Monodromy and faithful representability of Lie groupoids

TL;DR

This paper introduces a monodromy group associated with topological groupoids and demonstrates its utility in determining the existence of faithful representations, establishing invariance under Morita equivalence.

Contribution

It constructs a monodromy group for topological groupoids and shows its invariance under Morita equivalence, providing a new tool to analyze faithful representability.

Findings

Monodromy groups are constructed for topological groupoids.

Morita equivalent groupoids share the same monodromy groups.

Monodromy groups can test for the absence of faithful representations.

Abstract

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks faithful representations.

Equations

$$\partial :{\pi }_1(\mathcal{G};x)\to {\pi }_0(\mathrm{Aut}({\mathrm{K}}_x(\mathcal{G}))),$$

$$\mathrm{Mon}(\mathcal{G};\rho )$$

$$\chi \circ \mu =\chi$$

$${\pi }_0(\mathrm{Mon}(\mathcal{G};{\mathrm{id}}_{{\mathrm{K}}_x(\mathcal{G})})).$$

$$P{|}_{{\mathcal{H}}^{\prime }}=P{|}_{{\mathcal{H}}_0^{\prime }}={\pi }^{-1}({\mathcal{H}}_0^{\prime })$$

$${\gamma }_0,{\gamma }_1,\dots ,{\gamma }_n:[0,1]\to {\mathcal{H}}_0^{\prime }$$

$${\varsigma }_0\circ {\gamma }_0,{\varsigma }_1\circ {\gamma }_1,\dots ,{\varsigma }_n\circ {\gamma }_n$$

$$ϵ\circ {\varsigma }_0\circ {\gamma }_0,ϵ\circ {\varsigma }_1\circ {\gamma }_1,\dots ,ϵ\circ {\varsigma }_n\circ {\gamma }_n$$

$$h_k({\varsigma }_{k-1}({\gamma }_{k-1}(1)))={\varsigma }_k({\gamma }_k(0))g_k.$$

$${\mathsf{Gpd}}_{\ast }=(\ast \rightrightarrows \ast )\downarrow \mathsf{Gpd}$$

$$r:\mathcal{G}\to \mathrm{GL}(\mathsf{V})$$

$${\mathrm{c}}_g(h)=gh{g}^{-1}.$$

$$\mathrm{hom}(K,\mathcal{G})$$

$$(g\phi )(k)=({\mathrm{c}}_g\circ \phi )(k)=g\phi (k)g^{-1}$$

$${\mathcal{G}}^K=\mathcal{G}⋉\mathrm{hom}(K,\mathcal{G}),$$

$$\mathrm{hom}(K,\mathcal{G};\rho )$$

$$\mathrm{Pc}({\mathcal{G}}^K;\rho )=\mathcal{G}⋉\mathrm{hom}(K,\mathcal{G};\rho ).$$

$$\mathrm{hom}(K,\mathcal{G})\times \mathrm{hom}(L,K)\to \mathrm{hom}(L,\mathcal{G}).$$

$$\mathrm{hom}(K,\mathcal{G};\rho )\circ \alpha =\mathrm{hom}(L,\mathcal{G};\rho \circ \alpha ).$$

$$\mathrm{Mon}(\mathcal{G};\rho )=\{\mu \in \mathrm{Aut}(K)|\mathrm{hom}(K,\mathcal{G};\rho )\circ \mu =\mathrm{hom}(K,\mathcal{G};\rho )\}$$

$$\Sigma (\mathcal{G};\rho )=\omega (\mathrm{hom}(K,\mathcal{G};\rho )).$$

$$\mathrm{hom}(L,\mathcal{G};\rho \circ \alpha )\circ ({\alpha }^{-1}\circ \mu \circ \alpha )$$

$$=\mathrm{hom}(K,\mathcal{G};\rho )\circ \alpha =\mathrm{hom}(L,\mathcal{G};\rho \circ \alpha ),$$

$$\mathrm{hom}(K,\mathcal{G};\rho )=\mathrm{hom}(K,\mathcal{G};{\mathrm{c}}_{\rho (k)}\circ \rho )=\mathrm{hom}(K,\mathcal{G};\rho \circ {\mathrm{c}}_k)=\mathrm{hom}(K,\mathcal{G};\rho )\circ {\mathrm{c}}_k$$

$$P^K=P{\times }_{{\mathcal{H}}_0}^{\pi ,\omega }\mathrm{hom}(K,\mathcal{H})$$

$$(-{)}^K:\mathsf{GPD}\to \mathsf{GPD}.$$

$$P^K\otimes ⟨{\omega }_{\mathcal{G}}⟩\cong ⟨{\omega }_{\mathcal{H}}⟩\otimes P.$$

$$P_p\circ \sigma \circ \mu =\rho \circ \mu ,$$

$$\mathrm{Mon}(\mathcal{H};\sigma )\subset \mathrm{Mon}(\mathcal{G};P_p\circ \sigma ).$$

$$\mathrm{Mon}(\mathcal{H};\sigma )=\mathrm{Mon}(\mathcal{G};\rho ).$$

$${\varphi }^K(h,\vartheta )=(\varphi (h),\mathrm{hom}(K,\varphi )(\vartheta ))=(\varphi (h),\varphi \circ \vartheta )$$

Full text

This work was supported in part by the Slovenian Research Agency (ARRS)

1. Introduction

As we know from Peter-Weyl theorem, any compact Lie group has a faithful finite dimensional representation. It is a natural and a long-standing question how far can this result be extended to Lie groupoids. The structure of a Lie groupoid generalizes that of a Lie group, with examples including smooth manifolds, smooth bundles of Lie groups, action groupoids associated to smooth actions of Lie groups on smooth manifolds, holonomy groupoids of foliations, and orbifold atlases. Among Lie groupoids, proper Lie groupoids are considered to play the role of compact Lie groups. Action groupoids of proper smooth actions and orbifold atlases are all proper Lie groupoids. In fact, an orbifold can be equivalently described as a Morita equivalence class of proper étale Lie groupoids [9].

It turns out that a proper Lie groupoid admits a faithful representation on a vector bundle of finite rank if, and only if, it is Morita equivalent to the action groupoid of a smooth action of a compact Lie group on a smooth manifold (see, for example, [6, 17]). Effective orbifolds are all faithfully representable ([16], see also [4, 11]), but it is still an open question whether all non-effective orbifolds are faithfully representable as well. On the other hand, an example of a proper regular (non-étale) Lie groupoid which is not faithfully representable was found ([7], see also [17]). In [15], an obstruction to faithful representability of proper regular Lie groupoids was given and used to construct a large class of proper regular Lie groupoids which are not faithfully representable. This is based on the observation that the ineffective isotropy arrows $\mathord{\mathrm{K}}(\mathscr{G})$ of a proper regular Lie groupoid $\mathscr{G}$ form a locally trivial bundle of compact Lie groups over the manifold of objects of $\mathscr{G}$, and on an explicit computation of an associated long exact sequence of homotopy groups in which the obstruction can be found in the image of the boundary (monodromy) map

[TABLE]

for an object $x$ of $\mathscr{G}$. In this paper we generalize this obstruction to general Lie groupoids and construct examples of proper non-regular Lie groupoids which are not faithfully representable.

Let $K$ be a locally compact topological group. A homomorphism of topological groupoids $\rho:K\to\mathscr{G}$ is determined by an object $\omega(\rho)$ of $\mathscr{G}$ and a homomorphism of topological groups from $K$ to the isotropy group $\mathord{\mathrm{I}}_{\omega(\rho)}(\mathscr{G})$ of $\mathscr{G}$ at $\omega(\rho)$. The space $\hom(K,\mathscr{G})$ of all homomorphisms of topological groupoids from $K$ to $\mathscr{G}$ has a natural left $\mathscr{G}$-action along the map $\omega$ given by conjugation and a natural right action of the group $\mathord{\mathrm{Aut}}(K)$ of automorphisms of $K$ given by composition. For any $\rho\in\hom(K,\mathscr{G})$ we denote by $\hom(K,\mathscr{G};\rho)$ the space of objects of the path-component of the action groupoid $\mathscr{G}\ltimes\hom(K,\mathscr{G})$ which includes $\rho$, and define the monodromy group

[TABLE]

to be the subgroup of $\mathord{\mathrm{Aut}}(K)$ consisting of all automorphisms $\mu\in\mathord{\mathrm{Aut}}(K)$ which satisfy $\hom(K,\mathscr{G};\rho)\circ\mu=\hom(K,\mathscr{G};\rho)$. We show that the monodromy group includes the subgroup $\mathord{\mathrm{Inn}}(K)$ of inner automorphisms of $K$ and that Morita equivalent topological groupoids have the same monodromy groups.

If $K$ is a compact Lie group, if $\mathscr{G}$ is a faithfully representable Lie groupoid and if $\rho\in\hom(K,\mathscr{G})$ is a monomorphism, we prove that there exists a faithful character $\chi$ of the Lie group $K$ such that

[TABLE]

for any $\mu\in\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ (Theorem 12). This theorem shows that some automorphisms in a monodromy group of $\mathscr{G}$ can provide an obstruction to faithful representability of $\mathscr{G}$ (Example 13). The theorem also generalizes the obstruction given in [15], because for a proper regular Lie groupoid $\mathscr{G}$ with an object $x$, the image of the monodromy map $\partial:\pi_{1}(\mathscr{G};x)\to\pi_{0}(\mathord{\mathrm{Aut}}(\mathord{\mathrm{K}}_{x}(\mathscr{G})))$ is exactly the group

[TABLE]

We show that for a special class of locally trivial bundles of compact Lie groups, Theorem 12 provides not only necessary but also sufficient condition for faithful representability (Theorem 15). Finally, we use the linearization theorem for a proper Lie groupoid $\mathscr{G}$ to study the action Lie groupoid $\mathscr{G}\ltimes\hom(K,\mathscr{G};\rho)$ and the image $\Sigma(\mathscr{G};\rho)=\omega(\hom(K,\mathscr{G};\rho))$ locally. We show how closely the set $\Sigma(\mathscr{G};\rho)$ is related to the fixed point set of the action of the isotropy group $\mathord{\mathrm{I}}_{\omega(\rho)}(\mathscr{G})$ on a local slice.

2. Groupoids and representations

For the convenience of the reader and to fix the notations, we shall use this section to recall some basic definitions and facts about Lie groupoids and their representations. Detailed presentation with examples can be found in [11, 12, 8, 6].

2.1. Topological and Lie groupoids

A groupoid is a small category in which every arrow is invertible. In particular, a groupoid $\mathscr{G}$ consists of a set $\mathscr{G}_{1}$ of arrows and a set $\mathscr{G}_{0}$ of objects, any arrow $g\in\mathscr{G}_{1}$ has its source $\mathord{\mathrm{s}}(g)\in\mathscr{G}_{0}$ and its target $\mathord{\mathrm{t}}(g)\in\mathscr{G}_{0}$, the partial multiplication $\mathord{\mathrm{mlt}}(g^{\prime},g)=g^{\prime}g$ is defined for any pair of arrows $g^{\prime},g\in\mathscr{G}_{1}$ with $\mathord{\mathrm{s}}(g^{\prime})=\mathord{\mathrm{t}}(g)$, for any object $x\in\mathscr{G}_{0}$ there is the identity arrow $\mathord{\mathrm{uni}}(x)=1_{x}\in\mathscr{G}_{1}$ and any arrow $g\in\mathscr{G}_{1}$ has its inverse arrow $\mathord{\mathrm{inv}}(g)=g^{-1}\in\mathscr{G}_{1}$. We denote such a groupoid $\mathscr{G}$ also by $(\mathscr{G}_{1}\rightrightarrows\mathscr{G}_{0})$. The above mentioned maps $\mathord{\mathrm{s}},\mathord{\mathrm{t}}:\mathscr{G}_{1}\to\mathscr{G}_{0}$, $\mathord{\mathrm{mlt}}:\mathscr{G}_{1}\times^{\mathord{\mathrm{s}},\mathord{\mathrm{t}}}_{\mathscr{G}_{0}}\mathscr{G}_{1}\to\mathscr{G}_{1}$, $\mathord{\mathrm{uni}}:\mathscr{G}_{0}\to\mathscr{G}_{1}$ and $\mathord{\mathrm{inv}}:\mathscr{G}_{1}\to\mathscr{G}_{1}$ are called the structure maps of the groupoid $\mathscr{G}$. Note that the multiplication map $\mathord{\mathrm{mlt}}$ is defined on the pull-back with respect to the maps $\mathord{\mathrm{s}}$ and $\mathord{\mathrm{t}}$. We say that a groupoid $\mathscr{G}$ is a groupoid over $\mathscr{G}_{0}$ and we often denote the set $\mathscr{G}_{1}$ simply by $\mathscr{G}$. We write $\mathscr{G}(x,x^{\prime})$ for the set of arrows of $\mathscr{G}$ with source $x$ and target $x^{\prime}$, while $\mathscr{G}(x,\,\mbox{-}\,)=\mathord{\mathrm{s}}^{-1}(\{x\})$ and $\mathscr{G}(\,\mbox{-}\,,x^{\prime})=\mathord{\mathrm{t}}^{-1}(\{x^{\prime}\})$, for any $x,x^{\prime}\in\mathscr{G}_{0}$. The set $\mathscr{G}(x,x)=\mathord{\mathrm{I}}_{x}(\mathscr{G})$ is a group, called the isotropy group of $\mathscr{G}$ at $x$.

A topological groupoid is a groupoid $\mathscr{G}$, together with a topology on the set $\mathscr{G}_{1}$ and a topology on the set $\mathscr{G}_{0}$ such that all the structure maps are continuous. With continuous functors as homomorphisms, topological groupoids form a category $\mathord{\mathsf{Gpd}}$.

To make the terminology precise, we will say that a $C^{\infty}$-space is a topological space with a $C^{\infty}$-structure of constant dimension (given by a maximal $C^{\infty}$-atlas). In particular, a $C^{\infty}$-space is locally Euclidean of constant dimension without boundary and may not be Hausdorff. A smooth manifold is a $C^{\infty}$-space which is Hausdorff and second-countable. (If one prefers to allow manifolds and Lie groups to have uncountably many components - as some authors do - one could relax the assumption of second-countability of smooth manifolds by the condition of paracompactness.)

A $C^{\infty}$-groupoid is a topological groupoid $\mathscr{G}$, together with a $C^{\infty}$-structure on the space $\mathscr{G}_{1}$ and a $C^{\infty}$-structure on the space $\mathscr{G}_{0}$ such that the source map $\mathord{\mathrm{s}}$ is a surjective submersion and all the structure maps of $\mathscr{G}$ are smooth. A Lie groupoid is a $C^{\infty}$-groupoid $\mathscr{G}$ in which the space $\mathscr{G}_{0}$ and all the source-fibers $\mathscr{G}(x,\,\mbox{-}\,)$, $x\in\mathscr{G}_{0}$, are smooth manifolds. (In general theory, we do not assume that a Lie groupoid is a smooth manifold, in order to include many interesting examples.) The isotropy groups of a Lie groupoid are Lie groups. The $C^{\infty}$-groupoids form a category $\mathord{\mathsf{Gpd^{\infty}}}$ in which homomorphisms are smooth functors. The category $\mathord{\mathsf{LGpd}}$ of Lie groupoids is the corresponding full subcategory of $\mathord{\mathsf{Gpd^{\infty}}}$.

Example 1**.**

(1) Any smooth manifold $M$ can be viewed as a Lie groupoid $(M\rightrightarrows M)$, where both the source map and the target map are the identity. Any Lie group $K$ can be viewed as a Lie groupoid $(K\rightrightarrows\ast)$ over a one-point set.

(2) Let $\xi:\mathsf{V}\to M$ be a smooth complex vector bundle (of finite rank) over a smooth manifold $M$. The general linear Lie groupoid on $\mathsf{V}$ is the Lie groupoid $\mathord{\mathrm{GL}}(\mathsf{V})$ over $M$ of which the arrows with source $x\in M$ and target $x^{\prime}\in M$ are all linear isomorphisms from the fiber $\mathsf{V}_{x}=\xi^{-1}(x)$ to the fiber $\mathsf{V}_{x^{\prime}}=\xi^{-1}(x^{\prime})$. In the presence of a Hermitian metric on $\mathsf{V}$ we may consider also the unitary Lie groupoid $\mathord{\mathrm{U}}(\mathsf{V})$ over $M$, which is a closed subgroupoid of $\mathord{\mathrm{GL}}(\mathsf{V})$ consisting of all linear isometries between the fibers of $\mathsf{V}$.

(3) Let $\mathscr{G}$ be a Lie groupoid and let $U$ be an open subset of $\mathscr{G}_{0}$. Then the restriction $\mathscr{G}|_{U}$ is the Lie groupoid over $U$ which consists of all arrows of $\mathscr{G}$ with their source and their target in $U$. In fact, the restriction $\mathscr{G}|_{U}$ is an open subgroupoid of the Lie groupoid $\mathscr{G}$.

2.2. Actions

A continuous right action of a topological groupoid $\mathscr{G}$ on a topological space $P$ along a continuous map $\epsilon:P\to\mathscr{G}_{0}$ is a continuous map $P\times_{\mathscr{G}_{0}}^{\epsilon,\mathord{\mathrm{t}}}\mathscr{G}_{1}\to P$, $(p,g)\mapsto pg$, which satisfies the usual properties of an action. To such a right $\mathscr{G}$-action we associate the action groupoid $P\rtimes\mathscr{G}=(P\times_{\mathscr{G}_{0}}^{\epsilon,\mathord{\mathrm{t}}}\mathscr{G}_{1}\rightrightarrows P)$, a topological groupoid in which the source map is the action, the target map is the projection and the partial multiplication is given by $(p^{\prime},g^{\prime})(p,g)=(p^{\prime},g^{\prime}g)$. We write $p\mathscr{G}=\{pg\;|\;g\in\mathscr{G}(\,\mbox{-}\,,\epsilon(p))\}\subset P$ for the orbit through the point $p\in P$ and $P/\mathscr{G}$ for the quotient space of orbits of the $\mathscr{G}$-action. We say that a subset $X$ of $P$ is $\mathscr{G}$-saturated if it equals its saturation $\mathord{\mathrm{sat}}_{\mathscr{G}}(X)$, which is the union of all of those orbits of the $\mathscr{G}$-action that intersect $X$.

Similarly we define the notion of a left continuous action of a topological groupoid $\mathscr{H}$ on $P$ along a continuous map $\pi:P\to\mathscr{H}_{0}$, the associated action groupoid $\mathscr{H}\ltimes P$, the orbits $\mathscr{H}p$, the space of orbits $\mathscr{H}\backslash P$, the $\mathscr{H}$-saturated subsets of $P$ and the saturation $\mathord{\mathrm{sat}}_{\mathscr{H}}(Y)$ of a subset $Y$ of $P$.

We can also consider smooth actions of $C^{\infty}$-groupoids on $C^{\infty}$-spaces along smooth maps. The associated action groupoid is in this case a $C^{\infty}$-groupoid. The action groupoid associated to a Lie groupoid action on a smooth manifold is a Lie groupoid.

Note that any topological groupoid $\mathscr{G}$ acts continuously on $\mathscr{G}_{0}$ from the left along the target map and from the right along the source map. The orbit of $\mathscr{G}$ through a point $x\in\mathscr{G}_{0}$ is the orbit $\mathscr{G}x=x\,\mathscr{G}=\mathord{\mathrm{t}}(\mathord{\mathrm{s}}^{-1}(\{x\}))=\mathord{\mathrm{s}}(\mathord{\mathrm{t}}^{-1}(\{x\}))$. These actions are smooth if $\mathscr{G}$ is a $C^{\infty}$-groupoid. The orbits of a Lie groupoid $\mathscr{G}$ are immersed submanifolds of $\mathscr{G}_{0}$. There are natural isomorphisms (in the category $\mathord{\mathsf{Gpd}}$, or $\mathord{\mathsf{Gpd^{\infty}}}$) between $\mathscr{G}$, $\mathscr{G}\ltimes\mathscr{G}_{0}$ and $\mathscr{G}_{0}\rtimes\mathscr{G}$.

2.3. Morita maps

Let $\mathscr{G}$ and $\mathscr{H}$ be topological groupoids. A topological principal $\mathscr{G}$-bundle over $\mathscr{H}$ is a topological space $P$, together with a continuous right action of $\mathscr{G}$ along a continuous map $\epsilon$ and a continuous left action of $\mathscr{H}$ along a surjective continuous map $\pi$ with local sections such that $\pi(pg)=\pi(p)$, $\epsilon(hp)=\epsilon(p)$ and $h(pg)=(hp)g$, for all $p\in P$, $h\in\mathscr{H}(\pi(p),\,\mbox{-}\,)$ and $g\in\mathscr{G}(\,\mbox{-}\,,\epsilon(p))$, and such that the map $P\times_{\mathscr{G}_{0}}^{\epsilon,\mathord{\mathrm{t}}}\mathscr{G}_{1}\to P\times_{\mathscr{H}_{0}}^{\pi,\pi}P$, $(p,g)\mapsto(p,pg)$, is a homeomorphism.

As an important example, for any homomorphism $\phi:\mathscr{H}\to\mathscr{G}$ there is the associated principal $\mathscr{G}$-bundle $\langle\phi\rangle=\mathscr{H}_{0}\times_{\mathscr{G}_{0}}^{\phi,\mathord{\mathrm{t}}}\mathscr{G}_{1}$ over $\mathscr{H}$ equipped with the natural actions. For any principal $\mathscr{G}$-bundle $P$ over $\mathscr{H}$, any point $p\in P$ induces a homomorphism of topological groups $P_{p}:\mathord{\mathrm{I}}_{\pi(p)}(\mathscr{H})\to\mathord{\mathrm{I}}_{\epsilon(p)}(\mathscr{G})$, which is determined by the equation $hp=pP_{p}(h)$, $h\in\mathord{\mathrm{I}}_{\pi(p)}(\mathscr{H})$.

Let $P$ be a topological principal $\mathscr{G}$-bundle over $\mathscr{H}$ and let $Q$ be a topological principal $\mathscr{H}$-bundle over $\mathscr{K}$, where $\mathscr{K}$ is another topological groupoid. The tensor product $Q\otimes_{\mathscr{H}}P$ of $Q$ and $P$ is the space of orbits of the pull-back $Q\times_{\mathscr{H}_{0}}^{\epsilon,\pi}P$ with respect to the diagonal action of $\mathscr{H}$. With the natural actions, the tensor product $Q\otimes_{\mathscr{H}}P$ is a topological principal $\mathscr{G}$-bundle over $\mathscr{K}$.

An isomorphism between topological principal $\mathscr{G}$-bundles $P,P^{\prime}$ over $\mathscr{H}$ is a homeomorphism $P\to P^{\prime}$ which is equivariant for both the left and the right action. A topological Morita map from $\mathscr{H}$ to $\mathscr{G}$ is an isomorphism class of topological principal $\mathscr{G}$-bundles over $\mathscr{H}$. Such Morita maps form the Morita category $\mathord{\mathsf{GPD}}$ of topological groupoids, with composition induced by the tensor product of topological principal bundles.

For $C^{\infty}$-groupoids $\mathscr{G}$ and $\mathscr{H}$, a topological principal $\mathscr{G}$-bundle $P$ over $\mathscr{H}$ is smooth if it is equipped with a $C^{\infty}$-structure, both actions on $P$ are smooth, the map $\pi:P\to\mathscr{H}_{0}$ is a surjective submersion and the map $P\times_{\mathscr{G}_{0}}^{\epsilon,\mathord{\mathrm{t}}}\mathscr{G}_{1}\to P\times_{\mathscr{H}_{0}}^{\pi,\pi}P$, $(p,g)\mapsto(p,pg)$, is a diffeomorphism. The tensor product of two smooth principal bundles is again a smooth principal bundle. An isomorphism between smooth principal $\mathscr{G}$-bundles over $\mathscr{H}$ is a diffeomorphism which is equivariant for both actions. A smooth Morita map from $\mathscr{H}$ to $\mathscr{G}$ is an isomorphism class of smooth principal $\mathscr{G}$-bundles over $\mathscr{H}$. Such smooth Morita maps form the (smooth) Morita category $\mathord{\mathsf{GPD^{\infty}}}$ of $C^{\infty}$-groupoids, with composition induced by the tensor product of smooth principal bundles. The Morita category $\mathord{\mathsf{LGPD}}$ of Lie groupoids is the corresponding full subcategory of the category $\mathord{\mathsf{GPD^{\infty}}}$.

The isomorphisms in the Morita category (topological or smooth) are called Morita equivalences. Two topological (or $C^{\infty}$-) groupoids are Morita equivalent if they are isomorphic in the category $\mathord{\mathsf{GPD}}$ (respectively $\mathord{\mathsf{GPD^{\infty}}}$). A continuous (or smooth) functor $\phi:\mathscr{H}\to\mathscr{G}$ between topological (respectively $C^{\infty}$-) groupoids is a weak equivalence if $\langle\phi\rangle$ represents a Morita equivalence in the category $\mathord{\mathsf{GPD}}$ (respectively $\mathord{\mathsf{GPD^{\infty}}}$). For details, see [14, 12].

Example 2**.**

(1) The Lie groupoid $\mathord{\mathrm{GL}}(\mathsf{V})$, associated to a smooth complex vector bundle $\xi:\mathsf{V}\to M$ over a smooth manifold $M$, is Morita equivalent to the Lie group $\mathord{\mathrm{GL}}(\mathsf{V}_{x})$, for any $x\in M$. The natural inclusion $\mathord{\mathrm{GL}}(\mathsf{V}_{x})\to\mathord{\mathrm{GL}}(\mathsf{V})$ is a weak equivalence.

(2) Let $\mathscr{G}$ be a Lie groupoid and let $(U_{i})_{i\in I}$ be a countable open cover of $\mathscr{G}_{0}$. The natural map $f:\coprod_{i}U_{i}\to\mathscr{G}_{0}$ is a surjective local diffeomorphism, and we may define the pull-back Lie groupoid $f^{\ast}\mathscr{G}$ over $\coprod_{i}U_{i}$ with arrows given by the pull-back $(f^{\ast}\mathscr{G})_{1}=(\coprod_{i}U_{i}\times\coprod_{i}U_{i})\times_{\mathscr{G}_{0}\times\mathscr{G}_{0}}^{f\times f,(\mathord{\mathrm{s}},\mathord{\mathrm{t}})}\mathscr{G}_{1}$. The projection $(f^{\ast}\mathscr{G})_{1}\to\mathscr{G}_{1}$ gives us a smooth functor $f^{\ast}\mathscr{G}\to\mathscr{G}$ over $f$ which is a weak equivalence.

2.4. Path-components

Let $\mathscr{G}$ be a topological groupoid. A path-component of $\mathscr{G}$ is a full topological subgroupoid $\mathscr{G}^{\prime}$ of $\mathscr{G}$ such that $\mathscr{G}^{\prime}_{0}$ is a minimal non-empty union of path-components of the space $\mathscr{G}_{0}$ which is $\mathscr{G}$-saturated. Path-components of $\mathscr{G}$ form a partition of $\mathscr{G}$. A topological groupoid is path-connected if it is empty or it has only one path-component. Any path-component of a topological groupoid $\mathscr{G}$ is of course a path-connected topological groupoid and any path-connected topological subgroupoid of $\mathscr{G}$ is a subset of a path-component of $\mathscr{G}$.

If $\mathscr{H}^{\prime}$ is a path-connected topological subgroupoid of a topological groupoid $\mathscr{H}$ and $\phi:\mathscr{H}\to\mathscr{G}$ is a continuous functor, then the image $\phi(\mathscr{H}^{\prime})$ is a path-connected topological subgroupoid of $\mathscr{G}$. An analogous result holds for Morita maps as well:

Proposition 3**.**

Let $\mathscr{G}$ and $\mathscr{H}$ be topological groupoids and let $P$ be a topological principal $\mathscr{G}$-bundle over $\mathscr{H}$ with the structure maps $\pi:P\to\mathscr{H}_{0}$ and $\epsilon:P\to\mathscr{G}_{0}$. If $\mathscr{H}^{\prime}$ is a path-connected topological subgroupoid of $\mathscr{H}$, then the set $\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))$ is $\mathscr{G}$-saturated and the restriction $\mathscr{G}|_{\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))}$ is a path-connected full topological subgroupoid of $\mathscr{G}$. Moreover, the space

[TABLE]

is a topological principal $\mathscr{G}$-bundle over $\mathscr{H}^{\prime}$, a topological principal $\mathscr{G}|_{\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))}$-bundle over $\mathscr{H}^{\prime}$ and also a topological principal $\mathscr{G}^{\prime}$-bundle over $\mathscr{H}^{\prime}$, where $\mathscr{G}^{\prime}$ is the path-component of $\mathscr{G}$ with $\mathscr{G}|_{\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))}\subset\mathscr{G}^{\prime}$.

In particular, if $\mathscr{H}^{\prime}$ is a path-component of $\mathscr{H}$ and if $P$ represents a Morita equivalence form $\mathscr{H}$ to $\mathscr{G}$, then the restriction $P|_{\mathscr{H}^{\prime}}$ represents a Morita equivalence from $\mathscr{H}^{\prime}$ to $\mathscr{G}^{\prime}$.

Proof.

The subspace $\pi^{-1}(\mathscr{H}^{\prime}_{0})$ of $P$ is $\mathscr{G}$-saturated and $\epsilon:P\to\mathscr{G}_{0}$ is $\mathscr{G}$-equivariant, so the subspace $\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))$ of $\mathscr{G}_{0}$ is also $\mathscr{G}$-saturated. Take any two points $x,x^{\prime}\in\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))$. Choose $p,p^{\prime}\in\pi^{-1}(\mathscr{H}^{\prime}_{0})$ with $\epsilon(p)=x$ and $\epsilon(p^{\prime})=x^{\prime}$. The subgroupoid $\mathscr{H}^{\prime}$ is path-connected, so we can find continuous paths

[TABLE]

such that $\gamma_{0}(0)=\pi(p)$, $\gamma_{n}(1)=\pi(p^{\prime})$ and such that the points $\gamma_{k-1}(1)$ and $\gamma_{k}(0)$ are in the same $\mathscr{H}^{\prime}$-orbit for any $k=1,2,\ldots,n$. Since the map $\pi:P\to\mathscr{H}_{0}$ is surjective and has local sections, we may choose $\gamma_{0},\gamma_{1},\ldots,\gamma_{n}$ so that for any $k=0,1,\ldots,n$ the image of $\gamma_{k}$ lies inside an open subset $U_{k}$ of $\mathscr{H}_{0}$ on which there exists a local section $\varsigma_{k}:U_{k}\to P$ of $\pi$. We can choose the arrows $h_{1},\ldots,h_{n}\in\mathscr{H}^{\prime}$ such that $\mathord{\mathrm{s}}(h_{k})=\gamma_{k-1}(1)$ and $\mathord{\mathrm{t}}(h_{k})=\gamma_{k}(0)$ for any $k=1,2,\ldots,n$. Now

[TABLE]

are continuous paths in the space $\pi^{-1}(\mathscr{H}^{\prime}_{0})$, so

[TABLE]

are continuous paths in the space $\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))$.

There exist a unique $g_{0}\in\mathscr{G}(\epsilon(p),\epsilon(\varsigma_{0}(\gamma_{0}(0))))$ such that $p=\varsigma_{0}(\gamma_{0}(0))g_{0}$ and a unique $g_{n+1}\in\mathscr{G}(\epsilon(\varsigma_{n}(\gamma_{n}(1))),\epsilon(p^{\prime}))$ such that $\varsigma_{n}(\gamma_{n}(1))=p^{\prime}g_{n+1}$. Furthermore, for any $k=1,2,\ldots,n$ there exists a unique $g_{k}\in\mathscr{G}(\epsilon(\varsigma_{k-1}(\gamma_{k-1}(1))),\epsilon(\varsigma_{k}(\gamma_{k}(0))))$ such that

[TABLE]

The existence of the sequence of paths $\epsilon\circ\varsigma_{0}\circ\gamma_{0},\epsilon\circ\varsigma_{1}\circ\gamma_{1},\ldots,\epsilon\circ\varsigma_{n}\circ\gamma_{n}$ in $\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))$ therefore implies that the points $x=\epsilon(p)$ and $x^{\prime}=\epsilon(p^{\prime})$ are in the same path-component of the topological groupoid $\mathscr{G}|_{\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))}$. But since this is true for any points $x$ and $x^{\prime}$ in $\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))$, it follows that the topological groupoid $\mathscr{G}|_{\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))}$ is path-connected. The rest of the proposition is a direct consequence of the fact that both $\mathscr{G}|_{\epsilon(\pi^{-1}(\mathscr{H}^{\prime}_{0}))}$ and $\mathscr{G}^{\prime}$ are full topological subgroupoids of $\mathscr{G}$. ∎

A pointed topological groupoid is a topological groupoid $\mathscr{G}$ with a chosen point $x\in\mathscr{G}_{0}$. The point $x$ can be equivalently given by a homomorphism of topological groupoids $(\ast\rightrightarrows\ast)\to\mathscr{G}$, where $\ast$ denotes a fixed space with only one point. The category of pointed topological groupoids is the coslice category

[TABLE]

For any pointed topological groupoid $(\mathscr{G};x)$ denote by $\mathord{\mathrm{Pc}}(\mathscr{G};x)$ the path-component of $\mathscr{G}$ with $x\in\mathord{\mathrm{Pc}}(\mathscr{G};x)_{0}$. This definition clearly extends to a functor $\mathord{\mathrm{Pc}}:\mathord{\mathsf{Gpd}}_{\ast}\to\mathord{\mathsf{Gpd}}_{\ast}$.

Any Morita map $(\ast\rightrightarrows\ast)\to\mathscr{G}$ is also determined by a point $x\in\mathscr{G}_{0}$, but two points give the same Morita map if, and only if, they lie in the same $\mathscr{G}$-orbit. However, any $\mathscr{G}$-orbit is a subset of the space of objects of a path-component of $\mathscr{G}$. Together with Proposition 3 this implies that the functor $\mathord{\mathrm{Pc}}$ is well-defined on the coslice category of the Morita category, $\mathord{\mathrm{Pc}}:(\ast\rightrightarrows\ast)\downarrow\mathord{\mathsf{GPD}}\to(\ast\rightrightarrows\ast)\downarrow\mathord{\mathsf{GPD}}$.

Analogous results holds for smooth principal bundles; note that a path-component of a $C^{\infty}$-groupoid is an open $C^{\infty}$-subgroupoid.

2.5. Representations of Lie groupoids

Let $\mathscr{G}$ be a Lie groupoid. A representation of $\mathscr{G}$ (of finite rank) is a smooth complex vector bundle $\xi:\mathsf{V}\to\mathscr{G}_{0}$, together with a homomorphism of Lie groupoids

[TABLE]

which is the identity on objects. Sometimes we will denote such a representation simply by $\mathsf{V}$. Such a representation can be equivalently viewed as a smooth left action of $\mathscr{G}$ on $\mathsf{V}$, $(g,v)\mapsto gv=r(g)v$, along the map $\xi$, which is fiber-wise linear. Note that the fiber $\mathsf{V}_{x}$ of $\mathsf{V}$ is a representation of the Lie group $\mathord{\mathrm{I}}_{x}(\mathscr{G})$, for any $x\in\mathscr{G}_{0}$. A homomorphism between representations of $\mathscr{G}$ is a homomorphism of complex vector bundles over $\mathscr{G}_{0}$ which is $\mathscr{G}$-equivariant.

A unitary representation of $\mathscr{G}$ is a representation $r:\mathscr{G}\to\mathord{\mathrm{GL}}(\mathsf{V})$ of $\mathscr{G}$ such that $\mathsf{V}$ is equipped with a Hermitian metric and $r(\mathscr{G})\subset\mathord{\mathrm{U}}(\mathsf{V})$.

A representation $r:\mathscr{G}\to\mathord{\mathrm{GL}}(\mathsf{V})$ is faithful if it is injective. This is true if, and only if, the fiber $\mathsf{V}_{x}$ is a faithful representation of the Lie group $\mathord{\mathrm{I}}_{x}(\mathscr{G})$, for all $x\in\mathscr{G}_{0}$. A Lie groupoid is faithfully (unitary) representable if it admits a faithful (unitary) representation. A Lie groupoid is locally faithfully (unitary) representable if there exists an open cover $(U_{i})_{i\in I}$ of $\mathscr{G}_{0}$ such that the restriction $\mathscr{G}|_{U_{i}}$ is faithfully (unitary) representable for all $i\in I$. Note that any locally faithfully representable Lie groupoid is necessarily Hausdorff because $\mathord{\mathrm{GL}}(\mathsf{V})$ is Hausdorff.

Let $P$ be a smooth principal $\mathscr{G}$-bundle over another Lie groupoid $\mathscr{H}$. For any representation $\mathsf{V}$ of $\mathscr{G}$ there is the induced representation $P^{\ast}\mathsf{V}$ of $\mathscr{H}$. It is the quotient of the diagonal action of $\mathscr{G}$ on the pull-back vector bundle $P\times_{\mathscr{G}_{0}}^{\epsilon,\xi}\mathsf{V}$, with the obvious left action of $\mathscr{H}$. In particular, if $\phi:\mathscr{H}\to\mathscr{G}$ is a smooth functor, then $\langle\phi\rangle^{\ast}\mathsf{V}$ is given by the usual pull-back $\phi_{0}^{\ast}\mathsf{V}$. For another Lie groupoid $\mathscr{K}$ and a smooth principal $\mathscr{H}$-bundle $Q$ over $\mathscr{K}$ one can check that there is a natural isomorphism between the representations $(Q\otimes_{\mathscr{H}}P)^{\ast}\mathsf{V}$ and $Q^{\ast}(P^{\ast}\mathsf{V})$. Furthermore, for any $p\in P$ the representation $P^{\ast}(\mathsf{V})_{\pi(p)}$ of the Lie group $\mathord{\mathrm{I}}_{\pi(p)}(\mathscr{H})$ is naturally isomorphic to the composition of the representation $\mathord{\mathrm{I}}_{\epsilon(p)}(\mathscr{G})\to\mathord{\mathrm{GL}}(\mathsf{V}_{\epsilon(p)})$ with the homomorphism $P_{p}:\mathord{\mathrm{I}}_{\pi(p)}(\mathscr{H})\to\mathord{\mathrm{I}}_{\epsilon(p)}(\mathscr{G})$.

Suppose that $P$ is a Morita equivalence. This implies that $P_{p}:\mathord{\mathrm{I}}_{\pi(p)}(\mathscr{H})\to\mathord{\mathrm{I}}_{\epsilon(p)}(\mathscr{G})$ is an isomorphism of Lie groups, for any $p\in P$. In particular, if the representation $\mathsf{V}$ is faithful, then the representation $P^{\ast}\mathsf{V}$ is faithful. Hence, it follows that the faithful representability is a Morita-invariant property of Lie groupoids: if two Lie groupoids are Morita equivalent, then one is faithfully representable if, and only if, the other is. It is not difficult to show that the local faithful representability is a Morita-invariant property as well.

Example 4**.**

(1) Let $G$ be matrix Lie group. In other words, $G$ is a Lie subgroup of the general linear group $\mathord{\mathrm{GL}}(k,\mathbb{C})$, for a suitable natural number $k$. Suppose that we have a smooth right action of $G$ on a smooth manifold $M$. Then the associated action groupoid $M\rtimes G$ over $M$ is faithfully representable. Indeed, we can take the trivial vector bundle $\mathsf{V}=M\times\mathbb{C}^{k}$ over $M$ and define the left action of $M\rtimes G$ on $\mathsf{V}$ by $(y,A)(yA,v)=(y,Av)$. Analogously, the action groupoid associated to a smooth left action of a matrix Lie group on a smooth manifold is faithfully representable.

Since any compact Lie group is isomorphic to a Lie subgroup of the unitary group $\mathord{\mathrm{U}}(k)$, for some natural number $k$, it follows that the action Lie groupoid associated to a smooth action of a compact Lie group is faithfully unitary representable.

(2) A Lie groupoid $\mathscr{G}$ is proper if it is a smooth manifold and the map $(\mathord{\mathrm{s}},\mathord{\mathrm{t}}):\mathscr{G}_{1}\to\mathscr{G}_{0}\times\mathscr{G}_{0}$ is proper. For example, the action Lie groupoid of a smooth action of a compact Lie group on a smooth manifold is a proper Lie groupoid. Properness is a Morita-invariant property of Lie groupoids.

Any faithfully unitary representable proper Lie groupoid $\mathscr{G}$ is Morita equivalent to the action Lie groupoid of a smooth action of a unitary group on a smooth manifold. The proof of this well-known fact, essentially given by Satake [16] for effective orbifolds, can be found in [6] and goes as follows: Let $r:\mathscr{G}\to\mathord{\mathrm{U}}(\mathsf{V})$ be a faithful unitary representation and let $k$ be the rank of $\mathsf{V}$. Then $\mathscr{G}$ acts freely from the left on the unitary frame bundle $\mathord{\mathrm{UFr}}(\mathsf{V})$ of $\mathsf{V}$, so that the orbit space $M=\mathscr{G}\setminus\mathord{\mathrm{UFr}}(\mathsf{V})$ is a smooth manifold. The natural right action of $\mathord{\mathrm{U}}(k)$ on $\mathord{\mathrm{UFr}}(\mathsf{V})$ induces a right action on $M$, and the Lie groupoid $\mathscr{G}$ is Morita equivalent to the action Lie groupoid $M\rtimes\mathord{\mathrm{U}}(k)$.

Any proper Lie groupoid is locally linearizable and hence locally Morita equivalent to the action Lie groupoid of an action of a compact Lie group (see [2, 18, 19]). This implies that any proper Lie groupoid is locally faithfully unitary representable.

3. Monodromy group

Let $\mathscr{G}$ be a topological groupoid. The union $\mathord{\mathrm{I}}(\mathscr{G})=\cup\{\mathord{\mathrm{I}}_{x}(\mathscr{G})\;|\;x\in\mathscr{G}_{0}\}$ of the isotropy groups of $\mathscr{G}$ is a topological bundle of topological groups over $\mathscr{G}_{0}$ and a normal topological subgroupoid of $\mathscr{G}$. The groupoid $\mathscr{G}$ acts continuously on $\mathord{\mathrm{I}}(\mathscr{G})$ by conjugation: for any $g\in\mathscr{G}$ we have the associated isomorphism of topological groups $\mathord{\mathrm{c}}_{g}:\mathord{\mathrm{I}}_{\mathord{\mathrm{s}}(g)}(\mathscr{G})\to\mathord{\mathrm{I}}_{\mathord{\mathrm{t}}(g)}(\mathscr{G})$ given by

[TABLE]

We say that a function $\tau$, defined on $\mathord{\mathrm{I}}(\mathscr{G})$, is $\mathscr{G}$-conjugation-invariant if it is constant along the $\mathscr{G}$-conjugation classes, that is, if $\tau(ghg^{-1})=\tau(h)$ for all $g\in\mathscr{G}$ and all $h\in\mathord{\mathrm{I}}_{\mathord{\mathrm{s}}(g)}(\mathscr{G})$.

Let $K$ be a locally compact Hausdorff topological group. A homomorphism of topological groupoids $\varphi:K\to\mathscr{G}$ is given by a point $\omega(\varphi)$ in $\mathscr{G}_{0}$ and a continuous homomorphism of groups $\varphi:K\to\mathord{\mathrm{I}}_{\omega(\varphi)}(\mathscr{G})$. The set

[TABLE]

of all such homomorphisms is a subspace of the space $C(K,\mathscr{G})$ of all continuous maps from $K$ to $\mathscr{G}$ with the compact-open topology. With respect to this topology, the map $\omega:\hom(K,\mathscr{G})\to\mathscr{G}_{0}$ is continuous. There is a natural continuous left $\mathscr{G}$-action on $\hom(K,\mathscr{G})$ along $\omega$ defined by conjugation: for any $\varphi\in\hom(K,\mathscr{G})$ and any $g\in\mathscr{G}(\omega(\varphi),\,\mbox{-}\,)$, the homomorphism $g\varphi\in\hom(K,\mathscr{G})$ is given by

[TABLE]

and satisfies $\omega(g\varphi)=\mathord{\mathrm{t}}(g)$. We have the associated action groupoid

[TABLE]

while the $\mathscr{G}$-equivariant map $\omega:\hom(K,\mathscr{G})\to\mathscr{G}_{0}$ induces a continuous functor of topological groupoids $\omega=\omega_{\mathscr{G}}:\mathscr{G}^{K}\to\mathscr{G}$.

For any $\rho\in\hom(K,\mathscr{G})$ we write

[TABLE]

for the subspace of $\hom(K,\mathscr{G})$ which is the space of objects of the path-component $\mathord{\mathrm{Pc}}(\mathscr{G}^{K};\rho)$ of the topological groupoid $\mathscr{G}^{K}$ with $\rho\in\hom(K,\mathscr{G};\rho)$. In other words, the space $\hom(K,\mathscr{G};\rho)$ is the minimal union of path-components of the space $\hom(K,\mathscr{G})$ which includes the point $\rho$ and is saturated for the $\mathscr{G}$-action. The corresponding path-component of $\mathscr{G}^{K}$ is therefore the action groupoid

[TABLE]

If $L$ is another locally compact Hausdorff topological group, the composition gives us a continuous map

[TABLE]

For any $\alpha\in\hom(L,K)$, the continuous map $\hom(K,\mathscr{G})\to\hom(L,\mathscr{G})$, $\varphi\mapsto\varphi\circ\alpha$, is $\mathscr{G}$-equivariant, which implies $\hom(K,\mathscr{G};\rho)\circ\alpha\subset\hom(L,\mathscr{G};\rho\circ\alpha)$. In particular, if $\alpha$ is an isomorphism, then

[TABLE]

The topological group $\mathord{\mathrm{Aut}}(K)$ of automorphisms of the topological group $K$ acts continuously on $\hom(K,\mathscr{G})$ by composition from the right, and this action commutes with the left $\mathscr{G}$-action. In particular, the group $\mathord{\mathrm{Aut}}(K)$ also acts continuously on the action groupoid $\mathscr{G}^{K}$ by groupoid isomorphisms.

We define the monodromy group of the topological groupoid $\mathscr{G}$ at $\rho$ to be the subgroup

[TABLE]

of the group $\mathord{\mathrm{Aut}}(K)$. It follows that the monodromy group $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ acts continuously on the space $\hom(K,\mathscr{G};\rho)$ and also on the topological groupoid $\mathord{\mathrm{Pc}}(\mathscr{G}^{K};\rho)$ by groupoid isomorphisms. Let us also denote

[TABLE]

Since the left $\mathscr{G}$-action on $\hom(K,\mathscr{G};\rho)$ commutes with the right $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$-action, it induces a left $\mathscr{G}$-action on the quotient $\hom(K,\mathscr{G};\rho)/\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$, so that the quotient projection $\hom(K,\mathscr{G};\rho)\to\hom(K,\mathscr{G};\rho)/\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ is $\mathscr{G}$-equivariant. The map $\omega:\hom(K,\mathscr{G};\rho)\to\mathscr{G}_{0}$ factors as a surjective map $\hom(K,\mathscr{G};\rho)/\mathord{\mathrm{Mon}}(\mathscr{G};\rho)\to\Sigma(\mathscr{G};\rho)$ which is also $\mathscr{G}$-invariant.

If $\mathscr{G}$ is a Lie groupoid and $K$ a Lie group, then $\mathord{\mathrm{I}}(\mathscr{G})$ is a topological bundle of Lie groups over $\mathscr{G}_{0}$ and $\mathscr{G}$ acts on $\mathord{\mathrm{I}}(\mathscr{G})$ by isomorphisms of Lie groups. In this case, all the homomorphisms in $\hom(K,\mathscr{G})$ are homomorphisms of Lie groups and all the automorphisms in $\mathord{\mathrm{Aut}}(K)$ are automorphisms of the Lie group $K$. Indeed, recall that any continuous group homomorphism between Lie groups is necessarily smooth and hence a homomorphism of Lie groups.

Proposition 5**.**

Let $\mathscr{G}$ be a topological groupoid, let $K$ be a locally compact Hausdorff topological group and let $\rho\in\hom(K,\mathscr{G})$.

(i) For any $\varphi\in\hom(K,\mathscr{G};\rho)$ we have $\mathord{\mathrm{Mon}}(\mathscr{G};\varphi)=\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$.

(ii) For any $g\in\mathscr{G}(\omega(\rho),\,\mbox{-}\,)$ we have $\mathord{\mathrm{Mon}}(\mathscr{G};\mathord{\mathrm{c}}_{g}\circ\rho)=\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$.

(iii) For any continuous isomorphism $\alpha:L\to K$ between topological groups we have $\mathord{\mathrm{Mon}}(\mathscr{G};\rho\circ\alpha)=\alpha^{-1}\circ\mathord{\mathrm{Mon}}(\mathscr{G};\rho)\circ\alpha$.

(iv) If $\rho$ is a monomorphism, then $\rho\circ\mathord{\mathrm{Mon}}(\mathscr{G};\rho)=\mathord{\mathrm{Mon}}(\mathscr{G};\mathord{\mathrm{id}}_{\rho(K)})\circ\rho$.

(v) The group $\mathord{\mathrm{Inn}}(K)$ of inner automorphisms of $K$ is a normal subgroup the monodromy group $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$.

Proof.

Part (i) follows from the equality $\hom(K,\mathscr{G};\varphi)=\hom(K,\mathscr{G};\rho)$ and the definition of the monodromy group. Part (ii) is a special case of part (i), as the space $\hom(K,\mathscr{G};\rho)$ is saturated for the conjugation and therefore $\mathord{\mathrm{c}}_{g}\circ\rho\in\hom(K,\mathscr{G};\rho)$. Observe that for any $\mu\in\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ we have

[TABLE]

which yields $\alpha^{-1}\circ\mathord{\mathrm{Mon}}(\mathscr{G};\rho)\circ\alpha\subset\mathord{\mathrm{Mon}}(\mathscr{G};\rho\circ\alpha)$. The equality (iii) now follows by symmetry and directly implies (iv). To prove part (v), observe that

[TABLE]

for any $k\in K$, which yields $\mathord{\mathrm{Inn}}(K)\subset\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$. The group $\mathord{\mathrm{Inn}}(K)$ is in fact normal in $\mathord{\mathrm{Aut}}(K)$, so it is normal in $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ as well. ∎

It is well known that for a compact Lie group $K$, the group $\mathord{\mathrm{Aut}}(K)$ is a Lie group and that the subgroup $\mathord{\mathrm{Inn}}(K)$ is open in $\mathord{\mathrm{Aut}}(K)$ (for example, see [5, 13]). Proposition 5(v) thus immediately implies:

Proposition 6**.**

Let $\mathscr{G}$ be a topological groupoid, let $K$ be a compact Lie group and let $\rho\in\hom(K,\mathscr{G})$. Then the monodromy group $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ is an open subgroup of the Lie group $\mathord{\mathrm{Aut}}(K)$.

3.1. Functoriality

Let $K$ be a locally compact Hausdorff topological group, let $\mathscr{G}$ and $\mathscr{H}$ be topological groupoids and let $P$ be a topological principal $\mathscr{G}$-bundle over $\mathscr{H}$. The space

[TABLE]

is a topological principal $\mathscr{G}^{K}$-bundle over the groupoid $\mathscr{H}^{K}$, with the structure maps $P^{K}\to\hom(K,\mathscr{H})$, $(p,\vartheta)\mapsto\vartheta$, and $P^{K}\to\hom(K,\mathscr{G})$, $(p,\vartheta)\mapsto P_{p}\circ\vartheta$, with the left $\mathscr{H}^{K}$-action given by $(h,\vartheta)(p,\vartheta)=(hp,\mathord{\mathrm{c}}_{h}\circ\vartheta)$ and with the right $\mathscr{G}^{K}$-action given by $(p,\vartheta)(g,\mathord{\mathrm{c}}_{g^{-1}}\circ P_{p}\circ\vartheta)=(pg,\vartheta)$ (see [15]). This definition gives us a functor

[TABLE]

The group $\mathord{\mathrm{Aut}}(K)$ acts continuously on the space $P^{K}$ in the obvious way from the right such that both the structure maps $P^{K}\to\hom(K,\mathscr{H})$ and $P^{K}\to\hom(K,\mathscr{G})$, the left $\mathscr{H}^{K}$-action and the right $\mathscr{G}^{K}$-action are $\mathord{\mathrm{Aut}}(K)$-equivariant. There is a natural isomorphism of topological principal bundles

[TABLE]

Let $\sigma\in\hom(K,\mathscr{H})$, choose any $p\in P$ with $\pi(p)=\omega(\sigma)$ and put $\rho=P_{p}\circ\sigma$. Then $P^{K}|_{\hom(K,\mathscr{H};\sigma)}$ is a topological principal $\mathord{\mathrm{Pc}}(\mathscr{G}^{K};\rho)$-bundle over the groupoid $\mathord{\mathrm{Pc}}(\mathscr{H}^{K};\sigma)$. Of course, if we choose a different point $p^{\prime}\in P$ with $\pi(p^{\prime})=\omega(\sigma)$ and put $\rho^{\prime}=P_{p^{\prime}}\circ\sigma$, there is a unique $g\in\mathscr{G}(\,\mbox{-}\,,\epsilon(p))$ such that $pg=p^{\prime}$, which implies that $\rho^{\prime}=P_{pg}\circ\sigma=\mathord{\mathrm{c}}_{g^{-1}}\circ P_{p}\circ\sigma=\mathord{\mathrm{c}}_{g^{-1}}\circ\rho$ and hence $\hom(K,\mathscr{G};\rho)=\hom(K,\mathscr{G};\rho^{\prime})$.

The image of the set $P^{K}|_{\hom(K,\mathscr{H};\sigma)}$ along the structure map $P^{K}\to\hom(K,\mathscr{G})$ is a subset of $\hom(K,\mathscr{G};\rho)$. In particular, for any automorphism $\mu\in\mathord{\mathrm{Mon}}(\mathscr{H};\sigma)$ we have $\sigma\circ\mu\in\hom(K,\mathscr{H};\sigma)$ and hence $(p,\sigma\circ\mu)\in P^{K}|_{\hom(K,\mathscr{H};\sigma)}$. As the structure map $P^{K}\to\hom(K,\mathscr{G})$ maps $(p,\sigma\circ\mu)$ into

[TABLE]

we have $\rho\circ\mu\in\hom(K,\mathscr{G};\rho)$ and therefore $\mu\in\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$. We can conclude:

Proposition 7**.**

Let $K$ be a locally compact Hausdorff topological group, let $\mathscr{G}$ and $\mathscr{H}$ be topological groupoids and let $P$ be a topological principal $\mathscr{G}$-bundle over $\mathscr{H}$. For any $\sigma\in\hom(K,\mathscr{H})$ and any $p\in P$ with $\pi(p)=\omega(\sigma)$ we have

[TABLE]

If $P$ represents a Morita equivalence, then $P^{K}$ is a Morita equivalence from $\mathscr{H}^{K}$ to $\mathscr{G}^{K}$ and, by Proposition 3, the restriction $P^{K}|_{\hom(K,\mathscr{H};\sigma)}$ is a Morita equivalence from $\mathord{\mathrm{Pc}}(\mathscr{H}^{K};\sigma)$ to $\mathord{\mathrm{Pc}}(\mathscr{G}^{K};\rho)$. In this case we have

[TABLE]

In particular, any continuous functor $\phi:\mathscr{H}\to\mathscr{G}$ induces an $\mathord{\mathrm{Aut}}(K)$-equivariant continuous functor $\phi^{K}:\mathscr{H}^{K}\to\mathscr{G}^{K}$ given by

[TABLE]

and this gives us a functor $(-)^{K}:\mathord{\mathsf{Gpd}}\to\mathord{\mathsf{Gpd}}$ such that $\omega_{\mathscr{G}}\circ\phi^{K}=\phi\circ\omega_{\mathscr{H}}$. The functor $\phi^{K}$ restricts to a continuous functor $\mathord{\mathrm{Pc}}(\mathscr{H}^{K};\sigma)\to\mathord{\mathrm{Pc}}(\mathscr{G}^{K},\phi\circ\sigma)$ and we have

[TABLE]

If $\phi:\mathscr{H}\to\mathscr{G}$ is a weak equivalence, then $\phi^{K}:\mathscr{H}^{K}\to\mathscr{G}^{K}$ is a weak equivalence, its restriction $\mathord{\mathrm{Pc}}(\mathscr{H}^{K};\sigma)\to\mathord{\mathrm{Pc}}(\mathscr{G}^{K},\phi\circ\sigma)$ is also a weak equivalence and

[TABLE]

We may in fact consider the category of $K$-pointed topological groupoids, which is the coslice category

[TABLE]

The assignment $(\,\mbox{-}\,)^{K}$ gives us a functor $(\,\mbox{-}\,)^{K}:\mathord{\mathsf{Gpd}}_{K}\to\mathord{\mathsf{Gpd}}_{\ast}$, hence the composition

[TABLE]

is also a functor. A Morita map from $(K\rightrightarrows\ast)$ to $\mathscr{G}$ corresponds to a $\mathscr{G}$-orbit in $\hom(K,\mathscr{G})$, so the assignment $(\,\mbox{-}\,)^{K}$ also induces a functor

[TABLE]

and the composition $\mathord{\mathrm{Pc}}\circ(\,\mbox{-}\,)^{K}$ is also a functor $(K\rightrightarrows\ast)\downarrow\mathord{\mathsf{GPD}}\to(\ast\rightrightarrows\ast)\downarrow\mathord{\mathsf{GPD}}$. Furthermore, the assignment of the monodromy group can be seen as a functor

[TABLE]

to the category $\mathord{\mathsf{Grp}}$ of groups, which also factors as a functor

[TABLE]

The image of these two functors in fact lies in the lattice of subgroups of the group $\mathord{\mathrm{Aut}}(K)$ ordered by the inclusions.

Proposition 8**.**

Let $K$ be a locally compact Hausdorff topological group, let $\mathscr{G}$ be a topological groupoid and let $X$ be a subspace of $\mathscr{G}_{0}$ such that the inclusion $\mathscr{G}|_{X}\to\mathscr{G}$ is a weak equivalence. Then for any $\rho\in\hom(K,\mathscr{G})$ with $\omega(\rho)\in X$ we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Proof.

It is straightforward that $\hom(K,\mathscr{G}|_{X};\rho)\subset\hom(K,\mathscr{G};\rho)\cap\omega^{-1}(X)$. Recall that the inclusion $\mathscr{G}|_{X}\to\mathscr{G}$ is a weak equivalence if, and only if, the restriction $\mathord{\mathrm{t}}|_{\mathord{\mathrm{s}}^{-1}(X)}:\mathord{\mathrm{s}}^{-1}(X)\to\mathscr{G}_{0}$ is surjective with local sections. Since $\mathscr{G}|_{X}\to\mathscr{G}$ is a weak equivalence, the induced functor

[TABLE]

is also a weak equivalence. Hence for any $\varphi\in\hom(K,\mathscr{G};\rho)$ there exist $\psi\in\hom(K,\mathscr{G}|_{X};\rho)$ and an arrow $g\in\mathscr{G}(\omega(\psi),\omega(\varphi))$ such that $\varphi=\mathord{\mathrm{c}}_{g}\circ\psi$. If in addition we have $\omega(\varphi)\in X$, then $g\in\mathscr{G}|_{X}$ and therefore $\varphi\in\hom(K,\mathscr{G}|_{X};\rho)$. This proves the first two equations. The last two equations are direct consequences. ∎

4. Monodromy and representations

We shall now study the relation between monodromy groups and representability of Lie groupoids. In particular we will show that a monodromy group can present an obstruction for a Lie groupoid to be faithfully representable.

Proposition 9**.**

Let $\mathscr{G}$ be a Lie groupoid, let $K$ be a compact Lie group and let $\rho\in\hom(K,\mathscr{G})$. If $\tau:\mathord{\mathrm{I}}(\mathscr{G})\to\mathbb{C}$ is a continuous $\mathscr{G}$-conjugation-invariant function such that the restriction $\tau|_{\mathord{\mathrm{I}}_{x}(\mathscr{G})}$ is a character of the Lie group $\mathord{\mathrm{I}}_{x}(\mathscr{G})$, for all $x\in\mathscr{G}_{0}$, then

[TABLE]

for any $\varphi\in\hom(K,\mathscr{G};\rho)$ and any $k\in K$.

Proof.

For any $\varphi\in\hom(K,\mathscr{G})$ write $\theta(\varphi)=\tau\circ\varphi$. This gives us a continuous map $\theta$ from $\hom(K,\mathscr{G})$ to the space $C(K,\mathbb{C})$ of continuous complex functions on $K$ with compact-open topology. The image $\theta(\varphi)$ is a character of the Lie group $K$, because it is the composition of the Lie group homomorphism $\varphi:K\to\mathord{\mathrm{I}}_{\omega(\varphi)}(\mathscr{G})$ with the character $\tau|_{\mathord{\mathrm{I}}_{\omega(\varphi)}(\mathscr{G})}$ of the Lie group $\mathord{\mathrm{I}}_{\omega(\varphi)}(\mathscr{G})$.

The orthogonality relations imply that the characters of the compact Lie group $K$ form a discrete subset of the space $C(K,\mathbb{C})$ with respect to the $L^{2}$-norm given by the Haar measure. The compact-open topology on $C(K,\mathbb{C})$ coincides with the topology of uniform convergence, which is induced by the sup-norm. Since the $L^{2}$-norm is dominated by the sup-norm on $C(K,\mathbb{C})$, the characters of $K$ form a discrete subset of $C(K,\mathbb{C})$ with respect to the compact-open topology as well.

We may conclude that the continuous function $\theta:\hom(K,\mathscr{G})\to C(K,\mathbb{C})$ has a discrete image and is therefore constant on each path-component of $\hom(K,\mathscr{G})$. On the other hand, the map $\theta$ is also constant along the $\mathscr{G}$-orbits. Indeed, for any $\varphi\in\hom(K,\mathscr{G})$, any $g\in\mathscr{G}(\omega(\varphi),\,\mbox{-}\,)$ and any $k\in K$ we have

[TABLE]

It follows that $\theta$ is constant on $\hom(K,\mathscr{G};\rho)$. ∎

Proposition 10**.**

Let $\mathscr{G}$ be a locally faithfully representable Lie groupoid and let $K$ be a compact Lie group. If $\rho\in\hom(K,\mathscr{G})$ is a monomorphism, then any $\varphi\in\hom(K,\mathscr{G};\rho)$ is a monomorphism.

Proof.

First, assume that $\mathscr{G}$ is faithfully representable and choose a faithful representation $r:\mathscr{G}\to\mathord{\mathrm{GL}}(\mathsf{V})$ on a complex vector bundle $\mathsf{V}$ over $\mathscr{G}_{0}$. Let $\tau:\mathord{\mathrm{I}}(\mathscr{G})\to\mathbb{C}$ be given by $\tau(h)=\mathord{\mathrm{tr}}(r(h))$. By Proposition 9 we have $\tau\circ\varphi=\tau\circ\rho$ for any $\varphi\in\hom(K,\mathscr{G};\rho)$. Thus, the representations $r\circ\rho:K\to\mathord{\mathrm{GL}}(\mathsf{V}_{\omega(\rho})$ and $r\circ\varphi:K\to\mathord{\mathrm{GL}}(\mathsf{V}_{\omega(\varphi})$ have the same character, hence they are equivalent. Since $\rho$ is a monomorphism, the representation $r\circ\rho$ is faithful, so the equivalent representation $r\circ\varphi$ is faithful as well. This implies that $\varphi$ is a monomorphism.

Now let $\mathscr{G}$ be a general locally faithfully representable Lie groupoid. We can find an open cover $(U_{j})_{j\in J}$ of $\mathscr{G}_{0}$ such that the restrictions $\mathscr{G}|_{U_{j}}$ are all faithfully representable. For any $\varphi\in\hom(K,\mathscr{G};\rho)$ we can find a finite sequence $\varphi_{0},\varphi_{1},\ldots,\varphi_{n-1},\varphi_{n}$ in $\hom(K,\mathscr{G};\rho)$ with $\varphi_{0}=\rho$ and $\varphi_{n}=\varphi$ such that for any $k=1,2,\ldots,n$ there exists $j_{k}\in J$ satisfying $\omega(\varphi_{k-1}),\omega(\varphi_{k})\in U_{j_{k}}$ and

[TABLE]

From the first part of the proof we know that $\varphi_{k}$ is a monomorphism if $\varphi_{k-1}$ is. Since $\varphi_{0}$ is a monomorphism, it follows recursively that $\varphi_{n}$ is also a monomorphism. ∎

Corollary 11**.**

Let $\mathscr{G}$ be a locally faithfully representable Lie groupoid, let $K$ be a compact Lie group and let $\rho\in\hom(K,\mathscr{G})$ be a monomorphism.

(i) The action of the monodromy group $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ on $\hom(K,\mathscr{G};\rho)$ is free.

(ii) For any $\varphi,\psi\in\hom(K,\mathscr{G};\rho)$ we have $\varphi\circ\mathord{\mathrm{Mon}}(\mathscr{G};\rho)=\psi\circ\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ if, and only if, $\varphi(K)=\psi(K)$.

(iii) If $\sigma\in\hom(K,\mathscr{G})$ is a monomorphism with $\omega(\sigma)=\omega(\rho)$ such that the subgroup $\sigma(K)$ is conjugated to the subgroup $\rho(K)$ in $\mathord{\mathrm{I}}_{\omega(\rho)}(\mathscr{G})$, then

[TABLE]

and there exists an automorphism $\mu\in\mathord{\mathrm{Aut}}(K)$ so that

[TABLE]

Proof.

Part (i) follows directly from Proposition 10.

(ii) If $\varphi(K)=\psi(K)$, there exists $\kappa\in\mathord{\mathrm{Aut}}(K)$ such that $\varphi\circ\kappa=\psi$. Since $\varphi,\psi\in\hom(K,\mathscr{G};\rho)$, it follows that $\hom(K,\mathscr{G};\rho)\circ\kappa=\hom(K,\mathscr{G};\rho)$ and therefore $\kappa\in\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$.

(iii) Let $g$ be an element of $\mathord{\mathrm{I}}_{\omega(\rho)}(\mathscr{G})$ such that $\sigma(K)=\mathord{\mathrm{c}}_{g}(\rho(K))$. There exists a unique automorphism $\mu\in\mathord{\mathrm{Aut}}(K)$ so that $\mathord{\mathrm{c}}_{g}\circ\rho\circ\mu=\sigma$, and this implies

[TABLE]

Theorem 12**.**

Let $\mathscr{G}$ be a faithfully representable Lie groupoid, let $K$ be a compact Lie group and let $\rho\in\hom(K,\mathscr{G})$ be a monomorphism. Then there exists a faithful character $\chi$ of the Lie group $K$ such that

[TABLE]

for any $\mu\in\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$.

Proof.

Choose a faithful representation $r:\mathscr{G}\to\mathord{\mathrm{GL}}(\mathsf{V})$ on a complex vector bundle $\mathsf{V}$ over $\mathscr{G}_{0}$. Let $\tau:\mathord{\mathrm{I}}(\mathscr{G})\to\mathbb{C}$ be the given by $\tau(h)=\mathord{\mathrm{tr}}(r(h))$. The composition $\chi=\tau\circ\rho$ is a faithful character of the Lie group $K$. For any $\mu\in\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ we have $\rho\circ\mu\in\hom(K,\mathscr{G};\rho)$, by definition of the monodromy group. Proposition 9 implies $\chi\circ\mu=\tau\circ\rho\circ\mu=\tau\circ\rho=\chi$. ∎

This theorem generalizes the obstruction to faithful representability which was given in [15] for proper regular Lie groupoids. As a consequence it allows us to construct new examples of Lie groupoids which are not faithfully representable:

Example 13**.**

Let $M$ be a connected smooth manifold with a free proper right action of a discrete group $H$, and let $\mathscr{G}$ be a faithfully representable Lie groupoid equipped with a left action of the group $H$ by smooth functors. The left $H$-action on $\mathscr{G}$ is equivalently given by a homomorphism of groups

[TABLE]

The group $H$ then also acts diagonally from the right on the product Lie groupoid $M\times\mathscr{G}$ by smooth functors. The action on the manifold $M\times\mathscr{G}_{0}$ is free and proper, so the space of orbits $M\times_{H}\mathscr{G}_{0}=(M\times\mathscr{G}_{0})/H$ is a smooth manifold. Similarly, the space of orbits $M\times_{H}\mathscr{G}_{1}=(M\times\mathscr{G}_{1})/H$ is a Hausdorff $C^{\infty}$-space (and a smooth manifold if $\mathscr{G}_{1}$ is a smooth manifold). We have the quotient Lie groupoid

[TABLE]

which is clearly locally faithfully representable.

Let $K$ be a compact Lie subgroup of $\mathord{\mathrm{I}}_{y}(\mathscr{G})$, for some $y\in\mathscr{G}_{0}$, choose a point $x\in M$, and let $\rho\in\hom(K,\mathscr{B})$ be given by $\rho(k)=(x,k)H$. Consider the subgroup $H^{\prime}=\{h\in H\,|\,\eta(h)(K)=K\}$. It follows that for any $h\in H^{\prime}$ we have $\nu(h)=\eta(h)|_{K}\in\mathord{\mathrm{Aut}}(K)$ and that $\nu:H^{\prime}\to\mathord{\mathrm{Aut}}(K)$ is a homomorphism of groups. We claim that

[TABLE]

Indeed, for any $h\in H^{\prime}$ we can choose a continuous path $\gamma:[0,1]\to M$ from $x$ to $xh$ and lift this path to a continuous path $\bar{\gamma}:[0,1]\to\hom(K,\mathscr{B})$ by

[TABLE]

Observe that $\bar{\gamma}(0)=\rho$ and that $\bar{\gamma}(1)=\rho\circ\nu(h)$, as

[TABLE]

If the Lie groupoid $\mathscr{B}=M\times_{H}\mathscr{G}$ is faithfully representable, Theorem 12 implies that there exists a faithful character $\chi$ of the Lie group $K$ such that

[TABLE]

for any $h\in H^{\prime}$. However, this may be impossible, in particular if $K$ is abelian and $\nu(h)$ is of infinite multiplicative order. In this case, the Lie groupoid $M\times_{H}\mathscr{G}$ is an example of a locally faithfully representable Lie groupoid which is not (globally) faithfully representable.

As a concrete example, let $\mathscr{G}$ be the product $\mathscr{H}\times T^{n}=\mathscr{H}\times(T^{n}\rightrightarrows\ast)$ of an arbitrary non-empty proper Lie groupoid $\mathscr{H}$ with the $n$-torus $T^{n}$, for some $n\geq 2$, and we choose $M$, $H$ and $\eta$ so that for some $h\in H$ and some $y\in\mathscr{H}_{0}$, the automorphism $\eta(h)$ of $\mathscr{H}\times T^{n}$ restricts to an automorphism of the subgroup $K=\{1_{y}\}\times T^{n}\subset\mathord{\mathrm{I}}_{y}(\mathscr{H})\times T^{n}=\mathord{\mathrm{I}}_{(y,\ast)}(\mathscr{H}\times T^{n})$ of infinite multiplicative order. The associated groupoid $M\times_{H}(\mathscr{H}\times T^{n})$ is in this case a proper Lie groupoid which is not faithfully representable.

Note that we can choose $\mathscr{H}$ so that $\mathscr{B}$ in not regular. This example generalizes the one given in [15], where $\mathscr{G}=T^{n}$ and the proper Lie groupoid $\mathscr{B}$ is regular, in fact a bundle of compact Lie groups. Before that, only a special case with $n=2$ and $M=\mathbb{R}$ was known (see [7, 17]).

There is a simple, but important special case of Example 13 in which the constructed Lie groupoid is necessarily faithfully representable:

Proposition 14**.**

Let $M$ be a smooth connected manifold with a free proper right action of a discrete group $H$, let $G$ be a Lie group and let $q:H\to G$ be a homomorphism of groups. The composition of the homomorphism $q$ with the conjugation homomorphism $\mathord{\mathrm{c}}:G\to\mathord{\mathrm{Aut}}(G)$, $g\mapsto\mathord{\mathrm{c}}_{g}$, defines an action of $H$ on $G$.

(i) If the Lie group $G$ is faithfully representable, then the associated Lie groupoid $M\times_{H}G$ is faithfully representable.

(ii) If the Lie group $G$ is faithfully unitary representable, then the Lie groupoid $M\times_{H}G$ is faithfully unitary representable.

Proof.

(i) Since the group $G$ is faithfully representable, we may assume, without loss of generality, that $G$ is a Lie subgroup of the Lie group $\mathord{\mathrm{GL}}(m,\mathbb{C})$. Write $\iota:G\to\mathord{\mathrm{GL}}(m,\mathbb{C})$ for the inclusion. The image of the homomorphism $\mathord{\mathrm{c}}\circ q:H\to\mathord{\mathrm{Aut}}(G)$ lies inside the subgroup $\mathord{\mathrm{Inn}}(G)\subset\mathord{\mathrm{Aut}}(G)$ of inner automorphisms of the group $G$.

[TABLE]

We have the left action of $H$ on $\mathbb{C}^{m}$ given by the composition $\iota\circ q:H\to\mathord{\mathrm{GL}}(m,\mathbb{C})$, and with this action we construct the vector bundle

[TABLE]

over the smooth manifold $M/H$. The inclusion $\iota$ induces a faithful representation of the Lie groupoid $M\times_{H}G$ on the vector bundle $M\times_{H}\mathbb{C}^{m}$.

(ii) If $G$ is faithfully unitary representable, we can replace the group $\mathord{\mathrm{GL}}(m,\mathbb{C})$ in the proof of (i) with the unitary group $\mathord{\mathrm{U}}(m)$. The left $H$-action on $\mathbb{C}^{m}$ is now unitary and there is a natural Hermitian metric on the vector bundle $M\times_{H}\mathbb{C}^{m}$. ∎

In fact, in a special case we can show that the property given in Theorem 12 is not only necessary, but also sufficient for the given Lie groupoid to be faithfully representable:

Theorem 15**.**

Let $M$ be a smooth connected manifold with a proper free right action of a free discrete group $H$, let $K$ be a compact Lie group and let $\eta:H\to\mathord{\mathrm{Aut}}(K)$ be a homomorphism of groups. Then the proper Lie groupoid $M\times_{H}K$ is faithfully unitary representable if, and only if, there exists a faithful character $\chi$ of $K$ such that $\chi\circ\eta(h)=\chi$ for all $h\in H$.

Proof.

One implication is a direct consequence of Theorem 12. To prove the other implication, assume that $\chi$ is a faithful character of $K$ such that $\chi\circ\eta(h)=\chi$ for all $h\in H$, and choose a faithful unitary representation $r:K\to\mathord{\mathrm{U}}(m)$ with character $\chi$. As a consequence, the unitary representations $r\circ\eta(h)$ and $r$ are unitary equivalent, which means that there exists an element of the group $\mathord{\mathrm{U}}(m)$ which intertwines these two representations. In particular, this element lies of the normalizer $G=\mathord{\mathrm{N}}_{\mathord{\mathrm{U}}(m)}(r(K))$ of the subgroup $r(K)$ in the group $\mathord{\mathrm{U}}(m)$. We have the inclusion $\iota:G\to\mathord{\mathrm{U}}(m)$ and the homomorphism of groups $\nu:\mathord{\mathrm{Inn}}(G)\to\mathord{\mathrm{Aut}}(K)$ given by the composition of the restriction to $r(K)$ with the isomorphism $\mathord{\mathrm{Aut}}(K)\cong\mathord{\mathrm{Aut}}(r(K))$ induced by $r:K\to r(K)$.

Since the group $H$ is free, we can choose a homomorphism of groups $q:H\to G$ such that the diagram

[TABLE]

commutes, or explicitly, such that for any $h\in H$ and any $k\in K$ we have

[TABLE]

By Proposition 4, we obtain the associated faithful unitary representation

[TABLE]

of the Lie groupoid $M\times_{H}G$. The monomorphism $r:K\to G$ induces a monomorphism of Lie groupoids $M\times_{H}K\to M\times_{H}G$, and the composition of this monomorphism with the representation $r^{\prime}$ is a faithful unitary representation of the Lie groupoid $M\times_{H}K$. ∎

5. Monodromy and proper Lie groupoids

In this section we shall study the monodromy groups of proper Lie groupoids. Recall that any proper Lie groupoid $\mathscr{G}$ is locally faithfully representable, so by Proposition 10 all homomorphisms in $\hom(K,\mathscr{G};\rho)$ are monomorphisms, for any compact Lie group $K$ and any monomorphism $\rho\in\hom(K,\mathscr{G})$. In particular, the monodromy group $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$ acts freely on $\hom(K,\mathscr{G};\rho)$.

5.1. Compact Lie groups

Let $K$ and $G$ be compact Lie groups and let $\sigma:K\to G$ be a homomorphism of Lie groups. Recall that the space $\hom(K,G)$ is Hausdorff and that the $G$-conjugation orbits are open in $\hom(K,G)$ [1, 13]. This implies that the space $\hom(K,G;\sigma)$ is precisely the $G$-conjugation orbit through $\sigma$,

[TABLE]

which is homeomorphic to the manifold $G/\mathord{\mathrm{C}}_{G}(\sigma(K))$ of left cosets of the centralizer $\mathord{\mathrm{C}}_{G}(\sigma(K))$ of the subgroup $\sigma(K)$ in $G$. In particular, the space $\hom(K,G;\sigma)$ has a structure of a compact smooth manifold.

Assume now that $\sigma$ is a monomorphism. For an automorphism $\mu\in\mathord{\mathrm{Aut}}(K)$ we have $G\sigma\circ\mu=G\sigma$ if, and only if, there exists $g\in G$ such that $\sigma\circ\mu=\mathord{\mathrm{c}}_{g}\circ\sigma$. If this is the case, then $g$ lies in the normalizer $\mathord{\mathrm{N}}_{G}(\sigma(K))$ of the subgroup $\sigma(K)$ in $G$. It follows that we have a natural epimorphism

[TABLE]

which maps an element $g\in\mathord{\mathrm{N}}_{G}(\sigma(K))$ to the automorphism $\sigma^{-1}\circ\mathord{\mathrm{c}}_{g}|_{\sigma(K)}\circ\sigma$ of the Lie group $K$. The kernel of this epimorphism is exactly the centralizer $\mathord{\mathrm{C}}_{G}(\sigma(K))$ of the subgroup $\sigma(K)$ in $G$. Thus, we have constructed a natural isomorphism of groups

[TABLE]

If we choose $K=G$ and $\rho=\mathord{\mathrm{id}}$, we have

[TABLE]

and also

[TABLE]

where $\mathord{\mathrm{C}}_{G}(G)$ is the center of the Lie group $G$.

5.2. Actions of compact Lie groups

Let $G$ be a compact Lie group and let $M$ be a smooth manifold with a smooth left $G$-action. We write $G_{x}$ for the isotropy group of the $G$-action at a point $x\in M$. For a subgroup $H$ of $G$, we denote by

[TABLE]

the fixed point space of the subgroup $H\subset G$ in $M$, and by

[TABLE]

the path-component of the space $\mathord{\mathrm{fix}}_{H}(M)$ which includes a point $x\in\mathord{\mathrm{fix}}_{H}(M)$. The fixed point space $\mathord{\mathrm{fix}}_{H}(M)$ is a closed subspace of $M$. If the subgroup $H$ is compact, the Bochner linearization theorem [3] implies that the fixed point space $\mathord{\mathrm{fix}}_{H}(M)$ is locally a manifold, so $\mathord{\mathrm{fix}}_{H}(M)$ is locally path-connected and each path-component of $\mathord{\mathrm{fix}}_{H}(M)$ is a closed embedded submanifold of $M$. However, different path-components of $\mathord{\mathrm{fix}}_{H}(M)$ may have different dimensions.

We consider the action groupoid $G\ltimes M$, which is a proper Lie groupoid over the manifold $M$. Let $K$ be a compact Lie group. Note that the conjugation action of the Lie groupoid $G\ltimes M$ on the space $\hom(K,G\ltimes M)$ along $\omega:\hom(K,G\ltimes M)\to M$ is in this case simply given by the conjugation action of the Lie group $G$ on the space $\hom(K,G\ltimes M)$, and that the map $\omega$ is $G$-equivariant. Any homomorphism $\varphi\in\hom(K,G\ltimes M)$ is of the form $\varphi(k)=(\vartheta(k),x)$, for a point $x\in M$ and a homomorphism of Lie groups $\vartheta:K\to G_{x}$. Conversely, for any homomorphism $\sigma:K\to G$ of Lie groups and any point $y\in\mathord{\mathrm{fix}}_{\sigma(K)}(M)$ there is a homomorphism

[TABLE]

defined by $\bar{\sigma}(y)(k)=(\sigma(k),y)$, and this gives us the partially defined continuous section

[TABLE]

of the map $\omega:\hom(K,G\ltimes M)\to M$.

For any $x\in M$ we have the sequence of homomorphisms of Lie groupoids

[TABLE]

where the first homomorphism $\iota_{x}:G_{x}\to G\ltimes M$ is given by inclusion of the isotropy group $G_{x}$ at the point $x$, $\iota_{x}(g)=(g,x)$, while the second homomorphism is the projection $\mathord{\mathrm{pr}}_{G}:G\ltimes M\to G$. These homomorphisms induce the sequence of inclusions

[TABLE]

for any homomorphism of Lie groups $\sigma:K\to G_{x}$.

Example 16**.**

Let $K$ and $G$ be compact Lie groups, let $M$ be a smooth manifold with a smooth left $G$-action, and let $\sigma:K\to G$ be a homomorphism of Lie groups. It the $G$-action on $M$ has a fixed point $x\in M$, then $\mathord{\mathrm{Mon}}(G\ltimes M;\bar{\sigma}(x))=\mathord{\mathrm{Mon}}(G;\sigma).$

Choose a homomorphism of Lie groups $\sigma:K\to G$. Since the projection $\mathord{\mathrm{pr}}_{G}:G\ltimes M\to G$ is a homomorphism of Lie groupoids, the induced continuous map

[TABLE]

is $G$-equivariant. The orbit $G\sigma$ is $G$-invariant, open and closed in $\hom(K,G)$, thus the inverse image $\hom(K,\mathord{\mathrm{pr}}_{G})^{-1}(G\sigma)$ is $G$-saturated, open and closed in the space $\hom(K,G\ltimes M)$. Observe that we have

[TABLE]

We define a map

[TABLE]

by

[TABLE]

It is clear that this map is continuous, surjective and $G$-equivariant with respect to the left $G$-action on $G\times\mathord{\mathrm{fix}}_{\sigma(K)}(M)$ given by $g^{\prime}(g,y)=(g^{\prime}g,y)$. The centralizer

[TABLE]

is a compact subgroup of the normalizer $\mathord{\mathrm{N}}_{G}(\sigma(K))$ and acts by multiplication from the right on $G$, but also from the left on $\mathord{\mathrm{fix}}_{\sigma(K)}(M)$ because $\mathord{\mathrm{N}}_{G}(\sigma(K))\mathord{\mathrm{fix}}_{\sigma(K)}(M)=\mathord{\mathrm{fix}}_{\sigma(K)}(M)$. We combine these two actions to get a smooth right diagonal $H$-action on the space $G\times\mathord{\mathrm{fix}}_{\sigma(K)}(M)$. One may check that the map $\hat{\sigma}$ factors through the quotient projection G\times\mathord{\mathrm{fix}}_{\sigma(K)}(M)\to G\times_{H}\mathord{\mathrm{fix}}_{\sigma(K)}(M)=\big{(}G\times\mathord{\mathrm{fix}}_{\sigma(K)}(M)\big{)}/H as a continuous surjective $G$-equivariant map

[TABLE]

It is straightforward to check that $\tilde{\sigma}$ is a bijection. We claim:

Lemma 17**.**

For any compact Lie group $G$, any smooth manifold $M$ with a smooth left $G$-action and any homomorphism $\sigma:K\to G$ of Lie groups, the map $\tilde{\sigma}$ is a $G$-equivariant homeomorphism.

Proof.

We have already seen that the map $\tilde{\sigma}$ is a $G$-equivariant continuous bijection. We will show that the map $\hat{\sigma}$ is closed, which implies that $\tilde{\sigma}$ is closed and hence a homeomorphism. To this end, observe first that the map $f:G\to\hom(K,G)$, $g\mapsto\mathord{\mathrm{c}}_{g}\circ\sigma$, is a continuous map from a compact space to a Hausdorff space, which yields that the map $f\times\mathord{\mathrm{id}}_{M}:G\times M\to\hom(K,G)\times M$ is closed. Since the map $G\times M\to G\times M$, $(g,y)\mapsto(g,gy)$, is a diffeomorphism, the composition of $f\times\mathord{\mathrm{id}}_{M}$ with this diffeomorphism is also closed. In particular, the restriction of this composition to the closed subspace $G\times\mathord{\mathrm{fix}}_{\sigma(K)}(M)$ of $G\times M$ is a closed map $\beta:G\times\mathord{\mathrm{fix}}_{\sigma(K)}(M)\to\hom(K,G)\times M$, given by $\beta(g,y)=(\mathord{\mathrm{c}}_{g}\circ\sigma,gy)$. Now observe that we have a commutative diagram of continuous maps

[TABLE]

in which $\beta$ is closed and $(\hom(K,\mathord{\mathrm{pr}}_{G}),\omega)$ is injective. It follows that $\hat{\sigma}$ is closed. ∎

Proposition 18**.**

Let $K$ and $G$ be compact Lie groups, let $M$ be a smooth manifold with a smooth left $G$-action, let $x$ be a point in $M$ and let $\sigma:K\to G_{x}$ be a homomorphism of Lie groups. Denote $H=\mathord{\mathrm{C}}_{G}(\sigma(K))$.

(i) The space $\hom(K,G\ltimes M;\bar{\sigma}(x))$ is an open, closed and $G$-saturated subset of $\hom(K,G\ltimes M)$, and we have a $G$-equivariant homeomorphism

[TABLE]

given by $(g,y)H\mapsto(k\mapsto(\mathord{\mathrm{c}}_{g}(\sigma(k)),gy))$. In particular, we have a natural structure of a smooth manifold on the space $\hom(K,G\ltimes M;\bar{\sigma}(x))$.

(ii) We have

[TABLE]

and

[TABLE]

Proof.

(i) The smooth manifold $\mathord{\mathrm{fix}}_{\sigma(K)}(M;x)$ is a path-component of the space $\mathord{\mathrm{fix}}_{\sigma(K)}(M)$ which is locally a manifold, so the saturation $\mathord{\mathrm{sat}}_{H}(\mathord{\mathrm{fix}}_{\sigma(K)}(M;x))$ is also a smooth manifold and a union of path-components of the space $\mathord{\mathrm{fix}}_{\sigma(K)}(M)$. It follows that the product $G\times\mathord{\mathrm{sat}}_{H}(\mathord{\mathrm{fix}}_{\sigma(K)}(M;x))$ is a smooth manifold which is $H$-saturated, and since $H$ is a compact Lie group acting smoothly and freely on $G\times\mathord{\mathrm{sat}}_{H}(\mathord{\mathrm{fix}}_{\sigma(K)}(M;x))$, the space of orbits $G\times_{H}\mathord{\mathrm{sat}}_{H}(\mathord{\mathrm{fix}}_{\sigma(K)}(M;x))$ is a smooth manifold as well. Furthermore, the subset $G\times_{H}\mathord{\mathrm{sat}}_{H}(\mathord{\mathrm{fix}}_{\sigma(K)}(M;x))$ is clearly the minimal union of path-components of the space $G\times_{H}\mathord{\mathrm{fix}}_{\sigma(K)}(M)$ which is $G$-saturated and includes the point $(1,x)H$. The $G$-equivariant map $\tilde{\sigma}$, which is a homeomorphism by Lemma 17, therefore maps $G\times_{H}\mathord{\mathrm{sat}}_{H}(\mathord{\mathrm{fix}}_{\sigma(K)}(M;x))$ onto the space $\hom(K,G\ltimes M;\bar{\sigma}(x))$.

(ii) First equality is a direct consequence of part (i), while the second follows from the first because the map $\omega:\hom(K,G\ltimes M)\to M$ is $G$-equivariant. ∎

Example 19**.**

Let $G$ be a compact Lie group with an orthogonal representation $G\to\mathord{\mathrm{O}}(n)$. This means that we have an orthogonal left action of $G$ on the vector space $\mathbb{R}^{n}$. Let $M$ be an open ball in $\mathbb{R}^{n}$ centered at the origin $0\in\mathbb{R}^{n}$. Since $M$ is $G$-saturated, the action of $G$ on $\mathbb{R}^{n}$ restricts to an action on $M$.

Let $K$ be a compact Lie group and let $\sigma:K\to G$ be a monomorphism. The origin [math] is a fixed point of the $G$-action, so we have

[TABLE]

by Example 16 and Subsection 5.1. The fixed point set $\mathord{\mathrm{fix}}_{\sigma(K)}(\mathbb{R}^{n})$ is a linear subspace of $\mathbb{R}^{n}$, so the fixed point space $\mathord{\mathrm{fix}}_{\sigma(K)}(M)=\mathord{\mathrm{fix}}_{\sigma(K)}(\mathbb{R}^{n})\cap M$ is an open ball in the space $\mathord{\mathrm{fix}}_{\sigma(K)}(\mathbb{R}^{n})$. In particular we have

[TABLE]

It follows that

[TABLE]

and

[TABLE]

by Proposition 18.

5.3. Proper groupoids

Let $\mathscr{G}$ be a proper Lie groupoid. Recall that such a groupoid is locally Morita equivalent to the action Lie groupoid of an action of a compact Lie group. This is a consequence of the Weinstein-Zung linearization theorem for proper Lie groupoids [2, 18, 19].

In fact, for any point $x\in\mathscr{G}_{0}$ we can choose a slice $S$ through $x$, which is an embedded submanifold of $\mathscr{G}_{0}$ of dimension $m=\dim(\mathscr{G}_{0})-\dim(\mathscr{G}x)$ such that $S\cap\mathscr{G}x=\{x\}$ and so that for any $y\in S$ the orbit $\mathscr{G}y$ is transversal to $S$ at $y$. This implies that the restriction $\mathscr{G}|_{S}$ is a Lie groupoid, that $U=\mathord{\mathrm{sat}}_{\mathscr{G}}(S)$ is an open subset of $\mathscr{G}_{0}$ and that the embedding $\mathscr{G}|_{S}\to\mathscr{G}|_{U}$ is a weak equivalence. Furthermore, by linearization theorem we can choose $S$ so that there exist an orthogonal action of the isotropy group $G=\mathord{\mathrm{I}}_{x}(\mathscr{G})$ on $\mathbb{R}^{m}$, and open ball $M\subset\mathbb{R}^{m}$ centered at the origin and an isomorphism of Lie groupoids $G\ltimes M\to\mathscr{G}|_{S}$ which maps the origin $0\in M$ to the point $x$ and which is the projection when restricted to the isotropy group $\mathord{\mathrm{I}}_{x}(G\ltimes M)=G\times\{0\}$. If we compose this isomorphism with the inclusion $\mathscr{G}|_{S}\to\mathscr{G}|_{U}$, we obtain a weak equivalence

[TABLE]

Let $K$ be a compact Lie group, let $\sigma:K\to G$ be a monomorphism of Lie groups and let $\rho=\bar{\sigma}(0)\in\hom(K,G\ltimes M)$. Note that the composition $\lambda\circ\rho\in\hom(K,\mathscr{G}|_{U})$ is also a monomorphism. By Example 19 and Proposition 8 we therefore conclude that

[TABLE]

and

[TABLE]

while

[TABLE]

5.4. Proper regular groupoids

Recall that a Lie groupoid is regular if all its isotropy groups have the same dimension. For more details and a discussion on classification of such Lie groupoids, see [10].

Let $\mathscr{G}$ be a path connected regular proper Lie groupoid. The orbits of the left $\mathscr{G}$-action on $\mathscr{G}_{0}$ are leafs of a regular foliation of the manifold $\mathscr{G}_{0}$ and we have the induced representation of $\mathscr{G}$ on the normal bundle $\mathord{\mathrm{N}}\mathscr{G}$ of this foliation. The kernel of this representation is a locally trivial bundle $\mathord{\mathrm{K}}(\mathscr{G})$ of compact Lie groups over $\mathscr{G}_{0}$, a subbundle of the isotropy bundle $\mathord{\mathrm{I}}(\mathscr{G})$. The fibers of the bundle $\mathord{\mathrm{K}}(\mathscr{G})$ are called the ineffective isotropy groups of $\mathscr{G}$.

Choose a point $x\in\mathscr{G}_{0}$, put $K=\mathord{\mathrm{K}}(\mathscr{G})_{x}$ and let $\rho:K=\mathord{\mathrm{K}}(\mathscr{G})_{x}\to\mathscr{G}$ be the inclusion. By [15, Proposition 11] it follows that

[TABLE]

and any element of $\hom(K,\mathscr{G};\rho)$ is an isomorphism onto an ineffective isotropy group. In particular, the map $\omega:\hom(K,\mathscr{G};\rho)\to\mathscr{G}_{0}$ is a locally trivial bundle and

[TABLE]

The right $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$-action on $\hom(K,\mathscr{G};\rho)$ is free, but also transitive along the fibers of $\omega:\hom(K,\mathscr{G};\rho)\to\mathscr{G}_{0}$, by Corollary 11, and the map

[TABLE]

is a $\mathscr{G}$-equivariant principal $\mathord{\mathrm{Mon}}(\mathscr{G};\rho)$-bundle over $\mathscr{G}_{0}$. The associated functor

[TABLE]

is a Serre fibration of Lie groupoids [15] and hence there is a natural long exact sequence [15, Theorem 12] of homotopy groups:

[TABLE]

Note that for any $n\geq 1$ we have

[TABLE]

and

[TABLE]

by Proposition 6. Furthermore, the homomorphism of groups

[TABLE]

which we call the monodromy map, is surjective.

Example 20**.**

Let $M$ be a connected smooth manifold with a free proper right action of a discrete group $H$, let $G$ be a compact Lie group and let $\eta:H\to\mathord{\mathrm{Aut}}(G)$ be a homomorphism of groups. As in Example 13 we have the associated quotient Lie groupoid

[TABLE]

which is in this case a regular proper Lie groupoid, in fact a locally trivial bundle of Lie groups. Furthermore, we have $\mathord{\mathrm{K}}(\mathscr{B})=\mathord{\mathrm{I}}(\mathscr{B})=\mathscr{B}$. We have a natural isomorphism

[TABLE]

of Lie groupoids over $M/H$, where the left action of $H$ on $\hom(G,G)$ is given by $\eta$ and the composition, that is, for any $h\in H$ and any $\varphi\in\hom(G,G)$ we have $h\varphi=\eta(h)\circ\varphi$. Let $x$ be a point of $M$ and let $\rho\in\hom(G,\mathscr{B})$ be the monomorphism given by $\rho(g)=(x,g)H$. Under the isomorphism $\hom(G,\mathscr{B})\cong M\times_{H}\hom(G,G)$, the monomorphism $\rho$ corresponds to the $H$-orbit of the pair $(x,\mathord{\mathrm{id}}_{G})$.

For any $\varphi\in\hom(G,G)$, the set $G\varphi=\mathord{\mathrm{Inn}}(G)\circ\varphi$ is open, closed and $G$-saturated in $\hom(G,G)$. Note that $\eta(H)\circ\mathord{\mathrm{Inn}}(G)=\mathord{\mathrm{Inn}}(G)\circ\eta(H)$ because the subgroup $\mathord{\mathrm{Inn}}(G)$ is normal in $\mathord{\mathrm{Aut}}(G)$. In particular, the product of the subgroups $\eta(H)$ and $\mathord{\mathrm{Inn}}(G)$ in $\mathord{\mathrm{Aut}}(G)$ is a subgroup

[TABLE]

of the group $\mathord{\mathrm{Aut}}(G)$. It follows that the set $L\circ\varphi$ is open and closed $\hom(G,G)$, and is also the minimal subset of $\hom(G,G)$ which includes $\varphi$ and is both $G$-saturated and $H$-saturated.

This implies that the subset

[TABLE]

is open, closed and $\mathscr{B}$-saturated, and that the natural isomorphism $\hom(G,\mathscr{B})\cong M\times_{H}\hom(G,G)$ restricts to the isomorphism

[TABLE]

In particular, this implies that the monodromy group $\mathord{\mathrm{Mon}}(\mathscr{B};\rho)$ is the product of the subgroups $\eta(H)$ and $\mathord{\mathrm{Inn}}(G)$ in $\mathord{\mathrm{Aut}}(G)$,

[TABLE]