# Unipotent Ideals for Spin and Exceptional Groups

Lucas Mason-Brown  Mathematical Institute, University of Oxford, Oxford, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Dmytro
Matvieievskyi  Department of Mathematics, Northeastern University, Boston, MA
USA

###### Abstract.

In the monograph [LMM21], we define the notion of a unipotent representation
of a complex reductive group. The representations we define include, as a
proper subset, all special unipotent representations in the sense of [BV85]
and form the (conjectural) building blocks of the unitary dual. In [LMM21] we
provide combinatorial formulas for the infinitesimal characters of all
unipotent representations of linear classical groups. In this paper, we
establish analogous formulas for spin and exceptional groups, thus completing
the determination of the infinitesimal characters of all unipotent ideals.
Using these formulas, we prove an old conjecture of Vogan: all unipotent
ideals are maximal. For $G$ a real reductive Lie group (not necessarily
complex), we introduce the notion of a unipotent representation attached to a
rigid nilpotent orbit (in the complexified Lie algebra of $G$). Like their
complex group counterparts, these representations form the (conjectural)
building blocks of the unitary dual. Using the atlas software (and the work of
[AMLV]) we show that if $G$ is a real form of a simple group of exceptional
type, all such representations are unitary.

###### Key words and phrases:

unipotent representations, nilpotent orbits

###### 1991 Mathematics Subject Classification:

17B35, 17B08, 22E46

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Quantizations of conical symplectic singularities
    1. 2.1 Filtered quantizations
    2. 2.2 Conical symplectic singularities
    3. 2.3 Namikawa space
    4. 2.4 Structure of Namikawa space
    5. 2.5 Finite covers of conical symplectic singularities
    6. 2.6 Filtered quantizations of conical symplectic singularities
  3. 3 Nilpotent orbits and covers
    1. 3.1 Classification of nilpotent orbits and covers
    2. 3.2 Geometry of nilpotent orbits and covers
    3. 3.3 Induction of nilpotent orbits
    4. 3.4 Birational induction of nilpotent covers
    5. 3.5 Filtered quantizations of nilpotent covers
    6. 3.6 Description of partial Namikawa spaces
    7. 3.7 Geometric characterization of birationally rigid covers
    8. 3.8 Classification of birationally rigid orbits
    9. 3.9 Classification of birationally semi-rigid orbits
    10. 3.10 Some computational tools
      1. 3.10.1 Computation of $(L,\mathbb{O}_{L})$
      2. 3.10.2 Computation of $(M_{k},\mathbb{O}_{M_{k}})$
      3. 3.10.3 Computation of $\eta_{k}$
  4. 4 Unipotent ideals and their infinitesimal characters
    1. 4.1 Computing $\gamma_{0}(\widetilde{\mathbb{O}})$
      1. 4.1.1 Reduction to the case of birationally rigid orbits
      2. 4.1.2 Reduction to the case of rigid orbits
      3. 4.1.3 Case of rigid orbits
    2. 4.2 Spin groups
    3. 4.3 Exceptional groups
      1. 4.3.1 Birationally rigid orbits
      2. 4.3.2 Birationally rigid covers
  5. 5 Maximality of unipotent ideals
  6. 6 Real groups
    1. 6.1 Unitarity
  7. A Maximality computations
    1. A.1 Spin groups
    2. A.2 Exceptional groups

##  1\. Introduction

Let $G$ be a complex reductive algebraic group and let
$\mathfrak{g}=\mathrm{Lie}(G)$. Inspired by the orbit method for nilpotent and
solvable Lie groups, Vogan proposes the following in [Vog90]:

######  Conjecture 1.0.1.

For each (finite connected) cover $\widetilde{\mathbb{O}}$ of a nilpotent co-
adjoint $G$-orbit, there is a canonically defined completely prime primitive
ideal $I_{0}(\widetilde{\mathbb{O}})$ in the universal enveloping algebra
$U(\mathfrak{g})$.

The properties of the ideal $I_{0}(\widetilde{\mathbb{O}})$ and its relation
to $\widetilde{\mathbb{O}}$ are described in some detail in [Vog90] and also
in [Vog87, Chp 9]. In particular, the ideals $I_{0}(\widetilde{\mathbb{O}})$
should be maximal and should include, as a proper subset, all special
unipotent ideals in the sense of [BV85]. Conjecture 1.0.1 is called the
_quantization problem for nilpotent covers_ and the conjectured ideals are
called _unipotent_. We note that Conjecture 1.0.1 is intimately related to the
problem of classifying unitary representations. If we can define unipotent
ideals in $U(\mathfrak{g})$, we can define unipotent representations (of the
complex group $G$)—a $G$-representation is _unipotent_ if it is irreducible
and annihilated (on both sides) by a unipotent ideal. It is conjectured in
[Vog87, Chp 9] that unipotent representations are unitary, and in fact form
the ‘building blocks’ of the unitary dual.

Conjecture 1.0.1 has generated a large body of research over the past several
decades, with contributions from Barbasch ([Bar89]), Vogan
([Vog86],[Vog88],[Vog90]), Joseph ([Jos76]), McGovern ([McG94]), Brylinski
([Bry03]), and others. Previous approaches have made use of specialized
constructions to resolve Conjecture 1.0.1 in certain special cases (e.g. for
$\widetilde{\mathbb{O}}$ equal to the _minimal nilpotent orbit_ , as in
[Jos76], or for $G$ equal to a linear classical group, as in [Bar89],[McG94],
and [Bry03]). In the monograph [LMM21], we give a systematic solution to
Conjecture 1.0.1 in all cases, using the theory of filtered quantizations of
conical symplectic singularities. Our definition of
$I_{0}(\widetilde{\mathbb{O}})$ will be recalled in Section 4.

Write $\mathfrak{z}(U(\mathfrak{g}))$ for the center of $U(\mathfrak{g})$. If
$I\subset U(\mathfrak{g})$ is a primitive ideal, then
$I\cap\mathfrak{z}(U(\mathfrak{g}))$ is the kernel of an algebra homomorphism
$\mathfrak{z}(U(\mathfrak{g}))\stackrel{{\scriptstyle}}{{\to}}\mathbb{C}$,
called the _infinitesimal character_ of $I$. If we fix a Cartan subalgebra
$\mathfrak{h}\subset\mathfrak{g}$, we can use the Harish-Chandra isomorphism
$\mathfrak{z}(U(\mathfrak{g}))\simeq\mathbb{C}[\mathfrak{h}^{*}]^{W}$ to
identify infinitesimal characters with $W$-orbits on $\mathfrak{h}^{*}$. Write
$\gamma_{0}(\widetilde{\mathbb{O}})\in\mathfrak{h}^{*}/W$ for the
infinitesimal character of the unipotent ideal
$I_{0}(\widetilde{\mathbb{O}})$. One drawback of the approach taken in [LMM21]
is that the infinitesimal character $\gamma_{0}(\widetilde{\mathbb{O}})$ is
not easy to determine from the definition of $I_{0}(\widetilde{\mathbb{O}})$.
The computation of $\gamma_{0}(\widetilde{\mathbb{O}})$ requires a detailed
analysis of the birational geometry of $\widetilde{\mathbb{O}}$. In [LMM21],
we deduce combinatorial formulas for all $\gamma_{0}(\widetilde{\mathbb{O}})$
in the case of linear classical groups. In this paper, we establish analogous
formulas for spin and exceptional groups, thus completing the computation of
all unipotent ideals.

We will now provide a more detailed overview of the results contained in this
paper. One of the crucial ideas herein is _birational induction_. This is an
operation which takes nilpotent covers for a Levi subgroup $L\subset G$ to
nilpotent covers for $G$. A nilpotent cover is called _birationally rigid_ if
it cannot be obtained via birational induction from a proper Levi subgroup. In
Sections 3.8 and 3.9, we give a classification of birationally rigid nilpotent
covers for simple exceptional groups, see Propositions 3.8.3 and 3.9.5.

One important property of unipotent ideals is that their infinitesimal
characters are preserved under birational induction. More precisely, if
$\widetilde{\mathbb{O}}_{G}$ is birationally induced from
$\widetilde{\mathbb{O}}_{L}$, then

(1.0.1)
$\gamma_{0}(\widetilde{\mathbb{O}}_{G})=\gamma_{0}(\widetilde{\mathbb{O}}_{L}).$

In view of (1.0.1), it is enough to compute the infinitesimal characters
attached to birationally rigid covers. For simple exceptional groups, we
provide in Section 4.3 a complete list of such infinitesimal characters (see
Tables 4-13). For spin groups, we provide in Section 4.2 a combinatorial
formula (see Proposition 4.2.6).

These computations allow us to complete the proof of the following conjecture
of Vogan (see [Vog87, Conj 9.18]).

######  Theorem 1.0.2 (See Theorem 5.0.1 below).

Let $G$ be a complex connected reductive algebraic group and let
$\widetilde{\mathbb{O}}$ be a (finite connected) covering of a nilpotent co-
adjoint $G$-orbit. Then $I_{0}(\widetilde{\mathbb{O}})$ is a maximal ideal in
$U(\mathfrak{g})$.

In Section 6, we investigate the applicability of these ideas to real
reductive groups. Suppose $G$ is a real reductive Lie group (not necessarily
linear) and let $\mathbb{O}\subset\mathfrak{g}^{*}$ be a rigid nilpotent orbit
for the complexified Lie algebra of $G$. Let $K\subset G$ be a maximal compact
subgroup. We propose the following definition.

######  Definition 1.0.3 (See Definition 6.0.1 below).

A unipotent representation of $G$ attached to $\mathbb{O}$ is an irreducible
$(\mathfrak{g},K)$-module $\mathcal{B}$ such that
$\operatorname{Ann}_{U(\mathfrak{g})}(\mathcal{B})=I_{0}(\mathbb{O})$.

If $G$ is a real form of a simple exceptional group and $\mathbb{O}$ is a
special nilpotent orbit, then it will be shown in [AMLV] that all such
representations are unitary. In Section 6.1, we do the same for non-special
orbits, proving the following result.

######  Theorem 1.0.4 (See Theorem 6.1.1 below).

Suppose $G$ is a real form of a simple exceptional group and
$\mathbb{O}\subset\mathfrak{g}^{*}$ is a rigid nilpotent orbit. Then all
unipotent representations attached to $\mathbb{O}$ are unitary.

The proof of this result is a straightforward (albeit time-consuming)
computation using the atlas software.

The methods used in this paper mostly belong to the theory of filtered
quantizations of conical symplectic singularities. We review what is needed
from this theory in Section 2. The reader who is not interested in our methods
may prefer to regard this paper as a sort of ‘treasure map’ for interesting
unitary representations of spin and exceptional groups. Many of the
representations we describe have never been studied in the literature.

###  1.1. Acknowledgements

The authors would like to thank Ivan Losev and David Vogan for many helpful
conversations. Many of the computations in this paper were assisted, and in
some cases wholly carried out, using the atlas software. We would like to
acknowledge the whole atlas team, especially Jeffrey Adams, Annegret Paul,
Marc van Leeuwen, and David Vogan, for creating and maintaining such
fantastically useful software. The work of D.M. was partially supported by the
NSF under grant DMS-2001139.

##  2\. Quantizations of conical symplectic singularities

In this section, we review the theory of filtered quantizations of conical
symplectic singularities. The results in this section come from a variety of
sources, including [Bea00], [Kal06], [Los16], and [LMM21]. Our exposition
roughly follows Section 4 of [LMM21].

###  2.1. Filtered quantizations

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

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

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

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

For any algebra of this form, one can define the notion of a _filtered
quantization_.

######  Definition 2.1.1.

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, 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. 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 $A$ 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 setting (as above) we get a Lie algebra homomorphism

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

######  Definition 2.1.2.

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 $\mathrm{Quant}^{G}(A)$.

###  2.2. Conical symplectic singularities

Let $X$ be a normal Poisson variety.

######  Definition 2.2.1 ([Bea00], Def 1.1).

We say that $X$ has _symplectic singularities_ if

  * (i)

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

  * (ii)

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

In this paper, we will consider symplectic singularities of a very special
type. Let $d\in\mathbb{Z}_{>0}$ as in Section 2.1.

######  Definition 2.2.2.

A _conical symplectic singularity_ is a normal affine Poisson variety $X$ with
symplectic singularities and a contracting rational
$\mathbb{C}^{\times}$-action such that $\mathbb{C}[X]$ is a graded Poisson
algebra of degree $-d$.

######  Example 2.2.3.

The following are examples of conical symplectic singularities

  * (i)

Let $\Gamma\subset\mathrm{Sp}(2)$ be a finite subgroup. Then the Kleinian
singularity $\Sigma=\mathbb{C}^{2}/\Gamma$ is a conical symplectic
singularity, see [Bea00, Prop 2.4]. For $\rho$ we take the minimal resolution
$\mathfrak{S}\stackrel{{\scriptstyle}}{{\to}}\Sigma$.

  * (ii)

Let $\mathfrak{g}$ be a complex reductive Lie algebra and let
$\mathbb{O}\subset\mathfrak{g}^{*}$ be a nilpotent orbit. Then
$\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ is a conical symplectic
singularity, see [Bea00, Sec 2.5].

  * (iii)

In the setting of $(ii)$, let
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ be a
connected finite étale cover. Then
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ is a conical
symplectic singularity, see [Los21, Lem 2.5].

For an arbitrary variety $X$, define the subvarieties $X_{0},X_{1},X_{2},...$
as follows: $X_{0}:=X$ and $X_{k+1}:=X_{k}-X_{k}^{\mathrm{reg}}$. If $X$ is
Poisson, then all $X_{k}$ are Poisson subvarieties of $X$.

######  Definition 2.2.4.

We say that $X$ has finitely many (symplectic) leaves if
$X_{k}^{\mathrm{reg}}$ is a symplectic variety for all $k$. By a symplectic
leaf of $X$ we mean an irreducible (i.e. connected) component of
$X_{k}^{\mathrm{reg}}$ for some $k$.

######  Proposition 2.2.5 (Thm 2.3, [Kal06]).

Suppose $X$ has symplectic singularities. Then $X$ has finitely many leaves.

###  2.3. Namikawa space

Let $X$ be a normal Poisson variety with symplectic singularities. Recall that
a normal variety $Y$ is $\mathbb{Q}$-_factorial_ if every Weil divisor has a
(nonzero) integer multiple which is Cartier. The following is a consequence of
[BCHM10] (see [Los19, Prop 2.1] for a proof).

######  Proposition 2.3.1.

There is a birational projective morphism
$\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ such that

  * (i)

$Y$ is an irreducible, normal, Poisson variety (in particular, $Y$ has
symplectic singularities).

  * (ii)

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

  * $(iii)$

$Y$ has terminal singularities.

######  Remark 2.3.2.

Modulo (i), (iii) is equivalent to the condition that the singular locus of
$Y$ is of codimension $\geq 4$, see [Nam01, Main Thm]. In practice, the latter
condition is often easier to check.

The map $\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ in the proposition above (or
the variety $Y$ itself, if the map is understood) is called a
$\mathbb{Q}$-_factorial terminalization_. If $X$ is conical, then $Y$ admits a
$\mathbb{C}^{\times}$-action such that $\rho$ is
$\mathbb{C}^{\times}$-equivariant, see [Nam08, A.7].

######  Example 2.3.3.

Let $\mathfrak{g}$ be a complex reductive Lie algebra and let
$\mathcal{N}\subset\mathfrak{g}^{*}$ be its nilpotent cone. By Example
2.2.3(ii) (and the normality of $\mathcal{N})$, $X:=\mathcal{N}$ is a conical
symplectic singularity. For $\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$ we take
the Springer resolution $T^{*}(G/B)\stackrel{{\scriptstyle}}{{\to}}X$.

######  Definition 2.3.4.

Let $X$ be a conical symplectic singularity and $Y$ a $\mathbb{Q}$-factorial
terminalization of $X$. The _Namikawa space_ associated to $X$ is the complex
vector space

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

######  Remark 2.3.5.

It was shown in [LMM21] that $\mathfrak{P}^{X}$ depends only on $X$ up to
canonical isomorphism (i.e. it is independent of the choice of
$\mathbb{Q}$-terminalization), see [LMM21, Lem 4.6.6] and the discussion
following it. In particular, the notation $\mathfrak{P}^{X}$ is justified.

###  2.4. Structure of Namikawa space

Following [Nam11] and [Los16], we will provide a description of
$\mathfrak{P}^{X}$ in terms of the geometry of $X$. For this discussion, it is
convenient to fix a $\mathbb{Q}$-terminalization
$\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$. For each codimension 2 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
$\Sigma_{k}=\mathbb{C}^{2}/\Gamma_{k}$, see [Nam11]. 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 $H^{2}(\rho^{-1}(x),\mathbb{Z})\simeq\Lambda_{k}$, 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

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

The embedding $\rho^{-1}(x)\subset Y^{\mathrm{reg}}$ induces a map on
cohomology

$\mathfrak{P}^{X}:=H^{2}(Y^{\mathrm{reg}},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(\rho^{-1}(x),\mathbb{C})^{\pi_{1}(\mathfrak{L}_{k})}\simeq(\mathfrak{h}_{k}^{*})^{\pi_{1}(\mathfrak{L}_{k})}=:\mathfrak{P}^{X}_{k}.$

Also, define

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

The embedding $X^{\mathrm{reg}}\subset Y^{\mathrm{reg}}$ induces a map on
cohomology

$\mathfrak{P}^{X}:=H^{2}(Y^{\mathrm{reg}},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(X^{\mathrm{reg}},\mathbb{C})=:\mathfrak{P}^{X}_{0}.$

######  Proposition 2.4.1 ([Los16], Lem 2.8).

The maps $\mathfrak{P}^{X}\rightarrow\mathfrak{P}^{X}_{k}$ assemble into a
linear isomorphism

(2.4.1)
$\mathfrak{P}^{X}\simeq\bigoplus_{k=0}^{t}\mathfrak{P}^{X}_{k},\qquad\lambda\mapsto(\lambda_{0},\lambda_{1},...,\lambda_{t}).$

Finally, we define the _Namikawa Weyl group_ of $X$. The
$\pi_{1}(\mathfrak{L}_{k})$-action on $\Lambda_{k}$ induces a
$\pi_{1}(\mathfrak{L}_{k})$-action on $W_{k}$. Consider the subgroup
$W_{k}^{\pi_{1}(\mathfrak{L}_{k})}\subset W_{k}$. There is a natural action of
$W_{k}^{\pi_{1}(\mathfrak{L}_{k})}$ on
$\mathfrak{P}^{X}_{k}=(\mathfrak{h}_{k}^{*})^{\pi_{1}(\mathfrak{L}_{k})}$. The
_Namikawa Weyl group_ associated to $X$ is the product

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

We let $W$ act on $\mathfrak{P}$ via the isomorphism (2.4.1) (the action on
$\mathfrak{P}^{X}_{0}$ is trivial).

###  2.5. Finite covers of conical symplectic singularities

Let $X$ be a conical symplectic singularity. In this section, we will define
the notion of a _finite cover_ of $X$. Let
$p^{\prime}:\widetilde{X}^{\prime}\stackrel{{\scriptstyle}}{{\to}}X^{\mathrm{reg}}$
be a finite étale cover of the regular locus $X^{\mathrm{reg}}\subset X$.
Rescaling if necessary, we can arrange so that the
$\mathbb{C}^{\times}$-action on $X^{\mathrm{reg}}$ lifts to
$\widetilde{X}^{\prime}$. Consider the composition
$\widetilde{X}^{\prime}\overset{p^{\prime}}{\stackrel{{\scriptstyle}}{{\to}}}X^{\mathrm{reg}}\hookrightarrow
X$ and its Stein factorization

${\widetilde{X}^{\prime}}$${\widetilde{X}}$${X^{\mathrm{reg}}}$${X}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p}$

Note that $\widetilde{X}$ is affine and $\widetilde{X}^{\prime}$ embeds into
$\widetilde{X}$ as an open subvariety. Since
$\operatorname{codim}(X^{\mathrm{sing}},X)\geqslant 2$ and
$p:\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}X$ is finite, we have that
$\operatorname{codim}(\widetilde{X}-\widetilde{X}^{\prime},\widetilde{X})\geq
2$. Thus the algebra $\mathbb{C}[\widetilde{X}^{\prime}]$ is finitely
generated and
$\widetilde{X}=\operatorname{Spec}(\mathbb{C}[\widetilde{X}^{\prime}])$. In
particular, the $\mathbb{C}^{\times}$-action on $\widetilde{X}^{\prime}$
extends to $\widetilde{X}$. In fact, $\widetilde{X}$ is a conical symplectic
singularity, see [Los21, Lemma 2.5]. A map
$p:\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}X$ obtained in this fashion is
called a _finite cover_ of $X$. A finite cover
$p:\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}X$ is _Galois_ if it is Galois
over the regular locus $X^{\mathrm{reg}}$.

For each codimension 2 leaf $\mathfrak{L}_{k}\subset X$, choose a system of
fundamental weights

$\omega_{1}(k),\omega_{2}(k),...,\omega_{n(k)}(k)\in\mathfrak{h}_{k}^{*}$

To each fundamental weight $\omega_{i}(k)$ we assign a coefficient $a_{i}$,
which is the multiplicity of the corresponding simple root in the highest
weight for $\mathfrak{g}_{k}$ (in type $A$, $a_{i}=1$ for every $i$, and this
is the only case that will concern us). Define the element

$\epsilon_{k}:=|\Gamma_{k}|^{-1}\sum_{i}a_{i}\omega_{i}\in\mathfrak{h}_{k}^{*}.$

######  Lemma 2.5.1 (Prop 5.3.1,[LMM21]).

Suppose $X$ admits a finite Galois cover
$\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}X$ such that $\widetilde{X}$ has
no codimension 2 leaves. Then for each $\mathfrak{L}_{k}\subset X$, the
element $\epsilon_{k}$ is a fixed point for $\pi_{1}(\mathfrak{L}_{k})$ and
hence an element of the partial Namikawa space
$\mathfrak{P}_{k}=(\mathfrak{h}_{k}^{*})^{\pi_{1}(\mathfrak{L}_{k})}$.

In the setting of Lemma 2.5.1, define

(2.5.1)
$\epsilon:=(0,\epsilon_{1},\epsilon_{2},...,\epsilon_{t})\in\bigoplus_{i}\mathfrak{P}^{X}_{i}\simeq\mathfrak{P}^{X}$

We call $\epsilon$ the _weighted barycenter parameter_ for $X$. We note that
the Galois group $\Pi$ of $\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}X$
acts on the canonical quantization $\mathcal{A}_{0}^{\widetilde{X}}$, and
$(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}\simeq\mathcal{A}_{\epsilon}^{X}$, see
[LMM21, Prop 5.3.1].

###  2.6. 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 choose a $\mathbb{Q}$-terminalization
$\rho:Y\stackrel{{\scriptstyle}}{{\to}}X$. 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 2.6.1 ([Los16], Prop 3.1(1)).

The maps

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

are bijections.

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

######  Theorem 2.6.2 (Prop 3.3, Thm 3.4, [Los16]).

The following are true:

  * (i)

For every $\lambda\in\mathfrak{P}$, 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 $\lambda\in\mathfrak{P}$.

  * (iii)

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

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

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

There is an equivariant version of this result, which we will now state. Let
$G$ be a connected reductive algebraic group and suppose $A:=\mathbb{C}[X]$
admits a Hamiltonian $G$-action, see Section 2.1. Define the _extended
Namikawa space_

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

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

######  Proposition 2.6.3 (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

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

One consequence of this proposition is that there is always a distinguished
quantization of $X$.

######  Definition 2.6.4 (Def 5.0.1, [LMM21]).

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

##  3\. Nilpotent orbits and covers

In this section, we collect some basic facts about nilpotent orbits and their
(finite connected) covers.

###  3.1. Classification of nilpotent orbits and covers

In classical types, nilpotent orbits are classified by (decorated) integer
partitions.

######  Definition 3.1.1.

A partition $p$ is of _type C_ (resp type _B/D_) if every odd part (resp. even
part) occurs with even multiplicity.

The following result is well-known.

######  Proposition 3.1.2 (Section 5.1, [CM93]).

Suppose $\mathfrak{g}$ is classical. The set of nilpotent orbits
$\mathbb{O}\subset\mathfrak{g}^{*}$ is parameterized by (decorated) partitions
as follows

  1. (a)

If $\mathfrak{g}=\mathfrak{sl}(n)$, the set of nilpotent orbits is in one-to-
one correspondence with partitions of $n$.

  2. (b)

If $\mathfrak{g}=\mathfrak{so}(2n+1)$, the set of nilpotent orbits is in one-
to-one correspondence with partitions of $2n+1$ of type B/D.

  3. (c)

If $\mathfrak{g}=\mathfrak{sp}(2n)$, the set of nilpotent orbits is in one-to-
one correspondence with partitions of $2n$ of type C.

  4. (d)

If $\mathfrak{g}=\mathfrak{so}(2n)$, the set of nilpotent orbits is in one-to-
one correspondence with partitions of $2n$ of type B/D, except that each _very
even_ partition (i.e. a partition containing only even parts) corresponds to
two nilpotent orbits, labeled $\mathbb{O}^{I}$ and $\mathbb{O}^{II}$.

If $p$ is a partition of the appropriate type, we will denote the
corresponding nilpotent orbit by $\mathbb{O}_{p}$. In exceptional types, we
will use the Bala-Carter classification to label nilpotent orbits, see [CM93,
Sec 8.4] for an explanation.

By a ‘nilpotent cover’ we will mean a finite connected $G$-equivariant cover
of a nilpotent co-adjoint orbit. Up to isomorphism, nilpotent covers of
$\mathbb{O}$ are paramterized by conjugacy classes of subgroups of the
(necessarily finite) $G$-equivariant fundamental group
$\pi_{1}^{G}(\mathbb{O})$. A description of these fundamental groups can be
found in [CM93, Sec 6.1] (for classical groups) and [CM93, Sec 8.4] (for
simply connected exceptional groups). We will occasionally need a description
of $\pi_{1}^{G}(\mathbb{O})$ for more general classes of groups (for example,
the Levi subgroups of simply connected exceptional groups, which need not be
simply connected). In these cases, we use the atlas software to compute
$\pi_{1}^{G}(\mathbb{O})$.

###  3.2. Geometry of nilpotent orbits and covers

In this section, we will collect some basic facts about the geometry of
nilpotent orbits and covers. Suppose $\mathbb{O}$ is a nilpotent orbit. The
singular locus of $\overline{\mathbb{O}}$ coincides with the boundary
$\partial\mathbb{O}=\overline{\mathbb{O}}\setminus\mathbb{O}$, see [Nam04,
Proof of Prop 2.2]. Let $\mathbb{O}^{\prime}$ be a maximal $G$-orbit in
$\partial\mathbb{O}$. For each point $e^{\prime}\in\mathbb{O}^{\prime}$, there
is a transverse slice $S_{\mathbb{O},\mathbb{O}^{\prime}}$ to
$\mathbb{O}^{\prime}$ in $\overline{\mathbb{O}}$ (obtained, for example, by
intersecting the Slodowy slice to $\mathbb{O}^{\prime}$ at $e^{\prime}$ with
$\overline{\mathbb{O}}$). The variety $S_{\mathbb{O},\mathbb{O}^{\prime}}$ has
an isolated singularity at $e^{\prime}$ and a natural rational
$\mathbb{C}^{\times}$-action which is contracting onto $e^{\prime}$, see
[GG02, Sec 4]. We note that $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is
independent, up to algebraic isomorphism, of the choice of $e^{\prime}$ as
well as the $\mathfrak{sl}(2)$-triple $(e^{\prime},f^{\prime},h^{\prime})$
used to define the Slodowy slice. We call $S_{\mathbb{O},\mathbb{O}^{\prime}}$
the _singularity_ of the orbit
$\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$. The varieties
$S_{\mathbb{O}^{\prime},\mathbb{O}^{\prime}}$ were described in classical
types by Kraft and Procesi in [KP82] and in exceptional types by Fu, Juteau,
Levy, and Sommers in [FJLS15].

In this paper, we will restrict our attention to the singularities
corresponding to codimension 2 orbits
$\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$, i.e. to _dimension 2
singularities_ in $\overline{\mathbb{O}}$. The dimension 2 singularities in
nilpotent orbit closures can be rather complicated (sometimes non-normal)
varieties. We will briefly recall some of the standard conventions for
denoting them, from [Slo80],[KP82] and [FJLS15]. Fix a dimension 2 singularity
$S_{\mathbb{O},\mathbb{O}^{\prime}}$. If $S_{\mathbb{O},\mathbb{O}^{\prime}}$
is normal, it is isomorphic to a Kleinian singularity of type $A$, $D$, or
$E$. As explained in Section 2.4, the fundamental group
$\pi_{1}(\mathbb{O}^{\prime})$ acts on the Dynkin diagram of
$S_{\mathbb{O},\mathbb{O}^{\prime}}$ by a finite group $K$ of diagram
automorphisms. Following [Slo80] and [FJLS15], the pair
$(S_{\mathbb{O},\mathbb{O}^{\prime}},K)$ is denoted by

  * •

$B_{k}$, if $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is of type $A_{2k-1}$, and
$K=S_{2}$,

  * •

$C_{k}$, if $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is of type $D_{k+1}$, and
$K=S_{2}$,

  * •

$F_{4}$, if $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is of type $E_{6}$, and
$K=S_{2}$,

  * •

$G_{2}$, if $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is of type $D_{4}$, and
$K=S_{3}$,

  * •

$A_{2k}^{+}$, if $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is of type $A_{2k}$, and
$K=S_{2}$.

If $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is _non-normal_ and $\mathfrak{g}$ is
a classical Lie algebra, then $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is of the
following type:

  * •

$nA_{k}$: $n$ copies of the Kleinian singularity of type $A_{k}$, meeting at
the singular point.

If $\mathfrak{g}$ is exceptional, there are several additional non-normal
singularities which can appear. In the notation of [FJLS15], they are:

  * •

$nD_{k}$: $n$ copies of the Kleinian singularity of type $D_{k}$, meeting at
the singular point.

  * •

$m$: a non-normal 2-dimensional conical singularity admitting an
$\mathrm{SL}(2)$-action with an open orbit isomorphic to
$\mathbb{C}^{2}\setminus\\{0\\}$ and normalization isomorphic to
$\mathbb{C}^{2}$, see [FJLS15, Section 1.8.4] for details.

  * •

$\mu$: a non-normal 2-dimensional conical singularity with normalization
isomorphic to a Kleinian singularity of type $A_{3}$, see [FJLS15, Sec 1.8.4]
for details.

The next lemma, which asserts that the above singularities are in fact the
only possibilities, is immediate from [KP82] and [FJLS15].

######  Lemma 3.2.1.

Let $\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$ be a codimension 2
orbit, and assume that the singularity $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is
non-normal. Then $S_{\mathbb{O},\mathbb{O}^{\prime}}$ is of type $nA_{k}$,
$nD_{k}$ (for some $n$, $k$), $m$, or $\mu$.

Now let $\widetilde{\mathbb{O}}$ be a (finite connected) cover of $\mathbb{O}$
and let
$\widetilde{X}=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$.
Recall, (iii) of Example 2.2.3, that $\widetilde{X}$ is a conical symplectic
singularity. Let
$\mu:\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}\subset\mathfrak{g}^{*}$
denote the moment map. If $\mathfrak{L}\subset\widetilde{X}$ is a codimension
2 leaf, then $\mu(\overline{\mathfrak{L}})$ is the closure of a codimension 2
orbit $\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$, see [LMM21, Lem
4.6.1]. This defines a map

(3.2.1) $\\{\text{codimension 2 leaves in
}\widetilde{X}\\}\stackrel{{\scriptstyle}}{{\to}}\\{\text{codimension 2 orbits
in }\overline{\mathbb{O}}\\}$

Let $\Sigma$ denote the Kleinian singularity corresponding to the leaf
$\mathfrak{L}\subset\widetilde{X}$, see Section 2.4. There is a closed
embedding $\Sigma\subset\widetilde{X}$, constructed as follows. Since
$S_{\mathbb{O}^{\prime},\mathbb{O}}$ is transverse to $\mathbb{O}^{\prime}$,
$\mu^{-1}(S_{\mathbb{O}^{\prime},\mathbb{O}})$ is transverse to every leaf in
$\mu^{-1}(\mathbb{O}^{\prime})$. Thanks to the contracting
$\mathbb{C}^{\times}$-action on $S_{\mathbb{O}^{\prime},\mathbb{O}}$,
$\mu^{-1}(S_{\mathbb{O}^{\prime},\mathbb{O}})$ splits into a disjoint union of
connected components, indexed by points in
$\mu^{-1}(S_{\mathbb{O}^{\prime},\mathbb{O}}\cap\mathbb{O}^{\prime})$. Choose
a point in this set lying in $\mathfrak{L}$ and let $\Sigma$ be its connected
component in $\mu^{-1}(S_{\mathbb{O}^{\prime},\mathbb{O}})$. Then $\Sigma$ is
a Kleinian singularity and $\operatorname{Spec}(\mathbb{C}[\Sigma]^{\wedge})$
is a formal slice to $\mathfrak{L}$.

If we specialize to the case of nilpotent orbits, the map (3.2.1) is almost
always a bijection.

######  Lemma 3.2.2.

Suppose $\widetilde{\mathbb{O}}=\mathbb{O}$ is a nilpotent orbit. Then the
following are true:

  * (i)

The map (3.2.1) is injective.

  * (ii)

The map (3.2.1) is surjective unless $\overline{\mathbb{O}}$ contains a
dimension 2 singularity of type $m$.

###### Proof.

If $\overline{\mathbb{O}}$ is normal in codimension 2, then $\mu$ is an
isomorphism over every codimension 2 orbit in $\overline{\mathbb{O}}$. Hence,
(3.2.1) is a bijection.

If $\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$ is a codimension 2 orbit,
$\mu^{-1}(\mathbb{O}^{\prime})$ is a (union of) codimension 2 orbits.
Furthermore, by 3.2.1, $\mu^{-1}(\mathbb{O}^{\prime})$ lies in the singular
locus of $X$ if and only if the singularity
$S_{\mathbb{O},\mathbb{O}^{\prime}}$ is not of type $m$. This proves (ii) in
all cases. To prove (i) we consider all orbits $\mathbb{O}$ such that
$\overline{\mathbb{O}}$ contains a non-normal dimension 2 singularity not of
type $m$. The general idea is that if the singularity corresponds to a partial
Namikawa space of dimension $t$, then there are at most
$[\frac{\dim\mathfrak{P}^{X}}{t}]$ leaves with this singularity.

  * (i)

Let $\mathfrak{g}$ be a classical rank-$n$ Lie algebra not of type $A$.
Suppose that $\mathbb{O}_{1}\subset\overline{\mathbb{O}}$ corresponds to a
minimal degeneration of type (e) in the sense of [KP82, Table 1], and let
$\alpha$ and $\beta$ be the corresponding partitions. Let
$\mathbb{O}_{1},\ldots,\mathbb{O}_{t}\subset\overline{\mathbb{O}}$ be the
codimension 2 orbits, and let $k$ be the largest integer such that
$\alpha_{k}>\beta_{k}$. By loc.cit., $\alpha_{k}=\alpha_{k+1}+2t$ for some
$t$. Consider the Levi subalgebra
$\mathfrak{l}=\mathfrak{g}\mathfrak{l}(k)^{\times
t}\times\mathfrak{g}(n-2kt)$. Let $\mathbb{O}_{L}\subset\mathfrak{l}^{*}$ be
the nilpotent orbit corresponding to the partition $\gamma$ given by
$\gamma_{i}=\alpha_{i}-2t$ for $i\leq k$, and $\gamma_{i}=\alpha_{i}$ for
$i>k$. Set $\mathfrak{P}_{L}=\mathfrak{P}^{X_{L}}$. We note that $\mathbb{O}$
is birationally induced from $(L,\mathbb{O}_{L})$, and for each $j\neq 2$
there is a codimension $2$ orbit
$\mathbb{O}_{L,j}\subset\overline{\mathbb{O}}_{L}$ such that $\mathbb{O}_{j}$
is birationally induced from $\mathbb{O}_{L,j}$. By [Mat20, Lemma 4.16], the
singularities of $\mathbb{O}_{L,j}$ in $X_{L}$ and of $\mathbb{O}_{j}$ in $X$
are equivalent, and $\dim\mathfrak{P}_{j}^{X}=\dim\mathfrak{P}_{j}^{X_{L}}$.
We have $\dim\mathfrak{P}^{X}=\dim\mathfrak{P}^{X_{L}}+t$, and therefore
$\dim\mathfrak{P}_{1}^{X}=t$. It follows that there is only one codimension
$2$ leaf in $X$ over the orbit $\mathbb{O}_{1}$.

  * (ii)

Let $\mathfrak{g}$ be of type $F_{4}$, and set $\mathbb{O}=C_{3}(a_{1})$.
There is one codimension $2$ orbit
$\mathbb{O}^{\prime}=B_{2}\subset\overline{\mathbb{O}}$, and the corresponding
singularity is of type $2A_{1}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=1$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}^{\prime}$.

  * (iii)

Let $\mathfrak{g}$ be of type $F_{4}$, and set $\mathbb{O}=C_{3}$. There is
one codimension $2$ orbit
$\mathbb{O}^{\prime}=F_{4}(a_{1})\subset\overline{\mathbb{O}}$, and the
corresponding singularity is of type $4G_{2}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=2$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}^{\prime}$.

  * (iv)

Let $\mathfrak{g}$ be of type $E_{6}$, and set $\mathbb{O}=A_{4}$. There is
one codimension $2$ orbit
$\mathbb{O}^{\prime}=D_{4}(a_{1})\subset\overline{\mathbb{O}}$, and the
corresponding singularity is of type $3C_{2}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=3$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}^{\prime}$.

  * (v)

Let $\mathfrak{g}$ be of type $E_{7}$, and set $\mathbb{O}=D_{6}(a_{1})$.
There are two codimension $2$ orbits
$\mathbb{O}_{1}=E_{7}(a_{5})\subset\overline{\mathbb{O}}$ and
$\mathbb{O}_{2}=D_{5}\subset\overline{\mathbb{O}}$, and the corresponding
singularities are of type $3C_{2}$ and $A_{1}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=4$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}_{1}$.

  * (vi)

Let $\mathfrak{g}$ be of type $E_{7}$, and set $\mathbb{O}=A_{3}+A_{2}$. There
is one codimension $2$ orbit
$\mathbb{O}^{\prime}=D_{4}(a_{1})+A_{1}\subset\overline{\mathbb{O}}$, and the
corresponding singularity is of type $2A_{1}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=1$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}^{\prime}$.

  * (vii)

Let $\mathfrak{g}$ be of type $E_{7}$, and set
$\mathbb{O}=D_{4}(a_{1})+A_{1}$. There are two codimension $2$ orbits
$\mathbb{O}_{1}=D_{4}(a_{1})\subset\overline{\mathbb{O}}$ and
$\mathbb{O}_{2}=A_{3}+2A_{1}\subset\overline{\mathbb{O}}$, and the
corresponding singularities are of type $3A_{1}$ and $A_{1}$. Using [DE09,
Tables], $\dim\mathfrak{P}^{X}=2$, and therefore there is one codimension $2$
leaf over $\mathbb{O}_{1}$.

  * (viii)

Let $\mathfrak{g}$ be of type $E_{8}$, and set $\mathbb{O}=E_{7}(a_{1})$.
There is one codimension $2$ orbit
$\mathbb{O}^{\prime}=E_{8}(b_{5})\subset\overline{\mathbb{O}}$, and the
corresponding singularity is of type $3C_{5}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=5$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}^{\prime}$.

  * (ix)

Let $\mathfrak{g}$ be of type $E_{8}$, and set $\mathbb{O}=E_{7}(a_{1})$.
There are two codimension $2$ orbits
$\mathbb{O}_{1}=D_{6}(a_{1})\subset\overline{\mathbb{O}}$ and
$\mathbb{O}_{2}=A_{6}\subset\overline{\mathbb{O}}$, and the corresponding
singularities are of type $2A_{1}$ and $A_{1}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=2$. Since $\dim\mathfrak{P}_{2}^{X}=1$, it implies that
there is one codimension $2$ leaf over $\mathbb{O}_{1}$.

  * (x)

Let $\mathfrak{g}$ be of type $E_{8}$, and set $\mathbb{O}=D_{6}(a_{1})$.
There are two codimension $2$ orbits
$\mathbb{O}_{1}=E_{8}(a_{7})\subset\overline{\mathbb{O}}$ and
$\mathbb{O}_{2}=D_{5}+A_{1}\subset\overline{\mathbb{O}}$, and the
corresponding singularities are of type $10G_{2}$ and $A_{1}$. Using [DE09,
Tables], $\dim\mathfrak{P}^{X}=3$. Since $\dim\mathfrak{P}_{2}^{X}=1$, there
is one codimension $2$ leaf over $\mathbb{O}_{1}$.

  * (xi)

Let $\mathfrak{g}$ be of type $E_{8}$, and set $\mathbb{O}=A_{6}$. There is
one codimension $2$ orbit
$\mathbb{O}^{\prime}=E_{8}(a_{7})\subset\overline{\mathbb{O}}$, and the
corresponding singularity is of type $5G_{2}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=2$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}^{\prime}$.

  * (xii)

Let $\mathfrak{g}$ be of type $E_{8}$, and set $\mathbb{O}=E_{7}(a_{5})$.
There are two codimension $2$ orbits
$\mathbb{O}_{1}=D_{6}(a_{2})\subset\overline{\mathbb{O}}$ and
$\mathbb{O}_{2}=E_{6}(a_{3})+A_{1}\subset\overline{\mathbb{O}}$, and the
corresponding singularities are of types $2A_{1}$ and $m$. Using [DE09,
Tables], $\dim\mathfrak{P}^{X}=1$, and therefore there is one codimension $2$
leaf over $\mathbb{O}_{1}$.

  * (xiii)

Let $\mathfrak{g}$ be of type $E_{8}$, and set $\mathbb{O}=A_{3}+A_{2}$. There
is one codimension $2$ orbit
$\mathbb{O}^{\prime}=D_{4}(a_{1})+A_{1}\subset\overline{\mathbb{O}}$, and the
corresponding singularity is of type $3A_{1}$. Using [DE09, Tables],
$\dim\mathfrak{P}^{X}=1$, and therefore there is one codimension $2$ leaf over
$\mathbb{O}^{\prime}$.

∎

######  Remark 3.2.3.

Let $\mathbb{O}_{k}\subset\overline{\mathbb{O}}$ be a codimension 2 orbit such
that the corresponding singularity is not of type $m$. Then by Lemma 3.2.2,
there is a unique codimension 2 leaf
$\mathfrak{L}_{k}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ which
maps to $\mathbb{O}_{k}$ under (3.2.1).

###  3.3. Induction of nilpotent orbits

Let $M\subset G$ be a Levi subgroup, and let $\mathbb{O}_{M}$ be a nilpotent
$M$-orbit. Fix a parabolic subgroup $Q\subset G$ with a Levi decomposition
$Q=MU$. The annihilator of $\mathfrak{q}$ in $\mathfrak{g}^{*}$ is a
$Q$-stable subspace $\mathfrak{q}^{\perp}\subset\mathfrak{g}^{*}$. Choosing a
nondegenerate invariant symmetric form on $\mathfrak{g}$, we get a
$Q$-invariant identification $\mathfrak{q}^{\perp}\simeq\mathfrak{u}$. Form
the $G$-equivariant fiber bundle
$G\times^{Q}(\overline{\mathbb{O}}_{M}\times\mathfrak{q}^{\perp})$ over the
partial flag variety $G/Q$. There is a proper $G$-equivariant map

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

The image of $\mu$ is a closed irreducible $G$-invariant subset of
$\mathcal{N}$, and hence the closure in $\mathcal{N}$ of a nilpotent
$G$-orbit, denoted
$\mathrm{Ind}^{G}_{M}\mathbb{O}_{M}\subset\mathfrak{g}^{*}$. The
correspondence

$\mathrm{Ind}^{G}_{M}:\\{\text{nilpotent
}M\text{-orbits}\\}\stackrel{{\scriptstyle}}{{\to}}\\{\text{nilpotent
}G\text{-orbits}\\}$

is called _Lusztig-Spaltenstein_ induction. A nilpotent orbit is _rigid_ if it
cannot be induced from a proper Levi subgroup.

######  Proposition 3.3.1 ([LS79] or [CM93], Sec 7).

Lusztig-Spaltenstein induction has the following properties

  * (i)

$\mathrm{Ind}^{G}_{M}$ depends only on $M$ (and not on $Q$)

  * (ii)

If $L\subset M$ is a Levi subgroup of $M$, then

$\operatorname{Ind}^{G}_{L}=\operatorname{Ind}^{G}_{M}\circ\operatorname{Ind}^{M}_{L}.$

  * (iii)

If $\mathbb{O}$ is a nilpotent orbit, there is a Levi sugroup $M\subset G$ and
a rigid nilpotent $M$-orbit $\mathbb{O}_{M}$ such that

$\mathbb{O}=\operatorname{Ind}^{G}_{M}\mathbb{O}_{M}$

  * (iv)

If $\mathbb{O}_{M}\subset\mathfrak{m}^{*}$ is a nilpotent $M$-orbit and
$\mathbb{O}=\operatorname{Ind}^{G}_{M}\mathbb{O}_{M}$, then

$\mathrm{codim}(\mathbb{O}_{M},\mathcal{N}_{M})=\operatorname{codim}(\mathbb{O},\mathcal{N}).$

In classical types, a classification of rigid nilpotent orbits and a
description of induction in terms of partitions can be found in [CM93, Sec
7.3]. In exceptional types, this information can be found in the tables
appearing in [DE09, Sec 4]. For the explicit computations in Sections 3.9 and
4.3, we will make repeated use of these descriptions.

###  3.4. Birational induction of nilpotent covers

Choose a Levi subgroup $M\subset G$, a nilpotent $M$-orbit $\mathbb{O}_{M}$,
and a (finite, connected) $M$-equivariant cover $\widetilde{\mathbb{O}}_{M}$
of $\mathbb{O}_{M}$. Let
$\mathbb{O}=\operatorname{Ind}^{G}_{M}\mathbb{O}_{M}$. Consider the affine
variety
$\widetilde{X}_{M}:=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{M}])$.
There is an $M$-action on $\widetilde{X}_{M}$ (induced from the $M$-action on
$\widetilde{\mathbb{O}}_{M}$) and a finite surjective $M$-equivariant map
$\widetilde{X}_{M}\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}_{M}$.
Let $\widetilde{\mu}$ denote the composition

$G\times^{Q}(\widetilde{X}_{M}\times\mathfrak{q}^{\perp})\stackrel{{\scriptstyle}}{{\to}}G\times^{Q}(\overline{\mathbb{O}}_{M}\times\mathfrak{q}^{\perp})\overset{\mu}{\stackrel{{\scriptstyle}}{{\to}}}\overline{\mathbb{O}}.$

Note that
$\widetilde{\mu}^{-1}(\mathbb{O})\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$
is a (finite, connected) $G$-equivariant cover. The correspondence

$\displaystyle\mathrm{Bind}^{G}_{M}:\\{M\text{-eqvt nilpotent covers}\\}$
$\displaystyle\stackrel{{\scriptstyle}}{{\to}}\\{G\text{-eqvt nilpotent
covers}\\}$ $\displaystyle\widetilde{\mathbb{O}}_{M}$
$\displaystyle\mapsto\widetilde{\mu}^{-1}(\mathbb{O})$

is called _birational induction_. A nilpotent cover is _birationally rigid_ if
it cannot be birationally induced from a proper Levi subgroup.

The main properties of birational induction are catalogued below.

######  Proposition 3.4.1 (Prop 2.4.1, [LMM21]).

Birational induction has the following properties

  1. (i)

$\mathrm{Bind}^{G}_{M}$ depends only on $M$ (and not on $Q$)

  2. (ii)

If $L\subset M$ is a Levi subgroup of $M$, then

$\mathrm{Bind}^{G}_{L}=\mathrm{Bind}^{G}_{M}\circ\mathrm{Bind}^{M}_{L}.$

  3. (iii)

If $\widetilde{\mathbb{O}}$ is a $G$-equivariant nilpotent cover, there is a
Levi subgroup $M\subset G$ and a birationally rigid $M$-equivariant nilpotent
cover $\widetilde{\mathbb{O}}_{M}$ such that

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

The pair $(M,\widetilde{\mathbb{O}}_{M})$ is called a _birationally minimal
induction datum_ and is unique up to conjugation by $G$.

  4. (iv)

If we write $\deg(\widetilde{\mathbb{O}}_{M})$ for the degree of the covering
map
$\widetilde{\mathbb{O}}_{M}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}_{M}$,
then

$\deg(\widetilde{\mathbb{O}}_{M})\text{ divides
}\deg(\mathrm{Bind}^{G}_{M}(\widetilde{\mathbb{O}}_{M})).$

###  3.5. Filtered quantizations of nilpotent covers

Let $\widetilde{\mathbb{O}}$ be a nilpotent cover and consider the affine
variety
$\widetilde{X}:=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$. By
(iii) of Example 2.2.3, $\widetilde{X}$ is a conical symplectic singularity.
Fix the notation of Section 2.4, i.e. $\mathfrak{P}^{\widetilde{X}}$,
$W^{\widetilde{X}}$, $\mathfrak{P}_{k}^{\widetilde{X}}$, and so on. By Theorem
2.6.2, there is a canonical bijection

(3.5.1)
$\mathfrak{P}^{\widetilde{X}}/W^{\widetilde{X}}\xrightarrow{\sim}\mathrm{Quant}(\mathbb{C}[\widetilde{\mathbb{O}}])\qquad
W^{\widetilde{X}}\cdot\lambda\mapsto\mathcal{A}^{\widetilde{X}}_{\lambda}$

In this section, we will re-interpret the spaces
$\mathfrak{P}^{\widetilde{X}}$ and $\mathfrak{P}_{k}^{\widetilde{X}}$ in terms
of purely Lie-theoretic information.

Fix a birationally minimal induction datum $(L,\widetilde{\mathbb{O}}_{L})$
for $\widetilde{\mathbb{O}}$ (cf. Proposition 3.4.1(iii)) and let
$\widetilde{X}_{L}:=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])$.
Choose a parabolic subgroup $P\subset G$ with Levi factor $L$ and consider the
map

$\widetilde{\mu}:\widetilde{Y}:=G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$

defined in Section 3.4. Since
$\widetilde{\mu}^{-1}\simeq\widetilde{\mathbb{O}}$, $\widetilde{\mu}$ factors
through a partial resolution

(3.5.2) $\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}.$

######  Proposition 3.5.1.

The following are true:

  * (i)

The map (3.5.2) is a $\mathbb{Q}$-factorial terminalization.

  * (ii)

There is a linear isomorphism

$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\xrightarrow{\sim}H^{2}(G/P,\mathbb{C})\xrightarrow{\sim}H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})=:\mathfrak{P}^{\widetilde{X}},$

where the second map is the pullback along the natural projection
$\widetilde{Y}^{\mathrm{reg}}\stackrel{{\scriptstyle}}{{\to}}G/P.$

  * (iii)

$W^{\widetilde{X}}$ is identified with a normal subgroup of $N_{G}(L)/L$, with
its canonical action on
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$.

  * (iv)

Up to the action of $W^{\widetilde{X}}$ on the target, the map
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\xrightarrow{\sim}\mathfrak{P}^{\widetilde{X}}$
is independent of the choice of parabolic $P$.

###### Proof.

(i) is [Mat20, Cor 4.3]. For orbits, (ii) and (iii) are [Los16, Prop 4.7]. The
proofs there can be easily generalized to arbitrary nilpotent covers. (iv) is
[LMM21, Prop 7.2.5]. ∎

Combining (3.5.1) and (ii) of Proposition 3.5.1, we obtain a natural bijection

$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])/W^{\widetilde{X}}\xrightarrow{\sim}\mathrm{Quant}(\mathbb{C}[\widetilde{\mathbb{O}}])\qquad
W^{\widetilde{X}}\cdot\lambda\mapsto\mathcal{A}^{\widetilde{X}}_{\eta(\lambda)}$

Note that $G$ acts on $\mathbb{C}[\widetilde{\mathbb{O}}]$ by graded Poisson
automorphisms. There is a classical co-moment map
$\varphi:\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}\mathbb{C}[\widetilde{\mathbb{O}}]$
obtained from the map of varieties
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{g}^{*}$. The
map
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\xrightarrow{\sim}\mathfrak{P}^{\widetilde{X}}$
extends to an isomorphism (still denoted by $\eta$)

$\eta:\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\oplus\mathfrak{z}(\mathfrak{g})^{*}\xrightarrow{\sim}\mathfrak{P}^{\widetilde{X}}\oplus\mathfrak{z}(\mathfrak{g})^{*}=\overline{\mathfrak{P}}^{\widetilde{X}}.$

So by Proposition 2.6.3, we obtain a natural bijection

$\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}\mathrm{Quant}^{G}(\mathbb{C}[\widetilde{\mathbb{O}}])\qquad
W^{\widetilde{X}}\cdot(\lambda,\chi)\mapsto(\mathcal{A}_{\lambda}^{\widetilde{X}},\Phi_{\chi}^{\widetilde{X}}).$

###  3.6. Description of partial Namikawa spaces

In this seciton, we will give a Lie-theoretic description of the partial
Namikawa spaces $\mathfrak{P}_{k}^{\widetilde{X}}$ (under some conditions).
Passing to a covering group if necessary, we can (and will) assume that $G$ is
simply connected. Let $R_{x}$ denote the reductive part of the stabilizer of
$x\in\widetilde{\mathbb{O}}$ and let $\mathfrak{r}$ be its Lie algebra. We
note that $\mathfrak{r}$ does not depend on the choice of a point $x$, and the
adjoint action of $R_{x}$ on $\mathfrak{X}(\mathfrak{r})$ factors through
$R_{x}/R_{x}^{\circ}\simeq\pi^{G}_{1}(\widetilde{\mathbb{O}})$.

######  Lemma 3.6.1.

The following are true:

  * (i)

Restriction along $\widetilde{\mathbb{O}}\subset\widetilde{X}^{\mathrm{reg}}$
induces a linear isomorphism

$\mathfrak{P}_{0}^{\widetilde{X}}=H^{2}(\widetilde{X}^{\mathrm{reg}},\mathbb{C})\xrightarrow{\sim}H^{2}(\widetilde{\mathbb{O}},\mathbb{C})$

  * (ii)

There is a natural identification

$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})\xrightarrow{\sim}\mathfrak{X}(\mathfrak{r})^{\pi_{1}(\widetilde{\mathbb{O}})}$

A description of $\mathfrak{r}$ can be found in [CM93, Sec 6.1] (for classical
types) and [Car93, Sec 13.1] (for exceptional types).

######  Remark 3.6.2.

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 3.6.1. 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].

Assume for the remainder of this subsection that
$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})=0$ and fix a birationally minimal
induction datum $(L,\widetilde{\mathbb{O}}_{L})$ for $\widetilde{\mathbb{O}}$.
Suppose $Q\subset G$ is a parabolic subgroup with Levi factor $M$ and
$\widetilde{\mathbb{O}}_{M}$ is an $M$-equivariant nilpotent cover with
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{M}\widetilde{\mathbb{O}}_{M}$. The
triple $(Q,M,\widetilde{\mathbb{O}}_{M})$ gives rise to a projective
birational morphism (generalizing the map (3.5.2))

(3.6.1)
$\rho:G\times^{Q}(\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{M}])\times\mathfrak{q}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$

######  Proposition 3.6.3 (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 $M_{k}$-equivariant nilpotent cover
$\widetilde{\mathbb{O}}_{M_{k}}$ 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 partial
resolution (3.6.1) resolves $\Sigma_{k}$ and preserves $\Sigma_{j}$ for $j\neq
k$.

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

######  Proposition 3.6.4.

Let $\mathfrak{L}_{k}\subset\widetilde{X}$ be a codimension 2 leaf and let
$(M_{k},\widetilde{\mathbb{O}}_{M_{k}})$ be the $\mathfrak{L}_{k}$-adapted
resolution datum. Then the following are true:

  * (i)

The closed embedding $\Sigma_{k}\subset\widetilde{X}$ (cf. Section 3.2) lifts
to a closed embedding $\mathfrak{S}_{k}\subset\widetilde{Z}_{k}$.

  * (ii)

If $\mathcal{L}$ is a line bundle on $\widetilde{Z}_{k}$, then
$\mathcal{L}|_{\mathfrak{S}_{k}}$ is a $\pi_{1}(\mathfrak{L}_{k})$-equivariant
line bundle on $\mathfrak{S}_{k}$, i.e. there is a restriction map

(3.6.2)
$\operatorname{Pic}(\widetilde{Z}_{k})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\mathfrak{S}_{k})^{\pi_{1}(\mathfrak{L}_{k})}$

  * (iii)

There are natural group isomorphisms

$\operatorname{Pic}(\widetilde{Z}_{k})\simeq\mathfrak{X}(M_{k}),\qquad\operatorname{Pic}(\mathfrak{S}_{k})\simeq\Lambda_{k},$

i.e. (3.6.2) induces a group homomorphism

(3.6.3)
$\mathfrak{X}(M_{k})\stackrel{{\scriptstyle}}{{\to}}\Lambda_{k}^{\pi_{1}(\mathfrak{L}_{k})}$

  * (iv)

The complexification of (3.6.3) is a linear isomorphism

(3.6.4)
$\eta_{k}:\mathfrak{X}(\mathfrak{m}_{k})\xrightarrow{\sim}\mathfrak{P}_{k}^{\widetilde{X}},$

  * (v)

The following diagram commutes

${\mathfrak{X}(\mathfrak{l})}$${\mathfrak{P}^{\widetilde{X}}}$${\mathfrak{X}(\mathfrak{m}_{k})}$${\mathfrak{P}_{k}^{\widetilde{X}}}$$\scriptstyle{\eta}$$\scriptstyle{\eta_{k}}$

###### Proof.

(i) follows from (iii) of Proposition 3.6.3. (iii) follows from [LMM21, Prop
7.1.2]. (ii), (iv), and (v) follow from [LMM21, Lem 7.5.7]. ∎

The isomorphism $\eta_{k}$ was computed in [LMM21, Sec 7.7] under certain
conditions on $\widetilde{\mathbb{O}}$ and $\mathfrak{L}_{k}$. The relevant
statements will be recalled in Section 3.10.3.

###  3.7. Geometric characterization of birationally rigid covers

Combining Proposition 2.4.1 and (ii) of Proposition 3.5.1, we obtain the
following (purely geometric) characterization of birational rigidity.

######  Proposition 3.7.1.

Let $\widetilde{\mathbb{O}}$ be a nilpotent cover. 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.

Checking condition (i) of Proposition 3.7.1 is usually easy in view of Lemma
3.6.1. Checking condition (ii) is a subtler business in general. We will
develop some techniques for doing so in Section 3.9.

###  3.8. Classification of birationally rigid orbits

Let $\mathbb{O}$ be a nilpotent orbit.

######  Proposition 3.8.1.

Let $\mathbb{O}$ be a nilpotent orit. Then $\mathbb{O}$ is birationally rigid
if and only if the following conditions are satisfied:

  * (i)

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

  * (ii)

All dimension 2 singularities in $\overline{\mathbb{O}}$ are of type $m$.

###### Proof.

By Lemma 3.2.2, condition (ii) above is equivalent to condition (ii) of
Proposition 3.7.1. Now Proposition 3.8.1 follows at once from Proposition
3.7.1. ∎

An advantage of this formulation is that condition (ii) above is very easy to
check. In classical types, there are no singularities of type $m$. So (ii) is
equivalent to the condition that there are no codimension 2 orbits in
$\overline{\mathbb{O}}$. The set of codimension 2 orbits in
$\overline{\mathbb{O}}$ was described by Kraft and Procesi in [KP82] in terms
of the partition corresponding to $\mathbb{O}$. From this description, one
easily deduces the following.

######  Proposition 3.8.2 (Prop 7.6.3, [LMM21]).

Suppose $\mathfrak{g}$ is classical and let
$\mathbb{O}\subset\mathfrak{g}^{*}$ be a nilpotent orbit corresponding to a
partition $p$. Then $\mathbb{O}$ is birationally rigid if and only if one of
the following is true:

  * (i)

$\mathfrak{g}=\mathfrak{sl}(n)$ and $\mathbb{O}=\\{0\\}$.

  * (ii)

$\mathfrak{g}=\mathfrak{so}(2n+1)$ or $\mathfrak{sp}(2n)$ and $p$ satisfies

$p_{i}\leq p_{i+1}+1\qquad\forall i.$

  * (iii)

$\mathfrak{g}=\mathfrak{so}(2n)$, $p$ satisfies

$p_{i}\leq p_{i+1}+1\qquad\forall i,$

and $p$ is not of the form $(2^{m},1^{2})$ for some $m$.

In exceptional types, condition (ii) of Proposition 3.8.1 can be checked by
inspecting the incidence tables in [FJLS15, Sec 13]. One easily arrives at the
following classification.

######  Proposition 3.8.3.

The following is a complete list of birationally rigid orbits in simple
exceptional Lie algebras:

$\mathfrak{g}$ | Birationally rigid orbits
---|---
$G_{2}$ |  $\\{0\\}$, $A_{1}$, $\widetilde{A}_{1}$
$F_{4}$ |  $\\{0\\}$, $A_{1}$, $\widetilde{A}_{1}$, $A_{1}+\widetilde{A}_{1}$, $A_{2}+\widetilde{A}_{1}$, $\widetilde{A}_{2}+A_{1}$
$E_{6}$ |  $\\{0\\}$, $A_{1}$, $3A_{1}$, $2A_{2}+A_{1}$
$E_{7}$ |  $\\{0\\}$, $A_{1}$, $2A_{1}$, $(3A_{1})^{\prime}$, $4A_{1}$, $A_{2}+A_{1}$, $A_{2}+2A_{1}$, $2A_{1}+A_{1}$, $(A_{3}+A_{1})^{\prime}$, $A_{4}+A_{1}$
$E_{8}$ |  $\\{0\\}$, $A_{1}$, $2A_{1}$, $3A_{1}$, $4A_{1}$, $A_{2}+A_{1}$, $A_{2}+2A_{1}$, $A_{2}+3A_{1}$, $2A_{2}+A_{1}$, $A_{3}+A_{1}$, $2A_{2}+2A_{1}$,
|  $A_{3}+2A_{1}$, $D_{4}(a_{1})+A_{1}$, $A_{3}+A_{2}+A_{1}$, $A_{4}+A_{1}$,
$2A_{3}$, $A_{4}+A_{3}$, $A_{5}+A_{1}$, $D_{5}(a_{1})+A_{2}$

Three of these orbits are not rigid, namely:

$A_{2}+A_{1},A_{4}+A_{1}\subset E_{7},\qquad A_{4}+A_{1}\subset E_{8}.$

###### Proof.

In exceptional types, the cohomology groups $H^{2}(\mathbb{O},\mathbb{C})$
were computed by Biswas and Chatterjee in [BC12, Thms 5.11,5.12]. It was shown
there that $H^{2}(\mathbb{O},\mathbb{C})=0$ in all cases except for the
following nine orbits in type $E_{6}$:

$2A_{1},A_{2}+A_{1},A_{2}+2A_{1},A_{3},A_{3}+A_{1},A_{4},A_{4}+A_{1},A_{5},D_{5}(a_{1}).$

Thus by Proposition 3.8.1, $\mathbb{O}$ is birationally rigid if and only if
$\mathbb{O}$ is not one of these nine and all dimension 2 singularities in
$\overline{\mathbb{O}}$ are of type $m$. Inspecting the diagrams in [FJLS15,
Sec 13], one arrives at the list given in the statement of the proposition. A
list of rigid orbits in exceptional types is provided in [Els84]. ∎

###  3.9. Classification of birationally semi-rigid orbits

For the calculations in Section 4, birationally rigid nilpotent covers, and
the orbits which admit them, will play a central role. Make the following
definition.

######  Definition 3.9.1.

A nilpotent cover $\mathbb{O}$ is _birationally semi-rigid_ if

  * (i)

$\mathbb{O}$ admits a $G$-equivariant birationally rigid cover.

  * (ii)

$\mathbb{O}$ is not birationally rigid.

Below, we will give a classification of such orbits in simple exceptional
types. The following result from [LMM21] narrows the range of possibilities.

######  Proposition 3.9.2 (Prop 7.6.16,[LMM21]).

Suppose $\mathbb{O}$ is a birationally semi-rigid orbit in a simple
exceptional Lie algebra. Then all Kleinian singularities in
$\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ are of type $A_{1}$, with the
following four exceptions:

  * (i)

$\mathfrak{g}=E_{6}$ and $\mathbb{O}=2A_{2}$. There is a unique codimension 2
leaf, and the corresponding singularity is of type $A_{2}$.

  * (ii)

$\mathfrak{g}=E_{6}$ and $\mathbb{O}=A_{5}$. There is a unique codimension 2
leaf, and the corresponding singularity is of type $A_{2}$.

  * (iii)

$\mathfrak{g}=E_{6}$ and $\mathbb{O}=E_{6}(a_{3})$. There are two codimension
2 leaves, and the corresponding singularities are of types $A_{1}$ and
$A_{2}$.

  * (iv)

$\mathfrak{g}=E_{8}$ and $\mathbb{O}=E_{8}(b_{6})$. There are two codimension
2 leaves, and the corresponding singularities are of types $A_{1}$ and
$A_{2}$.

In the simple exceptional Lie algebras, there are 38 nilpotent orbits with
nontrivial $\pi_{1}(\mathbb{O})$ which satisfy the $A_{1}$ condition above.
However, not all such orbits are birationally semi-rigid (the simplest example
is the distinguished orbit $F_{4}(a_{2})$ in $F_{4}$). Our task in this
subsection is to determine precisely which of them are. For the most part, the
techniques we will employ were developed in [LMM21]. We will review some of
them here for the reader’s convenience.

If $\mathbb{O}$ is a nilpotent orbit, consider the finite set

(3.9.1)
$\mathcal{P}_{\mathrm{rig}}(\mathbb{O}):=\\{(M,\mathbb{O}_{M})\mid\mathbb{O}=\operatorname{Ind}^{G}_{M}\mathbb{O}_{M},\
\mathbb{O}_{M}\text{ is rigid}\\}/G.$

Let $m(\mathbb{O})$ denote the maximum value of
$\dim\mathfrak{z}(\mathfrak{m})$ for
$(M,\mathbb{O}_{M})\in\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$.

######  Lemma 3.9.3 (Prop 7.6.15, [LMM21]).

Let $\mathfrak{L}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ be a
codimension 2 leaf. Let $\Sigma=\mathbb{C}^{2}/\Gamma$ be the corresponding
Kleinian singularity, and $\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$
the corresponding codimension two $G$-orbit. Assume:

  * (i)

$\Gamma$ is a simple group.

  * (ii)

There is a strict inequality

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}^{\prime})|^{-1}>m(\mathbb{O})$

Then $\Sigma\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ is smoothened
under the covering map
$\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}])\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$.

Combining Lemma 3.9.3 and Proposition 3.7.1, we obtain the following useful
criterion.

######  Corollary 3.9.4.

Suppose

  * (i)

All Kleinian singularities in $\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$
are of type $A_{1}$.

  * (ii)

For each codimension 2 orbit
$\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$, there is a strict
inequality

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}^{\prime})|^{-1}>m(\mathbb{O})$

  * (iii)

The reductive part of the centralizer of $e\in\mathbb{O}$ is semisimple.

Then the universal cover $\widehat{\mathbb{O}}$ of $\mathbb{O}$ is
birationally rigid.

###### Proof.

By Lemma 3.9.3, conditions (i) and (ii) imply that
$\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}])$ has no codimension 2
leaves. By Lemma 3.6.1, condition (iii) implies that
$H^{2}(\widehat{\mathbb{O}},\mathbb{C})=0$. The corollary follows at once from
Proposition 3.7.1. ∎

######  Proposition 3.9.5.

The following is a complete list of birationally semi-rigid orbits in simple
exceptional Lie algebras:

$\mathfrak{g}$ | Birationally semi-rigid orbits
---|---
$G_{2}$ | $G_{2}(a_{1})$
$F_{4}$ |  $A_{2}$, $B_{2}$, $C_{3}(a_{1})$, $F_{4}(a_{3})$,
$E_{6}$ |  $A_{2}$, $D_{4}(a_{1})$, $2A_{2}$, $A_{5}$, $E_{6}(a_{3})$
$E_{7}$ |  $(3A_{1})^{\prime\prime}$, $A_{2}$, $A_{2}+3A_{1}$, $(A_{3}+A_{1})^{\prime\prime}$, $D_{4}(a_{1})$, $A_{3}+2A_{1}$, $D_{4}(a_{1})+A_{1}$,
|  $A_{3}+A_{2}+A_{1}$, $A_{5}+A_{1}$, $D_{5}(a_{1})+A_{1}$, $E_{7}(a_{5})$,
$E_{7}(a_{4})$
$E_{8}$ |  $A_{2}$, $2A_{2}$, $D_{4}(a_{1})$, $D_{4}(a_{1})+A_{2}$, $D_{4}+A_{2}$, $D_{6}(a_{2})$, $E_{6}(a_{3})+A_{1}$,
|  $E_{7}(a_{5})$, $E_{8}(a_{7})$, $E_{8}(b_{6})$

###### Proof.

By Proposition 3.9.2, we can restrict our attention to orbits with nontrivial
$\pi_{1}(\mathbb{O})$ and only $A_{1}$ singularities in
$\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$. Using the incidence diagrams in
[FJLS15, Sec 13], we find that there are 38 orbits with these properties. For
most of these orbits, Corollary 3.9.4 can be straightforwardly applied to show
that $\mathbb{O}$ is birationally semi-rigid. In some cases, a more elaborate
argument is required. On the other hand, 10 of these orbits are _not_ semi-
rigid. This is proved either by cohomology considerations, see Proposition
3.7.1, or by a counting argument involving Proposition 3.4.1.

We pause to introduce some notational conventions which will remain in place
for the rest of the paper. For $\mathfrak{g}$ a simple exceptional Lie
algebra, we number the simple roots as in Bourbaki ([Bou02]). In type $E_{n}$,
this means that the simple roots forming the subgraph of type $A_{n-1}$ are
labeled $\alpha_{1},\alpha_{3},...,\alpha_{n}$ from left to right and the
remaining simple root is labeled $\alpha_{2}$. In type $F_{4}$ the simple
roots are labeled $\alpha_{1},...,\alpha_{4}$ from left to right
($\alpha_{1}$, $\alpha_{2}$ are the long roots, and $\alpha_{3}$, $\alpha_{4}$
are short). In type $G_{2}$, $\alpha_{1}$ is the short root.

If $\mathfrak{g}$ has rank $n$ and
$\\{r_{1},...,r_{p}\\}\subseteq\\{1,...,n\\}$, there is a unique standard Levi
subalgebra with simple roots $\alpha_{r_{1}},...,\alpha_{r_{p}}$. We denote
this Levi subalgebra by $\mathfrak{l}(X;r_{1},...,r_{p})$, where $X$ is the
Lie type of the Levi ($X$ is included in the notation only for the reader’s
convenience—it is completely determined by the numbers $r_{1},...,r_{p}$). Two
standard Levis $\mathfrak{l}(X;r_{1},...,r_{p})$ and
$\mathfrak{l}(Y;s_{1},...,s_{q})$ are conjugate under
$\operatorname{Ad}(\mathfrak{g})$ if and only if $p=q$ and there is a Weyl
group element $w$ such that
$\\{\alpha_{r_{1}},...,\alpha_{r_{p}}\\}=w\\{\alpha_{s_{1}},...,\alpha_{s_{p}}\\}$.
Sometimes, the Levi subalgebra $\mathfrak{l}(X;r_{1},...,r_{p})$ is completely
determined by $X$. In such cases, we will often omit $r_{1},...,r_{p}$ from
the notation, writing simply $\mathfrak{l}(X)$.

The calculations below involve a number of elementary ‘micro-computations’,
which are carried out in each case in exactly the same fashion. To avoid
repeating references and explanations, we will catalogue them below:

  * •

Given a nilpotent orbit $\mathbb{O}$, determine the finite set
$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$, see (3.9.1). This is deducible in
every case from the tables in [DE09, Sec 4].

  * •

Given a nilpotent orbit $\mathbb{O}$, determine the codimension 2 orbits
$\mathbb{O}_{k}\subset\overline{\mathbb{O}}$. This is evident from the
incidence diagrams in [Spa82]. In some cases, we will also need to determine
the singularity of $\mathbb{O}_{k}$ and its normalization, $\Sigma_{k}$. This
can be deduced from the incidence diagrams in [FJLS15, Sec 13].

  * •

Determine the fundamental group $\pi_{1}(\mathbb{O})$ of a nilpotent orbit
$\mathbb{O}$. See [CM93, Sec 6.1] for classical types and [CM93, Sec 8.4] for
exceptional types.

  * •

Determine the reductive part $\mathfrak{r}$ of the centralizer of a nilpotent
element $e\in\mathbb{O}$. See [CM93, Sec 6.1] for classical types and [Car93,
Sec 13.1] for exceptional types.

$G_{2}(a_{1})\subset G_{2}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1};1),\\{0\\}),(L(A_{1};2),\\{0\\})\\}$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=\widetilde{A}_{1}$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{3}\qquad\pi_{1}(\mathbb{O}_{1})=1$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=6>1=m(\mathbb{O}).$

and $\mathfrak{r}=\\{0\\}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$A_{2}\subset F_{4}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(C_{3};2,3,4),\\{0\\}\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{1}+\widetilde{A}_{1}$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=A_{2}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Proposition 3.9.4.

$B_{2}\subset F_{4}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(C_{3};2,3,4),\mathbb{O}_{(2,1^{4})})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{2}+\widetilde{A}_{1}$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=2A_{1}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Proposition 3.9.4.

$C_{3}(a_{1})\subset F_{4}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(B_{3};1,2,3),\mathbb{O}_{(2^{2},1^{3})})\\},$

and therefore $m(\mathbb{O})=1$. There are two codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=B_{2}$ and
$\mathbb{O}_{2}=\widetilde{A}_{2}+A_{1}$. The singularity of
$\mathbb{O}_{2}\subset\overline{\mathbb{O}}$ is of type $m$ and is therefore
resolved under the normalization map
$X=\operatorname{Spec}(\mathbb{C}[\mathbb{O}])\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$.
Let $\widecheck{\mathbb{O}}_{1}\subset X$ be the preimage of $\mathbb{O}_{1}$.
The slice $\Sigma_{1}$ to $\mathbb{O}_{1}$ is of type $2A_{1}$, and therefore
the preimage of the $\Sigma_{1}$ is the disjoint union of two copies of
$\Sigma_{1}$. If $\widecheck{\mathbb{O}}_{1}$ is not connected, then there are
at least two symplectic leaves of codimension $2$ over $\mathbb{O}_{1}$ with
open $G$ orbits being the irreducible components of
$\widecheck{\mathbb{O}}_{1}$. That contradicts to Lemma 3.2.2. It follows that
$\widecheck{\mathbb{O}}_{1}$ is a $2$-fold connected cover of
$\mathbb{O}_{1}$. The fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=S_{2}\qquad\pi_{1}(\widecheck{\mathbb{O}}_{1})=1$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\widecheck{\mathbb{O}}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=A_{1}$. Analogously to Proposition 3.9.4, we see that
$\widehat{\mathbb{O}}$ is birationally rigid.

$F_{4}(a_{3})\subset F_{4}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1}+A_{2};1,3,4),\\{0\\}),(L(A_{2}+A_{1};1,2,4),\\{0\\}),(L(B_{2}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=2$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=C_{3}(a_{1})$. The fundamental
groups are as follows

$\pi_{1}(\mathbb{O})=S_{4}\qquad\pi_{1}(\mathbb{O}_{1})=S_{2}$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=12>2=m(\mathbb{O}),$

and $\mathfrak{r}=0$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Proposition 3.9.4.

$F_{4}(a_{2})\subset F_{4}$. Note that $\pi_{1}(\mathbb{O})=S_{2}$. Hence,
$\mathbb{O}$ admits 2 non-isomorphic covers (including the trivial one). We
have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(2A_{1};1,4),\\{0\\}),(L(B_{2};2,3),\mathbb{O}_{(2^{2},1)})\\}.$

By (iii) of Proposition 3.4.1, $\mathrm{Bind}^{G}_{L(2A_{1};1,4)}\\{0\\}$ and
$\mathrm{Bind}^{G}_{L(B_{2};2,3)}\mathbb{O}_{(2^{2},1)}$ are non-isomorphic
covers of $\mathbb{O}$. In particular, all covers of $\mathbb{O}$ are
birationally induced.

$A_{2}\subset E_{6}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{5}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=3A_{1}$. The fundamental
groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=2A_{2}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$D_{4}(a_{1})\subset E_{6}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(2A_{2}+A_{1}),\\{0\\}),(L(A_{3}+A_{1};1,2,4,5),\\{0\\}),(L(D_{4}),\mathbb{O}_{(2^{2},1^{4})})\\},$

and therefore $m(\mathbb{O})=2$. Consider the set
$\mathcal{P}_{1}(\mathbb{O})$ of pairs $(M,\mathbb{O}_{M})$ consisting of a
Levi subgroup $M\subset G$ of semisimple co-rank 1 and a nilpotent $M$-orbit
$\mathbb{O}_{M}$ such that
$\mathbb{O}=\operatorname{Ind}^{G}_{M}\mathbb{O}_{M}$, considered up to
$G$-conjugacy. Note that $(M,\mathbb{O}_{M})\in\mathcal{P}_{1}(\mathbb{O})$ if
and only if there is a pair
$(L,\mathbb{O}_{L})\in\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$ such that $L$ is
(conjugate to) a subgroup of $M$ and
$\mathbb{O}_{M}=\operatorname{Ind}^{M}_{L}\mathbb{O}_{L}$. Thus we have

$M$ | $\mathbb{O}_{M}$ | $\pi_{1}^{M}(\mathbb{O}_{M})$
---|---|---
$L(2A_{2}+A_{1})$ | $\\{0\\}$ | $1$
$L(D_{5})$ | $\mathbb{O}_{(3^{2},1^{4})}$ | $\mathbb{Z}_{2}$
$L(A_{4}+A_{1})$ | $\mathbb{O}_{(2,1^{3})}\times\\{0\\}$ | $1$
$L(A_{5})$ | $\mathbb{O}_{(2^{2},1^{2})}$ | $1$
Table 1. $\mathcal{P}_{1}(\mathbb{O})$

If $\widetilde{\mathbb{O}}$ is a birationally induced cover of $\mathbb{O}$,
then $\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{M}\widetilde{\mathbb{O}}_{M}$
for some pair $(M,\mathbb{O}_{M})\in\mathcal{P}_{1}(\mathbb{O})$ and
$M$-equivariant cover $\widetilde{\mathbb{O}}_{M}$ of $\mathbb{O}_{M}$. By the
table above, there are 5 such $(M,\widetilde{\mathbb{O}}_{M})$, up to
conjugation by $G$. However, the pairs
$(L(D_{5}),\mathbb{O}_{(3^{2},1^{4})}),(L(A_{4}+A_{1}),\mathbb{O}_{(2,1^{3})}\times\\{0\\}),(L(A_{5}),\mathbb{O}_{(2^{2},1^{2})})\in\mathcal{P}_{1}(\mathbb{O})$
are induced from a common element of $\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$,
namely $(L(A_{3}+A_{1};1,2,4,5),\\{0\\})$. Hence three of the five
$(M,\widetilde{\mathbb{O}}_{M})$ give rise to isomorphic covers of
$\mathbb{O}$. It follows that there are at most 3 non-isomorphic birationally
induced covers of $\mathbb{O}$. Since $\pi_{1}(\mathbb{O})\simeq S_{3}$, there
are a total of 4 non-isomorphic covers of $\mathbb{O}$. So at least one such
is birationally rigid.

$(3A_{1})^{\prime\prime}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{6}),\\{0\\})\\}$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=2A_{1}$. The fundamental
groups are as follows

$\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=F_{4}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$A_{2}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(D_{6}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=(3A_{1})^{\prime}$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=A_{5}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$A_{2}+3A_{1}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{6}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{2}+2A_{1}$. The fundamental
groups are as follows

$\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=G_{2}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$(A_{3}+A_{1})^{\prime\prime}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(D_{5}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=2$. There are two codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{3}$, and
$\mathbb{O}_{2}=2A_{2}$. The fundamental groups are as follows

$\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1\qquad\pi_{1}(\mathbb{O}_{2})=1.$

Let $\widehat{\mathbb{O}}$ be the universal cover of $\mathbb{O}$.

We will show that the singularity $\Sigma_{1}$ of $X$ is resolved under the
map $\widehat{X}\stackrel{{\scriptstyle}}{{\to}}X$. Note that
$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2=m(\mathbb{O})$. So if
$\Sigma_{1}$ is not resolved, both $\mathbb{O}$ and $\widehat{\mathbb{O}}$ are
birationally induced from a co-rank 2 Levi. Since
$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$ contains a single Levi (of co-rank
2), there is only one cover of $\mathbb{O}$ which can be induced from a corank
2 Levi, namely $\mathrm{Bind}_{L}^{G}(\\{0\\})$. We conclude that $\Sigma_{1}$
is resolved under the map $\widehat{X}\stackrel{{\scriptstyle}}{{\to}}X$.
Analogously, $\Sigma_{2}$ is resolved. We have $\mathfrak{r}=B_{3}$, and
therefore $\widehat{\mathbb{O}}$ is birationally rigid by Corollary 3.9.4.

$D_{4}(a_{1})\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1}+A_{5}),\\{0\\}),(L(D_{6}),\mathbb{O}_{(2^{4},1^{4})})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=(A_{3}+A_{1})^{\prime}$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{3}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=6>1=m(\mathbb{O}),$

and $\mathfrak{r}=3A_{1}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$A_{3}+2A_{1}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{6}),3A_{1})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=(A_{3}+A_{1})^{\prime}$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=2A_{1}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$D_{4}(a_{1})+A_{1}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{5};1,3,4,5,6),\\{0\\})\\},$

and therefore $m(\mathbb{O})=2$. There are two codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=D_{4}(a_{1})$ and
$\mathbb{O}_{2}=A_{3}+2A_{1}$. Consider the map
$X=\operatorname{Spec}(\mathbb{C}[\mathbb{O}])\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$.
Let $\widecheck{\mathbb{O}}_{1}\subset X$ be the preimage of $\mathbb{O}_{1}$.
The slice $\Sigma_{1}$ to $\mathbb{O}_{1}$ is of type $3A_{1}$, and therefore
the preimage of $\Sigma_{1}$ is a disjoint union of three copies of
$A_{1}$singularities. The singularity $\Sigma_{2}$ of the leaf
$\mathfrak{L}_{2}$ corresponding to $\mathbb{O}_{2}$ is of type $A_{1}$. If
$\widecheck{\mathbb{O}}_{1}$ is not connected, then there are at least two
symplectic leaves of codimension $2$ over $\mathbb{O}_{1}$ with open $G$
orbits being the irreducible components of $\widecheck{\mathbb{O}}_{1}$. That
contradicts to Lemma 3.2.2. It follows that $\widecheck{\mathbb{O}}_{1}$ is a
$3$-fold connected cover of $\mathbb{O}_{1}$. The fundamental groups are as
follows

$\pi_{1}(\mathbb{O})=S_{2}\times\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{1})=S_{3}\qquad\pi_{1}(\widecheck{\mathbb{O}}_{1})=\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{2})=\mathbb{Z}_{2}$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\widecheck{\mathbb{O}}_{1})|^{-1}=2=m(\mathbb{O}),\qquad|\pi_{1}(\mathbb{O})||\pi_{1}({\mathbb{O}}_{2})|^{-1}=2=m(\mathbb{O}).$

Let $\widehat{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ be the
universal cover of $\mathbb{O}$. If one of the singularities $\Sigma_{1}$,
$\Sigma_{2}$ is not resolved under
$\widehat{X}\stackrel{{\scriptstyle}}{{\to}}X$, then we have
$\dim\mathfrak{P}^{X}=2$ and $\dim\mathfrak{P}^{\widehat{X}}\geq 2$. Thus,
both $\mathbb{O}$ and $\widehat{\mathbb{O}}$ are birationally induced from a
corank $2$ Levi. However, there is only one cover that can be induced from a
corank $2$ Levi, namely $\mathrm{Bind}_{L}^{G}(\\{0\\})$. Therefore, both
$\Sigma_{1}$ and $\Sigma_{2}$ are resolved under
$\widehat{X}\stackrel{{\scriptstyle}}{{\to}}X$. We have $\mathfrak{r}=2A_{1}$,
and therefore $\widehat{\mathbb{O}}$ is birationally rigid by Corollary 3.9.4.

$A_{3}+A_{2}\subset E_{7}$. Note that $\pi_{1}(\mathbb{O})=S_{2}$. Hence,
$\mathbb{O}$ has two non-isomorphic covers (including the trivial one). We
have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1}+D_{5}),\\{0\\}\times\mathbb{O}_{(2^{2},1^{6})}),(L(D_{6}),\mathbb{O}_{(3,2^{2},1^{5})})\\}$

By (iii) of Proposition 3.4.1, $\mathrm{Bind}^{G}_{L(A_{1}+D_{5})}\\{0\\}$ and
$\mathrm{Bind}^{G}_{L(D_{6})}\mathbb{O}_{(3,2^{2},1^{5})}$ are non-isomorphic
covers of $\mathbb{O}$. In particular, all covers of $\mathbb{O}$ are
birationally induced.

$A_{3}+A_{2}+A_{1}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{4}+A_{2};1,2,3,4,6,7),\\{0\\})\\}.$

Since $L=L(A_{4}+A_{2};1,2,3,4,6,7)$ is of semisimple co-rank 1 and
$\mathbb{O}_{L}=\\{0\\}$, there is a unique birationally induced cover of
$\mathbb{O}$, namely $\mathrm{Bind}_{L}^{G}\\{0\\}$. On the other hand, since
$\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}$, there are two non-isomorphic covers of
$\mathbb{O}$. So one must be birationally rigid.

$D_{4}+A_{1}\subset E_{7}$. Note that $\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}$.
Hence, $\mathbb{O}$ has two non-isomorphic covers (including the trivial one).
We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(D_{6}),\mathbb{O}_{(3,2^{4},1)})\\}$

Write $(L,\mathbb{O}_{L})=(L(D_{6}),\mathbb{O}_{(3,2^{4},1)})$. An atlas
computation shows that $\pi_{1}^{L}(\mathbb{O}_{L})=\mathbb{Z}_{2}$. Let
$\widetilde{\mathbb{O}}_{L}$ denote the 2-fold $L$-equivariant cover of
$\mathbb{O}_{L}$. By (iv) of Proposition 3.4.1,
$\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$ is a 2-fold cover of
$\mathrm{Bind}^{G}_{L}\mathbb{O}_{L}$. In particular, both covers of
$\mathbb{O}$ are birationally induced.

$A_{5}+A_{1}\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{6}),2A_{2}+A_{1})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{4}+A_{2}$. The fundamental
groups are as follows

$\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=A_{1}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$D_{5}(a_{1})+A_{1}\subset E_{7}$. Note that
$\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}$. Hence, $\mathbb{O}$ has two non-
isomorphic covers (including the trivial one). We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{3}+A_{2}),\\{0\\})\\}$

We note that the induction from $(L(A_{3}+A_{2}),\\{0\\})$ is birational
(there are several ways to see this — one will be given in the
$D_{5}(a_{1})+A_{1}\subset E_{7}$ portion of Section 4.3). Recall the set
$\mathcal{P}_{1}(\mathbb{O})$ defined above. Since $\mathbb{O}$ is
birationally induced from $(L(A_{3}+A_{2}),\\{0\\})$, every pair
$(M,\mathbb{O}_{M})\in\mathcal{P}_{1}(\mathbb{O})$ is of the form
$\mathbb{O}_{M}=\operatorname{Ind}^{M}_{L}\\{0\\}$, where $M$ is a co-rank 1
Levi containing (a $G$-conjugate of) $L$. Thus we have

$M$ | $\mathbb{O}_{M}$ | $\pi_{1}^{M}(\mathbb{O}_{M})$
---|---|---
$A_{6}$ | $\mathbb{O}_{(2^{3},1^{3})}$ | 1
$D_{6}$ | $\mathbb{O}_{(3^{3},1^{3})}$ | 1
$A_{4}+A_{2}$ | $\mathbb{O}_{(2,1^{3})}\times\\{0\\}$ | 1
$A_{3}+A_{2}+A_{1}$ | $\\{0\\}\times\\{0\\}\times\mathbb{O}_{(2)}$ | 1
Table 2. $\mathcal{P}_{1}(\mathbb{O})$

Since $\mathbb{O}$ is birationally induced from $(L,\mathbb{O}_{L})$, it is
birationally induced from $(M,\mathbb{O}_{M})$ for all the pair
$(M,\mathbb{O}_{M})$ in the list above. It follows that the universal cover of
$\mathbb{O}$ is birationally rigid.

$D_{6}(a_{2})\subset E_{7}$. Note that $\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}$.
Hence, $\mathbb{O}$ admits two non-isomorphic covers (including the trivial
one). We have

$\displaystyle\mathbb{O}$
$\displaystyle=\operatorname{Ind}^{G}_{L(D_{5})}\mathbb{O}_{(3,2^{2},1^{3})}$
$\displaystyle=\operatorname{Ind}^{G}_{L(D_{5}+A_{1})}(\operatorname{Ind}^{L(D_{5}+A_{1})}_{L(D_{5})}\mathbb{O}_{(3,2^{2},1^{3})})$
$\displaystyle=\operatorname{Ind}^{G}_{L(D_{5}+A_{1})}\mathbb{O}_{(3,2^{2},1^{3})}\times\mathbb{O}_{(2)}$

Write
$(L,\mathbb{O}_{L})=(L(D_{5}+A_{1}),\mathbb{O}_{(3,2^{2},1^{3})}\times\mathbb{O}_{(2)})$.
An atlas computation shows that $\pi_{1}^{L}(\mathbb{O}_{L})=\mathbb{Z}_{2}$.
Let $\widetilde{\mathbb{O}}_{L}$ denote the two-fold $L$-equivariant cover of
$\mathbb{O}_{L}$. By (iv) of Proposition 3.4.1,
$\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$ is a two-fold cover of
$\mathrm{Bind}^{G}_{L}\mathbb{O}_{L}$. In particular, both covers of
$\mathbb{O}$ are birationally induced.

$E_{6}(a_{3})\subset E_{7}$. Note that $\pi_{1}(\mathbb{O})=S_{2}$. So
$\mathbb{O}$ admits two non-isomorphic covers (inluding the trivial one). We
have

$\displaystyle\mathbb{O}$
$\displaystyle=\operatorname{Ind}^{G}_{L(2A_{1}+A_{3};1,2,4,5,7)}\\{0\\}$
$\displaystyle=\operatorname{Ind}^{G}_{L(A_{1}+D_{5})}(\operatorname{Ind}^{L(A_{1}+D_{5})}_{L(2A_{1}+A_{3};1,2,4,5,7)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(A_{1}+D_{5})}\\{0\\}\times\mathbb{O}_{(3^{2},1^{4})}$

Write
$(L,\mathbb{O}_{L})=(L(A_{1}+D_{5}),\\{0\\}\times\mathbb{O}_{(3^{2},1^{4})})$.
An atlas computation shows that $\pi_{1}^{L}(\mathbb{O}_{L})=\mathbb{Z}_{2}$.
Let $\widetilde{\mathbb{O}}_{L}$ denote the two-fold $L$-equivariant cover of
$\mathbb{O}_{L}$. By (iv) of Proposition 3.4.1,
$\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$ is a two-fold cover of
$\mathrm{Bind}^{G}_{L}\mathbb{O}_{L}$. In particular, both covers of
$\mathbb{O}$ are birationally induced.

$E_{7}(a_{5})\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1}+2A_{2};1,2,3,5,6),\\{0\\}),(L(A_{1}+A_{3};1,2,4,5),\\{0\\}),(L(D_{4}),\mathbb{O}_{(2^{2},1^{4})})\\},$

and therefore $m(\mathbb{O})=3$. There are two codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=E_{6}(a_{3})$ and
$\mathbb{O}_{2}=E=D_{6}(a_{2})$. The fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{3}\times\mathbb{Z}_{2}\qquad\pi_{1}(\mathbb{O}_{1})=\pi_{1}(\mathbb{O}_{2})=\mathbb{Z}_{2}.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{k})|^{-1}=6>3=m(\mathbb{O})\qquad
k=1,2,$

and $\mathfrak{r}=0$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$E_{7}(a_{4})\subset E_{7}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1}+D_{4}),\\{0\\}\times\mathbb{O}_{(3,2^{2},1)}),(L(2A_{1}+A_{2};2,3,5,6),\\{0\\})\\},$

and therefore $m(\mathbb{O})=3$. There are three codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{6}$,
$\mathbb{O}_{2}=D_{5}+A_{1}$, and $\mathbb{O}_{3}=D_{6}(a_{1})$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\times\mathbb{Z}_{2},\qquad\pi_{1}(\mathbb{O}_{1})=1,\qquad\pi_{1}(\mathbb{O}_{2})=\pi_{1}(\mathbb{O}_{3})=\mathbb{Z}_{2}.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=4>3=m(\mathbb{O}),$

and therefore the singularity $\Sigma_{1}$ is resolved under the map
$\widehat{X}\stackrel{{\scriptstyle}}{{\to}}X$, see Lemma 3.9.3. Moreover, the
preimage of $\Sigma_{1}$ is two copies of $\mathbb{C}^{2}$, and the Galois
group $S_{2}\times\mathbb{Z}_{2}$ of the cover permutes the two copies. Thus,
we have a map $\pi_{1}(\mathbb{O})\stackrel{{\scriptstyle}}{{\to}}S_{2}$. Let
$K$ be its kernel, and let $\widetilde{\mathbb{O}}=\widehat{\mathbb{O}}/K$.
Then $\widetilde{X}$ has $2$ symplectic leaves over the leaf
$\mathfrak{L}_{1}\subset X$ corresponding to
$\mathbb{O}_{1}\subset\overline{\mathbb{O}}$. We claim that
$\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}X$ resolves $\Sigma_{2}$ and
$\Sigma_{3}$. Otherwise, we have $\dim\mathfrak{P}^{\widetilde{X}}\geq 3$, and
hence both $\mathbb{O}$ and $\widetilde{\mathbb{O}}$ are birationally induced
from a co-rank 3 Levi. Since $\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$ contains
a unique pair with the Levi of co-rank $\geq 3$, namely
$(L(2A_{1}+A_{2};2,3,5,6),\\{0\\})$, it follows that both $\mathbb{O}$ and
$\widetilde{\mathbb{O}}$ are birationally induced from
$(L(2A_{1}+A_{2};2,3,5,6),\\{0\\})$, which is a contradiction. Therefore,
$\widehat{\mathbb{O}}$ is $2$-leafless. We have $\mathfrak{r}=0$. So
$\widehat{\mathbb{O}}$ is birationally rigid by Corollary 3.9.4.

$A_{2}\subset E_{8}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{7}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=1$. There is one codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=3A_{1}$. The fundamental
groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=E_{6}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$2A_{2}\subset E_{8}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(D_{7}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=1$. There is a single codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{2}+3A_{1}$. The fundamental
groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=2G_{2}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Proposition 3.7.1.

$D_{4}(a_{1})\subset E_{8}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1}+E_{6}),\\{0\\}),(L(E_{7}),2A_{1})\\},$

and therefore $m(\mathbb{O})=1$. There is a single codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{3}+A_{1}$. The fundamnetal
groups are as follows

$\pi_{1}(\mathbb{O})=S_{3}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=6>1=m(\mathbb{O}),$

and $\mathfrak{r}=D_{4}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Proposition 3.7.1.

$A_{3}+A_{2}\subset E_{8}$. Recall that
$H^{2}(\widehat{\mathbb{O}},\mathbb{C})\simeq\mathfrak{z}(\mathfrak{r})$.
Since $\mathfrak{r}=B_{2}+T_{1}$, $H^{2}(\widehat{\mathbb{O}},\mathbb{C})\neq
0$. Hence, $\widehat{\mathbb{O}}$ is birationally induced by Proposition
3.7.1. On the other hand, $\widehat{\mathbb{O}}$ is the unique nontrivial
cover of $\mathbb{O}$, since $\pi_{1}(\mathbb{O})=S_{2}$.

$D_{4}(a_{1})+A_{2}\subset E_{8}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{7}),\\{0\\})\\},$

and therefore $m(\mathbb{O})=1$. There is a single codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{3}+A_{2}+A_{1}$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=A_{2}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Proposition 3.7.1.

$D_{4}+A_{2}\subset E_{8}$. Note that $\pi_{1}(\mathbb{O})=\mathbb{Z}_{2}$.
Hence, $\mathbb{O}$ has two non-isomorphic covers (including the trivial one).
We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{6};1,3,4,5,6,7),\\{0\\})\\}.$

We note that the induction from $(L(A_{6};1,3,4,5,6,7),\\{0\\})$ is
birational. Recall the set $\mathcal{P}_{1}(\mathbb{O})$ defined above. Since
$\mathbb{O}$ is birationally induced from $(L(A_{6};1,3,4,5,6,7),\\{0\\}))$,
every pair $(M,\mathbb{O}_{M})\in\mathcal{P}_{1}(\mathbb{O})$ is of the form
$\mathbb{O}_{M}=\operatorname{Ind}^{M}_{L}\\{0\\}$, where $M$ is a co-rank 1
Levi containing (a $G$-conjugate of) $L$. Thus we have

$M$ | $\mathbb{O}_{M}$ | $\pi_{1}^{M}(\mathbb{O}_{M})$
---|---|---
$D_{7}$ | $\mathbb{O}_{(2^{6},1^{2})}$ | 1
$A_{7}$ | $\mathbb{O}_{(2,1^{6})}$ | 1
$A_{6}+A_{1}$ | $\\{0\\}\times\mathbb{O}_{(2)}$ | 1
$E_{7}$ | $A_{2}+3A_{1}$ | 1
Table 3. $\mathcal{P}_{1}(\mathbb{O})$

Since $\mathbb{O}$ is birationally induced from $(L,\mathbb{O}_{L})$, it is
birationally induced from $(M,\mathbb{O}_{M})$ for all the pair
$(M,\mathbb{O}_{M})$ in the list above. It follows that the universal cover of
$\mathbb{O}$ is birationally rigid.

$D_{6}(a_{2})\subset E_{8}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(D_{7}),(3,2^{4},1))\\},$

and therefore $m(\mathbb{O})=1$. There are $2$ codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=D_{5}(a_{1})+A_{2}$ and
$\mathbb{O}_{2}=A_{5}+A_{1}$. The corresponding singularities are of types
$A_{1}$ and $m$. Thus, there is a unique codeminsion $2$ leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$. The
fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=2A_{1}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$E_{6}(a_{3})\subset E_{8}$. Note that $\pi_{1}(\mathbb{O})=S_{2}$. So
$\mathbb{O}$ admits two non-isomorphic covers (including the trivial one). We
have

$\mathbb{O}=\operatorname{Ind}^{G}_{L(D_{5}+A_{1})}\\{0\\}=\operatorname{Ind}^{G}_{L(D_{7})}(\operatorname{Ind}^{L(D_{7})}_{L(D_{5}+A_{1})}\\{0\\})=\operatorname{Ind}^{G}_{L(D_{7})}\mathbb{O}_{(3^{2},1^{8})}.$

Write $(L,\mathbb{O}_{L})=(L(D_{7}),\mathbb{O}_{(3^{2},1^{8})})$. An atlas
computation shows that $\pi_{1}^{L}(\mathbb{O}_{L})\simeq\mathbb{Z}_{2}$. Let
$\widetilde{\mathbb{O}}_{L}$ denote the two-fold $L$-equivariant cover of
$\mathbb{O}_{L}$. By (iv) of Proposition 3.4.1,
$\mathrm{Bind}^{G}_{M}\widetilde{\mathbb{O}}_{M}$ is a 2-fold cover of
$\mathrm{Bind}^{G}_{M}\mathbb{O}_{M}$. In particular, both covers of
$\mathbb{O}$ are birationally induced.

$E_{6}(a_{3})+A_{1}\subset E_{8}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{7}),A_{1}+2A_{2})\\},$

and therefore $m(\mathbb{O})=1$. There are $2$ codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=A_{5}+A_{1}$ and
$\mathbb{O}_{2}=D_{5}(a_{1})+A_{2}$. The corresponding singularities are of
types $A_{1}$ and $m$. Thus, there is a unique codeminsion $2$ leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$. The
fundamental groups are as follow

$\pi_{1}(\mathbb{O})=S_{2}\qquad\pi_{1}(\mathbb{O}_{1})=1.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=2>1=m(\mathbb{O}),$

and $\mathfrak{r}=A_{1}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$E_{7}(a_{5})\subset E_{8}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{7}),(A_{1}+A_{3})^{\prime}),(L(E_{6}+A_{1}),3A_{1}\times\\{0\\})\\},$

and therefore $m(\mathbb{O})=1$. There are two codimension 2 orbits in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=E_{6}(a_{3})+A_{1}$ and
$\mathbb{O}_{2}=D_{6}(a_{2})$. The fundamental groups are as follows

$\pi_{1}(\mathbb{O})=S_{3},\qquad\pi_{1}(\mathbb{O}_{1})=S_{2},\qquad\pi_{1}(\mathbb{O}_{2})=S_{2}.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{2})|^{-1}=3>1,$

and $\mathfrak{r}=A_{1}$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$E_{8}(a_{7})\subset E_{8}$. We have

$\displaystyle\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{$
$\displaystyle(L(A_{3}+A_{4}),\\{0\\}),(L(A_{2}+D_{5}),\\{0\\}\times\mathbb{O}_{(2^{2},1^{6})}),$
$\displaystyle(L(A_{1}+A_{5};1,2,4,5,6,7),\\{0\\}),(L(D_{6}),(2^{4},1^{4}))\\},$

and therefore $m(\mathbb{O})=2$. There is a single codimension 2 orbit in
$\overline{\mathbb{O}}$, namely $\mathbb{O}_{1}=E_{7}(a_{5})$. The fundamnetal
groups are as follows

$\pi_{1}(\mathbb{O})=S_{5}\qquad\pi_{1}(\mathbb{O}_{1})=S_{3}.$

Note that

$|\pi_{1}(\mathbb{O})||\pi_{1}(\mathbb{O}_{1})|^{-1}=20>2=m(\mathbb{O}),$

and $\mathfrak{r}=0$. So $\widehat{\mathbb{O}}$ is birationally rigid by
Corollary 3.9.4.

$E_{7}(a_{4})\subset E_{8}$. Note that $\pi_{1}(\mathbb{O})=S_{2}$. So
$\mathbb{O}$ admits two non-isomorphic covers (including the trivial one). We
have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{1}+D_{5};1,2,3,4,5,7),\\{0\\}\times\mathbb{O}_{(2^{2},1^{6})}),(L(D_{6}),\mathbb{O}_{(3,2^{2},1^{5})})\\}.$

By (iii) of Proposition 3.4.1,
$\mathrm{Bind}^{G}_{L(A_{1}+D_{5};1,2,3,4,5,7)}\\{0\\}$ and
$\mathrm{Bind}^{G}_{L(D_{6})}\mathbb{O}_{(3,2^{2},1^{5})}$ are non-isomorphic
covers of $\mathbb{O}$. In particular, both covers of $\mathbb{O}$ are
birationally induced.

$D_{5}+A_{2}\subset E_{8}$ . Recall that
$H^{2}(\widehat{\mathbb{O}},\mathbb{C})\simeq\mathfrak{z}(\mathfrak{r})$.
Since $\mathfrak{r}=T_{1}$, $H^{2}(\widehat{\mathbb{O}},\mathbb{C})\neq 0$. So
by Proposition 3.7.1, $\widehat{\mathbb{O}}$ is birationally induced. On the
other hand, $\widehat{\mathbb{O}}$ is the unique nontrivial cover of
$\mathbb{O}$, since $\pi_{1}(\mathbb{O})=S_{2}$. The singularity of a
codimension $2$ orbit $E_{7}(a_{4})\subset\overline{\mathbb{O}}$ is not of
type $m$ (namely, it is of type $A_{1}$), and therefore the Namikawa space
$\mathfrak{P}^{X}$ for $X=\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ is non-
trivial. Thus, $\mathbb{O}$ is birationally induced, and so all covers of
$\mathbb{O}$ are birationally induced.

$D_{7}(a_{1})\subset E_{8}$. Since there is a cosimension $2$ orbit
$D_{6}(a_{2})\subset\overline{\mathbb{O}}$ with the singularity not of type
$m$ (namely, it is of type $2A_{1}$), the argument above for
$D_{5}+A_{2}\subset E_{8}$ holds word for word. In particular, all covers of
$\mathbb{O}$ are birationally induced. ∎

###  3.10. Some computational tools

Let $\mathbb{O}$ be a nilpotent orbit such that
$H^{2}(\mathbb{O},\mathbb{C})=0$. Let
$X=\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and write
$\mathfrak{L}_{1},...,\mathfrak{L}_{t}\subset X$ for the codimension 2 leaves.
For the determination of unipotent infinitesimal characters (to be carried out
in Section 4), there are three separate computations one needs to perform:

  1. (1)

Compute the birationally minimal induction datum $(L,\mathbb{O}_{L})$ for
$\mathbb{O}$.

  2. (2)

Compute the $\mathfrak{L}_{k}$-adapted resolution datum
$(M_{k},\mathbb{O}_{M_{k}})$ for each codimension 2 leaf.

  3. (3)

Compute the isomorphism
$\eta_{k}:\mathfrak{X}(\mathfrak{m})\xrightarrow{\sim}\mathfrak{P}_{k}^{X}$
for each codimension 2 leaf.

We will use several techniques for each of these computations, which we will
explain and catalog below.

####  3.10.1. Computation of $(L,\mathbb{O}_{L})$

First, we explain two techniques for computing $(L,\mathbb{O}_{L})$. 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 event, we can define a parabolic subalgebra

(3.10.1)
$\mathfrak{p}_{\mathbb{O}}=\mathfrak{l}_{\mathbb{O}}\oplus\mathfrak{n}_{\mathbb{O}},\qquad\mathfrak{l}_{\mathbb{O}}:=\mathfrak{g}_{0},\qquad\mathfrak{n}_{\mathbb{O}}:=\bigoplus_{i\geq
1}\mathfrak{g}_{i}.$

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 $G$. It is very easy to determine
$\mathfrak{p}_{\mathbb{O}}$ from the weighted diagram for $\mathbb{O}$—it is
the parabolic subalgebra corresponding to the simple roots labeled ‘0.’ The
following result is well-known. The proof is contained in [Kos59], see also
[CM93, Thm 3.3.1].

######  Lemma 3.10.1.

If $\mathbb{O}$ is even, then
$\mathbb{O}=\mathrm{Bind}^{G}_{L_{\mathbb{O}}}\\{0\\}$.

We will also use the following.

######  Lemma 3.10.2.

Suppose

  * (i)

The set $\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$ contains a unique element
$(L_{0},\mathbb{O}_{L_{0}})$.

  * (ii)

$\dim\mathfrak{X}(\mathfrak{l}_{0}\cap[\mathfrak{g},\mathfrak{g}])\leq\dim\mathfrak{P}^{X}$.

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

###### Proof.

Choose a birationally minimal induction datum $(L,\mathbb{O}_{L})$ for
$\mathbb{O}$. Suppose for contradiction that $(L,\mathbb{O}_{L})$ is not
$G$-conjugate to $(L_{0},\mathbb{O}_{L_{0}})$. In particular, by condition
(i), $\mathbb{O}_{L}$ is not rigid. So there is a proper Levi subgroup
$M\subset L$ and a rigid $M$-orbit $\mathbb{O}_{M}$ such that
$\mathbb{O}_{L}=\operatorname{Ind}^{L}_{M}\mathbb{O}_{M}$. By the transitivity
of induction, $(M,\mathbb{O}_{M})\in\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$.
Hence, $(M,\mathbb{O}_{M})$ is $G$-conjugate to $(L_{0},\mathbb{O}_{L_{0}})$
and
$\dim\mathfrak{X}(\mathfrak{l}_{0}\cap[\mathfrak{g},\mathfrak{g}])>\dim\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$.
On the other hand, by (ii) of Proposition 3.5.1, there is a linear isomorphism
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\simeq\mathfrak{P}^{X}$.
So

$\dim\mathfrak{X}(\mathfrak{l}_{0}\cap[\mathfrak{g},\mathfrak{g}])>\dim\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])=\dim\mathfrak{P}^{X}.$

This contradicts condition (ii). ∎

####  3.10.2. Computation of $(M_{k},\mathbb{O}_{M_{k}})$

Fix a birationally minimal induction datum $(L,\mathbb{O}_{L})$ for
$\mathbb{O}$, and choose a codimension 2 leaf $\mathfrak{L}_{k}\subset X$. In
many cases we consider, $\mathfrak{L}_{k}=\mathfrak{L}_{1}$ is the _only_
codimension 2 leaf. In such cases, $(L,\mathbb{O}_{L})$ trivially satisfies
conditions (i)-(iii) of Proposition 3.6.3 and therefore

$(M_{1},\mathbb{O}_{M_{1}})=(L,\mathbb{O}_{L})$

In the remaining cases, we use the following result, which follows immediately
from [LMM21, Lem 7.5.10] and Lemma 3.2.2.

######  Lemma 3.10.3.

Suppose $M\subset G$ is a Levi subgroup containing $L$ and let
$\mathbb{O}_{M}=\operatorname{Ind}^{M}_{L}\mathbb{O}_{L}$. Then
$(M,\mathbb{O}_{M})=(M_{k},\mathbb{O}_{M_{k}})$ if the following conditions
are satisfied:

  * (i)

The semisimple corank of $M$ equals the dimension of $\mathfrak{P}_{k}$.

  * (ii)

For every $j\neq k$, there is a nilpotent $M$-orbit
$\mathbb{O}_{M,j}\subset\overline{\mathbb{O}}_{M}$ of codimension 2 such that

$\mathbb{O}_{j}=\mathrm{Ind}^{G}_{M}\mathbb{O}_{M,j}$

  * (iii)

For every $j\neq k$, such that the singularity of the orbit
$\mathbb{O}_{j}\subset\overline{\mathbb{O}}$ is not of type $m$, the
singularity of $\mathbb{O}_{M,j}\subset\overline{\mathbb{O}}_{M,j}$ is not of
type $m$, and moreover the singularities of $\mathfrak{L}_{j}\subset
Spec(\mathbb{C}[\mathbb{O}])$ and $\mathfrak{L}_{M,j}\subset
Spec(\mathbb{C}[\mathbb{O}_{M}])$ are equivalent.

######  Remark 3.10.4.

We note that condition (iii) of Lemma 3.10.3 is implied by the following
condition (which is often easier to check):

  * (iii’)

For every $j\neq k$ the singularities of
$\mathbb{O}_{j}\subset\overline{\mathbb{O}}$ and
$\mathbb{O}_{M,j}\subset\overline{\mathbb{O}}_{M}$ are equivalent.

####  3.10.3. Computation of $\eta_{k}$

As in the previous subsection, fix a birationally minimal induction datum
$(L,\mathbb{O}_{L})$ for $\mathbb{O}$, and choose a codimension 2 leaf
$\mathfrak{L}_{k}\subset X$. Assume

  * •

$\pi_{1}(\mathfrak{L}_{k})$ acts trivially on
$H^{2}(\mathfrak{S}_{k},\mathbb{C})$.

Under this condition, $\mathfrak{P}_{k}$ can be identified with the vector
space $\mathfrak{h}_{k}^{*}$, i.e. the dual Cartan subalgebra corresponding to
the Kleinian singularity $\Sigma_{k}$. In particular, $\mathfrak{P}_{k}$
admits a natural basis consisting of fundamental weights, denoted
$\\{\omega_{i}(k)\mid 1\leq i\leq n(k)\\}$. On the other hand,
$\mathfrak{X}(\mathfrak{m}_{k})$ admits a basis consisting of dominant
generators for the free abelian group $\mathfrak{X}(M_{k})$, denoted
$\\{\tau_{i}(k)\mid 1\leq i\leq n(k)\\}$. In cases when we have
$\mathfrak{P}=\mathfrak{P}_{1}$ we omit the index $k$ and write simply
$\omega_{i}$ and $\tau_{i}$. We wish to compute $\eta_{k}$ in terms of these
bases. In most cases, we will use one of the following two results,
established in [LMM21].

######  Proposition 3.10.5.

Suppose

  * (a1)

$\mathfrak{L}_{k}=\mathfrak{L}_{1}$ is the unique codimension 2 leaf in
$\widetilde{X}$.

  * (a2)

$\operatorname{Pic}(\mathbb{O})\simeq\mathfrak{X}(\Gamma_{1})$.

  * (a3)

$\operatorname{Pic}(\mathbb{O}_{M_{1}})=0$.

  * (a4)

Up to the action of $W^{\widetilde{X}}$, there is a unique parabolic subgroup
$P\subset G$ with Levi factor $L$.

Then $M_{1}=L$, $\eta_{1}=\eta$, and
$\eta_{1}\\{\tau_{i}\\}=\\{\omega_{i}\\}$.

######  Proposition 3.10.6.

Suppose

  * (b1)

$\Sigma_{k}$ is of type $A_{1}$.

Then $\operatorname{Pic}(\mathbb{O}_{M_{k}})$ is finite,

(3.10.2)
$c_{k}:=2|\operatorname{Pic}(\mathbb{O}_{M_{k}})||\operatorname{Pic}(\mathbb{O})|^{-1}$

is an integer, and $\eta_{k}(\tau_{1}(k))=c_{k}\omega_{1}(k)$.

In a few cases, neither proposition is straightforwardly applicable, and we
must provide a separate argument.

##  4\. Unipotent ideals and their infinitesimal characters

Let $G$ be a connected reductive algebraic group and let
$\widetilde{\mathbb{O}}$ be a $G$-equivariant nilpotent cover. Recall the
canonical quantization $(\mathcal{A}_{0},\Phi_{0})$ of
$\mathbb{C}[\widetilde{\mathbb{O}}]$, cf. Definition 2.6.4.

######  Definition 4.0.1.

The _unipotent ideal_ attached to $\widetilde{\mathbb{O}}$ is the two-sided
ideal

$I_{0}(\widetilde{\mathbb{O}}):=\ker{(\Phi_{0}:U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{0})}\subset
U(\mathfrak{g})$

We show in [LMM21] that $I_{0}(\widetilde{\mathbb{O}})$ is a completely prime
primitive ideal with associated variety $\overline{\mathbb{O}}$. We also give
a classification of unipotent ideals, which we will now recall. Write
$\widetilde{\mathbb{O}}^{1}\succeq\widetilde{\mathbb{O}}^{2}$ if there is a
$G$-equivariant morphism
$\widetilde{\mathbb{O}}^{1}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}^{2}$
such that the induced morphism
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}^{1}])\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}^{2}])$
is étale over the open subset in
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}^{2}])$ obtained by
removing all symplectic leaves of codimension $\geq 4$. Note that $\succeq$
defines a partial order on the set of ($G$-equivariant) nilpotent covers.
Consider the equivalence relation (on the same set) which is generated by
$\succeq$ and write $[\widetilde{\mathbb{O}}]$ for the equivalence class of
$\widetilde{\mathbb{O}}$.

######  Theorem 4.0.2 (Thm 6.5.5, [LMM21]).

Suppose $\widetilde{\mathbb{O}}^{1}$, $\widetilde{\mathbb{O}}^{2}$ are
$G$-equivariant nilpotent covers. Then
$I_{0}(\widetilde{\mathbb{O}}^{1})=I_{0}(\widetilde{\mathbb{O}}^{2})$ if and
only if $[\widetilde{\mathbb{O}}^{1}]=[\widetilde{\mathbb{O}}^{2}]$.

Let $\gamma_{0}(\widetilde{\mathbb{O}})\in\mathfrak{h}^{*}/W$ denote the
infinitesimal character of $I_{0}(\widetilde{\mathbb{O}})$. If
$(L,\widetilde{\mathbb{O}}_{L})$ is a birationally minimal indution datum,
then we have the following result.

######  Proposition 4.0.3 (Prop 8.1.1, [LMM21]).

There is an equality in $\mathfrak{h}^{*}/W$

$\gamma_{0}(\widetilde{\mathbb{O}})=\gamma_{0}(\widetilde{\mathbb{O}}_{L}).$

Thus, the computation of $\gamma_{0}(\widetilde{\mathbb{O}})$ can be reduced
to the case of birationally rigid covers. In [LMM21], we computed
$\gamma_{0}(\widetilde{\mathbb{O}})$ for such covers for linear classical
groups. In this section, we will compute $\gamma_{0}(\widetilde{\mathbb{O}})$
for all birationally rigid covers for spin and exceptional groups, effectively
completing the computation of unipotent infinitesimal characters for all
nilpotent covers.

###  4.1. Computing $\gamma_{0}(\widetilde{\mathbb{O}})$

Let $\widetilde{\mathbb{O}}$ be a birationally rigid nilpotent cover. In this
subsection, we will recall a general algorithm, developed in [LMM21], for
computing $\gamma_{0}(\widetilde{\mathbb{O}})$. This algorithm has three
separate components.

####  4.1.1. Reduction to the case of birationally rigid orbits

Choose a birationally minimal induction datum $(L,\mathbb{O}_{L})$ for
$\mathbb{O}$. Since $\widetilde{\mathbb{O}}$ is birationally rigid,
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ has no codimension 2
leaves, see Corollary 3.9.4. Let
$X=\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and let
$\epsilon\in\mathfrak{P}^{X}$ denote the barycenter parameter (cf. (2.5.1)).
Recall, Proposition 3.5.1, that there is a natural identification

$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\xrightarrow{\sim}\mathfrak{P}^{X}$

We define $\delta:=\eta^{-1}(\epsilon)$. This element can be computed using
the techniques of Section 3.10.3. The following is a special case of [LMM21,
Prop 8.1.3].

######  Proposition 4.1.1.

There is an equality in $\mathfrak{h}^{*}/W$

$\gamma_{0}(\widetilde{\mathbb{O}})=\gamma_{0}(\mathbb{O}_{L})+\delta$

This proposition reduces the computation of
$\gamma_{0}(\widetilde{\mathbb{O}})$ to the case of birationally rigid orbits.

####  4.1.2. Reduction to the case of rigid orbits

Let $\mathbb{O}$ be a birationally rigid orbit. Choose a Levi subgroup
$L\subset G$ and a rigid orbit $\mathbb{O}_{L}$ such that
$\mathbb{O}=\operatorname{Ind}^{G}_{L}\mathbb{O}_{L}$. The following is a
special case of [LMM21, Prop 8.1.3].

######  Proposition 4.1.2.

There is an equality in $\mathfrak{h}^{*}/W$

$\gamma_{0}(\mathbb{O})=\gamma_{0}(\mathbb{O}_{L}).$

####  4.1.3. Case of rigid orbits

If $\mathbb{O}$ is a rigid orbit, $\gamma_{0}(\mathbb{O})$ was computed in all
cases in [LMM21]. The argument proceeds as follows. With six exceptions (in
types $G_{2}$,$F_{4}$, $E_{7}$, and $E_{8}$), $I_{0}(\mathbb{O})$ is the
_unique_ primitive ideal with the following two properties

  1. (1)

The associated variety of $I_{0}(\mathbb{O})$ is $\overline{\mathbb{O}}$.

  2. (2)

$I_{0}(\mathbb{O})$ has multiplicity $1$ along $\overline{\mathbb{O}}$.

Such ideals were classified (i.e. their infinitesimal characters were
computed) by McGovern ([McG94]) in classical types and Premet ([Pre13]) in
exceptional types. For the six ‘bad’ orbits in exceptional types, there are
several ideals satisfying (1) and (2) above–some, though not all, were
computed by Premet in [Pre13]. In [LMM21, Appendix C] we use a technical
geometric argument to resolve this ambiguity, determing the unipotent ideal
$I_{0}(\widetilde{\mathbb{O}})$ in all cases.

###  4.2. Spin groups

Let $G=\mathrm{SO}(n)$, $\widetilde{G}=\mathrm{Spin}(n)$, and let
$\widetilde{\mathbb{O}}$ be a birationally rigid $\widetilde{G}$-equivariant
nilpotent cover. In this section, we will give a combinatorial formula for the
infinitesimal character $\gamma_{0}(\widetilde{\mathbb{O}})$.

We begin by recalling the classification of birationally semi-rigid nilpotent
orbits for $\widetilde{G}$.

######  Proposition 4.2.1 (Props 7.6.7, 7.6.11, [LMM21]).

Let $\mathbb{O}$ be a nilpotent $\widetilde{G}$-orbit corresponding to a
partition $p$ of $n$.

  * •

$\mathbb{O}$ admits a birationally rigid $G$-equivariant cover if and only if
the following conditions are satisfied:

    * (i)

if $p_{i}$ is odd, then $p_{i}\leq p_{i+1}+2$.

    * (ii)

If $p_{i}$ is even, then $p_{i}\leq p_{i+1}+1$.

  * •

$\mathbb{O}$ admits a birationally rigid $\widetilde{G}$-equivariant cover,
which is _not_ $G$-equivariant, if and only if the following conditions are
satisfied:

    * (ia)

$p$ is rather odd (i.e. every odd part occurs with multiplicity 1).

    * (iia)

$p_{i}\leq p_{i+1}+1$ if $p_{i}$ is even, and $p_{i}\leq p_{i+1}+4$ if $p_{i}$
is odd.

    * (iiia)

$p_{i}\neq p_{i+1}+3$ for all $i$.

    * (iva)

$p_{i}=p_{i+1}=4$ for some odd $p_{i}$.

Our main result in this subsection will require some additional notation. By a
$\frac{1}{2}\mathbb{Z}$-_partition_ of $n\in\mathbb{Z}$ we will mean a non-
increasing sequence $p=(p_{1},p_{2},...,p_{t})$ in $\frac{1}{2}\mathbb{Z}$
such that $\sum_{i=1}^{t}p_{i}=n$ (to prevent confusion, we will call ordinary
partitions ‘$\mathbb{Z}$-partitions’ in this section to emphasize the
distinction with $\frac{1}{2}\mathbb{Z}$-partitions).

######  Definition 4.2.2.

Let $p$ be a $\mathbb{Z}$-partition of $n$.

  * •

Define a $\mathbb{Z}$-partition (with no repeated parts)

$S_{4}(p)=\\{i:p_{i}=p_{i+1}+4\\}.$

  * •

If $x=(x_{1},...,x_{r})$ is a subpartition of $S_{4}(p)$, define a
$\mathbb{Z}$-partition $p\\#x$ by deleting the columns in $p$ numbered
$p_{x_{1}},p_{x_{1}}-1$, $p_{x_{2}}$, $p_{x_{2}}-1$, …, $p_{x_{r}}$,
$p_{x_{r}}-1$.

  * •

Define $\mathbb{Z}$-partitions $x(p)$, $y(p)$, and $z(p)$ by extracting all
parts of $p$ with multiplicity 1,2, and 4, respectively.

  * •

Define a $\mathbb{Z}$-partition $f(p)$ as follows: for every odd $i$ with
$p_{i}\geq p_{i+1}+2$, move one box down from $p_{i}$ to $p_{i+1}$.

  * •

If all parts in $p$ are of multiplicity 2, define a $\mathbb{Z}$-partition
$g(p)$ by replacing every pair $(p_{i},p_{i})$ with $(p_{i}+1,p_{i}-1)$.

  * •

If all parts in $p$ are of multiplicity 2, define a
$\frac{1}{2}\mathbb{Z}$-partition $h^{\prime}(p)$ by replacing every pair
$(p_{i},p_{i})$ with $(p_{i}+1/2,p_{i}-1/2)$.

  * •

If all parts in $p$ are of multiplicity 4, define
$\frac{1}{2}\mathbb{Z}$-partition $h(p)$ by replacing every quadruple
$(p_{i},p_{i},p_{i},p_{i})$ with $(p_{i}+1,p_{i}+1/2,p_{i}-1/2,p_{i}-1)$.

  * •

If all parts in $p$ are of even multiplicity, define $p^{1/2}$ by halving all
multiplicities.

######  Example 4.2.3.

If $p=(9,5,4^{2},3^{4},1)$, then

$S_{4}(p)=(1),\quad p\\#S_{4}(p)=(7,5,4^{2},3^{4},1),\quad x(p)=(9,5,1),\quad
y(p)=(4^{2}),\quad z(p)=(3^{4}).$

If $p=(5^{2},2^{2},1^{2})$, then $g(p)=(6,4,3,2,1)$. If $p=(5^{4},1^{4})$,
then $h(p)=(6,11/2,9/2,4,2,3/2,1/2)$.

######  Definition 4.2.4.

  * •

If $q$ is a $\mathbb{Z}$-partition of $n$, define a
$\lfloor\frac{n}{2}\rfloor$-tuple
$\rho(q)\in(\frac{1}{2}\mathbb{Z})^{\lfloor\frac{n}{2}\rfloor}$ by appending
the sequence

$(\frac{q_{i}-1}{2},\frac{q_{i}-3}{2},...,\frac{3-q_{i}}{2},\frac{1-q_{i}}{2})$

for each $i\geq 1$.

  * •

If $q$ is a $\frac{1}{2}\mathbb{Z}$-partition, define a
$\lfloor\frac{n}{2}\rfloor$-tuple
$\rho^{+}(q)\in(\frac{1}{4}\mathbb{Z})^{\lfloor\frac{n}{2}\rfloor}$ by
appending the positive elements of the sequence

$(\frac{q_{i}-1}{2},\frac{q_{i}-3}{2},...,\frac{3-q_{i}}{2},\frac{1-q_{i}}{2})$

for each $i\geq 1$, and then $0$’s as needed so that
$|\rho^{+}(p)|=\lfloor\frac{n}{2}\rfloor$ (if $q_{i}=1/2$, append nothing).

######  Example 4.2.5.

If $q=(3^{2},2)$, then

$\rho(q)=(1,0,-1,1,0,-1,1/2,-1/2).$

If $q=(5/2,5/2,2,3/2,1/2)$, then

$\rho^{+}(q)=(3/4,3/4,1/2,1/4,0).$

######  Proposition 4.2.6.

Let $\mathbb{O}$ be a nilpotent $\widetilde{G}$-orbit corresponding to a
partition $p$ of $n$. Assume $\mathbb{O}$ admits a birationally rigid
$\widetilde{G}$-equivariant nilpotent cover
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$. Let

$x:=x(p^{t}),\quad y:=y(p^{t}),\quad z:=z(p^{t}).$

By Proposition 4.2.1, $p^{t}=x\cup y\cup z$. In standard coordinates on
$\mathfrak{h}^{*}$

$\gamma_{0}(\widetilde{\mathbb{O}})=\rho^{+}(f(x)\cup g(y)\cup h(z))$

If $n$ is even, this weight contains (at least one) entry equal to $0$ (i.e.
the Weyl group acts on this weight by arbitrary permutations and sign
changes).

###### Proof.

Let
$\widehat{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$
denote the universal ($\widetilde{G}$-equivariant) cover. By Corollary 3.9.4,
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ has no codimension 2
leaves. So by Theorem 4.0.2,
$I_{0}(\widehat{\mathbb{O}})=I_{0}(\widetilde{\mathbb{O}})$, and therefore
$\gamma_{0}(\widehat{\mathbb{O}})=\gamma_{0}(\widetilde{\mathbb{O}})$. If
$\widehat{\mathbb{O}}$ is $G$-equivariant, then by Proposition 4.2.1,
$z=\emptyset$, i.e. $p^{t}=x\cup y$. Hence

$\rho^{+}(f(x)\cup g(y)\cup h(z))=\rho^{+}(f(x)\cup g(y)).$

This coincides with $\gamma_{0}(\widehat{\mathbb{O}})$ by [LMM21, Prop 8.2.8].
Thus, we can assume that $\widehat{\mathbb{O}}$ is not $G$-equivariant. In
particular, by Proposition 4.2.1, $p$ is rather odd. So
$\pi_{1}^{\widetilde{G}}(\mathbb{O})$ is a central $\mathbb{Z}_{2}$-extension
of $\pi_{1}^{G}(\mathbb{O})$, see [CM93, Cor 6.1.6], and
$\widehat{\mathbb{O}}$ is a 2-fold cover of the universal $G$-equivariant
cover $\widetilde{\mathbb{O}}$ of $\mathbb{O}$.

Form the partition $p\\#S_{4}(p)$ of $n-2|S_{4}(p)|$ as in Definition 4.2.2
and let $\widetilde{\mathbb{O}}_{p\\#S_{4}(p)}$ be the universal
$\mathrm{SO}(n-2|S_{4}(p)|)$ cover of $\mathbb{O}_{p\\#S_{4}(p)}$. We claim
first of all that the following pair is a birationally minimal induction datum
for $\widetilde{\mathbb{O}}$

$L=\prod_{k\in
S_{4}(p)}\mathrm{GL}(k)\times\mathrm{SO}(n-2|S_{4}(p)|),\qquad\widetilde{\mathbb{O}}_{L}=\\{0\\}\times...\\{0\\}\times\widetilde{\mathbb{O}}_{p\\#S_{4}(p)}.$

Indeed, this is a special case of [Mat20, Thm 4.17]. Let

$x^{\prime}=x((p\\#S_{4}(p))^{t}),\qquad y^{\prime}=y((p\\#S_{4}(p))^{t}),$

so that $(p\\#S_{4}(p))^{t}=x^{\prime}\cup y^{\prime}$. Applying [LMM21, Prop
8.2.8] to $\widetilde{\mathbb{O}}_{L}$ we obtain

$\gamma_{0}(\widetilde{\mathbb{O}}_{L})=(\rho(S_{4}(p)),\gamma_{0}(\widetilde{\mathbb{O}}_{p\\#S_{4}(p)}))=(\rho(S_{4}(p)),\rho^{+}(f(x^{\prime})\cup
g(y^{\prime}))).$

The shift $\delta=\eta^{-1}(\epsilon)\in\mathfrak{X}(\mathfrak{l})$ was
computed in [LMM21, Cor 7.7.7]. In standard coordinates it is the element

$\delta=(\underbrace{1/4,...,1/4}_{|S_{4}(p)|},0,...,0)\in\mathfrak{X}(\mathfrak{l})$

So by Proposition 4.1.1 we have

$\gamma_{0}(\widehat{\mathbb{O}})=(\rho(S_{4}(p)),\rho^{+}(f(x^{\prime})\cup
g(y^{\prime})))+(\underbrace{1/4,...,1/4}_{|S_{4}(p)|},0,...,0).$

Note that up to permutations and sign changes

$\rho(S_{4}(p))+(1/4,...,1/4)=\rho^{+}(h^{\prime}(z^{1/2})),$

So (up to permutations and sign changes)

$\gamma_{0}(\widehat{\mathbb{O}})=\rho^{+}(f(x^{\prime})\cup g(y^{\prime})\cup
h^{\prime}(S_{4}(p)\cup S_{4}(p))).$

By the construction of $S_{4}(p)$ and $p\\#S_{4}(p)$, we have

$x^{\prime}=x,\quad\text{and}\quad y^{\prime}=y\cup z^{1/2}.$

So (up to permutation)

(4.2.1) $\gamma_{0}(\widehat{\mathbb{O}})=\rho^{+}(f(x)\cup g(y)\cup
g(z^{1/2})\cup h^{\prime}(z^{1/2})).$

By definition, $h(z)=g(z^{1/2})\cup h^{\prime}(z^{1/2})$. So (4.2.1) becomes
(up to permutation)

$\gamma_{0}(\widehat{\mathbb{O}})=\rho^{+}(f(x)\cup g(y)\cup h(z)).$

∎

######  Example 4.2.7.

Let $n=6$ (so that $\widetilde{G}=\mathrm{Spin}(6)\simeq\mathrm{SL}(4)$), and
let $\mathbb{O}$ be the principal nilpotent orbit (so that $p=(5,1)$). By
Proposition 4.2.1, $\mathbb{O}$ admits a birationally rigid
$\widetilde{G}$-equivariant cover $\widetilde{\mathbb{O}}$ (which is not
$G$-equivariant). In the notation of Proposition 4.2.6, we have

$p^{t}=(2,1^{4}),\quad x=(2),\quad y=\emptyset,\quad z=(1^{4}).$

Hence

$f(x)=(1,1),\quad g(y)=\emptyset,\quad h(z)=(2,3/2,1/2).$

So

$\gamma_{0}(\widetilde{\mathbb{O}})=\rho^{+}(f(x)\cup g(y)\cup
h(z)))=(1/2,1/4,0)=\rho/4.$

######  Example 4.2.8.

Let $n=15$ (so that $\widetilde{G}=\mathrm{Spin}(15)$) and let $\mathbb{O}$ be
the nilpotent orbit corresponding to the partition $p=(9,5,1)$. By Proposition
4.2.1, $\mathbb{O}$ admits a birationally rigid $\widetilde{G}$-equivariant
cover $\widetilde{\mathbb{O}}$ (which is not $G$-equivariant). In the notation
of Proposition 4.2.6, we have

$p^{t}=(3,2^{4},1^{4}),\quad x=(3),\quad y=\emptyset,\quad z=(2^{4},1^{4}).$

Hence

$f(x)=(2,1),\quad g(y)=\emptyset,\quad h(z)=(3,5/2,2,3/2,3/2,1,1/2).$

So

$\gamma_{0}(\widetilde{\mathbb{O}})=\rho^{+}(3,5/2,2,2,3/2,3/2,1,1,1/2)=(1,3/4,1/2,1/2,1/4,1/4,0).$

###  4.3. Exceptional groups

In this section, we will produce a complete list of unipotent infinitesimal
characters attached to birationally rigid covers $\widetilde{\mathbb{O}}$ for
simple exceptional groups.

There are essentially two cases to consider:

  1. (1)

$\widetilde{\mathbb{O}}$ is (the trivial cover of) a birationally rigid orbit.

  2. (2)

$\widetilde{\mathbb{O}}$ is a nontrivial cover of a birationally induced
orbit.

Of course, it is also possible that $\widetilde{\mathbb{O}}$ is a _nontrivial_
cover of a birationally rigid orbit, but such covers bring nothing new into
the mix. Indeed, $I_{0}(\widetilde{\mathbb{O}})=I_{0}(\mathbb{O})$ by Theorem
4.0.2, so we can easily reduce to (1).

We pause to review our standing notational conventions for Levis and weights
in exceptional types. As explained in the proof of Proposition 3.9.5, simple
roots are denoted $\alpha_{1},...,\alpha_{n}$ and are numbered in accordance
with the Bourbaki conventions. We write $L(X;r_{1},...,r_{p})$ for the
standard Levi subgroup with simple roots $\alpha_{r_{1}},...,\alpha_{r_{p}}$
and Lie type $X$ (omitting $r_{1},...,r_{p}$ in some cases when $X$ determines
the Levi). All weights will be denoted in fundamental weight coordinates. For
example, $\rho$ is denoted, in every case, by the tuple $(1,1,...,1)$.

####  4.3.1. Birationally rigid orbits

In this section, we will produce a complete list of unipotent infinitesimal
characters attached to birationally rigid orbits. A list of such orbits is
provided in Proposition 3.8.3. For the rigid orbits, $\gamma_{0}(\mathbb{O})$
was computed in [LMM21]. This leaves only three (non-rigid) birationally rigid
orbits, namely $A_{2}+A_{1}$ and $A_{4}+A_{1}$ in $E_{7}$ and $A_{4}+A_{1}$ in
$E_{8}$. For these orbits we compute $\gamma_{0}(\mathbb{O})$ using
Proposition 4.1.2.

The calculations below involve a number of easy ‘micro-computations,’ which we
will catalogue here for the reader’s convenience (and to avoid repeating
references and explanations in the calculations below):

  1. (1)

Given a nilpotent orbit $\mathbb{O}$, determine the finite set
$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$ (see (3.9.1)). This is easily
deducible from the tables in [DE09, Sec 4].

  2. (2)

Determine the unipotent infinitesimal character $\gamma_{0}(\mathbb{O})$
attached to a birationally rigid orbit in a classical Lie algebra. This was
carried out in [LMM21, Sec 8], see [LMM21, Prop 8.2.3] for explicit formulas
in terms of partitions.

  3. (3)

Given a Levi subgroup $L\subset G$, express the fundamental weights for $L$ in
terms of fundamental weights for $G$. This is a somewhat tedious computation
involving the Cartan matrices for $L$ and $G$, which can be expedited using
atlas.

  4. (4)

Given an arbitrary weight $\lambda\in\mathfrak{h}^{*}$, compute the (unique)
dominant $W$-conjugate $\lambda^{+}\in\mathfrak{h}^{*}$. This is quite
difficult to do by hand, the atlas command ‘make_dominant’ is helpful.

We now begin the calculations:

  * •

$A_{2}+A_{1}\subset E_{7}$. By [DE09, Sec 4], $\mathbb{O}$ is induced from the
rigid orbit $\mathbb{O}_{L}=A_{1}$ of the Levi $L=L(E_{6})$. The infinitesimal
character $\gamma_{0}(\mathbb{O}_{L})$ was computed in [LMM21]. It is
$\rho(\mathfrak{l})-\varpi$, where $\varpi$ is the fundamental weight for
$\mathfrak{l}$ corresponding to the central node in $E_{6}$. In fundamental
weight coordinates for $\mathfrak{g}$

$\gamma_{0}(\mathbb{O}_{L})=(1,1,1,0,1,1,-6).$

Applying Proposition 4.1.2 and conjugating by $W$ we get

$\gamma_{0}(\mathbb{O})=(1,0,0,1,0,1,0).$

  * •