# Fields of invariants for unipotent radicals of parabolic subgroups

A. N. Panov111The work is supported by the RFBR grant 20-01-00091a

###### Abstract

The paper is devoted to the problem of finding free generators in the fields
of invariants for actions of unipotent groups on affine varieties. We consider
the case when the unipotent group is the unipotent radical in an arbitrary
parabolic subgroup in the reductive group of classical type
${\mathrm{GL}}(n)$, ${\mathrm{SL}}(n)$, $\mathrm{O}(n)$ or $\mathrm{Sp}(2n)$.
In the explicit form, we present a system of free generators in the field of
invariants for the action of the unipotent radical on the reductive group by
conjugation.

Keywords: theory of invariants, parabolic subgroup, unipotent radical,
determinant
2010 Mathematics Subject Classifications: 15A15, 15A72, 13A50

##  1 Introduction

Let $G$ be a split reductive group defined over a field $K$ of characteristic
zero. Let $U$ be the unipotent radical of an arbitrary parabolic subgroup $P$
in $G$. The group $U$ acts on $G$ by conjugation. There is a representation of
the group $U$ on the space ${\mathcal{A}}=K[G]$ by the formula
$\rho(g)f(x)=f(g^{-1}xg)$, $g\in U$, $x\in G$. This representation extends to
the action of $U$ on the field ${\mathcal{F}}=K(G)$ of rational functions on
$G$.

In the present paper we consider the cases when $G$ is one of the following
groups ${\mathrm{GL}}(n)$, ${\mathrm{SL}}(n)$, $\mathrm{O}(n)$ or
$\mathrm{Sp}(2n)$. We aim to construct a system of free generators of the
field of invariants ${\mathcal{F}}^{U}$ in an explicit form as polynomials in
matrix entries.

In the second section of this paper, we solve this problem for the case of the
general linear group ${\mathrm{GL}}(n)$. In Theorem 2, we construct the system
of free generators $\\{{\mathcal{J}}_{ij}:~{}(i,j)\in{\mathbb{S}}\\}$ for the
field ${\mathcal{F}}^{U}$, where ${\mathcal{J}}_{ij}$ are determinants of
special form. We obtain the similar system of generators for
${\mathrm{SL}}(n)$ (see Corollary 2). The case when $U$ is the unitriangular
group (i.e. the group of of upper triangular matrices with ones on the
diagonal) is treated by the author in the paper [1].

In the third section, we consider the case of $G=\mathrm{O}(n)$ or
$G=\mathrm{Sp}(2n)$. In Theorem 3, using the restrictions
$\\{{\mathcal{J}}_{ij}^{\circ}\\}$ of polynomials $\\{{\mathcal{J}}_{ij}\\}$
on $G$, we construct the system of free generators for the field of
$U$-invariants on the group $G$.

Since the group $U$ is unipotent, the field of invariants ${\mathcal{F}}^{U}$
is a field of fractions of the algebra of invariant regular functions
${\mathcal{A}}^{U}$ [3, Theorem 3.3]. The algebra ${\mathcal{A}}^{U}$ is
finitely generated [3, Theorem 3.13]. The problem of description of generators
for the algebra ${\mathcal{A}}^{U}$ is still an unsolved problem.

In conclusion of Introduction we emphasize a few papers which are devoted to
the theory of invariants of unipotent groups [2, 4, 5, 7].

##  2 Fields of invariants for radicals of parabolic subgroups in
${\mathrm{GL}}(n)$

Let $K$ be a field of characteristic zero and $P$ be a parabolic subgroup of
the general linear group $G={\mathrm{GL}}(n)$. There is a decomposition $P=LU$
of the group $P$ into a semidirect product of its unipotent radical $U$ and a
Levi subgroup $L$. Our aim is to construct a system of free generators of the
field $U$-invariants ${\mathcal{F}}^{U}$ explicitly.

The parabolic subgroup $P$ is defined by a partition of the integer segment
$[1,n]$ into a system of consecutive segments

$[1,n]=I_{1}\sqcup I_{2}\sqcup\ldots\sqcup I_{\ell}.$

Put $n_{i}=|I_{i}|$ for each $1\leqslant i\leqslant\ell$. The linear space
${\mathcal{M}}={\mathrm{Mat}}(n)$ of all $(n\times n)$ matrices is a direct
sum

${\mathcal{M}}=\bigoplus_{1\leqslant k,m\leqslant\ell}{\mathcal{M}}_{km},$

where ${\mathcal{M}}_{km}$ is spanned by the matrix units $E_{ij}$, $(i,j)\in
I_{k}\times I_{m}$. The Levi subgroup $L$ is isomorphic to the direct product

${\mathrm{GL}}(n_{1})\times\ldots\times{\mathrm{GL}}(n_{\ell}).$

The unipotent subgroup $U$ consists of matrices $E+A$, where $E$ is the
identity matrix and $A$ belongs to the sum of subspaces ${\mathcal{M}}_{km}$,
$k<m$.

For each $i\in[1,n]$, let $i^{\prime}=n+1-i$ be the symmetric number to $i$
with respect to the center of the segment. Respectively, for any
$T\subseteq[1,n]$, we define $T^{\prime}=\\{i^{\prime}:~{}i\in T\\}$.

We consider the set ${\mathbb{S}}$ of pairs $(i,j)$ such that

$i\in I_{k},~{}~{}j\in I_{m}^{\prime},~{}~{}k\geqslant m.$

Let $S$ be the intersection of the group ${\mathrm{GL}}(n)$ with the subspace
in ${\mathcal{M}}$ spanned by $E_{ij}$, $(i,j)\in{\mathbb{S}}$.

Proposition 2.1. The subset

$\bigcup_{g\in U}gSg^{-1}$ (1)

is dense in ${\mathrm{GL}}(n)$.
_Proof_. Let ${\mathcal{N}}={\mathrm{UT}}(n)$,
${\mathcal{N}}_{L}={\mathcal{N}}\cap L$ and $B$ be the subgroup of upper
triangular matrices in ${\mathrm{GL}}(n)$. The Bruhat cell
${\mathcal{N}}w_{0}B$, where $w_{0}$ is the element of greatest length in the
Weyl group, is dense in ${\mathrm{GL}}(n)$. The subgroup ${\mathcal{N}}$
decomposes into a product ${\mathcal{N}}=U\cdot{\mathcal{N}}_{L}$. Then
${\mathcal{N}}w_{0}B=U\left({\mathcal{N}}_{L}w_{0}B\right)$. By direct
calculations one can verify ${\mathcal{N}}_{L}w_{0}B\subset S$ (see Example
2). Then ${\mathcal{N}}w_{0}B\subset US$. Since $S$ is stable with respect to
the right multiplication by $U$, the subset $US$ coincides with the subset
(1). Therefore (1) is dense in ${\mathrm{GL}}(n)$. $\Box$

Denote by $\pi$ the restriction map ${\mathcal{A}}\to K[S]$. The map $\pi$
establishes an embedding ${\mathcal{A}}^{U}\to K[S]$ and it is extended to an
embedding $\pi:{\mathcal{F}}^{U}\to K(S)$.

Let $\\{x_{ij}\\}$ be the system of standard coordinate functions on
${\mathcal{M}}$. Define the matrix ${\mathbb{X}}=(x_{ij})_{i,j=1}^{n}$.
Consider the adjugate matrix

${\mathbb{X}}^{*}=(x^{*}_{ij})_{i,j=1}^{n}.$

We have
${\mathbb{X}}{\mathbb{X}}^{*}={\mathbb{X}}^{*}{\mathbb{X}}=\det({\mathbb{X}})E.$

Define the order relation on the set of pairs ${\mathbb{S}}$ as follows:

$(i_{1},j_{1})\prec(i,j)~{}~{}\mbox{if}~{}~{}j_{1}<j,~{}~{}\mbox{or}~{}~{}j_{1}=j~{}~{}\mbox{and}~{}~{}i_{1}>i.$
(2)

Denote $s_{ij}=\pi(x_{ij})$ for $(i,j)\in{\mathbb{S}}$.

Our aim is to attach the $U$-invariant ${\mathcal{J}}_{ij}$ in
$K[{\mathcal{M}}]$ to each pair $(i,j)\in{\mathbb{S}}$. Decompose
${\mathbb{S}}$ into two subsets
${\mathbb{S}}={\mathbb{S}}_{0}\sqcup{\mathbb{S}}_{1}$, where
${\mathbb{S}}_{0}$ consists of all $(i,j)\in{\mathbb{S}}$, which is lying on
or above the anti-diagonal, and ${\mathbb{S}}_{1}$ consists of all
$(i,j)\in{\mathbb{S}}$, which is lying strictly below the anti-diagonal.

Let $(i,j)\in{\mathbb{S}}_{0}$. Then $(i,j)$ is lying on or above the anti-
diagonal. In this case $i+j\leqslant n+1$ (equivalent to $i\leqslant
j^{\prime}$). Since $i\leqslant j^{\prime}$, the number $i$ is less than any
number from the segment $[j^{\prime}+1,n]$. Observe that
$[1,j-1]^{\prime}=[j^{\prime}+1,n]$. Consider the system of $j$ rows
$\\{i\\}\sqcup[j^{\prime}+1,n]$.

For each $(i,j)\in{\mathbb{S}}_{0}$, we define the polynomial
${\mathcal{J}}_{ij}$ as the minor of the matrix ${\mathbb{X}}$ defined by the
systems of columns $[1,j]$ and rows $\\{i\\}\cup[j^{\prime}+1,n]$.

Let $(i,j)\in{\mathbb{S}}_{1}$. Then $(i,j)$ is lying below the anti-diagonal.
In this case $i+j>n+1$ (equivalent to $j>i^{\prime}$ ). Then
$j=i^{\prime}+(j-i^{\prime})$. Consider the $(j\times j)$-matrix

${\mathbb{Y}}_{ij}=\left(\begin{array}[]{c}{\mathcal{X}}_{i^{\prime},j}\\\
\-----\\\ {\mathcal{X}}^{*}_{j-i^{\prime},j}\end{array}\right),$

where ${\mathcal{X}}_{i^{\prime},j}$ is the submatrix ${\mathbb{X}}$
associated with the columns $[1,j]$ and the last $i^{\prime}$ rows,
${\mathcal{X}}^{*}_{j-i^{\prime},j}$ is the submatrix of ${\mathbb{X}}^{*}$
associated with the columns $[1,j]$ and the last $j-i^{\prime}$ rows. For each
$(i,j)\in{\mathbb{S}}_{1}$, we define

${\mathcal{J}}_{ij}=\det{\mathbb{Y}}_{ij}.$

Proposition 2.2. The polynomials
$\\{{\mathcal{J}}_{ij}:~{}~{}(i,j)\in{\mathbb{S}}\\}$ are $U$-invariants.
_Proof_. Analogously to [1, Proposition 1].
Proposition 2.3. ${\mathcal{J}}_{ij}\neq 0$ for each $(i,j)\in{\mathbb{S}}$.
_Proof_. The statement is obvious for $(i,j)\in{\mathbb{S}}_{0}$. Let
$(i,j)\in{\mathbb{S}}_{1}$. Consider the matrix $A=(a_{st})$, where $a_{st}=1$
if $(s,t)$ lies on the anti-diagonal or $(s,t)=(i+k,j-k)$ for $k\in[0,n-i]$,
otherwise $a_{st}=0$. Then in the determinant ${\mathcal{J}}_{ij}(A)$ all
below the anti-diagonal entries of are zero and the anti-diagonal entries are
non-zero. Therefore ${\mathcal{J}}_{ij}(A)\neq 0$. For instance, in Example 2,
for $i=j=4$ we have

$A=\left(\begin{array}[]{ccccc}0&0&0&0&1\\\ 0&0&0&1&0\\\ 0&0&1&0&0\\\
0&1&0&1&0\\\ 1&0&1&0&0\\\ \end{array}\right),\qquad
A^{*}=\left(\begin{array}[]{ccccc}\times&\times&\times&\times&1\\\
\times&\times&\times&1&0\\\ \times&\times&1&0&0\\\ \times&1&0&0&0\\\
1&0&0&0&0\\\
\end{array}\right),\qquad{\mathcal{J}}_{44}(A)=\left|\begin{array}[]{cccc}0&1&0&1\\\
1&0&1&0\\\ \times&1&0&0\\\ 1&0&0&0\\\ \end{array}\right|=1\neq 0.~{}~{}\Box$

Corollary 2.4. $\pi({\mathcal{J}}_{ij})\neq 0$ for each
$(i,j)\in{\mathbb{S}}$.
The proof follows from the fact that $\pi$ is an embedding of
${\mathcal{F}}^{U}$ to $K(S)$.
Lemma 2.5. Let $M_{I,J}({\mathbb{X}}^{*})$ be the minor of ${\mathbb{X}}^{*}$
with systems of rows $I$ and columns $J$. Then if $I$ is a segment and it ends
in $n$, then $M_{I,J}({\mathbb{X}}^{*})$ is invariant under the right
multiplication of ${\mathbb{X}}$ by ${\mathrm{UT}}(n)$.
_Proof_. If $I$ is a segment and it ends in $n$, then for any $(n\times
n)$-matrix ${\mathbb{Y}}$ and $g\in{\mathrm{UT}}(n)$, we have
$M_{I,J}(g{\mathbb{Y}})=M_{I,J}({\mathbb{Y}})$. Therefore

$M_{I,J}\left(({\mathbb{X}}g)^{*}\right)=M_{I,J}(g^{*}{\mathbb{X}}^{*})=M_{I,J}({\mathbb{X}}^{*}).~{}~{}\Box$

The subset $S$ is stable under the right multiplication by ${\mathrm{UT}}(n)$.
Denote by $K(S)^{{\mathrm{UT}}(n)}$ the subfield of
${\mathrm{UT}}(n)$-invariants in $K(S)$.
Corollary 2.6. If $I$ is a segment and it ends in $n$, then
$\pi(M_{I,J}({\mathbb{X}}^{*}))$ belongs to $K(S)^{{\mathrm{UT}}(n)}$.
Definition 2.7. Let $\\{\xi_{\alpha}:\alpha\in{\mathfrak{A}}\\}$ and
$\\{\eta_{\alpha}:\alpha\in{\mathfrak{A}}\\}$ be two finite systems of free
generators of an extension $F$ of the field $K$. Let $\prec$ be a linear order
relation on ${\mathfrak{A}}$. We say that the second system of generators is
obtained from the first one by a triangular transformation if each
$\eta_{\alpha}$ can be presented in the form

$\eta_{\alpha}=\phi_{\alpha}\xi_{\alpha}+\psi_{\alpha},$ (3)

where $\phi_{\alpha}\neq 0$ and $\phi_{\alpha}$, $\psi_{\alpha}$ belong to the
subfield generated by $\\{\xi_{\beta}:~{}\beta\prec\alpha\\}$.

Denote by ${\mathcal{P}}_{0}$ the subfield of ${\mathcal{F}}^{U}$ generated by
$\\{{\mathcal{J}}_{ij}:~{}~{}(i,j)\in{\mathbb{S}}_{0}\\}$.

Proposition 2.8.
1) The subfield $K(S)^{{\mathrm{UT}}(n)}$ is freely generated by the elements

$\\{\pi({\mathcal{J}}_{ij}):~{}~{}(i,j)\in{\mathbb{S}}_{0}\\}.$

2) The map $\pi$ establishes an isomorphism of the subfield
${\mathcal{P}}_{0}$ of ${\mathcal{F}}^{U}$ to $K(S)^{{\mathrm{UT}}(n)}$.
_Proof_. Denote by $S_{0}$ the subset of $S$ that consists of matrices
$(s_{ij})$ with $s_{ij}=0$ for all $(i,j)\in{\mathbb{S}}_{1}$. The subset
$S_{0}\cdot{\mathrm{UT}}(n)$ is dense in $S$. The restriction map $\pi_{0}$
from $K[S]$ to $K[S_{0}]$ defines an embedding of the subfield
$K(S)^{{\mathrm{UT}}(n)}$ into the field $K(S_{0})$. Each polynomial in
$\\{\pi({\mathcal{J}}_{ij}):~{}(i,j)\in{\mathbb{S}}_{0}\\}$ is a
${\mathrm{UT}}(n)$-invariant and its restriction
$\pi_{0}\pi({\mathcal{J}}_{ij})$ on $S_{0}$ equals to

$\pm s_{n1}s_{n-1,2}\ldots s_{j^{\prime}+1,j-1}s_{ij}.$

For instance, in Example 2

$S=\left\\{\left(\begin{array}[]{ccccc}0&0&0&0&s_{15}\\\
0&0&s_{23}&s_{24}&s_{25}\\\ 0&0&s_{33}&s_{34}&s_{35}\\\
s_{41}&s_{42}&s_{43}&s_{44}&s_{45}\\\ s_{51}&s_{52}&s_{53}&s_{54}&s_{55}\\\
\end{array}\right)\right\\},\qquad
S_{0}=\left\\{\left(\begin{array}[]{ccccc}0&0&0&0&s_{15}\\\
0&0&s_{23}&s_{24}&0\\\ 0&0&s_{33}&0&0\\\ s_{41}&s_{42}&0&0&0\\\
s_{51}&0&0&0&0\\\ \end{array}\right)\right\\}.$

In the case $i=2$, $j=3$, we obtain

$\pi_{0}\pi({\mathcal{J}}_{23})=\left|\begin{array}[]{ccc}0&0&s_{23}\\\
s_{41}&s_{42}&0\\\ s_{51}&0&0\\\ \end{array}\right|=-s_{51}s_{42}s_{23}.$

The system of elements
$\\{\pi_{0}\pi({\mathcal{J}}_{ij}):~{}(i,j)\in{\mathbb{S}}_{0}\\}$ is obtained
by a triangular transformation with respect to the order relation $\prec$ (see
(2)) from the system $\\{s_{ij}:~{}(i,j)\in{\mathbb{S}}_{0}\\}$ of generators
of the field $K(S_{0})$. This implies $\pi_{0}$ is an isomorphism of
$K(S)^{{\mathrm{UT}}(n)}$ to the field $K(S_{0})$. $\Box$

Theorem 2.9. The system of polynomials
$\\{{\mathcal{J}}_{ij}:~{}(i,j)\in{\mathbb{S}}\\}$ freely generates over $K$
the field of invariants ${\mathcal{F}}^{U}$ for the group ${\mathrm{GL}}(n)$.
_Proof_.
Item 1. Let us show that the field $K(S)$ is freely generated over
$K(S)^{{\mathrm{UT}}(n)}$ by the system of elements
$\\{s_{ij}:~{}(i,j)\in{\mathbb{S}}_{1}\\}$.

For each $(i,j)\in{\mathbb{S}}_{0}$, we expand the minor ${\mathcal{J}}_{ij}$
along its first row. We get ${\mathcal{J}}_{ij}=\pm A_{ij}\cdot
x_{ij}+\Psi_{ij}$. The cofactor $A_{ij}$ and $\Psi_{ij}$, which is the sum of
the other cofactors multiplied by the respective entries, both are expressions
in the $x_{st}$, $(s,t)\prec(i,j)$. Therefore, the system of elements

$\\{\pi({\mathcal{J}}_{ij}):~{}(i,j)\in{\mathbb{S}}_{0}\\}\sqcup\\{s_{ij}:~{}(i,j)\in{\mathbb{S}}_{1}\\}$

is obtained by a triangular transformation from the system of free generators
$\\{s_{ij}:~{}(i,j)\in{\mathbb{S}}\\}$ of the field $K(S)$. This proves the
statement of Item 1.
Item 2. Let us show that the field ${\mathcal{F}}^{U}$ is freely generated
over the subfield ${\mathcal{P}}_{0}$ by the system of elements
$\\{{\mathcal{J}}_{ij}:~{}(i,j)\in{\mathbb{S}}_{1}\\}$. It is sufficient to
prove that the system of elements
$\\{\pi({\mathcal{J}}_{ij}):~{}(i,j)\in{\mathbb{S}}_{1}\\}$ is obtained from
$\\{s_{ij}:~{}(i,j)\in{\mathbb{S}}_{1}\\}$ by a triangular transformation over
the field $K(S)^{{\mathrm{UT}}(n)}$. We prove using the induction method that
the formula of type (3) is valid for the order relation (2).

For the smallest element $(n,1)$, we have ${\mathcal{J}}_{ij}=x_{n1}$, and
$\pi({\mathcal{J}}_{ij})=s_{n1}$ belongs to $K(S)^{{\mathrm{UT}}(n)}$.

Assume that the statement is true for all elements $\prec(i,j)$. Let us prove
for $(i,j)\in{\mathbb{S}}_{1}$. Expand the determinant ${\mathcal{J}}_{ij}$
along its first row. We get

${\mathcal{J}}_{ij}=\pm M\cdot x_{ij}+\Psi_{ij},$

where $M$ is the cofactor for $(i,j)$, and $\Psi_{ij}$ is the sum of the other
entries of the first row multiplying to their cofactors. For instance, in
Example 2 for $i=5$, $j=3$ we have

${\mathcal{J}}_{53}=\left|\begin{array}[]{ccc}x_{51}&x_{52}&x_{53}\\\
x^{*}_{41}&x^{*}_{42}&x^{*}_{43}\\\ x^{*}_{51}&x^{*}_{52}&x^{*}_{53}\\\
\end{array}\right|=Mx_{53}+\Psi_{53},$
$M=\left|\begin{array}[]{cc}x^{*}_{41}&x^{*}_{42}\\\ x^{*}_{51}&x^{*}_{52}\\\
\end{array}\right|,\qquad\Psi_{53}=-\left|\begin{array}[]{cc}x^{*}_{41}&x^{*}_{43}\\\
x^{*}_{51}&x^{*}_{53}\\\
\end{array}\right|x_{52}+\left|\begin{array}[]{cc}x^{*}_{42}&x^{*}_{43}\\\
x^{*}_{52}&x^{*}_{53}\\\ \end{array}\right|x_{51}.$

Restricting the above equality to $S$, we obtain

$\pi({\mathcal{J}}_{ij})=\pm\pi(M)\cdot s_{ij}+\pi(\Psi_{ij}).$ (4)

In the case $i<n$, we have $M={\mathcal{J}}_{i+1,j-1}$ and $M\neq 0$.
According to induction assumption, $\pi({\mathcal{J}}_{i+1,j-1})$ is an
expression over the field $K(S)^{{\mathrm{UT}}(n)}$ of $s_{kt}$,
$(k,t)\in{\mathbb{S}}_{1}$, $(k,t)\prec(i,j)$.

In the case $i=n$, the polynomial $M$ coincides with the minor
$M_{I,J}({\mathbb{X}}^{*})$ for $J=[1,j-1]$ and
$I=J^{\prime}=[j^{\prime}+1,n]$. The segment $I$ ends in $n$, therefore
$\pi(M_{I,J}({\mathbb{X}}^{*}))\in K(S)^{{\mathrm{UT}}(n)}$ (see Corollary 2).

The sum $\Psi_{ij}$ is an expression of $x_{kt}$, $(k,t)\prec(i,j)$, and
minors of the form $M_{I^{\prime},J^{\prime}}({\mathbb{X}}^{*})$, where the
segments of rows $I^{\prime}$ end in $n$. By the induction assumption and
Corollary 2, we verify that $\pi(\Psi_{ij})$ is an expression over the field
$K(S)^{{\mathrm{UT}}(n)}$ by $s_{kt}$, $(k,t)\in{\mathbb{S}}_{1}$,
$(k,t)\prec(i,j)$. The statement of Item 2 is proved. $\Box$
Corollary 2.10. The system of polynomials
$\\{{\mathcal{J}}_{ij}:~{}(i,j)\in{\mathbb{S}},~{}(i,j)\neq(1,n)\\}$ freely
generates over $K$ the field of invariants ${\mathcal{F}}^{U}$ for the group
${\mathrm{SL}}(n)$.

Example 2.11. Let $n=5$ and the parabolic subgroup $P$ be defined by the
partition $[1,5]=\\{1\\}\sqcup\\{2,3\\}\sqcup\\{4,5\\}$. In the $(5\times
5)$-table ${\mathbb{P}}$, by the symbol $"\times"\,$ (respectively, by the
symbol $"*"\,$), we mark the cells that correspond to the subgroup $L$
(respectively, to the subgroup $U$).

${\mathbb{P}}=\left(\begin{array}[]{ccccc}\times&*&*&*&*\\\
0&\times&\times&*&*\\\ 0&\times&\times&*&*\\\ 0&0&0&\times&\times\\\
0&0&0&\times&\times\\\
\end{array}\right),~{}~{}S=\left\\{\left(\begin{array}[]{ccccc}0&0&0&0&s_{15}\\\
0&0&s_{23}&s_{24}&s_{25}\\\ 0&0&s_{33}&s_{34}&s_{35}\\\
s_{41}&s_{42}&s_{43}&s_{44}&s_{45}\\\ s_{51}&s_{52}&s_{53}&s_{54}&s_{55}\\\
\end{array}\right)\right\\},~{}~{}S^{*}=\left\\{\left(\begin{array}[]{ccccc}s^{*}_{11}&s^{*}_{12}&s^{*}_{13}&s^{*}_{14}&s^{*}_{15}\\\
s^{*}_{21}&s^{*}_{22}&s^{*}_{23}&s^{*}_{24}&s^{*}_{25}\\\
s^{*}_{31}&s^{*}_{32}&s^{*}_{33}&0&0\\\
s^{*}_{41}&s^{*}_{42}&s^{*}_{43}&0&0\\\ s^{*}_{51}&0&0&0&0\\\
\end{array}\right)\right\\},$

${\mathcal{N}}_{L}=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&1&\times&0&0\\\
0&0&1&0&0\\\ 0&0&0&1&\times\\\ 0&0&0&0&1\\\
\end{array}\right),~{}~{}{\mathcal{N}}_{L}w_{0}B=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&1&\times&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&\times\\\ 0&0&0&0&1\\\
\end{array}\right)\left(\begin{array}[]{ccccc}0&0&0&0&\times\\\
0&0&0&\times&\times\\\ 0&0&\times&\times&\times\\\
0&\times&\times&\times&\times\\\ \times&\times&\times&\times&\times\\\
\end{array}\right)\subset S.$

The generators ${\mathcal{J}}_{ij}$, $(i,j)\in{\mathbb{S}}$, and their
restrictions $\pi({\mathcal{J}}_{i,j})$ on $S$ have the following form:
${\mathcal{J}}_{51}=x_{51}$, $\pi({\mathcal{J}}_{51})=s_{51}$,
${\mathcal{J}}_{41}=x_{41}$, $\pi({\mathcal{J}}_{41})=s_{41}$,
${\mathcal{J}}_{52}=\left|\begin{array}[]{cc}x_{51}&x_{52}\\\
x^{*}_{51}&x^{*}_{52}\\\ \end{array}\right|$,
$\pi({\mathcal{J}}_{52})=\left|\begin{array}[]{cc}s_{51}&s_{52}\\\
s^{*}_{51}&0\\\ \end{array}\right|$,
${\mathcal{J}}_{42}=\left|\begin{array}[]{cc}x_{41}&x_{42}\\\ x_{51}&x_{52}\\\
\end{array}\right|$,
$\pi({\mathcal{J}}_{42})=\left|\begin{array}[]{cc}s_{41}&s_{42}\\\
s_{51}&s_{52}\\\ \end{array}\right|$,
${\mathcal{J}}_{53}=\left|\begin{array}[]{ccc}x_{51}&x_{52}&x_{53}\\\
x^{*}_{41}&x^{*}_{42}&x^{*}_{43}\\\ x^{*}_{51}&x^{*}_{52}&x^{*}_{53}\\\
\end{array}\right|$,
$\pi({\mathcal{J}}_{53})=\left|\begin{array}[]{ccc}s_{51}&s_{52}&s_{53}\\\
s^{*}_{41}&s^{*}_{42}&s^{*}_{43}\\\ s^{*}_{51}&0&0\\\ \end{array}\right|$,
${\mathcal{J}}_{43}=\left|\begin{array}[]{ccc}x_{41}&x_{42}&x_{43}\\\
x_{51}&x_{52}&x_{53}\\\ x^{*}_{51}&x^{*}_{62}&x^{*}_{53}\\\
\end{array}\right|$,
$\pi({\mathcal{J}}_{43})=\left|\begin{array}[]{ccc}s_{41}&s_{42}&s_{43}\\\
s_{51}&s_{52}&s_{53}\\\ s^{*}_{51}&0&0\\\ \end{array}\right|$;
${\mathcal{J}}_{33}=\left|\begin{array}[]{ccc}x_{31}&x_{32}&x_{33}\\\
x_{41}&x_{42}&x_{43}\\\ x_{51}&x_{52}&x_{53}\\\ \end{array}\right|$,
$\pi({\mathcal{J}}_{33})=\left|\begin{array}[]{ccc}0&0&s_{33}\\\
s_{41}&s_{42}&s_{43}\\\ s_{51}&s_{52}&s_{53}\\\ \end{array}\right|$,

${\mathcal{J}}_{23}=\left|\begin{array}[]{ccc}x_{21}&x_{22}&x_{23}\\\
x_{41}&x_{42}&x_{43}\\\ x_{51}&x_{52}&x_{53}\\\ \end{array}\right|$,
$\pi({\mathcal{J}}_{23})=\left|\begin{array}[]{ccc}0&0&s_{23}\\\
s_{41}&s_{42}&s_{43}\\\ s_{51}&s_{52}&s_{53}\\\ \end{array}\right|$,

${\mathcal{J}}_{54}=\left|\begin{array}[]{cccc}x_{51}&x_{52}&x_{53}&x_{54}\\\
x^{*}_{31}&x^{*}_{32}&x^{*}_{33}&x^{*}_{34}\\\
x^{*}_{41}&x^{*}_{42}&x^{*}_{43}&x^{*}_{44}\\\
x^{*}_{51}&x^{*}_{52}&x^{*}_{53}&x^{*}_{54}\\\ \end{array}\right|$,
$\pi({\mathcal{J}}_{54})=\left|\begin{array}[]{cccc}s_{51}&s_{52}&s_{53}&s_{54}\\\
s^{*}_{31}&s^{*}_{32}&s^{*}_{33}&0\\\ s^{*}_{41}&s^{*}_{42}&s^{*}_{43}&0\\\
s^{*}_{51}&0&0&0\\\ \end{array}\right|$,

${\mathcal{J}}_{44}=\left|\begin{array}[]{cccc}x_{41}&x_{42}&x_{43}&x_{44}\\\
x_{51}&x_{52}&x_{53}&x_{54}\\\ x^{*}_{41}&x^{*}_{42}&x^{*}_{43}&x^{*}_{44}\\\
x^{*}_{51}&x^{*}_{52}&x^{*}_{53}&x^{*}_{54}\\\ \end{array}\right|$,
$\pi({\mathcal{J}}_{44})=\left|\begin{array}[]{cccc}s_{41}&s_{42}&s_{43}&s_{44}\\\
s_{51}&s_{52}&s_{53}&s_{54}\\\ s^{*}_{41}&s^{*}_{42}&s^{*}_{43}&0\\\
s^{*}_{51}&0&0&0\\\ \end{array}\right|$,

${\mathcal{J}}_{34}=\left|\begin{array}[]{cccc}x_{31}&x_{32}&x_{33}&x_{34}\\\
x_{41}&x_{42}&x_{43}&x_{44}\\\ x_{51}&x_{52}&x_{53}&x_{54}\\\
x^{*}_{51}&x^{*}_{52}&x^{*}_{53}&x^{*}_{54}\\\ \end{array}\right|$,
$\pi({\mathcal{J}}_{34})=\left|\begin{array}[]{cccc}0&0&s_{33}&s_{34}\\\
s_{41}&s_{42}&s_{43}&s_{44}\\\ s_{51}&s_{52}&s_{53}&s_{54}\\\
s^{*}_{51}&0&0&0\\\ \end{array}\right|$,

${\mathcal{J}}_{24}=\left|\begin{array}[]{cccc}x_{21}&x_{22}&x_{23}&x_{24}\\\
x_{31}&x_{32}&x_{33}&x_{34}\\\ x_{41}&x_{42}&x_{43}&x_{44}\\\
x_{51}&x_{52}&x_{53}&x_{54}\\\ \end{array}\right|$,
$\pi({\mathcal{J}}_{24})=\left|\begin{array}[]{cccc}0&0&s_{23}&s_{24}\\\
0&0&s_{33}&s_{34}\\\ s_{41}&s_{42}&s_{43}&s_{44}\\\
s_{51}&s_{52}&s_{53}&s_{54}\\\ \end{array}\right|$,

${\mathcal{J}}_{55}=\left|\begin{array}[]{c}{\mathcal{X}}_{15}\\\ \-----\\\
{\mathcal{X}}^{*}_{4,5}\end{array}\right|,$
$\pi({\mathcal{J}}_{55})=\left|\begin{array}[]{ccccc}s_{51}&s_{52}&s_{53}&s_{54}&s_{55}\\\
s^{*}_{21}&s^{*}_{22}&s^{*}_{23}&s^{*}_{24}&s^{*}_{25}\\\
s^{*}_{31}&s^{*}_{32}&s^{*}_{33}&0&0\\\
s^{*}_{41}&s^{*}_{42}&s^{*}_{43}&0&0\\\ s^{*}_{51}&0&0&0&0\\\
\end{array}\right|$,

${\mathcal{J}}_{45}=\left|\begin{array}[]{c}{\mathcal{X}}_{25}\\\ \-----\\\
{\mathcal{X}}^{*}_{3,5}\end{array}\right|,$
$\pi({\mathcal{J}}_{45})=\left|\begin{array}[]{ccccc}s_{41}&s_{42}&s_{43}&s_{44}&s_{45}\\\
s_{51}&s_{52}&s_{53}&s_{54}&s_{55}\\\ s^{*}_{31}&s^{*}_{32}&s^{*}_{33}&0&0\\\
s^{*}_{41}&s^{*}_{42}&s^{*}_{43}&0&0\\\ s^{*}_{51}&0&0&0&0\\\
\end{array}\right|$,

${\mathcal{J}}_{35}=\left|\begin{array}[]{c}{\mathcal{X}}_{35}\\\ \-----\\\
{\mathcal{X}}^{*}_{2,5}\end{array}\right|,$
$\pi({\mathcal{J}}_{35})=\left|\begin{array}[]{ccccc}0&0&s_{33}&s_{34}&s_{35}\\\
s_{41}&s_{42}&s_{43}&s_{44}&s_{45}\\\ s_{51}&s_{52}&s_{53}&s_{54}&s_{55}\\\
s^{*}_{41}&s^{*}_{42}&s^{*}_{43}&0&0\\\ s^{*}_{51}&0&0&0&0\\\
\end{array}\right|$,

${\mathcal{J}}_{25}=\left|\begin{array}[]{c}{\mathcal{X}}_{45}\\\ \-----\\\
{\mathcal{X}}^{*}_{1,5}\end{array}\right|,$
$\pi({\mathcal{J}}_{25})=\left|\begin{array}[]{ccccc}0&0&s_{23}&s_{24}&s_{25}\\\
0&0&s_{33}&s_{34}&s_{35}\\\ s_{41}&s_{42}&s_{43}&s_{44}&s_{45}\\\
s_{51}&s_{52}&s_{53}&s_{54}&s_{55}\\\ s^{*}_{51}&0&0&0&0\\\
\end{array}\right|$,

${\mathcal{J}}_{15}=\det{\mathbb{X}}$, $\pi({\mathcal{J}}_{15})=\det S$.

##  3 Fields of invariants for radicals of parabolic subgroups in the
orthogonal and symplectic subgroups

Let $K$ be a field of characteristic zero, $G$ be the orthogonal or symplectic
group over $K$, and $P$ be its parabolic subgroup. There is a decomposition of
the subgroup $P$ into a product $P=LU$ of a Levi subgroup $L$ and the
unipotent radical $U$. Let ${\mathcal{A}}=K[G]$ be the ring of regular
functions on $G$, and ${\mathcal{F}}=K(G)$ be the field of rational functions
on $G$. In this section, we construct a system of free generators in the field
of invariants ${\mathcal{F}}^{U}$ with respect to the action of $U$ on $G$ by
conjugation.

We introduce necessary notations. For each matrix $A$, we denote by $A^{t}$
(respectively, $A^{\sigma}$ ) its transpose (respectively, its transpose with
respect to the anti-diagonal). Let ${\mathbb{I}}_{N}$ be the matrix of size
$N\times N$ with ones on the anti-diagonal and zeros on other places. By
definition, the orthogonal group $\mathrm{O}(N)$ is the group of all
$g\in{\mathrm{GL}}(N)$ such that $g^{t}{\mathbb{I}}_{N}g={\mathbb{I}}_{N}$.

The symplectic group $\mathrm{Sp}(N)$, where $N=2n$, is defined analogously by
the matrix

${\mathbb{J}}_{2n}=\left(\begin{array}[]{cc}0&-{\mathbb{I}}_{n}\\\
{\mathbb{I}}_{n}&0\end{array}\right).$

Consider a decomposition of the integer segment $[1,N]$ into a system of
consecutive segments $[1,n]=I_{1}\sqcup I_{2}\sqcup\ldots\sqcup I_{\ell}$.
Denote $n_{k}=|I_{k}|$. Suppose that this decomposition is symmetric with
respect to the center of the segment, i.e. $I^{\prime}_{k}=I_{\ell-k+1}$.

Introduce notations $\ell_{0}=\left[\frac{\ell}{2}\right]$,
$N_{0}=n_{1}+\ldots+n_{\ell_{0}}$. The cases of even and odd $\ell$ separate.
If $\ell=2\ell_{0}$, then $N=2n$ and $N_{0}=n$. In the odd case
$\ell=2\ell_{0}+1$, denote $I_{0}=I_{\ell_{0}+1}$, $n_{0}=n_{\ell_{0}+1}$. Let
$G_{0}$ stand for the group $\mathrm{O}(n_{0})$ in the orthogonal case or the
group $\mathrm{Sp}(n_{0})$ in the symplectic case.

By this decomposition, we construct the parabolic subgroup $P=LU$ with the
Levi subgroup $L$ of block-symmetric matrices
${\mathrm{diag}}(A_{1},\ldots,A_{\ell})$, $A_{k}\in{\mathrm{GL}}(n_{k})$,
$A_{\ell-k+1}=(A_{k}^{\sigma})^{-1}$ for each $1\leqslant k\leqslant\ell_{0}$
and $A_{\ell_{0}+1}\in G_{0}$ (for $\ell=2\ell_{0}+1$).

If $\ell=2\ell_{0}$, then each $g\in P$ has the form

$g=\left(\begin{array}[]{cc}A&0\\\
0&(A^{\sigma})^{-1}\end{array}\right)\left(\begin{array}[]{cc}E&B\\\
0&E\end{array}\right)=\left(\begin{array}[]{cc}A&AB\\\
0&(A^{\sigma})^{-1}\end{array}\right),$

where $A$ is a matrix from the parabolic subgroup in ${\mathrm{GL}}(N_{0})$,
which relates to the decomposition $I_{1}\sqcup\ldots\sqcup I_{\ell_{0}}$ of
the segment $[1,N_{0}]$, and $B$ is a skew-symmetric (respectively, symmetric)
matrix with respect to the anti-diagonal in the orthogonal case (respectively,
in the symplectic case).

Consider the case $\ell=2\ell_{0}+1$. Denote
${\mathbb{I}}_{0}={\mathbb{I}}_{N_{0}}$,

${\mathbb{J}}_{0}=\left\\{\begin{array}[]{l}{\mathbb{I}}_{n_{0}},~{}~{}\mbox{in
~{}the ~{}case}~{}~{}\mathrm{O}(N),\\\ {\mathbb{J}}_{n_{0}},~{}~{}\mbox{in
~{}the ~{}case}~{}~{}\mathrm{Sp}(N).\end{array}\right.$

Each element $g\in P$ has the form

$g=\left(\begin{array}[]{ccc}A&0&0\\\ 0&A_{0}&0\\\
0&0&(A^{\sigma})^{-1}\end{array}\right)\left(\begin{array}[]{ccc}E&V&\frac{1}{2}VW\\\
0&E&W\\\ 0&0&E\end{array}\right)\left(\begin{array}[]{ccc}E&0&B\\\ 0&E&0\\\
0&0&E\end{array}\right)=$
$\left(\begin{array}[]{ccc}A&AV&A\left(B+\frac{1}{2}VW\right)\\\
0&A_{0}&A_{0}W\\\ 0&0&(A^{\sigma})^{-1}\end{array}\right),$

where $A$ and $B$ as above, $V$ is a matrix of size $N_{0}\times n_{0}$,
$W=-{\mathbb{J}}_{0}V^{t}{\mathbb{I}}_{0}$, and $A_{0}\in G_{0}$.

Denote $S^{\circ}=S\cap G$, where $S$ is the subset of ${\mathrm{GL}}(N)$
defined in the previous section. Arguing analogously Proposition 2, we obtain
that the subset

$\bigcup_{g\in U}gS^{\circ}g^{-1}$

is dense in $G$.

The restriction map

$\pi:{\mathcal{A}}\to K[S^{\circ}]$

establishes the embedding of the field of $U$-invariants ${\mathcal{F}}^{U}$
into $K(S^{\circ})$.

If $\ell=2\ell_{0}$, then $S^{\circ}$ consists of all matrices of the form

$\left(\begin{array}[]{cc}0&S_{12}\\\ S_{21}&S_{22}\\\
\end{array}\right)=\left(\begin{array}[]{cc}0&\pm{\mathbb{I}}_{0}(A^{\sigma})^{-1}\\\
{\mathbb{I}}_{0}A&{\mathbb{I}}_{0}AB\\\ \end{array}\right).$

If $\ell=2\ell_{0}+1$, then $S^{\circ}$ consists of all matrices of the form

$\left(\begin{array}[]{ccc}0&0&S_{13}\\\ 0&S_{22}&S_{23}\\\
S_{31}&S_{32}&S_{33}\end{array}\right)=\left(\begin{array}[]{ccc}0&0&\pm{\mathbb{I}}_{0}(A^{\sigma})^{-1}\\\
0&{\mathbb{J}}_{0}A_{0}&{\mathbb{J}}_{0}A_{0}W\\\
{\mathbb{I}}_{0}A&{\mathbb{I}}_{0}AV&{\mathbb{I}}_{0}A\left(B+\frac{1}{2}VW\right)\end{array}\right).$
(5)

Consider the set ${\mathbb{S}}^{\circ}$ of pairs $(i,j)$ such that
1) $i>N_{0}$ (for $\ell=2\ell_{0}$) or $i>N_{0}+n_{0}$ (for
$\ell=2\ell_{0}+1$),
2) if $i\in I_{k}$ and $j\in I^{\prime}_{m}$, then $k\geqslant m$,
3) $i>j$ in the orthogonal case or $i\geqslant j$ in the symplectic case.

Similarly to the previous section, let ${\mathbb{S}}^{\circ}_{0}$ consist of
all pairs $(i,j)\in{\mathbb{S}}^{\circ}$ lying on or above the anti-diagonal,
respectively, ${\mathbb{S}}^{\circ}_{1}$ consist of all pairs $(i,j)$ lying
below the anti-diagonal. Denote by ${\mathbb{G}}_{0}$ the set $I_{0}\times
I_{0}$.

Example 3.1. Let $G=\mathrm{Sp}(8)$ and the parabolic subgroup is defined by
the decomposition
$[1,8]=\\{1\\}\sqcup[2,3]\sqcup[4,5]\sqcup[6,7]\sqcup\\{8\\}$. On the
$(6\times 6)$-tables below, we mark by the symbol ”$\times$” the cells $(i,j)$
belonging to ${\mathbb{S}}^{\circ}$ (respectively, belonging to
${\mathbb{S}}^{\circ}_{0}$, ${\mathbb{S}}^{\circ}_{1}$, ${\mathbb{G}}_{0}$).
We have

${\mathbb{S}}^{\circ}=\left(\begin{array}[]{cccccccc}\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\times&\times&\times&\times&\times&\cdot&\cdot\\\
\cdot&\times&\times&\times&\times&\times&\times&\cdot\\\
\times&\times&\times&\times&\times&\times&\times&\times\\\
\end{array}\right),\qquad{\mathbb{G}}_{0}=\left(\begin{array}[]{cccccccc}\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\times&\times&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\times&\times&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\ \end{array}\right),$
${\mathbb{S}}^{\circ}_{0}=\left(\begin{array}[]{cccccccc}\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\times&\times&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\times&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\times&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\end{array}\right),\qquad{\mathbb{S}}^{\circ}_{1}=\left(\begin{array}[]{cccccccc}\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\\
\cdot&\cdot&\cdot&\times&\times&\times&\cdot&\cdot\\\
\cdot&\cdot&\times&\times&\times&\times&\times&\cdot\\\
\cdot&\times&\times&\times&\times&\times&\times&\times\\\ \end{array}\right).$

We attach to each $(i,j)\in{\mathbb{G}}_{0}$ the rational function $P_{ij}$ on
$G$ by the formula

$P_{ij}=\frac{M_{ij}}{M_{0}},$

where $M_{0}$ is the minor of order $N_{0}$ with the systems of rows
$I_{\ell_{0}+2}\sqcup\ldots\sqcup I_{\ell}$ and columns
$I_{1}\sqcup\ldots\sqcup I_{\ell_{0}}$, and $M_{ij}$ is the minor obtained
from $M_{0}$ by adding $i$th row and $j$th column. Easy to see that the minors
$M_{ij}$ and $M_{0}$ are invariants with respect to the action of $U$ on $G$
by conjugation. Therefore, $P_{ij}$ is also an invariant.

Consider the subfield ${\mathcal{F}}_{0}$ in ${\mathcal{F}}^{U}$ generated by
the system $\\{P_{ij}:~{}(i,j)\in{\mathbb{G}}_{0}\\}$. One can treat $G_{0}$
as a factor of the parabolic subgroup $P$. This makes it possible to consider
the field of rational functions $K(G_{0})$ as a subfield in the field $K(G)$
and also as a subfield in $K(S)$. The restriction map $\pi$ establishes an
isomorphism of the subfield ${\mathcal{F}}_{0}$ in ${\mathcal{F}}^{U}$ onto
the subfield $K(G_{0})$ in $K(S)$.

For each pair $(i,j)\in{\mathbb{S}}^{\circ}$, we denote by $s_{ij}$ the
restriction of the matrix element $x_{ij}$ on $S^{\circ}$.

Lemma 3.2. The field $K(S^{\circ})$ is freely generated over the subfield
$K(G_{0})$ by the system of matrix elements
$\\{s_{ij}:~{}~{}(i,j)\in{\mathbb{S}}^{\circ}\\}$.
_Proof_. Let us prove for $\ell=2\ell_{0}+1$. The case $\ell=2\ell_{0}$ is
treated similarly. We consider the system of matrix elements

$\\{a_{ij},v_{ij},b_{ij}\\}$

from (5), where $A=(a_{ij})$, $V=(v_{ij})$ and $\\{b_{ij}\\}$ are entries of
$B$ lying above the anti-diagonal in the case of $\mathrm{O}(N)$
(respectively, on or above the anti-diagonal in the case of $\mathrm{Sp}(N)$).
We treat these matrix elements as rational functions on $S^{\circ}$. The
formula (5) implies that the field $K(S^{\circ})$ is freely generated over the
subfield $K(G_{0})$ by the system $\\{a_{ij},~{}v_{ij},~{}b_{ij}\\}$. The
subfield generated by the matrix elements from $S_{31}$ and $S_{32}$ coincides
with the subfield generated by $\\{a_{ij},~{}v_{ij}\\}$. Recall that
$S_{33}={\mathbb{I}}_{0}A\left(B+\frac{1}{2}VW\right)$, where $A$ is a matrix
from the parabolic subgroup in ${\mathrm{GL}}(N_{0})$ defined by the
decomposition $I_{1}\sqcup\ldots\sqcup I_{\ell_{0}}$ of the segment
$[1,N_{0}]$, $B$ is the skew-symmetric (respectively, symmetric) matrix with
respect to the anti-diagonal in the orthogonal case (respectively, in the
symplectic case), and $W=-{\mathbb{J}}_{0}V^{t}{\mathbb{I}}_{0}$. Rewrite
$S_{33}$ in the form

$S_{33}={\mathbb{I}}_{0}A{\mathbb{I}}_{0}\cdot
C+\frac{1}{2}{\mathbb{I}}_{0}AVW,$

where $C={\mathbb{I}}_{0}B$ is a skew-symmetric matrix in the orthogonal case
and symmetric in the symplectic case. Consider the system of matrix elements
$\\{c_{ij}\\}$ from $C$ lying below the diagonal in the case $\mathrm{O}(N)$
(respectively, on or below the diagonal in the case of $\mathrm{Sp}(N)$). The
system $\\{c_{ij}\\}$ coincides with the system of $\\{b_{ij}\\}$ defined
above. Observe that ${\mathbb{I}}_{0}A{\mathbb{I}}_{0}$ is a lower block-
triangular matrix defined by the partition of the segment $[1,N_{0}]$ into
consecutive segments of sizes $n_{\ell_{0}},\ldots,n_{1}$. Present the matrix
${\mathbb{I}}_{0}A{\mathbb{I}}_{0}$ as a product of two matrices with rational
entries ${\mathbb{I}}_{0}A{\mathbb{I}}_{0}=RQ$, where $Q$ is a lower
triangular matrix and $R$ is an upper unitriangular matrix. Then the elements
$s_{ij}$, $(i,j)\in{\mathbb{S}}^{\circ}$ from $S_{33}$ are obtained from
$\\{c_{ij}\\}$ by a composition of two triangular transformations:

  * •

$C\to QC$ is a triangular transformation with respect to the order relation:
$(i_{1},j_{1})$ is less than $(i,j)$ if $j_{1}<j$ or $j_{1}=j$, $i_{1}<i$,

  * •

$QC\to RQC+\frac{1}{2}{\mathbb{I}}_{0}AVW$ is a triangular transformation with
respect to the order relation $\prec$ from (2).

The system $\\{s_{ij}:~{}~{}(i,j)\in{\mathbb{S}}^{\circ}\\}$ is obtained from
$\\{a_{ij},~{}v_{ij},~{}b_{ij}\\}$ by the composition of two triangular
transformations. $\Box$

Denote by ${\mathcal{J}}_{ij}^{\circ}$ the restriction of the determinant
${\mathcal{J}}_{ij}$ on the orthogonal or symplectic group $G$.
Proposition 3.3. ${\mathcal{J}}_{ij}^{\circ}\neq 0$ for each
$(i,j)\in{\mathbb{S}}^{\circ}$.
_Proof_. For $(i,j)\in{\mathbb{S}}_{0}^{\circ}$ the statement is obvious.
Consider the case $(i,j)\in{\mathbb{S}}_{1}^{\circ}$. Define the lower
triangular with respect to the anti-diagonal matrix $A=(a_{st})\in S^{\circ}$
that has nonzero entries on the anti-diagonal, has non-degenerate
$(i^{\prime},i^{\prime})$-block defined by the system of last $i^{\prime}$
rows and the columns $[j-i^{\prime}+1,j]$, and $a_{st}=0$ on other places.
Then the determinant ${\mathcal{J}}_{ij}^{\circ}(A)$ has the form

$\left|\begin{array}[]{cc}A_{11}&A_{12}\\\ A_{21}&0\end{array}\right|,$

where $A_{12}$ is the defined above non-degenerate
$(i^{\prime},i^{\prime})$-block, and $A_{21}$ is an upper triangular block
with non-zeros on the anti-diagonal. Therefore
${\mathcal{J}}_{ij}^{\circ}(A)\neq 0$. $\Box$
Corollary 3.4. $\pi({\mathcal{J}}^{\circ}_{ij})\neq 0$ for each
$(i,j)\in{\mathbb{S}}^{\circ}$.

Recall that the subfield ${\mathcal{F}}_{0}$ in ${\mathcal{F}}^{U}$ is
generated by $\\{P_{ij}:~{}(i,j)\in{\mathbb{G}}_{0}\\}$, and it is isomorphic
to $K(G_{0})$.
Theorem 3.5. For the orthogonal or symplectic group $G$, the system of
polynomials $\\{{\mathcal{J}}^{\circ}_{ij}:~{}(i,j)\in{\mathbb{S}}^{\circ}\\}$
freely generates over ${\mathcal{F}}_{0}$ the field of invariants
${\mathcal{F}}^{U}$.
_Proof_. Consider the extension ${\mathcal{P}}_{0}$ generated over
${\mathcal{F}}_{0}$ by

$\\{\pi({\mathcal{J}}_{ij}):~{}~{}(i,j)\in{\mathbb{S}}^{\circ}_{0}\\}.$

Let $Q$ be the parabolic subgroup in $G$ defined by the partition of the
segment $[1,N]$ into the system of consecutive segments of sizes
$(1,\ldots,1,n_{0},1,\ldots,1)$, and $U(Q)$ be its unipotent radical. The
group $U(Q)$ acts on $S^{\circ}$ by right multiplication. Similar to the proof
of Proposition 2, one can show that $\pi$ established isomorphism of the
subfield ${\mathcal{P}}_{0}$ onto the subfield of invariants in $K(S^{\circ})$
under the right multiplication by $U(Q)$. It follows from the proof of Theorem
2 that the system of polynomials
$\\{\pi({\mathcal{J}}_{ij}^{\circ}):~{}(i,j)\in{\mathbb{S}}^{\circ}\\}$ is
obtained by a triangular transformation from the system
$\\{s_{ij}:~{}(i,j)\in{\mathbb{S}}^{\circ}\\}$. Applying Lemma 3 we conclude
the proof. $\Box$

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