# Steinberg slices and group-valued moment maps

Ana Bălibanu  Department of Mathematics, Harvard University, 1 Oxford Street,
Cambridge, MA 02138, USA<EMAIL_ADDRESS>
###### Abstract.

We define a class of transversal slices in spaces which are quasi-Poisson for
the action of a complex semisimple group $G$. This is a multiplicative
analogue of Whittaker reduction. One example is the multiplicative universal
centralizer $\mathfrak{Z}$ of $G$, which is equipped with the usual symplectic
structure in this way. We construct a smooth partial compactification
$\overline{\mathfrak{Z}}$ by taking the closure of each centralizer fiber in
the wonderful compactification of $G$. By realizing this partial
compactification as a transversal in a larger quasi-Poisson variety, we show
that it is smooth and log-symplectic.

###### Contents

  1. 1 Quasi-Hamiltonian and quasi-Poisson structures
  2. 2 Steinberg slices
  3. 3 The wonderful compactification
  4. 4 The logarithmic double
  5. 5 The partial compactification of $\mathfrak{Z}$

## Introduction

Let $G$ be a simply-connected, complex semisimple group, and let
$G_{\operatorname{ad}}$ be its adjoint form. The group $G_{\operatorname{ad}}$
acts on $G$ by conjugation, and $G$ contains a transversal slice $\Sigma$ for
this action which was introduced by Steinberg [Ste]. The resulting
(multiplicative) universal centralizer is the smooth affine variety

$\mathfrak{Z}:=\left\\{(a,h)\in G_{\operatorname{ad}}\times\Sigma\mid
aha^{-1}=h\right\\}.$

This family of centralizers first appeared in work of Lusztig [Lus, Section
8.6]. When $G$ is simply-laced, Bezrukavnikov, Finkelberg, and Mirkovic [BFM]
have shown that its coordinate ring is isomorphic to the equivariant
$K$-theory of the affine Grassmannian of the Langlands dual group
$G^{\vee}$—therefore, $\mathfrak{Z}$ is a Coulomb branch in the sense of
Nakajima [Nak].

The natural symplectic structure on $\mathfrak{Z}$ is inherited from the
nondegenerate quasi-Poisson structure on the double
$\mathbf{D}_{G_{\operatorname{ad}}}:=G_{\operatorname{ad}}\times G$ as
described, up to a finite central quotient, by [BFM] and by Finkelberg and
Tsymbaliuk [FT]. We construct a smooth partial compactification
$\overline{\mathfrak{Z}}$ of $\mathfrak{Z}$, by taking the closure of each
centralizer fiber inside the wonderful compactification
$\overline{G_{\operatorname{ad}}}$. We show that the symplectic structure on
$\mathfrak{Z}$ extends to a log-symplectic Poisson structure on
$\overline{\mathfrak{Z}}$.

These results parallel the main theorem of [Bal]. That work considers the
principal slice $\mathcal{S}\subset\mathfrak{g}$ defined by Kostant [Kos],
which is a cross-section to the regular adjoint $G$-orbits on $\mathfrak{g}$.
The corresponding universal centralizer is the symplectic variety

$\mathcal{Z}:=\left\\{(a,x)\in
G_{\operatorname{ad}}\times\mathcal{S}\mid\operatorname{Ad}_{a}x=x\right\\},$

which is obtained from the cotangent bundle $T^{*}G_{\operatorname{ad}}$ by
Whittaker reduction. It has a smooth, log-symplectic partial compactification

$\overline{\mathcal{Z}}:=\left\\{(a,x)\in
G_{\operatorname{ad}}\times\mathcal{S}\mid
a\in\overline{G_{\operatorname{ad}}^{x}}\right\\},$

which is the Whittaker reduction of the log-contangent bundle
$T^{\ast}_{D}\overline{G_{\operatorname{ad}}}$ of the wonderful
compactification.

Identifying $\mathfrak{g}$ with $\mathfrak{g}^{*}$ under the Killing form
isomorphism, there is a commutative diagram of moment maps

${T^{*}G_{\operatorname{ad}}}$${T^{\ast}_{D}\overline{G_{\operatorname{ad}}}}$${\mathfrak{g}\times\mathfrak{g}.}$$\scriptstyle{\nu}$$\scriptstyle{\overline{\nu}}$

The varieties $\mathcal{Z}$ and $\overline{\mathcal{Z}}$ are simply the
preimages of the principal slice $\mathcal{S}\times(-\mathcal{S})$ under $\nu$
and $\overline{\nu}$. In particular, because $\mathcal{S}$ intersects every
adjoint orbit exactly once and transversally, $\mathcal{Z}$ and
$\overline{\mathcal{Z}}$ sit inside $T^{*}G_{\operatorname{ad}}$ and
$T^{\ast}_{D}\overline{G_{\operatorname{ad}}}$ as Poisson transversals—that
is, they intersect each symplectic leaf of the ambient space transversally and
symplectically. Their Poisson structures are therefore also obtained via
restriction in this way.

We give a multiplicative analogue of these results by considering manifolds
which are quasi-Poisson relative to the action of $G$. These can be viewed as
deformations of ordinary Poisson structures in which the Jacobi identity is
twisted by a canonical trivector field induced by the group action. They were
introduced in a series of papers by Alekseev, Malkin, and Meinrenken [AMM],
Alekseev and Kosmann-Schwarzbach [AKS], and Alekseev, Kosmann-Schwarzbach, and
Meinreken [AKSM]. These manifolds come equipped with group-valued momentum
maps, and they are foliated by nondegenerate leaves.

In this setting Kostant’s principal slice $\mathcal{S}$ is replaced by the
Steinberg cross-section $\Sigma$. We show that the preimage of this cross-
section under a quasi-Poisson moment map is a smooth manifold, which we call a
_Steinberg slice_. It has a natural Poisson structure which is “transverse” to
the quasi-Poisson structure on the ambient space, in the sense that it
intersects every nondegenerate leaf transversally and symplectically. In this
way, Steinberg slices can be viewed as a multiplicative counterpart to
Whittaker reduction. We use them to construct multiplicative analogues of
several Whittaker-type algebraic varieties.

In particular, the universal centralizer $\mathfrak{Z}$ sits as a Steinberg
slice in the double
$\mathbf{D}_{G_{\operatorname{ad}}}=G_{\operatorname{ad}}\times G$, which is
the quasi-Poisson analogue of the cotangent bundle of $G$. In fact, using the
identification $T^{*}G_{\operatorname{ad}}\cong
G_{\operatorname{ad}}\times\mathfrak{g}$, the cotangent bundle
$T^{*}G_{\operatorname{ad}}$ is a bundle of Lie algebras and
$\mathbf{D}_{G_{\operatorname{ad}}}$ is the simply-connected group scheme
which integrates it. We show that $\mathbf{D}_{G_{\operatorname{ad}}}$ extends
to a group scheme $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ over
$\overline{G_{\operatorname{ad}}}$ which integrates the log-cotangent bundle
$T^{\ast}_{D}\overline{G_{\operatorname{ad}}}$. We prove that this group
scheme is quasi-Poisson and “logarithmically nondegenerate” in a suitable
sense.

We then have a commutative diagram of group-valued moment maps

${\mathbf{D}_{G_{\operatorname{ad}}}}$${\mathbf{D}_{\overline{G_{\operatorname{ad}}}}}$${G\times
G.}$$\scriptstyle{\mu}$$\scriptstyle{\overline{\mu}}$

The varieties $\mathfrak{Z}$ and $\overline{\mathfrak{Z}}$ are exactly the
preimages of the Steinberg cross-section $\Sigma\times\iota(\Sigma)$, where
$\iota$ denotes the group inversion, under the moment maps $\mu$ and
$\overline{\mu}$. We show that this induces a log-symplectic Poisson structure
on the partial compactification $\overline{\mathfrak{Z}}$, whose unique open
dense symplectic leaf is $\mathfrak{Z}$.

In Section 1 we review quasi-Poisson manifolds as developed in [AKSM]. We also
outline how they fit into the framework of twisted Dirac structures, as in
[BC1] and [BC2]. In Section 2 we show that the preimage of the Steinberg
cross-section under a quasi-Poisson moment map has a natural induced Poisson
structure. As a variation on this result, we also construct a multiplicative
analogue of the twisted cotangent bundle of the base affine space. Then we
define the notion of log-nondegeneracy for quasi-Poisson manifolds, and we
show that Steinberg slices in log-nondegenerate quasi-Poisson manifolds are
log-symplectic.

In Section 3 we recall the multiplicative universal centralizer
$\mathfrak{Z}$, which is a Steinberg slice in the double
$\mathbf{D}_{G_{\operatorname{ad}}}$, and we review the wonderful
compactification $\overline{G_{\operatorname{ad}}}$. In Section 4 we use the
Vinberg monoid to construct the smooth group scheme
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$. Then we show that the quasi-
Poisson structure on $\mathbf{D}_{G_{\operatorname{ad}}}$ extends to a log-
nondegenerate structure on $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$.
Finally, in Section 5 we realize the partial compactification
$\overline{\mathfrak{Z}}$ as a Steinberg slice in
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, equipping it with a log-
symplectic Poisson structure. We give an explicit description of its
stratification by symplectic leaves.

Acknowledgements: During the completion of this work, the author was partially
supported by a National Science Foundation MSPRF under award DMS–1902921.

##  1\. Quasi-Hamiltonian and quasi-Poisson structures

We recall the basics of quasi-Hamiltonian and quasi-Poisson manifolds below,
and we refer to [AKSM] for more details. We then explain how to view quasi-
Poisson manifolds as twisted Dirac manifolds, following [BC1] and [BC2]. We
will use this formalism in Section 2.

###  1.1. Quasi-Poisson manifolds

Let $G$ be a simply-connected, semisimple complex group, let $\mathfrak{g}$ be
its Lie algebra, and write $(\cdot,\cdot)$ for the Killing form. Under the
isomorphism $\mathfrak{g}\cong\mathfrak{g}^{*}$ induced by this form, the
_Cartan $3$-tensor_ $\varphi\in\wedge^{3}\mathfrak{g}$ is the dual of the
invariant trilinear function $\eta\in\wedge^{3}\mathfrak{g}^{*}$ given by

$\eta(x,y,z)=\frac{1}{12}(x,[y,z])\qquad\text{for all }x,y,z\in\mathfrak{g}.$

If $\\{e_{i}\\}$ is a basis of $\mathfrak{g}$ which is orthonormal relative to
the Killing form,

$\varphi=\frac{1}{12}C_{ijk}e_{i}\wedge e_{j}\wedge e_{k}$

where $C_{ijk}=(e_{i},[e_{j},e_{k}])$ are the structure constants. Here and
throughout the paper we adopt the convention of summing over repeated indices.

If $G$ acts on a complex manifold $M$, we write $\xi_{M}$ for the polyvector
field induced by the infinitesimal action of an element
$\xi\in\wedge^{k}\mathfrak{g}$. In particular, the Cartan $3$-tensor $\varphi$
generates a trivector field $\varphi_{M}\in\Gamma(\wedge^{3}TM)$. A _quasi-
Poisson_ structure on the manifold $M$ is a $G$-invariant section
$\pi\in\Gamma(\wedge^{2}TM)$ such that

(1.1) $[\pi,\pi]=\varphi_{M},$

where the bracket on the left is the Schouten–Nijenhuis bracket. In the
special case where $G$ is abelian, the Cartan $3$-tensor is trivial, and a
quasi-Poisson structure on $M$ is simply a $G$-invariant Poisson structure.

######  Example 1.2.

[AKSM, Section 3] The group $G$, equipped with the conjugation action, has a
natural quasi-Poisson bivector given by

$\pi_{G}:=\frac{1}{2}e_{i}^{R}\wedge e_{i}^{L}.$

Here $e_{i}^{L}$ and $e_{i}^{R}$ are the invariant vector fields on $G$
corresponding to left- and right-multiplication. The bivector $\pi_{G}$ is
tangent to the conjugacy classes, and it induces a quasi-Poisson structure on
each one.

If $(M_{1},\pi_{1})$ and $(M_{2},\pi_{2})$ are quasi-Poisson $G$-manifolds, a
$G$-equivariant map $f:M_{1}\longrightarrow M_{2}$ is called _quasi-Poisson_
if the bivectors $\pi_{1}$ and $\pi_{2}$ are $f$-related. A quasi-Poisson
manifold $(M,\pi)$ is _Hamiltonian_ if it has a $G$-equivariant group-valued
moment map

$\Phi:M\longrightarrow G$

which satisfies a differential equation analogous to the usual moment map
condition [AKSM, Definition 2.2]. In particular, $\Phi$ is a quasi-Poisson map
when $G$ is equipped with the bivector $\pi_{G}$. In what follows all quasi-
Poisson manifolds will be Hamiltonian, so we will suppress this adjective.

######  Example 1.3.

[AKSM, Example 5.3] Consider the _internal fusion double_ $D(G):=G\times G$.
The group $G\times G$ acts on $D(G)$ by

$(g,h)\cdot(u,v)=\left(guh^{-1},hvg^{-1}\right)$

for $(g,h)\in G\times G$ and $(u,v)\in D(G)$. Let $\\{e_{i}^{1},e_{i}^{2}\\}$
be the induced orthonormal basis for the Lie algebra
$\mathfrak{g}\oplus\mathfrak{g}$. The manifold $D(G)$ has a quasi-Poisson
bivector

$\frac{1}{2}\left(e_{i}^{1L}\wedge e_{i}^{2R}+e_{i}^{1R}\wedge
e_{i}^{2L}\right).$

The associated moment map is

$\displaystyle D(G)\longrightarrow G\times G$
$\displaystyle(u,v)\longmapsto(uv,u^{-1}v^{-1}).$

In the subsequent sections we will often use the reparametrization of $D(G)$
given by setting $a=u$ and $b=vu$. This is analogous to the left-
trivialization of the cotangent bundle $T^{*}G$. In these coordinates the
$G\times G$-action is

(1.4) $(g,h)\cdot(a,b)=\left(gah^{-1},hbh^{-1}\right).$

At the point $(a,b)$ the quasi-Poisson bivector becomes

$\frac{1}{2}\left(e_{i}^{1L}\wedge e_{i}^{2R}+e_{i}^{2L}\wedge
e_{i}^{2R}+e_{i}^{1R}\wedge(\operatorname{Ad}_{a^{-1}}e_{i})^{2L}\right).$

Using the fact that $\operatorname{Ad}_{a^{-1}}$ is an orthogonal
transformation relative to the Killing form and summing once again over
repeated indices, the last term simplifies to

$e_{i}^{1R}\wedge(\operatorname{Ad}_{a^{-1}}e_{i})^{2L}=(\operatorname{Ad}_{a^{-1}}e_{i})^{1L}\wedge(\operatorname{Ad}_{a^{-1}}e_{i})^{2L}=e_{i}^{1L}\wedge
e_{i}^{2L}.$

Therefore the quasi-Poisson structure in these coordinates is

(1.5)
$\displaystyle\pi:=\frac{1}{2}\left(e_{i}^{1L}\wedge\left(e_{i}^{2L}+e_{i}^{2R}\right)+e_{i}^{2L}\wedge
e_{i}^{2R}\right),$

and the associated moment map is

(1.6) $\displaystyle\mu:\,$ $\displaystyle D(G)\longrightarrow G\times G$
$\displaystyle(a,b)\longmapsto(aba^{-1},b^{-1}).$

Quasi-Poisson structures are not compatible with restriction to the action of
a subgroup—that is, a quasi-Poisson $G$-manifold is not in general quasi-
Poisson for the action of a subgroup of $G$. An exception to this is the case
of diagonal subgroups, for which there is a procedure called _internal fusion_
[AKSM, Section 5] which we now describe.

Suppose that $(M,\pi)$ is a quasi-Poisson $G\times G$-manifold with group-
valued moment map

$\displaystyle\Phi:\,$ $\displaystyle M\longrightarrow G\times G$
$\displaystyle m\longmapsto(\Phi_{1}(m),\Phi_{2}(m)).$

Define a $2$-tensor

$\psi:=\frac{1}{2}e_{i}^{1}\wedge
e_{i}^{2}\in\wedge^{2}(\mathfrak{g}\oplus\mathfrak{g}),$

and consider the modified bivector

$\pi_{\text{fus}}:=\pi+\psi_{M}.$

Let $\Phi_{1}\Phi_{2}$ denote the pointwise product of the components of
$\Phi$. Then the triple

$(M,\pi_{\text{fus}},\Phi_{1}\Phi_{2})$

is a quasi-Poisson $G$-manifold relative to the diagonal action of $G$.

Fusion equips the category of quasi-Poisson $G$-manifolds with a monoidal
structure. Given two quasi-Poisson $G$-manifolds $(M_{1},\pi_{1},\Phi_{1})$
and $(M_{2},\pi_{2},\Phi_{2})$, their direct product $M_{1}\times M_{2}$ is
naturally a quasi-Poisson manifold for the action of $G\times G$. Fusing the
two sides of the $G$-action, we obtain a new quasi-Poisson $G$-manifold
denoted

$M_{1}\circledast M_{2},$

with bivector $(\pi_{1}+\pi_{2})_{\text{fus}}$ and moment map
$\Phi_{1}\Phi_{2}.$

###  1.2. Nondegenerate quasi-Poisson structures

Let $(M,\pi,\Phi)$ be a quasi-Poisson $G$-manifold. The bivector $\pi$ induces
a morphism of vector bundles

$\displaystyle\pi^{\\#}:T^{*}M$ $\displaystyle\longrightarrow TM$
$\displaystyle\alpha$ $\displaystyle\longmapsto\pi(\alpha,-)$

from the cotangent bundle $T^{*}M$ to the tangent bundle $TM$. The action of
$G$ differentiates to an infinitesimal action map

$\rho:M\times\mathfrak{g}\longrightarrow TM.$

The quasi-Poisson manifold $M$ is called _nondegenerate_ if the map

(1.7) $\displaystyle\pi^{\\#}\oplus\rho:T^{*}M\oplus\mathfrak{g}$
$\displaystyle\longrightarrow TM$ $\displaystyle(\alpha,\xi)$
$\displaystyle\longmapsto\pi^{\\#}(\alpha)+\rho(\xi)$

is surjective. For example, the double $D(G)$ defined in Example 1.3 is
nondegenerate.

Let $\theta^{L}$ and $\theta^{R}$ be the left- and right-invariant
Maurer–Cartan forms on $G$. These are $\mathfrak{g}$-valued $1$-forms defined
as follows: if $L_{h},R_{h}$ are the differentials of left- and right-
multiplication by the element $h\in G$, then for any $v\in T_{h}G$

$\theta^{L}_{h}(v)=L_{h^{-1}}v\qquad\text{and}\qquad\theta^{R}_{h}(v)=R_{h^{-1}}v.$

The bi-invariant $3$-form on $G$ induced by
$\eta\in\wedge^{3}\mathfrak{g}^{*}$, which we denote by the same symbol, is

(1.8)
$\eta=\frac{1}{12}\left(\theta^{L},[\theta^{L},\theta^{L}]\right)=\frac{1}{12}\left(\theta^{R},[\theta^{R},\theta^{R}]\right)\in\Gamma(\wedge^{3}T^{*}G).$

Every nondegenerate quasi-Poisson manifold $(M,\pi,\Phi)$ carries a
(potentially degenerate, non-closed) $2$-form $\omega$ which satifies the
following properties:

$\displaystyle\text{(Q1)}\quad d\omega=-\Phi^{*}\eta;$
$\displaystyle\text{(Q2)}\quad\iota_{\xi_{M}}\omega=\frac{1}{2}\Phi^{*}(\theta^{L}+\theta^{R},\xi)\quad\text{for
all }\xi\in\mathfrak{g};$
$\displaystyle\text{(Q3)}\quad\ker\omega_{m}=\left\\{\xi_{M}(m)\mid\xi\in\mathfrak{g}\text{
such that }\operatorname{Ad}_{\Phi(m)}\xi=-\xi\right\\}.$

This $2$-form gives $M$ the structure of a _quasi-Hamiltonian $G$-space_ in
the sense of [AMM]. We write $\theta_{i}^{L},\theta_{i}^{R}\in\Gamma(T^{*}G)$
for the components of $\theta^{L}$ and $\theta^{R}$ in the basis
$\\{e_{i}\\}$. At every point these $1$-forms are a dual basis to the left-
and right-invariant vector fields, so that

$\theta_{i}^{L}(e_{j}^{L})=\theta_{i}^{R}(e_{j}^{R})=\delta_{ij}.$

Define $C:TM\longrightarrow TM$ to be the morphism of vector bundles

(1.9) $C:=\text{Id}-\Phi^{*}(\theta_{i}^{L}-\theta_{i}^{R})\otimes e_{iM}.$

Then $\omega$ and $\pi$ satisfy the compatibility condition

$\pi^{\\#}\circ\omega^{\flat}=C,$

where $\omega^{\flat}:TM\longrightarrow T^{*}M$ is the vector bundle map given
by contraction with $\omega$.

######  Example 1.10.

The quasi-Hamiltonian $2$-form corresponding to the nondegenerate quasi-
Poisson manifold $D(G)$ from Example 1.3 is

(1.11)
$\omega=-\frac{1}{2}\left(\theta_{i}^{1L}\wedge\theta_{i}^{2R}+\theta_{i}^{1R}\wedge\theta_{i}^{2L}\right).$

######  Remark 1.12.

If the action of $G$ is trivial, the quasi-Poisson manifold $M$ is
nondegenerate if and only if $\pi^{\\#}$ is an isomorphism—that is, if and
only if $\pi$ is a nondegenerate Poisson structure. In this case $\omega$ is
exactly the corresponding symplectic form.

Even when $\pi$ is degenerate, the image of (1.7) is an integrable
distribution. Its integral submanifolds, which are $G$-stable, are called the
_nondegenerate leaves_ of $M$, because $\pi$ gives each the structure of a
nondegenerate quasi-Poisson manifold. In particular, each nondegenerate leaf
$S$ is equipped with a quasi-Hamiltonian $2$-form $\omega_{S}$.

######  Example 1.13.

The nondegenerate leaves of the quasi-Poisson structure $(G,\pi_{G})$ defined
in Example 1.2 are the conjugacy classes.

There is an analogue of Hamiltonian reduction for quasi-Poisson manifolds. Let
$(M,\pi,\Phi)$ be a quasi-Poisson $G$-manifold and fix a conjugacy class
$\mathcal{O}\subset G$. Then, if the quotient

$M\\!\sslash_{\mathcal{O}}\\!G:=\Phi^{-1}(\mathcal{O})/G$

is a manifold, it has a natural Poisson structure whose symplectic leaves are
precisely the reductions of the nondegenerate leaves of $M$. When
$\mathcal{O}=\\{1\\}$ is the identity element, we denote this quotient simply
by $M\\!\sslash\\!G$.

###  1.3. Twisted Dirac structures

Fix a closed $3$-form $\phi\in\Gamma(\wedge^{3}T^{*}M)$. A vector subbundle

$L\subset TM\oplus T^{*}M$

is called a _$\phi$ -twisted Dirac structure_ on $M$ if it satisfies the
following two conditions:

  * $\bullet$

$L$ is Lagrangian with respect to the symmetric pairing on $\Gamma(TM\oplus
T^{*}M)$ given by

$\langle(X,\alpha),(Y,\beta)\rangle=\beta(X)+\alpha(Y);$

  * $\bullet$

$\Gamma(L)$ is closed under the _$\phi$ -twisted Courant bracket_ on
$\Gamma(TM\oplus T^{*}M)$ defined by

$\llbracket(X,\alpha),(Y,\beta)\rrbracket_{\phi}=([X,Y],\mathcal{L}_{X}\beta-\iota_{Y}d\alpha+\iota_{X\wedge
Y}\phi).$

The projection of $L\subset TM\oplus T^{*}M$ onto the first summand is an
integrable distribution, and induces a foliation of $M$ by _presymplectic
leaves_. Each presymplectic leaf $S\subset M$ carries a (potentially
degenerate, non-closed) $2$-form $\omega_{S}$ such that
$d\omega_{S}=\phi_{|S}$.

######  Example 1.14.

  1. (1)

Any symplectic manifold $(M,\omega)$ corresponds to the $0$-twisted Dirac
structure

$L_{\omega}:=\\{(X,\omega^{\flat}(X))\mid X\in TM\\}\subset TM\oplus T^{*}M$

given by the graph of $\omega^{\flat}$. Conversely, a $0$-twisted Dirac
structure $L\subset TM\oplus T^{*}M$ is induced by a symplectic form if and
only if $L$ is transverse to both $TM$ and $T^{*}M,$ viewed as subbundles of
$TM\oplus T^{*}M$.

  2. (2)

Similarly, any Poisson manifold $(M,\pi)$ corresponds to the $0$-twisted Dirac
structure

$L_{\pi}:=\\{(\pi^{\\#}(\alpha),\alpha)\mid\alpha\in T^{*}M\\}\subset TM\oplus
T^{*}M$

given by the graph of $\pi^{\\#}$. Its projection onto the first coordinate is
the distribution whose integral submanifolds are the symplectic leaves of
$\pi$. Conversely, a $0$-twisted Dirac structure $L\subset TM\oplus T^{*}M$ is
induced by a Poisson bivector if and only if $L$ is transverse to $TM$.

  3. (3)

[BC1, Theorem 3.16] A quasi-Poisson $G$-manifold $(M,\pi,\Phi)$ corresponds to
the $-\Phi^{*}\eta$-twisted Dirac structure

$L=\left\\{\left(\pi^{\\#}(\alpha)+\rho(\xi),\,C^{*}(\alpha)+\Phi^{*}\sigma(\xi)\right)\mid\alpha\in
T^{*}M,\xi\in\mathfrak{g}\right\\}\subset TM\oplus T^{*}M.$

Here $C$ is as defined in (1.9) and $\sigma$ is given by

$\displaystyle\sigma:\mathfrak{g}$ $\displaystyle\longrightarrow T^{*}G$
$\displaystyle\xi$
$\displaystyle\longmapsto\frac{1}{2}\left(\xi^{L}+\xi^{R}\right)^{\vee},$

where $v^{\vee}$ is the dual of the vector $v\in TG$ under the isomorphism
$TG\cong T^{*}G$ induced by the Killing form.

This Dirac structure has the property that $\ker\Phi_{*}\cap L=0$. The
associated presymplectic foliation, given by projecting $L$ onto $TM$, is
exactly the foliation of $M$ by quasi-Hamiltonian leaves described in Section
1.2.

Let $(M,L_{M})$ and $(N,L_{N})$ be Dirac manifolds. A map $f:M\longrightarrow
N$ is _forward-Dirac_ if

$L_{N}=f_{*}L_{M}:=\\{(f_{*}X,\beta)\in TN\oplus T^{*}N\mid(X,f^{*}\beta)\in
L_{M}\\}.$

This notion generalizes the pushforward of vector fields, and all Poisson and
quasi-Poisson maps are forward-Dirac. In particular, if $(M,\pi,\Phi)$ is a
quasi-Poisson $G$-manifold, then the group-valued moment map $\Phi$ is
forward-Dirac when $M$ and $G$ are viewed as Dirac manifolds. Moreover, [BC1,
Theorem 3.16] shows that every $\phi$-twisted Dirac manifold $(M,L)$ equipped
with a forward-Dirac map $\Phi:M\longrightarrow G$ which satisfies

(1.15) $\phi=-\Phi^{*}\eta\qquad\text{and}\qquad\ker\Phi_{*}\cap L=0$

is a quasi-Poisson manifold. (In [BC2], such a map is called _strong_ forward-
Dirac.)

Conversely, the map $f$ is called _backward-Dirac_ if

$L_{M}=f^{*}L_{N}:=\\{(X,f^{*}\beta)\in TM\oplus T^{*}M\mid(f_{*}X,\beta)\in
L_{N}\\}.$

This is a generalization of the pullback of differential forms, and
symplectomorphisms, for instance, are backward-Dirac. We give the following
important example of a backward-Dirac map, which we will use repeatedly in the
next section.

######  Example 1.16.

[Bur, Proposition 5.6] Suppose that $(M,L)$ is a $\phi$-twisted Dirac
manifold. If $\imath:X\lhook\joinrel\longrightarrow M$ is a submanifold which
is transverse to the foliation of $M$ by presymplectic leaves, then

$\imath^{*}L=\\{(X,\imath^{*}\beta)\in TX\oplus
T^{*}X\mid(\imath_{*}X,\beta)\in L\\}$

is a $\imath^{*}\phi$-twisted Dirac structure on $X$, and $\imath$ is a
backward-Dirac map.

##  2\. Steinberg slices

In this section we show that any quasi-Poisson $G$-manifold $(M,\pi,\Phi)$ has
a distinguished submanifold $M_{\Sigma}$ which intersects each nondegenerate
leaf transversally and symplectically. This submanifold, which we call the
_Steinberg slice_ of $M$, is the preimage of the Steinberg cross-section of
$G$ under the moment map $\Phi$. It carries a Poisson structure whose
symplectic leaves are its intersections with the nondegenerate leaves of $M$.

###  2.1. Construction of $M_{\Sigma}$

Let $W$ be the Weyl group of $G$ corresponding to a maximal torus $T$, and let
$c\in W$ be a Coxeter element—that is, $c$ is the product of the simple
reflections, which is unique up to conjugation. Write $\dot{c}\in N_{G}(T)$
for a fixed group representative of $c$.

Fix a pair of opposite Borel subgroups $B$ and $B^{-}$ containing $T$, and let
$U$ and $U^{-}$ be their unipotent radicals. The _Steinberg cross-section_ of
$G$, which was introduced in [Ste], is the closed subvariety

$\Sigma:=U\dot{c}\cap\dot{c}\,U^{-}\subset G.$

It is an affine space which consists entirely of regular elements. Its
dimension is equal to the length of $c$ as an element of the Weyl group, which
is the rank of $G$.

Since $G$ is simply-connected, $\Sigma$ intersects every regular conjugacy
class in $G$ exactly once and transversally. (The proof of transversality does
not appear in [Ste], but can be found for example in [Sev, Proposition 2.3].)
If $\Xi:G\longrightarrow T/W$ is the quotient map induced by the Chevalley
isomorphism $\mathbb{C}[G]^{G}\cong\mathbb{C}[T]^{W}$, then the composition

(2.1) $\Sigma\lhook\joinrel\longrightarrow G\xlongrightarrow{\Xi}T/W$

is an isomorphism of affine varieties.

######  Theorem 2.2.

Let $(M,\pi,\Phi)$ be a quasi-Poisson $G$-manifold.

  1. _(a)_

$M_{\Sigma}:=\Phi^{-1}(\Sigma)$ is a smooth submanifold of $M$.

  2. _(b)_

The inclusion $\imath:M_{\Sigma}\lhook\joinrel\longrightarrow M$ induces a
Poisson structure $\pi_{\Sigma}$ on $M_{\Sigma}$.

  3. _(c)_

The symplectic leaves of $\pi_{\Sigma}$ are the connected components of
$M_{\Sigma}\cap S$, where $S$ varies over all nondegenerate leaves of $M$; the
symplectic form on each connected component of $M_{\Sigma}\cap S$ is the
restriction of the quasi-Hamiltonian $2$-form $\omega_{S}$.

###### Proof.

(a) Let $h\in\Sigma$ and $m\in\Phi^{-1}(h)$, and write $\mathcal{O}\subset G$
for the conjugacy class of $h$. Because $\Phi$ is $G$-equivariant,

$T_{h}\mathcal{O}=\Phi_{*}(T_{m}(G\cdot m))\subset\Phi_{*}(T_{m}M).$

Therefore, since $\Sigma$ is transverse to $\mathcal{O}$, it is transverse to
$\Phi$. It follows that $M_{\Sigma}=\Phi^{-1}(\Sigma)$ is a smooth submanifold
of $M$.

(b) Let $\jmath:\Sigma\lhook\joinrel\longrightarrow G$ be the inclusion.
First, note that for any $h\in\Sigma$, the image of

$\theta^{L}_{h}=(L_{h^{-1}})_{*}:T_{h}\Sigma\longrightarrow\mathfrak{g}$

is contained in $\mathfrak{b}=\text{Lie}(B)$, and
$(\mathfrak{b},[\mathfrak{b},\mathfrak{b}])=0$. In view of (1.8), the
restriction $\jmath^{*}\eta$ vanishes.

Since $\Sigma$ is transverse to the conjugacy classes of $G$,

$T_{m}M_{\Sigma}+T_{m}(G\cdot
m)=\Phi_{*}^{-1}(T_{h}\Sigma+T_{h}\mathcal{O})=T_{m}M.$

It follows that $M_{\Sigma}$ is transverse to the $G$-orbits on $M$, and
therefore also to the presymplectic leaves of $(M,L_{M})$. By Example 1.16,
the $-\Phi^{*}\eta$-twisted Dirac structure on $M$ pulls back to a
$-\imath^{*}(\Phi^{*}\eta)$-twisted Dirac structure $L_{M_{\Sigma}}$ on
$M_{\Sigma}$. The commutative diagram

${M_{\Sigma}}$${M}$${\Sigma}$${G}$$\scriptstyle{\imath}$$\scriptstyle{\Phi}$$\scriptstyle{\Phi}$$\scriptstyle{\jmath}$

implies that $-\imath^{*}(\Phi^{*}\eta)=-\Phi^{*}(\jmath^{*}\eta)=0.$
Therefore $L_{M_{\Sigma}}$ is a 0-twisted Dirac structure on $M_{\Sigma}$.

To show that $L_{M_{\Sigma}}$ is in fact Poisson, by Example 1.14(b) it is
sufficient to show that

$L_{M_{\Sigma}}\cap TM_{\Sigma}=0.$

First, let $L_{G}$ be the Dirac structure corresponding to the quasi-Poisson
structure $\pi_{G}$ on $G$. Since $\Sigma$ intersects each conjugacy class of
$G$ exactly once and transversally, $L_{G}$ pulls back to the trivial Poisson
structure

$L_{\Sigma}:=\jmath^{*}L_{G}=\\{(0,\beta)\mid\beta\in T^{*}\Sigma\\}\subset
T\Sigma\oplus T^{*}\Sigma$

on $\Sigma$, as in Example 1.16.

Since $\Phi$ is a moment map, it is forward-Dirac, and we have

$L_{\Sigma}=\Phi_{*}L_{M_{\Sigma}}=\\{(\Phi_{*}(X),\alpha)\in T\Sigma\oplus
T^{*}\Sigma\mid(X,\Phi^{*}\alpha)\in L_{M_{\Sigma}}\\}.$

Suppose that $(X,0)\in L_{M_{\Sigma}}\cap TM_{\Sigma}$. Then

$(\Phi_{*}(X),0)\in L_{\Sigma}\quad\Rightarrow\quad X\in\ker\Phi_{*}\cap
L_{M_{\Sigma}}.$

It follows from (1.15) that $X=0$, and therefore the Dirac structure
$L_{M_{\Sigma}}$ is a Poisson structure.

(c) This is immediate since we have shown that the Poisson structure
$\pi_{\Sigma}$ is the pullback of the Dirac structure $L_{M}$ to $M_{\Sigma}$.
∎

Our first example of a Steinberg slice is the group scheme of regular
centralizers of $G$, whose symplectic structure is constructed in essentially
the same way in [FT, Section 2].

######  Example 2.3.

Consider the double $D(G)$ of Example 1.3. Recall that its moment map is

$\displaystyle\mu:\,D(G)$ $\displaystyle\longrightarrow G\times G$
$\displaystyle(a,b)$ $\displaystyle\longmapsto(aba^{-1},b^{-1}),$

with image

(2.4) $\operatorname{im}(\mu)=\left\\{(g,h)\in G\times G\mid g\text{ is
conjugate to }h^{-1}\right\\}.$

Let

$\Sigma_{\Delta}:=\left\\{(h,h^{-1})\mid h\in\Sigma\right\\}\subset G\times G$

be the antidiagonal embedding of the Steinberg cross-section $\Sigma$. Since
two elements of $\Sigma$ are conjugate if and only if they are equal, we have

$\mu^{-1}(\Sigma_{\Delta})=\mu^{-1}(\Sigma\times\iota(\Sigma)),$

where $\iota:G\longrightarrow G$ is the inversion. Since
$\Sigma\times\iota(\Sigma)$ is a Steinberg cross-section in $G\times G$, it
follows from Proposition 2.2 that $\mu^{-1}(\Sigma_{\Delta})$ is a smooth
submanifold of $D(G)$ with an induced symplectic structure.

The fiber of $\mu$ above an antidiagonal point $(h,h^{-1})\in G\times G$ is
the $G$-centralizer of $h$, and therefore

$\mu^{-1}(\Sigma_{\Delta})=\\{(a,h)\in G\times\Sigma\mid aha^{-1}=h\\}.$

This space is the completion of the phase space of the open relativistic Toda
lattice, and this symplectic structure is precisely the one constructed in
[FT, Lemma 2.1].

###  2.2. Slices and the base affine space

We may also take the preimage of $\Sigma$ through only one component of the
moment map (1.6). This is the analogue of the one-sided Whittaker reduction of
$T^{*}G$, which gives the twisted cotangent bundle of the base affine space
$G/U$.

The preimage of $\iota(\Sigma)\subset G$ under the second component of the
moment map $\mu$ is

$D_{\Sigma}(G):=G\times\Sigma,$

and it carries a residual $G$-action

$g\cdot(a,h)=(ga,h),\qquad\text{for $g\in G$, $(a,h)\in D_{\Sigma}(G)$}.$

We write $\omega^{\prime}$ for the restriction of the quasi-Hamiltonian
$2$-form $\omega$ from (1.11) to $D_{\Sigma}(G)$, and

$\displaystyle\mu^{\prime}:\,D_{\Sigma}(G)$ $\displaystyle\longrightarrow G$
$\displaystyle(a,h)$ $\displaystyle\longrightarrow aha^{-1}$

for the $G$-equivariant map induced by the first component of $\mu$.

######  Remark 2.5.

Consider the affine space $\Theta:=U\dot{c}U$, which contains $\Sigma$. By
[Ste, Section 8.5], the conjugation action gives an isomorphism

$U\times\Sigma\xlongrightarrow{\sim}\Theta.$

(The proof in _loc. cit._ is omitted, but more general versions of this
statement are proved in [Sev, Proposition 2.1] or [LH, Theorem 3.6].) Using
this we can view $D_{\Sigma}(G)$ as a bundle of affine spaces

$G\times_{U}\Theta\longrightarrow G/U$

We show that it has a natural nondegenerate quasi-Poisson structure for the
action of $G$ on the left.

######  Proposition 2.6.

The embedding

$\imath:D_{\Sigma}(G)\lhook\joinrel\longrightarrow D(G)$

gives $D_{\Sigma}(G)$ the structure of a nondegenerate quasi-Poisson
$G$-manifold. The associated group-valued moment map is $\mu^{\prime}$ and the
corresponding quasi-Hamiltonian $2$-form is $\omega^{\prime}$.

###### Proof.

Let $(\eta^{1},\eta^{2})$ be the canonical bi-invariant $3$-form on $G\times
G$. The slice $D_{\Sigma}(G)$ is a smooth submanifold of $D(G)$, and the
quasi-Poisson structure on $D(G)$ is nondegenerate. Therefore, by Example
1.16, the $-\mu^{*}(\eta^{1},\eta^{2})$-twisted Dirac structure on $D(G)$
pulls back to a $-\imath^{*}\mu^{*}(\eta^{1},\eta^{2})$-twisted Dirac
structure on $D_{\Sigma}(G)$.

We have a commutative diagram

${D_{\Sigma}(G)}$${D(G)}$${G\times\Sigma}$${G\times
G,}$$\scriptstyle{\imath}$$\scriptstyle{\mu}$$\scriptstyle{\mu}$$\scriptstyle{\jmath}$

Since the restriction of $\eta$ to $\Sigma$ vanishes,

$-\imath^{*}\mu^{*}(\eta^{1},\eta^{2})=-\mu^{*}\jmath^{*}(\eta^{1},\eta^{2})=-\mu^{*}(\eta^{1},0)=-\mu^{\prime*}\eta.$

Therefore we obtain a $-\mu^{\prime*}\eta$-twisted Dirac structure on
$D_{\Sigma}(G)$ with a unique presymplectic leaf whose corresponding $2$-form
is precisely $\omega^{\prime}$.

Since $\mu$ is a quasi-Poisson moment map, it satisfies the second condition
of (1.15). Therefore, so does $\mu^{\prime}$. Applying [BC1, Theorem 3.16],
this induced Dirac structure on $M_{\Sigma}(G)$ corresponds to a quasi-Poisson
bivector $\pi^{\prime}$ for the action of $G$ on the left. It follows that
$(D_{\Sigma}(G),\pi^{\prime},\mu^{\prime})$ is a nondegenerate quasi-Poisson
$G$-manifold with quasi-Hamiltonian $2$-form $\omega^{\prime}$. ∎

The one-sided slice $D_{\Sigma}(G)$ is similar to the universal imploded
cross-section of [HJS], where the authors study real quasi-Hamiltonian
manifolds under the action of compact Lie groups. In this sense Steinberg
slices are a simplified counterpart to quasi-Hamiltonian implosion. Following
this analogy, we show that the Steinberg slice $M_{\Sigma}$ can always be
obtained as a quasi-Poisson reduction of the fusion product $M\circledast
D_{\Sigma}(G)$.

Let $(M,\pi,\Phi)$ be a quasi-Poisson $G$-manifold, and consider the embedding

$\displaystyle M$ $\displaystyle\lhook\joinrel\longrightarrow M\times D(G)$
$\displaystyle m$ $\displaystyle\longmapsto(m,1,\Phi(m)^{-1}).$

Its restriction to $M_{\Sigma}$ descends to an embedding

(2.7) $M_{\Sigma}\lhook\joinrel\longrightarrow M\times D_{\Sigma}(G).$

If we view the right-hand side as a fused quasi-Poisson $G$-manifold with
moment map

$\displaystyle J:M\circledast D_{\Sigma}(G)$ $\displaystyle\longrightarrow G$
$\displaystyle(m,a,x)$ $\displaystyle\longrightarrow\Phi(m)axa^{-1},$

the image of (2.7) is contained in the fiber $J^{-1}(1)$ above the identity.

######  Proposition 2.8.

The embedding (2.7) induces an isomorphism of Poisson manifolds

$M_{\Sigma}\cong(M\circledast D_{\Sigma}(G))\\!\sslash\\!G.$

###### Proof.

Since the diagonal action of $G$ on $M\circledast D_{\Sigma}(G)$ is free, each
$G$-orbit in $J^{-1}(1)$ contains a unique element of the form
$(m,1,\Phi(m)^{-1})$. Therefore the induced map

(2.9) $M_{\Sigma}\longrightarrow J^{-1}(1)/G=(M\circledast
D_{\Sigma}(G))\\!\sslash\\!G$

is an isomorphism. We only need to check that it is Poisson.

Because of the transversality condition (1.15), the inclusion

$J^{-1}(1)\lhook\joinrel\longrightarrow M\circledast D_{\Sigma}(G)$

equips $J^{-1}(1)$ with a Dirac structure via pullback. We get a diagram

${M_{\Sigma}}$${J^{-1}(1)}$${J^{-1}(1)/G,}$$\scriptstyle{\sim}$

where the horizontal arrow is backward-Dirac and the vertical arrow is
forward-Dirac [BC2, Proposition 4.1]. Since (2.9) is an isomorphism, the
diagram implies that it is also backward-Dirac, and therefore Poisson. ∎

###  2.3. Log-nondegenerate quasi-Poisson structures

Once again let $(M,\pi,\Phi)$ be a quasi-Poisson $G$-manifold, and let
$D\subset M$ be a $G$-stable divisor with simple normal crossings. The
_logarithmic tangent sheaf_ is the sheaf of logarithmic vector fields on
$M$—that is, vector fields which are tangent to the divisor $D$. Because $D$
has simple normal crossings, this sheaf is locally free. The associated vector
bundle $T_{D}M$ is called the _logarithmic tangent bundle_ of $M$, and its
dual $T^{*}_{D}M$ is the _logarithmic cotangent bundle_.

Suppose for the rest of this section that the bivector field $\pi$ is
logarithmic. Then it corresponds to a natural morphism of vector bundles

$\pi^{\\#}_{D}:T^{*}_{D}M\longrightarrow T_{D}M$

from the log-cotangent bundle of $M$ to the log-tangent bundle. Similarly, any
logarithmic $2$-form $\omega\in\Gamma(\wedge^{2}T^{*}_{D}M)$ corresponds to

$\omega_{D}^{\flat}:T_{D}M\longrightarrow T_{D}^{*}M.$

Since the action of $G$ on $M$ stabilizes the divisor $D$, there is also a
logarithmic infinitesimal action map

$\rho_{D}:M\times\mathfrak{g}\longrightarrow T_{D}M.$

######  Definition 2.10.

The quasi-Poisson $G$-manifold $M$ is _logarithmically nondegenerate_ if the
morphism of vector bundles

(2.11)
$\displaystyle\pi^{\\#}_{D}\oplus\rho_{D}:\,T^{*}_{D}M\oplus\mathfrak{g}$
$\displaystyle\longrightarrow T_{D}M$ $\displaystyle(\alpha,\xi)$
$\displaystyle\longmapsto\pi_{D}^{\\#}(\alpha)+\rho_{D}(\xi)$

is surjective.

######  Remark 2.12.

The pullback of $T_{D}M$ to the open dense locus $M^{\circ}:=M\backslash D$ is
just the ordinary cotangent bundle $TM^{\circ}$; similarly, the pullback of
$T^{*}_{D}M$ to $M^{\circ}$ is $T^{*}M^{\circ}$. Therefore, along $M^{\circ}$
the morphism $\eqref{lognon}$ agrees with the morphism of vector bundles
(1.7). In particular, if $M$ is log-nondegenerate then $M^{\circ}$ is its
unique open dense nondegenerate leaf.

Viewed as an automorphism of the tangent sheaf of $M$, the map defined in
(1.9) takes logarithmic vector fields to logarithmic vector fields. Therefore
it defines a morphism of vector bundles

(2.13) $C_{D}:T_{D}M\longrightarrow T_{D}M.$

Using this and [AKSM, Theorem 10.3], we give an equivalent condition for log-
nondegeneracy.

######  Proposition 2.14.

The quasi-Poisson manifold $(M,\pi,\Phi)$ is log-nondegenerate if and only if
there exists a logarithmic $2$-form $\omega\in\Gamma(\wedge^{2}T_{D}^{*}M)$
such that

(2.15) $\pi_{D}^{\\#}\circ\omega_{D}^{\flat}=C_{D}.$

###### Proof.

($\Rightarrow$) First suppose that $\pi$ is log-nondegenerate. Then its
restriction to $M^{\circ}$ is a nondegenerate quasi-Poisson bivector
$\pi^{\circ}$. By [AKSM, Theorem 10.3] there is a $2$-form

$\omega^{\circ}\in\Gamma(\wedge^{2}T^{*}M^{\circ})$

which satisfies conditions (Q1), (Q2), and (Q3). If $C^{\circ}$ is the
restriction of (2.13) to $M^{\circ}$, then

$\pi^{\circ\\#}\circ\omega^{\circ\flat}=C^{\circ}.$

Taking duals, we also obtain

$\omega^{\circ\flat}\circ\pi^{\circ\\#}=C^{\circ*}.$

If $\omega^{\circ}$ extends to a logarithmic $2$-form $\omega$ on $M$, then
condition (2.15) is automatically satisfied by continuity. Therefore it is
enough to show that $\omega^{\circ\flat}:TM^{\circ}\longrightarrow
T^{*}M^{\circ}$ extends to a morphism of vector bundles

$\omega_{D}^{\flat}:T_{D}M\longrightarrow T^{*}_{D}M.$

For any $v\in T_{D}^{*}M$, we define

$\omega_{D}^{\flat}(\pi_{D}^{\\#}(v)):=C^{*}_{D}(v).$

This extends $\omega_{D}^{\flat}$ to the entire image of $\pi^{\\#}_{D}$. On
the other hand, the condition (Q2) defines $\omega_{D}^{\flat}$ on the image
of $\rho_{D}$. By the log-nondegeneracy assumption (2.11), this determines
$\omega_{D}^{\flat}$ entirely, and we are done.

($\Leftarrow$) Conversely, suppose that there exists a logarithmic $2$-form
$\omega$ on $M$ such that (2.15) holds, and let $v\in T_{D}M$ be any
logarithmic vector. Then, in view of (1.9),

$\pi_{D}^{\\#}\circ\omega^{\flat}_{D}(v)=C_{D}(v)=v-\rho_{D}(\xi)$

for some $\xi\in\mathfrak{g}$. It follows that

$\pi_{D}^{\\#}\left(\omega^{\flat}_{D}(v)\right)+\rho_{D}(\xi)=v,$

and so $\pi^{\\#}_{D}\oplus\rho_{D}$ is surjective. Therefore $\pi$ is log-
nondegenerate. ∎

######  Remark 2.16.

Together with [AKSM, Theorem 10.3], Proposition 2.14 implies that any log-
nondegenerate quasi-Poisson manifold comes equipped with a unique logarithmic
$2$-form which satisfies logarithmic versions of conditions (Q1), (Q2), (Q3),
as well as the compatibility condition (2.15).

In the special case that the action of $G$ is trivial, $(M,\pi)$ is log-
nondegenerate if and only if $\pi_{D}^{\\#}$ is an isomorphism—that is, if and
only if $\pi$ is a log-symplectic Poisson structure. In this case $C_{D}$ is
the identity morphism and the logarithmic $2$-form $\omega_{D}$ is is exactly
the corresponding log-symplectic form.

The following proposition shows that Steinberg slices in log-nondegenerate
quasi-Poisson manifolds are log-symplectic.

######  Proposition 2.17.

Suppose that $(M,\pi,\Phi)$ is log-nondegenerate.

  1. _(a)_

$M_{\Sigma}\cap D$ is a simple normal crossing divisor in $M_{\Sigma}$.

  2. _(b)_

The induced bivector $\pi_{\Sigma}$ is tangent to $M_{\Sigma}\cap D$.

  3. _(c)_

$(M_{\Sigma},\pi_{\Sigma})$ is a log-symplectic Poisson manifold.

###### Proof.

(a) Let $D=D_{1}\cup\ldots\cup D_{l}$ be the smooth irreducible components of
the simple normal crossing divisor $D$. Since the bivector $\pi$ is tangent to
$D$ and since $D$ is $G$-stable, each partial intersection

$\qquad\bigcap_{i\in I}D_{i},\qquad I\subset\\{1,\ldots,l\\}$

is a union of nondegenerate leaves of $(M,\pi)$. Since $M_{\Sigma}$ is
transverse to these nondegenerate leaves, it is transverse to every partial
intersection of divisor components. It follows that $M_{\Sigma}\cap D$ is
again a simple normal crossing divisor.

(b) Fix a point $m\in M_{\Sigma}$ and a covector $\alpha\in
T^{*}_{m}M_{\Sigma}$, and let $\imath:M_{\Sigma}\longrightarrow M$ be the
inclusion map. Write $L_{M_{\Sigma}}$ and $L_{M}$ for the twisted Dirac
structures associated to $M_{\Sigma}$ and $M$. By Theorem 2.2,

$L_{M_{\Sigma}}=\imath^{*}L_{M}.$

Therefore, since $(\pi_{\Sigma}^{\\#}(\alpha),\alpha)\in L_{M_{\Sigma}}$,
there exists some $\beta\in T^{*}_{m}M$ such that

$\left(\pi_{\Sigma}^{\\#}(\alpha),\alpha\right)=\left(\pi_{\Sigma}^{\\#}(\alpha),\imath^{*}\beta\right)\qquad\text{and}\qquad\left(\imath_{*}\pi_{\Sigma}^{\\#}(\alpha),\beta\right)\in
L_{M}.$

Since $(M,\pi)$ is quasi-Poisson, Example 1.14(c) then implies that

$\imath_{*}\pi_{\Sigma}^{\\#}(\alpha)=\pi^{\\#}(\gamma)+\rho(\xi)$

for some $\gamma\in T_{m}^{*}M$ and $\xi\in\mathfrak{g}$. Since $\pi^{\\#}$ is
logarithmic and $D$ is $G$-stable, both terms on the right-hand side are
tangent to $D$. It follows that $\pi_{\Sigma}^{\\#}(\alpha)$ is tangent to
$M_{\Sigma}\cap D$, and therefore the bivector $\pi_{\Sigma}$ is logarithmic.

(c) Let $\omega$ be the logarithmic $2$-form on $M$ defined by Proposition
2.14. Write $\omega_{\Sigma}$ for its restriction to $M_{\Sigma}$, and
$\omega_{\Sigma}^{\circ}$ for its restriction to
$M_{\Sigma}^{\circ}:=M_{\Sigma}\cap M^{\circ}$. Since
$(M^{\circ},\pi^{\circ})$ is nondegenerate and $M_{\Sigma}^{\circ}\subset
M^{\circ}$ is a Steinberg slice, it follows from Theorem 2.2 that
$\omega_{\Sigma}^{\circ}$ is a symplectic form. Therefore

$\pi_{\Sigma}^{\circ\\#}\circ\omega_{\Sigma}^{\circ\flat}:TM_{\Sigma}^{\circ}\longrightarrow
TM_{\Sigma}^{\circ}$

is the identity map.

There is a morphism of vector bundles

$\pi_{\Sigma,D}^{\\#}\circ\omega_{\Sigma,D}^{\flat}:T_{D}M_{\Sigma}\longrightarrow
T_{D}M_{\Sigma}.$

For simplicity and since there is no risk of confusion, here we abuse notation
to write $T_{D}M_{\Sigma}$ for the log-tangent bundle of $M_{\Sigma}$ relative
to the normal crossing divisor $M_{\Sigma}\cap D$. This morphism agrees with
the identity map along $M_{\Sigma}^{\circ}$. Therefore it agrees with the
identity map everywhere, and $\pi_{\Sigma}$ is log-symplectic.∎

##  3\. The wonderful compactification

Let $Z_{G}$ be the center of the simply-connected group $G$, and let
$G_{\operatorname{ad}}:=G/Z_{G}$ be its adjoint form. A finite quotient of
Example 2.3 produces a smooth, symplectic family of centralizer subgroups of
$G_{\operatorname{ad}}$ over $\Sigma$. In the next sections we will compactify
the centralizer fibers of this family inside the wonderful compactification of
$G_{\operatorname{ad}}$. First we recall the construction of this universal
centralizer and of the wonderful compactification.

###  3.1. The multiplicative universal centralizer

The natural action of $G$ on itself by conjugation descends to an action of
$G_{\operatorname{ad}}$ on $G$, for which we use the same notation. For every
$h\in G$ we define the _adjoint centralizer_

$Z_{\operatorname{ad}}(h):=\\{a\in G_{\operatorname{ad}}\mid aha^{-1}=h\\}.$

Note that $Z_{\operatorname{ad}}(h)=Z_{G}(h)/Z_{G}$, where $Z_{G}$ is the
center of $G$, and we have the following simple lemma.

######  Lemma 3.1.

Suppose that $h\in G$ is a regular element. Then $Z_{\operatorname{ad}}(h)$ is
connected.

###### Proof.

Let $h=us$ be the Jordan decomposition of $h$ into a unipotent part $u$ and a
semisimple part $s$. Let $L=Z_{G}(s)$ be the centralizer of $s$ in $G$.
Because $G$ is simply-connected, the reductive group $L$ is connected.

Since $h$ is regular, the unipotent element $u$ is regular in $L$ and

$Z_{G}(h)=Z_{L}(u)=Z_{L}\times Z_{U_{L}}(u).$

Here $U_{L}$ is the unique maximal unipotent subgroup of $L$ which contains
$u$, and the second factor $Z_{U_{L}}(u)$ is connected by [Spr, Lemma 4.3].

Write $L_{\operatorname{ad}}:=L/Z_{G}\subset G_{\operatorname{ad}}$ for the
image of $L$ in $G_{\operatorname{ad}}$. We have $Z_{G}\subset Z_{L}$ and
$Z_{L}/Z_{G}=Z_{L_{\operatorname{ad}}}$. The center
$Z_{L_{\operatorname{ad}}}$ is connected because $G_{\operatorname{ad}}$ is of
adjoint type, and therefore

$Z_{\operatorname{ad}}(h)=Z_{G}(h)/Z_{G}\cong Z_{L_{\operatorname{ad}}}\times
Z_{U_{L}}(u)$

is also connected. ∎

######  Definition 3.2.

The (_multiplicative_) _universal centralizer_ associated to $G$ is the affine
variety

$\mathfrak{Z}:=\left\\{(a,h)\in G_{\operatorname{ad}}\times\Sigma\mid a\in
Z_{\operatorname{ad}}(h)\right\\}.$

We will consider the _double_

$\mathbf{D}_{G_{\operatorname{ad}}}:=G_{\operatorname{ad}}\times G,$

which is the quotient of the space $D(G)$ in Example 1.3 by the action of the
finite center $Z_{G}$ on the left. The $G\times G$-action (1.4), the bivector
$\pi$ (1.5), and the moment map $\mu$ (1.6) all descend to
$\mathbf{D}_{G_{\operatorname{ad}}}$. Keeping this notation,
$(\mathbf{D}_{G_{\operatorname{ad}}},\pi,\mu)$ is a nondegenerate quasi-
Poisson $G\times G$-variety.

######  Remark 3.3.

We may view $\mathbf{D}_{G_{\operatorname{ad}}}$ as a constant algebraic group
scheme over $G_{\operatorname{ad}}$. On the other hand, letting $\mathfrak{g}$
be the Lie algebra of $G_{\operatorname{ad}}$ and using the Killing form to
identify $\mathfrak{g}^{*}\cong\mathfrak{g}$, the cotangent bundle

$T^{*}G_{\operatorname{ad}}\cong G_{\operatorname{ad}}\times\mathfrak{g}$

becomes a bundle of Lie algebras. The double

$\mathbf{D}_{G_{\operatorname{ad}}}=G_{\operatorname{ad}}\times G$

is then its simply-connected integration.

In view of Example 2.3, the multiplicative universal centralizer

$\mathfrak{Z}=\mu^{-1}(\Sigma\times\iota(\Sigma))=\\{(a,h)\in
G_{\operatorname{ad}}\times\Sigma\mid aha^{-1}=h\\}$

sits inside $\mathbf{D}_{G_{\operatorname{ad}}}$ as a symplectic Steinberg
slice. In particular, as in [FT], through isomorphism (2.1) $\mathfrak{Z}$ is
equipped with an integrable system given by the invariant generators of
$\mathbb{C}[T]^{W}$.

###  3.2. The wonderful compactification

Let $l$ be the rank of $G$. The _wonderful compactification_
$\overline{G_{\operatorname{ad}}}$ is a canonical, smooth, $G\times
G$-equivariant compactification of $G_{\operatorname{ad}}$ which was
introduced by de Concini and Procesi [dCP]. We recall some of its structure
theory, following [EJ]. It is a smooth projective variety which contains
$G_{\operatorname{ad}}$ as an open dense subset and on which $G$ acts by
extensions of the left- and right-multiplication. The boundary

$D:=\overline{G_{\operatorname{ad}}}\backslash G_{\operatorname{ad}}$

is a simple normal crossing divisor with $l$ irreducible components
$D_{1},\ldots,D_{l}$, indexed by the simple roots.

The $G\times G$ orbits on $\overline{G_{\operatorname{ad}}}$ are in bijection
with subsets of the simple roots in the sense that, for any
$I\subset\\{1,\ldots,l\\}$, the closure of the orbit $\mathcal{O}_{I}$ is the
corresponding partial intersection of divisor components

$\overline{\mathcal{O}_{I}}=\bigcap_{i\not\in I}D_{i}.$

In particular, the closure of each orbit is smooth.

The subset $I\subset\\{1,\ldots,l\\}$ determines a “positive” parabolic
subgroup $P_{I}$, generated by the “positive” Borel $B$ and the simple root
spaces indexed by $I$. Write $P_{I}^{-}$ for the opposite parabolic and
$L_{I}$ for their common Levi component. Let $U^{\pm}_{I}\subset P^{\pm}_{I}$
be the unipotent radicals, and denote by $\mathfrak{p}^{\pm}_{I}$,
$\mathfrak{u}_{I}^{\pm}$, and $\mathfrak{l}_{I}$ the Lie algebras of these
subgroups. Each orbit $\mathcal{O}_{I}$ has a distinguished basepoint

$z_{I}\in\mathcal{O}_{I}$

whose $G\times G$-stabilizer is

(3.4) $\text{Stab}_{G\times G}(z_{I}):=\left\\{(us,vt)\in P_{I}\times
P_{I}^{-}\mid u\in U_{I},v\in U_{I}^{-},s,t\in L_{I},st^{-1}\in
Z_{L_{I}}\right\\}.$

It follows that $\mathcal{O}_{I}$ is a fiber bundle over the product of
partial flag varieties $G/P_{I}\times G/P_{I}^{-}$, with fiber isomorphic to
the adjoint group $L_{I}/Z_{L_{I}}$. This extends to a smooth fibration

${\overline{L_{I}/Z_{L_{I}}}}$${\overline{\mathcal{O}_{I}}}$${G/P_{I}\times
G/P_{I}^{-}}$

whose fiber is the wonderful compactification of $L_{I}/Z_{L_{I}}$.

The wonderful compactification $\overline{G_{\operatorname{ad}}}$ is _log-
homogeneous_ in the sense of [Bri]—that is, the logarithmic infinitesimal
action map

$\text{act}_{D}:\overline{G_{\operatorname{ad}}}\times\mathfrak{g}\times\mathfrak{g}\longrightarrow
T_{D}\overline{G_{\operatorname{ad}}}$

is surjective. In the short exact sequence of vector bundles over
$\overline{G_{\operatorname{ad}}}$

${0}$${\ker(\text{act}_{D})}$${\overline{G_{\operatorname{ad}}}\times\mathfrak{g}\times\mathfrak{g}}$${T_{D}\overline{G_{\operatorname{ad}}}}$${0,}$${\overline{G_{\operatorname{ad}}}}$$\scriptstyle{\text{act}_{D}}$

the kernel $\ker(\text{act}_{D})$ is Lagrangian relative to the Killing form
[Bri, Example 2.5]. It follows that

(3.5) $\ker(\text{act}_{D})\cong
T^{\ast}_{D}\overline{G_{\operatorname{ad}}}.$

This identifies the log-cotangent bundle
$T^{\ast}_{D}\overline{G_{\operatorname{ad}}}$ with a subbundle of the trivial
bundle $\overline{G_{\operatorname{ad}}}\times\mathfrak{g}\times\mathfrak{g}$,
extending the embedding

$\displaystyle T^{*}G_{\operatorname{ad}}\cong
G_{\operatorname{ad}}\times\mathfrak{g}$
$\displaystyle\lhook\joinrel\longrightarrow
G_{\operatorname{ad}}\times\mathfrak{g}\times\mathfrak{g}$
$\displaystyle(a,x)$
$\displaystyle\longrightarrow({a,\operatorname{Ad}_{a}x,x}).$

Under (3.5), the fiber of the log-cotangent bundle at the orbit basepoint
$z_{I}\in\mathcal{O}_{I}$ is

$T^{*}_{D,z_{I}}\overline{G_{\operatorname{ad}}}\cong\mathfrak{p}_{I}\times_{\mathfrak{l}_{I}}\mathfrak{p}^{-}_{I}.$

######  Remark 3.6.

Via (3.5), the log-cotangent bundle
$T^{\ast}_{D}\overline{G_{\operatorname{ad}}}$ is a bundle of Lie algebras
over $\overline{G_{\operatorname{ad}}}$. In analogy with Remark 3.3, we will
show in the next section that it integrates to a smooth subgroup scheme of the
constant group scheme

$\overline{G_{\operatorname{ad}}}\times G\times
G\longrightarrow\overline{G_{\operatorname{ad}}}.$

##  4\. The logarithmic double

In this section we recall the Vinberg monoid, and we use it to construct an
enlargement of the double $\mathbf{D}_{G_{\operatorname{ad}}}$ to a group
scheme $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ over the wonderful
compactification $\overline{G_{\operatorname{ad}}}$. The nondegenerate quasi-
Poisson structure on $\mathbf{D}_{G_{\operatorname{ad}}}$ will extend to a
log-nondegenerate quasi-Poisson structure on
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$.

###  4.1. Construction of $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$

The _Vinberg monoid_ $V_{G}$, introduced in [Vin], is a normal affine
algebraic semigroup whose locus of invertible elements is the _enhanced group_

$G_{\text{enh}}:=G\times_{Z_{G}}T.$

There are natural projections

${G_{\text{enh}}}$${G_{\operatorname{ad}}}$${T}$

—the first is a principal $T$-bundle, and the second is the abelianization of
the group $G_{\text{enh}}.$ These maps extend to

(4.1)
${V_{G}}$${\overline{G_{\operatorname{ad}}}}$${\overline{T}}$$\scriptstyle{\tau}$$\scriptstyle{\alpha}$

so that the first diagram is the pullback of the second along the inclusion
$G_{\text{enh}}\lhook\joinrel\longrightarrow V_{G}$. Here

$\overline{T}=\operatorname{Spec}\mathbb{C}[t^{\alpha_{1}},\ldots,t^{\alpha_{l}}]\cong\mathbb{C}^{l},$

where $\alpha_{1},\ldots,\alpha_{l}$ are the simple roots and
$t^{\lambda}\in\mathbb{C}[T]$ is the function on $T$ given by the weight
$\lambda$. The space $\overline{T}$ is an abelian monoid into which the
adjoint torus embeds as the group of units via the map

$t\longmapsto(\alpha_{1}(t),\ldots,\alpha_{l}(t)).$

The morphism $\alpha$ in (4.1) is the abelianization of $V_{G}$.

The monoid $V_{G}$ carries an action of $G\times G\times T$ that extends the
natural action on the enhanced group. The morphism $\tau$ is $T$-invariant,
and $\alpha$ is $G\times G$-invariant. In particular, every fiber of $\alpha$
contains an open dense $G\times G$-orbit. The _nondegenerate locus_
$\accentset{\circ}{V}_{G}\subset V_{G}$ is the quasi-affine open dense
subvariety whose intersection with each fiber of $\alpha$ is this maximal
orbit. Restricting diagram (4.1), we obtain

${\accentset{\circ}{V}_{G}}$${\overline{G_{\operatorname{ad}}}}$${\overline{T}.}$$\scriptstyle{\accentset{\circ}{\tau}}$$\scriptstyle{\accentset{\circ}{\alpha}}$

Now $\accentset{\circ}{\tau}$ and $\accentset{\circ}{\alpha}$ are smooth
morphisms, $\accentset{\circ}{\tau}$ is a principal $T$-bundle, and the
$G\times G$-stabilizer of any point $v\in\accentset{\circ}{\tau}^{-1}(z_{I})$
is

(4.2) $\text{Stab}_{G\times G}(v)=P_{I}\times_{L_{I}}P_{I}^{-}.$

Let $G\times G$ act on $\overline{G_{\operatorname{ad}}}\times G\times G$ via

$(g,h)\cdot(a,x,y)=(gah^{-1},gxg^{-1},hyh^{-1})$

for $(g,h)\in G\times G$ and $(a,x,y)\in\overline{G_{\operatorname{ad}}}\times
G\times G.$

######  Proposition 4.3.

There is a smooth, closed, $G\times G$-stable subgroup scheme
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}\subset\overline{G_{\operatorname{ad}}}\times
G\times G$ whose fiber over the basepoint
$z_{I}\in\overline{G_{\operatorname{ad}}}$ is

$P_{I}\times_{L_{I}}P_{I}^{-}.$

###### Proof.

Since $\accentset{\circ}{\alpha}$ is smooth, the fiber product
$\accentset{\circ}{V}_{G}\times_{\overline{T}}\accentset{\circ}{V}_{G}$ is a
smooth variety. The action morphism

(4.4) $\displaystyle\accentset{\circ}{V}_{G}\times G\times G$
$\displaystyle\longrightarrow\accentset{\circ}{V}_{G}\times_{\overline{T}}\accentset{\circ}{V}_{G}$
$\displaystyle(v,g,h)$ $\displaystyle\longrightarrow(v,gvh^{-1})$

is smooth and surjective, because every fiber of $\accentset{\circ}{\alpha}$
is a single $G\times G$-orbit. The preimage of the diagonal

$\accentset{\circ}{V}_{G}\lhook\joinrel\longrightarrow\accentset{\circ}{V}_{G}\times_{\overline{T}}\accentset{\circ}{V}_{G}$

under (4.4) is the smooth family of stabilizers

$\mathcal{S}=\left\\{(v,g,h)\in\accentset{\circ}{V}_{G}\times G\times
G\mid(g,h)\in\text{Stab}_{G\times G}(v)\right\\},$

defined for example in [DG, Appendix D].

Because the action of $G\times G$ commutes with the action of $T$, for any
$v\in\accentset{\circ}{V}_{G}$ and any $t\in T$ we have

$\text{Stab}_{G\times G}(v)=\text{Stab}_{G\times G}(t\cdot v).$

Therefore the group scheme of stabilizers $\mathcal{S}$ descends through the
principal $T$-bundle $\accentset{\circ}{\tau}$ to a smooth, closed, $G\times
G$-stable subvariety

$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}\subset\overline{G_{\operatorname{ad}}}\times
G\times G.$

By (4.2), the fiber of $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ over
$z_{I}\in\overline{G_{\operatorname{ad}}}$ is $P_{I}\times_{L_{I}}P_{I}^{-}.$
∎

The group scheme $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, which we
call the _logarithmic double_ , integrates the bundle of Lie algebras given by
the log-cotangent bundle

$T^{\ast}_{D}\overline{G_{\operatorname{ad}}}\subset\overline{G_{\operatorname{ad}}}\times\mathfrak{g}\times\mathfrak{g}$

described in (3.5). Its fiber at the identity element $1\in
G_{\operatorname{ad}}$ is the diagonal subgroup

$\\{(g,g)\mid g\in G\\}\subset G\times G.$

Since $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ is $G\times G$ stable,
it follows that its fiber at any point $a\in G_{\operatorname{ad}}$ is

$\\{(aga^{-1},g)\mid g\in G\\}.$

Therefore the logarithmic double
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ is the closure of the image of
the embedding

(4.5) $\displaystyle\mathbf{D}_{G_{\operatorname{ad}}}$
$\displaystyle\lhook\joinrel\longrightarrow\overline{G_{\operatorname{ad}}}\times
G\times G$ $\displaystyle(a,g)$ $\displaystyle\longmapsto(a,aga^{-1},g).$

The diagram

${\mathbf{D}_{G_{\operatorname{ad}}}}$${\mathbf{D}_{\overline{G_{\operatorname{ad}}}}}$${G_{\operatorname{ad}}}$${\overline{G_{\operatorname{ad}}},}$

is Cartesian, and $\mathbf{D}_{G_{\operatorname{ad}}}$ is exactly the
restriction of $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ to the open
dense copy of $G_{\operatorname{ad}}$ which sits inside
$\overline{G_{\operatorname{ad}}}$.

###  4.2. The quasi-Poisson structure on
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$

In view of the previous section, the nondegenerate quasi-Poisson variety
$(\mathbf{D}_{G_{\operatorname{ad}}},\pi,\mu)$ sits inside the logarithmic
double $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ as an open dense
subset. Its complement is a simple normal crossing divisor, and for simplicity
we abuse notation to denote it by $D$. We will show that the quasi-Poisson
bivector $\pi$ extends to a logarithmic bivector on
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, and that this gives
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ the structure of a log-
nondegenerate quasi-Poisson manifold in the sense of Section 2.3.

On $\overline{G_{\operatorname{ad}}}\times G\times G$, using the notation of
Section 1, define the bivector

(4.6)
$\overline{\pi}=\frac{1}{2}\left(e_{i}^{1L}\wedge(e_{i}^{2L}+e_{i}^{2R})+e_{i}^{1R}\wedge(e_{i}^{3L}+e_{i}^{3R})+e_{i}^{2L}\wedge
e_{i}^{2R}+e_{i}^{3R}\wedge e_{i}^{3L}\right),$

where once again we sum over repeated indices. Define the morphism
$\overline{\mu}$, which extends the moment map
$\mu:\mathbf{D}_{G_{\operatorname{ad}}}\longrightarrow G\times G$ first
defined in (1.6), to be the composition

(4.7)
${\mathbf{D}_{\overline{G_{\operatorname{ad}}}}}$${\overline{G_{\operatorname{ad}}}\times
G\times G}$${G\times G,}$$\scriptstyle{\overline{\mu}}$

where the vertical arrow is

$\displaystyle\overline{G_{\operatorname{ad}}}\times G\times G$
$\displaystyle\longrightarrow G\times G$ $\displaystyle(a,g,h)$
$\displaystyle\longmapsto(g,h^{-1}).$

######  Proposition 4.8.

The bivector $\overline{\pi}$ is tangent to
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, and
$(\mathbf{D}_{\overline{G_{\operatorname{ad}}}},\overline{\pi},\overline{\mu})$
is a quasi-Poisson variety whose unique open dense nondegenerate leaf is
$(\mathbf{D}_{G_{\operatorname{ad}}},\pi,\mu)$.

###### Proof.

It is enough to show that the restriction of $\overline{\pi}$ to

$\mathbf{D}_{G_{\operatorname{ad}}}\subset\overline{G_{\operatorname{ad}}}\times
G\times G$

agrees with $\pi$. This will imply that $\overline{\pi}$ is tangent to
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, which is the closure of
$\mathbf{D}_{G_{\operatorname{ad}}}$. Moreover, since $\pi$ satisfies the
quasi-Poisson condition (1.1) along $\mathbf{D}_{G_{\operatorname{ad}}}$,
$\overline{\pi}$ will satisfy (1.1) along
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$.

Recall that the embedding of $\mathbf{D}_{G_{\operatorname{ad}}}$ into
$\overline{G_{\operatorname{ad}}}\times G\times G$ fits into the commutative
diagram

${D(G)}$${G\times G\times
G}$${\mathbf{D}_{G_{\operatorname{ad}}}}$${\overline{G_{\operatorname{ad}}}\times
G\times G,}$

where $D(G)$ is as defined in Example 1.3. The top horizontal arrow is

$\displaystyle D(G)=G\times G$ $\displaystyle\lhook\joinrel\longrightarrow
G\times G\times G$ $\displaystyle(g,h)$ $\displaystyle\longmapsto(g,gh,hg).$

The bottom horizontal arrow is (4.5), and the vertical arrows are quotients by
the left action of the center $Z_{G}$. Therefore, from Example 1.3, it is
sufficient to check that the pushforward of

$\pi=\frac{1}{2}\left(e_{i}^{1L}\wedge e_{i}^{2R}+e_{i}^{1R}\wedge
e_{i}^{2L}\right)\in\Gamma\left(\wedge^{2}TD(G)\right)$

along the top arrow of this diagram agrees with (4.6).

At the point $(g,gh,hg)$ the vector fields which constitute $\pi$ push forward
to

$\displaystyle e_{i}^{1L}\longmapsto
e_{i}^{1L}+e_{i}^{2L}+(\operatorname{Ad}_{g}e_{i})^{3R}$ $\displaystyle
e_{i}^{1R}\longmapsto
e_{i}^{1R}+(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}+e_{i}^{3R}$ $\displaystyle
e_{i}^{2L}\longmapsto(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}+e_{i}^{3L}$
$\displaystyle e_{i}^{2R}\longmapsto
e_{i}^{2R}+(\operatorname{Ad}_{g}e_{i})^{3R}.$

Therefore, at $(g,gh,hg)$ the bivector $\pi$ is half the expression

(4.9) $\displaystyle e_{i}^{1L}\wedge e_{i}^{2R}$
$\displaystyle+e_{i}^{1L}\wedge(\operatorname{Ad}_{g}e_{i})^{3R}+e_{i}^{2L}\wedge
e_{i}^{2R}$ $\displaystyle+(\operatorname{Ad}_{g}e_{i})^{3R}\wedge
e_{i}^{2R}+e_{i}^{2L}\wedge(\operatorname{Ad}_{g}e_{i})^{3R}$
$\displaystyle+e_{i}^{1R}\wedge
e_{i}^{3L}+e_{i}^{1R}\wedge(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}+e_{i}^{3R}\wedge
e_{i}^{3L}$ $\displaystyle+(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}\wedge
e_{i}^{3L}+e_{i}^{3R}\wedge(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}.$

Since $\operatorname{Ad}_{g}$ and $\operatorname{Ad}_{g^{-1}}$ are orthogonal
operators relative to the Killing form, and since we are summing over repeated
indices, the second terms in the first and third lines simplify:

$\displaystyle
e_{i}^{1L}\wedge(\operatorname{Ad}_{g}e_{i})^{3R}=(\operatorname{Ad}_{g}e_{i})^{1R}\wedge(\operatorname{Ad}_{g}e_{i})^{3R}=e_{i}^{1R}\wedge
e_{i}^{3R};$ $\displaystyle
e_{i}^{1R}\wedge(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}=(\operatorname{Ad}_{g^{-1}}e_{i})^{1L}\wedge(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}=e_{i}^{1L}\wedge
e_{i}^{2L}.$

Moreover, applying orthogonality again, the terms in the last row become

$e_{i}^{3R}\wedge(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}=(\operatorname{Ad}_{g}e_{i})^{3R}\wedge
e_{i}^{2L}$

and

$(\operatorname{Ad}_{g^{-1}}e_{i})^{2L}\wedge
e_{i}^{3L}=e_{i}^{2L}\wedge(\operatorname{Ad}_{g}e_{i})^{3L}=e_{i}^{2R}\wedge(\operatorname{Ad}_{g}e_{i})^{3R}.$

Therefore the second and fourth lines of (4.9) sum to zero, and we see that
(4.9) agrees exactly with (4.6). ∎

######  Proposition 4.10.

The quasi-Poisson variety
$(\mathbf{D}_{\overline{G_{\operatorname{ad}}}},\overline{\pi},\overline{\mu})$
is log-nondegenerate.

###### Proof.

It is clear from (4.6) that $\overline{\pi}$ is a logarithmic bivector,
because the action of $G\times G$ on $\overline{G_{\operatorname{ad}}}$
preserves the boundary divisor. We will check that $\overline{\pi}$ satisfies
condition (2.11)—that the morphism of vector bundles

$\overline{\pi}^{\\#}_{D}\oplus\rho_{D}:T^{*}_{D}\mathbf{D}_{\overline{G_{\operatorname{ad}}}}\oplus\mathfrak{g}\oplus\mathfrak{g}\longrightarrow
T_{D}\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$

is surjective. By $G\times G$-equivariance, is sufficient to check this at a
point of the form
$(z_{I},x,y)\in\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$. We begin by
making a fixed choice of orthonormal basis.

Let $R_{0}$ be the set of weights of the $T$-action on $\mathfrak{g}$, with
multiplicity and including $0$. Write $R^{+}$ for the subset consisting of
positive roots. Choose a basis of generalized eigenvectors

$\mathcal{B}:=\\{E_{\alpha}\mid\alpha\in R_{0}\\}\subset\mathfrak{g}.$

By scaling $E_{\alpha}$ if necessary, we obtain an orthonormal basis

$\\{E_{\alpha}\mid\alpha=0\\}\cup\\{E_{\alpha}\pm E_{-\alpha}\mid\alpha\in
R^{+}\\}$

of $\mathfrak{g}$ relative to the Killing form. The bivector $\overline{\pi}$
from (4.6) becomes

$\overline{\pi}=E_{\alpha}^{1L}\wedge(E_{\alpha}^{2L}+E_{\alpha}^{2R})+E_{\alpha}^{1R}\wedge(E_{\alpha}^{3L}+E_{\alpha}^{3R})+E_{\alpha}^{2L}\wedge
E_{\alpha}^{2R}+E_{\alpha}^{3R}\wedge E_{\alpha}^{3L},$

where once again we sum over the repeated index $\alpha\in R_{0}$.

As in (3.5), the infinitesimal action map

$\mathfrak{g}\times\mathfrak{g}\longrightarrow
T_{D,z_{I}}\overline{G_{\operatorname{ad}}}$

is surjective with kernel
$\mathfrak{p}_{I}\times_{\mathfrak{l}_{I}}\mathfrak{p}_{I}^{-}$. Therefore the
image of $\rho_{D}$ at $(z_{I},x,y)$, which is spanned by the logarithmic
vectors

$\left\\{E_{\alpha}^{1L}+E_{\alpha}^{2L}-E_{\alpha}^{2R},\,E_{\alpha}^{1R}+E_{\alpha}^{3L}-E_{\alpha}^{3R}\right\\},$

contains a subspace of dimension $\dim G$ which is not parallel to the fiber.

Let $\\{\theta_{\alpha}\mid\alpha\in R_{0}\\}$ be the basis of
$\mathfrak{g}^{*}$ dual to $\mathcal{B}$. Since the logarithmic vector fields

$\left\\{E_{\alpha}^{1L}\mid
E_{\alpha}\in\mathfrak{p}_{I}^{-}\right\\}\subset\Gamma(T^{*}_{D}\mathbf{D}_{\overline{G_{\operatorname{ad}}}})$

are linearly independent at
$(z_{I},x,y)\in\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, the
corresponding $1$-forms

$\left\\{\theta_{\alpha}^{1L}\mid
E_{\alpha}\in\mathfrak{p}_{I}^{-}\right\\}\subset\Gamma(T^{*}\mathbf{D}_{G_{\operatorname{ad}}})$

extend to logarithmic $1$-forms in a neighborhood of
$(z_{I},x,y)\in\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$. By the same
argument, the same is true for

$\left\\{\theta_{\alpha}^{1R}\mid
E_{\alpha}\in\mathfrak{p}_{I}\right\\}\subset\Gamma(T^{*}\mathbf{D}_{G_{\operatorname{ad}}}).$

Applying $\overline{\pi}_{D}^{\\#}$ to these logarithmic $1$-forms at
$(z_{I},x,y)\in\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, we obtain

$\overline{\pi}_{D}^{\\#}(\theta_{\alpha}^{1L})=\begin{cases}E_{\alpha}^{2L}+E_{\alpha}^{2R},\quad\qquad\qquad\qquad\text{
if }E_{\alpha}\in\mathfrak{p}_{I}^{-}\backslash\mathfrak{l}_{I}\\\
E_{\alpha}^{2L}+E_{\alpha}^{2R}+E_{\alpha}^{3L}+E_{\alpha}^{3R},\quad\text{ if
}E_{\alpha}\in\mathfrak{l}_{I}\end{cases}$

and

$\overline{\pi}_{D}^{\\#}(\theta_{\alpha}^{1R})=\begin{cases}E_{\alpha}^{3L}+E_{\alpha}^{3R},\quad\qquad\qquad\qquad\text{
if }E_{\alpha}\in\mathfrak{p}_{I}\backslash\mathfrak{l}_{I}\\\
E_{\alpha}^{2L}+E_{\alpha}^{2R}+E_{\alpha}^{3L}+E_{\alpha}^{3R},\quad\text{ if
}E_{\alpha}\in\mathfrak{l}_{I}.\end{cases}$

This implies that the image of $\overline{\pi}^{\\#}_{D}$ contains a subspace
of dimension $\dim G$ which is parallel to the fiber. It follows that, at the
point $(z_{I},x,y)$,

$\dim\left(\operatorname{im}(\overline{\pi}^{\\#}_{D}\oplus\rho_{D})\right)=2\dim
G.$

Therefore this morphism of vector bundles is surjective.∎

##  5\. The partial compactification of $\mathfrak{Z}$

Consider the partially compactified universal centralizer

$\overline{\mathfrak{Z}}=\left\\{(a,h)\in\overline{G_{\operatorname{ad}}}\times\Sigma\mid
a\in\overline{Z_{\operatorname{ad}}(h)}\right\\}.$

By realizing $\overline{\mathfrak{Z}}$ as a Steinberg slice in
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, we will use the results of
the previous sections to show that it is a smooth algebraic variety whose
boundary is a simple normal crossing divisor, and that the symplectic
structure on $\mathfrak{Z}$ defined (up to a finite central quotient) in
Example 2.3 extends to a log-symplectic structure on
$\overline{\mathfrak{Z}}$. We will then describe the symplectic leaves of this
structure.

###  5.1. Construction of $\overline{\mathfrak{Z}}$

We begin by characterizing the image and fibers of the compactified moment map
$\overline{\mu}.$ In Section 2 we defined the quotient map
$\Xi:G\longrightarrow T/W$, whose fibers are the closures of the regular
conjugacy classes. In view of diagram (4.7), the map $\overline{\mu}$ is
proper, and we have the following description of its image.

######  Lemma 5.1.

The image of $\overline{\mu}$ is the closed subvariety

$\Delta:=\left\\{(g,h)\in G\times G\mid\Xi(g)=\Xi(h^{-1})\right\\}$

consisting of pairs of elements $(g,h)\in G\times G$ with the property that
$g$ and $h^{-1}$ lie in the closure of the same conjugacy class.

###### Proof.

Since $\overline{\mu}$ is proper, its image is closed, so it is the closure of
the image of $\mu$. As in (2.4), the image of $\mu$ is the collection of pairs

$\left\\{(g,h)\in G\times G\mid g\text{ is conjugate to }h^{-1}\right\\}.$

The closure of this set is precisely $\Delta$. ∎

######  Lemma 5.2.

The variety $\Delta$ is normal.

###### Proof.

Because $\Delta$ is the image of $\overline{\mu}$, it is irreducible of
dimension

$2\dim G-l.$

Let $f_{1},\ldots,f_{l}\in\mathbb{C}[G]^{G}$ be a set of generators for the
algebra of conjugation-invariant functions on $G$. Then

$\Delta=\left\\{(g,h)\in G\times G\mid f_{i}(g)=f_{i}(h^{-1})\text{ for all
}1\leq i\leq l\right\\}.$

In particular, $\Delta$ is the vanishing locus of exactly $l$ algebraically
independent functions on $G\times G$. Therefore it is a complete intersection.

The regular locus

$\Delta^{\text{r}}=\left\\{(g,h)\in\Delta\mid g\text{ and }h\text{ are
regular}\right\\}$

is a smooth open subset of $\Delta$ because the differentials
$df_{1},\ldots,df_{l}$ are linearly independent at every point of
$G^{\text{r}}$ [Ste, Theorem 1.5]. Moreover, the complement of
$\Delta^{\text{r}}$ in $\Delta$ has codimension at least two [Ste, Theorem
1.3]. It follows that $\Delta$ has no singularities in codimension one, so by
Serre’s criterion it is normal. ∎

######  Lemma 5.3.

The fibers of $\overline{\mu}$ are connected.

###### Proof.

A general fiber of $\overline{\mu}$ is the closure in
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ of a general fiber of $\mu$,
which is connected by Lemma 3.1. Moreover, $\overline{\mu}$ is proper and by
Lemma 5.2 its image is normal. Therefore, by Zariski’s main theorem, all the
fibers of $\overline{\mu}$ are connected. ∎

######  Theorem 5.4.

The variety $\overline{\mathfrak{Z}}$ is smooth and has a natural log-
symplectic Poisson structure whose open dense symplectic leaf is
$\mathfrak{Z}$.

###### Proof.

By Propositions 4.8 and 4.10, $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$
is a log-nondegenerate quasi-Poisson variety whose open dense leaf is the
double $\mathbf{D}_{G_{\operatorname{ad}}}$. There is a commutative diagram of
moment maps

(5.5)
${\mathbf{D}_{G_{\operatorname{ad}}}}$${\mathbf{D}_{\overline{G_{\operatorname{ad}}}}}$${G\times
G.}$$\scriptstyle{\mu}$$\scriptstyle{\overline{\mu}}$

Two elements of $\Sigma$ are in the closure of the same conjugacy class if and
only if they are equal. It follows from Lemma 5.1 that

$\overline{\mu}^{-1}(\Sigma\times\iota(\Sigma))=\overline{\mu}^{-1}(\Sigma_{\Delta}).$

Since $\Sigma\times\iota(\Sigma)$ is a Steinberg cross-section in $G\times G$,
Proposition 2.2 implies that the preimage
$\overline{\mu}^{-1}(\Sigma_{\Delta})$ is a smooth subvariety of
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ with a natural Poisson
structure whose symplectic leaves are the intersections of
$\overline{\mu}^{-1}(\Sigma_{\Delta})$ with the nondegenerate leaves of
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$. This Poisson structure is
log-symplectic by Proposition 2.17. It remains only to show that
$\overline{\mu}^{-1}(\Sigma_{\Delta})$ is isomorphic to
$\overline{\mathfrak{Z}}$.

By Proposition 5.3, the variety $\overline{\mu}^{-1}(\Sigma_{\Delta})$ is
connected. Since it is also smooth, it is irreducible, and therefore it is the
closure in $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ of
$\mu^{-1}(\Sigma_{\Delta})\subset\mathbf{D}_{G_{\operatorname{ad}}}$. In
particular, for any $h\in\Sigma$,

$\overline{\mu}^{-1}(h,h^{-1})=\overline{\mu^{-1}(h,h^{-1})}\cong\overline{Z_{\operatorname{ad}}(h)}\subset\overline{G_{\operatorname{ad}}}.$

It follows that

$\overline{\mu}^{-1}(\Sigma_{\Delta})=\left\\{(a,h,h^{-1})\in\overline{G_{\operatorname{ad}}}\times
G\times G\mid
h\in\Sigma,a\in\overline{Z_{\operatorname{ad}}(h)}\right\\}\cong\overline{\mathfrak{Z}}.$

We obtain a commutative diagram

${\mathfrak{Z}}$${\overline{\mathfrak{Z}}}$${\Sigma,}$

which is the pullback of (5.5) along the embedding
$\Sigma\cong\Sigma_{\Delta}\lhook\joinrel\longrightarrow G\times G$. Since the
horizontal arrow in this diagram is the restriction of a backward-Dirac map,
it is a Poisson morphism. In particular, $\mathfrak{Z}$ sits inside
$\overline{\mathfrak{Z}}$ as the unique open dense symplectic leaf. ∎

###  5.2. Symplectic leaves

By Proposition 2.17, the symplectic leaves of $\overline{\mathfrak{Z}}$ are
the connected components of the intersections of $\overline{\mathfrak{Z}}$
with the nondegenerate leaves of
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$. Therefore we first describe
the nondegenerate leaves of
$(\mathbf{D}_{\overline{G_{\operatorname{ad}}}},\overline{\pi},\overline{\mu})$.
For this we need to analyze image of the (non-logarithmic) bundle map

$\overline{\pi}^{\\#}\oplus\rho:\,T^{*}\mathbf{D}_{\overline{G_{\operatorname{ad}}}}\oplus\mathfrak{g}\oplus\mathfrak{g}\longrightarrow
T\mathbf{D}_{\overline{G_{\operatorname{ad}}}}.$

Fix an index set $I\subset\\{1,\ldots,l\\}$, and write

$\mathfrak{c}_{I}:P_{I}\longrightarrow P_{I}/[P_{I},P_{I}]=:A_{I}$

for the quotient of $P_{I}$ by its derived subgroup. The torus $A_{I}$ is the
“universal torus” associated to the standard parabolic $P_{I}$. We first give
a criterion for when two points in the fiber of
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$ above
$z_{I}\in\overline{G_{\operatorname{ad}}}$ are in the same nondegenerate leaf.

######  Proposition 5.6.

Let $(x,y),(x^{\prime},y^{\prime})\in P_{I}\times_{L_{I}}P_{I}^{-}$. Then
$(z_{I},x,y)$ and $(z_{I},x^{\prime},y^{\prime})$ are in the same
nondegenerate leaf of
$(\mathbf{D}_{\overline{G_{\operatorname{ad}}}},\overline{\pi},\overline{\mu})$
if and only if

$\mathfrak{c}_{I}(x)=\mathfrak{c}_{I}(x^{\prime}).$

######  Remark 5.7.

The value of $\mathfrak{c}_{I}(x)$ depends only on the $L_{I}$-component of
the element

$x\in P_{I}=L_{I}\ltimes U_{I}.$

Since points in $P_{I}\times_{L_{I}}P_{I}^{-}$ are pairs with the same Levi
component, the proposition could instead be stated in an equivalent way
relative to the second coordinate and the negative parabolic $P_{I}^{-}$.

###### Proof.

In order to determine the intersection of the fiber
$\\{z_{I}\\}\times(P_{I}\times_{L_{I}}P_{I}^{-})$ with each nondegenerate
leaf, we will find which vectors in the image of
$\overline{\pi}^{\\#}\oplus\rho$ are tangent to the fibers of
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$.

By (3.4), the kernel of the infinitesimal action map

$\mathfrak{g}\times\mathfrak{g}\longrightarrow
T_{z_{I}}\overline{G_{\operatorname{ad}}}$

is the subalgebra of pairs

$\\{(u+s,v+t)\in\mathfrak{p}_{I}\times\mathfrak{p}_{I}^{-}\mid
u\in\mathfrak{u}_{I},v\in\mathfrak{u}_{I}^{-},s,t\in\mathfrak{l}_{I},s-t\in
Z_{\mathfrak{l}_{I}}\\}.$

We use the same notation as in the proof of Proposition 4.10. Viewed as a
section of $\wedge^{2}T\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, at the
point $(z_{I},x,y)$ the value of the bivector $\overline{\pi}$ is

$\displaystyle\overline{\pi}=\sum_{E_{\alpha}\in\mathfrak{p}_{I}^{-}\backslash
Z_{\mathfrak{l}_{I}}}E_{\alpha}^{1L}\wedge$
$\displaystyle(E_{\alpha}^{2L}+E_{\alpha}^{2R})$
$\displaystyle+\sum_{E_{\alpha}\in\mathfrak{p}_{I}\backslash
Z_{\mathfrak{l}_{I}}}E_{\alpha}^{1R}\wedge(E_{\alpha}^{3L}+E_{\alpha}^{3R})$
$\displaystyle+\sum_{\alpha\in R_{0}}(E_{\alpha}^{2L}\wedge
E_{\alpha}^{2R}+E_{\alpha}^{3R}\wedge E_{\alpha}^{3L}).$

Therefore, the vectors in the image of $\overline{\pi}^{\\#}\oplus\rho$ which
are parallel to the fiber of $\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$
at $z_{I}\in\overline{G_{\operatorname{ad}}}$ are given by the span of

$\displaystyle\left\\{E_{\alpha}^{2L}+E_{\alpha}^{2R}\mid
E_{\alpha}\in\mathfrak{u}_{I}^{-}\right\\}\cup$
$\displaystyle\left\\{E_{\alpha}^{3L}+E_{\alpha}^{3R}\mid
E_{\alpha}\in\mathfrak{u}_{I}\right\\}$
$\displaystyle\qquad\quad\cup\left\\{E_{\alpha}^{2L}+E_{\alpha}^{2R}+E_{\alpha}^{3L}+E_{\alpha}^{3R}\mid
E_{\alpha}\in\mathfrak{l}_{I}\backslash Z_{\mathfrak{l}_{I}}\right\\}.$

At each point this is the tangent space to the fibers of the smooth morphism

$\displaystyle P_{I}\times_{L_{I}}P_{I}^{-}$ $\displaystyle\longrightarrow
A_{I}$ $\displaystyle(x,y)$ $\displaystyle\longrightarrow\mathfrak{c}_{I}(x).$

Since these fibers are connected, it follows that two points $(z_{I},x,y)$ and
$(z_{I},x^{\prime},y^{\prime})$ are in the same nondegenerate leaf if and only
if they have the same image under this map. ∎

Let $\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}$ be the preimage of
$\mathcal{O}_{I}\subset\overline{G_{\operatorname{ad}}}$ under the structure
map

$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}\longrightarrow\overline{G_{\operatorname{ad}}}.$

Since both $\pi$ and $\rho$ are tangent to the boundary of
$\mathbf{D}_{\overline{G_{\operatorname{ad}}}}$, each orbit preimage
$\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}$ is a union of nondegenerate
leaves. To extend the criterion of Proposition 5.6 to this preimage, we define
the following data.

For any $a\in\overline{G_{\operatorname{ad}}}$, there exist group elements
$g,h\in G$ such that $a=gz_{I}h^{-1}$. We associate to this point a
corresponding “positive” parabolic subgroup

$P_{a}:=gP_{I}g^{-1},$

which is well-defined in view of (3.4). There is a canonical identification of
tori

$P_{a}/[P_{a},P_{a}]\cong P_{I}/[P_{I},P_{I}]=A_{I},$

and we denote the corresponding quotient map by

$\mathfrak{c}_{a}:P_{a}\longrightarrow P_{a}/[P_{a},P_{a}]\cong A_{I}.$

The preimage $\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}$ is a locally
trivial $G\times G$-equivariant fiber bundle over $\mathcal{O}_{I}$. In other
words, there is an isomorphism

${\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}}$${(G\times
G)\times_{\text{Stab}_{G\times
G}(z_{I})}(P_{I}\times_{L_{I}}P_{I}^{-})}$${\mathcal{O}_{I},}$$\scriptstyle{\sim}$

Moreover, the map

$\displaystyle(G\times G)\times_{\text{Stab}_{G\times
G}(z_{I})}(P_{I}\times_{L_{I}}P_{I}^{-})$ $\displaystyle\longrightarrow A_{I}$
$\displaystyle[(g,h):(x,y)]$ $\displaystyle\longmapsto\mathfrak{c}_{I}(x)$

is well-defined. Composing it with the isomorphism above, we get a smooth
morphism

$\displaystyle\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}$
$\displaystyle\longrightarrow A_{I}$ $\displaystyle(a,x,y)$
$\displaystyle\longmapsto\mathfrak{c}_{a}(x).$

In the following proposition we show that its fibers are precisely the
nondegenerate quasi-Poisson leaves in
$\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}$.

######  Proposition 5.8.

Two points $(a,x,y),(b,w,z)\in\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}$
are in the same nondegenerate leaf of $\overline{\pi}$ if and only if

$\mathfrak{c}_{a}(x)=\mathfrak{c}_{b}(w).$

###### Proof.

There exist points

$(x^{\prime},y^{\prime}),(w^{\prime},z^{\prime})\in
P_{I}\times_{L_{I}}P_{I}^{-}$

such that $(a,x,y)$ is $G\times G$-conjugate to
$(z_{I},x^{\prime},y^{\prime})$ and $(b,w,z)$ is $G\times G$-conjugate to
$(z_{I},w^{\prime},z^{\prime})$. Since the nondegenerate leaves of
$(\mathbf{D}_{\overline{G_{\operatorname{ad}}}},\overline{\pi})$ are $G\times
G$-stable, $(a,x,y)$ and $(b,w,z)$ are in the same leaf if and only if their
translates $(z_{I},x^{\prime},y^{\prime})$ and $(z_{I},w^{\prime},z^{\prime})$
are in the same leaf. By Proposition 5.6, this occurs if and only if

$\mathfrak{c}_{I}(x^{\prime})=\mathfrak{c}_{I}(w^{\prime}).$

But now $\mathfrak{c}_{a}(x)=\mathfrak{c}_{I}(x^{\prime})$ and
$\mathfrak{c}_{b}(w)=\mathfrak{c}_{I}(w^{\prime})$, and the statement follows.
∎

The orbit stratification on $\overline{G_{\operatorname{ad}}}$ induces a
stratification

$\overline{\mathfrak{Z}}=\bigsqcup\overline{\mathfrak{Z}}_{I}$

on $\overline{\mathfrak{Z}}$, where

$\overline{\mathfrak{Z}}_{I}:=\overline{\mathfrak{Z}}\cap\mathbf{D}_{\overline{G_{\operatorname{ad}}},I}=\left\\{(a,h)\in\overline{G_{\operatorname{ad}}}\times\Sigma\mid
a\in\overline{Z_{\operatorname{ad}}(h)}\cap\mathcal{O}_{I}\right\\}.$

By Theorem 2.2(c), each stratum $\overline{\mathfrak{Z}}_{I}$ is a union of
symplectic leaves, and Proposition 5.8 has the following immediate corollary.

######  Corollary 5.9.

The symplectic leaves of $\overline{\mathfrak{Z}}_{I}$ are the fibers of the
smooth morphism

$\displaystyle\overline{\mathfrak{Z}}_{I}$ $\displaystyle\longrightarrow
A_{I}$ $\displaystyle(a,h)$ $\displaystyle\longmapsto\mathfrak{c}_{a}(h).$

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