The set of completely prime primitive ideals in $U(\mathfrak{g})$ includes all
primitive ideals of $\mathcal{W}$-dimension 1, see Corollary 3.5.6.
Conversely, every completely prime primitive ideal with integral infinitesimal
character is of $\mathcal{W}$-dimension 1 (with the possible exception of
primitive ideals attached to a single orbit in type $E_{8}$), see [Los10a],
[Los15]. At non-integral infinitesimal character, there are completely prime
primitive ideals with $\operatorname{\mathcal{W}-\dim}(I)>1$. For example, if
$\mathfrak{g}=\mathfrak{sp}(2n)$ and $n\geq 2$, the maximal ideal
$I_{\mathrm{max}}(\rho/2)$ is completely prime, whereas

$\operatorname{\mathcal{W}-\dim}(I_{\mathrm{max}}(\rho/2))=2^{\lfloor\frac{n-1}{2}\rfloor},$

see [Pre10], [LP21, Proposition 4.1]. However, as noted in Example 6.2.2, the
Harish-Chandra bimodules $U(\mathfrak{g})/I_{\mathrm{max}}(\rho/2)$ are non-
unitary in all cases we can check. These examples suggest the following
conjecture.

######  Conjecture 6.3.1.

The left and right annihilators of unitary Harish-Chandra bimodules have
$\mathcal{W}$-dimension $1$.

In view of this conjecture, and the examples supporting it, we restrict our
attention to primitive ideals of $\mathcal{W}$-dimension 1. This set includes
all ideals of the form $I_{\lambda}(\widetilde{\mathbb{O}})$, see Proposition
6.1.2. In classical types, all primitive ideals of $\mathcal{W}$-dimension 1
are of this form, see [Top21, Thm 1.2]. In exceptional types, there are
exceptions (a few will be encountered below). Nonetheless, ideals of the form
$I_{\lambda}(\widetilde{\mathbb{O}})$ constitute a particularly nice class of
primitive ideals of $\mathcal{W}$-dimension $1$.

####  6.3.2. Orbit method

The considerations above suggest that all unipotent ideals are of the form
$I_{\lambda}(\widetilde{\mathbb{O}})$. The orbit method philosophy suggests
that one should further restrict to $\lambda=0$.

######  Definition 6.3.2 ([Vog90], Def 2.1).

A _Dixmier algebra_ for $G$ is a pair $(\mathcal{A},\varphi)$ consisting of an
associative algebra $\mathcal{A}$ and an algebra homomorphism
$\varphi:U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}$ such that
$\mathcal{A}$, regarded as a $U(\mathfrak{g})$-bimodule via $\varphi$, is
Harish-Chandra (cf. Definition 3.4.1). An isomorphism of Dixmier algebras
$(\mathcal{A}_{1},\varphi_{1})\xrightarrow{\sim}(\mathcal{A}_{2},\varphi_{2})$
is an isomorphism of algebras
$\phi:\mathcal{A}_{1}\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{2}$ which
intertwines $\varphi_{1}$ and $\varphi_{2}$.

One version of the orbit method, articulated in [Vog90], is the following. For
each $G$-equivariant cover $\widetilde{\mathbb{O}}^{1}$ of a coadjoint orbit
$\mathbb{O}^{1}$ (not necessarily nilpotent), there should be an associated
Dixmier algebra, denoted $\mathrm{Dix}(\widetilde{\mathbb{O}}^{1})$. As a
representation of $G$, $\mathrm{Dix}(\widetilde{\mathbb{O}}^{1})$ should
coincide with the ring of regular functions
$\mathbb{C}[\widetilde{\mathbb{O}}^{1}]$. Finally, the passage from
$\widetilde{\mathbb{O}}^{1}$ to $\mathrm{Dix}(\widetilde{\mathbb{O}}^{1})$
should define an injective correspondence

$\mathrm{Dix}:\\{G\text{-eqvt covers of coadjoint
orbits}\\}\hookrightarrow\\{\text{Dixmier algebras for }G\\}.$

This existence of this injection, with the property described above, is known
as the _Dixmier conjecture_ , see [Vog90, Conj 2.3]. In the terminology of
[Vog90], a _unipotent Dixmier algebra_ is one which corresponds to a
_nilpotent_ cover under this (not-yet-defined) correspondence. A _unipotent
ideal_ is the kernel in $U(\mathfrak{g})$ of the co-moment map for a unipotent
Dixmier algebra.

In Section 7.8, we will give a definition of $\mathrm{Dix}$, extending a
result of the first-named author, see [Los16, Thm 5.3]. Our map has the
following property: If $\widetilde{\mathbb{O}}$ is a nilpotent cover, then
$\mathrm{Dix}(\widetilde{\mathbb{O}})$ is the canonical quantization
$\mathcal{A}_{0}^{\widetilde{X}}$ (regarded as a Dixmier algebra via the co-
moment map
$\Phi^{\widetilde{X}}_{0}:U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{0}^{\widetilde{X}}$).
Thus, Definition 6.0.1 is consistent with the orbit method philosophy.

####  6.3.3. Maximality

It is generally believed that unipotent ideals should be maximal (see
Desideratum (3) from Section 1.3 and the references therein). In Appendix B,
we will show that for linear classical groups, the ideal
$I_{0}(\widetilde{\mathbb{O}})$ is always maximal. This result will be
extended to arbitrary groups in a forthcoming paper [MM].

Now we show that the maximality of $I_{0}(\widetilde{\mathbb{O}})$ is
equivalent to the simplicity of $\mathcal{A}_{0}^{\widetilde{X}}$. The latter
is a general expectation of canonical quantizations of conical symplectic
singularities, see Section 5.5.

######  Proposition 6.3.3.

The following conditions are equivalent:

  1. (1)

$\mathcal{A}_{0}^{\widetilde{X}}$ is simple.

  2. (2)

$I_{0}(\widetilde{\mathbb{O}})$ is a maximal ideal.

###### Proof.

To simplify notation, let $I:=I_{0}(\widetilde{\mathbb{O}})$ and
$\mathcal{A}:=\mathcal{A}_{0}^{\widetilde{X}}$. We first prove
(1)$\Rightarrow$(2). Suppose $I$ is not maximal. Let $\mathcal{W}$ be the
$W$-algebra associated to $\mathbb{O}$ and let $R$ be the reductive part of
the centralizer of $e\in\mathbb{O}$. Consider the functors
$\bullet_{\dagger}:\operatorname{HC}^{G}(U(\mathfrak{g}))\stackrel{{\scriptstyle}}{{\to}}\operatorname{HC}^{R}(\mathcal{W})$
and
$\bullet^{\dagger}:\operatorname{HC}^{R}_{\mathrm{fin}}(\mathcal{W})\stackrel{{\scriptstyle}}{{\to}}\operatorname{HC}^{G}_{\overline{\mathbb{O}}}(U(\mathfrak{g}))$
constructed in Section 3.5. Note that $\mathcal{A}_{\dagger}$ is a completely
reducible $R$-equivariant $\mathcal{W}$-bimodule (in fact, it is the direct
sum of $1$-dimensional $\mathcal{W}$-bimodules, Lemma 6.1.1). Since
$U(\mathfrak{g})/I\subset\mathcal{A}$ is an inclusion of Harish-Chandra
bimodules, and $\bullet_{\dagger}$ is exact (Proposition 3.5.1),
$(U(\mathfrak{g})/I)_{\dagger}$ is an $R$-stable subbimodule of
$\mathcal{A}_{\dagger}$.

Consider the algebra
$\mathcal{A}^{\prime}=[(U(\mathfrak{g})/I)_{\dagger}]^{\dagger}$. Recall that
$\mathcal{A}=(\mathcal{A}_{\dagger})^{\dagger}$, see Lemma 6.1.1(iii). Thus
$\mathcal{A}^{\prime}$ embeds into $\mathcal{A}$ as a subalgebra and as a
$U(\mathfrak{g})$-bimodule direct summand. Moreover, the injective map
$U(\mathfrak{g})/I\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}=(\mathcal{A}_{\dagger})^{\dagger}$
factors through the natural map
$U(\mathfrak{g})/I\stackrel{{\scriptstyle}}{{\to}}([U(\mathfrak{g})/I]_{\dagger})^{\dagger}=\mathcal{A}^{\prime}$.
It follows that the latter map is injective, and thus by 3.5.1(iv) the
bimodule $\mathcal{A}^{\prime}$ is an extension of $U(\mathfrak{g})/I$ by a
bimodule supported on $\partial\mathbb{O}$. Note that for every ideal
$\widetilde{I}\subset U(\mathfrak{g})$ properly containing $I$ we have
$\mathcal{V}(U(\mathfrak{g})/\widetilde{I})\subsetneq\mathcal{V}(U(\mathfrak{g})/I)$,
since $I$ is a prime ideal, see (iii) of Proposition 3.1.1. Since
$U(\mathfrak{g})/I$ has finite length, see Lemma 3.4.2, there is a minimal
ideal $\widetilde{I}\subset U(\mathfrak{g})$ properly containing $I$.

We claim that $\widetilde{I}/I$ is a 2-sided ideal in $\mathcal{A}^{\prime}$.
Let us show that it is a right ideal (the proof that it is a left ideal is
analogous). The associated variety
$\mathcal{V}(\mathcal{A}^{\prime}/[\widetilde{I}/I])$ of the
$U(\mathfrak{g})$-bimodule $\mathcal{A}^{\prime}/(\widetilde{I}/I)$ is a
proper subvariety of
$\mathcal{V}(\mathcal{A}^{\prime})=\mathcal{V}(U(\mathfrak{g})/I)$. Hence
$\mathcal{V}(\widetilde{I}/I)=\mathcal{V}(\mathcal{A}^{\prime})$. If
$\mathcal{B}:=(\widetilde{I}/I)\otimes_{U(\mathfrak{g})}(\mathcal{A}^{\prime}/[\widetilde{I}/I])$
is nonzero, then
$\mathcal{V}(\mathcal{B})\subsetneq\mathcal{V}(\mathcal{A}^{\prime})$. The
simple bimodule $\widetilde{I}/I$ embeds into the locally finite part of
$\operatorname{Hom}_{U(\mathfrak{g})}(\mathcal{A}^{\prime}/[\widetilde{I}/I],\mathcal{B})$,
and the associated variety of the latter is contained in the associated
variety of $\mathcal{B}$, a contradiction. It follows that
$\mathcal{B}=\\{0\\}$,
$[\widetilde{I}/I]\mathcal{A}^{\prime}\subseteq\widetilde{I}/I$. Thus
$\widetilde{I}/I$ is a nonzero two-sided ideal in $\mathcal{A}^{\prime}$.

Since $\mathcal{A}^{\prime}$ is a direct bimodule summand of $\mathcal{A}$, we
have that $\mathcal{A}/[\mathcal{A}(\widetilde{I}/I)]\neq\\{0\\}$. Let
${I}^{\prime}$ be the left annihilator of
$\mathcal{A}/\mathcal{A}(\widetilde{I}/I)$. Note that
$\mathcal{V}(\mathcal{A}/\mathcal{A}(\widetilde{I}/I))\subset\mathcal{V}(U(\mathfrak{g})/\widetilde{I})\subsetneq\mathcal{V}(\mathcal{A})$.
So $I^{\prime}$ is a proper two-sided ideal in $\mathcal{A}$. Since
$\mathcal{A}/I^{\prime}$ is a HC bimodule over $U(\mathfrak{g})/\widetilde{I}$
we see that
$\mathcal{V}(\mathcal{A}/I^{\prime})\subset\mathcal{V}(U(\mathfrak{g})/\widetilde{I})$.
On the other hand,
$\mathcal{V}(U(\mathfrak{g})/\widetilde{I})\subsetneq\mathcal{V}(\mathcal{A})$.
It follows that $I^{\prime}\neq\\{0\\}$. Since $\mathcal{A}$ is simple, we get
a contradiction. This proves (1)$\Rightarrow$(2).

Next we prove (2)$\Rightarrow$(1). Suppose $\mathcal{A}$ is not simple. Then
there is a nonzero prime ideal $\widetilde{I}\subset\mathcal{A}$. Since
$\mathcal{A}$ is a Noetherian domain, the GK dimension of the quotient
$\mathcal{A}/\widetilde{I}$ is less than that of $\mathcal{A}$, and hence the
GK dimension of the image of $U(\mathfrak{g})/I$ in
$\mathcal{A}/\widetilde{I}$ is less than that of $U(\mathfrak{g})/I$. Since
$I$ is maximal, $U(\mathfrak{g})/I$ has no such algebra quotient, a
contradiction. This completes the proof. ∎

####  6.3.4. Applicability to real reductive groups

In this subsection, we will discuss the applicability of Definition 6.0.2 to
the more general setting of real reductive Lie groups. This issue is quite
subtle and deserving of careful consideration. It will be further explored in
[MM].

Let $G_{\mathbb{R}}$ be a real reductive Lie group. For this informal
discussion, the precise meaning of ‘real reductive Lie group’ will not matter
enormously. It should be broad enough to include non-linear covering groups,
such as $\mathrm{Mp}(2n,\mathbb{R})$, but narrow enough so that the theory of
Harish-Chandra modules is applicable. In particular, $G_{\mathbb{R}}$ should
have finitely many connected components and the identity component should be a
finite cover of an algebraic group. Let $\mathfrak{g}$ denote the
complexification of the Lie algebra $\mathfrak{g}_{\mathbb{R}}$ of
$G_{\mathbb{R}}$ and $K$ the complexification of a maximal compact subgroup
$K_{\mathbb{R}}\subset G_{\mathbb{R}}$. A $(\mathfrak{g},K)$-_module_ is a
finitely-generated $U(\mathfrak{g})$-module with with a compatible rational
$K$-action. A $(\mathfrak{g},K)$-module is _admissible_ if each irreducible
$K$-representation appears with finite multiplicity. Under suitable conditions
on $G_{\mathbb{R}}$, there is a bijection between the set of irreducible
admissible $G_{\mathbb{R}}$-representations and the set of irreducible
admissible $(\mathfrak{g},K)$-modules, see [Vog81, Chp 0.3] for an overview.
We note that if $G_{\mathbb{R}}$ is complex, then a $(\mathfrak{g},K)$-module
is the same thing as a $G_{\mathbb{R}}$-equivariant Harish-Chandra bimodule
for the (complex) Lie algebra $\mathfrak{g}_{\mathbb{R}}$. As explained in
Section 1, the set $\mathrm{Irr}_{u}(G_{\mathbb{R}})$ of irreducible unitary
$G_{\mathbb{R}}$-representations should contain a finite set of ‘building
blocks’ $\mathrm{Unip}(G_{\mathbb{R}})\subset\mathrm{Irr}_{u}(G_{\mathbb{R}})$
called ‘unipotent representations’. If $G_{\mathbb{R}}$ is complex, then
$\mathrm{Unip}(G_{\mathbb{R}})$ should be the set defined in Definition 6.0.2.
Our problem is to formulate a natural generalization of Definition 6.0.2 for
arbitrary $G_{\mathbb{R}}$.

If $G_{\mathbb{R}}$ is algebraic, there is a notion of a _special unipotent
representation_ , due to Adams, Barbasch, and Vogan. The definition in [ABV92,
Sec 27] is stated in the language of $L$-groups and Arthur parameters, but is
equivalent to the following (see [ABV92, Cor 27.13]).

######  Definition 6.3.4 ([ABV92]).

Suppose $G_{\mathbb{R}}$ is an algebraic group and let $\mathbb{O}^{\vee}$ be
a nilpotent orbit for the Langlands dual Lie algebra $\mathfrak{g}^{\vee}$. A
_special unipotent representation_ attached to $\mathbb{O}^{\vee}$ is an
irreducible $(\mathfrak{g},K)$-module $\mathcal{B}$ such that
$\operatorname{Ann}(\mathcal{B})=I_{\mathrm{max}}(h^{\vee}/2)$.

The following conjecture is essentially due to Arthur.

######  Conjecture 6.3.5 ([Art89]).

Suppose $G_{\mathbb{R}}$ is an algebraic group. Then all special unipotent
representations of $G_{\mathbb{R}}$ are unitary.

We summarize the cases in which Conjecture 6.3.5 is known

  * •

$G_{\mathbb{R}}$ is a split real form of a linear classical group ([Art13]).

  * •

$G_{\mathbb{R}}$ is a complex classical group, regarded as a real group by
restriction of scalars ([Bar89]).

  * •

$G_{\mathbb{R}}$ is a real form of a simple exceptional group ([AMLV]).

Notably, if we allow $G_{\mathbb{R}}$ to be a non-linear group, Conjecture
6.3.5 is false.

######  Example 6.3.6.

Let $G_{\mathbb{R}}=\mathrm{Mp}(4,\mathbb{R})$. Hence,
$\mathfrak{g}=\mathfrak{sp}(4)$ and $K$ is the ‘square-root of determinant’
cover of $GL(2)$

$K=\left\\{(g,z)\in GL(2)\times\mathbb{C}^{\times}\mid\det(g)=z^{2}\right\\}$

Let $\mathbb{O}^{\vee}$ be the subregular nilpotent orbit for
$\mathfrak{g}^{\vee}=\mathfrak{so}(5)$. Then in standard coordinates on
$\mathfrak{h}^{*}$, we have $\frac{1}{2}h^{\vee}=(1,0)$. There are two
degenerate principal series representations of $G_{\mathbb{R}}$ which are
annihilated by the ideal $I_{\mathrm{max}}(\frac{1}{2}h^{\vee})$. They are
irreducible by [KR90, Thm 1]. In the same (standard) coordinates, the
$K$-types are as follows:

$2\mathbb{Z}\times 2\mathbb{Z}+(\frac{1}{2},\frac{1}{2})\qquad\text{or}\qquad
2\mathbb{Z}\times 2\mathbb{Z}-(\frac{1}{2},\frac{1}{2}).$

By an easy application of the Dirac inequality, we see that neither
representation is unitary. We are grateful to Jing-Song Huang for suggesting
this approach.

We will show in Section 9 that the set of unipotent ideals includes (as a
proper subset) all special unipotent ideals. This suggests the following
generalization of Definition 6.3.4.

######  ‘Definition’ 6.3.7.

Suppose $G_{\mathbb{R}}$ is an algebraic group and let
$\widetilde{\mathbb{O}}$ be a finite cover of a nilpotent co-adjoint
$\operatorname{Ad}(\mathfrak{g})$-orbit. A _unipotent representation_ of
$G_{\mathbb{R}}$ attached to $\widetilde{\mathbb{O}}$ is an irreducible
$(\mathfrak{g},K)$-module $\mathcal{B}$ such that
$\mathrm{Ann}(\mathcal{B})=I_{0}(\widetilde{\mathbb{O}})$.

We use quotation marks to indicate that this is _not_ a reasonable
definition—if we take the ideal $I_{0}(\widetilde{\mathbb{O}})$ to be non-
special, there can be non-unitary $(\mathfrak{g},K)$-modules which are
annihilated by $I_{0}(\widetilde{\mathbb{O}})$ even if $G_{\mathbb{R}}$ is
algebraic. This problem is already apparent for $\mathrm{SL}(2,\mathbb{R})$.

######  Example 6.3.8.

Let $G_{\mathbb{R}}=\mathrm{SL}(2,\mathbb{R})$. Hence,
$\mathfrak{g}=\mathfrak{sl}(2)$ and $K=SO(2)$. The unipotent ideal attached to
the (universal) 2-fold cover of the principal nilpotent
$\operatorname{Ad}(\mathfrak{g})$-orbit is the maximal ideal in
$U(\mathfrak{g})$ of infinitesimal character $\frac{\rho}{2}$, see (v) of
Example 6.2.1. There are two irreducible $(\mathfrak{g},K)$-modules
annihilated by this ideal: one spherical and one non-spherical. The spherical
representation is unitary (it is the midpoint of a unitary complementary
series). The non-spherical representation, however, is _not_ unitary. In
particular, it should not be regarded as a unipotent representation of
$G_{\mathbb{R}}$.

We believe that ‘Definition’ 6.3.7 can be fixed in every case by replacing
$G_{\mathbb{R}}$ with an appropriate non-linear covering group. This covering
group should depend in a predictable way on the infinitesimal character of
$I_{0}(\widetilde{\mathbb{O}})$. In Example 6.3.8, the infinitesimal character
of $I_{0}(\widetilde{\mathbb{O}})$ is half-integral. This suggests that we
should look for genuine representations of the (non-linear) 2-fold covering
group $\mathrm{Mp}(2,\mathbb{R})$. There are two genuine irreducible
representations of $\mathrm{Mp}(2,\mathbb{R})$ annihilated by the ideal
$I_{0}(\widetilde{\mathbb{O}})$, namely the irreducible constituents of the
oscillator representation. Both are unitary, and (we believe) worthy of the
label ‘unipotent’.

If we assume that $\mathbb{O}$ satisfies the codimension condition

(6.3.1) $\operatorname{codim}(\partial\mathbb{O},\overline{\mathbb{O}})\geq
4,$

these subtleties seem to vanish. Many examples suggest that the following is
true.

######  Conjecture 6.3.9.

Suppose $\mathbb{O}$ is a nilpotent $\operatorname{Ad}(\mathfrak{g})$-orbit
satisfying condition (6.3.1). Then every irreducible $(\mathfrak{g},K)$-module
$\mathcal{B}$ with $\operatorname{Ann}(\mathcal{B})=I_{0}(\mathbb{O})$ is
unitary.

######  Example 6.3.10.

Let $G_{\mathbb{R}}$ be the split real form of the (unique) simple group of
type $E_{8}$ and consider the nilpotent orbits

$\mathbb{O}_{1}=2A_{2}+2A_{1},\qquad\mathbb{O}_{2}=A_{4}+A_{3}.$

Both orbits satisfy condition (6.3.1), see the incidence tables in [Car93]. We
will explain in Section 8 how the ideals $I_{0}(\mathbb{O})$ can be
computed—in these cases we have

$I_{0}(\mathbb{O}_{1})=I_{\mathrm{max}}(\rho/3),\qquad
I_{0}(\mathbb{O}_{2})=I_{\mathrm{max}}(\rho/5).$

An atlas computation shows that in each case, there is a unique irreducible
$(\mathfrak{g},K)$-module annihilated by $I_{0}(\mathbb{O})$, and it is
unitary.

######  Example 6.3.11.

Let $G_{\mathbb{R}}$ be the non-linear group $\mathrm{Mp}(2n,\mathbb{R})$ (for
$n\geq 2$) and let $\mathbb{O}$ be the minimal orbit. Since
$\dim(\mathbb{O})\geq 4$, $\mathbb{O}$ satisfies condition (6.3.1). As noted
in Example 6.2.1(iv), $I_{0}(\mathbb{O})$ is the Joseph ideal. There are two
irreducible $(\mathfrak{g},K)$-modules annihilated by this ideal—they are the
irreducible constituents of the oscillator representation. It is a classical
fact that both are unitary.

Conjecture 6.3.9 suggests that if $\mathbb{O}$ satisfies (6.3.1), ‘Definition’
6.3.7 is correct. We propose the following.

######  Definition 6.3.12.

Suppose $\mathbb{O}$ be a nilpotent $\operatorname{Ad}(\mathfrak{g})$-orbit
satisfying condition (6.3.1). A _unipotent representation_ of $G_{\mathbb{R}}$
attached to $\mathbb{O}$ is an irreducible $(\mathfrak{g},K)$-module
$\mathcal{B}$ such that $\mathrm{Ann}(\mathcal{B})=I_{0}(\mathbb{O})$. Write
$\mathrm{Unip}_{\mathbb{O}}(G_{\mathbb{R}})$ for the set of (equivalences
classes of) such representations.

######  Remark 6.3.13.

Note that all birationally rigid orbits (in particular, all rigid orbits)
satisfy condition (6.3.1). This is an easy consequence of Corollary 7.6.1
below.

As further vindication of Definition 6.3.12, we offer the following remark. It
will be shown in [LY] that the set
$\mathrm{Unip}_{\mathbb{O}}(G_{\mathbb{R}})$ admits a geometric
parameterization (analogous to our parameterization of unipotent bimodules,
see Theorem 6.6.2 below). More precisely, they will prove that
$\mathrm{Unip}_{\mathbb{O}}(G_{\mathbb{R}})$ is in bijection with pairs
$(\mathbb{O}_{K},\mathcal{L})$, where $\mathbb{O}_{K}$ is a $K$-orbit in
$\mathbb{O}\cap(\mathfrak{g}/\mathfrak{k})^{*}$ and $\mathcal{L}$ is an
irreducible twisted local system on $\mathbb{O}_{K}$ with half-canonical
twist.

###  6.4. Equivariant bimodules for Hamiltonian quantizations of nilpotent
covers

In this section we will establish some preliminary facts about Harish-Chandra
bimodules for quantizations of nilpotent covers, specializing (and modifying)
the general results and constructions of Sections 4.13, 4.14.

Let $\widetilde{\mathbb{O}}$ be a $G$-equivariant nilpotent cover and let
$(\mathcal{A},\Phi):=(\mathcal{A}^{\widetilde{X}}_{\lambda},\Phi^{\widetilde{X}}_{\lambda})$
be a Hamiltonian quantization of $\mathbb{C}[\widetilde{\mathbb{O}}]$. On any
Harish-Chandra bimodule $\mathcal{B}\in\operatorname{HC}(\mathcal{A})$, there
is an adjoint action of $\mathfrak{g}$, defined by

$\operatorname{ad}:\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}\operatorname{End}(\mathcal{B}),\qquad\operatorname{ad}(\xi)(b)=\Phi(\xi)b-b\Phi(\xi),\qquad\xi\in\mathfrak{g},b\in\mathcal{B}.$

######  Definition 6.4.1.

A Harish-Chandra bimodule $\mathcal{B}\in\operatorname{HC}(\mathcal{A})$ is
$G$-_equivariant_ if $\operatorname{ad}$ integrates to a rational action
$\operatorname{Ad}$ of $G$. Denote the full subcategory of $G$-equivariant
Harish-Chandra bimodules by
$\operatorname{HC}^{G}(\mathcal{A})\subset\operatorname{HC}(\mathcal{A})$ and
write $\overline{\operatorname{HC}}^{G}(\mathcal{A})$ for the quotient
category
$\operatorname{HC}^{G}(\mathcal{A})/\operatorname{HC}^{G}_{\partial}(\mathcal{A})$.

Let $\Gamma=\pi_{1}(\widetilde{\mathbb{O}})$. By (4.13.2), there is an
equivalence of categories

$\bullet_{\dagger}:\overline{\operatorname{HC}}(\mathcal{A})\xrightarrow{\sim}\Gamma/\Gamma(\lambda)\operatorname{-mod}$

In this section, we will provide an analogous description of the category
$\overline{\operatorname{HC}}^{G}(\mathcal{A})$. Let
$\Omega=\pi_{1}^{G}(\widetilde{\mathbb{O}})$ and let $\Omega(\lambda)$ be the
image of $\Gamma(\lambda)$ under the quotient map
$\Gamma\twoheadrightarrow\Omega$.

######  Proposition 6.4.2.

The functor
$\bullet_{\dagger}:\overline{\operatorname{HC}}(\mathcal{A})\xrightarrow{\sim}\Gamma/\Gamma(\lambda)\operatorname{-mod}$
restricts to an equivalence

$\bullet_{\dagger}:\overline{\operatorname{HC}}^{G}(\mathcal{A})\xrightarrow{\sim}\Omega/\Omega(\lambda)\operatorname{-mod}$

###### Proof.

First, note that the adjoint action of $\mathfrak{z}(\mathfrak{g})$, and hence
of $Z(G)^{\circ}$, on any bimodule
$\mathcal{B}\in\operatorname{HC}^{G}(\mathcal{A})$ is trivial. Replacing $G$
with $G/Z(G)^{\circ}$ if necessary, we can assume that $G$ is semisimple. Let
$G^{\mathrm{sc}}$ denote the universal cover of $G$, and $K$ the kernel of the
quotient map $G^{\mathrm{sc}}\stackrel{{\scriptstyle}}{{\to}}G$. An object in
$\overline{\operatorname{HC}}^{G}(\mathcal{A}_{\lambda})$ is a bimodule
$\mathcal{B}\in\overline{\operatorname{HC}}(\mathcal{A}_{\lambda})$ on which
the action of $K$ is trivial. Since $\widetilde{\mathbb{O}}$ is a
$G$-equivariant cover, we see that $K\subset G^{\mathrm{sc}}_{x}$ for all
$x\in\widetilde{\mathbb{O}}$. So $K$ maps to $\Gamma$ with cokernel equal to
$\Omega$.

We claim that the $K$-action on
$\mathcal{B}\in\operatorname{HC}(\mathcal{A}_{\lambda})$ is trivial if and
only if the $K$-action on $\mathcal{B}_{\dagger}$ is trivial. Clearly, $K$
acts trivially on $\mathcal{B}$ if and only if it acts trivially on
$\operatorname{gr}\mathcal{B}$. By (4.13.1), we have
$(\operatorname{gr}\mathcal{B})|_{X^{\mathrm{reg}}}\simeq(\mathcal{B}_{\dagger}\otimes\mathcal{O}_{\widehat{X}^{\mathrm{reg}}})^{\Gamma}$.
Since $\Gamma$ acts freely on $\mathcal{O}_{\widehat{X}^{\mathrm{reg}}}$, $K$
acts trivially on $(\operatorname{gr}\mathcal{B})|_{X^{\mathrm{reg}}}$ if and
only if it acts trivially on $\mathcal{B}_{\dagger}$. This completes the
proof. ∎

The next lemma will be useful for computing $\mathcal{B}_{\dagger}$ when
$\mathcal{B}\in\operatorname{HC}^{G}(\mathcal{A})$. Since the image of
$\mathfrak{g}$ in $\mathcal{A}$ lives in degree $\leqslant d$, every good
filtration on $\mathcal{B}$ is stable under $\operatorname{ad}(\mathfrak{g})$,
and hence $\operatorname{Ad}(G)$. In particular,
$\operatorname{gr}(\mathcal{B})$ is a $G$-equivariant
$\mathbb{C}[\widetilde{X}]$-module and the $G$-action is uniquely recovered
from the Poisson structure.

######  Lemma 6.4.3.

For any $\mathcal{B}\in\operatorname{HC}^{G}(\mathcal{A})$ (and any good
filtration), there is an isomorphism of $G$-equivariant coherent sheaves on
$\widetilde{\mathbb{O}}$

(6.4.1)
$(\operatorname{gr}\mathcal{B})|_{\widetilde{\mathbb{O}}}\simeq(\mathcal{B}_{\dagger}\otimes\mathcal{O}_{\widehat{\mathbb{O}}})^{\Omega}.$

This isomorphism uniquely determines the $\Omega$-module
$\mathcal{B}_{\dagger}$.

###### Proof.

We can assume in the proof that $G$ is semisimple and simply connected and
hence that
$\operatorname{HC}^{G}(\mathcal{A})=\operatorname{HC}(\mathcal{A})$. Recall
that for a smooth symplectic variety, a Poisson coherent sheaf is the same
thing as a local system, see Step 3 in the proof of [Los17, Lemma 3.9].

By (4.13.1) (and using the notation therein) there is a local system
isomorphism

(6.4.2)
$(\operatorname{gr}\mathcal{B})|_{\widetilde{X}^{\mathrm{reg}}}\simeq(\mathcal{B}_{\dagger}\otimes\mathcal{O}_{\widehat{X}^{\prime}})^{\Gamma}.$

On both sides of (6.4.2), the $\mathfrak{g}$-action is recovered from the
local system structure. It follows that the isomorphism (6.4.2) is
$\mathfrak{g}$\- and hence $G$-equivariant.

Since
$\operatorname{codim}(\widetilde{X}^{\mathrm{reg}}-\widetilde{\mathbb{O}},\widetilde{X}^{\mathrm{reg}})\geqslant
2$, (6.4.2) is equivalent to a $G$-equivariant local system isomorphism

(6.4.3)
$(\operatorname{gr}\mathcal{B})|_{\widetilde{\mathbb{O}}}\simeq(\mathcal{B}_{\dagger}\otimes\mathcal{O}_{\widehat{\mathbb{O}}})^{\Gamma}.$

This implies (6.4.1). It remains to show that (6.4.1) determines
$\mathcal{B}_{\dagger}$ uniquely. For this, it suffices to show that the
forgetful functor from the category of $G$-equivariant local systems on
$\widetilde{\mathbb{O}}$ to the category of $G$-equivariant coherent sheaves
on $\widetilde{\mathbb{O}}$ is a full embedding. Choose
$x\in\widetilde{\mathbb{O}}$. Via restriction to $x$, these categories are
equivalent to $G_{x}/G_{x}^{\circ}\operatorname{-mod}$ and
$G_{x}\operatorname{-mod}$, respectively. The forgetful functor is the
pullback under $G_{x}\twoheadrightarrow G_{x}/G_{x}^{\circ}$, which is indeed
a full embedding. This completes the proof. ∎

We will also need a $G$-equivariant version of Corollary 4.14.4. Let
$\widecheck{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$
be a $G$-equivariant Galois cover and let
$\Pi:=\operatorname{Aut}_{\widetilde{\mathbb{O}}}(\widecheck{\mathbb{O}})$.
Note that the $\Pi$-action on $\widecheck{\mathbb{O}}$ is by $G$-equivariant
Hamiltonian automorphisms. Suppose $\mathcal{A}^{\widecheck{X}}$ is a
$\Pi$-equivariant Hamiltonian quantization of
$\mathbb{C}[\widecheck{\mathbb{O}}]$ such that
$\mathcal{A}\simeq(\mathcal{A}^{\widecheck{X}})^{\Pi}$ as Hamiltonian
quantizations. In view of Proposition 6.4.2, the following result is
completely analogous to Corollary 4.14.4, and is proved in a similar fashion.

######  Corollary 6.4.4.

Suppose
$\overline{\operatorname{HC}}^{G}(\mathcal{A}^{\widecheck{X}})\simeq\operatorname{Vect}$.
Then $\bullet_{\dagger}$ induces an equivalence of monoidal categories

$\bullet_{\dagger}:\overline{\operatorname{HC}}^{G}(\mathcal{A})\xrightarrow{\sim}\Pi\operatorname{-mod}.$

The inverse equivalence, $\bullet^{\dagger}$, is the functor of _isotypic
components_

$V^{\dagger}\simeq(\mathcal{A}^{\widecheck{X}}\otimes V)^{\Pi},\qquad
V\in\Pi\operatorname{-mod}.$

Our next task is to relate the restriction and extension functors of Section
3.5 to those of Section 4.13. Let
$\mathcal{A}:=\mathcal{A}^{\widetilde{X}}_{\lambda},I:=I_{\lambda}(\widetilde{\mathbb{O}})\subset
U(\mathfrak{g})$ and $\underline{J}:=I_{\dagger}\subset\mathcal{W}$. Consider
the full subcategory of $\operatorname{HC}^{G}(U(\mathfrak{g}))$

(6.4.4)
$\operatorname{HC}^{G}(U(\mathfrak{g})/I):=\\{\mathcal{B}\in\operatorname{HC}^{G}(U(\mathfrak{g}))\mid
I\subseteq\mathrm{LAnn}(\mathcal{B}),\
I\subseteq\mathrm{RAnn}(\mathcal{B})\\}$

and the Serre subcategory

(6.4.5)
$\operatorname{HC}^{G}_{\partial}(U(\mathfrak{g})/I):=\\{\mathcal{B}\in\operatorname{HC}^{G}(U(\mathfrak{g}))\mid
I\subsetneq\mathrm{LAnn}(\mathcal{B}),\
I\subsetneq\mathrm{RAnn}(\mathcal{B})\\}\subset\operatorname{HC}^{G}(U(\mathfrak{g})/I).$

Note that
$\operatorname{HC}_{\partial}^{G}(U(\mathfrak{g})/I)=\operatorname{HC}^{G}_{\partial\mathbb{O}}(U(\mathfrak{g}))\cap\operatorname{HC}^{G}(U(\mathfrak{g})/I)$,
this follows from (iii) of Proposition 3.1.1. Form the quotient category

(6.4.6)
$\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I):=\operatorname{HC}^{G}(U(\mathfrak{g})/I)/\operatorname{HC}_{\partial}(U(\mathfrak{g})/I).$

Every bimodule $\mathcal{B}\in\operatorname{HC}^{G}(\mathcal{A})$ can be
regarded as a $G$-equivariant Harish-Chandra $U(\mathfrak{g})$-bimodule via
the co-moment map
$\Phi:U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}$. The
resulting bimodule is contained in the subcategory
$\operatorname{HC}^{G}(U(\mathfrak{g})/I)\subset\operatorname{HC}^{G}(U(\mathfrak{g}))$.
This defines a forgetful functor

$\Phi^{*}:\operatorname{HC}^{G}(\mathcal{A})\stackrel{{\scriptstyle}}{{\to}}\operatorname{HC}^{G}(U(\mathfrak{g})/I).$

This functor takes $\operatorname{HC}^{G}_{\partial}(\mathcal{A})$ to
$\operatorname{HC}^{G}_{\partial\mathbb{O}}(U(\mathfrak{g}))$ and thus
descends to a functor (still denoted by $\Phi^{*}$)

$\Phi^{*}:\overline{\operatorname{HC}}^{G}(\mathcal{A})\stackrel{{\scriptstyle}}{{\to}}\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I).$

On the other hand, there is a restriction functor

$\bullet_{\dagger}:\operatorname{HC}^{G}(U(\mathfrak{g}))\stackrel{{\scriptstyle}}{{\to}}\operatorname{HC}^{R}(\mathcal{W}),$

see Section 3.5. By 3.5.1, this functor gives rise to a full embedding
$\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)\hookrightarrow\operatorname{HC}^{R}(\mathcal{W}/\underline{J})$.
To avoid notational confusion, we will denote this embedding by
$\bullet_{\dagger^{\prime}}$ for the remainder of this subsection. The next
proposition shows that
$\bullet_{\dagger}:\overline{\operatorname{HC}}^{G}(\mathcal{A})\stackrel{{\scriptstyle}}{{\to}}\Omega\operatorname{-mod}$
and
$\bullet_{\dagger^{\prime}}:\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)\stackrel{{\scriptstyle}}{{\to}}\operatorname{HC}^{R}(\mathcal{W}/\underline{J})$
are compatible in a certain precise sense.

Choose $x\in\widetilde{\mathbb{O}}$ as in Lemma 6.1.1. Note that
$\Omega=R_{x}/R^{\circ}$. By Lemma 6.1.1(ii), we can choose an $\Omega$-stable
ideal $J\subset\mathcal{W}$ of codimension 1 such that $\underline{J}$ is the
intersection of the $R$-conjugates of $J$, and furthermore

$\mathcal{A}_{\dagger}\simeq\bigoplus_{r\in R/R_{x}}\mathcal{W}/rJ.$

The filtered algebra $\mathcal{A}_{\dagger}$ comes equipped with a natural
$R$-action as well as an $R$-equivariant algebra homomorphism
$\mathcal{W}/\underline{J}\hookrightarrow\mathcal{A}_{\dagger}$.

A representation $V$ of $\Omega$ is the same thing as an
$R_{x}/R^{\circ}$-equivariant module for $\mathcal{W}/J$. Consequently, there
is a uniquely defined $R$-equivariant $\mathcal{A}_{\dagger}$-module
$\mathsf{B}_{V}$ with fiber $V$ over the summand $\mathcal{W}/J$. We can view
$\mathsf{B}_{V}$ as an object in
$\operatorname{HC}^{R}(\mathcal{W}/\underline{J})$.

######  Lemma 6.4.5.

The following diagram of functors commutes

${\overline{\operatorname{HC}}^{G}(\mathcal{A})}$${\Omega\operatorname{-mod}}$${\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)}$${\operatorname{HC}^{R}(\mathcal{W}/\underline{J})}$$\scriptstyle{\Phi^{*}}$$\scriptstyle{\bullet_{\dagger}}$$\scriptstyle{\mathsf{B}_{\bullet}}$$\scriptstyle{\bullet_{\dagger^{\prime}}}$

###### Proof.

We can assume in the proof that $G$ is semisimple and simply connected: the
general case is obtained from this one by passing to full subcategories. In
particular, the $G$-equivariance condition for a Harish-Chandra
$\mathcal{A}$-bimodule is vacuous, and $\Omega=\Gamma$. Also all categories
involved are semisimple, so it is enough to establish an isomorphism on the
level of objects.

Recall, (4.13.2), that $\bullet_{\dagger}$ defines an equivalence
$\overline{\operatorname{HC}}(\mathcal{A})\xrightarrow{\sim}\Gamma/\Gamma(\lambda)\operatorname{-mod}$.
Define $\breve{X}$ and $\breve{\mathcal{A}}$ (for $\widetilde{X}$ and
$\mathcal{A}$) as in the discussion preceding Lemma 4.14.2. The functor
$V\mapsto(\breve{\mathcal{A}}\otimes V)^{\Gamma}$ is isomorphic to
$\bullet^{\dagger}$ by Lemma 4.14.2(i) and so is a quasi-inverse equivalence
for $\bullet_{\dagger}$, see (4.13.2). Thus, it suffices to exhibit a
functorial isomorphism (we omit $\Phi^{*}$ from the notation)

(6.4.7) $[(\breve{\mathcal{A}}\otimes
V)^{\Gamma}]_{\dagger^{\prime}}\xrightarrow{\sim}\mathsf{B}_{V}.$

Since the $\Gamma$-action on $\breve{\mathcal{A}}$ fixes the image of the
quantum comoment map, we have a $\Gamma$-equivariant isomorphism
$(\breve{\mathcal{A}}\otimes
V)_{\dagger^{\prime}}\xrightarrow{\sim}\breve{\mathcal{A}}_{\dagger^{\prime}}\otimes
V$ of objects in $\operatorname{HC}^{R}(\mathcal{W}/\underline{J})$. This
gives rise to an isomorphism

(6.4.8) $[(\breve{\mathcal{A}}\otimes
V)^{\Gamma}]_{\dagger^{\prime}}\xrightarrow{\sim}(\breve{\mathcal{A}}_{\underline{\dagger}^{\prime}}\otimes
V)^{\Gamma}.$

While $\breve{\mathcal{A}}$ is not, in general, a quantization of
$\mathbb{C}[\breve{X}]$, it quantizes an algebra whose normalization is
$\mathbb{C}[\breve{X}]$, see (ii) of Lemma 4.14.2. So (i) and (ii) of Lemma
6.1.1 hold for $\breve{\mathcal{A}}$ nonetheless. Thus

$\breve{\mathcal{A}}_{\dagger^{\prime}}\xrightarrow{\sim}\bigoplus_{r\in
R/\breve{R}}\mathcal{W}/rJ,$

where we write $\breve{R}$ for the preimage in $R$ of $\Gamma(\lambda)\subset
R/R^{\circ}$. From this description it is easy to see that the right hand side
of (6.4.8) coincides with $\mathsf{B}_{V}$.

∎

###  6.5. Classification of unipotent ideals

Let $\widetilde{\mathbb{O}}$ and $\widecheck{\mathbb{O}}$ be $G$-equivariant
covers of a common nilpotent orbit $\mathbb{O}$. As usual, let
$\widetilde{X}=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ and
$\widecheck{X}=\operatorname{Spec}(\mathbb{C}[\widecheck{\mathbb{O}}])$. If
$f:\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widecheck{\mathbb{O}}$
is a morphism of $G$-equivariant covers, there is an induced morphism of
spectra $\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}\widecheck{X}$. The
latter morphism is a finite $G$-equivariant cover, which is étale over the
subset $\widecheck{X}^{\mathrm{reg}}\subset\widecheck{X}$ by Lemma 4.6.1.
Recall, Definition 1.5.2, that this morphism is _almost étale_ if it is étale
over the locus

$\widecheck{X}^{\mathrm{reg}}\cup\bigcup_{k}\mathfrak{L}_{k}\subset\widecheck{X}.$

Define a binary relation $\succeq$ on the set of $G$-equivariant nilpotent
covers as follows

$\widetilde{\mathbb{O}}\succeq\widecheck{\mathbb{O}}\iff\exists\
f:\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widecheck{\mathbb{O}}\text{
such that }\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}\widecheck{X}\text{ is
an almost \'{e}tale cover}$

######  Definition 6.5.1.

Let $\sim$ be the equivalence relation on the isomorphism classes of
$G$-equivariant nilpotent covers defined by taking the symmetric closure of
$\succeq$. Denote the equivalence class of $\widetilde{\mathbb{O}}$ by
$[\widetilde{\mathbb{O}}]$.

Below, we will show that unipotent ideals are parameterized by equivalence
classes of covers. Our main result requires a bit of preparation.

######  Lemma 6.5.2.

Suppose $G$ is simple. Then $\widetilde{X}$ satisfies condition (5.0.1) except
in the following case: $G$ is the (unique) simple group of type $E_{8}$ and
$\widetilde{\mathbb{O}}$ is the principal nilpotent orbit.

###### Proof.

If
$\widehat{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$
is a $G$-equivariant cover, then the subgroups
$\Gamma_{k}^{\prime}\subset\mathrm{Sp}(2)$ corresponding to the codimension 2
leaves in $\widehat{X}$ are contained in the subgroups
$\Gamma_{k}\subset\mathrm{Sp}(2)$ for $\widetilde{X}$, see (4.6.3). Thus, we
can reduce to the case when $\widetilde{\mathbb{O}}$ is an orbit. The
singularities corresponding to the codimension 2 leaves in the closures of
nilpotent orbits were described in the classical cases by Kraft and Procesi
([KP81, Main Thm],[KP82, Thm 2]) and in the exceptional cases by Fu et.al.
([FJLS15, Sec 13]). The lemma follows by inspection of the references above. ∎

We note that it possible to give a more conceptual proof of Lemma 6.5.2, at
least in exceptional types. The proof requires some machinery which is
developed in Section 7.2. We refer the reader to Remark 7.2.6.

######  Lemma 6.5.3.

Let $[\widetilde{\mathbb{O}}]$ be an equivalence class of $G$-equivariant
nilpotent covers. Then

  * (i)

$[\widetilde{\mathbb{O}}]$ contains a unique maximal element
$\widehat{\mathbb{O}}$.

  * (ii)

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

  * (iii)

There is an equivalence of categories
$\overline{\operatorname{HC}}^{G}(\mathcal{A}_{0}^{\widehat{X}})\simeq\operatorname{Vect}$.

###### Proof.

For (i), it suffices to show that $\widetilde{X}$ admits a unique maximal
almost étale $G$-equivariant cover
$\widehat{X}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$. For $G$ simply
connected, this is a special case of Proposition 5.4.3. The general case is
proved in the same way. For similar reasons,
$\widehat{X}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$ is Galois, proving
(ii).

For (iii), note that $G$ decomposes as a product $G=H\times
E_{8}\times...\times E_{8}$, where $H$ is a reductive group containing no
simple factors of type $E_{8}$ and $E_{8}$ is the (unique) simple group of
type $E_{8}$. Indeed, every connected reductive group admits a central isogeny
from a product of simple groups and a torus, and the simply connected group of
type $E_{8}$ is adjoint. Thus, it suffices to prove (iii) separately for $G=H$
and $G=E_{8}$.

First suppose $G=H$. Then $\widehat{X}$ satisfies property (5.0.1) by Lemma
6.5.2. Let
$\Omega=\pi_{1}^{G}(\widehat{\mathbb{O}}),\Gamma=\pi_{1}(\widehat{\mathbb{O}})$.
Since $\widehat{\mathbb{O}}$ is maximal in its equivalence class,
$\Gamma=\Gamma(0)$, see Proposition 5.4.3 and Remark 5.4.4. Thus
$\Omega=\Omega(0)$ and by Proposition 6.4.2 there is an equivalence

$\overline{\operatorname{HC}}^{G}(\mathcal{A}_{0}^{\widehat{X}})\simeq\Omega/\Omega(0)\simeq\operatorname{Vect},$

proving (iii) in this case. Next suppose $G=E_{8}$. If $\mathbb{O}$ is the
principal nilpotent orbit, then $\pi_{1}(\mathbb{O})=1$. So
$\widehat{\mathbb{O}}=\mathbb{O}$ and $\Gamma=1$. By the results of Section
4.13, there is an embedding
$\overline{\operatorname{HC}}(\mathcal{A}_{0}^{\widehat{X}})\hookrightarrow\Gamma\operatorname{-mod}\simeq\operatorname{Vect}$.
(iii) follows at once. If $\widehat{\mathbb{O}}$ is not the principal orbit,
then $\widehat{X}$ satisfies property (5.0.1), see again Lemma 6.5.2. So we
can argue as in the case of $G=H$. ∎

######  Proposition 6.5.4.

Let $\widetilde{\mathbb{O}}$,
$\dot{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ be
$G$-equivariant covers. Then

$[\dot{\mathbb{O}}]=[\widetilde{\mathbb{O}}]\iff
I_{0}(\dot{\mathbb{O}})=I_{0}(\widetilde{\mathbb{O}}).$

Moreover, if $\widetilde{\mathbb{O}}$ is maximal in its equivalence class, and
$\Pi:=\operatorname{Aut}_{\mathbb{O}}(\widetilde{\mathbb{O}})$, then there is
a $G$-equivariant algebra isomorphism

(6.5.1)
$[(U(\mathfrak{g})/I_{0}(\widetilde{\mathbb{O}}))_{\dagger}]^{\dagger}\xrightarrow{\sim}(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}.$

###### Proof.

Note that every $G$-equivariant nilpotent cover is a homogeneous space for the
universal cover of the derived subgroup of $G$. Thus, we can reduce to the
case when $G$ is semisimple and simply connected. The proof has several steps.

Step 1. First we prove the easy implication
$[\dot{\mathbb{O}}]=[\widetilde{\mathbb{O}}]\Rightarrow
I_{0}(\dot{\mathbb{O}})=I_{0}(\widetilde{\mathbb{O}}).$ By Lemma 6.5.3, we can
assume $\dot{\mathbb{O}}$ is the maximal cover in $[\widetilde{\mathbb{O}}]$.
Hence, there is a Galois cover
$\dot{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widetilde{\mathbb{O}}$ such
that the induced map
$\dot{X}=\operatorname{Spec}(\mathbb{C}[\dot{\mathbb{O}}])\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])=\widetilde{X}$
is almost étale. By Proposition 5.4.5, there is a filtered algebra embedding
$i:\mathcal{A}_{0}^{\widetilde{X}}\hookrightarrow\mathcal{A}_{0}^{\dot{X}}$,
and by Lemma 4.11.2, $i\circ\Phi_{0}^{\widetilde{X}}=\Phi_{0}^{\dot{X}}$. Thus

$I_{0}(\widetilde{\mathbb{O}})=\ker{(\Phi_{0}^{\widetilde{X}})}=\ker{(i\circ\Phi_{0}^{\widetilde{X}})}=\ker{(\Phi_{0}^{\dot{X}})}=I_{0}(\dot{\mathbb{O}}),$

as desired.

Step 2. In steps 2-7, we will prove the hard implication
$I_{0}(\dot{\mathbb{O}})=I_{0}(\widetilde{\mathbb{O}})\Rightarrow[\dot{\mathbb{O}}]=[\widetilde{\mathbb{O}}]$.
By Step 1, it suffices to show that every $G$-equivariant nilpotent cover
$\widetilde{\mathbb{O}}$, maximal in its equivalence class, is determined by
the ideal $I_{0}(\widetilde{\mathbb{O}})$. Our proof is somewhat indirect. We
will first show that $I_{0}(\widetilde{\mathbb{O}})$ determines a pair
$(\widecheck{\mathbb{O}},\widecheck{\mathcal{A}})$ consisting of a nilpotent
cover $\widecheck{\mathbb{O}}$, covered by $\widetilde{\mathbb{O}}$, and a
filtered quantization $\widecheck{\mathcal{A}}$ of
$\mathbb{C}[\widecheck{\mathbb{O}}]$. The quantization
$\widecheck{\mathcal{A}}$ is constructed so that
$I_{0}(\widetilde{\mathbb{O}})$ coincides with kernel of the (unique) quantum
co-moment map
$U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\widecheck{\mathcal{A}}$ and
$\widecheck{\mathbb{O}}$ is the smallest cover of $\mathbb{O}$ admitting a
quantization with this property. Set
$\widecheck{X}:=\operatorname{Spec}(\mathbb{C}[\widecheck{\mathbb{O}}])$. The
pair $(\widecheck{X},\widecheck{\mathcal{A}})$ determines a unique Galois
cover $\breve{\mathbb{O}}$ of $\widecheck{\mathbb{O}}$ (see the discussion
preceding Lemma 4.14.2) such that the image of
$\overline{\operatorname{HC}}(\widecheck{\mathcal{A}})$ in
$\pi_{1}(\widecheck{\mathbb{O}})\operatorname{-mod}$ consists precisely of
representations with trivial restriction to $\pi_{1}(\breve{\mathbb{O}})$. To
prove that
$I_{0}(\widecheck{\mathbb{O}})=I_{0}(\widetilde{\mathbb{O}})\Rightarrow[\widecheck{\mathbb{O}}]=[\widetilde{\mathbb{O}}]$
we will show that $\breve{\mathbb{O}}\simeq\widetilde{\mathbb{O}}$.

Step 3. Let $R$ denote the reductive part of the centralizer of
$e\in\mathbb{O}$, and let $\Gamma=\pi_{1}(\widetilde{\mathbb{O}})$. Recall the
functors
$\bullet_{\dagger}:\operatorname{HC}^{G}(U(\mathfrak{g}))\stackrel{{\scriptstyle}}{{\to}}\operatorname{HC}^{R}(\mathcal{W})$
and
$\bullet^{\dagger}:\operatorname{HC}^{R}_{\mathrm{fin}}(\mathcal{W})\stackrel{{\scriptstyle}}{{\to}}\operatorname{HC}^{G}(U(\mathfrak{g}))$
from Section 3.5 and the related maps between the sets of ideals. By (iv) of
Lemma 6.1.1, applied to $\mathcal{A}_{0}^{\widetilde{X}}$, there is a
$\Gamma$-invariant ideal $J\subset\mathcal{W}$ of codimension 1 such that
$I_{0}(\widetilde{\mathbb{O}})=J^{\ddagger}$. Consider the finite index
subgroup $\operatorname{Stab}_{R}(J)\subset R$ and let
$\widecheck{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$ be the
cover corresponding to
$\operatorname{Stab}_{R}(J)/R^{\circ}\subset\pi_{1}(\mathbb{O})$. Thus,
$\pi_{1}(\widecheck{\mathbb{O}})\simeq\operatorname{Stab}_{R}(J)/R^{\circ}\subseteq
R/R^{\circ}\simeq\pi_{1}(\mathbb{O})$. Since $J$ is
$\pi_{1}(\widetilde{\mathbb{O}})$-stable, we get the inclusion
$\pi_{1}(\widetilde{\mathbb{O}})\subseteq\pi_{1}(\widecheck{\mathbb{O}})$.
Thus $\widetilde{\mathbb{O}}$ is a cover of $\widecheck{\mathbb{O}}$.

Step 4. Recall from Section 3.5 that $R^{\circ}$ preserves $J$. Let
$\underline{J}$ denote the intersection of the $R$-conjugates of $J$. This is
an $R$-stable ideal of finite codimension. Consider the Dixmier algebra
$\widecheck{\mathcal{A}}:=(\mathcal{W}/\underline{J})^{\dagger}$. We claim
that $\widecheck{\mathcal{A}}$ is a filtered quantization of
$\mathbb{C}[\widecheck{\mathbb{O}}]$. This will be proved in steps 5-6.

Step 5. By Lemma 3.5.5, $\operatorname{gr}\widecheck{\mathcal{A}}$ embeds into
$\mathbb{C}[\widecheck{\mathbb{O}}]$ as a $G$-stable subalgebra. This
embedding intertwines the natural homomorphisms
$\mathbb{C}[\mathfrak{g}^{*}]\stackrel{{\scriptstyle}}{{\to}}\operatorname{gr}\widecheck{\mathcal{A}}$
and
$\mathbb{C}[\mathfrak{g}^{*}]\stackrel{{\scriptstyle}}{{\to}}\mathbb{C}[\widecheck{\mathbb{O}}]$,
and hence the extension
$\operatorname{gr}\widecheck{\mathcal{A}}\hookrightarrow\mathbb{C}[\widecheck{\mathbb{O}}]$
is finite. By Proposition (v) of 3.5.1,
$\widecheck{\mathcal{A}}_{\dagger}\simeq\mathcal{W}/\underline{J}$. The
dimension of the right hand side is $|R/\operatorname{Stab}_{R}(J)|$. Thus
$\dim(\widecheck{\mathcal{A}}_{\dagger})$ coincides with the degree of the
covering map $\widecheck{\mathbb{O}}\rightarrow\mathbb{O}$. In particular, the
induced map
$\operatorname{Spec}(\mathbb{C}[\widecheck{\mathbb{O}}])\stackrel{{\scriptstyle}}{{\to}}\operatorname{Spec}(\operatorname{gr}(\widecheck{\mathcal{A}}))$
is birational. A finite birational morphism onto a normal variety is an
isomorphism. Thus,
$\operatorname{gr}(\widecheck{\mathcal{A}})\xrightarrow{\sim}\mathbb{C}[\widecheck{\mathbb{O}}]$
is equivalent to the normality of
$\operatorname{Spec}(\operatorname{gr}\widecheck{\mathcal{A}})$. This will be
established in Step 6.

Step 6. By Step 5, $\widecheck{\mathbb{O}}$ embeds as an open $G$-orbit in
$\operatorname{Spec}(\operatorname{gr}\widecheck{\mathcal{A}})$ with boundary
codimension $\geq 2$. Thus,
$\operatorname{Spec}(\operatorname{gr}\widecheck{\mathcal{A}})$ is normal if
$\operatorname{gr}\widecheck{\mathcal{A}}$ is Cohen-Macaulay. Note that the
homomorphism
$\mathcal{W}\rightarrow(\mathcal{A}_{0}^{\widetilde{X}})_{\dagger}$ factors
through $\mathcal{W}\rightarrow\mathcal{W}/\underline{J}$. Also
$\mathcal{W}/\underline{J}$, regarded as an object in
$\operatorname{HC}_{\mathrm{fin}}^{R}(\mathcal{W})$, embeds as a direct
summand in $(\mathcal{A}_{0}^{\widetilde{X}})_{\dagger}$. Therefore,
$\widecheck{\mathcal{A}}=(\mathcal{W}/\underline{J})^{\dagger}$, regarded as a
filtered $U(\mathfrak{g})$-bimodule, embeds as a direct summand in
$[(\mathcal{A}_{0}^{\widetilde{X}})_{\dagger}]^{\dagger}\simeq\mathcal{A}_{0}^{\widetilde{X}}$
(so that $\mathcal{A}_{0}^{\widetilde{X}}$ viewed as a filtered bimodule
decomposes into the direct sum of $\widecheck{\mathcal{A}}$ and another
filtered bimodule). The last isomorphism follows from Lemma 6.1.1(iii).
Passing to associated graded bimodules, we deduce that
$\operatorname{gr}\widecheck{\mathcal{A}}$, regarded as a
$\mathbb{C}[\mathfrak{g}^{*}]$-module, embeds as a direct summand into
$\operatorname{gr}\mathcal{A}_{0}^{\widetilde{X}}\simeq\mathbb{C}[\widetilde{\mathbb{O}}]$.
The latter, and hence the former, is a Cohen-Macaulay module. So
$\operatorname{gr}\widecheck{\mathcal{A}}$ is a Cohen-Macaulay algebra. As
noted above, this implies that $\widecheck{\mathcal{A}}$ is a filtered
quantization of $\mathbb{C}[\widecheck{\mathbb{O}}]$.

Step 7. Form $\breve{\mathbb{O}}$ and $\breve{\mathcal{A}}$ as in the
discussion preceding Lemma 4.14.2 (for
$\operatorname{Spec}(\mathbb{C}[\widecheck{\mathbb{O}}])$ and its quantization
$\widecheck{\mathcal{A}}$). The construction in Step 6 shows that
$\mathcal{A}_{0}^{\widetilde{X}}$ is a cover of $\widecheck{\mathcal{A}}$ in
the sense of Definition 4.14.1. Also, since $\widetilde{\mathbb{O}}$ is
maximal in its equivalence class, we have
$\operatorname{HC}(\mathcal{A}_{0}^{\widetilde{X}})\simeq\operatorname{Vect}$,
see (iii) of Lemma 6.5.3. Thus by Corollary 4.14.4, we have that
$\breve{\mathbb{O}}\simeq\widetilde{\mathbb{O}},\breve{\mathcal{A}}\simeq\mathcal{A}_{0}^{\widetilde{X}}$.
This proves the implication
$I_{0}(\dot{\mathbb{O}})=I_{0}(\widetilde{\mathbb{O}})\Rightarrow[\dot{\mathbb{O}}]=[\widetilde{\mathbb{O}}]$.

Step 8. It remains to prove (6.5.1). By Step 7,
$\widetilde{\mathbb{O}}=\breve{\mathbb{O}}$. By Proposition 4.14.3,
$\widetilde{\mathbb{O}}$ is a Galois cover of $\widecheck{\mathbb{O}}$. Let
$\Pi^{\prime}\subset\Pi$ be the Galois group of
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\widecheck{\mathbb{O}}$.
By Proposition 4.14.3,
$\widecheck{\mathcal{A}}=(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi^{\prime}}$. In
particular, there is an inclusion
$(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}\hookrightarrow\mathcal{A}$
intertwining the quantum comoment maps. Consider the chain of inclusions

(6.5.2)
$U(\mathfrak{g})/I\hookrightarrow(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}\hookrightarrow\widecheck{\mathcal{A}},$

where $I:=I_{0}(\widetilde{\mathbb{O}})$. Recall that the functor
$\bullet_{\dagger}$ is exact by (i) of Proposition 3.5.1. Applying
$\bullet_{\dagger}$ to (6.5.2) we obtain another chain of inclusions

$(U(\mathfrak{g})/I)_{\dagger}\hookrightarrow(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}_{\dagger}\hookrightarrow\widecheck{\mathcal{A}}_{\dagger}.$

By the construction of $\widecheck{\mathcal{A}}$ in Step 4, the composition of
these inclusions is an isomorphism. On the other hand, by (i) of Lemma 6.1.1,
we have

$\dim(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}_{\dagger}=\operatorname{deg}(\widetilde{\mathbb{O}}/\Pi),\qquad\dim\mathcal{A}_{\dagger}=\operatorname{deg}(\widecheck{\mathbb{O}}),$

where ‘deg’ denotes the degree of a finite cover of $\mathbb{O}$. We conclude
that $\Pi^{\prime}=\Pi$. This proves (6.5.1).

∎

The following is an immediate consequence of Proposition 6.5.4.

######  Theorem 6.5.5.

Let $\mathbb{O}$ be a fixed nilpotent $G$-orbit. The passage from
$\widetilde{\mathbb{O}}$ to $I_{0}(\widetilde{\mathbb{O}})$ defines a
bijection between the following two sets

  * •

Equivalence classes of $G$-equivariant covers
$\widetilde{\mathbb{O}}\stackrel{{\scriptstyle}}{{\to}}\mathbb{O}$.

  * •

Unipotent ideals $I\subset U(\mathfrak{g})$ with $V(I)=\overline{\mathbb{O}}$.

######  Corollary 6.5.6.

Suppose $\widetilde{\mathbb{O}}$ is maximal in its equivalence class and let
$\Pi:=\operatorname{Aut}_{\mathbb{O}}(\widetilde{\mathbb{O}})$. If
$I_{0}(\widetilde{\mathbb{O}})$ is a maximal ideal, then the quantum comoment
map $U(\mathfrak{g})\rightarrow\mathcal{A}_{0}^{\widetilde{X}}$ induces an
algebra algebra isomorphism

$U(\mathfrak{g})/I_{0}(\widetilde{\mathbb{O}})\xrightarrow{\sim}(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}.$

###### Proof.

Lemma 3.5.2 applied to
$\mathcal{B}=U(\mathfrak{g})/I_{0}(\widetilde{\mathbb{O}})$ implies
$U(\mathfrak{g})/I_{0}(\widetilde{\mathbb{O}})\xrightarrow{\sim}([U(\mathfrak{g})/I_{0}(\widetilde{\mathbb{O}})]_{\dagger})^{\dagger}$.
On the other hand,
$([U(\mathfrak{g})/I_{0}(\widetilde{\mathbb{O}})]_{\dagger})^{\dagger}\xrightarrow{\sim}(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}$
by (6.5.1). This completes the proof. ∎

###  6.6. Classification of unipotent bimodules

Let $\widetilde{\mathbb{O}}$ be a $G$-equivariant nilpotent cover, maximal in
its equivalence class, and let $I:=I_{0}(\widetilde{\mathbb{O}})$. Recall the
category $\operatorname{HC}^{G}(U(\mathfrak{g})/I)$ and its quotient
$\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)$ defined in (6.4.4) and
(6.4.6), respectively. Note that the irreducible objects in
$\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)$ are in bijection with
$\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$ — this is a consequence of
Definition 6.0.2 together with (iii) of Proposition 3.1.1.

Let $\Pi:=\operatorname{Aut}_{\mathbb{O}}(\widetilde{\mathbb{O}})$, and
$\widecheck{\mathbb{O}}:=\widetilde{\mathbb{O}}/{\Pi}$. By Proposition 5.1.1,
the $\Pi$-action on $\mathbb{C}[\widetilde{\mathbb{O}}]$ lifts to the
canonical quantization $\mathcal{A}_{0}^{\widetilde{X}}$. Let
$\widecheck{X}=\operatorname{Spec}(\mathbb{C}[\widecheck{\mathbb{O}}])$ and
choose $\delta\in\overline{\mathfrak{P}}^{\widecheck{X}}$ so that

$\mathcal{A}_{\delta}^{\widecheck{X}}:=(\mathcal{A}_{0}^{\widetilde{X}})^{\Pi}.$

An object
$\mathcal{B}\in\operatorname{HC}^{G}(\mathcal{A}_{\delta}^{\widecheck{X}})$
can be regarded as a $G$-equivariant Harish-Chandra bimodule for
$U(\mathfrak{g})/I$ via the co-moment map
$\Phi:=\Phi_{\delta}^{\widecheck{X}}:U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{\delta}^{\widecheck{X}}$.
As explained in Section 6.4, this gives rise to a forgetful functor between
the corresponding quotient categories

(6.6.1)
$\Phi^{*}:\overline{\operatorname{HC}}^{G}(\mathcal{A}_{\delta}^{\widecheck{X}})\stackrel{{\scriptstyle}}{{\to}}\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I).$

######  Lemma 6.6.1.

The functor (6.6.1) is an equivalence of monoidal categories.

###### Proof.

The quantization $\mathcal{A}^{\widecheck{X}}_{\delta}$ coincides with the
Dixmier algebra $\mathcal{A}$ constructed in Step 4 of the proof of
Proposition 6.5.4. This follows from the equality
$\breve{\mathcal{A}}=\mathcal{A}_{0}^{\widetilde{X}}$ established in Step 7 of
that proof and the definition of $\breve{\mathcal{A}}$ preceding Lemma 4.14.2.
By the construction of $\mathcal{A}$ we have
$(U(\mathfrak{g})/I)_{\dagger}\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{\dagger}$.
By (iii) of Proposition 3.5.1, the cokernel of the inclusion
$U(\mathfrak{g})/I\hookrightarrow\mathcal{A}$ is supported on
$\partial\mathbb{O}$. This shows that the functor
$\Phi^{*}:\overline{\operatorname{HC}}^{G}(\mathcal{A}_{\delta}^{\widecheck{X}})\stackrel{{\scriptstyle}}{{\to}}\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)$
is monoidal.

Next we construct a functor
$\Psi:\operatorname{HC}^{G}(U(\mathfrak{g})/I)\rightarrow\operatorname{HC}^{G}(\mathcal{A})$.
For $\mathcal{B}\in\operatorname{HC}^{G}(U(\mathfrak{g})/I)$, consider the
microlocalization $\mathcal{B}_{\widecheck{\mathbb{O}}}$ of $\mathcal{B}$ to
$\widecheck{\mathbb{O}}\subset\widecheck{X}$. Set
$\Psi(\mathcal{B}):=\Gamma(\mathcal{B}_{\widecheck{\mathbb{O}}})$. Of course,
$\Psi$ descends to a functor (still denoted by $\Psi$)

$\Psi:\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)\stackrel{{\scriptstyle}}{{\to}}\overline{\operatorname{HC}}^{G}(\mathcal{A})$

We claim that
$\Phi^{*}:\overline{\operatorname{HC}}^{G}(\mathcal{A}_{\delta}^{\widecheck{X}})\stackrel{{\scriptstyle}}{{\to}}\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)$
and
$\Psi:\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)\stackrel{{\scriptstyle}}{{\to}}\overline{\operatorname{HC}}^{G}(\mathcal{A})$
are quasi-inverse equivalences. This follows at once from two observations.
First, $\Phi^{*}$ commutes with microlocalization to $\widecheck{\mathbb{O}}$.
Second, for $\mathcal{B}\in\operatorname{HC}(\mathcal{A})$ or
$\operatorname{HC}(U(\mathfrak{g})/I)$, the kernel and cokernel of the natural
map $\mathcal{B}\rightarrow\Gamma(\mathcal{B}_{\widecheck{\mathbb{O}}})$ are
supported on $\partial\mathbb{O}$. Indeed, for any good filtration on
$\mathcal{B}$, there is an embedding
$\operatorname{gr}\Gamma(\mathcal{B}_{\widecheck{\mathbb{O}}})\hookrightarrow\Gamma(\operatorname{gr}\mathcal{B}|_{\widecheck{\mathbb{O}}})$
and the natural homomorphism
$\operatorname{gr}\mathcal{B}\rightarrow\Gamma(\operatorname{gr}\mathcal{B}|_{\widecheck{\mathbb{O}}})$
factors as the composition
$\operatorname{gr}\mathcal{B}\rightarrow\operatorname{gr}\Gamma(\mathcal{B}_{\widecheck{\mathbb{O}}})\hookrightarrow\Gamma(\operatorname{gr}\mathcal{B}|_{\widecheck{\mathbb{O}}})$.
∎

The next result provides a classification of
$\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$, compare to Theorem 1.5.4.

######  Theorem 6.6.2.

There is an equivalence of monoidal categories

$\Pi\operatorname{-mod}\xrightarrow{\sim}\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I),\qquad
V\mapsto\Phi^{*}(\mathcal{A}_{0}^{\widetilde{X}}\otimes V)^{\Pi}.$

In particular, there is a bijection

$\\{\text{irreducible representations of
}\Pi\\}\xrightarrow{\sim}\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G).$

###### Proof.

Since $\widetilde{\mathbb{O}}$ is maximal in its equivalence class,
$\overline{\operatorname{HC}}^{G}(\mathcal{A}_{0}^{\widetilde{X}})\simeq\operatorname{Vect}$
by Lemma 6.5.3. Thus, by Corollary 6.4.4, there is an equivalence of monoidal
categories

(6.6.2)
$\Pi\operatorname{-mod}\xrightarrow{\sim}\overline{\operatorname{HC}}^{G}(\mathcal{A}_{\delta}^{\widecheck{X}}),\qquad
V\mapsto(\mathcal{A}_{0}^{\widetilde{X}}\otimes V)^{\Pi}.$

We obtain the desired equivalence
$\Pi\operatorname{-mod}\xrightarrow{\sim}\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)$
by composing (6.6.2) with the equivalence of Lemma 6.6.1. ∎

######  Corollary 6.6.3.

Suppose $I$ is maximal. Then there is an equivalence of monoidal categories

$\Pi\operatorname{-mod}\xrightarrow{\sim}\operatorname{HC}^{G}(U(\mathfrak{g})/I),\qquad
V\mapsto\Phi^{*}(\mathcal{A}_{0}^{\widetilde{X}}\otimes V)^{\Pi}.$

In particular, every bimodule
$\mathcal{B}\in\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$ is of the form
$(\mathcal{A}_{0}^{\widetilde{X}}\otimes V)^{\Pi}$ for a uniquely defined
irreducible $V\in\Pi\operatorname{-mod}$.

###### Proof.

Since $I$ is maximal, $\operatorname{HC}^{G}_{\partial}(U(\mathfrak{g})/I)=0$,
see (6.4.5). Thus, the quotient functor
$\operatorname{HC}^{G}(U(\mathfrak{g})/I)\stackrel{{\scriptstyle}}{{\to}}\overline{\operatorname{HC}}^{G}(U(\mathfrak{g})/I)$
is an equivalence. Now the corollary follows from Theorem 6.6.2. ∎

###  6.7. A proof of a conjecture of Vogan

Recall from Section 3.4 that every Harish-Chandra bimodule
$\mathcal{B}\in\operatorname{HC}^{G}(U(\mathfrak{g}))$ can be regarded as a
$G$-representation via the adjoint action of $\mathfrak{g}$. If $\mathcal{B}$
is unipotent, then this $G$-representation is of a very special form.

######  Proposition 6.7.1.

Assume $I_{0}(\widetilde{\mathbb{O}})$ is a maximal ideal. Let
$\mathcal{B}\in\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$ and let
$e\in\mathbb{O}$. Then there is a finite-dimensional representation $\chi$ of
$G_{e}/G_{e}^{\circ}$ such that

$\mathcal{B}\simeq_{G}\mathrm{AlgInd}^{G}_{G_{e}}\chi.$

###### Proof.

Assume $\widetilde{\mathbb{O}}$ is maximal in its equivalence clas, and let
$\Pi=\operatorname{Aut}_{\mathbb{O}}(\widetilde{\mathbb{O}})$. By Corollary
6.6.3, there is a uniquely defined irreducible $V\in\Pi\operatorname{-mod}$
such that $\mathcal{B}\simeq\Phi^{*}(\mathcal{A}_{0}^{\widetilde{X}}\otimes
V)^{\Pi}$. The filtration on $\mathcal{A}_{0}^{\widetilde{X}}$ induces a good
filtration on $\mathcal{B}$, and there is a natural isomorphism of graded
$G$-equivariant $\mathbb{C}[\mathbb{O}]$-modules

(6.7.1)
$\operatorname{gr}(\mathcal{B})\simeq\operatorname{gr}(\mathcal{A}_{0}^{\widetilde{X}}\otimes
V)^{\Pi}\simeq(\mathbb{C}[\widetilde{\mathbb{O}}]\otimes V)^{\Pi}.$

Fix $\widecheck{\mathbb{O}}$ as in Section 6.6. Choose
$x\in\widecheck{\mathbb{O}}$ over $e$ and $y\in\widetilde{\mathbb{O}}$ over
$x$. Thus, $G_{y}$ is a normal subgroup of finite-index in $G_{x}$ and
$\Pi\simeq G_{x}/G_{y}$. Regard $V$ as an irreducible representation of
$G_{x}$, trivial on $G_{y}$, and form the $G$-equivariant vector bundle
$\mathcal{V}:=G\times^{G_{x}}V\stackrel{{\scriptstyle}}{{\to}}G/G_{x}\simeq\widecheck{\mathbb{O}}$.
Then $(\mathbb{C}[\widetilde{\mathbb{O}}]\otimes V)^{\Pi}$ can be identified
with $\Gamma(\widecheck{\mathbb{O}},\mathcal{V})$. As $G$-representations

$\Gamma(\widecheck{\mathbb{O}},\mathcal{V})\simeq_{G}\mathrm{AlgInd}^{G}_{G_{x}}V\simeq_{G}\mathrm{AlgInd}^{G}_{G_{e}}(\mathrm{AlgInd}^{G_{e}}_{G_{x}}V).$

Now set $\chi=\mathrm{AlgInd}^{G_{e}}_{G_{x}}V$ to complete the proof. ∎

######  Remark 6.7.2.

In Section 9, we will show that all special unipotent ideals
$I_{\mathrm{max}}(\frac{1}{2}h^{\vee})$ are unipotent. Thus, the conclusion of
Proposition 6.7.1 holds for all special unipotent bimodules. In particular,
Proposition 6.7.1 provides an affirmative answer to a conjecture of Vogan
([Vog91, Conj 12.1]) in the case of complex groups. Special cases of this
conjecture were previously established by [Mas18], [Bar17], and [Won18].

Proposition 6.7.1 is the ‘complex’ case of a more general conjecture about
irreducible representations of real reductive Lie groups, see [Vog91, Sec 12].
We conclude this section with a discussion of this more general conjecture.

Let $K$ be a symmetric subgroup of $G$, i.e. the identity component of the
fixed point locus of a nontrivial involution. One can define the notion of a
Harish-Chandra $(\mathcal{A}_{0}^{\widetilde{X}},K)$-module: this is a
$K$-equivariant $\mathcal{A}_{0}^{\widetilde{X}}$-module which has the
structure of a Harish-Chandra $(\mathfrak{g},K)$-module via the quantum co-
moment map
$\Phi:U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{0}^{\widetilde{X}}$.
Assume, for simplicity, that $\widetilde{\mathbb{O}}=\mathbb{O}$ is a
nilpotent orbit (rather than a cover), and suppose $\mathcal{B}$ is an
irreducible $(\mathcal{A}^{X}_{0},K)$-module. Suppose

(6.7.2) $\operatorname{codim}(\partial\mathbb{O},\overline{\mathbb{O}})\geq
4.$

Under this codimension condition, the associated variety of $\mathcal{B}$ is
the closure of a $K$-orbit $\mathbb{O}_{K}\subset\mathbb{O}$, which is an
irreducible component in $\mathbb{O}\cap(\mathfrak{g}/\mathfrak{k})^{*}$ and a
Lagrangian subvariety in $\mathbb{O}$, see [Vog91, Thm 4.6]. In particular,
(6.7.2) implies the (strictly weaker) condition

(6.7.3)
$\operatorname{codim}(\partial\mathbb{O}_{K},\overline{\mathbb{O}}_{K})\geq
2.$

In [Vog91, Sec 2], Vogan attaches to $\mathcal{B}$ a twisted local system
$\mathcal{L}_{\mathcal{B}}$ on $\mathbb{O}_{K}$. Since
$\mathcal{A}_{0}^{\widetilde{X}}$ is a canonical quantization, the ‘twist’
corresponds to a square root of the canonical bundle on $\mathbb{O}_{K}$, see
[Vog91, Thm 8.7]. In this setting, Vogan’s conjecture is as follows.

######  Conjecture 6.7.3.

Assume condition (6.7.2). Then there is an isomorphism of $K$-representations

$\mathcal{B}\simeq_{K}\Gamma(\mathbb{O}_{\mathcal{B}},\mathcal{L}_{\mathcal{B}})$

Under the codimension condition (6.7.3), it is possible to define a
distinguished good filtration on $\mathcal{B}$ such that

(6.7.4)
$\operatorname{gr}(\mathcal{B})\hookrightarrow\Gamma(\mathbb{O}_{K},\mathcal{L}_{\mathcal{B}}),$

with cokernel supported on $\partial\mathbb{O}_{K}$. This filtration can be
defined using certain generalized versions of the restriction and extension
functors defined in Section 3.5, see [Los15, Section 6.1]. Since $K$ is
reductive, $\operatorname{gr}(\mathcal{B})\simeq_{K}\mathcal{B}$. Hence,
Conjecture 6.7.3 holds if and only if (6.7.4) is an isomorphism.

In [LY] it will be shown that the support of the cokernel of the embedding
(6.7.4) is of codimension $\geq 3$. Now (6.7.4) is an isomorphism (and hence
Conjecture 6.7.3 holds) if the following conjecture is true.

######  Conjecture 6.7.4.

The $\mathbb{C}[\mathbb{O}_{K}]$-module
$\Gamma(\mathbb{O}_{K},\mathcal{L}_{\mathcal{B}})$ is Cohen-Macaulay.

Conjecture 6.7.4 was proved for complex groups in [Mas18]. Some additional
(real) cases were established in [Mas20].

If we drop the condition (6.7.2) but retain the condition (6.7.3), it is still
possible to define a distinguished filtration on $\mathcal{B}$ and Conjecture
6.7.4 still makes sense. However, it is no longer clear that Conjecture 6.7.4
implies Conjecture 6.7.3.

If we drop the weaker condition (6.7.3), all hell breaks loose. In particular,
the associated variety of $\mathcal{B}$ is no longer irreducible and the
$\mathbb{C}[\mathbb{O}_{K}]$-module $\Gamma(\mathbb{O}_{K},\mathcal{L}_{B})$
is no longer finitely-generated. Moreover, one can no longer define a
distinguished good filtration on $\mathcal{B}$ using our restriction and
extension functors. Still, it is reasonable to conjecture that there is a good
filtration on $\mathcal{B}$ with the property that
$\operatorname{gr}\mathcal{B}$ is Cohen-Macaulay. This is consistent with
other conjectures in the subject, see [SV11].

##  7\. Parabolic Induction of Hamiltonian Quantizations

Let $G$ be a connected reductive algebraic group. In Section 6, we defined,
for each $G$-equivariant nilpotent cover $\widetilde{\mathbb{O}}$, a
completely prime primitive ideal $I_{0}(\widetilde{\mathbb{O}})\subset
U(\mathfrak{g})$, Definition 6.0.1, and a finite set of irreducible Harish-
Chandra bimodules $\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$, Definition
6.0.2. We proved that the set $\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$ is
in natural bijection with irreducible representations of the (finite)
automorphism group of the maximal cover in the equivalence class
$[\widetilde{\mathbb{O}}]$, see Theorem 6.6.2, and that all bimodules in
$\mathrm{Unip}_{\widetilde{\mathbb{O}}}(G)$ are, as $G$-representations, of
the form conjectured by Vogan, see Proposition 6.7.1. However, three
properties of unipotent ideals and bimodules were left more or less
unaddressed, namely:

  * •

All unipotent bimodules are unitary.

  * •

All unipotent ideals are maximal.

  * •

All special unipotent bimodules are unipotent.

Compare to Desiderata (1),(3), and (6) from Section 1.3. We will verify these
properties in Sections 8-10. To do so, we will need a method for computing the
infinitesimal characters of unipotent ideals. This method will be developed in
Section 8. In this section, we establish some preliminary facts about filtered
quantizations of nilpotent covers which are needed for these computations.

We will summarize the method here in order to motivate what follows. Assume
for simplicity that $\widetilde{\mathbb{O}}$ is birationally rigid, see
Section 2.4, and maximal in its equivalence class (it is not particularly
difficult to reduce the computation to this special case and we will do so in
Section 8.1). These assumptions guarantee that $\widetilde{\mathbb{O}}$ is the
universal $G$-equivariant cover of $\mathbb{O}$. Taking
$\operatorname{Aut}_{\mathbb{O}}(\widetilde{\mathbb{O}})$-invariants in
$\mathcal{A}_{0}^{\widetilde{X}}$, we get a Hamiltonian quantization
$\mathcal{A}_{\epsilon}^{X}$ of $\mathbb{C}[\mathbb{O}]$ corresponding to a
special parameter $\epsilon\in\mathfrak{P}^{X}$ constructed in Example 5.3.2.
Moreover, $I_{\epsilon}(\mathbb{O})=I_{0}(\widetilde{\mathbb{O}})$. Choose a
Levi subgroup $L\subset G$ and a birationally rigid $L$-orbit $\mathbb{O}_{L}$
such that $\mathbb{O}=\mathrm{Bind}^{G}_{L}\mathbb{O}_{L}$. Since
$\mathbb{O}_{L}$ is birationally rigid, $\mathbb{C}[\mathbb{O}_{L}]$ admits a
unique filtered quantization, $\mathcal{A}_{0}^{X_{L}}$. In Section 7.3, we
will define the notion of parabolic induction for Hamiltonian quantizations.
For every ‘induction parameter’ $\lambda\in\mathfrak{X}(\mathfrak{l})$, we
construct an ‘induced’ quantization of $\mathbb{C}[\mathbb{O}]$, denoted
$\operatorname{Ind}^{G}_{L}\mathcal{A}_{\lambda}^{X_{L}}$. Every Hamiltonian
quantization of $\mathbb{C}[\mathbb{O}]$ arises in this fashion. More
precisely, there is a linear isomorphism
$\eta:\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}\overline{\mathfrak{P}}^{X}$
such that $\operatorname{Ind}^{G}_{L}\mathcal{A}_{\lambda}^{X_{L}}$ coincides
with the Hamiltonian quantization $\mathcal{A}^{X}_{\eta(\lambda)}$. Given an
induction parameter $\lambda\in\mathfrak{X}(\mathfrak{l})$ and the
infinitesimal character of $I_{0}(\mathbb{O}_{L})$, it is easy to determine
the infinitesimal character of $I_{\eta(\lambda)}(\mathbb{O})$. So we need to
determine the element $\eta^{-1}(\epsilon)\in\mathfrak{X}(\mathfrak{l})$.
Computing $\eta$ is a nontrivial problem, to which the bulk of this section is
devoted. In Sections 7.5-7.7, we solve this problem under certain conditions
on $\mathbb{O}$. This reduces the problem of computing unipotent infinitesimal
characters to the case of birationally rigid orbits. By a similar argument,
one can further reduce to the case of rigid nilpotent orbits. For such orbits,
we can compute the infinitesimal characters using the results of McGovern
([McG94]) and Premet ([Pre13]).

###  7.1. Partial resolutions of nilpotent covers

Choose a Levi subgroup $L\subset G$ and an $L$-equivariant cover
$\widetilde{\mathbb{O}}_{L}$ such that
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$. Fix
a parabolic subgroup $P\subset G$ with Levi decomposition $P=LN$, and form the
$G$-equivariant fiber bundle

$\pi:\widetilde{Y}:=G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}G/P,$

see Section 2.4. There is a proper map
$\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}$, which
factors through a partial resolution
$\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$.

In this preparatory section, we will compute the Picard group and class group
of $\widetilde{Y}$. We begin with a general observation.

######  Lemma 7.1.1.

Let $V$ be a normal affine cone. Then $\operatorname{Pic}(V)=0$.

###### Proof.

Choose a line bundle $\mathcal{L}\in\operatorname{Pic}(V)$. Since $V$ is
normal, $\mathcal{L}$ can be made $\mathbb{C}^{\times}$-equivariant, see e.g.
[KKLV89, Proposition 2.4]. Hence, $\mathcal{L}$ has the structure of a graded
$\mathbb{C}[V]$-module. Let $0\in V$ be the fixed point for the
$\mathbb{C}^{\times}$-action and consider the maximal ideal
$\mathfrak{m}_{0}\subset\mathbb{C}[V]$. Since $\mathcal{L}$ is a line bundle,
the fiber $\mathcal{L}_{0}=\mathcal{L}/\mathfrak{m}_{0}\mathcal{L}$ is one-
dimensional. Choose a nonzero element $\overline{s}\in\mathcal{L}_{0}$. This
element lifts to a section $s\in\mathcal{L}$ which generates $\mathcal{L}$ as
a $\mathbb{C}[V]$-module by the graded version of Nakayama’s lemma. The action
map defines a surjection
$\mathbb{C}[V]\stackrel{{\scriptstyle}}{{\to}}\mathcal{L}$, which is
manifestly injective, since $\mathcal{L}$ is torsion-free. ∎

Let $\mathfrak{X}(L)$ denote the character lattice of $L$. If
$\chi\in\mathfrak{X}(L)$, write $\mathcal{L}_{G/P}(\chi)$ for the
$G$-equivariant line bundle on $G/P$ with fiber $\chi$ at $[P]\in G/P$.

######  Proposition 7.1.2.

The following are true:

  * (i)

The map
$\pi^{*}:\operatorname{Pic}(G/P)\rightarrow\operatorname{Pic}(\widetilde{Y})$
is an isomorphism. If $G$ is semisimple and simply connected, there is a
further isomorphism
$\mathcal{L}_{G/P}:\mathfrak{X}(L)\xrightarrow{\sim}\operatorname{Pic}(G/P)$.

  * (ii)

There is a short exact sequence

$0\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\widetilde{Y})\overset{\mathrm{div}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Cl}(\widetilde{Y})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\widetilde{\mathbb{O}}_{L})\stackrel{{\scriptstyle}}{{\to}}0,$

where $\operatorname{Cl}$ denote the class group, and $\mathrm{div}$ is the
natural map from line bundles to divisors.

###### Proof.

First, we prove (ii). Let $C\subset G/P$ be the open Bruhat cell and let
$\widetilde{C}=\pi^{-1}(C)$. There is an $N^{-}$-invariant isomorphism

(7.1.1) $\widetilde{C}\simeq
N^{-}\times\widetilde{X}_{L}\times\mathfrak{p}^{\perp}$

Taking class groups, we get an identification
$\operatorname{Cl}(\widetilde{C})\simeq\operatorname{Cl}(\widetilde{X}_{L})$.
Since
$\operatorname{codim}(\widetilde{X}_{L}-\widetilde{\mathbb{O}}_{L},\widetilde{X}_{L})\geq
2$,
$\operatorname{Cl}(\widetilde{X}_{L})\simeq\operatorname{Cl}(\widetilde{\mathbb{O}}_{L})$
and since $\widetilde{\mathbb{O}}_{L}$ is smooth,
$\operatorname{Cl}(\widetilde{\mathbb{O}}_{L})\simeq\operatorname{Pic}(\widetilde{\mathbb{O}}_{L})$.
Thus,
$\operatorname{Cl}(\widetilde{C})\simeq\operatorname{Pic}(\widetilde{\mathbb{O}}_{L})$.

Since $\widetilde{C}$ is an open subset of $\widetilde{Y}$ there is a
surjective map
$\mathrm{res}:\operatorname{Cl}(\widetilde{Y})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Cl}(\widetilde{C})$.
We will show that the following sequence is exact

(7.1.2)
$0\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\widetilde{Y})\overset{\mathrm{div}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Cl}(\widetilde{Y})\overset{\mathrm{res}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Cl}(\widetilde{C})\stackrel{{\scriptstyle}}{{\to}}0.$

First, note that
$\mathrm{div}:\operatorname{Pic}(\widetilde{Y})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Cl}(\widetilde{Y})$
is injective, since $\widetilde{Y}$ is normal. Next, observe that the
composition

$\operatorname{Pic}(\widetilde{Y})\overset{\mathrm{div}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Cl}(\widetilde{Y})\overset{\mathrm{res}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Cl}(\widetilde{C})$

coincides with the composition

$\operatorname{Pic}(\widetilde{Y})\overset{\mathrm{res}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Pic}(\widetilde{C})\overset{\mathrm{div}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Cl}(\widetilde{C}),$

By (7.1.1),
$\operatorname{Pic}(\widetilde{C})\simeq\operatorname{Pic}(\widetilde{X}_{L})$,
and by Lemma 7.1.1, $\operatorname{Pic}(\widetilde{X}_{L})\simeq 0$. Thus,
$\operatorname{Pic}(\widetilde{C})\simeq 0$ and
$\mathrm{Im}(\mathrm{div})\subseteq\ker(\mathrm{res})$.

Next we prove the reverse inclusion:
$\mathrm{Im}(\mathrm{div})\supseteq\ker(\mathrm{res})$. Let $D=G/P-C$ and
$\widetilde{D}=\pi^{-1}(D)=\widetilde{Y}-\widetilde{C}$. Denote the
irreducible components of $D$ by $D_{1},...,D_{r}$. Note that each $D_{i}$ is
of codimension 1 and the set $\\{D_{1},...,D_{r}\\}$ is parameterized by
simple roots for $\mathfrak{g}$ which are not contained in $\mathfrak{l}$, see
[Bor69, Thm 21.14]. For each $D_{i}$, let $\widetilde{D}_{i}=\pi^{-1}(D_{i})$.
Since $\pi:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}G/P$ is a fiber bundle
with irreducible fibers, $\widetilde{D}_{1},...,\widetilde{D}_{r}$ are the
irreducible components of $\widetilde{D}$. If $[E]\in\ker(\mathrm{res})$, then
$[E]=\sum_{i=1}^{r}c_{i}[\widetilde{D}_{i}]$ for integers $c_{1},...,c_{r}$.
Hence

$[E]=\sum_{i=1}^{r}c_{i}[\widetilde{D}_{i}]=\mathrm{div}\left(\pi^{*}\mathcal{O}(\sum_{i=1}^{r}c_{i}[D_{i}])\right),$

where $\mathcal{O}(\sum_{i=1}^{r}c_{i}[D_{i}]))\in\operatorname{Pic}(G/P)$ is
the line bundle corresponding to the class
$\sum_{i=1}^{r}c_{i}[D_{i}]\in\operatorname{Cl}(G/P)$. In particular,
$\ker(\mathrm{res})\subseteq\mathrm{Im}(\mathrm{div})$. Thus, we have shown
that (7.1.2) is exact. Using the isomorphism
$\operatorname{Cl}(\widetilde{C})\simeq\operatorname{Pic}(\widetilde{\mathbb{O}}_{L})$
described in the first paragraph of the proof, we get a short exact sequence

$0\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\widetilde{Y})\overset{\mathrm{div}}{\stackrel{{\scriptstyle}}{{\to}}}\operatorname{Cl}(\widetilde{Y})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\widetilde{\mathbb{O}}_{L})\stackrel{{\scriptstyle}}{{\to}}0,$

as desired.

Next, we prove (i). For $G$ semisimple, the isomorphism
$\mathcal{L}_{G/P}:\mathfrak{X}(L)\simeq\operatorname{Pic}(G/P)$ is standard,
see e.g. [KKV89, Prop 3.2]. It remains to show that
$\pi^{*}:\operatorname{Pic}(G/P)\xrightarrow{\sim}\operatorname{Pic}(\widetilde{Y})$.

Restricting along the zero section $G/P\subset\widetilde{Y}$ defines a left
inverse for $\pi^{*}$. In particular, $\pi^{*}$ is injective. To prove it is
surjective, choose a line bundle
$\mathcal{L}\in\operatorname{Pic}(\widetilde{Y})$. By the proof of the
inclusion $\mathrm{Im}(\mathrm{div})\supseteq\ker(\mathrm{res})$ above,

$\mathrm{div}(\mathcal{L})=\sum_{i=1}^{r}c_{i}[\widetilde{D}_{i}]=\mathrm{div}\left(\pi^{*}\mathcal{O}(\sum_{i=1}^{r}c_{i}[D_{i}])\right).$

Since $\mathrm{div}$ is injective, this implies
$\mathcal{L}\simeq\pi^{*}\mathcal{O}(\sum_{i=1}^{r}c_{i}[D_{i}])$. Thus,
$\pi^{*}$ is surjective, as desired. This completes the proof. ∎

We say that a line bundle $\mathcal{L}\in\operatorname{Pic}(\widetilde{Y})$ is
_relatively ample_ if it is ample with respect to the projective morphism
$\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$. Write
$\operatorname{Pic}^{a}(\widetilde{Y})\subset\operatorname{Pic}(\widetilde{Y})$
for the semigroup of relatively ample line bundles and
$\mathfrak{X}(L)^{>0}\subset\mathfrak{X}(L)$ for the semigroup of (strictly)
$P$-dominant weights.

######  Proposition 7.1.3.

The isomorphism
$\pi^{*}:\operatorname{Pic}(G/P)\simeq\operatorname{Pic}(\widetilde{Y})$ of
Proposition 7.1.2 restricts to a semigroup isomorphism
$\operatorname{Pic}^{a}(G/P)\simeq\operatorname{Pic}^{a}(\widetilde{Y})$. If
$G$ is semisimple and simply connected, there is a further isomorphism
$\mathcal{L}_{G/P}:\mathfrak{X}(L)^{>0}\simeq\operatorname{Pic}^{a}(G/P)$.

###### Proof.

First, observe that there is a finite morphism

$\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}G\times^{P}\widetilde{X}\simeq
G/P\times\widetilde{X},$

and $\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$
coincides with the composition
$\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}G/P\times\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$
(where the second map is the projection). If
$\mathcal{L}\in\operatorname{Pic}(G/P)$ is ample, then $\pi^{*}\mathcal{L}$
coincides with the pullback to $\widetilde{Y}$ of the relatively ample line
bundle
$\mathcal{L}\otimes\mathcal{O}_{\widetilde{X}}\in\operatorname{Pic}(G/P\times\widetilde{X})$.
Since $\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}G/P\times\widetilde{X}$ is
finite, this pullback is relatively ample.

Conversely, suppose that $\pi^{*}\mathcal{L}$ is relatively ample. The zero
section $G/P\subset\widetilde{Y}$ is the preimage under
$\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$ of
$0\in\widetilde{X}$. Hence, $\mathcal{L}\simeq(\pi^{*}\mathcal{L})|_{G/P}$ is
ample.

Now suppose $G$ is semisimple and simply connected. It is a standard fact that
for a character $\chi\in\mathfrak{X}(L)$, the line bundle
$\mathcal{L}_{G/P}(\chi)$ is ample if and only if $\chi$ is strictly dominant.
Thus,
$\mathcal{L}_{G/P}:\mathfrak{X}(L)\xrightarrow{\sim}\operatorname{Pic}(G/P)$
restricts to an isomorphism
$\mathfrak{X}(L)^{>0}\xrightarrow{\sim}\operatorname{Pic}^{a}(G/P)$, as
asserted. ∎

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

Fix the notation of Section 7.1, i.e. $P=LN$, $\widetilde{\mathbb{O}}_{L}$,
$\widetilde{X}_{L}$,
$\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$,
$\pi:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}G/P$, and so on. For the
remainder of this subsection, we will also assume:

$\widetilde{\mathbb{O}}_{L}\text{ is birationally rigid}.$

Under this assumption, we will see below that the partial resolution
$\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$ is a
$\mathbb{Q}$-factorial terminalization. In the case when
$\widetilde{\mathbb{O}}$ is an orbit, this was proved by the first-named
author (see [Los16, Cor 4.6]). For universal covers in classical types, a
similar result was obtained by Namikawa (see [Nam20]). The result below is due
to the third-named author (see [Mat20, Cor 4.3]).

######  Theorem 7.2.1.

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

Theorem 7.2.1 will allow us to provide purely Lie-theoretic descriptions of
the Namikawa space $\mathfrak{P}^{\widetilde{X}}$ and Namikawa Weyl group
$W^{\widetilde{X}}$ attached to $\widetilde{X}$ (see Section 4.5 for
definitions). These descriptions are as follows. Let $\underline{\pi}$ denote
the restriction of $\pi:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}G/P$ to
$\widetilde{Y}^{\mathrm{reg}}$, and consider the pullback map on cohomology
$\underline{\pi}^{*}:H^{2}(G/P,\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})=\mathfrak{P}^{\widetilde{X}}$.
Note that $H^{2}(G/P,\mathbb{C})$ is identified with
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$. Consider the
composition

(7.2.1) $\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\simeq
H^{2}(G/P,\mathbb{C})\overset{\underline{\pi}^{*}}{\stackrel{{\scriptstyle}}{{\to}}}H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})=\mathfrak{P}^{\widetilde{X}},$

We will see below that this map is an isomorphism. To describe
$W^{\widetilde{X}}$, recall that $G$ acts on pairs of the form
$(L,\widetilde{\mathbb{O}}_{L})$, see the remarks preceding Proposition 2.4.1.
Consider the stabilizer $\mathrm{Stab}_{G}(L,\widetilde{\mathbb{O}}_{L})$ of
$(L,\widetilde{\mathbb{O}}_{L})$ under this $G$-action. This group can be
described explictly as follows. Choose an element
$x\in\widetilde{\mathbb{O}}_{L}$. Define the normalizer in $G$ of $L$,
$\mathbb{O}_{L}$, and $L_{x}$

$N_{G}(L,\mathbb{O}_{L},L_{x}):=\\{g\in N_{G}(L)\cap
N_{G}(L_{x})\mid\operatorname{Ad}(g)\mathbb{O}_{L}=\mathbb{O}_{L}\\}\subset G$

and the normalizer in $G$ of $L$ and $\widetilde{\mathbb{O}}_{L}$

(7.2.2)
$N_{G}(L,\widetilde{\mathbb{O}}_{L}):=LN_{G}(L,\mathbb{O}_{L},L_{x})\subset
G.$

It is easy to see that
$N_{G}(L,\widetilde{\mathbb{O}}_{L})=\mathrm{Stab}_{G}(L,\widetilde{\mathbb{O}}_{L})$.
In particular, $N_{G}(L,\widetilde{\mathbb{O}}_{L})$ depends only on $L$ and
$\widetilde{\mathbb{O}}_{L}$ (and not on $x\in\widetilde{\mathbb{O}}_{L}$).
Furthermore, $L$ coincides with the identity component of
$N_{G}(L,\widetilde{\mathbb{O}}_{L})$. There is a surjective homomorphism

$\kappa:N_{G}(L,\widetilde{\mathbb{O}}_{L})/L\twoheadrightarrow\operatorname{Aut}_{\mathbb{O}}(\widetilde{\mathbb{O}}).$

For nilpotent orbits, $\kappa$ was defined in Step (ii) of the proof of
[Los16, Prop 4.7]. For nilpotent covers, the definition is analogous.

######  Proposition 7.2.2.

The following are true:

  * (i)

The map
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\stackrel{{\scriptstyle}}{{\to}}\mathfrak{P}^{\widetilde{X}}$
is an isomorphism.

  * (ii)

$W^{\widetilde{X}}$ is identified with the normal subgroup

$\ker{\kappa}\subset N_{G}(L,\widetilde{\mathbb{O}}_{L})/L$

with its canonical action on
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$.

  * (iii)

The universal graded deformation $\widetilde{Y}_{\mathrm{univ}}$ (see
Proposition 4.7.1) is identified with the Poisson scheme
$G\times^{P}(\mathfrak{X}(\mathfrak{l})\times\widetilde{X}_{L}\times\mathfrak{p}^{\perp})$
over $\mathfrak{X}(\mathfrak{l})$ for a uniquely determined isomorphism
$\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})$.
This isomorphism coincides with $\eta$.

###### Proof.

(i) and (ii) are proved for nilpotent orbits in [Los16, Prop 4.7]. The proof
there can be generalized to arbitrary nilpotent covers. The isomorphism
$G\times^{P}(\mathfrak{X}(\mathfrak{l})\times\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\xrightarrow{\sim}\widetilde{Y}_{\mathrm{univ}}$
is proved in the same way as [Los16, Prop 4.7(2)], see also [Nam20a, Thm 9].
It remains to check that the isomorphism
$\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}\mathfrak{P}^{\widetilde{X}}$ in
Proposition 4.7.1 coincides with the map $\eta$ defined above. The variety
$\widetilde{X}^{\mathrm{reg}}=G\times^{P}(\widetilde{X}^{\mathrm{reg}}\times\mathfrak{p}^{\perp})$
is obtained as the Hamiltonian reduction of the product variety
$T^{*}G\times\widetilde{X}^{\mathrm{reg}}$ under the action of $P$ and the
deformation
$G\times^{P}(\mathfrak{X}(\mathfrak{l})\times\widetilde{X}_{L}\times\mathfrak{p}^{\perp})$
is realized as the universal Hamiltonian reduction for this action, i.e. as
$\mu_{P}^{-1}(\mathfrak{X}(\mathfrak{p}))/P$ (where $\mu_{P}$ is the moment
map). It remains to show that $\eta$ coincides with the period map from Remark
4.7.2. This is a special case of [Los12, Proposition 3.2.1]. ∎

######  Remark 7.2.3.

For universal covers (and under some additional mild conditions), an
equivalent description of $W^{\widetilde{X}}$ was obtained by Namikawa, see
the discussion following [Nam20a, Prop 5].

Next, we explain the classification of $\mathbb{Q}$-factorial terminalizations
of $\widetilde{X}$. Recall the subset
$\mathfrak{P}^{\mathrm{sing}}\subset\mathfrak{P}$ defined in Section 4.9. Let
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])_{\mathbb{R}}^{\geq
0}\subset\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])_{\mathbb{R}}$
denote the cone of $\mathfrak{p}$-dominant weights.

######  Lemma 7.2.4.

The following are true:

  * (i)

Under the identification
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\simeq\mathfrak{P}$,
the subset $\mathfrak{P}^{\mathrm{sing}}$ corresponds to the subset

$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])^{\mathrm{sing}}:=\\{\lambda\in\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\mid
L\subsetneq G_{\lambda}\\}.$

  * (ii)

Under the identification
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])_{\mathbb{R}}\simeq\mathfrak{P}_{\mathbb{R}}$,
the ample cone
$\mathrm{Amp}(G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp}))$
(defined by (4.9.2)) corresponds to the subset
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])_{\mathbb{R}}^{\geq
0}$.

  * (iii)

Every $\mathbb{Q}$-terminalization of $\widetilde{X}$ is of the form
$G\times^{P_{1}}(\widetilde{X}_{L}\times\mathfrak{p}_{1}^{\perp})$, where
$P_{1}$ is a parabolic subgroup of $G$ with Levi subgroup $L$. Two parabolic
subgroups give rise to the same $\mathbb{Q}$-factorial terminalization if and
only if their dominant cones are $W^{\widetilde{X}}$-conjugate.

###### Proof.

By definition $\mathfrak{P}^{\mathrm{sing}}$ consists of all parameters
$\lambda\in\mathfrak{P}$ for which $\widetilde{Y}_{\lambda}$ is not affine.
Viewing $\lambda$ as an element of $\mathfrak{X}(\mathfrak{l})$, we have
$\widetilde{Y}_{\lambda}=G\times^{P}(\\{\lambda\\}\times\mathfrak{p}^{\perp}\times\widetilde{X}_{L})$.
If $G_{\lambda}=L$, this variety is affine. If $G_{\lambda}\neq L$, then
$\widetilde{Y}_{\lambda}$ contains the projective variety
$G_{\lambda}/(G_{\lambda}\cap P)$ and is therefore not affine. This completes
the proof of (i). (ii) follows immediately from Proposition 7.1.3. Cones of
the form $\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])^{\geq
0}_{\mathbb{R}}$ partition $\mathfrak{P}_{\mathbb{R}}$. Now (iii) follows from
Theorem 4.9.1. ∎

Our definition of $\eta$ depends on the choice of $P$. The next result is
essential for our computation of $\eta$.

######  Proposition 7.2.5.

For two choices of $P$, the corresponding maps
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\xrightarrow{\sim}\mathfrak{P}^{\widetilde{X}}$
coincide up to the action of $W^{\widetilde{X}}$ on the target.

###### Proof.

Replacing $G$ with its commutator subgroup, we can assume that $G$ is
semisimple. The proof has several steps.

Step 1. Let $P_{1},P_{2}$ be parabolics with Levi factor $L$ and write
$\eta_{1},\eta_{2}$ for the corresponding isomorphisms. We identify
$\mathbb{C}[G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})]$ with
$\mathbb{C}[\widetilde{X}]$ via the pullback map under the terminalization
morphism
$G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\twoheadrightarrow\widetilde{X}$.

Step 2. Choose a Zariski-generic element
$\lambda\in\mathfrak{X}(\mathfrak{l})$. Thanks to (iii) of Theorem 4.7.5, it
suffices to show that $\mathbb{C}[\widetilde{X}_{\eta_{1}(\lambda)}]$ and
$\mathbb{C}[\widetilde{X}_{\eta_{2}(\lambda)}]$ are isomorphic as filtered
Poisson deformations of $\mathbb{C}[\widetilde{X}]$.

Step 3. Let $\ell=\mathbb{C}\lambda$. Consider the variety
$\widetilde{Y}_{\ell,i}:=G\times^{P_{i}}(\ell\times\widetilde{X}_{L}\times\mathfrak{p}_{i}^{\perp})$
and the natural map
$\rho_{i}:\widetilde{Y}_{\ell,i}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{g}^{*}$.
Let $\ell^{\times}:=\ell-\\{0\\}$ and
$\widetilde{Y}^{\times}_{\ell,i}:=\ell^{\times}\times_{\ell}\widetilde{Y}_{\ell,i}$.
There is a $\mathbb{C}^{\times}\times G$-equivariant identification
$\widetilde{Y}^{\times}_{\ell,i}\simeq
G\times^{L}(\ell^{\times}\times\widetilde{X}_{L})$ intertwining the maps to
$\mathfrak{g}^{*}$.

Step 4. The pullback $\widetilde{X}_{\eta_{i}(\ell)}$ of the universal
deformation $\widetilde{X}_{\mathrm{univ}}$ (see Proposition 4.7.7) to
$\eta_{i}(\ell)$ coincides with
$\operatorname{Spec}(\mathbb{C}[\eta_{i}(\ell)\times_{\mathfrak{P}}\widetilde{Y}_{\mathrm{univ}}])$
as a Poisson scheme over $\ell$ with $\mathbb{C}^{\times}$-action. So there is
the Stein factorization
$Y^{\times}_{\ell,i}\rightarrow\widetilde{X}_{\eta_{i}(\ell)}\rightarrow\mathfrak{g}^{*}$.
This gives rise to a $\mathbb{C}^{\times}$-equivariant isomorphism of Poisson
$\ell$-schemes
$\widetilde{X}_{\eta_{1}(\ell)}\xrightarrow{\sim}\widetilde{X}_{\eta_{2}(\ell)}$.
By the construction in Step 1, this isomorphism gives the identity on
$\mathbb{C}[\widetilde{X}]$, i.e. is an isomorphism of graded Poisson
deformations. It follows that $\mathbb{C}[\widetilde{X}_{\eta_{1}(\lambda)}]$
and $\mathbb{C}[\widetilde{X}_{\eta_{2}(\lambda)}]$ are isomorphic as filtered
Poisson deformations, completing the proof. ∎

By Proposition 4.5.1, $\mathfrak{P}^{\widetilde{X}}$ decomposes into partial
Namikawa spaces

(7.2.3)
$\mathfrak{P}^{\widetilde{X}}\simeq\mathfrak{P}^{\widetilde{X}}_{0}\oplus\bigoplus_{k=1}^{t}\mathfrak{P}^{\widetilde{X}}_{k}.$

Here,
$\mathfrak{P}_{0}^{\widetilde{X}}=H^{2}(\widetilde{X}^{\mathrm{reg}},\mathbb{C})$
and $\mathfrak{P}_{k}^{\widetilde{X}}$ is the space of
$\pi_{1}(\mathfrak{L}_{k})$-invariants in the dual Cartan subalgebra
$\mathfrak{h}_{k}^{*}$ associated to the Kleinian singularity $\Sigma_{k}$.

######  Remark 7.2.6.

We note that Proposition 7.2.2 together with (7.2.3) provides a more
conceptual proof of Lemma 6.5.2, at least in exceptional types. Let
$\mathbb{O}$ be a nilpotent orbit in a simple exceptional Lie algebra and
suppose $X=\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ contains a codimension
2 leaf $\mathfrak{L}_{k}\subset X$ such that the corresponding singularity is
of type $E_{8}$. Note that $E_{8}$ has no nontrivial diagram automorphisms.
Hence, $\dim\mathfrak{P}^{X}_{k}=\dim\mathfrak{h}_{k}^{*}=8$. Choose a Levi
subgroup $L\subset G$ and a birationally rigid $L$-orbit $\mathbb{O}_{L}$ such
that $\mathbb{O}=\mathrm{Bind}^{G}_{L}\mathbb{O}_{L}$. Then by Proposition
7.2.2 and the decomposition (7.2.3), we have
$\dim(\mathfrak{X}(\mathfrak{l}))=\dim\mathfrak{P}^{X}\geq\dim\mathfrak{P}^{X}_{k}=8$.
Thus, $G=E_{8}$ and $(L,\mathbb{O}_{L})=(T,\\{0\\})$. That is, $\mathbb{O}$ is
the principal nilpotent orbit in $E_{8}$.

For our computation of $\eta$, we will need Lie-theoretic descriptions of the
spaces $\mathfrak{P}_{k}^{\widetilde{X}}$ analogous to Proposition 7.2.2.
First, we will describe the space $\mathfrak{P}_{0}^{\widetilde{X}}$. We will
describe the spaces $\mathfrak{P}_{k}^{\widetilde{X}}$, $k\geq 1$, in Section
7.5, under some additional restrictions. For the following lemma, suppose $G$
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.
Note that 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 7.2.7.

Suppose that $G$ is semisimple and simply connected. The following are true:

  * (i)

There are natural identifications
$\operatorname{Pic}(\widetilde{X}^{\mathrm{reg}})\xrightarrow{\sim}\operatorname{Pic}(\widetilde{\mathbb{O}})$
and
$\operatorname{Pic}(\widetilde{\mathbb{O}})\xrightarrow{\sim}\mathfrak{X}(R_{x})$.

  * (ii)

There are isomorphisms

$\operatorname{Pic}(\widetilde{\mathbb{O}})\otimes_{\mathbb{Z}}\mathbb{C}\xrightarrow{\sim}H^{2}(\widetilde{\mathbb{O}},\mathbb{C})\xrightarrow{\sim}\mathfrak{X}(R_{x})\otimes_{\mathbb{Z}}\mathbb{C}=\mathfrak{X}(\mathfrak{r})^{\pi_{1}(\widetilde{\mathbb{O}})},$

where the first map is $c_{1}$.

  * (iii)

Restriction along the embedding
$\widetilde{\mathbb{O}}\hookrightarrow\widetilde{X}^{\mathrm{reg}}$ induces an
isomorphism
$H^{2}(\widetilde{X}^{\mathrm{reg}},\mathbb{C})\xrightarrow{\sim}H^{2}(\widetilde{\mathbb{O}},\mathbb{C})$.

###### Proof.

(i): the first isomorphism follows from the inequality
$\operatorname{codim}(\widetilde{X}^{\mathrm{reg}}-\widetilde{\mathbb{O}},\widetilde{X}^{\mathrm{reg}})\geqslant
2$ and the fact that $\widetilde{X}^{\mathrm{reg}}$ is smooth. The second
isomorphism is standard, see e.g. [KKV89, Proposition 3.2].

(ii): The isomorphism
$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})\xrightarrow{\sim}\mathfrak{X}(\mathfrak{r})^{\pi_{1}(\widetilde{\mathbb{O}})}$
is also standard, see e.g. [BC12, Theorem 3.3]. Under the identifications
$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})\xrightarrow{\sim}\mathfrak{X}(\mathfrak{r})^{\pi_{1}(\widetilde{\mathbb{O}})}$
and
$\operatorname{Pic}(\widetilde{\mathbb{O}})\xrightarrow{\sim}\mathfrak{X}(R_{x})$,
the Chern class map
$c_{1}:\operatorname{Pic}(\widetilde{\mathbb{O}})\rightarrow
H^{2}(\widetilde{\mathbb{O}},\mathbb{C})$ coincides with the natural map
$\mathfrak{X}(R_{x})\rightarrow\mathfrak{X}(R_{x})\otimes_{\mathbb{Z}}\mathbb{C}=\mathfrak{X}(\mathfrak{r})^{\pi_{1}(\widetilde{\mathbb{O}})}$.
(ii) follows at once.

(iii): For $i\geq 0$, let $\widetilde{X}^{i}$ denote the union of
$\widetilde{\mathbb{O}}\subset\widetilde{X}$ and all $G$-orbits in
$\widetilde{X}$ of codimension $\leq 2i$. Since $G$ acts on $\widetilde{X}$
with finitely many orbits, $\widetilde{X}=\widetilde{X}^{N}$ for some $N\geq
1$. Thus, it suffices to show that
$H^{2}(\widetilde{X}^{i+1},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(\widetilde{X}^{i},\mathbb{C})$
is an isomorphism for all $i$. Let $Y_{1},...,Y_{m}\subset\widetilde{X}^{i+1}$
denote the $G$-orbits of codimension $2i$, so that the pullback map
$\widetilde{X}^{i+1}=\widetilde{X}^{i}\cup\bigcup_{k=1}^{m}Y_{k}$. For each
$k$, choose a tubular neighborhood $U_{k}\subset\widetilde{X}^{i+1}$ of
$Y_{k}$ and set $U_{k}^{\times}:=U_{k}\cap\widetilde{X}^{i}$. There is a
Mayer-Vietoris sequence

$\displaystyle\ldots\stackrel{{\scriptstyle}}{{\to}}H^{1}(\widetilde{X}^{i},\mathbb{C})\oplus\bigoplus
H^{1}(U_{k},\mathbb{C})$
$\displaystyle\stackrel{{\scriptstyle}}{{\to}}\bigoplus
H^{1}(U_{k}^{\times},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{2}(\widetilde{X}^{i+1},\mathbb{C})$
$\displaystyle\stackrel{{\scriptstyle}}{{\to}}H^{2}(\widetilde{X}^{i},\mathbb{C})\oplus\bigoplus
H^{2}(U_{k},\mathbb{C})$
$\displaystyle\stackrel{{\scriptstyle}}{{\to}}\bigoplus
H^{2}(U_{k}^{\times},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}\ldots$

Note that $U_{k}$ and $U_{k}^{\times}$ are fibrations over $Y_{k}$ with fibers
$D_{2i}$ and $D_{2i}^{\times}$ respectively, where $D_{2i}$ is the
$2i$-dimensional disc and $D_{2}^{\times}$ is its puncture. Since

$H^{1}(D_{2i},\mathbb{Q})=H^{2}(D_{2i},\mathbb{C})=H^{1}(D_{2i}^{\times},\mathbb{C})=H^{2}(D_{2i}^{\times},\mathbb{C})=0,$

the maps

$H^{j}(Y_{k},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{j}(U_{k},\mathbb{C}),\quad
H^{j}(Y_{k},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{j}(U_{k}^{\times},\mathbb{C}),\qquad
j=1,2$

are isomorphisms. It follows that the restriction maps
$H^{j}(U_{i},\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}H^{j}(U_{i}^{\times},\mathbb{C})$
are isomorphisms, and therefore
$H^{2}(\widetilde{X}^{i+1},\mathbb{C})\xrightarrow{\sim}H^{2}(\widetilde{X}^{i},\mathbb{C})$,
as asserted. ∎

###  7.3. Parabolic induction of Hamiltonian quantizations

In this section, we will define the notion of parabolic induction for
Hamiltonian quantizations of nilpotent covers.

Choose a parabolic subgroup $P\subset G$ with Levi decomposition $P=LN$. Let
$\widetilde{\mathbb{O}}_{L}$ be an $L$-equivariant nilpotent cover and let
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$.
Consider the partial resolution
$\rho:\widetilde{Y}:=G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$
defined in Section 7.1.

Starting with a Hamiltonian quantization $\mathcal{A}_{L}$ of
$\mathbb{C}[\widetilde{\mathbb{O}}_{L}]$, we will produce a Hamiltonian
quantization $\operatorname{Ind}^{G}_{L}\mathcal{A}_{L}$ of
$\mathbb{C}[\widetilde{\mathbb{O}}]$, thus defining a map

$\operatorname{Ind}^{G}_{L}:\mathrm{Quant}^{L}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])\stackrel{{\scriptstyle}}{{\to}}\mathrm{Quant}^{G}(\mathbb{C}[\widetilde{\mathbb{O}}])$

called _parabolic induction_.

Our construction requires a bit of additional notation.

  * •

Let
$\mu_{L}:\widetilde{X}_{L}\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}_{L}\subset\mathfrak{l}^{*}$
denote the (classical) moment map for the right $L$-action on
$\widetilde{X}_{L}$.

  * •

Let $\Phi_{L}:\mathfrak{l}\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{L}$
denote the (quantum) co-moment map for the right $L$-action on
$\mathcal{A}_{L}$.

  * •

Let $\mathcal{D}_{L}$ denote the microlocalization of $\mathcal{A}_{L}$ over
$\widetilde{X}_{L}$.

  * •

Let $\mathfrak{D}_{G/N}$ denote the sheaf of differential operators on $G/N$
(regarded as a sheaf in the conical topology on $T^{*}(G/N)$). Since $L$
normalizes $N$, there is a right $L$-action on $G/N$ and hence on $T^{*}(G/N)$
and $\mathfrak{D}_{G/N}$.

  * •

Let

$\mu_{G/N}:T^{*}(G/N)\simeq
G\times^{N}(\mathfrak{g}/\mathfrak{n})^{*}\stackrel{{\scriptstyle}}{{\to}}(\mathfrak{g}/\mathfrak{n})^{*}\stackrel{{\scriptstyle}}{{\to}}(\mathfrak{q}/\mathfrak{n})^{*}\simeq\mathfrak{l}^{*}$

denote the (classical) moment map for the right $L$-action on $T^{*}(G/N)$.

  * •

Let
$\Phi^{0}_{G/N}:\mathfrak{l}\stackrel{{\scriptstyle}}{{\to}}\Gamma(T^{*}(G/N),\mathfrak{D}_{G/N})$
denote the usual (quantum) co-moment map for the right $L$-action on
$\mathfrak{D}_{G/N}$, i.e. $\Phi^{0}_{G/N}$ takes $\xi\in\mathfrak{l}$ to the
right-invariant vector field on $G/N$ determined by $\xi$.

  * •

Let $\Phi_{G/N}:=\Phi_{G/N}^{0}+\rho_{\mathfrak{n}}$, where
$\rho_{\mathfrak{n}}$ denotes one-dimensional representation of $\mathfrak{l}$
defined in (3.4.2).

Consider the product variety $T^{*}(G/N)\times\widetilde{X}_{L}$ with the
diagonal left $L$-action

$l(d,x):=(dl^{-1},lx),\qquad l\in L,\ d\in T^{*}(G/N),\
x\in\widetilde{X}_{L}.$

This action has a moment map

$\mu:=-\mu_{G/N}+\mu_{\widetilde{X}_{L}}:T^{*}(G/N)\times\widetilde{X}_{L}\stackrel{{\scriptstyle}}{{\to}}\mathfrak{l}^{*}.$

Note that

$\mu^{-1}(0)\simeq
G\times^{N}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\subset
G\times^{N}(\widetilde{X}_{L}\times\mathfrak{n}^{\perp})\simeq
T^{*}(G/N)\times\widetilde{X}_{L}.$

and therefore

$\mu^{-1}(0)/L\simeq
G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})=\widetilde{Y}.$

Form the completed tensor product

$\mathcal{D}^{\prime}:=\mathfrak{D}_{G/N}\ \widehat{\otimes}\
\mathcal{D}_{L}.$

Here, the completion is defined with respect to the filtrations on the
factors. Note that $\mathcal{D}^{\prime}$ is an $L$-equivariant filtered
quantization of $T^{*}(G/N)\times\widetilde{X}_{L}$ with a naturally defined
quantum co-moment map

$\Phi:=-\Phi_{G/N}\otimes
1+1\otimes\Phi_{L}:\mathfrak{l}\stackrel{{\scriptstyle}}{{\to}}\Gamma(T^{*}(G/N)\times\widetilde{X}_{L},\mathcal{D}^{\prime}).$

Let $\mathrm{Ind}^{G}_{L}\mathcal{D}_{L}$ denote the quantum Hamiltonian
reduction of $\mathcal{D}^{\prime}$ at $0\in\mathfrak{l}^{*}$. More precisely,
form the left ideal

$I:=\mathcal{D}^{\prime}\
\mathrm{Span}\\{\Phi(\xi)\mid\xi\in\mathfrak{l}\\}\subset\mathcal{D}^{\prime}.$

The quotient $\mathcal{D}^{\prime}/I$ is an $L$-equivariant sheaf of
$\mathcal{D}^{\prime}$-modules on $T^{*}(G/N)\times\widetilde{X}_{L}$, set-
theoretically supported on $\mu^{-1}(0)$. Consider the quotient morphism
$q:\mu^{-1}(0)\stackrel{{\scriptstyle}}{{\to}}\widetilde{Y}$ and set

$\operatorname{Ind}^{G}_{L}\mathcal{D}_{L}:=\left(q_{*}[\mathcal{D}^{\prime}/I]\right)^{L}.$

Since the $L$-action on $\mu^{-1}(0)$ is free,
$\operatorname{Ind}^{G}_{L}\mathcal{D}_{L}$ is a filtered quantization of
$\widetilde{Y}$, see e.g. [Los12, Lem 3.3.1]. It has a left $G$-action (coming
from the left $G$-action on $\mathfrak{D}_{G/N}$) and a quantum co-moment map
(coming from the quantum co-moment map $\Phi_{G,G/N}\otimes
1:\mathfrak{g}\stackrel{{\scriptstyle}}{{\to}}\Gamma(\mathfrak{D}_{G/N}\widehat{\otimes}\mathcal{D}_{L})$,
where $\Phi_{G,G/N}:\mathfrak{g}\rightarrow\Gamma(\mathfrak{D}_{G/N})$ is the
quantum comoment map for the $G$-action, taking $\xi\in\mathfrak{g}$ to the
corresponding vector field on $G/N$).

Finally, define

$\operatorname{Ind}^{G}_{L}\mathcal{A}_{L}:=\Gamma(\widetilde{Y},\operatorname{Ind}^{G}_{L}\mathcal{D}_{L}).$

We now show that $\operatorname{Ind}^{G}_{L}\mathcal{A}_{L}$ is a quantization
of $\mathbb{C}[\widetilde{\mathbb{O}}]=\mathbb{C}[\widetilde{X}]$. The
following lemma is standard.

######  Lemma 7.3.1.

Let $X_{1},X_{2}$ be normal graded Poisson varieties and let
$\rho:X_{1}\rightarrow X_{2}$ be a $\mathbb{C}^{\times}$-equivariant Poisson
morphism such that
$\mathcal{O}_{X_{2}}\xrightarrow{\sim}\rho_{*}\mathcal{O}_{X_{1}}$ and
$R^{1}\rho_{*}\mathcal{O}_{X_{1}}=0$. Then for any filtered quantization
$\mathcal{D}_{1}$ of $X_{1}$, $\rho_{*}\mathcal{D}_{1}$ is a filtered
quantization of $X_{2}$.

We provide a proof for the reader’s convenience.

###### Proof.

The proof reduces to the case when $X_{2}$ is affine. The algebra
$\Gamma(X_{1},\mathcal{D}_{1})$ comes with a complete and separated filtration
induced from the filtration on $\mathcal{D}_{1}$. We wish to show that the
embedding
$\operatorname{gr}\Gamma(\mathcal{D}_{1})\hookrightarrow\mathbb{C}[X_{1}](=\mathbb{C}[X_{2}])$
is an isomorphism or, equivalently, that every homogeneous element lies in the
image. Cover $X_{1}$ by $\mathbb{C}^{\times}$-stable open affine subvarieties
$U_{1},\ldots,U_{k}$. For a homogeneous element $f\in\mathbb{C}[X_{2}]$ of
degree $d$ we can find elements
$\widetilde{f}_{i}\in\Gamma(U_{i},\mathcal{D}_{1,\leqslant d})$ whose
principal symbols satisfy
$\operatorname{gr}\widetilde{f}_{i}=\rho^{*}(f)|_{U_{i}}$. The elements
$\widetilde{f}_{i}-\widetilde{f}_{j}$ lie in $\Gamma(U_{i}\cap
U_{j},\mathcal{D}_{1,\leqslant d-1})$. Let $f_{ij}\in\mathbb{C}[U_{i}\cap
U_{j}]$ denote the degree $d-1$ component of
$\widetilde{f}_{i}-\widetilde{f}_{j}$. It is a 1-cocycle. Since
$H^{1}(X_{1},\mathcal{O}_{X_{1}})=0$, it is a coboundary: we can find degree
$d-1$ elements $g_{i}\in\mathbb{C}[U_{i}]$ with $f_{ij}=g_{i}-g_{j}$. Lift
$g_{i}$ to an element
$\widetilde{g}_{i}\in\Gamma(U_{i},\mathcal{D}_{1,\leqslant d-1})$. Replacing
$\widetilde{f}_{i}$ with $\widetilde{f}_{i}-\widetilde{g}_{i}$ if necessary we
can arrange so that the elements $\widetilde{f}_{i}-\widetilde{f}_{j}$ live in
degree $\leqslant d-2$. We can iterate this procedure, each time reducing the
degree by $1$. This process converges because the filtration on
$\mathcal{D}_{1}$ is complete and separated. Thus we can assume that
$\widetilde{f}_{i}-\widetilde{f}_{j}=0$, i.e. that the elements
$\widetilde{f}_{i}$ glue to $\widetilde{f}\in\Gamma(\mathcal{D}_{1})$. The
principal symbol of this element is $f$ and we are done. ∎

By Lemma 4.3.2, we have
$\mathcal{O}_{\widetilde{X}}\xrightarrow{\sim}\rho_{*}\mathcal{O}_{\widetilde{Y}}$
and $R^{i}\rho_{*}\mathcal{O}_{\widetilde{Y}}=0$ for $i>0$. Hence by Lemma
7.3.1, the filtered algebra $\operatorname{Ind}^{G}_{L}\mathcal{A}_{L}$ is a
quantization of $\mathbb{C}[\widetilde{\mathbb{O}}]$. Since
$\operatorname{Ind}^{G}_{L}\mathcal{D}_{L}$ has a Hamiltonian $G$-action, so
does $\operatorname{Ind}^{G}_{L}\mathcal{A}_{L}$. Thus, we have defined the
desired map
$\operatorname{Ind}^{G}_{L}:\mathrm{Quant}^{L}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])\stackrel{{\scriptstyle}}{{\to}}\mathrm{Quant}^{G}(\mathbb{C}[\widetilde{\mathbb{O}}])$.

Next, we will show that parabolic induction of Hamiltonian quantizations is
transitive. Choose a parabolic subgroup $Q=MU\subset G$ such that $P\subset Q$
and $L\subset M$. Let
$\widetilde{\mathbb{O}}_{M}=\mathrm{Bind}^{M}_{L}\widetilde{\mathbb{O}}_{L}$
and
$\widetilde{X}_{M}=\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{M}])$.

######  Lemma 7.3.2.

Let $\mathcal{A}_{L}$ be a Hamiltonian quantization of
$\mathbb{C}[\widetilde{\mathbb{O}}_{L}]$. Then there is an isomorphism of
Hamiltonian quantizations

$\operatorname{Ind}^{G}_{L}\mathcal{A}_{L}\simeq\operatorname{Ind}^{G}_{M}(\operatorname{Ind}^{M}_{L}\mathcal{A}_{L}).$

###### Proof.

It is clear from the construction that the central part of the co-moment map
is preserved under induction. Thus, we can reduce to the case when
$\mathfrak{g}$ is semisimple. In this case, the Hamiltonian structure is
determined uniquely, see Lemma 4.11.2. So it suffices to exhibit an
isomorphism of filtered quantizations.

Consider the partial resolution
$\overline{\rho}:\widetilde{Z}:=G\times^{Q}(\widetilde{X}_{M}\times\mathfrak{q}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$.
Note that $\rho:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$
factors through a proper birational map
$\rho^{\prime}:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{Z}$. By
the transitivity of pushforwards, it suffices to show that

(7.3.1)
$\rho^{\prime}_{*}(\operatorname{Ind}^{G}_{L}\mathcal{D}_{L})\simeq\operatorname{Ind}^{G}_{M}\mathcal{D}_{M}.$

as filtered quantizations. Let $\rho_{M}:\widetilde{Y}_{M}=M\times^{M\cap
P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}_{M}$
denote the partial resolution. Consider the variety
$T^{*}(G/U)\times\widetilde{Y}_{M}$ and the filtered quantization

(7.3.2) $\mathfrak{D}_{G/U}\ \widehat{\otimes}\
\operatorname{Ind}_{L}^{M}\mathcal{D}_{L}.$

The quantization $\mathfrak{D}_{G/U}\widehat{\otimes}\mathcal{D}_{M}$
appearing in the construction of $\operatorname{Ind}^{G}_{M}\mathcal{D}_{M}$
is the pushforward of (7.3.2) under
$\operatorname{id}_{T^{*}(G/U)}\times\rho_{M}$. On the other hand, (7.3.2) is
the reduction of

(7.3.3) $\mathfrak{D}_{G/U}\ \widehat{\otimes}\ \mathfrak{D}_{M/M\cap N}\
\widehat{\otimes}\ \mathcal{D}_{L}$

under the diagonal action of $L$ on the second and third factors.

Note that the $M$-action on (7.3.3) is trivial on the third factor. Thus, if
we reduce (7.3.3) under the $M$-action, we recover the filtered quantization
$\mathfrak{D}_{G/N}\ \widehat{\otimes}\ \mathcal{D}_{L}$. The quantum co-
moment map for the $L$-action on $\mathfrak{D}_{G/N}\ \widehat{\otimes}\
\mathcal{D}_{L}$ is induced from the quantum co-moment map on (7.3.3), and is
therefore shifted by the character
$\rho_{\mathfrak{n}}=\rho_{\mathfrak{u}}|_{\mathfrak{l}}+\rho_{\mathfrak{u}\cap\mathfrak{m}}$.
So if we reduce (7.3.3) under the action of $M\times L$ we recover the
quantization $\operatorname{Ind}^{G}_{L}\mathcal{D}_{L}$.

Thus, the left hand side of (7.3.1) is obtained from (7.3.2) by first pushing
forward along the map $\operatorname{id}_{T^{*}(G/U)}\times\rho_{M}$ and then
reducing under the $M$-action. The right hand side of (7.3.1) is obtained from
(7.3.2) by first reducing under the $M$-action and then pushing forward along
$\rho^{\prime}$. Note that $\rho^{\prime}$ is the morphism induced by
$\widehat{\rho}:=\operatorname{id}_{T^{*}(G/U)}\times\rho_{M}$ under classical
Hamiltonian reduction. The group $M$ acts freely on both
$T^{*}(G/U)\times\widetilde{X}_{M}$ and $T^{*}(G/U)\times\widetilde{Y}_{M}$.
Note that the morphisms $\widehat{\rho}$ and $\rho^{\prime}$ satisfy the
conditions of Lemma 7.3.1. Write $\bullet/\\!/\\!/M$ for the quantum
Hamiltonian reduction of a Hamiltonian quantization by a free action of $M$.
To prove (7.3.1), it suffices to show for an arbitrary Hamiltonian
quantization $\mathcal{D}$ of $T^{*}(G/U)\times\widetilde{Y}_{M}$, there is an
isomorphism of quantizations
$(\widehat{\rho}_{*}\mathcal{D})/\\!/\\!/M\xrightarrow{\sim}\rho^{\prime}_{*}(\mathcal{D}/\\!/\\!/M)$.
We note that Hamiltonian reduction sends quantizations to quantizations
because the action of $M$ is free, while
$\rho^{\prime}_{*},\widehat{\rho}_{*}$ send quantizations to quantizations
thanks to Lemma 7.3.1. Note that there is a natural homomorphism
$(\widehat{\rho}_{*}\mathcal{D})/\\!/\\!/M\rightarrow\rho^{\prime}_{*}(\mathcal{D}/\\!/\\!/M)$
with associated graded equal to the identity map on
$\mathcal{O}_{\widetilde{Z}}$. Since both sides are filtered quantizations,
this homomorphism is an isomorphism.

∎

Now suppose $\widetilde{\mathbb{O}}_{L}$ is birationally rigid. By Theorem
4.7.6, quantizations of $\mathbb{C}[\widetilde{\mathbb{O}}]$ are parameterized
by $W^{\widetilde{X}}$-orbits on $\mathfrak{P}^{\widetilde{X}}$. By
Proposition 7.2.2, the latter space is identified with
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])$. Thus, we get a
bijection

(7.3.4)
$\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])/W^{\widetilde{X}}\xrightarrow{\sim}\mathrm{Quant}(\mathbb{C}[\widetilde{\mathbb{O}}]),\qquad
W^{\widetilde{X}}\cdot\beta\mapsto\mathcal{A}_{\eta(\beta)}^{\widetilde{X}}.$

Abusing notation slightly, we will often abbreviate
$\mathcal{A}_{\eta(\beta)}^{\widetilde{X}}$ by writing
$\mathcal{A}_{\beta}^{\widetilde{X}}$. The isomorphism
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\simeq\mathfrak{P}^{\widetilde{X}}$
of Proposition 7.2.2 extends to an isomorphism (still denoted by $\eta$)

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

Hence by Proposition 4.11.3, there is a bijection

(7.3.5)
$\mathfrak{X}(\mathfrak{l})/W^{\widetilde{X}}\xrightarrow{\sim}\mathrm{Quant}^{G}(\mathbb{C}[\widetilde{\mathbb{O}}]),\qquad
W^{\widetilde{X}}\cdot\beta\mapsto(\mathcal{A}^{\widetilde{X}}_{\beta},\Phi^{\widetilde{X}}_{\beta})$

Here,
$\Phi_{\beta}^{\widetilde{X}}:U(\mathfrak{g})\stackrel{{\scriptstyle}}{{\to}}\mathcal{A}_{\beta}^{\widetilde{X}}$
is the (unique) quantum co-moment map satisfying

$\Phi_{\beta}^{\widetilde{X}}|_{\mathfrak{z}(\mathfrak{g})}=\beta|_{\mathfrak{z}(\mathfrak{g})},$

see Section 4.11. The next proposition shows that the quantization parameter
$\beta$ is preserved under $\operatorname{Ind}^{G}_{M}$.

######  Proposition 7.3.3.

For each $\beta\in\mathfrak{X}(\mathfrak{l})$, there is an isomorphism of
Hamiltonian quantizations

(7.3.6)
$\operatorname{Ind}^{G}_{M}\mathcal{A}_{\beta}^{\widetilde{X}_{M}}\simeq\mathcal{A}_{\beta}^{\widetilde{X}}.$

###### Proof.

As in the proof of Lemma 7.3.2, we can assume that $\mathfrak{g}$ is
semisimple. Thanks to Lemma 7.3.2, we can also assume that $M=L$. Consider the
$\mathbb{Q}$-factorial terminalization
$\widetilde{Y}:=G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$,
and its universal graded Poisson deformation

$\widetilde{Y}_{\mathrm{univ}}=G\times^{P}(\mathfrak{X}(\mathfrak{l})\times\widetilde{X}_{L}\times\mathfrak{p}^{\perp}),$

see Proposition 7.2.2(iii). Let $\mathcal{D}^{\widetilde{Y}_{\mathrm{univ}}}$
(resp. $\mathcal{D}^{\widetilde{X}_{L}}$) denote the canonical quantization of
$\widetilde{Y}_{\mathrm{univ}}$ (resp. $\widetilde{X}_{L}$), see Proposition
4.7.4. Consider the filtered quantization
$\overline{\mathcal{D}}^{\widetilde{Y}_{\mathrm{univ}}}$ of
$\widetilde{Y}_{\mathrm{univ}}$ defined by

$\overline{\mathcal{D}}^{\widetilde{Y}_{\mathrm{univ}}}:=\left([\mathfrak{D}_{G/N}\
\widehat{\otimes}\ \mathcal{D}^{{\widetilde{X}_{L}}}]/[\mathfrak{D}_{G/N}\
\widehat{\otimes}\
\mathcal{D}^{\widetilde{X}_{L}}]\Phi([\mathfrak{l},\mathfrak{l}])\right)^{L},$

By [Los12, Theorem 5.4.1],
$\overline{\mathcal{D}}^{\widetilde{Y}_{\mathrm{univ}}}$ is an even
quantization (more precisely, its restriction to
$\widetilde{Y}^{\mathrm{reg}}_{\mathrm{univ}}$ is the specialization of a
graded even quantization). Thus, by the results of [Los12, Section 2.3], there
is an isomorphism of filtered quantizations

$\overline{\mathcal{D}}^{\widetilde{Y}_{\mathrm{univ}}}\simeq\mathcal{D}^{\widetilde{Y}_{\mathrm{univ}}}$

Specializing both sides to $\beta\in\mathfrak{X}(\mathfrak{l})$, we get an
isomorphism of filtered quantizations

$\operatorname{Ind}^{G}_{L}\mathcal{D}_{\beta}^{\widetilde{X}_{L}}\simeq\mathcal{D}_{\beta}^{\widetilde{Y}}.$

Now, the result follows by taking global sections.

∎

Using the classification of filtered quantizations of
$\mathbb{C}[\widetilde{X}]$, see Theorem 4.7.6, and Proposition 7.3.3 we
obtain the following.

######  Corollary 7.3.4.

Let $\mathcal{A}_{M}$ be a Hamiltonian quantization of $\widetilde{X}_{M}$.
Then, up to an isomorphism of Hamiltonian quantizations,
$\operatorname{Ind}^{G}_{M}\mathcal{A}_{M}$ depends only on $M$ (and not on
$Q$).

###  7.4. Namikawa spaces vs parabolic induction

In this section we begin the task of computing the isomorphism
$\eta:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{g},\mathfrak{g}])\xrightarrow{\sim}\mathfrak{P}^{\widetilde{X}}$.
Here we establish some basic facts about the behavior of Namikawa spaces under
parabolic induction.

Choose a parabolic subgroup $P=LN\subset G$ and a birationally rigid
$L$-equivariant nilpotent cover $\widetilde{\mathbb{O}}_{L}$ such that
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$.
Choose also a parabolic subgroup $Q=MU\subset G$ such that $P\subset Q$ and
$L\subset M$. Let
$\widetilde{\mathbb{O}}_{M}=\mathrm{Bind}^{M}_{L}\widetilde{\mathbb{O}}_{L}$
and $P_{M}=P\cap M$, a parabolic in $M$. Consider the
$\mathbb{Q}$-terminalizations

$\rho:\widetilde{Y}=G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X},\qquad\rho_{M}:\widetilde{Y}_{M}=M\times^{P_{M}}(\widetilde{X}_{L}\times\mathfrak{p}_{M}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}_{M}$

Note that $\widetilde{Y}_{M}$ is an $M$-equivariant fiber bundle over
$M/P_{M}=Q/P$. Hence
$G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})$ is a
$G$-equivariant fiber bundle over $G/P$. Also
$G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})$ has a
$G$-equivariant projective birational morphism to
$G\times^{Q}(\widetilde{X}_{M}\times\mathfrak{q}^{\perp})$ and hence to
$\widetilde{X}$. Note that both varieties
$\widetilde{Y},G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})$ come
with natural $G\times\mathbb{C}^{\times}$-actions (on
$G\times^{Q}(\widetilde{X}_{M}\times\mathfrak{q}^{\perp})$ the factor
$\mathbb{C}^{\times}$ acts fiberwise via $t.(y,z)=(ty,t^{d}z)$ for
$t\in\mathbb{C}^{\times},y\in\widetilde{Y}_{M},z\in\mathfrak{q}^{\perp}$,
where $d$ is the weight of the Poisson bracket on $\widetilde{Y}_{M}$), and
the morphisms to $\widetilde{X}$ are $G\times\mathbb{C}^{\times}$-equivariant.

######  Lemma 7.4.1.

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

$\widetilde{Y}\xrightarrow{\sim}G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})$

intertwining the morphisms to $\widetilde{X}$ and to $G/P$.

###### Proof.

The variety $G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})$ is
$\mathbb{Q}$-factorial terminal and so is a $\mathbb{Q}$-factorial
terminalization of $\widetilde{X}$. The fiber of $0\in\widetilde{X}$ is $G/P$
so restricting a relatively ample line bundle from
$G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})$ to $G/P$ gives an
ample line bundle. Thus, by Proposition 7.1.3, the ample cone of
$G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})$ is contained in
that of $\widetilde{Y}$. By Lemma 7.2.4, there is an isomorphism of
$\mathbb{Q}$-factorial terminalizations of $\widetilde{X}$

(7.4.1)
$\widetilde{Y}\xrightarrow{\sim}G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp}).$

Such an isomorphism is necessarily unique because the terminalization
morphisms are birational. In particular, it is
$G\times\mathbb{C}^{\times}$-equivariant. The morphisms
$\widetilde{Y},G\times^{Q}(\widetilde{Y}_{M}\times\mathfrak{q}^{\perp})\rightarrow
G/P$ are recovered as the categorical quotient morphisms for the action of
$\mathbb{C}^{\times}$. Hence, they are intertwined by (7.4.1) as well. ∎

For the next result, we will assume that $G$ is semisimple in order to
simplify the statements. Let ${\pi}^{\prime}$ denote the composition of
$\pi:\widetilde{Y}^{\mathrm{reg}}\rightarrow G/P$ with the projection
$G/P\rightarrow G/Q$. Consider the composition

(7.4.2) ${\eta}^{\prime}:\mathfrak{X}(\mathfrak{m})\simeq
H^{2}(G/Q,\mathbb{C})\overset{{\pi}^{{}^{\prime}*}}{\stackrel{{\scriptstyle}}{{\to}}}H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})=\mathfrak{P}^{\widetilde{X}}.$

Note that ${\eta}^{\prime}$ coincides with the composition

(7.4.3)
$\mathfrak{X}(\mathfrak{m})\hookrightarrow\mathfrak{X}(\mathfrak{l})\xrightarrow{\eta}\mathfrak{P}^{\widetilde{X}}.$

Thanks to Lemma 7.4.1, we can view $\widetilde{Y}^{\mathrm{reg}}$ as a fiber
bundle over $G/Q$ with fiber
$\widetilde{Y}_{M}^{\mathrm{reg}}\times\mathfrak{q}^{\perp}$. In particular,
there are pullback maps

(7.4.4) $H^{2}(G/Q,\mathbb{C})\rightarrow
H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C}),\qquad
H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})\rightarrow
H^{2}(\widetilde{Y}_{M}^{\mathrm{reg}},\mathbb{C}).$

We wish to describe (7.4.4) under the identifications

(7.4.5)
$\mathfrak{X}(\mathfrak{m})\xrightarrow{\sim}H^{2}(G/Q,\mathbb{C}),\quad\eta:\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C}),\quad\eta_{M}:\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])\xrightarrow{\sim}H^{2}(\widetilde{Y}_{M}^{\mathrm{reg}},\mathbb{C}).$

######  Proposition 7.4.2.

Under the identifications in (7.4.5), the pullback maps in (7.4.4) correspond
to the inclusion
$\mathfrak{X}(\mathfrak{m})\hookrightarrow\mathfrak{X}(\mathfrak{l})$ and the
projection
$\mathfrak{X}(\mathfrak{l})\twoheadrightarrow\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])$,
respectively.

###### Proof.

Since $\eta^{\prime}$ coincides with (7.4.3), $H^{2}(G/Q)\rightarrow
H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})$ is indeed the inclusion
$\mathfrak{X}(\mathfrak{m})\hookrightarrow\mathfrak{X}(\mathfrak{l})$. Next we
show that $H^{2}(\widetilde{Y}^{\mathrm{reg}},\mathbb{C})\rightarrow
H^{2}(\widetilde{Y}_{M}^{\mathrm{reg}},\mathbb{C})$ coincides with
$\mathfrak{X}(\mathfrak{l})\twoheadrightarrow\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])$.
There is a commutative diagram

${\widetilde{Y}_{M}^{\mathrm{reg}}}$${\widetilde{Y}^{\mathrm{reg}}}$${M/P_{M}}$${G/P}$

Here the vertical maps are the projections, while the horizontal maps are the
fiber inclusions (for fiber bundles over $G/Q$). We have
$H^{2}(G/P,\mathbb{C})\simeq\mathfrak{X}(\mathfrak{l}),H^{2}(M/P_{M},\mathbb{C})\simeq\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])$,
and the pullback map for the bottom arrow is the projection
$\mathfrak{X}(\mathfrak{l})\twoheadrightarrow\mathfrak{X}(\mathfrak{l}\cap[\mathfrak{m},\mathfrak{m}])$.
Since the pullback maps for the vertical arrows are $\eta_{M}$ and $\eta$, we
are done.

∎

###  7.5. Partial resolutions from codimension 2 leaves

In this section, we will provide a Lie-theoretic description of the partial
Namikawa spaces $\mathfrak{P}_{k}^{\widetilde{X}}$, $k\geq 1$, under some
additional restrictions. Assume $G$ is semisimple in order to simplify the
statements.

Choose a Levi subgroup $L\subset G$ and a birationally rigid $L$-equivariant
nilpotent cover $\widetilde{\mathbb{O}}_{L}$ such that
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{L}\widetilde{\mathbb{O}}_{L}$.
Choose a codimension 2 leaf $\mathfrak{L}_{k}\subset\widetilde{X}$ and let
$\Sigma_{k}$ be the corresponding Kleinian singularity. There is a closed
embedding $\Sigma_{k}\hookrightarrow\widetilde{X}$, constructed as follows.
The image of $\overline{\mathfrak{L}}_{k}$ under the moment map
$\mu:\widetilde{X}\stackrel{{\scriptstyle}}{{\to}}\overline{\mathbb{O}}\subset\mathfrak{g}^{*}$
is the closure of a codimension $2$ orbit
$\mathbb{O}^{\prime}\subset\overline{\mathbb{O}}$. Consider the Slodowy slice
$S^{\prime}$ to $\mathbb{O}^{\prime}$ in $\mathfrak{g}^{*}$. Recall from
Section 3.5 that $S^{\prime}$ comes with a natural contracting
$\mathbb{C}^{\times}$-action. Since $S^{\prime}$ is transverse to
$\mathbb{O}^{\prime}$, $\mu^{-1}(S^{\prime})$ is transverse to every leaf in
$\mu^{-1}(\mathbb{O}^{\prime})$. Thanks to the contracting
$\mathbb{C}^{\times}$-action on $S^{\prime}$, $\mu^{-1}(S^{\prime})$ splits
into a disjoint union of connected components indexed by the points in
$\mu^{-1}(S^{\prime}\cap\mathbb{O}^{\prime})$. Choose a point in this set
lying in $\mathfrak{L}_{k}$ and let $\Sigma_{k}$ be its connected component.
Note that $\Sigma_{k}$ is a Kleinian singularity and
$\operatorname{Spec}(\mathbb{C}[\Sigma_{k}]^{\wedge})$ is a formal slice to
$\mathfrak{L}_{k}$.

Assume for the remainder of this subsection that
$H^{2}(\widetilde{\mathbb{O}},\mathbb{C})=0$. By Lemma 7.2.7, this is
equivalent to the condition $\mathfrak{P}_{0}^{\widetilde{X}}=0$. Under this
assumption, we will define, for each codimension 2 leaf
$\mathfrak{L}_{k}\subset\widetilde{X}$, a Levi subgroup $M_{k}\subset G$
containing $L$ which is ‘adapted’ to $\mathfrak{L}_{k}$. This Levi subgroup is
uniquely determined by the following property: for any parabolic $Q\subset G$
with Levi factor $M_{k}$, there is a partial resolution
$\rho_{k}:\widetilde{Z}_{k}=G\times^{Q}(\widetilde{X}_{M_{k}}\times\mathfrak{q}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$
such that:

  * •

$\rho_{k}$ is a minimal resolution over $\Sigma_{k}\subset\widetilde{X}$.

  * •

$\rho_{k}$ is an isomorphism over $\Sigma_{j}\subset\widetilde{X}$ $j\neq k$.

In other words, $\rho_{k}$ resolves $\Sigma_{k}$ and preserves all other
slices. Replacing $G$ with a covering group if necessary, we can (and will)
assume that $G$ is simply connected. To simplify the notation we write
$\mathfrak{P}$ instead of $\mathfrak{P}^{\widetilde{X}}$ and
$\mathfrak{P}_{k}$ instead of $\mathfrak{P}^{\widetilde{X}}_{k}$.

Recall, see Proposition 7.2.2, that a choice of parabolic $P\subset G$ with
Levi factor $L$ gives rise to an identification
$\eta:\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}\mathfrak{P}$. Different
parabolics $P$ give rise to different identifications, but these
identifications coincide up to the $W^{\widetilde{X}}$-action on the target,
see Proposition 7.2.5.

######  Lemma 7.5.1.

There is a parabolic $P$ with Levi factor $L$ such that the intersection
$\mathfrak{C}:=\mathfrak{X}(\mathfrak{l})_{\mathbb{R}}^{\geqslant
0}\cap\eta^{-1}(\mathfrak{P}_{\mathbb{R},k}^{\geqslant 0})$ is a face in
$\mathfrak{X}(\mathfrak{l})_{\mathbb{R}}^{\geqslant 0}$ of dimension
$\dim\mathfrak{P}_{k}$.

###### Proof.

Since $\mathfrak{P}_{\mathbb{R},0}\simeq
H^{2}(\widetilde{\mathbb{O}},\mathbb{R})=0$, the product
$\prod_{i}\mathfrak{P}_{\mathbb{R},i}^{\geq 0}$ is a fundamental chamber for
the $W^{\widetilde{X}}$-action on $\mathfrak{P}_{\mathbb{R}}$, and
$\mathfrak{P}_{\mathbb{R},k}$ is a face of this chamber. By Theorem 4.9.1, the
ample cones of all $\mathbb{Q}$-terminalizations of $\widetilde{X}$ partition
the fundamental chamber. Hence, there is a $\mathbb{Q}$-terminalization
$\widetilde{Y}$ of $\widetilde{X}$ such that the intersection
$\eta(\mathfrak{C}):=\mathrm{Amp}(\widetilde{Y})\cap\mathfrak{P}_{\mathbb{R},k}$
is of dimension $\dim\mathfrak{P}_{\mathbb{R},k}$. Note that
$\eta(\mathfrak{C})$ is automatically a face of $\mathrm{Amp}(\widetilde{Y})$.
By Lemma 7.2.4,
$\widetilde{Y}=G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})$ for a
parabolic $P$ with Levi factor $L$ and
$\mathrm{Amp}(\widetilde{Y})=\eta\mathfrak{X}(\mathfrak{l})_{\mathbb{R}}^{\geq
0}$. Hence, $\mathfrak{C}=\mathfrak{X}(\mathfrak{l})_{\mathbb{R}}^{\geq
0}\cap\eta^{-1}(\mathfrak{P}_{\mathbb{R},k}^{\geq 0})$ is a face of
$\mathfrak{X}(\mathfrak{l})_{\mathbb{R}}^{\geq 0}$ of dimension
$\dim\mathfrak{P}_{\mathbb{R},k}$. ∎

Choose $\mathfrak{C}$ as in Lemma 7.5.1. Let $\mathfrak{m}_{k}$ be the Levi
subalgebra

(7.5.1)
$\mathfrak{m}_{k}:=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Delta(\mathfrak{C})}\mathfrak{g}_{\alpha},\qquad\Delta(\mathfrak{C}):=\\{\alpha\in\Delta\mid\langle\alpha^{\vee},\mathfrak{C}\rangle=0\\},$

and let $M_{k}$ be the corresponding Levi subgroup of $G$. By construction,
$\mathfrak{X}(\mathfrak{m}_{k})_{\mathbb{R}}=\mathbb{R}\mathfrak{C}$ and
$\eta(\mathbb{R}\mathfrak{C})=\mathfrak{P}_{\mathbb{R},k}$. Thus, $\eta$
restricts to an isomorphism

(7.5.2)
$\eta_{k}:\mathfrak{X}(\mathfrak{m}_{k})\xrightarrow{\sim}\mathfrak{P}_{k}$

######  Remark 7.5.2.

Note that the Levi subgroup $M_{k}\subset G$ is uniquely determined by the
triple $(L,\widetilde{\mathbb{O}}_{L},\mathfrak{L}_{k})$. Indeed,
$\mathfrak{m}_{k}$ is the Levi subalgebra corresponding to the co-roots which
vanish on the subspace $\eta^{-1}(\mathfrak{P}_{k})$. And by Proposition
7.2.5, this subspace is independent of $P$.

Next, we will give a geometric characterization of the Levi subgroup $M_{k}$.
It will be convenient to introduce some additional terminology.

######  Definition 7.5.3.

A _resolution datum_ for $L$ is a triple $(P,M,Q)$ consisting of a parabolic
$P\subset G$ with Levi factor $L$, a Levi subgroup $M\subset G$ containing
$L$, and a parabolic $Q\subset G$ containing $P$ with Levi factor $M$. A
resolution datum is said to be _adapted to the leaf_
$\mathfrak{L}_{k}\subset\widetilde{X}$ if $M=M_{k}$.

######  Remark 7.5.4.

If $M$ and $Q$ are fixed, we can always find a resolution datum $(P,M,Q)$.
However, if $M$ and $P$ are fixed, it may not be possible to do so.
Nonetheless, we can always find a resolution datum $(P^{\prime},M,Q)$ such
that the identification
$\eta^{\prime}:\mathfrak{X}(\mathfrak{l})\xrightarrow{\sim}\mathfrak{P}$
corresponding to $P^{\prime}$ coincides with $\eta$. Indeed, $\eta$ is
determined by a choice of fundamental chamber for the
$W^{\widetilde{X}}$-action on $\mathfrak{X}(\mathfrak{l})_{\mathbb{R}}$. Take
$P^{\prime}$ such that $\mathfrak{X}(\mathfrak{l})_{\mathbb{R}}^{\geq 0}$
intersects $\mathfrak{X}(\mathfrak{m}_{k})$. Since every fundamental chamber
intersects $\mathfrak{X}(\mathfrak{m}_{k})$, such a $P^{\prime}$ exists.

A choice of resolution datum $(P,M,Q)$ gives rise to the following package of
data:

  * •

A $\mathbb{Q}$-terminalization
$\rho:\widetilde{Y}:=G\times^{P}(\widetilde{X}_{L}\times\mathfrak{p}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$.

  * •

A projection $\pi:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}G/P$.

  * •

An $M$-equivariant nilpotent cover
$\widetilde{\mathbb{O}}_{M}:=\mathrm{Bind}^{M}_{L}\widetilde{\mathbb{O}}_{L}$.

  * •

A partial resolution
$\overline{\rho}:\widetilde{Z}:=G\times^{Q}(\widetilde{X}_{M}\times\mathfrak{q}^{\perp})\stackrel{{\scriptstyle}}{{\to}}\widetilde{X}$.

  * •

A projection
$\overline{\pi}:\widetilde{Z}\stackrel{{\scriptstyle}}{{\to}}G/Q$.

  * •

A $\mathbb{Q}$-terminalization
$\rho^{\prime}:\widetilde{Y}\stackrel{{\scriptstyle}}{{\to}}\widetilde{Z}$
such that $\rho=\overline{\rho}\circ\rho^{\prime}$.

For an $\mathfrak{L}_{k}$-adapted resolution datum $(P,M_{k},Q)$, we write
$\widetilde{Z}_{k}$ for $\widetilde{Z}$, $\rho_{k}$ for $\overline{\rho}$,
$\pi_{k}$ for $\overline{\pi}$, and $\rho_{k}^{\prime}$ for $\rho^{\prime}$.

######  Remark 7.5.5.

Suppose $(P,M,Q)$ is a resolution datum for $L$. Under the identifications
$\operatorname{Pic}(\widetilde{Z})\simeq\mathfrak{X}(M)$ and
$\operatorname{Pic}(\widetilde{Y})\simeq\mathfrak{X}(L)$ constructed in
Section 7.1, the pullback map
$\rho^{{}^{\prime}*}:\operatorname{Pic}(\widetilde{Z})\rightarrow\operatorname{Pic}(\widetilde{Y})$
coincides with the inclusion $\mathfrak{X}(M)\hookrightarrow\mathfrak{X}(L)$
induced by the restriction of characters.

######  Proposition 7.5.6.