Cuspidal modules for the derivation Lie algebra over a rational quantum torus

TL;DR

This paper provides an explicit description of cuspidal modules for the derivation Lie algebra over a rational quantum torus, including their structure and the action of the center, advancing understanding in quantum algebra representations.

Contribution

It introduces a detailed structural characterization of cuspidal modules for derivation Lie algebras over rational quantum tori, incorporating the center's action.

Findings

Explicit module structures characterized

Inclusion of the center's action in the module description

Enhanced understanding of derivation Lie algebra representations

Abstract

Let $\mathbb C_Q$ denote a rational quantum torus with $d$ variables, and $\mathcal Z$ be the centre of $\mathbb C_Q$. In this paper we give a explicit description of the structure of the cuspidal modules for the derivation Lie algebra $\mathcal D$ over $\mathbb C_Q$, with an extra associative $\mathcal Z$-action.

Equations

$$[x^{\mathbf{m}}{x}_i{\frac{\partial }{\partial x}}_i,x^{\mathbf{n}}{x}_j{\frac{\partial }{\partial x}}_j]=x^{\mathbf{m}+\mathbf{n}}(n_i{x}_j{\frac{\partial }{\partial x}}_j-m_j{x}_i{\frac{\partial }{\partial x}}_i).$$

$$t_i{t}_j=q_{ij}{t}_j{t}_i\text{ for all }1\le i\ne j\le d.$$

$$\sigma (\mathbf{m},\mathbf{n})=\prod _{1\le i<j\le d}{q}_{ji}^{n_j{m}_i}\text{ and }R=\{\mathbf{m}\in {\mathbb{Z}}^d\mid \sigma (\mathbf{m},\mathbf{n})=\sigma (\mathbf{n},\mathbf{m})\text{ for any }\mathbf{n}\in {\mathbb{Z}}^d\}.$$

$$\{t^{\mathbf{m}}{\partial }_i,\mathrm{ad}{t}^{\mathbf{s}}\mid \mathbf{m}\in R,\mathbf{s}\notin R\},$$

$$R=\underset{i=1}{\overset{z}{⨁}}\left(\mathbb{Z}{k}_i{ϵ}_{2i-1}\oplus \mathbb{Z}{k}_i{ϵ}_{2i}\right)\oplus \underset{l>2z}{⨁}\mathbb{Z}{ϵ}_l,$$

$$t_1^{k_1},t_2^{k_1},t_3^{k_2},t_4^{k_2},\cdots ,t_{2z-1}^{k_z},t_{2z}^{k_z},t_{2z+1},\cdots ,t_d.$$

$$\sigma (\mathbf{m},\mathbf{r})=\sigma (\mathbf{r},\mathbf{m}),t^{\mathbf{m}}{t}^{\mathbf{r}}=t^{\mathbf{m}+\mathbf{r}}\text{ for all }\mathbf{m}\in R,\mathbf{r}\in {\mathbb{Z}}^d.$$

$$[\partial (\mathbf{u},\mathbf{m}),\partial (\mathbf{v},\mathbf{n})]=(\mathbf{u}\mid \mathbf{n})\partial (\mathbf{v},\mathbf{m}+\mathbf{n})-(\mathbf{v}\mid \mathbf{m})\partial (\mathbf{u},\mathbf{m}+\mathbf{n});$$

$$[\partial (\mathbf{u},\mathbf{m}),t^{\mathbf{s}}]=(\mathbf{u}\mid \mathbf{s})t^{\mathbf{m}+\mathbf{s}};[t^{\mathbf{r}},t^{\mathbf{s}}]=(\sigma (\mathbf{r},\mathbf{s})-\sigma (\mathbf{s},\mathbf{r}))t^{\mathbf{r}+\mathbf{s}},$$

$$X_{2i-1}=E_{1,1}+q_i{E}_{2,2}+\cdots +q_i^{k_1-1}{E}_{k_i,k_i},$$

$$X_{2i}=E_{1,2}+E_{2,3}+\cdots +E_{k_i-1,k_i}+E_{k_i,1},$$

$$[X^{\mathbf{m}},X^{\mathbf{n}}]=(\sigma (\mathbf{m},\mathbf{n})-\sigma (\mathbf{n},\mathbf{m}))X^{\mathbf{m}+\mathbf{n}}.$$

$${\Gamma }_0=\{\mathbf{n}\in {\mathbb{Z}}^d\mid 0<n_{2i-1}\le k_i,0<n_{2i}\le k_i,1\le i\le z,\text{ and }{n}_l=0,2z<l\le d\},$$

$$\{\partial (\mu ,\mathbf{m}),t^{\mathbf{s}}\mid \mathbf{m}\in R,\mathbf{s}\notin R\}.$$

$$\{\partial (\mathbf{u},\mathbf{m})\mid \mathbf{u}\in {\mathbb{C}}^d,\mathbf{m}\in R\}$$

$$\partial (\mathbf{u},\mathbf{m})\mapsto x^{\mathbf{n}}\sum _{i=1}^d{u}_i{k}_i{x}_i\frac{\partial }{\partial x_i},$$

$$d_{\mathbf{u}}=\sum _{i=1}^d{u}_i\frac{\partial }{\partial x_i}+\sum _{i=1}^d{u}_i{t}_i\frac{\partial }{\partial t_i}\in \mathrm{Der}(AC).$$

$$\{x^{\mathbf{p}}{d}_{\mathbf{u}},x^{\mathbf{l}}{t}^{\overline{\mathbf{s}}}\mid \mathbf{p}\in {\mathbb{N}}^d\backslash \{0\},\mathbf{l}\in {\mathbb{N}}^d,\overline{\mathbf{s}}\in \Gamma ,\mathbf{u}\in {\mathbb{C}}^d\}.$$

$$[x^{\mathbf{m}}{d}_{\mathbf{u}},x^{\mathbf{n}}{d}_{\mathbf{v}}]=\sum _{i=1}^d{u}_i{n}_i{x}^{\mathbf{m}+\mathbf{n}-{ϵ}_i}{d}_{\mathbf{v}}-\sum _{i=1}^d{v}_i{m}_i{x}^{\mathbf{m}+\mathbf{n}-{ϵ}_i}{d}_{\mathbf{u}};$$

$$[x^{\mathbf{m}}{d}_{\mathbf{u}},x^{\mathbf{l}}{t}^{\overline{\mathbf{s}}}]=\sum _{i=1}^d{u}_i{n}_i{x}^{\mathbf{m}+\mathbf{n}-{ϵ}_i}{t}^{\overline{\mathbf{s}}}+(\mathbf{u}\mid \mathbf{s})x^{\mathbf{m}+\mathbf{s}}{t}^{\overline{\mathbf{s}}};$$

$$[x^{\mathbf{p}}{t}^{\overline{\mathbf{r}}},x^{\mathbf{l}}{t}^{\overline{\mathbf{s}}}]=(\sigma (\mathbf{r},\mathbf{s})-\sigma (\mathbf{s},\mathbf{r}))x^{\mathbf{m}+\mathbf{s}}{t}^{\overline{\mathbf{r}+\mathbf{s}}},$$

$$\mathcal{G}(\overline{\mathbf{s}})={\mathrm{span}}_{\mathbb{C}}\left\{x^{\mathbf{n}}{t}^{\overline{\mathbf{s}}}\mid \mathbf{n}\in {\mathbb{N}}^d\right\};$$

$$\mathcal{G}(\overline{0})={\mathrm{span}}_{\mathbb{C}}\left\{x^{\mathbf{n}}{t}^{\overline{0}},x^{\mathbf{p}}{d}_{\mathbf{u}}\mid \mathbf{n}\in {\mathbb{N}}^d,\mathbf{p}\in {\mathbb{N}}^d\backslash \{0\},\mathbf{u}\in {\mathbb{C}}^d\right\}.$$

$${\mathcal{G}}^{(x)}={\mathrm{span}}_{\mathbb{C}}\left\{x^{\mathbf{p}}{d}_{\mathbf{u}}\mid \mathbf{p}\in {\mathbb{N}}^d\backslash \{0\},\mathbf{u}\in {\mathbb{C}}^d\right\}$$

$${\mathcal{G}}^{(t)}={\mathrm{span}}_{\mathbb{C}}\left\{x^{\mathbf{n}}{t}^{\overline{\mathbf{s}}}\mid \mathbf{n}\in {\mathbb{N}}^d,\overline{\mathbf{s}}\in \Gamma \right\}.$$

$${\mathcal{G}}_i^{(x)}={\mathrm{span}}_{\mathbb{C}}\left\{x^{\mathbf{p}}{d}_{\mathbf{u}}\mid |\mathbf{p}|=i+1,\mathbf{u}\in {\mathbb{C}}^d\right\},{\mathcal{G}}_i^{(t)}={\mathrm{span}}_{\mathbb{C}}\left\{x^{\mathbf{n}}{t}^{\overline{\mathbf{s}}}\mid |\mathbf{n}|=i,\overline{\mathbf{s}}\in \Gamma \right\}.$$

$$[x^{\mathbf{m}}{d}_{\mathbf{u}},t^{\overline{\mathbf{s}}}]=(\mathbf{u}\mid \mathbf{s})x^{\mathbf{m}}{t}^{\overline{\mathbf{s}}}$$

$$X^{\mathbf{r}}\mapsto t^{\overline{\mathbf{r}}}+{\mathcal{G}}_{+};E_{ij}\mapsto x_i{d}_{{ϵ}_j}+{\mathcal{G}}_{+},$$

$$M\cong \underset{\overline{\mathbf{s}}}{⨁}{U}_{\overline{\mathbf{s}}}\otimes \mathcal{Z},$$

$$[D(\mathbf{u},\mathbf{m}),D(\mathbf{v},\mathbf{n})]=(\mathbf{u}\mid \mathbf{n})(D(\mathbf{v},\mathbf{m}+\mathbf{n})-D(\mathbf{v},\mathbf{n}))-(\mathbf{v}\mid \mathbf{m})(D(\mathbf{u},\mathbf{m}+\mathbf{n})-D(\mathbf{u},\mathbf{m}));$$

$$[D(\mathbf{u},\mathbf{m}),L(\mathbf{n},\mathbf{s})]=(\mathbf{u}\mid \mathbf{n}+\mathbf{s})L(\mathbf{m}+\mathbf{n},\mathbf{s})-(\mathbf{u}\mid \mathbf{n})L(\mathbf{n},\mathbf{s});$$

$$[L(\mathbf{m},\mathbf{r}),L(\mathbf{n},\mathbf{s})]=(\sigma (\mathbf{r},\mathbf{s})-\sigma (\mathbf{s},\mathbf{r}))L(\mathbf{m}+\mathbf{n},\mathbf{r}+\mathbf{s}),$$

$$\partial (\mathbf{u},\mathbf{m})(t^{\mathbf{n}}v)=(\mathbf{u}\mid \mathbf{n})t^{\mathbf{m}+\mathbf{n}}v+t^{\mathbf{m}+\mathbf{n}}D(\mathbf{u},\mathbf{m})v;$$

$$t^{\mathbf{m}+\mathbf{r}}(t^{\mathbf{n}}v)=t^{\mathbf{n}}(t^{\mathbf{m}+\mathbf{r}}v)=t^{\mathbf{m}+\mathbf{n}}(L(\mathbf{m},\mathbf{r})v),$$

Full text

Cuspidal modules for the derivation Lie algebra

over a rational quantum torus

Chengkang Xu*∗*

School of Mathematical Science, Shangrao Normal University, Shangrao, China

\par\par* Corresponding author. The author is supported by the National Natural Science Foundation of China(No. 11626157, 11801375). E-mail: xiaoxiongxu@126.com.

Abstract: Let ${\mathbb{C}}_{Q}$ denote a rational quantum torus with $d$ variables, and $\mathcal{Z}$ be the centre of ${\mathbb{C}}_{Q}$. In this paper we give a explicit description of the structure of the cuspidal modules for the derivation Lie algebra $\mathcal{D}$ over ${\mathbb{C}}_{Q}$, with an extra associative $\mathcal{Z}$-action.

Keywords: derivation Lie algebra; cuspidal module; rational quantum torus; module of tensor fields.

1 Introduction

In the past few decades, the representation theory of the derivation algebra over a torus has attracted many mathematicians and physicists. Let $d>1$ be a positive integer. Denote by $A={\mathbb{C}}[x_{1}^{\pm 1},\cdots,x_{d}^{\pm 1}]$ the commuting torus in variables $x_{1},\cdots,x_{d}$, and by $W_{d}$ the derivation algebra over $A$. Irreducible modules with finite dimensional weight spaces over $W_{d}$ were classified in [BF1]. They are modules of tensor fields constructed independently by Shen[S] and Larsson[Lar], and modules of highest weight type given in [BB]. For the derivation algebra $\mathcal{D}$ over a rational quantum torus ${\mathbb{C}}_{Q}$, although the modules of tensor fields were constructed in [LT] and [LiuZ2] and the modules of highest weight type were given in [Xu], we are still far from the complete classification of irreducible modules with finite dimensional weight spaces for $\mathcal{D}$. However, some partial classifications are known to us. For example, irreducible $\mathcal{D}$-modules with certain ${\mathbb{C}}_{Q}$-action were classified in [RBS]. In [LiuZ2], Liu and Zhao classified irreducible cuspidal $\mathcal{D}$-modules with an extra associative action of the centre $\mathcal{Z}$ of ${\mathbb{C}}_{Q}$. These modules were constructed using a finite dimensional irreducible $\mathfrak{gl}_{d}$-module $V$ and a finite dimensional ${\mathbb{Z}}^{d}/R$-graded irreducible $\mathfrak{gl}_{N}$-module $W$. Here $N$ is a positive number closely related to the structure of ${\mathbb{C}}_{Q}$, and $R$ is a subgroup of ${\mathbb{Z}}^{d}$ corresponding to the center $\mathcal{Z}$. To achieve their classification result in [LiuZ2], Liu and Zhao used heavy computations(totally five pages of them). Moreover, the appearance of the algebra $\mathfrak{gl}_{N}$ in the proof seems a little farfetched, and it is not clear what is the relation between the $\mathfrak{gl}_{d}$-module $V$ and the ${\mathbb{Z}}^{d}/R$-graded $\mathfrak{gl}_{N}$-module $W$.

In this paper we study $\mathcal{D}$-modules(not necessarily irreducible) with an extra associative $\mathcal{Z}$-action. We will simply call such a module a $\mathcal{Z}\mathcal{D}$-module. Our main result is that cuspidal $\mathcal{Z}\mathcal{D}$-modules with support lying in one coset $\alpha+{\mathbb{Z}}^{d},\alpha\in{\mathbb{C}}^{d}$, are in a one-to-one correspondence with finite dimensional $\Gamma$-graded modules over a subalgebra $\mathcal{G}$ of $\mathrm{Der}(AC)$, where $AC={\mathbb{C}}[x_{1},\cdots,x_{d}]\oplus{\mathbb{C}}_{Q}/\mathcal{Z}$. The algebra $\mathcal{G}$ has a quotient isomorphic to $\mathfrak{gl}_{d}\oplus\mathfrak{gl}_{N}$. We will use some results about the solenoidal Lie algebras over $A$ and ${\mathbb{C}}_{Q}$, which were given in [BF2] and [Xu2] separately. As a corollary, we reprove the classification of irreducible cuspidal $\mathcal{Z}\mathcal{D}$-modules, in a more conceptional way. From our proof one can see clearly that the algebras $\mathfrak{gl}_{d},\mathfrak{gl}_{N}$ and their modules appear naturally. We mention that Liu and Zhao classified irreducible cuspidal $\mathcal{D}$-modules with no other restriction in [LiuZ] using the classification result in [LiuZ2] and a method introduced in [BF1].

The present paper is arranged as follows. In Section 2 we recall some results about the quantum torus ${\mathbb{C}}_{Q}$, the algebras $\mathcal{D}$, the solenoidal Lie algebra $W_{\mu}$ over $A$ and the solenoidal Lie algebra $\mathfrak{g}_{\mu}$ over ${\mathbb{C}}_{Q}$. Section 3 is devoted to finite dimensional $\mathcal{G}$-modules. In Section 4 we prove our main theorem about the cuspidal $\mathcal{Z}\mathcal{D}$-modules and classify irreducible cuspidal $\mathcal{Z}\mathcal{D}$-modules once again.

Throughout this paper, ${\mathbb{C}},{\mathbb{Z}},{\mathbb{N}},{{\mathbb{Z}}_{+}}$ refer to the set of complex numbers, integers, nonnegative integers and positive integers respectively. For a Lie algebra $\mathcal{G}$, we denote by ${\mathcal{U}}(\mathcal{G})$ the universal enveloping algebra of $\mathcal{G}$. Fix $1<d\in{{\mathbb{Z}}_{+}}$ and a standard basis ${\epsilon}_{1},\cdots,{\epsilon}_{d}$ for the space ${\mathbb{C}}^{d}$. Denote by $(\cdot\mid\cdot)$ the inner product on ${\mathbb{C}}^{d}$.

2 Notations and Preliminaries

For a Lie algebra, a weight module is called a cuspidal module if all weight spaces are uniformly bounded. The support of a weight module is defined to be the set of all weights.

For ${\bf m}=(m_{1},\cdots,m_{d})^{T}\in{\mathbb{Z}}^{d}$ we denote by $x^{\bf m}$ the monomial $x_{1}^{m_{1}}\cdots x_{d}^{m_{d}}$ in $A$. Then the derivation Lie algebra $W_{d}$ over $A$ has a basis $\{x^{\bf m}x_{i}\frac{\partial}{\partial{x}}_{i}\mid{\bf m}\in{\mathbb{Z}}^{d},1\leq i\leq d\}$ subject to the Lie bracket

[TABLE]

Let $\mu=(\mu_{1},\cdots,\mu_{d})^{T}\in{\mathbb{C}}^{d}$ be generic, which means that $\mu_{1},\cdots,\mu_{d}$ are linearly independent over the field of rational numbers. The solenoidal Lie algebra $W_{\mu}$ over $A$ is the subalgebra of $W_{d}$ spanned by $\{x^{\bf m}\sum_{i=1}^{d}\mu_{i}x_{i}\frac{\partial}{\partial{x}}_{i}\mid{\bf m}\in{\mathbb{Z}}^{d}\}$. Many others call $W_{\mu}$ (centerless) higher rank Virasoro algebra.

Let $Q=(q_{ij})$ be a $d\times d$ complex matrix with all $q_{ij}$ being roots of unity and satisfying $q_{ii}=1,\ q_{ij}q_{ji}=1\text{ for all }1\leq i\neq j\leq d.$ The rational quantum torus relative to $Q$ is the unital associative algebra ${\mathbb{C}}_{Q}={\mathbb{C}}[t_{1}^{\pm 1},\cdots,t_{d}^{\pm 1}]$ with multiplication

[TABLE]

For ${{\bf m}}=(m_{1},\cdots,m_{d})^{T}\in{\mathbb{Z}}^{d}$ we denote $t^{\bf m}=t_{1}^{m_{1}}\cdots t_{d}^{m_{d}}$. For ${\bf m},{\bf n}\in{\mathbb{Z}}^{d}$, set

[TABLE]

Then the center $\mathcal{Z}$ of ${\mathbb{C}}_{Q}$ is spanned by $\{t^{\bf m}\mid{\bf m}\in R\}$. By [BGK], the derivation Lie algebra $\mathcal{D}$ over ${\mathbb{C}}_{Q}$ has a basis

[TABLE]

where $\partial_{i}$ is the degree derivation with respect to $t_{i}$ defined by $\partial_{i}t^{\bf r}=r_{i}t^{\bf r}$, and $\mathrm{ad}t^{\bf s}$ is the inner derivation given by $\mathrm{ad}t^{\bf s}\cdot t^{\bf r}=[t^{\bf s},t^{\bf r}]$. For convenience we will simply write $t^{\bf s}=\mathrm{ad}t^{\bf s}$.

By Theorem 4.5 in [N], up to an isomorphism of ${\mathbb{C}}_{Q}$, we may assume that $q_{2i,2i-1}=q_{i},q_{2i-,2i}=q_{i}^{-1}$ for $1\leq i\leq z$, and other entries of $Q$ are all 1, where $z\in{{\mathbb{Z}}_{+}}$ with $2z\leq d$ and the orders $k_{i}$ of $q_{i}$ as roots of unity satisfy $k_{i+1}\mid k_{i}$ for $1\leq i\leq z$. In this paper we always assume that $Q$ is of this form, and keep the notation $q_{i},z$ and $k_{i}$. Then the subgroup $R$ of ${\mathbb{Z}}^{d}$ has the simple form

[TABLE]

and the center $\mathcal{Z}$ becomes a commuting torus on variables

[TABLE]

Moreover, we have

[TABLE]

Set $\partial({\bf u},{\bf m})=\sum_{i=1}^{d}u_{i}t^{\bf m}\partial_{i}$ for ${\bf u}=(u_{1},\cdots,u_{d})^{T}\in{\mathbb{C}}^{d}$ and ${\bf m}\in R$. Then the Lie bracket of $\mathcal{D}$ is

[TABLE]

where ${\bf u},{\bf v}\in{\mathbb{C}}^{d},{\bf m},{\bf n}\in R$ and ${\bf r},{\bf s}\notin R$.

Denote by $\mathcal{I}$ the ideal of the associative algebra ${\mathbb{C}}_{Q}$ generated by elements $t^{{\bf n}+{\bf r}}-t^{\bf r},{\bf n}\in R,{\bf r}\in{\mathbb{Z}}^{d}$. By [N], we have ${\mathbb{C}}_{Q}/\mathcal{I}\cong\bigotimes_{i=1}^{z}M_{k_{i}}({\mathbb{C}})\cong M_{N}({\mathbb{C}})$, where $N=\prod_{i=1}^{z}k_{i}$ and $M_{n}({\mathbb{C}})$ denotes the associative algebra of all $n\times n$ complex matrices. It is well known that $M_{k_{i}}({\mathbb{C}})$ can be generated as an associative algebra by

[TABLE]

where $E_{kl}$ is the $k_{i}\times k_{i}$ matrix with 1 at the $(k,l)$-entry and 0 elsewhere. Denote $X^{{\bf n}}=\bigotimes_{i=1}^{z}X_{2i-1}^{n_{2i-1}}X_{2i}^{n_{2i}}$ for ${\bf n}\in{\mathbb{Z}}^{d}$. It is easy to see that for any ${\bf n}\in R$, $X^{\bf n}$ is the identity matrix in $M_{N}({\mathbb{C}})$, and $X^{\bf r}X^{\bf s}=\sigma({\bf r},{\bf s})X^{{\bf r}+{\bf s}}$ for any ${\bf r},{\bf s}\in{\mathbb{Z}}^{d}$. Then the general linear Lie algebra $\mathfrak{gl}_{N}=M_{N}({\mathbb{C}})$ has Lie bracket

[TABLE]

Set $\Gamma={\mathbb{Z}}^{d}/R$ and let $\overline{{\bf n}}$ denote the image of ${\bf n}\in{\mathbb{Z}}^{d}$. Clearly, $\Gamma$ has order $N$ and the algebra $\mathfrak{gl}_{N}$ has a $\Gamma$-gradation $\mathfrak{gl}_{N}=\bigoplus_{\overline{{\bf n}}\in\Gamma}(\mathfrak{gl}_{N})_{\overline{{\bf n}}}$ where $(\mathfrak{gl}_{N})_{\overline{{\bf n}}}={\mathbb{C}}X^{\bf n}$. In this paper by a $\Gamma$-gradation on $\mathfrak{gl}_{N}$ we always mean this gradation. Set

[TABLE]

which is a complete set of representatives for $\Gamma$. When a representative of $\overline{{\bf n}}$ is needed we always choose the one ${\bf n}\in\Gamma_{0}$.

Let $\mu\in{\mathbb{C}}^{d}$ be generic. A solenoidal Lie algebra $\mathfrak{g}_{\mu}$ over ${\mathbb{C}}_{Q}$ is the subalgebra of $\mathcal{D}$ spanned by

[TABLE]

We mention that the subalgebra $\mathcal{D}_{R}$ spanned by

[TABLE]

is isomorphic to the Lie algebra $W_{d}$ through the map defined by

[TABLE]

where ${\bf m}=B{\bf n}$ and $B$ is the $d\times d$ matrix $\mathrm{diag}\{k_{1},k_{1},k_{2},k_{2},\cdots,k_{z},k_{z},1,\cdots,1\}$.

3 Finite dimensional $\mathcal{G}$-modules

In this section we introduce the Lie algebra $\mathcal{G}$ and study its modules of finite dimension. For ${\bf s}\notin R$ we will denote by $t^{\bf\overline{s}}$ the image of $t^{\bf s}$ in ${\mathbb{C}}_{Q}/\mathcal{Z}$. Let $AC={\mathbb{C}}[x_{1}\cdots,x_{d}]\oplus{\mathbb{C}}_{Q}/\mathcal{Z}$ and $x_{i}t_{j}=t_{j}x_{i}$ for all $1\leq i,j\leq d$. Then $AC$ becomes an associative algebra. For ${\bf u}\in{\mathbb{C}}^{d}$ set

[TABLE]

Denote by $\mathcal{G}$ the Lie subalgebra of $\mathrm{Der}(AC)$ spanned by

[TABLE]

The Lie algebra $\mathcal{G}$ has Lie bracket

[TABLE]

where ${\bf u},{\bf v}\in{\mathbb{C}}^{d},{\bf m},{\bf n}\in{\mathbb{N}}^{d}\backslash\{{\bf 0}\},{\bf p},{\bf l}\in{\mathbb{N}}^{d}$ and ${\bf r},{\bf s}\in\Gamma_{0}$. It is easy to see that $\mathcal{G}$ is $\Gamma$-graded, i.e., $\mathcal{G}=\bigoplus_{\overline{{\bf s}}\in\Gamma}\mathcal{G}(\overline{{\bf s}})$, where

[TABLE]

On the other hand, $\mathcal{G}$ may be decomposed into the sum of two subalgebras

[TABLE]

and

[TABLE]

Set $\deg{x_{i}}=1,\ \deg{\frac{\partial}{\partial{x_{i}}}}=-1$ for all $1\leq i\leq d$, and for ${\bf m}\in{\mathbb{N}}^{d}$ write $|{\bf m}|=\sum_{i=1}^{d}m_{i}$. This gives ${\mathbb{Z}}$-gradations on the algebras $\mathcal{G}^{(x)}$ and $\mathcal{G}^{(t)}$. For $i\in{\mathbb{N}}$, set

[TABLE]

Then we have $\mathcal{G}^{(x)}=\bigoplus_{i\in{\mathbb{N}}}\mathcal{G}^{(x)}_{i}$ and $\mathcal{G}^{(t)}=\bigoplus_{i\in{\mathbb{N}}}\mathcal{G}^{(t)}_{i}$.

Set $\mathcal{G}_{i}=\mathcal{G}^{(x)}_{i}+\mathcal{G}^{(t)}_{i}$, which is a subspace of $\mathcal{G}$. These subspaces $\mathcal{G}_{i}$ do not make the whole algebra $\mathcal{G}\ {\mathbb{Z}}$-graded, since the derivations $d_{{\bf u}},\ {\bf u}\in{\mathbb{C}}^{d}$, are not homogeneous. However, $\mathcal{G}_{+}=\bigoplus_{i\geq 1}\mathcal{G}_{i}$ still makes a $\Gamma$-graded ideal of $\mathcal{G}$. Write $\mathcal{G}^{(x)}_{+}=\bigoplus_{i\geq 1}\mathcal{G}^{(x)}_{i}$ and $\mathcal{G}^{(t)}_{+}=\bigoplus_{i\geq 1}\mathcal{G}^{(t)}_{i}$. The main result in this section is the following

Theorem 3.1**.**

*(1) The commutator $[\mathcal{G}^{(x)},\mathcal{G}^{(x)}]=[\mathcal{G}^{(x)}_{0},\mathcal{G}^{(x)}_{0}]\oplus\mathcal{G}^{(x)}_{\geq 1}$;

(2) For every finite dimensional $\mathcal{G}^{(x)}$-module $U$, there exists an integer $p\gg 0$ such that $\mathcal{G}^{(x)}_{p}U=0$.

(3) For every finite dimensional $\mathcal{G}$-module $U$, there exists an integer $p\gg 0$ such that $\mathcal{G}_{p}U=0$.

(4) The ideal $\mathcal{G}_{+}$ annihilates any finite dimensional irreducible $\mathcal{G}$-module.*

Proof. (1) is easy to check by equation (3.1). For (2) we choose a basis $\mu_{1},\cdots,\mu_{d}$, of ${\mathbb{C}}^{d}$ with all $\mu_{i}$ being generic. Then $\mathcal{G}^{(x)}$ has a decomposition $\mathcal{G}^{(x)}=\bigoplus_{i=1}^{d}\mathcal{G}^{(x,\mu_{i})}$, where $\mathcal{G}^{(x,\mu_{i})}=\mathrm{span}_{\mathbb{C}}\left\{x^{\bf m}d_{\mu_{i}}\mid{\bf m}\in{\mathbb{N}}^{d}\backslash\{{\bf 0}\}\right\}$ are subalgebras of $\mathcal{G}^{(x)}$. Notice that for each $1\leq i\leq d$, the subalgebra $\mathcal{G}^{(x,\mu_{i})}$ is exactly the infinite dimensional Lie algebra $\mathcal{L}_{+}$ studied in [BF1]. By Theorem 3.1 in [BF1] we see that there exists an integer $p_{i}\gg 0$ such that $(\mathcal{G}^{(x,\mu_{i})}\cap\mathcal{G}^{(x)}_{p_{i}})U=0$. Then (2) stands for the integer $p=\max\{p_{1},\cdots,p_{d}\}$.

To prove (3) we consider $U$ as an $\mathcal{G}^{(x)}$-module and let $p$ be as in (2). For ${\bf s}\in\Gamma_{0}$ choose ${\bf u}\in{\mathbb{C}}^{d}$ such that $({\bf u}\mid{\bf s})\neq 0$. Then the equation

[TABLE]

implies that $(x^{\bf m}t^{\bf\overline{s}})U=0$ if $|{\bf m}|\geq p$. This proves (3).

Let $V$ be a finite dimensional irreducible $\mathcal{G}$-module with representation map $\rho$. Since $\mathcal{G}_{+}$ is an ideal of $\mathcal{G}$, the subspace $V^{\prime}=\{v\in V\mid\mathcal{G}_{+}v=0\}$ is a $\mathcal{G}$-submodule of $V$. By (3) and equation (3.1) we see that $\rho(\mathcal{G}_{+})$ is nilpotent. Then the Lie’s Theorem implies that $V^{\prime}$ is nonzero, hence $V^{\prime}=V$ by the irreducibility of $V$. So $\mathcal{G}_{+}V=0$, proving (4). $\Box$

At last we mention that the quotient $\mathcal{G}/\mathcal{G}_{+}$ is isomorphic to the direct sum $\mathfrak{gl}_{d}\oplus\mathfrak{gl}_{N}$ as Lie algebras. The isomorphism map is given by

[TABLE]

where ${\bf r}\in\Gamma_{0},1\leq i,j\leq d$ and $E_{ij}$ is the $d\times d$ matrix with 1 at the $(i,j)$-th entry and 0 elsewhere. Through this map we also have $\mathcal{G}^{(x)}/\mathcal{G}^{(x)}_{+}\cong\mathfrak{gl}_{d}$ and $\mathcal{G}^{(t)}/\mathcal{G}^{(t)}_{+}\cong\mathfrak{gl}_{N}$.

4 Cuspidal $\mathcal{Z}\mathcal{D}$-modules

In this section we study cuspidal $\mathcal{Z}\mathcal{D}$-modules with support lying in one coset $\alpha+{\mathbb{Z}}^{d}$ for some $\alpha\in{\mathbb{C}}^{d}$. Fix $M=\bigoplus_{{\bf s}\in{\mathbb{Z}}^{d}}M_{\alpha+{\bf s}}$ such a $\mathcal{Z}\mathcal{D}$-module, where $M_{\alpha+{\bf s}}$ is the weight space with weight $\alpha+{\bf s}$. Clearly we have $t^{\bf r}M_{\alpha+{\bf s}}\subseteq M_{\alpha+{\bf r}+{\bf s}}$. Since $M$ is a free $\mathcal{Z}$-module, we may write

[TABLE]

where $U_{\overline{{\bf s}}}=M_{\alpha+{\bf s}}$ for ${\bf s}\in\Gamma_{0}$. Set $U=\bigoplus_{{\bf s}\in\Gamma_{0}}U_{\overline{{\bf s}}}$, which is a $\Gamma$-graded space.

We should introduce some operators in ${\mathcal{U}}(\mathcal{D}\ltimes\mathcal{Z})$ that act on $U$. Set $D({{\bf u}},{{\bf m}})=t^{\bf-m}\partial({\bf u},{\bf m})$ for ${\bf u}\in{\mathbb{C}}^{d}$ and ${\bf m}\in R$, acts on each $U_{\overline{{\bf s}}}$, hence on $U$. For ${\bf r}\in\Gamma_{0},{\bf n}\in R$, define the operator $L({{\bf n},{\bf r}})$ to be the restriction of $t^{\bf-n}t^{{\bf n}+{\bf r}}$ mapping $U_{\overline{{\bf s}}}$ to $U_{\bf\overline{r+s}}$ for each $\overline{{\bf s}}\in\Gamma$. The commutators among these operators are

[TABLE]

where ${\bf u},{\bf v}\in{\mathbb{C}}^{d},{\bf m},{\bf n}\in R,{\bf r},{\bf s}\in\Gamma_{0}$. Since for $v\in U_{\overline{{\bf s}}},\ {\bf n}\in R$, we have

[TABLE]

the operators $D({{\bf u}},{{\bf m}}),L({{\bf m},{\bf r}})$ completely determine the $\mathcal{D}$-action on $M$. Furthermore, the operators $D({{\bf u}},{{\bf m}})$ may be restricted to $U_{\overline{{\bf s}}}$, and $L({{\bf n},{\bf r}})$ may be considered as an operator from $U_{\overline{{\bf s}}}$ to $U_{\bf\overline{r+s}}$ for each $\overline{{\bf s}}\in\Gamma$. In other words, $D({{\bf u}},{{\bf m}})$ are of degree $\overline{{\bf 0}}$ and $L({{\bf n},{\bf r}})$ are of degree $\overline{{\bf r}}$ in $\mathrm{End}(U)$.

Let $\mu_{1},\cdots,\mu_{d}$ be a basis of ${\mathbb{C}}^{d}$ with all $\mu_{i}$ being generic. Recall the solenoidal Lie algebra $\mathfrak{g}_{\mu}$ over ${\mathbb{C}}_{Q}$, the solenoidal Lie algebra $W_{\mu}$ over the commuting torus $A={\mathbb{C}}[x_{1}^{\pm 1},\cdots,x_{d}^{\pm 1}]$. Then the algebra $\mathcal{D}$ may be decomposed into direct sum of subalgebras

[TABLE]

Here $W_{\mu_{i}}$ are solenoidal Lie algebras over the commuting torus $\mathcal{Z}$. By results from [BF1] and [Xu2], we know that the operators $D({{\bf\mu}_{i}},{{\bf m}}),1\leq i\leq d$ and $L({{\bf m},{\bf r}}),{\bf r}\in\Gamma_{0}$, have polynomial dependence on ${\bf m}\in R$ with coefficients in $\mathrm{End}(U)$. Moreover, the constant term of $D({{\bf\mu}_{i}},{{\bf m}})$ on each $U_{\overline{{\bf s}}}$ is $({\bf u}\mid\alpha+{\bf s})Id$. Hence $D({{\bf u}},{{\bf m}})$ are polynomials on ${\bf m}$, since they are linear combinations of $D({{\bf\mu}_{i}},{{\bf m}})$.

Set

[TABLE]

where $f_{{\bf u}}({\bf p})$ are operators in $\mathrm{End}(U)$ of degree $\overline{{\bf 0}}$ and $g_{{\bf r}}({\bf p})\in\mathrm{End}(U)$ are operators of degree $\overline{{\bf r}}$.

Now by expanding the equations in (4.1) and comparing coefficients at both sides, we get the following commutators

[TABLE]

where ${\bf u},{\bf v}\in{\mathbb{C}}^{d},{\bf p},{\bf l}\in{\mathbb{N}}^{d}$ and ${\bf r},{\bf s}\in\Gamma_{0}$. Recall the Lie bracket of the algebra $\mathcal{G}$ from equation (3.1) and we see from equations in (4.3) that the operators

[TABLE]

yield a finite dimensional $\Gamma$-graded representation for $\mathcal{G}$ on $U$. Denote by $\rho$ this representation map. Then combining equation (4.2) we obtain

Theorem 4.1**.**

There exists an equivalence between the category of finite dimensional $\Gamma$-graded $\mathcal{G}$-modules and the category of cuspidal $\mathcal{Z}\mathcal{D}$-modules with support lying in some coset $\alpha+{\mathbb{Z}}^{d}$. This equivalence functor associates to a finite dimensional $\Gamma$-graded $\mathcal{G}$-module $U=\bigoplus_{\overline{{\bf s}}\in\Gamma}U_{\overline{{\bf s}}}$ an $\mathcal{Z}\mathcal{D}$-module

[TABLE]

with the $\mathcal{D}$-action

[TABLE]

where ${\bf m},{\bf n}\in R,{\bf r}\in\Gamma_{0}\backslash R,{\bf s}\in\Gamma_{0}$ and $v_{\overline{{\bf s}}}\in U_{\overline{{\bf s}}}$.

As a consequence we classify irreducible cuspidal $\mathcal{Z}\mathcal{D}$-modules, which was first done in [LiuZ2]. First we recall modules of tensor fields over the algebra $\mathcal{D}$. Let $\alpha\in{\mathbb{C}}^{d}$ and let $V$ be a finite dimensional $\mathfrak{gl}_{d}$-module, $W=\bigoplus_{\overline{{\bf s}}\in\Gamma}W_{\overline{{\bf s}}}$ a finite dimensional $\Gamma$-graded $\mathfrak{gl}_{N}$-module. On the tensor space $\bigotimes_{{\bf s}\in{\mathbb{Z}}^{d}}(V\otimes W_{\overline{{\bf s}}}\otimes t^{\bf s})$ there is a $\mathcal{D}$-module structure defined by

[TABLE]

for ${\bf u}\in{\mathbb{C}}^{d},{\bf m}\in R,{\bf r}\notin R,{\bf s}\in{\mathbb{Z}}^{d},v\in V$ and $w\in W_{\overline{{\bf s}}}$. We denote this module by ${\mathcal{V}^{\alpha}(V,W)}$, called a module of tensor fields for $\mathcal{D}$. Clearly, ${\mathcal{V}^{\alpha}(V,W)}$ becomes a $\mathcal{Z}\mathcal{D}$-module provided the $\mathcal{Z}$-action

[TABLE]

Moreover, ${\mathcal{V}^{\alpha}(V,W)}$ is irreducible as a $\mathcal{Z}\mathcal{D}$-module as long as $V,W$ are irreducible.

Theorem 4.2**.**

Any irreducible cuspidal $\mathcal{Z}\mathcal{D}$-module is isomorphic to ${\mathcal{V}^{\alpha}(V,W)}$ for some $\alpha\in{\mathbb{C}}^{d}$, finite dimensional irreducible $\mathfrak{gl}_{d}$-module $V$ and finite dimensional $\Gamma$-graded irreducible $\mathfrak{gl}_{N}$-module $W$.

By applying Theorem 4.1, we will prove this theorem again in a rather conceptional way. Let $M=\bigoplus_{{\bf s}\in{\mathbb{Z}}^{d}}M_{\alpha+{\bf s}}$ be an irreducible cuspidal $\mathcal{Z}\mathcal{D}$-module. By Theorem 4.1, $M\cong M^{\prime}\otimes\mathcal{Z}$, where $M^{\prime}=\bigoplus_{\overline{{\bf s}}\in\Gamma}M_{\alpha+{\bf s}}$ is a finite dimensional irreducible $\Gamma$-graded $\mathcal{G}$-module. By Theorem 3.1(4) we have $\mathcal{G}_{+}M^{\prime}=0$. Notice that the ideal $\mathcal{G}_{+}$ is still $\Gamma$-graded. This reduces $M^{\prime}$ to a $\Gamma$-graded and irreducible module over $\mathcal{G}/\mathcal{G}_{+}$, which is isomorphic to $\mathfrak{gl}_{d}\oplus\mathfrak{gl}_{N}$. The following lemma is from [Li].

Lemma 4.3**.**

Let $A_{1},A_{2}$ be two associative algebras with identity and $V$ be an irreducible module over $A_{1}\otimes A_{2}$. Suppose that $A_{1}$ is of countable dimension. Then $V$ is isomorphic to an $A_{1}\otimes A_{2}$-module of the form $V_{1}\otimes V_{2}$, where $V_{1}$ is an irreducible $A_{1}$-submodule of $V$, and $V_{2}=\mathrm{Hom}_{A_{1}}(V_{1},V)$ is a natural irreducible $A_{2}$-module given by

[TABLE]

Denote $\mathfrak{gl}=\mathfrak{gl}_{d}\oplus\mathfrak{gl}_{N}$, which is $\Gamma$-graded with graded spaces $\mathfrak{gl}_{\overline{{\bf 0}}}=\mathfrak{gl}_{d}\oplus{\mathbb{C}}X^{\bf 0}$ and $\mathfrak{gl}_{\overline{{\bf s}}}={\mathbb{C}}X^{\bf s}$ for $\overline{{\bf s}}\in\Gamma\backslash\{\overline{{\bf 0}}\}$. Fix a $\overline{{\bf s}}\in\Gamma$ such that $M_{\alpha+{\bf s}}\neq 0$. Notice that $M_{\alpha+{\bf s}}$ is a $\mathfrak{gl}_{d}$-module through the $\mathcal{G}^{(x)}/\mathcal{G}^{(x)}_{+}$-action. Fix $V$ an irreducible $\mathfrak{gl}_{d}$-submodule of $M_{\alpha+{\bf s}}$ and set $W_{\overline{{\bf r}}}=\mathrm{Hom}_{\mathfrak{gl}_{d}}(V,M_{\alpha+{\bf r}+{\bf s}})$ for any $\overline{{\bf r}}\in\Gamma$. Let

[TABLE]

and define

[TABLE]

where $1\otimes X^{\bf p}$ is an element in ${\mathcal{U}}(\mathfrak{gl}_{d})\otimes{\mathcal{U}}(\mathfrak{gl}_{N})={\mathcal{U}}(\mathfrak{gl}_{d}\oplus\mathfrak{gl}_{N})$. Clearly this makes $W$ a $\Gamma$-graded $\mathfrak{gl}_{N}$-module. By Lemma 4.3 we have $M^{\prime}\cong V\otimes W$. Moreover, $W$ is irreducible as a $\mathfrak{gl}_{N}$-module since $M^{\prime}$ is irreducible as a $\mathfrak{gl}_{d}\oplus\mathfrak{gl}_{N}$-module. This proves Theorem 4.2.

From the proof above one can see that all irreducible $\mathfrak{gl}_{d}$-submodules of $M^{\prime}$ are isomorphic to $V$.