[labelstyle=]

# A higher-dimensional Chevalley restriction theorem for orthogonal groups

Lei Song School of Mathematics, Sun Yat-sen University
No. 135 Xingang Xi Road, Guangzhou, Guangdong 510275, P. R. China
<EMAIL_ADDRESS>,  Xiaopeng Xia School of Mathematical Sciences,
University of Science and Technology of China
No. 96 Jinzhai Road, Hefei, Anhui 230026, P. R. China<EMAIL_ADDRESS>and  Jinxing Xu School of Mathematical Sciences, University of Science and
Technology of China
No. 96 Jinzhai Road, Hefei, Anhui 230026, P. R. China<EMAIL_ADDRESS>
###### Abstract.

We prove a higher-dimensional Chevalley restriction theorem for orthogonal
groups, which was conjectured by Chen and Ngô for reductive groups. In
characteristic $p>2$, we also prove a weaker statement. In characteristic $0$,
the theorem implies that the categorical quotient of a commuting scheme by the
diagonal adjoint action of the group is integral and normal. As applications,
we deduce some trace identities and a certain multiplicative property of the
Pfaffian over an arbitrary commutative algebra.

###### Key words and phrases:

Chevalley Restriction Theorem, Invariant Theory, Commuting Scheme, Orthogonal
Groups, Pfaffians

###### 2020 Mathematics Subject Classification:

14L30 (primary), 20G05 (secondary).

##  1\. Introduction

Let $G$ be a reductive group over an algebraically closed field $\mathbb{K}$
with Lie algebra $\mathfrak{g}$. For an integer $d\geq 2$, let
$\mathfrak{C}^{d}_{\mathfrak{g}}\subset\mathfrak{g}^{d}$ be the commuting
scheme, which is defined as the scheme-theoretic fiber of the commutator map
over the zero

$\mathfrak{g}^{d}\rightarrow\prod\limits_{i<j}\mathfrak{g},\ \
(x_{1},\cdots,x_{d})\mapsto\prod_{i<j}[x_{i},x_{j}].$

Its underlying variety (the reduced induced closed subscheme)
$\mathfrak{C}^{d}_{\mathfrak{g},red}$ is called the commuting variety. As a
set, $\mathfrak{C}^{d}_{\mathfrak{g},red}$ consists of $d$-tuples
$(x_{1},\cdots,x_{d})\in\mathfrak{g}^{d}$ such that $[x_{i},x_{j}]=0$, for all
$1\leq i,j\leq d$. It is a long-standing open question whether or not
$\mathfrak{C}^{d}_{\mathfrak{g}}$ is reduced, that is
$\mathfrak{C}^{d}_{\mathfrak{g}}=\mathfrak{C}^{d}_{\mathfrak{g},red}$. When
the characteristic of $\mathbb{K}$ ($\rm{char}\ \mathbb{K}$ for short) is
zero, Charbonnel [References] recently claims a proof for
$\mathfrak{C}^{2}_{\mathfrak{g}}=\mathfrak{C}^{2}_{\mathfrak{g},red}$.
Although there is no adequate evidence to expect that
$\mathfrak{C}^{d}_{\mathfrak{g}}$ is reduced for general $d$, one can study
the categorical quotient $\mathfrak{C}^{d}_{\mathfrak{g}}/\\!\\!/G$ and ask
the same question. Here $G$ acts on $\mathfrak{g}^{d}$ by the diagonal adjoint
action, and the action leaves $\mathfrak{C}^{d}_{\mathfrak{g}}$ stable.

Let $T$ be a maximal torus of $G$ and $\mathfrak{t}$ be the Lie algebra of
$T$. Then the Weyl group $W:=N_{G}(T)/T$ acts on $\mathfrak{t}^{d}$
diagonally. The embedding $\mathfrak{t}^{d}\hookrightarrow\mathfrak{g}^{d}$
factors through $\mathfrak{C}^{d}_{\mathfrak{g}}$ and induces the natural
morphism

$\Phi:\mathfrak{t}^{d}/\\!\\!/W\rightarrow\mathfrak{C}^{d}_{\mathfrak{g}}/\\!\\!/G.$

In studying the Hitchin morphism from the moduli stack of principal $G$-Higgs
bundles on a proper smooth variety $X$ of dimension $d\geq 2$, Chen and Ngô
[4] are led to

######  Conjecture 1.1 (Chen-Ngô).

The morphism
$\Phi:\mathfrak{t}^{d}/\\!\\!/W\rightarrow\mathfrak{C}^{d}_{\mathfrak{g}}/\\!\\!/G$
is an isomorphism.

When $d=1$ and $\rm{char}\ \mathbb{K}=0$, Conjecture 1.1 is simply the
classical Chevalley restriction theorem. Since in the context of Higgs
bundles, $d$ is the dimension of the underlying variety $X$, we view the
conjecture as a higher-dimensional analogue of Chevalley restriction theorem.
Note when $d=2$ and $\rm{char}\ \mathbb{K}=0$, this conjecture is a special
case (degree zero part) of a more general conjecture proposed by Berest et al.
[References].

If $\rm{char}\ \mathbb{K}=0$, Conjecture 1.1 is known to hold for
$G=GL_{n}(\mathbb{K})$ (Vaccarino [References], Domokos [References], and
later Chen-Ngô [References] independently; see also Gan-Ginzburg [References]
for case $d=2$) and for $G=Sp_{n}(\mathbb{K})$ (Chen-Ngô [References]). A
weaker version
$\mathfrak{t}^{d}/\\!\\!/W\xrightarrow{\sim}\mathfrak{C}^{d}_{\mathfrak{g},red}/\\!\\!/G$
is proved by Hunziker [References] if $G$ is of type $A,B,C,D$ or $G_{2}$.

If $\rm{char}\ \mathbb{K}>0$, Conjecture 1.1 is largely open. However,
Vaccarino [References] proved the weaker version
$\mathfrak{t}^{d}/\\!\\!/W\xrightarrow{\sim}\mathfrak{C}^{d}_{\mathfrak{g},red}/\\!\\!/G$
for $G=GL_{n}(\mathbb{K})$.

The main purpose of the article is to prove Conjecture 1.1 for orthogonal
groups in case $\rm{char}\ \mathbb{K}=0$ and to prove a weaker version of this
conjecture in case $\rm{char}\ \mathbb{K}>2$. To be more precise, our main
result is the following (see Theorems 3.4, 3.5, 4.3 and 4.4):

######  Theorem 1.2.

Suppose $n\geq 2$, $d\geq 1$, and $G$ is an orthogonal group
$O_{n}(\mathbb{K})$ or a special orthogonal group $SO_{n}(\mathbb{K})$.

  1. (1)

If $\rm{char}\ \mathbb{K}=0$, then
$\Phi:\mathfrak{t}^{d}/\\!\\!/W\rightarrow\mathfrak{C}^{d}_{\mathfrak{g}}/\\!\\!/G$
is an isomorphism.

  2. (2)

If $\rm{char}\ \mathbb{K}>2$, then
$\Phi:\mathfrak{t}^{d}/\\!\\!/W\rightarrow\mathfrak{C}^{d}_{\mathfrak{g},red}/\\!\\!/G$
is an isomorphism.

Our proof can also treat in a uniform way the case $G=Sp_{n}(\mathbb{K})$ ($n$
even), which is due to Chen-Ngô [References] if $\rm{char}\ \mathbb{K}=0$.

Since $W$ is finite and $\mathfrak{t}^{d}$ is an affine space, the theorem
implies

######  Corollary 1.3.

If $\rm{char}\ \mathbb{K}=0$ and $G=O_{n}(\mathbb{K})$ or
$SO_{n}(\mathbb{K})$, the quotient $\mathfrak{C}^{d}_{\mathfrak{g}}/\\!\\!/G$
is integral (i.e. reduced and irreducible) and normal. ∎

We will divide the proof into two parts, according to the type of the Weyl
group $W$. In the first part, $W$ is of type $B$, and this includes cases
$G=O_{n}(\mathbb{K})$, $SO_{n}(\mathbb{K})$ ($n$ odd), and
$Sp_{n}(\mathbb{K})$ ($n$ even). Based on the results in the first part
(Theorems 3.4 and 3.5), we prove Theorem 4.3 and Theorem 4.4 in the second
one, which corresponds to the $W$ of type $D$ ($G=SO_{n}(\mathbb{K})$, $n$
even).

Next let us explain the strategy of our proof of Theorem 3.4. Let
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$ (resp.
$\mathbb{K}[\mathfrak{t}^{d}]$) be the coordinate ring of
$\mathfrak{C}^{d}_{\mathfrak{g}}$ (resp. $\mathfrak{t}^{d}$). In order to show
the restriction homomorphism
$\Phi:\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$
is an isomorphism, it suffices to construct a $\mathbb{K}$-linearly spanning
set of $\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]^{G}$, which are mapped
bijectively to a $\mathbb{K}$-linear basis of
$\mathbb{K}[\mathfrak{t}^{d}]^{W}$.

The results of Procesi [References] give a set of generators of
$\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G}$, which induces a set of generators of
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$ under the surjective
homomorphism
$\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G}\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$.
We shall encode these generators in the determinant of a universal matrix. In
(3.0.1), we define the formal power series

$F_{\mathfrak{g}}=\det(I_{n}+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots i_{d}}),$

with
$F_{\mathfrak{g}}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}[[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$, where the index set $S$ is

$\\{(i_{1},\cdots,i_{d})\>|\>i_{1},\cdots,i_{d}\in\mathbb{Z}_{\geq
0},(i_{1},\cdots,i_{d})\neq(0,\cdots,0),i_{1}+\cdots+i_{d}\textmd{ is
even}\\},$

and $X(1),\cdots,X(d)$ are “generic” commuting skew symmetric $n\times n$
matrices over $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$. Based on
Procesi’s results, we can show that a $\mathbb{K}$-linearly spanning set of
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$ is given by the coefficients
of $F_{\mathfrak{g}}$, and in turn by the coefficients of
$\sqrt{F_{\mathfrak{g}}}$, the unique square root of $F_{\mathfrak{g}}$ with
constant coefficient $1$.

Under the restriction homomorphism
$\Phi:\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$,
we deduce that

$\Phi(\sqrt{F_{\mathfrak{g}}})=\sqrt{F_{\mathfrak{t}}}.$

See (3.0.1), (3.0.2) for the definition of $F_{\mathfrak{t}}$ and
$\sqrt{F_{\mathfrak{t}}}$. Now explicit computations show
$\deg\sqrt{F_{\mathfrak{t}}}\leq\lfloor\frac{n}{2}\rfloor$, that is,
$\sqrt{F_{\mathfrak{t}}}$ can be written as

$\sqrt{F_{\mathfrak{t}}}=1+\sum\limits_{w\in\mathcal{M}(\lfloor\frac{n}{2}\rfloor)}c_{w}w,$

where $c_{w}\in\mathbb{K}[\mathfrak{t}^{d}]^{W}$ and
$\mathcal{M}(\lfloor\frac{n}{2}\rfloor)$ is the set of non-empty monomials in
variables $\\{T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S\\}$ whose degree
is less than or equal to $\lfloor\frac{n}{2}\rfloor$. We can also show that
the coefficients of $\sqrt{F_{\mathfrak{t}}}$ form a $\mathbb{K}$-linear basis
of $\mathbb{K}[\mathfrak{t}^{d}]^{W}$. So to finish, all we need to show is
$\deg\sqrt{F_{\mathfrak{g}}}\leq\lfloor\frac{n}{2}\rfloor$. This turns out to
be the technical heart of this paper.

To prove $\deg\sqrt{F_{\mathfrak{g}}}\leq\lfloor\frac{n}{2}\rfloor$, we
explicitly construct a power series $N$ with constant coefficient $1$ such
that $N^{2}=F_{\mathfrak{g}}$. By the uniqueness of square roots with constant
coefficient $1$, we see $N=\sqrt{F_{\mathfrak{g}}}$. So the degree bound
follows from the explicit construction of $N$.

If $G=Sp_{n}(\mathbb{K})$, our construction of $N$ is essentially the same as
that presented in Chen-Ngô [References]. However, if $G=O_{n}(\mathbb{K})$,
the construction of $N$ is much more involved and poses significant
challenges. We provide the full details in Section 5.2.

### Acknowledgments

L. S. thanks Jianxun Hu for the encouragement. J. X. would like to thank
Xiaowen Hu and Mao Sheng for helpful discussions related to this work. The
authors would like to thank Hao Sun for the helpful comments which improve the
manuscript. We are very grateful to the anonymous referee for a careful
reading and for providing valuable references. During the preparation of the
article, L. S. was partially supported by the Guangdong Basic and Applied
Basic Research Foundation 2020A1515010876, X. X. was partially supported by
the Innovation Program for Quantum Science and Technology (2021ZD0302902), and
J. X. was partially supported by the National Key R and D Program of China
(2020YFA0713100), CAS Project for Young Scientists in Basic Research
(YSBR-032) and NSFC (12271495).

##  2\. Notations and preliminaries

In this section we fix some notations and record some useful lemmas that will
be used frequently in the subsequent sections.

Throughout the paper, $\mathbb{K}$ is an algebraically closed field of
characteristic not equal to $2$. All rings are commutative, unless otherwise
specified.

Given a ring $R$, we denote by $M_{n}(R)$ the set of $n\times n$ matrices over
$R$, and for a matrix $M$, we denote by $M^{t}$ its transpose, and by $M(i,j)$
the $(i,j)$-entry of $M$. We denote the coordinate ring of an affine
$\mathbb{K}$-scheme $X$ by $\mathbb{K}[X]$. In particular, if $V$ is a
$\mathbb{K}$-linear space, $\mathbb{K}[V]$ means the $\mathbb{K}$-algebra of
polynomial functions on $V$. If a group $G$ acts $\mathbb{K}$-linearly on $V$,
then $G$ acts naturally on $\mathbb{K}[V]$ by $g\cdot f\ (v)=f(g^{-1}\cdot
v)$, for $g\in G,\ f\in\mathbb{K}[V],\ v\in V$. The $\mathbb{K}$-algebra of
$G$-invariant polynomials on $V$ is denoted by $\mathbb{K}[V]^{G}$. For
$n\in\mathbb{Z}$, we denote by $\lfloor n\rfloor$ the maximal integer less
than or equal to $n$.

Here are some lemmas on square roots of formal power series.

######  Lemma 2.1.

Suppose $R=\oplus_{i=0}^{\infty}R_{i}$ is a graded $\mathbb{K}$-algebra. Let
$f_{1},f_{2}\in R$ and let $c_{j}\in R_{0}$ be the degree zero part of
$f_{j}$, $j=1,2$. If $f_{1}^{2}=f_{2}^{2}$, $c_{1}=c_{2}$, and if $c_{1}$ is
not a zero-divisor in $R$, then $f_{1}=f_{2}$.

###### Proof.

Put $f_{1}=c_{1}+g_{1}$ and $f_{2}=c_{2}+g_{2}$. Since $f_{1}^{2}=f_{2}^{2}$
and $c_{1}=c_{2}$, we see

$(c_{1}+c_{2}+g_{1}+g_{2})(g_{1}-g_{2})=0.$

Since $c_{1}+c_{2}=2c_{1}$ is not a zero-divisor in $R$, it is easy to see
$c_{1}+c_{2}+g_{1}+g_{2}$ is not a zero-divisor in $R$. From this, we deduce
$g_{1}=g_{2}$ and hence $f_{1}=f_{2}$. ∎

######  Lemma 2.2.

Let $R$ be a $\mathbb{K}$-algebra, and $R[[t_{1},t_{2},\cdots]]$ be the formal
power series ring with countable variables.

  1. (1)

Suppose $f_{1},f_{2}\in R[[t_{1},t_{2},\cdots]]$, and let
$c_{j}=f_{j}(0,0,\cdots)\in R$ be the constant coefficient of $f_{j}$. If
$f_{1}^{2}=f_{2}^{2}$, $c_{1}=c_{2}$, and if $c_{1}$ is not a zero-divisor in
$R$, then $f_{1}=f_{2}$.

  2. (2)

Suppose $g\in R[[t_{1},t_{2},\cdots]]$ has constant coefficient $1$, then
there exists a unique $f\in R[[t_{1},t_{2},\cdots]]$ whose constant
coefficient is $1$ and which satisfies $f^{2}=g$. Moreover, the
$\mathbb{K}$-subalgebra of $R$ generated by the coefficients of $g$ coincides
with that generated by the coefficients of $f$.

###### Proof.

  1. (1)

The proof is the same as that for the previous lemma.

  2. (2)

The uniqueness of $f$ follows from $(1)$. In order to show its existence, let
$\mathcal{T}$ be the set of all nonempty monomials in the variables
$\\{t_{1},t_{2},\cdots\\}$. For $w_{1},\ w_{2}\in\mathcal{T}$, define
$w_{1}\leq w_{2}$ if there exists $w\in\\{1\\}\cup\mathcal{T}$ such that
$w_{2}=w_{1}w$. This gives a partial order on $\mathcal{T}$. Write
$g=1+\sum_{w\in\mathcal{T}}a_{w}w$, with $a_{w}\in R$. For each
$w\in\mathcal{T}$, we will define by induction $b_{w}\in R$, so that
$f:=1+\sum_{w\in\mathcal{T}}b_{w}w$ satisfies $f^{2}=g$.

Suppose $w\in\mathcal{T}$ and suppose for any $v\in\mathcal{T}$ with $v<w$,
the element $b_{v}\in R$ has been defined. Let
$\mathcal{T}_{w}:=\\{(v_{1},v_{2})\in\mathcal{T}^{2}|v_{1}<w,\
v_{2}<w,v_{1}v_{2}=w\\}$. Then define

$b_{w}:=\frac{1}{2}(a_{w}-\sum_{(v_{1},v_{2})\in\mathcal{T}_{w}}b_{v_{1}}b_{v_{2}}).$

By direct computations, we can see that $f:=1+\sum_{w\in\mathcal{T}}b_{w}w$
satisfies $f^{2}=g$. Moreover, it follows from the explicit expressions that
the $\mathbb{K}$-subalgebras $\mathbb{K}[a_{w}|w\in\mathcal{T}]$ and
$\mathbb{K}[b_{w}|w\in\mathcal{T}]$ of $R$ coincide.

∎

######  Lemma 2.3.

Let $R=\oplus_{i=0}^{\infty}R_{i}$ be a graded $\mathbb{K}$-algebra, with
$R_{0}=\mathbb{K}$. Suppose $f,g\in R[[y,z_{0},z_{1},\cdots]]$ satisfy
$f^{2}=y^{2n}g$ and $g=z_{0}^{2m}+g_{1}+g_{2}+\cdots$, for $g_{i}\in
R_{i}[[y,z_{0},z_{1},\cdots]]$, with $n,m\geq 0$, then there exists
$\tilde{f}\in R[[y,z_{0},z_{1},\cdots]]$ satisfying $f=y^{n}\tilde{f}$.
Moreover, if $f^{2}=y^{2n}$, then $f=\pm y^{n}$.

###### Proof.

Let $f=f_{0}+f_{1}+\cdots$, with $f_{i}\in R_{i}[[y,z_{0},z_{1},\cdots]]$. By
$f^{2}=y^{2n}g$ we get $f_{0}^{2}=y^{2n}z_{0}^{2m}$ in
$R_{0}[[y,z_{0},z_{1},\cdots]]=\mathbb{K}[[y,z_{0},z_{1},\cdots]]$. This
implies $f_{0}=\pm y^{n}z_{0}^{m}$. Assume without loss of generality that
$f_{0}=y^{n}z_{0}^{m}$. Now suppose $i\geq 1$ and suppose for each $1\leq
j<i$, there exists $\tilde{f}_{j}\in R_{j}[[y,z_{0},z_{1},\cdots]]$ such that
$f_{j}=y^{n}\tilde{f}_{j}$. By comparing the $R_{i}[[y,z_{0},z_{1},\cdots]]$
part of $f^{2}$ and $y^{2n}g$, we get
$\sum_{j=0}^{i}f_{j}f_{i-j}=y^{2n}g_{i}$. Since
$\sum_{j=0}^{i}f_{j}f_{i-j}=2f_{0}f_{i}+\sum_{j=1}^{i-1}f_{j}f_{i-j}=2y^{n}z_{0}^{m}f_{i}+y^{2n}\sum_{j=1}^{i-1}\tilde{f}_{j}\tilde{f}_{i-j}$,
we obtain
$2y^{n}z_{0}^{m}f_{i}=y^{2n}(g_{i}-\sum_{j=1}^{i-1}\tilde{f}_{j}\tilde{f}_{i-j})$.
As neither $z_{0}$ nor $y$ is a zero-divisor in $R[[y,z_{0},z_{1},\cdots]]$,
this implies
$2z_{0}^{m}f_{i}=y^{n}(g_{i}-\sum_{j=1}^{i-1}\tilde{f}_{j}\tilde{f}_{i-j})$,
and then $f_{i}=y^{n}\tilde{f}_{i}$ for some $\tilde{f}_{i}\in
R_{i}[[y,z_{0},z_{1},\cdots]]$. By induction on $i$ we see that for each
$i\geq 1$, there exists $\tilde{f}_{i}\in R_{i}[[y,z_{0},z_{1},\cdots]]$
satisfying $f_{i}=y^{n}\tilde{f}_{i}$. Since $f\in R[[y,z_{0},z_{1},\cdots]]$,
the formal sum $\tilde{f}:=z_{0}^{m}+\sum_{i=1}^{\infty}\tilde{f}_{i}$ can be
viewed as an element of $R[[y,z_{0},z_{1},\cdots]]$ and it satisfies
$f=y^{n}\tilde{f}$.

If $f^{2}=y^{2n}$, then $\tilde{f}^{2}=1$, and this implies $\tilde{f}\in
R_{0}=\mathbb{K}$. So $\tilde{f}=\pm 1$ and $f=\pm y^{n}$. ∎

##  3\. Main theorems: type $B$ case

Suppose $n\geq 2,\ d\geq 1$ are positive integers. Let

$O_{n}(\mathbb{K}):=\\{A\in M_{n}(\mathbb{K})\>|\>AA^{t}=I_{n}\\}$

be the orthogonal group, and if $n$ is even, let

$Sp_{n}(\mathbb{K}):=\\{A\in M_{n}(\mathbb{K})\>|\>A^{t}JA=I_{n}\\}$

be the symplectic group, where $J=\begin{pmatrix}0&I_{\frac{n}{2}}\\\
-I_{\frac{n}{2}}&0\end{pmatrix}$.

Throughout this section, $G$ is one of the following groups:

$G=\begin{cases}O_{n}(\mathbb{K}),\\\ Sp_{n}(\mathbb{K}),&n\textmd{ even},\\\
SO_{n}(\mathbb{K}),&n\textmd{ odd}.\end{cases}$

Let $T$ be a maximal torus of $G$ and $W=N_{G}(T)/T$ be the Weyl group. If
$n\geq 4$ is even, the Weyl groups of $O_{n}(\mathbb{K})$ and
$Sp_{n}(\mathbb{K})$ are isomorphic, and both are of type $B$. As usual,
$\mathfrak{g}$ denotes the Lie algebra of $G$. Explicitly,

$\mathfrak{g}=\begin{cases}\mathfrak{so}_{n},\ \textmd{ if }G\ \textmd{ is
}O_{n}(\mathbb{K})\textmd{ or }SO_{n}(\mathbb{K});\\\ \mathfrak{sp}_{n},\
\textmd{ if }G\ \textmd{ is }Sp_{n}(\mathbb{K}),\end{cases}$

where we fix the realizations of the Lie algebras as matrices:

$\displaystyle\mathfrak{so}_{n}$ $\displaystyle=$ $\displaystyle\\{A\in
M_{n}(\mathbb{K}\>)|\>A+A^{t}=0\\},$ $\displaystyle\mathfrak{sp}_{n}$
$\displaystyle=$ $\displaystyle\\{A\in M_{n}(\mathbb{K})\>|\>A^{t}J+JA=0\\}.$

The Lie algebra $\mathfrak{t}$ of $T$ is a Cartan subalgebra of
$\mathfrak{g}$. Via the diagonal adjoint representation, the group $G$ acts on
$\mathfrak{g}^{d}$ and this induces an action of $W$ on $\mathfrak{t}^{d}$.
For $1\leq k\leq d$, $1\leq i,j\leq n$, let $x(k)_{ij}$ be the polynomial
function of $\mathfrak{g}^{d}$ whose value at a point
$(A_{1},\cdots,A_{d})\in\mathfrak{g}^{d}$ is the $(i,j)$-entry of the matrix
$A_{k}\in M_{n}(\mathbb{K})$.

Over the ring $\mathbb{K}[\mathfrak{g}^{d}]$, consider the “generic” $n\times
n$ matrices $X(1)$, $X(2)$, $\cdots$, $X(d)$, such that the $(i,j)$-entry of
$X(k)$ is $x(k)_{ij}$. Let $I$ be the ideal of $\mathbb{K}[\mathfrak{g}^{d}]$
generated by all of the entries of the matrices
$[X(k),X(l)]:=X(k)X(l)-X(l)X(k),\ 1\leq k,l\leq d$, and define the quotient
ring

$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]:=\mathbb{K}[\mathfrak{g}^{d}]/I.$

This ring can be viewed as the coordinate ring of the commuting scheme
$\mathfrak{C}^{d}_{\mathfrak{g}}$. Since $I$ is obviously a homogeneous ideal
of the polynomial ring $\mathbb{K}[\mathfrak{g}^{d}]$, the quotient ring
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]=\oplus_{i=0}^{\infty}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]_{i}$
is a graded $\mathbb{K}$-algebra. Moreover, the degree zero part
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]_{0}=\mathbb{K}$, and the degree
one part $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]_{1}$ is
$\mathbb{K}$-linearly spanned by $x(k)_{ij},\ 1\leq k\leq d,\ 1\leq i,j\leq
n$. From now on, we view $X(k)$ $(1\leq k\leq d)$ as matrices over
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$. Note these matrices are
mutually commutative by the definition of $I$, and
$(X(1),\cdots,X(d))\in\mathfrak{C}^{d}_{\mathfrak{g}}(\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}])$
can also be viewed as the tautological
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$-valued point of the commuting
scheme $\mathfrak{C}^{d}_{\mathfrak{g}}$.

Since obviously $I$ is $G$-invariant, we have the induced action of $G$ on
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$. Moreover, the action preserves
the degrees on $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$, so that the
invariant subring $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$ is still a
graded $\mathbb{K}$-algebra whose degree zero part is equal to $\mathbb{K}$.
Any polynomial function on $\mathfrak{g}^{d}$ restricts to a polynomial
function on $\mathfrak{t}^{d}$ through the inclusion
$\mathfrak{t}^{d}\subset\mathfrak{g}^{d}$, and the restriction homomorphism
$\mathbb{K}[\mathfrak{g}^{d}]\rightarrow\mathbb{K}[\mathfrak{t}^{d}]$ factors
through $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$. This induces the
following restriction homomorphism between the invariant rings:

$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}.$

For $1\leq k\leq d$, let $Y(k)$ be the image of $X(k)$ under the restriction
homomorphism $M_{n}(\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}])\rightarrow
M_{n}(\mathbb{K}[\mathfrak{t}^{d}])$. Then $Y(1),\cdots,Y(d)$ are commuting
$n\times n$ matrices over $\mathbb{K}[\mathfrak{t}^{d}]$. We define the
following formal power series:

(3.0.1)
$\begin{split}F_{\mathfrak{g}}&:=\det(I_{n}+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots i_{d}}),\\\
F_{\mathfrak{t}}&:=\det(I_{n}+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}Y(1)^{i_{1}}Y(2)^{i_{2}}\cdots Y(d)^{i_{d}}T_{i_{1}\cdots
i_{d}}),\end{split}$

with
$F_{\mathfrak{g}}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ and
$F_{\mathfrak{t}}\in\mathbb{K}[\mathfrak{t}^{d}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$. Here and throughout this paper, $S$
denotes the index set

$\\{(i_{1},\cdots,i_{d})\>|\>i_{1},\cdots,i_{d}\in\mathbb{Z}_{\geq
0},(i_{1},\cdots,i_{d})\neq(0,\cdots,0),i_{1}+\cdots+i_{d}\textmd{ is
even}\\}.$

Since determinants are invariant under conjugations, we see in fact
$F_{\mathfrak{g}}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}[[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ and
$F_{\mathfrak{t}}\in\mathbb{K}[\mathfrak{t}^{d}]^{W}[[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$. Moreover, under the restriction
homomorphism, we have $\Phi(F_{\mathfrak{g}})=F_{\mathfrak{t}}$.

By Lemma 2.2, there exists a unique
$\sqrt{F_{\mathfrak{g}}}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}[[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ with constant coefficient $1$ and
satisfying $\sqrt{F_{\mathfrak{g}}}^{2}=F_{\mathfrak{g}}$. Similarly, we
denote
$\sqrt{F_{\mathfrak{t}}}\in\mathbb{K}[\mathfrak{t}^{d}]^{W}[[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ as the unique square root of
$F_{\mathfrak{t}}$ with constant coefficient $1$. Define $\mathcal{M}$ to be
the set of non-empty monomials in the variables $\\{T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S\\}$. We expand these two formal power series
as follows:

(3.0.2) $\sqrt{F_{\mathfrak{g}}}=1+\sum\limits_{w\in\mathcal{M}}d_{w}w,\ \ \
\sqrt{F_{\mathfrak{t}}}=1+\sum\limits_{w\in\mathcal{M}}c_{w}w,$

with $d_{w}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$,
$c_{w}\in\mathbb{K}[\mathfrak{t}^{d}]^{W}$.

For a commutative ring $R$, and for a nonzero formal power series $f\in
R[[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S]]$, we write
$f=a_{0}+\sum\limits_{w\in\mathcal{M}}a_{w}w$ with $a_{0},\ a_{w}\in R$.
Define the degree $\deg f\in\mathbb{Z}_{\geq 0}\cup\\{+\infty\\}$ of $f$ by

$\deg f:=\begin{cases}0,\ \textmd{if }f=a_{0};\\\ \sup\\{\deg
w|w\in\mathcal{M},a_{w}\neq 0\\},\ \textmd{if }f\neq a_{0}.\end{cases}$

We say $f$ is of finite degree if $\deg f<+\infty$.

Let $\bigoplus\limits_{S}\mathbb{K}$ be the direct sum of $\mathbb{K}$ indexed
by $S$. Note that for an element $\underline{t}=(t_{i_{1}\cdots
i_{d}})\in\bigoplus\limits_{S}\mathbb{K}$, the components $t_{i_{1}\cdots
i_{d}}=0$ for all but finitely many $(i_{1},\cdots,i_{d})\in S$. Then if a
formal power series $f\in R[[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in
S]]$ is of finite degree, we get a well-defined element $f(\underline{t})\in
R$ by evaluating $f$ at $T_{i_{1}\cdots i_{d}}=t_{i_{1}\cdots i_{d}}$. Note
$f(\underline{0})\in R$ is the constant coefficient of $f$, where
$\underline{0}$ is the zero element in $\bigoplus\limits_{S}\mathbb{K}$.

Let $G$ act on $M_{n}(\mathbb{K})^{d}$ via simultaneous conjugation:
$g\cdot(A_{1},\cdots,A_{d})=(gA_{1}g^{-1},\cdots,gA_{d}g^{-1})$, for $g\in G$,
$(A_{1},\cdots,A_{d})\in M_{n}(\mathbb{K})^{d}$. Then the inclusion
$\mathfrak{g}^{d}\subset M_{n}(\mathbb{K})^{d}$ induces a restriction
homomorphism
$\varphi:\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G}\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$,
and we have the following diagram:

$\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G}\xrightarrow{\varphi}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}\xrightarrow{\Phi}\mathbb{K}[\mathfrak{t}^{d}]^{W}.$

######  Proposition 3.1.

The image of $\varphi$ is $\mathbb{K}$-linearly spanned by the coefficients
$\\{1\\}\cup\\{d_{w}|w\in\mathcal{M}\\}$ of $\sqrt{F_{\mathfrak{g}}}$.

###### Proof.

Let $S^{\prime}:=\\{(i_{1},\cdots,i_{d})|i_{1},\cdots,i_{d}\in\mathbb{Z}_{\geq
0},i_{1}+\cdots+i_{d}\textmd{ odd}\\}$, and define the following $n\times n$
matrix

$A:=I_{n}+\sum_{(i_{1},\cdots,i_{d})\in S\cup S^{\prime}}X(1)^{i_{1}}\cdots
X(d)^{i_{d}}T_{i_{1}\cdots i_{d}}.$

Let $F^{\prime}:=\det A$, a formal power series of the variables
$\\{T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S\cup S^{\prime}\\}$. By
Procesi [References] (char. $0$ case) and Zubkov [References] (char. $p>2$
case), $\mathrm{Im}\ \varphi$ is generated as a $\mathbb{K}$-algebra by the
coefficients of $\mathrm{tr}\ A$. From this or directly from Lopatin
[References, Corollary 2], we deduce that $\mathrm{Im}\ \varphi$ is generated
by the coefficients of $F^{\prime}$, and hence by the coefficients of
$(F^{\prime})^{2}$ according to Lemma 2.2 (2). Note if $G=O_{n}(\mathbb{K})$
or $SO_{n}(\mathbb{K})$, the matrix $X(1)^{i_{1}}\cdots X(d)^{i_{d}}$ is
symmetric for $(i_{1},\cdots,i_{d})\in S$ and skew-symmetric for
$(i_{1},\cdots,i_{d})\in S^{\prime}$, so in these cases

$\displaystyle(F^{\prime})^{2}$ $\displaystyle=\det(AA^{t})$
$\displaystyle=\det[(I_{n}+\sum_{(i_{1},\cdots,i_{d})\in S}X(1)^{i_{1}}\cdots
X(d)^{i_{d}}T_{i_{1}\cdots i_{d}})^{2}-$
$\displaystyle(\sum_{(i_{1},\cdots,i_{d})\in S^{\prime}}X(1)^{i_{1}}\cdots
X(d)^{i_{d}}T_{i_{1}\cdots i_{d}})^{2}].$

Similarly if $G=Sp_{n}(\mathbb{K})$, we have

$\displaystyle(F^{\prime})^{2}$ $\displaystyle=\det(A(JAJ)^{t})$
$\displaystyle=\det[(I_{n}+\sum_{(i_{1},\cdots,i_{d})\in S}X(1)^{i_{1}}\cdots
X(d)^{i_{d}}T_{i_{1}\cdots i_{d}})^{2}-$
$\displaystyle(\sum_{(i_{1},\cdots,i_{d})\in S^{\prime}}X(1)^{i_{1}}\cdots
X(d)^{i_{d}}T_{i_{1}\cdots i_{d}})^{2}].$

In either case, each coefficient of $(F^{\prime})^{2}$ is a
$\mathbb{K}$-linear combination of the coefficients of
$F_{\mathfrak{g}}=\det(I_{n}+\sum_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots i_{d}})$. Then $\mathrm{Im}\
\varphi$ is generated by the coefficients of $F_{\mathfrak{g}}$, or
equivalently, by the evaluations $F_{\mathfrak{g}}(\underline{t})$
$(\underline{t}\in\bigoplus\limits_{S}\mathbb{K})$. By the multiplicative
property of determinants, for any
$\underline{t}_{1},\underline{t}_{2}\in\bigoplus\limits_{S}\mathbb{K}$, there
exists $\underline{t}_{3}\in\bigoplus\limits_{S}\mathbb{K}$, such that

(3.1.1)
$F_{\mathfrak{g}}(\underline{t}_{1})F_{\mathfrak{g}}(\underline{t}_{2})=F_{\mathfrak{g}}(\underline{t}_{3}).$

So $\mathrm{Im}\ \varphi$ is $\mathbb{K}$-linearly spanned by
$F_{\mathfrak{g}}(\underline{t})$
$(\underline{t}\in\bigoplus\limits_{S}\mathbb{K})$.

By Lemma 2.2 (2), the $\mathbb{K}$-subalgebra of
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$ generated by the
coefficients of $\sqrt{F_{\mathfrak{g}}}$ coincides with that generated by the
coefficients of $F_{\mathfrak{g}}$, and hence is equal to $\mathrm{Im}\
\varphi$. Moreover, the equality (3.1.1) implies that
$(\sqrt{F_{\mathfrak{g}}}(\underline{t}_{1})\sqrt{F_{\mathfrak{g}}}(\underline{t}_{2}))^{2}=(\sqrt{F_{\mathfrak{g}}}(\underline{t}_{3}))^{2}$.
By applying Lemma 2.1 to the graded $\mathbb{K}$-algebra
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$, we obtain
$\sqrt{F_{\mathfrak{g}}}(\underline{t}_{1})\sqrt{F_{\mathfrak{g}}}(\underline{t}_{2})=\sqrt{F_{\mathfrak{g}}}(\underline{t}_{3})$.
So finally $\mathrm{Im}\ \varphi$ is $\mathbb{K}$-linearly spanned by the
coefficients of $\sqrt{F_{\mathfrak{g}}}$. ∎

The following two lemmas are important, but technical in nature, so their
proofs will be postponed to Section 5. Here and throughout this paper,
$\mathcal{M}(\lfloor\frac{n}{2}\rfloor):=\\{w\in\mathcal{M}|\deg
w\leq\lfloor\frac{n}{2}\rfloor\\}$ is the subset of $\mathcal{M}$ consisting
of monomials with degree less than or equal to $\lfloor\frac{n}{2}\rfloor$.

######  Lemma 3.2.

$\deg\sqrt{F_{\mathfrak{t}}}\leq\lfloor\frac{n}{2}\rfloor$, and
$\\{1\\}\cup\\{c_{w}|w\in\mathcal{M}(\lfloor\frac{n}{2}\rfloor)\\}$ is a
$\mathbb{K}$-linear basis of $\mathbb{K}[\mathfrak{t}^{d}]^{W}$.

######  Lemma 3.3.

$\deg\sqrt{F_{\mathfrak{g}}}\leq\lfloor\frac{n}{2}\rfloor$.

Assuming these lemmas, we can now prove one of our main theorems.

######  Theorem 3.4.

If $\rm{char}\ \mathbb{K}=0$, then the restriction homomorphism
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$
is an isomorphism of $\mathbb{K}$-algebras.

###### Proof.

By Lemma 3.2 and 3.3, the expansions (3.0.2) reduce to the following:

(3.4.1)
$\sqrt{F_{\mathfrak{g}}}=1+\sum\limits_{w\in\mathcal{M}(\lfloor\frac{n}{2}\rfloor)}d_{w}w,\
\ \
\sqrt{F_{\mathfrak{t}}}=1+\sum\limits_{w\in\mathcal{M}(\lfloor\frac{n}{2}\rfloor)}c_{w}w.$

Since
$\Phi(\sqrt{F_{\mathfrak{g}}})^{2}=\Phi(F_{\mathfrak{g}})=F_{\mathfrak{t}}=(\sqrt{F_{\mathfrak{t}}})^{2}$,
and both the constant coefficients of $\Phi(\sqrt{F_{\mathfrak{g}}})$,
$\sqrt{F_{\mathfrak{t}}}$ are $1$, we deduce from Lemma 2.2 that
$\Phi(\sqrt{F_{\mathfrak{g}}})=\sqrt{F_{\mathfrak{t}}}$. So

(3.4.2) $\Phi(d_{w})=c_{w},\quad\text{ for all
}w\in\mathcal{M}(\lfloor\frac{n}{2}\rfloor).$

Since $\rm{char}\ \mathbb{K}=0$, the reductive group $G$ is linearly
reductive. Then
$\varphi:\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G}\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$
is surjective, because it is induced from the surjective restriction
homomorphism
$\mathbb{K}[M_{n}(\mathbb{K})^{d}]\twoheadrightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$.
By Proposition 3.1 and (3.4.1),
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$ is $\mathbb{K}$-linearly
spanned by
$\\{1\\}\cup\\{d_{w}|w\in\mathcal{M}(\lfloor\frac{n}{2}\rfloor)\\}$. Then by
Lemma 3.2 and (3.4.2), the homomorphism $\Phi$ maps a $\mathbb{K}$-linearly
spanning set bijectively to a $\mathbb{K}$-linear basis, hence it is an
isomorphism.

∎

Let
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]:=\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]/\sqrt{(0)}$
be the quotient of $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$ by its
nilpotent radical. This is the coordinate ring of the commuting variety
$\mathfrak{C}^{d}_{\mathfrak{g},red}$. As $\mathbb{K}[\mathfrak{t}^{d}]^{W}$
is reduced,
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$
factors through $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}$.

######  Theorem 3.5.

If $\rm{char}\ \mathbb{K}=p>2$, then the restriction homomorphism
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$
is an isomorphism of $\mathbb{K}$-algebras.

###### Proof.

Let
$\pi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}$
be the homomorphism induced by the natural quotient homomorphism
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]$.
Consider the composition of homomorphisms:

$\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G}\xrightarrow{\varphi}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}\xrightarrow{\pi}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}.$

By the same arguments as above, we can see that when restricted on the
subspace $\mathrm{Im}\ (\pi\circ\varphi)$, $\Phi$ induces an isomorphism
$\mathrm{Im}\
(\pi\circ\varphi)\xrightarrow{\sim}\mathbb{K}[\mathfrak{t}^{d}]^{W}$ of
$\mathbb{K}$-linear spaces. In particular,
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$
is surjective.

For any $f\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}$, it follows
from Mumford-Fogarty-Kirwan [References, Lemma A.1.2] that there exists $m\geq
1$, such that $f^{m}\in\mathrm{Im}~{}(\pi\circ\varphi)$. If $\Phi(f)=0$, then
$\Phi(f^{m})=0$. Since $f^{m}\in\mathrm{Im}~{}(\pi\circ\varphi)$ and
$\Phi|_{\mathrm{Im}~{}(\pi\circ\varphi)}$ is an isomorphism, we obtain
$f^{m}=0$. This implies $f=0$ as the ring
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}$ is reduced. So we obtain
$\Phi$ is injective. This in turn implies that
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$
is an isomorphism.

∎

##  4\. Main theorems: type $D$ case

In this section, keeping the same notations as in Section 3, we assume
furthermore that $n\geq 2$ is even and $G=O_{n}(\mathbb{K})$. Let
$G^{\prime}:=SO_{n}(\mathbb{K})$, and define the corresponding Weyl group by
$W^{\prime}:=N_{G^{\prime}}(T)/T$. Note both $G^{\prime}\subset G$ and
$W^{\prime}\subset W=N_{G}(T)/T$ are subgroups of index two. We have the
following commutative diagram.

$\begin{diagram}$

Take $w_{0}\in W$ which generates the quotient group
$W/W^{\prime}\simeq\mathbb{Z}/2\mathbb{Z}$. Note $W/W^{\prime}$ acts naturally
on $\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}$ and we have the eigen-subspace
decomposition

$\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}=\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(0)}\oplus\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)},$

where
$\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(0)}=\mathbb{K}[\mathfrak{t}^{d}]^{W}$
is the invariant part and

$\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}=\\{v\in\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}|w_{0}v=-v\\}.$

Similarly, the action of $G/G^{\prime}\simeq\mathbb{Z}/2\mathbb{Z}$ on
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}$ induces the eigen-
subspace decomposition
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}=\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}\oplus\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$,
with
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}=\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G}$
and
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}=\\{v\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}|g_{0}v=-v\\}$,
where $g_{0}$ is a generator of $G/G^{\prime}$. Clearly the restriction
homomorphism $\Phi$ preserves these decompositions:
$\Phi(\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(i)})\subset\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(i)}$,
$i=0,1$.

Let $S^{\prime}:=\\{(i_{1},\cdots,i_{d})|i_{1},\cdots,i_{d}\in\mathbb{Z}_{\geq
0},i_{1}+\cdots+i_{d}\textmd{ odd}\\}$, and
$\mathcal{M}^{\prime}(\frac{n}{2})$ be the set of non-empty monomials in the
variables $\\{T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S^{\prime}\\}$
whose degree is less than or equal to $\frac{n}{2}$. Define the following
formal power series:

$\begin{split}H_{\mathfrak{g}}&:={\rm Pf}(\sum\limits_{(i_{1},\cdots,i_{d})\in
S^{\prime}}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots
i_{d}}),\\\ H_{\mathfrak{t}}&:={\rm Pf}(\sum\limits_{(i_{1},\cdots,i_{d})\in
S^{\prime}}Y(1)^{i_{1}}Y(2)^{i_{2}}\cdots Y(d)^{i_{d}}T_{i_{1}\cdots
i_{d}}),\end{split}$

with
$H_{\mathfrak{g}}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S^{\prime}]]$,
$H_{\mathfrak{t}}\in\mathbb{K}[\mathfrak{t}^{d}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S^{\prime}]]$. Here ${\rm Pf}(A)$ means the
Pfaffian of a skew symmetric matrix $A$. Note for any $P\in
G=O_{n}(\mathbb{K})$,

$P\cdot H_{\mathfrak{g}}={\rm Pf}(\sum\limits_{(i_{1},\cdots,i_{d})\in
S^{\prime}}P^{-1}X(1)^{i_{1}}\cdots X(d)^{i_{d}}PT_{i_{1}\cdots i_{d}})=\det
P\cdot H_{\mathfrak{g}},$

so
$H_{\mathfrak{g}}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]_{(1)}^{G^{\prime}}[[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$. In a similar way, we can see that
$H_{\mathfrak{t}}\in\mathbb{K}[\mathfrak{t}^{d}]_{(1)}^{W^{\prime}}[[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S^{\prime}]]$. By constructions, $\deg
H_{\mathfrak{g}}\leq\frac{n}{2}$ and $\deg H_{\mathfrak{t}}\leq\frac{n}{2}$,
so we can write

(4.0.1)
$H_{\mathfrak{g}}=\sum\limits_{w\in\mathcal{M}^{\prime}(\frac{n}{2})}d^{\prime}_{w}w,\
\ \
H_{\mathfrak{t}}=\sum\limits_{w\in\mathcal{M}^{\prime}(\frac{n}{2})}c^{\prime}_{w}w,$

with
$d^{\prime}_{w}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]_{(1)}^{G^{\prime}}$,
$c^{\prime}_{w}\in\mathbb{K}[\mathfrak{t}^{d}]_{(1)}^{W^{\prime}}$.

We postpone the proof of the next lemma to Section 5.

######  Lemma 4.1.

The set $\\{c^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$ is a
$\mathbb{K}$-linear basis of
$\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$.

Now consider the composition of homomorphisms:

$\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G^{\prime}}\xrightarrow{\ \varphi\
}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}\xrightarrow{\
\pi_{1}\ }\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)},$

where $\varphi$ is the restriction homomorphism and $\pi_{1}$ is the
projection under the decomposition
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}=\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}\oplus\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$.

######  Proposition 4.2.

The image of $\pi_{1}\circ\varphi$ is $\mathbb{K}$-linearly spanned by
$\\{d^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$.

###### Proof.

Note that $\mathrm{Im}\ \varphi=\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}\oplus\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$,
since $\varphi$ is $G/G^{\prime}$-equivariant. We can deduce from Lopatin
[References, Corollary 2] that $\mathrm{Im}\ \varphi$ is generated as a
$\mathbb{K}$-algebra by $d^{\prime}_{w}\
(w\in\mathcal{M}^{\prime}(\frac{n}{2}))$ and $\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}$.
Note also
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(i)}\cdot\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(j)}\subset\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(i+j)}$
for $i,j\in\mathbb{Z}/2\mathbb{Z}=\\{0,1\\}$. It follows that $\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$ is
$\mathbb{K}$-linearly spanned by

$\\{a\cdot d^{\prime}_{w}|a\in\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)},\
w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}.$

By the proof of Proposition 3.1, $\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}=\varphi(\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G})$
is $\mathbb{K}$-linearly spanned by the evaluations
$F_{\mathfrak{g}}(\underline{t})$
$(\underline{t}\in\bigoplus\limits_{S}\mathbb{K})$. On the other hand, it is
easy to see the $\mathbb{K}$-linear subspace spanned by the coefficients
$\\{d^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$ coincides with
the $\mathbb{K}$-linear subspace spanned by the evaluations
$H_{\mathfrak{g}}(\underline{t})$
$(\underline{t}\in\bigoplus\limits_{S^{\prime}}\mathbb{K})$. It follows then
that $\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$ is
$\mathbb{K}$-linearly spanned by $\\{F_{\mathfrak{g}}(\underline{t}_{1})\cdot
H_{\mathfrak{g}}(\underline{t}_{2})|\underline{t}_{1}\in\bigoplus\limits_{S}\mathbb{K},\
\underline{t}_{2}\in\bigoplus\limits_{S^{\prime}}\mathbb{K}\\}$.

As $\det A\cdot{\rm Pf}(B)={\rm Pf}(ABA)$ for any $n\times n$ symmetric matrix
$A$ and skew symmetric matrix $B$, we can see for any
$\underline{t}_{1}\in\bigoplus\limits_{S}\mathbb{K}$ and any
$\underline{t}_{2}\in\bigoplus\limits_{S^{\prime}}\mathbb{K}$, there exists
$\underline{t}_{3}\in\bigoplus\limits_{S^{\prime}}\mathbb{K}$, such that
$F_{\mathfrak{g}}(\underline{t}_{1})\cdot
H_{\mathfrak{g}}(\underline{t}_{2})=H_{\mathfrak{g}}(\underline{t}_{3})$. So
$\mathrm{Im}\ \pi_{1}\circ\varphi=\mathrm{Im}\
\varphi\cap\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$ is
$\mathbb{K}$-linearly spanned by
$\\{H_{\mathfrak{g}}(\underline{t})|\underline{t}\in\bigoplus\limits_{S^{\prime}}\mathbb{K}\\}$,
and hence by the coefficients
$\\{d^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$. ∎

######  Theorem 4.3.

If $\rm{char}\ \mathbb{K}=0$, then the restriction homomorphism
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}$
is an isomorphism of $\mathbb{K}$-algebras.

###### Proof.

By Theorem 3.4, $\Phi$ induces an isomorphism
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}\xrightarrow{\
\ \sim\ \ }\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(0)}$. Since $\rm{char}\
\mathbb{K}=0$, the special orthogonal group $G^{\prime}$ is linearly
reductive, and then
$\varphi:\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G^{\prime}}\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}$
is surjective as it is induced from the surjection
$\mathbb{K}[M_{n}(\mathbb{K})^{d}]\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$.
So by Proposition 4.2,
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$ is
$\mathbb{K}$-linearly spanned by
$\\{d^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$. Note
$\Phi(H_{\mathfrak{g}})=H_{\mathfrak{t}}$, so
$\Phi(d^{\prime}_{w})=c^{\prime}_{w}$, for any
$w\in\mathcal{M}^{\prime}(\frac{n}{2})$. By Lemma 4.1, the set
$\\{c^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$ is a
$\mathbb{K}$-linear basis of
$\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$. It follows that $\Phi$ maps
the $\mathbb{K}$-linearly spanning set
$\\{d^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$ of
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}$ bijectively
to the $\mathbb{K}$-linear basis
$\\{c^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$ of
$\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$, and hence $\Phi$ induces a
$\mathbb{K}$-linear isomorphism
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(1)}\xrightarrow{\
\ \sim\ \ }\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$. Combining with
the isomorphism
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}_{(0)}\xrightarrow{\
\ \sim\ \ }\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(0)}$, we finish the
proof. ∎

######  Theorem 4.4.

If $\rm{char}\ \mathbb{K}=p>2$, then the restriction homomorphism

$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}$

is an isomorphism of $\mathbb{K}$-algebras.

###### Proof.

By Theorem 3.5, $\Phi$ induces an isomorphism
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(0)}\xrightarrow{\
\ \sim\ \ }\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(0)}$. So it suffices to
show
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(1)}\xrightarrow{\
\ \Phi\ \ }\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$ is an isomorphism.

Let $V$ be the image of the following composition of homomorphisms:

$\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G^{\prime}}\xrightarrow{\varphi}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]^{G^{\prime}}\xrightarrow{}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}\rightarrow\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(1)}.$

By Proposition 4.2, $V$ is $\mathbb{K}$-linearly spanned by
$\\{d^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$. The same
arguments as above show that the restriction
$\Phi|_{V}:V\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}$ is an
isomorphism. In particular,
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(1)}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$
is surjective.

Since
$\mathbb{K}[M_{n}(\mathbb{K})^{d}]\xrightarrow{}\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]$
is surjective, it follows from Mumford-Fogarty-Kirwan [References, Lemma
A.1.2] that for any
$f\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}$, a positive
power of $f$ is in the image of
$\mathbb{K}[M_{n}(\mathbb{K})^{d}]^{G^{\prime}}$. So for any
$f\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(1)}$,
there exists $m\geq 1$, such that
$f^{m}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(0)}$
($m$ even) or $f^{m}\in V$ ($m$ odd). If $\Phi(f)=0$, then $\Phi(f^{m})=0$.
Since either
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(0)}\xrightarrow{}\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(0)}$
or $\Phi|_{V}:V\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}$ is
injective, we obtain $f^{m}=0$, and then $f=0$ by the reducedness of
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}$. This shows
$\Phi:\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g},red}]^{G^{\prime}}_{(1)}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$
is injective, and we have finished the proof. ∎

##  5\. Proofs of some lemmas

The section is the most technical part of the paper. The reader is advised to
skip the section at the first reading. We keep the same notations as in
Section 3, 4, and let $m=\lfloor\frac{n}{2}\rfloor$.

###  5.1. Proofs of Lemma 3.2 and 4.1

Since for any two Cartan subalgebras $\mathfrak{t}_{1}$, $\mathfrak{t}_{2}$ of
$\mathfrak{g}$, there exists an element $h\in G$, such that
$Ad(h)\mathfrak{t}_{1}=\mathfrak{t}_{2}$ and since
$Ad(h)F_{\mathfrak{g}}=F_{\mathfrak{g}}$, Lemma 3.2 and 4.1 hold for
$\mathfrak{t}_{1}$ if and only if they hold for $\mathfrak{t}_{2}$. As a
result, we will choose a specific Cartan subalgebra $\mathfrak{t}$ for the
proofs.

If $n$ is even and $G=Sp_{n}(\mathbb{K})$, let

$\mathfrak{t}=\\{\mathrm{diag}(x_{1},\cdots,x_{m},-x_{1},\cdots,-x_{m})\>|\>x_{i}\in\mathbb{K},1\leq
i\leq m\\}.$

If $G=O_{n}(\mathbb{K})$ or $SO_{n}(\mathbb{K})$, let

$\mathfrak{t}=\\{\textmd{SK }(x_{1},\cdots,x_{m})\>|\>x_{i}\in\mathbb{K},1\leq
i\leq m\\},$

where $\textmd{SK }(x_{1},\cdots,x_{m})$ is the $n\times n$ skew symmetric
matrix:

(5.0.1) $\begin{pmatrix}0&\sqrt{-1}x_{1}&&&\\\ -\sqrt{-1}x_{1}&0&&&\\\
&&0&\sqrt{-1}x_{2}&\\\ &&-\sqrt{-1}x_{2}&0&\\\ &&&&\ddots\end{pmatrix}.$

In other words, for $1\leq i,j\leq n$, the $(i,j)$-entry of $\textmd{SK
}(x_{1},\cdots,x_{m})$ is

$\begin{cases}\sqrt{-1}x_{p}&\textmd{ if }(i,j)=(2p-1,2p)\textmd{ and }1\leq
p\leq m,\\\ -\sqrt{-1}x_{p}&\textmd{ if }(i,j)=(2p,2p-1)\textmd{ and }1\leq
p\leq m,\\\ 0&\textmd{ otherwise}.\end{cases}$

###### Proof of Lemma 3.2.

Let $V=\mathbb{K}^{m}$, we identify $V$ and $\mathfrak{t}$ by:

$\begin{cases}(x_{1},\cdots,x_{m})\mapsto\mathrm{diag}(x_{1},\cdots,x_{m},-x_{1},\cdots,-x_{m}),\
\textmd{ if }n\textmd{ even},G=Sp_{n}(\mathbb{K});\\\
(x_{1},\cdots,x_{m})\mapsto\mathrm{SK}(x_{1},\cdots,x_{m}),\ \textmd{ if
}G=O_{n}(\mathbb{K})\textmd{ or }SO_{n}(\mathbb{K}).\end{cases}$

Under this identification, the Weyl group $W$ acts on $V$ by permuting the
coordinates $x_{1},\cdots,x_{m}$ and sign changing $x_{i}\mapsto-x_{i}$. Let
$x_{ij}$ be the linear function on $V^{d}$ whose value at a point
$(v_{1},\cdots,v_{d})$ is the $i$-th component of $v_{j}$. A direct
computation shows that under the identification of $V$ and $\mathfrak{t}$, we
have $F_{\mathfrak{t}}=N_{\mathfrak{t}}^{2}$, where

$N_{\mathfrak{t}}=\prod_{k=1}^{m}(1+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}x_{k1}^{i_{1}}x_{k2}^{i_{2}}\cdots x_{kd}^{i_{d}}T_{i_{1}\cdots i_{d}}).$

The constant coefficient of $N_{\mathfrak{t}}$ is $1$, so by the uniqueness of
square root (Lemma 2.2), $N_{\mathfrak{t}}=\sqrt{F_{\mathfrak{t}}}$, and hence
$\deg\sqrt{F_{\mathfrak{t}}}=\deg N_{\mathfrak{t}}\leq
m=\lfloor\frac{n}{2}\rfloor$.

Next we will show that the coefficients of $N_{\mathfrak{t}}$ form a
$\mathbb{K}$-linear basis of $\mathbb{K}[V^{d}]^{W}$. If $\rm{char}\
\mathbb{K}=0$, this is a direct consequence of Hunziker [References, Lemma
2.2]. In the following we shall adapt Hunziker’s proof slightly so that it
works for all the $\rm{char}\ \mathbb{K}\neq 2$ cases.

We denote by $\Lambda$ the following set of nonzero $m\times d$ matrices:

$\Lambda:=\\{\lambda=\begin{pmatrix}\lambda_{11}&\cdots&\lambda_{1d}\\\
\vdots&\ddots&\vdots\\\
\lambda_{m1}&\cdots&\lambda_{md}\end{pmatrix}|\lambda_{ij}\in\mathbb{Z}_{\geq
0},\ \lambda\neq 0\\}.$

Let $\Lambda_{even}:=\\{\lambda\in\Lambda|\sum_{j=1}^{d}\lambda_{ij}\textmd{
is even, }\text{for}\ 1\leq i\leq m\\}$.

The symmetric group $S_{m}$ acts on $\Lambda$ by permuting the rows. Let
$\Lambda^{+}_{even}\subset\Lambda_{even}$ be the subset of all
$\lambda\in\Lambda_{even}$ such that $\lambda_{1}\geq\cdots\geq\lambda_{m}$
with respect to the lexicographic order on the rows, where $\lambda_{i}$ is
the $i$-th row of $\lambda$. Then for $\lambda\in\Lambda_{even}$, the orbit
$S_{m}\cdot\lambda\subset\Lambda_{even}$ contains a unique element in
$\Lambda^{+}_{even}$.

To each $\lambda\in\Lambda$ corresponds to the monomials
$x^{\lambda}:=\prod_{k=1}^{m}x_{k1}^{\lambda_{k1}}x_{k2}^{\lambda_{k2}}\cdots
x_{kd}^{\lambda_{kd}}$ and
$T_{\lambda}:=\prod_{k=1}^{m}T_{\lambda_{k1},\lambda_{k2},\cdots,\lambda_{kd}}$.
Here by convention $T_{0,\cdots,0}=1$. For $\lambda\in\Lambda^{+}_{even}$ we
put

$a_{\lambda}:=\sum\limits_{\mu\in S_{m}\cdot\lambda}x^{\mu}.$

Using these notations, we can rewrite $N_{\mathfrak{t}}$ as

$N_{\mathfrak{t}}=1+\sum\limits_{\lambda\in\Lambda^{+}_{even}}a_{\lambda}T_{\lambda}.$

It is direct to see $a_{\lambda}\in\mathbb{K}[V^{d}]^{W}$, and $a_{\lambda}$
($\lambda\in\Lambda^{+}_{even}$) are linearly independent. Suppose
$f\in\mathbb{K}[V^{d}]^{W}$ is nonzero and write
$f=b_{0}+\sum_{\lambda\in\Lambda}b_{\lambda}x^{\lambda}$, with $b_{0},\
b_{\lambda}\in\mathbb{K}$. From $wf=f$ for all sign changes $w$ we deduce
$\lambda\in\Lambda_{even}$ if $b_{\lambda}\neq 0$. Then
$\sum_{\lambda\in\Lambda_{even}}b_{\lambda}x^{w\lambda}=\sum_{\lambda\in\Lambda_{even}}b_{\lambda}x^{\lambda}$,
for $w\in S_{m}$. This implies $b_{w\lambda}=b_{\lambda}$, for any $w\in
S_{m}$. So $f$ is a linear combination of $1$ and $a_{\lambda}$,
$\lambda\in\Lambda^{+}_{even}$. Finally $\\{1\\}$ $\cup$
$\\{a_{\lambda}|\lambda\in\Lambda^{+}_{even}\\}$, i.e., the coefficients of
$N_{\mathfrak{t}}=\sqrt{F_{\mathfrak{t}}}$, are a $\mathbb{K}$-linear basis of
$\mathbb{K}[V^{d}]^{W}$. ∎

###### Proof of Lemma 4.1.

Keep notations as above. Define

$\Lambda_{odd}:=\\{\lambda\in\Lambda|\sum_{j=1}^{d}\lambda_{ij}\textmd{ is odd
},\text{for}\ 1\leq i\leq m\\},$

and let $\Lambda^{+}_{odd}\subset\Lambda_{odd}$ be the subset of all
$\lambda\in\Lambda_{odd}$ such that $\lambda_{1}\geq\cdots\geq\lambda_{m}$
with respect to the lexicographic order on the rows, where $\lambda_{i}$ is
the $i$-th row of $\lambda$. Then for $\lambda\in\Lambda_{odd}$, the orbit
$S_{m}\cdot\lambda\subset\Lambda_{odd}$ contains a unique element in
$\Lambda^{+}_{odd}$. For $\lambda\in\Lambda^{+}_{odd}$ we put

$a_{\lambda}=\sum\limits_{\mu\in S_{m}\cdot\lambda}x^{\mu}.$

Then under the identification of $V$ and $\mathfrak{t}$, a direct computation
shows that

$H_{\mathfrak{t}}=(\sqrt{-1})^{m}\sum\limits_{\lambda\in\Lambda^{+}_{odd}}a_{\lambda}T_{\lambda}.$

The Weyl group $W$ of $G$ acts on $V$ by permuting the coordinates
$x_{1},\cdots,x_{d}$ and sign changing $\tau_{i}:x_{i}\mapsto-x_{i}$. The Weyl
group $W^{\prime}$ of $G^{\prime}$ is then the index two subgroup of $W$,
generated by the permutations and $\tau_{i}\circ\tau_{j}$, $1\leq i,j\leq d$.
It follows that

$a_{\lambda}\in\mathbb{K}[V^{d}]^{W^{\prime}}_{(1)},\ \text{for all
}\lambda\in\Lambda^{+}_{odd},$

where $\mathbb{K}[V^{d}]^{W^{\prime}}_{(1)}$ is the $(-1)$-eigen subspace of
$\mathbb{K}[V^{d}]^{W^{\prime}}$ under the action of $W/W^{\prime}$.

Moreover, it is easy to see $a_{\lambda}$ $(\lambda\in\Lambda^{+}_{odd})$ are
linearly independent. Similar to the proof of Lemma 3.2, we can see
$\mathbb{K}[V^{d}]^{W^{\prime}}_{(1)}$ is linearly spanned by $a_{\lambda}$
$(\lambda\in\Lambda^{+}_{odd})$. So the coefficients of $H_{\mathfrak{t}}$
form a basis of $\mathbb{K}[V^{d}]^{W^{\prime}}_{(1)}$. Under the
identification $V=\mathfrak{t}$, this means
$\\{c^{\prime}_{w}|w\in\mathcal{M}^{\prime}(\frac{n}{2})\\}$ is a
$\mathbb{K}$-linear basis of
$\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}_{(1)}$, as asserted.

∎

###  5.2. Proof of Lemma 3.3

If $n$ is even and $G=Sp_{n}(\mathbb{K})$, the following matrix

$A:=J+J\sum_{(i_{1},\cdots,i_{d})\in S}X(1)^{i_{1}}\cdots
X(d)^{i_{d}}T_{i_{1}\cdots i_{d}}$

is skew-symmetric. According to Chen-Ngô [References], let

$N:={\rm Pf}(A){\rm
Pf}(J)^{-1}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]].$

Then the constant coefficient of $N$ is $1$ and

$N^{2}=\det(I_{n}+\sum_{(i_{1},\cdots,i_{d})\in S}X(1)^{i_{1}}\cdots
X(d)^{i_{d}}T_{i_{1}\cdots i_{d}})=F_{\mathfrak{g}}.$

By Lemma 2.2, we get $N=\sqrt{F_{\mathfrak{g}}}$. Since $\deg
N\leq\frac{n}{2}$ by its explicit construction, we see
$\deg\sqrt{F_{\mathfrak{g}}}\leq\frac{n}{2}$. This gives a proof of Lemma 3.3
in the $G=Sp_{n}(\mathbb{K})$ case.

In the remaining of this subsection, we assume $G$ is one of the following
orthogonal groups.

$G=\begin{cases}O_{n}(\mathbb{K}),\\\ SO_{n}(\mathbb{K}),&n\textmd{
odd}.\end{cases}$

We will need the following lemma, see e.g. [10], [14].

######  Lemma 5.1.

Let $R$ be a ring, and $X_{ij}\in M_{l}(R)$, for $1\leq i,j\leq k$. Let
$\tilde{X}\in M_{kl}(R)$ be the following block matrix

$\begin{pmatrix}X_{11}&X_{12}&\cdots&X_{1k}\\\ X_{21}&X_{22}&\cdots&X_{2k}\\\
\vdots&\vdots&\ddots&\vdots\\\ X_{k1}&X_{k2}&\cdots&X_{kk}\end{pmatrix}.$

If $X_{ij}$ commute pairwise, then

$\det\tilde{X}=\det(\sum\limits_{\sigma\in
S_{k}}(\mathrm{sgn}~{}\sigma)X_{1\sigma(1)}X_{2\sigma(2)}\cdots
X_{k\sigma(k)}).$

Suppose $d\geq 1$ and consider the following $(2d+3)\times(2d+3)$ skew
symmetric matrix $T$ over the polynomial ring $\mathbb{K}[t_{ij}|1\leq i<j\leq
2d+3]$:

$T(i,j)=\begin{cases}t_{ij},\ 1\leq i<j\leq 2d+3;\\\ -t_{ji},\ 1\leq j<i\leq
2d+3;\\\ 0,\ 1\leq i=j\leq 2d+3.\end{cases}$

For any $k\geq 1$ and any $1\leq i_{1},\cdots,i_{k}\leq 2d+3$, let
$T(\hat{i}_{1},\cdots,\hat{i}_{k})$ be the matrix obtained from $T$ by
deleting the $i_{j}$-th row and the $i_{j}$-th column for all $1\leq j\leq k$,
and let $h_{i_{1},\cdots,i_{k}}=\mathrm{Pf}\
T(\hat{i}_{1},\cdots,\hat{i}_{k})$ be the corresponding Pfaffian. Define the
$2d\times 2d$ matrix $A$ over $\mathbb{K}[t_{ij}|1\leq i<j\leq 2d+3]$ by
$A(i,j):=h_{i,j,2d+2}^{2}+h_{i,2d+1,2d+2}^{2}+h_{j,2d+1,2d+2}^{2}$, for $1\leq
i,j\leq 2d$.

######  Lemma 5.2.

The following system of equations for the variables $t_{ij}$ ($1\leq i<j\leq
2d+3$) has a solution in $\mathbb{K}$:

$\begin{cases}h_{2d+2}=0;\\\ h_{2d+3}\det A\neq 0.\end{cases}$

###### Proof.

It suffices to show that in the polynomial ring $\mathbb{K}[t_{ij}|1\leq
i<j\leq 2d+3]$, the polynomial $h_{2d+3}\det A$ is not contained in the ideal
$\sqrt{(h_{2d+2})}$. Since the Pfaffians $h_{2d+2}$ and $h_{2d+3}$ are coprime
irreducible polynomials (cf. Goodman-Wallach [8, Lemma B.2.10]), we only need
to prove $\det A\notin(h_{2d+2})$. We will proceed by induction on $d$.

If $d=1$, by direct computations, we have

$h_{2d+2}=h_{4}=t_{12}t_{35}-t_{13}t_{25}+t_{15}t_{23},$

and

$\displaystyle\det A$ $\displaystyle=$ $\displaystyle
4h_{1,3,4}^{2}h_{2,3,4}^{2}-(h_{1,2,4}^{2}+h_{1,3,4}^{2}+h_{2,3,4}^{2})^{2}$
$\displaystyle=$
$\displaystyle-(t_{35}^{2}+(t_{25}+t_{15})^{2})(t_{35}^{2}+(t_{25}-t_{15})^{2}).$

Since $h_{4}$ is irreducible, we can directly verify that $\det
A\notin(h_{4})$.

Suppose $d\geq 2$ and the statement holds for $d-1$. Suppose to the contrary
that $\det A\in(h_{2d+2})$, so there exists $g\in\mathbb{K}[t_{ij}|1\leq
i<j\leq 2d+3]$ such that

(5.2.1) $\det A=g\ h_{2d+2}.$

On the one hand the degree of $\det A$ with respect to the variable $t_{12}$
is at most $4d-4$, and the coefficient of $t_{12}^{4d-4}$ is given by

$\det\begin{pmatrix}2h_{1,2d+1,2d+2}^{2}&h_{1,2,2d+2}^{2}+h_{1,2d+1,2d+2}^{2}+h_{2,2d+1,2d+2}^{2}\\\
h_{1,2,2d+2}^{2}+h_{1,2d+1,2d+2}^{2}+h_{2,2d+1,2d+2}^{2}&2h_{2,2d+1,2d+2}^{2}\end{pmatrix}\cdot\det
A^{\prime},$

where $A^{\prime}$ is the $(2d-2)\times(2d-2)$ matrix over
$\mathbb{K}[t_{ij}|3\leq i<j\leq 2d+3]$ with

$A^{\prime}(i,j)=h_{1,2,i+2,j+2,2d+2}^{2}+h_{1,2,i+2,2d+1,2d+2}^{2}+h_{1,2,j+2,2d+1,2d+2}^{2}.$

On the other hand, the degree of $h_{2d+2}$ with respect to $t_{12}$ is $1$,
and the coefficient of $t_{12}$ is $h_{1,2,2d+2}$. By comparing the
coefficient of $t_{12}^{4d-4}$, we obtain from (5.2.1) the following equality:

$\displaystyle=g_{1}\ h_{1,2,2d+2},$

where $g_{1}\in\mathbb{K}[t_{ij}|1\leq i<j\leq 2d+3]$. A further
simplification yields that

$(h_{1,2d+1,2d+2}^{2}-h_{2,2d+1,2d+2}^{2})^{2}\ \det A^{\prime}=g_{2}\
h_{1,2,2d+2},$

for some $g_{2}\in\mathbb{K}[t_{ij}|1\leq i<j\leq 2d+3]$. Applying the
induction hypothesis to the skew symmetric matrix $T(\hat{1},\hat{2})$, we
obtain

$\det A^{\prime}\notin(h_{1,2,2d+2}).$

Then $(h_{1,2d+1,2d+2}^{2}-h_{2,2d+1,2d+2}^{2})^{2}\in(h_{1,2,2d+2})$, since
$h_{1,2,2d+2}$ is irreducible. Note the variable $t_{3,2d+1}$ is absent from
$(h_{1,2d+1,2d+2}^{2}-h_{2,2d+1,2d+2}^{2})^{2}$, and the degree of
$h_{1,2,2d+2}$ with respect to $t_{3,2d+1}$ is $1$. This obviously contradicts
the relation $(h_{1,2d+1,2d+2}^{2}-h_{2,2d+1,2d+2}^{2})^{2}\in(h_{1,2,2d+2})$.
So $\det A\notin(h_{2d+2})$, as desired. ∎

We consider the ring
$\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}][T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$, by adding a formal variable $T_{0}$.

######  Lemma 5.3.

There exists
$\tilde{N}\in\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}][T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ satisfying:

$\tilde{N}^{2}=\det(T_{0}^{2}I_{n}+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots i_{d}}).$

###### Proof.

For ease of notation, set $R=\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$.
Note to begin with that

$T_{0}^{2}I_{n}+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots
i_{d}}=T_{0}^{2}I_{n}+\sum\limits_{j=1}^{d}X(j)F(j),$

where

$F(j)=\sum\limits_{(0,\cdots,0,i_{j},\cdots,i_{d})\in S,i_{j}\geq
1}X(j)^{i_{j}-1}X(j+1)^{i_{j+1}}\cdots X(d)^{i_{d}}T_{0\cdots 0i_{j}\cdots
i_{d}}.$

By Lemma 5.2, we can take a skew symmetric matrix $T\in M_{2d+3}(\mathbb{K})$
such that $h_{2d+2}=0$ and $h_{2d+3}\det A\neq 0$. Here recall
$h_{i_{1},\cdots,i_{k}}=\mathrm{Pf}\ T(\hat{i}_{1},\cdots,\hat{i}_{k})$, and
$A\in M_{2d}(\mathbb{K})$ whose $(i,j)$-entry is
$a_{ij}:=h_{i,j,2d+2}^{2}+h_{i,2d+1,2d+2}^{2}+h_{j,2d+1,2d+2}^{2}$. By scaling
$T$ if necessary, we can assume $h_{2d+3}=1$.

Over the polynomial ring $\mathbb{K}[y,x_{0},x_{i}|1\leq i\leq 2d]$, let
$v:=(0,\cdots,0,1,yx_{0})$ be the $2d+3$-tuple. Define the
$(2d+3)\times(2d+3)$ matrix $B(T)$ by

$B(T):=\mathrm{diag}(yx_{1},yx_{2},\cdots,yx_{2d},\sum\limits_{i=1}^{2d}yx_{i},0,0)+T,$

and the $(2d+4)\times(2d+4)$ matrix $M(T)$ by

$M(T):=\left(\begin{array}[]{c|c}B(T)&v^{t}\\\
\hline\cr-v&0\end{array}\right).$

By Lemma 5.4 below,

$\begin{split}&\det M(T)=(h_{2d+2}-h_{2d+3}\
yx_{0})^{2}+\sum\limits_{i=1}^{2d}y^{2}x_{i}^{2}(h_{i,2d+1,2d+2}-h_{i,2d+1,2d+3}yx_{0})^{2}\\\
&+\sum\limits_{1\leq i<j\leq
2d}y^{2}x_{i}x_{j}[(h_{i,j,2d+2}-h_{i,j,2d+3}yx_{0})^{2}+(h_{i,2d+1,2d+2}-h_{i,2d+1,2d+3}yx_{0})^{2}\\\
&+(h_{j,2d+1,2d+2}-h_{j,2d+1,2d+3}yx_{0})^{2}]+y^{4}\ g,\end{split}$

where $g\in\mathbb{K}[y,x_{0},x_{i}|1\leq i\leq 2d]$ and
$g\in(x_{1},\cdots,x_{2d})$. Since $h_{2d+2}=0$, $h_{2d+3}=1$, and
$h_{i,j,2d+2}^{2}+h_{i,2d+1,2d+2}^{2}+h_{j,2d+1,2d+2}^{2}=a_{ij}$, the above
expression can be simplified as

(5.3.1) $\det
M(T)=y^{2}(x_{0}^{2}+\frac{1}{2}\sum\limits_{i=1}^{2d}\sum\limits_{j=1}^{2d}a_{ij}x_{i}x_{j})+y^{3}\
g_{1},$

where $g_{1}\in\mathbb{K}[y,x_{0},x_{i}|1\leq i\leq 2d]$ and
$g_{1}\in(x_{1},\cdots,x_{2d})$. Since $A=(a_{ij})\in M_{2d}(\mathbb{K})$ is
symmetric with $\det A\neq 0$, there exists an invertible $P\in
M_{2d}(\mathbb{K})$ such that if we let
$(y_{1},\cdots,y_{2d})=(x_{1},\cdots,x_{2d})\cdot P$, then

$\frac{1}{2}\sum\limits_{i=1}^{2d}\sum\limits_{j=1}^{2d}a_{ij}x_{i}x_{j}=\sum\limits_{i=1}^{d}y_{2i-1}y_{2i}=y_{1}y_{2}+y_{3}y_{4}+\cdots+y_{2d-1}y_{2d}.$

In the matrix $M(T)$, we replace $x_{0}$ by $T_{0}I_{n}$, $y_{2i-1}$ by the
skew symmetric matrix $X(i)$, $y_{2i}$ by $F(i)$, $1\leq i\leq d$, and any
constant number $a\in\mathbb{K}$ by $aI_{n}$. In this way, we obtain a
$(2d+4)n\times(2d+4)n$ matrix $\tilde{M}$ over $R[y,T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$. Note $\tilde{M}$ is skew symmetric.
Define $\tilde{Q}_{y}:={\rm Pf}\ \tilde{M}\in R[y,T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]].$ Then by (5.3.1) and Lemma 5.1,

(5.3.2) $\tilde{Q}_{y}^{2}=\det\
\tilde{M}=y^{2n}\det(T_{0}^{2}I_{n}+\sum\limits_{i=1}^{d}X(i)F(i)+y\ g_{2}),$

where $g_{2}$ is an $n\times n$ matrix over $R[y,T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$. Note $R=\oplus_{i=0}^{\infty}R_{i}$ is a
graded $\mathbb{K}$-algebra with $R_{0}=\mathbb{K}$, and the entries of $X(i)$
are all in $R_{1}$. Since $g_{1}\in(x_{1},\cdots,x_{2d})$, we can see

$\det(T_{0}^{2}I_{n}+\sum\limits_{i=1}^{d}X(i)F(i)+y\
g_{2})-T_{0}^{2n}\in\oplus_{i=1}^{\infty}R_{i}[y,T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]].$

Now in $R[[y,T_{0},T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S]]$, we can
apply Lemma 2.3 to the equality (5.3.2) and obtain $\tilde{N}_{y}\in
R[[y,T_{0},T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ satisfying
$\tilde{Q}_{y}=y^{n}\tilde{N}_{y}$. Moreover, $\tilde{N}_{y}\in
R[y,T_{0}][[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ since
$\tilde{Q}_{y}\in R[y,T_{0}][[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in
S]]$. By (5.3.2), we see

(5.3.3)
$\tilde{N}_{y}^{2}=\det(T_{0}^{2}I_{n}+\sum\limits_{i=1}^{d}X(i)F(i)+y\
g_{2}).$

Now let $\tilde{N}\in R[T_{0}][[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in
S]]$ be the evaluation of $\tilde{N}_{y}$ at $y=0$, we deduce from (5.3.3) the
desired equation:

$\begin{split}\tilde{N}^{2}&=\det(T_{0}^{2}I_{n}+\sum\limits_{i=1}^{d}X(i)F(i))\\\
&=\det(T_{0}^{2}I_{n}+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots
i_{d}}).\end{split}$

∎

Now we begin to complete the proof of Lemma 3.3. Still let
$R=\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$.

By Lemma 5.3, we can find $\tilde{N}\in R[T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ satisfying:

(5.3.4)
$\tilde{N}^{2}=\det(T_{0}^{2}I_{n}+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots i_{d}}).$

Let $\tilde{N}(\underline{0})\in R[T_{0}]$ be the valuation of $\tilde{N}$ at
$T_{i_{1}\cdots i_{d}}=0$, $(i_{1},\cdots,i_{d})\in S$. Then
$\tilde{N}(\underline{0})^{2}=\det(T_{0}^{2}I_{n})=T_{0}^{2n}$. Recall that
$R=\oplus_{i=0}^{\infty}R_{i}$ is a graded $\mathbb{K}$-algebra with
$R_{0}=\mathbb{K}$. By Lemma 2.2, we get $\tilde{N}(\underline{0})=\pm
T_{0}^{n}$. We assume without loss of generality that
$\tilde{N}(\underline{0})=T_{0}^{n}$.

Let $N\in R[[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S]]$ be the
evaluation of $\tilde{N}$ at $T_{0}=1$. Obviously the constant coefficient of
$N$ is $1$, and

$N^{2}=\det(1+\sum\limits_{(i_{1},\cdots,i_{d})\in
S}X(1)^{i_{1}}X(2)^{i_{2}}\cdots X(d)^{i_{d}}T_{i_{1}\cdots
i_{d}})=F_{\mathfrak{g}}.$

By applying Lemma 2.2 to $R[[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in
S]]$, we get $\sqrt{F_{\mathfrak{g}}}=N$.

For any non-zero $\lambda\in\mathbb{K}^{*}$, let $\tilde{N}_{\lambda}\in
R[T_{0}][[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in S{}]]$ be the image of
$\tilde{N}$ under the automorphism of $R$-algebras:

$\begin{split}R[T_{0}][[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in
S{}]]&\rightarrow R[T_{0}][[T_{i_{1}\cdots i_{d}}|(i_{1},\cdots,i_{d})\in
S]]\\\ T_{0}&\mapsto\lambda T_{0}\\\ T_{i_{1}\cdots
i_{d}}&\mapsto\lambda^{2}T_{i_{1}\cdots i_{d}}\end{split}$

By (5.3.4),
$\tilde{N}_{\lambda}^{2}=\lambda^{2n}\tilde{N}^{2}=(\lambda^{n}\tilde{N})^{2}$.
Since $\tilde{N}(\underline{0})=T_{0}^{n}$, we see
$\tilde{N}_{\lambda}(\underline{0})=\lambda^{n}\tilde{N}(\underline{0})=\lambda^{n}T_{0}^{n}$
is not a zero-divisor in $R[T_{0}][[T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S{}]]$. Then
$\tilde{N}_{\lambda}=\lambda^{n}\tilde{N}$ by Lemma 2.2. From this we deduce
that, by requiring $\deg T_{0}=1$ and $\deg T_{i_{1}\cdots i_{d}}=2$ for all
$(i_{1},\cdots,i_{d})\in S$, $\tilde{N}$ is a degree $n$ weighted homogeneous
formal power series with respect to the variables $\\{T_{0},T_{i_{1}\cdots
i_{d}}|(i_{1},\cdots,i_{d})\in S\\}$. Thus $\deg\sqrt{F_{\mathfrak{g}}}=\deg
N\leq\lfloor\frac{n}{2}\rfloor$, and this completes the proof of Lemma 3.3.

######  Lemma 5.4.

Let $R$ be a $\mathbb{K}$-algebra. Suppose $n\geq 2$, and $T=(t_{ij})\in
M_{2n}(R)$ is a skew symmetric matrix. Over the polynomial ring
$R[x_{1},\cdots,x_{2n-3}]$, let

$M:=T+\mathrm{diag}(x_{1},\cdots,x_{2n-3},0,0,0)$

be the sum of $T$ and a diagonal matrix. Suppose $t_{i,2n}=0$, for
$i=1,\cdots,2n-3$. Then the determinant $\det M$ has the following expansion
as a polynomial in $x_{1},\cdots,x_{2n-3}$:

$\begin{split}\det
M&=(h_{2n-2,2n}~{}t_{2n-2,2n}-h_{2n-1,2n}~{}t_{2n-1,2n})^{2}\\\
&+\sum\limits_{1\leq i<j\leq
2n-3}x_{i}x_{j}(h_{i,j,2n-2,2n}~{}t_{2n-2,2n}-h_{i,j,2n-1,2n}~{}t_{2n-1,2n})^{2}\\\
&+\sum\limits_{k=2}^{n-2}f_{2k}.\end{split}$

Here for even $m$ and for $1\leq i_{1}<i_{2}<\cdots<i_{m}\leq 2n$, the symbol
$h_{i_{1},\cdots,i_{m}}$ means ${\rm Pf}~{}T(\hat{i}_{1},\cdots,\hat{i}_{m})$,
the Pfaffian of the matrix obtained from $T$ by deleting the $i_{j}$-th row
and the $i_{j}$-th column for all $1\leq j\leq m$, and $f_{2k}$ means a degree
$2k$ homogeneous polynomial in $x_{1},\cdots,x_{2n-3}$.

###### Proof.

By expanding $\det M$ along the diagonal, we obtain

$\det M=\det T+\sum\limits_{m=1}^{2n-3}\sum\limits_{1\leq
i_{1}<i_{2}<\cdots<i_{m}\leq 2n-3}\det T(\hat{i}_{1},\cdots,\hat{i}_{m})\
x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}.$

Note when $m$ is odd, $\det T(\hat{i}_{1},\cdots,\hat{i}_{m})=0$ because
$T(\hat{i}_{1},\cdots,\hat{i}_{m})$ is a skew symmetric matrix of odd order.
So the above expansion reduces to

(5.4.1) $\det M=\det
T+\sum\limits_{k=1}^{2n-4}\sum\limits_{i_{1},\cdots,i_{2k}}\det
T(\hat{i}_{1},\cdots,\hat{i}_{2k})\ x_{i_{1}}x_{i_{2}}\cdots x_{i_{2k}},$

where the sum $\sum\limits_{i_{1},\cdots,i_{2k}}$ is over

$\\{{(i_{1},\cdots,i_{2k})|\ 1\leq i_{1}<i_{2}<\cdots<i_{2k}\leq 2n-3}\\}.$

For a $2m\times 2m$ skew symmetric matrix $A=(a_{ij})$, the Pfaffian ${\rm
Pf}~{}A$ can be expanded with respect to the last column (cf. Cayley
[References]) as follows:

${\rm Pf}~{}A=\sum\limits_{i=1}^{2m-1}(-1)^{i+1}{\rm
Pf}~{}A(\hat{i},\widehat{2m})~{}a_{i,2m}.$

Applying this expansion to the skew symmetric matrix
$T(\hat{i}_{1},\cdots,\hat{i}_{2k})$, we obtain

$\begin{split}{\rm Pf}~{}T(\hat{i}_{1},\cdots,\hat{i}_{2k})=&-t_{2n-2,2n}{\rm
Pf}~{}T(\hat{i}_{1},\cdots,\hat{i}_{2k},\widehat{2n-2},\widehat{2n})\\\
&+t_{2n-1,2n}{\rm
Pf}~{}T(\hat{i}_{1},\cdots,\hat{i}_{2k},\widehat{2n-1},\widehat{2n}).\end{split}$

Combining this with (5.4.1), we get the desired expansion of $\det M$. ∎

##  6\. Applications

In this section, we apply the restriction isomorphism to obtain some
identities about polynomial functions on commuting skew symmetric matrices.

From now on, we suppose $\rm{char}\ \mathbb{K}=0$, and let
$G=O_{n}(\mathbb{K})$ be the orthogonal group, so that its Lie algebra
$\mathfrak{g}$ is the space of $n\times n$ skew symmetric matrices. We also
fix the Cartan subalgebra

$\mathfrak{t}=\\{\textmd{SK
}(x_{1},\cdots,x_{\lfloor\frac{n}{2}\rfloor})|x_{i}\in\mathbb{K},\text{ for
all }\ 1\leq i\leq\lfloor\frac{n}{2}\rfloor\\}$

as in Section 5.1, where recall $\textmd{SK
}(x_{1},\cdots,x_{\lfloor\frac{n}{2}\rfloor})$ is the $n\times n$ skew
symmetric matrix defined in (5.0.1). Recall
$(X(1),\cdots,X(d))\in\mathfrak{C}^{d}_{\mathfrak{g}}(\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}])$
is the tautological $\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}]$-valued point
of $\mathfrak{C}^{d}_{\mathfrak{g}}$. Under the restriction homomorphism
$M_{n}(\mathbb{K}[\mathfrak{C}^{d}_{\mathfrak{g}}])\rightarrow
M_{n}(\mathbb{K}[\mathfrak{t}^{d}])$, the skew symmetric matrices $X(i)$ are
mapped to $Y(i)$.

Let $R$ be a $\mathbb{K}$-algebra.

######  Corollary 6.1.

Suppose $n\geq 3$ is odd and let $X_{1},\cdots,X_{d}\in M_{n}(R)$ be commuting
skew symmetric matrices. For any $f\in\mathbb{K}[x_{1},\cdots,x_{d}]$, if
$f(0,\cdots,0)=0$, then $\det f(X_{1},\cdots,X_{d})=0$.

###### Proof.

Clearly there exists a $\mathbb{K}$-algebra homomorphism
$\varphi:\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]\rightarrow R$ such that
$\varphi$ maps the matrix $X(i)$ to $X_{i}$, $i=1,\cdots,d$. Then $\det
f(X_{1},\cdots,X_{d})=\varphi(\det f(X(1),\cdots,X(d)))$. Since $\det
f(X(1),\cdots,X(d))\in\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]^{G}$, we can
apply the restriction homomorphism

$\Phi:\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]^{G}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W}$

to obtain

$\Phi(\det f(X(1),\cdots,X(d)))=\det f(Y(1),\cdots,Y(d)).$

By the definition of $Y(i)$ we see $\det f(Y(1),\cdots,Y(d))=0$. Since $\Phi$
is an isomorphism by Theorem 3.4, we obtain $\det f(X(1),\cdots,X(d))=0$ and
hence $\det f(X_{1},\cdots,X_{d})=0$. ∎

######  Corollary 6.2.

Suppose $n\geq 2$ is even. Let $X_{1},X_{2},X_{3}\in M_{n}(R)$ be commuting
skew symmetric matrices. Then

$\mathrm{Pf}(X_{1}X_{2}X_{3})=(-1)^{\frac{n}{2}}\mathrm{Pf}(X_{1})\mathrm{Pf}(X_{2})\mathrm{Pf}(X_{3}).$

###### Proof.

The case $n=2$ is trivial, so we assume $n\geq 4$ and take $d=3$. Let
$G^{\prime}=SO_{n}(\mathbb{K})$ be the special orthogonal group, which has the
same Lie algebra as $G=O_{n}(\mathbb{K})$. There exists a $\mathbb{K}$-algebra
homomorphism $\varphi:\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]\rightarrow
R$ such that $\varphi(X(i))=X_{i}$, $i=1,2,3$. Then

$\begin{split}&\mathrm{Pf}(X_{1}X_{2}X_{3})-(-1)^{\frac{n}{2}}\mathrm{Pf}(X_{1})\mathrm{Pf}(X_{2})\mathrm{Pf}(X_{3})\\\
&=\varphi(\mathrm{Pf}(X(1)X(2)X(3))-(-1)^{\frac{n}{2}}\mathrm{Pf}(X(1))\mathrm{Pf}(X(2))\mathrm{Pf}(X(3))).\end{split}$

Let
$r=\mathrm{Pf}(X(1)X(2)X(3))-(-1)^{\frac{n}{2}}\mathrm{Pf}(X(1))\mathrm{Pf}(X(2))\mathrm{Pf}(X(3))$,
we see $r\in\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]^{G^{\prime}}$. Then we
apply the restriction homomorphism
$\Phi:\mathbb{K}[\mathfrak{C}_{\mathfrak{g}}^{d}]^{G^{\prime}}\rightarrow\mathbb{K}[\mathfrak{t}^{d}]^{W^{\prime}}$
to obtain

$\Phi(r)=\mathrm{Pf}(Y(1)Y(2)Y(3))-(-1)^{\frac{n}{2}}\mathrm{Pf}(Y(1))\mathrm{Pf}(Y(2))\mathrm{Pf}(Y(3)).$

By direct computations, we can see $\Phi(r)=0$. Since $\Phi$ is an isomorphism
by Theorem 4.3, we get finally $r=0$. ∎

For a positive integer $m$, write $\mathcal{P}_{m}$ for the set of partitions
$\lambda=\lambda_{1}\cup\cdots\cup\lambda_{h}$ of the set $\\{1,\cdots,m\\}$
into the disjoint union of non-empty subsets $\lambda_{i}$, and denote
$h(\lambda)=h$ the number of parts of the partition $\lambda$.

######  Corollary 6.3.

Suppose $n\geq 2$, $d\geq 1$. Let $m=\lfloor\frac{n}{2}\rfloor+1$. Suppose
$X_{1},\cdots,X_{d}\in M_{n}(R)$ are commuting skew symmetric matrices. For
$j=1,\cdots,m$, let $Y_{j}=\prod_{i=1}^{d}X_{i}^{a_{ij}}\in M_{n}(R)$ be a
monomial of $X_{1},\cdots,X_{d}$, with $\sum_{i=1}^{d}a_{ij}>0$ even. Then

$\sum\limits_{\lambda\in\mathcal{P}_{m}}(\frac{-1}{2})^{h(\lambda)}\prod\limits_{i=1}^{h(\lambda)}((|\lambda_{i}|-1)!\cdot\mathrm{tr}\prod\limits_{s\in\lambda_{i}}Y_{s})=0.$

###### Proof.

In a similar way as above, it suffices to verify the identity under the
assumption $R=\mathbb{K}[\mathfrak{t}^{d}]$ and $X_{i}=Y(i)$, $i=1,\cdots,d$.
Then the required trace identity is just a reformulation of Domokos [6,
Proposition 2.3]. ∎

######  Example 6.4.

As an illustration, we take $n=4$, so $m=3$. All of the partitions of
$\\{1,2,3\\}$ are:

$\\{1,2,3\\},\ \\{1,2\\}\cup\\{3\\},\ \\{1,3\\}\cup\\{2\\},\
\\{2,3\\}\cup\\{1\\},\ \\{1\\}\cup\\{2\\}\cup\\{3\\}.$

Then according to Corollary 6.3, for any $d\geq 1$ commuting skew symmetric
$4\times 4$ matrices $X_{1},\cdots,X_{d}\in M_{4}(R)$, and for
$Y_{1},Y_{2},Y_{3}$ which are monomials of $X_{i}$ of even degree, we have the
following trace identity:

$\displaystyle
2\rm{tr}(Y_{1})\rm{tr}(Y_{2}Y_{3})+2\rm{tr}(Y_{2})\rm{tr}(Y_{1}Y_{3})+2\rm{tr}(Y_{3})\rm{tr}(Y_{1}Y_{2})$
$\displaystyle=8\rm{tr}(Y_{1}Y_{2}Y_{3})+\rm{tr}(Y_{1})tr(Y_{2})tr(Y_{3}).$

To be more specific, let $Y_{i}=X_{i}^{2}$, $i=1,2,3$, then this identity
reduces to:

$\displaystyle
2\rm{tr}(X_{1}^{2})tr(X^{2}_{2}X^{2}_{3})+2\rm{tr}(X^{2}_{2})tr(X^{2}_{1}X^{2}_{3})+2\rm{tr}(X^{2}_{3})tr(X^{2}_{1}X^{2}_{2})$
$\displaystyle=8\rm{tr}(X^{2}_{1}X^{2}_{2}X^{2}_{3})+\rm{tr}(X^{2}_{1})tr(X^{2}_{2})tr(X^{2}_{3}).$

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