Irreducible twisted Heisenberg-Virasoro modules from tensor products

TL;DR

This paper constructs new irreducible modules for the twisted Heisenberg-Virasoro algebra using tensor products of polynomial modules and induces modules, providing conditions for their irreducibility and novelty.

Contribution

It introduces a new class of irreducible modules for the twisted Heisenberg-Virasoro algebra via tensor products and induction, with explicit irreducibility criteria.

Findings

New irreducible modules constructed from tensor products.

Necessary and sufficient conditions for irreducibility and isomorphism.

Identification of these modules as novel in the representation theory context.

Abstract

In this paper, we realize polynomial $\H$-modules $\Omega(\lambda,\alpha,\beta)$ from irreducible twisted Heisenberg-Virasoro modules $\A_{\alpha,\beta}$. It follows from $\H$-modules $\Omega(\lambda,\alpha,\beta)$ and $\mathrm{Ind}(M)$ that we obtain a class of natural non-weight tensor product modules $\big(\bigotimes_{i=1}^m\Omega(\lambda_i,\alpha_i,\beta_i)\big)\otimes \mathrm{Ind}(M)$. Then we give the necessary and sufficient conditions under which these modules are irreducible and isomorphic, and also give that the irreducible modules in this class are new.

Equations

$$\bar{\mathcal{H}}:=\left\{f(t)\frac{d}{dt}+g(t)\,\Big{|}\,f(t),g(t)\in\mathbb{C}[t,t^{-1}]\right\}.$$

$$[L_m,L_n]=(n-m)L_{m+n}+{\delta }_{m+n,0}\frac{m^3-m}{12}{C}_1,$$

$$[L_m,I_n]=n{I}_{m+n}+{\delta }_{m+n,0}(m^2+m)C_2,$$

$$[I_m,I_n]=n{\delta }_{m+n,0}{C}_3,$$

$$[\mathcal{H},C_1]=[\mathcal{H},C_2]=[\mathcal{H},C_3]=0.$$

$$\mathcal{K}/(\mathcal{K}\cap\big{(}\mathbb{C}(t)[\partial]\mu)\big{)}$$

$$t^m{\partial }^n={\lambda }^m(\partial -m{)}^n,\partial {\partial }^n={\partial }^{n+1}$$

$$\mbox{$\sum\limits_{m\in\mathbb{Z}_{+}}$}(\mathbb{C}L_{m}\oplus\mathbb{C}I_{m-e})\oplus\mathbb{C}C_{1}\oplus\mathbb{C}C_{2}\oplus\mathbb{C}C_{3}.$$

$$\mathrm{Ind}(M):=\mathcal{U}(\mathcal{H}){\otimes }_{\mathcal{U}({\mathcal{H}}_e)}M.$$

$$\sum _{i=1}^s\sum _{j=0}^k{\mu }_i^m{f}_{i,j}(m)v_{i,j}\in P_{1}forK<m\in \mathbb{Z}(Kanyfixedelementin\mathbb{Z}\cup \{-\infty \})$$

$$\displaystyle L_{m}v=\big{(}t^{m}\partial+m\alpha t^{m}\big{)}v,\ I_{m}v=\beta t^{m}v,\ C_{i}v=0$$

$$\displaystyle(L_{m}I_{n}-I_{n}L_{m})v=\beta\big{(}t^{m}\partial+m\alpha t^{m}\big{)}t^{n}v-\beta t^{n}\big{(}t^{m}\partial+m\alpha t^{m}\big{)}v=nI_{m+n}v.$$

$$\phi (t^{m}v)=t^m\phi (v)$$

$$\phi ({\partial }^{m}v)={\partial }^m\phi (v)$$

$$0=\varphi(L_{m}v)-L_{m}\varphi(v)=\varphi\big{(}(t^{m}\partial+m\alpha_{1}t^{m})v\big{)}-(t^{m}\partial+m\alpha_{2}t^{m})\partial\varphi(v)=m(\alpha_{1}-\alpha_{2})t^{m}\varphi(v).$$

$$t^m\cdot {\partial }^n={\lambda }^m(\partial -m{)}^n,\partial \cdot {\partial }^m={\partial }^{m+1}\mathrm{for}m\in \mathbb{Z},n\in {\mathbb{Z}}_{+}.$$

$$\displaystyle L_{m}\partial^{n}=\lambda^{m}\big{(}\partial+m(\alpha-1)\big{)}(\partial-m)^{n},\ I_{m}\partial^{n}=\lambda^{m}\beta(\partial-m)^{n}\quad{\rm for}\ m\in\mathbb{Z},n\in\mathbb{Z}_{+}.$$

$$\mathcal{A}=\mathcal{K}/\big{(}\mathcal{K}\cap(\mathbb{C}(t)[\partial]\mu)\big{)}=\mathcal{K}/(\mathcal{K}\mu)$$

$$\partial \cdot t^n=t^n(\gamma +n),t^m\cdot t^n=t^{m+n}\mathrm{for}m,n\in \mathbb{Z}.$$

$$L_m{t}^n=(\gamma +n+m\alpha )t^{m+n},I_m{t}^n=\beta t^{m+n}\mathrm{for}m,n\in \mathbb{Z}.$$

$$\mathcal{A}=\mathcal{K}/\big{(}\mathcal{K}\cap(\mathbb{C}(t)[\partial]\mu)\big{)}=\mathcal{K}/(\mathcal{K}\mu),$$

$$t^m\cdot t^n=t^{m+n},t^m\cdot (t^n\partial )=t^{m+n}\partial ,$$

$$\partial \cdot t^n=t^n(\partial +n),\partial \cdot (t^n\partial )=t^n(f(t)+n\partial ),$$

$$L_m\cdot t^n=t^{m+n}(n+m\alpha +\partial ),L_m\cdot (t^n\partial )=t^{m+n}(f(t)+m\alpha +n\partial ),$$

$$I_m\cdot t^n=\beta t^{m+n},I_m\cdot (t^n\partial )=\beta t^{m+n}\partial .$$

$$\mathcal{A}=\mathcal{K}/(\mathcal{K}\cap (\mathbb{C}(t)[\partial ]\mu ))=\mathcal{K}/(\mathcal{K}\mu ),$$

$$t^k\cdot (t^r(\frac{d}{dt}{)}^m)=t^{k+r}(\frac{d}{dt}{)}^m\mathrm{for}k,r\in \mathbb{Z},0\le m\le n-1,$$

$$\frac{d}{dt}\cdot (t^r(\frac{d}{dt}{)}^m)=r{t}^{r-1}(\frac{d}{dt}{)}^m+t^r(\frac{d}{dt}{)}^{m+1}\mathrm{for}r\in \mathbb{Z},0\le m<n-1,$$

$$\frac{d}{dt}\cdot (t^r(\frac{d}{dt}{)}^{n-1})=r{t}^{r-1}(\frac{d}{dt}{)}^{n-1}+t^{r+1}\mathrm{for}r\in \mathbb{Z}.$$

$$L_k\cdot (t^r(\frac{d}{dt}{)}^m)=(r{t}^{k+r}+\alpha k{t}^{k+r+1})(\frac{d}{dt}{)}^m+t^{k+r+1}(\frac{d}{dt}{)}^{m+1},$$

$$L_k\cdot (t^r(\frac{d}{dt}{)}^{n-1})=(r{t}^{k+r}+\alpha k{t}^{k+r+1})(\frac{d}{dt}{)}^{n-1}+t^{k+r+2},$$

$$I_k\cdot (t^r(\frac{d}{dt}{)}^m)=\beta t^{k+r}(\frac{d}{dt}{)}^m,$$

$$L_{k}\partial_{i}^{n}=\lambda_{i}^{k}\big{(}\partial_{i}+k\alpha_{i})(\partial_{i}-k)^{n},\ I_{k}\partial_{i}^{n}=\lambda_{i}^{k}\beta_{i}(\partial_{i}-k)^{n},\ C_{j}\partial_{i}^{n}=0$$

$$\{{\partial }_1^{p_1}\cdots {\partial }_m^{p_m}\otimes v\mid \mathbf{P}=(p_1,\dots ,p_m)\in {\mathbb{Z}}_{+}^m,m\in \mathbb{N},0\ne v\in \mathrm{Ind}(M)\},$$

$${\partial }_1^{p_1}\cdots {\partial }_m^{p_m}\otimes u\prec {\partial }_1^{q_1}\cdots {\partial }_m^{q_m}\otimes v⟺\exists k\in \mathbb{N}\mathrm{such}\mathrm{that}{p}_k<q_k\mathrm{and}{p}_n=q_n\mathrm{for}n<k.$$

Full text

Irreducible twisted Heisenberg-Virasoro modules from tensor products

Haibo Chen, Yucai Su

††​​​​​​​​​​Haibo Chen: School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China. e-mail: rebel1025@126.com. Yucai Su: School of Mathematical Sciences, Tongji University, Shanghai 200092, China. e-mail: ycsu@tongji.edu.cn. Key words: twisted Heisenberg-Virasoro algebra, tensor product module, irreducible module

Mathematics Subject Classification: 17B10 $\cdot$ 17B65 $\cdot$ 17B68

ABSTRACT. In this paper, we realize polynomial $\mathcal{H}$-modules $\Omega(\lambda,\alpha,\beta)$ from irreducible twisted Heisenberg-Virasoro modules $\mathcal{A}_{\alpha,\beta}$. It follows from $\mathcal{H}$-modules $\Omega(\lambda,\alpha,\beta)$ and $\mathrm{Ind}(M)$ that we obtain a class of natural non-weight tensor product modules \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M). Then we give the necessary and sufficient conditions under which these modules are irreducible and isomorphic, and also give that the irreducible modules in this class are new.

1 Introduction

It is well-known that the twisted Heisenberg-Virasoro algebra $\mathcal{H}$ [1] is the universal central extension of the Lie algebra $\bar{\mathcal{H}}$ of differential operators of order at most one on the Laurent polynomial algebra $\mathbb{C}[t,t^{-1}]$, where

[TABLE]

More precisely, $\mathcal{H}$ is an infinite dimensional Lie algebra with $\mathbb{C}$-basis $\{L_{m},I_{m},C_{i}\mid m\in\mathbb{Z},\,i=1,2,3\}$ subject to the Lie bracket as follows:

[TABLE]

It is clear that the subspaces spanned by $\{I_{m},C_{3}\mid 0\neq m\in\mathbb{Z}\}$ and by $\{L_{m},C_{1}\mid m\in\mathbb{Z}\}$ are respectively the Heisenberg algebra and the Virasoro algebra. Notice that the center of $\mathcal{H}$ is spanned by $\{I_{0},C_{i}\mid i=1,2,3\}$.

The representation theory on the twisted Heisenberg-Virasoro algebra has attracted a lot of attention from mathematicians and physicists. The theory of weight twisted Heisenberg-Virasoro modules with finite-dimensional weight spaces is fairly well-developed (see [15, 9, 12]). While weight modules with an infinite dimensional weight spaces were also studied (see [4, 14]). In the last few years, various families of non-weight irreducible twisted Heisenberg-Virasoro modules were investigated (see, e.g., [2, 3, 6, 5, 4, 10]). These are basically various versions of Whittaker modules and $\mathcal{U}(\mathbb{C}L_{0})$-free modules constructed using different tricks. However, the theory of representation over the twisted Heisenberg-Virasoro algebra is far more from being well-developed.

In the present paper, we construct a class of non-weight $\mathcal{H}$-modules by taking tensor products of a finite number of irreducible $\mathcal{H}$-modules $\Omega(\lambda,\alpha,\beta)$ with irreducible $\mathcal{H}$-modules $\mathrm{Ind}(M)$. At the same time, inspired by [11], a class of $\mathcal{H}$-modules $\mathcal{A}_{\alpha,\beta}$ are given. Then many interesting examples for such irreducible twisted Heisenberg-Virasoro modules with different features are provided. In particular, a class of irreducible polynomial modules $\Omega(\lambda,\alpha,\beta)$ over the twisted Heisenberg-Virasoro algebra are defined.

We briefly give a summary of the paper below. In Sections $2$ and $3$, we recall some known results and construct a class of modules $\mathcal{A}_{\alpha,\beta}$ over the twisted Heisenberg-Virasoro algebra. In Section $4$, we determine the necessary and sufficient conditions for \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) to be irreducible. In Section $5$, we present the necessary and sufficient conditions for $\mathcal{H}$-modules \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) to be isomorphic. Finally, we prove $\mathcal{H}$-modules \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) are new.

Throughout this paper, we respectively denote by $\mathbb{C},\mathbb{C}^{*},\mathbb{Z},\mathbb{Z}_{+}$ and $\mathbb{N}$ the sets of complex numbers, nonzero complex numbers, integers, nonnegative integers and positive integers. All vector spaces are assumed to be over $\mathbb{C}$.

2 Some known results

In this section, we recall some definitions and known results.

Let $\mathbb{C}[t]$ be the (associative) polynomial algebra. Denote $\partial:=t\frac{d}{dt}$. We know that $\partial t^{n}=t^{n}(\partial+n)$ for $n\in\mathbb{Z}$. It is clear that the associative algebra $\widetilde{\mathcal{A}}=\mathbb{C}[t,\partial]$ is a proper subalgebra of the rank $1$ Weyl algebra $\mathbb{C}[t,\frac{d}{dt}]$. We note that $\widetilde{\mathcal{A}}$ is the universal enveloping algebra of the $2$-dimensional solvable Lie algebra $\mathfrak{a}_{1}=\mathbb{C}L_{0}\oplus\mathbb{C}L_{1}$, which subjects to $[L_{0},L_{1}]=L_{1}$. Let $\mathcal{K}=\mathbb{C}[t,t^{-1},\partial]$ be the Laurent polynomial differential operator algebra.

Definition 2.1**.**

Let $\mathcal{P}$ be an associative or Lie algebra over $\mathbb{C}$ and $\mathcal{Q}$ be a subspace of $\mathcal{P}$. A module $V$ over $\mathcal{P}$ is called $\mathcal{Q}$-torsion if there exists a nonzero $q\in\mathcal{Q}$ such that $qv=0$ for some nonzero $v\in V$; otherwise $V$ is called $\mathcal{Q}$-torsion-free.

Let us recall some results about irreducible modules over the associative algebra $\mathcal{K}$.

Lemma 2.2**.**

[11]* Let $V$ be any $\mathbb{C}[t]$-torsion-free irreducible module over the associative algebra $\widetilde{\mathcal{A}}$. Then $V$ can be extended into a module over the associative algebra $\mathcal{K}=\mathbb{C}[t,t^{-1},\frac{d}{dt}]$, i.e., the action of $\widetilde{\mathcal{A}}$ on $V$ is a restriction of an irreducible $\mathbb{C}[t,t^{-1}]$-torsion-free $\mathcal{K}$-module.*

Lemma 2.3**.**

[11]* Let $\mu$ be an irreducible element in the associative algebra $\mathbb{C}(t)[\partial]$. Then*

[TABLE]

is a $\mathbb{C}[t,t^{-1}]$-torsion-free irreducible module over the associative algebra $\mathcal{K}$. Moreover any $\mathbb{C}[t,t^{-1}]$-torsion-free irreducible module over the associative algebra $\mathcal{K}$ can be obtained in this way.

For any $\lambda\in\mathbb{C}^{*}$, define a $\mathcal{K}$-module structure on the space $\Omega(\lambda)=\mathbb{C}[\partial]$, the polynomial algebra in $\partial$, by

[TABLE]

for all $n\in\mathbb{Z}_{+},m\in\mathbb{Z}$. Then $\Omega(\lambda)$ is an irreducible module over the associative algebra $\mathcal{K}$ for any $\lambda\in\mathbb{C}^{*}$ (see [11]).

Lemma 2.4**.**

[11]* Let $V$ be an irreducible module over the associative algebra $\mathcal{K}$ on which $\mathbb{C}[t,t^{-1}]$ is torsion. Then $V\cong\Omega(\lambda)$ for some $\lambda\in\mathbb{C}^{*}$.*

Combining Lemmas 2.3 and 2.4, a classification for all irreducible modules over the associative algebra $\mathcal{K}$ are obtained.

Now let us recall a large class of irreducible $\mathcal{H}$-modules, which includes the known irreducible modules such as highest weight modules and Whittaker modules. For any $e\in\mathbb{Z}_{+}$, denote by $\mathcal{H}_{e}$ the subalgebra

[TABLE]

Take $M(c_{0},c_{1},c_{2},c_{3})$ to be an irreducible $\mathcal{H}_{e}$-module such that $I_{0},C_{1},C_{2}$ and $C_{3}$ act on it as scalars $c_{0},c_{1},c_{2},c_{3}$ respectively. For convenience, we denote $M(c_{0},c_{1},c_{2},c_{3})$ by $M$ and form the induced $\mathcal{H}$-module

[TABLE]

Theorem 2.5**.**

[2]* Let $e\in\mathbb{Z}_{+}$ and $M$ be a simple $\mathcal{H}_{e}$-module with $c_{3}=0$. Assume there exists $k\in\mathbb{Z}_{+}$ such that*

• (1)

\left\{\begin{array}[]{llll}\mbox{the\ action\ of}\ I_{k}\ \mbox{on}\ M\ \mbox{is\ injective}&\mbox{if \ }k\neq 0,\\[4.0pt] c_{0}+(n-1)c_{2}\neq 0\quad\mathrm{for\ all}\ n\in\mathbb{Z}\setminus\{0\}&\mbox{if \ }k=0,\end{array}\right.**

• (2)

$I_{n}M=L_{m}M=0$* for all $n>k$ and $m>k+e$.*

Then

• (i)

$\mathrm{Ind}(M)$* is a simple $\mathcal{H}$-module**;***

• (ii)

the actions of $I_{n},L_{m}$ on $\mathrm{Ind}(M)$ for all $n>k$ and $m>k+e$ are locally nilpotent.

The following result will be used in the following (see [13]).

Proposition 2.6**.**

Let $P$ be a vector space over $\mathbb{C}$ and $P_{1}$ a subspace of $P$. Suppose that $\mu_{1},\mu_{2},\ldots,\mu_{s}\in\mathbb{C}^{*}$ are pairwise distinct, $v_{i,j}\in P$ and $f_{i,j}(t)\in\mathbb{C}[t]$ with $\mathrm{deg}\,f_{i,j}(t)=j$ for $i=1,2,\ldots,s;j=0,1,2,\ldots,k.$ If

[TABLE]

then $v_{i,j}\in P_{1}$ for all $i,j$.

3 Realize $\mathcal{H}$-module $\Omega(\lambda,\alpha,\beta)$

Let $\mathcal{A}$ be an irreducible module over the associative algebra $\mathcal{K}$. For any $\alpha,\beta\in\mathbb{C}$, we define the action of $\mathcal{H}$ on $\mathcal{A}$ as follows

[TABLE]

for $i\in\{1,2,3\},m\in\mathbb{Z},v\in\mathcal{A}$. Denote the above action by $\mathcal{A}_{\alpha,\beta}$.

Proposition 3.1**.**

For any $\alpha,\beta\in\mathbb{C}$, we obtain that $\mathcal{A}_{\alpha,\beta}$ is an $\mathcal{H}$-module under the action given in (3.1).

Proof.

It follows from (3.1) that we have

[TABLE]

That is, $L_{m}I_{n}-I_{n}L_{m}=nI_{n+m}$ holds on $\mathcal{A}_{\alpha,\beta}$. By [11], $L_{m}L_{n}-L_{n}L_{m}=(n-m)L_{n+m}$ holds on $\mathcal{A}_{\alpha,\beta}$. Finally, the relation $I_{m}I_{n}-I_{n}I_{m}=0$ on $\mathcal{A}_{\alpha,\beta}$ is trivial. Thus, we obtain that $\mathcal{A}_{\alpha,\beta}$ is an $\mathcal{H}$-module. ∎

Now we recall the necessary and sufficient conditions for $\mathcal{A}_{\alpha,\beta}$ to be irreducible (see [11]).

Theorem 3.2**.**

Let $\alpha,\beta\in\mathbb{C}$ and $\mathcal{A}$ be an irreducible module over the association algebra $\mathcal{K}$. Then $\mathcal{A}_{\alpha,\beta}$ as an irreducible $\mathcal{H}$-modules if and only if one of the following holds

• (1)

$\alpha\notin\{0,1\}$* or $\beta\neq 0$.*

• (2)

$\alpha=1,\beta=0$* and $\partial\mathcal{A}=\mathcal{A}$.*

• (3)

$\alpha=\beta=0$* and $\mathcal{A}$ is not isomorphic to the natural $\mathcal{K}$ module $\mathbb{C}[t,t^{-1}]$.*

The isomorphism results for modules $\mathcal{A}_{\alpha,\beta}$ as follows.

Theorem 3.3**.**

Let $\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\in\mathbb{C}$ and $\mathcal{A},\mathcal{B}$ be irreducible modules over the associative algebra $\mathcal{K}$. Then $\mathcal{A}_{\alpha_{1},\beta_{1}}\cong\mathcal{B}_{\alpha_{2},\beta_{2}}$ as $\mathcal{H}$-modules if and only if one of the following holds

• (1)

$\mathcal{A}\cong\mathcal{B}$* as $\mathcal{K}$-modules, $\alpha_{1}=\alpha_{2}$ and $\beta_{1}=\beta_{2}$.*

• (2)

$\mathcal{A}\cong\mathcal{B}$* as $\mathcal{K}$-modules, $\alpha_{1}=1,\alpha_{2}=0,\beta_{1}=\beta_{2}=0$ and $\partial\mathcal{A}=\mathcal{A}$.*

• (3)

$\mathcal{A}\cong\mathcal{B}$* as $\mathcal{K}$-modules, $\alpha_{1}=0,\alpha_{2}=1,\beta_{1}=\beta_{2}=0$ and $\partial\mathcal{B}=\mathcal{B}$.*

Proof.

(1) The sufficiency of the conditions is clear. Now suppose that $\varphi:\mathcal{A}_{\alpha_{1},\beta_{1}}\rightarrow\mathcal{A}_{\alpha_{2},\beta_{2}}$ is an $\mathcal{H}$-module isomorphism. For any $v\in\mathcal{A}$, we have $\varphi(I_{0}v)=I_{0}\varphi(v)$, which gives $\beta_{1}=\beta_{2}$. In particular, $\beta_{1}\neq 0$. We note that $\varphi(I_{m}v)=I_{m}\varphi(v)$, which implies

[TABLE]

for any $m\in\mathbb{Z}$. From $\varphi(L_{0}^{m}v)=L_{0}^{m}\varphi(v)$, we have

[TABLE]

for $m\in\mathbb{Z}$. Combining (3.2) and (3.3), we obtain $\mathcal{A}\cong\mathcal{B}$ as $\mathcal{K}$-modules. From (3.2) and (3.3), it is easy to get

[TABLE]

Then $\alpha_{1}=\alpha_{2}$. If $\beta_{1}=0$, these modules reduce to the Virasoro modules (see [11]). This is (1).

By the [11, Theorem 12], we get (2) and (3). ∎

Now we realize $\mathcal{H}$-modules $\Omega(\lambda,\alpha,\beta)$ from $\mathcal{A}_{\alpha,\beta}$. Let $\lambda\in\mathbb{C}^{*}$ and $\alpha,\beta\in\mathbb{C}$. Then we get the irreducible $\mathcal{K}$-module $\Omega(\lambda)$, which has a basis $\{\partial^{k}:k\in\mathbb{Z}_{+}\}$, and the $\mathcal{K}$-actions are given by

[TABLE]

According to (3.1) we have $\mathcal{H}$-modules $\Omega(\lambda,\alpha,\beta)=\mathbb{C}[\partial]$ with the action:

[TABLE]

Then $\Omega(\lambda,\alpha,\beta)$ is irreducible if and only if $\alpha\neq 1$ or $\beta\neq 0$ (see [3]). In the following sections, we will consider a class of tensor product $\mathcal{H}$-modules related to $\Omega(\lambda,\alpha,\beta)$.

Now we describe some other examples about irreducible $\mathcal{H}$-modules $\mathcal{A}_{\alpha,\beta}$, such as intermediate series modules, degree two modules and degree $n$ modules.

Example 3.4**.**

Let $\gamma\in\mathbb{C}[t,t^{-1}],\beta\in\mathbb{C}$ and $\mu=\partial-\gamma$ in Lemma 2.3. Then we obtain the irreducible $\mathcal{K}$-module

[TABLE]

with a basis $\{t^{k}:k\in\mathbb{Z}\}$. We see that the $\mathcal{K}$-actions on $\mathcal{A}$ are given by

[TABLE]

It follows from (3.1) that we get $\mathcal{H}$-modules $\mathcal{A}_{\gamma,\alpha,\beta}=\mathbb{C}[t,t^{-1}]$ with the action:

[TABLE]

If $\gamma\in\mathbb{C}\setminus\mathbb{Z}$ or $\alpha\notin\{0,1\}$ or $\beta\neq 0$, then $\mathcal{A}_{\gamma,\alpha,\beta}$ is an irreducible $\mathcal{H}$-module (see [9, 8]). In particular, $\gamma\in\mathbb{C}$ these modules $\mathcal{A}_{\gamma,\alpha,\beta}$ are the intermediate series modules of $\mathcal{H}$ (see [9, 7]).

Some degree two irreducible elements in $\mathbb{C}(t)[\partial]$ were first constructed in [11].

Example 3.5**.**

Let $f(t)\in\mathbb{C}[t,t^{-1}]$ be such that $\partial^{2}-f(t)$ is irreducible in $\mathbb{C}(t)[\partial]$. Take $\mu=\partial^{2}-f(t)$ in Lemma 2.3. Then one obtain the irreducible $\mathcal{K}$-module

[TABLE]

which has a basis $\{t^{k},t^{k}\partial:k\in\mathbb{Z}\}$. The $\mathcal{K}$-actions on $\mathcal{A}$ are given by

[TABLE]

where $m,n\in\mathbb{Z}$. From (3.1), for $\alpha\neq 1$ or $\beta\neq 0$, we have irreducible $\mathcal{H}$-modules $\mathcal{A}_{\alpha,\beta}=\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}[t,t^{-1}]\partial$ with the action:

[TABLE]

Some degree $n$ irreducible elements in $\mathbb{C}(t)[\partial]$ were first constructed in [11].

Example 3.6**.**

For any $n\in\mathbb{Z}_{+}\setminus\{0\}$, let $\mu=(\frac{d}{dt})^{n}-t$ in Lemma 2.3. Then we have the irreducible $\mathcal{K}$-module

[TABLE]

which has a basis $\{t^{k}(\frac{d}{dt})^{m}:k\in\mathbb{Z},m=0,1,\ldots,n-1\}$. The actions of $\mathcal{K}=\mathbb{C}[t,t^{-1}][\frac{d}{dt}]$ are given by

[TABLE]

Using (3.1), for $\alpha\neq 1$ or $\beta\neq 0$, we obtain irreducible $\mathcal{H}$-modules $\mathcal{A}_{\alpha,\beta}=\mathbb{C}[t,t^{-1}]\times(\Sigma_{i=0}^{n-1}\mathbb{C}(\frac{d}{dt})^{i})$ with the action:

[TABLE]

where $k,r\in\mathbb{Z},0\leq m<n-1$.

4 Irreducibilities

In this section, we will determine the irreducibility of \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M).

Now we introduce some notations. Let $m\in\mathbb{N},\lambda_{i},\alpha_{i},\beta_{i}\in\mathbb{C}$ for $i=1,2,\ldots,m$. Denote $\Omega(\lambda_{i},\alpha_{i},\beta_{i})=\mathbb{C}[\partial_{i}]$. The actions of $\mathcal{H}$ on $\Omega(\lambda_{i},\alpha_{i},\beta_{i})$ are

[TABLE]

for $k\in\mathbb{Z},n\in\mathbb{Z}_{+},i=1,2,\ldots,m,j=1,2,3$. Then $\Omega(\lambda_{i},\alpha_{i},\beta_{i})$ is irreducible if and only if $\alpha_{i}\neq 0$ or $\beta_{i}\neq 0$ for $i=1,\ldots,m.$ For convenience, we write $\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})=\mathbb{C}[\partial_{1},\partial_{2},\ldots,\partial_{m}]$ for $m\in\mathbb{N}$.

Now we consider the tensor product \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M). Define a total order $``\prec"$ on the subset

[TABLE]

by

[TABLE]

Then each non-zero element $w$ in \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) can be (uniquely) written as follows

[TABLE]

where $I$ is a finite subset of $\mathbb{Z}_{+}^{m}$ and the $v_{\mathbf{p}}$ are nonzero vectors of $\mathrm{Ind}(M)$. Now we define $\mathrm{deg}(w)=(p_{1},\ldots,p_{m})$, where $\partial_{1}^{p_{1}}\cdots\partial_{m}^{p_{m}}\otimes v_{\mathbf{p}}$ is the term with maximal order in the sum. Notice that $\mathrm{deg}(1\otimes v)=\mathbf{0}=(\underbrace{0,0,\ldots,0}_{m})$.

Lemma 4.1**.**

Let $S=\{1,2\},S^{\prime}\subseteq S,\lambda\in\mathbb{C}^{*},\beta_{i},\alpha_{j}\in\mathbb{C}^{*},\beta_{j}=0$ for $i\in S^{\prime},j\in S\setminus S^{\prime}$ and $s\in\mathbb{Z}_{+}$. Denote $W_{s}$ the vector subspace of $\Omega(\lambda,\alpha_{1},\beta_{1})\otimes\Omega(\lambda,\alpha_{2},\beta_{2})$ spanned by $\{f(\partial_{1})(\partial_{1}+\partial_{2})^{n}\mid n\in\mathbb{Z}_{+},0\leq\mathrm{deg}(f)\leq s\}$ or $\{(\partial_{1}+\partial_{2})^{n}f(\partial_{2})\mid n\in\mathbb{Z}_{+},0\leq\mathrm{deg}(f)\leq s\}$. Then $W_{s}$ is a submodule of $\Omega(\lambda,\alpha_{1},\beta_{1})\otimes\Omega(\lambda,\alpha_{2},\beta_{2})$.

Proof.

Without loss of generality, we may assume $\lambda=1.$ For any $n\in\mathbb{Z}_{+},f(\partial_{1})\in W_{s},k\in\mathbb{Z}$, it is easy to get

[TABLE]

By Theorem 9 of [16], we have L_{k}\big{(}f(\partial_{1})(\partial_{1}+\partial_{2})^{n}\big{)}\in W_{s}. By the similar method, we obtain L_{k}\big{(}(\partial_{1}+\partial_{2})^{n}f(\partial_{2})\big{)}\in W_{s} and I_{k}\big{(}(\partial_{1}+\partial_{2})^{n}f(\partial_{2})\big{)}\in W_{s}. Thus, $W_{s}$ is a submodule of $\Omega(\lambda,\alpha_{1},\beta_{1})\otimes\Omega(\lambda,\alpha_{2},\beta_{2})$, completing the proof. ∎

Corollary 4.2**.**

Let $S=\{1,2\},S^{\prime}\subseteq S,\lambda\in\mathbb{C}^{*},\beta_{i},\alpha_{j}\in\mathbb{C}^{*},\beta_{j}=0$ for $i\in S^{\prime},j\in S\setminus S^{\prime}$ and $s\in\mathbb{Z}_{+}$. Assume that $W_{s}$ is the subspace of $\Omega(\lambda,\alpha_{1},\beta_{1})\otimes\Omega(\lambda,\alpha_{2},\beta_{2})$, where $W_{s}$ is spanned by $\{f(\partial_{1})(\partial_{1}+\partial_{2})^{n}\mid n\in\mathbb{Z}_{+},0\leq\mathrm{deg}(f)\leq s\}$ or $\{(\partial_{1}+\partial_{2})^{n}f(\partial_{2})\mid n\in\mathbb{Z}_{+},0\leq\mathrm{deg}(f)\leq s\}$. Then $\Omega(\lambda,\alpha_{1},\beta_{1})\otimes\Omega(\lambda,\alpha_{2},\beta_{2})$ has a series of $\mathcal{L}$-submodules

[TABLE]

such that W_{s}/W_{s-1}\cong\Omega\big{(}\lambda,s+\alpha_{1}+\alpha_{2},\beta_{1}+\beta_{2}\big{)} as $\mathcal{L}$-module for each $s\geq 1$.

Proof.

For $s,n\in\mathbb{Z}_{+},k\in\mathbb{Z}$, it follows from Lemma 4.1 that we check

[TABLE]

From Corollary 10 of [16], we get

[TABLE]

By the similar method, we have I_{k}\big{(}\partial_{1}^{s}(\partial_{1}+\partial_{2})^{n}\big{)}\equiv\lambda^{k}(\beta_{1}+\beta_{2})\partial_{1}^{s}(\partial_{1}+\partial_{2}-k)^{n}(\mathrm{mod}\ W_{s-1}). and L_{k}\big{(}\partial_{1}^{s}(\partial_{1}+\partial_{2})^{n}\big{)}\equiv\lambda^{k}\partial_{1}^{s}\big{(}\partial_{1}+\partial_{2}-k(s+\alpha_{0}+\alpha_{1})\big{)}(\partial_{1}+\partial_{2}-k)^{n}(\mathrm{mod}\ W_{s-1}).

These show that the $\mathcal{L}$-module isomorphism $W_{s}/W_{s-1}\cong\Omega(\lambda,s+\alpha_{1}+\alpha_{2},\beta_{1}+\beta_{2}).$ ∎

By the similar method in Lemma 3 of [16], we get the following results.

Lemma 4.3**.**

Let $m\in\mathbb{N},\lambda_{i}\in\mathbb{C}^{*},\alpha_{i},\beta_{i}\in\mathbb{C}$ for $i=1,2,\ldots,m$ with the $\lambda_{i}$ pairwise distinct. Then $\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes v$ generates the $\mathcal{H}$-module \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M).

Now we are ready to prove the irreducibility of $\mathcal{H}$-module \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M).

Theorem 4.4**.**

Let $m\in\mathbb{N},S=\{1,\ldots,m\},S^{\prime}\subseteq S$ and $\lambda_{i}\in\mathbb{C}^{*}$ for $i\in S$ with the $\lambda_{i}$ pairwise distinct. Let $\beta_{i},\alpha_{j}\in\mathbb{C}^{*},\beta_{j}=0$ for $i\in S^{\prime},j\in S\setminus S^{\prime}$. Assume $\mathrm{Ind}(M)$ is an $\mathcal{H}$-module defined by (2.1) for which $M$ satisfies the conditions in Theorem 2.5. Then the tensor product \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) is an irreducible $\mathcal{H}$-module.

Proof.

For any $w\in\mathrm{Ind}(M)$, there exists $K(w)\in\mathbb{Z}_{+}$ such that $L_{k}\cdot w=I_{k}\cdot w=0$ for all $k\geq K(w)$ by Theorem 2.5. Suppose $W$ is a nonzero submodule of \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M). Choose a nonzero element $u\in W$ with minimal degree. We claim that $\mathrm{deg}(u)=0$. If not, we assume

[TABLE]

where $I$ is a finite subset of $\mathbb{Z}_{+}^{m}$ and $w_{\mathbf{p}}$ are nonzero vectors of $\mathrm{Ind}(M)$. Let $\partial_{1}^{p_{1}}\cdots\partial_{m}^{p_{m}}\otimes w_{\mathbf{p}}$ be maximal among the terms in the sum with respect to $``\prec"$ and let $i^{\prime}$ be minimal such that $p_{i^{\prime}}>0$.

First we consider $i^{\prime}\in S^{\prime}$. For enough large $k\in\mathbb{Z}$, we obtain

[TABLE]

where $I$ is a finite subset of $\mathbb{Z}_{+}^{m}$ and $w_{\mathbf{p}}$ are nonzero vectors of $\mathrm{Ind}(M)$. Now we consider $i^{\prime}\in S\setminus S^{\prime}$. For enough large $k\in\mathbb{Z}$, one can easily to get

[TABLE]

where $I$ is a finite subset of $\mathbb{Z}_{+}^{m}$ and $w_{\mathbf{p}}$ are nonzero vectors of $\mathrm{Ind}(M)$.

By Proposition 2.6, we respectively consider the coefficient of $\lambda_{i^{\prime}}^{k}k^{p_{i^{\prime}}}$ and $\lambda_{i^{\prime}}^{k}k^{p_{i^{\prime}}+1}$ in (4.1) and (4.2), one has

[TABLE]

where $m\in\mathbb{N},w_{{\mathbf{p}}}$ are nonzero vectors of $\mathrm{Ind}(M)$. Then

[TABLE]

has lower degree than $u$, which is contrary to the choice of $u$. Hence, $\mathrm{deg}(u)=0$.

By Lemma 4.3, we see that \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes w_{\mathbf{0}} can be generated by $\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes w_{\mathbf{0}}$. It follows that \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathcal{U}(\mathcal{H})w_{\mathbf{0}}\subseteq W. Thus, W=\big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M), since the nonzero $\mathcal{H}$-submodule $\mathcal{U}(\mathcal{H})w_{\mathbf{0}}$ of ${\rm Ind}(M)$ generated by $w_{\mathbf{0}}$ is equal to ${\rm Ind}(M)$ by the irreducibility of ${\rm Ind}(M)$. This completes the proof of Theorem 4.4. ∎

Remark 4.5**.**

When $S^{\prime}={\O}$ in Theorem 4.4, it was studied in [16].

It follows from Lemma 4.1 and Theorem 4.4 that we have the following remark.

Remark 4.6**.**

Let $m\in\mathbb{N},S=\{1,\ldots,m\},S^{\prime}\subseteq S$ and $\lambda_{i}\in\mathbb{C}^{*}$ for $i\in S$. Let $\beta_{i},\alpha_{j}\in\mathbb{C}^{*},\beta_{j}=0$ for $i\in S^{\prime},j\in S\setminus S^{\prime}$. Assume $\mathrm{Ind}(M)$ is an $\mathcal{H}$-module defined by (2.1) for which $M$ satisfies the conditions in Theorem 2.5. Then the tensor product \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) is an irreducible $\mathcal{H}$-module if and only if the $\lambda_{i}$ pairwise distinct.

5 Isomorphism classes

In this section, we will give isomorphism results for modules \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M). We denote the number of elements in set $A$ by $\mathrm{card}(A)$.

Theorem 5.1**.**

Let $m,n\in\mathbb{N},S=\{1,\ldots,m\},T=\{1,\ldots,n\},S^{\prime}\subseteq S,T^{\prime}\subseteq T,\lambda_{i},\mu_{j}\in\mathbb{C}^{*}$ with the $\lambda_{i}$ pairwise distinct as well as $\mu_{j}$ pairwise distinct for $i\in S,j\in T$. Let $\beta_{i^{\prime}},\alpha_{i}\in\mathbb{C}^{*},\beta_{i}=0$ and $d_{j^{\prime}},c_{j}\in\mathbb{C}^{*},d_{j}=0$ for $i^{\prime}\in S^{\prime},{i}\in S\setminus S^{\prime},j^{\prime}\in T^{\prime},j\in T\setminus T^{\prime}$. Assume $\mathrm{Ind}(M_{1})$ and $\mathrm{Ind}(M_{2})$ are $\mathcal{H}$-modules defined by (2.1) for which $M_{1}$ and $M_{2}$ satisfy the conditions in Theorem 2.5. Then \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M_{1}) and \big{(}\bigotimes_{j=1}^{n}\Omega(\mu_{j},c_{j},d_{j})\big{)}\otimes\mathrm{Ind}(M_{2}) are isomorphic as $\mathcal{H}$-modules if and only if $m=n,\mathrm{card}(S^{\prime})=\mathrm{card}(T^{\prime}),\mathrm{Ind}(M_{1})\cong\mathrm{Ind}(M_{2})$ as $\mathcal{H}$-modules and $(\lambda_{i},\alpha_{i},\beta_{i})=(\mu_{i^{\prime}},c_{i^{\prime}},d_{i^{\prime}})$ and $(\lambda_{j},\alpha_{j})=(\mu_{j^{\prime}},c_{j^{\prime}}),\beta_{j}=d_{j^{\prime}}=0$ for $i\in S^{\prime},i^{\prime}\in T^{\prime},j\in S\setminus S^{\prime},j^{\prime}\in T\setminus T^{\prime}$ \mathrm{(}$$\varphi_{1}:S^{\prime}\rightarrow T^{\prime}\ and\ \varphi_{2}:S\setminus S^{\prime}\rightarrow T\setminus T^{\prime} are both bijections $\mathrm{)}$.

Proof.

The sufficiency is trivial. We denote \Omega(\lambda_{i},\alpha_{i},\beta_{i})=\mathbb{C}[\partial_{i}],\Omega(\mu_{j},c_{j},d_{j})=\mathbb{C}[\widetilde{\partial}_{j}],V_{1}=\big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M_{1}) and V_{2}=\big{(}\bigotimes_{j=1}^{n}\Omega(\mu_{j},c_{j},d_{j})\big{)}\otimes\mathrm{Ind}(M_{2}), respectively.

Let $\phi$ be an isomorphism from $V_{1}$ to $V_{2}$. Take a nonzero element $\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes w\in V_{1}$. Assume

[TABLE]

where $I$ is a finite subset of $\mathbb{Z}_{+}^{n}$ and $v_{\mathbf{p}}$ are nonzero vectors of $V_{2}$. There exists a positive integer $K=\mathrm{max}\{K(w),K(v_{\mathbf{p}})\mid\mathbf{p}\in\mathbb{Z}_{+}^{n}\}$ such that $I_{m}\cdot w=I_{m}\cdot v_{\mathbf{p}}=L_{m}\cdot w=L_{m}\cdot v_{\mathbf{p}}=0$ for all integers $m\geq K$ and $\mathbf{p}\in\mathbb{Z}_{+}^{n}$.

First consider $d_{j^{\prime}}\in\mathbb{C}^{*},d_{j}=0$ for $j^{\prime}\in T^{\prime},j\in T\setminus T^{\prime}$. We note that $m=\mathrm{card}(S^{\prime})+\mathrm{card}(S\setminus S^{\prime}),n=\mathrm{card}(T^{\prime})+\mathrm{card}(T\setminus T^{\prime})$. For enough large $k\in\mathbb{Z}$, we know that

[TABLE]

According to Proposition 2.6 in (5.2), we get $p_{j^{\prime}}=0,\lambda_{j}=\mu_{j^{\prime}},\mathrm{card}(T^{\prime})=\mathrm{card}(S^{\prime})$, where $j\in S^{\prime},j^{\prime}\in T^{\prime}$ and $\varphi_{1}:S^{\prime}\rightarrow T^{\prime}$ is bijection. Then $\phi(\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes w)=\sum\limits_{\widehat{\mathbf{p}}\in I}\widetilde{\partial_{1}}^{\widehat{p_{1}}}\cdots\widetilde{\partial_{n}}^{\widehat{p_{n}}}\otimes v_{\widehat{\mathbf{p}}}$, where $\widehat{p_{j^{\prime}}}=0$ for $j^{\prime}\in T^{\prime}$. Now we consider $c_{j}\in\mathbb{C}^{*}$ for $j\in T\setminus T^{\prime}$. For enough large $k\in\mathbb{Z}$, it follows from \phi\big{(}L_{k}(\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes w)\big{)}=L_{k}\phi(\underbrace{1\otimes\cdots\otimes 1}_{n}\otimes w) that we have

[TABLE]

Using Proposition 2.6 in (5.3), one can easily to check that $\widehat{p_{j}}=0,\lambda_{i}=\mu_{j},\mathrm{card}(T\setminus T^{\prime})=\mathrm{card}(S\setminus S^{\prime})$, where $i\in S\setminus S^{\prime},j\in T\setminus T^{\prime}$ and $\varphi_{2}:S\setminus S^{\prime}\rightarrow T\setminus T^{\prime}$ is bijection. Then $m=\mathrm{card}(T\setminus T^{\prime})+\mathrm{card}(T)=\mathrm{card}(S\setminus S^{\prime})+\mathrm{card}(S)=n$ and

[TABLE]

Thus, (5.2) can be rewritten as

[TABLE]

we obtain $\beta_{i}=d_{i^{\prime}}$ for $i\in S^{\prime},i^{\prime}\in T^{\prime}$, which can be obtained by \phi\big{(}\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes w\big{)}\neq 0,\lambda_{i}=\mu_{i^{\prime}} and Proposition 2.6. We note that $\beta_{j}=d_{j^{\prime}}=0$ for $j\in S\setminus S^{\prime},j^{\prime}\in T\setminus T^{\prime}$. Then for enough large $k\in\mathbb{Z}$, by \phi\big{(}L_{k}(\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes w)\big{)}=L_{k}(\underbrace{1\otimes\cdots\otimes 1}_{m}\otimes v_{\mathbf{0}}) and $\lambda_{i}=\mu_{i^{\prime}},\lambda_{j}=\mu_{j^{\prime}}$, we can easily check that $\alpha_{i}=c_{i^{\prime}},\alpha_{j}=c_{j^{\prime}}$ for $i\in S^{\prime},i^{\prime}\in T^{\prime},j\in S\setminus S^{\prime},j^{\prime}\in T\setminus T^{\prime}$.

There exists a linear bijection $\tau:V_{1}\rightarrow V_{2}$ such that

[TABLE]

for all $v\in V_{1}.$ Meanwhile we conclude that $\tau(L_{k}v)=L_{k}\tau(v)$ for all $k\in\mathbb{Z},v\in V_{1}.$ Then from

[TABLE]

and

[TABLE]

we see that $\tau(I_{k}v)=I_{k}\tau(v)$ and $\tau(C_{i}v)=C_{i}\tau(v)$ for $i=1,2,3,k\in\mathbb{Z}$, respectively. Thus, $V_{1}\cong V_{2}$ as $\mathcal{H}$-modules for $d_{j^{\prime}},c_{j}\in\mathbb{C}^{*},d_{j}=0$ for $j^{\prime}\in T^{\prime},j\in T\setminus T^{\prime}$.

To sum up, we obtain $m=n,V_{1}\cong V_{2},(\lambda_{i},\alpha_{i},\beta_{i})=(\mu_{i^{\prime}},c_{i^{\prime}},d_{i^{\prime}}),(\lambda_{j},\alpha_{j})=(\mu_{j^{\prime}},c_{j^{\prime}})$ and $\beta_{j}=d_{j^{\prime}}=0$ for $i\in S^{\prime},i^{\prime}\in T^{\prime},j\in S\setminus S^{\prime},j^{\prime}\in T\setminus T^{\prime}$. This completes the proof. ∎

Remark 5.2**.**

When $S^{\prime}=T^{\prime}={\O}$ in Theorem 5.1, it was investigated in [16].

6 New irreducible modules

In this section, we shall show that \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) is not isomorphic to $\mathrm{Ind}(M)$ or \mathcal{M}\big{(}V,\Omega(\lambda,\alpha,\beta)\big{)} or $\widetilde{\mathcal{M}}(W,\gamma(t))$ or $\mathcal{A}_{\alpha,\beta}$.

For any $l,m\in\mathbb{Z}$, $s\in\mathbb{Z}_{+}$, define a sequence of operators $T_{l,m}^{(s)}$ as follows

[TABLE]

For $d\in\{0,1\},r\in\mathbb{Z}_{+}$, denote by $\mathcal{H}_{r,d}$ the Lie subalgebra of $\mathcal{H}_{+,d}=\mathrm{span}_{\mathbb{C}}\{L_{i},I_{j}\mid i\geq 0,j\geq d\}$ generated by $L_{i},I_{j}$ for all $i>r,j>r+d$. Now we write $\bar{\mathcal{H}}_{r,d}$ the quotient algebra $\mathcal{H}_{+,d}/\mathcal{H}_{r,d}$, and $\bar{L}_{i},\bar{I}_{i+d}$ the respective images of $L_{i},I_{i+d}$ in $\bar{\mathcal{H}}_{r,d}$.

Let $d\in\{0,1\},r\in\mathbb{Z}_{+}$ and $V$ be an $\bar{\mathcal{H}}_{r,d}$-module. For any fixed $\gamma(t)=\sum_{i}c_{i}t^{i}\in\mathbb{C}[t,t^{-1}]$, define the action of $\mathcal{H}$ on $V\otimes\mathbb{C}[t,t^{-1}]$ as follows

[TABLE]

Then $V\otimes\mathbb{C}[t,t^{-1}]$ carries the structure of an $\mathcal{H}$-module under the above given actions, which is denoted by $\widetilde{\mathcal{M}}(V,\gamma(t))$. Note that $\widetilde{\mathcal{M}}(V,\gamma(t))$ is a weight $\mathcal{H}$-module if and only if $\gamma(t)\in\mathbb{C}$ and also that the $\mathcal{H}$-module $\widetilde{\mathcal{M}}(V,\gamma(t))$ for $\gamma(t)\in\mathbb{C}[t,t^{-1}]$ is irreducible if and only if $V$ is irreducible (see [4]).

Let $d\in\{0,1\},r\in\mathbb{Z}_{+}$ and $V$ be an $\bar{\mathcal{H}}_{r,d}$-module. For any $\lambda,\alpha,\beta\in\mathbb{C}$, define an $\mathcal{H}$-action on the vector space \mathcal{M}\big{(}V,\Omega(\lambda,\alpha,\beta)\big{)}:=V\otimes\mathbb{C}[t] as follows

[TABLE]

We note that \mathcal{M}\big{(}V,\Omega(\lambda,\alpha,\beta)\big{)} is reducible if and only if $V\cong V_{\alpha,\delta_{d,0}\tau}$ for some $\tau\in\mathbb{C}$ such that $\delta_{d,0}\beta\tau=0$ (see [5]).

Lemma 6.1**.**

Let $\lambda_{i}\in\mathbb{C}^{*},\alpha,\beta,\alpha_{i},\beta_{i}\in\mathbb{C},s\in\mathbb{Z}_{+},d\in\{0,1\}$ and $M$ be an irreducible $\mathcal{H}_{e}$-module satisfying the conditions in Theorem 2.5. Assume that $r^{\prime}$ is the maximal nonnegative integer such that $\bar{I}_{r^{\prime}+d}V\neq 0$. Then

• (i)

the action of $L_{m}$ for $m$ sufficiently large is not locally nilpotent on \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M);

• (ii)

the action of $T_{l,m}^{(s)}$ on $\mathcal{A}_{\alpha,\beta}$ is trivial for $l,m\in\mathbb{Z}$;

• (iii)

$T_{l,m}^{(1)}$* acts nontrivially on \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) whenever $m\ll 0$ and $l\ll m$;*

• (iv)

The action of $T_{l,m}^{(s)}$ on \mathcal{M}\big{(}V,\Omega(\lambda,\alpha,\beta)\big{)} and $\widetilde{\mathcal{M}}(V,\gamma(t))$ are trivial for $s>2(r^{\prime}+d)$.

Proof.

(i) follows from the local nilpotency of $L_{m}$ on $\mathrm{Ind}(M)$ by Theorem 2.5 for $m$ sufficiently large and its non-local nilpotency on $\Omega(\lambda,\alpha,\beta)$. (ii) follows from (3.1). (iii) can be obtained by the similar compute in Lemma 5.1 (v) of [5]. (iv) follows from [4, Lemma 3.3]. ∎

We are now ready to state the main result of this section.

Proposition 6.2**.**

Let $d\in\{0,1\}$, $r,e\in\mathbb{Z}_{+},$ $\alpha,\beta\in\mathbb{C},$ $M$ be an irreducible $\mathcal{H}_{e}$-module satisfying the conditions in Theorem 2.5. Then \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) is not isomorphic to $\mathrm{Ind}(M^{\prime})$ for any simple $\mathcal{H}_{e}$-module $M^{\prime}$ satisfying the conditions in Theorem 2.5, or \mathcal{M}\big{(}V,\Omega(\lambda,\alpha^{\prime},\beta^{\prime})\big{)} for any $\bar{\mathcal{H}}_{r,d}$-module $V$, $\lambda\in\mathbb{C}^{*},\alpha^{\prime},\beta^{\prime}\in\mathbb{C}$, or $\widetilde{\mathcal{M}}(W,\gamma(t))$ for any $\bar{\mathcal{H}}_{r,d}$-module $W$ and $\gamma(t)=\sum_{i}c_{i}t^{i}\in\mathbb{C}[t,t^{-1}]$, or $\mathcal{A}_{\alpha,\beta}$ for $\alpha,\beta\in\mathbb{C}$.

Proof.

From Lemma 6.1 (i), we have \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M)\ncong\mathrm{Ind}(M^{\prime}). Let $m\ll 0,l\ll m$ that $I_{l-m},I_{m}\notin\mathcal{H}_{e}$ and $s>2(r^{\prime}+d)$. For any 1\otimes v\in\big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M), noting that the action of $I_{m}$ on $1$ is scalar for any $m\in\mathbb{Z}$, we deduce that

[TABLE]

Then by Lemma 6.1 (iv), we obtain that \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) is not isomorphic to \mathcal{M}\big{(}V,\Omega(\lambda,\alpha,\beta)\big{)}, or $\widetilde{\mathcal{M}}(W,\gamma(t))$.

At last, by Lemma 6.1 (ii) and (iii), we get that \big{(}\bigotimes_{i=1}^{m}\Omega(\lambda_{i},\alpha_{i},\beta_{i})\big{)}\otimes\mathrm{Ind}(M) is not isomorphic to $\mathcal{A}_{\alpha,\beta}$. ∎

Acknowledgements

This work was partially supported by the NSFC (11801369, 11431010). We would like to thank Prof. Jianzhi Han for his useful discussions.