$c_{x}$. If $\Lambda_{0}$ is a subgroup of $\Lambda_{x}$, the cohomology class
$[\omega_{0}]\in H^{2}(\Lambda_{0},\mathbb{T})$ is called the restriction of
the cohomology class $c_{x}$ on $\Lambda_{0}$, and is denoted by
$c_{x,\Lambda_{0}}$.

Obviously, $c_{x,\Lambda_{0}}$ depends on $\Lambda_{0}$ and $x$, and
$c_{x,\Lambda_{0}}=c_{x}$ if $\Lambda_{0}=\Lambda_{x}$. To apply the character
theory of projective representations, we need to suitably rescale the
projective representations in question so that they share the same cocycle
(and not merely the same cohomology class for their cocycles). In the case
where the representation
$u\in\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})$ of
$\mathbb{G}$ is irreducible, and
$V\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathscr{H})$ is a unitary
projective representation satisfying the covariance condition (8.3), such a
rescaling is implicit in the choice of
$V(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot u,u)$ for each
$r_{0}\in\Lambda_{0}$. However, Remark 8.4 tells us we should take extra care
if we want to talk about equivalence class of these projective representations
once we do the rescaling.

We finish this section with an easy result.

######  Proposition 8.7.

Let $x\in\operatorname{Irr}(\mathbb{G})$, $u\in x$, $\Lambda_{0}$ a subgroup
of $\Lambda_{x}$, $c_{0}\in H^{2}(\Lambda_{0},\mathbb{T})$ is the image of the
cohomology class $c_{x}\in H^{2}(\Lambda_{x},\mathbb{T})$ associated with $x$
under $H^{2}(\Lambda_{0}\hookrightarrow\Lambda_{x},\mathbb{T})$. Then for any
$2$-cocycle $\omega_{0}\in c_{0}$, there exists a unitary projective
representation $V$ of the isotropy subgroup $\Lambda_{0}$ with cocycle
$\omega_{0}$, such that $V$ and $u$ are covariant, and such $V$ is unique up
to rescaling by a character of $\Lambda_{0}$.

###### Proof.

This is clear from Proposition 8.5, Lemma 8.2 and Lemma 8.3. ∎

##  9\. Pure, stable, distinguished CSRs and representation parameters

Recall that for any finite dimensional representation $u$ of $\mathbb{G}$, the
support of $u$, denoted by $\operatorname{supp}(u)$, is the set

$\left\\{x\in\operatorname{Irr}(\mathbb{G})\;:\;\dim_{\mathbb{G}}\operatorname{Mor}_{\mathbb{G}}(x,[u])\neq
0\right\\}$

where $[u]$ is the class of unitary representations of $\mathbb{G}$ equivalent
to $u$. We call $u$ _pure_ if $\operatorname{supp}(u)$ is a singleton.

######  Definition 9.1.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$,
$\mathbf{S}=(\mathscr{H},u,w)\in\mathcal{CSR}_{\Lambda_{0}}$, we call
$\mathbf{S}$

  * •

pure, if $u$ is pure;

  * •

stable, if $r\cdot[u](=[r\cdot u])=[u]$ for all $r\in\Lambda_{0}$;

  * •

stably pure, if it is both pure and stable;

  * •

maximally stable, if

$\Lambda_{0}=\left\\{r\in\Lambda\;:\;r\cdot[u]=[u]\right\\};$

  * •

simple, if $\mathbf{S}$ is a simple object in $\mathcal{CSR}_{\Lambda_{0}}$;

  * •

distinguished, if it is maximally stable, pure and simple.

As remarked earlier, while it is not reasonable for the moment to hope for a
satisfactory description of all simple CSRs in $\mathcal{CSR}_{\Lambda_{0}}$,
it is possible to describe simple CSRs that are stably pure using unitary
projective representations of $\Lambda_{0}$. Somewhat surprisingly, one can
even describe all stably pure CSRs, even the non-simple ones, in this way. To
achieve the latter, we introduce the following definitions, which are closely
related to the materials in § 8.

######  Definition 9.2.

Let $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Suppose $u$ is
a unitary representation of $\mathbb{G}$ on some finite dimensional Hilbert
space $\mathscr{H}$, and
$V\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathscr{H})$ is a unitary
projective representation of $\Lambda_{0}$. We say $u$ and $V$ are covariant
if they satisfy the covariance condition (8.3), or equivalently
$V(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot u,u)$ for all
$r_{0}\in\Lambda_{0}$.

######  Definition 9.3.

Let $x\in\operatorname{Irr}(\mathbb{G})$,
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$ with
$\Lambda_{0}\subseteq\Lambda_{x}$, $u\in x$, $\omega_{0}\in c_{x,\Lambda_{0}}$
(see Definition 8.6), then a unitary projective representation $V$ of
$\Lambda_{0}$ that is covariant with $u$ is said to be a covariant projective
$\Lambda_{0}$-representation of $u$ (with cocycle $\omega_{0}$).

######  Remark 9.4.

In the setting of Definition 9.3, fix any covariant projective
$\Lambda_{0}$-representation $V$ of $u$ with cocycle $\omega_{0}$, the set of
covariant projective $\Lambda_{0}$-representations of $u$ with multiplier
$\omega_{0}$ is in bijective correspondence with the group of characters of
$\Lambda_{0}$, via $\mathsf{b}\mapsto\mathsf{b}V$ (see Lemma 8.2 and Lemma
8.3).

######  Proposition 9.5 (Structure of stably pure CSR).

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, let
$\mathbf{S}=(\mathscr{H},u,w)$ be a stably pure CSR in
$\mathcal{CSR}_{\Lambda_{0}}$ $x\in\operatorname{Irr}(\mathbb{G})$ is the
support point of $u$, $u_{0}\in x$ a representation on some finite dimensional
Hilbert space $\mathscr{H}_{0}$, $n$ is the multiplicity of $u_{0}$ in $u$,
$V_{0}$ a covariant projective $\Lambda_{0}$-representation of $u$, then there
exists a unique unitary projective representation
$v_{0}\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathbb{C}^{n})$ of
$\Lambda_{0}$ on $\mathbb{C}^{n}$, such that the following hold:

  1. (a)

$V_{0}$ and $v_{0}$ have opposing cocycles;

  2. (b)

$\mathbf{S}_{0}=(\mathbb{C}^{n}\otimes\mathscr{H}_{0},\epsilon_{n}\times
u_{0},v_{0}\times V_{0})$ is a CSR in $\mathcal{CSR}_{\Lambda_{0}}$, where
$\epsilon_{n}$ is the trivial representation of $\mathbb{G}$ on
$\mathbb{C}^{n}$;

  3. (c)

$\mathbf{S}_{0}$ and $\mathbf{S}$ are isomorphic in
$\mathcal{CSR}_{\Lambda_{0}}$.

###### Proof.

Uniqueness is almost clear once we finish the proof of existence, which we do
now. Let $U$ be a unitary intertwiner from $u$ to $\epsilon_{n}\otimes u_{0}$.
Noting that $u$ is pure and replacing $\mathbf{S}$ with $U\mathbf{S}U^{\ast}$
if necessary, we may assume $\mathscr{H}=\mathbb{C}^{n}\otimes\mathscr{H}_{0}$
and $u=\epsilon_{n}\times u_{0}={(u_{0})}_{23}$. For any
$r_{0}\in\Lambda_{0}$, we claim that there exists a unique
$v_{0}(r_{0})\in\mathcal{B}(\mathbb{C}^{n})$ such that
$w(r_{0})=v_{0}(r_{0})\otimes V_{0}(r_{0})$. Admitting the claim for the
moment, the unitarity of $v_{0}(r_{0})$ follows from the unitarity of
$w(r_{0})$ and $V_{0}(r_{0})$, and $w$ being a representation and $V_{0}$
being a projective representation force $v_{0}$ to be a unitary projective
representation with a cocycle opposing to the cocycle of $V_{0}$. Thus the
proposition follows from the claim, which we now prove. Since
$\mathcal{B}(\mathbb{C}^{n}\otimes\mathscr{H}_{0})=\mathcal{B}(\mathbb{C}^{n})\otimes\mathcal{B}(\mathscr{H}_{0})$
by the usual identification, there exists an $m\in\mathbb{N}$,
$A_{1},\ldots,A_{m}\in\mathcal{B}(\mathbb{C}^{n})$ and
$B_{1},\ldots,B_{m}\in\mathcal{B}(\mathscr{H}_{0})$, such that

(9.1) $w(r_{0})=\sum_{i=1}^{m}A_{i}\otimes B_{i}.$

Furthermore, we can and do choose these operators so that $A_{1},\ldots,A_{m}$
are linearly independent in $\mathcal{B}(\mathbb{C}^{n})$. Since $u$ and $w$
are covariant, we have

(9.2) $\bigl{(}w(r_{0})\otimes
1\bigr{)}u=\bigl{[}(\operatorname{id}_{\mathscr{H}}\otimes\alpha^{\ast}_{r_{0}})u\bigr{]}\bigl{(}w(r_{0})\otimes
1\bigr{)}.$

Substituting $u={(u_{0})}_{23}$ and (9.1) in (9.2) yields

(9.3) $\begin{split}&\hskip 11.99998pt\sum_{i=1}^{m}A_{i}\otimes[(B_{i}\otimes
1)u_{0}]\\\
&=\sum_{i=1}^{m}A_{i}\otimes\Bigl{(}\bigl{[}(\operatorname{id}_{\mathscr{H}_{0}}\otimes\alpha^{\ast}_{r_{0}})u_{0}\bigr{]}\bigl{(}B_{i}\otimes
1\bigr{)}\Bigr{)}\in\mathcal{B}(\mathbb{C}^{n})\otimes\mathcal{B}(\mathscr{H}_{0})\otimes\operatorname{Pol}(\mathbb{G}).\end{split}$

Since $A_{1},\ldots,A_{m}$ are linearly independent, there exists linear
functionals $l_{1},\ldots,l_{m}$ on $\mathcal{B}(\mathbb{C}^{n})$ such that
$l_{i}(A_{j})=\delta_{i,j}$. Applying
$l_{i}\otimes\operatorname{id}_{\mathscr{H}_{0}}\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}$
on (9.3) shows that for each $i=1,\ldots,m$,

(9.4) $(B_{i}\otimes
1)u_{0}=\bigl{[}(\operatorname{id}\otimes\alpha^{\ast}_{r_{0}})u_{0}\bigr{]}(B_{i}\otimes
1),$

or equivalently

(9.5) $B_{i}\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot
u_{0},u_{0})=\mathbb{C}V_{0}(r_{0}).$

Now the claim follows from (9.1) and (9.5). ∎

Conversely, we have

######  Proposition 9.6.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$,
$x\in\operatorname{Irr}(\mathbb{G})$ with $\Lambda_{0}\subseteq\Lambda_{x}$.
Take a $u\in x$ acting on some finite dimensional Hilbert space $\mathscr{H}$,
and a covariant projective $\Lambda_{0}$-representation $V$ of $u$, then for
any unitary projective representation
$v\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathscr{K})$ of $\Lambda_{0}$ with
cocycle opposing the cocycle of $V$, the unitary representation $v\times V$ of
$\Lambda_{0}$ is covariant with the unitary representation
$\operatorname{id}_{\mathscr{K}}\otimes u=\epsilon_{\mathscr{K}}\times u$ of
$\mathbb{G}$, where $\epsilon_{\mathscr{K}}$ is the trivial representation of
$\mathbb{G}$ on $\mathscr{K}$, i.e.
$(\mathscr{K}\otimes\mathscr{H},\epsilon_{\mathscr{K}}\times u,v\times V)$ is
a stably pure CSR in $\mathcal{CSR}_{\Lambda_{0}}$.

###### Proof.

Since $V$ and $u$ are covariant, for any $r_{0}\in\Lambda_{0}$, we have

(9.6) $\bigl{(}V(r_{0})\otimes
1\bigr{)}u=\bigl{[}(\operatorname{id}\otimes\alpha^{\ast}_{r_{0}})u\bigr{]}\bigl{(}V(r_{0})\otimes
1\bigr{)}.$

The proposition follows by tensoring $v(r_{0})$ on the left in (9.6). ∎

By Proposition 9.5 and Proposition 9.6, we now have a satisfactory description
of stably pure CSRs in $\mathcal{CSR}_{\Lambda_{0}}$—from any irreducible
representation $u$ of $\mathbb{G}$ on $\mathscr{H}$ such that
$\Lambda_{0}\cdot[u]=[u]$, one choose a covariant projective
$\Lambda_{0}$-representation $V$ of $u$ with some cocycle $\omega$, then any
unitary projective representation $v$ of $\Lambda_{0}$ with cocycle
$\omega^{-1}=\overline{\omega}$ gives rise to a stably pure CSR in
$\mathcal{CSR}_{\Lambda_{0}}$, namely
$\mathcal{S}(u,V,v)=(\mathscr{K}\otimes\mathscr{H},\epsilon_{\mathscr{K}}\times
u,v\times V)$; and all stably pure CSRs in $\mathcal{CSR}_{\Lambda_{0}}$ arise
in this way up to isomorphism.

######  Remark 9.7.

Using the above notations, while it is true that $V$ is determined by $u$ to a
great extent due to the restriction of Schur’s lemma, it is still not
completely determined (see Proposition 8.7), and a choice of this $V$ is
vitally relevant as is demonstrated by Remark 8.4 applied to $v$. This is why
$V$ can _not_ be suppressed in our notation $\mathcal{S}(u,V,v)$.

######  Definition 9.8.

Let $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. A triple
$(u,V,v)$ is called a representation parameter for $\mathbb{G}\rtimes\Lambda$
associated with $\Lambda_{0}$, if it the following hold:

  * •

$u$ is an irreducible unitary representation of $\mathbb{G}$ on some finite
dimensional Hilbert space $\mathscr{H}$;

  * •

$V$ is a covariant projective $\Lambda_{0}$-representation of $u$;

  * •

$v$ is a unitary projective representation of $\Lambda_{0}$ (possibly on
Hilbert spaces other than $\mathscr{H}$), such that $v$ and $V$ have opposing
cocycles.

If $(u,V,v)$ is a representation parameter, the stably pure CSR
$\mathcal{S}(u,V,v)$ in $\mathcal{CSR}_{\Lambda_{0}}$ is called the CSR
parametrized by the representation parameter $(u,V,v)$. If furthermore the
unitary projective representation $v$ is irreducible, we say the
representation parameter $(u,V,v)$ is _irreducible_.

Thus Proposition 9.5 immediately implies the following corollary.

######  Corollary 9.9.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, then every
stably pure CSR associated with $\Lambda_{0}$ is parameterised by some
representation parameter associated with $\Lambda_{0}$. ∎

######  Definition 9.10.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Let $u$ be an
irreducible unitary representation of $\mathbb{G}$ such that
$\Lambda_{0}\cdot[u]=[u]$, $V_{1}$ and $V_{2}$ are two covariant projective
$\Lambda_{0}$-representations of $u$, the unique mapping
$\mathsf{b}\colon\Lambda_{0}\rightarrow\mathbb{T}$ such that
$V_{2}=\mathsf{b}V_{1}$ is called the $u$-transitional mapping from $V_{1}$ to
$V_{2}$ (note that we do _not_ require $V_{1}$ and $V_{2}$ to have the same
cocycle here).

######  Proposition 9.11.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. For $i=1,2$,
let $(u_{i},V_{i},v_{i})$ be a representation parameter associated with
$\Lambda_{0}$, $U_{i}$ denote the unitary representation
$\mathscr{R}_{\Lambda_{0}}\Bigl{(}\mathcal{S}(u_{i},V_{i},v_{i})\Bigr{)}$ of
$\mathbb{G}\rtimes\Lambda_{0}$, then the following holds:

  1. (a)

if $[u_{1}]\neq[u_{2}]$ in $\operatorname{Irr}(\mathbb{G})$, then
$\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{2})=0$;

  2. (b)

if $u_{1}=u_{2}=u$, and $\mathsf{b}\colon\Lambda_{0}\rightarrow\mathbb{T}$ the
$u$-transitional map from $V_{1}$ to $V_{2}$, then

(9.7)
$\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{2})=\dim\operatorname{Mor}_{\Lambda_{0}}\left(v_{1},\mathsf{b}v_{2}\right).$

###### Proof.

Let $h$ be the Haar state of $\mathbb{G}$, by (1.7), the Haar state
$h_{\Lambda_{0}}$ of $\mathbb{G}\rtimes\Lambda_{0}$ is the linear functional
on $A\otimes C(\Lambda_{0})$ defined by
$a\otimes\delta_{r_{0}}\mapsto\left\lvert\Lambda_{0}\right\rvert^{-1}h(a)$,
where $a\in A$, $r_{0}\in\Lambda_{0}$ (recall that $A=C(\mathbb{G})$).

Suppose $[u_{1}]\neq[u_{2}]$. For any $i=1,2$, by choosing a Hilbert space
basis for the representation of $u_{i}$, one can write $u_{i}$ as a square
matrix $\left(u^{(i)}_{jk}\right)$ over
$\operatorname{Pol}(\mathbb{G})\subseteq A$, and $V_{i}$ as a matrix
$\left(V^{(i)}_{jk}\right)$ over $C(\Lambda_{0})$ of the same size of
$\left(u^{(i)}_{jk}\right)$. Then the character $\chi_{i}$ of $U_{i}$ is given
by

(9.8)
$\chi_{i}=\sum_{r_{0}\in\Lambda_{0}}\sum_{j=1}^{n_{i}}\operatorname{Tr}(v_{i})\left(\sum_{k=1}^{n_{i}}V^{(i)}_{kj}(r_{0})u^{(i)}_{jk}\right)\otimes\delta_{r_{0}}\in\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0}).$

The orthogonality relation for the nonequivalent irreducible representations
$u_{1}$ and $u_{2}$ implies that

(9.9) $\forall j_{1},k_{1},j_{2},k_{2},\quad
h\left({\bigl{(}u^{(1)}_{j_{1}k_{1}}\bigr{)}}^{\ast}u^{(2)}_{j_{2}k_{2}}\right)=0.$

Hence, by (9.8) and (9.9),

(9.10) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{2})=h_{\Lambda_{0}}\bigl{(}\overline{\chi_{1}}\chi_{2}\bigr{)}\\\
&=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\sum_{j_{1},k_{1}=1}^{n_{1}}\sum_{j_{2},k_{2}=1}^{n_{2}}\overline{\operatorname{Tr}\bigl{(}v_{1}(r_{0})\bigr{)}}\operatorname{Tr}\bigl{(}v_{2}(r_{0})\bigr{)}\\\
&\hskip
30.00005pt\overline{V^{(1)}_{k_{1}j_{1}}(r_{0})}V^{(2)}_{k_{2}j_{2}}(r_{0})h\left({\bigl{(}u^{(1)}_{j_{1}k_{1}}\bigr{)}}^{\ast}u^{(2)}_{j_{2}k_{2}}\right)\\\
&=0.\end{split}$

This proves (a).

Under the hypothesis of (b), using the same notations as in the previous
paragraph, we have $n_{1}=n_{2}=\dim U$. We may assume that
$e^{(1)}_{j}=e^{(2)}_{j}=e_{j}$, hence $u_{jk}:=u^{(1)}_{jk}=u^{(2)}_{jk}$ for
all possible $j,k$. Note that $V_{2}=\mathsf{b}V_{1}$, and
$\mathcal{S}(u,V_{2},v_{2})=\mathcal{S}(u,V_{1},\mathsf{b}v_{2})$ because
$\mathsf{b}V_{1}\times v_{2}=V_{1}\times\mathsf{b}v_{2}$, we may assume that
$V_{2}=V_{1}=V$ and $\mathsf{b}=1$, with $V_{jk}:=V^{(1)}_{jk}=V^{(2)}_{jk}\in
C(\Lambda_{0})$ for all possible $j,k$. Let $\rho$ be the unique invertible
positive operator in $\operatorname{Mor}_{\mathbb{G}}(u,u^{cc})$ such that
$\operatorname{Tr}(\cdot\,\rho)=\operatorname{Tr}(\cdot\,\rho^{-1})$ on
$\operatorname{End}_{\mathbb{G}}(u)$. With these assumptions, by (9.10), the
orthogonality relation takes the form

(9.11)
$h(u_{ij}^{\ast}u_{kl})=\frac{\delta_{j,l}{\bigl{(}\rho^{-1}\bigr{)}}_{ki}}{\dim_{q}U}$

where $\dim_{q}U=\operatorname{Tr}(\rho)=\operatorname{Tr}(\rho^{-1})$ is the
quantum dimension of $U$ (see [MR3204665, §1.4]). Since $\rho$ is positive, we
might choose the basis $e_{1},\ldots,e_{n}$ to diagonize $\rho$, so that
$\rho_{ki}={\bigl{(}\rho^{-1}\bigr{)}}_{ki}=0$ whenever $k\neq i$. Using this
basis, (9.11) and (9.10), we have

(9.12) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{2})\\\
&=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\sum_{j_{1},k_{1}=1}^{n}\sum_{j_{2},k_{2}=1}^{n}\overline{\operatorname{Tr}\bigl{(}v_{1}(r_{0})\bigr{)}}\operatorname{Tr}\bigl{(}v_{2}(r_{0})\bigr{)}\\\
&\hskip
120.00018pt\cdot\overline{V_{k_{1}j_{1}}(r_{0})}V_{k_{2}j_{2}}(r_{0})h\left({\bigl{(}u_{j_{1}k_{1}}\bigr{)}}^{\ast}u_{j_{2}k_{2}}\right)\\\
&=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\sum_{j_{1},k_{1}=1}^{n}\sum_{j_{2},k_{2}=1}^{n}\overline{\operatorname{Tr}\bigl{(}v_{1}(r_{0})\bigr{)}}\operatorname{Tr}\bigl{(}v_{2}(r_{0})\bigr{)}\overline{V_{k_{1}j_{1}}(r_{0})}V_{k_{2}j_{2}}(r_{0})\\\
&\hskip
120.00018pt\cdot\frac{\delta_{j_{1},j_{2}}\delta_{k_{1},k_{2}}{\bigl{(}\rho^{-1}\bigr{)}}_{j_{2}j_{1}}}{\dim_{q}U}\\\
&=\left\lvert\Lambda_{0}\right\rvert^{-1}\overline{\operatorname{Tr}\bigl{(}v_{1}(r_{0})\bigr{)}}\operatorname{Tr}\bigl{(}v_{2}(r_{0})\bigr{)}\sum_{r_{0}\in\Lambda_{0}}\sum_{j=1}^{n}\left\\{\sum_{k=1}^{n}\overline{V_{kj}(r_{0})}V_{kj}(r_{0})\right\\}\frac{{\bigl{(}\rho^{-1}\bigr{)}}_{jj}}{\dim_{q}U}\\\
&\hskip 11.99998pt(\text{Note that $V(r_{0})$ is unitary})\\\
&=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\overline{\operatorname{Tr}\bigl{(}v_{1}(r_{0})\bigr{)}}\operatorname{Tr}\bigl{(}v_{2}(r_{0})\bigr{)}\frac{\sum_{j=1}^{n}{\bigl{(}\rho^{-1}\bigr{)}}_{jj}}{\dim_{q}U}\\\
&=\left\lvert\Lambda_{0}\right\rvert^{-1}\sum_{r_{0}\in\Lambda_{0}}\overline{\operatorname{Tr}\bigl{(}v_{1}(r_{0})\bigr{)}}\operatorname{Tr}\bigl{(}v_{2}(r_{0})\bigr{)}=\dim\operatorname{Mor}_{\Lambda_{0}}(v_{1},v_{2}).\end{split}$

This proves (b). ∎

The following corollary is now clear.

######  Corollary 9.12.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Let $(u,V,v)$
be a representation parameter associated with $\Lambda_{0}$, then the
representation $\mathscr{R}_{\Lambda_{0}}\Bigl{(}\mathcal{S}(u,V,v)\Bigr{)}$
of $\mathbb{G}\rtimes\Lambda_{0}$ is irreducible if and only if the
representation parameter $(u,V,v)$ is irreducible. ∎

##  10\. Distinguished representation parameters and distinguished
representations

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. For any
unitary projective representation
$V\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathscr{H})$ of $\Lambda_{0}$, and
any $r\in\Lambda$, define $r\cdot V$ to be the unitary projective
representation of $r\Lambda_{0}r^{-1}$ on $\mathscr{H}$ sending
$s_{0}=rr_{0}r^{-1}\in r\Lambda_{0}r^{-1}$ to
$(V\circ\operatorname{Ad}_{r^{-1}})(s_{0})=V(r_{0})$. Then $(rs)\cdot
V=r\cdot(s\cdot V)$ for all $r,s\in\Lambda$ with $1_{\Lambda}\cdot V=V$, in
other words, this defines an action of the group $\Lambda$ on the class of all
unitary projective representations of general isotropy subgroups of $\Lambda$.

It is easy to see from Proposition 3.9 that whenever
$\mathbf{S}=(\mathscr{H},u,w)\in\mathcal{CSR}_{\Lambda_{0}}$, the triple
$r\cdot\mathbf{S}=(\mathscr{H},r\cdot u,r\cdot w)$ is a CSR in
$\mathcal{CSR}_{r\Lambda_{0}r^{-1}}$. If
$U=\mathscr{R}_{\Lambda_{0}}(\mathbf{S})$ is the unitary representation of
$\mathbb{G}\rtimes\Lambda_{0}$, then it is easy to see by restriction that
$\mathscr{R}_{r\Lambda_{0}r^{-1}}(r\cdot\mathbf{S})$ is the unitary
representation $r\cdot
U=(\operatorname{id}\otimes\alpha^{\ast}_{r^{-1}}\otimes\operatorname{Ad}^{\ast}_{r^{-1}})(U)$
of $\mathbb{G}\rtimes r\Lambda_{0}r^{-1}$, as described in Proposition 5.1.
Thus by Corollary 4.4, we see that $\operatorname{Ind}(U)$ and
$\operatorname{Ind}(r\cdot U)$ are equivalent representations of
$\mathbb{G}\rtimes\Lambda$.

Similarly, for any representation parameter $(u,V,v)$ associated with
$\Lambda_{0}$ and any $r\in\Lambda$, the triple $(r\cdot u,r\cdot V,r\cdot v)$
is a representation parameter associated with $r\Lambda_{0}r^{-1}$, which we
denoted by $r\cdot(u,V,v)$. This clearly defines an $\Lambda$-action on the
proper class of all representation parameters associated with any group in
some conjugacy class of a general isotropy subgroup of $\Lambda$. A simple
calculation shows that (recall $\mathcal{S}(u,V,v)$ is the CSR parameterized
by $(u,V,v)$)

(10.1) $\forall r\in\Lambda,\quad
r\cdot\mathcal{S}(u,V,v)=\mathcal{S}\bigl{(}r\cdot(u,V,v)\bigr{)}.$

######  Definition 10.1.

Let $(u,V,v)$ be a representation parameter associated with some
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, the induced
representation
$\operatorname{Ind}\Bigl{(}\mathscr{R}_{\Lambda_{0}}\bigl{(}\mathcal{S}(u,V,v)\bigr{)}\Bigr{)}$
of $\mathbb{G}\rtimes\Lambda$ is called the representation of
$\mathbb{G}\rtimes\Lambda$ parameterized by $(u,V,v)$.

######  Proposition 10.2.

Let $(u,V,v)$ be a representation parameter associated with some
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Then for any
$r\in\Lambda$, the representation parameters $(u,V,v)$ and $r\cdot(u,V,v)$
parameterize equivalent representations of $\mathbb{G}\rtimes\Lambda$.

###### Proof.

Since $\mathscr{R}_{\Lambda_{0}}\bigl{(}\mathcal{S}(u,V,v)\bigr{)}$ and
$\mathscr{R}_{r\Lambda_{0}r^{-1}}\bigl{(}r\cdot\mathcal{S}(u,V,v)\bigr{)}$
induces equivalent representations of $\mathbb{G}\rtimes\Lambda$, the
proposition now follows from equation (10.1) and Definition 10.1. ∎

######  Proposition 10.3.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Let $(u,V,v)$
be an irreducible representation parameter associated with $\Lambda_{0}$, $U$
denote the representation
$\mathscr{R}_{\Lambda_{0}}\Bigl{(}\mathcal{S}(u,V,v)\Bigr{)}$. If
$\Lambda_{0}=\Lambda_{[u]}$, then the the induced representation
$\operatorname{Ind}(U)$ of $\mathbb{G}\rtimes\Lambda$ is irreducible.

###### Proof.

By Corollary 6.4, the proposition amounts to show that

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

where $\Lambda(r,s)=r\Lambda_{0}r^{-1}\cap s\Lambda_{0}s^{-1}$. Since
$\Lambda_{0}=\Lambda_{[u]}$, by the definition of $\Lambda_{[u]}$, we have
$[r\cdot u]\neq[s\cdot u]$ whenever $r^{-1}s\notin\Lambda_{0}$. Now condition
(10.2) holds by Proposition 9.11. ∎

######  Definition 10.4.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, an
_irreducible_ representation parameter $(u,V,v)$ associated with $\Lambda_{0}$
is called distinguished if $\Lambda_{0}=\Lambda_{[u]}$. When this is the case,
the irreducible unitary representation $\operatorname{Ind}(U)$ of
$\mathbb{G}\rtimes\Lambda$ is called distinguished, where $U$ is the unitary
representation $\mathscr{R}_{\Lambda_{0}}\Bigl{(}\mathcal{S}(u,V,v)\Bigr{)}$
of $\mathbb{G}\rtimes\Lambda_{0}$.

######  Remark 10.5.

The associated group of a distinguished representation parameter must be an
isotropy subgroup of $\Lambda$ for the action
$\Lambda\curvearrowright\operatorname{Irr}(\mathbb{G})$. More precisely, a
representation parameter $(u,V,v)$ is distinguished if and only if its
associated group is exactly the isotropy subgroup of
$[u]\in\operatorname{Irr}(\mathbb{G})$ under the action
$\Lambda\curvearrowright\operatorname{Irr}(\mathbb{G})$. As we will see
presently, in the formulation of our results on the classification of
irreducible representations of $\mathbb{G}\rtimes\Lambda$ and the conjugation
on $\operatorname{Irr}(\mathbb{G})$, only distinguished representation
parameters are needed. This makes one wonder why we pose the family of general
isotropy subgroup $\operatorname{\mathcal{G}_{iso}}(\Lambda)$ instead of only
isotropy subgroups. The main reason we need general isotropy subgroups of
$\Lambda$ is that in proving these results, as well as the formulation and the
proof of the fusion rules, we need to express the dimensions of various
intertwiner spaces. The calculation of the dimensions of these intertwiner
spaces will rely on Proposition 6.3, which clearly requires us to consider the
intersections of isotropy subgroups, i.e. general isotropy subgroups.

######  Definition 10.6.

Let $\Lambda_{0}$ be an isotropy subgroup of $\Lambda$ for the action
$\Lambda\curvearrowright\operatorname{Irr}(\mathbb{G})$. Suppose
$(u_{1},V_{1},v_{1})$ and $(u_{2},V_{2},v_{2})$ are two distinguished
representation parameters associated with $\Lambda_{0}$. If the CSRs
$\mathcal{S}(u_{1},V_{1},v_{1})$ and $\mathcal{S}(u_{2},V_{2},v_{2})$ are
isomorphic in $\mathcal{CSR}_{\Lambda_{0}}$, we say $(u_{1},V_{1},v_{1})$ and
$(u_{2},V_{2},v_{2})$ are equivalent.

The following proposition serves to characterize equivalence of distinguished
representation parameters in some more concrete ways.

######  Proposition 10.7.

Let $\Lambda_{0}$ be an isotropy subgroup of $\Lambda$ for the action
$\Lambda\curvearrowright\operatorname{Irr}(\mathbb{G})$, $(u_{1},V_{1},v_{1})$
and $(u_{2},V_{2},v_{2})$ two distinguished representation parameters
associated with $\Lambda_{0}$. The following are equivalent:

  1. (a)

$(u_{1},V_{1},v_{1})$ and $(u_{2},V_{2},v_{2})$ are equivalent;

  2. (b)

$(u_{1},V_{1},v_{1})$ and $(u_{2},V_{2},v_{2})$ parameterize equivalent
representations of $\mathbb{G}\rtimes\Lambda_{0}$;

  3. (c)

there exists a mapping $\mathsf{b}\colon\Lambda_{0}\rightarrow\mathbb{T}$ such
that $\mathsf{b}V_{1}$ and $V_{2}$ share the same cocycle, and both
$\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}V_{1},V_{2})$
and $\operatorname{Mor}_{\Lambda_{0}}(v_{1},\mathsf{b}v_{2})$ are nonzero;

  4. (d)

there exists a mapping $\mathsf{b}\colon\Lambda_{0}\rightarrow\mathbb{T}$ such
that $\mathsf{b}V_{1}$ and $V_{2}$ share the same cocycle, and both
$\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}V_{1},V_{2})$
and $\operatorname{Mor}_{\Lambda_{0}}(v_{1},\mathsf{b}v_{2})$ contain unitary
operators.

###### Proof.

The equivalence of (a) and (b) follows directly from the definitions. It is
also clear that (d) implies (c). If (c) holds, and

(10.3) $\begin{gathered}0\neq
S\in\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}V_{1},V_{2}),\\\
\text{and}\quad 0\neq
T\in\operatorname{Mor}_{\Lambda_{0}}(v_{1},\mathsf{b}v_{2})=\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}^{-1}v_{1},v_{2}),\end{gathered}$

then both $S$ and $T$ are invertible by Schur’s lemma as
$u_{1},u_{2},\mathsf{b}^{-1}v_{1},v_{2}$ are all irreducible. Since
$u_{1},u_{2},\mathsf{b}V_{1},V_{2},v_{1},\mathsf{b}v_{2}$ are all unitary, we
have

(10.4)
$0\neq\Upsilon_{S}\in\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}V_{1},V_{2}),\quad\text{and}\quad
0\neq\Upsilon_{T}\in\operatorname{Mor}_{\Lambda_{0}}(v_{1},\mathsf{b}v_{2}),$

where $S=\Upsilon_{S}\left\lvert S\right\rvert$ is the polar decomposition of
$S$, and $T=\Upsilon_{T}\left\lvert T\right\rvert$ the polar decomposition of
$T$. As $S,T$ are invertible, $\Upsilon_{S}$ and $\Upsilon_{T}$ are unitary.
This proves that (c) implies (d).

Let $\mathscr{K}_{i}$ be the representation space of $v_{i}$ for $i=1,2$. By
definition,
$\mathcal{S}(u_{i},V_{i},v_{i})=(\operatorname{id}_{\mathscr{K}_{i}}\otimes
u,v_{i}\times V_{i})$, and
$\mathsf{b}^{-1}v_{i}\times\mathsf{b}V_{i}=v_{i}\times V_{i}$ for any mapping
$\mathsf{b}\colon\Lambda_{0}\rightarrow\mathbb{T}$. If (c) holds, let $S$, $T$
be operators as in (10.3), then

(10.5) $T\otimes
S\in\operatorname{Mor}_{\mathbb{G}}(\operatorname{id}_{\mathscr{K}_{1}}\otimes
u_{1},\operatorname{id}_{\mathscr{K}_{2}}\otimes
u_{2})\cap\operatorname{Mor}_{\Lambda_{0}}(v_{1}\times V_{1},v_{2}\times
V_{2}).$

Now (a) follows from (10.5), Proposition 7.4 and the fact that both $S$ and
$T$ are invertible. Thus (c) implies (a).

We conclude the proof by showing (a) implies (d). By Schur’s lemma, and the
irreducibility of $u_{1}$ and $u_{2}$, it is easy to see that

(10.6)
$\operatorname{Mor}_{\mathbb{G}}(\operatorname{id}_{\mathscr{K}_{1}}\otimes
u_{1},\operatorname{id}_{\mathscr{K}_{2}}\otimes
u_{2})=\mathcal{B}(\mathscr{K}_{1},\mathscr{K}_{2})\otimes\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2}).$

Suppose (a) holds. Then the intertwiner space given by the intersection in
(10.5) is nonzero, and

(10.7) $\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2})=\mathbb{C}W_{r}$

for some unitary operator $W_{r}$. By (10.6) and (a), there exists a unitary
$W_{l}\in\mathcal{B}(\mathscr{K}_{1},\mathscr{K}_{2})$ such that

(10.8) $W_{l}\otimes W_{r}\in\operatorname{Mor}_{\Lambda_{0}}(v_{1}\times
V_{1},v_{2}\times
V_{2})=\operatorname{Mor}_{\Lambda_{0}}\bigl{(}(\mathsf{b}^{-1}v_{1})\times(\mathsf{b}V_{1}),v_{2}\times
V_{2}\bigr{)}.$

By (10.7), both $W_{r}V_{1}W_{r}^{\ast}$ and $V_{2}$ are covariant projective
$\Lambda_{0}$-representations of $u_{2}$. Thus we can take a
$u_{2}$-transitional mapping $\mathsf{b}$ from $W_{r}V_{1}W_{r}^{\ast}$ to
$V_{2}$ (see Definition 9.10), i.e. a mapping
$\mathsf{b}\colon\Lambda_{0}\rightarrow\mathbb{T}$ such that

(10.9)
$W_{r}(\mathsf{b}V_{1})W_{r}^{\ast}=\mathsf{b}(W_{r}V_{1}W_{r}^{\ast})=V_{2},$

which forces the cocycles of $\mathsf{b}V_{1}$ and $V_{2}$ coincide, and

(10.10)
$W_{r}\in\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}V_{1},V_{2})\cap\operatorname{Mor}_{\mathbb{G}}(u_{1},u_{2}).$

Now (10.8) and (10.10) forces

(10.11)
$W_{l}\in\operatorname{Mor}_{\Lambda_{0}}(\mathsf{b}^{-1}v_{1},v_{2})=\operatorname{Mor}_{\Lambda_{0}}(v_{1},\mathsf{b}v_{2}).$

Thus (d) holds by (10.10) and (10.11). ∎

##  11\. Density of matrix coefficients of distinguished representations

The aim of this section is to show that the linear span of matrix coefficients
of distinguished representations of $\mathbb{G}\rtimes\Lambda$ is exactly
$\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$, hence is dense in
$C(\mathbb{G}\rtimes\Lambda)=A\otimes C(\Lambda)$ in particular. As a
consequence, any irreducible unitary representation of
$\mathbb{G}\rtimes\Lambda$ is equivalent to a distinguished one.

The following lemma essentially establishes the density of the linear span of
matrix coefficients of distinguished representations of
$\mathbb{G}\rtimes\Lambda$ in $C(\mathbb{G}\rtimes\Lambda)=A\otimes
C(\Lambda)$.

######  Lemma 11.1.

Let $u$ be an irreducible unitary representation of $\mathbb{G}$ on some
finite dimensional Hilbert space $\mathscr{H}$,
$x=[u]\in\operatorname{Irr}(\mathbb{G})$, $V$ the covariant projective
$\Lambda_{x}$-representation of $u$ with cocycle $\omega$. Let $M(u)$ denote
the linear subspace of $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$
spanned by matrix coefficients of distinguished representations of
$\mathbb{G}\rtimes\Lambda$ parameterized by distinguished representation
parameters of the form $(u,V,v)$, where $v$ runs through all irreducible
unitary projective representations of $\Lambda_{x}$ with cocycle
$\omega^{-1}=\overline{\omega}$. For any $r\in\Lambda$, suppose $M_{c}(r\cdot
u)$ is the linear subspace of $\operatorname{Pol}(\mathbb{G})$ spanned by
matrix coefficients of $r\cdot u$, then

(11.1) $M(u)=\sum_{r\in\Lambda}M_{c}(r\cdot u)\otimes
C(\Lambda)=\left(\sum_{r\in\Lambda}M_{c}(r\cdot u)\right)\otimes C(\Lambda).$

###### Proof.

Take any irreducible unitary projective representation $v$ of $\Lambda_{x}$ on
some finite dimensional Hilbert space $\mathscr{K}$ with cocycle
$\overline{\omega}$, then $(u,V,v)$ is a distinguished representation
parameter. The distinguished CSR $\mathcal{S}(u,V,v)$ subordinate to
$\Lambda_{x}$ parameterized by $(u,V,v)$ is given by

(11.2)
$\mathcal{S}(u,V,v)=(\mathscr{K}\otimes\mathscr{H},\operatorname{id}_{\mathscr{K}}\otimes
u,v\times V)$

by definition. Let
$U=\mathscr{R}_{\Lambda_{0}}\bigl{(}\mathcal{S}(u,V,v)\bigr{)}$, then the
distinguished representation $W=\operatorname{Ind}(U)$ of
$\mathbb{G}\rtimes\Lambda$ parameterized by $(u,V,v)$ is obtained as follows
by the construction of induced representations presented in § 4. First we
define a unitary representation

(11.3)
$\begin{split}\widetilde{W}=\sum_{r,s\in\Lambda}e_{rs^{-1},r}\otimes\operatorname{id}_{\mathscr{K}}\otimes[(\operatorname{id}_{\mathscr{H}}\otimes\alpha^{\ast}_{rs^{-1}})(u)]\otimes\delta_{s}\\\
\in\mathcal{B}(\ell^{2}(\Lambda))\otimes\mathcal{B}(\mathscr{K})\otimes\mathcal{B}(\mathscr{H})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)\end{split}$

of $\mathbb{G}\rtimes\Lambda$ on
$\ell^{2}(\Lambda)\otimes\mathscr{K}\otimes\mathscr{H}$. The subspace

(11.4)
$\mathscr{H}_{(u,V,v)}=\left\\{\sum_{r\in\Lambda}\delta_{r}\otimes\zeta_{r}\;:\;\begin{array}[]{r}\zeta_{r}\in\mathscr{K}\otimes\mathscr{H},\text{
and }\;\zeta_{r_{0}r}=\bigl{(}v(r_{0})\otimes V(r_{0})\bigr{)}\zeta_{r}\\\
\text{ for all }r_{0}\in\Lambda_{0},r\in\Lambda\end{array}\right\\}$

of $\ell^{2}(\Lambda)\otimes\mathscr{K}\otimes\mathscr{H}$ is invariant under
$\widetilde{W}$ and $W$ is the subrepresentation $\mathscr{H}_{(u,V,v)}$ of
$\widetilde{W}$. Recall (Lemma 4.5) that the projection
$\pi\in\mathcal{B}(\ell^{2}(\Lambda)\otimes\mathscr{K}\otimes\mathscr{H})$
with range $\mathscr{H}_{u,V,v}$ is given by

(11.5)
$\pi=\frac{1}{\left\lvert\Lambda_{x}\right\rvert}\sum_{r_{0}\in\Lambda_{x}}\sum_{s\in\Lambda}e_{r_{0}s,s}\otimes
v(r_{0})\otimes V(r_{0}).$

Since vectors of the form $\delta_{r}\otimes\xi\otimes\eta$, $r\in\Lambda$,
$\xi\in\mathscr{K}$, $\eta\in\mathscr{H}$ span
$\ell^{2}(\Lambda)\otimes\mathscr{K}\otimes\mathscr{H}$, the matrix
coefficients of $W$ is spanned by elements of
$\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$ of the form

(11.6) $\begin{split}&\hskip
11.99998ptc(v;r,s,\xi_{1},\xi_{2},\eta_{1},\eta_{2})\\\
&=(\omega_{\pi(\delta_{r}\otimes\xi_{1}\otimes\eta_{1}),\pi(\delta_{s}\otimes\xi_{2}\otimes\eta_{2})}\otimes\operatorname{id}_{\operatorname{Pol}(\mathbb{G})}\otimes\operatorname{id}_{C(\Lambda)})(W)\\\
&=(\omega_{\delta_{r}\otimes\xi_{1}\otimes\eta_{1},\delta_{s}\otimes\xi_{2}\otimes\eta_{2}}\otimes\operatorname{id}\otimes\operatorname{id})\bigl{(}(\pi\otimes
1\otimes 1)\widetilde{W}(\pi\otimes 1\otimes 1)\bigr{)}\\\
&=(\omega_{\delta_{r}\otimes\xi_{1}\otimes\eta_{1},\delta_{s}\otimes\xi_{2}\otimes\eta_{2}}\otimes\operatorname{id}\otimes\operatorname{id})\bigl{(}\widetilde{W}(\pi\otimes
1\otimes 1)\bigr{)},\end{split}$

where the last equality follows from Lemma 4.5, and $\omega_{x,y}$ is the
linear form $\left\langle\,\cdot\,x\,,\,y\right\rangle$.

By (11.3) and (11.5), we see that

(11.7) $\begin{split}&\hskip
11.99998pt\left\lvert\Lambda_{0}\right\rvert\cdot\bigl{[}\widetilde{W}(\pi\otimes
1\otimes 1)\bigr{]}(\delta_{r}\otimes\xi_{1}\otimes\eta_{1}\otimes 1\otimes
1)\\\ &=\sum_{\begin{subarray}{c}r^{\prime},s^{\prime},t\in\Lambda,\\\
r_{0}\in\Lambda_{x}\end{subarray}}\left[e_{r^{\prime}s^{\prime-1},r^{\prime}}e_{r_{0}t,t}\otimes
v(r_{0})\otimes\Bigl{(}\bigl{(}(s^{\prime}r^{\prime-1})\cdot
u\bigr{)}\bigl{(}V(r_{0})\otimes
1\bigr{)}\Bigr{)}\otimes\delta_{s^{\prime}}\right]\\\ &\hskip
120.00018pt\cdot(\delta_{r}\otimes\xi_{1}\otimes\eta_{1}\otimes 1\otimes 1)\\\
&\hskip 11.99998pt(\text{Only terms with $t=r$, and $r^{\prime}=r_{0}t=r_{0}r$
can be nonzero})\\\
&=\sum_{s^{\prime}\in\Lambda}\sum_{r_{0}\in\Lambda_{x}}\delta_{r_{0}rs^{\prime-1}}\otimes[v(r_{0})\xi_{1}]\otimes\Bigl{[}\bigl{(}(s^{\prime}r^{-1}r_{0}^{-1})\cdot
u\bigr{)}\bigl{(}V(r_{0})\eta_{1}\otimes
1\bigr{)}\Bigr{]}\otimes\delta_{s^{\prime}}.\end{split}$

Note that $r_{0}rs^{\prime-1}=s\iff s^{\prime}=s^{-1}r_{0}r\iff
s^{\prime}r^{-1}r_{0}^{-1}=s^{-1}$, by (11.6) and (11.7), we have

(11.8)
$c(v;r,s,\xi_{1},\xi_{2},\eta_{1},\eta_{2})=\sum_{r_{0}\in\Lambda_{x}}\omega_{\xi_{1},\xi_{2}}\bigl{(}v(r_{0})\bigr{)}\bigl{[}(\omega_{V(r_{0})\eta_{1},\eta_{2}}\otimes\operatorname{id})(s^{-1}\cdot
u)\bigr{]}\otimes\delta_{s^{-1}r_{0}r}.$

For any $r_{0}\in\Lambda_{x}$, we have

(11.9)
$\omega_{\xi_{1},\xi_{2}}\bigl{(}v(r_{0})\bigr{)}\in\mathbb{C}\quad\text{and}\quad\bigl{[}(\omega_{V(r_{0})\eta_{1},\eta_{2}}\otimes\operatorname{id})(s^{-1}\cdot
u)\bigr{]}\in M_{c}(s^{-1}\cdot u).$

By (11.8) and (11.9), we have

(11.10) $c(v;r,s,\xi_{1},\xi_{2},\eta_{1},\eta_{2})\in M_{c}(s^{-1}\cdot
u)\otimes C(\Lambda),$

which proves that

(11.11) $M(u)\subseteq\sum_{r^{\prime}\in\Lambda}M_{c}(r^{\prime}\cdot
u)\otimes C(\Lambda)=\left(\sum_{r^{\prime}\in\Lambda}M_{c}(r^{\prime}\cdot
u)\right)\otimes C(\Lambda).$

It remains to establish the reverse inclusion, which is easily seen to be
equivalent to show that for any $r_{1},r_{2}\in\Lambda$, we have

(11.12) $M(u)\supseteq M_{c}(r_{1}\cdot u)\otimes\delta_{r_{2}}.$

By the general theory of projective representations, there exists irreducible
unitary projective representations $v_{1},\ldots,v_{m}$ on
$\mathscr{K}_{1},\ldots,\mathscr{K}_{m}$ respectively, all with cocycle
$\overline{\omega}$, and $\xi^{(i)}_{1},\xi^{(i)}_{2}\in\mathscr{K}_{i}$, such
that

(11.13)
$\sum_{i=1}^{m}\left(\omega_{\xi^{(i)}_{1},\xi^{(i)}_{2}}\otimes\operatorname{id}\right)(v_{i})=\delta_{e}\in
C(\Lambda_{x}).$

By (11.8) and (11.13), we see that for any $r,s\in\Lambda$, and any
$\eta_{1},\eta_{2}\in\mathscr{H}$, $M(u)$ contains

(11.14) $\begin{split}&\hskip
11.99998pt\sum_{i=1}^{n}c(v_{i};r,s\xi^{(i)}_{1},\xi^{(i)}_{2},\eta_{1},\eta_{2})\\\
&=\sum_{r_{0}\in\Lambda_{x}}\delta_{e}(r_{0})\bigl{[}(\omega_{V(r_{0})\eta_{1},\eta_{2}}\otimes\operatorname{id})(s^{-1}\cdot
u)\bigr{]}\otimes\delta_{s^{-1}r_{0}r}\\\ &\hskip 11.99998pt(\text{Only terms
with $r_{0}=e$ can be nonzero, and $V(e)=\operatorname{id}_{\mathscr{H}}$})\\\
&=\bigl{[}(\omega_{\eta_{1},\eta_{2}}\otimes\operatorname{id})(s^{-1}\cdot
u)\bigr{]}\otimes\delta_{s^{-1}r}.\end{split}$

Taking $s=r_{1}^{-1}$ and $r=sr_{2}=r_{1}^{-1}r_{2}$ in (11.14) proves (11.12)
and finishes the proof of the lemma. ∎

######  Proposition 11.2.

The linear span of matrix coefficients of distinguished representations of
$\mathbb{G}\rtimes\Lambda$ in $\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda)$ is $\operatorname{Pol}(\mathbb{G})\otimes C(\Lambda)$ itself. In
particular, every unitary irreducible representation of
$\mathbb{G}\rtimes\Lambda$ is unitarily equivalent to a distinguished one.

###### Proof.

The first assertion follows from Lemma 11.1, and the second assertion follows
from the first and the orthogonality relations of irreducible representations
of $\mathbb{G}\rtimes\Lambda$. ∎

##  12\. Classification of irreducible representations of
$\mathbb{G}\rtimes\Lambda$

For each isotropy subgroup $\Lambda_{0}$ of $\Lambda$, let
$\mathfrak{D}_{\Lambda_{0}}$denotes the collection of equivalence classes of
distinguished representation parameters associated with $\Lambda_{0}$. By
Proposition 10.7, the mapping

(12.1)
$\begin{split}\Psi_{\Lambda_{0}}\colon\mathfrak{D}_{\Lambda_{0}}&\to\operatorname{Irr}(\mathbb{G}\rtimes\Lambda)\\\
[(u,V,v)]&\mapsto\mathscr{R}_{\Lambda_{0}}\bigl{(}\mathcal{S}(u,V,v)\bigr{)}\end{split}$

is well-defined and injective. In particular, $\mathfrak{D}_{\Lambda_{0}}$ is
a set (instead of a proper class). Let $\mathfrak{D}$ be the collection of
equivalence classes of distinguished representation parameters associated with
any isotropy subgroup of $\Lambda$. By definition, $\mathfrak{D}$ is the
disjoint union of $\mathfrak{D}_{\Lambda_{0}}$ as $\Lambda_{0}$ runs through
all isotropy subgroups of $\Lambda$, hence $\mathfrak{D}$ is also a set. For
any $[(u,V,v)]\in\mathfrak{D}_{\Lambda_{0}}$ and any $r\in\Lambda$,
$r\cdot[(u,V,v)]=[r\cdot(u,V,v)]$ is a well-defined class in
$\mathfrak{D}_{r\Lambda_{0}r^{-1}}$. This defines an action of $\Lambda$ on
$\mathfrak{D}$. We are now ready to state and prove the classification of
irreducible representations of $\mathbb{G}\rtimes\Lambda$.

######  Theorem 12.1 (Classification of irreducible representations of
$\mathbb{G}\rtimes\Lambda$).

The mapping

(12.2)
$\begin{split}\Psi\colon\mathfrak{D}&\rightarrow\operatorname{Irr}(\mathbb{G}\rtimes\Lambda)\\\
[(u,V,v)]\in\mathfrak{D}_{\Lambda_{0}}&\mapsto\Psi_{\Lambda_{0}}\bigl{(}[(u,V,v)]\bigr{)}=\operatorname{Ind}\Bigl{(}\mathscr{R}_{\Lambda_{0}}\bigl{(}\mathcal{S}(u,V,v)\bigr{)}\Bigr{)}\end{split}$

is surjective, and the fibers of $\Psi$ are exactly the $\Lambda$-orbits in
$\mathfrak{D}$.

###### Proof.

By Proposition 11.2, $\Psi$ is surjective. By Corollary 5.2 and (10.1), each
$\Lambda$-orbits in $\mathfrak{D}$ maps to the same point under $\Psi$. It
remains to show that if $(u_{i},V_{i},v_{i})$ is a distinguished
representation parameter with associated subgroup $\Lambda_{i}$ for $i=1,2$,
and

(12.3)
$\Psi\bigl{(}[(u_{1},V_{1},v_{1})]\bigr{)}=\Psi\bigl{(}[(u_{2},V_{2},v_{2})]\bigr{)},$

then there exists an $r_{0}\in\Lambda$, such that

(12.4)
$r_{0}\cdot[(u_{1},V_{1},v_{1})]=[(u_{2},V_{2},v_{2})]\in\mathfrak{D}_{\Lambda_{2}}.$

Let $\mathbf{S}_{i}=\mathcal{S}(u_{i},V_{i},v_{i})$,
$U_{i}=\mathscr{R}_{\Lambda_{i}}(\mathbf{S}_{i})$ for $i=1,2$. If
$[u_{2}]\notin\Lambda\cdot[u_{1}]$, then by Proposition 9.11, we have

(12.5) $\forall
r,s\in\Lambda,\;\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigl{(}(r\cdot
U_{1})|_{\mathbb{G}\rtimes\Lambda(r,s)},(s\cdot
U_{2})|_{\mathbb{G}\rtimes\Lambda(r,s)}\Bigr{)}=0,$

where $\Lambda(r,s)=r\Lambda_{1}r^{-1}\cap s\Lambda_{2}s^{-1}$. This is
because $(r\cdot U_{1})|_{\mathbb{G}\rtimes\Lambda(r,s)}$ is parameterized by
the representation parameter
$(u_{1},V_{1}|_{\Lambda(r,s)},v_{1}|_{\Lambda(r,s)})$ associated with
$\Lambda(r,s)$, and a similar assertion holds for $(s\cdot
U_{2})|_{\mathbb{G}\rtimes\Lambda(r,s)}$. Thus

(12.6)
$\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}\bigl{(}\operatorname{Ind}(U_{1}),\operatorname{Ind}(U_{2})\bigr{)}=0$

by Proposition 6.3, which contradicts (12.3).

Thus $[u_{2}]\in\Lambda\cdot[u_{1}]$, by replacing $[(u_{1},V_{1},v_{1})]$
with $r_{0}\cdot[(u_{1},V_{1},v_{1})]$ for some $r_{0}\in\Lambda$ if
necessary, we may assume without loss of generality that
$[u_{1}]=[u_{2}]\in\operatorname{Irr}(\mathbb{G})$, and
$\Lambda_{1}=\Lambda_{2}$, which we now denote by $\Lambda_{0}$. It remains to
prove that under this assumption, we have

(12.7)
$[(u_{1},V_{1},v_{1})]=[(u_{2},V_{2},v_{2})]\in\mathfrak{D}_{\Lambda_{0}}$

Since when $r^{-1}s\notin\Lambda_{0}$ if and only if $r\cdot[u_{1}]\neq
s\cdot[u_{2}]$, we have

(12.8) $\forall r,s\in\Lambda,\quad
r^{-1}s\notin\Lambda_{0}\implies\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda(r,s)}\bigl{(}(r\cdot
U_{1})|_{\mathbb{G}\rtimes\Lambda(r,s)},(r\cdot
U_{2})|_{\mathbb{G}\rtimes\Lambda(r,s)}\bigr{)}=0.$

Note that when $r^{-1}s\in\Lambda_{0}$, we have
$\Lambda(r,s)=r\Lambda_{0}r^{-1}=s\Lambda_{0}s^{-1}$, and
$[\Lambda\colon\Lambda(r,s)]=[\Lambda\colon\Lambda_{0}]$. By (12.3), (12.8)
and Proposition 6.3, we have

(12.9)
$1=\frac{1}{\left\lvert\Lambda_{0}\right\rvert^{2}[\Lambda\colon\Lambda_{0}]}\sum_{\begin{subarray}{c}r,s\in\Lambda,\\\
r^{-1}s\in\Lambda_{0}\end{subarray}}\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U_{1},s\cdot U_{2}\bigr{)}.$

Since $r\cdot U_{1}$, $s\cdot U_{2}$ are both irreducible, we have

(12.10)
$r^{-1}s\in\Lambda_{0}\implies\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U_{1},s\cdot U_{2}\bigr{)}=0\;\text{or}\;1.$

Note that there are
$\left\lvert\Lambda_{0}\right\rvert^{2}[\Lambda\colon\Lambda_{0}]=\left\lvert\Lambda\right\rvert\cdot\left\lvert\Lambda_{0}\right\rvert$
terms on the right side of (12.9), (12.10) forces

(12.11)
$r^{-1}s\in\Lambda_{0}\implies\dim\operatorname{Mor}_{\mathbb{G}\rtimes
r\Lambda_{0}r^{-1}}\bigl{(}r\cdot U_{1},s\cdot U_{2}\bigr{)}=1.$

In particular, taking $r=s=1_{\Lambda}$ in (12.11) shows that $U_{1}$ and
$U_{2}$ are equivalent, hence (12.7) holds by Proposition 10.7. This finishes
the proof of the theorem. ∎

##  13\. The conjugate representation of distinguished representations

We now study the conjugation of irreducible representations of
$\mathbb{G}\rtimes\Lambda$ in terms of the classification presented in Theorem
12.1. There is a small complication here in the non-Kac type case, where the
contragredient of a unitary representation need not be unitary. Resolving this
kind of question involves the modular operator, just as in Proposition 7.7.

We begin with a simple lemma on linear operators.

######  Lemma 13.1.

Let $\mathscr{H}$ be a Hilbert space, $U,P\in\mathcal{B}(\mathscr{H})$ such
that $U$ is unitary, $P$ is invertible and positive, if $PUP^{-1}$ is unitary,
then $PUP^{-1}=U$, i.e. $P$ commutes with $U$.

###### Proof.

Let $V=PUP^{-1}$. We have

(13.1) $PU^{\ast}P^{-1}=PU^{-1}P^{-1}=V^{-1}=V^{\ast}=P^{-1}U^{\ast}P.$

Thus $U^{\ast}$ commutes with the positive operator $P^{2}$. Hence $U^{\ast}$
commutes with ${(P^{2})}^{1/2}=P$, i.e. $U^{\ast}P=PU^{\ast}$. Taking adjoints
of this proves $PU=UP$. ∎

######  Proposition 13.2.

Let $u$ be an irreducible unitary representation of $\mathbb{G}$,
$\Lambda_{0}$ a subgroup of the isotropy subgroup $\Lambda_{[u]}$, $V$ a
covariant projective $\Lambda_{0}$-representation of $u$. Then any operator
$\rho\in\operatorname{Mor}_{\mathbb{G}}(u,u^{cc})$ commutes with $V$ (i.e.
$\rho V(r_{0})=V(r_{0})\rho$ for all $r_{0}\in\Lambda_{0}$).

###### Proof.

Since $u$ is irreducible, $\operatorname{Mor}_{\mathbb{G}}(u,u^{cc})$ is a one
dimensional space spanned by an invertible positive operator ([MR3204665,
Lemma 1.3.12]). By definition (see [MR3204665, Proposition 1.4.4 and
Definition 1.4.5]), the conjugation $\overline{u}$ of $u$ is given by

(13.2) $\overline{u}=\bigl{(}{j(\rho_{u})}^{1/2}\otimes
1\bigr{)}u^{c}\bigl{(}{j(\rho_{u})}^{-1/2}\otimes 1\bigr{)},$

where $\rho_{u}$ is the unique _positive_ operator in
$\operatorname{Mor}_{\mathbb{G}}(u,u^{cc})$ with
$\operatorname{Tr}(\rho_{u})=\operatorname{Tr}(\rho_{u}^{-1})$. Since
$\operatorname{Mor}_{\mathbb{G}}(u,u^{cc})=\mathbb{C}\rho_{u}$, it suffices to
show that $\rho_{u}$ commutes with $V$.

Since $u$, $V$ are covariant, we have

(13.3) $\forall r_{0}\in\Lambda_{0},\;\bigl{(}V(r_{0})\otimes
1\bigr{)}(r_{0}\cdot u)=u\bigl{(}V(r_{0})\otimes 1\bigr{)}.$

Taking the adjoint of both sides of (13.3) then applying
$j\otimes\operatorname{id}$, we get

(13.4) $\forall r_{0}\in\Lambda_{0},\;\bigl{(}V^{c}(r_{0})\otimes
1\bigr{)}(r_{0}\cdot u^{c})=u^{c}\bigl{(}V^{c}(r_{0})\otimes 1\bigr{)},$

where

(13.5)
$V^{c}=(j\otimes\operatorname{id})(V^{-1})=(j\otimes\operatorname{id})(V^{\ast})$

is the contragredient of $V$, and

(13.6)
$u^{c}=(j\otimes\operatorname{id})(u^{-1})=(j\otimes\operatorname{id})(u^{\ast})$

the contragredient of $u$. We pose

(13.7) $\overline{V}=\bigl{(}{j(\rho_{u})}^{1/2}\otimes
1\bigr{)}V^{c}\bigl{(}{j(\rho_{u})}^{-1/2}\otimes 1\bigr{)},$

then by (13.4) and (13.2), we have

(13.8) $\forall r_{0}\in\Lambda_{0},\;\bigl{(}\overline{V}(r_{0})\otimes
1\bigr{)}(r_{0}\cdot\overline{u})=\overline{u}\bigl{(}\overline{V}(r_{0})\otimes
1\bigr{)}.$

Thus for any $r_{0}\in\Lambda_{0}$,
$\overline{V}(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot\overline{u},\overline{u})$,
which is a one dimensional space spanned by a unitary operator since both
$r_{0}\cdot\overline{u}$ and $\overline{u}$ are irreducible unitary
representations of $\mathbb{G}$. Note that
$V^{c}(r_{0})=j\bigl{(}{V(r_{0})}^{\ast}\bigr{)}$ is unitary, by (13.7), we
have

(13.9)
$\det\bigl{(}\overline{V}(r_{0})\bigr{)}=\det\bigl{(}{j(\rho_{u})}^{1/2}V^{c}(r_{0}){j(\rho_{u})}^{-1/2}\bigr{)}=\det\bigl{(}V^{c}(r_{0})\bigr{)}\in\mathbb{T}.$

This forces $\overline{V}(r_{0})$ to be unitary since it is a scalar multiple
of a unitary operator. Applying Lemma 13.1 to (13.7) (evaluated on each
$r_{0}\in\Lambda_{0}$), we see that

(13.10) $V^{c}=\overline{V}=\bigl{(}{j(\rho_{u})}^{1/2}\otimes
1\bigr{)}V^{c}\bigl{(}{j(\rho_{u})}^{-1/2}\otimes 1\bigr{)}.$

Applying $j\otimes\operatorname{id}$ to the inverse of both sides of (13.10)
and note that $V^{cc}=V$, we see that

(13.11) $V=V^{cc}=(\rho_{u}^{1/2}\otimes 1)V^{cc}(\rho_{u}^{-1/2}\otimes
1)=(\rho_{u}^{1/2}\otimes 1)V(\rho_{u}^{-1/2}\otimes 1),$

i.e. $\rho_{u}^{1/2}$ (hence $\rho_{u}$) commutes with $V$. ∎

######  Proposition 13.3.

Let $(u,V,v)$ be a representation parameter associated with some
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, $U$ is the unitary
representation of $\mathbb{G}\rtimes\Lambda_{0}$ parameterized by $(u,V,v)$,
then the following hold:

  1. (a)

$(\overline{u},V^{c},v^{c})$ is also a representation parameter;

  2. (b)

$\rho_{U}=\operatorname{id}_{\mathscr{H}_{v}}\otimes\rho_{u}$, where
$\rho_{U}$ (resp. $\rho_{u}$) is the modular operator for the representation
$U$ (resp. $u$);

  3. (c)

$\overline{U}$ is parameterized by $(\overline{u},V^{c},v^{c})$.

###### Proof.

As we’ve seen in Proposition 13.2 and its proof, we have
$\overline{V}(r_{0})\in\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot u,u)$ for
all $r_{0}\in\Lambda_{0}$, thus $V^{c}=\overline{V}$ is covariant with
$\overline{u}$. Since

(13.12) $\begin{split}\forall r_{0}\in\Lambda_{0},\qquad(v^{c}\times
V^{c})(r_{0})&=j\bigl{(}{[v(r_{0})]}^{-1}\otimes{[V(r_{0})]}^{-1}\bigr{)}\\\
&=j\bigl{(}\\{[{(v\times V)}](r_{0})\\}^{-1}\bigr{)}\end{split}$

$v^{c}\times V^{c}$ is the contragredient of the unitary representation
$v\times V$ of $\Lambda_{0}$, hence is a unitary representation itself. Thus
$v^{c}$ and $V^{c}$ are unitary projective representations with opposing
cocycles. This proves (a).

To prove (b), by the characterizing property of $\rho_{U}$, it suffices to
show that the invertible positive operator $\operatorname{id}\otimes\rho_{u}$
satisfies

(13.13)
$\operatorname{id}\otimes\rho_{u}\in\operatorname{Mor}_{\mathbb{G}}(U,U^{cc})$

and (by Proposition 7.4 and Schur’s lemma applied to the irreducible
representation $u$)

(13.14) $\begin{split}&\hskip
11.99998pt\operatorname{Tr}((\cdot)(\operatorname{id}\otimes\rho_{u}))=\operatorname{Tr}((\cdot)(\operatorname{id}\otimes\rho_{u}^{-1}))\\\
&\in\operatorname{End}_{\mathbb{G}\rtimes\Lambda_{0}}(U)=\operatorname{End}_{\mathbb{G}}(\operatorname{id}\otimes
u)\cap\operatorname{End}_{\Lambda_{0}}(v\times
V)\subseteq\mathcal{B}(\mathscr{H}_{v})\otimes\mathbb{C}\operatorname{id}.\end{split}$

Since $\operatorname{Tr}(\rho_{u})=\operatorname{Tr}(\rho_{u}^{-1})$, (13.14)
holds. We now prove (13.13). As is seen in the proof of Proposition 13.2,
condition (13.3) holds, and a similar calculation by applying
$j\otimes\operatorname{id}$ to the inverse of both sides of (13.3) yields
(note that $V^{cc}=V$),

(13.15) $\forall r_{0}\in\Lambda_{0},\;\bigl{(}V(r_{0})\otimes
1\bigr{)}(r_{0}\cdot u^{cc})=u^{cc}\bigl{(}V(r_{0})\otimes 1\bigr{)}.$

By definition, we have

(13.16) $\begin{split}U&={(\operatorname{id}\otimes u)}_{123}{(v\times
V)}_{124}=(\operatorname{id}\otimes u\otimes 1)v_{14}V_{24}\\\
&\in\mathcal{B}(\mathscr{H}_{v})\otimes\mathcal{B}(\mathscr{H}_{u})\otimes\operatorname{Pol}(\mathbb{G})\otimes
C(\Lambda_{0}).\end{split}$

Thus

(13.17) $U^{c}=(j\otimes
j\otimes\operatorname{id}\otimes\operatorname{id})(U^{-1})=(\operatorname{id}\otimes
u^{c}\otimes 1)v^{c}_{14}V^{c}_{24},$

and

(13.18) $U^{cc}=(\operatorname{id}\otimes u^{cc}\otimes
1)v^{cc}_{14}V^{cc}_{24}=(\operatorname{id}\otimes u^{cc}\otimes
1)v_{14}V_{24}.$

By (13.16), (13.18) and Proposition 13.2, we have

(13.19) $\begin{split}&\hskip
11.99998pt(\operatorname{id}\otimes\rho_{u}\otimes 1\otimes
1)U=(\operatorname{id}\otimes\rho_{u}\otimes 1\otimes
1)(\operatorname{id}\otimes u\otimes 1)v_{14}V_{24}\\\
&=(\operatorname{id}\otimes u^{cc}\otimes
1)v_{14}[(\operatorname{id}\otimes\rho_{u}\otimes 1\otimes 1)V_{24}]\\\
&=(\operatorname{id}\otimes u^{cc}\otimes
1)v_{14}V_{24}(\operatorname{id}\otimes\rho_{u}\otimes 1\otimes 1)\\\
&=U^{cc}(\operatorname{id}\otimes\rho_{u}\otimes 1\otimes 1).\end{split}$

This proves (13.13) and finishes the proof of (b).

By Proposition 7.7 and (b), $\overline{U}$ corresponds to the CSR
$(\overline{\mathscr{H}_{u}},u^{\prime},w^{\prime})$ in
$\mathcal{CSR}_{\Lambda_{0}}$, where $\mathscr{H}_{u}$ is the underlying
finite dimensional Hilbert space of $u$,

(13.20) $u^{\prime}=({\operatorname{id}\otimes j(\rho)}^{1/2}\otimes
1)(\operatorname{id}\otimes
u^{c})(\operatorname{id}\otimes{j(\rho)}^{-1/2}\otimes
1)=\operatorname{id}\otimes\overline{u},$

and

(13.21)
$\begin{split}w^{\prime}&=(\operatorname{id}\otimes{j(\rho)}^{1/2}\otimes
1)(v^{c}_{13}V^{c}_{23})(\operatorname{id}\otimes{j(\rho)}^{-1/2}\otimes 1)\\\
&=v^{c}_{13}[(\operatorname{id}\otimes{j(\rho)}^{1/2}\otimes
1)V^{c}_{23}(\operatorname{id}\otimes{j(\rho)}^{1/2}\otimes 1)]\\\
&=v^{c}_{13}V^{c}_{23}=v^{c}\times V^{c}.\end{split}$

Thus the CSR $(\overline{\mathscr{H}_{u}},u^{\prime},w^{\prime})$, and
consequently $\overline{U}$, is indeed parameterized by
$(\overline{u},v^{c},V^{c})$, which proves (c). ∎

Proposition 13.2 motivates the following definition.

######  Definition 13.4.

Let $(u,V,v)$ be a representation parameter associated with some
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, the representation
parameter $(\overline{u},V^{c},v^{c})$ is called the conjugate of $(u,V,v)$.

By Proposition 10.7 and Corollary 9.12, it is clear that the conjugation of an
irreducible representation parameter is irreducible, and
$\overline{[(u,V,v)]}=[(\overline{u},V^{c},v^{c})]$ gives a well-defined
mapping $\overline{(\cdot)}:\mathfrak{D}\rightarrow\mathfrak{D}$. The
following theorem describes how the conjugate representation of irreducible
(unitary) representation of $\mathbb{G}\rtimes\Lambda$ looks like in terms of
the classification given in Theorem 12.1.

######  Theorem 13.5.

Let $[(u,V,v)]\in\mathfrak{D}$,
$x=\Psi([(u,V,v)])\in\operatorname{Irr}(\mathbb{G}\rtimes\Lambda)$, then

(13.22)
$\overline{x}=\Psi(\overline{[(u,V,v)]})=\Psi([(\overline{u},V^{c},v^{c})]).$

###### Proof.

This follows immediately from Proposition 13.3 and the character formula (5.3)
for representations induced from representations of principal subgroups of
$\mathbb{G}\rtimes\Lambda$. ∎

##  14\. The incidence numbers

We now turn our attention to the fusion rules of $\mathbb{G}\rtimes\Lambda$.
We define and study incidence numbers in this section, and use these numbers
to express the fusion rules in § 15.

######  Definition 14.1.

For $i=1,2,3$, let $\Lambda_{i}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$.
Suppose $U_{i}$ is a unitary representation of $\mathbb{G}\rtimes\Lambda_{i}$,
and $r_{i}\in\Lambda$, then the incidence number of $(r_{1},r_{2},r_{3})$
relative to $(U_{1},U_{2},U_{3})$, denoted by
$m_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3})$, is defined by

(14.1) $\begin{split}&\hskip
11.99998ptm_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3})\\\
&=\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}\left((r_{1}\cdot
U_{1})|_{\mathbb{G}\rtimes\Lambda_{0}},(r_{2}\cdot
U_{2})|_{\mathbb{G}\rtimes\Lambda_{0}}\times(r_{3}\cdot
U_{3})|_{\mathbb{G}\rtimes\Lambda_{0}}\right),\end{split}$

where $\Lambda_{0}=\cap_{i=1}^{3}r_{i}\Lambda_{i}r_{i}^{-1}$.

We now aim to express the incidence numbers in terms of characters. Let
$\Theta,\Xi$ be two subgroups of $\Lambda$ with $\Theta\subseteq\Xi$. Recall
that $C(\mathbb{G})=A$. Suppose $F=\sum_{r\in\Xi}a_{r}\otimes\delta_{r}$,
$a_{r}\in A$ is an element of $C(\mathbb{G})\otimes C(\Xi)=A\otimes C(\Xi)$.
We use $F|_{\mathbb{G}\rtimes\Theta}$ to denote the element
$\sum_{r\in\Theta}a_{r}\otimes\delta_{r}$ in $\mathbb{G}\rtimes\Theta$, and
call it the restriction of $F$ to $\mathbb{G}\rtimes\Theta$. A simple
calculation shows that this restriction operation gives a surjective unital
morphism of $C^{\ast}$-algebras from $C(\mathbb{G}\rtimes\Xi)=A\otimes C(\Xi)$
to $C(\mathbb{G}\rtimes\Theta)=A\otimes C(\Theta)$ that also preserves
comultiplication, thus allows us to view $\mathbb{G}\rtimes\Theta$ as a closed
subgroup of $\mathbb{G}\rtimes\Xi$ in the sense of Definition 3.1. Recall that
we also have the extension morphism $E_{\Lambda_{0}}\colon
C(\Lambda_{0})\rightarrow C(\Lambda)$, $\delta_{r_{0}}\mapsto\delta_{r_{0}}$
for every subgroup $\Lambda_{0}$ of $\Lambda$, which simply sends each
function in $C(\Lambda_{0})$ to its unique extension in $C(\Lambda)$ that
vanishes outside $\Lambda_{0}$. Finally, we use $h^{\Lambda_{0}}$ to denote
the Haar state on $\mathbb{G}\rtimes\Lambda_{0}$. For $i=1,2,3$, let
$\Lambda_{i}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Suppose $U_{i}$ is
a unitary representation of $\mathbb{G}\rtimes\Lambda_{i}$, $\chi_{i}$ is the
character of $U_{i}$. Let
$\Lambda_{0}=\cap_{i=1}^{3}r_{i}\Lambda_{i}r_{i}^{-1}$. Then we have the
following formula to calculate the incidence numbers in terms of characters.

(14.2) $\begin{split}\forall r_{1},r_{2},r_{3}\in\Lambda,\quad&\hskip
11.99998ptm_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3})\\\
&=h^{\Lambda_{0}}\left(\overline{(r_{1}\cdot\chi_{1})|_{\mathbb{G}\rtimes\Lambda_{0}}}(r_{2}\cdot\chi_{2})|_{\mathbb{G}\rtimes\Lambda_{0}}(r_{3}\cdot\chi_{3})|_{\mathbb{G}\rtimes\Lambda_{0}}\right).\end{split}$

######  Proposition 14.2.

Using the above notations, the incidence number
$m_{U_{1},U_{2},U_{3}}(s_{1},s_{2},s_{3})$ depends only on the classes
$[U_{1}]$, $[U_{2}]$, $[U_{3}]$ of equivalent unitary representations and the
left cosets $r_{1}\Lambda_{1}$, $r_{2}\Lambda_{2}$, $r_{3}\Lambda_{3}$.

###### Proof.

Note that for any $i=1,2,3$,
$s_{i}\Lambda_{i}s_{i}^{-1}=r_{i}\Lambda_{i}r_{i}^{-1}$ whenever
$r_{i}^{-1}s_{i}\in\Lambda_{i}$. The proposition follows from (14.2) and Lemma
5.3 (b). ∎

By Proposition 14.2, we see immediately that the following definition is well-
defined.

######  Definition 14.3.

For $i=1,2,3$, let $\Lambda_{i}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$.
Suppose $x_{i}$ is a class of equivalent unitary representations of
$\mathbb{G}\rtimes\Lambda_{i}$, and $z_{i}\in\Lambda/\Lambda_{i}$ is a left
coset of $\Lambda_{i}$ in $\Lambda$, then the incidence number of
$(z_{1},z_{2},z_{3})$ relative to $(x_{1},x_{2},x_{3})$, denoted by
$m_{x_{1},x_{2},x_{3}}(z_{1},z_{2},z_{3})$, is defined by

(14.3)
$m_{x_{1},x_{2},x_{3}}(z_{1},z_{2},z_{3})=m_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3})$

where $U_{i}\in x_{i}$, $r_{i}\in z_{i}$ for $i=1,2,3$.

The rest of this section is devoted to the calculation of the incidence number
(14.3) in terms of more basic ingredients when
$x_{i}=\Phi_{\Lambda_{i}}(\mathfrak{p}_{i})$ for some
$\mathfrak{p}_{i}\in\mathfrak{D}_{\Lambda_{i}}$ (see § 12), as this will be
the case we need in the calculation of fusion rules for
$\mathbb{G}\rtimes\Lambda$ in § 15. We begin with a result on the structure of
unitary projective representations of some
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$ that are covariant
with some unitary representation of $\mathbb{G}$.

######  Lemma 14.4.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Let $u_{0}$
be an irreducible unitary representation of $\mathbb{G}$,
$[u_{0}]\in\operatorname{Irr}(\mathbb{G})$ the class of $u_{0}$, such that
$\Lambda_{0}\subseteq\Lambda_{[u_{0}]}$. Suppose $u$ is a unitary
representation of $\mathbb{G}$,
$V\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathscr{H}_{u})$ is a unitary
projective representation covariant with $u$, $p$ is the minimal central
projection in $\operatorname{End}_{\mathbb{G}}(u)$ corresponding to the
maximal pure subrepresentation of $u$ supported by
$[u_{0}]\in\operatorname{Irr}(\mathbb{G})$. Let $q=1-p$, then $V$ is
diagonalizable along $p$ in the sense that

(14.4) $\begin{split}&(p\otimes 1)V=V(p\otimes 1),\quad(q\otimes
1)V=V(q\otimes 1),\\\ &\text{and }\;(p\otimes 1)V(q\otimes 1)=(q\otimes
1)V(p\otimes 1)=0.\end{split}$

###### Proof.

Since $V$ and $u$ are covariant, we have

(14.5) $\forall r_{0}\in\Lambda_{0},\;\bigl{(}V(r_{0})\otimes
1\bigr{)}(r_{0}\cdot u)=u\bigl{(}V(r_{0})\otimes 1\bigr{)}.$

Note that
$p\in\operatorname{End}_{\mathbb{G}}(u)=\operatorname{End}_{\mathbb{G}}(r_{0}\cdot
u)$ (see (8.1)), then for every $r_{0}\in\Lambda_{0}$, it follows that

(14.6) $\begin{split}&\hskip 11.99998pt\bigl{(}[pV(r_{0})q]\otimes
1\bigr{)}[(q\otimes 1)(r_{0}\cdot u)]=(p\otimes 1)\bigl{(}V(r_{0})\otimes
1\bigr{)}(q\otimes 1)(r_{0}\cdot u)\\\ &=(p\otimes 1)\bigl{(}V(r_{0})\otimes
1\bigr{)}(r_{0}\cdot u)(q\otimes 1)=(p\otimes 1)u\bigl{(}V(r_{0})\otimes
1\bigr{)}(q\otimes 1)\\\ &=[(p\otimes 1)u](p\otimes 1)\bigl{(}V(r_{0})\otimes
1\bigr{)}(q\otimes 1)=[(p\otimes 1)u]\bigl{(}[pV(r_{0})q]\otimes
1\bigr{)}.\end{split}$

Let $u_{p}$ (resp. $u_{q}$) be the subrepresentation of $u$ corresponding to
$p$ (resp. $q$), then $r_{0}^{-1}\cdot u_{p}$ is equivalent to $u_{p}$ for all
$r_{0}\in\Lambda_{0}$ since $\Lambda_{0}\subseteq\Lambda_{[u]}$, and

(14.7) $\operatorname{Mor}_{\mathbb{G}}(r_{0}\cdot
u_{q},u_{p})=\operatorname{Mor}_{\mathbb{G}}(u_{q},r_{0}^{-1}\cdot
u_{p})=\operatorname{Mor}_{\mathbb{G}}(u_{q},u_{p})=0.$

By (14.6), the operator $pV(r_{0})q$, when viewed as an operator from
$p(\mathscr{H}_{u})$ to $q(\mathscr{H}_{u})$, intertwines $r_{0}\cdot u_{q}$
and $u_{p}$. Thus by (14.7),

(14.8) $\forall r_{0}\in\Lambda_{0},\quad pV(r_{0})q=0.$

Similarly,

(14.9) $\forall r_{0}\in\Lambda_{0},\quad qV(r_{0})p=0.$

Hence

(14.10) $pV(r_{0})=pV(r_{0})(p+q)=pV(r_{0})p=(p+q)V(r_{0})p=V(r_{0})p,$

and similarly,

(14.11) $qV(r_{0})=qV(r_{0})(p+q)=qV(r_{0})q=(p+q)V(r_{0})q=V(r_{0})q.$

Now (14.4) follows from equations (14.8), (14.9), (14.10), and (14.11). ∎

We also need to generalize the notion of representation parameter a little, as
the natural candidate of the “tensor product” of two representation parameters
need not be a representation parameter, but it still possesses the same
covariant property.

######  Definition 14.5.

Let $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, we call a
triple $(u,V,v)$ a generalized representation parameter ((GRP) for short)
associated with $\Lambda_{0}$, if the following hold:

  1. (a)

$V$ is a unitary projective representation of $\Lambda_{0}$ on
$\mathscr{H}_{u}$, such that

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

  2. (b)

$v$ is a unitary projective representation (on some other finite dimensional
Hilbert space $\mathscr{H}_{v}$) of $\Lambda_{0}$, such that the cocycles of
$v$ and $V$ are opposite to each other.

######  Proposition 14.6.

If $(u,V,v)$ is a GRP associated with some
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, then
$(\mathscr{H}_{v}\otimes\mathscr{H}_{u},\operatorname{id}\otimes u,v\times
V)\in\mathcal{CSR}_{\Lambda_{0}}$.

###### Proof.

The proof of Proposition 9.6 applies almost verbatim here. ∎

######  Definition 14.7.

If $(u,V,v)$ is a GRP associated with
$\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, then the CSR
$\mathbf{S}:=(\mathscr{H}_{v}\otimes\mathscr{H}_{u},\operatorname{id}\otimes
u,v\times V)$ associated with $\Lambda_{0}$ and the unitary representation
$\mathscr{R}_{\Lambda_{0}}(\mathbf{S})$ of $\mathbb{G}\rtimes\Lambda_{0}$ are
said to be parameterized by $(u,V,v)$.

We now describe a reduction process for generalized representation parameters,
which leads to our desired calculation of the incidence numbers using more
basic ingredients—the dimension of a certain intertwiner space of two
projective representations of some generalized isotropy subgroup of $\Lambda$.

######  Proposition 14.8.

Fix a $\Lambda_{0}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$. Let $(u,V,v)$
be a GRP associated with $\Lambda_{0}$, $x\in\operatorname{Irr}(\mathbb{G})$
such that $\Lambda_{0}\subseteq\Lambda_{x}$, and $u_{0}\in x$. Suppose $p$ is
the minimal central projection of $\operatorname{End}_{\mathbb{G}}(u)$
corresponding to the maximal pure subrepresentation of $u$ supported by $x$.
The following holds:

  1. (a)

$(u_{p},V_{p},v)$ is a GRP, where $u_{p}$ (resp. $V_{p}$) is the
subrepresentation of $u$ (resp. $V$) on $p(\mathscr{H}_{u})$;

  2. (b)

let $n\in\mathbb{N}$ be the multiplicity of $x$ in $u$, $V_{0}$ a covariant
projective $\Lambda_{0}$-representation of $u_{0}$, then _up to unitary
equivalence_ , there exists a unique unitary projective representation $V_{1}$
of $\Lambda_{0}$ on $\mathbb{C}^{n}$, such that $V_{p}$ is unitarily
equivalent to $V_{1}\times V_{0}$;

  3. (c)

$(u_{0},V_{0},v\times V_{1})$ is representation parameter, and the CSR
$(\mathscr{H}_{v}\otimes p(\mathscr{H}_{u}),\operatorname{id}\otimes
u_{p},v\times V_{p})$ parameterized by $(u_{p},V_{p},v)$ is isomorphic to the
CSR $\mathcal{S}(u_{0},V_{0},v\times V_{1})$ parameterized by
$(u_{0},V_{0},v\times V_{1})$ in the category $\mathcal{CSR}_{\Lambda_{0}}$.
In particular, the representation parameter $(u_{0},V_{0},v\times V_{1})$ and
the GRP $(u_{p},V_{p},v\times V_{p})$ parameterize equivalent unitary
representations of $\mathbb{G}\rtimes\Lambda_{0}$.

###### Proof.

By Lemma 14.4, $u_{p}$ and $V_{p}$ are covariant. Since $V_{p}$ is a
subrepresentation of $V$, it has the same cocycle as $V$, hence $V_{p}$ and
$v$ have opposing cocycles. This proves (a).

The proof of (b) parallels that of Proposition 9.5. Since $u_{p}$ is
equivalent to a direct sum of $n$ copies of $u_{0}$, thus there exists a
unitary operator
$U\in\operatorname{Mor}_{\mathbb{G}}(\operatorname{id}_{\mathbb{C}^{n}}\otimes
u_{0},u_{p})$. Replace $(u_{p},V_{p},v)$ with
$(U^{\ast}u_{p}U,U^{\ast}V_{p}U,v)$ if necessary, we may assume
$u_{p}=\mathbb{C}^{n}\otimes u_{0}$. Repeat the proof of Proposition 9.5 with
the small modification of replacing the unitary representation $w$ there with
the unitary projective representation $V_{p}$, we see that there exists a
unique unitary projective representation
$V_{1}\colon\Lambda_{0}\rightarrow\mathcal{U}(\mathbb{C}^{n})$, such that
$V_{p}=V_{1}\times V_{0}$. This proves (b).

By (b) and its proof, we may suppose
$u_{p}=\operatorname{id}_{\mathbb{C}^{n}}\otimes u_{0}$. Note that the CSR
parameterized by $(u_{p},V_{p},v)$ is exactly
$(\operatorname{id}_{\mathscr{H}_{v}}\otimes\operatorname{id}_{\mathbb{C}^{n}}\otimes
u_{0},v\times V_{p})$, which coincides exactly with the CSR parameterized by
$(\operatorname{id}_{\mathbb{C}^{n}\otimes\mathscr{H}_{v}}\otimes
u_{0},V_{0},v\times V_{1})$ since $v\times V_{p}=v\times V_{1}\times V_{0}$.
This proves (c). ∎

######  Definition 14.9.

Using the notation of Proposition 14.8, the representation parameter
$(u_{0},V_{0},v\times V_{1})$ is called a reduction of the GRP $(u,V,v)$ along
$(u_{0},V_{0})$.

######  Remark 14.10.

Since $V_{1}$ is determined up to unitary equivalence, so is the reduction
$(u_{0},V_{0},v\times V_{1})$.

The following result describes the incidence numbers
$m_{[U_{1}],[U_{2}],[{U_{3}}]}(z_{1},z_{2},z_{3})$ in terms of the dimension
of the intertwiner space of some projective representations of $\Lambda_{0}$.

######  Proposition 14.11.

Suppose we are given the following data for each $i=1,2,3$:

  * •

a $\Lambda_{i}\in\operatorname{\mathcal{G}_{iso}}(\Lambda)$, a left coset
$z_{i}$ in $\Lambda/\Lambda_{i}$ and a $r_{i}\in z_{i}$;

  * •

a representation parameter $(u_{i},V_{i},v_{i})$ associated with
$\Lambda_{i}$;

  * •

the unitary representation $U_{i}$ of $\mathbb{G}\rtimes\Lambda_{i}$
parameterized by $(u_{i},V_{i},v_{i})$.

Let
$\Lambda_{0}=\cap_{i=1}^{3}r_{i}\Lambda_{i}r_{i}^{-1}=\cap_{i=1}^{3}z_{i}\Lambda_{i}z_{i}^{-1}$.
Suppose

$\bigl{(}r_{1}\cdot u_{1},(r_{1}\cdot V_{1})|_{\Lambda_{0}},(r_{2}\cdot
v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot v_{3})|_{\Lambda_{0}}\times V\bigr{)}$

is the reduction of the GRP

$\bigl{(}(r_{2}\cdot u_{2})\times(r_{3}\cdot u_{3}),(r_{2}\cdot
V_{2})|_{\Lambda_{0}}\times(r_{3}\cdot V_{3})|_{\Lambda_{0}},(r_{2}\cdot
v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot v_{3})|_{\Lambda_{0}}\bigr{)}$

along $\bigl{(}r_{1}\cdot u_{1},(r_{1}\cdot V_{1})|_{\Lambda_{0}}\bigr{)}$.
Then the unitary projective representations $(r_{1}\cdot
v_{1})|_{\Lambda_{0}}$ and

$(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}\times V$

of $\Lambda_{0}$ have the same cocycle, and

(14.13) $\begin{split}&\hskip
11.99998ptm_{[U_{1}],[U_{2}],[U_{3}]}(z_{1},z_{2},z_{3})\\\
&=\dim\operatorname{Mor}_{\Lambda_{0}}\bigl{(}(r_{1}\cdot
v_{1})|_{\Lambda_{0}},(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}\times V\bigr{)}.\end{split}$

###### Proof.

It is easy to check that $\bigl{(}(r_{2}\cdot u_{2})\times(r_{3}\cdot
u_{3}),(r_{2}\cdot V_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
V_{3})|_{\Lambda_{0}},(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}\bigr{)}$ is indeed a generalized representation
parameter. Take the minimal central projection $p$ of
$\operatorname{End}_{\mathbb{G}}\bigl{(}(r_{2}\cdot u_{2})\times(r_{3}\cdot
u_{3})\bigr{)}$ corresponding to the maximal pure subrepresentation $u_{p}$ of
$(r_{2}\cdot u_{2})\times(r_{3}\cdot u_{3})$ that is supported by $[r_{1}\cdot
u_{1}]\in\operatorname{Irr}(\mathbb{G})$. Suppose $q=1-p$. By Lemma 14.4, $q$
also corresponds to a subrepresentation $u_{q}$ of $(r_{2}\cdot
u_{2})\times(r_{3}\cdot u_{3})$ on
$q(\mathscr{H}_{u_{2}}\otimes\mathscr{H}_{u_{3}})$. Similarly, let $V_{p}$
(resp. $V_{q}$) be the subrepresentation of the unitary projective
representation $(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}$ on $p(\mathscr{H}_{u_{2}}\otimes\mathscr{H}_{u_{3}})$
(resp. $q(\mathscr{H}_{u_{2}}\otimes\mathscr{H}_{3})$). Let $U_{p}$ (resp.
$U_{q}$) be the representation of $\mathbb{G}\rtimes\Lambda_{0}$ parameterized
by the GRP $(u_{p},V_{p},(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}})$ (resp. $(u_{q},V_{q},(r_{2}\cdot
v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot v_{3})|_{\Lambda_{0}})$). By
construction, the unitary representation $U$ of $\mathbb{G}\rtimes\Lambda_{0}$
parameterized by $\bigl{(}(r_{2}\cdot u_{2})\times(r_{3}\cdot
u_{3}),(r_{2}\cdot V_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
V_{3})|_{\Lambda_{0}},(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}\bigr{)}$ is the direct sum of $U_{p}$ and $U_{q}$. By
definition,

(14.14)
$\begin{split}m_{[U_{1}],[U_{2}],[U_{3}]}(z_{1},z_{2},z_{3})&=\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U)\\\
&=\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{p})+\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{q}).\end{split}$

From our construction, the matrix coefficients of $u_{p}$ and $u_{q}$ are
orthogonal with respect to the Haar state $h$ of $\mathbb{G}$. Thus the proof
of Proposition 9.11 (a) applies almost verbatim, and shows that

(14.15)
$\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{q})=0.$

On the other hand, the cocycles of both $(r_{1}\cdot v_{1})|_{\Lambda_{0}}$
and $(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}\times V$ are both opposite to that of $(r_{1}\cdot
V_{1})|_{\Lambda_{0}}$ by the reduction process described above, hence these
cocycles coincide. By Proposition 9.11 (b) and Proposition 14.8 (c), we have

(14.16)
$\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda_{0}}(U_{1},U_{p})=\dim\operatorname{Mor}_{\Lambda_{0}}\bigl{(}(r_{1}\cdot
v_{1})|_{\Lambda_{0}},(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}\times V\bigr{)}.$

Now (14.13) follows from (14.15) and (14.16). ∎

##  15\. Fusion rules

We now calculate the fusion rules of $\mathbb{G}\rtimes\Lambda$. From the
classification theorem (Theorem 12.1), up to unitary equivalence, all unitary
irreducible representations of $\mathbb{G}\rtimes\Lambda$ are distinguished.
Thus the task falls to the calculation of

(15.1)
$\dim\operatorname{Mor}_{\mathbb{G}}\bigl{(}\operatorname{Ind}(U_{1}),\operatorname{Ind}(U_{2})\times\operatorname{Ind}(U_{3})\bigr{)},$

where, for $i=1,2,3$, $U_{i}$ is the irreducible unitary representation of
$\mathbb{G}\rtimes\Lambda_{i}$ parameterized (see Definition 10.1 and
Definition 10.4) by some _distinguished_ representation parameter
$(u_{i},V_{i},v_{i})$ associated with $\Lambda_{i}$ (recall that
$\Lambda_{i}=\Lambda_{[u_{i}]}$ since $(u_{i},V_{i},v_{i})$ is distinguished).
Let $h$ be the Haar state on $C(\mathbb{G})=A$. For any subgroup $\Lambda_{0}$
of $\Lambda$, we use $h^{\Lambda_{0}}$ to denote the Haar state on
$C(\mathbb{G}\rtimes\Lambda_{0})=A\otimes C(\Lambda_{0})$, and
$E_{\Lambda_{0}}\colon C(\Lambda_{0})\rightarrow C(\Lambda)$ denotes the
linear embedding such that $\delta_{r_{0}}\in
C(\Lambda_{0})\mapsto\delta_{r_{0}}\in C(\Lambda)$(the extension of functions
in $C(\Lambda_{0})$ to functions in $C(\Lambda)$ that vanishes outside
$\Lambda_{0}$). In particular, $h^{\Lambda}$ is the Haar state on
$C(\mathbb{G}\rtimes\Lambda)=A\otimes C(\Lambda)$. For $i=1,2,3$, let
$\chi_{i}=(\operatorname{Tr}\otimes\operatorname{id})(U_{i})\in A\otimes
C(\Lambda_{i})$ be the character of $U_{i}$, and $r\cdot\chi_{i}$ is defined
to be the character of the representation $r\cdot U_{i}$ of $\mathbb{G}\rtimes
r\Lambda_{i}r^{-1}$.

Using these notations, by Proposition 5.1, we have the following formula for
the character of $\operatorname{Ind}(U_{i})$,

(15.2) $\forall
i=1,2,3,\quad\chi(\operatorname{Ind}(U_{i}))=\left\lvert\Lambda_{i}\right\rvert^{-1}\sum_{r_{i}\in\Lambda}(\operatorname{id}\otimes
E_{r_{i}\Lambda_{i}r_{i}^{-1}})(r_{i}\cdot\chi_{i}).$

Thus

(15.3) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}\bigl{(}\operatorname{Ind}(U_{1}),\operatorname{Ind}(U_{2})\times\operatorname{Ind}(U_{3})\bigr{)}\\\
&=h^{\Lambda}\bigl{(}\overline{\chi(\operatorname{Ind}(U_{1}))}[\chi(\operatorname{Ind}(U_{2}))][\chi(\operatorname{Ind}(U_{3}))]\bigr{)}\\\
&=\sum_{r_{1},r_{2},r_{3}}{h^{\Lambda}\bigl{(}\chi(r_{1},r_{2},r_{3})\bigr{)}\over\left\lvert\Lambda_{1}\right\rvert\cdot\left\lvert\Lambda_{2}\right\rvert\cdot\left\lvert\Lambda_{3}\right\rvert},\end{split}$

where

(15.4) $\begin{split}&\hskip 11.99998pt\chi(r_{1},r_{2},r_{3})\\\
&=\overline{(\operatorname{id}\otimes
E_{r_{1}\Lambda_{1}r_{1}^{-1}})(r_{1}\cdot\chi_{1})}[(\operatorname{id}\otimes
E_{r_{2}\Lambda_{2}r_{2}^{-1}})(r_{2}\cdot\chi_{2})][(\operatorname{id}\otimes
E_{r_{3}\Lambda_{3}r_{3}^{-1}})(r_{3}\cdot\chi_{3})].\end{split}$

If $\Theta,\Xi$ are subgroups of $\Lambda$ with $\Theta\subseteq\Xi$, and
$\sum_{r\in\Xi}a_{r}\otimes\delta_{r}$ is an arbitrary element of $A\otimes
C(\Xi)$ with all $a_{r}\in A$, we call the element
$\sum_{r\in\Theta}a_{r}\otimes\delta_{r}$ of $A\otimes C(\Theta)$ the
restriction of $\sum_{r\in\Xi}a_{r}\otimes\delta_{r}$ and denote it by
${\left(\sum_{r\in\Xi}a_{r}\otimes\delta_{r}\right)}|_{\mathbb{G}\rtimes\Theta}$.
Recall that

(15.5) $h^{\Lambda_{0}}=[\Lambda\colon\Lambda_{0}]\cdot
h^{\Lambda}\circ(\operatorname{id}\otimes E_{\Lambda_{0}})$

for any subgroup $\Lambda_{0}$ of $\Lambda$, posing

(15.6)
$\Lambda(r_{1},r_{2},r_{3})=\bigcap_{i=1}^{3}r_{i}\Lambda_{i}r_{i}^{-1},$

we have

(15.7) $\begin{split}&\hskip
11.99998pth^{\Lambda}\bigl{(}\chi(r_{1},r_{2},r_{3})\bigr{)}=h^{\Lambda}\left(\chi(r_{1},r_{2},r_{3})|_{\mathbb{G}\rtimes\Lambda(r_{1},r_{2},r_{3})}\right)\\\
&={h^{\Lambda(r_{1},r_{2},r_{3})}\left(\overline{(r_{1}\cdot\chi_{1})|_{\mathbb{G}\rtimes\Lambda(r_{1},r_{2},r_{3})}}(r_{2}\cdot\chi_{2})|_{\mathbb{G}\rtimes\Lambda(r_{1},r_{2},r_{3})}(r_{3}\cdot\chi_{3})|_{\mathbb{G}\rtimes\Lambda(r_{1},r_{2},r_{3})}\right)\over[\Lambda\colon\Lambda(r_{1},r_{2},r_{3})]}\\\
&={[\Lambda\colon\Lambda(r_{1},r_{2},r_{3})]}^{-1}m_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3}),\end{split}$

where $m_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3})$ is the incidence number of
$(r_{1},r_{2},r_{3})$ relative to $(U_{1},U_{2},U_{3})$.

By (15.3) and (15.7), we have

(15.8) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}\bigl{(}\operatorname{Ind}(U_{1}),\operatorname{Ind}(U_{2})\times\operatorname{Ind}(U_{3})\bigr{)}\\\
&=\sum_{r_{1},r_{2},r_{3}\in\Lambda}\frac{m_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3})}{\left\lvert\Lambda_{1}\right\rvert\cdot\left\lvert\Lambda_{2}\right\rvert\cdot\left\lvert\Lambda_{3}\right\rvert\cdot[\Lambda\colon\Lambda(r_{1},r_{2},r_{3})]}.\end{split}$

As we’ve seen in Definition 14.3 and the discussion before it, we have

(15.9) $\begin{split}&(\forall i=1,2,3,r_{i}\in
z_{i}\in\Lambda/\Lambda_{i})\\\ &\implies
m_{[U_{1}],[U_{2}],[U_{3}]}(z_{1},z_{2},z_{3})=m_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3}),\end{split}$

where $\Lambda(z_{1},z_{2},z_{3}):=\cap_{i=1}^{3}r_{i}\Lambda_{i}r_{i}^{-1}$
does not depend on the choices for $r_{i}\in z_{i}$, $i=1,2,3$. Thus (15.8)
can be written more succinctly as

(15.10) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}\bigl{(}\operatorname{Ind}(U_{1}),\operatorname{Ind}(U_{2})\times\operatorname{Ind}(U_{3})\bigr{)}\\\
&=\sum_{z_{1}\in\Lambda/\Lambda_{1}}\sum_{z_{2}\in\Lambda/\Lambda_{2}}\sum_{z_{3}\in\Lambda/\Lambda_{3}}\frac{m_{[U_{1}],[U_{2}],[U_{3}]}(z_{1},z_{2},z_{3})}{[\Lambda\colon\Lambda(z_{1},z_{2},z_{3})]}.\end{split}$

We formalize the above calculation as the following theorem, which describes
the fusion rules of $\mathbb{G}\rtimes\Lambda$ in terms of the more basic
ingredients of incidence numbers, which in turn is completely determined by
the representation theory of $\mathbb{G}$, the action of $\Lambda$ on
$\operatorname{Irr}(\mathbb{G})$, and various unitary projective
representations of some naturally appeared subgroups in
$\operatorname{\mathcal{G}_{iso}}(\Lambda)$.

######  Theorem 15.1.

The fusion rules for $\mathbb{G}\rtimes\Lambda$ is given as the following. For
$i=1,2,3$, let $W_{i}$ be an irreducible representation of
$\mathbb{G}\rtimes\Lambda$. Suppose $U_{i}$ is the distinguished
representation parameterized by some distinguished representation parameter
$(u_{i},V_{i},v_{i})$ associated with some isotropy subgroup $\Lambda_{i}$ of
$\Lambda$, such that $W_{i}$ is equivalent to $\operatorname{Ind}(U_{i})$,
then

(15.11) $\begin{split}&\hskip
11.99998pt\dim\operatorname{Mor}_{\mathbb{G}\rtimes\Lambda}(W_{1},W_{2}\times
W_{3})\\\
&=\sum_{z_{1}\in\Lambda/\Lambda_{1}}\sum_{z_{2}\in\Lambda/\Lambda_{2}}\sum_{z_{3}\in\Lambda/\Lambda_{3}}\frac{m_{[U_{1}],[U_{2}],[U_{3}]}(z_{1},z_{2},z_{3})}{[\Lambda\colon\Lambda(z_{1},z_{2},z_{3})]}.\end{split}$

Here the incidence numbers

(15.12) $\begin{split}&\hskip
11.99998ptm_{[U_{1}],[U_{2}],[U_{3}]}(z_{1},z_{2},z_{3})=m_{U_{1},U_{2},U_{3}}(r_{1},r_{2},r_{3})\\\
&=\dim\operatorname{Mor}_{\Lambda(z_{1},z_{2},z_{3})}\bigl{(}(r_{1}\cdot
v_{1})|_{\Lambda_{0}},(r_{2}\cdot v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot
v_{3})|_{\Lambda_{0}}\times V\bigr{)},\end{split}$

where $r_{i}\in z_{i}$ for $i=1,2,3$, and the unitary projective
representation $V$ of $\Lambda(z_{1},z_{2},z_{3})$ is taken from the reduction

$\bigl{(}r_{1}\cdot u_{1},(r_{1}\cdot V_{1})|_{\Lambda_{0}},(r_{2}\cdot
v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot v_{3})|_{\Lambda_{0}}\times V\bigr{)}$

of the generalized representation parameter

$\bigl{(}(r_{2}\cdot u_{2})\times(r_{3}\cdot u_{3}),(r_{2}\cdot
V_{2})|_{\Lambda_{0}}\times(r_{3}\cdot V_{3})|_{\Lambda_{0}},(r_{2}\cdot
v_{2})|_{\Lambda_{0}}\times(r_{3}\cdot v_{3})|_{\Lambda_{0}}\bigr{)}$

along $\bigl{(}r_{1}\cdot u_{1},(r_{1}\cdot V_{1})|_{\Lambda_{0}}\bigr{)}$.

###### Proof.

The above calculation proves (15.11), and (15.12) follows from Proposition
14.11. ∎

## References