# Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits

Mikhail V. Ignatev  Matvey A. Surkov

###### Abstract

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie
algebra with root system $\Phi$. A subset $D$ of the set $\Phi^{+}$ of
positive roots is called a rook placement if it consists of roots with
pairwise non-positive scalar products. To each rook placement $D$ and each map
$\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one
can naturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space
$\mathfrak{n}^{*}$. By definition, $\Omega_{D,\xi}$ is the orbit of
$f_{D,\xi}$, where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^{*}$
multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint
orbits studied at the moment have such a form for certain $D$ and $\xi$.) It
follows from the results of Andrè that if $\xi_{1}$ and $\xi_{2}$ are distinct
maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_{1}}$ and
$\Omega_{D,\xi_{2}}$ do not coincide for classical root systems $\Phi$. We
prove that this is true if $\Phi$ is of type $G_{2}$, or if $\Phi$ is of type
$F_{4}$ and $D$ is orthogonal.

Keywords: coadjoint orbit, the orbit method, rook placement, orthogonal
subset, root system.
AMS subject classification: 17B08, 17B10, 17B22, 17B25, 17B30, 17B35.

1\. Introduction and the main result

Let $\mathfrak{g}$ be a simple complex Lie algebra, $\mathfrak{b}$ be a Borel
subalgebra of $\mathfrak{g}$, $\Phi$ be the root system of $\mathfrak{g}$,
$\Phi^{+}$ be the set of positive roots corresponding to $\mathfrak{b}$,
$\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, $N=\exp(\mathfrak{n})$ be
the corresponding nilpotent algebraic group, and $\mathfrak{n}^{*}$ be the
dual space to $\mathfrak{n}$. The group $N$ acts on $\mathfrak{n}$ by the
adjoint action; the dual action of $N$ on the space $\mathfrak{n}^{*}$ is
called coadjoint; we will denote the result of this action by $g.\lambda$ for
$g\in N$, $\lambda\in\mathfrak{n}^{*}$. Accordingly to the orbit method
discovered by A.A. Kirillov in 1962, coadjoint orbits play a key role in
representation theory of $N$ [Ki62], [Ki04]. We will consider a special class
of coadjoint orbits defined below.

Definition 1.1. A subset $D$ of $\Phi^{+}$ is called a _rook placement_ if
$(\alpha,\beta)\leq 0$ for all distinct $\alpha,\beta\in D$.

The root vectors $e_{\alpha}$, $\alpha\in\Phi^{+}$ form a basis of
$\mathfrak{n}$; we denote by $\\{e_{\alpha}^{*},~{}\alpha\in D\\}$ the dual
basis of $\mathfrak{n}^{*}$. Given a rook placement $D$ and a map $\xi\colon
D\to\mathbb{C}^{\times}$, we put

$f_{D,\xi}=\sum_{\alpha\in D}\xi(\alpha)e_{\alpha}^{*}\in\mathfrak{n}^{*}.$

Definition 1.2. We say that the coadjoint orbit $\Omega_{D,\xi}$ of the linear
form $f_{D,\xi}$ is _associated_ with the rook placement $D$ and the map
$\xi$.

It turns out that almost all coadjoint orbit studied to the moment are
associated with certain $D$ and $\xi$, see, e.g., [An95], [AN06], [Ko12],
[Ko13], [Pa08], [IPa09], [Ig09], [Ig11], [Ig12]. On the other hand, C.A.M.
Andrè discovered that, for the case of $A_{n-1}$, rook placements themselves
provide a nice splitting of $\mathfrak{n}^{*}$ into a disjoint union of
$N$-stable affine subvarieties called basic subvarieties. (We will recall
Andrè’s results in details in Section Rook placements in $G_{2}$ and $F_{4}$
and associated coadjoint orbits, because we will use them for the case of
$F_{4}$.) By definition, the _basic subvariety_ $\mathcal{O}_{D,\xi}$
corresponding to a rook placement $D$ and a map $\xi\colon
D\to\mathbb{C}^{\times}$ is

$\mathcal{O}_{D,\xi}=\sum_{\alpha\in D}\Omega_{\\{\alpha\\},\xi_{\alpha}},$

where $\xi_{\alpha}$ is the restriction of $\xi$ to $\\{\alpha\\}$.
Accordingly to [An95, Theorem 1], for $A_{n-1}$,
$\mathfrak{n}^{*}=\bigsqcup_{D,\xi}\mathcal{O}_{D,\xi}$, and all
$\mathcal{O}_{D,\xi}$’s are affine subvarieties of $\mathfrak{n}^{*}$.

Even for $B_{n}$, $C_{n}$ and $D_{n}$, the similar question is still open.
Nevertheless, we may formulate the following conjecture for an arbitrary root
system. (Non-singularity of a rook placement $D$ means that if
$\alpha,~{}\beta\in D$ and $\alpha\neq\beta$ then
$\alpha-\beta\notin\Phi^{+}$; for $A_{n-1}$, all rook placements are
automatically non-singular.)

Conjecture 1.3. _Each basic subvariety $\mathcal{O}_{D,\xi}$ is an affine
subvariety of $\mathfrak{n}^{*}$, and_

$\mathfrak{n}^{*}=\bigsqcup_{D,\xi}\mathcal{O}_{D,\xi},$

_where the union is taken over all non-singular rook placements $D$ and all
maps $\xi\colon D\to\mathbb{C}^{\times}$._

Direct computations show that this conjecture is true for classical root
systems of low rank. In the paper, we check that this conjecture is true for
the case of $G_{2}$. This is our first main result. (In fact, given $D$ and
$\xi$, we present an explicit system of equations describing
$\mathcal{O}_{D,\xi}$.)

Theorem 1.4. _Let $\Phi=G_{2}$. Then each basic subvariety
$\mathcal{O}_{D,\xi}$ is an affine subvariety of $\mathfrak{n}^{*}$, and_

$\mathfrak{n}^{*}=\bigsqcup_{D,\xi}\mathcal{O}_{D,\xi},$

_where the union is taken over all non-singular rook placements $D$ and all
maps $\xi\colon D\to\mathbb{C}^{\times}$._

It turned out that, for $A_{n-1}$, if $D$ is a rook placement and $\xi_{1}$,
$\xi_{2}$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then the
associated orbits $\Omega_{D,\xi_{1}}$ and $\Omega_{D,\xi_{2}}$ do not
coincide. (It follows immediately from Andrè’s theory, since
$\Omega_{D,\xi}\subset\mathcal{O}_{D,\xi}$, see Section Rook placements in
$G_{2}$ and $F_{4}$ and associated coadjoint orbits.) For other classical root
systems this fact can be obtained as a corollary of the case of $A_{n-1}$ (see
also [AN06]). This was used by M.V. Ignatyev and I. Penkov in [IPe16] and
[Ig19] for explicit classification of centrally generated primitive ideals in
the universal enveloping algebra $U(\mathfrak{n})$ for classical root systems.

In [IS21], M.V. Ignatyev and A.A. Shevchenko, classifying centrally generated
primitive ideals in $U(\mathfrak{n})$ for exceptional root systems, proved
that the similar is true for certain orthogonal rook placements in $F_{4}$ and
$E_{6}$, $E_{7}$, $E_{8}$. This allows us formulate the second conjecture for
an arbitrary root system.

Conjecture 1.5. _Let $D$ be a non-singular rook placement and $\xi_{1}$,
$\xi_{2}$ be distinct maps from $D$ to $\mathbb{C}^{\times}$. Then the
associated coadjoint orbits $\Omega_{D,\xi_{1}}$ and $\Omega_{D,\xi_{2}}$ do
not coincide._

Our second main result is that this conjecture is true for $F_{4}$ if $D$ is
orthogonal (i.e., if all roots from $D$ are pairwise orthogonal).

Theorem 1.6. _Let $\Phi=F_{4}$, $D$ be an orthogonal non-singular rook
placement and $\xi_{1}$, $\xi_{2}$ be distinct maps from $D$ to
$\mathbb{C}^{\times}$. Then the associated coadjoint orbits
$\Omega_{D,\xi_{1}}$ and $\Omega_{D,\xi_{2}}$ do not coincide._

The paper is organized as follows.

Funding. The work performed under the development program of Volga Region
Mathematical Center (agreement no. 075–02–2021–1393).

Acknowledgements. We thank Prof. A.N. Panov for useful discussions.

2\. Andrè’s theory

In this section, we briefly recall Andrè’s results from [An95], which will be
needed in the sequel. Throughout this section, $\Phi$ will be of type
$A_{n-1}$. As usual, we identify the set of positive roots with the following
subset of the Euclidean space $\mathbb{R}^{n}$ with the standard inner
product:

$A_{n-1}^{+}=\\{\varepsilon_{i}-\varepsilon_{j},~{}1\leq i<j\leq n\\}.$

Here $\varepsilon_{1},\ldots,\varepsilon_{n}$ is the standard basis of
$\mathbb{R}^{n}$.

In this case, $\mathfrak{n}$ can be identified with the space of strictly
upper-triangular $n\times n$ matrices with zeroes on the diagonal. Given
$\alpha=\varepsilon_{i}-\varepsilon_{j}\in\Phi^{+}$, one can pick the
$(i,j)$th elementary matrix $e_{i,j}$ as a root vector $e_{\alpha}$, so that
$[e_{\alpha},e_{\beta}]=\pm e_{\alpha+\beta}$ for $\alpha,\beta\in\Phi^{+}$
(we put $e_{\alpha+\beta}=0$ if $\alpha+\beta\notin\Phi^{+}$). We will
identify the dual space $\mathfrak{n}^{*}$ with the space $\mathfrak{n}^{-}$
of strictly lower-triangular $n\times n$ matrices with zeroes on the diagonal
via the formula $\langle\lambda,x\rangle=\mathrm{tr}\,(\lambda x)$ for
$x\in\mathfrak{n}$, $\lambda\in\mathfrak{n}^{-}$. The root vectors
$e_{\alpha}$, $\alpha\in\Phi^{+}$ form a basis of $\mathfrak{n}$; let
$\\{e_{\alpha}^{*},~{}\alpha\in\Phi^{+}\\}$ be the dual basis of
$\mathfrak{n}^{*}$ (in fact, $e_{i,j}^{*}=e_{j,i}$).

The group $N$ is the group of all strictly upper-triangular $n\times n$
matrices with $1$’s on the diagonal. It acts on its Lie algebra $\mathfrak{n}$
via the adjoint action $\mathrm{Ad}_{g}(x)=gxg^{-1}$, $g\in N$,
$x\in\mathfrak{n}$. The dual action of $N$ on the space $\mathfrak{n}^{*}$ is
called coadjoint; we will denote the result of this action by $g.\lambda$,
$g\in N$, $\lambda\in\mathfrak{n}^{*}$. It is easy to check that, after the
identification of $\mathfrak{n}^{*}$ with $\mathfrak{n}^{-}$, this action has
the form $g.\lambda=(g\lambda g^{-1})_{\mathrm{low}}$. Here, given an $n\times
n$ matrix $a$, we set

$(a_{\mathrm{low}})_{i,j}=\begin{cases}a_{i,j},&\text{if }i>j,\\\
0&\text{otherwise}.\end{cases}$

Definition 2.1. Pick a number $k$ from $1$ to $n$. We call the sets

$\mathcal{R}_{k}=\\{\varepsilon_{j}-\varepsilon_{k},~{}1\leq
j<k\\},~{}\mathcal{C}_{k}=\\{\varepsilon_{k}-\varepsilon_{i},~{}k<i\leq n\\}$

the _$k$ th row_ and the _$k$ th column_ of $\Phi^{+}$ respectively. We say
that the number $i$ (respectively, the number $j$) is the _row_ (respectively,
the _column_) of a root $\alpha=\varepsilon_{i}-\varepsilon_{j}$.

Example 2.2. Let $n=6$. On the picture below boxes from
$\mathcal{R}_{5}\cup\mathcal{C}_{2}$ are grey. Here we identify a root
$\varepsilon_{i}-\varepsilon_{j}\in\Phi^{+}$ with the box $(j,i)$.

To each rook placement $D\subset\Phi^{+}$ and each map $\xi\colon
D\to\mathbb{C}^{\times}$, one can assign the linear form

$f_{D,\xi}=\sum_{\alpha\in D}\xi(\alpha)e_{\alpha}^{*}\in\mathfrak{n}^{*}.$

Example 2.3. Let $n=8$,
$D=\\{\varepsilon_{1}-\varepsilon_{3},~{}\varepsilon_{2}-\varepsilon_{6},~{}\varepsilon_{3}-\varepsilon_{7},~{}\varepsilon_{4}-\varepsilon_{5},~{}\varepsilon_{6}-\varepsilon_{8}\\}$.
On the picture below we schematically draw the linear form $f_{D,\xi}$ putting
symbols $\otimes$ in the boxes corresponding to the roots from $D$.

It follows immediately from the definition of a rook placement that

$|D\cap\mathcal{R}_{k}|\leq 1,~{}|D\cap\mathcal{C}_{k}|\leq 1$

for all $k$. This explains the term “rook placement”: if we identify symbols
$\otimes$ from $f_{D,\xi}$ with rooks on the lower-triangular chessboard, then
these rooks do not hit each other.

Now, given a root $\alpha\in D$, we denote by $\xi_{\alpha}$ the restriction
of the map $\xi$ to the subset $\\{\alpha\\}$, and put

$\mathcal{O}_{D,\xi}=\sum_{\alpha\in
D}\mathcal{O}_{\\{\alpha\\},\xi_{\alpha}}.$

Clearly, $\Omega_{D,\xi}\subset\mathcal{O}_{D,\xi}$.

Definition 2.4. The set $\mathcal{O}_{D,\xi}$ is called a _basic subvariety_
of $\mathfrak{n}^{*}$ corresponding to the rook placement $D$ and the map
$\xi$.

Accordingly to [An95, Theorem 1], $\mathfrak{n}^{*}$ is a disjoint unions of
basic subvarieties:

$\mathfrak{n}^{*}=\bigsqcup_{D,\xi}\mathcal{O}_{D,\xi},$

where the union is taken over all rook placements in $A_{n-1}^{+}$ and all
maps $\xi\colon D\to\mathbb{C}^{\times}$. (Formally, Andrè considered the case
of finite ground field, but his proofs are valid over $\mathbb{C}$, too.)
Furthermore, each basic subvariety $\mathcal{O}_{D,\xi}$ is in fact an affine
subvariety of $\mathfrak{n}^{*}$, and Andrè presented an explicit set of
defining equations for it. To describe this set, we need some more notation.

Definition 2.5. A root $\alpha\in\Phi^{+}$ is called $\beta$-_singular_ for a
root $\beta\in\Phi^{+}$ if $\beta-\alpha\in\Phi^{+}$. The set of all
$\beta$-singular roots is denoted by $S(\beta)$.

Let $D$ be a rook placement and $\xi\colon D\to\mathbb{C}^{\times}$ be a map.
We denote

$S(D)=\bigcup_{\beta\in D}S(\beta)$

and $R(D)=\Phi^{+}\setminus S(D)$. Obviously, $D\subset R(D)$. Roots from
$S(D)$ (respectively, from $R(D)$) are called $D$-_singular_ (respectively,
$D$-_regular_).

Example 2.6. Let $n=10$,
$D=\\{\varepsilon_{1}-\varepsilon_{6},~{}\varepsilon_{3}-\varepsilon_{10},~{}\varepsilon_{5}-\varepsilon_{8}\\}$.
On the picture below, boxes corresponding to the roots from $D$ are filled by
$\otimes$’s, as above, while the boxes corresponding by the $D$-singular roots
are marked gray.

It turns out that to each $D$-regular root $\alpha$ one can assign the
defining equation of $\mathcal{O}_{D,\xi}$. Namely, there is a natural partial
order on $\Phi^{+}$: we write $\alpha\leq\beta$ if $\beta-\alpha$ is a sum of
positive roots. Evidently,
$\varepsilon_{i}-\varepsilon_{j}\leq\varepsilon_{r}-\varepsilon_{s}$ if and
only if $s\leq j$ and $r\geq i$. (In other words, on our pictures
$\varepsilon_{r}-\varepsilon_{s}$ is located non-strictly to the South-West
from $\varepsilon_{i}-\varepsilon_{j}$.) Given $\alpha\in R(D)$, we set
$D(\alpha)=\\{\alpha\\}\cup\\{\beta\in D\mid\beta\geq\alpha\\}.$ Now, let
$R_{D}(\alpha)$ (respectively $C_{D}(\alpha)$) be the set of all rows
(respectively, of all columns) of the roots from $D(\alpha)$. Finally, for a
matrix $\lambda\in\mathfrak{n}^{*}$, we denote by
$\Delta^{D}_{\alpha}(\lambda)$ the minor of the matrix $\lambda$ with the set
of rows $R_{D}(\alpha)$ and the set of columns $C_{D}(\alpha)$. (We assume
that the numbers of rows and columns are taken in the increasing order).

For instance, in the previous example, if
$\alpha=\varepsilon_{4}-\varepsilon_{5}$ then
$D(\alpha)=\\{\varepsilon_{1}-\varepsilon_{6},~{}\varepsilon_{3}-\varepsilon_{10},~{}\varepsilon_{4}-\varepsilon_{5}\\}$,
hence $R_{D}(\alpha)=\\{5,~{}6,~{}10\\}$, $C_{D}(\alpha)=\\{1,~{}3,~{}4\\}$
and

$\Delta^{D}_{\alpha}(\lambda)=\begin{vmatrix}\lambda_{5,1}&\lambda_{5,3}&\lambda_{5,4}\\\
\lambda_{6,1}&\lambda_{6,3}&\lambda_{6,4}\\\
\lambda_{10,1}&\lambda_{10,3}&\lambda_{10,4}\\\ \end{vmatrix}.$

On the other hand, for $\alpha=\varepsilon_{5}-\varepsilon_{8}\in D$, one has
$D(\alpha)=\\{\varepsilon_{3}-\varepsilon_{10},~{}\varepsilon_{5}-\varepsilon_{8}\\}$,
$R_{D}(\alpha)=\\{8,~{}10\\}$, $C_{D}(\alpha)=\\{3,~{}5\\}$, and,
consequently,

$\Delta^{D}_{\alpha}(\lambda)=\begin{vmatrix}\lambda_{8,3}&\lambda_{8,5}\\\
\lambda_{10,3}&\lambda_{10,5}\\\ \end{vmatrix}.$

Thanks to [An95, Proposition 2], a matrix $\lambda\in\mathfrak{n}^{*}$ belongs
to $\mathcal{O}_{D,\xi}$ if and only if

$\Delta^{D}_{\alpha}(\lambda)=\Delta^{D}_{\alpha}(f_{D,\xi})\text{ for all
}\lambda\in R(D).$

Precisely, $\Delta^{D}_{\alpha}(\lambda)=0$ for all $\alpha\in R(D)\setminus
D$ and $\Delta^{D}_{\alpha}(\lambda)=\pm\prod_{\beta\in D(\alpha)}\xi(\beta)$
for $\alpha\in D$. It follows immediately that

$\dim\mathcal{O}_{D,\xi}=|S(D)|.$

Remark 2.7. Actually, the Andrè’s proof of the fact that each
$\lambda\in\mathfrak{n}^{*}$ belongs to a certain basic subvariety
$\mathcal{O}_{D,\xi}$ is very straightforward. Namely, there is a total order
$\leq_{t}$ on $\Phi^{+}$ refining the partial order $\leq$ defined above. By
definition,
$\varepsilon_{i}-\varepsilon_{j}<_{t}\varepsilon_{r}-\varepsilon_{s}$ if $s<j$
or $s=j$, $i<r$. Now, given $\lambda\in\mathfrak{n}^{*}$, we construct
$\mathcal{O}_{D,\xi}$ containing $\lambda$ inductively. If $\lambda=0$, then
$D=\varnothing$ and $\xi$ is the unique empty map from $\varnothing$ to
$\mathbb{C}^{\times}$. If $\lambda\neq 0$ then we find the smallest (with
respect to $\leq_{t}$) root $\alpha_{1}$ from $\Phi^{+}$ such that
$\lambda(e_{\alpha_{1}})\neq 0$, and put $D=\\{\alpha_{1}\\}$,
$\xi(\alpha_{1})=\lambda(\alpha_{1})$. If $\lambda\in\mathcal{O}_{D,\xi}$,
then we are done. Otherwise, let $\alpha_{2}$ be the smallest root such that
$\Delta^{D}_{\alpha_{2}}(\lambda)\neq\Delta^{D}_{\alpha_{2}}(f_{D,\xi})$. Then
we add $\alpha_{2}$ to $D$ and define $\xi(\alpha_{2})$ in the obvious way.
Now, one can repeat this procedure to obtain the required basic subvariety
$\mathcal{O}_{D,\xi}$.

3\. Case $\Phi=G_{2}$

In this section, we prove our first main result, Theorem Rook placements in
$G_{2}$ and $F_{4}$ and associated coadjoint orbits. First, we briefly recall
some basic facts about the simple Lie algebra $\mathfrak{g}$ of type $G_{2}$
and its maximal nilpotent subalgebra $\mathfrak{n}$. By definition, the root
system $\Phi=G_{2}$ has the form $\Phi=\Phi^{+}\cup\Phi^{-}$, where
$\Phi^{+}=\\{\alpha,\beta,\alpha+\beta,2\alpha+\beta,3\alpha+\beta,3\alpha+2\beta\\}$,
$\Phi^{-}=-\Phi^{+}$, and $\alpha$, $\beta$ are vectors from $\mathbb{R}^{2}$
such that $||\alpha||^{2}=1,~{}||\beta||^{2}=3$ and the angle between $\alpha$
and $\beta$ equals $5\pi/6$. There is a Cartan decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{n}\oplus\mathfrak{n}_{-}$, where
$\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, and $\mathfrak{n}$
has a basis consisting of the root vectors $e_{\gamma}$, $\gamma\in\Phi^{+}$.

It is well known that there exists nonzero scalars $c_{i}$, $1\leq i\leq 5$,
such that $[e_{\alpha},e_{\beta}]=c_{1}\cdot e_{\alpha+\beta}$,
$[e_{\alpha},e_{\alpha+\beta}]=c_{2}\cdot e_{2\alpha+\beta}$,
$[e_{\alpha},e_{2\alpha+\beta}]=c_{3}\cdot e_{3\alpha+\beta}$,
$[e_{3\alpha+\beta},e_{\beta}]=c_{4}\cdot e_{3\alpha+2\beta}$,
$[e_{\alpha+\beta},e_{2\alpha+\beta}]=c_{5}\cdot e_{3\alpha+2\beta}$. (In
fact, one can choose the root vectors so that $c_{1}=1$, $c_{2}=2$, $c_{3}=3$,
$c_{4}=1$, $c_{5}=3$, but we will not use these explicit values in the
sequel.) One can immediately check that $c_{1}c_{5}=c_{3}c_{4}$ for an
arbitrary choice of the root vectors.

Recall the definition of the group $N=\exp(\mathfrak{n})$ and the coadjoint
action of $N$ on the dual space $\mathfrak{n}^{*}$. It is straightforward to
check that this action has the form

$(\exp(x).\lambda)(y)=\lambda(\exp(-\mathrm{ad}_{x})(y))=\lambda(y)-\lambda([x,y])+\dfrac{1}{2!}\lambda([x,[x,y]])-\ldots,~{}x,y\in\mathfrak{n},~{}\lambda\in\mathfrak{n}^{*}.$

Now, let $D$ be a non-singular rook placement in $\Phi^{+}$. Recall that non-
singularity means that $\gamma\notin S(\delta)$ for all distinct
$\gamma,~{}\delta\in D$, where $S(\delta)$ denotes the set of
$\delta$-singular roots in $\Phi^{+}$. Fix a map $\xi\colon
D\to\mathbb{C}^{\times}$, and recall that, by definition, $\Omega_{D,\xi}$ is
the coadjoint orbit of the linear form $f_{D,\xi}$. It follows immediately
that if $\gamma$ is a maximal (with respect to the partial order $\leq$ on
$\Phi^{+}$) among all roots from $D$ then
$\lambda(e_{\gamma})=f_{D,\xi}(e_{\gamma})=\xi(\gamma)$ for all
$\lambda\in\Omega_{D,\xi}$. Similarly, $\lambda(e_{\gamma})=0$ for all
$\lambda\in\Omega_{D,\xi}$, if there are no $\delta\in D$ such that
$\delta\geq\gamma$. Recall also the definition of $\mathcal{O}_{D,\xi}$.

Given $\gamma\in\Phi^{+}$, we write $\lambda_{\gamma}=\lambda(e_{\gamma})$, so
that $\lambda=\sum_{\gamma\in\Phi^{+}}\lambda_{\gamma}e_{\gamma}^{*}$. We will
prove Theorem Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint
orbits as an immediate corollary of the following key proposition:

Proposition 3.1. _Let $D$ be a non-singular rook placement in $\Phi^{+}$,
$\xi\colon D\to\mathbb{C}^{\times}$ be a map. Pick a linear form
$\lambda\in\mathfrak{n}^{*}$. Then $\lambda\in\mathcal{O}_{D,\xi}$ if and only
if $\lambda$ satisfy the following system of equations. _

|  | $D$
---
| System
---
of equations for __ $\mathcal{O}_{D,\xi}$
|  | $D$
---
| System
---
of equations for __ $\mathcal{O}_{D,\xi}$
1 | $\varnothing$ |  |  $\lambda_{\gamma}=0$_for all_ $\gamma\in\Phi^{+}$
---
2 | $\alpha$ |  |  $\lambda_{\alpha}=\xi(\alpha)$_,_
---
$\lambda_{\gamma}=0$_for_ $\gamma\neq\alpha$
3 | $\beta$ |  |  $\lambda_{\beta}=\xi(\beta)$_,_
---
$\lambda_{\gamma}=0$_for_ $\gamma\neq\beta$
4 | $\alpha+\beta$ |  |  $\lambda_{\alpha+\beta}=\xi(\alpha+\beta)$_,_
---
$\lambda_{2\alpha+\beta}=\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0$
5 | $2\alpha+\beta$ |  |  $\lambda_{2\alpha+\beta}=\xi(2\alpha+\beta)$_,_
---
$2c_{2}\lambda_{\beta}\lambda_{2\alpha+\beta}-c_{1}\lambda_{\alpha+\beta}^{2}=0$_,_
$\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0$
6 | $3\alpha+\beta$ |  |  $6c_{3}^{2}\lambda_{\beta}\lambda_{3\alpha+\beta}^{2}-c_{1}c_{2}\lambda_{2\alpha+\beta}^{3}=0$_,_
---
$2c_{3}\lambda_{\alpha+\beta}\lambda_{3\alpha+\beta}-c_{2}\lambda_{2\alpha+\beta}^{2}=0$_,_
$\lambda_{3\alpha+\beta}=\xi(3\alpha+\beta)$_,_ $\lambda_{3\alpha+2\beta}=0$
7 | $3\alpha+2\beta$ |  | $2c_{5}\lambda_{\alpha}\lambda_{3\alpha+2\beta}-2c_{3}\lambda_{\alpha+\beta}\lambda_{3\alpha+\beta}+$
---
$c_{2}\lambda_{2\alpha+\beta}^{2}=0$_,_
$\lambda_{3\alpha+2\beta}=\xi(3\alpha+2\beta)$
8 | $\alpha,~{}\beta$ |  |  $\lambda_{\alpha}=\xi(\alpha)$_,_ $\lambda_{\beta}=\xi(\beta)$,
---
$\lambda_{\gamma}=0$_for_ $\gamma\neq\alpha,~{}\beta$
9 | $\beta,~{}2\alpha+\beta$ |  |  $\lambda_{2\alpha+\beta}=\xi(2\alpha+\beta)$_,_
---
$2c_{2}\lambda_{\beta}\lambda_{2\alpha+\beta}-c_{1}\lambda_{\alpha+\beta}^{2}=$
$2c_{2}\xi(\beta)\xi(2\alpha+\beta)$_,_
$\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0$
10 | $\beta,~{}3\alpha+\beta$ |  | $6c_{3}^{2}\lambda_{\beta}\lambda_{3\alpha+\beta}^{2}-c_{1}c_{2}\lambda_{2\alpha+\beta}^{3}=$
---
$6c_{3}^{2}\xi(\beta)\xi(3\alpha+\beta)^{2}$_,_
$2c_{3}\lambda_{\alpha+\beta}\lambda_{3\alpha+\beta}-c_{2}\lambda_{2\alpha+\beta}^{2}=0$_,_
$\lambda_{3\alpha+\beta}=\xi(3\alpha+\beta)$_,_
$\lambda_{3\alpha+2\beta}=0$
11 | $\alpha+\beta,~{}3\alpha+\beta$ |  | $2c_{3}\lambda_{\alpha+\beta}\lambda_{3\alpha+\beta}-c_{2}\lambda_{2\alpha+\beta}^{2}=$
---
$2c_{3}\xi(\alpha+\beta)\xi(3\alpha+\beta)$_,_
$\lambda_{3\alpha+\beta}=\xi(3\alpha+\beta)$_,_
$\lambda_{3\alpha+2\beta}=0$
12 | $\alpha,~{}3\alpha+2\beta$ |  | $2c_{5}\lambda_{\alpha}\lambda_{3\alpha+2\beta}-2c_{3}\lambda_{\alpha+\beta}\lambda_{3\alpha+\beta}+$
---
$c_{2}\lambda_{2\alpha+\beta}^{2}=c_{5}\xi(\alpha)\xi(3\alpha+2\beta)$_,_
$\lambda_{3\alpha+2\beta}=\xi(3\alpha+2\beta)$

Proof. The proof will be performed for all rook placements in $\Phi^{+}$
subsequently. First, assume that $|D|=1$, then
$\Omega_{D,\xi}=\mathcal{O}_{D,\xi}$. Pick a linear form
$\lambda\in\mathcal{O}_{D,\xi}$, then there exists
$x=\sum_{\gamma\in\Phi^{+}}x_{\gamma}e_{\gamma}\in\mathfrak{n}$ such that
$\lambda=\exp(x).f_{D,\xi}$. Let us proceed case-by-case. Cases 1, 2, 3 from
the table above are evident, so we start from case 4.

4\. $D=\\{\alpha+\beta\\}$. It follows immediately from the paragraph before
Proposition Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint
orbits that $\lambda_{\alpha+\beta}=\xi(\alpha+\beta)$ and
$\lambda_{2\alpha+\beta}=\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0$.
To compute $\lambda_{\alpha}$, we note that, obviously,
$(\alpha+\beta)-\alpha$ can be uniquely represented as a sum of positive
roots: $(\alpha+\beta)-\alpha=\beta$. Therefore,

$\lambda_{\alpha}=\lambda(e_{\alpha})=\xi(\alpha+\beta)e_{\alpha+\beta}^{*}(e_{\alpha}-[x,e_{\alpha}])=\xi(\alpha+\beta)e_{\alpha+\beta}^{*}(-[x_{\beta}e_{\beta},e_{\alpha}])=\xi(\alpha+\beta)c_{1}x_{\beta}.$

Similarly, $\lambda_{\beta}=-\xi(\alpha+\beta)c_{1}x_{\alpha}.$ Since
$x_{\alpha}$ and $x_{\beta}$ can be arbitrary, we obtain the required system
of equations.

5\. $D=\\{2\alpha+\beta\\}$. Here $\lambda_{2\alpha+\beta}=\xi(2\alpha+\beta)$
and $\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0$. Now,
$(2\alpha+\beta)-(\alpha+\beta)=\alpha$ is the unique representation of
$(2\alpha+\beta)-(\alpha+\beta)$ as a sum of positive roots, hence

$\lambda_{\alpha+\beta}=\lambda(e_{\alpha+\beta})=\xi(2\alpha+\beta)e_{2\alpha+\beta}^{*}(e_{\alpha+\beta}-[x,e_{\alpha+\beta}])=-\xi(2\alpha+\beta)c_{2}x_{\alpha}.$

Next, since $(2\alpha+\beta)-\beta=2\alpha$ is the unique representation of
$(2\alpha+\beta)-\beta$ as a sum of positive roots, we obtain

$\begin{split}\lambda_{\beta}&=\lambda(e_{\beta})=\xi(2\alpha+\beta)e_{2\alpha+\beta}^{*}(e_{\beta}-[x,e_{\beta}]+\frac{1}{2!}[x,[x,e_{\beta}]])=\xi(2\alpha+\beta)e_{2\alpha+\beta}^{*}(\frac{1}{2}[x_{\alpha}e_{\alpha},[x_{\alpha}e_{\alpha},e_{\beta}]])\\\
&=\frac{1}{2}\xi(2\alpha+\beta)c_{1}x_{\alpha}^{2}e_{2\alpha+\beta}^{*}([e_{\alpha},e_{\alpha+\beta}])=\frac{1}{2}\xi(2\alpha+\beta)c_{1}c_{2}x_{\alpha}^{2}.\end{split}$

Finally, since $2\alpha\notin\Phi^{+}$, we can obtain $2\alpha+\beta$ either
by adding to $\alpha$ the roots $\beta$ and $\alpha$ subsequently, or by
adding to $\alpha$ the root $\alpha+\beta$. So,

$\lambda_{\alpha}=\lambda(e_{\alpha})=\xi(2\alpha+\beta)e_{2\alpha+\beta}^{*}(e_{\alpha}-[x,e_{\alpha}]+\frac{1}{2!}[x,[x,e_{\alpha}]])=\xi(2\alpha+\beta)(c_{2}x_{\alpha+\beta}-\frac{1}{2}c_{1}c_{2}x_{\alpha}x_{\beta}).$

Thus, $\lambda_{\alpha}$ can be arbitrary, while
$2c_{2}\lambda_{2\alpha+\beta}\lambda_{\beta}=c_{1}\lambda_{\alpha+\beta}^{2}$,
as required.

6\. $D=\\{3\alpha+\beta\\}$. Arguing as above, we see that
$\lambda_{3\alpha+\beta}=\xi(3\alpha+\beta)$, $\lambda_{3\alpha+2\beta}=0$,

$\begin{split}&\lambda_{2\alpha+\beta}=-\xi(3\alpha+\beta)c_{3}x_{\alpha},~{}\lambda_{\alpha+\beta}=\frac{1}{2}\xi(3\alpha+\beta)c_{2}c_{3}x_{\alpha}^{2},~{}\lambda_{\beta}=-\frac{1}{6}\xi(3\alpha+\beta)c_{1}c_{2}c_{3}x_{\alpha}^{3},\\\
&\lambda_{\alpha}=\xi(3\alpha+\beta)(c_{3}x_{2\alpha+\beta}-\frac{1}{2}c_{2}c_{3}x_{\alpha}x_{\alpha+\beta}+\frac{1}{6}c_{1}c_{2}c_{3}x_{\alpha}^{2}x_{\beta}).\end{split}$

Now, it is clear that the equations from the table above define
$\mathcal{O}_{D,\xi}$.

7\. $D=\\{3\alpha+2\beta\\}$. Here
$\lambda_{3\alpha+2\beta}=\xi(3\alpha+2\beta)$,
$\lambda_{3\alpha+\beta}=\xi(3\alpha+2\beta)c_{4}x_{\beta},$

$\begin{split}&\lambda_{2\alpha+\beta}=\xi(3\alpha+2\beta)(-c_{5}x_{\alpha+\beta}-\frac{1}{2}c_{3}c_{4}x_{\alpha}x_{\beta}),\\\
&\lambda_{\alpha+\beta}=\xi(3\alpha+2\beta)(c_{5}x_{2\alpha+\beta}+\frac{1}{2}c_{2}c_{5}x_{\alpha}x_{\alpha+\beta}+\frac{1}{6}c_{2}c_{3}c_{4}x_{\alpha}^{2}x_{\beta}),\\\
&\lambda_{\beta}=\xi(3\alpha+2\beta)(-c_{4}x_{3\alpha+\beta}-\frac{1}{2}c_{1}c_{5}x_{\alpha}x_{2\alpha+\beta}-\frac{1}{6}c_{1}c_{2}c_{5}x_{\alpha}^{2}x_{\alpha+\beta}-\frac{1}{24}c_{1}c_{2}c_{3}c_{4}x_{\alpha}^{3}x_{\beta}),\\\
&\lambda_{\alpha}=\xi(3\alpha+2\beta)(c_{1}c_{5}x_{\beta}x_{2\alpha+\beta}-\frac{1}{2}c_{2}c_{5}x_{\alpha+\beta}^{2}+\frac{1}{24}c_{1}c_{2}c_{3}c_{4}x_{\alpha}^{2}x_{\beta}^{2}).\end{split}$

One can immediately check that $\lambda$ satisfy the required system of
equations. On the other hand, given arbitrary $\lambda_{2\alpha+\beta}$,
$\lambda_{\alpha+\beta}$, $\lambda_{\beta}$, one can put
$x_{\beta}=\dfrac{\lambda_{3\alpha+\beta}}{c_{4}\xi(3\alpha+2\beta)}$,

$\begin{split}x_{\alpha+\beta}&=-\frac{\lambda_{2\alpha+\beta}+\frac{1}{2}c_{3}c_{4}x_{\alpha}x_{\beta}\xi(3\alpha+2\beta)}{c_{5}\xi(3\alpha+2\beta)}=-\frac{\lambda_{2\alpha+\beta}+\frac{1}{2}c_{3}\lambda_{3\alpha+\beta}x_{\alpha}}{c_{5}\xi(3\alpha+2\beta)},\\\
x_{2\alpha+\beta}&=\frac{\lambda_{\alpha+\beta}-\frac{1}{2}c_{2}c_{5}x_{\alpha}x_{\alpha+\beta}\xi(3\alpha+2\beta)-\frac{1}{6}c_{2}c_{3}c_{4}x_{\alpha}^{2}x_{\beta}\xi(3\alpha+2\beta)}{c_{5}\xi(3\alpha+2\beta)}\\\
&=\frac{\lambda_{\alpha+\beta}+\frac{1}{2}c_{2}x_{\alpha}(\lambda_{2\alpha+\beta}+\frac{1}{2}c_{3}\lambda_{3\alpha+\beta}x_{\alpha})-\frac{1}{6}c_{2}c_{3}\lambda_{3\alpha+\beta}x_{\alpha}^{2}}{c_{5}\xi(3\alpha+2\beta)}\\\
&=\frac{\lambda_{\alpha+\beta}+\frac{1}{2}c_{2}x_{\alpha}\lambda_{2\alpha+\beta}+\frac{1}{12}c_{2}c_{3}\lambda_{3\alpha+\beta}x_{\alpha}^{2}}{c_{5}\xi(3\alpha+2\beta)}.\end{split}$

It is straightforward to check that, for these values of $x_{\beta}$,
$x_{\alpha+\beta}$ and $x_{2\alpha+\beta}$, one has

$\lambda_{\alpha}=\dfrac{c_{3}\lambda_{\alpha+\beta}\lambda_{3\alpha+\beta}}{c_{5}\lambda_{3\alpha+2\beta}}-\dfrac{c_{2}\lambda_{2\alpha+\beta}^{2}}{2c_{5}\lambda_{3\alpha+2\beta}}.$

Thus, $\mathcal{O}_{D,\xi}=\Omega_{D,\xi}$ is exactly the set of solutions of
the required system of equations.

Cases 8–12 can be considered uniformly. (In all these cases $|D|=2$.) In all
cases, except case 11, $D$ contains a basis root $\gamma$ ($\gamma=\alpha$ or
$\gamma=\beta$). Since the coadjoint orbit of $\xi(\gamma)e_{\gamma}^{*}$ is
$\\{\xi(\gamma)e_{\gamma}^{*}\\}$, everything is evident. Case 11 is an easy
exercise.$\square$

We are ready now to prove our first main result, Theorem Rook placements in
$G_{2}$ and $F_{4}$ and associated coadjoint orbits, which claims that, for
$\Phi=G_{2}$, $\mathfrak{n}^{*}=\bigsqcup_{D,\xi}\mathcal{O}_{D,\xi}$, where
the union is taken over all non-singular rook placements $D$ and all maps
$\xi\colon D\to\mathbb{C}^{\times}$.

Proof of Theorem Rook placements in $G_{2}$ and $F_{4}$ and associated
coadjoint orbits. Using Proposition Rook placements in $G_{2}$ and $F_{4}$ and
associated coadjoint orbits, it is straightforward to check that each
$\lambda\in\mathfrak{n}^{*}$ belongs to exactly one $\mathcal{O}_{D,\xi}$.
Namely, pick a linear form $\lambda\in\mathfrak{n}^{*}$. Then exactly one of
the following cases can occur.

  * •

$\lambda_{\alpha}=\lambda_{\beta}=\lambda_{\alpha+\beta}=\lambda_{2\alpha+\beta}=\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0$.
Then $\lambda\in\mathcal{O}_{\varnothing,\xi}$.

  * •

$\lambda_{\beta}=\lambda_{\alpha+\beta}=\lambda_{2\alpha+\beta}=\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0,\lambda_{\alpha}\neq
0$. Then $\lambda\in\mathcal{O}_{\\{\alpha\\},\xi}$.

  * •

$\lambda_{\alpha+\beta}=\lambda_{2\alpha+\beta}=\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0,\lambda_{\beta}\neq
0$. If $\lambda_{\alpha}=0$ then $\lambda\in\mathcal{O}_{\\{\beta\\},\xi}$, if
$\lambda_{\alpha}\neq 0$ then
$\lambda\in\mathcal{O}_{\\{\alpha,\beta\\},\xi}$.

  * •

$\lambda_{2\alpha+\beta}=\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0,\lambda_{\alpha+\beta}\neq
0$. Then $\lambda\in\mathcal{O}_{\\{\alpha+\beta\\},\xi}$.

  * •

$\lambda_{3\alpha+\beta}=\lambda_{3\alpha+2\beta}=0,\lambda_{2\alpha+\beta}\neq
0$. If
$\lambda_{\beta}=\dfrac{c_{1}\lambda_{\alpha+\beta}^{2}}{2c_{2}\lambda_{2\alpha+\beta}}$
then $\lambda\in\mathcal{O}_{\\{2\alpha+\beta\\},\xi}$,
otherwise$\lambda\in\mathcal{O}_{\\{\beta,2\alpha+\beta\\},\xi}$.

  * •

$\lambda_{3\alpha+2\beta}=0,\lambda_{3\alpha+\beta}\neq 0$. If
$\lambda_{\beta}=\dfrac{c_{1}c_{2}\lambda_{2\alpha+\beta}^{2}}{6c_{3}^{2}\lambda_{3\alpha+\beta}^{2}}$
and
$\lambda_{\alpha+\beta}=\dfrac{c_{2}\lambda_{2\alpha+\beta}^{2}}{2c_{3}\lambda_{3\alpha+\beta}}$
then $\lambda\in\mathcal{O}_{\\{3\alpha+\beta\\},\xi}$. If
$\lambda_{\beta}\neq\dfrac{c_{1}c_{2}\lambda_{2\alpha+\beta}^{2}}{6c_{3}^{2}\lambda_{3\alpha+\beta}^{2}}$
and
$\lambda_{\alpha+\beta}=\dfrac{c_{2}\lambda_{2\alpha+\beta}^{2}}{2c_{3}\lambda_{3\alpha+\beta}}$
then $\lambda\in\mathcal{O}_{\\{\beta,3\alpha+\beta\\},\xi}$. If
$\lambda_{\alpha+\beta}\neq\dfrac{c_{2}\lambda_{2\alpha+\beta}^{2}}{2c_{3}\lambda_{3\alpha+\beta}}$
then $\lambda\in\mathcal{O}_{\\{\alpha+\beta,3\alpha+\beta\\},\xi}$.

  * •

$\lambda_{3\alpha+2\beta}\neq 0$. If
$\lambda_{\alpha}=\dfrac{c_{3}\lambda_{\alpha+\beta}\lambda_{3\alpha+\beta}}{c_{5}\lambda_{3\alpha+2\beta}}-\dfrac{c_{2}\lambda_{2\alpha+\beta}^{2}}{2c_{5}\lambda_{3\alpha+2\beta}}$
then $\lambda\in\mathcal{O}_{\\{3\alpha+2\beta\\},\xi}$, otherwise
$\lambda\in\mathcal{O}_{\\{\alpha,3\alpha+2\beta\\},\xi}$.

The proof is complete.$\square$

Remark 3.2. i) In fact, $\mathcal{O}_{D,\xi}=\Omega_{D,\xi}$ for all $D$ (and
$\xi$), except case 11. ii) Actually, there exists exactly one singular rook
placement in $G_{2}^{+}$, namely, $D=\\{\alpha,~{}\alpha+\beta\\}$. We do not
consider this rook placement because
$\Omega_{D,\xi}=\mathcal{O}_{D,\xi}=\Omega_{\\{\alpha+\beta\\},\xi_{\alpha+\beta}}=\mathcal{O}_{\\{\alpha+\beta\\},\xi_{\alpha+\beta}}$,
where $\xi_{\alpha+\beta}={\left.\kern-1.2pt\xi\right|_{\\{\alpha+\beta\\}}}$.
iii) It follows from Proposition Rook placements in $G_{2}$ and $F_{4}$ and
associated coadjoint orbits that $\dim\mathcal{O}_{D,\xi}=|S(D)|$ does not
depend on $\xi$, as for $A_{n-1}^{+}$.

4\. Case $\Phi=F_{4}$

In this section we prove our second main result, Theorem Rook placements in
$G_{2}$ and $F_{4}$ and associated coadjoint orbits. To do this, we firstly
prove the following simple lemma. Let $D$ be a non-singular orthogonal rook
placement in $\Phi^{+}$, where $\Phi=F_{4}$, and $\xi_{1},~{}\xi_{2}\colon
D\to\mathbb{C}^{\times}$ be a map. Assume that there is the unique maximal
root $\beta_{0}$ in $D$ (with respect to the natural order on $\Phi^{+}$).

Lemma 4.1. _Let $\beta\in D\setminus\\{\beta_{0}\\}$ be such that
$\gamma\ngtr\beta$ for all $\gamma\in D\setminus\\{\beta_{0}\\}$. Assume that
$\beta_{0}-\beta=\gamma_{1}+\ldots+\gamma_{k}$ can be uniquely expressed as a
sum of positive roots $\gamma_{j}$, $1\leq j\leq k$. Further, assume that_

$\beta+\sum_{j\in J}\gamma_{j}\in\Phi^{+}$

_for each subset $J\subset\\{1,~{}\ldots,~{}k\\}$. If
$\xi_{1}(\beta)\neq\xi_{2}(\beta)$ then $\Omega_{D,\xi_{1}}$ and
$\Omega_{D,\xi_{2}}$ do not coincide._

Proof. Suppose that $\Omega_{D,\xi_{1}}=\Omega_{D,\xi_{2}}$. Then
$\xi_{1}(\beta_{0})=\xi_{2}(\beta_{0})$. Let $\xi\colon
D\to\mathbb{C}^{\times}$ be a map. Pick an element $x\in\mathfrak{n}$ and
denote $\mu=\exp{x}.f_{D,\xi}$. One has

$\begin{split}\mu(e_{\beta_{0}-\gamma_{j}})&=(\exp
x.f_{D,\xi})(e_{\beta_{0}-\gamma_{j}})=f_{D,\xi}(e_{\beta_{0}-\gamma_{j}}-[x,e_{\beta_{0}-\gamma_{j}}]+\ldots)\\\
&=a-f_{D,\xi}([x_{\gamma_{j}}e_{\gamma_{j}},e_{\beta_{0}-\gamma_{j}}])=a-x_{\gamma_{j}}f_{D,\xi}([e_{\gamma_{j}},e_{\beta_{0}-\gamma_{j}}])=a-x_{\gamma_{j}}\cdot
c_{j}\cdot\xi(\beta_{0}),\end{split}$

where $c_{j}$ is the nonzero scalar such that
$[e_{\gamma_{j}},e_{\beta_{0}-\gamma_{j}}]=c_{j}e_{\beta_{0}}$, while

$a=\begin{cases}\xi(\beta),&\text{if }\beta_{0}-\gamma_{j}=\beta,\\\
0&\text{otherwise}.\end{cases}$

Hence, all $x_{\gamma_{j}}$ are uniquely defined by $\mu$. Now, let $S_{k}$ be
the symmetric group on $k$ letters. We obtain

$\begin{split}(\exp
x.f_{D,\xi})(e_{\beta})&=f_{D,\xi}\left(e_{\beta}+(-1)^{k}\cdot\frac{1}{k!}\underbrace{[x,[x,\ldots[x,e_{\beta}]\ldots]]}_{\text{$k$
commutators}}\right)\\\
&=\xi(\beta)+f_{D,\xi}\left((-1)^{k}\cdot\frac{1}{k!}\left[\sum\limits_{j=1}^{k}x_{\gamma_{j}}e_{\gamma_{j}},\left[\sum\limits_{j=1}^{k}x_{\gamma_{j}}e_{\gamma_{j}},\ldots\left[\sum\limits_{j=1}^{k}x_{\gamma_{j}}e_{\gamma_{j}},e_{\beta}\right]\ldots\right]\right]\right)\\\
&=\xi(\beta)+(-1)^{k}\cdot\frac{1}{k!}\prod_{j=1}^{k}x_{\gamma_{j}}\cdot
f_{D,\xi}\left(\sum\limits_{\delta\in
S_{k}}[e_{\gamma_{\delta(1)}},[e_{\gamma_{\delta(2)}},\ldots[e_{\gamma_{\delta(k)}},e_{\beta}]\ldots]]\right).\end{split}$

Denote the second summand by $F$. Then $F$ is uniquely defined by $\mu$,
because $x_{\gamma_{j}}$ and $\xi(\beta_{0})$ are uniquely defined by $\mu$.
If $\Omega_{D,\xi_{1}}=\Omega_{D,\xi_{2}}$ then there exist $x_{1}$, $x_{2}$,
for which $\exp x_{1}.f_{D,\xi_{1}}=\exp x_{2}.f_{D,\xi_{2}}$. So, $(\exp
x_{1}.f_{D,\xi_{1}})(e_{\beta})=(\exp x_{2}.f_{D,\xi_{2}})(e_{\beta})$, i.e.,
$\xi_{1}(\beta)+F=\xi_{2}(\beta)+F$, hence $\xi_{1}(\beta)=\xi_{2}(\beta)$, a
contradiction.$\square$

Now, we need some general construction, which can be applied to an arbitrary
root system.

Namely, recall the notion of $\mathfrak{g}$, $\mathfrak{b}$ and
$N=\exp(\mathfrak{n})$ from the introduction. Let $\mathfrak{h}$ be the Cartan
subalgebra of $\mathfrak{g}$ such that
$\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{h}\oplus\mathfrak{n}^{-}$, where
$\mathfrak{n}^{-}$ is the nilradical of the Borel subalgebra opposite to
$\mathfrak{b}$. Let $\Phi^{-}$ be the set of negative roots, then the root
vectors $e_{\alpha}$, $\alpha\in\Phi^{-}$, form a basis of $\mathfrak{n}^{-}$.
Further, let $\alpha_{1},~{}\ldots,~{}\alpha_{n}$ be the simple roots from
$\Phi$, and $h_{\alpha_{i}}$, $1\leq i\leq n$, be a basis of $\mathfrak{h}$
such that $\\{e_{\alpha},\alpha\in\Phi\\}\cup\\{h_{\alpha_{i}},~{}1\leq i\leq
n\\}$ is a Chevalley basis of $\mathfrak{g}$.

We fix a total order $\leq_{t}$ on this basis such that
$e_{\alpha}<_{t}h_{\alpha_{i}}<_{t}e_{-\beta}$ for all
$\alpha,\beta\in\Phi^{+}$, $1\leq i\leq n$, and $e_{\alpha}<_{t}e_{\beta}$ if
$\alpha,\beta\in\Phi$ and $\alpha>\beta$. This identifies
$\mathfrak{gl}(\mathfrak{g})$ with the Lie algebra
$\mathfrak{gl}_{\dim\mathfrak{g}}(\mathbb{C})$, and
$\mathrm{ad}(\mathfrak{n})$ with a subalgebra of the Lie algebra
$\mathfrak{u}$ of all upper-triangular matrices from
$\mathfrak{gl}_{\dim\mathfrak{g}}(\mathbb{C})$ with zeroes on the diagonal.

Let $\mathrm{GL}(V)$ be the group of all invertible linear operators on a
vector space $V$. Since we fixed a basis in $\mathfrak{g}$, the group
$\mathrm{GL}(\mathfrak{g})$ is identified with the group
$\mathrm{GL}_{\dim\mathfrak{g}}(\mathbb{C})$, and
$\exp\mathrm{ad}(\mathfrak{n})\cong N$ is identified with a subgroup of the
group $U$ of all upper-triangular matrices from
$\mathrm{GL}_{\dim\mathfrak{g}}(\mathbb{C})$ with $1$’s on the diagonal.
Furthermore, using the Killing form on $\mathfrak{g}$ and the trace form on
$\mathfrak{gl}(\mathfrak{g})$, one can identify $\mathfrak{n}^{*}$ with the
space $\mathfrak{n}_{-}=\langle
e_{-\alpha},~{}\alpha\in\Phi^{+}\rangle_{\mathbb{C}}$ and $\mathfrak{u}^{*}$
with the space $\mathfrak{u}_{-}=\mathfrak{u}^{T}$, where the superscript $T$
denote the transposed matrix. Under all these identifications, it is enough to
check that the coadjoint $U$-orbits of the linear forms
$\widetilde{f}_{D,\xi_{1}}$ and $\widetilde{f}_{D,\xi_{2}}$ are distinct.
Here, given a map $\xi\colon D\to\mathbb{C}^{\times}$, we denote by
$\widetilde{f}_{D,\xi}$ the matrix

$\widetilde{f}_{D,\xi}=\left(\mathrm{ad}\left(\sum\nolimits_{\beta\in
D}\xi(\beta)e_{\beta}\right)\right)^{T}\in\mathfrak{u}_{-}\cong\mathfrak{u}^{*}.$

To do this, we will study the matrix $f=\widetilde{f}_{D,\xi}$ in more
details. The rows and the columns of matrices from
$\mathfrak{gl}(\mathfrak{g})$ are now indexed by the elements of the Chevalley
basis fixed above. Given a matrix $x$ from $\mathfrak{gl}(\mathfrak{g})$ and
two basis elements $a,~{}b$, we will denote by $x_{a,b}$ the entry of $x$
lying in the $a$th row and the $b$th column. The following proposition was
proved in [IS21]. For the reader’s convenience, we reproduce the proof here,
because our main technical tool used in the proof of Theorem Rook placements
in $G_{2}$ and $F_{4}$ and associated coadjoint orbits is based on similar
ideas.

Proposition 4.2. __[_IS21, Proposition 4.2] Let $\Phi$ be an irreducible root
system, and $D$ be a non-singular rook placement in $\Phi^{+}$. Let
$\beta_{0}$ be a root in $D$, $\xi_{1}$ and $\xi_{2}$ be maps from $D$ to
$\mathbb{C}^{\times}$ for which $\xi_{1}(\beta_{0})\neq\xi_{2}(\beta_{0})$.
Assume that there exists a simple root $\alpha_{0}\in\Delta$ satisfying
$(\alpha_{0},\beta_{0})\neq 0$ and $(\alpha_{0},\beta)=0$ for all $\beta\in D$
such that $\beta\nless\beta_{0}$. Then
$\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$._

Proof. Since

$\mathrm{ad}_{e_{\beta_{0}}}(h_{\alpha_{0}})=[e_{\beta_{0}},h_{\alpha_{0}}]=-\dfrac{2(\alpha_{0},\beta_{0})}{(\alpha_{0},\alpha_{0})}e_{\beta_{0}},$

we obtain
$f_{h_{\alpha_{0}},e_{\beta_{0}}}=-\xi(\beta_{0})\dfrac{2(\alpha_{0},\beta_{0})}{(\alpha_{0},\alpha_{0})}\neq
0$. One may assume without loss of generality that
$h_{\alpha_{0}}>_{t}h_{\alpha_{i}}$ for all $\alpha_{i}\neq\alpha_{0}$. We
claim that

$f_{h_{\alpha_{0}},e_{\alpha}}=f_{e_{-\gamma},e_{\beta_{0}}}=0\text{ for all
}e_{\alpha}<_{t}e_{\beta_{0}}\text{ and all
}e_{-\gamma},~{}\alpha,\gamma\in\Phi^{+}.$ (1)

Indeed, if $\alpha\notin D$ then, evidently,
$f_{h_{\alpha_{0}},e_{\alpha}}=0$. If $\alpha=\beta\in D$ and
$e_{\beta}<_{t}e_{\beta_{0}}$ then $\beta\nless\beta_{0}$, hence

$f_{h_{\alpha_{0}},e_{\beta}}=-\xi(\beta)\dfrac{2(\alpha_{0},\beta)}{(\alpha_{0},\alpha_{0})}=0,$

because $(\alpha_{0},\beta)=0$. On the other hand, if
$f_{e_{-\gamma},e_{\beta_{0}}}\neq 0$ for some $\gamma\in\Phi^{+}$ then
$\beta_{0}=\beta-\gamma$. This contradicts the condition $\beta_{0}\notin
S(\beta)$.

Thus, $(\widetilde{f}_{D,\xi_{1}})_{h_{\alpha_{0}},e_{\alpha}}$ and
$(\widetilde{f}_{D,\xi_{2}})_{h_{\alpha_{0}},e_{\alpha}}$ are different
nonzero scalars, and (1) is satisfied both for $f=\widetilde{f}_{D,\xi_{1}}$
and for $f=\widetilde{f}_{D,\xi_{2}}$. Now it follows immediately from the
proof of [An95, Proposition 3] (or from Remark Rook placements in $G_{2}$ and
$F_{4}$ and associated coadjoint orbits) that the coadjoint $U$-orbits of
these matrices are distinct, and, consequently,
$\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$, as required.$\square$

Our main technical tool generalizes the proposition above in the following
way. Fix an order $\\{\beta_{1},~{}\ldots,~{}\beta_{m}\\}$ on $D$ and an order
$\\{\alpha_{1},~{}\ldots,~{}\alpha_{n}\\}$ on the simple roots in $\Phi^{+}$
such that $h_{\alpha_{i}}<_{t}h_{\alpha_{j}}$ and
$e_{\beta_{i}}<_{t}e_{\beta_{j}}$ for $i<j$. Note that

$f_{h_{\alpha_{i}},e_{\beta_{j}}}=-\xi(\beta_{j})\dfrac{2(\alpha_{i},\beta_{j})}{(\alpha_{i},\alpha_{i})}.$

Given $J\in\\{1,~{}\ldots,~{}m\\}$ and $I\subset\\{1,~{}\ldots,~{}n\\}$ with
$|I|=|J|$, denote by $\Delta_{I}^{J}(\xi)$ the minor of the matrix $f$ with
the set of rows $\\{h_{\alpha_{i}},~{}i\in I\\}$ and the set of columns
$\\{e_{\beta_{j}},~{}j\in J\\}$. Furthermore, let $\widetilde{\Delta}_{I}^{J}$
be the determinant of the matrix, which $(i,j)$th element equals
$p_{i,j}=\dfrac{2(\alpha_{i},\beta_{j})}{(\alpha_{i},\alpha_{i})}$, so
that$\Delta_{I}^{J}(\xi)=\pm\prod_{j\in
J}\xi(\beta_{j})\widetilde{\Delta}_{I}^{J}$.

Proposition 4.3. _Assume that there exist an $m$-tuple
$I=(i_{1},~{}\ldots,~{}i_{m})$ such that, for all $1\leq k\leq m$,
$\widetilde{\Delta}_{I_{k}}^{J_{k}}\neq 0$, where $I_{k}=\\{i_{l},l\leq
k,~{}i_{l}\geq i_{k}\\}$ and $J_{k}=\\{j\mid i_{j}\in I_{k}\\}$. Assume also
that, for all $1\leq k\leq m$,
$\widetilde{\Delta}_{I_{l}^{\prime}}^{J_{l}^{\prime}}=0$ for
$l\notin\\{i_{1},~{}\ldots,~{}i_{k-1}\\}$, $l>i_{k}$, where
$I_{l}^{\prime}=\\{l\\}\cup\\{i_{s}\mid s<k,~{}i_{s}>l\\}$ and
$J_{l}^{\prime}=\\{k\\}\cup\\{j\mid i_{j}\in
I_{l}^{\prime}\setminus\\{l\\}\\}$. Let $\xi_{1}$ and $\xi_{2}$ be maps from
$D$ to $\mathbb{C}^{\times}$. If $\xi_{1}\neq\xi_{2}$ then
$\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$._

Proof. For simplicity, we denote $\Phi^{+}_{1}=\\{\delta\mid
e_{\delta}<_{t}e_{\beta}\text{ for all }\beta\in D\\}$ and
$\Phi^{+}_{2}=\Phi^{+}\setminus(D\cup\Phi^{+}_{1})$.

First, note that $f_{e_{-\gamma},e_{\beta_{j}}}=0$ and
$f_{h_{\alpha_{i}},e_{\delta}}=0$ for all $\gamma,~{}\delta\in\Phi^{+}_{1}$,
$\alpha_{i}\in\Delta$, $\beta_{j}\in D$. Indeed,
$f_{e_{-\gamma},e_{\beta_{j}}}$ equals the coefficient at $e_{\beta_{j}}$ in
the expression $\sum_{\beta\in D}\xi(\beta)[e_{\beta},e_{-\gamma}]$. But if
this coefficient is nonzero then $\beta-\gamma=\beta_{j}$ for some $\beta\in
D$, which contradicts the non-singularity of $D$. On the other hand,
$f_{h_{\alpha_{i}},e_{\delta}}$ equals the coefficient at $e_{\delta}$ in the
expression $\sum_{\beta\in D}\xi(\beta)[e_{\beta},h_{\alpha_{i}}]$ which is
clearly zero, because $[e_{\beta},h_{\alpha_{i}}]$ is parallel to $e_{\beta}$
for each $\beta\in D$, while $\delta\notin D$. On the picture below we draw
schematically the matrix $f$. Marks $\Phi^{+}_{1}$, $D$, $\Phi^{+}_{2}$,
$\Delta$, $\Phi^{-}$ mean that the corresponding rows and columns of the
matrix $f$ are indexed by $e_{\delta}$ for $\delta\in\Phi^{+}_{1}$,
$e_{\beta_{j}}$ for $\beta_{j}\in D$, $e_{\gamma}$ for
$\gamma\in\Phi^{+}_{2}$, $h_{\alpha_{i}}$ for $\alpha_{i}\in\Delta$,
$e_{\alpha}$ for $\alpha\in\Phi^{-}$ respectively. We replaced by big zeroes
the blocks $\Delta\times\Phi^{+}_{1}$ and $\Phi^{-}\times D$ filled in zero
entries. The minors $\Delta_{I_{l}}^{J_{l}}$ are the determinants of
submatrices of the grey block $\Delta\times D$.

The Lie algebra $\mathfrak{u}$ corresponds to the root system $A_{N-1}$, where
$N=|\Phi|+\mathrm{rk}\,{\Phi}$. Let $\widetilde{D}_{i}$ be the subset of
$A_{N-1}^{+}$ and
$\widetilde{\xi}_{i}\colon\widetilde{D}_{i}\to\mathbb{C}^{\times}$ be the map
such that $f_{i}=\widetilde{f}_{D,\xi_{i}}$ (as an element of
$\mathfrak{u}^{*}$) belongs to the basic subvariety
$\mathcal{O}_{\widetilde{D}_{i},\widetilde{\xi}_{i}}$ of $\mathfrak{u}^{*}$
defined in Section Rook placements in $G_{2}$ and $F_{4}$ and associated
coadjoint orbits, $i=1,~{}2$. Put
$\mathcal{J}=\\{e_{\alpha},~{}\alpha\in\Phi\\}\cup\\{h_{\alpha_{i}},~{}\alpha_{i}\in\Delta\\}$.
Each pair $(x,y)\in\mathcal{J}\times\mathcal{J}$ such that the $(x,y)$th entry
of $f_{i}$ lies under the diagonal corresponds to the unique root
$\varepsilon_{y}-\varepsilon_{x}\in A_{N-1}^{+}$. We denote the inverse map
from $A_{N-1}^{+}$ to $\mathcal{J}\times\mathcal{J}$ by $\tau$.

Put
$L=\\{e_{\alpha},~{}\alpha\in\Phi^{-}\\}\times\\{e_{\delta,~{}\delta\in\Phi^{+}_{1}}\\}$
and $\widetilde{D}_{i}^{L}=\tau(\widetilde{D}_{i})\cap L$. Accordingly to the
Andrè’s theory, we may assume without loss of generality that
$\widetilde{D}_{1}^{L}=\widetilde{D}_{2}^{L}$ and
$\widetilde{\xi}_{1}(\varepsilon_{y}-\varepsilon_{x})=\widetilde{\xi}_{2}(\varepsilon_{y}-\varepsilon_{x})$
for each $(x,y)\in\widetilde{D}_{i}^{L}$. (If not, then
$\mathcal{O}_{\widetilde{D}_{1},\widetilde{\xi}_{1}}\neq\mathcal{O}_{\widetilde{D}_{2},\widetilde{\xi}_{2}}$
and, consequently, $\Omega_{D,\xi_{1}}\neq\Omega_{D,\xi_{2}}$.) We will prove
that $\xi_{1}(\beta_{j})=\xi_{2}(\beta_{j})$ for all $1\leq j\leq m$ by
induction on $j$. The case $j=0$ (with $I_{0}=\varnothing$ can be considered
as an evident inductive base.)

Let $j\geq 1$. Note that each $\beta_{l}$, $1\leq l\leq m$, belongs to
$\widetilde{D}_{i}$, and the intersection of $\tau(\widetilde{D}_{i})$ with
$\\{h_{\alpha_{i}},~{}i\in\Delta\\}\times\\{e_{\beta},~{}\beta\in D\\}$ (i.e.,
with the “grey” area) coincides with
$\\{(h_{\alpha_{i_{l}}},\beta_{l})\\}_{l=1}^{m}$. (This follows immediately
from Remark Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint
orbits.) Furthermore, recall the notion of
$\Delta_{\widetilde{\alpha}}^{\widetilde{D}_{i}}(f_{i})$ for
$\widetilde{\alpha}\in A_{N-1}^{+}$ from Section Rook placements in $G_{2}$
and $F_{4}$ and associated coadjoint orbits, where $\widetilde{f}_{i}$ is
considered as an element of $\mathfrak{u}^{*}$. It follows from Remark Rook
placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits also that,
for each $l$ from $1$ to $m$ and for $i=1,~{}2$,

$\Delta_{\tau^{-1}(h_{\alpha_{i_{l}}},e_{\beta_{l}})}^{\widetilde{D}_{i}}=\pm\prod\nolimits_{\widetilde{\alpha}\in\widetilde{D}_{i},~{}\tau(\widetilde{\alpha})\in\widetilde{D}_{i}^{L}}\widetilde{\xi}_{i}(\widetilde{\alpha})\Delta_{I_{l}}^{J_{l}}=\mathrm{const}_{i,l}\widetilde{\Delta}_{I_{l}}^{J_{l}},$

where $\mathrm{const}_{i,l}$ is a scalar depending only on
$D_{i,l}=\widetilde{D}_{i}^{L}\cup\\{(h_{\alpha_{i_{s}}},~{}e_{\beta_{s}}),~{}s<l,~{}h_{\alpha_{i_{s}}}>_{t}h_{\alpha_{i_{l}}}\\}$
and on ${\left.\kern-1.2pt\widetilde{\xi}_{i}\right|_{D_{i,l}}}$. By the
inductive assumption, $D_{1,j}=D_{2,j}$ and
${\left.\kern-1.2pt\widetilde{\xi}_{2}\right|_{D_{i,j}}}={\left.\kern-1.2pt\widetilde{\xi}_{2}\right|_{D_{2,j}}}$.
We conclude that
$\widetilde{\xi}_{1}(\beta_{j})=\widetilde{\xi}_{2}(\beta_{j})$, as required.
The proof is complete.$\square$

Remark 4.4. It follows from the conditions of Proposition Rook placements in
$G_{2}$ and $F_{4}$ and associated coadjoint orbits that if such an $m$-tuple
$I$ exists then it is unique.

From now on, let $\Phi=F_{4}$. Recall that the set $\Delta$ of simple roots
can be identified with the following subset of $\mathbb{R}^{4}$:

$\Delta=\\{\alpha_{1}=\varepsilon_{2}-\varepsilon_{3},~{}\alpha_{2}=\varepsilon_{3}-\varepsilon_{4},~{}\alpha_{3}=\varepsilon_{4},~{}\alpha_{4}=\dfrac{1}{2}(\varepsilon_{1}-\varepsilon_{2}-\varepsilon_{3}-\varepsilon_{4})\\}.$

Here $\\{\varepsilon_{i}\\}_{i=1}^{4}$ is the standard basis of
$\mathbb{R}^{4}$ (with the standard inner product). The set of positive roots
is as follows:

$\begin{split}\Phi^{+}=\\{&\alpha_{1},~{}\alpha_{2},~{}\alpha_{3},~{}\alpha_{4},~{}\alpha_{1}+\alpha_{2},~{}\alpha_{2}+\alpha_{3},~{}\alpha_{2}+2\alpha_{3},~{}\alpha_{3}+\alpha_{4},~{}\alpha_{1}+\alpha_{2}+\alpha_{3},~{}\alpha_{1}+\alpha_{2}+2\alpha_{3},\\\
&\alpha_{1}+2\alpha_{2}+2\alpha_{3},~{}\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},\\\
&\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},\\\
&\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\alpha_{4},~{}\alpha_{1}+2\alpha_{2}+3\alpha_{3}+2\alpha_{4},~{}\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},\\\
&2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}\\}.\end{split}$

We will apply Proposition Rook placements in $G_{2}$ and $F_{4}$ and
associated coadjoint orbits above to the following rook placements.

Proposition 4.5. _Let $\Phi=F_{4}$, and
$D=\\{\beta_{1},~{}\ldots,~{}\beta_{m}\\}$ be one of rook placements from the
table below. Then the orders on $\Delta$ and $D$ and the sets
$I_{j}=\\{i_{1},~{}\ldots,~{}i_{j}\\}$, $1\leq j\leq m$, from the table below
satisfy the conditions of Proposition Rook placements in $G_{2}$ and $F_{4}$
and associated coadjoint orbits._

|  _Rook placement_ $D$ |  _Order on_ $\Delta$ | $i_{1},~{}\ldots,~{}i_{m}$
---|---|---|---
1 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$
$\alpha_{1},~{}\alpha_{4},~{}\alpha_{2},~{}\alpha_{3}$ | $3,~{}2,~{}1$
2 |  |  $\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}$
---
$\alpha_{2},~{}\alpha_{1},~{}\alpha_{4},~{}\alpha_{3}$ | $3,~{}2$
3 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$_,_
$\beta_{4}=\alpha_{1}+\alpha_{2}$
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{4},~{}\alpha_{3}$ | $2,~{}3,~{}4,~{}1$
4 |  |  $\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}$
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{4},~{}\alpha_{3}$ | $3,~{}4,~{}2$
5 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}$
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{3},~{}\alpha_{4}$ | $2,~{}4,~{}3$
6 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}$
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{3},~{}\alpha_{4}$ | $2,~{}3,~{}1$
7 |  |  $\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}$
---
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{3},~{}\alpha_{4}$ | $3,~{}2$
8 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}$_,_ $\beta_{4}=\alpha_{1}$
$\alpha_{2},~{}\alpha_{3},~{}\alpha_{4},~{}\alpha_{1}$ | $2,~{}3,~{}1,~{}4$
9 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}$
$\alpha_{3},~{}\alpha_{2},~{}\alpha_{1},~{}\alpha_{4}$ | $2,~{}1,~{}3$
10 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{3}+\alpha_{4}$
---
$\alpha_{4},~{}\alpha_{3},~{}\alpha_{2},~{}\alpha_{1}$ | $3,~{}2$
11 |  |  $\beta_{1}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{2}+\alpha_{3}+\alpha_{4}$,
---
$\beta_{3}=\alpha_{2}+2\alpha_{3}$
$\alpha_{1},~{}\alpha_{3},~{}\alpha_{4},~{}\alpha_{2}$ | $1,~{}4,~{}3$
12 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4}$_,_ $\beta_{2}=\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
---
$\beta_{3}=\alpha_{2}+2\alpha_{3}$
$\alpha_{1},~{}\alpha_{4},~{}\alpha_{3},~{}\alpha_{2}$ | $4,~{}2,~{}3$
13 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$
$\alpha_{3},~{}\alpha_{4},~{}\alpha_{1},~{}\alpha_{2}$ | $4,~{}3,~{}2$
14 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$,
---
$\beta_{3}=\alpha_{3}+\alpha_{4}$
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{3},~{}\alpha_{4}$ | $2,~{}4,~{}3$
15 |  |  $\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}$, $\beta_{3}=\alpha_{4}$
---
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{3},~{}\alpha_{4}$ | $2,~{}3,~{}4$
16 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}$
$\alpha_{2},~{}\alpha_{3},~{}\alpha_{1},~{}\alpha_{2}$ | $2,~{}4,~{}3$
17 |  |  $\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{2}+\alpha_{3}+\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}$
$\alpha_{4},~{}\alpha_{1},~{}\alpha_{2},~{}\alpha_{3}$ | $3,~{}4,~{}2$
18 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}$
$\alpha_{3},~{}\alpha_{2},~{}\alpha_{4},~{}\alpha_{1}$ | $2,~{}4,~{}3$
19 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{2}+2\alpha_{3}+\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}$
$\alpha_{1},~{}\alpha_{2},~{}\alpha_{4},~{}\alpha_{3}$ | $2,~{}4,~{}3$
20 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$
---
$\alpha_{2},~{}\alpha_{3},~{}\alpha_{1},~{}\alpha_{4}$ | $2,~{}1$
21 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{3}+\alpha_{4}$, $\beta_{3}=\alpha_{1}$
---
$\alpha_{3},~{}\alpha_{4},~{}\alpha_{2},~{}\alpha_{1}$ | $3,~{}2,~{}4$
22 |  |  $\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4}$_,_ $\beta_{2}=\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
---
$\beta_{3}=\alpha_{2}+2\alpha_{3}$
$\alpha_{2},~{}\alpha_{1},~{}\alpha_{3},~{}\alpha_{4}$ | $3,~{}4,~{}2$
23 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4}$,
---
$\beta_{3}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$
$\alpha_{2},~{}\alpha_{1},~{}\alpha_{3},~{}\alpha_{4}$ | $3,~{}4,~{}2$
24 |  |  $\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+2\alpha_{4}$_,_ $\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}$, $\beta_{3}=\alpha_{1}+\alpha_{2}$
---
$\alpha_{2},~{}\alpha_{1},~{}\alpha_{3},~{}\alpha_{4}$ | $4,~{}3,~{}2$

Proof. The proof is case-by-case and is completely straightforward. For
instance, consider the 17th rook placement $D$.

Clearly, the root $\beta_{3}$ (respectively, $\beta_{1}$) is orthogonal to the
unique simple root, namely, to $\alpha_{3}$ (respectively, to $\alpha_{4}$).
There are no simple roots orthogonal to $\beta_{2}$. Write out the minor of
the matrix $f$, which rows correspond to $h_{\alpha_{i}}$,
$\alpha_{i}\in\Delta$, and columns correspond to $e_{\beta_{j}}$,
$\beta_{j}\in D$. Recall the notion of $p_{i,j}$ introduced before Proposition
Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits.

$\begin{matrix}&\beta_{3}&\beta_{2}&\beta_{1}\\\
\alpha_{4}&p_{1,1}&p_{1,2}&0\\\ \alpha_{1}&p_{2,1}&p_{2,2}&p_{2,3}\\\
\alpha_{2}&p_{3,1}&p_{3,2}&p_{3,3}\\\
\alpha_{3}&0&p_{4,2}&p_{4,3}\end{matrix}$

Obviously, $\widetilde{\Delta}_{\\{4\\}}^{\\{1\\}}=|p_{4,1}|=0$, while, for
$i_{1}=3$,

$\widetilde{\Delta}_{I_{1}}^{J_{1}}=|p_{3,1}|=\dfrac{2(\alpha_{2},\beta_{3})}{(\alpha_{2},\alpha_{2})}=-1\neq
0.$

Hence, in fact we have the only possibility for $i_{1}$: $i_{1}=3$. Next, for
$i_{2}=4$, one has

$\widetilde{\Delta}_{I_{2}}^{J_{2}}=|p_{4,2}|=\dfrac{2(\alpha_{3},\beta_{2})}{(\alpha_{3},\alpha_{3})}=-1\neq
0.$

Therefore, we have to put $i_{2}=4$. Finally, for $i_{3}=2$, we obtain

$\begin{split}\widetilde{\Delta}_{I_{3}}^{J_{3}}=\begin{vmatrix}p_{2,1}&p_{2,2}&p_{2,3}\\\
p_{3,1}&p_{3,2}&p_{3,3}\\\
0&p_{4,2}&p_{4,3}\end{vmatrix}&=\dfrac{8}{(\alpha_{1},\alpha_{1})(\alpha_{2},\alpha_{2})(\alpha_{3},\alpha_{3})}\begin{vmatrix}(\alpha_{1},\beta_{3})&(\alpha_{1},\beta_{2})&(\alpha_{1},\beta_{1})\\\
(\alpha_{2},\beta_{3})&(\alpha_{2},\beta_{2})&(\alpha_{2},\beta_{1})\\\
0&(\alpha_{3},\beta_{2})&(\alpha_{3},\beta_{1})\end{vmatrix}\\\
&=2\begin{vmatrix}1&-1&1\\\ -1&1&1\\\ 0&-\frac{1}{2}&-1\end{vmatrix}=4\neq
0.\end{split}$

Thus, there is the only candidate for $i_{3}$: $i_{3}=2$. It is easy to check
that the sequence $(3,~{}4,~{}2)$ satisfy the conditions of Proposition Rook
placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits.

All other rook placements from the table above can be considered
similarly.$\square$

We are ready now to prove our second main result, Theorem Rook placements in
$G_{2}$ and $F_{4}$ and associated coadjoint orbits, which claims that, for a
non-singular orthogonal rook placement $D\subset F_{4}^{+}$ and two distinct
maps $\xi_{1}$, $\xi_{2}$ from $D$ to $\mathbb{C}^{\times}$, the associated
coadjoint orbits $\Omega_{D,\xi_{1}}$, $\Omega_{D,\xi_{2}}$ do not coincide.

Proof of Theorem Rook placements in $G_{2}$ and $F_{4}$ and associated
coadjoint orbits. The proof is more or less case-by-case. Namely, we split the
rook placements in $F_{4}^{+}$ into several “classes” and then apply Lemma
Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits and
Propositions Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint
orbits, Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits
to these classes. We start with the maximal (possibly, singular) rook
placements. It is easy to check that there are 24 maximal rook placements in
$F_{4}^{+}$:

$\begin{split}D_{1}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3},~{}\beta_{3}=\alpha_{2}+\alpha_{3},~{}\beta_{4}=\alpha_{3}\\},\\\
D_{2}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3},~{}\beta_{3}=\alpha_{2}+2\alpha_{3},~{}\beta_{4}=\alpha_{2}\\},\\\
D_{3}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3},~{}\beta_{3}=\alpha_{3},~{}\beta_{4}=\alpha_{1}\\},\\\
D_{4}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{1}+\alpha_{2}+\alpha_{3},~{}\beta_{4}=\alpha_{2}+\alpha_{3}\\},\\\
D_{5}=\\{&\beta_{1}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{2}+\alpha_{3},~{}\beta_{4}=\alpha_{3}\\},\\\
D_{6}=\\{&\beta_{1}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{2}+2\alpha_{3},~{}\beta_{4}=\alpha_{2}\\},\\\
D_{7}=\\{&\beta_{1}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{4},~{}\beta_{4}=\alpha_{2}\\},\\\
D_{8}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{4}=\alpha_{2}\\},\\\
D_{9}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\beta_{4}=\alpha_{3}+\alpha_{4}\\},\\\
D_{10}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{4}=\alpha_{4}\\},\\\
D_{11}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3},~{}\beta_{3}=\alpha_{2}+\alpha_{3},~{}\beta_{4}=\alpha_{1}+\alpha_{2}\\},\\\
D_{12}=\\{&\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{1}+\alpha_{2}+\alpha_{3},~{}\beta_{4}=\alpha_{3}\\},\\\
D_{13}=\\{&\beta_{1}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{3}+\alpha_{4},~{}\beta_{4}=\alpha_{2}+2\alpha_{3}\\},\\\
D_{14}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{4}=\alpha_{2}+2\alpha_{3}\\},\\\
D_{15}=\\{&\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{1}+\alpha_{2}+2\alpha_{3},~{}\beta_{4}=\alpha_{1}+\alpha_{2}\\},\\\
D_{16}=\\{&\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{1}+\alpha_{2}+2\alpha_{3},~{}\beta_{4}=\alpha_{3}+\alpha_{4}\\},\\\
D_{17}=\\{&\beta_{1}=\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{1}+\alpha_{2},~{}\beta_{4}=\alpha_{4}\\},\\\
D_{18}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{2}+2\alpha_{3}+\alpha_{4},\\\
&\beta_{4}=\alpha_{1}+\alpha_{2}+2\alpha_{3}\\},\\\
D_{19}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\beta_{4}=\alpha_{1}+\alpha_{2}\\},\\\
D_{20}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{1}+2\alpha_{2}+2\alpha_{3},~{}\beta_{4}=\alpha_{1}\\},\\\
D_{21}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{1}+2\alpha_{2}+2\alpha_{3},\\\
&\beta_{4}=\alpha_{2}+\alpha_{3}+\alpha_{4}\\},\\\
D_{22}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+4\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{4},~{}\beta_{4}=\alpha_{1}\\},\\\
D_{23}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{2}+2\alpha_{3}+\alpha_{4},\\\
&\beta_{4}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}\\},\\\
D_{24}=\\{&\beta_{1}=\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\alpha_{4},~{}\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{3}=\alpha_{3}+\alpha_{4},~{}\beta_{4}=\alpha_{1}\\}.\end{split}$

The first root $\beta_{1}$ is maximal among all roots in each of these rook
placements. In the rook placements

$D_{8},~{}D_{14},~{}D_{18},~{}D_{19},~{}D_{24}$

the second root $\beta_{2}$ is maximal, too. As we mentioned above, if $D$ is
a subset of $D_{i}$ containing a maximal root $\beta$ from $D_{i}$ and
$\Omega_{D,\xi_{1}}=\Omega_{D,\xi_{2}}$ then $\xi_{1}(\beta)=\xi_{2}(\beta)$
(here $\xi_{1}$, $\xi_{2}$ are maps from $D$ to $\mathbb{C}^{\times}$).

Next, it is straightforward to check that the following maximal rook
placements $D_{i}$ (together with a simple root $\alpha_{0}$ and a
distinguished root $\beta_{0}\in D_{i}$) satisfy the conditions of Proposition
Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits, except
the non-singularity of $D_{i}$:

$D_{2},~{}\beta_{0}=\beta_{3},~{}\alpha_{0}=\alpha_{1}$; |  $D_{2},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{2}$; |  $D_{3},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{2}$;
---|---|---
$D_{3},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{1}$; |  $D_{4},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{4}$; |  $D_{4},~{}\beta_{0}=\beta_{3},~{}\alpha_{0}=\alpha_{1}$;
$D_{5},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{4}$; |  $D_{5},~{}\beta_{0}=\beta_{3},~{}\alpha_{0}=\alpha_{2}$; |  $D_{6},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{4}$;
$D_{6},~{}\beta_{0}=\beta_{3},~{}\alpha_{0}=\alpha_{3}$; |  $D_{6},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{2}$; |  $D_{7},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{3}$;
$D_{7},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{2}$; |  $D_{8},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{2}$; |  $D_{11},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{1}$;
$D_{12},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{2}$; |  $D_{13},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{2}$; |  $D_{15},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{4}$;
$D_{15},~{}\beta_{0}=\beta_{3},~{}\alpha_{0}=\alpha_{3}$; |  $D_{17},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{3}$; |  $D_{20},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{4}$;
$D_{20},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{1}$; |  $D_{21},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{1}$; |  $D_{22},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{1}$;
$D_{23},~{}\beta_{0}=\beta_{2},~{}\alpha_{0}=\alpha_{1}$; |  $D_{24},~{}\beta_{0}=\beta_{4},~{}\alpha_{0}=\alpha_{1}$. |

This implies that if $D$ is a non-singular rook placement contained in one of
these maximal rook placements and containing the root $\beta_{0}$, then $D$,
$\beta_{0}$, $\alpha_{0}$ satisfy the conditions of Proposition Rook
placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits. Hence, if
$\Omega_{D,\xi_{1}}=\Omega_{D,\xi_{2}}$ then
$\xi_{1}(\beta_{0})=\xi_{2}(\beta_{0})$.

Now, let $D$ a non-singular subset of one of the rook placements
$D_{1},~{}\ldots,~{}D_{10}$. Assume that $D\subset D_{1}$. Note that
$\beta_{i}\in S(\beta_{j})$ for all $2\leq i\leq 4$ and $1\leq j<i$. Hence,
$|D|=1$, and there is nothing to prove. If $D\subset D_{2}$ contains
$\beta_{1}$ then $\beta_{2}\notin D$, because $\beta_{2}\in S(\beta_{1})$. On
the other hand, if $\beta_{1}\notin D$ and $\beta_{2}\in D$ then $\beta_{2}$
is maximal in $D$. For $\beta_{3}$, $\beta_{4}$ see the previous paragraph.
Another example: assume that $D\subset D_{3}$. If $\beta_{1}\in D$ then
$\beta_{3}\notin D$, because $\beta_{3}\in S(\beta_{1})$. If $\beta_{1}\notin
D$ and $\beta_{3}\in D$ then $D$, $\beta_{0}=\beta_{3}$,
$\alpha_{0}=\alpha_{3}$ satisfy the condition of Proposition Rook placements
in $G_{2}$ and $F_{4}$ and associated coadjoint orbits. For the roots
$\beta_{2}$, $\beta_{4}$, see the previous paragraph. All other rook
placements $D_{4},~{}\ldots,~{}D_{10}$ can be considered in a similar way.

Most of the remaining rook placements (i.e., non-singular subsets of
$D_{11},~{}\ldots,~{}D_{24}$) can be considered by completely similar
arguments. The exceptions are 24 rook placements from Proposition Rook
placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits and the 8
following rook placements:

$\begin{split}&D_{25}=\\{\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3},~{}\beta_{2}=\alpha_{2}+\alpha_{3},~{}\beta_{3}=\alpha_{1}+\alpha_{2}\\};\\\
&D_{26}=\\{\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3},~{}\beta_{2}=\alpha_{2}+\alpha_{3}\\};\\\
&D_{27}=\\{\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3},~{}\beta_{2}=\alpha_{1}+\alpha_{2}\\};\\\
&D_{28}=\\{\beta_{1}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{1}+\alpha_{2}+2\alpha_{3}\\};\\\
&D_{29}=\\{\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3},~{}\beta_{3}=\alpha_{1}\\};\\\
&D_{30}=\\{\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}\\};\\\
&D_{31}=\\{\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{1}+\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}\\};\\\
&D_{32}=\\{\beta_{1}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+2\alpha_{4},~{}\beta_{2}=\alpha_{2}+2\alpha_{3}+\alpha_{4},~{}\beta_{3}=\alpha_{1}+2\alpha_{2}+2\alpha_{3}\\}.\end{split}$

Proposition Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint
orbits completes the proof for 24 rook placements from Proposition Rook
placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits. For the
rook placements $D_{25},~{}\ldots,~{}D_{32}$, one can apply Lemma Rook
placements in $G_{2}$ and $F_{4}$ and associated coadjoint orbits with
$\beta=\beta_{2}$ or $\beta_{3}$ for $D_{25}$, $\beta=\beta_{2}$ for $D_{26}$,
$\beta=\beta_{2}$ for $D_{27}$, $\beta=\beta_{2}$ or $\beta_{3}$ for $D_{28}$,
$\beta=\beta_{2}$ for $D_{29}$, $\beta=\beta_{2}$ for $D_{30}$,
$\beta=\beta_{3}$ for $D_{31}$, $\beta=\beta_{3}$ for $D_{32}$. All other
roots from these rook placements either are maximal or satisfy the conditions
of Proposition Rook placements in $G_{2}$ and $F_{4}$ and associated coadjoint
orbits for an appropriate simple root $\alpha_{0}$, as we mentioned above.
This completes the proof. $\square$

Mikhail V. Ignatev: Samara National Research University, Ak. Pavlova 1,Samara
443011, Russia

_E-mail address_<EMAIL_ADDRESS>
Matvey A. Surkov: Samara National Research University, Ak. Pavlova 1,Samara
443011, Russia

_E-mail address_<EMAIL_ADDRESS>
##  References

  * [An95] C.A.M. Andrè. Basic sums of coadjoint orbits of the unitriangular group. J. Algebra 176 (1995), 959–1000.
  * [AN06] C.A.M Andrè. and A.M. Neto. Super-characters of finite unipotent groups of types $B_{n}$, $C_{n}$ and $D_{n}$. J. Algebra 305 (2006), 394–429.
  * [Ig09] M.V. Ignat’ev. Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups. Math. Notes 86 (2009), no. 1, 65–80; also arXiv: math.RT/0904.2841.
  * [Ig11] M.V. Ignat’ev. Orthogonal subsets of root systems and the orbit method. St. Petersburg Math. J., 22 (2011), no. 5, 777–794; arXiv: math.RT/1007.5220.
  * [Ig12] M.V. Ignatyev. Combinatorics of $B$-orbits and Bruhat–Chevalley order on involutions. Transformation Groups 17 (2012), no. 3, 747–780; arXiv: math.RT/1101.2189.
  * [Ig19] M.V. Ignatyev. Centrally generated ideals of $U(\mathfrak{n})$ in types $B$ and $D$. Transformation Groups 24 (2019), no. 4, 1067–1093; arXiv: math.RT/1907.04219.
  * [IPa09] M.V. Ignatyev, A.N. Panov. Coadjoint orbits of the group $\mathrm{UT}(7,K)$. J. Math. Sci. 156 (2009), no. 2, 292–312; arXiv: math.RT/0603649.
  * [IPe16] M.V. Ignatyev, I. Penkov. Infinite Kostant cascades and centrally generated primitive ideals of $U(\mathfrak{n})$ in types $A_{\infty}$, $C_{\infty}$. J. Algebra 447 (2016), 109–134; arXiv: math.RT/1502.05486.
  * [IS21] M.V. Ignatyev, A.A. Shevchenko. Centrally generated primitive ideals of $U(\mathfrak{n})$ for exceptional types. J. Algebra 565 (2021), 627–650; arXiv: math.RT/1907.04219.
  * [Ki62] A.A. Kirillov. Unitary representations of nilpotent Lie groups. Russian Math. Surveys 17 (1962), 53–110.
  * [Ki04] A.A. Kirillov. Lectures on the orbit method. Grad. Stud. in Math. 64, AMS, 2004.
  * [Ko12] B. Kostant. The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group, Moscow Math. J. 12 (2012), no. 3, 605–620.
  * [Ko13] B. Kostant. Center of $U(\mathfrak{n})$, cascade of orthogonal roots and a construction of Lipsman–Wolf. In: Lie groups: structure, actions and representations, Progr. in Math. 306. Birkh$\ddot{\mathrm{a}}$user, 2013, 163–174.
  * [Pa08] A.N. Panov. Involutions in $S_{n}$ and associated coadjoint orbits. J. Math. Sci. 151 (2008), 3018–3031.