between the groups of connected components of $H$ and $G$,

  * •

a maximally toric $H$-representation $(V,\omega_{V})$.

Associated to this is a Hamiltonian $G$-space:

(106)
$J_{\textrm{mod}}:(S_{\textrm{mod}},\omega_{\textrm{mod}})\to\mathfrak{g}^{*},$

where:

  * •

$(S_{\textrm{mod}},\omega_{\textrm{mod}})$ is the symplectic manifold obtained
from symplectic reduction (at value $0$) of the (right) Hamiltonian $H$-space:

$(G\times\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})\times(V,\omega_{V})\to\mathfrak{h}^{*},\quad(g,\alpha,v)\mapsto\alpha|_{\mathfrak{h}}-J_{V}(v),$

with $J_{V}$ as in (16) and with the $H$-action given by $(g,\alpha,v)\cdot
h=(gh,h^{-1}\cdot\alpha,h^{-1}\cdot v)$,

  * •

the $G$-action on $S_{\textrm{mod}}$ is that inherited from the $G$-action on
$G\times\mathfrak{g}^{*}\times V$ by left translation on the $G$-component,

  * •

the momentum map $J_{\textrm{mod}}$ is the map induced by the projection map
$\textrm{pr}_{\mathfrak{g}^{*}}:G\times\mathfrak{g}^{*}\times
V\to\mathfrak{g}^{*}$.

It follows from a straighforward verification that:

######  Proposition 4.24.

Given the data above, consider the regular and proper symplectic groupoid
$(\mathcal{G},\Omega):=(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$
(the cotangent groupoid, as in Example 3). The Hamiltonian
$(\mathcal{G},\Omega)$-space corresponding to the Hamiltonian $G$-space (106)
is a faithful multiplicity-free $(\mathcal{G},\Omega)$-space.

Up to Morita equivalence, this construction provides a local model for
faithful multiplicity-free actions of general regular proper symplectic
groupoids. In view of this, it will be useful to first single out the
following case of the backward implication in Theorem 4.20.

######  Proposition 4.25.

Suppose that we are given $G$, $H$ and $(V,\omega_{V})$ as above, so that as
for (95) there is an induced map:

(107) $I^{1}(H,T_{H})\to I^{1}(G,T_{G}),$

where $T_{H}$ and $T_{G}$ denote the respective identity components of $H$ and
$G$. Consider the associated faithful multiplicity-free
$(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$-space
${J_{\textrm{mod}}}$ (as in Proposition 4.24). The following hold.

  * a)

The image of ${J_{\textrm{mod}}}$ is
$\pi_{\mathfrak{h}^{*}}^{-1}(\Delta_{V})$, where $\Delta_{V}$ is the image of
the momentum map (16) and
$\pi_{\mathfrak{h}^{*}}:\mathfrak{g}^{*}\to\mathfrak{h}^{*}$ is the canonical
projection.

  * b)

The map (107) sends $e(V,\omega_{V})\in I^{1}(H,T_{H})$ to the ext-invariant
$e({J_{\textrm{mod}}})_{0}\in I^{1}(G,T_{G})$ of ${J_{\textrm{mod}}}$ at the
origin in $\mathfrak{g}^{*}$.

  * c)

The collection of ext-classes of ${J_{\textrm{mod}}}$ is centered (in the
sense of Definition 4.17).

###### Proof.

Part $a$ readily follows from surjectivity of
${\varGamma}_{H}\to{\varGamma}_{G}$. For the remainder, let:

$q:(G\times\mathfrak{g}^{*})\times_{\mathfrak{h}^{*}}V\to{S_{\textrm{mod}}}$

denote the quotient map. Part $b$ follows from the observation that the
symplectic normal representation at $[1,0,0]\in{J_{\textrm{mod}}^{-1}(0)}$ is
canonically isomorphic to $(V,\omega_{V})$, via the $H$-equivariant linear
symplectomorphism:

$(V,\omega_{V})\xrightarrow{\sim}(\mathcal{S}\mathcal{N}_{[1,0,0]},({\omega_{\textrm{mod}}})_{[1,0,0]}),\quad
v\mapsto[\textrm{d}q_{(1,0,0)}(0,0,v)].$

We turn to part $c$. Fix an isomorphism of symplectic $T_{H}$-representations:

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

and let $\mathcal{W}=\\{\alpha_{1},...,\alpha_{n}\\}$ denote the set of
weights. First notice that (by part $b$) the class $e({J_{\textrm{mod}}})_{0}$
is represented by the unique $1$-cocycle ${c}_{0}:G\to T_{G}$ that restricts
to the identity map on $T_{G}$ and that restricts to ${c}_{\psi}$ on $H$.
Next, let $\alpha\in{J_{\textrm{mod}}(S_{\textrm{mod}})}$. Then
$\alpha|_{\mathfrak{h}}\in\Delta_{V}$ by part $a$. From the description (21)
it is clear that:

$\alpha|_{\mathfrak{h}}=\sum_{\alpha_{i}\in\mathcal{W}}t_{\alpha_{i}}\alpha_{i}$

for unique $t_{\alpha_{i}}\in\mathbb{R}_{\geq 0}$. Let
$\mathcal{W}_{\alpha}=\\{\alpha_{i}\in\mathcal{W}\mid t_{\alpha_{i}}=0\\}$ and
let $p\in J_{V}^{-1}(\alpha|_{\mathfrak{h}})$ be the element with component
$\psi(p)_{\alpha_{i}}\in\mathbb{C}_{\alpha_{i}}$ equal to
$\sqrt{t_{\alpha_{i}}}$ for each $\alpha_{i}\in\mathcal{W}$. Consider
$[1,\alpha,p]\in J_{\textrm{mod}}^{-1}(\alpha)$. As is readily verified, the
map:

$(\mathcal{S}\mathcal{N}_{p},\omega_{p})\to(\mathcal{S}\mathcal{N}_{[1,\alpha,p]},({\omega_{\textrm{mod}}})_{[1,\alpha,p]}),\quad[v]\mapsto[\textrm{d}q_{(1,\alpha,p)}(0,0,v)],$

is a symplectic linear isomorphism. Furthermore, it is equivariant with
respect to the action of $H_{[1,\alpha,p]}=H_{\alpha}\cap H_{p}$. As the
identity component of $H_{\alpha}\cap H_{p}$ is the identity component
$T_{H_{p}}$ of $H_{p}$, it follows that:

$e\left(\mathcal{S}\mathcal{N}_{[1,\alpha,p]},({\omega_{\textrm{mod}}})_{[1,\alpha,p]}\right)=[{c}_{\psi_{p}}|_{H_{\alpha}\cap
H_{p}}]\in I^{1}(H_{\alpha}\cap H_{p},T_{H_{p}}).$

Therefore, $e({J_{\textrm{mod}}})_{\alpha}$ is represented by the unique
$1$-cocycle ${c}_{\alpha}:G_{\alpha}\to T_{G}$ that restricts to the identity
map on $T_{G}$ and to ${c}_{\psi_{p}}|_{H_{\alpha}\cap H_{p}}$ on
$H_{\alpha}\cap H_{p}$. To complete the proof, we will now show that:

(108) ${c}_{0}|_{G_{\alpha}}={c}_{\alpha}.$

Using (21) one computes that:

$\psi(\textrm{Ker}(\textrm{d}J_{V})_{p})=\left(\bigoplus_{\alpha_{i}\in\mathcal{W}_{\alpha}}\mathbb{C}_{\alpha_{i}}\right)\oplus\left(\bigoplus_{\alpha_{i}\in\mathcal{W}-\mathcal{W_{\alpha}}}\sqrt{-1}\cdot\mathbb{R}_{\alpha_{i}}\right).$

On the other hand, it is clear that:

$\psi(T_{p}\mathcal{O})=\bigoplus_{\alpha_{i}\in\mathcal{W}-\mathcal{W_{\alpha}}}\sqrt{-1}\cdot\mathbb{R}_{\alpha_{i}}.$

It follows from this that $\psi$ induces a $(T_{H})_{p}$-equivariant
symplectic linear isomorphism:

$\psi_{p}:(\mathcal{S}\mathcal{N}_{p},\omega_{p})\xrightarrow{\sim}\bigoplus_{\alpha_{i}\in\mathcal{W}_{\alpha}}\mathbb{C}_{\alpha_{i}},\quad\psi_{p}([v])_{\alpha_{i}}=\psi(v)_{\alpha_{i}}.$

Hence, the weights of the symplectic normal representation at $p$ are given by
$\alpha_{i}|_{\mathfrak{t}_{p}}\in\mathfrak{t}_{p}^{*}$ for
$\alpha_{i}\in\mathcal{W}_{\alpha}$ and from (27) it follows that the
$1$-cocycle ${c}_{\psi_{p}}:H_{p}\to T_{H_{p}}$ representing
$e(\mathcal{S}\mathcal{N}_{p},\omega_{p})\in I^{1}(H_{p},T_{H_{p}})$ has the
property that for each $h\in H_{p}$,
$v\in\psi^{-1}(\bigoplus_{\alpha_{i}\in\mathcal{W}_{\alpha}}\mathbb{C}_{\alpha_{i}})$
and each $\alpha_{i}\in\mathcal{W}_{\alpha}$:

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

The same property is satisfied by the $1$-cocycle ${c}_{\psi}:H\to T_{H}$
representing $e(V,\omega)\in I^{1}(H,T_{H})$. Therefore, we conclude that:

(109)
$\chi_{\alpha_{i}}\circ{c}_{\psi_{p}}=\chi_{\alpha_{i}}\circ{c}_{\psi}|_{H_{p}},\quad\forall\text{
}\alpha_{i}\in\mathcal{W}_{\alpha}.$

On the other hand, it holds that:

(110)
$\chi_{\alpha_{i}}\circ{c}_{\psi_{p}}=1=\chi_{\alpha_{i}}\circ{c}_{\psi}|_{H_{p}},\quad\forall\text{
}\alpha_{i}\in\mathcal{W}-\mathcal{W}_{\alpha}.$

Indeed, the first equality in (110) follows from the observation that, because
${c}_{\psi_{p}}$ takes values in $(T_{H})_{p}$, for all $h\in H_{p}$ and
$\alpha_{i}\in\mathcal{W}$ we have:

$\chi_{\alpha_{i}}({c}_{\psi_{p}}(h))\cdot\psi(p)_{\alpha_{i}}=({c}_{\psi_{p}}(h)\cdot\psi(p))_{\alpha_{i}}=\psi(p)_{\alpha_{i}}.$

For the latter equality, notice first that, since each $h\in H_{p}$ fixes
$\alpha$ (as $J_{V}$ is $H$-equivariant), it holds that
$t_{[h]\cdot\alpha_{i}}=t_{\alpha_{i}}$, and so
$\psi(p)_{[h]\cdot\alpha_{i}}=\psi(p)_{\alpha_{i}}$ for each
$\alpha_{i}\in\mathcal{W}$. Using this and (27), we find that for such $h$ and
$\alpha_{i}$:

$\chi_{\alpha_{i}}(c_{\psi}(h))\cdot\psi(p)_{\alpha_{i}}=\psi(h\cdot
p)_{[h]\cdot\alpha_{i}}=\psi(p)_{[h]\cdot\alpha_{i}}=\psi(p)_{\alpha_{i}},$

from which we conclude that the second equality in (110) holds. Together,
(109) and (110) imply that:

(111) ${c}_{\psi_{p}}={c}_{\psi}|_{H_{p}}.$

We conclude from this that (108) indeed holds, since
${c}_{0}|_{G_{\alpha}}:G_{\alpha}\to T_{G}$ is a $1$-cocycle that restricts to
the identity on $T_{G}$, and that restricts to
${c}_{\psi_{p}}|_{H_{\alpha}\cap H_{p}}$ on $H_{\alpha}\cap H_{p}$ by (111). ∎

We are now ready to complete the proof.

###### End of the proof of Theorem 4.20 and Remark 4.22.

First, in the setting of Theorem 4.20, suppose that there is an invariant open
$U$ in $M$ around $\mathcal{O}\in\underline{\Delta}$ and a faithful
multiplicity-free $(\mathcal{G},\Omega)$-space $J:(S,\omega)\to M$ such that
$J(S)=U\cap\Delta$ and the collection of its ext-invariants is
$\sigma|_{\underline{U}\cap\underline{\Delta}}$. We will not only show that
$\sigma$ is flat at $\mathcal{O}$, but that for every symplectic Morita
equivalence as after Lemma 4.16, with $W\subset U$, the invariant local
section of (103) corresponding to
$\sigma|_{\underline{W}\cap\underline{\Delta}}$ via (101) is centered on some
neighbourhood of the origin. Suppose that we are given any such symplectic
Morita equivalence:

$(\mathcal{G},\Omega)|_{W}$$W$$(P,\omega_{P})$$W_{\mathfrak{g}^{*}}$$(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})|_{W_{\mathfrak{g}^{*}}}$$\text{
}\text{ }\alpha_{1}$$\alpha_{2}$

After choosing a $p\in\alpha_{1}^{-1}(\mathcal{O})$ and composing the Morita
equivalence with the automorphism of
$(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$ associated to
the isomorphism of isotropy groups $\varphi_{p}:G\to G$ induced, as in (37),
by the Morita equivalence, we can assume without loss of generality that there
is a $p\in\alpha_{1}^{-1}(\mathcal{O})$ such that $G$ is the isotropy group of
$\mathcal{G}$ at $x:=\alpha_{1}(p)$ and such that $\varphi_{p}$ is the
identity map of $G$. Then the set-germ at the origin of the invariant subset
of $W_{\mathfrak{g}^{*}}$ corresponding to $W\cap\Delta$ is that of the
$G$-invariant smooth polyhedral cone
$C_{\mathfrak{g}^{*}}:=((\rho_{\Omega})_{x}^{*})^{-1}(C_{x}(\Delta))$. This
follows by combining Lemma 1.38, Proposition 1.39 and the observation that
$(\rho_{-\textrm{d}\lambda_{\textrm{can}}})_{0}^{*}:\mathfrak{g}^{*}\to
T_{0}\mathfrak{g}^{*}$ (which is the canonical isomorphism of the vector space
$\mathfrak{g}^{*}$ with its tangent space at $0$) represents the map germ
$\log_{0}$. Note here that the polyhedral cone $C_{\mathfrak{g}^{*}}$ is
indeed $G$-invariant, as follows from Corollary 1.41 and $G$-equivariance of
$(\rho_{\Omega})_{x}^{*}:\mathfrak{g}^{*}\to\mathcal{N}_{x}{\mathcal{O}}$ with
respect to the coadjoint action of $G$ on $\mathfrak{g}^{*}$ (which follows
from Lemma 1.38, applied to the identity equivalence). Let $H$ and
$(V,\omega_{V})$ be the isotropy group and the symplectic normal
representation of the $(\mathcal{G},\Omega)$-space $J$ at some point in
$J^{-1}(x)$, and consider the faithful multiplicity-free
$(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$-space
$J_{\textrm{mod}}$ associated to this data, as in Proposition 4.24 (which
applies because of Proposition 1.43). In view of the above, (98), Proposition
1.43$c$ and Proposition 4.25, the
$(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$-spaces
$J_{\textrm{mod}}$ and $P_{*}(J)$ satisfy the criteria in Proposition 4.10 at
the origin in $\mathfrak{g}^{*}$. Hence, there is an invariant open
neighbourhood of the origin in $\mathfrak{g}^{*}$ such that the restrictions
of the $(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$-spaces
$J_{\textrm{mod}}$ and $P_{*}(J)$ to this open neighbourhood are isomorphic.
Since the ext-invariant of $J_{\textrm{mod}}$ is centered (Proposition 4.25),
the same holds for the restriction of $P_{*}(J)$ to this open neighbourhood.
Because of (98), the collection of ext-invariants of $P_{*}(J)$ is the
invariant local section of (103) corresponding to
$\sigma|_{\underline{W}\cap\underline{\Delta}}$ via (101), which is therefore
centered on a neighbourhood of the origin, as we wanted to show.

Having shown this, to complete the proof of both Theorem 4.20 and Remark 4.22,
it remains to prove the forward implication in Theorem 4.20. To this end,
suppose that $\sigma$ is flat at $\mathcal{O}$. By flatness and the same
reasoning as above, there is a symplectic Morita equivalence
$((P,\omega_{P}),\alpha_{1},\alpha_{2})$ with a
$p\in\alpha_{1}^{-1}(\mathcal{O})$ as above, with the additional property that
the invariant local section corresponding to
$\sigma|_{\underline{W}\cap\underline{\Delta}}$ via (101) is centered. Since
the set-germ at the origin of the invariant subset of $W_{\mathfrak{g}^{*}}$
corresponding to $W\cap\Delta$ is that of the $G$-invariant smooth polyhedral
cone $C_{\mathfrak{g}^{*}}$ (defined as above), after possibly shrinking $W$
and $W_{\mathfrak{g}^{*}}$ we can further arrange that $W\cap\Delta$ is
related to the invariant subset $W_{\mathfrak{g}^{*}}\cap
C_{\mathfrak{g}^{*}}$. To construct a faithful multiplicity-free
$(\mathcal{G},\Omega)$-space with momentum image $W\cap\Delta$ and ext-
invariant $\sigma|_{\underline{W}\cap\underline{\Delta}}$, we will construct:

  * •

a closed subgroup $H$ of $G$ with the property that
${\varGamma}_{H}\to{\varGamma}_{G}$ is bijective,

  * •

a maximally toric $H$-representation $(V,\omega_{V})$ such that
$\pi_{\mathfrak{h}^{*}}^{-1}(\Delta_{V})=C_{\mathfrak{g}^{*}}$ and such that
$e(V,\omega_{V})\in I^{1}(H,T_{H})$ is mapped to $\sigma(x)\in I^{1}(G,T_{G})$
by (107) (for $x:=\alpha_{1}(p)$ as above).

The restriction to $W_{\mathfrak{g}^{*}}$ of the faithful multiplicity-free
space $(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$-space
$J_{\textrm{mod}}$ associated to this data (as in Proposition 4.24) would have
momentum map image $W_{\mathfrak{g}^{*}}\cap C_{\mathfrak{g}^{*}}$ and
collection of ext-invariants equal to the invariant local section
corresponding to $\sigma|_{\underline{W}\cap\underline{\Delta}}$ via (101)
(because both are centered and they have the same value at the origin).
Therefore, the $(\mathcal{G},\Omega)|_{W}$-space associated to this by the
symplectic Morita equivalence $((P,\omega_{P}),\alpha_{1},\alpha_{2})$ would
have momentum image $W\cap\Delta$ and ext-invariant
$\sigma|_{\underline{W}\cap\underline{\Delta}}$.

So, to complete the proof it remains to construct $H$ and $(V,\omega_{V})$
satisfying the requirements listed above. For this, let ${c}:G\to T_{G}$ be a
$1$-cocycle such that $[{c}]=\sigma(x)\in I^{1}(G,T_{G})$. Further, let us
(suggestively) denote by $\mathfrak{h}^{0}$ the largest linear subspace of
$\mathfrak{g}^{*}$ that is contained in $C_{\mathfrak{g}^{*}}$ and let
$\mathfrak{h}\subset\mathfrak{g}$ be the annihilator of $\mathfrak{h}^{0}$.
Since the polyhedral cone $C_{\mathfrak{g}^{*}}$ is smooth in
$(\mathfrak{g}^{*},\Lambda_{T_{G}}^{*})$, the lattice
$\mathfrak{h}\cap\Lambda_{T_{G}}$ has full rank in $\mathfrak{h}$ and so
$T_{H}:=\exp_{G}(\mathfrak{h})$ is a subtorus of $T_{G}$ with Lie algebra
$\mathfrak{h}$. Since $C_{\mathfrak{g}^{*}}$ is invariant under the coadjoint
action of $G$, $T_{H}$ is invariant under conjugation by elements of $G$. From
this it readily follows that ${\varGamma}_{G}\times T_{H}$ is a subgroup of
${\varGamma}_{G}\ltimes T_{G}$. Now, let $H$ be the subgroup of $G$
corresponding to ${\varGamma}_{G}\times T_{H}$ under the isomorphism of Lie
groups:

$G\xrightarrow{\sim}{\varGamma}_{G}\ltimes T_{G},\quad g\mapsto([g],{c}(g)).$

Then $H$ is a closed Lie subgroup of $G$ with Lie algebra $\mathfrak{h}$ and
${\varGamma}_{H}\to{\varGamma}_{G}$ is bijective. Further notice that the
$1$-cocycle ${c}$ restricts to a $1$-cocycle ${c}|_{H}:H\to T_{H}$, that
restricts to the identity map on $T_{H}$. Next, we construct the desired
representation of $H$. Let us (suggestively) denote by $\Delta_{V}$ the image
of $C_{\mathfrak{g}^{*}}$ under the projection
$\pi_{\mathfrak{h}^{*}}:\mathfrak{g}^{*}\to\mathfrak{h}^{*}$. By construction,
$\Delta_{V}$ is a smooth and pointed polyhedral cone in the integral affine
vector space $(\mathfrak{h}^{*},\Lambda_{T_{H}}^{*})$. Furthermore,
$\Delta_{V}$ is ${\varGamma}_{H}$-invariant, because $\pi_{\mathfrak{h}^{*}}$
is $H$-equivariant and $C_{\mathfrak{g}^{*}}$ is ${\varGamma}_{G}$-invariant.
Hence, by Theorem 1.3 there is a maximally toric $H$-representation
$(V,\omega_{V})$ with momentum image $\Delta_{V}$ and
$e(V,\omega_{V})=[{c}|_{H}]\in I^{1}(H,T_{H})$. As is clear from their
construction, $H$ and $(V,\omega_{V})$ satisfy the requirements. ∎

###  4.2. The third structure theorem

####  4.2.1. Proof of the third structure theorem

We now turn to the proof of Theorem 6, starting with:

######  Proposition 4.26.

The injection (12) is compatible with the actions in the first and second
structure theorems.

###### Proof.

After restricting to a complete transversal, the proof reduces to the case in
which $\mathcal{B}$ is etale. Suppose we are in that case. Let
$\underline{\textrm{c}}\in H^{1}(\underline{\Delta},\underline{\mathcal{L}})$
and let $J:(S,\omega)\to M$ be a faithful multiplicity-free
$(\mathcal{G},\Omega)$-space with momentum image $\Delta$. Via the natural
isomorphism from Čech to sheaf cohomology we can represent
$\underline{\textrm{c}}$ by a Čech $1$-cocycle
$\underline{\tau}\in\check{C}^{1}_{q(\text{$\mathcal{U}$})}(\underline{\Delta},\underline{\mathcal{L}})$
with respect to an open cover of the form
$q(\text{$\mathcal{U}$}):=\\{q(U)\mid U\in\text{$\mathcal{U}$}\\}$, where
$q_{\Delta}:\Delta\to\underline{\Delta}$ is the orbit projection and
$\mathcal{U}$ is some basis of $\Delta$ for which
$\mathcal{L}:=\mathcal{L}_{\Delta}$ is $\mathcal{U}$-acyclic. The composition:

$\check{H}^{1}_{q(\text{$\mathcal{U}$})}(\underline{\Delta},\underline{\mathcal{L}})\to\check{H}^{1}(\underline{\Delta},\underline{\mathcal{L}})\xrightarrow{\sim}H^{1}(\underline{\Delta},\underline{\mathcal{L}})\to
H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})\to\check{H}^{1}_{\text{$\mathcal{U}$}}(\mathcal{B}|_{\Delta},\mathcal{L})$

is induced by the map of complexes given in degree $n$ by:

$(q_{\Delta})^{*}:\check{C}^{n}_{q(\text{$\mathcal{U}$})}(\underline{\Delta},\underline{\mathcal{L}})\to\check{C}^{n}_{\text{$\mathcal{U}$}}(\mathcal{B}|_{\Delta},\mathcal{L}),\quad(q_{\Delta})^{*}(c)(U_{0}\xleftarrow{\sigma_{1}}...\xleftarrow{\sigma_{n}}U_{n}):=c(q(U_{1}),...,q(U_{n}))|_{U_{n}}\in\mathcal{L}(U_{n}).$

So, denoting
$\tau:=(q_{\Delta})^{*}(\underline{\tau})\in\check{C}^{1}_{\text{$\mathcal{U}$}}(\mathcal{B}|_{\Delta},\mathcal{L})$,
to prove the proposition we ought to give an isomorphism of faithful
multiplicity-free $(\mathcal{G},\Omega)$-spaces:

${(S_{\tau},\omega_{\tau})}$${(S_{\underline{\tau}},\omega_{\underline{\tau}})}$${M}$$\scriptstyle{\psi}$$\scriptstyle{J_{\tau}}$$\scriptstyle{J_{\underline{\tau}}}$

where the left-hand $(\mathcal{G},\Omega)$-space is that defined in the proof
of Theorem 3, whereas the right-hand one is that defined as in the proof of
Theorem 2.8, but with $\mathcal{L}$ replaced by $\underline{\mathcal{L}}$.
Consider the map $\psi:S_{\tau}\to S_{\underline{\tau}}$ given by
$\psi([p,U])=[p,\widehat{U}]$, where $\widehat{U}$ denotes the
$\mathcal{G}$-saturation of $U$. A straightforward verification shows that
this is well-defined, bijective and intertwines the maps $J_{\tau}$ and
$J_{\underline{\tau}}$. Further notice that, for each
$U\in\text{$\mathcal{U}$}$, the pre-composition of $\psi$ with the inclusion
(72) coincides with the composition:

$(J^{-1}(U),\omega)\hookrightarrow(J^{-1}(\widehat{U}),\omega)\hookrightarrow(S_{\underline{\tau}},\omega_{\underline{\tau}}),$

which is a symplectomorphism onto an open in
$(S_{\underline{\tau}},\omega_{\underline{\tau}})$. Therefore, $\psi$ is a
symplectomorphism. Finally, the fact that ${\underline{\tau}}(q(U),q(V))$ is
$\mathcal{B}$-invariant for all $U,V\in\text{$\mathcal{U}$}$ implies that
$\psi$ is $\mathcal{G}$-equivariant. So, $\psi$ is an isomorphism of faithful
multiplicity-free $(\mathcal{G},\Omega)$-spaces. ∎

###### Proof of Theorem 6.

The second structure theorem and Proposition 4.26 imply that the action in the
first structure theorem descends to an action of the quotient group (14) on
the image of (13), uniquely determined by the fact that (13) becomes
equivariant with respect to the
$\check{H}^{1}(\mathcal{B}|_{\Delta},\mathcal{L})$-action on its image
corresponding to the quotient group action. It follows by combining the first
and second structure theorem with Proposition 4.26 that this action is free
and transitive. ∎

####  4.2.2. On the action in the third structure theorem

In this subsection we give a more direct description of the action in the
third structure theorem and show that the ext-sheaf comes with a free and
transitive action of a more familiar sheaf on $\underline{\Delta}$: the first
right-derived functor $\mathcal{H}^{1}:=R^{1}(q_{\Delta})_{*}(\mathcal{L})$ of
the push-forward $(q_{\Delta})_{*}$ (as in Remark 4.3) applied to
$\mathcal{L}:=\mathcal{L}_{\Delta}$.

Let $(\mathcal{G},\Omega)\rightrightarrows M$ be a regular and proper
symplectic groupoid with associated orbifold groupoid
$\mathcal{B}:=\mathcal{G}/\mathcal{T}$ and let
$\underline{\Delta}\subset\underline{M}$ be a Delzant subspace. There is a
Grothendieck spectral sequence [17] associated to the commutative triangle of
functors:

${\textsf{Sh}(\mathcal{B}|_{\Delta})}$Ab${\textsf{Sh}(\underline{\Delta})}$$\scriptstyle{\Gamma(\cdot)^{\mathcal{B}}}$$\scriptstyle{(q_{\Delta})_{*}}$$\scriptstyle{\Gamma(\cdot)}$

which is a version of the usual Leray spectral sequence for the canonical map
of topological groupoids $q_{\Delta}$ from $\mathcal{B}|_{\Delta}$ to the unit
groupoid of $\underline{\Delta}$. The associated exact sequence in low degrees
(15) has as first few terms:

(112) $0\to H^{1}(\underline{\Delta},\underline{\mathcal{L}})\to
H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})\to\mathcal{H}^{1}(\underline{\Delta}).$

Concretely, $\mathcal{H}^{1}$ is the sheafification of the pre-sheaf on
$\underline{\Delta}$ that assigns to an open $\underline{U}$ the abelian group
$H^{1}(\mathcal{B}|_{U},\mathcal{L})$ (with the restriction maps induced by
pull-back along the inclusion
$\mathcal{B}|_{V}\hookrightarrow\mathcal{B}|_{U}$ for opens
$\underline{V}\subset\underline{U}$) and the map:

(113)
$H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})\to\mathcal{H}^{1}(\underline{\Delta})$

in the above sequence takes a global section of this pre-sheaf to the induced
global section of its sheafification. To define the $\mathcal{H}^{1}$-action
on the ext-sheaf
$\mathcal{I}^{1}:=\mathcal{I}^{1}_{((\mathcal{G},\Omega),\underline{\Delta})}$,
let $x\in\Delta$. Given an open $\underline{U}$ in $\underline{\Delta}$ such
that $x\in U$, we consider the map:

(114)
$\Phi_{x}:=(\textrm{ev}_{x})_{*}\circ(i_{x})^{*}:H^{\bullet}(\mathcal{B}|_{U},\mathcal{L})\xrightarrow{(i_{x})^{*}}H^{\bullet}(\mathcal{B}_{x},\mathcal{L}_{x})\xrightarrow{(\textrm{ev}_{x})_{*}}H^{\bullet}(\mathcal{B}_{x},\mathcal{T}_{x}),$

where $i_{x}:\mathcal{B}_{x}\hookrightarrow\mathcal{B}$ is the inclusion of
the isotropy group $\mathcal{B}_{x}$ at $x$ (viewed as groupoid over
$\\{x\\}$) and $\textrm{ev}_{x}:\mathcal{L}_{x}\to\mathcal{T}_{x}$ is the
evaluation map from the stalk of $\mathcal{L}$ at $x$ into the fiber of
$\mathcal{T}$ over $x$. The last two groups denote group cohomology. Combining
the map $\Phi_{x}$ (in degree $1$) and the action in Remark 1.5, we obtain an
$H^{1}(\mathcal{B}|_{U},\mathcal{L})$-action on the set
$I^{1}(\mathcal{G}_{x},\mathcal{T}_{x})$ —the stalk of $\mathcal{I}^{1}$ at
the orbit $\mathcal{O}_{x}$ of $\mathcal{G}$ through $x$ (see Remark 4.19).
For varying $\underline{U}$ the actions on this set are compatible with
restriction of opens and so descend to an action of the stalk of
$\mathcal{H}^{1}$ at $\mathcal{O}_{x}$.

######  Theorem 4.27.

The stalk-wise actions given above define an action of the sheaf of abelian
groups $\mathcal{H}^{1}$ on the ext-sheaf $\mathcal{I}^{1}$ that is free and
transitive (meaning that for each open the associated group action is free and
transitive). Moreover, via the map (113), the induced
$\mathcal{H}^{1}(\underline{\Delta})$-action on
$\mathcal{I}^{1}(\underline{\Delta})$ restricts to the action in Theorem 6 on
the image of (13).

To prove this we will use the lemma below.

######  Lemma 4.28.

Let $G$ be an infinitesimally abelian compact Lie group and consider the
regular and proper symplectic groupoid
$(\mathcal{G},\Omega):=(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$
over $M:=\mathfrak{g}^{*}$, with associated orbifold groupoid
$\mathcal{B}:=\mathcal{G}/\mathcal{T}=\Gamma\ltimes\mathfrak{g}^{*}$, where
$\Gamma:=G/T$ with $T$ the identity component of $G$. Suppose that
$\underline{\Delta}\subset\underline{M}=\mathfrak{g}^{*}/G$ is a Delzant
subspace with the property that the corresponding invariant subspace $\Delta$
in $\mathfrak{g}^{*}$ is convex. Let $x\in\Delta^{\Gamma}$ be a $\Gamma$-fixed
point in $\Delta$.

  * a)

The map as in (114):

$\Phi_{x}:H^{n}(\Gamma\ltimes\Delta,\mathcal{L})\to H^{n}(\Gamma,T)$

is an isomorphism in all degrees $n>0$.

  * b)

For any $y\in\Delta$, the maps $\Phi_{x}$ and $\Phi_{y}$ are related via
restriction, meaning that these fit into a commutative diagram:

${H^{n}(\Gamma\ltimes\Delta,\mathcal{L})}$${H^{n}(\Gamma,T)}$${H^{n}(\Gamma_{y},T)}$$\scriptstyle{\Phi_{x}}$$\scriptstyle{\Phi_{y}}$$\scriptstyle{(i_{\Gamma_{y}})^{*}}$

for each $n>0$, where $i_{\Gamma_{y}}:\Gamma_{y}\hookrightarrow{\Gamma}$
denotes the inclusion.

###### Proof.

Let $\mathcal{U}$ be the basis of $\Delta$ consisting of all convex opens. By
assumption, $\Delta\in\text{$\mathcal{U}$}$. Moreover, by Lemma 3.5,
$\mathcal{L}$ is $\mathcal{U}$-acyclic. To prove part $a$, we will show that
the composition:

(115)
$\check{H}^{n}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\mathcal{L})\xrightarrow{(\ref{eqn:cechtoderivedmorph:equivshf})}H^{n}(\Gamma\ltimes\Delta,\mathcal{L})\xrightarrow{\Phi_{x}}H^{n}(\Gamma,T)$

is an isomorphism. Consider the $\Gamma$-module $\mathcal{L}(\Delta)$ of
global section of $\mathcal{L}$, with action given by:

$(\tau\cdot\gamma)(y)=\tau(\gamma\cdot
y)\cdot(\gamma,y)\in\mathcal{T},\quad\gamma\in\Gamma,\quad y\in\Delta.$

The map (115) is equal to the composition:

$\check{H}^{n}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\mathcal{L})\xrightarrow{(i_{\Delta})^{*}}H^{n}(\Gamma,\mathcal{L}(\Delta))\xrightarrow{(\textrm{ev}_{x})_{*}}H^{n}(\Gamma,T),$

where $\textrm{ev}_{x}:\mathcal{L}(\Delta)\to T$ is the map of
$\Gamma$-modules given by evaluation at $x$ and $i_{\Delta}$ is the functor:

$\Gamma\to\textsf{Emb}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta),\quad\gamma\mapsto(\Delta\xleftarrow{\sigma_{\gamma}}\Delta),$

where we view $\Gamma$ as category with a single object and $\sigma_{\gamma}$
denotes the bisection with $\Gamma$-component constantly equal to $\gamma$.
Consider the short exact sequence of $(\Gamma\ltimes\Delta)$-sheaves:

(116)
$0\to\mathcal{C}^{\infty}(\Lambda)\to\Omega^{1}_{\textrm{cl}}\xrightarrow{(\ref{exptorbun})}\mathcal{L}\to
0,$

where $\mathcal{C}^{\infty}(\Lambda)$ is the sheaf of smooth local sections of
$\Lambda|_{\Delta}$, with $\Lambda$ the lattice bundle associated to
$(\mathcal{G},\Omega)$, and $\Omega^{1}_{\textrm{cl}}$ is the sheaf of closed
$1$-forms on the manifold with corners $\Delta$, each equipped with the
$(\Gamma\ltimes\Delta)$-action defined like that on $\mathcal{L}$ (see Example
3.1). Since $\mathcal{U}$ consists of convex opens, any section of
$\mathcal{L}$ over an open $U\in\text{$\mathcal{U}$}$ lifts to a closed
$1$-form on $U$ along the universal covering map
$T^{*}\mathfrak{g}^{*}\to\mathcal{T}$. It follows from this that global
sections of (116) form an exact sequence of $\Gamma$-modules and the sequence
of pre-sheaves on $\textsf{Emb}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta)$
associated to (116) is exact too. Moreover, we have a short exact sequence of
$\Gamma$-modules:

$0\to\Lambda_{T}\to\mathfrak{g}\to T\to 0.$

The associated long exact sequences in cohomology give rise to a commutative
diagram:

${\check{H}^{n}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\Omega^{1}_{\textrm{cl}})}$${\check{H}^{n}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\mathcal{L})}$${\check{H}^{n+1}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\mathcal{C}^{\infty}(\Lambda))}$${\check{H}^{n+1}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\Omega^{1}_{\textrm{cl}})}$${H^{n}(\Gamma,\Omega^{1}_{\textrm{cl}}(\Delta))}$${H^{n}(\Gamma,\mathcal{L}(\Delta))}$${H^{n+1}(\Gamma,\mathcal{C}^{\infty}(\Lambda)(\Delta))}$${H^{n+1}(\Gamma,\Omega^{1}_{\textrm{cl}}(\Delta))}$${H^{n}(\Gamma,\mathfrak{g})}$${H^{n}(\Gamma,T)}$${H^{n+1}(\Gamma,\Lambda_{T})}$${H^{n+1}(\Gamma,\mathfrak{g})}$$\scriptstyle{(i_{\Delta})^{*}}$$\scriptstyle{(i_{\Delta})^{*}}$$\scriptstyle{(i_{\Delta})^{*}}$$\scriptstyle{(i_{\Delta})^{*}}$$\scriptstyle{(\textrm{ev}_{x})_{*}}$$\scriptstyle{(\textrm{ev}_{x})_{*}}$$\scriptstyle{(\textrm{ev}_{x})_{*}}$$\scriptstyle{(\textrm{ev}_{x})_{*}}$

with exact rows. The outer left and right groups in the middle and bottom row
vanish, because the group cohomology of an $\mathbb{R}$-linear representation
of a finite group vanishes in all degrees $n>0$. The outer left and right
groups in the top row vanish as well. One way to see this is to note that an
explicit null-homotopy in strictly positive degree is given by:

$h:\check{C}^{\bullet+1}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\Omega^{1}_{\textrm{cl}})\to\check{C}^{\bullet}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\Omega^{1}_{\textrm{cl}}),\quad
h(c)(U_{0}\xleftarrow{\sigma_{1}}\dots\xleftarrow{\sigma_{n}}U_{n})=\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}c(\Delta\xleftarrow{\sigma_{\gamma}}U_{0}\xleftarrow{\sigma_{1}}\dots\xleftarrow{\sigma_{n}}U_{n}).$

In view of this, it is enough to show that each of the maps in the
composition:

$\check{H}^{n+1}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\mathcal{C}^{\infty}(\Lambda))\xrightarrow{(i_{\Delta})^{*}}H^{n+1}(\Gamma,\mathcal{C}^{\infty}(\Lambda)(\Delta))\xrightarrow{(\textrm{ev}_{x})_{*}}H^{n+1}(\Gamma,\Lambda_{T})$

is an isomorphism. For this, note that
$\Lambda=\Lambda_{T}\times\mathfrak{g}^{*}$ (viewed as subset of
$\mathfrak{g}\times\mathfrak{g}^{*}=T^{*}\mathfrak{g}^{*}$). In view of this,
the map of $\Gamma$-modules
$\textrm{ev}_{x}:\mathcal{C}^{\infty}(\Lambda)(\Delta)\to\Lambda_{T}$ is an
isomorphism (since $\Delta$ is connected), hence so is the induced map of
cohomology groups $(\textrm{ev}_{x})_{*}$. Moreover, the pre-sheaf on
$\textsf{Emb}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta)$ induced by
$\mathcal{C}^{\infty}(\Lambda)$ is the pull-back of the $\Gamma$-module
$\mathcal{C}^{\infty}(\Lambda)(\Delta)$ along the functor
$r_{\Delta}:\textsf{Emb}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta)\to\Gamma$
that sends an arrow $V\xleftarrow{\sigma}U$ to the value of the
$\Gamma$-component of $\sigma$ (which is constant as $U$ is connected). So,
there is the morphism:

$(r_{\Delta})^{*}:H^{\bullet}(\Gamma,\mathcal{C}^{\infty}(\Lambda)(\Delta))\to\check{H}_{\text{$\mathcal{U}$}}^{\bullet}(\Gamma\ltimes\Delta,\mathcal{C}^{\infty}(\Lambda)).$

This is inverse to $(i_{\Delta})^{*}$. Indeed, on the level of complexes
$(i_{\Delta})^{*}\circ(r_{\Delta})^{*}$ is the identity map (since
$r_{\Delta}\circ i_{\Delta}=\textrm{id}_{\Gamma}$) and
$(r_{\Delta})^{*}\circ(i_{\Delta})^{*}$ is homotopic to the identity map via
the homotopy $H$ given by:

$H(c)(U_{0}\xleftarrow{\sigma_{1}}\dots\xleftarrow{\sigma_{n}}U_{n})=\sum_{i=0}^{n}(-1)^{i}c(\Delta\xleftarrow{\sigma_{\gamma_{1}}}\dots\xleftarrow{\sigma_{\gamma_{i}}}\Delta\hookleftarrow
U_{i}\xleftarrow{\sigma_{i+1}}\dots\xleftarrow{\sigma_{n}}U_{n}),$

where $\Delta\hookleftarrow U_{i}$ is the arrow with underlying bisection the
unit map and $\gamma_{i}$ is the constant value of the $\Gamma$-component of
$\sigma_{i}$ (so that $\sigma_{i}=\sigma_{\gamma_{i}}|_{U_{i}}$). So, both
$(i_{\Delta})^{*}$ and $(\textrm{ev}_{x})_{*}$ are isomorphisms. This proves
part $a$. For part $b$, note that the composition:

$\check{H}^{n}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\mathcal{L})\xrightarrow{(\ref{eqn:cechtoderivedmorph:equivshf})}H^{n}(\Gamma\ltimes\Delta,\mathcal{L})\xrightarrow{\Phi_{y}}H^{n}(\Gamma_{y},T)$

is equal to the composition:

$\check{H}^{n}_{\text{$\mathcal{U}$}}(\Gamma\ltimes\Delta,\mathcal{L})\xrightarrow{(i_{\Delta})^{*}}H^{n}(\Gamma,\mathcal{L}(\Delta))\xrightarrow{(i_{\Gamma_{y}})^{*}}H^{n}(\Gamma_{y},\mathcal{L}(\Delta))\xrightarrow{(\textrm{ev}_{y})_{*}}H^{n}(\Gamma_{y},T),$

where $\textrm{ev}_{y}:\mathcal{L}(\Delta)\to T$ is the map of
$\Gamma_{y}$-modules given by evaluation at $y$ and $i_{\Delta}$ is as before.
So, it suffices to show that:

${H^{n}(\Gamma,\mathcal{L}(\Delta))}$${H^{n}(\Gamma,T)}$${H^{n}(\Gamma_{y},\mathcal{L}(\Delta))}$${H^{n}(\Gamma_{y},T)}$$\scriptstyle{(\textrm{ev}_{x})_{*}}$$\scriptstyle{(i_{\Gamma_{y}})^{*}}$$\scriptstyle{(i_{\Gamma_{y}})^{*}}$$\scriptstyle{(\textrm{ev}_{y})_{*}}$

commutes. Consider the map of $\Gamma$-modules
$\tau_{\Delta}:T\to\mathcal{L}(\Delta)$ that sends $t\in T$ to the section of
$\mathcal{T}=T\times\mathfrak{g}^{*}$ with $T$-component constantly equal to
$t$. Since $\textrm{ev}_{x}\circ\tau_{\Delta}=\textrm{id}_{T}$, the induced
map:

$(\tau_{\Delta})_{*}:H^{n}(\Gamma,T)\to H^{n}(\Gamma,\mathcal{L}(\Delta))$

is the inverse of the isomorphism $(\textrm{ev}_{x})_{*}$. Therefore, it is
enough to check the commutativity of:

${H^{n}(\Gamma,\mathcal{L}(\Delta))}$${H^{n}(\Gamma,T)}$${H^{n}(\Gamma_{y},\mathcal{L}(\Delta))}$${H^{n}(\Gamma_{y},T)}$$\scriptstyle{(i_{\Gamma_{y}})^{*}}$$\scriptstyle{(i_{\Gamma_{y}})^{*}}$$\scriptstyle{(\tau_{\Delta})_{*}}$$\scriptstyle{(\textrm{ev}_{y})_{*}}$

and this is readily verified on the level of cocycles, using the fact that
$\textrm{ev}_{y}\circ\tau_{\Delta}=\textrm{id}_{T}$. ∎

We will also use the lemma below, which we leave for the reader to verify.

######  Lemma 4.29.

Suppose we are given a symplectic Morita equivalence between regular and
proper symplectic groupoids $(\mathcal{G}_{1},\Omega_{1})\rightrightarrows
M_{1}$ and $(\mathcal{G}_{2},\Omega_{2})\rightrightarrows M_{2}$ that relates
a Delzant subspace $\underline{\Delta}_{1}\subset\underline{M}_{1}$ to a
Delzant subspace $\underline{\Delta}_{2}\subset\underline{M}_{2}$. For any
$p\in P$, the isomorphism in cohomology induced (as in Corollary 3.3) by the
associated integral affine Morita equivalence (Example 1.37) fits into a
commutative square:

${H^{1}(\mathcal{B}|_{\Delta_{1}},\mathcal{L}_{1})}$${H^{1}(\mathcal{B}_{x_{1}},\mathcal{T}_{x_{1}})}$${H^{1}(\mathcal{B}_{2}|_{\Delta_{2}},\mathcal{L}_{2})}$${H^{1}(\mathcal{B}_{x_{2}},\mathcal{T}_{x_{2}})}$$\scriptstyle{\sim}$$\scriptstyle{\Phi_{x_{1}}}$$\scriptstyle{(\varphi_{p})_{*}}$$\scriptstyle{\sim}$$\scriptstyle{\Phi_{x_{2}}}$

where $x_{1}=\alpha_{1}(p)$, $x_{2}=\alpha_{2}(p)$ and
$\varphi_{p}:\mathcal{G}_{x_{1}}\xrightarrow{\sim}\mathcal{G}_{x_{2}}$ is the
isomorphism of isotropy groups (37).

###### Proof of Theorem 4.27.

Given an open $\underline{U}$ in $\underline{\Delta}$, the
$H^{1}(\mathcal{B}|_{U},\mathcal{L})$-actions on the sets
$I^{1}(\mathcal{G}_{x},\mathcal{T}_{x})$ for $x\in U$ (described before the
statement of Theorem 4.27) induce an action on the set
$\mathcal{I}^{1}_{\textrm{Set}}(\underline{U})$ (denoting
$\mathcal{I}^{1}_{\textrm{Set}}:=\mathcal{I}^{1}_{\textrm{Set},(\mathcal{G},\Omega,\underline{\Delta})}$
as in Definition 4.14) given by:

$(\textrm{c}\cdot\sigma)(x):=\Phi_{x}(\textrm{c})\cdot\sigma(x),\quad\textrm{c}\in
H^{1}(\mathcal{B}|_{U},\mathcal{L}),\quad\sigma\in\mathcal{I}^{1}_{\textrm{Set}}(\underline{U}),\quad
x\in U,$

where on the right we use the action in Remark 1.5. Note here that
$\textrm{c}\cdot\sigma$ is indeed $\mathcal{B}$-invariant and hence defines an
element of $\mathcal{I}^{1}_{\textrm{Set}}(\underline{U})$. This can be seen
using Lemma 4.29, applied to the identity equivalence. For varying
$\underline{U}$, these actions are compatible with restriction of opens and so
descend to an action of the sheaf $\mathcal{H}^{1}$ on the sheaf
$\mathcal{I}^{1}_{\textrm{Set}}$. As we will now show, this action is free in
the sense that the $\mathcal{H}^{1}(\underline{U})$-action on
$\mathcal{I}^{1}_{\textrm{Set}}(\underline{U})$ is free for each open
$\underline{U}$ in $\underline{\Delta}$. For this it is enough to show that
each $\mathcal{O}\in\underline{U}$ admits an open neighbourhood
$\underline{V}$ in $\underline{U}$ such that the
$\mathcal{H}^{1}(\underline{V})$-action on
$\mathcal{I}^{1}_{\textrm{Set}}(\underline{V})$ is free. To this end, note
that for each such $\mathcal{O}$ there is (as in Subsection 4.1.3) an
invariant open $W$ around $\mathcal{O}$ in $M$, together with an
infinitesimally abelian compact Lie group $G$, a $G$-invariant open
$W_{\mathfrak{g}^{*}}$ around the origin in $\mathfrak{g}^{*}$, and a
symplectic Morita equivalence:

$(\mathcal{G},\Omega)|_{W}$$W$$(P,\omega_{P})$$W_{\mathfrak{g}^{*}}$$(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})|_{W_{\mathfrak{g}^{*}}}$$\text{
}\text{ }\alpha_{1}$$\alpha_{2}$

that relates the orbit $\mathcal{O}$ of $\mathcal{G}$ to the origin in
$\mathfrak{g}^{*}$. After possibly shrinking $W$, we can further arrange that
$V:=W\cap\Delta\subset U$ and that the invariant subset
$\Delta_{\mathfrak{g}^{*}}$ of $\mathfrak{g}^{*}$ related to $V$ is convex. By
Lemma 4.29, for any choice of $x\in\mathcal{O}$ and $p\in\alpha_{1}^{-1}(x)$
we have a commutative square:

${H^{1}(\mathcal{B}|_{V},\mathcal{L})}$${H^{1}(\mathcal{B}_{x},\mathcal{T}_{x})}$${H^{1}(\Gamma\ltimes\Delta_{\mathfrak{g}^{*}},\mathcal{L}_{\Delta_{\mathfrak{g}^{*}}})}$${H^{1}(\Gamma,T)}$$\scriptstyle{\sim}$$\scriptstyle{\Phi_{x}}$$\scriptstyle{(\varphi_{p})_{*}}$$\scriptstyle{\sim}$$\scriptstyle{\Phi_{x}}$

in which the vertical maps are isomorphisms. This and part $a$ of Lemma 4.28
show that:

(117) $\Phi_{x}:H^{1}(\mathcal{B}|_{V},\mathcal{L})\to
H^{1}(\mathcal{B}_{x},\mathcal{T}_{x})$

is an isomorphism as well. It follows that if $\textrm{c}\in
H^{1}(\mathcal{B}|_{V},\mathcal{L})$ and
$\sigma\in\mathcal{I}^{1}_{\textrm{Set}}(\underline{V})$ are such that
$\textrm{c}\cdot\sigma=\sigma$, then
$\Phi_{x}(\textrm{c})\cdot\sigma(x)=\sigma(x)$, hence $\Phi_{x}(\textrm{c})=0$
(since the action in Remark 1.5 is free), and so $\textrm{c}=0$. So, the
$\mathcal{H}^{1}(\underline{V})$-action on
$\mathcal{I}^{1}_{\textrm{Set}}(\underline{V})$ is free. This shows freeness
of the $\mathcal{H}^{1}$-action on $\mathcal{I}^{1}_{\textrm{Set}}$.

Next, we will show that it restricts to a transitive action on
$\mathcal{I}^{1}$. This would prove the first part of the theorem, since
stalk-wise this $\mathcal{H}^{1}$-action on $\mathcal{I}^{1}$ would be given
by the actions described before the statement of Theorem 4.27. To show that
the action restricts, it is enough to show that for each open $\underline{U}$
the $H^{1}(\mathcal{B}|_{U},\mathcal{L})$-action on
$\mathcal{I}^{1}_{\textrm{Set}}(\underline{U})$ preserves
$\mathcal{I}^{1}(\underline{U})$. Let $\sigma\in\mathcal{I}(\underline{U})$
and $\mathcal{O}\in\underline{U}$. Since $\sigma$ is flat, there is a
symplectic Morita equivalence as above that relates $\sigma|_{\underline{V}}$
(via Lemma 4.16) to a centered section (Definition 4.17). In view of Lemma
4.29 and part $b$ of Lemma 4.28, for any choice of $x\in\mathcal{O}$,
$p\in\alpha_{1}^{-1}(x)$ and $q\in Q$ the diagram:

${H^{1}(\mathcal{B}|_{V},\mathcal{L})}$${H^{1}(\mathcal{B}_{x},\mathcal{T}_{x})}$${H^{1}(\Gamma,T)}$${H^{1}(\mathcal{B}_{y},\mathcal{T}_{y})}$${H^{1}(\Gamma_{\alpha},T)}$$\scriptstyle{\Phi_{x}}$$\scriptstyle{\Phi_{y}}$$\scriptstyle{(\varphi_{p})_{*}}$$\scriptstyle{(i_{\Gamma_{\alpha}})^{*}}$$\scriptstyle{(\varphi_{q})_{*}}$

commutes, where $y:=\alpha_{1}(q)$ and $\alpha:=\alpha_{2}(q)$. Because of
this, the section related to $(\textrm{c}\cdot\sigma)|_{\underline{V}}$ by the
Morita equivalence is centered as well, and so $\textrm{c}\cdot\sigma$ is flat
at $\mathcal{O}$, for any $\textrm{c}\in H^{1}(\mathcal{B}|_{U},\mathcal{L})$.
So, the $H^{1}(\mathcal{B}|_{U},\mathcal{L})$-action indeed preserves
$\mathcal{I}^{1}(\underline{U})$. For transitivitity, let
$\sigma_{1},\sigma_{2}\in\mathcal{I}^{1}(\underline{U})$ and
$\mathcal{O}\in\underline{U}$. In view of Remark 4.22, there is a symplectic
Morita equivalence as above that relates both $\sigma_{1}|_{\underline{V}}$
and $\sigma_{2}|_{\underline{V}}$ to centered sections. Consider an
$x\in\mathcal{O}$. By surjectivity of (117) and transitivity of the
$H^{1}(\mathcal{B}_{x},\mathcal{T}_{x})$-action on
$I^{1}(\mathcal{G},\mathcal{T}_{x})$, there is a $\textrm{c}_{V}\in
H^{1}(\mathcal{B}|_{V},\mathcal{L})$ such that
$\Phi_{x}(\textrm{c}_{V})\cdot\sigma_{1}(x)=\sigma_{2}(x)$. From commutativity
of the last diagram above and the fact that both $\sigma_{1}|_{\underline{V}}$
and $\sigma_{2}|_{\underline{V}}$ are related to centered sections by the
Morita equivalence, it follows that in fact
$\textrm{c}_{V}\cdot{\sigma_{1}}|_{\underline{V}}=\sigma_{2}|_{\underline{V}}$.
This shows that there is an open cover $\underline{\mathcal{V}}$ of
$\underline{U}$ with a collection $\\{\textrm{c}_{V}\in
H^{1}(\mathcal{B}|_{V},\mathcal{L})\mid\underline{V}\in\underline{\mathcal{V}}\\}$
such that
$\textrm{c}_{V}\cdot{\sigma_{1}}|_{\underline{V}}=\sigma_{2}|_{\underline{V}}$
for each $\underline{V}\in\underline{\mathcal{V}}$. Since the
$\mathcal{H}^{1}$-action is free, for any
$\underline{V}_{1},\underline{V}_{2}$ the $\mathcal{H}^{1}$-germs of
$\textrm{c}_{V_{1}}$ and $\textrm{c}_{V_{2}}$ must coincide at all points in
$\underline{V}_{1}\cap\underline{V}_{2}$. So, these glue to a section
$\textrm{c}\in\mathcal{H}^{1}(\underline{U})$ such that
$\textrm{c}\cdot\sigma_{1}=\sigma_{2}$. This shows that the
$\mathcal{H}^{1}$-action on $\mathcal{I}^{1}$ is indeed transitive.

It remains to prove the second part of the theorem. We ought to show that the
map (13) is $H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})$-equivariant with
respect to the action in the first structure theorem and that on
$\mathcal{I}^{1}(\underline{\Delta})$ obtained from the
$\mathcal{H}^{1}(\underline{\Delta})$-action via the group homomorphism (113).
After restricting to a complete transversal the proof reduces to the case in
which $\mathcal{B}$ is etale. To prove the equivariance in that case, suppose
that $\mathcal{B}$ is etale, that $\textrm{c}\in
H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})$ and that $J:(S,\omega)\to M$ is a
faithful multiplicity-free $(\mathcal{G},\Omega)$-space with momentum map
image $\Delta$. Let $\mathcal{U}$ be a basis of $\Delta$ for which
$\mathcal{L}$ is $\mathcal{U}$-acyclic, consisting of connected opens.
Consider a $1$-cocycle $\tau$ representing the class
$[\tau]\in\check{H}^{1}_{\text{$\mathcal{U}$}}(\mathcal{B}|_{\Delta},\mathcal{L})$
corresponding to c via the isomorphism (67). Let $x\in\Delta$. In view of
Lemma 1.46, there is a $U\in\text{$\mathcal{U}$}$ with the property that for
each $\gamma\in\mathcal{B}_{x}$ there is a (necessarily unique) smooth
bisection $\sigma_{\gamma}:U\to\mathcal{B}|_{U}$ such that $\sigma(x)=\gamma$,
and the map $\mathcal{B}_{x}\times U\to\mathcal{B}|_{U}$,
$(\gamma,y)\mapsto\sigma_{\gamma}(y)$ is surjective. Then, in fact, for each
$\gamma\in\mathcal{B}|_{U}$ there is a unique smooth bisection
$\sigma_{\gamma}:U\to\mathcal{B}|_{U}$ that maps $s(\gamma)$ to $\gamma$. The
composition:

$\check{H}^{1}_{\text{$\mathcal{U}$}}(\mathcal{B}|_{\Delta},\mathcal{L})\xrightarrow{(\ref{eqn:cechtoderivedmorph:equivshf})}H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})\xrightarrow{\Phi_{x}}H^{1}(\mathcal{B}_{x},\mathcal{T}_{x})$

is equal to the composition:

$\check{\Phi}_{x}:=(\textrm{ev}_{x})_{*}\circ(i_{U})^{*}:\check{H}^{1}_{\text{$\mathcal{U}$}}(\mathcal{B}|_{\Delta},\mathcal{L})\xrightarrow{(i_{U})^{*}}H^{1}(\mathcal{B}_{x},\mathcal{L}(U))\xrightarrow{(\textrm{ev}_{x})_{*}}H^{1}(\mathcal{B}_{x},\mathcal{T}_{x}),$

with
$i_{U}:\mathcal{B}_{x}\to\textsf{Emb}_{\text{$\mathcal{U}$}}(\mathcal{B}|_{\Delta})$
the functor (where we view $\mathcal{B}_{x}$ as category over $\\{x\\}$) that
sends $\\{x\\}$ to $U$ and that sends an arrow $\gamma\in\mathcal{B}_{x}$ to
the arrow $U\xleftarrow{\sigma_{\gamma}}U$. So, we have to show that:

(118) $\check{\Phi}_{x}([\tau])\cdot e(J)_{x}=e(J_{\tau})_{x}.$

Consider the map:

$\varphi_{\tau}:(\mathcal{G},\Omega)|_{U}\to(\mathcal{G},\Omega)|_{U},\quad
g\mapsto g\tau(U\xleftarrow{\sigma_{{[g]}}}U)(s(g))^{-1}.$

This is an automorphism of symplectic groupoids with corners. To see this, let
$\gamma\in\mathcal{B}|_{U}$. The image $\sigma_{\gamma}(U)$ in
$\mathcal{B}|_{\Delta}$ is open, hence the pre-image of this under the
projection $\mathcal{G}|_{\Delta}\to\mathcal{B}|_{\Delta}$ is an open in
$\mathcal{G}|_{U}$ on which the map $\varphi_{\tau}$ is given by the smooth
map:

$g\mapsto g\tau(U\xleftarrow{\sigma_{\gamma}}U)(s(g))^{-1}.$

Since $\mathcal{G}|_{U}$ can be covered by such opens, $\varphi_{\tau}$ is
smooth. Furthermore, the fact that each $\tau(U\xleftarrow{\sigma_{\gamma}}U)$
is Lagrangian implies that $\varphi_{\tau}$ is symplectic, and $\tau$ being a
$1$-cocycle implies that $\varphi_{\tau}$ a morphism of groupoids. Moreover,
$\varphi_{-\tau}$ is inverse to $\varphi_{\tau}$. So, $\varphi_{\tau}$ is
indeed an automorphism. The pair $(\varphi_{\tau},j_{U})$ consisting of
$\varphi_{\tau}$ and the symplectomorphism (72) is compatible with the
$\mathcal{G}|_{U}$-actions, in the sense that for all $g\in\mathcal{G}|_{U}$
and $p\in J^{-1}(U)$ such that $s(g)=J(p)$ it holds that:

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

Consequently, this pair induces an equivalence of symplectic representations:

$(\varphi_{\tau},(\textrm{d}j_{U})_{p}):(\mathcal{G}_{p},(\mathcal{S}\mathcal{N}_{p},\omega_{p}))\xrightarrow{\sim}(\mathcal{G}_{[p,U]},(\mathcal{S}\mathcal{N}_{[p,U]},(\omega_{\tau})_{[p,U]}))$

for each $p\in S$ such that $J(p)=x$. Using Lemma 1.15 and the observation
that $\varphi_{\tau}$ restricts to the identity on $\mathcal{T}$, (118)
readily follows from this. ∎

####  4.2.3. The structure groups for the case of group actions

In this subsection we give the remaining details for Examples 5 and 6. The
facts used in Example 5 that remain to be proved are collected in the
following proposition.

######  Proposition 4.30.

Let $G$ be an infinitesimally abelian compact Lie group with identity
component $T$ and group of connected components $\Gamma:=G/T$. Consider the
cotangent groupoid
$(\mathcal{G},\Omega):=(G\ltimes\mathfrak{g}^{*},-\textrm{d}\lambda_{\textrm{can}})$
and let $\underline{\Delta}$ be a non-empty Delzant subspace of
$\underline{M}=\mathfrak{g}^{*}/G$ such that the corresponding invariant
subspace $\Delta$ of $\mathfrak{g}^{*}$ is convex. Then the following hold.

  * a)

For each $n>0$, there is a canonical group isomorphism:

$H^{n}(\mathcal{B}|_{\Delta},\mathcal{L})\xrightarrow{\sim}H^{n}(\Gamma,T)$

  * b)

$H^{1}(\underline{\Delta},\underline{\mathcal{L}})=0$.

  * c)

There is a canonical bijection between the set
$\mathcal{I}^{1}(\underline{\Delta})$ of global sections of the ext-sheaf (8)
and $I^{1}(G,T)$, which together with the above group isomorphism is
compatible with the action of $H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})$ on
$\mathcal{I}^{1}(\underline{\Delta})$ and the canonical action of
$H^{1}(\varGamma,T)$ on $I^{1}(G,T)$.

###### Proof.

Since the finite group $\Gamma$ acts linearly on $\mathfrak{g}^{*}$ and
$\Delta$ is convex and non-empty, there is a $\Gamma$-fixed point
$x\in\Delta^{\Gamma}$ (as follows by averaging over $\Gamma$). The map
$\Phi_{x}:H^{n}(\mathcal{B}|_{\Delta},\mathcal{L})\to H^{n}(\Gamma,T)$ is an
isomorphism by part $a$ of Lemma 4.28, and by part $b$ of the same lemma this
does not depend on the choice of $x\in\Delta^{\Gamma}$. This proves part $a$.
For part $b$, note that $\Phi_{x}$ factors as:

${H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})}$${\mathcal{H}^{1}(\underline{\Delta})}$${H^{1}(\Gamma,T)}$$\scriptstyle{\Phi_{x}}$$\scriptstyle{\widehat{\Phi}_{x}}$

with the horizontal map as in the sequence (112) and with $\widehat{\Phi}_{x}$
given by composition of the evaluation at the stalk of $\mathcal{H}^{1}$ at
$\mathcal{O}_{x}$ with the map from this stalk into $H^{1}(\Gamma,T)$ induced
by the maps (114). As (112) is exact, part $b$ follows. For part $c$, consider
the evaluation map:

(119) $\textrm{ev}_{x}:\mathcal{I}^{1}(\underline{\Delta})\to I^{1}(G,T).$

Clearly, this and the map $\Phi_{x}$ are compatible with the action of
$H^{1}(\mathcal{B}|_{\Delta},\mathcal{L})$ and that of $H^{1}(\Gamma,T)$. To
prove part $c$, we will show that this is a bijection that does not depend on
the choice of $x\in\Delta^{\Gamma}$. For injectivity, suppose that
$\sigma_{1},\sigma_{2}\in\mathcal{I}^{1}(\underline{\Delta})$ are such that
$\sigma_{1}(x)=\sigma_{2}(x)$. Let $y\in\Delta$. Then $(1-t)x+ty\in\Delta$ for
all $t\in[0,1]$, by convexity. Since $x$ is fixed by $\Gamma$ and the action
is linear, it holds that $\Gamma_{(1-t)x+ty}=\Gamma_{y}$ for all $t\in]0,1]$.
Hence, the interval $]0,1]$ is partitioned by the sets:

$S([c])=\\{t\in]0,1]\mid\sigma_{1}((1-t)x+ty)=[c]\\},\quad[c]\in
I^{1}(\Gamma_{y},T).$

From flatness of $\sigma_{1}$ and $\sigma_{2}$ and the fact that
$\sigma_{1}(x)=\sigma_{2}(x)$, it follows that $S(\sigma_{2}(y))$ is non-
empty. Moreover, $S([c])$ is open in $]0,1]$ for each $[c]\in I^{1}(G,T)$ due
to flatness of $\sigma_{1}$. So, by connectedness of $]0,1]$ it must hold that
$\sigma_{1}(y)=\sigma_{2}(y)$, which shows that (119) is indeed injective. For
the remainder, consider the map:

$I^{1}(G,T)\to\mathcal{I}^{1}(\underline{\Delta})$

that sends $[c]\in I^{1}(G,T)$ to the global section of $\mathcal{I}^{1}$
defined by the section of (103) with value at $y\in\Delta$ given by
$[c|_{G_{y}}]\in I^{1}(G_{y},T)$ (which is indeed invariant, as follows from
Lemma 4.13). Since this is a section of (119), we conclude that (119) is a
bijection with this map as its inverse. As its inverse is independent of the
choice of $x\in\Delta^{\Gamma}$, so is the map (119). ∎

Next, we return to Example 6.

######  Example 4.31.

Let $G$ be any compact Lie group. 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 $G$. Consider the regular and proper
symplectic groupoid
$(\mathcal{G},\Omega):=(G\ltimes\mathfrak{g}^{*}_{\textrm{reg}},-\textrm{d}\lambda_{\textrm{can}})$
over $M:=\mathfrak{g}^{*}_{\textrm{reg}}$. Since $\mathfrak{c}$ is a complete
transversal, $(\mathcal{G},\Omega)$ is canonically Morita equivalent to its
restriction $(\mathcal{G},\Omega)|_{\mathfrak{c}}$, which coincides with the
restriction to $\mathfrak{c}$ of cotangent symplectic groupoid of
$N(\mathfrak{c})$. This Morita equivalence induces a bijection, as in (75),
between isomorphism classes of compact, connected, regular, multiplicity-free
Hamiltonian $G$-spaces (in the sense of Example 6) and isomorphism classes of
compact, connected, multiplicity-free Hamiltonian $N(\mathfrak{c})$-spaces
with momentum map image contained in $\mathfrak{c}$. Explicitly, this is given
by associating to a regular Hamiltonian $G$-space its restriction to
$\mathfrak{c}$. To a Hamiltonian $N(\mathfrak{c})$-space
$J:(S,\omega)\to\mathfrak{t}^{*}$ as above we can associate the triple
$(K,\Delta,e)$ consisting of:

  * •

the kernel $K$ of the morphism $T\to\textrm{Diff}(S)$ obtained by restricting
the $N(\mathfrak{c})$-action to $T$,

  * •

the image $\Delta:=J(S)$,

  * •

the ext-invariant $e$ of the induced maximally toric Hamiltonian
$N(\mathfrak{c})/K$-action (as in Example 5), with momentum map
$\mu-x:(S,\omega)\to(\mathfrak{t}/\mathfrak{k})^{*}$, for any
$N(\mathfrak{c})$-fixed point $x\in\Delta$.

Note here that $e$ does not depend on the choice of $N(\mathfrak{c})$-fixed
point. All together, this gives a bijection between isomorphism classes of
compact, connected, regular, multiplicity-free Hamiltonian $G$-spaces and
triples as in Example 6.

Finally, using the above proposition we can also complete the argument in
Example 4.23.

######  Example 4.32.

To show that $H^{1}(\underline{\Delta},\underline{\mathcal{L}})$ vanishes in
Example 4.23, consider the open cover
$\underline{\text{$\mathcal{U}$}}=\\{\underline{U}^{+},\underline{U}^{-}\\}$
with $U^{\pm}:=W^{\pm}\cap\Delta$ for
$W^{\pm}:=\\{(x,\mu)\in\mathbb{R}\times\mathbb{S}^{1}\mid\mu\neq\pm 1\\}$.
Since $(\mathcal{G},\Omega)|_{W^{+}}$ and $(\mathcal{G},\Omega)|_{W^{-}}$ are
both isomorphic to restrictions of symplectic groupoids of the form in
Proposition 4.30 via isomorphisms that identify $U^{+}$ and $U^{-}$ with
convex sets, the cohomology groups
$H^{1}(\underline{U}^{+},\underline{\mathcal{L}})$ and
$H^{1}(\underline{U}^{-},\underline{\mathcal{L}})$ vanish. So, in view of the
Mayer-Vietoris sequence, to show that
$H^{1}(\underline{\Delta},\underline{\mathcal{L}})=0$ it suffices to show that
every $\tau\in\underline{\mathcal{L}}(\underline{U}^{+}\cap\underline{U}^{-})$
can be written as the difference of a section of $\underline{\mathcal{L}}$ on
$\underline{U}^{+}$ and one on $\underline{U}^{-}$. For this, note that by
$\Gamma$-invariance of $\tau$, after possibly subtracting the section
$[\frac{1}{2}\textrm{d}x]\in\underline{\mathcal{L}}(\underline{\Delta})$
(obtained from the closed $1$-form $\frac{1}{2}\textrm{d}x$ by projection
along $T^{*}M\to\mathcal{T}_{\Lambda}$), we can assume that
$\tau(0,i)=[c\,\textrm{d}\theta_{(0,i)}]$ for some $c\in\mathbb{R}$. Let
$V^{\pm}:=\\{(x,\mu)\in\Delta\mid\pm\textrm{Im}(\mu)>0\\}$ denote the
connected components of $U^{+}\cap U^{-}$. Since $V^{+}$ is simply-connected,
there is a unique $f\in C^{\infty}(V^{+})$ such that $\textrm{d}f$ projects to
$\tau|_{V^{+}}$ along the covering map $T^{*}M\to\mathcal{T}_{\Lambda}$ and
such that $\textrm{d}f_{(0,i)}=c\,\textrm{d}\theta_{(0,i)}$ and $f(0,i)=0$.
Since $(0,i)$ and $\textrm{d}f_{(0,i)}$ are fixed by
$\mathbb{Z}_{2}\times\\{1\\}$, the function $f$ is
$\mathbb{Z}_{2}\times\\{1\\}$-invariant. Hence, the function $\widehat{f}\in
C^{\infty}(U^{+}\cap U^{-})$ given by:

$\widehat{f}(x,\mu)=\begin{cases}f(x,\mu)\quad&\text{ if }(x,\mu)\in V^{+}\\\
f(x,\mu^{-1})\quad&\text{ if }(x,\mu)\in V^{-}\end{cases}$

is $\Gamma$-invariant. Choose a smooth function $\rho\in
C^{\infty}(\mathbb{S}^{1})$ that has constant value $1$ on a neighbourhood of
$1$, that is supported in $\mathbb{S}^{1}-\\{-1\\}$ and that is
$\mathbb{Z}_{2}$-invariant with respect to the action given by
$\varepsilon\cdot\mu=\mu^{\varepsilon}$, for $\varepsilon\in\mathbb{Z}_{2}$
and $\mu\in\mathbb{S}^{1}$. The respective functions:

$(x,\mu)\mapsto\rho(\mu)\widehat{f}(x,\mu)\quad\&\quad(x,\mu)\mapsto(\rho(\mu)-1)\widehat{f}(x,\mu)$

on $U^{+}\cap U^{-}$ extend by zero to $\Gamma$-invariant smooth functions
$f_{+}$ and $f_{-}$ on $U^{+}$ and $U^{-}$, and the closed $1$-forms
$\textrm{d}f_{+}$ and $\textrm{d}f_{-}$ project to respective sections
$\tau_{+}\in\underline{\mathcal{L}}(\underline{U}^{+})$ and
$\tau_{-}\in\underline{\mathcal{L}}(\underline{U}^{-})$ with the property that
$\tau_{+}|_{U^{+}\cap U^{-}}-\tau_{-}|_{U^{+}\cap U^{-}}=\tau$. This proves
the claim.

##  Appendix A On the multiplicity-free condition

In this appendix we use results on the Poisson geometry of the orbit space of
a Hamiltonian action to derive the following (in the same spirit as [48,
Proposition A.1]).

######  Proposition A.1.

Let $(\mathcal{G},\Omega)\rightrightarrows M$ be a regular and proper
symplectic groupoid. A Hamiltonian $(\mathcal{G},\Omega)$-action along
$J:(S,\omega)\to M$ is faithful multiplicity-free if and only if the following
four conditions hold.

  * i)

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

  * ii)

The equality:

(120) $\text{dim}(S)=2\text{dim}(M)-\textrm{rank}(\pi)$

holds, where $\pi$ is the Poisson structure on $M$ induced by
$(\mathcal{G},\Omega)$.

  * iii)

The momentum map $J$ has connected fibers.

  * iv)

The transverse momentum map $\underline{J}:\underline{S}\to\underline{M}$ is
closed as a map into its image.

To prove Proposition A.1, let us recall some facts on the Poisson geometry of
Hamiltonian actions. Let $(\mathcal{G},\Omega)$ be a proper symplectic
groupoid and let $J:(S,\omega)\to M$ be a Hamiltonian
$(\mathcal{G},\Omega)$-space.

  * •

The sheaf of smooth functions $\mathcal{C}^{\infty}_{\underline{S}}$ on the
orbit space $\underline{S}:=S/\mathcal{G}$ is the sheaf of
$\mathbb{R}$-algebras consisting of $\mathcal{G}$-invariant smooth functions
on invariant opens in $S$. This can naturally be viewed as a subsheaf of the
sheaf of continuous functions on $\underline{S}$. For each invariant open $U$
in $S$, the subalgebra $\mathcal{C}^{\infty}_{\underline{S}}(\underline{U})$
of $\mathcal{C}^{\infty}_{S}(U)$ is a Poisson subalgebra with respect to the
Poisson bracket associated to the symplectic form $\omega$ on $S$. These
Poisson brackets make $\mathcal{C}^{\infty}_{\underline{S}}$ into a sheaf of
Poisson algebras, and as for smooth manifolds these brackets are uniquely
determined by the single Poisson bracket on the algebra of global smooth
functions on $\underline{S}$. The stratification
$\mathcal{S}_{\textrm{Gp}}(\underline{S})$ of $\underline{S}$ induced by the
$\mathcal{G}$-action has the property that each stratum $\underline{\Sigma}$
admits a natural structure of Poisson manifold, uniquely determined by the
fact that restriction along the inclusion
$i:\underline{\Sigma}\hookrightarrow\underline{S}$ induces a surjective map of
sheaves
$i^{*}:\mathcal{C}^{\infty}_{\underline{S}}|_{\underline{\Sigma}}\to\mathcal{C}^{\infty}_{\underline{\Sigma}}$
that respects the Poisson brackets.

  * •

The complexity C($J$) of the Hamiltonian $(\mathcal{G},\Omega)$-action is, by
definition, half of the maximum of the dimensions of the symplectic leaves on
all of these strata. The dimension of the symplectic leaves in $\underline{S}$
is locally non-decreasing ([38, Proposition 2.92]). Therefore, the union of
the symplectic leaves in $\underline{S}$ of maximal dimension is open in
$\underline{S}$. This can be used to deduce (using for instance [38,
Proposition 2.69 and Remark 2.93]) that, if the set of points in $S$ at which
the momentum map $J$ is a submersion is dense in $S$, then the complexity of
the Hamiltonian action is:

(121) $\textrm{C}(J)=\frac{1}{2}\left(\text{dim}(S)-2\text{dim}(M)+\max_{x\in
J(S)}\textrm{rank}(\pi_{x})\right),$

where $\pi$ is the Poisson structure on $M$ induced by $(\mathcal{G},\Omega)$.

  * •

The symplectic leaves of the orbit space $\underline{S}$ can be described in
terms of the symplectic reduced spaces of the Hamiltonian action. As
topological spaces, the reduced spaces are the subspaces of $\underline{S}$ of
the form:

$J^{-1}(\mathcal{O})/\mathcal{G}=\underline{J}^{-1}(\mathcal{O}),$

where $\mathcal{O}$ is an orbit of $\mathcal{G}$ in $M$. The reduced spaces
can naturally be stratified into symplectic manifolds (generalizing the main
result of [26]) and the symplectic leaves of the orbit space $\underline{S}$
coincide with the symplectic strata of the reduced spaces.

The facts mentioned above are probably well-known for Hamiltonian actions of
compact Lie groups. Such Hamiltonian actions that satisfy the equivalent
conditions in the proposition below are called multiplicity-free [18].

######  Proposition A.2.

Let $(\mathcal{G},\Omega)$ be a proper symplectic groupoid and let
$J:(S,\omega)\to M$ be a Hamiltonian $(\mathcal{G},\Omega)$-space. Then the
following are equivalent.

  * a)

The induced Poisson bracket on $C^{\infty}(\underline{S})$ is the zero
bracket.

  * b)

The Hamiltonian action has complexity zero.

  * c)

All of the reduced spaces are discrete topological subspaces of
$\underline{S}$.

###### Proof.

As mentioned above, the Poisson brackets on the algebras associated to the
sheaf of Poisson algebras $\mathcal{C}^{\infty}_{\underline{S}}$ are uniquely
determined by the single Poisson bracket on the algebra
$C^{\infty}(\underline{S})$. So, if $a$ holds, then the Poisson bracket on
$\mathcal{C}^{\infty}_{\underline{S}}(\underline{U})$ is zero for each
invariant open $U$ in $S$. The Poisson structure on each stratum of
$\mathcal{S}_{\textrm{Gp}}(\underline{S})$ must then also be zero, so that $b$
holds. Conversely, from $b$ it is immediate that the Poisson bracket on each
stratum is zero. Since the inclusion of each stratum is a Poisson map, the
bracket on $C^{\infty}(\underline{S})$ is zero as well, as follows from
pointwise evaluation. So, $b$ implies $a$. This proves the equivalence between
$a$ and $b$. The equivalence between $b$ and $c$ is clear from the description
of the symplectic leaves as strata of the symplectic reduced spaces. ∎

We now turn to the proof of the main result of this appendix.

###### Proof of Proposition A.1.

First notice that both sets of conditions contain the assumption that the
induced $\mathcal{T}$-action is free on a dense subset of $S$. The momentum
map $J$ is a submersion at all points in $S$ at which the $\mathcal{T}$-action
is free. Therefore, under both sets of conditions, the set of points in $S$ at
which $J$ is a submersion is dense in $S$, which implies that the complexity
of the Hamiltonian action is given by (121). Since $(\mathcal{G},\Omega)$ is
regular, this means that:

$\textrm{C}(J)=\frac{1}{2}\left(\text{dim}(S)-2\text{dim}(M)+\textrm{rank}(\pi)\right).$

Now, suppose that all properties in Proposition A.1 are satisfied. By the
above equation for the complexity of the Hamiltonian action, the second
condition in Proposition A.1 means that the action has complexity zero, which
by Proposition A.2 means that for every $x\in M$, the subspace
$\underline{J}^{-1}({\mathcal{O}}_{x})$ of $\underline{S}$ is discrete. Since
$\underline{J}^{-1}(\mathcal{O}_{x})$ is (canonically) homeomorphic to the
quotient $J^{-1}(x)/\mathcal{G}_{x}$ and the $J$-fibers are assumed to be
connected, it follows that both $J^{-1}(x)/\mathcal{G}_{x}$ and
$\underline{J}^{-1}(\mathcal{O}_{x})$ consist of a single point. Firstly, it
follows from this that the $\mathcal{G}_{x}$-orbit (which is embedded, seeing
as $\mathcal{G}_{x}$ is compact) is the subspace $J^{-1}(x)$ of $S$. Since
$J^{-1}(x)$ is connected, the $\mathcal{T}_{x}$-orbit must then also be the
entire space $J^{-1}(x)$. Secondly, it follows that $\underline{J}$ is
injective (its fibers being points). So, since it is assumed to be closed as
map into its image, it must be a topological embedding. This proves that the
Hamiltonian action is faithful multiplicity-free.

Next, suppose that the action is faithful multiplicity-free. Then clearly the
transverse momentum map is closed as map into its image. Furthermore, since
the $J$-fibers coincide with the $\mathcal{T}$-orbits, they are connected and
for each $x\in M$ the quotient $J^{-1}(x)/\mathcal{G}_{x}$ consists of a
single point. So, for every $x\in M$ the set
$\underline{J}^{-1}(\mathcal{O}_{x})$ consists of a single point as well,
which by Proposition A.2 implies that the Hamiltonian action has complexity
zero. By the above equation for the complexity, this proves that the
Hamiltonian action satisfies all conditions in Proposition A.1. ∎

##  Appendix B Background on manifolds with corners

Let $X$ be a topological space. Given integers $n\geq 0$ and
$k\in\\{0,...,n\\}$, we use the notation:

$\mathbb{R}^{n}_{k}:=[0,\infty[^{k}\times\mathbb{R}^{n-k}.$

By an $n$-dimensional chart with corners for $X$ we mean a pair $(U,\chi)$
consisting of an open $U$ in $X$ and a homeomorphism $\chi$ from $U$ onto an
open in $\mathbb{R}^{n}_{k}$, for some $k\in\\{0,...,n\\}$. Given two subsets
$A\subset\mathbb{R}^{n}$ and $B\subset\mathbb{R}^{m}$, we say that a map
$f:A\to B$ is smooth if for every $x\in A$ there is an open $U_{x}$ in
$\mathbb{R}^{n}$ around $x$ and a smooth map $U_{x}\to\mathbb{R}^{m}$ that
coincides with $f$ on $U_{x}\cap A$. Two charts with corners $(U,\chi)$ and
$(V,\varphi)$ on $X$ are called smoothly compatible if both transition maps
between them are smooth maps in this sense. Just as for manifolds without
corners, this leads to a notion of smooth atlas consisting of charts with
corners for $X$ (that we require to consist of charts of a fixed dimension)
and every such atlas is contained in a unique maximal one. We refer to a
maximal such atlas as a smooth structure with corners on $X$.

######  Definition B.1.

A smooth manifold with corners is a second countable and Hausdorff space $X$
together with a smooth structure with corners on $X$. Henceforth, by a
manifold with corners we always mean a smooth manifold with corners and we
omit the smooth structure from the notation. Furthermore, we use the following
terminology and notation.

  * •

The common dimension of the charts for $X$ is called the dimension of $X$,
denoted $\text{dim}(X)$.

  * •

By a smooth map $f:X\to Y$ between two manifolds with corners we will mean a
continuous map with the property that for any chart with corners $(U,\chi)$
for $X$ and any chart with corners $(V,\varphi)$ for $Y$, the coordinate
representation:

$\varphi\circ f\circ\chi^{-1}:\chi(U\cap f^{-1}(V))\to\varphi(V)$

is smooth in the sense above.

  * •

A homeomorphism $f:X\to Y$ between manifolds with corners is called a
diffeomorphism if $f$ and $f^{-1}$ are smooth.

######  Remark B.2.

The above definition of smooth map coincides with that used in [40, 21, 39].
For a further comparison to the literature on manifolds with corners and maps
between them, see [20] (where a smooth map in the above sense is called weakly
smooth).

######  Remark B.3.

Any open subspace of a manifold with corners $X$ inherits a smooth structure
with corners and the $\mathbb{R}$-valued smooth functions on these opens form
a sheaf of algebras $\mathcal{C}^{\infty}_{X}$ on $X$. Two smooth structures
with corners coincide if their associated sheaves of smooth functions
coincide.

######  Remark B.4.

A manifold with corners $X$ is a reduced differentiable space with structure
sheaf $\mathcal{C}^{\infty}_{X}$, in the sense of [16] (also see [38] for a
more direct introduction to these). Furthermore, a continuous map $f:X\to Y$
between manifolds with corners is smooth if and only if it is a morphism of
the underlying reduced differentiable spaces (meaning that for every smooth
function on an open in $Y$, the pull-back along $f$ is again smooth). As for
any second countable and Hausdorff reduced differentiable space, there exist
$\mathcal{C}^{\infty}_{X}$-partitions of unity subordinate to any open cover.

Let $X$ be an $n$-dimensional manifold with corners. Given $x\in X$, let
$\textrm{Charts}_{x}(X)$ denote the set of charts for $X$ around $x$. As for
manifolds without corners, one can define the tangent space $T_{x}X$ of $X$ at
$x$ as the $n$-dimensional real vector space of maps:

$\textrm{Charts}_{x}(X)\to\mathbb{R}^{n}$

that are compatible with coordinate changes (so that any such map is
determined by its value on a single chart), and for every smooth map $f:X\to
Y$ between manifolds with corners one can define its differential
$\textrm{d}f_{x}:T_{x}X\to T_{f(x)}Y$ at $x$. The composition of two smooth
maps is again smooth, and the chain rule still holds.

######  Remark B.5.

The tangent space $T_{x}X$ is naturally isomorphic to the vector space of
derivations at $x$ of the stalk of $\mathcal{C}^{\infty}_{X}$ at $x$, and to
that of the algebra of global smooth functions on $X$.

######  Remark B.6.

On manifolds with corners one can define smooth vector fields and differential
forms, the pull-back of differential forms along smooth maps, their wedge-
product and their exterior derivative, as for manifolds without corners. The
Poincaré Lemma still holds: every closed differential form is locally exact,
by arguments as in the case without corners (e.g. as in [4]).

Unlike for manifolds without corners, there may be tangent vectors that cannot
be realized as the derivative of a smooth curve in $X$. A tangent vector $v\in
T_{x}X$ is called inward pointing if there is an $\varepsilon>0$ and a smooth
curve $\gamma:[0,\varepsilon[\to X$ such that $v=\dot{\gamma}(0)$. The inward
pointing tangent vectors form a polyhedral cone $C_{x}X$ in $T_{x}X$, that we
call the tangent cone of $X$ at $x$. We let $F_{x}X$ denote the largest linear
subspace of $T_{x}X$ that is contained in $C_{x}X$. This consists of those
$v\in T_{x}X$ for which there is an $\varepsilon>0$ and a smooth curve
$\gamma:]-\varepsilon,\varepsilon[\to X$ such that $v=\dot{\gamma}(0)$. For
any smooth map $f:X\to Y$ and $x\in X$, it holds that:

(122) $\textrm{d}f_{x}(C_{x}X)\subset
C_{f(x)}Y\quad\&\quad\textrm{d}f_{x}(F_{x}X)\subset F_{f(x)}Y.$

For any chart $(U,\chi)$ for $X$ onto an open in $\mathbb{R}^{n}_{k}$ that
sends $x\in U$ to the origin, the differential
$\textrm{d}\chi_{x}:T_{x}X\xrightarrow{\sim}\mathbb{R}^{n}$ identifies
$C_{x}X$ with $\mathbb{R}^{n}_{k}$ and $F_{x}X$ with
$\\{0\\}\times\mathbb{R}^{n-k}$. The depth of $x\in X$ is:

$\textrm{depth}_{X}(x):=\text{dim}(X)-\text{dim}(F_{x}X).$

For any chart $(U,\chi)$ for $X$ around $x$ mapping onto an open in
$\mathbb{R}^{n}_{k}$, the depth of $x$ equals the number of
$j\in\\{1,...,k\\}$ such that $\chi^{j}(x)=0$.

Next, we turn to embeddings. Following [21], we use the definition below.

######  Definition B.7.

We call a topological embedding $i:X\to Y$ between manifolds with corners a
smooth embedding if it is smooth and at each point in $X$ its differential is
injective.

######  Example B.8.

Given manifolds with corners $X$ and $Y$, the product $X\times Y$ inherits a
natural structure of smooth manifold with corners. For any smooth map $f:X\to
Y$, the graph map:

$X\to X\times Y,\quad x\mapsto(x,f(x)),$

is a smooth embedding.

######  Remark B.9.

A topological embedding $i:X\to Y$ between manifolds with corners is a smooth
embedding if and only if
$i:(X,\mathcal{C}^{\infty}_{X})\to(Y,\mathcal{C}^{\infty}_{Y})$ is an
embedding of reduced differentiable spaces, meaning that $i$ is smooth and for
every smooth function $g$ on an open $U$ in $X$ and every $x\in U$ there is a
smooth function $\widehat{g}$ on an open $U_{x}$ in $Y$ around $i(x)$ such
that $\widehat{g}\circ i$ coincides with $g$ on $U\cap i^{-1}(U_{x})$. Here,
the forward implication follows using the immersion theorem for smooth maps
between opens in Euclidean spaces, while the backwards implication is clear
from Remark B.5. In view of Remark B.4 we conclude from this characterization
that: if $i:X\to Y$ is a smooth embedding of manifolds with corners, then a
map $f:Z\to X$ from another manifold with corners into $X$ is smooth if and
only if $i\circ f$ is smooth.

######  Remark B.10.

In view of Remark B.3 and Remark B.9 it holds that: given a topological
embedding $i:X\to Y$ from a topological space $X$ into a manifold with corners
$Y$, there is at most one smooth structure with corners for $X$ that makes
$i:X\to Y$ a smooth embedding.

######  Definition B.11.

We call a subspace of a manifold with corners an embedded submanifold if it
admits a (necessarily unique) smooth structure with corners that makes the
inclusion map a smooth embedding.

######  Example B.12.

Each of the subspaces:

$X_{k}:=\\{x\in X\mid\textrm{depth}_{X}(x)=k\\}$

is an embedded submanifold of $X$ without corners, with tangent space at $x\in
X_{k}$ equal to $F_{x}X$. Their connected components – called the open faces –
form a stratification of $X$. The open and dense subset $X_{0}$ of $X$ is the
regular part of this stratification (that is, the union of all open strata).
We usually denote $X_{0}$ as $\mathring{X}$.

We now turn to submersions. Following [40, 39], we use the notion below.

######  Definition B.13.

By a submersion $f:X\to Y$ between manifolds with corners we mean a smooth map
with the property that, for each $x\in X$, the differential
$\textrm{d}f_{x}:T_{x}X\to T_{f(x)}Y$ is surjective. Such a submersion is
called tame if in addition, for each $x\in X$ it holds that:

$\textrm{d}f_{x}^{-1}(C_{f(x)}Y)=C_{x}X.$

In other words: if $v\in T_{x}X$ and $\textrm{d}f_{x}(v)$ is an inward
pointing, then $v$ is inward pointing.

Note here that tame submersions are simply called submersions in [40].

######  Example B.14.

The prototypical example of a tame submersion is the following. Let $f:M\to N$
be a submersion between smooth manifolds without corners and let $Z$ be an
embedded submanifold with corners in $N$. Then it follows from the submersion
theorem that the pre-image $f^{-1}(Z)$ is an embedded submanifold with corners
in $M$, with tangent space and tangent cone:

$T_{x}f^{-1}(Z)=\textrm{d}f_{x}^{-1}(T_{f(x)}Z)\quad\&\quad
C_{x}f^{-1}(Z)=\textrm{d}f_{x}^{-1}(C_{f(x)}Z),\quad\quad x\in f^{-1}(Z),$

and the restriction $f:f^{-1}(Z)\to Z$ is a tame submersion.

The following shows that tame submersions behave much like submersions between
manifolds without corners.

######  Proposition B.15.

Let $f:X\to Y$ be a submersion between manifolds with corners. The following
are equivalent.

  * a)

The submersion $f$ is tame.

  * b)

The submersion $f$ preserves depth. That is, for each $x\in X$ it holds that:

$\textrm{depth}_{Y}(f(x))=\textrm{depth}_{X}(x).$

  * c)

For each $x\in X$, there is a chart $(U,\chi)$ around $x$ onto an open in
$\mathbb{R}^{n}_{k}$ that sends $x$ to the origin and there is a chart
$(V,\varphi)$ around $f(x)$ onto an open in $\mathbb{R}^{m}_{k}$ that sends
$f(x)$ to the origin, such that $f(U)=V$ and the coordinate representation of
$f$ is:

$\varphi\circ
f\circ\chi^{-1}:\chi(U)\to\varphi(V),\quad(x_{1},...,x_{n})\mapsto(x_{1},...,x_{m}).$

Here $n=\text{dim}(X)$, $m=\text{dim}(Y)$ and $k$ is the depth of $x$ and
$f(x)$.

###### Proof.

For the implication from $a$ to $b$ notice that, since $f$ is tame, the linear
subspace $\textrm{d}f_{x}^{-1}(F_{f(x)}Y)$ of $T_{x}X$ is contained in
$C_{x}X$ and hence it must be contained in the largest such subspace $F_{x}X$.
Combining this with (122), we conclude that
$\textrm{d}f_{x}^{-1}(F_{f(x)}Y)=F_{x}X$. Using this and surjectivity of
$\textrm{d}f_{x}$, it follows from the rank-nullity theorem that
$\textrm{depth}_{Y}(f(x))=\textrm{depth}_{X}(x)$.

For the implication from $b$ to $c$, we essentially follow the proof of [39,
Lemma 1.3]. To this end, suppose that $f$ preserves depth and let $x\in X$.
Since $\textrm{depth}_{X}(x)=\textrm{depth}_{Y}(f(x))$, we can assume (after
fixing appropriate charts with corners for $X$ and $Y$ around $x$ and $f(x)$)
that $f$ is a depth-preserving submersion between opens $U$ in
$\mathbb{R}^{n}_{k}$ and $V$ in $\mathbb{R}^{m}_{k}$ around the respective
origins, and that $x$ and $f(x)$ are the respective origins. After possibly
further shrinking $U$, we can assume that there is an open $W$ in
$\mathbb{R}^{n}$ and a smooth map $\widehat{f}:W\to\mathbb{R}^{m}$ such that
$U=W\cap\mathbb{R}^{n}_{k}$ and $\widehat{f}$ coincides with $f$ on $U$. Then
the differential of $\widehat{f}$ at the origin coincides with that of $f$, so
that it is surjective. Hence, by the submersion theorem for maps between opens
in Euclidean spaces, we can (after possibly shrinking $U$ and $W$) further
arrange for there to be a diffeomorphism
$\chi:W\to]-\varepsilon,\varepsilon[^{n}$, for some $\varepsilon>0$, that maps
the origin to the origin and is such that
$\widehat{f}\circ\chi^{-1}:]-\varepsilon,\varepsilon[^{n}\to\mathbb{R}^{m}$ is
the projection onto the first $m$ coordinates. To prove part $b$, it is now
enough to show that
$\chi(U)=]-\varepsilon,\varepsilon[^{n}\cap\mathbb{R}^{n}_{k}$. The inclusion
from left to right is clear. For the other inclusion, suppose that
$y\in]-\varepsilon,\varepsilon[^{n}\cap\mathbb{R}^{n}_{k}$. To show that
$\chi^{-1}(y)\in\mathbb{R}^{n}_{k}$, we fix an
$\widetilde{x}\in\mathring{U}=U\cap\mathring{\mathbb{R}}^{n}_{k}$ (meaning
that its first $k$ components are strictly positive; cf. Rem B.12) and we show
that the curve:

$\gamma:[0,1]\to W,\quad\gamma(t)=\chi^{-1}((1-t)\chi(\widetilde{x})+ty).$

takes values in $\mathbb{R}^{n}_{k}$. For this, by continuity of $\gamma$ it
is enough to show that $\gamma(t)\in\mathbb{R}^{n}_{k}$ for all $t\in[0,1[$.
The set of $t\in[0,1[$ such that $\gamma(t)\in\mathbb{R}^{n}_{k}$ is non-empty
(for $\gamma(0)\in\mathbb{R}^{n}_{k}$) and is clearly closed in $[0,1[$. Next,
we will show that for $t\in[0,1[$: $\gamma(t)\in\mathbb{R}^{n}_{k}$ if and
only if $\gamma(t)\in\mathring{\mathbb{R}}^{n}_{k}$, so that the set of
$t\in[0,1[$ such that $\gamma(t)\in\mathbb{R}^{n}_{k}$ is open in $[0,1[$ as
well ($\mathring{\mathbb{R}}^{n}_{k}$ being open in $\mathbb{R}^{n}$) and
hence it must be all of $[0,1[$, by connectedness. To this end, let
$t\in[0,1[$ such that $\gamma(t)\in\mathbb{R}^{n}_{k}$. Then $\gamma(t)\in U$,
and so:

$f(\gamma(t))=\widehat{f}(\gamma(t))=(1-t)(\chi^{1}(\widetilde{x}),...,\chi^{m}(\widetilde{x}))+t(y_{1},...,y_{m}).$

Since $f$ preserves depth, the first $k$ components of
$(\chi^{1}(\widetilde{x}),...,\chi^{m}(\widetilde{x}))=f(\widetilde{x})$ are
strictly positive. So, because $y\in\mathbb{R}^{n}_{k}$, the first $k$
components of $f(\gamma(t))$ are strictly positive as well. Therefore, the
depth of $f(\gamma(t))$ in $V$ is zero, hence so is the depth of $\gamma(t)$
in $U$, meaning that $\gamma(t)$ indeed belongs to
$\mathring{\mathbb{R}}^{n}_{k}$. This proves that $b$ implies $c$.
Furthermore, the implication from $c$ to $a$ is clear and so we conclude that
$a$, $b$ and $c$ are indeed equivalent. ∎

######  Corollary B.16.

Every tame submersion $f:X\to Y$ has the following properties.

  * a)

The map $f:X\to Y$ is open and for every $x\in X$ there is a smooth local
section of $f$, defined on an open around $f(x)$ in $Y$, that maps $f(x)$ to
$x$.

  * b)

For each $y\in Y$ the fiber $f^{-1}(y)$ is an embedded submanifold of $X$
without corners, with tangent space:

$T_{x}f^{-1}(y)=\textrm{Ker}(\textrm{d}f_{x}).$

  * c)

For every smooth map $g:Z\to Y$ from another manifold with corners into $Y$,
the set-theoretic fiber product $X\times_{Y}Z$ is an embedded submanifold with
corners of $X\times Z$, with tangent space:

$T_{(x,z)}(X\times_{Y}Z)=\\{(v,w)\in T_{x}X\times
T_{z}Z\mid\textrm{d}f_{x}(v)=\textrm{d}g_{z}(w)\\},$

and tangent cone:

$\displaystyle C_{(x,z)}(X\times_{Y}Z)$ $\displaystyle=\\{(v,w)\in
C_{x}X\times C_{z}Z\mid\textrm{d}f_{x}(v)=\textrm{d}g_{z}(w)\\}$
$\displaystyle=\\{(v,w)\in T_{x}X\times
C_{z}Z\mid\textrm{d}f_{x}(v)=\textrm{d}g_{z}(w)\\}.$

The analogous statement holds when interchanging the roles of $X$ and $Z$.

###### Proof.

We leave it to the reader to derive property $a$ from Proposition B.15 and $b$
from $c$. To verify property $c$, let $g:Z\to Y$ be a smooth map from another
manifold with corners into $Y$. By Remark B.10 it is enough to show that every
point $(x,z)\in X\times_{Y}Z$ admits an open neighbourhood in $X\times_{Y}Z$
that is an embedded submanifold of $X\times Z$, with the prescribed tangent
space and tangent cone at $(x,z)$. To this end, let $(x,z)\in X\times_{Y}Z$
and for this $x\in X$ consider charts $(U,\chi)$ and $(V,\varphi)$ as in
Proposition B.15, such that:

$\displaystyle\chi(U)$
$\displaystyle=[0,\varepsilon[^{k}\times]-\varepsilon,\varepsilon[^{m-k}\times]-\varepsilon,\varepsilon[^{n-m},$
$\displaystyle\varphi(V)$
$\displaystyle=[0,\varepsilon[^{k}\times]-\varepsilon,\varepsilon[^{m-k},$

for some $\varepsilon>0$. Then $U\times_{Y}g^{-1}(V)$ is an open in
$X\times_{Y}Z$ around $(x,z)$ and the map:

$i:]-\varepsilon,\varepsilon[^{n-m}\times g^{-1}(V)\to X\times
Z,\quad(p,q)\mapsto(\chi^{-1}(\varphi(g(q)),p),q)$

is a smooth embedding (it is essentially a graph map, cf. Example B.8) with
image $U\times_{Y}g^{-1}(V)$. Therefore $U\times_{Y}g^{-1}(V)$ is an embedded
submanifold of $X\times Z$, with tangent space:

$\displaystyle T_{(x,z)}(U\times_{Y}g^{-1}(V))$
$\displaystyle=\textrm{d}i_{(0,z)}(T_{0}\mathbb{R}^{n-m}\oplus T_{z}Z),$
$\displaystyle=\\{(v,w)\in T_{x}X\times
T_{z}Z\mid\textrm{d}f_{x}(v)=\textrm{d}g_{z}(w)\\}$

and tangent cone:

$\displaystyle C_{(x,z)}(U\times_{Y}g^{-1}(V))$
$\displaystyle=\textrm{d}i_{(0,z)}(T_{0}\mathbb{R}^{n-m}\oplus C_{z}Z),$
$\displaystyle=\\{(v,w)\in C_{x}X\times
C_{z}Z\mid\textrm{d}f_{x}(v)=\textrm{d}g_{z}(w)\\},$
$\displaystyle=\\{(v,w)\in T_{x}X\times
C_{z}Z\mid\textrm{d}f_{x}(v)=\textrm{d}g_{z}(w)\\},$

where the last equality follows from (122) (applied to $g$) and the equality
$\textrm{d}f_{x}^{-1}(C_{f(x)}Y)=C_{x}X$. This concludes the proof of property
$c$. ∎

######  Example B.17.

Unlike for manifolds without corners, for the conclusion of Proposition
B.15$c$ to hold at a given point $x\in X$, it is not enough to require the
conditions in Definition B.13 just at that point. To see this, consider for
instance the map:

$f:[0,\infty[^{2}\to[0,\infty[^{2},\quad(x,y)\mapsto(x,y+x^{2}).$

This is a submersion that satisfies the tameness condition in Definition B.13
at the origin, but does not satisfy the conclusion of Proposition B.15$c$
there.

Now, we turn to Lie groupoids with corners.

######  Definition B.18 ([40]).

A (Hausdorff) Lie groupoid with corners $\mathcal{G}\rightrightarrows X$ is a
groupoid for which $\mathcal{G}$ and $X$ are manifolds with corners, the
source and target map are tame submersions and all structure maps are smooth
as maps between manifolds with corners.

Note here that, because the source and target of $\mathcal{G}$ are tame
submersions, by Corollary B.16$c$ the space of composable arrows
$\mathcal{G}^{(2)}$ is an embedded submanifold with corners of
$\mathcal{G}\times\mathcal{G}$, so that (as usual) the requirement for the
multiplication map of $\mathcal{G}$ to be smooth makes sense.

######  Remark B.19.

Let $\mathcal{G}\rightrightarrows X$ be a Lie groupoid with corners. In view
of Proposition B.15, for each integer $0\leq k\leq\text{dim}(X)$, the embedded
submanifold $\mathcal{G}_{k}$ of $\mathcal{G}$ (as in Example B.12) coincides
with both $s^{-1}(X_{k})$ and $t^{-1}(X_{k})$, so that $X_{k}$ is
$\mathcal{G}$-invariant and the structure maps of $\mathcal{G}$ restrict to
give $\mathcal{G}_{k}$ the structure of Lie groupoid without corners over
$X_{k}$. From this and the standard theory of Lie groupoids without corners we
conclude that for each $x\in X$ the following hold.

  * a)

The isotropy group $\mathcal{G}_{x}$ of $\mathcal{G}$ is an embedded
submanifold of $\mathcal{G}$ without corners and, as such, it is a Lie group.

  * b)

The source-fiber $s^{-1}(x)$ is an embedded submanifold of $\mathcal{G}$
without corners, and the orbit $\mathcal{O}_{x}$ is an initial submanifold of
$X_{k}$ without corners, for $k=\textrm{depth}_{X}(x)$, with smooth manifold
structure uniquely determined by the fact that:

$t:s^{-1}(x)\to\mathcal{O}_{x},$

is a (right) principal $\mathcal{G}_{x}$-bundle.

As in the case without corners, we can define Morita equivalences.

######  Definition B.20.

Let $\mathcal{G}_{1}\rightrightarrows X_{1}$ and
$\mathcal{G}_{2}\rightrightarrows X_{2}$ be Lie groupoid with corners. A
Morita equivalence from $\mathcal{G}_{1}$ to $\mathcal{G}_{2}$ is a principal
$(\mathcal{G}_{1},\mathcal{G}_{2})$-bibundle $(P,\alpha_{1},\alpha_{2})$. This
consists of:

  * •

A manifold with corners $P$ with two surjective tame submersions
$\alpha_{i}:P\to X_{i}$.

  * •

A smooth left action of $\mathcal{G}_{1}$ along $\alpha_{1}$ that is free and
the orbits of which coincide with the $\alpha_{2}$-fibers.

  * •

A smooth right action of $\mathcal{G}_{2}$ along $\alpha_{2}$ that is free and
the orbits of which coincide with the $\alpha_{1}$-fibers.

Furthermore, the two actions are required to commute.

Here, smoothness of the actions means that the action maps
$\mathcal{G}_{1}\times_{X_{1}}P\to P$ and $P\times_{X_{2}}\mathcal{G}_{2}\to
P$ are smooth as maps between manifolds with corners (which makes sense in
view of Corollary B.16$c$). As in the case without corners, the following
holds.

######  Proposition B.21.

The respective conditions on the left and right action above are equivalent to
the requirement that the respective maps:

(123) $\displaystyle\mathcal{G}_{1}\times_{X_{1}}P$ $\displaystyle\to
P\times_{X_{2}}P,\quad(g,p)\mapsto(g\cdot p,p),$ (124) $\displaystyle
P\times_{X_{2}}\mathcal{G}_{2}$ $\displaystyle\to
P\times_{X_{1}}P,\quad(p,g)\mapsto(p,p\cdot g),$

are well-defined diffeomorphisms of manifolds with corners.

###### Proof.

The respective conditions on the left and right action above mean that the
respective maps (123) and (124) are well-defined, smooth and bijective. So, we
ought to show that if (123) respectively (124) is a smooth bijection, then it
is in fact a diffeomorphism. In view of Corollary B.16$a$ it is enough to show
for each of the respective maps that if it is a smooth bijection, then at
every point its differential is bijective and it is tame. Bijectivity of the
differential follows as in the case without corners and tameness is immediate
from the second description of the tangent cone in Corollary B.16$c$. ∎

As in the case without corners, Morita equivalences can be adapted to the
(pre-)symplectic setting [49, 50, 5]. Explicitly:

######  Definition B.22.

A pre-symplectic form $\omega$ on a manifold with corners $P$ is a closed
differential $2$-form. Such a pre-symplectic form is called symplectic if it
is non-degenerate. We call $(P,\omega)$ a (pre-)symplectic manifold with
corners. A pre-symplectic groupoid with corners
$(\mathcal{G},\Omega)\rightrightarrows X$ is a Lie groupoid with corners
equipped with a pre-symplectic form $\Omega$ on $\mathcal{G}$, satisfying:

  * i)

$\text{dim}(\mathcal{G})=2\text{dim}(X)$,

  * ii)

the form $\Omega$ is multiplicative, in the sense that:

$m^{*}\Omega=(\textrm{pr}_{1})^{*}\Omega+(\textrm{pr}_{2})^{*}\Omega,$

where we denote by:

$m,\textrm{pr}_{1},\textrm{pr}_{2}:\mathcal{G}^{(2)}\to\mathcal{G}$

the multiplication and projection maps from the space of composable arrows
$\mathcal{G}^{(2)}$ to $\mathcal{G}$,

  * iii)

the form $\Omega$ satisfies the non-degeneracy condition:

$\textrm{Ker}(\Omega)_{1_{x}}\cap\textrm{Ker}(\textrm{d}s)_{1_{x}}\cap\textrm{Ker}(\textrm{d}t)_{1_{x}}=0,\quad\forall
x\in X.$

This is called a symplectic groupoid with corners if $\Omega$ is symplectic. A
Morita equivalence $(P,\omega_{P},\alpha_{1},\alpha_{2})$ between
(pre-)symplectic groupoids with corners
$(\mathcal{G}_{1},\Omega_{1})\rightrightarrows X_{1}$ and
$(\mathcal{G}_{2},\Omega_{2})\rightrightarrows X_{2}$ consists of:

  * •

a (pre-)symplectic manifold with corners $(P,\omega_{P})$,

  * •

a Morita equivalence $(P,\alpha_{1},\alpha_{2})$ between the underlying Lie
groupoids with corners, with the additional property that both actions are
Hamiltonian, in the sense that:

$(m^{L}_{P})^{*}\omega_{P}=(\textrm{pr}_{\mathcal{G}_{1}})^{*}\Omega_{1}+(\textrm{pr}_{P}^{L})^{*}\omega_{P}\quad\&\quad(m_{P}^{R})^{*}\omega_{P}=(\textrm{pr}_{\mathcal{G}_{2}})^{*}\Omega_{2}+(\textrm{pr}_{P}^{R})^{*}\omega_{P}$

where we denote by:

$\displaystyle m_{P}^{L},\textrm{pr}_{P}^{L}:\mathcal{G}_{1}\times_{X_{1}}P\to
P,\quad\textrm{pr}_{\mathcal{G}_{1}}:\mathcal{G}_{1}\times_{X_{1}}P\to\mathcal{G}_{1},$
$\displaystyle m_{P}^{R},\textrm{pr}_{P}^{R}:P\times_{X_{2}}\mathcal{G}_{2}\to
P,\quad\textrm{pr}_{\mathcal{G}_{2}}:P\times_{X_{2}}\mathcal{G}_{2}\to\mathcal{G}_{2},$

the maps defining the actions and the projections onto $P$, $\mathcal{G}_{1}$
and $\mathcal{G}_{2}$.

######  Remark B.23.

A pre-symplectic Morita equivalence $((P,\omega_{P}),\alpha_{1},\alpha_{2})$
between two symplectic groupoids with corners is automatically symplectic.

######  Proposition B.24.

Morita equivalence between Lie or (pre-)symplectic groupoids with corners is
an equivalence relation.

###### Proof.

This proposition follows from the observation that, as in the case without
corners, for Lie and pre-symplectic groupoids with corners there is the
identity Morita equivalence and Morita equivalences can be inverted and
composed. The only extra technicality arising here is in the construction of
composition of two Morita equivalences, for it involves quotients by actions
of Lie groupoids with corners. To be more precise, suppose that we are given
two Morita equivalences between Lie groupoids with corners:

$\mathcal{G}_{1}$$X_{1}$$P$$X_{2}$$\mathcal{G}_{2}$$\mathcal{G}_{2}$$X_{2}$$Q$$X_{3}$$\mathcal{G}_{3}$$\quad\text{
}\alpha_{1}$$\alpha_{2}$ $\quad\text{ }\beta_{2}$$\beta_{3}$

Consider the induced left anti-diagonal $\mathcal{G}_{2}$-action along
$\alpha_{2}\circ\textrm{pr}_{P}:P\times_{X_{2}}Q\to X_{2}$. We will show that
the topological quotient space (which is second countable and Hausdorff):

(125) $P\ast_{\mathcal{G}_{2}}Q:=\frac{P\times_{X_{2}}Q}{\mathcal{G}_{2}}$

admits a unique smooth structure with corners with respect to which the
quotient map:

(126) $P\times_{X_{2}}Q\to P\ast_{\mathcal{G}_{2}}Q$

is a tame submersion. As for Lie groupoids without corners, one can then
define the composite Morita equivalence:

$\mathcal{G}_{1}$$X_{1}$$P\ast_{\mathcal{G}_{2}}Q$$X_{3}$$\mathcal{G}_{3}$$\quad\quad\quad{}\text{
}\underline{\alpha}_{1}\circ\underline{\textrm{pr}}_{P}$$\underline{\beta}_{3}\circ\underline{\textrm{pr}}_{Q}$

Moreover, if the given groupoids and Morita equivalences are (pre-)symplectic
with corners, with (pre-)symplectic forms $\omega_{P}$ and $\omega_{Q}$, then
as for (pre-)symplectic groupoids without corners (see [49, 50]) the form
$\omega_{P}\oplus\omega_{Q}$ on $P\times_{X_{2}}Q$ descends to a
(pre-)symplectic form on $P\ast_{\mathcal{G}_{2}}Q$ that makes the composite
Morita equivalence (pre-)symplectic.

To prove that (125) indeed admits a unique smooth structure with corners with
respect to which (126) is a tame submersion, we give an adaptation of the
proof of [45, Lemma B.1.4] (in fact, we believe that Lie groupoids with
corners fall into the much more general framework developed in that thesis).
Uniqueness follows from Corollary B.16$a$. By this uniqueness, to prove
existence it is enough to show that every $[p,q]\in P\ast_{\mathcal{G}_{2}}Q$
admits an open neighbourhood with a smooth structure with corners, with
respect to which the restriction of (126) is a tame submersion. To this end,
let $[p,q]\in P\ast_{\mathcal{G}_{2}}Q$. By Corollary B.16$a$ there is a
smooth local section $\sigma:U\to Q$ of $\beta_{3}$, defined on an open
neighbourhood $U$ of $\beta_{3}(q)$ in $X_{3}$, that maps $\beta_{3}(q)$ to
$q$. This induces a diffeomorphism:

$\Phi_{\sigma}:\mathcal{G}_{2}\times_{X_{2}}U\to\beta_{3}^{-1}(U),\quad(g,x)\mapsto
g\cdot\sigma(x),$

where the fiber-product $\mathcal{G}_{2}\times_{X_{2}}U$ is taken with respect
to the source-map of $\mathcal{G}_{2}$ and the map $\beta_{2}\circ\sigma:U\to
X_{2}$. To see that the inverse of this map is indeed smooth, consider the
division map:

$Q\times_{M_{3}}Q\to\mathcal{G}_{2},\quad(q_{1},q_{2})\mapsto[q_{1}:q_{2}],$

that assigns the unique element of $\mathcal{G}_{2}$ satisfying
$[q_{1}:q_{2}]\cdot q_{2}=q_{1}$. This is smooth, as a consequence of
Proposition B.21. Therefore, so is $\Phi_{\sigma}^{-1}$, for it is given by:

$\Phi_{\sigma}^{-1}:\beta_{3}^{-1}(U)\to\mathcal{G}_{2}\times_{X_{2}}U,\quad
q\mapsto([q:(\sigma\circ\beta_{3})(q)],\beta_{3}(q)).$

Now consider the composition:

(127)
$P\times_{X_{2}}\beta_{3}^{-1}(U)\xrightarrow{\textrm{Id}_{P}\times\Phi_{\sigma}^{-1}}P\times_{X_{2}}\mathcal{G}_{2}\times_{X_{2}}U\xrightarrow{(\ref{moreqrightprinbunmap})\times\textrm{Id}_{U}}P\times_{X_{1}}P\times_{X_{2}}U\xrightarrow{\textrm{pr}_{P,2}\times\textrm{Id}_{U}}P\times_{X_{2}}U,$

where the last fiber-product is taken with respect to $\alpha_{2}:P\to X_{2}$
and $\beta_{2}\circ\sigma:U\to X_{2}$. This being a composition of two
diffeomorphisms and a tame submersion, the composite (127) is a tame
submersion. It factors through a homeomorphism from
$P\ast_{\mathcal{G}_{2}}\beta_{3}^{-1}(U)$ to $P\times_{X_{2}}U$, with
inverse:

(128) $P\times_{X_{2}}U\to
P\ast_{\mathcal{G}_{2}}\beta_{3}^{-1}(U),\quad(p,x)\mapsto[p,\sigma(x)].$

Since the composite (127) is a tame submersion, this homeomorphism induces a
smooth structure with corners on the open
$P\ast_{\mathcal{G}_{2}}\beta_{3}^{-1}(U)$ around $[p,q]$, with respect to
which the restriction of (126) to $P\times_{X_{2}}\beta_{3}^{-1}(U)$ is a tame
submersion, as was to be constructed. ∎

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