega_{j})$ be the simple highest weight $U(\mathfrak{sl}_{m})$-module of highest weight $\omega_{j}$. Let $v_{0}\in L(\omega_{j})$ be a highest weight vector, then we have $e_{i}\cdot v_{0}=0$ and $H_{i}\cdot v_{0}=\delta_{ij}v_{0}$ ($1\leq i\leq m-1$) by the definition. Recall that $L(\omega_{j})^{\mathbf{{ev}}_{\gamma}}$ is the evaluation module of $L(\omega_{j})$ at $\gamma\in\mathbb{C}$. From the definition, we see that

[TABLE]

in $L(\omega_{j})^{\mathbf{{ev}}_{\gamma}}$.

For (\bm{\varphi},\bm{\beta})=((\varphi_{i},\beta_{i}))_{1\leq i\leq m-1}\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}, we consider the $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module

[TABLE]

where $n_{j}$ and $\gamma_{j,k}$ ($1\leq k\leq n_{j}$) are determined by $\varphi_{j}=(x-\gamma_{j,1})(x-\gamma_{j,2})\dots(x-\gamma_{j,n_{j}})$ for each $j=1,2,\dots,m-1$, and $\bm{\beta}=(\beta_{i})_{1\leq i\leq m-1}$. Let $v_{0}^{(j,k)}\in L(\omega_{j})^{\mathbf{{ev}}_{\gamma_{j,k}}}$ ($1\leq j\leq m-1$, $1\leq k\leq n_{j}$) be a highest weight vector, and $\mathcal{L}^{\bm{\beta}}=\mathbb{C}w_{0}$. Put $v_{(\bm{\varphi},\bm{\beta})}=(\otimes_{j=1}^{m-1}\otimes_{k=1}^{n_{j}}v_{0}^{(j,k)})\otimes w_{0}\in\mathcal{N}_{(\bm{\varphi},\bm{\beta})}$, then we have

[TABLE]

by (6.3.2). Let $\mathcal{N}^{\prime}_{(\bm{\varphi},\bm{\beta})}$ be the $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-submodule of $\mathcal{N}_{(\bm{\varphi},\bm{\beta})}$ generated by $v_{(\bm{\varphi},\bm{\beta})}$. Then (6.3.3) implies that $\mathcal{N}^{\prime}_{(\bm{\varphi},\bm{\beta})}$ is a finite dimensional highest weight module of highest weight $\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta})$. Then we obtain the following classification of finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules.

Theorem 6.4**.**

For (\bm{\varphi},\bm{\beta})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}, the highest weight simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $\mathcal{L}(\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta}))$ of highest weight $\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta})$ is finite dimensional, and we have that

[TABLE]

for (\bm{\varphi},\bm{\beta}),(\bm{\varphi}^{\prime},\bm{\beta}^{\prime})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)} Moreover,

[TABLE]

gives a complete set of isomorphism classes of finite dimensional simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules.

7. Finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules

In this section, we classify the finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules. By Proposition 1.4 (iii), $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ is a Lie subalgebra of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$. The difference of representations of $\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ from one of $\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x]$ is given by the family of $1$-dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules $\{\widetilde{\mathcal{L}}^{\mathbf{h}}\mid\mathbf{h}\in\prod_{t\geq 0}\mathbb{C}\}$. We remark that $\widetilde{\mathcal{L}}^{\mathbf{h}}$ ($\mathbf{h}\in\prod_{t\geq 0}\mathbb{C}$) is isomorphic to the trivial representation as a $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module when we restrict the action.

7.1. $1$-dimensional representations. For $\bm{\beta}=(\beta_{i})_{1\leq i\leq m-1}\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}$, by checking the defining relations, we can define the $1$-dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $\widetilde{\mathcal{L}}^{\bm{\beta}}=\mathbb{C}v$ by

[TABLE]

Note that $\mathcal{J}_{j,t}=\mathcal{I}_{j,t}-\mathcal{I}_{j+1,t}$ in $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$, we see that $\widetilde{\mathcal{L}}^{\beta}\cong\mathcal{L}^{\beta}$ as $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules when we restrict the action on $\widetilde{\mathcal{L}}^{\beta}$ to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$ through the injective homomorphism $\Upsilon$ in the proposition 1.4 (iii).

For $\mathbf{h}=(h_{t})_{t\geq 0}\in\prod_{t\geq 0}\mathbb{C}$, we can also define the $1$-dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $\widetilde{\mathcal{L}}^{\mathbf{h}}=\mathbb{C}v$ by

[TABLE]

We see that $\widetilde{\mathcal{L}}^{\mathbf{h}}\cong\mathcal{L}^{\mathbf{0}}$ as $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules when we restrict the action on $\widetilde{\mathcal{L}}^{\mathbf{h}}$ to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$， where $\mathbf{0}=(0)_{1\leq i\leq m-1}\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}$ (i.e. $\mathcal{L}^{\mathbf{0}}$ is the trivial representation). Then we have the following classification of $1$-dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle})$-modules.

Lemma 7.2**.**

Any $1$-dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to $\widetilde{\mathcal{L}}^{\bm{\beta}}\otimes\widetilde{\mathcal{L}}^{\mathbf{h}}$ for some $\bm{\beta}\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}$ and $\mathbf{h}\in\prod_{t\geq 0}\mathbb{C}$. We have that

[TABLE]

Moreover, we see that $\widetilde{\mathcal{L}}^{\bm{\beta}}\otimes\widetilde{\mathcal{L}}^{\mathbf{h}}\cong\mathcal{L}^{\bm{\beta}}$ as $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules when we restrict the action on $\widetilde{\mathcal{L}}^{\bm{\beta}}\otimes\widetilde{\mathcal{L}}^{\mathbf{h}}$ to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$.

Proof.

Let $\mathcal{L}=\mathbb{C}v$ be a $1$-dimensional $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle})$-module. By restricting the action on $\mathcal{L}$ to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle})$ through the injective homomorphism $\Upsilon$ in the proposition 1.4 (iii), we have

[TABLE]

for some $\bm{\beta}=(\beta_{i})\in\prod_{i=1}^{m-1}\mathbb{B}^{\langle Q_{i}\rangle}$ by Lemma 6.2.

On the other hand, for $t\in\mathbb{Z}_{\geq 0}$, there exists $h_{t}\in\mathbb{C}$ such that

[TABLE]

since $\dim\mathcal{L}=1$. Then (7.2.1) and (7.2.2) imply taht

[TABLE]

Then we see that $\mathcal{L}\cong\widetilde{\mathcal{L}}^{\bm{\beta}}\otimes\widetilde{\mathcal{L}}^{\mathbf{h}}$. The remaining statements are clear. ∎

7.3. For $\widetilde{\mathbf{u}}=(\widetilde{u}_{j,t})\in\prod_{j=1}^{m}\prod_{t\geq 0}\mathbb{C}$, let $v_{0}$ be a highest weight vector of the simple highest weight $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $\mathcal{L}(\widetilde{\mathbf{u}})$. By restricting the action on $\mathcal{L}(\widetilde{\mathbf{u}})$ to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$, Theorem 6.4 implies that

[TABLE]

for some $(\bm{\varphi},\bm{\beta})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle})$ if $\mathcal{L}(\widetilde{\mathbf{u}})$ is finite dimensional.

For $t\in\mathbb{Z}_{\geq 0}$, let $h_{t}\in\mathbb{C}$ be such that

[TABLE]

By (7.3.1) and (7.3.2), we have

[TABLE]

for some $(\bm{\varphi},\bm{\beta})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle})$ and $\mathbf{h}=(h_{t})\in\prod_{t\geq 0}\mathbb{C}$ if $\mathcal{L}(\widetilde{\mathbf{u}})$ is finite dimensional.

For $(\bm{\varphi},\bm{\beta},\mathbf{h})=((\varphi_{i},\beta_{i})_{1\leq i\leq m-1},(h_{t})_{t\geq 0})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle})\times\prod_{t\geq 0}\mathbb{C}$, we define

[TABLE]

by

[TABLE]

From the definition, we see that

[TABLE]

for $(\bm{\varphi},\bm{\beta},\mathbf{h}),(\bm{\varphi}^{\prime},\bm{\beta}^{\prime},\mathbf{h}^{\prime})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle})\times\prod_{t\geq 0}\mathbb{C}$. By the above argument, any finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module is isomorphic to $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ for some $(\bm{\varphi},\bm{\beta},\mathbf{h})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle})\times\prod_{t\geq 0}\mathbb{C}$.

On the other hand, for each $(\bm{\varphi},\bm{\beta},\mathbf{h})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle})\times\prod_{t\geq 0}\mathbb{C}$, we can construct a finite dimensional highest weight $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module of highest weight $\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h})$ as follows.

Let $P=\bigoplus_{i=1}^{m}\mathbb{Z}\varepsilon_{i}$ be the weight lattice of $\mathfrak{gl}_{m}$. Put $\widetilde{\omega}_{l}=\varepsilon_{1}+\varepsilon_{2}+\dots+\varepsilon_{l}$ for $l=1,2,\dots,m-1$. Let $L(\widetilde{\omega}_{l})$ be the simple highest weight $U(\mathfrak{gl}_{m})$-module of highest weight $\widetilde{\omega}_{l}$, and $v_{0}\in L(\widetilde{\omega}_{l})$ be a highest weight vector. Then, we have

[TABLE]

Recall that $L(\widetilde{\omega}_{l})^{\widetilde{\mathbf{{ev}}}_{\gamma}}$ is the evaluation module of $L(\widetilde{\omega}_{l})$ at $\gamma\in\mathbb{C}$. From the definition, we see that

[TABLE]

in $L(\widetilde{\omega}_{l})^{\widetilde{\mathbf{{ev}}}_{\gamma}}$. (We remark that $L(\widetilde{\omega}_{l})^{\widetilde{\mathbf{{ev}}}_{\gamma}}\cong L(\omega_{l})^{\mathbf{{ev}}_{\gamma}}$ as $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules when we restrict the action on $L(\widetilde{\omega}_{l})^{\widetilde{\mathbf{{ev}}}_{\gamma}}$ to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$.)

For $(\bm{\varphi},\bm{\beta},\mathbf{h})=((\varphi_{i},\beta_{i})_{1\leq i\leq m-1},(h_{t})_{t\geq 0})\in\prod_{i=1}^{m-1}(\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle})\times\prod_{t\geq 0}\mathbb{C}$, we consider the $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module

[TABLE]

where $n_{l}$ and $\gamma_{l,k}$ ($1\leq k\leq n_{l}$) are determined by $\varphi_{l}=(x-\gamma_{l,1})(x-\gamma_{l,2})\dots(x-\gamma_{l,n_{l}})$ for each $l=1,2,\dots,m-1$, and we put $\bm{\beta}=(\beta_{i})_{1\leq i\leq m-1}$ and $\mathbf{h}=(h_{t})_{t\geq 0}$. Let $v_{0}^{(l,k)}\in L(\widetilde{\omega}_{l})^{\widetilde{\mathbf{{ev}}}_{\gamma_{l,k}}}$ ($1\leq l\leq m-1$, $1\leq k\leq n_{l}$) be a highest weight vector, $\widetilde{\mathcal{L}}^{\bm{\beta}}=\mathbb{C}w_{0}$ and $\widetilde{\mathcal{L}}^{\mathbf{h}}=\mathbb{C}z_{0}$. Put $v_{(\bm{\varphi},\bm{\beta},\mathbf{h})}=(\otimes_{l=1}^{m-1}\otimes_{k=1}^{n_{l}}v_{0}^{(l,k)})\otimes w_{0}\otimes z_{0}\in\widetilde{\mathcal{N}}_{(\bm{\varphi},\bm{\beta})}$, then we have

[TABLE]

for $1\leq i\leq m-1$, $1\leq j\leq m$ and $t\geq 0$ by (7.3.3). Let $\widetilde{\mathcal{N}}^{\prime}_{(\bm{\varphi},\bm{\beta},\mathbf{h})}$ be the $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-submodule of $\widetilde{\mathcal{N}}_{(\bm{\varphi},\bm{\beta},\mathbf{h})}$ generated by $v_{(\bm{\varphi},\bm{\beta},\mathbf{h})}$. Then (7.3.4) implies that $\widetilde{\mathcal{N}}^{\prime}_{(\bm{\varphi},\bm{\beta},\mathbf{h})}$ is a finite dimensional highest weight module of highest weight $\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h})$. Now we obtain the following classification of finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules.

Theorem 7.4**.**

For (\bm{\varphi},\bm{\beta},\mathbf{h})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}\times\prod_{t\geq 0}\mathbb{C}, the highest weight simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ of highest weight $\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h})$ is finite dimensional, and we have that

[TABLE]

for (\bm{\varphi},\bm{\beta},\mathbf{h}),(\bm{\varphi}^{\prime},\bm{\beta}^{\prime},\mathbf{h}^{\prime})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}\times\prod_{t\geq 0}\mathbb{C}. Moreover,

[TABLE]

gives a complete set of isomorphism classes of finite dimensional simple $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-modules.

We also have the following corollary.

Corollary 7.5**.**

For (\bm{\varphi},\bm{\beta},\mathbf{h})\in\prod_{i=1}^{m-1}\big{(}\mathbb{C}[x]^{\langle Q_{i}\rangle}_{\operatorname{monic}}\times\mathbb{B}^{\langle Q_{i}\rangle}\big{)}\times\prod_{t\geq 0}\mathbb{C}, we have

[TABLE]

when we restrict the action on $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$.

Proof.

We prove that $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ is also simple when we restrict the action to $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$. Then the isomorphism follows from the definitions of $\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h})$ and $\mathbf{u}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta})$.

Let $v_{0}\in\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ be a highest weight vector as the $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module. Then we have

[TABLE]

by the triangular decomposition in Proposition 1.4 (iv). This implies that

[TABLE]

Assume that $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ is not simple as a $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module by the restriction, then $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ contains a non-zero proper simple $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-submodule which is a highest weight $U(\mathfrak{sl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-module. This implies that there exist an element $w_{0}\in\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$ such that $\mathcal{X}_{i,t}^{+}\cdot w_{0}=0$ ($1\leq i\leq m-1,\,t\geq 0)$ and $w_{0}\not\in\mathbb{C}v_{0}$. Then $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])\cdot w_{0}$ turns out to be a non-zero proper $U(\mathfrak{gl}_{m}^{\langle\mathbf{Q}\rangle}[x])$-submodule of $\mathcal{L}(\widetilde{\mathbf{u}}^{\langle\mathbf{Q}\rangle}(\bm{\varphi},\bm{\beta},\mathbf{h}))$. This is a contradiction. ∎

Appendix A Some combinatorics

A.1. Let $\mathbb{Z}[x_{1},\dots,x_{n}]$ be the ring of polynomials in independent variables $x_{1},\dots,x_{n}$ over $\mathbb{Z}$. For $k\in\mathbb{Z}_{>0}$, put

[TABLE]

Namely, $p_{k}(x_{1},\dots,x_{n})$ is the power sum symmetric polynomial of degree $k$, and $e_{k}(x_{1},\dots,x_{n})$ is the elementary symmetric polynomial of degree $k$. We also put $e_{0}(x_{1},\dots,x_{n})=1$. Then, for $k>0$, we have

[TABLE]

by [M, §1 (2.11’)]. For $s>n$, we have

[TABLE]

where we note that $e_{s-z}(x_{1},\dots,x_{n})=0$ if $z<s-n$. Put $w=z-s+n$, we have

[TABLE]

for $s>n$.

Lemma A.2**.**

For $n\in\mathbb{Z}_{>0}$ and $u_{1},u_{2},\dots,u_{n}\in\mathbb{C}$, the simultaneous equations

[TABLE]

has a solution in $\mathbb{C}$.

Proof.

We prove the lemma by the induction on $n$. In the case where $n=1$, it is clear. If $n>1$, the equations (A.2.1) are equivalent to the equations

[TABLE]

By (A.1.1), we have

[TABLE]

where we note that $e_{n}(x_{1},x_{2},\dots,x_{n-1})=0$. On the other hand, we can write

[TABLE]

for some $\alpha_{\lambda}\in\mathbb{C}$, where $p_{\lambda}(x_{1},x_{2},\dots,x_{n-1})=\prod_{j=1}^{\ell(\lambda)}p_{\lambda_{j}}(x_{1},x_{2},\dots,x_{n-1})$ for $\lambda=(\lambda_{1},\lambda_{2},\dots)\vdash n-i$. Thus we have

[TABLE]

(Note that $\{p_{\mu}(x_{1},x_{2},\dots,x_{n-1})\mid\mu\vdash k\}$ is not linearly independent if $k\geq n$. For an example, we have $p_{(3)}(x_{1},x_{2})=\frac{3}{2}p_{(2,1)}(x_{1},x_{2})-\frac{1}{2}p_{(1,1,1)}(x_{1},x_{2})$.)

Then the equations (A.2.2) are equivalent to the equations

[TABLE]

Let $\beta_{n}$ be a solution of the equation $(\ast 1)$ for the variable $x_{n}$. By the assumption of the induction, the simultaneous equations

[TABLE]

for variables $x_{1},x_{2},\dots,x_{n-1}$ has a solution. We denote it by $(x_{1},x_{2},\dots,x_{n-1})=(\beta_{1},\beta_{2},\dots,\beta_{n-1})$. Then $(x_{1},x_{2},\dots,x_{n})=(\beta_{1},\beta_{2},\dots,\beta_{n})$ gives a solution of (A.2.3). ∎

A.3. We consider some modifications of the formulas (A.1.1) and (A.1.2) as follows. Let $\mathbf{b}=(b_{1},\dots,b_{n})$ be $n$ independent variables, and we consider the ring of polynomials $\mathbb{Z}[x_{1},\dots,x_{n}][b_{1},\dots,b_{n}]$. For $k\in\mathbb{Z}_{>0}$, put

[TABLE]

and

[TABLE]

We also put $e_{0}^{(\mathbf{b})}=1$. Note that $e_{k}^{(\mathbf{b})}(x_{1},\dots,x_{n})=0$ if $k>n$. Put $\mathbf{1}=(1,1,\dots,1)$, then we have $e_{k}^{(\mathbf{1})}(x_{1},\dots,x_{n})=ke_{k}(x_{1},\dots,x_{n})$ and $p_{k}^{(\mathbf{1})}(x_{1},\dots,x_{n})=p_{k}(x_{1},\dots,x_{n})$.

We consider the generating functions $E(t)$, $E^{(\mathbf{b})}(t)$ and $P^{(\mathbf{b})}(t)$ by

[TABLE]

Then, we have

[TABLE]

and

[TABLE]

This implies that, for $k\geq 0$,

[TABLE]

In the case where $k=n$, we have

[TABLE]

since $e_{n+1}^{(\mathbf{b})}(x_{1},\dots,x_{n})=0$. This implies that

[TABLE]