# Generalized Projective Representations for sl(n+1)111Research supported by
NSFC Grant 10701002

$\mbox{Yufeng Zhao}^{1}$ and $\mbox{Xiaoping Xu}^{2}$

1 LMAM, School of Mathematical Sciences
Peking University, Beijing, 100871, P. R. China

2 Hua Loo-Keng Key Mathematical Laboratory
Institute of Mathematics, Academy of Mathematics and Systems Sciences
Chinese Academy of Sciences, Beijing, 100190, P. R. China

###### Abstract

It is well known that $n$-dimensional projective group gives rise to a non-
homogenous representation of the Lie algebra $sl(n+1)$ on the polynomial
functions of the projective space. Using Shen’s mixed product for Witt
algebras (also known as Larsson functor), we generalize the above
representation of $sl(n+1)$ to a non-homogenous representation on the tensor
space of any finite-dimensional irreducible $gl(n)$-module with the polynomial
space. Moreover, the structure of such a representation is completely
determined by employing projection operator techniques and well-known
Kostant’s characteristic identities for certain matrices with entries in the
universal enveloping algebra. In particular, we obtain a new one parameter
family of infinite-dimensional irreducible $sl(n+1)$-modules, which are in
general not highest-weight type, for any given finite-dimensional irreducible
$sl(n)$-module. The results could also be used to study the quantum field
theory with the projective group as the symmetry.

##  1 Introduction

A projective transformation on $\mathbb{F}^{n}$ for a field $\mathbb{F}$ is
given by

$u\mapsto\frac{Au+\vec{b}}{\vec{c}\>^{t}u+d}\qquad\mbox{for}\;\;u\in\mathbb{F}^{n},$
$None$

where all the vectors in $\mathbb{F}^{n}$ are in column form and

$\left(\begin{array}[]{cc}A&\vec{b}\\\ \vec{c}\>^{t}&d\end{array}\right)\in
GL(n).$ $None$

It is well-known that a transformation of mapping straight lines to lines must
be a projective transformation. The group of projective transformations is the
fundamental symmetry of $n$-dimensional projective geometry. Physically, the
group with $n=4$ and $\mathbb{F}=\mathbb{R}$ consists of all the
transformations of keeping free particles including light signals moving with
constant velocities along straight lines (e.g., cf. [GWZ1-2]). Based on the
embeddings of the poincaré group and De Sitter group into the projective group
with $n=4$ and $\mathbb{F}=\mathbb{R}$, Guo, Wu and Zhou [GWZ1-2] proposed
three kinds of special relativity.

In this paper, we give a representation-theoretic exploration on the impact of
projective transformations. Note that the Lie algebra of $n$-dimensional
projective group is spanned by the following differential operations

$\\{\partial_{x_{j}},x_{i}\partial_{x_{j}},x_{i}\sum_{r=1}^{n}x_{r}\partial_{x_{r}}\mid
i,j=1,2,...,n\\},$ $None$

which is isomorphic to the special linear Lie algebra $sl(n+1)$. Through the
above operators, we obtain a representation of $sl(n+1)$ on the polynomial
algebra $\mathbb{F}[x_{1},...,x_{n}]$. The non-homogeneity of (1.3) motivates
us to generalize the above representation of $sl(n+1)$ to a non-homogenous
representation on the tensor space of any finite-dimensional irreducible
$gl(n)$-module with $\mathbb{F}[x_{1},...,x_{n}]$ via Shen’s mixed product for
Witt algebras (cf. [Sg1-3]) (also known as Larsson functor (cf. [La])). It
turns out that the structure of such generalized projective representations
can be completely determined by employing projection operator techniques (cf.
[Gm1]) and well-known Kostant’s characteristic identities for certain matrices
with entries in the universal enveloping algebra (cf. [K]). In particular, we
obtain a new one parameter family of infinite-dimensional irreducible
$sl(n+1)$-modules for any given finite-dimensional irreducible $sl(n)$-module.

Denote by $\mathbb{Z}$ the ring of integers and by $\mathbb{N}$ the additive
semigroup of nonnegative integers. For any two integers $m$ and $n$, we denote

$\overline{m,n}=\left\\{\begin{array}[]{ll}\\{m,m+1,\cdots,n\\}&\mbox{if}\;m\leq
n,\\\ \emptyset&\mbox{otherwise}.\end{array}\right.$ $None$

Let ${\cal A}$ be a commutative associative algebra over $\mathbb{F}$. If
$\\{D_{i}\ |\ i\in\overline{1,n}\\}$ is a set of commuting derivations of
${\cal A}$, the set of derivations ${\cal
W}(n)=\\{\sum\limits_{i=1}^{n}a_{i}D_{i}|\ a_{i}\in{\cal A}\\}$ forms a Lie
algebra via the following Lie brackets:

$[\sum\limits_{i=1}^{n}a_{i}D_{i},\sum\limits_{i=1}^{n}b_{i}D_{i}]=\sum\limits_{i,j=1}^{n}(a_{j}D_{j}(b_{i})-b_{j}D_{j}(a_{i}))D_{i}.$
$None$

Let $E_{r,s}$ be the square matrix with 1 as its $(r,s)$-entry and 0 as the
others. The general linear Lie algebra $gl(n)$ is the Lie algebra of $n\times
n$ matrices over ${\mathbb{F}}$ with a vector space basis $\\{E_{i,j}\mid
i,j\in\overline{1,n}\\}$. For any $gl(n)$-module $V$, we define an action
$\pi$ of the Lie algebra ${\cal W}(n)$ on ${\cal A}\otimes_{\mathbb{F}}V$ by

$\pi(\sum\limits_{i=1}^{n}a_{i}D_{i})=\sum_{i,j=1}^{n}D_{i}(a_{j})\otimes
E_{i,j}+\sum\limits_{i=1}^{n}a_{i}D_{i}\otimes\mbox{Id}_{V}.$ $None$

Then $\pi$ gives a representation of ${\cal W}(n)$ (cf. [Sg1-3]) and the
functor from $gl(n)$-modules to ${\cal W}(n)$-modules was also later known as
Larsson functor (cf. [L]). The structure of the module ${\cal
A}\otimes_{\mathbb{F}}V$ was determined by Rao [R] when ${\cal
A}=\mathbb{F}[x_{1}^{\pm 1},...,x_{n}^{\pm 1}]$, $D_{i}=\partial_{x_{i}}$ and
$V$ is a finite-dimensional irreducible $gl(n)$-module. Lin and Tan [LT] did
the similar thing when ${\cal A}$ is the algebra of quantum torus. The first
author of this paper [Z] determined the ${\cal W}(n)$-module structure of
${\cal A}\otimes_{\mathbb{F}}V$ in the case that ${\cal A}$ is a certain semi-
group algebra and $D_{i}$ are locally-finite derivations in [X]. Throughout
this paper, we always assume $\mbox{char}\>\mathbb{F}=0$.

Take ${\cal H}=\sum\limits_{i=1}^{n}\mathbb{F}E_{i,i}$ as a Cartan subalgebra
of $gl(n)$. Assume ${\cal A}=\mathbb{F}[x_{1},...,x_{n}]$ and let
$D_{i}=\partial_{x_{i}}$. Embed $sl(n+1)$ into ${\cal W}(n)$ via (1.3). Then
the space ${\cal A}\otimes_{\mathbb{F}}V$ forms an $sl(n+1)$-module with the
representation $\pi|_{sl(n+1)}$, which we call a generalized projective
representation.

Characteristic identities have a long history. The first person to exploit
them was Dirac [D], who wrote down what amounts to the characteristic identity
for the Lie algebra $so(1,3)$. This particular example is intimately connected
with the problem of describing the structure of relativistically invariant
wave equations. Such identities have been shown to be powerful tools for the
analysis of finite dimensional representations of Lie groups (cf. [BB], [BG],
[F]). It has been shown by Kostant [K] (also cf. [Gm4]) that the
characteristic identities for semi-simple Lie algebras also hold for infinite
dimensional representations. Moreover, one may construct projection operators
analogous to the projection operators of Green [G], and Bracken and Green [BG]
in finite dimensions. We refer [Gh], [Gm1-Gm3], [Ha], [L], [LG], [M], [OCC]
and [O] for the other works on the identities. Using projection operator
techniques and Kostant’s characteristic identities, we prove:

Main Theorem. Let $V$ be a finite-dimensional irreducible $gl(n)$-module with
highest weight $\mu$. We have the following conclusions:

(i) The space ${\cal A}\otimes_{\mathbb{F}}V$ is an irreducible
$sl(n+1)$-module if and only if

$\mu(E_{1,1})+\sum\limits_{j=1}^{n}\mu(E_{j,j})\not\in-\mathbb{N}\bigcup\overline{2,1+\mu(E_{1,1}-E_{2,2})}$
$None$

and

$\mu(E_{i,i})+\sum\limits_{j=1}^{n}\mu(E_{j,j})-i\not\in\overline{1,\mu(E_{i,i}-E_{i+1,i+1})}\qquad\mbox{for}\;i\in\overline{2,n-1}.$
$None$

(ii) If one of the conditions in (1.7) and (1.8) fails, then both
$U(sl(n+1))(1\otimes V)$ and $({\cal
A}\otimes_{\mathbb{F}}V)/(U(sl(n+1))(1\otimes V))$ are irreducible
$sl(n+1)$-modules .

Note that all $\mu(E_{i,i}-E_{i+1,i+1})$ are nonnegative integers, which
determine the corresponding $sl(n)$-module $V$ uniquely. Moreover, the
identity matrix in $gl(n)$ are allowed to be any constant map. The above
theorem says that given a finite-dimensional irreducible $sl(n)$-module, we
can construct a new one-parameter family of explicit irreducible
$sl(n+1)$-modules via its projective representation and Shen’s mixed product.

A quantum field is an operator value function on a certain Hilbert space,
which is often a direct sum of infinite-dimensional irreducible modules of a
certain Lie algebra (group). The Lie algebra of two-dimensional conformal
group is exactly the Virasoro algebra, which is infinite-dimensional. The
minimal models of two-dimensional conformal field theory were constructed from
direct sums of certain infinite-dimensional irreducible modules of the
Virasoro algebra, where a distinguished module gives rise to a vertex operator
algebra. When $n>2$, the $n$-dimension conformal group is finite-dimensional,
whose Lie algebra is exactly isomorphic to $so(n,2)$. It is still unknown what
should a higher-dimensional conformal field theory be. Part of reason is that
we lack of enough knowledge on the infinite-dimensional irreducible
$so(n,2)$-modules that are compatible to the natural conformal representation
of $so(n,2)$. This motivates us to study explicit infinite-dimensional
irreducible modules of finite-dimensional simple Lie algebras by using non-
homogeneous polynomial representations and Shen’s mixed product for Witt
algebras. This paper is the first work in this direction.

As we mentioned earlier, projective groups are important groups in physics. In
comparison with the minimal models of two-dimensional conformal field theory,
the underlying module of the projective representation of $sl(n+1)$ should be
the distinguished module in the possible quantum field theory with the
projective group as the symmetry. The other modules would make the theory more
substantial.

The paper is organized as follows. In Section 2, we slightly generalize
Kostant’s characteristic identities and recall some facts about projection
operators based on Kostant’s work [K] and Gould’s works [Gm1, Gm4]. In Section
3, we prove (i) and (ii) in the theorem.

Acknowledgement: We would like to thank Professor Han-Ying Guo for his
interesting talk that motivates this work.

##  2 Characteristic Identities and Projection Operators

In order to keep the paper self-contained, we will first prove certain
characteristic identities for $gl(n)$, which will be used to study the
irreducibility of the generalized projective representations for $sl(n+1)$ .
Then we will recall some facts about the projection operators for $gl(n)$,
although part of the results in this section has appeared in [Gm1, Gm4], [OCC]
and [K].

###  2.1 Some Standard Facts for $gl(n)$ and $sl(n)$

Recall that $gl(n)$ is the Lie algebra of $n\times n$ matrices over
${\mathbb{F}}$ with a basis $\\{E_{i,j}\mid i,j\in\overline{1,n}\\}$ and the
Lie bracket:

$[E_{i,j},E_{k,l}]=\delta_{k,j}E_{i,l}-\delta_{i,l}E_{k,j}\qquad\mbox{for}\;\;i,j,k,l\in\overline{1,n}.$
$None$

Note that $gl(n)$ is reductive with the following decomposition of ideals
$gl(n)=sl(n)\oplus{\mathbb{F}}\mbox{I}$, where $sl(n)$ is the special linear
Lie algebra of matrices with zero trace and
$\mbox{I}=\sum\limits_{i=1}^{n}E_{i,i}$ is the identity matrix, which is
central.

Take ${\cal H}=\sum\limits_{i=1}^{n}\mathbb{F}E_{i,i}$ as a Cartan subalgebra
of $gl(n)$. If $\lambda\in{\cal{H}}^{*}$ is a weight of $gl(n)$, we identify
$\lambda$ with the $n$-tuple $\lambda=(\lambda_{1},\cdots,\lambda_{n})$, where
$\lambda_{i}=\lambda(E_{i,i})$. For $\lambda,\mu\in{\cal{H}}^{*}$, we define

$(\lambda,\mu)=\sum\limits_{i=1}^{n}\lambda_{i}\mu_{i}.$ $None$

Denote by $\varepsilon_{i}$ the weight with 1 as its $i$th coordinate and 0 as
the others, i.e.

$\varepsilon_{i}=(0,\cdots,0,\stackrel{{\scriptstyle i}}{{1}},0,\cdots,0).$
$None$

The set $\Phi^{+}=\\{\varepsilon_{i}-\varepsilon_{j}\mid 1\leq i<j\leq n\\}$
forms a set of positive roots of $gl(n)$. In this case, the half-sum of the
positive roots is given by

$\delta=\frac{1}{2}\sum\limits_{i<j}(\varepsilon_{i}-\varepsilon_{j})=\frac{1}{2}\sum\limits_{i=1}^{n}(n+1-2i)\varepsilon_{i}.$
$None$

Moreover, there exists a one-to-one correspondence between the set of finite-
dimensional irreducible $gl(n)$-modules and the set of $n$ tuples
$\lambda=(\lambda_{1},\cdots,\lambda_{n})$ such that
$\lambda_{i}-\lambda_{i+1}\in{\mathbb{N}}\ \mbox{for}\ i\in\overline{1,n-1}$.
Such an $n$ tuple $\lambda$ is called the highest weight of the corresponding
module which we denote as $V(\lambda)$.

Let $U$ denote the universal enveloping algebra of $gl(n)$ and let $Z$ be the
center of $U$. Set

$\sigma_{1}=\mbox{I},\
\sigma_{r}=\sum\limits_{i_{1},..,i_{r}=1}^{n}E_{i_{1},i_{2}}E_{i_{2},i_{3}}\cdots
E_{i_{r-1},i_{r}}E_{i_{r},i_{1}}\qquad\mbox{for}\;r\in\overline{2,n}.$ $None$

Then the center

$Z=\mathbb{F}[\sigma_{1},\cdots,\sigma_{n}].$ $None$

The subspace ${\cal H}_{0}={\cal H}\bigcap sl(n)$ is a Cartan subalgebra of
$sl(n)$ with the standard basis $\\{\alpha_{i}^{\vee}=E_{i,i}-E_{i+1,i+1}\mid
i\in\overline{1,n-1}\\}.$ Denote the dual vector space of ${\cal H}_{0}$ by
${\cal H}_{0}^{*}$ and let $\omega_{1},\omega_{2},\cdots,\omega_{n-1}$ be the
fundamental integral dominant weights in ${\cal H}_{0}^{*}$ defined by
$\omega_{i}(\alpha_{j}^{\vee})=\delta_{i,j}.$

Let $V(\psi)$ be the finite dimensional irreducible $sl(n)$-module with the
highest weight $\psi$. We can make $V(\psi)$ as a $gl(n)$-module $V(\psi,b)$
by letting the central element I act as the scalar map $b\mbox{Id}_{V(\psi)}$.

For $\vec{a}=(a_{1},...,a_{n-1})\in\mathbb{N}^{n-1}$ and $0<k\in\mathbb{Z}$,
we denote

$\displaystyle\hskip 39.83368ptI(\vec{a},k)$ $\displaystyle=$
$\displaystyle\\{(a_{1}+c_{1}-c_{2},a_{2}+c_{2}-c_{3},...,a_{n-1}+c_{n-1}-c_{n})\mid
c_{i}\in\mathbb{N}$ $\displaystyle\mbox{such
that}\;\sum_{i=1}^{n}c_{i}=k\;\mbox{and}\;c_{s+1}\leq
a_{s}\;\mbox{for}\;s\in\overline{1,n-1}\\}.\hskip 85.35826pt(2.7)$

Moreover, we set

$\omega_{\vec{a}}=\sum_{i=1}^{n-1}a_{i}\omega_{i}\qquad\mbox{for}\;\vec{a}\in\mathbb{N}^{n-1}.$
$None$

Lemma 2.1.1 (e.g., cf. Proposition 15.25 in [FH])  For any
$\vec{a}\in\mathbb{N}^{n-1}$, the tensor product of $sl(n)$-module
$V(\omega_{\vec{a}})$ with $V(k\omega_{1})$ decomposes into a direct sum:

$V(\omega_{\vec{a}})\otimes_{\mathbb{F}}V(k\omega_{1})=\bigoplus_{\vec{b}\in
I(\vec{a},k)}V(\omega_{\vec{b}}).$ $None$

Lemma 2.1.2 Let $\Pi$ be the weight set of $gl(n)$-module $V(\psi,b)$. Assume
$\psi=\sum\limits_{i=1}^{n-1}a_{i}\omega_{i}$,
$\nu=(\nu_{1},\cdots,\nu_{n})\in\Pi$ and
$(\nu_{1}-\nu_{2},\cdots,\nu_{n-1}-\nu_{n})=\psi-\sum\limits_{i=1}^{n-1}k_{i}\alpha_{i}$.
Then

$\nu_{1}=\sum\limits_{i=1}^{n-1}a_{i}+\frac{b-\sum\limits_{i=1}^{n-1}ia_{i}}{n}-k_{1},\
\nu_{n}=\frac{b-\sum\limits_{i=1}^{n-1}ia_{i}}{n}+k_{n-1},$
$\nu_{j}=\sum\limits_{i=j}^{n-1}a_{i}+\frac{b-\sum\limits_{i=1}^{n-1}ia_{i}}{n}+k_{j-1}-k_{j},\;\;j\in\overline{2,n-1}.$
$None$

Denote

$\underline{k}=(k_{1},k_{2},...,k_{n})\in\mathbb{N}^{n},\quad|\underline{k}|=\sum_{i=1}^{n}k_{i},$
$I(\mu,j)=\\{\underline{c}=(c_{1},\cdots,c_{n})\ |\ c_{i}\in\mathbb{N}\ ,\
|\underline{c}|=j,\
c_{s+1}\leq\mu_{s}-\mu_{s+1}\;\mbox{for}\;s\in\overline{1,n-1}\\}.$ $None$

It is easy to deduce the following lemma from the above two lemmas.

Lemma 2.1.3 The tensor product of $gl(n)$-module $V(\mu)$ with
$V(k\varepsilon_{1})$ decomposes into a direct sum:

$V(\mu)\otimes_{\mathbb{F}}V(k\varepsilon_{1})=\bigoplus_{\underline{c}\in
I(\mu,k)}V(\mu+\underline{c}).$ $None$

###  2.2 Characteristic Identities and Projection Operators for $gl(n)$

Now we will introduce the characteristic identities for $gl(n)$. We know that
the universal enveloping algebra $U$ of $gl(n)$ can be imbedded into $U\otimes
U$ by the associative algebra homomorphism $d:U\rightarrow U\otimes U$
determined by

$d(u)=u\otimes 1+1\otimes u\qquad\mbox{ for}\ u\in gl(n).$ $None$

Let $V(\lambda)$ be a fixed finite-dimensional $gl(n)$-module with highest
weight $\lambda$ and let $\pi_{\lambda}$ be the corresponding representation.
Kostant [K] considered the map

$\partial:U\rightarrow(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}U;$ $\
u\mapsto 1\otimes u+\pi_{\lambda}(u)\otimes 1$ $None$

for $u\in gl(n)$ and extended $\partial$ to an associative algebra
homomorphism from $U$ to $(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}U$. More
generally, if $d(u)=\sum\limits_{r}u_{r}\otimes v_{r}$, we have
$\partial(u)=\sum\limits_{r}\pi_{\lambda}(u_{r})\otimes v_{r}$. For $z\in Z$,
we denote

$\tilde{z}=-\frac{1}{2}[\partial(z)-\pi_{\lambda}(z)\otimes 1-1\otimes z],$
$None$

which may be viewed as an $m\times m$ ($m=\mbox{dim}V(\lambda)$) matrix with
entries in $U$.

Denote by $\chi_{\zeta}$ the central character of a highest weight
$gl(n)$-module with highest weight $\zeta$. Suppose now that $W$ is another
$gl(n)$-module admitting the central character $\chi_{\mu}$ and $\pi_{\mu}$ is
the corresponding representation. We extend $\pi_{\mu}$ to an algebra
homomorphism

$\tilde{\pi_{\mu}}:(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}U\rightarrow(\mbox{End}\>V(\lambda))\otimes_{\mathbb{F}}(\mbox{End}\>W);$
$\ \sum\limits_{i}\rho_{i}\otimes
u_{i}\mapsto\sum\limits_{i}\rho_{i}\otimes\pi_{\mu}(u_{i}),$ $None$

where $\rho_{i}\in\mbox{End}\>V(\lambda)$ and $u_{i}\in U$. In particular, we
have

$\tilde{\pi_{\mu}}(\tilde{z})=-\frac{1}{2}[(\pi_{\lambda}\otimes\pi_{\mu})(z)-\pi_{\lambda}(z)\otimes
1-1\otimes\pi_{\mu}(z)]$ $None$

for $z\in Z$. Clearly, $\tilde{\pi_{\mu}}(\tilde{z})$ is a linear operator on
$V(\lambda)\otimes_{\mathbb{F}}W$ which may be viewed as an $m\times m$ matrix
with entries from $\mbox{End}\>W$ under a basis of $V(\lambda)$. For
$\nu\in{\cal H}^{\ast}$, we define

$f_{\nu}=-\frac{1}{2}(\chi_{\mu+\nu}-\chi_{\lambda}-\chi_{\mu}).$ $None$

Denote by $\Pi_{\lambda}$ the weight set of $V(\lambda)$.

Lemma 2.2.1 (cf. [K], [G], [OCC] ) On the space $W$, the matrix $\tilde{z}$
satisfies the following characteristic identity:

$\prod\limits_{\nu\in\Pi_{\lambda}}(\tilde{z}-f_{\nu}(z))=0\qquad\mbox{for}\;\;z\in
Z.$ $None$

By varying the module $V(\lambda)$ and the central element $z$, we obtain a
series of characteristic identities. In particular, the following
characteristic identity will be used in the proof of the main theorem:

Corollary 2.2.2 Take $V(\lambda)$ to be the dual module of $gl(n)$-module
$V(2,1,\cdots,1)$. Then the matrix $\tilde{\sigma_{2}}$ satisfies the
following characteristic identity on $W$:

$\prod\limits_{i=1}^{n}(\tilde{\sigma_{2}}-m_{i})=0,\ \
m_{i}=\frac{1}{2}(\lambda,\lambda+2\delta)-\frac{1}{2}(\lambda_{i},\lambda_{i}+2(\mu+\delta)),$
$None$

where $\lambda_{i}=(-1,\cdots,\stackrel{{\scriptstyle i}}{{-2}},\cdots,-1),\
i\in\overline{1,n}$.

Proof Note that the module $V(2,1,\cdots,1)$ has a basis
$\\{e_{1},\cdots,e_{n}\\}$ such that

$E_{i,i}(e_{k})=(1+\delta_{i,k})e_{k},\ E_{i,j}(e_{k})=\delta_{j,k}e_{i}\
(i\neq j).$ $None$

Let $\pi^{*}$ be the dual module of $gl(n,{\mathbb{F}})$-module
$V(2,1,\cdots,1)$. Then

$\pi^{*}(E_{i,i})=-(E_{i,i}+\mbox{I}),\ \pi^{*}(E_{i,j})=-E_{j,i}\ (i\neq j).$
$None$

Obviously, the representation $\pi^{*}$ is $n$-dimensional and all its weights

$\\{\lambda_{1}=(-2,-1,\cdots,-1),...,\lambda_{i}=(-1,\cdots,\stackrel{{\scriptstyle
i}}{{-2}},\cdots,-1),\cdots,\lambda_{n}=(-1,\cdots,-1,-2)\\}$ $None$

occur with multiplicity one. The matrix $\tilde{\sigma_{2}}$ in this case is
given by

$\tilde{\sigma_{2}}=-\sum\limits_{i,j}^{n}\pi^{*}(E_{j,i})E_{i,j}.$ $None$

This is the matrix

$\begin{array}[]{rcl}\tilde{\sigma_{2}}&=&\left[\begin{array}[]{cccc}\mbox{I}+E_{1,1}&E_{1,2}&\cdots&E_{1,n}\\\
E_{2,1}&\mbox{I}+E_{2,2}&\cdots&E_{2,n}\\\ \vdots&\vdots&\cdots&\vdots\\\
E_{n,1}&E_{n,2}&\cdots&\mbox{I}+E_{n,n}\end{array}\right].\end{array}$ $None$

By Lemma 2.2.1, the operator $\tilde{\sigma_{2}}$ satisfies the characteristic
identity on $W$:

$\prod\limits_{i=1}^{n}(\tilde{\sigma_{2}}-m_{i})=0,\ \
m_{i}=\frac{1}{2}(\lambda,\lambda+2\delta)-\frac{1}{2}(\lambda_{i},\lambda_{i}+2(\mu+\delta)),\
i\in\overline{1,n}.$ $None$

$\Box$

In the rest of this section, we will recall some facts about projection
operators appeared in [Gm2].

Take $gl(n)$-module $V(\lambda)=V(\varepsilon_{1})^{*}$ (resp.
$V(\varepsilon_{1})$). Then the corresponding matrices $\tilde{\sigma_{2}}$
are

$M=(E_{i,j})_{i,j=1}^{n},\ \tilde{M}=-M^{T}\in U(gl(n)),$ $None$

respectively. On the space $W$, the matrices $M$ and $\tilde{M}$ satisfy the
following characteristic identities:

$\prod\limits_{i=1}^{n}(M-d_{i})=0,\ d_{i}=\mu_{i}+n-i;$ $None$
$\prod\limits_{i=1}^{n}(\tilde{M}-\tilde{d}_{i})=0,\ \tilde{d}_{i}=n-1-d_{i},\
i\in\overline{1,n}.$ $None$

For $r\in\overline{1,n}$,

$P_{r}=\prod\limits_{r\neq
l\in\overline{1,n}}(\frac{M-d_{l}}{d_{r}-d_{l}}),\qquad\tilde{P}_{r}=\prod\limits_{r\neq
l\in\overline{1,n}}(\frac{\tilde{M}-\tilde{d}_{l}}{\tilde{d}_{r}-\tilde{d}_{l}})$
$None$

are called projection operators, which project the tensor product space
$V(\varepsilon_{1})^{*}{\otimes}_{\mathbb{F}}W$ (resp.
$V(\varepsilon_{1})\otimes_{\mathbb{F}}W$ ) onto the irreducible module
$W_{r}=P_{r}(V(\varepsilon_{1})^{*}\otimes_{\mathbb{F}}W)$ (resp.
$\tilde{W}_{r}=\tilde{P}_{r}(V(\varepsilon_{1})\otimes_{\mathbb{F}}W)$) with
central character $\chi_{\nu-\varepsilon_{r}}$ (resp.
$\chi_{\nu+\varepsilon_{r}}$).

##  3 Generalized Projective Representations

In this section, we will give the detailed construction of generalized
projective representations for the special linear Lie algebra $sl(n+1)$ and
study their irreducibility.

###  3.1 Construction of the Representations

Let ${\cal A}=\mathbb{F}[x_{1},\cdots,x_{n}]$. The operator
$p_{i}=x_{i}\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}$ is called pseudo-
translation operator on ${\cal A}$ in physics. Note that the Lie algebra of
$n$-dimensional projective group

$L_{n+1}=\sum\limits_{i,j=1}^{n}\mathbb{F}x_{i}\partial_{x_{j}}+\sum\limits_{i=1}^{n}\mathbb{F}\partial_{x_{i}}+\sum\limits_{i=1}^{n}\mathbb{F}p_{i},$
$None$

forms a Lie subalgebra of Witt algebra

${\cal W}(n)=\\{\sum\limits_{i=1}^{n}f_{i}\partial_{x_{i}}\ |\ f_{i}\in{\cal
A}\\}.$ $None$

Moreover, we have the following Lie brackets:

$[\partial_{x_{j}},p_{i}]=\left\\{\begin{array}[]{llll}x_{i}\partial_{x_{j}},&i\neq
j,\\\
\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{i}\partial_{x_{i}},&i=j,\end{array}\right.$
$None$ $[x_{i}\partial_{x_{j}},p_{k}]=\delta_{j,k}{p}_{i},\
[x_{i}\partial_{x_{j}},\partial_{x_{k}}]=-\delta_{i,k}\partial_{x_{j}},\
[x_{i}\partial_{x_{j}},x_{k}\partial_{x_{l}}]=\delta_{j,k}x_{i}\partial_{x_{l}}-\delta_{i,l}x_{k}\partial_{x_{j}}.$
$None$

To abbreviate, we denote

$P=\sum\limits_{i=1}^{n}\mathbb{F}p_{i},\
S=\sum\limits_{i=1}^{n}\mathbb{F}\partial_{x_{i}},\
\overline{L}_{n}=\sum\limits_{i,j=1}^{n}\mathbb{F}x_{i}\partial_{x_{j}},\
\overline{L}_{n}^{\prime}=[\overline{L}_{n},\overline{L}_{n}].$ $None$

Then

$L_{n+1}=P\oplus S\oplus\overline{L}_{n},\ [P,P]=\\{0\\},\ [S,S]=\\{0\\}.$
$None$

Moreover, $\overline{L}_{n}$ (resp. $\overline{L}_{n}^{\prime}$) is isomorphic
to $gl(n)$ (resp. $sl(n)$). It is easy to verify:

Lemma 3.1.1  The special linear Lie algebra $sl(n+1)$ is isomorphic to
$L_{n+1}$ with the following identification of Chevalley generators:

$h_{i}=x_{i}\partial_{x_{i}}-x_{i+1}\partial_{x_{i+1}},\qquad
h_{n}=\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{n}\partial_{x_{n}},$
$None$ $e_{i}=x_{i}\partial_{x_{i+1}},\;\;f_{i}=x_{i+1}\partial_{x_{i}},\qquad
e_{n}=p_{n},\;\;f_{n}=-\partial_{x_{n}},$ $None$

for $i\in\overline{1,n-1}$.

For any finite dimensional $gl(n)$-module $V$ with highest weight
$\mu=(\mu_{1},\cdots,\mu_{n})$, we define an action $\pi$ of Witt Lie algebra
${\cal W}(n)$ on ${\cal A}\otimes_{\mathbb{F}}V$ by

$\pi(\sum\limits_{i=1}^{n}a_{i}D_{i})=\sum_{i,j=1}^{n}D_{i}(a_{j})\otimes
E_{i,j}+\sum\limits_{i=1}^{n}a_{i}D_{i}\otimes\mbox{Id}_{V}.$ $None$

Embed $sl(n+1)$ into ${\cal W}(n)$ by (3.7) and (3.8). The space ${\cal
A}\otimes_{\mathbb{F}}V$ forms an $sl(n+1)$-module with the representation
$\pi|_{sl(n+1)}$, which we call a generalized projective representation of
$sl(n+1)$.

It follows from (3.7)-(3.9) that the explicit $L_{n+1}$-module ${\cal
A}\otimes_{\mathbb{F}}V$ structure is given by:

$(x_{i}\partial_{x_{j}}).(f\otimes v)=(x_{i}\partial_{x_{j}})f\otimes
v+f\otimes E_{i,j}.v,$ $None$ $\partial_{x_{i}}.(f\otimes
v)=\partial_{x_{i}}(f)\otimes v,$ $None$ $p_{i}.(f\otimes v)=p_{i}(f)\otimes
v+x_{i}f\otimes\sum\limits_{i=1}^{n}E_{i,i}.v+\sum\limits_{j=1}^{n}x_{j}f\otimes
E_{i,j}.v$ $None$

for $i,j\in\overline{1,n}$, where $f\in{\cal A}$ and $v\in V$.

###  3.2 Irreducibility Criteria for Generalized Projective Representations

In this section, we will give two irreducibility criteria for $L_{n+1}$-module
${\cal A}\otimes_{\mathbb{F}}V$ (cf. Proposition 3.2.3 and Proposition 3.2.5).

Lemma 3.2.1 The vector space $U(P)(1\otimes_{\mathbb{F}}V)$ is an irreducible
$L_{n+1}$-submodule of ${\cal A}\otimes_{\mathbb{F}}V$, where $U(P)$ denote
the universal enveloping algebra of abelian Lie algebra $P$.

Proof Denote

$x^{\underline{c}}=x_{1}^{c_{1}}x_{2}^{c_{2}}\cdots
x_{n}^{c_{n}}\qquad\mbox{for}\
\underline{c}=(c_{1},\cdots,c_{n})\in\mathbb{N}^{n}.$ $None$

Recall that $\Pi_{\mu}$ denotes the weight set of $V$. By (3.10), we have

$(x_{i}\partial_{x_{i}}).(x^{\underline{c}}\otimes
v_{\nu})=(c_{i}+\nu_{i})x^{\underline{c}}\otimes v_{\nu},$ $None$
$(\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}).(x^{\underline{c}}\otimes
v_{\nu})=(|\underline{c}|+\sum\limits_{i=1}^{n}\nu_{i})x^{\underline{c}}\otimes
v_{\nu},$ $None$

for any $x^{\underline{c}}\in{\cal A}$ and $\nu\in\Pi_{\mu}$.

Assume that $\\{v_{1},\cdots,v_{\ell}\\}$ is a basis of $V$. For $1\leq
k\in\mathbb{N}$, we set

$U(P)(1\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=\mbox{Span}_{\mathbb{F}}\\{p_{i_{1}}p_{i_{2}}\cdots
p_{i_{k}}(1\otimes v_{j})\ |\ 1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n,j\in\overline{1,d}\\},$ $None$ $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=\mbox{Span}_{\mathbb{F}}\\{x^{\underline{c}}\otimes
v_{j}\ |\ x^{\underline{c}}\in{\cal A},\ |\underline{c}|=k,\
j\in\overline{1,l}\\}.$ $None$

Then (3.14) implies that

$U(P)(1\otimes_{\mathbb{F}}V)=\bigoplus\limits_{k\in\mathbb{N}}(U(P)(1\otimes
V))_{{\langle}k\rangle}.$ $None$

Denote

$\triangle_{i,j}^{k}=\left\\{\begin{array}[]{llll}x_{j}\partial_{x_{i}}&\mbox{if}\;i\neq
j,\\\
\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{i}\partial_{x_{i}}+k-1&\mbox{if}\;i=j,\end{array}\right.$
$None$

By induction, we can easily verify the following fomula:

$\partial_{x_{i}}.p_{i_{1}}p_{i_{2}}\cdots p_{i_{k}}(1\otimes
v_{j})=\sum\limits_{s=1}^{n}p_{i_{1}}p_{i_{2}}\cdots\hat{p}_{i_{s}}\cdots
p_{i_{k}}\triangle_{i,i_{s}}^{k}(1\otimes v_{j}),$ $None$
$\displaystyle(x_{i}\partial_{x_{l}}).p_{i_{1}}p_{i_{2}}\cdots
p_{i_{k}}(1\otimes v_{j})$ $\displaystyle=$
$\displaystyle\sum\limits_{s=1}^{n}\delta_{l,i_{s}}p_{i_{1}}p_{i_{2}}\cdots\hat{p}_{i_{s}}\cdots
p_{i_{k}}p_{i}(1\otimes
v_{j})+p_{i_{1}}p_{i_{2}}\cdots{\cal}{p}_{i_{k}}(1\otimes
E_{i,l}.v_{j}),\hskip 73.97733pt(3.21)$
${p}_{i}.{\cal}{p}_{i_{1}}{p}_{i_{2}}\cdots p_{i_{k}}(1\otimes
v_{j})={p}_{i}p_{i_{1}}p_{i_{2}}\cdots{p}_{i_{k}}(1\otimes v_{j}).$ $None$

So (3.20)-(3.22) imply that $U(P)(1\otimes_{\mathbb{F}}V)$ is an
$L_{n+1}$-submodule of ${\cal A}\otimes_{\mathbb{F}}V$. Furthermore, it is
easy to verify that

$\bigcap\limits_{i=1}^{n}\mbox{Ker}\ \partial_{x_{i}}|_{({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}}=\\{0\\}\ \mbox{for \ any }\
1\leq k\in\mathbb{N}.$ $None$

Thus any non-trivial submodule of $U(P)(1\otimes_{\mathbb{F}}V)$ must contain
$1\otimes V$. The irreducibility of $U(P)(1\otimes_{\mathbb{F}}V)$ follows.
$\Box$

In the rest of this section, we will investigate the condition for ${\cal
A}\otimes_{\mathbb{F}}V=U(P)(1\otimes_{\mathbb{F}}V)$.

It is obvious that ${\cal
A}\otimes_{\mathbb{F}}V=U(P)(1\otimes_{\mathbb{F}}V)$ if and only if $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}k\rangle}$
holds for any $k\in\mathbb{N}$.

Suppose that $B_{i}$ is an ordered basis for $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}i\rangle}$. Denote by ${P}_{i+1,i}^{j}$ the
matrix of the linear map ${p}_{j}|_{({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}i\rangle}}$ (cf. (3.12)) with respect to
the bases $B_{i}$ and $B_{i+1}$. For any $0<j\in\mathbb{Z}$, we denote

$\Gamma_{j}=\\{\hat{i}=(i_{1},i_{2},...,i_{j})\mid
i_{s}\in\overline{1,n};i_{1}\leq i_{2}\leq\cdots\leq i_{j}\\}.$ $None$

Set

$P_{\hat{i}}={P}_{j,j-1}^{i_{1}}{P}_{j-1,j-2}^{i_{2}}\cdots{P}_{1,0}^{i_{j}}.$
$None$

We order

$\Gamma_{j}=\\{\hat{k}^{1},\hat{k}^{2},...,\hat{k}^{\ell_{j}}\\}$ $None$

lexically. In particular,

$\hat{k}^{1}=(1,1,...,1),\qquad\hat{k}^{\ell_{j}}=(n,n,...,n).$ $None$

Then $(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$ is, as a vector
space, isomorphic to the column space of the $\ell\ell_{j}\times\ell\ell_{j}$
matrix

$M_{j}=[P_{\hat{k}^{1}},P_{\hat{k}^{2}},...,P_{\hat{k}^{\ell_{j}}}]$ $None$

(recall that $\ell=\dim V$). Denote

$I_{1}=\\{1\\}\bigcup\\{i\in\overline{2,n}\ |\ \mu_{i-1}-\mu_{i}\geq 1\\}.$
$None$

By means of the characteristic identity in Corollary 2.2.2, we get the
following necessary but not sufficient condition for the irreducibility of
$L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$:

Lemma 3.2.2  For any $1\leq k\in\mathbb{N}$. If $\mu_{i}+|\mu|-i+s\neq 0$ for
any $s\in\overline{1,k},\ i\in I_{1}$, then $L_{n+1}$-module ${\cal
A}\otimes_{\mathbb{F}}V$ is irreducible.

Proof The Lemma is based on the following result:

Claim. For any $1\leq k\in\mathbb{N}$. If $\mu_{i}+|\mu|-i+s\neq 0,\ \forall\
s\in\overline{1,k},\ i\in I_{1}$, then $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}k\rangle}$
holds.

We will prove this claim by induction on $k$. For $k=1$,
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}1\rangle}$ is isomorphic as vector
space to the column space of $\ell n\times\ell n$ matrix

$\begin{array}[]{rcl}&&[{P}_{1,0}^{1},{P}_{1,0}^{2},\cdots,{P}_{1,0}^{n}]\end{array}\mbox{with}\
\begin{array}[]{rcl}{P}_{1,0}^{k}&=&\left[\begin{array}[]{c}E_{k,1}|_{V}\\\
E_{k,2}|_{V}\\\ \vdots\\\ (\mbox{I}+E_{k,k})|_{V}\\\ \vdots\\\
E_{k,n}|_{V}\end{array}\right],\end{array}$ $None$

which is exactly $\tilde{\sigma_{2}}|_{V}$ (cf. (2.25)). By Corollary 2.2.2,
the matrix $\tilde{\sigma_{2}}|_{V}$ is diagonalizable and it has full rank if
and only if all its eigenvalues are not zero, i.e.

$m_{i}=\frac{1}{2}(\lambda,\lambda+2\delta)-\frac{1}{2}(\lambda_{i},\lambda_{i}+2(\mu+\delta))=\mu_{i}+|\mu|-i+1\neq
0$ $None$

for any $i\in I_{1}$.

Now suppose that the lemma holds for $k=\iota-1$. Assume $k=\iota$. Note that
the eigenvalues of the matrix

$\left[\begin{array}[]{cccc}(\mbox{I}+E_{1,1}+s-1)|_{V}&E_{1,2}|_{V}&\cdots&E_{1,n}|_{V}\\\
E_{2,1}|_{V}&(\mbox{I}+E_{2,2}+s-1)|_{V}&\cdots&E_{2,n}|_{V}\\\
\vdots&\vdots&\cdots&\vdots\\\
E_{n,1}|_{V}&E_{n,2}|_{V}&\cdots&(\mbox{I}+E_{n,n}+s-1)|_{V}\end{array}\right]$
$None$

are $\mu_{i}+\sum\limits_{j=1}^{n}\mu_{j}-i+s\ (i\in I_{1})$ by (3.31). Thus
it is invertible if and only if $\mu_{i}+|\mu|-i+s\neq 0,\ \forall\ i\in
I_{1}$.

Observe that $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}k\rangle}$
if and only if the vectors

$\\{p_{i_{1}}\cdots p_{i_{k}}(1\otimes v_{j})\ |\ 1\leq i_{1}\leq\cdots\leq
i_{k}\leq n,j\in\overline{1,\ell}\\}$ $None$

are linearly independent. Equivalently, the corresponding
$\ell\ell_{k}\times\ell\ell_{k}$ system of homogeneous linear equations

$M_{k}X=0$ $None$

has only zero solution by (3.31).

Suppose

$\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n,j\in\overline{1,d}}a_{i_{1},i_{2},\cdots,i_{k}}^{j}p_{i_{1}}p_{i_{2}}\cdots
p_{i_{k}}(1\otimes v_{j})=0,\ a_{i_{1},i_{2},\cdots,i_{k}}^{j}\in\mathbb{F}.$
$None$

It can be written as

$\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n}p_{i_{1}}p_{i_{2}}\cdots p_{i_{k}}(1\otimes
w_{i_{1},i_{2},\cdots,i_{k}})=0,\ w_{i_{1},i_{2},\cdots,i_{k}}\in V.$ $None$

Then

$\partial_{x_{l}}.\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n}p_{i_{1}}p_{i_{2}}\cdots p_{i_{k}}(1\otimes
w_{i_{1},i_{2},\cdots,i_{k}})=0,\ \forall\ l\in\overline{1,n}.$ $None$

Equivalently,

$\sum\limits_{1\leq i_{1}\leq i_{2}\leq\cdots\leq i_{k}\leq
n}\sum\limits_{s=1}^{n}p_{i_{1}}p_{i_{2}}\cdots\hat{p}_{i_{s}}\cdots
p_{i_{k}}\triangle_{l,i_{s}}^{k}(1\otimes w_{i_{1},i_{2},\cdots,i_{k}})=0$
$None$

by (3.20). Moreover, it can be written as the form

$\sum\limits_{1\leq j_{1}\leq j_{2}\leq\cdots\leq j_{k-1}\leq
n}p_{j_{1}}p_{j_{2}}\cdots p_{j_{k-1}}(1\otimes
u_{j_{1},j_{2},\cdots,j_{k-1}})=0,$ $None$

where

$\displaystyle 1\otimes u_{j_{1},j_{2},\cdots,j_{k-1}}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{j_{1}}\triangle_{l,i}^{k}(1\otimes
w_{i,j_{1},j_{2},\cdots,j_{k-1}})+\sum\limits_{i=j_{1}}^{j_{2}}\triangle_{l,i}^{k}(1\otimes
w_{j_{1},i,j_{2},\cdots,j_{k-1}})+\cdots$
$\displaystyle+\sum\limits_{i=j_{s-1}}^{j_{s}}\triangle_{l,i}^{k}(1\otimes
w_{j_{1},j_{2},\cdots,j_{s-1},i,j_{s},\cdots j_{k-1}})+\cdots$
$\displaystyle+\sum\limits_{i=j_{k-1}}^{n}\triangle_{l,i}^{k}(1\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},i})=N_{k-1}\varpi_{n}\hskip 142.26378pt(3.40)$

with

$N_{k-1}={\small\left[\begin{array}[]{cccc}\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{1}\partial_{x_{1}}+k-1&x_{2}\partial_{x_{1}}&\cdots&x_{n}\partial_{x_{1}}\\\
x_{1}\partial_{x_{2}}&\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{2}\partial_{x_{2}}+k-1&\cdots&x_{n}\partial_{x_{2}}\\\
\vdots&\vdots&\cdots&\vdots\\\
x_{1}\partial_{x_{n}}&x_{2}\partial_{x_{n}}&\cdots&\sum\limits_{i=1}^{n}x_{i}\partial_{x_{i}}+x_{n}\partial_{x_{n}}+k-1\end{array}\right]}$
$None$

and

$\varpi_{n}=\left[\begin{array}[]{c}X_{1,j_{1}-1}\\\ X_{j_{1},j_{2}-1}\\\
X_{j_{2},j_{3-1}}\\\ \vdots\\\ X_{j_{k-1},n}\end{array}\right],$ $None$

in which

$X_{1,j_{1}-1}=\left[\begin{array}[]{c}1\otimes
w_{1,j_{1},j_{2},\cdots,j_{k-1}}\\\ 1\otimes
w_{2,j_{1},j_{2},\cdots,j_{k-1}}\\\ \vdots\\\ 1\otimes
w_{j_{1}-1,j_{1},j_{2},\cdots,j_{k-1}}\end{array}\right],\;\;X_{j_{1},j_{2}-1}=\left[\begin{array}[]{c}2\otimes
w_{j_{1},j_{1},j_{2},\cdots,j_{k-1}}\\\ 1\otimes
w_{j_{1},j_{1}+1,j_{2},\cdots,j_{k-1}}\\\ \vdots\\\ 1\otimes
w_{j_{1},j_{2}-1,j_{2},\cdots,j_{k-1}}\end{array}\right],$ $None$
$X_{j_{2},j_{3-1}}=\left[\begin{array}[]{c}2\otimes
w_{j_{1},j_{2},j_{2},\cdots,j_{k-1}}\\\ 1\otimes
w_{j_{1},j_{2},j_{2}+1,\cdots,j_{k-1}}\\\ \vdots\\\ 1\otimes
w_{j_{1},j_{2},j_{3}-1,\cdots,j_{k-1}}\end{array}\right],\cdots,\;\;X_{j_{k-1},n}=\left[\begin{array}[]{c}2\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},j_{k-1}}\\\ 1\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},j_{k-1}+1}\\\ \vdots\\\ 1\otimes
w_{j_{1},j_{2},\cdots,j_{k-1},n}\end{array}\right].$ $None$

We know that $N_{k-1}\varpi_{n}=0$ has only zero solution if and only if
$\mu_{i}+|\mu|-i+k\neq 0,\ \forall\ i\in I_{1}$. By Lemma 3.2.1 and the claim,
the Lemma is followed. $\Box$

Next, we study the structure
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$ as an
$\bar{L}_{n}$-module and finally get a necessary and sufficient condition for
the irreducibility of $L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$ (cf.
Proposition 3.2.5 ).

Note that $\bar{L}_{n}\simeq gl(n)$ ( resp. $\bar{L}_{n}^{\prime}\simeq sl(n)$
) according to (3.5). Obviously, (3.12) implies that as
$\overline{L}_{n}$-module $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$ is isomorphic to tensor product
module $V(j\epsilon_{1})\otimes_{\mathbb{F}}V$

$({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\cong
V(j\varepsilon_{1})\otimes_{\mathbb{F}}V(\mu)=\bigoplus_{\underline{c}\in
I(\mu,j)}V(\mu+\underline{c})$ $None$

(cf. Lemma 2.1.3). Denote

$p^{\underline{c}}=p_{1}^{c_{1}}p_{2}^{c_{2}}\cdots
p_{n}^{c_{n}}\qquad\mbox{for}\
\underline{c}=(c_{1},\cdots,c_{n})\in\mathbb{N}^{n}.$ $None$

Define the linear map

$\varphi_{j}:({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\rightarrow(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$
$\sum\limits_{|\underline{l}|=j,\
q\in\overline{1,l}}a_{\underline{l},q}x^{\underline{l}}\otimes
v_{q}\mapsto\sum\limits_{|\underline{l}|=j,\
q\in\overline{1,l}}a_{\underline{l},q}p^{\underline{l}}.(1\otimes v_{q}).$
$None$

It is easy to verify that $\varphi_{j}$ is an $\overline{L}_{n}$-module
homomorphism by (3.21).

For any $\underline{c}\in I(\mu,j)$ (cf. (2.11)), let $\xi_{\underline{c}}$ be
a maximal vector for highest weight module
$V(\mu+\underline{c})\subseteq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$. Then

$\hat{\xi}_{\underline{c}}=\varphi_{j}(\xi_{\underline{c}})$ $None$

is also a maximal vector of
$V(\mu+\underline{c})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}$.
Write

$\hat{\xi}_{\underline{c}}=q_{\underline{c}}\xi_{\underline{c}},\;\ \
q_{\underline{c}}\in\mathbb{F}.$ $None$

We know that

$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}\bigcap
V(\mu+\underline{c})\neq\\{0\\}\Rightarrow
V(\mu+\underline{c})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}.$
$None$

So

$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}\bigcap
V(\mu+\underline{c})\neq\\{0\\}\qquad\mbox{iff}\qquad q_{\underline{c}}\neq
0.$ $None$

In the following Lemma, we will calculate $q_{\underline{c}}$ explicitly:

Lemma 3.2.3  For any $\underline{c}\in I(\mu,j)$, we have

$q_{\underline{c}}=\prod\limits_{s=1}^{n}\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i),$
$None$

where we treat $\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i)=1$ if $c_{s}=0$.

Proof Let $v_{\mu}$ be a highest weight vector of the given $gl(n)$-module
$V$.

For any $\underline{c}\in I(\mu,j)$, we write a maximal vector
$\xi_{\underline{c}}$ for highest weight module $V(\mu+\underline{c})$ as

$\xi_{\underline{c}}=a_{\underline{c}}x^{{\underline{c}}}\otimes
v_{\mu}+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}x^{\underline{l}}\otimes
v_{\nu}^{i},\qquad 0\neq a_{\underline{c}},\
a_{\nu,\underline{l}}^{i}\in\mathbb{F},$ $None$

where $m({\nu})$ denotes the multiplicity of weight $\nu\in\Pi_{\mu}$.

Case 1. $c_{n}\neq 0$.

Note that $[x_{s}\partial_{x_{t}},\partial_{x_{n}}]=0,\ \forall\ 1\leq s<t\leq
n$. We know

$0\neq\partial_{x_{n}}.\xi_{\underline{c}}=a_{\underline{c}}c_{n}x^{{\underline{c}}-\varepsilon_{n}}\otimes
v_{\mu}+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}l_{n}x^{\underline{l}-\varepsilon_{n}}\otimes
v_{\nu}^{i}$ $None$

is a maximal vector for the $\overline{L}_{n}$\- highest weight module
$V(\mu+\underline{c}-\varepsilon_{n})$. Set

$\xi_{\underline{c}-\varepsilon_{n}}=\partial_{x_{n}}.\xi_{\underline{c}}.$
$None$

Obviously,

$\hat{\xi}_{\underline{c}}=a_{\underline{c}}p^{{\underline{c}}}.(1\otimes
v_{\mu})+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}p^{\underline{l}}.(1\otimes
v_{\nu}^{i}),$ $None$
$\hat{\xi}_{\underline{c}-\varepsilon_{n}}=\varphi_{j}(\partial_{x_{n}}.\xi_{\underline{c}})=a_{\underline{c}}c_{n}p^{{\underline{c}}-\varepsilon_{n}}.(1\otimes
v_{\mu})+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}l_{n}p^{\underline{l}-\varepsilon_{n}}.(1\otimes
v_{\nu}^{i}).$ $None$

Write

$\partial_{x_{n}}.\hat{\xi}_{\underline{c}}=b_{\underline{c}}(n)\hat{\xi}_{\underline{c}-\varepsilon_{n}},\qquad
b_{\underline{c}}(n)\in\mathbb{F}.$ $None$

By (3.21) and (3.54), we have

$\displaystyle\partial_{x_{n}}.\hat{\xi}_{\underline{c}}=\partial_{x_{n}}.\varphi_{j}(\xi_{\underline{c}})$
$\displaystyle=$ $\displaystyle
c_{n}a_{\underline{c}}p^{{\underline{c}}-\varepsilon_{n}}\Delta_{n,n}^{j}(1\otimes
v_{\mu})$
$\displaystyle+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}[\sum\limits_{s=1}^{n-1}l_{s}p^{\underline{l}-\varepsilon_{s}}(1\otimes
E_{s,n}v_{\nu}^{i})+l_{n}p^{\underline{l}-\varepsilon_{n}}\Delta_{n,n}^{j}(1\otimes
v_{\nu}^{i})].\hskip 28.45274pt(3.59)$

Denote

$E_{s,n}.v_{\mu+\varepsilon_{n}-\varepsilon_{s}}^{i}=\Im_{i}v_{\mu},\qquad\Im_{i}\in\mathbb{F}.$
$None$

Then, we have

$b_{\underline{c}}(n)=\frac{c_{n}a_{\underline{c}}(j-1+\mu_{n}+|\mu|)+\sum\limits_{s=1}^{n-1}(1+c_{s})\sum\limits_{i=1}^{m(\mu-\varepsilon_{s}+\varepsilon_{n})}a_{\mu-\varepsilon_{s}+\varepsilon_{n},\underline{c}+\varepsilon_{s}-\varepsilon_{n}}^{i}\Im_{i}}{c_{n}a_{\underline{c}}}$
$None$

by (3.58)-(3.61). For any $s\in\overline{1,n-1}$, we get

$\displaystyle 0=x_{s}\partial_{x_{n}}.\xi_{\underline{c}}$ $\displaystyle=$
$\displaystyle
c_{n}a_{\underline{c}}x^{\underline{c}+\varepsilon_{s}-\varepsilon_{n}}\otimes
v_{\mu}+\sum\limits_{(\nu,\underline{l})\neq(\mu,\underline{c}),{\underline{c}}+\mu={\underline{l}}+\nu,\
i\in\overline{1,m({\nu})}}a_{\nu,\underline{l}}^{i}[l_{n}x^{\underline{l}+\varepsilon_{s}-\varepsilon_{n}}\otimes
v_{\nu}^{i}+x^{\underline{l}}\otimes E_{s,n}.v_{\nu}^{i}].\hskip
5.69046pt(3.62)$

Therefore,

$c_{n}a_{\underline{c}}+\sum\limits_{i=1}^{m(\mu-\varepsilon_{s}+\varepsilon_{n})}a_{\mu-\varepsilon_{s}+\varepsilon_{n},\underline{c}+\varepsilon_{s}-\varepsilon_{n}}^{i}\Im_{i}=0$
$None$

Hence, we have

$b_{\underline{c}}(n)=c_{n}-n+\mu_{n}+|\mu|.$ $None$

Furthermore, by (3.55), we obtain

$\partial_{x_{n}}.\hat{\xi}_{\underline{c}}=q_{\underline{c}}\partial_{x_{n}}.\xi_{\underline{c}}=q_{\underline{c}}\xi_{\underline{c}-\varepsilon_{n}}=b_{\underline{k}}(n)\hat{\xi}_{\underline{c}-\varepsilon_{n}}=b_{\underline{k}}(n)q_{\underline{c}-\varepsilon_{n}}\xi_{\underline{c}-\varepsilon_{n}}$
$None$

which implies that

$q_{\underline{c}}=b_{\underline{c}}(n)q_{\underline{c}-\varepsilon_{n}}$
$None$

Case 2. $c_{n}=0$.

Suppose $c_{n-1}\neq 0$. We claim that $l_{n}=0$ for any $\underline{l}$ of
$a_{\nu,\underline{l}}^{i}\neq 0$ in (3.54). Indeed, we can write

$(\nu_{1}-\nu_{2},\cdots,\nu_{n-1}-\nu_{n})=\sum\limits_{i=1}^{n-1}(\mu_{i}-\mu_{i+1})\varpi_{i}-\sum\limits_{i=1}^{n-1}k_{i}\alpha_{i}.$
$None$

Since $\nu_{n}+l_{n}=\mu_{n}+c_{n}$, we have
$l_{n}=\mu_{n}-\nu_{n}=\mu_{n}-(\mu_{n}+k_{n-1})$ by (2.9). Thus we get
$l_{n}+k_{n-1}=0$. So $l_{n}=0$.

Hence,

$x_{n-1}\partial_{x_{n}}.\partial_{x_{n-1}}.\xi_{\underline{c}}=-\partial_{x_{n}}.\xi_{\underline{c}}+\partial_{x_{n-1}}.x_{n-1}\partial_{x_{n}}.\xi_{\underline{c}}=0.$
$None$

This implies that $\partial_{x_{n-1}}.\xi_{\underline{c}}$ is a maximal vector
for the highest weight $\overline{L}_{n}$-module
$V(\mu+\underline{c}-\varepsilon_{n-1})$. Repeating the process from (3.56) to
(3.64), we get

$q_{\underline{c}}=(\mu_{n-1}+|\mu|-(n-1)+c_{n-1})q_{\underline{c}-\varepsilon_{n-1}}.$
$None$

Then induction implies that

$q_{\underline{c}}=\prod\limits_{s=1}^{n}\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i).$
$None$

Thus the lemma is proved. $\Box$

By Lemma 3.2.3, we know that
$(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}=({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$ iff $q_{\underline{c}}\neq 0$
for any $\underline{c}\in I(\mu,j),\ 0<j\in\mathbb{N}$. Thus we get the
following sufficient and necessary condition for the irreducibility of
$L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$:

Proposition 3.2.4  The $L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$ is
irreducible iff $\ q_{\underline{c}}\neq 0$ for any $\underline{c}\in
I(\mu,j),\ 0<j\in\mathbb{N}$.

###  3.3 Jordan-Holder Series for $L_{n+1}$-module ${\cal
A}\otimes_{\mathbb{F}}V$

In this section, we will assume that ${\cal A}\otimes_{\mathbb{F}}V\neq
U(P)(1\otimes_{\mathbb{F}}V)$ and prove the irreducibility of the quotient
module $({\cal A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$.

First, we study the structure of $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ as an
$\overline{L}_{n}$-module (cf. Lemma 3.3.1 and Lemma 3.3.2). Recall the
$\overline{L}_{n}$-module homomorphism $\varphi_{j}$ defined by (3.47). Denote
$\mbox{Ker}(\varphi_{j})$ by ${\cal{R}}_{{\langle}j\rangle}$ and set

$I(\mu,j)^{\prime}=\\{\underline{c}\in I(\mu,j)\ |\ \ q_{\underline{c}}=0\\}.$
$None$

According to Lemma 3.2.3, we have the following result:

Lemma 3.3.1 For any $1\leq j\in\mathbb{N}$, we have
${\cal{R}}_{{\langle}j\rangle}=\bigoplus\limits_{\underline{c}\in
I(\mu,j)^{\prime}}V(\mu+\underline{c})$ and the following direct sum of
submodules for $\overline{L}_{n}$: $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}=(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j\rangle}\oplus{\cal{R}}_{{\langle}j\rangle}.$

Let $1\leq k\in\mathbb{N}$. Suppose that

$({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}i\rangle}=(U(P)(1\otimes
V))_{{\langle}i\rangle}\;\;\mbox{for any }\ \;i\in\overline{0,k},$ $None$

but

$({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\neq(U(P)(1\otimes
V))_{{\langle}j\rangle}\;\;\mbox{when}\ j\geq k+1.$ $None$

Then by Lemma 3.3.1, as $\overline{L}_{n}$-modules,

$({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)\cong\bigoplus\limits_{j=k+1}^{\infty}{\cal{R}}_{{\langle}j\rangle}.$
$None$

Lemma 3.3.2 Let $1\leq k\in\mathbb{N}$ such that (3.72) and (3.73) hold. Then
$\bar{L}_{n}$-module ${\cal{R}}_{{\langle}k+1\rangle}$ is irreducible.

Proof Since $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}=(U(P)(1\otimes
V))_{{\langle}k\rangle}$, Lemma 3.2.3 implies

$q_{\underline{c}}\neq 0\qquad\forall\ \underline{c}\in I(\mu,k).$ $None$

For convenience, we denote

$q_{s}(\underline{c})=\prod\limits_{i=1}^{c_{s}}(\mu_{s}+|\mu|-s+i).$ $None$

So

$q_{\underline{c}}=\prod\limits_{s=1}^{n}q_{s}(\underline{c}).$ $None$

Assume $\underline{m}\in I(\mu,k+1)^{\prime}$, i.e. $q_{\underline{m}}=0$.
Since the $gl(n)$-modules

$V((k+1)\varepsilon_{1})\otimes_{\mathbb{F}}V\simeq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k+1\rangle}\subset
V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}\simeq
V(\varepsilon_{1})\otimes_{\mathbb{F}}(V(k\varepsilon_{1})\otimes_{\mathbb{F}}V)$
$None$

in the sense of monomorphism, we know that

$\exists\quad\underline{t}\in{\mathbb{N}}^{n}\ \mbox{and}\
r\in\overline{1,n},\ \mbox{such \ that}\ \underline{t}\in I(\mu,k)\
\mbox{and}\ \underline{m}=\underline{t}+\varepsilon_{r}.$ $None$

Therefore, $m_{r}=t_{r}+1$. The fact $q_{\underline{m}}=0$ implies that
$q_{r}(\underline{m})=0$ because
$q_{s}(\underline{m})=q_{s}(\underline{t})\neq 0$ for $s\neq r$. Then
$q_{r}(\underline{t})\neq 0$ and
$q_{r}(\underline{m})=0=q_{r}(\underline{t})(\mu_{r}+|\mu|-r+t_{r}+1)$ imply
that

$\mu_{r}+|\mu|-r+t_{r}+1=0=\mu_{r}+|\mu|-r+m_{r}.$ $None$

Obviously, $1\leq m_{r}=t_{r}+1\leq|\underline{m}|=k+1$ and $\underline{m}\in
I(\mu,k+1)^{\prime}\subset I(\mu,k+1)$ imply that

$m_{r}\leq\mu_{r-1}-\mu_{r}$ $None$

by (2.11). Assume $1\leq m_{r}<k$. Then by (2.11) and (3.81), we know there
exists some $\underline{l}\in{\mathbb{N}}^{n}$ satisfying $l_{r}=m_{r}$ and
$\underline{l}\in I(\mu,k)$. So $q_{r}(\underline{l})=q_{r}(\underline{m})=0$.
Furthermore, $q_{\underline{l}}=0$, which contradicts (3.75). Therefore,
$m_{r}=k+1$, i.e. $\underline{m}=(k+1)\epsilon_{r}$.

Suppose there exists another $(k+1)\varepsilon_{s}\in I(\mu,k+1)^{\prime}$ but
$s\neq r$. Then $0=\mu_{r}+|\mu|-r+k+1=\mu_{s}+|\mu|-s+k+1$. This is
impossible, since $\mu_{s}\geq\mu_{r}$ whenever $s<r$. Thus we prove that
$|I(\mu,k+1)^{\prime}|=1$, i.e. $\bar{L}_{n}$-module
${\cal{R}}_{{\langle}k+1\rangle}$ is irreducible. $\Box$

Set

$I_{s}=\\{1\\}\bigcup\\{j\in\overline{2,n}\ |\ \mu_{j-1}-\mu_{j}\geq s\\}.$
$None$

By the above two lemmas, we can give the proof of (i) in the Main Theorem:

Proposition 3.3.3 The vector space ${\cal A}\otimes_{\mathbb{F}}V$ is an
irreducible $sl(n+1)$-module if and only if

$\forall\ 1\leq s\in\mathbb{N},\ \mu_{i}+|\mu|-i+s\neq 0\qquad\mbox{for\
any}\;\;i\in I_{s}.$ $None$

Proof  Assume that there exist $1\leq s\in\mathbb{N}$ and $i\in I_{s}$
satisfying $\mu_{i}+|\mu|-i+s=0$. Then
$V(\mu+s\varepsilon_{i})\subseteq{\cal{R}}_{{\langle}s\rangle}$ by (2.11),
(3.71), (3.82) and Lemma 3.3.1. Hence, $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s\rangle}\neq(U(P)(1\otimes
V))_{{\langle}s\rangle}$, i.e. $sl(n+1)$-module ${\cal
A}\otimes_{\mathbb{F}}V$ is reducible.

Suppose that $sl(n+1)$-module ${\cal A}\otimes_{\mathbb{F}}V$ is reducible.
Let $k$ satisfying (3.72) and (3.73). By Lemma 3.3.2, we have
$V(\mu+(k+1)\epsilon_{s})={\cal{R}}_{{\langle}k+1\rangle}$ for some $s\in
I_{k+1}$. So Lemma 3.2.3 implies $\mu_{s}+|\mu|-s+k+1=0$. $\Box$

Remark 3.3.4 The condition (3.83) is equivalent to (1.7) and (1.8) given in
the Main Theorem.

In the rest of this section, we study the relationship between any
$\bar{L}_{n}$-module ${\cal{R}}_{{\langle}j\rangle}$ and $\bar{L}_{n}$-module
${\cal{R}}_{{\langle}j+1\rangle}$ for any $j\geq k+1$ based on the
decomposition of tensor module and the projection operator techniques for
$gl(n)$ (Recall Lemma 2.1.3 and projection operators appeared in Section 2.2).
And we finally prove the irreducibility of quotient module $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ (cf. Proposition 3.3.8)
.

Lemma 3.3.5 Let $k+1\leq s\in\mathbb{N}$. For $\nu=\mu+\underline{l}$ with
$\underline{l}\in I(\mu,s+1)^{\prime}$, there exist
$\nu^{\prime}=\mu+\underline{m}$ with $\underline{m}\in I(\mu,s)^{\prime}$ and
$r\in\overline{1,n}$ such that $\nu=\nu^{\prime}+\varepsilon_{r}$ and
$\mu_{r}+|\mu|-r+m_{r}+1\neq 0$.

Proof Suppose $\nu=\mu+\underline{l}$ with $\underline{l}\in
I(\mu,s+1)^{\prime}$. Then $q_{\underline{l}}=0$.

Claim. There exists $r\in\overline{1,n}$ such that $\mu_{r}+|\mu|-r+l_{r}\neq
0$ and $l_{r}\geq 1$.

Set

$I_{\underline{l}}=\\{t\in\overline{1,n}\ |\ \mu_{t}+|\mu|-t+l_{t}=0\\}.$
$None$

It follows that $|I_{\underline{l}}|=0,1$. Otherwise,
$\mu_{t}+|\mu|-t+l_{t}=0=\mu_{q}+|\mu|-q+l_{q}$ for some $t<q$. This is
impossible because $\mu+\underline{l}$ is a highest weight implies that
$l_{t}+\mu_{t}\geq l_{q}+\mu_{q}$ whenever $t<q$. It is obvious that the claim
holds when $|I_{\underline{l}}|=0$.

Now assume $|I_{\underline{l}}|=1$ and $r_{0}\in I_{\underline{l}}$. If the
claim does not hold, then $\underline{l}=(s+1)\varepsilon_{r_{0}}$ and
$\mu_{t}+|\mu|-t\neq 0$ for any $t\neq r_{0}$. From Lemma 3.3.2, we know
${\cal{R}}_{{\langle}k+1\rangle}=V(\mu+(k+1)\varepsilon_{t})$ for some $t\in
I_{k+1}$. Thus, Lemma 3.2.3 implies
$q_{(k+1)\epsilon_{t}}=\prod\limits_{i=1}^{k+1}(\mu_{t}+|\mu|-t+i)=0$. Assume
$\mu_{t}+|\mu|-t+r=0$ for some $r\in\overline{1,k+1}$. On the other hand,
$\mu_{r_{0}}+|\mu|-r_{0}+s+1=0$ by (3.84) due to $r_{0}\in I_{\underline{l}}$.
Hence, we have $\mu_{r_{0}}-\mu_{t}=r_{0}+r-(t+s+1)$. If $r_{0}\leq t$, then
$\mu_{r_{0}}-\mu_{t}\geq 0$; which contradicts
$r_{0}+r-(t+s+1)=r_{0}-t+r-(s+1)<0$. If $r_{0}>t$, then
$\mu_{t}+l_{t}=\mu_{t}\geq\mu_{r_{0}}+l_{r_{0}}=\mu_{r_{0}}+s+1$ because
$\mu+\underline{l}=\mu+(s+1)\varepsilon_{r_{0}}$ is a highest weight.
Therefore, $\mu_{t}-\mu_{r_{0}}\geq s+1$; i.e. $t+s+1-(r_{0}+r)\geq s+1$.
Hence, $r_{0}-t+r\leq 0$. A contradiction arises. Thus the claim holds.

Suppose that $r$ satisfies the claim. Take $\nu^{\prime}=\nu-\varepsilon_{r},\
\underline{m}=\underline{l}-\varepsilon_{r}$. We claim that $\underline{m}\in
I(\mu,s)^{\prime}$. In fact, $\underline{m}\in I(\mu,s)$ by (2.11) because
$l_{r}-1\leq s$ and $l_{r}\leq\mu_{r-1}-\mu_{r}$ implies
$l_{r}-1<\mu_{r-1}-\mu_{r}$. Furthermore,
$q_{\underline{l}}=(\mu_{r}+|\mu|-r+l_{r})q_{\underline{m}}=0$ implies that
$q_{\underline{m}}=0$. Therefore, $\underline{m}\in I(\mu,s)^{\prime}$. Thus
the Lemma follows. $\Box$

Lemma 3.3.6 We have
$\partial_{l}({\cal{R}}_{{\langle}j\rangle})\not\subseteq(U(P)(1\otimes
V))_{{\langle}j-1\rangle}$ for any $l\in\overline{1,n}$ and $j>k+1.$

Proof Assume

$\partial_{l}({\cal{R}}_{{\langle}j\rangle})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j-1\rangle}\
\mbox{for \ some}\ l\in\overline{1,n}\;\mbox{and}\;j>k+1.$ $None$

By (3.11), we know that

$\partial_{l}(U(P)(1\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})\subseteq(U(P)(1\otimes_{\mathbb{F}}V))_{{\langle}j-1\rangle}.$
$None$

Then (3.85), (3.86) and Lemma 3.3.1 imply that

$\partial_{l}(({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})\subseteq(U(P)(1\otimes
V))_{{\langle}j-1\rangle},$ $None$

that is,

$\partial_{l}(({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})\varsubsetneq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j-1\rangle}.$ $None$

This contradicts the fact

$\partial_{l}(({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle})=({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j-1\rangle}\;\;\ \mbox{for any}\
l\in\overline{1,n}.$ $None$

Thus the lemma follows. $\Box$

Let $\\{e_{1}^{\prime},\cdots,e_{n}^{\prime}\\}$ be a basis for
$\overline{L}_{n}$-module $V(\varepsilon_{1})$. For any $1\leq
j\in\mathbb{N}$, we define the following linear map:

$T_{j}:V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}\rightarrow({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j+1\rangle}$ $T_{j}(e_{i}^{\prime}\otimes
v)=p_{i}.v,\ \forall\ v\in({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}.$ $None$

Lemma 3.3.7  The linear map $\ T_{j}\ $ is an intertwining operator from the
tensor module $V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$ to $({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j+1\rangle}$ for $\overline{L}_{n}$.

Proof For any $s,t,i\in\overline{1,n}$ and $v\in({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}j\rangle}$, we have

$\displaystyle T_{j}(x_{s}\partial_{x_{t}}.(e_{i}^{\prime}\otimes v))$
$\displaystyle=$ $\displaystyle
T_{j}(x_{s}\partial_{x_{t}}(e_{i}^{\prime})\otimes v+e_{i}^{\prime}\otimes
x_{s}\partial_{x_{t}}.v)=T_{j}(\delta_{t,i}e_{s}^{\prime}\otimes
v+e_{i}^{\prime}\otimes x_{s}\partial_{x_{t}}.v)$ $\displaystyle=$
$\displaystyle\delta_{t,i}p_{s}.v+p_{i}.x_{s}\partial_{x_{t}}.v=x_{s}\partial_{x_{t}}.p_{i}.v=x_{s}\partial_{x_{t}}.T_{j}(e_{i}^{\prime}\otimes
v)\hskip 145.10922pt(3.91)$

by (3.21). Thus the lemma follows. $\Box$

Based on the Lemma 3.3.5, Lemma 3.3.6 and Lemma 3.3.7, we can prove (ii) of
the Main Theorem in the following:

Proposition 3.3.8  If ${\cal A}\otimes_{\mathbb{F}}V\neq
U(P)(1\otimes_{\mathbb{F}}V)$, then $\\{0\\}\subset
U(P)(1\otimes_{\mathbb{F}}V)\subset{\cal A}\otimes_{\mathbb{F}}V$ is a Jordan-
Holder Series for $L_{n+1}$-module ${\cal A}\otimes_{\mathbb{F}}V$.

Proof Suppose that $W+U(P)(1\otimes_{\mathbb{F}}V)$ is any nonzero submodule
of quotient module $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$, where
$W=\bigoplus\limits_{j\geq k+1}W\bigcap{\cal{R}}_{{\langle}j\rangle}$ is a
weighted subspace of ${\cal A}\otimes_{\mathbb{F}}V$. By Lemma 3.3.6, we get
$W\bigcap{\cal{R}}_{{\langle}k+1\rangle}\neq\\{0\\}$.

By Lemma 3.3.5, we know that for any $s\geq k+1$ and $\nu=\mu+\underline{l}$
with $\underline{l}\in I(\mu,s+1)^{\prime}$, there exist
$\nu^{\prime}=\mu+\underline{m}$ with $\underline{m}\in I(\mu,s)^{\prime}$ and
$r\in\overline{1,n}$ such that $\nu=\nu^{\prime}+\varepsilon_{r}$ and
$\mu_{r}+|\mu|-r+m_{r}+1\neq 0$. Therefore, the highest weight module
$V(\nu)\;(\subseteq{\cal{R}}_{{\langle}s+1\rangle})$ of highest weight $\nu$
appears in the decomposition of $\overline{L}_{n}$-tensor module
$V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime})$ ($\subseteq
V(\varepsilon_{1})\otimes_{\mathbb{F}}{\cal{R}}_{{\langle}s\rangle}\subseteq
V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s\rangle}$).

Claim. There exists some maximal vector $v_{\nu}$ (resp.
$\xi_{\underline{m}+\varepsilon_{r}}$ ) of highest weight module
$V(\nu)\subseteq V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime})$ (resp.
$V(\nu)\subseteq{\cal{R}}_{{\langle}s+1\rangle}$ ) satisfying

$T_{s}(v_{\nu})=(\mu_{r}+|\mu|-r+m_{r}+1)\xi_{\underline{m}+\varepsilon_{r}}\neq
0.$ $None$

Assume that $w_{\nu}$ is a maximal vector of irreducible module
$V(\nu)\subseteq V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime})$. Since
$T_{s}:V(\varepsilon_{1})\otimes_{\mathbb{F}}({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s\rangle}\rightarrow({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s+1\rangle}$ is an intertwining operator
for $\overline{L}_{n}$, we know $T_{s}(w_{\nu})$ is also a maximal vector of
$\overline{L}_{n}$-module
$V(\nu)\subseteq{\cal{R}}_{{\langle}s+1\rangle}\;(\subseteq({\cal
A}\otimes_{\mathbb{F}}V)_{{\langle}s+1\rangle})$. Since the maximal vector
$w_{\nu}$ must take the following form:

$w_{\nu}=a_{r}e_{r}^{\prime}\otimes\varpi_{\nu^{\prime}}+\sum_{j=1}^{r-1}\sum_{i=1}^{m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r})}a_{j}^{i}e_{j}^{\prime}\otimes
v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i},\qquad 0\neq a_{r},\
a_{j}^{i}\in\mathbb{F},$ $None$

where $\varpi_{\nu^{\prime}}$ is a maximal vector of the highest weight module
$V(\nu^{\prime})$ and the set
$\\{v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i}\mid
i\in\overline{1,m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}))}\\}$ is a
basis of the weight subspace
$[V(\nu^{\prime})]_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}$. For
convenience, we write

$v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i}=\sum\limits_{\varepsilon_{j}-\varepsilon_{r}=\sum\limits_{k=1}^{m}i_{k}(\varepsilon_{s_{k}}-\varepsilon_{t_{k}});\>s_{i}>t_{i}}a_{\underline{s},\underline{t}}^{\underline{i}}(x_{s_{1}}\partial_{t_{1}})^{i_{1}}.\cdots.(x_{s_{m}}\partial_{t_{m}})^{i_{m}}.\varpi_{\nu^{\prime}}.$
$None$

Therefore,

$T_{s}(w_{\nu})=a_{r}p_{r}.\varpi_{\nu^{\prime}}+\sum_{j=1}^{r-1}\sum_{i=1}^{m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r})}a_{j}^{i}p_{j}.v_{\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r}}^{i}.$
$None$

So (3.94) and (3.95) imply that

$T_{s}(w_{\nu})=n_{\varepsilon_{r}}.\varpi_{\nu^{\prime}}\qquad\mbox{for \
some}\ n_{\varepsilon_{r}}\in U(P\oplus\bar{L}_{n}).$ $None$

Let $\varpi_{\nu^{\prime}}={\xi}_{\underline{m}}$ (resp.
$\varpi_{\nu^{\prime}}=\hat{\xi}_{\underline{m}}$ ) in (3.96), where

$\xi_{\underline{m}}=a_{\underline{m}}x^{{\underline{m}}}\otimes
v_{\mu}+\sum\limits_{(\eta,\underline{l})\neq(\mu,\underline{m}),{\underline{m}}+\mu={\underline{l}}+\eta,\
i\in\overline{1,m({\eta})}}a_{\eta,\underline{l}}^{i}x^{\underline{l}}\otimes
v_{\eta}^{i},\qquad 0\neq a_{\underline{m}},\
a_{\eta,\underline{l}}^{i}\in\mathbb{F};$ $None$
$\hat{\xi}_{\underline{m}}=a_{\underline{m}}p^{{\underline{m}}}.(1\otimes
v_{\mu})+\sum\limits_{(\eta,\underline{l})\neq(\mu,\underline{m}),{\underline{m}}+\mu={\underline{l}}+\eta,\
i\in\overline{1,m({\eta})}}a_{\eta,\underline{l}}^{i}p^{\underline{l}}.(1\otimes
v_{\eta}^{i}),\qquad 0\neq a_{\underline{m}},\
a_{\eta,\underline{l}}^{i}\in\mathbb{F}.$ $None$

Take

$\hat{\xi}_{\underline{m}+\varepsilon_{r}}=n_{\varepsilon_{r}}.\hat{\xi}_{\underline{m}}.$
$None$

Then by Lemma 3.2.3, we have

$\hat{\xi}_{\underline{m}+\varepsilon_{r}}=q_{\underline{m}+\varepsilon_{r}}{\xi}_{\underline{m}+\varepsilon_{r}}=n_{\varepsilon_{r}}.\hat{\xi}_{\underline{m}}=q_{\underline{m}}n_{\varepsilon_{r}}.{\xi}_{\underline{m}}.$
$None$

Hence, we have

$(\mu_{r}+|\mu|-r+m_{r}+1){\xi}_{\underline{m}+\varepsilon_{r}}=n_{\varepsilon_{r}}.{\xi}_{\underline{m}}.$
$None$

Now we take

$\displaystyle v_{\nu}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{r-1}\sum_{i=1}^{m(\nu^{\prime}-\varepsilon_{j}+\varepsilon_{r})}\sum\limits_{\varepsilon_{j}-\varepsilon_{r}=\sum\limits_{k=1}^{m}i_{k}(\varepsilon_{s_{k}}-\varepsilon_{t_{k}});\>s_{i}>t_{i}}a_{j}^{i}a_{\underline{s},\underline{t}}^{\underline{i}}e_{j}^{\prime}\otimes(x_{s_{1}}\partial_{t_{1}})^{i_{1}}.\cdots.(x_{s_{m}}\partial_{t_{m}})^{i_{m}}.{\xi}_{\underline{m}}$
$\displaystyle+a_{r}e_{r}^{\prime}\otimes{\xi}_{\underline{m}}.\hskip
318.67078pt(3.102)$

Then
$T_{s}(v_{\nu})=(\mu_{r}+|\mu|-r+m_{r}+1)\xi_{\underline{m}+\varepsilon_{r}}\neq
0$ by (3.101) and (3.102). Therefore, (2.30) and (3.92) imply that

$\\{0\\}\neq(T_{s}|_{V(\varepsilon_{1})\bigotimes_{\mathbb{F}}V(\nu^{\prime})}\circ\tilde{P}_{r})(V(\varepsilon_{1})\otimes_{\mathbb{F}}V(\nu^{\prime}))=V(\nu),$
$None$

where

$\tilde{P}_{r}=\prod\limits_{l\neq
r}(\frac{\tilde{M}-\tilde{d}_{l}}{\tilde{d}_{r}-\tilde{d}_{l}}),\
\tilde{d}_{i}=i-1-m_{i}-\mu_{i},$ $None$

and $\tilde{M}$ is the matrix in (2.27). Hence,
${\cal{R}}_{{\langle}s\rangle}\subseteq W$ for any $s\geq k+1$. Thus we prove
$({\cal A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ is irreducible.
$\Box$

Suppose ${\cal{R}}_{{\langle}k+1\rangle}=V(\mu+(k+1)\varepsilon_{r})$ for some
$r\in I_{k+1}$. From the proof of Proposition 3.3.8, we know $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ is generated by
$\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}={\xi}_{\mu+(k+1)\varepsilon_{r}}+U(P)(1\otimes_{\mathbb{F}}V)$,
i.e. $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)=U(L_{n+1}).\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}$.
Hence, we get the following result:

Proposition 3.3.9 The $L_{n+1}$-module $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ is isomorphic to the
irreducible module $U(P)(1\otimes_{\mathbb{F}}M)$, where $M$ is a
$gl(n)$-irreducible module admitting the character
$\chi_{\mu+(k+1)\varepsilon_{r}}$.

Proof Since $\partial_{i}.{\xi}_{\mu+(k+1)\varepsilon_{r}}\in(U(P)(1\otimes
V))_{{\langle}k\rangle}=({\cal A}\otimes_{\mathbb{F}}V)_{{\langle}k\rangle}$,
we get $\partial_{i}.\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}=0$ for any
$i\in\overline{1,n}$. It follows from Proposition 3.3.8 that both $({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)$ and
$U(P)(1\otimes_{\mathbb{F}}M)$ are irreducible $L_{n+1}$-modules. So it is
easy to verify that

$\vartheta:({\cal
A}\otimes_{\mathbb{F}}V)/U(P)(1\otimes_{\mathbb{F}}V)\rightarrow
U(P)(1\otimes_{\mathbb{F}}M);x.\bar{\xi}_{\mu+(k+1)\varepsilon_{r}}\mapsto
x.(1\otimes v_{\mu+(k+1)\varepsilon_{r}})$ $None$

is an $L_{n+1}$-module isomorphism, where $x\in U(P\oplus\bar{L}_{n})$ and
$v_{\mu+(k+1)\varepsilon_{r}}$ is a maximal vector of $M$. $\Box$

From Proposition 3.3.3, Proposition 3.3.8 and Proposition 3.3.9, we get the
Main Theorem.

Remark 3.3.10 The irreducible module $U(P)(1\otimes_{\mathbb{F}}V)$ is cyclic,
i.e. it is generated by one vector. From Lemma 3.1.1 and (3.12), we know that
$U(P)(1\otimes_{\mathbb{F}}V)$ is in general not a highest weight module. We
can easily verify the following result from the Main Theorem:

Corollary 3.3.11 Assume one of the conditions in (1.7) and (1.8) fails. Denote

$i_{0}=\mbox{min}\\{i\in\overline{1,n}\ |\
\mu_{i}+|\mu|-i+1\in-\mathbb{N}\\}.$ $None$

We have:

(i) The integer $k=-\mu_{i_{0}}-|\mu|+i_{0}-1$ satisfies (3.72) and (3.73).

(ii) The irreducible module $U(P)(1\otimes_{\mathbb{F}}V)$ is finite
dimensional highest weight module with highest weight
$k\omega_{1}+\sum\limits_{i=2}^{n}m_{i-1}\omega_{i}$ iff $\ i_{0}=1$, where
$m_{i}=\mu_{i}-\mu_{i+1}$ for $i\in\overline{1,n-1}$.

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