# Demazure Filtrations of Tensor Product Modules and Character Formula

Divya Setia  and  Tanusree Khandai  Indian Institute of Science Education and
Research Mohali, Knowledge City, Sector 81, S.A.S. Nagar 140306, Punjab, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the structure of the finite-dimensional representations of
$\mathfrak{sl}_{2}[t]$, the current Lie algebra type of $A_{1}$, which are
obtained by taking tensor products of special Demazure modules. We show that
these representations admit a Demazure flag and obtain a closed formula for
the graded multiplicities of the level 2 Demazure modules in the filtration of
the tensor product of two local Weyl modules for $\mathfrak{sl}_{2}[t]$.
Furthermore, we derive an explicit expression for graded character of the
tensor product of a local Weyl module with an irreducible
$\mathfrak{sl}_{2}[t]$ module. In conjunction with the results of [7], our
findings provide evidence for the conjecture in [5] that the tensor product of
Demazure modules of levels m and n respectively has a filtration by Demazure
modules of level m + n.

##  1\. INTRODUCTION

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over the complex
field $\mathbb{C}$ and $\mathfrak{g}[t]$ be its associated current algebra,
which is the Lie algebra of polynomimal mappings from $\mathbb{C}$ to
$\mathfrak{g}$. Due to its connections with combinatorics, number theory and
mathematical physics, the theory of finite-dimensional representations of
current algebras has garnered significant attention in the past two decades.
Motivated by a conjecture in [5] that suggests tensor products of Demazure
modules with levels $m$ and $n$, respectively, have a filtration by Demazure
modules of level $m+n$, in this paper we consider the class of finite-
dimensional representations, namely those of $\mathfrak{sl}_{2}[t]$, which are
derived from tensor products of certain Demazure modules of
$\mathfrak{sl}_{2}[t]$ and prove results that provide evidence in support of
the conjecture.

Let $\hat{\mathfrak{g}}$ be an affine Kac-Moody Lie algebra and
$\hat{\mathfrak{b}}$ be a standard positive Borel subalgebra of
$\hat{\mathfrak{g}}$. Given an integrable highest weight irreducible
representation $\mathcal{V}$ of $\hat{\mathfrak{g}}$, a Demazure module
associated with $\mathcal{V}$, is defined as the $\hat{\mathfrak{b}}$-module
generated by an extremal weight vector $u_{0}$ of $\mathcal{V}$. These modules
are $\mathfrak{g}$-stable when the restriction of the weight of $u_{0}$ to a
Cartan subalgebra of $\mathfrak{g}$ is anti-dominant and in this case the
corresponding Demazure module is a module of the current algebra
$\mathfrak{g}[t]$. A Demazure module is said to be of level $\ell$ if the
central element of $\hat{\mathfrak{g}}$ acts on $\mathcal{V}$ by the scalar
$\ell$. A $\hat{\mathfrak{b}}$-module is said to admit a Demazure flag if it
has a filtration by submodules whose successive quotients are Demazure
modules. While it is known that the tensor product of two Demazure modules
does not in general have a Demazure flag, in [16, 20, 17] it was proved that
for simply-laced Kac–Moody Lie algebras, the tensor product of a one-
dimensional Demazure module with an arbitrary one admits a Demazure flag. Our
study here extends this question to tensor products of level 1 Demazure
modules with Demazure modules of arbitrary levels for current algebras of type
$A_{1}$.

The notion of Weyl modules for affine Kac-Moody Lie algebras was introduced in
[9]. In the case when $\mathfrak{g}=\mathfrak{sl}_{2}$, it was proven that
these modules are the classical limit ($q\rightarrow 1$) of standard modules
of the quantum affine algebras. Furthermore, it was shown that for a dominant
integral weight $\lambda$, $W_{loc}(\lambda)$, the local Weyl module with
highest weight $\lambda$, is a finite dimensional graded
$\mathfrak{g}[t]$-module generated by a highest weight vector of weight
$\lambda$ and every finite-dimensional graded $\mathfrak{g}[t]$-module with
highest weight $\lambda$ is a quotient of $W_{loc}(\lambda)$. Subsequently it
was demonstrated ([12, 10, 15, 21]) that for current algebra of type ADE, the
local Weyl modules are in fact Demazure modules of level 1 and their Demazure
characters coincide with non-symmetric Macdonald polynomials, specialized at t
= 0. On the other hand, it was demonstrated in [20] that the local Weyl
modules for current algebras of non-simply laced Lie algebras $\mathfrak{g}$
have filtrations by Demazure modules whose multiplicity coincides with the
multiplicity of certain Demazure modules in local Weyl modules of type $A_{r}$
for suitable $r$.

In [8], a family of finite-dimensional quotients of the local Weyl modules,
often referred to as Chari-Ventakesh Modules (in short CV-modules) was
introduced. It was shown that these modules subsume many disparate classes of
finite-dimensional graded representations of current algebras. By definition,
a CV module $V(\xi)$ is associated to a family of partitions
${\mathbb{\xi}}=\\{\xi_{\alpha}\\}_{\alpha}$, which is indexed by the set of
positive roots of $\mathfrak{g}$. It was proven in [8, Theorem 2] that the
Demazure modules in various levels can be realized as CV-modules associated to
a set of rectangular or special near rectangular partitions. The
interpretation of Demazure modules as CV-module greatly simplified their
defining relations and aided in the study of the structure of fusion product
modules for $\mathfrak{sl}_{2}[t]$. In [7] a necessary and sufficient
condition was obtained for the existence of level $\ell$ Demazure flags in an
arbitrary CV-module for $\mathfrak{sl}_{2}[t]$ and combinatorial descriptions
of the graded multiplicities of Demazure modules in $V(\xi)$ was given.
Ensuing research ([3, 2]) on the generating functions associated with the
graded multiplicities of Demazure modules in special CV-modules established
connections between these modules, number theory and combinatorics.

Motivated by the study initiated in [5] on the structure of tensor product of
two local Weyl modules for $\mathfrak{sl}_{2}[t]$, we begin by obtaining a
presentation of the representations of $\mathfrak{sl}_{2}[t]$ that arise from
taking the tensor product of a local Weyl module with a CV module $V(\xi)$.
Specializing the partition $\xi$, we then explore the tensor product of local
Weyl modules with irreducible $\mathfrak{sl}_{2}[t]$-modules and local Weyl
modules for $\mathfrak{sl}_{2}[t]$. In each of these cases, we observe that
the corresponding tensor product possesses a filtration by CV-modules. Using
[7, Theorem 3.3], we thus conclude that each such module has a filtration by
Demazure modules at appropriate levels.

For a given positive integer $m$, let $W_{loc}(m)$ denote the local Weyl
module of $\mathfrak{sl}_{2}[t]$ with the highest weight $m$. Through explicit
construction, we demonstrate that when $\xi=(n)$, the module
$W_{loc}(m)\otimes V(\xi)$ admits a filtration by CV-modules with hook-type
partitions. Utilizing this filtration, we derive expressions for the outer
multiplicities in the tensor product module $W_{loc}(m\omega)\otimes
ev_{0}V(n\omega)$. As a consequence we are able to express the product of a
Schur polynomial with a specialized Macdonald polynomial in terms of Schur
polynomials.

Finally we consider the tensor product of a local Weyl module with certain
special Demazure modules. In [18], it had been shown that the truncated local
Weyl module of $\mathfrak{sl}_{2}[t]$ of highest weight $r\omega$ can be
realized as a CV-module. Interestingly, we observe that the tensor product of
two local Weyl modules have a filtration by truncated local Weyl modules of
suitable weights. We determine the graded character of such truncated local
Weyl modules in terms of level 2 Demazure modules and give the graded
multicilities of level 2 Demazure modules in $W_{loc}(m)\otimes W_{loc}(n)$.

The character formulas for certain tensor product modules had been obtained in
[5]. Using an uniform approach we give a direct proof for the character
formula of the modules which we consider. Then, using the Chari-Pressley-
Loktev basis ([9, Section 6], [10, Theorem 2.1.3]) and Chari-Venkatesh basis
([8, Theorem 5]) for local Weyl modules for $\mathfrak{sl}_{2}[t]$, the
presentation of the tensor products modules we obtained and the character
formulas obtained, we show that in the cases under consideration, the tensor
product modules have a Demazure filtration of appropriate level.

The paper is structured as follows: In Section 2, we set the notations and
recall the basic definitions and results essential for our paper. Section 3
begins with a recap of the definition of CV modules. We then proceed to give a
presentation via generators and relations for the tensor product modules
$W_{loc}(m)\otimes V(\xi)$. Moving on to Section 4, we systematically
construct a decreasing chain of submodules of $W_{loc}(m\omega)\otimes
ev_{0}V(n\omega)$ such that the successive quotients in this chain are
isomorphic to $V(\xi)$-module, where $\xi$ is a partition with a hook shape.
We determine the graded character of these modules. In section 5, we prove the
existence of a Demazure flag for modules arising from tensor product of local
Weyl module with Demazure modules of level 1 and determine the graded
character formula for the tensor product of two local Weyl modules in terms of
level 2 Demazure modules.

##  2\. PRELIMINARIES

In this section we set the notation for the paper and recall the definitions
that are willl be used throughout.

### 2.1.

Throughout this paper, $\mathbb{C}$ will denote the field of complex numbers,
$\mathbb{Z}$ (resp. $\mathbb{Z_{+}}$), the set of integers (resp. non-negative
integers) and $\mathbb{C}[t]$, the polynomial algebra in an indeterminate $t$.
For $n,r\in\mathbb{Z}_{+}$, set

$[n]_{q}=\frac{1-q^{n}}{1-q},\quad\begin{bmatrix}n\\\
r\end{bmatrix}_{q}=\frac{[n]_{q}[n-1]_{q}\dots[n-r+1]_{q}}{[r]_{q}[r-1]_{q}\dots[1]_{q}}$
$\begin{bmatrix}n\\\ 0\end{bmatrix}_{q}=1,\quad\begin{bmatrix}n\\\
1\end{bmatrix}_{q}=\frac{1-q^{n}}{1-q},\quad\begin{bmatrix}n\\\
r\end{bmatrix}_{q}=0,\quad\text{ unless }\\{n,r,n-r\\}\subset\mathbb{Z}_{+}.$

### 2.2.

Given any complex Lie algebra $\mathfrak{a}$, denote by $U({\mathfrak{a}})$
the corresponding universal enveloping algebra and by $\mathfrak{a}[t]$ the
associated current Lie-algebra with underlying vector space
$\mathfrak{a}\otimes\mathbb{C}[t]$ and Lie-bracket :

$[x\otimes f,y\otimes g]=[x,y]\otimes fg\quad\forall x,y\in\mathfrak{a}\text{
and }f,g\in\mathbb{C}[t]$

The natural grading on $\mathbb{C}[t]$ defines a $\mathbb{Z}_{+}$-grading on
$\mathfrak{a}[t]$ and $U(\mathfrak{a}[t])$. With respect to it, any element of
$U(\mathfrak{a}[t])$ is of the form $(x_{1}\otimes t^{a_{1}})(x_{2}\otimes
t^{a_{2}})\dots(x_{n}\otimes t^{a_{n}})$ and has grade
$a_{1}+a_{2}+\dots+a_{n}$.

With respect to an ordered basis of $\mathfrak{a}[t]$, the PBW filtration on
$U(\mathfrak{a}[t])$ is given as follows:

$U(\mathfrak{a}[t])^{\leq r}=\\{a_{1}^{s_{1}}a_{2}^{s_{2}}\cdots
a_{l}^{s_{l}}:0\leq\sum\limits_{j=1}^{l}js_{j}\leq r\\}$

We say length of $X\in U(\mathfrak{a}[t])$ is $r$ if $X\in
U(\mathfrak{a}[t])^{\leq r}\backslash U(\mathfrak{a}[t])^{\leq r-1}$.

### 2.3.

Let $\mathfrak{sl}_{2}$ be the Lie algebra of two by two trace zero matrices
with entries in $\mathbb{C}$ and $\mathfrak{sl_{2}}[t]$ the associated current
algebra. Let $\\{x,h,y\\}$ be the standard basis, $\alpha$ (resp. $\omega$)
the unique simple root (resp. fundamental weight) of $\mathfrak{sl}_{2}$.

### 2.4.

For $n\in\mathbb{Z}_{+}$, let $V(n)$ be the $n+1$-dimensional irreducible
$\mathfrak{sl}_{2}$-module generated by a highest weight vector $v_{n}$ with
the following defining relations:

$xv_{n}=0,\quad hv_{n}=nv_{n},\quad y^{n+1}v_{n}=0.$

Any finite-dimensional $\mathfrak{sl}_{2}$-module $V$ can be decomposed as:

$V=\bigoplus_{k\in\mathbb{Z}}V_{k},\qquad V_{k}=\\{v\in V\mid hv=kv\\}.$

The formal character of a finite-dimensional $\mathfrak{sl}_{2}$-module $V$,
denoted as $ch_{\mathfrak{sl}_{2}}V$ is an element of $\mathbb{Z}[e(\omega)]$
such that

$V=\bigoplus_{k\in\mathbb{Z}_{+}}V(k)^{\oplus
m_{k}}\qquad\Longrightarrow\qquad ch_{\mathfrak{sl}_{2}}\
V=\sum\limits_{k\in\mathbb{Z}_{+}}m_{k}\ ch_{\mathfrak{sl}_{2}}\ V(k),$

where $ch_{\mathfrak{sl}_{2}}\ V(k)=\sum\limits_{i=0}^{k}e((k-2i)\omega)$ for
$k\in\mathbb{Z}_{+}.$

### 2.5.

A graded $\mathfrak{sl}_{2}[t]$-module $V$ is a $\mathbb{Z}$-graded vector
space $V=\bigoplus\limits_{k\in\mathbb{Z}}V[k]$, which admits a
$\mathfrak{sl}_{2}[t]$ action in the following way:

$(X\otimes t^{r})V[k]\subseteq V[k+r],\quad
r\in\mathbb{Z}_{+},k\in\mathbb{Z}.$

For each $k$, $V[k]$ is a $\mathfrak{sl}_{2}$-submodule of $V$. If $V$ and $W$
are graded $\mathfrak{sl}_{2}[t]$-modules, then so is $V\otimes W$, with the
following $\mathfrak{sl}-2[t]$-action:

$(X\otimes t^{r})(v\otimes w)=(X\otimes t^{r})v\otimes w+v\otimes(X\otimes
t^{r})w$

Thus in this case we have, $(V\otimes
W)[k]=\bigoplus\limits_{j\in\mathbb{Z}_{+}}V[j]\otimes W[k-j]$, where $W[j]=0$
if $j<0$.

Given a $\mathfrak{sl}_{2}$-module $W$, we define a
$\mathfrak{sl}_{2}[t]$-structure on $W$ as follows:

$(X\otimes t^{r})w=\delta_{r,0}Xw,$

We denote such a module by $ev_{0}(W)$. Clearly it is $\mathbb{Z}$-graded
$\mathfrak{sl}_{2}[t]$-module with $ev_{0}(W)[k]=\delta_{k,0}W$. For $r\in Z$,
let $\tau_{r}$ be the shift operater given $(\tau_{r}W)[k]=W[k-r].$ Given
$(n,r)\in\mathbb{Z}_{+}^{2}$, we denote the irreducible
$\mathfrak{sl}_{2}[t]$-module $\tau_{r}\ ev_{0}V(n)$ by $V(n,r)$ and in
particular denote $ev_{0}V(n)$ by $V(n,0)$. It was shown in [6] that any
irreducible, graded, finite-dimensional module is isomorphic to $V(n,r)$ for a
pair $(n,r)\in\mathbb{Z}_{+}^{2}.$

The graded character of a graded $\mathfrak{sl}_{2}[t]$-module
$V=\bigoplus\limits_{k\in\mathbb{Z}_{\geq 0}}V[k]$ is given by:

$ch_{gr}\ V=\sum\limits_{k\geq 0}ch_{\mathfrak{sl}_{2}}\ V[k]\
q^{k}\in\mathbb{Z}[e(\omega)][q],\qquad\text{ where }q\text{ is an
indeterminate}.$

### 2.6.

For $\lambda\in\mathbb{Z}_{+}$, the local Weyl module $W_{loc}(\lambda)$ is a
finite-dimensional, $\mathbb{Z}_{+}$-graded $\mathfrak{sl}_{2}[t]$-module
generated by an element $w_{\lambda}$ with the following defining relations:

$(x\otimes t^{r})w_{\lambda}=0,\quad(h\otimes
t^{r})w_{\lambda}=\lambda\delta_{r,0}w_{\lambda},\quad(y\otimes
1)^{\lambda+1}w_{\lambda}=0,\qquad\quad\forall r\in\mathbb{Z}_{\geq 0}.$

By definition, $W_{loc}(\lambda)$ is a highest weight representation of
$\mathfrak{sl}_{2}[t]$ with highest weight $\lambda\omega$ and any finite-
dimensional integrable $\mathfrak{sl}_{2}[t]$-module with highest weight
$\lambda\omega$ is a quotient of $W_{loc}(\lambda)$. In particular,
$ev_{0}V(\lambda)$ is the unique irreducible quotient of $W_{loc}(\lambda)$.
It was proven in [7] that the graded character of $W_{loc}(m)$ is given as
follows:

$ch_{gr}\ W_{loc}(m)=\sum\limits_{l=0}^{m}(\begin{bmatrix}m\\\
l\end{bmatrix}_{q}-\begin{bmatrix}m\\\ l-1\end{bmatrix}_{q})\ ch_{gr}\
ev_{0}V(m-2l).$

##  3\. PRESENTATION OF $W_{loc}(m)\otimes V(\mathbb{n})$

In [8], a family of graded representations of the current Lie algebras, which
we shall refer to as CV-modules, was introduced. It was shown that local Weyl
modules, Demazure modules of various levels, truncated Weyl modules and fusion
product modules for $\mathfrak{sl}_{2}[t]$ can be realized as CV-modules [8,
18]. In this section we recall the definition and properties of CV-modules and
using them give a presentation of the $\mathfrak{sl}_{2}[t]$-modules that are
obtained by taking tensor products of local Weyl modules with certain Demazure
modules.

### 3.1.

We first recall the definition of a CV-module $V(\xi)$ associated to a
partition $\xi$.

######  Definition 3.1.1.

Given a partition $\mathbb{n}=(n_{1}\geq n_{2}\geq\cdots\geq n_{k}>0)$ of
$n\in\mathbb{Z}_{+}$, let $V(\mathbb{n})$ be a cyclic
$\mathfrak{sl}_{2}[t]$-module generated by an element $v_{\mathbb{n}}$ which
satisfies the following relations:

$(x\otimes t^{r})v_{\mathbb{n}}=0,\quad(h\otimes
t^{r})v_{\mathbb{n}}=n\delta_{r,0}v_{\mathbb{n}},\quad(y\otimes
1)^{n+1}v_{\mathbb{n}}=0$ (3.1) $(x\otimes t)^{s}(y\otimes
1)^{r+s}v_{\mathbb{n}}=0\quad\forall\,r+s\geq 1+rl+\sum_{j\geq
l+1}n_{j},\text{ for some }l\in\mathbb{N}$ (3.2)

Here, $r\in\mathbb{Z}_{+}$ in (3.1) and $r,s\in\mathbb{N}$ in (3.2).

Defining the grade of $v_{\mathbb{n}}$ to be zero, we see that $V(\mathbb{n})$
is a finite-dimensional graded $\mathfrak{sl}_{2}[t]$-quotient of
$W_{loc}(n)$. The CV-modules are defined in [8] for an arbitrary current
algebra, but for our purposes, we need the definition only for
$\mathfrak{sl}_{2}[t]$.

Certain special cases of $V(\mathbb{n})$ discussed in [8] are as follows:

$\begin{array}[]{ll}V(\mathbb{n})\cong W_{loc}(n),&\text{for
}\,\mathbb{n}=(1^{n}),\\\ V(\mathbb{n})\cong ev_{0}(V(n))&\text{for
}\,\mathbb{n}=(n),\\\ V(\mathbb{n})\cong D(l,lr+n_{0})&\text{for
}\,\mathbb{n}=(l^{r},n_{0}),\end{array}$ (3.3)

where $D(l,lr+n_{0})$ denotes the Demazure module of level $l$ with highest
weight $(lr+n_{0})\omega=n\omega$.

### 3.2.

Define a power series in the indeterminate $u$ as follows:

$\displaystyle H(u)=\exp\left(-\sum_{r=1}^{\infty}\frac{h\otimes
t^{r}}{r}u^{r}\right)$

Let $P(u)_{k}$ denote the coefficient of $u^{k}$ in $H(u)$. For
$s,r\in\mathbb{Z}_{+}$, $x\in\mathfrak{g}$, define elements $x(r,s)\in
U(\mathfrak{g}[t])$ as follows:

$x(r,s)=\sum\limits_{(b_{p})_{p\geq 0}}(x\otimes 1)^{(b_{0})}(x\otimes
t^{1})^{(b_{1})}\cdots(x\otimes t^{s})^{(b_{s})}$

with $p\in\mathbb{Z},b_{p}\in\mathbb{Z}_{+}$ such that $\sum\limits_{p\geq
0}b_{p}=r$, $\sum\limits_{p\geq 1}pb_{p}=s$ and for any $X\in\mathfrak{g}$,
$X^{(p)}=\dfrac{X^{p}}{p!}$. The following result was proved in [13] and
reformulated in its present form in [9].

######  Lemma 3.2.1.

Given $s\in N,r\in\mathbb{Z}_{+}$, we have

$(x\otimes t)^{(s)}(y\otimes 1)^{(r+s)}-(-1)^{s}\big{(}\ \underset{k\geq
0}{\sum}\ y(r,s-k){P(u)}_{k}\big{)}\in U(\mathfrak{g}[t])\mathfrak{n}^{+}[t].$
(3.4)

Using Lemma 3.4 and (3.1), relation (3.2) in definition of $V(\mathbb{n})$ can
be replaced by the following:

$y(r,s)v_{\mathbb{n}}=0,\text{ if }r+s\geq 1+rl+\sum_{j\geq l+1}n_{j}.$

### 3.3.

Using the presenation of $W_{loc}(n)$ as a $CV$-module, we now give an
alternate presentation of $W_{loc}(n)$ as a lowest weight module. In what
follows, we use the notation : $a\otimes t^{r}:=a_{r},$ for all
$a\in\mathfrak{sl}_{2},\ r\in\mathbb{Z}_{+}$.

######  Lemma 3.3.1.

Given $n\in\mathbb{Z}_{+}$, let $W(-n)$ be the $\mathfrak{sl}_{2}[t]$-module
generated by a vector $w_{-n}$ which satisfies the following relations:

$\begin{array}[]{lll}y_{r}w_{-n}&=0,&\forall\,r\geq 0,\\\
h_{r}w_{-n}&=\delta_{0,r}(-n)w_{-n},&\forall\,r\geq 0,\\\
x(r,s)w_{-n}&=0,&\text{if }r+s\geq 1+rl+n-l.\end{array}$

Then, $W(-n)$ is isomorphic to $W_{loc}(n).$

###### Proof.

Let $W_{loc}(n)$ be generated by a vector $w_{n}$ which satisfies the
relations (3.1) and (3.2) for $\mathbb{n}=(1^{n}).$ Then $y_{0}^{n}w_{n}$ is a
non-zero vector in $W_{loc}(n)$ such that

$\begin{array}[]{llll}y_{r}y_{0}^{n}w_{n}=0,&\forall\,r\geq 0,\quad\quad
y(n,r).w_{n}=0,&\forall\,r\geq 1.\end{array}$

Further, as

$\begin{array}[]{c}y(n,1)=Ah_{1}y^{n}_{0}w_{n},\\\
y(n,r)w_{n}=A_{r}h_{r}y_{0}^{n}w_{n}+A_{r-1}h_{r-1}y(n,1)w_{n}+\cdots+{A_{1}h_{1}y(n,r-1)w_{n}},\,\forall\,r\geq
2\end{array}$

with $A,A_{1},\cdots,A_{r}\in\mathbb{Q}$, using induction and the above
relations we see that

$h_{r}y^{n}_{0}w_{n}=\delta_{r,0}(-n)w_{n}.$

Now by applying to (3.4) the $\mathfrak{sl}_{2}[t]$-isomorphism that maps
$y_{s}\rightarrow x_{s}$, $x_{s}\rightarrow y_{s}$ and $h_{s}\rightarrow-
h_{s}$ for $s\geq 0$, using the above relations we get,

$(y\otimes t)^{(s)}(x\otimes
1)^{(r+s)}y_{0}^{n}w_{\mathbb{n}}=(-1)^{s}x(r,s)(-n)y_{0}^{n}w_{\mathbb{n}}$
(3.5)

As, $x_{0}^{(r+s)}.y_{0}^{n}v_{\mathbb{n}}=0,$ for $r+s\geq n+1$, it follows
that,

$x(r,s)y_{0}^{n}w_{\mathbb{n}}=0,\,\qquad\text{ when }\hskip 7.11317ptr+s\geq
1+rk+n-k.$

Thus, the vector $y_{0}^{n}w_{n}\in W_{loc}(n)$ satisfies the relations

$\begin{array}[]{lll}(y\otimes t^{r})y_{0}^{n}w_{n}&=0,&\forall\,r\geq 0,\\\
(h\otimes t^{r})y_{0}^{n}w_{n}&=\delta_{0,r}(-n)y_{0}^{n}w_{n},&\forall\,r\geq
0,\\\ x(r,s)y_{0}^{n}w_{n}&=0,&\text{if }r+s\geq 1+rl+n-l.\end{array}$

Therefore by PBW theorem,
$W_{loc}(n)=\mathbb{U}(\mathfrak{n}^{+}[t])y_{0}^{n}w_{n}$.

Let $\phi:W(-n)\rightarrow W_{loc}(n)$ be a map such that
$\phi(w_{-n})=y_{0}^{n}w_{n}$. Then it is clear from above that $\phi$ defines
a surjective $\mathfrak{sl}_{2}[t]$-homomorphism from $W(-n)$ to $W_{loc}(n).$
Since by definition, $W(-n)=U(\mathfrak{n}^{+}[t])w_{-n}$, it follows from
above that $\phi$ is in fact an isomorphism. ∎

### 3.4.

The following results on the $\mathfrak{sl}_{2}[t]$-modules $W{loc}(m)$ and
$V(\mathbb{n})$ were proven in [9] and [8]. We record them for future
reference.

######  Lemma 3.4.1.

Let $m,n\in\mathbb{N}$.

  * i.

For $m\in\mathbb{N}$, $\dim W_{loc}(m\omega)=2^{m}.$

  * ii.

Given a partition $\mathbb{n}=(n_{1}\geq n_{2}\geq\cdots\geq n_{k})$ of $n$,
let

$\mathbb{n}^{+}=(n_{1},n_{2}\cdots,n_{k-1}+1,n_{k}-1);\quad\mathbb{n}^{-}=(n_{1}\geq
n_{2}\geq\cdots\geq n_{k-1}-n_{k}).$

For $k>1$, there exists a short exact sequence of
$\mathfrak{sl}_{2}[t]$-modules,

$0\rightarrow\tau_{(k-1)n_{k}}V(\mathbb{n}^{-})\xrightarrow{\phi_{-}}V(\mathbb{n})\xrightarrow{{{}_{+}}\phi}V(\mathbb{n}^{+})\rightarrow
0.$

Furthermore, $\dim V(\mathbb{n})=\prod\limits_{i=1}^{k}(n_{i}+1)$ and

$\mathbb{B}(\mathbb{n})=\\{v_{\mathbb{n}},(y_{0})^{i_{1}}(y_{1})^{i_{2}}\cdots(y_{k-1})^{i_{k}}v_{\mathbb{n}}:(i_{1},i_{2},\dots,i_{k})\in
J(\mathbb{n})\\}$

is a basis of $V(\mathbb{n})$, where

$J(\mathbb{n})=\\{(i_{1},i_{2},\dots,i_{k})\in\mathbb{Z}_{\geq
0}^{k}:(ji_{r-1}+(j+1)i_{r})+2\sum_{l=r+1}^{n}i_{l}\leq\sum_{p=r-j}^{k}n_{p},2\leq
r\leq k+1,1\leq j\leq r-1\\}.$

######  Lemma 3.4.2.

Given $m,n\in\mathbb{N}$, let $\mathbb{n}=(n_{1}\geq\cdots\geq n_{k})$ be a
partition of $n$. Let $W_{loc}(m)$ and $V(\mathbb{n})$ be the graded
$\mathfrak{sl}_{2}[t]$-modules with generators $w_{m}$ and $v_{\mathbb{n}}$,
as defined above. Then $\mathfrak{sl}_{2}[t]$-module $W_{loc}(m\omega)\otimes
V(\mathbb{n})$ is a cyclic module generated by $y_{0}^{m}w_{m}\otimes
v_{\mathbb{n}}$ and

$\dim W(m\omega)\otimes V(\mathbb{n})=2^{m}\prod\limits_{i=1}^{k}(n_{i}+1).$
(3.6)

.

###### Proof.

Given non-zero vectors $v_{\mathbb{n}}$ and $w_{m}$ satisfying the given
conditions, let $S$ be the submodule of $W_{loc}(m)\otimes V(\mathbb{n})$
generated by the vector $y_{0}^{m}w_{m}\otimes v_{\mathbb{n}}.$ Since
$\mathfrak{n}^{-}.y_{0}^{m}w_{m}=0$ and
$V(\mathbb{n})=U(\mathfrak{n}^{-}[t])v_{\mathbb{n}}$,

$S\supseteq U(\mathfrak{n}^{-}[t])y_{0}^{m}w_{m}\otimes
v_{\mathbb{n}}=y_{0}^{m}w_{m}\otimes
U(\mathfrak{n}^{-}[t])v_{\mathbb{n}}=y_{0}^{m}w_{m}\otimes V(\mathbb{n}).$

On the other hand, by Lemma 3.3.1,
$W_{loc}(m)=U(\mathfrak{n}^{+}[t])y_{0}^{m}w_{m}$. Hence, for $r\in\mathbb{N}$
and $v\in V(\mathbb{n})$,

$x_{r}.y_{0}^{m}w_{m}\otimes V(\mathbb{n})=x_{r}y_{0}^{m}w_{m}\otimes v\mod
y_{0}^{m}w_{m}\otimes V(\mathbb{n}),$

implying $x_{r}y_{0}^{m}w_{m}\otimes V(\mathbb{n})\subseteq S$. Now using an
obvious induction argument on the length of $X\in U(\mathfrak{n}^{+}[t])$ for
all basis elements $Xy_{0}^{m}w_{m}$ of $W_{loc}(m)$, we see that
$W_{loc}(m)\otimes V(\mathbb{n})\subseteq S$. This shows that
$W_{loc}(m)\otimes V(\mathbb{n})$ is generated by the element
$y_{0}^{m}w_{m}\otimes v_{\mathbb{n}}$.

Using Lemma 3.4.1, it follows from 2.5 that $W_{loc}(m)\otimes V(\mathbb{n})$
is a finite-dimensional, graded $\mathfrak{sl}_{2}[t]$-module such that (3.6)
holds. ∎

### 3.5.

Let $\mathfrak{b}^{\pm}$ be the subalgebra $\mathfrak{h}+\mathfrak{n}^{\pm}$
of $\mathfrak{g}$. Then
$\mathfrak{g}=\mathfrak{b}^{+}\oplus\mathfrak{n}^{-}=\mathfrak{b}^{-}\oplus\mathfrak{n}^{+}$
and by the Poincaré-Birkhoff-Witt theorem,

$U(\mathfrak{g}[t])=U(\mathfrak{n}^{-}[t])\oplus
U(\mathfrak{g}[t])\mathfrak{b}^{+}[t]=U(\mathfrak{n}^{+}[t])\oplus
U(\mathfrak{g}[t])\mathfrak{b}^{-}[t].$ (3.7)

Let $\operatorname{\mathbb{pr}}^{\pm}:U(\mathfrak{g}[t])\rightarrow
U(\mathfrak{n}^{\pm}[t])$ be the projections with respect to decompositions
(3.7).

######  Lemma 3.5.1.

Let $\mathbb{n}=(n_{1}\geq n_{2}\geq\cdots\geq n_{k})$ be a partition of a
natural number $n$ and let $V(\mathbb{n})$ be the associated $CV$-module
generated by a highest weight vector $v_{\mathbb{n}}$. Let $J_{\mathbb{n}}$ be
the ideal of $U(\mathfrak{n}^{-}[t])$ generated by the elements

$\\{\operatorname{\mathbb{pr}}^{-}((x_{1})^{s}(y_{0})^{r+s}):s+r\geq
1+rl+\sum\limits_{j\geq l+1}n_{j},\,\text{ for some}\ k\in\mathbb{N}\\}.$
(3.8)

The map $U(\mathfrak{n}^{-}[t])\rightarrow V(\mathbb{n})$ which sends $g$ to
$gv_{\mathbb{n}}$ induces a vector space isomorphism

$U(\mathfrak{n}^{-}[t])/J_{\mathbb{n}}\cong V(\mathbb{n}).$ (3.9)

###### Proof.

Given the partition $\mathbb{n}$ of $n$, for all $Y\in
U(\mathfrak{n}^{-}[t]\oplus\mathfrak{b}^{+}t[t])$ we have,

$Yv_{\mathbb{n}}=\operatorname{\mathbb{pr}}^{-}(Y)v_{\mathbb{n}}.$

Thus by definition of the CV-module $V(\mathbb{n})$, the kernel of the map
$U(\mathfrak{n}^{-}[t])\rightarrow V(\mathbb{n})$ which sends $g$ to
$gv_{\mathbb{n}}$ is generated by the set (3.8).

Using the defining relations of the $CV$-module $V(\mathbb{n})$ we have

$y(r,s)v_{\mathbb{n}}=0,\quad\text{ for }r+s\geq 1+r(k-1)+n_{k}.$

Hence $y_{k-1}^{i_{k}}v_{\mathbb{n}}=0$ whenever $i_{k}>n_{k}$.

Furthermore, as $\mathbb{n}^{+}$ is the partition $(n_{1}^{+}\geq\cdots\geq
n_{k-1}^{+}\geq n_{k}-1)$ with

$n_{j_{0}}^{+}=n_{k-1}+1,\quad\text{ if }j_{0}=\min\\{j:n_{j}=n_{k}-1,1\leq
j\leq k-1\\}$

and $n_{j}^{+}=n_{j}$ for $1\leq j\leq k-1$, $j\neq j_{0}$, using the short
exact sequence in Lemma 3.4.1(b) repeatedly, we see that for $0\leq
i_{k}<n_{k}$, if
$\mathbb{n}^{+}(i_{k})=(\bar{n}_{1}\geq\cdots\geq\bar{n}_{k-1}\geq i_{k})$ is
a partition of $n$ such that
${{{}_{+}}\phi}^{n_{k}-i_{k}}(V(\mathbb{n}))=V(\mathbb{n}^{+}(i_{k}))$, then
from the definition of the map ${{}_{+}}\phi$, it follows that

$\bar{n}_{k-1}\leq\min\\{n_{k-1}+n_{k}-i_{k},n_{j}+\frac{n_{k}-i_{k}-\sum\limits_{s=1}^{k-j}(s)(n_{k-s-1}-n_{k-s})}{j},1\leq
j\leq k-2\\}.$

Therefore, using the short exact sequence for $V(\mathbb{n}^{+}(i_{k}))$, we
see that

$\begin{array}[]{rl}y_{k-2}^{i_{k-1}}y_{k-1}^{i_{k}}v_{\mathbb{n}}=0,&\text{
if }i_{k-1}>\bar{n}_{k-1}-i_{k},\\\ i.e.,\qquad\quad
y_{k-2}^{i_{k-1}}y_{k-1}^{i_{k}}v_{\mathbb{n}}=0,&\text{ if
}ji_{k-1}+(j+1)i_{k}>\sum\limits_{r=k-j}^{k}n_{r},\quad\text{ for some }j\leq
k-1.\end{array}$

As $\sum\limits_{j=1}^{k-1}\bar{n}_{j}=\sum\limits_{j=1}^{k}n_{j}-i_{k}$,
using the arguments recursively, we see that for a $k$-tuple
$(s_{1},\cdots,s_{k})$,

$y_{r-1}^{s_{r-1}}y_{r}^{s_{r}}\cdots
y_{k-1}^{s_{k}}v_{\mathbb{n}}=0\quad\text{ if
}(ji_{r-1}+(j+1)i_{r})+2\sum\limits_{l=r+1}^{n}i_{l}>\sum\limits_{p=r-j}^{k}n_{p},$

for some $0<r-1\leq k-1$ and $1\leq j\leq r-1$. This shows that all monomials
of the form $y_{0}^{s_{1}}\cdots y_{r-1}^{s_{r-1}}y_{r}^{s_{r}}\cdots
y_{k-1}^{s_{k}}\in J_{\mathbb{n}}$, whenever
$(ji_{r-1}+(j+1)i_{r})+2\sum\limits_{l=r+1}^{n}i_{l}>\sum\limits_{p=r-j}^{k}n_{p}$
for some $0<r-1\leq k-1$ and $1\leq j\leq r-1$. Consequently,
$U(\mathfrak{n}^{-}[t])/J_{\mathbb{n}}$ is spanned by cosets of the elements
$(y_{0})^{i_{1}}(y_{1})^{i_{2}}\cdots(y_{k-1})^{i_{k}}$ with
$(i_{1},i_{2},\cdots,i_{k})\in J(\mathbb{n})\cup(0,\cdots,0).$

On the other hand by Lemma 3.4.1,
$\\{v_{\mathbb{n}},(y_{0})^{i_{1}}(y_{1})^{i_{2}}\cdots(y_{k-1})^{i_{k}}v_{\mathbb{n}}:(i_{1},i_{2},\dots,i_{k})\in
J(\mathbb{n})\\}$ is a basis of $V(\mathbb{n})$. Thus it follows that the map
$U(\mathfrak{n}^{-}[t])\rightarrow V(\mathbb{n})$ gives a vector space
isomorphism between $U(\mathfrak{n}^{-}[t])/J_{\mathbb{n}}$ and
$V(\mathbb{n})$ . ∎

######  Remark 3.5.2.

Similarly, it can be shown that if $I_{m}$ is the ideal of
$U(\mathfrak{n}^{+}[t])$ generated by the set
$\\{\operatorname{\mathbb{pr}}^{+}((y_{1})^{k}(x_{0})^{\ell}):\ell>m\\}$, then
the map $U(\mathfrak{n}^{+}[t])\rightarrow W_{loc}(m)$ given by $g\mapsto
gy_{0}^{m}w_{m}$ gives a vector space isomorphism between
$U(\mathfrak{n}^{+}[t])/I_{m}$ and $W_{loc}(m)$. In this case
$U(\mathfrak{n}^{+}[t])/I_{m}$ is spanned by the set

$\\{(x_{0})^{i_{1}}(x_{1})^{i_{2}}\cdots(x_{m-1})^{i_{m}}:(i_{1},i_{2},\cdots,i_{m})\in
J(1^{n})\cup(0,\cdots,0)\\}.$

###  3.6. The module $W(m,\mathbb{n})$

Given $n,m\in\mathbb{N}$ and a partition $\mathbb{n}=(n_{1}\geq
n_{2}\geq\cdots\geq n_{k})$ of $n$, define $W(m,\mathbb{n})$ to be the
$\mathfrak{sl}_{2}[t]$-module generated by an element $w_{\mathbb{n}}^{m}$
which satisfies the following relations:

$\displaystyle(h\otimes t^{r})w_{\mathbb{n}}^{m}$
$\displaystyle=(n-m)\delta_{r,0}w_{\mathbb{n}}^{m}$ (3.10) $\displaystyle
x(r,s)w_{\mathbb{n}}^{m}$ $\displaystyle=0,\text{ if }r+s\geq 1+rl+m-l\text{
for some }l\in\mathbb{N}$ (3.11) $\displaystyle y(r,s)w_{\mathbb{n}}^{m}$
$\displaystyle=0,\text{ if }r+s\geq 1+rl+\sum_{j\geq l+1}n_{j}\text{ for some
}l\in\mathbb{N}$ (3.12)

The following is the main result of this section :

######  Theorem 3.6.1.

Given $n,m\in\mathbb{N}$ and $\mathbb{n}=(n_{1}\geq n_{2}\geq\cdots\geq
n_{k})$, a partition of $n$, the $\mathfrak{sl}_{2}[t]$-module
$W(m,\mathbb{n})$ is isomorphic to $W_{loc}(m)\otimes V(\mathbb{n})$.

Observe that Theorem 3.6.1 gives a presentation of the
$\mathfrak{sl}_{2}[t]$-module $W_{loc}(m)\otimes V(\mathbb{n})$. The following
lemmas help prove the theorem.

######  Lemma 3.6.2.

There exists a surjective, $\mathfrak{sl}_{2}[t]$-module homomorphism from
$W(m,\mathbb{n})$ to $W_{loc}(m)\otimes V(\mathbb{n})$.

###### Proof.

By Lemma 3.4.2, we know that if $w_{m}$ and $v_{\mathbb{n}}$ are the highest
weight generators of $W_{loc}(m)$ and $V(\mathbb{n})$ respectively, then
$y_{0}^{m}w_{m}\otimes v_{\mathbb{n}}$ generates $W_{loc}(m)\otimes
V(\mathbb{n}).$

Define a map $\phi:W(m,\mathbb{n})\rightarrow W_{loc}(m)\otimes
V(\mathbb{n})$, such that, $\phi(w_{\mathbb{n}}^{m})=y_{0}^{m}w_{m}\otimes
v_{\mathbb{n}}$. Since $y_{0}^{m}w_{m}\otimes v_{\mathbb{n}}$ generates
$W_{loc}(m)\otimes V(\mathbb{n})$ and for $r\geq 0$, using Lemma 3.3.1 we
have,

$\displaystyle(h\otimes 1)(y_{0}^{m}w_{m}\otimes v_{\mathbb{n}})$
$\displaystyle=h_{0}y_{0}^{m}w_{m}\otimes v_{\mathbb{n}}+y_{0}^{m}w_{m}\otimes
h_{0}v_{\mathbb{n}}=(n-m)(y_{0}^{m}w_{m}\otimes v_{\mathbb{n}})$
$\displaystyle x(r,s)(y_{0}^{m}w_{m}\otimes v_{\mathbb{n}})$
$\displaystyle=x(r,s)y_{0}^{m}w_{m}\otimes v_{\mathbb{n}}=0\quad\text{ if
}r+s\geq 1+rl+m-l$ $\displaystyle y(r,s)(y_{0}^{m}w_{m}\otimes
v_{\mathbb{n}})$ $\displaystyle=y_{0}^{m}w_{m}\otimes
y(r,s)v_{\mathbb{n}}=0\quad\text{ if }r+s\geq 1+rl+\sum_{j\geq
l+1}n_{j}\quad(\text{By \eqref{CV relations}})$

and for $r\geq 1$,

$\displaystyle(h\otimes t^{r})(y_{0}^{m}w_{m}\otimes v_{\mathbb{n}})$
$\displaystyle=h_{r}y_{0}^{m}w_{m}\otimes v_{\mathbb{n}}=0.\quad$

This shows that $\phi$ is a well-defined surjective
$\mathfrak{sl}_{2}[t]$-module homomorphism. ∎

######  Lemma 3.6.3.

$dim(W(m,\mathbb{n}))\leq 2^{m}\prod\limits_{j=1}^{k}(n_{j}+1)$.

###### Proof.

By definition, $W(m,\mathbb{n})=U(\mathfrak{sl}_{2}[t])w_{\mathbb{n}}^{m}$,
where $w_{\mathbb{n}}^{m}$ is a non-zero vector satisfying the relations
(3.10)-(3.12). Using Poincaré-Birkhoff-Witt theorem and (3.10), we can write

$U(\mathfrak{sl}_{2}[t])w_{\mathbb{n}}^{m}=U(\eta^{+}[t])U(\eta^{-}[t])U(\mathfrak{h}[t])w_{\mathbb{n}}^{m}=U(\eta^{+}[t])U(\eta^{-}[t])w_{\mathbb{n}}^{m}.$

For the partition $\mathbb{n}$, let $J_{\mathbb{n}}$ be the ideal of
$U(\mathfrak{n}^{-}[t])$ as described in Lemma 3.5.1. As a consequence of the
defining relations (3.10) and (3.12), we see that the same arguments as those
used in Lemma 3.5.1 show that $Yw^{m}_{\mathbb{n}}=0$ for all $Y\in
J_{\mathbb{n}}$. Hence it follows that if

${\mathcal{B}}_{\mathbb{n}}=\\{y_{0}^{i_{1}}y_{1}^{i_{2}}\dots
y_{k-1}^{i_{k}}|(i_{1},i_{2},\dots,i_{k})\in
J(\mathbb{n})\cup(0,0,\cdots,0)\\},$

then

$W(m,\mathbb{n})\subseteq\sum_{Y\in{\mathcal{B}}_{\mathbb{n}}}U(\mathfrak{n}^{+}[t])Yw^{m}_{\mathbb{n}}.$

For $Y\in U(\mathfrak{n}^{-}[t]$, let $\ell{n}(Y)$ be the PBW length of $Y$.
For $Y\in U(\mathfrak{n}^{-}[t])$, when $\ell{n}(Y)=0$, we have,
$U(\mathfrak{n}^{+}[t])Yw^{m}_{\mathbb{n}}=U(\mathfrak{n}^{+}[t])w^{m}_{\mathbb{n}}.$
In this case, as a consequence of the defining relations (3.10) and (3.11),
the similar arguments to those used in Lemma 3.5.1 show that if $I_{m}$ is the
ideal of $U(\mathfrak{n}^{+}[t])$ defined in Remark 3.5.2, then, as vector
spaces $U(\mathfrak{n}^{+}[t])/I_{m}$ is isomorphic to
$U(\mathfrak{n}^{+}[t])w^{m}_{n}$. Hence $\dim
U(\mathfrak{n}^{+}[t])w^{m}_{\mathbb{n}}\leq 2^{m}.$

Given $Y\in{\mathcal{B}}_{\mathbb{n}}$, by applying induction on $\ell{n}(Y)$,
we now prove that

$U(\mathfrak{n}^{+}[t])Yw^{m}_{\mathbb{n}}\subseteq
YU(\mathfrak{n}^{+}[t])w^{m}_{\mathbb{n}}+\sum\limits_{\\{Y^{\prime}\in{\mathcal{B}}_{\mathbb{n}}:\ell{n}(Y^{\prime})<\ell{n}Y\\}}Y^{\prime}U(\mathfrak{n}^{+}[t])w^{m}_{\mathbb{n}}.$
(3.13)

When $\ell{n}(Y)=1$, $Y=y_{j}$ with $0\leq j\leq m-1.$ In this case, given
$x^{a_{0}}_{0}x^{a_{1}}_{1}\dots x^{a_{p}}_{p}\in U(\mathfrak{n}^{+}[t])$, we
have,

$x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}}_{p}y_{j}w_{\mathbb{n}}^{m}=x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}y_{j}x_{p}w_{\mathbb{n}}^{m}+x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}h_{j+p}w_{\mathbb{n}}^{m}.$

Since $x_{r}w_{\mathbb{n}}^{m}=0$ for $r\geq m$, we see that
$x^{a_{0}}_{0}x^{a_{1}}_{1}\dots x^{a_{p}}_{p}y_{j}x_{p}w_{\mathbb{n}}^{m}=0$
whenever $p\geq m$. Therefore, assuming $p\leq m-1$, we get,

$\displaystyle x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}}_{p}y_{j}w_{\mathbb{n}}^{m}$
$\displaystyle=x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}y_{j}x_{p}w_{\mathbb{n}}^{m}+x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}h_{j+p}w_{\mathbb{n}}^{m},$
$\displaystyle=x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-2}_{p}y_{j}x^{2}_{p}w_{\mathbb{n}}^{m}+x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-2}_{p}h_{j+p}x_{p}w_{\mathbb{n}}^{m}+x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}h_{j+p}w_{\mathbb{n}}^{m}$ $\displaystyle\vdots$
$\displaystyle=y_{j}x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}}_{p}w_{\mathbb{n}}^{m}+u,\quad\text{where }u\in
U(\mathfrak{n}^{+}[t])w_{\mathbb{n}}^{m}.$

Thus (3.13) holds $\ell{n}(Y)=1$. Now assume that (3.13) holds for all
$Y\in{\mathcal{B}}_{\mathbb{n}}$ with $\ell{n}(Y)<r$ and consider
$x^{a_{0}}_{0}x^{a_{1}}_{1}\dots x^{a_{p}}_{p}Yw_{\mathbb{n}}^{m}$, with
$Y=y^{i_{1}}_{0}y^{i_{2}}_{1}\dots y^{i_{k}}_{k-1}$ with $\ell{n}(Y)=r$. Then,

$\begin{array}[]{l}x^{a_{0}}_{0}x^{a_{1}}_{1}\cdots
x^{a_{p}}_{p}y^{i_{1}}_{0}\cdots y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}\\\
=(x^{a_{0}}_{0}x^{a_{1}}_{1}\cdots
x^{a_{p}-1}_{p}y_{0}x_{p}y^{i_{1}-1}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1}+x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}h_{p}y^{i_{1}-1}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1})w_{\mathbb{n}}^{m}\\\ =x^{a_{0}}_{0}x^{a_{1}}_{1}\cdots
x^{a_{p}-1}_{p}y_{0}x_{p}y^{i_{1}-1}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}+(i_{1}-1){x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}}y_{0}^{i_{1}-2}y^{i_{2}}_{1}\dots y_{p}^{i_{p}+1}\cdots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}\\\
+\sum\limits_{j=2}^{k-p-1}(i_{j}){x^{a_{0}}_{0}x^{a_{1}}_{1}\cdots
x^{a_{p}-1}_{p}}y^{i_{1}-1}_{0}y^{i_{2}}_{1}\dots y^{i_{j}-1}\dots
y_{j+p}^{i_{j+p}+1}\cdots y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}.\end{array}$

Using the vector space isomorphism between
$U(\mathfrak{n}^{-}[t])w_{\mathbb{n}}^{m}$ and
$U(\mathfrak{n}^{-}[t])/J_{\mathbb{n}}$, it follows from Lemma 3.5.1 that,

$x^{a_{0}}_{0}x^{a_{1}}_{1}\dots x^{a_{p}}_{p}y^{i_{1}}_{0}\cdots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}=x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-1}_{p}y_{0}x_{p}y^{i_{1}-1}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}+u_{r-1},$

where
$u_{r-1}\in\sum\limits_{\\{Y^{\prime}\in\mathcal{B}_{\mathbb{n}}:\ell{n}(Y^{\prime})=r-1\\}}U(\eta^{+}[t])\cdot
Y^{\prime}w_{\lambda}^{m}$. Thus by induction hypothesis,

$u_{r-1}\in\sum\limits_{Y\in\mathcal{B}_{\mathbb{n}}}Y\cdot
U(\eta^{+}[t])\cdot w_{\mathbb{n}}^{m}.$

Now, repeating the proceedure we get,

$x^{a_{0}}_{0}x^{a_{1}}_{1}\dots x^{a_{p}}_{p}y^{i_{1}}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}=x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}-2}_{p}y_{0}x^{2}_{p}y^{i_{1}-1}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}+u_{r-2}+u_{r-1},$

where
$u_{r-2}\in\sum\limits_{\\{Y\in\mathcal{B}_{\mathbb{n}}\\}}YU(\mathfrak{n}^{+}[t])w_{\mathbb{n}}^{m}.$
Continuing this way, we get

$x^{a_{0}}_{0}x^{a_{1}}_{1}\dots x^{a_{p}}_{p}y^{i_{1}}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}=y_{0}x^{a_{0}}_{0}x^{a_{1}}_{1}\dots
x^{a_{p}}_{p}y^{i_{1}-1}_{0}y^{i_{2}}_{1}\dots
y^{i_{k}}_{k-1}w_{\mathbb{n}}^{m}+u,$

where
$u\in\sum\limits_{\\{Y^{\prime}\in\mathcal{B}_{\mathbb{n}}\\}}Y^{\prime}\cdot
U(\eta^{+}[t])w_{\mathbb{n}}^{m})$. This shows that $W(m,\mathbb{n})$ is
spanned by elements of the form
$\\{YXw_{\mathbb{n}}^{m}:X\in\mathcal{B}_{{\bf{1}}^{m}}\,Y\in\mathcal{B}_{\mathbb{n}}\\}$.
Hence it follows that

$\qquad\qquad\qquad\dim
W(m,\mathbb{n})\leq|\mathcal{B}_{{\bf{1}}^{m}}||\mathcal{B}_{\mathbb{n}}|=2^{m}\prod\limits_{j=1}^{k}(n_{j}+1).\qquad\qquad\qquad\qquad\qquad\qed$

We conclude this section with the proof of Theorem 3.6.1.

###### Proof.

From Lemma 3.6.2 and Lemma 3.4.1, it follows that $\dim W(m,\mathbb{n})\geq
2^{m}\prod\limits_{i=1}^{k}(n_{i}+1).$
On the other hand by Lemma 3.6.3 , $\dim W(m,\mathbb{n})\leq
2^{m}\prod\limits_{i=1}^{k}(n_{i}+1).$
Hence it follows that $\dim W(m,\mathbb{n})\geq
2^{m}\prod\limits_{i=1}^{k}(n_{i}+1)=\dim W(m)\otimes V(\mathbb{n})$.
This completes the proof of the theorem. ∎

##  4\. $CV$-filtration and Graded character of $W_{loc}(m\omega)\otimes V(n)$

Given a finite-dimensional graded $\mathfrak{sl}_{2}[t]$-module $U$, we say
$U$ admits a $CV$-filtration, if there exist an decreasing chain of
$\mathfrak{sl}_{2}[t]$-submodules

$U=U_{0}\supset U_{1}\supset U_{2}\supset U_{3}\supset\dots\supset
U_{r}\supset(0)$

such that for each $i$, $\frac{U_{i}}{U_{i+1}}\cong V(\xi^{i})$, where
$\xi^{i}$ is a partition. On the other hand, a filtration of a representation
whose successive quotients are isomorphic to Demazure modules is called an
excellent filltration. It is known that if $V=V_{1}\supseteq
V_{2}\supseteq\cdots\supseteq V_{n}\supset 0$ is a chain of graded
$\mathfrak{g}[t]$-modules whose successive quotients have excellent filtration
then $V$ has an excellent filtration. In this context the following result was
proved in [7, Theorem 3.3].

######  Proposition 4.0.1.

Let $n\in\mathbb{N}$. Given a partition $\mathbb{n}=(n_{1}\geq
n_{2}\geq\cdots\geq n_{k})$ of $n$, the $\mathfrak{sl}_{2}[t]$-module
$V(\mathbb{n})$ has a Demazure flag of level $l$ if and only if $n_{1}\leq l.$

We use this result to show that the $\mathfrak{sl}_{2}[t]$-modules of the form
$W_{loc}(m)\otimes V(n)$ admit a Demazure flag of level $k$ for $k\geq n+1$.
We also deduce the fermionic version of the character formula for
$W_{loc}(m)\otimes V(n)$.

### 4.1.

Given $k,r\in\mathbb{N}$, with $k\geq r$, let $\lceil(k,r):=(k-r+1,1^{r-1})$
denote the hook type partition of $k$ into $r$ parts. In this section we prove
the following:

######  Theorem 4.1.1.

Let $m,n\in\mathbb{N}$. The $\mathfrak{sl}_{2}[t]$-module,
$W_{loc}(m\omega)\otimes V(n)$ admits a $CV$-filtration whose successive
quotients are isomorphic to

$\displaystyle V(\lceil(m-n+2i,m-n+i))$ $\displaystyle,\quad 0\leq i\leq
n,\quad\text{when }m>n;$ $\displaystyle V(\lceil(n-m+2i,i),V(n-m)$
$\displaystyle,\quad 1\leq i\leq m,\quad\text{when }m\leq n.$

We need the following lemmas to prove the theorem.

######  Lemma 4.1.2.

Let $r,i\in\mathbb{N}$.
If $r\geq i$, then the $i\times i$ matrix $A(r,i)$ given by

$A(r,i)=\begin{bmatrix}1&1&\cdots&1\\\
{{r}\choose{1}}&{{r+1}\choose{1}}&\cdots&{{r+i-1}\choose{1}}\\\
\vdots&\vdots&\cdots&\vdots\\\
{{r}\choose{i-1}}&{{r+1}\choose{i-1}}&\cdots&{{r+i-1}\choose{i-1}}\end{bmatrix}$

is invertible.
If $r<i$, then the $i\times i$ matrix $B(r,i)$ given by

$B(r,i)=\begin{bmatrix}1&1&1&\cdots&1&1&\cdots&1\\\
{{r}\choose{1}}&{{r+1}\choose{1}}&{{r+2}\choose{1}}&\cdots&{{i-2}\choose{1}}&{{i-1}\choose{1}}&\cdots&{{r+i-1}\choose{1}}\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\\
{{r}\choose{r}}&{{r+1}\choose{r}}&{{r+2}\choose{r}}&\cdots&{{i-2}\choose{r}}&{{i-1}\choose{r}}&\cdots&{{r+i-1}\choose{r}}\\\
0&{{r+1}\choose{r+1}}&{{r+2}\choose{r+1}}&\cdots&{{i-2}\choose{r+1}}&{{i-1}\choose{r+1}}&\cdots&{{r+i-1}\choose{r+1}}\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots&\cdots&\vdots\\\
0&0&0&\cdots&0&{{i-1}\choose{i-1}}&\cdots&{{r+i-1}\choose{i-1}}\end{bmatrix}$

is invertible.

###### Proof.

We prove the lemma by applying induction on $i$.
When $i=1$, $A(r,1)=[1]$ which is invertible.
When $i=2$, $A(r,2)=\begin{bmatrix}1&1\\\ r&r+1\end{bmatrix}$ and $\det
A(r,2)=1$, impying that $A(r,i)$ is invertible.
Assume that the result holds for all $k<i$ and consider the matrix $A(r,i)$ .
On applying column transformation to $A(r,i)$ we can get the matrix

$\tilde{A}(r)=\begin{bmatrix}1&0&\cdots&0\\\ {{r}\choose{1}}&1&\cdots&1\\\
{{r}\choose{2}}&{{r}\choose{1}}&\cdots&{{r+i-2}\choose{1}}\\\
\vdots&\vdots&\cdots&\vdots\\\
{{r}\choose{i-1}}&{{r}\choose{i-2}}&\cdots&{{r+i-2}\choose{i-2}}\end{bmatrix}$

Clearly $\det A(r,i)=\det\tilde{A}(r)=\det A(r,i-1).$ By induction hypothesis
the matrix $A(r,i-1)$ is invertible. Hence, it follows that $A(r,i)$ is
invertible.
Similar arguments show that the matrix $B(r,i)$ is also invertible when $r<i$.
∎

######  Lemma 4.1.3.

Let $m,n\in\mathbb{N}$.
(i) For $m>n$, $\sum\limits_{i=0}^{n}\dim V(\lceil(m-n+2i,m-n+i)=2^{m}(n+1)$.
(ii) For $m\leq n$, $\sum\limits_{i=1}^{m}\dim V(\lceil(n-m+2i,i)+\dim
V(n-m)=2^{m}(n+1)$ .

###### Proof.

By Lemma 3.4.1,

$\dim V(\lceil(m+n-2i,m-i))=2^{m-i-1}(n-i+1+1)=2^{m-i-1}(n-i+2).$

(i) When $m>n$, we have,

$\displaystyle\sum\limits_{i=0}^{n}\dim V(\lceil(m-n+2i,m-n+i)$
$\displaystyle=\sum\limits_{i=0}^{n}(i+2)2^{m-n+i-1}=2^{m-n-1}\sum\limits_{i=0}^{n}(i+2)2^{i}$

Putting $s_{n}=\sum\limits_{i=0}^{n}(i+2)2^{i},$ we see that
$s_{n}=\sum\limits_{i=1}^{n}i2^{i}+\sum\limits_{i=1}^{n+1}2^{i}.$
As
$\sum\limits_{i=1}^{n}i2^{i}+(n+1)2^{n+1}-2(\sum\limits_{i=1}^{n}i2^{i})=2(2^{n+1}-1)$,

$s_{n}=(n+1)2^{n+1}-2(2^{n+1}-1)+\sum\limits_{i=1}^{n+1}2^{i}=(n+1)2^{n+1}.$

Hence, $\sum\limits_{i=0}^{n}\dim
V(\lceil(m+n-2i,m-i)=2^{m-n-1}s_{n}=2^{m}(n+1).$

(ii) When $m\leq n$, we have,

$\displaystyle\sum\limits_{i=1}^{m}\dim V(\lceil(n-m+2i,i))+\dim
V(n-m)=\sum_{i=1}^{m}(n-m+2+i)2^{i-1}+(n-m+1)$
$\displaystyle=\sum_{i=0}^{m-1}2^{i}(n-m+3+i)+(n-m+1)=(n-m+1)\sum_{i=0}^{m-1}2^{i}+\sum_{i=0}^{m-1}(i+2)2^{i}+(n-m+1)$

Using part(i), we thus get,

$\displaystyle\sum\limits_{i=0}^{m-1}\dim V(\lceil(n+m-2i,m-i))+\dim V(n-m)$
$\displaystyle=(n-m+1)(2^{m}-1)+m2^{m}+(n-m+1)=2^{m}(n+1).$

This completes the proof of the lemma. ∎

We now prove Theorem 4.1.1.

###### Proof.

Let $w_{m}$ and $v_{n}$ be the highest weight generators of the modules
$W_{loc}(m)$ and $V(n)$ respectively. By Lemma 3.4.2, $W_{loc}(m)\otimes
V(n)=U(\mathfrak{sl}_{2}[t]).w_{m}\otimes y^{(n)}v_{n}$.

When $m>n\geq 1$, set

$U_{i}=W_{n-i}=U(\mathfrak{sl}_{2}[t])(w_{m}\otimes y_{0}^{(n-i)}v_{n}),\qquad
0\leq i\leq n.$

As $x_{r}.w_{m}=0$ for all $r\in\mathbb{Z}_{+}$, using representation theory
of $\mathfrak{sl}_{2}(\mathbb{C})$ [14, Section 7], we have

$x_{0}.(w_{m}\otimes y_{0}^{(i)}v_{n})=w_{m}\otimes
x_{0}y_{0}^{(i)}v_{n}=(n-i+1)w_{m}\otimes y_{0}^{(i-1)}v_{n}.$

generates $W_{loc}(m\omega)\otimes V(n)$. Hence,

$W_{loc}(m\omega)\otimes V(n)=U_{0}\supset U_{1}\supset\cdots U_{n}\supset 0,$

is an decreasing chain of submodules of $W_{loc}(m\omega)\otimes V(n)$. To
prove the result we show:

$\frac{U_{i}}{U_{i+1}}\cong
V(\lceil(m-n+2i,m-n+i)=V(i+1,1^{m-n+i-1}),\qquad\text{ for }\,0\leq i\leq n,$

or equivalently, $\dfrac{W_{i}}{W_{i-1}}\cong
V(\lceil(m+n-2i,m-i)=V(n+1-i,1^{m-i-1})$ for $0\leq i\leq n.$

By definition (Section 2.5),

$g_{r}.v=0,\qquad\text{ for all }\,v\in
V(n),\,g\in\mathfrak{g},\,r\in\mathbb{N}.$

Hence for $r,s\in\mathbb{Z}_{+}$,

$\begin{array}[]{ll}(-1)^{s}y(r,s)(w_{m}\otimes
y_{0}^{(i)}v_{n})&=x_{1}^{(s)}y_{0}^{(r+s)}(w_{m}\otimes y_{0}^{(i)}v_{n})\\\
&=\frac{x_{1}^{(s)}}{(r+s)!i!}\left(\sum\limits_{l=0}^{r+s}{r+s\choose
l}y_{0}^{r+s-l}w_{m}\otimes y_{0}^{l+i}v_{n}\right)\\\
&=\,\,x_{1}^{(s)}\quad\left(\sum\limits_{l=0}^{r+s}{l+i\choose
l}x_{1}^{(s)}y_{0}^{(r+s-l)}w_{m}\otimes y_{0}^{(l+i)}v_{n}\right)\\\
&=\sum\limits_{l=0}^{r+s}\,\,{l+i\choose
l}\,\,x_{1}^{(s)}y_{0}^{(r+s-l)}w_{m}\otimes y_{0}^{(l+i)}v_{n}\\\
&=\sum\limits_{l=0}^{r-1}\,\,{l+i\choose l}\,\,y(r-l,s)w_{m}\otimes
y_{0}^{(l+i)}v_{n}\end{array}$ (4.1)

Further, as $y(r,s)w_{m}=0$ whenever $r+s>m$,

$y(r,s)(w_{m}\otimes y_{0}^{(i)}v_{n})=0\qquad\forall\,s\geq m,\,r\geq
1,\,i\geq 0.$

Therefore,

$\begin{array}[]{c}x_{r}.(w_{m}\otimes v_{n})=0,\qquad h_{r}.(w_{m}\otimes
v_{n})=\delta_{r,0}(n+m)(w_{m}\otimes
v_{n}),\qquad\forall\,r\in\mathbb{Z}_{+},\\\ \\\ \qquad y(r,m)(w_{m}\otimes
v_{n})=0\quad\forall\,1\leq r\leq n+1,\end{array}$

implying that $W_{0}=U_{n}$ is a quotient of $V(\lceil(m+n,m)).$

Now, using standard $\mathfrak{sl}_{2}$-representation theory ([14, Section
7]) we see that for $i>0$,

$\begin{array}[]{ll}x_{r}.(w_{m}\otimes
y_{0}^{(i)}v_{n})=\delta_{r,0}(n-i+1)\,w_{m}\otimes y_{0}^{(i-1)}v_{n}\in
U_{n-i+1}\\\ h_{r}.(w_{m}\otimes
y_{0}^{(i)}v_{n})=\delta_{r,0}(m+n-2i)\,w_{m}\otimes
y_{0}^{(i)}v_{n},\end{array}$

and putting $s=m-i$ in (4.1), we get,

$y(r,m-i)(w_{m}\otimes
y_{0}^{(i)}v_{n})=\sum\limits_{l=0}^{r-1}\,\,{l+i\choose
l}\,\,y(r-l,m-i)w_{m}\otimes y_{0}^{(l+i)}v_{n}.$ (4.2)

Once again using the relation $y(r,s)w_{m}=0$ whenever $r+s>m$, we see that
for $r\geq i$, (4.2) reduces to:

$\begin{array}[]{ll}(x\otimes t)^{(m-i)}(y\otimes 1)^{(r+m-i)}(w_{m}\otimes
y_{0}^{(i)}v_{n})&=\sum\limits_{l=0}^{r-1}\,\,{l+i\choose
l}\,\,y(r-l,m-i)w_{m}\otimes y_{0}^{(l+i)}v_{n}\\\
&=\sum\limits_{l=r-i}^{r-1}\,\,{l+i\choose l}\,\,y(r-l,m-i)w_{m}\otimes
y_{0}^{(l+i)}v_{n}\\\
&=\sum\limits_{l=0}^{i-1}{{r+l}\choose{i}}y(i-l,m-i)w_{m}\otimes
y_{0}^{(r+l)}v_{n});\end{array}$ (4.3)

and for $r<i$, putting $r=i-p$, (4.2) reduces to:

$\begin{array}[]{ll}y(r,m-i)(w_{m}\otimes
y_{0}^{(i)}v_{n})&=\sum\limits_{l=0}^{r-1}\,\,{l+i\choose
l}\,\,y(i-p-l,m-i)w_{m}\otimes y_{0}^{(l+i)}v_{n}\\\
&=\sum\limits_{k=p}^{r+p-1}\,\,{k-p+i\choose i}\,\,y(i-k,m-i)w_{m}\otimes
y_{0}^{(k-p+i)}v_{n}\\\ &=\sum\limits_{k=p}^{i-1}\,\,{k+r\choose
i}\,\,y(i-k,m-i)w_{m}\otimes y_{0}^{(k+r)}v_{n}.\end{array}$ (4.4)

Having established (4.3) and (4.4), to prove that $\dfrac{W_{i}}{W_{i-1}}$ is
a quotient of $V(\lceil(m+n-2i,m-i))$ it suffices to show that

$\\{y(i-l,m-i)w_{m}\otimes y_{0}^{(r+l)}v_{n}:0\leq l\leq i-1\\}\subset
W_{i-1},\quad\text{ for }1\leq r\leq n-i+1$ (4.5)

Since $w_{m}\otimes y_{0}^{(j)}v_{n}\in W_{i-1}$ for $0\leq j\leq i-1,$ $0\leq
i\leq n$,

$y(r+i-j,m-i)(w_{m}\otimes y_{0}^{(j)}v_{n})\in W_{i-1},\qquad\forall\,\,0\leq
j\leq i-1.$

Further, for $r\geq i>j\geq 0$ we have,

$\begin{array}[]{ll}(-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r-j)}(w_{m}\otimes
y_{0}^{(j)}v_{n})&=y(i+r-j,m-i)(w_{m}\otimes y_{0}^{(j)}v_{n}),\\\
&=\sum\limits_{l=r-j}^{m+r-j-1}{{j+l}\choose{j}}y(r-j+i-l,m-i)w_{m}\otimes
y_{0}^{(j+l)}v_{n}\\\
&=\sum\limits_{s=0}^{i-1}{{r+s}\choose{j}}y(i-s,m-i)w_{m}\otimes
y_{0}^{(r+s)}v_{n}.\end{array}$ (4.6)

Setting, $X:=\begin{bmatrix}y(i,m-i)w_{m}\otimes y_{0}^{(r)}v_{n}\\\
y(i-1,m_{i})w_{m}\otimes y_{0}^{(r+1)}v_{n}\\\ \\\ \vdots\\\
y(1,m-i)w_{m}\otimes y_{0}^{(r+i-1)}v_{n}\\\ \end{bmatrix}$ and
$Y:=\begin{bmatrix}(-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r)}(w_{m}\otimes
y_{0}v_{n})\\\ (-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r-1)}(w_{m}\otimes
y_{0}^{(1)}v_{n})\\\ \\\ \vdots\\\
(-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r-i+1)}(w_{m}\otimes y_{0}^{(i-1)}v_{n})\\\
\end{bmatrix}$

we can write the set of equations (4.6), for $0\leq j\leq i-1$, as
$Y=A(r,i)X$. As the entries of the colummn matrix $Y$ are contained in
$W_{i-1}$ and by Lemma 4.1.2, the coeffient matrix $A(r,i)$ is invertible, it
follows that the entries of $X$ lie in $W_{i-1}$. Hence, (4.5) holds implying
that $y(r,m-i)(w_{m}\otimes y_{0}^{(i)}v_{n})\in U_{i-1}$ for $i\leq r\leq
n-i+1$.

On the other hand, for $r=i-p<i$, $j=r+s\leq i-1$, $1\leq s\leq p-1$, we have

$\begin{array}[]{ll}(-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r-j)}(w_{m}\otimes
y_{0}^{(j)}v_{n})&=y(i+r-j,m-i)(w_{m}\otimes y_{0}^{(j)}v_{n}),\\\
&=\sum\limits_{l=0}^{m+r-j-1}{{r+s+l}\choose{r+s}}y(i-s-l,m-i)w_{m}\otimes
y_{0}^{(r+s+l)}v_{n}\\\
&=\sum\limits_{k=j-r}^{i-1}{{r+k}\choose{j}}y(i-k,m-i)w_{m}\otimes
y_{0}^{(r+k)}v_{n}.\end{array}$ (4.7)

Now setting,

$\bar{Y}:=\begin{bmatrix}(-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r)}(w_{m}\otimes
v_{n})\\\ \vdots\\\ (-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m)}(w_{m}\otimes
y_{0}^{(r)}v_{n})\\\ (-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r-(r+1)}(w_{m}\otimes
y_{0}^{(r+1)}v_{n})\\\ \vdots\\\
(-1)^{m-i}x_{1}^{(m-i)}y_{0}^{(m+r-(i-1)}(w_{m}\otimes y_{0}^{(i-1)}v_{n})\\\
\end{bmatrix}\qquad\text{and}\qquad\bar{X}:=\begin{bmatrix}y(i,m-i)w_{m}\otimes
y_{0}^{(r)}v_{n}\\\ y(i-1,m-i)w_{m}\otimes y_{0}^{(r+1)}v_{n}\\\ \\\ \vdots\\\
y(1,m-i)w_{m}\otimes y_{0}^{(r+i-1)}v_{n}\\\ \end{bmatrix}$

we can write the equations (4.6) for $0\leq j\leq r$ and (4.7) for $r<j\leq
i-1$, as $\bar{Y}=B(r,i)\bar{X}$, where and $B(r,i)$ is the $i\times i$ matrix
defined in Lemma 4.1.2. As in the case above, using the fact that $B(r,i)$ is
invertible (Lemma 4.1.2), we see that the entries of $X$ lie in the space
spanned by the vectors $\\{x_{1}^{(m-i)}y_{0}^{(m+r-j)}(w_{m}\otimes
y_{0}^{(j)}v_{n}):0\leq j\leq i-1\\}\subset W_{i-1}.$

If $\overline{w_{m}\otimes y_{0}^{(i)}v_{n}}$ is the image of $w_{m}\otimes
y_{0}^{(i)}v_{n}$ in $\dfrac{W_{i}}{W_{i-1}}$ and
$\Phi_{i}:V(\lceil(m+n-2i,m-i)\rightarrow\dfrac{W_{i}}{W_{i-1}}$ is a map such
that $\Phi_{i}(v_{\lceil(m+n-2i,m-i)})=\overline{w_{m}\otimes
y_{0}^{(i)}v_{n}},$ then it follows from above that $\Phi_{i}$ is a surjective
$\mathfrak{sl}_{2}[t]$-module homomorphism. Therefore,

$\dim W(m\omega)\otimes
V(n)=\sum\limits_{i=0}^{n}\dim\dfrac{W_{i}}{W_{i-1}}\leq\sum\limits_{i=0}^{n}\dim
V(\lceil(m+n-2i,m-i)).$

Setting $W_{-1}=0$ and using Lemma 3.4.2 and Lemma 4.1.3(i), we thus see that

$\sum\limits_{i=0}^{n}\dim\dfrac{W_{i}}{W_{i-1}}=2^{m}(n+1).$

However, this is possible only if the surjection $\Phi_{i}$ is an isomorphism
for $0\leq i\leq n$. This completes the proof of the theorem in this case.

When $m\leq n$, considering the generator $y_{0}^{m}w_{m}\otimes v_{n}$ of
$W_{loc}(m)\otimes V(n)$ and setting
$U_{i}=U(\mathfrak{sl}_{2}[t])(y_{0}^{(m-i)}w_{m}\otimes v_{n})$ for $0\leq
i\leq m$, we see that there exists a decreasing chain of submodules

$W_{loc}(m\omega)\otimes V(n)=U_{0}\supset U_{1}\supset\cdots\supset
U_{m}\supset 0.$

Using similar arguments as above and Lemma 4.1.3(ii) we see that even in this
case, $\dfrac{U_{i}}{U_{i+1}}$ is isomorphic to $V(\lceil(n-m+2i,i))$ for
$1\leq i\leq m$ and $\dfrac{U_{0}}{U_{1}}$ is isomorphic to $V(n-m)$. This
completes the proof of the theorem. ∎

The following result is an immediate consequence of Theorem 4.1.1

######  Corollary 4.1.4.

Given two positive integers $m,n$, we have,

$\begin{array}[]{l}ch_{gr}\,W_{loc}(m)\otimes V(n)\\\
\,\qquad=\left\\{\begin{array}[]{ll}\sum\limits_{i=0}^{n}ch_{gr}\,V(i+1,1^{m-n+i-1})&\text{for
}m>n,\\\
\sum\limits_{i=1}^{m}ch_{gr}\,V(n-m+i+1,1^{i-1})+ch_{gr}\,V(n-m,0)&\text{for
}m\leq n.\end{array}\right.\end{array}$

######  Remark 4.1.5.

Combinatorial formulas for determining the multiplicity of level $\ell$
Demazure modules in CV-modules associated with hook-type partitions have been
obtained in [2, Theorem 4] and [7, Theorem 4.3]. Using the latter in
combination with Corollary 4.1.4, one can deduce a combinatorial formula for
the graded multiplicities of level $\ell$ Demazure modules in
$W_{loc}(m\omega)\otimes ev_{0}V(n\omega)$ for all positive integers $m,n$.

### 4.2.

In view of Theorem 3.6.1, we shall refer to $W_{loc}(m)\otimes V(n)$ as
$W(m,n)$ in the rest of this section. As $W(m,n)$ is a graded
$\mathfrak{sl}_{2}[t]$-module,

$ch_{gr}W(m,n)=\sum\limits_{k\in\mathbb{Z}_{+}}ch_{\mathfrak{sl}_{2}}\
W(m,n)[k]\ q^{k},$

where $W(m,n)[k]$ is a finite-dimensional $\mathfrak{sl}_{2}$-module.
Define $[W(m,n):V(r)]_{q}$ as the polynomial in indeterminate $q$ given by

$[W(m,n):V(r)]_{q}=\sum\limits_{k\geq 0}\,[W(m,n)[k]:V(r,k)]q^{k},$

where $[W(m,n)[k]:V(r,k)]$, denotes the multiplicity of the irreducible
$\mathfrak{sl}_{2}[t]$-module $V(r,k)$ in $W(m,n)[k]$ for
$r\in\mathbb{Z}_{+}$. Using Theorem 4.1.1, we now obtain the polynomials
$[W(m,n):V(r)]_{q}$.

######  Theorem 4.2.1.

Let $m,n\in\mathbb{N}$, and $i\in\mathbb{Z}_{+}$ be such that $m+n-2i\geq 0$.
If $m\leq n$,

$[W(m,n):V(m+n-2i)]_{q}=\begin{bmatrix}m\\\ i\end{bmatrix}_{q},\quad 0\leq
i\leq m,$

If $m>n$,

$\begin{array}[]{l}[W(m,n):V(m+n-2i)]_{q}=\left\\{\begin{array}[]{ll}\begin{bmatrix}m\\\
i\end{bmatrix}_{q},\quad 0\leq i\leq n,\\\ \begin{bmatrix}m\\\
i\end{bmatrix}_{q}-\begin{bmatrix}m\\\ i-n-1\end{bmatrix}_{q},\quad n+1\leq
i\leq\lfloor{\frac{n+m}{2}}\rfloor\end{array}\right.\end{array}$

The proof of the theorem occupies the rest of this section.

At the onset we record three $q-$identities that we shall use.

$\displaystyle q^{r}\begin{bmatrix}n-1\\\
r\end{bmatrix}_{q}+\begin{bmatrix}n-1\\\
r-1\end{bmatrix}_{q}=\begin{bmatrix}n\\\ r\end{bmatrix}_{q},$
$\displaystyle\qquad\text{ for }n\in\mathbb{N},\,r\in\mathbb{Z}_{+}.$ (4.8)
$\displaystyle\qquad\begin{bmatrix}n\\\
r\end{bmatrix}_{q}+q^{n-r+1}\begin{bmatrix}n\\\
r-1\end{bmatrix}_{q}=\begin{bmatrix}n+1\\\ r\end{bmatrix}_{q},$
$\displaystyle\qquad\text{ for }n\in\mathbb{N},\,r\in\mathbb{Z}_{+}.$ (4.9)
$\displaystyle\sum\limits_{i=0}^{n}q^{i}\begin{bmatrix}m+i\\\
i\end{bmatrix}_{q}=\begin{bmatrix}n+m+1\\\ n\end{bmatrix}_{q}$
$\displaystyle\qquad\text{ for }m\in\mathbb{N},\,n\in\mathbb{Z}_{+}.$ (4.10)

We need the following results to prove Theorem 4.2.1.

######  Lemma 4.2.2.

Given $k\in\mathbb{N}\text{ and }r\in Z_{+}$, we have,

$\begin{array}[]{lll}ch_{gr}V(k+r,1^{k})&=&ch_{gr}V(k+r+1,1^{k-1})+q^{k}\Big{(}ch_{\mathfrak{sl}_{2}}V(r,0)+\sum\limits_{i=2}^{k}ch_{gr}V(i+r,1^{i-2})\Big{)}\\\
&\\\ ch_{gr}V(k,1^{k+r})&=&ch_{gr}\
V(k+1,1^{k+r-1})+q^{k+r}\Big{(}ch_{gr}W_{loc}(r)+\sum\limits_{i=2}^{k}ch_{gr}\
V(i,1^{i+r-2})\Big{)}\end{array}$

###### Proof.

This is a direct consequence of [2, Theorem 18].∎

######  Lemma 4.2.3.

The graded characters of $V(\xi)$ corresponding to hook type partitions $\xi$
are as follows:

$\begin{array}[]{lll}ch_{gr}V(k+r,1^{k})&=&\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2(k-p))\\\ &&\\\
ch_{gr}V(k,1^{k+1})&=&\sum\limits_{p=0}^{k}q^{k-p}\begin{bmatrix}k+1\\\
p+1\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2p)\\\ &&\\\
ch_{gr}V(k,1^{k+r})&=&\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k+r\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2(k-p))\\\
&+&\sum\limits_{p=k+1}^{\lfloor\frac{2k+r}{2}\rfloor}q^{p}(\begin{bmatrix}k+r\\\
p\end{bmatrix}_{q}-q^{2k+r+1-2p}\begin{bmatrix}k+r\\\
p-k-1\end{bmatrix})ch_{\mathfrak{sl}_{2}}V(r-2(p-k))\qquad\text{ for }r\geq
2.\end{array}$

###### Proof.

Applying induction on $k$, we prove the character formula of $V(k+r,1^{k})$,
$r\geq 0$.
When $k=1$, by Lemma 3.4.1(b) we have the following short exact sequence :

$0\rightarrow\tau_{1}(V(n-1))\rightarrow
V(n,1)\xrightarrow{\phi}V(n+1,0)\rightarrow 0$

Therefore,

$ch_{gr}V(n,1)=ch_{\mathfrak{sl}_{2}}V(n+1)+q\
ch_{\mathfrak{sl}_{2}}V(n-1)=\sum\limits_{p=0}^{1}q^{p}\begin{bmatrix}1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}\ V(n+1-2p).$

Assume that the given character formula holds for all $k<l$ and consider the
module $V(l+r,l)$. Using Lemma 4.2.2 and applying the inductive hypothesis, we
get

$\displaystyle ch_{gr}\
V(l+r,1^{l})=ch_{gr}V(l+r+1,1^{l-1})+q^{l}ch_{\mathfrak{sl}_{2}}V(r)+q^{l}\big{(}\sum\limits_{i=2}^{l}ch_{gr}V(i-2+r+2,1^{i-2})\big{)}$
$\displaystyle=\sum\limits_{p=0}^{l-1}q^{p}\begin{bmatrix}l-1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2+2(l-p))+q^{l}\Big{(}ch_{\mathfrak{sl}_{2}}V(r)+\sum\limits_{i=0}^{l-2}\big{(}\sum\limits_{s=0}^{i}q^{s}\begin{bmatrix}i\\\
s\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}\ V(r+2+2i-2s)\big{)}\Big{)}$
$\displaystyle=\sum\limits_{p=0}^{l-1}q^{p}\begin{bmatrix}l-1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2(l-p+1))+q^{l}\Big{(}ch_{\mathfrak{sl}_{2}}V(r)+\sum\limits_{j=0}^{l-2}\big{(}\sum\limits_{s=0}^{l-2-j}q^{s}\begin{bmatrix}j+s\\\
s\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}\ V(r+2+2j)\big{)}\Big{)}$
$\displaystyle=\sum\limits_{j=1}^{l}q^{l-j}\begin{bmatrix}l-1\\\
l-j\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2(j+1))+\sum\limits_{j=0}^{l-2}q^{l}\begin{bmatrix}l-1\\\
j+1\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}\
V(r+2(j+1))+q^{l}ch_{\mathfrak{sl}_{2}}V(r),\quad\text{ by }\eqref{q-combn
identity.2}$ $\displaystyle=\sum_{p=0}^{l}q^{p}\begin{bmatrix}l\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2(l-p))\qquad\quad\text{ by
}\eqref{q-combn identity}.$

This establishes the character formula for the module $V(k+r,1^{k})$.

When $r=1$, using Lemma 4.2.2 and the character formula for $V(k+1,1^{k})$, we
have

$\displaystyle
ch_{gr}(V(k,1^{k+1}))=ch_{gr}V(k+1,1^{k})+q^{k+1}\big{(}\sum\limits_{i=1}^{k}ch_{gr}V(i,1^{i-1})\big{)}$
$\displaystyle=\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2(k-p))+q^{k+1}\Big{(}\sum\limits_{j=0}^{k-1}\big{(}\sum\limits_{s=0}^{j}q^{s}\begin{bmatrix}j\\\
s\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2(j-s))\big{)}\Big{)}$
$\displaystyle(\text{ Rearranging the terms and using \eqref{q-combn
identity.2} we get })$
$\displaystyle=\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2(k-p))+q^{k+1}\Big{(}\sum\limits_{i=0}^{k-1}\begin{bmatrix}k\\\
i+1\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2i)\Big{)}$
$\displaystyle=\sum\limits_{j=0}^{k}q^{k-j}\Big{(}\begin{bmatrix}k\\\
j\end{bmatrix}_{q}+q^{j+1}\begin{bmatrix}k\\\
j+1\end{bmatrix}_{q}\Big{)}ch_{\mathfrak{sl}_{2}}V(1+2j)=\sum\limits_{j=0}^{k}q^{k-j}\begin{bmatrix}k+1\\\
j+1\end{bmatrix}_{q}V(1+2j),\qquad(\text{using \eqref{q-combn identity.3}}).$

To to establish the character formula of $V(k,1^{k+r})$ for $r\geq 2$, we
apply induction on $r$.
When $r=2$, using Lemma 4.2.2 and the character formula for $V(k,1^{k})$ we
have,

$\displaystyle
ch_{gr}(V(k,1^{k+2}))=ch_{gr}V(k+1,1^{k+1})+q^{k+2}\big{(}\sum\limits_{i=2}^{k}ch_{gr}V(i,1^{i-1})\big{)}$
$\displaystyle=\sum\limits_{p=0}^{k+1}\begin{bmatrix}k+1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2(k+1-p))+q^{k+2}\Big{(}\sum\limits_{i=1}^{k}\sum\limits_{s=0}^{i}\begin{bmatrix}i\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2(i-p))\Big{)}$

Rearranging the terms in the summation and using (4.10) we have,

$\displaystyle
ch_{gr}(V(k,1^{k+2}))=\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k+1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2+2(k-p))$
$\displaystyle+\sum\limits_{j=1}^{k}q^{k+2}\begin{bmatrix}k+1\\\
j+1\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2j)+(q^{k+1}+q^{k+3})\begin{bmatrix}k\\\
1\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(0)$
$\displaystyle=V(2k+2)+(q^{k+1}+q^{k+3})\begin{bmatrix}k\\\
1\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(0)+\sum\limits_{j=1}^{k}q^{k-j+1}\big{(}\begin{bmatrix}k+1\\\
j\end{bmatrix}_{q}+q^{j+1}\begin{bmatrix}k+1\\\
j+1\end{bmatrix}_{q}\big{)}ch_{\mathfrak{sl}_{2}}V(2j)$ Using (4.8) and
substituting $k-j+1=p$ we get,
$\displaystyle=\sum\limits_{j=1}^{k}q^{p}\begin{bmatrix}k+2\\\
k-p+2\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2+2(k-p))+q^{k+1}\big{(}\begin{bmatrix}k+2\\\
1\end{bmatrix}_{q}-q\big{)}ch_{\mathfrak{sl}_{2}}V(0).$

This proves that the formula for $V(k,1^{k+r})$ holds when $r=2$. Similarly,
we can prove the formula for $r=3$ separately. Now, assume that the character
formula for $V(k,1^{k+r})$ holds for all $4\leq r<l$. Then

$\displaystyle
ch_{gr}V(k,1^{k+l})=ch_{gr}V(k+1,1^{k+l-1})+q^{k+l}\Big{(}\sum\limits_{i=1}^{k}ch_{gr}V(i,1^{i+l-2})\Big{)}$
$\displaystyle=\sum\limits_{p=0}^{k+1}q^{p}\begin{bmatrix}k+l-1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2k+l-2p)+q^{k+l}\Big{(}\sum\limits_{i=1}^{k}\big{(}\sum\limits_{p=0}^{i}q^{p}\begin{bmatrix}i+l-2\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(l+2(i-p-1))\Big{)}$
$\displaystyle+\sum\limits_{p=k+2}^{\lfloor\frac{2k+l}{2}\rfloor}q^{p}(\begin{bmatrix}k+l-1\\\
p\end{bmatrix}_{q}-q^{2k+l+1-2p}\begin{bmatrix}k+l-1\\\
p-k\end{bmatrix})ch_{\mathfrak{sl}_{2}}V(2k+l-2p)$
$\displaystyle+q^{k+l}\Big{(}\sum\limits_{p=i+1}^{\lfloor\frac{2i+l-2}{2}\rfloor}q^{p}(\begin{bmatrix}i+l-2\\\
p\end{bmatrix}_{q}-q^{2i+l-1-2p}\begin{bmatrix}i+l-2\\\
p-i-1\end{bmatrix})ch_{\mathfrak{sl}_{2}}V(2i+l-2-2p)\big{)}\Big{)}$
$\displaystyle=\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k+l\\\
p\end{bmatrix}_{q}ch_{gr}V(2k+l-2p,0)$
$\displaystyle+\sum_{p=k+1}^{\lfloor{\frac{2k+l}{2}}\rfloor}q^{p}(\begin{bmatrix}k+l\\\
p\end{bmatrix}_{q}-q^{2k+l+1-2p}\begin{bmatrix}k+l\\\
p-k-1\end{bmatrix}_{q})ch_{gr}V(2k+l-2p,0),$

where the last line follows after rearranging the terms and using (4.8) and
(4.9). ∎

We now complete the proof of Theorem 4.2.1.

###### Proof.

We first consider the case when $m\leq n$. Suppose that $n=m+r$.
Then from Corollary 4.1.4 and Lemma 4.2.3 it follows that :

$\displaystyle ch_{gr}\ W(m,m+r)$
$\displaystyle=\sum\limits_{i=1}^{m}ch_{gr}V(r+2+i-1,1^{i-1})+ch_{\mathfrak{sl}_{2}}\
V(r)$
$\displaystyle=\sum\limits_{k=0}^{m-1}\Big{(}\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}\
V(r+2+2(k-p))\Big{)}+ch_{\mathfrak{sl}_{2}}V(r)$
$\displaystyle\text{Rearranging the terms we thus get},$
$\displaystyle=\sum\limits_{i=1}^{m}\Big{(}\sum\limits_{p=0}^{m-i}q^{p}\begin{bmatrix}p+i-1\\\
p\end{bmatrix}_{q}\Big{)}ch_{\mathfrak{sl}_{2}}\
V(r+2+2i)+ch_{\mathfrak{sl}_{2}}V(r)$
$\displaystyle=\sum_{i=0}^{m}\begin{bmatrix}m\\\
i\end{bmatrix}_{q}Ch_{gr}V(r+2i),\quad(\text{ by }\eqref{q-combn
identity.2}).$

We now consider the case when $m>n$, i.e., $m=n+r$. It follows from Corollary
4.1.4 that,

$ch_{gr}W(n+r,n)=\sum\limits_{i=0}^{n}ch_{gr}V(i+1,1^{r+i-1})=\sum\limits_{k=1}^{n+1}ch_{gr}V(k,1^{r-2+k}).$

Hence, using Lemma 4.2.3, for $r>3$, we have,

$\begin{array}[]{l}ch_{gr}W(n+r,n)=\sum\limits_{k=1}^{n+1}\big{(}\sum\limits_{p=0}^{k}q^{p}\begin{bmatrix}k+r-2\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r-2+2(k-p))\big{)}\\\
\quad+\sum\limits_{k=1}^{n+1}\Big{(}\sum\limits_{p=k+1}^{\lfloor\frac{2k+r-2}{2}\rfloor}q^{p}\big{(}\begin{bmatrix}k+r-2\\\
p\end{bmatrix}_{q}-q^{r-1-2(p-k)}\begin{bmatrix}k+r-2\\\
p-k-1\end{bmatrix}\big{)}ch_{\mathfrak{sl}_{2}}V(r-2-2(p-k))\Big{)}\\\
=\sum\limits_{i=-1}^{n}\big{(}\sum\limits_{p=0}^{n-i}q^{p}\begin{bmatrix}p+i+r-1\\\
p\end{bmatrix}_{q}\big{)}ch_{\mathfrak{sl}_{2}}V(r+2i)\\\
+\sum\limits_{i=1}^{\lfloor\frac{r}{2}\rfloor-1}\sum\limits_{p=i+1}^{i+n+1}q^{p}\big{(}\begin{bmatrix}p-i+r-2\\\
p\end{bmatrix}_{q}-q^{r-1-2i}\begin{bmatrix}p-i+r-2\\\
i-1\end{bmatrix}_{q}\big{)}ch_{\mathfrak{sl}_{2}}V(r-2(i+1))\\\
=\sum\limits_{i=-1}^{n}\begin{bmatrix}n+r\\\
n-i\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2i)\\\
+\sum\limits_{i=1}^{\lfloor\frac{r}{2}\rfloor-1}\Big{(}\begin{bmatrix}n+r\\\
n+i+1\end{bmatrix}_{q}-\begin{bmatrix}r-1\\\
i\end{bmatrix}_{q}-\big{(}\sum\limits_{j=0}^{n}q^{r-i+j}\begin{bmatrix}r-1+j\\\
i-1\end{bmatrix}_{q}\big{)}\Big{)}ch_{\mathfrak{sl}_{2}}V(r-2(i+1))\quad(\text{
by }\eqref{q-combn identity.2})\\\
=\sum\limits_{i=-1}^{n}\begin{bmatrix}n+r\\\
n-i\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(r+2i)+\sum\limits_{i=1}^{\lfloor\frac{r}{2}\rfloor-1}\Big{(}\begin{bmatrix}n+r\\\
n+i+1\end{bmatrix}_{q}-\begin{bmatrix}r+n\\\
i-1\end{bmatrix}_{q}\Big{)}ch_{\mathfrak{sl}_{2}}V(r-2(i+1))\quad(\text{ by
}\eqref{q-combn identity})\end{array}$

The proof of the theorem in the case when $r=3$, is similar except for the
fact that here we need to use the formula for $ch_{gr}V(k,1^{k+1})$ as given
in Lemma 4.2.3.

For $m=n+1$, using Corollary 4.1.4, Lemma 4.2.3 and identities (4.8)-(4.10),
we have:

$\displaystyle
ch_{gr}W(n+1,n)=\sum\limits_{i=0}^{n}ch_{gr}V(i+1,1^{i})=\sum\limits_{i=0}^{n}\sum\limits_{p=0}^{i}q^{p}\begin{bmatrix}i\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2(i-p))$
$\displaystyle=\sum\limits_{j=0}^{n}\sum\limits_{p=0}^{n-j}q^{p}\begin{bmatrix}p+j\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2j)=\sum\limits_{j=0}^{n}\begin{bmatrix}n+1\\\
n-j\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(1+2j).$

Likewise, for $m=n+2$, we have,

$\displaystyle
ch_{gr}W(n+2,n)=\sum\limits_{i=0}^{n}ch_{gr}V(i+1,1^{i+1})=\sum\limits_{i=0}^{n}\sum\limits_{p=0}^{i+1}q^{p}\begin{bmatrix}i+1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2+2(i-p))$
$\displaystyle=\sum\limits_{j=0}^{n}\sum\limits_{p=0}^{n-j}q^{p}\begin{bmatrix}p+j+1\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2+2j)+\sum\limits_{p=1}^{n+1}q^{p}\begin{bmatrix}p\\\
p\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(0)$
$\displaystyle=\sum\limits_{j=0}^{n}\begin{bmatrix}n+2\\\
n-j\end{bmatrix}_{q}ch_{\mathfrak{sl}_{2}}V(2+2j)+\big{(}\begin{bmatrix}n+2\\\
1\end{bmatrix}_{q}-1\big{)}ch_{\mathfrak{sl}_{2}}V(0).$

This completes the proof of Theorem 4.2.1.∎

##  5\. The module $W_{loc}(n\omega)\otimes W_{loc}(m\omega)$ and Demazure
flags

In this section we recall the definition of truncated local Weyl modules and
show that the tensor product of two local Weyl modules has a filtration by
truncated local Weyl modules. Then using Proposition 4.0.1, we establish these
tensor product modules have an excellent filtration and give a closed formula
for the graded multiplicity of level two Demazure modules in the tensor
product of level one Demazure modules. In addition we show that the tensor
product of a level one Demazure module with a level two Demazure module admits
a filtration by level 3 Demazure modules.

###  5.1. Truncated local Weyl Modules

We begin by recalling the definition of truncated local Weyl modules.
Given a pair of integers $(m,N)$, the truncated local Weyl module
$W_{loc}([m],N)$ is a quotient of the local Weyl module $W_{loc}(m)$ generated
by an element $w_{m,N}$ satisfying the following relations:

$\begin{array}[]{c}(x\otimes t^{r})w_{m,N}=0,\qquad(h\otimes
t^{r})w_{m,N}=m\delta_{r,0}w_{m,N},\qquad(y\otimes 1)^{m+1}w_{m,N}=0,\\\
y\otimes t^{s}.w_{m,N}=0,\quad\forall\,s\geq N.\end{array}$ (5.1)

The truncated local Weyl modules were studied in [18, Theorem 4.3] and it was
shown that if $m=Nk+r$ with $0\leq r<N$, then the
$\mathfrak{sl}_{2}[t]$-module $W_{loc}([m],N)$ is isomorphic to the fusion
product module $V(k+1)^{\ast r}\ast V(k)^{\ast N-r}$. Hence by [8, Theorem
5](ii), it is known that the module $W_{loc}([m],N)$ is isomorphic to the CV-
module $V(\xi(m,N))$ where $\xi(m,N)=((k+1)^{r},k^{N-r}).$

In particular, given non-negative integers $a$ and $b$, since
$2a+b=1\cdot(a+b)+a$, the module $W_{\text{loc}}([2a+b],a+b)$ is isomorphic to
$V(2^{a},1^{b})$ and hence has a Demazure flag of level $2$, as per
Proposition 4.0.1. The following result gives the character formula for such
truncated local Weyl modules in terms of level 1 and level 2 Demazure modules.

######  Lemma 5.1.1.

Give two non-negative integers $a,b$, we have,
i. $ch_{gr}V(2^{a},1^{b})=\sum\limits_{k=0}^{a}(-1)^{k}\begin{bmatrix}a\\\
k\end{bmatrix}_{q}q^{k(a+b)-\frac{k(k-1)}{2}}\ ch_{gr}\ W_{loc}(b+2a-2k)$

ii. The truncated local Weyl module $W_{loc}([2a+b],a+b)$ has a level two
Demazure flag and

$ch_{gr}V(2^{a},1^{b})=\sum\limits_{k=0}^{\lfloor{\frac{b}{2}}\rfloor}q^{k(a+\lceil{\frac{b}{2}}\rceil)}\begin{bmatrix}{\lfloor{\frac{b}{2}}\rfloor}\\\
k\end{bmatrix}_{q}\,ch_{gr}D(2,2a+b-2k)$ (5.2)

###### Proof.

i. We prove part (i) of the lemma by applying induction on $a$.
For $a=1$, from the short exact sequence of $W_{loc}((b+2)\omega)$ given in
Lemma 3.4.1(b), we deduce that

$ch_{gr}\ V(2,1^{b})=ch_{gr}W_{loc}(b+2)-q^{(b+1)}ch_{gr}W_{loc}(b)$

Thus, the result holds for $a=1$.
Assume that the result holds when $a<l$ and consider the case when $a=l$.
Once again, from Lemma 3.4.1(b) we deduce that

$\displaystyle ch_{gr}V(2^{l},1^{b})$ $\displaystyle=ch_{gr}\
V(2^{l-1},1^{b+2})-q^{(l+b)}ch_{gr}\ V(2^{l-1},1^{b})$ Applying induction
hypothesis we have,
$\displaystyle=\sum\limits_{k=0}^{l-1}(-1)^{k}\begin{bmatrix}l-1\\\
k\end{bmatrix}_{q}q^{k(l+b+1)-\frac{k(k-1)}{2}}ch_{gr}\ W_{loc}(b+2(l-k))$
$\displaystyle-q^{l+b}\sum\limits_{k=0}^{l-1}(-1)^{k}\begin{bmatrix}l-1\\\
k\end{bmatrix}_{q}q^{k(l+b-1)-\frac{k(k-1)}{2}}ch_{gr}W_{loc}(b+2(l-k-1))$
$\displaystyle=\sum\limits_{k=0}^{l-1}(-1)^{k}\begin{bmatrix}l-1\\\
k\end{bmatrix}_{q}q^{k(l+b+1)-\frac{k(k-1)}{2}}ch_{gr}\ W_{loc}(b+2(l-k))$
$\displaystyle-\sum\limits_{k=0}^{l-1}(-1)^{k}\begin{bmatrix}l-1\\\
k\end{bmatrix}_{q}q^{(k+1)(l+b)-\frac{k(k+1)}{2}}ch_{gr}\ W_{loc}(b+2(-k-1))$
$\displaystyle\text{Reindexing the indices we get},$ $\displaystyle
ch_{gr}V(2^{l},1^{b})$
$\displaystyle=\sum\limits_{k=0}^{l-1}(-1)^{k}\begin{bmatrix}l-1\\\
k\end{bmatrix}_{q}q^{k(l+b+1)-\frac{k(k-1)}{2}}ch_{gr}W_{loc}(b+2(l-k))$
$\displaystyle+\sum\limits_{k=1}^{l}(-1)^{k}\begin{bmatrix}l-1\\\
k-1\end{bmatrix}_{q}q^{k(l+b)-\frac{k(k-1)}{2}}ch_{gr}W_{loc}(b+2(l-k))$
$\displaystyle=\sum\limits_{k=0}^{l}(-1)^{k}\begin{bmatrix}l\\\
k\end{bmatrix}_{q}q^{k(l+b)-\frac{k(k-1)}{2}}ch_{gr}W_{loc}(b+2(l-k)).\qquad\qquad$

ii. We prove part(ii) of the lemma by applying induction on $b$.
First note that for any partition $\mathbb{n}$ of $n\in\mathbb{N}$, the
multiplicity of a Demazure module $D(m,r)$ with highest weight $r$ is zero
whenever $n-r\notin 2\mathbb{Z}_{+},$ and

$V(2^{a})\cong D(2,2a),\qquad V(2^{a},1)\cong D(2,2a+1).$

Hence there is nothing to prove for $b=0$ and 1.
Now suppose $2a+b\in 2\mathbb{Z}+1$ and $b=2d+1$.
For $d=1$, using (3.4.1), we have the following short exact sequence,

$0\rightarrow\tau_{a+1}V(2^{a},1)\rightarrow
V(2^{a},1^{3})\xrightarrow{\phi}V(2^{a+1},1)\rightarrow 0$

Hence,

$\begin{array}[]{ll}ch_{gr}V(2^{a},1^{3})&=ch_{gr}D(2,2(a+1)+1)+q^{a+d+1}ch_{gr}D(2,2a+1)\\\
&=\sum\limits_{k=0}^{1}q^{k(a+d+1)}\begin{bmatrix}d\\\
k\end{bmatrix}_{q}ch_{gr}D(2,2a+3-2k)\end{array}$

Assume that the result holds for all $d<l$ and consider module
$V(2^{a},1^{2l+1})$.
Once again, using (3.4.1), we have the following short exact sequence,

$0\rightarrow\tau_{a+2b-1}V(2^{a},1^{2l-1})\rightarrow
V(2^{a},1^{2l+1})\xrightarrow{\phi}V(2^{a+1},1^{2l-1})\rightarrow 0$
$\displaystyle ch_{gr}V(2^{a},1^{2l+1})$
$\displaystyle=ch_{gr}V(2^{a+1},1^{2l-1})+q^{a+2l}ch_{gr}V(2^{a},1^{2l-1})$
$\displaystyle=\sum\limits_{k=0}^{l-1}q^{k(a+l+1)}\begin{bmatrix}l-1\\\
k\end{bmatrix}_{q}ch_{gr}D(2,1+2(a+l-k))\quad\text{(By induction hypothesis)}$
$\displaystyle+\sum_{r=0}^{l-1}q^{r(a+l)+a+2l}\begin{bmatrix}l-1\\\
r\end{bmatrix}_{q}ch_{gr}D(2,1+2(a+l-k-1))$
$\displaystyle=\sum_{k=0}^{l}q^{k(a+\lceil\frac{b}{2}\rceil)}\begin{bmatrix}\lfloor\frac{b}{2}\rfloor\\\
k\end{bmatrix}_{q}ch_{gr}D(2,2a+b-2k),$

where we obtain the last equality by using the q-combinatorial identity (4.9).
This proves part (ii) when $2a+b$ is an odd integer. The proof is similar in
the case when $2a+b$ is an even integer. ∎

###  5.2. Character formula of $W_{loc}(m)\otimes W_{loc}(n)$.

The following result on the character formula for the tensor product of two
local Weyl modules of $\mathfrak{sl}_{2}[t]$ was given in [5]. Since this
result plays an important role in the proof of the main result of this
section, Theorem 5.4.1, for the sake of completeness, we include its
independent proof here.

######  Lemma 5.2.1.

Given $m,n\in\mathbb{Z}_{+}$,

$ch_{gr}W_{loc}(n)\otimes
W_{loc}(m)=\sum_{i=0}^{\min\\{n,m\\}}\begin{bmatrix}n\\\
i\end{bmatrix}_{q}\begin{bmatrix}m\\\
i\end{bmatrix}_{q}(1-q)\dots(1-q^{i})ch_{gr}W_{loc}(n+m-2i).$ (5.3)

###### Proof.

The result is a consequence of [1, Proposition 3.2, Proposition 5.13].

######  Remark 5.2.2.

In Appendix we give an alternate proof of the lemma using Pieri rule.

######  Lemma 5.2.3.

For $m,n\in\mathbb{Z}_{+}$ with $n\geq m$,

$ch_{gr}(W_{loc}(m)\otimes W_{loc}(n))=\sum_{k=0}^{m}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}ch_{gr}V(2^{m-k},1^{n-m}).$ (5.4)

###### Proof.

Given (5.3), to prove (5.4) it suffices to show that

$\begin{array}[]{l}\sum\limits_{i=0}^{m}\begin{bmatrix}n\\\
i\end{bmatrix}_{q}\begin{bmatrix}m\\\
i\end{bmatrix}_{q}(1-q)\dots(1-q^{i})\,ch_{gr}W_{loc}(n+m-2i)=\sum\limits_{k=0}^{m}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}ch_{gr}V(2^{m-k},1^{n-k}),\end{array}$

or equivalently:

$\begin{array}[]{l}\sum\limits_{i=0}^{m}\begin{bmatrix}n\\\
i\end{bmatrix}_{q}\begin{bmatrix}m\\\
i\end{bmatrix}_{q}(1-q)\dots(1-q^{i})\,ch_{gr}W_{loc}(n+m-2i)\\\
=\sum\limits_{k=0}^{m}\sum\limits_{i=0}^{k}(-1)^{i}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}\begin{bmatrix}k\\\
i\end{bmatrix}_{q}q^{i(k+n-m)-\frac{i(i-1)}{2}}\,ch_{gr}W_{loc}(n-m+2k-2i)\end{array}$
(5.5)

Using Lemma 5.1.1. Rearranging the terms on the right hand side of (5.5) we
have:

$\begin{array}[]{l}\sum\limits_{k=0}^{m}\sum\limits_{i=0}^{k}(-1)^{i}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}\begin{bmatrix}k\\\
i\end{bmatrix}_{q}q^{i(k+n-m)-\frac{i(i-1)}{2}}\,ch_{gr}W_{loc}(n-m+2(k-i))\\\
=\sum\limits_{i=0}^{m}\sum\limits_{k=i}^{m}(-1)^{i}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}\begin{bmatrix}k\\\
i\end{bmatrix}_{q}q^{i(k+n-m)-\frac{i(i-1)}{2}}\,ch_{gr}W_{loc}(n-m+2(k-i))\\\
=\sum\limits_{i=0}^{m}\sum\limits_{k=0}^{i}(-1)^{k}\begin{bmatrix}m\\\
m-i+k\end{bmatrix}_{q}\begin{bmatrix}m-i+k\\\
k\end{bmatrix}_{q}q^{k(n-i+k)-\frac{k(k-1)}{2}}\,ch_{gr}W_{loc}(n+m-2i)\\\
=\sum\limits_{i=0}^{m}\begin{bmatrix}m\\\
i\end{bmatrix}_{q}(\sum\limits_{k=0}^{i}(-1)^{k}\begin{bmatrix}i\\\
k\end{bmatrix}_{q}q^{k(n-i+k)-\frac{k(k-1)}{2}}\,)ch_{gr}W_{loc}(n+m-2i)\end{array}$

To complete the proof of the lemma it suffices to show

$\sum_{k=0}^{i}(-1)^{k}\begin{bmatrix}i\\\
k\end{bmatrix}_{q}q^{k(n-i+k)-\frac{k(k-1)}{2}}=\begin{bmatrix}n\\\
i\end{bmatrix}_{q}(1-q)\dots(1-q^{i}).$ (5.6)

We prove (5.6) by using the following combinatorial identity:

$\sum\limits_{k=0}^{i}(-1)^{i-k}\begin{bmatrix}i\\\
k\end{bmatrix}_{q}q^{{i-k\choose 2}}x^{k}=(x-1)(x-q)\cdots(x-q^{i-1})$ (5.7)

Putting $x=q^{n}$ in (5.7), we get

$\begin{array}[]{rl}\sum\limits_{k=0}^{i}(-1)^{i-k}\begin{bmatrix}i\\\
k\end{bmatrix}_{q}q^{{i-k\choose
2}}q^{nk}&=(q^{n}-1)(q^{n}-q)\cdots(q^{n}-q^{i-1})\\\
\sum\limits_{k=0}^{i}(-1)^{i-k}\begin{bmatrix}i\\\
k\end{bmatrix}_{q}q^{(i-k)(i-k-1)/2}q^{nk}&=(-1)^{i}q^{i(i-1)/2}(1-q^{n})(1-q^{n-1})\cdots(1-q^{n-i+1})\\\
\sum\limits_{k=0}^{i}(-1)^{k}\begin{bmatrix}i\\\
k\end{bmatrix}_{q}q^{k(n-i+k)-k(k-1)/2}&=(1-q^{n})\cdots(1-q^{n-i+1})=\begin{bmatrix}n\\\
i\end{bmatrix}_{q}(1-q)\dots(1-q^{i}).\end{array}$

This establishes (5.6) and hence the lemma. ∎

###  5.3. Ordered basis of the graded weight spaces of $W_{loc}(m)$

We have seen in Lemma 3.4.1(b), for $m\in\mathbb{Z}_{+}$, $W_{loc}(m)$ has a
basis $\mathbb{B}(1^{n})$ which is indexed by the set
$J(1^{n})\cup\\{\emptyset\\}$. Define a function
$|.|:J(1^{n})\cup\\{\emptyset\\}\rightarrow\mathbb{Z}$ such that

$|\emptyset|=0,\qquad\quad|\mathbb{i}|:=\sum\limits_{k=1}^{m}i_{k},\quad\forall\,\mathbb{i}=(i_{1},\cdots,i_{m})\in
J(1^{m}).$

Defining in $U(\mathfrak{n}^{-}[t])$ the elements

$y({{0}},\emptyset)=1,\qquad\quad
y(|\mathbb{i}|,\mathbb{i}):=(y_{0})^{i_{1}}(y_{1})^{i_{2}}\cdots(y_{m-1})^{i_{m}},\quad\forall\,\mathbb{i}=(i_{1},\cdots,i_{m})\in
J(1^{m}),$

we see that,
$\mathbb{B}(1^{n})=\\{y(|\mathbb{i}|,\mathbb{i})w_{m}:\mathbb{i}\in
J(1^{m})\cup\\{\emptyset\\}\\}.$ We define an ordering on
$J(1^{m})\cup\\{\emptyset\\}$ as follows. By fiat, $\emptyset>\mathbb{i}$ for
all $\mathbb{i}\in J(1^{m})$ and given $\mathbb{i},\mathbb{j}\in J(1^{m})$, we
say, $\mathbb{i}>\mathbb{j}$, if either $|\mathbb{i}|<|\mathbb{j}|$ or
$|\mathbb{i}|=|\mathbb{j}|$ and there exists $1\leq k\leq m$ such that
$i_{k}>j_{k}$ and $i_{s}=j_{s}$ for $k+1\leq s\leq m$. This clearly induces an
ordering in $\mathbb{B}(1^{m})$, thereby making $\mathbb{B}(1^{m})$ an ordered
basis of $W_{loc}(m)$.
On the other hand it was proved in [10] that :

######  Proposition 5.3.1.

Given a positive integer $m$, if

$F(m)=\\{(k,\underline{s}):k\in\mathbb{N},\underline{s}=(s_{1},\cdots,s_{k})\in{\mathbb{Z}}^{k},0\leq
s_{i}\leq m-k\text{ for }1\leq i\leq k\\}.$

Then the set $\mathbb{B}(m)=\\{y(k,\underline{s})w_{m}:(k,\underline{s})\in
F(m)\cup\\{(0,\emptyset)\\}\\}$ is a basis of $W_{loc}(m)$, where
$y(k,\underline{s})=(y\otimes t^{s_{1}})\cdots(y\otimes t^{s_{k}})$ and
$y(0,\emptyset)w_{m}=w_{m}$.

Thus using standard q-binomial theory, it follows from Proposition 5.3.1 that
the number of $l$-graded elements of weight $m\omega-k\alpha$ in
$\mathbb{B}(m)$ and hence in $\mathbb{B}(1^{m})$ is equal to the coeffient of
$q^{l}$ in the polynomial $\begin{bmatrix}m\\\ k\end{bmatrix}_{q}$.

###  5.4. Filtration of $W_{loc}(n)\otimes W_{loc}(m)$

The following is the main result of this section.

######  Theorem 5.4.1.

Let $n,m\in\mathbb{N}$ and $n\geq m$. The $\mathfrak{sl}_{2}[t]$-module
$W_{loc}(n)\otimes W_{loc}(m)$ admits a filtration whose successive quotients
are isomorphic to truncated local Weyl modules

$\tau_{k_{r}}W_{loc}([m+n-2r],n-r),\quad 0\leq r\leq m,\,0\leq k_{r}\leq
r(m-r).$

###### Proof.

Given $n,m\in\mathbb{N}$ with $n\geq m$, let $w_{n}$ be the generator of
$W_{loc}(n)$ and $\mathbb{B}(1^{m})$ the ordered basis of $W_{loc}(m)$ as
described in Section 5.3.1. For each element $\mathbb{i}\in
J(1^{m})\cup\\{\emptyset\\}$, set

$\begin{array}[]{c}v_{n,m}(k,\mathbb{i})=w_{n}\otimes
y(k,\mathbb{i})w_{m},\qquad\text{ where }k=|\mathbb{i}|,\\\ \\\
\mathcal{B}(m,n)^{\mathbb{i}\leq}=\\{v_{n,m}(l,\mathbb{j}):\mathbb{j}\in
J(1^{m})\cup\\{\emptyset\\},\mathbb{j}\geq\mathbb{i}\\}\end{array}$

and let $T(k,\mathbb{i})$ be the submodule of $W_{loc}(n)\otimes W_{loc}(m)$
generated by $\mathcal{B}(m,n)^{\mathbb{i}\leq}$.

Let $\mathbb{e}_{i}[m]$ denote the $m$-tuple with 1 in the $i^{th}$-position
and zero elsewhere. By Lemma 3.4.2,

$T(m,m\mathbb{e}_{1}[m])=W_{loc}(n)\otimes W_{loc}(m),$

and from the definition of the submodules $T(|\mathbb{i}|,\mathbb{i})$ it is
clear that

$T(m,m\mathbb{e}_{1}[m])\supset
T(m-1,(m-1)\mathbb{e}_{1}[m])\supset\cdots\supset
T(1,\mathbb{e}_{m-1}[m])\supset T(1,\mathbb{e}_{m}[m])\supset
T(0,\emptyset)\supset 0,$

is a descending chain of submodules of $W_{loc}(n)\otimes W_{loc}(m)$.

Additionally, since $x.y(k,\mathbb{i}).w_{m}$ lies in the subspace of
$W_{loc}(m)$ spanned by the vectors $\\{y(k-1,\mathbb{j})w_{m}:\mathbb{j}\in
J(1^{m})\\}$ and for $r>0$ , $\mathbb{i}\in J(1^{m})$ with $|\mathbb{i}|=k$,

$(h\otimes t^{r})y(k,\mathbb{i}).w_{m}=\sum\limits_{\\{1\leq j\leq
m-r\,:\,0<i_{j}<m-r\\}}-2y(k,\mathbb{i}+(\mathbb{e}_{i_{j}+r}[m]-\mathbb{e}_{i_{j}}[m])).w_{m},$

using the fact that $x.w_{n}=0,$ and $h\otimes t^{r}.w_{n}=0$, for $r>0$, it
follows that

$\begin{array}[]{c}x.v_{n,m}(1,\mathbb{i})=0,\qquad\quad
x.v_{n,m}(k,\mathbb{i})\in T(k-1,(k-1)\mathbb{e}_{1}[m])\quad\text{ when
}k>1,\\\ \\\ (h\otimes
t^{r})v_{n,m}(k,\mathbb{i})\in\sum\limits_{\mathbb{j}>\mathbb{i}}\,T(k,\mathbb{i}).\\\
\end{array}$ (5.8)

As $J(1^{m})$, and hence $\mathbb{B}(1^{m})$ is totally ordered, given
$\mathbb{i}\in J(1^{m})$ with $|\mathbb{i}|=k$ there exists a unique pair
$(\hat{k},\hat{\mathbb{i}})\in\mathbb{Z}\times J(1^{m})$ such that
$y(\hat{k},\hat{\mathbb{i}})>y(k,\mathbb{i})$ and if
$y(l,\mathbb{j})\in\mathbb{Z}\times J(1^{m})$ is such that
$y(l,\mathbb{j})>y(k,\mathbb{i})$, then $y(l,\mathbb{j})\geq
y(\hat{k},\hat{\mathbb{i}}).$ In particular, as
$y(k-1,(k-1)\mathbb{e}_{1}[m])>y(k,\mathbb{i})$, in view of (5.8) we see that,
$\dfrac{T(k,\mathbb{i})}{T(\hat{k},\hat{\mathbb{i}})}$ is a highest weight
module generated by the image of $v_{n,m}(k,\mathbb{i})$ under the natural
surjection from $T(k,\mathbb{i})$ onto
$\dfrac{T(k,\mathbb{i})}{T(\hat{k},\hat{\mathbb{i}})}.$

Since $n\geq m$, from the defining relations of a local Weyl module we have
$y_{n}v_{n,m}(k,\mathbb{i})=0$. Further,

$(h\otimes 1)v_{n,m}(k,\mathbb{i})=h.w_{n}\otimes
y(k,\mathbb{i})w_{m}+w_{n}\otimes h.y(k,\mathbb{i})w_{m}=(n+m-2k)(w_{n}\otimes
y(k,\mathbb{i})w_{m}).$

Hence, for all $\mathbb{i}\in J(1^{m})$, the module
$\dfrac{T(k,\mathbb{i})}{T(\hat{k},\hat{\mathbb{i}})}$ is a quotient of the
truncated Weyl module $\tau^{\ast}_{g_{\mathbb{i}}}W_{loc}(m+n-2k,n)$, where
$g_{\mathbb{i}}=\sum\limits_{k=1}^{m}(k-1)i_{k}$, $T(0,\emptyset)$ is a
quotient of $W_{loc}(n+m,n)$ and

$\,ch_{gr}W_{loc}(n)\otimes
W_{loc}(m)=ch_{gr}T(0,\emptyset)+\sum\limits_{\mathbb{i}\in
J(1^{m})}q^{g_{\mathbb{i}}}\,ch_{gr}\dfrac{T(|\mathbb{i}|,\mathbb{i})}{T(|\hat{\mathbb{i}}|,\hat{\mathbb{i}})}.$
(5.9)

Now comparing with the character formulas (5.4) and (5.9), we see that
$T(0,\emptyset)$ is isomorphic $V(2^{n},1^{m-n})$ which by Section 5.1 is
isomorphic to $W_{loc}(m+n,n)$.

On the other hand for $\mathbb{i}\in J(1^{m})$, with $|\mathbb{i}|=k$ and
$g_{\mathbb{i}}=l$, as $y(|\mathbb{i}|,\mathbb{i})w_{m}$ is a $l$-graded
vector of weight $m\omega-k\alpha$, the quotient module,
$\dfrac{T(|\mathbb{i}|,\mathbb{i})}{T(|\hat{\mathbb{i}}|,\hat{\mathbb{i}})}$
is a quotient of $\tau_{l}W(m+n-2k,n)$. By Section 5.3.1, the number of
l-graded vectors of weight $m\omega-k\alpha$ in $\mathbb{B}(1^{m})$ is equal
to the coefficient of $q^{k}$ in the polynomial $\begin{bmatrix}m\\\
k\end{bmatrix}_{q}$. Hence comparing the character formulas (5.4) and (5.9)
and the grades of the highest weight generating vectors the modules
$\dfrac{T(|\mathbb{i}|,\mathbb{i})}{T(|\hat{\mathbb{i}}|,\hat{\mathbb{i}})}$,
we see that for any $0\leq k\leq m$,
$\dfrac{T(k,\mathbb{i})}{T(\hat{k},\hat{\mathbb{i}})}$ is isomorphic to the CV
module $V(2^{m-k},1^{n-m})$ which, by Section 5.1, is a quotient of
$W_{loc}(n+m-2k,n)$and is isomorphic to the truncated Weyl module
$W_{loc}(n+m-2k,n-k)$. This completes the proof of the theorem. ∎

#### 5.4.1.

For $m,n,s\in\mathbb{N}$, such that $m+n-2s\geq 0$, define a polynomial in
indeterminate $q$ by:

$[W_{loc}(n)\otimes W_{loc}(m):D(2,m+n-2s)]_{q}=\sum\limits_{p\geq
0}[W_{loc}(n)\otimes W_{loc}(m):\tau_{p}D(2,m+n-2s)]q^{p},$

where $[W_{loc}(n)\otimes W_{loc}(m):\tau_{p}D(l,m+n-2s)]$ is the multiplicity
of $\tau_{p}D(l,m+n-2s)$ in $W_{loc}(n)\otimes W_{loc}(m)$. The polynomial
$[W_{loc}(n)\otimes W_{loc}(m):D(2,m+n-2s)]_{q}$ is called the graded
multiplicity of $D(2,m+n-2s)$ in level $2$ Demazure flag of $W_{loc}(n)\otimes
W_{loc}(m).$

######  Corollary 5.4.2.

Given two positive integers $m,n$, the $\mathfrak{sl}_{2}[t]$-module
$W_{loc}(n)\otimes W_{loc}(m)$ has a level 2 Demazure flag and

$\begin{array}[]{l}[W_{loc}(n)\otimes W_{loc}(m):D(2,m+n-2s)]_{q}\\\
=\left\\{\begin{array}[]{ll}\sum\limits_{k=0}^{\min\\{s,m\\}}q^{(s-k)(m-k+\lceil\frac{n-m}{2}\rceil)}\begin{bmatrix}m\\\
k\end{bmatrix}\begin{bmatrix}\lfloor\frac{n-m}{2}\rfloor\\\
s-k\end{bmatrix}_{q},&0\leq s\leq\lfloor\frac{n-m}{2}\rfloor,\\\
\sum\limits_{k=0}^{\min\\{m-j,\lfloor\frac{n-m}{2}\rfloor\\}}q^{(\lfloor\frac{n-m}{2}\rfloor-k)(m-k-j+\lceil\frac{n-m}{2}\rceil)}\begin{bmatrix}m\\\
k+j\end{bmatrix}\begin{bmatrix}\lfloor\frac{n-m}{2}\rfloor\\\
\lfloor\frac{n-m}{2}\rfloor-k\end{bmatrix}_{q},&s=j+\lfloor\frac{n-m}{2}\rfloor,\,1\leq
j\leq m\end{array}\right.\end{array}$

for $m+n-2s\geq 0$.

###### Proof.

By Theorem 5.4.1, the tensor product product module $W_{loc}(n)\otimes
W_{loc}(m)$ has a filtration by submodules whose successive quotients are
isomorphic to translates of modules of the form $V(2^{a},1^{b})$. Hence by
Theorem 4.0.1, $W_{loc}(n\omega)\otimes W_{loc}(m\omega)$ has a level 2
Demazure flag. Now using Lemma 5.2.3 and Lemma 5.1.1, we see that

$\begin{array}[]{ll}ch_{gr}W_{loc}(n){\otimes}W_{loc}(m)&=\sum\limits_{k=0}^{m}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}ch_{gr}\,V(2^{m-k},1^{n-m})\\\
&=\sum\limits_{k=0}^{m}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}\left(\sum\limits_{r=0}^{\lfloor\frac{n-m}{2}\rfloor}q^{r(m-k+\lceil\frac{n-m}{2}\rceil)}\begin{bmatrix}\lfloor\frac{n-m}{2}\rfloor\\\
r\end{bmatrix}_{q}ch_{gr}D(2,n+m-2(k+r))\right)\\\ \end{array}$

Hence, we have

$\begin{array}[]{ll}[W_{loc}(n\omega)\otimes
W_{loc}(m\omega):D(2,m+n-2s)]_{q}&=\sum\limits_{k=0}^{\min\\{s,m\\}}q^{(s-k)(m-k+\lceil\frac{n-m}{2}\rceil)}\begin{bmatrix}m\\\
k\end{bmatrix}\begin{bmatrix}\lfloor\frac{n-m}{2}\rfloor\\\
s-k\end{bmatrix}_{q},\end{array}$

if $s\in\mathbb{N}$ is such that $0\leq s\leq\lfloor\frac{n-m}{2}\rfloor$ and

$\begin{array}[]{l}[W_{loc}(n\omega)\otimes
W_{loc}(m\omega):D(2,m+n-2s)]_{q}\\\
=\sum\limits_{k=0}^{\min\\{m-j,\lfloor\frac{n-m}{2}\rfloor\\}}q^{(\lfloor\frac{n-m}{2}\rfloor-k)(m-k-j+\lceil\frac{n-m}{2}\rceil)}\begin{bmatrix}m\\\
k+j\end{bmatrix}\begin{bmatrix}\lfloor\frac{n-m}{2}\rfloor\\\
\lfloor\frac{n-m}{2}\rfloor-k\end{bmatrix}_{q},\end{array}$

when $s=j+\lfloor\frac{n-m}{2}\rfloor$ for some $1\leq j\leq m$. ∎

In [5] the graded character formula for $D(2,2n+r)\otimes W_{loc}(m)$ has been
obtained when $n\geq m$. We can extend the methods employed in proving Theorem
5.4.1 to show that the tensor product of a level 2 Demazure module with a
local Weyl module has a decreasing chain of submodules, with each successive
quotient isomorphic to a CV-module of the form $V(3^{a},2^{b},1^{c})$.
Consequently, it follows from Proposition 4.0.1 that the module
$D(2,2n+r)\otimes W_{\text{loc}}(m)$ admits a level 3-Demazure flag.

##  6\. Appendix

In the appendix we give a proof of Lemma 5.2.1 using Pieri formulas ([19,
Chapter VI]).

Let $t,q$ be independent indeterminates and $\mathbb{C}(q,t)$ be the field of
rational functions in $q$ and $t$. Let $\\{x_{i}:1\leq i\leq n+1\\}$ be a set
of indeterminates and $\mathbb{C}(q,t)[x_{1},\cdots,x_{n+1}]$ be the ring of
polynomials in $\\{x_{j}:1\leq j\leq n+1\\}$ with coefficients in
$\mathbb{C}(q,t)$. Let $\bm{x}=(x_{1},x_{2},\dots,x_{n+1})$ be a basis for the
ring of symmetric polynomials in $\mathbb{C}(q,t)[x_{1},x_{2},\dots,x_{n+1}]$.

For $\lambda\in P^{+}$, a family of orthogonal symmetric polynomials
$P_{\lambda}(\bm{x};q,t)$ was introduced in [19], and it was shown in [21]
that the character of a level one Demazure module of highest weight $\lambda$
is given by the specialized Macdonald polynomial $P_{\lambda}(\bm{x};q,0)$. On
the other hand it was proved in [12] that a local Weyl module for a simply-
laced current Lie algebra is isomorphic to a Demazure module of level one.
Thus in case of $\mathfrak{sl}_{2}[t]$, we see that for $m\in\mathbb{Z}_{+}$,
$ch_{gr}W_{loc}(m)=P_{(m,0)}(\bm{x};q,0)$. We use this fact to give an
alternate proof of Lemma 5.2.1.

Set

$\displaystyle(q;q)_{n}$ $\displaystyle=(1-q)(1-q^{2})\dots(1-q^{n})\quad$
$\displaystyle(t;q)_{n}$
$\displaystyle=(1-t)(1-tq)\dots(1-tq^{n-1})\quad\text{for }|q|<1$
$\displaystyle(t;q)_{\infty}$ $\displaystyle=\prod_{i=0}^{\infty}(1-tq^{i})$

By [19, Chapter VI, Equation 4.9],

$P_{(m)}(\bm{x};q,t)=\frac{(q;q)_{m}}{(t;q)_{m}}g_{m}(\bm{x},q,t)$ (6.1)

where $g_{m}(\bm{x};q,t)$ denotes the coefficient of $y^{m}$ in the power-
series expansion of the infinite product

$\prod_{i\geq 1}\frac{(tx_{i}y;q)_{\infty}}{(x_{i}y;q)_{\infty}}=\sum_{m\geq
0}g_{m}(\bm{x};q,t)y^{m}$

Putting $t=0$ in (6.1), we get

$P_{(m)}(\bm{x};q,0)=(q;q)_{m}g_{m}(\bm{x},q,0)$ (6.2)

We need the following result on Pieri formula for our proof of Lemma 5.2.1.

######  Lemma 6.0.1.

[19, Theorem 6.24] Let $m,n\in\mathbb{Z}_{+}$, with $n\geq m$. Given a
partition $\lambda$ of $m+n$ containing the partition $(n)$ and having atmost
two parts, we have,

$P_{n}(\bm{x};q,0)g_{m}(\bm{x};q,0)=\sum_{\lambda-(n)=(m)}\phi_{\lambda/(n)}P_{\lambda}(\bm{x};q,0)$

and the coefficients are given by

$\phi_{\lambda/(n)}=\prod_{s\in
C_{\lambda/(n)}}\frac{b_{\lambda}(s)}{b_{(n)}(s)}$

where $C_{\lambda/(n)}$ denote the union of columns that intersect
$\lambda-(n)$, and

$b_{\lambda}(s)=b_{\lambda}(s;q,t)=\begin{cases}\frac{1-q^{a_{\lambda}(s)}t^{l_{\lambda}(s)+1}}{1-q^{a_{\lambda}(s)+1}t^{l_{\lambda}(s)}}&\quad\text{if
}s\in\lambda,\\\ 1&\quad\text{otherwise}\end{cases}$

### Proof of Lemma 5.2.1.

Since for $m\in\mathbb{Z}_{+}$, $ch_{gr}W_{loc}(m)=P_{(m,0)}(\bm{x};q,0)$, we
have

$\displaystyle ch_{gr}(W_{loc}(n)\otimes W_{loc}(m))$
$\displaystyle=ch_{gr}W_{loc}(n)ch_{gr}W_{loc}(m)$
$\displaystyle=P_{(n,0)}(\bm{x};q,0)P_{(m,0)}(\bm{x};q,0)$
$\displaystyle=(q;q)_{m}P_{(n,0)}(\bm{x};q,0)g_{m}(\bm{x};q,0)\quad(\text{Using
\ref{P in terms of g}})$
$\displaystyle=(q;q)_{m}\sum_{\lambda-(n)=(m)}\phi_{\lambda/(n)}(q,0)P_{\lambda}(\bm{x};q,0)$
$\displaystyle=(q;q)_{m}\sum_{\lambda-(n)=(m)}\prod_{s\in
C_{\lambda/(n)}}\frac{b_{\lambda}(s;q,0)}{b_{(n)}(s;q,0)}P_{\lambda}(\bm{x};q,0)$
$\displaystyle=(q;q)_{m}(\frac{1}{(1-q)(1-q^{2})\dots(1-q^{m})}P_{(n+m,0)}(\bm{x};q,0)$
$\displaystyle+\frac{1-q^{n}}{(1-q)(1-q^{2})\dots(1-q^{m-1})(1-q)}P_{(n+m-1,1)}(\bm{x};q,0)$
$\displaystyle+\frac{(1-q^{n})(1-q^{n-1})}{(1-q)(1-q^{2})\dots(1-q^{m-2})(1-q)(1-q^{2})}P_{(n+m-2,2)}(\bm{x};q,0)$
$\displaystyle+\dots+\frac{(1-q^{n})(1-q^{n-1})\dots(1-q^{n-m+1})}{(1-q)(1-q^{2})\dots(1-q^{m})}P_{(n-m,m)}(\bm{x};q,0))$
$\displaystyle=P_{(n+m,0)}(\bm{x};q,0)+\frac{(1-q^{n})(1-q^{m})}{(1-q)}P_{(n+m-1,1)}(\bm{x};q,0)$
$\displaystyle+\frac{(1-q^{n})(1-q^{n-1})(1-q^{m})(1-q^{m-1})}{(1-q)(1-q^{2})}P_{(n+m-2,2)}(\bm{x};q,0)$
$\displaystyle+\dots+\frac{(1-q^{n})(1-q^{n-1})\dots(1-q^{n-m+1})(1-q)(1-q^{2})\dots(1-q^{m})}{(1-q)(1-q^{2})\dots(1-q^{m})}$
$\displaystyle P_{(n-m,m)}(\bm{x};q,0)$

Hence,

$\displaystyle ch_{gr}(W_{loc}(n)\otimes W_{loc}(m))$
$\displaystyle=\sum_{k=0}^{m}\begin{bmatrix}n\\\
k\end{bmatrix}_{q}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}(1-q)(1-q^{2})\dots(1-q^{k})P_{(n+m-k,k)}(\bm{x};q,0)$
$\displaystyle=\sum_{k=0}^{m}\begin{bmatrix}n\\\
k\end{bmatrix}_{q}\begin{bmatrix}m\\\
k\end{bmatrix}_{q}(1-q)\dots(1-q^{k})ch_{gr}W_{loc}(n+m-2k).\qquad\qquad\qed$

## Acknowledgements

The first author would like to thank Professor S.Vishwanath and Professor
Vyjayanthi Chari for useful discussions. The first author is grateful to IISER
Mohali for PhD fellowship and support.

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