# Invariants and dualities of a certain parabolic group

Bin Liu  School of Mathematical Sciences, East China Normal University,
Shanghai, 200241, China<EMAIL_ADDRESS>
###### Abstract.

In this note, I will prove a conjecture in [1], which is related to the
invariants of a maximal parabolic subgroup of $\text{GL}_{n+1}$. Consequently,
the natural tensor invariants of this typical maximal parabolic subgroup of
$\text{GL}_{n+1}$ are determined when $n\geq r$.

###### Key words and phrases:

enhanced group, Schur-Weyl duality, degenerate double Hecke algebra

######  Mathematics Subject Classification:

This work is partially supported by the NSF of China (Grant: 12071136).

## 0\. Introduction

The purpose of this note is to prove Conjecture (0.1) proposed in [1], which
is actually much related to tensor invariants of an distinguished maximal
parabolic subgroup of $\text{GL}_{n+1}$. The authors in [1] defined an
enhanced reductive algebraic group $\underline{G}$, which is naturally a semi-
reductive group. In order to illustrate the tensor invariants of
$\underline{\text{GL}_{n}}=\text{GL}_{n}\ltimes\mathbb{C}^{n}$ in
$\text{End}_{\mathbb{C}}(\underline{V}^{\otimes r})$ with
$\underline{V}=\mathbb{C}^{n+1}$, they introduced a finite-dimensional
degenerate double Hecke algebra $D(n,r)$, and got a Levi Schur-Weyl duality
for $\underline{V}=\mathbb{C}^{n+1}$ in [1] (see [1, Theorem 4.3]) which is an
analog of classical Schur-Weyl duality.

By [1, Example 5.2],
$\mathbb{C}\Psi(\mathfrak{S}_{r})\subsetneq\text{End}_{\underline{\text{GL}_{n}}\rtimes\textbf{G}_{m}}(\underline{V}^{\otimes
r})$ when $n<r$. However in the case $n\geq r$, there is a conjecture

$\text{End}_{\underline{\text{GL}_{n}}\rtimes\textbf{G}_{m}}(\underline{V}^{\otimes
r})=\mathbb{C}\Psi(\mathfrak{S}_{r})$ (0.1)

(see [1, Conjecture 5.4]). In §2 of this article, I will give a proof of
(0.1). Then it is a position to investigate the enhanced tensor invariants,
and this is listed in §3.

Note that $\underline{\text{GL}_{n}}\rtimes\textbf{G}_{m}$ is a canonical
maximal parabolic subgroup of $\text{GL}_{n+1}$. It is worth mentioning that
the study of invariants beyond reductive groups is a challenge (see [3]-[5]).
Consequently the invariant property of $\underline{G}\rtimes\textbf{G}_{m}$
has its own interest.

##  1\. Levi Schur-Weyl duality.

###  1.1. A general setting-up.

In this note, all vector spaces are over a complex number field of
$\mathbb{C}$. Let $G$ be a connected reductive algebraic group over
$\mathbb{C}$, and $(V,\rho)$ be a finite-dimensional rational representation
of $G$ with representation space $V$ over $\mathbb{C}$. We can define an
enhanced reductive algebraic group $\underline{G}=G\times_{\rho}V$ as follows
(see [1] explicitly):

Regard $\underline{G}$ as a set, we have $\underline{G}=G\times V$; For any
$(g_{1},v_{1}),(g_{2},v_{2})\in G\times V$,

$\displaystyle(g_{1},v_{1})\cdot(g_{2},v_{2}):=(g_{1}g_{2},\rho(g_{1})(v_{2})+v_{1}).$
(1.1)

From now on, we will write down $e^{v}$ for $(e,v)$ (where $e\in G$ is the
identity), and always suppose that $\textsf{{dim}}V=n$, i.e.
$V\cong\mathbb{C}^{n}$. Let $G_{1}$ and $G_{2}$ be subgroups of $G$, we denote
by $\langle G_{1},G_{2}\rangle$ the subgroup of $G$ generated by $G_{1}$ and
$G_{2}$. All representations for algebraic groups are always assumed to be
rational.

In particular, assume that $G$ is a closed subgroup of $\text{GL}(V)$. Since
$\text{GL}(V)$ can naturally act on $V$, we can defined the enhanced reductive
algebraic group $\underline{G}$ as above (where $\rho$ is just the natural
representation of $G$ on $V$). Note that $V$ is a closed subgroup of
$\underline{G}$, then an irreducible $G$-module becomes naturally an
irreducible module of $\underline{G}$ with trivial $V$-action. The isomorphism
classes of irreducible rational representations of $\underline{G}$ coincide
with the ones of $G$.

Let $\underline{V}:=V\oplus\mathbb{C}\eta\cong\mathbb{C}^{n+1}$, which is
called the enhanced space of $V$. Naturally $\underline{V}$ becomes a
$\underline{G}$-module that is defined for any
$\underline{g}=(g,v)\in\underline{G}$ and
$\underline{u}=u+a\eta\in\underline{V}$ with $g\in G$, $u,v\in V$ and
$a\in\mathbb{C}$, via

$\displaystyle\underline{g}\cdot\underline{u}:=\rho(g)(u)+av+a\eta.$ (1.2)

It is easy to see that this module is a rational module of $\underline{G}$.

In particular, if $G$ is a classical group, then $G\subseteq\text{GL}(V)$.
Since $\text{GL}(V)$ can naturally act on $V$, we can defined $\underline{G}$
and $\underline{G}$-module $\underline{V}$ as above.

Since $\text{GL}(V)\cong\text{GL}_{n}$ and
$\text{GL}(\underline{V})\cong\text{GL}_{n+1}$, we can regard $\text{GL}(V)$
and $\text{GL}(\underline{V})$ as $\text{GL}_{n}$ and $\text{GL}_{n+1}$
respectively. There are canonical imbeddings of algebraic groups
$\text{GL}_{n}\hookrightarrow\text{GL}_{n+1}$ given by

$\displaystyle g\mapsto\text{diag}(g,1):=\left(\begin{matrix}g&0\\\
0&1\end{matrix}\right),\forall g\in\text{GL}_{n}.$ (1.3)

and $\textbf{G}_{m}\hookrightarrow\text{GL}_{n+1}$ given by

$\displaystyle c\mapsto\text{diag}(1,\cdots,1,c):=\left(\begin{matrix}I_{n}\\\
&c\\\ \end{matrix}\right),\forall c\in\textbf{G}_{m}=\mathbb{C}^{\times}.$
(1.4)

where $I_{n}$ is $n$ identity matrix and $\textbf{G}_{m}$ is the
multiplicative algebraic group. In this sense, both $\text{GL}_{n}$ and
$\textbf{G}_{m}$ can be viewed as closed subgroups of $\text{GL}_{n+1}$.

Obviously, the closed subgroups $\langle\text{GL}(V),\textbf{G}_{m}\rangle$
and $\langle\underline{\text{GL}(V)},\textbf{G}_{m}\rangle$ are isomorphic to
$\text{GL}(V)\times{\textbf{G}}_{m}$ and
$\underline{\text{GL}(V)}\rtimes{\textbf{G}}_{m}$ respectively. We identify
all throughout the note.

Note that $\underline{\text{GL}(V)}\rtimes{\textbf{G}}_{m}$ is actually a
parabolic subgroup of $\text{GL}(\underline{V})$ associated with the Levi
subgroup $\text{GL}(V)\times{\textbf{G}}_{m}$. Clearly,
$\underline{\text{GL}(V)}$ is a closed subgroup of
$\underline{\text{GL}(V)}\rtimes{\textbf{G}}_{m}$, then
$e^{v}\in\underline{\text{GL}(V)}\rtimes\textbf{G}_{m}$ for all $v\in V$.

The fact that $\text{GL}(\underline{V})$ can naturally act on $\underline{V}$
implies that $\underline{V}$ can be viewed as a natural
$\text{GL}(\underline{V})$-module. Note that
$\text{GL}(V),\underline{\text{GL}(V)},\text{GL}(V)\times\textbf{G}_{m}$ and
$\underline{\text{GL}(V)}\rtimes\textbf{G}_{m}$ are all closed subgroups of
$\text{GL}(\underline{V})$, hence $\underline{V}$ can be viewed as a natural
$\text{GL}({V})$-module, $\underline{\text{GL}(V)}$-module,
$\text{GL}(V)\times\textbf{G}_{m}$-module and
$\underline{\text{GL}(V)}\rtimes\textbf{G}_{m}$-module respectively. It is
easy to see that the natural $\underline{\text{GL}(V)}$-module $\underline{V}$
coincides with (1.2), where $\rho$ is just the natural representation of
$\text{GL}(V)$ on $V$.

###  1.2. Classical Schur-Weyl duality.

We can define a representation $(\underline{V}^{\otimes r},\Phi)$ of
$\text{GL}(\underline{V})$ on $\underline{V}^{\otimes r}$

$\Phi(g)(v_{1}\otimes v_{2}\otimes\cdots\otimes v_{r})=g(v_{1})\otimes
g(v_{2})\otimes\cdots\otimes g(v_{r}).$ (1.5)

for any $g\in\text{GL}(\underline{V})$ and any monomial tensor product
$v_{1}\otimes v_{2}\otimes\cdots\otimes v_{r}\in\underline{V}^{\otimes r}$.

In the meanwhile, $(\underline{V}^{\otimes r},\Psi)$ naturally becomes a
representation of $\mathfrak{S}_{r}$ with following permutation action

$\Psi(\sigma)(v_{1}\otimes v_{2}\otimes\cdots\otimes
v_{r})=v_{\sigma^{-1}(1)}\otimes v_{\sigma^{-1}(2)}\otimes\cdots\otimes
v_{\sigma^{-1}(r)}.$ (1.6)

for any $\sigma\in\mathfrak{S}_{r}$ and any monomial tensor product
$v_{1}\otimes v_{2}\otimes\cdots\otimes v_{r}\in\underline{V}^{\otimes r}$.

The classical Schur-Weyl duality implies that the images of $\Phi$ and $\Psi$
are double centralizers in $\text{End}_{\mathbb{C}}(\underline{V}^{\otimes
r})$, namely

$\text{End}_{\text{GL}(\underline{V})}(V^{\otimes
r})=\mathbb{C}\Psi(\mathfrak{S}_{r}),\text{End}_{\mathfrak{S}_{r}}(\underline{V}^{\otimes
r})=\mathbb{C}\Phi(\mathfrak{\text{GL}}(\underline{V})).$ (1.7)

Here $\mathbb{C}\Psi(\mathfrak{S}_{r})$ and
$\mathbb{C}\Phi(\mathfrak{\text{GL}}(\underline{V}))$ are the subalgebras of
$\text{End}_{\mathbb{C}}(\underline{V}^{\otimes r})$ that generated by
$\Psi(\mathfrak{S}_{r})$ and $\Phi(\mathfrak{\text{GL}}(\underline{V}))$
respectively.

###  1.3. Degenerate double Hecke algebras.

From now on, we always assume that $G=\text{GL}(V)\cong\text{GL}_{n}$.

For given positive integers $r$ and $l$ with $r>l$, we define the $l$th
degerate double Hecke algebra $\mathcal{H}^{l}_{r}$ of $\mathfrak{S}_{r}$,
which is generated by
$\\{{\rm{\textbf{x}}}_{\sigma}\mid\sigma\in\mathfrak{S}_{l}\\}$ and
$\\{\rm{\textbf{s}}_{\textit{i}}\mid\textit{i}=1,2,\cdots,\textit{r}-1\\}$
with relations as follows.

$\displaystyle\rm{\textbf{s}}_{\textit{i}}^{2}=1,\rm{\textbf{s}}_{\textit{i}}\rm{\textbf{s}}_{\textit{j}}=\rm{\textbf{s}}_{\textit{j}}\rm{\textbf{s}}_{\textit{i}}\text{
for }0<\textit{i}\neq\textit{j}\leq\textit{r}-1,|\textit{j}-\textit{i}|>1;$
(1.8)
$\displaystyle\rm{\textbf{s}}_{\textit{i}}\rm{\textbf{s}}_{\textit{j}}\rm{\textbf{s}}_{\textit{i}}=\rm{\textbf{s}}_{\textit{j}}\rm{\textbf{s}}_{\textit{i}}\rm{\textbf{s}}_{\textit{j}}\text{
for }0<\textit{i}\neq\textit{j}\leq\textit{r}-1,|\textit{j}-\textit{i}|=1;$
(1.9)
$\displaystyle{\rm{\textbf{x}}}_{\sigma}{\rm{\textbf{x}}}_{\mu}={\rm{\textbf{x}}}_{\sigma\circ\mu}\text{
for }\sigma,\mu\in\mathfrak{S}_{l};$ (1.10)
$\displaystyle\rm{\textbf{s}}_{\textit{i}}{\rm{\textbf{x}}}_{\sigma}={\rm{\textbf{x}}}_{\rm{\textbf{s}}_{\textit{i}}\circ\sigma},{\rm{\textbf{x}}}_{\sigma}\rm{\textbf{s}}_{\textit{i}}={\rm{\textbf{x}}}_{\sigma\circ\rm{\textbf{s}}_{\textit{i}}}\text{
for }\sigma\in\mathfrak{S}_{\textit{{l}}},\textit{i}<\textit{l};$ (1.11)
$\displaystyle\rm{\textbf{s}}_{\textit{i}}{\rm{\textbf{x}}}_{\sigma}={\rm{\textbf{x}}}_{\sigma}={\rm{\textbf{x}}}_{\sigma}\rm{\textbf{s}}_{\textit{i}}\text{
for }\sigma\in\mathfrak{S}_{\textit{{l}}},\textit{i}>\textit{l}.$ (1.12)

This is an infinite-dimensional associative algebra.

The degenerate double Hecke algebra $\mathcal{H}_{r}$ of $\mathfrak{S}_{r}$ is
an associative algebra with generators $\rm{\textbf{s}}_{\textit{i}}$
($i=1,2,\cdots,r-1$) and ${\rm{\textbf{x}}}_{\sigma}^{(l)}$
($\sigma\in\mathfrak{S}_{r},l=0,1,\cdots,r$) with relations as (1.8)-(1.12) in
which ${\rm{\textbf{x}}}_{\sigma},{\rm{\textbf{x}}}_{\mu}$ are replaced by
${\rm{\textbf{x}}}_{\sigma}^{(l)},{\rm{\textbf{x}}}_{\mu}^{(l)}$, and
additional ones:

$\displaystyle{\rm{\textbf{x}}}_{\delta}^{(l)}{\rm{\textbf{x}}}_{\gamma}^{(k)}=0\text{
for }\delta\in\mathfrak{S}_{l},\gamma\in\mathfrak{S}_{k}\text{ with }l\neq k.$
(1.13)

For any $m\in\mathbb{Z}_{>0}$, we define $\underline{m}:=\\{1,2,\cdots,m\\}$.
Suppose $V=\bigoplus_{i=1}^{n}\mathbb{C}\eta_{i}$, then
$\\{\eta_{1},\cdots,\eta_{n},\eta\\}$ is a basis of $\underline{V}$. Clearly,
all $\eta_{\textbf{i}}:=\eta_{i_{1}}\otimes\cdots\otimes\eta_{i_{r}}$ for
$\textbf{i}=\\{i_{1},\cdots,i_{r}\\}\in\underline{(n+1)}^{r}$ form a basis of
$\underline{V}^{\otimes r}$ where $\eta_{n+1}:=\eta$. For any
$I\subseteq\underline{r}$, if $i\in I$, we set $V_{i}:=V$, otherwise
$V_{i}:=\mathbb{C}\eta$. Now we define a subspace $\underline{V}_{I}^{\otimes
r}$ of $\underline{V}^{\otimes r}$ as $\underline{V}_{I}^{\otimes
r}:=V_{1}\otimes\cdots\otimes V_{r}$. In particular,
$\underline{V}_{\phi}^{\otimes
r}=\mathbb{C}\eta\otimes\cdots\otimes\mathbb{C}\eta$ with $r$ copies. Suppose
$I=\\{i_{1},\cdots,i_{l}\\}$ where $i_{1}<i_{2}<\cdots<i_{l}$, it is not hard
to see that $\underline{V}_{I}^{\otimes r}$ has a basis as follows

$\displaystyle\\{\eta_{\textbf{j}}:=\eta_{j_{1}}\otimes\cdots\otimes\eta_{j_{r}}\mid
j_{i_{k}}\in\underline{n},k=1,\cdots,l;j_{d}=n+1\text{ for }d\neq i_{k}\\}.$

Let $\underline{V}_{l}^{\otimes
r}:=\bigoplus\limits_{I\subseteq\underline{r},~{}\sharp
I=l}\underline{V}_{I}^{\otimes r}$, thus $\underline{V}^{\otimes r}$ has a
decomposition as a linear space $\underline{V}^{\otimes
r}=\bigoplus_{l=0}^{r}\underline{V}_{l}^{\otimes r}$.

Given $\sigma\in\mathfrak{S}_{l}$ and $I=\\{i_{1},\cdots,i_{l}\\}$ where
$i_{1}<i_{2}<\cdots<i_{l}$, assume that $v_{1}\otimes\cdots\otimes
v_{r}\in\underline{V}_{I}^{\otimes r}$. Let
$w_{i_{k}}:=v_{i_{\sigma^{-1}(k)}}$ with $1\leq k\leq l$, while for $j\notin
I$, set $w_{j}:=v_{j}$. We can define
$x_{\sigma}\in\text{End}_{\mathbb{C}}(\underline{V}_{I}^{\otimes r})$ with

$\displaystyle x_{\sigma}(v_{1}\otimes\cdots\otimes
v_{r})=w_{1}\otimes\cdots\otimes w_{r}.$

Now we extend $x_{\sigma}$ to an element $x_{\sigma}^{I}$ of
$\text{End}_{\mathbb{C}}(\underline{V}_{l}^{\otimes r})$ by annihilating any
other summand $\underline{V}_{J}^{\otimes r}$ with $J\neq I$ and $\sharp J=l$.

[1, Lemma 3.3] gives a representation of the degenerate double Hecke algebra
$\mathcal{H}_{r}$ on $\underline{V}^{\otimes r}$.

######  Lemma 1.1.

There is a representation
$\Xi:\mathcal{H}_{r}\rightarrow\text{End}_{\mathbb{C}}(\underline{V}^{\otimes
r})$ defined via:

  * (1)

$\Xi|_{\mathbb{C}\mathfrak{S}_{r}}=\Psi$;

  * (2)

For any ${\rm{\mathbf{x}}}_{\sigma}^{(l)}\in\mathcal{H}_{r}^{l}$ with
$\sigma\in\mathfrak{S}_{l}$, $l=0,1,\cdots,r$,
$\Xi({\rm{\mathbf{x}}}_{\sigma}^{(l)})|_{\underline{V}_{k}^{\otimes
r}}=\left\\{\begin{aligned} 0,k\neq l;\\\
x_{\sigma}^{\underline{l}},k=l.\end{aligned}\right.$

######  Remark 1.2.

Lemma 1.1 implies that $\Xi(\tau)(\underline{V}_{l}^{\otimes
r})\subseteq\underline{V}_{l}^{\otimes r}$ for all
$\tau\in\mathcal{H}_{r}^{l}$, so we can give a representation of
$\mathcal{H}^{l}_{r}$ on $\underline{V}_{l}^{\otimes r}$.

By (1), for $\rm{\mathbf{s}}_{\textit{i}}\in\mathcal{H}_{r}$ with
$i=1,2,\cdots,r-1$, we have
$\Xi(\mathbf{s}_{\textit{i}})=\Psi(\mathbf{s}_{\textit{i}})$.

For $0\leq l\leq r$, we define a representation
$\Xi^{(l)}:\mathcal{H}_{r}^{l}\rightarrow\text{End}_{\mathbb{C}}(\underline{V}^{\otimes
r})$ via:

$\displaystyle\text{For }\tau\in\mathbb{C}\mathfrak{S}_{r},\text{ set
}\Xi^{(l)}(\tau)|_{\underline{V}_{k}^{\otimes r}}:=\left\\{\begin{aligned}
\Psi(\tau)|_{\underline{V}_{l}^{\otimes r}},\text{ }k=l;\\\ 0,\text{ }k\neq
l.\end{aligned}\text{ }\right.$ (1.14) $\displaystyle\text{For any
}{\rm{\mathbf{x}}}_{\sigma}^{(l)}\in\mathcal{H}_{r}^{l}\text{ with
}\sigma\in\mathfrak{S}_{l},\text{ set
}\Xi^{(l)}({\rm{\mathbf{x}}}_{\sigma}^{(l)}):=\Xi({\rm{\mathbf{x}}}_{\sigma}^{(l)}).$
(1.15)

Clearly, $\Xi^{(l)}$ can be viewed as a representation on
$\underline{V}_{l}^{\otimes r}$, i.e. homomorphism
$\Xi^{(l)}:\mathcal{H}_{r}^{l}\rightarrow\text{End}_{\mathbb{C}}(\underline{V}_{l}^{\otimes
r})$.

###  1.4. Levi Schur-Weyl duality

Set $D(n,r):=\Xi(\mathcal{H}_{r})$, which is called the finite-dimensional
degenerate double Hecke algebra (DDHA for short) of $\mathfrak{S}_{r}$. We
should note that $\mathcal{H}_{r}$ is infinite-dimensional.

Let $D(n,r)_{l}:=\Xi^{(l)}(\mathcal{H}^{l}_{r})$ for $l=0,1,\cdots,r$. By [1,
Lemma 3.5(1)], we have a decomposition of $D(n,r)$ as a vector space

$\displaystyle D(n,r)=\bigoplus_{l=0}^{r}D(n,r)_{l}.$ (1.16)

Now we list the Levi Schur-Weyl duality and Parabolic Schur-Weyl duality in
[1].

######  Theorem 1.3.

([1, Theorem 4.3]) The following duality holds:

$\displaystyle\text{End}_{\text{GL}_{n}\times\textbf{G}_{m}}(\underline{V}^{\otimes
r})=D(n,r);$ (1.17) $\displaystyle\text{End}_{D(n,r)}(\underline{V}^{\otimes
r})=\mathbb{C}\Psi(\text{GL}_{n}\times\textbf{G}_{m}).$ (1.18)

######  Theorem 1.4.

([1, Theorem 5.3]) Set $D(n,r)^{V}=\\{\phi\in
D(n,r)\mid\Phi(e^{v})\circ\phi=\phi\circ\Phi(e^{v}),\forall v\in V\\}$. Then
the following holds:

$\displaystyle\text{End}_{\underline{\text{GL}_{n}}\rtimes\textbf{G}_{m}}(\underline{V}^{\otimes
r})=D(n,r)^{V}.$ (1.19)

##  2\. Tensor invariant of enhanced reductive group
$\textbf{G}_{m}\ltimes\underline{\text{GL}(V)}$.

Keep the notations as in Section 1, and in this section, we assume that
$G=\text{GL}(V)\cong\text{GL}_{n}$. For the symmetric group
$\mathfrak{S}_{r}$, we denote by $(i,i+1)$ for $i=1,\cdots,r-1$, the
transposition just interchanging $i$ and $i+1$, and fixing the others.

Given $\sigma\in\mathbb{C}\Psi(\mathfrak{S}_{r})$ and
$I\subseteq\underline{r}$, define
$\sigma^{[I]}\in\text{End}_{\mathbb{C}}(\underline{V}^{\otimes r})$ as below:

$\displaystyle\sigma^{[I]}|_{\underline{V}_{J}^{\otimes
r}}:=\left\\{\begin{aligned} \sigma|_{\underline{V}_{I}^{\otimes r}},\text{
}I=J;\\\ 0,\text{ },I\neq J.\end{aligned}\right.$ (2.1)

By definition, $\sigma^{[I]}(\underline{V}_{l}^{\otimes
r})\subseteq\underline{V}_{l}^{\otimes r}$. For $I=\\{i_{1},\cdots i_{l}\\}$
and $J=\\{j_{1},\cdots j_{l}\\}$ with $i_{1}<\cdots<i_{l}$,
$j_{1}<\cdots<j_{l}$ and $0\leq l\leq r$, there exists
$\varepsilon_{J,I}\in\mathfrak{S}_{r}$ satisfying
$\varepsilon_{J,I}(i_{t})=j_{t}$ for all $1\leq t\leq l$. Now we define
$E_{J,I}:=(\Psi(\varepsilon_{J,I}))^{[I]}$, and note that
$E_{J,I}=\text{id}^{[I]}$ when $l=0$ (i.e. $I=J=\phi$). Although the element
$\varepsilon_{J,I}$ in $\mathfrak{S}_{r}$ satisfying
$\varepsilon_{J,I}(i_{t})=j_{t}$ ($\forall 1\leq t\leq l$) is not unique,
$(\Psi(\varepsilon_{J,I}))^{[I]}$ does not depend on the choice of
$\varepsilon_{J,I}$. It is easy to see that
$E_{J,I}(\underline{V}_{I}^{\otimes r})\subseteq\underline{V}_{J}^{\otimes
r}$. Let
$D_{[I]}:=\\{\sigma^{[I]}\mid\sigma\in\mathbb{C}\Psi(\mathfrak{S}_{r})\\}$, it
is clear that $\sum\limits_{I\subseteq\underline{r},~{}\sharp I=l}D_{[I]}$ is
a direct sum, i.e. $\bigoplus\limits_{I\subseteq\underline{r},~{}\sharp
I=l}D_{[I]}$. Now we suppose $n\geq r$.

######  Lemma 2.1.

$D(n,r)_{l}=\bigoplus\limits_{I\subseteq\underline{r},~{}\sharp I=l}D_{[I]}$.

###### Proof.

Since $n\geq r$, [1, Remark 3.6] implies that $D(n,r)_{l}$ has a basis as
follows

$\displaystyle\\{E_{J,I}\circ x_{\sigma}^{I}\mid
I,J\subseteq\underline{r},\sharp I=\sharp J=l,\sigma\in\mathfrak{S}_{l}\\}.$

For $x_{\sigma}^{I}$ with $\sharp I=l$, there exists
$\sigma^{\prime}\in\mathfrak{S}_{r}$ such that
$(\Psi(\sigma^{\prime}))^{[I]}=x_{\sigma}^{I}$. Thus
$E_{J,I}x_{\sigma}^{I}=(\Psi(\varepsilon_{J,I}))^{[I]}\cdot(\Psi(\sigma^{\prime}))^{[I]}=(\Psi(\varepsilon_{J,I}\cdot\sigma^{\prime}))^{[I]}\in
D_{[I]}$, which yields that
$D(n,r)_{l}\subseteq\bigoplus\limits_{I\subseteq\underline{r},~{}\sharp
I=l}D_{[I]}$.

Conversely, if $\tau\in\mathfrak{S}_{r}$, in order to conclude
$D_{[I]}\subseteq D(n,r)_{l}$ with $\sharp I=l$, it suffices to show
$(\Psi(\tau))^{[I]}\in D(n,r)_{l}$. Suppose $I=\\{i_{1},\cdots,i_{l}\\}$ with
$i_{1}<\cdots<i_{l}$ and $\tau(I)=\\{j_{1},\cdots,j_{l}\\}$ with
$j_{1}<\cdots<j_{l}$. Moreover, assume that $m_{1},\cdots,m_{l}$ are different
numbers in $\underline{l}$ satisfying $\tau(i_{t})=j_{m_{t}}$ ($\forall
t\in\underline{l}$). There exist $\sigma^{\prime}\in\mathfrak{S}_{r}$
satisfying $\sigma^{\prime}(i_{t})=i_{m_{t}}$ ($\forall 1\leq t\leq l$) and
set $\sigma=\begin{pmatrix}1&\cdots&l\\\
m_{1}&\cdots&m_{l}\end{pmatrix}\in\mathfrak{S}_{l}$. Recall that
$\varepsilon_{\tau(I),I}(i_{t})=j_{t}$ for all $1\leq t\leq l$ (i.e.
$\varepsilon_{\tau(I),I}(i_{m_{t}})=j_{m_{t}}$ for all $1\leq t\leq l$). It is
not hard to see that
$(\Psi(\tau))^{[I]}=(\Psi(\varepsilon_{\tau(I),I}\cdot\sigma^{\prime}))^{[I]}=(\Psi(\varepsilon_{\tau(I),I}))^{[I]}\cdot(\Psi(\sigma^{\prime}))^{[I]}=E_{\tau(I),I}\cdot
x_{\sigma}^{I}\in D(n,r)_{l}$.

For all $I\subseteq\underline{r}$ with $\sharp I=l$, we have $D_{[I]}\subseteq
D(n,r)_{l}$. Hence $\bigoplus\limits_{I\subseteq\underline{r},~{}\sharp
I=l}D_{[I]}\subseteq D(n,r)_{l}$. ∎

######  Remark 2.2.

Lemma 2.1 and (1.16) implies that
$D(n,r)=\bigoplus\limits_{I\subseteq\underline{r}}D_{[I]}$.

Keep in mind that $n\geq r$. Fix a basis $\\{v_{1},\cdots,v_{n}\\}$ of $V$,
and recall that $\underline{V}=V\oplus\mathbb{C}\eta$. Given $w\in V$ and
$J\subseteq\underline{r}$, we define $a_{i}\in\underline{V}$ ($1\leq i\leq r$)
and $b_{i}\in\underline{V}$ ($1\leq i\leq r$) as follows:

$\displaystyle\text{If }i\in J,\text{ set }a_{i}:=w+\eta;\text{ if }i\notin
J,\text{ set }a_{i}:=v_{i}.$ $\displaystyle\text{If }i\in J,\text{ set
}b_{i}:=\eta;\text{ if }i\notin J,\text{ set }b_{i}:=v_{i}.$

Let $A_{w}^{J}:=a_{1}\otimes\cdots\otimes a_{r}-b_{1}\otimes\cdots\otimes
b_{r}\in\underline{V}^{\otimes r}$, we have the following Lemma.

######  Lemma 2.3.

Suppose $n\geq r$. For $\delta\in\mathbb{C}\Psi(\mathfrak{S}_{r})$ and
$\phi\neq J\subseteq\underline{r}$, if $\forall w\in V$, we always have
$\delta(A_{w}^{J})=0$, then $\delta(\underline{V}_{\underline{r}\backslash
J}^{\otimes r})=0$.

###### Proof.

We assume that $\sharp J=r-k$ (i.e. $\sharp(I\backslash J)=k$). Obviously, it
is enough for us to prove the case
$J=\underline{r}\backslash\underline{k}=\\{k+1,\cdots,r\\}$.

Now we supoose $J=\\{k+1,\cdots,r\\}$ and take $w=v_{k+1}+\cdots+v_{r}$, then
$A_{w}^{J}=v_{1}\otimes\cdots\otimes
v_{k}\otimes(v_{k+1}+\cdots+v_{r}+\eta)^{\otimes(r-k)}-v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)}$. Set $G_{1}:=\langle(i,i+1)\mid 1\leq i\leq
k-1\rangle$ and $G_{2}:=\langle(i,i+1)\mid k+1\leq i\leq r-1\rangle$, then
$G_{1}\cong\mathfrak{S}_{k}$ and $G_{2}\cong\mathfrak{S}_{r-k}$. For all
$s\in\mathfrak{S}_{r}$, we define $\underline{s}:=\\{s(x)\mid
x\in\underline{k}\text{ and }k+1\leq s(x)\leq r\\}$ and $\bar{s}:=\\{s(x)\mid
x\in\underline{r}\backslash\underline{k}\text{ and }1\leq s(x)\leq k\\}$. It
is clear that
$\underline{s}=(\underline{r}\backslash\underline{k})\backslash\\{{s(x)\mid
x\in\underline{r}\backslash\underline{k}\text{ and
}s(x)\in\underline{r}\backslash\underline{k}}\\}$,
$\bar{s}=\underline{k}\backslash\\{s(x)\mid x\in\underline{k}\text{ and
}s(x)\in\underline{k}\\}$ and $\sharp\underline{s}=\sharp\bar{s}$. For
$B\subseteq\underline{k}$ and
$A\subseteq\underline{r}\backslash\underline{k}=\\{k+1,\cdots,r\\}$, if
$\sharp A=\sharp B$, set
$\Psi_{A,B}:=\sum\limits_{s\in\mathfrak{S}_{r}\atop\underline{s}=A,~{}\bar{s}=B}\mathbb{C}\Psi(s)$;
if $\sharp A\neq\sharp B$, there does not exist $s\in\mathfrak{S}_{r}$
satisfying $\underline{s}=A$ and $\bar{s}=B$, then we set $\Psi_{A,B}:=0$.
Obviously,
$\mathbb{C}\Psi(\mathfrak{S}_{r})=\sum\limits_{A\subseteq\underline{r}}\sum\limits_{B\subseteq\underline{r}}\Psi_{A,B}=\sum\limits_{A\subseteq\underline{r}\backslash\underline{k},~{}B\subseteq\underline{k}\atop\sharp
A=\sharp B}\Psi_{A,B}$.

Clearly, for $s\in\mathfrak{S}_{r}$,
$\underline{s}=\phi\Leftrightarrow\bar{s}=\phi\Leftrightarrow s\in
G_{1}G_{2}$. Given $A\subseteq\underline{r}\backslash\underline{k}$ and
$B\in\underline{k}$ with $\sharp A=\sharp B=q\neq 0$ ($\Rightarrow
A\neq\phi,B\neq\phi$), assume that $A=\\{i_{1},\cdots i_{q}\\}$ and
$B=\\{j_{1},\cdots j_{q}\\}$, now we set
$\theta=(i_{1},j_{1})\cdots(i_{q},j_{q})$. For all
$\varepsilon\in\mathfrak{S}_{r}$ with $\underline{\varepsilon}=A$ and
$\bar{\varepsilon}=B$, we have $\theta\cdot\varepsilon\in G_{1}G_{2}=\langle
G_{1},G_{2}\rangle$.

Hence fix $A\subseteq\underline{r}\backslash\underline{k}$ and
$B\subseteq\underline{k}$ with $\sharp A=\sharp B$,
$\forall\varepsilon\in\mathfrak{S}_{r}$, we have $\theta\cdot\varepsilon\in
G_{1}G_{2}$ (Here $\theta:=1=\text{id}$, if $A=B=\phi$). Namely,

$\displaystyle\varepsilon=\theta^{-1}\cdot\varepsilon^{\prime}=\theta\cdot\varepsilon^{\prime},\text{
where }\varepsilon^{\prime}=\theta\varepsilon\in G_{1}G_{2},\text{ and note
that }\theta=\theta^{-1}\in\mathfrak{S}_{r}\backslash(G_{1}G_{2}).$ (2.2)

The terms in $A_{w}^{J}$ containing $v_{1},\cdots,v_{r}$ can be written as
$\Psi(s)(v_{1}\otimes\cdots\otimes v_{r})$, where $s$ ranges over $G_{2}$.
Hence $\delta(A_{w}^{J})=0$ implies that $\delta(\sum_{s\in
G_{2}}\Psi(s)(v_{1}\otimes\cdots\otimes v_{r}))=0$. Thanks to
$\mathbb{C}\Psi(\mathfrak{S}_{r})=\sum\limits_{A\subseteq\underline{r}}\sum\limits_{B\subseteq\underline{r}}\Psi_{A,B}$
and (2.2), there exist $m\in\mathbb{Z}_{>0}$,
$\phi_{1},\cdots,\phi_{m}\in\mathfrak{S}_{r}\backslash(G_{1}G_{2})$ and
$T_{0},T_{1},\cdots,T_{m}\in\mathbb{C}\Psi(G_{1}G_{2})$, such that
$\delta=T_{0}+\sum_{i=1}^{m}\Psi(\phi_{i})\circ
T_{i}=\sum_{i=0}^{m}\Psi(\phi_{i})\circ T_{i}$ (Here $\phi_{0}=1$), and
satisfying

$\displaystyle\Psi(\phi_{i})\circ T_{i}\in\Psi_{A_{i},B_{i}}\text{ and
}\Psi(\phi_{i})\in\Psi_{A_{i},B_{i}}\text{, with }0\leq i\leq
m,A_{i}\subseteq\underline{r}\backslash\underline{k},B_{i}\subseteq\underline{k},\sharp
A_{i}=\sharp B_{i}\neq 0.$ (2.3)

as well as $A_{i}\neq A_{j}$ or $B_{i}\neq B_{j}$, $\forall 0\leq i\neq j\leq
m$, i.e.

$\displaystyle(A_{i},B_{i})\neq(A_{j},B_{j})\text{ for all }0\leq i\neq j\leq
m.$ (2.4)

By (2.3), we see that

$\displaystyle\Psi(\phi_{i})\circ T_{i}\circ(\sum_{s\in
G_{2}}\Psi(s))\in\Psi_{A_{i},B_{i}}.$ (2.5)

We have $(T_{0}\circ(\sum_{s\in G_{2}}\Psi(s)))(v_{1}\otimes\cdots\otimes
v_{r})+(\Psi(\phi_{1})\circ T_{1}\circ(\sum_{s\in
G_{2}}\Psi(s)))(v_{1}\otimes\cdots\otimes v_{r})+\cdots+(\Psi(\phi_{m})\circ
T_{m}\circ(\sum_{s\in G_{2}}\Psi(s)))(v_{1}\otimes\cdots\otimes v_{r})=0$. Due
to (2.4) and (2.5), it yields that

$\displaystyle(\Psi(\phi_{i})\circ T_{i}\circ(\sum_{s\in
G_{2}}\Psi(s)))(v_{1}\otimes\cdots\otimes v_{r})=0,\text{ }\forall
i=0,1,\cdots,m.$

Thus we have

$\displaystyle(\Psi(\phi_{i}^{-1})\circ\Psi(\phi_{i})\circ
T_{i}\circ(\sum_{s\in G_{2}}\Psi(s)))(v_{1}\otimes\cdots\otimes
v_{r})=0,\text{ }\forall i=0,1,\cdots,m.$

Namely

$\displaystyle(T_{i}\circ(\sum_{s\in G_{2}}\Psi(s)))(v_{1}\otimes\cdots\otimes
v_{r})=0,\text{ }\forall i=0,1,\cdots,m.$ (2.6)

Since $T_{i}\in\mathbb{C}\Psi(G_{1}G_{2})$ and $n\geq r$, there exist
$l\in\mathbb{N}^{*}$, $k_{j}\in\mathbb{C}$, $g_{t}^{(j)}\in G_{t}$ with
$t=1,2$ and $j\in\underline{l}=\\{1,\cdots,l\\}$, such that
$T_{i}=\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)})\circ\Psi(g_{2}^{(j)})$. Hence

$\displaystyle T_{i}\circ(\sum_{s\in
G_{2}}\Psi(s))=(\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)})\circ\Psi(g_{2}^{(j)}))\circ(\sum_{s\in
G_{2}}\Psi(s))=\sum_{j=1}^{l}\sum_{s\in
G_{2}}k_{j}\Psi(g_{1}^{(j)})\circ\Psi(g_{2}^{(j)})\circ\Psi(s)$
$\displaystyle=\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)})(\sum_{s\in
G_{2}}\Psi(g_{2}^{(j)}\cdot
s))=(\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)}))(\sum_{s\in G_{2}}\Psi(s)).$

So $(\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)}))\circ(\sum_{s\in
G_{2}}\Psi(s))(v_{1}\otimes\cdots\otimes v_{r})=0$. Thanks to $G_{1}\cap
G_{2}=\\{1\\}$, we can conclude that

$\displaystyle(\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)}))(v_{1}\otimes\cdots\otimes
v_{r})=0.$ (2.7)

Since $\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)})\in\mathbb{C}\Psi(G_{1})$, there
exists $h_{i}\in G_{1},c_{i}\in\mathbb{C}$ with $1\leq i\leq t$ satisfying
$h_{p}\neq h_{q}$ for all $1\leq p<q\leq t$, such that
$\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)})=\sum_{i=1}^{t}c_{i}\Psi(h_{i})$. By
(2.7), we see that $\sum_{i=1}^{t}c_{i}\Psi(h_{i})(v_{1}\otimes\cdots\otimes
v_{r})=0$. Due to $h_{1},\cdots,h_{t}$ are all different,
$\Psi(h_{1})(v_{1}\otimes\cdots\otimes
v_{r}),\cdots,\Psi(h_{t})(v_{1}\otimes\cdots\otimes v_{r})$ are linear
independent vectors, which implies that $c_{1}=\cdots=c_{t}=0$. Hence

$\displaystyle\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)})=0.$ (2.8)

Note that $\underline{V}_{\underline{k}}^{\otimes r}=V^{\otimes
k}\otimes(\mathbb{C}\eta)^{\otimes(r-k)}$, thus
$\Psi(g_{2}^{(j)})|_{\underline{V}_{\underline{k}}^{\otimes
r}}=\text{id}_{\underline{V}_{\underline{k}}^{\otimes r}}$. This yields that

$\displaystyle T_{i}|_{\underline{V}_{\underline{k}}^{\otimes
r}}=\sum_{j=1}^{l}(k_{j}\Psi(g_{1}^{(j)})\circ\Psi(g_{2}^{(j)}))|_{\underline{V}_{\underline{k}}^{\otimes
r}}=\sum_{j=1}^{l}k_{j}\Psi(g_{1}^{(j)})|_{\underline{V}_{\underline{k}}^{\otimes
r}}\xlongequal{\text{By }(\ref{0})}0$

Thus $T_{i}|_{\underline{V}_{\underline{k}}^{\otimes r}}=0$ is proved. ∎

Now we begin to prove [1, Conjecture 5.4].

######  Theorem 2.4.

([1, Conjecture 5.4]) If $n\geq r$, we have
$D(n,r)^{V}=\mathbb{C}\Psi(\mathfrak{S}_{r})$, i.e.
$\text{End}_{\underline{G}\rtimes\textbf{G}_{m}}(\underline{V}^{\otimes
r})=\mathbb{C}\Psi(\mathfrak{S}_{r})$.

###### Proof.

Clearly, Schur-Weyl duality implies that
$\mathbb{C}\Psi(\mathfrak{S}_{r})=\text{End}_{\text{GL}(\underline{V})}(\underline{V}^{\otimes
r})\subseteq\text{End}_{\underline{G}\rtimes\textbf{G}_{m}}(\underline{V}^{\otimes
r})$, i.e. $\mathbb{C}\Psi(\mathfrak{S}_{r})\subseteq D(n,r)^{V}$. Now we just
need to show that $D(n,r)^{V}\subseteq\mathbb{C}\Psi(\mathfrak{S}_{r})$.

$\forall\sigma\in D(n,r)^{V}$, Remark 2.2 implies that there exist
$\tau_{I}\in\mathbb{C}\Psi(\mathfrak{S}_{r})$ with all
$I\subseteq\underline{r}$, such that
$\sigma=\sum_{I\subseteq\underline{r}}\tau_{I}^{[I]}$ (Note that
$\tau_{I}^{[I]}$ is defined in (2.1)). Now we use induction to prove.

If $\sharp I=r$, then it must have $I=\underline{r}$. So we assume that there
exist $\tau\in\mathbb{C}\Psi(\mathfrak{S}_{r})$ with $0\leq k\leq r-1$, such
that $\forall I\subseteq\underline{r}$ with $\sharp I\geq k+1$ we have
$\tau_{I}^{[I]}=\tau^{[I]}$ ($\tau^{[I]}$ is defined in (2.1)). Then
$\sigma=\sum\limits_{I\subseteq\underline{r},~{}\sharp I\leq
k}\tau_{I}^{[I]}+\sum\limits_{I\subseteq\underline{r},~{}\sharp
I>k}\tau^{[I]}$. Recall that $\\{v_{1},\cdots,v_{n}\\}$ is a basis of $V$ and
$\underline{V}=V\oplus\mathbb{C}\eta$, it is easy to see $\forall w\in
V,e^{w}(\eta)=w+\eta$ and $e^{w}|_{V}=\text{id}_{V}$.

For $w\in V$, we have $v_{1}\otimes\cdots\otimes
v_{k}\otimes(w+\eta)^{\otimes(r-k)}=v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)}+A_{w}^{\underline{r}\backslash\underline{k}}$,
where
$A_{w}^{\underline{r}\backslash\underline{k}}=A_{w}^{\\{k+1,\cdots,r\\}}$ is
defined as in Lemma 2.3.

$\displaystyle(\sigma\circ\Phi(e^{w}))(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)})=\sigma(v_{1}\otimes\cdots\otimes
v_{k}\otimes(w+\eta)^{\otimes(r-k)})$
$\displaystyle=(\sum\limits_{I\subseteq\underline{r},~{}\sharp I\leq
k}\tau_{I}^{[I]}+\sum\limits_{I\subseteq\underline{r},~{}\sharp
I>k}\tau^{[I]})(v_{1}\otimes\cdots\otimes
v_{k}\otimes(w+\eta)^{\otimes(r-k)})$
$\displaystyle=(\tau_{\underline{k}}^{[\underline{k}]}+\sum\limits_{I\subseteq\underline{r},~{}\sharp
I>k}\tau^{[I]})(v_{1}\otimes\cdots\otimes
v_{k}\otimes(w+\eta)^{\otimes(r-k)})$
$\displaystyle=\tau_{\underline{k}}^{[\underline{k}]}(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)})+\tau(A_{w}^{\underline{r}\backslash\underline{k}})$
$\displaystyle=\tau_{\underline{k}}(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)})+\tau(A_{w}^{\underline{r}\backslash\underline{k}}).$
(2.9) $\displaystyle(\Phi(e^{w})\circ\sigma)(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)})=\Phi(e^{w})(\tau_{\underline{k}}^{[\underline{k}]}(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)}))$
$\displaystyle=\Phi(e^{w})(\tau_{\underline{k}}(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)}))$
$\displaystyle=\tau_{\underline{k}}(\Phi(e^{w})(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)}))\text{ }(\text{By Schur-Weyl duality})$
$\displaystyle=\tau_{\underline{k}}(v_{1}\otimes\cdots\otimes
v_{k}\otimes(w+\eta)^{\otimes(r-k)})$
$\displaystyle=\tau_{\underline{k}}(v_{1}\otimes\cdots\otimes
v_{k}\otimes\eta^{\otimes(r-k)})+\tau_{\underline{k}}(A_{w}^{\underline{r}\backslash\underline{k}}).$
(2.10)

Since $\sigma\in D(n,r)^{V}$, i.e.
$\Phi(e^{w})\circ\sigma=\sigma\circ\Phi(e^{w})$, (2.9) and (2.10) yields that
$\forall w\in V$,
$\tau(A_{w}^{\underline{r}\backslash\underline{k}})=\tau_{\underline{k}}(A_{w}^{\underline{r}\backslash\underline{k}})$.
Similarly, $\forall w\in V$ and $\forall J\subseteq\underline{r}$ with $\sharp
J=r-k$, we have $\tau(A_{w}^{J})=\tau_{\underline{r}\backslash J}(A_{w}^{J})$,
namely $(\tau-\tau_{\underline{r}\backslash J})(A_{w}^{J})=0$. Due to Lemma
2.3 (note that $k\leq r-1$), we conclude that
$(\tau-\tau_{\underline{r}\backslash
J})(\underline{V}_{\underline{r}\backslash J}^{\otimes r})=0$. Thus $\forall
I\subseteq\underline{r}$ with $\sharp I=k$, we have
$(\tau-\tau_{I})(\underline{V}_{I}^{\otimes r})=0$, i.e.
$\tau|_{\underline{V}_{I}^{\otimes r}}=\tau_{I}|_{\underline{V}_{I}^{\otimes
r}}$, which implies that $\tau^{[I]}=\tau_{I}^{[I]}$. By induction, we have
$\tau^{[I]}=\tau_{I}^{[I]}$ for all $I\subseteq\underline{r}$. Thus
$\sigma=\sum\limits_{I\subseteq\underline{r}}\tau_{I}^{[I]}=\tau\in\mathbb{C}\Psi(\mathfrak{S}_{r})$,
i.e. $D(n,r)^{V}\subseteq\mathbb{C}\Psi(\mathfrak{S}_{r})$. ∎

##  3\. Enhanced tensor invariants.

Keep the notations as in §1. In this section, we still set
$G=GL(V)\cong\text{GL}_{n}$, and identify $\underline{V}^{*\otimes r}$ with
$(\underline{V}^{\otimes r})^{*}$. For convenience, we denote
$\underline{n+1}$ by $\mathcal{N}$. Then there is a natural
$\underline{G}\rtimes\textbf{G}_{m}$-module isomorphism
$T:\underline{V}^{\otimes r}\otimes\underline{V}^{*\otimes
r}\rightarrow\text{End}_{\mathbb{C}}(\underline{V}^{\otimes r})$. Due to the
parabolic Schur-Weyl duality (i.e. Theorem 2.4), it is time to describe the
enhanced tensor invariants.

Now we still let $\eta_{1},\cdots,\eta_{n}$ be a basis of $V$ and
$\underline{V}=V\oplus\mathbb{C}\eta_{n+1}$ with $\eta_{n+1}:=\eta$. Then
$\\{\eta_{i}\mid i=1,2,\cdots,n+1\\}$ constitute a basis of $\underline{V}$.
Let $\\{\eta_{i}^{*}\mid i=1,2,\cdots,n+1\\}$ be the dual basis of
$\underline{V}^{*}$. Given a multi-index
$\textbf{i}=(i_{1},\cdots,i_{r})\in\mathcal{N}^{r}$, we still set
$\eta_{\textbf{i}}:=\eta_{i_{1}}\otimes\cdots\otimes\eta_{i_{r}}$ and
$\eta_{\textbf{i}}^{*}:=\eta_{i_{1}}^{*}\otimes\cdots\otimes\eta_{i_{r}}^{*}$.
Then $\eta_{\textbf{i}}$ and $\eta_{\textbf{i}}^{*}$ with i ranges over
$\mathcal{N}^{r}$ form a basis of $\underline{V}^{\otimes r}$ and
$\underline{V}^{*\otimes r}$ respectively. Moreover,
$\\{\eta_{\textbf{i}}\otimes\eta_{\textbf{j}}^{*}\mid(\textbf{i},\textbf{j})\in\mathcal{N}^{r}\times\mathcal{N}^{r}\\}$
is a basis of $\underline{V}^{\otimes r}\otimes\underline{V}^{*\otimes r}$.

For $\sigma\in\mathfrak{S}_{r}$, define a mixed tensor $C_{\sigma}$ as follows

$\displaystyle
C_{\sigma}=\sum_{\textbf{i}\in\mathcal{N}^{r}}\eta_{\sigma.\textbf{i}}\otimes\eta_{\textbf{i}}^{*}.$
(3.1)

######  Proposition 3.1.

([1, Proposition 5.5]) When $D(n,r)^{V}=\mathbb{C}\Psi(\mathfrak{S}_{r})$,
then the space of $\underline{G}\rtimes\textbf{G}_{m}$-invariants in the
mixed-tensor product $\underline{V}^{\otimes r}\otimes\underline{V}^{*\otimes
r}$ is generated by $C_{\sigma}$ defined in (3.1) with
$\sigma\in\mathfrak{S}_{r}$.

Obviously, Theorem 2.4 and Propsition 3.1 yield the following proposition.

######  Proposition 3.2.

If $n\geq r$, then the space of
$\underline{G}\rtimes\textbf{G}_{m}$-invariants in the mixed-tensor product
$\underline{V}^{\otimes r}\otimes\underline{V}^{*\otimes r}$ is generated by
$C_{\sigma}$ with $\sigma\in\mathfrak{S}_{r}$.

## Acknowledgment

The author thanks Bin Shu for his guidance, and thanks Yufeng Yao for his
helpful comments and discussions.

## References

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