# Weights for $\ell$-local compact groups

Jason Semeraro

###### Abstract.

In this short note, we initiate the study of $\mathcal{F}$-weights for an
$\ell$-local compact group $\mathcal{F}$ over a discrete $\ell$-toral group
$S$ with discrete torus $T$. Motivated by Alperin’s Weight Conjecture for
simple groups of Lie-type, we conjecture that when $\mathcal{F}$ is
(algebraically) connected, that is every element of $S$ is $\mathcal{F}$-fused
into $T$, the number of weights of $\mathcal{F}$ is equal to the number of
ordinary irreducible characters of its Weyl group. By combining the structure
theory of $\mathcal{F}$ with the theory of blocks with cyclic defect group, we
are able to give a proof of this conjecture in the case when $\mathcal{F}$ is
simple and $|S:T|=\ell$. We also propose and give evidence for an analogue of
the Alperin–McKay conjecture in this setting.

###### Key words and phrases:

fusion systems, weight conjectures, blocks, $p$-local compact groups

###### 2010 Mathematics Subject Classification:

20C33, 20D20, 20D06, 55R35

##  1\. Introduction

An $\ell$-compact group is, broadly speaking, the $\ell$-local homotopy
theoretic analogue of a compact Lie group with a ‘Weyl group’ $W$ which is a
$\mathbb{Z}_{\ell}$-reflection group. In [13], the authors introduce and prove
a version of Alperin’s Weight Conjecture for fusion systems associated to
homotopy fixed point spaces of connected $\ell$-compact groups under the
action of unstable Adams operations. For such a fusion system $\mathcal{F}$
associated to an $\ell$-compact group with Weyl group $W$, this conjecture
asserts that, if $\ell$ is a good prime (see [13, Definition 2.4]), then
$\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}(W)|$ where $\mathbf{w}(-)$
counts the number of weights (defect zero characters of $\mathcal{F}$-centric
radical subgroups) of $\mathcal{F}$ up to conjugacy. One consequence of this
result is that the number of weights appears to be an invariant of the
underlying $\ell$-compact group, as opposed to the space of fixed points. In
an attempt to understand this, we introduce the study of weights for
$\ell$-compact groups by appealing to the more general theory of $\ell$-local
compact groups [3]. Here, we view an $\ell$-local compact group as a fusion
system $\mathcal{F}$ on an $\ell$-group which contains a finite index infinite
torus whose $\mathcal{F}$-automorphism group we define to be the Weyl group of
$\mathcal{F}$ (see Section 2). Since $\mathcal{F}$ has finitely many classes
of $\mathcal{F}$-centric radical subgroups, each with a finite
$\mathcal{F}$-outer automorphism group (see Proposition 2.2), we obtain an
integer invariant $\mathbf{w}(\mathcal{F})$ which, by analogy with the finite
case, we call the ‘number of weights of $\mathcal{F}$’ (see Definition 2.3).

An $\ell$-local compact group $\mathcal{F}$ is simple if it possesses no
normal subsystems and connected if every element of the underlying
$\ell$-group is $\mathcal{F}$-conjugate to an element of the maximal torus
(see Section 2 for precise definitions). Based on [13, Theorem 1] for finite
fusion systems, and observations made in the present paper, we make the
following conjecture which may be regarded as a version of Alperin’s Weight
Conjecture for connected $\ell$-local compact groups.

######  Conjecture 1.1 (Alperin’s Weight Conjecture).

Let $\mathcal{F}$ be a connected $\ell$-local compact group with Weyl group
$W$. Then $\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}(W)|$.

Suppose $\mathcal{F}$ is as in Conjecture 1.1 and $v_{\ell}(|W|)=0$ where
$v_{\ell}(-)$ is the $\ell$-adic valuation. Then there is a unique
$\mathcal{F}$-centric radical subgroup $T$ in $\mathcal{F}$ (the maximal
torus) and $W={{\operatorname{Out}}}_{\mathcal{F}}(T)$ is an
$\ell^{\prime}$-group, so $\mathcal{F}$ is connected and Conjecture 1.1
follows easily. Our main result considers the case $v_{\ell}(|W|)=1$.

######  Theorem 1.2.

Conjecture 1.1 holds if $\mathcal{F}$ is simple and $v_{\ell}(|W|)=1$.

Markus Linckelmann has asked for conditions on a fusion system $\mathcal{G}$
which imply the existence of a subgroup $T$ with
$\mathbf{w}(\mathcal{G})=|{{\operatorname{Irr}}}({{\operatorname{Out}}}_{\mathcal{G}}(T))|$
so Conjecture 1.1 would provide a partial answer to this. We hope to provide
further evidence in a future paper with Radha Kessar which considers the case
when $\mathcal{F}$ is induced by a simply connected $\ell$-compact group and
$\ell$ is good.

Next, we formulate a version of the Alperin–McKay conjecture for connected
$\ell$-local compact groups. Conjecture 1.3 below is best understood as a
generalisation of [13, Conjecture 6.12] for fusion systems associated to
homotopy point spaces of connected $\ell$-compact groups with Weyl group $W$
where $d=v_{\ell}(|S|)$, $\phi=1$, $\ell$ is a good prime and $q\equiv
1\pmod{\ell}$.

######  Conjecture 1.3 (Alperin–McKay).

Let $\mathcal{F}$ be a connected $\ell$-local compact group with Weyl group
$W$ and set

$\mathcal{M}(\mathcal{F})=\\{(\psi,\chi)\mid\psi\in{{\operatorname{Irr}}}(S^{ab})/{{\operatorname{Out}}}_{\mathcal{F}}(S),\chi\in{{\operatorname{Irr}}}(C_{{{\operatorname{Out}}}_{\mathcal{F}}(S)}(\psi)\\}.$

Then there is an ${{\operatorname{Aut}}}(\mathcal{F})$-equivariant bijection

$\mathcal{M}(\mathcal{F})\longleftrightarrow\\{(s,\chi)\mid s\in
Z(S)/\mathcal{F},\chi\in{{\operatorname{Irr}}}_{0}(W(s))\\}$

where for each $s\in S$, $W(s)$ denotes the Weyl group of
$C_{\mathcal{F}}(s)$.

Here ${{\operatorname{Aut}}}(\mathcal{F})\leq{{\operatorname{Aut}}}(S)$
denotes the group of $\mathcal{F}$-fusion preserving automorphisms of $S$, so
${{\operatorname{Aut}}}(\mathcal{F})$ contains
${{\operatorname{Aut}}}_{\mathcal{F}}(S)$ and hence acts on
$\mathcal{M}(\mathcal{F})$. Moreover, since
${{\operatorname{Aut}}}(\mathcal{F})$ permutes the set
$\\{C_{\mathcal{F}}(s)\mid s\in Z(S)/\mathcal{F}\\}$ of
$\mathcal{F}$-centralisers of $\mathcal{F}$-classes of $Z(S)$, it also
permutes their Weyl groups and hence acts on the right hand side. The
${{\operatorname{Aut}}}(\mathcal{F})$-equivariance requirement in Conjecture
1.3 is partially inspired by the inductive McKay condition  for simple groups
due to Isaacs–Malle–Navarro [10]; indeed it may be that the natural map
${{\operatorname{Out}}}(G)\rightarrow{{\operatorname{Aut}}}(\mathcal{F})/{{\operatorname{Aut}}}_{\mathcal{F}}(S)$
(see [1, Section 4]) can be used to exhibit a precise connection with that
condition when $\mathcal{F}$ is realisable by a simple group $G$, but we do
not attempt that here. Our second main result supplies some evidence for
Conjecture 1.3 in the case $v_{\ell}(|W|)=1$.

######  Theorem 1.4.

Let $\mathcal{F}$ be a simple $\ell$-local compact group with Weyl group $W$
and assume $v_{\ell}(|W|)=1$. Then
$\mathcal{F}=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}\mathcal{F}_{n}$ where for each $n\geq 1$, $\mathcal{F}_{n}$
is a finite fusion system on $S_{n}$ and there is a bijection

$\mathcal{M}(\mathcal{F}_{n})\longleftrightarrow\\{(s,\chi)\mid s\in
Z(S_{n})/\mathcal{F}_{n},\chi\in{{\operatorname{Irr}}}_{0}(W(s))\\}.$

Several results in the literature suggest that Theorem 1.4 would follow from
Conjecture 1.3. In [11], Junod, Levi and Libman prove that for any
$\ell$-local compact group $\mathcal{F}$ with torus $T$ and any
$\zeta\in{\mathbb{Z}}_{\ell}^{\times}$ with $v_{\ell}(\zeta-1)>\\!\\!>0$,
${{\operatorname{Aut}}}(\mathcal{F})$ contains an unstable Adams operation
$\Psi=\Psi(\zeta)$ with the property that $\Psi|_{T}(g)=g^{\zeta}$ for each
$g\in T$. Given such an operation $\Psi$, computations of González [9] seem to
indicate that the homotopy fixed points of powers of $\Psi$ stratify
$\mathcal{F}$ as an indirect limit of finite fusion systems
$\\{\mathcal{F}_{n}\mid n\geq 1\\}$ where $\mathcal{F}_{n}$ is the fusion
subsystem of $\mathcal{F}$ fixed by $\Psi^{n}$ (so
$\mathcal{F}=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}\mathcal{F}_{n}$). Thus the
${{\operatorname{Aut}}}(\mathcal{F})$-equivariance in Conjecture 1.3 should
imply the existence of bijections between sets fixed by
$\Psi^{n}\in{{\operatorname{Aut}}}(\mathcal{F})$ as in Theorem 1.4. Note that
when $v_{\ell}(|W|)=0$, $S_{n}$ is abelian, and the Glauberman correspondence
implies there is a $W$-equivariant bijection between $S_{n}$ and
${{\operatorname{Irr}}}(S_{n})$.

We prove Theorems 1.2 and 1.4 in Section 5. The proofs only require the
structure of blocks with cyclic defect group and particular information
concerning the $\mathcal{F}$-automorphisms of $\mathcal{F}$-centric radical
subgroups mostly established in [14] (see Theorems 4.1 and 4.5). We observe
that $\mathcal{F}$ is connected, and this is crucial to ensuring that there
are a sufficiently small number of classes of $\mathcal{F}$-centric radical
subgroups.

We end by remarking that we expect Conjecture 1.3 to be a special case of a
conjecture which generalises [13, Conjecture 6.12] to to the setting of
$\ell$-local compact groups for all values of $d$. This would involve the
natural action of ${{\operatorname{Aut}}}(\mathcal{F})$ on normal chains of
$\ell$-subgroups in $\mathcal{F}$-automizers of $\mathcal{F}$-centric radical
subgroups, but we leave the formulation of a precise statement to future work.

###  1.1. Acknowledgements

The author(s) would like to thank the Isaac Newton Institute for Mathematical
Sciences, Cambridge, for support and hospitality during the programme “Groups,
representations and applications: new perspectives” (supported by EPSRC grant
EP/R014604/1.#34) where work on this paper was undertaken. Thanks in
particular are extended to Gunter Malle, Radha Kessar and Bob Oliver for
helpful discussions. The author also gratefully acknowledges personal support
from the Heilbronn Institute for Mathematical Research and the EPSRC (grant
EP/W028794/1).

##  2\. $\ell$-local compact groups and weights

For the rest of the paper $\ell$ is a prime and $k$ is an algebraically closed
field of characteristic $\ell$. We refer the reader to [2] for basic notation
and results in the theory of fusion systems. We begin by introducing the class
of $\ell$-groups over which we will consider fusion systems.

######  Definition 2.1.

Write ${\mathbb{Z}}/\ell^{\infty}$ for the infinite union of the cyclic
$\ell$-groups ${\mathbb{Z}}/\ell^{n}$ under the obvious inclusions. A group
isomorphic to a direct product of finitely many copies of
${\mathbb{Z}}/\ell^{\infty}$ is called a discrete $\ell$-torus. A discrete
$\ell$-toral group is a group $S$ which contains a discrete $\ell$-torus $T$
as a normal subgroup of finite $\ell$-power index.

As observed in [3, Lemma 1.3], any quotient, subgroup or extension of an
infinite discrete $\ell$-toral group $S$ is a discrete $\ell$-toral group and
$S$ contains only finitely many conjugacy classes of subgroups of order
$\ell^{n}$ for every $n\geq 0$ ([3, Lemma 1.4]). A fusion system $\mathcal{F}$
over $S$ with discrete torus $T$ is defined by analogy with the finite case
[3, Definition 2.1]. What it means for $\mathcal{F}$ to be saturated is almost
analogous, provided one makes an additional assumption that morphisms behave
well with respect to filtrations of subgroups (see [3, Definition 2.2 (III)]).
The definitions of fully $\mathcal{F}$-centralised, fully
$\mathcal{F}$-normalised, $\mathcal{F}$-centric and $\mathcal{F}$-radical are
unchanged from those for fusion systems on finite groups. We write
$\mathcal{F}^{cr}$ for the set of $\mathcal{F}$-centric radical subgroups. As
in [14, Definition 1.4], we say that $\mathcal{F}$ is simple if it possesses
no non-trivial normal subsystems, and based on [8, Definition 3.1.4] we say
that $\mathcal{F}$ is connected if every element of $S$ is
$\mathcal{F}$-conjugate to an element of $T$. Using a certain “bullet
construction” first considered by Benson (see [3, Definition 3.1]) it is
possible to prove the following result.

######  Proposition 2.2.

Let $\mathcal{F}$ be a saturated fusion system over a discrete $\ell$-toral
group $S$. Then ${{\operatorname{Out}}}_{\mathcal{F}}(P)$ is finite for all
$P\leq S$. Moreover, $\mathcal{F}$ has only finitely many conjugacy classes of
$\mathcal{F}$-centric, $\mathcal{F}$-radical subgroups.

###### Proof.

See [3, Proposition 2.3 and Corollary 3.5]. ∎

In particular $W:={{\operatorname{Out}}}_{\mathcal{F}}(T)$ is finite and we
refer to $W$ as the Weyl group of $\mathcal{F}$. We are interested in the
following invariant.

######  Definition 2.3.

Let $\mathcal{F}$ be a saturated fusion system over a discrete $\ell$-toral
group $S$. The number of weights $\mathbf{w}(\mathcal{F})$ of $\mathcal{F}$ is
given by:

$\mathbf{w}(\mathcal{F}):=\sum_{P\in\mathcal{F}^{cr}/\mathcal{F}}z(k{{\operatorname{Out}}}_{\mathcal{F}}(P)),$

where the sum runs over a set of $\mathcal{F}$-conjugacy class representatives
of $\mathcal{F}$-centric, $\mathcal{F}$-radical subgroups and for a group $H$,
$z(kH)$ denotes the number of projective simple $kH$-modules up to
isomorphism.

By Proposition 2.2, $\mathbf{w}(\mathcal{F})$ is finite.

Now, if $S$ is an abelian discrete $\ell$-toral group then $S=T_{1}\times
T_{2}$ where $T_{2}$ is a finite abelian $\ell$-group and
$T_{1}=({\mathbb{Z}}/\ell^{\infty})^{r}$ is a discrete $\ell$-torus. Thus
$S=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}S_{n}$ where $S_{n}$ is the $\ell^{n}$-torsion subgroup of
$S$. Now define
${{\operatorname{Irr}}}(S)=\displaystyle\lim_{\begin{subarray}{c}\longrightarrow\\\
n\geq 1\end{subarray}}{{\operatorname{Irr}}}(S_{n})$ and observe that
${{\operatorname{Aut}}}(S)$ acts on ${{\operatorname{Irr}}}(S)$.

######  Definition 2.4.

Let $\mathcal{F}$ be a saturated fusion system over a discrete $\ell$-toral
group. The set $\mathcal{M}(\mathcal{F})$ of ordinary weights of height $0$ is
given by

$\mathcal{M}(\mathcal{F})=\\{(\psi,\chi)\mid\psi\in{{\operatorname{Irr}}}(S^{ab})/{{\operatorname{Out}}}_{\mathcal{F}}(S),\chi\in{{\operatorname{Irr}}}(C_{{{\operatorname{Out}}}_{\mathcal{F}}(S)}(\psi))\\}.$

Observe that when $S$ is finite we have

$|\mathcal{M}(\mathcal{F})|=\mathbf{m}(\mathcal{F},v_{\ell}(|S|))=\sum_{\chi\in{{\operatorname{Irr}}}_{0}(S)/{{\operatorname{Out}}}_{\mathcal{F}}(S)}|{{\operatorname{Irr}}}(C_{{{\operatorname{Out}}}_{\mathcal{F}}(S)}(\chi))|=|{{\operatorname{Irr}}}(S^{ab}:{{\operatorname{Out}}}_{\mathcal{F}}(S))|$

where for $d\in{\mathbb{N}}$, $\mathbf{m}(\mathcal{F},d)$ is as defined in
[12, Section 2]. The latter equality follows from Lemma 3.2 in the present
paper.

##  3\. Some character theory

We need an elementary consequence of the theory of blocks with cyclic defect
group.

######  Lemma 3.1.

Let $W$ be a finite group with a cyclic Sylow $\ell$-subgroup $U$ of order
$\ell$. Then

$|{{\operatorname{Irr}}}(W)|-z(kW)=|{{\operatorname{Irr}}}(N_{W}(U))|=|{{\operatorname{Irr}}}_{0}(N_{W}(U))|=|{{\operatorname{Irr}}}_{0}(W)|.$

###### Proof.

This follows from the existence of a bijection between blocks of $W$ and
$N_{W}(U)$ with non-trivial defect which preserves the number of irreducible
characters (see [7, VII.2.12]). Hence summing over all such blocks we obtain,

$|{{\operatorname{Irr}}}(W)|-z(kW)=|{{\operatorname{Irr}}}(N_{W}(U))|-z(kN_{W}(U))=|{{\operatorname{Irr}}}(N_{W}(U))|,$

since $z(kN_{W}(U))=0$, as required.

∎

The following is sometimes referred to as the “Method of Little Groups.”

######  Lemma 3.2.

Let $H\unlhd G$ be finite groups and $\Theta:={{\operatorname{Irr}}}(H)/G$ be
a set of $G$-orbit representatives for ${{\operatorname{Irr}}}(H)$. Suppose
that each $\theta\in{{\operatorname{Irr}}}(H)$ extends to a character
$\widehat{\theta}\in{{\operatorname{Irr}}}(I_{G}(\theta))$, where
$I_{G}(\theta)$ denotes the inertia subgroup of $\theta$ in $G$. Then there is
a bijection

$\\{(\theta,\beta)\mid\theta\in\Theta,\beta\in{{\operatorname{Irr}}}(I_{G}(\theta)/H)\\}\longrightarrow{{\operatorname{Irr}}}(G),\mbox{
given by }(\theta,\beta)\mapsto(\hat{\theta}\beta)_{I_{G}(\theta)}^{G}.$

###### Proof.

See, for example, [5, Theorem 11.5]. ∎

The next lemma concerning characters of the extension of a normal subgroup by
an $\ell^{\prime}$-group is well-known.

######  Lemma 3.3.

Let $G$ be a finite group with $N\unlhd G$. Suppose that $V$ is an inertial
projective simple $kN$-module and that $G/N$ is a cyclic
$\ell^{\prime}$-group. Then $G$ has exactly $|G:N|$ projective simple modules
lying over $V$. In particular if $N=\operatorname{SL}_{2}(\ell)\leq
G\leq\operatorname{GL}_{2}(\ell)$, then
$z(kG)=|G:\operatorname{SL}_{2}(\ell)|$.

###### Proof.

See, for example, [12, Lemma 4.12]. ∎

To end this section, we record the following technical result which is used in
our calculation of $\mathbf{w}(\mathcal{F})$ in Section 4.

######  Lemma 3.4.

Let $X_{1}$ be a finite $\ell^{\prime}$-group and $X_{2}$ be any finite group
and suppose $H$ is a normal subgroup of $X_{1}\times X_{2}$ with $(X_{1}\times
X_{2})/H$ cyclic of order $e\mid\ell-1$. The following hold:

  * (1)

If $X_{2}=\operatorname{GL}_{2}(\ell)$ and
$O^{\ell^{\prime}}(H)=\operatorname{SL}_{2}(\ell)$ then
$z(kH)=|{{\operatorname{Irr}}}(X_{1})|\cdot(\ell-1)/e$.

  * (2)

If $X_{2}=N_{\operatorname{GL}_{2}(\ell)}(U)$ for some Sylow $\ell$-subgroup
$U$ of $\operatorname{SL}_{2}(\ell)$ and $O^{\ell^{\prime}}(H)=U$ then
$|{{\operatorname{Irr}}}(H)|=|{{\operatorname{Irr}}}(X_{1})|\cdot\ell\cdot(\ell-1)/e$.

###### Proof.

Part (1) follows immediately from Lemma 3.3. Alternatively, under the
hypotheses of (1), since $(X_{1}\times X_{2})/H$ is cyclic of
$\ell^{\prime}$-order, every element of ${{\operatorname{Irr}}}(H)$ extends to
$X_{1}\times X_{2}$, so $z(kH)\cdot
e=|{{\operatorname{Irr}}}(X_{1})|\cdot(\ell-1)$ by Lemma 3.2. Part (2) is
proved similarly: in this case every element of ${{\operatorname{Irr}}}(H)$
extends to $X_{1}\times X_{2}\cong
X_{1}\times(C_{\ell}\rtimes(C_{\ell-1})^{2})$, and this group has
$|{{\operatorname{Irr}}}(X_{1})|\ell(\ell-1)$ irreducible characters. Now
Lemma 3.2 implies that
$|{{\operatorname{Irr}}}(X_{1})|\ell(\ell-1)=|{{\operatorname{Irr}}}(H)|e$, as
needed. ∎

##  4\. Fusion systems on $\ell$-groups with an abelian maximal subgroup

In this section we assume that $\mathcal{F}$ is a simple fusion system on a
discrete $\ell$-toral group $S$ with discrete torus $T$ of index $\ell$.

We adopt the notation of [14] to describe the structure of $\mathcal{F}$. Set

$\mathcal{H}=\\{Z(S)\langle x\rangle\mid x\in S\backslash T\\}\mbox{ and
}\mathcal{B}=\\{Z_{2}(S)\langle x\rangle\mid x\in S\backslash T\\},$

where $Z(S)$ and $Z_{2}(S)$ denote the first and second centres of $S$
respectively. We have the following result when $T$ is not
$\mathcal{F}$-centric radical.

######  Theorem 4.1.

Assume that $T$ is not $\mathcal{F}$-centric radical. Then $\mathcal{F}$ is
connected, $T$ has rank $\ell-2$, $\mathcal{F}^{cr}=\\{S\\}\cup\mathcal{H}$
and for any prime $q\neq\ell$, $\mathcal{F}$ is isomorphic to the unique
subsystem of the $\ell$-fusion system of
$\Gamma=\operatorname{PSL}_{\ell}(\overline{\mathbb{F}_{q}})$ with

  * (1)

${{\operatorname{Aut}}}_{\mathcal{F}}(S)={{\operatorname{Aut}}}_{\Gamma}(S)\cong
C_{\ell-1}$;

  * (2)

${{\operatorname{Aut}}}_{\mathcal{F}}(T)={{\operatorname{Aut}}}_{N_{\Gamma}(S)}(T)\cong
C_{\ell}\rtimes C_{\ell-1}$; and

  * (3)

${{\operatorname{Aut}}}_{\mathcal{F}}(P)={{\operatorname{Aut}}}_{\Gamma}(P)\cong\operatorname{SL}_{2}(\ell)$,
for $P\in\mathcal{H}$.

###### Proof.

Parts (1), (2) and (3) follow from [14, Theorem 5.12]. If $x\notin T$ then $x$
is ${{\operatorname{Aut}}}_{\mathcal{F}}(P)$-conjugate to an element of
$Z(S)\leq T$ where $P=Z(S)\langle x\rangle\in\mathcal{H}$ and $\mathcal{F}$ is
connected. ∎

In particular, we see from Lemma 3.4(1) (with $X_{1}=1$) that for the fusion
system $\mathcal{F}$ described in Theorem 4.1 and any $P\in\mathcal{H}$,

$\mathbf{w}(\mathcal{F})=z(k{{\operatorname{Out}}}_{\mathcal{F}}(S))+z(k{{\operatorname{Out}}}_{\mathcal{F}}(P))=(\ell-1)+1=\ell=|{{\operatorname{Irr}}}({{\operatorname{Aut}}}_{\mathcal{F}}(T))|,$

and Theorem 1.2 holds in this case. Thus for the remainder of this subsection
we may assume that $T$ is an $\mathcal{F}$-centric radical subgroup of $S$.

The following notation from [14] provides a convenient way to describe the
actions of ${{\operatorname{Out}}}_{\mathcal{F}}(S)$ and
${{\operatorname{Out}}}_{\mathcal{F}}(T)$ on $S/T$ and $Z(S)\cap[S,S]$, as
well as the two-way traffic between them. Define:

${{\operatorname{Aut}}}^{\vee}(S):=\\{\alpha\in{{\operatorname{Aut}}}(S)\mid[\alpha,Z(S)]\leq
Z(S)\cap[S,S]\\}\mbox{ and
}{{\operatorname{Aut}}}^{\vee}(T)=\\{\alpha|_{T}\mid\alpha\in{{\operatorname{Aut}}}^{\vee}(S)\\}$

and let

${{\operatorname{Aut}}}^{\vee}_{\mathcal{F}}(S):={{\operatorname{Aut}}}^{\vee}(S)\cap{{\operatorname{Aut}}}_{\mathcal{F}}(S)\mbox{
and
}{{\operatorname{Aut}}}^{\vee}_{\mathcal{F}}(T):={{\operatorname{Aut}}}^{\vee}(T)\cap{{\operatorname{Aut}}}_{\mathcal{F}}(T).$

Hence,

${{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T)=\\{\beta\in
N_{{{\operatorname{Aut}}}_{\mathcal{F}}(T)}({{\operatorname{Aut}}}_{S}(T))\mid[\beta,Z(S)]\leq
Z(S)\cap[S,S]\\}$

(see [14, Notation 2.9]). Observe that
${{\operatorname{Inn}}}(S)\leq{{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(S)$
and
${{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(S)\unlhd{{\operatorname{Aut}}}_{\mathcal{F}}(S)$
(it is the kernel of a homomorphism to
${{\operatorname{Aut}}}(Z(S)/(Z(S)\cap[S,S]))$ so we may define
${{\operatorname{Out}}}_{\mathcal{F}}^{\vee}(S):={{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(S)/{{\operatorname{Inn}}}(S).$
Now set,

$\Delta:=({\mathbb{Z}}/\ell)^{\times}\times({\mathbb{Z}}/\ell)^{\times},\mbox{
and }\Delta_{n}:=\\{(r,r^{i})\mid
r\in({\mathbb{Z}}/\ell)^{\times}\\}\leq\Delta\mbox{ for $i\in{\mathbb{Z}}$ },$

define

$\mu:{{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(S)\longrightarrow\Delta\mbox{
via }\mu(\alpha)=(r,s)\mbox{ if }\begin{cases}x\alpha\in x^{r}T&\mbox{ for
$x\in S\backslash T$ }\\\ g\alpha=g^{s}&\mbox{ for $g\in
Z(S)\cap[S,S],$}\end{cases}$

and let

$\mu_{T}:{{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T)\longrightarrow\Delta\mbox{
and
}\hat{\mu}:{{\operatorname{Out}}}_{\mathcal{F}}^{\vee}(S)\longrightarrow\Delta$

be given by $\mu_{T}(\alpha|_{T})=\mu(\alpha)$ and
$\hat{\mu}([\alpha])=\mu(\alpha)$ if
$\alpha\in{{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(S)$ (here $[\alpha]$
denotes the class of $\alpha$ in
${{\operatorname{Out}}}_{\mathcal{F}}^{\vee}(S)$).

We begin by appealing to results in [4, 14] to describe particular groups of
automorphisms of subgroups which turn out to be $\mathcal{F}$-centric radical.
The following result is crucial to our analysis.

######  Lemma 4.2.

Let $A$ be an abelian discrete $\ell$-toral group and
$G\leq{{\operatorname{Aut}}}(A)$ be finite. Assume $v_{\ell}(|G|)=1$,
$O_{\ell}(G)=1$ and that $|[x,A]|=\ell$ for every element $x\in G$ of order
$\ell$. Then, there is a factorization $A=A_{1}\times A_{2}$ with $A_{2}\cong
C_{\ell}\times C_{\ell}$ such that ${{\operatorname{Aut}}}_{G}(A_{2})$
contains $\operatorname{SL}_{2}(\ell)$ and $G$ is a normal subgroup of index
dividing $\ell-1$ in
${{\operatorname{Aut}}}_{G}(A_{1})\times{{\operatorname{Aut}}}_{G}(A_{2})$.

###### Proof.

See [14, Proposition A.7]. ∎

We first deal with the subgroups $Q\in\mathcal{B}\cup\mathcal{H}$ described
above.

######  Lemma 4.3.

Let $Q\in\mathcal{B}\cup\mathcal{H}$. There are unique subgroups
$\tilde{Z}\leq Z(S)$ and $\tilde{Q}\geq Q\cap[S,S]$ such that

  * (1)

if $Q\in\mathcal{H}$ then $Q=\tilde{Z}\times\tilde{Q}$ and $\tilde{Q}\cong
C_{\ell}\times C_{\ell}$; and

  * (2)

if $Q\in\mathcal{B}$ then $\tilde{Z}=Z(S)$, $Q=\tilde{Z}\tilde{Q}$ and
$\tilde{Q}\cong\ell^{1+2}_{+}$.

In either case if $Q$ is $\mathcal{F}$-centric radical there is a unique
subgroup $\Theta\leq{{\operatorname{Aut}}}(Q)$ containing
${{\operatorname{Inn}}}(Q)$ which acts trivially on $\tilde{Z}$, normalises
$\tilde{Q}$ and is such that
$\Theta/{{\operatorname{Inn}}}(Q)\cong\operatorname{SL}_{2}(\ell)$. Moreover,
there is an $\ell^{\prime}$-subgroup $X$ of
${{\operatorname{Out}}}_{\mathcal{F}}(Q)$ such that

${{\operatorname{Out}}}_{\mathcal{F}}(Q)=X(\Theta/{{\operatorname{Inn}}}(Q))\mbox{
and
}N_{{{\operatorname{Out}}}_{\mathcal{F}}(Q)}({{\operatorname{Out}}}_{S}(Q))=X(C_{\ell}\rtimes
C_{\ell-1}).$

###### Proof.

Parts (1) and (2) are shown in [14, Lemma 5.9]. The existence of the subgroup
$\Theta$ with the stated properties is argued in the proof of [14, Theorem
5.11] where it is also shown that $\Theta$ has $\ell^{\prime}$-index in
${{\operatorname{Aut}}}_{\mathcal{F}}(Q)$. From Lemma 4.2 applied with $Q$
(case (1)) or $Q/(Z(S)\cap[S,S])$ (case (2)) and $\Theta$ in the roles of $A$
and $G$ respectively we have

$X_{1}(\Theta/{{\operatorname{Inn}}}(Q))\leq{{\operatorname{Out}}}_{\mathcal{F}}(Q)\leq
X_{1}\times X_{2}$

where $X_{2}\cong\operatorname{GL}_{2}(\ell)$ (see also [14, Lemma
5.9(b)(iv)]). This description of ${{\operatorname{Out}}}_{\mathcal{F}}(Q)$
implies that there must exist a group $X$ with the stated properties. The
lemma follows. ∎

We next deal with automorphisms of $S$ and $T$.

######  Lemma 4.4.

Suppose $Q\in\mathcal{B}\cup\mathcal{H}$ is $\mathcal{F}$-centric radical and
$X$ is as in Lemma 4.3. For some $t\in\\{0,-1\\}$ the following hold:

  * (1)

There is an $\ell^{\prime}$-subgroup
$Y\leq{{\operatorname{Out}}}_{\mathcal{F}}(S)$ with $Y\cong X$ such that

${{\operatorname{Out}}}_{\mathcal{F}}(S)=Y\times{{\operatorname{Out}}}_{\mathcal{F}}^{\vee}(S)\cap\hat{\mu}^{-1}(\Delta_{t}),$

where
${{\operatorname{Out}}}_{\mathcal{F}}^{\vee}(S)\cap\hat{\mu}^{-1}(\Delta_{t})\cong
C_{\ell-1}$.

  * (2)

There is an $\ell^{\prime}$-subgroup $Z\leq
N_{{{\operatorname{Aut}}}_{\mathcal{F}}(T)}({{\operatorname{Aut}}}_{S}(T))$
where $Z\cong X$ such that

$N_{{{\operatorname{Aut}}}_{\mathcal{F}}(T)}({{\operatorname{Aut}}}_{S}(T))=Z({{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T)\cap\mu_{T}^{-1}(\Delta_{t})),$

with
${{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T)\cap\mu_{T}^{-1}(\Delta_{t})\cong
C_{\ell}\rtimes C_{\ell-1}$.

###### Proof.

By [14, Corollary 2.6], we may choose $x\in S\backslash T$ of order $\ell$ and
$Q\in\mathcal{B}\cup\mathcal{H}$ so that $x\in Q$. Plainly, each element of
${{\operatorname{Aut}}}_{\mathcal{F}}(S)$ normalises $S/T\cong\langle
x\rangle$, so in particular the restriction to $Q$ of any
$\ell^{\prime}$-element in ${{\operatorname{Aut}}}_{\mathcal{F}}(S)$ lies in
$N_{{{\operatorname{Aut}}}_{\mathcal{F}}(Q)}({{\operatorname{Aut}}}_{S}(Q))$.
By the Schur–Zassenhaus theorem, ${{\operatorname{Inn}}}(S)$ and
${{\operatorname{Aut}}}_{S}(Q)$ are complemented by $\ell^{\prime}$-groups
$Y_{1}$ and $Y_{2}$ in ${{\operatorname{Aut}}}_{\mathcal{F}}(S)$ and
$N_{{{\operatorname{Aut}}}_{\mathcal{F}}(Q)}({{\operatorname{Aut}}}_{S}(Q))$
respectively. We claim that the restriction map
$\psi:{{\operatorname{Aut}}}_{\mathcal{F}}(S)\longrightarrow
N_{{{\operatorname{Aut}}}_{\mathcal{F}}(Q)}({{\operatorname{Aut}}}_{S}(Q))$
restricts to an isomorphism from $Y_{1}$ to $Y_{2}$. If
$\alpha\in\ker(\psi|_{Y_{1}})$ then $\alpha$ acts trivially on $Q$ so
$\alpha\in{{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(S)$,
$\hat{\mu}([\alpha])=(0,0)$ and hence $[\alpha]=1$ since
$\hat{\mu}|_{{{\operatorname{Out}}}_{\mathcal{F}}^{\vee}(S)}$ is injective (by
[14, Lemma 4.3(a)]). We conclude that $\alpha=1$ whence $\psi|_{Y_{1}}$ is
injective. Conversely, since $\mathcal{F}$ is saturated, every element of
$Y_{2}$ must extend to an element of $Y_{2}$ and $\psi|_{Y_{1}}$ is also
surjective. We conclude that $Y_{1}\cong Y_{2}$. By [14, Lemma 5.9(b)(iii)],
the restriction to $Q$ of elements in
${{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(S)\cap\mu^{-1}(\Delta_{t})$ lies
in $N_{\Theta}({{\operatorname{Aut}}}_{S}(Q))\cong C_{\ell}:C_{\ell-1}$.
Hence, by Lemma 4.3, we must have
${{\operatorname{Out}}}_{\mathcal{F}}(S)=Y\times{{\operatorname{Out}}}_{\mathcal{F}}^{\vee}(S)\cap\widehat{\mu}^{-1}(\Delta_{t})$
where

$X\cong
Y:=\\{\alpha\in{{\operatorname{Out}}}_{\mathcal{F}}(S)\mid[\alpha,Z(S)]\nleq
Z(S)\cap[S,S]]\\},$

(with $X$ is as in Lemma 4.3). Hence (1) holds. We next prove (2). Since every
element of
$N_{{{\operatorname{Aut}}}_{\mathcal{F}}(T)}({{\operatorname{Aut}}}_{S}(T))$
extends to an automorphism of $S$, the description of this group follows from
(1) with $Z=\\{\alpha|_{T}\mid\alpha\in Y\\}$. Moreover,
${{\operatorname{Aut}}}_{S}(T)\leq{{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T),\mu_{T}^{-1}(\Delta_{t})$
since $[x,Z(S)]\leq Z(S)\cap[S,S]$ for $x\in S\backslash T$ so
${{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T)\cap\mu_{T}^{-1}(\Delta_{t})\cong
C_{\ell}\rtimes C_{\ell-1}$ and (2) holds. ∎

The structure of $\mathcal{F}$ is now given as follows.

######  Theorem 4.5.

Let $\mathcal{F}$ be a simple fusion system over an infinite non-abelian
discrete $\ell$-toral group $S$ with discrete torus $T$ of index $\ell$ which
is $\mathcal{F}$-centric radical. Then $\mathcal{F}$ is connected and for some
$t\in\\{0,-1\\}$ one of the following holds:

  * (1)

$\dim(\Omega_{1}(T))=\ell-1$,
$\mu_{T}({{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T))\geq\Delta_{t}$ and
${{\operatorname{Aut}}}_{\mathcal{F}}(T)=O^{\ell^{\prime}}({{\operatorname{Aut}}}_{\mathcal{F}}(T))\mu_{T}^{-1}(\Delta_{t})$;

  * (2)

$\dim(\Omega_{1}(T))\geq\ell$, $t=0$,
$\mu_{T}({{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T))=\Delta_{0}$ and
${{\operatorname{Aut}}}_{\mathcal{F}}(T)=O^{\ell^{\prime}}({{\operatorname{Aut}}}_{\mathcal{F}}(T)){{\operatorname{Aut}}}_{\mathcal{F}}^{\vee}(T)$.

In either case,

$\mathcal{F}^{cr}=\begin{cases}\\{S,T\\}\cup\mathcal{H}&\mbox{ if $t=-1$ }\\\
\\{S,T\\}\cup\mathcal{B}&\mbox{ if $t=0$. }\end{cases}$

where the sets $\mathcal{H}$ and $\mathcal{B}$ each form a single
$S$-conjugacy class and there is an $\ell^{\prime}$-group $X$ such that for
each $Q\in\mathcal{H}$ (if $t=-1$) or $Q\in\mathcal{B}$ (if $t=0$),

${{\operatorname{Out}}}_{\mathcal{F}}(S)\cong C_{\ell-1}\times X,\hskip
5.69054ptN_{{{\operatorname{Aut}}}_{\mathcal{F}}(T)}({{\operatorname{Aut}}}_{S}(T))\cong(C_{\ell}:C_{\ell-1})X,\hskip
5.69054pt\mbox{ and }\hskip
5.69054pt{{\operatorname{Out}}}_{\mathcal{F}}(Q)\cong\operatorname{SL}_{2}(p)X.$

###### Proof.

Parts (1) and (2) and the description of $\mathcal{F}^{cr}$ follow from [14,
Theorem B(a)]. $\mathcal{H}$ and $\mathcal{B}$ each consist of one
$S$-conjugacy class by [14, Lemma 5.4]. The descriptions of
${{\operatorname{Out}}}_{\mathcal{F}}(Q)$,
${{\operatorname{Out}}}_{\mathcal{F}}(S)$ and
$N_{{{\operatorname{Aut}}}_{\mathcal{F}}(T)}({{\operatorname{Aut}}}_{S}(T))$
follow from more precise descriptions in Lemmas 4.3 and 4.4. Moreover, any
$x\in S\backslash T$ is $\mathcal{F}$-conjugate to an element of
$Q\cap[S,S]<T$ via an $\mathcal{F}$-automorphism of $Q=Z(S)\langle x\rangle$
$(t=-1)$ or $Q=Z_{2}(S)\langle x\rangle$ $(t=0)$. Hence $\mathcal{F}$ is
connected. ∎

We end this section by showing that $\mathcal{F}$ can be expressed as direct
limit of finite fusion systems.

######  Proposition 4.6.

Let $\mathcal{F}$ be a simple fusion system over over an infinite non-abelian
discrete $\ell$-toral group $S$ with discrete torus $T$ of index $\ell$. Then
$\displaystyle\mathcal{F}=\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}\mathcal{F}_{n}$ for saturated fusion systems
$\mathcal{F}_{n}$ on $S_{n}\leq S$ with $\displaystyle
S=\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq 1\end{subarray}}S_{n}$

###### Proof.

If $T$ is not $\mathcal{F}$-essential then for any prime $q$ with
$v_{\ell}(q-1)=1$ the $\ell$-fusion system of
$\operatorname{PSL}_{\ell}(\overline{\mathbb{F}_{q}})$ is a direct limit

$\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}\mathcal{F}_{S_{n}}(\operatorname{PSL}_{\ell}(q^{\ell^{n-1}})),\mbox{
where }\displaystyle S=\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}S_{n}\mbox{ with
}S_{n}\in{{\operatorname{Syl}}}_{\ell}(\operatorname{PSL}_{\ell}(q^{\ell^{n-1}})).$

Setting $\mathcal{F}_{n}$ to be the unique subsystem of
$\mathcal{F}_{S_{n}}(\operatorname{PSL}_{\ell}(q^{\ell^{n-1}}))$ with

${{\operatorname{Aut}}}_{\mathcal{F}_{n}}(S_{n})=C_{\ell-1},\hskip
5.69054pt{{\operatorname{Aut}}}_{\mathcal{F}_{n}}(T_{n})\cong C_{\ell}\rtimes
C_{\ell-1}\mbox{ and
}{{\operatorname{Aut}}}_{\mathcal{F}_{n}}(P_{n})\cong\operatorname{SL}_{2}(\ell)\mbox{
for $P_{n}\in\mathcal{H}$ },$

we realise $\mathcal{F}$ as a direct limit of fusion systems.

If $T$ is $\mathcal{F}$-essential, $P\in\mathcal{F}^{fcr}$ and for each
$n\in{\mathbb{N}}$ we let $P_{n}$ be the $\ell^{n}$-torsion subgroup of $P$.
Then $\displaystyle P=\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}P_{n}$ and we may define

$\mathcal{F}_{n}:=\langle\\{\varphi|_{P_{n}}\mid\varphi\in{{\operatorname{Out}}}_{\mathcal{F}}(P)\\}\mid
P\in\mathcal{F}^{fcr}\rangle.$

By [3, Proposition 2.3],
${{\operatorname{Out}}}_{\mathcal{F}_{n}}(P_{n})={{\operatorname{Out}}}_{\mathcal{F}}(P_{n})$
is finite and by [4, Theorem 2.8], $\mathcal{F}_{n}$ is a saturated fusion
system uniquely determined by
${{\operatorname{Aut}}}_{\mathcal{F}_{n}}(T_{n})$, $T_{n}$,
${{\operatorname{Aut}}}_{S_{n}}(T_{n})$ and
$\mathcal{F}_{n}^{cr}\backslash\\{S_{n},T_{n}\\}\subseteq\mathcal{H}_{n}\cup\mathcal{B}_{n}$
where

$\mathcal{H}_{n}=\\{Z(S_{n})\langle x\rangle\mid x\in S_{n}\backslash
T_{n}\\}\mbox{ and }\mathcal{B}_{n}=\\{Z_{2}(S_{n})\langle x\rangle\mid x\in
S_{n}\backslash T_{n}\\}.$

Moreover $\mathcal{F}_{n}\subseteq\mathcal{F}_{n+1}$ for all $n\geq 1$ by
construction. The result follows. ∎

##  5\. Proofs of Theorems 1.2 and 1.4

Combining the results in Sections 3 and 4 we can now prove:

######  Theorem 5.1.

Let $\mathcal{F}$ be a simple fusion system over an infinite non-abelian
discrete $\ell$-toral group $S$ with discrete torus $T$ of index $\ell$. Then
$\mathcal{F}$ is connected and
$\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}({{\operatorname{Aut}}}_{\mathcal{F}}(T))|$.

###### Proof.

By the remarks following Theorem 4.1 we may assume that $T$ is
$\mathcal{F}$-centric radical. By Theorem 4.5, $\mathcal{F}$ is connected and
either $\mathcal{F}^{cr}=\\{S,T\\}\cup\mathcal{H}$ or
$\mathcal{F}^{cr}=\\{S,T\\}\cup\mathcal{B}$. Set
$W={{\operatorname{Aut}}}_{\mathcal{F}}(T)$ and
$U={{\operatorname{Aut}}}_{S}(T)$ for short. By Theorem 4.5 and Lemma 3.1, for
any representative $Q\in\mathcal{H}$ or $Q\in\mathcal{B}$ we have,

$\begin{array}[]{rcl}|{{\operatorname{Irr}}}(W)|-\mathbf{w}(\mathcal{F})&=&|{{\operatorname{Irr}}}(W)|-(z(kW)+z(k{{\operatorname{Out}}}_{\mathcal{F}}(Q))+z(k{{\operatorname{Out}}}_{\mathcal{F}}(S))\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
&=&|{{\operatorname{Irr}}}(N_{W}(U))|-|{{\operatorname{Irr}}}({{\operatorname{Out}}}_{\mathcal{F}}(S))|-z(k{{\operatorname{Out}}}_{\mathcal{F}}(Q)).\end{array}$

The descriptions of the groups
$N_{W}(U),{{\operatorname{Out}}}_{\mathcal{F}}(S)$ and
${{\operatorname{Out}}}_{\mathcal{F}}(Q)$ in Theorem 4.5 combined with Lemma
3.4 then yield:

$\begin{array}[]{rcl}|{{\operatorname{Irr}}}(W)|-\mathbf{w}(\mathcal{F})&=&\ell|{{\operatorname{Irr}}}(X)|-(\ell-1)|{{\operatorname{Irr}}}(X)|-z(k(\operatorname{SL}_{2}(\ell)X))\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
&=&|{{\operatorname{Irr}}}(X)|-z(k(\operatorname{SL}_{2}(\ell)X))\vskip 6.0pt
plus 2.0pt minus 2.0pt\\\ &=&0,\end{array}$

where $X$ is as in Lemma 4.3. The result follows from this. ∎

######  Theorem 5.2.

Let $\mathcal{F}$ be a simple $\ell$-local compact group with
$v_{\ell}(|S|)=1$. Then
$\displaystyle\mathcal{F}=\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq
1\end{subarray}}\mathcal{F}_{n}$ for saturated fusion systems
$\mathcal{F}_{n}$ on $S_{n}\leq S$ with $\displaystyle
S=\lim_{\begin{subarray}{c}\longrightarrow\\\ n\geq 1\end{subarray}}S_{n}$ and
for each $i$ there is a bijection

${{\operatorname{Irr}}}(S_{n}^{ab}:{{\operatorname{Out}}}_{\mathcal{F}}(S_{n}))\longleftrightarrow\\{(s,\chi)\mid
s\in Z(S_{n})/\mathcal{F}_{n},\chi\in{{\operatorname{Irr}}}_{0}(W(s))\\}$

###### Proof.

As above, set $W={{\operatorname{Aut}}}_{\mathcal{F}}(T)$ and
$U={{\operatorname{Aut}}}_{S}(T)$. Fix $n\in{\mathbb{N}}$ and define

$R_{n}:=\\{(s,\chi)\mid s\in
Z(S_{n})/\mathcal{F}_{n},\chi\in{{\operatorname{Irr}}}_{0}(W(s))\\}.$

By Theorems 4.1 and 4.5, $TN_{W}(U)\cong
S:{{\operatorname{Out}}}_{\mathcal{F}}(S)$ and hence Lemma 3.2 implies
${{\operatorname{Irr}}}(S^{ab}:{{\operatorname{Out}}}_{\mathcal{F}}(S))$ is in
bijection with the set of classes of pairs

$L_{n}:=\\{(\psi,\chi)\mid\psi\in{{\operatorname{Irr}}}(T_{n}/[S_{n},S_{n}])/N_{W}(U),\chi\in{{\operatorname{Irr}}}(C_{N_{W}(U)}(\psi))\\}\\}.$

From Lemma 3.1, we see that all characters in $C_{N_{W}(U)}(\psi)$ have height
$0$. Moreover, any $W$-equivariant bijection between $T_{n}$ and
${{\operatorname{Irr}}}(T_{n})$ must send $Z(S_{n})=C_{T_{n}}(U)$ to
${{\operatorname{Irr}}}(T_{n}/[S_{n},S_{n}])$. These facts imply there is a
bijection

$L_{n}\longleftrightarrow\\{(s,\chi)\mid s\in
Z(S_{n})/N_{W}(U),\chi\in{{\operatorname{Irr}}}(C_{N_{W}(U)}(s))\\}.$

Now every morphism between elements of $Z(S_{n})$ extends to an
$\mathcal{F}_{n}$-automorphism of $S_{n}$, so the $N_{W}(U)$-classes of
elements in $Z(S_{n})$ are exactly the $\mathcal{F}_{n}$-classes of such
elements. Moreover,
$|{{\operatorname{Irr}}}(C_{N_{W}(U)}(s))|=|{{\operatorname{Irr}}}_{0}(C_{W}(s))|$
for each $s\in Z(S_{n})$ by Lemma 3.1. These facts together imply we have a
bijection

$L_{n}\longleftrightarrow\\{(s,\chi)\mid s\in
Z(S_{n})/\mathcal{F}_{n},\chi\in{{\operatorname{Irr}}}_{0}(C_{W}(s))\\}=R_{n},$

whence the result. ∎

## References

  * [1] K. Andersen, B. Oliver and J. Ventura, Reduced, tame and exotic fusion systems. _Proc. Lond. Math. Soc. 105_ (2012), 87–152.
  * [2] M. Aschbacher, R. Kessar, and B. Oliver, _Fusion Systems in Algebra and Topology_. London Mathematical Society Lecture Note Series, 391, Cambridge University Press, Cambridge, 2011.
  * [3] C. Broto, R. Levi and B. Oliver, Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups. _Geom. Topol. 11_ (2007), 315–427.
  * [4] D. Craven, B. Oliver, and J. Semeraro, Reduced fusion systems over $p$-groups with abelian subgroup of index $p$: II. _Adv. Math. 322_ (2017), 201–268.
  * [5] C. W. Curtis and I. Reiner, Methods of representation theory with applications to finite groups and orders Vol. I. _Wiley Classics Library, John Wiley & Sons Inc._ (1990).
  * [6] E. Dade, Counting characters in blocks with cyclic defect groups, I. _J. Algebra 186_ (1996), 934–969.
  * [7] W. Feit, The Representation Theory of Finite Groups, _Elsevier._ (1982).
  * [8] A. González, The structure of $p$-local compact groups, _Ph.D Thesis, Universitat Autònoma de Barcelona.,_ (2010).
  * [9] A. González, Unstable Adams operations acting on $p$–local compact groups and fixed points. _Algebr. Geom. Topol. 12_ (2012), 867–908.
  * [10] I. M. Isaacs, G. Malle, and G. Navarro, A reduction theorem for the McKay conjecture. _Invent. Math. 170_ (2007), 33–101.
  * [11] F. Junod, R. Levi, and A. Libman, Unstable Adams operations on $p$–local compact groups. _Algebr. Geom. Topol. 12_ (2012), 49–74.
  * [12] R. Kessar, M. Linckelmann, J. Lynd and J. Semeraro, Weight conjectures for fusion systems. _Adv. Math. 357_ (2019).
  * [13] R. Kessar, G. Malle and J. Semeraro, Weight conjectures for $\ell$-compact groups and spetses. arXiv:2003.07213.
  * [14] B. Oliver and A. Ruiz, Reduced fusion systems over $p$-groups with abelian subgroup of index $p$: III, _Proc. Royal Soc. Edinb. 150_ (2019), 1–53.