# Minimal special degenerations and duality

Daniel Juteau, Paul Levy and Eric Sommers

###### Abstract.

This paper includes the classification, in a simple Lie algebra
${\mathfrak{g}}$, of the singularities of Slodowy slices between special
nilpotent orbits that are adjacent in the partial order on nilpotent orbits.
The irreducible components of most singularities are (up to normalization)
either a simple surface singularity or the closure of a minimal special
nilpotent orbit in a smaller rank Lie algebra. Besides those cases, there are
some exceptional cases that arise as quotients of the closure of a minimal
orbit in types $D_{n}$ by $V_{4}$, in type $A_{2}$ by ${\mathfrak{S}}_{2}$ or
in type $D_{4}$ by ${\mathfrak{S}}_{4}$. We also consider the action on the
slice of the fundamental group of the smaller orbit. With this action, we
observe that under Lusztig-Spaltenstein duality, in most cases, a singularity
of simple surface singularity is interchanged with the closure of a minimal
special orbit of Langlands dual type (or a cover of it with action). Lusztig’s
canonical quotient helps explain when this duality fails. This empirical
observation generalizes an observation of Kraft and Procesi in type $A_{n}$,
where all nilpotent orbits are special. We also resolve a conjecture of
Lusztig that concerns the intersection cohomology of slices between special
nilpotent orbits.

##  1\. Introduction

###  1.1. Minimal degenerations

Let $G$ be a simple algebraic group over $\mathbb{C}$ and ${\mathfrak{g}}$ its
Lie algebra. Let ${\mathcal{N}}_{o}:={\mathcal{N}}({\mathfrak{g}})/G$ be the
set of nilpotent orbits in ${\mathfrak{g}}$. The partial order on
${\mathcal{N}}_{o}$ is defined so that $\mathcal{O}^{\prime}{<}\mathcal{O}$
whenever $\mathcal{O}^{\prime}\subsetneq\overline{\mathcal{O}}$ for
$\mathcal{O},\mathcal{O}\in{\mathcal{N}}_{o}$, where $\overline{\mathcal{O}}$
is the closure of $\mathcal{O}$. A pair $\mathcal{O}^{\prime}{<}\mathcal{O}$
is called a degeneration. If $\mathcal{O}$ and $\mathcal{O}^{\prime}$ are
adjacent in the partial order (that is, there is no orbit strictly between
them), then the pair is called a minimal degeneration. There are two minimal
degenerations at either extreme of the poset ${\mathcal{N}}_{o}$: the regular
and subregular nilpotent orbits give a minimal degeneration, as does the
minimal nilpotent orbit and the zero orbit.

Given $e\in{\mathcal{N}}_{o}$, let ${\mathfrak{s}}\subset{\mathfrak{g}}$ be an
${\mathfrak{sl}}_{2}$-triple $\\{e,h,f\\}$ through $e$. Then
$\mathcal{S}_{e}:=e+{\mathfrak{g}}^{f}$, where ${\mathfrak{g}}^{f}$ is the
centralizer of $f$ in ${\mathfrak{g}}$, is called a Slodowy slice. Associated
to any degeneration $\mathcal{O}^{\prime}{<}\mathcal{O}$ is a smooth
equivalence class of singularities ${\rm
Sing}(\mathcal{O},\mathcal{O}^{\prime})$ [KP82], which can be represented by
the intersection
$\mathcal{S}_{\mathcal{O},e}:={\mathcal{S}}_{e}\cap\overline{\mathcal{O}}$,
where $e\in\mathcal{O}^{\prime}$. We call $\mathcal{S}_{\mathcal{O},e}$ a
Slodowy slice singularity.

The singularities $\mathcal{S}_{\mathcal{O},e}$ of minimal degenerations are
known in the classical types by [KP81] and [KP82] and in the exceptional types
by [FJLS23] and [FJLS17], up to normalization for a few cases in $E_{7}$ and
$E_{8}$. These results can be summarized as:

  * •

the irreducible components of $\mathcal{S}_{\mathcal{O},e}$ are pairwise
isomorphic;

  * •

if $\dim(\mathcal{S}_{\mathcal{O},e})=2$, then the normalization of an
irreducible component of $\mathcal{S}_{\mathcal{O},e}$ is isomorphic to
$\mathbb{C}^{2}/\Gamma$ where $\Gamma\subset\operatorname{SL}_{2}(\mathbb{C})$
is a finite subgroup, possibly trivial. Such a variety is called a simple
surface singularity when $\Gamma$ is non-trivial.

  * •

if $\dim(\mathcal{S}_{\mathcal{O},e})\geq 4$, then an irreducible component of
$\mathcal{S}_{\mathcal{O},e}$ is isomorphic to the closure of a minimal
nilpotent orbit in some simple Lie algebra, or else is one of four exceptional
cases, denoted $m^{\prime}$, $\tau$, $\chi$, or $a_{2}/{\mathfrak{S}}_{2}$ in
[FJLS17] and each appearing exactly one time.

###  1.2. Action on slices

A simple surface singularity $X=\mathbb{C}^{2}/\Gamma$ corresponds to the
Dynkin diagram of a simply-laced Lie algebra (e.g., $A_{n}$, $D_{n}$, $E_{n}$)
either by using the irreducible representations of $\Gamma$ as done by McKay,
or by looking at the exceptional fiber of the minimal resolution of $X$, which
is union of projective lines, whose arrangement yields the Dynkin diagram.
Slodowy defined an action on $X$ by using a normalizing subgroup
$\Gamma^{\prime}$ of $\Gamma$ in $\operatorname{SL}_{2}(\mathbb{C})$ [Slo80,
III.6]. Looking at the image of the action of $\Gamma^{\prime}$ on the Dynkin
diagram, he introduced the notation $B_{n}$ (resp. $C_{n}$, $F_{4}$, $G_{2}$)
to denote a simple surface singularity $A_{2n-1}$ (resp. $D_{n+1}$, $E_{6}$,
$D_{4}$) singularity with an “outer” action of $\mathfrak{S}_{2}$ (resp.
$\mathfrak{S}_{2}$, $\mathfrak{S}_{2}$, $\mathfrak{S}_{3}$). Here, “outer”
refers to the fact that on the corresponding Lie algebra these come from outer
automorphisms. It is also possible to do the same thing for the simple surface
singularity $A_{2n}$, where we used the notation $A_{2n}^{+}$ in [FJLS17],
when the outer action is included. Note, however, that this arises from a
cyclic group of order four acting on $X$.

The centralizer $G^{e}$ of $e$ in $G$ has a reductive part
$C({\mathfrak{s}})$, given by the centralizer of ${\mathfrak{s}}$ in $G$. Then
$C({\mathfrak{s}})$ acts on $\mathcal{S}_{\mathcal{O},e}$ and we are
interested in the image of $C({\mathfrak{s}})$ in $\rm
Aut(\mathcal{S}_{\mathcal{O},e})$. Slodowy [Slo80, IV.8] showed for the
regular/subregular minimal degeneration, that $\mathcal{S}_{\mathcal{O},e}$
with the action induced from $C({\mathfrak{s}})$ is exactly the simple surface
singularity denoted by the type of ${\mathfrak{g}}$. This explains his choice
of notation.

Let $a_{n},b_{n},\dots,g_{2}$ denote the closure of the minimal nilpotent
orbit according to the type of ${\mathfrak{g}}$. In [FJLS17], we introduced
the notation $a_{n}^{+}$, $d_{n}^{+}$, $e_{6}^{+}$, $d_{4}^{++}$ to denote
these varieties with the outer action of $\mathfrak{S}_{2}$,
$\mathfrak{S}_{2}$, $\mathfrak{S}_{2}$, $\mathfrak{S}_{3}$, respectively,
coming from the outer automorphisms of ${\mathfrak{g}}$. In op. cit., using
these two notions of action, we studied the action of $C({\mathfrak{s}})$ on
$\mathcal{S}_{\mathcal{O},e}$ for all minimal degenerations, where we found
that $C({\mathfrak{s}})$ acts transitively on the irreducible components of
$\mathcal{S}_{\mathcal{O},e}$ and in some sense acts as non-trivially as
possible on $\mathcal{S}_{\mathcal{O},e}$ given the size of the component
group $A(e):=C({\mathfrak{s}})/C^{\circ}({\mathfrak{s}})$. In this paper one
of our results is to repeat this calculation for the classical groups (see
§5).

###  1.3. Minimal Special Degenerations

Lusztig defined the notion of special representations of the Weyl group $W$ of
$G$ [Lus79], which led him to define the special nilpotent orbits, denoted
${\mathcal{N}}^{sp}_{o}$, via the Springer correspondence. The regular,
subregular, and zero nilpotent orbits are always special, but the minimal
nilpotent orbit is only special when ${\mathfrak{g}}$ is simply-laced (types
$A_{n}$, $D_{n}$, or $E_{n}$). In the other types, there is always a unique
minimal (nonzero) special nilpotent orbit. We denote the closure of the
minimal special nilpotent orbits (which are not minimal nilpotent) by
$b_{n}^{sp},c_{n}^{sp},f_{4}^{sp},$ and $g_{2}^{sp}$, according to the type of
${\mathfrak{g}}$.

In this paper, we classify the Slodowy slice singularities
$\mathcal{S}_{\mathcal{O},e}$ when $\mathcal{O}$ and $\mathcal{O}^{\prime}$
are adjacent special orbits, i.e., there is no special orbit strictly between
them. We call these minimal special degenerations. Since
$\dim(\mathcal{S}_{\mathcal{O},e})=2$ implies the degeneration is already a
minimal degeneration, we are left only to classify the cases where
$\dim(\mathcal{S}_{\mathcal{O},e})\geq 4$. Our main result on the
classification of minimal special degenerations is summarized as:

  * •

the irreducible components of $\mathcal{S}_{\mathcal{O},e}$ are pairwise
isomorphic;

  * •

if $\dim(\mathcal{S}_{\mathcal{O},e})=2$, then the normalization of an
irreducible component of $\mathcal{S}_{\mathcal{O},e}$ is isomorphic to
$\mathbb{C}^{2}/\Gamma$ where $\Gamma\subset\operatorname{SL}_{2}(\mathbb{C})$
is a finite, non-trivial subgroup.

  * •

if $\dim(\mathcal{S}_{\mathcal{O},e})\geq 4$, then an irreducible component of
$\mathcal{S}_{\mathcal{O},e}$ is isomorphic to the closure of a minimal
special nilpotent orbit in some simple Lie algebra, or else is isomorphic to
one of the following quotients of the closure of a minimal (special) nilpotent
orbit: $a_{2}/{\mathfrak{S}}_{2}$, $d_{n+1}/V_{4}$ or
$d_{4}/{\mathfrak{S}}_{4}$.

The singularities $a_{2}/{\mathfrak{S}}_{2}$ and $d_{4}/{\mathfrak{S}}_{4}$
arose in [FJLS23] and along with $d_{n+1}/V_{4}$, they also appear in the
physics literature [HKK23].

In the case where $\dim(\mathcal{S}_{\mathcal{O},e})\geq 4$, the singularities
of $\mathcal{S}_{\mathcal{O},e}$ are mostly controlled by the simple factors
of ${\mathfrak{c}}({\mathfrak{s}})$ (see Corollary 4.12 and the remarks after
it), just as occurs for most of the minimal degenerations of dimension four or
more.

For dimension two, there is a single slice where one of its irreducible
components is known not to be normal, namely the $\mu$ singularity from
[FJLS23], which occurs once in $E_{8}$ (it is irreducible). We expect the
other components of slices of dimension two all to be normal in the case of
minimal special degenerations, unlike the case of minimal degenerations. The
components of slices of dimension at least four are all known to be normal.

The irreducible minimal special degenerations in the classical types $B$, $C$,
$D$, are listed in Tables 1 and 2, in analogy with the classification of Kraft
and Procesi for minimal degenerations [KP82, Table 1]. The minimal special
degenerations of codimension two are already minimal degenerations and so are
contained in [KP82], except for the action of $A(e)$. The notation of
$[2B_{n}]^{+},$ means that the image of $C({\mathfrak{s}})$ acts by a Klein
$4$-group $V_{4}$ on $\mathcal{S}_{\mathcal{O},e}$, where one generator
switches the two components of the exceptional fiber and a second generator
preserves both components of type $A_{2n-1}$, but acts by outer automorphism
on each one. The table assumes that $G$ is the orthogonal group $\rm O(2n)$
for type $D_{n}$, hence making use of the outer ${\mathfrak{S}}_{2}$-action of
$D_{n}$.

In type $D_{n}$ without this outer action, we would get these same
singularities but without some or all of the action on
$\mathcal{S}_{\mathcal{O},e}$. Specifically, $D_{k}$ and $d_{k}$ arise,
without the ${\mathfrak{S}}_{2}$-action. The singularity $[2B_{k}]^{+}$ will
become $B_{k}$ for the minimal degenerations where $\mathcal{O}$ is a very
even orbit. We discuss this further in §8.

Table 1. Minimal special degenerations of codimension two Name of singularity | $a$ | $b$ | $c$ | $d$ | $e$
---|---|---|---|---|---
Lie algebra | $\mathfrak{sp}_{2}$ | $\mathfrak{sp}_{2n}$ | $\mathfrak{so}_{2n+1}$ | $\mathfrak{sp}_{4n+2}$ | $\mathfrak{so}_{4n}$
|  | $n\geq 2$ | $n\geq 1$ | $n\geq 1$ | $n\geq 1$
$l$ rows removed | $l\equiv\epsilon^{\prime}$ | any | $l\not\equiv\epsilon^{\prime}$ | $l\equiv\epsilon^{\prime}$ | $l\equiv\epsilon^{\prime}$
$s$ columns removed | $s\not\equiv\epsilon$ | $s\not\equiv\epsilon$ | $s\equiv\epsilon$ | $s\not\equiv\epsilon$ | $s\equiv\epsilon$
$\lambda$ | $[2]$ | $[2n]$ | $[2n{+}1]$ | $[2n{+}1,2n{+}1]$ | $[2n,2n]$
$\mu$ | $[1,1]$ | $[2n{-}2,2]$ | $[2n{-}1,1,1]$ | $[2n,2n,2]$ | $[2n{-}1,2n{-}1,1,1]$
Singularity | $C_{1}$ | $C_{n}$ | $B_{n}$ | $B_{n}$ | $[2B_{n}]^{+}$
Table 2. Minimal Special Degenerations of codimension 4 or more Name of singularity | $g_{sp}$ | $h$ | $f^{1}_{sp}$ | $f^{2}_{sp}$ | $h_{sp}$
---|---|---|---|---|---
Lie algebra | $\mathfrak{sp}_{2n}$ | $\mathfrak{so}_{2n}$ | $\mathfrak{so}_{2n+1}$ | $\mathfrak{sp}_{4n+2}$ | $\mathfrak{sp}_{4n}$
| $n\geq 2$ | $n\geq 3$ | $n\geq 2$ | $n\geq 2$ | $n\geq 2$
$l$ rows removed | $l\equiv\epsilon^{\prime}$ | $l\equiv\epsilon^{\prime}$ | $l\not\equiv\epsilon^{\prime}$ | $l\equiv\epsilon^{\prime}$ | $l\not\equiv\epsilon^{\prime}$
$s$ columns removed | $s\not\equiv\epsilon$ | $s\equiv\epsilon$ | $s\equiv\epsilon$ | $s\not\equiv\epsilon$ | $s\not\equiv\epsilon$
$\lambda$ | $[2^{2},1^{2n-4}]$ | $[2^{2},1^{2n-4}]$ | $[3,1^{2n-2}]$ | $[3^{2},2^{2n-2}]$ | $[4,2^{2n-2}]$
$\mu$ | $[1^{2n}]$ | $[1^{2n}]$ | $[1^{2n+1}]$ | $[2^{2n+1}]$ | $[2^{2n}]$
codimension | $4n\\!-\\!2$ | $4n\\!-\\!6$ | $4n\\!-\\!2$ | $4n\\!-\\!2$ | $4n\\!-\\!2$
Singularity | $c^{sp}_{n}$ | $d_{n}^{+}$ | $b^{sp}_{n}$ | $b^{sp}_{n}$ | $d_{n+1}/V_{4}$

######  Remark 1.1.

The $h$ singularity for $n=2$ is $d_{2}^{+}$, which coincides with the $e$
singularity for $n=1$. We use $d_{2}^{+}$ in the graphs for the classical
groups since the action of $A(e)$ for the $e$-singularity with $n=1$ is
actually only by ${\mathfrak{S}}_{2}$.

The proof that these tables give the classification of minimal special
degenerations is given in §3. In §4, we establish that the singularities in
classical types are as given in Table 2, and in §4.4 we complete the story in
the exceptional groups. In §5.1 and §5.7, we establish the $A(e)$-action both
for minimal special degenerations and minimal degenerations. The graphs at the
end of the paper give the results for the exceptional groups and several
examples in the classical groups §11.

###  1.4. Duality

Using the Springer correspondence, Lusztig defined two maps, which are order-
reversing involutions: $d:{\mathcal{N}}_{o}^{sp}\to{\mathcal{N}}_{o}^{sp}$ and
$d_{LS}:{\mathcal{N}}_{o}^{sp}\to{{}^{L}}{\mathcal{N}}_{o}^{sp}$ (see
[Car93]).

For $G=GL_{n}$ all nilpotent orbits are special and Kraft and Procesi [KP81]
computed the singularity type of $\mathcal{S}_{\mathcal{O},e}$ for minimal
degenerations (hence, minimal special degenerations). The singularity is
either of type $A_{k}$ or $a_{k}$ for some $k$. Kraft and Procesi observed
that if the singularity of $(\mathcal{O},\mathcal{O}^{\prime})$ is of type
$A_{k}$ then the singularity of $(d(\mathcal{O}^{\prime}),d(\mathcal{O}))$ is
of type $a_{k}$. In the case of $GL_{n}$, each orbit is given by a partition
and the dualities $d=d_{LS}$ are given by taking the transpose partition.

Our duality is a generalization of the Kraft-Procesi observation, but with
some wrinkles. It says that typically an irreducible component of a simple
surface singularity (with $A(e)$-action) is interchanged with the minimal
special orbit of Langlands dual type (after taking the quotient of the
$A(e)$-action). More explicitly, $d_{LS}$ exchanges the following
singularities.

$\displaystyle A_{n}$ $\displaystyle\leftrightarrow$ $\displaystyle a_{n}$
$\displaystyle B_{n}$ $\displaystyle\leftrightarrow$ $\displaystyle
a_{2n-1}^{+}\text{ or }c_{n}^{sp}$ $\displaystyle C_{n}$
$\displaystyle\leftrightarrow$ $\displaystyle d_{n+1}^{+}\text{ or
}b_{n}^{sp}$ $\displaystyle D_{n}$ $\displaystyle\leftrightarrow$
$\displaystyle d_{n}$ $\displaystyle G_{2}$ $\displaystyle\leftrightarrow$
$\displaystyle d_{4}^{++}\text{ or }g_{2}^{sp}$ $\displaystyle F_{4}$
$\displaystyle\leftrightarrow$ $\displaystyle e_{6}^{+}\text{ or }f_{4}^{sp}$
$\displaystyle E_{n}$ $\displaystyle\leftrightarrow$ $\displaystyle e_{n}$

The only interchange of dimension two with dimension two is when both slices
have irreducible components of type $A_{1}$. The fact that for each dual pair
of orbits, one of the pairs yields a slice of dimension two was observed by
Lusztig [Lus22]. For the cases with two options on the right, notice that the
first option arises as cover of the second (see e.g. [FJLS23]). Indeed we
expect this cover to occur intrinsically since in all these cases
$\mathcal{O}$ itself admits such a cover. We could also alternatively say that
the second option is a quotient of the first by the $A(e)$-action.

There are three families of situations that do not obey this relationship.

  1. (1)

Sometimes

$C_{n+1}\leftrightarrow c_{n}^{sp}\text{ or }a_{2n-1}^{+}$

  2. (2)

When $d_{n+1}/V_{4}$ or $d_{4}/{\mathfrak{S}}_{4}$ occurs in a dual pair of
orbits, we always have

$\displaystyle C_{n}$ $\displaystyle\leftrightarrow$ $\displaystyle
d_{n+1}/V_{4}$ $\displaystyle G_{2}$ $\displaystyle\leftrightarrow$
$\displaystyle d_{4}/S_{4}$

  3. (3)

For the three exceptional special orbits in $E_{7}$ and $E_{8}$,

$\displaystyle A_{2}^{+}$ $\displaystyle\leftrightarrow$ $\displaystyle
a_{2}^{+}\text{ or }a_{2}/S_{2}$ $\displaystyle A_{4}^{+}$
$\displaystyle\leftrightarrow$ $\displaystyle a_{4}^{+}$

In the first case, Lusztig’s canonical quotient of $A(e)$ is playing a role.
Namely, the kernel of the map from $A(e)$ to the canonical quotient
$\bar{A}(e)$ is acting by outer action on $\mathcal{S}_{\mathcal{O},e}$. We
denote this property by adding a $*$ to the singularity, $C_{n+1}^{*}$. This
phenomenon is described in §6. In the second case, there is an impact of the
canonical quotient, see again §6. In the third case, these cases arise because
the only representative of an order two element in $A(e)$ is an order $4$
element in $C({\mathfrak{s}})$ (see [FJLS17]). We gather the duality results
into one theorem in §10.

###  1.5. Full automorphism group of ${\mathfrak{g}}$

We also consider, building on work of Slodowy, the case where $G=\rm
Aut({\mathfrak{g}})$. For $A_{n},E_{6}$, and $D_{4}$, we leave this for §8. We
find that in type $A_{n}$, all singularities acquire the expected outer action
and thus, for example, $A_{k}^{+}\leftrightarrow a_{k}^{+}$ for the full
automorphism group of ${\mathfrak{g}}$.

To get more uniform statements for type $D_{n}$, we use $G=\rm O(2n)$ at the
beginning and then explain what changes when $G=\operatorname{SO}(2n)$ in §5.6
and §5.8.

###  1.6. Three quartets of singularities in classical types $B$, $C$, $D$

The duality of the last section has a richer structure in types $B$, $C$, $D$.
The internal duality $d$ for $B_{n}$ and $C_{n}$, together with $d_{LS}$ and
the composition $f:=d\circ d_{LS}$, yield $4$ related special orbits (see
Figure 1). Applying these $3$ maps to a minimal special degeneration, we find
there are only three possible outputs for the four singularities that arise
(see Figure 2).

There is also a story that involves $D_{n}$. As mentioned above, we work with
$G=\rm O(2n)$. Then there is a subset of the nilpotent orbits in type $C_{n}$
that is a slight modification of the special orbits, by changing the parity
condition in the definition. We call these the alternative special nilpotent
orbits in type $C$ and denote them by $\mathcal{N}_{o}^{C,asp}$ in §2.2. Its
minimal element is the minimal orbit in type $C_{n}$ of dimension $2n$. There
is a bijection between $\mathcal{N}^{D,sp}_{o}$ and $\mathcal{N}_{o}^{C,asp}$,
also denoted $f$, that preserves the partial order and codimensions (more
precisely, it sends an orbit of dimension $N$ to one of dimension $N+2n$).
This bijection, together with $d_{LS}$ and $d=f\circ d_{LS}$, also gives rise
to the same three quartets of singularities as in Figure 2. An example is
given in Figure 3. This is also the first case where all three quartets arise.

Figure 1.
Dualities${\mathcal{N}^{B,sp}_{o}}$${\mathcal{N}^{C,sp}_{o}}$${\mathcal{N}^{B,sp}_{o}}$${\mathcal{N}^{C,sp}_{o}}$$f$$f$$d$$d$$d_{LS}$$d_{LS}$${\mathcal{N}^{D,sp}_{o}}$${\mathcal{N}^{C,asp}_{o}}$${\mathcal{N}^{C,asp}_{o}}$${\mathcal{N}^{D,sp}_{o}}$$f$$f$$d$$d$$d_{LS}$$d_{LS}$
Figure 2. The three quartets of possible singularities in classical
groups${C_{n}}$${B_{n}\
}$${c^{\mathrm{sp}}_{n}}$${b^{\mathrm{sp}}_{n}}$$f$$f$$d$$d$$d_{LS}$$d_{LS}$${C_{n}}$${C_{n+1}^{*}}$${c^{\mathrm{sp}}_{n}}$${d^{+}_{n+1}}$$f$$f$$d$$d$$d_{LS}$$d_{LS}$${C_{n}}$${[2B_{n}]^{+}}$${c^{\mathrm{sp}}_{n}}$${d_{n+1}/V_{4}}$$f$$f$$d$$d$$d_{LS}$$d_{LS}$
Figure 3. Duality between $\mathcal{P}_{D}^{sp}(10)$ and
$\mathcal{P}_{C}^{asp}(10)$
${[7,1]}$${[5,3]}$${[4^{2}]}$${[5,1^{3}]}$${[3^{2},1^{2}]}$${[2^{4}]}$${[3,1^{5}]}$${[2^{2},1^{4}]}$${[1^{8}]}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{3}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}[2B_{2}]^{+}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c^{\mathrm{sp}}_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b^{\mathrm{sp}}_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{2}^{+}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{4}^{+}}$
$D_{4}$ Minimal Special Degenerations
${[8]}$${[6,2]}$${[4^{2}]}$${[6,1^{2}]}$${[4,2^{2}]}$${[2^{4}]}$${[4,1^{4}]}$${[2^{3},1^{2}]}$${[2,1^{6}]}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C^{*}_{4}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C^{*}_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{3}/V_{4}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c^{\mathrm{sp}}_{2}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{1}}$${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c^{\mathrm{sp}}_{3}}$
$C_{4}$ Alternative Minimal Special Degenerations

###  1.7. Lusztig’s Weyl group conjecture

In [Lus22, §0.4], Lusztig attached a Weyl group $W^{\prime}$ to each minimal
special degeneration. He then made a conjecture relating the exponents of
$W^{\prime}$ to what amounts to the $C({\mathfrak{s}})$-invariant part of the
intersection homology $\rm IH^{*}(\mathcal{S}_{\mathcal{O},e})$ when
$\dim(\mathcal{S}_{\mathcal{O},e})\geq 4$. In §9 we prove his conjecture,
which is open in types $B$, $C$, $D$, although we have to modify slightly the
$W^{\prime}$ that he attaches to those minimal degenerations $(\lambda,\mu)$
in type $D_{n}$ where there is a single odd part in $\mu$.

###  1.8. Acknowledgments

This work, major parts of which were sketched in 2012, is a continuation of
the papers [FJLSa], [FJLSb] that were jointly authored with Baohua Fu. We
thank him for his vital contribution to the project from its inception.

##  2\. Background material in the classical group case

###  2.1. Notation on partitions

In the classical groups it will be helpful to have a description of the
elements of ${\mathcal{N}}_{o}$ and the map $d$ in terms of partitions. We
introduce that notation following the references [CM93], [Car93], [Spa82].

Let $\mathcal{P}(N)$ denote the set of partitions of $N$. For
$\lambda\in\mathcal{P}(N)$, we write
$\lambda=[\lambda_{1},\dots,\lambda_{k}]$, where
$\lambda_{1}\geq\dots\geq\lambda_{k}>0$ and $|\lambda|:=\sum\lambda_{j}$ is
equal to $N$. Define

$m_{\lambda}(s)=\\#\\{j\ |\ \lambda_{j}=s\\},$

the multiplicity of the part $s$ in $\lambda$. We use $m(s)$ if the partition
is clear. Sometimes we write $[\dots,s^{m(s)},\dots]$ instead of

$[\dots,\overbrace{s,s,\dots,s}^{m(s)},\dots]$

for a part $s$ in $\lambda$. The set of nilpotent orbits ${\mathcal{N}}_{o}$
in ${\mathfrak{g}}=\mathfrak{sl}_{n}$ under the adjoint action of $G=SL_{n}$
is in bijection with $\mathcal{P}(n)$.

For $\epsilon\in\\{0,1\\}$, let $V=V_{\epsilon}$ be a vector space, of
dimension $N$, with a nondegenerate bilinear form satisfying $\langle
v,v^{\prime}\rangle=(-1)^{\epsilon}\langle v^{\prime},v\rangle$ for
$v,v^{\prime}\in V$. Let ${\mathfrak{g}}(V)$ be the Lie algebra associated to
the form on $V$, so that ${\mathfrak{g}}(V)=\mathfrak{so}_{N}$ when
$\epsilon=0$ and ${\mathfrak{g}}(V)=\mathfrak{sp}_{N}$ when $\epsilon=1$ and
$N$ is even.

Let

$\mathcal{P}_{\epsilon}(N):=\\{\lambda\in\mathcal{P}(N)\ |\ m(s)\equiv 0\text{
whenever }s\equiv\epsilon\\},$

where all congruences are modulo $2$. Then the set of nilpotent orbits
${\mathcal{N}}_{o}$ in ${\mathfrak{g}}(V)$ under the group $G=G(V)$ preserving
the form is given by $\mathcal{P}_{1}(2n)$ when ${\mathfrak{g}}$ is of type
$C_{n}$; by $\mathcal{P}_{0}(2n+1)$ when ${\mathfrak{g}}$ is of type $B_{n}$;
and by $\mathcal{P}_{0}(2n)$ when ${\mathfrak{g}}$ is of type $D_{n}$, except
that those partitions with all even parts correspond to two orbits in
${\mathcal{N}}_{o}$ (called the very even orbits, where there are two orbits
interchanged by the orthogonal group). We will also refer to
$\mathcal{P}_{1}(2n)$ as $\mathcal{P}_{C}(2n)$; to $\mathcal{P}_{0}(2n+1)$ as
$\mathcal{P}_{B}(2n+1)$; and to $\mathcal{P}_{0}(2n)$ as
$\mathcal{P}_{D}(2n)$. We sometimes call a partition
$\lambda\in\mathcal{P}_{\epsilon}(N)$ an $\epsilon$-partition. For
$\lambda\in\mathcal{P}(N)$ or $\lambda\in\mathcal{P}_{\epsilon}(N)$, we denote
by $\mathcal{O}_{\lambda}$ the corresponding nilpotent orbit in
${\mathfrak{g}}$.

Define the height of a part $s$ in $\lambda$ to be the number

$h_{\lambda}(s):=\\#\\{\lambda_{j}\,|\,\lambda_{j}\geq s\\}.$

We write $h(s)$ if the partition is clear. In terms of Young diagrams, the
position $(s,h(s))$ is a corner of the diagram, writing each part
$\lambda_{i}$ as the boxes with upper right corner
$(1,i),\dots(\lambda_{i},i)$. In other words, we have $\lambda_{h(s)}=s$ and
$\lambda_{h(s)+1}<\lambda_{h(s)}$.

The dual or transpose partition of $\lambda$, denoted $\lambda^{*}$, is
defined by

$(\lambda^{*})_{i}=\\#\\{j\ |\ \lambda_{j}\geq i\\}.$

If we set $j=h(s)$, then $\lambda^{*}$ is the partition with part $h(s)$
occurring $\lambda_{j}-\lambda_{j+1}=s-\lambda_{j+1}$ times.

The set $\mathcal{P}(N)$ is partially ordered by the dominance order on
partitions, where $\mu\preceq\lambda$ whenever
$\sum_{i=1}^{k}\mu_{i}\leq\sum_{i=1}^{k}\lambda_{i}$ for all $k$. This induces
a partial ordering on the sets $\mathcal{P}_{C}(2n)$, $\mathcal{P}_{B}(2n+1)$,
and $\mathcal{P}_{D}(2n)$ and these partial orderings coincide with the
partial ordering on nilpotent orbits given by the closure ordering. We will
refer to nilpotent orbits and partitions interchangeably in the classical
groups (with the caveat mentioned earlier for the very even orbits in type
$D$).

Let $X=B$, $C$, or $D$. Let $N$ be even (resp. odd) if $X$ is of type $C$ or
$D$ (resp. $B$). The $X$-collapse of $\lambda\in\mathcal{P}(N)$ is the
partition $\lambda_{X}\in\mathcal{P}_{X}(N)$ satisfying
$\lambda_{X}\preceq\lambda$ and such that if $\mu\in\mathcal{P}_{X}(N)$ and
$\mu\preceq\lambda$, then $\mu\preceq\lambda_{X}$. The $X$-collapse always
exists and is unique.

###  2.2. Special partitions and the duality maps

The special nilpotent orbits were defined by Lusztig [Lus79]. Denote by
${\mathcal{N}}^{sp}_{o}$ the special nilpotent orbits in
${\mathfrak{g}}={\mathfrak{g}}(V)$. All nilpotent orbits are special in type
$A$. Here we describe the special nilpotent orbits in types $B$, $C$, and $D$,
as well as introduce a second subset of ${\mathcal{N}}_{o}$, which behaves
like special orbits. We define four sets of partitions, with
$\epsilon^{\prime}\in\\{0,1\\}$, as follows

(1)
$\mathcal{P}_{\epsilon,\epsilon^{\prime}}(N):=\\{\lambda\in\mathcal{P}_{\epsilon}(N)\
|\ h(s)\equiv\epsilon^{\prime}\text{ whenever }s\equiv\epsilon\\}.$

Because of the $s=0$ case, for $N$ odd, the set is nonempty only when
$(\epsilon,\epsilon^{\prime})=(0,1)$. For $N$ even, the set is nonempty for
$(\epsilon,\epsilon^{\prime})\in\\{(0,0),(1,0),(1,1)\\}$. Then the partitions
for the special orbits in type $B_{n},C_{n},D_{n}$ are given by
$\mathcal{P}^{sp}_{B}(2n+1):=\mathcal{P}_{0,1}(2n+1)$,
$\mathcal{P}^{sp}_{C}(2n):=\mathcal{P}_{1,0}(2n)$, and
$\mathcal{P}^{sp}_{D}(2n):=\mathcal{P}_{0,0}(2n)$. The fourth case leads to a
second subset of $\mathcal{P}_{C}(2n)$, which is
$\mathcal{P}^{asp}_{C}(2n):=\mathcal{P}_{1,1}(2n)$. We refer to these
nilpotent orbits in type $C$ as the alternative special nilpotent orbits.

Each of $\mathcal{P}^{sp}_{B}(2n+1)$, $\mathcal{P}^{sp}_{C}(2n)$,
$\mathcal{P}^{sp}_{D}(2n)$, and $\mathcal{P}^{asp}_{C}(2n)$ inherits the
partial order from the set of all partitions and this agrees with the one
coming from inclusion of closures of the corresponding nilpotent orbits.

The sets $\mathcal{P}^{sp}_{B}(2n\\!+\\!1)$ and $\mathcal{P}^{sp}_{C}(2n)$ are
in bijection (see [Spa82], [KP89]), and also the sets
$\mathcal{P}^{sp}_{D}(2n)$ and $\mathcal{P}^{asp}_{C}(2n)$ are in bijection,
as we now describe.

Given $\lambda=[\lambda_{1}\geq\dots\geq\lambda_{k-1}\geq\lambda_{k}>0]$, let

$\lambda^{-}=[\lambda_{1}\geq\dots\geq\lambda_{k-1}\geq\lambda_{k}-1]$

and

$\lambda^{+}=[\lambda_{1}+1\geq\dots\geq\lambda_{k-1}\geq\lambda_{k}].$

Then the bijections are given as follows, using $f$ for each map:

(2) $\displaystyle
f_{BC}:\mathcal{P}^{sp}_{B}(2n+1)\to\mathcal{P}^{sp}_{C}(2n)\text{ given by
}f(\lambda)=(\lambda^{-})_{C}$ $\displaystyle
f_{CB}:\mathcal{P}^{sp}_{C}(2n)\to\mathcal{P}^{sp}_{B}(2n+1)\text{ given by
}f(\lambda)=(\lambda^{+})_{B}$ $\displaystyle
f_{DC}:\mathcal{P}^{sp}_{D}(2n)\to\mathcal{P}^{asp}_{C}(2n)\text{ given by
}f(\lambda)=((\lambda^{+})^{-})_{C}$ $\displaystyle
f_{CD}:\mathcal{P}^{asp}_{C}(2n)\to\mathcal{P}^{sp}_{D}(2n)\text{ given by
}f(\lambda)=\lambda_{D}$

Note that in general $f$ maps $\mathcal{P}_{\epsilon,\epsilon^{\prime}}$ to
$\mathcal{P}_{1-\epsilon,1-\epsilon^{\prime}}$.

Each of these maps respects the partial order. The first two maps are
dimension preserving (and codimension preserving). The second two maps are
codimension preserving (as we shall see). More precisely, the $f_{DC}$ map
sends an orbit of dimension $N$ to one of dimension $N+2n$. The shift is
because the minimal orbit in $C_{n}$ is the minimal element in
$\mathcal{P}^{asp}_{C}(2n)$.

Write $d(\lambda)$ for $\lambda^{*}$. It is known (or easy to show) that $d$
determines bijections between the following sets:

(3) $\displaystyle\mathcal{P}^{sp}_{B}(2n\\!+\\!1)$
$\displaystyle\xrightarrow{\text{ \, d \, }}\mathcal{P}^{sp}_{B}(2n\\!+\\!1)$
$\displaystyle\mathcal{P}^{sp}_{C}(2n)$ $\displaystyle\xrightarrow{\text{ \, d
\, }}\mathcal{P}^{sp}_{C}(2n)$ $\displaystyle\mathcal{P}^{sp}_{D}(2n)$
$\displaystyle\xleftrightarrow{\text{ \, d \, }}\mathcal{P}^{asp}_{C}(2n)$

It is order-reversing since this holds for all partitions. We refer to $d$ as
the internal duality or just transpose.

It is known that $d\circ f=f\circ d$ and the duality $d_{LS}$ of Lusztig-
Spaltenstein is given by $d_{LS}=d\circ f=f\circ d$. We have squares relating
the three kinds of maps between the orbits (or their corresponding partitions)
as shown in Figure 1.

###  2.3. Explicit description of $f$ maps

We now describe more specifically how the $f$ maps work.

Let $X$ be one of the types $B$, $C$, $D$. Let $\lambda\in\mathcal{P}(N)$ with
$N$ even for type $C$ and $D$ and odd for type $B$. We want to find
$\lambda_{X}$. Set $\epsilon=\epsilon_{X}$. List the parts $s$ in $\lambda$
with $s\equiv\epsilon$ and $m(s)$ odd as $a_{1}>a_{2}>\dots>a_{2n} \geq 0$.
For $X=C$, these are the odd parts, so there is an even number of them. But
for $X=B$ or $X=D$, since $\epsilon=0$, we will add a single part equal to
$0$, if necessary, so that there is an even number of even parts with odd
multiplicity. Next, between $a:=a_{2i-1}$ and $c:=a_{2i}$, list the parts
$s\equiv\epsilon$ as $b_{1}>b_{2}>\dots>b_{j}$, which necessarily have
$m(b_{i})$ even. Ignoring the parts not congruent to $\epsilon$ , then
$\lambda$ will look locally like

$a^{m(a)},b_{1}^{m(b_{1})},\dots,b_{j}^{m(b_{j})},c^{m(c)}.$

Then under the collapse of $\lambda$ to $\lambda_{X}$, these values will
change to

$a^{m(a)-1},a{-}1,b_{1}{+}1,b_{1}^{m(b_{1})-2},b_{1}{-}1,\dots,b_{j}{+}1,b_{j}^{m(b_{j})-2},b_{j}{-}1,c{+}1,c^{m(c)-1}$

so that the multiplicities of the parts congruent to $\epsilon$ are now even,
as required to be in $\mathcal{P}_{X}(N)$. The other parts of $\lambda$ are
unaffected under the collapse. As a result of this rule, there is a formula
for the collapse for a part $s$ based on its height $h(s)$ and its
multiplicity $m(s)$.

######  Lemma 2.1.

Let $X$ be $B,C,$ or $D$ and $\epsilon=\epsilon_{X}$. Let
$\lambda\in\mathcal{P}(N)$ with $N$ as above. Assume $m(s)$ is even if
$s\not\equiv\epsilon$. Let $s\equiv\epsilon$ be a part in $\lambda$. Then
$[s^{m(s)}]$ in $\lambda$ changes to the following in $\lambda_{X}$:

$\displaystyle[s^{m(s)-1},s{-}1]$ $\displaystyle\text{ if }h(s)\equiv
1,m(s)\equiv 1$ $\displaystyle[s{+}1,s^{m(s)-1}]$ $\displaystyle\text{ if
}h(s)\equiv 0,m(s)\equiv 1$ $\displaystyle[s{+}1,s^{m(s)-2},s{-}1]$
$\displaystyle\text{ if }h(s)\equiv 1,m(s)\equiv 0$ $\displaystyle[s^{m(s)}]$
$\displaystyle\text{ if }h(s)\equiv 0,m(s)\equiv 0$

###### Proof.

Since $h(s)=\sum_{j\geq s}m(j)$, it is clear that $h(s)\equiv\\#\\{m(j)\ |\
m(j)\text{ is odd and }j\geq s\\}$. Since any part $s$ with
$s\not\equiv\epsilon$ has $m(s)$ even, it follows that

$h(s)\equiv\\#\\{m(j)\ |\ m(j)\equiv 1,j\geq s,\text{ and
}j\equiv\epsilon\\}.$

So the four conditions in the lemma specify whether the part $s$ plays the
role of some $a_{2i-1}$, $a_{2i}$, $b_{k}$, or a part between some $a_{2i}$
and $a_{2i+1}$ and hence unaffected by the collapse. ∎

Let $X$ be one of the types $B,C,D$ or $C^{\prime}$. Here $C^{\prime}$, refers
to the alternative special setting. Now we can say what happens under the $f$
maps passing from $X$ to type $f(X)$.

######  Lemma 2.2.

Let $\lambda\in\mathcal{P}^{sp}_{X}(N)$. Let $s\not\equiv\epsilon_{X}$ be a
part in $\lambda$. Then $f(\lambda)$ is computed by replacing each occurrence
of $[s^{m(s)}]$ by

(4) $\displaystyle[s^{m(s)-1},s{-}1]$ $\displaystyle\text{ if
}h(s)\equiv\epsilon^{\prime}_{X},m(s)\equiv 1$ (5)
$\displaystyle[s{+}1,s^{m(s)-1}]$ $\displaystyle\text{ if
}h(s)\not\equiv\epsilon^{\prime}_{X},m(s)\equiv 1$ (6)
$\displaystyle[s{+}1,s^{m(s)-2},s{-}1]$ $\displaystyle\text{ if
}h(s)\equiv\epsilon^{\prime}_{X},m(s)\equiv 0$ (7) $\displaystyle[s^{m(s)}]$
$\displaystyle\text{ if }h(s)\not\equiv\epsilon^{\prime}_{X},m(s)\equiv 0$

###### Proof.

First, if $s\not\equiv\epsilon_{X}$, then $s\equiv\epsilon_{f(X)}$. For
$f_{BC}$ and $f_{CD}$ the map is just the ordinary collapse (except for the
smallest two parts in type $B$). In these cases, $\epsilon^{\prime}_{X}=1$ and
we are in the situation of the previous lemma when performing the collapse in
type $f(X)$. In type $B$, there are a couple of cases to check that the effect
of $\lambda_{k}$ being replaced by $\lambda_{k}{-}1$ is consistent with the
above cases.

On the other hand, for $f_{DC}$ and $f_{CB}$ we have
$\epsilon^{\prime}_{X}=0$. For the $f$ map, we first increase $\lambda_{1}$ by
1 and then perform the collapse for type $f(X)$. This $1$, under the collapse,
moves down to the first part $x$ with $x\not\equiv\epsilon_{X}$. By the
assumption that the parts congruent to $\epsilon_{X}$ have even multiplicity,
we have that $h(s)$ odd. So the rule is correct for the part $x$. Call this
new partition, where $x$ changes to $x+1$, $\lambda^{\prime}$. Then

$h_{\lambda^{\prime}}(s)\not\equiv\\#\\{m_{\lambda^{\prime}}(j)\ |\
m_{\lambda^{\prime}}(j)\equiv 1,j\geq s,\text{ and
}j\equiv\epsilon_{f(X)}\\}.$

Since $\epsilon^{\prime}_{X}=0$, the previous lemma gives the result again for
the collapse of $\lambda^{\prime}$, which is $f(\lambda)$. For $f_{DC}$, there
are a couple of cases to check that the effect of $\lambda_{k}$ being replaced
by $\lambda_{k}-1$ is consistent with the above cases. ∎

###  2.4. Special pieces result

Spaltenstein [Spa82] showed that each non-special orbit $\mathcal{O}^{\prime}$
belongs to the closure of a unique special orbit $\mathcal{O}$, which is
minimal among all special orbits whose closure contains
$\mathcal{O}^{\prime}$. That is, if a special orbit contains
$\mathcal{O}^{\prime}$ in its closure, then it contains $\mathcal{O}$ in its
closure.

We now describe the process for finding the partition $\lambda$ for
$\mathcal{O}$ given the partition $\nu$ for $\mathcal{O}^{\prime}$. Let $X$ be
one of the four types $B,C,D,C^{\prime}$. Let $\nu\in\mathcal{P}_{X}(N)$ be
non-special. Let $S$ be the collection of parts $s$ in $\nu$ such that
$s\equiv\epsilon_{X}$ and $h(s)\not\equiv\epsilon^{\prime}_{X}$. These are the
parts that fail the condition for $\nu$ to be in
$\mathcal{P}_{\epsilon,\epsilon^{\prime}}$ as required by (1). Note that
$s\equiv\epsilon_{X}$ means that $m(s)$ is even, so $m(s)\geq 2$. Let
$\lambda$ be obtained from $\nu$ by replacing the subpartition $[s^{m(s)}]$ in
$\nu$ by $[s{+}1,s^{m(s)-2},s{-}1]$, for each $s\in S$. It is clear that
$\lambda\in\mathcal{P}_{X}(N)$ and that it satisfies the special condition in
(1), so lies in $\mathcal{P}^{sp}_{X}(N)$. In fact, $\lambda$ is the partition
for $\mathcal{O}$ by [KP89] for the cases of $B,C,D$ (the case of $C^{\prime}$
is similar).

###  2.5. Removing rows and columns from a partition

In [KP81] and [KP82], Kraft and Procesi defined two operations that take a
pair of partitions $(\lambda,\mu)\in\mathcal{P}(N)$ to another pair of
partitions. Viewing the partitions as Young diagrams, the first operation is
removing common initial rows of $\lambda$ and $\mu$ and the second operation
is removing common initial columns.

####  2.5.1. Type $A$

More precisely, we say $(\lambda,\mu)$ is leading-row-equivalent (after
removing $r$ rows) to $(\lambda^{\prime},\mu^{\prime})$ if
$\lambda_{i}=\mu_{i}$ for $i\leq r$, while
$\lambda_{i}=\lambda^{\prime}_{i-r}$ and $\mu_{i}=\mu^{\prime}_{i-r}$ for
$i>r$. We say $(\lambda,\mu)$ is column-equivalent (after removing $s$
columns) to $(\lambda^{\prime},\mu^{\prime})$ if $\lambda_{i}=\mu_{i}$ for
$i>\ell$ and $\lambda_{i}=\lambda^{\prime}_{i}+s$ and
$\mu_{i}=\mu^{\prime}_{i}+s$ for $i\leq\ell$, where $\ell=\max\\{i\ |\
\lambda_{i}>s\\}$. In both cases, $|\lambda^{\prime}|=|\mu^{\prime}|$, so
$\lambda^{\prime}$ and $\mu^{\prime}$ are partitions of the same integer. We
say $(\lambda,\mu)$ is equivalent to $(\lambda^{\prime},\mu^{\prime})$ if they
are related by a sequence of these two equivalences, and it follows in that
case when $\lambda\preceq\mu$ that

  1. (1)

$\lambda^{\prime}\preceq\mu^{\prime}$

  2. (2)

$\operatorname{codim}_{\bar{\mathcal{O}}_{\lambda}}\mathcal{O}_{\mu}=\operatorname{codim}_{\bar{\mathcal{O}}_{\lambda^{\prime}}}\mathcal{O}_{\mu^{\prime}}$

  3. (3)

The singularity of $\bar{\mathcal{O}}_{\lambda}$ at $\mathcal{O}_{\mu}$ is
smoothly equivalent to the singularity of
$\bar{\mathcal{O}}_{\lambda^{\prime}}$ at $\mathcal{O}_{\mu^{\prime}}$.

for the corresponding nilpotent orbits in $\mathfrak{sl}_{n}$ [KP81].

####  2.5.2. Other classical types

For $\epsilon\in\\{0,1\\}$ and ${\mathfrak{g}}={\mathfrak{g}}(V_{\epsilon})$
as in §2.1, similar results hold as above hold when we cancel $r$ leading rows
and $s$ columns, with an additional condition. Let
$\lambda,\mu\in\mathcal{P}_{\epsilon}(N)$ and assume when we cancel $r$
leading rows that

(8) $[\lambda_{1},\dots,\lambda_{r}]\text{ is an }\epsilon\text{-partition.}$

This condition always holds if we choose the maximal possible number of rows
to cancel between $\lambda$ and $\mu$. If (8) holds, then $\lambda^{\prime}$
and $\mu^{\prime}$ are ${\tilde{\epsilon}}$-partitions, with
$\tilde{\epsilon}\equiv\epsilon+s$, where $s$ is the number of columns
canceled. Then the above three results hold when the nilpotent orbits are
considered in ${\mathfrak{g}}$ [KP82, §13]. A pair of partitions
$(\lambda,\mu)$ is irreducible if no common rows or columns can be canceled.

Next, we say $(\lambda,\mu)\in\mathcal{P}_{\epsilon}(N)$ is $\epsilon$-row-
equivalent to $(\lambda^{\prime},\mu^{\prime})\in\mathcal{P}_{\epsilon}(m)$,
if the latter is obtained from the former by canceling some leading and some
trailing rows of the Young diagram. Namely, there exist
$r,r^{\prime}\in\mathbb{N}$ so that $\lambda_{i}=\mu_{i}$ for $i\leq r$ and
$i\geq r^{\prime}$, while $\lambda_{i}=\lambda^{\prime}_{i-r}$ and
$\mu_{i}=\mu^{\prime}_{i-r}$ for $r<i<r^{\prime}$. We pad the partitions by
adding zeros so that both partitions have the same number of parts. If we set
$\nu=[\lambda_{1},\dots,\lambda_{r},\lambda_{r^{\prime}},\lambda_{r^{\prime}+1},\dots]$,
then $\nu$ is also an $\epsilon$-partition. We also say $(\lambda,\mu)$ is
locally of the form $(\lambda^{\prime},\mu^{\prime})$.

Now suppose that $(\lambda,\mu)$ is $\epsilon$-row-equivalent to
$(\lambda^{\prime},\mu^{\prime})$. Let $V=V_{\epsilon}$. Then, as in [KP82,
§13.4], there is an orthogonal decomposition $V=V_{1}\oplus V_{2}$, with $\dim
V_{1}=|\lambda^{\prime}|=|\mu^{\prime}|$ and $\dim V_{2}=|\nu|$ and the
$V_{i}$ carry a nondegenerate $\epsilon$-form by restriction from $V$.
Moreover, $\lambda^{\prime},\mu^{\prime}\in\mathcal{P}_{\epsilon}(\dim V_{1})$
and $\nu\in\mathcal{P}_{\epsilon}(\dim V_{2})$, so we can pick nilpotent
elements $x_{1},e_{1}\in{\mathfrak{g}}(V_{1})$ with partitions
$\lambda^{\prime},\mu^{\prime}$, respectively, and
$e_{2}\in{\mathfrak{g}}(V_{2})$ with partition $\nu$. Then $x=x_{1}+e_{2}$ has
partition $\lambda$ and $e=e_{1}+e_{2}$ has partition $\mu$. The arguments in
§13 in [KP82] give

######  Proposition 2.3.

Choose an $\mathfrak{sl}_{2}$-triple for $x_{1}$ in ${\mathfrak{g}}(V_{1})$.
Then the natural map of ${\mathfrak{g}}(V_{1})$ to
${\mathfrak{g}}(V_{1})+e_{2}\subset{\mathfrak{g}}$ gives an isomorphism of the
slice ${\mathcal{S}}_{\mathcal{O}_{\lambda^{\prime}},e_{1}}$ in
${\mathfrak{g}}(V_{1})$ to the slice ${\mathcal{S}}_{\mathcal{O}_{\lambda},e}$
in ${\mathfrak{g}}$.

The key ideas in the proof are that both slices have the same dimension (by
the codimension result) and the fact that the closure of any nilpotent orbit
in $\mathfrak{gl}_{N}$ is normal.

We note that if $(\lambda^{\prime},\mu^{\prime})$ are obtained from
$(\lambda,\mu)$ by removing $r$ leading rows and $s$ columns and if condition
(8) holds, then $(\lambda,\mu)$ is $\epsilon$-row-equivalent to :

(9)
$(\lambda^{\prime\prime},\mu^{\prime\prime}):=([\lambda^{\prime}_{1}+s,\lambda^{\prime}_{2}+s,\dots],[\mu^{\prime}_{1}+s,\mu^{\prime}_{2}+s,\dots])$

Finally, we call $(\lambda,\mu)$ and
$(\lambda^{\prime\prime},\mu^{\prime\prime})$ locally equivalent, or say
locally $(\lambda,\mu)$ is equal to
$(\lambda^{\prime\prime},\mu^{\prime\prime})$.

In the next section §3, we show that each pair of partitions corresponding to
a minimal special degeneration in orthogonal and symplectic type is equivalent
to a unique pair of partitions $(\lambda,\mu)$ of $N$ for a unique smallest
$N$. These pairs are irreducible in the sense of Kraft and Procesi: the
maximal possible number of common rows and columns has been removed from the
original pair of partitions to obtain $(\lambda,\mu)$.

##  3\. Combinatorial classification of minimal special degenerations in $B$,
$C$, $D$

######  Theorem 3.1.

Let $(\lambda,\mu)\in\mathcal{P}_{\epsilon,\epsilon^{\prime}}$ be partitions
corresponding to a minimal special degeneration in the corresponding classical
Lie algebra. Then $(\lambda,\mu)$ is equivalent to a unique entry in Table 1
or Table 2.

###### Proof.

If a minimal special degeneration $(\lambda,\mu)$ is not already minimal, then
there exists a non-special orbit $\mathcal{O}_{\nu}$ such that
$\mathcal{O}_{\mu}\leq\mathcal{O}_{\nu}\leq\mathcal{O}_{\lambda}$, and such
that $(\nu,\mu)$ is a minimal degeneration. Hence, the latter would be one of
the entries in the Kraft-Procesi list [KP82, §3.4]. We need to show that
$(\lambda,\mu)$ must be equivalent to one of the five cases in Table 2.

First, since $\mathcal{O}_{\nu}$ is not special, there is a unique special
orbit whose closure contains $\mathcal{O}_{\nu}$ and which is contained in the
closure of all other special orbits whose closure contains $\mathcal{O}_{\nu}$
(see §2.4). Consequently, $\mathcal{O}_{\lambda}$ must be this orbit, as we
are assuming the degeneration $(\lambda,\mu)$ is minimal among special
degenerations.

Next, we will show that $(\nu,\mu)$ cannot be one of the cases in Table 1. Let
$X$ be the type of the Lie algebra and $\epsilon=\epsilon_{X}$,
$\epsilon^{\prime}=\epsilon^{\prime}_{X}$.

If $(\nu,\mu)$ is type $a$, then locally it is $([s{+}2,s],[(s{+}1)^{2}])$
where $s\not\equiv\epsilon$. Since $s{+}1\equiv\epsilon$ and $s{+}1$ appears
exactly twice, the heights satisfy $h_{\nu}(x)\equiv h_{\mu}(x)$ for all
$x\equiv\epsilon$. This means that $\nu$ must be special since $\mu$ is
special. Therefore the type $a$ minimal degeneration cannot occur between a
larger orbit that is not special and a smaller orbit that is special.

If $(\nu,\mu)$ is type $b$, then locally it is
$([s{+}2n,s],[s{+}2n{-}2,s{+}2])$ where $s\not\equiv\epsilon$. Hence, all four
of $s{+}2n,s,s{+}2n{-}2,$ and $s{+}2$ are not congruent to $\epsilon$. As in
the previous case, $h_{\nu}(x)\equiv h_{\mu}(x)$ for all $x\equiv\epsilon$,
and again this forces $\nu$ to be special too, a contradiction.

If $(\nu,\mu)$ is of type $c,d$ or $e$, it will be possible for $\nu$ to be
non-special, but we will show that then the degeneration $(\lambda,\mu)$ is
not minimal among degenerations between special orbits.

For type $c$, the pair $(\nu,\mu)$ is locally
$([s\\!+\\!2n\\!+\\!1,s^{2}],[s\\!+\\!2n\\!-\\!1,(s\\!+\\!1)^{2}])$ where
$s\equiv\epsilon$. In this case $\nu$ will be non-special, as noted in the
Table 1, exactly when the number of rows $l$ removed is congruent to
$\epsilon^{\prime}$. This means $h_{\nu}(s)=l+3\not\equiv\epsilon^{\prime}$
(and necessarily $s\geq 1$). If that is the case, then by §2.4, $\lambda$ must
locally be $[s\\!+\\!2n\\!+\\!1,s\\!+\\!1,s\\!-\\!1]$ . But then $\lambda$
degenerates to the partition $\nu^{\prime}$ that is locally
$[s\\!+\\!2n\\!-\\!1,s\\!+\\!3,s\\!-\\!1]$, which is also special and
degenerates to $\mu$. Hence the degeneration $(\lambda,\mu)$ is not a minimal
special degeneration, which is what we wanted to show.

For type $d$, the pair $(\nu,\mu)$ is locally

$([s\\!+\\!2n\\!+\\!1,s\\!+\\!2n\\!+\\!1,s],[s\\!+\\!2n,s\\!+\\!2n,s\\!+\\!2])$

where $s\not\equiv\epsilon$ . In this case $\nu$ will be non-special exactly
when $h_{\nu}(s\\!+\\!2n\\!+\\!1)\not\equiv\epsilon^{\prime}$. If that is the
case, then $\lambda$ must locally be $[s\\!+\\!2n\\!+\\!2,s\\!+\\!2n,s]$ from
§2.4. But $\lambda$ also degenerates to the partition $\nu^{\prime}$ that is
locally $[s\\!+\\!2n\\!+\\!2,s\\!+\\!2n{-}2,s\\!+\\!2]$, which is also special
and degenerates to $\mu$. Hence the degeneration $(\lambda,\mu)$ is not a
minimal special degeneration.

For type $e$, the pair $(\nu,\mu)$ is locally

$([s\\!+\\!2n,s\\!+\\!2n,s,s],[s\\!+\\!2n\\!-\\!1,s\\!+\\!2n\\!-\\!1,(s\\!+\\!1)^{2}])$

where $s\equiv\epsilon$. In this case $\nu$ will be non-special exactly when
$h_{\nu}(s\\!+\\!2n)\not\equiv\epsilon^{\prime}$ (and $s\geq 1$ is forced).
Then $\lambda$ must locally be
$[s{+}2n\\!+\\!1,s\\!+\\!2n\\!-\\!1,s\\!+\\!1,s\\!-\\!1]$ by §2.4. But
$\lambda$ degenerates to the partition $\nu^{\prime}$ that is locally
$[s\\!+\\!2n\\!-\\!1,s\\!+\\!2n{-}1,s\\!+\\!3,s\\!-\\!1]$, whenever $n\geq 2$.
This orbit is special since $\mu$ is. Moreover, $\nu^{\prime}$ degenerates to
$\mu$, so $(\lambda,\mu)$ is not a minimal special degeneration.

This shows, for any minimal special degeneration $(\lambda,\mu)$, which is not
already a minimal degeneration, that there exists an intermediate orbit $\nu$
such that $(\nu,\mu)$ is a minimal degeneration of codimension at least $4$,
unless $\nu$ is of type $e$ with $n=1$. In Kraft-Procesi’s classification
[KP82, Table I], the minimal degenerations of dimension at least $4$ are
labeled $f,g$ and $h,$ and are given by the minimal nilpotent orbit closures
in types $B$, $C$ and $D$, respectively.

Starting with type $g$, where $n\geq 2$, the pair $(\nu,\mu)$ is locally

$([s\\!+\\!2,(s\\!+\\!1)^{2n-2},s],[(s\\!+\\!1)^{2n}])$

with $s\not\equiv\epsilon$. Then $\nu$ is never special since $\mu$, being
special, forces $h_{\nu}(s{+}1)=h_{\mu}(s{+}1){+}1$ to fail the special
condition. Then $\lambda$ is forced, locally, to equal
$[(s\\!+\\!2)^{2},(s\\!+\\!1)^{2n-4},s^{2}]$ by §2.4. Because $\mu$ is
special, the number of rows $l$ removed is congruent to $\epsilon^{\prime}$.
After removing $s$ columns, we see that $(\lambda,\mu)$ has the type of
$g_{sp}$, and the latter is indeed a minimal special degeneration, containing
only the (non-special) orbit between $\lambda$ and $\mu$.

For type $f$, the pair $(\nu,\mu)$ is locally

$([(s\\!+\\!2)^{2},(s\\!+\\!1)^{2n-3},s^{2}],[(s\\!+\\!1)^{2n+1}])$

with $s\equiv\epsilon$ and $n\geq 2$. This is never special since
$h_{\nu}(s+2)$ and $h_{\nu}(s)$ have different parities, so exactly one of
them fails the special condition. In the former case, $\lambda$ is locally
equal to $[s\\!+\\!3,(s\\!+\\!1)^{2n-2},s^{2}]$ and the degeneration is given
by $f^{1}_{sp}$ in the Table 2. That this is a minimal such degeneration
follows since $\nu$ is the only (non-special) orbit between $\lambda$ and
$\mu$. In the latter case, which forces $s\geq 1$, then $\lambda$ is locally
equal to $[(s\\!+\\!2)^{2},(s\\!+\\!1)^{2n-2},s\\!-\\!1]$ and this is the
minimal special degeneration $f^{2}_{sp}$. Again, $\nu$ is the only (non-
special) orbit between $\lambda$ and $\mu$.

Finally, assume the pair $(\nu,\mu)$ is locally

$([(s\\!+\\!2)^{2},(s\\!+\\!1)^{2n-4},s^{2}],[(s\\!+\\!1)^{2n}])$

with $s\equiv\epsilon$ for $n\geq 2$. This is type $e$ if $n=2$ (for $n=1$ in
the table for $e$) and type $h$ for $n\geq 3$. Observe that the special
condition $h_{\nu}(s+2)\equiv\epsilon^{\prime}$ is satisfied if and only if
$h_{\nu}(s)\equiv\epsilon^{\prime}$, since these heights differ by the even
number $2n-4$. If both conditions are met, then $\nu$ will be special since
$\mu$ is special and this is handled by the minimal degeneration cases.

Otherwise, both the pairs $(s{+}2,s{+}2)$ and $(s,s)$ in $\nu$ cause $\nu$ to
fail to be special, which implies $s\geq 1$. Then $\lambda$ takes the form
locally $[s\\!+\\!3,(s\\!+\\!1)^{2n-2},s\\!-\\!1]$ by §2.4. This is the form
of $h_{sp}$ in Table 2 after removing $s{-}1$ columns. This is a minimal
special degeneration containing $3$ (non-special) orbits between $\lambda$ and
$\mu$:

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The four unlabeled edges all have an $A_{1}=C_{1}$ singularity (type $a$).

We have therefore shown that every minimal special degeneration is either
minimal or takes the form in Table 2. ∎

Next we will show that each degeneration in Table 2 has the given singularity
type.

##  4\. Determining the singularities in Table 2

For each type in Table 2, we need to show that the degeneration is as
promised. The case of type $h$ was done in [KP82]. We begin with the $g_{sp}$
case.

###  4.1. Type $g_{sp}$ case

As discussed in §2, for the classical Lie algebras $\mathfrak{so}_{2n+1}$,
$\mathfrak{sp}_{2n}$, and $\mathfrak{so}_{2n}$, the nilpotent orbits under the
groups $\rm O(2n+1)$, $\operatorname{Sp}(2n)$, and $\rm O(2n)$ are
parametrized by partitions in $\mathcal{P}_{B}(2n+1)$, $\mathcal{P}_{C}(2n)$,
and $\mathcal{P}_{D}(2n)$. This occurs via the Jordan-canonical form of the
matrix in the ambient general linear Lie algebra.

Let $e\in{\mathfrak{g}}$ be nilpotent. Fix an $\mathfrak{sl}_{2}$-subalgebra
${\mathfrak{s}}$ through $e$ and let ${\mathfrak{c}}({\mathfrak{s}})$ be the
centralizer of ${\mathfrak{s}}$ in ${\mathfrak{g}}$, which is a maximal
reductive subalgebra of the centralizer of $e$ in ${\mathfrak{g}}$. Let
$C({\mathfrak{s}})$ be the centralizer in $G$. Then $C({\mathfrak{s}})$ is a
product of orthogonal and symplectic groups, with each part $s$ of $\lambda$
contributing a factor $G^{s}$, which is isomorphic to $\mathrm{O}(m(s))$ when
$s\not\equiv\epsilon$ and isomorphic to $\operatorname{Sp}(m(s))$ when
$s\equiv\epsilon$. Denote by ${\mathfrak{g}}^{s}$ the Lie algebra of $G^{s}$.
See [CM93] for this background material.

Let $V$ denote the defining representation of ${\mathfrak{g}}$ via the ambient
general linear Lie algebra. If $\lambda$ is the partition corresponding to
$e$, then under ${\mathfrak{s}}$, the representation $V$ decomposes as a
direct sum

$\bigoplus_{s}V(s{-}1)^{\oplus m(s)}$

over the distinct parts $s$ of $\lambda$. Here $V(m)$ is the irreducible
$\mathfrak{sl}_{2}$-representation of highest weight $m$.

Now let $e_{0}\in{\mathfrak{c}}({\mathfrak{s}})$ be nilpotent. Then
$e_{0}=\sum_{s}e^{(s)}_{0}$ for some nilpotent
$e^{(s)}_{0}\in{\mathfrak{g}}^{s}$. Choose an $\mathfrak{sl}_{2}$-subalgebra
through $e^{(s)}_{0}$ in ${\mathfrak{g}}^{s}$ and let ${\mathfrak{s}}_{0}$ be
the diagonal $\mathfrak{sl}_{2}$-subalgebra for

$e_{0}=\sum_{s}e^{(s)}_{0}.$

Each $e^{(s)}_{0}$ corresponds to a partition $\mu^{(s)}$ of $m(s)$, using the
defining representation of ${\mathfrak{g}}^{s}$.

Under the sum ${\mathfrak{s}}\oplus{\mathfrak{s}}_{0}$, $V$ decomposes as

$\bigoplus_{s,j}V(s{-}1)\otimes V({\mu^{(s)}_{j}{-}1})$

where $s$ runs over the distinct parts of $\lambda$ and $j$ indexes the parts
of $\mu_{s}$.

Now consider the diagonal $\mathfrak{sl}_{2}$-subalgebra for $e+e_{0}$ in
${\mathfrak{s}}+{\mathfrak{s}}_{0}$. An application of the Clebsch-Gordan
formula immediately gives

######  Lemma 4.1.

The nilpotent element $e+e_{0}$ in ${\mathfrak{g}}$ has partition equal to the
union of the partitions

$\displaystyle[s{+}\mu^{(s)}_{j}{-}1,$ $\displaystyle
s{+}\mu^{(s)}_{j}{-}3,\dots,|s-\mu^{(s)}_{j}|{+}1]$

for each distinct part $s$ in $\lambda$ and each part $\mu^{(s)}_{j}$ of
$\mu^{(s)}$.

Suppose that $e_{0}\in{\mathfrak{c}}({\mathfrak{s}})$ is a nilpotent element
such that each $e^{(s)}_{0}\in{\mathfrak{g}}^{s}$ has partition of the form

(10) $\mu^{(s)}=[2^{a_{s}},1^{b_{s}}]$

for some positive integers $a_{s}$ and $b_{s}$ with $2a_{s}+b_{s}=m(s)$. Then
the partition $\nu$ of $e+e_{0}$ equals the union of the partitions

$\displaystyle[(s+1)^{a_{\tiny i}},s^{b_{i}},(s-1)^{a_{i}}]$

for each part $s$ in $\lambda$. This follows immediately from the previous
lemma since the part $s$ contributes $[s{+}1,s{-}1]$ to $\nu$ when
$\mu^{(s)}_{j}=2$ and it contributes $[s]$ when $\mu^{(s)}_{j}=1$.

######  Proposition 4.2.

Let $e_{0}\in{\mathfrak{c}}({\mathfrak{s}})$ be a nilpotent element satisfying
(10). Let $\mathcal{O}$ be the orbit through $e+e_{0}$. Then the slice
${\mathcal{S}}_{\mathcal{O},e}$ is isomorphic to

(11) $\prod_{s}\overline{\mathcal{O}}_{\mu^{(s)}}$

where the product is over the distinct parts $s$ of $\lambda$. Here,
$\mathcal{O}_{\mu^{(s)}}$ is the orbit with partition $\mu^{(s)}$ in
$\mathfrak{sp}(m(s))$ if $s\equiv\epsilon$ and in $\mathfrak{so}(m(s))$ if
$s\not\equiv\epsilon$.

###### Proof.

The partition of $e^{(s)}_{0}$ in ${\mathfrak{g}}$ (rather than
${\mathfrak{g}}^{i}$) is equal to

$[2^{s\cdot a_{s}},1^{N-2s\cdot a_{s}}]$

where $N$ is the dimension of $V$. Setting $a=\sum_{s}a_{s}$, the partition of
$e_{0}$ is equal to

$[2^{a},1^{N-2a}],$

which is of height $2$ in ${\mathfrak{g}}$. Then Corollary 4.9 in [FJLS17]
implies that ${\mathcal{S}}_{\mathcal{O},e}$ and
$\prod_{s}\overline{\mathcal{O}}_{\mu^{s}}$ have the same dimension. The
latter is isomorphic to

$f+\overline{C({\mathfrak{s}})\cdot(e+e_{0})}=f+e+\overline{C({\mathfrak{s}})\cdot
e_{0}},$

which is a subvariety of ${\mathcal{S}}_{\mathcal{O},e}$. The result follows
from [KP82, Cor 13.3] if we can show that $\overline{\mathcal{O}}$ is normal
at $e$.

Since the only minimal degeneration of $[2^{a_{s}},1^{b_{s}}]$ in
${\mathfrak{g}}^{s}$ is $[2^{a_{s}-1},1^{b_{s}+2}]$ when $a_{s}>1$ is of
minimal type (that is, types $a$, $f$, $g$, or $h$ in [KP82, Table 1]), the
only minimal degenerations of $\mathcal{O}$ that contain $e$ are also of
minimal type. The argument in [KP82, Thm 16.2] then shows that
$\overline{\mathcal{O}}$ is normal at $e$. ∎

We can now prove the $g_{sp}$ case.

######  Corollary 4.3.

Let $\mathcal{O}=\mathcal{O}_{\lambda}$ and $e\in\mathcal{O}_{\mu}$, where
$(\lambda,\mu)$ are of type $g_{sp}$ in Table 2. Then
${\mathcal{S}}_{\mathcal{O},e}$ is isomorphic to
$\overline{\mathcal{O}}_{[2^{2},1^{2n-4}]}$, the closure of the minimal
special orbit in $\mathfrak{sp}_{2n}$.

###### Proof.

If $k$ is the number of columns removed, then $s=k+1$ is the relevant part of
$\mu$ and $m(s)=2n$. We take $e_{0}$ to be exactly equal to
$e^{(s)}_{0}\in{\mathfrak{g}}^{s}$ with partition $\mu^{s}=[2^{2},1^{2n-4}]$.
Then the result follows from the proposition. Since $k\not\equiv\epsilon$ for
the $g_{sp}$ case, we have $s\equiv\epsilon$ and
${\mathfrak{g}}^{s}=\mathfrak{sp}_{2n}$. ∎

######  Remark 4.4.

The $h$ case in the table, already done in [KP82], corresponds to the
situation where $s\not\equiv\epsilon$ and $m(s)$ is even, as well as having
$\mathcal{O}_{\lambda}$ is special (so the condition on $l$ holds). Of course,
the slice is still of type $h$ even if the degeneration is not a minimal
special one.

###  4.2. Type $f_{sp}^{1}$ and $f_{sp}^{2}$

These cases occur when $(\lambda,\mu)$ is $\epsilon$-row equivalent to
$(\lambda^{(i)},[a^{N}])$ where $N$ is odd and $a\not\equiv\epsilon$ and

$\displaystyle\lambda^{(1)}=[a{+}2,a^{N-3},(a{-}1)^{2}]$ $\displaystyle\text{
(type }f_{sp}^{1})$ $\displaystyle\lambda^{(2)}=[(a{+}1)^{2},a^{N-3},a{-}2]$
$\displaystyle\text{ (type }f_{sp}^{2})$

with $a\geq 1$ when $i=1$ and $a\geq 2$ when $i=2$. By Proposition 2.3, it is
enough to assume that $\mu=[a^{N}]$ and $\lambda$ equals $\lambda^{(1)}$ or
$\lambda^{(2)}$. Since we will need it for the next section,we consider the
more general case $N$ is any integer with $N\geq 3$.

######  Proposition 4.5.

Let ${\mathfrak{g}}=\mathfrak{so}_{aN}$ if $a$ is odd and $\mathfrak{sp}_{aN}$
if $a$ is even. Let $e\in\mathcal{O}_{\mu}$. For $i\in\\{1,2\\}$, let
$\mathcal{O}=\mathcal{O}_{\lambda^{(i)}}$. Then there is an isomorphism

${\mathcal{S}}_{\mathcal{O},e}\simeq\overline{\mathcal{O}}_{[3,1^{N-3}]}$

where the orbit closure is in $\mathfrak{so}_{N}$.

###### Proof.

If $a=1$, the $\lambda^{(1)}$ case is clear since there is equality between
the slice and the orbit closure, so assume $a\geq 2$. The situation is very
similar to §11.2 in [FJLS17]. Let $I_{N}$ be the $N\times N$ identity matrix.
Let us define the form on $V$ explicitly to be the one defined by the $a\times
a$ block anti-diagonal matrix $J$ with

$J=\begin{pmatrix}0&0&\dots&0&0&I_{N}\\\ 0&0&\dots&0&-I_{N}&0\\\
0&0&\dots&I_{N}&0&0\\\ \dots\\\ (-1)^{a-1}I_{N}&0&\dots&0&0&0\end{pmatrix}.$

The bilinear form defined by $J$ is nondegenerate and is symmetric if $a$ is
odd and symplectic if $a$ is even. Since $a\not\equiv\epsilon$, this is the
correct form for defining ${\mathfrak{g}}={\mathfrak{g}}(V_{\epsilon})$.

The $a\times a$-block-matrices $e$ and $f$ given by

(12) $\displaystyle e=\begin{pmatrix}0&0&\dots&0&0\\\
c_{1}I_{N}&0&\dots&0&0\\\ 0&c_{2}I_{N}&\dots&0&0\\\ &&\dots&&\\\
0&0&\dots&c_{N-1}I_{N}&0\end{pmatrix},$ $\displaystyle
f=\begin{pmatrix}0&0&\dots&0&0\\\ I_{N}&0&\dots&0&0\\\ 0&I_{N}&\dots&0&0\\\
&&\dots&&\\\ 0&0&\dots&I_{N}&0\end{pmatrix}^{T}$

with $c_{j}=j(N-j)$ lie in ${\mathfrak{g}}$ and $e$ and $f$ are both nilpotent
with partition $\mu$. They complete to an ${\mathfrak{sl}}_{2}$-triple as in
[FJLS17, §11.2]. The centralizer ${\mathfrak{g}}^{f}$ is the set of block
upper triangular matrices of the form

$X=\begin{pmatrix}Y_{1}&Y_{2}&Y_{3}&...\text{ }Y_{a-2}&Y_{a-1}&Y_{a}\\\
0&Y_{1}&Y_{2}&\dots&Y_{a-2}&Y_{a-1}\\\ 0&0&Y_{1}&\dots&Y_{a-3}&Y_{a-2}\\\
&&&\dots&&\\\ 0&0&0&\dots&Y_{2}&Y_{3}\\\ 0&0&0&\dots&Y_{1}&Y_{2}\\\
0&0&0&\dots&0&Y_{1}\end{pmatrix}.$

Then $X$ lies in ${\mathfrak{g}}$ if and only if $Y_{i}=(-1)^{i}Y^{T}_{i}$.
Let $\Sigma_{N}$ denote the set of $N\times N$ symmetric matrices and set

$\phi:\mathfrak{so}_{N}\times\Sigma_{N}\times\mathfrak{so}_{N}\times\dots\to{\mathfrak{g}}^{f}$

denote the map where $\phi(Y_{1},Y_{2},\dots)$ is given by the matrix $X$
above. The reductive centralizer
${\mathfrak{c}}({\mathfrak{s}})\simeq\mathfrak{so}_{N}$ is given by $Y_{i}=0$
for $i\geq 2$; similarly $C({\mathfrak{s}})\simeq\rm O(N)$. An element $g\in
C({\mathfrak{s}})$ acts on $e+X\in\mathcal{S}_{e}$ by sending $Y_{i}$ to
$gY_{i}g^{T}$. The $\mathbb{C}^{*}$-action on $\mathcal{S}_{e}$ is given by
$t.Y_{i}=t^{2i}Y_{i}$ for $t\in\mathbb{C}^{*}$.

Let $e_{0}\in{\mathfrak{c}}({\mathfrak{s}})$ be a nilpotent element with
partition $[3,1^{N-3}]$. Pick an $\mathfrak{sl}_{2}$-triple
${\mathfrak{s}}_{0}$ through $e_{0}$ in ${\mathfrak{c}}({\mathfrak{s}})$ and
assume that the semisimple element $h_{0}$ is a diagonal matrix. By Lemma 4.1
or computing $h+h_{0}$ directly, we see that $e+e_{0}$ has partition
$\nu:=[a{+}2,a^{N-2},a{-}2]$ since $a\geq 2$.

Let $N=3$. By [KP82] we have
${\mathcal{S}}_{\mathcal{O}_{\lambda^{(i)}},e}\simeq\overline{\mathcal{O}}_{[3]}$
since for $i{=}1$ the degeneration is type $c$ and for $i{=}2$ it is type $d$
in Table 1. Set $A=e_{0}$, then $A^{2}\in\Sigma_{N}$. Then
$\phi(0,A^{2},0,\dots)$ is an eigenvector for both $\operatorname{ad}(h)$ and
$\operatorname{ad}(h_{0})$, with eigenvalue $-2$ and $4$, respectively. Since
the absolute values of the eigenvalues of $\operatorname{ad}(h_{0})$ on
${\mathfrak{z}}(e)$ are at most $4$ and the eigenvalue $4$ only occurs once in
$\Sigma_{N}$, there are no other exceptional pairs in the sense of [FJLS17,
§4]. It follows that
$e+\phi(A,z_{i}A^{2},0,\dots,0)\in\mathcal{O}_{\lambda^{(i)}}$ for a unique
$z_{i}\in\mathbb{C}^{*}$. Since
${\mathcal{S}}_{\mathcal{O}_{\lambda^{(i)}},e}$ has dimension two and is
irreducible, this means it is exactly the set of elements
$e+\phi(A,z_{i}A^{2},0,\dots,0)$ where $A\in\mathfrak{so}_{3}$ is nilpotent,
giving the isomorphism to $\overline{\mathcal{O}}_{[3]}$ explicitly.

Now consider the general $N$ case. We can embed $\mathfrak{so}_{3}$ into
${\mathfrak{c}}({\mathfrak{s}})\simeq\mathfrak{so}_{N}$ via the first $3$
coordinates and similarly for the rest of the centralizer of $e$ in the $N=3$
case. Clearly, for $A\in\mathcal{O}_{[3]}$, then
$\phi(A,0\dots,0)\in{\mathfrak{c}}({\mathfrak{s}})$ lies in
$\mathcal{O}_{[3,1^{N-3}]}$, but also
$e+\phi(A,z_{i}A^{2},0,\dots,0)\in{\mathcal{S}}_{e}$ lies in
$\mathcal{O}_{\lambda^{(i)}}$, by observing the action of this element on the
standard basis of $V$. It follows, using the action of $C({\mathfrak{s}})$,
that ${\mathcal{S}}_{\mathcal{O}_{\lambda^{(i)}},e}$ contains $e+A+z_{i}A^{2}$
for $A\in\overline{\mathcal{O}}_{[3,1^{N-3}]}$.

Next, $\dim\mathcal{O}_{\lambda_{1}}=\dim\mathcal{O}_{\lambda_{2}}$ since both
orbits are minimal degenerations from $\mathcal{O}_{\nu}$ of type $a$, hence
they are codimension two in $\overline{{\mathcal{\mathcal{O}}_{\nu}}}$. The
pair $(\lambda_{1},\mu)$ is equivalent to $([3,1^{N-3}],[1^{N}])$ after
canceling $a{-}1$ columns, thus the codimension of $\mathcal{O}_{\mu}$ in
$\overline{\mathcal{O}_{\lambda^{(i)}}}$ equals the dimension of
$\overline{\mathcal{O}}_{[3,1^{N-3}]}$ for both $i=1$ and $i=2$. The only
minimal degeneration from $\mathcal{O}_{\lambda^{(i)}}$ that contains
$\mathcal{O}_{\mu}$ is to the partition $[(a{+}1)^{2},a^{N-4},(a{-}1)^{2}]$,
which an $A_{1}$ singularity for both $i=1$ and $i=2$. Hence, as in §4.1, we
have $\overline{\mathcal{O}_{\lambda^{(i)}}}$ is unibranch at $e$. Thus
${\mathcal{S}}_{\mathcal{O}_{\lambda^{(i)}},e}\simeq\overline{\mathcal{O}}_{[3,1^{N-3}]}$.
∎

###  4.3. Type $h_{sp}$

As in the previous subsection, we are reduced by Proposition 9 to the case
where $\lambda=[a{+}2,a^{N-2},a{-}2]$ and $\mu=[a^{N}]$. We have the same
description for $e\in\mathcal{O}_{\mu}$, ${\mathfrak{s}}$, etc. as above. In
the previous subsection, ${\mathcal{S}}_{\mathcal{O}_{\lambda^{(i)}},e}$ was
the closure of a $C({\mathfrak{s}})$-orbit. We will first show that
${\mathcal{S}}_{\mathcal{O}_{\lambda},e}$ is the closure of a
$C({\mathfrak{s}})\times\mathbb{C}^{*}$-orbit for $N\geq 2$, where the
$\mathbb{C}^{*}$-action is as above.

The arguments from [FJLS17, §11.2.2] apply. First, we use them to show that
for $M\in{\mathcal{S}}_{\mathcal{O}_{\lambda},e}$, the matrices
$Y_{3},Y_{4},\dots$ are equal to sums of products of $Y_{1}$ and $Y_{2}$. This
follows since $\mathrm{rank}(M^{i})\leq N(a-i)$ for $i=1,\dots,a-2$ as in loc.
cit. However, $\mathrm{rank}(M^{a-1})\leq N+1$ unlike in loc. cit. and this
implies that

(13) $\mathrm{rank}(Y_{2}-dY_{1}^{2})\leq 1$

for some $d\in\mathbb{C}^{*}$. The condition $M^{a+2}=0$ yields the equation,
in the block lower left corner,

(14) $d_{1}Y_{3}+d_{2}(Y_{1}Y_{2}+Y_{2}Y_{1})+d_{3}Y_{1}^{3}=0$

where $d_{1}=\frac{(a+2)!(a-1)!}{6}$ and $d_{3}=\frac{(a-1)(a-2)}{5}\cdot
d_{1}$. It follows that the
$C({\mathfrak{s}})\times\mathbb{C}^{*}$-equivariant map from
${\mathcal{S}}_{\mathcal{O}_{\lambda},e}$ to the space $e+X$ where $Y_{i}=0$
for $i\geq 3$ is an isomorphism.

Now for $N\geq 3$, there are actually two equations involving a linear term
for $Y_{3}$. The one from the lower left corner of $M^{a+2}=0$ and the one
from the $\mathrm{rank}(M^{a-2})=N$ condition:

(15) $c_{1}Y_{3}+c_{2}(Y_{1}Y_{2}+Y_{2}Y_{1})+c_{3}Y_{1}^{3}=0.$

where $c_{1}=\frac{a!(a-1)!^{2}(a-2)!^{2}(a-3)!}{24}$ and
$c_{3}=\frac{(a+1)(a+2)}{5}\cdot c_{1}$. The equations (14) and (15) are not
multiples of each other since $\frac{d_{3}}{d_{1}}\neq\frac{c_{3}}{c_{1}}$ for
$a>0$. It follows, by canceling the $Y_{3}$ term, that

(16) $tY_{1}^{3}=Y_{1}Y_{2}+Y_{2}Y_{1}$

for some nonzero $t$.

Consider the $N=2$ case. Conjugating in $GL_{2}$ so that $C({\mathfrak{s}})$
becomes the diagonal torus in $SL_{2}$, we can represent
$Y_{1}=\left(\begin{smallmatrix}x&0\\\ 0&-x\end{smallmatrix}\right)$ and
$Y_{2}=\left(\begin{smallmatrix}y&z\\\ w&y\end{smallmatrix}\right)$ for
$x,y,z,w\in\mathbb{C}$. Then (16) implies that either $x$ is identically $0$,
i.e., $Y_{1}\equiv 0$ or $y=\frac{t}{2}x^{2}$, i.e., $\rm
tr(Y_{2}-tY_{1}^{2})=0$. By (13), $\det(Y_{2}-dY_{1}^{2})=0$. If $Y_{1}\equiv
0$, then $\det(Y_{2})=y^{2}-zw=0$ and $x=0$ are the conditions defining the
slice. If $t=d$, the condition is $zw=0$, with $x$ arbitrary. Since
$([a{+}2,a{-}2],[a^{2}])$ is a minimal degeneration of type $b$, the slice is
isomorphic to the $A_{3}$-simple surface singularity, so neither of these
cases hold. Instead, $d\neq t$ and the defining equations are
$y=\frac{t}{2}x^{2}$ and $t^{2}x^{4}=4wz$, which is indeed an
$A_{3}$-singularity. Moreover, the points with $x\neq 0$ form a single
$C({\mathfrak{s}})\times\mathbb{C}^{*}$-orbit, each of which has finite
stabilizer, so this orbit is dense in the slice. Let
$v=e+\phi(v_{1},v_{2},\dots)$ be such a point in the slice.

Next, as in §4.5, we bootstrap up to the general case by embedding the slice
for the $N=2$ case into the general slice by using the first two coordinates
in each block. Since the coefficients in the equations given above continue to
hold for ${\mathcal{S}}_{\mathcal{O}_{\lambda},e}$ in ${\mathcal{S}}_{e}$,
independent of $N$, we do indeed have a
$\operatorname{SO}_{2}\times\mathbb{C}^{*}$-equivariant embedding of the $N=2$
case. Note that $v_{1}\in\operatorname{\mathfrak{so}}_{N}$ is a multiple of a
$C({\mathfrak{s}})$-conjugate of $h_{0}$ from §4.5. Is stabilizer in
$C({\mathfrak{s}})$ is
$\operatorname{SO}_{2}\times\operatorname{SO}_{N-2}\subset C({\mathfrak{s}})$.
From the $N=2$ case it follows that the connected stabilizer in
$C({\mathfrak{s}})\times\mathbb{C}^{*}$of $v$ is $\operatorname{SO}_{N-2}$.
Hence, the $C({\mathfrak{s}})\times\mathbb{C}^{*}$-orbit through $v$ has
dimension $(N(N-1)/2+1)-(N-2)(N-3)/2=2N-2$. This is also the dimension
${\mathcal{S}}_{\mathcal{O}_{\lambda},e}$. Since
$\overline{\mathcal{O}}_{\lambda}$ is unibranch at $e$, we conclude

######  Proposition 4.6.

The slice ${\mathcal{S}}_{\mathcal{O}_{\lambda},e}$ is isomorphic to the
closure of a $C({\mathfrak{s}})\times\mathbb{C}^{*}$-orbit through
$(A,B)\in\operatorname{\mathfrak{so}}_{N}\times\Sigma_{N}$ where
$A=h_{0}\in\operatorname{\mathfrak{so}}_{N}$, and $B\in\Sigma_{N}$ satisfies
$\mathrm{rank}(B-dA^{2})=1$ and $\rm tr(B-tA^{2})=0$ for some nonzero $d,t$
with $d\neq t$.

Our goal now is to identify the subvariety of
$\operatorname{\mathfrak{so}}_{N}\times\Sigma_{N}$ in the proposition with the
quotient of closure of the orbit $\mathcal{O}_{[3,1^{N-2}]}$ in
$\operatorname{\mathfrak{so}}_{N+1}$. First, by employing the
$C({\mathfrak{s}})\times\mathbb{C}^{*}$-equivariant isomorphism of
$\operatorname{\mathfrak{so}}_{N}\times\Sigma_{N}$ to itself sending
$(A,B)\to(A,B-dA^{2})$, it follows that the slice is isomorphic to the closure
of the $C({\mathfrak{s}})\times\mathbb{C}^{*}$-orbit through
$(A,B)\in\operatorname{\mathfrak{so}}_{N}\times\Sigma_{N}$ where $B$ is now a
matrix of rank $1$ and $\rm tr(B-tA^{2})=0$ for some nonzero $t$.

Next, we recall material from [FJLS23, §2]. Let
$X\in\operatorname{\mathfrak{so}}_{N+1}$ be written as

$\begin{pmatrix}M&u\\\ -u^{T}&0\end{pmatrix}$

where $M\in\mathfrak{so}_{N}$ and $u\in\mathbb{C}^{N}$ is a column vector. Let
$\theta$ be the involution of $\operatorname{\mathfrak{so}}_{N+1}$ given by
conjugation of ${\rm diag}(1,1,\ldots,1,-1)\in{\rm O}_{N+1}$. Identifying
$\operatorname{\mathfrak{so}}_{N+1}$ with the set of pairs $(M,u)$, we see
that $\theta$ maps $(M,u)\mapsto(M,-u)$. Then the map
$\varphi:\operatorname{\mathfrak{so}}_{N+1}\to\operatorname{\mathfrak{so}}_{N}\times\Sigma_{N}$
sending $X$ to $(M,uu^{T})$ induces an ${\rm
O}_{n}\times\mathbb{C}^{*}$-equivariant isomorphism of
$\mathfrak{so}_{n+1}/\langle\theta\rangle$ with $\mathfrak{so}_{n}\times\Xi$
where $\Xi$ is the cone of elements of $\Sigma_{N}$ of rank at most 1.

We can now state

######  Proposition 4.7.

The slice ${\mathcal{S}}_{\mathcal{O}_{\lambda},e}$ is isomorphic to

$\overline{\mathcal{O}}_{[3,1^{N-2}]}/\langle\theta\rangle.$

###### Proof.

By [FJLS23, Corollary 2,.2], if $Y=\overline{\mathcal{O}}_{[3,1^{N-2}]}$, then
$Y/\langle\theta\rangle\simeq\varphi(Y)$. As before we can use the $N=2$ case.
In that case, $X=\begin{pmatrix}0&a&b\\\ -a&0&c\\\ -b&0&-c\end{pmatrix}$ and

$\varphi(X)=\left(\left(\begin{smallmatrix}0&a\\\
{-}a&0\end{smallmatrix}\right),\left(\begin{smallmatrix}b^{2}&bc\\\
bc&c^{2}\end{smallmatrix}\right)\right).$

The condition for $X$ to be nilpotent is $a^{2}+b^{2}+c^{2}=0$ and so the
image is exactly the matrices $(A,B)$ where $\det(B)=0$ and $\rm
tr(B+A^{2})=0$. In the general case, we embed
$\operatorname{\mathfrak{so}}_{3}$ into the lower right corner. It follows
from the discussion above and the proof of Proposition 4.6 that $\varphi(Y)$
is isomorphic to the closure of the
$C({\mathfrak{s}})\times\mathbb{C}^{*}$-orbit through $(A,B)$ with $A\neq 0$,
and hence isomorphic to the slice by Proposition 4.6. ∎

Let the Klein four-group $V_{4}$ act on $\mathfrak{so}_{N+2}$ via the pair of
commuting involutions $\theta_{1},\theta_{2}$ given by conjugation by $\rm
diag(1,\ldots,1,-1)$ and $\rm diag(1,\ldots,1,-1,1)$, respectively. Let
$\overline{{\mathcal{O}}}_{[2^{2},1^{N-2}]}$ be the minimal orbit in
$\mathfrak{so}_{N+2}$ . Then by [FJLS23, Corollary 2.5], for example, it
follows that

$\overline{{\mathcal{O}}}_{[3,1^{N-2}]}\simeq\overline{{\mathcal{O}}}_{[2^{2},1^{N-2}]}/\langle\theta_{1}\rangle.$

######  Corollary 4.8.

We have the isomorphism

${\mathcal{S}}_{\mathcal{O}_{\lambda},e}\simeq\overline{{\mathcal{O}}}_{[2^{2},1^{N-2}]}/\langle\theta_{1},\theta_{2}\rangle.$

and hence the minimal special degeneration $h_{sp}$ in Table 2 is
$d_{n+1}/V_{4}$.

###### Proof.

The $h_{sp}$ degeneration is covered by the case when $N$ is even with
$n=N/2$. ∎

######  Remark 4.9.

a) The special case $n=3$ was already observed in [Fu07]. In that case we have
$\overline{\mathcal{O}_{\rm{min}}(\mathfrak{so}_{5})}\cong{\mathbb{C}}^{4}/\\{\pm
1\\}$ and so we obtain isomorphisms of
$\mathcal{S}_{\mathcal{O},e}\cap\overline{{\mathcal{O}}^{\prime}}$ with (i)
${\mathbb{C}}^{4}/W(B_{2})$; (ii)
${\mathcal{N}}(\mathfrak{so}_{4})/\mathfrak{S}_{2}=({\mathcal{N}}(\mathfrak{sl}_{2})\times{\mathcal{N}}(\mathfrak{sl}_{2}))/\langle\theta\rangle$
where $\theta$ swaps the two copies of $\mathfrak{sl}_{2}$.

b) The orbits which intersect non-trivially with $\mathcal{S}_{\mathcal{O},e}$
are the nilpotent orbits lying between $\mathcal{O}$ and
$\mathcal{O}^{\prime}$ in the partial order. If $N\geq 4$ then there are five
of these, as indicated by the following diagram:

$\textstyle{\mathcal{O}_{\rm{min}}(\mathfrak{so}_{N+2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\rm{min}}(\mathfrak{so}_{N+1}^{(1)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\rm{min}}(\mathfrak{so}_{N+1}^{(2)})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\rm{min}}(\mathfrak{so}_{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\\{0\\}}$

where $\mathfrak{so}_{N+1}^{(1)}$ and $\mathfrak{so}_{N+1}^{(2)}$ are the two
fixed point subalgebras for the two proper parabolic subgroups of
$\mathfrak{S}_{2}\mathfrak{S}_{2}$. For $N=3$ there is no orbit with partition
$[3^{2},2^{N-4},1^{2}]$, equivalently, $\mathfrak{so}_{3}$ contains no
elements of $\mathcal{O}_{\rm{min}}(\mathfrak{so}_{5})$.

a) For $N$ odd, the singularity
$\overline{{\mathcal{O}}}_{[2^{2},1^{N-2}]}/\langle\theta_{1},\theta_{2}\rangle$
arises as a slice, but never for a minimal special degeneration. This is
because the $f_{sp}$ singularities arise in this case as the minimal special
degenerations.

###  4.4. Minimal special degenerations in the exceptional Lie algebras

There are three unexpected singularities that arise in the exceptional Lie
algebra: (i) $\mu$ (with normalization $A_{3}$); (ii)
$a_{2}/\mathfrak{S}_{2}$; (iii) $d_{4}/\mathfrak{S}_{4}$, which are dealt with
in [FJLS17], [FJLS23]. They appear once, once, and twice, respectively. We
will show in this subsection that all remaining singularities associated to
minimal special degenerations in exceptional types are unions of simple
surface singularities or minimal special singularities.

The case of $G_{2}$ is clear. Most of the minimal special degenerations are
minimal degenerations and hence were dealt with in [FJLS17] or [FJLS23]. There
are three (resp. three, eight, ten) minimal special degenerations which are
not minimal degenerations in type $F_{4}$ (resp. $E_{6}$, $E_{7}$, $E_{8}$).
These cases, with two exceptions, are covered by the following proposition.

######  Proposition 4.10.

Let $\mathcal{O}^{\prime}$ be a special nilpotent orbit in an exceptional Lie
algebra such that the reductive centralizer ${\mathfrak{c}}({\mathfrak{s}})$
contains a non-simply-laced simple component ${\mathfrak{c}}_{0}={\rm
Lie}(C_{0})$.

(a) There is a unique special orbit $\mathcal{O}>\mathcal{O}^{\prime}$ such
that $\operatorname{codim}_{\overline{\mathcal{O}}}\mathcal{O}^{\prime}$ is
equal to the dimension of the minimal special nilpotent $C_{0}$-orbit
$\mathcal{O}_{0}$ in ${\mathfrak{c}}_{0}$.

(b) If $\mathcal{O}^{\prime}=\mathcal{O}_{2A_{2}}$ in type $E_{8}$ then there
are two such simple components ${\mathfrak{c}}_{0}$, both of type $G_{2}$, and
$\mathcal{S}_{\mathcal{O},e}$ is a union of two copies of
$\overline{\mathcal{O}_{0}}$. The two copies are interchanged by
$C({\mathfrak{s}})$. Other than this case, there is exactly one such
${\mathfrak{c}}_{0}$ and
$\mathcal{S}_{\mathcal{O},e}\simeq\overline{\mathcal{O}_{0}}$.

###### Proof.

Statement (a) is a straightforward check using the tables of nilpotent orbits
and Hasse diagrams in [Car93].

The singularities in (b) can be classified using the arguments in [FJLS17,
§4.3]. Indeed, several of these are discussed there, see [FJLS17, §11, Table
13]. Let $e_{0}\in\mathcal{O}_{0}$. We claim that, with the sole exception of
$\mathcal{O}^{\prime}=\mathcal{O}_{A_{2}+3A_{1}}$ in type $E_{7}$,
$e+e_{0}\in\mathcal{O}$. By unibranchness and dimensions, it follows that
$\mathcal{S}_{\mathcal{O},e}=\overline{f+C_{0}\cdot
e_{0}}\cong\overline{\mathcal{O}_{0}}$. By [FJLS17, Prop. 4.8], it suffices to
verify the following condition: let $\langle
h_{0},e_{0},f_{0}\rangle={\mathfrak{s}}_{0}\subset{\mathfrak{c}}_{0}$ be an
$\mathfrak{sl}_{2}$-subalgebra, then all irreducible
${\mathfrak{s}}_{0}$-summands in ${\mathfrak{g}}^{e}(i)$ have dimension
$\leq(i+1)$. This can be checked by inspecting the tables in [LT11]. If
${\mathfrak{c}}_{0}$ is of type $B$, then all non-trivial simple summands for
the action on the centralizer of $e$ are natural modules or spin modules; a
short root element acts with Jordan blocks of size 2 on the spin module and of
size $\leq 3$ on the natural module, so we only need to check that no natural
modules occur in ${\mathfrak{g}}^{f}(1)$. When ${\mathfrak{c}}_{0}$ is of type
$G_{2}$ (excluding $\mathcal{O}^{\prime}=\mathcal{O}_{A_{2}+3A_{1}}$ in type
$E_{7}$), all non-trivial summands are isomorphic to the minimal faithful
representation for ${\mathfrak{c}}_{0}$; $e_{0}$ acts on the minimal faithful
representation with Jordan blocks of size $\leq 3$, so we only need to check
that the minimal representation doesn’t appear in ${\mathfrak{g}}^{e}(1)$.
Finally, ${\mathfrak{c}}_{0}$ of type $C$ occurs once, when
$\mathcal{O}^{\prime}=\mathcal{O}_{D_{4}}$ in type $E_{7}$ and
${\mathfrak{c}}_{0}=\mathfrak{sp}_{6}$; here one has to check that $e_{0}$ has
no Jordan blocks of size $>7$ on $V(\varpi_{2})$, hence on the alternating
square of the natural module, which is straightforward.

This only leaves the case $\mathcal{O}^{\prime}=\mathcal{O}_{A_{2}+3A_{1}}$ in
$E_{7}$. Here ${\mathfrak{c}}_{0}={\mathfrak{g}}^{e}\cap{\mathfrak{g}}^{h}$ is
simple of type $G_{2}$ and the positive graded parts of ${\mathfrak{g}}^{f}$
are:

${\mathfrak{g}}^{f}(2)=V(2\varpi_{1})\oplus\mathbb{C}e,\quad{\mathfrak{g}}^{f}(4)=V_{\mathrm{min}},$

where $V_{\rm min}=V(\varpi_{1})$ is the minimal faithful representation for
${\mathfrak{c}}_{0}$. Note that the action of ${\mathfrak{c}}_{0}$ on
$V_{\mathrm{min}}$ induces an embedding in $\mathfrak{so}_{7}$, and
$\mathfrak{sl}_{7}$ decomposes over
${\mathfrak{c}}_{0}\subset\mathfrak{so}_{7}$ as $\mathfrak{so}_{7}\oplus
V(2\varpi_{1})$. (In the notation of §4.5, $V(2\varpi_{1})=\Sigma_{7}$.)
Furthermore, the matrix square operation on $\mathfrak{gl}_{7}$ determines a
quadratic map $\mathfrak{so}_{7}\rightarrow V(2\varpi_{1})$ which restricts to
${\mathfrak{c}}_{0}$ to give a $C_{0}$-equivariant map
$\psi:{\mathfrak{c}}_{0}\rightarrow
V(2\varpi_{1})\subset{\mathfrak{g}}^{e}(2)$. In particular, if
$x=e_{\beta_{2}}+e_{3\beta_{1}+\beta_{2}}$ then $\psi(x)$ is non-zero of
weight $3\beta_{1}+2\beta_{2}$. We checked using GAP that (with this notation)
there exists an element of $\mathcal{O}=\mathcal{O}_{D_{4}(a_{1})}$ of the
form: $e+x+\psi(x)$, where $x$ is in the subregular nilpotent orbit in
${\mathfrak{c}}_{0}$. It follows that
$\mathcal{S}_{\mathcal{O},e}=e+\overline{C_{0}\cdot(x+\psi(x))}$, hence is
isomorphic to the closure of the $C_{0}$-orbit through $x$, which completes
our proof. ∎

######  Proposition 4.11.

All minimal special degenerations in exceptional Lie algebras are either: (1)
minimal degenerations, (2) covered by the above proposition, or (3) isomorphic
to $d_{4}/\mathfrak{S}_{4}$, which occurs for the two cases of
$\mathcal{O}_{F_{4}(a_{3})}>\mathcal{O}_{A_{2}}$ in $F_{4}$ and
$\mathcal{O}_{E_{8}(a_{7})}>\mathcal{O}_{D_{4}+A_{2}}$ in $E_{8}$.

###### Proof.

We checked that all the minimal special degenerations, which are not minimal
degenerations, are covered by the proposition, except for the two cases
listed. The exceptional special degeneration
$\mathcal{O}_{F_{4}(a_{3})}>\mathcal{O}_{A_{2}}$ in type $F_{4}$ was dealt
with in [FJLS23, Theorem 4.11]. Hence it remains to show that the Slodowy
slice singularity from $D_{4}+A_{2}$ to $E_{8}(a_{7})$ in $E_{8}$ is also
isomorphic to $d_{4}/\mathfrak{S}_{4}$. To do this, we first consider
$\mathcal{O}^{\prime}=\mathcal{O}_{D_{4}}$. Let ${\mathcal{S}}_{D_{4}}$ be the
Slodowy slice at $f\in\mathcal{O}^{\prime}$. Repeating the calculation in the
proof of Proposition 4.10, we see that the condition of [FJLS17, Prop. 4.8]
holds for an element of ${\mathfrak{c}}$ of type $F_{4}(a_{3})$. It follows
that
${\mathcal{S}}_{D_{4}}\cap\overline{\mathcal{O}_{E_{8}(a_{7})}}=f+\overline{C_{0}\cdot
e_{2}}$ where $e_{2}$ belongs to the $F_{4}(a_{3})$ orbit in
${\mathfrak{c}}_{0}$. By the same calculation (or by direct observation),
${\mathcal{S}}_{D_{4}}\cap\overline{\mathcal{O}_{D_{4}+A_{2}}}=f+\overline{C_{0}\cdot
e_{1}}$ where $e_{1}$ is in the $A_{2}$ orbit in ${\mathfrak{c}}_{0}$. Now we
use the following fact (which follows from equality of dimensions): if
$\\{e_{1},h_{1},f_{1}\\}$ is an $\mathfrak{sl}_{2}$-triple in
${\mathfrak{c}}_{0}$ such that $\dim C_{0}\cdot e_{1}$ equals the codimension
of $\mathcal{O}^{\prime}$ in $\overline{G\cdot(f+f_{1})}$, then the
centralizer of $e+e_{1}$ equals
${\mathfrak{g}}^{e}\cap{\mathfrak{g}}^{e_{1}}$. Hence the Slodowy slice at
$f+f_{1}$ is contained in the Slodowy slice at $f$. It follows that
${\mathcal{S}}_{D_{4}+A_{2}}\cap\overline{\mathcal{O}_{E_{8}(a_{7})}}$ is
isomorphic to the Slodowy slice singularity in $F_{4}$ from
$\mathcal{O}_{A_{2}}$ to $\mathcal{O}_{F_{4}(a_{3})}$, hence is isomorphic to
$d_{4}/\mathfrak{S}_{4}$. ∎

The following is true in both the classical and exceptional types.

######  Corollary 4.12.

Let $\mathcal{O}^{\prime}=\mathcal{O}^{\prime}_{e}$ be special. The action of
$C({\mathfrak{s}})$ on ${\mathfrak{c}}({\mathfrak{s}})$ induces an action of
$A(e)$ on the set of simple components of ${\mathfrak{c}}({\mathfrak{s}})$.
Each $A(e)$-orbit of simple components ${\mathfrak{c}}_{0}$ corresponds to a
unique special nilpotent orbit $\mathcal{O}$ in ${\mathfrak{g}}$ such that
$(\mathcal{O},\mathcal{O}^{\prime})$ is a minimal special degeneration.
Moreover, $\mathcal{S}_{\mathcal{O},e}$ contains a subvariety isomorphic to
the minimal special nilpotent orbit closure in ${\mathfrak{c}}_{0}$. All
minimal special degenerations of codimension at least $4$ arise in this way

###### Proof.

We just showed this in the exceptional types when ${\mathfrak{c}}_{0}$ is not
simply-laced, but it also holds when ${\mathfrak{c}}_{0}$ is simply-laced
where it gives a minimal degeneration. It also holds in the cases of
$d_{4}/{\mathfrak{S}}_{4}$ and $a_{2}/{\mathfrak{S}}_{2}$ from [FJLS17]. In
the classical types, we showed that each simple factor of
${\mathfrak{c}}({\mathfrak{s}})$ leads to a unique minimal special
degeneration. The $A(e)$-orbits on the simple factors of
${\mathfrak{c}}({\mathfrak{s}})$ are singletons except for the case where
${\mathfrak{c}}({\mathfrak{s}})$ contains a copy of
$\operatorname{\mathfrak{so}}_{4}$. This corresponds to the case of
$[2A_{1}]^{+}=d_{2}^{+}$. ∎

##  5\. $A(e)$-action on slices

In this section we compute the action of $A(e)$ on the slice
$\mathcal{S}_{\mathcal{O},e}$ for both minimal degenerations and minimal
special degenerations in the classical types, and determine when the action is
outer. This was done in the exceptional groups in [FJLS17] for minimal
degenerations. There is only a single case of a minimal special degeneration
not covered by those results: the case of $e\in A_{2}$ from Proposition 4.10,
which we now denote as $[2g_{2}^{sp}]^{+}$.

###  5.1. Union of simple surface singularities

Recall that $C({\mathfrak{s}})$ acts on $\mathcal{S}_{\mathcal{O},e}$. In the
case of a simple surface singularity, as discussed in the introduction, we use
Slodowy’s notion of action, which amounts to the action on the projective
lines in the exceptional fiber. Even when $\mathcal{S}_{\mathcal{O},e}$ is not
irreducible, we want to describe how $C({\mathfrak{s}})$ permutes the
projective lines in the fiber, something we did in the exceptional groups.
Since $C^{\circ}({\mathfrak{s}})$ acts trivially, we get a permutation action
of $A(e)\simeq C({\mathfrak{s}})/C^{\circ}({\mathfrak{s}})$ on the
$\mathbb{P}^{1}$’s. We call this the outer action of $A(e)$ on the slice.

To compute the action for $\dim(\mathcal{S}_{\mathcal{O},e})=2$, we use
[FJLS17, Lemma 5.8]. We do not assume that the orbits are special, so the set-
up is a minimal degeneration $({\mathcal{O}}_{\lambda},{\mathcal{O}}_{\mu})$
in the classical groups where $\dim(\mathcal{S}_{\mathcal{O},e})=2$ for
$e\in\mathcal{O}_{\mu}$, and where $\lambda,\mu$ are the appropriate
partitions indexing the nilpotent orbits. Let ${\mathfrak{n}}_{P}$ denote the
nilradical of the Lie algebra of a parabolic subgroup $P$ of $G$ such that
$\mathcal{O}_{\lambda}$ is Richardson for ${\mathfrak{n}}_{P}$. Then we have
the proper, surjective map $\pi:G\times^{P}\
{\mathfrak{n}}_{P}\to\overline{{\mathcal{O}}}_{\lambda}$, which is generically
finite. Below, we will always choose ${\mathfrak{n}}_{P}$ so that $\pi$ is
birational.

Next, assume that the reductive centralizer for an element in
$\mathcal{O}_{\lambda}$ is semisimple. Let
$\mathcal{O}_{1},\mathcal{O}_{2},...\mathcal{O}_{t}$ be the maximal orbits in
the complement of $\mathcal{O}_{\lambda}$ in its closure. Assume that all
$\mathcal{O}_{i}$ are codimension two in $\bar{{\mathcal{O}}}_{\lambda}$. Let
$e_{i}\in\mathcal{O}_{i}$. Let $r_{i}$ equal the number of $A(e_{i})$-orbits
on $\pi^{-1}(e_{i})$. Then as in [FJLS17, Lemma 5.8], if $G$ is connected, we
have $\sum_{i}r_{i}$ is equal to rank of ${\mathfrak{g}}$ minus the rank of
the Levi subgroup of $P$. The quantities $r_{i}$ will be enough to determine
the outer action.

Remarkably, in types $B$ and $C$, the actions are large as possible as they
were in the exceptional types (at least given the size of $A(e)$).

######  Proposition 5.1.

In the classical groups $B,C,D$ (working in the full orthogonal group for
$D$),

  1. (1)

If $\mathcal{S}_{\mathcal{O},e}$ is a simple surface singularity of type
$D_{k+1}$ or $A_{2k-1}$, then the $A(e)$-action upgrades these singularities
to $C_{k}$ and $B_{k}$, respectively.

  2. (2)

If $\mathcal{S}_{\mathcal{O},e}$ is a union of two branches of type
$A_{2k-1}$, the $A(e)$-action is $[2B_{k}]^{+}$ as described in §1.3.

The proof will occupy the remainder of this section. For the moment let $G=\rm
O(V)$ or $\rm Sp(V)$, so that, as noted in §4, a reductive subgroup of the
centralizer $G^{e}$ of $e$ in $G$ is $C({\mathfrak{s}})$, which is a product
of orthogonal and symplectic groups.

Then the component group $A(e):=G^{e}/(G^{e})^{\circ}$ of $e$ with partition
$\mu$ is generated by the corners of Young diagram corresponding to parts $s$
with $s\not\equiv\epsilon$. Each such part $s$ determines a copy of an
orthogonal group in $C({\mathfrak{s}})$ and we denote by $x_{s}$ an element of
determinant $-1$ in each orthogonal group. Then $A(e)$ is elementary abelian
${\mathbf{Z}}_{2}^{r}$ where $r$ is the number of parts $s$ with
$s\not\equiv\epsilon$.

###  5.2. Type $b$ degeneration

This is the case of a simple surface singularity of type $D_{k+1}$ and it
arises whenever $(\lambda,\mu)$ is locally
$(\lambda^{\prime},\mu^{\prime}):=([a\\!+\\!2k,a],[a\\!+\\!2k\\!-\\!2,a\\!+\\!2])$,
by [KP82]. Here $k\geq 2$. This is a valid pair of partitions when $a$ is even
if ${\mathfrak{g}}$ is of type $C$ and odd if ${\mathfrak{g}}$ is of types $B$
or $D$. By Proposition 2.3, we can replace $(\lambda,\mu)$ by
$(\lambda^{\prime},\mu^{\prime})$. We note that the centralizer of $e_{1}$ in
$G(V_{1})$ is a subgroup of the centralizer of $e$ in $G$. This gives an
embedding of the component group fo $e_{1}$ of $G(V_{1})$, which is the Klein
$4$-group $V_{4}$, into $A(e)$, given by sending $A(e_{1})$ to the subgroup of
$A(e)$ generated by $x_{a\\!+\\!2k\\!-\\!2}$ and $x_{a\\!+\\!2}$. The other
parts contributing to $A(e)$ act trivially on ${\mathfrak{g}}(V_{1})$ and
hence trivially on the slice.

####  5.2.1. $G$ is of type $C$, $a$ even

The weighted Dynkin diagram for ${\mathcal{O}}_{\lambda}$ is

$\overbrace{2\dots 2}^{k}\overbrace{0202\dots 02}^{a/2}$

where the final node corresponds to the long simple root. Taking the
associated parabolic subgroup $P$, the map $\pi$ above is birational.

If $a=0$, we are in type $C_{k}$ and $\mathcal{O}_{\lambda}$ is regular. There
is a unique minimal degeneration to $\mathcal{O}_{\mu}$, the subregular orbit.
Hence, using [FJLS17, Lemma 5.8], there are exactly $k$ orbits for $A(e)$ on
the $\mathbb{P}^{1}$’s in the fiber, which implies the action on $D_{k+1}$
must be $C_{k}$. Indeed, the sole $A(e)$-orbit of size two is coming from the
orbital variety corresponding to the long root. (We could use knowledge of the
Springer fiber in this case too).

Next if $a>0$, which means $a\geq 2$ since $a$ is even, there is the
degeneration of $\lambda$ to $\mu$ but also to
$\mu^{\prime}=[a{+}2k,a{-}2,2]$. The latter minimal degeneration is equivalent
to $([a],[a{-}2,2])$, which is a simple surface singularity of type
$D_{\frac{a}{2}+1}$ with action of $A(e_{\mu^{\prime}})$ having $\frac{a}{2}$
orbits, by induction. Since the total number of component group orbits on the
fiber is $k+\frac{a}{2}$, that leaves $k$ orbits corresponding to the
degeneration to $e=e_{\mu}$. This forces the action on $D_{k+1}$ to be non-
trivial and must be $C_{k}$, as desired. Indeed, we could explicitly see by
using instead the parabolic $P$ for the diagram

$\overbrace{0202\dots 02}^{a/2}\overbrace{2\dots 2}^{k},$

which is also birational to $\mathcal{O}_{\lambda}$. Then the orbital
varieties for $\mathcal{O}_{\mu}$ correspond to the last $k$ two’s. The last
node gives the $A(e)$-orbit with two elements.

Finally, the element $x_{a{+}2k}x_{a}$ acts trivially on the fibers, since it
belongs to the center of $G$. So both $x_{a{+}2k}$ and $x_{a}$ will yield the
outer action on the slice.

####  5.2.2. $G$ is of type $D$, $a$ odd

The weighted Dynkin diagram for ${\mathcal{O}}_{\lambda}$ is

$\overbrace{2\dots 2}^{k-1}\overbrace{0202\dots 02}^{(a-1)/2}2$

where the two final nodes correspond to orthogonal simple roots and the first
$k-1$ nodes form a subsystem of type $A_{k-1}$. Taking associated parabolic
subgroup $P$, the map $\pi$ above is birational. This is similar to the type
$C$ case. If we work in the full orthogonal group then $A(e)$ permutes the two
${\mathbf{P}}^{1}$’s corresponding to the tails of the Dynkin diagram.
Finally, the element $x_{a{+}2k}x_{a}$ acts trivially on the fiber, since it
belongs to the center of $G$. So both $x_{a{+}2k}$ and $x_{a}$ will yield the
outer action on the slice.

###  5.3. Type $c$ singularity

This is a simple surface singularity of type $A_{2k-1}$ and it arises whenever
$(\lambda,\mu)$ is equivalent to

$([a\\!+\\!2k\\!+\\!1,a,a],[a\\!+\\!2k\\!-\\!1,a\\!+\\!1,a\\!+\\!1]).$

Here, $a$ is even for types $B,D$ and odd for type $C$. As in §5.2 using
Proposition 2.3, we can first reduce to the case of
$([a\\!+\\!2k\\!+\\!1,a,a],[a\\!+\\!2k\\!-\\!1,a\\!+\\!1,a\\!+\\!1])$ where
$G$ is type $B$ for $a$ even and type $C$ for $a$ odd.

The $A_{2k-1}$ simple surface singularity arises from the diagonal cyclic
group $\Gamma$ of order $2k$ in $\operatorname{SL}_{2}(\mathbb{C})$. The
centralizer of $\Gamma$ in $\operatorname{SL}_{2}(\mathbb{C})$ is the diagonal
one-dimensional torus, leading to an invariant of degree two for the action of
$\Gamma$ on $\mathbb{C}^{2}$. Since the isomorphism to the slice is
$\mathbb{C}^{*}$-equivariant, we see that the slice, upon projection to
${\mathfrak{c}}({\mathfrak{s}})$ must be isomorphic to the Lie algebra of the
torus for $\operatorname{O}(2)$ corresponding to the part $a{+}1$ in $\mu$.
The outer automorphism on $\mathbb{C}^{2}/\Gamma$ acts non-trivially on the
diagonal torus, we see that $x_{a{+}1}$ gives rise to the action, while
$x_{a\\!+\\!2k\\!-\\!1}$ acts trivially.

###  5.4. Type $d$ degeneration

This is again a simple surface singularity of type $A_{2k-1}$ and it arises
whenever $(\lambda,\mu)$ is equivalent to

$([a{+}2k{+}1,a{+}2k{+}1,a],[a{+}2k,a{+}2k,a{+}2]).$

This is a valid pair of partitions when $a$ is even in type $C$ and odd in
types $B$ or $D$.

As in the previous case, it is enough to work it out for the case
$\lambda=[a{+}2k{+}1,a{+}2k{+}1,a]$ and $\mu=[a{+}2k,a{+}2k,a{+}2]$. when $G$
of type $C$ for $a$ even and type $B$ when $a$ is odd. As before, we can
detect the action by looking at the action of
${\mathfrak{c}}({\mathfrak{s}})$. Thus $x_{a+2k}$ acts by outer action and
$x_{a+2}$ acts trivially.

###  5.5. Type $e$ degeneration

This is a union of simple surface singularities $A_{2k-1}\cup A_{2k-1}$ and it
arises whenever $(\lambda,\mu)$ is equivalent to

$([a{+}2k,a{+}2k,a,a],[a{+}2k{-}1,a{+}2k{-}1,a{+}1,a{+}1]).$

Here, $a$ is odd in type $C$ and even in types $B$ or $D$. As before, we are
reduced to the case of $\lambda=[a{+}2k,a{+}2k,a,a]$ and
$\mu=[a{+}2k{-}1,a{+}2k{-}1,a{+}1,a{+}1]$ in type $D$ for $a$ even and type
$C$ for $a$ odd. Here $C({\mathfrak{s}})\simeq{\rm O}(2)\times{\rm O}(2)$.

The full automorphism group of the singularity is dihedral of order eight. We
want to show $A(e)$ embeds as the Klein $4$-group generated by the reflections
through the midpoints of edges of the square. This will follow if we show that
there is at least one orbit of size $4$ of $A(e)$ on the fiber over $e$. This
will force there to be $k-1$ orbits of size $4$ on the $4k-2$ projective lines
and one orbit of size $2$,

By the method of the previous two sections, the element
$x_{a{+}2k{-}1}x_{a+1}$ must fix each irreducible component and act by outer
automorphism on each one individually. This is because it is acting by $-1$ on
the two-dimensional space ${\mathfrak{c}}({\mathfrak{s}})$. The action
$x_{a{+}2k{-}1}$ and $x_{a{+}1}$ can be determined in each case separately.
Both of them will interchange the two irreducible components.

####  5.5.1. C case

The Dynkin diagram of ${\mathcal{O}}_{\lambda}$ is

$\overbrace{0202\dots 02}^{k}\overbrace{00020002\dots 0002}^{(a-1)/2}00.$

Using the method of §5.2, if $a=0$, we find there $k$ orbits for the unique
minimal degeneration to $\mathcal{O}_{\mu}$. At the same time, there are
$4k-2$ projective lines in the fiber over $e$. Since $A(e)$ is isomorphic to
$V_{4}$, the possible orbit sizes are $1$, $2$, and $4$. The only way for this
to work is for there to be $k-1$ orbits of size $4$ and one orbit of size $2$.
Therefore the action is as desired.

When $a>0$, there is another minimal degeneration to
$\mathcal{O}_{\mu}^{\prime}=[(a{+}2k)^{2},(a{-}1)^{2},2]$. Then
$(\lambda,\mu^{\prime})$ is equivalent to $([a,a],[(a{-}1)^{2},2])$, which is
of a type $d$ generation and has the form $B_{\frac{a-1}{2}}$. This
degeneration therefore accounts for $\frac{a-1}{2}$ of the $k+\frac{a-1}{2}$
orbits, leaving $k$ for the studied minimal degeneration and the result
follows as in the $a=0$ case.

####  5.5.2. D case

If $G$ is the full orthogonal group, there is a single orbit
${\mathcal{O}}_{\lambda}$ with the given singularity. Working in the special
orthogonal group, there are two very even orbits with the given partition,
interchanged by the action of any element of $O(N)$ not in
$\operatorname{SO}(N)$. This is where the two irreducible components are
coming from, as they both degenerate to $\mu$, which has an element fixed by
action. Hence, the result follows.

###  5.6. $G$ is special orthogonal

When $G$ is special orthogonal, there are two situations where the component
group action changes.

For the type $b$ singularity when $\mu$ has exactly two odd parts (e.g.,
$\mu=[8,8,5,3]$ or $\mu=[8,8,5,5]$ ). In this case the component group is
trivial. If there were more than two odd parts for this degeneration, there
would have to be at least 3 distinct odd parts, which would guarantee the non-
trivial action of $A(e)$.

For the type $e$ singularity when $\mu$ again has only the odd parts that
appear in the local version of $\mu$ in Table 2. Otherwise, $\mu$ would have
at least two additional odd parts (possibly equal), which would ensure the
same action by $V_{4}$. Now if $\mu$ has only the odd parts, say
$[(a{+}2k{-}1)^{2},(a{+}1)^{2}]$, then since its others parts are even, the
partition $\lambda$ must be very even. Then there are two orbits corresponding
to $\lambda$ and $A(e)\simeq{\mathfrak{S}}_{2}$ acts by outer automorphism on
each degeneration to $\mu$, so both are of type $B_{k}$.

###  5.7. Dimension four or greater

In [FJLS17], we studied the image of $C({\mathfrak{s}})$ in $\rm
Aut({\mathfrak{c}}({\mathfrak{s}}))$ via the adjoint action in the exceptional
groups, and then restricted the action to orbits of simple factors of
${\mathfrak{c}}({\mathfrak{s}})$. We observed using [Som98] (also computable
using [Ale79]) that $C({\mathfrak{s}})$ tends to act by outer automorphisms of
simple factors of ${\mathfrak{c}}({\mathfrak{s}})$ that admit outer
automorphisms. As in Corollary 4.12, the minimal (and minimal special
degenerations) are controlled by ${\mathfrak{c}}({\mathfrak{s}})$ for most
cases when $\dim(\mathcal{S}_{\mathcal{O},e})\geq 4$. We then recorded this
outer action on minimal singularities $a_{n}$, $d_{n}$, $d_{4}$, and $e_{6}$,
when they arose.

A more intrinsic framework is to use the intersection homology
$IH^{*}(\mathcal{S}_{\mathcal{O},e})$ of $\mathcal{S}_{\mathcal{O},e}$ under
the induced action of $A(e)$. Let $\rm p(X)=\sum_{i}\dim(IH^{2i}(X))q^{i}$.
When $\mathcal{S}_{\mathcal{O},e}\simeq\overline{\mathcal{O}_{\rm min.sp}}$
for the minimal special orbit in the simple Lie algebra ${\mathfrak{c}}_{0}$,
then we have

$p(\mathcal{S}_{\mathcal{O},e})=q^{e_{1}-1}+q^{e_{2}-1}+\dots+q^{e_{k}}$

where $e_{i}$ are the exponents of ${\mathfrak{c}}_{0}$ (see [Lus22]).

Let ${\mathfrak{c}}_{0}$ be of type $A_{k}$, $D_{k}$, or $E_{6}$ and $\theta$
be an outer involution and denote by
${\mathfrak{c}}_{0}^{\prime}:={\mathfrak{c}}_{0}^{\langle\theta\rangle}$ the
fixed subalgebra. Then ${\langle\theta\rangle}$ acts trivially on the part of
$IH^{*}(\overline{\mathcal{O}}_{\rm min.sp})$ corresponding to exponents of
${\mathfrak{c}}_{0}^{\prime}$ and by the sign representation on the remaining
part. In other words,

$IH^{*}(\overline{\mathcal{O}}_{\rm
min.sp})^{\langle\theta\rangle}=IH^{*}(\overline{\mathcal{O}}_{\rm
min.sp}/{\langle\theta\rangle}).$

In the case of ${\mathfrak{S}}_{3}$ acting by outer automorphisms when
${\mathfrak{c}}_{0}$ is of type $D_{4}$, the ${\mathfrak{S}}_{3}$-invariants
on $IH^{*}(\overline{\mathcal{O}}_{\rm min.sp})$ correspond to the exponents
of $G_{2}$ (namely, $1$ and $5$) and ${\mathfrak{S}}_{3}$ acts by the
reflection representation on the two-dimensional space
$IH^{4}(\overline{\mathcal{O}}_{\rm min.sp})$ for the two exponents of $D_{4}$
equal to $3$ and again

$IH^{*}(\overline{\mathcal{O}}_{\rm
min.sp})^{K}=IH^{*}(\overline{\mathcal{O}}_{\rm min.sp}/K).$

Since $C^{\circ}({\mathfrak{s}})$ acts trivially on
$IH^{*}(\mathcal{S}_{\mathcal{O},e})$, there is an action of $A(e)$ on
$IH^{*}(\mathcal{S}_{\mathcal{O},e})$ and this gives an intrinsic way to see
the outer action when the slice is isomorphic to the closure of a minimal
special orbit, rather than appealing to the action on ${\mathfrak{c}}_{0}$
itself, when ${\mathfrak{c}}_{0}$ is the relevant factor of
${\mathfrak{c}}({\mathfrak{s}})$ as in Corollary 4.12.

###  5.8. Type $h$ singularity

This corresponds to the closure of the minimal nilpotent orbit in type
$D_{k}$. The local action of the reductive centralizer coincides with the
orthogonal group $\rm O(2k)$, which contains an outer involution of
$\operatorname{\mathfrak{so}}_{2k}$ and so the $A(e)$ acts by outer action and
coincides with the $d_{k}^{+}$.

In the case of $G=\operatorname{SO}(2N)$, the component group $A(e)$ will
still act by outer involution in this way, except for those cases where the
partition $\mu$ contains exactly one odd part (of even multiplicity $2k$).

###  5.9. Exceptional degenerations

####  5.9.1. The case of $d_{n+1}/V_{4}$

From §4.3, the $V_{4}$ is acting on $d_{n+1}$ with $\theta_{1}$ outer and
$\theta_{2}$ inner. Hence,

$IH^{*}(d_{n+1}/V_{4})\simeq IH^{*}(d_{n+1}/\theta_{1})\simeq
IH^{*}(b^{sp}_{n}).$

Let $\mathcal{S}_{\mathcal{O},e}$ be a slice of type $h_{sp}$. Recall that
there is a natural $\rm O(2n)$ action on $d_{n+1}/V_{4}$ which is the fixed
points of the $V_{4}$-action on $O(2n)$. Under the isomorphism to
$\mathcal{S}_{\mathcal{O},e}$, the $\rm O(2n)$-action becomes the action of
$C({\mathfrak{s}})$ on $\mathcal{S}_{\mathcal{O},e}$.

Since the action of $O(2n)$ on $d_{n+1}$ is also inner, we find that $O(2n)$
acts trivially on $IH^{*}(d_{n+1}/V_{4})$, and hence $C({\mathfrak{s}})$ acts
trivially on $d_{n+1}/V_{4}$.

On the other hand, it seems relevant that if we take the minimal degeneration
to $\mu$ in $\mathcal{S}_{\mathcal{O},e}$, which is of type $h$, then indeed
$A(e)$ acts by outer action on this $d_{n}$.

####  5.9.2. The case of $d_{4}/S_{4}$

From the proof of [FJLS23, Theorem 4.11], $S_{4}$ acts on $d_{4}$ by the semi-
direct product of an inner $V_{4}$ group and an outer $S_{3}$ group. Hence,

$IH^{*}(d_{4}/S_{4})\simeq IH^{*}(d_{4}/S_{3})\simeq IH^{*}(g^{sp}_{2}).$

Let $\mathcal{S}_{\mathcal{O},e}$ be one of the two slices of type
$d_{4}/S_{4}$. There is a natural action of $SL_{3}\rtimes{\mathfrak{S}}_{2}$
on $d_{4}/S_{4}$, which is the fixed points of $S_{4}$ on the adjoint group of
type $D_{4}$. This action, as in the previous section, corresponds to
$C({\mathfrak{s}})$ on $\mathcal{S}_{\mathcal{O},e}$ under the equivariant
isomorphism. The action of the ${\mathfrak{S}}_{2}$ is inner, so we again find
that $A(e)$ acts trivially on $\mathcal{S}_{\mathcal{O},e}$.

The minimal degeneration in $\mathcal{S}_{\mathcal{O},e}$ corresponds to an
$a_{2}$ singularity coming from ${\mathfrak{c}}({\mathfrak{s}})$ and we note
again that the action on this singularity is outer, $a_{2}^{+}$.

##  6\. Action of the canonical quotient

Let $\mathcal{S}_{\mathcal{O},e}$ be the slice for a minimal special
degeneration. In this section we explain how the kernel $H$ of the
homomorphism from $A(e)$ to Lusztig’s canonical quotient $\bar{A}(e)$ acts on
the slice $\mathcal{S}_{\mathcal{O},e}$. When $H$ acts by outer action, the
exchange of singularities under the duality is not as expected.

###  6.1. Exceptional groups

######  Proposition 6.1.

Assume $G$ is connected of exceptional type and $H$ is nontrivial for $A(e)$.
Then there exists a unique minimal special degeneration to $e$ and the
degeneration is $C_{k}$ for $k\geq 2$, $\mu$, or $d_{4}/{\mathfrak{S}}_{4}$.

In the $C_{k}$ cases, $H=A(e)={\mathfrak{S}}_{2}$ acts by outer automorphism
on $\mathcal{S}_{\mathcal{O},e}$.

In the one case $(D_{7}(a_{1}),E_{8}(b_{6}))$, where the singularity is of
type $\mu$ (which is $C_{2}$ upon normalization), $H$ acts trivially on
$\mathcal{S}_{\mathcal{O},e}$ and the induced action of $\bar{A}(e)$ is by
outer automorphism.

In the two cases where $\mathcal{S}_{\mathcal{O},e}$ is
$d_{4}/{\mathfrak{S}}_{4}$, the action of $H$ is trivial on
$IH^{*}(\mathcal{S}_{\mathcal{O},e})$, however the action of $H$ on the
minimal degeneration to $e$ is outer.

###### Proof.

The cases in the exceptional groups where $H$ is nontrivial for $A(e)$ can be
read off from [Som01]. They are $A_{2}$ and $F_{4}(a_{2})$ in type $F_{4}$;
$A_{3}+A_{2}$ and $E_{7}(a_{4})$ in type $E_{7}$; and $A_{3}+A_{2}$,
$D_{4}+A_{2}$, $E_{7}(a_{4})$, $D_{5}+A_{2}$, $E_{8}(b_{6})$, $D_{7}(a_{1})$,
and $E_{8}(b_{4})$ in type $E_{8}$. In all these cases, there is a unique
$\mathcal{O}$ such that $(\mathcal{O},\mathcal{O}^{\prime}_{e})$ is a minimal
(special) degeneration.

If $e$ does not belong to the $E_{8}(b_{6})$ orbit and is not of type $A_{2}$
in $F_{4}$ or $D_{4}+A_{2}$ in $E_{8}$, then $A(e)\simeq{\mathfrak{S}}_{2}$
and $H=A(e)$, so that $\bar{A}(e)$ is trivial. Since we already know from
[FJLS17] that $A(e)$ is acting by outer action, we see that $H$ does too.

If $e$ is not of type $A_{2}$ in $F_{4}$ or $D_{4}+A_{2}$ in $E_{8}$, then
there exists a unique $\mathcal{O}$ where $\mathcal{O}$ is a minimal (special)
degeneration to $e$ and $A(e)$ acts non-trivially
$\mathcal{S}_{\mathcal{O},e}$. The singularity of
$\mathcal{S}_{\mathcal{O},e}$ is a simple surface singularity $D_{k+1}$,
yielding a $C_{k}$ singularity. It follows that $H$ itself acts non-trivially
on $IH^{*}(\mathcal{S}_{\mathcal{O},e})$.

For the $E_{8}(b_{6})$ orbit, we have $A(e)\simeq{\mathfrak{S}}_{3}$ and $H$
is the cyclic group of order $3$. The slice of $D_{7}(a_{1})$ at $e$ is of
type $\mu$, which is not normal, but has normalization of type $A_{3}=D_{3}$
and we previously computed that $A(e)$ acts by outer action upon the
normalization, so that the normalization is $C_{2}$. Since the elements of $H$
cannot give an outer action, the outer action descends to $\bar{A}(e)$.

If $e$ is of type $A_{2}$ in $F_{4}$ or $D_{4}+A_{2}$ in $E_{8}$, then
$\mathcal{S}_{\mathcal{O},e}$ is isomorphic to $d_{4}/S_{4}$. This occurs for
$(F_{4}(a_{3}),A_{2})$ and $(E_{8}(a_{7}),D_{4}+A_{2})$. By §5.7, the action
of $A(e)$ on $IH^{*}(\mathcal{S}_{\mathcal{O},e})$ is trivial. On the other
hand, the action of $A(e)$ is non-trivial on
$IH^{*}(\mathcal{S}_{\mathcal{O}^{\prime\prime},e})$, where
$\mathcal{O}^{\prime\prime}$ is the minimal non-special orbit between
$\mathcal{O}$ and $\mathcal{O}_{e}$. ∎

###  6.2. Classical types

Let $X$ be of type $B,C,D,$ or $C^{\prime}$. Let $\epsilon$ and
$\epsilon^{\prime}$ be defined for the given type. For a partition $\mu$,
define

$R:=\\{s\ |\ s\not\equiv\epsilon,m_{\mu}(s)\neq
0,h_{\mu}(s)\not\equiv\epsilon^{\prime}\\}.$

For $s\in R$, define $s^{\prime}$ to satisfy $s^{\prime}\not\equiv\epsilon$,
$m(s^{\prime})\neq 0$ and maximal for this property with $s^{\prime}<s$. Set
$s^{\prime}=0$ if no such $s^{\prime}$ exists and set $x_{0}=1$ in $A(e)$.
Define $H$ to the be subgroup of $A(e)$ generated by the following elements of
$A(e)$:

(17) $H:=\langle x_{s}x_{s^{\prime}}\ |\ s\in R\rangle.$

By [Som01], the quotient of $A(e)$ by $H$ gives Lusztig’s canonical quotient
in $B$,$C$. In $D$ we get an extra factor of ${\mathbf{Z}}_{2}$, as opposed to
working in the special orthogonal group. In type $C^{\prime}$, we get
something new and we take this as the definition of the canonical quotient (we
can give a definition that is simliar to the characterization in op. cit.)
Let $r=\\#R$. Then the canonical quotient ${\bar{A}}(e)$ is elementary abelian
with $r$ generators in types $C,D,C^{\prime}$ and $r-1$ in type $B$.

Let $G$ be classical group.

######  Proposition 6.2.

If the type of the minimal special degeneration is type $C_{n}$ for $n\geq 2$
and $l\equiv\epsilon^{\prime}$ in Table 1, then $H$ acts non-trivially on
slice. Otherwise, $H$ acts trivially on $\mathcal{S}_{\mathcal{O},e}$.

When the slice $\mathcal{S}_{\mathcal{O},e}\simeq d_{n}/V_{4}$, $H$ acts by
outer automorphism on the $\mathcal{S}_{\mathcal{O}^{\prime\prime},e}$, where
$\mathcal{O}^{\prime\prime}$ is the minimal non-special orbit between
$\mathcal{O}$ and $\mathcal{O}_{e}$.

###### Proof.

In Table 1, the element $e\in\mathcal{O}_{\mu}$. For the type $B_{n}$
singularities:

  * •

Type $c$. The elements acting non-trivially on the slice involve $x_{a+1}$.
But the part $a{+}2n{-}1$ has height $l+1$ and the part $a+1$ has height
$l+3$, both of which are congruent to $\epsilon^{\prime}$. Hence, all the
elements in $H$ do not involve $x_{a+1}$ and $H$ acts trivially.

  * •

Type $d$. The elements acting non-trivially on the slice involve $x_{a+2n}$.
But the part $a{+}2n$ has height $l+2$, which is congruent to
$\epsilon^{\prime}$. For $s$ minimal for $s>a{+}2n$ and $s\not\equiv\epsilon$,
we must have $h(s)$ even since the parts between $s$ and $a{+}2n$ are
congruent to $\epsilon$ and so come with even multiplicity. Hence none of the
elements generating $H$ involve $x_{a+2n}$ and $H$ acts trivially.

  * •

Type $e$. The elements acting non-trivially on the slice involve $x_{a+2n-1}$
or $x_{a+1}$. Both of these parts have height congruent to
$\epsilon^{\prime}$, so as in the type $d$, none of the elements generating
$H$ involve either part and $H$ acts trivially.

Next, we treat the case of type $b$. Here, $H$ acts non-trivially if either
$x_{a+2n-2}$ or $x_{a+2}$ are involved in a generator of $H$, but not both.
The height of $a{+}2n{-}2$ is $l+1$ and $a+2$ is $l+2$. If
$l\equiv\epsilon^{\prime}$, then $x_{a+2n-2}x_{a+2}$ is in $H$, but no other
generator involves $a{+}2n{-}2$ since $s$ minimal for $s>a{+}2n{-}2$ and
$s\not\equiv\epsilon$ must have $s\equiv\epsilon^{\prime}$. So $H$ acts
trivially. But if $l\not\equiv\epsilon^{\prime}$, then some element of the
form $x_{a+2}x_{s^{\prime}}$ is in $H$ and $H$ does not act trivially on the
slice. This happens exactly when the second diagram in Table 2 occurs, for the
upper right $C_{n+1}$ singularity. Hence it is denoted $C_{n+1}^{*}$.

For exceptional type $g_{sp}$ there is no $A(e)$ outer action. However, $H$
acts non-trivially on the $\mathcal{S}_{\mathcal{O}^{\prime\prime},e}$. The
proof is similar to the above cases, as is the proof that $H$ acts trivially
for type $h$. ∎

##  7\. Combinatorial statement of duality, classical groups

Let $(\lambda,\mu)$ be a minimal special degeneration in types $B,C,D$ or
$C^{\prime}$. Then all three of $(f(\lambda),f(\mu))$, $(d(\mu),d(\lambda))$,
and $(d_{LS}(\mu),d_{LS}(\lambda))$ are minimal special degenerations. We now
prove that the four types of singularities are given by the Figure 2.

There is a bit more going on in the first quartet where the partition pattern
of type $c$ will get interchanged under internal duality with that of type
$f^{1}_{sp}$ and that of type $d$ is interchanged under internal duality with
type $f^{2}_{sp}$. Let internal duality map from type $X$ to $d(X)$.

######  Lemma 7.1.

The vertical arrows in Figure 2 are correct.

In particular, if $l\equiv\epsilon^{\prime}_{X}$, then the singularity of type
$b$ is interchanged with the minimal special $g_{sp}$.

If $l\not\equiv\epsilon^{\prime}_{X}$, then the singularity of type $b$ is
interchanged with the minimal special $h$.

Each of the two types of $B_{n}$ singularities switches with a corresponding
type of $b_{n}^{sp}$ singularity: type $c$ with $f^{1}_{sp}$ and type $d$ with
$f^{2}_{sp}$. The type $e$ singularity switches with type $h_{sp}$ and type
$a$ goes to type $a$.

###### Proof.

If $(\lambda,\mu)$ becomes $(\lambda^{\prime},\mu^{\prime})$ after removing
$l$ rows and $s$ columns, then clearly $(d(\mu),d(\lambda))$ becomes
$(d(\mu^{\prime}),d(\lambda^{\prime}))$ after removing $s$ rows and $l$
columns. So it is sufficient to work with the irreducible forms in Tables 1
and 2 to understand how a pair of partitions behaves under $d$.

A quick check shows that, under the transpose operation on partitions, the
partition in the table of type $c$ is interchanged with the one of type
$f^{1}_{sp}$; the type $d$ partition is interchanged with the one of type
$f^{2}_{sp}$; the one of type $e$ is interchanged with the one of type
$h_{sp}$; the one of type $b$ is interchanged with the partition found in
ether $g_{sp}$ and $h$; and type $a$ is self-dual.

The behavior of $\epsilon$ and $\epsilon^{\prime}$ under $d$ is also follows:

$(\epsilon,\epsilon^{\prime})_{X}+(\epsilon^{\prime},\epsilon)_{d(X)}\equiv(1,1).$

As a result, for the interchange of the first three partition types described
above, the switching of $l$ and $s$ upon going from $(\lambda,\mu)$ to
$(d(\mu),d(\lambda))$ agrees with the restriction on $l$ and $s$ in the dual
type $d(X)$ given in the Tables. However, for type $b$, when
$l\equiv\epsilon^{\prime}_{X}$, the interchange is with type $g_{sp}$ and when
$l\not\equiv\epsilon^{\prime}_{X}$, the interchange is with type $h$. The
self-dual type $a$ is clear. ∎

Next we want to write down the rules for the horizontal arrows in Figure 2.
First, we start with the case of type $b$, the $C_{n}$ singularity from
[KP82]. In that case, we can write $(\lambda,\mu)$ locally as
$([(2n\\!+\\!s)^{t},s^{u}],(2n\\!+\\!s)^{t-1},2n\\!-\\!2\\!+\\!s,s\\!+\\!2,s^{u-1}]$
for positive integers $t$ and $u$, where $s\not\equiv\epsilon_{X}$ where $f$
maps $X$ to $f(X)$.

######  Lemma 7.2.

Assume the degeneration is of type $b$ for a given $n$. When
$l\equiv\epsilon^{\prime}_{X}$, under the $f$ map, the degeneration
$(\lambda,\mu)$ is carried to a singularity of type

$\displaystyle e$ $\displaystyle\text{ if }t\geq 2,u\geq 2$ $\displaystyle d$
$\displaystyle\text{ if }t\geq 2,u=1$ $\displaystyle c$ $\displaystyle\text{
if }t=1,u\geq 2$ $\displaystyle C_{n+1}$ $\displaystyle\text{ if }t=u=1$

When $l\not\equiv\epsilon^{\prime}_{X}$, then type $b$ is exchanged with type
$b$ with $n$ replaced by $n{-}1$, that is $C_{n-1}$.

###### Proof.

Since $l$ rows are removed, $h_{\lambda}(2n{+}s)=l{+}1$ and
$h_{\lambda}(s)=l+1+m_{\lambda}(s)$. Also, $h_{\mu}(2n{-}2+s)=l{+}1$ and
$h_{\mu}(s{+}2)=l{+}2$ and $m_{\mu}(2n{-}2+s)=m_{\mu}(s{+}2)=1$ if $n>2$; and
$h_{\mu}(2n{-}2+s)=h_{\mu}(s{+}2)=l{+}2$ and $m_{\mu}(s{+}2)=2$ if $n=2$. All
these parts are not congruent to $\epsilon_{X}$ since $s\not\equiv\epsilon$.

If $l\not\equiv\epsilon^{\prime}_{X}$,then
$h_{\lambda}(2n{+}s)\equiv\epsilon^{\prime}_{X}$ and
$h_{\lambda}(s)+m_{\lambda}(s)\equiv\epsilon^{\prime}_{X}$. In particular, in
Lemma 2.2, $2n{+}s$ obeys line 1 or 3, and $s$ obeys lines 2 or 3. Hence the
partition $[2n{+}s,s]$ in $\lambda$ gets replaced by $[2n{+}s{-}1,s{+}1]$ in
$f(\lambda)$ regardless of the parities of $m_{\lambda}(2n{+}s)$ and
$m_{\lambda}(s)$. Moreover, $[2n{-}2{+}s,s{+}2]$ in $\mu$ goes to
$[2n{-}3{+}s,s{+}3]$ since $s{+}2$ obeys line 2 if $n>2$ and line $3$ if
$n=2$. Hence we end of with $f(\lambda),f(\mu)$ locally equal to

$([2n{-}1{+}s,s{+}1],[2n{-}3{+}s,s{+}3]),$

which is of type $C_{n-1}$ with $s+1$ rows removed. Note that
$s+1\not\equiv\epsilon_{f(X)}$ since $\epsilon_{X}$ and $\epsilon_{f(X)}$ have
different parities.

Now if $l\equiv\epsilon^{\prime}$, there are four cases to consider depending
on $t=m_{\lambda}(2n{+s})$ and $u=m_{\lambda}(s)$: if $t=u=1$, $\lambda$ is
locally $[2n{+}s,s]$

Now if $l\equiv\epsilon^{\prime}$, then
$h_{\lambda}(2n{+}s)\not\equiv\epsilon^{\prime}_{X}$ so if $m(2n{+}s)\geq 2$,
the last two values of $2n{+}s$ in $\lambda$ are unchanged in $f(\lambda)$
since lines two or four apply in Lemma 2.2. But if $m(2n{+}s)=1$, then line
two applies and $2n{+}s$ becomes $2n{+}s+1$. Now we have
$h_{\lambda}(s)+m_{\lambda}(s)\not\equiv\epsilon^{\prime}_{X}$, so we are in
the setting of lines one and four of the lemma. Thus the first two values of
$s$ are unchanged if $m(s)\geq 2$ and the solve values of $s$ changes to $s-1$
if $m(s)=1$. Thus $f(\lambda)$ is locally $[2n{+}s,2n{+}s,s,s]$,
$[2n{+}s,2n{+}s,s{-}1]$, $[2n{+}s{+}1,s,s]$, or $[2n{+}s{+}1,s{-}1]$ when
$(t,u)$ is $(\geq 2,\geq 2)$, $(\geq 2,1)$, $(1,\geq 2)$, $(1,1)$,
respectively, after removing $l{-}1$, $l{-}1$, $l$ or $l$ rows, respectively.

In the four cases, $\mu$ looks locally like

$[2n{+}s,2n{+}s{-}2,s{+}2,s],[2n{+}s,2n{+}s{-}2,s{+}2],[2n{+}s{-}2,s{+}2,s],\text{
and }[2n{+}s{-}2,s{+}2],$

respectively. Using that $h_{\mu}(2n{+}s{-}2)\not\equiv\epsilon^{\prime}_{X}$
and $m_{\mu}(2n{+}s{-}2)=1$ and $h_{\mu}(s)+m_{\mu}(s)\equiv\epsilon^{\prime}$
we get using Lemma 2.2, that $f(\mu)$ is locally
$[2n{+}s{-}1,2n{+}s{-}1,s{+}1,s{+}1]$ $[2n{+}s{-}1,2n{+}s{-}1,s{+}1]$
$[2n{+}s{-}1,s{+}1,s{+}1]$ $[2n{+}s{-}1,s{+}1]$, after removing $l{-}1$,
$l{-}1$, $l$ or $l$ rows, respectively.

Hence, after removing $s$, $s{-}1$, $s$, or $s{-}1$ rows, respectively, we
find that $(f(\lambda),f(\mu))$ is of type $e$, $d$, $c$, for the same value
of $n$ or type $b$ with $C_{n+1}$. ∎

######  Proposition 7.3.

The four singularities behave as in the three quartets.

###### Proof.

The same ideas in the previous lemma, or the fact that $f\circ f=\text{id}$,
shows that the type $c$,$d$,$e$ partitions for rank $n$ are mapped to the
$C_{n}$ singularity.

Now given any of the singularities in Table 1 or 2, either the singularity is
codimension two or the degeneration obtained using $d$ is. Either this
singularity is type $C_{n}$ or applying $f$ to the degeneration gives a type
$C_{n}$ singularity with $l\equiv\epsilon^{\prime}_{X}$. Putting this
singularity in the upper left corner of the square of singularities, the two
lemmas show that the three corners are as shown in Figure 2 after we note that
if $C_{n}$ is carried to $C_{n+1}$ under $f$, the value of $l$ stays the same
and satisfies $l\not\equiv\epsilon^{\prime}_{f(X)}$. This means that applying
$d$ leads to the $h$ singularity according to Lemma 7.1. In effect, the three
quartets (or four if we include the two different ways to obtain $B_{n}$ and
$b_{n}^{sp}$) are controlled by the four possibilities for $t$ and $u$ in the
$l\equiv\epsilon^{\prime}_{X}$ case in Lemma 7.2. ∎

######  Remark 7.4.

We can also write the specific conditions that describe the action under $f$
where the singularity starts as $c_{n}^{sp}$. Writing $\mu$ locally as
$[(s{+}2)^{t},(s{+}1)^{n},s^{u}]$ where $t=m(s{+}2)$ and $u=m(s)$, then
$c_{n}^{sp}$ maps to $h,f^{1}_{sp},f^{2}_{sp},$ or $h_{sp}$ when $(t,u)$ is
$(\geq 1,\geq 1)$, $(0,\geq 1)$, $(\geq 1,0)$, $(0,0)$. respectively. Here,
$s=0$ is considered a part in type $C$. These conditions are exactly the
conditions from the four cases in Lemma 7.2 for $C_{n}$, under $d$ when
$l\equiv\epsilon^{\prime}_{X}$.

##  8\. Outer automorphisms of ${\mathfrak{g}}$

###  8.1. Outer automorphisms for $A_{n}$ and $E_{6}$

Besides considering the automorphism group for $D_{n}$, we can do the same for
$A_{n}$ and $E_{6}$ (and the full automorphism group for $D_{4}$). The ideas
follow [Slo80, §7.5] where the case of the regular and subregular orbits were
handled. Let $CA({\mathfrak{s}})$, called the outer reductive centralizer in
loc. cit., denote the centralizer of ${\mathfrak{s}}$ in $\rm
Aut({\mathfrak{g}})$. There is a surjective map $\pi:\rm
Aut({\mathfrak{g}})\to\rm Aut(\Delta)$ where $\Delta$ is the Dynkin diagram of
${\mathfrak{g}}$. Let $e\in{\mathcal{N}}_{o}$ and ${\mathfrak{s}}$ be an
${\mathfrak{sl}}_{2}$-triple $\\{e,h,f\\}$ for $e$. Assume that the weighted
Dynkin diagram for $e$ is invariant under $\rm Aut(\Delta)$. Then $\sigma(h)$
is conjugate to $g.h$ for some $g\in G$. This assumption holds as long as $e$
is not very even in $D_{2n}$ and is not $[5,1^{3}]$ or $[3,1^{5}]$ in $D_{4}$.
Then the proof of Lemma 2 in loc. cit. still applies.

######  Lemma 8.1.

For all ${\mathfrak{sl}}_{2}$-triples ${\mathfrak{s}}$ as above, the map $\pi$
restricted to $CA({\mathfrak{s}})$ is surjective onto $\rm Aut(\Delta)$. In
particular, $CA({\mathfrak{s}})/C({\mathfrak{s}})\simeq\rm Aut(\Delta)$.

As before, notice that $CA({\mathfrak{s}})$ acts on
${\mathfrak{c}}({\mathfrak{s}})$ and we are interested in this action. Let
${\mathfrak{c}}_{0}\subset{\mathfrak{c}}({\mathfrak{s}})$ be a simple factor
or the central toral subalgebra of ${\mathfrak{c}}({\mathfrak{s}})$. The rest
of Lemma 2 in loc. cit. generalizes as follows:

######  Lemma 8.2.

Suppose that ${\mathfrak{g}}$ has type $A_{n},D_{2n+1},$ or $E_{6}$. Then
there is an element $\phi\in CA({\mathfrak{s}})$ that stabilizes
${\mathfrak{c}}_{0}$ and acts by $-1$ on some maximal toral subalgebra of
${\mathfrak{c}}_{0}$. In particular, if ${\mathfrak{c}}_{0}$ is simple of type
$A_{k},D_{2k+1},$ or $E_{6}$, then the image of $CA({\mathfrak{s}})$ in $\rm
Aut({\mathfrak{c}}_{0})$ is an outer automorphism of order two.

###### Proof.

Fix a toral subalgebra ${\mathfrak{h}}$ of ${\mathfrak{g}}$. Let $\sigma\in\rm
Aut({\mathfrak{g}})$ be an automorphism of ${\mathfrak{g}}$ that acts by $-1$
on ${\mathfrak{h}}$. Choose ${\mathfrak{m}}$ to be a standard Levi subalgebra
of ${\mathfrak{g}}$ relative to ${\mathfrak{h}}$ so that $e$ is distinguished
in ${\mathfrak{m}}$ [Car93]. Pick ${\mathfrak{s}}$ so that
${\mathfrak{s}}\subset{\mathfrak{m}}$. Then
$\sigma({\mathfrak{m}})={\mathfrak{m}}$ and since $e\in{\mathfrak{m}}$ is
distinguished (and so satisfies the assumption above), there exists $g\in G$
(specifically in the subgroup $M$ with Lie algebra ${\mathfrak{m}}$) such that
$\rm Int(g)\circ\sigma$ is the identity on ${\mathfrak{s}}$. That is,
$\phi:=\rm Int(g)\circ\sigma\in CA({\mathfrak{s}})$.

Next, by [Car93] the center ${\mathfrak{t}}$ of ${\mathfrak{m}}$, which lies
in ${\mathfrak{h}}$, is a maximal toral subalgebra of
${\mathfrak{c}}({\mathfrak{s}})$. Since $M$ acts trivially on
${\mathfrak{t}}$, it follows that $\phi$ acts by $-1$ on ${\mathfrak{t}}$. In
particular, $\phi({\mathfrak{c}}_{0})={\mathfrak{c}}_{0}$ and $\phi$ acts by
$-1$ on its maximal toral subalgebra ${\mathfrak{c}}_{0}\cap{\mathfrak{t}}$.
Since $-1$ is not in the Weyl groups of $A_{k},D_{2k+1},$ or $E_{6}$, the
induced automorphism of ${\mathfrak{c}}_{0}$ must by outer (since $-w_{0}$ i s
then clearly outer). ∎

######  Remark 8.3.

We want to apply this when $-1$ is not in the Weyl group of ${\mathfrak{g}}$,
but it is also applies when $-1$ is in the Weyl group and
${\mathfrak{c}}({\mathfrak{s}})$ has simple factors of type $a_{k}$, where it
forces $C({\mathfrak{s}})$ to contain an element of order two. So in
$F_{4},E_{7}$ and $E_{8}$, this explains why the action is upgraded to
$a_{k}^{+}$ always. There are no type $D_{k}$ simple factors of
${\mathfrak{c}}({\mathfrak{s}})$ for $k>4$ in the exceptional groups, except
for the sole case of $d_{6}$ the minimal orbit in $E_{7}$. It remains (along
with its dual), the only case where a natural outer action exists on the
slice, but the induced action of $CA({\mathfrak{s}})$ does not realize it.

######  Corollary 8.4.

Let $G=\rm Aut({\mathfrak{g}})$.

In type $A_{n}$, any singularity for a minimal degeneration, which are not
type $A_{1}$, acquires an outer action (that is, becomes $A_{k}^{+}$ or
$a_{k}^{+}$, when $k\geq 2$).

In type $E_{6}$, the singularities with no outer action using
$C({\mathfrak{s}})$, all acquire the natural outer action.

###### Proof.

For the minimal orbit types, we can use the previous lemma since the simple
factors of ${\mathfrak{c}}({\mathfrak{s}})$ are all of type $A$ or $E_{6}$.

For the simple surface singularities, we can use the action on the center of
${\mathfrak{c}}({\mathfrak{s}})$ as in §5 (the subregular case already follows
from Slodowy), since they are simple surface singularities of type $A_{k}$.

Finally, the case of $[2a_{2}]^{+}$ becomes $[2a^{+}_{2}]^{+}$ since the outer
automorphism preserves each simple factor of ${\mathfrak{c}}({\mathfrak{s}})$.
∎

In §11 we include the diagram for the $D_{4}$ case with the
${\mathfrak{S}}_{3}$-action.

##  9\. A conjecture of Lusztig

In [Lus22, §0.4], Lusztig associated to each minimal special degeneration in
${\mathfrak{g}}$ a certain Weyl group $W^{\prime}$. We describe it in the
classical groups.

Let $h:=\dim(\mathcal{S}_{\mathcal{O},e})/2+1$ for the slice
$\mathcal{S}_{\mathcal{O},e}$ associated to the degeneration. In types
$B$,$C$,$D$, the slice $\mathcal{S}_{\mathcal{O},e}$ is either dimension two
or the dimension of the minimal special orbit for a Lie algebra
${\mathfrak{c}}_{0}$ of type $B$, $C$, $D$. The dimension of the latter is
$2h^{\prime}-2$ where $h^{\prime}$ is the Coxeter number of
${\mathfrak{c}}_{0}$. Since $h^{\prime}$ is even for types $B_{k}$, $C_{k}$,
$D_{k}$, respectively equal to $2k$, $2k$, $2k-2$, the number $h$ is even when
${\mathfrak{g}}$ has type $B,C,D$.

Here is the assignment of $W^{\prime}$ to the degeneration, which varies in
type $D$ from Lusztig’s assignment. For ${\mathfrak{g}}$ of type $A$, the Weyl
group $W^{\prime}$ associated to the degeneration is the Weyl group of type
$A_{h-1}$; for ${\mathfrak{g}}$ of types $B/C$, it is of type $B_{h/2}$; and
for ${\mathfrak{g}}$ of type $D$, it is of type $D_{h/2}+1$ when
$\mathcal{S}_{\mathcal{O},e}$ is of type $d_{k}$ when
$G=\operatorname{SO}(2N)$, and it is of type $B_{h/2}$, otherwise.

Lusztig worked with the associated special representations of $W$, the Weyl
group of ${\mathfrak{g}}$. Our definition of $W^{\prime}$ is slightly