# On induced completely prime primitive ideals in enveloping algebras of
classical Lie algebras

Simon M. Goodwin, Lewis Topley and Matthew Westaway  School of Mathematics,
University of Birmingham, Birmingham, B15 2TT, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematical Sciences, University of
Bath, Bath, BA2 7AY, UK<EMAIL_ADDRESS>
###### Abstract.

A distinguished family of completely prime primitive ideals in the universal
enveloping algebra of a reductive Lie algebra ${\mathfrak{g}}$ over
${\mathbb{C}}$ are those ideals constructed from one-dimensional
representations of finite $W$-algebras. We refer to these ideals as
Losev–Premet ideals. For ${\mathfrak{g}}$ simple of classical type, we prove
that for a Losev–Premet ideal $I$ in $U({\mathfrak{g}})$, there exists a
Losev–Premet ideal $I_{0}$ for a certain Levi subalgebra ${\mathfrak{g}}_{0}$
of ${\mathfrak{g}}$ such that associated variety of $I_{0}$ is the closure of
a rigid nilpotent orbit in ${\mathfrak{g}}_{0}$ and $I$ is obtained from
$I_{0}$ by parabolic induction. This is deduced from the corresponding
statement about one-dimensional representations of finite $W$-algebras.

##  1\. Introduction

Let $G$ be a connected reductive algebraic group over ${\mathbb{C}}$ and let
${\mathfrak{g}}=\operatorname{Lie}G$. The study of the spectrum
$\operatorname{Prim}U({\mathfrak{g}})$ of primitive ideals in the universal
enveloping algebra $U({\mathfrak{g}})$ of ${\mathfrak{g}}$ is one of the most
venerable topics in representation theory. These primitive ideals serve as a
tractable approximation to the irreducible representations of
$U({\mathfrak{g}})$ and have generated an enormous amount of interest from
leading mathematicians. Amongst the primitive ideals of $U({\mathfrak{g}})$,
an especially important role is played by the spectrum
$\operatorname{Prim}^{1}U({\mathfrak{g}})$ of completely prime primitive
ideals. This importance is largely due to the close relationship between these
ideals and the unitary representations of Lie groups (see [LMM21] and the
references therein).

The classification of $\operatorname{Prim}U({\mathfrak{g}})$ as a set was
established in the early 1980s following important contributions from
Barbasch–Vogan, Duflo, Joseph and others; we refer to [Jo83] and [Jo84] for an
overview with references. Despite the classification of
$\operatorname{Prim}U({\mathfrak{g}})$ being established 40 years ago, the
classification problem for $\operatorname{Prim}^{1}U({\mathfrak{g}})$ has
proved to be very challenging and remains open. A classification is known in
the case $G$ is of type ${\mathsf{A}}$, where it was established by Mœglin in
[Mœ87]. There is also work dealing with some small rank cases, for instance
the work of Borho in [Bo76] for $G$ of type ${\mathsf{B}}_{2}$, and there have
been many further important contributions to the general classification
problem. We refer to [PT14, §1.2] for some discussion and references.

A general paradigm in classification problems in Lie theoretic representation
theory is to use a process of parabolic induction. For the case of
$\operatorname{Prim}^{1}U({\mathfrak{g}})$, parabolic induction of ideals, as
recalled in §2.6, can be used. Of importance for this approach is the theorem
of Conze in [Co74], which states that the induced ideal of a completely prime
primitive ideal is itself a completely prime primitive ideal. Thus to classify
completely prime primitive ideals a key problem is to determine the so-called
rigid completely prime primitive ideals, namely those that cannot be non-
trivially obtained by parabolic induction. The main result of this paper
solves this problem, in the case $G$ is of classical type, for an important
family of completely prime primitive ideals that arise naturally from the
theory of finite $W$-algebras. Before stating this result in Theorem 1.1, we
need to recall some more background.

Following the classification of $\operatorname{Prim}U({\mathfrak{g}})$, an
important problem was to determine the associated variety ${\mathcal{VA}}(I)$
of a primitive ideal $I$; the definition of ${\mathcal{VA}}(I)$ is recalled in
§2.6. This problem was made precise by Joseph in [Jo85]: by identifying
${\mathfrak{g}}$ with its dual via a nondegenerate invariant symmetric
bilinear form, ${\mathcal{VA}}(I)$ is viewed as a closed subvariety of
${\mathfrak{g}}$, and the main result of op. cit. states that
${\mathcal{VA}}(I)$ is the closure $\overline{\mathbb{O}}_{e}$ of the
$G$-orbit ${\mathbb{O}}_{e}$ of some nilpotent element $e\in{\mathfrak{g}}$.
This leads to the decomposition
$\operatorname{Prim}U({\mathfrak{g}})=\bigsqcup\operatorname{Prim}_{{\mathbb{O}}}U({\mathfrak{g}})$,
where the (finite, disjoint) union is taken over the nilpotent $G$-orbits
${\mathbb{O}}$ in ${\mathfrak{g}}$ and
$\operatorname{Prim}_{{\mathbb{O}}}U({\mathfrak{g}}):=\\{I\in\operatorname{Prim}U({\mathfrak{g}})\mid{\mathcal{VA}}(I)=\overline{\mathbb{O}}\\}$.
Consequently, problems regarding $\operatorname{Prim}U({\mathfrak{g}})$ can
broken down to problems about
$\operatorname{Prim}_{{\mathbb{O}}}U({\mathfrak{g}})$, and those about
$\operatorname{Prim}^{1}U({\mathfrak{g}})$ can be reduced to
$\operatorname{Prim}^{1}_{{\mathbb{O}}}U({\mathfrak{g}}):=\operatorname{Prim}^{1}U({\mathfrak{g}})\cap\operatorname{Prim}_{{\mathbb{O}}}U({\mathfrak{g}})$.

The introduction of finite $W$-algebras to the mathematical literature by
Premet in [Pr02] led to a resurgence in interest in the primitive ideals of
$U({\mathfrak{g}})$. In §4, we recall some background on the finite
$W$-algebra $U({\mathfrak{g}},e)$ associated to a nilpotent element
$e\in{\mathfrak{g}}$. In particular, we recall the close relationship between
$\operatorname{Prim}_{{\mathbb{O}}_{e}}U({\mathfrak{g}})$ and the finite
dimensional irreducible representations of the finite $W$-algebra
$U({\mathfrak{g}},e)$. This gives an approach to
$\operatorname{Prim}_{{\mathbb{O}}_{e}}U({\mathfrak{g}})$ via the
representation theory of $U({\mathfrak{g}},e)$, which has led to spectacular
progress. For instance, it leads to a resolution of the problem discussed in
[BJ01] that $\operatorname{Prim}U({\mathfrak{g}})$ can be described as a
countable union of varieties; the work of Losev in [Lo15] made a breakthrough
in the programme proposed by Joseph in [Jo80] to determine Goldie rank
polynomials; and the longstanding problem that
$\operatorname{Prim}^{1}_{{\mathbb{O}}_{e}}U({\mathfrak{g}})\neq\varnothing$
has been resolved, with the proof completed by Premet in [Pr14].

The theory of finite $W$-algebras picks out a distinguished class of
completely prime primitive ideals, namely those corresponding to one-
dimensional representations of finite $W$-algebras. We name these ideals
Losev–Premet ideals, reflecting that major progress on their theory has been
made in the work of both Losev and Premet. Thanks to [Lo10a, Theorem 1.2.2]
every Losev–Premet ideal coming from a one-dimensional representation of
$U({\mathfrak{g}},e)$ lies in
$\operatorname{Prim}^{1}_{{\mathbb{O}}_{e}}U({\mathfrak{g}})$. It is now well-
established that the Losev–Premet ideals form an important family in
$\operatorname{Prim}^{1}_{{\mathbb{O}}_{e}}U({\mathfrak{g}})$. For instance,
it is proved in [Lo15, Corollary 1.2] that for primitive ideals with integral
central character the Losev–Premet ideals give all completely prime primitive
ideals (with a possible exception for $G$ of type ${\sf E}_{8}$); it is known
in type A that this statement holds without the restriction on central
character (see the remarks following [Pr11, Theorem B]); it is proved in
[To23, Theorem 1.2] that, for $G$ of classical type, the Losev–Premet ideals
are precisely the ideals in the image of Losev’s orbit map introduced in
[Lo22]; and it is expected that the left and right annihilators of unitary
Harish-Chandra bimodules for $U({\mathfrak{g}})$ are Losev–Premet ideals, see
[LMM21, Conjecture 6.3.1]. We note that it is in general false that every
ideal in $\operatorname{Prim}^{1}U({\mathfrak{g}})$ is a Losev–Premet ideal
and refer to the work of Losev–Panin in [LP21] for an important recent
development regarding this defect.

We now state our main theorem which, as explained after its statement,
completes the classification of rigid Losev–Premet ideals for $G$ of classical
type. In the statement we refer to Lusztig–Spaltenstein induction of nilpotent
orbits, which is recapped in §2.4.

######  Theorem 1.1.

Let $G$ be a simple algebraic group over ${\mathbb{C}}$ of classical type and
let ${\mathbb{O}}$ be a nilpotent orbit in
${\mathfrak{g}}=\operatorname{Lie}G$. Let $I\subseteq U({\mathfrak{g}})$ be a
Losev–Premet ideal with ${\mathcal{VA}}(I)=\overline{{\mathbb{O}}}$. Then
there exists a Levi subalgebra ${\mathfrak{g}}_{0}$ of ${\mathfrak{g}}$, a
parabolic subalgebra ${\mathfrak{p}}$ of ${\mathfrak{g}}$ with Levi factor
${\mathfrak{g}}_{0}$, a rigid nilpotent orbit ${\mathbb{O}}_{0}$ in
${\mathfrak{g}}_{0}$, and a Losev–Premet ideal $I_{0}\subseteq
U({\mathfrak{g}}_{0})$ such that ${\mathbb{O}}$ is obtained from
${\mathbb{O}}_{0}$ by Lusztig–Spaltenstein induction,
${\mathcal{VA}}(I_{0})=\overline{\mathbb{O}}_{0}$, and $I$ is induced from
$({\mathfrak{p}},I_{0})$.

This theorem implies that the rigid Losev–Premet ideals, for $G$ of classical
type, are precisely those whose associated variety is the closure of a rigid
nilpotent orbit. By [PT14, Theorem 1] it is known that, for $G$ simple of
classical type and ${\mathbb{O}}$ a rigid nilpotent orbit in ${\mathfrak{g}}$,
there is a unique Losev–Premet ideal $I$ with
${\mathcal{VA}}(I)=\overline{\mathbb{O}}$. Moreover, the highest weight of a
simple highest weight $U({\mathfrak{g}})$-module with annihilator $I$ can be
determined using [LMM21, Proposition 8.2.3]. Thus Theorem 1.1 completes the
classification of rigid Losev–Premet ideals for $G$ of classical type.

Theorem 1.1 can be viewed as an extension of [PT14, Theorem 5] and our methods
build on the methods developed there, whilst also making use of recent results
from [To23]. The proof deduces the theorem from the corresponding statement
for finite $W$-algebras. In Theorem 4.1 we prove that, with the set up in
Theorem 1.1, any one-dimensional $U({\mathfrak{g}},e)$-module can be obtained
from a one-dimensional $U({\mathfrak{g}}_{0},e_{0})$-module via the parabolic
induction functor introduced by Losev in [Lo11].

We give now a brief overview of the strategy of the proof of Theorem 4.1.
First we need to establish some notation. The variety of one-dimensional
$U({\mathfrak{g}},e)$-modules is denoted ${\mathcal{E}}({\mathfrak{g}},e)$,
see (4.2) for a formal definition. We say that
$({\mathfrak{g}}_{0},{\mathbb{O}}_{0})$ is a rigid induction datum for
$({\mathfrak{g}},{\mathbb{O}})$ if ${\mathbb{O}}$ is induced from
${\mathbb{O}}_{0}$ and ${\mathbb{O}}_{0}$ is rigid in
${\mathfrak{g}}_{0}$.111In the body of the paper we prefer to use this term to
denote a $G$-orbit of such pairs, but we elide this subtlety in the
introduction. Parabolic induction for finite $W$-algebras gives a finite
morphism
${\mathcal{E}}({\mathfrak{g}}_{0},e_{0})\to{\mathcal{E}}({\mathfrak{g}},e)$
for any rigid induction datum $({\mathfrak{g}}_{0},{\mathbb{O}}_{e_{0}})$ for
$({\mathfrak{g}},{\mathbb{O}}_{e})$. In Proposition 5.5 it is shown that the
multiset of dimensions of irreducible components of the disjoint union
$\bigsqcup{\mathcal{E}}({\mathfrak{g}}_{i},{\mathbb{O}}_{e_{i}})$ over the
($G$-orbits of) rigid induction data
$({\mathfrak{g}}_{i},{\mathbb{O}}_{e_{i}})$ for
$({\mathfrak{g}},{\mathbb{O}}_{e})$ is equal to the multiset of dimensions of
irreducible components of ${\mathcal{E}}({\mathfrak{g}},e)$. Combining this
with Proposition 5.4, which shows that distinct irreducible components of
$\bigsqcup{\mathcal{E}}({\mathfrak{g}}_{i},e_{i})$ are mapped by the parabolic
induction morphism to distinct irreducible components of
${\mathcal{E}}({\mathfrak{g}},e)$, we can deduce Theorem 4.1. An important
result along the way is Theorem 3.3, which shows that for distinct rigid
induction data $({\mathfrak{g}}_{1},{\mathbb{O}}_{e_{1}})$ and
$({\mathfrak{g}}_{2},{\mathbb{O}}_{e_{2}})$ for
$({\mathfrak{g}},{\mathbb{O}}_{e})$ we have
${\mathfrak{g}}_{1}\not\subseteq{\mathfrak{g}}_{2}$ and
${\mathfrak{g}}_{2}\not\subseteq{\mathfrak{g}}_{1}$; we refer also to Remark
3.4 for the extension of this to any reductive $G$.

We remark that the computer calculations reported in [BG18] show that the
analogue of Theorem 1.1 for $G$ of exceptional type is not true. Those
calculations show that there are Losev–Premet ideals in $U({\mathfrak{g}})$
for $G$ of type $F_{4}$ or $E_{6}$, with associated variety a non-rigid
nilpotent orbit, that are not parabolically induced. It is not known whether
there are such ideals for $G$ of type $E_{7}$ or $E_{8}$, though this seems
quite plausible.

To end this introduction we draw attention to the remarkable recent work of
Losev in [Lo22] and Losev–Mason-Brown–Matvieievskyi in [LMM21]. An orbit
method for Lie algebras was introduced by Losev in [Lo22, §5], which gives a
natural embedding from the set of coadjoint orbits of $G$ to
$\operatorname{Prim}U({\mathfrak{g}})$. It was conjectured that the ideals
obtained in this way for classical Lie algebras are precisely the Losev–Premet
ideals, and as noted above this has now been proved by the second author in
[To23]. In [LMM21, Theorem 7.8.1], Losev’s orbit method is extended to define
an injection from isomorphism classes of $G$-equivariant covers of coadjoint
orbits to isomorphism classes of $G$-equivariant filtered quantizations of
affinizations of $G$-equivariant covers of nilpotent orbits. Each of these
quantizations gives rise to a primitive ideal, completing Vogan’s orbit method
program, which in turn has important applications to the theory of unipotent
representations of Lie groups. Furthermore, their methods give another
approach to understanding Losev–Premet ideals via birationally rigid induction
data, see for instance [LMM21, Remark 8.1.2].

This paper is organised as follows. In §2 we recall preliminaries on reductive
groups, Levi subgroups, nilpotent orbits, Lusztig-Spaltenstein induction,
sheets and primitive ideals; we also use this section to introduce notation
used throughout the paper. Then, in §3 we prove Theorem 3.3 by means of a
combinatorial argument involving the Kempken-Spaltenstein algorithm. In §4 we
introduce finite $W$-algebras and establish the framework needed to state
Theorem 4.1; the paper concludes in §5 with the proof of this theorem.

### Acknowledgements

We would like to thanks Sasha Premet for a useful comment on the first version
of this paper. All three authors were supported during this research by the
EPSRC grant EP/R018952/1. The research of the second author is supported by a
UKRI Future Leaders Fellowship, grant numbers MR/S032657/1, MR/S032657/2,
MR/S032657/3. The third author gratefully acknowledges funding from the Royal
Commission for the Exhibition of 1851.

##  2\. Reductive groups, Levi subgroups, nilpotent orbits,
Lusztig–Spaltenstein induction, sheets and primitive ideals

###  2.1. Reductive groups

Throughout this paper $G$ is a connected reductive algebraic group over
${\mathbb{C}}$. We often specialize to the case where $G$ is simple of
classical type, by which we mean $G$ is one of
$\mathrm{SL}_{n}({\mathbb{C}})$, $\mathrm{SO}_{N}({\mathbb{C}})$, or
$\mathrm{Sp}_{2n}({\mathbb{C}})$, where $n,N\in{\mathbb{Z}}_{>0}$. The results
of our paper do not depend on the isogeny class of $G$, which justifies the
restriction to these choices of $G$ for types ${\mathsf{A}}$, ${\mathsf{B}}$,
${\mathsf{C}}$ and ${\mathsf{D}}$. As we work entirely over ${\mathbb{C}}$ in
this paper we allow ourselves to abbreviate notation and just write
$\mathrm{SL}_{n}$, $\mathrm{SO}_{N}$, or $\mathrm{Sp}_{2n}$ for these groups;
we write ${\mathfrak{sl}}_{n}$, ${\mathfrak{so}}_{N}$ and
${\mathfrak{sp}}_{2n}$ for their Lie algebras. Explicitly we take
$\mathrm{SO}_{N}=\\{x\in\mathrm{SL}_{N}\mid x^{\mathrm{tr}}J_{N}x=J_{N}\\}$,
where

$J_{N}=\left(\begin{array}[]{ccc}&&1\\\ &\iddots&\\\ 1&&\end{array}\right)$

is the $N\times N$ matrix with $(i,j)$ entry equal to $\delta_{i,N+1-j}$; and
we take $\mathrm{Sp}_{2n}=\\{x\in\mathrm{SL}_{2n}\mid
x^{\mathrm{tr}}\tilde{J}_{2n}x=\tilde{J}_{2n}\\}$ where $\tilde{J}_{2n}$ has
block form

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

For a closed subgroup $H$ of $G$ we write ${\mathfrak{h}}=\operatorname{Lie}H$
for the Lie algebra of $H$ and ${\mathfrak{z}}({\mathfrak{h}})$ for the centre
of ${\mathfrak{h}}$. We let $(\cdot\,,\cdot)$ be a nondegenerate $G$-invariant
symmetric bilinear form on ${\mathfrak{g}}$, which allows us to identify
${\mathfrak{g}}\cong{\mathfrak{g}}^{*}$ as $G$-modules. We have that $G$ acts
on ${\mathfrak{g}}$ via the adjoint action. Let $x\in{\mathfrak{g}}$, let
$X\subseteq{\mathfrak{g}}$ and let $H$ be a subgroup of $G$. We write $H\cdot
x$ for the $H$-orbit of $x$, though often use the notation ${\mathbb{O}}_{x}$
for $G\cdot x$. The centralizer of $x$ in $H$ is denoted $H^{x}$ and the
centralizer of $x$ in ${\mathfrak{h}}$ is denoted ${\mathfrak{h}}^{x}$. We
write $H\cdot X$ for the $H$-saturation of $X$, i.e. $H\cdot X=\bigcup_{x\in
X}H\cdot x$. The closure of $X$ with respect to the Zariski topology on
${\mathfrak{g}}$ is denoted $\overline{X}$.

We fix a maximal torus $T$ of $G$ and a Borel subgroup $B$ of $G$ with
$T\subseteq B$, and let $U$ denote the unipotent radical of $B$. We denote by
${\mathfrak{t}}$, ${\mathfrak{b}}$ and ${\mathfrak{u}}$ the respective Lie
algebras of these subgroups. Let $\Phi$ be the root system of $G$ with respect
to $T$, let $\Phi^{+}$ be the set of positive roots determined by $B$ and let
$\Pi$ be the corresponding set of simple roots. We write
$h_{\alpha}\in{\mathfrak{t}}$ for the coroot corresponding to $\alpha\in\Phi$.

The Weyl group of $G$ with respect to $T$ is denoted $W$. We choose
$\rho\in{\mathfrak{t}}^{*}$ such that $\rho(h_{\alpha})=1$ for all
$\alpha\in\Pi$; for instance we could take $\rho$ to be the half sum of the
roots in $\Phi^{+}$. The dot action of $W$ on ${\mathfrak{t}}^{*}$ is defined
by $w\bullet\lambda:=w(\lambda+\rho)-\rho$, and we note that this does not
depend on the choice of $\rho$.

For $\alpha\in\Phi$, we write ${\mathfrak{g}}_{\alpha}\subseteq{\mathfrak{g}}$
for the corresponding root subspace. Given a closed subgroup $H$ of $G$
containing $T$, we define $\Phi({\mathfrak{h}})$ to be the subset of $\Phi$
such that
${\mathfrak{h}}={\mathfrak{t}}\oplus\bigoplus_{\alpha\in\Phi({\mathfrak{h}})}{\mathfrak{g}}_{\alpha}$.
Given a subset $\Gamma$ of $\Pi$, we define $\Phi_{\Gamma}$ to be the root
subsystem of $\Phi$ generated by $\Gamma$, more precisely
$\Phi_{\Gamma}=\\{\sum_{\alpha\in\Gamma}a_{\alpha}\alpha\in\Phi\mid
a_{\alpha}\in{\mathbb{Z}}\\}$.

###  2.2. Parabolic subgroups and Levi subgroups

We recall some results about parabolic subgroups and their Levi factors that
can be found in [CM93, §3.8], for example.

First we recap that a subgroup $P$ of $G$ is a parabolic subgroup if it
contains a conjugate of $B$. We say that $P$ is a standard parabolic subgroup
if $B\subseteq P$, so every parabolic subgroup is conjugate to a standard
parabolic subgroup. For a standard parabolic subgroup $P$ with unipotent
radical $U_{P}$, there is a subset $\Gamma$ of $\Pi$ such that
$\Phi({\mathfrak{p}})=\Phi_{\Gamma}\sqcup\Phi({\mathfrak{u}}_{P})$, and we use
the notation $P=P_{\Gamma}$. For $\Gamma,\Delta\subseteq\Pi$, we have that
$P_{\Gamma}$ is conjugate to $P_{\Delta}$ if and only if $\Gamma=\Delta$. Thus
the conjugacy classes of parabolic subgroups of $G$ are in bijection with
subsets of $\Pi$.

A parabolic subgroup $P$ of $G$ has a Levi decomposition $P=G_{0}U_{P}$, where
$U_{P}$ is the unipotent radical of $P$ and $G_{0}$ is a Levi factor. By a
Levi subgroup of $G$ we mean a Levi factor $G_{0}$ of some parabolic subgroup,
and we refer to ${\mathfrak{g}}_{0}$ as a Levi subalgebra.

We recall that Levi subalgebras can equivalently be defined as centralizers of
semisimple elements. More precisely, for any semisimple $s\in{\mathfrak{g}}$,
we have that ${\mathfrak{g}}^{s}$ is a Levi subalgebra of ${\mathfrak{g}}$ and
all Levi subalgebras are of the form ${\mathfrak{g}}^{s}$ for some semisimple
$s\in{\mathfrak{g}}$. In particular, this implies that if ${\mathfrak{g}}_{1}$
and ${\mathfrak{g}}_{2}$ are Levi subalgebras of ${\mathfrak{g}}$ with
${\mathfrak{g}}_{1}\subseteq{\mathfrak{g}}_{2}$, then ${\mathfrak{g}}_{1}$ is
a Levi subalgebra of ${\mathfrak{g}}_{2}$.

We recall that for a standard parabolic subgroup $P$ there is a unique Levi
factor $G_{0}$ containing $T$, which we refer to as a standard Levi subgroup,
and then we refer to ${\mathfrak{g}}_{0}$ as a standard Levi subalgebra. We
note that any Levi subgroup is conjugate to a standard Levi subgroup. For a
standard Levi subgroup $G_{0}$, we have that
$\Phi({\mathfrak{g}}_{0})=\Phi_{\Gamma}$ for some $\Gamma\subseteq\Pi$, and we
use the notation $G_{\Gamma}$ for $G_{0}$. For $\Gamma,\Delta\subseteq\Pi$ we
have $G_{\Gamma}$ is conjugate to $G_{\Delta}$ if and only if there exists
$w\in W$ such that $w\cdot\Gamma=\Delta$.

We make the following observation for later. Suppose that $G_{1}$ and $G_{2}$
are standard Levi subgroups of $G$ with Lie algebras ${\mathfrak{g}}_{1}$ and
${\mathfrak{g}}_{2}$ respectively, and that there exists a $G$-conjugate
$\tilde{\mathfrak{g}}_{1}$ of ${\mathfrak{g}}_{1}$ such that
$\tilde{\mathfrak{g}}_{1}\subseteq{\mathfrak{g}}_{2}$. Then
$\tilde{\mathfrak{g}}_{1}$ is a Levi subalgebra of ${\mathfrak{g}}_{2}$, and
thus is $G_{2}$-conjugate (and hence $G$-conjugate) to a standard Levi
subalgebra $\widehat{{\mathfrak{g}}}_{1}$ of ${\mathfrak{g}}_{2}$. It is
straightforward to see that standard Levi subalgebras of ${\mathfrak{g}}_{2}$
are standard Levi subalgebras of ${\mathfrak{g}}$. We therefore conclude that
any inclusion of Levi subalgebras of ${\mathfrak{g}}$ is $G$-conjugate to an
inclusion of standard Levi subalgebras of ${\mathfrak{g}}$.

For a Levi subgroup $G_{0}$, we define
${\mathfrak{z}}({\mathfrak{g}}_{0})^{\mathrm{reg}}:=\\{x\in{\mathfrak{z}}({\mathfrak{g}}_{0})\mid({\mathfrak{g}}_{0})^{x}={\mathfrak{g}}_{0}\\}$.
For $G_{0}=G_{\Gamma}$ we have
${\mathfrak{z}}({\mathfrak{g}}_{0})=\\{x\in{\mathfrak{t}}\mid\gamma(x)=0\text{
for all }\gamma\in\Gamma\\}$ and
${\mathfrak{z}}({\mathfrak{g}}_{0})^{\mathrm{reg}}=\\{x\in{\mathfrak{t}}\mid\gamma(x)=0\text{
for all }\gamma\in\Gamma\text{ and }\alpha(x)\neq 0\text{ for all
}\alpha\in\Pi\setminus\Gamma\\}$.

We explain the conjugacy classes of Levi subgroups for $G$ of classical type
explicitly. The $W$-orbits of subsets $\Gamma$ of $\Pi$ are determined by the
Dynkin type of $\Phi_{\Gamma}$ with root lengths taken into account and with
care needed to not identify ${\mathsf{D}}_{2}$ with
${\mathsf{A}}_{1}+{\mathsf{A}}_{1}$ or ${\mathsf{D}}_{3}$ with
${\mathsf{A}}_{3}$; except in the case where $G$ has type ${\mathsf{D}}_{n}$
and $\Phi_{\Gamma}$ has type
${\mathsf{A}}_{i_{1}-1}+\dots+{\mathsf{A}}_{i_{s}-1}$ with all $i_{j}$ even
and $i_{1}+\dots+i_{s}=n$, in which case there are two $W$-orbits.

Below we give these standard Levi subgroups explicitly, up to conjugacy, when
$T$ is the maximal torus of diagonal matrices in $G$ and $B$ is the Borel
subgroup of upper triangular matrices in $G$. To do this we define an inc-
sequence to be a sequence $\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ with
$i_{j}\in{\mathbb{Z}}_{>0}$, $i_{1}\leq\dots\leq i_{s}$. For an inc-sequence
$i$ we define $|\text{\boldmath$i$}|=i_{1}+\dots+i_{s}$.

It is easier to explain the standard Levi subgroups of ${\mathrm{GL}}_{n}$
than those of $\mathrm{SL}_{n}$, and these are of the form

${\mathrm{GL}}_{\text{\boldmath$i$}}:={\mathrm{GL}}_{i_{1}}\times\dots\times{\mathrm{GL}}_{i_{s}},$
(2.1)

for $i$ an inc-sequence with $|\text{\boldmath$i$}|=n$. The standard Levi
subgroups of $\mathrm{SL}_{n}$ are given by

$\mathrm{SL}_{\text{\boldmath$i$}}:={\mathrm{GL}}_{\text{\boldmath$i$}}\cap\mathrm{SL}_{n}$
(2.2)

for $i$ an inc-sequence with $|\text{\boldmath$i$}|=n$. For $j$ an inc-
sequence with $|\text{\boldmath$j$}|=n$, we have that
$\mathrm{SL}_{\text{\boldmath$i$}}$ is conjugate to
$\mathrm{SL}_{\text{\boldmath$j$}}$ if and only if
$\text{\boldmath$i$}=\text{\boldmath$j$}$.

The standard Levi subgroups of $\mathrm{Sp}_{2n}$ are of the form

$\mathrm{Sp}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}:={\mathrm{GL}}_{i_{1}}\times\dots\times{\mathrm{GL}}_{i_{s}}\times\mathrm{Sp}_{2n_{\text{\boldmath$i$}}},$
(2.3)

for $i$ an inc-sequence with $|\text{\boldmath$i$}|\leq n$, and with
$n_{\text{\boldmath$i$}}:=n-|\text{\boldmath$i$}|$. For $j$ an inc-sequence
with $|\text{\boldmath$j$}|\leq n$, we have that
$\mathrm{Sp}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ is conjugate to
$\mathrm{Sp}_{\text{\boldmath$j$},2n_{\text{\boldmath$j$}}}$ if and only if
$\text{\boldmath$i$}=\text{\boldmath$j$}$.

The standard Levi subgroups of $\mathrm{SO}_{N}$ are of the form

$\mathrm{SO}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}}:={\mathrm{GL}}_{i_{1}}\times\dots\times{\mathrm{GL}}_{i_{s}}\times\mathrm{SO}_{N_{\text{\boldmath$i$}}},$
(2.4)

where $i$ is an inc-sequence with $2|\text{\boldmath$i$}|\leq N$, and with
$N_{\text{\boldmath$i$}}:=N-2|\text{\boldmath$i$}|\neq 2$. In the case that
$N=2|\text{\boldmath$i$}|$ and all $i_{j}$ are even, there are two conjugacy
classes of Levi subgroups of the form $\mathrm{SO}_{\text{\boldmath$i$},0}$.
In all other cases, for an inc-sequence $j$ with $2|\text{\boldmath$j$}|\leq
N$ and $N-2|\text{\boldmath$j$}|\neq 2$, we have that
$\mathrm{SO}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}}$ is conjugate to
$\mathrm{SO}_{\text{\boldmath$j$},N_{\text{\boldmath$j$}}}$ if and only if
$\text{\boldmath$i$}=\text{\boldmath$j$}$. We note that the case
$N_{\text{\boldmath$i$}}=2$ is excluded as in this case we have
$\mathrm{SO}_{2}\cong{\mathrm{GL}}_{1}$, and then
$\mathrm{SO}_{\text{\boldmath$i$},2}$ is conjugate to
$\mathrm{SO}_{\overline{\text{\boldmath$i$}},0}$, where
$\overline{\text{\boldmath$i$}}$ is obtained from $i$ by appending a 1 to the
start.

We write ${\mathfrak{gl}}_{\text{\boldmath$i$}}$,
${\mathfrak{sl}}_{\text{\boldmath$i$}}$,
${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ and
${\mathfrak{so}}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}}$ for the Lie
algebras of ${\mathrm{GL}}_{\text{\boldmath$i$}}$,
$\mathrm{SL}_{\text{\boldmath$i$}}$,
$\mathrm{Sp}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ and
$\mathrm{SO}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}}$ respectively.

###  2.3. Nilpotent orbits

We refer to $G$-orbits of nilpotent elements in ${\mathfrak{g}}$ as nilpotent
$G$-orbits. We recap the parameterization of nilpotent orbits for $G$ of
classical type, as can be found for instance in [CM93, Chapter 5]. To do this
we define a partition $\lambda$ to be a sequence
$\lambda=(\lambda_{1},\lambda_{2},\dots)$ with finitely many non-zero
$\lambda_{i}\in{\mathbb{Z}}_{\geq 0}$, and $\lambda_{i}\geq\lambda_{i+1}$ for
each $i$. Often we write $\lambda=(\lambda_{1},\dots,\lambda_{r})$ where $r$
is maximal such that $\lambda_{r}\neq 0$. We say that $\lambda$ is a partition
of $|\lambda|:=\lambda_{1}+\dots+\lambda_{r}$.

For $G=\mathrm{SL}_{n}$ the nilpotent orbits are classified by their Jordan
type, which is the partition $\lambda$ of $n$ giving the sizes of the Jordan
blocks of an element in the orbit.

For $G=\mathrm{Sp}_{2n}$ the nilpotent orbits are classified by their Jordan
type, as each nilpotent ${\mathrm{GL}}_{2n}$-orbit that intersects
${\mathfrak{sp}}_{2n}$ does so in a single $\mathrm{Sp}_{2n}$-orbit. The
nilpotent ${\mathrm{GL}}_{2n}$-orbits that have a non-empty intersection with
${\mathfrak{sp}}_{2n}$ are those with Jordan type $\lambda$, where $\lambda$
is a partition of $2n$ such that every odd part of $\lambda$ occurs with even
multiplicity; we refer to such partitions as symplectic partitions.

For $G=\mathrm{SO}_{N}$, there is a minor complication as the intersection of
a ${\mathrm{GL}}_{N}$-orbit with ${\mathfrak{so}}_{N}$ may split into two
$\mathrm{SO}_{N}$-orbits; such an intersection is a single
$\mathrm{O}_{N}$-orbit, but splits into two $\mathrm{SO}_{N}$-orbits if the
centralizer (in ${\mathrm{GL}}_{N}$) of an element of the orbit is contained
in $\mathrm{SO}_{N}$. The Jordan types $\lambda$ of the
${\mathrm{GL}}_{N}$-orbits having non-empty intersection with
${\mathfrak{so}}_{N}$ are those such that every positive even part of
$\lambda$ occurs with even multiplicity; we refer to such partitions as
orthogonal partitions. This intersection is a single $\mathrm{SO}_{N}$-orbit
except when $\lambda$ is a very even partition, i.e. when all parts of
$\lambda$ are even. For a very even partition $\lambda$, it is customary to
label the two $\mathrm{SO}_{N}$-orbits by Roman numerals I and II.

For $G$ of classical type and $\lambda$ a partition corresponding to a
nilpotent $G$-orbit in ${\mathfrak{g}}$, we use the notation
${\mathbb{O}}_{\lambda}$ for this nilpotent orbit. When this notation is used
it is always clear from the context which $G$ is being considered, and in the
case when $G=\mathrm{SO}_{N}$ and $\lambda$ is a very even partition it refers
to a fixed choice of one of the two orbits corresponding to this partition.

###  2.4. Lusztig–Spaltenstein induction and rigid nilpotent orbits

We recap Lusztig–Spaltenstein induction of nilpotent orbits as introduced in
[LS79]. In fact in [LS79] the theory was developed for unipotent classes, but
is also valid for nilpotent classes, see for instance [CM93, Chapter 7]. We
mention that this procedure has been extended from nilpotent orbits to
equivariant covers of orbits in [LMM21, §2.3].

Let $G_{0}$ be a Levi subgroup of $G$ and choose a parabolic subgroup $P$ with
$G_{0}$ as a Levi factor. Then we have the decomposition
${\mathfrak{p}}={\mathfrak{g}}_{0}\oplus{\mathfrak{u}}_{P}$ where
${\mathfrak{u}}_{P}$ is the Lie algebra of the unipotent radical of $P$. Given
a nilpotent $G_{0}$-orbit ${\mathbb{O}}_{0}$, there is a unique nilpotent
$G$-orbit which has dense intersection with
${\mathbb{O}}_{0}+{\mathfrak{u}}_{P}$. We denote this orbit by
$\operatorname{Ind}_{{\mathfrak{g}}_{0}}^{\mathfrak{g}}({\mathbb{O}}_{0})$,
and say that ${\mathbb{O}}$ is Lusztig–Spaltenstein induced from
${\mathbb{O}}_{0}$. As the notation indicates, Remarkably,
Lusztig–Spaltenstein induction depends only on the Levi subalgebra
${\mathfrak{g}}_{0}$, and not on the parabolic subalgebra ${\mathfrak{p}}$
containing ${\mathfrak{g}}_{0}$.

We note that $G$ acts on the set of pairs
$({\mathfrak{g}}_{0},{\mathbb{O}}_{0})$, where ${\mathfrak{g}}_{0}$ is the Lie
algebra of a Levi subgroup $G_{0}$ of $G$ and ${\mathbb{O}}_{0}$ is a
nilpotent $G_{0}$-orbit. We denote the $G$-orbit of
$({\mathfrak{g}}_{0},{\mathbb{O}}_{0})$ by
$[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$ and note that the induced orbit
$\operatorname{Ind}_{{\mathfrak{g}}_{0}}^{\mathfrak{g}}({\mathbb{O}}_{0})$
does not depend on the choice of representative for this orbit. We refer to
$[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$ as an induction datum for the induced
orbit
${\mathbb{O}}=\operatorname{Ind}_{{\mathfrak{g}}_{0}}^{\mathfrak{g}}({\mathbb{O}}_{0})$.
Since there are finitely many conjugacy classes of Levi subgroups of $G$ and
finitely many nilpotent orbits for each Levi subgroup, there are finitely many
induction data.

We say that a nilpotent $G$-orbit ${\mathbb{O}}$ is rigid in ${\mathfrak{g}}$
if it cannot be obtained by induction from a nilpotent orbit in a proper Levi
subalgebra. We say that the induction datum
$[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$ is rigid if ${\mathbb{O}}_{0}$ is
rigid in ${\mathfrak{g}}_{0}$. For a fixed nilpotent $G$-orbit ${\mathbb{O}}$,
we define the set of rigid induction data for ${\mathbb{O}}$ to be

${\mathcal{R}}({\mathfrak{g}},{\mathbb{O}}):=\\{[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]\mid[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]\text{
is a rigid induction datum for }{\mathbb{O}}\\}.$

To state what the rigid induction data are, we need to know the rigid
nilpotent orbits in Levi subalgebras of ${\mathfrak{g}}$. For this we note
that a nilpotent orbit ${\mathbb{O}}_{0}$ in a Levi subalgebra
${\mathfrak{g}}_{0}$ of ${\mathfrak{g}}$ is rigid if and only if the
intersection of ${\mathbb{O}}_{0}$ with any simple factor ${\mathfrak{h}}$ of
${\mathfrak{g}}_{0}$ is rigid in ${\mathfrak{h}}$. Below we explain what the
rigid induction data are for $G$ simple of classical type. For this we use the
notation for Levi subgroups given in (2.2), (2.3) and (2.4).

For $G=\mathrm{SL}_{n}$, the only rigid nilpotent orbit is the zero orbit.
Thus the rigid induction data are
$[{\mathfrak{sl}}_{\text{\boldmath$i$}},\\{0\\}]$ for
$\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ an inc-sequence with
$|\text{\boldmath$i$}|=n$. We add here that it is well-known that
$\operatorname{Ind}_{{\mathfrak{sl}}_{\text{\boldmath$i$}}}^{{\mathfrak{sl}}_{n}}(\\{0\\})={\mathbb{O}}_{\text{\boldmath$i$}^{\mathrm{t}}}$,
where $\text{\boldmath$i$}^{\mathrm{t}}$ is the dual of $i$, defined by
$(\text{\boldmath$i$}^{t})_{j}=|\\{k\in\\{1,\dots,s\\}\mid i_{k}\geq j\\}|$.
In particular, this implies that there is a unique rigid induction datum for
each nilpotent orbit ${\mathbb{O}}$ in ${\mathfrak{g}}={\mathfrak{sl}}_{n}$.

For $G=\mathrm{Sp}_{2n}$, a symplectic partition $\lambda$ of $2n$ is the
Jordan type of a rigid nilpotent orbit if and only if
$\lambda_{i}-\lambda_{i+1}\leq 1$ for all $i$ and even $\lambda_{i}$ do not
have multiplicity $2$ in $\lambda$; we refer to such $\lambda$ as a rigid
symplectic partition. The rigid induction data are
$[{\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{0}]$
for $\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ an inc-sequence with
$|\text{\boldmath$i$}|\leq n$ and ${\mathbb{O}}_{0}$ a rigid nilpotent orbit
in ${\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}$. To clarify notation we
explain that $\\{0\\}\times{\mathbb{O}}_{0}$ denotes the nilpotent
$\mathrm{Sp}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$-orbit in
${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ that projects
to $\\{0\\}$ in each of the ${\mathfrak{gl}}_{i_{j}}$ factors for
$j=1,\dots,s$ and projects to ${\mathbb{O}}_{0}$ in the
${\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}$ factor.

For $G=\mathrm{SO}_{N}$, an orthogonal partition $\lambda$ of $N$ is the
Jordan type of a rigid nilpotent orbit if and only if
$\lambda_{i}-\lambda_{i+1}\leq 1$ for all $i$ and odd $\lambda_{i}$ do not
have multiplicity $2$ in $\lambda$; we refer to such $\lambda$ as a rigid
orthogonal partition. The rigid induction data are
$[{\mathfrak{so}}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{0}]$
for $\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ an inc-sequence with
$2|\text{\boldmath$i$}|\leq N$ and $2|\text{\boldmath$i$}|\neq N-2$, and
${\mathbb{O}}_{0}$ a rigid nilpotent orbit in
${\mathfrak{so}}_{N_{\text{\boldmath$i$}}}$.

###  2.5. Sheets and decomposition classes

We recap some aspects of the theory of sheets and decomposition classes. The
material we cover here is contained in [BK79] and [Bo81], we also refer to
[PT14, §3.1] for an overview with references.

For each induction datum $[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$ the
corresponding decomposition class is

${\mathcal{D}}({\mathfrak{g}}_{0},{\mathbb{O}}_{0}):=G\cdot({\mathfrak{z}}({\mathfrak{g}}_{0})^{{\mathrm{reg}}}+{\mathbb{O}}_{0}).$

Decomposition classes give a stratification

${\mathfrak{g}}=\bigsqcup_{[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]}{\mathcal{D}}({\mathfrak{g}}_{0},{\mathbb{O}}_{0})$

where the (disjoint) union is taken over the finitely many induction data for
all nilpotent orbits in ${\mathfrak{g}}$.

The rank strata of ${\mathfrak{g}}$ are defined to be

${\mathfrak{g}}^{(j)}:=\\{x\in{\mathfrak{g}}\mid\dim G\cdot x=j\\}$

for $j\in{\mathbb{Z}}_{\geq 0}$. The sheets of ${\mathfrak{g}}$ are the
irreducible components of the rank strata.

Each decomposition class ${\mathcal{D}}$ is irreducible and there exists
$j_{\mathcal{D}}\in{\mathbb{Z}}_{\geq 0}$ such that
${\mathcal{D}}\subseteq{\mathfrak{g}}^{(j_{\mathcal{D}})}$. Therefore, each
sheet contains a dense decomposition class. Given a decomposition class
${\mathcal{D}}$ we let
$\overline{{\mathcal{D}}}^{{\mathrm{reg}}}:=\overline{{\mathcal{D}}}\cap{\mathfrak{g}}^{(j_{\mathcal{D}})}$
be the union of the orbits of maximal dimension in $\overline{\mathcal{D}}$.
Given decomposition classes
${\mathcal{D}}_{1}:={\mathcal{D}}({\mathfrak{g}}_{1},{\mathbb{O}}_{1}),$ and
${\mathcal{D}}_{2}:={\mathcal{D}}({\mathfrak{g}}_{2},{\mathbb{O}}_{2})$ we
have ${\mathcal{D}}_{1}\subseteq\overline{{\mathcal{D}}}_{2}^{{\mathrm{reg}}}$
if and only if there are representatives
$({\mathfrak{g}}_{1},{\mathbb{O}}_{1})$ and
$({\mathfrak{g}}_{2},{\mathbb{O}}_{2})$ such that
${\mathfrak{g}}_{2}\subseteq{\mathfrak{g}}_{1}$ and
${\mathbb{O}}_{2}=\operatorname{Ind}_{{\mathfrak{g}}_{1}}^{{\mathfrak{g}}_{2}}({\mathbb{O}}_{1})$.
From this we can deduce that the sheets of ${\mathfrak{g}}$ are precisely the
$\overline{{\mathcal{D}}}^{{\mathrm{reg}}}({\mathfrak{g}}_{0},{\mathbb{O}}_{0})$:=$\overline{{\mathcal{D}}({\mathfrak{g}}_{0},{\mathbb{O}}_{0})}^{{\mathrm{reg}}}$
as we vary over all rigid induction data
$[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$, and moreover that
$\operatorname{Ind}_{{\mathfrak{g}}_{0}}^{\mathfrak{g}}{\mathbb{O}}_{0}$ is
the unique nilpotent orbit lying in
$\overline{{\mathcal{D}}}^{{\mathrm{reg}}}({\mathfrak{g}}_{0},{\mathbb{O}}_{0})$.
Finally, we note for future reference that
$\dim\overline{{\mathcal{D}}}^{{\mathrm{reg}}}({\mathfrak{g}}_{0},{\mathbb{O}}_{0})=\dim{\mathfrak{z}}({\mathfrak{g}}_{0})+\dim\operatorname{Ind}_{{\mathfrak{g}}_{0}}^{\mathfrak{g}}{\mathbb{O}}_{0}$
(see [BK79, Satz 4.5]).

###  2.6. Primitive ideals of universal enveloping algebras and their central
characters

We recall the required background on primitive ideals and central characters
of the universal enveloping algebra $U({\mathfrak{g}})$ of ${\mathfrak{g}}$.
We refer to [BJ01] as a general reference on primitive ideals and to [Hu08,
Chapter 1] as a general reference for central characters.

An ideal $I$ of $U({\mathfrak{g}})$ is called primitive if
$I=\operatorname{Ann}_{U({\mathfrak{g}})}(E)$ for some simple
$U({\mathfrak{g}})$-module $E$, where
$\operatorname{Ann}_{U({\mathfrak{g}})}(E)$ denotes the annihilator of $E$ in
$U({\mathfrak{g}})$. We denote the spectrum of primitive ideals by
$\operatorname{Prim}U({\mathfrak{g}})$. An ideal $I$ of $U({\mathfrak{g}})$ is
called completely prime if $U({\mathfrak{g}})/I$ is a domain. The spectrum of
completely prime primitive ideals is denoted
$\operatorname{Prim}^{1}U({\mathfrak{g}})$.

The Poincaré–Birkhoff–Witt (PBW) filtration of $U({\mathfrak{g}})$ induces a
filtration on an ideal $I$ of $U({\mathfrak{g}})$ and we write
$\operatorname{gr}I\subseteq\operatorname{gr}U({\mathfrak{g}})\cong
S({\mathfrak{g}})$ for the associated graded ideal. We identify
$S({\mathfrak{g}})$ with ${\mathbb{C}}[{\mathfrak{g}}]$ via our choice of
nondegenerate invariant symmetric bilinear form $(\cdot\,,\cdot)$ on
${\mathfrak{g}}$. We recall that the associated variety ${\mathcal{VA}}(I)$ of
an ideal $I$ of $U({\mathfrak{g}})$ is the vanishing locus in ${\mathfrak{g}}$
of $\operatorname{gr}I\subseteq{\mathbb{C}}[{\mathfrak{g}}]$. As mentioned in
the introduction it is known that for
$I\in\operatorname{Prim}U({\mathfrak{g}})$ we have that ${\mathcal{VA}}(I)$ is
the closure of a nilpotent orbit in ${\mathfrak{g}}$ by Joseph’s
irreducibility theorem [Jo85]. For a nilpotent $G$-orbit ${\mathbb{O}}$ we
define
$\operatorname{Prim}_{\mathbb{O}}U({\mathfrak{g}})=\\{I\in\operatorname{Prim}(U({\mathfrak{g}}))\mid{\mathcal{VA}}(I)=\overline{\mathbb{O}}\\}$
and
$\operatorname{Prim}^{1}_{\mathbb{O}}U({\mathfrak{g}})=\operatorname{Prim}^{1}U({\mathfrak{g}})\cap\operatorname{Prim}_{\mathbb{O}}U({\mathfrak{g}})$.

Let $P$ be a parabolic subgroup of $G$, let $G_{0}$ be a Levi factor of $P$,
and let $U_{P}$ be the unipotent radical of $P$. We have the decomposition
${\mathfrak{p}}={\mathfrak{g}}_{0}\oplus{\mathfrak{u}}_{P}$. Let
$I_{0}\in\operatorname{Prim}U({\mathfrak{g}}_{0})$ and choose a simple
$U({\mathfrak{g}}_{0})$-module $E$ such that
$I_{0}=\operatorname{Ann}_{U({\mathfrak{g}}_{0})}(E)\subseteq
U({\mathfrak{g}}_{0})$.

We define the induced ideal

$\operatorname{Ind}_{{\mathfrak{p}}}^{\mathfrak{g}}(I_{0}):=\operatorname{Ann}_{U({\mathfrak{g}})}(U({\mathfrak{g}})\otimes_{U({\mathfrak{p}})}E)$

where $E$ is viewed as a $U({\mathfrak{p}})$-module via the projection
$U({\mathfrak{p}})\twoheadrightarrow U({\mathfrak{g}}_{0})$. It is well-known
and straightforward to show that the induced ideal
$\operatorname{Ind}_{{\mathfrak{p}}}^{\mathfrak{g}}(I_{0})$ can also be
described as the largest two-sided ideal contained in the left ideal
$U({\mathfrak{g}})(I_{0}+{\mathfrak{u}}_{P})$. From this alternative
description it follows that
$\operatorname{Ind}_{{\mathfrak{p}}}^{\mathfrak{g}}(I_{0})$ does not depend on
the choice of simple $U({\mathfrak{g}}_{0})$-module $E$ with
$I_{0}=\operatorname{Ann}_{U({\mathfrak{g}}_{0})}(E)$. (We remark that there
appears to be no general result in the literature about the dependence of
$\operatorname{Ind}_{{\mathfrak{p}}}^{\mathfrak{g}}(I_{0})$ on the choice of
parabolic subalgebra ${\mathfrak{p}}$ with Levi factor ${\mathfrak{g}}_{0}$.)

The induced ideal $\operatorname{Ind}_{{\mathfrak{p}}}^{\mathfrak{g}}(I_{0})$
is in general not primitive. However, if
$I_{0}\in\operatorname{Prim}^{1}U({\mathfrak{g}}_{0})$, then
$\operatorname{Ind}_{{\mathfrak{p}}}^{\mathfrak{g}}(I_{0})\in\operatorname{Prim}^{1}U({\mathfrak{g}})$
as is proved in [Co74, Theorem 3.1].

Although $\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}(I_{0})$ is not
primitive in general its associated variety is the closure of a nilpotent
orbit and is given by

${\mathcal{VA}}(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}(I_{0}))=\overline{\operatorname{Ind}_{{\mathfrak{g}}_{0}}^{\mathfrak{g}}({\mathbb{O}}_{0})}$
(2.5)

where $\overline{{\mathbb{O}}}_{0}={\mathcal{VA}}(I_{0})$. A proof of (2.5) is
given in within the proof of [BJ01, Lemma 6.1] and another proof using results
of Losev is explained within [BG18, §5].

Let ${\mathbb{O}}_{0}$ be a nilpotent $G_{0}$-orbit and let
${\mathbb{O}}=\operatorname{Ind}_{{\mathfrak{g}}_{0}}^{\mathfrak{g}}({\mathbb{O}}_{0})$.
From the discussion above it follows that parabolic induction restricts to a
map

$\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}:\operatorname{Prim}^{1}_{{\mathbb{O}}_{0}}U({\mathfrak{g}}_{0})\to\operatorname{Prim}^{1}_{{\mathbb{O}}}U({\mathfrak{g}}).$

We write $Z({\mathfrak{g}})$ for the centre of $U({\mathfrak{g}})$, and refer
to a character $\chi:Z({\mathfrak{g}})\to{\mathbb{C}}$ as a central character
of $U({\mathfrak{g}})$. By Quillen’s lemma we know that $Z({\mathfrak{g}})$
acts on a simple $U({\mathfrak{g}})$-module $E$ by a central character
$\chi_{E}:Z({\mathfrak{g}})\to{\mathbb{C}}$. Thus,
$\operatorname{Ann}_{U({\mathfrak{g}})}(E)\cap Z({\mathfrak{g}})$ is an ideal
of $Z({\mathfrak{g}})$ of codimension $1$. Setting
$I=\operatorname{Ann}_{U({\mathfrak{g}})}(E)$, we write
$\chi_{I}:Z({\mathfrak{g}})\to{\mathbb{C}}$ for the corresponding central
character. It is then clear that $\chi_{I}=\chi_{E^{\prime}}$ for any simple
$U({\mathfrak{g}})$-module $E^{\prime}$ with
$I=\operatorname{Ann}_{U({\mathfrak{g}})}(E^{\prime})$.

We write $U({\mathfrak{g}})_{0}$ for the subalgebra of $U({\mathfrak{g}})$
given by the zero weight space for the adjoint action of ${\mathfrak{t}}$ on
$U({\mathfrak{g}})$. Then $Z({\mathfrak{g}})\subseteq U({\mathfrak{g}})_{0}$.
We have that
$U({\mathfrak{g}})_{0}=S({\mathfrak{t}})\oplus(U({\mathfrak{g}}){\mathfrak{u}}\cap
U({\mathfrak{g}})_{0})$ with $U({\mathfrak{g}}){\mathfrak{u}}\cap
U({\mathfrak{g}})_{0}$ being a 2-sided ideal of $U({\mathfrak{g}})_{0}$, where
we recall that ${\mathfrak{u}}$ is the Lie algebra of the unipotent radical of
$B$. Thus the projection $U({\mathfrak{g}})_{0}\to S({\mathfrak{t}})$ defined
from the decomposition
$U({\mathfrak{g}})_{0}=S({\mathfrak{t}})\oplus(U({\mathfrak{g}}){\mathfrak{u}}\cap
U({\mathfrak{g}})_{0})$ is a homomorphism. The homomorphism
$Z({\mathfrak{g}})\to S({\mathfrak{t}})$ given by restriction is known to be
an isomorphism onto its image and this image is
$S({\mathfrak{t}})^{W,\bullet}:=\\{u\in S({\mathfrak{t}})\mid w\bullet
u=u\text{ for all }w\in W\\}$, where to define the action of $W$ on
$S({\mathfrak{t}})$ we identity $S({\mathfrak{t}})$ with the algebra of
polynomial functions on ${\mathfrak{t}}^{*}$ and recall the dot action on
${\mathfrak{t}}^{*}$ as defined in §2.1. This isomorphism
$\psi:Z({\mathfrak{g}})\to S({\mathfrak{t}})^{W,\bullet}$ is called the
Harish-Chandra isomorphism. Thus as
$S({\mathfrak{t}})^{W,\bullet}\cong{\mathbb{C}}[{\mathfrak{t}}^{*}/(W,\bullet)]$,
where ${\mathfrak{t}}^{*}/(W,\bullet)$ denotes the quotient of
${\mathfrak{t}}^{*}$ under the dot-action of $W$, we may identify central
characters of primitive ideals of $U({\mathfrak{g}})$ with
$(W,\bullet)$-orbits in ${\mathfrak{t}}^{*}$. We write

$\operatorname{ch}:\operatorname{Prim}U({\mathfrak{g}})\to{\mathfrak{t}}^{*}/(W,\bullet)$

for this central character map.

In fact the map $\operatorname{ch}$ can be extended to all ideals $I$ of
$U({\mathfrak{g}})$ for which $I\cap Z({\mathfrak{g}})$ is of codimension $1$
in $Z({\mathfrak{g}})$; such ideals are said to admit a central character.
Given a highest weight $U({\mathfrak{g}})$-module $M$, we can see that the
action of $Z({\mathfrak{g}})$ on a highest weight vector of $M$ must be given
by a central character, and thus $Z({\mathfrak{g}})$ acts on all of $M$ by
this central character. A useful observation is that the map
$\operatorname{ch}$ is set up so that if
$I=\operatorname{Ann}_{U({\mathfrak{g}})}(M)$ where $M$ is a highest weight
$U({\mathfrak{g}})$-module with highest weight $\lambda\in{\mathfrak{t}}^{*}$
with respect to ${\mathfrak{b}}$, then $\operatorname{ch}(I)=W\bullet\lambda$.

We now suppose that $P$ is a standard parabolic subgroup of $G$ and $G_{0}$ is
the standard Levi factor of $P$. We note that $B_{0}:=B\cap G_{0}$ is a Borel
subgroup of $G_{0}$. The Weyl group of $G_{0}$ with respect to $T$ is denoted
$W_{0}$.

We have the Harish-Chandra isomorphism $\psi_{0}:Z({\mathfrak{g}}_{0})\to
S({\mathfrak{t}})^{W_{0},\bullet}$; here we note that we can use the same
$\rho$ for $G_{0}$ as was used for $G$ to define the dot action of $W_{0}$.
Therefore, we have a central character map

$\operatorname{ch}_{0}:\operatorname{Prim}U({\mathfrak{g}}_{0})\to{\mathfrak{t}}^{*}/(W_{0},\bullet).$

We write

$\pi:{\mathfrak{t}}^{*}/(W_{0},\bullet)\twoheadrightarrow{\mathfrak{t}}^{*}/(W,\bullet)$

for the projection map.

Let $I_{0}\in\operatorname{Prim}U({\mathfrak{g}}_{0})$. By Duflo’s theorem we
can assume $I_{0}=\operatorname{Ann}_{U({\mathfrak{g}}_{0})}(L_{0}(\lambda))$
for some $\lambda\in{\mathfrak{t}}^{*}$, where $L_{0}(\lambda)$ denotes the
simple highest weight $U({\mathfrak{g}}_{0})$-module with highest weight
$\lambda$ with respect to the Borel subalgebra ${\mathfrak{b}}_{0}$. Then we
have $\operatorname{ch}_{0}(I_{0})=W_{0}\bullet\lambda$. Let
$I=\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}(I_{0})$, so we have
$I=\operatorname{Ann}_{U({\mathfrak{g}})}(U({\mathfrak{g}})\otimes_{U({\mathfrak{p}})}L_{0}(\lambda))$.
Since $U({\mathfrak{g}})\otimes_{U({\mathfrak{p}})}L_{0}(\lambda)$ is a
highest weight $U({\mathfrak{g}})$-module of highest weight $\lambda$, we see
that $\operatorname{ch}(I)=W\bullet\lambda$. Summarising this discussion we
have that

$\pi(\operatorname{ch}_{0}(I_{0}))=\operatorname{ch}(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}}(I_{0})).$
(2.6)

##  3\. On Lusztig–Spaltenstein induction for simple algebraic groups of
classical type

For this section we restrict to the case where $G$ is simple of classical
type, and in fact restrict just to $G=\mathrm{Sp}_{2n}$ or $G=\mathrm{SO}_{N}$
until Theorem 3.3. Our main goal is to prove Theorem 3.3, which gives an
important property about rigid induction data and is a key ingredient in our
proof of Theorem 4.1.

Central to our approach is the Kempken–Spaltenstein algorithm (or KS-algorithm
for short) which was introduced in [PT14, §3.1] and determines the rigid
induction data attached to a nilpotent orbit in ${\mathfrak{g}}$. We give an
overview of this algorithm for the case $G=\mathrm{Sp}_{2n}$ and then explain
the minor adaptations required for $G=\mathrm{SO}_{N}$. We refer to [PT14,
§3–4] for all the claims about the KS-algorithm made below. We begin with a
symplectic partition $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ of $2n$. Then,
as we define and explain next, the KS algorithm makes a sequence of reductions
to $\lambda$ (called type 1 reductions and type 2 reductions) corresponding to
an admissible sequence $\text{\boldmath$i$}=(i_{1},\dots,i_{s})$, where $1\leq
i_{j}\leq r$ for each $j=1,\dots,s$.

We say that a type 1 reduction can be made to $\lambda$ at position $i$ if
$\lambda_{i}\geq\lambda_{i+1}+2$. Applying this reduction gives a partition
$\lambda^{i}$ of $2n-2i$ defined by

$\lambda^{i}_{j}=\begin{cases}\lambda_{j}-2&\text{if $j\leq i$,}\\\
\lambda_{j}&\text{if $j>i$}.\end{cases}$

We say that a type 2 reduction can be made to $\lambda$ at position $i$ if
$\lambda_{i}$ is even and
$\lambda_{i-1}>\lambda_{i}=\lambda_{i+1}>\lambda_{i+2}$ (here we consider the
inequality $\lambda_{0}>\lambda_{1}$ to automatically hold). Applying this
reduction gives a partition $\lambda^{i}$ of $2n-2i$ defined by

$\lambda^{i}_{j}=\begin{cases}\lambda_{j}-2&\text{if $j<i$,}\\\
\lambda_{j}-1&\text{if $i\leq j\leq i+1$,}\\\ \lambda_{j}&\text{if
$j>i+1$}.\end{cases}$

We note that at a given position $i$, it is not possible that both a type 1
reduction and a type 2 reduction can be made to $\lambda$, so there is no
ambiguity in the notation $\lambda^{i}$. We say that $i$ is an admissible
position for $\lambda$ if a type 1 or type 2 reduction can be made to
$\lambda$ at position $i$.

We define recursively what it means for
$\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ to be an admissible sequence for
$\lambda$. First we introduce the notation that for $k=1,\dots,s$ we write
$\text{\boldmath$i$}_{k}=(i_{1},\dots i_{k})$. We say that
$\text{\boldmath$i$}_{1}$ is an admissible sequence for $\lambda$ if $i_{1}$
is an admissible position for $\lambda$; we then set
$\lambda^{\text{\boldmath$i$}_{1}}=\lambda^{i_{1}}$. We then recursively say
that $\text{\boldmath$i$}_{k}$ is an admissible sequence for $\lambda$, and
define
$\lambda^{\text{\boldmath$i$}_{k}}:=(\lambda^{\text{\boldmath$i$}_{k-1}})^{i_{k}}$,
if $i_{k}$ is an admissible position for
$\lambda^{\text{\boldmath$i$}_{k-1}}$. We say that $i$ is a maximal admissible
sequence if there is no admissible position for
$\lambda^{\text{\boldmath$i$}}$.

To summarise, the KS-algorithm takes as input a symplectic partition $\lambda$
and an admissible sequence $i$, and outputs the symplectic partition
$\lambda^{\text{\boldmath$i$}}$. The sequence of type 1 and type 2 reductions
given by $i$ are used to determine $\lambda^{\text{\boldmath$i$}}$.

To explain the significance of the outcome of this algorithm we require some
notation. We fix an admissible sequence
$\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ for $\lambda$. We write
${\mathbb{O}}_{\lambda}$ for the nilpotent orbit in
${\mathfrak{g}}={\mathfrak{sp}}_{2n}$ with Jordan type $\lambda$ and
${\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}$ for the nilpotent orbit in
${\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}$ with Jordan type
$\lambda^{\text{\boldmath$i$}}$. The upshot of the KS-algorithm is that
$[{\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}]$
is an induction datum for ${\mathbb{O}}_{\lambda}$, see [PT14, Proposition 7].
Here we note that although in (2.3) we just consider the definition of
${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ for $i$ an
inc-sequence, the definition makes sense for any admissible sequence $i$.
Furthermore, $\\{0\\}\times{\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}$ is a
rigid nilpotent orbit in
${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ if and only
if $i$ is a maximal admissible sequence. Moreover, every rigid induction datum
for ${\mathbb{O}}_{\lambda}$ is of the form
$[{\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}]$
for some maximal admissible sequence $i$. Thus to obtain all rigid induction
data for ${\mathbb{O}}_{\lambda}$ we can apply the KS-algorithm for all
maximal admissible sequences for $\lambda$.

Now we discuss the modification needed to the algorithm for
$G=\mathrm{SO}_{N}$. This is simply that we replace “even” with “odd” in the
description of a type 2 reduction. With this version of the algorithm all of
the results given in the symplectic case above have analogues for the
orthogonal case. In particular, a maximal admissible sequence
$\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ gives a rigid induction datum
$[{\mathfrak{so}}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}]$
for ${\mathbb{O}}_{\lambda}$, where now ${\mathbb{O}}_{\lambda}$ is a
nilpotent orbit in ${\mathfrak{so}}_{N}$ and
${\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}$ is a nilpotent orbit in
${\mathfrak{so}}_{N_{\text{\boldmath$i$}}}$; and, moreover, all rigid
induction data are obtained in this way.

There is a subtlety in the orthogonal case that if there are two
$\mathrm{SO}_{N}$-orbits of Jordan type $\lambda$, then either there are two
$\mathrm{SO}_{N_{\text{\boldmath$i$}}}$-orbits of Jordan type
$\lambda^{\text{\boldmath$i$}}$, or there are two conjugacy classes of Levi
subgroups $\mathrm{SO}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}}$. In
these cases it is implicit that we have to choose the correct
$\mathrm{SO}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}}$-orbits with Jordan
type $\lambda^{\text{\boldmath$i$}}$, or the correct
$\mathrm{SO}_{N}$-conjugacy class of Levi subalgebras of the form
${\mathfrak{so}}_{\text{\boldmath$i$},N_{\text{\boldmath$i$}}}$.

We recall a property of the KS-algorithm that is important in this paper as it
reduces the number of maximal admissible sequences that have to be considered.
In the statement $\lambda$ is a symplectic partition or an orthogonal
partition depending on which case is being considered.

######  KS-property 1.

Any reordering of the entries in a maximal admissible sequence $i$ for
$\lambda$ gives a maximal admissible sequence for $\lambda$, and the partition
$\lambda^{\text{\boldmath$i$}}$ does not depend on the order the entries of
$i$.

This property follows from [PT14, Corollary 7] and [PT14, Proposition 7]. We
remark here that for the orthogonal case the statement of [PT14, Corollary 7]
excludes the possibility that $N_{\text{\boldmath$i$}}=2$. We note, however,
that in the case $N_{\text{\boldmath$i$}}=2$, we have
$\lambda^{\text{\boldmath$i$}}=(1^{2})$ and we can make a final type 2
reduction at position $1$, and thus $i$ is not a maximal admissible sequence.

As a consequence of KS-property 1 we can restrict to considering maximal
admissible sequences $i$ that are inc-sequences, and we refer to such $i$ as a
maximal admissible inc-sequence.

Before embarking on the proof of Lemma 3.2, which provides the main step for
the proof of Theorem 3.3, we choose to first include an example. This should
be helpful for the reader as the proof of Lemma 3.2 is combinatorial and a
little technical, and the example demonstrates some of the ideas in the proof.

######  Example 3.1.

We consider the nilpotent orbit with Jordan type $\lambda=(6,4,2)$ in
${\mathfrak{sp}}_{12}$. We determine all maximal admissible inc-sequences,
showing that there are just three of them. Let
$\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ be a maximal admissible inc-
sequence. We proceed by considering cases based on the values of $i_{1}$ and
$i_{2}$.

Case $i_{1}=3$. We have
$\lambda^{\text{\boldmath$i$}_{1}}=\lambda^{3}=(4,2,0)$ and $i_{2}\geq 3$.
Since $\lambda^{3}$ is not a rigid symplectic partition and there are no
admissible positions $i\geq 3$ for $\lambda^{3}$, we deduce that $i_{1}=3$ is
not possible.

Case $i_{1}=2$. We have
$\lambda^{\text{\boldmath$i$}_{1}}=\lambda^{2}=(4,2,2)$ and $i_{2}\geq 2$.

Case $(i_{1},i_{2})=(2,3)$. We have
$\lambda^{\text{\boldmath$i$}_{2}}=\lambda^{(2,3)}=(2,0,0)$ and $i_{3}\geq 3$.
Since $\lambda^{(2,3)}$ is not a rigid symplectic partition and there are no
admissible positions $i\geq 3$ for $\lambda^{(2,3)}$, we deduce that $i_{2}=3$
is not possible.

Case $(i_{1},i_{2})=(2,2)$. We have
$\lambda^{\text{\boldmath$i$}_{2}}=\lambda^{(2,2)}=(2,1,1)$ is a rigid
symplectic partition. Thus $\text{\boldmath$i$}=(2,2)$ is a maximal admissible
inc-sequence for $\lambda$.

Case $i_{1}=1$. We have
$\lambda^{\text{\boldmath$i$}_{1}}=\lambda^{1}=(4,4,2)$.

Case $(i_{1},i_{2})=(1,3)$. We have
$\lambda^{\text{\boldmath$i$}_{2}}=\lambda^{(1,3)}=(2,2,0)$ and $i_{3}\geq 3$.
Since $\lambda^{(1,3)}$ is not a rigid symplectic partition and there are no
admissible positions $i\geq 3$ for $\lambda^{(1,3)}$, we can deduce that
$i_{2}=3$ is not possible.

Case $(i_{1},i_{2})=(1,2)$. We have
$\lambda^{\text{\boldmath$i$}_{2}}=\lambda^{(1,2)}=(2,2,2)$. The only
admissible position for $(2,2,2)$ is 3, so we must have $i_{3}=3$ and then
$\lambda^{\text{\boldmath$i$}_{3}}=\lambda^{(1,2,3)}=(0,0,0)$ is a rigid
symplectic partition. Thus $\text{\boldmath$i$}=(1,2,3)$ is a maximal
admissible inc-sequence for $\lambda$.

Case $(i_{1},i_{2})=(1,1)$. We have
$\lambda^{\text{\boldmath$i$}_{2}}=\lambda^{(1,1)}=(3,3,2)$. The only
admissible position for $(3,3,2)$ is 3, so we must have $i_{3}=3$ and then
$\lambda^{\text{\boldmath$i$}_{3}}=\lambda^{(1,1,3)}=(1,1,0)$ is a rigid
symplectic partition. Thus $\text{\boldmath$i$}=(1,1,3)$ is a maximal
admissible inc-sequence for $\lambda$.

We have seen that there are three maximal admissible inc-sequences which we
label as $\text{\boldmath$i$}^{1}=(1,1,3)$, $\text{\boldmath$i$}^{2}=(1,2,3)$
and $\text{\boldmath$i$}^{3}=(2,2)$. We label the rigid induction data
corresponding to $\text{\boldmath$i$}^{j}$ by
$[{\mathfrak{g}}_{j},{\mathbb{O}}_{j}]$ for $j=1,2,3$, and these are given as
follows.

  * •

For $\text{\boldmath$i$}^{1}$ we have
${\mathfrak{g}}_{1}={\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{3}\times{\mathfrak{sp}}_{2}$
and ${\mathbb{O}}_{1}$ is the zero orbit.

  * •

For $\text{\boldmath$i$}^{2}$ we have
${\mathfrak{g}}_{2}={\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{2}\times{\mathfrak{gl}}_{3}$
and ${\mathbb{O}}_{2}$ is the zero orbit.

  * •

For $\text{\boldmath$i$}^{3}$ we have
${\mathfrak{g}}_{3}={\mathfrak{gl}}_{2}\times{\mathfrak{gl}}_{2}\times{\mathfrak{sp}}_{4}$
and ${\mathbb{O}}_{3}=\\{0\\}^{2}\times{\mathbb{O}}_{(2,1,1)}$.

In preparation for the proof of Lemma 3.2 it is helpful for us to observe that
for $i,j\in\\{1,2,3\\}$ with $i\neq j$ there is no inclusion
$\tilde{\mathfrak{g}}_{i}\subseteq{\mathfrak{g}}_{j}$ for
$\tilde{\mathfrak{g}}_{i}$ a $G$-conjugate of ${\mathfrak{g}}_{i}$. To do this
we recall from §2.2 that we may assume $\tilde{\mathfrak{g}}_{i}$ is a
standard Levi subalgebra and thus of the form described in §2.2.

To demonstrate this lack of inclusions, first consider the standard Levi
subalgebras of ${\mathfrak{g}}_{1}$. These are of the form
${\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{\text{\boldmath$j$}^{1}}\times{\mathfrak{sp}}_{\text{\boldmath$j$}^{2},2(1_{\text{\boldmath$j$}^{2}})}$,
where $\text{\boldmath$j$}^{1}$ and $\text{\boldmath$j$}^{2}$ are inc-
sequences with $|\text{\boldmath$j$}^{1}|=3$ and
$|\text{\boldmath$j$}^{2}|\leq 1$. Therefore,
${\mathfrak{gl}}_{\text{\boldmath$j$}^{1}}$ is one of ${\mathfrak{gl}}_{3}$,
${\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{2}$ or
${\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{1}\times{\mathfrak{gl}}_{1}$, and
${\mathfrak{sp}}_{\text{\boldmath$j$}^{2},2(1_{\text{\boldmath$j$}^{2}})}$ is
one of ${\mathfrak{sp}}_{2}$ or ${\mathfrak{gl}}_{1}$. Since none of these
Levi subalgebras involves both a ${\mathfrak{gl}}_{2}$ and a
${\mathfrak{gl}}_{3}$, we see as above that no $G$-conjugate of
${\mathfrak{g}}_{2}$ can be a subalgebra of ${\mathfrak{g}}_{1}$, and since
none of these involve an ${\mathfrak{sp}}_{4}$ we see that no $G$-conjugate of
${\mathfrak{g}}_{3}$ can be a subalgebra of ${\mathfrak{g}}_{1}$.

We can similarly consider the forms of all standard Levi subalgebras of
${\mathfrak{g}}_{2}$ and note that none of these involve a symplectic
subalgebra (containing a root subspace for a long root), so that no conjugates
of either ${\mathfrak{g}}_{1}$ nor ${\mathfrak{g}}_{3}$ are contained in
${\mathfrak{g}}_{2}$. For ${\mathfrak{g}}_{3}$ we can see that no standard
Levi subalgebra can involve a ${\mathfrak{gl}}_{3}$, so that no conjugate of
either ${\mathfrak{g}}_{1}$ or ${\mathfrak{g}}_{2}$ is contained in
${\mathfrak{g}}_{3}$.

We pick out a key property of the KS-algorithm, which is demonstrated in
Example 3.1.

######  KS-property 2.

Let $\lambda$ be a symplectic partition or an orthogonal partition (depending
on which case we are considering). Let $i$ be a maximal admissible inc-
sequence for $\lambda$, and let $i\in{\mathbb{Z}}_{\geq 1}$ with $i_{1}\geq
i$. Suppose that $\lambda_{i}=\lambda_{i+1}+2$. Then $i_{1}=i$ or $i_{1}=i+1$,
and moreover, if $i_{1}=i+1$ and $k$ is maximal such that $i_{k}=i+1$, then
the $k$th reduction determined by $i$ is of type 2.

###### Proof.

Since $i$ is a maximal admissible sequence, we must have
$\lambda^{\text{\boldmath$i$}}_{i}\leq\lambda^{\text{\boldmath$i$}}_{i+1}+1$
and thus that
$\lambda^{\text{\boldmath$i$}_{j}}_{i}-\lambda^{\text{\boldmath$i$}_{j}}_{i+1}<\lambda^{\text{\boldmath$i$}_{j-1}}_{i}-\lambda^{\text{\boldmath$i$}_{j-1}}_{i+1}$
for some $j$. The only way we can get that
$\lambda^{\text{\boldmath$i$}_{j}}_{i}-\lambda^{\text{\boldmath$i$}_{j}}_{i+1}<\lambda^{\text{\boldmath$i$}_{j-1}}_{i}-\lambda^{\text{\boldmath$i$}_{j-1}}_{i+1}$
is for $i_{j}=i$, or for $i_{j}=i+1$ corresponding to a type 2 reduction.
Since $i$ is an admissible inc-sequence we deduce that $i_{1}=i$ or
$i_{1}=i+1$. Furthermore, in the latter case there must be a type 2 reduction
at position $i+1$, which implies that for the maximal $k$ such that
$i_{k}=i+1$, we have that the reduction corresponding to $i_{k}$ is of type 2.
∎

We move on to state and prove Lemma 3.2, which covers a key special case of
Theorem 3.3.

######  Lemma 3.2.

Let $r\in{\mathbb{Z}}_{>0}$.

  1. (a)

Let $G=\mathrm{Sp}_{r(r+1)}$ and let $\lambda=(2r,2r-2,\dots,4,2)$. Let
$[{\mathfrak{g}}_{1},{\mathbb{O}}_{1}]$ and
$[{\mathfrak{g}}_{2},{\mathbb{O}}_{2}]$ be rigid induction data for
${\mathbb{O}}_{\lambda}$, and let $({\mathfrak{g}}_{1},{\mathbb{O}}_{1})$ and
$({\mathfrak{g}}_{2},{\mathbb{O}}_{2})$ be representatives thereof. Suppose
that ${\mathfrak{g}}_{1}\subseteq{\mathfrak{g}}_{2}$. Then
${\mathfrak{g}}_{1}={\mathfrak{g}}_{2}$ and
${\mathbb{O}}_{1}={\mathbb{O}}_{2}$.

  2. (b)

Let $G=\mathrm{SO}_{r^{2}}$ and let $\lambda=(2r-1,2r-3,\dots,3,1)$. Let
$[{\mathfrak{g}}_{1},{\mathbb{O}}_{1}]$ and
$[{\mathfrak{g}}_{2},{\mathbb{O}}_{2}]$ be rigid induction data for
${\mathbb{O}}_{\lambda}$, and let $({\mathfrak{g}}_{1},{\mathbb{O}}_{1})$ and
$({\mathfrak{g}}_{2},{\mathbb{O}}_{2})$ be representatives thereof. Suppose
that ${\mathfrak{g}}_{1}\subseteq{\mathfrak{g}}_{2}$. Then
${\mathfrak{g}}_{1}={\mathfrak{g}}_{2}$ and
${\mathbb{O}}_{1}={\mathbb{O}}_{2}$.

###### Proof.

(a) We assume that $r>3$ as the case $r=3$ is covered in Example 3.1, and the
cases $r=1$ and $r=2$ can be dealt with easily.

Arguing as in §2.2 and Example 3.1, we may assume that ${\mathfrak{g}}_{1}$
and ${\mathfrak{g}}_{2}$ are standard Levi subalgebras with
${\mathfrak{g}}_{1}$ contained in ${\mathfrak{g}}_{2}$. There are maximal
admissible inc-sequences $\text{\boldmath$i$}=(i_{1},\dots,i_{s})$ and
$\text{\boldmath$j$}=(j_{1},\dots,j_{t})$ such that
${\mathfrak{g}}_{1}={\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$
and ${\mathbb{O}}_{1}={\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}$, and
${\mathfrak{g}}_{2}={\mathfrak{sp}}_{\text{\boldmath$j$},2n_{\text{\boldmath$j$}}}$
and ${\mathbb{O}}_{2}={\mathbb{O}}_{\lambda^{\text{\boldmath$j$}}}$. As
explained in §2.2, ${\mathfrak{g}}_{1}$ is a standard Levi subalgebra of
${\mathfrak{g}}_{2}$. Now by considering the form of Levi subalgebras given in
(2.1) and (2.3) we deduce that there is a function
$h:\\{1,\dots,s\\}\to\\{1,\dots,t,t+1\\}$ such that for each
$k\in\\{1,\dots,t\\}$ we have $\sum_{l\in h^{-1}(k)}i_{l}=j_{k}$. The idea
here is that the inclusion ${\mathfrak{g}}_{1}\subseteq{\mathfrak{g}}_{2}$ is
given by inclusions $\bigoplus_{l\in
h^{-1}(k)}{\mathfrak{gl}}_{i_{l}}\subseteq{\mathfrak{gl}}_{j_{k}}$ and an
inclusion $(\bigoplus_{l\in
h^{-1}(t+1)}{\mathfrak{gl}}_{i_{l}})\oplus{\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}\subseteq{\mathfrak{sp}}_{2n_{\text{\boldmath$j$}}}$.
Moreover, for such a function $h$ we may (and shall) assume that for $l\leq
l^{\prime}$ with $i_{l}=i_{l^{\prime}}$ we have $h(l)\leq h(l^{\prime})$.

We prove by induction on $k$ that
$\text{\boldmath$i$}_{k}=\text{\boldmath$j$}_{k}$ and that $h(l)=l$ for
$l=1,\dots,k$. For the case $k=s$ this implies that
$\text{\boldmath$i$}=\text{\boldmath$j$}$ and thus that
${\mathfrak{g}}_{1}={\mathfrak{g}}_{2}$ and
${\mathbb{O}}_{1}={\mathbb{O}}_{2}$ as required

We could take our base case to be $k=0$, which is trivial, but we find it more
instructive to include $k=1$ for the base case. We are led to consider the
possibilities for $(i_{1},i_{2},i_{3})$ and the possibilities for
$(j_{1},j_{2},j_{3})$. The analysis of cases below is very similar to that in
Example 3.1.

By KS-property 2, we have $i_{1}=1$ or $i_{1}=2$. We consider these cases
separately.

Case $i_{1}=2$. We must have $i_{2}=2$ by KS-property 2. Then we have
$\lambda^{(2,2)}=(2r-4,2r-5,2r-5,2r-6,2r-8,\dots,2)$ and we see that
$i_{3}\geq 4$.

Case $i_{1}=1$. We have $\lambda^{1}=(2r-2,2r-2,2r-4,2r-6,\dots,2)$. By using
KS-property 2 we see that the only possibilities for $i_{2}$ are $i_{2}=1$,
$i_{2}=2$ or $i_{2}=3$, and we consider these cases.

Case $(i_{1},i_{2})=(1,3)$. Using KS-property 2 again we have that $i_{3}=3$.
Then we have $\lambda^{(1,3,3)}=(2r-6,2r-6,2r-7,2r-7,2r-8,\dots,2)$. By
considering the subsequent reductions we see that
$\lambda^{\text{\boldmath$i$}_{k}}_{1}$ is even and
$\lambda^{\text{\boldmath$i$}_{k}}_{1}=\lambda^{\text{\boldmath$i$}_{k}}_{2}>\lambda^{\text{\boldmath$i$}_{k}}_{3}$
for each $k$. This is not possible as $\lambda^{\text{\boldmath$i$}}$ is a
rigid symplectic partition.

Case $(i_{1},i_{2})=(1,2)$. We have
$\lambda^{(1,2)}=(2r-4,2r-4,2r-4,2r-6,\dots,2)$ and then by KS-property 2 we
have that $i_{3}\geq 3$.

Case $(i_{1},i_{2})=(1,1)$. We have
$\lambda^{(1,1)}=(2r-3,2r-3,2r-4,2r-6,\dots,2)$ and then by KS-property 2 we
have that $i_{3}\geq 3$.

Summarising this case analysis we deduce that $(i_{1},i_{2},i_{3})$ is one of:
$(2,2,a)$ where $a\geq 4$; $(1,2,b)$ where $b\geq 3$; or $(1,1,c)$ where
$c\geq 3$. Similarly $(j_{1},j_{2},j_{3})$ must be one of these possibilities.

Suppose that $i_{1}=2$ and $j_{1}=1$. Then we have $h(l)>1$ for all $l$, as
$2=i_{1}>j_{1}=1$, and $\sum_{l\in h^{-1}(1)}i_{l}=0\neq j_{1}$. Thus
$i_{1}=2$ and $j_{1}=1$ is not possible.

Suppose that $i_{1}=1$ and $j_{1}=2$, so $(j_{1},j_{2},j_{3})$ is of the form
$(2,2,a)$ and $(i_{1},i_{2},i_{3})$ is one of the two possibilities with
$i_{1}=1$. We have that $\sum_{l\in h^{-1}(1)}i_{l}=2$, and thus the multiset
$\\{i_{l}\mid l\in h^{-1}(1)\\}$ is either $\\{1,1\\}$ or $\\{2\\}$.
Similarly, the multiset $\\{i_{l}\mid l\in h^{-1}(2)\\}$ is either $\\{1,1\\}$
or $\\{2\\}$. For this to be possible, one of the following possibilities must
occur:

  * •

$1$ occurs with multiplicity at least 4 in $i$;

  * •

$1$ occurs with multiplicity at least 2 and 2 occurs with multiplicity at
least 1 in $i$; or

  * •

$2$ occurs with multiplicity at least 2 in $i$.

By considering the two possibilities for $(i_{1},i_{2},i_{3})$ with $i_{1}=1$,
we see that none of the above three conditions hold for $i$. Thus $i_{1}=1$
and $j_{1}=2$ is not possible.

Therefore, $i_{1}=j_{1}$. Then we must also have that $h(1)=1$, as $i$ is an
increasing sequence and, by assumption, $h(1)\leq h(l)$ for any $l$ such that
$i_{1}=i_{l}$. This completes the base case.

We move on to the inductive step so we assume, for some $k\leq s$, that
$\text{\boldmath$i$}_{k-1}=\text{\boldmath$j$}_{k-1}$ and $h(l)=l$ for all
$l<k$.

First, we make here a couple of useful observations that we exploit several
times below. For any $i$ the multiplicity of $i$ in $i$ is at most 2, and if
the multiplicity is 2 then the first reduction at position $i$ is a type 1
reduction and the second is a type 2 reduction. This is necessary as
$\lambda^{\text{\boldmath$i$}}_{i}\geq\lambda^{\text{\boldmath$i$}}_{i+1}$,
and for parity reasons we can have at most one type 2 reduction at position
$i$. Moreover, if there is a type 2 reduction at position $i$, then there are
no reductions at position $i+1$. To explain this suppose that
$i_{k^{\prime}}=i$ and this corresponds to a type $2$ reduction. Then we have
$\lambda^{\text{\boldmath$i$}_{k^{\prime}}}_{i+1}=\lambda^{\text{\boldmath$i$}_{k^{\prime}}}_{i+2}+1$,
so that $i+1$ is not an admissible position for
$\lambda^{\text{\boldmath$i$}_{k^{\prime}}}$, and thus $i_{k^{\prime}+1}>i+1$.

Let $i=i_{k}$ and $j=j_{k}$. We note that $j\geq i$, because if $j<i$, then
there are no $l$ such that $h(l)=k$, and thus $\sum_{l\in
h^{-1}(k)}i_{l}=0\neq j$. We aim to show that $i=j$, which implies that
$h(k)=k$, to complete the inductive step. Thus we assume that $j>i$ and we aim
for a contradiction. We also note that if $k=s$, then we must have $i=j$, so
we assume that $k<s$.

We proceed to consider two cases.

Case $i_{k-1}<i$. We first note that we cannot have $i_{k-1}=i-1$ with this
reduction being of type $2$ otherwise $i$ is not an admissible position for
$\lambda^{\text{\boldmath$i$}_{k-1}}$. Thus in the first $k-1$ reductions
given by $i$ there have been no reductions at position $i$ and no type 2
reduction at position $i-1$, so none of these reductions have decreased
$\lambda_{i^{\prime}}$ for $i^{\prime}\geq i$. Therefore,
$(\lambda^{\text{\boldmath$i$}_{k-1}}_{i},\lambda^{\text{\boldmath$i$}_{k-1}}_{i+1},\lambda^{\text{\boldmath$i$}_{k-1}}_{i+2})=(2r^{\prime},2r^{\prime}-2,2r^{\prime}-4)$,
where $r^{\prime}=r-i+1$, and
$(\lambda^{\text{\boldmath$i$}_{k}}_{i},\lambda^{\text{\boldmath$i$}_{k}}_{i+1},\lambda^{\text{\boldmath$i$}_{k}}_{i+2})=(2r^{\prime}-2,2r^{\prime}-2,2r^{\prime}-4)$.
Here we can exclude the possibility that $r^{\prime}=1$ as then we would have
$k=s$, so we assume that $r^{\prime}\geq 2$.

As $j_{k}>i$ we have that $j_{k}=j_{k+1}=i+1$ by KS-property 2. Then we get
$(\lambda^{\text{\boldmath$j$}_{k+1}}_{i},\lambda^{\text{\boldmath$j$}_{k+1}}_{i+1},\lambda^{\text{\boldmath$j$}_{k+1}}_{i+2})=(2r^{\prime}-4,2r^{\prime}-5,2r^{\prime}-5)$,
and thus $(j_{k},j_{k+1},j_{k+2})=(i+1,i+1,a)$ with $a\geq i+3$. From this we
can exclude the case $r^{\prime}=2$, and thus assume that $r^{\prime}\geq 3$.

We consider the possibilities for $i_{k+1}$. By using KS-property 2 (for
“$i=i+1$”) we have that $i_{k+1}$ is $i$, $i+1$ or $i+2$. Then:

  * •

If $i_{k+1}=i+2$, then by KS-property 2 we also have $i_{k+2}=i+2$. (We note
that this case is not possible for $r^{\prime}=3$.)

  * •

If $i_{k+1}=i+1$, then we have
$(\lambda^{\text{\boldmath$i$}_{k+1}}_{i},\lambda^{\text{\boldmath$i$}_{k+1}}_{i+1},\lambda^{\text{\boldmath$i$}_{k+1}}_{i+2})=(2r^{\prime}-4,2r^{\prime}-4,2r^{\prime}-4)$
and see that $i_{k+2}\geq i+2$.

  * •

If $i_{k+1}=i$, then we have
$(\lambda^{\text{\boldmath$i$}_{k+1}}_{i},\lambda^{\text{\boldmath$i$}_{k+1}}_{i+1},\lambda^{\text{\boldmath$i$}_{k+1}}_{i+2})=(2r^{\prime}-3,2r^{\prime}-3,2r^{\prime}-4)$
and see that $i_{k+2}\geq i+2$.

Summarising we get that $(i_{k},i_{k+1},i_{k+2})$ is one of $(i,i+2,i+2)$,
$(i,i+1,b)$, where $b\geq i+2$, or $(i,i,c)$ where $c\geq i+2$.

We now consider $h^{-1}(k)$, which is a subset of $\\{k,k+1,\dots,s\\}$ such
that $\sum_{l\in h^{-1}(k)}i_{l}=i+1$. Similarly, $h^{-1}(k+1)$ is a subset of
$\\{k,k+1,\dots,s\\}$ such that $\sum_{l\in h^{-1}(k+1)}i_{l}=i+1$. We see
from the above possibilities for $(i_{k},i_{k+1},i_{k+2})$ that this is not
possible.

Therefore, we have shown that $i_{k-1}<i$ is impossible, and we have our
contradiction in this case.

Case $i_{k-1}=i$. Then we must have $i_{k-1}=i$ corresponding to a type 1
reduction, and $i_{k}=i$ corresponding to a type 2 reduction. For the type 2
reduction to be possible we require that
$\lambda^{\text{\boldmath$i$}_{k-1}}_{i-1}>\lambda^{\text{\boldmath$i$}_{k-1}}_{i}$.
We then have that
$(\lambda^{\text{\boldmath$i$}_{k-1}}_{i},\lambda^{\text{\boldmath$i$}_{k-1}}_{i+1},\lambda^{\text{\boldmath$i$}_{k-1}}_{i+2})=(2r^{\prime}-2,2r^{\prime}-2,2r^{\prime}-4)$
and
$(\lambda^{\text{\boldmath$i$}_{k}}_{i},\lambda^{\text{\boldmath$i$}_{k}}_{i+1},\lambda^{\text{\boldmath$i$}_{k}}_{i+2})=(2r^{\prime}-3,2r^{\prime}-3,2r^{\prime}-4)$,
where $r^{\prime}=r-i+1$.

Thus we have that $i_{k+1}\geq i+2$. Since
$\lambda^{\text{\boldmath$j$}_{k-1}}=\lambda^{\text{\boldmath$i$}_{k-1}}$ and
$j_{k}>i_{k}$ we must have that $(j_{k},j_{k+1})$ is one of $(i+2,i+2)$ or
$(i+1,b)$ where $b\geq i+2$.

In the case $(j_{k},j_{k+1})=(i+1,b)$ we see that $h^{-1}(k)$ is a subset of
$\\{k,k+1,\dots,s\\}$ such that $\sum_{l\in h^{-1}(k)}i_{l}=i+1$. Since
$i_{k+1}>i+1$ the only possible element of $h^{-1}(k)$ is $k$, but as
$i_{k}=i<i+1$ we see that this is not possible.

We show that the case $(j_{k},j_{k+1})=(i+2,i+2)$ is in fact also not
possible. For this case we would have
$(\lambda^{\text{\boldmath$j$}_{k+1}}_{i},\lambda^{\text{\boldmath$j$}_{k+1}}_{i+1},\lambda^{\text{\boldmath$j$}_{k+1}}_{i+2})=(2r^{\prime}-6,2r^{\prime}-6,2r^{\prime}-7)$,
and also that $\lambda^{\text{\boldmath$j$}_{k+1}}_{i-1}>2r^{\prime}-6$ and
$\lambda^{\text{\boldmath$j$}_{k+1}}_{i+3}=2r^{\prime}-7$. Thus we see that
$j_{l}>i+2$ for all $l>k+1$. It follows that
$\lambda^{\text{\boldmath$j$}}_{i-1}>\lambda^{\text{\boldmath$j$}}_{i}=\lambda^{\text{\boldmath$j$}}_{i+1}>\lambda^{\text{\boldmath$j$}}_{i+2}$
and that $\lambda^{\text{\boldmath$j$}}_{i}$ is even. This is a contradiction
to $\lambda^{\text{\boldmath$j$}}$ being a rigid symplectic partition.

Therefore, we have shown that $i_{k-1}=i$ is also impossible, and we have our
contradiction in this case.

The contradictions in the two cases above imply that $i=j$ and therefore
complete our inductive step.

(b) The proof for $G=\mathrm{SO}_{r^{2}}$ is very similar. We note that the
case $r\leq 4$ where $\lambda$ is one of $(7,5,3,1)$, $(5,3,1)$ or $(3,1)$ can
be dealt with similarly to Example 3.1. So the assumption $r>4$ can be made.
The proof now proceeds in the same way just with $2r^{\prime}+1$ in place of
$2r^{\prime}$. ∎

We strengthen Lemma 3.2 in Theorem 3.3. For the proof of this theorem it is
helpful to pick out the following useful property of the KS-algorithm.

######  KS-property 3.

Let $\lambda$ be a symplectic partition or an orthogonal partition depending
on which case we are considering. Let $i$ be a maximal admissible inc-sequence
for $\lambda$, and let $i\in{\mathbb{Z}}_{\geq 1}$ with $i_{1}\geq i$. Suppose
that $\lambda_{i}>\lambda_{i+1}+2$. Then $i_{1}=i$.

###### Proof.

Suppose $i_{1}>i$. For $j\in\\{1,\dots,s\\}$ the only way we can have that
$\lambda^{\text{\boldmath$i$}_{j}}_{i}-\lambda^{\text{\boldmath$i$}_{j}}_{i+1}<\lambda^{\text{\boldmath$i$}_{j-1}}_{i}-\lambda^{\text{\boldmath$i$}_{j-1}}_{i+1}$
is if $i_{j}=i+1$ and this corresponds to a type 2 reduction in the KS-
algorithm. Then we have
$\lambda^{\text{\boldmath$i$}_{j}}_{i}-\lambda^{\text{\boldmath$i$}_{j}}_{i+1}=\lambda^{\text{\boldmath$i$}_{j-1}}_{i}-\lambda^{\text{\boldmath$i$}_{j-1}}_{i+1}-1$.
By considering the parity of $\lambda^{\text{\boldmath$i$}_{j}}_{i+1}$ there
can be only one such type 2 reduction. Thus we have that
$\lambda^{\text{\boldmath$i$}}_{i}\geq\lambda^{\text{\boldmath$i$}}_{i+1}+2$,
which is contrary to $\lambda^{\text{\boldmath$i$}}$ being a rigid partition.
Hence, we can conclude that $i_{1}=i$. ∎

We are now ready to state and prove Theorem 3.3 and note that the idea of
proof of the theorem is to reduce to a case given in Lemma 3.2.

######  Theorem 3.3.

Let $G$ be a simple algebraic group of classical type and let ${\mathbb{O}}$
be a nilpotent $G$-orbit. Let $[{\mathfrak{g}}_{1},{\mathbb{O}}_{1}]$ and
$[{\mathfrak{g}}_{2},{\mathbb{O}}_{2}]$ be rigid induction data for
${\mathbb{O}}$, and let $({\mathfrak{g}}_{1},{\mathbb{O}}_{1})$ and
$({\mathfrak{g}}_{2},{\mathbb{O}}_{2})$ be representatives thereof. Suppose
that ${\mathfrak{g}}_{1}\subseteq{\mathfrak{g}}_{2}$. Then
${\mathfrak{g}}_{1}={\mathfrak{g}}_{2}$ and
${\mathbb{O}}_{1}={\mathbb{O}}_{2}$.

###### Proof.

We first cover the case $G=\mathrm{SL}_{n}$. There is a unique rigid induction
datum $[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$ for ${\mathbb{O}}$, as explained
in §2.4. Thus trivially ${\mathfrak{g}}_{1}={\mathfrak{g}}_{2}$ and
${\mathbb{O}}_{1}={\mathbb{O}}_{2}$ in this case.

We move on to consider the case $G=\mathrm{Sp}_{2n}$ so
${\mathfrak{g}}={\mathfrak{sp}}_{2n}$. Let
$\lambda=(\lambda_{1},...,\lambda_{r})$ be the symplectic partition such that
${\mathbb{O}}={\mathbb{O}}_{\lambda}$.

By [To23, Lemma 8.4] we can find $\text{\boldmath$j$}=(j_{1},\dots,j_{t})$
such that the orbit $\\{0\\}\times{\mathbb{O}}_{\lambda}$ in
${\mathfrak{sp}}_{\text{\boldmath$j$},2n}={\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}$
induces to an orbit ${\mathbb{O}}_{\tilde{\lambda}}$ in
${\mathfrak{sp}}_{2|\text{\boldmath$j$}|+2n}$ such that the symplectic
partition
$\tilde{\lambda}=(\tilde{\lambda}_{1},...,\tilde{\lambda}_{\tilde{r}})$ of
$2|\text{\boldmath$j$}|+2n$ satisfies that $\tilde{\lambda}_{i}$ is even for
all $i$ and $\tilde{\lambda}_{i}>\tilde{\lambda}_{i+1}$ for all
$i=1,...,\tilde{r}$. We have that
${\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{1}$ and
${\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{2}$ are Levi
subalgebras of ${\mathfrak{sp}}_{\text{\boldmath$j$},2n}$ and thus Levi
subalgebras of ${\mathfrak{sp}}_{2|\text{\boldmath$j$}|+2n}$. It follows that
$[{\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{1},\\{0\\}\times{\mathbb{O}}_{1}]$
and
$[{\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{2},\\{0\\}\times{\mathbb{O}}_{2}]$
are rigid induction data for ${\mathbb{O}}_{\tilde{\lambda}}$.

We have
${\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{1}\subseteq{\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{2}$.
Moreover ${\mathfrak{g}}_{1}={\mathfrak{g}}_{2}$ and
${\mathbb{O}}_{1}={\mathbb{O}}_{2}$ if and only if
${\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{1}={\mathfrak{gl}}_{\text{\boldmath$j$}}\times{\mathfrak{g}}_{2}$
and $\\{0\\}\times{\mathbb{O}}_{1}=\\{0\\}\times{\mathbb{O}}_{2}$. By
replacing $\lambda$ by $\tilde{\lambda}$, we can therefore assume that
$\lambda$ satisfies that $\lambda_{i}$ is even for all $i$ and
$\lambda_{i}>\lambda_{i+1}$ for all $i=1,...,r$.

Let
$\text{\boldmath$i$}^{\prime}=(i^{\prime}_{1},...,i^{\prime}_{s^{\prime}})$ be
the inc-sequence in which $i\in\\{1,...,r\\}$ occurs with multiplicity
$(\lambda_{i}-\lambda_{i+1})/2-1$. Then we have that
$\text{\boldmath$i$}^{\prime}$ is an admissible sequence for $\lambda$, for
which reductions of the KS-algorithm are of type 1. Further, we have that
$\lambda^{\text{\boldmath$i$}^{\prime}}=(2r,2r-2,\dots,4,2)$ is a partition as
in Lemma 3.2.

Let $i$ be a maximal admissible sequence for $\lambda$. Using KS-property 1 we
may first assume that $i$ is an inc-sequence without affecting
$\lambda^{\text{\boldmath$i$}}$ or
${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ up to
conjugacy. Then we observe that $i$ must start with at least
$l:=(\lambda_{1}-\lambda_{2})/2-1$ entries equal to $1$ by using KS-property
3. We see that the maximal $k$ such that $i_{k}=1$ is $l$, $l+1$ or $l+2$, and
then the next $(\lambda_{2}-\lambda_{3})/2-1$ entries in $i$ must be equal to
$2$ by using KS-property 3. Continuing this argument we see that for each $i$
there must be at least $(\lambda_{i}-\lambda_{i+1})/2-1$ entries equal to $i$
in $i$. In other words, $\text{\boldmath$i$}^{\prime}$ is a subsequence of
$i$. Another application of KS-property 1 allows us reorder $i$ to assume that
$\text{\boldmath$i$}_{s^{\prime}}=\text{\boldmath$i$}^{\prime}$ whilst not
affecting ${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ or
$\lambda^{\text{\boldmath$i$}}$.

It now follows that there is a bijection from the rigid induction data for
${\mathbb{O}}_{\lambda}\subseteq{\mathfrak{sp}}_{2n}$ to the rigid induction
data for
${\mathbb{O}}_{\lambda^{\text{\boldmath$i$}^{\prime}}}\subseteq{\mathfrak{sp}}_{2n_{\text{\boldmath$i$}^{\prime}}}$.
The bijection is given by
$[{\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},\\{0\\}^{s}\times{\mathbb{O}}_{\lambda^{\text{\boldmath$i$}}}]\mapsto[{\mathfrak{sp}}_{\text{\boldmath$i$}^{\prime\prime},2(n_{\text{\boldmath$i$}^{\prime}})_{\text{\boldmath$i$}^{\prime\prime}}},\\{0\\}^{s^{\prime\prime}}\times{\mathbb{O}}_{(\lambda^{\text{\boldmath$i$}^{\prime}})^{\text{\boldmath$i$}^{\prime\prime}}}]$,
where
$\text{\boldmath$i$}^{\prime\prime}=(i^{\prime\prime}_{1},...,i^{\prime\prime}_{s^{\prime\prime}})$
is such that
$\text{\boldmath$i$}=(i^{\prime}_{1},...,i^{\prime}_{s^{\prime}},i^{\prime\prime}_{1},...,i^{\prime\prime}_{s^{\prime\prime}})$;
so that
$(n_{\text{\boldmath$i$}^{\prime}})_{\text{\boldmath$i$}^{\prime\prime}}=n_{\text{\boldmath$i$}}$
and
$(\lambda^{\text{\boldmath$i$}^{\prime}})^{\text{\boldmath$i$}^{\prime\prime}}=\lambda^{\text{\boldmath$i$}}$.

We observe that $[{\mathfrak{g}}_{1},{\mathbb{O}}_{1}]$ and
$[{\mathfrak{g}}_{2},{\mathbb{O}}_{2}]$ are rigid induction data for
${\mathbb{O}}_{\lambda^{\text{\boldmath$i$}^{\prime}}}\subseteq{\mathfrak{sp}}_{\text{\boldmath$i$}^{\prime},n_{\text{\boldmath$i$}^{\prime}}}$.
Thus by Lemma 3.2 we obtain that ${\mathfrak{g}}_{1}={\mathfrak{g}}_{2}$ and
${\mathbb{O}}_{1}={\mathbb{O}}_{2}$. This completes the proof for the
symplectic case.

We end the proof by considering the case $G=\mathrm{SO}_{N}$. Here we can
proceed exactly as in the symplectic case. The only minor changes required are
to replace all occurrences of ${\mathfrak{sp}}$ with ${\mathfrak{so}}$, to
take all entries of $\tilde{\lambda}$ to be odd then assume that all
$\lambda_{i}$ are odd and distinct, and to say that
$\lambda^{\text{\boldmath$i$}^{\prime}}$ is equal to $(2r-1,2r-3,\dots,3,1)$.
∎

######  Remark 3.4.

We note that the statement of Theorem 3.3 holds more generally for any
reductive algebraic group $G$ over ${\mathbb{C}}$. First, from the definition
of Lusztig–Spaltenstein induction it is straightforward to see that the
theorem reduces to the case where $G$ is simple and that it is independent of
the isogeny class of $G$. So we are just left to consider $G$ simple of
exceptional type.

The description of induced orbits and the Levi subalgebras from which they are
induced for $G$ of exceptional type can be found in [EdG09, Tables 6–10]. The
majority of induced orbits have a unique rigid induction datum, and so the
statement of Theorem 3.3 holds trivially for these orbits. For all the other
orbits a case by case analysis shows that there can never be containment of
Levi subalgebras for distinct rigid induction data. Thus the statement of
Theorem 3.3 does hold for $G$ of exceptional type.

We also comment here that it would be interesting to find a general conceptual
argument to prove this more general version of Theorem 3.3.

##  4\. Finite $W$-algebras and Losev–Premet ideals

For this section $G$ is any connected reductive algebraic group over
${\mathbb{C}}$. We let $e\in{\mathfrak{g}}$ be a nilpotent element and let
${\mathbb{O}}={\mathbb{O}}_{e}$. We recall the required background on the
finite $W$-algebra $U({\mathfrak{g}},e)$, move on to discuss Losev’s parabolic
induction and then provide the setup required to state Theorem 4.1. As a
general reference for this material on finite $W$-algebras we refer for
example to [Lo10b], and note that this covers all of the fundamental results
recapped below.

To define $U({\mathfrak{g}},e)$ we extend $e$ to an
${\mathfrak{sl}}_{2}$-triple $(e,h,f)$ in ${\mathfrak{g}}$ and consider the
$\operatorname{ad}h$-grading
${\mathfrak{g}}=\bigoplus_{i\in{\mathbb{Z}}}{\mathfrak{g}}(i)$ of
${\mathfrak{g}}$. We write $\chi:=(e,\cdot)\in{\mathfrak{g}}^{*}$, where we
recall that $(\cdot\,,\cdot)$ is a nondegenerate invariant symmetric bilinear
form on ${\mathfrak{g}}$. There is a symplectic form on ${\mathfrak{g}}(-1)$
defined by $(x,y)\mapsto\chi([x,y])$, and we choose a Lagrangian subspace
${\mathfrak{l}}\subseteq{\mathfrak{g}}(-1)$. We let
${\mathfrak{m}}:={\mathfrak{l}}\oplus\bigoplus_{i<-1}{\mathfrak{g}}(i)$ and
${\mathfrak{m}}_{\chi}:=\\{x-\chi(x)\mid x\in{\mathfrak{m}}\\}\subseteq
U({\mathfrak{g}})$. Let
$Q_{\chi}:=U({\mathfrak{g}})/U({\mathfrak{g}}){\mathfrak{m}}_{\chi}$, which is
(left) $U({\mathfrak{g}})$-module called the generalised Gelfand–Graev module
associated to $\chi$. The finite $W$-algebra corresponding to ${\mathfrak{g}}$
and $e$ is defined as

$U({\mathfrak{g}},e):=\operatorname{End}_{U({\mathfrak{g}})}(Q_{\chi})^{\mathrm{op}}.$

It is known that up to isomorphism $U({\mathfrak{g}},e)$ does not depend on
the choice of ${\mathfrak{l}}$ or on the choice of
${\mathfrak{sl}}_{2}$-triple, and in fact only depends on the $G$-orbit of
$e$.

The generalised Gelfand-Graev module $Q_{\chi}$ is a left
$U({\mathfrak{g}})$-module and a right $U({\mathfrak{g}},e)$-module. Let
$\operatorname{Wh}_{\chi}(U({\mathfrak{g}}))$ denote the full subcategory of
$U({\mathfrak{g}})$-modules on which ${\mathfrak{m}}_{\chi}$ acts locally
nilpotently. The functor

$Q_{\chi}\otimes_{U({\mathfrak{g}},e)}(-):U({\mathfrak{g}},e)\operatorname{-mod}\overset{\sim}{\longrightarrow}\operatorname{Wh}_{\chi}(U({\mathfrak{g}}))$
(4.1)

is an equivalence of categories known as Skryabin’s equivalence. This leads to
a close relationship between finite dimensional simple
$U({\mathfrak{g}},e)$-modules and the set
$\operatorname{Prim}_{\mathbb{O}}U({\mathfrak{g}})$, as established within
[Lo10a, Theorem 1.2.2] and which we now describe.

For $M$ a finite dimensional simple $U({\mathfrak{g}},e)$-module,
$\operatorname{Ann}_{U({\mathfrak{g}})}(Q_{\chi}\otimes_{U({\mathfrak{g}},e)}M)$
lies in $\operatorname{Prim}_{\mathbb{O}}U({\mathfrak{g}})$; conversely, any
$I\in\operatorname{Prim}_{\mathbb{O}}(U({\mathfrak{g}}))$ is of the form
$\operatorname{Ann}_{U({\mathfrak{g}})}(Q_{\chi}\otimes_{U({\mathfrak{g}},e)}M)$
for some finite dimensional simple $U({\mathfrak{g}},e)$-module $M$. Moreover,
two finite dimensional simple $U({\mathfrak{g}},e)$-modules $M$ and $N$
satisfy
$\operatorname{Ann}_{U({\mathfrak{g}})}(Q_{\chi}\otimes_{U({\mathfrak{g}},e)}M)=\operatorname{Ann}_{U({\mathfrak{g}})}(Q_{\chi}\otimes_{U({\mathfrak{g}},e)}N)$
if and only if $M$ is isomorphic to a twist of $N$ by an action of the
component group of $(G^{e})^{h}$ on $U({\mathfrak{g}},e)$; we note that an
alternative definition of $U({\mathfrak{g}},e)$ is required to see the action
of $(G^{e})^{h}$ on $U({\mathfrak{g}},e)$. Lastly, we recall that for a
$1$-dimensional $U({\mathfrak{g}},e)$-module $M$ it is known that
$\operatorname{Ann}_{U({\mathfrak{g}})}(Q_{\chi}\otimes_{U({\mathfrak{g}},e)}M)$
is a completely prime primitive ideal, i.e. it lies in
$\operatorname{Prim}_{\mathbb{O}}^{1}(U({\mathfrak{g}}))$.

The $1$-dimensional representations of $U({\mathfrak{g}},e)$ correspond to the
points of the affine variety

${\mathcal{E}}({\mathfrak{g}},e):=\operatorname{Spec}_{\mathrm{max}}(U({\mathfrak{g}},e)^{\mathrm{ab}}),$
(4.2)

where $U({\mathfrak{g}},e)^{\mathrm{ab}}:=U({\mathfrak{g}},e)/([a,b]\mid
a,b\in U({\mathfrak{g}},e))$. It follows from the discussion above that there
is a map

$\mathscr{S}=\mathscr{S}({\mathfrak{g}},e):=\operatorname{Ann}_{U({\mathfrak{g}})}(Q_{\chi}\otimes_{U({\mathfrak{g}},e)}(-)):{\mathcal{E}}({\mathfrak{g}},e)\to\operatorname{Prim}_{\mathbb{O}}^{1}(U({\mathfrak{g}})),$
(4.3)

whose fibres are orbits under the action of the component group of
$(G^{e})^{h}$.

######  Definition.

An ideal $I$ in $U({\mathfrak{g}})$ is called a Losev-Premet ideal if it lies
in the image of the map $\mathscr{S}$.

To discuss parabolic induction of Losev-Premet ideals, we first need to refine
our earlier discussion of rigid induction data. Recall (see §2.4) that
${\mathcal{R}}({\mathfrak{g}},{\mathbb{O}})$ denotes the finite set of rigid
induction data for ${\mathbb{O}}$. Let us now enumerate this set as
${\mathcal{R}}({\mathfrak{g}},{\mathbb{O}})=\\{[{\mathfrak{g}}_{i},{\mathbb{O}}_{i}]\mid
i=0,...,m\\}$. From now on, we consider each rigid induction datum
$[{\mathfrak{g}}_{i},{\mathbb{O}}_{i}]$ to come equipped with a fixed
representative $({\mathfrak{g}}_{i},{\mathbb{O}}_{i})$ such that
${\mathfrak{g}}_{i}$ is a standard Levi subalgebra, a fixed element
$e_{i}\in{\mathbb{O}}_{i}$, and a parabolic subalgebra ${\mathfrak{p}}_{i}$ of
${\mathfrak{g}}$ of which ${\mathfrak{g}}_{i}$ is a Levi factor.

Now, fix $i\in\\{0,\ldots,m\\}$. In [Lo11, Theorem 1.2.1] Losev established a
dimension preserving parabolic induction functor from finite dimensional
$U({\mathfrak{g}}_{i},e_{i})$-modules to finite dimensional
$U({\mathfrak{g}},e)$-modules. At the level of $1$-dimensional modules this
induces a finite morphism

$\Phi_{i}:{\mathcal{E}}({\mathfrak{g}}_{i},e_{i})\to{\mathcal{E}}({\mathfrak{g}},e),$
(4.4)

see [Lo11, Theorem 6.5.2]. The construction of this morphism depends on a
choice of a parabolic subalgebra; in our set-up, we always use
${\mathfrak{p}}_{i}$. This map $\Phi_{i}$ fits into the commutative diagram

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60.21796pt\raise-35.98038pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.2713pt\raise-41.37204pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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73.2713pt\raise-41.37204pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
103.2713pt\raise-41.37204pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
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(4.5)

where $\mathscr{S}_{i}:=\mathscr{S}({\mathfrak{g}}_{i},e_{i})$. Note that the
commutativity of this diagram particularly implies that Losev-Premet ideals
parabolically induce to Losev-Premet ideals (induction of ideals is recapped
in §2.6).

Combining the diagrams (4.5) for each $i=0,\ldots,m$, we obtain the
commutative diagram

$\displaystyle\begin{array}[]{c}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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(4.11)

Here we write $\Phi:=\bigsqcup_{i=0}^{m}\Phi_{i}$,
$\mathscr{S}_{\operatorname{rig}}:=\bigsqcup_{i=0}^{m}\mathscr{S}_{i}$, and
$\operatorname{Ind}^{{\mathfrak{g}}}:=\bigsqcup_{i=0}^{m}\operatorname{Ind}_{{\mathfrak{p}}_{i}}^{{\mathfrak{g}}}$.
Note that $\Phi$ in diagram (4.11) is a finite morphism. The following theorem
is the main technical result of the paper, and immediately implies Theorem
1.1.

######  Theorem 4.1.

If $G$ is a simple algebraic group over ${\mathbb{C}}$ of classical type, then
$\Phi$ in (4.11) is surjective.

We prove Theorem 4.1 in full generality in §5, but to introduce some of the
ideas of the proof we consider first the case of $m=0$, which is already known
due to [PT14, Theorem 5]. Note that this case corresponds to ${\mathbb{O}}$
having a unique rigid induction datum; by §2.5, this is the same as
${\mathbb{O}}$ lying on a unique sheet. The argument follows [PT14, Theorem
5].

######  Proposition 4.2.

Let $G$ be a simple algebraic group over ${\mathbb{C}}$ of classical type and
let ${\mathbb{O}}$ be a nilpotent orbit in ${\mathfrak{g}}$ with unique rigid
induction datum $[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$. Fix
$e\in{\mathbb{O}}$. Then $U({\mathfrak{g}},e)^{\mathrm{ab}}$ is a polynomial
ring in $\dim{\mathfrak{z}}({\mathfrak{g}}_{0})$ variables and the map
$\Phi_{0}$ in (4.4) is surjective.

###### Proof.

By [PT14, Theorem 1] and [PT14, Theorem 4(ii)],
$U({\mathfrak{g}},e)^{\mathrm{ab}}$ is a polynomial algebra in $c(e)$
variables, where
$c(e)=\dim\overline{{\mathcal{D}}}^{{\mathrm{reg}}}({\mathfrak{g}}_{0},{\mathbb{O}}_{0})-\dim{\mathbb{O}}$.
This is equal to $\dim{\mathfrak{z}}({\mathfrak{g}}_{0})$ by the discussion at
the end of §2.5 (citing [BK79, Satz 4.5]), and thus the first claim is proved.

On the other hand it follows from a definition of the finite $W$-algebra that
$U({\mathfrak{g}},e_{0})\cong S({\mathfrak{z}}({\mathfrak{g}}_{0}))\otimes
U([{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}],e_{0})$ and thus
$U({\mathfrak{g}},e_{0})^{\mathrm{ab}}\cong
S({\mathfrak{z}}({\mathfrak{g}}_{0}))\otimes
U([{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}],e_{0})^{\mathrm{ab}}$. Applying
[PT14, Theorem 1] again, we see that
$U([{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}],e_{0})^{\mathrm{ab}}\cong{\mathbb{C}}$
and thus that $U({\mathfrak{g}},e_{0})^{\mathrm{ab}}\cong
S({\mathfrak{z}}({\mathfrak{g}}_{0}))$ is a polynomial algebra in
$\dim{\mathfrak{z}}({\mathfrak{g}}_{0})$ variables.

We therefore conclude that
$\Phi_{0}:{\mathcal{E}}({\mathfrak{g}}_{0},e_{0})\to{\mathcal{E}}({\mathfrak{g}},e)$
is a finite morphism between affine spaces of the same dimension. Since finite
morphisms are closed, $\Phi$ is surjective. ∎

As explained in §2.4, a nilpotent orbit ${\mathbb{O}}$ in
${\mathfrak{sl}}_{n}$ has a unique rigid induction datum and so lies in a
unique sheet. Proposition 4.2 is thus sufficient to prove Theorem 4.1 in type
A and we therefore get the following result.

######  Corollary 4.3.

If $G$ is a simple algebraic group over ${\mathbb{C}}$ of type A then $\Phi$
in (4.11) is surjective.

For types B, C and D we may have $m>0$ and so we need to work a bit harder; we
do so in the next section.

##  5\. Proof of Theorem 4.1

Thanks to Corollary 4.3, what remains is to prove Theorem 4.1 when $G$ is
$\mathrm{Sp}_{2n}$ or $\mathrm{SO}_{N}$. For brevity we tackle only
$G=\mathrm{Sp}_{2n}$ in this section, but the argument works almost
identically for $G=\mathrm{SO}_{N}$.

###  5.1. Preliminary observations

Before getting into the substance of the proof of Theorem 4.1 it will be
useful to make a handful of observations about one-dimensional representations
and central characters in the case of ${\mathfrak{g}}={\mathfrak{sp}}_{2n}$
and its Levi subalgebras (analogous results hold for ${\mathfrak{so}}_{N}$).
We do so here.

As discussed in §2.2, the standard Levi subalgebras of ${\mathfrak{sp}}_{2n}$
are all of the form

${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}={\mathfrak{gl}}_{i_{1}}\times\cdots\times{\mathfrak{gl}}_{i_{s}}\times{\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}$

for $\text{\boldmath$i$}=(i_{1},\ldots,i_{s})$ an inc-sequence with
$|\text{\boldmath$i$}|\leq n$, and with
$n_{\text{\boldmath$i$}}:=n-|\text{\boldmath$i$}|$. In particular, each
${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$ contains
${\mathfrak{t}}=\operatorname{Lie}(T)$, where $T$ is the maximal torus
consisting of diagonal matrices in $\mathrm{Sp}_{2n}$. Furthermore, the rigid
induction data are of the form
$[{\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{0}]$,
where ${\mathbb{O}}_{0}$ is a rigid nilpotent orbit in
${\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}$.

For such a rigid induction datum let $e_{0}\in{\mathbb{O}}_{0}$; we sometimes
abuse notation to identify $e_{0}\in{\mathbb{O}}_{0}$ with
$(0,e_{0})\in\\{0\\}\times{\mathbb{O}}_{0}$, thus viewing it as a nilpotent
element in ${\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}$.
As algebras, we have

$\displaystyle
U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})\cong
U({\mathfrak{gl}}_{i_{1}})\otimes\cdots\otimes
U({\mathfrak{gl}}_{i_{s}})\otimes
U({\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}},e_{0})$ (5.1)

and thus

$\displaystyle
U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})^{\mathrm{ab}}\cong
S({\mathfrak{z}}({\mathfrak{gl}}_{i_{1}}))\otimes\cdots\otimes
S({\mathfrak{z}}({\mathfrak{gl}}_{i_{s}}))\otimes
U({\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}},e_{0})^{\mathrm{ab}}.$ (5.2)

Noting that
${\mathfrak{z}}({\mathfrak{sp}}_{\text{\boldmath$i$},n_{\text{\boldmath$i$}}})={\mathfrak{z}}({\mathfrak{gl}}_{i_{1}})\times\cdots\times{\mathfrak{z}}({\mathfrak{gl}}_{i_{s}})\times\\{0\\}$
and that [PT14, Theorem 1] implies that
$U({\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}},e_{0})^{\mathrm{ab}}\cong{\mathbb{C}}$
(since $e_{0}$ is rigid in the simple Lie algebra
${\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}$), we can simplify this as

$\displaystyle
U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})^{\mathrm{ab}}\cong
S({\mathfrak{z}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})).$
(5.3)

In particular,
$U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})^{\mathrm{ab}}$
is a polynomial algebra in
$\dim{\mathfrak{z}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})=s$
variables and
${\mathcal{E}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})$
is $s$-dimensional affine space.

It will be helpful for us to give a more explicit description of
${\mathcal{E}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})$;
we do so by turning our attention to the central characters corresponding to
one-dimensional
$U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})$-modules.
We have seen that there is a unique ${\mathbb{C}}$-algebra homomorphism
$\eta=\eta(\text{\boldmath$i$},e_{0}):U({\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}},e_{0})\to{\mathbb{C}}$,
and we denote the corresponding one-dimensional
$U({\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}},e_{0})$-module by
${\mathbb{C}}_{\eta}$. Thanks to (5.2), for each
$\lambda\in{\mathfrak{z}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})^{*}$
there is a unique one-dimensional representation
${\mathbb{C}}_{\eta}^{\lambda}$ of
$U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})$ on
which $U({\mathfrak{gl}}_{i_{1}})\otimes\cdots\otimes
U({\mathfrak{gl}}_{i_{s}})$ acts via $\lambda$ and
$U({\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}},e_{0})$ acts via $\eta$.
Therefore,

${\mathcal{E}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})=\\{{\mathbb{C}}_{\eta}^{\lambda}\mid\lambda\in{\mathfrak{z}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})^{*}\\}.$
(5.4)

The map
$\mathscr{S}_{0}:=\mathscr{S}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},e_{0})$
then defines for each
$\lambda\in{\mathfrak{z}}({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})^{*}$
a Losev-Premet ideal

$I_{\eta}^{\lambda}:=\mathscr{S}_{0}({\mathbb{C}}_{\eta}^{\lambda})\subseteq
U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}}).$

Using Duflo’s theorem, there exists
$\mu=\mu(\text{\boldmath$i$},e_{0})\in{\mathfrak{t}}^{*}$ such that
$I_{\eta}^{0}=\operatorname{Ann}_{U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})}L(\mu)$.
It is not hard to see that

$\displaystyle
I_{\eta}^{\lambda}=\operatorname{Ann}_{U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})}(L(\mu)\otimes{\mathbb{C}}_{\lambda}),$
(5.5)

where ${\mathbb{C}}_{\lambda}$ is the one-dimensional
$U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})$-module
upon which $U({\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}})$ acts trivially and
$U({\mathfrak{gl}}_{i_{1}})\otimes\cdots\otimes U({\mathfrak{gl}}_{i_{s}})$
acts via $\lambda$. Let $W_{\text{\boldmath$i$}}$ be the Weyl group
corresponding to
$(\mathrm{Sp}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},T)$ and let
$\operatorname{ch}_{\text{\boldmath$i$}}$ be the corresponding central
character map from §2.6. We observe that $L(\mu)\otimes{\mathbb{C}}_{\lambda}$
is a highest weight
$U({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}})$-module
with highest weight $\mu+\lambda$; as discussed in §2.6, this implies that
$\operatorname{ch}_{\text{\boldmath$i$}}(I_{\eta}^{\lambda})=W_{\text{\boldmath$i$}}\bullet(\mu+\lambda)$.
Thus, using (2.6), we have:

######  Lemma 5.1.

$\operatorname{ch}(\operatorname{Ind}_{{\mathfrak{p}}_{\text{\boldmath$i$}}}^{{\mathfrak{sp}}_{2n}}(I_{\eta}^{\lambda}))=W\bullet(\mu+\lambda).$

######  Remark 5.2.

When ${\mathfrak{g}}$ is any simple Lie algebra over ${\mathbb{C}}$ with rigid
induction datum $[{\mathfrak{g}}_{0},{\mathbb{O}}_{0}]$, the decomposition
${\mathfrak{g}}_{0}={\mathfrak{z}}({\mathfrak{g}}_{0})\times[{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}]$
can be used to obtain analogues of the above results so long as
$U([{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}],e_{0})^{\mathrm{ab}}\cong{\mathbb{C}}$.
Reducing to the case where $[{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}]$ is
simple, this holds in all classical cases by [PT14] and in all but 6
exceptional types by [Pr14] – using Bala-Carter notation, these 6 cases
correspond to the nilpotent orbits $\tilde{A}_{1}$ in $G_{2}$,
$\tilde{A}_{2}+A_{1}$ in $F_{4}$, $(A_{3}+A_{1})^{\prime}$ in $E_{7}$, and
$A_{3}+A_{1}$, $A_{5}+A_{1}$ and $D_{5}(a_{1})+A_{2}$ in $E_{8}$. In each of
these cases,
$U([{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}],e_{0})^{\mathrm{ab}}\cong{\mathbb{C}}\times{\mathbb{C}}$
by [PS22, Theorem A] and there are thus two one-dimensional representations
$\eta_{1},\eta_{2}:U([{\mathfrak{g}}_{0},{\mathfrak{g}}_{0}],e_{0})\to{\mathbb{C}}$.
One then gets that

${\mathcal{E}}({\mathfrak{g}}_{0},e_{0})=\\{{\mathbb{C}}_{\eta_{i}}^{\lambda}\mid
i=1,2\mbox{ and }\lambda\in{\mathfrak{z}}({\mathfrak{g}}_{0})^{*}\\}.$

All the other observations work as for ${\mathfrak{sp}}_{2n}$, except that
$\mu$ will depend on which $\eta_{i}$ is being considered.

The final result we need is the following simple lemma. Recalling that $W$
acts on ${\mathfrak{t}}^{*}$ via the dot-action, let us write $V/(W,\bullet)$
for the image of any subset $V$ of ${\mathfrak{t}}^{*}$ under the surjection
${\mathfrak{t}}^{*}\twoheadrightarrow{\mathfrak{t}}^{*}/(W,\bullet)$.

######  Lemma 5.3.

Let $\mu_{0},\ldots,\mu_{m}\in{\mathfrak{t}}^{*}$ and let
${\mathfrak{z}}_{0},\ldots,{\mathfrak{z}}_{m}\subseteq{\mathfrak{t}}^{*}$ be
vector subspaces. If $(\mu_{0}+{\mathfrak{z}}_{0})/(W,\bullet)$ is contained
in $\bigcup_{i=1}^{m}(\mu_{i}+{\mathfrak{z}}_{i})/(W,\bullet)$ then there
exists $1\leq i_{0}\leq m$ and $w_{0}\in W$ such that
${\mathfrak{z}}_{0}\subseteq w_{0}({\mathfrak{z}}_{i_{0}})$.

###### Proof.

The inclusion
$(\mu_{0}+{\mathfrak{z}}_{0})/(W,\bullet)\subseteq\bigcup_{i=1}^{m}(\mu_{i}+{\mathfrak{z}}_{i})/(W,\bullet)$
implies that

$\mu_{0}+{\mathfrak{z}}_{0}\subseteq\bigcup_{i=1}^{m}\bigcup_{w\in
W}w\bullet(\mu_{i}+{\mathfrak{z}}_{i}).$

The right hand side is an affine variety with irreducible components of the
form $w\bullet(\mu_{i}+{\mathfrak{z}}_{i})$, where $w\in W$ and
$i\in\\{1,...,m\\}$. Since $\mu_{0}+{\mathfrak{z}}_{0}$ is an irreducible
subvariety there exist $1\leq i_{0}\leq m$ and $w_{0}\in W$ such that
$\mu_{0}+{\mathfrak{z}}_{0}\subseteq
w_{0}\bullet(\mu_{i_{0}}+{\mathfrak{z}}_{i_{0}}).$ Now,
${\mathfrak{z}}_{0}\subseteq
w_{0}\bullet\mu_{i_{0}}-\mu_{0}+w_{0}({\mathfrak{z}}_{i_{0}})$ and so
$0\in{\mathfrak{z}}_{0}$ implies
$w_{0}\bullet\mu_{i_{0}}-\mu_{0}+w_{0}({\mathfrak{z}}_{i_{0}})=w_{0}({\mathfrak{z}}_{i_{0}})$.
The result follows. ∎

###  5.2. Main proof

We are now in a position to proceed to the main steps in the proof of Theorem
4.1. From now on, we recall the refined set-up of rigid induction data from §4
and our notation is consistent with that section. In particular, we label our
rigid induction data by $[{\mathfrak{g}}_{i},{\mathbb{O}}_{i}]$,
$i=0,\ldots,m$, rather than by
$[{\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{0}]$
with $i$ a maximal admissible inc-sequence and ${\mathbb{O}}_{0}$ a rigid
nilpotent orbit in ${\mathfrak{sp}}_{2n_{\text{\boldmath$i$}}}$. The reader
should nonetheless recall that our chosen representative of each
$[{\mathfrak{g}}_{i},{\mathbb{O}}_{i}]$ is selected to be of the form
$({\mathfrak{sp}}_{\text{\boldmath$i$},2n_{\text{\boldmath$i$}}},\\{0\\}\times{\mathbb{O}}_{0})$
so that we may use the results of §5.1.

######  Proposition 5.4.

For $i=0,...,m$, we have
$\Phi({\mathcal{E}}({\mathfrak{g}}_{i},e_{i}))\nsubseteq\bigcup_{j\neq
i}\Phi({\mathcal{E}}({\mathfrak{g}}_{j},e_{j}))$.

###### Proof.

For each $i$, denote by $W_{i}$ the Weyl group associated to $(G_{i},T)$.
Proceeding as in §5.1, we find $\eta_{i}\in{\mathfrak{t}}^{*}$ and write
${\mathcal{E}}({\mathfrak{g}}_{i},e_{i})=\\{{\mathbb{C}}_{\eta_{i}}^{\lambda}\mid\lambda\in{\mathfrak{z}}({\mathfrak{g}}_{i})^{*}\\}$.
We pick $\mu_{i}\in{\mathfrak{t}}^{*}$ as in §5.1 so that
$\operatorname{ch}_{i}(I_{\eta_{i}}^{0})=W_{i}\bullet\mu_{i}$ (where
$\operatorname{ch}_{i}$ denotes the obvious central character map); as before,
$\operatorname{ch}_{i}(I_{\eta_{i}}^{\lambda})=W_{i}\bullet(\mu_{i}+\lambda)$
for $\lambda\in{\mathfrak{z}}({\mathfrak{g}}_{i})^{*}$.

Suppose now for a contradiction that
$\Phi({\mathcal{E}}({\mathfrak{g}}_{i},e_{i}))\subseteq\bigcup_{j\neq
i}\Phi({\mathcal{E}}({\mathfrak{g}}_{j},e_{j})$. Since Diagram (4.11) commutes
we deduce that

$\operatorname{ch}\operatorname{Ind}_{{\mathfrak{p}}_{i}}^{{\mathfrak{g}}}\mathscr{S}_{i}({\mathcal{E}}({\mathfrak{g}}_{i},e_{i}))\subseteq\bigcup_{j\neq
i}\operatorname{ch}\operatorname{Ind}_{{\mathfrak{p}}_{j}}^{{\mathfrak{g}}}\mathscr{S}_{j}({\mathcal{E}}({\mathfrak{g}}_{j},e_{j}))\subseteq{\mathfrak{t}}^{*}/(W,\bullet).$

Making the notation
${\mathfrak{z}}_{k}:=\operatorname{Ann}_{{\mathfrak{t}}^{*}}([{\mathfrak{g}}_{k},{\mathfrak{g}}_{k}])$
(which we easily identify with ${\mathfrak{z}}({\mathfrak{g}}_{k})^{*}$) and
using Lemma 5.1 we deduce that

$(\mu_{i}+{\mathfrak{z}}_{i})/(W,\bullet)\subseteq\bigcup_{j\neq
i}(\mu_{j}+{\mathfrak{z}}_{j})/(W,\bullet).$

By Lemma 5.3 we deduce that ${\mathfrak{z}}_{i}$ is $W$-conjugate to a
subspace of ${\mathfrak{z}}_{j}$ for some $j\neq i$. This is equivalent to
saying that ${\mathfrak{z}}({\mathfrak{g}}_{i})$ is $W$-conjugate to a
subspace of ${\mathfrak{z}}({\mathfrak{g}}_{j})$ for some $j\neq i$ which, in
turn, implies that ${\mathfrak{g}}_{j}$ is $G$-conjugate to a subset of
${\mathfrak{g}}_{i}$. This contradicts Theorem 3.3, and the contradiction
completes the proof. ∎

Recalling from §2.5 that sheets of ${\mathfrak{g}}$ containing ${\mathbb{O}}$
are in bijection with the rigid induction data
$[{\mathfrak{g}}_{i},{\mathbb{O}}_{i}]$, $i=0,\ldots,m$, let us denote by
${\mathcal{S}}_{i}$ the sheet corresponding to
$[{\mathfrak{g}}_{i},{\mathbb{O}}_{i}]$. As in §2.5, we can write
${\mathcal{S}}_{i}=\overline{{\mathcal{D}}}^{{\mathrm{reg}}}({\mathfrak{g}}_{i},{\mathbb{O}}_{i})$.

Now, recall the ${\mathfrak{sl}}_{2}$-triple $(e,h,f)$ from the definition of
the finite $W$-algebra $U({\mathfrak{g}},e)$. The Katsylo variety is defined
as

$e+X:=(e+{\mathfrak{g}}^{f})\cap\bigcup_{i=0}^{m}{\mathcal{S}}_{i}.$

Key to the proof of Theorem 4.1 is the following result on the irreducible
components of $e+X$; given a variety $Y$, we denote by $\operatorname{Comp}Y$
the set of irreducible components of $Y$.

######  Proposition 5.5.

There are dimension preserving bijections

$\operatorname{Comp}{\mathcal{E}}({\mathfrak{g}},e)\overset{\sim}{\longrightarrow}\operatorname{Comp}(e+X)\overset{\sim}{\longrightarrow}\operatorname{Comp}\bigsqcup_{i=0}^{m}{\mathcal{E}}({\mathfrak{g}}_{i},e_{i}).$

###### Proof.

The first bijection is given in [To23, Theorem 1.1].

To establish the second bijection we first observe the following three facts
about $e+X$ which allow us to describe $\operatorname{Comp}(e+X)$:

  1. (i)

The intersections $(e+{\mathfrak{g}}^{f})\cap{\mathcal{S}}_{i}$ are all
irreducible, thanks to [IH05, Theorem 6.2].

  2. (ii)

There are no inclusions between the varieties
$\\{(e+{\mathfrak{g}}^{f})\cap{\mathcal{S}}_{i}\mid i=0,...,m\\}$; this can be
immediately deduced from the fact that
$G\cdot((e+{\mathfrak{g}}^{f})\cap{\mathcal{S}}_{i})={\mathcal{S}}_{i}$ by
[IH05, Lemma 2.8(i)] and that, by definition, no sheet can admit an inclusion
into another.

  3. (iii)

The dimension of $(e+{\mathfrak{g}}^{f})\cap{\mathcal{S}}_{i}$ is
$\dim{\mathfrak{z}}({\mathfrak{g}}_{i})$ (see [IH05, Lemma 2.8(2)], for
example).

The result then follows from the observation in (5.3) that, for
$G=\mathrm{Sp}_{2n}$, each ${\mathcal{E}}({\mathfrak{g}}_{i},e_{i})$ is an
affine space of dimension $\dim{\mathfrak{z}}({\mathfrak{g}}_{i})$. ∎

###### Proof of Theorem 4.1.

When $G$ is of type ${\sf A}$ this is Corollary 4.3. For $G$ of type ${\sf
B}$, ${\sf C}$ or ${\sf D}$ we easily reduce to $G=\mathrm{Sp}_{2n}$ or
$\mathrm{SO}_{N}$ and use Proposition 5.4 and Proposition 5.5 to argue as
follows.

We know that $\Phi$ as in (4.11) is a finite morphism. Thus, its image
$\Phi\left(\bigsqcup_{i}{\mathcal{E}}({\mathfrak{g}}_{i},e_{i})\right)\subseteq{\mathcal{E}}({\mathfrak{g}},e)$
is closed and for any irreducible component $Y$ of
$\bigsqcup_{i}{\mathcal{E}}({\mathfrak{g}}_{i},e_{i})$ we have
$\dim\Phi(Y)=\dim Y$. Thanks to Proposition 5.4 and Proposition 5.5 we see
that the multiset of dimensions of irreducible components of
$\Phi\left(\bigsqcup_{i}{\mathcal{E}}({\mathfrak{g}}_{i},e_{i})\right)$ is
equal to the multiset of dimensions of the irreducible components of
${\mathcal{E}}({\mathfrak{g}},e)$. This implies that $\Phi$ is surjective,
completing the proof of Theorem 4.1. ∎

###### Proof of Theorem 1.1.

This follows directly from Theorem 4.1 using the maps $\mathscr{S}$ and
$\mathscr{S}_{\operatorname{rig}}$ (which are surjective onto the sets of
Losev-Premet ideals with the appropriate associated varieties) and the
commutativity of Diagram (4.11). ∎

## References

  * [Bo76] W. Borho, Primitive vollprime Ideale in der Enhüllenden von ${\mathfrak{so}}(5,{\mathbb{C}})$, J. Algebra 43 (1976), 619–654.
  * [Bo81] by same author, Über Schichten halbeinfacher Lie-Algebren, Invent. Math. 65, (1981), 283–317.
  * [BJ01] W. Borho and A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 (2001), 76–167.
  * [BK79] W. Borho and H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), 61–104.
  * [BG18] J. Brown and S. M. Goodwin, On the variety of 1-dimensional representations of finite $W$-algebras in low rank, J. Algebra 511 (2018), 499–515.
  * [CM93] D.H. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York, 1993.
  * [Co74] N. Conze, Algèbres d’opérateurs différentiels et quotients des algèbres enveloppantes, Bull. Soc. Math. France, 102 (1974), 379–415.
  * [EdG09] W. A. de Graaf and A. Elashvili, Induced nilpotent orbits of the simple Lie algebras of exceptional type, Georgian Math. J. 2 (2009), 257–278.
  * [Hu08] J. E. Humphreys, Representations of Semisimple Lie algebras in the BGG category $\mathcal{O}$, Graduate Studies in Mathematics 94, American Mathematical Society, 2008.
  * [IH05] A. Im Hof, The Sheets of a Classical Lie Algebra, PhD thesis, Universität Basel, 2005, available at http://edoc.unibas.ch
  * [Jo80] A. Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra, II. J. Algebra 65 (1980), 284–306.
  * [Jo83] by same author, On the Classification of Primitive Ideals in the Enveloping Algebra of a Semisimple Lie Algebra: Lie Group Representations I, Springer Lecture Notes 1024, 1983.
  * [Jo84] by same authorPrimitive ideals in envloping algebras, Proceedings of the International Congress of Mathematicians, Vol. 1,2 (Warsaw, 1983), PWN, Warsaw, 1984, 403–414.
  * [Jo85] by same author, On the associated variety of a primitive ideal, J. Algebra 93 (1985), 509–523.
  * [Lo10a] I. Losev, Quantized symplectic actions and $W$-algebras, J. Amer. Math. Soc. 23 (2010), 35–59.
  * [Lo10b] by same author, Finite $W$-algebras, Proceedings of the International Congress of Mathematicians. Volume III (Hyperabad 2010), Hindustan Book Agency, New Delhi, 2010, 1281–1307.
  * [Lo11] by same author, 1-dimensional representations and parabolic induction for $W$-algebras, Adv. Math. 226 (2011), 4841–4883.
  * [Lo15] by same author, Dimensions of irreducible modules over $W$-algebras and Goldie ranks, Invent. Math. 200 (2015), 849–923.
  * [Lo22] by same author, Deformations of symplectic singularities and orbit method for semisimple Lie algebras, Selecta Math. 28 (2022), Art. no. 30, 52pp.
  * [LMM21] I. Losev, D. Matvieievskyi and L. Mason-Brown, Unipotent ideals and Harish-Chandra bimodules, arXiv:2108.03453, 2021.
  * [LP21] I. Losev, I. Panin, Goldie ranks of primitive ideals and indexes of equivariant Azumaya algebras, Mosc. Math. J. 21 (2021), 383–399.
  * [LS79] G. Lusztig and N. Spaltenstein, Induced unipotent classes, J. London Math. Soc. 19 (1979), 41–52.
  * [Mœ87] C. Mœglin, Idéaux complètement premiers de l’alg‘ebre enveloppante de ${\mathfrak{gl}}_{n}({\mathbb{C}})$, J. Algebra 106 (1987) 287–366.
  * [Pr02] A. Premet, Special transverse slices and their enveloping algebras, Adv. Math. 170 (2002), 1–55. (with an Appendix by S. Skryabin).
  * [Pr11] by same author, Enveloping algebras of Slodowy slices and Goldie rank, Transform. Groups 16 (2011), 857–888.
  * [Pr14] by same author, Multiplicity-free primitive ideals associated with rigid nilpotent orbits, Transform. Groups 19 (2014), 569–641.
  * [PS22] A. Premet and D. I. Stewart, The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits, arXiv:2211.14581, 2022.
  * [PT14] A. Premet and L. Topley, Derived subalgebras of centralisers and finite $W$-algebras, Compos. Math. 150 (2014), 1485–1548.
  * [To23] L. Topley, One dimensional representations of finite $W$-algebras, Dirac reduction and the orbit method, To appear in Invent. Math. (2023).