Simple weak modules for some subalgebras of the Heisenberg vertex algebra and Whittaker vectors

TL;DR

This paper classifies certain simple weak modules over singlet vertex operator algebras, focusing on modules with specific eigenvector properties related to the conformal vector, advancing understanding of their module structure.

Contribution

It provides a classification of simple weak modules for singlet vertex operator algebras with particular eigenvector conditions, revealing new module structures.

Findings

Classification of simple weak modules with specified eigenvector properties

Identification of modules with eigenvalues for conformal vector modes

Enhanced understanding of module structure for singlet vertex algebras

Abstract

Let ${\mathscr M}(p)$ $(p=2,3,\ldots)$ be the singlet vertex operator algebra and $\omega$ its conformal vector. We classify the simple weak ${\mathscr M}(p)$-modules with a non-zero element $u$ such that for some integer $s\geq 2$, $\omega_i u\in{\mathbb C} u$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s} u\in{\mathbb C}^{\times} u$, and $\omega_i u=0$ for all $i>s$.

Equations

$$\displaystyle\omega_{i}u=\left\{\begin{array}[]{ll}\lambda_{i}u,&i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s.\\ 0,&\mbox{if }i>s,\end{array}\right.$$

$$\displaystyle\big{(}\dfrac{1}{4p}\sum_{\begin{subarray}{l}j,k\in{\mathbb{N}}\\ j+k=i-1\end{subarray}}\zeta_{j}\zeta_{k}-\delta_{i,r+1}\dfrac{(p-1)(r+1)}{2p}\zeta_{r}\big{)}_{i=r+1}^{2r+1}$$

$$H_{i}u$$

$$H_{(s+1)(2p-1)/2-1}u$$

$$a_{k}M(j)\subset M(\mathrm{wt}a+j-k-1)$$

$${\omega }_{i_1}\cdots {\omega }_{i_n}u(n\in \mathbb{N},i_1,\dots ,i_n\in \mathbb{Z},u\in U).$$

$${\mathbb{Z}}_{<i}$$

$$\hat{\mathfrak{h}}$$

$$[h\otimes t^m,h^{\prime }\otimes t^n]$$

$${\hat{\mathfrak{h}}}_{\ge 0}$$

$$⟨\alpha ,\alpha ⟩=2p.$$

$$\alpha (i)\cdot u_{\zeta }$$

$$M(1,\zeta )$$

$$\omega$$

$${\omega }_i{u}_{\zeta }$$

$$\displaystyle\big{(}\dfrac{1}{4p}\sum_{\begin{subarray}{l}j,k\in{\mathbb{N}}\\ j+k=i-1\end{subarray}}\zeta_{j}\zeta_{k}-\delta_{i,r+1}\dfrac{(p-1)(r+1)}{2p}\zeta_{r}\big{)}_{i=r+1}^{2r+1}$$

$${\mathbb{C}}^r\times {\mathbb{C}}^{\times }$$

$$({\zeta }_0,\dots ,{\zeta }_r)$$

$${\zeta }_i+{\zeta }_i^{\prime }$$

$$\frac{1}{4p}\sum _{\begin{array}{l}j,k\in \mathbb{N}\\ j+k=i-1\end{array}}{\zeta }_j{\zeta }_k-{\delta }_{i,r+1}\frac{(p-1)(r+1)}{2p}{\zeta }_r$$

$$\displaystyle=\left\{\begin{array}[]{ll}\dfrac{1}{4p}\zeta_{r}^{2}&\mbox{if }i=2r+1,\\ \dfrac{1}{2p}\zeta_{r}\zeta_{i-1-r}+\dfrac{1}{4p}\sum\limits_{j=i-1-r+1}^{r-1}\zeta_{j}\zeta_{i-1-j}&\\ \quad{}-\delta_{i,r+1}\dfrac{(p-1)(r+1)}{2p}\zeta_{r}&\mbox{if }r+1\leq i<2r+1,\end{array}\right.$$

$$L(n)$$

$$L(n+r)$$

$$+\frac{1}{2p}\sum _{\begin{array}{l}i\ge r+1,\\ j\in \mathbb{Z},i+j=n+r\end{array}}\alpha (j)\alpha (i)$$

$$+\frac{1}{4p}\sum _{\begin{array}{l}0\le j\le r-1,\\ i<0,i+j=n+r\end{array}}\alpha (i)\alpha (j)+\frac{1}{4p}\sum _{\begin{array}{l}i,j<0,\\ i+j=n+r\end{array}}\alpha (i)\alpha (j)$$

$$+\frac{p-1}{2p}(-n-r-1)\alpha (n+r)$$

$$L(n+r)u_{\zeta }$$

$$+\frac{1}{4p}\sum _{\begin{array}{l}0\le j\le r-1,\\ n+r-j<0\end{array}}{\zeta }_j\alpha (n+r-j)u_{\zeta }+\frac{1}{4p}\sum _{\begin{array}{l}i,j<0,\\ i+j=n+r\end{array}}\alpha (i)\alpha (j)u_{\zeta }$$

$$+\frac{p-1}{2p}(-n-r-1)\alpha (n+r)u_{\zeta }.$$

$$[L(j),\alpha (i)]$$

$$=-i\alpha (i+j)+(-2p+2)\left(\genfrac{}{}{0ex}{}{j+1}{2}\right)1_{i+j-1,-1}.$$

$$\alpha (i)L(j_1)\cdots L(j_m)$$

$$\in \sum _{l\ge 0}\sum _{1\le k_1<\cdots <k_l\le m}\mathbb{C}L(j_{k_1})\dots L(j_{k_l})$$

$$+\sum _{l\ge 0}\sum _{1\le k_1<\cdots <k_l\le m}\sum _{\begin{array}{l}T\subset \{1,\dots ,m\}\\ T\cap \{k_1,\dots ,k_l\}=\varnothing \end{array}}\mathbb{C}L(j_{k_1})\dots L(j_{k_l})\alpha (i+\sum _{t\in T}{j}_t).$$

$$S_j$$

Full text

Mathematics Subject Classification. 17B69

Key Words. vertex operator algebra, weak module, Whittaker vector.

1 Introduction

The singlet vertex operator algebra $\mathcal{M}({p})$ $(p=2,3,\ldots)$ is a subalgebra of the Heisenberg vertex algebra $M(1)$ of rank $1$ with two generators, and is known as ${\mathscr{W}}(2,2p-1)$-algebra in the physics literature (cf. [7, Section V], [16]). Since $\mathcal{M}({p})$ admits infinitely many non-isomorphic simple (irreducible) modules, it is non-$C_{2}$-cofinite and non-rational. The vertex operator algebra $\mathcal{M}({p})$ has been studied from various perspectives (cf. [1],[3],[4],[8],[9],[10],[25],[26]), partly because of its connection to the triplet vertex algebra. For the representation of $\mathcal{M}({p})$, Adamović [1] classifies the simple $\mathcal{M}({p})$-modules. The notation $\overline{M(1)}$ is used there instead of $\mathcal{M}({p})$. So, one of the next task is to investigate simple weak $\mathcal{M}({p})$-modules. The purpose of this paper is to classify the simple weak $\mathcal{M}({p})$-modules with a Whittaker vector for $\omega$ in the sense of [24, (1.1)].

Whittaker modules (Whittaker vectors) are non-weight modules defined over various Lie algebras, first appeared in [5] for $sl_{2}$. Whittaker modules for any finite dimensional complex simple Lie algebra are systematically studied in [17] and applied to the study of the Toda lattice in [18]. Results in [18] are generalized to affine Lie algebras or quantum groups in [12] and [23]. Whittaker modules are also studied for the Virasoro algebra in [22], [21] and [13], and for the affine Kac–Moody algebra $A_{1}^{(1)}$ in [2]. Whittaker modules for the Virasoro algebra also appear in the study of two-dimensional conformal field theory in physics(cf.[15]). Whittaker modules for a general vertex operator algebra $V$ are not defined, however, we note that for the conformal vector (the Virasoro element) $\omega$ of $V$, $\omega_{n+1}=L(n),n\in{\mathbb{Z}}$ satisfy the Virasoro algebra relations (cf. [19, (1.3.4)]). Thus, based on the definition of Whittaker vectors for the Virasoro algebra in [22] and [21], I introduced the following notion in [24]: for a weak $V$-module $M$, a non-zero element $u$ of $M$ is called a Whittaker vector for $\omega$ if there exists an integer $s$ with $s\geq 2$ and ${\lambda}=(\lambda_{\lfloor s/2\rfloor+1},\lambda_{\lfloor s/2\rfloor+2},\ldots,\lambda_{s})\in{\mathbb{C}}^{s-\lfloor s/2\rfloor}$ with $\lambda_{s}\neq 0$ such that

[TABLE]

where $\lfloor s/2\rfloor=\max\{i\in{\mathbb{Z}}\ |\ i\leq s/2\}$. We call ${\lambda}$ the type of $u$. We note that if we regard a weak $V$-module $M$ as a module for the Virasoro algebra, then the Whittaker vectors in $M$ for $\omega$ coincide with the Whittaker vectors in $M$ for the Virasoro algebra. It is well known that for each non-negative integer $r$ and $\zeta=(\zeta_{0},\ldots,\zeta_{r})\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$, the Heisenberg vertex algebra $M(1)$ has a simple weak module $M(1,\zeta)$ (See (2.11)), which is also a Whittaker module for the Heisenberg algebra. Let $\omega$ be the conformal vector of $\mathcal{M}({p})$. As we shall see later in Corollary 2.2, for every $\zeta\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$ the weak $\mathcal{M}({p})$-module $M(1,\zeta)$ is simple and contains a Whittaker vector $u_{\zeta}$ of type

[TABLE]

for $\omega$. The following is the main result of this paper, which implies that the converse is true:

Theorem 1.1**.**

The set $\{M(1,{\zeta})\ |\ r\in{\mathbb{Z}}_{\geq 1},{\zeta}\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}\}$ is a complete set of representatives of equivalence classes of simple weak $\mathcal{M}({p})$-modules with a Whittaker vector for $\omega$.

It is worth mentioning that the simple weak $M(1)^{+}$-modules with a Whittaker vector for its conformal vector are already classified in [24, Theorem 1.1], where $M(1)^{+}$ is the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. Applying a slight modification of the methods in [24] to $\mathcal{M}({p})$, we can show Theorem 1.1. Let us explain the basic idea briefly. It is shown in [1, Theorem 3.2] that $\mathcal{M}({p})$ is generated by the conformal vector $\omega$ and homogeneous $H\in\mathcal{M}({p})$ of weight $2p-1$. Let $M$ be a weak $\mathcal{M}({p})$-module generated by a Whittaker vector $u$ of type $\lambda=(\lambda_{i})_{i=\lfloor s/2\rfloor+1}^{s}$ for $\omega$. Fist we find two relations for $\omega$ and $H$ in $\mathcal{M}({p})$ (See (2.48) and (2.70)). Using these relations, we have that $s$ is an odd integer and

[TABLE]

and that $H_{(s+1)(2p-1)/2-1}u\in{\mathbb{C}}u$ if and only if $M$ is simple (See Lemma 3.1 (4) and (5)). In this case

[TABLE]

where $q$ is a square root of $(4p)^{2p-1}\lambda_{s}^{2p-1}/((2p-1)!)^{2}$, $M$ is generated by $u$ as a module for the Virasoro algebra associated to $\omega$, and the actions of $H$ on $M$ are uniquely determined by the scalar $q$ and the actions of $\omega$ on $M$. In particular, if $M$ is simple, then $M$ is a Whittaker module for the Virasoro algebra in the sense of [21, 22]. Since any Whittaker module for the Virasoro algebra is uniquely determined by its type by [22, Proposition 3.2] and [13, Theorem 2.3], every simple weak $\mathcal{M}({p})$-module with a Whittaker vector $u$ for $\omega$ is uniquely determined by the type of $u$ for $\omega$ and $q\in{\mathbb{C}}$ in (1.6). Thus, the theorem follows from (1.4) and the computation of $H_{(s+1)(2p-1)/2-1}u_{\zeta}$ in $M(1,\zeta)$.

The organization of the paper is as follows. In Section 2 we recall some basic properties of the Heisenberg vertex algebra $M(1)$, the singlet vertex algebra $\mathcal{M}({p})$, and the weak $M(1)$-modules $M(1,\zeta)$ where $r\in{\mathbb{Z}}_{>0}$ and $\zeta\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$. We show that $M(1,\zeta)$, $r\in{\mathbb{Z}}_{>0}$ and $\zeta\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$, are pairwise non-isomorphic simple weak $\mathcal{M}({p})$-modules. We also find two relations for $\omega$ and $H$ in $\mathcal{M}({p})$. In Section 3 we give a proof of Theorem 1.1.

2 Preliminary

We assume that the reader is familiar with the basic knowledge on vertex algebras as presented in [6, 14, 19].

Throughout this paper, $p$ is an integer with $p\geq 2$, ${\mathbb{N}}$ denotes the set of all non-negative integers, ${\mathbb{C}}^{\times}=\{z\in{\mathbb{C}}\ |\ z\neq 0\}$ and $(V,Y,{\mathbf{1}},\omega)$ is a vertex operator algebra. Recall that $V$ is the underlying vector space, $Y(-,x)$ is the linear map from $V\otimes_{{\mathbb{C}}}V$ to $V(\!({x})\!)$, ${\mathbf{1}}$ is the vacuum vector, and $\omega$ is the conformal vector. A weak $V$-module $M$ (cf. [20, p.157]) is called ${\mathbb{N}}$-graded if $M$ admits a decomposition $M=\oplus_{j=0}^{\infty}M(j)$ such that

[TABLE]

for homogeneous $a\in V$, $j\in{\mathbb{N}}$, and $k\in{\mathbb{Z}}$. For a weak $V$-module $M$ and $U\subset M$, $\langle\omega\rangle U$ denotes the set of linear span of the following elements:

[TABLE]

When $U=\{u\}$, we simply write $\langle\omega\rangle u$ for $\langle\omega\rangle U$. For $i\in{\mathbb{Z}}$, we define

[TABLE]

We also define ${\mathbb{Z}}_{\leq i}$ and ${\mathbb{Z}}_{\geq i}$ similarly.

We recall the vertex operator algebra $M(1)$ associated to the Heisenberg algebra of rank $1$ and some weak $M(1)$-modules. Let ${\mathfrak{h}}$ be a one dimensional vector space equipped with a nondegenerate symmetric bilinear form $\langle-,-\rangle$. Set a Lie algebra

[TABLE]

with the Lie bracket relations

[TABLE]

for $h,h^{\prime}\in{\mathfrak{h}}$ and $m,n\in{\mathbb{Z}}$. For $h\in{\mathfrak{h}}$ and $n\in{\mathbb{Z}}$, $h(n)$ denotes $h\otimes t^{n}\in\widehat{H}$. Set two Lie subalgebras of $\widehat{{\mathfrak{h}}}$:

[TABLE]

We take $\alpha\in{\mathfrak{h}}$ such that

[TABLE]

For a non-negative integer $r$ and an $r+1$-tuple ${\zeta}=(\zeta_{0},\ldots,\zeta_{r})\in{\mathbb{C}}^{r+1}$, ${\mathbb{C}}u_{{\zeta}}$ denotes a one dimensional $\widehat{{\mathfrak{h}}}_{{\geq 0}}$-module uniquely determined by

[TABLE]

We take an $\widehat{{\mathfrak{h}}}$-module

[TABLE]

where ${\mathscr{U}}({\mathfrak{g}})$ is the universal enveloping algebra of a Lie algebra ${\mathfrak{g}}$. Then, $M(1)=M(1,(0))$ has a vertex algebra structure and $M(1,{\zeta})$ is a simple weak $M(1)$-module for any ${\zeta}\in{\mathbb{C}}^{r+1}$. The vertex operator algebra $M(1)$ is called the vertex operator algebra associated to the Heisenberg algebra $\oplus_{0\neq n\in{\mathbb{Z}}}{\mathfrak{h}}\otimes t^{n}\oplus{\mathbb{C}}K$. If $r=0$, then $M(1,(\lambda_{0}))$ is a simple $M(1)$-module. We define

[TABLE]

Then $\langle\omega\rangle{\mathbf{1}}$ is the simple Virasoro vertex operator algebra with central charge $1-6(p-1)^{2}/p$. Since for $i\in{\mathbb{Z}}$

[TABLE]

$u_{{\zeta}}$ is a Whittaker vector of type

[TABLE]

for $\omega$. If $r\geq 1$ and $\zeta_{r}\neq 0$, then $u$ is an eigenvector for $\omega_{2r+1}$ with eigenvalue $\zeta_{r}^{2}/(4p)$ and hence $M(1,{\zeta})$ is not an ${\mathbb{N}}$-graded weak $M(1)$-module by (2.1). We also note that the map

[TABLE]

is onto and the images of ${\zeta}=(\zeta_{i})_{i=0}^{r},\zeta^{\prime}=(\zeta^{\prime}_{i})_{i=0}^{r}\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$ under this map are equal if and only if

[TABLE]

since

[TABLE]

for $i=r+1,r+2,\ldots,2r+1$. In particular, this map is two-to-one. Regarding the weak $M(1)$-module $M(1,\zeta)$ as a weak $\langle\omega\rangle{\mathbf{1}}$-module, we have the following results:

Lemma 2.1**.**

1. (1)

For every $\zeta\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$, $M(1,\zeta)$ is a simple weak $\langle\omega\rangle{\mathbf{1}}$-module. 2. (2)

For $\zeta=(\zeta_{i})_{i=0}^{r}\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$ and $\zeta=(\zeta^{\prime}_{i})_{i=0}^{r^{\prime}}\in{\mathbb{C}}^{r^{\prime}}\times{\mathbb{C}}^{\times}$, $M(1,{\zeta})\cong M(1,{\zeta}^{\prime})$ as weak $\langle\omega\rangle{\mathbf{1}}$-modules if and only if $r=r^{\prime}$ and (2.18) holds.

Proof.

1. (1)

For convenience, we define

[TABLE]

for $n\in{\mathbb{Z}}$. For $n\in{\mathbb{Z}}_{<0}$, we have

[TABLE]

and therefore

[TABLE]

Since $L(1)\alpha(-1)=(-2p+2){\mathbf{1}}$, for $i,j\in{\mathbb{Z}}$ we have

[TABLE]

Thus, for $i\in{\mathbb{Z}}$, $m\in{\mathbb{Z}}_{\geq 0}$ and $j_{1},\ldots,j_{m}\in{\mathbb{Z}}$, an inductive argument on $m$ shows that

[TABLE]

For $j\in{\mathbb{Z}}$, we define

[TABLE]

We note that $S_{j}\subset S_{j-1}$ for all $j\in{\mathbb{Z}}$. We shall show that

[TABLE]

for all $n\in{\mathbb{Z}}$. If $n\geq 0$, then (2.32) follows from (2.10). For $n<0$, we shall show (2.32) by induction on $n=0,-1,-2,\ldots$. We have already shown that (2.32) holds for $n=0$. Let $n\in{\mathbb{Z}}_{<0}$. Let $i,j\in{\mathbb{Z}}_{<0}$ with $i+j=n+r$. We have $\alpha(j)u_{\zeta}\in S_{j+r}$ and $\alpha(i+k)u_{\zeta}\in S_{i+k+r}$ for $k\geq n-i+1$ by the induction hypothesis. Since $j+r=n+2r-i\geq n+1+r$ and $i+j+r=n+2r\geq n+1+r$, it follows from (1) that

[TABLE]

Thus, (2.32) follows from (1) and the induction hypothesis. It follows from (1) and (2.32) that $M(1,\zeta)=\langle\omega\rangle u_{\zeta}$ and therefore $M(1,\zeta)$ is simple by (2.14), [22, Corollary 4.2] and [21, Theorem 7]. 2. (2)

It follows from (1) that $M(1,\zeta)=\langle\omega\rangle u_{\zeta}$. Since any Whittaker vector of the type given in (2.14) for $\omega$ in $M(1,\zeta)$ is a non-zero scalar multiple of $u_{\zeta}$ by [22, Proposition 3.2] and [13, Theorem 2.3], (2) follows from (2.14)–(2.18).

∎

Let

[TABLE]

be the generalized vertex algebra associated to the lattice ${\mathbb{Z}}(\alpha/p)$ (See [11]). We define a linear map

[TABLE]

and denote by $\mathcal{M}({p})$ the kernel of $\widetilde{Q}$ in $M(1)$:

[TABLE]

For $i\in{\mathbb{N}}$, we also denote by $S_{i}(\alpha)$ the $i$-th coefficient of $t^{i}$ in the expansion of $\exp(\sum_{j=1}^{\infty}\alpha(-j)t^{j}/j)$:

[TABLE]

It is shown in [1, Theorem 3.2] that $\mathcal{M}({p})$ is generated by $\omega$ and homogeneous

[TABLE]

of weight $2p-1$. A direct computation shows that

[TABLE]

and, in particular,

[TABLE]

It is shown in [1, p.122] that $H$ is a primary vector of weight $2p-1$ for $\omega$, so $\omega$ and $H$ satisfies

[TABLE]

The following result is a direct consequence of Lemma 2.1 and (2.42):

Corollary 2.2**.**

The weak $\mathcal{M}({p})$-modules $M(1,\zeta)$,$r\in{\mathbb{Z}}_{\geq 1}$ and $\zeta\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$, are simple and pairwise non-isomorphic. In particular, for any odd integer $s\geq 3$, $\lambda=(\lambda_{(s+1)/2},\lambda_{(s+1)/2+1},\ldots,\lambda_{s})\in{\mathbb{C}}^{(s-1)/2}\times{\mathbb{C}}^{\times}$, and $q\in{\mathbb{C}}$ such that

[TABLE]

there exists a weak $\mathcal{M}({p})$-module with a Whittaker vector $u$ of type ${\lambda}$ for $\omega$ such that

[TABLE]

Proof.

It follows from Lemma 2.1 (1) that $M(1,\zeta)$ is a simple weak $\mathcal{M}({p})$-module for every $r\in{\mathbb{Z}}_{\geq 1}$ and $\zeta\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$. Let $\zeta\in{\mathbb{C}}^{r}\times{\mathbb{C}}^{\times}$ and $\zeta^{\prime}\in{\mathbb{C}}^{r^{\prime}}\times{\mathbb{C}}^{\times}$ such that $M(1,\zeta)\cong M(1,\zeta^{\prime})$ as weak $\mathcal{M}({p})$-modules. Since any Whittaker vector of the type given in (2.14) for $\omega$ in $M(1,\zeta)=\langle\omega\rangle u_{\zeta}$ is a non-zero scalar multiple of $u_{\zeta}$ by [22, Proposition 3.2] and [13, Theorem 2.3], Lemma 2.1 (2) shows that $r=r^{\prime}$ and (2.18) holds. Therefore, $\zeta=\zeta^{\prime}$ by (2.42). The equality (2.47) follows from (2) and (2.42). ∎

The following relation for $\omega$ and $H$ will be used in Lemma 3.1.

Lemma 2.3**.**

[TABLE]

Proof.

For

[TABLE]

we have

[TABLE]

Using (2.50) repeatedly, we have

[TABLE]

Thus

[TABLE]

and therefore

[TABLE]

Since $\omega_{0}^{i}H=i!H_{-1-i}{\mathbf{1}}$ for $i\in{\mathbb{Z}}_{\geq 0}$, the assertion follows from a direct computation. ∎

The same argument as in the proof of [24, Lemma 3.2] shows the following result:

Lemma 2.4**.**

Let $s\in{\mathbb{Z}}$ with $s\geq 2$. Let $M$ be a weak $\langle\omega\rangle{\mathbf{1}}$-module and $u\in M$ such that $\omega_{i}u=0$ for all $i>s$. Let $p,q,i_{1},\ldots,i_{p},i_{p+1},\ldots,i_{q}\in{\mathbb{Z}}$ such that $0\leq p\leq q$, $i_{1},\ldots,i_{p}\geq 0,i_{p+1},\ldots,i_{q}<0$, and $sq\leq i_{1}+\cdots+i_{q}$. If $p<q$ or $(i_{1},\ldots,i_{p})\neq(s,\ldots,s)$, then

[TABLE]

The following results will be used in Lemmas 2.6.

Lemma 2.5**.**

Let $s\in{\mathbb{Z}}$ with $s\geq 2$, $M$ a weak $\mathcal{M}({p})$-module, and $u\in M$ such that $\omega_{i}u=0$ for all $i>s$. Let $q\in{\mathbb{Z}}_{\geq 0}$ and $j_{1},\ldots,j_{q}\in{\mathbb{Z}}_{\leq-1}$, and define

[TABLE]

Let $m\in{\mathbb{Z}}_{\geq(s+1)k/2-1}$.

1. (1)

If $k\neq 2q$ or $m>(s+1)k/2-1$, then

[TABLE] 2. (2)

If $k=2q$, then

[TABLE]

and

[TABLE] 3. (3)

For $i,j\in{\mathbb{Z}}$ with $i+j=(s+1)(2p-1)-2$, we have

[TABLE]

Proof.

It follows from (2.55) that $2q\leq k$. Thus, for $i_{1},\ldots,i_{q}\in{\mathbb{Z}}$ such that $i_{1}+\cdots+i_{q}=j_{1}+\cdots+j_{q}+m+1$, we have

[TABLE]

It follows from Lemma 2.4 that if $(i_{1},\ldots,i_{q})\neq(s,\ldots,s)$, then

[TABLE]

Here, for $i_{1},\ldots,i_{q}\in{\mathbb{Z}}$ we define $\begin{subarray}{c}\circ\\ \circ\end{subarray}\omega_{i_{1}}\cdots\omega_{i_{q}}\begin{subarray}{c}\circ\\ \circ\end{subarray}$ inductively by

[TABLE]

1. (1)

If $k>2q$ or $m>(s+1)k/2-1$, then

[TABLE]

by (2). Thus, (2.56) follows from (2.61). 2. (2)

Suppose $k=2q$. It follows from (2.55) that $j_{1}=\cdots=j_{k}=-1$. It follows from (2) and (2.61) that

[TABLE] 3. (3)

It is shown in [1, Lemma 3.2] that $H_{k}H\in\langle\omega\rangle{\mathbf{1}}$ for all $k\geq-2p$. Thus, for $k=0,1,\ldots$ since

[TABLE]

we have

[TABLE]

by (1) and therefore

[TABLE]

∎

By Lemma 2.5, we have the following relation for $\omega$ and $H$, which will be used in Lemma 3.1 together with (2.48).

Lemma 2.6**.**

[TABLE]

where $a$ is an element of

[TABLE]

Proof.

By [1, Lemma 3.2], we can write

[TABLE]

where $c\in{\mathbb{C}}$ and $a$ is an element of (2.74). By (2.42) and Lemma 2.5 (3) with $s=2r+1$, we have

[TABLE]

By Lemma 2.5 (1), we also have

[TABLE]

It follows from (2.13) and Lemma 2.5 (2) that

[TABLE]

By (2.42) and (2), we have $c=(4p)^{2p-1}/((2p-1)!)^{2}$. ∎

Remark 2.7**.**

Lemma 2.6 also follows from the proof of [1, Lemma 6.1].

3 Weak $\mathcal{M}({p})$-modules with Whittaker vectors.

In this section, we will show Theorem 1.1. The following is a key result to show Theorem 1.1.

Lemma 3.1**.**

Let $s,t\in{\mathbb{Z}}$ with $2\leq t\leq s$, ${\lambda}=(\lambda_{t},\ldots,\lambda_{s})\in{\mathbb{C}}^{s-t+1}$ with $\lambda_{s}\neq 0$, and $q\in{\mathbb{C}}$ such that

[TABLE]

and define

[TABLE]

Let $M$ be a weak $\mathcal{M}({p})$-module and $u\in M$ a non-zero element such that $\omega_{i}u=\lambda_{i}u$ for $i=t,\ldots,s$ and $\omega_{i}u=0$ for all $i>s$.

1. (1)

The integer $s$ is odd and

[TABLE]

Moreover, the submodule of $M$ generated by $u$ is equal to $\langle\omega\rangle({\mathbb{C}}u+{\mathbb{C}}H_{m}u)$. 2. (2)

If $H_{m}u\not\in{\mathbb{C}}u$, then two nonzero elements $v_{\pm}=H_{m}u\pm qu\in M$ satisfy

[TABLE]

and

[TABLE] 3. (3)

Suppose that $H_{m}u\in{\mathbb{C}}u$. Then

[TABLE]

and

[TABLE]

where $\mu_{i}$ is a polynomial in $\lambda_{s-m+i},\lambda_{s-m+i+1},\ldots,\lambda_{s-1},\lambda_{s}^{-1}$ and $q$ for each $i=m-s+t,m-s+t+1,\ldots,m$. Moreover, the submodule of $M$ generated by $u$ is equal to $\langle\omega\rangle u$. 4. (4)

If $t=(s+1)/2$ and $H_{m}u\in{\mathbb{C}}u$, then the submodule of $M$ generated by $u$ is simple. 5. (5)

If $t=(s+1)/2$ and $M$ is simple, then $H_{m}u\in{\mathbb{C}}u$.

Proof.

1. (1)

For $n\in{\mathbb{Z}}$, by (2.48) we have

[TABLE]

Here $\begin{subarray}{c}\circ\\ \circ\end{subarray}\omega_{i}H_{j}\begin{subarray}{c}\circ\\ \circ\end{subarray}$ is defined to be

[TABLE]

Since $\mathcal{M}({p})$ is simple by [3, Theorem 4.3], it follows from [11, Proposition 11.9] that there exists $m^{\prime}\in{\mathbb{Z}}$ such that $H_{m^{\prime}}u\neq 0$ and $H_{j}u=0$ for all $j>m^{\prime}$. It follows from (1) that $s$ is an odd integer and $m^{\prime}=(s+1)(2p-1)/2-1=m$. By (2.43) and (1), an inductive argument shows that

[TABLE]

and therefore the submodule of $M$ generated by $u$ is equal to $\langle\omega\rangle({\mathbb{C}}u+{\mathbb{C}}H_{m}u)$. By using Lemma 2.6, the same argument as in the proof of Lemma 2.6 shows that

[TABLE] 2. (2)

The assertion follows from (2.43) and (1). 3. (3)

Equation (3.8) follows from (1). By (2.43) and (2.48), we have

[TABLE]

Thus, an inductive argument on $t=s,s-1,\ldots$ shows (3.9). It follows from (1) that the submodule of $M$ generated by $u$ is equal to $\langle\omega\rangle u$. 4. (4)

Since $u$ is a Whittaker vector of type $(\lambda_{i})_{i=(s+1)/2}^{s}$ for $\omega$, the assertion follows from (3), [22, Corollary 4.2], and [21, Theorem 7]. 5. (5)

Suppose $H_{m}u\not\in{\mathbb{C}}$. Let $v_{\pm}\in M$ be two nonzero vectors defined in (2). We note that $v_{\pm}$ are Whittaker vectors for $\omega$. Since $M$ is simple we have $M=\langle\omega\rangle v_{+}=\langle\omega\rangle v_{-}$ by (3). Since $v_{\pm}$ are Whittaker vectors of type $(\lambda_{i})_{i=(s+1)/2}^{s}$ for $\omega$ in $M$, $v_{+}$ is a non-zero scalar multiple of $v_{-}$ by [22, Proposition 3.2] and [13, Theorem 2.3]. This contradicts to (3.7).

∎

Now we give a proof of Theorem 1.1.

(Proof of Theorem 1.1).

The following argument is almost the same as in the proof of [24, Theorem 1.1]. Let $s$ be an odd integer with $s\geq 2$, $\lambda=(\lambda_{(s+1)/2},\lambda_{(s+1)/2+1},\ldots,\lambda_{s})\in{\mathbb{C}}^{(s-1)/2}\times{\mathbb{C}}^{\times}$, $m\in{\mathbb{Z}}$ defined by (3.2), and $q\in{\mathbb{C}}$ which satisfies (2.44). Taking a quotient space of the tensor algebra of $\mathcal{M}({p})\otimes{\mathbb{C}}[t,t^{-1}]$ by the two sided ideal generated by the Borcherds identity, $H\otimes t^{i}\ (i>m)$, $H\otimes t^{m}-q({\mathbf{1}}\otimes 1)$, $\omega\otimes t^{i}\ (i>s)$, and $\omega\otimes t^{i}-\lambda_{i}({\mathbf{1}}\otimes 1)\ ((s+1)/2\leq i\leq s)$, we obtain a pair $(N(\lambda,q),v(\lambda,q))$ of a weak $\mathcal{M}({p})$-module $N(\lambda,q)$ and a Whittaker vector $v(\lambda,q)\in N(\lambda,q)$ of type ${\lambda}$ for $\omega$ such that $Y_{N(\lambda,q)}(H,x)v(\lambda,q)\in x^{-m-1}N(\lambda,q)[[x]]$ and $H_{m}v(\lambda,q)=qv(\lambda,q)$ with the following universal property: for any pair $(U,u)$ of a weak $\mathcal{M}({p})$-module $U$ and a Whittaker vector $u\in U$ of type ${\lambda}$ such that $Y_{U}(H,x)u\in x^{-m-1}U[[x]]$ and $H_{m}u=qu$, there exists a unique weak $\mathcal{M}({p})$-module homomorphism $N(\lambda,q)\rightarrow U$ which maps $v(\lambda,q)$ to $u$. The weak module $N(\lambda,q)$ is not zero by Corollary 2.2 and simple by Lemma 3.1 (4).

Let $M$ be a simple weak $\mathcal{M}({p})$-module with a Whittaker vector $u$ of type $\lambda$ for $\omega$. By Lemma 3.1 (3) and (5), $M$ is isomorphic to a quotient weak module of $N(\lambda,q)$ or $N(\lambda,-q)$ and, moreover, $N(\lambda,q)\cong M$ or $N(\lambda,-q)\cong M$ since $N(\lambda,\pm q)$ are simple. Thus, $M$ is isomorphic to one of the weak $\mathcal{M}({p})$-modules listed in Theorem 1.1 by Corollary 2.2. The proof is complete. ∎