# Irreducible modules of toroidal Lie algebras arising from
$\phi_{\epsilon}$-coordinated modules of vertex algebras

Fulin Chen  School of Mathematical Sciences, Xiamen University, Xiamen,
361005, China<EMAIL_ADDRESS>,  Huansheng Li  School of Mathematical
Sciences, Xiamen University, Xiamen, 361005, China<EMAIL_ADDRESS>and
Nina Yu  School of Mathematical Sciences, Xiamen University, Xiamen, 361005,
China<EMAIL_ADDRESS>
###### Abstract.

In this paper, for every $\epsilon\in\mathbb{Z}$, we introduce an extension of
the 2-toroidal Lie algebra by certain derivations. Based on the
$\phi_{\epsilon}$-coordinated modules theory for vertex algebras, we give an
explicit realization of a class of irreducible highest weight modules for this
extended toroidal Lie algebra. When $\epsilon=1$, this affords a realization
of certain irreducible modules for the toroidal extended affine Lie algebras
first constructed by Billig.

###### Key words and phrases:

toroidal Lie algebra, vertex algebra, $\phi_{\epsilon}$-coordinated module,
highest weight module

###### 2010 Mathematics Subject Classification:

17B67, 17B69

##  1\. Introduction

Toroidal Lie algebras are natural multi-variable generalizations of affine Lie
algebras. One starts with a finite-dimensional complex simple Lie algebra
$\mathfrak{g}$ and forms the multi-loop algebra
$\mathcal{R}\otimes\mathfrak{g}$, where $\mathcal{R}=\mathbb{C}[t_{0}^{\pm
1},t_{1}^{\pm 1},\dots,t_{N}^{\pm 1}]$ is the ring of Laurent polynomials in
the variables $t_{0},t_{1},\dots,t_{N}$. The $(N+1)$-toroidal Lie algebra
$\mathfrak{t}(\mathfrak{g})$ is by definition the universal central extension
of $\mathcal{R}\otimes\mathfrak{g}$ [F, MRY, EM]. When $N=0$, this yields the
usual affine Lie algebra. In the literature, various extensions of
$\mathfrak{t}(\mathfrak{g})$ obtained by adding certain subalgebras of
$\mathrm{Der}(\mathcal{R})$ (the Lie algebra of derivations over
$\mathcal{R}$) have been studied extensively. For examples, the subalgebra
$\sum_{i=0}^{N}\mathbb{C}\mathrm{d}_{i}(:=t_{i}\frac{\partial}{\partial
t_{i}})$ of degree-zero derivations [E1, E2], the subalgebra
$\mathcal{R}\mathrm{d}_{0}\oplus\cdots\oplus\mathcal{R}\mathrm{d}_{N-1}\oplus\mathbb{C}\mathrm{d}_{N}$
[EM, FM, BB, JM], the subalgebra of divergence zero (or skew) derivations [B2,
CLT1, ESB, CLT2], and the algebra $\mathrm{Der}(\mathcal{R})$ itself [B1, EJ].

In this paper we consider a sequence of new extensions

$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}=\mathfrak{t}(\mathfrak{g})\oplus\mathcal{D}^{\epsilon}\quad(\epsilon\in\mathbb{Z})$

of the $2$-toroidal Lie algebra $\mathfrak{t}(\mathfrak{g})$, where

(1.1)
$\displaystyle\mathcal{D}^{\epsilon}=\bigg{\\{}f_{0}t_{0}^{\epsilon}\frac{\partial}{\partial
t_{0}}+f_{1}\mathrm{d}_{1}\mid f_{0},f_{1}\in\mathcal{R},\
t_{0}^{\epsilon}\frac{\partial}{\partial
t_{0}}(f_{0})+\mathrm{d}_{1}(f_{1})=0\bigg{\\}}$

is a Lie subalgebra of $\mathrm{Der}(\mathcal{R})$. The Lie algebras
$\mathcal{D}^{\epsilon}$ are closely related to the Lie algebras
$\mathcal{B}(q)$, $q\in\mathbb{C}$ of Block type studied in [CGZ, SXX1, SXX2,
XZ]. In fact, when $q=\epsilon$, $\mathcal{B}(q)$ is a “half” of
$\mathcal{D}^{\epsilon}$, which consists of those derivations in (1.1)
satisfying the condition that $f_{0},f_{1}$ are spanned by the monomials
$t_{0}^{m_{0}}t_{1}^{m_{1}}$ with $m_{0},m_{1}\in\mathbb{Z}$ and $m_{1}\geq
0$. In particular, when $\epsilon=1$, $\mathcal{D}^{\epsilon}$ is the algebra
of divergence zero derivations and
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$ is a nullity $2$ extended
affine Lie algebra in the sense of [AABGP], which is commonly called toroidal
extended affine Lie algebras [B2].

One of the most important classes of modules for affine Kac-Moody algebras is
the highest weight modules. Their analogs for various extensions of toroidal
Lie algebras have been extensively studied in [B1, B2, BB, BBS, CLT1, CLT2,
E1, E2, EJ, ESB, FM, JM]. In this paper we will give an explicit realization
of a class of irreducible highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules. When
$\epsilon=1$, this construction have been previously obtained by Billig in
[B2].

The key ingredient in our realization is the theory of $\phi$-coordinated
modules for vertex algebras, which was first introduced in [Li] for the
purpose of associating quantum affine algebras with (quantum) vertex algebra.
Here, $\phi$ is an associate of the $1$-dimensional additive formal group
(law) $F(z,w)=z+w$, which is by definition a formal series
$\phi(w,z)\in\mathbb{C}((w))[[z]]$, satisfying the conditions

$\phi(w,0)=w\quad\text{and}\quad\phi(\phi(w,z_{1}),z_{2})=\phi(w,z_{1}+z_{2}).$

It was proved therein that for every $p(w)\in\mathbb{C}((w))$,
$\phi(w,z)=e^{zp(w)\frac{d}{dw}}(w)$ is an associate of $F(z,w)$ and every
associate is of this form. When $p(w)=1$, the $\phi$-coordinated modules are
just the usual modules. When $p(w)=w^{\epsilon}$ with $\epsilon\in\mathbb{Z}$,
set

$\phi_{\epsilon}(w,z)=e^{zw^{\epsilon}\frac{d}{dw}}(w).$

The structure of $\phi_{\epsilon}$-coordinated modules for vertex algebras was
further studied in [BLP]. In particular, a Jacobi-like identity for
$\phi_{\epsilon}$-coordinated modules was established therein.

Now we give an outline of the structure of this paper. In Section 2, we first
introduce the extended toroidal Lie algebra
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$ and give the commutators
in terms of certain generating functions on
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$ (see Proposition 2.3). We
will work in a more general framework with $\mathfrak{g}$ an arbitrary Lie
algebra equipped with an invariant symmetric bilinear form. Meanwhile, in the
process of adding $\mathcal{D}^{\epsilon}$ to $\mathfrak{t}(\mathfrak{g})$, an
abelian two-cocycle depending on any complex number $\mu$ will be involved.

In Section 3, we first recall some facts about affine-Virasoro algebras
$\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$ associated to $\mathfrak{g}$. Then
we give the first main result of this paper in Theorem 3.2. Namely, we
construct a class of irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules

(1.2) $\displaystyle
L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$

where $\alpha,\beta,\ell\in\mathbb{C}$ with $\ell\neq 0$, $U$ is an
irreducible $\mathfrak{g}$-module,
$L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)$ is
an irreducible highest weight module for the affine-Virasoro algebra
$\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$, and
$V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$ is a highest
weight module for the Heisenberg algebra $\widehat{\mathfrak{h}}$ associated
to a $2$-dimensional abelian Lie algebra $\mathfrak{h}=\mathbb{C}{\bf
k}\oplus\mathbb{C}{\bf d}$. The second main result of this paper is a
characterization of these irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules in terms of the
highest weight modules (see Theorem 3.4).

Sections 4 and 5 are devoted to the proof of Theorem 3.2. In Section 4 we
first recall some results on $\phi_{\epsilon}$-coordinated modules. Let $V$ be
a vertex algebra. A fundamental property for a $V$-module $(W,Y_{W})$ is that
for $u,v\in V$, $Y_{W}(u_{-1}v,z)$ is just the normally ordered product
${}_{\circ}^{\circ}Y_{W}(u,z)Y_{W}(v,z)_{\circ}^{\circ}$ (see [LL] for
example). We prove in Proposition 4.6 that for a $\phi_{\epsilon}$-coordinated
$V$-module $(W,Y_{W}^{\epsilon})$,

(1.3) $\displaystyle
Y_{W}^{\epsilon}(u_{-1}v,z)=_{\circ}^{\circ}Y_{W}^{\epsilon}(u,z)Y_{W}^{\epsilon}(v,z)_{\circ}^{\circ}+\sum_{n\geq
0}c_{n}z^{(\epsilon-1)(n+1)}Y_{W}^{\epsilon}(u_{n}v,z),$

where $c_{n}$ are some explicitly determined complex numbers. We also
determine the $\phi_{\epsilon}$-coordinated modules for vertex algebras
arising from vertex Lie algebras introduced in [DLM, K2, P] (see Proposition
4.9).

In Section 5, we first introduce a subalgebra
$\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$ of
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$ such that
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}=\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}\oplus\mathbb{C}t_{0}^{\epsilon-1}\mathrm{d}_{1}$.
When $\epsilon=0$, $\widehat{\mathfrak{t}}(\mathfrak{g})^{0}$ is a vertex Lie
algebra and therefore there is a vertex algebra
$V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$ associated to
$\widehat{\mathfrak{t}}(\mathfrak{g})^{0}$ for any nonzero complex number
$\ell$ [DLM]. Based on a result of [B1], it was proved in [CLiT] that there is
a surjective vertex algebra homomorphism

$\Theta:\quad
V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})\longrightarrow
V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)},$

where $V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)$ is the
affine-Virasoro vertex algebra and $V_{(\mathfrak{h},L)}$ is the lattice
vertex algebra associated to the pair $(\mathfrak{h},L:=\mathbb{Z}{\bf k})$
[B1, LW]. Note that $V_{(\mathfrak{h},L)}$ can be realized as a quotient of
the affine vertex algebra associated to an affine Lie algebra
$\widehat{\mathfrak{p}}$ [LW]. By applying Proposition 4.9, we deduce in
Section 5.1 the following three correspondences:

  * •

restricted $\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules of level
$\ell$$\scriptstyle{1-1}$$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$-modules

  * •

certain restricted
$\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$-modules$\scriptstyle{1-1}$$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)$-modules

  * •

certain restricted
$\widehat{\mathfrak{p}}$-modules$\scriptstyle{1-1}$$\phi_{\epsilon}$-coordinated
$V_{(\mathfrak{h},L)}$-modules

The second correspondence implies that there is an irreducible
$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)$-module
structure on the $\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$-module
$L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)$ (see
Proposition 5.2). Meanwhile, the third correspondence allows us to give an
irreducible $\phi_{\epsilon}$-coordinated $V_{(\mathfrak{h},L)}$-module
structure on the $\widehat{\mathfrak{h}}$-module
$V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$ (see
Proposition 5.5). Thus, the tensor product space (1.2) becomes an irreducible
$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module. Via the homomorphism $\Theta$, this
$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module carries a canonical (irreducible)
$\phi_{\epsilon}$-coordinate
$V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$-module
structure. Then by applying the first correspondence above, the space (1.2)
automatically admits an irreducible
$\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module structure. This
irreducible $\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module can be
extended to a $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module (see
Theorem 5.7). And, based on the formula (1.3), we are able to compute the
module action explicitly, which is exactly what we state in Theorem 3.2.

In the rest of this paper, we let $\mathbb{Z},\ \mathbb{Z}^{\times},\
\mathbb{N},\ \mathbb{C}$ and $\mathbb{C}^{\times}$ be the sets of integers,
nonzero integers, nonnegative integers, complex numbers and nonzero complex
numbers, respectively. And, let $z,w,z_{0},z_{1},z_{2},\dots$ be mutually
commuting independent formal variables. For a vector space $W$,
$W[[z_{1},z_{2},\dots,z_{r}]]$ denotes the space of formal (possibly doubly
infinite) power series in $z_{1},z_{2},\dots,z_{r}$ with coefficients in $W$,
and $W((z_{1},z_{2},\dots,z_{r}))$ denotes the space of lower truncated
Laurent power series in $z_{1},z_{2},\dots,z_{r}$ with coefficients in $W$.

##  2\. Toroidal Lie algebras

In this section we introduce the extended toroidal Lie algebras we concern
about in this paper.

###  2.1. Toroidal Lie algebras
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$

Let $\mathcal{R}=\mathbb{C}[t_{0}^{\pm 1},t_{1}^{\pm 1}]$ be the Laurent
polynomial ring in the variables $t_{0}$ and $t_{1}$. Denote by

$\Omega_{\mathcal{R}}^{1}=\mathcal{R}\mathrm{d}t_{0}\oplus\mathcal{R}\mathrm{d}t_{1}=\mathcal{R}\mathrm{k}_{0}\oplus\mathcal{R}\mathrm{k}_{1}$

the space of $1$-forms on $\mathcal{R}$, where
$\mathrm{k}_{0}:=t_{0}^{-1}\mathrm{d}t_{0}$ and
$\mathrm{k}_{1}:=t_{1}^{-1}\mathrm{d}t_{1}$. Set

$\mathrm{d}({\mathcal{R}})=\\{\mathrm{d}(f)\,|\,f\in\mathcal{R}\\},$

the space of exact $1$-forms, where

$\mathrm{d}(f):=\frac{\partial f}{\partial
t_{0}}\mathrm{d}t_{0}+\frac{\partial f}{\partial
t_{1}}\mathrm{d}t_{1}=t_{0}\frac{\partial f}{\partial
t_{0}}\mathrm{k}_{0}+t_{1}\frac{\partial f}{\partial
t_{1}}\mathrm{k}_{1}\in\Omega_{\mathcal{R}}^{1}$

is the differential of $f$. Form the quotient vector space

$\displaystyle\mathcal{K}=\Omega_{\mathcal{R}}^{1}/\mathrm{d}({\mathcal{R}}).$

For $m_{0},m_{1}\in\mathbb{Z}$, set

$\mathrm{k}_{m_{0},m_{1}}=\begin{cases}\frac{1}{m_{1}}t_{0}^{m_{0}}t_{1}^{m_{1}}\mathrm{k}_{0},\
&\textnormal{{if}}\ m_{1}\neq 0\\\
-\frac{1}{m_{0}}t_{0}^{m_{0}}\mathrm{k}_{1},\ &\textnormal{{if}}\ m_{1}=0,\
m_{0}\neq 0\\\ 0,\ &\textnormal{{if}}\ m_{0}=m_{1}=0.\end{cases}$

Then the set

(2.1)
$\displaystyle\\{\mathrm{k}_{0},\mathrm{k}_{1}\\}\cup\\{\mathrm{k}_{m_{0},m_{1}}\mid
m_{0},m_{1}\in\mathbb{Z}\ \textnormal{{with}}\ (m_{0},m_{1})\neq(0,0)\\}$

is a basis of $\mathcal{K}$, noting that these elements in
$\Omega_{\mathcal{R}}^{1}$ should be understood as their images in
$\mathcal{K}$.

Let $\mathfrak{g}$ be a Lie algebra equipped with a symmetric invariant
bilinear form ${\langle}\cdot,\cdot{\rangle}$. Form a central extension of the
double loop algebra $\mathcal{R}\otimes\mathfrak{g}$:

$\displaystyle\mathfrak{t}(\mathfrak{g})=(\mathcal{R}\otimes\mathfrak{g})\oplus\mathcal{K},$

called the $2$-toroidal Lie algebra, where $\mathcal{K}$ is central and

(2.2) $\begin{split}[t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes
u,t_{0}^{n_{0}}t_{1}^{n_{1}}\otimes
v]=t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\otimes[u,v]+{\langle}u,v{\rangle}\sum_{r=0,1}m_{r}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\mathrm{k}_{r}\end{split}$

for $u,v\in\mathfrak{g}$, $m_{0},n_{0},m_{1},n_{1}\in\mathbb{Z}$. If
$\mathfrak{g}$ is a finite-dimensional simple Lie algebra and
${\langle}\cdot,\cdot{\rangle}$ is nondegenerate, $\mathfrak{t}(\mathfrak{g})$
is known as a universal central extension of $\mathcal{R}\otimes\mathfrak{g}$
(see [MRY]).

We denote by

$\mathrm{Der}(\mathcal{R})=\mathcal{R}\frac{\partial}{\partial
t_{0}}\oplus\mathcal{R}\frac{\partial}{\partial
t_{1}}=\mathcal{R}\mathrm{d}_{0}\oplus\mathcal{R}\mathrm{d}_{1}$

the derivation Lie algebra on $\mathcal{R}$, where
$\mathrm{d}_{0}:=t_{0}\frac{\partial}{\partial t_{0}}$ and
$\mathrm{d}_{1}:=t_{1}\frac{\partial}{\partial t_{1}}$. By adding
$\mathrm{Der}(\mathcal{R})$ to $\mathfrak{t}(\mathfrak{g})$, we obtain the
full $2$-toroidal Lie algebra (see [B1])

$\displaystyle\mathcal{T}(\mathfrak{g})=\mathcal{T}(\mathfrak{g})_{\mu}:=\mathfrak{t}(\mathfrak{g})\oplus\mathrm{Der}(\mathcal{R})=(\mathcal{R}\otimes\mathfrak{g})\oplus\mathcal{K}\oplus\mathrm{Der}(\mathcal{R}),$

with Lie relations

$\displaystyle\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\mathrm{d}_{i},t_{0}^{n_{0}}t_{1}^{n_{1}}\otimes
x\right]$ $\displaystyle=n_{i}(t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\otimes
x),$
$\displaystyle\left[t_{0}^{m_{0}}t_{1}^{m_{1}}\mathrm{d}_{i},t_{0}^{n_{0}}t_{1}^{n_{1}}\mathrm{k}_{j}\right]$
$\displaystyle=n_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\mathrm{k}_{j}+\delta_{i,j}\sum_{r=0,1}m_{r}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\mathrm{k}_{r},$
(2.3)
$\displaystyle\left[pt_{0}^{m_{0}}t_{1}^{m_{1}}\mathrm{d}_{i},t_{0}^{n_{0}}t_{1}^{n_{1}}\mathrm{d}_{j}\right]$
$\displaystyle=n_{i}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\mathrm{d}_{j}-m_{j}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\mathrm{d}_{i}$
$\displaystyle-\mu
m_{j}n_{i}\left(m_{0}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\mathrm{k}_{0}+m_{1}t_{0}^{m_{0}+n_{0}}t_{1}^{m_{1}+n_{1}}\mathrm{k}_{1}\right)$

for $x\in\mathfrak{g}$, $m_{0},n_{0},m_{1},n_{1}\in\mathbb{Z}$ and
$i,j\in\\{0,1\\}$, where $\mu$ is a fixed complex number throughout this
paper. The algebra $\mathcal{T}(\mathfrak{g})$ is $\mathbb{Z}$-graded with
respect to the adjoint action of $-\mathrm{d}_{0}$, i.e.,

$\mathcal{T}(\mathfrak{g})=\oplus_{n\in\mathbb{Z}}\mathcal{T}(\mathfrak{g})_{(n)},$

where

$\displaystyle\mathcal{T}(\mathfrak{g})_{(n)}=\\{x\in\mathcal{T}(\mathfrak{g})\mid[\mathrm{d}_{0},x]=-nx\\}.$

For a given integer $\epsilon$, set

$\displaystyle\mathcal{D}^{\epsilon}=\bigg{\\{}f_{0}t_{0}^{\epsilon}\frac{\partial}{\partial
t_{0}}+f_{1}\mathrm{d}_{1}\mid f_{0},f_{1}\in\mathcal{R},\
t_{0}^{\epsilon}\frac{\partial}{\partial
t_{0}}(f_{0})+\mathrm{d}_{1}(f_{1})=0\bigg{\\}}\subset\mathrm{Der}(\mathcal{R}).$

For $m_{0},m_{1}\in\mathbb{Z}$, write

$\tilde{\mathrm{d}}_{m_{0},m_{1}}^{\epsilon}=(m_{0}-\epsilon+1)t_{0}^{m_{0}}t_{1}^{m_{1}}\mathrm{d}_{1}-m_{1}t_{0}^{m_{0}}t_{1}^{m_{1}}\mathrm{d}_{0},$

which is an element of $\mathcal{D}^{\epsilon}$. A simple fact is that the set

(2.4)
$\displaystyle\\{t_{0}^{\epsilon-1}\mathrm{d}_{0},t_{0}^{\epsilon-1}\mathrm{d}_{1}\\}\cup\\{\tilde{\mathrm{d}}^{\epsilon}_{m_{0},m_{1}}\mid(m_{0},m_{1})\in(\mathbb{Z}\times\mathbb{Z})\backslash\\{(\epsilon-1,0)\\}\\}$

is a basis of $\mathcal{D}^{\epsilon}$. Furthermore, $\mathcal{D}^{\epsilon}$
is a Lie subalgebra of $\mathrm{Der}(\mathcal{R})$ with the relations

$\displaystyle[t_{0}^{\epsilon-1}\mathrm{d}_{0},\tilde{\mathrm{d}}^{\epsilon}_{m_{0},m_{1}}]$
$\displaystyle=\left(m_{0}-\epsilon+1\right)\tilde{\mathrm{d}}^{\epsilon}_{m_{0}+\epsilon-1,m_{1}},\quad[t_{0}^{\epsilon-1}\mathrm{d}_{1},\tilde{\mathrm{d}}^{\epsilon}_{m_{0},m_{1}}]=m_{1}\tilde{\mathrm{d}}^{\epsilon}_{m_{0}+\epsilon-1,m_{1}},$
$\displaystyle[\tilde{\mathrm{d}}^{\epsilon}_{m_{0},m_{1}},\tilde{\mathrm{d}}^{\epsilon}_{n_{0},n_{1}}]$
$\displaystyle=\left((m_{0}-\epsilon+1)n_{1}-m_{1}(n_{0}-\epsilon+1)\right)\tilde{\mathrm{d}}^{\epsilon}_{m_{0}+n_{0},m_{1}+n_{1}},$

where $m_{0},m_{1},n_{0},n_{1}\in\mathbb{Z}$. Set

$\displaystyle\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$
$\displaystyle=\mathfrak{t}(\mathfrak{g})\oplus\mathcal{D}^{\epsilon}\subset\mathcal{T}(\mathfrak{g}),$

which is the Lie algebra we study in this paper.

######  Remark 2.1.

When $\epsilon=1$, the form ${\langle}\cdot,\cdot{\rangle}$ on $\mathfrak{g}$
can be extended to an invariant symmetric bilinear form $(\cdot,\cdot)$ on the
algebra $\widetilde{\mathfrak{t}}(\mathfrak{g})^{1}$ such that the nontrivial
values are as follows:

$(t_{0}^{m_{0}}t_{1}^{m_{1}}\otimes u,t_{0}^{-m_{0}}t_{1}^{-m_{1}}\otimes
v)={\langle}u,v{\rangle},\quad(\mathrm{k}_{n_{0},n_{1}},\mathrm{d}_{-n_{0},-n_{1}})=-1,\quad(\mathrm{k}_{i},\mathrm{d}_{i})=1$

where $u,v\in\mathfrak{g},m_{0},m_{1},n_{0},n_{1}\in\mathbb{Z}$ with
$(n_{0},n_{1})\neq(0,0)$ and $i=0,1$. Furthermore, if $\mathfrak{g}$ is a
finite-dimensional simple Lie algebra equipped with a Cartan subalgebra $H$,
then the triple

$\left(\widetilde{\mathfrak{t}}(\mathfrak{g})^{1},\widetilde{H}:=H+\mathbb{C}\mathrm{k}_{0}+\mathbb{C}\mathrm{k}_{1}+\mathbb{C}\mathrm{d}_{0}+\mathbb{C}\mathrm{d}_{1},(\cdot,\cdot)\right)$

is an extended affine Lie algebra in the sense of [AABGP], and is often
referred as the nullity-2 toroidal extended affine Lie algebra [B2, CLT1].

###  2.2. Relations of generating functions on
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$

Consider the following subalgebra of
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$:

$\displaystyle\widetilde{\mathfrak{g}}_{1}=\left(\mathbb{C}[t_{1},t_{1}^{-1}]\otimes\mathfrak{g}\right)\oplus\mathbb{C}\mathrm{k}_{1}\oplus\mathbb{C}\mathrm{d}_{1},$

which is isomorphic to the affine Kac-Moody algebra associated to
$\mathfrak{g}$ [K1]. For any $m\in\mathbb{Z}$ and $u=t_{1}^{n}\otimes
x+bk_{1}+cd_{1}\in\widetilde{\mathfrak{g}}_{1}$ with $x\in\mathfrak{g}$,
$n\in\mathbb{Z}$ and $b,c\in\mathbb{C}$, we will often write

$t_{0}^{m}u:=t_{0}^{m}t_{1}^{n}\otimes
x+bt_{0}^{m}k_{1}+ct_{0}^{m}d_{1}\in\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}.$

In addition, for $m,n\in\mathbb{Z}$, set

$\displaystyle\mathrm{d}_{n,m}^{\epsilon}:=\tilde{\mathrm{d}}_{n,m}^{\epsilon}+\mu(1-\epsilon)\left(n+\frac{1}{2}(1-\epsilon)\right)m^{2}\mathrm{k}_{n,m}.$

Let $\mathcal{B}$ be a vector space with a designated basis
$\\{\mathrm{K}_{n},\,\mathrm{D}_{n}\mid n\in\mathbb{Z}^{\times}\\}.$ Set

(2.5)
$\displaystyle\mathcal{B}_{\mathfrak{g}}=\widetilde{\mathfrak{g}}_{1}\oplus\mathcal{B}=\left(\mathbb{C}[t_{1},t_{1}^{-1}]\otimes\mathfrak{g}\right)\oplus\mathbb{C}\mathrm{k}_{1}\oplus\mathbb{C}\mathrm{d}_{1}\oplus\sum_{n\in\mathbb{Z}^{\times}}\left(\mathbb{C}\mathrm{K}_{n}\oplus\mathbb{C}\mathrm{D}_{n}\right).$

Define a linear map

$\psi_{\mathfrak{g}}^{\epsilon}:\mathcal{B}_{\mathfrak{g}}\rightarrow\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}[[z,z^{-1}]],\quad
a\mapsto a^{\epsilon}(z)$

by letting

$\displaystyle u^{\epsilon}(z)$
$\displaystyle=\sum_{n\in\mathbb{Z}}(t_{0}^{n}u)z^{\epsilon-n-1},\quad{\mathrm{D}_{m}^{\epsilon}(z)=\sum_{n\in\mathbb{Z}}\mathrm{d}^{\epsilon}_{n,m}z^{2\epsilon-n-2},\quad\mathrm{K}_{m}^{\epsilon}(z)=\sum_{n\in\mathbb{Z}}\mathrm{k}_{n,m}z^{-n},}$

where $u\in\widetilde{\mathfrak{g}}_{1},\ m\in\mathbb{Z}^{\times}$. Recall
that $\mathrm{k}_{m,0}=-\frac{1}{m}t_{0}^{n}\mathrm{k}_{1}$ for
$m\in\mathbb{Z}^{\times}$ and
$\mathrm{d}^{\epsilon}_{n,0}=\tilde{\mathrm{d}}^{\epsilon}_{n,0}=(n-\epsilon+1)t_{0}^{n}\mathrm{d}_{1}$
for $n\in\mathbb{Z}$. It follows from (2.1) and (2.4) that the coefficients of
$a^{\epsilon}(z)$ for $a\in\mathcal{B}_{\mathfrak{g}}$ together with the
elements $\mathrm{k}_{0}$ and $t_{0}^{\epsilon-1}\mathrm{d}_{0}$ linearly span
the algebra $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$.

For $a\in\mathbb{C}$ and $r\in\mathbb{N}$, set

$\displaystyle a^{(r)}_{\epsilon}=\prod_{s=0}^{r-1}(a+s(\epsilon-1)).$

The following result will be used later:

######  Lemma 2.2.

Let $p\in\mathbb{N}$, $a,b,\alpha,\beta\in\mathbb{C}$. Then

$\displaystyle\sum_{r=0}^{p}\binom{p}{r}\alpha^{p-r}{a_{\epsilon}}^{(p-r)}(-\beta)^{r}{b_{\epsilon}}^{(r)}=\sum_{s=0}^{p}\binom{p}{s}(\alpha+\beta)^{p-s}a^{(p-s)}_{\epsilon}\beta^{s}(-a-b-(p-1)(\epsilon-1))^{(s)}_{\epsilon}.$

###### Proof.

Note that we have the following identity

(2.6)
$\displaystyle(a+b)^{(p)}_{\epsilon}=\sum_{i=0}^{p}\binom{p}{i}a^{(i)}_{\epsilon}b^{(p-i)}_{\epsilon},$

which can be proved by induction on $p$. Using (2.6) and the facts

$\binom{p-r}{t}\binom{p}{r}=\binom{p}{r+t}\binom{r+t}{r}\quad\left(0\leqslant
r\leqslant p,\ 0\leqslant t\leqslant p-r\right),$

and

$a_{\epsilon}^{(p-r)}=a_{\epsilon}^{(p-s)}\left(a+(p-s)(\epsilon-1)\right)_{\epsilon}^{(s-r)}\quad\left(0\leqslant
r\leqslant s\leqslant p\right),$

we get

$\displaystyle\sum_{r=0}^{p}\binom{p}{r}\alpha^{p-r}a_{\epsilon}^{(p-r)}(-\beta)^{r}b_{\epsilon}^{(r)}$
$\displaystyle=\ $
$\displaystyle\sum_{r=0}^{p}\sum_{t=0}^{p-r}\binom{p-r}{t}\binom{p}{r}(\alpha+\beta)^{p-r-t}a_{\epsilon}^{(p-r)}(-\beta)^{r+t}b_{\epsilon}^{(r)}$
$\displaystyle=\ $
$\displaystyle\sum_{r=0}^{p}\sum_{t=0}^{p-r}\binom{p}{r+t}\binom{r+t}{r}(\alpha+\beta)^{p-r-t}a_{\epsilon}^{(p-r)}(-\beta)^{r+t}b_{\epsilon}^{(r)}$
$\displaystyle=\ $
$\displaystyle\sum_{s=0}^{p}\sum_{r=0}^{s}\binom{p}{s}\binom{s}{r}(\alpha+\beta)^{p-s}a_{\epsilon}^{(p-r)}(-\beta)^{s}b_{\epsilon}^{(r)}$
$\displaystyle=\ $
$\displaystyle\sum_{s=0}^{p}\sum_{r=0}^{s}\binom{p}{s}\binom{s}{r}(\alpha+\beta)^{p-s}a_{\epsilon}^{(p-s)}(a+(p-s)(\epsilon-1))_{\epsilon}^{(s-r)}(-\beta)^{s}b_{\epsilon}^{(r)}$
$\displaystyle=\ $
$\displaystyle\sum_{s=0}^{p}\binom{p}{s}(\alpha+\beta)^{p-s}a_{\epsilon}^{(p-s)}(a+b+(p-s)(\epsilon-1))_{\epsilon}^{(s)}(-\beta)^{s}$
$\displaystyle=\ $
$\displaystyle\sum_{s=0}^{p}\binom{p}{s}(\alpha+\beta)^{p-s}a_{\epsilon}^{(p-s)}(-a-b-(p-1)(\epsilon-1))_{\epsilon}^{(s)}\beta^{s}.$

∎

The following proposition collects the Lie relations among these generating
functions.

######  Proposition 2.3.

For $u,v\in\mathfrak{g}$, $a\in\mathcal{B}_{\mathfrak{g}}$, $m,n\in\mathbb{Z}$
and $k,l\in\mathbb{Z}^{\times}$, we have

$\displaystyle\ \textnormal{{(1)}}\quad[(t_{1}^{m}\otimes
u)^{\epsilon}(z),(t_{1}^{n}\otimes
v)^{\epsilon}(w)]=\left(t_{1}^{m+n}\otimes[u,v]\right)^{\epsilon}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$
$\displaystyle\quad\quad\
+{\langle}u,v{\rangle}m\left(w^{\epsilon}\frac{\partial}{\partial
w}\mathrm{K}^{\epsilon}_{m+n}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+{\langle}u,v{\rangle}(m+n)\mathrm{K}^{\epsilon}_{m+n}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$ $\displaystyle\quad\quad\
+\delta_{m+n,0}{\langle}u,v{\rangle}\left(m\mathrm{k}_{1}^{\epsilon}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+\mathrm{k}_{0}w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)\right),$ $\displaystyle\
\textnormal{{(2)}}\quad[\mathrm{k}_{1}^{\epsilon}(z),(t_{1}^{n}\otimes
u)^{\epsilon}(w)]=0=[\mathrm{K}^{\epsilon}_{k}(z),(t_{1}^{n}\otimes
u)^{\epsilon}(w)],$ $\displaystyle\
\textnormal{{(3)}}\quad[\mathrm{D}_{k}^{\epsilon}(z),(t_{1}^{n}\otimes
u)^{\epsilon}(w)]=k\left(w^{\epsilon}\frac{\partial}{\partial
w}(t_{1}^{k+n}\otimes
u)^{\epsilon}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$
$\displaystyle\quad\quad\ +(k+n)(t_{1}^{k+n}\otimes
u)^{\epsilon}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$ $\displaystyle\
\textnormal{{(4)}}\quad[\mathrm{d}_{1}^{\epsilon}(z),(t_{1}^{n}\otimes
u)^{\epsilon}(w)]=n(t_{1}^{n}\otimes
u)^{\epsilon}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$ $\displaystyle\
\textnormal{{(5)}}\quad[\mathrm{k}_{1}^{\epsilon}(z),\mathrm{k}_{1}^{\epsilon}(w)]=[\mathrm{K}^{\epsilon}_{k}(z),\mathrm{K}^{\epsilon}_{l}(w)]=[\mathrm{k}_{1}^{\epsilon}(z),\mathrm{K}^{\epsilon}_{l}(w)]=0,$
$\displaystyle\
\textnormal{{(6)}}\quad[\mathrm{D}_{k}^{\epsilon}(z),\mathrm{K}^{\epsilon}_{l}(w)]=k\left(w^{\epsilon}\frac{\partial}{\partial
w}\mathrm{K}^{\epsilon}_{k+l}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+(k+l)\mathrm{K}^{\epsilon}_{k+l}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$ $\displaystyle\quad\quad\
+\delta_{k+l,0}\left(k\mathrm{k}_{1}^{\epsilon}(z)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+\mathrm{k}_{0}w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)\right),$ $\displaystyle\
\textnormal{{(7)}}\quad[\mathrm{D}_{k}^{\epsilon}(z),\mathrm{k}_{1}^{\epsilon}(w)]=kw^{\epsilon}\frac{\partial}{\partial
w}\left(\left(w^{\epsilon}\frac{\partial}{\partial
w}\mathrm{K}^{\epsilon}_{k}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+\mathrm{K}^{\epsilon}_{k}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)\right),$ $\displaystyle\
\textnormal{{(8)}}\quad[\mathrm{d}^{\epsilon}_{1}(z),\mathrm{K}^{\epsilon}_{n}(w)]=n\mathrm{K}^{\epsilon}_{n}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$
$\displaystyle\
\textnormal{{(9)}}\quad[\mathrm{d}_{1}^{\epsilon}(z),\mathrm{k}_{1}^{\epsilon}(w)]=\mathrm{k}_{0}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),\quad\quad[\mathrm{d}_{1}^{\epsilon}(z),\mathrm{d}_{1}^{\epsilon}(w)]=0,$
$\displaystyle\ \textnormal{{(10)}}\ \
[\mathrm{D}^{\epsilon}_{k}(z),\mathrm{D}^{\epsilon}_{l}(w)]=k\left(w^{\epsilon}\frac{\partial}{\partial
w}\mathrm{D}^{\epsilon}_{k+l}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+(k+l)\mathrm{D}^{\epsilon}_{k+l}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$ $\displaystyle\quad\quad\
-k\delta_{k+l,0}\left(\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{2}\mathrm{d}_{1}^{\epsilon}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+\mu
k^{3}\delta_{k+l,0}\left(\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{2}\mathrm{k}_{1}^{\epsilon}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$
$\displaystyle\quad\quad\ +\mu\sum_{r=0}^{3}{3\choose
r}\left(\left(kw^{\epsilon}\frac{\partial}{\partial
w}\right)^{r}\mathrm{K}^{\epsilon}_{k+l}(w)\right)\left((k+l)w^{\epsilon}\frac{\partial}{\partial
w}\right)^{3-r}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$ $\displaystyle\
\textnormal{{(11)}}\quad[\mathrm{d}^{\epsilon}_{1}(z),\mathrm{D}^{\epsilon}_{l}(w)]=l\mathrm{D}^{\epsilon}_{l}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+\mu
l^{3}\mathrm{K}^{\epsilon}_{l}(w)\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{2}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$ $\displaystyle\
\textnormal{{(12)}}\quad[t_{0}^{\epsilon-1}\mathrm{d}_{0},a^{\epsilon}(z)]=-z^{\epsilon}\frac{d}{dz}a^{\epsilon}(z),$

where $\mathrm{K}_{0}^{\epsilon}(z)$ and $\mathrm{D}_{0}^{\epsilon}(z)$ are
understood as the zero formal series.

###### Proof.

We will only verify relation (10), which is the most complicated one. The
verification of other relations are straightforward. Let
$i,j,k,l\in\mathbb{Z}$. Firstly, from (2.3) we have

$\displaystyle[\tilde{\mathrm{d}}_{i,k}^{\epsilon},\tilde{\mathrm{d}}_{j,l}^{\epsilon}]$
$\displaystyle=\ $
$\displaystyle[(i-\epsilon+1)t_{0}^{i}t_{1}^{k}\mathrm{d}_{1}-kt_{0}^{i}t_{1}^{k}\mathrm{d}_{0},(j-\epsilon+1)t_{0}^{j}t_{1}^{l}\mathrm{d}_{1}-lt_{0}^{j}t_{1}^{l}\mathrm{d}_{0}]$
$\displaystyle=\ $
$\displaystyle((i-\epsilon+1)l-k(j-\epsilon+1))(1-\delta_{k+l,0})\tilde{\mathrm{d}}_{i+j,k+l}^{\epsilon}-k\delta_{k+l,0}(i+j-\epsilon+1)(i+j-2\epsilon+2)t_{0}^{i+j}\mathrm{d}_{1}$
$\displaystyle+\mu((i-\epsilon+1)l-jk)(il-(j-\epsilon+1)k)(il-
jk)\mathrm{k}_{i+j,k+l}+\delta_{k+l,0}(kt_{0}^{i+j}\mathrm{k}_{1}+i\delta_{i+j,0}\mathrm{k}_{0})$
$\displaystyle=\ $
$\displaystyle((i-\epsilon+1)l-k(j-\epsilon+1))(1-\delta_{k+l,0})\mathrm{d}_{i+j,k+l}^{\epsilon}-\mu(1-\epsilon)(i+j+\frac{1}{2}(1-\epsilon))(k+l)^{2}\mathrm{k}_{i+j,k+l}$
$\displaystyle-k\delta_{k+l,0}(i+j-\epsilon+1)(i+j-2\epsilon+2)t_{0}^{i+j}\mathrm{d}_{1}$
$\displaystyle+\mu\delta_{k+l,0}((i+j-\epsilon+1)^{2}k^{3}t_{0}^{i+j}\mathrm{k}_{1}+(1-\epsilon)^{2}k^{2}i\delta_{i+j,0}\mathrm{k}_{0})$
$\displaystyle+\mu((i-\epsilon+1)l-jk)(il-(j-\epsilon+1)k)(il-
jk)(1-\delta_{k+l,0})\mathrm{k}_{i+j,k+l}$ $\displaystyle=\ $
$\displaystyle((i-\epsilon+1)l-k(j-\epsilon+1))(1-\delta_{k+l,0})\mathrm{d}_{i+j,k+l}^{\epsilon}-k\delta_{k+l,0}(i+j-\epsilon+1)(i+j-2\epsilon+2)t_{0}^{i+j}\mathrm{d}_{1}$
$\displaystyle+\mu(1-\delta_{k+l,0})C_{k,l}^{i,j}\mathrm{k}_{i+j,k+l}+\mu\delta_{k+l,0}((i+j-\epsilon+1)^{2}k^{3}t_{0}^{i+j}\mathrm{k}_{1}+(1-\epsilon)^{2}k^{2}i\delta_{i+j,0}\mathrm{k}_{0}),$

where $C_{k,l}^{i,j}$ stands for the following number

$\displaystyle
l^{3}(i-\epsilon+1)\left(i^{2}-(1-\epsilon)(i+j+\frac{1}{2}(1-\epsilon))\right)$
$\displaystyle-k^{3}(j-\epsilon+1)\left(j^{2}-(1-\epsilon)(i+j+\frac{1}{2}(1-\epsilon))\right)$
$\displaystyle-
kl^{2}\left(3(i-\epsilon+1)i(j-\epsilon+1)-(j+\frac{1}{2}(1-\epsilon))(j+\epsilon-1)\right)$
$\displaystyle+k^{2}l\left(3(i-\epsilon+1)i(j-\epsilon+1)-(i+\frac{1}{2}(1-\epsilon))(i+\epsilon-1)\right).$

Next, using the fact

$[\tilde{\mathrm{d}}^{\epsilon}_{i,k},\mathrm{k}_{j,l}]=((i-\epsilon+1)l-k(j+\epsilon-1))\mathrm{k}_{i+j,k+l},$

we obtain

$\displaystyle[\tilde{\mathrm{d}}_{i,k}^{\epsilon},\mu(1-\epsilon)(j+\frac{1}{2}(1-\epsilon))l^{2}\mathrm{k}_{j,l}]-[\tilde{\mathrm{d}}_{j,l}^{\epsilon},\mu(1-\epsilon)(i+\frac{1}{2}(1-\epsilon))k^{2}\mathrm{k}_{i,k}]$
$\displaystyle=\ $
$\displaystyle\mu(1-\epsilon)(1-\delta_{k+l,0})\bigg{(}((i-\epsilon+1)l-k(j+\epsilon-1))(j+\frac{1}{2}(1-\epsilon))l^{2}$
$\displaystyle-((j-\epsilon+1)k-(i+\epsilon-1)l)(i+\frac{1}{2}(1-\epsilon))k^{2}\bigg{)}\mathrm{k}_{i+j,k+l}$
$\displaystyle+\mu(1-\epsilon)\delta_{k+l,0}\bigg{(}(j+\frac{1}{2}(1-\epsilon))l^{2}((i-\epsilon+1)\delta_{i+j,0}\mathrm{k}_{0}+kt_{0}^{i+j}\mathrm{k}_{1})$
$\displaystyle-(i+\frac{1}{2}(1-\epsilon))k^{2}((j-\epsilon+1)\delta_{i+j,0}\mathrm{k}_{0}+lt_{0}^{i+j}\mathrm{k}_{1})\bigg{)}$
$\displaystyle=\ $
$\displaystyle\mu(1-\epsilon)(1-\delta_{k+l,0})\bigg{(}((i-\epsilon+1)l-k(j+\epsilon-1))(j+\frac{1}{2}(1-\epsilon))l^{2}$
$\displaystyle-((j-\epsilon+1)k-(i+\epsilon-1)l)(i+\frac{1}{2}(1-\epsilon))k^{2}\bigg{)}\mathrm{k}_{i+j,k+l}$
$\displaystyle+\mu(1-\epsilon)\delta_{k+l,0}(k^{3}(i+j-\epsilon+1)t_{0}^{i+j}\mathrm{k}_{1}-k^{2}i(1-\epsilon)\delta_{i+j,0}\mathrm{k}_{0}).$

By Lemma 2.2 with $p=3$, we have

$\displaystyle\sum_{r=0}^{3}\binom{3}{r}l^{3-r}(i-\epsilon+1)_{\epsilon}^{(3-r)}(-k)^{r}(j-\epsilon+1)^{(r)}_{\epsilon}$
$\displaystyle=\,$
$\displaystyle\sum_{s=0}^{3}\binom{3}{s}(k+l)^{3-s}(i-\epsilon+1)_{\epsilon}^{(3-s)}k^{s}(-i-j)^{(s)}_{\epsilon}.$

Now by summarizing the above results, we find

$\displaystyle[\mathrm{d}_{i,k}^{\epsilon},\mathrm{d}_{j,l}^{\epsilon}]$
$\displaystyle=\ $
$\displaystyle[\tilde{\mathrm{d}}_{i,k}^{\epsilon}+\mu(1-\epsilon)\left(i+\frac{1}{2}(1-\epsilon)\right)k^{2}\mathrm{k}_{i,k},\tilde{\mathrm{d}}_{j,l}^{\epsilon}+\mu(1-\epsilon)\left(j+\frac{1}{2}(1-\epsilon)\right)l^{2}\mathrm{k}_{j,l}]$
$\displaystyle=\ $
$\displaystyle(1-\delta_{k+l,0})((i-\epsilon+1)l-k(j-\epsilon+1))\mathrm{d}_{i+j,k+l}^{\epsilon}$
$\displaystyle+\delta_{k+l,0}(i+j-\epsilon+1)(i+j-2\epsilon+2)\left(-kt_{0}^{i+j}\mathrm{d}_{1}+\mu
k^{3}t_{0}^{i+j}\mathrm{k}_{1}\right)$
$\displaystyle+\mu\left(1-\delta_{k+l,0}\right)\sum_{r=0}^{3}\binom{3}{r}l^{3-r}(i-\epsilon+1)_{\epsilon}^{(3-r)}(-k)^{r}(j-\epsilon+1)_{\epsilon}^{(r)}\mathrm{k}_{i+j,k+l}$
$\displaystyle=\ $
$\displaystyle(1-\delta_{k+l,0})((i-\epsilon+1)l-k(j-\epsilon+1))\mathrm{d}_{i+j,k+l}^{\epsilon}$
$\displaystyle+\delta_{k+l,0}(i+j-\epsilon+1)(i+j-2\epsilon+2)\left(-kt_{0}^{i+j}\mathrm{d}_{1}+\mu
k^{3}t_{0}^{i+j}\mathrm{k}_{1}\right)$
$\displaystyle+\mu(1-\delta_{k+l,0})\sum_{s=0}^{3}\binom{3}{s}(k+l)^{3-s}(i-\epsilon+1)_{\epsilon}^{(3-s)}k^{s}(-i-j)_{\epsilon}^{(s)}\mathrm{k}_{i+j,k+l}.$

According to the above commutator relation and the fact that

$(i-\epsilon+1)l-k(j-\epsilon+1)=k(2\epsilon-i-j-2)+(i-\epsilon+1)(k+l),$

we have

$\displaystyle[\mathrm{D}_{k}^{\epsilon}(z),\mathrm{D}_{l}^{\epsilon}(w)]$
$\displaystyle=\ $
$\displaystyle\sum_{i,j\in\mathbb{Z}}[\mathrm{d}_{i,k}^{\epsilon},\mathrm{d}_{j,l}^{\epsilon}]z^{2\epsilon-i-2}w^{2\epsilon-j-2}$
$\displaystyle=\ $ $\displaystyle
k(1-\delta_{k+l,0})\sum_{i,j\in\mathbb{Z}}(2\epsilon-
i-j-2)\mathrm{d}_{i+j,k+l}^{\epsilon}w^{3\epsilon-
i-j-3}z^{2\epsilon-i-2}w^{i-\epsilon+1}$
$\displaystyle+(k+l)\sum_{i,j\in\mathbb{Z}}\mathrm{d}_{i+j,k+l}^{\epsilon}w^{2\epsilon-
i-j-2}(i-\epsilon+1)z^{-i+2\epsilon-2}w^{i}$
$\displaystyle-k\delta_{k+l,0}\sum_{i,j\in\mathbb{Z}}(\epsilon-
i-j-1)_{\epsilon}^{(2)}t_{0}^{i+j}\mathrm{d}_{1}w^{3\epsilon-
i-j-3}z^{2\epsilon-i-2}w^{i-\epsilon+1}$
$\displaystyle+\mu(1-\delta_{k+l,0})\sum_{i,j\in\mathbb{Z}}\sum_{r=0}^{3}\binom{3}{r}k^{r}(-i-j)_{\epsilon}^{(r)}\mathrm{k}_{i+j,k+l}w^{-i-j+r(\epsilon-1)}$
$\displaystyle\cdot(k+l)^{3-r}(i-\epsilon+1)_{\epsilon}^{(3-r)}z^{2\epsilon-i-2}w^{i+(2-r)(\epsilon-1)}$
$\displaystyle+\mu k^{3}\delta_{k+l,0}\sum_{i,j\in\mathbb{Z}}(\epsilon-
i-j-1)_{\epsilon}^{(2)}t_{0}^{i+j}\mathrm{k}_{1}w^{3\epsilon-
i-j-3}z^{2\epsilon-i-2}w^{i-\epsilon+1}$ $\displaystyle=\ $ $\displaystyle
kw^{\epsilon}\frac{\partial}{\partial
w}\mathrm{D}_{k+l}^{\epsilon}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+(k+l)\mathrm{D}_{k+l}^{\epsilon}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$
$\displaystyle-k\delta_{k+l,0}w^{\epsilon}\frac{\partial}{\partial
w}^{2}\mathrm{d}_{1}^{\epsilon}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$
$\displaystyle+\mu\sum_{r=0}^{3}{3\choose
r}kw^{\epsilon}\frac{\partial}{\partial
w}^{r}\mathrm{K}_{k+l}^{\epsilon}(w)(k+l)w^{\epsilon}\frac{\partial}{\partial
w}^{3-r}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$ $\displaystyle+\mu
k^{3}\delta_{k+l,0}w^{\epsilon}\frac{\partial}{\partial
w}^{2}\mathrm{k}_{1}^{\epsilon}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$

proving (10). ∎

##  3\. Realization of irreducible highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules

In this section, as the main result of the paper, we present an explicit
realization of certain irreducible highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules.

###  3.1. Highest weight modules for affine-Virasoro algebras

Let $\mathfrak{b}$ be a Lie algebra equipped with an invariant symmetric
bilinear form ${\langle}\cdot,\cdot{\rangle}$. We denote by

$\displaystyle\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}=\left(\mathrm{Der}\,\mathbb{C}[t,t^{-1}]\right)\ltimes\left(\mathbb{C}[t,t^{-1}]\otimes\mathfrak{b}\right)\oplus\mathbb{C}\mathrm{k}\oplus\mathbb{C}\mathrm{k}_{\mathsf{Vir}},$

the affine-Virasoro algebra associated to $\mathfrak{b}$, where $\mathrm{k}$
and $\mathrm{k}_{\mathsf{Vir}}$ are central elements and

(3.1)
$\begin{split}[L(m),L(n)]&=(m-n)L(m+n)+\frac{m^{3}-m}{12}\delta_{m+n,0}\mathrm{k}_{\mathsf{Vir}},\\\
[u(m),v(n)]&=[u,v](m+n)+m\delta_{m+n,0}{\langle}u,v{\rangle}\mathrm{k},\\\
[L(m),u(n)]&=-nu(m+n)\end{split}$

for $u,v\in\mathfrak{b}$, $m,n\in\mathbb{Z}$ and

$L(m):=-t^{m+1}\frac{d}{dt},\quad u(m):=t^{m}\otimes u\quad\text{ for
}m\in\mathbb{Z},\ u\in\mathfrak{b}.$

Note that $\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}$ contains the Virasoro
algebra

$\mathsf{Vir}=\mathrm{Der}\,\mathbb{C}[t,t^{-1}]+\mathbb{C}\mathrm{k}_{\mathsf{Vir}}$

and the affine algebras

$\widehat{\mathfrak{b}}=\left(\mathbb{C}[t,t^{-1}]\otimes\mathfrak{b}\right)\oplus\mathbb{C}\mathrm{k},\quad\widetilde{\mathfrak{b}}=\widehat{\mathfrak{b}}\oplus\mathbb{C}t\frac{d}{dt}$

as subalgebras. For any $\epsilon\in\mathbb{Z}$, set

(3.2) $\displaystyle a^{\epsilon}(z)$
$\displaystyle=\sum_{n\in\mathbb{Z}}\left(a\otimes
t^{n}\right)z^{\epsilon-n-1},\quad\forall a\in\mathfrak{b},$ (3.3)
$\displaystyle L^{\epsilon}(z)$
$\displaystyle=\sum_{n\in\mathbb{Z}}L(n)z^{2\epsilon-n-2}+\frac{\epsilon^{2}-2\epsilon}{24}z^{2(\epsilon-1)}\mathrm{k}_{\mathsf{Vir}}.$

When $\epsilon=0$, we also write $a(z)=a^{0}(z)$ and $L(z)=L^{0}(z)$. To
emphasize the dependence on $\mathfrak{b}$, sometimes we will write
$L_{\mathfrak{b}}(n)$, $L^{\epsilon}_{\mathfrak{b}}(z)$ and
$L_{\mathfrak{b}}(z)$, instead of $L(n),L^{\epsilon}(z)$ and $L(z)$,
respectively.

We fix a $\mathbb{Z}$-gradation on $\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}$
with respect to the adjoint action of $L(0)$:

$\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}=\oplus_{n\in\mathbb{Z}}(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(n)}.$

This $\mathbb{Z}$-gradation naturally induces a triangular decomposition of
$\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}$:

$\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}=(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(+)}\oplus(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(0)}\oplus(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(-)},$

where
$(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(0)}=\mathfrak{b}\oplus\mathbb{C}L(0)\oplus\mathbb{C}\mathrm{k}\oplus\mathbb{C}\mathrm{k}_{\mathsf{Vir}},$
and

$\displaystyle(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(\pm)}=\oplus_{n>0}(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(\mp
n)}={\rm Span}\\{u(\pm n),\ L(\pm n)\mid u\in\mathfrak{b},\ n>0\\}.$

Let $U$ be a $\mathfrak{b}$-module and $\beta,\ell,c\in\mathbb{C}$. View $U$
as a $(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(0)}$-module with $L(0)$,
$\mathrm{k}$ and $\mathrm{k}_{\mathsf{Vir}}$ act respectively as the scalars
$\beta$, $\ell$ and $c$, and extend it to a
$((\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(+)}+(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(0)})$-module
by letting $(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(+)}$ acts trivially.
Then we have the induced $\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}$-module:

(3.4) $\displaystyle
V_{\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}}(\ell,c,U,\beta)=\mathcal{U}(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})\otimes_{\mathcal{U}\left((\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(+)}+(\widehat{\mathfrak{b}}\rtimes\mathsf{Vir})_{(0)}\right)}U.$

When the $\mathfrak{b}$-module $U$ is irreducible,
$V_{\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}}(\ell,c,U,\beta)$ has a unique
irreducible quotient, called
$L_{\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}}(\ell,c,U,\beta)$. Note that the
$\mathbb{Z}$-grading on $\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}$ affords
naturally $\mathbb{N}$-grading structures on
$V_{\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}}(\ell,c,U,\beta)$ and
$L_{\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}}(\ell,c,U,\beta)$ with $\deg
U=0$.

Similarly, we have a $\mathbb{Z}$-grading
$\widehat{\mathfrak{b}}=\bigoplus\widehat{\mathfrak{b}}_{(n)}$ and a
triangular decompositions
$\widehat{\mathfrak{b}}=\widehat{\mathfrak{b}}_{(+)}\oplus\widehat{\mathfrak{b}}_{(0)}\oplus\widehat{\mathfrak{b}}_{(-)}$
of the affine Lie algebra $\widehat{\mathfrak{b}}$, where
$\mathfrak{b}_{(n)}=t^{-n}\otimes\mathfrak{b}+\delta_{n,0}\mathbb{C}\mathrm{k}$
and $\mathfrak{b}_{(\pm)}=t^{\pm 1}\mathbb{C}[t^{\pm 1}]\otimes\mathfrak{b}$.
Extend the $\mathfrak{b}$-module $U$ to a
$(\widehat{\mathfrak{b}}_{(+)}\oplus\widehat{\mathfrak{b}}_{(0)})$-module such
that $\widehat{\mathfrak{b}}_{(+)}$ acts trivially and $\mathrm{k}$ acts as
the scalar $\ell$. Then we have the Verma type highest weight
$\widehat{\mathfrak{b}}$-module

(3.5) $\displaystyle
V_{\widehat{\mathfrak{b}}}(\ell,U)=\mathcal{U}(\widehat{\mathfrak{b}})\otimes_{\mathcal{U}(\widehat{\mathfrak{b}}_{(+)}+\widehat{\mathfrak{b}}_{(0)})}U$

and its irreducible quotient $L_{\widehat{\mathfrak{b}}}(\ell,U)$ provided
that $U$ is irreducible. Note that both $V_{\widehat{\mathfrak{b}}}(\ell,U)$
and $L_{\widehat{\mathfrak{b}}}(\ell,U)$ are naturally $\mathbb{N}$-graded
with $U$ as the degree $0$ subspace.

We also have a triangular decomposition
$\mathsf{Vir}=\mathsf{Vir}_{(+)}\oplus\mathsf{Vir}_{(0)}\oplus\mathsf{Vir}_{(-)}$
of the Virasoro algebra $\mathsf{Vir}$, where
$\mathsf{Vir}_{(\pm)}=\oplus_{m>0}\mathbb{C}L(\pm m)$ and
$\mathsf{Vir}_{(0)}=\mathbb{C}L(0)+\mathbb{C}\mathrm{k}_{\mathsf{Vir}}$. View
$\mathbb{C}$ as a $(\mathsf{Vir}_{(+)}\oplus\mathsf{Vir}_{(0)})$-module such
that $\mathsf{Vir}_{(+)}$ acts trivially, $L(0)$ acts as the scalar $\beta$
and $\mathrm{k}_{\mathsf{Vir}}$ acts as the scalar $c$. Then we have the Verma
type highest weight module

(3.6) $\displaystyle
V_{\mathsf{Vir}}(c,\beta)=\mathcal{U}(\mathsf{Vir})\otimes_{\mathcal{U}(\mathsf{Vir}_{(+)}+\mathsf{Vir}_{(0)})}\mathbb{C}$

and its irreducible quotient $L_{\mathsf{Vir}}(c,\beta)$.

######  Remark 3.1.

_Assume that $\mathfrak{b}$ is finite-dimensional simple,
${\langle}\cdot,\cdot{\rangle}$ is the normalized form, and $U$ is finite
dimensional. By the Sugawara construction (see [K1]), when $\ell$ is not the
negative dual Coxeter number, the irreducible
$\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}$-module
$L_{\widehat{\mathfrak{b}}\rtimes\mathsf{Vir}}(\ell,c,U,\beta)$ can be
realized as the tensor product of the $\widehat{\mathfrak{b}}$-module
$L_{\widehat{\mathfrak{b}}}(\ell,U)$ and the $\mathsf{Vir}$-module
$L_{\mathsf{Vir}}(c^{\prime},\beta^{\prime})$ for some suitable
$c^{\prime},\beta^{\prime}\in\mathbb{C}$. _

###  3.2. The main construction

In this subsection we construct a class of irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules.

Let $\mathfrak{h}=\mathbb{C}{\bf k}+\mathbb{C}{\bf d}$ be a vector space
equipped with a designated basis $\\{{\bf k},{\bf d}\\}$ and a symmetric
bilinear form ${\langle}\cdot,\cdot{\rangle}$ determined by

(3.7) $\displaystyle{\langle}{\bf k},{\bf k}{\rangle}=0={\langle}{\bf d},{\bf
d}{\rangle},\quad\quad{\langle}{\bf k},{\bf d}{\rangle}=1.$

Set $L=\mathbb{Z}{\bf k}\subset\mathfrak{h}$ and $\mathfrak{h}_{{\bf
k}}=\mathbb{C}\otimes_{\mathbb{Z}}L=\mathbb{C}{\bf k}$. Denote by

$\mathbb{C}[L]=\oplus_{\gamma\in
L}\mathbb{C}e^{\gamma}\quad\text{and}\quad\mathbb{C}[\mathfrak{h}_{{\bf
k}}]=\oplus_{h\in\mathfrak{h}_{{\bf k}}}\mathbb{C}e^{h}$

the group algebras of $L$ and $\mathfrak{h}_{\bf k}$, respectively. We endow
$\mathbb{C}[\mathfrak{h}_{{\bf k}}]$ with an $\mathfrak{h}$-module structure:
$he^{h^{\prime}}={\langle}h,h^{\prime}{\rangle}e^{h^{\prime}}$ for
$h\in\mathfrak{h},h^{\prime}\in\mathfrak{h}_{\bf k}$. Note that for each
$\alpha\in\mathbb{C}$, $e^{\alpha{\bf k}}\mathbb{C}[L]$ is an
$\mathfrak{h}$-submodule of $\mathbb{C}[\mathfrak{h}_{{\bf k}}]$. Then for any
$\ell\in\mathbb{C}^{\times}$, we have a Verma type highest weight
$\widehat{\mathfrak{h}}$-module $V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])$ (see (3.5)). Note that we have the following vector spaces
identification:

$V_{\widehat{\mathfrak{h}}}\left(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L]\right)=V_{\widehat{\mathfrak{h}}}(\ell,0)\otimes
e^{\alpha{\bf k}}\mathbb{C}[L].$

It is known (see [LL]) that the highest weight $\widehat{\mathfrak{h}}$-module
$V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$ can be
extended to an $\widehat{\mathfrak{h}}\rtimes\mathsf{Vir}$-module with
$\mathrm{k}_{\mathsf{Vir}}=2$ and

(3.8) $\displaystyle L_{\mathfrak{h}}(z)=\frac{1}{\ell}\ _{\circ}^{\circ}{\bf
k}(z){\bf d}(z)_{\circ}^{\circ}.$

Let $W$ be a vector space and $u,v\in W$. For any two formal series
$u(z)=\sum_{n\in\mathbb{Z}}u_{n}z^{-n-1}$ and
$v(z)=\sum_{n\in\mathbb{Z}}v_{n}z^{-n-1}$ in $W[[z,z^{-1}]]$, we denote by

${}_{\circ}^{\circ}u(z_{1})v(z_{2})_{\circ}^{\circ}\
=u^{+}(z_{1})v(z_{2})+v(z_{2})u^{-}(z_{1})$

the normally ordered product of $u(z_{1})$ and $v(z_{2})$, where

$u^{+}(z)=\sum_{n\textless 0}u_{n}z^{-n-1}\quad\textnormal{{and}}\quad
u^{-}(z)=\sum_{n\geqslant 0}u_{n}z^{-n-1}.$

Form the direct sum

$\mathfrak{f}=\mathfrak{g}\oplus\mathfrak{h}$

of the Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$. Let $U$ be an
irreducible $\mathfrak{g}$-module and $\alpha,\beta,\ell\in\mathbb{C}$ with
$\ell\neq 0$. We define an $\widehat{\mathfrak{f}}\rtimes\mathsf{Vir}$-module
structure on the tensor space

$L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$

by letting $\mathrm{k}=\ell$, $\mathrm{k}_{\mathsf{Vir}}=24\mu\ell$ and

(3.9) $\displaystyle u(z)=u(z)\otimes 1,\quad h(z)=1\otimes h(z),\quad
L_{\mathfrak{f}}(z)=L_{\mathfrak{g}}(z)\otimes 1+1\otimes
L_{\mathfrak{h}}(z),$

for $u\in\mathfrak{g}$ and $h\in\mathfrak{h}$.

Let $\mathbb{C}[L]$ act on $V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])$ by

$e^{\gamma}(v\otimes e^{\alpha{\bf k}+\gamma^{\prime}})=v\otimes e^{\alpha{\bf
k}+\gamma^{\prime}+\gamma},\quad\forall\text{$\gamma,\gamma^{\prime}\in L$,
$v\in V_{\widehat{\mathfrak{h}}}(\ell,0)$}.$

For $\gamma\in L$, introduce the following operator in
$\mathrm{End}\left(V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])\right)[[z,z^{-1}]]$:

(3.10) $\displaystyle
E^{\gamma}(z)=\textnormal{{exp}}\left(\frac{1}{\ell}\sum_{m=1}^{\infty}\frac{\gamma(-m)}{m}z^{m}\right)\textnormal{{exp}}\left(-\frac{1}{\ell}\sum_{m=1}^{\infty}\frac{\gamma(m)}{m}z^{-m}\right)e^{\gamma}.$

For convenience, we also set

(3.11) $\displaystyle E^{\gamma}(z)=1\otimes
E^{\gamma}(z)\in\mathrm{End}\left(L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])\right)[[z,z^{-1}]].$

The following theorem is the first main result of this paper. We will prove
this theorem in Section 5.

######  Theorem 3.2.

Let $\ell,\alpha,\beta\in\mathbb{C}$ with $\ell\neq 0$ and let $U$ be an
irreducible $\mathfrak{g}$-module. Then there is an irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module structure on the
$\widehat{\mathfrak{f}}\rtimes\mathsf{Vir}$-module

$L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L]),$

where $\mathrm{k}_{0}=\ell$,
$t_{0}^{\epsilon-1}\mathrm{d}_{0}=-L_{\mathfrak{f}}(\epsilon-1)+\delta_{\epsilon,1}\mu\ell$,
and

(3.12) $\begin{split}&(t_{1}^{m}\otimes
u)^{\epsilon}(z)=u^{\epsilon}(z)E^{m{\bf k}}(z),\
\mathrm{k}_{1}^{\epsilon}(z)={\bf k}^{\epsilon}(z),\
\mathrm{d}_{1}^{\epsilon}(z)={\bf d}^{\epsilon}(z),\
\mathrm{K}^{\epsilon}_{n}(z)=\frac{\ell}{n}E^{n{\bf k}}(z),\\\
&\mathrm{D}^{\epsilon}_{n}(z)=n_{\circ}^{\circ}L^{\epsilon}_{\mathfrak{f}}(z)E^{n{\bf
k}}(z)_{\circ}^{\circ}+\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}E^{n{\bf
k}}(z)-z^{\epsilon}\frac{d}{dz}\ _{\circ}^{\circ}{\bf d}^{\epsilon}(z)E^{n{\bf
k}}(z)_{\circ}^{\circ}\\\
&\qquad\qquad+n^{2}(\mu-\frac{1}{\ell})\left(z^{\epsilon}\frac{d}{dz}{\bf
k}^{\epsilon}(z)\right)E^{n{\bf k}}(z)\end{split}$

for $u\in\mathfrak{g},\ m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\times}$.

######  Remark 3.3.

When $\epsilon=1$, the
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module
$L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$ was first
constructed by Billig in [B2] (see also [CLiT]). And, when $\epsilon=0$, this
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module was previously
constructed in [CLiT].

###  3.3. Realization of irreducible highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules

In this subsection we give a characterization of the irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules constructed in
Theorem 3.2.

Note that
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}=\oplus_{n\in\mathbb{Z}}\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(n)}$
is a $\mathbb{Z}$-graded subalgebra of $\mathcal{T}(\mathfrak{g})$, where
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(n)}=\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}\cap\mathcal{T}(\mathfrak{g})_{(n)}$.
Explicitly, we have

$\widetilde{\mathfrak{t}}(\mathfrak{g})_{(n)}^{\epsilon}=\textnormal{{Span}}\\{t_{0}^{-n}u,\,\mathrm{d}_{-n,m}^{\epsilon},\,\mathrm{k}_{-n,m}\mid
m\in\mathbb{Z},u\in\widetilde{\mathfrak{g}}_{1}\\}\oplus\mathbb{C}\delta_{n,1-\epsilon}t_{0}^{\epsilon-1}\mathrm{d}_{0},$

if $n\neq 0$, and

$\widetilde{\mathfrak{t}}(\mathfrak{g})_{(0)}^{\epsilon}={\tilde{\mathfrak{g}}}_{1}+\sum_{n\in\mathbb{Z}^{\times}}(\mathbb{C}\mathrm{d}_{0,n}^{\epsilon}+\mathbb{C}\mathrm{k}_{0,n})+\mathbb{C}\mathrm{k}_{0}+\mathbb{C}\delta_{\epsilon,1}\mathrm{d}_{0}.$

This $\mathbb{Z}$-grading induces a natural triangular decomposition of
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$:

$\displaystyle\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}=\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(+)}\oplus\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}\oplus\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(-)},$

where

$\widetilde{\mathfrak{t}}(\mathfrak{g})_{(\pm)}^{\epsilon}=\oplus_{n>0}\widetilde{\mathfrak{t}}(\mathfrak{g})_{(\mp
n)}^{\epsilon}.$

As usual, we say that a
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module $W$ is a highest
weight module if there exists a vector $w\in W$ such that
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(+)}w=0$ and
$\mathcal{U}(\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon})w=W$. Let $T$
be an irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-module. Then we have
the Verma type highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module

$V_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T)=\mathcal{U}\left(\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}\right)\otimes_{\mathcal{U}\left(\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(+)}+\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}\right)}T,$

where $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(+)}$ acts trivially
on $T$. The $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module
$V_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T)$ admits an
$\mathbb{N}$-grading structure determined by $\deg T=0$. Denote by
$L_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T)$ the quotient of
$V_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T)$ modulo by its
maximal graded submodule. Note that
$L_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T)$ is an irreducible
highest weight $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module, and
conversely any irreducible highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module is of this form.

Let $U$ be an irreducible $\mathfrak{g}$-module and $\ell,a,b\in\mathbb{C}$.
Associated to these data, we will construct an irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-module structure on
the loop space $\mathbb{C}[t,t^{-1}]\otimes U$. Assume first that
$\epsilon\neq 1$. One can check that in this case

$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}=\mathbb{C}[t_{1},t_{1}^{-1}]\otimes\mathfrak{g}+\mathbb{C}\mathrm{k}_{1}+\sum_{m\in\mathbb{Z}}\left(\mathbb{C}\mathrm{d}_{0,m}^{\epsilon}+\mathbb{C}t_{1}^{m}\mathrm{k}_{0}\right),$

where $\mathrm{k}_{1}$ is central and

$\displaystyle[t_{1}^{m}\otimes u,t_{1}^{n}\otimes
v]=t_{1}^{m+n}\otimes[u,v]+m\delta_{m+n,0}{\langle}u,v{\rangle}\mathrm{k}_{1},\
[t_{1}^{m}\mathrm{k}_{0},t_{1}^{n}\mathrm{k}_{0}]=0,\
[t_{1}^{m}\mathrm{k}_{0},t_{1}^{n}\otimes u]=0,$
$\displaystyle[\mathrm{d}^{\epsilon}_{0,m},t_{1}^{n}\mathrm{k}_{0}]=(1-\epsilon)nt_{1}^{m+n}\mathrm{k}_{0}-m^{2}\delta_{m+n,0}\mathrm{k}_{1},\
[\mathrm{d}^{\epsilon}_{0,m},t_{1}^{n}\otimes
u]=(1-\epsilon)nt_{1}^{m+n}\otimes u,$
$\displaystyle[\mathrm{d}^{\epsilon}_{0,m},\mathrm{d}^{\epsilon}_{0,n}]=(1-\epsilon)(n-m)\mathrm{d}^{\epsilon}_{0,m+n}+2\mu
m^{3}(\epsilon-1)^{2}\delta_{m+n,0}\mathrm{k}_{1}$

for $m,n,k,l\in\mathbb{Z}$ and $u,v\in\mathfrak{g}$. This implies that the
assignment ($m\in\mathbb{Z},u\in\mathfrak{g}$)

(3.13) $\displaystyle\mathrm{k}_{1}\mapsto
0,\quad\mathrm{d}^{\epsilon}_{0,m}\mapsto(1-\epsilon)\,t^{m+1}\frac{d}{dt},\quad
t_{1}^{m}\otimes u\mapsto t^{m}\otimes u,\quad t_{1}^{m}\mathrm{k}_{0}\mapsto
t^{m}\otimes\mathrm{k}_{0}$

determines a surjective homomorphism from
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$ to the centerless
affine-Virasoro algebra

$\mathcal{W}_{\mathfrak{g}}:=(\mathrm{Der}\,\mathbb{C}[t,t^{-1}])\ltimes\left(\mathbb{C}[t,t^{-1}]\otimes(\mathfrak{g}+\mathbb{C}\mathrm{k}_{0})\right)$

associated to $\mathfrak{g}\oplus\mathbb{C}\mathrm{k}_{0}$. Extend $U$ to a
$(\mathfrak{g}\oplus\mathbb{C}\mathrm{k}_{0})$-module such that
$\mathrm{k}_{0}$ acts as a scalar $\ell$. Following [B3], there is an
irreducible $\mathcal{W}_{\mathfrak{g}}$-module structure on the loop space
$\mathbb{C}[t,t^{-1}]\otimes U$ with

$(t^{m}\otimes v)(t^{n}\otimes
u)=t^{m+n}\otimes(vu),\quad\left(t^{m+1}\frac{d}{dt}\right)(t^{n}\otimes
u)=(n+a+bm)t^{m+n}\otimes u$

for $x\in\mathfrak{g}\oplus\mathbb{C}\mathrm{k}_{0}$, $m,n\in\mathbb{Z}$ and
$u\in U$. Via the homomorphism (3.13), $\mathbb{C}[t,t^{-1}]\otimes U$ becomes
a $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-module, which we
denote by $T_{U,\ell,a,b}$.

When $\epsilon=1$ we have

$\widetilde{\mathfrak{t}}(\mathfrak{g})^{1}_{(0)}=\mathbb{C}[t_{1},t_{1}^{-1}]\otimes\mathfrak{g}+\sum_{m\in\mathbb{Z}}\left(\mathbb{C}t_{1}^{m}\mathrm{d}_{0}+\mathbb{C}t_{1}^{m}\mathrm{k}_{0}\right)+\mathbb{C}\mathrm{k}_{1}+\mathbb{C}\mathrm{d}_{1},$

which is isomorphic to the affine Kac-Moody algebra $\widetilde{\mathfrak{f}}$
associated to $\mathfrak{f}$ with

$t_{1}^{m}\otimes u\mapsto u(m),\ t_{1}^{m}\mathrm{k}_{0}\mapsto{\bf k}(m),\
t_{1}^{m}\mathrm{d}_{0}\mapsto{\bf d}(m),\ \mathrm{k}_{1}\mapsto\mathrm{k},\
\mathrm{d}_{1}\mapsto t\frac{d}{dt}$

for $m\in\mathbb{Z},u\in\mathfrak{g}$. Extend $U$ to an $\mathfrak{f}$-module
such that ${\bf d}$ and ${\bf k}$ act as the scalars $b$ and $\ell$,
respectively. Following [CP], we have the loop module
$\mathbb{C}[t,t^{-1}]\otimes U$ for the affine Lie algebra
$\widetilde{\mathfrak{f}}$ with the actions given by

$\displaystyle x(m)(t^{n}\otimes u)=t^{m+n}\otimes(xu),\quad
t\frac{d}{dt}(t^{n}\otimes u)=(n+a)t^{n}\otimes u,\quad\mathrm{k}=0$

for $x\in\mathfrak{f}$, $m,n\in\mathbb{Z}$ and $u\in U$. Since
$\widetilde{\mathfrak{t}}(\mathfrak{g})_{(0)}^{1}$ is isomorphism to
$\widetilde{\mathfrak{f}}$, the loop $\widetilde{\mathfrak{f}}$-module
$\mathbb{C}[t,t^{-1}]\otimes U$ becomes a
$\widetilde{\mathfrak{t}}(\mathfrak{g})_{(0)}^{1}$-module, which is also
denoted by $T_{U,\ell,a,b}$.

Now, as the second main result of this paper, we give a characterization of
the irreducible $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules
constructed in Theorem 3.2.

######  Theorem 3.4.

Let $\ell,\alpha,\beta\in\mathbb{C}$ with $\ell\neq 0$ and let $U$ be an
irreducible $\mathfrak{g}$-module. Then the
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module
$L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$ is isomorphic
to the irreducible highest weight module
$L_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T_{\ell,U,\alpha,b})$,
where
$b=\frac{\beta+(\epsilon^{2}-2\epsilon)\mu\ell}{1-\epsilon}+\frac{\epsilon}{2}$
if $\epsilon\neq 1$ and $b=\beta-\mu\ell$ if $\epsilon=1$.

###### Proof.

Write

$W=L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L]).$

It is clear that $W=\oplus_{k\in\mathbb{Z}}W_{(k)}$ is a $\mathbb{Z}$-graded
$\widehat{\mathfrak{f}}\rtimes\mathsf{Vir}$-module such that $W_{(0)}=U\otimes
e^{\alpha{\bf k}}\mathbb{C}[L]$ and $W_{(k)}=0$ for $k<0$. For
$n\in\mathbb{Z}$, set

$E^{n{\bf k}}(z)=\sum_{m\in\mathbb{Z}}E^{n{\bf k}}(m)z^{-m}.$

Then we have $E^{n{\bf k}}(m)W_{(k)}\subset W_{(m+k)}$ for
$n,m,k\in\mathbb{Z}$. This together with (LABEL:eq:mainaction) implies that
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(m)}W_{(k)}=W_{(m+k)}$ for
$m,k\in\mathbb{Z}$. That is, $W$ is an irreducible $\mathbb{N}$-graded
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module. Thus, $W_{(0)}$ is
an irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-module and we need
to prove that

$W_{(0)}(=U\otimes e^{\alpha{\bf k}}\mathbb{C}[L])\quad\text{is isomorphic
to}\quad T_{\ell,U,\alpha,b}(=\mathbb{C}[t,t^{-1}]\otimes U)$

as $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-modules, where
$\ell,U,\alpha,b$ are as given in the theorem.

It is straightforward to check that in this case the
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-module actions on
$T_{\ell,U,\alpha,b}$ can be written in the following expressions:

(3.14) $\begin{split}&(t_{1}^{m}\otimes a)(t^{n}\otimes u)=t^{m+n}\otimes
au,\quad(t_{1}^{m}\mathrm{k}_{0})(t^{n}\otimes u)=\ell t^{m+n}\otimes u,\\\
&\mathrm{d}_{1}(t^{n}\otimes u)=(n+\alpha)t^{n}\otimes
u,\quad\mathrm{k}_{1}(t^{n}\otimes u)=0,\\\
&\mathrm{d}^{\epsilon}_{0,k}(t^{n}\otimes
u)=\left(k(\beta+(\epsilon^{2}-2\epsilon)\mu\ell)+(1-\epsilon)(\alpha+n+\frac{1}{2}k\epsilon)\right)t^{k+n}\otimes
u,\\\ &\mathrm{d}_{0}(t^{n}\otimes u)=(\mu\ell-\beta)(t^{n}\otimes
u)\quad(\textnormal{{when }}\epsilon=1),\end{split}$

for $m,n\in\mathbb{Z},\ k\in\mathbb{Z}^{\times},\ a\in\mathfrak{g}$ and $u\in
U$.

Now we are ready to show that
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-module actions on
$W_{(0)}$ coincide with that of $T_{\ell,U,\alpha,b}$ given in
(LABEL:tg0action). For $u\in U$ and $r\in\mathbb{Z}$, set

$u(r):=u\otimes e^{(\alpha+r){\bf k}}.$

Then we have

(3.15) $\displaystyle\mathrm{k}_{1}(u(r))$ $\displaystyle={\bf
k}(0)(u(r))=u\otimes{\bf k}(0)e^{(\alpha+r){\bf k}}={\langle}{\bf
k},(\alpha+r){\bf k}{\rangle}u\otimes e^{(\alpha+r){\bf k}}=0,$ (3.16)
$\displaystyle\mathrm{d}_{1}(u(r))$ $\displaystyle={\bf
d}(0)(u(r))={\langle}{\bf d},(\alpha+r){\bf k}{\rangle}u(r)=(\alpha+r)u(r).$

When $\epsilon=1$, we also have

(3.17) $\displaystyle\mathrm{d}_{0}(u(r))=$
$\displaystyle(-L(0)-\sum_{i\textless 1}{\bf k}_{i}{\bf d}_{-i}-\sum_{i\geq
1}{\bf d}_{-i}{\bf k}_{i}+\mu\ell)(u(r))$ $\displaystyle=$
$\displaystyle(-L(0)+\mu\ell)(u(r))=(-\beta+\mu\ell)u(r).$

On the other hand, for $m\in\mathbb{Z}$, we have

$\displaystyle E^{m{\bf k}}(0)u(r)=u\otimes e^{m{\bf k}}e^{(\alpha+r){\bf
k}}=u(m+r).$

In view of this, we have

(3.18) $\displaystyle(t_{1}^{n}\mathrm{k}_{0})(u(r))=$ $\displaystyle
n\mathrm{k}_{0,n}(u(r))=\ell E^{n{\bf k}}(0)(u(r))=\ell(u(n+r)),$ (3.19)
$\displaystyle(t_{1}^{m}\otimes a)(u(r))=$
$\displaystyle\sum_{i\in\mathbb{Z}}a(i)u\otimes E^{m{\bf
k}}(-i)E^{(\alpha+r){\bf k}}$ $\displaystyle=\,$ $\displaystyle a(0)u\otimes
E^{m{\bf k}}(0)e^{(\alpha+r){\bf k}}=(au)(m+r),$

for $n\in\mathbb{Z}^{\times}$, $a\in\mathfrak{g}$ and $m\in\mathbb{Z}$.

For the action of $\mathrm{d}_{0,n}^{\epsilon}$ $(n\in\mathbb{Z}^{\times})$ on
$u(r)$, note firstly that

$\displaystyle[L_{\mathfrak{h}}(i),E^{n{\bf k}}(-i)]e^{(\alpha+r){\bf
k}}=\left(\frac{n}{\ell}\sum_{j\in\mathbb{Z}}{\bf k}(j)E^{n{\bf
k}}(-j)\right)e^{(\alpha+r){\bf k}}=0\quad(i\in\mathbb{Z}),$

as ${\bf k}(i)$ commutes with $E^{n{\bf k}}(j)$ for $i,j\in\mathbb{Z}$, and
that $L_{\mathfrak{h}}(i)e^{(\alpha+r){\bf k}}=0$ for $i\geq 0.$ Using these
and (3.3) we have

$\displaystyle\textnormal{{Res}}_{z}z^{1-2\epsilon}\
_{\circ}{}^{\circ}L^{\epsilon}_{\mathfrak{f}}(z)E^{n{\bf
k}}(z)_{\circ}^{\circ}u(r)$ $\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z}\left(\sum_{i\leq
2\epsilon-2}L_{\mathfrak{f}}(i)z^{-i-1}E^{n{\bf k}}(z)+E^{n{\bf
k}}(z)\sum_{i>2\epsilon-2}L_{\mathfrak{f}}(i)z^{-i-1}+\frac{\epsilon^{2}-2\epsilon}{24}z^{-1}\mathrm{k}_{\mathsf{Vir}}E^{n{\bf
k}}(z)\right)u(r)$ $\displaystyle=\ $ $\displaystyle\left(\sum_{i\leq
2\epsilon-2}L_{\mathfrak{f}}(i)E^{n{\bf k}}(-i)+\sum_{i>2\epsilon-2}E^{n{\bf
k}}(-i)L_{\mathfrak{f}}(i)+(\epsilon^{2}-2\epsilon)\mu\ell(E^{n{\bf
k}})(0)\right)u(r)$ $\displaystyle=\ $
$\displaystyle\sum_{i\in\mathbb{Z}}\left(L_{\mathfrak{g}}(i)u\otimes E^{n{\bf
k}}(-i)e^{(\alpha+r){\bf k}}+u\otimes L_{\mathfrak{h}}(i)E^{n{\bf
k}}(-i)e^{(\alpha+r){\bf k}}\right)+(\epsilon^{2}-2\epsilon)\mu\ell(E^{n{\bf
k}})(0)u(r)$ $\displaystyle=\ $ $\displaystyle L_{\mathfrak{g}}(0)u\otimes
E^{n{\bf k}}(0)e^{(\alpha+r){\bf k}}+(\epsilon^{2}-2\epsilon)\mu\ell\,u(n+r)$
$\displaystyle=\ $
$\displaystyle(\beta+(\epsilon^{2}-2\epsilon)\mu\ell)\,u(n+r).$

Secondly, we have

$\displaystyle\textnormal{{Res}}_{z}z^{1-2\epsilon}\left(z^{2\epsilon-2}E^{n{\bf
k}}(z)\right)u(r)=(E^{n{\bf k}})(0)u(r)=u(n+r).$

Thirdly, from the fact that ${\bf k}(i)$ commutes with $E^{n{\bf k}}(j)$ for
$i,j\in\mathbb{Z}$, we have

$\displaystyle\textnormal{{Res}}_{z}z^{1-2\epsilon}\left(z^{\epsilon}\frac{d}{dz}{\bf
k}^{\epsilon}(z)\right)E^{n{\bf k}}(z)u(r)$ $\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z}\sum_{i\in\mathbb{Z}}(\epsilon-i-1){\bf
k}(i)z^{-i-1}E^{n{\bf k}}(z)u(r)$ $\displaystyle=\ $
$\displaystyle\sum_{i\in\mathbb{Z}}(\epsilon-i-1){\bf k}(i)E^{n{\bf
k}}(-i)(u\otimes e^{(\alpha+r){\bf k}})=0.$

Fourthly, using the relation $[{\bf d}(i),E^{n{\bf k}}(-i)]=nE^{n{\bf k}}(0)$
for $i\in\mathbb{Z}$, we have

$\displaystyle\textnormal{{Res}}_{z}z^{1-2\epsilon}\left(z^{\epsilon}\frac{d}{dz}\
_{\circ}^{\circ}{\bf d}^{\epsilon}(z)E^{n{\bf
k}}(z)_{\circ}^{\circ}\right)u(r)$ $\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z}z^{1-\epsilon}\frac{d}{dz}\
_{\circ}^{\circ}{\bf d}^{\epsilon}(z)E^{n{\bf k}}(z)_{\circ}^{\circ}\ u(r)$
$\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z}z^{-\epsilon}(\epsilon-1)_{\circ}^{\circ}{\bf
d}^{\epsilon}(z)E^{n{\bf k}}(z)_{\circ}^{\circ}\ u(r)$ $\displaystyle=\ $
$\displaystyle(\epsilon-1)\textnormal{{Res}}_{z}\left(\sum_{i\leq\epsilon-1}{\bf
d}(i)z^{-i-1}E^{n{\bf k}}(z)+\sum_{i\textgreater\epsilon-1}E^{n{\bf k}}(z){\bf
d}(i)z^{-i-1}\right)u(r)$ $\displaystyle=\ $
$\displaystyle(\epsilon-1)\left(\sum_{i\leq\epsilon-1}{\bf d}(i)E^{n{\bf
k}}(-i)+\sum_{i\textgreater\epsilon-1}E^{n{\bf k}}(-i){\bf d}(i)\right)u(r)$
$\displaystyle=\ $ $\displaystyle\begin{cases}(\epsilon-1)\sum_{0\leq
i\leq\epsilon-1}{\bf d}(i)E^{n{\bf k}}(-i)u(r),&\textnormal{{if }}\
\epsilon\geq 1\\\ (\epsilon-1)\sum_{\epsilon-1\textless i\leq 0}E^{n{\bf
k}}(-i){\bf d}(i)u(r),&\textnormal{{if }}\ \epsilon<1\end{cases}$
$\displaystyle=\ $ $\displaystyle(\epsilon-1)(\alpha+r+n\epsilon)\,u(n+r).$

Recall that

$\displaystyle\mathrm{d}_{0,n}^{\epsilon}$
$\displaystyle=\textnormal{{Res}}_{z}z^{1-2\epsilon}\mathrm{D}_{n}^{\epsilon}(z)$
$\displaystyle=\textnormal{{Res}}_{z}z^{1-2\epsilon}\Big{(}n\
_{\circ}^{\circ}L_{\mathfrak{f}}^{\epsilon}(z)E^{n{\bf k}}(z)\
_{\circ}^{\circ}+\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}E^{n{\bf
k}}(z)$ $\displaystyle\ \ \ -z^{\epsilon}\frac{d}{dz}\ _{\circ}^{\circ}{\bf
d}^{\epsilon}(z)E^{n{\bf
k}}(z)_{\circ}^{\circ}+n^{2}(\mu-\frac{1}{\ell})z^{\epsilon}\frac{d}{dz}{\bf
k}^{\epsilon}(z)E^{n{\bf k}}(z)\Big{)}.$

Then by combining the above identities together, we obtain

(3.20)
$\displaystyle\mathrm{d}_{0,n}^{\epsilon}(u(r))=(n(\beta+(\epsilon^{2}-2\epsilon)\mu\ell)+(1-\epsilon)(\alpha+r+\frac{1}{2}n\epsilon))(u(n+r)).$

Finally, by comparing (3.15)-(3.20) with (LABEL:tg0action), we conclude that
the map

$T_{\ell,U,\alpha,b}=\mathbb{C}[t,t^{-1}]\otimes U\rightarrow W_{(0)}=U\otimes
e^{\alpha{\bf k}}\mathbb{C}[L],\quad t^{r}\otimes u\mapsto
u(r)\quad(r\in\mathbb{Z},u\in U)$

is a $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}_{(0)}$-module
isomorphism. This completes the proof. ∎

######  Remark 3.5.

Assume that $\mathfrak{g}$ is finite-dimensional and simple. Note that
$\mathrm{d}_{1}$ acts semisimply on any irreducible highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module
$L_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T)$. We say that
$L_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T)$ is bounded if every
graded subspace of it is a direct sum of finite dimensional
$\mathrm{d}_{1}$-eigenvalue spaces. For example, if $U$ is finite-dimensional,
then from Theorem 3.4 we find that every
$L_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T_{\ell,U,\alpha,b})$
is bounded. When $\epsilon=0$, it was proved in [CLiT] that every irreducible
bounded highest weight $\widetilde{\mathfrak{t}}(\mathfrak{g})^{0}$-module has
such a form. However, when $\epsilon=1$, a large class of different
irreducible bounded highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{1}$-modules was constructed in
[CLT1]. Thus, when $\epsilon\neq 0,1$, it is interesting to classify the
irreducible bounded highest weight
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules, which we believe
should have the form
$L_{\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}}(T_{\ell,U,\alpha,b})$
with $U$ finite-dimensional.

##  4\. Basics on $\phi_{\epsilon}$-coordinated modules for vertex algebras

In this section, we collect some results on $\phi_{\epsilon}$-coordinated
modules for later use. We denote a vertex algebra by $V=(V,Y,\mathbf{1})$
[FHL, LL], where ${\mathbf{1}}$ is the vacuum vector, and

$Y(\cdot,z):V\rightarrow\mathrm{Hom}(V,V((z))),\
v\mapsto\sum_{n\in\mathbb{Z}}v_{n}z^{-n-1}$

is the vertex operator.

###  4.1. Basics on $\phi_{\epsilon}$-coordinated modules

As in Introduction, set

$\phi_{\epsilon}(z_{2},z_{0})=e^{z_{0}\left(z_{2}^{\epsilon}\frac{d}{dz_{2}}\right)}z_{2},$

a particular associate of the one-dimensional additive formal group
$F\left(z,w\right)=z+w$ as defined in [Li]. Now we recall the notion of
$\phi_{\epsilon}$-coordinated module for a vertex algebra (see [Li, BLP]).

######  Definition 4.1.

Let $V$ be a vertex algebra. A $\phi_{\epsilon}$-coordinated $V$-module
$\left(W,Y_{W}^{\epsilon}\right)$ is a vector space $W$ equipped with a linear
map

$Y_{W}^{\epsilon}(\cdot,z):V\rightarrow\mathrm{Hom}(W,W((z))),\quad v\mapsto
Y_{W}^{\epsilon}(v,z)$

satisfying the conditions that $Y_{W}^{\epsilon}({\bf 1},z)=1_{W}$ and that
for $u,v\in V$, there exists $k\in\mathbb{N}$ such that

(4.1)
$\displaystyle(z_{1}-z_{2})^{k}Y_{W}^{\epsilon}(u,z_{1})Y_{W}^{\epsilon}(v,z_{2})$
$\displaystyle\in\mathrm{Hom}(W,W((z_{1},z_{2}))),$ (4.2)
$\displaystyle(\phi_{\epsilon}(z_{2},z_{0})-z_{2})^{k}Y_{W}^{\epsilon}(Y(u,z_{0})v,z_{2})$
$\displaystyle=\left((z_{1}-z_{2})^{k}Y_{W}^{\epsilon}(u,z_{1})Y_{W}^{\epsilon}(v,z_{2})\right)|_{z_{1}=\phi_{\epsilon}(z_{2},z_{0})}.$

We denote by $\mathcal{D}$ the canonical derivation on $V$ defined by
$v\mapsto v_{-2}\mathbf{1}$ for $v\in V$. The following result was proved in
[Li].

######  Lemma 4.2.

For a $\phi_{\epsilon}$-coordinated $V$-module
$\left(W,Y_{W}^{\epsilon}\right)$, we have

$\displaystyle
Y_{W}^{\epsilon}\left(\mathcal{D}v,z\right)=z\frac{d}{dz}Y_{W}^{\epsilon}\left(v,z\right),\quad\forall
v\in V.$

The following Borcherds formula and Jacobi-type identity were obtained in
[BLP]:

######  Proposition 4.3.

Let $(W,Y_{W}^{\epsilon})$ be a $\phi_{\epsilon}$-coordinated $V$-module. Then
for $u,v\in V$, we have

(4.3) $\displaystyle[Y_{W}^{\epsilon}(u,z),Y_{W}^{\epsilon}(v,w)]=\sum_{j\geq
0}\frac{1}{j!}Y_{W}^{\epsilon}(u_{j}v,w)\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{j}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$

and

(4.4)
$\begin{split}&(z_{2}z)^{-1}\delta\left(\frac{z_{1}-z_{2}}{z_{2}z}\right)Y_{W}^{\epsilon}(u,z_{1})Y_{W}^{\epsilon}(v,z_{2})-(z_{2}z)^{-1}\delta\left(\frac{z_{2}-z_{1}}{-z_{2}z}\right)Y_{W}^{\epsilon}(v,z_{2})Y_{W}^{\epsilon}(u,z_{1})\\\
=\
&z_{1}^{-1}\delta\left(\frac{z_{2}(1+z)}{z_{1}}\right)Y_{W}^{\epsilon}(Y(u,f_{\epsilon}(z_{2},z))v,z_{2}),\end{split}$

where

$f_{\epsilon}(z_{2},z)=\begin{cases}z_{2}^{1-\epsilon}\cdot\frac{(1+z)^{1-\epsilon}-1}{1-\epsilon},\
&\text{for }\ \epsilon\neq 1\\\ \textnormal{{log}}(1+z),\ &\text{for }\
\epsilon=1.\end{cases}$

As in the module case, we have the following result whose proof is
straightforward and is omitted (cf. [FHL]).

######  Lemma 4.4.

Let $V,V^{\prime}$ be two vertex algebras, $(W,Y_{W}^{\epsilon})$ a
$\phi_{\epsilon}$-coordinated $V$-module, and
$(W^{\prime},Y_{W^{\prime}}^{\epsilon})$ a $\phi_{\epsilon}$-coordinated
$V^{\prime}$-module. Then $(W\otimes W^{\prime},Y_{W}^{\epsilon}\otimes
Y_{W^{\prime}}^{\epsilon})$ is a $\phi_{\epsilon}$-coordinated module the
tensor product vertex algebra $V\otimes V^{\prime}$.

In what follows, for $u,v\in V$, we consider the expressions of
$Y_{W}^{\epsilon}(u_{0}v,z)$ and $Y_{W}^{\epsilon}(u_{-1}v,z)$ in a
$\phi_{\epsilon}$-coordinated $V$-module $(W,Y_{W}^{\epsilon})$.

######  Lemma 4.5.

Let $(W,Y_{W}^{\epsilon})$ be a $\phi_{\epsilon}$-coordinated $V$-module. Then
for $u,v\in V$,

(4.5) $\displaystyle
Y_{W}^{\epsilon}(u_{0}v,w)=[\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(u,z),Y_{W}^{\epsilon}(v,w)].$

###### Proof.

By taking $\textnormal{{Res}}_{z}z^{-\epsilon}$ in the both sides of (4.3), we
have

$\displaystyle[\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(u,z),Y_{W}^{\epsilon}(v,w)]=\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(u_{0}v,w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)=Y_{W}^{\epsilon}(u_{0}v,w),$

noting that for $j>0$

$\textnormal{{Res}}_{z}z^{-\epsilon}\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{j}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)=\textnormal{{Res}}_{z}z^{-\epsilon}\left(-z^{\epsilon}\frac{\partial}{\partial
z}\right)^{j}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)=0.$

∎

Set $h_{0}=1$ and

$h_{n}=\sum_{\begin{subarray}{c}r_{1},r_{2},\dots,r_{n}\geq 0,\\\
r_{1}+2r_{2}+\cdots+nr_{n}=n\end{subarray}}(-1)^{\sum\limits_{m=1}^{n}r_{m}}\left(\sum_{m=1}^{n}r_{m}\right)!\prod_{m=1}^{n}\frac{1}{r_{m}!}\left(\frac{\epsilon^{(m)}}{(m+1)!}\right)^{r_{m}},\
\forall n\geq 1,$

where $\epsilon^{(m)}=\prod_{s=0}^{m-1}(\epsilon+s(\epsilon-1))$. We have

######  Proposition 4.6.

Let $(W,Y_{W}^{\epsilon})$ be a $\phi_{\epsilon}$-coordinated $V$-module. Then
for $u,v\in V$,

(4.6) $\displaystyle
Y_{W}^{\epsilon}(u_{-1}v,z)=_{\circ}^{\circ}Y_{W}^{\epsilon}(u,z)Y_{W}^{\epsilon}(v,z)_{\circ}^{\circ}-\sum_{n\geq
0}\sum_{i=0}^{n+1}\frac{1}{i!}\epsilon^{(i)}h_{n-i+1}z^{(n+1)(\epsilon-1)}Y_{W}^{\epsilon}(u_{n}v,z).$

###### Proof.

Note that we have

$\displaystyle\phi_{\epsilon}(z_{2},z_{0})-z_{2}=\sum_{n\geq
1}\frac{1}{n!}z_{0}^{n}\left(z_{2}^{\epsilon}\frac{d}{dz_{2}}\right)^{n}z_{2}=z_{0}z_{2}^{\epsilon}h(z_{2},z_{0}),$

where

$h(z_{2},z_{0})=\sum_{n\geq
0}\frac{1}{(n+1)!}\epsilon^{(n)}z_{2}^{n(\epsilon-1)}z_{0}^{n}\in\mathbb{C}((z_{2}))[[z_{0}]].$

Since $h(z_{2},0)=1\neq 0$, the inverse $h(z_{2},z_{0})^{-1}$ of
$h(z_{2},z_{0})$ exists in $\mathbb{C}((z_{2}))[[z_{0}]]$. Explicitly, we have

$\displaystyle h(z_{2},z_{0})^{-1}=\left(1+\sum_{i\geq
1}\frac{\epsilon^{(i)}}{(i+1)!}z_{2}^{i(\epsilon-1)}z_{0}^{i}\right)^{-1}=\sum_{j\geq
0}\left(-\frac{\epsilon^{(i)}}{(i+1)!}z_{2}^{i(\epsilon-1)}z_{0}^{i}\right)^{j}=\sum_{n\geq
0}h_{n}z_{2}^{n(\epsilon-1)}z_{0}^{n}.$

Next we compute the residue
$\textnormal{{Res}}_{z}z^{-1}Y_{W}^{\epsilon}(Y(u,f_{\epsilon}(z_{2},z))v,z_{2})$.
By substituting $z$ by $\frac{\phi_{\epsilon}(z_{2},z_{0})}{z_{2}}-1$ and
using the facts that (see [BLP])

$f_{\epsilon}\left(z_{2},\frac{\phi_{\epsilon}(z_{2},z_{0})}{z_{2}}-1\right)=z_{0}\quad\textnormal{{and}}\quad\frac{\partial}{\partial
z_{0}}\phi_{\epsilon}(z_{2},z_{0})=\phi_{\epsilon}(z_{2},z_{0})^{\epsilon},$

we have

$\displaystyle\textnormal{{Res}}_{z}z^{-1}Y_{W}^{\epsilon}(Y(u,f_{\epsilon}(z_{2},z))v,z_{2})$
$\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z_{0}}\left(\frac{\phi_{\epsilon}(z_{2},z_{0})}{z_{2}}-1\right)^{-1}\left(\frac{\partial}{\partial
z_{0}}\left(\frac{\phi_{\epsilon}(z_{2},z_{0})}{z_{2}}-1\right)\right)Y_{W}^{\epsilon}(Y(u,z_{0})v,z_{2})$
$\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z_{0}}\left(\phi_{\epsilon}(z_{2},z_{0})-z_{2}\right)^{-1}\phi_{\epsilon}(z_{2},z_{0})^{\epsilon}Y_{W}^{\epsilon}(Y(u,z_{0})v,z_{2})$
$\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z_{0}}z_{0}^{-1}z_{2}^{-\epsilon}h(z_{2},z_{0})^{-1}e^{z_{0}\left(z_{2}^{\epsilon}\frac{\partial}{\partial
z_{2}}\right)}\left(z_{2}^{\epsilon}\right)Y_{W}^{\epsilon}(Y(u,z_{0})v,z_{2})$
$\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z_{0}}\sum_{n\geqslant-1}z_{0}^{-n-1}z_{0}^{-1}z_{2}^{-\epsilon}h(z_{2},z_{0})^{-1}e^{z_{0}\left(z_{2}^{\epsilon}\frac{\partial}{\partial
z_{2}}\right)}\left(z_{2}^{\epsilon}\right)Y_{W}^{\epsilon}(u_{n}v,z_{2})$
$\displaystyle=\ $ $\displaystyle
Y_{W}^{\epsilon}(u_{-1}v,z_{2})+\sum_{n\geqslant
0}\sum_{i=0}^{n+1}\frac{1}{i!}\epsilon^{(i)}h_{n-i+1}z_{2}^{(n+1)(\epsilon-1)}Y_{W}^{\epsilon}(u_{n}v,z_{2}).$

On the other hand, by taking
$\textnormal{{Res}}_{z}\textnormal{{Res}}_{z_{1}}z^{-1}$ in the Jacobi-type
identity (LABEL:eq:jacobi), we find

$\displaystyle\text{Res}_{z}z^{-1}Y_{W}^{\epsilon}(Y\left(u,f_{\epsilon}(z_{2},z)\right)v,z_{2})$
$\displaystyle=\ $
$\displaystyle\textnormal{{Res}}_{z_{1}}\big{(}(z_{1}-z_{2})^{-1}Y_{W}^{\epsilon}(u,z_{1})Y_{W}^{\epsilon}(v,z_{2})-\left(z_{2}-z_{1}\right)^{-1}Y_{W}^{\epsilon}(v,z_{2})Y_{W}^{\epsilon}(u,z_{1})\big{)}$
$\displaystyle=\ $
${}_{\circ}^{\circ}Y_{W}^{\epsilon}(u,z_{2})Y_{W}^{\epsilon}(v,z_{2})_{\circ}^{\circ}.$

This completes the proof of the proposition. ∎

######  Remark 4.7.

_Here we give the first four terms in the expression ( 4.6) for later use:_

$\displaystyle Y_{W}^{\epsilon}(u_{-1}v,z)=$
${}_{\circ}^{\circ}Y_{W}^{\epsilon}(u,z)Y_{W}^{\epsilon}(v,z)_{\circ}^{\circ}-\frac{1}{2}\epsilon
z^{\epsilon-1}Y_{W}^{\epsilon}(u_{0}v,z)-\frac{5\epsilon-4}{12}\epsilon
z^{2(\epsilon-1)}Y_{W}^{\epsilon}(u_{1}v,z)$
$\displaystyle-\frac{3\epsilon^{2}-5\epsilon+2}{8}\epsilon
z^{3(\epsilon-1)}Y_{W}^{\epsilon}(u_{2}v,z)+\cdots.$

###  4.2. On $\epsilon$-deformation of vertex Lie algebras

We start with a notion of vertex Lie algebra introduced in [DLM] (see also
[K2, P]). A vertex Lie algebra is a quadruple
$(\mathcal{L},\mathcal{A},\mathcal{Z},\eta)$, where $\mathcal{L}$ is a Lie
algebra, $\mathcal{A}$ and $\mathcal{Z}$ are two vector spaces, and

(4.7) $\displaystyle\eta:\
\left(\mathbb{C}[t,t^{-1}]\otimes\mathcal{A}\right)\oplus{\mathcal{Z}}\rightarrow\mathcal{L},\quad
t^{n}\otimes a+c\mapsto
a(n)+c\quad(n\in\mathbb{Z},a\in\mathcal{A},c\in\mathcal{Z})$

is a linear bijection, satisfying the condition that $\eta(\mathcal{Z})$ is
central in $\mathcal{L}$ and for $a,b\in{\mathcal{A}}$, there exist finitely
many elements $a_{(i,j)}b\in\mathcal{A}$ and $a_{(j)}b\in\mathcal{Z}$ for
$i,j\in\mathbb{N}$ such that

(4.8)
$\displaystyle[a(z),b(w)]=\sum_{i,j\in\mathbb{N}}\left(\left(\frac{\partial}{\partial
w}\right)^{i}(a_{(i,j)}b)(w)+(a_{(j)}b)\right)\frac{1}{j!}\left(\frac{\partial}{\partial
w}\right)^{j}z^{-1}\delta\left(\frac{w}{z}\right),$

where $a(z)=\sum_{n\in\mathbb{Z}}a(n)z^{-n-1}$ for $a\in\mathcal{A}$.

Let $(\mathcal{L},\mathcal{A},\mathcal{Z},\eta)$ be a vertex Lie algebra. Note
that $\mathcal{L}^{+}=\eta(\mathbb{C}[t]\otimes\mathcal{A}\oplus\mathcal{Z})$
is a subalgebra of $\mathcal{L}$. For any $\gamma\in\mathcal{Z}^{*}$, denote
by $\mathbb{C}_{\gamma}$ the one dimensional $\mathcal{L}^{+}$-module such
that $\eta(\mathbb{C}[t]\otimes\mathcal{A})$ acts trivially and
$c\in\mathcal{Z}$ act as the scalars $\gamma(c)$. Form the induced
$\mathcal{L}$-module

$V_{\mathcal{L}}(\gamma)=\mathcal{U}(\mathcal{L})\otimes_{\mathcal{U}(\mathcal{L})}\mathbb{C}_{\gamma}.$

Set ${\bm{1}}=1\otimes 1\in V_{\mathcal{L}}(\gamma)$ and identify
$\mathcal{A}$ as a subspace of $V_{\mathcal{L}}(\gamma)$ with
$a=a(-1){\bm{1}}$ for $a\in\mathcal{A}$. It was proved in [DLM] that there is
a unique vertex algebra structure on $V_{\mathcal{L}}(\gamma)$ with ${\bm{1}}$
as the vacuum vector and with $Y(a,z)=a(z)$ for $a\in\mathcal{A}$.

In what follows we define an $\epsilon$-deformation Lie algebra
$\mathcal{L}_{\epsilon}$ of $\mathcal{L}$. As a vector space,
$\mathcal{L}_{\epsilon}$ is a linear copy of $\mathcal{L}$ equipped with a
linear bijection

(4.9) $\displaystyle\eta_{\epsilon}:\
\left(\mathbb{C}[t,t^{-1}]\otimes\mathcal{A}\right)\oplus{\mathcal{Z}}\rightarrow\mathcal{L}_{\epsilon},\quad
t^{n}\otimes a+c\mapsto
a^{(\epsilon)}(n)+c^{(\epsilon)}\quad(n\in\mathbb{Z},a\in\mathcal{A},c\in\mathcal{Z}).$

We define a multiplication on $\mathcal{L}_{\epsilon}$ such that
$\eta_{\epsilon}(\mathcal{Z})$ is central in $\mathcal{L}_{\epsilon}$ and

(4.10) $\displaystyle[a^{(\epsilon)}(z),b^{(\epsilon)}(w)]$ $\displaystyle=\,$
$\displaystyle\sum_{i,j\in\mathbb{N}}\left(\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{i}(a_{(i,j)}b)^{(\epsilon)}(w)+(a_{(j)}b)^{(\epsilon)}\right)\frac{1}{j!}\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{j}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),$

where
$a^{(\epsilon)}(z)=\sum_{n\in\mathbb{Z}}a^{(\epsilon)}(n)z^{\epsilon-n-1}$ for
$a\in\mathcal{A}$.

We say that an $\mathcal{L}_{\epsilon}$-module $W$ is restricted if
$a^{(\epsilon)}(z)\in\mathrm{Hom}(W,W((z)))$ for any $a\in\mathcal{A}$, and is
of level $\gamma\in\mathcal{Z}^{*}$ if $c^{(\epsilon)}$ act as the scalars
$\gamma(c)$ for $c\in\mathcal{Z}$.

The main goal of this subsection is to prove the following two propositions.

######  Proposition 4.8.

Let $(\mathcal{L},\mathcal{A},\mathcal{Z},\eta)$ be a vertex Lie algebra. Then
$\mathcal{L}_{\epsilon}$ is a Lie algebra.

######  Proposition 4.9.

Let $(\mathcal{L},\mathcal{A},\mathcal{Z},\eta)$ be a vertex Lie algebra and
$\gamma\in\mathcal{Z}^{*}$. Then the restricted
$\mathcal{L}_{\epsilon}$-modules $W$ of level $\gamma$ are exactly
$\phi_{\epsilon}$-coordinated $V_{\mathcal{L}}(\gamma)$-modules
$(W,Y_{W}^{\epsilon})$ with $Y_{W}^{\epsilon}(a,z)=a^{(\epsilon)}(z)$ for
$a\in\mathcal{A}$.

Now we prove the above two propositions. We will need to use a result on
$\phi_{\epsilon}$-coordinated modules for the universal enveloping vertex
algebra of a Lie conformal algebra from [CLTW]. Recall from [K2] that a
conformal algebra is a $\mathbb{C}[\partial]$-module ${\mathcal{C}}$ endowed
with $\mathbb{C}$-bilinear products $(a,b)\mapsto a_{j}b$ for
$a,b\in{\mathcal{C}}$ and $j\in\mathbb{N}$ such that $a_{j}b=0$ for $j\gg 0$
and

(4.11) $\displaystyle(\partial a)_{j}b=-ja_{j-1}b,\quad a_{j}(\partial
b)=\partial(a_{j}b)+ja_{j-1}b.$

For a conformal algebra ${\mathcal{C}}$, we define a multiplication on its
loop space $\mathbb{C}[t,t^{-1}]\otimes{\mathcal{C}}$ by

(4.12) $\displaystyle[a^{(\epsilon)}(z),b^{(\epsilon)}(w)]=\sum_{i\geq
0}\frac{1}{i!}(a_{i}b)^{(\epsilon)}(w)\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{i}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)$

where $a,b\in{\mathcal{C}}$ and
$a^{(\epsilon)}(z)=\sum_{n\in\mathbb{Z}}(t^{n}\otimes a)z^{\epsilon-n-1}$. One
can easily check that
$\mathrm{Im}\,(1\otimes\partial+t^{\epsilon}\frac{d}{dt}\otimes 1)$ is a two-
sided ideal of $\mathbb{C}[t,t^{-1}]\otimes{\mathcal{C}}$ under this
multiplication. Form the quotient algebra

$\displaystyle\widehat{{\mathcal{C}}}_{\epsilon}:=\mathbb{C}[t,t^{-1}]\otimes{\mathcal{C}}/\mathrm{Im}\,(1\otimes\partial+t^{\epsilon}\frac{d}{dt}\otimes
1),$

and denote by
$\rho_{\epsilon}:\mathbb{C}[t,t^{-1}]\otimes{\mathcal{C}}\rightarrow\widehat{{\mathcal{C}}}_{\epsilon}$
the natural homomorphism. For each $a\in{\mathcal{C}}$, we will still denote
the generating function $\sum_{n\in\mathbb{Z}}\rho_{\epsilon}(t^{n}\otimes
a)z^{\epsilon-n-1}$ in $\widehat{{\mathcal{C}}}_{\epsilon}[[z,z^{-1}]]$ by
$a^{(\epsilon)}(z)$.

A Lie conformal algebra is a conformal algebra ${\mathcal{C}}$ such that for
$a,b,c\in{\mathcal{C}}$ and $m,n\in\mathbb{N}$,

$\displaystyle
a_{n}b=-\sum_{j=0}^{\infty}\frac{(-1)^{j+n}}{j!}\partial^{j}(b_{n+j}a)\quad\textnormal{{and}}\quad
a_{m}(b_{n}c)=\sum_{j=0}^{m}\binom{m}{j}(a_{j}b)_{m+n-j}c+b_{n}(a_{m}c).$

The following results are from in [P, K2].

######  Lemma 4.10.

Let ${\mathcal{C}}$ be a conformal algebra. Then ${\mathcal{C}}$ is a Lie
conformal algebra if and only if $\widehat{{\mathcal{C}}}_{0}$ is a Lie
algebra. Furthermore, in this case each $\widehat{{\mathcal{C}}}_{\epsilon}$
is also a Lie algebra.

Let ${\mathcal{C}}$ be a Lie conformal algebra. Set
$\widehat{{\mathcal{C}}}_{0}^{+}=\rho_{0}(\mathbb{C}[t]\otimes{\mathcal{C}})$,
a subalgebra of $\widehat{{\mathcal{C}}}_{0}$. Let $\mathbb{C}$ be the trivial
$\widehat{{\mathcal{C}}}_{0}^{+}$-module and form the induced
$\widehat{{\mathcal{C}}}_{0}$-module

$V_{{\mathcal{C}}}=\mathcal{U}(\widehat{{\mathcal{C}}}_{0})\otimes_{\mathcal{U}(\widehat{{\mathcal{C}}}_{0}^{+})}\mathbb{C}.$

Set $\bm{1}=1\otimes 1$ and identify ${\mathcal{C}}$ as a subspace of
$V_{{\mathcal{C}}}$ with $a\mapsto\rho_{0}(t^{-1}\otimes a)\bm{1}$ for
$a\in{\mathcal{C}}$. It is known that there is a unique vertex algebra
structure on $V_{{\mathcal{C}}}$ with ${\bm{1}}$ as the vacuum vector and with
$Y(a,z)=a^{0}(z)$ for $a\in{\mathcal{C}}$ [P]. We say that a
$\widehat{{\mathcal{C}}}_{\epsilon}$-module $W$ is restricted if
$a^{(\epsilon)}(z)\in\mathrm{Hom}(W,W((z)))$ for any $a\in{\mathcal{C}}$. The
following result was from [CLTW].

######  Proposition 4.11.

Let ${\mathcal{C}}$ be a Lie conformal algebra. Then the restricted
$\widehat{{\mathcal{C}}}_{\epsilon}$-modules $W$ are exactly the
$\phi_{\epsilon}$-coordinated $V_{{\mathcal{C}}}$-modules
$(W,Y_{W}^{\epsilon})$ with $a^{(\epsilon)}(z)=Y_{W}^{\epsilon}(a,z)$ for
$a\in{\mathcal{C}}$.

Proof of Propositions 4.8 and 4.9. Associated to the vertex Lie algebra
$\mathcal{L}$, we introduce a vector space as follows:

$\displaystyle{\mathcal{CL}}=\left(\mathbb{C}[\partial]\otimes\mathcal{A}\right)\oplus\mathcal{Z}.$

Endow $\mathcal{CL}$ with a $\mathbb{C}[\partial]$-module structure such that
$\partial(\partial^{m}\otimes a)=\partial^{m+1}\otimes a$ and $\partial(c)=0$
for $m\in\mathbb{N},a\in\mathcal{A}$ and $c\in\mathcal{Z}$. Furthermore,
define $j$-products $(j\in\mathbb{N})$ on $\mathcal{CL}$ such that

$a_{j}b=a_{(j)}b+\sum_{i\geq 0}\partial^{i}(a_{(i,j)}b),\quad
a_{j}c=c_{j}a=c_{j}c^{\prime}=0$

for $a,b\in\mathcal{A}$ and $c,c^{\prime}\in\mathcal{Z}$, and then extend them
to the whole space $\mathcal{CL}$ via the rule (4.11). This gives a conformal
algebra structure on $\mathcal{CL}$. Note that in
$\widehat{\mathcal{CL}}_{\epsilon}$ we have $(\partial
a)^{(\epsilon)}(z)=z^{\epsilon}\frac{\partial}{\partial z}a^{(\epsilon)}(z)$
for $a\in\mathcal{A}$ and $(\partial
c)^{(\epsilon)}(z)=z^{\epsilon}\frac{\partial}{\partial z}c^{(\epsilon)}(z)=0$
for $c\in\mathcal{Z}$. This implies that there is a linear isomorphism from
$\widehat{\mathcal{CL}}_{\epsilon}$ to $\mathcal{L}_{\epsilon}$ such that

(4.13) $\displaystyle\rho_{\epsilon}(t^{n}\otimes a)\mapsto
a^{(\epsilon)}(n),\quad\rho_{\epsilon}(t^{\epsilon-1}\otimes c)\mapsto
c^{(\epsilon)}$

for $n\in\mathbb{Z},a\in\mathcal{A}$ and $c\in\mathcal{Z}$. Furthermore, under
this linear isomorphism, the multiplication (4.12) on
$\widehat{\mathcal{CL}}_{\epsilon}$ coincides with the multiplication (4.10)
on $\mathcal{L}_{\epsilon}$.

Since $\mathcal{L}_{0}=\mathcal{L}$ is a Lie algebra,
$\widehat{\mathcal{CL}}_{0}$ is a Lie algebra as well. This together with
Lemma 4.10 gives that $\mathcal{CL}$ is a Lie conformal algebra and
$\widehat{\mathcal{CL}}_{\epsilon}$ is a Lie algebra. Thus
$\mathcal{L}_{\epsilon}$ is a Lie algebra, which proves the Proposition 4.8.
For the Proposition 4.9, by identifying $\widehat{\mathcal{CL}}_{0}$ with
$\mathcal{L}$, one can easily check that the vertex algebra
$V_{\mathcal{L}}(\gamma)$ is isomorphic to the quotient vertex algebra of
$V_{\mathcal{CL}}$ modulo the ideal generated by $c-\gamma(c)$ for
$c\in\mathcal{Z}$. According to Proposition 4.11,
$\phi_{\epsilon}$-coordinated $V_{\mathcal{CL}}$-modules are exactly
restricted $\mathcal{L}_{\epsilon}$-modules, and hence
$\phi_{\epsilon}$-coordinated $V_{\mathcal{L}}(\gamma)$-modules are exactly
restricted $\mathcal{L}_{\epsilon}$-modules of level $\gamma$. This finishes
the proof of proposition 4.9.

##  5\. Proof of theorem 3.2

Here we finish the proof of Theorem 3.2.

###  5.1. $\phi_{\epsilon}$-coordinated modules for certain vertex algebras

In this subsection we study the $\phi_{\epsilon}$-coordinated modules for
three types of vertex algebras.

First note that the derived subalgebra of $\mathcal{D}^{\epsilon}$ is

$[\mathcal{D}^{\epsilon},\mathcal{D}^{\epsilon}]=\textnormal{{Span}}_{\mathbb{C}}\\{\tilde{\mathrm{d}}_{m,n}^{\epsilon}\mid
m,n\in\mathbb{Z}\\}.$

Set

$\displaystyle\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}=\mathfrak{t}(\mathfrak{g})+[\mathcal{D}^{\epsilon},\mathcal{D}^{\epsilon}]+\mathbb{C}t_{0}^{\epsilon-1}\mathrm{d}_{1},$

which is a subalgebra of $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$
such that
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}=\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}\oplus\mathbb{C}t_{0}^{\epsilon-1}\mathrm{d}_{0}$.
Recall the space $\mathcal{B}_{\mathfrak{g}}$ defined in (2.5). We define a
linear isomorphism
$\theta_{\epsilon}:\left(\mathbb{C}[t,t^{-1}]\otimes\mathcal{B}_{\mathfrak{g}}\right)\oplus\mathbb{C}\mathrm{k}_{0}\rightarrow\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$
as follows:

(5.1) $\displaystyle t^{n}\otimes u\mapsto t_{0}^{n}u,\
t^{n}\otimes\mathrm{D}_{m}\mapsto\mathrm{d}_{n+\epsilon-1,m},\
t^{n}\otimes\mathrm{K}_{m}\mapsto\mathrm{K}_{n-\epsilon+1,m},\
\mathrm{k}_{0}\mapsto\mathrm{k}_{0}$

for $u\in\widetilde{\mathfrak{g}}_{1},n\in\mathbb{Z}$ and
$m\in\mathbb{Z}^{\times}$.

In view of Proposition 2.3, we see that the quadruple

$\displaystyle(\mathcal{L}=\widehat{\mathfrak{t}}(\mathfrak{g})^{0},\mathcal{A}=\mathcal{B}_{\mathfrak{g}},\mathcal{Z}=\mathbb{C}\mathrm{k}_{0},\eta=\theta_{0})$

is a vertex Lie algebra. Furthermore, the map
$\eta_{\epsilon}\circ\theta_{\epsilon}^{-1}$ (see (4.9)) is a Lie isomorphism
from $\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$ to
$\widehat{\mathfrak{t}}(\mathfrak{g})^{0}_{\epsilon}$. Then for any complex
number $\ell$, we have a vertex algebra
$V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$, where
$\gamma_{\ell}\in(\mathbb{C}\mathrm{k}_{0})^{*}$ defined by
$\gamma_{\ell}(\mathrm{k}_{0})=\ell$. We say that a
$\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module $W$ is restricted if
for any $a\in\mathcal{B}_{\mathfrak{g}}$,
$a^{\epsilon}(z)\in\mathrm{Hom}(W,W((z)))$, and is of level
$\ell\in\mathbb{C}$ if $\mathrm{k}_{0}$ acts as the scalar $\ell$ on $W$. Then
by Proposition 4.9 and the Lie isomorphism
$\eta_{\epsilon}\circ\theta_{\epsilon}^{-1}:\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}\rightarrow\widehat{\mathfrak{t}}(\mathfrak{g})^{0}_{\epsilon}$,
we immediately have the following result.

######  Proposition 5.1.

For any $\ell\in\mathbb{C}$, the restricted
$\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-modules of level $\ell$ are
exactly the $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$-modules
$(W,Y_{W}^{\epsilon})$ with $Y_{W}^{\epsilon}(a,z)=a^{\epsilon}(z)$ for
$a\in\mathcal{B}_{\mathfrak{g}}$.

Next, recall the affine-Virasoro algebra
$\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$ defined in Section 3. Form the
vector spaces $\mathcal{A}=\mathfrak{g}\oplus\mathbb{C}\omega_{\mathfrak{g}}$
and
$\mathcal{Z}=\mathbb{C}\mathrm{k}\oplus\mathbb{C}\mathrm{k}_{\mathsf{Vir}}$.
Define a linear bijection

$\eta:\left(\mathbb{C}[t,t^{-1}]\otimes\mathcal{A}\right)\oplus\mathcal{Z}\rightarrow\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$

by letting

$\displaystyle t^{n}\otimes u\mapsto u(n),\quad
t^{n}\otimes\omega_{\mathfrak{g}}\mapsto
L(n-1),\quad\mathrm{k}\mapsto\mathrm{k},\quad\mathrm{k}_{\mathsf{Vir}}\mapsto\mathrm{k}_{\mathsf{Vir}}$

for $n\in\mathbb{Z}$, $u\in\mathfrak{g}$. Note that in terms of generating
functions

$\displaystyle u(z)=\sum_{n\in\mathbb{Z}}u(n)z^{-n-1}\
(u\in\mathfrak{g})\quad\text{and}\quad\omega_{\mathfrak{g}}(z)=\sum_{n\in\mathbb{Z}}\omega_{\mathfrak{g}}(n)z^{-n-1}=\sum_{n\in\mathbb{Z}}L(n)z^{-n-2},$

the Lie relations in (3.1) can be rewritten as follows:

(5.2)
$\begin{split}[\omega_{\mathfrak{g}}(z),\omega_{\mathfrak{g}}(w)]&=\left(\frac{\partial}{\partial
w}\omega_{\mathfrak{g}}(w)\right)z^{-1}\delta\left(\frac{w}{z}\right)+2\omega_{\mathfrak{g}}(w)\frac{\partial}{\partial
w}z^{-1}\delta\left(\frac{w}{z}\right)+\frac{1}{12}\mathrm{k}_{\mathsf{Vir}}\left(\frac{\partial}{\partial
w}\right)^{3}z^{-1}\delta\left(\frac{w}{z}\right),\\\
[u(z),v(w)]&=[u,v](w)z^{-1}\delta\left(\frac{w}{z}\right)+{\langle}u,v{\rangle}\mathrm{k}\frac{\partial}{\partial
w}z^{-1}\delta\left(\frac{w}{z}\right),\\\
[\omega_{\mathfrak{g}}(z),u(w)]&=\left(\frac{\partial}{\partial
w}u(w)\right)z^{-1}\delta\left(\frac{w}{z}\right)+u(w)\frac{\partial}{\partial
w}z^{-1}\delta\left(\frac{w}{z}\right)\end{split}$

for $u,v\in\mathfrak{g}$. This implies that the quadruple

$(\mathcal{L}=\widehat{\mathfrak{g}}\rtimes\mathsf{Vir},\mathcal{A}=\mathfrak{g}\oplus\mathbb{C}\omega_{\mathfrak{g}},\mathcal{Z}=\mathbb{C}\mathrm{k}\oplus\mathbb{C}\mathrm{k}_{\mathsf{Vir}},\eta)$

is a vertex Lie algebra.

For $\ell,c\in\mathbb{C}$, denote by $\gamma_{\ell,c}$ the linear functional
on $\mathcal{Z}^{*}$ such that $\gamma_{\ell,c}(\mathrm{k})=\ell$ and
$\gamma_{\ell,c}(\mathrm{k}_{\mathsf{Vir}})=c$. Then we have the (universal)
affine-Virasoro vertex algebra

$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,c):=V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\gamma_{\ell,c}),$

which is equipped with a conformal vector $\omega_{\mathfrak{g}}$. We say that
a $\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$-module $W$ is restricted if for
any $w\in W,a\in\mathfrak{g}$, one has $a(n)w=0=L(n)w$ for $n\gg 0$, is of
level $\ell\in\mathbb{C}$ if $\mathrm{k}$ acts as the scalar $\ell$, and is of
central charge $c$ if $\mathrm{k}_{\mathsf{Vir}}$ acts as the scalar $c$.
Recall the generating functions $a^{\epsilon}(z)$ for $a\in\mathfrak{g}$ and
$L^{\epsilon}(z)$ defined in (3.2) and (3.3), respectively. Then we have:

######  Proposition 5.2.

For $\ell,c\in\mathbb{C}$, the restricted
$\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$-modules $W$ of level $\ell$ and
central charge $c$ are exactly the $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,c)$-modules
$(W,Y_{W}^{\epsilon})$ with

(5.3) $\displaystyle Y_{W}^{\epsilon}(a,z)=a^{\epsilon}(z),\forall
a\in\mathfrak{g};\quad
Y_{W}^{\epsilon}(\omega_{\mathfrak{g}},z)=L_{\mathfrak{g}}^{\epsilon}(z).$

###### Proof.

By Proposition 4.8 we have an $\epsilon$-deformation Lie algebra

$(\widehat{\mathfrak{g}}\rtimes\mathsf{Vir})_{\epsilon}=\text{Span}_{\mathbb{C}}\\{u^{(\epsilon)}(n),\omega_{\mathfrak{g}}^{(\epsilon)}(n),\mathrm{k}^{(\epsilon)},\mathrm{k}_{\mathsf{Vir}}^{(\epsilon)}\mid
u\in\mathfrak{g},n\in\mathbb{Z}\\}$

of $\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$ with the Lie brackets given by

$\begin{split}[\omega_{\mathfrak{g}}^{(\epsilon)}(z),\omega_{\mathfrak{g}}^{(\epsilon)}(w)]&=\left(w^{\epsilon}\frac{\partial}{\partial
w}\omega_{\mathfrak{g}}^{(\epsilon)}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+2\omega_{\mathfrak{g}}^{(\epsilon)}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)\\\ &\ \
+\frac{1}{12}\mathrm{k}_{\mathsf{Vir}}^{(\epsilon)}\left(w^{\epsilon}\frac{\partial}{\partial
w}\right)^{3}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),\\\
[u^{(\epsilon)}(z),v^{(\epsilon)}(w)]&=[u,v]^{(\epsilon)}(w)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+{\langle}u,v{\rangle}\mathrm{k}^{(\epsilon)}w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right),\\\
[\omega_{\mathfrak{g}}^{(\epsilon)}(z),u^{(\epsilon)}(w)]&=\left(w^{\epsilon}\frac{\partial}{\partial
w}u^{(\epsilon)}(w)\right)z^{\epsilon-1}\delta\left(\frac{w}{z}\right)+u^{(\epsilon)}(w)w^{\epsilon}\frac{\partial}{\partial
w}z^{\epsilon-1}\delta\left(\frac{w}{z}\right)\end{split}$

for $u,v\in\mathfrak{g}$.

It is straightforward to check that the linear map

$u^{(\epsilon)}(n)\mapsto u(n),\ \omega_{\mathfrak{g}}^{(\epsilon)}(n)\mapsto
L(n+\epsilon-1)+\delta_{n+\epsilon-1,0}\frac{\epsilon^{2}-2\epsilon}{24}\mathrm{k}_{\mathsf{Vir}},\
\mathrm{k}^{(\epsilon)}\mapsto\mathrm{k},\
\mathrm{k}_{\mathsf{Vir}}^{(\epsilon)}\mapsto\mathrm{k}_{\mathsf{Vir}}$

where $u\in\mathfrak{g}$ and $n\in\mathbb{Z}$, is a Lie algebra isomorphism
from $(\widehat{\mathfrak{g}}\rtimes\mathsf{Vir})_{\epsilon}$ to
$\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$. Then the assertion follows
immediately from Proposition 4.9. ∎

Finally, recall from Section 3.2 that we have an
$\widehat{\mathfrak{h}}$-module

$V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])=V_{\widehat{\mathfrak{h}}}(\ell,0)\otimes e^{\alpha{\bf
k}}\mathbb{C}[L].$

When $\ell=1$ and $\alpha=0$, it is known that there is a vertex algebra
structure on

$V_{(\mathfrak{h},L)}:=V_{\widehat{\mathfrak{h}}}(1,\mathbb{C}[L])=V_{\widehat{\mathfrak{h}}}(1,0)\otimes\mathbb{C}[L]$

with $Y(h,z)=h(z)$ for $h\in\mathfrak{h}$ and $Y(e^{\gamma},z)=E^{\gamma}(z)$
(see [B1] or [LW] for example). Note that $\omega_{\mathfrak{h}}={\bf
k}_{-1}{\bf d}$ is a conformal vector of $V_{(\mathfrak{h},L)}$ of rank $2$,
which is also a conformal vector of the Heisenberg vertex subalgebra
$V_{\widehat{\mathfrak{h}}}(1,0)$.

View $\mathfrak{h}$ and $\mathbb{C}[L]$ as abelian Lie algebras and let
$\mathfrak{h}$ act on $\mathbb{C}[L]$ by derivations with

$he^{\gamma}={\langle}h,\gamma{\rangle}e^{\gamma}$

for $h\in H,\gamma\in L$. Form the semiproduct Lie algebra

$\mathfrak{p}=\mathfrak{h}\ltimes\mathbb{C}[L],$

on which we have

$[h,h^{\prime}]=0=[e^{\gamma},e^{\gamma^{\prime}}],\quad[h,e^{\gamma}]={\langle}h,\gamma{\rangle}e^{\gamma}$

for $h,h^{\prime}\in\mathfrak{h},\gamma,\gamma^{\prime}\in L$. Extend the form
${\langle}\cdot,\cdot{\rangle}$ on $\mathfrak{h}$ (see (3.7)) to
$\mathfrak{p}$ by letting ${\langle}\mathfrak{p},\mathbb{C}[L]{\rangle}=0$,
which is still symmetric and invariant. Recall that $\widehat{\mathfrak{p}}$
is the affine Lie algebra associated to the pair
$(\mathfrak{p},{\langle}\cdot,\cdot{\rangle})$, which is obviously a vertex
Lie algebra with $\mathcal{A}=\mathfrak{p}$ and
$\mathcal{Z}=\mathbb{C}\mathrm{k}$. Take $\gamma\in(\mathbb{C}\mathrm{k})^{*}$
such that $\gamma(\mathrm{k})=1$. Then we have the universal affine vertex
algebra

$V_{\widehat{\mathfrak{p}}}(1,0):=V_{\widehat{\mathfrak{p}}}(\gamma).$

The following relation between the vertex algebras $V_{(\mathfrak{h},L)}$ and
$V_{\widehat{\mathfrak{p}}}(1,0)$ was proved in [LW].

######  Lemma 5.3.

The vertex algebra $V_{(\mathfrak{h},L)}$ is isomorphic to the quotient vertex
algebra of $V_{\widehat{\mathfrak{p}}}(1,0)$ modulo the ideal generated by the
set

(5.4) $\displaystyle\\{e^{0}-{\bm{1}},\
e^{\gamma}e^{\lambda}-e^{\gamma+\lambda},\
\mathcal{D}e^{\lambda}-\lambda(-1)e^{\lambda}\mid\gamma,\lambda\in L\\}.$

Recall that $u^{\epsilon}(z)=\sum_{n\in\mathbb{Z}}u(n)z^{\epsilon-n-1}$ for
$u\in\mathfrak{p}$. In view of the above lemma, we have:

######  Proposition 5.4.

The $\phi_{\epsilon}$-coordinated $V_{(\mathfrak{h},L)}$-modules
$(W,Y_{W}^{\epsilon})$ are exactly the restricted
$\widehat{\mathfrak{p}}$-modules $W$ of level $1$ satisfying the conditions
that

(5.5)
$\displaystyle\left(e^{0}\right)^{\epsilon}(z)=1,\quad\left(e^{\gamma}\right)^{\epsilon}(z)\left(e^{\lambda}\right)^{\epsilon}(z)=\left(e^{\gamma+\lambda}\right)^{\epsilon}(z),\quad
z^{\epsilon}\frac{d}{dz}\left(e^{\gamma}\right)^{\epsilon}(z)=\gamma^{\epsilon}(z)\left(e^{\gamma}\right)^{\epsilon}(z)$

for $\gamma,\lambda\in L$. And the correspondence is given by
$Y_{W}^{\epsilon}(h,z)=h^{\epsilon}(z)$ and
$Y_{W}^{\epsilon}(e^{\gamma},z)=\left(e^{\gamma}\right)^{\epsilon}(z)$ for
$h\in\mathfrak{h}$ and $\gamma\in L$.

###### Proof.

It is routine to see that the $\epsilon$-deformation
$\widehat{\mathfrak{p}}_{\epsilon}$ of $\widehat{\mathfrak{p}}$ is isomorphic
to $\widehat{\mathfrak{p}}$ itself with

$u^{(\epsilon)}(n)\mapsto
u(n)\quad\text{and}\quad\mathrm{k}^{(\epsilon)}\mapsto\mathrm{k}$

for $u\in\mathfrak{p},n\in\mathbb{Z}$. Thus from Proposition 4.9 it follows
that the $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{p}}}(1,0)$-modules $(W,Y_{W}^{\epsilon})$ are exactly
the restricted $\widehat{\mathfrak{p}}$-modules $W$ of level $1$ with

(5.6) $\displaystyle Y_{W}^{\epsilon}(u,z)=u^{\epsilon}(z),\quad\forall\
u\in\mathfrak{p}.$

Lemma 5.3 implies that the $\phi_{\epsilon}$-coordinated
$V_{(\mathfrak{h},L)}$-modules are exactly those $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{p}}}(1,0)$-modules $(W,Y_{W}^{\epsilon})$ such that
$Y_{W}^{\epsilon}(v,z)=0$ for all $v$ in the set $\eqref{genvhL}$. Thus, by
the correspondence (5.6), the $\phi_{\epsilon}$-coordinated
$V_{(\mathfrak{h},L)}$-modules are exactly the restricted
$\widehat{\mathfrak{p}}$-modules $W$ of level $1$ such that the following
relations hold:

$\displaystyle Y_{W}^{\epsilon}(e^{0}-{\bm{1}},z)$
$\displaystyle=(e^{0})^{\epsilon}(z)-1=0,$ $\displaystyle
Y_{W}^{\epsilon}(e_{-1}^{\gamma}e^{\lambda}-e^{\lambda+\gamma},z)$
$\displaystyle=_{\circ}^{\circ}Y_{W}^{\epsilon}(e^{\gamma},z)Y_{W}^{\epsilon}(e^{\lambda},z)_{\circ}^{\circ}-Y_{W}^{\epsilon}(e^{\gamma+\lambda},z)$
$\displaystyle=\left(e^{\gamma}\right)^{\epsilon}(z)\left(e^{\lambda}\right)^{\epsilon}(z)-\left(e^{\gamma+\lambda}\right)^{\epsilon}(z)=0,$
$\displaystyle
Y_{W}^{\epsilon}(\mathcal{D}e^{\gamma}-\gamma_{-1}e^{\gamma},z)$
$\displaystyle=z^{\epsilon}\frac{d}{dz}Y_{W}^{\epsilon}(e^{\gamma},z)-_{\circ}^{\circ}Y_{W}^{\epsilon}(\gamma,z)Y_{W}^{\epsilon}(e^{\gamma},z)_{\circ}^{\circ}$
$\displaystyle=z^{\epsilon}\frac{d}{dz}\left(e^{\gamma}\right)^{\epsilon}(z)-\gamma^{\epsilon}(z)\left(e^{\gamma}\right)^{\epsilon}(z)=0,$

for $\gamma,\lambda\in L$. Here the second relation follows from (4.6) and the
fact that $e^{\gamma}$ commutes with $e^{\lambda}$, and the third relation
follows from Lemma 4.2, (4.6) and the fact that $\gamma$ commutes with
$e^{\gamma}$. This completes the proof. ∎

Recall the operator $E^{\gamma}(z)$ defined in (3.10). The following result
shows that there is a canonical $\phi_{\epsilon}$-coordinated
$V_{(\mathfrak{h},L)}$-module structure on the $\widehat{\mathfrak{h}}$-module
$V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$. Similar
results for lattice vertex algebras have been obtained in [JKLT].

######  Proposition 5.5.

For every $\alpha\in\mathbb{C}$ and $\ell\in\mathbb{C}^{\times}$, the
$\widehat{\mathfrak{h}}$-module
$W:=V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$ admits an
irreducible $\phi_{\epsilon}$-coordinated $V_{(\mathfrak{h},L)}$-module
structure $Y^{\epsilon}_{W}$, which is uniquely determined by

(5.7) $\displaystyle Y^{\epsilon}_{W}({\bf k},z)=\frac{1}{\ell}{\bf
k}^{\epsilon}(z),\quad Y^{\epsilon}_{W}({\bf d},z)={\bf
d}^{\epsilon}(z),\quad\mathrm{and}\quad
Y^{\epsilon}_{W}(e^{\gamma},z)=E^{\gamma}(z),\ \forall\gamma\in L.$

Furthermore, the action of the conformal vector $\omega_{\mathfrak{h}}$ is
(see (3.3) and (3.8))

(5.8) $\displaystyle
Y^{\epsilon}_{W}(\omega_{\mathfrak{h}},z)=L_{\mathfrak{h}}^{\epsilon}(z).$

###### Proof.

We first claim that the assignment

(5.9) $\displaystyle{\bf k}(z)\mapsto\frac{1}{\ell}{\bf k}(z),\quad{\bf
d}(z)\mapsto{\bf d}(z),\quad\mathrm{k}\mapsto 1\quad\textnormal{{and}}\quad
e^{\gamma}(z)\mapsto z^{-\epsilon}E^{\gamma}(z)\ (\gamma\in L)$

gives an irreducible restricted $\widehat{\mathfrak{p}}$-module structure on
the $\widehat{\mathfrak{h}}$-module
$V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$ and on which
the three conditions stated in (5.5) hold.

Indeed, the commutators among those operators in (5.9) are as follows (see
[LL, Proposition 6.5.2] for example)

$\displaystyle[{\bf k}(z),{\bf d}(w)]=\frac{\partial}{\partial
w}z^{-1}\delta\left(\frac{w}{z}\right),\quad[{\bf k}(z),{\bf k}(w)]=0=[{\bf
d}(z),{\bf d}(w)],$ $\displaystyle[{\bf k}(z),E^{\gamma}(w)]=0,\quad[{\bf
d}(z),w^{-\epsilon}E^{\gamma}(w)]={\langle}{\bf
d},\gamma{\rangle}w^{-\epsilon}E^{\gamma}(w)z^{-1}\delta\left(\frac{w}{z}\right).$

This implies that $V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])$ is a restricted $\widehat{\mathfrak{p}}$-module of level
$1$ (with the action (5.9)). Meanwhile, it is known that (see [LL, (6.5.64)]
for example)

$E^{\lambda}(z)E^{\gamma}(z)=E^{\lambda+\gamma}(z),\quad\frac{d}{dz}E^{\lambda}(z)=\lambda(z)E^{\lambda}(z)$

for $\gamma,\lambda\in L$. From this and the fact that
$\left(e^{\gamma}\right)^{\epsilon}(z)=E^{\gamma}(z)$, one can conclude that
this $\widehat{\mathfrak{p}}$-module satisfies the conditions in (5.5), as
desired.

In view of Proposition 5.4, the above claim implies that the assignment (5.9)
determines a $\phi_{\epsilon}$-coordinated $V_{(\mathfrak{h},L)}$-module
structure on $V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])$. The irreducibility of this $V_{(\mathfrak{h},L)}$-module
is obvious, and it remains to deduce the action $\omega_{\mathfrak{h}}$ on
$W$.

Note that in $V_{\mathfrak{h},L}$ we have ${\bf k}_{n}{\bf d}=0$ for $n\geq 0$
unless $n=1$, in which case ${\bf k}_{1}{\bf d}={\bm{1}}$. Then from Remark
4.7 we have

$\displaystyle Y_{W}^{\epsilon}(\omega_{\mathfrak{h}},z)$
$\displaystyle=Y^{\epsilon}({\bf k}_{-1}{\bf d},z)$
$\displaystyle=_{\circ}^{\circ}Y_{W}^{\epsilon}({\bf
k},z)Y_{W}^{\epsilon}({\bf
d},z)_{\circ}^{\circ}-\frac{5\epsilon-4}{12}\epsilon z^{2(\epsilon-1)}$
$\displaystyle=\frac{1}{\ell}\ _{\circ}^{\circ}{\bf k}^{\epsilon}(z){\bf
d}^{\epsilon}(z)_{\circ}^{\circ}-\frac{5\epsilon-4}{12}\epsilon
z^{2(\epsilon-1)}.$

Recall from (3.8) that $L_{\mathfrak{h}}(z)=\frac{1}{\ell}\
_{\circ}^{\circ}{\bf k}^{\epsilon}(z){\bf d}^{\epsilon}(z)_{\circ}^{\circ}$,
and from (3.3) that

$L_{\mathfrak{h}}^{\epsilon}(z)=z^{2\epsilon}L_{\mathfrak{h}}(z)+\frac{\epsilon^{2}-2\epsilon}{12}z^{2(\epsilon-1)}=\frac{1}{\ell}z^{2\epsilon}\
_{\circ}{}^{\circ}{\bf k}(z){\bf
d}(z)_{\circ}^{\circ}+\frac{\epsilon^{2}-2\epsilon}{12}z^{2(\epsilon-1)}.$

By the definition of normally ordered products, in the
$\widehat{\mathfrak{h}}$-module $V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf
k}}\mathbb{C}[L])$ we have

$\displaystyle\frac{1}{\ell}\ _{\circ}^{\circ}{\bf k}^{\epsilon}(z){\bf
d}^{\epsilon}(z)_{\circ}^{\circ}$
$\displaystyle=\frac{1}{\ell}\sum_{n<\epsilon}{\bf k}(n)z^{\epsilon-n-1}{\bf
d}^{\epsilon}(z)+{\bf d}^{\epsilon}(z)\sum_{n\geq\epsilon}{\bf
k}(n)z^{\epsilon-n-1}$
$\displaystyle=z^{2\epsilon}\frac{1}{\ell}\sum_{n<\epsilon}{\bf
k}(n)z^{-n-1}{\bf d}(z)+{\bf d}(z)\sum_{n\geq\epsilon}{\bf k}(n)z^{-n-1}.$

This implies that

$\begin{split}&\frac{1}{\ell}\ _{\circ}^{\circ}{\bf k}^{\epsilon}(z){\bf
d}^{\epsilon}(z)_{\circ}^{\circ}-z^{2\epsilon}\frac{1}{\ell}\
_{\circ}^{\circ}{\bf k}(z){\bf d}(z)_{\circ}^{\circ}\\\
=\,&\begin{cases}z^{2\epsilon-2}\cdot\sum_{0\leq
n<\epsilon}\frac{1}{\ell}[{\bf k}(n),{\bf d}(-n)],\quad&\textnormal{{if}}\ \
\epsilon>0\\\ 0,\quad&\textnormal{{if}}\ \ \epsilon=0\\\
z^{2\epsilon-2}\cdot\sum_{\epsilon\leq n<0}\frac{1}{\ell}[{\bf d}(-n),{\bf
k}(n)],\quad&\textnormal{{if}}\ \ \epsilon<0\end{cases}\\\
=\,&z^{2\epsilon-2}\frac{1}{2}\epsilon(\epsilon-1).\end{split}$

Summarizing the above relations we obtain

$\displaystyle Y_{W}^{\epsilon}(\omega_{\mathfrak{h}},z)$
$\displaystyle=\frac{1}{\ell}\ _{\circ}^{\circ}{\bf k}^{\epsilon}(z){\bf
d}^{\epsilon}(z)_{\circ}^{\circ}-\frac{5\epsilon-4}{12}\epsilon
z^{2(\epsilon-1)}$ $\displaystyle=z^{2\epsilon}\frac{1}{\ell}\
_{\circ}^{\circ}{\bf k}(z){\bf
d}(z)_{\circ}^{\circ}+z^{2\epsilon-2}\frac{1}{2}\epsilon(\epsilon-1)-\frac{5\epsilon-4}{12}\epsilon
z^{2(\epsilon-1)}$ $\displaystyle=\frac{1}{\ell}z^{2\epsilon}\
_{\circ}{}^{\circ}{\bf k}(z){\bf
d}(z)_{\circ}^{\circ}+\frac{\epsilon^{2}-2\epsilon}{12}z^{2(\epsilon-1)}=L_{\mathfrak{h}}^{\epsilon}(z).$

This finishes the proof. ∎

###  5.2. Proof of theorem 3.2

This section is devoted to a proof of Theorem 3.2.

By taking the tensor product of the latter two vertex algebras considered in
last subsection, for every nonzero complex number $\ell$, we have a conformal
vertex algebra

$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)},$

which has a conformal vector

$\omega=\omega_{\mathfrak{g}}\otimes\mathbf{1}+\mathbf{1}\otimes\omega_{\mathfrak{h}}.$

We will often view
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)$ and
$V_{(\mathfrak{h},L)}$ as subalgebras of
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$ in the canonical way. The following result was obtained
in [CLiT, Corollary 5.9]:

######  Proposition 5.6.

Let $\ell\in\mathbb{C}^{\times}$. Then there exists a vertex algebra
epimorphism

$\Theta:V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})\rightarrow
V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)},$

which is uniquely determined by

(5.10) $\displaystyle\Theta(t_{1}^{m}\otimes u)=u\otimes e^{m{\bf
k}},\quad\Theta(\mathrm{k}_{1})=\ell{\bf k},\quad\Theta(\mathrm{d}_{1})={\bf
d},\quad\Theta(\mathrm{K}_{n})=\frac{\ell}{n}e^{n{\bf k}},$ (5.11)
$\displaystyle\quad\qquad\Theta(\mathrm{D}_{n})=n\omega_{-1}e^{n{\bf
k}}-\omega_{0}{\bf d}_{-1}e^{n{\bf k}}+n^{2}(\mu\ell-1){\bf k}_{-2}e^{n{\bf
k}}$

for $u\in\mathfrak{g},\ m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\times}$.

The following result asserts that every (irreducible)
$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module admits an (irreducible)
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module structure.

######  Theorem 5.7.

Let $(W,Y_{W}^{\epsilon}(\cdot,z))$ be a $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module with $\ell\in\mathbb{C}^{\times}$. Then there is
a $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module structure on $W$
with the actions given by

(5.12) $\displaystyle t_{0}^{\epsilon-1}\mathrm{d}_{0}$
$\displaystyle=-\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(\omega,z),\quad\mathrm{k}_{0}=\ell,\quad(t_{1}^{m}\otimes
u)^{\epsilon}(z)=Y_{W}^{\epsilon}(u\otimes e^{m{\bf k}},z),$ (5.13)
$\displaystyle\mathrm{k}_{1}^{\epsilon}(z)$ $\displaystyle=\ell
Y_{W}^{\epsilon}({\bf
k},z),\quad\mathrm{d}_{1}^{\epsilon}(z)=Y_{W}^{\epsilon}({\bf
d},z),\quad\mathrm{K}_{n}^{\epsilon}(z)=\frac{\ell}{n}Y_{W}^{\epsilon}(e^{n{\bf
k}},z),$ (5.14) $\displaystyle\mathrm{D}_{n}^{\epsilon}(z)$
$\displaystyle=n_{\circ}^{\circ}Y_{W}^{\epsilon}(\omega,z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-z^{\epsilon}\frac{d}{dz}\
_{\circ}^{\circ}Y_{W}^{\epsilon}({\bf d},z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}$ $\displaystyle\
+\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}Y_{W}^{\epsilon}(e^{n{\bf
k}},z)+n^{2}(\mu\ell-1)z^{\epsilon}\frac{d}{dz}Y_{W}^{\epsilon}({\bf
k},z)Y_{W}^{\epsilon}(e^{n{\bf k}},z)$

for $u\in\mathfrak{g},\ m\in\mathbb{Z}$ and $n\in\mathbb{Z}^{\times}$.
Furthermore, if the $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module $(W,Y_{W}^{\epsilon}(\cdot,z))$ is irreducible,
then $W$ is also irreducible as a
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module.

###### Proof.

Let $(W,Y_{W}^{\epsilon}(\cdot,z))$ be a $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module with $\ell\neq 0$. For every $v\in
V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$, set

$Y_{W}^{\epsilon}[v,z]:=Y_{W}^{\epsilon}(\Theta(v),z)\in\mathrm{Hom}(W,W((z))).$

From Proposition 5.6, it follows that $(W,Y_{W}^{\epsilon}[\cdot,z])$ is a
$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$-module. Thus, by
applying Proposition 5.1, $W$ becomes a restricted
$\widehat{\mathfrak{t}}(\mathfrak{g})_{\mu,\epsilon}$-module of level $\ell$
with

$a^{\epsilon}(z)=Y_{W}^{\epsilon}[a,z],\quad\forall
a\in\mathcal{B}_{\mathfrak{g}}.$

Assume further that $(W,Y_{W}^{\epsilon}(\cdot,z))$ is irreducible. Then the
$\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{t}}(\mathfrak{g})^{0}}(\gamma_{\ell})$-module
$(W,Y_{W}^{\epsilon}[\cdot,z])$ is also irreducible as the map $\Theta$ is
surjective. This together with Proposition 5.1 gives that as a
$\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module, $W$ is still
irreducible.

For every $v\in
V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$, it follows from Lemma 4.5 and Lemma 4.2 that

$\displaystyle[\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(\omega,z),Y_{W}^{\epsilon}(v,w)]=Y_{W}^{\epsilon}(\omega_{0}v,w)=Y_{W}^{\epsilon}(\mathcal{D}v,w)=w^{\epsilon}\frac{d}{dw}Y_{W}^{\epsilon}(v,w).$

In particular, for $a\in\mathcal{B}_{\mathfrak{g}}$, we have

$[-\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(\omega,z),Y_{W}^{\epsilon}[a,w]]=-w^{\epsilon}\frac{d}{dw}Y_{W}^{\epsilon}[a,w].$

This together with Proposition 2.3 (12) implies that the
$\widehat{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module $W$ can be extended
to a $\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module with

$t_{0}^{\epsilon-1}\mathrm{d}_{0}=-\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(\omega,z).$

Thus it remains to prove that the operators $Y_{W}^{\epsilon}[a,z]$,
$a\in\mathcal{B}_{\mathfrak{g}}$ have the form as given in theorem. The case
that
$a\in\widetilde{\mathfrak{g}}_{1}\oplus\sum_{n\in\mathbb{Z}^{\times}}\mathbb{C}\mathrm{K}_{n}$
is obvious, and so we only need to prove that the operators
$Y_{W}^{\epsilon}[\mathrm{D}_{n},z]$, $n\in\mathbb{Z}^{\times}$ coincide with
the right hand side of (5.14).

Indeed, note that for $i\geq 0$ and $n\in\mathbb{Z}$, we have

${\bf d}_{i}e^{n{\bf k}}=n\delta_{i,0}e^{n{\bf k}},\quad{\bf k}_{i}e^{n{\bf
k}}=0.$

This implies that for $j>0$, $(\omega_{\mathfrak{h}})_{j}e^{n{\bf k}}=({\bf
k}_{-1}{\bf d})_{j}e^{n{\bf k}}=0$ and hence $\omega_{j}e^{n{\bf k}}=0$. Thus,
by Remark 4.7 and Lemma 4.2, we have

$\displaystyle Y_{W}^{\epsilon}(\omega_{-1}e^{n{\bf k}},z)$
$\displaystyle=_{\circ}^{\circ}Y_{W}^{\epsilon}(\omega,z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-\frac{\epsilon}{2}z^{\epsilon-1}Y_{W}(\omega_{0}e^{n{\bf
k}},z)$
$\displaystyle=_{\circ}^{\circ}Y_{W}^{\epsilon}(\omega,z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-\frac{\epsilon}{2}z^{2\epsilon-1}\frac{d}{dz}Y_{W}(e^{n{\bf
k}},z).$

Similarly, from Remark 4.7 and Lemma 4.2, it follows that

$\displaystyle Y_{W}^{\epsilon}\left(\omega_{0}({\bf d}_{-1}e^{n{\bf
k}}),z\right)$ $\displaystyle\ =z^{\epsilon}\frac{d}{dz}Y_{W}^{\epsilon}({\bf
d}_{-1}e^{n{\bf k}},z)$ $\displaystyle\ =z^{\epsilon}\frac{d}{dz}\
_{\circ}^{\circ}Y_{W}^{\epsilon}({\bf d},z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-\frac{1}{2}\epsilon
z^{\epsilon-1}Y_{W}^{\epsilon}(ne^{n{\bf k}},z)$ $\displaystyle\
=z^{\epsilon}\frac{d}{dz}\ _{\circ}^{\circ}Y_{W}^{\epsilon}({\bf
d},z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}Y_{W}^{\epsilon}(e^{n{\bf
k}},z)-\frac{1}{2}n\epsilon
z^{2\epsilon-1}\frac{d}{dz}Y_{W}^{\epsilon}(e^{n{\bf k}},z),$

and that

$\displaystyle Y_{W}^{\epsilon}({\bf k}_{-2}e^{n{\bf k}},z)$
$\displaystyle=Y_{W}^{\epsilon}((\omega_{0}{\bf k})_{-1}e^{n{\bf
k}},z)=_{\circ}^{\circ}Y_{W}^{\epsilon}(\omega_{0}{\bf
k},z)Y_{W}^{\epsilon}(e^{n{\bf k}},z)_{\circ}^{\circ}$
$\displaystyle=z^{\epsilon}\frac{d}{dz}Y_{W}^{\epsilon}({\bf
k},z)Y_{W}^{\epsilon}(e^{n{\bf k}},z),$

where we used the fact that $(\omega_{0}{\bf k})_{i}e^{n{\bf k}}$ for $i\geq
0$ as ${\bf k}_{i}e^{n{\bf k}}=0$.

Finally, by summarizing the above together, we obtain

$\displaystyle Y_{W}^{\epsilon}[\mathrm{D}_{n},z]$
$\displaystyle=Y_{W}^{\epsilon}(\Theta(\mathrm{D}_{n}),z)$
$\displaystyle=Y_{W}^{\epsilon}(n\omega_{-1}e^{n{\bf k}}-\omega_{0}{\bf
d}_{-1}e^{n{\bf k}}+n^{2}(\mu\ell-1){\bf k}_{-2}e^{n{\bf k}},z)$
$\displaystyle=n\
_{\circ}^{\circ}Y_{W}^{\epsilon}(\omega,z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-\frac{1}{2}n\epsilon
z^{2\epsilon-1}\frac{d}{dz}Y_{W}^{\epsilon}(e^{n{\bf k}},z)$ $\displaystyle\
-z^{\epsilon}\frac{d}{dz}\ _{\circ}^{\circ}Y_{W}^{\epsilon}({\bf
d},z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}+\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}Y_{W}^{\epsilon}(e^{n{\bf
k}},z)$ $\displaystyle\ +\frac{1}{2}n\epsilon
z^{2\epsilon-1}\frac{d}{dz}Y_{W}^{\epsilon}(e^{n{\bf
k}},z)+n^{2}(\mu\ell-1)z^{\epsilon}\frac{d}{dz}Y_{W}^{\epsilon}({\bf
k},z)Y_{W}^{\epsilon}(e^{n{\bf k}},z)$ $\displaystyle=n\
_{\circ}^{\circ}Y_{W}^{\epsilon}(\omega,z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-z^{\epsilon}\frac{d}{dz}\
_{\circ}^{\circ}Y_{W}^{\epsilon}({\bf d},z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}$ $\displaystyle\
+\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}Y_{W}^{\epsilon}(e^{n{\bf
k}},z)+n^{2}(\mu\ell-1)z^{\epsilon}\frac{d}{dz}Y_{W}^{\epsilon}({\bf
k},z)Y_{W}^{\epsilon}(e^{n{\bf k}},z),$

as desired. ∎

Proof of Theorem 3.2. Now we are ready to complete the proof of Theorem 3.2.
Let $U,\ell,\alpha,\beta\in\mathbb{C}$ be as in Theorem 3.2. By Proposition
5.2, the irreducible highest weight
$\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}$-module

$W_{1}:=L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)$

is naturally an irreducible $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)$-module.
Meanwhile, by Proposition 5.5, the $\widehat{\mathfrak{h}}$-module

$W_{2}:=V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$

is naturally an irreducible $\phi_{\epsilon}$-coordinated
$V_{(\mathfrak{h},L)}$-module. Thus, by taking tensor product (see Lemma 4.4)
we obtain an irreducible $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module

$W:=W_{1}\otimes
W_{2}=L_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2,U,\beta)\otimes
V_{\widehat{\mathfrak{h}}}(\ell,e^{\alpha{\bf k}}\mathbb{C}[L])$

with the actions given by

$\displaystyle Y_{W}^{\epsilon}(u,z)=Y_{W_{1}}^{\epsilon}(u,z)\otimes
1=u^{\epsilon}(z)\otimes 1=u^{\epsilon}(z),$ $\displaystyle
Y_{W}^{\epsilon}({\bf k},z)=1\otimes Y_{W_{2}}^{\epsilon}({\bf
k},z)=1\otimes\frac{1}{\ell}{\bf k}^{\epsilon}(z)=\frac{1}{\ell}{\bf
k}^{\epsilon}(z),$ $\displaystyle Y_{W}^{\epsilon}({\bf d},z)=1\otimes
Y_{W_{2}}^{\epsilon}({\bf d},z)=1\otimes{\bf d}^{\epsilon}(z)={\bf
d}^{\epsilon}(z),$ $\displaystyle Y_{W}^{\epsilon}(e^{n{\bf k}},z)=1\otimes
Y_{W_{2}}^{\epsilon}(e^{n{\bf k}},z)=1\otimes E^{n{\bf k}}(z)=E^{n{\bf
k}}(z),$ $\displaystyle
Y_{W}^{\epsilon}(\omega,z)=Y_{W_{1}}^{\epsilon}(\omega_{\mathfrak{g}},z)\otimes
1+1\otimes
Y^{\epsilon}_{W_{2}}(\omega_{\mathfrak{h}},z)=L_{\mathfrak{g}}^{\epsilon}(z)\otimes
1+1\otimes L_{\mathfrak{h}}^{\epsilon}(z)=L_{\mathfrak{f}}^{\epsilon}(z),$

for $u\in\mathfrak{g},\ n\in\mathbb{Z}$, where we have used the actions (5.3),
(5.7), the fact (5.8), and the conventions (3.9), (3.11).

According to Theorem 5.7, the $\phi_{\epsilon}$-coordinated
$V_{\widehat{\mathfrak{g}}\rtimes\mathsf{Vir}}(\ell,24\mu\ell-2)\otimes
V_{(\mathfrak{h},L)}$-module $W$ is also an irreducible
$\widetilde{\mathfrak{t}}(\mathfrak{g})^{\epsilon}$-module with
$\mathrm{k}_{0}=\ell$ and

$\displaystyle t_{0}^{\epsilon-1}\mathrm{d}_{0}$
$\displaystyle=-\textnormal{{Res}}_{z}z^{-\epsilon}Y_{W}^{\epsilon}(\omega,z)=-L_{\mathfrak{f}}(\epsilon-1)+\delta_{\epsilon,1}\mu\ell,$
$\displaystyle(t_{1}^{m}\otimes u)^{\epsilon}(z)$
$\displaystyle=Y_{W}^{\epsilon}(u\otimes e^{m{\bf
k}},z)=(Y_{W_{1}}^{\epsilon}(u,z)\otimes 1)(1\otimes
Y_{W_{2}}^{\epsilon}(e^{m{\bf k}},z))=u^{\epsilon}(z)E^{m{\bf k}}(z),$
$\displaystyle\mathrm{k}_{1}^{\epsilon}(z)$ $\displaystyle=\ell
Y_{W}^{\epsilon}({\bf k},z)={\bf
k}^{\epsilon}(z),\quad\mathrm{d}_{1}^{\epsilon}(z)=Y_{W}^{\epsilon}({\bf
d},z)={\bf d}^{\epsilon}(z),$ $\displaystyle\mathrm{K}_{n}^{\epsilon}(z)$
$\displaystyle=\frac{\ell}{n}Y_{W}^{\epsilon}(e^{n{\bf
k}},z)=\frac{\ell}{n}E^{n{\bf k}}(z),$
$\displaystyle\mathrm{D}_{n}^{\epsilon}(z)$ $\displaystyle=n\
_{\circ}^{\circ}Y_{W}^{\epsilon}(\omega,z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}-z^{\epsilon}\frac{d}{dz}\
_{\circ}^{\circ}Y_{W}^{\epsilon}({\bf d},z)Y_{W}^{\epsilon}(e^{n{\bf
k}},z)_{\circ}^{\circ}$
$\displaystyle\quad+\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}Y_{W}^{\epsilon}(e^{n{\bf
k}},z)+n^{2}(\mu\ell-1)z^{\epsilon}\frac{d}{dz}Y_{W}^{\epsilon}({\bf
k},z)Y_{W}^{\epsilon}(e^{n{\bf k}},z)$ $\displaystyle=n\
_{\circ}^{\circ}L_{\mathfrak{f}}^{\epsilon}(z)E^{n{\bf
k}}(z)_{\circ}^{\circ}+\frac{1}{2}n\epsilon(\epsilon-1)z^{2\epsilon-2}E^{n{\bf
k}}(z)-z^{\epsilon}\frac{d}{dz}\ _{\circ}^{\circ}{\bf d}^{\epsilon}(z)E^{n{\bf
k}}(z)_{\circ}^{\circ}$
$\displaystyle\quad+n^{2}(\mu-\frac{1}{\ell})z^{\epsilon}\frac{d}{dz}{\bf
k}^{\epsilon}(z)E^{n{\bf k}}(z).$

This finishes the proof of Theorem 3.2.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China
(Nos. 11971396, 11971397, 12131018, 12161141001), and the Fundamental Research
Funds for the Central Universities (No. 20720200067).

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