Higher Specht Polynomials and Modules over the Weyl algebra

Ibrahim Nonkané,
Léonard  Todjihounde

Departement d'économie et de mathématiques appliquées, IUFIC, Université Thomas Sankara, Burkina faso

Institut de Mathé,matiques et de Sciences Physiques (IMSP), Université d'Abomey-Calavi, Benin

In this paper, we study an irreducible  decomposition structure of the $\Dc$-module direct image $\pi_+(\Oc_{ \bC^n})$
for  the finite map $\pi: \bC^n  \to  \bC^n/ ({\Sc_{n_1}\times \cdots \times \Sc_{n_r}}).$

We explicitly construct the simple component of $\pi_+(\Oc_{\bC^n})$ by providing their generators  and  their multiplicities.
Using an equivalence categories and the higher Specht polynomials, we describe a $\D$-module decomposition of the  of the polynomial ring localized  at the discriminant of $\pi$. Furthermore we study the action invariants differential operators on the higher Specht polynomials.

keywords: Direct image, Direct product, Group representation theory, Higher Specht polynomials, Partitions,  Primitive idempotents, Semisimplicity, Symmetric group , Young diagram.

Mathematics Subject Classification:  Primary 13N10, Secondary 20C30.

[Acknowledgements This  work was done while the first author was visiting IMSP at Benin,  under the  Staff Exchange program of the German  Office of Academic Exchange  (DAAD).
He warmly thanks the DAAD for the financial support.

§ INTRODUCTION

It is well-known that the ring $\Oc_X:={\mathbb C}[x_1,...,x_n]$  of the polynomials in $n$ indeterminates is a simple module over the Wyel algebra $\Dc_X$ associated with the $\Oc_X.$
The direct image of a simple module under a proper map $\pi$ is semisimple by the Kashiwara's decomposition Theorem [8].
The simplest case is when the map $\pi: X \to Y$ is finite, in which case it is easy to give an elementary and wholly algebraic proof, using essentially the generic correspondence with the differential Galois group, which equals the ordinary Galois  group $G$. The irreducible submodule of the direct image are in one-to-one correspondence with the irreducible representations of $G$ (see [21]).
The goal of this paper is to find the simple component of the direct image $\pi_+(\Oc_X)$ of the polynomial ring $\Oc_X$ as a $\Dc$-module  under the map $  \pi: \spec \Oc_X \to \spec \Oc_Y$ where $ \Oc_Y= \Oc_X^{\Sc_{n_1}\times \cdots \times \Sc_{n_r}};$ the ring of invariant polynomials under the action of ${\Sc_{n_1}\times \cdots \times \Sc_{n_r}}.$
We describe The generators of the simple components of $\pi_+(\Oc_X)$ and their multiplicities as in [21].
We then  give the decomposition of the structure of $\pi_+(\Oc_X)$ by  giving a basis generated by the higher  Specht polynomials.  This proof uses the fact  that the irreducible $\D$-submodules of $\pi_+(\Oc_X)$ are in one-to-one correspondence with irreducible representations of $ {\Sc_{n_1}\times \cdots \times \Sc_{n_r}}$.

Secondly we  elaborate a $\D$-module decomposition of the polynomial ring localized  at the discriminant of $\pi.$
Finally we describe the action invariants differential operators on  higher Specht polynomials
The higher Specht polynomials (introduced combinatorially by several authors [28], [3]), are adapted to the $\D$-module structure.

This  paper  generalizes results on modules over the Weyl algebra appeared in [21] and [22].
The case $r=2$ have been presented at 10th International Conference on Mathematical Modeling in Physical Sciences in order to describe the action of the rational quantum Calogero-Moser system on polynomials  [23].

§ PRELIMINARIES

§.§ Direct image

We briefly recall the definition of the direct image of a $\D$-module [10].

Let $K$ be a field of characteristic zero,  put $X=K^n$. The polynomial ring $K[x_1,\ldots, x_n]$ will be denoted by $K[X];$ and the Weyl algebra generated by $x_i$'s and $\frac{\partial}{\partial x_i}$'s by $\D_X$. The $n$-tuple $(x_1, \ldots,x_n)$ will be denoted by $X$. Similar conventions will holds for $Y=K^m$, with polynomial ring $K[Y]$ and Weyl algebra $\D_Y$.

Let $\pi  : X \to Y$ be a polynomial map, with $\pi=(\pi_1,\ldots,\pi_m)$. Let $M$ be a left $\D_Y$-module. The inverse image of $M$ under the map  $\pi$ is $\pi^+( M)= K[X] \otimes_{K[Y]}M.$ This is a $K[X]$-module. It becomes a $\D_X$-module with $\partial_{x_i}$ acting according to the formula

$$\frac{ \partial }{\partial x_i} (h \otimes u) = \frac{\partial h}{\partial{x_i}}\otimes u + \sum_{j=1}^m \frac{\partial  \pi_j}{\partial x_i} \otimes \frac{\partial }{\partial y_j} u, \ h \in K[X], u \in M.$$

Since $\D_Y \otimes_{\D_Y} M \cong M,$ we have that

$$ \pi^+ (M) \cong K[X] \otimes_{K[Y]} \D_{Y} \otimes_{\D_Y}M = \pi^+(K[Y])  \otimes_{\D_Y}M.$$

Writing $D_{X \to Y}$ for $\pi^+( K[Y])$, on has that $\pi^+(M)= \D_{X \to Y}  \otimes_{\D_Y}M.$
Note that $\D_{X\to Y}$ is $\D_X$- $\D_Y$-bimodule.

Let $N$ be a right $D_X$-module. The tensor product

$$\pi_+(N) =N \otimes_{\D_X} \D_{X \to Y}$$
is a right $\Dc_Y$-module, which is called the direct image of $N$ under the polynomial map $\pi$.
Let us consider the standard transposition $ \tau : \D_X \to \D_X$ defined by $ \tau (h \partial ^{\alpha} )= (-1)^{|\alpha|} \partial^{\alpha} h,$ where $h \in K[X]$ and $\alpha \in \bN^n.$ If $N$ is a right $\D_X$-module then we define a left $\D_X$-module $N^t$ as follows. As an abelian group, $N^t=N.$ If $a \in \D_X$ and $ u \in N^t$ then the left action of $a$ on $u$ is defined by $a \star u= u \tau(a).$ Using the standard transposition for $\D_Y$ and $\D_X$, put $D_{ Y\leftarrow X} = (D_{ X \to Y} )^t$, this is a $\D_Y$-$\D_X$-bimodule. Let $M$ be a left $\D_X$-module. The direct image of $M$ under $\pi$ is defined by the formula

$$ \pi_+(M)= D_{ Y \leftarrow X} \otimes_{\D_X} M.$$
It is clear that $\pi_+(M)$ is  a $\D_Y$-module.

The following is the Kashiwara decomposition theorem

Let $\pi: X\to Y$ be a polynomial map. If $M$ is a a simple (holonomic) module over $\D_X$. Then $\pi_+ (M)$ is a semisimple $\D_Y$-module. we have

$$ \pi_+(M)= \oplus  M_i^{\alpha_i},$$
where the $M_i$ are inequivalent irreducible $\D_Y$-submodules.

§.§ Higher Specht polynomials

In this subsection we recall some notions about irreducibles of representations of product of symmetric groups (see [3] for more details).

Let $\Oc_X$ be the algebra of polynomials of $n$ variables $x_1,\ldots,x_n$ with complex coefficients, on which the symmetric group $\Sc_n$ acts by permutation of variables:

$$( \sigma f)(x_1,\ldots,x_n)=f( x_{\sigma(1)}, \ldots, x_{\sigma(n)}), \sigma \in \Sc_n , f \in \Oc_X.$$

Let $n_1,\ldots,n_r$ be natural numbers such that $n=\sum_{i=1}^r n_i.$ Then the product of symmetric groups $\Sc_{n_1} \times \cdots \times \Sc_{n_r}$ is naturally embedded in $\Sc_n.$

A partition $\lambda$ is a non-increasing finite sequence of positive  integers $\lambda \geq \cdots \geq \lambda_l > 0.$ We write $\lambda \vdash n$ when  $\sum_{i=1}^l \lambda_i=n$, and $n$ is called the size of $\lambda.$ To every partition  corresponds  a Young diagram <cit.>.
Let  $r$ be a positive integer and $\lambda = (\lambda^1,\ldots, \lambda^r)$ a $r$-tuple of partitions (Young diagrams), with $\lambda^1 \vdash n_1,\ldots, \lambda^r \vdash n_r$ ,  $\lambda$ is called an $r$-diagram. The sequence $(n_1,\ldots, n_r)$ is called the type $\lambda$ and denoted by $type(\lambda)$ and $n$ called the size of $\lambda$. The irreducible representations of $\Sc_{n_1} \times \cdots \times \Sc_{n_r}$ are indexed by the set of $r$-diagrams of type $(n_1,\ldots,n_r).$ By filling each "box" with a non-negative integer, we obtain an $r$-tableau from an $r$-diagram. The original $r$-diagram is called the shape of the $r$-diagram. An $r$-tableau $ T= (T^1,\dots, T^r)$ is said to be standard if the written sequence on each column and row of $T^i\ (1 \leq i \leq r)$ is strictly increasing, and each number from 1 to $n$ appears exactly once. The set of all standard $r$-tableaux of shape $\lambda$ is denoted by $ST(\lambda)$.
A standard $r$-tableau $ T= (T^1,\ldots,T^r)$ is said to be natural if and only if the set of numbers written in $T^{i}$ is $\{ n_1+\cdots +n_{i-1}+1, \ldots, n_1+\ldots+n_{i} \}$. The set of natural standard $r$-tableaux of shape $\lambda$ is denoted by $NST(\lambda)$.

For a standard $r$-tableau $T$, we associate a word  $w(T)$ in the following way.  First we read each column of the tableau $T^1$ from the bottom to the top starting from the left. We continue this procedure for the tableau $T^2$ and so on.
We define the index $i(w(T))$ of $w(T)$ as follows. The number 1 in the word $w(T)$ has index 0. If $k$ in the word has index $p$, then $k+1$ has index $p$ or $p+1$ according as it lies to the right or the left of $k$.
Assigning the index $i(w(T))$ to the corresponding of $w(T)$ to the corresponding box, we get a new $r$-tableau $i(T)$ which is called the index $r$-tableau of $T$.

§.§ Example

For   $n=8, r=2 \  n_1=5 ,\  n_2=3,\  \lambda^1=(3,2),\  \lambda^2=(2,1) $

$$  \lambda = \begin{pmatrix} \begin{Young}
&  &  \cr
&  \cr
\end{Young},
\ \begin{Young}
&  \cr
\cr
\end{Young}
\end{pmatrix}

$$  T =\begin{pmatrix}  \begin{Young}
1 &4  &6  \cr
2&7 \cr
\end{Young},
\ \begin{Young}
3 & 8 \cr
5 \cr
\end{Young}
\end{pmatrix}

$$  i (T)= \begin{pmatrix} \begin{Young}
$0$ & $2$ & $3$ \cr
$1$ & $4$ \cr
\end{Young},\ \begin{Young}
$1$ & $4$   \cr
$2$   \cr
\end{Young}
\end{pmatrix}

For words $u=(u_1,\ldots,u_n)$ and $v=(v_1,\ldots,v_n)$, we define $\displaystyle x_v^u= x_{v_1}^{u_1} \cdots x_{v_n}^{u_n}$. For standard $2$-tableaux $S, T,$ we define $x_{T}^{i(S)}= x_{w(T)}^{i(w(S))}$.

Let $ T=(T^1,\ldots, T^r)$ be a standard $r$-tableau of shape $\lambda$. For each component $T^i\ (1\leq i \leq r),$ The Young symmetrizer $\eb_{T^i}$ of $T^i$ is defined by

\begin{equation}
\eb_{T^i}= \frac{f^{\la_i}}{ {n_i}!} \sum_{\sigma \in R(T^i)\ \tau \in C(T^i)} \sgn (\tau) \tau \sigma \in \bC[\Sc_{n_i}],
\end{equation}

where $f^{\la_i}$ is the number of standard tableau of shape $\lambda^i$,  $R(T^i)$ and $C(T^i)$ are the row-stabilizer and colomn-stabilizer of $T^i,$ respectively.

We set
\begin{equation} \eb_{T}= \eb_{T^1} \cdots \eb_{T^r}. \end{equation}
For $T, S \in ST(\lambda)$, Ariki, Terasoma and Yamada have defined the higher Specht polynomial for $(T, S)$ in [3] by

\begin{equation}
F_{T}^{S} = F_{T}^{S}(x_1,\ldots,x_n)= \eb_{T} (x_{T}^{i(S)}).
\end{equation}

Let $\Pc_{r,n}$ be the set $r$-tuples of Young diagrams $\lambda$ of type $n$

For an $r$-diagram $\lambda$ of type $(n_1,\ldots,n_r)$ and  $S \in ST(\lambda)$, the set  $\{ F_{T}^{S} | T \in NST(\lambda) \}$ forms a $\bC$-basis of a $\bC[\Sc_{n_1} \times \cdots \times  \Sc_{n_r}]$-submodule denoted by $V^{S}(\lambda)$, which affords  an irreducible representation of  $\Sc_{n_1} \times \cdots \times  \Sc_{n_r}$ corresponding to $\lambda$. All the other irreducible representation of $\Sc_{n_1} \times \cdots \times  \Sc_{n_r}$ are obtained by same procedure.

§ DECOMPOSITION THEOREM

We are interested in studying the decomposition structure of $\pi_+(M)$, where $M=\Oc_X,\ \pi :X= \spec(\Oc_X)\to Y=\spec(\Oc_X^{ \Sc_{n_1} \times \cdots \times \Sc_{n_r}}) $. Since $\Oc_X$ is a holonomic $\D_X$-module <cit.>, $\pi_+(\Oc_X)$ is a semisimple $\D_Y$-module by the Kashiwara decomposition theorem. We  construct the simple components of  $\pi_+(\Oc_X)$ and provide their multiplicities.  Let us recall some useful facts from [21].

Let $\Delta:= Jac( (\pi)$ be the Jacobian of $\pi$, $\Delta^2$ the discriminant of $\pi$ we denote the complement of the branch locus and the discriminant by $U$ and $V$, respectively.
Assume now that $U,V$ are such that the respective canonical modules are generated by volume forms $dx$, and $dy$,  related by $dx=\Delta dy$, where  $\Delta$ is the Jacobian of $\pi$.

(i) There is an isomorphism of $\D_V$-modules
$$T:\pi_+(O_U)\cong O_U,\quad r(  dy^{-1}\otimes dx)\mapsto r \Delta ^{-1}.$$

$T(\pi_+(O_X))$ is isomorphic as a $\D_Y$-module to $\pi_+(\Oc_X)$.

See <cit.>.

It is more convenient to study $T(\pi_+(O_X))\cong \pi_+(O_X)$, as a submodule of $O_U$, than using the definition of $\pi_+(O_X)$. Therefore to reach our goal, we will first  study the decomposition of $O_U$ into irreducible components as a $\D_V$-module.

The following proposition enables us to reduce the study of the decomposition factors of $\pi_+(O_X)$
to the behavior of the direct image over the complement to the branch locus, or even over the generic point. Let $j:U\hookrightarrow X$  and $i:V\hookrightarrow Y$ be the inclusions.

Let $\pi: X\to Y$ be a finite map. Then

$\pi_+(O_X)$ is semi-simple as a $\D_Y$-module.

(ii) If $\pi_+(O_X)=\oplus M_i,\ i\in I$ is a decomposition into simple
(non-zero) $D_Y$-modules, then $\pi_+(O_U)=\oplus i^+(M_i),\ i\in I$, is a decomposition of  $\pi_+(O_U)$ into simple (non-zero) $\Dc_{V}$-modules.

See <cit.>.

§.§ Notation

Let $\displaystyle \Dc_X:=\bC \langle x_1, \ldots, x_n, \frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n} \rangle$ be the Weyl algebra associated with the polynomial ring $\Oc_X$, and $\Oc_Y:=\Oc_X^{\Sc_{n_1} \times \cdots \times \Sc_{n_r}}= \bC[y_1,\ldots,y_n]$ be the ring of invariant polynomials under the group $\Sc_{n_1} \times \cdots \times \Sc_{n_r}$. We denote by $\Dc_Y= \bC \langle y_1, \ldots, y_n, \frac{\partial}{\partial y_1}, \ldots, \frac{\partial}{\partial y_n} \rangle $ the Weyl algebra associated with  $\Oc_Y$.
We have
$\displaystyle \OO =\bC [x_1,\ldots,x_n, \Delta^{-1}],\  \displaystyle \Oc_{V} =\bC [y_1,\ldots,y_n, \Delta^{-2}],$ and
$$  \DD = \bC \langle y_1, \ldots, y_n, \frac{\partial}{\partial y_1}, \ldots, \frac{\partial}{\partial y_n}, \Delta^{-2} \rangle.$$

We adopt the following notations  for $i =1,\ldots, r$

$$ \Delta_i = \prod_{  n_1+ \cdots + n_{i-1} +1 \leq i<j \leq { n_1+ \cdots + n_i}} (x_i-x_j)$$

$$\displaystyle \Oc_{X_i} :=\bC [x_{ n_1+ \cdots + n_{i-1} +1}, \ldots, x_{ n_1+ \cdots + n_i} \Delta_i^{-1}],$$

$$\displaystyle \Oc_{Y_i} :=\bC [y_{ n_1+ \cdots + n_{i-1} +1}, \ldots, y_{ n_1+ \cdots + n_i} \Delta_i^{-2}],\ \mbox{and}$$

$$   \Dc_{Y_i}:= \bC \langle y_{ n_1+ \cdots + n_{i-1} +1}, \ldots, y_{ n_1+ \cdots + n_i}, \frac{\partial}{\partial y_{ n_1+ \cdots + n_{i-1} +1}}, \ldots, \frac{\partial}{\partial y_{ n_1+ \cdots + n_i}}, \Delta_i^{-2} \rangle.$$

Then  we have $ \OO=\Oc_{X_1} \otimes \cdots \otimes  \Oc_{X_r},$  $ \Oc_V=\Oc_{Y_1} \otimes \cdots \otimes  \Oc_{Y_r},$ $\DD = \Dc_{Y_1} \otimes \cdots \otimes  \Dc_{Y_r}$ and $\displaystyle \Delta= \prod \Delta_i.$

For $M_i$ a $\Dc_{Y_i}$-module $i=1,\ldots,r$, we make $M_1\otimes \cdots \otimes M_r$ into a $\DD $-module by setting

\begin{equation} (D_1 \otimes \cdots \otimes  D_r) (m_1 \otimes \cdots \otimes m_r)= D_1 m_1 \otimes \cdots \otimes D_r m_r \end{equation}

for $D_i \in \Dc_{Y_i}$ and $m_i \in \Oc_{X_i}, i=1\ldots,r.$

$\OO$ is a $\DD$-module.

Since by <cit.> $\Oc_{X_i}$ is a $\Dc_{Y_i}$-module for $i=1, \ldots, r$, it clearly  follows from equality (3.1).

Let $V_i $ be a $\bC[\Sc_{n_i}]$-module $(i=1,\ldots,r)$, we define the action $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_2}]$ on   $V_1\otimes \cdots \otimes V_r$  by

$$(s_1 \times \cdots \times s_r) (v_1 \otimes \cdots  \otimes  v_r)= s_1 v_1 \otimes  \cdots \otimes s_r v_r,$$
for $s_i \in \Sc_{n_i}, v_i \in V_i, 1,\ldots ,r.$
This makes $V_1\otimes \cdots \otimes  V_r$  into $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}].$ If $V_i $ is an irreducible $\bC[\Sc_{n_i}]$-module $(i=1,\ldots, r)$, then $V_1\otimes \cdots \otimes V_r$ is an irreducible  $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$-module and all irreducible  $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$-modules have this form<cit.>.

We   know  by  <cit.> that:

\begin{equation}  \bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]  \cong \bC[\Sc_{n_1}]  \otimes \cdots \otimes \bC[\Sc_{n_r}].
\end{equation}
A basis over $\bC$ of $ \bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$ is given
$\C=\{ F_{T}^{S}; S \in ST(\lambda), T \in NST(\lambda) | \lambda \in \Pc_{r,n} \}$. For every couple $(\la, S) \in \Pc_{r,n} \times ST(\la)$ corresponds an irreducible  $\Sc_{n_1} \times \cdots \times \Sc_{n_r}$-representation $V^S(\la).$  For $i=1,\ldots, r$ a basis over $\bC$ of $\bC[\Sc_{n_i}]$ is given by $\C_i=\{ F_{T^i}^{S^i}; S^i , T^i \in ST(\lambda) | \lambda \vdash n_i \}.$ For every couple $(\la_i, S^i) \in \Pc_{1,n_i} \times ST(\la_i)$ corresponds an irreducible  $\Sc_{n_i}$-representation $V^{S^i}(\la_i),\ i=1\ldots, r$,  [28].

For $T=(T^1 \ldots, T^r) \in NST(\la)$ and $S=(S^1, \ldots, S^r) \in ST(\la)$, define the linear map $\varphi: \bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}] \to  \bC[\Sc_{n_1}]  \otimes \cdots \otimes \bC[\Sc_{n_r}]$ by $\varphi (F_T^S)= F_{T^1}^{S^1}\otimes \cdots \otimes F_{T^r}^{S^r}.$ Then $\varphi$ is a $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$-isomorphism.
Moreover we have
$$\varphi(V^S(\la))= V^{S^1}(\la_1) \otimes \cdots \otimes V^{S^r}(\la_r).$$

It obvious that $\varphi$ is a $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$ which is a bijection.

From now we will use the map $\varphi$ in Lemma 3.4 to identify elements of $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}] $ with elements of $\bC[ \Sc_{n_1}] \otimes \cdots \otimes \bC[\Sc_{n_r} ].$

§.§ Simple components and their multiplicities

Let  $\lambda \in \Pc_{r,n}$ be an $r$-diagram of size $n$ and  $ T \in NST(\Lambda)$ a natural standard tableau and let the  $\eb_{T}$ be as in (2,2). The element $\eb_T$ is a primitive idempotents $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$ and each primitive idempotent of $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$ is associated with a natural standard tableau <cit.>. $\{ \eb_T ; T\in NST(\lambda), \lambda \in \Pc_{r,n}  \}$ is the complete list of all primitive idempotents of $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}].$

For $i =1,\ldots, r$ let $\lambda_i \vdash n_i$;  the canonical standard tableau $S_0^i$ of shape $\lambda_i$  is the unique  $\lambda_i$-tableau  whose  cells are numbered from the left to the right in successive rows, starting from the top. Let $T^i$ be a $\lambda_i$-standard tableau, we denote by $F_{T^i}$ the ordinary Specht polynomial associated with $T^i$ [24]. Then  the higher Specht polynomial  $F_{T^i}^{S_0^i}$ is proportional to the Specht polynomial $F_{T^i}$  [28] .

The following theorem is the analog  of <cit.> for product of symmetric groups.

Let   $\lambda \in \Pc_{r,n}$ be an $r$-diagram of size $n$, $T \in NST(\lambda)$ a natural standard tableau of shape $\lambda$, and  $\eb_{T}$ is the  primitive idempotent associated with $T$. Then we have :

* $ \eb_{T}\OO$ is a nontrivial  $\DD$-submodule of $\OO,$

* The $\DD$-module $\eb_{T}\OO$ is simple,

* There exist a  $ S  \in ST(\lambda)$ and a higher  Specht polynomial $F_{T}^{S}$ for $(T, S)$   such that $\eb_{T} \OO= \DD F_{T}^{S}.$

Let $\lambda \in \Pc_{r,n}$ be an $r$-diagram of size $n$ and  $T \in NST(\lambda)$ , There exist $n_1, \ldots, n_r \in \mathbb N, \lambda^i \vdash n_i,$ and  $T_i\in ST(\lambda_i),\ i=1,\ldots,r$ such that
$$ \sum n_i=n,\ \lambda= (\lambda^1, \ldots, \lambda^r) \ \mbox{and}\  \eb_{T}= \eb_{T^1} \cdots \eb_{T^r}.$$

* We know that $e_{T^i}$ is an primitive idempotent for $\Cb[\Sc_{n_i}], i=1,\ldots,r$ and    $\eb_{T} \OO = (\eb_{T^1} \cdots \eb_{T^r} )\OO= \eb_{T^1} \Oc_{X_1} \otimes \cdots \otimes \eb_{T^r} \Oc_{X_r}.$ By <cit.>,  $\eb_{T^i} \Oc_{X_i}$  is a nontrivial  $\Dc_{Y_i}$ module for $i=1,\ldots,r$. Hence $\eb_{T} \OO$ is a nontrivial module over $\DD.$

* Since $e_{T^i}$  being a primitive idempotent for $\Cb[\Sc_{n_i}], i=1,\ldots, r$, by <cit.>, we have that $\eb_{T^i} \Oc_{X_i}$ is a simple $\Dc_{Y_i}$-modules for $i=1,\ldots,r$. Then $\eb_{T^1} \Oc_{X_1} \otimes \cdots \otimes \eb_{T^r} \Oc_{X_r}$ is an irreducible $\Dc_{Y_1} \otimes \cdots \otimes  \Dc_{Y_r}$-module. Hence $\eb_{T} \OO$ is a simple $\DD$-module.

* Let   $S_0^i$ be the canonical standard tableau of  shape $\lambda_i, i=1,\ldots, r$ and we know that  the higher Specht polynomial $F_{T^i}^{S_0^i},$  is proportional to the Specht polynomial $F_{T^i}$ of $T^i,\ i=1,\ldots, r.$ Then  By <cit.> we have that  $\eb_{T^i} \Oc_{X_i}=  \Dc_{Y_i} F_{T^i}, i=1,\ldots,r$ so that,

\begin{eqnarray*}
\eb_{T} \OO&=&\eb_{T^1}  \Oc_{X_1} \otimes \cdots \otimes  \eb_{T^r} \Oc_{X_r} \\
&=& \Dc_{Y_1} F_{T^1} \otimes \cdots \otimes    \Dc_{Y_r} F_{T^r} \  \mbox{by \cite[Theorem 3 (iii)]{RTG}} \\
&=&   \Dc_{Y_1} F_{T^1}^{S_0^1} \otimes \cdots \otimes    \Dc_{Y_r} F_{T^r}^{S_0^r} \\
&=& (\Dc_{Y_1} \otimes \cdots \otimes  \Dc_{Y_r} ) ( F_{T^1}^{S_0^1} \otimes \cdots \otimes   F_{T^r}^{S_0^r}) \\
&=& \DD F_{T}^{S_0} \  \  \mbox{by\ the identification\ map\ Lemma\ 3.3}
\end{eqnarray*}

$\mbox{where}\  T=(T^1, \ldots, T^r) \  \mbox{and}\  S_0=(S_0^1, \ldots, S_0^r).$

From now we adopt the following notation.
Let  $\lambda$ be an $r$ -diagram of size $n$, $ T \in NST(\lambda) $, put $F_T:=F_T^{S_0}$ where $S_0=(S_0^1, \ldots, S_0^r)$ so that $\eb_T \OO =\DD F_T.$ We denote by $F_{\lambda}$ the unique higher Specht polynomial $F_{S_0}^{S_0}.$

With the above notations, $\eb_{T_1}\OO \cong_{\DD} \eb_{T_2} \OO$ if $T_1$ and $T_2$  have the same shape i.e. if there is a $r$-diagram $\lambda $ of size $n$ such that $ T_1, T_2 \in NST(\lambda ).$

Let $T_1, T_2 \in NST(\lambda)$ with $T_1= (T_1^1,\ldots,T_1^r)$ and $ T_2=(T_2^1, \ldots, T_2^r)$. Then $\eb_{T_1}= \eb_{T_1^1} \cdots \eb_{T_1^r}$ and $\eb_{T_2}= \eb_{T_2^1} \cdots \eb_{T_2^r}$, so that $ \eb_{T_k}\OO  \cong \eb_{T_k^1} \Oc_{X_1} \otimes \cdots \otimes  \eb_{T_k^r} \Oc_{X_r},$ $k=1,2.$ By <cit.>, we know that
$\eb_{T_1^i} \Oc_{X_i} \cong_{\Dc_{Y_i}} \eb_{T_2^i} \Oc_{X_i}$ if $T_1^i$ and $T_2^i$  have the same shape, $i=1,\ldots,r$. Hence $\eb_{T_1}\OO \cong_{\DD} \eb_{T_2} \OO$ if $T_1$ and $T_2$  have the same shape.

Let  $\lambda=(\la^1, \ldots, \la^r)$ be an $r$ -diagram of size $n$, $ T \in NST(\lambda) $, and $\eb_{T}$ the  primitive idempotent associated  with $T$
Then with the notation above, we have:

\begin{equation}
\displaystyle \OO=  \bigoplus_{\lambda \in \Pc_{r,n}} \bigg( \bigoplus_{{ T\in NST(\lambda)}}  \DD F_{T} \bigg),
\end{equation}

\begin{equation}
\displaystyle \OO \cong  \bigoplus_{\lambda \in \Pc_{r,n}}    f^\lambda \DD F_{\lambda},
\end{equation}

where $\displaystyle f^\la = \dim_{\bC} (V^{S_0}(\la)).$

* Since  $ \displaystyle 1= \sum_{ \lambda \in \Pc_{r,n}} \sum_{ T \in NST(\lambda)}  \eb_{T}$, we have   $\displaystyle\OO =\sum_{\lambda \in \Pc_{r,n}}\sum_{ T \in NST(\lambda)}  \eb_{T} \OO$. Let $m \in \eb_{T_1} \OO \cap \eb_{T_2} \OO$ with $T_1 \neq T_2$ so that $m = \eb_{T_1} m_1$ and $ m=\eb_{T_2} m_2.$ Then $\eb_{T_1} m= \eb_{T_1} \eb_{T_2} m_2 =0$, hence $m=0$. It is clear that $ \eb_{T_1} \OO \cap \eb_{T_2} \OO = \{0\}$ and

$$  \OO = \bigoplus_{\lambda } \bigg( \bigoplus_{{T\in NST(\lambda)}} \eb_{T} \OO \bigg),$$

where the $\eb_{T} \OO$ are simple $\DD$-modules. Since to each an $r$-tableau ${T}$ corresponds a higher Spect polynomial  $F_T$ such  that $\eb_{T} \OO= \DD F_{T}$ then $\displaystyle \OO= \bigoplus_{\lambda \in \Pc_{r,n}} \bigg( \bigoplus_{{T\in NST(\lambda)}}  \DD F_T \bigg).$

* By Corollary 3.6, $ \DD F_{T_j} \cong  \DD F_{T_j}$ if $T_i,T_j \in NST(\lambda)$ for some $\lambda \in \Pc_{r,n}$ and so we have $f^\la$ isomorphic copies of $ \DD F_{\la}$ in the direct sum (3.3).

Using Proposition 3.1 and Proposition 3.2 we get the next theorem.

(i) $N_T:=\Dc_Y F_T$ is an irreducible $D_Y$-submodule of $\pi_+( \Oc_X)$.

(ii) There is a direct sum decomposition

\begin{equation}
\label{eq:direct sum}
\pi_+ (\Oc_X)=\bigoplus_{\lambda \in \Pc_{r,n} }\bigoplus_{T\in NST(\lambda)}N_T
\end{equation}

We get in Theorm 3.8 a decomposition of the $\pi_+(\Oc_X)$ into irreducible $\Dc_Y$ modules generated by the higher Specht polynomials.

§.§ Using correspondence between $G$-representations and $D$-modules

Recall that if $M$ is a semi-simple module over a ring $R$, and $N$ is simple $R$-module, then the isotopic component of $M$ associated with $N$ is the sum $\sum N' \subset M$ of all $N'\subset M$ such that $N' \cong N.$

For $i=1,\ldots,r$, let $V({\lambda^i})$ be the Specht module corresponding to the partition $\lambda^i \vdash n_i$, $T^i$ a $\lambda^i$-standard tableau and  $M_i:=\Oc_{X_i}$ and $M_i^{\lambda^i}$ the isotopic component of $M_i$ (as $\Oc_{Y_i}$-module) associated with $V({\lambda^i})$. Then

(i)  $\eb_{T^i}(V({\lambda^i}))= \{ \eb_{T^i}(m) | m \in V({\lambda^i} )\}$  is a one dimensional $\bC$-vector space.

(ii)  $M_i^{\lambda^i}$ is $\Dc_{Y_i} $-module.

(iii) $\eb_{T^i} (M_i^{\lambda^i})$ si a $\Dc_{Y_i}$-module

(i) In fact we have

\begin{eqnarray*}
\eb_{T^i} V({\lambda^i}) \cong \eb_{T^i} \bC[\Sc_{n_i}] \eb_{T^i} \cong  \bC \eb_{T^i} \  \mbox{by \ \cite[Theorem 3.9]{Boerner}}
\end{eqnarray*}

(ii) We only have to prove that $\Dc_{Y_i}  M_i^{\lambda^i} \subset M_i^{\lambda^i}.$ Let $ D \in \Dc_{Y_i}$ and $N$  be a $\bC[\Sc_{n_i}]$-module isomorphic to $V({\lambda^i})$, since by <cit.> $D$ commute with the elements of the group algebra $\bC[\Sc_{n_i}]$, $D$ is an $\bC[\Sc_{n_i}]$-homomorphism from $N$ into $D(N)$. Then by virtue of the Schur lemma $D(N)=0$ or $D(N)\cong N$ as a $\bC[\Sc_{n_i}]$-module, and $D(N) \subset M_i^{\lambda^i}$. Hence $\Dc_{Y_i}  M_i^{\lambda_i} \subset M_i^{\lambda^i}.$

(iii) Let  $D \in \Dc_{Y_i},$ we have $ D(\eb_{T^i}(M_i^{\lambda_i})) = \eb_{T^i} (D(M_i^{\lambda^i})) \subset \eb_{T^i} (M_i^{\lambda^i})$, so that  $\eb_{T^i} (M_i^{\lambda^i})$ is a $\Dc_{Y_i}$-module.

Let us recall  the correspondence between $G$-representations and $D$-modules <cit.>.
Let $L$ and $K$ be  two extensions fields a field $k$, denote by $T_{K/k}$ the $k$-linear derivations of $K$. We  say that a $T_{K/k}$-module $M$
is  $L$-trivial  if $L\otimes_KM \cong L ^n $ as
$T_{L/k}$-modules. Denote by  $\Mod^{L}(T_{K/k})$ the full subcategory of finitely generated $T_{K/k}$-modules that are $L$-trivial.
It is immediate that it is closed under taking submodules and quotient modules. Using a lifting $\phi$ , $L$ may be thought of as a $T_{K/k}$-module.
If $G$ is a finite group let $ \Mod (k[G]) $ be the category of finite-dimensional representations of $k[G]$.
Let now $k\to K \to L$ be a tower of fields such that $K= L^G$.
Note that the action of $T_{K/k}$ commutes with the action of $G$. If $V$ is a $k[G]$-module,  $L\otimes_kV$ is a $T_{K/k}$-module by $D(l\otimes v)=D(l)\otimes v,\quad D\in T_{K/k}$, and  $(L\otimes_kV)^G$ is a $T_{K/k}$-submodule.

The functor

∇:     (k[G]) →(T_K/k),   V↦(L⊗_k V)^G

is fully faithful, and defines an equivalence of
$$ \Mod (k[G]) \to \Mod^{L}(T_{K/k}).$$
The quasi-inverse
of $\nabla$ is the functor

Loc : ^L(T_K/k) →(k[G]),  Loc (M) = (L⊗_K M)^ϕ(T_K/k).

see <cit.>

In the following proposition we take $G=\Sc_{n_i}$, $ K $ the field of fractions of $\Oc_{Y_i}$ and $L$ the field of fractions of $\Oc_{X_i}$ so that $K=L^{\Sc_{n_i}}.$ It is clear that $L$ is a Galois extension of $K$ with Galois $\Sc_{n_i}$, $i=1,\ldots,r$.

For $i=1,\ldots, r$, let $T^i$ be a $\lambda^i$-standard tableau where $\lambda^i \vdash n_i$,  $M_{T^i}:=\eb_{T^i} \Oc_{X_i}$, $V(\lambda^i):=V^{S_0^i} (\lambda^i)$ and $ST(n_i) = \bigcup_{\lambda^i \vdash n_i} ST(\lambda^i).$ Then we have:

* $M_{T^i}= \nabla(V({\lambda^i}))$,

* $M_{T^i}= \eb_{T^i} (M_i^{\lambda^i})$ is simple $\Dc_{Y_i}$-module;

* $M_i^{\lambda^i} =\displaystyle \bigoplus_{T^i \in ST(n_i)} \eb_{T^i}(M_i^{\lambda^i}).$

* Let us consider  the right $\bC[\Sc_{n_i}]$-module $V=\eb_{T^i} \bC[\Sc_{n_i}]$ where $T^i$ is a $\lambda^i$-standard tableau. This is the image of $\bC[\Sc_{n_i}]$ by right multiplication map $\eb_{T^i}: \bC[\Sc_{n_i}] \to \bC[\Sc_{n_i}]$.  By <cit.>, we may turn this map into a left multiplication $ \bC[\Sc_{n_i}]^r \to \bC[\Sc_{n_i}]^r$ and get an image which is isomorphic to $V({\lambda^i})$. Then we have an induced map

$$ \nabla (\bC[\Sc_{n_i}]^r ) \to \nabla (V({\lambda^i})) \subset \nabla (\bC[\Sc_{n_i}]^r ),$$
which is a multiplication by $\eb_{T^i}$ according to <cit.>. Then $\nabla(V({\lambda^i}))$ is egal to $\eb_{T^i} \Oc_{X_i} = M_{T^i}.$

* Since $V({\lambda^i})$ is a simple $\bC[\Sc_{n_i}]$-module, $\nabla (V({\lambda^i}))$ is also a simple $\Dc_{Y_i}$-module.

* follows from the fact that $1=\sum_{T \in ST(n_i)} \eb_{T}$ and $\eb_{T} (M^{\lambda^i})=0 $ if $T$ is not a $\lambda^i$-tableau.

For $i=1,\ldots, r$, let $T^i$ be a $\lambda^i$-standard tableau where $\lambda^i \vdash n_i$,  let $M_{T^i}:=\eb_{T^i} \Oc_{X_i}$.

* $ M_{T^i}=\displaystyle \bigoplus_{ S^i\in ST(\lambda^ii)}\Oc_{Y_i}  F_{T^i}^{S^i}$ as $\Dc_{Y_i}$-module,

* $\Oc_{X_i} = \displaystyle \bigoplus_{\lambda^i \vdash n_i}  \bigg( \bigoplus_{ S^i, T^i\in ST(\lambda^i)}\Oc_{Y_i}  F_{T^i}^{S^i} \bigg) $ as a $\Dc_{Y_i}$-module.

* For a fixed $S^i \in ST(\lambda^i)$, we know that the polynomial $F_{T^i}^{S^i}$ generate a cyclic  $\bC[\Sc_{n_i}]$-module inside $\Oc_{X_i}$ which is isomorphic to $V({\lambda^i}).$ Then $F_{T^i}^{S^i}\in M_i^{\lambda^i}$ and $M_i^{\lambda^i}= \displaystyle \bigoplus_{S^i,T^i \in ST(\lambda^i)} \bC[\Sc_{n_i}] F_{T^i}^{S^i} \Oc_{Y_i}$ by [28]. Moreover $\eb_{T^i}( F_{T^i}^{S^i})= c F_{T^i}^{S^i}, c\in \bC$ and by Lemma 3.9  $\eb_{T^i} (\bC[\Sc_{n_i}] F_{T^i}^{S^i})=\bC F_{T^i}^{S^i}.$ Hence $M_{T^i}=\eb_{T^i}(M_i^{\lambda^i}) =\displaystyle \bigoplus_{S^i \in ST(\lambda^i)} \Oc_{Y_i}  F_{T^i}^{S^i}.$

* follows from Proposition 3.11 and  <cit.>.

Let  $\lambda \in \Pc_{r,n}$ be an $r$-diagram of size $n$, $T \in NST(\lambda)$ and $M_{T}:=\eb_{T} \OO$. Then

* $ M_{T}=\displaystyle \bigoplus_{ S\in ST(\lambda)} \Oc_{V}  F_{T}^{S}$ as $\DD$-module,

* $\OO = \displaystyle \bigoplus_{\lambda \in \Pc_{r,n}}  \bigg( \bigoplus_{ S \in STab (\lambda) T\in NSTab(\lambda)}\Oc_{V}  F_{T}^{S} \bigg) $ as a $\DD$-module.

Suppose  that $\lambda= (\lambda^1, \ldots, \lambda^r),  T=(T^1,\ldots, T^r)$, with $\lambda_i \vdash n_i, T^i \in ST(\lambda^i), i=1,\ldots, r$ and $ \sum n_i=n$
We have that

\begin{eqnarray*}
M_{T} &=& \eb_{T} \OO \\
&=& (\eb_{T^1} \times \ldots \times \eb_{T^r}) (\Oc_{X_1} \otimes \cdots \otimes \Oc_{X_r}) \\
&=& \eb_{T^1} \Oc_{X_1} \otimes \cdots \otimes  \eb_{T^r} \Oc_{X_r} \\
&=& M_{T^1} \otimes \cdots \otimes  M_{T^r} \\
&=&\displaystyle (\bigoplus_{ S^1\in ST(\lambda^1)} \Oc_{Y_1}  F_{T^1}^{S^1}) \otimes \cdots \otimes ( \bigoplus_{ S^2\in ST(\lambda^r)} \Oc_{Y_r}  F_{T^r}^{S^r} ) \  \mbox{by\ Proposition 3.12}\\
&=& \displaystyle \bigoplus_{ S^i\in ST(\lambda^i)} \bigg( \Oc_{Y_1}   \otimes \cdots \otimes \Oc_{Y_r} \bigg) \bigg(F_{T^1}^{S^1} \otimes \cdots \otimes  F_{T^r}^{S^r} \bigg)\  \\
&=& \displaystyle \bigoplus_{ S \in ST(\lambda)} \Oc_{V}  F_{T}^{S} \   \mbox{by\ Lemma 3.4}\  \mbox{with}\  S= (S^1,\ldots ,S^r).
\end{eqnarray*}

* follows from the fact that $\OO =\Oc_{X_1} \otimes \cdots \otimes \Oc_{X_r}$ and

$\Oc_{X_i}= \displaystyle \bigoplus_{\lambda^i \vdash n_i}  \bigg( \bigoplus_{ S^i, T^i\in ST(\lambda^i)} \Oc_{Y_i}  F_{T^i}^{S^i} \bigg)$

§.§  Invariant differential operators and higher Specht polynomials for the symmetric group

In this subsection we investigate the action of invariant differential operators on higher Specht polynomials.
Let $\lambda \vdash n$,  $T$ a $\lambda$-tableau, and let  $C(T)$  be column stabilizer of $T,$ by  <cit.>  we know that  for every derivation  ${\bf D}$   such that ${\bf D}(F_T) \neq 0$ then there exists a polynomial  $G$ in  $\mathbb C[x_1,\ldots,x_n]^{C(T)}$, the polynomial ring invariant under the subgroup $C(T)$, such that ${\bf D}(F_T) =F_T G$, we will show that this is also true for the higher Specht polynomials.

For $i=1,\ldots, r$, let $\lambda^i \vdash n_i$,  $S^i\in  ST(\lambda^i)$ and $T^i \in ST(\lambda^i)$,  we have that
$ \mbox{for\ all}\  \sigma \in C(T^i), \sigma(F_{T^i})= \sgn(\sigma) F_{T^i}$ and $\sigma(F^{S^i}_{T^i})= \sgn(\sigma) F^{S^i}_{T^i}.$

For $i=1,\ldots, r$, let $\lambda^i \vdash n_i,$ $T^i, S^i \in  ST(\lambda^i).$

Then there exists a polynomial $\displaystyle G \in \Oc_{X_i}^{C(T^i)}$ such that $ F_{T^i}^{S^i} =F_{T^i} G.$

Let  us consider the linear application $\varphi : V(\lambda^i) \to V^{S^i}(\lambda^i)$ defined by $\varphi (F_{T^i})= F_{T^i}^{S^i}$. For every $\sigma \in \bC[\Sc_{n_i}],$ we have that $\varphi(\sigma F_{T^i})= \sigma \varphi (F_{T^i})$, so that  $\varphi$ is a $\bC[\Sc_{n_i}]$-homomorphism  and by the Schur' lemma  $\varphi$ is a $\bC[\Sc_{n_i}]$-isomorphism. Suppose that $x_k$ and $x_l$ occur in the same column of $T^i$, and let $\pi= (k,l)$ the transposition of $k$ and $l$. Then

$$\pi F_{T^i}^{S^i}=\pi \varphi (F_{T^i}) = \varphi(\pi F_{T^i})= \varphi(-F_{T^i})= -F_{T^i}^{S^i}.$$

This implies that $(x_k-x_l)$ is a factor of $F_{T^i}^{S^i}.$ This holds for each linear factor of $F_{T^i}$, so that $F_{T^i}$ divides $F_{T^i}^{S^i}$. Hence there exists a polynomial  $\displaystyle G \in \mathbb C[x_{n_1+\cdots +n_{i-1}+1},\ldots,x_{n_1 +\cdots +n_i}] $ such that $F_{T^i}^{S^i}= F_{T^i} G $.
Let now $ \sigma \in C(T^i)$, we get
$$ \sigma G=\sigma \bigg( \frac{F_{T^i}^{S^i}}{F_{T^i}}\bigg)  =\frac{\sigma F_{T^i}^{S^i}}{\sigma F_{T^i}}= \frac{\sgn \sigma F_{T^i}^{S^i}}{\sgn \sigma F_{T^i}}= G.$$

Then $ \displaystyle G \in \mathbb C[x_{n_1+\cdots +n_{i-1}+1},\ldots,x_{n_1 +\cdots +n_i}]^{C(T^i)}.$

For $i=1,\ldots, r$, let $\lambda^i \vdash n_i,$  $ T^i, S^i \in  ST(\lambda^i),$

and ${\bf D}$ a derivation in $\Dc_{Y_i}$ such that ${\bf D}(F_{T^i}^{S^i}) \neq 0$. Then there exists a polynomial  $G \in \Oc_{X_i}^{C(T)}$ such that ${\bf D}(F_{T^i}^{S^i}) =F_{T^i}G.$

Let ${\bf D}$ a derivation in $ \Dc_{Y_i}$, we have that

\begin{eqnarray*}
{\bf D} (F_{T^i}^{S^i})
&=&  {\bf D}(F_{T^i} G') \  \mbox{where}\ G' \mbox{is\ a\ polynomial\ in } \Oc_{X_i}^{C(T^i)} \mbox{by\ Lemma 3.14}  \\ %\\ \in \mathbb C[x_1,\ldots,x_n]^{C(T)} \  }\\
&=&  {\bf D}(F_T) G' + F_T {\bf D}(G') \\
&=&F_T G''G' + F_T {\bf D}(G') \  \mbox{where}\ G''\in \Oc_{X_i}^{C(T^i)} \  \mbox{by \cite[Lemma 4.4]{DOS}}\\
&=&F_T (G'' G' +  {\bf D}(G') ) \  \mbox{with}\ G', G''  \in \Oc_{X_i}^{C(T^i)}.
\end{eqnarray*}

Now let $ \pi \in {C(T^i)}$ we have

\begin{eqnarray*}
\pi (G''G' +  {\bf D}(G'))&=& \pi (G'')\pi(G') + \pi {\bf D}(G') \\
&=& G''G' +  {\bf D}(\pi G')\ \mbox{since}\ G', G''  \in \Oc_{X_i}^{C(T^i)} \\
&=& G''G' +  {\bf D}(G')\
\end{eqnarray*}

Then   $G''G' +  {\bf D}(G') \in \Oc_{X_i}^{C(T^i)}.$ Set $G= G''G' +  {\bf D}(G')$ and we get  ${\bf D}(F_{T^i}^{S^i}) =F_{T^i}G.$

let $\lambda \in \Pc_{r,n}$ be an $r$-diagram, $T, S \in ST(\lambda)$ and ${\bf D}$ a derivation in $\DD$ such that ${\bf D}(F_{T}^{S}) \neq 0$. Then there exists a polynomial  $G \in \OO^{C(T)}$, where $C(T)= C(T^1) \times \cdots \times  C(T^r)$ such that ${\bf D}(F_{T}^{S}) =F_{T}G.$

For $i=1\ldots r$, there exists a derivation ${D}_i \in \Dc_{Y_i}$ with $D_i(F_{T^i}^{S^i}) \neq 0,$ such that $ {\bf D}= D_1 \otimes \cdots \otimes D_r.$ Then

\begin{eqnarray*}
{\bf D} (F_T^S)  &=& (D_1 \otimes \cdots \otimes D_r) (F_T^S) \\
&=& (D_1 \otimes \cdots \otimes D_r) (F_{T^1}^{S^1} \otimes \cdots \otimes F_{T^r}^{S^r}) \mbox{by\ Lemma 3.4}\\
&=& D_1(F_{T^1}^{S^1}) \otimes \cdots \otimes D_r(F_{T^r}^{S^r})  \\
&=& F_{T^1} G_1 \otimes \cdots \otimes F_{T^r}G_r\  \mbox{where}\ G_i \in \Oc_{X_i}^{C(T^i)}\  \mbox{by Lemma 3.15} \\
&=& (F_{T^1} \otimes \cdots \otimes F_{T^r})( G_1 \otimes \cdots \otimes G_r)\  \mbox{with}\ G_1\otimes \cdots \otimes G_r \in \OO^{C(T)} \\
&=& F_T G\  \mbox{where}\  G= G_1 \otimes \cdots \otimes G_r\  \mbox{by\ Lemma 3.4}
\end{eqnarray*}

For $i=1,\ldots, r$, let $\lambda^i \vdash n_i,$  $ T^i, S^i \in  ST(\lambda^i)$ and ${\bf D} \in \Dc_{Y_i}$ such that ${\bf D}(F_{T^i}^{S^i}) \neq 0$ for $S^i, T^i \in ST(\lambda^i)$. Then the image of the $\bC[\Sc_{n_i}]$-module $V^{S^i}({\lambda^i})$  by ${\bf D}$ is an $\bC[\Sc_{n_i}]$-module  isomorphic to $V^{S^i}({\lambda^i})$.

Let  $\lambda^i \vdash n_i, {\bf D} \in \Dc_{Y_i}$ such that ${\bf D}(F_{T^i}^{S^i}) \neq 0$ for $S^i,T^i \in ST(\lambda^i)$ and set $W^{S^i}_{\bf D}({\lambda^i}):= {\bf D} (V^{S^i}({\lambda^i}))$ the image of the  module $V^{S^i}({\lambda^i})$  under the map ${\bf D}$. Since the $\bC$-vector space $V^{S^i}({\lambda^i})$ is  equipped with a basis $\C^{S^i}(\lambda^i) = \{ F_{T^i}^{S^i}; T^i\in ST (\lambda^i) \}$,  $W^{S^i}_{\bf D}({\lambda^i})$ is the vector space  spanned by the set $\{ {\bf D}(F_{T^i}^{S^i});\  T^i\in ST(\lambda^i) \}$. The elements of $\{ F_{T^i}^{S^i} ; T^i\in ST(\lambda^i) \}$ are linearly independent over $\Dc_{Y_i}$, otherwise the direct sums in  Proposition 3.12 cannot hold. It follows that the elements  in $\{ {\bf D}(F_{T^i}^{S^i});\ T^i \in ST(\lambda^i) \}$ are linear independent over $\bC$.  Hence $\{ {\bf D}(F_{T^i}^{S^i}) ;\  T^i \in ST(\lambda^i) \}$ is a basis of $W^{S^i}_{\bf D}({\lambda^i})$ over $\bC$. Since ${\bf D}$ commute with  elements of $\bC[\Sc_{n_i}],\ W^{S^i}_{\bf D}({\lambda^i})$ is an $\bC[\Sc_{n_i}]$-module  isomorphic to $V^{S^i}({\lambda^i}).$

Let  $\lambda \in \Pc_{r,n}$ be an $r$-diagram of size $n$, $T \in NST(\lambda)$  and ${\bf D} \in \DD$ such that ${\bf D}(F_{T}^{S}) \neq 0$ for $S \in ST(\lambda)$. Then the image of the $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$-module $V^{S}({\lambda})$  by ${\bf D}$ is a $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$-module  isomorphic to $V^{S}({\lambda})$. In others words, the action of the differential operators of $\DD$ on the higher Specht polynomials generate  isomorphic copies of the corresponding  module.

Let  $\lambda$ be an $r$-diagram of size $n$, $T \in NST(\lambda)$  and ${\bf D} \in \DD$ such that ${\bf D}(F_{T}^{S}) \neq 0$ for $S \in ST(\lambda)$. Then ${\bf D}$ may be written as ${\bf D}= { D_1}\otimes \cdots \otimes  { D_r}$ where ${ D_i} \in \Dc_{Y_i},
\ i=1,\ldots, r $.

{\bf D} (F_{T}^{S}) = ({ D_1}\otimes \cdots \otimes  { D_r}) (F_{T^1}^{S^1} \otimes \cdots \otimes  F_{T^r}^{S^r})
=  D_1(F_{T^1}^{S^1}) \otimes \cdots \otimes  D_r(F_{T^r}^{S^r})  \neq 0,$$

so that  ${ D_i} F_{T^i}^{S^i} \neq 0,\ i=1\ldots, r.$ Then by Proposition 3.17, ${ D_i} F_{T^i}^{S^i}$  generate a $\bC[\Sc_{n_i}]$-module isomorphic to $V^{S^i} (\lambda^i),\ i=1,\ldots, r$. Hence $D_1(F_{T^1}^{S^1}) \otimes \cdots \otimes  D_r(F_{T^r}^{S^r}) $ generate a $\bC[\Sc_{n_1} \times \cdots \times \Sc_{n_r}]$-module isomorphic to $V^{S^1} (\lambda^1) \otimes \cdots \otimes  V^{S^r} (\lambda^r) \cong  V^{S}({\lambda})$ by LEmma 3.4.

[3]
Ariki, S.  Terasoma, T. and Yamada, H.,
Higher Specht polynomials,
Hiroshima Math, 27  (1997),
no. 1, 177-188.

[7]
Representations of Groups with special Consideration for the Needs of Modern Physics, North-Holland, New York, NY, (1970).

[8]
de Cataldo, M. A.A and  Migliorini. L.,
The decomposition theorem, perverse sheaves and the topology
of algebraic maps,
Bull. Amer. Math. Soc. (N.S.)  46 (2009), no. 4, 535-633.

[10]
Coutinho, S. C.,
A primer of algebraic $D$-modules,
Cambridge University Press,

[11]
Curtis, W,C  and Reiner, I.,
Representation Theory of Finite groups and Associative Algebras ,
New York,

[12]
Endelman, R. and Mukherjee, M.,
Primitive central idempotents of the group algebra.,
arXiv:0803.1336v2 [math.RT]

[19]
Milies, P.C. and Sehgal, K.S,
An introduction to group rings,
volume 1 of Algebras and Applications.
Kluwer Academic Publishers,

[20]
Nonkané, I.,
Differential operators and the symmetric groups,
Journal of Physics: Conference Series 1730(1):012129,

[21]
Nonkané, I.,
Specht polynomials and modules over the Weyl algebra,
Afr. Mat,
30, Issue 11 pp 279-290

[22]
Nonkané, I.,
Representation Theory of Groups and $\mathcal{D}$-Modules,
International Journal of Mathematics and Mathematical Sciences, vol. 2021, Article ID 6613869,
7 pages,

[23]
Nonkané, I. and Todjihounde, L.,
Differential operators and higher Specht polynomials.,
Proceedings of 10th International Conference on Mathematical Modeling in Physical Science (2021),
accepted to  appear in J of phys.: confer.ser.

[24]
Peel, M.H,
Specht modules and symmetric groups ,
J. Algebra,
no. 1, 88-97

[25]
Sagan, Bruce. E.,
The symmetric group. Representations, combinatorial
algorithms, and symmetric functions,
Second edition. Graduate Texts in Mathematics,
New York,

[28]
Terasoma, T. and Yamada, H.,
Higher Specht polynomials for the symmetric group,
Proc Japan Acad,

[30]
Weyl, Hermann,
The theory of groups  and quantum mechanics,
New York