#  Gelfand-Kirillov Dimensions of the $\mathbb{Z}$-graded Oscillator
Representations of $\mathfrak{o}(n,\mathbb{C})$ and
$\mathfrak{sp}(2n,\mathbb{C})$

Zhanqiang Bai  Institute of Mathematics, Academy of Mathematics and System
Sciences, Chinese Academy of Sciences, Beijing 100080, China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we compute the Gelfand-Kirillov dimensions for the infinite-
dimensional explicit irreducible $\mathfrak{o}(n,\mathbb{C})$-modules and
$\mathfrak{sp}(2n,\mathbb{C})$-modules that appeared in the
$\mathbb{Z}$-graded oscillator generalizations of the classical theorem on
harmonic polynomials established by Luo and Xu. We also found that some of the
modules have the secondly minimal GK-dimension, and some of them have the
larger GK-dimension than the maximal GK-dimension of unitary highest-weight
modules.

Key Words: Gelfand-Kirillov dimension; Highest-weight module; Oscillator
representation; Universal enveloping algebra.

2010 Mathematics Subject Classification. Primary 22E47; Secondary 17B10

##  1\. Introduction

In 1960s, Gelfand-Kirillov [5] introduced a quantity to measure the rate of
growth of an algebra in terms of any generating set, which is now known as
Gelfand-Kirillov dimension. In 1970s, Vogan [17] and Joseph [7] used the
Gelfand-Kirillov dimension to measure the size of the infinite-dimensional
modules. Gelfand-Kirillov dimension has been an important invariant in the
theory of algebras over a field for the past forty years.

A module of a finite-dimensional simple Lie algebra is called a weight module
if it is a direct sum of its weight subspaces. The classification of weight
modules had been completed by Mathieu [14] after many mathematician’s work.
But we don’t have many results about the distribution of Gelfand-Kirillov
dimensions of weight modules. Let $M$ be an irreducible highest-weight module
for a finite-dimensional simple Lie algebra $\mathfrak{g}$. Then $M$ is
naturally a weight module with finite-dimensional weight subspaces. Denote by
$d_{M}$ its Gelfand-Kirillov dimension. We fix a Cartan subalgebra
$\mathfrak{h}$, a root system $\Delta\subset\mathfrak{h}^{*}$ and a set of
positive roots $\Delta_{+}\subset\Delta$. Let $\rho$ be half the sum of all
positive roots. Suppose that $\beta$ is the highest root. It is well known
that $d_{M}=0$ if and only if $M$ is finite-dimensional, in which case
irreducible modules are classified by the highest-weight theory. From Vogan
[18] and Wang [20], we know that the next smallest integer occurring is
$d_{M}=(\rho,\beta^{\vee})$. We call them the _minimal Gelfand-Kirillov
dimension module_. These small modules are of great interest in representation
theory. A general introduction can be found in Vogan [18]. Recently we [2]
studied the GK-dimensions of unitary highest-weight modules. We found that the
secondly minimal GK-dimension of a unitary highest-weight module is
$2((\rho,\beta^{\vee})-C)$ and the maximal GK-dimension of a unitary highest-
weight module is $r((\rho,\beta^{\vee})-(r-1)C)$, where $C$ and $r$ are
constants only depending on the type of Lie algebras and given by Enright,
Howe and Wallach in [4]. Does any irreducible weight module have larger GK-
dimension than $r((\rho,\beta^{\vee})-(r-1)C)$? The answer is ’Yes’ since we
can find examples in this paper.

In classical harmonic analysis, a fundamental theorem says that the spaces of
homogeneous harmonic polynomials are irreducible modules of the corresponding
orthogonal Lie group (algebra) and the whole polynomial algebra is a free
module over the invariant polynomials generated by harmonic polynomials. Bases
of these irreducible modules can be obtained easily (e.g., cf. [21]). The
algebraic beauty of the above theorem is that Laplace equation characterizes
the irreducible submodules of the polynomial algebra and the corresponding
quadratic invariant gives a decomposition of the polynomial algebra into a
direct sum of irreducible submodules, namely, the complete reducibility.
Recently Luo and Xu [11] established the $\mathbb{Z}^{2}$-graded oscillator
generalizations of the above theorem for $\mathfrak{sl}(n,\mathbb{C})$, where
the irreducible submodules are $\mathbb{Z}^{2}$-graded homogeneous polynomial
solutions of deformed Laplace equations. In [1], we find an exact formula of
Gelfand-Kirillov dimensions for these $\mathfrak{sl}(n,\mathbb{C})$-modules.
It turns out that their Gelfand-Kirillov dimensions are independent of the
double grading and three infinite subfamilies of these modules have the
minimal Gelfand-Kirillov dimension. In [12, 13], by using the results in [11],
Luo and Xu established the structure of the corresponding two-parameter
$\mathbb{Z}$-graded oscillator representations of $\mathfrak{o}(n,\mathbb{C})$
and $\mathfrak{sp}(2n,\mathbb{C})$. It turned out that theses modules are
irreducible weight modules. In this paper, we will compute the Gelfand-
Kirillov dimensions of these modules. Below we give a more detailed
introduction for theses modules.

For convenience, we will use the notion
$\overline{{i,i+j}}=\\{i,i+1,i+2,...,i+j\\}$ for integers $i$ and $j$ with
$i\leq j$. Denote by $\mathbb{N}$ the additive semigroup of nonnegative
integers. Let $E_{r,s}$ be the square matrix with $1$ as its $(r,s)$-entry and
$0$ as the others. The orthogonal Lie algebra

$\mathfrak{o}(2n,\mathbb{C})=\sum_{i,j=1}^{n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{1\leq
i<j\leq n}[\mathbb{C}(E_{i,n+j}-E_{j,n+i})+\mathbb{C}(E_{n+j,i}-E_{n+i,j})]$

Denote ${\mathcal{B}}=\mathbb{C}[x_{1},...,x_{n},y_{1},...,y_{n}]$. Fix
$n_{1},n_{2}\in\overline{1,n}$ with $n_{1}\leq n_{2}$. We have the following
non-canonical oscillator representation of $\mathfrak{o}(2n,\mathbb{C})$ on
${\mathcal{B}}$ determined by

(1.1)
$(E_{i,j}-E_{n+j,n+i})|_{\mathcal{B}}=E_{i,j}^{x}-E_{j,i}^{y}\qquad\text{for}~{}i,j\in\overline{1,n}$

with

(1.2)
$E_{i,j}^{x}|_{\mathcal{B}}=\left\\{\begin{array}[]{ll}-x_{j}\partial_{x_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{1,n_{1}},\\\
\partial_{x_{i}}\partial_{x_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{n_{1}+1,n},\\\
-x_{i}x_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{1,n_{1}},\\\
x_{i}\partial_{x_{j}}&\mbox{if}\;i,j\in\overline{n_{1}+1,n}\end{array}\right.$

and

(1.3)
$E_{i,j}^{y}|_{\mathcal{B}}=\left\\{\begin{array}[]{ll}y_{i}\partial_{y_{j}}&\mbox{if}\;i,j\in\overline{1,n_{2}},\\\
-y_{i}y_{j}&\mbox{if}\;i\in\overline{1,n_{2}},\;j\in\overline{n_{2}+1,n},\\\
\partial_{y_{i}}\partial_{y_{j}}&\mbox{if}\;i\in\overline{n_{2}+1,n},\;j\in\overline{1,n_{2}},\\\
-y_{j}\partial_{y_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{n_{2}+1,n}\end{array}\right.$

and

(1.4)
$E_{i,n+j}|_{\mathcal{B}}=\left\\{\begin{array}[]{ll}\partial_{x_{i}}\partial_{y_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\,j\in\overline{1,n_{2}},\\\
-y_{j}\partial_{x_{i}}&\mbox{if}\;i\in\overline{1,n_{1}},\,j\in\overline{n_{2}+1,n},\\\
{x_{i}}\partial_{y_{j}}&\mbox{if}\;i\in\overline{n_{1}+1,n},\,j\in\overline{1,n_{2}},\\\
-x_{i}y_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\,j\in\overline{n_{2}+1,n}\end{array}\right.$

and

(1.5)
$E_{n+i,j}|_{\mathcal{B}}=\left\\{\begin{array}[]{ll}-{x_{j}}{y_{i}}&\mbox{if}\;j\in\overline{1,n_{1}},\,i\in\overline{1,n_{2}},\\\
-x_{j}\partial_{y_{i}}&\mbox{if}\;j\in\overline{1,n_{1}},\,i\in\overline{n_{2}+1,n},\\\
{y_{i}}\partial_{x_{j}}&\mbox{if}\;j\in\overline{n_{1}+1,n},\,i\in\overline{1,n_{2}},\\\
\partial_{x_{j}}\partial_{y_{i}}&\mbox{if}\;j\in\overline{n_{1}+1,n},\,i\in\overline{n_{2}+1,n}.\end{array}\right.$

The related variated Laplace operator becomes

(1.6)
${\mathcal{D}}=\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}-\sum_{r=n_{1}+1}^{n_{2}}\partial_{x_{r}}\partial_{y_{r}}+\sum_{s=n_{2}+1}^{n}y_{s}\partial_{x_{s}}.$

Set

(1.7) ${\mathcal{B}}_{\langle
k^{\prime}\rangle}=\mbox{Span}\\{x^{\alpha}y^{\beta}\mid\alpha,\beta\in\mathbb{N}\>^{n},\sum_{r=n_{1}+1}^{n}\alpha_{r}-\sum_{i=1}^{n_{1}}\alpha_{i}+\sum_{i=1}^{n_{2}}\beta_{i}-\sum_{r=n_{2}+1}^{n}\beta_{r}=k^{\prime}\\}$

for $k^{\prime}\in\mathbb{Z}$. Define

(1.8) ${\mathcal{H}}_{\langle k^{\prime}\rangle}=\\{f\in{\mathcal{B}}_{\langle
k^{\prime}\rangle}\mid{\mathcal{D}}(f)=0\\}.$

The following is the first main theorem of this paper.

######  Theorem 1.1.

For any $k^{\prime}\in\mathbb{Z}$, if the $\mathfrak{o}(2n,\mathbb{C})$-module
${\mathcal{H}}_{\langle k^{\prime}\rangle}$ is irreducible, then it has the
Gelfand-Kirillov dimension

(1.9) $d=\left\\{\begin{array}[]{ll}2n-1,&\hbox{\text{\emph{if~{}}}
$1=n_{1}<n_{2}<n-1$, \emph{or~{}} $3\leq n_{1}<n_{2}=n$,}\\\
&\hbox{\emph{or~{}}$1<n_{1}<n_{2}\leq n-1$, \emph{or~{}}$n_{1}=n_{2}$
\emph{when~{}} $n\geq 5$;}\\\ 2n-2,&\hbox{\emph{if~{}} $1=n_{1}<n_{2}=n-1,n$,
\emph{or~{}} $2=n_{1}<n_{2}=n$}\\\ &\hbox{\emph{or~{}} $n_{1}=n_{2}$
\emph{when~{}} $n=4$;}\\\ 2n-3,&\hbox{\emph{if~{}} $n_{1}=n_{2}$
\emph{when~{}} $n=2,3$.}\end{array}\right.$

######  Remark 1.1.

For this case, the minimal GK-dimension is $2n-3$. From our paper [2], the
secondly minimal GK-dimension and the maximal GK-dimension are the same, i.e.,
$2n-2$. So $2n-1$ is larger than the maximal GK-dimension of any unitary
highest-weight modules.

We observe that the orthogonal Lie algebra

$\mathfrak{o}(2n+1,\mathbb{C})=\mathfrak{o}(2n,\mathbb{C})\oplus\bigoplus_{i=1}^{n}[\mathbb{C}(E_{0,i}-E_{n+i,0})+\mathbb{C}(E_{0,n+i}-E_{i,0})].$

Let
${\mathcal{B}}^{\prime}=\mathbb{C}[x_{0},x_{1},...,x_{n},y_{1},...,y_{n}]$. We
define a non-canonical oscillator representation of
$\mathfrak{o}(2n+1,\mathbb{C})$ on ${\mathcal{B}}^{\prime}$ by the
differential operators in (1.1)-(1.5) and

$E_{0,i}|_{{\mathcal{B}}^{\prime}}=\left\\{\begin{array}[]{ll}-x_{0}x_{i}&\mbox{if}\;i\in\overline{1,n_{1}},\\\
x_{0}\partial_{x_{i}}&\mbox{if}\;i\in\overline{n_{1}+1,n},\\\
x_{0}\partial_{y_{i-n}}&\mbox{if}\;i\in\overline{n+1,n+n_{2}},\\\
-x_{0}y_{i-n}&\mbox{if}\;i\in\overline{n+n_{2}+1,2n}\end{array}\right.$

and

$E_{i,0}|_{{\mathcal{B}}^{\prime}}=\left\\{\begin{array}[]{ll}\partial_{x_{0}}\partial_{x_{i}}&\mbox{if}\;i\in\overline{1,n_{1}},\\\
x_{i}\partial_{x_{0}}&\mbox{if}\;i\in\overline{n_{1}+1,n},\\\
y_{i-n}\partial_{x_{0}}&\mbox{if}\;i\in\overline{n+1,n+n_{2}},\\\
\partial_{x_{0}}\partial_{y_{i-n}}&\mbox{if}\;i\in\overline{n+n_{2}+1,2n}.\end{array}\right.$

Now the variated Laplace operator becomes

${\mathcal{D}}^{\prime}=\partial_{x_{0}}^{2}-2\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}+2\sum_{r=n_{1}+1}^{n_{2}}\partial_{x_{r}}\partial_{y_{r}}-2\sum_{s=n_{2}+1}^{n}y_{s}\partial_{x_{s}}.$

Set

${\mathcal{B}}^{\prime}_{\langle
k\rangle}=\sum_{i=0}^{\infty}{\mathcal{B}}_{\langle
k-i\rangle}x_{0}^{i},\qquad{\mathcal{H}}^{\prime}_{\langle
k\rangle}=\\{f\in{\mathcal{B}}^{\prime}_{\langle
k\rangle}\mid{\mathcal{D}}^{\prime}(f)=0\\}.$

The following is the second main theorem of this paper.

######  Theorem 1.2.

For any $k^{\prime}\in\mathbb{Z}$, the irreducible
$\mathfrak{o}(2n+1,\mathbb{C})$-module ${\mathcal{H}^{\prime}}_{\langle
k^{\prime}\rangle}$ has the Gelfand-Kirillov dimension

(1.10) $d=\left\\{\begin{array}[]{ll}2n,&\hbox{\text{\emph{if~{}}} $1\leq
n_{1}<n_{2}<n-1$, \emph{or~{}} $3\leq n_{1}<n_{2}=n$,}\\\
&\hbox{\emph{or~{}}$1<n_{1}<n_{2}=n-1$, \emph{or~{}}$n_{1}=n_{2}$
\emph{when~{}} $n\geq 5$,}\\\ &\hbox{\emph{or~{}}$n_{1}=n_{2}=n=3$,
\emph{or~{}}$n_{1}=n_{2}>1$ \emph{when~{}} $n=4$;}\\\ 2n-1,&\hbox{\emph{if~{}}
$1=n_{1}<n_{2}=n-1,n$, \emph{or~{}} $2=n_{1}<n_{2}=n$},\\\ &\hbox{\emph{or~{}}
$n_{1}=n_{2}=2$ \emph{when~{}} $n=2,3$, \emph{or~{}} $n_{1}=n_{2}=1$
\emph{when~{}} $n=1,4$;}\\\ 2n-2,&\hbox{\emph{if~{}}
$1=n_{1}=n_{2}<n=2,3$.}\end{array}\right.$

######  Remark 1.2.

For this case, the minimal GK-dimension is $2n-2$. From our paper [2], the
secondly minimal GK-dimension and the maximal GK-dimension are the same, i.e.,
$2n-1$. So $2n-1$ is larger than the maximal GK-dimension of any unitary
highest-weight modules.

The symplectic Lie algebra

$\displaystyle\hskip 28.45274pt\mathfrak{sp}(2n,\mathbb{C})$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{i=1}^{n}(\mathbb{C}E_{i,n+i}+\mathbb{C}E_{n+i,i})$
$\displaystyle+\sum_{1\leq i<j\leq
n}[\mathbb{C}(E_{i,n+j}+E_{j,n+i})+\mathbb{C}(E_{n+i,j}+E_{n+j,i})].$

We define the two-parameter $\mathbb{Z}$-graded oscillator representation of
$\mathfrak{sp}(2n,\mathbb{C})$ on ${\mathcal{B}}$ via (1.1)-(1.5).

The related variated Laplace operator becomes

(1.11)
${{D}}=\sum_{r=n_{1}+1}^{n}x_{r}\partial_{x_{r}}-\sum_{i=1}^{n_{1}}x_{i}\partial_{x_{i}}+\sum_{i=1}^{n_{2}}y_{i}\partial_{y_{i}}-\sum_{r=n_{2}+1}^{n}y_{r}\partial_{y_{r}}.$

Set

(1.12) ${\mathcal{B}}_{\langle
k^{\prime}\rangle}=\mbox{Span}\\{x^{\alpha}y^{\beta}\mid\alpha,\beta\in\mathbb{N}\>^{n},\sum_{r=n_{1}+1}^{n}\alpha_{r}-\sum_{i=1}^{n_{1}}\alpha_{i}+\sum_{i=1}^{n_{2}}\beta_{i}-\sum_{r=n_{2}+1}^{n}\beta_{r}=k^{\prime}\\}$

for $k^{\prime}\in\mathbb{Z}$. Then ${\mathcal{B}}_{\langle
k^{\prime}\rangle}=\\{f\in{\mathcal{B}}\mid{D}(f)=k^{\prime}f\\}$.

The following is the third main theorem of this paper.

######  Theorem 1.3.

For any $k^{\prime}\in\mathbb{Z}$, if the
$\mathfrak{sp}(2n,\mathbb{C})$-module ${\mathcal{B}}_{\langle
k^{\prime}\rangle}$ is irreducible, then it has the Gelfand-Kirillov dimension

(1.13) $d=2n-1.$

When $n_{1}=n_{2}$, the $\mathfrak{sp}(2n,\mathbb{C})$-module
${\mathcal{B}}_{\langle 0\rangle}$ also has the Gelfand-Kirillov dimension
$d=2n-1.$ When $n_{1}=n_{2}=n$, the two irreducible components of
${\mathcal{B}}_{\langle 0\rangle}$ also have the Gelfand-Kirillov dimension
$d=2n-1.$

######  Remark 1.3.

For this case, the minimal GK-dimension is $n$. From our paper [2], the
secondly minimal GK-dimension and is $2n-1$. So all the modules in the above
theorem have the secondly minimal GK-dimension.

In Section 2 we recall the basic definition of the Gelfand-Kirillov dimension.
In Section 3, 4, 5, we give a case-by-case proof for our three theorems.

##  2\. Preliminaries on Gelfand-Kirillov Dimension

In this section we recall the definition and some properties of the Gelfand-
Kirillov dimension. The details can be found in Refs.[3, 6, 10, 16, 17, 19].

######  Definition 2.1.

Let $A$ be an algebra (not necessarily associative) generated by a finite-
dimensional subspace $V$. Let $V^{n}$ denote the linear span of all products
of length at most $n$ in elements of $V$. The _Gelfand-Kirillov dimension_ of
$A$ is defined by:

$GKdim(A)=\limsup_{n\rightarrow\infty}\frac{\log\mathrm{dim}(V^{n})}{\log n}$

######  Remark 2.1.

It is well-known that the above definition is independent of the choice of the
finite dimensional generating subspace $V$ (see Ref.[3, 10]). Clearly
$GKdim(A)=0$ if and only if $\mathrm{dim}(A)<\infty$.

The notion of Gelfand-Kirillov dimension can be extended for left $A$-modules.
In fact, we have the following definition.

######  Definition 2.2.

Let $A$ be an algebra (not necessarily associative) generated by a finite-
dimensional subspace $V$. Let $M$ be a left $A$-module generated by a finite-
dimensional subspace $M_{0}$. Let $V^{n}$ denote the linear span of all
products of length at most $n$ in elements of $V$. The _Gelfand-Kirillov
dimension_ $GKdim(M)$ of $M$ is defined by

$GKdim(M)=\limsup_{n\rightarrow\infty}\frac{\log\mathrm{dim}(V^{n}M_{0})}{\log
n}.$

In particular, let $\mathfrak{g}$ be a complex Lie algebra. Let
$A=\mathcal{U}(\mathfrak{g})$ be the enveloping algebra of $\mathfrak{g}$,
with the standard filtration given by $A_{n}=\mathcal{U}_{n}(\mathfrak{g})$,
the subspace of $\mathcal{U}(\mathfrak{g})$ spanned by products of at most
$n$-elements of $\mathfrak{g}$. By the Poincaré-Birkhoff-Witt theorem (see
Knapp [9, Prop. 3.16]), the graded algebra
$\text{gr}(\mathcal{U}(\mathfrak{g}))$ is canonically isomorphic to the
symmetric algebra $S(\mathfrak{g})$. Suppose $M$ is a
$\mathcal{U}(\mathfrak{g})$-module generated by a finite-dimensional subspace
$M_{0}$. We set $M_{n}=\mathcal{U}_{n}(\mathfrak{g})M_{0}$. Denote
$\text{gr}M=\bigoplus\limits_{n=0}^{\infty}\text{gr}_{n}M$, where
$\text{gr}_{n}M=M_{n}/M_{n-1}$. Then $\text{gr}M$ becomes a graded
$S(\mathfrak{g})$-module. We denote $\dim(M_{n})$ by $\varphi_{M}(n)$. Then we
have the following lemma.

######  Lemma 2.1.

_(Hilbert-Serre[23, Chapter VII. Th.41] and [22])_

  1. (1)

With the notations as above, there exists a unique polynomial
$\tilde{\varphi}_{M}(n)$ such that $\varphi_{M}(n)=\tilde{\varphi}_{M}(n)$ for
large $n$. The leading term of $\tilde{\varphi}_{M}(n)$ is

$\frac{c(M)}{(d_{M})!}n^{d_{M}},$

where $c(M)$ is an integer.

  2. (2)

The degree $d_{M}$ of this polynomial $\tilde{\varphi}_{M}(n)$ is equal to the
dimension of the _associated variety_

$\mathscr{V}(M)=\\{X\in\mathfrak{g}^{*}|p(X)=0,\forall
p\in\mathrm{Ann}_{S(\mathfrak{g})}(\text{\emph{gr}}M)\\},$

where $\mathrm{Ann}_{S(\mathfrak{g})}(\text{\emph{gr}}M)=\\{D\in
S(\mathfrak{g})|Dv=0,\forall v\in\text{\emph{gr}}M\\}$ is the annihilator
ideal of $\mathrm{gr}M$ in $S(\mathfrak{g})$, and $S(\mathfrak{g})$ is
identified with the polynomial ring over $\mathfrak{g}^{*}$ through the
Killing form of $\mathfrak{g}$.

######  Remark 2.2.

From the definition of Gelfand-Kirillov dimension, we know

$GKdim(M)=\limsup_{n\rightarrow\infty}\frac{\log\dim(U_{n}(\mathfrak{g})M_{0})}{\log
n}=\limsup_{n\rightarrow\infty}\frac{\log\tilde{\varphi}_{M}(n)}{\log
n}=d_{M}=\dim\mathscr{V}(M).$

######  Example 2.1.

Let $M=\mathbb{C}[x_{1},...,x_{k}]$. Then $M$ is an algebra generated by the
finite-dimensional subspace $V=Span_{\mathbb{C}}\\{x_{1},...,x_{k}\\}$. So
$M_{n}=V^{n}=\bigoplus\limits_{0\leq q\leq n}P_{q}[x_{1},...,x_{k}]$ is the
subset of homogeneous polynomials of degree $\leq n$. Then

$\displaystyle\varphi_{M}(n)=$ $\displaystyle\sum\limits_{0\leq q\leq
n}\dim_{\mathbb{C}}(P_{q}[x_{1},...,x_{k}])$ $\displaystyle=$
$\displaystyle\sum\limits_{0\leq q\leq n}\binom{k+q-1}{q}$ $\displaystyle=$
$\displaystyle\binom{k+n}{n}$ $\displaystyle=$
$\displaystyle\frac{n^{k}}{k!}+O(n^{k-1}).$

Then we have $GKdim(M)=k.$

##  3\. Proof of the main theorem for $\mathfrak{o}(2n,\mathbb{C})$

We keep the same notations with the introduction. Through the paper we always
take $\mathcal{K}=\sum\limits_{i,j=1}^{n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})$, and
$\mathcal{K}_{+}=\sum_{1\leq i<j\leq n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})$. A
weight vector $v$ in $\mathcal{B}$ is called a _$\mathcal{K}$ -singular
vector_ if $\mathcal{K}_{+}(v)=0$.

We simply write $E_{i,j}|_{\mathcal{B}}$ as $E_{i,j}$. Take

(3.1) $\mathfrak{h}=\sum_{i=1}^{n}\mathbb{C}(E_{i,i}-E_{n+i,n+i})$

as a Cartan subalgebra of $\mathfrak{o}(2n,\mathbb{C})$ and the subspace
spanned by positive root vectors:

(3.2) $\mathfrak{o}(2n,\mathbb{C})_{+}=\sum_{1\leq i<j\leq
n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{1\leq i<j\leq
n}\mathbb{C}(E_{i,n+j}-E_{j,n+i}).$

Correspondingly, we have

(3.3) $\mathfrak{o}(2n,\mathbb{C})_{-}=\sum_{1\leq i<j\leq
n}\mathbb{C}(E_{j,i}-E_{n+i,n+j})+\sum_{1\leq i<j\leq
n}\mathbb{C}(E_{n+j,i}-E_{n+i,j}).$

If we take $\mathcal{P}_{+}=\sum_{1\leq i<j\leq
n}\mathbb{C}(E_{i,n+j}-E_{j,n+i})$, then
$\mathfrak{o}(2n,\mathbb{C})_{+}=\mathcal{K}_{+}+\mathcal{P}_{+}$. From the
PBW theorem we know that the irreducible $\mathfrak{o}(2n,\mathbb{C})$-module
${\mathcal{H}}_{\langle
k\rangle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g_{-}+\mathcal{P}_{+}})v_{\mathcal{K}}$
for any $\mathcal{K}$-singular vector $v_{\mathcal{K}}$. In the following we
will compute the Gelfand-Kirillov dimension of $\mathcal{H}_{\langle
k\rangle}$ in a case-by-case way.

Firstly we need the following two well-known lemmas.

######  Lemma 3.1.

_(Multinomial theorem)_
Let $n,m$ be two positive integers,then

(3.4)
$\left|\\{(k_{1},k_{2},...,k_{m})\in\mathbb{N}^{m}|\sum\limits_{i=1}^{m}k_{i}=n\\}\right|={n+m-1\choose
m-1}.$

######  Lemma 3.2.

Let $p,n$ be two positive integers, then

$\sum\limits_{i=0}^{n}i^{p}=\frac{(n+1)^{p+1}}{p+1}+\sum\limits_{k=1}^{p}\frac{B_{k}}{p-k+1}{p\choose
k}(n+1)^{p-k+1},$

where $B_{k}$ denotes a Bernoulli number.

From these two lemmas, we can get several propositions.

######  Proposition 3.1.

Let $k\in\mathbb{N}$ and we denote

$M_{k}=\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq t\leq
n\end{subarray}}({x_{i}x_{t}})^{p_{it}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ n_{1}+1\leq t\leq
n\end{subarray}}p_{it}=k,p_{it}\in\mathbb{N}\right\\}.$

Then

$d_{k}=\dim Span_{\mathbb{R}}M_{k}={n_{1}+k-1\choose k}{n-n_{1}+k-1\choose
k}\approx ak^{n-2},$

for some constant _a_.

###### Proof.

From the definition of $M_{k}$, we know that all the elements in $M_{k}$ are
monomials and they must form a basis for $Span_{\mathbb{R}}M_{k}$. Thus

$\displaystyle d_{k}=$ $\displaystyle\dim
Span_{\mathbb{R}}M_{k}=\\#\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{1}+1\leq t\leq
n\end{subarray}}({x_{i}x_{t}})^{p_{it}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ n_{1}+1\leq t\leq
n\end{subarray}}p_{it}=k,p_{it}\in\mathbb{N}\right\\}$ $\displaystyle=$
$\displaystyle\\#\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\end{subarray}}(x_{i})^{\sum_{n_{1}+1\leq t\leq
n}p_{it}}\prod\limits_{\begin{subarray}{c}n_{1}+1\leq t\leq
n\end{subarray}}(x_{t})^{\sum_{1\leq i\leq
n_{1}}p_{it}}|\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\ n_{1}+1\leq
t\leq n\end{subarray}}p_{it}=k,p_{it}\in\mathbb{N}\right\\}$ $\displaystyle=$
$\displaystyle{n_{1}+k-1\choose k}{n-n_{1}+k-1\choose k}\approx
ak^{n-2},\emph{~{}~{}for some constant~{}}\emph{a}.$

∎

The idea of the proof for the following propositions are very simple: Denote
${\mathcal{B}}=\mathbb{C}[x_{1},...,x_{n},y_{1},...,y_{n}]$. We define a
partial order(i.e, dictionary order) on the monomials of ${\mathcal{B}}$:

$x_{1}^{p_{1}}...x_{n}^{p_{n}}y_{1}^{p_{n+1}}...y_{n}^{p_{2n}}\preceq
x_{1}^{p^{\prime}_{1}}...x_{n}^{p^{\prime}_{n}}y_{1}^{p^{\prime}_{n+1}}...y_{n}^{p^{\prime}_{2n}}$

if there exists $1\leq m\leq 2n$, such that $p_{i}=p^{\prime}_{i}$ for any
$i<m$ and $p_{m}<p^{\prime}_{m}$. We can also interchange the place of $x_{1}$
and some $x_{j}$ (or $y_{j}$), then define a similar partial order.

Suppose $I=\cup\\{i\\}$ is a given index set and $P$ is a set of homogeneous
polynomials which are products of some binomials $(f_{i}-g_{i})^{a_{i}}$
($f_{i}$ and $g_{i}$ are monomials of degree $2$ in ${\mathcal{B}}$, and
$\sum\limits_{i\in I}a_{i}=k$ is a constant), i.e., $P=\\{\prod\limits_{i\in
I}(f_{i}-g_{i})^{a_{i}}|\sum a_{i}=k\\}$. We fix $i\in\overline{1,n}.$ We
choose two subsets $I_{1}$ and $I_{2}$ in $I$, such that $I_{1}\cap
I_{2}=\emptyset$, and $I_{1}\cup I_{2}=\cup\\{i\\}=I$. Suppose $f_{i_{p}}$ is
a multiple of $x_{i}$ and $g_{i_{p}}$ is not a multiple of $x_{i}$ when
$i_{p}\in I_{1}$, and $g_{i_{l}}$ is a multiple of $x_{i}$ and $f_{i_{l}}$ is
not a multiple of $x_{i}$ when $i_{l}\in I_{2}$.

Then we have

$\dim Span_{\mathbb{R}}P\geq\\#\\{\prod\limits_{i_{p}\in
I_{1}}(f_{i_{p}})^{a_{i_{p}}}\cdot\prod\limits_{i_{l}\in
I_{2}}(g_{i_{l}})^{a_{i_{l}}}|I=I_{1}\sqcup I_{2},\sum a_{i_{p}}+\sum
a_{i_{l}}=k\\},$

here we interchange the place of $x_{1}$ and $x_{i}$. So the monomial
$\prod\limits_{i_{p}\in
I_{1}}(f_{i_{p}})^{a_{i_{p}}}\cdot\prod\limits_{i_{l}\in
I_{2}}(g_{i_{l}})^{a_{i_{l}}}$ which contain the largest power of $x_{i}$ is
the leading term in the expression of $\prod\limits_{i\in
I}(f_{i}-g_{i})^{a_{i}}$.

######  Proposition 3.2.

  1. (1)

$(n_{1}=n_{2}=1)$ Let $k\in\mathbb{N}$ and we denote

$T_{k}=\left\\{\prod\limits_{\begin{subarray}{c}2\leq p<t\leq
n\end{subarray}}({x_{p}y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot\prod\limits_{2\leq
t\leq n}(x_{1}x_{t}-y_{1}y_{t})^{g_{1t}}|\sum\limits_{1\leq p<t\leq
n}g_{pt}=k,g_{pt}\in\mathbb{N}\right\\}.$

Then we have

$\displaystyle d_{k}=\dim
Span_{\mathbb{R}}T_{k}\approx\left\\{\begin{array}[]{ll}c_{0}k^{2n-4},&\text{\emph{if}~{}}{n=2\mathrm{~{}or~{}}n=3;}\\\
c_{1}k^{2n-3},&\text{\emph{if}~{}}{n=4;}\\\
c_{2}k^{2n-2},&\text{\emph{if}~{}}{n\geq 5.}\end{array}\right.$

Here $c_{0},c_{1}$ and $c_{2}$ are some positive constants which are
independent of $k$.

  2. (2)

$(n_{1}=n_{2}=n-1)$ Let $k\in\mathbb{N}$ and we denote

$S_{k}=\left\\{\prod\limits_{\begin{subarray}{c}1\leq i<r\leq
n-1\end{subarray}}({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}\cdot\prod\limits_{1\leq
i\leq n-1}(x_{i}x_{n}-y_{i}y_{n})^{f_{in}}|\sum\limits_{1\leq i<r\leq
n}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}.$

Then we have

$\displaystyle d_{k}=\dim
Span_{\mathbb{R}}S_{k}\approx\left\\{\begin{array}[]{ll}a_{0}k^{2n-4},&\text{\emph{if}~{}}{n=2\mathrm{~{}or~{}}n=3;}\\\
a_{1}k^{2n-3},&\text{\emph{if}~{}}{n=4;}\\\
a_{2}k^{2n-2},&\text{\emph{if}~{}}{n\geq 5.}\end{array}\right.$

Here $a_{0},a_{1}$ and $a_{2}$ are some positive constants which are
independent of $k$.

  3. (3)

$(n_{1}=n_{2}=n)$Let $k\in\mathbb{N}$ and we denote

$R_{k}=\left\\{\prod\limits_{\begin{subarray}{c}1\leq i<r\leq
n\end{subarray}}({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}|\sum\limits_{1\leq i<r\leq
n}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}.$

Then we have

$\displaystyle d_{k}=\dim
Span_{\mathbb{R}}R_{k}\approx\left\\{\begin{array}[]{ll}b_{0}k^{2n-4},&\text{\emph{if}~{}}{n=2\mathrm{~{}or~{}}n=3;}\\\
b_{1}k^{2n-3},&\text{\emph{if}~{}}{n=4;}\\\
b_{2}k^{2n-2},&\text{\emph{if}~{}}{n\geq 5.}\end{array}\right.$

Here $b_{0},b_{1}$ and $b_{2}$ are some positive constants which are
independent of $k$.

  4. (4)

$(1<n_{1}=n_{2}<n-1)$Suppose $1<n_{1}<n-1$. Let $k\in\mathbb{N}$ and we denote

$\displaystyle U_{k}$
$\displaystyle=\left\\{\prod\limits_{\begin{subarray}{c}n_{1}+1\leq p<t\leq
n\end{subarray}}({x_{p}y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot\prod\limits_{\begin{subarray}{c}1\leq
i<r\leq n_{1}\end{subarray}}({x_{i}y_{r}}-x_{r}y_{i})^{g_{ir}}\right.$
$\displaystyle\left.\cdot\prod\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq t\leq
n\end{subarray}}(x_{i}x_{t}-y_{i}y_{t})^{g_{1t}}|\sum\limits_{1\leq p<t\leq
n}g_{pt}=k,g_{pt}\in\mathbb{N}\right\\}.$

Then we have

$\displaystyle d_{k}=\dim
Span_{\mathbb{R}}U_{k}\approx\left\\{\begin{array}[]{ll}e_{1}k^{2n-3},&\text{\emph{if}~{}}{n=4;}\\\
e_{2}k^{2n-2},&\text{\emph{if}~{}}{n\geq 5.}\end{array}\right.$

Here $e_{0},e_{1}$ and $e_{2}$ are some positive constants which are
independent of $k$.

###### Proof.

The statements $(1)$ and $(2)$ are dual to each other. The proof of $(4)$ is
similar to $(1)$. So we only need to give the proof for $(1)$ and $(3)$.

Proof of $(1)$:

When $n=3$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}T_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{({x_{1}x_{2}}-y_{1}y_{2})^{g_{12}}({x_{1}x_{3}}-y_{1}y_{3})^{g_{13}}(x_{2}y_{3}-x_{3}y_{2})^{g_{23}}|\sum
g_{pt}=k\right\\}$ $\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{({x_{1}x_{2}})^{g_{12}}(x_{1}x_{3})^{g_{13}}(x_{2}y_{3})^{g_{23}}|\sum
g_{pt}=k\right\\}$ $\displaystyle=\dim
Span_{\mathbb{R}}\left\\{({x_{1}})^{g_{12}+g_{13}}(y_{3})^{g_{23}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\cdot(x_{2})^{g_{12}+g_{23}}(x_{3})^{g_{13}}\left.|\sum
g_{pt}=k\right\\}$ $\displaystyle\approx c_{00}k^{2},\emph{~{}for some
constant }c_{00}.$

On the other hand, we have $d_{k}=\dim Span_{\mathbb{R}}T_{k}\leq
c_{01}k^{3-1}=c_{01}k^{2},$ for some positive constant $c_{01}$. So we must
have $d_{k}=\dim Span_{\mathbb{R}}T_{k}\approx c_{0}k^{2}=c_{0}k^{2n-4},$ for
some positive constant $c_{0}$.

When $n=4$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}T_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{({x_{1}x_{2}}-y_{1}y_{2})^{g_{12}}({x_{1}x_{3}}-y_{1}y_{3})^{g_{13}}({x_{1}x_{4}}-y_{1}y_{4})^{g_{14}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.(x_{2}y_{3}-x_{3}y_{2})^{g_{23}}(x_{2}y_{4}-x_{4}y_{2})^{g_{24}}(x_{3}y_{4}-x_{4}y_{3})^{g_{34}}|\sum
g_{pt}=k\right\\}$ $\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{({y_{1}y_{2}})^{g_{12}}(x_{1}x_{3})^{g_{13}}(x_{1}x_{4})^{g_{14}}(x_{3}y_{2})^{g_{23}}(x_{2}y_{4})^{g_{24}}(x_{4}y_{3})^{g_{34}}|\sum
g_{pt}=k\right\\}$ $\displaystyle=\dim
Span_{\mathbb{R}}\left\\{({y_{2}})^{g_{12}+g_{23}}(x_{1})^{g_{13}+g_{14}}(y_{3})^{g_{34}}(y_{4})^{g_{24}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\cdot({y_{1}})^{g_{12}}(x_{3})^{g_{13}+g_{23}}(x_{4})^{g_{14}+g_{34}}(x_{2})^{g_{24}}\left.|\sum
g_{pt}=k\right\\}$ $\displaystyle\approx c_{10}k^{5},\emph{~{}for some
constant }c_{10}.$

On the other hand, we have $d_{k}=\dim Span_{\mathbb{R}}T_{k}\leq
c_{11}k^{6-1}=c_{11}k^{5},$ for some positive constant $c_{11}$. So we must
have $d_{k}=\dim Span_{\mathbb{R}}T_{k}\approx c_{1}k^{5}=c_{1}k^{2n-3},$ for
some positive constant $c_{1}$.

When $n\geq 5$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}T_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}2\leq p<t\leq
4\end{subarray}}({x_{p}y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot\prod\limits_{2\leq
t\leq 4}(x_{1}x_{t}-y_{1}y_{t})^{g_{1t}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.\cdot\prod\limits_{\begin{subarray}{c}2\leq
p<t\leq n\\\ t\geq
5\end{subarray}}({x_{p}y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot\prod\limits_{t\geq
5}(x_{1}x_{t}-y_{1}y_{t})^{g_{1t}}|\sum\limits_{1\leq p<t\leq
n}g_{pt}=k,\right\\}$ $\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{({y_{2}})^{g_{12}+g_{23}}(x_{1})^{g_{13}+g_{14}}(y_{3})^{g_{34}}(y_{4})^{g_{24}}\prod\limits_{5\leq
t\leq n}y_{t}^{g_{it}+g_{2t}}\prod\limits_{3\leq
p<n}y_{p}^{\sum\limits_{p<t\leq n}g_{pt}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\cdot({y_{1}})^{g_{12}+\sum\limits_{5\leq
t\leq
n}g_{1t}}(x_{3})^{g_{13}+g_{23}}(x_{4})^{g_{14}+g_{34}}(x_{2})^{g_{24}+\sum\limits_{5\leq
t\leq n}g_{2t}}\prod\limits_{5\leq t\leq n}x_{t}^{\sum\limits_{3\leq
p<t}g_{pt}}$ $\displaystyle\quad\quad\quad\quad\quad\quad\left.|\sum
g_{pt}=k\right\\}$ $\displaystyle\approx c_{20}k^{2n-2},\emph{~{}for some
constant }c_{20}.$

On the other hand, we have

$\displaystyle d_{k}=$ $\displaystyle\dim Span_{\mathbb{R}}T_{k}$
$\displaystyle\leq$ $\displaystyle\dim
Span_{\mathbb{R}}\left\\{\prod\limits({x_{1}})^{p_{1}}\prod({y_{t}})^{q_{t}}\cdot\prod({x_{t}})^{l_{t}}\prod(y_{1})^{f_{1}}|\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.p_{1}+\sum\limits_{2\leq t\leq
n}q_{t}=f_{1}+\sum\limits_{2\leq t\leq n}l_{t}=k\right\\}$
$\displaystyle\approx$ $\displaystyle c_{21}k^{2n-2},\emph{~{}for some
constant }c_{21}.$

So we must have $d_{k}=\dim Span_{\mathbb{R}}T_{k}\approx c_{2}k^{2n-2},$ for
some positive constant $c_{2}$.

Proof of $(3)$:

When $n=2$, we have $d_{k}=\dim Span_{\mathbb{R}}R_{k}=\dim
Span_{\mathbb{R}}\left\\{({x_{1}y_{2}}-x_{2}y_{1})^{k}\right\\}=1$.

When $n=3$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}R_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{({x_{1}y_{2}}-x_{2}y_{1})^{f_{12}}(x_{1}y_{3}-x_{3}y_{1})^{f_{13}}(x_{2}y_{3}-x_{3}y_{2})^{f_{23}}|f_{12}+f_{13}+f_{23}=k\right\\}$
$\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{({x_{1}y_{2}})^{f_{12}}(x_{3}y_{1})^{f_{13}}(x_{2}y_{3})^{f_{23}}|f_{12}+f_{13}+f_{23}=k\right\\}$
$\displaystyle={3+k-1\choose k}$ $\displaystyle\approx\frac{1}{2}k^{2}.$

On the other hand, we have $d_{k}=\dim Span_{\mathbb{R}}R_{k}\leq
b_{00}k^{3-1}=b_{00}k^{2},$ for some positive constant $b_{00}$. So we must
have $d_{k}=\dim Span_{\mathbb{R}}R_{k}\approx b_{0}k^{2}=b_{0}k^{2n-4},$ for
some positive constant $b_{0}$.

When $n=4$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}R_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i<r\leq
4\end{subarray}}({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}|\sum\limits_{1\leq i<r\leq
4}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}$ $\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{({x_{1}y_{2}})^{f_{12}}({x_{3}y_{1}})^{f_{13}}({x_{1}y_{4}})^{f_{14}}({x_{2}y_{3}})^{f_{23}}({x_{4}y_{2}})^{f_{24}}({x_{3}y_{4}})^{f_{34}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.|\sum\limits_{1\leq i<r\leq
4}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}$ $\displaystyle=\dim
Span_{\mathbb{R}}\left\\{({x_{1}})^{f_{12}+f_{14}}({x_{3}})^{f_{13}+f_{34}}({x_{4}})^{f_{24}}({x_{2}})^{f_{23}}\cdot({y_{2}})^{f_{12}+f_{24}}({y_{1}})^{f_{13}}({y_{4}})^{f_{14}+f_{34}}({y_{3}})^{f_{23}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.|\sum\limits_{1\leq i<r\leq
4}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}$ $\displaystyle\approx
b_{11}k^{5},\emph{~{}for some constant }b_{11}.$

On the other hand, we have $d_{k}=\dim Span_{\mathbb{R}}R_{k}\leq
b_{12}k^{6-1}=b_{12}k^{5},$ for some positive constant $b_{12}$. So we must
have $d_{k}=\dim Span_{\mathbb{R}}R_{k}\approx b_{1}k^{5}=b_{1}k^{2n-3},$ for
some positive constant $b_{1}$.

When $n\geq 5$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}R_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i<r\leq
4\end{subarray}}({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}\cdot\prod\limits_{\begin{subarray}{c}1\leq
i<r\leq n\\\ r\geq
5\end{subarray}}({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}|\sum\limits_{1\leq i<r\leq
n}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}$ $\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i<r\leq
4\end{subarray}}({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}\cdot\prod\limits_{5\leq
r\leq
n}({x_{1}y_{r}})^{f_{1r}}({x_{2}y_{r}})^{f_{2r}}({x_{r}y_{3}})^{f_{3r}}({x_{r}y_{4}})^{f_{4r}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.|\sum\limits_{1\leq
i<r\leq n}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}$ $\displaystyle=\dim
Span_{\mathbb{R}}\left\\{({x_{1}})^{f_{12}+f_{14}+\sum
f_{1r}}({x_{3}})^{f_{13}+f_{34}}({x_{4}})^{f_{24}}({x_{2}})^{f_{23}+\sum
f_{2r}}\prod\limits_{5\leq r\leq n}(x_{r})^{f_{3r}+f_{4r}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.\cdot({y_{2}})^{f_{12}+f_{24}}({y_{1}})^{f_{13}}({y_{4}})^{f_{14}+f_{34}+\sum
f_{4r}}({y_{3}})^{f_{23}+\sum f_{3r}}\prod\limits_{5\leq r\leq
n}(y_{r})^{f_{1r}+f_{2r}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.|\sum\limits_{1\leq
i<r\leq n}f_{ir}=k,f_{ir}\in\mathbb{N}\right\\}$ $\displaystyle\approx
b_{21}k^{2n-2},\emph{~{}for some constant }b_{21}.$

On the other hand, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}R_{k}$
$\displaystyle\leq\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq n\\\
1\leq r\leq n\end{subarray}}{x_{i}}^{a_{i}}y_{r}^{b_{r}}|\sum a_{i}=\sum
b_{r}=k\right\\}$ $\displaystyle\approx b_{22}k^{2n-2},$

for some positive constant $b_{22}$. So we must have $d_{k}=\dim
Span_{\mathbb{R}}R_{k}\approx b_{2}k^{2n-2},$ for some positive constant
$b_{2}$. ∎

######  Proposition 3.3.

  1. (1)

$(1<n_{1}<n_{2}=n-1)$Suppose $1<n_{1}<n-1$. Let $k\in\mathbb{N}$ and we denote

$\displaystyle V_{k}$
$\displaystyle=\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits(x_{s}y_{n})^{u_{s}}\prod\limits(y_{s}y_{n})^{q_{s}}\prod\limits({x_{i}x_{n}}-y_{i}y_{n})^{h_{i}}\prod\limits({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}|\right.$
$\displaystyle\quad\left.\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq s\leq
n-1\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq s\leq
n-1\end{subarray}}l_{is}+\sum\limits_{\begin{subarray}{c}n_{1}+1\leq s\leq
n-1\end{subarray}}u_{s}+\sum\limits_{\begin{subarray}{c}n_{1}+1\leq s\leq
n-1\end{subarray}}q_{s}\right.$
$\displaystyle\quad\quad\left.+\sum\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\end{subarray}}h_{i}+\sum\limits_{1\leq i<r\leq n_{1}}=k\right\\},$

then

$d_{k}=\dim Span_{\mathbb{R}}V_{k}\approx bk^{2n-2},$

for some constant $b$.

  2. (2)

$(1=n_{1}<n_{2}<n-1)$Suppose $1<n_{2}<n-1$. Let $k\in\mathbb{N}$ and we denote

$\displaystyle W_{k}$
$\displaystyle=\left\\{\prod\limits(x_{1}x_{s})^{p_{s}}\prod\limits(x_{1}y_{s})^{l_{s}}\prod\limits(x_{s}y_{t})^{u_{st}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits({x_{1}x_{t}}-y_{1}y_{t})^{h_{t}}\prod\limits({x_{p}y_{t}}-x_{t}y_{p})^{g_{pt}}|\right.$
$\displaystyle\quad\left.\sum\limits_{2\leq s\leq
n_{2}}p_{s}+\sum\limits_{2\leq s\leq
n_{2}}l_{s}+\sum\limits_{\begin{subarray}{c}2\leq s\leq n_{2}\\\ n_{2}+1\leq
t\leq n\end{subarray}}u_{st}+\sum\limits_{\begin{subarray}{c}2\leq s\leq
n_{2}\\\ n_{2}+1\leq t\leq
n\end{subarray}}q_{st}+\sum\limits_{\begin{subarray}{c}n_{2}+1\leq t\leq
n\end{subarray}}h_{t}+\sum\limits_{n_{2}+1\leq p<t\leq n}g_{pt}=k\right\\},$

then

$d_{k}=\dim Span_{\mathbb{R}}W_{k}\approx\beta k^{2n-2},$

for some constant $\beta$.

  3. (3)

$(1<n_{1}<n_{2}=n)$

Suppose $1<n_{1}<n$. Let $k\in\mathbb{N}$ and we denote

$\displaystyle Z_{k}$
$\displaystyle=\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}|\right.$
$\displaystyle\quad\quad\left.\sum\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{1}+1\leq s\leq
n\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq s\leq n\end{subarray}}l_{is}+\sum\limits_{\begin{subarray}{c}1\leq
i<r\leq n_{1}\end{subarray}}f_{ir}=k\right\\},$

then

$\displaystyle d_{k}=\dim
Span_{\mathbb{R}}Z_{k}\approx\left\\{\begin{array}[]{ll}\alpha_{0}k^{2n-3},&\text{\emph{if}~{}}{n_{1}=2<n;}\\\
\alpha_{1}k^{2n-2},&\text{\emph{if}~{}}{3\leq n_{1}<n.}\end{array}\right.$

Here $\alpha_{0}$ and $\alpha_{1}$ are some positive constants which are
independent of $k$.

  4. (4)

$(1<n_{1}<n_{2}<n-1)$

Let $k\in\mathbb{N}$. Suppose $1<n_{1}<n_{2}<n-1$ and we denote

$\displaystyle N^{\prime}_{k}$
$\displaystyle=\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits(x_{s}y_{t})^{u_{st}}\right.$
$\displaystyle\quad\left.\cdot\prod\limits({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}}\prod\limits({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}\prod\limits({x_{p}y_{t}}-x_{t}y_{p})^{g_{pt}}\right.$
$\displaystyle\quad\left.\mid\sum\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{1}+1\leq s\leq
n_{2}\end{subarray}}(p_{is}+l_{is})+\sum\limits_{\begin{subarray}{c}n_{1}+1\leq
s\leq n-1\\\ n_{2}+1\leq t\leq n\end{subarray}}(u_{st}+q_{st})\right.$
$\displaystyle\quad\left.+\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{2}+1\leq t\leq n\end{subarray}}h_{it}+\sum\limits_{1\leq i<r\leq
n_{1}}f_{ir}+\sum\limits_{n_{2}+1\leq p<t\leq n_{1}}g_{pt}=k\right\\},$

then

$d_{k}=\dim Span_{\mathbb{R}}N^{\prime}_{k}\approx ck^{2n-2},$

for some constant $c$.

###### Proof.

The statements $(1)$ and $(2)$ are dual to each other. Their proofs are
similar to the proof of $(4)$. So we only need to prove $(3)$ and $(4)$.

Proof of $(3)$: When $n_{1}=2$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}({x_{1}y_{2}}-x_{2}y_{1})^{f_{12}}|\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.\sum\limits_{\begin{subarray}{c}1\leq
i\leq 2\\\ n_{1}+1\leq s\leq
n\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}1\leq i\leq 2\\\ 3\leq
s\leq n\end{subarray}}l_{is}+f_{12}=k\right\\}$ $\displaystyle=$
$\displaystyle\dim Span_{\mathbb{R}}\left\\{\prod\limits_{1\leq i\leq
2}(x_{i})^{\sum\limits_{3\leq s\leq n}p_{is}+l_{is}}\cdot\prod\limits_{3\leq
s\leq n}(x_{s})^{\sum\limits_{1\leq i\leq 2}p_{is}}\prod\limits_{3\leq s\leq
n}(y_{s})^{\sum\limits_{1\leq i\leq 2}l_{is}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.\cdot\prod\limits({x_{1}y_{2}}-x_{2}y_{1})^{f_{12}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 2\\\ 3\leq s\leq
n\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}1\leq i\leq 2\\\ 3\leq
s\leq n\end{subarray}}l_{is}+f_{12}=k\right\\}$
$\displaystyle\approx\alpha_{0}k^{2n-3},\emph{~{}for some constant
}\alpha_{0}.$

When $n_{1}>2$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}|\right.$
$\displaystyle\quad\quad\left.\sum\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{1}+1\leq s\leq
n\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq s\leq n\end{subarray}}l_{is}+\sum\limits_{\begin{subarray}{c}1\leq
i<r\leq n_{1}\end{subarray}}f_{ir}=k\right\\}$ $\displaystyle\geq$
$\displaystyle\dim Span_{\mathbb{R}}\left\\{\prod\limits_{1\leq i\leq
n_{1}}(x_{i})^{\sum\limits_{n_{1}+1\leq s\leq
n}p_{is}+l_{is}}\cdot\prod\limits_{n_{1}+1\leq s\leq
n}(x_{s})^{\sum\limits_{1\leq i\leq n_{1}}p_{is}}\prod\limits_{n_{1}+1\leq
s\leq n}(y_{s})^{\sum\limits_{1\leq i\leq n_{1}}l_{is}}\right.$
$\displaystyle\quad\quad\quad\left.\cdot\prod\limits_{2\leq i<r\leq
n_{1}}(x_{r}y_{i})^{f_{ir}}\cdot(x_{1}y_{n_{1}})^{f_{1n_{1}}}\cdot\prod\limits_{2\leq
r<n_{1}}(x_{r}y_{1})^{f_{1r}}\right.$
$\displaystyle\left.\quad\quad|\sum\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{1}+1\leq s\leq
n\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq s\leq n\end{subarray}}l_{is}+\sum\limits_{\begin{subarray}{c}1\leq
i<r\leq n_{1}\end{subarray}}f_{ir}=k\right\\}$ $\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{1\leq i\leq
n_{1}}(x_{i})^{\sum\limits_{n_{1}+1\leq s\leq
n}p_{is}+l_{is}}\cdot\prod\limits_{2<r\leq
n_{1}}(x_{r})^{\sum\limits_{i<r}f_{ir}}\cdot(x_{1})^{f_{1n_{1}}}\cdot\prod\limits_{2\leq
r<n_{1}}(x_{r})^{f_{1r}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.\cdot\prod\limits_{n_{1}+1\leq
s\leq n}(x_{s})^{\sum\limits_{1\leq i\leq
n_{1}}p_{is}}\prod\limits_{n_{1}+1\leq s\leq n}(y_{s})^{\sum\limits_{1\leq
i\leq n_{1}}l_{is}}\cdot\prod\limits_{2\leq
i<n_{1}}(y_{i})^{\sum\limits_{i<r\leq n_{1}}f_{ir}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.\cdot\prod\limits_{2\leq
r<n_{1}}(y_{1})^{f_{1r}}\cdot(y_{n_{1}})^{f_{1n_{1}}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 2\\\ 3\leq s\leq
n\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}1\leq i\leq 2\\\ 3\leq
s\leq n\end{subarray}}l_{is}+f_{12}=k\right\\}$
$\displaystyle\approx\alpha_{10}k^{2n-2},\emph{~{}for some constant
}\alpha_{10}.$

On the other hand, we have

$\displaystyle d_{k}=$ $\displaystyle\dim Span_{\mathbb{R}}Z_{k}$
$\displaystyle\leq$ $\displaystyle\dim
Span_{\mathbb{R}}\left\\{\prod\limits({x_{i}})^{p_{i}}\cdot\prod(x_{s})^{a_{s}}\prod(y_{s})^{b_{s}}\prod(y_{i})^{f_{i}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.|\sum p_{i}=\sum f_{i}+\sum
a_{s}+\sum b_{s}=k\right\\}$ $\displaystyle\approx$
$\displaystyle\alpha_{11}k^{2n-2},\emph{~{}for some constant }\alpha_{11}.$

So we must have $d_{k}=\dim Span_{\mathbb{R}}Z_{k}\approx\alpha_{1}k^{2n-2},$
for some positive constant $\alpha_{1}$.

Proof of $(4)$:

When $n_{1}=2<n_{2}<n-1$, we have

$\displaystyle d_{k}=\dim
Span_{\mathbb{R}}\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits(x_{s}y_{t})^{u_{st}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.\prod\limits({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}}\prod\limits({x_{i}y_{r}}-x_{r}y_{i})^{f_{ir}}\prod\limits({x_{p}y_{t}}-x_{t}y_{p})^{g_{pt}}|\right.$
$\displaystyle\left.\quad\quad\quad\quad\quad\quad\quad\sum\limits
p_{is}+\sum\limits q_{st}+\sum\limits l_{is}+\sum\limits u_{st}+\sum\limits
h_{it}+\sum\limits f_{ir}+\sum\limits g_{pt}=k\right\\}$
$\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits(x_{i}y_{s})^{l_{is}}\prod\limits(x_{s}y_{t})^{u_{st}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.\prod\limits_{n_{2}+2\leq
t\leq
n}({x_{1}x_{t}})^{h_{1t}}\cdot(y_{1}y_{n_{2}+1})^{h_{1,n_{2}+1}}\cdot(x_{2}x_{n_{2}+1})^{h_{2,n_{2}+1}}\cdot\prod\limits_{n_{2}+2\leq
t\leq n}(y_{2}y_{t})^{h_{2t}}\right.$
$\displaystyle\left.\quad\quad\quad\quad\quad\quad\quad\prod\limits({x_{1}y_{2}}-x_{2}y_{1})^{f_{12}}\prod\limits(x_{t}y_{n_{2}+1})^{g_{n_{2}+1,t}}|\right.$
$\displaystyle\left.\quad\quad\quad\quad\quad\quad\quad\sum\limits
p_{is}+\sum\limits q_{st}+\sum\limits l_{is}+\sum\limits u_{st}+\sum\limits
h_{it}+\sum\limits f_{ir}+\sum\limits g_{pt}=k\right\\}$
$\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{\left(({x_{2}})^{h_{2,n_{2}+1}+\sum(p_{2s}+l_{2s})}(x_{1})^{\sum(p_{1s}+l_{1s})+f_{12}}(\prod\limits_{n_{2}+2\leq
t\leq n}(x_{1})^{h_{1t}})\right.\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.\left.\prod\limits_{n_{2}+2\leq
t\leq
n}(y_{t})^{h_{2t}}(y_{t})^{\sum(q_{st}+u_{st})}({y_{n_{2}+1}})^{h_{1,n_{2}+1}+\sum
g_{n_{2}+1,t}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.\cdot\left(\prod\limits(x_{s})^{p_{1s}+p_{2s}+\sum
u_{st}}(x_{n_{2}+1})^{h_{2,n_{2}+1}}(\prod\limits_{n_{2}+2\leq t\leq
n}(x_{t})^{h_{1t}+g_{n_{2}+1,t}})\right.\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.\left.(\prod\limits(y_{s})^{\sum
q_{st}+\sum l_{is}})({y_{1}})^{h_{1,n_{2}+1}}\prod\limits_{n_{2}+2\leq t\leq
n}(y_{2})^{h_{2t}}y_{2}^{f_{12}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.|\sum\limits
p_{is}+\sum\limits q_{st}+\sum\limits l_{is}+\sum\limits u_{st}+\sum\limits
h_{it}+\sum\limits f_{ir}+\sum\limits g_{pt}=k\right\\}$ $\displaystyle\approx
c_{0}k^{2n-2},\emph{~{}for some constant }c_{0}.$

On the other hand, we have

(3.5) $\displaystyle d_{k}=$ $\displaystyle\dim
Span_{\mathbb{R}}N^{\prime}_{k}$ (3.6) $\displaystyle\leq$ $\displaystyle\dim
Span_{\mathbb{R}}\left\\{\prod\limits({x_{i}})^{p_{i}}\prod({y_{t}})^{q_{t}}\cdot\prod(x_{s})^{a_{s}}\prod(y_{s})^{b_{s}}\prod({x_{t}})^{l_{t}}\prod(y_{i})^{f_{i}}|\right.$
(3.7) $\displaystyle\quad\quad\quad\quad\quad\left.\sum p_{i}+\sum q_{t}=\sum
l_{t}+\sum f_{i}+\sum a_{s}+\sum b_{s}=k\right\\}$ (3.8)
$\displaystyle\approx$ $\displaystyle c_{00}k^{2n-2},\emph{~{}for some
constant }c_{00}.$

So we must have $d_{k}=\dim Span_{\mathbb{R}}N^{\prime}_{k}\approx ck^{2n-2},$
for some positive constant $c$.

When $n_{1}>2$, we have a similar argument. And for these cases we still have
$d_{k}=\dim Span_{\mathbb{R}}N^{\prime}_{k}\approx ck^{2n-2},$ for some
positive constant $c$.

∎

Next, we will compute the Gelfand-Kirillov dimensions of our modules in a
case-by-case way.

Luo and Xu [12] proved that for any $n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}\geq
k^{\prime}\in\mathbb{Z}$, ${\mathcal{H}}_{\langle k^{\prime}\rangle}$ is an
irreducible $\mathfrak{o}(2n,\mathbb{C})$-module. Moreover, the homogeneous
subspace $\mathcal{B}_{\langle
k^{\prime}\rangle}=\bigoplus_{i=0}^{\infty}\eta^{i}(\mathcal{H}_{\langle
k^{\prime}-2i\rangle})$ is a direct sum of irreducible submodules. The module
${\mathcal{H}}_{\langle k^{\prime}\rangle}$ under the assumption is of
highest-weight type only if $n_{2}=n$, in which case $x_{n_{1}}^{-k^{\prime}}$
is a highest-weight vector with weight
$-k^{\prime}\lambda_{n_{1}-1}+(k^{\prime}-1)\lambda_{n_{1}}+[(k^{\prime}-1)\delta_{n_{1},n-1}-2k^{\prime}\delta_{n_{1},n}]\lambda_{n}$.

###  3.1. Case 1. $n_{1}+1\leq n_{2}$ and $n_{1}-n_{2}+1\geq
k^{\prime}\in\mathbb{Z}$.

In this case we have:

(3.9) $\displaystyle(E_{r,i}-E_{n+i,n+r})|_{\mathcal{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}$ for
$\displaystyle 1\leq i<r\leq n_{1},$ (3.10)
$\displaystyle(E_{s,i}-E_{n+i,n+s})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}x_{s}}-{y_{i}}\partial_{y_{s}}$ for $\displaystyle
i\in\overline{1,n_{1}},\;s\in\overline{n_{1}+1,n_{2}},$ (3.11)
$\displaystyle(E_{t,i}-E_{n+i,n+t})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}x_{t}}+y_{i}y_{t}$ for $\displaystyle
i\in\overline{1,n_{1}},\;t\in\overline{n_{2}+1,n},$ (3.12)
$\displaystyle(E_{s,j}-E_{n+j,n+s})|_{\mathcal{B}}$
$\displaystyle=x_{s}\partial_{x_{j}}-y_{j}\partial_{y_{s}}$ for $\displaystyle
n_{1}<j<s\leq n_{2},$ (3.13)
$\displaystyle(E_{t,s}-E_{n+s,n+t})|_{\mathcal{B}}$
$\displaystyle=x_{t}\partial_{x_{s}}+y_{s}y_{t}$ for $\displaystyle
s\in\overline{n_{1}+1,n_{2}},\;t\in\overline{n_{2}+1,n},$ (3.14)
$\displaystyle(E_{t,p}-E_{n+p,n+t})|_{\mathcal{B}}$
$\displaystyle=x_{t}\partial_{x_{p}}+y_{t}\partial_{y_{p}}$ for $\displaystyle
n_{2}+1\leq p<t\leq n,$ (3.15)
$\displaystyle(E_{i,n+r}-E_{r,n+i})|_{\mathcal{B}}$
$\displaystyle=\partial_{x_{i}}\partial_{y_{r}}-\partial_{x_{r}}\partial_{y_{i}}$
for $\displaystyle 1\leq i<r\leq n_{1},$ (3.16)
$\displaystyle(E_{n+r,i}-E_{n+i,r})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}}{y_{r}}+{x_{r}}{y_{i}}$ for $\displaystyle 1\leq
i<r\leq n_{1},$ (3.17) $\displaystyle(E_{i,n+s}-E_{s,n+i})|_{\mathcal{B}}$
$\displaystyle=\partial_{x_{i}}\partial_{y_{s}}-{x_{s}}\partial_{y_{i}}$ for
$\displaystyle i\in\overline{1,n_{1}},\;s\in\overline{n_{1}+1,n_{2}},$ (3.18)
$\displaystyle(E_{n+s,i}-E_{n+i,s})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}}{y_{s}}-{y_{i}}\partial_{x_{s}}$ for $\displaystyle
i\in\overline{1,n_{1}},\;s\in\overline{n_{1}+1,n_{2}},$ (3.19)
$\displaystyle(E_{i,n+t}-E_{t,n+i})|_{\mathcal{B}}$
$\displaystyle=-y_{t}\partial_{x_{i}}-x_{t}\partial_{y_{i}}$ for
$\displaystyle i\in\overline{1,n_{1}},\;t\in\overline{n_{2}+1,n},$ (3.20)
$\displaystyle(E_{n+t,i}-E_{n+i,t})|_{\mathcal{B}}$
$\displaystyle=-x_{i}\partial_{y_{t}}-y_{i}\partial_{x_{t}}$ for
$\displaystyle i\in\overline{1,n_{1}},\;t\in\overline{n_{2}+1,n},$ (3.21)
$\displaystyle(E_{j,n+s}-E_{s,n+j})|_{\mathcal{B}}$
$\displaystyle=x_{j}\partial_{y_{s}}-x_{s}\partial_{y_{j}}$ for $\displaystyle
n_{1}<j<s\leq n_{2},$ (3.22)
$\displaystyle(E_{n+j,s}-E_{n+s,j})|_{\mathcal{B}}$
$\displaystyle=-y_{s}\partial_{x_{j}}+y_{j}\partial_{x_{s}}$ for
$\displaystyle n_{1}<j<s\leq n_{2},$ (3.23)
$\displaystyle(E_{s,n+t}-E_{t,n+s})|_{\mathcal{B}}$
$\displaystyle=-x_{s}{y_{t}}-x_{t}\partial_{y_{s}}$ for $\displaystyle
s\in\overline{n_{1}+1,n_{2}},\;t\in\overline{n_{2}+1,n},$ (3.24)
$\displaystyle(E_{n+s,t}-E_{n+t,s})|_{\mathcal{B}}$
$\displaystyle=-\partial_{x_{s}}\partial_{y_{t}}-y_{s}\partial_{x_{t}}$ for
$\displaystyle s\in\overline{n_{1}+1,n_{2}},\;t\in\overline{n_{2}+1,n},$
(3.25) $\displaystyle(E_{p,n+t}-E_{t,n+p})|_{\mathcal{B}}$
$\displaystyle=-x_{p}{y_{t}}+x_{t}{y_{p}}$ for $\displaystyle n_{2}+1\leq
p<t\leq n,$ (3.26) $\displaystyle(E_{n+p,t}-E_{n+t,p})|_{\mathcal{B}}$
$\displaystyle=-\partial_{x_{p}}\partial_{y_{t}}+\partial_{x_{t}}\partial_{y_{p}}$
for $\displaystyle n_{2}+1\leq p<t\leq n.$

Then the above root elements form a basis for the subalgebra
$\mathfrak{g}(\mathcal{P}_{+})_{-}:=\mathfrak{o}(2n,\mathbb{C})_{-}+\mathcal{P}_{+}$.

From Luo-Xu [12] we know that the $\mathcal{K}$-singular vectors in
${\mathcal{H}}_{\langle k^{\prime}\rangle}$ are:

(3.27)
$x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}\qquad\mbox{with}\;-(m_{1}+m_{2})=k^{\prime},$
(3.28)
$x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{m_{2}}\qquad\mbox{with}\;m_{1}-m_{2}=k^{\prime},$
(3.29)
$x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}}\qquad\mbox{with}\;-m_{1}+m_{2}=k^{\prime},$

for all possible $m_{1},m_{2}\in\mathbb{N}$. When $n_{1}+1=n_{2}=n$, the
$\mathcal{K}$-singular vectors in ${\mathcal{H}}_{\langle k^{\prime}\rangle}$
are those in $(\ref{n})$.

Let $\mathfrak{g}_{1}$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$
spanned by the root vectors in the following set:

$I_{1}:=\\{(\ref{ri}),(\ref{sj}),(\ref{tp}),(\ref{inr}),(\ref{ins}),(\ref{int}),(\ref{nti}),(\ref{jns}),(\ref{njs}),(\ref{nst}),(\ref{npt})\\}.$

Let $\mathfrak{g}_{2}$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$
spanned by the root vectors in the following set:

$I_{2}:=\\{(\ref{si}),(\ref{ti}),(\ref{ts}),(\ref{nri}),(\ref{nsi}),(\ref{snt}),(\ref{pnt})\\}.$

So we get
$U(\mathfrak{g}(\mathcal{P}_{+})_{-})=U(\mathfrak{g}_{2})U(\mathfrak{g}_{1}).$
From the construction of the root vectors we have the following lemma.

######  Lemma 3.3.

Every root vector in $\mathfrak{g}_{1}$ acts locally nilpotently on
$\mathcal{H}_{\langle k^{\prime}\rangle}$ and every root vector in
$\mathfrak{g}_{2}$ acts torsion-freely (injectively) on $\mathcal{H}_{\langle
k^{\prime}\rangle}$.

We take a $\mathcal{K}$-singular vector
$v_{\mathcal{K}}=x_{n_{1}}^{-k^{\prime}}$, and set
$M_{0}=U(\mathfrak{g}_{1})x_{n_{1}}^{-k^{\prime}}$. Then $M_{0}$ is finite-
dimensional from the above lemma.

Thus

${\mathcal{H}}_{\langle
k^{\prime}\rangle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g_{-}+\mathcal{P}_{+}})v_{\mathcal{K}}=U(\mathfrak{g}_{2})M_{0}.$

Let $k$ be any positive integer. We want to compute
$\mathrm{dim}(U_{k}(\mathfrak{g}_{2})M_{0})$, and then get the Gelfand-
Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$.

The cardinality of a set $A$ is usually denoted by $|A|$.

We use $E_{p}$ to stand for the root vector in the equation $(p)$. And denote

$\displaystyle N_{0}(k)=$ $\displaystyle\left\\{\left(\prod
E_{\ref{si}}^{p_{is}}\prod E_{\ref{ti}}^{h_{it}}\prod
E_{\ref{ts}}^{q_{st}}\prod E_{\ref{nri}}^{f_{ir}}\prod
E_{\ref{nsi}}^{l_{is}}\prod E_{\ref{snt}}^{u_{st}}\prod
E_{\ref{pnt}}^{g_{pt}}\right)v_{\mathcal{K}}\right.$ $\displaystyle\left.|\sum
p_{is}+\sum h_{it}+\sum q_{st}+\sum f_{ir}+\sum l_{is}+\sum u_{st}+\sum
g_{pt}=k,\right.$
$\displaystyle\left.p_{is},h_{it},q_{st},f_{ir},l_{is},u_{st},g_{pt}\in\mathbb{N}\right\\}.$

From the definition we know

$\displaystyle\left(\prod E_{\ref{si}}^{p_{is}}\prod
E_{\ref{ti}}^{h_{it}}\prod E_{\ref{ts}}^{q_{st}}\prod
E_{\ref{nri}}^{f_{ir}}\prod E_{\ref{nsi}}^{l_{is}}\prod
E_{\ref{snt}}^{u_{st}}\prod E_{\ref{pnt}}^{g_{pt}}\right)v_{\mathcal{K}}$
$\displaystyle=$
$\displaystyle\left(\prod(-{x_{i}x_{s}}-{y_{i}}\partial_{y_{s}})^{p_{is}}\prod(-{x_{i}x_{t}}+y_{i}y_{t})^{h_{it}}\prod(y_{s}y_{t})^{q_{st}}\prod(-x_{i}y_{r}+x_{r}y_{i})^{f_{ir}}\right.$
$\displaystyle\left.\cdot\prod(-x_{i}y_{s}-y_{i}\partial_{x_{s}})^{l_{is}}\prod(-x_{s}y_{t}-x_{t}\partial_{y_{s}})^{u_{st}}\right)\cdot
x_{n_{1}}^{-k^{\prime}}$ $\displaystyle=$
$\displaystyle\left(\prod(-{x_{i}x_{s}})^{p_{is}}\prod(-{x_{i}x_{t}}+y_{i}y_{t})^{h_{it}}\prod(y_{s}y_{t})^{q_{st}}\prod(-x_{i}y_{r}+x_{r}y_{i})^{f_{ir}}\right.$
$\displaystyle\left.\cdot\prod(-x_{i}y_{s})^{l_{is}}\prod(-x_{s}y_{t})^{u_{st}}\right)\cdot
x_{n_{1}}^{-k^{\prime}}$ $\displaystyle+\text{lower degree polynomials of
}~{}y_{s}\text{~{}and~{}}x_{s}.$

Now we suppose $1<n_{1}<n_{2}<n-1$. Then we must have

$\dim Span_{\mathbb{R}}N_{0}(m)\geq d_{m}=\dim
Span_{\mathbb{R}}N^{\prime}_{m}.$

Using the same idea with the proof of Proposition 3.3 (4), we can also get

$\dim Span_{\mathbb{R}}N_{0}(m)\leq d_{m}=\dim
Span_{\mathbb{R}}N^{\prime}_{m}.$

Thus $\dim Span_{\mathbb{R}}N_{0}(m)=d_{m}=\dim
Span_{\mathbb{R}}N^{\prime}_{m}.$

Then using Proposition 3.3 (4), we can get

$\displaystyle\dim Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq k}N_{0}(m))$
$\displaystyle=$ $\displaystyle\sum\limits_{0\leq m\leq k}\dim
Span_{\mathbb{R}}N^{\prime}_{m}$ $\displaystyle=$ $\displaystyle
c\sum\limits_{0\leq m\leq k}m^{2n-2}$ $\displaystyle=$ $\displaystyle
c^{\prime}k^{2n-1}$

Also we have

$\dim Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq
k}N_{0}(m))\leq\mathrm{dim}(U_{k}(\mathfrak{g}_{2})M_{0})\leq\dim{M_{0}}\dim
Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq k}N_{0}(m)).$

Then from the definition, we know that the Gelfand-Kirillov dimension of
$U(\mathfrak{g}_{2})M_{0}$ is

$d=2n-1,\text{if $1<n_{1}<n_{2}<n-1$.}$

When $n_{1}=1<n_{2}<n-1$ or $1<n_{1}<n_{2}=n-1$, through a similar argument
and using Proposition 3.3 (1) and 3.3(2), we find that the Gelfand-Kirillov
dimension of $U(\mathfrak{g}_{2})M_{0}$ is

$d=2n-1.$

When $1=n_{1}<n_{2}=n-1$ or $1=n_{1}<n_{2}=n$, through a similar argument and
using Proposition 3.1, we find that the Gelfand-Kirillov dimension of
$U(\mathfrak{g}_{2})M_{0}$ is

$d=2n-2.$

When $1<n_{1}<n_{2}=n$, through a similar argument and using Proposition 3.3
(3), we find that the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$
is

$d=\left\\{\begin{array}[]{ll}2n-2,&\hbox{if $n_{1}=2<n_{2}=n$;}\\\
2n-1,&\hbox{if $3\leq n_{1}<n_{2}=n$.}\end{array}\right.$

Therefore, for this case $n_{1}+1\leq n_{2}$, the Gelfand-Kirillov dimension
of $U(\mathfrak{g}_{2})M_{0}$ is

$\displaystyle d=\left\\{\begin{array}[]{ll}2n-2,&\hbox{if $n_{1}=1,n_{2}=n-1$
~{}or~{} $n_{1}=1,2,n_{2}=n$;}\\\ 2n-1,&\hbox{if $n_{1}=1<n_{2}<n-1$ ~{}or~{}
$1<n_{1}<n_{2}\leq n-1$ ~{}or~{} $3\leq n_{1}<n_{2}=n$.}\end{array}\right.$

###  3.2. Case 2. $n_{1}=n_{2}$ _and_ $0\geq k^{\prime}\in\mathbb{Z}$.

In this case we have:

(3.30) $\displaystyle(E_{r,i}-E_{n+i,n+r})|_{\mathcal{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}$ for
$\displaystyle 1\leq i<r\leq n_{1},$ (3.31)
$\displaystyle(E_{t,i}-E_{n+i,n+t})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}x_{t}}+y_{i}y_{t}$ for $\displaystyle
i\in\overline{1,n_{1}},\;t\in\overline{n_{1}+1,n},$ (3.32)
$\displaystyle(E_{t,p}-E_{n+p,n+t})|_{\mathcal{B}}$
$\displaystyle=x_{t}\partial_{x_{p}}+y_{t}\partial_{y_{p}}$ for $\displaystyle
n_{1}+1\leq p<t\leq n,$ (3.33)
$\displaystyle(E_{i,n+r}-E_{r,n+i})|_{\mathcal{B}}$
$\displaystyle=\partial_{x_{i}}\partial_{y_{r}}-\partial_{x_{r}}\partial_{y_{i}}$
for $\displaystyle 1\leq i<r\leq n_{1},$ (3.34)
$\displaystyle(E_{n+r,i}-E_{n+i,r})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}}{y_{r}}+{x_{r}}{y_{i}}$ for $\displaystyle 1\leq
i<r\leq n_{1},$ (3.35) $\displaystyle(E_{i,n+t}-E_{t,n+i})|_{\mathcal{B}}$
$\displaystyle=-y_{t}\partial_{x_{i}}-x_{t}\partial_{y_{i}}$ for
$\displaystyle i\in\overline{1,n_{1}},\;t\in\overline{n_{1}+1,n},$ (3.36)
$\displaystyle(E_{n+t,i}-E_{n+i,t})|_{\mathcal{B}}$
$\displaystyle=-x_{i}\partial_{y_{t}}-y_{i}\partial_{x_{t}}$ for
$\displaystyle i\in\overline{1,n_{1}},\;t\in\overline{n_{1}+1,n},$ (3.37)
$\displaystyle(E_{p,n+t}-E_{t,n+p})|_{\mathcal{B}}$
$\displaystyle=-x_{p}{y_{t}}+x_{t}{y_{p}}$ for $\displaystyle n_{1}+1\leq
p<t\leq n,$ (3.38) $\displaystyle(E_{n+p,t}-E_{n+t,p})|_{\mathcal{B}}$
$\displaystyle=-\partial_{x_{p}}\partial_{y_{t}}+\partial_{x_{t}}\partial_{y_{p}}$
for $\displaystyle n_{1}+1\leq p<t\leq n.$

Then the above root elements form a basis for the subalgebra
$\mathfrak{g}(\mathcal{P}_{+})_{-}:=\mathfrak{o}(2n,\mathbb{C})_{-}+\mathcal{P}_{+}$.

Suppose $n_{1}=n_{2}<n-1$. From Luo-Xu [12] we know that the
$\mathcal{K}$-singular vectors in ${\mathcal{H}}_{\langle k^{\prime}\rangle}$
are:

(3.39)
$x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}\qquad\mbox{with}\;-(m_{1}+m_{2})=k^{\prime},$
(3.40)
$x_{n_{1}}^{-k^{\prime}}\zeta_{1}^{m+1}\qquad\mbox{with}\;m\in\mathbb{N},$
(3.41)
$y_{n_{1}+1}^{-k^{\prime}}\zeta_{2}^{m+1}\qquad\mbox{with}\;m\in\mathbb{N},$

where $\zeta_{1}=x_{n_{1}-1}y_{n_{1}}-x_{n_{1}}y_{n_{1}-1}$ and
$\zeta_{2}=x_{n_{1}+1}y_{n_{1}+2}-x_{n_{1}+2}y_{n_{1}+1}$.

Let $\mathfrak{g}_{1}$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$
spanned by the root vectors in the following set:

$I_{1}:=\\{(\ref{ri-}),(\ref{tp-}),(\ref{inr-}),(\ref{int-}),(\ref{nti-}),(\ref{npt-})\\}.$

Let $\mathfrak{g}_{2}$ be the subalgebra of $\mathfrak{o}(2n,\mathbb{C})$
spanned by the root vectors in the following set:

$I_{2}:=\\{(\ref{ti-}),(\ref{nri-}),(\ref{pnt-})\\}.$

So we get
$U(\mathfrak{g}(\mathcal{P}_{+})_{-})=U(\mathfrak{g}_{2})U(\mathfrak{g}_{1}).$

We take a $\mathcal{K}$-singular vector
$v_{\mathcal{K}}=x_{n_{1}}^{-k^{\prime}}$, and set
$M_{0}=U(\mathfrak{g}_{1})x_{n_{1}}^{-k^{\prime}}$. Then $M_{0}$ is finite-
dimensional.

Thus

${\mathcal{H}}_{\langle
k^{\prime}\rangle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g_{-}+\mathcal{P}_{+}})v_{\mathcal{K}}=U(\mathfrak{g}_{2})M_{0}.$

Let $k$ be any positive integer. We want to compute
$\mathrm{dim}(U_{k}(\mathfrak{g}_{2})M_{0})$, and then get the Gelfand-
Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$.

The argument for this case is similar to case 1, and using Proposition 3.2, we
find that the Gelfand-Kirillov dimension of $U(\mathfrak{g}_{2})M_{0}$ is

(3.45) $\displaystyle d=\left\\{\begin{array}[]{ll}2n-3,&\hbox{if $n=2,3$;}\\\
2n-2,&\hbox{if $n=4$;}\\\ 2n-1,&\hbox{if $n\geq 5$.}\end{array}\right.$

When $n_{1}=n_{2}=n-1,n$, the arguments are similar with the above, and the
conclusion is the same with 3.45.

##  4\. Proof of the main theorem for $\mathfrak{o}(2n+1,\mathbb{C})$

We keep the same notations with the introduction. We know

$\mathfrak{o}(2n+1,\mathbb{F})=\mathfrak{o}(2n,\mathbb{C})\oplus\bigoplus_{i=1}^{n}[\mathbb{C}(E_{0,i}-E_{n+i,0})+\mathbb{C}(E_{0,n+i}-E_{i,0})]$

and $\mathcal{B}^{\prime}=\mathbb{C}[x_{0},x_{1},...,x_{n},y_{1},...,y_{n}]$.

Luo and Xu [12] proved that for any $k^{\prime}\in\mathbb{Z}$,
${\mathcal{H}^{\prime}}_{\langle k^{\prime}\rangle}$ is an irreducible
$\mathfrak{o}(2n+1,\mathbb{C})$-module. Moreover, the homogeneous subspace
${\mathcal{B}}^{\prime}=\bigoplus_{k\in\mathbb{Z}}\bigoplus_{i=0}^{\infty}(\eta^{\prime})^{i}({\mathcal{H}}^{\prime}_{\langle
k^{\prime}\rangle})$ is a decomposition of irreducible submodules. The module
${\mathcal{H}^{\prime}}_{\langle k^{\prime}\rangle}$ under the assumption is
of highest-weight type only if $n_{2}=n$, in which case
$x_{n_{1}}^{-k^{\prime}}$ is a highest-weight vector with weight
$-k^{\prime}\lambda_{n_{1}-1}+(k^{\prime}-1)\lambda_{n_{1}}+[(k^{\prime}-1)\delta_{n_{1},n-1}-2k^{\prime}\delta_{n_{1},n}]\lambda_{n}.$

###  4.1. Case 1. $n_{1}<n_{2}$ _and_ $k^{\prime}\in\mathbb{N}$.

The representation of $\mathfrak{o}(2n+1,\mathbb{C})$ on
$\mathcal{B}^{\prime}$ by the differential operators in (3.9)-(3.26) and
$\mathcal{K}_{+}$ with $|_{\mathcal{B}}$ is replaced by
$|_{\mathcal{B}^{\prime}}$ and also contains the following:

(4.1) $\displaystyle(E_{0,i}-E_{n+i,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=-x_{0}x_{i}-y_{i}\partial_{x_{0}}$ for $\displaystyle
i\in\overline{1,n_{1}},$ (4.2)
$\displaystyle(E_{0,s}-E_{n+s,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=x_{0}\partial_{x_{s}}-y_{s}\partial_{x_{0}}$ for $\displaystyle
s\in\overline{n_{1}+1,n_{2}},$ (4.3)
$\displaystyle(E_{0,t}-E_{n+t,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=x_{0}\partial_{x_{t}}-\partial_{x_{0}}\partial_{y_{t}}$ for
$\displaystyle t\in\overline{n_{2}+1,n},$ (4.4)
$\displaystyle(E_{0,n+i}-E_{i,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=x_{0}\partial_{y_{i}}-\partial_{x_{0}}\partial_{x_{i}}$ for
$\displaystyle i\in\overline{1,n_{1}},$ (4.5)
$\displaystyle(E_{0,n+s}-E_{s,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=x_{0}\partial_{y_{s}}-{x_{s}}\partial_{x_{0}}$ for
$\displaystyle s\in\overline{n_{1}+1,n_{2}},$ (4.6)
$\displaystyle(E_{0,n+t}-E_{t,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=-x_{0}{y_{t}}-{x_{t}}\partial_{x_{0}}$ for $\displaystyle
t\in\overline{n_{2}+1,n}.$

Now we want to compute the Gelfand-Kirillov dimensions of the
$\mathfrak{o}(2n+1,\mathbb{C})$-module ${\mathcal{H}^{\prime}}_{\langle
k^{\prime}\rangle}$ and ${\mathcal{H}^{\prime}}_{\langle-k^{\prime}\rangle}$
for this case. From Luo-Xu [12] we know that ${\mathcal{H}^{\prime}}_{\langle
k^{\prime}\rangle}$ is an irreducible
$\mathfrak{o}(2n+1,\mathbb{C})$-submodule generated by
$x_{n_{1}+1}^{k^{\prime}}$, and
${\mathcal{H}^{\prime}}_{\langle-k^{\prime}\rangle}$ is an irreducible
$\mathfrak{o}(2n+1,\mathbb{C})$-submodule generated by
$x_{n_{1}}^{k^{\prime}}$. Then similar to the computation of
$\mathfrak{o}(2n+1,\mathbb{C})$, the Gelfand-Kirillov dimension of
${\mathcal{H}^{\prime}}_{\langle k^{\prime}\rangle}$ is

$\displaystyle d=\left\\{\begin{array}[]{ll}2n-1,&\hbox{if $2=n_{1}<n_{2}=n$
or $1=n_{1}<n_{2}=n-1,n$;}\\\ 2n,&\hbox{if $3\leq n_{1}<n_{2}=n$ or
$1<n_{1}<n_{2}=n-1$ or $1\leq n_{1}<n_{2}<n-1$.}\end{array}\right.$

${\mathcal{H}^{\prime}}_{\langle-k^{\prime}\rangle}$ has the same Gelfand-
Kirillov dimension with ${\mathcal{H}^{\prime}}_{\langle k^{\prime}\rangle}$.

###  4.2. Case 2. $n_{1}=n_{2}$ _and_ $k^{\prime}\in\mathbb{N}$.

The representation of $\mathfrak{o}(2n+1,\mathbb{C})$ on
$\mathcal{B}^{\prime}$ by the differential operators in (3.9)-(3.26) and
$\mathcal{K}_{+}$ with $|_{\mathcal{B}}$ is replaced by
$|_{\mathcal{B}^{\prime}}$ and also contains the following:

(4.7) $\displaystyle(E_{0,i}-E_{n+i,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=-x_{0}x_{i}-y_{i}\partial_{x_{0}}$ for $\displaystyle
i\in\overline{1,n_{1}},$ (4.8)
$\displaystyle(E_{0,t}-E_{n+t,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=x_{0}\partial_{x_{t}}-\partial_{x_{0}}\partial_{y_{t}}$ for
$\displaystyle t\in\overline{n_{2}+1,n},$ (4.9)
$\displaystyle(E_{0,n+i}-E_{i,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=x_{0}\partial_{y_{i}}-\partial_{x_{0}}\partial_{x_{i}}$ for
$\displaystyle i\in\overline{1,n_{1}},$ (4.10)
$\displaystyle(E_{0,n+t}-E_{t,0})|_{\mathcal{B}^{\prime}}$
$\displaystyle=-x_{0}{y_{t}}-{x_{t}}\partial_{x_{0}}$ for $\displaystyle
t\in\overline{n_{2}+1,n}.$

Now we want to compute the Gelfand-Kirillov dimensions of the
$\mathfrak{o}(2n+1,\mathbb{C})$-module ${\mathcal{H}^{\prime}}_{\langle
k^{\prime}\rangle}$ and ${\mathcal{H}^{\prime}}_{\langle-k^{\prime}\rangle}$
for this case. From Luo-Xu [12] we know that
${\mathcal{H}^{\prime}}_{\langle-k^{\prime}\rangle}$ is an irreducible
$\mathfrak{o}(2n+1,\mathbb{C})$-submodule generated by
$x_{n_{1}}^{k^{\prime}}$, and ${\mathcal{H}^{\prime}}_{\langle
k^{\prime}\rangle}$ is an irreducible
$\mathfrak{o}(2n+1,\mathbb{C})$-submodule generated by
$T_{1}(y_{n_{1}}^{k^{\prime}-1})$ (here
$T_{1}=\sum\limits_{i=0}^{\infty}\frac{(-2)^{i}x_{0}^{2i+1}{\mathcal{D}}^{i}}{(2i+1)!}$
and
$\mathcal{D}=-\sum\limits_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}+\sum\limits_{s=n_{1}+1}^{n_{2}}\partial_{x_{s}}\partial_{y_{s}}-\sum\limits_{t=n_{2}+1}^{n}y_{t}\partial_{x_{t}}$).
Then similar to the computation of $\mathfrak{o}(2n+1,\mathbb{C})$, the
Gelfand-Kirillov dimension of
${\mathcal{H}^{\prime}}_{\langle-k^{\prime}\rangle}$ is

$\displaystyle d=\left\\{\begin{array}[]{ll}2n-2,&\hbox{if
$1=n_{1}=n_{2}<n=2,3$;}\\\ 2n-1,&\hbox{if $n_{1}=n_{2}=2$ when $n=2,3$ or
$n_{1}=n_{2}=1$ when $n=1,4$;}\\\ 2n,&\hbox{if $n_{1}=n_{2}=n=3$ or $2\leq
n_{1}=n_{2}\leq n=4$ or $1\leq n_{1}=n_{2}\leq n$ when $n\geq
5$.}\end{array}\right.$

${\mathcal{H}^{\prime}}_{\langle k^{\prime}\rangle}$ has the same Gelfand-
Kirillov dimension with ${\mathcal{H}^{\prime}}_{\langle-k^{\prime}\rangle}$.

##  5\. Proof of the main theorem for $\mathfrak{sp}(2n,\mathbb{C})$

We keep the same notations with the introduction. Recall the symplectic Lie
algebra

$\displaystyle\hskip 28.45274pt\mathfrak{sp}(2n,\mathbb{C})$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})+\sum_{i=1}^{n}(\mathbb{C}E_{i,n+i}+\mathbb{C}E_{n+i,i})$
$\displaystyle+\sum_{1\leq i<j\leq
n}[\mathbb{C}(E_{i,n+j}+E_{j,n+i})+\mathbb{C}(E_{n+i,j}+E_{n+j,i})].$

Again we take the Cartan subalgebra
$\mathfrak{h}=\sum_{i=1}^{n}\mathbb{C}(E_{i,i}-E_{n+i,n+i})$ and the subspace
spanned by positive root vectors

$\mathfrak{sp}(2n,\mathbb{C})_{+}=\sum_{1\leq i<j\leq
n}[\mathbb{C}(E_{i,j}-E_{n+j,n+i})+\mathbb{C}(E_{i,n+j}+E_{j,n+i})]+\sum_{i=1}^{n}\mathbb{C}E_{i,n+i}.$

Correspondingly, we have

$\mathfrak{sp}(2n,\mathbb{C})_{-}=\sum_{1\leq i<j\leq
n}[\mathbb{C}(E_{j,i}-E_{n+i,n+j})+\mathbb{C}(E_{n+i,j}+E_{n+j,i})]+\sum_{i=1}^{n}\mathbb{C}E_{n+i,i}.$

Fix $1\leq n_{1}\leq n_{2}\leq n$. We have the following two-parameter
$\mathbb{Z}$-graded oscillator representation of
$\mathfrak{sp}(2n,\mathbb{C})$ on
$\mathcal{B}=\mathbb{C}[x_{1},...,x_{n},y_{1},...,y_{n}]$ determined by

$(E_{i,j}-E_{n+j,n+i})|_{\mathcal{B}}=E_{i,j}^{x}-E_{j,i}^{y}.$

In particular we have

(5.1) $\displaystyle(E_{s,i}-E_{n+i,n+s})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}x_{s}}-{y_{i}}\partial_{y_{s}}$ for $\displaystyle
i\in\overline{1,n_{1}},\;s\in\overline{n_{1}+1,n_{2}},$ (5.2)
$\displaystyle(E_{t,i}-E_{n+i,n+t})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}x_{t}}+y_{i}y_{t}$ for $\displaystyle
i\in\overline{1,n_{1}},\;t\in\overline{n_{2}+1,n},$ (5.3)
$\displaystyle(E_{t,s}-E_{n+s,n+t})|_{\mathcal{B}}$
$\displaystyle=x_{t}\partial_{x_{s}}+y_{s}y_{t}$ for $\displaystyle
s\in\overline{n_{1}+1,n_{2}},\;t\in\overline{n_{2}+1,n},$ (5.4)
$\displaystyle(E_{n+r,i}+E_{n+i,r})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}}{y_{r}}-{x_{r}}{y_{i}}$ for $\displaystyle 1\leq
i<r\leq n_{1},$ (5.5) $\displaystyle(E_{n+s,i}+E_{n+i,s})|_{\mathcal{B}}$
$\displaystyle=-{x_{i}}{y_{s}}+{y_{i}}\partial_{x_{s}}$ for $\displaystyle
i\in\overline{1,n_{1}},\;s\in\overline{n_{1}+1,n_{2}},$ (5.6)
$\displaystyle(E_{s,n+t}+E_{t,n+s})|_{\mathcal{B}}$
$\displaystyle=-x_{s}{y_{t}}+x_{t}\partial_{y_{s}}$ for $\displaystyle
s\in\overline{n_{1}+1,n_{2}},\;t\in\overline{n_{2}+1,n},$ (5.7)
$\displaystyle(E_{p,n+t}+E_{t,n+p})|_{\mathcal{B}}$
$\displaystyle=-x_{p}{y_{t}}-x_{t}{y_{p}}$ for $\displaystyle n_{2}+1\leq
p<t\leq n,$ (5.8) $\displaystyle(E_{n+i,i})|_{\mathcal{B}}$
$\displaystyle=-x_{i}{y_{i}}$ for $\displaystyle i\in\overline{1,n_{1}},$
(5.9) $\displaystyle(E_{t,n+t})|_{\mathcal{B}}$ $\displaystyle=-x_{t}{y_{t}}$
for $\displaystyle t\in\overline{n_{2}+1,n}.$

Then the above root elements form a subalgebra for
$\mathfrak{sp}(2n,\mathbb{C})$, denoted by $\mathfrak{g}_{2}$. The rest root
elements form another subalgebra for $\mathfrak{sp}(2n,\mathbb{C})$, denoted
by $\mathfrak{g}_{1}$.

Luo and Xu [12] proved that for any $k^{\prime}\in\mathbb{Z}$, when
$n_{1}<n_{2}$ or $k^{\prime}\neq 0$, ${\mathcal{B}}_{\langle
k^{\prime}\rangle}$ is an irreducible weight
$\mathfrak{sp}(2n,\mathbb{C})$-module. Moreover, the module
${\mathcal{B}}_{\langle k^{\prime}\rangle}$ under the assumption is of
highest-weight type only if $n_{2}=n$, in which case for $m\in\mathbb{N}$,
$x_{n_{1}}^{-m}$ is a highest-weight vector of
${\mathcal{B}}_{\langle-m\rangle}$ with weight
$-m\lambda_{n_{1}-1}+(m-1)\lambda_{n_{1}}$, $x_{n_{1}+1}^{m+1}$ is a highest-
weight vector of ${\mathcal{B}}_{\langle m+1\rangle}$ with weight
$-(m+2)\lambda_{n_{1}}+(m+1)\lambda_{n_{1}+1}+(m+1)\delta_{n_{1},n-1}\lambda_{n}$
if $n_{1}<n_{2}=n$, and $y_{n}^{m+1}$ is a highest-weight vector of
${\mathcal{B}}_{\langle m+1\rangle}$ with weight
$(m+1)\lambda_{n-1}-2(m+1)\lambda_{n}$ if $n_{1}=n_{2}=n$. When $n_{1}=n_{2}$,
the subspace ${\mathcal{B}}_{\langle 0\rangle}$ is a direct sum of two
irreducible weight $\mathfrak{sp}(2n,\mathbb{C})$-submodules. If
$n_{1}=n_{2}=n$, they are highest-weight modules with a highest-weight vector
$1$ of weight $-2\lambda_{n}$ and with a highest-weight vector
$x_{n-1}y_{n}-x_{n}y_{n-1}$ of weight
$(1-\delta_{n,2})\lambda_{n-2}-4\lambda_{n}$, respectively.

We take $\mathcal{K}=\sum\limits_{i,j=1}^{n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})$,
and $\mathcal{K}_{+}=\sum_{1\leq i<j\leq n}\mathbb{C}(E_{i,j}-E_{n+j,n+i})$. A
weight vector $v$ in $\mathcal{B}$ is called a _$\mathcal{K}$ -singular
vector_ if $\mathcal{K}_{+}(v)=0$.

From the PBW theorem we have

${\mathcal{B}}_{\langle
k^{\prime}\rangle}=U(\mathfrak{g})v_{\mathcal{K}}=U(\mathfrak{g}_{2})U(\mathfrak{g}_{1})v_{\mathcal{K}}$

for any fixed $\mathcal{K}$-singular vector $v_{\mathcal{K}}$. If we denote
$M_{0}:=U(\mathfrak{g}_{1})v_{\mathcal{K}}$, then $M_{0}$ is finite-
dimensional and ${\mathcal{B}}_{\langle
k^{\prime}\rangle}=U(\mathfrak{g}_{2})M_{0}$.

Similar to the $\mathfrak{o}(2n,\mathbb{C})$ case, we can compute the Gelfand-
Kirillov dimension of $\mathcal{B}_{\langle k^{\prime}\rangle}$ in a case-by-
case way. Actually the Gelfand-Kirillov dimension is equal to

$2n-1$

for any irreducible $\mathfrak{sp}(2n,\mathbb{C})$-module
$\mathcal{B}_{\langle k^{\prime}\rangle}$.

Acknowledgment. The author would like to thank Prof. Xiaoping Xu for several
helpful discussions and his financial support from NSFC Grant 11171324.

## References

  * [1] Bai, Z.Q.: Gelfand-Kirillov dimensions of the $\mathbb{Z}^{2}$-graded oscillator representations of $\mathfrak{sl}(n)$, accepted by Acta Mathematica Sinica, English Series.
  * [2] Bai, Z.Q., Hunziker, Bai.: On the Gelfand-Kirillov dimension of a unitary highest weight module, accepted by Science China Mathematics.
  * [3] Borho, W., Kraft, H.: Über die Gelfand-Kirillov dimension, Math. Ann., 220, 1-24 (1976)
  * [4] Enright, T. J., Howe, R., Wallach, N. R.: A classification of unitary highest weight modules, in Progr Math, 40, 97-143 (1983)
  * [5] Gelfand, I. G., Kirillov, A. A.: Sur les corps liés aux algébres enveloppantes des algébres de Lie, Publ. Math. IHES., 31, 5-19 (1966)
  * [6] Jantzen, J. C.: Einhüllende Algebren halbeinfacher Lie-Algebren. (German) [Enveloping algebras of semisimple Lie algebras], Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 3, Springer-Verlag, Berlin, 1983
  * [7] Joseph, A.: Gelfand-Kirillov dimension for the annihilators of simple quotients of Verma modules, J. London Math. Soc. (2), 18, 50-60 (1978)
  * [8] Kazhdan, D.; Lusztig, G.: Representations of Coxeter groups and Hecke algebras, Invent. Math., 53, no. 2, 165-184 (1979)
  * [9] Knapp, A. W.: Lie groups beyond an Introduction, 2nd ed., Progress in Mathematics, Vol. 140, Birkhäuser Boston, Boston, 2002.
  * [10] Krause, G., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov dimension, Research Notes in Math. Pitman Adv. Publ. Program, Vol. 116, 1985.
  * [11] Luo, C., Xu, X.: $\mathbb{Z}^{2}$-graded oscillator representations of $\mathfrak{sl}(n)$, Commun. Algebra, 41, 3147-3173 (2013)
  * [12] Luo, C., Xu, X.: Graded Oscillator Generalizations of the Classical Theorem on Harmonic Polynomials, J. Lie Theory, 23, 979-1003 (2013)
  * [13] Luo, C., Xu, X.: $\mathbb{Z}$-graded oscillator representations of symplectic Lie algebras, J. Algebra, 403, 401-425 (2014)
  * [14] Mathieu, O.: Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble), 50, 537-592 (2000)
  * [15] Matumoto, H.: The homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras, Duke Math. J., 131, no.1, 75-119 (2006)
  * [16] Nishiyama, K., Ochiai, H., Taniguchi, K., Yamashita, H., Kato, S.: Nilpotent orbits, associated cycles and Whittaker models for highest weight representations, Astérisque, 273, 1-163 (2001)
  * [17] Vogan, D.: Gelfand-Kirillov dimension for Harish-Chandra modules, Invent.Math., 48, 75-98 (1978)
  * [18] Vogan, D.: Singular unitary representations, in: Noncommutative harmonic analysis and Lie groups, Lecture Notes in Mathamatics, Vol. 880, pp. 506-535, Springer, Berlin-New York, 1981
  * [19] Vogan, D.: Associated varieties and unipotent representations, in: Harmonic Analysis on Reductive Groups, Progr. Math. Vol. 101, pp. 315–388, Birkhäuser, Boston, 1991
  * [20] Wang, W.: Dimension of a minimal nilpotent orbit, Proc. Amer. Math. Soc., 127, 935-936 (1999)
  * [21] Xu, X.: Flag partial differential equations and representations of Lie algebras, Acta. Appl. Math., 102, 249–280 (2008)
  * [22] Yamashita, H.: Criteria for the finiteness of restriction of $U(\mathfrak{g})$-modules to subalgebras and applications to Harish-Chandra modules: a study in relation to the associated varieties, J. Funct. Anal., 121, 296-329 (1994)
  * [23] Zariski, O., Samuel, P.: Commutative algebra, Vol. II, Springer-Verlag, New York/Berlin(World Publishing Corporation, China), 1975