# On the classification of multiplicity-free Hamiltonian actions by regular
proper symplectic groupoids

Maarten Mol

###### Abstract.

In this paper we study a natural generalization of symplectic toric manifolds
in the context of regular Poisson manifolds of compact types. To be more
precise, we consider a class of multiplicity-free Hamiltonian actions by
regular proper symplectic groupoids that we call faithful. Given such a
groupoid, we classify its faithful multiplicity-free Hamiltonian actions in
terms of what we call Delzant subspaces of its orbit space – certain
‘suborbifolds with corners’ satisfying the Delzant condition relative to the
integral affine orbifold structure of the orbit space. This encompasses both
the classification of symplectic toric manifolds (due to Delzant) in terms of
Delzant polytopes and the classification of proper Lagrangian fibrations over
an integral affine base manifold (due to Duistermaat) in terms of a sheaf
cohomology group. Each Delzant subspace comes with an orbifold version of this
cohomology, the degree one part of which classifies faithful multiplicity-free
Hamiltonian actions with momentum map image equal to the Delzant subspace,
provided there exists such an action. The obstruction to existence is encoded
by a degree two class in this cohomology: the Lagrangian Dixmier-Douady class.
In addition to the above, we introduce another invariant, which leads to a
variation of our classification result involving only classical sheaf
cohomology and the group cohomology of certain modules for the isotropy groups
of the groupoid.

###### Contents

  1. 1 Delzant subspaces and the momentum map image
    1. 1.1 Classification of maximally toric representations
      1. 1.1.1 Statement of the classification theorem
      2. 1.1.2 Isomorphism versus equivalence of symplectic representations
      3. 1.1.3 Maximally toric representations of tori
      4. 1.1.4 Splittings of abelian group extensions
      5. 1.1.5 The classification of maximally toric representations
    2. 1.2 Delzant subspaces of integral affine orbifolds
      1. 1.2.1 Background on orbifolds
      2. 1.2.2 Background on integral affine structures on orbifolds
      3. 1.2.3 Background on the integral affine orbifold associated to a regular proper symplectic groupoid
      4. 1.2.4 Delzant subspaces
    3. 1.3 On the image of the momentum map
      1. 1.3.1 The symplectic normal representations and the momentum image
      2. 1.3.2 Proofs of Theorem 2 and Proposition 1.43
  2. 2 Classification in the case of symplectic torus bundle actions
    1. 2.1 The sheaf of Lagrangian sections and the structure theorem
      1. 2.1.1 The sheaves of automorphisms and Lagrangian sections
      2. 2.1.2 The structure theorem
    2. 2.2 Constructing a (natural) toric space out of a Delzant subspace
      1. 2.2.1 The result of the construction
      2. 2.2.2 The topology and the action
      3. 2.2.3 The smooth and symplectic structure
      4. 2.2.4 The natural and local dependence
  3. 3 The first structure theorem and the splitting theorem
    1. 3.1 On equivariant sheaves for topological groupoids
      1. 3.1.1 Equivariant sheaves and their cohomology
      2. 3.1.2 Bisections, embedding categories and their cohomology
    2. 3.2 Proof of the first structure theorem
      1. 3.2.1 Lifting to a Lagrangian cocycle
      2. 3.2.2 The rest of the proof
    3. 3.3 Proof of the splitting theorem
      1. 3.3.1 Overview of the proof
      2. 3.3.2 Existence of a faithful multiplicity-free $(\mathcal{B}\Join\mathcal{T},\textrm{pr}_{\mathcal{T}}^{*}\Omega)$-space
      3. 3.3.3 Morita equivalences between pre-symplectic groupoids with corners
      4. 3.3.4 The Lagrangian Dixmier-Douady class
      5. 3.3.5 End of the proof: constructing a principal Hamiltonian bundle out of a Lagrangian cocycle
  4. 4 The second and third structure theorems
    1. 4.1 The ext-invariant and the second structure theorem
      1. 4.1.1 The sheaves of automorphisms and invariant isotropic sections
      2. 4.1.2 The point-wise ext-invariant and proof of the second structure theorem
      3. 4.1.3 The ext-invariant as global section of the ext-sheaf
      4. 4.1.4 Proof of Theorem 4.20 and Remark 4.22
    2. 4.2 The third structure theorem
      1. 4.2.1 Proof of the third structure theorem
      2. 4.2.2 On the action in the third structure theorem
      3. 4.2.3 The structure groups for the case of group actions
  5. A On the multiplicity-free condition
  6. B Background on manifolds with corners

## Introduction

This paper concerns the classification of multiplicity-free Hamiltonian
actions of regular proper symplectic groupoids. Recall that the notion of
Hamiltonian action for symplectic groupoids (introduced in [30]) unifies
various momentum map theories appearing in Poisson geometry, with the common
feature that in each case the momentum map is a Poisson map:

$J:(S,\omega)\to(M,\pi)$

from a symplectic manifold into a specified Poisson manifold $(M,\pi)$,
equipped with a Hamiltonian action of a natural symplectic groupoid
integrating $(M,\pi)$ (see e.g. [46, 52] and the references therein). Here, we
focus on such actions for which the Poisson structure is regular and the
symplectic groupoid is proper, using the framework developed in [7] for such
symplectic groupoids and their associated Poisson structures. Furthermore, the
actions that we consider are multiplicity-free, in the sense of [18]. The main
point of this paper is to give a classification of a large class of such
actions, that we call faithful.

To explain this classification we first focus on actions of symplectic torus
bundles, which form a class of symplectic groupoids analogous to the class of
Lie groups formed by tori. Like tori, these admit a particularly concrete
description in terms of lattices. Indeed, a symplectic torus bundle
$(\mathcal{T},\Omega)$ induces an integral affine structure on its base $M$,
encoded by a Lagrangian lattice bundle $\Lambda$ in the cotangent bundle
$T^{*}M$, and $(\mathcal{T},\Omega)$ is entirely encoded by $\Lambda$. More
precisely, it is canonically isomorphic to the symplectic torus bundle
$(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$ over $M$, where
$\mathcal{T}_{\Lambda}:=T^{*}M/\Lambda$ denotes the bundle of tori associated
to $\Lambda$ (a groupoid over $M$ with multiplication given by fiberwise
addition) and $\Omega_{\Lambda}$ is the symplectic form induced by the
canonical symplectic form $\textrm{d}\lambda_{\textrm{can}}$ on the cotangent
bundle $T^{*}M$ (see e.g. [7, Proposition 3.1.6]). The starting point for our
classification in the case of $(\mathcal{T},\Omega)$-spaces (Theorem 1 below)
was to find the connection between two classical results: the classification
of symplectic toric manifolds in terms of Delzant polytopes (due to Delzant
[12]) and the classification of proper Lagrangian fibrations via their
Lagrangian Chern class (implicit in Duistermaat’s work [14]), recalled in
Examples 1 and 2 below. Inspired by these, we consider Hamiltonian
$(\mathcal{T},\Omega)$-spaces $J:(S,\omega)\to M$ with the three properties
below.

  * i)

The $\mathcal{T}$-action is free on a dense subset of $S$.

  * ii)

$\text{dim}(S)=2\text{dim}(M)$ and the fibers of $J$ are connected.

  * iii)

The momentum map $J:S\to M$ is proper as a map into its image.

We call these faithful toric $(\mathcal{T},\Omega)$-spaces. An equivalent set
of conditions consists of i) together with the two conditions below (cf.
Appendix A).

  * ii’)

The $\mathcal{T}$-orbits coincide with the fibers of $J$.

  * iii’)

The map $S/\mathcal{T}\to M$ induced by $J$ is a topological embedding.

Note that the last condition here is purely topological, as the second
condition means that the induced map $S/\mathcal{T}\to M$ is injective. Like
symplectic toric manifolds, faithful toric $(\mathcal{T},\Omega)$-spaces are
closely related to integrable systems. For instance, in local coordinates for
$M$, the momentum map $J$ of such a $(\mathcal{T},\Omega)$-space is an
integrable system with only elliptic singularities.

######  Theorem 1 (Classification of faithful toric spaces).

Let $(\mathcal{T},\Omega)$ be a symplectic torus bundle over $M$ with induced
integral affine structure $\Lambda$ on $M$.

  * a)

For any faithful toric $(\mathcal{T},\Omega)$-space $J:(S,\omega)\to M$, the
image of the momentum map $J$ is a Delzant subspace of the integral affine
manifold $(M,\Lambda)$ (Definition 1.22; cf. Example 1).

  * b)

For any Delzant subspace $\Delta$ in $(M,\Lambda)$, there is a canonical
bijection:

$\left\\{\begin{aligned} &\text{ }\text{ }\textrm{Isomorphism classes of}\\\
&\textrm{ {faithful} toric $(\mathcal{T},\Omega)$-spaces }\\\ &\textrm{ with
momentum image $\Delta$}\end{aligned}\right\\}\longleftrightarrow
H^{1}(\Delta,\mathcal{L}),$

defined by associating to such a $(\mathcal{T},\Omega)$-space its ‘Lagrangian
Chern class’ (cf. Example 2). Here the right-hand side denotes the degree one
cohomology of the sheaf $\mathcal{L}$ on $\Delta$ consisting of Lagrangian
sections of $\mathcal{T}$ over $\Delta$ (Definition 2.2).

So, there is a canonical bijection:

(1) $\left\\{\begin{aligned} &\text{ }\text{ }\textrm{Isomorphism classes
of\text{ }}\\\ &\textrm{{faithful} toric $(\mathcal{T},\Omega)$-spaces
}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned} &\text{Pairs
$(\Delta,\kappa)$ consisting of a Delzant subspace $\Delta$}\\\ &\quad\text{
of $(M,\Lambda)$ and an element $\kappa$ of $H^{1}(\Delta,\mathcal{L})$
}\end{aligned}\right\\}.$

This theorem is an amalgam of the classifications in the work of Delzant and
Duistermaat mentioned before. The Delzant subspaces appearing in part $a$ are
codimension zero submanifolds with corners that are ‘fully compatible with the
integral affine structure’. The Delzant polytopes in Delzant’s classification
are examples of these (see Example 1 below). On the other hand, the idea
behind the bijection in part $b$ is the same as that in the classification in
Duistermaat’s work (briefly recalled in Example 2 below). As in those
classifications, the integral affine structure plays a key role in Theorem 1.

######  Example 1.

Delzant’s classification of toric symplectic manifolds can be recovered from
Theorem 1, as follows. Let $T$ be a torus with Lie algebra $\mathfrak{t}$ and
character lattice $\Lambda_{T}^{*}$ in $\mathfrak{t}^{*}$. Recall that a
Delzant polytope in the integral affine vector space
$(\mathfrak{t}^{*},\Lambda_{T}^{*})$ is a convex polytope $\Delta$ in
$\mathfrak{t}^{*}$ with the property that at each vertex $x$ the polyhedral
cone $C_{x}(\Delta)$ spanned by the edges meeting at $x$ is generated by a
$\mathbb{Z}$-basis of $\Lambda_{T}^{*}$. Delzant showed that there is a
canonical bijection:

$\left\\{\begin{aligned} &\quad\textrm{ Isomorphism classes of compact, \text{
}}\\\ &\textrm{ connected toric symplectic $T$-spaces
}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned}
\textrm{Delzant polytopes in
$(\mathfrak{t}^{*},\Lambda_{T}^{*})$}\end{aligned}\right\\},$

defined by assigning to such a $T$-space the image of its momentum map. This
can be recovered from the theorem above, applied to the cotangent groupoid
(see e.g. [30, Section 3]):

$(\mathcal{T},\Omega)=(T{\times}\mathfrak{t}^{*},\Omega_{\textrm{can}}),$

by means of the following two observations:

  * •

$\\{\textrm{Delzant polytopes in
$(\mathfrak{t}^{*},\Lambda_{T}^{*})$}\\}=\\{\textrm{Compact, connected Delzant
subspaces of $(\mathfrak{t}^{*},\Lambda_{T}^{*})$}\\}$,

  * •

for any Delzant polytope $\Delta$ the cohomology $H^{1}(\Delta,\mathcal{L})$
vanishes, due to convexity of $\Delta$.

######  Example 2.

For any integral affine manifold $(M,\Lambda)$, $\Delta:=M$ itself is a
Delzant subspace. In this case Theorem 1 recovers the aforementioned
classification of proper Lagrangian fibrations. To explain this, recall that a
proper Lagrangian fibration is a symplectic manifold $(S,\omega)$ together
with a proper surjective submersion $J:(S,\omega)\to M$ with Lagrangian
fibers. We will always assume such fibrations to have connected fibers,
without further notice. Any such fibration over $M$ induces an integral affine
structure $\Lambda$ on $M$. Duistermaat implicitly showed that, given any
fixed integral affine manifold $(M,\Lambda)$, there is a canonical bijection:

(2) $\left\\{\begin{aligned} &\quad\quad\quad\quad\textrm{ Isomorphism classes
of}\\\ &\quad\textrm{ proper Lagrangian fibrations over $M$ }\\\ &\textrm{
with induced integral affine structure $\Lambda$
}\end{aligned}\right\\}\longleftrightarrow H^{1}(M,\mathcal{L})$

that associates to $[J:(S,\omega)\to M]$ its so-called Lagrangian Chern class
(called Lagrangian class in [51, 44]). The key insight leading to (2) is that
fibrations as in (2) can be viewed as certain principal
$\mathcal{T}_{\Lambda}$-bundles, called symplectic
$(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$-torsors in [44, 7]. Indeed, any
such fibration is a Poisson map into $(M,0)$ that admits a (necessarily
unique) Hamiltonian $(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$-action along
it, which turns it into a symplectic
$(\mathcal{T}_{\Lambda},\Omega_{\Lambda})$-torsor (cf. [7, pg. 23, pg. 88]).
From that point of view the construction of the bijection (2) is analogous to
that in the usual classification of principal $G$-bundles in terms of Čech
cohomology. Given a symplectic torus bundle $(\mathcal{T},\Omega)$ over $M$,
the notion of a symplectic $(\mathcal{T},\Omega)$-torsor coincides with that
of a faithful toric $(\mathcal{T},\Omega)$-space with momentum image $M$. So,
in the setting of Theorem 1 there is a canonical bijection:

$\left\\{\begin{aligned} &\quad\quad\quad\quad\textrm{Isomorphism classes
of}\\\ &\quad\textrm{proper Lagrangian fibrations over $M$ }\\\ &\textrm{ with
induced integral affine structure $\Lambda$
}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned} &\text{
}\text{ }\text{ }\textrm{Isomorphism classes of}\\\ &\textrm{ {faithful} toric
$(\mathcal{T},\Omega)$-spaces }\\\ &\text{}\textrm{ with momentum image
$M$}\end{aligned}\right\\}.$

In this way Theorem 1 recovers the classification of proper Lagrangian
fibrations above.

Next, we explain our classification in full generality. Throughout this paper,
we use the following as a working definition for the class of actions that we
consider.

######  Definition.

Let $(\mathcal{G},\Omega)\rightrightarrows M$ be a regular and proper
symplectic groupoid and let $\mathcal{T}$ be the torus bundle over $M$ with
fibers the identity components of the isotropy groups of $\mathcal{G}$ (also
see Subsection 1.2.3). By a faithful multiplicity-free
$(\mathcal{G},\Omega)$-space we mean a Hamiltonian
$(\mathcal{G},\Omega)$-space $J:(S,\omega)\to M$ with the following three
properties.

  * i)

The induced $\mathcal{T}$-action is free on a dense subset of $S$.

  * ii)

The $\mathcal{T}$-orbits coincide with the fibers of $J$.

  * iii)

The transverse momentum map $\underline{J}:\underline{S}\to\underline{M}$ is a
topological embedding.

Here and throughout:

  * •

$\underline{S}:=S/\mathcal{G}$ denotes the orbit space of the
$\mathcal{G}$-action,

  * •

$\underline{M}:=M/\mathcal{G}$ denotes the orbit space of $\mathcal{G}$,

  * •

$\underline{J}$ denotes the map induced by $J$, which we refer to as the
transverse momentum map.

Of course, when $(\mathcal{G},\Omega)$ is a symplectic torus bundle this boils
down to the notion of faithful toric space considered before. As in that case,
the last condition above is purely topological, since $\underline{J}$ is
injective by the second condition. Moreover, there is an equivalent set of
conditions (given in Proposition A.1) of the same type as the first set of
conditions for faithful toric spaces. From these conditions it is clearer that
faithful multiplicity-free $(\mathcal{G},\Omega)$-spaces are closely related
to non-commutative integrable systems (e.g. in the sense of [25]).

For the rest of this introduction, let $(\mathcal{G},\Omega)\rightrightarrows
M$ and $\mathcal{T}$ be as in the above definition. The fact that symplectic
torus bundles induce an integral affine structure on their base generalizes as
follows. The symplectic groupoid $(\mathcal{G},\Omega)$ induces an integral
affine structure, not on $M$, but on its orbit space $\underline{M}$, which
has the structure of an orbifold encoded by the orbifold groupoid
$\mathcal{B}:=\mathcal{G}/\mathcal{T}\rightrightarrows M$. The induced
integral affine structure on this orbifold is encoded by a
$\mathcal{B}$-invariant lattice bundle $\Lambda$ in the co-normal bundle
$\mathcal{N}^{*}\mathcal{F}$ to the orbits of $\mathcal{G}$. This is shown in
[7] and recalled in Subsection 1.2.3. Part $a$ of Theorem 1 generalizes as
follows.

######  Theorem 2.

Let $(\mathcal{G},\Omega)$ be a regular and proper symplectic groupoid. For
any faithful multiplicity-free $(\mathcal{G},\Omega)$-space $J:(S,\omega)\to
M$, the image of the transverse momentum map:

$\underline{J}(\underline{S})\subset\underline{M}$

is a Delzant subspace of the orbit space $\underline{M}$ (in the sense of
Definition 1.28, with respect to the integral affine orbifold structure
mentioned above).

Here, intuitively, one can think of Delzant subspaces as codimension zero
‘suborbifolds with corners’ that are ‘fully compatible with the integral
affine structure’. We split the generalization of part $b$ of Theorem 1 into
two theorems. The first explains to which extent the momentum map image
determines the isomorphism class of a given faithful multiplicity-free
$(\mathcal{G},\Omega)$-space.

######  Theorem 3 (First structure theorem).

Let $(\mathcal{G},\Omega)\rightrightarrows M$ be a regular and proper
symplectic groupoid, let $\underline{\Delta}$ be a Delzant subspace of the
orbit space $\underline{M}$ (in the same sense as in Theorem 2) and let
$\Delta$ be the corresponding invariant subspace of $M$. If:

(3) $\left\\{\begin{aligned} &\quad\quad\text{ }\text{ }\text{
}\textrm{Isomorphism classes of}\\\ &\textrm{ faithful multiplicity-free
$(\mathcal{G},\Omega)$-spaces }\\\ &\quad\text{ }\text{ }\textrm{with momentum
map image $\Delta$}\end{aligned}\right\\}\neq\emptyset,$

then it is a torsor with abelian structure group:

(4) $H^{1}(\mathcal{B}|_{\Delta},\mathcal{L}),$

the degree one cohomology of the $\mathcal{B}|_{\Delta}$-sheaf $\mathcal{L}$
consisting of isotropic sections of $\mathcal{T}$ over $\Delta$ (defined in
Subsection 3.1.1). This action is natural with respect to Morita equivalences
(in the sense of Proposition 3.9).

The second addresses the condition (3).

######  Theorem 4 (Splitting theorem).

In the setting of Theorem 3, the following are equivalent.

  * a)

The condition (3) holds.

  * b)

As pre-symplectic groupoid with corners over $\Delta$ (see Remark 1.29 and
Definition B.22), $(\mathcal{G},\Omega)|_{\Delta}$ and
$(\mathcal{B}\Join\mathcal{T},\textrm{pr}_{\mathcal{T}}^{*}\Omega_{\mathcal{T}})|_{\Delta}$
are Morita equivalent.

  * c)

The Lagrangian Dixmier-Douady class (see Definition 3.14):

(5) $\emph{{c}}_{2}(\mathcal{G},\Omega,\underline{\Delta})\in
H^{2}(\mathcal{B}|_{\Delta},\mathcal{L})$

vanishes.

Here $\mathcal{B}\Join\mathcal{T}$ denotes the semi-direct product groupoid
over $M$ associated to the $\mathcal{B}$-action along $\mathcal{T}\to M$ via
conjugation in $\mathcal{G}$. Moreover, $\Omega_{\mathcal{T}}$ is the
restriction to $\mathcal{T}$ of the symplectic form $\Omega$ on $\mathcal{G}$.

######  Remark 1.

Theorem 3 and the proof of Theorem 4 lead to a sharper conclusion in the
setting of Theorem 1. Indeed, Theorem 2.11 shows that for any Delzant subspace
$\Delta$ of $(M,\Lambda)$ there is a canonically associated faithful toric
$(\mathcal{T},\Omega)$-space $J_{\Delta}:(S_{\Delta},\omega_{\Delta})\to M$
with momentum image $\Delta$. So, there is a canonical section of the map:

$\left\\{\begin{aligned} &\quad\textrm{Isomorphism classes of}\\\ &\textrm{
faithful toric $(\mathcal{T},\Omega)$-spaces
}\end{aligned}\right\\}\longrightarrow\\{\textrm{Delzant subspaces of
$(M,\Lambda)$}\\}$

defined by sending such a $(\mathcal{T},\Omega)$-space to its momentum map
image. Combined with Theorem 3 this leads to the bijection (1), where a pair
$(\Delta,\kappa)$ corresponds to the isomorphism class obtained by letting
$\kappa$ act on the isomorphism class of $J_{\Delta}$. More generally, Theorem
3 leads to such a bijection when $(\mathcal{G},\Omega)$ is a semi-direct
product symplectic groupoid
$(\mathcal{B}\Join\mathcal{T},\textrm{pr}_{\mathcal{T}}^{*}\Omega_{\mathcal{T}})$,
with $\mathcal{B}$ an etale orbifold groupoid acting on a symplectic torus
bundle $(\mathcal{T},\Omega_{\mathcal{T}})$ as in Subsection 3.3.2 (by the
same reasoning, using Proposition 3.10 in addition to Theorem 2.11).

######  Example 3.

Let $G$ be an infinitesimally abelian compact Lie group, meaning that its Lie
algebra $\mathfrak{g}$ is abelian. For Hamiltonian $G$-spaces, the outcome of
Theorems 3 and 4 can be rephrased in terms of the group $G$. To be more
precise, recall that Hamiltonian $G$-spaces correspond to Hamiltonian
$(\mathcal{G},\Omega)$-spaces for the cotangent groupoid:

$(\mathcal{G},\Omega):=(G\ltimes\mathfrak{g}^{*},\Omega_{\textrm{can}})$

over $\mathfrak{g}^{*}$ (see e.g. [30]). A compact and connected Hamiltonian
$G$-space $J:(S,\omega)\to\mathfrak{g}^{*}$ corresponds to a faithful
multiplicity-free $(\mathcal{G},\Omega)$-space if and only if
$J:(S,\omega)\to\mathfrak{g}^{*}$ is a toric symplectic manifold with respect
to the induced action of the identity component $T$ of $G$ (a torus). So,
these are toric symplectic manifolds with additional symmetry, which is
reflected by the fact that the Delzant polytope corresponding to such a
$T$-space (its momentum map image) is invariant with respect to the induced
action of the finite group $\varGamma:=G/T$ on $\mathfrak{g}^{*}$. Let us call
such Hamiltonian $G$-spaces maximally toric. Our theorems lead to the
following conclusions, which (somewhat surprisingly) we could not find in the
literature. Let $\Delta$ be a $\varGamma$-invariant Delzant polytope in
$(\mathfrak{g}^{*},\Lambda_{T}^{*})$ and let $J:(S,\omega)\to\mathfrak{g}^{*}$
be a toric symplectic $T$-space with momentum map image $\Delta$. The latter
is unique up to $T$-equivariant symplectomorphism and any isomorphism class of
maximally toric Hamiltonian $G$-spaces with momentum map image $\Delta$ can be
represented by a Hamiltonian $G$-action on $(S,\omega)$ with momentum map $J$
(cf. Example 1).

  * •

The condition (3) can be rephrased as requiring that the symplectic $T$-action
on $(S,\omega)$ extends to such an action of $G$ that is compatible with the
$\varGamma$-action on $\Delta$. It follows from Theorem 4 that this holds if
and only if the short exact sequence of groups:

(6) $1\to T\to G\to\varGamma\to 1$

admits a right splitting (i.e. the extension is trivial up to isomorphism). Of
course, there is a natural group cohomology class in $H^{2}(\varGamma,T)$ that
encodes the obstruction to the existence of such a splitting, where $T$ is
viewed as $\varGamma$-module via the action by conjugation in $G$. This class
is recovered by the Lagrangian Dixmier-Douady class via a canonical
isomorphism between $H^{2}(\varGamma,T)$ and
$H^{2}(\mathcal{B}|_{\Delta},\mathcal{L})$ (see Proposition 4.30$a$).

  * •

Theorem 3 explains in which different ways the $T$-action on $(S,\omega)$
extends to a $G$-action as above. Namely: if the $T$-action extends, then up
to isomorphism of Hamiltonian $G$-spaces there is exactly one such $G$-action
for each group cohomology class in $H^{1}(\varGamma,T)$. This is because the
structure group (4) is isomorphic to $H^{1}(\varGamma,T)$ (see Proposition
4.30$a$). Explicitly, given such a $G$-action, the $H^{1}(\varGamma,T)$-action
of Theorem 3 twists the $G$-action by a group $1$-cocycle ${c}:\varGamma\to
T$:

$g\cdot_{{c}}p=g{c}([g])\cdot p,\quad g\in G,\quad p\in S.$

So, the $T$-action extends in different ways, each corresponding to an
automorphism of the extension (6), modulo automorphisms given by conjugation
by elements of $T$.

######  Example 4.

Example 2 generalizes as follows: for source-connected
$(\mathcal{G},\Omega)\rightrightarrows M$, faithful multiplicity-free
$(\mathcal{G},\Omega)$-spaces with momentum image $\Delta=M$ correspond to
certain proper isotropic fibrations over $M$. To elaborate, recall that proper
isotropic fibrations are defined by replacing ‘Lagrangian’ by ‘isotropic’ in
Example 2, with the additional requirement that the symplectic orthogonal
$\textrm{Ker}(\textrm{d}J)^{\omega}$ is an involutive distribution (these are
symplectically complete isotropic fibrations in the sense of [11]). Any such
fibration $J:(S,\omega)\to M$ induces a regular Poisson structure $\pi$ on
$M$, uniquely determined by the property that $J:(S,\omega)\to(M,\pi)$ is a
Poisson map (as a consequence of Libermann’s theorem [28]). Moreover, as shown
in [11], such a fibration induces a transverse integral affine structure to
the foliation $\mathcal{F}_{\pi}$ on $M$ by symplectic leaves of $\pi$,
encoded by a lattice bundle $\Lambda$ in the co-normal bundle
$\mathcal{N}^{*}\mathcal{F}_{\pi}$. Suppose now that $\mathcal{F}_{\pi}$ is of
proper type, meaning that its holonomy groupoid
$\text{Hol}(M,\mathcal{F}_{\pi})$ is proper. It is shown in [7] that there is
a natural source-connected proper symplectic groupoid:

(7) $(\textrm{Hol}_{J}(M),\Omega_{J})\rightrightarrows M$

associated to $J:(S,\omega)\to M$, with the following properties.

  * •

It integrates $(M,\pi)$.

  * •

The (transverse) integral affine structure induced by
$(\textrm{Hol}_{J}(M),\Omega)$ is that induced by $J$.

  * •

The associated orbifold structure on $\underline{M}$ is that encoded by
$\text{Hol}(M,\mathcal{F}_{\pi})$, because the associated orbifold groupoid is
the integration $\textrm{Hol}(M,\mathcal{F}_{\pi})$ of $\mathcal{F}_{\pi}$.

The map $J:(S,\omega)\to M$ admits a canonical Hamiltonian
$(\textrm{Hol}_{J}(M),\Omega)$-action, which is makes it a faithful
multiplicity-free $(\textrm{Hol}_{J}(M),\Omega)$-space. More generally, for
any orbifold groupoid $\mathcal{B}$ integrating $\mathcal{F}_{\pi}$, there is
a natural integration $(\mathcal{G},\Omega)$ of $(M,\pi)$ associated to $J$,
for which the associated orbifold groupoid $\mathcal{G}/\mathcal{T}$ is the
integration $\mathcal{B}$ of $\mathcal{F}_{\pi}$. This is called the
$\mathcal{B}$-integration of $(M,\pi)$ relative to $J$ ([7, pg. 82]). It also
induces the same (transverse) integral affine structure as $J$ and acts
canonically along $J$, making it a faithful multiplicity-free
$(\mathcal{G},\Omega)$-space. This explains how such proper isotropic
fibrations can naturally be viewed as faithful multiplicity-free spaces. For
the converse, let $(\mathcal{G},\Omega)$ be any source-connected, regular and
proper symplectic groupoid, let $\mathcal{B}=\mathcal{G}/\mathcal{T}$ and let
$\pi$ be the induced Poisson structure on its base $M$. Then for any faithful
multiplicity-free $(\mathcal{G},\Omega)$-space $J:(S,\omega)\to M$ with
$J(S)=M$, the momentum map is a proper isotropic fibration that induces this
same Poisson structure on $M$. In fact, $(\mathcal{G},\Omega)$ is the
$\mathcal{B}$-integration of $(M,\pi)$ relative to $J$. An isomorphism of two
such proper isotropic fibrations:

${(S_{1},\omega_{1})}$${(S_{2},\omega_{2})}$${M}$$\scriptstyle{\sim}$$\scriptstyle{J_{1}}$$\scriptstyle{J_{2}}$

is automatically $\mathcal{G}$-equivariant, or in other words, it is an
isomorphism of Hamiltonian $(\mathcal{G},\Omega)$-spaces. Hence, there is a
canonical bijection:

$\left\\{\begin{aligned} &\quad\quad\quad\textrm{ Isomorphism classes of }\\\
&\textrm{proper isotropic fibrations over $(M,\pi)$}\\\
&\quad\quad\quad\textrm{with $\mathcal{B}$-integration
$(\mathcal{G},\Omega)$}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned}
&\quad\quad\quad\text{ }\textrm{ Isomorphism classes of}\\\ &\textrm{ faithful
multiplicity-free $(\mathcal{G},\Omega)$-spaces }\\\
&\quad\quad\quad\textrm{with momentum image $M$}\end{aligned}\right\\}.$

This explains the relationship between faithful multiplicity-free
$(\mathcal{G},\Omega)$-spaces with momentum image $\Delta=M$ and proper
isotropic fibrations over $M$. Finally, let us point out: by the splitting
theorem, $(\mathcal{G},\Omega)$ is the $\mathcal{B}$-integration relative to
some proper isotropic fibration if and only if $(\mathcal{G},\Omega)$ is
Morita equivalent to
$(\mathcal{B}\Join\mathcal{T},\textrm{pr}_{\mathcal{T}}^{*}\Omega_{\mathcal{T}})$.
This recovers a result of [7].

Our final two main theorems provide an alternative way of listing the elements
of (3), by means of an additional invariant of faithful multiplicity-free
$(\mathcal{G},\Omega)$-spaces that we call the ext-invariant (Definition
4.21). This invariant is a global section of what we call the ext-sheaf
(Definition 4.18):

(8)
$\mathcal{I}^{1}=\mathcal{I}^{1}_{(\mathcal{G},\Omega,\underline{\Delta})}.$

This is a sheaf of sets on $\underline{\Delta}$ naturally associated to
$(\mathcal{G},\Omega)$. Its stalk at an orbit $\mathcal{O}_{x}$ of
$\mathcal{G}$ through a point $x\in\Delta$ is the set:

$I^{1}(\mathcal{G}_{x},\mathcal{T}_{x}),$

consisting of right-splittings of the short exact sequence of isotropy groups:

(9) $1\to\mathcal{T}_{x}\to\mathcal{G}_{x}\to\mathcal{B}_{x}\to 1,$

modulo the $\mathcal{T}_{x}$-action by conjugation (cf. Definition 1.4 and
Proposition 1.11).

######  Theorem 5 (Second structure theorem).

Suppose that we are in the setting of Theorem 3 and let $e$ be a global
section of the ext-sheaf (8). If:

(10) $\left\\{\begin{aligned} &\quad\quad\text{ }\text{ }\textrm{Isomorphism
classes of}\\\ &\textrm{ faithful multiplicity-free
$(\mathcal{G},\Omega)$-spaces }\\\ &\quad\textrm{ with momentum map image
$\Delta$}\\\ &\quad\quad\textrm{ and with ext-invariant $e$
}\end{aligned}\right\\}\neq\emptyset,$

then it is a torsor with abelian structure group:

(11) $H^{1}(\underline{\Delta},\underline{\mathcal{L}}),$

the degree one cohomology of the sheaf $\underline{\mathcal{L}}$ on
$\underline{\Delta}$ associated to the $\mathcal{B}|_{\Delta}$-sheaf
$\mathcal{L}$ by considering $\mathcal{B}$-invariant sections (see Subsection
4.1.1). This action is natural with respect to Morita equivalences (in the
sense explained in Subsection 4.1.2).

The structure groups in the first and second structure theorem are related by
a natural injective group homomorphism:

(12) $H^{1}(\underline{\Delta},\underline{\mathcal{L}})\hookrightarrow
H^{1}(\mathcal{B}|_{\Delta},\mathcal{L}),$

which is compatible with the actions in these theorems. This leads to:

######  Theorem 6 (Third structure theorem).

In the setting of Theorem 3, if (3) holds, then the image of the map:

(13) $\left\\{\begin{aligned} &\quad\quad\quad\text{
}\text{}\textrm{Isomorphism classes of}\\\ &\textrm{ faithful multiplicity-
free $(\mathcal{G},\Omega)$-spaces }\\\ &\quad\quad\textrm{with momentum map
image
$\Delta$}\end{aligned}\right\\}\longrightarrow\mathcal{I}^{1}(\underline{\Delta})$

that associates to an isomorphism class of such a $(\mathcal{G},\Omega)$-space
its ext-invariant, is a torsor with abelian structure group the quotient:

(14)
$\frac{H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})}{H^{1}(\underline{\Delta},\underline{\mathcal{L}})}.$

This action is natural with respect to Morita equivalences as well.

######  Remark 2.

The injection (12) fits in an exact sequence:

(15) $0\to H^{1}(\underline{\Delta},\underline{\mathcal{L}})\to
H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})\to\mathcal{H}^{1}(\underline{\Delta})\to
H^{2}(\underline{\Delta},\underline{\mathcal{L}})\to
H^{2}(\mathcal{B}|_{\Delta},\mathcal{L}),\quad\mathcal{H}^{1}:=R^{1}(q_{\Delta})_{*}(\mathcal{L}),$

which is the exact sequence in low degrees of a Leray-type spectral sequence
for the canonical map of topological groupoids $q_{\Delta}$ from
$\mathcal{B}|_{\Delta}$ to the unit groupoid of $\underline{\Delta}$. Here
$\mathcal{H}^{1}$ denotes the first right-derived functor of the push-forward
along $q_{\Delta}$ applied to the $\mathcal{B}|_{\Delta}$-sheaf $\mathcal{L}$.
In Subsection 4.2.2 we show that there is a natural free and transitive action
of the sheaf of abelian groups $\mathcal{H}^{1}$ on $\mathcal{I}^{1}$.
Moreover, we show that the action of (14) on the image of (13) in Theorem 6
(which is inherited from the action of (4) in Theorem 3) factors through this
$\mathcal{H}^{1}$-action via the middle map in (15).

Together, the second and third structure theorem provide a way of listing the
isomorphism classes of faithful multiplicity-free
$(\mathcal{G},\Omega)$-spaces different from that in the first structure
theorem.

######  Example 5.

To illustrate this difference we return to Example 3. From the second and
third structure theorem and the splitting theorem we obtain a canonical
bijection:

$\left\\{\begin{aligned} &\text{ Isomorphism classes of compact, connected,
}\\\ &\quad\text{ {maximally} toric Hamiltonian $G$-spaces
}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned}
&\text{$\varGamma$-invariant Delzant}\\\ &\text{polytopes in
$(\mathfrak{g}^{*},\Lambda_{T}^{*})$}\end{aligned}\right\\}\times I^{1}(G,T),$

that associates to a class $[J:(S,\omega)\to\mathfrak{g}^{*}]$ the pair
consisting of the image $\Delta$ of $J$ and the germ of its ext-invariant at
any $\varGamma$-fixed point in $\Delta$. This is because for any
$\varGamma$-invariant Delzant polytope $\Delta$ in
$(\mathfrak{g}^{*},\Lambda_{T}^{*})$ the following hold.

  * •

The structure group (11) is trivial, due to convexity of $\Delta$ (see
Proposition 4.30$b$).

  * •

As mentioned before: the structure group (4) is naturally isomorphic to the
degree one group cohomology $H^{1}(\varGamma,T)$.

  * •

The set of global sections of (8) is naturally in bijection with the set
$I^{1}(G,T)$ consisting of right splittings of (6) modulo the $T$-action by
conjugation. Moreover, under this and the previous identification the action
of (4) on the set of global sections of (8) is identified with the free and
transitive $H^{1}(\varGamma,T)$-action on $I^{1}(G,T)$ obtained by viewing
$H^{1}(\varGamma,T)$ as the group of automorphisms of the extension (6) modulo
the subgroup of automorphisms given by conjugation by elements of $T$ (see
Proposition 4.30$c$).

To further illustrate the use of the second and third structure theorem, in
Subsection 4.1.3 we give a more concrete example (Example 4.23) of a
symplectic groupoid and a Delzant subspace to which we apply these theorems,
in order to obtain an explicit list of all isomorphism classes of faithful
multiplicity-free actions of that symplectic groupoid with that Delzant
subspace as their momentum map image.

In fact, the above theorems apply to a more general class of Hamiltonian group
actions than that in Example 5, to give the following classification.

######  Example 6.

Let $G$ be any compact Lie group. A compact and connected Hamiltonian
$G$-space is called multiplicity-free if all its symplectic reduced spaces are
points. Let us call a Hamiltonian $G$-space regular if its momentum map image
is contained in $\mathfrak{g}^{*}_{\textrm{reg}}$ —the union of coadjoint
orbits of maximal dimension. Fix a maximal torus $T$ and an open Weyl chamber
$\mathfrak{c}$ in $\mathfrak{t}^{*}$. Further, let $N(\mathfrak{c})$ be the
normalizer of $\mathfrak{c}$ in $G$. It follows from the second and third
structure theorem and the splitting theorem (see Example 4.31) that compact,
connected, regular, multiplicity-free Hamiltonian $G$-spaces are classified up
to isomorphism by triples of:

  * •

a closed normal subgroup $K$ of $N(\mathfrak{c})$ that is contained in $T$,

  * •

an $N(\mathfrak{c})$-invariant polytope $\Delta$ in $\mathfrak{c}$ that is
Delzant as polytope in an integral affine subspace of
$(\mathfrak{t}^{*},\Lambda_{T}^{*})$ parallel to
$((\mathfrak{t}/\mathfrak{k})^{*},\Lambda_{T/K}^{*})$,

  * •

an element of $I^{1}(N(\mathfrak{c})/K,T/K)$.

Whilst for connected $G$ Examples 5 and 6 are a simple case of the existing
classification of all multiplicity-free Hamiltonian actions of such Lie groups
[12, 13, 48, 22] (since in that case $N(\mathfrak{c})=T$), for disconnected
$G$ it seems that there are no such classifications in the literature.

Finally, let us mention that the main theorems above readily generalize to
quasi-Hamiltonian actions of regular, proper, twisted pre-symplectic groupoids
(in the sense of [50, 5]), essentially because every regular twisted pre-
symplectic groupoid is Morita equivalent to a regular (non-twisted) symplectic
groupoid. In particular, for any compact Lie group $G$, this applies to a
class of quasi-Hamiltonian $G$-spaces (in the sense of [1]) for which the
momentum map image is contained in the regular part of $G$. This provides some
new insight into the classification problem for multiplicity-free quasi-
Hamiltonian $G$-spaces, for in the current results on this [23] the Lie group
$G$ is assumed to be simply-connected.

Brief outline: In Part 1 we introduce Delzant subspaces and prove that the
image of the momentum map of a faithful multiplicity-free Hamiltonian action
is such a subspace (Theorem 2). The classification for the case of symplectic
torus bundles (Theorem 1) is proved in Part 2. In Part $3$ we prove the first
structure theorem and the splitting theorem (Theorem 3 and Theorem 4) and in
Part 4 we introduce the ext-invariant and the ext-sheaf, and prove the second
and third structure theorems (Theorem 5 and Theorem 6). Apart from the section
on the third structure theorem, Part 4 can be read independently of Part 3.
More detailed outlines can be found at the beginning of each of these parts.

Acknowledgements: First, I wish to thank my PhD supervisor Marius Crainic for
suggesting this project to me, for useful discussions on the cohomology of
orbifold sheaves and for commenting on an earlier version of this text. The
latter has surely helped me to improve the exposition. Furthermore, I would
like to thank Rui Loja Fernandes for sharing a private note on proper
isotropic fibrations with me. That short note contains a statement similar to
the outcome of the first structure theorem above when applied to the
particular case of Example 4, which was a source of inspiration for the
statement of the first structure theorem. This work was supported by NWO (Vici
Grant no. 639.033.312) and the Max Planck Institute for Mathematics.

Conventions: Throughout, we require smooth manifolds (with or without corners)
to be both Hausdorff and second countable and we require the same for both the
base and the space of arrows of a Lie groupoid (with or without corners).
Furthermore, given a groupoid $\mathcal{G}\rightrightarrows X$ we use the
notation $\underline{X}:=X/\mathcal{G}$ for its orbit space and we denote
subsets of $\underline{X}$ as $\underline{A}$, where $A$ denotes the
corresponding invariant subset of $X$.

##  1\. Delzant subspaces and the momentum map image

This part concerns the properties of the image of the momentum map and its
relation to those of the Hamiltonian action.

In Section 1.1 we formulate and prove linear versions of the classification
results in Example 3 and Example 5, which illustrate our classification
results in a particularly simple case. Moreover, these form the foundation for
the construction of the ext-invariant and the proof of Theorem 5, given in
Part 4. In Section 1.2 we introduce Delzant subspaces, present some of their
basic features and recall the necessary background on integral affine
orbifolds. This can be read independently from Section 1.1, except for the
fact that some of the terminology on linear integral affine geometry used in
Section 1.2 is introduced in Subsection 1.1.3. Lastly, in Section 1.3 we prove
Theorem 2 and Proposition 1.43, which connect the objects studied in Section
1.1 to those in Section 1.2 and explain how to read off some invariants of the
Hamiltonian action from the momentum map image.

###  1.1. Classification of maximally toric representations

####  1.1.1. Statement of the classification theorem

Let us first introduce some terminology.

######  Definition 1.1.

By a symplectic representation $(V,\omega)$ of a Lie group $H$ we mean finite-
dimensional real symplectic vector space $(V,\omega)$ together with a morphism
$H\to\textrm{Sp}(V,\omega)$ into the Lie group of linear symplectic
automorphisms of $(V,\omega)$.

For any such representation $(V,\omega)$, the $H$-action is Hamiltonian with
equivariant momentum map:

(16) $J_{V}:(V,\omega)\to\mathfrak{h}^{*},\quad\langle
J_{V}(v),\xi\rangle=\frac{1}{2}\omega(\xi\cdot v,v).$

######  Definition 1.2.

We call a Lie group $H$ infinitesimally abelian if its Lie algebra is abelian.
Given an infinitesimally abelian compact Lie group $H$ with identity component
$T$, we call a symplectic representation $H\to\textrm{Sp}(V,\omega)$ maximally
toric if the induced torus representation $T\to\textrm{Sp}(V,\omega)$ is
faithful and $\text{dim}(V)=2\text{dim}(T)$.

The representations in this definition are linear versions of the Hamiltonian
group actions considered in Example 3 and Example 5. The theorem below is a
linear version of the classification result in Example 5. It will also be
important later for understanding the local properties of general faithful
multiplicity-free actions.

######  Theorem 1.3.

Let $H$ be an infinitesimally abelian compact Lie group with identity
component $T$ and let $\varGamma:=H/T$ be the group of connected components of
$H$. There is a canonical bijection:

$\left\\{\begin{aligned} &\quad\quad\text{Isomorphism classes of\text{ }}\\\
&\text{ maximally toric $H$-representations
}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned} &\text{
\quad\quad Smooth \& $\varGamma$-invariant }\\\ &\text{ pointed polyhedral
cones in $(\mathfrak{t}^{*},\Lambda_{T}^{*})$ }\end{aligned}\right\\}\times
I^{1}(H,T)$

defined by sending the class of such an $H$-representation $(V,\omega)$ to the
pair consisting of the image $\Delta_{V}$ of its momentum map (16) and its
ext-class $\textrm{e}(V,\omega)$ (see Definition 1.13).

In this section we explain and prove this statement, which is a modest
extension of a well-known result for representations of tori (Proposition 1.9
recalled below). First, we elaborate on the set $I^{1}(H,T)$. Notice that $T$
is a torus and the quotient ${\varGamma}=H/T$ is a finite group. Furthermore,
we have a canonical short exact sequence of Lie groups:

(17) $1\to T\to H\to\varGamma\to 1,$

via which we view $H$ as an extension of $\varGamma$ by the abelian group $T$
and we view $T$ both as an $H$-module and as a $\varGamma$-module, equipped
with the actions by conjugation in $H$. We denote by $H^{1}(H,T)$ the degree
one group cohomology of the $H$-module $T$, represented throughout this
section as the abelian group of maps ${c}:H\to T$ satisfying the cocycle
condition:

${c}(h_{1}h_{2})=({c}(h_{1})\cdot h_{2}){c}(h_{2}),\quad h_{1},h_{2}\in H,$

modulo the subgroup of those ${c}:H\to T$ for which there is a $t\in T$ such
that ${c}$ is given by:

${c}(h)=(t\cdot h)t^{-1},\quad h\in H.$

######  Definition 1.4.

Let $H$ be an infinitesimally abelian compact Lie group with identity
component $T$. We let $I^{1}(H,T)$ denote the subset of $H^{1}(H,T)$
consisting of cohomology classes whose representatives restrict to the
identity map on $T$.

######  Remark 1.5.

If $I^{1}(H,T)$ is non-empty, then it is naturally a torsor with abelian
structure group $H^{1}(\varGamma,T)$ (the degree one group cohomology), which
makes it amenable to computation in explicit examples. Here, the action is
defined by assigning to $[\kappa]\in H^{1}(\varGamma,T)$ and $[{c}]\in
I^{1}(H,T)$ the class $[\kappa]\cdot[{c}]$ represented by the $1$-cocycle
$H\to T$ that maps $h\in H$ to $\kappa([h]){c}(h)\in T$.

The set $I^{1}(H,T)$ is non-empty if and only if $H$ is split (by Proposition
1.11), in the sense below.

######  Definition 1.6.

We will say that an infinitesimally abelian compact Lie group $H$ is split if
the short exact sequence of groups (17) admits a right-splitting.

In Subsection 1.1.4 we will prove the proposition below, which is a linear
version of the forward implication in the Theorem 4. In view of this
proposition, the statement of Theorem 1.3 is trivially true when $H$ is not
split, since in this case both sets in the bijection are empty.

######  Proposition 1.7.

Let $H$ be an infinitesimally abelian compact Lie group. If $H$ admits a
maximally toric representation, then it is split.

The proof of this will reveal the group cohomology class (28). In the coming
subsections we will explain the remaining parts of the statement and give a
proof of Theorem 1.3.

####  1.1.2. Isomorphism versus equivalence of symplectic representations

Throughout, we will use the following notions of equivalence between
symplectic representations.

######  Definition 1.8.

An isomorphism of symplectic $H$-representations is an $H$-equivariant
symplectic linear isomorphism. Given two Lie groups $H_{1}$ and $H_{2}$, by an
equivalence between a symplectic $H_{1}$-representation $(V_{1},\omega_{1})$
and a symplectic $H_{2}$-representation $(V_{2},\omega_{2})$ we mean a pair of
maps:

(18) $(\varphi,\psi):(H_{1},(V_{1},\omega_{1}))\to(H_{2},(V_{2},\omega_{2}))$

consisting of an isomorphism of Lie groups $\varphi:H_{1}\to H_{2}$ and a
symplectic linear isomorphism $\psi:(V_{1},\omega_{1})\to(V_{2},\omega_{2})$
that are compatible with the actions, in the sense that:

$\varphi(h)\cdot\psi(v)=\psi(h\cdot v),\quad h\in H_{1},\text{ }v\in V_{1}.$

Notice that the momentum map (16) is an invariant of a symplectic
representation, in the sense that an equivalence (18) induces an
identification:

${(V_{1},\omega_{1})}$${(V_{2},\omega_{2})}$${\mathfrak{h}_{1}^{*}}$${\mathfrak{h}_{2}^{*}}$$\scriptstyle{\psi}$$\scriptstyle{J_{V_{1}}}$$\scriptstyle{J_{V_{2}}}$$\scriptstyle{\varphi_{*}}$

####  1.1.3. Maximally toric representations of tori

If an infinitesimally abelian compact Lie group is connected, then it is
simply a torus. In this case, Theorem 1.3 boils down to:

######  Proposition 1.9 ([12]).

Let $T$ be an $n$-dimensional torus. Then:

  1. a)

A symplectic representation of $T$ is maximally toric if and only if its
weight-tuple forms a basis of the character lattice $\Lambda_{T}^{*}$ in
$\mathfrak{t}^{*}$.

  2. b)

The map that associates to each unordered basis
$\\{\alpha_{1},...,\alpha_{n}\\}$ of $\Lambda_{T}^{*}$ the polyhedral cone
generated by $(\alpha_{1},...,\alpha_{n})$ is a bijection from the set of such
$n$-tuples to the set of smooth pointed polyhedral cones in
$(\mathfrak{t}^{*},\Lambda_{T}^{*})$.

Therefore we have a bijection:

$\left\\{\begin{aligned} &\quad\quad\text{ Isomorphism classes of }\\\ &\text{
{maximally} toric $T$-representations
}\end{aligned}\right\\}\longleftrightarrow\left\\{\begin{aligned}
&\quad\quad\text{ Smooth pointed}\\\ &\text{ polyhedral cones in
}(\mathfrak{t}^{*},\Lambda_{T}^{*})\text{ }\end{aligned}\right\\}$

that associates to such a $T$-representation $(V,\omega)$ the image
$\Delta_{V}$ of the momentum map $(\ref{quadsympmommap})$.

Let us briefly recall the meaning of the various notions appearing in this
statement. The character lattice $\Lambda_{T}^{*}$ of $T$ in
$\mathfrak{t}^{*}$ is the dual of the full rank lattice
$\Lambda_{T}:=\textrm{Ker}(\exp_{T})$ in $\mathfrak{t}$. This gives
$\mathfrak{t}^{*}$ the structure of an integral affine vector space, by which
we mean a pair $(V,\Lambda)$ consisting of a finite-dimensional real vector
space $V$ and a full rank lattice $\Lambda$ in $V$. Any such lattice $\Lambda$
admits a basis (that is, a basis of $V$ consisting of vectors that
$\mathbb{Z}$-span $\Lambda$).

Recall that symplectic $T$-representations are classified by their weight-
tuples, as follows. Associated to $\alpha\in\Lambda_{T}^{*}$ is the
irreducible symplectic representation
$T\to\textrm{Sp}(\mathbb{C},\omega_{\textrm{st}})$ defined via the character:

(19)
$\chi_{\alpha}:T\to\mathbb{S}^{1},\quad\chi_{\alpha}(\exp_{T}(\xi))=e^{2\pi
i\langle\alpha,\xi\rangle},$

and the $\mathbb{S}^{1}$-action on $\mathbb{C}$ by complex multiplication.
Here $(\mathbb{C},\omega_{\textrm{st}})$ is viewed as real vector space
equipped with the standard linear symplectic form:

$\omega_{\textrm{st}}(w,z)=\frac{1}{2\pi
i}(w\overline{z}-\overline{w}z)\in\mathbb{R},\quad w,z\in\mathbb{C}.$

We denote this symplectic $T$-representation associated to
$\alpha\in\Lambda_{T}^{*}$ as:

$(\mathbb{C}_{\alpha},\omega_{\textrm{st}}).$

The weight-tuple of a symplectic $T$-representation $(V,\omega)$ is the unique
unordered tuple $(\alpha_{1},...,\alpha_{n})$ such that:

(20)
$(V,\omega)\cong(\mathbb{C}_{\alpha_{1}},\omega_{\textrm{st}})\oplus...\oplus(\mathbb{C}_{\alpha_{n}},\omega_{\textrm{st}})$

as symplectic $T$-representation (for details on this, see for instance the
appendices of [27, 21]). This explains the terminology in part $a$ of the
proposition. Turning to part $b$, by a polyhedral cone in a finite-dimensional
real vector space $V$, we mean a subset of the form:

$\textrm{Cone}(v_{1},...,v_{k}):=\left\\{\sum_{i=1}^{k}s_{i}v_{i}\mid
s_{i}\geq 0\right\\},$

with $v_{1},...,v_{k}\in V$. The tuple $(v_{1},...,v_{k})$ is said to generate
the polyhedral cone. We call a polyhedral cone pointed if it does not contain
a line through the origin in $V$. Finally, we call a polyhedral cone $C$ in an
integral affine vector space $(V,\Lambda)$ smooth if there is a basis
$\\{v_{1},...,v_{n}\\}$ of the lattice $\Lambda$ in $V$ and a $k\leq n$ such
that:

$C=\textrm{Cone}(v_{1},...,v_{k},v_{k+1},-v_{k+1},...,v_{n},-v_{n}).$

This explains the terminology in part $b$. For the remainder, notice that
under an identification of symplectic $T$-representations (20) the momentum
map $J_{V}$ is identified with:

(21) $J_{V}(z_{1},...,z_{n})=\sum_{i=1}^{n}|z_{i}|^{2}\alpha_{i},$

where $z_{i}$ denotes the standard complex linear coordinate on
$\mathbb{C}_{\alpha_{i}}$. Therefore, the image $\Delta_{V}$ of $J_{V}$ is the
polyhedral cone in $\mathfrak{t}^{*}$ generated by the weight-tuple
$(\alpha_{1},...,\alpha_{n})$, which is smooth and pointed if the symplectic
$T$-representation is maximally toric.

######  Remark 1.10.

Let $H$ be an infinitesimally abelian compact Lie group with identity
component $T$ and let $(V,\omega)$ be a symplectic $H$-representation. For
future reference, let us point out that the representation
$H\to\textrm{Sp}(V,\omega)$ is maximally toric if and only if the map of Lie
groups induced by the weights of the induced representation
$T\to\textrm{Sp}(V,\omega)$:

(22) $(\chi_{\alpha_{1}},...,\chi_{\alpha_{n}}):T\to\mathbb{T}^{n}$

is an isomorphism.

####  1.1.4. Splittings of abelian group extensions

The main point of this subsection will be to prove Proposition 1.7. It will be
useful to have some alternative descriptions of right splittings of abelian
group extensions. Let $A$ and $K$ be groups and suppose that $A$ is abelian.
Recall the following.

  * •

An extension of $K$ by $A$ is a short exact sequence of groups:

$1\to A\to G\to K\to 1.$

When the maps are clear, we simply refer to $G$ as an extension of $K$ by $A$.
Given any such extension, conjugation yields an action of $G$ on $A$ by group
automorphisms, which descends to an action of $K$ on $A$.

  * •

A morphism of two extensions of $K$ by $A$:

$\displaystyle 1\to A\to G_{1}\to K\to 1$ $\displaystyle 1\to A\to G_{2}\to
K\to 1$

is a morphism of groups $\varphi:G_{1}\to G_{2}$ that makes the diagram:

${1}$${A}$${G_{1}}$${K}$${1}$${1}$${A}$${G_{2}}$${K}$${1}$$\scriptstyle{\varphi}$

commute. By a version of the five-lemma, any such morphism $\varphi$ must be
an isomorphism.

  * •

A right splitting $\sigma:K\to G$ of an extension as above is a group
homomorphism that is a section of the map $G\to K$.

  * •

Given an action of $K$ on $A$ by group automorphisms, the semi-direct product
$K\ltimes A$ is canonically an extension of $K$ by $A$.

  * •

Given an extension as above, a $1$-cocycle (with respect to the right action
of $G$ on $A$ by conjugation) is a map ${c}:G\to A$ satisfying:

${c}(g_{1}g_{2})=({c}(g_{1})\cdot g_{2}){c}(g_{2}),\quad g_{1},g_{2}\in G.$

We can now give the desired characterizations of right splittings.

######  Proposition 1.11.

Consider an extension

$1\to A\to G\to K\to 1$

of a group $K$ by an abelian group $A$. There are natural bijections between:

  * i)

The set of right splittings $K\to G$.

  * ii)

The set of isomorphisms of extensions:

$K\ltimes A\to G,$

where $K$ acts on $A$ via conjugation in $G$.

  * iii)

The set of $1$-cocycles $G\to A$ that restrict to the identity map on $A$.

  * iv)

The set of subgroups $K_{G}$ in $G$ such that the composite
$K_{G}\hookrightarrow G\to K$ is an isomorphism.

###### Proof.

We will define a square of maps:

(i)(ii)(iv)(iii)

Firstly, given a right splitting $\sigma:K\to G$, the corresponding
isomorphism of extensions is:

(23) $K\ltimes A\to G,\quad(k,a)\mapsto\sigma(k)a.$

Secondly, given an isomorphism of extensions $\varphi:K\ltimes A\to G$, the
corresponding $1$-cocycle is the composition:

$G\xrightarrow{\varphi^{-1}}K\ltimes A\xrightarrow{\textrm{pr}_{A}}A.$

Thirdly, given a $1$-cocycle ${c}:G\to A$ that restricts to the identity map
on $A$, the subset:

(24) $K_{{c}}:=\\{g\in G\mid{c}(g)=1\\}$

is a subgroup of $G$ and, because ${c}$ restricts to the identity map on $A$,
the composite $K_{{c}}\hookrightarrow G\to K$ is an isomorphism. Finally,
given a subgroup $K_{G}$ of $G$ with the property that the composite
$K_{G}\hookrightarrow G\to K$ is an isomorphism, the inverse to this map
yields the corresponding right splitting:

$K\to K_{G}\hookrightarrow G,$

which completes the square. We leave it to the reader to verify that each of
the four maps in this triangle is inverse to the composition by the other
three, so as to complete the proof. ∎

With this and the lemma below, we are ready to prove Proposition 1.7 .

######  Lemma 1.12.

A maximally toric representation of a torus has a unique decomposition into
irreducible symplectic subrepresentations.

###### Proof.

The existence of such a decomposition holds for all symplectic representations
of tori. Now suppose $T\to\text{Sp}(V,\omega)$ is a maximally toric
representation of an $n$-dimensional torus. Let
$V=V_{\alpha_{1}}\oplus...\oplus V_{\alpha_{n}}$ be a decomposition into
irreducible symplectic subrepresentations indexed by the corresponding weights
$(\alpha_{1},...,\alpha_{n})$. By Proposition 1.9$a$ the weights are linearly
independent. So, since after identifying $V_{\alpha_{i}}$ with
$\mathbb{C}_{\alpha_{i}}$ as symplectic $T$-representations the momentum map
is given by (21), each subspace $V_{\alpha_{i}}$ coincides with
$(J_{V})^{-1}(\mathbb{R}_{+}\cdot\alpha_{i})$. Hence, the decomposition is
indeed unique. ∎

###### Proof of Proposition 1.7.

Let $H\to\text{Sp}(V,\omega)$ be a maximally toric representation. After a
choice of isomorphism of symplectic $T$-representations, we can assume that:

(25)
$(V,\omega)=(\mathbb{C}_{\alpha_{1}},\omega_{\textrm{st}})\oplus...\oplus(\mathbb{C}_{\alpha_{n}},\omega_{\textrm{st}}),$

as symplectic $T$-representations. Let $h\in H$. The pair of maps:

$(C_{h},m_{h}):(T,(V,\omega))\to(T,(V,\omega)),$

consisting of conjugation and multiplication by $h$, is a self-equivalence of
the induced symplectic representation of $T$ on $(V,\omega)$. Therefore,
$[h]\in\varGamma$ (which acts on $\mathfrak{t}^{*}$ via the coadjoint
$H$-action) permutes the weights $(\alpha_{1},...,\alpha_{n})$ and, by
uniqueness of the decomposition into irreducibles, it follows that $h$ acts as
a symplectic $\mathbb{R}$-linear map
$\mathbb{C}_{\alpha_{i}}\to\mathbb{C}_{[h]\cdot\alpha_{i}}$ for each $i$.
Since $J_{V}$ is given by (21) and is $H$-equivariant, any $h$ maps the unit
circle $(J_{V})^{-1}(\alpha_{i})$ to the unit circle
$(J_{V})^{-1}([h]\cdot\alpha_{i})$, and it follows that
$h:\mathbb{C}_{\alpha_{i}}\to\mathbb{C}_{[h]\cdot\alpha_{i}}$ acts by a
rotation determined by an element $h_{i}\in\mathbb{S}^{1}$. So, we can
associate to every $h\in H$ an element $(h_{1},...,h_{n})\in\mathbb{T}^{n}$.
Via the isomorphism (22) we obtain a map ${c}:H\to T$. This is a $1$-cocycle
for the extension (17) that restricts to the identity map on $T$. So, in view
of Proposition 1.11, $H$ is split. ∎

####  1.1.5. The classification of maximally toric representations

We will now address the associated group cohomology class appearing in Theorem
1.3. Let $(V,\omega)$ be a maximally toric $H$-representation. As in the proof
of Proposition 1.7, the choice of an isomorphism of symplectic
$T$-representations:

(26)
$\psi:(V,\omega)\xrightarrow{\sim}(\mathbb{C}_{\alpha_{1}},\omega_{\textrm{st}})\oplus...\oplus(\mathbb{C}_{\alpha_{n}},\omega_{\textrm{st}})$

induces a $1$-cocycle ${c}_{\psi}:H\to T$, determined by the fact that for
each $h\in H$, $v\in V$ and each $\alpha_{i}$:

(27) $\chi_{\alpha_{i}}({c}_{\psi}(h))\cdot\psi(v)_{\alpha_{i}}=\psi(h\cdot
v)_{[h]\cdot\alpha_{i}}.$

This $1$-cocycle restricts to the identity map on $T$ and for different
choices of $\psi$ the resulting cocycles are cohomologous. Hence, the class:

(28) $e(V,\omega):=[{c}_{\psi}]\in I^{1}(H,T)$

is independent of the choice of $\psi$.

######  Definition 1.13.

We call (28) the ext-class of the maximally toric $H$-representation
$(V,\omega)$.

In fact, this class is the same for any two maximally toric representations
that are isomorphic as symplectic $H$-representations. This explains the
remaining part of the statement of Theorem 1.3. We now turn to its proof.

###### Proof of Theorem 1.3.

For injectivity, let $(V_{1},\omega_{1})$ and $(V_{2},\omega_{2})$ be
maximally toric $H$-representation such that $\Delta_{V_{1}}=\Delta_{V_{2}}$
and $e(V_{1},\omega_{1})=e(V_{2},\omega_{2})$. Then since
$\Delta_{V_{1}}=\Delta_{V_{2}}$, it follows from Proposition 1.9 that the two
symplectic representations have the same weight-tuple, say
$(\alpha_{1},...,\alpha_{n})$. Hence, there are isomorphisms of symplectic
$T$-representations:

$(V_{1},\omega_{1})\xrightarrow{\psi_{1}}(\mathbb{C}_{\alpha_{1}},\omega_{\textrm{st}})\oplus...\oplus(\mathbb{C}_{\alpha_{n}},\omega_{\textrm{st}})\xleftarrow{\psi_{2}}(V_{2},\omega_{2}).$

Since $e(V_{1},\omega_{1})=e(V_{2},\omega_{2})$, there is a $t\in T$ such that
for each $h\in H$:

(29) ${c}_{\psi_{1}}(h)=(t\cdot h)t^{-1}{c}_{\psi_{2}}(h).$

Consider:

$\psi_{t}:(V_{1},\omega_{1})\to(\mathbb{C}_{\alpha_{1}},\omega_{\textrm{st}})\oplus...\oplus(\mathbb{C}_{\alpha_{n}},\omega_{\textrm{st}}),\quad\psi_{t}(v)=t^{-1}\cdot\psi_{1}(v),$

which is again an isomorphism of symplectic $T$-representations. As one
readily verifies, it follows from (29) that
$\psi_{2}^{-1}\circ\psi_{t}:(V_{1},\omega_{1})\to(V_{2},\omega_{2})$ is
$H$-equivariant, and hence it is an isomorphism of symplectic
$H$-representations. This proves injectivity.

For surjectivity, let $\Delta$ be a $\varGamma$-invariant and smooth pointed
polyhedral cone in $(\mathfrak{t}^{*},\Lambda_{T})$, and let ${c}:H\to T$ be a
$1$-cocycle that restricts to the identity on $T$. Let
$(\alpha_{1},...,\alpha_{n})$ be a tuple that generates $\Delta$ and forms a
basis of $\Lambda_{T}^{*}$. Consider the maximally toric $T$-representation:

$(V,\omega):=(\mathbb{C}_{\alpha_{1}},\omega_{\textrm{st}})\oplus...\oplus(\mathbb{C}_{\alpha_{n}},\omega_{\textrm{st}}).$

Since $\varGamma$ leaves both $\Delta$ and $\Lambda_{T}^{*}$ invariant, by
Proposition 1.9$b$ it must permute the ordered tuple
$(\alpha_{1},...,\alpha_{n})$. This induces an action of $\varGamma$ on $V$ by
permuting the components indexed by this tuple. Using this, the
$T$-representation extends to a representation $r:\varGamma\ltimes
T\to\textrm{Sp}(\mathbb{C}^{n},\omega_{\textrm{st}})$ by setting
$(\gamma,t)\cdot z=\gamma\cdot(t\cdot z)$. By construction, this
representation is maximally toric and has momentum image $\Delta$. Next,
consider the map:

$\varphi_{{c}}:H\to\varGamma\ltimes T,\quad\varphi_{{c}}(h)=([h],{c}(h)).$

This is the isomorphism of extensions of $\varGamma$ by $T$ corresponding to
${c}$ via the bijection in Proposition 1.11. Composing $\varphi_{{c}}$ with
$r$, we obtain a maximally toric $H$-representation for which $\Delta$ is the
momentum image. Moreover, the cohomology class associated to this is
represented by the $1$-cocycle ${c}_{\psi}$ where we may pick $\psi$ to be the
identity map on $\mathbb{C}^{n}$. Using (27), this $1$-cocycle is readily seen
to coincide with the given $1$-cocycle ${c}$. So we have constructed a
maximally toric $H$-representation with momentum image $\Delta$ and associated
group cohomology class $[{c}]$. ∎

######  Remark 1.14.

The analogue of Example 3 holds as well in this linear setting: by Theorem 1.3
and Remark 1.5 the set of isomorphism of class of maximally toric
$H$-representations with momentum image a prescribed $\Delta$ is a torsor with
structure group $H^{1}(\varGamma,T)$, provided $H$ is split, and (using the
lemma below) it is readily verified that this action can be described more
explicitly, as follows. Consider the isomorphism between the abelian group of
$1$-cocycles ${c}:\varGamma\to T$ and the group of automorphisms of the
extension (17) (with group structure given by composition of maps), that
associates to a $1$-cocycle ${c}$ the automorphism:

$\varphi_{{c}}:H\to H,\quad h\mapsto h{c}([h]).$

This descends to an isomorphism between $H^{1}(\varGamma,T)$ and the group of
automorphisms of (17) modulo the subgroup of automorphisms given by
conjugation by elements of $T$. Now, the $H^{1}(\varGamma,T)$-action on the
set of isomorphism classes of maximally toric $H$-representations is given by:

$[{c}]\cdot[(V,\omega)]:=[(V_{{c}},\omega)],$

where $(V_{{c}},\omega)$ is equal to $(V,\omega)$ as symplectic vector space,
but equipped with the linear symplectic action given by
$h\cdot_{{c}}v=\varphi_{{c}}(h)\cdot v$, for $h\in H$ and $v\in V$, where the
right-hand dot denotes the original action of $H$ on $V$.

In the above remark we referred to:

######  Lemma 1.15.

Suppose that we are given an equivalence of maximally toric representations of
infinitesimally abelian compact Lie groups:

$(\varphi,\psi):(H_{1},(V_{1},\omega_{1}))\to(H_{2},(V_{2},\omega_{2})).$

Then the induced isomorphism:

$\varphi_{*}:I^{1}(H_{1},T_{1})\xrightarrow{\sim}I^{1}(H_{2},T_{2})$

sends $e(V_{1},\omega_{1})$ to $e(V_{2},\omega_{2})$.

This lemma is straightforward to verify. It will also be useful for later
reference.

###  1.2. Delzant subspaces of integral affine orbifolds

####  1.2.1. Background on orbifolds

Following [7], we use the terminology below for orbifolds.

  * •

An orbifold groupoid is a proper foliation groupoid (that is, a proper Lie
groupoid with discrete isotropy groups).

  * •

By an orbifold atlas on a topological space $B$ we mean an orbifold groupoid
$\mathcal{B}\rightrightarrows M$, together with a homeomorphism $p$ between
the orbit space $\underline{M}$ and $B$.

  * •

By an orbifold we mean a pair consisting of a topological space $B$ together
with an orbifold atlas $(\mathcal{B},p)$ on $B$.

  * •

We call two orbifold atlases on $B$ equivalent if there is a Morita
equivalence between the given orbifold groupoids that intertwines the
respective homeomorphisms between their orbit spaces and $B$.

This approach to orbifolds (using groupoids instead of atlases of charts) is
in the spirit of [34].

####  1.2.2. Background on integral affine structures on orbifolds

An integral affine atlas on a manifold $M$ is one for which the coordinate
changes are (restrictions of) integral affine transformations of
$\mathbb{R}^{n}$. A maximal such atlas is called an integral affine structure
on $M$. Such a structure can be encoded globally, as follows. Given a vector
bundle $E\to M$, a smooth lattice in $E$ is a subbundle $\Lambda$ with the
property that, for each $x_{0}\in M$, there is a local frame $e$ of $E$
defined on an open neighbourhood $U$ of $x_{0}$, such that for all $x\in U$:

$\Lambda_{x}=\mathbb{Z}(e_{1})_{x}\oplus...\oplus\mathbb{Z}(e_{n})_{x}.$

In particular, $\Lambda_{x}$ is a full rank lattice in $E_{x}$ for each $x\in
M$. The data of an integral affine structure on $M$ is equivalent to that of a
smooth lattice $\Lambda$ in the cotangent bundle $T^{*}M$ satisfying the
integrability condition that $\Lambda$ is locally spanned by closed $1$-forms,
or equivalently, that $\Lambda$ is Lagrangian as submanifold of
$(T^{*}M,\Omega_{\textrm{can}})$. The corresponding lattice is locally spanned
by the coframe associated to any integral affine chart. See [7] for more
details.

######  Example 1.16.

The $n$-torus $\mathbb{T}^{n}$ has an integral affine structure with lattice
$\mathbb{Z}\textrm{d}\theta_{1}\oplus...\oplus\mathbb{Z}\textrm{d}\theta_{n}$.

######  Remark 1.17.

Besides the $2$-torus, the only other compact surface that admits an integral
affine structure is the Klein bottle [2, 31] and the various such structures
on these have been classified [32, 43]. There are also classification results
for integral affine structures on three-dimensional compact manifolds [24].

The global description is well-suited for a generalization of integral affine
structures to orbifolds. First of all, the notion of vector bundle
generalizes: a vector bundle over an orbifold $(B,\mathcal{B},p)$ is a
representation of $\mathcal{B}$, meaning that it is a vector bundle $E\to M$
(in the sense of manifolds) equipped with a fiberwise linear action of
$\mathcal{B}$. For example, the tangent bundle of an orbifold
$(B,\mathcal{B},p)$ is the canonical representation of $\mathcal{B}$ on the
normal bundle $\mathcal{N}\mathcal{F}$ to the foliation $\mathcal{F}$ on $M$
by connected components of the orbits of $\mathcal{B}$. Explicitly, this
representation is given by:

(30) $g\cdot[v]=[\textrm{d}t(\widehat{v})],\quad g\in\mathcal{B},\quad
v\in\mathcal{N}_{s(g)}\mathcal{F},$

where $\widehat{v}\in T_{g}\mathcal{B}$ is any choice of tangent vector such
that $\textrm{d}s(\widehat{v})=v$. If $\mathcal{B}$ is source-connected, then
this coincides with the linear holonomy representation. The cotangent bundle
of the orbifold is the dual representation of $\mathcal{B}$ on the co-normal
bundle:

$\mathcal{N}^{*}\mathcal{F}=T\mathcal{F}^{0}\subset T^{*}M.$

Now, the definition of an integral affine structure generalizes to orbifolds,
as follows.

######  Definition 1.18.

An integral affine structure on an orbifold $(B,\mathcal{B},p)$ is a smooth
lattice $\Lambda$ in $\mathcal{N}^{*}\mathcal{F}$ with the property that
$\Lambda$ is Lagrangian as submanifold of $(T^{*}M,\Omega_{\textrm{can}})$ and
$\Lambda$ is invariant with respect to the co-normal representation of
$\mathcal{B}$. We call $(B,\mathcal{B},p,\Lambda)$ an integral affine
orbifold.

######  Example 1.19.

For a proper action of a countable discrete group $\Gamma$ on a manifold $M$,
the data of an integral affine structure on the orbit space $M/\Gamma$ (viewed
as orbifold with orbifold groupoid the action groupoid $\Gamma\ltimes M$) is
that of a $\Gamma$-invariant integral affine structure on $M$.

######  Example 1.20.

Given a foliated manifold $(M,\mathcal{F})$, the data of a smooth Lagrangian
lattice in $\mathcal{N}^{*}\mathcal{F}$ is the same as that of a transverse
integral affine structure on $(M,\mathcal{F})$. If an orbifold groupoid
$\mathcal{B}\rightrightarrows M$ is source-connected, then every smooth
Lagrangian lattice in $\mathcal{N}^{*}\mathcal{F}$ is automatically
$\mathcal{B}$-invariant, so that in this case the data of an integral affine
structure on $(B,\mathcal{B},p)$ is simply that of a transverse integral
affine structure on the associated foliation $\mathcal{F}$ on $M$.

####  1.2.3. Background on the integral affine orbifold associated to a
regular proper symplectic groupoid

Let $(\mathcal{G},\Omega)\rightrightarrows M$ be a regular and proper
symplectic groupoid. Let $\mathcal{F}$ denote the foliation of $M$ by
connected components of the $\mathcal{G}$-orbits, and let
$\underline{M}:=M/\mathcal{G}$ denote the orbit space of $\mathcal{G}$. It
follows from [35, Proposition 2.5] that there is a canonical short exact
sequence of Lie groupoids over $M$:

(31) $1\to\mathcal{T}\to\mathcal{G}\to\mathcal{B}\to 1$

where $\mathcal{T}$ is the bundle of Lie groups with fiber $\mathcal{T}_{x}$
the identity component of the isotropy group $\mathcal{G}_{x}$ of
$\mathcal{G}$ at $x\in M$, and $\mathcal{B}=\mathcal{G}/\mathcal{T}$ is an
orbifold groupoid over $M$. The fibers of $\mathcal{T}$ are in fact tori and
the orbifold $(\underline{M},\mathcal{B},\textrm{Id}_{\underline{M}})$ comes
with a natural integral affine structure. To see this, recall first that, as
for any symplectic groupoid, the conormal space
$\mathcal{N}^{*}_{x}\mathcal{F}$ at $x\in M$ can be canonically identified
with the isotropy Lie algebra $\mathfrak{g}_{x}$ of $\mathcal{G}$ at $x$ via
the isomorphism of Lie algebroids:

(32)
$\rho_{\Omega}:T_{\pi}^{*}M\to\text{Lie}(\mathcal{G}),\quad\iota_{\rho_{\Omega}(\alpha)}\Omega_{1_{x}}=(\textrm{d}t_{1_{x}})^{*}\alpha,\quad\alpha\in
T^{*}_{x}M,\text{ }x\in M,$

where $T^{*}_{\pi}M$ denotes the cotangent bundle equipped with the Lie
algebroid structure associated to the Poisson structure $\pi$ on $M$ induced
by $(\mathcal{G},\Omega)$. Since the Poisson structure $\pi$ is regular, its
isotropy Lie algebras are abelian. Hence, so are the isotropy Lie algebras of
$\mathcal{G}$. Since $\mathcal{G}$ is proper, its isotropy groups are compact.
Therefore, $\mathcal{T}$ is a bundle of tori and the kernel of each
exponential map $\mathfrak{g}_{x}\to\mathcal{G}_{x}^{0}=\mathcal{T}_{x}$
determines a full rank lattice $\Lambda_{x}$ in
$\mathcal{N}^{*}_{x}\mathcal{F}$. All together, this yields a map of Lie
groupoids:

(33) $\mathcal{N}^{*}\mathcal{F}\to\mathcal{T}$

with kernel the desired smooth lattice $\Lambda$ in
$\mathcal{N}^{*}\mathcal{F}$.

######  Remark 1.21.

The map (33) factors through an isomorphism of Lie groupoids:

(34)
$\mathcal{T}_{\Lambda}:=\mathcal{N}^{*}\mathcal{F}/\Lambda\xrightarrow{\sim}\mathcal{T},$

and the co-normal representation of $\mathcal{B}$ on
$\mathcal{N}^{*}\mathcal{F}$ descends to an action of $\mathcal{B}$ on
$\mathcal{T}_{\Lambda}$, which under the above isomorphism is identified with
the action of $\mathcal{B}$ on $\mathcal{T}$ by conjugation. The symplectic
form $\Omega$ on $\mathcal{G}$ restricts to a pre-symplectic form
$\Omega_{\mathcal{T}}$ on $\mathcal{T}$, which makes:

(35) $(\mathcal{T},\Omega_{\mathcal{T}})\to M$

into a pre-symplectic torus bundle. On the other hand, the canonical
symplectic form on $T^{*}M$ restricts to a pre-symplectic form on
$\mathcal{N}^{*}\mathcal{F}$, which in turn descends to a pre-symplectic form
$\Omega_{\Lambda}$ on $\mathcal{T}_{\Lambda}$. The map (34) identifies
$\Omega_{\mathcal{T}}$ with $\Omega_{\Lambda}$. So, (35) is fully encoded by
the integral affine orbifold associated to $(\mathcal{G},\Omega)$.

####  1.2.4. Delzant subspaces

We define Delzant subspaces of integral affine manifolds as follows.

######  Definition 1.22.

A Delzant subspace $\Delta$ of an integral affine manifold $(M,\Lambda)$ is a
subset of $M$ with the property that for every $x\in\Delta$ and every (or
equivalently some) integral affine chart $(U,\chi)$ around $x$ into
$\mathbb{R}^{n}$, there is a smooth polyhedral cone $C^{\chi}_{x}(\Delta)$ in
$(\mathbb{R}^{n},\mathbb{Z}^{n})$ (in the sense of Subsection 1.1.3) such that
the germ of $\chi(U\cap\Delta)$ at $\chi(x)$ in $\mathbb{R}^{n}$ is that of
$\chi(x)+C^{\chi}_{x}(\Delta)$ at $\chi(x)$.

As mentioned in the introduction: Delzant subspaces are embedded submanifolds
with corners of codimension zero (as in Definition B.11), Delzant polytopes
are examples of Delzant subspaces, and so are integral affine manifolds (being
Delzant subspaces of themselves). Below we give a few other examples.

######  Example 1.23.

Let
$(M,\Lambda):=(\mathbb{R}\times\mathbb{S}^{1},\mathbb{Z}\textrm{d}x\oplus\mathbb{Z}\textrm{d}\theta)$
—a cylinder equipped with its standard integral affine structure. The cylinder
with boundary $\Delta:=[0,1]\times\mathbb{S}^{1}$ is a Delzant subspace.

######  Example 1.24.

Let
$(M,\Lambda):=(\mathbb{R}\times\mathbb{S}^{1},\mathbb{Z}(\textrm{d}x+(\theta-\frac{1}{2})\textrm{d}\theta)\oplus\mathbb{Z}\textrm{d}\theta)$
—a cylinder equipped with a non-standard integral affine structure. The
subspace:

$\Delta:=\\{(x,e^{2\pi i\theta})\in\mathbb{R}\times\mathbb{S}^{1}\mid
x+\theta(\theta-1)/2\geq 0,\text{ }\theta\in[0,1[\\}$

is a Delzant subspace with a single vertex and a single stratum of codimension
one (see Figure 1).

Figure 1. A sketch of the boundary $\partial\Delta$ (coloured in green) of the
Delzant subspace in Example 1.24.

To extend this definition to orbifolds (in the sense of Subsection 1.2.1) it
is convenient to have a coordinate-free description of Delzant subspaces of
integral affine manifolds. To this end, notice that around each point in an
integral affine manifold $(M,\Lambda)$ there is one natural choice of integral
affine ‘chart’. More precisely, around each $x\in M$ there is unique map germ:

$\log_{x}\in\textrm{Germ}_{x}(M;T_{x}M)$

induced by an integral affine isomorphism $\iota$ from $(U,\Lambda)$ onto an
open in $(T_{x}M,\Lambda_{x}^{*})$, that maps $x$ to the origin in $T_{x}M$
and the derivative of which at $x$ is the identity map on $T_{x}M$. Given a
subset $\Delta$ of $M$ and an $x\in\Delta$, there is an associated set germ
$\log_{x}(\Delta)$ at the origin in $T_{x}M$, defined as the germ of
$\iota(U\cap\Delta)$ at the origin, which is independent of the choice of
$\iota$ as above. Now, $\Delta$ is a Delzant subspace of $(M,\Lambda)$ if and
only if for every $x\in\Delta$ the set germ $\log_{x}(\Delta)$ is the germ of
a smooth polyhedral cone in the integral affine vector space
$(T_{x}M,\Lambda_{x}^{*})$ (in the sense of Subsection 1.1.3). To define
Delzant subspaces of integral affine orbifolds we will now generalize this
characterization, starting with:

######  Proposition 1.25.

Let $(M,\mathcal{F},\Lambda)$ be a foliated manifold with a transverse
integral affine structure and let $x\in M$. There is a unique map germ:

$\log_{x}\in\textrm{Germ}_{x}(M;\mathcal{N}_{x}\mathcal{F})$

induced by a submersion $\nu$ defined on an open $U$ around $x$ in $M$, with
the following properties.

  * i)

The tangent distribution to the fibers of $\nu$ coincides with that of the
foliation $\mathcal{F}$ over $U$.

  * ii)

It maps $x$ to the origin in $\mathcal{N}_{x}\mathcal{F}$ and its differential
at $x$ is the projection $T_{x}M\to\mathcal{N}_{x}\mathcal{F}$.

  * iii)

It is compatible with the integral affine structure, in the sense that for
each $y\in U$:

$\textrm{d}\nu_{y}:(\mathcal{N}_{y}\mathcal{F},\Lambda_{y}^{*})\to(\mathcal{N}_{x}\mathcal{F},\Lambda_{x}^{*})$

is an isomorphism of integral affine vector spaces.

###### Proof.

First, we prove existence. Since $\Lambda$ is a smooth lattice in
$\mathcal{N}^{*}\mathcal{F}$, we can choose a local frame $\alpha$ of
$\mathcal{N}^{*}\mathcal{F}$, defined on an open $U$ around $x$, such that:

$\Lambda|_{U}=\mathbb{Z}\text{ }\alpha_{1}\oplus...\oplus\mathbb{Z}\text{
}\alpha_{n}.$

Since $\Lambda$ is Lagrangian in $T^{*}M$, all of its local sections are
closed $1$-forms. So, by the Poincaré Lemma, we can (after shrinking $U$)
arrange that $\alpha_{i}=\textrm{d}f_{i}$ for some $f_{i}\in C^{\infty}(U)$
such that $f_{i}(x)=0$. Consider:

$f=(f_{1},...,f_{n}):U\to\mathbb{R}^{n}.$

Then
$(\underline{\textrm{d}f})_{x}:\mathcal{N}_{x}\mathcal{F}\to\mathbb{R}^{n}$ is
a linear isomorphism, so that we can define

$\nu=(\underline{\textrm{d}f})_{x}^{-1}\circ
f:U\to\mathcal{N}_{x}\mathcal{F}.$

As is readily verified, this has the desired properties. To prove uniqueness,
let $\nu_{1}:U_{1}\to\mathcal{N}_{x}\mathcal{F}$ and
$\nu_{2}:U_{2}\to\mathcal{N}_{x}\mathcal{F}$ be two submersions as above.
Since both $\nu_{1}$ and $\nu_{2}$ satisfy property i), we can find an open
neighbourhood $U$ of $x$ in $U_{1}\cap U_{2}$, together with a connected
transversal $\Sigma$ to $\mathcal{F}$ through $x$, with the property that:

  * •

$\Sigma$ is contained in $U$ and every leaf of the foliation on $U$ induced by
$\mathcal{F}$ intersects $\Sigma$,

  * •

both $\nu_{1}|_{\Sigma}$ and $\nu_{2}|_{\Sigma}$ are open embeddings into
$\mathcal{N}_{x}\mathcal{F}$.

The transversal $\Sigma$ inherits an honest integral affine structure
$\Lambda_{\Sigma}$ from the transverse integral affine structure $\Lambda$,
and by property iii) both $\nu_{1}|_{\Sigma}$ and $\nu_{2}|_{\Sigma}$ are
isomorphisms of integral affine manifolds onto their image in
$(\mathcal{N}_{x}\mathcal{F},\Lambda_{x}^{*})$ with respect to
$\Lambda_{\Sigma}$. Therefore,
$\nu_{1}|_{\Sigma}\circ(\nu_{2}|_{\Sigma})^{-1}$ is a morphism of integral
affine manifolds between connected opens in the integral affine vector space
$(\mathcal{N}_{x}\mathcal{F},\Lambda_{x}^{*})$. By the lemma below, this means
that it must be the restriction of an integral affine transformation of
$(\mathcal{N}_{x}\mathcal{F},\Lambda_{x}^{*})$. So, it is determined by its
value and its derivative at the origin in $\mathcal{N}_{x}\mathcal{F}$. By
property ii), $\nu_{1}|_{\Sigma}\circ(\nu_{2}|_{\Sigma})^{-1}$ fixes the
origin and its derivative at the origin is the identity map. Hence,
$\nu_{1}|_{\Sigma}\circ(\nu_{2}|_{\Sigma})^{-1}$ must be the restriction of
the identity map on $\mathcal{N}_{x}\mathcal{F}$ to $\nu_{2}(\Sigma)$, so that
$\nu_{1}|_{\Sigma}=\nu_{2}|_{\Sigma}$. It follows from this and property i)
that in fact $\nu_{1}|_{U}=\nu_{2}|_{U}$, because every leaf of the foliation
on $U$ induced by $\mathcal{F}$ intersects $\Sigma$. So, $\nu_{1}$ and
$\nu_{2}$ have the same germ at $x$, as was to be shown. ∎

######  Lemma 1.26.

Let $(V_{1},\Lambda_{1})$ and $(V_{2},\Lambda_{2})$ be integral affine vector
spaces. Then every morphism of integral affine manifolds from a connected open
in $(V_{1},\Lambda_{1})$ into $(V_{2},\Lambda_{2})$ is of the form $v\mapsto
Av+b$ for some linear map $A:V_{1}\to V_{2}$ that maps $\Lambda_{1}$ into
$\Lambda_{2}$ and some $b\in V_{2}$.

###### Proof.

After a choosing bases for $\Lambda_{1}$ and $\Lambda_{2}$ we can assume that
$(V_{1},\Lambda_{1})=(\mathbb{R}^{n_{1}},\mathbb{Z}^{n_{1}})$ and
$(V_{2},\Lambda_{2})=(\mathbb{R}^{n_{2}},\mathbb{Z}^{n_{2}})$. Let $f$ be a
morphism of integral affine manifolds from a connected open $U_{1}$ in
$(\mathbb{R}^{n_{1}},\mathbb{Z}^{n_{1}})$ into
$(\mathbb{R}^{n_{2}},\mathbb{Z}^{n_{2}})$, meaning that its partial
derivatives take values in $\mathbb{Z}$. Since $U_{1}$ is connected and
$\mathbb{Z}$ is a discrete subspace of $\mathbb{R}$, the Jacobian of $f$ must
be constant. Fix a $u_{1}\in U_{1}$. Since any two points in $U_{1}$ can be
connected to $u_{1}$ by a smooth path, it follows by integrating along such
paths that $f(v)=A(v)+b$ for all $v\in U_{1}$, where $A$ is the constant value
of the Jacobian of $f$ and $b=f(u_{1})-A(u_{1})$. ∎

Now, let $(M,\mathcal{F},\Lambda)$ be a foliated manifold with a transverse
integral affine structure, let $\Delta$ be a subset of $M$ and let
$x\in\Delta$. Then, as for integral affine manifolds, there is an associated
set germ $\log_{x}(\Delta)$ at the origin in $\mathcal{N}_{x}\mathcal{F}$. To
define this, let us call a submersion $\nu:U\to\mathcal{N}_{x}\mathcal{F}$
representing $\log_{x}$ $\Delta$-adapted if:

$\nu^{-1}(\nu(U\cap\Delta))=U\cap\Delta.$

The set-germ of $\nu(U\cap\Delta)$ at the origin in
$\mathcal{N}_{x}\mathcal{F}$ is independent of the choice of $\Delta$-adapted
submersion $\nu:U\to\mathcal{N}_{x}\mathcal{F}$ representing $\log_{x}$.
Moreover, if $\Delta$ is $\mathcal{F}$-invariant, we can always find a small
enough open $U$ around $x$ in $M$ that admits a $\Delta$-adapted submersion
$\nu:U\to\mathcal{N}_{x}\mathcal{F}$ representing $\log_{x}$. Therefore, it
makes sense to define:

######  Definition 1.27.

Let $(M,\mathcal{F},\Lambda)$ be a foliated manifold with a transverse
integral affine structure, let $\Delta$ be an $\mathcal{F}$-invariant subset
of $M$ and let $x\in\Delta$. We define $\log_{x}(\Delta)$ to be the set germ
of $\nu(U\cap\Delta)$ at the origin in $\mathcal{N}_{x}\mathcal{F}$, for any
$\Delta$-adapted submersion $\nu:U\to\mathcal{N}_{x}\mathcal{F}$ representing
$\log_{x}$.

We are now ready to define Delzant subspaces of integral affine orbifolds.

######  Definition 1.28.

Let $(B,\mathcal{B},p,\Lambda)$ be an integral affine orbifold. A Delzant
subspace $\underline{\Delta}$ is a subset of $B$ with the property that for
every $x\in\Delta$ (the corresponding invariant subset of $M$), the set germ
$\log_{x}(\Delta)$ is the germ of a smooth polyhedral cone in the integral
affine vector space $(\mathcal{N}_{x}\mathcal{F},\Lambda_{x}^{*})$ (in the
sense of Subsection 1.1.3). For each $x\in\Delta$, we denote this polyhedral
cone in $\mathcal{N}_{x}\mathcal{F}$ (which is necessarily unique) by
$C_{x}(\underline{\Delta})$ and call it the cone of $\underline{\Delta}$ at
$x$.

######  Remark 1.29.

Let $(B,\mathcal{B},p,\Lambda)$ be an integral affine orbifold and let
$\underline{\Delta}$ be a Delzant subspace. Then $\Delta$ is an embedded
submanifold with corners of $M$ of codimension zero (as in Definition B.11),
with tangent cone $C_{x}(\Delta)$ the pre-image of $C_{x}(\underline{\Delta})$
under the projection $T_{x}M\to\mathcal{N}_{x}\mathcal{F}$. The restriction
$\mathcal{B}|_{\Delta}$ is a Lie groupoid with corners (as in Definition B.18;
cf. Example B.14). If $\underline{M}$ is the orbit space of a regular and
proper symplectic groupoid $(\mathcal{G},\Omega)$ (equipped with the
associated integral affine orbifold structure), then
$(\mathcal{G},\Omega)|_{\Delta}$ is a symplectic groupoid with corners (as in
Definition B.22).

######  Example 1.30.

Let $\mathcal{B}\rightrightarrows(M,\Lambda)$ be an etale integral affine
orbifold groupoid. For any $\mathcal{B}$-invariant Delzant subspace $\Delta$
of the integral affine manifold $(M,\Lambda)$ (in the sense of Definition
1.22), the quotient $\underline{\Delta}$ is a Delzant subspace of
$\underline{M}$. In fact, any Delzant subspace of $\underline{M}$ is of this
form.

######  Example 1.31.

Let $G$ be a compact Lie group. Consider the open and dense subset
$\mathfrak{g}^{*}_{\textrm{reg}}$ of $\mathfrak{g}^{*}$ consisting of
coadjoint $G$-orbits of maximal dimension and consider the regular and proper
symplectic groupoid
$(\mathcal{G},\Omega):=(G\ltimes\mathfrak{g}^{*}_{\textrm{reg}},-\textrm{d}\lambda_{\textrm{can}})$
over $M:=\mathfrak{g}^{*}_{\textrm{reg}}$ (the restriction of the cotangent
symplectic groupoid of $G$). Fix a maximal torus $T$ in $G$ and let
$\mathfrak{c}$ be the interior of a Weyl chamber in $\mathfrak{t}^{*}$. As is
common, we identify $\mathfrak{t}^{*}$ with the $T$-fixed point set in
$\mathfrak{g}^{*}$. Let $N(\mathfrak{c})$ be the normalizer of $\mathfrak{c}$
in $G$. Delzant subspaces of $\underline{M}=\mathfrak{g}^{*}_{\textrm{reg}}/G$
are naturally in bijection with Delzant subspaces of the integral affine
vector space $(\mathfrak{t}^{*},\Lambda_{T})$ that are contained in
$\mathfrak{c}$ and that are invariant with respect to the natural action of
the finite group $N(\mathfrak{c})/T$ on $\mathfrak{c}$. If $G$ is connected,
then $N(\mathfrak{c})=T$, so that the invariance condition becomes vacuous. A
conceptual explanation for this correspondence will be given in Example 1.42,
using Corollary 1.40 below.

We think of a Delzant subspace as what should be an integral affine
suborbifold with corners, without making this precise. In particular, these
objects should be well-behaved with respect to equivalences of the ambient
integral affine orbifold that respect the integral affine structure. The
latter we will make precise, for besides its conceptual value it will be of
use throughout.

######  Definition 1.32.

By an integral affine orbifold groupoid
$\mathcal{B}\rightrightarrows(M,\Lambda)$ we mean an orbifold groupoid
$\mathcal{B}\rightrightarrows M$ with an integral affine structure $\Lambda$
on the orbifold ($\underline{M},\mathcal{B},\textrm{Id}_{\underline{M}}$) (its
orbit space). We call a Delzant subspace $\underline{\Delta}$ of this integral
affine orbifold simply a Delzant subspace of $\underline{M}$.

######  Definition 1.33.

By an integral affine Morita equivalence between integral affine orbifold
groupoids $\mathcal{B}_{1}\rightrightarrows(M_{1},\Lambda_{1})$ and
$\mathcal{B}_{2}\rightrightarrows(M_{2},\Lambda_{2})$ we mean a Morita
equivalence:

$\mathcal{B}_{1}$$M_{1}$$P$$M_{2}$$\mathcal{B}_{2}$$\text{ }\text{
}\alpha_{1}$$\alpha_{2}$

with the additional property that
$\alpha_{1}^{*}(\Lambda_{1})=\alpha_{2}^{*}(\Lambda_{2})$ as subbundles of
$\alpha_{1}^{*}(\mathcal{N}^{*}\mathcal{F}_{1})=\alpha_{2}^{*}(\mathcal{N}^{*}\mathcal{F}_{2})$.

######  Remark 1.34.

A Morita equivalence as above is integral affine if and only if for each $p\in
P$, writing $x_{1}=\alpha_{1}(p)$ and $x_{2}=\alpha_{2}(p)$, the induced
linear isomorphism:

(36)
$\psi_{p}:\mathcal{N}_{x_{1}}\mathcal{F}_{1}\xrightarrow{\sim}\mathcal{N}_{x_{2}}\mathcal{F}_{2},\quad[v]\mapsto[\textrm{d}\alpha_{2}(\widehat{v})],$

where $\widehat{v}\in T_{p}P$ is any tangent vector with the property that
$\textrm{d}\alpha_{1}(\widehat{v})=v$, is an isomorphism of integral affine
vector spaces:

$\psi_{p}:(\mathcal{N}_{x_{1}}\mathcal{F}_{1},(\Lambda_{1})^{*}_{x_{1}})\xrightarrow{\sim}(\mathcal{N}_{x_{2}}\mathcal{F}_{2},(\Lambda_{2})^{*}_{x_{2}}).$

In particular, for each such $p\in P$ there is an induced isomorphism of tori:

$(\psi_{p})_{*}:(\mathcal{T}_{\Lambda_{1}})_{x_{1}}\xrightarrow{\sim}(\mathcal{T}_{\Lambda_{2}})_{x_{2}}.$

######  Remark 1.35.

Given a Morita equivalence between orbifold groupoids
$\mathcal{B}_{1}\rightrightarrows M_{1}$ and $\mathcal{B}_{2}\rightrightarrows
M_{2}$, and an integral affine structure $\Lambda_{1}$ on the orbifold
$(\underline{M}_{1},\mathcal{B}_{1},\textrm{Id}_{\underline{M}_{1}})$, there
is a unique integral affine structure $\Lambda_{2}$ on the orbifold
$(\underline{M}_{2},\mathcal{B}_{2},\textrm{Id}_{\underline{M}_{2}})$ with
respect to which the given Morita equivalence becomes integral affine.

######  Example 1.36.

Let $\mathcal{B}\rightrightarrows(M,\Lambda)$ be an integral affine orbifold
groupoid and $\Sigma$ a transversal for $\mathcal{B}$ (by which we mean a
transversal to the foliation $\mathcal{F}$ by connected components of
$\mathcal{B}$-orbits, of complementary dimension). There is a canonical Morita
equivalence:

$\mathcal{B}|_{\widehat{\Sigma}}$$\widehat{\Sigma}$$s_{\mathcal{B}}^{-1}(\Sigma)$$\Sigma$$\mathcal{B}|_{\Sigma}$$\text{
}\text{ }t_{\mathcal{B}}$$s_{\mathcal{B}}$

where $\widehat{\Sigma}:=t(s^{-1}(\Sigma))$ denotes the saturation of $\Sigma$
with respect to $\mathcal{B}$ (which is open in $M$). The manifold $\Sigma$
inherits an honest integral affine structure $\Lambda_{\Sigma}$ from
$\Lambda$, $\mathcal{B}|_{\Sigma}\rightrightarrows(\Sigma,\Lambda_{\Sigma})$
is an etale integral affine orbifold groupoid and the above Morita equivalence
becomes integral affine.

######  Example 1.37.

Let $(\mathcal{G}_{1},\Omega_{1})\rightrightarrows M_{1}$ and
$(\mathcal{G}_{2},\Omega_{2})\rightrightarrows M_{2}$ be regular and proper
symplectic groupoids and let
$\mathcal{B}_{1}\rightrightarrows(M_{1},\Lambda_{1})$ and
$\mathcal{B}_{2}\rightrightarrows(M_{2},\Lambda_{2})$ be the associated
integral affine orbifold groupoids (as in the previous subsection). A
symplectic Morita equivalence:

$(\mathcal{G}_{1},\Omega_{1})$$M_{1}$$(P,\omega_{P})$$M_{2}$$(\mathcal{G}_{2},\Omega_{2})$$\text{
}\text{ }\alpha_{1}$$\alpha_{2}$

induces an integral affine Morita equivalence:

$\mathcal{B}_{1}$$(M_{1},\Lambda_{1})$$\underline{P}$$(M_{2},\Lambda_{2})$$\mathcal{B}_{2}$$\text{
}\text{ }\underline{\alpha}_{1}$$\underline{\alpha}_{2}$

where $\underline{P}=P/\mathcal{T}_{1}=P/\mathcal{T}_{2}$. To see this, let
$p\in P$ and denote $x_{1}=\alpha_{1}(p)$ and $x_{2}=\alpha_{2}(p)$. The given
Morita equivalence induces an isomorphism of Lie groups:

(37) $\varphi_{p}:(\mathcal{G}_{1})_{x_{1}}\to(\mathcal{G}_{2})_{x_{2}},$

uniquely determined by the property that for each
$g\in(\mathcal{G}_{1})_{x_{1}}$:

$g\cdot p=p\cdot\varphi_{p}(g).$

Since $\varphi_{p}$ is an isomorphism of Lie groups, it identifies the
identity component of $(\mathcal{G}_{1})_{x_{1}}$ with the identity component
of $(\mathcal{G}_{2})_{x_{2}}$. In other words, it identifies
$(\mathcal{T}_{1})_{x_{1}}$ with $(\mathcal{T}_{2})_{x_{2}}$ and hence it
follows that the $\mathcal{T}_{1}$-orbit through $p$ coincides with the
$\mathcal{T}_{2}$-orbit through $p$. This shows that
$P/\mathcal{T}_{1}=P/\mathcal{T}_{2}$. From the lemma below, it is clear that
the induced Morita equivalence is integral affine.

######  Lemma 1.38.

Suppose that we are given a symplectic Morita equivalence:

$(\mathcal{G}_{1},\Omega_{1})$$M_{1}$$(P,\omega_{P})$$M_{2}$$(\mathcal{G}_{2},\Omega_{2})$$\text{
}\text{ }\alpha_{1}$$\alpha_{2}$

For each $p\in P$, writing $x_{1}=\alpha_{1}(p)$ and $x_{2}=\alpha_{2}(p)$, we
have a commutative square:

${\mathcal{N}_{x_{2}}{\mathcal{O}_{2}}}$${\mathcal{N}_{x_{1}}{\mathcal{O}_{1}}}$${(\mathfrak{g}_{2})^{*}_{x_{2}}}$${(\mathfrak{g}_{1})^{*}_{x_{1}}}$$\scriptstyle{\psi_{p}^{-1}}$$\scriptstyle{\rho_{\Omega_{2}}^{*}}$$\scriptstyle{({\varphi_{p}})^{*}}$$\scriptstyle{\rho_{\Omega_{1}}^{*}}$

with horizontal arrows defined as in (36), respectively (37), and vertical
arrows defined as in (32).

###### Proof.

Notice (by dualizing and unravelling the definition of the upper horizontal
map) that we ought to prove the commutativity of the diagram:

${\mathcal{N}_{p}^{*}{\mathcal{O}}}$${\mathcal{N}_{x_{1}}^{*}\mathcal{O}_{1}}$${\mathcal{N}_{x_{2}}^{*}\mathcal{O}_{2}}$${(\mathfrak{g}_{1})_{x_{1}}}$${(\mathfrak{g}_{2})_{x_{2}}}$$\scriptstyle{(\textrm{d}\alpha_{1})^{*}}$$\scriptstyle{(\textrm{d}\alpha_{2})^{*}}$$\scriptstyle{(\rho_{\Omega_{1}})^{-1}}$$\scriptstyle{(\varphi_{p})_{*}}$$\scriptstyle{(\rho_{\Omega_{2}})^{-1}}$

First notice that for all $\xi\in(\mathfrak{g}_{1})_{x_{1}}$:

(38) $\exp(\xi)\cdot p=p\cdot\exp((\varphi_{p})_{*}(\xi)).$

Now consider the respective Lie algebroid actions:

$a_{L}:\alpha_{1}^{*}(T^{*}M_{1})\to TP\quad\&\quad
a_{R}:\alpha_{2}^{*}(T^{*}M_{2})\to TP$

induced by the $(\mathcal{G}_{1},\Omega_{1})$-action via $\rho_{\Omega_{1}}$
and by the $(\mathcal{G}_{2},\Omega_{2})$-action via $\rho_{\Omega_{2}}$. By
definition of $a_{L}$ and $a_{R}$, for every
$\xi\in(\mathfrak{g}_{1})_{x_{1}}$ and $\eta\in(\mathfrak{g}_{2})_{x_{2}}$ we
have:

$a_{L}\left((\rho_{\Omega_{1}})^{-1}(\xi)\right)=\left.\frac{\textrm{d}}{\textrm{d}t}\right|_{t=0}\exp(t\xi)\cdot
p\quad\&\quad
a_{R}\left((\rho_{\Omega_{2}})^{-1}(\eta)\right)=\left.\frac{\textrm{d}}{\textrm{d}t}\right|_{t=0}p\cdot\exp(-t\eta),$

which combined with (38) gives:

(39)
$a_{L}\left((\rho_{\Omega_{1}})^{-1}(\xi)\right)=-a_{R}\left((\rho_{\Omega_{2}})^{-1}((\varphi_{p})^{*}(\xi))\right).$

Since the (left) $(\mathcal{G}_{1},\Omega_{1})$-action and the (right)
$(\mathcal{G}_{2},\Omega_{2})$-action are Hamiltonian, $a_{L}$ and $a_{R}$
satisfy the momentum map condition. This means that for all
$\beta_{1}\in\Omega^{1}(M_{1})$ and $\beta_{2}\in\Omega^{1}(M_{2})$:

$\iota_{a_{L}(\beta_{1})}\omega_{P}=\alpha_{1}^{*}(\beta_{1})\quad\&\quad\iota_{a_{R}(\beta_{2})}\omega_{P}=-\alpha_{2}^{*}(\beta_{2}).$

Combined with (39) this implies the desired commutativity, which concludes the
proof. ∎

Below we give a precise meaning to the statement that Delzant subspaces are
well-behaved with respect to integral affine Morita equivalences of the
ambient integral affine orbifold.

######  Proposition 1.39.

Suppose that we are given an integral affine Morita equivalence:

$\mathcal{B}_{1}$$M_{1}$$P$$M_{2}$$\mathcal{B}_{2}$$\text{ }\text{
}\alpha_{1}$$\alpha_{2}$

that relates a given subset $\underline{\Delta}_{1}$ of $\underline{M}_{1}$ to
a subset $\underline{\Delta}_{2}$ of $\underline{M}_{2}$. Then for each $p\in
P$ such that $x_{1}:=\alpha_{1}(p)\in\Delta_{1}$, and
$x_{2}:=\alpha_{2}(p)\in\Delta_{2}$, it holds that:

$\psi_{p}(\log_{x_{1}}(\Delta_{1}))=\log_{x_{2}}(\Delta_{2}),$

where $\psi_{p}$ is defined as in (36).

######  Corollary 1.40.

In the setting of Proposition 1.39, $\underline{\Delta}_{1}$ is a Delzant
subspace of $\underline{M}_{1}$ if and only if $\underline{\Delta}_{2}$ is a
Delzant subspace of $\underline{M}_{2}$. In this case, for each $p\in P$ such
that $x_{1}:=\alpha_{1}(p)\in\Delta_{1}$, and
$x_{2}:=\alpha_{2}(p)\in\Delta_{2}$, their cones at $x_{1}$ and $x_{2}$ are
related as:

$\psi_{p}(C_{x_{1}}(\underline{\Delta}_{1}))=C_{x_{2}}(\underline{\Delta}_{2}),$

where $\psi_{p}$ is defined as in (36).

###### Proof of Proposition 1.39.

First notice that the respective foliations $\mathcal{F}_{1}$ and
$\mathcal{F}_{2}$ pull back along $\alpha_{1}$ and $\alpha_{2}$ to the same
foliation $\mathcal{F}$ on $P$. Moreover, since the Morita equivalence is
integral affine, the smooth lattices $\Lambda_{1}$ and $\Lambda_{2}$ pull back
to the same smooth lattice $\Lambda$ in $\mathcal{N}^{*}\mathcal{F}$. One
readily verifies that this lattice $\Lambda$ is Lagrangian in
$(T^{*}P,\Omega_{\textrm{can}})$. So, it defines a transverse integral affine
structure on the foliated manifold $(P,\mathcal{F})$. Being related by the
Morita equivalence, the respective invariant subsets $\Delta_{1}$ in $M_{1}$
and $\Delta_{2}$ in $M_{2}$, corresponding to $\underline{\Delta}_{1}$ and
$\underline{\Delta}_{2}$, have the same pre-image $\Delta$ in $P$ under
$\alpha_{1}$ and $\alpha_{2}$. Let $p\in\Delta$. Notice that, to prove the
proposition, it is enough to show that for both $i\in\\{1,2\\}$ the linear
isomorphism
$(\underline{\textrm{d}\alpha}_{i})_{p}:\mathcal{N}_{p}\mathcal{F}\to\mathcal{N}_{x_{i}}\mathcal{F}_{i}$
identifies the set germ $\log_{p}(\Delta)$ with the set-germ
$\log_{x_{i}}(\Delta_{i})$. To see that this is indeed the case, observe that
if $\nu_{i}:U_{i}\to\mathcal{N}_{x_{i}}\mathcal{F}_{i}$ is a
$\Delta_{i}$-adapted submersion as in Proposition 1.25, with respect to
$(M_{i},\mathcal{F}_{i},\Lambda_{i})$, then the composition:

$\alpha_{i}^{-1}(U_{i})\xrightarrow{\nu_{i}\circ\alpha_{i}}\mathcal{N}_{x_{i}}\mathcal{F}_{i}\xrightarrow{(\underline{\textrm{d}\alpha}_{i})^{-1}_{p}}\mathcal{N}_{p}\mathcal{F}$

is a $\Delta$-adapted submersion as in Proposition 1.25, with respect to
$(P,\mathcal{F},\Lambda)$. ∎

######  Corollary 1.41.

Let $(B,\mathcal{B},p,\Lambda)$ be an integral affine orbifold and let
$\underline{\Delta}$ be a Delzant subspace. For each $x\in\Delta$ the cone
$C_{x}(\underline{\Delta})$ of $\underline{\Delta}$ at $x$ is
$\mathcal{B}_{x}$-invariant.

###### Proof of Corollary 1.41.

Apply Corollary 1.40 to the identity equivalence. ∎

######  Example 1.42.

The description of the Delzant subspaces in Example 1.31 can be obtained as
follows. Recall that $\mathfrak{c}$ is a complete transversal for the
foliation of $\mathfrak{g}^{*}_{\textrm{reg}}$ by coadjoint $G^{0}$-orbits,
where $G^{0}$ denotes the identity component of $G$. The transverse integral
affine structure $\Lambda_{\mathcal{G}}$ induced by
$(\mathcal{G},\Omega):=(G\ltimes\mathfrak{g}^{*}_{\textrm{reg}},-\textrm{d}\lambda_{\textrm{can}})$
restricts to the same integral affine structure on $\mathfrak{c}$ as that
inherited from the integral affine vector space
$(\mathfrak{t}^{*},\Lambda_{T})$ (of which $\mathfrak{c}$ is an open subset).
Moreover, the restriction of the orbifold groupoid
$\mathcal{B}=\mathcal{G}/\mathcal{T}$ to $\mathfrak{c}$ is canonically
isomorphic to the action groupoid $N(\mathfrak{c})/T\ltimes\mathfrak{c}$ (cf.
[15, Proposition 3.15.1]). So, via Example 1.35 we obtain an integral affine
Morita equivalence:

$\mathcal{B}$$(\mathfrak{g}^{*}_{\textrm{reg}},\Lambda_{\mathcal{G}})$$s_{\mathcal{B}}^{-1}(\mathfrak{c})$$(\mathfrak{c},\Lambda_{T})$$N(\mathfrak{c})/T\ltimes\mathfrak{c}$

Applying Corollary 1.40 to this leads to the bijection in Example 1.31.

###  1.3. On the image of the momentum map

####  1.3.1. The symplectic normal representations and the momentum image

The objects in the previous two sections are related via the proposition
below, which we prove in the next subsection along with Theorem 2.

######  Proposition 1.43.

Let $(\mathcal{G},\Omega)$ be a regular and proper symplectic groupoid and let
$J:(S,\omega)\to M$ be a faithful multiplicity-free Hamiltonian
$(\mathcal{G},\Omega)$-space. Further, let $p\in S$ and $x:=J(p)$.

  * a)

The symplectic normal representation $(\mathcal{S}\mathcal{N}_{p},\omega_{p})$
at $p$ is maximally toric (Definition 1.2).

  * b)

The inclusion of the isotropy group $\mathcal{G}_{p}$ of the action into the
isotropy group $\mathcal{G}_{x}$ of $\mathcal{G}$ induces an isomorphism
between their groups of connected components:

(40)
$\varGamma_{\mathcal{G}_{p}}\xrightarrow{\sim}\varGamma_{\mathcal{G}_{x}}.$

  * c)

The cone at $x$ of $\underline{\Delta}:=\underline{J}(\underline{S})$ (which
is a Delzant subpace by Theorem 2) is given by:

(41)
$C_{x}(\underline{\Delta})=(\rho_{\Omega})_{x}^{*}(\pi_{\mathfrak{g}_{p}^{*}}^{-1}(\Delta_{J_{\mathcal{S}\mathcal{N}_{p}}})),$

where:

    * –

$(\rho_{\Omega})_{x}:\mathcal{N}_{x}^{*}\mathcal{F}\to\mathfrak{g}_{x}$ is
defined as in (32),

    * –

$\pi_{\mathfrak{g}_{p}^{*}}:\mathfrak{g}_{x}^{*}\to\mathfrak{g}_{p}^{*}$
denotes the canonical projection (dual to the inclusion
$\mathfrak{g}_{p}\hookrightarrow\mathfrak{g}_{x}$),

    * –

$J_{\mathcal{S}\mathcal{N}_{p}}:(\mathcal{S}\mathcal{N}_{p},\omega_{p})\to\mathfrak{g}_{p}^{*}$
denotes the quadratic momentum map of the symplectic normal representation at
$p$, as defined in (16), and $\Delta_{J_{\mathcal{S}\mathcal{N}_{p}}}$ denotes
its image.

Recall here that, given a Hamiltonian $(\mathcal{G},\Omega)$-space
$J:(S,\omega)\to M$, for each $p\in S$ there is a naturally associated
symplectic representation:

$(\mathcal{S}\mathcal{N}_{p},\omega_{p})\in\textrm{SympRep}(\mathcal{G}_{p}),$

of the isotropy group $\mathcal{G}_{p}$ of the action at $p$. We call this the
symplectic normal representation at $p$. Explicitly, this consists of the
symplectic normal space to the $\mathcal{G}$-orbit
$\mathcal{O}\subset(S,\omega)$ through $p$:

$\mathcal{S}\mathcal{N}_{p}:=\frac{T_{p}\mathcal{O}^{\omega}}{T_{p}\mathcal{O}\cap
T_{p}\mathcal{O}^{\omega}},$

equipped with the linear symplectic form induced by $\omega$ and with the
$\mathcal{G}_{p}$-action:

$g\cdot[v]=[\textrm{d}m_{(g,p)}(0,v)],\quad g\in\mathcal{G}_{p},\quad v\in
T_{p}\mathcal{O}^{\omega},$

where $m:\mathcal{G}\ltimes S\to S$ denotes the action map. Here
$T_{p}\mathcal{O}^{\omega}$ denotes the $\omega$-orthogonal to the tangent
space $T_{p}\mathcal{O}$ of the orbit at $p$. This generalizes the so-called
symplectic normal (or slice) representations for Hamiltonian Lie group
actions. For further details on this definition in the generality of
symplectic groupoid actions we refer to [38].

######  Remark 1.44.

As for symplectic toric manifolds, there is a natural way to read off the
isomorphism type of the isotropy groups (and so, also the canonical
stratification of the orbit space) of a faithful multiplicity-free Hamiltonian
action from its momentum image. Indeed, in view of Proposition 1.7 and
Proposition 1.43, the isotropy group $\mathcal{G}_{p}$ at $p\in S$ is
isomorphic to the semi-direct product of $\Gamma_{\mathcal{G}_{x}}$ with the
$\Gamma_{\mathcal{G}_{x}}$-invariant subtorus of $(\mathcal{T}_{\Lambda})_{x}$
with Lie algebra given by the annihilator in $T^{*}_{x}M$ of the face of the
polyhedral cone $C_{x}(\underline{\Delta})$ through the origin.

####  1.3.2. Proofs of Theorem 2 and Proposition 1.43

For torus actions, Theorem 2 and Proposition 1.43 boil down to the known facts
below about symplectic toric manifolds (which follow from an argument using
the Marle-Guillemin-Sternberg normal form; see [12] and [21, Lemma B.3]).

######  Proposition 1.45.

Let $T$ be a torus and let $J:(S,\omega)\to\mathfrak{t}^{*}$ be a possibly
non-compact and non-connected symplectic toric manifold for which $J$ descends
to a topological embedding $S/T\to\mathfrak{t}^{*}$. Let $p\in S$ and
$x=J(p)$.

  * a)

The symplectic normal representation $(\mathcal{S}\mathcal{N}_{p},\omega_{p})$
at $p$ is maximally toric.

  * b)

The isotropy group $T_{p}$ of the $T$-action is connected.

  * c)

The germ at $x$ of the image of $J$ is equal to that of the subset $x+C_{x}$,
where $C_{x}$ is the pre-image of the polyhedral cone
$\Delta_{J_{\mathcal{S}\mathcal{N}_{p}}}\subset\mathfrak{t}_{p}^{*}$ under the
canonical projection $\mathfrak{t}^{*}\to\mathfrak{t}_{p}^{*}$.

Our strategy of proof will be to reduce to this case. For this we will use:

######  Lemma 1.46.

Let $\mathcal{B}\rightrightarrows M$ be a proper etale Lie groupoid. Every
$x\in M$ admits a connected open neighbourhood $U$ such that for each
$\gamma\in\mathcal{B}_{x}$ there exists a (necessarily unique) smooth local
section $\sigma_{\gamma}:U\to\mathcal{B}|_{U}$ of the source-map that sends
$x$ to $\gamma$, and such that the map:

$\mathcal{B}_{x}\times
U\to\mathcal{B}|_{U},\quad(\gamma,y)\mapsto\sigma_{\gamma}(y)$

is surjective. This defines an isomorphism of Lie groupoids between the action
groupoid $\mathcal{B}_{x}\ltimes U$ of the $\mathcal{B}_{x}$-action on $U$
given by $\gamma\cdot y:=t(\sigma_{\gamma}(y))$ and $\mathcal{B}|_{U}$. In
particular, $\mathcal{B}|_{U}$ is source-proper.

###### Proof.

See [36, Proposition 5.30]. ∎

######  Lemma 1.47.

Let $(\mathcal{G},\Omega)$ be regular and proper symplectic groupoid and let
$J:(S,\omega)\to M$ be a faithful multiplicity-free
$(\mathcal{G},\Omega)$-space. If the associated orbifold groupoid
$\mathcal{B}=\mathcal{G}/\mathcal{T}$ is etale, then $J:(S,\omega)\to M$ is a
faithful toric $(\mathcal{T},\Omega_{\mathcal{T}})$-space with respect to the
induced $\mathcal{T}$-action.

###### Proof.

Note that if $\mathcal{B}$ is etale, then $(\mathcal{T},\Omega_{\mathcal{T}})$
is symplectic. To see that the induced Hamiltonian
$(\mathcal{T},\Omega_{\mathcal{T}})$-action is faithful toric, the only thing
to show is that the map $\underline{J}_{\mathcal{T}}:S/{\mathcal{T}}\to M$ is
a topological embedding. This map clearly being a continuous injection, it
remains to show that $\underline{J}_{\mathcal{T}}:S/{\mathcal{T}}\to\Delta$ is
closed, or equivalently, that $J:S\to\Delta$ is closed. For this, it suffices
to show that every $x\in\Delta$ admits an open neighbourhood $U$ in $\Delta$
such that $J:J^{-1}(U)\to U$ is closed. Let $x\in\Delta$, take $U_{M}$ to be
an open in $M$ around $x$ as Lemma 1.46 and let $U=U_{M}\cap\Delta$. Then
$\mathcal{B}|_{U_{M}}$ is source-proper, hence so is $\mathcal{G}|_{U_{M}}$
and so is the action groupoid of the restriction of the $\mathcal{G}$-action
to $U_{M}$. Therefore, the quotient map $q_{S}:S\to\underline{S}$ restricts to
a proper map from $J^{-1}(U)$ onto its image in $\underline{S}$. Combined with
the fact that $\underline{J}:\underline{S}\to\underline{M}$ is a topological
embedding, it follows that $J:J^{-1}(U)\to U$ is proper. Because any
continuous proper map into a first-countable and Hausdorff space is closed
(see e.g. [41]), we conclude that $J^{-1}(U)\to U$ is indeed closed. ∎

We are now ready to complete the proof.

###### End of the proof of Theorem 2 and Proposition 1.43.

First suppose that the orbifold groupoid $\mathcal{B}=\mathcal{G}/\mathcal{T}$
is etale, so that the integral affine structure $\Lambda$ induced by
$(\mathcal{G},\Omega)$ is an honest integral affine structure on $M$ and (by
Lemma 1.47) $J:(S,\omega)\to M$ is a faithful toric
$(\mathcal{T},\Omega_{\mathcal{T}})$-space with respect to the induced action.
Let $p\in S$, $x:=J(p)$ and fix an integral affine isomorphism $\iota$ from an
open $U$ in $(M,\Lambda)$ into an open $V$ in $(T_{x}M,\Lambda^{*}_{x})$ that
maps $x$ to the origin and the derivative of which at $x$ is the identity map
on $T_{x}M$. This induces an isomorphism of symplectic torus bundles:

(42)
$(\mathcal{T},\Omega_{\mathcal{T}})|_{U}\xrightarrow{(\ref{exptorbun2})}(\mathcal{T}_{\Lambda},\Omega_{\Lambda})|_{U}\xrightarrow{\iota_{*}}(T\ltimes\mathfrak{t}^{*},-\textrm{d}\lambda_{\textrm{can}})|_{V},\quad
T:=(\mathcal{T}_{\Lambda})_{x}.$

Consider the Hamiltonian
$(T\ltimes\mathfrak{t}^{*},-\textrm{d}\lambda_{\textrm{can}})|_{V}$-action
along $\iota\circ J:(J^{-1}(U),\omega)\to V$ corresponding to the restriction
to $U$ of the Hamiltonian $(\mathcal{T},\Omega)$-action along $J$. The data of
such a Hamiltonian action is the same as that of a Hamiltonian $T$-action with
momentum map:

(43) $\iota\circ J:(J^{-1}(U),\omega)\to T_{x}M=\mathfrak{t}^{*}.$

Since $J$ is a faithful multiplicity-free Hamiltonian
$(\mathcal{G},\Omega)$-space, (43) is a Hamiltonian $T$-space as in
Proposition 1.45. Further notice that
$T_{p}=\mathcal{G}_{p}\cap\mathcal{T}_{x}$ and that the symplectic normal
representation at $p$ of the Hamiltonian $T$-space (43) coincides with the
symplectic $T_{p}$-representation induced by the symplectic normal
representation
$\mathcal{G}_{p}\to\textrm{Sp}(\mathcal{S}\mathcal{N}_{p},\omega_{p})$ of the
$(\mathcal{G},\Omega)$-space $J$. So, it follows from parts $a$ and $b$ of
Proposition 1.45 that the symplectic normal representation at $p$ of the
$(\mathcal{G},\Omega)$-space $J$ is maximally toric as well and that
$\mathcal{G}_{p}\cap\mathcal{T}_{x}$ is connected. The latter means that
$\mathcal{G}_{p}\cap\mathcal{T}_{x}$ is the identity component of
$\mathcal{G}_{p}$, or in other words, (40) is injective. For surjectivity,
note that given $g\in\mathcal{G}_{x}$, both $p$ and $g^{-1}\cdot p$ belong to
the same fiber of $J$ over $x$. So, since the $J$-fibers coincide with the
$\mathcal{T}$-orbits, there is a $t\in\mathcal{T}_{x}$ such that $g^{-1}\cdot
p=t\cdot p$. Then $gt\in\mathcal{G}_{p}$, and
$[gt]\in\varGamma_{\mathcal{G}_{p}}$ is send to
$[g]\in\varGamma_{\mathcal{G}_{x}}$. This proves parts $a$ and $b$ of
Proposition 1.43. From part $c$ of Proposition 1.45 it follows that the germ
of $\iota(U\cap\Delta)$ at the origin in $T_{x}M$ is that of the polyhedral
cone $\pi_{\mathfrak{g}_{p}}^{-1}(\Delta_{J_{\mathcal{S}\mathcal{N}_{p}}})$,
where $\Delta:=J(S)$. This polyhedral cone in $(T_{x}M,\Lambda_{x}^{*})$
smooth, since the polyhedral cone $\Delta_{J_{\mathcal{S}\mathcal{N}_{p}}}$ is
smooth in $(\mathfrak{g}_{p}^{*},\Lambda_{T_{p}})$ and
$\mathfrak{g}_{p}^{0}\cap\Lambda_{x}$ is a full-rank lattice in
$\mathfrak{g}_{p}^{0}$ (because $\mathfrak{g}_{p}$ is the Lie algebra of a
subtorus of $\mathcal{T}_{x}$). So, Theorem 2 and Proposition 1.43 hold when
$\mathcal{B}=\mathcal{G}/\mathcal{T}$ is etale.

When $\mathcal{B}$ is not etale, we reduce to the etale case as follows. As
before, let $p\in S$ and $x:=J(p)$. Choose a transversal $\Sigma$ for
$\mathcal{B}$ through $x$. Then $\Sigma$ is a Poisson transversal in
$(M,\pi)$, where $\pi$ is the Poisson structure on $M$ induced by
$(\mathcal{G},\Omega)$. Therefore, any Poisson map from a symplectic manifold
into $(M,\pi)$ is transversal to $\Sigma$ and the pre-image of $\Sigma$ under
such a map is a symplectic submanifold of its domain. So, $J^{-1}(\Sigma)$ is
a symplectic submanifold of $(S,\omega)$ and $(\mathcal{G},\Omega)|_{\Sigma}$
is a symplectic subgroupoid of $(\mathcal{G},\Omega)$. The
$(\mathcal{G},\Omega)$-action along $J$ restricts to a Hamiltonian
$(\mathcal{G},\Omega)|_{\Sigma}$-action along:

$J_{\Sigma}:(J^{-1}(\Sigma),\omega|_{J^{-1}(\Sigma)})\to\Sigma.$

This is again a faithful multiplicity-free Hamiltonian action and its
symplectic normal representation at $p$ is canonically isomorphic to that of
the $(\mathcal{G},\Omega)$-action along $J$. The symplectic groupoids
$(\mathcal{G},\Omega)$ and $(\mathcal{G},\Omega)|_{\Sigma}$ are Morita
equivalent via a bibundle like that in Example 1.36 and the induced the
integral affine Morita equivalence (Example 1.37) is that in Example 1.36. In
view of Corollary 1.40, since the orbifold groupoid $\mathcal{B}|_{\Sigma}$
associated to $\mathcal{G}|_{\Sigma}$ is etale, this reduces the proof of
Theorem 2 and Proposition 1.43 to the etale case. ∎

##  2\. Classification in the case of symplectic torus bundle actions

In this part we complete the proof of Theorem 1 (the classification in the
case of symplectic torus bundle actions), for which it remains to prove part
$b$. One reason for treating this case separately is that the classification
simplifies, whilst the most of the main ideas of the classification are
already present. The orbifold structure plays no role in this case (it is
trivial) and the three structure theorems boil down to one and the same
theorem, that we prove in Section 2.1. In Section 2.2 we complete the proof of
Theorem 1 by showing that for any Delzant subspace of an integral affine
manifold, there is a canonical faithful toric space with that Delzant subspace
as the image of its momentum map. This construction (and its naturality, in
particular) will be key as well in the proofs of Theorem 3 (the first
structure theorem) and Theorem 4 (the splitting theorem) in the next part.

###  2.1. The sheaf of Lagrangian sections and the structure theorem

####  2.1.1. The sheaves of automorphisms and Lagrangian sections

In this subsection we explain and prove the theorem below, which will be key
for the proof of Theorem 1$b$.

######  Theorem 2.1.

Let $(\mathcal{T},\Omega)$ be a symplectic torus bundle and let
$J:(S,\omega)\to M$ be a faithful toric $(\mathcal{T},\Omega)$-space with
associated Delzant subspace $\Delta:=J(S)$. There is an isomorphism of sheaves
on $\Delta$ —induced by the map (45) below —between the sheaf
$\textrm{Aut}_{\mathcal{T}}(J,\omega)$ of automorphisms of the
$(\mathcal{T},\Omega)$-space $J:(S,\omega)\to M$ and the sheaf $\mathcal{L}$
of Lagrangian sections of $\mathcal{T}|_{\Delta}$ (as in Definition 2.2
below).

The sheaf of Lagrangian sections referred to here and in Theorem 1 is defined
as follows.

######  Definition 2.2.

Let $(\mathcal{T},\Omega)\to M$ be a symplectic torus bundle with induced
integral affine structure $\Lambda$ on $M$ and let $\Delta$ a Delzant subspace
of $(M,\Lambda)$. We denote by:

  * •

$\mathcal{C}^{\infty}_{\Delta}(\mathcal{T})$ the sheaf on $\Delta$ of smooth
sections of the bundle $\mathcal{T}|_{\Delta}\to\Delta$ (where smooth is meant
as maps between manifolds with corners),

  * •

$\mathcal{L}:=\mathcal{L}_{\Delta}$ the subsheaf of
$\mathcal{C}^{\infty}_{\Delta}(\mathcal{T})$ consisting of those sections
${\tau}$ that are Lagrangian, in the sense that ${\tau}^{*}\Omega=0$.

######  Remark 2.3.

Given a submersion $f:M\to N$ between manifolds without corners and an
embedded submanifold with corners $\Delta$ in $N$ (see Definition B.11), a
local section $\sigma:U\to f^{-1}(\Delta)$ of $f$, defined on an open $U$ in
$\Delta$, is smooth as map of manifolds with corners if and only if for every
$x\in U$ there is a smooth local section $U_{x}\to M$ of $f$, defined on an
open $U_{x}$ in $N$ around $x$, that coincides with $\sigma$ on $U\cap U_{x}$
(also see Remark B.9). This can be taken as a working definition.

######  Remark 2.4.

The sheaf $\mathcal{L}$ is a sheaf of abelian groups (with group structure
induced by that on the fibers of $\mathcal{T}$). So, automorphisms of a
faithful toric $(\mathcal{T},\Omega)$-space commute.

To explain the definition of the isomorphism in Theorem 2.1, consider the
sheaf of groups $\textrm{Aut}_{\mathcal{T}}(J)$ on $\Delta$ that assigns to an
open $U$ in $\Delta$ the group:

(44) $\textrm{Aut}_{\mathcal{T}}(J)(U)$

consisting of $\mathcal{T}$-equivariant diffeomorphisms:

${J^{-1}(U)}$${J^{-1}(U)}$${U}$$\scriptstyle{\cong}$$\scriptstyle{J}$$\scriptstyle{J}$

The sheaf $\textrm{Aut}_{\mathcal{T}}(J,\omega)$ in Theorem 2.1 is the
subsheaf of $\textrm{Aut}_{\mathcal{T}}(J)$ that assigns to such an open the
subgroup of $\mathcal{T}$-equivariant symplectomorphisms. To relate these to
the sheaves in Definition 2.2, consider the map of sheaves (of groups) on
$\Delta$:

(45)
$\mathcal{C}^{\infty}_{\Delta}(\mathcal{T})\to\textrm{Aut}_{\mathcal{T}}(J),\quad{\tau}\mapsto\psi_{{\tau}},$

where, for ${\tau}\in\mathcal{C}^{\infty}_{\Delta}(\mathcal{T})(U)$, we
define:

$\psi_{{\tau}}:J^{-1}(U)\to J^{-1}(U),\quad\psi_{{\tau}}(p)={\tau}(J(p))\cdot
p.$

Notice that (because the $\mathcal{T}$-action is free on a dense subset):

######  Proposition 2.5.

The map of sheaves (45) is injective.

Furthermore, we have:

######  Proposition 2.6.

A section ${\tau}\in\mathcal{C}^{\infty}_{\Delta}(\mathcal{T})(U)$ is
Lagrangian if and only if $\psi_{{\tau}}$ is a symplectomorphism.

###### Proof.

Using that the $(\mathcal{T},\Omega)$-action is Hamiltonian, we deduce:

$\displaystyle(\psi_{{\tau}})^{*}\omega$ $\displaystyle=({\tau}\circ
J,\textrm{id}_{S})^{*}(m_{S})^{*}\omega$ $\displaystyle=({\tau}\circ
J,\textrm{id}_{S})^{*}\left((\textrm{pr}_{S})^{*}\omega+(\textrm{pr}_{\mathcal{T}})^{*}\Omega\right)$
$\displaystyle=\omega+J^{*}({\tau}^{*}\Omega).$

Therefore, $\psi_{{\tau}}$ is a symplectomorphism if and only if
$J^{*}({\tau}^{*}\Omega)=0$. If the $\mathcal{T}$-action is free at a point
$p\in J^{-1}(U)$, then $J$ is a submersion at $p$, so that
$J^{*}({\tau}^{*}\Omega)_{p}=0$ if and only if $({\tau}^{*}\Omega)_{J(p)}=0$.
As the set of points where $\mathcal{T}$ acts freely is dense in $S$, it
follows from continuity that $J^{*}({\tau}^{*}\Omega)=0$ if and only if
${\tau}^{*}\Omega=0$. This proves the proposition. ∎

In view of these propositions, the map (45) induces injective maps of sheaves:

(46) $\displaystyle\mathcal{C}^{\infty}_{\Delta}(\mathcal{T})$
$\displaystyle\to\textrm{Aut}_{\mathcal{T}}(J),$ (47)
$\displaystyle\mathcal{\mathcal{L}}_{\Delta}$
$\displaystyle\to\textrm{Aut}_{\mathcal{T}}(J,\omega).$

Theorem 2.1 says that (47) is an isomorphism. To prove this, by Proposition
2.6 it suffices to show:

######  Theorem 2.7.

The map (46) is an isomorphism of sheaves.

###### Proof of Theorem 2.7 (and hence of Theorem 2.1).

It remains to show that (46) is stalk-wise surjective. Let $x\in\Delta$ and
let $\psi\in\textrm{Aut}_{\mathcal{T}}(J)(U)$ for some open $U$ around $x$ in
$\Delta$. We have to show that there is a smooth section ${\tau}$ of
$\mathcal{T}|_{\Delta}$, defined on an open neighbourhood of $x$ in $\Delta$,
such that the germ of $\psi_{{\tau}}$ at $x$ coincides with that of $\psi$.
The proof of this reduces to the case in which $M=\mathfrak{t}^{*}$, $x$ is
the origin and $(\mathcal{T},\Omega)$ is the symplectic torus bundle
$(T\times\mathfrak{t}^{*},-\textrm{d}\lambda_{\textrm{can}})$ associated to a
torus $T$, by using an isomorphism (42) as in the proof of Theorem 2 and
Proposition 1.43. In that case, the data of a faithful complexity
$(\mathcal{T},\Omega)$-space $J:(S,\omega)\to\mathfrak{t}^{*}$ is the same as
that of a (possibly non-compact and non-connected) symplectic toric
$T$-manifold $(S,\omega)$ with momentum map $J$ that descends to a topological
embedding $\underline{J}:S/T\to\mathfrak{t}^{*}$. By appealing to the standard
normal form theorem for such Hamiltonian $T$-spaces (see e.g. [21, Lemma B.3])
and using that any $T$-invariant open in $S$ is the pre-image under $J$ of an
open in $\mathfrak{t}^{*}$ ($\underline{J}$ being a topological embedding),
the proof further reduces to the subcase in which:

  * •

$T=\mathbb{T}^{n}$ (the standard torus),

  * •

$S$ is an open in $\mathbb{T}^{n-k}\times\mathbb{R}^{n-k}\times\mathbb{C}^{k}$
of the form $\mathbb{T}^{n-k}\times B^{n-k}\times B^{k}$, where
$B^{n-k}\subset\mathbb{R}^{n-k}$ and $B^{k}\subset\mathbb{C}^{k}$ are open
balls of the same radius and centered around the respective origins, and
$\mathbb{T}^{n-k}\times\mathbb{R}^{n-k}\times\mathbb{C}^{k}$ is equipped with
the standard symplectic form,

  * •

the $T$-action on $S$ is given by:

$(\lambda_{1},...,\lambda_{n})\cdot(t_{1},...,t_{n-k},x_{1},...,x_{n-k},z_{1},...,z_{k})=(\lambda_{k+1}t_{1},...,\lambda_{n}t_{n-k},x_{1},...,x_{n-k},\lambda_{1}z_{1},...,\lambda_{k}z_{k}),$

  * •

$J$ is the restriction of the map:

$\displaystyle\mathbb{T}^{n-k}\times\mathbb{R}^{n-k}\times\mathbb{C}^{k}$
$\displaystyle\to\mathbb{R}^{n},$
$\displaystyle(t_{1},...,t_{n-k},x_{1},....,x_{n-k},z_{1},...,z_{k})$
$\displaystyle\mapsto(|z_{1}|^{2},...,|z_{k}|^{2},x_{1},...,x_{n-k}),$

  * •

$\psi$ is a $\mathbb{T}^{n}$-equivariant self-diffeomorphism of
$\mathbb{T}^{n-k}\times B^{n-k}\times B^{k}$ such that $J\circ\psi=J$.

To conclude the proof we will show that there is map
${\tau}:\Delta\to\mathbb{T}^{n}$, defined on the image $\Delta$ of
$\mathbb{T}^{n-k}\times B^{n-k}\times B^{k}$ under $J$, such that $\psi$ is
given by:

(48)
$\psi(t,x,z)={\tau}(J(t,x,z))\cdot(t,x,z),\quad(t,x,z)\in\mathbb{T}^{n-k}\times
B^{n-k}\times B^{k},$

and such that ${\tau}$ extends to a smooth map from an open in
$\mathbb{R}^{n}$ into $\mathbb{T}^{n}$. For this we use a variation of the
argument used to prove [12, Lemma 2.6]. Let $\varphi_{j}$ and $\psi_{j}$
denote the $j^{\textrm{th}}$ component of $\psi$ in $\mathbb{T}^{n-k}$ and
$\mathbb{C}^{k}$, respectively. Using equivariance of $\psi$ and the fact that
$\psi$ preserves $J$, we find:

(49)
$\psi(t,x,z)=(t_{1}\cdot\varphi_{1}(1,x,z),...,t_{n-k}\cdot\varphi_{n-k}(1,x,z),x,\psi_{1}(1,x,z),...,\psi_{k}(1,x,z)).$

Using equivariance of $\psi$ once more, it further follows that for all
$(x,z)\in B^{n-k}\times B^{k}$ and $\lambda\in\mathbb{T}^{k}$:

$\displaystyle\varphi_{j}(1,x,\lambda\cdot z)=\varphi_{j}(1,x,z),$
$\displaystyle\psi_{j}(1,x,z)=\lambda_{j}\psi_{j}(1,x,z).$

In particular, $(x,u)\mapsto\psi_{j}(1,x,u)$ restricts to a smooth function on
$B^{n-k}\times(B^{k}\cap\textrm{Re}(\mathbb{C}^{k}))$, which is odd in the
$u_{j}$ variable and even in the other $u$-variables, while
$(x,u)\mapsto\varphi_{j}(1,x,u)$ restricts to a smooth function on this domain
as well, but is even in all $u$-variables. Therefore, a theorem due to Whitney
[47] (a particular case of Schwarz’ theorem for finite groups [3, 42]) implies
that there are continuous functions:

$f_{j},g_{j}:\Delta\to\mathbb{C}$

that satisfy:

$\displaystyle\varphi_{j}(1,x,u)=f_{j}(u_{1}^{2},...,u_{k}^{2},x)$
$\displaystyle\psi_{j}(1,x,u)=u_{j}g_{j}(u_{1}^{2},...,u_{k}^{2},x),$

for all $(x,u)\in B^{n-k}\times(B^{k}\cap\textrm{Re}(\mathbb{C}^{k}))$, and
extend to smooth functions on some open neighbourhood of $\Delta$ in
$\mathbb{R}^{n}$. The fact that $\psi$ preserves $J$ implies that:

$u_{j}^{2}|g_{j}(u_{1}^{2},...,u_{k}^{2},x)|^{2}=|\psi_{j}(1,x,u)|^{2}=u_{j}^{2}$

for all $(x,u)\in B^{n-k}\times(B^{k}\cap\textrm{Re}(\mathbb{C}^{k}))$.
Therefore $g_{j}$ takes values in $\mathbb{S}^{1}$ on the interior of $\Delta$
in $\mathbb{R}^{n}$, hence it must do so on all of $\Delta$ by a density
argument. Note that $f_{j}$ does so as well, since $\varphi_{j}$ does.
Finally, observe that for $(x,z)\in B^{n-k}\times B^{k}$, writing
$z_{j}=e^{i\theta_{j}}|z_{j}|$ one finds:

(50) $\displaystyle\psi_{j}(1,x,z)$
$\displaystyle=e^{i\theta_{j}}\psi_{j}(1,x,|z_{1}|,...,|z_{k}|)$
$\displaystyle=z_{j}g_{j}(|z_{1}|^{2},...,|z_{k}|^{2},x),$

by equivariance of $\psi$. Similarly:

(51) $\varphi_{j}(1,x,z)=f_{j}(|z_{1}|^{2},...,|z_{k}|^{2},x).$

Now define:

$f:\Delta\to\mathbb{T}^{n-k}\quad\&\quad g:\Delta\to\mathbb{T}^{k}$

to have $j^{\textrm{th}}$ component $f_{j}$ and $g_{j}$, respectively, and
consider:

${\tau}:=(g,f):\Delta\to\mathbb{T}^{k}\times\mathbb{T}^{n-k}.$

Combining (49), (50) and (51) we find that (48) holds. Moreover, by
construction of $f$ and $g$, ${\tau}$ extends to a smooth map into
$\mathbb{C}^{n}$ on an open neighbourhood of $\Delta$ in $\mathbb{R}^{n}$.
Since none of its components vanish on a small enough such neighbourhood, by
normalizing them we can find such an extension that maps smoothly into
$\mathbb{T}^{n}$. So, ${\tau}$ has the desired properties. ∎

####  2.1.2. The structure theorem

For symplectic torus bundle actions the three structure theorems boil down to
the statement below, which we prove in this subsection.

######  Theorem 2.8.

Let $(\mathcal{T},\Omega)$ be a symplectic torus bundle over $M$ with induced
integral affine structure $\Lambda$ and let $\Delta$ be a Delzant subspace of
$(M,\Lambda)$. There is a natural free and transitive action of the group
$H^{1}(\Delta,\mathcal{L})$ (as in Theorem 1) on the set of isomorphism
classes of faithful toric $(\mathcal{T},\Omega)$-spaces with momentum image
$\Delta$.

To prove Theorem 2.8, we will first show the following.

######  Proposition 2.9.

Let $(\mathcal{T},\Omega)$ be a symplectic torus bundle and suppose that we
are given faithful toric $(\mathcal{T},\Omega)$-spaces
$J_{1}:(S_{1},\omega_{1})\to M$ and $J_{2}:(S_{2},\omega_{2})\to M$. Let $x\in
J_{1}(S_{1})\cap J_{2}(S_{2})$. Then there is an open neighbourhood $U$ of $x$
in $M$ and a $\mathcal{T}$-equivariant symplectomorphism:

${(J_{1}^{-1}(U),\omega_{1})}$${(J_{2}^{-1}(U),\omega_{2})}$${M}$$\scriptstyle{\cong}$$\scriptstyle{J_{1}}$$\scriptstyle{J_{2}}$

if and only if the germs at $x$ in $M$ of $J_{1}(S_{1})$ and $J_{2}(S_{2})$
are equal.

For this we use the theorem below, which is a generalization of a well-known
theorem due to Marle [29] and Guillemin and Sternberg [19] for Hamiltonian
actions of compact Lie groups.

######  Theorem 2.10 ([38]).

Let $(\mathcal{G},\Omega)\rightrightarrows M$ be a proper symplectic groupoid.
Suppose we are given two Hamiltonian $(\mathcal{G},\Omega)$-spaces
$J_{1}:(S_{1},\omega_{1})\to M$ and $J_{2}:(S_{2},\omega_{2})\to M$. Let
$p_{1}\in S_{1}$ and $p_{2}\in S_{2}$ be such that
$J_{1}(p_{1})=J_{2}(p_{2})$, $\mathcal{G}_{p_{1}}=\mathcal{G}_{p_{2}}$ and
$(\mathcal{S}\mathcal{N}_{p_{1}},\omega_{p_{1}})\cong(\mathcal{S}\mathcal{N}_{p_{2}},\omega_{p_{2}})$
are isomorphic symplectic representations. Then there are
$\mathcal{G}$-invariant open neighbourhoods $U_{i}$ of $p_{i}$ in $S_{i}$ and
an equivariant symplectomorphism that sends $p_{1}$ to $p_{2}$:

${(U_{1},\omega_{1},p_{1})}$${(U_{2},\omega_{2},p_{2})}$${M}$$\scriptstyle{\sim}$$\scriptstyle{J_{1}}$$\scriptstyle{J_{2}}$

For actions of symplectic torus bundles, Theorem 2.10 can be proved by
reducing to the case of Hamiltonian torus actions via a trivialization of the
symplectic torus bundle as in (42).

###### Proof of Proposition 2.9.

The implication from left to right is straightforward. For the other
implication, let $x\in J_{1}(S_{1})\cap J_{2}(S_{2})$ such that the germs at
$x$ of $J_{1}(S_{1})$ and $J_{2}(S_{2})$ coincide. Since
$\underline{J}:\underline{S}\to M$ is a topological embedding, every invariant
$\mathcal{T}$-invariant open in $S$ is of the form $J^{-1}(U)$ where $U$ is
some open in $M$. So, in view of Theorem 2.10, it remains to show that there
are $p_{1}\in J_{1}^{-1}(x)$ and $p_{2}\in J_{2}^{-1}(x)$ such that
$\mathcal{T}_{p_{1}}=\mathcal{T}_{p_{2}}$ and the symplectic representations
$(\mathcal{S}\mathcal{N}_{p_{1}},\omega_{p_{1}})$ and
$(\mathcal{S}\mathcal{N}_{p_{2}},\omega_{p_{2}})$ are isomorphic. Fix any
$p_{1}\in J_{1}^{-1}(x)$ and $p_{2}\in J_{2}^{-1}(x)$. Since the germs at $x$
of $J_{1}(S_{1})$ and $J_{2}(S_{2})$ coincide, Proposition 1.43$c$ implies
that:

(52)
$(\pi_{\mathfrak{t}^{*}_{p_{1}}})^{-1}(\Delta_{\mathcal{S}\mathcal{N}_{p_{1}}})=(\pi_{\mathfrak{t}^{*}_{p_{2}}})^{-1}(\Delta_{\mathcal{S}\mathcal{N}_{p_{2}}}).$

Since the symplectic normal representations at $p_{1}$ and $p_{2}$ are
maximally toric (Proposition 1.43$a$), the polyhedral cones
$\Delta_{\mathcal{S}\mathcal{N}_{p_{1}}}$ and
$\Delta_{\mathcal{S}\mathcal{N}_{p_{2}}}$ are pointed. So, (52) implies that
$\textrm{Ker}(\pi_{\mathfrak{t}^{*}_{p_{1}}})=\textrm{Ker}(\pi_{\mathfrak{t}^{*}_{p_{2}}})$.
Seeing as $\textrm{Ker}(\pi_{\mathfrak{t}^{*}_{p_{i}}})$ is the annihilator of
$\mathfrak{t}_{p_{i}}=\textrm{Lie}(\mathcal{T}_{p_{i}})$ in
$\mathfrak{t}_{x}^{*}$, this means that
$\mathfrak{t}_{p_{1}}=\mathfrak{t}_{p_{2}}$, and hence
$\mathcal{T}_{p_{1}}=\mathcal{T}_{p_{2}}$. In view of $(\ref{eqpolcon})$, we
further conclude that
$\Delta_{\mathcal{S}\mathcal{N}_{p_{1}}}=\Delta_{\mathcal{S}\mathcal{N}_{p_{2}}}$.
So, it follows from Proposition 1.9 that the symplectic normal representations
at $p_{1}$ and $p_{2}$ are isomorphic. ∎

###### Proof of Theorem 2.8.

The construction of the action uses a Čech model for sheaf cohomology, in much
the same way as in the usual classification of principal torus bundles. Since
the natural group homomorphism from Čech into sheaf cohomology:

(53) $\check{H}^{1}(\Delta,\mathcal{L})\to H^{1}(\Delta,\mathcal{L})$

is an isomorphism, given $\textrm{c}\in H^{1}(\Delta,\mathcal{L})$ we can
consider a Čech $1$-cocycle
${\tau}\in\check{C}^{1}_{\text{$\mathcal{U}$}}(\Delta,\mathcal{L})$ with
respect to some open cover $\mathcal{U}$ that represents c via (53). Write
$\psi_{UV}:=\psi_{{\tau}(U,V)}$ as in (45). Furthermore, let $J:(S,\omega)\to
M$ be a faithful toric $(\mathcal{T},\Omega)$-space with momentum image
$\Delta$. Then we can define another $(\mathcal{T},\Omega)$-space
$J_{{\tau}}:(S_{{\tau}},\omega_{{\tau}})\to M$, as follows. As topological
space, define:

$S_{{\tau}}:=\frac{\left(\bigsqcup_{U\in\text{$\mathcal{U}$}}J^{-1}(U)\right)}{\sim_{{\tau}}}$

where $(p,U)\sim_{{\tau}}(q,V)$ if and only if $J(p)=J(q)$ and
$p=\psi_{UV}(q)$. This indeed defines an equivalence relation, because
${\tau}$ is a $1$-cocycle. Furthermore, because
$\psi_{UV}\in\textrm{Aut}_{\mathcal{T}}(J,\omega)(U\cap V)$ for all
$U,V\in\text{$\mathcal{U}$}$, there is a unique smooth structure on
$S_{{\tau}}$, a unique symplectic structure $\omega_{{\tau}}$ on $S_{{\tau}}$
and a unique $\mathcal{T}$-action along the canonical map
$J_{{\tau}}:S_{{\tau}}\to M$ such that, for each $U\in\text{$\mathcal{U}$}$,
the canonical inclusion:

${(J^{-1}(U),\omega)}$${(S_{{\tau}},\omega_{{\tau}})}$${M}$$\scriptstyle{J}$$\scriptstyle{J_{{\tau}}}$

is a smooth, symplectic and $\mathcal{T}$-equivariant embedding. Clearly,
$J_{{\tau}}:(S_{{\tau}},\omega_{{{\tau}}})\to M$ is a faithful toric
$(\mathcal{T},\Omega)$-space with momentum image equal to $\Delta$. Now,
define:

$\textrm{c}\cdot[J:(S,\omega)\to
M]:=[J_{{\tau}}:(S_{{\tau}},\omega_{{\tau}})\to M].$

As one readily verifies, this does not depend on the choice of representative
${\tau}$ or cover $\mathcal{U}$, and defines an action of
$H^{1}(\Delta,\mathcal{L})$. To see that this action is free, suppose that:

$[J_{{\tau}}:(S_{{\tau}},\omega_{{\tau}})\to M]=[J:(S,\omega)\to M],$

meaning that there is an isomorphism of $(\mathcal{T},\Omega)$-spaces:

${(S_{{\tau}},\omega_{{\tau}})}$${(S,\omega)}$${M}$$\scriptstyle{\psi}$$\scriptstyle{J_{{\tau}}}$$\scriptstyle{J}$

For each $U\in\text{$\mathcal{U}$}$, consider:

$\psi_{U}:=\psi|_{J^{-1}(U)}\in\textrm{Aut}_{\mathcal{T}}(J,\omega)(U).$

Since $\psi$ is well-defined, for each $q\in J^{-1}(U\cap V)$ it must hold
that $\psi_{V}(q)=\psi_{U}(\psi_{UV}(q))$. Hence:

$\psi_{UV}=\psi_{U}^{-1}\circ\psi_{V}.$

Letting $\sigma(U):U\to\mathcal{T}$ be the Lagrangian section such that
$\psi_{\sigma(U)}=\psi_{U}$ (which exists by Theorem 2.1), we find that:

${\tau}(U,V)=\sigma(V)-\sigma(U),$

hence ${\tau}$ is a Čech $1$-coboundary. This shows that the action is indeed
free. To verify transitivity, let $J_{1}:(S_{1},\omega_{1})\to M$ and
$J_{2}:(S_{2},\omega_{2})\to M$ be faithful toric
$(\mathcal{T},\Omega)$-spaces with momentum image $\Delta$. By Proposition 2.9