The category of weight modules for symplectic oscillator Lie algebras

TL;DR

This paper classifies simple weight modules with finite-dimensional weight spaces over symplectic oscillator Lie algebras, establishing an equivalence with modules over symplectic Lie algebras when a certain parameter is non-zero.

Contribution

It provides a classification of simple weight modules for symplectic oscillator Lie algebras and links their category to that of symplectic Lie algebras via an equivalence.

Findings

Equivalence between categories of modules when z ≠ 0

Classification of simple weight modules with finite-dimensional weight spaces

Use of localization and highest weight techniques

Abstract

The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $\mathfrak{g}_n$. When $\dot z\neq 0$, it is shown that there is an equivalence between the full subcategory $\mathcal{O}_{\mathfrak{g}_n}[\dot z]$ of the BGG category $\mathcal{O}_{\mathfrak{g}_n}$ for $\mathfrak{g}_n$ and the BGG category $\mathcal{O}_{\mathfrak{sp}_{2n}}$ for $\mathfrak{sp}_{2n}$. Then using the technique of localization and the structure of generalized highest weight modules, we also give the classification of simple weight modules over $\mathfrak{g}_n$ with finite-dimensional weight spaces.

Equations

$$[e_i,e_{n+i}]=z,[z,H_n]=0.$$

$$\mathfrak{g}={\mathfrak{sp}}_{2n}⋉H_n.$$

$$[X,v]=Xv,[X,z]=0,$$

$$S=\left(\begin{array}[]{cc}0&I_{n}\\ -I_{n}&0\\ \end{array}\right).$$

$$\left(\begin{array}[]{cc}A&B\\ C&-A^{T}\\ \end{array}\right)$$

$${\mathfrak{h}}_n=\mathbb{C}{h}_1\oplus \cdots \oplus \mathbb{C}{h}_n\oplus \mathbb{C}z$$

$$\Delta =\{\pm {ϵ}_i\pm {ϵ}_j,\pm {ϵ}_j|1\le i,j\le n\}\setminus \{0\}={\Delta }_{{\mathfrak{sp}}_{2n}}\cup \{\pm {ϵ}_j|1\le j\le n\}$$

$${\Delta }_{+}=\{{ϵ}_i-{ϵ}_j,{ϵ}_k+{ϵ}_l,{ϵ}_k,\mid 1\le i<j\le n,1\le k,l\le n\}.$$

$$\begin{array}[]{r c l | c}\lx@intercol\hfil\text{Root vector}\hfil\lx@intercol&\text{Root}\\ \hline\cr X_{\epsilon_{i}}&:=&e_{i}&\epsilon_{i}\\ X_{-\epsilon_{i}}&:=&e_{n+i}&-\epsilon_{i}\\ X_{\epsilon_{i}+\epsilon_{j}}&:=&e_{i,n+j}+e_{j,n+i}&\epsilon_{i}+\epsilon_{j}\\ X_{-\epsilon_{i}-\epsilon_{j}}&:=&e_{n+i,j}+e_{n+j,i}&-\epsilon_{i}-\epsilon_{j}\\ X_{\epsilon_{i}-\epsilon_{j}}&:=&e_{i,j}-e_{n+j,n+i}&\epsilon_{i}-\epsilon_{j}\\ \end{array}$$

$$B:=\{X_{\alpha }|\alpha \in \Delta \}\cup \{h_i,z|1\le i\le n\}.$$

$${\mathfrak{n}}_{\pm }:=\underset{\alpha \in {\Delta }_{\pm }}{⨁}{\mathfrak{g}}_{\alpha }.$$

$${\mathfrak{g}}_n={\mathfrak{n}}_{-}\oplus {\mathfrak{h}}_n\oplus {\mathfrak{n}}_{+}$$

$$U({\mathfrak{g}}_n)\cong U({\mathfrak{n}}_{-})\otimes U({\mathfrak{h}}_n)\otimes U({\mathfrak{n}}_{+}).$$

$$M={\oplus }_{\lambda \in {\mathfrak{h}}_n^{\ast }}{M}_{\lambda },$$

$$\mathrm{Supp}(M):=\{\lambda \in {\mathfrak{h}}_n^{\ast }\mid M_{\lambda }\ne 0\}.$$

$${\mathfrak{n}}_{+}{\mathbb{C}}_{\lambda ,\dot{z}}=0,z\cdot v_{\lambda }=\dot{z}{v}_{\lambda },h\cdot v_{\lambda }=\lambda (h)v_{\lambda }\text{ for all }h\in \mathfrak{h}.$$

$$M(\dot{z},\lambda ):={\mathrm{Ind}}_{\mathfrak{b}}^{{\mathfrak{g}}_n}{\mathbb{C}}_{\lambda ,\dot{z}}\cong U({\mathfrak{g}}_n)\underset{U(\mathfrak{b})}{⨂}{\mathbb{C}}_{\lambda ,\dot{z}}.$$

$$0=M_0\subset M_1\subset \cdots \subset M_m=M$$

$$[{\partial }_i,{\partial }_j]=[t_i,t_j]=0,[{\partial }_i,t_j]={\delta }_{i,j},1\le i,j\le n.$$

$$X_{{ϵ}_i-{ϵ}_j}\mapsto X_{{ϵ}_i-{ϵ}_j}\otimes 1+1\otimes t_i{\partial }_j,(i\ne j),$$

$$X_{{ϵ}_i+{ϵ}_j}\mapsto X_{{ϵ}_i+{ϵ}_j}\otimes 1+1\otimes t_i{t}_j,$$

$$X_{-{ϵ}_i-{ϵ}_j}\mapsto X_{-{ϵ}_i-{ϵ}_j}\otimes 1-1\otimes {\partial }_i{\partial }_j,$$

$$h_i\mapsto h_i\otimes 1+1\otimes t_i{\partial }_i+\frac{1}{2},$$

$$X_{{ϵ}_i}\mapsto 1\otimes \sqrt{\dot{z}}{t}_i,$$

$$X_{-{ϵ}_i}\mapsto -1\otimes \sqrt{\dot{z}}{\partial }_i,$$

$$X_{{ϵ}_i-{ϵ}_j}\mapsto t_i{\partial }_j,(i\ne j),$$

$$X_{{ϵ}_i+{ϵ}_j}\mapsto t_i{t}_j,$$

$$X_{-{ϵ}_i-{ϵ}_j}\mapsto -{\partial }_i{\partial }_j,$$

$$h_i\mapsto t_i{\partial }_i+\frac{1}{2},$$

$$X_{{ϵ}_i}\mapsto \sqrt{\dot{z}}{t}_i,$$

$$X_{-{ϵ}_i}\mapsto -\sqrt{\dot{z}}{\partial }_i,$$

$$X_{{ϵ}_i-{ϵ}_j}\mapsto X_{{ϵ}_i-{ϵ}_j}\otimes 1+1\otimes t_i{\partial }_j,(i\ne j),$$

$$X_{{ϵ}_i+{ϵ}_j}\mapsto X_{{ϵ}_i+{ϵ}_j}\otimes 1+1\otimes t_i{t}_j,$$

$$X_{-{ϵ}_i-{ϵ}_j}\mapsto X_{-{ϵ}_i-{ϵ}_j}\otimes 1-1\otimes {\partial }_i{\partial }_j,$$

$$h_i\mapsto h_i\otimes 1+1\otimes t_i{\partial }_i+\frac{1}{2},$$

$$X_{{ϵ}_i}\mapsto \sqrt{\dot{z}}{t}_i,X_{-{ϵ}_i}\mapsto -\sqrt{\dot{z}}{\partial }_i,\forall 1\le i\le n.$$

$$U(\mathfrak{sp}(2n))\otimes {\mathfrak{D}}_n={\iota }_1(U({\mathfrak{sp}}_{2n}))\otimes {\iota }_2(U(H_n)/⟨z-\dot{z}⟩),$$

Full text

1. Introduction

Many important Lie algebras in mathematical physics are finite dimensional but not semi-simple, such as Schrödinger algebras [8], conformal Galilei algebras [17, 22], symplectic oscillator Lie algebras [3, 21], Euclidean algebras [20] and so on. Unlike finite dimensional semi-simple Lie algebras, the representation theory of those Lie algebras is still not well developed [19, 25]. In this paper we will establish the representation theory for symplectic oscillator Lie algebras.

The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_{n}$ is the the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_{n}$. This algebra is also called Jacobi Lie algebra in the literature, see [3, 2]. The universal enveloping algebra of $\mathfrak{g}_{n}$ is an infinitesimal Hecke algebra, see [13]. In mathematics, the Jacobi group is the semidirect product of the symplectic group and the Heisenberg group [3]. The Jacobi group is an important object in connection with quantum mechanics, geometric quantization, optics. The Jacobi groups were used to describe the “squeezed coherent states” of quantum optics [1]. In Number Theory, automorphic forms on the Jacobi group are called Jacobi forms which has close relationship with the modular forms, see [3, 12].

In this paper, we study the BGG category $\mathcal{O}_{\mathfrak{g}_{n}}$ for $\mathfrak{g}_{n}$ and give the classification of simple Harish-Chandra modules for $\mathfrak{g}_{n}$.

A classification of simple Harish-Chandra modules over the following Lie algebras have been obtained, the Virasoro algebra [24], finite dimensional simple Lie algebras [14, 25], conformal Galilei algebras [22], the twisted Heisenberg Virasoro algebra [23], and affine Kac-Moody algebras [16, 10] (the zero central charge case was claimed in [10]).

The paper is organized as follows. In Section 2 we provide the related definitions and notations.

In Section 3 we first we establish an isomorphism from $U(\mathfrak{g}_{n})/\langle z-\dot{z}\rangle$ to the associative algebra $U(\mathfrak{sp}_{2n})\otimes\mathfrak{D}_{n}$ for $\dot{z}\neq 0$ (Proposition 3), and show that any module in $\mathcal{O}_{\mathfrak{g}_{n}}[\dot{z}]$ with $\dot{z}\neq 0$ is completely reducible over $H_{n}$ (Lemma 5). Then we prove that the full subcategory $\mathcal{O}_{\mathfrak{g}_{n}}[\dot{z}]$ of $\mathcal{O}_{\mathfrak{g}_{n}}$ for $\mathfrak{g}_{n}$ with nonzero $\dot{z}$ is equivalent to the BGG category $\mathcal{O}_{\mathfrak{sp}_{2n}}$ for $\mathfrak{sp}_{2n}$ (Theorem 7).

The classification of all simple weight modules with finite dimensional weight spaces for $\mathfrak{g}_{n}$ is obtained in Section 4. Theorem 14 gave all such $\mathfrak{g}_{n}$-modules with $z$ acts trivially, which are actually simple parabolically induced $\mathfrak{sp}_{2n}$-modules and simple cuspidal $\mathfrak{sp}_{2n}$-modules (See [25]). Theorem 15 gave all such $\mathfrak{g}_{n}$-modules with $z$ acts non-trivially, which consists of three classes: cuspidal $\mathfrak{g}_{n}$-modules in (a), parabolically induced $\mathfrak{g}_{n}$-modules in (b), and a third class described in (c). We can see that the third class in Theorem 15 (c) does not appear for finite-dimensional simple Lie algebras. The representation theory of polynomial differential operator algebras, technique of localization and the structure of parabolically induced modules are widely used in our proofs.

Throughout this paper, we denote by $\mathbb{Z}$, $\mathbb{Z}_{+}$, $\mathbb{N}$, $\mathbb{C}$ and $\mathbb{C}^{*}$ the sets of all integers, nonnegative integers, positive integers, complex numbers, and nonzero complex numbers, respectively. For any Lie algebra $\mathfrak{g}$, we denote its universal enveloping algebra by $U(\mathfrak{g})$.

2. Definitions and notations

2.1. The symplectic oscillator Lie algebra $\mathfrak{g}_{n}$

We know that the symplectic Lie algebra $\mathfrak{sp}_{2n}$ has the natural representation on $\mathbb{C}^{2n}$ by left matrix multiplication. Let $\{e_{1},e_{2},\cdots,e_{2n}\}$ be the standard basis of $\mathbb{C}^{2n}$. The Heisenberg Lie algebra $H_{n}=\mathbb{C}^{2n}\oplus\mathbb{C}z$ is the Lie algebra with Lie bracket given by

[TABLE]

Recall that the symplectic oscillator algebra $\mathfrak{g}_{n}$ is the the semidirect product Lie algebra

[TABLE]

Explicitly, the semidirect relations are

[TABLE]

for all $X\in\mathfrak{sp}_{2n},v\in\mathbb{C}^{2n}$. When $n=1$, $\mathfrak{g}_{1}$ is the Schrödinger algebra studied in [8]. The representations of $\mathfrak{g}_{1}$ were studied in [9, 8, 11].

The universal enveloping algebra $U(\mathfrak{g}_{n})$ of the Jacobi Lie algebra $\mathfrak{g}_{n}$ is an infinitesimal Hecke algebra, see Example 4.11 in [13].

2.2. Root space decomposition of $\mathfrak{g}_{n}$

Recall that $\mathfrak{sp}_{2n}$ is the Lie subalgebra of $\mathfrak{gl}_{2n}$ consisting of all $2n\times 2n$-matrices $X$ satisfying $SX=-X^{T}S$ where

[TABLE]

Equivalently, $\mathfrak{sp}_{2n}$ consists of all $2n\times 2n$-matrices with block form

[TABLE]

such that $B=B^{T}$, $C=C^{T}$. Let $e_{ij}$ denote the matrix unit whose $(i,j)$-entry is $1$ and [math] elsewhere. Then $\mathfrak{h}=\text{span}\{h_{i}:=e_{i,i}-e_{n+i,n+i}\mid 1\leq i\leq n\}$ is the standard Cartan subalgebra of $\mathfrak{sp}_{2n}$ and

[TABLE]

is a Cartan subalgebra of $\mathfrak{g}_{n}$. Let $\{\epsilon_{i}\}\subset\mathfrak{h}_{n}^{*}$ be such that $\epsilon_{i}(h_{k})=\delta_{i,k}$ and $\epsilon_{i}(z)=0$. The root system of $\mathfrak{g}_{n}$ is precisely

[TABLE]

where $\Delta_{\mathfrak{sp}_{2n}}$ is the root system of ${\mathfrak{sp}_{2n}}$. The positive root system is

[TABLE]

We list root vectors in $\mathfrak{g}_{n}$ as follows. The indices $i,j$ are integers between $1$ and $n$, with $i\neq j$ when we encounter $\epsilon_{i}-\epsilon_{j}$.

[TABLE]

Then we obtain a basis of $\mathfrak{g}_{n}$ as follows

[TABLE]

Set

[TABLE]

Then the decomposition

[TABLE]

is a triangular decomposition of $\mathfrak{\mathfrak{g}}_{n}$ which tells us that

[TABLE]

The Lie subalgebra $\mathfrak{b}:=\mathfrak{h}_{n}\oplus\mathfrak{n}_{+}$ is a Borel subalgebra of $\mathfrak{g}_{n}$.

2.3. Weight modules

A $\mathfrak{g}_{n}$-module $M$ is called a weight module if $\mathfrak{h}_{n}$ acts diagonally on $M$, i.e.

[TABLE]

where $M_{\lambda}:=\{v\in M\mid hv=\lambda v\}.$ Denote

[TABLE]

A weight module is called a Harish-Chandra module if all its weight spaces are finite dimensional. For a weight module $M$, a weight vector $v\in M_{\lambda}$ is called a highest weight vector if $\mathfrak{n}_{+}v=0$. A module is called a highest weight module if it is generated by a highest weight vector. A module is called to be uniformly bounded if the dimensions of all weight spaces are bounded by a fixed integer.

2.4. Verma modules

For $\lambda\in\mathfrak{h}^{*}$ and $\dot{z}\in\mathbb{C}$ denote by $\mathbb{C}_{\lambda,\dot{z}}$ the one-dimensional $\mathfrak{b}$-module with the generator $v_{\lambda}$ and the action given by

[TABLE]

The Verma module is defined, as usual, as follows:

[TABLE]

For convenience, we also denote $1\otimes v_{\lambda}$ of $M(\dot{z},\lambda)$ simply by $v_{\lambda}$. Let $K(\dot{z},\lambda)$ be the unique maximal proper submodule of $M(\dot{z},\lambda)$. The quotient module $L(\dot{z},\lambda)=M(\dot{z},\lambda)/K(\dot{z},\lambda)$ is the unique simple quotient module of $M(\dot{z},\lambda)$. Similarly, we have the modules $M_{\mathfrak{sp}_{2n}}(\lambda)$ and $L_{\mathfrak{sp}_{2n}}(\lambda)$ for $\mathfrak{sp}_{2n}$.

3. BGG category

In this section, we study the BGG category $\mathcal{O}_{\mathfrak{g}_{n}}$ for $\mathfrak{g}_{n}$.

3.1. The BGG category $\mathcal{O}_{\mathfrak{g}_{n}}$

Definition 1**.**

The BGG category $\mathcal{O}_{\mathfrak{g}_{n}}$ for $\mathfrak{g}_{n}$ is the full subcategory of $\emph{Mod}\,U(\mathfrak{g}_{n})$ (the category of all left $U(\mathfrak{g}_{n})$-modules) whose objects $M$ are the modules satisfying the following three conditions.

1. $($a$)$

$M$* is a finitely generated $U(\mathfrak{g}_{n})$-module.* 2. $($b$)$

$M$* is a weight module.* 3. $($c$)$

$M$* is locally $\mathfrak{n}_{+}$-finite, i.e., for each $v\in M$, the subspace $U(\mathfrak{n}_{+})v$ is finite dimensional.*

Similarly, we have the BGG category $\mathcal{O}_{\mathfrak{sp}_{2n}}$ for $\mathfrak{sp}_{2n}$.

For information on the BGG category $\mathcal{O}$ for semisimple Lie algebras, one can see [4, 19]. By the standard arguments, we can prove that the category $\mathcal{O}_{\mathfrak{g}_{n}}$ has the following property.

Lemma 2**.**

Let $M$ be any object in $\mathcal{O}_{\mathfrak{g}_{n}}$.

1. $($1$)$

The module $M$ has a finite filtration

[TABLE]

with each factor $M_{j}/M_{j-1}$ for $1\leq j\leq m$ being a highest weight module. 2. $($2$)$

Each weight space of $M$ is finite dimensional. 3. $($3$)$

Any simple object in $\mathcal{O}_{\mathfrak{g}_{n}}$ is isomorphic to some $L(\dot{z},\lambda)$ for $\lambda\in\mathfrak{h}^{*},\dot{z}\in\mathbb{C}$.

So highest weight modules are the basic elements of the category $\mathcal{O}_{\mathfrak{g}_{n}}$.

Next we will introduce the Shale-Weil representation for $\mathfrak{g}_{n}$, which is important for our later arguments on the category $\mathcal{O}_{\mathfrak{g}_{n}}$.

3.2. Shale-Weil representation

For $n\in\mathbb{N}$, denote by $\mathfrak{D}_{n}$ the algebra of polynomial differential operators of $\mathbb{C}[t_{1},\cdots,t_{n}]$, called the Weyl algebra of rank $n$. Namely, $\mathfrak{D}_{n}$ is the associative algebra over $\mathbb{C}$ generated by $2n$-indeterminates $t_{1},\dots,t_{n}$, $\partial_{1},\dots,\partial_{n}$ subject to the relations

[TABLE]

A $\mathfrak{D}_{n}$-module $V$ is a weight module if all $t_{i}\partial_{i}$ are semisimple on $V$. For a weight module $V$, a weight vector $v\in V_{\lambda}$ is called a highest weight vector if $\mathbb{C}[t_{1},t_{2},\cdots,t_{n}]v=0$. A $\mathfrak{D}_{n}$-module $V$ is called a highest weight module if it is generated by a highest weight vector.

Proposition 3**.**

For any nonzero scalar $\dot{z}$, we have the associative algebra isomorphism $\phi_{\dot{z}}:\,\,U(\mathfrak{g}_{n})/\langle z-\dot{z}\rangle\rightarrow U(\mathfrak{sp}(2n))\otimes\mathfrak{D}_{n}$ defined by

[TABLE]

where $1\leq i,j\leq n$.

Proof.

We can directly verify that the linear map $f_{\dot{z}}:\,\,\mathfrak{g}_{n}\rightarrow(\mathfrak{D}_{n},[,])$ defined by

[TABLE]

for $1\leq i,j\leq n$, is a Lie algebra homomorphism. (Note that the above embedding for $\mathfrak{sp}_{2n}$ is similar to [6]). Using PBW Theorem we have the surjective associative algebra homomorphism $\theta_{\dot{z}}:\,\,U(\mathfrak{g}_{n})\rightarrow\mathfrak{D}_{n}$ defined by extending $f_{\dot{z}}$.

It is clear that $U(\mathfrak{g}_{n})/\langle z-\dot{z}\rangle=U(\mathfrak{sp}_{2n})\otimes U(H_{n})/\langle z-\dot{z}\rangle$ where the right-hand side is just considered as a vector space tensor product, not tensor product of the two associative algebras.

We have the algebra homomorphism $\iota_{1}:\,\,U(\mathfrak{sp}_{2n})\rightarrow U(\mathfrak{sp}_{2n})\otimes\mathfrak{D}_{n}$ defined by

[TABLE]

where $1\leq i,j\leq n$. Note that $\iota_{1}$ is injective. We also have the algebra isomorphism $\iota_{2}:\,\,U(H_{n})/\langle z-\dot{z}\rangle\rightarrow\mathfrak{D}_{n}$ defined by

[TABLE]

We can eaily see that $\phi_{\dot{z}}$ is a surjective algebra homomorphism, and

[TABLE]

where the right-hand side is just considered as a vector space tensor product. Then the linear map (not considered as algebra homomorphism)

[TABLE]

is a vector space isomorphism. Therefore $\phi_{\dot{z}}$ is an algebra isomorphism. The proof is complete. ∎

Note that the restriction of $\phi_{\dot{z}}:U(H_{n})/\langle z-\dot{z}\rangle\to\mathfrak{D}_{n}$ is an algebra isomorphism for $\dot{z}\neq 0$. Later in this paper will identify these two associative algebras.

For any $U(\mathfrak{sp}_{2n})$-module $V$, $\mathfrak{D}_{n}$-module $N$, via the homomorphism $\phi_{\dot{z}}$, $V\otimes N$ can be viewed as a $\mathfrak{g}_{n}$-module which is denoted by ${V}\otimes_{\dot{z}}{N}$. In particular, the simple $\mathfrak{D}_{n}$-module

[TABLE]

becomes a $\mathfrak{g}_{n}$-module through $\theta_{\dot{z}}$, denoted by $S_{\dot{z}}$, which is isomorphic to the simple highest weight $\mathfrak{g}_{n}$-module $L(\dot{z},-\frac{1}{2}(\epsilon_{1}+\cdots+\epsilon_{n}))$. This module is called the Shale-Weil module over $\mathfrak{g}_{n}$ (see [25]).

3.3. The structure of highest weight modules over $\mathfrak{g}_{n}$

Proposition 4**.**

Let $\dot{z}\in\mathbb{C},\lambda\in\mathfrak{h}^{*}$.

1. $($a$)$

If $\dot{z}=0$, then $H_{n}L(\dot{z},\lambda)=0$. 2. $($b$)$

If $\dot{z}\neq 0$, then we have that

[TABLE]

[TABLE]

where $\lambda^{\prime}=\lambda+\frac{1}{2}(\epsilon_{1}+\cdots+\epsilon_{n})$. Consequently, $M(\dot{z},\lambda)$ is simple if and only if $M_{\mathfrak{sp}_{2n}}(\lambda+\frac{1}{2}(\epsilon_{1}+\cdots+\epsilon_{n}))$ is simple.

Proof.

(a) In $M(\dot{z},\lambda)$ we have

[TABLE]

and

[TABLE]

yielding that

[TABLE]

which is a proper submodule of $M(\dot{z},\lambda)$. So $X_{-\epsilon_{k}}v_{\lambda}\in K(\dot{z},\lambda)$, i.e, $X_{-\epsilon_{k}}v_{\lambda}=0$ in $L(\dot{z},\lambda)$, for any $k\in\{1,\dots,n\}$. By the fact that $H_{n}$ is an ideal of $\mathfrak{g}_{n}$ and the simplicity of $L(\dot{z},\lambda)$, we have $H_{n}L(\dot{z},\lambda)=0$.

(b) Let $v_{\lambda^{\prime}}$ be a highest weight vector of $M_{\mathfrak{sp}_{2n}}(\lambda^{\prime})$. It is not hard to see that ${M}_{\mathfrak{sp}_{2n}}(\lambda^{\prime})\otimes_{\dot{z}}S_{\dot{z}}$ is a highest weight $\mathfrak{g}_{n}$-module with highest weight $\lambda^{\prime}-\frac{1}{2}(\epsilon_{1}+\cdots+\epsilon_{n})$. Next we prove that

[TABLE]

Take $X\in U(\mathfrak{n}_{-})$ with

[TABLE]

Using a PBW basis, we can write

[TABLE]

where $f_{\alpha}(X_{\epsilon_{i}-\epsilon_{j}},X_{-\epsilon_{i}-\epsilon_{j}})\in U(\mathfrak{n}_{-})\cap U(\mathfrak{sp}_{2n})$. From (4) and Proposition 3, by induction on $|\alpha|$ we deduce that $f_{\alpha}(X_{\epsilon_{i}-\epsilon_{j}},X_{-\epsilon_{i}-\epsilon_{j}})=0$ for all $\alpha$. Thus $X=0$, yielding that ${M}_{\mathfrak{sp}_{2n}}(\lambda^{\prime})\otimes_{\dot{z}}S\simeq M(\dot{z},\lambda^{\prime}-\frac{1}{2}(\epsilon_{1}+\cdots+\epsilon_{n}))$.

Since ${L}_{\mathfrak{sp}_{2n}}(\lambda^{\prime})\otimes S$ is a simple $U(\mathfrak{sp}_{2n})\otimes\mathfrak{D}_{n}$-module, from the definition of ${L}_{\mathfrak{sp}_{2n}}(\lambda^{\prime})\otimes_{\dot{z}}S$ and Proposition 3 we see that it is a simple $\mathfrak{g}_{n}$-module. Thus the second isomorphism in (b) follows. ∎

3.4. The characterizations of $\mathcal{O}_{\mathfrak{g}_{n}}[\dot{z}]$

We denote by $\mathcal{O}_{\mathfrak{g}_{n}}[\dot{z}]$ the full subcategory of $\mathcal{O}_{\mathfrak{g}_{n}}$ consisting of all modules $M\in\mathcal{O}_{\mathfrak{g}_{n}}$ such that $z$ acts on $M$ as a scalar $\dot{z}$.

Lemma 5**.**

Let $M\in\mathcal{O}_{\mathfrak{g}_{n}}[\dot{z}]$ with $\dot{z}\neq 0$. Let $v$ be a nonzero weight vector in $M$. Then $U(H_{n})v=V_{1}\oplus\cdots\oplus V_{k}$, with $V_{i}\cong S_{\dot{z}}$ for each $i\in\{1,\dots,k\}$.

Proof.

Since $U(H_{n})/\langle z-\dot{z}\rangle\cong\mathfrak{D}_{n}$, we can consider that $U(H_{n})v=\mathfrak{D}_{n}v$ which is still a weight module with respect to $\mathfrak{h}$. Since $M$ has finite dimensional weight spaces, we see that

[TABLE]

Note that as vectors spaces

[TABLE]

Thus the $\mathfrak{D}_{n}$-module $\mathfrak{D}_{n}v$ has finite composition length $k$. We will prove the statement by induction on $k$.

If $k=1$, then $\mathfrak{D}_{n}v$ is a simple highest weight module, i.e., $\mathfrak{D}_{n}v\cong S_{\dot{z}}$. Let $V$ be a maximal $\mathfrak{D}_{n}$-submodule of $\mathfrak{D}_{n}v$. By the induction hypothesis, $V=V_{1}\oplus\cdots\oplus V_{k-1}$, with $V_{i}\cong S_{\dot{z}}$ for each $1\leq i\leq k-1$, and there is an epimorphism $\gamma:\mathfrak{D}_{n}v\rightarrow S_{\dot{z}}$ such that $\ker(\gamma)=V$. Since $\mathfrak{D}_{n}v/V$ is a highest weight module over $\mathfrak{D}_{n}$, we take a highest weight vector $\bar{w}\in\mathfrak{D}_{n}v/V$ with $w\in\mathfrak{D}_{n}$ being a weight vector. Then $t_{i}w\in V$ for any $1\leq i\leq n$.

Claim: There exists $v^{\prime}\in V$ such that $t_{i}(w+v^{\prime})=0$ for any $1\leq i\leq n$.

If $t_{1}w\neq 0$, we choose the smallest integer $l$ such that $t_{1}^{l+1}w=0$. From

[TABLE]

we have that

[TABLE]

Note that $\sum_{i=1}^{l}\frac{1}{i!}\partial_{1}^{i}t_{1}^{i}w\in V$. By repeating the above arguments, we can have $v^{\prime}\in V$ such that $t_{i}(w+v^{\prime})=0$ for any $1\leq i\leq n$. The claim follows.

Then $\mathfrak{D}_{n}(w+v^{\prime})\cong S_{\dot{z}}$. By the simplicity of $\mathfrak{D}_{n}(w+v^{\prime})$, we see that $\mathfrak{D}_{n}v=V\oplus\mathfrak{D}_{n}(w+v^{\prime})\cong V\oplus S_{\dot{z}}$. Then the proof is complete. ∎

Proposition 6**.**

Let $\dot{z}\neq 0$. For any $M\in\mathcal{O}_{\mathfrak{g}_{n}}[\dot{z}]$, there is an $N\in\mathcal{O}_{\mathfrak{sp}_{2n}}$ such that $M\cong{N}\otimes_{\dot{z}}S$.

Proof.

Consider $M$ as a $U(\mathfrak{sp}_{2n})\otimes\mathfrak{D}_{n}$-module via the isomorphism $\phi_{\dot{z}}$ in Proposition 3 where actually $\phi_{\dot{z}}(U(H_{n})/\langle z-\dot{z}\rangle)=\mathfrak{D}_{n}$. By Lemma 5, we know that $M$ is a direct sum of irreducible highest weight $\mathfrak{D}_{n}$-modules all isomorphic.

Let $V$ be the subspace of $M$ consisting all highest weight vectors over $\mathfrak{D}_{n}$. We see that $U(\mathfrak{sp}_{2n})V=V$. On the other hand we have $M=V\otimes S$. Thus going back to the $\mathfrak{g}_{n}$-module $M$ we see that $M=V\otimes_{\dot{z}}S$.∎

Theorem 7**.**

If $\dot{z}\neq 0$, then the functor

[TABLE]

is an equivalence of the two categories.

Proof.

Denote $\mathcal{F}=-\otimes_{\dot{z}}S$. Note that the automorphisms of simple $\mathfrak{D}_{n}$-modue $S$ is $\mathbb{C}^{*}$. From Proposition 5, we see that the homomorphism

[TABLE]

is isomorphism, for any modules $V,W\in\mathcal{O}_{\mathfrak{sp}_{2n}}$. Again from Proposition 6, $\mathcal{F}$ is an equivalence of the two categories. ∎

Remark: When $\dot{z}=0$, the Verma module $M(0,\lambda)$ has infinite composition length. Then subcategory $\mathcal{O}_{\mathfrak{g}_{n}}[0]$ has more complicated structure. For $n=1$, the discussion of the category $\mathcal{O}_{\mathfrak{g}_{n}}[0]$ can be referred to [11]. Some properties of the category $\mathcal{O}$ over a deformation of the symplectic oscillator algebra were given in [18].

4. Classification of simple Harish-Chandra $\mathfrak{g}_{n}$-modules

Before giving the classification of simple Harish-Chandra modules over $\mathfrak{g}_{n}$, we will first study generalized highest weight modules.

4.1. Generalized highest weight modules

Denote

[TABLE]

Then we have a different triangular decomposition

[TABLE]

Let $V$ be a simple $\mathfrak{g}_{n}^{0}\oplus\mathfrak{g}_{n}^{+}$-module with $\mathfrak{g}_{n}^{+}V=0$ and $zv=\dot{z}v$ for any $v\in V$, where $\dot{z}\in\mathbb{C}$. Note that $\mathfrak{g}_{n}^{0}=\mathfrak{gl}_{n}\oplus\mathbb{C}z$. So $V$ is actually a simple $\mathfrak{gl}_{n}$-module. The induced $\mathfrak{g}_{n}$-module

[TABLE]

is called a generalized Verma module over $\mathfrak{g}_{n}$. We denote its unique irreducible quotient module by $L(\dot{z},V)$. By the analogous construction, we have the $\mathfrak{sp}_{2n}$-modules $M_{\mathfrak{sp}_{2n}}(V)$ and $L_{\mathfrak{sp}_{2n}}(V)$, for any simple $\mathfrak{gl}_{n}$-module $V$.

Similar to Proposition 4, we have the next result.

Proposition 8**.**

1. $($a$)$

If $\dot{z}=0$, then $H_{n}L(\dot{z},V)=0$. 2. $($b$)$

If $\dot{z}\neq 0$, then we have that

[TABLE]

and

[TABLE]

Proof.

(a) The proof is similar to that for Proposition 4 (a). We omit the details.

(b) It is not hard to see that ${M}_{\mathfrak{sp}_{2n}}(V)\otimes_{\dot{z}}S$ is a generalized highest weight $\mathfrak{g}_{n}$-module. Next we prove that

[TABLE]

It is enough to show that, for any linearly independent $v_{k}\in V$ ($k=1,2,\cdots,r$) and $X_{k}\in U(\mathfrak{g}_{n}^{-})$ with

[TABLE]

we can deduce that $X_{k}=0$ for all $k$. Using a PBW basis, we can write

[TABLE]

where $f^{(k)}_{\alpha}(X_{-\epsilon_{i}-\epsilon_{j}})\in U(\mathfrak{g}_{n}^{-})\cap U(\mathfrak{sp}_{2n})$. From (5) and Proposition 3, by induction on $|\alpha|$ we deduce that $f^{(k)}_{\alpha}(X_{-\epsilon_{i}-\epsilon_{j}})=0$ for all $\alpha$. Thus all $X_{k}=0$, yielding that ${M}_{\mathfrak{sp}_{2n}}(V)\otimes_{\dot{z}}S$ is a generalized Verma module over $\mathfrak{g}_{n}$, i.e., $M(\dot{z},V)\cong{M}_{\mathfrak{sp}_{2n}}(V)\otimes_{\dot{z}}S.$

Since ${L}_{\mathfrak{sp}_{2n}}(V)\otimes_{\dot{z}}S$ is a simple $U(\mathfrak{sp}_{2n})\otimes\mathfrak{D}_{n}$-module, from the definition of ${L}_{\mathfrak{sp}_{2n}}(V)\otimes_{\dot{z}}S$ and Proposition 3 we see that it is a simple $\mathfrak{g}_{n}$-module. Thus the second isomorphism in (b) follows. ∎

By Lemma 3.3 in [7], we have the following Lemma.

Lemma 9**.**

Let $M$ be a weight $\mathfrak{g}_{n}$-module with all finite dimensional weight spaces, $\alpha\in\Delta_{\mathfrak{sp}_{2n}}$. The following two conditions are equivalent.

1. $($a$)$

For each $\lambda\in\text{Supp}M$, the space $M_{\lambda+k\alpha}$ is zero for all but finitely many $k>0$. 2. $($b$)$

The element $X_{\alpha}$ acts locally nilpotently on $M$.

Lemma 10**.**

[7]** If $M$ is a simple weight $\mathfrak{g}_{n}$-module, then the action of $X_{\alpha}$ on $M$ is injective or locally nilpotent, for any $\alpha\in\Delta$.

Next, we will introduce a lemma which will be used in studying the structure of generalized highest modules.

Lemma 11**.**

Let $M$ be any simple Harish-Chandra $\mathfrak{g}_{n}$-module, on which $z$ acts as some $\dot{z}\in\mathbb{C}$.

1. $($a$)$

If $X_{2\epsilon_{i}}$ acts locally nilpotently on $M$, then so does $X_{\epsilon_{i}}$. If $\dot{z}\neq 0$, then the converse is also true. 2. $($b$)$

If $X_{2\epsilon_{i}}$ and $X_{2\epsilon_{j}}$ acts locally nilpotently on $M$ with $i\neq j$, then $X_{\epsilon_{i}+\epsilon_{j}}$ also acts locally nilpotently on $M$.

Proof.

(a) If $X_{2\epsilon_{i}}$ acts locally nilpotently on $M$, then by Lemma 9, for each $\lambda\in\text{Supp}M$, the space $M_{\lambda+2k\epsilon_{i}}$ is zero for all but finitely many $k>0$. Thus $X_{\epsilon_{i}}$ acts locally nilpotently on $M$.

Conversely, if $\dot{z}\neq 0$, $X_{\epsilon_{i}}$ acts nilpotently on $M$ but $X_{2\epsilon_{i}}$ acts injectively on $M$, then for each $\lambda\in\text{Supp}M$, the space $M_{\lambda+2k\epsilon_{i}}\neq 0$ for any $k\in\mathbb{Z}_{+}$. By the nilpotence of $X_{\epsilon_{i}}$, we have an infinite sequence of positive integers $k_{1}<k_{2}<\cdots<k_{l}<\cdots$ and nonzero $v_{k_{l}}\in M_{\lambda+k_{l}\epsilon_{i}}$ such that $X_{\epsilon_{i}}v_{k_{l}}=0$ for each $l\in\mathbb{N}$. Using that $\dot{z}\neq 0$, we can see that $\{X_{-\epsilon_{i}}^{k_{l}}v_{k_{l}}\mid l\in\mathbb{N}\}$ is a linearly independent infinite subset of $M_{\lambda}$, contradicting the fact that $\dim M_{\lambda}<\infty$.

(b) Suppose that $X_{\epsilon_{i}+\epsilon_{j}}$ acts injectively on $M$.

We claim that both $X_{-2\epsilon_{i}}$ and $X_{\epsilon_{j}-\epsilon_{i}}$ acts nilpotently on $M$. If $X_{\epsilon_{j}-\epsilon_{i}}$ acts injectively on $M$, then from $[X_{\epsilon_{j}-\epsilon_{i}},X_{\epsilon_{i}+\epsilon_{j}}]=X_{2\epsilon_{j}}$ and Lemma 9, we see that $X_{2\epsilon_{j}}$ acts injectively on $M$, contradicting the nilpotence of $X_{2\epsilon_{j}}$. If $X_{-2\epsilon_{i}}$ acts injectively on $M$, then from $[X_{\epsilon_{i}+\epsilon_{j}},X_{-2\epsilon_{i}}]=2X_{\epsilon_{j}-\epsilon_{i}}$ and Lemma 9, we see that $X_{\epsilon_{j}-\epsilon_{i}}$ acts injectively on $M$, contradicting the nilpotence of $X_{\epsilon_{j}-\epsilon_{i}}$.

Let $\mathfrak{s}_{i}$ be the subalgebra generated by $X_{2\epsilon_{i}}$, $X_{-2\epsilon_{i}}$ which is isomorphic to $\mathfrak{sl}_{2}$. The nilpotence of $X_{2\epsilon_{i}}$ and $X_{-2\epsilon_{i}}$ tells us that $M$ is a sum of finite dimensional irreducible $\mathfrak{s}_{i}$-modules. So the weight set of $M$ is invariant under the action of the reflection $r_{2\epsilon_{i}}$. Since $X_{\epsilon_{i}+\epsilon_{j}}$ acts injectively on $M$, for each $\lambda\in\text{Supp}M$, the space $M_{\lambda+k\epsilon_{i}+k\epsilon_{j}}\neq 0$ for any $k\in\mathbb{N}$. Then

[TABLE]

for any $k\in\mathbb{N}$. This implies that $X_{\epsilon_{j}-\epsilon_{i}}$ acts injectively on $M$, a contradiction. So (b) follows. ∎

Proposition 12**.**

Let $M$ be a simple weight $\mathfrak{g}_{n}$-module with all finite dimensional weight spaces. If $X_{2\epsilon_{i}}$ acts locally nilpotently on $M$ for any $1\leq i\leq n$, then $M\cong L(\dot{z},V)$ for some simple Harish-Chandra $\mathfrak{gl}_{n}$-module $V$.

Proof.

By Lemma 11, $X_{\epsilon_{i}},X_{\epsilon_{i}+\epsilon_{j}}$ act locally nilpotently on $M$ for any $1\leq i\leq n$. Then we can find a weight vector $v$ which is annihilated by $\mathfrak{g}_{n}^{+}$. Let $V=\{v\in M\mid\mathfrak{g}_{n}^{+}v=0\}$ which has to be a simple $\mathfrak{g}_{n}^{0}$-module. Consequently, $M\cong L(\dot{z},V)$. Since $M$ is a Harish-Chandra $\mathfrak{g}_{n}$-module, $V$ has to be a Harish-Chandra $\mathfrak{gl}_{n}$-module. ∎

4.2. Simple Harish-Chandra modules

In this subsection, we will give the classification of simple weight $\mathfrak{g}_{n}$-modules with finite dimensional weight spaces. Our method is quite different from those in [25, 7] where they broke the whole system into different subsets according to actions’ nilpotency of root vectors. We need only to use all the root vectors with respect to long roots $\pm 2\epsilon_{i}$ as you will see next.

For any simple Harish-Chandra $\mathfrak{g}_{n}$-module $M$, we set

[TABLE]

Then $I_{M}\cup F_{M}\cup F^{+}_{M}\cup F^{-}_{M}=\{1,\dots,n\}$.

Consider the multiplicative subset

[TABLE]

which is an Ore subset of $U(\mathfrak{g}_{n})$, and hence we have the corresponding Ore localization $U(\mathfrak{g}_{n})_{S(I)}$, see [5, 25]. Assume that $I_{M}=\{i_{1},\dots,i_{k}\}$.

For $b=(b_{1},\dots,b_{k})\in\mathbb{C}^{k}$, there is an isomorphism $\theta_{b}$ of $U(\mathfrak{g}_{n})_{S(I)}$ such that

[TABLE]

for any $u\in U(\mathfrak{g}_{n})_{S}$. In particular, we can check that

[TABLE]

[TABLE]

for any $1\leq l\leq k$.

For a $U(\mathfrak{g}_{n})_{S(I)}$-module $M$, it can be twisted by $\theta_{b}$ to be a new $U(\mathfrak{g}_{n})_{S(I)}$-module $M^{\theta_{b}}$. As vector spaces $M^{\theta_{b}}=M$. For $v\in M$, $x\in\mathfrak{g}_{n}$, $x\cdot v=\theta_{b}(x)v$. The modules $M$ and $M^{\theta_{b}}$ are said to be equivalent.

Lemma 13**.**

Let $M$ be a simple Harish-Chandra $\mathfrak{g}_{n}$-module. If both $X_{\alpha}$ and $X_{-\alpha}$ act on $M$ injectively, then both act on $M$ bijectively.

Proof.

Consider the two injective linear maps $X_{\alpha}:M_{\lambda}\to M_{\lambda+\alpha}$ and $X_{-\alpha}:M_{\lambda+\alpha}\to M_{\lambda}$. Since $\dim M_{\lambda}<\infty$ and $\dim M_{\lambda+\alpha}<\infty$, we see that $\dim M_{\lambda}=\dim M_{\lambda+\alpha}$, and both $X_{\alpha}$ and $X_{-\alpha}$ act on $M$ bijectively. ∎

Theorem 14**.**

Let $M$ be a simple Harish-Chandra $\mathfrak{g}_{n}$-module, on which $z$ acts as zero. Then $H_{n}M=0$, i.e., $M$ is a simple Harish-Chandra $\mathfrak{sp}_{2n}$-module.

Proof.

We can now consider $M$ as a module over $\mathfrak{sp}_{2n}\ltimes\mathbb{C}^{2n}$. By the structure of the Weyl group of $\mathfrak{sp}_{2n}$, there is an inner automorphism $\sigma$ of $\mathfrak{sp}_{2n}$ such that

[TABLE]

and

[TABLE]

Since $\sigma$ is an inner automorphism, we can extend $\sigma$ to be an automorphism $\tilde{\sigma}$ of $\mathfrak{sp}_{2n}\ltimes\mathbb{C}^{2n}$ such that $\tilde{\sigma}(\mathbb{C}^{2n})=\mathbb{C}^{2n}$. The module $M$ can be twisted by $\tilde{\sigma}$ to be a new $\mathfrak{g}_{n}$-module $M^{\tilde{\sigma}}$. Then $H_{n}M=0$ if and only if $H_{n}\cdot M^{\tilde{\sigma}}=0$. For $M^{\tilde{\sigma}}$,

[TABLE]

For convenience, we can assume that

[TABLE]

Let $L$ be the subalgebra of $\mathfrak{g}_{n}$ generated by

[TABLE]

which is isomorphic to $\mathfrak{g}_{k}$.

If $F_{M}\cup F^{+}_{M}=\emptyset$, let $N=M$. Otherwise set

[TABLE]

Since $X_{\epsilon_{i}},X_{2\epsilon_{i}}$ act locally nilpotent on $M$ and $[X_{\epsilon_{i}},X_{2\epsilon_{i}}]=0$ for any $i\in F_{M}\cup F^{+}_{M}$, the space $N$ is nonzero. Note that $N$ is a uniformly bounded $\mathfrak{sp}_{2n}$-module. Then $N$ is a uniformly bounded $L$-module. Choose a weight vector $v$ which is a common eigenvalue of $X_{2\epsilon_{i}}X_{-2\epsilon_{i}}$ for any $i\in I_{M}$. Assume that $X_{2\epsilon_{i}}X_{-2\epsilon_{i}}v=a_{i}v,a_{i}\in\mathbb{C}$. Let $w=X_{-2\epsilon_{i_{1}}}\cdots X_{-2\epsilon_{i_{k}}}v$. Take $b_{i}$ so that $2b_{l}(b_{l}-1-2h_{i_{l}})w=a_{i_{l}}w$. Then by the formula (7), we see that $\theta_{b}(X_{2\epsilon_{i}})w=0$ for any $i\in I_{M}$. Now consider the $L$-module $N^{\theta_{b}}$. We can check that

[TABLE]

is a nonzero $L$-submodule of $N^{\theta_{b}}$. Being uniformly bounded, $N^{\prime}$ has a simple $L$-submodule $N^{\prime\prime}$. By (a) in Lemma 11, $X_{\epsilon_{i}}$ acts locally nilpotent on $N^{\prime\prime}$ for any $i\in I_{M}$. Then by Proposition 12, $N^{\prime\prime}$ is a simple generalized highest weight $L$-module. By Proposition 8 (a), we deduce that

[TABLE]

By the formula (6), we see that

[TABLE]

Consequently $X_{\epsilon_{i}}N^{\prime\prime}=0$, and hence $X_{\epsilon_{i}}$ acts locally nilpotent on $M$ for any $i=1,\dots,n$. Similarly, we can prove $X_{-\epsilon_{i}}$ acts locally nilpotent on $M$ for any $i=1,\dots,n$. Then

[TABLE]

is nonzero, which is a $\mathfrak{g}_{n}$-submodule of $M$. The simplicity of $M$ tells us that $W=M$, i.e., $H_{n}M=0$. ∎

It is known (see [15]) that any irreducible weight module over the Weyl algebra $\mathfrak{D}_{n}$ is isomorphic to some irreducible subquotient of the module

[TABLE]

for some $a=(a_{1},\dots,a_{n})\in\mathbb{C}^{n},$ where $t^{a}=t_{1}^{a_{1}}\cdots t_{n}^{a_{n}}$. Since $F(a)\cong F(a+\beta)$ for any $\beta\in\mathbb{Z}^{n}$, we always assume that $a_{i}=0$ if $a_{i}\in\mathbb{Z}$. Set

[TABLE]

Assume that $\text{Int}_{a}=\{j_{1},\dots,j_{l}\}$. Let

[TABLE]

We can see that $G(a)$ is a simple $\mathfrak{D}_{n}$-module.

For any simple Harish-Chandra $\mathfrak{g}_{n}$-module $M$, we have defined the set

[TABLE]

Theorem 15**.**

Let $M$ be a simple Harish-Chandra $\mathfrak{g}_{n}$-module with a nonzero central charge, say $\dot{z}\in\mathbb{C}^{*}$.

1. $($a$)$

If $I_{M}=\{1,\dots,n\}$, then $M$ is equivalent to $N\otimes_{\dot{z}}F(a)$ for some finite dimensional simple $\mathfrak{sp}_{2n}$-module $N$ and $a=(a_{1},\dots,a_{n})\in\mathbb{C}^{n}$ with $a_{1},\dots,a_{n}\not\in\mathbb{Z}$. 2. $($b$)$

If $I_{M}=\emptyset$, then there is a simple Harish-Chandra $\mathfrak{gl}_{n}$-module $V$ such that $M$ is equivalent (under an inner automorphism of $\mathfrak{g}_{n}$) to the generalized highest weight module $L(\dot{z},V)$ defined in subsection 4.1. 3. $($c$)$

If $I_{M}$ is a nonempty proper subset of $\{1,\dots,n\}$, then there is a simple Harish-Chandra $\mathfrak{gl}_{n}$-module $V$ and $a=(a_{1},\dots,a_{n})\in\mathbb{C}^{n}$ such that $M$ is equivalent (under an inner automorphism of $\mathfrak{g}_{n}$) to $L_{\mathfrak{sp}_{2n}}(V)\otimes_{\dot{z}}G(a)$, where $\text{Int}_{a}=\{1,\dots,n\}\setminus I_{M}$.

Proof.

For each $i:1\leq i\leq n$, either $X_{\epsilon_{i}}$ or $X_{-\epsilon_{i}}$ acts injectively on $M$. Otherwise $\dot{z}=0$ since there is no nontrivial finite-dimensional $\mathfrak{D}_{n}$-module, which contradicts our assumption. So $F_{M}=\emptyset$.

(a) By Lemma 11, both $X_{\epsilon_{i}}$ and $X_{-\epsilon_{i}}$ act injectively on $M$. Let $v\in M$ be a nonzero weight vector which is a common eigenvector of $X_{\epsilon_{i}}X_{-\epsilon_{i}}$ for all $i$. Then by the classification of simple weight modules over the Weyl algebra $\mathfrak{D}_{n}$ (see [15]), there is an $a=(a_{1},\dots,a_{n})\in\mathbb{C}^{n}$ with $a_{1},\dots,a_{n}\not\in\mathbb{Z}$ such that $U(H_{n})v\cong F(a)$. Note that $F(a)$ is a $U(H_{n})$-module via the map $\theta_{\dot{z}}$. By the isomorphism $\phi_{\dot{z}}$ in Proposition 3 and the dimension finiteness of weight spaces, we must have $M\cong{N}\otimes_{\dot{z}}F(a)$ for some finite dimensional simple $\mathfrak{sp}_{2n}$-module $N$.

(b) Since $I_{M}=\emptyset$, after twisting $M$ by an inner automorphism, we can assume that $F^{+}_{M}=\{1,\dots,n\}$, that is, $X_{2\epsilon_{i}}$ acts locally nilpotently on $M$ for any $1\leq i\leq n$. By Proposition 12, there is a simple Harish-Chandra $\mathfrak{gl}_{n}$-module $V$ such that $M$ is equivalent to the generalized highest weight module $L(\dot{z},V)$.

(c) Suppose that $I_{M}=\{i_{1},\dots,i_{k}\}$. After twisting $M$ by an inner automorphism (similar arguments to those in the proof of Proposition 14), we can assume that

[TABLE]

Here we have replaced our old $M$ with an equivalent module.

Choose a nonzero weight vector $v\in M$ such that $X_{\epsilon_{i}}v=0$ for any $i\in F^{+}_{M}$. We can further choose $v$ so that it is a common eigenvector of $X_{\epsilon_{i}}X_{-\epsilon_{i}}$ for all $i$. Then we can see that there is an $a=(a_{1},\dots,a_{n})\in\mathbb{C}^{n}$ such that $U(H_{n})v\cong G(a)$ with $a_{i}=0$ for $i\in F^{+}_{M}$.

Using the isomorphism in Proposition 3, there is a simple $\mathfrak{sp}_{2n}$-module $N$ such that $M$ is equivalent to $N\otimes_{\dot{z}}G(a)$. Note that

[TABLE]

Since all the weight spaces of $N\otimes_{\dot{z}}G(a)$ are finite dimensional, there is a weight $\lambda$ of $N$ such that $\lambda+2\epsilon_{i}\not\in\text{Supp}(N)$ for any $i\in\{1,\dots,n\}$. This implies that there are nonzero vectors in $N$ which are annihilated by $\mathfrak{g}_{n}^{+}\cap\mathfrak{sp}_{2n}$. Therefore $N$ is isomorphic to the generalized highest weight module $L_{\mathfrak{sp}_{2n}}(V)$. See Proposition 12. ∎

So far we have obtained the classification for all simple Harish-Chandra modules over the symplectic oscillator Lie algebra $\mathfrak{g}_{n}$ for all $n$. Theorem 14 gave all such $\mathfrak{g}_{n}$-modules with $z$ acts trivially, which are actually simple parabolically induced $\mathfrak{sp}_{2n}$-modules and simple cuspidal $\mathfrak{sp}_{2n}$-modules (See [25]). Theorem 15 gave all such $\mathfrak{g}_{n}$-modules with $z$ acts non-trivially, which consists of three classes: cuspidal $\mathfrak{g}_{n}$-modules in (a), parabolically induced $\mathfrak{g}_{n}$-modules in (b), and a third class described in (c). We can see that the third class in Theorem 15 (c) does not appear for finite-dimensional simple Lie algebras.

**Acknowledgements. ** This research is partially supported by NSFC (11771122, 11871190) and NSERC (311907-2015).