# Representation theory of semidirect products of a compact quantum group with
a finite group

Hua Wang<EMAIL_ADDRESS>Institut de mathématiques de Jussieu – Paris
Rive Gauche

###### Abstract.

We study unitary representations of semidirect products of a compact quantum
group with a finite group. We give a classification of all irreducible unitary
representations, a description of the conjugate representation of irreducible
unitary representations in terms of this classification, and the fusion rules
for the semidirect product.

###### Contents

  1. 1 Semidirect product of a compact quantum group with a finite group
  2. 2 A first look at unitary representations of $\mathbb{G}\rtimes\Lambda$
  3. 3 Principal subgroups of $\mathbb{G}\rtimes\Lambda$
  4. 4 Induced representations of principal subgroups
  5. 5 Some character formulae
  6. 6 Dimension of the intertwiner space of induced representations
  7. 7 The $C^{\ast}$-tensor category $\mathcal{CSR}_{\Lambda_{0}}$
  8. 8 Group actions and projective representations
  9. 9 Pure, stable, distinguished CSRs and representation parameters
  10. 10 Distinguished representation parameters and distinguished representations
  11. 11 Density of matrix coefficients of distinguished representations
  12. 12 Classification of irreducible representations of $\mathbb{G}\rtimes\Lambda$
  13. 13 The conjugate representation of distinguished representations
  14. 14 The incidence numbers
  15. 15 Fusion rules

## Introduction

It is often the case that one can retrieve significant information about
representations of a group $G$ from representations of some subgroups of $G$.
As a trivial example, the study of representations of a direct product
$G\times H$ of groups of $G$ and $H$ can be easily reduced to the study of
representations of $G$ and $H$ separately. However, when one replaces direct
products with the more ubiquitous semidirect products, the situation quickly
becomes complicated. To get a taste of this complication, the classic
[MR0450380, §8.2] treats representations of a semidirect product $G\rtimes H$
in the special case where $G,H$ are both finite and $G$ is abelian.

In the setting of locally compact groups and their unitary representations,
via the theories of systems of imprimitivity, induced representations,
projective representations (a.k.a. ray representations), etc., George Mackey
developed a heavy machinery of techniques, often referred as Mackey’s
analysis, Mackey’s machine or the little group method (which is also due to
Wigner), to attack such kind of problems. Subsequent works based on Mackey’s
analysis emerge rapidly, making it one of the most powerful tools to study
unitary representations of locally compact groups. For an introduction of this
development, we refer the reader to [MR0098328, MR44536, MR0031489, MR936629,
MR3012851] among the large volumes of literature on this subject.

The author’s own interest of this subject comes from the joint work
[2018arXiv181204078F] with Pierre Fima. In [2018arXiv181204078F], we
systematically studied the permanence of property (RD) and polynomial growth
of the dual of a bicrossed product of a matched pair consisting of a second
countable compact group and a countable discrete group. The natural subsequent
question of constructing examples of nontrivial bicrossed products with or
without (RD) leads one to study closely the representation theory of
semidirect products $G\rtimes\Lambda$ of a compact group $G$ with a finite
group $\Lambda$. More precisely, as required by the study of length functions
relevant to these properties, we need a classification of all irreducible
unitary representations of $G\rtimes\Lambda$, the conjugate (which, when we
adopt the point of view of topological quantum groups as in this paper, is
also the contragredient since classic groups are of Kac type) of irreducible
representations in terms of this classification, and most importantly, the
fusion rules of $G\rtimes\Lambda$, i.e. how the tensor product of two
irreducible representations decomposes into a direct sum of irreducible
representations. While the first two questions can be settled using Mackey’s
machine as mentioned above, the fusion rules, however, to the best of the
author’s limited knowledge, are never calculated in the literature.

This paper treats these questions in the more general setting of semidirect
products of the form $\mathbb{G}\rtimes\Lambda$, where $\mathbb{G}$ is a
compact quantum group and $\Lambda$ a finite group. However, instead of using
systems of imprimitivity, we introduce the notion of representation parameters
(see Definition 9.8), which appears naturally when we try to analyze the rigid
$C^{\ast}$-tensor category $\operatorname{Rep}(\mathbb{G}\rtimes\Lambda)$.
Roughly speaking, a representation parameter is a triple $(u,V,v)$, where $u$
is an irreducible representation of $\mathbb{G}$ on some finite dimensional
Hilbert space $\mathscr{H}$, $V$ is a unitary projective representation of a
certain subgroup $\Lambda_{0}$ of $\Lambda$ on the same space $\mathscr{H}$,
and $v$ is a unitary projective representation of the same $\Lambda_{0}$ on
some other finite dimensional Hilbert space, such that $V$ is covariant with
$u$ in a certain sense, and $V$ and $v$ have opposing cocycles. Here, the
subgroup $\Lambda_{0}$ arises as an isotropy subgroup of a natural action
$\Lambda\curvearrowright\operatorname{Irr}(\mathbb{G})$, and the projective
representation $V$ is then determined by Schur’s lemma on irreducible
representations.

As the precise formulation of our main results are long and complicated, we
give here only a crude summary of these results in terms of representation
parameters (see Definition 9.8) mentioned above.

  1. (A)

Up to equivalence, irreducible unitary representations of
$\mathbb{G}\rtimes\Lambda$ are classified by (equivalence classes of)
representation parameters (see Theorem 12.1 for the precise formulation);

  2. (B)

The classification in (A) is compatible with the conjugate operation— the
conjugate111The conjugate should _not_ be confused with the contragredient,
with the contragredient not necessarily unitary if the quantum group is not
unimodular (of Kac-type).of an irreducible representation of
$\mathbb{G}\rtimes\Lambda$ parameterized by some representation parameter
$(u,V,v)$ is itself parameterized by the conjugate of $(u,V,v)$ (see Theorem
13.5 for the precise formulation);

  3. (C)

The fusion rules of $\mathbb{G}\rtimes\Lambda$ is calculated by summing a
series of incidence numbers, where all of these numbers can be calculated
using unitary projective representations of some suitable subgroup of
$\Lambda$ through an explicit reduction procedure (see Theorem 15.1 for the
precise formulation), where the reduction procedure itself is determined by
the representation theory of $\mathbb{G}$ and the action of $\Lambda$ acting
on $\mathbb{G}$, with respect to which we form the semidirect product.

While (A) and (B) may well be regarded as the quantum analogue of the
corresponding results of Mackey’s analysis in the classical case of groups,
our result (C) is new, even in the case where $\mathbb{G}$ is another finite
group. We should mention that our main idea of this paper starts with
reformulating $\operatorname{Rep}(\mathbb{G})$ as a semisimple rigid
$C^{\ast}$-tensor category for an arbitrary compact quantum group
$\mathbb{G}$, which is the modern point of view; however, Mackey’s ingenious
ideas, such as studying the dynamics of a naturally appeared group action on
the representations of a normal subgroup, and using projective representations
of the isotropy subgroups of this action, still play an essential part in the
development of this theory.

We now describe the organization of this paper. The numerous sections of this
paper are roughly divided into the following four parts. In the first part (§
1 and § 2), we lay out the basic properties and constructions of the objects
to be studied in this paper—semidirect products of a compact quantum group by
a finite group and their unitary representations. In § 2, the problem of
describing unitary representations of these semidirect products is reduced to
the study of the so-called covariant pair of representations for each of the
factors. The second part (§§ 3\- 6) gives a self-contained treatment of
induced representation which will be used later in this paper. We are aware
that there are already much more general theory for induced representations in
the quantum setting, e.g. [MR1934609] based on the classic work [MR0353003].
Moreover S. Vaes has generalizes a large part of Mackey’s theory of
imprimitivity to locally compact quantum groups in [MR2182592]. Besides the
obvious reason for fixing the notations, the treatment of the induced
representation here is specially tailored to the various calculations in the
later half of this paper. The third part (§§ 7-11) is the technical core of
this paper. The treatment here is largely inspired by Woronowicz’s Krein-
Tannaka reconstruction [MR943923] of a compact quantum group from its
representation category. Here instead of directly attacking the representation
category $\operatorname{Rep}{\mathbb{G}\rtimes\Lambda}$ of the semidirect
product, we introduce and study a family of rigid $C^{\ast}$-tensor categories
(called the category of covariant systems of representations and denoted by
$\mathcal{CSR}_{\Lambda_{0}}$ with respect to some suitable subgroup of
$\Lambda$), each of which has a simpler structure. Combining the information
we have on these simpler $C^{\ast}$-tensor categories allows us not only to
classify the irreducible unitary representations of
$\mathbb{G}\rtimes\Lambda$, but also to calculate the fusion rules of
$\mathbb{G}\rtimes\Lambda$. The details of this classification and calculation
are given in the fourth part (§§ 12-15).

Before we proceed further, we feel that we should say a little more about § 1
for the experts. We emphasize our construction of semidirect products as the
axiomatically more elaborate _algebraic_ compact quantum groups, the theory of
which is developed by van Daele [MR1658585, MR1378538, MR1220906], instead of
the more modern and standard formulation, due to Woronowicz [MR1616348,
MR943923, MR901157], using $C^{\ast}$-algebras. Of course, these two
approaches are essentially equivalent—one passes from Woronowicz’s approach to
van Daele’s via the Peter-Weyl theory for compact quantum groups, and from van
Daele’s approach to Woronowicz’s via the famous GNS construction with respect
to the Haar integral. The reasons we prefer van Daele’s algebraic theory here
are two-fold: on the one hand, one has the advantage of having direct access
to the Haar state and the antipode, as well as the polynomial algebra, which
are powerful tools for our purposes of studying the representations of these
objects (or corepresentations if one insists on viewing these essentially
analytic objects as Hopf algebras); on the other hand, when one tries to
restrict representations to certain (quantum) subgroups of these semidirect
products, as we will do later, one will need to use the counit, which is
always everywhere defined in the more elaborate algebraic approach of van
Daele, but is merely densely defined if the compact quantum group in the sense
of Woronowicz is not universal. We also point out here that the term
_semidirect product_ in the quantum setting has an unfortunate ambiguity.
Nowadays many use this term to refer to the crossed product, as first defined
and studied by S. Wang [MR1347410]. In the case of classical groups, it is
long known that this crossed product construction yields the _convolution
algebra_ of the semidirect of groups. So if we believe classic compact groups
are exactly compact quantum groups whose algebra is commutative, then this is
not the correct formulation for semidirect products, even though these are
closely related via the convolution operation (which is a manifestation of the
quantum version of Pontryagin’s duality as developed in [MR1832993], preceded
by many important works along the lines of the Kac’s program, a history of
which is described in the introduction of the above article). To be more
precise, this crossed product of S. Wang is in fact a special case of the
bicrossed product as described in [MR1970242] where one of the actions for the
matched pair is trivial; what we call semidirect in this paper is a special
case of the double crossed product as described in [MR2115071] where again one
of the actions for the matched pair is trivial. We don’t pursue the full
generality of the bicrossed product construction and double crossed product
construction here, but merely point out that they are all based on the notion
of matched pair of (quantum) groups ([MR1045735, MR1092128],[MR611561]). We
also mention in passing works such as [MR1235438], [MR1098985], [MR320056],
[MR1745504], and [MR1970242] in the direction of bicrossed products, and works
such as [MR1235438], [MR1098985] and [MR2115071] in the direction of double
crossed products. We hope these backgrounds provide some justification of our
choice of terminology for semidirect products by its consistency with the
classical group case. We also note that representation theory for compact
bicrossed products (which includes the crossed product as a special case) of a
matched pair of classical groups are thoroughly investigated in the author’s
joint work with P. Fima [2018arXiv181204078F], and as one can see by comparing
the results there and the results of this paper, the representation theory for
semidirect products are significantly more delicate than crossed products,
even for classical finite groups.

We conclude this introduction by making some conventions. All representations
and projective representations in this paper are finite dimensional. All of
them are unitary, except the contragredient of a unitary representation, which
may not be unitary when the compact quantum group is not of Kac-type. We also
assume all (projective) representations are over a finite dimensional Hilbert
space instead of a mere complex vector space. Terminologies and notations
concerning compact quantum groups and $C^{\ast}$-tensor categories are largely
in consistent with those in [MR3204665]. We also use freely the Peter-Weyl
theory for projective representations of finite groups as presented in
[MR3299063]. We also freely use the Heyenmann-Sweedler notation in performing
calculations on comultiplications. The unitary group of unitary
transformations from a Hilbert space $\mathscr{H}$ to itself is denoted by
$\mathcal{U}(\mathscr{H})$. From § 8 on, $\mathbb{T}$ denotes the circle
group, i.e. the abelian compact group $\left\\{z\in\mathbb{C}\;:\;\left\lvert
z\right\rvert=1\right\\}$ viewed as a subgroup of $\mathbb{C}^{\times}$. Since
we often view a representation of compact quantum groups as an operator, we
denote the tensor product of representations using $\times$ instead of
$\otimes$, as the latter is reserved to denote tensor products of spaces,
algebras, linear operators, etc. Finally, throughout this paper, we fix a
compact quantum group $\mathbb{G}=(A,\Delta)$, a finite group $\Lambda$, and
an antihomomorphism of groups
$\alpha^{\ast}\colon\Lambda\rightarrow\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$,
where $\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$ is the
subgroup222Note that the notation $\operatorname{Aut}(\mathbb{G})$ has a
certain ambiguity which we try to avoid: one the one hand, if we let
$\mathbb{G}$ to be a classical compact group, then elements of
$\operatorname{Aut}(\mathbb{G})$ are group automorphisms, and the group law of
$\operatorname{Aut}(\mathbb{G})$ is given by composition of _set-theoretic_
mappings; on the other hand, if we view $\mathbb{G}$ as a
Hopf-$C^{\ast}$-algebra, say $\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$, then
$\operatorname{Aut}(\mathbb{G})$ can also be mean the automorphism group of
this Hopf-$C^{\ast}$-algebra, whose group law is given by composition of _Hopf
algebraic_ -morphisms. This is the reason we prefer the more cumbersome
notation $\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$ instead of
the ambiguous but more succinct $\operatorname{Aut}(\mathbb{G})$. of
$\operatorname{Aut}\bigl{(}C(\mathbb{G})\bigr{)}$ consisting of automorphisms
of the $C^{\ast}$-algebra $C(\mathbb{G})$ that intertwines the
comultiplication $\Delta$.

Acknowledgment. This work is funded by the ANR Project (No. ANR-19-CE40-0002),
to which the author expresses his sincere gratitude. The author especially
thanks Pierre Fima for his encouragement and various helpful discussions
duringthe production of this work.

##  1\. Semidirect product of a compact quantum group with a finite group

Let $\mathbb{G}=(A,\Delta)$ be a compact quantum group, $\Lambda$ a finite
group. An action of $\Lambda$ on $\mathbb{G}$ via quantum automorphisms is an
antihomomorphism
$\alpha^{\ast}\colon\Lambda\rightarrow\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$.
One can then form the semi-direct $\mathbb{G}\rtimes_{\alpha^{\ast}}\Lambda$,
or simply $\mathbb{G}\rtimes\Lambda$ if the action $\alpha^{\ast}$ is clear
from the context, which is again a compact quantum group. The underlying
$C^{\ast}$-algebra $\mathscr{A}$ of $\mathbb{G}\rtimes\Lambda$ is $A\otimes
C(\Lambda)$, and the comultiplication $\widetilde{\Delta}$ on $\mathscr{A}$ is
determined by

(1.1)
$\widetilde{\Delta}(a\otimes\delta_{r})=\sum_{s\in\Lambda}{\left[(\operatorname{id}_{A}\otimes\alpha^{\ast}_{s})\Delta(a)\right]}_{13}{(\delta_{s}\otimes\delta_{s^{-1}r})}_{24}\in
A\otimes C(\Lambda)\otimes A\otimes C(\Lambda)$

for any $a\in A$ and $r\in\Lambda$. As we’ve mentioned at the end of the
introduction, from now on, $\mathbb{G}$, $\Lambda$ and the action
$\alpha^{\ast}$ are fixed until the end of the paper.

It is clear that $\widetilde{\Delta}$ is a unital $\ast$-morphism. We now
check that in the six-fold tensor product $A\otimes C(\Lambda)\otimes A\otimes
C(\Lambda)\otimes A\otimes C(\Lambda)$, we have

(1.2) $\forall a\in
A,r\in\Lambda,\qquad(\operatorname{id}\otimes\operatorname{id}\otimes\widetilde{\Delta})[\widetilde{\Delta}(a\otimes\delta_{r})]=(\widetilde{\Delta}\otimes\operatorname{id}\otimes\operatorname{id})[\widetilde{\Delta}(a\otimes\delta_{r})],$

i.e. our new comultiplication $\widetilde{\Delta}$ is coassociative. Indeed,
put
$\Delta^{(2)}:=(\operatorname{id}\otimes\Delta)\Delta=(\Delta\otimes\operatorname{id})\Delta$,
since
$\alpha^{\ast}_{s}\in\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$
for all $s\in\Lambda$, we have
$(\alpha_{s}^{\ast}\otimes\alpha_{s}^{\ast})\circ\Delta=\Delta\circ\alpha_{s}^{\ast}$,

(1.3) $\begin{split}&\hskip
11.99998pt(\operatorname{id}\otimes\operatorname{id}\otimes\widetilde{\Delta})[\widetilde{\Delta}(a\otimes\delta_{r})]\\\
&=(\operatorname{id}\otimes\operatorname{id}\otimes\widetilde{\Delta})\left(\sum_{s\in\Lambda}{[(\operatorname{id}\otimes\alpha_{s}^{\ast})\Delta(a)]}_{13}{(\delta_{s}\otimes\delta_{s^{-1}r})}_{24}\right)\\\
&=\sum_{s,t\in\Lambda}{\Bigl{\\{}\bigl{[}\bigl{(}\operatorname{id}\otimes\alpha_{s}^{\ast}\otimes(\alpha_{t}^{\ast}\circ\alpha_{s}^{\ast})\bigr{)}(\operatorname{id}\otimes\Delta)\Delta\bigr{]}(a)\Bigr{\\}}}_{135}{(\delta_{s}\otimes\delta_{t}\otimes\delta_{t^{-1}s^{-1}r})}_{246}\\\
&=\sum_{s,t\in\Lambda}{\Bigl{\\{}\bigl{[}(\operatorname{id}\otimes\alpha_{s}^{\ast}\otimes\alpha_{st}^{\ast})\Delta^{(2)}\bigr{]}(a)\Bigr{\\}}}_{135}{(\delta_{s}\otimes\delta_{t}\otimes\delta_{t^{-1}s^{-1}r})}_{246}.\end{split}$

On the other hand,

(1.4) $\begin{split}&\hskip
11.99998pt(\widetilde{\Delta}\otimes\operatorname{id}\otimes\operatorname{id})[\widetilde{\Delta}(a\otimes\delta_{r})]\\\
&=(\widetilde{\Delta}\otimes\operatorname{id}\otimes\operatorname{id})\left(\sum_{s\in\Lambda}{[(\operatorname{id}\otimes\alpha_{s}^{\ast})\Delta(a)]}_{13}{(\delta_{s}\otimes\delta_{s^{-1}r})}_{24}\right)\\\
&=\sum_{s,t\in\Lambda}\Bigl{\\{}\bigl{[}(\operatorname{id}\otimes\alpha_{t}^{\ast}\otimes\alpha_{s}^{\ast})(\Delta\otimes\operatorname{id})\Delta\bigr{]}(a)\Bigr{\\}}_{135}{(\delta_{t}\otimes\delta_{t^{-1}s}\otimes\delta_{s^{-1}r})}_{246}\\\
&\hskip 11.99998pt(s^{\prime}=t,t^{\prime}=t^{-1}s\iff
s=s^{\prime}t^{\prime},t=s^{\prime})\\\
&=\sum_{s^{\prime},t^{\prime}\in\Lambda}\Bigl{\\{}\bigl{[}(\operatorname{id}\otimes\alpha_{s^{\prime}}^{\ast}\otimes\alpha_{s^{\prime}t^{\prime}}^{\ast})\Delta^{(2)}\bigr{]}(a)\Bigr{\\}}_{135}{(\delta_{s^{\prime}}\otimes\delta_{t^{\prime}}\otimes\delta_{t^{\prime-1}s^{\prime-1}r})}_{246}.\end{split}$

Now (1.2) follows from (1.3) and (1.4).

Since $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$ is dense in $A\otimes
C(\Lambda)$, in order to prove that $\mathbb{G}\rtimes\Lambda$ is indeed a
compact quantum group, it suffices to show that
$(\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda),\widetilde{\Delta})$ is an
algebraic compact quantum group, i.e. a Hopf $\ast$-algebra with an invariant
state (called the Haar state or Haar integral).

First of all, since
$\alpha_{s}^{\ast}\in\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$
for all $s\in\Lambda$, we have
$\alpha_{s}^{\ast}\bigl{(}\operatorname{Pol}(\mathbb{G})\bigr{)}=\operatorname{Pol}(\mathbb{G})$,
and $\widetilde{\Delta}$ indeed restricts to a well-defined comultiplication
on $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$.

Let $\epsilon,S$ be the counit and the antipode respectively for the Hopf
$\ast$-algebra $\operatorname{Pol}(\mathbb{G})$. Denoting the neutral element
of the group $\Lambda$ by $e$, we define

(1.5)
$\begin{split}\widetilde{\epsilon}\colon\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)&\rightarrow\mathbb{C}\\\
\sum_{r\in\Lambda}x_{r}\otimes\delta_{r}&\mapsto\epsilon(x_{e}),\end{split}$

and

(1.6) $\begin{split}\widetilde{S}\colon\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)&\rightarrow\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)\\\
\sum_{r\in\Lambda}x_{r}\otimes\delta_{r}&\mapsto\sum_{r\in\Lambda}\alpha_{r}^{\ast}(S(x_{r}))\otimes\delta_{r^{-1}}=\sum_{r\in\Lambda}S(\alpha_{r^{-1}}^{\ast}(x_{r^{-1}}))\otimes\delta_{r}.\end{split}$

Since $\epsilon$ is a $\ast$-morphism of algebras, so is
$\widetilde{\epsilon}$. Moreover, for any $x\in\operatorname{Pol}(\mathbb{G})$
and $r\in\Lambda$, we have

$\displaystyle\hskip
11.99998pt(\widetilde{\epsilon}\otimes\operatorname{id})\widetilde{\Delta}(x\otimes\delta_{r})=(\widetilde{\epsilon}\otimes\operatorname{id})\sum_{s\in\Lambda}\sum
x_{(1)}\otimes\delta_{s}\otimes\alpha_{s}^{\ast}(x_{(2)})\otimes\delta_{s^{-1}r}$
$\displaystyle=\sum\epsilon(x_{(1)})\alpha_{e}^{\ast}(x_{(2)})\otimes\delta_{r}=\sum\epsilon(x_{(1)})x_{(2)}\otimes\delta_{r}=x\otimes\delta_{r}$
$\displaystyle=\sum x_{(1)}\epsilon(x_{(2)})\otimes\delta_{r}=\sum
x_{(1)}\epsilon(\alpha_{r}^{\ast}(x_{(2)}))\otimes\delta_{r}$
$\displaystyle=(\operatorname{id}\otimes\widetilde{\epsilon})\sum_{s\in\Lambda}\sum
x_{(1)}\otimes\delta_{s}\otimes\alpha_{s}^{\ast}(x_{(2)})\otimes\delta_{s^{-1}r}=(\operatorname{id}\otimes\widetilde{\epsilon})\widetilde{\Delta}(x\otimes\delta_{r}).$

Hence $\widetilde{\epsilon}$ is the counit for $\widetilde{\Delta}$. Let
$m\colon\operatorname{Pol}(G)\otimes\operatorname{Pol}(G)\rightarrow\operatorname{Pol}(G)$
be the multiplication map, and $\widetilde{m}$ the multiplication map on
$\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$, then

$\displaystyle\widetilde{m}(\widetilde{S}\otimes\operatorname{id})\widetilde{\Delta}(x\otimes\delta_{r})$
$\displaystyle=\widetilde{m}(\widetilde{S}\otimes\operatorname{id})\sum_{s\in\Lambda}\sum
x_{(1)}\otimes\delta_{s}\otimes\alpha_{s}^{\ast}(x_{(2)})\otimes\delta_{s^{-1}r}$
$\displaystyle=\widetilde{m}\sum_{s\in\Lambda}\sum\alpha_{s}^{\ast}(S(x_{(1)}))\otimes\delta_{s^{-1}}\otimes\alpha_{s}^{\ast}(x_{(2)})\otimes\delta_{s^{-1}r}$
$\displaystyle=\sum_{s\in\Lambda}\left[m(S\otimes\operatorname{id})(\alpha_{s}^{\ast}\otimes\alpha_{s}^{\ast})\Delta(x)\right]\otimes\delta_{s^{-1}}\cdot\delta_{s^{-1}r}$
$\displaystyle=\delta_{e,r}\sum_{s\in\Lambda}\left[m(S\otimes\operatorname{id})\Delta(\alpha_{s}^{\ast}(x))\right]\otimes\delta_{s^{-1}}$
$\displaystyle=\delta_{e,r}\sum_{s\in\Lambda}\epsilon(\alpha_{s}^{\ast}(x))1_{A}\otimes\delta_{s^{-1}}$
$\displaystyle=\delta_{e,r}\epsilon(x)1_{A}\otimes\sum_{s\in\Lambda}\delta_{s^{-1}}$
$\displaystyle=\delta_{e,r}\epsilon(x)1_{A}\otimes
1_{C(\Lambda)}=\widetilde{\epsilon}(x\otimes\delta_{r})1_{A}\otimes
1_{C(\Lambda)}.$

Similarly, since for any $s\in\Lambda$,

$\alpha_{s^{-1}r}^{\ast}S\alpha_{s}^{\ast}=\alpha_{s^{-1}r}^{\ast}\alpha_{s}^{\ast}S=\alpha_{r}^{\ast}S,$

we have

$\displaystyle\widetilde{m}(\operatorname{id}\otimes\widetilde{S})\widetilde{\Delta}(x\otimes\delta_{r})$
$\displaystyle=\widetilde{m}(\operatorname{id}\otimes\widetilde{S})\sum_{s\in\Lambda}\sum
x_{(1)}\otimes\delta_{s}\otimes\alpha_{s}^{\ast}(x_{(2)})\otimes\delta_{s^{-1}r}$
$\displaystyle=\widetilde{m}\sum_{s\in\Lambda}\sum
x_{(1)}\otimes\delta_{s}\otimes(\alpha_{s^{-1}r}^{\ast}S\alpha_{s}^{\ast})(x_{(2)})\otimes\delta_{r^{-1}s}$
$\displaystyle=\widetilde{m}\sum_{s\in\Lambda}\sum
x_{(1)}\otimes\delta_{s}\otimes(\alpha_{r}^{\ast}S)(x_{2})\otimes\delta_{r^{-1}s}$
$\displaystyle=\delta_{e,r}\sum_{s\in\Lambda}\sum
x_{(1)}[S(x_{(2)})]\otimes\delta_{s}=\delta_{e,r}\sum_{s\in\Lambda}\epsilon(x)1_{A}\otimes\delta_{s}$
$\displaystyle=\delta_{e,r}\epsilon(x)1_{A}\otimes
1_{C(\Lambda)}=\widetilde{\epsilon}(x\otimes\delta_{r})1_{A}\otimes
1_{C(\Lambda)}.$

Therefore, $\widetilde{S}$ is the antipode for
$(\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda),\widetilde{\Delta})$.

It remains to construct the Haar state on the Hopf $\ast$-algebra
$\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$. Suppose $h\colon
A\rightarrow\mathbb{C}$ is the Haar state on $\mathbb{G}$, define

(1.7) $\begin{split}\widetilde{h}\colon\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)&\to\mathbb{C}\\\
\sum_{r}x_{r}\otimes\delta_{r}&\mapsto\left\lvert\Lambda\right\rvert^{-1}\sum_{r\in\Lambda}h(x_{r}).\end{split}$

It is obvious that $\widetilde{h}$ is a state. For any
$x\in\operatorname{Pol}(\mathbb{G})$, $r\in\Lambda$,

$\displaystyle(\widetilde{h}\otimes\operatorname{id})\widetilde{\Delta}(x\otimes\delta_{r})$
$\displaystyle=\left\lvert\Lambda\right\rvert^{-1}\sum_{s\in\Lambda}\sum
h(x_{(1)})\alpha_{s}^{\ast}(x_{(2)})\otimes\delta_{s^{-1}r}$
$\displaystyle=\left\lvert\Lambda\right\rvert^{-1}\sum_{s\in\Lambda}\alpha_{s}^{\ast}\bigl{(}\sum
h(x_{(1)})x_{(2)}\bigr{)}\otimes\delta_{s^{-1}r}$
$\displaystyle=\left\lvert\Lambda\right\rvert^{-1}\sum_{s\in\Lambda}\alpha_{s}^{\ast}(h(x)1_{A})\otimes\delta_{s^{-1}r}$
$\displaystyle=\left\lvert\Lambda\right\rvert^{-1}h(x)\sum_{s\in\Lambda}1_{A}\otimes\delta_{s^{-1}r}$
$\displaystyle=\widetilde{h}(x\otimes\delta_{r})1_{A}\otimes 1_{C(\Lambda)}.$

The uniqueness of the Haar state implies that $h\circ\alpha_{s}^{\ast}=h$ for
any $s\in\Lambda$, hence

$\displaystyle(\operatorname{id}\otimes\widetilde{h})\widetilde{\Delta}(x\otimes\delta_{r})$
$\displaystyle=\left\lvert\Lambda\right\rvert^{-1}\sum_{s\in\Lambda}\sum
x_{(1)}h\bigl{(}\alpha_{s}^{\ast}(x_{(2)})\bigr{)}\otimes\delta_{s}$
$\displaystyle=\left\lvert\Lambda\right\rvert^{-1}\sum_{s\in\Lambda}\sum
x_{(1)}h(x_{(2)})\otimes\delta_{s}$
$\displaystyle=\left\lvert\Lambda\right\rvert^{-1}h(x)1_{A}\otimes\sum_{s\in\Lambda}\delta_{s}$
$\displaystyle=\widetilde{h}(x\otimes\delta_{r})1_{A}\otimes 1_{C(\Lambda)}.$

Therefore, $\widetilde{h}$ is indeed the Haar state on
$(\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda),\widetilde{\Delta})$. So
far, we’ve established that $(\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda),\widetilde{\Delta})$ is an algebraic compact quantum group (cf.
[MR2397671, paper $3$]).

Now the density of $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$ in
$A\otimes C(\Lambda)$ implies that $(A\otimes C(\Lambda),\widetilde{\Delta})$
is indeed a compact quantum group, with

(1.8)
$\operatorname{Pol}(\mathbb{G}\rtimes\Lambda)=\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda),$

and Haar state (which we still denote by $\widetilde{h}$)

(1.9) $\begin{split}\widetilde{h}:A\otimes C(\Lambda)&\rightarrow\mathbb{C}\\\
\sum
x_{r}\otimes\delta_{r}&\mapsto\left\lvert\Lambda\right\rvert^{-1}\sum_{r\in\Lambda}h(x_{r}).\end{split}$

Furthermore, we’ve seen that the counit $\widetilde{\epsilon}$ and the
antipode $\widetilde{S}$ of the Hopf $\ast$-algebra
$\operatorname{Pol}(\mathbb{G}\rtimes\Lambda)$ are given by (1.5) and (1.6)
respectively (cf. [MR2397671, §5.4.2]).

######  Definition 1.1.

Using the above notations, it is well-known that the analytic compact quantum
group $(\mathscr{A},\widetilde{\Delta})$ and the algebraic compact quantum
group $(A\otimes C(\Lambda),\widetilde{\Delta})$ are equivalent descriptions
of the same object, which we call the semidirect product of $\mathbb{G}$ and
$\Lambda$ with respect to the action $\alpha^{\ast}$, and is denoted by
$\mathbb{G}\rtimes_{\alpha^{\ast}}\Lambda$, or simply
$\mathbb{G}\rtimes\Lambda$ if the underlying action $\alpha^{\ast}$ is clear
from context.

######  Remark 1.2.

There is a faster way of establishing $\mathbb{G}\rtimes\Lambda$ as a compact
quantum group, which we refer to as the analytic approach. Namely, one might
use (1.1) directly to define a comultiplication on the $C^{\ast}$-algebra
$A\otimes C(\Lambda)$ and show that this comultiplication satisfy the density
condition in the definition of a compact quantum group in the sense of
Woronowicz (cf. [MR1616348]). We prefer the more algebraic approach presented
above as it provides more insight for our purpose of studying representations
of $\mathbb{G}\rtimes\Lambda$. As an illustration, from our treatment, one
knows immediately that
$\operatorname{Pol}(\mathbb{G}\rtimes\Lambda)=\operatorname{Pol}(\mathbb{G})\rtimes\Lambda$,
a fact that is not clear from the faster analytic approach.

######  Remark 1.3.

When $\mathbb{G}$ comes from a genuine compact group $G$, it is easy to check
via Gelfand theory that the antihomomorphism
$\alpha^{\ast}\colon\Lambda\rightarrow\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$
comes from the pull-back of a group morphism
$\alpha\colon\Lambda\rightarrow\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$,
and $\mathbb{G}\rtimes\Lambda$ is exactly the compact group
$G\rtimes_{\alpha}\Lambda$ viewed as a compact quantum group, where the group
law on $G\times\Lambda$ is defined by

(1.10) $\forall g,h\in
G,\,r,s\in\Lambda,\qquad(g,r)(h,s)=\bigl{(}g\alpha_{r}(h),rs\bigr{)}.q$

In treating the dual objects of some rigid $C^{\ast}$-tensor to be presented
later, the following result will be useful.

######  Proposition 1.4.

The compact quantum group $\mathbb{G}\rtimes\Lambda$ is of Kac type if and
only if $\mathbb{G}$ is of Kac type.

###### Proof.

Of the many equivalent characterization for a compact quantum group to be of
Kac type333see e.g. [MR3204665, §1.7], we use the fact that such a quantum
group is of Kac type if and only if the antipode of its polynomial algebra
preserves adjoints. The proposition now becomes trivial in view of (1.6). ∎

##  2\. A first look at unitary representations of $\mathbb{G}\rtimes\Lambda$

A unitary representation $U$ of a classic compact semidirect product
$G\rtimes\Lambda$ is determined by the restrictions $U_{G}$ and $U_{\Lambda}$
on the subgroups $G\times 1_{\Lambda}\simeq G$ and
$1_{G}\times\Lambda\simeq\Lambda$ respectively. It is easy to see that (cf.
(1.10))

(2.1) $\begin{split}\forall g\in G,r\in\Lambda,&\hskip
11.99998ptU_{G}(\alpha_{r}(g))U_{\Lambda}(r)=U(\alpha_{r}(g),r)\\\
&=U((1,r)(g,1))=U_{\Lambda}(r)U_{G}(g).\end{split}$

Conversely, suppose $U_{G}$, $U_{\Lambda}$ are unitary representations on the
same Hilbert space of $G$ and $\Lambda$ respectively, if (2.1) is satisfied,
then $U(g,r):=U_{G}(g)U_{\Lambda}(r)$ defines a unitary representation of
$G\rtimes\Lambda$. When $G$ is replaced by a general compact quantum group
$\mathbb{G}$, even though the “elements” of $\mathbb{G}$ are no longer
available, one can still establish a reasonable quantum analogue. We begin
with a simple lemma.

######  Lemma 2.1.

Let $\epsilon$ be the counit for $\operatorname{Pol}(\mathbb{G})$,
$\epsilon_{\Lambda}$ the counit for $C(\Lambda)$, then
$\epsilon\otimes\operatorname{id}_{C(\Lambda)}$ is a Hopf $\ast$-algebra
morphism from $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$ onto
$C(\Lambda)$, and
$\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\epsilon_{\Lambda}$
is a Hopf $\ast$-algebra morphism from $\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)$ onto $\operatorname{Pol}(\mathbb{G})$.

###### Proof.

Since the antipodes are $\ast$-morphisms of involutive algebras, it suffices
to check that both morphisms preserve comultiplication.

Take any $a\in\operatorname{Pol}(\mathbb{G})$, $r\in\Lambda$, we have

$\displaystyle\hskip
11.99998pt[(\epsilon\otimes\operatorname{id})\otimes(\epsilon\otimes\operatorname{id})]\Delta_{\mathbb{G}\rtimes\Lambda}(a\otimes\delta_{r})$
$\displaystyle=\sum_{s\in\Lambda}\sum\epsilon(a_{(1)})\epsilon(\alpha_{s}^{\ast}(a_{(2)}))\delta_{s}\otimes\delta_{s^{-1}r}$
$\displaystyle=\sum_{s\in\Lambda}\sum\epsilon(a_{(1)})\epsilon(a_{(2)})\delta_{s}\otimes\delta_{s^{-1}r}$
$\displaystyle=\sum_{s\in\Lambda}\epsilon(a)\delta_{s}\otimes\delta_{s^{-1}t}=\Delta_{\Lambda}(\epsilon\otimes\operatorname{id})(a\otimes\delta_{r}),$

where $\Delta_{\Lambda}$ is the comultiplication for $\Lambda$ viewed as a
compact quantum group. Thus $\epsilon\otimes\operatorname{id}$ preserves
comultiplication. On the other hand, note that
$\epsilon_{\Lambda}(\delta_{r})=\delta_{r,1_{\Lambda}}$, we have

$\displaystyle\hskip
11.99998pt[(\operatorname{id}\otimes\epsilon_{\Lambda})\otimes(\operatorname{id}\otimes\epsilon_{\Lambda})]\Delta_{\mathbb{G}\otimes\Lambda}(a\otimes\delta_{r})$
$\displaystyle=\sum_{s\in\Lambda}\delta_{s,1_{\Lambda}}\delta_{s^{-1}r,1_{\Lambda}}\sum
a_{(1)}\otimes\alpha_{s}^{\ast}(a_{(2)})$
$\displaystyle=\delta_{r,1_{\Lambda}}\sum a_{(1)}\otimes a_{(2)}$
$\displaystyle=\delta_{r,1_{\Lambda}}\Delta(a)=\Delta[(\operatorname{id}\otimes\epsilon_{\Lambda})(a\otimes\delta_{r})].$

Thus $\operatorname{id}\otimes\epsilon_{\Lambda}$ preserves comultiplication
too. ∎

Let $U\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)$ be a finite dimensional unitary representation of
$\mathbb{G}\rtimes\Lambda$. Define the unitaries

$\operatorname{Res}_{\mathbb{G}}(U)\colon=(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\epsilon_{\Lambda})(U)\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G}),$

and

$\operatorname{Res}_{\Lambda}(U)\colon=(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\epsilon_{\mathbb{G}}\otimes\operatorname{id}_{C(\Lambda)})(U)\in\mathcal{B}(\mathscr{H})\otimes
C(\Lambda).$

Then by Lemma 2.1, we see that $\operatorname{Res}_{\mathbb{G}}(U)$ is a
finite dimensional unitary representation of $\mathbb{G}$ and
$\operatorname{Res}_{\Lambda}(U)$ a finite dimensional unitary representation
of $\Lambda$. We call $\operatorname{Res}_{\mathbb{G}}(U)$ (resp.
$\operatorname{Res}_{\Lambda}(U)$) the restriction of $U$ to $\mathbb{G}$
(resp. $\Lambda$). For reasons to be explained presently, we also write
$U_{\mathbb{G}}$ for $\operatorname{Res}_{\mathbb{G}}(U)$ and $U_{\Lambda}$
for $\operatorname{Res}_{\Lambda}(U)$.

######  Proposition 2.2.

Using the above notations, we have

(2.2) $\forall r_{0}\in\Lambda,\quad(U_{\Lambda}(r_{0})\otimes
1_{A})U_{\mathbb{G}}=[(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\alpha_{r_{0}}^{\ast})(U_{\mathbb{G}})](U_{\Lambda}(r_{0})\otimes
1_{A})$

in $\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})$. Moreover,

(2.3)
$U={(U_{\mathbb{G}})}_{12}{(U_{\Lambda})}_{13}\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda).$

Conversely, suppose $U_{\mathbb{G}}$ and $U_{\Lambda}$ are finite dimensional
unitary representations of $\mathbb{G}$ and $\Lambda$ respectively on the same
Hilbert space $\mathscr{H}$, if $U_{\mathbb{G}}$ and $U_{\Lambda}$ satisfy
(2.2), then (2.3) defines a finite dimensional unitary representation $U$ of
$\mathbb{G}\rtimes\Lambda$ on $\mathscr{H}$. Moreover,

(2.4a)
$U_{\mathbb{G}}=(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\epsilon_{\Lambda})(U)\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G}),$
(2.4b)
$U_{\Lambda}=(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\epsilon_{\mathbb{G}}\otimes\operatorname{id}_{C(\Lambda)})(U)\in\mathcal{B}(\mathscr{H})\otimes
C(\Lambda).$

###### Proof.

Let $d=\dim\mathscr{H}$, and fix a Hilbert basis $(e_{1},\ldots,e_{d})$ for
$\mathscr{H}$. Let $(e_{ij},i,j=1,\ldots,d)$ be the corresponding matrix units
(i.e. $e_{ij}\in\mathcal{B}(\mathscr{H})$ is characterized by
$e_{ij}(e_{k})=\delta_{j,k}e_{i}$). Then there is a unique
$U_{ij}\in\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$ for each pair of
$i,j$, such that

$U=\sum_{i,j}e_{ij}\otimes U_{ij},$

with each $U_{ij}$ decomposed further as
$U_{ij}=\sum_{r\in\Lambda}U_{ij,r}\otimes\delta_{r}$, where each
$U_{ij,r}\in\operatorname{Pol}(\mathbb{G})$. Since $U$ is a finite dimensional
unitary representation of $\mathbb{G}\rtimes\Lambda$, for any
$i,j\in\left\\{1,\ldots,d\right\\}$, we have

(2.5) $\Delta_{\mathbb{G}\rtimes\Lambda}(U_{ij})=\sum_{k=1}^{d}U_{ik}\otimes
U_{kj},$

where in $\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)\otimes\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$, we have

(2.6)
$\begin{split}\Delta_{\mathbb{G}\rtimes\Lambda}(U_{ij})&=\sum_{\begin{subarray}{c}r,s,t\in\Lambda,\\\
r=st\end{subarray}}{\left[(\operatorname{id}_{A}\otimes\alpha_{s}^{\ast})\Delta(U_{ij,r})\right]}_{13}(1_{A}\otimes\delta_{s}\otimes
1_{A}\otimes\delta_{t})\\\
&=\sum_{s,t\in\Lambda}{\left[(\operatorname{id}_{A}\otimes\alpha_{s}^{\ast})\Delta(U_{ij,st})\right]}_{13}(1_{A}\otimes\delta_{s}\otimes
1_{A}\otimes\delta_{t})\end{split}$

and

(2.7) $\sum_{k=1}^{d}U_{ik}\otimes
U_{kj}=\sum_{k=1}^{d}\sum_{s,t\in\Lambda}U_{ik,s}\otimes\delta_{s}\otimes
U_{kj,t}\otimes\delta_{t}.$

Comparing (2.5), (2.6) and (2.7), we get

(2.8)
$(\operatorname{id}_{A}\otimes\alpha_{s}^{\ast})\Delta(U_{ij,st})=\sum_{k=1}^{d}U_{ik,s}\otimes
U_{kj,t}\in A\otimes A$

or equivalently (by applying
$(\operatorname{id}_{A}\otimes\alpha_{s^{-1}}^{\ast})$ on both sides)

(2.9)
$\Delta(U_{ij,st})=\sum_{k=1}^{d}U_{ik,s}\otimes\alpha_{s^{-1}}^{\ast}(U_{kj,t})$

for every $s,t\in\Lambda$. Since
$(\operatorname{id}\otimes\epsilon)\Delta=\operatorname{id}=(\epsilon\otimes\operatorname{id})\Delta$,
we have

(2.10)
$U_{ij,st}=\sum_{k=1}^{d}\epsilon(U_{ik,s})\alpha_{s^{-1}}^{\ast}(U_{kj,t})=\sum_{k=1}^{d}\epsilon(U_{kj,s})U_{ik,t}$

for any $i,j\in\left\\{1,\ldots,d\right\\}$, $s,t\in\Lambda$.

We have $\epsilon_{\Lambda}(\delta_{r})=\delta_{r,1_{\Lambda}}$, thus by
definition

(2.11) $U_{\mathbb{G}}=\sum_{i,j=1}^{d}e_{ij}\otimes
U_{ij,1_{\Lambda}}\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G}).$

Similarly,

(2.12)
$U_{\Lambda}=\sum_{r\in\Lambda}\sum_{i,j=1}^{d}\epsilon(U_{ij,r})e_{ij}\otimes\delta_{r}\in\mathcal{B}(\mathscr{H})\otimes
C(\Lambda).$

Thus

(2.13)
$U_{\Lambda}(r_{0})=\sum_{i,j=1}^{d}\epsilon(U_{ij,r_{0}})e_{ij}\in\mathcal{B}(\mathscr{H}).$

Hence,

(2.14) $\begin{split}(U_{\Lambda}(r_{0})\otimes
1_{A})U_{\mathbb{G}}&=\sum_{i,j,k,l=1}^{d}\delta_{j,k}\epsilon(U_{ij,r_{0}})e_{il}\otimes
U_{kl,1_{\Lambda}}\\\
&=\sum_{i,l=1}^{d}e_{il}\otimes\sum_{k=1}^{d}\epsilon(U_{ik,r_{0}})U_{kl,1_{\Lambda}}\\\
&=\sum_{i,l=1}^{d}e_{il}\otimes\alpha_{r_{0}}^{\ast}\left(\sum_{k=1}^{d}\epsilon(U_{ik,r_{0}})\alpha_{r_{0}^{-1}}^{\ast}(U_{kl,1_{\Lambda}})\right)\\\
&=\sum_{i,l=1}^{d}e_{il}\otimes\alpha_{r_{0}}^{\ast}(U_{il,r_{0}})\end{split}$

where the last equality follows from (2.10); and

(2.15)
$\begin{split}[(\operatorname{id}\otimes\alpha_{r_{0}}^{\ast})U_{\mathbb{G}}](U_{\Lambda}(r_{0})\otimes
1_{A})&=\sum_{i,j,k,l=1}^{d}\delta_{j,k}\epsilon(U_{kl,r_{0}})e_{il}\otimes\alpha_{r_{0}}^{\ast}(U_{ik,1_{\Lambda}})\\\
&=\sum_{i,l=1}e_{il}\otimes\sum_{k=1}^{d}\epsilon(U_{kl,r_{0}})\alpha_{r_{0}}^{\ast}(U_{ik,1_{\Lambda}})\\\
&=\sum_{i,l=1}e_{il}\otimes\alpha_{r_{0}}^{\ast}\left(\sum_{k=1}^{d}\epsilon(U_{ik,r_{0}})U_{kj,1_{\Lambda}}\right)\\\
&=\sum_{i,l=1}e_{il}\otimes\alpha_{r_{0}}^{\ast}(U_{il,r_{0}})\end{split}$

where (2.10) is used again in the last equality.

Combining (2.14) and (2.15) finishes the proof of (2.2).

By (2.11), (2.12) and (2.10), one has

(2.16)
$\begin{split}{(U_{\mathbb{G}})}_{12}{(U_{\Lambda})}_{13}&=\sum_{i,j,k,l=1}^{d}\sum_{r\in\Lambda}\delta_{j,k}\epsilon(U_{kl,r})e_{il}\otimes
U_{ij,1_{\Lambda}}\otimes\delta_{r}\\\
&=\sum_{i,l=1}^{d}e_{il}\otimes\sum_{r\in\Lambda}\left(\sum_{k=1}^{d}\epsilon(U_{kl,r})U_{ik,1_{\Lambda}}\right)\otimes\delta_{r}\\\
&=\sum_{i,l=1}^{d}e_{il}\otimes\sum_{r\in\Lambda}U_{il,r}\otimes\delta_{r}=U\end{split}$

in $\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)$. This proves (2.3).

Conversely, suppose $U_{\mathbb{G}}$ and $U_{\Lambda}$ are unitary
representations on some finite dimensional Hilbert space $\mathscr{H}$. We
still use $(e_{1},\ldots,e_{d})$ to denote a Hilbert basis for $\mathscr{H}$,
where $d=\dim\mathscr{H}$, and $(e_{ij},i,j=1,\ldots,d)$ the corresponding
matrix unit of $\mathcal{B}(\mathscr{H})$. Then for each pair $i,j$, one has a
unique $u_{ij}\in\operatorname{Pol}(\mathbb{G})$ and a unique $f_{ij}\in
C(\Lambda)$, such that $U_{\mathbb{G}}=\sum_{i,j}e_{ij}\otimes u_{ij}$,
$U_{\Lambda}=\sum_{ij}e_{ij}\otimes f_{ij}$. By suitably choosing the basis
$(e_{1},\ldots,e_{d})$, we may and do assume $\epsilon(u_{ij})=\delta_{i,j}$.
Since these are representations, we have

(2.17a) $\Delta(u_{ij})=\sum_{k=1}^{d}u_{ik}\otimes u_{kj},$ (2.17b)
$\Delta_{\Lambda}{(f_{ij})}=\sum_{k=1}^{d}f_{ik}\otimes f_{kj}.$

By definition,

(2.18) $U=\sum_{i,j,k,l=1}^{d}\delta_{jk}e_{il}\otimes u_{ij}\otimes
f_{kl}=\sum_{i,j=1}^{d}e_{ij}\otimes U_{ij}$

with

(2.19) $U_{ij}=\sum_{k=1}^{d}u_{ik}\otimes
f_{kj}=\sum_{r\in\Lambda}\sum_{k=1}^{d}f_{kj}(r)u_{ik}\otimes\delta_{r}.$

Since $U_{\mathbb{G}}$ and $U_{\Lambda}$ are unitary, so is $U$. Using
$\epsilon(u_{ij})=\delta_{i,j}$, one has

(2.20)
$(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\epsilon\otimes\operatorname{id}_{C(\Lambda)})(U)=\sum_{i,j=1}^{d}e_{ij}\otimes\sum_{k=1}^{d}\delta_{i,k}f_{kj}=\sum_{i,j}^{d}e_{ij}\otimes
f_{ij}=U_{\Lambda}.$

This proves (2.4b). The proof of (2.4a) is more involved and must resort to
condition (2.2), which using the above notations, translates to

(2.21) $\forall
r\in\Lambda,\quad\sum_{i,j}e_{ij}\otimes\sum_{k}f_{ik}(r)u_{kj}=\sum_{i,j}e_{ij}\otimes\sum_{k}f_{kj}(r)\alpha_{r}^{\ast}(u_{ik}),$

or equivalently,

(2.22) $\forall
r\in\Lambda,i,j\in\left\\{1,\ldots,d\right\\},\quad\sum_{k=1}^{d}f_{ik}(r)u_{kj}=\sum_{k=1}^{d}f_{kj}(r)\alpha_{r}^{\ast}(u_{ik}).$

Since $U_{\Lambda}(1_{\Lambda})=\operatorname{id}_{\mathscr{H}}$, one has
$f_{ij}(1_{\Lambda})=\delta_{i,j}$. Taking $r=1_{\Lambda}$ in (2.22) yields

(2.23) $\begin{split}&\hskip
11.99998pt(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\epsilon_{\Lambda})(U)=\sum_{i,j=1}^{d}e_{ij}\otimes\sum_{k=1}^{d}f_{kj}(1_{\Lambda})u_{ik}\\\
&=\sum_{i,j=1}^{d}e_{ij}\otimes\sum_{k=1}^{d}\delta_{k,j}u_{ik}=\sum_{i,j=1}^{d}e_{ij}\otimes
u_{ij}=U_{\mathbb{G}},\end{split}$

which proves (2.4a). To finishes the proof of the proposition, it remains to
check that the unitary $U$ is indeed a representation of
$\mathbb{G}\rtimes\Lambda$.

Using (2.17a), (2.19) and (2.22), one has

(2.24)
$\begin{split}\Delta_{\mathbb{G}\rtimes\Lambda}(U_{ij})&=\sum_{r\in\Lambda}\sum_{s\in\Lambda}{\left[(\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\alpha_{s}^{\ast})\Delta\left(\sum_{k=1}^{d}f_{kj}(r)u_{ik}\right)\right]}_{13}{\left(\delta_{s}\otimes\delta_{s^{-1}1}\right)}_{24}\\\
&=\sum_{r,s\in\Lambda}\sum_{k,l=1}^{d}f_{kj}(r)u_{il}\otimes\delta_{s}\otimes\alpha_{s}^{\ast}(u_{lk})\otimes\delta_{s^{-1}r}\\\
&=\sum_{r,s\in\Lambda}\sum_{l=1}^{d}u_{il}\otimes\delta_{s}\otimes\left[\sum_{k=1}^{d}f_{kj}(r)\alpha_{s}^{\ast}(u_{lk})\right]\otimes\delta_{s^{-1}r}\\\
&=\sum_{s,t\in\Lambda}\sum_{k,l=1}^{d}u_{il}\otimes\delta_{s}\otimes\left[f_{kj}(st)\alpha_{s}^{\ast}(u_{lk})\right]\otimes\delta_{t}\\\
&=\sum_{s,t\in\Lambda}\sum_{k,l=1}^{d}u_{il}\otimes\delta_{s}\otimes\left[\sum_{h=1}^{d}f_{hj}(t)f_{kh}(s)\alpha_{s}^{\ast}(u_{lk})\right]\otimes\delta_{t}\\\
&=\sum_{s,t\in\Lambda}\sum_{h,l=1}^{d}u_{il}\otimes\delta_{s}\otimes\left[f_{hj}(t)\sum_{k=1}^{d}f_{kh}(s)\alpha_{s}^{\ast}(u_{lk})\right]\otimes\delta_{t}\\\
&=\sum_{s,t\in\Lambda}\sum_{h,l=1}^{d}u_{il}\otimes\delta_{s}\otimes\left[f_{hj}(t)\sum_{k=1}^{d}f_{lk}(s)u_{kh}\right]\otimes\delta_{t}\\\
&=\sum_{h,k,l=1}^{d}u_{il}\otimes f_{lk}\otimes u_{kh}\otimes f_{hj}\\\
&=\sum_{h=1}^{d}\left(\sum_{l=1}^{d}u_{il}\otimes
f_{lk}\right)\otimes\left(\sum_{k=1}^{d}u_{kh}\otimes
f_{hj}\right)=\sum_{h=1}^{d}U_{ih}\otimes U_{hj}.\end{split}$

Thus $U$ is indeed a (unitary) representation. ∎

######  Definition 2.3.

Let
$U_{\mathbb{G}}\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})$
be a finite dimensional unitary representation of $\mathbb{G}$,
$U_{\Lambda}\in\mathcal{B}(\mathscr{H})\otimes C(\Lambda)$ a finite
dimensional unitary representation of $\Lambda$ on the same space
$\mathscr{H}$, we say $U_{\mathbb{G}}$ and $U_{\Lambda}$ are covariant if they
satisfy condition (2.2).

We track here a simple criterion for two representations to be covariant using
matrix units and matrix coefficients.

######  Proposition 2.4.

Let
$U_{\mathbb{G}}\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})$,
$U_{\Lambda}\in\mathcal{B}(\mathscr{H})\otimes C(\Lambda)$ be finite-
dimensional unitary representations of $\mathbb{G}$ and $\Lambda$
respectively. Let $(e_{1},\ldots,e_{d})$ be a Hilbert basis of $\mathscr{H}$,
$e_{ij}\in\mathcal{B}(\mathscr{H})$ the operator with
$e_{ij}(e_{k})=\delta_{j,k}e_{i}$, and $U_{\mathbb{G}}=\sum_{i,j}e_{ij}\otimes
u_{ij}$, $U_{\Lambda}=\sum_{i,j}e_{ij}\otimes f_{ij}$, then $U_{\mathbb{G}}$
and $U_{\Lambda}$ are covariant if and only if

(2.25) $\forall
r\in\Lambda,i,j\in\left\\{1,\ldots,d\right\\},\quad\sum_{k=1}^{d}f_{ik}(r)u_{kj}=\sum_{k=1}^{d}f_{kj}(r)\alpha_{r}^{\ast}(u_{ik}).$

###### Proof.

This is just a restatement of condition (2.2). ∎

By Proposition 2.2, unitary representations of $\mathbb{G}\rtimes\Lambda$, at
least the finite dimensional ones, correspond bijectively to pairs of
covariant unitary representations of $\mathbb{G}$ and $\Lambda$.

##  3\. Principal subgroups of $\mathbb{G}\rtimes\Lambda$

######  Definition 3.1.

Let $\mathbb{H}=(B,\Delta_{B})$, $\mathbb{K}=(C,\Delta_{C})$ be compact
quantum groups, we say $\mathbb{K}$ is isomorphic to a closed quantum subgroup
of $\mathbb{H}$, or simply $\mathbb{K}$ is a closed subgroup of $\mathbb{H}$,
if there exists a _surjective_ mapping
$\varphi:\operatorname{Pol}(\mathbb{H})\to\operatorname{Pol}(\mathbb{K})$ such
that $\varphi$ is a morphism of Hopf $\ast$-algebras.

When $\mathcal{\mathbb{H}}$ is universal, then Definition 3.1 can be
reformulated as the existence of a surjective unital $C^{\ast}$-algebra
morphism $\varphi:B\to C$ such that
$(\varphi\otimes\varphi)\Delta_{B}=\Delta_{C}\varphi$.

In the context of compact quantum groups, we will use the terms “quantum
closed subgroup” and “closed subgroup”, sometimes even “subgroup”,
interchangeably without further explanation.

######  Remark 3.2.

If $\mathbb{H}$ and $\mathbb{K}$ are commutative, i.e. they come from genuine
compact groups, then $\mathbb{K}$ being isomorphic to a closed subgroup, says
exactly that there exists a continuous injective map $\varphi_{\ast}$ from
$\operatorname{Spec}(C)$, the underlying space of the compact group
$\mathbb{K}$, into $\operatorname{Spec}(B)$, the underlying space of the
compact group $\mathbb{H}$, such that $\varphi_{\ast}$ preserves
multiplication. Thus the above definition for closed (quantum) subgroups is
consistent with the classical case of compact groups.

Recall that $\mathbb{G}=(A,\Delta)$, $C(\mathbb{G}\rtimes\Lambda)=A\otimes
C(\Lambda)$, and
$\operatorname{Pol}(\mathbb{G}\rtimes\Lambda)=C(\mathbb{G})\rtimes
C(\Lambda)$.

######  Proposition 3.3.

Let $\Lambda_{0}$ be a subgroup of $\Lambda$, then the mapping

$\displaystyle\varphi\colon A\otimes C(\Lambda)$ $\displaystyle\rightarrow
A\otimes C(\Lambda_{0})$
$\displaystyle\sum_{r\in\Lambda}a_{r}\otimes\delta_{r}$
$\displaystyle\mapsto\sum_{r\in\Lambda_{0}}a_{r}\otimes\delta_{r}$

is a unital surjective morphism444Note that $\delta_{r}$ has different
meanings when viewed as functions in $C(\Lambda)$ and in $C(\Lambda_{0})$ of
$C^{\ast}$-algebras that also intertwines the comultiplications on
$\mathbb{G}\rtimes\Lambda_{0}$ and $\mathbb{G}\rtimes\Lambda$. In particular,
$\mathbb{G}\rtimes\Lambda_{0}$ is a closed subgroup of
$\mathbb{G}\rtimes\Lambda$.

###### Proof.

Obviously $\varphi$ is a unital surjective morphism of $C^{\ast}$-algebras. We
need to show that $\varphi$ intertwines the comultiplication
$\widetilde{\Delta}$ on $\mathbb{G}\rtimes\Lambda$ and the comultiplication
$\widetilde{\Delta}_{0}$ on $\mathbb{G}\rtimes\Lambda_{0}$. For this, by
density, it suffices to prove that the restriction

(3.1) $\begin{split}\varphi\colon\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)&\rightarrow\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda_{0})\\\
\sum_{r\in\Lambda}a_{r}\otimes\delta_{r}&\mapsto\sum_{r\in\Lambda_{0}}a_{r}\otimes\delta_{r}\end{split}$

intertwines the comultiplications. Indeed, given an arbitrary
$a_{r}\in\operatorname{Pol}(G)$ for any $r\in\Lambda$, note that for any
$a\in\operatorname{Pol}(\mathbb{G})$ and $\lambda\in\Lambda$,
$\varphi(a\otimes\delta_{\lambda})=0$ whenever $\lambda\notin\Lambda_{0}$, we
have

(3.2) $\begin{split}&\hskip
11.99998pt(\varphi\otimes\varphi)\widetilde{\Delta}\left(\sum_{r\in\Lambda}a_{r}\otimes\delta_{r}\right)\\\
&=(\varphi\otimes\varphi)\sum_{r\in\Lambda}\sum_{s\in\Lambda}\sum{(a_{r})}_{(1)}\otimes\delta_{s}\otimes\alpha_{s}^{\ast}\bigl{(}{(a_{r})}_{(2)}\bigr{)}\otimes\delta_{s^{-1}r}\\\
&=\sum_{r\in\Lambda_{0}}\sum_{s\in\Lambda_{0}}\sum{(a_{r})}_{(1)}\otimes\delta_{s}\otimes\alpha_{s}^{\ast}\bigl{(}{(a_{r})}_{(2)}\bigr{)}\otimes\delta_{s^{-1}r}\\\
&\text{(Since $s,s^{-1}r\in\Lambda_{0}$ implies
$r=s(s^{-1}r)\in\Lambda_{0}$)}\\\
&=\widetilde{\Delta}_{0}\left(\sum_{r\in\Lambda_{0}}a_{r}\otimes\delta_{r}\right)=\widetilde{\Delta}_{0}\varphi\left(\sum_{r\in\Lambda}a_{r}\otimes\delta_{r}\right).\end{split}$

This shows that $\varphi$ indeed intertwines comultiplications and finishes
the proof. ∎

######  Definition 3.4.

A closed subgroup of $\mathbb{G}\rtimes\Lambda$ of the form
$\mathbb{G}\rtimes\Lambda_{0}$, where $\Lambda_{0}$ is a subgroup $\Lambda$,
is called a principal subgroup of $\mathbb{G}\rtimes\Lambda$.

######  Remark 3.5.

If we let $p_{0}=\sum_{r\in\Lambda_{0}}\delta_{r}\in C(\Lambda)$, then $p_{0}$
is a projection in $C(\Lambda)$, thus $1\otimes p_{0}$ is a central projection
in $A\otimes C(\Lambda)$. The morphism $\varphi$ is in fact given by the
“compression” map $(1\otimes p_{0})(\cdot)(1\otimes p_{0})$. Essentially,
these data says that the principal subgroup $\mathbb{G}\rtimes\Lambda_{0}$ is
in fact an open subgroup of $\mathbb{G}\rtimes\Lambda$. As we don’t really
need the general theory of open subgroups of topological quantum groups in
this paper, we won’t recall the relevant notions here and refer the interested
reader to the articles [MR2980506, MR3552528] for a treatment in the more
general setting of locally compact quantum groups.

######  Corollary 3.6.

Using the notations in Proposition 3.3, if
$U\in\mathcal{B}(\mathscr{H})\otimes A\otimes C(\Lambda)$ is a (unitary)
representation of $\mathbb{G}\rtimes\Lambda$, then
$(\operatorname{id}\otimes\varphi)(U)$ is a (unitary) representation of
$\mathbb{G}\rtimes\Lambda_{0}$.

###### Proof.

This follows directly from the fact that the restriction of the mapping
$\varphi$ as specified in (3.1) is a morphism of Hopf $\ast$-algebras. ∎

######  Definition 3.7.

Using the above notations, the representation
$(\operatorname{id}\otimes\varphi)(U)$ is called the restriction of $U$ to
$\mathbb{G}\rtimes\Lambda_{0}$, and is denoted by
$U|_{\mathbb{G}\rtimes\Lambda_{0}}$.

######  Remark 3.8.

Again, when $\mathbb{G}$ is an classical compact group $G$, we recover the
classical notion of restriction of a representation of $G\rtimes\Lambda$ to
the subgroup $G\rtimes\Lambda_{0}$.

There is a natural “conjugate” relation between principal subgroups of the
form $\mathbb{G}\rtimes\Lambda_{0}$ where $\Lambda_{0}$ is a subgroup of
$\Lambda$, which will be used to simplify some calculations in our later
treatment of representations. This relation is described in the following
proposition.

######  Proposition 3.9.

Let $\Lambda_{0}$ be a subgroup of $\Lambda$, $r\in\Lambda$,
$\operatorname{Ad}_{r}\colon\Lambda_{0}\rightarrow r\Lambda_{0}r^{-1}$ the
isomorphism $s\mapsto rsr^{-1}$. Then
$\alpha_{r}^{\ast}\otimes\operatorname{Ad}_{r}^{\ast}$ is an isomorphism of
compact quantum groups from $\mathbb{G}\rtimes\Lambda_{0}$ to
$\mathbb{G}\rtimes r\Lambda_{0}r^{-1}$.

###### Proof.

By density, it suffices to prove that the unital $\ast$-isomorphism

$\alpha_{r}^{\ast}\otimes\operatorname{Ad}_{r}^{\ast}\colon\operatorname{Pol}(\mathbb{G})\otimes
C(r\Lambda_{0}r^{-1})\rightarrow\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0})$

of involutive algebras preserves comultiplication. To fix the notations, let
$\Delta_{0}$ (resp. $\Delta_{r}$) be the comultiplication on
$\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda_{0})$ (resp.
$\operatorname{Pol}(\mathbb{G})\otimes C(r\Lambda_{0}r^{-1})$). For any
$x\in\operatorname{Pol}(\mathbb{G})$, $\lambda\in\Lambda_{0}$, we have

(3.3) $\begin{split}&\hskip
11.99998pt(\alpha_{r}^{\ast}\otimes\operatorname{Ad}_{r}^{\ast}\otimes\alpha_{r}^{\ast}\otimes\operatorname{Ad}_{r}^{\ast})\Delta_{r}(x\otimes\delta_{r\lambda
r^{-1}})\\\
&=\sum_{\mu\in\Lambda_{0}}{\bigl{[}\bigl{(}\alpha_{r}^{\ast}\otimes(\alpha_{r}^{\ast}\alpha_{r\mu
r^{-1}}^{\ast})\bigr{)}\Delta(x)\bigr{]}}_{13}{((\operatorname{Ad}_{r}^{\ast}\delta_{r\mu
r^{-1}})\otimes(\operatorname{Ad}_{r}^{\ast}\delta_{r\mu^{-1}\lambda
r^{-1}}))}_{24}\\\
&=\sum_{\mu\in\Lambda_{0}}{\bigl{[}\bigl{(}\alpha_{r}^{\ast}\otimes\alpha_{r\mu}^{\ast}\bigr{)}\Delta(x)\bigr{]}}_{13}{((\operatorname{Ad}_{r}^{\ast}\delta_{r\mu
r^{-1}})\otimes(\operatorname{Ad}_{r}^{\ast}\delta_{r\mu^{-1}\lambda
r^{-1}}))}_{24}\\\
&=\sum_{\mu\in\Lambda_{0}}{\bigl{[}\bigl{(}(\operatorname{id}\otimes\alpha_{\mu}^{\ast})[(\alpha_{r}^{\ast}\otimes\alpha_{r}^{\ast})\Delta(x)]\bigr{)}\bigr{]}}_{13}{(\delta_{\mu}\otimes\delta_{\mu^{-1}\lambda})}_{24}\\\
&=\sum_{\mu\in\Lambda_{0}}{\bigl{[}\bigl{(}(\operatorname{id}\otimes\alpha_{\mu}^{\ast})\Delta(\alpha_{r}^{\ast}(x))\bigr{)}\bigr{]}}_{13}{(\delta_{\mu}\otimes\delta_{\mu^{-1}\lambda})}_{24}\\\
&=\Delta_{0}\bigl{(}\alpha_{r}^{\ast}(x)\otimes\delta_{\lambda}\bigr{)}=[\Delta_{0}(\alpha_{r}^{\ast}\otimes\operatorname{Ad}_{r}^{\ast})](x\otimes\delta_{r\lambda
r^{-1}}).\end{split}$

Thus $\alpha_{r}^{\ast}\otimes\operatorname{Ad}_{r}^{\ast}$ indeed preserves
comultiplication. ∎

##  4\. Induced representations of principal subgroups

We begin by describing an outline of our approach to induced representations
of principal subgroups of $\mathbb{G}\rtimes\Lambda$. Let $\Lambda_{0}$ be a
subgroup of $\Lambda$,
$U\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0})$ a finite dimensional unitary representation of
$\mathbb{G}\rtimes\Lambda_{0}$. We want to construct the induced
representation
$\operatorname{Ind}^{\mathbb{G}\rtimes\Lambda}_{\mathbb{G}\rtimes\Lambda_{0}}(U)$
of the larger quantum group $\mathbb{G}\rtimes\Lambda$. The idea of the
construction goes as follows: by the results in § 2, we know $U$ is determined
by its restrictions $U_{\mathbb{G}}=\operatorname{Res}_{\mathbb{G}}(U)$ and
$U_{\Lambda_{0}}=\operatorname{Res}_{\Lambda_{0}}(U)$. While one may not be
able to directly extend the representation $U_{\Lambda_{0}}$ of $\Lambda_{0}$
to a representation of $\Lambda$ on the same space $\mathscr{H}$, we do have
the right-regular representation $\widetilde{W}_{\Lambda}$ of $\Lambda$ on
$\ell^{2}(\Lambda)\otimes\mathscr{H}$ using the group structure of $\Lambda$.
On the other hand, the direct sum $\widetilde{W}_{\mathbb{G}}$ of various
copies of $U_{\mathbb{G}}$ placed suitably in
$\ell^{2}(\Lambda)\otimes\mathscr{H}$ will give a representation of
$\mathbb{G}$ on $\ell^{2}(\Lambda)\otimes\mathscr{H}$. It is then easy to
check that $\widetilde{W}_{\mathbb{G}}$ and $\widetilde{W}_{\Lambda}$ are
covariant, thus determine a representation $\widetilde{W}$ of
$\mathbb{G}\rtimes\Lambda$ on $\ell^{2}(\Lambda)\otimes\mathscr{H}$. To
retrieve the information of $U_{\Lambda_{0}}$, which is implicitly encoded in
the $\mathscr{H}$ factor of $\ell^{2}(\Lambda)\otimes\mathscr{H}$, we consider
the subspace $\mathscr{K}$ of $\ell^{2}(\Lambda)\otimes\mathscr{H}$ consisting
of vectors which behave in a covariant way with the representation
$U_{\Lambda_{0}}$ on $\mathscr{H}$. More precisely, $\mathscr{K}$ is defined
by

(4.1)
$\mathscr{K}=\left\\{\sum_{r\in\Lambda}\delta_{r}\otimes\xi_{r}\;:\;\forall
r_{0}\in\Lambda_{0},\forall
r\in\Lambda,\,\xi_{r_{0}r}=U_{\Lambda_{0}}(r_{0})\xi_{r}\right\\}.$

One checks that $\mathscr{K}$ is an invariant subspace for both
$\widetilde{W}_{\Lambda}$ and $\widetilde{W}_{\mathbb{G}}$, hence
$\mathscr{K}$ is a subrepresentation $W$ of $\widetilde{W}$, and we define $W$
to be the induced representation $\operatorname{Ind}(U)$. We now proceed to
carry out this idea precisely.

######  Definition 4.1.

Let $U$, $\mathscr{H}$, $\Lambda_{0}$ retain their meanings as above, and let
$(e_{r,s};\,r,s\in\Lambda)$ be the matrix unit of
$\mathcal{B}(\ell^{2}(\Lambda))$ associated with the standard Hilbert basis
$(\delta_{r};\,r\in\Lambda)$ for $\ell^{2}(\Lambda)$, i.e.
$e_{r,s}\delta_{t}=\delta_{s,t}\delta_{r}$ for all $r,s,t\in\Lambda$. The
right regular representation $\widetilde{W}_{\Lambda}$ of $\Lambda$ on
$\ell^{2}\otimes\mathscr{H}$ is an operator in
$\mathcal{B}(\ell^{2}(\Lambda))\otimes\mathcal{B}(\mathscr{H})\otimes
C(\Lambda)$ defined by

(4.2)
$\widetilde{W}_{\Lambda}=\sum_{r,s\in\Lambda}e_{rs^{-1},r}\otimes\operatorname{id}_{\mathscr{H}}\otimes\delta_{s}.$

It is easy to see that if we regard $\ell^{2}(\Lambda)\otimes\mathscr{H}$ as
$\ell^{2}(\Lambda,\mathscr{H})$, then for any $s\in\Lambda$,
$\widetilde{W}_{\Lambda}(s)$ is the operator in
$\mathcal{B}(\ell^{2}(\Lambda,\mathscr{H}))$ sending each
$F\colon\Lambda\rightarrow\mathscr{H}$ to $F\circ R_{s}$, where
$R_{s}\colon\Lambda\rightarrow\Lambda$ is the right multiplication by $s$.
Hence $\widetilde{W}_{\Lambda}$ is indeed a unitary representation of
$\Lambda$ on $\ell^{2}(\Lambda)\otimes\mathscr{H}$. By definition, for any
$s\in\Lambda$, the unitary operator
$\widetilde{W}_{\Lambda}(s)\in\mathcal{U}(\ell^{2}(\Lambda)\otimes\mathscr{H})$
is characterized by

(4.3)
$\begin{split}\widetilde{W}_{\Lambda}(s)\colon\ell^{2}(\Lambda)\otimes\mathscr{H}&\rightarrow\ell^{2}(\Lambda)\otimes\mathscr{H}\\\
\delta_{r}\otimes\xi&\mapsto\delta_{rs^{-1}}\otimes\xi,\end{split}$

or equivalently

(4.4)
$\begin{split}\widetilde{W}_{\Lambda}(s)\colon\ell^{2}(\Lambda)\otimes\mathscr{H}&\rightarrow\ell^{2}(\Lambda)\otimes\mathscr{H}\\\
\sum_{r\in\Lambda}\delta_{r}\otimes\xi_{r}&\mapsto\sum_{r\in\Lambda}\delta_{rs^{-1}}\otimes\xi_{r}=\sum_{r\in\Lambda}\delta_{r}\otimes\xi_{rs}.\end{split}$

######  Proposition 4.2.

Using the above notations, the unitary operator

(4.5)
$\widetilde{W}_{\mathbb{G}}:=\sum_{s\in\Lambda}e_{s,s}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{s})(U_{\mathbb{G}})]\in\mathcal{B}(\ell^{2}(\Lambda))\otimes\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})$

is a unitary representation of $\mathbb{G}$ on
$\ell^{2}(\Lambda)\otimes\mathscr{H}$. Furthermore, for every $s\in\Lambda$,
$\delta_{s}\otimes\mathscr{H}$ is invariant under
$\widetilde{W}_{\mathbb{G}}$, and the subrepresentation
$\delta_{s}\otimes\mathscr{H}$ of $\widetilde{W}_{\mathbb{G}}$ is unitarily
equivalent to the unitary representation
$(\operatorname{id}\otimes\alpha^{\ast}_{s})(U_{\mathbb{G}})$ of $\mathbb{G}$.
In particular,
$\widetilde{W}_{\mathbb{G}}\simeq\bigoplus_{s\in\Lambda}(\operatorname{id}\otimes\alpha_{s}^{\ast})(U_{\mathbb{G}})$.

###### Proof.

For each $s\in\Lambda$, since
$\alpha^{\ast}_{s}\in\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$,
the unitary operator
$(\operatorname{id}\otimes\alpha^{\ast}_{s})(U_{\mathbb{G}})$ is indeed a
representation of $\mathbb{G}$ on $\mathscr{H}$. It is easy to see that

$e_{s,s}(\ell^{2}(\Lambda))\otimes\mathscr{H}=\mathbb{C}\delta_{s}\otimes\mathscr{H}=\delta_{s}\otimes\mathscr{H},$

hence $e_{s,s}\otimes\operatorname{id}_{\mathscr{H}}$ is the orthogonal
projection in $\mathcal{B}(\ell^{2}(\Lambda)\otimes\mathscr{H})$ onto the
subspace $\delta_{s}\otimes\mathscr{H}$ of
$\ell^{2}(\Lambda)\otimes\mathscr{H}$ (Here and below, we abuse the notation
$\delta_{s}\otimes\mathscr{H}$ to denote the subspace
$\\{\delta_{s}\otimes\xi\;:\;\xi\in\mathscr{H}\\}$ of
$\ell^{2}(\Lambda)\otimes\mathscr{H}$). We also have the intertwining relation

(4.6) $(e_{s,s}\otimes\operatorname{id}_{\mathscr{H}}\otimes
1_{A})\widetilde{W}_{\mathbb{G}}=e_{s,s}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{s})(U_{\mathbb{G}})]=\widetilde{W}_{\mathbb{G}}(e_{s,s}\otimes\operatorname{id}_{\mathscr{H}}\otimes
1_{A}).$

Now the theorem follows from (4.6), the direct sum decomposition

(4.7)
$\ell^{2}(\Lambda)\otimes\mathscr{H}=\bigoplus_{s\in\Lambda}e_{s,s}(\ell^{2}(\Lambda))\otimes\mathscr{H}=\bigoplus_{s\in\Lambda}\delta_{s}\otimes\mathscr{H},$

and the obvious fact that the unitary operator
$\delta_{s}\otimes\mathscr{H}\to\mathscr{H}$, $\delta_{s}\otimes\xi\mapsto\xi$
intertwines the representation
$(\operatorname{id}\otimes\delta_{s})(U_{\mathbb{G}})$ and the
subrepresentation of $\widetilde{W}_{\mathbb{G}}$ determined by the subspace
$\delta_{s}\otimes\mathscr{H}$ of $\ell^{2}(\Lambda)\otimes\mathscr{H}$. ∎

######  Proposition 4.3.

The representations $\widetilde{W}_{\mathbb{G}}$ and $\widetilde{W}_{\Lambda}$
are covariant.

###### Proof.

For any $s\in\Lambda$, by definition,

(4.8)
$\widetilde{W}_{\Lambda}(s)=\sum_{r\in\Lambda}e_{rs^{-1},r}\otimes\operatorname{id}_{\mathscr{H}}\in\mathcal{B}(\ell^{2}(\Lambda))\otimes\mathcal{B}(\mathscr{H})=\mathcal{B}(\ell^{2}(\Lambda)\otimes\mathscr{H}).$

Thus

(4.9) $\begin{split}&\hskip 11.99998pt\left(\widetilde{W}_{\Lambda}(s)\otimes
1\right)\widetilde{W}_{\mathbb{G}}\\\
&=\left(\sum_{r\in\Lambda}e_{rs^{-1},r}\otimes\operatorname{id}_{\mathscr{H}}\otimes
1_{A}\right)\sum_{t\in\Lambda}e_{t,t}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{t})(U_{\mathbb{G}})]\\\
&=\sum_{r,t\in\Lambda}\delta_{r,t}e_{rs^{-1},t}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{t})(U_{\mathbb{G}})]=\sum_{r\in\Lambda}e_{rs^{-1},r}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{r})(U_{\mathbb{G}})]\\\
&=(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{s})\sum_{r\in\Lambda}e_{rs^{-1},r}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{rs^{-1}})(U_{\mathbb{G}})]\\\
&=(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{s})\left[\left(\sum_{t\in\Lambda}e_{t,t}\otimes[(\operatorname{id}_{\mathscr{H}}\otimes\alpha^{\ast}_{t})(U_{\mathbb{G}})]\right)\left(\sum_{r\in\Lambda}e_{rs^{-1},r}\otimes\operatorname{id}_{\mathscr{H}}\otimes
1_{A}\right)\right]\\\
&=[(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{s})(\widetilde{W}_{\mathbb{G}})]\bigl{(}\widetilde{W}_{\Lambda}(s)\otimes
1\bigr{)}.\end{split}$

This proves that $\widetilde{W}_{\mathbb{G}}$ and $\widetilde{W}_{\Lambda}$
are indeed covariant. ∎

######  Corollary 4.4.

The unitary operator

(4.10)
$\begin{split}\widetilde{W}:&={(\widetilde{W}_{\mathbb{G}})}_{123}{(\widetilde{W}_{\Lambda})}_{124}=\sum_{r,s,t\in\Lambda}e_{t,t}e_{rs^{-1},r}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{t})(U_{\mathbb{G}})]\otimes\delta_{s}\\\
&=\sum_{r,s\in\Lambda}e_{rs^{-1},r}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{rs^{-1}})(U_{\mathbb{G}})]\otimes\delta_{s}\\\
&\in\mathcal{B}(\ell^{2}(\Lambda))\otimes\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)\end{split}$

is a representation of $\mathbb{G}\rtimes\Lambda$ on
$\ell^{2}(\Lambda)\otimes\mathscr{H}$.

###### Proof.

This follows from Proposition 2.2 and Proposition 4.3. ∎

We now proceed to prove the invariance of the subspace $\mathscr{K}$ defined
in (4.1) under $\widetilde{W}_{\mathbb{G}}$ and $\widetilde{W}_{\Lambda}$.

######  Lemma 4.5.

Using the above notations, the following hold:

  1. (a)

the orthogonal projection
$\pi\in\mathcal{B}(\ell^{2}(\Lambda)\otimes\mathscr{H})$ with range
$\mathscr{K}$ is given by555Recall that we’ve identified
$\mathcal{B}(\ell^{2}(\Lambda)\otimes\mathscr{H})$ with
$\mathcal{B}(\ell^{2}(\Lambda))\otimes\mathcal{B}(\mathscr{H})$ the following
formula:

(4.11)
$\begin{split}\pi\colon\ell^{2}(\Lambda)\otimes\mathscr{H}&\rightarrow\ell^{2}(\Lambda)\otimes\mathscr{H}\\\
\delta_{r}\otimes\xi&\mapsto\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\delta_{r_{0}r}\otimes
U_{\Lambda_{0}}(r_{0})\xi.\end{split}$

In other words,

(4.12)
$\pi=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\sum_{s\in\Lambda}e_{r_{0}s,s}\otimes
U_{\Lambda_{0}}(r_{0});$

  2. (b)

$\mathscr{K}$ is invariant under both $\widetilde{W}_{\mathbb{G}}$ and
$\widetilde{W}_{\Lambda}$, i.e.

(4.13a) $(\pi\otimes
1)\widetilde{W}_{\mathbb{G}}=\widetilde{W}_{\mathbb{G}}(\pi\otimes
1)=(\pi\otimes 1)\widetilde{W}_{\mathbb{G}}(\pi\otimes 1),$ (4.13b)
$(\pi\otimes 1)\widetilde{W}_{\Lambda}=\widetilde{W}_{\Lambda}(\pi\otimes
1)=(\pi\otimes 1)\widetilde{W}_{\Lambda}(\pi\otimes 1).$

In particular, we have

(4.14) $(\pi\otimes 1\otimes 1)\widetilde{W}=\widetilde{W}(\pi\otimes 1\otimes
1)=(\pi\otimes 1\otimes 1)\widetilde{W}(\pi\otimes 1\otimes 1).$

###### Proof.

It is easy to see that $\pi(\ell^{2}(\Lambda)\otimes\mathscr{H})$ is precisely
$\mathscr{K}$ and $\pi_{\mathscr{K}}=\operatorname{id}_{\mathscr{K}}$. To
finish the proof of (a), it suffices to check that $\pi$ is self-adjoint (or
even stronger, positive). Since

(4.15)
$\begin{split}\left\lparen\pi(\delta_{r}\otimes\xi_{r}),\delta_{r}\otimes\xi_{r}\right\rparen&=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\left\lparen\delta_{r_{0}r}\otimes
U_{\Lambda_{0}}(r_{0})\xi,\delta_{r}\otimes\xi\right\rparen\\\
&=\left\lvert\Lambda_{0}\right\rvert^{-1}\left\lVert\xi\right\rVert^{2}\geq
0,\end{split}$

$\pi$ is indeed positive.

We now prove (b). The invariance of $\mathscr{K}$ under
$\widetilde{W}_{\Lambda}$ (equation (4.13b)) follows from (4.1) and (4.4). We
now prove the invariance of $\mathscr{K}$ under $\widetilde{W}_{\mathbb{G}}$
(equation (4.13a)). By the definitions of $\pi$ and
$\widetilde{W}_{\mathbb{G}}$, we have

(4.16) $\begin{split}&\hskip
11.99998pt\left\lvert\Lambda_{0}\right\rvert(\pi\otimes
1)\widetilde{W}_{\mathbb{G}}\\\
&=\sum_{r_{0}\in\Lambda_{0}}\sum_{r,s\in\Lambda}\bigl{(}e_{r_{0}s,s}\otimes
U_{\Lambda_{0}}(r_{0})\otimes
1\bigr{)}\bigl{(}e_{r,r}\otimes[(\operatorname{id}\otimes\alpha^{\ast}_{r})(U_{\mathbb{G}})]\bigr{)}\\\
&=\sum_{r\in\Lambda}(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{r})\left(\sum_{r_{0}\in\Lambda_{0}}\sum_{s\in\Lambda}e_{r_{0}s,s}e_{r,r}\otimes\bigl{[}\bigl{(}U_{\Lambda_{0}}(r_{0})\otimes
1\bigr{)}U_{\mathbb{G}}\bigr{]}\right)\\\
&=\sum_{r\in\Lambda}(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{r})\left(\sum_{r_{0}\in\Lambda_{0}}e_{r_{0}r,r}\otimes\bigl{[}\bigl{(}U_{\Lambda_{0}}(r_{0})\otimes
1\bigr{)}U_{\mathbb{G}}\bigr{]}\right);\end{split}$

and

(4.17) $\begin{split}&\hskip
11.99998pt\left\lvert\Lambda_{0}\right\rvert\widetilde{W}_{\mathbb{G}}(\pi\otimes
1)\\\
&=\sum_{r_{0}\in\Lambda_{0}}\sum_{r,s\in\Lambda}\Bigl{(}e_{r,r}\otimes\bigl{[}(\operatorname{id}\otimes\alpha^{\ast}_{r})(U_{\mathbb{G}})\bigr{]}\Bigr{)}\bigl{(}e_{r_{0}s,s}\otimes
U_{\Lambda_{0}}(r_{0})\otimes 1\bigr{)}\\\
&=\sum_{r_{0}\in\Lambda_{0}}\sum_{r,s\in\Lambda}(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{r})\left(e_{r,r}e_{r_{0}s,s}\otimes\bigl{[}U_{\mathbb{G}}(U_{\Lambda_{0}}(r_{0})\otimes
1)\bigr{]}\right)\\\
&=\sum_{r_{0}\in\Lambda_{0}}\sum_{s\in\Lambda}(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{r_{0}s})\left(e_{r_{0}s,s}\otimes\bigl{[}U_{\mathbb{G}}(U_{\Lambda_{0}}(r_{0})\otimes
1)\bigr{]}\right)\\\
&=\sum_{s\in\Lambda}(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{s})\left[(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{r_{0}})\left(\sum_{r_{0}\in\Lambda_{0}}e_{r_{0}s,s}\otimes\bigl{[}U_{\mathbb{G}}(U_{\Lambda_{0}}(r_{0})\otimes
1)\bigr{]}\right)\right]\\\
&=\sum_{s\in\Lambda}(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{s})\left(\sum_{r_{0}\in\Lambda_{0}}e_{r_{0}s,s}\otimes\Bigl{(}\bigl{[}(\operatorname{id}\otimes\alpha^{\ast}_{r_{0}})(U_{\mathbb{G}})\bigr{]}(U_{\Lambda_{0}}(r_{0})\otimes
1)\Bigr{)}\right)\\\
&=\sum_{s\in\Lambda}(\operatorname{id}\otimes\operatorname{id}\otimes\alpha^{\ast}_{s})\left(\sum_{r_{0}\in\Lambda_{0}}e_{r_{0}s,s}\otimes\bigl{[}\bigl{(}U_{\Lambda_{0}}(r_{0})\otimes
1\bigr{)}(U_{\mathbb{G}})\bigr{]}\right),\end{split}$

where the last equality used the covariance of $U_{\mathbb{G}}$ and
$U_{\Lambda}$. Combining (4.16) and (4.17) proves

(4.18) $(\pi\otimes
1)\widetilde{W}_{\mathbb{G}}=\widetilde{W}_{\mathbb{G}}(\pi\otimes 1),$

from which (4.13a) follows by noting that $\pi$ is a projection. Now (4.14)
follows from (4.13a), (4.13b) and (4.10). This proves (b). ∎

######  Proposition 4.6.

Using the above notations, let
$c_{\pi}\colon\mathcal{B}(\ell^{2}(\Lambda)\otimes\mathscr{H})\to\mathcal{B}(\mathscr{K})$
be the compression by the projection $\pi$ (i.e. the graph of $c_{\pi}(A)$ is
the intersection of the graph of $\pi A\pi$ with
$\mathscr{K}\times\mathscr{K}$), then the following holds:

  1. (a)

the unitary operator

$W=\left(c_{\pi}\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\operatorname{id}_{C(\Lambda)}\right)\left(\widetilde{W}_{\mathbb{G}}\right)\in\mathcal{B}(\mathscr{K})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)$

is a unitary representation of $\mathbb{G}\rtimes\Lambda$ on $\mathscr{K}$;

  2. (b)

The subrepresentation $\mathscr{K}$ of $\widetilde{W}_{\mathbb{G}}$ (resp.
$\widetilde{W}_{\Lambda}$) is given by
$W_{\mathbb{G}}=(c_{\pi}\otimes\operatorname{id})\left(\widetilde{W}_{\mathbb{G}}\right)$
(resp.
$W_{\Lambda}=(c_{\pi}\otimes\operatorname{id})\left(\widetilde{W}_{\Lambda}\right)$),
and

(4.19) $W_{\mathbb{G}}=\operatorname{Res}_{\mathbb{G}}(W),\quad
W_{\Lambda}=\operatorname{Res}_{\Lambda}(W).$

###### Proof.

This follows from Proposition 2.2, Corollary 4.4, Lemma 4.5 and the definition
of subrepresentations. ∎

######  Definition 4.7.

Using the above notations, we call $W$ the induced representation of $U$, and
denote it by
$\operatorname{Ind}^{\mathbb{G}\rtimes\Lambda}_{\mathbb{G}\rtimes\Lambda_{0}}(U)$,
or simply $\operatorname{Ind}(U)$ when the underlying compact quantum groups
$\mathbb{G}\rtimes\Lambda_{0}$ and $\mathbb{G}\rtimes\Lambda$ are clear from
context.

##  5\. Some character formulae

Let $\Lambda_{0}$ be a subgroup of $\Lambda$, $U$ a finite dimensional unitary
representation of $\mathbb{G}\rtimes\Lambda_{0}$,
$\operatorname{Ind}^{\mathbb{G}\rtimes\Lambda}_{\mathbb{G}\rtimes\Lambda_{0}}(U)$
the induced representation of the global compact quantum group
$\mathbb{G}\rtimes\Lambda$. In this section, we aim to calculate the character
of the induced representation
$\operatorname{Ind}^{\mathbb{G}\rtimes\Lambda}_{\mathbb{G}\rtimes\Lambda_{0}}(U)$.
The approach adopted here emphasizes the underlying group action of $\Lambda$
on the characters of the conjugacy class of the principal subgroup
$\mathbb{G}\rtimes\Lambda_{0}$ as described in Proposition 3.9.

For any subgroup $\Lambda_{1}$ and any $f_{0}\in C(\Lambda_{1})$, we use
$E_{\Lambda_{1}}(f_{0})$ to denote the function in $C(\Lambda)$ with
$[E_{\Lambda_{1}}(f_{0})](r)=0$ if $r\notin\Lambda_{1}$ and
$[E_{\Lambda_{1}}(f_{0})](r)=f_{0}(r)$ if $r\in\Lambda_{1}$. Then
$E_{\Lambda_{1}}\colon C(\Lambda_{1})\rightarrow C(\Lambda)$ is a morphism of
$C^{\ast}$-algebras, which is not unital unless $\Lambda_{1}=\Lambda$, in
which case $E_{\Lambda_{1}}=\operatorname{id}_{C(\Lambda)}$. By Proposition
3.9, we have an action

(5.1) $\begin{split}\Lambda&\curvearrowright\left\\{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}\;:\;r\in\Lambda\right\\}\\\
s&\mapsto\left\\{\mathbb{G}\rtimes r\Lambda_{0}r^{-1}\mapsto\mathbb{G}\rtimes
sr\Lambda_{0}{(sr)}^{-1}\right\\}\end{split}$

of $\Lambda$ on the set of subgroups of $\mathbb{G}\rtimes\Lambda$ conjugate
to $\mathbb{G}\rtimes\Lambda_{0}$ via elements in $\Lambda$ (the term
conjugate is justified by considering the case when $\mathbb{G}$ is a genuine
compact group).

Our main result in this section is the following proposition.

######  Proposition 5.1.

Let $\Lambda_{0}$ be a subgroup of $\Lambda$,
$U\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0})$ a finite dimensional unitary representation of
$\mathbb{G}\rtimes\Lambda_{0}$, $W$ the induced representation
$\operatorname{Ind}^{\mathbb{G}\rtimes\Lambda}_{\mathbb{G}\rtimes\Lambda_{0}}(U)$.
Suppose $\chi$ is the character of the unitary representation $U$ of
$\mathbb{G}\rtimes\Lambda_{0}$, and for each $r$, define

(5.2) $r\cdot
U:=(\operatorname{id}_{\mathscr{H}}\otimes\alpha_{r^{-1}}^{\ast}\otimes\operatorname{Ad}_{r^{-1}}^{\ast})(U)\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(r\Lambda_{0}r^{-1}).$

Then $r\cdot U$ is a unitary representation of $\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}$ with $1\cdot U=U$, and $(rs)\cdot U=r\cdot(s\cdot U)$ for
all $r,s\in\Lambda$. Denote the character of $r\cdot U$ by $\chi_{r}$ (so
$\chi_{1_{\Lambda}}=\chi$), then

(5.3)
$\chi_{W}=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r\in\Lambda}(\operatorname{id}_{A}\otimes
E_{r\Lambda_{0}r^{-1}})\chi_{r},$

where $\chi_{W}$ is the character of $W$.

###### Proof.

That $r\cdot U$ is a finite dimensional unitary representation of
$\mathbb{G}\rtimes r\Lambda_{0}r^{-1}$ follows from the fact (Proposition 3.9)
that

$\alpha_{r}^{\ast}\otimes\operatorname{Ad}_{r}^{\ast}\colon A\otimes
r^{-1}\Lambda_{0}r\rightarrow A\otimes\Lambda_{0}$

is an isomorphism of compact quantum groups for any $r\in\Lambda$. The
identities $1_{\Lambda}\cdot U=U$ and $r\cdot(s\cdot U)=(rs)\cdot U$ follows
directly from definitions. We proceed to prove the character formula (5.3).

For any $r\in\Lambda$, let ${(r\cdot U)}_{\mathbb{G}}$ be the restriction of
$r\cdot U$ to $\mathbb{G}$, and ${(r\cdot U)}_{r\Lambda_{0}r^{-1}}$ the
restriction of $r\cdot U$ to $r\Lambda_{0}r^{-1}$. We denote the character of
${(r\cdot U)}_{\mathbb{G}}$ (resp. ${(r\cdot U)}_{r\Lambda_{0}r^{-1}}$) by
$\chi_{r,\mathbb{G}}$ (resp. $\chi_{r,r\Lambda_{0}r^{-1}}$). One easily checks
that
$\chi_{r,\mathbb{G}}=\alpha_{r^{-1}}^{\ast}(\chi_{1_{\Lambda},\mathbb{G}})$
and
$\chi_{r,r\Lambda_{0}r^{-1}}=\operatorname{Ad}_{r^{-1}}^{\ast}(\chi_{1_{\Lambda},\Lambda_{0}})$.
Fix a Hilbert basis $(e_{1},\ldots,e_{d})$ for $\mathscr{H}$, and let
$(e_{ij},i,j=1,\ldots,d)$ be the corresponding matrix unit for
$\mathcal{B}(\mathscr{H})$. Using this matrix unit, we can write

(5.4a) $U_{\mathbb{G}}=\sum_{i,j=1}^{d}e_{i,j}\otimes
u_{ij},\;u_{ij}\in\operatorname{Pol}(\mathbb{G});$ (5.4b)
$U_{\Lambda_{0}}=\sum_{r_{0}\in\Lambda_{0}}U_{\Lambda_{0}}(r_{0})\otimes\delta_{r_{0}}.$

Let $e_{r,s}$, $\pi$, $\mathscr{K}$, $\widetilde{W}_{\mathbb{G}}$,
$\widetilde{W}_{\Lambda}$, $W_{\mathbb{G}}$ and $W_{\Lambda}$ have the same
meaning as in § 4, then the construction in § 4 tells us that

(5.5)
$\begin{split}\chi_{W}&=(\operatorname{Tr}_{\ell^{2}(\Lambda)}\otimes\operatorname{Tr}_{\mathscr{H}}\otimes\operatorname{id}_{A}\otimes\operatorname{id}_{C(\Lambda)})\left[\pi_{12}\cdot{(\widetilde{W}_{\mathbb{G}})}_{123}\cdot\pi_{12}\cdot{(\widetilde{W}_{\Lambda})}_{124}\cdot\pi_{12}\right].\end{split}$

In the following calculations, we often omit the subscripts of the trace
functions $\operatorname{Tr}$ on $\ell^{2}(\Lambda)$ or on $\mathscr{H}$, and
also the subscripts for the multiplicative neutral element $1$ of various
algebras, whenever it is a trivial task to decipher to which trace and
multiplicative neutral element we are referring. The same goes with
$\operatorname{id}$ without subscripts.

Note that for any $r,s\in\Lambda$,
$\operatorname{Ad}_{r}^{\ast}(\delta_{s})=\delta_{r^{-1}sr}$. With these
preparations, we now have

(5.6)
$\begin{split}\chi_{r}&=(\alpha_{r^{-1}}^{\ast}\otimes\operatorname{Ad}_{r^{-1}}^{\ast})(\chi)\\\
&=(\alpha_{r^{-1}}^{\ast}\otimes\operatorname{Ad}_{r^{-1}}^{\ast})\left(\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\Bigl{(}e_{i,j}U_{\Lambda_{0}}(r_{0})\Bigr{)}u_{ij}\otimes\delta_{r_{0}}\right)\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\Bigl{(}e_{i,j}U_{\Lambda_{0}}(r_{0})\Bigr{)}\alpha_{r^{-1}}^{\ast}(u_{ij})\otimes\delta_{rr_{0}r^{-1}}.\end{split}$

By (4.4), (4.10) and (4.12), we deduce from (5.5) that

(5.7)
$\begin{split}\left\lvert\Lambda_{0}\right\rvert^{3}\chi_{W}&=\sum_{a_{0},b_{0},c_{0}\in\Lambda_{0}}\sum_{a,b,c\in\Lambda}\sum_{r,s,t\in\Lambda}\sum_{i,j=1}^{d}\operatorname{Tr}(e_{a_{0}a,a}e_{r,r}e_{b_{0}b,b}e_{st^{-1},s}e_{c_{0}c,c})\\\
&\hskip
34.44434pt\operatorname{Tr}\Bigl{(}U_{\Lambda_{0}}(a_{0})e_{i,j}U_{\Lambda_{0}}(b_{0})U_{\Lambda_{0}}(c_{0})\Bigr{)}\alpha_{r}^{\ast}(u_{ij})\otimes\delta_{t}.\end{split}$

On the right side of the above sum, the first trace doesn’t vanish if and only
if it is $1$, which happens exactly when

(5.8)
$\begin{split}&\phantom{\iff}a=r=b_{0}b,\;b=st^{-1},\;s=c_{0}c,\;a_{0}a=c\\\
&\iff
b=b_{0}^{-1}a,\;c=a_{0}a,\;r=a,\;s=c_{0}a_{0}a,\;t=b^{-1}s=a^{-1}b_{0}c_{0}a_{0}a.\end{split}$

Using this condition in (5.7), we get

(5.9) $\begin{split}&\hskip
11.99998pt\left\lvert\Lambda_{0}\right\rvert^{3}\chi_{W}\\\
&=\sum_{a_{0},b_{0},c_{0}\in\Lambda_{0}}\sum_{a\in\Lambda}\sum_{i,j=1}^{d}\operatorname{Tr}\Bigl{(}U_{\Lambda_{0}}(a_{0})e_{i,j}U_{\Lambda_{0}}(b_{0})U_{\Lambda_{0}}(c_{0})\Bigr{)}\alpha_{a}^{\ast}(u_{ij})\otimes\delta_{a^{-1}b_{0}c_{0}a_{0}a}\\\
&=\sum_{a_{0},b_{0},c_{0}\in\Lambda_{0}}\sum_{a\in\Lambda}\sum_{i,j=1}^{d}\operatorname{Tr}\Bigl{(}e_{i,j}U_{\Lambda_{0}}(b_{0})U_{\Lambda_{0}}(c_{0})U_{\Lambda_{0}}(a_{0})\Bigr{)}\alpha_{a}^{\ast}(u_{ij})\otimes\delta_{a^{-1}b_{0}c_{0}a_{0}a}\\\
&=\sum_{a_{0},b_{0},c_{0}\in\Lambda_{0}}\sum_{a\in\Lambda}\sum_{i,j=1}^{d}\operatorname{Tr}\Bigl{(}e_{i,j}U_{\Lambda_{0}}(b_{0}c_{0}a_{0})\Bigr{)}\alpha_{a}^{\ast}(u_{ij})\otimes\delta_{a^{-1}b_{0}c_{0}a_{0}a}\\\
&=\left\lvert\Lambda_{0}\right\rvert^{2}\sum_{a\in\Lambda}\sum_{r_{0}\in\Lambda_{0}}\sum_{i,j=1}^{d}\operatorname{Tr}\Bigl{(}e_{i,j}U_{\Lambda_{0}}(r_{0})\Bigr{)}\alpha_{a}^{\ast}(u_{ij})\otimes\delta_{a^{-1}r_{0}a}\\\
&=\left\lvert\Lambda_{0}\right\rvert^{2}\sum_{r\in\Lambda}(\operatorname{id}\otimes
E_{r^{-1}\Lambda_{0}r})(\chi_{r}),\end{split}$

where the last line uses (5.6) and the change of variable $r=a^{-1}$. Dividing
$\left\lvert\Lambda_{0}\right\rvert^{3}$ on both sides of (5.9) proves (5.3).
∎

######  Corollary 5.2.

Using the notations in Proposition 5.1, $U$ and $r\cdot U$ induce equivalent
unitary representations of $\mathbb{G}\rtimes\Lambda$ for all $r\cdot U$.

###### Proof.

By Proposition 5.1, we see that $\operatorname{Ind}(U)$ and
$\operatorname{Ind}(r\cdot U)$ have the same character. ∎

It is worth pointing out that there are in fact many repetitions in the terms
of the right side of formula (5.3), as is shown by the following lemma.

######  Lemma 5.3.

Using the notations of Proposition 5.1, the following holds:

  1. (a)

for any $r\in\Lambda$, we have

(5.10) $(\operatorname{id}\otimes
E_{r\Lambda_{0}r^{-1}})\chi_{r}=\bigl{(}\alpha_{r^{-1}}^{\ast}\otimes\operatorname{Ad}_{r^{-1}}^{\ast}\bigr{)}\left[(\operatorname{id}\otimes
E_{\Lambda_{0}})(\chi)\right];$

in $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$;

  2. (b)

for any $r,s\in\Lambda$, if $r^{-1}s\in\Lambda_{0}$, i.e.
$r\Lambda_{0}=s\Lambda_{0}$ and $r\Lambda_{0}r^{-1}=s\Lambda_{0}s^{-1}$, then

(5.11) $(\operatorname{id}\otimes
E_{r\Lambda_{0}r^{-1}})\chi_{r}=(\operatorname{id}\otimes
E_{s\Lambda_{0}s^{-1}})\chi_{s}$

in $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$. In particular,

(5.12) $\chi_{r}=\chi_{s},$

or equivalently, $r\cdot U$ and $s\cdot U$ are unitarily equivalent unitary
representations of the same compact quantum group $\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}$.

###### Proof.

Using the same notations as in the proof of Proposition 5.1, it is clear that

(5.13a) ${(r\cdot
U)}_{\mathbb{G}}=\sum_{i,j=1}^{d}e_{i,j}\otimes\alpha_{r^{-1}}^{\ast}(u_{ij}),$
(5.13b) ${(r\cdot
U)}_{r\Lambda_{0}r^{-1}}=\sum_{r_{0}\in\Lambda_{0}}U_{\Lambda_{0}}(r_{0})\otimes\delta_{rr_{0}r^{-1}}.$

Calculating in $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$, we have

(5.14) $\begin{split}(\operatorname{id}\otimes
E_{r\Lambda_{0}r^{-1}})\chi_{r}&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\bigl{(}e_{i,j}U_{\Lambda_{0}}(r_{0})\bigr{)}\otimes\alpha_{r^{-1}}^{\ast}(u_{ij})\otimes\delta_{rr_{0}r^{-1}}\\\
&=\bigl{(}\alpha_{r^{-1}}^{\ast}\otimes\operatorname{Ad}_{r^{-1}}^{\ast}\bigr{)}\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\bigl{(}e_{i,j}U_{\Lambda_{0}}(r_{0})\bigr{)}\otimes
u_{ij}\otimes\delta_{r_{0}}\\\
&=\bigl{(}\alpha_{r^{-1}}^{\ast}\otimes\operatorname{Ad}_{r^{-1}}\bigr{)}\left[(\operatorname{id}\otimes
E_{\Lambda_{0}})\chi\right].\end{split}$

This proves (a).

By (a), to establish (b), it suffices to show that

(5.15) $\forall s_{0}\in\Lambda_{0},\quad(\operatorname{id}\otimes
E_{\Lambda_{0}})\chi=\left(\alpha_{s_{0}}^{\ast}\otimes\operatorname{Ad}_{s_{0}}^{\ast}\right)\left[(\operatorname{id}\otimes
E_{\Lambda_{0}})\chi\right].$

Calculating the right side gives

(5.16) $\begin{split}&\hskip
11.99998pt\bigl{(}\alpha_{s_{0}}^{\ast}\otimes\operatorname{Ad}_{s_{0}}^{\ast}\bigr{)}\left[(\operatorname{id}\otimes
E_{\Lambda_{0}})\chi\right]\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\bigl{(}e_{i,j}U_{\Lambda_{0}}(r_{0})\bigr{)}\otimes\alpha_{s_{0}}^{\ast}(u_{ij})\otimes\delta_{s_{0}^{-1}r_{0}s_{0}}\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\bigl{(}e_{i,j}U_{\Lambda_{0}}(s_{0}r_{0}s_{0}^{-1})\bigr{)}\otimes\alpha_{s_{0}}^{\ast}(u_{ij})\otimes\delta_{r_{0}}\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\Bigl{(}U(r_{0})U_{\Lambda_{0}}(s_{0}^{-1})e_{i,j}U(s_{0})\Bigr{)}\otimes\alpha_{s_{0}}^{\ast}(u_{ij})\otimes\delta_{r_{0}}.\end{split}$

Since $U_{\Lambda_{0}}$ and $U_{\mathbb{G}}$ are covariant, we have

(5.17) $\sum_{i,j}^{d}U(s_{0})e_{i,j}\otimes
u_{ij}=\sum_{i,j=1}^{d}e_{i,j}U(s_{0})\otimes\alpha^{\ast}_{s_{0}}(u_{ij}).$

Combining (5.16) and (5.17), we have

(5.18) $\begin{split}&\hskip
11.99998pt\bigl{(}\alpha_{s_{0}}^{\ast}\otimes\operatorname{Ad}_{s_{0}}^{\ast}\bigr{)}\left[(\operatorname{id}\otimes
E_{\Lambda_{0}})\chi\right]\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\Bigl{(}U(r_{0})U_{\Lambda_{0}}(s_{0}^{-1})e_{i,j}U(s_{0})\Bigr{)}\otimes\alpha_{s_{0}}^{\ast}(u_{ij})\otimes\delta_{r_{0}}\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\Bigl{(}U(r_{0})U_{\Lambda_{0}}(s_{0}^{-1})U(s_{0})e_{i,j}\Bigr{)}\otimes
u_{ij}\otimes\delta_{r_{0}}\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\Bigl{(}U(r_{0})e_{i,j}\Bigr{)}\otimes
u_{ij}\otimes\delta_{r_{0}}\\\
&=\sum_{i,j=1}^{d}\sum_{r_{0}\in\Lambda_{0}}\operatorname{Tr}\Bigl{(}e_{i,j}U(r_{0})\Bigr{)}\otimes
u_{ij}\otimes\delta_{r_{0}}\\\ &=(\operatorname{id}\otimes
E_{\Lambda_{0}})\chi.\end{split}$

This establishes (5.15) and proves (b). ∎

######  Remark 5.4.

By Lemma 5.3 (b) and Proposition 5.1, one can in fact choose any complete set
$L\subseteq\Lambda$ of representatives of the left coset space
$\Lambda/\Lambda_{0}$, and the character formula (5.3) can then be written
more concisely as

(5.19) $\chi_{W}=\sum_{r\in L}(\operatorname{id}_{A}\otimes
E_{r\Lambda_{0}r^{-1}})\chi_{r}.$

In the classical case where $\mathbb{G}$ is a genuine compact group, one can
easily check that the usual character formula for the representation induced
by a representation of an open subgroup takes the form (5.19). The reason we
prefer (5.3) is that it does not involve a seemingly arbitrary choice of a
complete set of representatives $L$ for $\Lambda/\Lambda_{0}$, and thus, in
the author’s opinion, is more aesthetically pleasing. One might also use this
choice of left coset representatives to fabric the induced representation.
However, in our more symmetric approach (cf. § 4), everything seems more
natural, and the underlying group action of $\Lambda$ on the various
characters $\chi_{r},r\in\Lambda$ becomes more transparent in (5.3), and we
hope this hidden symmetry will keep the reader from losing himself/herself in
the details of the tedious calculations to be presented later.

##  6\. Dimension of the intertwiner space of induced representations

Let $\Theta,\Xi$ be subgroups of $\Lambda$,
$U\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Theta)$ a finite dimensional unitary representation of
$\mathbb{G}\rtimes\Theta$,
$W\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Xi)$ a finite dimensional unitary representation of $\mathbb{G}\rtimes\Xi$.
For the sake of brevity, we denote the induced representation
$\operatorname{Ind}_{\mathbb{G}\rtimes\Theta}^{\mathbb{G}\rtimes\Lambda}(U)$
simply by $\operatorname{Ind}(U)$, and $\operatorname{Ind}(W)$ has the similar
obvious meaning. Equipped with the character formula established in § 5, one
naturally wonders how can we calculate
$\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}\bigl{(}\operatorname{Ind}(U),\operatorname{Ind}(W)\bigr{)}$
in terms of some simpler data. This section focuses on this calculation, and
the result here will play an important role in proving the irreducibility of
some induced representations (as it turns out, these are all irreducible
representations of $\mathbb{G}\rtimes\Lambda$ up to equivalence) as well as
our later calculation of the fusion rules.

For any representation $\rho$, we use $\chi(\rho)$ to denote the character of
the representation. We denote the Haar state on $\mathbb{G}$ by $h$, and the
Haar state on $\mathbb{G}\rtimes\Lambda_{0}$ by $h^{\Lambda_{0}}$ whenever
$\Lambda_{0}$ is a subgroup of $\Lambda$.

By the general representation theory of compact quantum groups, we have

(6.1)
$\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}\bigl{(}\operatorname{Ind}(U),\operatorname{Ind}(W)\bigr{)}=h^{\Lambda}\bigl{(}{[\chi(\operatorname{Ind}(U))]}^{\ast}{[\chi(\operatorname{Ind}(W))]}\bigr{)}.$

By Proposition 5.1, for each $r\in\Lambda$, we have a representation $r\cdot
U$ (resp. $r\cdot W$) of $\mathbb{G}\rtimes r\Theta r^{-1}$ (resp.
$\mathbb{G}\rtimes r\Xi r^{-1}$), and combined with (6.1), we have

(6.2) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}}(\operatorname{Ind}(U),\operatorname{Ind}(W))\\\
&=\frac{1}{\left\lvert\Theta\right\rvert\cdot\left\lvert\Xi\right\rvert}\sum_{r,s\in\Lambda}h^{\Lambda}\left({[(\operatorname{id}\otimes
E_{r\Theta r^{-1}})\chi(r\cdot U)]}^{\ast}{[(\operatorname{id}\otimes E_{s\Xi
s^{-1}})\chi(s\cdot W)]}\right).\end{split}$

######  Notations 6.1.

To simplify our notations, let $\Lambda(r,s):=r\Theta r^{-1}\cap s\Xi s^{-1}$
for any $r,s\in\Lambda$.

######  Lemma 6.2.

Using the above notations, for any $r,s\in\Lambda$, we have

(6.3) $\begin{split}&\hskip
11.99998pth^{\Lambda}\left({[(\operatorname{id}\otimes E_{r\Theta
r^{-1}})\chi(r\cdot U)]}^{\ast}{[(\operatorname{id}\otimes E_{s\Xi
s^{-1}})\chi(s\cdot W)]}\right)\\\
&=\frac{1}{[\Lambda\colon\Lambda(r,s)]}\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\left((r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
W)|_{\mathbb{G}\rtimes\Lambda(r,s)}\right).\end{split}$

###### Proof.

For any subgroup $\Lambda_{0}$ of $\Lambda$, whenever
$f\in\operatorname{Pol}(\mathbb{G})$, $r_{0}\in\Lambda_{0}$, by (1.7) in § 1,
we have

(6.4)
$h^{\Lambda}(f\otimes\delta_{r_{0}})=\frac{1}{\left\lvert\Lambda\right\rvert}h(f)=\frac{1}{[\Lambda\colon\Lambda_{0}]}h^{\Lambda_{0}}(f\otimes\delta_{r_{0}}).$

Hence,

(6.5) $h^{\Lambda}\circ(\operatorname{id}\otimes
E_{\Lambda_{0}})=\frac{1}{[\Lambda\colon\Lambda_{0}]}h^{\Lambda_{0}}.$

By definition and a straightforward calculation, we have

(6.6a) $(\operatorname{id}\otimes E_{r\Theta r^{-1}})\chi(r\cdot U)=\sum_{t\in
r\Theta r^{-1}}(\operatorname{Tr}\otimes\operatorname{id})\left({(r\cdot
U)}_{\mathbb{G}}\bigl{(}{(r\cdot U)}_{r\Theta r^{-1}}(t)\otimes
1\bigr{)}\right)\otimes\delta_{t},$ (6.6b) $(\operatorname{id}\otimes E_{s\Xi
s^{-1}})\chi(s\cdot W)=\sum_{t\in s\Xi
s^{-1}}(\operatorname{Tr}\otimes\operatorname{id})\left({(s\cdot
W)}_{\mathbb{G}}\bigl{(}{(s\cdot W)}_{s\Xi s^{-1}}(t)\otimes
1\bigr{)}\right)\otimes\delta_{t}.$

It follows from (6.6a) and (6.6b) that

(6.7) $\begin{split}&\hskip 11.99998pt{[(\operatorname{id}\otimes E_{r\Theta
r^{-1}})\chi(r\cdot U)]}^{\ast}{[(\operatorname{id}\otimes E_{s\Xi
s^{-1}})\chi(s\cdot W)]}\\\
&=\sum_{t\in\Lambda(r,s)}\Big{\\{}{\left[(\operatorname{Tr}\otimes\operatorname{id})\left({(r\cdot
U)}_{\mathbb{G}}\bigl{(}{(r\cdot U)}_{r\Theta r^{-1}}(t)\otimes
1\bigr{)}\right)\right]}^{\ast}\\\ &\hskip
30.00005pt\left[(\operatorname{Tr}\otimes\operatorname{id})\left({(s\cdot
W)}_{\mathbb{G}}\bigl{(}{(s\cdot W)}_{s\Xi s^{-1}}(t)\otimes
1\bigr{)}\right)\right]\Big{\\}}\otimes\delta_{t}\\\
&=(\operatorname{id}\otimes E_{\Lambda(r,s)})\left({\left[\chi\left((r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\right)\right]}^{\ast}\left[\chi\left((s\cdot
W)|_{\mathbb{G}\rtimes\Lambda(r,s)}\right)\right]\right).\end{split}$

Taking $\Lambda_{0}=\Lambda(r,s)$ in (6.5) and combining with (6.7) proves
(6.3). ∎

######  Proposition 6.3.

Using the above notations, we have

(6.8) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}\bigl{(}\operatorname{Ind}(U),\operatorname{Ind}(W)\bigr{)}\\\
&=\frac{1}{\left\lvert\Theta\right\rvert\cdot\left\lvert\Xi\right\rvert}\sum_{r,s\in\Lambda}\frac{1}{[\Lambda\colon\Lambda(r,s)]}\\\
&\hskip
30.00005pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\left({(r\cdot
U)}|_{\mathbb{G}\rtimes\Lambda(r,s)},{(s\cdot
W)}|_{\mathbb{G}\rtimes\Lambda(r,s)}\right).\end{split}$

###### Proof.

This follows directly from the formula (6.2) and Lemma 6.2. ∎

######  Corollary 6.4.

Let $\Lambda_{0}$ be a subgroup of $\Lambda$, $U$ a unitary representation of
$\mathbb{G}\rtimes\Lambda_{0}$, then the following are equivalent:

  1. (a)

the unitary representation $\operatorname{Ind}(U)$ of
$\mathbb{G}\rtimes\Lambda$ is irreducible;

  2. (b)

for any $r,s\in\Lambda$, posing $\Lambda(r,s)=r\Lambda_{0}r^{-1}\cap
s\Lambda_{0}s^{-1}$, we have

(6.9) $\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}=\delta_{r\Lambda_{0},s\Lambda_{0}};$

  3. (c)

$U$ is irreducible, and

(6.10) $\begin{split}&\forall r,s\in\Lambda,\quad r^{-1}s\notin\Lambda_{0}\\\
&\implies\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}=0.\end{split}$

In particular, if any of the above conditions holds, then $U$ itself is
irreducible.

###### Proof.

If $r^{-1}s\in\Lambda_{0}$, then $r\Lambda_{0}r^{-1}=s\Lambda_{0}s^{-1}$, so
$\Lambda(r,s)=r\Lambda_{0}r^{-1}=s\Lambda_{0}s^{-1}$. By Proposition 3.9, we
see that

(6.11) $\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U,r\cdot
U\bigr{)}=\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U,U).$

By Proposition 6.3, Lemma 5.3, and the above, we have

$\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}}\bigl{(}\operatorname{Ind}(U),\operatorname{Ind}(U)\bigr{)}\\\
&=\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\in\Lambda_{0}\end{subarray}}\frac{1}{[\Lambda\colon
r\Lambda_{0}r^{-1}]}\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U,s\cdot U\bigr{)}\\\ &\hskip
30.00005pt+\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\notin\Lambda_{0}\end{subarray}}\frac{1}{[\Lambda\colon\Lambda(r,s)]}\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}\\\
&=\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\in\Lambda_{0}\end{subarray}}\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U,r\cdot U\bigr{)}\\\ &\hskip
30.00005pt+\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\notin\Lambda_{0}\end{subarray}}\frac{1}{[\Lambda\colon\Lambda(r,s)]}\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}\\\
&=\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\in\Lambda_{0}\end{subarray}}\frac{1}{[\Lambda\colon\Lambda_{0}]}\dim_{\mathbb{G}\rtimes\Lambda_{0}}(U,U)\\\
&\hskip
30.00005pt+\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\notin\Lambda_{0}\end{subarray}}\frac{1}{[\Lambda\colon\Lambda(r,s)]}\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}\\\
&=\frac{\left\lvert\Lambda\right\rvert\cdot\left\lvert\Lambda_{0}\right\rvert}{\left\lvert\Lambda_{0}\right\rvert^{2}\cdot[\Lambda\colon\Lambda_{0}]}\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U,U)\\\
&\hskip
30.00005pt+\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\notin\Lambda_{0}\end{subarray}}\frac{1}{[\Lambda\colon\Lambda(r,s)]}\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}.\end{split}$

Since
$\left\lvert\Lambda\right\rvert\cdot\left\lvert\Lambda_{0}\right\rvert=\left\lvert\Lambda_{0}\right\rvert^{2}\cdot[\Lambda\colon\Lambda_{0}]$
and

(6.12) $\begin{split}\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U,s\cdot
U\bigr{)}&=\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U,r\cdot U\bigr{)}\\\
&=\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U,U)=\dim\operatorname{End}_{\mathbb{G}\rtimes\Lambda_{0}}(U)\end{split}$

whenever $r^{-1}s\in\Lambda_{0}$ by Lemma 5.3 and Proposition 3.9, the above
calculation yields

(6.13)
$\dim\operatorname{End}_{\mathbb{G}\rtimes\Lambda}\left(\operatorname{Ind}(U)\right)=\dim\operatorname{End}_{\mathbb{G}\rtimes\Lambda_{0}}(U)+\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\notin\Lambda_{0}\end{subarray}}{d(r,s)\over[\Lambda:\Lambda(r,s)]},$

where

(6.14)
$d(r,s):=\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U)|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}.$

The corollary follows from (6.12) (6.13), (6.14) and the fact that a
representation is irreducible if and only if the dimension of the space of its
self-intertwiners is $1$. ∎

######  Remark 6.5.

Corollary 6.4 is the quantum analogue for Mackey’s criterion for
irreducibility.

##  7\. The $C^{\ast}$-tensor category $\mathcal{CSR}_{\Lambda_{0}}$

We begin by recalling the notations in Proposition 5.1: for any unitary
representation
$U_{\mathbb{G}}\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})$
of $\mathbb{G}$ on some finite dimensional Hilbert space $\mathscr{H}$, and
any $r\in\Lambda$, let $r\cdot U_{\mathbb{G}}$ be the unitary representation
$(\operatorname{id}_{\mathscr{H}}\otimes\alpha^{\ast}_{r^{-1}})(U_{\mathbb{G}})$
of $\mathbb{G}$ on the same space $\mathscr{H}$. It is easy to see that this
defines a left group action of $\Lambda$ on the proper class of all unitary
representations of $\mathbb{G}$, and by passing to quotients, this
representation induces an action of $\Lambda$ on
$\operatorname{Irr}(\mathbb{G})$. From now on, whenever we talk about
$\Lambda$ acting on a unitary representation $U_{\mathbb{G}}$ of $\mathbb{G}$,
or on some class $x\in\operatorname{Irr}(\mathbb{G})$, we always mean these
actions.

######  Definition 7.1.

A subgroup $\Lambda_{0}$ of $\Lambda$ is called a general isotropy subgroup if
there is some $n\in\mathbb{N}$, such that $\Lambda_{0}$ is an isotropy
subgroup (subgroup of stabilizer for some point) for the $n$-fold product
${[\operatorname{Irr}(\mathbb{G})]}^{n}$ as a $\Lambda$-set; in other words,
if there exists an $n$-tuple $(x_{1},\ldots,x_{n})$ with all
$x_{i}\in\operatorname{Irr}(\mathbb{G})$, such that

$\Lambda_{0}=\left\\{r\in\Lambda\;:\;\forall i=1,\ldots,n,\quad r\cdot
x_{i}=x_{i}\right\\}=\cap_{i=1}^{n}\Lambda_{x_{i}}.$

The finite (recall that $\Lambda$ is finite) family of all general isotropy
subgroups of $\Lambda$ is denoted by
$\operatorname{\mathcal{G}_{iso}}(\Lambda)$.

The following proposition is an immediate consequence of properties of
$\Lambda$-sets and Definition 7.1.

######  Proposition 7.2.

The family $\operatorname{\mathcal{G}_{iso}}(\Lambda)$ is stable under
intersection and conjugation by elements of $\Lambda$. ∎

######  Definition 7.3.

Let $\Lambda_{0}$ be a general isotropy subgroup of $\Lambda$. A covariant
system of representations (or CSR for short) subordinate to $\Lambda_{0}$ is a
triple $(\mathscr{H},u,w)$, where

  * •

$\mathscr{H}$ is a finite dimensional Hilbert space;

  * •

$u$ is a unitary representation of $\mathbb{G}$ on $\mathscr{H}$;

  * •

$w$ is a unitary representation of $\Lambda_{0}$ on $\mathscr{H}$,

such that $u$ and $w$ are covariant. In this paper, CSRs are often denoted by
bold faced uppercase letters like
$\mathbf{A},\mathbf{B},\mathbf{C},\ldots$(mostly $\mathbf{S}$) with possible
subscripts.

By Proposition 2.2, the covariant systems of representations subordinate to a
general isotropy subgroup $\Lambda_{0}$ correspond bijectively to the class of
unitary representations of $\mathbb{G}\rtimes\Lambda_{0}$, via

$(\mathscr{H},u,w)\mapsto u_{12}w_{13}$

in one direction, and

$U_{\mathscr{H}}\mapsto(\mathscr{H},U_{\mathscr{H},\mathbb{G}},U_{\mathscr{H},\Lambda_{0}})$

in the other, where $\mathscr{H}$ is the underlying space of the
representation $U_{\mathscr{H}}$ of $\mathbb{G}\rtimes\Lambda_{0}$, and
$U_{\mathscr{H},\mathbb{G}}$, $U_{\mathscr{H},\Lambda_{0}}$ are the
restrictions of $U_{\mathscr{H}}$ to $\mathbb{G}$ and $\Lambda_{0}$
respectively. Using this bijection, we can transport the rigid
$C^{\ast}$-tensor category structure on
$\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})$–the category of all finite
dimensional unitary representations of $\mathbb{G}\rtimes\Lambda_{0}$, to the
class of covariant systems of representations subordinate to $\Lambda_{0}$,
thereby getting a rigid $C^{\ast}$-tensor category
$\mathcal{CSR}_{\Lambda_{0}}$ whose objects are CSRs subordinate to
$\Lambda_{0}$.

To make this transport of categorical structures less tautological, we make a
convenient characterization of the morphisms in $\mathcal{CSR}_{\Lambda_{0}}$.

######  Proposition 7.4.

Fix a general isotropy subgroup $\Lambda_{0}$ of $\Lambda$. For $i=1,2$, let
$\mathbf{S}_{i}=(\mathscr{H}_{i},u_{i},w_{i})$ be a CSR subordinate to
$\Lambda_{0}$, $U_{i}={(u_{i})}_{12}{(w_{i})}_{13}$ the corresponding unitary
representation of $\mathbb{G}\rtimes\Lambda_{0}$,
$S\in\mathcal{B}(\mathscr{H}_{1},\mathscr{H}_{2})$. Then
$S\in\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}\bigl{(}U_{1},U_{2}\bigr{)}$
if and only if

(7.1)
$S\in\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(w_{1},w_{2}).$

###### Proof.

The condition is easily seen to be sufficient. Indeed, if condition (7.1)
holds, then

(7.2) $(S\otimes 1)w_{1}=w_{2}(S\otimes 1),\quad\\\ (S\otimes
1)u_{1}=u_{2}(S\otimes 1).$

Thus

(7.3) $\begin{split}(S\otimes 1\otimes 1)U_{1}&=(S\otimes 1\otimes
1){(u_{1})}_{12}{(w_{1})}_{13}={(u_{2})}_{12}(S\otimes 1\otimes
1){(w_{1})}_{13}\\\ &={(u_{2})}_{12}{(w_{2})}_{13}(S\otimes 1\otimes
1)=U_{2}(S\otimes 1\otimes 1).\end{split}$

This means exactly $S\in\operatorname{Mor}_{\mathbb{G}}(U_{1},U_{2})$.

To show the necessity of this condition, let
$\epsilon_{\mathbb{G}}\colon\operatorname{Pol}(\mathbb{G})\rightarrow\mathbb{C}$
be the counit of the Hopf-$\ast$-algebra, $\epsilon_{\Lambda_{0}}\colon
C(\Lambda_{0})\rightarrow\mathbb{C}$ the counit for the Hopf $\ast$-algebra
$C(\Lambda_{0})$. Since
$U_{i}\in\mathcal{B}(\mathscr{H}_{i})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0})$ for $i=1,2$ and
$S\in\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{2})$, we have

(7.4) $(S\otimes 1\otimes 1)U_{1}=U_{2}(S\otimes 1\otimes 1).$

Applying
$\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}}$ on
both sides of (7.4) yields

(7.5) $(S\otimes 1)u_{1}=u_{2}(S\otimes 1),$

which means $S\in\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2})$. Applying
$\operatorname{id}\otimes\epsilon_{\mathbb{G}}\otimes\operatorname{id}$ on
both sides of (7.4) yields

(7.6) $(S\otimes 1)w_{1}=w_{1}(S\otimes 1),$

which means $S\in\operatorname{Mor}_{\Lambda_{0}}(w_{1},w_{2})$. ∎

We now define a pair of functors,

$\mathscr{R}_{\Lambda_{0}}\colon\mathcal{CSR}_{\Lambda_{0}}\rightarrow\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})\qquad\text{
and
}\qquad\mathscr{S}_{\Lambda_{0}}\colon\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})\rightarrow\mathcal{CSR}_{\Lambda_{0}}$

between $\mathcal{CSR}_{\Lambda_{0}}$ and
$\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})$ that reflects the transport
of categorical structures discussed above. On the object level, for any
$(\mathscr{H},u,w)\in\mathcal{CSR}_{\Lambda_{0}}$, let
$\mathscr{R}_{\Lambda_{0}}(u,w)$ be the representation $u_{12}w_{13}$ of
$\mathbb{G}\rtimes\Lambda_{0}$ on $\mathscr{H}$; for any unitary
representation $U\in\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})$ on
$\mathscr{H}_{U}$, let $\mathscr{S}_{\Lambda_{0}}(U)$ be the CSR
$(\mathscr{H}_{U},U_{\mathbb{G}},U_{\Lambda_{0}})$ where $U_{\mathbb{G}}$
(resp. $U_{\Lambda_{0}}$) is the restriction of $U$ onto $\mathbb{G}$ (resp.
$\Lambda_{0}$). On the morphism level, both $\mathscr{R}_{\Lambda_{0}}$ and
$\mathscr{S}_{\Lambda_{0}}$ act as identity. By Proposition 7.4 and
Proposition 2.2, $\mathscr{R}_{\Lambda_{0}}$ and $\mathscr{S}_{\Lambda_{0}}$
are indeed well-defined functors inverses to each other, and they are fiber
functors (exact unitary tensor functors [MR3204665, §§2.1, 2.2]) simply
because the rigid $C^{\ast}$-tensor category structure on
$\mathcal{CSR}_{\Lambda_{0}}$ is transported from that of
$\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})$ via
$\mathscr{S}_{\Lambda_{0}}$.

######  Proposition 7.5.

For $i=1,2$, let
$\mathbf{S}_{i}=(\mathscr{H}_{i},u_{i},w_{i})\in\mathcal{CSR}_{\Lambda_{0}}$,
$U_{i}=\mathscr{R}_{\Lambda_{0}}(\mathbf{S}_{i})\in\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})$,
then

(7.7) $\mathscr{S}_{\Lambda_{0}}(U_{1}\times
U_{2})=(\mathscr{H}_{1}\otimes\mathscr{H}_{2},u_{1}\times u_{2},w_{1}\times
w_{2})=\mathbf{S}_{1}\otimes\mathbf{S}_{2}.$

###### Proof.

By definition of the tensor product of representations, $U_{1}\times U_{2}$ is
the representation of $\mathbb{G}\rtimes\Lambda_{0}$ defined by

(7.8) $U_{1}\times
U_{2}={(U_{1})}_{134}{(U_{2})}_{234}\in\mathcal{B}(\mathscr{H}_{1})\otimes\mathcal{B}(\mathscr{H}_{2})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0}),$

where we identified
$\mathcal{B}(\mathscr{H}_{1})\otimes\mathcal{B}(\mathscr{H}_{2})$ with
$\mathcal{B}(\mathscr{H}_{1}\otimes\mathscr{H}_{2})$ canonically.

The restriction of $U_{1}\times U_{2}$ onto $\mathbb{G}$ is

(7.9) $\begin{split}&\hskip
11.99998pt(\operatorname{id}\otimes\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}})(U_{1}\times
U_{2})\\\
&=(\operatorname{id}\otimes\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}})({(U_{1})}_{134})(\operatorname{id}\otimes\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}})({(U_{2})}_{234})\\\
&={(u_{1})}_{13}{(u_{2})}_{23}=u_{1}\times
u_{2}\in\mathcal{B}(\mathscr{H}_{1})\otimes\mathcal{B}(\mathscr{H}_{2})\otimes\operatorname{Pol}(\mathbb{G}).\end{split}$

Similarly, the restriction of $U_{1}\times U_{2}$ onto $\Lambda_{0}$ is

(7.10) $\begin{split}&\hskip
11.99998pt(\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\mathbb{G}}\otimes\operatorname{id})(U_{1}\times
U_{2})\\\
&=(\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\mathbb{G}}\otimes\operatorname{id})({(U_{1})}_{134})(\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\mathbb{G}}\otimes\operatorname{id})({(U_{2})}_{234})\\\
&={(w_{1})}_{13}{(w_{(2)})}_{23}=w_{1}\times
w_{2}\in\mathcal{B}(\mathscr{H}_{1})\otimes\mathcal{B}(\mathscr{H}_{2})\otimes
C(\Lambda_{0}).\end{split}$

Now (7.7) follows from (7.9), (7.10) and the definition of the tensor product
in $\mathcal{CSR}_{\Lambda_{0}}$. ∎

######  Proposition 7.6.

For $i=1,2$, let
$\mathbf{S}_{i}=(\mathscr{H},u_{i},w_{i})\in\mathcal{CSR}_{\Lambda_{0}}$,
$U_{i}:=\mathscr{R}_{\Lambda_{0}}(\mathbf{S}_{i})\in\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})$,
then

(7.11) $\mathscr{S}_{\Lambda_{0}}(U_{1}\oplus
U_{2})=(\mathscr{H}_{1}\oplus\mathscr{H}_{2},u_{1}\oplus u_{2},w_{1}\oplus
w_{2})=\mathbf{S}_{1}\oplus\mathbf{S}_{2}.$

###### Proof.

The proof use the same restriction technique as in the proofs of Proposition
7.4 and Proposition 7.5, which is even simpler in this case. ∎

Until now, we’ve shown that the morphisms, tensor products, and direct sums
all behave as expected in $\mathcal{CSR}_{\Lambda_{0}}$. The description of
the dual of a CSR when $\mathbb{G}$ is of non-Kac type requires a bit further
work on the so-called modular operator, as we presently discuss.

Recall that the contragredient representation $U^{c}$ of a _unitary_
representation $U$ of $\mathbb{G}\rtimes\Lambda_{0}$ on some finite
dimensional Hilbert space $\mathscr{H}$ is defined as
$U^{c}=(j\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0})})(U^{\ast})$, where
$j\colon\mathcal{B}(\mathscr{H})\rightarrow\mathcal{B}(\overline{\mathscr{H}})$
is defined as $T\mapsto\overline{T^{\ast}}$, with $\overline{\mathscr{H}}$
being the conjugate Hilbert space of $\mathscr{H}$, and $\overline{T^{\ast}}$
meaning $T^{\ast}$ viewed as a linear mapping from $\overline{\mathscr{H}}$ to
$\overline{\mathscr{H}}$. Note that
$j:\mathcal{B}(\mathscr{H})\rightarrow\mathcal{B}(\overline{\mathscr{H}})$,
$T\mapsto\overline{T^{\ast}}$ is linear, antimultiplicative and positive (in
particular, it preserves adjoints). If $\mathbb{G}$ is of non-Kac type, so is
$\mathbb{G}\rtimes\Lambda_{0}$ by Proposition 1.4, in which case $U^{c}$ might
not be unitary, which is exactly why the “modular” operator $\rho_{U}$ is
necessary to express the dual object of $\mathscr{S}_{\Lambda_{0}}(U)$ in
$\mathcal{CSR}_{\Lambda_{0}}$ as presented in Proposition 7.7.

######  Proposition 7.7.

Let $\mathbf{S}=(\mathscr{H},u,w)\in\mathcal{CSR}_{\Lambda_{0}}$,
$U=\mathscr{R}_{\Lambda_{0}}(\mathbf{S})\in\operatorname{Rep}(\mathbb{G}\rtimes\Lambda_{0})$,
$U^{c}$ the contragredient representation of $U$ on the conjugate space
$\overline{\mathscr{H}}$ of $\mathscr{H}$. If $\rho_{U}$ is the unique
invertible positive operator in
$\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U,U^{cc})$ (which we call
modular operator) such that
$\operatorname{Tr}(\cdot\,\rho_{U})=\operatorname{Tr}(\cdot\,\rho_{U}^{-1})$
on $\operatorname{End}_{\mathbb{G}\rtimes\Lambda_{0}}(U)$, so that

(7.12) $\overline{U}={\bigl{\\{}{[j(\rho_{U})]}^{1/2}\otimes
1_{\operatorname{Pol}(\mathbb{G})}\otimes
1_{C(\Lambda_{0})}\bigr{\\}}}U^{c}{\bigl{\\{}{[j(\rho_{U})]}^{-1/2}\otimes
1_{\operatorname{Pol}(\mathbb{G})}\otimes 1_{C(\Lambda_{0})}\bigr{\\}}}$

is the conjugate representation of $U$, then the dual of $\mathbf{S}$ is given
by $\overline{\mathbf{S}}=(\overline{\mathscr{H}},u^{\prime},w^{\prime})$,
where

(7.13) $\begin{split}u^{\prime}&=({j(\rho_{U})}^{1/2}\otimes
1)u^{c}({j(\rho_{U})}^{-1/2}\otimes 1),\\\
w^{\prime}&=({j(\rho_{U})}^{1/2}\otimes 1)w^{c}({j(\rho_{U})}^{-1/2}\otimes
1).\end{split}$

Note that $w^{c}=\overline{w}$ as $\Lambda$ is a finite (compact) group. In
particular, if $\mathbb{G}$ is of Kac-type, then $u^{c}=\overline{u}$,
$\rho_{U}=1$, and
$\overline{\mathbf{S}}=(\overline{\mathscr{H}},\overline{u},\overline{w})$.

###### Proof.

By definition, $\overline{S}=\mathscr{S}_{\Lambda_{0}}(\overline{U})$, thus

$\begin{split}u^{\prime}&=(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\epsilon_{\Lambda_{0}})(\overline{U})\\\
&=(\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}})\bigl{[}({j(\rho_{U})}^{1/2}\otimes
1\otimes 1)U^{c}({j(\rho_{U})}^{-1/2}\otimes 1\otimes 1)\bigr{]}\\\
&=(\operatorname{id}\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}})\Bigl{(}({j(\rho_{U})}^{1/2}\otimes
1\otimes 1)\\\ &\hskip
80.00012pt[(j\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\operatorname{id}_{C(\Lambda_{0})})(U^{\ast})]({j(\rho_{U})}^{-1/2}\otimes
1\otimes 1)\Bigr{)}\\\ &=({j(\rho_{U})}^{1/2}\otimes
1)\bigl{[}(j\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}})(U^{\ast})\bigr{]}({j(\rho_{U})}^{-1/2}\otimes
1)\\\ &=({j(\rho_{U})}^{1/2}\otimes
1){\bigl{[}(j\otimes\operatorname{id}\otimes\epsilon_{\Lambda_{0}})(U)\bigr{]}}^{\ast}({j(\rho_{U})}^{-1/2}\otimes
1)\\\ &=({j(\rho_{U})}^{1/2}\otimes
1){\bigl{[}(j\otimes\operatorname{id})(u)\bigr{]}}^{\ast}({j(\rho_{U})}^{-1/2}\otimes
1)\\\ &=({j(\rho_{U})}^{1/2}\otimes
1)\bigl{[}(j\otimes\operatorname{id})(u^{\ast})\bigr{]}({j(\rho_{U})}^{-1/2}\otimes
1)\\\ &=({j(\rho_{U})}^{1/2}\otimes 1)u^{c}({j(\rho_{U})}^{-1/2}\otimes
1).\end{split}$

The expression for $w^{\prime}$ is proved analogously by applying
$\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\epsilon_{\operatorname{Pol}(\mathbb{G})}\otimes\operatorname{id}_{C(\Lambda_{0})}$
on (7.12). Finally, if $\mathbb{G}$ is of Kac-type, then
$\rho_{U}=\operatorname{id}_{\mathscr{H}}=1$. ∎

######  Remark 7.8.

The “modular” operator $\rho_{U}$ of the representation $U$ is derived from
the representation theory of $\mathbb{G}\rtimes\Lambda_{0}$ instead of the
representation theory of $\mathbb{G}$ and (projective) representation theory
of $\Lambda_{0}$. This makes the description of $\overline{\mathbf{S}}$ in
Proposition 7.7 quite unsatisfactory in the non-unimodular case. This being
said, we point out that as far as the fusion rules of
$\mathbb{G}\rtimes\Lambda_{0}$ are concerned, the duals of a sufficiently
large family of CSRs admit a much more satisfactory description (see
Proposition 13.3).

Of course, the description of the dual in $\mathcal{CSR}_{\Lambda_{0}}$ is
much easier if $\mathbb{G}$ is of Kac-type, as is clearly seen from the last
part of Proposition 7.7.

##  8\. Group actions and projective representations

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Via the
functors $\mathscr{S}_{\Lambda_{0}}$ and $\mathscr{R}_{\Lambda_{0}}$, we see
that the problem classifying of irreducible representations of
$\mathbb{G}\rtimes\Lambda_{0}$ are essentially the same as classifying simple
CSRs in $\mathcal{CSR}_{\Lambda_{0}}$. Thus for the moment, it might be too
much to hope there exists a satisfactory description of _all_ simple CSRs in
$\mathcal{CSR}_{\Lambda_{0}}$. However, as we will see in § 9, if we restrict
our attention to the so-called _stably pure_ simple CSRs in
$\mathcal{CSR}_{\Lambda_{0}}$, then such a description is indeed achievable
via the theory of unitary projective representations of $\Lambda_{0}$. This
section studies how such projective representations arise naturally from the
action of $\Lambda$ on irreducible representations of $\mathbb{G}$, as well as
establishes some basic properties of these projective representations. The
results here will be used in § 9 to describe the structure of stably pure CSRs
in $\mathcal{CSR}_{\Lambda_{0}}$ (Proposition 9.5 and Proposition 9.6).

We begin with a simple observation which is a trivial quantum analogue of one
of the most basic ingredients of the Mackey analysis. Let $U_{\mathbb{G}}$ be
a unitary representation of $\mathbb{G}$ on some finite dimensional Hilbert
space $\mathscr{H}$. Since
$\alpha^{\ast}\colon\Lambda\rightarrow\operatorname{Aut}\bigl{(}C(\mathbb{G}),\Delta\bigr{)}$
is an antihomomorphism of groups, we know that
${(\operatorname{id}_{\mathcal{B}(\mathscr{H})}\otimes\alpha_{r^{-1}}^{\ast})}(U_{\mathbb{G}})$
is again a unitary representation of $\mathbb{G}$ on the same space
$\mathscr{H}$, and we denote this new representation by $r\cdot
U_{\mathbb{G}}$ as we did in Proposition 5.1. One checks that $(rs)\cdot
U_{\mathbb{G}}=r\cdot(s\cdot U_{\mathbb{G}})$. Thus this defines a left action
of the group $\Lambda$ on the (proper) class of all unitary representation of
$\mathbb{G}$, which is easily seen to preserve irreducibility and pass to a
well-defined action of $\Lambda$ on the _set_ $\operatorname{Irr}(\mathbb{G})$
by letting $r\cdot[u]=[r\cdot u]$, where $r\in\Lambda$, $u$ is an irreducible
unitary representation of $\mathbb{G}$ and $[u]$ is the equivalence class of
$u$ in $\operatorname{Irr}(\mathbb{G})$. Take another unitary representation
$W_{\mathbb{G}}$ of $\mathbb{G}$ on some other finite dimensional Hilbert
space $\mathscr{K}$. For any $r,s\in\Lambda$ and any
$T\in\mathcal{B}(\mathscr{H},\mathscr{K})$, we have

(8.1) $\begin{split}&T\in\operatorname{Mor}_{\mathbb{G}}(r\cdot
U_{\mathbb{G}},W_{\mathbb{G}})\\\ \iff&W_{\mathbb{G}}(T\otimes 1)=(T\otimes
1)(\operatorname{id}\otimes\alpha_{r^{-1}}^{\ast})(U_{\mathbb{G}})\\\
\iff&[(\operatorname{id}\otimes\alpha_{s^{-1}}^{\ast})(W_{\mathbb{G}})](T\otimes
1)=(T\otimes
1)(\operatorname{id}\otimes\alpha_{{(sr)}^{-1}}^{\ast})(U_{\mathbb{G}})\\\
\iff&T\in\operatorname{Mor}_{\mathbb{G}}(sr\cdot U_{\mathbb{G}},s\cdot
W_{\mathbb{G}}).\end{split}$

Now take any irreducible unitary representation $u$ of $\mathbb{G}$ on some
finite dimensional Hilbert space $\mathscr{H}$. Let
$x=[u]\in\operatorname{Irr}(\mathbb{G})$, and

$\Lambda_{x}=\left\\{r\in\Lambda\;:\;r\cdot x=x\right\\},$

i.e. $\Lambda_{x}$ is the isotropy subgroup of $\Lambda$ fixing $x$. Then for
any $r_{0}\in\Lambda_{x}$, $u$ and $r_{0}\cdot u$ are equivalent by
definition, hence there exists a unitary $V(r_{0})\in\mathcal{U}(\mathscr{H})$
intertwining $r_{0}\cdot u$ and $u$, in other words,

(8.2) $\bigl{(}V(r_{0})\otimes
1\bigr{)}(\operatorname{id}\otimes\alpha^{\ast}_{r_{0}^{-1}})(u)=u\bigl{(}V(r_{0})\otimes
1\bigr{)},$

which is clearly equivalent to

(8.3) $\forall r_{0}\in\Lambda_{x},\quad\bigl{(}V(r_{0})\otimes
1\bigr{)}u=[(\operatorname{id}\otimes\alpha_{r_{0}}^{\ast})(u)]\bigl{(}V(r_{0})\otimes
1\bigr{)}.$

It is remarkable that (8.3) takes exactly the same form as the covariance
condition (2.2) when we define covariant representations in § 2. Now if we
choose a

(8.4) $V(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot
u,u)\cap\mathcal{U}(\mathscr{H})$

for each $r_{0}\in\Lambda_{x}$, then for any $s_{0}\in\Lambda_{x}$, by (8.1),
we have

$V(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(s_{0}r_{0}\cdot u,s_{0}\cdot
u),\quad V(s_{0})\in\operatorname{Mor}_{\mathbb{G}}(s_{0}\cdot u,u),\quad
V(s_{0}r_{0})\in\operatorname{Mor}_{\mathbb{G}}(s_{0}r_{0}\cdot u,u),$

thus

(8.5) $\forall r_{0},s_{0}\in\Lambda_{x},\quad
V(s_{0}r_{0}){[V(r_{0})]}^{\ast}{[V(s_{0})]}^{\ast}\in\operatorname{Mor}_{\mathbb{G}}(u,u)\cap\mathcal{U}(\mathscr{H})=\mathbb{T}\cdot\operatorname{id}_{\mathscr{H}}.$

This means that $V\colon\Lambda_{x}\rightarrow\mathcal{U}(\mathscr{H})$ is a
unitary projective representation (Definition 8.1) of $\Lambda_{x}$ on
$\mathscr{H}$, which satisfies the covariant condition (8.3) for each
$r_{0}\in\Lambda_{x}$.

To facilitate our discussion, we digress now to give a brief summary of some
basic terminologies of the theory of group cohomology which we will use (cf.
[MR1324339]). We regard $\mathbb{T}$ as a trivial module over any finite group
when considering unitary projective representations of finite groups. For any
finite group $\Gamma$, an $n$-cochain on $\Gamma$ with coefficients in
$\mathbb{T}$, or simply an $n$-cochain (on $\Gamma$), as we won’t consider
coefficient module other that the trivial module $\mathbb{T}$, is a mapping
from the $n$-fold product $\Gamma^{n}=\Gamma\times\cdots\times\Gamma$ to
$\mathbb{T}$. Let $C^{n}(\Gamma,\mathbb{T})$ be the abelian group of
$n$-cochains on $\Gamma$, $Z^{2}(\Gamma,\mathbb{T})$ the subgroup of
$2$-cocycles on $\Gamma$, i.e. mappings
$\omega\colon\Gamma\times\Gamma\rightarrow\mathbb{T}$ satisfying the cocycle
condition

(8.6) $\forall
r,s,t\in\Gamma,\quad\omega(r,st)\omega(s,t)=\omega(r,s)\omega(rs,t).$

The mapping

(8.7) $\begin{split}\delta\colon C^{1}(\Gamma,\mathbb{T})&\rightarrow
Z^{2}(\Gamma,\mathbb{T})\\\
\mathsf{b}&\mapsto\left\\{(r,s)\in\Gamma\times\Gamma\mapsto\frac{\mathsf{b}(r)\mathsf{b}(s)}{\mathsf{b}(rs)}\right\\}\end{split}$

is easily checked to be a well-defined group morphism. We use
$B^{2}(\Gamma,\mathbb{T})$ to denote the image of $\delta$, and the
$2$-cocycles in $B^{2}(\Gamma,\mathbb{T})$ are called $2$-coboundaries of
$\Gamma$. The quotient group
$Z^{2}(\Gamma,\mathbb{T})/B^{2}(\Gamma,\mathbb{T})$ is called the second
cohomology group of $\Gamma$ with coefficients in the trivial $\Gamma$-module
$\mathbb{T}$, and is denoted by $H^{2}(\Gamma,\mathbb{T})$. Elements in
$H^{2}(\Gamma,\mathbb{T})$ are called cohomology class. Note that
$\ker(\delta)$ is exactly the group of characters on $\Gamma$, i.e. group
morphisms from $\Gamma$ to $\mathbb{T}$.

######  Definition 8.1.

Let $\Gamma$ be a group, $\mathscr{H}$ a finite dimensional Hilbert space, a
projective representation of $\Gamma$ on $\mathscr{H}$ is a mapping
$V:\Gamma\to\mathcal{U}(\mathscr{H})$ such that
$V(e_{\Gamma})=\operatorname{id}_{\mathscr{H}}$, and there exists a
$2$-cochain $\omega\in C^{2}(\Gamma,\mathbb{T})$, such that

(8.8) $\forall r,s\in\Gamma,\qquad\omega(r,s)V(r,s)=V(r)V(s).$

It is easy to check that such $\omega$ is uniquely determined by $V$, and it
is in fact a $2$-cocylce, with the additional property (which follows from our
assumption $V(e_{\Gamma})=\operatorname{id}_{\mathscr{H}}$) that

(8.9)
$\forall\gamma\in\Gamma,\qquad\omega(e_{\Gamma},\gamma)=\omega(\gamma,e_{\Gamma})=1\in\mathbb{T}.$

We call $\omega$ the cocylce (or Schur multiplier after Schur who introduced
them in his work on projective representations [schur1904darstellung]) of the
projective representation $V$.

We will freely use the character theory and the Peter Weyl theory of
projective representations of finite groups, and we refer the reader to
[MR3299063] for the proofs.

We track here the following easy results for convenience of the reader.

######  Lemma 8.2.

Let $\Gamma$ be a finite group,
$V\colon\Gamma\rightarrow\mathcal{U}(\mathscr{H})$ a finite dimensional
unitary projective representation of $\Gamma$ with cocycle $\omega$. If
$\omega^{\prime}\in[\omega]\in H^{2}(\Gamma,\mathbb{T})$, then there exists a
mapping $\mathsf{b}\colon\Gamma\rightarrow\mathbb{T}$, such that
$\mathsf{b}V\colon\Gamma\rightarrow\mathcal{U}(\mathscr{H})$,
$\gamma\mapsto\mathsf{b}(\gamma)V(\gamma)$ is a unitary projective
representation with cocycle $\omega^{\prime}$.

###### Proof.

Since $\omega^{\prime}\in[\omega]$, there is a mapping
$\mathsf{b}\colon\Gamma\rightarrow\mathbb{T}$ such that
$\omega^{\prime}=(\delta\mathsf{b})\omega$, and obviously, $\mathsf{b}V$ is a
unitary projective representation with
$(\delta\mathsf{b})\omega=\omega^{\prime}$ as its cocycle. ∎

######  Lemma 8.3.

Let $\Gamma$ be a finite group,
$V\colon\Gamma\rightarrow\mathcal{U}(\mathscr{H})$ a finite dimensional
unitary projective representation of $\Gamma$ with cocycle $\omega$, and let
$\mathsf{b}\colon\Gamma\rightarrow\mathbb{T}$ be an arbitrary mapping. The
following hold:

  1. (a)

$\mathsf{b}V\colon\Gamma\rightarrow\mathcal{U}(\mathscr{H})$,
$\gamma\mapsto\mathsf{b}(\gamma)V(\gamma)$ is a projective representation with
cocycle $(\delta\mathsf{b})\omega$;

  2. (b)

$\mathsf{b}V$ and $V$ have the same cocycle if and only if
$\mathsf{b}\in\ker(\delta)$, i.e. $\mathsf{b}$ is a character of $\Gamma$;

  3. (c)

$\mathsf{b}V$ is irreducible if and only if $V$ is irreducible.

###### Proof.

It is clear that (a) and (b) are direct consequences of the relevant
definitions. We now prove (c). If we denote the character of $V$ by
$\chi_{V}$, then the character of $\mathsf{b}V$ is $\mathsf{b}\chi_{V}$. Hence

(8.10) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\Gamma}(\mathsf{b}V,\mathsf{b}V)=\frac{1}{\left\lvert\Gamma\right\rvert}\sum_{\gamma\in\Gamma}\overline{\mathsf{b}(\gamma)\chi_{V}(\gamma)}\mathsf{b}(\gamma)\chi_{V}(\gamma)\\\
&=\frac{1}{\left\lvert\Gamma\right\rvert}\sum_{\gamma\in\Gamma}\overline{\chi_{V}(\gamma)}\chi_{V}(\gamma)=\dim\operatorname{Mor}_{\Gamma}(V,V),\end{split}$

and $\mathsf{b}V$ is irreducible if and only if $V$ is. ∎

######  Remark 8.4.

If $\mathsf{b}$ is a character of $\Gamma$, and
$V\colon\Gamma\rightarrow\mathcal{U}(\mathscr{H})$ an irreducible unitary
projective representation, then $\mathsf{b}V$ is also an irreducible unitary
projective representation with the same cocycle as that of $V$. Note that
$\left\lvert\mathsf{b}(\gamma)\right\rvert=1$ for all $\gamma\in\Gamma$, we
have

(8.11) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\Gamma}(\mathsf{b}V,V)=\frac{1}{\left\lvert\Gamma\right\rvert}\sum_{\gamma\in\Gamma}\overline{\mathsf{b}(\gamma)}\chi_{V}(\gamma)\overline{\chi_{V}(\gamma)}\\\
&\leq\frac{1}{\left\lvert\Gamma\right\rvert}\sum_{\gamma\in\Gamma}\chi_{V}(\gamma)\overline{\chi_{V}(\gamma)}=\dim\operatorname{Mor}_{\Gamma}(V,V)=1.\end{split}$

with equality holds if and only if $\mathsf{b}(\gamma)=1$ whenever
$\chi_{V}(\gamma)\neq 0$. If equality doesn’t hold in (8.11), then
$\dim\operatorname{Mor}_{\Gamma}(\mathsf{b}V,V)$ must be $0$ since it is a
natural number. Therefore, whenever $\Gamma$ is not trivial, it is possible
that $\mathsf{b}V$ and $V$ are irreducible unitary projective representations
with the same cocycle but not equivalent. Thus one must be careful not to
confuse our definition of projective representation with the more naive
definition where one simply replaces $\operatorname{GL}(\mathscr{H})$ by
$\operatorname{PGL}(\mathscr{H})$ as the target model group. For us, how we
lift from $\operatorname{PGL}(\mathscr{H})$ to
$\operatorname{GL}(\mathscr{H})$ _does_ matter, even if we keep the cocycle in
the process.

After this digression, we now resume our discussion. Using terminologies in
the theory of group cohomology, and regarding $\mathbb{T}$ as the trivial
$\Lambda_{x}$-module, we see that the $2$-cocycle $\omega_{x}\in
C^{2}(\Lambda_{x},\mathbb{T})$ of the unitary projective representation $V$ of
$\Lambda_{x}$ is determined up to a $2$-boundary in
$B^{2}(\Lambda_{x},\mathbb{T})$, because each unitary operator $V(r_{0})$,
$r_{0}\in\Lambda_{x}$ is uniquely determined up to a scalar multiple in
$\mathbb{T}$ (Schur’s lemma plus the unitarity of $V(r_{0})$). In other words,
$[\omega_{x}]\in H^{2}(\Lambda_{x},\mathbb{T})$ is a well-defined cohomology
class of $\Lambda_{x}$ with coefficients in $\mathbb{T}$.

Conversely, let $u$ be an _irreducible_ unitary representation of $\mathbb{G}$
on some finite dimensional Hilbert space $\mathscr{H}$, and
$x=[u]\in\operatorname{Irr}(\mathbb{G})$. If $\Lambda_{0}$ is a subgroup of
$\Lambda$, $V\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathscr{H})$ a unitary
projection representation of $\Lambda_{0}$ such that $u$ and $V$ satisfy the
covariance condition (8.3), then

$\forall r_{0}\in\Lambda_{0},\quad
V(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot u,u).$

In particular, $\Lambda_{0}$ fixes $x=[u]$ under the action
$\Lambda\curvearrowright\operatorname{Irr}(\mathbb{G})$. Repeat the above
reasoning shows that (8.5) still holds.

We summarize the above discussion in the following proposition, which proves
slightly more.

######  Proposition 8.5.

Let $u$ be an irreducible unitary representation of $\mathbb{G}$ on some
finite dimensional Hilbert space $\mathscr{H}$,
$x=[u]\in\operatorname{Irr}(\mathbb{G})$, $\Lambda_{x}$ the isotropy group
fixing $x$ (under the action
$\Lambda\curvearrowright\operatorname{Irr}(\mathbb{G})$). For any
$r_{0}\in\Lambda_{x}$, choose a unitary $V(r_{0})$ according to (8.4). Then

  1. (a)

$V\colon\Lambda_{x}\rightarrow\mathcal{U}(\mathscr{H})$, $r_{0}\mapsto
V(r_{0})$ is a unitary projective representation satisfying the covariance
condition (8.3);

  2. (b)

let $\omega\in C^{2}(\Lambda_{0},\mathbb{T})$ be the $2$-cocycle of $V$, then
the cohomology class $c_{x}\colon=[\omega]\in H^{2}(\Lambda_{x},\mathbb{T})$
depends only on $x$, i.e. it does not depend on any particular choice of $u\in
x$.

Conversely, if $V_{0}\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathscr{H})$ is
a unitary projective representation of some subgroup $\Lambda_{0}$ of
$\Lambda$ that satisfies the covariance condition (8.3), then

  1. (c)

for every $r_{0}\in\Lambda_{0}$, the condition (8.4) holds;

  2. (d)

$\Lambda_{0}\subseteq\Lambda_{x}$;

  3. (e)

there is a choice of $V\colon\Lambda_{x}\rightarrow\mathcal{U}(\mathscr{H})$
satisfying (8.3) such that $V|_{\Lambda_{0}}=V_{0}$;

  4. (f)

let $\omega_{0}\in C^{2}(\Lambda_{0},\mathbb{T})$ be the $2$-cocycle of
$V_{0}$, then $[\omega_{0}]$ is the image of $c_{x}$ under the morphism of
groups

$H^{2}(\Lambda_{0}\hookrightarrow\Lambda_{x})\colon
H^{2}(\Lambda_{x},\mathbb{T})\rightarrow H^{2}(\Lambda_{0},\mathbb{T}).$

###### Proof.

The above discussion already establishes (a), (c) and (d). Assertion (e)
follows from (a) and (c), while (f) follows from (e). Moreover, we’ve seen
that $[\omega]\in H^{2}(\Lambda_{x},\mathbb{T})$ does not depend on the choice
of $V$. For any $w\in x$, there exists a unitary intertwiner
$U\in\operatorname{Mor}_{\mathbb{G}}(u,w)$. It is trivial to check that
$V_{w}(r_{0})=UV(r_{0})U^{\ast}$ defines a unitary projective representation
of $\Lambda_{x}$ such that

$V_{w}(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot w,w).$

Since $V_{w}$ and $V$ are unitarily equivalent projective representations of
$\Lambda_{x}$, the $2$-cocycle of $V_{w}$ coincides with $\omega$—the
$2$-cocycle of $V$. This proves that $c_{x}=[\omega]\in
H^{2}(\Lambda_{x},\mathbb{T})$ indeed depends only on $x$ and not on any
particular choice of $u\in x$. This proves (b) and finishes the proof of the
proposition. ∎

######  Definition 8.6.

Using the notations in Proposition 8.5, we call the cohomology class
$[\omega]\in H^{2}(\Lambda_{x},\mathbb{T})$ the cohomology class associated
with $x=[u]\in\operatorname{Irr}(\mathbb{G})$, and we denote $[\omega]$ by