Irreducible Twisted Heisenberg–Virasoro Modules from Tensor Products

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 tensor product modules $\big (\bigotimes _{i=1}^m\Omega (\lambda _i,\alpha _i,\beta _i)\big )\otimes \mathrm {Ind}(M)$. 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

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

More precisely, $\mathcal {H}$ is an infinite-dimensional Lie algebra with $\mathbb {C}$-basis $\{L_m=t^{m+1}\frac{d}{dt},I_m=t^m,C_i \mid m\in \mathbb {Z},\,i=1,2,3\}$ subject to the Lie bracket as follows:

$$\begin{aligned} \begin{aligned}&[L_m,L_n]= (n-m)L_{m+n}+\delta _{m+n,0}\frac{m^{3}-m}{12}C_1,\\&[L_m,I_n]=n I_{m+n}+\delta _{m+n,0}(m^{2}+m)C_2,\\ {}&[I_m,I_n]=n\delta _{m+n,0}C_3, \\&[\mathcal {H},C_1]=[\mathcal {H},C_2]= [\mathcal {H},C_3]=0. \end{aligned} \end{aligned}$$

It is clear that the subspaces spanned by $\{I_m, C_3 \mid 0\ne m\in \mathbb {Z}\}$ 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 twisted Heisenberg–Virasoro algebra $\mathcal {H}$ has been studied by Arbarello et al. (see [1]), where a connection is established between the second cohomology of certain moduli spaces of curves and the second cohomology of the Lie algebra of differential operators of order at most one. Furthermore, when the central element of the Heisenberg subalgebra acts in a non-zero way, an irreducible highest weight module for $\mathcal {H}$ is isomorphic to the tensor product of an irreducible module for the Virasoro algebra and an irreducible module for the infinite-dimensional Heisenberg algebra was proved. A more general result for this was given in [14].

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 [9, 12, 16]). While weight modules with an infinite dimensional weight spaces were also studied (see [4, 15]). In the last few years, various families of non-weight irreducible twisted Heisenberg–Virasoro modules were investigated (see, e.g., [2,3,4,5,6, 10]). These are basically various versions of Whittaker modules and $\mathcal {U}(\mathbb {C}L_0)$-free modules constructed using different tricks. Whittaker modules for $\mathcal {H}$ were defined in [10], but the correct classification of irreducible modules appeared in [2]. A more general setting for Whittaker modules over $\mathcal {H}$ was given in [14]. 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 $\mathrm {Ind}(M)$. We present the necessary and sufficient conditions under which these modules are irreducible and also determine all the equivalent irreducible modules in this class. The irreducibility and isomorphism problem of the $\mathcal {H}$-modules $\big (\bigotimes _{i=1}^m\Omega (\lambda _i,\alpha _i,\beta _i)\big )\otimes \mathrm {Ind}(M)$ were also given in Examples 8, 10 and Theorem 35 in [14]. 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 Sects. 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 Sect. 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 Sect. 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. At last, we present that $\mathcal {H}$-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. For convenience, we denote $\partial :=t\frac{d}{dt}$. We see that $\partial t^n=t^n(\partial +n)$ for $n\in \mathbb {Z}$. It is obvious that associative algebra $\widetilde{\mathcal {A}}=\mathbb {C}[t,\partial ]$ is a proper subalgebra of the rank 1 Weyl algebra $\mathbb {C}[t,\frac{d}{dt}]$. Then $\widetilde{\mathcal {A}}$ is the universal enveloping algebra of the 2-dimensional solvable Lie algebra ${\mathfrak {b}}_1=\mathbb {C}L_0\oplus \mathbb {C}L_1$, which subjects to $[L_0,L_1]=L_1$. Now, assume that $\mathcal {K}=\mathbb {C}[t, t^{-1}, \partial ]$ is the Laurent polynomial differential operator algebra.

Definition 2.1

Assume that $\mathcal {P}$ is an associative or Lie algebra over $\mathbb {C}$ and $\mathcal {Q}$ is a subspace of $\mathcal {P}$. If there exists $0\ne q\in \mathcal {Q}$ such that $qv=0$ for some $0\ne v\in V$, the $\mathcal {P}$-module V is called $\mathcal {Q}$ -torsion; 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] Assume that V is a $\mathbb {C}[t]$-torsion-free irreducible module over the associative algebra $\widetilde{\mathcal {A}}$. Then V can be extended into a $\mathcal {K}=\mathbb {C}[t,t^{-1}, \frac{d}{dt}]$-module, 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] Assume that $\mu$ is an irreducible element in the associative algebra $\mathbb {C}(t)[\partial ]$. Then $\mathcal {K}/(\mathcal {K}\cap \big (\mathbb {C}(t)[\partial ]\mu )\big )$ is a $\mathbb {C}[t,t^{-1}]$-torsion-free irreducible $\mathcal {K}$-module. In this way any $\mathbb {C}[t,t^{-1}]$-torsion-free irreducible $\mathcal {K}$-module can be realized.

For any $\lambda \in \mathbb {C}^*$, it follows from $t^m\partial ^n=\lambda ^m(\partial -m)^n, \partial \partial ^n=\partial ^{n+1}$ for all $n\in \mathbb {Z}_+,m\in \mathbb {Z}$ that we can define a $\mathcal {K}$-module structure on the space $\Omega (\lambda )=\mathbb {C}[\partial ]$. 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] Assume that V is 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

$$\begin{aligned} {\sum \limits _{m\in \mathbb {Z}_+}}(\mathbb {C}L_{m}\oplus \mathbb {C}I_{m-e})\oplus \mathbb {C}C_1\oplus \mathbb {C}C_2\oplus \mathbb {C}C_3. \end{aligned}$$

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

$$\begin{aligned} \mathrm {Ind}(M):=\mathcal {U}(\mathcal {H})\otimes _{\mathcal {U}(\mathcal {H}_{e})}M. \end{aligned}$$

(2.1)

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) $\left\{ \begin{array}{llll} \text{ the } \text{ action } \text{ of }\ I_{k} \ \text{ on }\ M \ \text{ is } \text{ injective }\ \quad &{}\text{ if } k\ne 0,\\ c_0+(n-1)c_2\ne 0\quad \mathrm {for\ all}\ n\in \mathbb {Z}{\setminus }\{0\} &{}\text{ if } k=0, \end{array}\right.$

• (2) $I_nM=L_mM=0$ 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

$$\begin{aligned}&\sum _{i=1}^{s}\sum _{j=0}^{k}\mu _i^mf_{i,j}(m)v_{i,j}\in P_1\\&\quad { for}\ K< m\in \mathbb {Z}\ (K \ { any\ fixed\ element\ in}\ \mathbb {Z}\cup \{-\infty \}) \end{aligned}$$

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

$$\begin{aligned} L_mv=\big (t^{m}\partial +m\alpha t^m\big )v,\ I_mv=\beta t^mv,\ C_iv=0 \end{aligned}$$

(3.1)

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

$$\begin{aligned} (L_mI_n-I_nL_m)v =\beta \big (t^{m}\partial + m\alpha t^m\big )t^nv -\beta t^n\big (t^{m}\partial + m\alpha t^m\big )v =nI_{m+n}v. \end{aligned}$$

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

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 $\beta \ne 0$.

• (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 $\varphi (I_0v)=I_0\varphi (v)$, which gives $\beta _1=\beta _2$. In particular, $\beta _1\ne 0$. We note that $\varphi (I_mv)=I_m\varphi (v)$, which implies

$$\begin{aligned} \varphi (t^mv)=t^m\varphi (v) \end{aligned}$$

(3.2)

for any $m\in \mathbb {Z}$. From $\varphi (L_0^mv)=L_0^m\varphi (v)$, we have

$$\begin{aligned} \varphi (\partial ^mv)=\partial ^m\varphi (v) \end{aligned}$$

(3.3)

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

$$\begin{aligned} 0= & {} \varphi (L_mv)-L_m\varphi (v)=\varphi \big ((t^{m}\partial +m\alpha _1 t^m)v\big )-(t^{m}\partial +m\alpha _2 t^m)\partial \varphi (v)\\= & {} m(\alpha _1-\alpha _2)t^{m}\varphi (v). \end{aligned}$$

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). $\square$

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

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

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

$$\begin{aligned}&L_m \partial ^n=\lambda ^m\big (\partial +m(\alpha -1)\big )(\partial -m)^{n},\ I_m \partial ^n= \lambda ^m\beta (\partial -m)^{n}\\&\quad \mathrm{for}\ m\in \mathbb {Z},n\in \mathbb {Z}_+. \end{aligned}$$

Then $\Omega (\lambda ,\alpha ,\beta )$ is irreducible if and only if $\alpha \ne 1$ or $\beta \ne 0$ (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

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

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

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

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

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

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

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

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

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

$$\begin{aligned}&t^m\cdot t^n=t^{m+n},\ t^m\cdot (t^n\partial )=t^{m+n}\partial ,\\&\partial \cdot t^n=t^{n}(\partial +n),\ \partial \cdot (t^n\partial )=t^{n}(f(t)+n\partial ), \end{aligned}$$

where $m,n\in \mathbb {Z}$. From (3.1), for $\alpha \ne 1$ or $\beta \ne 0$, 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:

$$\begin{aligned}&L_m\cdot t^n=t^{m+n}(n+m\alpha +\partial ),\ L_m\cdot (t^n\partial )=t^{m+n}(f(t)+m\alpha +n\partial ),\\&I_m\cdot t^n=\beta t^{m+n},\ I_m\cdot (t^n\partial )=\beta t^{m+n}\partial . \end{aligned}$$

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

Example 3.6

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

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

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

$$\begin{aligned}&t^k\cdot \left( t^r\left( \frac{d}{dt}\right) ^m\right) =t^{k+r}(\frac{d}{dt})^m\quad \mathrm {for} \ k,r\in \mathbb {Z},\ 0\le m\le n-1,\\&\frac{d}{dt}\cdot \left( t^r\left( \frac{d}{dt}\right) ^m\right) =rt^{r-1}\left( \frac{d}{dt}\right) ^m+t^r\left( \frac{d}{dt}\right) ^{m+1}\quad \mathrm {for} \ r\in \mathbb {Z},\ 0\le m< n-1,\\&\frac{d}{dt}\cdot \left( t^r\left( \frac{d}{dt}\right) ^{n-1}\right) =rt^{r-1}\left( \frac{d}{dt}\right) ^{n-1}+t^{r+1}\quad \mathrm {for} \ r\in \mathbb {Z}. \end{aligned}$$

Using (3.1), for $\alpha \ne 1$ or $\beta \ne 0$, 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:

$$\begin{aligned}&L_k\cdot \left( t^r\left( \frac{d}{dt}\right) ^m\right) =(rt^{k+r}+\alpha kt^{k+r+1})\left( \frac{d}{dt}\right) ^m+t^{k+r+1}\left( \frac{d}{dt}\right) ^{m+1},\\&L_k\cdot \left( t^r\left( \frac{d}{dt}\right) ^{n-1}\right) =(rt^{k+r}+\alpha kt^{k+r+1})\left( \frac{d}{dt}\right) ^{n-1}+t^{k+r+2},\\&I_k\cdot \left( t^r\left( \frac{d}{dt}\right) ^m\right) =\beta t^{k+r}\left( \frac{d}{dt}\right) ^m, \end{aligned}$$

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

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

$$\begin{aligned} L_k \partial _i^n=\lambda _i^k\big (\partial _i+k \alpha _i)(\partial _i-k)^{n},\ I_k \partial _i^n= \lambda _i^k\beta _i (\partial _i-k)^{n},\ C_j\partial _i^n=0 \end{aligned}$$

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 $\alpha _i\ne 0$ or $\beta _i\ne 0$ 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

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

by

$$\begin{aligned}&\partial _1^{p_1}\cdots \partial _m^{p_m}\otimes u\prec \partial _1^{q_1}\cdots \partial _m^{q_m}\otimes v \\&\quad \Longleftrightarrow \exists k\in \mathbb {N}\ \mathrm {such\ that} \ p_k<q_k\ \mathrm {and} \ p_n=q_n \ \mathrm {for} \ n<k. \end{aligned}$$

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

$$\begin{aligned} w=\sum _{{\mathbf {p}}\in I} \partial _1^{p_1}\cdots \partial _m^{p_m}\otimes v_{\mathbf {p}}, \end{aligned}$$

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 $\{f(\partial _1)(\partial _1+\partial _2)^n\mid n\in \mathbb {Z}_+,0\le \mathrm {deg}(f)\le s\}$ or $\{(\partial _1+\partial _2)^nf(\partial _2)\mid n\in \mathbb {Z}_+,0\le \mathrm {deg}(f)\le s\}$. 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

$$\begin{aligned}&I_k\big (f(\partial _1)(\partial _1+\partial _2)^n\big ) =I_k\big (\sum _{i=0}^n\left( {\begin{array}{c}n\\ i\end{array}}\right) f(\partial _1)\partial _1^{i}\partial _2^{n-i}\big ) \\&\quad =\sum _{i=0}^n\left( {\begin{array}{c}n\\ i\end{array}}\right) \Big (\beta _1f(\partial _1-k)(\partial _1-k)^{i}\partial _2^{n-i}+\beta _2f(\partial _1)\partial _1^{i}(\partial _2-k)^{n-i}\Big ) \\&\quad =\big (\beta _1f(\partial _1-k)+\beta _2f(\partial _1)\big )(\partial _1+\partial _2-k)^n\in W_s. \end{aligned}$$

By Theorem 9 of [17], we have $L_k\big ( f(\partial _1)(\partial _1+\partial _2)^n\big ) \in W_s.$ By the similar method, we obtain $L_k\big ((\partial _1+\partial _2)^nf(\partial _2)\big )\in W_s$ and $I_k\big ((\partial _1+\partial _2)^nf(\partial _2)\big )\in W_s$. Thus, $W_s$ is a submodule of $\Omega (\lambda ,\alpha _1,\beta _1)\otimes \Omega (\lambda ,\alpha _2,\beta _2)$, completing the proof.

$\square$

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 $\{f(\partial _1)(\partial _1+\partial _2)^n\mid n\in \mathbb {Z}_+,0\le \mathrm {deg}(f)\le s\}$ or $\{(\partial _1+\partial _2)^nf(\partial _2)\mid n\in \mathbb {Z}_+,0\le \mathrm {deg}(f)\le s\}$. Then $\Omega (\lambda ,\alpha _1,\beta _1)\otimes \Omega (\lambda ,\alpha _2,\beta _2)$ has a series of $\mathcal {L}$-submodules

$$\begin{aligned} W_1\subset W_2\subset \cdots \subset W_s\subset \cdots \end{aligned}$$

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 $s\ge 1$.

Proof

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

$$\begin{aligned}&I_k\big (\partial _1^s(\partial _1+\partial _2)^n\big ) =I_k\left( \sum _{i=0}^n\left( {\begin{array}{c}n\\ i\end{array}}\right) \partial _1^{i+s}\partial _2^{n-i}\right) \\&\quad \equiv \lambda ^k(\beta _1+\beta _2)\partial _1^s(\partial _1+\partial _2-k)^n\quad (\mathrm {mod}\ W_{s-1}). \end{aligned}$$

From Corollary 10 of [17], we get

$$\begin{aligned}&L_k\big (\partial _1^s(\partial _1+\partial _2)^n\big )\\&\quad \equiv \lambda ^k\partial _1^s\big (\partial _1+\partial _2-k(s+\alpha _0+\alpha _1)\big )(\partial _1+\partial _2-k)^n\quad (\mathrm {mod}\ W_{s-1}). \end{aligned}$$

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 $W_s/W_{s-1}\cong \Omega (\lambda ,s+\alpha _1+\alpha _2,\beta _1+\beta _2).$ $\square$

By the similar method in Lemma 3 of [17], 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 $k\ge K(w)$ 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

$$\begin{aligned} u=\sum _{\mathbf {p}\in I} \partial _1^{p_1}\cdots \partial _m^{p_m}\otimes w_{\mathbf {p}}\in W, \end{aligned}$$

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

$$\begin{aligned} I_k\left( \sum _{\mathbf {p}\in I} \partial _1^{p_1}\cdots \partial _m^{p_m}\otimes w_{\mathbf {p}}\right) =\sum _{i=1}^m\sum _{\mathbf {p}\in I} \partial _1^{p_1}\cdots \lambda _i^k\beta _i(\partial _i-k)^{p_i}\cdots \partial _m^{p_m}\otimes w_{\mathbf {p}}\in W, \end{aligned}$$

(4.1)

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

$$\begin{aligned}&L_k\left( \sum _{\mathbf {p}\in I} \partial _1^{p_1}\cdots \partial _m^{p_m}\otimes w_{\mathbf {p}}\right) \nonumber \\&\quad =\sum _{i=1}^m\sum _{\mathbf {p}\in I} \partial _1^{p_1}\cdots \lambda _i^k(\partial _i+k\alpha _i)(\partial _i-k)^{p_i}\cdots \partial _m^{p_m}\otimes w_{\mathbf {p}}\in W, \end{aligned}$$

(4.2)

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 $\lambda _{i^\prime }^kk^{p_{i^\prime }}$ and $\lambda _{i^\prime }^kk^{p_{i^\prime }+1}$ in (4.1) and (4.2), one has

$$\begin{aligned} \partial _1^{{p}_1}\cdots \partial _{i^\prime -1}^{{p}_{i^\prime -1}}\partial _{i^\prime +1}^{{p}_{i^\prime +1}}\cdots \partial _m^{{p}_m}\otimes w_{{\mathbf {p}}}\in W, \end{aligned}$$

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

$$\begin{aligned} \partial _1^{{p}_1}\cdots \partial _{i^\prime -1}^{{p}_{i^\prime -1}}\partial _{i^\prime +1}^{{p}_{i^\prime +1}}\cdots \partial _m^{{p}_m}\otimes w_{{\mathbf {p}}}+\mathrm {lower\ terms} \end{aligned}$$

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 $\mathrm{Ind}(M)$ generated by $w_\mathbf {0}$ is equal to $\mathrm{Ind}(M)$ by the irreducibility of $\mathrm{Ind}(M)$. This completes the proof of Theorem 4.4. $\square$

Remark 4.5

When in Theorem 4.4, it was studied in [17].

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. The irreducibility of the $\mathcal {H}$-modules $\big (\bigotimes _{i=1}^m\Omega (\lambda _i,\alpha _i,\beta _i)\big )\otimes \mathrm {Ind}(M)$ was also given in Examples 8 and 10 in [14].

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

$$\begin{aligned} \phi \left( \underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\right) =\sum \limits _{\mathbf {p}\in I}\widetilde{\partial _1}^{p_1}\cdots \widetilde{\partial _n}^{p_n}\otimes v_{\mathbf {p}}, \end{aligned}$$

(5.1)

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 $m\ge K$ 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

$$\begin{aligned}&\sum _{j=1}^m \lambda _j^k\beta _j\phi \big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\big ) =\phi \big (I_k\big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\big )\big )= I_k\phi \big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\big )\nonumber \\&\quad = \sum _{j^\prime =1}^n\sum \limits _{\mathbf {p}\in \mathbb {Z}_+^n}\widetilde{\partial _1}^{p_1}\cdots \mu _{j^\prime }^kd_{j^\prime }(\widetilde{\partial _{j^\prime }}-k)^{p_{j^\prime }}\cdots \widetilde{\partial _n}^{p_n}\otimes v_{\mathbf {p}}. \end{aligned}$$

(5.2)

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 $\phi (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w)=\sum \nolimits _{\widehat{\mathbf {p}}\in I}\widetilde{\partial _1}^{\widehat{p_1}}\cdots \widetilde{\partial _n}^{\widehat{p_n}}\otimes v_{\widehat{\mathbf {p}}}$, 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

$$\begin{aligned}&\sum _{i=1}^m \Big (\lambda _i^k\phi (1\otimes \cdots \otimes \partial _i\otimes \cdots \otimes 1\otimes w)+\lambda _i^kk\alpha _i\phi \Big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\Big )\Big )\nonumber \\&\quad =\sum _{j=1}^n\sum \limits _{\widehat{\mathbf {p}}\in \mathbb {Z}_+^n}\widetilde{\partial _1}^{\widehat{p_1}}\cdots \mu _j^k(\widetilde{\partial _j}+kc_j) (\widetilde{\partial _j}-k)^{\widehat{p_j}}\cdots \widetilde{\partial _n}^{\widehat{p_n}}\otimes v_{\widehat{\mathbf {p}}}. \end{aligned}$$

(5.3)

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

$$\begin{aligned} \phi \big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\big ) =\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes v_{\mathbf {0}} \end{aligned}$$

Thus, (5.2) can be rewritten as

$$\begin{aligned} \sum _{i=1}^m \lambda _i^k\beta _i\phi \big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\big ) =\sum _{i=1}^m\mu _i^kd_i\big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes v_{\mathbf {0}}\big ), \end{aligned}$$

we obtain $\beta _i=d_{i^\prime }$ for $i\in S^\prime ,i^\prime \in T^\prime$, which can be obtained by $\phi \big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes w\big )\ne 0,\lambda _i=\mu _{i^\prime }$ 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

$$\begin{aligned} \phi \left( \underbrace{1\otimes \cdots \otimes 1}_{m}\otimes v\right) =\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes \tau (v) \end{aligned}$$

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

$$\begin{aligned} \phi \big (I_k\big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes v\big )\big )=I_k\phi \big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes v\big ) \end{aligned}$$

and

$$\begin{aligned} \phi \big (C_i\big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes v\big )\big )=C_i\phi \big (\underbrace{1\otimes \cdots \otimes 1}_{m}\otimes v\big ), \end{aligned}$$

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. $\square$

Remark 5.2

When in Theorem 5.1, it was investigated in [17].

The isomorphism problem of the $\mathcal {H}$-modules $\big (\bigotimes _{i=1}^m\Omega (\lambda _i,\alpha _i,\beta _i)\big )\otimes \mathrm {Ind}(M)$ was also given in Theorem 35 in [14].

6 New Irreducible Modules

In this section, we prove 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 $\mathcal {A}_{\alpha ,\beta }$, i.e., $\big (\bigotimes _{i=1}^m\Omega (\lambda _i,\alpha _i,\beta _i)\big )\otimes \mathrm {Ind}(M)$ is a class of new irreducible $\mathcal {H}$-modules.

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

$$\begin{aligned} T_{l,m}^{(s)}=\sum _{i=0}^s(-1)^{s-i} \left( {\begin{array}{c}s\\ i\end{array}}\right) I_{l-m-i}I_{m+i}. \end{aligned}$$

For $d\in \{0,1\},r\in \mathbb {Z}_+$, denote by $\mathcal {H}_{r,d}$ the Lie subalgebra of $\mathcal {H}_{+,d}=\mathrm {span}_{\mathbb {C}}\{L_i,I_j\mid i\ge 0,j\ge d\}$ 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 ${\bar{\mathcal {H}}}_{r,d}$.

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

$$\begin{aligned}&L_m \circ (v\otimes t^n)=(L_m+\sum _{i}c_iI_{m+i})(v\otimes t^n),\\&I_m\circ (v\otimes t^n)=I_m(v\otimes t^n),\ C_i\circ (v\otimes t^n)=0, \end{aligned}$$

where $m,n\in \mathbb {Z}, v\in V$ and $i=1,2,3$. Then $V\otimes \mathbb {C}[t,t^{-1}]$ is made into an $\mathcal {H}$-module, which is denoted by $\widetilde{{\mathcal {M}}}(V,\gamma (t))$. It is clear that $\widetilde{{\mathcal {M}}}(V,\gamma (t))$ is a weight $\mathcal {H}$-module if and only if $\gamma (t)\in \mathbb {C}$. Moreover, 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

$$\begin{aligned}&L_m\big (v\otimes f(t)\big )=v\otimes \lambda ^m(t-m\alpha )f(t-m)+\sum _{i=0}^r\Big (\frac{m^{i+1}}{(i+1)!}\bar{L}_i\Big )v\otimes \lambda ^mf(t-m),\\&I_m\big (v\otimes f(t)\big )=\sum _{i=0}^r\Big (\frac{m^{i+d}}{(i+d)!}\bar{I}_{i+d}\Big )v\otimes \lambda ^m\beta f(t-m),\ C_i\big (v\otimes f(t)\big )=0, \end{aligned}$$

where $i\in \{1,2,3\}, m\in \mathbb {Z},v\in V, f(t)\in \mathbb {C}[t]$. 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

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

• (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]. $\square$

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

Proposition 6.2

Assume that $d\in \{0,1\}$, $r,e\in \mathbb {Z}_+,\alpha ,\beta \in \mathbb {C},$ M is an irreducible $\mathcal {H}_e$-module satisfying the conditions in Theorem 2.5. Then we have $\big (\bigotimes _{i=1}^m\Omega (\lambda _i,\alpha _i,\beta _i)\big )\otimes \mathrm {Ind}(M)$ is not isomorphic to $\mathrm {Ind}(M^\prime )$, or $\mathcal {M}\big (V,\Omega (\lambda ,\alpha ^\prime ,\beta ^\prime )\big )$, or $\widetilde{{\mathcal {M}}}(W,\gamma (t))$, or $\mathcal {A}_{\alpha ,\beta }$.

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

$$\begin{aligned}&T_{l,m}^{(s)}(1\otimes v) \\&\quad =\sum _{i=0}^s(-1)^{s-i} \left( {\begin{array}{c}s\\ i\end{array}}\right) \big (I_{l-m-i}I_{m+i}(1)\otimes v+I_{m+i}(1)\otimes I_{l-m-i}v\\&\qquad +I_{l-m-i}(1)\otimes I_{m+i}v+1\otimes I_{l-m-i}I_{m+i}v\big )\ne 0. \end{aligned}$$

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 $\mathcal {M}\big (V,\Omega (\lambda ,\alpha ^\prime ,\beta ^\prime )\big )$, 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 $\mathcal {A}_{\alpha ,\beta }$. $\square$