# Unipotent Representations of Complex Groups and Extended Sommers Duality

Lucas Mason-Brown, Dmytro Matvieievskyi, and Shilin Yu

###### Abstract.

Let $G$ be a complex reductive algebraic group. In [LMM21], we have defined a
finite set of irreducible admissible representations of $G$ called _unipotent
representations_ , generalizing the _special unipotent representations_ of
Arthur ([Art83]) and Barbasch-Vogan ([BV85]). These representations are
defined in terms of filtered quantizations of symplectic singularities and are
expected to form the building blocks of the unitary dual of $G$. In this
paper, we provide a description of these representations in terms of the
Langlands dual group $G^{\vee}$. To this end, we construct a duality map $D$
from the set of pairs $(\mathbb{O}^{\vee},\bar{C})$ consisting of a nilpotent
orbit $\mathbb{O}^{\vee}\subset\mathfrak{g}^{\vee}$ and a conjugacy class
$\bar{C}$ in Lusztig’s canonical quotient $\bar{A}(\mathbb{O}^{\vee})$ to the
set of finite covers of nilpotent orbits in $\mathfrak{g}^{*}$.

##  1\. Introduction

Let $G$ be a complex reductive algebraic group with Lie algebra
$\mathfrak{g}$. A _nilpotent cover_ is a finite connected $G$-equivariant
cover of a nilpotent co-adjoint orbit $\mathbb{O}\subset\mathfrak{g}^{*}$.
Write $\mathsf{Cov}(G)$ for the (finite) set of isomorphism classes of
nilpotent covers for $G$. In [LMM21], we attach to each nilpotent cover
$\widetilde{\mathbb{O}}$ a finite set
$\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$ of irreducible unitary
representations of $G$ called _unipotent representations_. Such
representations possess a variety of special properties and are conjectured to
form the building blocks of the unitary dual. They include, as a proper
subset, all special unipotent representations in the sense of Arthur ([Art83])
and Barbasch-Vogan ([BV85]). The purpose of this article is to give a
description of the sets $\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$ in terms
of the Langlands dual group $G^{\vee}$.

The dual-group parameters which appear in our description are called _Lusztig-
Achar data_. A Lusztig-Achar datum for $G^{\vee}$ is a pair
$(\mathbb{O}^{\vee},\bar{C})$ consisting of a nilpotent orbit
$\mathbb{O}^{\vee}\subset\mathfrak{g}^{\vee}$ and a conjugacy class $\bar{C}$
in Lusztig’s canonical quotient $\bar{A}(\mathbb{O}^{\vee})$ of the
$G^{\vee}$-equivariant fundamental group of $\mathbb{O}^{\vee}$, see Section
2.9. Such objects have previously appeared in the work of Achar [Ach03]. Write
$\mathsf{LA}(G^{\vee})$ for the (finite) set of Lusztig-Achar data for
$G^{\vee}$. We will also consider the subset
$\mathsf{LA}^{*}(G^{\vee})\subset\mathsf{LA}(G^{\vee})$ of so-called _special
Lusztig-Achar data_ , defined in [Ach03, Section 3]. This subset includes all
pairs of the form $(\mathbb{O}^{\vee},1)$ (among others).

Choose a Cartan subalgebra $\mathfrak{h}\subset\mathfrak{g}$ and let $W$
denote the Weyl group. Let $\mathfrak{h}_{\mathbb{R}}^{*}$ denote the real
form of the dual space $\mathfrak{h}^{*}$ spanned by the roots in $G$. To each
Lusztig-Achar datum $(\mathbb{O}^{\vee},\bar{C})$, we attach a $W$-invariant
subset $S(\mathbb{O}^{\vee},\bar{C})\subset\mathfrak{h}_{\mathbb{R}}^{*}$. If
$(\mathbb{O}^{\vee},\bar{C})$ is special we show in Theorem 4.8 that there is
a unique minimal-length $W$-orbit $\gamma(\mathbb{O}^{\vee},\bar{C})\subset
S(\mathbb{O}^{\vee},\bar{C})$. This $W$-orbit determines an infinitesimal
character for $U(\mathfrak{g})$ by the Harish-Chandra isomorphism (which we
also denote by $\gamma(\mathbb{O}^{\vee},\bar{C})$), and hence a maximal ideal
$I(\mathbb{O}^{\vee},\bar{C})$ in $U(\mathfrak{g})$. We consider the finite
set

$\displaystyle\mathrm{Unip}_{(\mathbb{O}^{\vee},\bar{C})}(G)=\\{$ irreducible
$G$-equivariant Harish-Chandra $U(\mathfrak{g})$-bimodules
$\displaystyle\text{which are annihilated on both sides by
}I(\mathbb{O}^{\vee},\bar{C})\\}.$

The notation suggests a relationship between
$\mathrm{Unip}_{(\mathbb{O}^{\vee},\bar{C})}(G)$ and
$\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$. To make this relationship
precise, we define in Section 4 a natural duality map

$D:\mathsf{LA}^{*}(G^{\vee})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Cov}(G)$

This map generalizes the duality maps of Barbasch-Vogan-Lusztig-Spaltenstein
([BV85]), Sommers ([Som01]), and Mason-Brown-Matvieievskyi-Losev ([LMM21]),
and is equivalent (under some nontrivial identifications) to the duality map
of Achar ([Ach03]). It enjoys various nice properties. For example, it is
injective, maps distinguished Lusztig-Achar data to birationally rigid covers,
and intertwines saturation of Lusztig-Achar data with birational induction of
nilpotent covers. Our first main result is the following.

######  Theorem 1.1 (See Theorem 4.8 below).

Let $(\mathbb{O}^{\vee},\bar{C})\in\mathsf{LA}^{*}(G^{\vee})$ and let
$\widetilde{\mathbb{O}}=D(\mathbb{O}^{\vee},\bar{C})$. Then there is an
equality

$\mathrm{Unip}_{(\mathbb{O}^{\vee},\bar{C})}(G)=\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G).$

Our second main result gives a parameterization of the set
$\mathrm{Unip}_{(\mathbb{O}^{\vee},\bar{C})}(G^{\vee})$ in terms of the
Langlands dual group. For each special Lusztig-Achar datum
$(\mathbb{O}^{\vee},\bar{C})\in\mathsf{LA}^{*}(G^{\vee})$, define

$\gamma=\gamma(\mathbb{O}^{\vee},\bar{C}),\quad s=\exp(2\pi i\gamma),\quad
R^{\vee}=Z_{G^{\vee}}(s)^{\circ},\quad L^{\vee}=Z_{G^{\vee}}(\gamma).$

Note that $R^{\vee}$ is a pseudo-Levi subgroup of $G^{\vee}$ and $L^{\vee}$ is
a Levi subgroup of $R^{\vee}$. Hence, we can consider the Richardson orbit
$\mathbb{O}_{R^{\vee}}$ for $R^{\vee}$ corresponding to $L^{\vee}$ (this is
the nilpotent orbit for $R^{\vee}$ obtained by Lusztig-Spaltenstein induction
from the $0$ orbit for $L^{\vee}$).

######  Theorem 1.2 (See Corollary 4.16 below).

Assume $G$ is adjoint and let
$(\mathbb{O}^{\vee},\bar{C})\in\mathrm{LA}^{*}(G^{\vee})$. Then there is a
natural bijection

$\mathrm{Unip}_{(\mathbb{O}^{\vee},\bar{C})}(G)\simeq\\{\text{irreducible
representations of }\bar{A}(\mathbb{O}_{R^{\vee}})\\}.$

######  Remark 1.3.

In fact, the representations in
$\mathrm{Unip}_{(\mathbb{O}^{\vee},\bar{C})}(G)$ are the irreducible objects
in the monoidal category
$\operatorname{HC}^{G}(U(\mathfrak{g})/I(\mathbb{O}^{\vee},\bar{C}))$ of
$G$-equivariant Harish-Chandra $U(\mathfrak{g})$-modules annihilated on both
sides by $I(\mathbb{O}^{\vee},\bar{C})$, see Section 3.13. The bijection in
Theorem 1.2 comes from a monoidal equivalence of categories

$\operatorname{HC}^{G}(U(\mathfrak{g})/I(\mathbb{O}^{\vee},\bar{C}))\simeq\bar{A}(\mathbb{O}_{R^{\vee}})\operatorname{-mod}$

We note that the main definitions in this paper (the duality map $D$, the the
set $S(\mathbb{O}^{\vee},\bar{C})\subset\mathfrak{h}_{\mathbb{R}}^{*}$, and
the pair $(R^{\vee},\mathbb{O}_{R^{\vee}})$) are formulated in a canonical
(i.e. case-free) manner, although many of our proofs require case-by-case
anaylsis.

To deduce Theorem 1.2, we were forced to prove several new results on conical
symplectic singularities and nilpotent covers, which may be of independent
interest (see Section 3, in particular Sections 3.10-3.14). The main such
result is Theorem 3.48 — it states that birational induction of nilpotent
covers takes maximal covers to maximal covers (in the sense of [LMM21, Section
6.5], see also Lemma 2.5).

Now fix a nilpotent adjoint orbit
$\mathbb{O}^{\vee}\subset\mathfrak{g}^{\vee}$. Choose an
$\mathfrak{sl}(2)$-triple $(e^{\vee},f^{\vee},h^{\vee})$ with
$e^{\vee}\in\mathbb{O}^{\vee}$. The corresponding homomorphism
$\psi_{\mathbb{O}^{\vee}}:SL(2)\stackrel{{\scriptstyle}}{{\to}}G^{\vee}$ is a
unipotent Arthur parameter for the complex group $G$ and the corresponding
Arthur packet coincides with the set
$\mathrm{Unip}_{(\mathbb{O}^{\vee},1)}(G)$. So in fact Theorem 1.2 provides,
as a special case, a parameterization of the elements of the unipotent Arthur
packets for a complex reductive group (such a parameterization was previously
obtained in [BV85] and [Won23] only in the special case when
$\mathbb{O}^{\vee}$ is special).

We conclude with a few remarks regarding symplectic duality, which appears to
be connected, in somewhat mysterious ways, to the main results of this paper.
In [LMM21, Section 9.3] it was conjectured that the nilpotent Slodowy slice
$S^{\vee}$ to the nilpotent orbit
$\mathbb{O}^{\vee}\subset\mathfrak{g}^{\vee}$ is symplectically dual to the
affinization of the nilpotent cover $D(\mathbb{O}^{\vee},1)$. In ongoing work,
Finkelberg, Hanany, and Nakajima produce for each nilpotent orbit
$\mathbb{O}^{\vee}$ in $\mathfrak{so}(2n)$ or $\mathfrak{sp}(2n)$ an ortho-
symplectic quiver gauge theory $Q$ with Higgs branch $S^{\vee}$ and Coulomb
branch isomorphic to the affinization of a certain cover of the special
nilpotent orbit $d(\mathbb{O}^{\vee})$ with Galois group
$\bar{A}(\mathbb{O}^{\vee})$. It is reasonable to conjecture that this
nilpotent cover is isomorphic to $D(\mathbb{O}^{\vee},1)$. The results of this
paper offer strong evidence for this. Indeed, it follows from Theorem 4.14
that $D(\mathbb{O}^{\vee},1)$ is a Galois cover of $d(\mathbb{O}^{\vee})$ with
Galois group $\bar{A}(\mathbb{O}^{\vee})$.

To each conjugacy class $\bar{C}$ in $\bar{A}(\mathbb{O}^{\vee})$ we can
associate a certain finite group $\Pi$ depending on $\bar{C}$ which acts on
both $S^{\vee}$ and $\operatorname{Spec}(\mathbb{C}[D(\mathbb{O}^{\vee},1)])$
by graded Poisson automorphisms. Based on the observations in [KP89], we
expect that the variety of fixed points
$\operatorname{Spec}(\mathbb{C}[D(\mathbb{O}^{\vee},1)])^{\Pi}$ is identified
with $\operatorname{Spec}(\mathbb{C}[D(\mathbb{O}^{\vee},\bar{C})])$. We
believe that the assignment

$(S^{\vee},\Pi)\mapsto\operatorname{Spec}(\mathbb{C}[D(\mathbb{O}^{\vee},\bar{C})])$

should be regarded as a special case of a (still highly conjectural)
_equivariant_ version of symplectic duality. This topic will be explored in a
future paper.

###  1.1. Structure of paper

In Section 2, we recall some preliminaries related to nilpotent orbits and Lie
theory, including (birational) induction, Lusztig’s canonical quotient,
primitive ideals in the unversal enveloping algebra, and Sommers duality. In
Section 3, we recall some preliminaries related to symplectic singularities
and unipotent ideals. This section also includes several new results. In
Section 4 we state our main results on unipotent representations. The proofs
of these results appear in Sections 5-8. The classical cases are proved in
Sections 5 and 6; the exceptional cases are handled in Sections 7 and 8.

###  1.2. Acknowledgments

We would like to thank Jeffrey Adams, Dan Barbasch, Ivan Losev, Hiraku
Nakajima, Alexander Premet, Eric Sommers, and David Vogan for helpful
discussions. The third author is grateful to Chen Jiang and Yoshinori Namikawa
for numerous discussions on birational geometry, and to Binyong Sun for his
hospitality and inspiring discussions during his visits to the Institute for
Advanced Study at Zhejiang University.

The work of S.Yu was partially supported by China NSFC grants (Grants No.
12001453 and 12131018) and Fundamental Research Funds for the Central
Universities (Grants No. 20720200067 and 20720200071).

###### Contents

  1. 1 Introduction
    1. 1.1 Structure of paper
    2. 1.2 Acknowledgments
  2. 2 Preliminaries
    1. 2.1 Saturation
    2. 2.2 Nilpotent covers
    3. 2.3 Induction
    4. 2.4 Birational induction and equivariant fundamental groups
    5. 2.5 Primitive ideals
    6. 2.6 BVLS duality
    7. 2.7 Truncated induction
    8. 2.8 Sommers duality
    9. 2.9 Lusztig’s canonical quotient
    10. 2.10 Lusztig-Achar data
    11. 2.11 Special Lusztig-Achar data
  3. 3 Symplectic singularities and unipotent ideals
    1. 3.1 Poisson deformations
    2. 3.2 Filtered quantizations
    3. 3.3 Symplectic singularities and $\mathbb{Q}$-factorial terminalizations
    4. 3.4 The Namikawa space and Weyl group
    5. 3.5 Filtered quantizations of conical symplectic singularities
    6. 3.6 $\mathbb{Q}$-factorial terminalizations of nilpotent covers
    7. 3.7 Universal Poisson deformations of conical symplectic singularities
    8. 3.8 Universal Poisson deformations of nilpotent covers
    9. 3.9 Graded Poisson automorphisms of conical symplectic singularities
    10. 3.10 The extended Namikawa Weyl groups of nilpotent covers
    11. 3.11 Extended Namikawa Weyl group vs parabolic induction
    12. 3.12 Unipotent ideals
    13. 3.13 Unipotent bimodules
    14. 3.14 Birational induction preserves maximal covers
  4. 4 Main results
  5. 5 Combinatorics in classical types
    1. 5.1 Nilpotent orbits
    2. 5.2 The group $A(\mathbb{O})$
    3. 5.3 Levi subalgebras
    4. 5.4 Maximal pseudo-Levi subalgebras
    5. 5.5 Saturation of nilpotent orbits
    6. 5.6 (Birational) induction of nilpotent orbits
    7. 5.7 Birationally rigid covers
    8. 5.8 BVLS duality
    9. 5.9 Lusztig-Achar data
    10. 5.10 Special Lusztig-Achar data
    11. 5.11 Saturation of conjugacy data
    12. 5.12 Saturation of Lusztig-Achar data
    13. 5.13 Sommers duality
    14. 5.14 Block decompositions
    15. 5.15 Unipotent infinitesimal characters
  6. 6 Proofs of main results in classical types
    1. 6.1 Proof of Proposition 4.3
    2. 6.2 Proof of Theorem 4.8
    3. 6.3 Proof of Proposition 4.11
    4. 6.4 Proof of Theorem 4.14
  7. 7 Proofs of main results in exceptional types
    1. 7.1 Proof of Proposition 4.3 and Theorem 4.8
    2. 7.2 Proof of Proposition 4.11 and Theorem 4.14
      1. 7.2.1 Computations of $\Gamma$
  8. 8 Tables

##  2\. Preliminaries

Let $G$ be a complex connected reductive algebraic group with Lie algebra
$\mathfrak{g}$. A _nilpotent orbit_ for $G$ is a co-adjoint orbit
$\mathbb{O}\subset\mathfrak{g}^{*}$ which is stable under scaling. Let

$\mathsf{Orb}(G):=\\{\text{nilpotent orbits
$\mathbb{O}\subset\mathfrak{g}^{*}$}\\}$

It is well-known that $\mathsf{Orb}(G)$ is finite and independent of isogeny.
Let $\mathcal{N}\subset\mathfrak{g}^{*}$ denote the union of all nilpotent
orbits in $\mathfrak{g}^{*}$.

For each $\mathbb{O}\in\mathsf{Orb}(G)$, choose an element $e\in\mathbb{O}$,
and let $A(\mathbb{O})$ denote the (finite) component group of the centralizer
of $e$ in $G$. Note that $A(\mathbb{O})$ is independent (up to conjugacy) of
the choice of $e$ in $\mathbb{O}$.

A _conjugacy datum_ for $G$ is a pair $(\mathbb{O},C)$ consisting of a
nilpotent orbit $\mathbb{O}\in\mathsf{Orb}(G)$ and a conjugacy class $C$ in
$A(\mathbb{O})$. Let

$\mathsf{Conj}(G):=\\{\text{conjugacy data $(\mathbb{O},C)$ for $G$}\\}$

Since $\mathsf{Orb}(G)$ is finite and $A(\mathbb{O})$ is finite (for each
$\mathbb{O}\in\mathsf{Orb}(G)$), $\mathsf{Conj}(G)$ is finite.

A _pseudo-Levi subgroup_ of $G$ is the identity component of the centralizer
of a semisimple element $s\in G$. A _McNinch-Sommers datum_ for $G$ is a
triple $(M,tZ^{\circ},\mathbb{O}_{M})$ consisting of a pseudo-Levi subgroup
$M\subset G$, a coset $tZ^{\circ}$ in the component group $Z/Z^{\circ}$ of
$Z=Z(M)$, and a nilpotent orbit $\mathbb{O}_{M}\in\mathsf{Orb}(M)$ such that
$M=Z_{G}(tZ^{\circ})^{\circ}$. Note that $G$ acts by conjugation on the set of
McNinch-Sommers data. Let

$\mathsf{MS}(G):=\\{\text{McNinch-Sommers data $(M,tZ^{\circ},\mathbb{O}_{M})$
for $G$}\\}/G.$

###  2.1. Saturation

Suppose $L\subset G$ is a Levi subgroup. If
$\mathbb{O}_{L}\in\mathsf{Orb}(L)$, then
$G\cdot\mathbb{O}_{L}\in\mathsf{Orb}(G)$. This defines a map

(2.1.1)
$\mathrm{Sat}^{G}_{L}:\mathsf{Orb}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G),\qquad\mathrm{Sat}^{G}_{L}\mathbb{O}_{L}=G\cdot\mathbb{O}_{L}.$

Now choose $e\in\mathbb{O}_{L}$. The inclusion $Z_{L}(e)\subset Z_{G}(e)$
induces a group homomorphism

(2.1.2)
$\iota:A(\mathbb{O}_{L})\stackrel{{\scriptstyle}}{{\to}}A(\mathrm{Sat}^{G}_{L}\mathbb{O}_{L}).$

(in fact, $\iota$ is injective, but we will not use this fact). Thus we get a
map

(2.1.3)
$\mathrm{Sat}^{G}_{L}:\mathsf{Conj}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Conj}(G),\qquad\mathrm{Sat}^{G}_{L}(\mathbb{O}_{L},C_{L})=(\mathrm{Sat}^{G}_{L}\mathbb{O}_{L},\iota(C_{L})).$

Finally, suppose $(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(L)$. Since
$Z_{G}(tZ^{\circ})\subseteq Z_{G}(Z^{\circ})\subseteq Z_{G}(Z(L)^{\circ})=L$,
we have $Z_{G}(tZ^{\circ})=Z_{L}(tZ^{\circ})^{\circ}=M$. So there is a
tautological map

$\operatorname{Sat}^{G}_{L}:\mathsf{MS}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{MS}(G),\qquad\operatorname{Sat}^{G}_{L}(M,tZ^{\circ},\mathbb{O}_{M})=(M,tZ^{\circ},\mathbb{O}_{M})$

A nilpotent orbit (resp. conjugacy datum, resp. McNinch-Sommers datum) is
_distinguished_ if cannot be obtained by saturation from a proper Levi
subgroup. Let $\mathfrak{z}(\mathfrak{a})$ denote the center of a Lie algebra
$\mathfrak{a}$.

######  Lemma 2.1.

Let $(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(G)$. Then the following are
equivalent:

  * (i)

$(M,tZ^{\circ},\mathbb{O}_{M})$ is distinguished.

  * (ii)

$M$ is not contained in a proper Levi subgroup of $G$.

  * (iii)

$\mathfrak{z}(\mathfrak{g})=\mathfrak{z}(\mathfrak{m})$.

  * (iv)

$M$ is of maximal semisimple rank.

###### Proof.

Clearly (ii) implies (i). To see that (i) implies (ii), suppose that $M$ is
contained in a proper Levi subgroup $L$ of $G$. Then
$Z_{L}(tZ^{\circ})^{\circ}=Z_{G}(tZ^{\circ})^{\circ}\cap L=M\cap L=M$. So
$(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(L)$, i.e.
$(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(G)$ is not distinguished.

The equivalence of (ii) and (iii) is an immediate consequence of the following
well-known facts: if $L$ is a Levi subgroup of $G$, then
$L=Z_{G}(\mathfrak{z}(\mathfrak{l}))$ and $L\neq G$ if and only if
$\mathfrak{z}(\mathfrak{l})\neq\mathfrak{z}(\mathfrak{g})$. (iii) and (iv) are
equivalent by definition. ∎

We say that a McNinch-Sommers datum $(M,tZ^{\circ},\mathbb{O}_{M})$ is _large_
if $\mathbb{O}_{M}$ is distinguished. Denote the set of conjugacy classes of
large McNinch-Sommers data by $\mathsf{MS}^{large}(G)$. It is clear that
largeness is preserved under saturation. Let
$(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(G)$. Choose $e\in\mathbb{O}_{M}$.
Then $Z^{\circ}\subset Z_{G}(e)^{\circ}$ and $t\in Z_{G}(e)$. Thus, we get a
map

$\pi:\mathsf{MS}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Conj}(G),\qquad\pi(M,tZ^{\circ},\mathbb{O}_{M})=(G\cdot
e,tZ_{G}(e)^{\circ}).$

We will sometimes write $\pi^{G}$ instead of $\pi$ when we wish to emphasize
the dependence on $G$.

######  Lemma 2.2.

The following are true

  * (i)

The restriction of $\pi$ to $\mathsf{MS}^{large}(G)$ is a bijection onto
$\mathsf{Conj}(G)$.

  * (ii)

If $L\subset G$ is a Levi subgroup, then the following diagram commutes

${\mathsf{MS}(L)}$${\mathsf{MS}(G)}$${\mathsf{Conj}(L)}$${\mathsf{Conj}(G)}$$\scriptstyle{\pi^{L}}$$\scriptstyle{\mathrm{Sat}^{G}_{L}}$$\scriptstyle{\pi^{G}}$$\scriptstyle{\mathrm{Sat}^{G}_{L}}$

  * (iii)

If $(\mathbb{O},C)\in\mathsf{Conj}(G)$ is distinguished, then
$\pi^{-1}(\mathbb{O},C)$ consists of large, distinguished McNinch-Sommers
data.

###### Proof.

(i) is the content of [MS03, Theorem 1]. (ii) is immediate from the
definitions. For (iii), suppose $(\mathbb{O},C)\in\mathsf{Conj}(G)$ is
distinguished. It follows from (ii) that $\pi^{-1}(\mathbb{O},C)$ consists of
distinguished McNinch-Sommers data. Suppose
$(M,tZ(M)^{\circ},\mathbb{O}_{M})\in\pi^{-1}(\mathbb{O},C)$ is not large, i.e.
$\mathbb{O}_{M}$ is not distinguished. Then there is a proper Levi subgroup
$L\subset M$ and a nilpotent orbit $\mathbb{O}_{L}\in\mathsf{Orb}(L)$ such
that $\mathbb{O}_{M}=\operatorname{Sat}^{G}_{L}\mathbb{O}_{L}$. A Levi
subgroup of a pseudo-Levi subgroup of $G$ is a pseudo-Levi subgroup of $G$. So
$L$ is a pseudo-Levi subgroup of $G$. Since $L\subset M$, there are inclusions
$Z(M)\subset Z(L)$ and $Z(M)^{\circ}\subset Z(L)^{\circ}$. Hence,
$tZ(M)^{\circ}$ determines a coset $tZ(L)^{\circ}$ in $Z(L)/Z(L)^{\circ}$.
Since $tZ(M)^{\circ}\subseteq tZ(L)^{\circ}$, we have

$Z_{G}(tZ(L)^{\circ})^{\circ}\subseteq Z_{G}(tZ(M)^{\circ})^{\circ}=M$

Also

$Z_{G}(tZ(L)^{\circ})^{\circ}\subseteq Z_{G}(Z(L)^{\circ})$

Thus,

$Z_{G}(tZ(L)^{\circ})^{\circ}\subseteqq Z_{M}(Z(L)^{\circ})=L$

On the other hand, clearly $L\subseteq Z_{G}(tZ(L)^{\circ})^{\circ}$. So in
fact, $L=Z_{G}(tZ(L)^{\circ})^{\circ}$, i.e.
$(L,tZ(L)^{\circ},\mathbb{O}_{L})\in\mathsf{MS}(G)$. By the definition of
$\pi$, we have $\pi(L,tZ(L)^{\circ},\mathbb{O}_{L})$, so
$(L,tZ(L)^{\circ},\mathbb{O}_{L})$ must be distinguished by the argument
above. And yet, since $L\subset M$ is a proper Levi subgroup, we have
$\mathfrak{z}(\mathfrak{g})\subseteq\mathfrak{z}(\mathfrak{m})\subsetneq\mathfrak{z}(\mathfrak{l})$,
so $(L,tZ(L)^{\circ},\mathbb{O}_{L})$ is not distinguished by Lemma 2.1. This
is a contradiction. ∎

Write

$\displaystyle\mathsf{Orb}_{0}(G)$
$\displaystyle:=\\{(L,\mathbb{O}_{L})\mid\mathbb{O}_{L}\in\mathsf{Orb}(L)\text{
distinguished}\\}/G$ $\displaystyle\mathsf{Conj}_{0}(G)$
$\displaystyle:=\\{(L,(\mathbb{O}_{L},C_{L}))\mid(\mathbb{O}_{L},C_{L})\in\mathsf{Conj}(L)\text{
distinguished}\\}/G$ $\displaystyle\mathsf{MS}_{0}(G)$
$\displaystyle:=\\{(L,(M,tZ^{\circ},\mathbb{O}_{M}))\mid(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(L)\text{
distinguished}\\}/G$ $\displaystyle\mathsf{MS}^{large}_{0}(G)$
$\displaystyle=\\{(L,(M,tZ^{\circ},\mathbb{O}_{M}))\mid(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}^{large}(L)\text{
distinguished}\\}/G$

where $L$ runs over all Levi subgroups of $G$. Saturation gives rise to
surjective maps

$\mathsf{Orb}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G),\quad\mathsf{Conj}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Conj}(G),\quad\mathsf{MS}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{MS}(G),\quad\mathsf{MS}^{large}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{MS}^{large}(G)$

At the risk of abusing notation, we denote all four maps by
‘$\operatorname{Sat}$’.

######  Proposition 2.3.

The maps

$\mathsf{Orb}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G),\quad\mathsf{Conj}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Conj}(G),\quad\mathsf{MS}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{MS}(G),\quad\mathsf{MS}^{large}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{MS}^{large}(G)$

are bijections.

###### Proof.

The assertion for $\mathsf{Orb}(G)$ is the classical Bala-Carter theorem, see
[CM93, Theorem 8.2.12] for a proof. By Lemma 2.1, a McNinch-Sommers datum
$(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(G)$ is distinguished if and only
if $\mathfrak{z}(\mathfrak{g})=\mathfrak{z}(\mathfrak{m})$. On the other hand,
there is a unique Levi subgroup $L\subset G$ containing $M$ such that
$\mathfrak{z}(\mathfrak{m})=\mathfrak{z}(\mathfrak{l})$, namely
$L:=Z_{G}(\mathfrak{z}(\mathfrak{m}))$. This proves this assertion for both
$\mathsf{MS}(G)$ and $\mathsf{MS}^{large}(G)$. The assertion for
$\mathsf{Conj}(G)$ follows from the assertion for $\mathsf{MS}^{large}(G)$ and
Lemma 2.2. ∎

###  2.2. Nilpotent covers

A _nilpotent cover_ for $G$ is a finite étale $G$-equivariant cover of a
nilpotent co-adjoint $G$-orbit. A _morphism of nilpotent covers_ is a finite
étale map
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$
which intertwines the covering maps
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ and
$\widehat{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ (any such map
is automatically $G$-equivariant, see Example 3.8). Let
$\operatorname{Aut}(\widetilde{\mathbb{O}},\mathbb{O})$ denote the set of
invertible endomorphisms of
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$. If we fix
a morphism of covers
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$,
we can similarly define
$\operatorname{Aut}(\widetilde{\mathbb{O}},\widehat{\mathbb{O}})$. In this
case, we call
$\operatorname{Aut}(\widetilde{\mathbb{O}},\widehat{\mathbb{O}})$ the _Galois
group_ of the cover
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$.
We say that the moprhism
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$
is _Galois_ if it induces an isomorphism
$\widehat{\mathbb{O}}\simeq\widetilde{\mathbb{O}}/\operatorname{Aut}(\widetilde{\mathbb{O}},\widehat{\mathbb{O}})$.
Let

$\mathsf{Cov}(G):=\\{\text{isomorphism classes of nilpotent covers
$\widetilde{\mathbb{O}}$ for $G$}\\}$

If $\mathbb{O}\in\mathsf{Orb}(G)$, then (isomorphism classes of) nilpotent
covers of $\mathbb{O}$ are in one-to-one correspondence with (conjugacy
classes of) subgroups of $A(\mathbb{O})$. In particular, $\mathsf{Cov}(G)$ is
finite. Occasionally, we will also consider the category $\mathcal{C}ov(G)$
consisting of nilpotent covers for $G$, equipped with morphisms of nilpotent
covers.

Following [LMM21], we will define an equivalence relation $\sim$ on
$\mathsf{Cov}(G)$. Suppose
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$
is a morphism of covers. Then there is an induced map of affine varieties
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}])$.
We say this map is almost étale if it is étale over all $G$-orbits in
$\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}])$ of codimension $2$ (it
is automatically étale over the open $G$-orbit $\widehat{\mathbb{O}}$). We
write $\widetilde{\mathbb{O}}\geqslant\widehat{\mathbb{O}}$ if there exists a
morphism
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$
such that the induced map
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}])$
has this property. This defines a partial order on $\mathsf{Cov}(G)$.

######  Definition 2.4 (Definition 6.5.1, [LMM21]).

Let $\sim$ be the equivalence relation on $\mathsf{Cov}(G)$ defined by taking
the symmetric closure of $\geqslant$. For
$\widetilde{\mathbb{O}}\in\mathsf{Cov}(G)$, let $[\widetilde{\mathbb{O}}]$
denote the equivalence class of $\widetilde{\mathbb{O}}$.

We will need some basic facts about equivalence classes in $\mathsf{Cov}(G)$.

######  Lemma 2.5 (Lemma 6.5.3, [LMM21]).

Let $[\widetilde{\mathbb{O}}]\subset\mathsf{Cov}(G)$ be an equivalence class.
Then the following are true:

  * (i)

$[\widetilde{\mathbb{O}}]$ contains a unique maximal cover
$\widetilde{\mathbb{O}}_{max}$.

  * (ii)

$\widetilde{\mathbb{O}}_{max}$ is Galois over every cover in
$[\widetilde{\mathbb{O}}]$.

###  2.3. Induction

Suppose $L\subset G$ is a Levi subgroup and let
$\mathbb{O}_{L}\in\mathsf{Orb}(L)$. Fix a parabolic subgroup $P\subset G$ with
Levi decomposition $P=LU$. The annihilator of $\mathfrak{p}$ in
$\mathfrak{g}^{*}$ is a $P$-stable subspace
$\mathfrak{p}^{\perp}\subset\mathfrak{g}^{*}$. Form the $G$-equivariant fiber
bundle $G\times^{P}(\overline{\mathbb{O}}_{L}\times\mathfrak{p}^{\perp})$ over
the partial flag variety $G/P$. There is a proper $G$-equivariant map

$\mu:G\times^{P}(\overline{\mathbb{O}}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\mathfrak{g}^{*}\qquad\mu(g,\xi)=\operatorname{Ad}^{*}(g)\xi$

The image of $\mu$ is a closed irreducible $G$-invariant subset of
$\mathcal{N}$, and hence the closure in $\mathfrak{g}^{*}$ of a nilpotent
$G$-orbit, denoted $\mathrm{Ind}^{G}_{L}\mathbb{O}_{L}\in\mathsf{Orb}(G)$. It
is a standard fact that $\operatorname{Ind}^{G}_{L}\mathbb{O}_{L}$ is
independent of the choice of parabolic $P$. The correspondence

$\mathrm{Ind}^{G}_{L}:\mathsf{Orb}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G)$

is called _Lusztig-Spaltenstein_ induction ([LS79]).

Now let $\widetilde{\mathbb{O}}_{L}\in\mathsf{Cov}(L)$ and form the affine
variety $\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])$. There
is an $L$-action on
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])$ (induced from
the $L$-action on $\widetilde{\mathbb{O}}_{L}$) and a finite surjective
$L$-equivariant map
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}_{L}$.
Let $\widetilde{\mu}$ denote the composition

$G\times^{P}(\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}G\times^{P}(\overline{\mathbb{O}}_{L}\times\mathfrak{p}^{\perp})\overset{\mu}{\stackrel{{\scriptstyle}}{{\to}}}\mathfrak{g}^{*}.$

The image of $\widetilde{\mu}$ is the closure of
$\operatorname{Ind}^{G}_{L}\mathbb{O}_{L}$ and the preimage of
$\operatorname{Ind}^{G}_{L}\mathbb{O}_{L}$ is a finite étale $G$-equivariant
cover, denoted
$\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}\in\mathsf{Cov}(G)$. Again,
$\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$ is independent of the choice
of parabolic $P$ (see [LMM21, Proposition 2.4.1(i)]). The correspondence

$\mathrm{Bind}^{G}_{L}:\mathsf{Cov}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Cov}(G)$

is called _birational induction_. We will need several basic facts about
birational induction.

######  Proposition 2.6.

Birational induction has the following properties:

  1. (i)

Suppose
$\widetilde{\mathbb{O}}_{L},\widehat{\mathbb{O}}_{L}\in\mathcal{C}ov(L)$ and
let $\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$,
$\widehat{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widehat{\mathbb{O}}_{L}$. Then any
$L$-equivariant morphism
$p_{L}:\widehat{\mathbb{O}}_{L}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}_{L}$
induces a canonically defined finite $G$-equivariant morphism
$p=\mathrm{Bind}_{L}^{G}(p_{L}):\widehat{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$
with $\deg p=\deg p_{L}$. Therefore birational induction induces a well-
defined functor
$\mathcal{B}ind_{L}^{G}:\mathcal{C}ov(L)\stackrel{{\scriptstyle}}{{\to}}\mathcal{C}ov(G)$,
$\widetilde{\mathbb{O}}_{L}\mapsto\mathrm{Bind}_{L}^{G}\widetilde{\mathbb{O}}_{L}$.

  2. (ii)

Suppose the morphism
$p_{L}:\widehat{\mathbb{O}}_{L}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}_{L}$
in (i) is a finite Galois $L$-equivariant covering. Then the induced covering
$p=\mathrm{Bind}_{L}^{G}(p_{L}):\widehat{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$
is a finite Galois $G$-equivariant covering. Moreover, the induced group
homomorphism

$\operatorname{Aut}(\widehat{\mathbb{O}}_{L},\widetilde{\mathbb{O}}_{L})\xrightarrow{\sim}\operatorname{Aut}(\widehat{\mathbb{O}},\widetilde{\mathbb{O}})$

is an isomorphism.

  3. (iii)

Suppose $M\subset G$ is a Levi subgroup containing $L$. Then there is a
canonical natural equivalence of functors

$\mathcal{B}ind^{G}_{L}\simeq\mathcal{B}ind^{G}_{M}\circ\mathcal{B}ind^{M}_{L}.$

  4. (iv)

For any $\widetilde{\mathbb{O}}_{L}\in\mathsf{Cov}(L)$,
$\deg(\widetilde{\mathbb{O}}_{L}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}_{L})$
divides
$\deg(\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}\stackrel{{\scriptstyle}}{{\to}}\operatorname{Ind}^{G}_{L}\mathbb{O}_{L})$.

###### Proof.

Let
$\widetilde{Y}=G\times^{P}(\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])\times\mathfrak{p}^{\perp})$
and
$\widehat{Y}=G\times^{P}(\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}_{L}])\times\mathfrak{p}^{\perp})$.
Extend the morphism $p_{L}$ to
$p_{L}:\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}_{L}])\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])$.
$p_{L}$ induces a natural finite morphism
$\widehat{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{Y}$ which restricts to
the morphism
$p=\mathrm{Bind}_{L}^{G}(p_{L}):\widehat{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$
as claimed in (i). The equality $\deg p=\deg p_{L}$ is clear by construction.
The independence of $p=\mathrm{Bind}_{L}^{G}(p_{L})$ on the parabolic $P$
follows from the argument of [Los22a, Lemma 4.1].

For (ii), we argue as follows. Every automorphism $\gamma$ of
$\widehat{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{Y}$ restricts to an
automorphism
$\widetilde{\gamma}|_{\widetilde{\mathbb{O}}}\in\operatorname{Aut}(\widehat{\mathbb{O}},\widetilde{\mathbb{O}})$.
Thus, we obtain an injective homomorphism
$\operatorname{Aut}(\widehat{\mathbb{O}}_{L},\widetilde{\mathbb{O}}_{L})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Aut}(\widehat{\mathbb{O}},\widetilde{\mathbb{O}})$.
We wish to show that this homomorphism is surjective. Note that

$|\operatorname{Aut}(\widehat{\mathbb{O}},\widetilde{\mathbb{O}})|\leq\deg
p=\deg
p_{L}=|\operatorname{Aut}(\widehat{\mathbb{O}}_{L},\widetilde{\mathbb{O}}_{L})|$

and equality holds if and only if the cover is Galois. The statement follows.

(iii) and (iv) are (ii) and (iv) of [LMM21, Proposition 2.4.1]. ∎

A nilpotent orbit (resp. nilpotent cover) is _rigid_ (resp. _birationally
rigid_) if it cannot be obtained by induction (resp. birational induction)
from a proper Levi subgroup. Note that if $\mathbb{O}\in\mathsf{Orb}(G)$ is
rigid and $\widetilde{\mathbb{O}}$ covers $\mathbb{O}$, then
$\widetilde{\mathbb{O}}$ is birationally rigid. Let

$\mathsf{Cov}_{0}(G):=\\{(L,\widetilde{\mathbb{O}}_{L})\mid\widetilde{\mathbb{O}}_{L}\in\mathsf{Cov}(L)\text{
is birationally rigid}\\}/G$

where $L$ runs over all Levi subgroups of $G$.

######  Proposition 2.7 (Proposition 2.4.1(iii), [LMM21]).

The map

$\mathrm{Bind}:\mathsf{Cov}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Cov}(G),\qquad\mathrm{Bind}(L,\widetilde{\mathbb{O}}_{L})=\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$

is a bijection.

We will see in Section 3.12 that birational induction preserves the
equivalence relation on nilpotent covers and takes maximal covers to maximal
covers.

We conclude this subsection by describing a large class of orbits which are
birationally induced from $\\{0\\}$. Suppose $\mathbb{O}\in\mathsf{Orb}(G)$.
Using an $\operatorname{Ad}(\mathfrak{g})$-invariant identification
$\mathfrak{g}\simeq\mathfrak{g}^{*}$, we can regard $\mathbb{O}$ as a
nilpotent $G$-orbit in $\mathfrak{g}$. Choose an element $e\in\mathbb{O}$ and
an $\mathfrak{sl}(2)$-triple $(e,f,h)$. The operator $\operatorname{ad}(h)$
defines a $\mathbb{Z}$-grading on $\mathfrak{g}$

$\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}}\mathfrak{g}_{i},\qquad\mathfrak{g}_{i}:=\\{\xi\in\mathfrak{g}\mid\operatorname{ad}(h)(\xi)=i\xi\\}.$

We say that $\mathbb{O}$ is _even_ if $\mathfrak{g}_{i}=0$ for every odd
integer $i$. In any case, we can define a parabolic subalgebra

(2.3.1)
$\mathfrak{p}_{\mathbb{O}}=\mathfrak{l}_{\mathbb{O}}\oplus\mathfrak{n}_{\mathbb{O}},\qquad\mathfrak{l}_{\mathbb{O}}:=\mathfrak{g}_{0},\qquad\mathfrak{n}_{\mathbb{O}}:=\bigoplus_{i\geq
1}\mathfrak{g}_{i}.$

We call $\mathfrak{p}_{\mathbb{O}}$ (resp. $\mathfrak{l}_{\mathbb{O}}$) the
_Jacobson-Morozov_ parabolic (resp. Levi) associated to $\mathbb{O}$. Both
$\mathfrak{p}_{\mathbb{O}}$ and $\mathfrak{l}_{\mathbb{O}}$ are well-defined
up to conjugation by $G$. The following result is well-known. The proof is
contained in [Kos59], see also [CM93, Thm 3.3.1].

######  Proposition 2.8.

Suppose $\mathbb{O}$ is an even nilpotent $G$-orbit. Then

$\mathbb{O}=\mathrm{Bind}^{G}_{L_{\mathbb{O}}}\\{0\\}.$

###  2.4. Birational induction and equivariant fundamental groups

Fix the notation of Section 2.3, e.g. $L$, $P$, $\widetilde{\mathbb{O}}_{L}$,
$\widetilde{\mathbb{O}}$,
$\widetilde{\mu}:G\times^{P}(\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$
and so on. Let $U\subset P$ denote the unipotent radical. In this section, we
will construct a surjective homomorphism
$\phi^{G}_{L}(\widetilde{\mathbb{O}}_{L}):\pi_{1}^{G}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})$
between the equivariant fundamental groups of $\widetilde{\mathbb{O}}$ and
$\widetilde{\mathbb{O}}_{L}$.

Let
$\widetilde{Z}^{0}:=G\times^{P}(\widetilde{\mathbb{O}}_{L}\times\mathfrak{p}^{\perp})$,
a fiber bundle over the partial flag variety $G/P$. The inclusion
$i:\widetilde{\mathbb{O}}=\widetilde{\mu}^{-1}(\mathbb{O})\subset\widetilde{Z}^{0}$
induces a group homomorphism

(2.4.1)
$i_{*}:\pi_{1}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\pi_{1}(\widetilde{Z}^{0}).$

Since $i:\widetilde{\mathbb{O}}\hookrightarrow\widetilde{Z}^{0}$ is an open
embedding of smooth complex manifolds, the complement
$\widetilde{Z}^{0}-i(\widetilde{\mathbb{O}})$ is of real codimension $\geq 2$.
Hence, the homomorphism (2.4.1) is surjective.

Note that $\widetilde{Z}^{0}$ is a vector bundle over the homogeneous space
$G\times^{P}\widetilde{\mathbb{O}}_{L}$ with fiber $\mathfrak{p}^{\perp}$. So
there is a natural isomorphism

$\pi_{1}(\widetilde{Z}^{0})\simeq\pi_{1}(G\times^{P}\widetilde{\mathbb{O}}_{L}).$

If we fix a base point $x\in\widetilde{\mathbb{O}}$, we get a fibration
$G\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$ with fiber $G_{x}$.
This fibration gives rise to an exact sequence of homotopy groups

$\pi_{1}(G)\stackrel{{\scriptstyle}}{{\to}}\pi_{1}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\pi_{0}(G_{x})\stackrel{{\scriptstyle}}{{\to}}1.$

The final (nontrivial) term is isomorphic to
$\pi_{1}^{G}(\widetilde{\mathbb{O}})$.

Similarly, if we fix a base point $(1,y)\in
G\times^{P}\widetilde{\mathbb{O}}_{L}$, we get a fibration
$G\stackrel{{\scriptstyle}}{{\to}}G\times^{P}\widetilde{\mathbb{O}}_{L}$ with
fiber $P_{y}$. This fibration gives rise to an exact sequence of homotopy
groups

$\pi_{1}(G)\stackrel{{\scriptstyle}}{{\to}}\pi_{1}(G\times^{P}\widetilde{\mathbb{O}}_{L})\stackrel{{\scriptstyle}}{{\to}}\pi_{0}(P_{y})\stackrel{{\scriptstyle}}{{\to}}1.$

Since $U\subset P_{y}$, we have $P_{y}=L_{y}\ltimes U$. It follows that
$\pi_{0}(P_{y})\simeq\pi_{0}(L_{y})\simeq\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})$.

Now choose $x\in\widetilde{\mathbb{O}}$ and $y\in\widetilde{\mathbb{O}}_{L}$
such that $x=(1,\bar{x})$, where
$\bar{x}\in\widetilde{\mathbb{O}}_{L}\times\mathfrak{p}^{\perp}$, and
$\bar{x}$ is mapped to $y$ under the projection
$\widetilde{\mathbb{O}}_{L}\times\mathfrak{p}^{\perp}\twoheadrightarrow\widetilde{\mathbb{O}}_{L}$.
Then we have group isomorphisms

$\pi_{0}(L_{\bar{x}})\simeq\pi_{0}(P_{\bar{x}})\simeq\pi_{0}(G_{x})=\pi^{G}_{1}(\widetilde{\mathbb{O}})$

induced by the inclusions $L\subset P\subset G$. There is a commutative
diagram of groups

${\pi_{1}(G)}$${\pi_{1}(\widetilde{\mathbb{O}})}$${\pi_{1}^{G}(\widetilde{\mathbb{O}})}$${1}$${\pi_{1}(G)}$${\pi_{1}(\widetilde{Z}^{0})}$${\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})}$${1}$$\scriptstyle{i_{*}}$

where the fundamental groups $\pi_{1}(G)$, $\pi_{1}(\widetilde{\mathbb{O}})$,
$\pi_{1}(\widetilde{Z}^{0})$, and $\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})$
are defined with respect to the base points $1\in G$,
$x\in\widetilde{\mathbb{O}}$, $x\in\widetilde{Z}^{0}$, and
$y\in\widetilde{\mathbb{O}}_{L}$, respectively. We define

$\phi_{L}^{G}(\widetilde{\mathbb{O}}_{L}):\pi_{1}^{G}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})$

to be the unique homomorphism which makes the above diagram commute. Sometimes
we simply write $\phi$ for $\phi_{L}^{G}(\widetilde{\mathbb{O}}_{L})$ when
there is no ambiguity.

The following lemma about properties of $\phi$ is standard and the proof is
left to the reader.

######  Lemma 2.9.

With the notations above, the following are true:

  * (i)

Let $\widetilde{\mathbb{O}}_{L},\widehat{\mathbb{O}}_{L}\in\mathcal{C}ov(L)$
and let
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$,
$\widehat{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widehat{\mathbb{O}}_{L}$. Let
$p_{L}:\widehat{\mathbb{O}}_{L}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}_{L}$
be an $L$-equivariant covering map, inducing a $G$-equivariant covering map
$p=\mathrm{Bind}_{L}^{G}(p_{L}):\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$
as in 2.6. Then we have the following commutative diagram

${\pi_{1}^{G}(\widetilde{\mathbb{O}})}$${\pi_{1}^{G}(\widehat{\mathbb{O}})}$${\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})}$${\pi_{1}^{L}(\widehat{\mathbb{O}}_{L})}$$\scriptstyle{p_{*}}$$\scriptstyle{\phi}$$\scriptstyle{\phi}$$\scriptstyle{(p_{L})_{*}}$

Moreover, we have
$\phi^{-1}(\pi_{1}^{G}(\widetilde{\mathbb{O}}_{L}))=\pi_{1}^{G}(\widetilde{\mathbb{O}})$.

  * (ii)

Under the group isomorphisms
$\pi_{0}(L_{\bar{x}})\simeq\pi^{G}_{1}(\widetilde{\mathbb{O}})$ and
$\pi_{0}(L_{y})\simeq\pi^{L}_{1}(\widetilde{\mathbb{O}}_{L})$, the map
$\phi:\pi^{G}_{1}(\widehat{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\pi^{L}_{1}(\widetilde{\mathbb{O}}_{L})$
is identified with the map
$\pi_{0}(L_{\bar{x}})\stackrel{{\scriptstyle}}{{\to}}\pi_{0}(L_{y})$ induced
by the natural inclusion $L_{\bar{x}}\subset L_{y}$.

  * (iii)

Suppose $M\subset G$ is a Levi subgroup containing $L$. Let
$\widetilde{\mathbb{O}}_{L}\in\mathsf{Cov}(L)$ and
$\widetilde{\mathbb{O}}_{M}=\mathrm{Bind}_{L}^{M}\widetilde{\mathbb{O}}_{L}$
and
$\widetilde{\mathbb{O}}=\mathrm{Bind}_{M}^{G}\widetilde{\mathbb{O}}_{M}=\mathrm{Bind}_{L}^{G}\widetilde{\mathbb{O}}_{L}$.
Then the map
$\phi_{L}^{G}(\mathbb{O}_{L}):\pi_{1}^{G}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})$
is equal to the composition of
$\phi_{L}^{M}(\widetilde{\mathbb{O}}_{L}):\pi_{1}^{M}(\widetilde{\mathbb{O}}_{M})\stackrel{{\scriptstyle}}{{\to}}\pi_{1}^{L}(\widetilde{\mathbb{O}}_{L})$
and
$\phi_{M}^{G}(\widetilde{\mathbb{O}}_{M}):\pi_{1}^{G}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\pi_{1}^{M}(\widetilde{\mathbb{O}}_{M})$.

###  2.5. Primitive ideals

Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$.
Recall that $U(\mathfrak{g})$ has a natural natural filtration and there is a
$G$-equivariant isomorphism $\operatorname{gr}U(\mathfrak{g})\simeq
S(\mathfrak{g})$. Now let $I\subset U(\mathfrak{g})$ be a two-sided ideal.
Then $\operatorname{gr}(I)$ corresponds under the isomorphism
$\operatorname{gr}U(\mathfrak{g})\simeq S(\mathfrak{g})$ to a $G$-invariant
ideal in $S(\mathfrak{g})$. The _associated variety_ of $I$ is defined to be
the vanishing locus $V(I)\subset\mathfrak{g}^{*}$ of this ideal.

A two-sided ideal $I\subset U(\mathfrak{g})$ is said to be _primitive_ if it
is the annihilator of a simple left $U(\mathfrak{g})$-module. If $I$ is
primitive, then $V(I)$ is irreducible (see [Jos85]) and hence the closure of a
(unique) nilpotent orbit.

By Quillen’s lemma, the intersection of $I$ with the center
$\mathfrak{Z}(\mathfrak{g})$ of $U(\mathfrak{g})$ is the kernel of an algebra
homomorphism
$\mathfrak{Z}(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathbb{C}$, called
the _infinitesimal character_ of $I$. Such homomorphisms are identified via
the Harish-Chandra isomorphism with $W$-orbits on $\mathfrak{h}^{*}$. For each
$\gamma\in\mathfrak{h}^{*}/W$, there is a _unique_ maximal (primitive) ideal
in $U(\mathfrak{g})$ with infinitesimal character $\gamma$, which we will
denote by $J(\gamma)\subset U(\mathfrak{g})$.

###  2.6. BVLS duality

Let $G^{\vee}$ be the Langlands dual group of $G$ and let
$\mathfrak{g}^{\vee}$ be its Lie algebra. If we fix a Cartan subalgebra
$\mathfrak{h}\subset\mathfrak{g}$, we get a Cartan subalgebra
$\mathfrak{h}^{\vee}\subset\mathfrak{g}^{\vee}$ and a canonical identification
$\mathfrak{h}^{\vee}\simeq\mathfrak{h}^{*}$. To each nilpotent
$G^{\vee}$-orbit $\mathbb{O}^{\vee}\in\mathsf{Orb}(G^{\vee})$ we can associate
a maximal ideal in $U(\mathfrak{g})$ in the following manner. First, using an
$G^{\vee}$-invariant isomorphism
$(\mathfrak{g}^{\vee})^{*}\simeq\mathfrak{g}^{\vee}$, we can identify
$\mathbb{O}^{\vee}$ with a nilpotent $G^{\vee}$-orbit in $\mathfrak{g}^{\vee}$
(still denoted $\mathbb{O}^{\vee}$). Next, we choose an
$\mathfrak{sl}(2)$-triple $(e^{\vee},f^{\vee},h^{\vee})$ with
$e^{\vee}\in\mathbb{O}^{\vee}$ and
$h^{\vee}\in\mathfrak{h}^{\vee}\simeq\mathfrak{h}^{*}$. The element
$\frac{1}{2}h^{\vee}$ is well-defined up to the $W$-action on
$\mathfrak{h}^{*}$, and hence determines an infinitesimal character for
$U(\mathfrak{g})$, which we will denote by $\gamma_{\mathbb{O}^{\vee}}$.

######  Definition 2.10.

Let $\mathbb{O}^{\vee}\in\mathsf{Orb}(G^{\vee})$. Then the _special unipotent
ideal_ attached to $\mathbb{O}^{\vee}$ is the unique maximal ideal
$J(\gamma_{\mathbb{O}^{\vee}})\subset U(\mathfrak{g})$ with infinitesimal
character $\gamma_{\mathbb{O}^{\vee}}$.

For each nilpotent orbit $\mathbb{O}^{\vee}\in\mathsf{Orb}(G^{\vee})$, let
$d(\mathbb{O}^{\vee})$ denote the (unique) open $G$-orbit in
$V(J(\gamma_{\mathbb{O}^{\vee}}))$. This defines a map

$d:\mathsf{Orb}(G^{\vee})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G)$

called _Barbasch-Vogan-Lusztig-Spaltenstein (BVLS) duality_. We will sometimes
write $d^{G}$ instead of $d$, when we wish to emphasize the dependence on $G$.

######  Proposition 2.11 (Proposition A2, [BV85]).

The map $d$ enjoys the following properties:

  * (i)

$d$ is order-reversing (with respect to the clsoure orderings on
$\mathsf{Orb}(G)$ and $\mathsf{Orb}(G^{\vee})$).

  * (ii)

$d^{3}=d$.

  * (iii)

If $L\subset G$ is a Levi subgroup, then

$d^{G}\circ\operatorname{Sat}^{G^{\vee}}_{L^{\vee}}=\operatorname{Ind}^{G}_{L}\circ
d^{L}$

An orbit $\mathbb{O}\in\mathsf{Orb}(G)$ is said to be _special_ if it lies in
the image of $d^{G}$. We write $\mathsf{Orb}^{*}(G)$ for the set of special
nilpotent orbits.

###  2.7. Truncated induction

Let $M^{\vee}\subset G^{\vee}$ be a pseudo-Levi subgroup and let $M$ be the
Langlands dual group of $M^{\vee}$. In [Lus84, Chapter 13.3], Lusztig defines
a map called _truncated induction_

$j^{G}_{M}:\mathsf{Orb}^{*}(M)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G)$

This map generalizes Lusztig-Spaltenstein induction in the following sense: if
$M$ is a Levi subgroup of $G$, then $j^{G}_{M}$ coincides with the restriction
of $\operatorname{Ind}^{G}_{M}$ to $\mathsf{Orb}^{*}(M)$.

The definition of $j^{G}_{M}$ is not particularly relevant for the purposes of
this paper, so we will be brief in recalling it. Choose a Cartan subalgebra
$\mathfrak{h}\subset\mathfrak{m}$ and let $W$ (resp. $W_{M}$) denote the Weyl
group of $G$ (resp. $M$). There is a surjective map

(2.7.1) $\\{(\mathbb{O},\psi)\mid\mathbb{O}\in\mathsf{Orb}(G),\
\psi\in\widehat{A(\mathbb{O})}\\}\twoheadrightarrow\widehat{W}\sqcup\\{0\\},\qquad(\mathbb{O},\psi)\mapsto
E_{(\mathbb{O},\psi)}.$

called the _Springer correspondence_ (see [Spr78]). We say that an irreducible
representation $\psi$ of $A(\mathbb{O})$ is of _Springer type_ if
$E_{(\mathbb{O},\psi)}\neq 0$ (the trivial representation of $A(\mathbb{O})$
is always of Springer type). For each irreducible representation $E$ of $W$,
Lusztig defines a nonnegative integer $b_{E}$ called the _b-value_ or _fake
degree_ of $E$. If $\mathbb{O}\in\mathsf{Orb}^{*}(M)$, it is shown in [Lus84,
Chapter 13.3] that $\operatorname{Ind}^{W}_{W_{M}}E_{\mathbb{O},1}$ contains a
unique irreducible subrepresentation $E^{\prime}$ such that
$b_{E^{\prime}}=b_{E_{(\mathbb{O},1)}}$ and
$E^{\prime}=E_{(\mathbb{O}^{\prime},1)}$ for some
$\mathbb{O}^{\prime}\in\mathsf{Orb}(G)$. Then $j^{G}_{M}$ is defined by
$j^{G}_{M}\mathbb{O}=\mathbb{O}^{\prime}$.

###  2.8. Sommers duality

Consider the map

$\underline{d}_{S}:\mathsf{MS}(G^{\vee})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G),\qquad\underline{d}_{S}(M^{\vee},tZ^{\circ},\mathbb{O}_{M})=j^{G}_{M}d(\mathbb{O}_{M^{\vee}})$

_Sommers duality_ is defined to be the composition

$d_{S}:\mathsf{Conj}(G^{\vee})\overset{\pi^{-1}}{\stackrel{{\scriptstyle}}{{\to}}}\mathsf{MS}^{large}(G^{\vee})\overset{\underline{d}_{S}}{\stackrel{{\scriptstyle}}{{\to}}}\mathsf{Orb}(G)$

where
$\pi:\mathsf{MS}^{large}(G^{\vee})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Conj}(G^{\vee})$
is the bijection of Lemma 2.2(i). We will sometimes write $d_{S}^{G}$ instead
of $d_{S}$ when we wish to emphasize the dependence on $G$.

######  Proposition 2.12 (Section 6, [Som01]).

The map
$d_{S}:\mathsf{Conj}(G^{\vee})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G)$
has the following properties:

  * (i)

$d_{S}$ is surjective.

  * (ii)

If $L\subset G$ is a Levi subgroup, then

$d_{S}^{G}\circ\operatorname{Sat}^{G^{\vee}}_{L^{\vee}}=\operatorname{Ind}^{G}_{L}\circ
d_{S}^{L}$

  * (iii)

$d_{S}(\mathbb{O}^{\vee},1)=d(\mathbb{O}^{\vee})$, for every
$\mathbb{O}^{\vee}\in\mathsf{Orb}(G^{\vee})$.

###  2.9. Lusztig’s canonical quotient

For each $\mathbb{O}\in\mathsf{Orb}(G)$, there is a canonically defined
quotient group $\bar{A}(\mathbb{O})$ of $A(\mathbb{O})$, called _Lusztig’s
canonical quotient_. In [Lus84], this quotient is defined in the case when
$\mathbb{O}$ is _special_ , but in fact Lusztig’s definition is valid for
arbitrary $\mathbb{O}$. An alternative description of $\bar{A}(\mathbb{O})$ is
provided in [Som01, Section 5], which we briefly recall here.

For each conjugacy datum $(\mathbb{O},C)\in\mathsf{Conj}(G)$, Sommers defines
a nonnegative integer $\tilde{b}_{(\mathbb{O},C)}$ called the
$\tilde{b}$_-value_ of $(\mathbb{O},C)$ as follows. If
$(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(G)$ is such that
$\pi(M,tZ^{\circ},\mathbb{O}_{M})=(\mathbb{O},C)$, then

$\tilde{b}_{(\mathbb{O},C)}:=\frac{1}{2}(\dim
M-\dim(d^{M}(\mathbb{O}_{M}))-\operatorname{rank}G),$

Equivalently, $\tilde{b}_{(\mathbb{O},C)}$ is the dimension of the Springer
fiber attached to any element in $d^{M}(\mathbb{O}_{M})$. Sommers shows that
$\tilde{b}_{(\mathbb{O},C)}$ only depends on $(\mathbb{O},C)$ (i.e. is
independent of the lift $(M,tZ^{\circ},\mathbb{O}_{M})$) [Som01, Proposition
1]. In fact, $\tilde{b}_{(\mathbb{O},C)}$ coincides with the dimension of the
Springer fibers of the orbit $d_{S}(\mathbb{O},C)$ in $\mathfrak{g}^{\vee}$
(see [Som01, Section 6]). The $\tilde{b}$-values satisfy
$\tilde{b}_{\mathbb{O},C}\geq\tilde{b}_{\mathbb{O},1}$ for any conjugacy class
$C$ of $A(\mathbb{O})$ ([Som01, Proposition 3]).

Let $N^{\prime}$ be the union of all conjugacy classes $C$ in $A(\mathbb{O})$
such that $\tilde{b}_{\mathbb{O},C}=\tilde{b}_{\mathbb{O},1}$. A case-by-case
calculation shows that $N^{\prime}$ is a subgroup of $A(\mathbb{O})$ and
coincides with a subgroup $N$ which Lusztig defines using Springer theory and
generic degrees of irreducible Weyl group representations, so that
$\bar{A}(\mathbb{O})=A(\mathbb{O})/N=A(\mathbb{O})/N^{\prime}$ ([Som01,
Theorem 6]).

###  2.10. Lusztig-Achar data

A _Lusztig-Achar datum_ for $G$ is a pair $(\mathbb{O},\bar{C})$ consisting of
a nilpotent orbit $\mathbb{O}\in\mathsf{Orb}(G)$ and a conjugacy class
$\bar{C}$ in $\bar{A}(\mathbb{O})$. Let

$\mathsf{LA}(G):=\\{\text{Lusztig-Achar data $(\mathbb{O},\bar{C})$ for
$G$}\\}$

Since the groups $\bar{A}(\mathbb{O})$ are independent of isogeny, so is the
set $\mathsf{LA}(G)$. Of course, there is a natural projection
$\mathsf{Conj}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{LA}(G)$. The
following is a consequence of [Ach03, Theorems 5.1,6.1].

######  Lemma 2.13.

Let $\mathbb{O}\in\mathsf{Orb}(G)$ and let $C_{1}$ and $C_{2}$ be conjugacy
classes in $A(\mathbb{O})$. Then the following are equivalent

  * (i)

$d_{S}(\mathbb{O},C_{1})=d_{S}(\mathbb{O},C_{2})$.

  * (ii)

$C_{1}$ and $C_{2}$ have the same image in $\bar{A}(\mathbb{O})$.

######  Corollary 2.14.

The map
$d_{S}:\mathsf{Conj}(G^{\vee})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Orb}(G)$
factors through the projection
$\mathsf{Conj}(G^{\vee})\stackrel{{\scriptstyle}}{{\to}}\mathsf{LA}(G^{\vee})$.

We wish to define a notion of ‘saturation’ for Lusztig-Achar data analogous to
(2.1.3). So let $L\subset G$ be a Levi subgroup and let
$\mathbb{O}_{L}\in\mathsf{Orb}(L)$.

######  Proposition 2.15.

The homomorphism
$\iota:A(\mathbb{O}_{L})\stackrel{{\scriptstyle}}{{\to}}A(\operatorname{Sat}^{G}_{L}\mathbb{O}_{L})$
(cf. (2.1.2)) descends to a homomorphism
$\bar{\iota}:\bar{A}(\mathbb{O}_{L})\stackrel{{\scriptstyle}}{{\to}}\bar{A}(\operatorname{Sat}^{G}_{L}\mathbb{O}_{L})$.

###### Proof.

Let $\mathbb{O}=\operatorname{Sat}^{G}_{L}\mathbb{O}_{L}$. It follows from the
definition of $\tilde{b}$-value and Proposition 2.12(ii) that

$\tilde{b}(\operatorname{Sat}^{G}_{L}(\mathbb{O}_{L},C_{L}))=\tilde{b}(\mathbb{O}_{L},C_{L})$

for any conjugacy class $C_{L}$ in $A(\mathbb{O}_{L})$. Therefore
$\iota(H^{\prime}_{\mathbb{O}_{L}})\subset H^{\prime}_{\mathbb{O}}$ and the
proposition holds. ∎

Using Proposition 2.15, we can define a map

$\mathrm{Sat}^{G}_{L}:\mathsf{LA}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{LA}(G),\qquad\mathrm{Sat}^{G}_{L}(\mathbb{O}_{L},\bar{C}_{L})=(\mathrm{Sat}^{G}_{L}\mathbb{O}_{L},\bar{\iota}(\bar{C}_{L}))$

A Lusztig-Achar datum $(\mathbb{O},\bar{C})$ is _distinguished_ if it cannot
be obtained by saturation from a proper Levi subgroup. Let

$\mathsf{LA}_{0}(G):=\\{(L,(\mathbb{O}_{L},\bar{C}_{L}))\mid(\mathbb{O}_{L},\bar{C}_{L})\text{
is distinguished}\\}/G$

where, as usual, $L$ runs over all Levi subgroups of $G$.

######  Lemma 2.16.

Suppose $Z(G)$ is connected. Then for each $\mathbb{O}\in\mathsf{Orb}(G)$,
there is a subgroup $K(\mathbb{O})\subset A(\mathbb{O})$ such that

  * (i)

The quotient map
$A(\mathbb{O})\stackrel{{\scriptstyle}}{{\to}}\bar{A}(\mathbb{O})$ restricts
to an isomorphism $K(\mathbb{O})\xrightarrow{\sim}\bar{A}(\mathbb{O})$.

  * (ii)

The natural map from conjugacy classes in $K(\mathbb{O})$ to conjugacy classes
in $A(\mathbb{O})$ is injective.

  * (iii)

If $L\subset G$ is a Levi subgroup and $\mathbb{O}_{L}\in\mathsf{Orb}(L)$ such
that $\mathbb{O}=\mathrm{Sat}^{G}_{L}\mathbb{O}_{L}$, then the homomorphism
$\iota$ defined in (2.1.2) maps $K(\mathbb{O}_{L})$ to $K(\mathbb{O})$.

###### Proof.

For $G$ a classical group, subgroups with these properties were exhibited in
[Som01, Section 5] (see also Section 5.12 for the case of distinguished
Lusztig-Achar data). For $G$ an adjoint exceptional group, there are three
possibilities:

  1. (1)

$A(\mathbb{O})\simeq\bar{A}(\mathbb{O})$.

  2. (2)

$A(\mathbb{O})\simeq S_{2}$, $\bar{A}(\mathbb{O})\simeq 1$.

  3. (3)

$G$ is of type $E_{8}$, $\mathbb{O}$ is the distinguished orbit
$E_{8}(b_{6})$, $A(\mathbb{O})\simeq S_{3}$, and $\bar{A}(\mathbb{O})\simeq
S_{2}$.

In the first case, we take $K(\mathbb{O})=A(\mathbb{O})$. In the second case,
we take $K(\mathbb{O})=\\{1\\}$. In the third case, we take $K(\mathbb{O})$ to
be any order $2$ subgroup of $A(\mathbb{O})$. It is an easy exercise to check
that conditions (i), (ii), and (iii) are satisfied for these choices of
$K(\mathbb{O})$. ∎

######  Proposition 2.17.

The map

$\mathrm{Sat}:\mathsf{LA}_{0}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{LA}(G),\qquad\mathrm{Sat}(L,(\mathbb{O}_{L},\bar{C}_{L}))=\operatorname{Sat}^{G}_{L}(\mathbb{O}_{L},\bar{C}_{L})$

is a bijection.

###### Proof.

Since $\mathsf{LA}(G)$ is independent of isogeny, we can assume in the proof
that $Z(G)$ is connected. Let $\bar{C}$ be a conjugacy class in
$\bar{A}(\mathbb{O})$ and let $C^{\prime}$ be the preimage of $\bar{C}$ under
the isomorphism $K(\mathbb{O})\xrightarrow{\sim}\bar{A}(\mathbb{O})$ of Lemma
2.16(i). By Lemma 2.16(ii), there is a unique conjugacy class $C$ in
$A(\mathbb{O})$ such that $C\cap K(\mathbb{O})=C^{\prime}$. Taking
$(\mathbb{O},\bar{C})$ to $(\mathbb{O},C)$ defines an injective map
$\eta:\mathsf{LA}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Conj}(G)$, right-
inverse to the projection
$\mathsf{Conj}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{LA}(G)$.

Now suppose $L\subset G$ is a Levi subgroup. Then $Z(L)$ is connected (see
e.g. [PN21, Lemma 9]) and the following diagrams commute

(2.10.1)
${\mathsf{Conj}(L)}$${\mathsf{Conj}(G)}$${\mathsf{LA}(L)}$${\mathsf{LA}(G)}$$\scriptstyle{\operatorname{Sat}^{G}_{L}}$$\scriptstyle{\operatorname{Sat}^{G}_{L}}$${\mathsf{LA}(L)}$${\mathsf{LA}(G)}$${\mathsf{Conj}(L)}$${\mathsf{Conj}(G)}$$\scriptstyle{\operatorname{Sat}^{G}_{L}}$$\scriptstyle{\eta^{L}}$$\scriptstyle{\eta^{G}}$$\scriptstyle{\operatorname{Sat}^{G}_{L}}$

(the first diagram commutes by definition, and the second by Lemma 2.16(iii)).

Suppose
$(L_{1},(\mathbb{O}_{L_{1}},\bar{C}_{L_{1}})),(L_{2},(\mathbb{O}_{L_{2}},\bar{C}_{L_{2}}))\in\mathsf{LA}_{0}(G)$
such that
$\operatorname{Sat}(\mathbb{O}_{L_{1}},\bar{C}_{L_{1}}))=\operatorname{Sat}(\mathbb{O}_{L_{2}},\bar{C}_{L_{2}}))=(\mathbb{O},\bar{C})$.
Let $(\mathbb{O}_{L_{1}},C_{L_{1}})=\eta(\mathbb{O}_{L_{1}},\bar{C}_{L_{1}})$,
$(\mathbb{O}_{L_{2}},C_{L_{2}})=\eta(\mathbb{O}_{L_{2}},\bar{C}_{L_{2}})$, and
$(\mathbb{O},C)=\eta(\mathbb{O},\bar{C})$. By the first commutative diagram
the conjugacy data $(\mathbb{O}_{L_{1}},C_{L_{1}})$ and
$(\mathbb{O}_{L_{2}},C_{L_{2}})$ are distinguished. By the second commutative
diagram
$\operatorname{Sat}(L_{1},(\mathbb{O}_{L_{1}},C_{L_{1}}))=\operatorname{Sat}(L_{2},(\mathbb{O}_{L_{2}},C_{L_{2}})=(\mathbb{O},C)$.
So by Proposition 2.3, $(L_{1},(\mathbb{O}_{L_{1}},C_{L_{1}}))$ and
$(L_{2},(\mathbb{O}_{L_{2}},C_{L_{2}}))$ are conjugate. Hence,
$(L_{1},(\mathbb{O}_{L_{1}},\bar{C}_{L_{1}}))$ and
$(L_{2},(\mathbb{O}_{L_{2}},\bar{C}_{L_{2}}))$ are conjugate. This completes
the proof. ∎

We conclude by describing a useful parameterization of $\mathsf{LA}(G)$. A
_Sommers datum_ for $G$ is a pair $(M,\mathbb{O}_{M})$ consisting of a pseudo-
Levi subgroup $M\subset G$ and a distinguished nilpotent orbit
$\mathbb{O}_{M}\in\mathsf{Orb}(M)$. Note that $G$ acts by conjugation on the
set of Sommers data. Let

$\mathsf{Som}(G):=\\{\text{Sommers data $(M,\mathbb{O}_{M})$ for $G$}\\}/G$

If $L\subset G$ is a Levi subgroup, there is a tautological map

$\operatorname{Sat}^{G}_{L}:\mathsf{Som}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Som}(G),\qquad\operatorname{Sat}^{G}_{L}(M,\mathbb{O}_{M})=(M,\mathbb{O}_{M}).$

which is clearly compatible with
$\operatorname{Sat}^{G}_{L}:\mathsf{MS}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{MS}(G)$
under the forgetful maps
$\mathsf{MS}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Som}(G)$ and
$\mathsf{MS}(L)\stackrel{{\scriptstyle}}{{\to}}\mathsf{Som}(L)$.

We define an equivalence relation $\sim$ on $\mathsf{Som}(G)$ as follows:
$(M_{1},\mathbb{O}_{M_{1}})\sim(M_{2},\mathbb{O}_{M_{2}})$ if and only if
$\operatorname{Sat}^{G}_{M_{1}}\mathbb{O}_{M_{1}}=\operatorname{Sat}^{G}_{M_{2}}\mathbb{O}_{M_{2}}$
and
$j^{G^{\vee}}_{M_{1}^{\vee}}d(\mathbb{O}_{M_{1}^{\vee}})=j^{G^{\vee}}_{M_{2}^{\vee}}d(\mathbb{O}_{M_{2}^{\vee}})$.

######  Lemma 2.18.

Let $G_{ad}$ denote the adjoint form of $G$. Then the following are true:

  * (i)

The forgetful map

$\mathsf{MS}^{large}(G_{ad})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Som}(G_{ad})=\mathsf{Som}(G)$

is a bijection.

  * (ii)

The fibers of the surjective map

$\mathsf{Som}(G)\xrightarrow{\sim}\mathsf{MS}^{large}(G_{ad})\xrightarrow{\sim}\mathsf{Conj}(G_{ad})\twoheadrightarrow\mathsf{LA}(G_{ad})=\mathsf{LA}(G)$

(here, the first map is the inverse of the forgetful map in (i), the second
map is $\pi^{G_{ad}}$, and the third map is the natural quotient map) are
exactly the equivalence classes in $\mathsf{Som}(G)$. In particular, there is
a natural bijection

$\mathsf{Som}(G)/\sim\,\xrightarrow{\sim}\mathsf{LA}(G).$

  * (iii)

Suppose $(\mathbb{O},\bar{C})\in\mathsf{LA}(G)$ corresponds under the
bijection in (ii) to an equivalence class
$\\{(M_{1},\mathbb{O}_{M_{1}}),...,(M_{k},\mathbb{O}_{M_{k}})\\}\subset\mathsf{Som}(G)$.
Then $(\mathbb{O},\bar{C})$ is distinguished if and only if one (equivalently,
all) of the pseudo-Levi subgroups $M_{1},...,M_{k}$ is of maximal semisimple
rank. In this case, the elements in the equivalence class in $\mathsf{Som}(G)$
correspond bijectively via the bijection
$\mathsf{Som}(G)\xrightarrow{\sim}\mathsf{Conj}(G_{ad})$ in (ii) to the
elements in the preimage of $(\mathbb{O},\bar{C})$ under the projection
$\mathsf{Conj}(G_{ad})\twoheadrightarrow\mathsf{LA}(G)$.

  * (iv)

Suppose $(\mathbb{O},\bar{C})\in\mathsf{LA}(G)$ is distinguished and
$(M,tZ^{\circ},\mathbb{O}_{M})\in\mathsf{MS}(G)$ maps to
$(\mathbb{O},\bar{C})$ under
$\mathsf{MS}(G)\overset{\pi}{\stackrel{{\scriptstyle}}{{\to}}}\mathsf{Conj}(G)\stackrel{{\scriptstyle}}{{\to}}\mathsf{LA}(G)$.
Then $(M,\mathbb{O}_{M})$ belongs to the equivalence class in
$\mathsf{Som}(G)$ corresponding to $(\mathbb{O},\bar{C})$ under the bijection
in (ii).

###### Proof.

(i) is [MS03, Proposition 35]. (ii) is immediate from Lemma 2.13. (iii) and
(iv) follow from part (i) and (iii) of Lemma 2.2 and Lemma 2.1. Indeed, the
first commtative diagram in (2.10.1) implies that any lift $(\mathbb{O},C)$ of
a distinguished pair $(\mathbb{O},\bar{C})$ in $\mathsf{Conj}(G_{ad})$ is also
distinguished. Then Lemma 2.2 (iii) implies that the preimage
$\pi^{-1}(\mathbb{O},C)$ of $(\mathbb{O},C)$ under the map
$\pi:\mathsf{MS}(G_{ad})\stackrel{{\scriptstyle}}{{\to}}\mathsf{Conj}(G_{ad})$
is included in $\mathsf{MS}^{large}(G_{ad})$. Therefore by Lemma 2.2 (i),
$\pi^{-1}(\mathbb{O},C)$ consists of only one element, which corresponds to a
unique element $(M,\mathbb{O}_{M})\in\mathsf{Som}(G)$, where $M$ is of maximal
semisimple rank by Lemma 2.1. ∎

###  2.11. Special Lusztig-Achar data

######  Definition 2.19.

Let $(\mathbb{O},\bar{C})\in\mathsf{LA}(G)$ and let
$\mathbb{O}^{\vee}=d_{S}^{G^{\vee}}(\mathbb{O},\bar{C})$. Following Achar
([Ach03]) we say that $(\mathbb{O},\bar{C})$ is _special_ if there is a
conjugacy class $\bar{C}^{\prime}\subset\bar{A}(\mathbb{O}^{\vee})$ such that

$d_{S}^{G}(\mathbb{O}^{\vee},\bar{C}^{\prime})=\mathbb{O}.$

Let

$\mathsf{LA}^{*}(G):=\\{\text{special Lusztig-Achar data
$(\mathbb{O},\bar{C})$ for $G$}\\}$

and

$\mathsf{LA}^{*}_{0}(G):=\\{(L,(\mathbb{O}_{L},\bar{C}_{L}))\mid(\mathbb{O}_{L},\bar{C}_{L})\text{
is special and distinguished}\\}/G$

where, as usual, $L$ runs over all Levi subgroups of $G$. We will need several
basic facts about special Achar data.

######  Proposition 2.20.

The following are true:

  * (i)

For any orbit $\mathbb{O}\in\mathsf{Orb}(G)$, the Achar datum $(\mathbb{O},1)$
is special.

  * (ii)

The bijection of Proposition 2.17 induces an injection

$\operatorname{Sat}^{-1}:\mathsf{LA}^{*}(G)\hookrightarrow\mathsf{LA}^{*}_{0}(G)$

###### Proof.

(i) is [Ach03, Proposition 2.7]. We proceed to proving (ii). In classical
types all distinguished Achar data are special, see [Ach03, Section 5.2]. So
(ii) is immediate. In exceptional types, a list of (special) Achar data is
given in [Ach03, Section 6]. There are only two non-special distinguished
Achar data in exceptional types, the (unique) nontrivial conjugacy class for
the nilpotent orbit $A_{4}+A_{1}$ in $E_{7}$ and the (unique) nontrivial
conjugacy class for the nilpotent orbit $E_{6}(a_{1})+A_{1}$ in $E_{8}$.
Saturation to $E_{8}$ takes the former to the (unique) nontrivial conjugacy
class for the nilpotent orbit $A_{4}+A_{1}$ in $E_{8}$, which is non-special
by [Ach03, Section 6]. This completes the proof of (ii). ∎

##  3\. Symplectic singularities and unipotent ideals

In this section, we will review some preliminary facts about conical
symplectic singularities and their Poisson deformations and filtered
quantizations. We will also establish several new facts about conical
symplectic singularities and nilpotent covers. The main new result is Theorem
3.48. It states that birational induction takes the maximal cover in an
equivalence class to the maximal cover in an equivalence class. This result is
essential for our parameterization of unipotent representations (Theorem
4.14).

###  3.1. Poisson deformations

In this subsection, we will recall the definitions of (graded, formal) Poisson
deformations of (graded) Poisson varieties.

A _Poisson scheme_ is a scheme $X$ equipped with a Poisson bracket on its
structure sheaf $\mathcal{O}_{X}$. If $S$ is an affine scheme, a _Poisson
$S$-scheme_ is a scheme $X$ equipped with a morphism
$f:X\stackrel{{\scriptstyle}}{{\to}}S$ and a $f^{-1}\mathcal{O}_{S}$-linear
Poisson bracket on $\mathcal{O}_{X}$. These definitions have obvious formal
counterparts (a _formal Poisson scheme_ is a formal scheme with a Poisson
bracket on its structure sheaf, and so on).

######  Definition 3.1.

Let $X$ be a Poisson variety and let $S$ be a (possibly formal) affine scheme
with a distinguished closed point $0$. Then a _(formal) Poisson deformation of
$X$ over $S$_ is a flat (formal) Poisson $S$-scheme $\mathcal{X}_{S}$ equipped
with a Poisson isomorphism
$\iota:\mathcal{X}_{S}\times_{S}\\{0\\}\xrightarrow{\sim}X$. If $S$ is a
formal scheme complete at $0\in S$, we say that $\mathcal{X}_{S}$ is a formal
Poisson deformation of $X$.

An isomorphism between Poisson deformations $(\mathcal{X}_{S},\iota)$ and
$(\mathcal{X}^{\prime}_{S},\iota^{\prime})$ over $S$ is an isomorphism of
Poisson schemes
$\mathcal{X}_{S}\stackrel{{\scriptstyle}}{{\to}}\mathcal{X}^{\prime}_{S}$ over
$S$ such that the induced isomorphism
$\mathcal{X}_{S}\times_{S}\\{0\\}\xrightarrow{\sim}\mathcal{X}^{\prime}_{S}\times_{S}\\{0\\}$
intertwines $\iota$ and $\iota^{\prime}$.

Let $X$ be a Poisson variety. Let $\mathcal{A}rt_{\mathbb{C}}$ denote the
category of local Artinian $\mathbb{C}$-algebras with residue field
$\mathbb{C}$. For $R\in\mathcal{A}rt_{\mathbb{C}}$, define
$\mathrm{PD}_{X}(R)$ to be the set of isomorphism classes of Poisson
deformations over $\operatorname{Spec}(R)$. This defines a functor
$\mathrm{PD}_{X}:\mathcal{A}rt_{\mathbb{C}}\stackrel{{\scriptstyle}}{{\to}}\mathcal{S}et$.
Let $\mathbb{C}\\{\epsilon\\}:=\mathbb{C}[\epsilon]/(\epsilon^{2})$ denote the
ring of dual numbers over $\mathbb{C}$. Then
$\mathrm{PD}_{X}(\mathbb{C}\\{\epsilon\\})$ is the tangent space of
$\mathrm{PD}_{X}$.

Next, we recall the notion of a graded Poisson deformation. Let $X$ be a
graded normal Poisson variety. By this, we mean a normal variety equipped with
two additional structures:

  * •

A rational $\mathbb{C}^{\times}$-action. Let $\mathcal{O}_{X}$ denote the
structure sheaf of $X$ with respect to the conical topology on $X$ (i.e. the
topology with open subsets equal to $\mathbb{C}^{\times}$-invariant Zariski
open subsets);

  * •

A Poisson bracket
$\\{\cdot,\cdot\\}:\mathcal{O}_{X}\otimes\mathcal{O}_{X}\stackrel{{\scriptstyle}}{{\to}}\mathcal{O}_{X}$
of degree $-d$.

######  Definition 3.2.

Let $X$ be a graded normal Poisson variety and let $S$ be an affine scheme
with a rational $\mathbb{C}^{\times}$-action contracting onto a distinguished
closed point $0\in S$. Then a _graded Poisson deformation of $X$ over $S$_ is
a Poisson deformation $(\mathcal{X}_{S},\iota)$ of $X$ over $S$ (cf.
Definition 3.1) equipped with a rational $\mathbb{C}^{\times}$-action on
$\mathcal{X}_{S}$ such that

  * (i)

The action rescales the Poisson bracket by $t\mapsto t^{-d}$.

  * (ii)

The action is compatible with the action on $S$.

  * (iii)

$\iota$ is $\mathbb{C}^{\times}$-equivariant.

An isomorphism between graded Poisson deformations $(\mathcal{X}_{S},\iota)$
and $(\mathcal{X}_{S}^{\prime},\iota^{\prime})$ over $S$ is an isomorphism of
Poisson deformations which is $\mathbb{C}^{\times}$-equivariant.

###  3.2. Filtered quantizations

In this subsection, we will recall the definitions of filtered quantizations
of graded Poisson algebras and graded Poisson varieties.

Let $A$ be a _graded Poisson algebra_ of degree $-d\in\mathbb{Z}_{<0}$. By
this, we mean a finitely-generated commutative associative algebra equipped
with two additional structures: an algebra grading

$A=\bigoplus_{i=-\infty}^{\infty}A_{i};$

and a Poisson bracket $\\{\cdot,\cdot\\}$ of degree $-d$

$\\{A_{i},A_{j}\\}\subset A_{i+j-d},\qquad i,j\in\mathbb{Z}.$

######  Definition 3.3.

A _filtered quantization_ of $A$ is a pair $(\mathcal{A},\theta)$ consisting
of

  * (i)

an associative algebra $\mathcal{A}$ equipped with a complete and separated
filtration by subspaces

$\mathcal{A}=\bigcup_{i=-\infty}^{\infty}\mathcal{A}_{\leq
i},\qquad...\subseteq\mathcal{A}_{\leq-1}\subseteq\mathcal{A}_{\leq
0}\subseteq\mathcal{A}_{\leq 1}\subseteq...$

such that

$[\mathcal{A}_{\leq i},\mathcal{A}_{\leq j}]\subseteq\mathcal{A}_{\leq
i+j-d}\qquad i,j\in\mathbb{Z},$

and

  * (ii)

an isomorphism of graded Poisson algebras

$\theta:\operatorname{gr}(\mathcal{A})\xrightarrow{\sim}A,$

where the Poisson bracket on $\operatorname{gr}(\mathcal{A})$ is defined by

$\\{a+\mathcal{A}_{\leq i-1},b+\mathcal{A}_{\leq
j-1}\\}=[a,b]+\mathcal{A}_{\leq i+j-d-1},\qquad a\in\mathcal{A}_{\leq i},\
b\in\mathcal{A}_{\leq j}.$

An isomorphism of filtered quantizations
$(\mathcal{A}_{1},\theta_{1})\xrightarrow{\sim}(\mathcal{A}_{2},\theta_{2})$
is an isomorphism of filtered algebras
$\phi:\mathcal{A}_{1}\xrightarrow{\sim}\mathcal{A}_{2}$ such that
$\theta_{1}=\theta_{2}\circ\operatorname{gr}(\phi)$. Denote the set of
isomorphism classes of quantizations of $A$ by $\mathrm{Quant}(A)$.

Often, the isomorphism $\theta$ is clear from the context, and will be omitted
from the notation. However, the reader should keep in mind that a filtered
quantization $(\mathcal{A},\theta)$ is _not_ determined up to isomorphism by
$\mathcal{A}$ alone.

Now let $X$ be a graded normal Poisson variety.

######  Definition 3.4.

A _filtered quantization_ of $X$ is a pair $(\mathcal{D},\theta)$ consisting
of

  * (i)

a sheaf $\mathcal{D}$ of associative algebras in the conical topology on $X$,
equipped with a complete and separated filtration by subsheaves of vector
spaces

$\mathcal{D}=\bigcup_{i=-\infty}^{\infty}\mathcal{D}_{\leq
i},\qquad...\subseteq\mathcal{D}_{\leq-1}\subseteq\mathcal{D}_{\leq
0}\subseteq\mathcal{D}_{\leq 1}\subseteq...$

such that

$[\mathcal{D}_{\leq i},\mathcal{D}_{\leq j}]\subseteq\mathcal{D}_{\leq
i+j-d}\qquad i,j\in\mathbb{Z},$

and

  * (ii)

an isomorphism of sheaves of graded Poisson algebras

$\theta:\operatorname{gr}(\mathcal{D})\xrightarrow{\sim}\mathcal{O}_{X},$

where the Poisson bracket on $\operatorname{gr}(\mathcal{D})$ is defined by

$\\{a+\mathcal{D}_{\leq i-1},b+\mathcal{D}_{\leq
j-1}\\}=[a,b]+\mathcal{D}_{\leq i+j-d-1},\qquad a\in\mathcal{D}_{\leq i},\
b\in\mathcal{D}_{\leq j}.$

An isomorphism of filtered quantizations
$(\mathcal{D}_{1},\theta_{1})\xrightarrow{\sim}(\mathcal{D}_{2},\theta_{2})$
is an isomorphism of sheaves of filtered algebras
$\phi:\mathcal{D}_{1}\xrightarrow{\sim}\mathcal{D}_{2}$ such that
$\theta_{1}=\theta_{2}\circ\operatorname{gr}(\phi)$. Denote the set of
isomorphism classes of quantizations of $X$ by $\mathrm{Quant}(X)$.

We conclude this subsection by recalling the definition of a _Hamiltonian
quantization_ of a graded Poisson algebra with Hamiltonian $G$-action. Let $A$
be a graded Poisson algebra of degree $-d$. Suppose $G$ is an algebraic group
which acts rationally on $A$ by graded Poisson automorphisms. Write
$\mathrm{Der}(A)$ for the Lie algebra of derivations of $A$. The $G$-action on
$A$ induces by differentiation a Lie algebra homomorphism

$\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}\mathrm{Der}(A),\qquad\xi\mapsto\xi_{A},$

We say that $A$ (or the $G$-action on $A$) is _Hamiltonian_ if there is a
$G$-equivariant map
$\varphi:\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}A_{d}$ (called a
_classical co-moment map_) such that

$\\{\varphi(\xi),a\\}=\xi_{A}(a),\qquad\xi\in\mathfrak{g},\quad a\in A.$

A filtered quantization $(\mathcal{A},\theta)$ is $G$-_equivariant_ if $G$
acts rationally on $\mathcal{A}$ by filtered algebra automorphisms and the
isomorphism $\theta:\operatorname{gr}(\mathcal{A})\xrightarrow{\sim}A$ is
$G$-equivariant. In this case, we get a Lie algebra homomorphism

$\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}\mathrm{Der}(\mathcal{A}),\qquad\xi\mapsto\xi_{\mathcal{A}}.$

######  Definition 3.5.

Suppose $A$ is a graded Poisson algebra equipped with a Hamiltonian
$G$-action. A _Hamiltonian_ quantization of $A$ is a triple
$(\mathcal{A},\theta,\Phi)$ consisting of

  * (i)

a $G$-equivariant filtered quantization $(\mathcal{A},\theta)$ of $A$, and

  * (ii)

a $G$-equivariant map
$\Phi:\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{\leq d}$
(called a _quantum co-moment map_) such that

$[\Phi(\xi),a]=\xi_{\mathcal{A}}(a),\qquad\xi\in\mathfrak{g},\quad
a\in\mathcal{A}.$

An isomorphism
$(\mathcal{A}_{1},\theta_{1},\Phi_{1})\xrightarrow{\sim}(\mathcal{A}_{2},\theta_{2},\Phi_{2})$
of Hamiltonian quantizations of $A$ is a $G$-equivariant isomorphism of
filtered algebras
$\phi:\mathcal{A}_{1}\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{2}$ such
that $\theta_{1}=\theta_{2}\circ\operatorname{gr}(\phi)$ and
$\Phi_{2}=\phi\circ\Phi_{1}$. Denote the set of isomorphism classes of
Hamiltonian quantizations of $A$ by $\mathrm{Quant}^{G}(A)$.

###  3.3. Symplectic singularities and $\mathbb{Q}$-factorial terminalizations

Let $X$ be a normal Poisson variety.

######  Definition 3.6 ([Bea00], Definition 1.1).

We say that $X$ has _symplectic singularities_ if

  * (i)

The regular locus $X^{reg}\subset X$ is symplectic; denote the symplectic form
by $\omega$.

  * (ii)

There is a resolution of singularities
$\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ such that $\rho^{*}\omega$ extends
to a regular (not necessarily symplectic) $2$-form on $Y$.

Now let $X$ be a graded normal Poisson variety.

######  Definition 3.7.

We say that $X$ is a _conical symplectic singularity_ if $X$ has symplectic
singularities and the $\mathbb{C}^{\times}$-action on $X$ is contracting onto
a point.

We note that every conical symplectic singularity is automatically affine.

######  Example 3.8.

Let $G$ be a complex connected reductive algebraic group and let
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ be a finite
étale $G$-equivariant cover of a nilpotent co-adjoint $G$-orbit
$\mathbb{O}\subset\mathfrak{g}^{*}$. There is a natural
$\mathbb{C}^{\times}$-action on $\widetilde{\mathbb{O}}$ defined in the
following manner (see [BK94, Section 1] for details and proofs). Consider the
‘doubled’ dilation action of $\mathbb{C}^{\times}$ on $\mathfrak{g}^{*}$, i.e.

$z\cdot\xi=z^{2}\xi,\qquad z\in\mathbb{C}^{\times},\ \xi\in\mathfrak{g}^{*}.$

This action preserves $\mathbb{O}$, since $\mathbb{O}$ is a nilpotent orbit,
and lifts to a unique $\mathbb{C}^{\times}$-action on
$\widetilde{\mathbb{O}}$, which commutes with the $G$-action on
$\widetilde{\mathbb{O}}$. It follows that any morphism of covers
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$
is automatically ${\mathbb{C}^{\times}}$-equivariant. The
${\mathbb{C}^{\times}}$-action on $\widetilde{\mathbb{O}}$ induces a non-
negative grading on the ring of regular functions
$\mathbb{C}[\widetilde{\mathbb{O}}]$.

There is also a natural $G$-equivariant symplectic form on
$\widetilde{\mathbb{O}}$, obtained by pullback from the Kirillov-Kostant-
Souriau form on $\mathbb{O}$. This form induces a Poisson bracket on
$\mathbb{C}[\widetilde{\mathbb{O}}]$, and this bracket is of degree $-2$ with
respect to the grading defined above (see again [BK94, Section 1]). It is
known that $X=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ is a
conical symplectic singularity (see [Los21, Lemma 2.5]).

The natural map $\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}X$ is
an open embedding, and the $G$-action on $\widetilde{\mathbb{O}}$ extends to a
$G$-action on $X$. The $G$-equivariant covering map
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ extends to
a finite $G$-equivariant surjection
$X\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$. The $G$-action on
$\widetilde{\mathbb{O}}$ (resp. $X$) is Hamiltonian; the moment map is the
composition
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}\subset\mathfrak{g}^{*}$
(resp.
$X\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}\subset\mathfrak{g}^{*}$).
Note that any morphism of covers
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widehat{\mathbb{O}}$
is symplectic and intertwines the moment maps, hence is automatically
$G$-(hence $G\times{\mathbb{C}^{\times}}$-)equivariant, since $G$ is
connected.

Recall that a normal variety $Y$ is $\mathbb{Q}$_-factorial_ if every Weil
divisor has a nonzero integer multiple which is Cartier.

######  Proposition 3.9 (Proposition 2.1, [Los19]).

Let $X$ be a Poisson variety with symplectic singularities. Then there is a
birational projective morphism $\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ such
that

  * (i)

$Y$ is an irreducible Poisson variety.

  * (ii)

$Y$ is $\mathbb{Q}$-factorial.

  * (iii)

The singular locus of $Y$ is of codimension $\geq 4$.

The morphism $\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ in the proposition
above (or the variety $Y$, if the morphism is understood) is called a
$\mathbb{Q}$_-factorial terminalization_ of $X$. If $X$ is a conical
symplectic singularity, then there is a $\mathbb{C}^{\times}$-action on $Y$
such that $\rho$ is $\mathbb{C}^{\times}$-equivariant, see [Nam08, A.7]. By
[Nam15], every conical symplectic singularity has only finitely-many
isomorphism classes of $\mathbb{Q}$-factorial terminalizations.

###  3.4. The Namikawa space and Weyl group

Let $X$ be a conical symplectic singularity. Associated to $X$ are two
important invariants: a finite-dimensional complex vector space
$\mathfrak{P}=\mathfrak{P}^{X}$ called the _Namikawa space_ and a finite
Coxeter group $W=W^{X}$ called the _Namikawa Weyl group_. In this section, we
will recall several equivalent definitions of these objects.

######  Definition 3.10.

Let $X$ be a conical symplectic singularity and let
$\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ be a $\mathbb{Q}$-factorial
terminalization. The _Namikawa space_ associated to $X$ is the finite-
dimensional complex vector space

$\mathfrak{P}=\mathfrak{P}^{X}:=H^{2}(Y^{reg},\mathbb{C})$

We will see in a moment that $\mathfrak{P}$ depends only on $X$ (and not on
the choice of $\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$). Since $X$ is a
symplectic singularity, it contains finitely-many symplectic leaves ([Kal06,
Theorem 2.3]). Let $\mathfrak{L}_{k}$, $k=1,2,\ldots,t$, denote the symplectic
leaves of codimension 2. For each such leaf $\mathfrak{L}_{k}\subset X$, the
formal slice to $\mathfrak{L}_{k}\subset X$ is identified with the formal
neighborhood at $0$ in a Kleinian singularity
$\Sigma_{k}=\mathbb{C}^{2}/\Gamma_{k}$. Under the McKay correspondence,
$\Gamma_{k}$ corresponds to a complex simple Lie algebra $\mathfrak{g}_{k}$ of
type A, D, or E. Fix a Cartan subalgebra
$\mathfrak{h}_{k}\subset\mathfrak{g}_{k}$. Write
$\Lambda_{k}\subset\mathfrak{h}_{k}^{*}$ for the weight lattice and $W_{k}$
for the Weyl group. If we choose a point $x\in\mathfrak{L}_{k}$, there is a
natural identification
$H^{2}(\widetilde{\mu}^{-1}(x),\mathbb{Z})\simeq\Lambda_{k}$, and
$\pi_{1}(\mathfrak{L}_{k})$ acts on $\Lambda_{k}$ by diagram automorphisms.
The _partial Namikawa space_ corresponding to $\mathfrak{L}_{k}$ is the
subspace

$\mathfrak{P}_{k}=\mathfrak{P}_{k}^{X}:=(\mathfrak{h}_{k}^{*})^{\pi_{1}(\mathfrak{L}_{k})}.$

Also define
$\mathfrak{P}_{0}=\mathfrak{P}_{0}^{X}:=H^{2}(X^{reg},\mathbb{C})$.

######  Proposition 3.11 ([Los22a], Lem 2.8).

There is a linear isomorphism

(3.4.1) $\mathfrak{P}\xrightarrow{\sim}\bigoplus_{k=0}^{t}\mathfrak{P}_{k}.$

For each codimension 2 leaf $\mathfrak{L}_{k}\subset X$, consider the subgroup
of monodromy invariants $W_{k}^{\pi_{1}(\mathfrak{L}_{k})}\subset W_{k}$. Note
that there is a natural action of $W_{k}^{\pi_{1}(\mathfrak{L}_{k})}$ on
$\mathfrak{P}_{k}=(\mathfrak{h}_{k}^{*})^{\pi_{1}(\mathfrak{L}_{k})}$.

######  Definition 3.12.

The _Namikawa Weyl group_ associated to $X$ is the finite group

$W=W^{X}:=\prod_{k=1}^{t}W_{k}^{\pi_{1}(\mathfrak{L}_{k})}.$

Note that $W$ acts on $\mathfrak{P}$ via the isomorphism (3.4.1) (the action
on $\mathfrak{P}_{0}$ is trivial).

###  3.5. Filtered quantizations of conical symplectic singularities

In this section, we will recall the classification of filtered quantizations
of conical symplectic singularities. Let $X$ be a conical symplectic
singularity and let $\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ be a graded
$\mathbb{Q}$-factorial terminalization of $X$. For any graded smooth
symplectic variety $V$, there is a (non-commutative) _period map_

$\mathrm{Per}:\mathrm{Quant}(V)\stackrel{{\scriptstyle}}{{\to}}H^{2}(V,\mathbb{C}),$

see [BK04, Sec 4], [Los12, Sec 2.3].

######  Proposition 3.13 ([Los22a], Prop 3.1(1)).

The maps

$\mathrm{Quant}(Y)\overset{|_{Y^{\mathrm{reg}}}}{\stackrel{{\scriptstyle}}{{\to}}}\mathrm{Quant}(Y^{\mathrm{reg}})\overset{\mathrm{Per}}{\stackrel{{\scriptstyle}}{{\to}}}H^{2}(Y^{\mathrm{reg}},\mathbb{C})=\mathfrak{P}^{X}$

are bijections.

For $\lambda\in\mathfrak{P}^{X}$, let $\mathcal{D}_{\lambda}$ denote the
corresponding filtered quantization of $Y$ and let
$\mathcal{A}_{\lambda}:=\Gamma(Y,\mathcal{D}_{\lambda})$.

######  Theorem 3.14 (Prop 3.3, Thm 3.4, [Los22a]).

The following are true:

  * (i)

For every $\lambda\in\mathfrak{P}^{X}$, the algebra $\mathcal{A}_{\lambda}$ is
a filtered quantization of $\mathbb{C}[X]$.

  * (ii)

Every filtered quantization of $\mathbb{C}[X]$ is isomorphic to
$\mathcal{A}_{\lambda}$ for some $\lambda\in\mathfrak{P}^{X}$.

  * (iii)

For every $\lambda,\lambda^{\prime}\in\mathfrak{P}^{X}$, we have
$\mathcal{A}_{\lambda}\simeq\mathcal{A}_{\lambda^{\prime}}$ if and only if
$\lambda^{\prime}\in W^{X}\cdot\lambda$.

Hence, the map $\lambda\mapsto\mathcal{A}_{\lambda}$ induces a bijection

$\mathfrak{P}^{X}/W^{X}\simeq\mathrm{Quant}(\mathbb{C}[X]),\qquad
W^{X}\cdot\lambda\mapsto\mathcal{A}_{\lambda}.$

We conclude by recalling an equivariant version of Theorem 3.14 from [LMM21].
Let $G$ be a connected reductive algebraic group and suppose
$A:=\mathbb{C}[X]$ admits a Hamiltonian $G$-action, see Section 3.2. Define
the _extended Namikawa space_

$\overline{\mathfrak{P}}^{X}:=\mathfrak{P}^{X}\oplus\mathfrak{z}(\mathfrak{g})^{*}$

This space should be viewed as the equivariant counterpart of
$\mathfrak{P}^{X}$. Let $W^{X}$ act on $\overline{\mathfrak{P}}^{X}$ via the
decomposition above (the $W^{X}$-action on the second factor is defined to be
trivial).

######  Proposition 3.15 (Lem 4.11.2, [LMM21]).

Let $G$ be a connected reductive algebraic group and suppose
$A:=\mathbb{C}[X]$ admits a Hamiltonian $G$-action. Then the following are
true:

  * (i)

There is a unique classical co-moment map
$\varphi:\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}A_{d}$.

  * (ii)

Every filtered quantization $\mathcal{A}\in\mathrm{Quant}(A)$ has a unique
$G$-equivariant structure.

  * (iii)

For every $\mathcal{A}\in\mathrm{Quant}(A)$ and
$\chi\in\mathfrak{z}(\mathfrak{g})^{*}$, there is a unique quantum co-moment
map $\Phi_{\chi}:\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{\leq
d}$ such that $\Phi|_{\mathfrak{z}(\mathfrak{g})}=\chi$.

In particular, there is a canonical bijection

$\overline{\mathfrak{P}}^{X}/W^{X}\xrightarrow{\sim}\mathrm{Quant}^{G}(A)\qquad
W^{X}\cdot(\lambda,\chi)\mapsto(\mathcal{A}_{\lambda},\Phi_{\chi}).$

######  Definition 3.16 (Def 5.0.1, [LMM21]).

The _canonical quantization_ of $\mathbb{C}[X]$ is the Hamiltonian
quantization corresponding to the parameter $0\in\overline{\mathfrak{P}}^{X}$.

###  3.6. $\mathbb{Q}$-factorial terminalizations of nilpotent covers

Let $\widetilde{\mathbb{O}}\in\mathsf{Cov}(G)$ and let
$X=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$. Then $X$ is a
conical symplectic singularity (cf. Example 3.8). In this section, we will
give explicit (Lie-theoretic) descriptions of the $\mathbb{Q}$-factorial
terminalizations of $X$, the Namikawa space
$\mathfrak{P}(\widetilde{\mathbb{O}}):=\mathfrak{P}^{X}$, and the partial
Namikawa spaces
$\mathfrak{P}_{k}(\widetilde{\mathbb{O}}):=\mathfrak{P}_{k}^{X}$.

Choose $(L,\widetilde{\mathbb{O}}_{L})\in\mathsf{Cov}_{0}(G)$ such that
$\mathrm{Bind}(L,\widetilde{\mathbb{O}}_{L})=\widetilde{\mathbb{O}}$. By
Proposition 2.7, $(L,\widetilde{\mathbb{O}}_{L})$ is unique (up to conjugation
by $G$). Let
$X_{L}=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])$. Choose a
parabolic subgroup $P\subset G$ with Levi decomposition $P=LN$. Form the
variety $Y=G\times^{P}(X_{L}\times\mathfrak{p}^{\perp})$ and consider the
proper $G$-equivariant map
$\widetilde{\mu}:Y\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$
defined in Section 2.3. Then $\widetilde{\mu}$ admits a Stein factorization
$Y\xrightarrow{\rho}X\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$.
The first map is projective birational and the second is finite.

######  Theorem 3.17 ([Mat20, Cor 4.3], [LMM21, Lemma 7.2.4]).

The following are true.

  1. (i)

The morphism $\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ is a
$\mathbb{Q}$-factorial terminalization of $X$.

  2. (ii)

Any $\mathbb{Q}$-factorial terminalization $Y$ of $X$ is of this form.

If $\mathfrak{a}$ is a Lie algebra, we write $\mathfrak{X}(\mathfrak{a})$ for
the vector space of Lie algebra homomorphisms
$\mathfrak{a}\stackrel{{\scriptstyle}}{{\to}}\mathbb{C}$. Let
$\pi:Y\stackrel{{\scriptstyle}}{{\to}}G/P$ be the natural projection map which
makes $Y$ into a fiber bundle over $G/P$. Let $\underline{\pi}$ denote the
restriction of $\pi$ to $Y^{reg}$. Note that $H^{2}(G/P,\mathbb{C})$ is
identified with $\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$.
Consider the composite map

(3.6.1) $\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\simeq
H^{2}(G/P,\mathbb{C})\xrightarrow{\underline{\pi}^{*}}H^{2}(Y^{reg},\mathbb{C})=\mathfrak{P}(\widetilde{\mathbb{O}}),$

where
$\underline{\pi}^{*}:H^{2}(G/P,\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(Y^{reg},\mathbb{C})=\mathfrak{P}(\widetilde{\mathbb{O}})$
is the pullback map on cohomology induced by $\underline{\pi}$.

######  Proposition 3.18 (Proposition 7.2.2, [LMM21]).

The map
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\xrightarrow{\sim}\mathfrak{P}(\widetilde{\mathbb{O}})$
is an isomorphism.

We can also describe the spaces $H^{2}(\widetilde{\mathbb{O}},\mathbb{C})$ and
$\mathfrak{P}_{k}(\widetilde{\mathbb{O}}):=\mathfrak{P}_{k}^{X}$ in terms of
Lie-theoretic data. We begin by describing
$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})$. Assume for simplicity that $G$ is
simply connected and semisimple. Let $R$ denote the reductive part of the
stabilizer of $e\in\mathbb{O}$ and let $\mathfrak{r}$ be its Lie algebra. We
note that $\mathfrak{r}$ does not depend on the choice of $e$ in $\mathbb{O}$,
and the adjoint action of $R$ on $\mathfrak{z}(\mathfrak{r})$ factors through
$R/R^{\circ}\simeq\pi_{1}(\widetilde{\mathbb{O}})$.

######  Lemma 3.19 (Lemma 7.2.7, [LMM21]).

There is a natural identification

$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})\simeq\mathfrak{z}(\mathfrak{r})^{\pi_{1}(\widetilde{\mathbb{O}})}$

######  Remark 3.20.

If $\widetilde{\mathbb{O}}=\widehat{\mathbb{O}}$ is the _universal_ cover of
$\mathbb{O}$, then
$H^{2}(\widehat{\mathbb{O}},\mathbb{C})\simeq\mathfrak{z}(\mathfrak{r})$ by
Lemma 3.19. In particular, $H^{2}(\widehat{\mathbb{O}},\mathbb{C})=0$ if and
only if $\mathfrak{r}$ is semisimple. On the other hand, if
$\widetilde{\mathbb{O}}=\mathbb{O}$, then
$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})\simeq\mathfrak{z}(\mathfrak{r})^{\pi_{1}(\mathbb{O})}$
was computed in every case by Biswas and Chatterjee in [BC12].

We proceed to describing the partial Namikawa spaces
$\mathfrak{P}_{k}(\widetilde{\mathbb{O}})$ for $k\geq 1$. Assume that
$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})=0$. Suppose $Q\subset G$ is a
parabolic subgroup with Levi factor $M$ and
$\widetilde{\mathbb{O}}_{M}\in\mathsf{Cov}(M)$ satisfies
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{M}\widetilde{\mathbb{O}}_{M}$. The
triple $(Q,M,\widetilde{\mathbb{O}}_{M})$ gives rise to a projective
birational morphism

(3.6.2)
$\rho;:G\times^{Q}(\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{M}])\times\mathfrak{q}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$

######  Proposition 3.21 (Prop 7.5.6, [LMM21]).

For each codimension 2 leaf $\mathfrak{L}_{k}\subset\widetilde{X}$, there is a
unique pair $(M_{k},\widetilde{\mathbb{O}}_{M_{k}})$ consisting of a Levi
subgroup $M_{k}\subset G$ and a nilpotent cover
$\widetilde{\mathbb{O}}_{M_{k}}\in\mathsf{Cov}(M_{k})$ such that

  * (i)

$L\subset M_{k}$.

  * (ii)

$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{M_{k}}\widetilde{\mathbb{O}}_{M_{k}}$.

  * (iii)

For any parabolic $Q\subset G$ with Levi factor $M_{k}$, the morphism (3.6.2)
resolves $\Sigma_{k}$ and preserves $\Sigma_{j}$ for $j\neq k$.

The pair $(M_{k},\widetilde{\mathbb{O}}_{M_{k}})$ appearing in Proposition
3.21 is called the $\mathfrak{L}_{k}$-_adapted resolution datum._

######  Proposition 3.22 (Proposition 3.6.4, [MM23]).

Let
$\mathfrak{L}_{k}\subset\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$
be a codimension 2 leaf and let $(M_{k},\widetilde{\mathbb{O}}_{M_{k}})$ be
the $\mathfrak{L}_{k}$-adapted resolution datum. Then the isomorphism
$\mathfrak{P}\simeq\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$
of Proposition 3.18 restricts to an isomorphism
$\mathfrak{P}_{k}\simeq\mathfrak{X}(\mathfrak{m}_{k}\cap[\mathfrak{g},\mathfrak{g}])$.

Comparing Propositions 3.18 and 3.11, we arrive at the following (purely
geometric) criterion for birational rigidity.

######  Proposition 3.23 (Proposition 3.7.1, [MM23]).

Let $\widetilde{\mathbb{O}}\in\mathsf{Cov}(G)$. Then $\widetilde{\mathbb{O}}$
is birationally rigid if and only if the following conditions hold:

  * (i)

$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})=0$.

  * (ii)

$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ has no codimension 2
leaves.

We conclude by sketching a simple criterion for birational induction. For any
$\mathbb{O}\in\mathsf{Orb}(G)$, let $\mathcal{P}_{rig}(\mathbb{O})$ denote the
set of $G$-conjugacy classes of pairs $(L,\mathbb{O}_{L})$ consisting of a
Levi subgroup $L\subset G$ and a rigid nilpotent orbit
$\mathbb{O}_{L}\in\mathsf{Orb}(L)$ such that
$\mathbb{O}=\operatorname{Ind}^{G}_{L}\mathbb{O}_{L}$. This set can be
computed in classical types using the results [CM93, Section 7] and in
exceptional types using the tables in [DE09, Section 4]. Consider the integer

(3.6.3)
$m(\mathbb{O})=\max\\{\dim\mathfrak{z}(\mathfrak{l})\mid(L,\mathbb{O}_{L})\in\mathcal{P}_{rig}(\mathbb{O})\\}.$

######  Lemma 3.24.

Suppose $m(\mathbb{O})=\dim\mathfrak{P}$ and there is a unique pair
$(L,\mathbb{O}_{L})\in\mathcal{P}_{rig}(\mathbb{O})$ such that
$\dim(\mathfrak{z}(\mathfrak{l}))=m(\mathbb{O})$. Then
$\mathbb{O}=\mathrm{Bind}^{G}_{L}\mathbb{O}_{L}$.

###### Proof.

Suppose $\mathbb{O}$ is birationally induced from a birationally rigid orbit
$\mathbb{O}_{M}\in\mathsf{Orb}(M)$ for a Levi subgroup $M\subset G$. By
Proposition 3.18, $\dim(\mathfrak{z}(\mathfrak{m}))=m(\mathbb{O})$. Suppose
$\mathbb{O}_{M}$ is not rigid. Then there is a proper Levi subgroup $K\subset
M$ and a rigid orbit $\mathbb{O}_{K}\in\mathsf{Orb}(K)$ such that
$\mathbb{O}=\operatorname{Ind}^{G}_{K}\mathbb{O}_{K}$. So
$(K,\mathbb{O}_{K})\in\mathcal{P}_{rig}(\mathbb{O})$ and
$\dim(\mathfrak{z}(\mathfrak{k}))>\dim(\mathfrak{z}(\mathfrak{m}))=m(\mathbb{O})$,
a contradiction. Hence, $(M,\mathbb{O}_{M})\in\mathcal{P}_{rig}(\mathbb{O})$
and so $G$-conjugate to $(L,\mathbb{O}_{L})$. ∎

###  3.7. Universal Poisson deformations of conical symplectic singularities

Let $X$ be an affine symplectic singularity. Let $X^{\diamond}\subset X$
denote the complement in $X$ to the union of all symplectic leaves of
codimension $\geq 4$. Then

$X^{\diamond}=X^{reg}\sqcup\bigsqcup_{k=1}^{t}\mathfrak{L}_{k},$

where $\mathcal{L}_{k}$ are the codimension 2 leaves in $X$ as in Section 3.4.
As observed by Namikawa [Nam11, p. 52], the variety $X^{\diamond}$ admits a
unique $\mathbb{Q}$-factorial terminalization
$\pi:\widetilde{X}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}X^{\diamond}$ (up
to isomorphism), and $\widetilde{X}^{\diamond}$ is smooth. Now we take a
$\mathbb{Q}$-factorial terminalization
$\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ of $X$. Set
$Y^{\diamond}=\rho^{-1}(X^{\diamond})$. Then $Y^{\diamond}$ lies in the
regular locus $Y^{reg}$ of $Y$ and the restriction
$\rho|_{Y^{\diamond}}:Y^{\diamond}\stackrel{{\scriptstyle}}{{\to}}X^{\diamond}$
is also a $\mathbb{Q}$-factorial terminalization of $X^{\diamond}$. Hence
there is a unique isomorphism between $Y^{\diamond}$ and
$\widetilde{X}^{\diamond}$ as varieties over $X^{\diamond}$. By (2.4) in the
proof of [Los22a, Proposition 2.14], we have
$\operatorname{codim}_{Y}Y\backslash Y^{\diamond}\geqslant 2$ and so
$\operatorname{codim}_{Y^{reg}}Y^{reg}\backslash Y^{\diamond}\geqslant 2$.
This implies that the restriction map
$\mathfrak{P}=H^{2}(Y^{reg},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(Y^{\diamond},\mathbb{C})\simeq
H^{2}(\widetilde{X}^{\diamond},\mathbb{C})$ is an isomorphism by a standard
argument involving a long exact sequence of cohomology groups (see e.g. the
proof of [Nam11a, Theorem 5.1(i)]).

By the proof of [Nam11a, Theorems 5.1, 5.2], the inclusions
$X^{\diamond}\subset X$ and $Y^{\diamond}\subset Y$ induce natural
transformations
$\mathrm{PD}_{X}\stackrel{{\scriptstyle}}{{\to}}\mathrm{PD}_{X^{\diamond}}$
and
$\mathrm{PD}_{Y}\stackrel{{\scriptstyle}}{{\to}}\mathrm{PD}_{Y^{\diamond}}$.
The birational maps $\pi$, $\rho$ and $\rho|_{Y^{\diamond}}$ induce natural
transformations

$\pi_{*}:\mathrm{PD}_{\widetilde{X}^{\diamond}}\stackrel{{\scriptstyle}}{{\to}}\mathrm{PD}_{X^{\diamond}},\quad\rho_{*}:\mathrm{PD}_{Y}\stackrel{{\scriptstyle}}{{\to}}\mathrm{PD}_{X}\quad\text{and}\quad(\rho|_{X^{\diamond}})_{*}:\mathrm{PD}_{Y^{\diamond}}\stackrel{{\scriptstyle}}{{\to}}\mathrm{PD}_{X^{\diamond}}.$

These natural transformations fit into the following commutative diagram:

(3.7.1)
${\mathrm{PD}_{Y}}$${\mathrm{PD}_{Y^{\diamond}}}$${\mathrm{PD}_{\widetilde{X}^{\diamond}}}$${\mathrm{PD}_{X}}$${\mathrm{PD}_{X^{\diamond}}}$${\mathrm{PD}_{X^{\diamond}}}$$\scriptstyle{\rho_{*}}$$\scriptstyle{\sim}$$\scriptstyle{(\rho|_{X^{\diamond}})_{*}}$$\scriptstyle{\pi_{*}}$

######  Theorem 3.25 (Theorems 5.1, 5.2, [Nam11a]).

The following are true:

  1. (i)

The horizontal transformations
$\mathrm{PD}_{X}\stackrel{{\scriptstyle}}{{\to}}\mathrm{PD}_{X^{\diamond}}$
and
$\mathrm{PD}_{Y}\stackrel{{\scriptstyle}}{{\to}}\mathrm{PD}_{Y^{\diamond}}$ in
(3.7.1) are isomorphisms.

  2. (ii)

We have
$\mathrm{PD}_{Y}(\mathbb{C}\\{\epsilon\\})\simeq\mathrm{PD}_{Y^{\diamond}}(\mathbb{C}\\{\epsilon\\})\simeq\mathrm{PD}_{\widetilde{X}^{\diamond}}(\mathbb{C}\\{\epsilon\\})\simeq\mathfrak{P}$.

  3. (iii)

All deformation functors in (3.7.1) are unobstructed and pro-representable.
$\mathrm{PD}_{Y}\simeq\mathrm{PD}_{Y^{\diamond}}\simeq\mathrm{PD}_{\widetilde{X}^{\diamond}}$
are pro-represented by
$\mathbb{C}[\mathfrak{P}]^{\wedge}=\mathbb{C}[\mathfrak{P}^{\wedge}]$, the
completion of $\mathbb{C}[\mathfrak{P}]$ at the maximal ideal corresponding to
$0\in\mathfrak{P}$.

By Theorem 3.25, each Poisson variety appearing in (3.7.1) admits a universal
formal Poisson deformation. Now assume that $X$ is a conical symplectic
singularity. Then by [Nam08, Lemma 20], the ${\mathbb{C}^{\times}}$-action on
$X$ induces ${\mathbb{C}^{\times}}$-actions on the universal formal Poisson
deformations of the Poisson varieties in (3.7.1). Using these
$\mathbb{C}^{\times}$-actions, Namikawa shows that the universal formal
Poisson deformations of $X$ and $Y$ can be algebraized to graded Poisson
deformations ([Nam08, Lemma 22]). And Losev shows that these graded Poisson
deformations are _universal_ in the following sense ([Los22a, Section
2.2-2.4]).

######  Definition 3.26.

Let $X$ be a graded Poisson variety. A _universal graded Poisson deformation
of $X$_ is a graded Poisson deformation $\mathcal{X}_{univ}$ of $X$ over some
conical affine scheme $S$ satisfying the following universal property: let
$\mathcal{X}_{B}$ be any graded Poisson deformation of $X$ over a conical
affine scheme $B$. Then there is a unique ${\mathbb{C}^{\times}}$-equivariant
morphism $B\stackrel{{\scriptstyle}}{{\to}}S$ and a (not necessarily unique)
isomorphism of graded Poisson deformations
$\mathcal{X}_{B}\xrightarrow{\sim}\mathcal{X}_{univ}\times_{S}B$ over $B$.

The following results are proved in [Nam11a, Nam10, Nam08] and [Los22a]. We
use the formulation in [LMM21, Section 4.7].

######  Theorem 3.27.

Let $X$ be a conical symplectic singularity with Poisson bracket of degree
$d>0$. Then there exist universal graded Poisson deformations
$\mathcal{X}_{univ}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{P}/W$ and
$\mathcal{Y}_{univ}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{P}$ of $X$ and
$Y$, respectively. There exists a surjective projective Poisson morphism
$\tilde{\rho}:\mathcal{Y}_{univ}\stackrel{{\scriptstyle}}{{\to}}\mathcal{X}_{univ}$
such that:

  1. (i)

The following diagram commutes and all maps are
${\mathbb{C}^{\times}}$-equivariant:

${\mathcal{Y}_{univ}}$${\mathfrak{P}}$${\mathcal{X}_{univ}}$${\mathfrak{P}/W}$$\scriptstyle{\tilde{\rho}}$

Here, $\mathfrak{P}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{P}/W$ is the
quotient map and ${\mathbb{C}^{\times}}$ acts linearly on $\mathfrak{P}$ by
$t.v\mapsto t^{d}v$, i.e., all vectors in $\mathfrak{P}$ are eigenvectors of
weight $-d$. The induced sequence of morphisms
$\mathcal{Y}_{univ}\stackrel{{\scriptstyle}}{{\to}}\mathcal{X}_{univ}\times_{\mathfrak{P}/W}\mathfrak{P}\stackrel{{\scriptstyle}}{{\to}}\mathcal{X}_{univ}$
is the Stein factorization of $\tilde{\rho}$.

  2. (ii)

The algebra $\mathbb{C}[\mathcal{Y}_{univ}]$ carries an action of $W$ by
graded Poisson algebra automorphisms, which makes the map
$\mathbb{C}[\mathfrak{P}]\stackrel{{\scriptstyle}}{{\to}}\mathbb{C}[\mathcal{Y}_{univ}]$
$W$-equivariant. The induced action on $\mathbb{C}[Y]$ is trivial.

  3. (iii)

The map $\tilde{\rho}$ induces an isomorphism
$\mathbb{C}[\mathcal{X}_{univ}]\xrightarrow{\sim}\mathbb{C}[\mathcal{Y}_{univ}]^{W}$.

Set
$\mathcal{X}_{\mathfrak{P}}:=\mathcal{X}_{univ}\times_{\mathfrak{P}/W}\mathfrak{P}$.
Then $\mathcal{X}_{\mathfrak{P}}$ is a Poisson scheme over $\mathfrak{P}$ and
(i) of Theorem 3.27 says that $\tilde{\rho}$ induces a morphism
$\mathcal{Y}_{univ}\stackrel{{\scriptstyle}}{{\to}}\mathcal{X}_{\mathfrak{P}}$
of Poisson schemes over $\mathfrak{P}$.

For any $\lambda\in\mathfrak{P}$, let $X_{\lambda}$ (resp. $Y_{\lambda}$)
denote the fiber of the universal deformation $\mathcal{X}_{univ}$ (resp.
$\mathcal{Y}_{univ}$) over $W\lambda$ (resp. $\lambda$). By Theorem 3.27, the
morphism
$\tilde{\rho}:\mathcal{Y}_{univ}\stackrel{{\scriptstyle}}{{\to}}\mathcal{X}_{univ}$
restricts to a projective birational surjective morphism
$Y_{\lambda}\stackrel{{\scriptstyle}}{{\to}}X_{\lambda}$ with connected
fibers, which induces an isomorphism
$\mathbb{C}[X_{\lambda}]\xrightarrow{\sim}\mathbb{C}[Y_{\lambda}]$ of Poisson
algebras. Let $\mathfrak{P}^{reg}\subset\mathfrak{P}$ denote the subset
consisting of all $\lambda\in\mathfrak{P}$ for which the morphism
$Y_{\lambda}\stackrel{{\scriptstyle}}{{\to}}X_{\lambda}$ is an isomorphism (or
equivalently, $Y_{\lambda}$ is affine) and let
$\mathfrak{P}^{sing}:=\mathfrak{P}\backslash\mathfrak{P}^{reg}$.

######  Theorem 3.28 ([Nam15, Main Theorem, (i)]).

The subset $\mathfrak{P}^{sing}\subset\mathfrak{P}$ is a $W$-stable union of
finitely many rational hyperplanes in $\mathfrak{P}$, including the walls
corresponding to the $W$-action. In particular, $\mathfrak{P}^{reg}$ is a
Zariski-dense open subset of $\mathfrak{P}$.

###  3.8. Universal Poisson deformations of nilpotent covers

Continue with the notation of Section 3.7. In this subsection, we will give an
explicit description of $\mathcal{Y}_{univ}$ in the case of nilpotent covers.
Let $\widetilde{\mathbb{O}}\in\mathsf{Cov}(G)$. Choose
$(L,\widetilde{\mathbb{O}}_{L})\in\mathsf{Cov}_{0}(G)$ such that
$\mathrm{Bind}(L,\widetilde{\mathbb{O}}_{L})=\widetilde{\mathbb{O}}$. Let
$X=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ and
$X_{L}=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])$. Recall
that by Theorem 3.17, any $\mathbb{Q}$-factorial terminalization $Y$ of $X$ is
of the form $Y=G\times^{P}(X_{L}\times\mathfrak{p}^{\perp})$, where $P\subset
G$ is a parabolic subgroup with Levi decomposition $P=LN$.

Before we continue let us introduce the general notion of an induced variety.
Let $Z$ be a Poisson variety equipped with a Hamiltonian $L$-action and a
moment map $\mu_{L}:Z\stackrel{{\scriptstyle}}{{\to}}\mathfrak{l}^{*}$. Let
$Z\times_{\mathfrak{l}^{*}}\mathfrak{n}^{\perp}$ be the pullback of
$\mu_{L}:Z\stackrel{{\scriptstyle}}{{\to}}\mathfrak{l}^{*}$ along the
projection
$\mathfrak{n}^{\perp}=(\mathfrak{g}/\mathfrak{n})^{*}\twoheadrightarrow(\mathfrak{p}/\mathfrak{n})^{*}=\mathfrak{l}^{*}$.
Let $P$ act on $Z$ via the quotient morphism $P\twoheadrightarrow P/N=L$ and
on $\mathfrak{n}^{\perp}$ in the natural way. These actions induce a
$P$-action on $Z\times_{\mathfrak{l}^{*}}\mathfrak{n}^{\perp}$. We define

$\operatorname{Ind}_{P}^{G}Z:=G\times^{P}\left(Z\times_{\mathfrak{l}^{*}}\mathfrak{n}^{\perp}\right).$

Note that $\operatorname{Ind}^{G}_{P}Z\simeq
G\times^{P}(Z\times\mathfrak{p}^{\perp})$. In particular, if $Z=X_{L}$, then
$\operatorname{Ind}^{G}_{P}Z$ coincides with the variety
$Y=G\times^{P}(X_{L}\times\mathfrak{p}^{\perp})$. Finally, we note that
$\operatorname{Ind}^{G}_{P}Z$ is naturally a graded Poisson variety with a
Hamiltonian $G$-action, see [LMM21, Section 7.3] for details.

Set

$\mathfrak{z}:=\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}]),\quad\mathfrak{z}^{\circ}=\\{\chi\in\mathfrak{z}\,|\,G_{\chi}=L\\},\quad\text{and}\quad
X_{L}^{e}:=X_{L}\times\mathfrak{z}.$

(To make sense of the stabilizer $G_{\chi}$ in the definition of
$\mathfrak{z}^{\circ}$, we use the isomorphism
$\mathfrak{g}*\simeq\mathfrak{g}$ induced by the Killing form to identify
$\mathfrak{z}$ with
$\mathfrak{z}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$). The Poisson
structure on $X_{L}$ induces a Poisson structure on $X_{L}^{e}$. The
$L$-action on $X_{L}$ and the natural (trivial) $L$-action on $\mathfrak{z}$
induces a Hamiltonian $L$-action on $X_{L}^{e}$, such that the moment map
$\mu_{L}^{e}:X_{L}^{e}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{l}^{*}$ is the
composition of the projection map $X_{L}^{e}\twoheadrightarrow X_{L}$ with the
moment map $X_{L}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{l}^{*}$. Hence we
can define

$\mathcal{Y}_{\mathfrak{z}}:=\operatorname{Ind}_{P}^{G}X_{L}^{e}=G\times^{P}\left(\mathfrak{z}\times
X_{L}\times\mathfrak{p}^{\perp}\right).$

Then we have a natural $L$-equivariant projection map
$X_{L}^{e}\twoheadrightarrow\mathfrak{z}$, which induces a projection map
$\mathcal{Y}_{\mathfrak{z}}\twoheadrightarrow
G\times^{P}\mathfrak{z}\twoheadrightarrow\mathfrak{z}$ so that
$\mathcal{Y}_{\mathfrak{z}}$ is a graded Poisson scheme over $\mathfrak{z}$.
Note that we have a Poisson isomorphism

$\mathcal{Y}_{\mathfrak{z}}\times_{\mathfrak{z}}\\{0\\}\simeq
Y=G\times^{P}\left(X_{L}\times\mathfrak{p}^{\perp}\right)$

and hence $\mathcal{Y}_{\mathfrak{z}}$ is a graded Poisson deformation of the
$\mathbb{Q}$-factorial terminalization $Y$ of $X$. Set
$\mathcal{X}_{\mathfrak{z}}:=\mathbb{C}[\mathcal{Y}_{\mathfrak{z}}]$. This is
a graded affine Poisson scheme over $\mathfrak{z}$.

######  Proposition 3.29 ([LMM21, Proposition 7.2.2]).

The following are true:

  1. (i)

The universal graded deformation $\mathcal{Y}_{univ}$ of $Y$ in Theorem 3.27
can be identified with $\mathcal{Y}_{\mathfrak{z}}$ over $\mathfrak{z}$ as
graded Poisson deformations of $Y$ under the isomorphism
$\eta:\mathfrak{z}\xrightarrow{\sim}\mathfrak{P}$ of Proposition 3.18.

  2. (ii)

The isomorphism $\mathcal{Y}_{\mathfrak{z}}\simeq\mathcal{Y}_{univ}$ in (i)
induces a graded $G$-equivariant isomorphism
$\mathcal{X}_{\mathfrak{z}}\simeq\mathcal{X}_{univ}\times_{\mathfrak{P}/W}\mathfrak{P}$
of Poisson schemes under the isomorphism
$\eta:\mathfrak{z}\xrightarrow{\sim}\mathfrak{P}$ (cf. Theorem 3.27, (i)).

  3. (iii)

Under the isomorphism $\eta:\mathfrak{z}\xrightarrow{\sim}\mathfrak{P}$, the
open subset $\mathfrak{P}^{reg}\subset\mathfrak{P}$ in Theorem 3.28
corresponds to the open subset $\mathfrak{z}^{\circ}\subset\mathfrak{z}$.

###### Proof.

(i) is [LMM21, Proposition 7.2.2, (iii)]; (ii) follows from Theorem 3.27;
(iii) is [LMM21, Lemma 7.2.4, (i)]. ∎

We can describe the base change of $\mathcal{X}_{\mathfrak{z}}$ and
$\mathcal{Y}_{\mathfrak{z}}$ along the open inclusion
$\mathfrak{z}^{\circ}\hookrightarrow\mathfrak{z}$ explicitly as follows.
Define open subschemes

$(X_{L}^{e})^{\circ}:=X_{L}\times\mathfrak{z}^{\circ}\subset
X_{L}^{e},\quad\mathcal{X}_{\mathfrak{z}}^{\circ}:=\mathcal{X}_{\mathfrak{z}}\times_{\mathfrak{z}}\mathfrak{z}^{\circ}\subset\mathcal{X}_{\mathfrak{z}}\quad\text{and}\quad\mathcal{Y}_{\mathfrak{z}}^{\circ}:=\mathcal{Y}_{\mathfrak{z}}\times_{\mathfrak{z}}\mathfrak{z}^{\circ}\subset\mathcal{Y}_{\mathfrak{z}}$

Note that $(X_{L}^{e})^{\circ}$ is stable under the $L$-action on $X_{L}^{e}$.
Then

$\mathcal{Y}_{\mathfrak{z}}^{\circ}=G\times^{P}\left((X_{L}^{e})^{\circ}\times_{\mathfrak{l}^{*}}\mathfrak{n}^{\perp}\right)=G\times^{P}\left(\mathfrak{z}^{\circ}\times
X_{L}\times\mathfrak{p}^{\perp}\right).$

There is a canonical $G\times\mathbb{C}^{\times}$-equivariant isomorphism

(3.8.1)
$G\times^{L}(X_{L}^{e})^{\circ}=G\times^{P}\left(N\times(X_{L}^{e})^{\circ}\right)\xrightarrow{\sim}\mathcal{Y}_{\mathfrak{z}}^{\circ}=G\times^{P}\left((X_{L}^{e})^{\circ}\times_{\mathfrak{l}^{*}}\mathfrak{n}^{\perp}\right)$

induced by the isomorphism

(3.8.2) $\displaystyle N\times(X_{L}^{e})^{\circ}$
$\displaystyle\xrightarrow{\hskip 2.5pt\sim\hskip
2.5pt}(X_{L}^{e})^{\circ}\times_{\mathfrak{l}^{*}}\mathfrak{n}^{\perp}$
$\displaystyle(n,x^{e})\quad$
$\displaystyle\longmapsto\,n.(x^{e},\mu_{L}^{e}(x^{e}))$

for any $n\in N$ and $x^{e}\in(X_{L}^{e})^{\circ}$. Here we regard
$\mu_{L}^{e}(x^{e})$ as a vector in $\mathfrak{n}^{\perp}$ using the
$L$-equivariant section $\mathfrak{l}^{*}\hookrightarrow\mathfrak{n}^{\perp}$
of the natural projection
$\mathfrak{n}^{\perp}\twoheadrightarrow\mathfrak{l}^{*}$ induced by the
triangular decomposition
$\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{l}\oplus\mathfrak{n}^{-}$. This
means that, even though $\mathcal{Y}_{\mathfrak{z}}$ depends on the choice of
the parabolic $P$, over $\mathfrak{z}^{\circ}$ there are canonical isomophisms
between various $\mathcal{Y}_{\mathfrak{z}}^{\circ}$ for different $P$ which
preserve the Hamiltonian actions. Moreover, by Proposition 3.29, the map
$\mathcal{Y}_{\mathfrak{z}}^{\circ}\stackrel{{\scriptstyle}}{{\to}}\mathcal{X}_{\mathfrak{z}}^{\circ}$
is an isomorphism.

###  3.9. Graded Poisson automorphisms of conical symplectic singularities

Let $X$ be an affine symplectic singularity and define the open subvariety
$X^{\diamond}\subset X$ and its unique $\mathbb{Q}$-factorial terminalization
$\widetilde{X}^{\diamond}$ as in Section 3.7. Let $\theta$ be a Poisson
automorphism of $X$. Note that $\theta$ preserves $X^{\diamond}$, hence
$\theta$ lifts uniquely to a Poisson automorphism $\tilde{\theta}$ of
$\widetilde{X}^{\diamond}$, so that the following diagram commutes

${\widetilde{X}^{\diamond}}$${\widetilde{X}^{\diamond}}$${X^{\diamond}}$${X^{\diamond}}$$\scriptstyle{\tilde{\theta}}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\theta}$

Correspondingly, we have a commutative diagram of functors:

${\mathrm{PD}_{\widetilde{X}^{\diamond}}}$${\mathrm{PD}_{\widetilde{X}^{\diamond}}}$${\mathrm{PD}_{X^{\diamond}}}$${\mathrm{PD}_{X^{\diamond}}}$$\scriptstyle{\tilde{\theta}_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\tilde{\theta}_{*}}$

By the (formal) universality of $\mathrm{PD}_{X^{\diamond}}$ and
$\mathrm{PD}_{\widetilde{X}^{\diamond}}$, this in turn corresponds to a
commutative diagram

(3.9.1)
${\mathfrak{P}^{\wedge}}$${\mathfrak{P}^{\wedge}}$${(\mathfrak{P}/W)^{\wedge}}$${(\mathfrak{P}/W)^{\wedge}}$$\scriptstyle{\tilde{\theta}_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\pi_{*}}$$\scriptstyle{\tilde{\theta}_{*}}$

of completions of affine schemes at their distinguished points.

######  Proposition 3.30.

Assume that $X$ is a conical symplectic singularity and $\theta$ is a graded
(i.e., ${\mathbb{C}^{\times}}$-equivariant) Poisson automorphism of $X$. Then
the commutative diagram (3.9.1) can be obtained by completion from a
commutative diagram of linear spaces

(3.9.2) ${\mathfrak{P}}$${\mathfrak{P}}$${\mathfrak{P}/W}$${\mathfrak{P}/W}$

where the top horizontal map is linear and the vertical ones are quotient
maps. The map $\mathfrak{P}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{P}$ is
given by the pullback map
$(\tilde{\theta}^{-1})^{*}:H^{2}(\widetilde{X}^{\diamond},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(\widetilde{X}^{\diamond},\mathbb{C})$
induced by $\tilde{\theta}^{-1}$.

###### Proof.

$X^{\diamond}$ is ${\mathbb{C}^{\times}}$-stable and the
${\mathbb{C}^{\times}}$-action lifts to a ${\mathbb{C}^{\times}}$-action on
$\widetilde{X}^{\diamond}$. These ${\mathbb{C}^{\times}}$-actions induce those
on $\mathfrak{P}^{\wedge}$ and $(\mathfrak{P}/W)^{\wedge}$ and their
completions, so that the morphisms in (3.9.1) are
${\mathbb{C}^{\times}}$-equivariant. By taking ${\mathbb{C}^{\times}}$-finite
part of the coordinate algebra, we can algebraize the diagram (3.9.1) and get
(3.9.2), whose morphisms are all ${\mathbb{C}^{\times}}$-equivariant. Let
$\gamma:\mathfrak{P}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{P}$ denote the
top horizontal map in (3.9.2).

Since $\mathfrak{P}$ is a vector space, the Zariski tangent space
$T_{0}\mathfrak{P}$ of $\mathfrak{P}$ at $0$ is canonically identified with
$\mathfrak{P}$ itself. Under this identification, it is easy to see that the
differential
$d\gamma:T_{0}\mathfrak{P}\stackrel{{\scriptstyle}}{{\to}}T_{0}\mathfrak{P}$
of $\gamma$ coincides with $\gamma$ itself since $\gamma$ is
${\mathbb{C}^{\times}}$-equivariant. Indeed, by Theorem part (i) of 3.27,
${\mathbb{C}^{\times}}$ acts linearly on $\mathfrak{P}$ by $t.v\mapsto
t^{d}v$. Therefore $\gamma$ being ${\mathbb{C}^{\times}}$-equivariant just
means that $\gamma$ commutes with the scalar mulitiplication on
$\mathfrak{P}$, and hence

$d\gamma(v)=\frac{d}{dt}\gamma(tv)=\frac{d}{dt}[t\gamma(v)]=\gamma(v),\quad\forall\,v\in
T_{0}\mathfrak{P}\simeq\mathfrak{P}.$

In particular, $\gamma$ is a linear automorphism of $\mathfrak{P}$. On the
other hand,
$T_{0}\mathfrak{P}=\mathrm{PD}_{\widetilde{X}^{\diamond}}(\mathbb{C}\\{\epsilon\\})\simeq
H^{2}(\widetilde{X}^{\diamond},\mathbb{C})$ and
$d\gamma=(\tilde{\theta}^{-1})^{*}$. ∎

For any graded Poisson varity $X$, let $\operatorname{Aut}(X)$ denote the
group of graded Poisson automorphisms of $X$. Note that if
$X=\operatorname{Spec}\mathbb{C}[\widetilde{\mathbb{O}}]$ for
$\widetilde{\mathbb{O}}\in\mathsf{Cov}(G)$, then
$\operatorname{Aut}(X)=\operatorname{Aut}(\widetilde{\mathbb{O}})$. For a
general conical symplectic singularity $X$, Proposition 3.30 yields an action
of $\operatorname{Aut}(X)$ on
$\mathfrak{P}=H^{2}(\widetilde{X}^{\diamond},\mathbb{C})$.

######  Proposition 3.31.

Let $X$ be a conical symplectic singularity. Then the action of
$\operatorname{Aut}(X)$ on $\mathfrak{P}=\mathfrak{P}^{X}$ enjoys the
following properties:

  1. (i)

It normalizes $W=W^{X}\subset GL(\mathfrak{P})$, so that the quotient map
$\mathfrak{P}\twoheadrightarrow\mathfrak{P}/W$ is
$\operatorname{Aut}(X)$-equivariant.

  2. (ii)

It lifts to an $\operatorname{Aut}(X)$-action on $X_{\mathfrak{P}}$ by graded
Poisson automorphisms, which normalizes the $W$-action. The induced action on
$X_{0}\simeq X$ is just the defining action of $\operatorname{Aut}(X)$ on $X$.

###### Proof.

(i) is an immediate consequence of Proposition 3.30. For (ii), note that by
the universal property we have a natural $\operatorname{Aut}(X)$-action on the
universal family $\mathcal{X}_{univ}$ over $\mathfrak{P}/W$ by graded Poisson
automorphisms, and this action is compatible with the
$\operatorname{Aut}(X)$-action on $\mathfrak{P}/W$. Then by (i), the fiber
product
$X_{\mathfrak{P}}=\mathcal{X}_{univ}\times_{\mathfrak{P}/W}\mathfrak{P}$
admits an action of $\operatorname{Aut}(X)$. Now $W$ can be identified with
the group of graded Poisson automorphisms of $X_{\mathfrak{P}}$ whose
restrictions to the central fiber
$X_{\mathfrak{P}}\times_{\mathfrak{P}}\\{0\\}\simeq X$ is identity map (this
will appear in [LMM21, Section 7], see also [Los22]). Clearly the
$\operatorname{Aut}(X)$-action on $X_{\mathfrak{P}}$ normalizes $W$. ∎

We conclude this section with an easy lemma.

######  Lemma 3.32.

Let $X_{1}$ and $X_{2}$ be two affine symplectic singularities and define open
subvarieties $X_{i}^{\diamond}\subset X_{i}$, $i=1,2$, as in Section 3.7.
Suppose $p:X_{1}\stackrel{{\scriptstyle}}{{\to}}X_{2}$ is a finite almost
étale (Poisson) morphism, such that its restriction
$p|_{X_{1}^{\diamond}}:X_{1}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}X_{2}^{\diamond}$
is a Galois covering with Galois group $\Gamma$. Then $p$ induces a canonical
injective linear map
$p^{*}:\mathfrak{P}^{X_{2}}\hookrightarrow\mathfrak{P}^{X_{1}}$ whose image is
the the space $(\mathfrak{P}^{X_{1}})^{\Gamma}$ of $\Gamma$-fixed vectors of
$\mathfrak{P}^{X_{1}}$.

###### Proof.

Let
$\widetilde{X}_{2}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}X_{2}^{\diamond}$
be the unique (smooth) $\mathbb{Q}$-factorial terminalization of
$X_{2}^{\diamond}$ and set
$\widetilde{X}_{1}^{\diamond}=\widetilde{X}_{2}^{\diamond}\times_{X_{2}^{\diamond}}X_{1}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}_{2}^{\diamond}$
to be the base change of
$\widetilde{X}_{2}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}X_{2}^{\diamond}$
along the morphism
$p|_{X_{1}^{\diamond}}:X_{1}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}X_{2}^{\diamond}$.
Then
$\widetilde{X}_{1}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}_{2}^{\diamond}$
is also a Galois covering with Galois group $\Gamma$ and the canonical
projection
$\widetilde{X}_{1}^{\diamond}\stackrel{{\scriptstyle}}{{\to}}X_{1}^{\diamond}$
is a (smooth) $\mathbb{Q}$-factorial terminalization of $X_{1}^{\diamond}$.
Therefore the pullback map
$p^{*}:H^{2}(\widetilde{X}_{2}^{\diamond},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(\widetilde{X}_{1}^{\diamond},\mathbb{C})$
satisfies the claim. ∎

###  3.10. The extended Namikawa Weyl groups of nilpotent covers

Fix all of the notation of Section 3.8, e.g. $\widetilde{\mathbb{O}}$, $X$,
$(L,\widetilde{\mathbb{O}}_{L})$, $X_{L}$, $\mathfrak{z}$, and so on. In this
section, we will give an explicit description of the Namikawa Weyl group
$W(\widetilde{\mathbb{O}}):=W^{X}$ in terms of
$(L,\widetilde{\mathbb{O}}_{L})$. This will require some additional notation.
Let $N_{G}(L)$ denote the normalizer of $L$ in $G$. Let
$\operatorname{Ad}^{*}$ denote the co-adjoint action of $N_{G}(L)$ on
$\mathfrak{l}^{*}$ and let
$\mu_{L}:X_{L}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{l}^{*}$ denote the
moment map. Consider the group

$N_{G}(L,\widetilde{\mathbb{O}}_{L}):=\\{(n,\zeta)\in
N_{G}(L)\times\operatorname{Aut}(X_{L})\,|\,\operatorname{Ad}(n)\circ\mu_{L}=\mu_{L}\circ\zeta\\}.$

We can regard $L$ as a normal subgroup of
$N_{G}(L,\widetilde{\mathbb{O}}_{L})$ via the natural embedding

$L\hookrightarrow N_{G}(L,\widetilde{\mathbb{O}}_{L}),\qquad l\mapsto(l,l).$

######  Definition 3.33.

The extended Namikawa-Weyl group associated to $\widetilde{\mathbb{O}}$ is the
finite group

$\widetilde{W}(\widetilde{\mathbb{O}}):=N_{G}(L,\widetilde{\mathbb{O}}_{L})/L.$

We note that $\widetilde{W}(\widetilde{\mathbb{O}})$ acts on
$\mathfrak{z}=\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$ via
the natural map
$\widetilde{W}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}N_{G}(L)/L$.

Next, we will describe an action of $\widetilde{W}(\widetilde{\mathbb{O}})$ on
$\mathcal{X}_{\mathfrak{z}}$. For a general graded Poisson variety $X$ with a
Hamiltonian $G$-action that commutes with the ${\mathbb{C}^{\times}}$-action,
let $\operatorname{Aut}^{G}(X)\subset\operatorname{Aut}(X)$ denote the
subgroup consisting of $G\times{\mathbb{C}^{\times}}$-equivariant Poisson
automorphisms of $X$. Note that if
$X=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ for some
$\widetilde{\mathbb{O}}\in\mathsf{Cov}(G)$, we have
$\operatorname{Aut}^{G}(X)=\operatorname{Aut}^{G}(\widetilde{\mathbb{O}})=\operatorname{Aut}(\widetilde{\mathbb{O}},\mathbb{O})$
by Example 3.8.

######  Theorem 3.34.

With notations as in Section 3.8, the following are true:

  1. (i)

There is a natural action of $\widetilde{W}(\widetilde{\mathbb{O}})$ on
$\mathcal{X}_{\mathfrak{z}}$ by $G\times{\mathbb{C}^{\times}}$-equivariant
Poisson automorphisms preserving the moment map and lifting the action on
$\mathfrak{z}$. Moreover, the action preserves
$\mathcal{X}_{\mathfrak{z}}^{\circ}$ and its restriction on
$\mathcal{X}_{\mathfrak{z}}^{\circ}$ induces an action on
$G\times^{L}(X_{L}^{e})^{\circ}$ via the isomorphism (3.8.1), which descends
from the $N_{G}(L,\widetilde{\mathbb{O}}_{L})$-action on
$G\times^{L}(X_{L}^{e})^{\circ}$ given explicitly by

$[g,x,\beta]\mapsto[gn^{-1},\zeta(x),n.\beta],$

for any $g\in G$,
$(x,\beta)\in(X_{L}^{e})^{\circ}=X_{L}\times\mathfrak{z}^{\circ}$ and
$(n,\zeta)\in N_{G}(L,\widetilde{\mathbb{O}}_{L})$. The action of
$\widetilde{W}(\widetilde{\mathbb{O}})$ on $\mathcal{X}_{\mathfrak{z}}$ is
uniquely characterized by this property.

  2. (ii)

Let $\varphi:X_{\chi}\stackrel{{\scriptstyle}}{{\to}}X_{\chi^{\prime}}$,
$\chi,\chi^{\prime}\in\mathfrak{z}^{\circ}$, be a Hamiltonian isomorphism,
i.e., a $G$-equivariant Poisson isomorphism intertwining the moment maps to
$\mathfrak{g}^{*}$. Then $\varphi$ is induced by a unique element of
$\widetilde{W}(\widetilde{\mathbb{O}})$.

  3. (iii)

There is a short exact sequence of groups,

(3.10.1)
$1\stackrel{{\scriptstyle}}{{\to}}W(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\widetilde{W}(\widetilde{\mathbb{O}})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Aut}^{G}(X)\stackrel{{\scriptstyle}}{{\to}}1,$

where the surjective homomorphism
$\widetilde{W}(\widetilde{\mathbb{O}})\twoheadrightarrow\operatorname{Aut}^{G}(X)$
is given by the restriction of the
$\widetilde{W}(\widetilde{\mathbb{O}})$-action on $\mathcal{X}_{\mathfrak{z}}$
described in (i) to the fiber
$\mathcal{X}_{\mathfrak{z}}\times_{\mathfrak{z}}\\{0\\}\simeq X$.

###### Proof.

The last statement of (i) follows from the fact that
$\mathcal{X}_{\mathfrak{z}}$ is reduced and separated. The rest will appear in
[LMM21, Section 7], see also [Los22]. ∎

In view of 3.31, we can form the semi-direct product

$\widetilde{W}^{X}_{G}:=W^{X}\rtimes\operatorname{Aut}^{G}(X),$

This is a finite group which naturally acts on $\mathfrak{P}$,
$\mathcal{X}_{univ}$ and $\mathcal{X}_{\mathfrak{P}}$. Note that the
definitions of $\widetilde{W}^{X}_{G}$ and its actions are independent of
choices. The following will appear in [LMM21, Section 7], see also [Los22].

######  Proposition 3.35.

A choice of a parabolic subgroup $P=LN\subset G$ determines an isomorphism
$\kappa_{P}:\widetilde{W}(\widetilde{\mathbb{O}})\xrightarrow{\sim}\widetilde{W}^{X}_{G}$,
such that

  1. (1)

$\kappa_{P}$ is compatible with the short exact sequence (3.10.1), i.e., it
restricts to an isomorphism from
$W(\widetilde{\mathbb{O}})\subset\widetilde{W}(\widetilde{\mathbb{O}})$ onto
$W^{X}\subset\widetilde{W}^{X}_{G}$ and the induced map between the quotients
is the identity map of $\operatorname{Aut}^{G}(X)$. In particular, (3.10.1)
splits.

  2. (2)

The isomorphism $\eta:\mathfrak{z}\xrightarrow{\sim}\mathfrak{P}$ intertwines
the action of $\widetilde{W}(\widetilde{\mathbb{O}})$ on $\mathfrak{z}$ and
the action of $\widetilde{W}^{X}_{G}$ on $\mathfrak{P}$ under the
identification
$\kappa_{P}:\widetilde{W}(\widetilde{\mathbb{O}})\xrightarrow{\sim}\widetilde{W}^{X}_{G}$.

###  3.11. Extended Namikawa Weyl group vs parabolic induction

Fix all of the notation of Section 3.8, e.g. $\widetilde{\mathbb{O}}$, $X$,
$(L,\widetilde{\mathbb{O}}_{L})$, $X_{L}$, $\mathfrak{z}$, and so on. Suppose
$M$ is a Levi subgroup of $G$ containing $L$. Set
$\widetilde{\mathbb{O}}_{M}=\mathrm{Bind}_{L}^{M}\widetilde{\mathbb{O}}_{L}$
and $X_{M}=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{M}])$. We
want to compare the Namikawa spaces and (extended) Namikawa groups of
$\widetilde{\mathbb{O}}$ and $\widetilde{\mathbb{O}}_{M}$. To simplify the
statements, we assume in this section that $G$ is semisimple.

Choose parabolic subgroups $P=LN\subset G$ and $Q=MU\subset G$, such that
$L\subset M$ and $P\subset Q$. Let $P_{M}=P\cap M=LN_{M}$, a parabolic
subgroup of $M$ with Levi factor $L$ and unipotent radical $N_{M}=N\cap M$.
Note that $\mathfrak{n}=\mathfrak{n}_{M}\oplus\mathfrak{u}$. We have an
inclusion
$\mathfrak{X}(\mathfrak{m})\hookrightarrow\mathfrak{X}(\mathfrak{l})$ and a
projection
$\mathfrak{X}(\mathfrak{l})\twoheadrightarrow\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])$
induced by restriction of characters. We write

$\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}:=\mathfrak{X}(\mathfrak{m}),\quad\mathfrak{z}_{\mathfrak{l}}^{\mathfrak{m}}:=\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}]).$

Note that we have natural isomorphisms
$\mathfrak{z}\simeq\mathfrak{z}(\mathfrak{l})^{*}$,
$\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}\simeq\mathfrak{z}(\mathfrak{m})^{*}$
and
$\mathfrak{z}_{\mathfrak{l}}^{\mathfrak{m}}\simeq\mathfrak{z}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])^{*}$.
Moreover, we have a direct sum decomposition
$\mathfrak{z}(\mathfrak{l})=\mathfrak{z}(\mathfrak{m})\oplus\mathfrak{z}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])$.
The latter induces a direct sum decomposition
$\mathfrak{z}=\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}\oplus\mathfrak{z}_{\mathfrak{l}}^{\mathfrak{g}}$.
Set $\mathcal{Y}_{\mathfrak{z}}:=\operatorname{Ind}_{P}^{G}X_{L}^{e}$,
$\mathcal{Y}_{\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}}^{M}:=\operatorname{Ind}_{P_{M}}^{M}(X_{L}\times\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}})$,
$\mathcal{Y}_{\mathfrak{z}}^{M}:=\operatorname{Ind}_{P_{M}}^{M}X_{L}^{e}$ and
$Y_{M}=\operatorname{Ind}_{P_{M}}^{M}X_{L}$. As before, set
$\mathcal{X}_{\mathfrak{z}}:=\mathbb{C}[\mathcal{Y}_{\mathfrak{z}}]$ and
similarly
$\mathcal{X}_{\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}}^{M}:=\mathbb{C}[\mathcal{Y}_{\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}}^{M}]$,
$\mathcal{X}_{\mathfrak{z}}^{M}:=\mathbb{C}[\mathcal{Y}_{\mathfrak{z}}^{M}]$.
Then
$\mathcal{Y}_{\mathfrak{z}}^{M}=\mathcal{Y}_{\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}}^{M}\times\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}$
and
$\mathcal{X}_{\mathfrak{z}}^{M}=X_{\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}}^{M}\times\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}$.

There is a canonical $G\times\mathbb{C}^{\times}$-equivariant isomorphism

(3.11.1)
$\mathcal{Y}_{\mathfrak{z}}\xrightarrow{\sim}\operatorname{Ind}_{Q}^{G}\mathcal{Y}_{\mathfrak{z}}^{M}$

or

(3.11.2) $\displaystyle
G\times^{P}\left(X_{L}^{e}\times_{\mathfrak{l}^{*}}\mathfrak{n}^{\perp}\right)$
$\displaystyle\xrightarrow{\>\sim\>}G\times^{Q}\left(M\times^{P_{M}}\left(X_{L}^{e}\times_{\mathfrak{l}^{*}}(\mathfrak{m}/\mathfrak{n}_{M})^{*}\right)\times_{\mathfrak{m}^{*}}(\mathfrak{g}/\mathfrak{u})^{*}\right)$
$\displaystyle[g,x^{e},\alpha]\qquad$
$\displaystyle\longmapsto\qquad[g,1_{M},x^{e},\pi_{\mathfrak{m}}(\alpha),\iota_{\mathfrak{u}}(\alpha)]$

for any $g\in G$, $x^{e}\in X_{L}^{e}$ and $\alpha\in\mathfrak{n}^{\perp}$,
where $1_{M}$ denotes the identity element of $M$. Here we are using the
canonical isomorphism

$\mathfrak{n}^{\perp}\xrightarrow{\sim}(\mathfrak{m}/\mathfrak{n}_{M})^{*}\times_{\mathfrak{m}^{*}}(\mathfrak{g}/\mathfrak{u})^{*},\quad\alpha\mapsto(\pi_{\mathfrak{m}}(\alpha),\iota_{\mathfrak{u}}(\alpha)),$

of $L$-representations, which is given by the pullback diagram

${\mathfrak{n}^{\perp}}$${(\mathfrak{g}/\mathfrak{u})^{*}}$${(\mathfrak{m}/\mathfrak{n}_{M})^{*}}$${\mathfrak{m}^{*}}$$\scriptstyle{\iota_{\mathfrak{u}}}$$\scriptstyle{\pi_{\mathfrak{m}}}$

where $\iota_{\mathfrak{u}}$ is the dual map of the projection
$\mathfrak{g}/\mathfrak{u}\twoheadrightarrow\mathfrak{g}/\mathfrak{n}$ and
$\pi_{\mathfrak{m}}$ is the dual map of the inclusion
$\mathfrak{m}/\mathfrak{n}_{M}\hookrightarrow\mathfrak{g}/\mathfrak{n}$. It
can be checked that (3.11.1) is well-defined and intertwines the maps to
$\mathfrak{g}^{*}\times\mathfrak{z}$.

Recall that $\mathcal{Y}_{\mathfrak{z}}\times_{\mathfrak{z}}\\{0\\}$ is
isomorphic to
$Y=G\times^{P}\left(X_{L}\times(\mathfrak{g}/\mathfrak{p})^{*}\right)$.
Therefore restricting the isomorphism (3.11.1) to the fibers over
$0\in\mathfrak{z}$ gives an isomorphism

(3.11.3)
$Y=G\times^{P}(X_{L}\times\mathfrak{p}^{\perp})\xrightarrow{\sim}\operatorname{Ind}_{P}^{G}Y_{M}=G\times^{Q}(Y_{M}\times\mathfrak{q}^{\perp}).$

(cf. [LMM21, Lemma 7.4.1]). Thus we can view $Y^{reg}$ as a fiber bundle over
$G/Q$ with fiber $Y_{M}^{reg}\times\mathfrak{q}^{\perp}$, so that there are
pullback maps

(3.11.4)
$H^{2}(G/Q,\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(Y^{reg},\mathbb{C}),\qquad\mathfrak{P}(\widetilde{\mathbb{O}})=H^{2}(Y^{reg},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}\mathfrak{P}(\widetilde{\mathbb{O}}_{M})=H^{2}(Y^{reg}_{M},\mathbb{C}).$

On the other hand, we have identifications

(3.11.5)
$\mathfrak{X}(\mathfrak{m})\xrightarrow{\sim}H^{2}(G/Q,\mathbb{C}),\quad\eta:\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}H^{2}(Y^{reg},\mathbb{C}),\quad\eta_{M}:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])\xrightarrow{\sim}H^{2}(Y^{reg}_{M},\mathbb{C}),$

where the latter two are from Proposition 3.18. The following result is
Proposition 7.4.2 of [LMM21].

######  Proposition 3.36.

Under the isomorphisms in (3.11.5), the pullback maps in (3.11.4) correspond
to the inclusion
$\mathfrak{X}(\mathfrak{m})\hookrightarrow\mathfrak{X}(\mathfrak{l})$ and the
projection
$\mathfrak{X}(\mathfrak{l})\twoheadrightarrow\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])$,
respectively.

Next we compare the extended Namikawa groups
$\widetilde{W}(\widetilde{\mathbb{O}})$ and
$\widetilde{W}(\widetilde{\mathbb{O}}_{M})$. Note that the natural inclusion
$N_{M}(L)\subset N_{G}(L)$ induces an inclusion
$N_{M}(L,\widetilde{\mathbb{O}}_{L})\subset
N_{G}(L,\widetilde{\mathbb{O}}_{L})$ and hence an injective group homomorphism
$\iota_{\widetilde{W}}:\widetilde{W}(\widetilde{\mathbb{O}}_{M})\hookrightarrow\widetilde{W}(\widetilde{\mathbb{O}})$.
The restriction of the standard action of
$\widetilde{W}(\widetilde{\mathbb{O}})$ on
$\mathfrak{z}=\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}\oplus\mathfrak{z}_{\mathfrak{l}}^{\mathfrak{g}}$
to $\widetilde{W}(\widetilde{\mathbb{O}}_{M})$ is the product of the standard
action of $\widetilde{W}(\widetilde{\mathbb{O}}_{M})$ on
$\mathfrak{z}_{\mathfrak{l}}^{\mathfrak{g}}$ and the trivial one on
$\mathfrak{z}_{\mathfrak{m}}^{\mathfrak{g}}$.

Note that there is also a natural inclusion
$\iota_{\operatorname{Aut}}:\operatorname{Aut}^{M}(X_{M})\hookrightarrow\operatorname{Aut}^{G}(X)$
of groups induced by the natural action of
$\operatorname{Aut}^{M}(X_{M})\simeq\operatorname{Aut}^{M}(\widetilde{\mathbb{O}}_{M})$
on
$\mathrm{Bind}_{M}^{G}\widetilde{\mathbb{O}}_{M}=\mathrm{Bind}_{M}^{G}\mathrm{Bind}_{L}^{M}\widetilde{\mathbb{O}}_{L}=\mathrm{Bind}_{L}^{G}\widetilde{\mathbb{O}}_{L}=\widetilde{\mathbb{O}}$
(cf. Proposition 2.6). We will also need the following result, which is
immediate from the definition of the $\operatorname{Aut}(X)$-action on the
Namikawa space $\mathfrak{P}$.

######  Lemma 3.37.

The pullback map
$H^{2}(Y^{reg},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(Y_{M}^{reg},\mathbb{C})$
in (3.11.4) is $\operatorname{Aut}^{M}(X_{M})$-equivariant, where