# Quotients of commuting schemes associated to Symmetric Pairs

Santosh Nadimpalli, Santosha Pattanayak

###### Abstract.

Let $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ be a
$\mathbb{Z}_{2}$-grading of a classical Lie algebra such that
$(\mathfrak{g},\mathfrak{g}_{0})$ is a classical symmetric pair. Let $G$ be a
classical group with Lie algebra $\mathfrak{g}$ and let $G_{0}$ be the
connected subgroup of $G$ with ${\rm Lie}(G_{0})=\mathfrak{g}_{0}$. For $d\geq
2$, let $\mathfrak{C}^{d}(\mathfrak{g}_{1})$ be the $d$-th commuting scheme
associated with the symmetric pair $(\mathfrak{g},\mathfrak{g}_{0})$. In this
article, we study the categorical quotient
$\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}$ via the Chevalley restriction
map. As a consequence we show that the categorical quotient scheme
$\mathfrak{C}^{d}(\mathfrak{g}_{1})//G_{0}$ is normal and reduced. As a part
of the proof, we describe a generating set for the algebra
$k[\mathfrak{g}_{1}^{d}]^{G_{0}}$, which are of independent interest.

##  1\. Introduction

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed
field $k$ of characteristic zero or of large prime characteristic, and let
$\theta$ be an involution of $\mathfrak{g}$. The eigen spaces of $\theta$
determine a $\mathbb{Z}_{2}$-grading of $\mathfrak{g}$, we have
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$. We say that
$(\mathfrak{g},\mathfrak{g}_{0})$ is a symmetric pair. Let $G$ be a reductive
group with Lie algebra $\mathfrak{g}$ and $G_{0}$ the connected subgroup of
$G$ with Lie algebra $\mathfrak{g}_{0}$. Then $\mathfrak{g}_{1}$ is a
$\mathfrak{g}_{0}$-module. Let $\mathfrak{c}\subset\mathfrak{g}_{1}$ be a
maximal subspace consisting of pairwise commuting semisimple elements. Any
such subspace is called a Cartan subspace. All Cartan subspaces are
$G_{0}$-conjugate. The number $\dim_{k}\mathfrak{c}$ is called the rank of the
symmetric pair. Let $N(\mathfrak{c})$ and $Z(\mathfrak{c})$ be the normaliser
and the centraliser of $\mathfrak{c}$ in $G_{0}$ respectively. Then
$Z(\mathfrak{c})$ is a normal subgroup of $N(\mathfrak{c})$ and the quotient
$W_{\mathfrak{c}}:=N(\mathfrak{c})/Z(\mathfrak{c})$ is a finite group, called
the little Weyl group of the pair $(G,G_{0})$. From [Vin76], it is known that
the action of $W_{\mathfrak{c}}$ on $\mathfrak{c}$ is generated by
transformations fixing a hyperplane in $\mathfrak{c}$, that the restriction
map $k[\mathfrak{g}_{1}]^{G_{0}}\rightarrow
k[\mathfrak{c}]^{W_{\mathfrak{c}}}$ is an isomorphism.

For $d\geq 2$, the $d$-th commuting scheme associated with the involution
$\theta$, or with the symmetric pair $(\mathfrak{g},\mathfrak{g}_{0})$, is
defined as

$\mathfrak{C}^{d}(\mathfrak{g}_{1}):=\\{(x_{1},x_{2},\cdots,x_{d})\in\mathfrak{g}_{1}^{d}:[x_{i},x_{j}]=0,\
\text{for all}\ i,j\in\\{1,2\dots,n\\}\\}.$

The scheme $\mathfrak{C}^{d}(\mathfrak{g}_{1})$ naturally generalises the
commuting scheme defined by

$\mathfrak{C}^{d}(\mathfrak{g})=\\{(x_{1},x_{2},\cdots,x_{d})\in\mathfrak{g}^{d}:[x_{i},x_{j}]=0,\
\text{for all}\ i,j\in\\{1,2\dots,n\\}\\}.$

Richardson in the article [Ric88], proved that the scheme
$\mathfrak{C}^{2}(\mathfrak{g})$ is irreducible. Iarrobino’s work on punctual
Hilbert schemes on $\mathbb{A}^{d}$, with $d\geq 3$ ([Iar72]), it is known
that $\mathfrak{C}^{d}(\mathfrak{g})$ is not irreducible for $d\geq 3$. It is
a well known conjecture that the commuting scheme
$\mathfrak{C}^{2}(\mathfrak{g})$ is reduced. For $d\geq 3$ it is not expected
that $\mathfrak{C}^{d}(\mathfrak{g})$ is reduced, as we do not understand its
irreducible components.

However, the categorical quotient $\mathfrak{C}^{d}(\mathfrak{g})//G:={\rm
Spec}(k[\mathfrak{C}^{d}(\mathfrak{g})]^{G})$ behaves better. A weak version
of the Chevalley restriction theorem for the commuting scheme is proved by
Hunziker in [Hun97]. Hunziker proved that the morphism
$i:\mathfrak{h}^{d}//W\rightarrow\mathfrak{C}^{d}(\mathfrak{g})//G$ of
quotient schemes induced by the inclusion map
$\mathfrak{h}^{d}\rightarrow\mathfrak{C}^{d}(\mathfrak{g})$ is a finite
bijective morphism, where $\mathfrak{h}$ is a Cartan subalgebra of
$\mathfrak{g}$. In other words, $i$ is a universal homeomorphism of schemes.
In particular, $\mathfrak{h}^{d}//W$ is the normalisation of the underlying
reduced subscheme $(\mathfrak{C}^{d}(\mathfrak{g})//G)^{red}$. This also
follows from a result of Luna (see [Lun75]). Since $\mathfrak{h}^{d}//W$ is
irreducible, the categorical quotient $\mathfrak{C}^{d}(\mathfrak{g})//G$ is
also irreducible. In [GG06], Gan and Ginzburg proved that the quotient scheme
$\mathfrak{C}^{2}(\mathfrak{g})//G$ is reduced in the case where $G={\rm
GL}_{n}(k)$. In [CN20], Chen and Ng$\rm{\hat{o}}$ conjectured that the
morphism $i:\mathfrak{h}^{d}//W\rightarrow\mathfrak{C}^{d}(\mathfrak{g})//G$
is an isomorphism of schemes. Since $\mathfrak{h}^{d}//W$ is reduced and
normal, the conjecture implies that the categorical quotient
$\mathfrak{C}^{d}(\mathfrak{g})//G$ is reduced and normal. They proved a
weaker version of the conjecture for $G={\rm GL}_{n}(k)$ and that was
sufficient to show that the quotient $\mathfrak{C}^{d}(\mathfrak{g})//G$ is
reduced. For $G={\rm GL}_{n}$, Vaccarino ([Vac07]) also proved that the map
$i:\mathfrak{h}^{d}//W\rightarrow\mathfrak{C}^{d}(\mathfrak{g})//G$ is an
isomorphism. Both the proofs use the spectral data map constructed by Deligne
which we recall in section 3.

The irreducibility problem for the commuting varieties associated to symmetric
pairs was first considered by Panyushev in [Pan94], and he observed that
$\mathfrak{C}^{2}(\mathfrak{g}_{1})$ can be reducible. He showed that
$\mathfrak{C}^{2}(\mathfrak{g}_{1})$ is irreducible if and only if
$\mathfrak{C}^{2}(\mathfrak{g}_{1})=\overline{G_{0}(\mathfrak{c}\times\mathfrak{c})}$.
Using this he concluded that if $(\mathfrak{g},\mathfrak{g}_{0})$ is a
symmetric pair of maximal rank (i.e.,
rk($(\mathfrak{g},\mathfrak{g}_{0}))=rk(\mathfrak{g})$), then
$\mathfrak{C}^{2}(\mathfrak{g}_{1})$ is an irreducible normal complete
intersection and the ideal of $\mathfrak{C}^{2}(\mathfrak{g}_{1})$ in
$k[\mathfrak{g}_{1}\times\mathfrak{g}_{1}]$ is generated by quadrics. Later
(see [PY07] and [Pan04]) he extended the irreducibility results for some more
symmetric pairs including the pairs that we are interested in this paper. See
also [SY02] and [SY06] for some more results in this direction. In [Pan94]
Panyushev showed that for a symmetric pair $(\mathfrak{g},\mathfrak{g}_{0})$
of maximal rank, the quotient variety
$\mathfrak{C}^{2}(\mathfrak{g}_{1})//G_{0}$ is isomorphic to
$\mathfrak{h}^{2}//W$, where $\mathfrak{h}$ is a Cartan subalgebra of
$\mathfrak{g}$ and $W$ is the Weyl group of $\mathfrak{g}$. In this note,
following the work of Chen and Ng$\rm{\hat{o}}$ in [CC21] we consider all
classical symmetric pairs except the pairs
$(\mathfrak{so}_{2n}(k),\mathfrak{gl}_{n}(k))$,
$(\mathfrak{sp}_{2(n+m)}(k),\mathfrak{sp}_{2n}(k)\times\mathfrak{sp}_{2m}(k))$
and $(\mathfrak{so}_{n}(k)\times\mathfrak{so}_{n}(k),\mathfrak{so}_{n}(k))$
and show that the induced morphism
$\mathfrak{i}:\mathfrak{c}^{d}//W_{\mathfrak{c}}\rightarrow\mathfrak{C}^{d}(\mathfrak{g}_{1})//G_{0}$
is an isomorphism of affine schemes. Since
$\mathfrak{c}^{d}//W_{\mathfrak{c}}$ is normal and reduced, the isomorphism
implies that the quotient scheme $\mathfrak{C}^{d}(\mathfrak{g}_{1})//G_{0}$
is normal and reduced. The main idea of the proofs is to construct a section
of the Chevalley restriction map using either the determinant map or the
Pfaffian norm map defined by Chen and Ng$\rm{\hat{o}}$ for the symplectic
group.

###  1.1. Classical Symmetric Pairs

The classical symmetric pairs are (see for example [Hel78]) the following:

Diagonal:
$(\mathfrak{gl}_{n}(k)\times\mathfrak{gl}_{n}(k),\mathfrak{gl}_{n}(k))$,
$(\mathfrak{so}_{n}(k)\times\mathfrak{so}_{n}(k),\mathfrak{so}_{n}(k))$,
$(\mathfrak{sp}_{2n}(k)\times\mathfrak{sp}_{2n}(k),\mathfrak{sp}_{2n}(k))$,

Bilinear Form:
$(AI):(\mathfrak{gl}_{n}(k),\mathfrak{so}_{n}(k)),\,\,\,\,(AII):(\mathfrak{gl}_{2n}(k),\mathfrak{sp}_{2n}(k))$,

Polarization: $(DIII):(\mathfrak{so}_{2n}(k),\mathfrak{gl}_{n}(k))$,
$(CI):(\mathfrak{sp}_{2n}(k),\mathfrak{gl}_{n}(k))$,

Direct Sum:
$(AIII):(\mathfrak{gl}_{m+n}(k),\mathfrak{gl}_{m}(k)\times\mathfrak{gl}_{n}(k))$,
$(BDI):(\mathfrak{so}_{m+n}(k),\mathfrak{so}_{m}(k)\times\mathfrak{so}_{n}(k))$,
$(CII):(\mathfrak{sp}_{2(m+n)}(k),\mathfrak{sp}_{2m}(k)\times\mathfrak{sp}_{2n}(k))$.

The Diagonal symmetric pairs corresponding to $\mathfrak{gl}_{n}(k)$ and
$\mathfrak{sp}_{2n}(k)$ are considered by Chen and Ng$\rm{\hat{o}}$ in the
articles [CN20] and [CC21] respectively.

Let $(\mathfrak{g},\mathfrak{g}_{0})$ be a symmetric pair except
$(\mathfrak{so}_{2n}(k),\mathfrak{gl}_{n}(k))$,
$(\mathfrak{sp}_{2(n+m)}(k),\mathfrak{sp}_{2n}(k)\times\mathfrak{sp}_{2m}(k))$
and $(\mathfrak{so}_{n}(k)\times\mathfrak{so}_{n}(k),\mathfrak{so}_{n}(k))$.
The detailed structure of these pairs are given in Sections 5. The main
theorem of this paper is the following:

######  Theorem 1.1.

The Chevalley restriction map
$\mathfrak{i}:\mathfrak{c}^{d}//W_{\mathfrak{c}}\rightarrow\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}$
is an isomorphism of affine schemes.

Since $\mathfrak{c}^{d}//W_{\mathfrak{c}}$ is normal and reduced, as an
immediate corollary we get the following:

######  Corollary 1.2.

The categorical quotient scheme $\mathfrak{C}^{d}(\mathfrak{g}_{1})//G_{0}$ is
normal and reduced.

###  1.2. Presentation

The presentation of the paper is as follows. In section 2 we recall the notion
of polynomial laws. In section 3 we recall Deligne’s construction of the
spectral data map. In Section 4 we describe the ring of $G_{0}$ invariants of
$\mathfrak{C}^{d}(\mathfrak{g}_{1})$ for many pairs which are new and of
independent interest. In section 5 we give a proof of the main theorem in case
by case by constructing the respective spectral data map.

##  2\. Polynomial Laws

### 2.1.

Let $A$ be a commutative ring and let ${\rm Alg}_{A}$ be the category of
$A$-algebras. Let $V$ be an $A$-module and let $V_{A}$ be the functor from
${\rm Alg}_{A}$ to the category of sets given by $R\mapsto V\otimes_{A}R$. For
two $A$-modules $V$ and $W$, we denote by $P(V,W)$, the set of natural
transformations between the functors $V_{A}$ and $W_{A}$. If $W=A$, then we
denote by $P(V)$ the set $P(V,A)$. The set $P(V)$ is called the set of
polynomial laws on $V$. Let $v_{1},v_{2},\dots,v_{n}$ be any sequence of
elements in $V$. Let $S_{A}$ be the polynomial ring
$A[X_{1},X_{2},\dots,X_{n}]$. Let $f\in P(V)$, then we get a map
$f_{S_{A}}:V\otimes_{A}S_{A}\rightarrow S_{A}$. Thus for any $f\in P(V)$, and
a finite set of elements $v_{1},\dots,v_{n}$, we associate a polynomial
$P_{f}\in S_{A}$ given by $f_{S_{A}}(v_{1}X_{1}+v_{2}X_{2}+\dots+v_{n}X_{n})$.
A polynomial law $f\in P(V)$ is called homogenous of degree $d$ if
$f(uv)=u^{n}f(v)$, for all $u\in R^{\times}$ and $v\in V\otimes_{A}R$. Let
$P_{n}(V)$ be the set of all degree $n$ homogenous polynomial laws on $V$.

### 2.2.

Let $V$ be an $A$-module, and let $T^{n}(V)$ be the space of $n$-fold tensor
product of the $A$-module $V$. We denote by $TS^{n}(V)$, the $A$-submodule
fixed by the action of $S_{n}$ on $T^{n}(V)$ and by $S^{n}(V)$, the
$S_{n}$-coinvariants of $T^{n}(V)$. Roby (see [Rob63]) showed that the
homogenous polynomial laws on $V$ of degree $n$ are in canonical bijection
with homogenous degree $1$ polynomial laws on $TS^{n}(V)$ given by the
relation

$f(v)=h(v^{\otimes n}),f\in P_{n}(V),h\in P_{1}(TS^{n}(V)),$

where $P_{1}(TS^{n}(V))$ is the space of degree $1$ polynomial laws on
$TS^{n}(V)$. Moreover, if $V$ is an $A$-algebra, which is free as an
$A$-module, and $f$ is a multiplicative homogenous polynomial law of degree
$n$ on $V$, then the degree $1$ polynomial law $h$ associated to $f$ is a
homomorphism of algebras

$TS^{n}(V)\rightarrow A.$

##  3\. Spectral data map

In this section, we first recall Deligne’s construction of spectral data map
for ${\rm GL}_{n}(k)$ (see [Del73, Section 6.3.1]) which plays a pivotal role
in the proof of the main theorem. As the name suggests it assigns the set of
common eigenvalues of a set of commuting matrices. In fact, we construct the
spectral data map for the relevant symmetric pairs and show that it is
actually a section of the Chevalley restriction map. For convenience, we
recall the Chevalley restriction map here. Let $G$ be a reductive group over
$k$ with Lie algebra $\mathfrak{g}$. Let $\mathfrak{h}$ be a Cartan subalgebra
of $\mathfrak{g}$ and let $W$ be the Weyl group of $\mathfrak{g}$. The
diagonal adjoint action of $G$ on $\mathfrak{g}^{d}$ leaves the commuting
scheme $\mathfrak{C}^{d}(\mathfrak{g})$ invariant. The inclusion
$\mathfrak{h}^{d}\rightarrow\mathfrak{g}^{d}$ factors through the commuting
scheme $\mathfrak{C}^{d}(\mathfrak{g})$ and it induces a homomorphism of
$k$-algebras $\mathfrak{i}:k[\mathfrak{C}^{d}(\mathfrak{g})]^{G}\rightarrow
k[\mathfrak{h}^{d}]^{W}$ because the restriction of a $G$-invariant function
to $\mathfrak{h}^{d}$ is also $W$-invariant. In other words, we have a
morphism of affine schemes (which we still denote by $\mathfrak{i}$):

$\mathfrak{i}:\mathfrak{h}^{d}//W\rightarrow\mathfrak{C}^{d}(\mathfrak{g})//G.$

### 3.1.

Let $V$ be an $n$-dimensional $k$-vector space and and let $R$ be any
$k$-algebra. Let $\mathfrak{C}^{d}(\mathfrak{gl}(V))$ be the pair wise
commuting scheme of $\mathfrak{gl}(V)$ whose $R$-points are given by the set

$\\{(x_{1},\dots,x_{d})\in\mathfrak{gl}(V\otimes R)^{d}:[x_{i},x_{j}]=0\\}.$

Let $A$ be a $k$-algebra representing the functor
$\mathfrak{C}^{d}(\mathfrak{gl}(V))$. Let $(x_{1},x_{2},\dots,x_{d})$ be the
universal point in $\mathfrak{C}^{d}(A)$. We then have a map from

$p:k[X_{1},\dots,X_{d}]\otimes R\rightarrow\mathfrak{gl}(A\otimes R)$

given by $p(X_{1},\dots,X_{d})=(x_{1},\dots,x_{d})$. The map $\det\circ p$
gives a homogenous polynomial law on the algebra $k[X_{1},\dots,X_{d}]$. Since
$\det\circ p$ is multiplicative, using Roby’s result we get an algebra
homomorphism from

$\mathfrak{s}:(k[X_{1},X_{2},\dots X_{d}]^{\otimes n})^{S_{n}}\rightarrow A$

such that $\det p((f))=\mathfrak{s}(f^{\otimes n})$ for all $f\in
k[X_{1},X_{2},\dots X_{d}]$. This implies that the image of the map
$\mathfrak{s}$ belongs to the algebra of $GL(V)$-invariants $A^{{\rm GL}(V)}$.
Thus we obtain a map of schemes

$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{gl}(V))//{\rm
GL}(V)\rightarrow\mathfrak{h}^{d}//{S_{n}}.$

Here, $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{gl}(V)$. This map
$\mathfrak{s}$ is called the spectral data map. Note that in this case the
Weyl group is $S_{n}$ and the Cartan subalgebra is the standard representation
of $S_{n}$. The compositions $\mathfrak{s}\circ\mathfrak{i}$ and
$\mathfrak{i}\circ\mathfrak{s}$ are verified to be identity on a generating
set given by trace functions due to Processi (see [Pro76]). As a result it was
concluded that the quotient scheme $\mathfrak{C}^{d}(\mathfrak{gl}(V))//GL(V)$
is normal and reduced (see [CN20] for details).

##  4\. Invariants

Let $(\mathfrak{g},\mathfrak{g}_{0})$ be a classical symmetric pair so that
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$. In this section, we
describe a convenient generating set for the $k$-algebra
$k[\mathfrak{g}_{1}^{d}]^{G_{0}}$. These invariants will be used to check that
the spectral data map is the section of the Chevalley restriction.

### 4.1.

Let $W(n,m,d)$ be the space $M_{n\times m}^{d}(k)\oplus M^{d}_{m\times n}(k)$.
Let $G$ be the group ${\rm GL}_{n}(k)\times{\rm GL}_{m}(k)$ via left and right
action on $W(n,m,d)$. We define

$\mu:W(n,m,d)\rightarrow M^{d^{2}}_{n\times
n}(k);((M_{1},\dots,M_{d}),(N_{1},\dots,N_{d}))\mapsto(A_{ij}),$

where $A_{ij}=M_{i}N_{j}$. Note that the map $\mu^{\ast}:k[M^{d^{2}}_{n\times
n}(k)]\rightarrow k[M_{n\times m}^{d}(k)\oplus M^{d}_{m\times n}(k)]^{{\rm
GL}_{m}(k)}$ is surjective from the first fundamental theorem for $GL_{n}(k)$
(see Theorem 5.2.1 of [GW98]). Hence, we get that the map

$\mu^{\ast}:k[M^{d^{2}}_{n\times n}(k)]^{{\rm GL}_{n}(k)}\rightarrow
k[M_{n\times m}^{d}(k)\oplus M^{d}_{m\times n}(k)]^{G}$ (4.1)

is surjective. Since $k[M^{d^{2}}_{n\times n}(k)]^{{\rm GL}_{n}(k)}$ is
generated by monomials in $A_{i}$, $i\in[d^{2}]$ (see Theorem 3.4 of [Pro76]),
from (4.1) we get that polynomials of the form

${\rm Tr}(M_{1}M_{2}\dots M_{l})$

where $M_{i}=Q_{n_{i}}R_{m_{i}}$ with $Q_{n_{i}},R_{m_{i}}^{t}\in M_{n\times
m}(k)$ and $n_{i},m_{i}\in[d]$ for all $i\in[l]$ generate the algebra
$k[W(n,m,d)]^{G}$.

### 4.2.

Let $V(n,m,d)$ be the space $M^{d}_{n\times m}(k)$, and let $G$ be the group
$SO_{n}(k)\times SO_{m}(k)$. The group $G$ acts on $V(m,n,d)$ by setting

$(A,B)(M_{1},M_{2},\dots,M_{d})=(AM_{1}B^{t},AM_{2}B^{t},\dots,AM_{d}B^{t}).$

Let $\mu:M^{d}_{n\times m}(k)\rightarrow({\rm Sym}^{2}k^{n})^{d}$ be the map

$(M_{1},M_{2},\dots,M_{d})\mapsto(M_{1}M_{1}^{t},M_{2}M_{2}^{t},\dots,M_{d}M_{d}^{t}).$

The map $\mu^{\ast}:k[{\rm Sym}^{2}(k^{n})^{d}]\rightarrow k[M^{d}_{n\times
m}(k)]^{{\rm O}_{n}(k)}$ is a surjective map from the first fundamental
theorem for orthogonal group (see Theorem 5.2.2 of [GW98]). Since $SO_{m}(k)$
is reductive, we get that the map

$\mu^{\ast}:k[{\rm Sym}^{2}(k^{n})^{d}]^{{\rm SO}_{m}(k)}\rightarrow
k[M^{d}_{n\times m}(k)]^{O_{n}(k)\times SO_{m}(k)}$ (4.2)

is surjective. The algebra $k[{\rm Sym}^{2}(k^{n})^{d}]^{{\rm SO}_{n}(k)}$ is
spanned by ${\rm Tr}(M_{n_{1}}M_{n_{2}}\dots M_{n_{l}})$, for some tuple
$(n_{1},n_{2},\dots,n_{l})$ such that $n_{i}\in[d]$ for all $i\in[l]$, and
here, $M_{k}$ is an $n\times n$ symmetric matrix (see [ATZ95]). Note that in
[ATZ95], the authors showed that for the action of $SO_{n}(k)$ on $d$-tuple of
$n\times n$ matrices, the ring of invariants is generated by traces of
monomials and some polarized Pfaffians. The Pfaffians become zero when we
restrict them to $k[{\rm Sym}^{2}(k^{n})^{d}]$. Hence from (4.2), we get that
the algebra $k[M^{d}_{n\times m}(k)]^{O_{n}(k)\times SO_{m}(k)}$ is spanned by
elements of the form

${\rm Tr}(M_{n_{1}}M_{n_{1}}^{t}M_{n_{2}}M_{n_{2}}^{t}\dots
M_{n_{k}}M_{n_{l}}^{t}),$ (4.3)

for some tuple $(n_{1},n_{2},\dots,n_{l})$ such that $n_{i}\in[d]$ for all
$i\in[l]$. If $n$ is odd then we get that the generators of $k[M^{d}_{n\times
m}(k)]^{SO_{n}(k)\times SO_{m}(k)}$ are given by the invariants in (4.3). If
$n$ is even and $n=m$, then the algebra $k[M^{d}_{n\times
m}(k)]^{SO_{n}(k)\times SO_{m}(k)}$ is strictly bigger than $k[M^{d}_{n\times
m}(k)]^{O_{n}(k)\times SO_{m}(k)}$; for instance the elements

$\det(T_{1}\otimes A_{1}+T_{2}\otimes A_{2}+\cdots+T_{d}\otimes A_{d})$

for any $(T_{1},T_{2},\dots,T_{d})\in M^{d}_{l\times l}(k)$, where $l\geq 1$,
are invariant for ${\rm SO}_{n}(k)\times{\rm SO}_{m}(k)$.

### 4.3.

Let $\mu:M_{n\times m}(k)\rightarrow{\rm Sym}^{2}(k^{m})$ be the map $X\mapsto
X^{t}X$ and let $\mu^{\vee}:M_{m\times n}(k)\rightarrow{\rm Sym}^{2}(k^{m})$
be the map $X\mapsto XX^{t}$. Let $\lambda$ be the map
$\mu^{d}\oplus(\mu^{\vee})^{d}$. We then have

$\lambda^{\ast}:k[({\rm Sym}^{2}(k^{m})^{d}\oplus({\rm
Sym}^{2}(k^{m})^{\ast})^{d}]\rightarrow k[M^{d}_{n\times m}(k)\oplus
M^{d}_{m\times n}(k)]^{{\rm O}_{n}(k)}$

If $n\geq m$ then using SFT of invariant theory for orthogonal groups the
above map is an isomorphism (see Corollary 5.2.5 of [GW98]). We then have

$\lambda^{\ast}:k[(({\rm Sym}^{2}(k^{m}))^{d}\oplus({\rm
Sym}^{2}(k^{m})^{\ast})^{d}]^{{\rm GL}_{m}(k)}\simeq k[M^{d}_{n\times
m}(k)\oplus M^{d}_{m\times n}(k)]^{{\rm O}_{n}(k)\times{\rm GL}_{m}(k)}.$

Note that the map

$\kappa:W(m,n,d)\rightarrow M^{d^{2}}_{n\times
n}(k);((M_{1},\dots,M_{d}),(N_{1},\dots,N_{d}))\rightarrow(A_{ij}),$

where $A_{ij}=M_{i}N_{j}$, induces a surjective map

$\kappa^{\ast}:k[M^{d^{2}}_{n\times n}(k)]\rightarrow k[M_{n\times
m}^{d}(k)\oplus M^{d}_{m\times n}(k)]^{{\rm GL}_{m}(k)}.$

Hence, we get that the map

$\kappa^{\ast}:k[M^{d^{2}}_{n\times n}(k)]^{{\rm O}_{n}(k)}\rightarrow
k[M^{d}_{n\times m}(k)\oplus M^{d}_{m\times n}(k)]^{{\rm O}_{n}(k)\times{\rm
GL}_{m}(k)}$

is surjective. Let $S$ be the set of polynomials

${\rm Tr}(M_{1}M_{2}\dots M_{l})$ (4.4)

such that $M_{i}=Q_{n_{i}}R_{m_{i}}$ with $n_{i},m_{i}\in[d]$ and $Q_{n_{i}}$
and $R_{m_{i}}$ are symmetric $n\times n$ matrices. The set
$\lambda^{\ast}(S)$ is the set of polynomials ${\rm
Tr}(M_{1}^{\prime}M_{2}^{\prime}\dots M_{l}^{\prime})$, where
$M_{i}^{\prime}=A_{n_{i}}A_{n_{i}}^{t}B_{m_{i}}^{t}B_{m_{i}}$ for some
$n_{i},m_{i}\in[d]$ and $A_{n_{i}}$ and $B_{m_{i}}$ are $n\times n$ matrices.
The image of $\kappa^{\ast}$ is generated by polynomials of the form

${\rm Tr}(\prod_{i=1}^{l}A_{n_{i}}B_{m_{i}}),$ (4.5)

where $A_{n_{i}}$ is an $n\times m$ matrix and $B_{m_{i}}$ is an $m\times n$
matrix (see Theorem 7.3 of [Pro76]). Since the algebra generated by
$\lambda^{\ast}(S)$ coincides with the algebra generated by the polynomials in
4.5, we get that $S$ is a generating set for $k[({\rm
Sym}^{2}(k^{m})^{d}\oplus({\rm Sym}^{2}(k^{m})^{\ast})^{d}]^{{\rm
GL}_{m}(k)}$.

##  5\. Construction of Spectral data map and proof of the main theorem

In this section we construct the spectral data map for the relevant classical
symmetric pairs and we show that in each case the spectral data map is a
section of the Chevalley restriction map.

###  5.1. The symmetric pair $AI$

Let $V$ be a $k$-vector space of dimension $n$, and let $\omega:V\times
V\rightarrow k$ be a non-degenerate symmetric bilinear form on $V$. Let
$\theta:\mathfrak{gl}(V)\rightarrow\mathfrak{gl}(V)$ be the involution defined
by $\theta(X)=-X^{\ast}$, where $X^{\ast}$ is given by:

$\omega(Xv,w)=\omega(v,X^{\ast}w).$

Let $\mathfrak{g}_{0}$ and $\mathfrak{g}_{1}$ be the $1$ and $-1$ eigenspaces
of $\theta$. Then the algebra $\mathfrak{g}_{0}$ can be identified with the
Lie algebra of the connected component $G_{0}$ of ${\rm O}(V,\omega)$
containing the identity element. With respect to an orthogonal basis
$(v_{1},v_{2},\dots,v_{n})$ of $V$ a Cartan subspace
$\mathfrak{c}\subset\mathfrak{g}_{1}$ is given by the following subspace:

$\\{\operatorname{diag}(b_{1},b_{2},\dots,b_{n}):b_{i}\in k,i\in[n]\\}.$

The little Weyl group $W_{\mathfrak{c}}$ is isomorphic to $S_{n}$–which is
identified with the group of permutations of the coordinates of
$\mathfrak{c}$.

Let $\mathfrak{C}^{d}(\mathfrak{g}_{1})$ be the $d$-fold commuting scheme
associated with the symmetric pair $(\mathfrak{gl}(V),\mathfrak{g}_{0})$. We
have the Chevalley restriction map

$\mathfrak{i}:\mathfrak{c}^{d}//W_{\mathfrak{c}}\rightarrow\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}},$

which is induced from the inclusion
$\mathfrak{c}^{d}\rightarrow\mathfrak{g}_{1}^{d}$. We will construct a section
of this map.

Let $R$ be any $k$-algebra, and let $A$ be the coordinate ring of
$\mathfrak{C}_{d}(\mathfrak{g}_{1})$. Let $(x_{1},x_{2},\dots,x_{d})$ be the
universal point of $\mathfrak{C}_{d}(\mathfrak{g}_{1})$. For any $k$-algebra
$R$, we define the map

$p:R[X_{1},X_{2},\dots,X_{d}]\rightarrow\mathfrak{gl}_{n}(A\otimes R),\
X_{i}\mapsto x_{i},i\in[d].$

Note that the map $\det\circ\ p$ is a degree $n$ multiplicative map. Hence, by
Roby’s theorem we get an algebra map

$\tilde{\mathfrak{s}}:TS^{n}(k[X_{1},X_{2},\dots,X_{d}])\rightarrow A$

such that

$\tilde{\mathfrak{s}}(\theta^{\otimes n})=\det\circ\ p(\theta)$

for all $\theta\in k[X_{1},X_{2},\dots,X_{d}]$. Since $\det$ is
$G_{0}$-invariant, the image of the map $\mathfrak{s}$ is is contained in
$A^{G_{0}}$.

Let $(y_{1},\dots,y_{d})$ be the tautological point of $\mathfrak{c}(B)$,
where $B=k[\mathfrak{c}^{d}]$. Note that $B=k[\mathfrak{c}^{d}]$ is a
polynomial algebra in the variables $b_{j}(y_{i})$, $1\leq i\leq d$ and $1\leq
j\leq n$. Let

$\beta:TS^{n}(k[X_{1},X_{2},\dots,X_{d}])\rightarrow k[\mathfrak{c}^{d}]$

be the isomorphism given by $\beta(X_{i,j})=b_{j}(y_{i})$, where $X_{i,j}$ is
the variable $X_{i}$ in the $j$-th copy, for $i\in[d]$ and $j\in[n]$. We get a
map

$\mathfrak{s}:k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}\rightarrow A^{G_{0}}$
(5.1)

such that $\mathfrak{s}\circ\beta(\theta^{\otimes n})=\det\circ p(\theta)$.
Thus we obtain a map of schemes
$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}\rightarrow\mathfrak{c}^{d}//W_{\mathfrak{c}}$.
We show that the map
$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}\rightarrow\mathfrak{c}^{d}//W_{\mathfrak{c}}$
is the inverse of the Chevalley restriction map
$\mathfrak{i}:\mathfrak{c}^{d}//W_{\mathfrak{c}}\rightarrow\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}$.

For the diagonal action of $G_{0}$ on the $d$-copies of $\mathfrak{gl}(V)$,
the ring of invariants $k[\mathfrak{gl}(V)^{d}]^{G_{0}}$ is generated by the
elements $Tr(M)$ and some polarized Pfaffians, where $M$ is a monomial in
$X_{j},X_{j}^{*}$, $j=1,2,\cdots,d$ (see [ATZ95]). Recall that the space
$\mathfrak{g}_{1}$ is the $-1$ eigen space of the map $\theta(X)=-X^{*}$. So
the elements in $\mathfrak{g}_{1}$ satisfy $X^{*}=X$ and hence the polarized
Pfaffians become zero when we restrict them to $\mathfrak{g}_{1}^{d}$.

Since $\mathfrak{C}^{d}(\mathfrak{g}_{1})$ is a closed subscheme of
$\mathfrak{gl}(V)^{d}$ and $G_{0}$ is (linearly) reductive, the restriction
map $k[\mathfrak{gl}(V)^{d}]\rightarrow k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]$
induces a surjective map $k[\mathfrak{gl}(V)^{d}]^{G_{0}}\rightarrow
k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$. So the restrictions of the
functions $Tr(M)$, where $M$ is a monomial in $X_{j}$, $j=1,2,\cdots,d$ form a
generating set for $k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$. Let
$(x_{1},x_{2},\cdots,x_{d})$ be the tautological point of the commuting scheme
$\mathfrak{C}^{d}(\mathfrak{g}_{1})$. Then the set
$\\{\phi_{\underline{a}}={\rm Tr}(x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}}):\underline{a}=(a_{1},\dots,a_{d})\in\mathbb{Z}^{d}_{\geq 0}\\}$
is a generating set for $k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G}$.

Note that

$\mathfrak{i}(\phi_{\underline{a}})=\sum_{j=1}^{n}\prod_{i}b_{j}(y_{i})^{a_{i}}.$

From the above equation, we get that $\mathfrak{i}(\phi_{\underline{a}})$
generate the $k$-algebra $k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$. We set
$\psi_{\underline{a}}=\mathfrak{i}(\phi_{\underline{a}})$. Let
$\theta_{\underline{a}}\in k[t]\otimes[X_{1},X_{2},\cdots,X_{d}]$ be the
polynomial $t-X_{1}^{a_{1}}X_{2}^{a_{2}}\dots X_{n}^{a_{n}}$. Then
$p(\theta_{\underline{a}})=tI-x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}}\in\mathfrak{g}_{1}(A\otimes k[t])$, where $I$ is the identity
matrix. Using (5.1) we have

$\det(p(\theta_{\underline{a}}))=\mathfrak{s}(\beta(\theta_{\underline{a}}^{\otimes
n})).$

By the definition of the map $\beta$ we have
$\beta(\theta_{\underline{a}}^{\otimes
n})=\prod_{j=1}^{n}(t-\prod_{i=1}^{d}b_{j}(y_{i})^{a_{i}})$. Hence, we get
that

$\det(tI-x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}})=\mathfrak{s}(\prod_{j=1}^{n}(t-\prod_{i=1}^{d}b_{j}(y_{i})^{a_{i}})).$
(5.2)

Note that $\det(tI-x_{1}^{a_{1}}x_{2}^{a_{2}}\dots x_{d}^{a_{d}})$ is the
characteristic polynomial of the matrix $x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}}$ and hence

$\mathfrak{s}(\det(tI-x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}}))=t^{n}-\mathfrak{s}({\rm Tr}(x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}}))t^{n-1}+\ \text{lower degree terms in}\ t.$ (5.3)

Comparing the coefficients of $t^{2n-1}$ in equations (5.2) and (5.3) we get
that $\mathfrak{s}(\mathfrak{i}(\phi_{\underline{a}}))=\phi_{\underline{a}}$.
Thus, we conclude that $\mathfrak{s}\circ\mathfrak{i}$ and
$\mathfrak{i}\circ\mathfrak{s}$ are both identities.

Hence, we get that the Chevalley restriction map
$\mathfrak{i}:\mathfrak{c}^{d}//W_{\mathfrak{c}}\rightarrow\mathfrak{C}^{d}(\mathfrak{g}_{1})//G$
is an isomorphism.

###  5.2. The symmetric pair $AII$

Let $V$ be a $k$-vector space of dimension $2n$, and let $\omega:V\times
V\rightarrow k$ be a non-degenerate skew symmetric bilinear form on $V$. Let
$\theta:\mathfrak{gl}(V)\rightarrow\mathfrak{gl}(V)$ be the involution defined
by $\theta(X)=-X^{\ast}$, where $X^{\ast}$ is given by:

$\omega(Xv,w)=\omega(v,X^{\ast}w).$

Let $\mathfrak{g}_{0}$ and $\mathfrak{g}_{1}$ be the $1$ and $-1$ eigenspaces
of $\theta$. Then the algebra $\mathfrak{g}_{0}$ can be identified with the
Lie algebra of the connected component $G_{0}$ of ${\rm Sp}(V,\omega)$
containing the identity. Let

$(w_{-n},w_{-n+1},\dots,w_{n-1},w_{n})$

be a Witt basis of $V$, and in this basis a Cartan subspace
$\mathfrak{c}\subset\mathfrak{g}_{1}$ is given by the following subspace:

$\\{\operatorname{diag}(b_{n},\dots b_{1},b_{1},\dots,b_{n}):b_{i}\in
k,i\in[n]\\}.$

The little Weyl group $W_{\mathfrak{c}}$ is isomorphic to $S_{n}$-which is
identified with the group of permutations of the coordinates of
$\mathfrak{c}$.

In [CC21] for any $k$-algebra $R$, the authors defined a map
$N_{+}:\mathfrak{g}_{1}(R)\rightarrow R$, called it the Pfaffian norm map
which satisfies that $\det(x)=N_{+}(x)^{2}$ for any $x\in\mathfrak{g}_{1}(R)$.
More interestingly, they showed that the map $N_{+}$ is multiplicative on the
coordinate ring of the commuting subscheme $\mathfrak{C}({\mathfrak{g}_{1}})$
of $\mathfrak{g}_{1}\times\mathfrak{g}_{1}$.

Let $A$ be the coordinate ring of $\mathfrak{C}^{d}(\mathfrak{g}_{1})$, and
let $(x_{1},x_{2},\dots,x_{d})$ be the universal point of this scheme. For any
$k$-algebra $R$, we define the map

$p:R[X_{1},X_{2},\dots,X_{d}]\rightarrow\mathfrak{gl}_{2n}(A\otimes
R),X_{i}\mapsto x_{i},i\in[d].$

The image of $p$ is contained in $\mathfrak{g}_{1}(A\otimes R)$. Then we have
the map

$R[X_{1},X_{2},\dots,X_{d}]\xrightarrow{p}\mathfrak{g}_{1}(A\otimes
R)\xrightarrow{N_{+}}A\otimes R.$ (5.4)

Since $N_{+}$ is multiplicative, the composition $N_{+}\circ p$ is
multiplicative and homogeneous of degree $n$. Thus by Roby’s theorem we get a
homomorphism of $k$-algebras:

$\tilde{\mathfrak{s}}:TS^{n}(k[X_{1},\dots,X_{d}])\rightarrow A$

such that

$\tilde{\mathfrak{s}}(q^{\otimes n})=N_{+}(p(\theta)),\,\,\text{for
all}\,\,\theta\in k[X_{1},\dots,X_{d}].$ (5.5)

Since $N_{+}$ is a $G$-invariant map, the map $\mathfrak{s}$ is also
$G_{0}$-invariant and hence we get that the image of $s$ is contained in
$A^{G_{0}}$.

Let $(y_{1},\dots,y_{d})$ be the tautological point of $\mathfrak{c}(B)$,
where $B=k[\mathfrak{c}^{d}]$. Note that $B=k[\mathfrak{c}^{d}]$ is a
polynomial algebra in the variables $b_{j}(y_{i})$, $1\leq i\leq d$ and $1\leq
j\leq n$. Let

$\beta:TS^{n}(k[X_{1},X_{2},\dots,X_{d}])\rightarrow k[\mathfrak{c}^{d}]$

be the isomorphism given by $\beta(X_{i,j})=b_{j}(y_{i})$, where $X_{i,j}$ is
the variable $X_{i}$ in the $j$-th copy, for $i\in[d]$ and $j\in[n]$. Thus, we
get a map $\mathfrak{s}:k[\mathfrak{c}^{d}]\rightarrow A^{G_{0}}$ such that

$\mathfrak{s}\circ\beta(\theta^{\otimes n})=N_{+}\circ p(\theta),$ (5.6)

for all $\theta\in k[X_{1},\dots,X_{d}]$. Thus we obtain a map of schemes
$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}\rightarrow\mathfrak{c}^{d}//W_{\mathfrak{c}}.$
We show that the map
$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}\rightarrow\mathfrak{c}^{d}//W_{\mathfrak{c}}$
is the inverse of the restriction map
$\mathfrak{i}:\mathfrak{c}^{d}//W_{\mathfrak{c}}\rightarrow\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}$.

For the diagonal action of the symplectic group ${G_{0}}$ on the $d$-copies of
$\mathfrak{gl}(V)$, the ring of invariants $k[\mathfrak{gl}(V)^{d}]^{G_{0}}$
is generated by the elements $Tr(M)$, where $M$ is a monomial in
$X_{j},X_{j}^{*}$, $j=1,2,\cdots,d$ of degree less than or equal to $2^{n}-1$
(see Theorem 10.1 of [Pro76]). Recall that the space $\mathfrak{g}_{1}$ is the
$-1$ eigen space of the map $\theta(X)=-X^{*}$. So the elements in
$\mathfrak{g}_{1}$ satisfy $X^{*}=X$. Since
$\mathfrak{C}^{d}(\mathfrak{g}_{1})$ is a closed subscheme of
$\mathfrak{gl}(V)^{d}$ and $G_{0}$ is (linearly) reductive, the restriction
map $k[\mathfrak{gl}(V)^{d}]\rightarrow k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]$
induces a surjective map $k[\mathfrak{gl}(V)^{d}]^{G_{0}}\rightarrow
k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$. So the restrictions of the
functions $Tr(M)$, where $M$ is a monomial in $X_{j}$, $j=1,2,\cdots,d$ form a
generating set for $k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$. Let
$(x_{1},x_{2},\cdots,x_{d})$ be the tautological point of the commuting scheme
$\mathfrak{C}^{d}(\mathfrak{g}_{1})$. Then the set
$\\{\phi_{\underline{a}}={\rm Tr}(x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}}):\underline{a}=(a_{1},\dots,a_{d})\in\mathbb{Z}^{d}_{\geq 0}\\}$
is a generating set for $k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$.

Note that

$\mathfrak{i}(\phi_{\underline{a}})=2\sum_{j=1}^{n}\prod_{i}b_{j}(y_{i})^{a_{i}}.$

From the above equation, we get that $\mathfrak{i}(\phi_{\underline{a}})$
generate the $k$-algebra $k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$. We set
$\psi_{\underline{a}}=\mathfrak{i}(\phi_{\underline{a}})$, for all $n$-tuples
$\underline{a}\in\mathbb{Z}^{n}_{\geq 0}$. In order to show that
$\mathfrak{c}$ and $\mathfrak{s}$ are inverses of each other we need to show
that $\mathfrak{s}(\psi_{\underline{a}})=\phi_{\underline{a}}$, for all
$n$-tuples $\underline{a}\in\mathbb{Z}^{n}_{\geq 0}$.

Let $\theta\in k[X_{1},X_{2},\cdots,X_{d}]\otimes k[t]$ be the polynomial
$t-X_{1}^{a_{1}}X_{2}^{a_{2}}\dots X_{n}^{a_{n}}$. Then $p(\theta)=tI-
x_{1}^{a_{1}}x_{2}^{a_{2}}\dots x_{d}^{a_{d}}\in\mathfrak{g}_{1}(A\otimes
k[t])$, where $I$ is the identity matrix. Since $\det(x)=N_{+}(x)^{2}$ for any
$x\in\mathfrak{g}_{1}(R)$, where $R$ is a $k$-algebra, we have

$\det(p(\theta))=N_{+}(p(\theta))^{2}.$

Using (5.6) we also have

$N_{+}(p(\theta))=\mathfrak{s}(\beta(\theta^{\otimes n})).$

By the definition of the map $\beta$ we have $\beta(\theta_{a}^{\otimes
n})=\prod_{j=1}^{n}(t-\prod_{i=1}^{d}b_{j}(y_{i})^{a_{i}})$.

Hence, we get that

$\displaystyle\det(tI-x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}})=\mathfrak{s}(\prod_{j=1}^{n}(t-\prod_{i=1}^{d}b_{j}(y_{i})^{a_{i}}))^{2}.$

Note that $\det(tI-x_{1}^{a_{1}}x_{2}^{a_{2}}\dots x_{d}^{a_{d}})$ is the
characteristic polynomial of the matrix $x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}}$ and hence $\det(tI-x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}})=t^{2n}-Tr(x_{1}^{a_{1}}x_{2}^{a_{2}}\dots
x_{d}^{a_{d}})t^{2n-1}+$ lower degree terms in $t$. On the other hand

$\mathfrak{s}(\prod_{j=1}^{n}(t-\prod_{i=1}^{d}b_{j}(y_{i})^{a_{i}}))^{2}=t^{2n}+2\sum_{j=1}^{n}\prod_{i}b_{j}(y_{i})^{a_{i}}t^{2n+1}+\
\text{lower degree terms in}\ t.$

Comparing the coefficients of $t^{2n-1}$ we get that
$\mathfrak{s}(\mathfrak{i}(\phi_{a}))=\phi_{a}$.

Hence, we get that the Chevalley restriction map
$\mathfrak{i}:\mathfrak{c}^{d}//W_{\mathfrak{c}}\rightarrow\mathfrak{C}^{d}(\mathfrak{g}_{1})//{G_{0}}$
is an isomorphism.

###  5.3. The symmetric pair $AIII$

Let $m,n$ be positive integers such that $m\geq n$. Let $\mathfrak{g}$ be the
Lie algebra $\mathfrak{gl}_{n+m}$ and let $\mathfrak{g}_{0}$ be the Lie
algebra

$\left\\{\begin{pmatrix}A&0\\\ 0&B\end{pmatrix}:X\in
A,\in\mathfrak{gl}_{n},B\in\mathfrak{gl}_{m}\right\\},$

and let $\mathfrak{g}_{1}$ be the space

$\left\\{\begin{pmatrix}0&X\\\ Y&0\end{pmatrix}:X,Y^{t}\in{\rm M}_{n\times
m}(k)\right\\}.$

The pair $(\mathfrak{g},\mathfrak{g}_{0})$ is a symmetric pair for an
involution $\theta$ with $\mathfrak{g}_{0}$, the space of invariants and
$\mathfrak{g}_{1}$, the $-1$ eigenspace. The adjoint action of the group
$G_{0}={\rm GL}_{n}(k)\times{\rm GL}_{m}(k)$ on the space $\mathfrak{g}_{1}$
is given by

$(g_{1},g_{2})(X,Y)=(g_{1}Xg_{2}^{-1},g_{2}Yg_{1}^{-1}).$

A cartan subspace, denoted by $\mathfrak{c}$, of the symmetric pair
$(\mathfrak{g},\mathfrak{g}_{0})$ is given by the space consisting of matrices
of the form

$\begin{pmatrix}0&X\\\ X^{t}&0\end{pmatrix},$

where $X=[\operatorname{diag}(b_{1},b_{2},\dots,b_{n}),0_{n\times m-n}]$ for
some $b_{i}\in k$ for all $i\in[n]$. The little Weyl group $W_{\mathfrak{c}}$
in this case is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{n}\rtimes S_{n}$.

Let $\mathfrak{C}_{d}(\mathfrak{g}_{1})$ be the commuting scheme associated to
the pair $(\mathfrak{g},\mathfrak{g}_{0})$. Let
$A=k[\mathfrak{C}_{d}(\mathfrak{g}_{1})]$ be the coordinate ring of
$\mathfrak{C}_{d}(\mathfrak{g}_{1})$ and let $(x_{1},\dots,x_{d})$ be the
universal point of $\mathfrak{C}_{d}(\mathfrak{g}_{1})(A)$. We set

$x_{i}=\begin{pmatrix}0&Q_{i}\\\
R_{i}&0\end{pmatrix},Q_{i},R_{i}^{t}\in\mathfrak{gl}_{n\times m}(A).$

Let $R$ be a $k$-algebra and let
$p:R[X_{1},\dots,X_{d}]\rightarrow\mathfrak{gl}_{n+m}(A\otimes R)$ be the map
sending $X_{i}\mapsto x_{i}$ for all $i\in[d]$. Let $R[X_{1},\dots,X_{d}]^{+}$
be the algebra spanned by even degree monomials. Note that the image of the
map $p:R[X_{1},\dots,X_{d}]^{+}\rightarrow\mathfrak{gl}_{n+m}(A\otimes R)$ is
contained in $\mathfrak{gl}_{n}(A\otimes R)\times\mathfrak{gl}_{m}(A\otimes
R)$. Let $q_{1}$ be the first projection of $\mathfrak{gl}_{n}(A\otimes
R)\times\mathfrak{gl}_{m}(A\otimes R)$. Note that the composite map

$R[X_{1},\dots,X_{d}]^{+}\xrightarrow{p}\mathfrak{gl}_{n}(A\otimes
R)\times\mathfrak{gl}_{m}(A\otimes
R)\xrightarrow{q_{1}}\mathfrak{gl}_{n}(A\otimes R)\xrightarrow{{\rm
det}}{A\otimes R}$

has degree $n$. By Roby’s theorem we get a map

$\tilde{\mathfrak{s}}:TS^{n}(k[X_{1},\dots,X_{d}]^{+})\rightarrow A$

such that $\tilde{\mathfrak{s}}(\theta^{\otimes n})=\det\ \circ\ q_{1}\ \circ
p(\theta)$, for all $\theta\in k[X_{1},\dots,X_{d}]^{+}$. Note that the image
of the map $\tilde{\mathfrak{s}}$ is contained in $A^{G_{0}}$.

The algebra $TS^{n}k[X_{1},\dots,X_{d}]^{+}$ can be identified with the
subalgebra of $T^{n}k[X_{1},\dots,X_{d}]$ of fixed points under
$W_{\mathfrak{c}}=S_{n}\ltimes(\mathbb{Z}/2\mathbb{Z})^{n}$. Now, we give an
isomorphism of $TS^{n}k[X_{1},\dots,X_{d}]^{+}$ with
$k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$. Let $(y_{1},\dots,y_{d})$ be the
tautological point of $\mathfrak{c}(k[\mathfrak{c}^{d}])$. Note that
$k[\mathfrak{c}^{d}]$ is a polynomial algebra in the variables $b_{j}(y_{i})$,
$1\leq i\leq d$ and $1\leq j\leq n$. Let

$\beta:T^{n}k[X_{1},\dots,X_{d}]\rightarrow k[\mathfrak{c}^{d}]$

be the isomorphism of algebras defined by $\beta(X_{ij})=b_{j}(y_{i})$, for
$i\in[d]$ and $j\in[n]$. Here, $X_{ij}$ is the $i$-th variable in the $j$-th
copy of $k[X_{1},\dots,X_{n}]$. By restriction we have an isomorphism of
algebras

$\beta:TS^{n}k[X_{1},\dots,X_{d}]^{+}\rightarrow
k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}.$

Then it follows that we have an algebra map

$\mathfrak{s}:k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}\rightarrow A^{G_{0}}$

such that

$\mathfrak{s}\beta(\theta^{\otimes n})=\det\circ q_{1}\circ
p(\theta),\,\,\text{for all}\,\,\theta\in k[X_{1},\dots,X_{d}]^{+}.$

Thus we obtain a map of schemes
$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{g}_{1})//G_{0}\rightarrow\mathfrak{c}^{d}//W_{\mathfrak{c}}$.

Let $\mathfrak{i}:k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}\rightarrow
k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$ be the restriction map. We will show
that $\mathfrak{s}$ and $\mathfrak{i}$ are inverses of each other.

The ring of invariants $k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$ is
generated by the images of the following polynomials, via the restriction map
$k[\mathfrak{g}_{1}^{d}]^{G_{0}}\rightarrow
k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$:

${\rm Tr}(M_{1}M_{2}\dots M_{k})$ (5.7)

where $M_{i}=Q_{n_{i}}R_{m_{i}}$ for some $n_{i},m_{i}\in[d]$; and we denote
the polynomial in (5.7) by $P$ (see Section 4.1). We then have

$\mathfrak{i}(P)=\sum_{j=1}^{n}\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}}).$

From the above equation, we get that $\mathfrak{i}(\phi_{\underline{a}})$
generate the $k$-algebra $k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$. Let $R$ be
the algebra $k[t]$, and consider the element
$\theta=t-\prod_{(n_{i},m_{i})}X_{n_{i}}X_{m_{i}}$ in
$R[X_{1},X_{2},\dots,X_{d}]^{+}$. We then have

$\beta(\theta^{\otimes
n})=\prod_{j=1}^{n}(t-\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}})).$

We then observe that $\det\circ q_{1}\circ p(\theta)$ is equal to

$\det(t-\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}})).$

Since $\mathfrak{s}\circ\beta(\theta^{\otimes n})=\det\circ q_{1}\circ
p(\theta)$, we get that

$t^{n}-\mathfrak{s}\left(\sum_{j=1}^{n}\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}})\right)t^{n-1}+\cdots=t^{n}-{\rm
Tr}(\prod_{(n_{i},m_{i})}Q_{n_{i}}R_{m_{i}})t^{n-1}+\cdots$

Comparing the coefficients of $t^{n-1}$ we get that
$\mathfrak{s}\circ\mathfrak{\iota}(P)=P$. Thus, we get that $\mathfrak{s}$ is
a section of the map $\mathfrak{i}$.

###  5.4. The symmetric pair $BDI$

Let $m,n$ be positive integers such that $m\geq n$ and let $\mathfrak{g}$ be
the Lie algebra $\mathfrak{so}_{n+m}$. Let $\mathfrak{g}_{0}$ be the Lie
algebra

$\left\\{\begin{pmatrix}A&0\\\ 0&B\end{pmatrix}:X\in
A,\in\mathfrak{so}_{n},B\in\mathfrak{so}_{m}\right\\},$

and let $\mathfrak{g}_{1}$ be the space

$\left\\{\begin{pmatrix}0&X\\\ -X^{t}&0\end{pmatrix}:X\in{\rm M}_{n\times
m}(k)\right\\}.$

The pair $(\mathfrak{g},\mathfrak{g}_{0})$ is a symmetric pair for an
involution with $\mathfrak{g}_{0}$ the space of invariants and
$\mathfrak{g}_{1}$ the $-1$ eigenspace. The adjoint action of the group
$G_{0}={\rm SO}_{n}(k)\times{\rm SO}_{m}(k)$ on the space $\mathfrak{g}_{1}$
is given by

$(g_{1},g_{2})(X,Y)=(g_{1}Xg_{2}^{-1},g_{2}Yg_{1}^{-1}).$

A cartan subspace, denoted by $\mathfrak{c}$, of the symmetric pair
$(\mathfrak{g},\mathfrak{g}_{0})$ is given by the space consisting of matrices
of the form

$\begin{pmatrix}0&X\\\ -X^{t}&0\end{pmatrix},$

where $X$ is a matrix of the form $[A,0_{n\times(m-n)}]$, with
$A=\operatorname{diag}(b_{1},b_{2},\dots,b_{n})$ for some $b_{i}\in k$ for all
$i\in[n]\\}$. The little Weyl group $W_{\mathfrak{c}}$ is of the form
$\mathbb{Z}/2\mathbb{Z}^{n}\rtimes S_{n}$. This case is similar to the case of
$AIII$, however since the space $\mathfrak{g}_{1}$ differs from its counter
part in $AIII$, we briefly describe the section for the Chevalley restriction
map.

Let $\mathfrak{C}_{d}(\mathfrak{g}_{1})$ be the commuting scheme associated to
the pair $(\mathfrak{g},\mathfrak{g}_{0})$. Let
$A=k[\mathfrak{C}_{d}(\mathfrak{g}_{1})]$, and let $(x_{1},\dots,x_{d})$ be
the universal point in $\mathfrak{C}_{d}(\mathfrak{g}_{1})(A)$. We set

$x_{i}=\begin{pmatrix}0&Q_{i}\\\
-Q_{i}^{t}&0\end{pmatrix},Q_{i}\in\mathfrak{gl}_{n\times m}(A).$

Let $R$ be a $k$-algebra and let
$p:R[X_{1},\dots,X_{d}]\rightarrow\mathfrak{gl}_{n+m}(A\otimes R)$ be the map
sending $X_{i}\mapsto x_{i}$ for all $i\in[d]$. Let $R[X_{1},\dots,X_{d}]^{+}$
be the algebra spanned by the even degree monomials, and note that the image
of the map $p:R[X_{1},\dots,X_{d}]^{+}\rightarrow\mathfrak{gl}_{n+m}(A\otimes
R)$ is contained in $\mathfrak{gl}_{n}(A\otimes
R)\times\mathfrak{gl}_{m}(A\otimes R)$. Let $q_{1}$ be the first projection of
$\mathfrak{gl}_{n}(A\otimes R)\times\mathfrak{gl}_{m}(A\otimes R)$. Note that
the composite map

$R[X_{1},\dots,X_{d}]^{+}\xrightarrow{p}\mathfrak{gl}_{n}(A\otimes
R)\times\mathfrak{gl}_{m}(A\otimes
R)\xrightarrow{q_{1}}\mathfrak{gl}_{n}(A\otimes R)\xrightarrow{{\rm
det}}{A\otimes R}$

is multiplicative and has degree $n$. So by Roby’s theorem we get a map

$\tilde{\mathfrak{s}}:TS^{n}(k[X_{1},\dots,X_{d}]^{+})\rightarrow A$

such that $\tilde{\mathfrak{s}}(\theta^{\otimes n})=\det\ \circ\ q_{1}\ \circ
p(\theta)$, for all $\theta\in k[X_{1},\dots,X_{d}]^{+}$. We note that the
image of $\tilde{\mathfrak{s}}$ belongs to $A^{G_{0}}$.

Let $(y_{1},\dots y_{d})$ be a tautological point of $k[\mathfrak{c}^{d}](B)$,
where $B=k[\mathfrak{c}^{d}]$. Let

$\beta:TS^{n}(k[X_{1},\dots,X_{d}]^{+})\rightarrow k[\mathfrak{c}^{d}]$

be the map $\beta(X_{ij})=b_{j}(y_{i})$, for all $i\in[d]$ and $j\in[n]$.
Here, $X_{ij}$ be the variable $X_{j}$ in the $i$-th copy in
$TS^{n}(k[X_{1},\dots,X_{d}])$. As in the case of $AIII$, $\beta$ restricts to
an isomorphism of algebras

$\beta:TS^{n}k[X_{1},\dots,X_{d}]^{+}\rightarrow
k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}.$

Then it follows that we have an algebra map

$\mathfrak{s}:k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}\rightarrow A^{G_{0}}$

such that

$\mathfrak{s}\beta(\theta^{\otimes n})=\det\circ q_{1}\circ
p(\theta),\,\,\text{for all}\,\,\theta\in k[X_{1},\dots,X_{d}]^{+}.$

Thus we obtain a map of schemes
$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{g}_{1})//G_{0}\rightarrow\mathfrak{c}^{d}//W_{\mathfrak{c}}$.

We assume that $m$ is odd. The ring of invariants
$k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$ is generated by the images of
the following polynomials, via the restriction map
$k[\mathfrak{g}_{1}^{d}]^{G_{0}}\rightarrow
k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$:

${\rm Tr}(M_{1}M_{2}\dots M_{k})$ (5.8)

where $M_{i}=Q_{n_{i}}Q^{t}_{n_{i}}$ for some $n_{i},m_{i}\in[d]$; and we
denote the polynomial in (5.8) by $P$ (see Subsection 4.2). We then have

$\mathfrak{i}(P)=\sum_{j=1}^{n}\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})^{2}.$

From the above equation, we get that $\mathfrak{i}(\phi_{\underline{a}})$
generate the $k$-algebra $k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$. Let $R$ be
the algebra $k[t]$, and set $\theta=t-\prod_{n_{i}}X_{n_{i}}^{2}$ in the
algebra $R[X_{1},X_{2},\dots,X_{d}]^{+}$. We then have

$\beta(\theta^{\otimes
n})=\prod_{j=1}^{n}(t-\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})^{2}).$

We then observe that $\det\circ q_{1}\circ p(\theta)$ is equal to

$\det(t-\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})^{2}).$

Since $\mathfrak{s}\circ\beta(\theta^{\otimes n})=\det\circ q_{1}\circ
p(\theta)$, we get that

$t^{n}-\mathfrak{s}\left(\sum_{j=1}^{n}\prod_{n_{i}}b_{j}(y_{n_{i}})^{2}\right)t^{n-1}+\cdots=t^{n}-{\rm
Tr}(\prod_{n_{i}}Q_{n_{i}}Q^{t}_{n_{i}})t^{n-1}+\cdots$

Comparing the coefficients of $t^{n-1}$ we get that
$\mathfrak{s}\circ\mathfrak{\mathfrak{i}}(P)=P$. Thus, we get that
$\mathfrak{s}$ is a section of the map $\mathfrak{i}$.

######  Remark 5.1.

When $m=n$ is even, the ring of invariants
$k[\mathfrak{C}_{d}(\mathfrak{g}_{1})]^{G_{0}}$ has additional invariants of
the form

$\det(T_{1}\otimes A_{1}+T_{2}\otimes A_{2}+\cdots+T_{d}\otimes A_{d})$ (5.9)

where $T=(T_{1},\dots,T_{d})$ is an element of $M^{d}_{r\times r}$ and $r\geq
1$. We denote by $\psi_{T}$ the invariant in (5.9). Consider the element

$\theta=\det(T_{1}\otimes X_{1}I_{n\times n}+T_{2}\otimes X_{2}I_{n\times
n}+\cdots+T_{d}\otimes X_{d}I_{n\times n})$

in the ring $R[X_{1},\dots,X_{d}]^{+}$ and $R=k[M^{d}_{r\times r}]$. The
identity $\mathfrak{s}\circ\beta(\theta^{\otimes n})=\det\circ q_{1}\circ
p(\theta)$ implies that $\mathfrak{s}\circ\mathfrak{i}(\psi_{T})=\psi_{T}$.
Now, under the assumption that the invariants in (5.9) and (5.8) generate the
algebra $k[\mathfrak{C}_{d}(\mathfrak{g}_{1})]^{G_{0}}$, we get that the map
$\mathfrak{s}$ is the section of the Chevalley restriction map.

###  5.5. The symmetric pair $CI$

Let $V$ be a $2n$-dimensional $k$-vector space and let $\omega$ be a non-
degenerate skew symmetric bilinear form on $V$. Let $(v_{1},v_{2},\dots
v_{n},v_{-1},\dots,v_{-n})$ be a Witt basis for the pair $(V,\omega)$. The
matrix of the form $\omega$ is equal to

$J=\begin{pmatrix}0&I_{n}\\\ I_{n}&0\end{pmatrix},$

where $I_{n}$ is the $n\times n$ identity matrix. Let $V_{+}$ and $V_{-}$ be
the spaces $\langle v_{1},\dots,v_{n}\rangle$ and $\langle
v_{-1},\dots,v_{-n}\rangle$ respectively. In this basis we identify
$\mathfrak{g}=\mathfrak{sp}(V,\omega)$ as a Lie subalgebra of
$\mathfrak{gl}_{2n}$. Let $T\in GL(V)$ be such that $T|_{V_{+}}=id$ and
$T|_{V_{-}}=-id$. Then the conjugation by $T$ defines an involution of
$\mathfrak{g}$ such that the Lie algebra $\mathfrak{g}_{0}$ is identified with
the Lie algebra

$\\{\begin{pmatrix}A&0\\\ 0&-A^{t}\end{pmatrix}:A\in\mathfrak{gl}_{n}\\},$

and $\mathfrak{g}_{1}$ is the space

$\\{\begin{pmatrix}0&X\\\
Y&0\end{pmatrix}:X,Y\in\mathfrak{gl}_{n},X^{t}=X,Y^{t}=Y\\}.$

Let $G_{0}$ be the connected subgroup of ${\rm Sp}(V,\omega)$ with
$\mathfrak{g}_{0}$ as its Lie algebra. We identify $G_{0}$ with ${\rm
GL}_{n}(k)$. As a $G_{0}$ module $\mathfrak{g}_{1}$ is isomorphic to ${\rm
Sym}^{2}(k^{n})\oplus({\rm Sym}^{2}(k^{n})^{*}$. We choose the Cartan subspace
$\mathfrak{c}$ to be the subspace

$\\{\begin{pmatrix}0&X\\\
X&0\end{pmatrix}:X=\operatorname{diag}(b_{1},\dots,b_{n}),b_{i}\in
k,i\in[n]\\}$

of $\mathfrak{g}_{1}$. Then the little Weyl group, denoted by
$W_{\mathfrak{c}}$, is isomorphic to
$S_{n}\ltimes(\mathbb{Z}/2\mathbb{Z})^{n}$.

Let $\mathfrak{C}^{d}(\mathfrak{g}_{1})$ be the $d$-fold commuting scheme
associated with the symmetric pair $(\mathfrak{g},\mathfrak{g}_{0})$ and let
$A=k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]$ be the coordinate ring. Let
$(x_{1},\dots,x_{d})$ be the universal point in
$\mathfrak{C}^{d}(\mathfrak{g}_{1})(A)$. Let $Q_{i}$ and $R_{i}$ be matrices
with values in $A$ such that

$x_{i}=\begin{pmatrix}0&Q_{i}\\\ R_{i}&0\end{pmatrix},i\in[d].$ (5.10)

Note that $Q_{i}^{t}=Q_{i}$ and $R_{i}^{t}=R_{i}$, for all $i\in[d]$. Let $R$
be a $k$ algebra and let
$p:R[X_{1},\dots,X_{d}]\rightarrow\mathfrak{gl}_{2n}(A\otimes R)$ be the map
defined by $X_{i}\mapsto x_{i}$. Let $R[X_{1},\dots,X_{d}]^{+}$ be the
$R$-subalgebra generated by even degree monomials. Note that the image of the
map $p$ is contained in $\mathfrak{gl}_{n}(A\otimes
R)\times\mathfrak{gl}_{n}(A\otimes R)$. Let $q_{1}$ and $q_{2}$ be the first
and second projections of $\mathfrak{gl}_{n}(A\otimes
R)\times\mathfrak{gl}_{n}(A\otimes R)$ respectively. We have

$R[X_{1},\dots,X_{d}]^{+}\xrightarrow{p}\mathfrak{gl}_{n}(A\otimes
R)\times\mathfrak{gl}_{n}(A\otimes R)\xrightarrow{det\ \circ\ q_{1}}A\otimes
R.$

Since $\det$ is multiplicative, the composition $\det\circ q_{1}\circ p$ is
multiplicative and homogeneous of degree $n$. Thus by Roby’s theorem we get a
homomorphism of $k$-algebras:

$\tilde{\mathfrak{s}}:TS^{n}(k[X_{1},\dots,X_{d}]^{+})\rightarrow A$

such that

$\tilde{\mathfrak{s}}(\theta^{\otimes n})=det\circ
q_{1}(p(\theta)),\,\,\text{for all}\,\,\theta\in k[X_{1},\dots,X_{d}].$ (5.11)

Note that the image of the map $\tilde{\mathfrak{s}}$ is contained in
$A^{G_{0}}$.

The subalgebra $TS^{n}k[X_{1},\dots,X_{d}]^{+}$ can be identified with the
subalgebra of $T^{n}k[X_{1},\dots,X_{d}]$ of fixed points under
$S_{n}\ltimes(\mathbb{Z}/2\mathbb{Z})^{n}$. Now, we give an isomorphism of
$TS^{n}k[X_{1},\dots,X_{d}]^{+}$ with
$k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$. Let $(y_{1},\dots,y_{d})$ be the
tautological point of $\mathfrak{c}(k[\mathfrak{c}^{d}])$. Note that
$k[\mathfrak{c}^{d}]$ is a polynomial algebra in the variables $b_{j}(y_{i})$,
$1\leq i\leq d$ and $1\leq j\leq n$. Let

$\beta:T^{n}k[X_{1},\dots,X_{d}]\rightarrow k[\mathfrak{c}^{d}]$

be the isomorphism of algebras defined by $\beta(X_{ij})=b_{j}(y_{i})$, for
$i\in[d]$ and $j\in[n]$. Here, $X_{ij}$ is the $i$-th variable in the $j$-th
copy of $k[X_{1},\dots,X_{d}]$. By restriction we have an isomorphism of
algebras

$\beta:TS^{n}k[X_{1},\dots,X_{d}]^{+}\rightarrow
k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}.$

Then it follows that we have an algebra map

$\mathfrak{s}:k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}\rightarrow A^{G_{0}}$

such that

$\mathfrak{s}\beta(\theta^{\otimes n})=\det\circ q_{1}\circ
p(\theta),\,\,\text{for all}\,\,\theta\in k[X_{1},\dots,X_{d}]^{+}.$

Thus we obtain a map of schemes
$\mathfrak{s}:\mathfrak{C}^{d}(\mathfrak{g}_{1})//G_{0}\rightarrow\mathfrak{c}^{d}//W_{\mathfrak{c}}$.

Let $\mathfrak{i}:k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}\rightarrow
k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$ be the restriction map. We will show
that $\mathfrak{s}$ and $\mathfrak{i}$ are inverses of each other.

Recall that $\mathfrak{g}_{1}$ as a $G_{0}$-module is isomorphic to ${\rm
Sym}^{2}(k^{n})\oplus({\rm Sym}^{2}(k^{n}))^{*}$. The ring of invariants
$k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$ are generated by the images of
the following polynomials, via the restriction map
$k[\mathfrak{g}_{1}^{d}]^{G_{0}}\rightarrow
k[\mathfrak{C}^{d}(\mathfrak{g}_{1})]^{G_{0}}$:

${\rm Tr}(M_{1}M_{2}\dots M_{k})$ (5.12)

where $M_{i}=Q_{n_{i}}R_{m_{i}}$ for some $n_{i},m_{i}\in[d]$; and we denote
the polynomial in (5.12) by $P$ (see Subsection 4.3). Here $Q_{n_{i}}$ and
$R_{n_{i}}$ are as defined in (5.10). We then have

$\mathfrak{i}(P)=\sum_{j=1}^{n}\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}}).$

From the above equation, we get that $\mathfrak{i}(\phi_{\underline{a}})$
generate the $k$-algebra $k[\mathfrak{c}^{d}]^{W_{\mathfrak{c}}}$. Let $R$ be
the algebra $k[t]$, and consider the element
$\theta=t-\prod_{(n_{i},m_{i})}X_{n_{i}}X_{m_{i}}$ in
$R[X_{1},X_{2},\dots,X_{d}]^{+}$. We then have

$\beta(\theta^{\otimes
n})=\prod_{j=1}^{n}(t-\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}})).$

We then observe that $\det\circ q_{1}\circ p(\theta)$ is equal to

$\det(t-\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}})).$

Since $\mathfrak{s}\circ\beta(\theta^{\otimes n})=\det\circ q_{1}\circ
p(\theta)$, we get that

$t^{n}-\mathfrak{s}\left(\sum_{j=1}^{n}\prod_{(n_{i},m_{i})}b_{j}(y_{n_{i}})b_{j}(y_{m_{i}})\right)t^{n-1}+\cdots=t^{n}-{\rm
Tr}(\prod_{(n_{i},m_{i})}Q_{n_{i}}R_{m_{i}})t^{n-1}+\cdots$

Comparing the coefficients of $t^{n-1}$ we get that
$\mathfrak{s}\circ\mathfrak{i}(P)=P$. Thus, we get that $\mathfrak{s}$ is a
section of the map $\mathfrak{i}$.

This concludes the proof of Theorem 1.1.

######  Remark 5.2.

Among the classical symmetric pairs, our results do not cover the cases:
$(\mathfrak{so}_{n}(k)\times\mathfrak{so}_{n}(k),\mathfrak{so}_{n}(k))$,
$(\mathfrak{so}_{2n}(k),\mathfrak{gl}_{n}(k))$ and
$(\mathfrak{sp}_{2(n+m)}(k),\mathfrak{sp}_{2n}(k)\times\mathfrak{sp}_{2m}(k))$.
Let $(x_{1},x_{2},\dots,x_{d})$ be the universal point of
$\mathfrak{C}_{d}(\mathfrak{g}_{1})(A)$. We use a multiplicative map on the
image of $p$ or some projection of $p$ to get a section of the Chevalley
restriction map. Here,
$p:R[X_{1},X_{2},\dots,X_{d}]\rightarrow\mathfrak{gl}_{k}(A\otimes R)$ sending
$X_{i}\mapsto x_{i}$. In the above mentioned examples, we do not know any such
multiplicative map with degree equal to the rank of the symmetric pair.

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Department of Mathematics and Statistics,
Indian Institute of Technology Kanpur,
U.P. India, 208016.
email<EMAIL_ADDRESS><EMAIL_ADDRESS>