$\Theta:{\mathfrak{v}}\rightarrow\left(\mathfrak{g}_{0}\oplus S^{2}V\oplus
S^{2}V^{*}\right)\times S^{2}{\mathfrak{w}}\times
S^{3}{\mathfrak{w}},\quad(x,w_{1},w_{2})\mapsto(x,w_{1}\otimes
w_{2},w_{1}^{\oplus 3}+w_{2}^{\oplus 3}).$

###### Proof.

Part (a) is just a summary of the above observation on the
$\mathfrak{S}_{3}$-action. Part (b) follows from the same argument as in
§2.3.5. ∎

The rather cumbersome notation of Lemma 4.5 is only a temporary necessity: our
main result in this section is that, when restricted to $Z$, the quotient
factors through the projection to $\mathfrak{g}_{0}\oplus S^{2}V\oplus
S^{2}V^{*}$. Using the notation of Lemma 4.4, this is summarized in the
following:

######  Theorem 4.6.

The quotient variety $\overline{\mathcal{O}_{\mathrm{min}}}/\mathfrak{S}_{4}$
is isomorphic to the closure of the image of the morphism
$\psi:\operatorname{SL}_{3}\times{\mathfrak{z}}({\mathfrak{l}})\rightarrow\mathfrak{sl}_{3}\oplus
S^{2}V\oplus S^{2}V^{*}$,

$(g,z=(\lambda_{1},\lambda_{2},\lambda_{3}))\mapsto\left(g\Delta(z)g^{-1},\sum_{i}(\lambda_{i}-\lambda_{i+1})^{2}gv_{i}\otimes
gv_{i},\sum_{i}(\lambda_{i}-\lambda_{i-1})^{2}\chi_{i}g^{-1}\otimes\chi_{i}g^{-1}\right).$

###### Proof.

The proof uses similar arguments to Lemma 4.4, but is significantly more
involved. The details will therefore be dealt with in a sequence of Lemmas.
Denote the projection ${\mathfrak{v}}\rightarrow{\mathfrak{g}}_{0}\oplus
S^{2}V\oplus S^{2}V^{*}$ by $\Theta_{0}$. Let
$A=g\Delta(z)g^{-1}\in\mathfrak{g}_{0}$ and for $i\in{\mathbb{Z}}/(3)$ let
$C_{i}=(\lambda_{i}-\lambda_{i+1})^{2}gv_{i}\otimes gv_{i}$,
$D_{i}=(\lambda_{i}-\lambda_{i-1})^{2}\chi_{i}g^{-1}\otimes\chi_{i}g^{-1}$.
Depending on the context, we consider $C_{i}$ to be an element of $S^{2}V$ or
a morphism $G_{0}\times{\mathfrak{z}}({\mathfrak{l}})\rightarrow S^{2}V$, and
similarly for $A$, $D_{i}$. In the notation of Lemma 4.4,
$\pi(g,z)=(A,z,(C_{i},D_{i})_{i\in{\mathbb{Z}}/(3)})$.

We often think of $C_{i}$ and $D_{i}$ in terms of symmetric $3\times 3$
matrices, although the action of $g\in\operatorname{SL}_{3}$ on the former is
via $g\cdot C_{i}=gC_{i}g^{T}$ and on the latter is $g\cdot
D_{i}=(g^{T})^{-1}D_{i}g^{-1}$. Since $\mathfrak{S}_{3}$ acts by
simultaneously permuting the indices in $\lambda_{i}$, $C_{i}$, $D_{i}$, the
invariants for this action can be easily described via polarisations. This is
best illustrated by considering the action of $\mathfrak{S}_{3}$, which
permutes the columns of:

$\begin{pmatrix}\lambda_{1}&\lambda_{2}&\lambda_{3}\\\ C_{1}&C_{2}&C_{3}\\\
D_{1}&D_{2}&D_{3}\end{pmatrix}.$

Then $\Theta_{0}(\pi(g,z))$ therefore equals
$(A,C_{1}+C_{2}+C_{3},D_{1}+D_{2}+D_{3})$. Let $C=C_{1}+C_{2}+C_{3}$,
$D=D_{1}+D_{2}+D_{3}$. The remaining components in $\Theta$ are:

(i) quadratic term $\sum_{i}\lambda_{i}^{2}$ and its polarisations
$\sum_{i}C_{i}\otimes C_{i}$, $\sum_{i}D_{i}\otimes D_{i}$,
$\sum_{i}\lambda_{i}C_{i}$, $\sum_{i}\lambda_{i}D_{i}$, $\sum_{i}C_{i}\otimes
D_{i}$;

(ii) cubic term $\sum_{i}\lambda_{i}^{3}$ and its polarisations
$\sum_{i}C_{i}\otimes C_{i}\otimes C_{i}$, $\sum_{i}D_{i}\otimes D_{i}\otimes
D_{i}$, $\sum_{i}\lambda_{i}^{2}C_{i}$, etc.

In order to prove Thm. 4.6 we have to show that the various terms
$\sum_{i}C_{i}\otimes C_{i}$, $\sum_{i}C_{i}\otimes D_{i}$ etc. (which we now
think of as morphisms $G_{0}\times{\mathfrak{z}}({\mathfrak{l}})\rightarrow
S^{2}(S^{2}V)$, $S^{2}V\otimes S^{2}V^{*}$ etc.) factor through
$\Theta_{0}\circ\pi$. In other words, we want to express them in terms of
$A,C,D$. We will say that each of the terms $\sum_{i}C_{i}\otimes C_{i}$ etc.
depends algebraically on $A$, $C$, and $D$. We first make the trivial
observations that $\sum_{i}\lambda_{i}^{2}={\rm Tr}\,A^{2}$ and
$\sum_{i}\lambda_{i}^{3}=3\lambda_{1}\lambda_{2}\lambda_{3}=3\,{\rm det}\,A$
depend algebraically on $A$. Next, we note that the action of
$\mathfrak{gl}_{3}$ on each factor in the tensor product induces an action by
derivations of $\mathfrak{sl}_{3}$ on the symmetric square $S^{2}V$. Then
$A\cdot C=2\,\sum_{i}\lambda_{i}C_{i}$ depends algebraically on $A$ and $C$,
and similarly for $\sum_{i}\lambda_{i}^{2}C_{i}$, $\sum_{i}\lambda_{i}D_{i}$,
$\sum_{i}\lambda_{i}^{2}D_{i}$. Using the same argument to account for
$\sum_{i}\lambda_{i}C_{i}\otimes C_{i}$, $\sum_{i}\lambda_{i}D_{i}\otimes
D_{i}$ and (letting $A$ act on $S^{2}V$ only) $\sum_{i}\lambda_{i}C_{i}\otimes
D_{i}$, it therefore suffices to prove that the quadratic and cubic terms only
involving $C_{i}$ and $D_{i}$ depend algebraically on $A$, $C$, $D$. This is
established in Lemmas 4.7, 4.8, 4.9 and 4.10, which will therefore complete
the proof. ∎

In the following lemmas, we consider various $G_{0}$-equivariant maps from
$\mathfrak{g}_{0}\oplus S^{2}V\oplus S^{2}V^{*}$ to $S^{2}V\otimes
S^{2}V^{*}$, $S^{2}(S^{2}V)$ etc. We want to check in each case that the map
sends $\Theta_{0}(\pi(g,z))$ to (a scalar multiple of) $\sum_{i}C_{i}\otimes
D_{i}$, $\sum_{i}C_{i}\otimes C_{i}$ etc. Because of the $G_{0}$-invariance,
it is enough to check this fact when $g=1$. For ease of notation, we will
therefore only consider the values of $A,C_{i},D_{i}$ when $g=1$.

######  Lemma 4.7.

The mixed term $\sum_{i}C_{i}\otimes D_{i}\in S^{2}V\otimes S^{2}V^{*}$
depends algebraically on $A$.

###### Proof.

We will construct a $G_{0}$-equivariant quadratic map from $\mathfrak{sl}_{3}$
to $S^{2}V\otimes S^{2}V^{*}$ which sends $A$ to $\sum C_{i}\otimes D_{i}$. In
terms of symmetric tensors, we have

$\sum_{i}C_{i}\otimes D_{i}=\sum_{i}\prod_{j\neq
i}(\lambda_{i}-\lambda_{j})^{2}(v_{i}\otimes
v_{i})\otimes(\chi_{i}\otimes\chi_{i}).$

There is a natural $\operatorname{GL}_{3}$-equivariant linear map
$\mathfrak{gl}_{3}\otimes\mathfrak{gl}_{3}\rightarrow S^{2}V\otimes
S^{2}V^{*}$, by identifying $\mathfrak{gl}_{3}$ with $V\otimes V^{*}$ and
projecting from $V\otimes V^{*}\otimes V\otimes V^{*}\cong(V\otimes
V)\otimes(V^{*}\otimes V^{*})$ to $S^{2}V\otimes S^{2}V^{*}$. Clearly this
induces a linear map $\rho:S^{2}(\mathfrak{gl}_{3})\rightarrow S^{2}V\otimes
S^{2}V^{*}$. We have $\left(\prod_{j\neq
i}(\lambda_{i}-\lambda_{j})\right)v_{i}\chi_{i}=\prod_{j\neq
i}(A-\lambda_{j})$ for each $i$ and hence

$\sum_{i}C_{i}\otimes D_{i}=\rho\left(\sum_{i}\left(\prod_{j\neq
i}(A-\lambda_{j})\right)\otimes\left(\prod_{j\neq
i}(A-\lambda_{j})\right)\right).$

The expression in the parentheses is a symmetric tensor which can be
considered symbolically as an element of
$R[\lambda_{1},\lambda_{2},\lambda_{3}]^{\mathfrak{S}_{3}}$, where
$R={\mathbb{C}}[A\otimes 1,1\otimes A]$ and with the action given by permuting
the $\lambda_{i}$. Hence
$R[\lambda_{1},\lambda_{2},\lambda_{3}]^{\mathfrak{S}_{3}}$ is generated over
$R$ by $\sum\lambda_{i}=0$, $\sum\lambda_{i}^{2}={\rm Tr}\,A^{2}$ and
$\lambda_{1}\lambda_{2}\lambda_{3}=\det A$. Therefore the sum in the
parentheses depends algebraically on $A$. (This can also be checked directly.)
∎

Before we prove our next result, we describe an $\operatorname{SL}_{3}$-linear
inclusion $\sigma:S^{2}V\rightarrow S^{2}(S^{2}V^{*})$ as follows. There is a
non-zero linear map $S^{2}V\otimes S^{2}V\otimes
S^{2}V\rightarrow{\mathbb{C}}$ which sends $(a_{1}\otimes
a_{2})\otimes(b_{1}\otimes b_{2})\otimes(c_{1}\otimes c_{2})$ to

$\sum_{1\leq i,j\leq
2}\det(a_{1}\;b_{i}\;c_{j})\det(a_{2}\;b_{3-i}\;c_{3-j}),$

where the $a_{i},b_{i},c_{i}$ are here considered as column vectors. Clearly
this induces a map $S^{2}V\otimes S^{2}(S^{2}V)\rightarrow{\mathbb{C}}$, and
hence induces a map $\sigma:S^{2}V\rightarrow(S^{2}(S^{2}V))^{*}\cong
S^{2}(S^{2}V^{*})$. Similarly (by means of the dual construction), we obtain a
linear map $\tau:S^{2}(S^{2}V^{*})\rightarrow S^{2}V$, which for convenience
we normalize such that $\tau\circ\sigma=3\,{\rm Id}_{S^{2}V}$.

It will be useful for us to record the effect on the elements of the standard
basis. For ease of notation, denote $v_{i_{1}}\otimes\ldots\otimes
v_{i_{r}}\in S^{r}V$ as $v_{i_{1}\ldots i_{r}}$, and similarly for elements of
$S^{r}V^{*}$. For the map $S^{2}V\rightarrow S^{2}(S^{2}V^{*})$, after
multiplying by a scalar if necessary we have, for $i,j\in{\mathbb{Z}}/(3)$
(where all tensors are symmetric):

$\sigma:v_{ij}\mapsto\chi_{i+1,j+1}\otimes\chi_{i-1,j-1}-\chi_{i+1,j-1}\otimes\chi_{j+1,i-1}$

and:

$\tau:\chi_{ij}\otimes\chi_{ll}\mapsto\left\\{\begin{array}[]{rl}0&\mbox{if
$i=l$ or $j=l$,}\\\ 2v_{l+1,l+1}&\mbox{if $i=j=l-1$,}\\\ 2v_{l-1,l-1}&\mbox{if
$i=j=l+1$,}\\\ -2v_{l-1,l+1}&\mbox{if
$\\{i,j\\}=\\{l-1,l+1\\}$,}\end{array}\right.$

while
$\tau(\chi_{il}\otimes\chi_{jl})=-\frac{1}{2}\tau(\chi_{ij}\otimes\chi_{ll})$.

######  Lemma 4.8.

The term $\sum_{i}C_{i}\otimes C_{i}$ (resp. $\sum_{i}D_{i}\otimes D_{i}$)
depends algebraically on $A$, $C$ and $D$.

###### Proof.

Using the dual maps to $\sigma$ and $\tau$, it is easy to see that there is an
isomorphism of $\operatorname{SL}_{3}$-modules: $S^{2}(S^{2}V)\cong
S^{4}V\oplus S^{2}V^{*}$. Clearly $\sum_{i}C_{i}\otimes C_{i}\in S^{4}V$, and
hence it will be enough to show that the projection of $C\otimes
C-\sum_{i}C_{i}\otimes C_{i}$ to $S^{4}V$ depends algebraically on $A$ and
$D$.

To prove our assertion we consider an $\operatorname{SL}_{3}$-equivariant
linear map $S^{2}(\mathfrak{gl}_{3})\otimes S^{2}V^{*}\rightarrow S^{4}V$ as
follows. With the aid of the map $\rho:S^{2}(\mathfrak{gl}_{3})\rightarrow
S^{2}V\otimes S^{2}V^{*}$ constructed in the proof of Lemma 4.7, there is a
sequence of linear maps:

$S^{2}(\mathfrak{gl}_{3})\otimes S^{2}V^{*}\rightarrow(S^{2}V\otimes
S^{2}V^{*})\otimes S^{2}V^{*}\rightarrow S^{2}V\otimes
S^{2}(S^{2}V^{*})\rightarrow S^{2}V\otimes S^{2}V\rightarrow S^{4}V$

where the penultimate map is $1\otimes\tau$. One now observes that the
composed linear map sends $\frac{1}{2}(A\otimes A)\otimes D$ to

$\sum_{i}(\lambda_{i-1}-\lambda_{i+1})^{2}(\lambda_{i}-\lambda_{i-1})^{2}v_{i-1,i-1}\otimes
v_{i+1,i+1}$

which is nothing but the projection of $\frac{1}{2}\left(C\otimes
C-\sum_{i}C_{i}\otimes C_{i}\right)=C_{1}\otimes C_{2}+C_{1}\otimes
C_{3}+C_{2}\otimes C_{3}$ to $S^{4}V$. This establishes the required statement
for $\sum_{i}C_{i}\otimes C_{i}$. The same argument proves our assertion
regarding $\sum_{i}D_{i}\otimes D_{i}$. ∎

######  Lemma 4.9.

The term $\sum_{i}C_{i}\otimes C_{i}\otimes D_{i}$ (resp.
$\sum_{i}C_{i}\otimes D_{i}\otimes D_{i}$) depends algebraically on $A$ and
$C$ (resp. $A$ and $D$).

###### Proof.

Let $\hat{\rho}:S^{2}V\otimes S^{2}(\mathfrak{gl}_{3})\rightarrow
S^{2}(S^{2}V)\otimes S^{2}V^{*}$ be induced by the linear map
$\rho:S^{2}(\mathfrak{gl}_{3})\rightarrow S^{2}V\otimes S^{2}V^{*}$ and the
canonical map $S^{2}V\otimes S^{2}V\rightarrow S^{2}(S^{2}V)$. Since
$C_{i}\otimes D_{i}=\rho\left(\left(\prod_{j\neq
i}(A-\lambda_{j})\right)\otimes\left(\prod_{j\neq
i}(A-\lambda_{j})\right)\right)$ we have:

$\hat{\rho}\left(\sum_{i}C_{i}\otimes\left(\prod_{j\neq
i}(A-\lambda_{j})\right)\otimes\left(\prod_{j\neq
i}(A-\lambda_{j})\right)\right)=\sum_{i}C_{i}\otimes C_{i}\otimes D_{i}.$

In this case, the expression to which we apply $\hat{\rho}$ is symmetric in
the second and third terms and can be considered symbolically as an element of
$R[C_{1},C_{2},C_{3},\lambda_{1},\lambda_{2},\lambda_{3}]^{\mathfrak{S}_{3}}$,
where $R={\mathbb{C}}[1\otimes A\otimes 1,1\otimes 1\otimes A]$ and
$\mathfrak{S}_{3}$ acts by simultaneously permuting the $C_{i}$ and the
$\lambda_{i}$. So our assertion in this case follows from the same fact about
$\mathfrak{S}_{3}$-invariants used earlier. (Alternatively we can directly
compute the expression in terms of $\sum_{i}C_{i}$,
$\sum_{i}\lambda_{i}C_{i}$, $\sum_{i}\lambda_{i}^{2}C_{i}$.) Replacing $V$ by
$V^{*}$ establishes the corresponding result for $\sum_{i}C_{i}\otimes
D_{i}\otimes D_{i}$. ∎

######  Lemma 4.10.

The tensors $\sum_{i}C_{i}\otimes C_{i}\otimes C_{i}$ and
$\sum_{i}D_{i}\otimes D_{i}\otimes D_{i}$ depend algebraically on $A$, $C$,
$D$.

###### Proof.

Similarly to Lemma 4.8, we observe that $\sum_{i}C_{i}\otimes C_{i}\otimes
C_{i}\in S^{6}V$ and that there is a direct sum decomposition
$S^{3}(S^{2}V)\cong S^{6}V\oplus U$ where $U$ is the kernel of the projection
$S^{3}(S^{2}V)\rightarrow S^{6}V$. Noting further that

$C\otimes C\otimes C-3\left(\sum_{i}C_{i}\otimes C_{i}\right)\otimes
C+2\sum_{i}C_{i}\otimes C_{i}\otimes C_{i}=6\,C_{1}\otimes C_{2}\otimes C_{3}$

(where all tensors are symmetric), it will therefore suffice to show that the
projection of $C_{1}\otimes C_{2}\otimes C_{3}$ to $S^{6}V$ depends
algebraically on $A$. We have

${\rm pr}_{S^{6}V}\left(C_{1}\otimes C_{2}\otimes
C_{3}\right)=\left(\prod_{i<j}(\lambda_{i}-\lambda_{j})^{2}\right)v_{112233}.$

To show that the right-hand side can be expressed in terms of $A$, we consider
the linear map $\mu:S^{2}(S^{2}\mathfrak{gl}_{3})\rightarrow S^{6}V$ obtained
via the composition

$S^{2}\mathfrak{gl}_{3}\otimes
S^{2}\mathfrak{gl}_{3}\rightarrow\left(S^{2}V\otimes
S^{2}V^{*}\right)\otimes\left(S^{2}V\otimes S^{2}V^{*}\right)\rightarrow
S^{2}(S^{2}V)\otimes S^{2}(S^{2}V^{*})\rightarrow S^{2}(S^{2}V)\otimes
S^{2}V\rightarrow S^{6}V$

where the penultimate map is $1\otimes\tau$. We note that the map
$S^{2}(\mathfrak{gl}_{3})\rightarrow S^{2}V\otimes S^{2}V^{*}$ sends
$e_{ii}\otimes e_{jj}$ to $v_{ij}\otimes\chi_{ij}$. It follows, similarly to
the description of $\tau(\chi_{il}\otimes\chi_{jl})$, that

$\mu:(e_{ii}\otimes e_{jj})\otimes(e_{ll}\otimes
e_{ll})\mapsto\left\\{\begin{array}[]{rl}0&\mbox{if $i=l$ or $j=l$,}\\\
2v_{112233}&\mbox{if $i=j=l\pm 1$,}\\\
-2v_{112233}&\\{i,j\\}=\\{l-1,l+1\\},\end{array}\right.$

and $\mu((e_{ii}\otimes e_{ll})\otimes(e_{jj}\otimes
e_{ll}))=-\frac{1}{2}\mu((e_{ii}\otimes e_{jj})\otimes(e_{ll}\otimes e_{ll}))$
for all $i,j,l$. A routine calculation therefore shows that

$\mu\left((\Delta\otimes\Delta)\otimes(\Delta^{2}\otimes\Delta^{2})\right)=-2(\lambda_{1}-\lambda_{2})^{2}(\lambda_{1}-\lambda_{3})^{2}(\lambda_{2}-\lambda_{3})^{2}v_{112233}$

and hence $\mu\left(-\frac{1}{2}(A\otimes A)\otimes(A^{2}\otimes
A^{2})\right)={\rm pr}_{S^{6}V}C_{1}\otimes C_{2}\otimes C_{3}$. Symmetry
establishes the remaining case $\sum_{i}D_{i}\otimes D_{i}\otimes D_{i}$. ∎

###  4.4. A transverse slice in type $F_{4}$

With the benefit of Thm. 4.6, we can appeal to some straightforward
computations in GAP to describe the singularity associated to the last
remaining minimal degeneration of special orbits in type $F_{4}$. Now that
Thm. 4.6 has been established, we can switch notation. Thus let $G$ be a
simple group of type $F_{4}$, let $\mathfrak{g}=\operatorname{Lie}(G)$ with a
fixed Chevalley basis and let $\\{h,e,f\\}$ be an $\mathfrak{sl}_{2}$-triple
with $f$ lying in the orbit $\mathcal{O}^{\prime}=\mathcal{O}_{A_{2}}$. Let
${\mathcal{S}}=f+\mathfrak{g}^{e}$ be the Slodowy slice at $f$ and let
$\mathcal{O}$ be the nilpotent orbit with label $F_{4}(a_{3})$.

######  Theorem 4.11.

With the above notation, ${\mathcal{S}}\cap\overline{\mathcal{O}}$ is
isomorphic to
$\overline{\mathcal{O}_{\mathrm{min}}(\mathfrak{so}_{8})}/\mathfrak{S}_{4}$
(in shorter form, $d_{4}/\mathfrak{S}_{4}$).

###### Proof.

This follows from Thm. 4.6 and a computation in GAP. With the standard
numbering of the roots, we can assume that $f=2f_{\alpha_{1}}+2f_{\alpha_{2}}$
and $e=e_{\alpha_{1}}+e_{\alpha_{2}}$. The reductive part $C:=G^{e}\cap G^{h}$
of the centralizer of $e$ has connected component isomorphic to
$\operatorname{SL}_{3}$. The standard maximal torus of $G$ contains a maximal
torus of $C$, and
$\\{\alpha_{4},\alpha_{1}+2\alpha_{2}+3\alpha_{3}+\alpha_{4}\\}=:\\{\beta_{1},\beta_{2}\\}$
is a basis of simple roots of $C^{\circ}$. Let $V$ be the natural module for
$C^{\circ}=\operatorname{SL}_{3}$. The nilradical of ${\mathfrak{g}}^{e}$ is
isomorphic as a $C$-module to $S^{2}V\oplus S^{2}V^{*}\oplus{\mathbb{C}}^{2}$,
where the trivial component is spanned by $e$ and $e_{\alpha_{1}+\alpha_{2}}$.
By abuse of notation, identify $\\{v_{i}\otimes v_{j}:i\leq j\\}$ and
$\\{\chi_{i}\otimes\chi_{j}:i\leq j\\}$ with bases for these copies of
$S^{2}V$ and $S^{2}V^{*}$. Let
$\mathfrak{c}=\mathfrak{g}^{e}\cap\mathfrak{g}^{h}=\operatorname{Lie}(C^{\circ})$.
Let $\lambda_{1},\lambda_{2},\lambda_{3}\in{\mathbb{C}}$ be distinct such that
$\sum_{i}\lambda_{i}=0$, and let
$\Delta=\lambda_{1}\beta_{1}^{\vee}+(\lambda_{1}+\lambda_{2})\beta_{2}^{\vee}$,
which identifies with $\Delta(\lambda_{1},\lambda_{2},\lambda_{3})$ as an
element of $\mathfrak{c}=\mathfrak{sl}_{3}$. Now we observe computationally
that for any such $\lambda_{1},\lambda_{2},\lambda_{3}$, the following element
of $\mathfrak{g}$:

$f+\Delta+\frac{1}{2}\sum_{i=1}^{3}(\lambda_{i}-\lambda_{i+1})^{2}v_{i}\otimes
v_{i}+\frac{1}{2}\sum_{i=1}^{3}(\lambda_{i}-\lambda_{i-1})^{2}\chi_{i}\otimes\chi_{i}-\frac{1}{4}\sum_{i=1}^{3}\lambda_{i}^{2}e+\frac{5}{2}\lambda_{1}\lambda_{2}\lambda_{3}e_{\alpha_{1}+\alpha_{2}}$

belongs to $\overline{\mathcal{O}}$. It therefore follows, by Thm. 4.6 and
projecting onto $\mathfrak{c}\oplus S^{2}V\oplus S^{2}V^{*}$, that there is a
subvariety of ${\mathcal{S}}\cap\overline{\mathcal{O}}$ which is isomorphic to
$\overline{\mathcal{O}_{\mathrm{min}}(\mathfrak{so}_{8})}/\mathfrak{S}_{4}$.
By dimensions and unibranchness, this subvariety must equal the whole
intersection, and we are done. ∎

Note that one consequence of Thm. 4.11 is that ${\rm
Sing}(\mathcal{O},\mathcal{O}^{\prime})$ is equal to the smooth equivalence
class of ${\mathbb{C}}^{6}/\mathfrak{S}_{4}$. In the next section we establish
that the corresponding Slodowy slice singularity is isomorphic to
${\mathbb{C}}^{6}/\mathfrak{S}_{4}$ (via a computational proof).

######  Remark 4.12.

a) All of the above arguments can be adapted to positive characteristic $p$,
along similar lines to Remarks 2.11(ii) and 3.14. If $p>3$ then the arguments
in §4.2 go through verbatim. For the subsequent section, we once more need to
ensure that all relevant Weyl modules are irreducible and that any sums of
such modules remain direct in characteristic $p$. This issue occurs:

$\bullet$ in Lemma 4.8, where the projection from $S^{2}(S^{2}V^{*})$ to
$S^{4}V$ is used. The modules $S^{4}V$ and $S^{2}V^{*}$ are irreducible in
arbitrary characteristic.

$\bullet$ in Lemma 4.10, the inclusion of $S^{6}V$ as a direct summand in
$S^{3}(S^{2}V)$ is used. The complement is
$\Delta(2\varpi_{1}+2\varpi_{2})\oplus{\mathbb{C}}$. The Weyl modules with
highest weights $6\varpi_{1}$ and $2\varpi_{1}+2\varpi_{2}$ remain irreducible
in characteristic $>5$.

Observing the representatives found in the Slodowy slice in the proof, it
therefore follows that the conclusion of Thm. 4.11 holds (at least) in
characteristic $>5$.

b) Our description of ${\mathcal{S}}\cap\overline{\mathcal{O}}$ in Thm. 4.11
has a parallel in type $E_{8}$ which we do not fully understand: in that case,
a slice from $\mathcal{O}_{A_{4}}$ to $\mathcal{O}_{E_{8}(a_{7})}$ has an open
subset which is a fibre bundle (with fibres of dimension 4) over the regular
semisimple elements of $\mathfrak{sl}_{5}$. Thm. 5.1 and Thm. 5.3 can be
deduced (with some effort in the latter case) from these fibre bundle
descriptions.

c) The component group of $C$ in Thm. 4.11 is $\mathfrak{S}_{2}$. It follows
that $C$ acts on $d_{4}/\mathfrak{S}_{4}$ as the full automorphism group (i.e.
the $\mathfrak{S}_{4}$-fixed point subgroup in the adjoint group of type
$D_{4}$).

##  5\. Computational results

In §§2-4, we described various quotients $\overline{\mathcal{O}}/\Gamma$ where
$\mathcal{O}$ was (usually) a nilpotent orbit in a Lie algebra
$\mathfrak{g}={\rm Lie}(G)$, and $\Gamma$ was a finite group of automorphisms
of $G$. We exploited as much as possible the action of the group
$G_{0}:=G^{\Gamma}$ on the quotient. This line of attack is powerless when the
$G_{0}$-orbits have large codimension. For example, we have no hope of
generalizing Lemma 3.8 to the action of $\mathfrak{S}_{r}$ on
${\mathfrak{h}}_{r-1}\oplus{\mathfrak{h}}_{r-1}^{*}$ when $r\geq 4$: in this
case $\dim G_{0}=3$ is too small compared to the dimension $2(r-1)\geq 6$ of
the quotient. However, the cases $r=4$ and $r=5$ are of particular interest to
us, since we have asserted (and will now prove) that they are the Slodowy
slice singularities associated to the degenerations
$\mathcal{O}_{F_{4}(a_{3})}>\mathcal{O}_{A_{2}+\tilde{A}_{1}}$ in type $F_{4}$
and $\mathcal{O}_{E_{8}(a_{7})}>\mathcal{O}_{A_{4}+A_{3}}$ in type $E_{8}$.
Note that $\mathcal{O}_{F_{4}(a_{3})}$ (resp. $\mathcal{O}_{E_{8}(a_{7})}$) is
special with component group $\mathfrak{S}_{4}$ (resp. $\mathfrak{S}_{5}$),
and the minimal orbit in the special piece is
$\mathcal{O}_{A_{2}+\tilde{A}_{1}}$ (resp. $\mathcal{O}_{A_{4}+A_{3}}$). Hence
these results are a crucial part of our Main Theorem. In the third subsection
we apply a similar computational approach to prove that the Slodowy slice
singularity for the degeneration
$\mathcal{O}_{E_{8}(a_{6})}>\mathcal{O}_{E_{8}(b_{6})}$ is
$a_{2}/\mathfrak{S}_{4}$, for an appropriate action of $\mathfrak{S}_{4}$.
There is one orbit intermediate between $\mathcal{O}_{E_{8}(a_{6})}$ and
$\mathcal{O}_{E_{8}(b_{6})}$, with Bala-Carter label $D_{7}(a_{1})$. One
corollary of this rather surprising result is that the singularity associated
to the minimal degeneration
$\mathcal{O}_{D_{7}(a_{1})}>\mathcal{O}_{E_{8}(b_{6})}$ is the non-normal
surface singularity labelled $\mu$ in [FJLS17].

###  5.1. Structure of
$({\mathfrak{h}}_{n-1}\oplus{\mathfrak{h}}_{n-1}^{*})/\mathfrak{S}_{n}$

In the next two subsections we consider the invariants for $\mathfrak{S}_{n}$
(with $n=4,5$) acting on a sum of two copies of the reflection representation
${\mathfrak{h}}_{n-1}$. We note that
${\mathfrak{h}}_{n-1}^{*}\cong{\mathfrak{h}}_{n-1}$, so this is the quotient
singularity appearing in the statement of our Main Theorem. It is well known
that the invariant polynomials on a single copy of ${\mathfrak{h}}_{n-1}$ form
a polynomial ring on $(n-1)$ generators, of degrees $2,3,\ldots,n$. The
generators can be easily understood by identifying ${\mathfrak{h}}_{n-1}$ with
the traceless diagonal $n\times n$ matrices $x$, where $\mathfrak{S}_{n}$ acts
by permuting the entries. Then the functions $x\mapsto{\rm tr}(x^{i})$, $2\leq
i\leq n$ form a system of (free) generators.

It is also well known that the invariant polynomials on a finite sum of copies
of ${\mathfrak{h}}_{n-1}$ are generated by polarisations of these invariants,
as we explained in §2.3.5. For pairs $(x,y)$ of traceless diagonal $n\times n$
matrices, the polarisations are the functions $g_{ij}:(x,y)\mapsto{\rm
tr}(x^{i}y^{j})$ for $2\leq i+j\leq n$. Relations between these generators are
known [DP12], but appear to be rather complicated for $n\geq 4$. We could not
find a result in the literature describing the number of generators of the
ideal of relations; indeed, the problem may be open (see [Hai04, Problem
2.12]). We are specifically interested in a sum of two copies of
${\mathfrak{h}}_{n-1}$ when $n=4$ or $n=5$.

(i) For $n=4$ there are twelve generators, with three generators of degree
$2$, four of degree $3$ and five of degree $4$. We used GAP to find fifteen
relations: three of degree $6$ and six each of degrees $7$ and $8$. We were
able to verify in Magma that this ideal is radical (of height $6$), so this is
a complete set.

(ii) For $n=5$ we add six more generators, of degree $5$. We found thirty five
relations: there are $4$ (resp. $9$, $12$, $10$) of degree $7$ (resp. $8$,
$9$, $10$). We have not been able to confirm that this is a complete set.

(iii) For $n=6$, there are twenty five generators in total. We used GAP to
find seventy relations: there are $5$ (resp. $12$, $18$, $20$, $15$) of deree
$8$ (resp. $9$, $10$, $11$, $12$). We cannot confirm that this is a complete
set.

On the basis of these results (and those for $n=2,3$), we conjecture that for
general $n$ there are $\begin{pmatrix}n+2\\\ 4\end{pmatrix}$ relations in
total, and that $\tfrac{1}{2}j(j+1)(n-j)$ of them are of degree $n+j+1$, for
$1\leq j\leq n$.

###  5.2. The exceptional special slice singularity in $F_{4}$

Let $\mathfrak{g}$ be a simple Lie algebra of type $F_{4}$, let
$\mathcal{O}^{\prime}$ (resp. $\mathcal{O}$) be the nilpotent orbit in
$\mathfrak{g}$ with Bala-Carter label $A_{2}+\tilde{A}_{1}$ (resp.
$F_{4}(a_{3})$), let $\\{h,e,f\\}$ be an $\mathfrak{sl}_{2}$-triple with
$e\in\mathcal{O}^{\prime}$ and let ${\mathcal{S}}=f+\mathfrak{g}^{e}$ be the
Slodowy slice. Our goal is to show that the intersection
${\mathcal{S}}\cap\overline{\mathcal{O}}$ is isomorphic to
$({\mathfrak{h}}_{3}\oplus{\mathfrak{h}}_{3}^{*})/\mathfrak{S}_{4}$. Let
$\rho:{\mathfrak{g}}\rightarrow\mathfrak{so}_{26}$ be the minimal faithful
representation of $\mathfrak{g}$.

######  Theorem 5.1.

Let $\mathfrak{g},\mathcal{O}^{\prime},\mathcal{O},{\mathcal{S}}$ be as above.
Then ${\mathcal{S}}\cap\overline{\mathcal{O}}$ is isomorphic to
$({\mathfrak{h}}_{3}\oplus{\mathfrak{h}}_{3}^{*})/\mathfrak{S}_{4}$.

###### Proof.

It is easy to check that $x$ belongs to $\overline{\mathcal{O}}$ if and only
if $\rho(x)^{5}=0$. (That this is a sufficient condition follows from
comparing Jordan block sizes on $\mathfrak{g}$ and on the Lie algebra of type
$E_{6}$: the orbits with labels $B_{3}$ and $C_{3}$ have blocks on $\rho$ of
size $7$ and $9$ respectively.) Let $\\{h,e,f\\}$ be an
$\mathfrak{sl}_{2}$-triple with $e\in\mathcal{O}^{\prime}$. The centralizer
${\mathfrak{g}}^{e}$ has dimension $18$, and decomposes over the reductive
centralizer $\mathfrak{c}$ as $V(2)\oplus V(3)\oplus V(4)\oplus W$, where $W$
is a reducible representation for $\mathfrak{c}$ of dimension six. (Note that
$\mathfrak{c}$ is the copy of $V(2)$.) Let $f+z\in{\mathcal{S}}$, where
$z\in\mathfrak{g}^{e}$; we first observed that the conditions ${\rm
tr}(\rho((f+z)^{2}))=0$ and $\rho(f+z)^{5}=0$ imply certain conditions of the
form $(\chi+p)(z)=0$, where $\chi\in W^{*}$ and $p$ is a polynomial in
$V(2)+V(3)+V(4)$. In particular, we were able to construct an explicit
subvariety $Y$, which contains ${\mathcal{S}}\cap\overline{\mathcal{O}}$ and
is isomorphic to $V(2)\oplus V(3)\oplus V(4)$. We could then check that the
remaining conditions $\rho((f+z)^{5})$, when restricted to $Y$, give precisely
the fifteen relations satisfied by the generators of
${\mathbb{C}}[{\mathfrak{h}}_{3}\oplus{\mathfrak{h}}_{3}^{*}]^{\mathfrak{S}_{4}}$.
The statement in the theorem follows.

Our proof turns out to be more transparent in reverse. Thus we can use the
above to construct a morphism from
$({\mathfrak{h}}_{3}\oplus{\mathfrak{h}}_{3}^{*})/\mathfrak{S}_{4}$ to the
intersection ${\mathcal{S}}\cap\overline{\mathcal{O}}$. Since all of our
generating invariants $g_{ij}$ appear, this map is a closed immersion. By
general results on branching and dimension, the image equals the intersection
${\mathcal{S}}\cap\overline{\mathcal{O}}$. One can verify that the image lies
in $\overline{\mathcal{O}}$ by checking the condition $\rho(x)^{5}=0$. ∎

######  Remark 5.2.

a) It is possible to argue from Thm. 4.11 that
${\mathcal{S}}\cap\overline{\mathcal{O}}$ is isomorphic to
${\mathbb{C}}^{6}/\mathfrak{S}_{4}$. (A smooth equivalence of singularities
follows more or less immediately; our result establishes an isomorphism as
varieties.) We have included the above computational proof because it helps to
illustrate the general approach, which we will also employ in the following
subsection.

b) The symplectic leaves in
$({\mathfrak{h}}_{3}\oplus{\mathfrak{h}}_{3}^{*})/\mathfrak{S}_{4}$ are in
one-to-one correspondence with the parabolic subgroups of $\mathfrak{S}_{4}$
up to conjugacy. This is summarized in the following diagram:

$\textstyle{F_{4}(a_{3})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{3}(a_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\tilde{A}_{2}+A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{2}\mathfrak{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}+\tilde{A}_{1}}$$\textstyle{\mathfrak{S}_{4}}$

###  5.3. The exceptional special slice singularity in $E_{8}$

The notation is as in the previous subsection, except that $\mathfrak{g}$ is
simple of type $E_{8}$ and $\mathcal{O}$ has label $E_{8}(a_{7})$ and
$\mathcal{O}^{\prime}$ has label $A_{4}+A_{3}$. We used a computer to show
that ${\mathcal{S}}\cap\overline{\mathcal{O}}$ has the form predicted by the
Main Theorem.

######  Theorem 5.3.

The slice ${\mathcal{S}}\cap\overline{\mathcal{O}}$ is isomorphic to
$({\mathfrak{h}}_{4}\oplus{\mathfrak{h}}_{4}^{*})/\mathfrak{S}_{5}$.

###### Proof.

The weighted Dynkin diagram of $\mathcal{O}$ is nonzero only on the fifth
simple root, where its value is $2$. The highest root has coefficient $5$ on
this simple root, which implies that $10$ is the largest eigenvalue for
$\operatorname{ad}(h)$, where $h$ is the semisimple element of the
$\operatorname{\mathfrak{sl}_{2}}2$-subalgebra for $\mathcal{O}$. It follows
that $({\rm ad}\,x)^{11}=0$ for all $x$ in the closure of $\mathcal{O}$. There
are two orbits, of type $D_{5}+A_{1}$ and $A_{6}$, which contain $\mathcal{O}$
in their closure and have a minimal degeneration to $\mathcal{O}$. Both of
these have a largest eigenvalue for their $\operatorname{ad}(h)$ that exceeds
$11$ (namely, $15$ and $13$, respectively). It follows that the zero set of
the equations $({\rm ad}\,x)^{11}$ for $x\in\mathfrak{g}$ is exactly
$\overline{\mathcal{O}}$.

The centralizer $\mathfrak{z}$ in $\mathfrak{g}$ of an element in
$\mathcal{O}^{\prime}$ has dimension $48$. The reductive part $\mathfrak{c}$
of $\mathfrak{z}$ is isomorphic to $\operatorname{\mathfrak{sl}_{2}}2$. Let
$V(i)$ denote the irreducible representation of
$\operatorname{\mathfrak{sl}_{2}}2$ of highest weight $i$. As a
$\mathfrak{c}$-module, $\mathfrak{z}$ decomposes as $V(2)\oplus V(3)\oplus
V(4)\oplus V(5)\oplus W$, where $W$ is a $30$-dimensional reducible
$\operatorname{\mathfrak{sl}_{2}}2$-representation. We found in Magma a basis
for $\mathfrak{z}$ that simultaneously respected this
$\operatorname{\mathfrak{sl}_{2}}2$-decomposition and the Kazhdan grading.
(For our calculations in GAP, we used the basis constructed in [LT11].) Then
by restricting the generic matrix $({\rm ad}\,x)^{11}$ to $\mathcal{S}$ using
the dual basis, we located several equations that contained linear terms in
the dual basis, allowing us to eliminate those variables. After eliminating
those variables (also using the condition ${\rm tr}(({\rm ad}\,x)^{2})=0$), we
again found more variables we could eliminate. Continuing in this way, we were
able to eliminate all $30$ variables coming from $W$.

We thus constructed a subvariety $Y$ of ${\mathcal{S}}$, which is isomorphic
to $V(2)\oplus V(3)\oplus V(4)\oplus V(5)$ and which contains
${\mathcal{S}}\cap\overline{\mathcal{O}}$. Because we do not know that we have
obtained a complete set of relations for
$({\mathfrak{h}}_{4}\oplus{\mathfrak{h}}_{4}^{*})/\mathfrak{S}_{5}$, we could
not repeat the next step in the proof of Thm. 5.1. Instead, we used some
additional equations in $({\rm ad}\,x)^{11}$ to parametrize an $8$-dimensional
${\mathbb{G}}_{m}$-stable subvariety $X$ of $Y$ that could potentially lie in
the zero set of $({\rm ad}\,x)^{11}$ and be isomorphic to
$({\mathfrak{h}}_{4}\oplus{\mathfrak{h}}_{4}^{*})/\mathfrak{S}_{5}$ and also
respect the action of $\mathfrak{c}$. However, we could not easily show that
$X\subset\overline{\mathcal{O}}$. We have two different methods to accomplish
this fact. One way was to check the condition $({\rm ad}\,x)^{11}=0$ in GAP,
column by column, for $x\in X$. This was very time consuming, requiring a few
hours for each column. A faster way was to pick a small set of the matrix
equations and show that they vanish on $X$. Then we checked that the zero set
of this small set of equations on $\mathcal{S}$ at a regular point of $X$ has
dimension $8$. (We ported the small set of equations used in the proof to
Singular and found the ideal they generate has a minimal set of $35$
generators, and that the degrees match those of the relations for
${\mathbb{C}}[{\mathfrak{h}}_{4}\oplus{\mathfrak{h}}_{4}^{*}]^{{\mathfrak{S}}_{5}}$.)
Since the full ideal of equations on $\mathcal{S}$ has zero set of dimension
$8$ and is irreducible since $\overline{\mathcal{O}}$ is unibranch at
$\mathcal{O}^{\prime}$, we know that the zero set must be exactly $X$,
completing the proof. ∎

######  Remark 5.4.

a) Theorems 5.1 and 5.3 give new proofs of the appearance of the non-normal
singularity $m$ for the degenerations
$(A_{2}+\tilde{A}_{1},\tilde{A}_{2}+A_{1})$ in $F_{4}$ and
$(A_{4}+A_{3},A_{5}+A_{1})$ and $(A_{4}+A_{3},D_{5}(a_{1})+A_{1})$ in $E_{8}$.
For example, the first singularity is given by the image in the quotient of
all points with stabilizer equal to $\mathfrak{S}_{3}\subset\mathfrak{S}_{4}$.
This corresponds to a pair of diagonal matrices of the form: $(x,y)=({\rm
diag}\,(s,s,s,-3s),{\rm diag}\,(t,t,t,-3t))$. It is easy to see that the
values of the invariants ${\rm tr}(x^{i}y^{j})$ induce functions generating
the ring ${\mathbb{C}}[s^{2},st,t^{2},s^{3},s^{2}t,st^{2},t^{3}]$.

b) Similarly to the $F_{4}$ case, we have a poset equivalence between the
orbits in ${\mathcal{P}}(\mathcal{O})$ and the parabolic subgroups of
$\mathfrak{S}_{5}$ up to conjugacy:

$\textstyle{E_{8}(a_{7})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{7}(a_{5})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{6}(a_{3})+A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{6}(a_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{2}\mathfrak{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{5}+A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{5}(a_{1})+A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{3}\mathfrak{S}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{S}_{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{4}+A_{3}}$$\textstyle{\mathfrak{S}_{5}}$

###  5.4. An $\mathfrak{S}_{4}$-quotient from outer space

In this subsection, let $\mathfrak{g}=\mathfrak{sl}_{3}$. Recall that $a_{2}$
denotes the closure of the minimal nilpotent orbit in $\mathfrak{g}$. There
are two natural embeddings of $\mathfrak{S}_{4}$ in the automorphism group of
$\mathfrak{g}$. The first of these embeds $\mathfrak{S}_{4}$ in
$\operatorname{GL}_{3}$, hence in the inner automorphism group of
$\mathfrak{sl}_{3}$, via the $3$-dimensional reflection representation. (It is
easy to see that twisting by the sign representation has no impact on the
inclusion in ${\rm Aut}(\mathfrak{sl}_{3})$.) Concretely, we consider the
(unique) Klein four subgroup $\Gamma_{4}$ of $\operatorname{SL}_{3}$ contained
in the diagonal maximal torus. The Weyl group $\mathfrak{S}_{3}$ acts on
$\Gamma_{4}$ via the permutation action on the three non-identity elements.
Choosing the permutation matrices as representatives of the Weyl group, we
obtain a subgroup of $\operatorname{GL}_{3}$ isomorphic to
$\mathfrak{S}_{3}\cong\mathfrak{S}_{4}/\Gamma_{4}$, which together with
$\Gamma_{4}$ generates a subgroup of the inner automorphism group isomorphic
to $\mathfrak{S}_{4}$. For short, we call this inner $\mathfrak{S}_{4}$.

All elements of $\mathfrak{S}_{4}$ in the above construction belong to ${\rm
O}_{3}(\mathbb{C})$ and hence commute with the outer automorphism
$\gamma:g\mapsto g^{-T}$. Replacing each odd permutation of inner
$\mathfrak{S}_{4}$ by its composition with $\gamma$, we obtain a different
subgroup of ${\rm Aut}(\mathfrak{g})$ which we call outer $\mathfrak{S}_{4}$.
Let us fix this action of $\mathfrak{S}_{4}$ on $\mathfrak{g}$ (and hence on
$a_{2}$). Let $\mathcal{O}$ (resp. $\mathcal{O}^{\prime}$) be the nilpotent
orbit with label $E_{8}(a_{6})$ (resp. $E_{8}(b_{6})$) and let ${\mathcal{S}}$
be the Slodowy slice at an element of $\mathcal{O}^{\prime}$. We will show
that ${\mathcal{S}}\cap\overline{\mathcal{O}}$ is isomorphic to
$a_{2}/{\mathfrak{S}}_{4}$.

The action on $a_{2}$ is not free: each $4$-cycle has a 2-dimensional fixed
point subset, which is equal to the set of nilpotent elements in a proper Levi
subalgebra. This can be seen by a direct calculation, using the much easier
fact that each element of $\Gamma_{4}$ fixes an $\mathfrak{sl}_{2}$. Note
however that transpositions act freely on the minimal nilpotent orbit. (This
contrasts with inner $\mathfrak{S}_{4}$, where the nilpotent elements in a
proper Levi subalgebra are fixed by a pair of commuting transpositions, with a
$4$-cycle acting by an involution of $\mathfrak{sl}_{2}$.)

####  5.4.1. Generators and relations for $a_{2}/\mathfrak{S}_{4}$

Our first step is to determine the generators and relations of the coordinate
ring of $a_{2}/{\mathfrak{S}}_{4}$. It will be useful to consider the larger
set $X$ of elements of $\mathfrak{g}$ of rank at most 1, which contains
$a_{2}$ as a divisor. The elements of $X$ can be written in the form $xy^{T}$
where $x,y\in{\mathbb{C}}^{3}$. Hence the coordinate ring of $X$ can be
identified with $\mathbb{C}[x_{i}y_{j}:1\leq i,j\leq
3]\subset\mathbb{C}[x_{1},\ldots,y_{3}]$. With this description, the ideal of
$a_{2}$ is principal, generated by $z:=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}$. It
will be convenient to write the indices $i,j$ modulo $3$. Note that each non-
identity element of $\Gamma_{4}$ fixes the diagonal entries $x_{i}y_{i}$ and
one of the pairs $x_{i}y_{i+1},x_{i+1}y_{i}$, and acts as $-1$ on all other
$x_{i}y_{j}$. The same argument as Lemma 4.1 therefore gives us:

######  Lemma 5.5.

a) The $\Gamma_{4}$-invariants on ${\mathbb{C}}[X]$ are generated by:

$\\{b_{i}=x_{i}y_{i},c_{i}=x_{i+1}^{2}y_{i-1}^{2},d_{i}=x_{i-1}^{2}y_{i+1}^{2}:i\in{\mathbb{Z}}/(3)\\}.$

b) Let $\lambda_{i}=b_{i+1}-b_{i-1}$ for $i\in{\mathbb{Z}}/(3)$. (In
particular, $\sum_{i}\lambda_{i}=0$.) The action of
$\mathfrak{S}_{4}/\Gamma_{4}\cong\mathfrak{S}_{3}$ on
${\mathbb{C}}[X]^{\Gamma_{4}}$ is as follows:

$\bullet$ $z=b_{1}+b_{2}+b_{3}$ spans a copy of the sign representation;

$\bullet$ $\mathfrak{S}_{3}$ acts by permuting the columns of the matrix:

$\begin{pmatrix}\lambda_{1}&\lambda_{2}&\lambda_{3}\\\ c_{1}&c_{2}&c_{3}\\\
d_{1}&d_{2}&d_{3}\end{pmatrix}.$

Note the similarity of the description in (b) with the proof of Thm. 4.6. We
can now easily list generators for ${\mathbb{C}}[a_{2}]^{{\mathfrak{S}}_{4}}$:
these come from the elementary invariants in the $\lambda_{i}$ and their
polarisations $\sum_{i}c_{i}$, $\sum_{i}\lambda_{i}c_{i}$,
$\sum_{i}c_{i}^{2}d_{i}$ etc. In fact, several of these turn out to be
redundant. To obtain the relations between them, we exploit the isomorphism
${\mathbb{C}}[a_{2}]^{{\mathfrak{S}}_{4}}\cong{\mathbb{C}}[X]^{{\mathfrak{S}}_{4}}/{\mathbb{C}}[X]^{{\mathfrak{S}}_{4}}\cap(z)$.
The ideal ${\mathbb{C}}[X]^{{\mathfrak{S}}_{4}}\cap(z)$ is simply the set of
all $zh$ with $h\in{\mathbb{C}}[X]^{{\mathfrak{A}}_{4}}$ a semi-invariant for
${\mathfrak{S}}_{4}$.

Thus we work in the ring ${\mathbb{C}}[X]^{{\mathfrak{A}}_{4}}$. It is not
difficult to obtain an exhaustive list of generators. Those which are
invariant for ${\mathfrak{S}}_{4}$ can be obtained by polarisations as above;
there are 18 of them. Those which are semi-invariant for ${\mathfrak{S}}_{4}$
can be obtained similarly, by considering eigenvectors for a $3$-cycle. Apart
from the linear term $z$, there are quadratic semi-invariants such as
$\sum\lambda_{i}(b_{i+1}-b_{i-1})$, and cubic terms
$\prod_{i<j}(\lambda_{i}-\lambda_{j})$ along with similar mixed cubic terms in
$\lambda_{i}$, $c_{i}$, $d_{i}$. There are $14$ generating semi-invariants.

Calculations in GAP allowed us to eliminate ten of the invariants and seven of
the semi-invariants. Hence ${\mathbb{C}}[X]^{{\mathfrak{A}}_{4}}$ (resp.
${\mathbb{C}}[a_{2}]^{{\mathfrak{S}}_{4}}$) has fifteen (resp. eight)
generators. Note that there is an action of a cyclic group of order 3 on
$a_{2}/\mathfrak{S}_{4}$, coming from the element ${\rm
diag}(\omega,1,\omega^{-1})\in\operatorname{SL}_{3}$ (which normalizes
$\mathfrak{S}_{4}$ in ${\rm Aut}(\mathfrak{g})$); then we can choose
homogeneous generators of ${\mathbb{C}}[a_{2}]^{{\mathfrak{S}}_{4}}$ which are
eigenvectors for this action. Denote the generators

$f_{0},f_{1},f_{2},g_{0},g_{1},g_{2},h_{1},h_{2}$

where the $f_{i}$ (resp. $g_{i}$, $h_{i}$) are of degree $4$ (resp. $6$, $8$)
and the subscript $i$ indicates a $\omega^{i}$-eigenvector for the above
element of order 3.

We used GAP to find relations among the generators of
${\mathbb{C}}[X]^{{\mathfrak{A}}_{4}}$. Setting $z=0$, we thus obtained the
following nine elements of the ideal of relations:

$5f_{0}^{3}+108f_{0}f_{1}f_{2}-216f_{1}h_{2}-216f_{2}h_{1}+24g_{0}^{2}-216g_{1}g_{2},\quad
8f_{0}^{2}f_{1}-15f_{0}h_{1}-18f_{2}h_{2}+6g_{0}g_{1}+18g_{2}^{2},$
$8f_{0}^{2}f_{2}-15f_{0}h_{2}-18f_{1}h_{1}+6g_{0}g_{2}+18g_{1}^{2},\quad
f_{0}^{2}g_{1}-10f_{0}f_{1}g_{0}+18f_{0}f_{2}g_{2}+12f_{2}^{2}g_{0}+12g_{0}h_{1}-36g_{2}h_{2},$
$f_{0}^{2}g_{0}-4f_{1}f_{2}g_{0}-12g_{1}h_{2}-12g_{2}h_{1},\quad
f_{0}^{2}g_{2}-10f_{0}f_{2}g_{0}+18f_{0}f_{1}g_{1}+12f_{1}^{2}g_{0}+12g_{0}h_{2}-36g_{1}h_{1},$
$f_{0}^{4}-68f_{0}^{2}f_{1}f_{2}-48f_{0}g_{1}g_{2}+144f_{1}^{2}h_{1}-144f_{1}g_{1}^{2}+144g_{2}^{2}h_{2}-144f_{2}g_{2}^{2}+144h_{1}h_{2},$
$9f_{0}^{3}f_{1}+32f_{0}^{2}f_{2}^{2}-36f_{0}f_{1}^{2}f_{2}-18f_{0}^{2}h_{1}-120f_{0}f_{2}h_{2}+24f_{0}g_{2}^{2}+72f_{1}^{2}h_{1}+72f_{1}g_{1}g_{2}+72f_{2}g_{1}^{2}+72h_{2}^{2},$
$9f_{0}^{3}f_{2}+32f_{0}^{2}f_{1}^{2}-36f_{0}f_{1}f_{2}^{2}-18f_{0}^{2}h_{2}-120f_{0}f_{1}h_{1}+24f_{0}g_{1}^{2}+72f_{2}^{2}h_{1}+72f_{2}g_{1}g_{2}+72f_{1}g_{2}^{2}+72h_{1}^{2}.$

We then checked using Magma that the ideal generated by these relations is
prime, of height 4. Hence this is a complete list of relations. (We note the
remarkable coincidence that, like ${\mathbb{C}}[a_{2}]$, there are eight
generators and nine relations.) This preparation allowed us to computationally
verify the following:

######  Theorem 5.6.

Let $\mathcal{O}^{\prime}$ and $\mathcal{O}$ be the nilpotent orbits in
$E_{8}$ with respective Bala-Carter labels $E_{8}(b_{6})$ and $E_{8}(a_{6})$,
and let ${\mathcal{S}}$ be the Slodowy slice to an element
$f\in\mathcal{O}^{\prime}$. Then ${\mathcal{S}}\cap\overline{\mathcal{O}}$ is
isomorphic to $a_{2}/\mathfrak{S}_{4}$.

###### Proof.

To verify this, we used the fact that an element $x\in{\mathcal{S}}$ belongs
to $\overline{\mathcal{O}}$ if and only if $(\operatorname{ad}x)^{19}=0$.
Although the Slodowy slice is of dimension $28$, it was not difficult (in GAP)
to find an $8$-dimensional subset, isomorphic to ${\mathbb{A}}^{8}$, which
must contain ${\mathcal{S}}\cap\overline{\mathcal{O}}$. By carefully choosing
coordinate functions on the sub-slice and identifying them with
$f_{0},\ldots,h_{2}$, we were able to determine that the relations above are
necessary conditions for the $19$-th power to equal zero. This implies that
the relations are also sufficient conditions, since $\mathcal{O}^{\prime}$ is
of codimension $4$ in $\overline{\mathcal{O}}$ (and $\overline{\mathcal{O}}$
is unibranch at $f$).

The phrase ‘must contain’ in the above paragraph is perhaps slightly
unsatisfactory. To be slightly more transparent, we have also verified in GAP
that the conditions are sufficient. View $f_{0},\ldots,h_{2}$ as functions on
$a_{2}$, hence we have a map $\Theta:a_{2}\rightarrow{\mathcal{S}}$ which
factors through the quotient by $\mathfrak{S}_{4}$. We want to check that the
image of $\Theta$ is contained in $\overline{\mathcal{O}}$. Similarly to Thm.
5.3, GAP runs out of memory when computing the $19$-th power of this matrix;
we have verified column by column that the $19$-th power equals zero. A much
quicker probabilistic proof can be obtained by checking that $\Theta(g)=0$ for
a few specific $g$ in $a_{2}$. ∎

####  5.4.2. A non-normal minimal degeneration

As in the previous subsection, let $\mathcal{O}$ (resp.
$\mathcal{O}^{\prime}$) be the orbit with Bala-Carter label $E_{8}(a_{6})$
(resp. $E_{8}(b_{6})$). Recall that
$\mathcal{O}^{\prime\prime}=\mathcal{O}_{D_{7}(a_{1})}$ is the only orbit
intermediate between $\mathcal{O}^{\prime}$ and $\mathcal{O}$. In [FJLS17] we
asserted that ${\mathcal{S}}\cap\overline{\mathcal{O}^{\prime\prime}}$ is
isomorphic to the non-normal singularity $\mu={\rm
Spec}\,{\mathbb{C}}[s^{2}t^{2},s^{3}t^{3},s^{4},t^{4},s^{5}t,st^{5}]$. Let us
show how this follows from Thm. 5.6. The nilpotent elements of $a_{2}$ with
non-trivial stabilizer subgroup in $\mathfrak{S}_{4}$ are precisely those
fixed by the 4-cycles, hence are the nilpotent elements lying in the three
type $A_{1}$ Levi subalgebras. It follows that the singular locus of
$a_{2}/\mathfrak{S}_{4}$ is isomorphic to the image of the quotient
homomorphism applied to (the nilpotent elements in) one of these Levi
subalgebras. We fix the following choice:

$N_{st}=\begin{pmatrix}st&s^{2}&0\\\ -t^{2}&-st&0\\\
0&0&0\end{pmatrix}:s,t\in{\mathbb{C}}.$

Recalling the generators of the $\Gamma_{4}$-invariants, we can see that we
have

$\begin{pmatrix}\lambda_{1}&\lambda_{2}&\lambda_{3}\\\ c_{1}&c_{2}&c_{3}\\\
d_{1}&d_{2}&d_{3}\end{pmatrix}=\begin{pmatrix}-st&-st&2st\\\ 0&0&s^{4}\\\
0&0&t^{4}\end{pmatrix}.$

In particular, we obtain the following values of the symmetric functions in
the columns of the matrix:

$\sum_{i}\lambda_{i}^{2}=6s^{2}t^{2},\quad\sum_{i}\lambda_{i}^{3}=6s^{3}t^{3},\quad\sum_{i}c_{i}=s^{4},\quad\sum_{i}d_{i}=t^{4},\quad\sum_{i}\lambda_{i}c_{i}=2s^{5}t,\quad\sum_{i}\lambda_{i}d_{i}=2st^{5},$

and it is easy to see that all other symmetric functions, evaluated at
$N_{st}$, belong to
${\mathbb{C}}[s^{2}t^{2},s^{3}t^{3},s^{4},t^{4},s^{5}t,st^{5}]$.

######  Corollary 5.7.

With the notation of Thm. 5.6,
${\mathcal{S}}\cap\overline{\mathcal{O}^{\prime\prime}}$ is isomorphic to
$\mu$.

###### Proof.

This follows from the fact that the singular locus of $\overline{\mathcal{O}}$
equals
$\overline{\mathcal{O}}\setminus\mathcal{O}=\overline{\mathcal{O}^{\prime\prime}}$.
∎

The normalization of $\mu$ is easily seen to be ${\rm
Spec}\,{\mathbb{C}}[st,s^{4},t^{4}]$, i.e. an $A_{3}$ surface singularity.
This is the only known degeneration of special orbits with a non-normal
unibranch singularity.

####  5.4.3. Monodromy action

The component group $\mathfrak{S}_{3}$ of $\mathcal{O}^{\prime}$ acts on the
Slodowy slice, and therefore on $a_{2}/\mathfrak{S}_{4}$. In fact, $N_{{\rm
Aut}{\mathfrak{g}}}(\mathfrak{S}_{4})/\mathfrak{S}_{4}$ is isomorphic to
$\mathfrak{S}_{3}$, and the monodromy action coincides with the action of the
normalizer. Concretely, ${\mathfrak{A}}_{3}$ is generated by conjugation by
${\rm diag}(\omega,1,\omega^{2})$, and $g\mapsto-g^{T}$ is a transposition in
$\mathfrak{S}_{3}$.

Similarly, we have an action of $\mathfrak{S}_{3}$ on $\mu$: taking the
normalization, this is nothing other than the action of $\mathfrak{S}_{3}$ on
a simple singularity of type $A_{3}$, via the inclusion of a cyclic subgroup
of $\operatorname{SL}_{2}$ of order $4$ in a binary dihedral subgroup of order
$24$. It is easy to check that $\mu/\mathfrak{S}_{3}$ is normal.

##  6\. Slodowy slices in special pieces

We are now ready to prove the local version of Lusztig’s special pieces
conjecture. Let ${\mathcal{P}}(\mathcal{O})$ be a special piece in a simple
Lie algebra, let $\mathcal{O}_{m}$ be the unique minimal orbit in
${\mathcal{P}}(\mathcal{O})$ and let ${\mathcal{S}}_{m}$ be the Slodowy slice
at an element of $\mathcal{O}_{m}$. Denote by ${\mathfrak{h}}_{n-1}$ the
reflection representation for $\mathfrak{S}_{n}$.

Table 1. The non-trivial special pieces in the exceptional types $(n,k)$ | sing. | $\mathfrak{g}$ | pairs $(\mathcal{O},\mathcal{O}_{m})$
---|---|---|---
$(2,1)$ | $A_{1}$ | $E_{6}$ |  $(E_{6}(a_{3}),A_{5})$, $(A_{2},3A_{1})$
|  | $E_{7}$ |  $(E_{7}(a_{3}),D_{6})$, $(E_{6}(a_{3}),A^{\prime}_{5})$, $(D_{4}(a_{1})+A_{1},A_{3}+2A_{1})$, $(A_{2},(3A_{1})^{\prime})$
|  | $E_{8}$ |  $(E_{8}(a_{3}),E_{7})$, $(E_{8}(a_{5}),D_{7})$, $(E_{8}(b_{6}),A_{7})$, $(D_{6}(a_{1}),D_{5}+A_{1})$,
|  |  |  $(E_{6}(a_{3}),A_{5})$, $(D_{4}(a_{1})+A_{2},A_{3}+A_{2}+A_{1})$, $(2A_{2},A_{2}+3A_{1})$, $(A_{2},3A_{1})$
$(2,2)$ | $c_{2}$ | $E_{7}$ | $(D_{5}(a_{1}),D_{4}+A_{1})$
|  | $E_{8}$ |  $(E_{7}(a_{3}),D_{6})$, $(A_{4}+2A_{1},2A_{3})$
$(2,3)$ | $c_{3}$ | $F_{4}$ | $(\tilde{A}_{1},A_{1})$
|  | $E_{7}$ | $(A_{2}+A_{1},4A_{1})$
|  | $E_{8}$ | $(D_{5}(a_{1}),D_{4}+A_{1})$
$(2,4)$ | $c_{4}$ | $E_{8}$ | $(A_{2}+A_{1},4A_{1})$
$(3,1)$ | ${\mathbb{C}}^{4}/\mathfrak{S}_{3}$ | $G_{2}$ | $(G_{2}(a_{1}),A_{1})$
|  | $E_{6}$ | $(D_{4}(a_{1}),2A_{2}+A_{1})$
|  | $E_{7}$ |  $(E_{7}(a_{5}),A_{5}+A_{1})$, $(D_{4}(a_{1}),2A_{2}+A_{1})$
|  | $E_{8}$ |  $(E_{8}(b_{5}),E_{6}+A_{1})$, $(D_{4}(a_{1}),2A_{2}+A_{1})$
$(3,2)$ | ${\mathbb{C}}^{8}/\mathfrak{S}_{3}$ | $E_{8}$ | $(D_{4}(a_{1})+A_{1},2A_{2}+2A_{1})$
$(4,1)$ | ${\mathbb{C}}^{6}/\mathfrak{S}_{4}$ | $F_{4}$ | $(F_{4}(a_{3}),A_{2}+\tilde{A}_{1})$
$(5,1)$ | ${\mathbb{C}}^{8}/\mathfrak{S}_{5}$ | $E_{8}$ | $(E_{8}(a_{7}),A_{4}+A_{3})$

###### Main Theorem.

For $\mathfrak{g}$ exceptional, the intersection
${\mathcal{S}}_{m}\cap{\mathcal{P}}(\mathcal{O})$ is isomorphic to

$({\mathfrak{h}}_{n-1}\oplus{\mathfrak{h}}_{n-1}^{*})^{k}/\mathfrak{S}_{n}$

for some $k$ and $n$.

For $\mathfrak{g}$ classical, let
$\mathcal{O}_{1},\mathcal{O}_{2},\dots\mathcal{O}_{r}$ be minimal
degenerations of $\mathcal{O}$ contained in ${\mathcal{P}}(\mathcal{O})$. Then
$H$ is a product of $r$ copies of $\mathfrak{S}_{2}$. Let
$k_{i}=\frac{1}{2}\operatorname{codim}_{\overline{\mathcal{O}}}\mathcal{O}_{i}$.
Then ${\mathcal{S}}_{m}\cap{\mathcal{P}}(\mathcal{O})$ is isomorphic to

$\prod_{i=1}^{r}({\mathfrak{h}}_{1}\oplus{\mathfrak{h}}_{1}^{*})^{k_{i}}/\mathfrak{S}_{2}.$

In both cases, the slice is isomorphic to $\mathbb{C}^{d}/H$ where $d$ is the
codimension of $\mathcal{O}_{m}$ in $\overline{\mathcal{O}}$ and the action of
$H$ is the one described in the above isomorphisms.

###### Proof.

Case: $\mathfrak{g}$ exceptional. When $\mathfrak{g}$ is of exceptional type,
the group appearing in Lusztig’s special pieces conjecture, and hence in our
Main Theorem, is $\mathfrak{S}_{n}$ for some $n\leq 5$. There is nothing to
prove if ${\mathcal{P}}(\mathcal{O})=\mathcal{O}$ (in which case $H$ is
trivial). If $H=\mathfrak{S}_{2}$ then there are two orbits in
${\mathcal{P}}(\mathcal{O})$, hence $\mathcal{O}>\mathcal{O}_{m}$ is a minimal
degeneration and we only need to check that the Slodowy slice singularity is a
minimal singularity of type $C$. This follows from our earlier work, see
[FJLS17, §13]. The statement when $H=\mathfrak{S}_{4}$ (resp.
$\mathfrak{S}_{5}$) is Thm. 5.1 (resp. Thm. 5.3). If
$\mathcal{O}=\mathcal{O}_{D_{4}(a_{1})+A_{1}}$ in type $E_{8}$ then
$H=\mathfrak{S}_{3}$ and this was proved in Lemma 3.12. (This is the only case
with $n>2$ and $k>1$.) If $\mathcal{O}$ is the subregular orbit in type
$G_{2}$ then this is Lemma 3.9. This leaves one special piece in $E_{6}$, two
in $E_{7}$ and two in $E_{8}$. We show that all of the remaining Slodowy slice
singularities are isomorphic to the slice from the minimal orbit to the
subregular orbit closure in $G_{2}$ (hence to
$({\mathfrak{h}}_{2}\oplus{\mathfrak{h}}_{2}^{*})/\mathfrak{S}_{3}$). To see
this, we note that there are always three orbits in the special piece and (for
these cases) there always exists a minimal special degeneration
$\mathcal{O}>\mathcal{O}^{\prime}$ such that
$\mathcal{O}_{m}>\mathcal{O}^{\prime}$ is a minimal degeneration. Furthermore,
we can assume that $\mathcal{O}^{\prime}$ is not $A_{2}+3A_{1}$ in type
$E_{7}$. (When $\mathcal{O}=\mathcal{O}_{D_{4}(a_{1})}$ we choose
$\mathcal{O}^{\prime}=\mathcal{O}_{2A_{2}}$ instead.) This ensures the
following: if ${\mathcal{S}}=f+\mathfrak{g}^{e}$ is the Slodowy slice at an
element $f\in\mathcal{O}^{\prime}$, then
${\mathcal{S}}\cap\overline{\mathcal{O}}=f+\overline{C_{0}\cdot e_{0}}$, where
$e_{0}$ is a subregular element of $\mathfrak{c}_{0}$, the simple component of
$\mathfrak{g}^{e}\cap\mathfrak{g}^{h}$ of type $G_{2}$. Note in particular
that if $\\{e_{m},h_{m},f_{m}\\}\subset\mathfrak{c}_{0}$ is an
$\mathfrak{sl}_{2}$-triple with $e_{m}$ a long root element of
$\mathfrak{c}_{0}$, then $f+f_{0}\in\mathcal{O}_{m}$ and, by the argument in
[FJLS17, Prop. 4.8], $f+f_{m}+\mathfrak{g}^{e}\cap\mathfrak{g}^{e_{m}}$ is a
Slodowy slice at $f+f_{0}$. It follows that the Slodowy slice singularity from
$\mathcal{O}_{m}$ to $\overline{\mathcal{O}}$ is isomorphic to the Slodowy
slice singularity from the minimal orbit to the subregular orbit closure in
$G_{2}$.

Case: $\mathfrak{g}$ classical. For generalities on the classification of
nilpotent orbits via $\epsilon$-partitions, including criteria for
specialness, we refer to [CM93, Ch. 5]. For a partition $\lambda$ of $n$ and
an integer $s$, let $m(s)$ (resp. $h(s)$) denote the multiplicity (resp.
height, i.e. $\sum_{j\geq s}m(j)$) of $s$. Let $\mathcal{O}$ be a special
orbit and let $\lambda$ be the partition of $\mathcal{O}$. Let
$\mathcal{O}^{\prime}$ be a non-special orbit in the piece, with partition
$\nu$. Pick $e\in\mathcal{O}^{\prime}$. As will be explained in [JLSb], it
follows from the classification of special orbits (detailed in [CM93, §6.3])
that $\lambda$ is obtained from $\nu$ by replacing occurrences of $[s^{m(s)}]$
in $\nu$ with parts $[s{+}1,s^{m(s)-2},s{-}1]$ in $\lambda$, where
$s\equiv\epsilon$ and $h(s)\not\equiv\epsilon^{\prime}$. Note that for each
such part $[s^{m(s)}]$ there is a simple factor $\mathfrak{sp}_{m}(s)$ of the
reductive centralizer of $e$. Choose a minimal nilpotent element in each such
simple factor, and let $e_{0}$ be their sum. Then $e_{0}$ is of height 2 in
$\mathfrak{g}$, hence by [FJLS17, Cor. 4.9 and Lemma 4.3] apply. It is also
now clear (by considering $\mathfrak{sl}_{2}$-triples) that $f+e_{0}$ belongs
to $\mathcal{O}$, hence the slice is equal to
$\overline{C(e)\cdot(f+e_{0})}\cong\overline{C(e)\cdot e_{0}}$. The result
therefore holds for any orbit in the special piece, including
$\mathcal{O}_{m}$, since the minimal orbit closure in $\mathfrak{sp}(2k_{i})$
is isomorphic to $\mathbb{C}^{2k_{i}}/\mathfrak{S}_{2}$ with the nontrivial
elements acting by $-1$, as noted before. In the case of $\mathcal{O}_{m}$, we
have $r$ is the number of factors of $\mathfrak{S}_{2}$ in $H$, completing the
proof. ∎

In Table 1, we list values of $n$ and $k$ for all special pieces with more
than one orbit.

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Baohua Fu ()

AMSS, HLM and MCM, Chinese Academy of Sciences, 55 ZhongGuanCun East Road,
Beijing, 100190, China and School of Mathematical Sciences, University of
Chinese Academy of Sciences, Beijing, China

Daniel Juteau ( )

LAMFA, Université de Picardie Jules Verne, CNRS, Amiens, France

Paul Levy ()

Department of Mathematics and Statistics Fylde College, Lancaster University,
Lancaster, LA1 4YF, United Kingdom

Eric Sommers()

Department of Mathematics and Statistics, University of Massachusetts Amherst,
Amherst, MA 01003-4515, USA