# On the Alperin–McKay conjecture for $2$-blocks of maximal defect

Julian Brough and Lucas Ruhstorfer  School of Mathematics and Natural Sciences
University of Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany
Fachbereich Mathematik, TU Kaiserslautern, 67653 Kaiserslautern, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we show that the Alperin–McKay conjecture holds for $2$-blocks
of maximal defect. A major step in the proof is the verification of the
inductive Alperin–McKay condition for the principal $2$-block of groups of Lie
type in odd characteristic.

##  1\. Introduction

In the representation theory of finite groups some of the most important
conjectures predict a very strong relationship between the representations of
a finite group $G$ and certain representations of its $\ell$-local subgroups,
where $\ell$ is a prime dividing the order of $G$. One of these conjectures is
the Alperin–McKay conjecture. For an $\ell$-block $b$ of $G$ we denote by
$\operatorname{Irr}_{0}(G,b)$ its set of height zero characters.

######  Conjecture 1.1 (Alperin–McKay).

Let $b$ be an $\ell$-block of $G$ with defect group $Q$ and $B$ its Brauer
correspondent in $\mathrm{N}_{G}(Q)$. Then

$|\operatorname{Irr}_{0}(G,b)|=|\operatorname{Irr}_{0}(\mathrm{N}_{G}(Q),B)|.$

In this article we show that the Alperin–McKay conjecture holds for $2$-blocks
of maximal defect.

######  Theorem 1.2.

Let $b$ be a $2$-block of a finite group $G$ whose defect group is a Sylow
$2$-subgroup $Q$ and $B$ its Brauer correspondent in $\mathrm{N}_{G}(Q)$. Then

$|\operatorname{Irr}_{0}(G,b)|=|\operatorname{Irr}_{0}(\mathrm{N}_{G}(Q),B)|.$

Späth [26, Theorem C] showed that the Alperin–McKay conjecture holds for the
prime $\ell$ if the so-called inductive Alperin–McKay condition holds for all
blocks of all finite quasi-simple groups with respect to the prime $\ell$. It
is therefore possible to approach the Alperin–McKay conjecture through the
classification of finite simple groups. Thanks to the work of several authors
the inductive condition has been shown for all finite simple groups except in
the case where $G$ is a group of Lie type defined over a field of
characteristic $p\neq\ell$. Hence, we will focus on the case where $G$ is a
group of Lie type defined over a field of odd characteristic. In their seminal
paper, Malle–Späth [19] showed that in this case $G$ is McKay-good for the
prime $2$. For this they constructed a bijection
$\operatorname{Irr}_{2^{\prime}}(G)\to\operatorname{Irr}_{2^{\prime}}(M)$
between the set of irreducible odd degree characters of $G$ and the
corresponding set of characters of a well chosen subgroup $M$ of $G$
containing $\operatorname{N}_{G}(Q)$, for $Q$ a Sylow $2$-subgroup of $G$.
Based on their bijection we are able to construct an explicit bijection
between the height zero characters in the principal blocks of $G$ and
$\operatorname{N}_{G}(Q)$ and show the following.

######  Theorem 1.3.

Let $G$ be a quasi-simple group of Lie type defined over a field of odd
characteristic. Then the principal $2$-block of $G$ satisfies the AM-
condition.

In a previous article, the second author has reduced the verification of the
inductive Alperin–McKay condition to quasi-isolated blocks of $G$ [23] and
then subsequently for groups of type $A$ to unipotent blocks [24]. One major
hurdle that arises when making use of this reduction, in its current form, is
that the possibility to choose a suitable subgroup $M$ as done in Malle-Späth
no longer holds. As a consequence of the bijection explicitly constructed to
prove Theorem 1.3 and the classification of quasi-isolated elements we obtain
the following:

######  Corollary 1.4.

Let $G$ be a quasi-simple group of classical Lie type. Then every $2$-block of
$G$ satisfies the AM-condition.

Unfortunately, if $G$ is a group of Lie type with exceptional root system
there are many quasi-isolated $2$-blocks. However, one can show that the
principal $2$-block is the unique quasi-isolated $2$-block of maximal defect.

######  Corollary 1.5.

Let $G$ be a quasi-simple group of exceptional Lie type. Then every $2$-block
of maximal defect of $G$ satisfies the AM-condition.

Using this we are able to settle the AM-condition for blocks of maximal defect
of finite quasi-simple groups, which is then enough to establish Theorem 1.2.

### Structure of the paper

In Section 2 we derive some fundamental results on the structure of
normalisers of Sylow $2$-subgroups of groups of Lie type. This will be used in
Section 3 to provide a description of the height zero characters in the
principal block of this normaliser. In the same section we moreover give a
parametrisation of the height zero characters of the principal block of $G$ in
terms of the $1$-Harish-Chandra series. In Section 4 and Section 5 we study
the action of group automorphisms of $G$ on our parametrisation of characters.
This will be used in Section 6 to prove the AM-condition for the principal
block. In Section 7 we deal with the remaining finite simple groups and in
Section 8 we prove our main results.

### Acknowledgments

The authors would like to thank both Britta Späth and Gunter Malle for
discussions relating to previous drafts of this article. This paper is a
contribution via the second author to the SFB TRR 195, and the first author is
supported by the DFG (Project: BR 6142/1-1).

##  2\. Sylow $2$-subgroups

###  2.A. Weyl groups

It is well known that the Sylow $2$-subgroups of the symmetric group are self-
normalising. That is for $P\in{\rm Syl}_{2}(\mathfrak{S}_{n})$, we have that
$\operatorname{N}_{\mathfrak{S}_{n}}(P)=P$. It turns out for all Weyl groups
of irreducible type that the Sylow $2$-subgroups will be self-normalising. In
the following we denote by $\operatorname{C}_{n}$ the cyclic group of order
$n$, while $C_{n}$ denotes the root system of type $C$ with $n$ nodes.

######  Lemma 2.1.

Let $W$ be a Weyl group. Then every Sylow $2$-subgroup of $W$ is self-
normalising.

###### Proof.

Since any Weyl group is a direct product of irreducible Weyl groups we can
assume that $W$ is irreducible. The case $W(A_{n})\cong\mathfrak{S}_{n+1}$ is
well-known, which moreover implies the case
$W(C_{n})\cong\operatorname{C}_{2}\wr\mathfrak{S}_{n}$. The group $W(D_{n})$
also follows from the symmetric group as it arises as a normal subgroup of
index 2 in $W(C_{n})$ isomorphic to
$\operatorname{C}_{2}^{n-1}\rtimes\mathfrak{S}_{n}$ (which can be constructed
as the quotient of $W(C_{n})$ by the kernel of the homomorphism
$\operatorname{C}_{2}^{n}\rightarrow\operatorname{C}_{2}$ which maps
$(g_{1},\dots,g_{n})$ to the product $g_{1}\dots g_{n}$). This only leaves the
exceptional cases. The result is immediate for $W(G_{2})\cong{\rm Dih}_{12}$.
For the remaining cases the description of these groups provided in [10,
Section 2.12] will be taken.

Observe that $W(F_{4})$ arises as the semidirect product of $W(D_{4})$ with
the automorphism group of the Dynkin diagram of type $D_{4}$. The group
$W(D_{4})\cong(\operatorname{C}_{2})^{3}\rtimes\mathfrak{S}_{4}$ is generated
by signed permutations $g_{1}=(1,2)(-1,-2)$ $g_{2}=(2,3)(-2,-3)$,
$g_{3}=(3,4)(-3,-4)$ and $g_{4}=(3,-4)(-3,4)$. Set $\gamma_{1}$ to be the
automorphism of order $2$ fixing both $g_{1},g_{2}$ and interchanging $g_{3}$
and $g_{4}$, while $\gamma_{2}$ denotes the automorphism of order $3$ which
fixes $g_{2}$ and permutes $g_{1},g_{3}$ and $g_{4}$ cyclically. Then
$W(F_{4})\cong W(D_{4})\rtimes\langle\gamma_{1},\gamma_{2}\rangle$. The group
$W(D_{4})$ has three Sylow $2$-subgroups one of which must be fixed by
$\gamma_{1}$. Moreover only one Sylow $2$-subgroup of $W(D_{4})$ contains
$g_{2}$ and thus all three subgroups are fixed by $\gamma_{2}$. In particular,
$W(D_{4})$ has a Sylow $2$-subgroup $Q$ which is fixed by both automorphisms
$\gamma_{1}$ and $\gamma_{2}$. Set $P:=\langle Q,\gamma_{1}\rangle$ which is a
Sylow $2$-subgroup of $W(F_{4})$. As $\operatorname{N}_{W(F_{4})}(Q)=\langle
Q,\gamma_{1},\gamma_{2}\rangle$ and $P^{\gamma_{2}}=\langle
Q,\gamma_{1}^{\gamma_{2}}\rangle\neq P$, it follows that
$\operatorname{N}_{W(F_{4})}(P)=P$.

The group $W(E_{6})$ contains a subgroup
$W^{+}(E_{6})\cong\operatorname{SU}_{4}(2)$ of index two. In
$\operatorname{SU}_{4}(2)$ the normaliser of a Sylow $2$-subgroup $Q$ is a
Borel subgroup $B$, but $B=Q$ as $q=2$. Hence $W^{+}(E_{6})$ and thus
$W(E_{6})$ has self-normalising Sylow $2$-subgroups. The same argument proves
the case of $W(E_{7})\cong\operatorname{C}_{2}\times\operatorname{Sp}_{6}(2)$.
While for $W(E_{8})$ the index two subgroup $W^{+}(E_{8})$ surjects onto
$\Omega_{8}^{+}(2)$ with kernel $Z(W(E_{8}))$ of order $2$. Thus for $G$ the
universal cover of $\Omega_{8}^{+}(2)$ with
$Z(G)\cong\operatorname{C}_{2}\times\operatorname{C}_{2}$, the same argument
as used in $E_{6}$ shows that $G$ and consequently also the groups
$\Omega_{8}^{+}(2)$, $W^{+}(E_{8})$ and $W(E_{8})$ have self-normalising Sylow
$2$-subgroups. ∎

###  2.B. Normalisers of Sylow $2$-subgroups

Let $H$ be a finite group and $Q$ be a Sylow $2$-subgroup of $H$. In this
section we consider when $\operatorname{N}_{H}(Q)=\operatorname{C}_{H}(Q)Q$,
for $H$ a group of Lie type. The following remark will be helpful in answering
this question.

######  Remark 2.2.

Let $H$ and $Q$ be as above. By Schur-Zassenhaus, we have
$\operatorname{N}_{H}(Q)=Q\rtimes K$ for some subgroup $K$ of
$\operatorname{N}_{H}(Q)$. In particular,
$\operatorname{N}_{H}(Q)=\operatorname{C}_{H}(Q)Q$ if and only if
$K\lhd\operatorname{N}_{H}(Q)$.

For any central subgroup $Z\leq\operatorname{Z}(H)$ let $\overline{U}$ denote
the image of any subgroup $U$ of $H$ in the quotient $H/Z$. Observe that $K$
is the unique Hall $2^{\prime}$-subgroup of $KZ$ and thus a characteristic
subgroup of $KZ$. In particular,
$\operatorname{N}_{H}(K)=\operatorname{N}_{H}(KZ)$ and similarly
$\operatorname{N}_{H}(Q)=\operatorname{N}_{H}(QZ)$. Thus $\overline{K}$ is a
complement to $\overline{Q}$ in
$\operatorname{N}_{\overline{H}}(\overline{Q})=\operatorname{N}_{H}(QZ)/Z$. As
$KZ\lhd\operatorname{N}_{H}(Q)$ if and only if
$\overline{K}\lhd\operatorname{N}_{\overline{H}}(\overline{Q})$, it follows
that $\operatorname{N}_{H}(Q)=\operatorname{C}_{H}(Q)Q$ if and only if
$\operatorname{N}_{\overline{H}}(\overline{Q})=\operatorname{C}_{\overline{H}}(\overline{Q})\overline{Q}$.

We use the following theorem by Malle [16, Theorem 5.19] which is based on
work by Aschbacher.

######  Theorem 2.3.

Let ${{\mathbf{H}}}$ be a simple algebraic group and
$F:{{\mathbf{H}}}\to{{\mathbf{H}}}$ a Frobenius endomorphism defining an
$\mathbb{F}_{q}$-structure on ${{\mathbf{H}}}$. Let $d$ be the order of $q$
modulo $4$ and $\mathbf{S}$ a Sylow $d$-torus of $({{\mathbf{H}}},F)$. Assume
that ${{\mathbf{H}}}^{F}$ is not isomorphic to $\operatorname{Sp}_{2n}(q)$
with $n\geq 1$ and $q\equiv 3,5$ $\mathrm{mod}$ $8$. Then there exists a Sylow
$2$-subgroup $Q$ of ${{\mathbf{H}}}^{F}$ with
$\operatorname{N}_{{{\mathbf{H}}}^{F}}(Q)\leq\operatorname{N}_{{{\mathbf{H}}}^{F}}(\mathbf{S})$.

We can now answer the question posed at the beginning of this section. Note
that a similar result to the following corollary was obtained in [14, Theorem
1].

######  Corollary 2.4.

Keep the assumption of Theorem 2.3 and let $Q$ be a Sylow $2$-subgroup of
$H:={{\mathbf{H}}}^{F}$. Then
$\operatorname{N}_{H}(Q)=\operatorname{C}_{H}(Q)Q$. Moreover, for
${{\mathbf{H}}}\hookrightarrow\tilde{{{\mathbf{H}}}}$ a regular embedding and
$\widetilde{Q}$ a Sylow $2$-subgroup of
$\tilde{H}:=\tilde{{{\mathbf{H}}}}^{F}$ with $Q=\widetilde{Q}\cap H$, then
$\operatorname{N}_{\tilde{H}}(Q)=\operatorname{N}_{\tilde{H}}(\tilde{Q})=\operatorname{C}_{\tilde{H}}(\tilde{Q})\tilde{Q}$.

###### Proof.

As in Remark 2.2, take $K$ a complement to $Q$ in $\operatorname{N}_{H}(Q)$.
According to Theorem 2.3, $K\leq\operatorname{N}_{H}({\mathbf{T}})$, where
${\mathbf{T}}=\operatorname{C}_{{{\mathbf{G}}}}({\mathbf{S}})$ is a maximal
torus of ${{\mathbf{H}}}$, see [13, Lemma 3.17]. In particular, $K$ normalises
$Q{\mathbf{T}}^{F}$. As $\mathbf{S}$ is $d$-split with $d\in\\{1,2\\}$, the
group $W$, where $W:=\operatorname{N}_{\mathbf{H}^{F}}({\bf
T})/{\mathbf{T}^{F}}$, is again isomorphic to a Weyl group (use [4, page 121]
and [20, Corollary B.23]). Hence $Q{\mathbf{T}}^{F}/{\mathbf{T}}^{F}$, which
is a Sylow $2$-subgroup of $W$, is self-normalising in $W$ by Lemma 2.1. Thus
$K\leq Q{\mathbf{T}}^{F}={\mathbf{T}}^{F}_{2^{\prime}}\rtimes Q$. As $K$ is a
$2^{\prime}$-group, then $K\leq{\mathbf{T}}^{F}_{2^{\prime}}$ and so
$[K,Q]\leq Q\cap{\mathbf{T}}^{F}_{2^{\prime}}=1$. In other words
$K\leq\operatorname{C}_{H}(Q)$. This proves the first statement.

Next observe that
$\tilde{{{\mathbf{H}}}}^{F}/\operatorname{Z}(\tilde{{{\mathbf{H}}}})^{F}\cong{{\mathbf{H}}}_{\mathrm{ad}}^{F}$
and the assumption of Theorem 2.3 is always satisfied for
${{\mathbf{H}}}_{\mathrm{ad}}^{F}$. Thus by applying Remark 2.2 it follows
that
$\operatorname{N}_{\tilde{H}}(\tilde{Q})=\operatorname{C}_{\tilde{H}}(\tilde{Q})\tilde{Q}$.
Therefore it remains to show that
$\operatorname{N}_{\tilde{H}}(Q)=\operatorname{N}_{\tilde{H}}(\tilde{Q})$. As
any two Sylow $2$-subgroups above $Q$ must be conjugate by an element of
$\operatorname{N}_{\widetilde{H}}(Q)$, it suffices to consider a fixed
$\widetilde{Q}\in{\rm Syl}_{2}(\widetilde{H})$ lying above $Q$.

For groups of type $A$ this follows from [14, Theorem 1]. In the remaining
cases $\tilde{H}/H\operatorname{Z}(\tilde{H})$ is either a $2$\- or a
$2^{\prime}$-group. Note that if
$\widetilde{H}/H\operatorname{Z}(\widetilde{H})$ is a $2^{\prime}$-group, then
$\widetilde{Q}=Q\operatorname{Z}(\widetilde{H})_{2}$ is the unique Sylow
$2$-subgroup of $\tilde{H}$ containing $Q$ and so
$\operatorname{N}_{\widetilde{H}}(\widetilde{Q})=\operatorname{N}_{\widetilde{H}}(Q)$.
Thus assume that $\widetilde{H}/H\operatorname{Z}(\widetilde{H})$ is a
$2$-group. For $\widetilde{\bf T}:=\bf
T\operatorname{Z}(\tilde{{{\mathbf{G}}}})$ a maximal torus of $\widetilde{\bf
H}$, we have $\widetilde{Q}:=\widetilde{{\mathbf{T}}}^{F}_{2}Q$ is a Sylow
$2$-subgroup of $\widetilde{H}=H\operatorname{Z}(\widetilde{H})\widetilde{Q}$
and $[K,\widetilde{{\mathbf{T}}}^{F}]=1$. Thus
$\operatorname{N}_{\widetilde{H}}(Q)=\operatorname{N}_{H}(Q)\operatorname{Z}(\widetilde{H})\widetilde{Q}=K\operatorname{Z}(\widetilde{H})\widetilde{Q}\leq\operatorname{C}_{\widetilde{H}}(\widetilde{Q})\widetilde{Q}$.
∎

###  2.C. Groups of Lie type

The following section is used to introduce the setup which will be in place
for the remainder of this article. Let ${{\mathbf{G}}}$ be a simple algebraic
group of simply connected type defined over an algebraic closure of
$\mathbb{F}_{p}$ for some odd prime $p$. We adopt the notation of [19, Section
2.B]. In particular, $F_{0}:{{\mathbf{G}}}\to{{\mathbf{G}}}$ denotes a field
endomorphism inducing an $\mathbb{F}_{p}$-structure on ${{\mathbf{G}}}$ and
for every symmetry of the Dynkin diagram associated to ${{\mathbf{G}}}$ we
have a graph automorphism $\gamma:{{\mathbf{G}}}\to{{\mathbf{G}}}$. We
consider a Frobenius endomorphism $F:=F_{0}^{m}\gamma$ with $\gamma$ a
(possibly trivial) graph automorphism of ${{\mathbf{G}}}$ such that $F$
defines an $\mathbb{F}_{q}$-structure on ${{\mathbf{G}}}$, where $q=p^{m}$. In
addition, we let ${{\mathbf{G}}}\hookrightarrow\tilde{{{\mathbf{G}}}}$ be the
regular embedding constructed in [19, Section 2.B].

We will also assume until Section 7 that ${{\mathbf{G}}}^{F}$ is not of type
$C_{n}(q)$, $n\geq 1$, or ${}^{3}D_{4}(q)$ whenever $q\not\equiv
1\,\mathrm{mod}\,8$.

Denote by $d$ the order of $q$ modulo $4$. We let ${\mathbf{T}}$ be a
maximally split torus of ${{\mathbf{G}}}$ with corresponding Weyl group
$\mathbf{W}$. We set ${{\mathbf{V}}}:=\langle
n_{\alpha}(1)\mid\alpha\in\Phi\rangle\subset\operatorname{N}_{{\mathbf{G}}}({\mathbf{T}})$,
and ${{\mathbf{H}}}:={{\mathbf{V}}}\cap{\mathbf{T}}$. We define $v:=1$ if
$d=1$ and $v:=\tilde{w_{0}}$ if $d=2$, where $\tilde{w_{0}}$ is the canonical
representative in ${{\mathbf{V}}}$ of the longest element $w_{0}\in\mathbf{W}$
as defined in [19, Section 3.A]. We recall [19, Notation 3.3]:

######  Notation 2.5.

As before let $F:=F_{0}^{m}\gamma$ be a fixed Frobenius endomorphism of
${{\mathbf{G}}}$. Let $E_{1}$ be the subgroup of
$\operatorname{Aut}({{\mathbf{G}}})$ generated by the graph automorphisms
which commute with $\gamma$. Set
$e:=\mathrm{o}(\gamma)\mathrm{exp}(E_{1})\mathrm{o}(v)$. Let
$E:=\operatorname{C}_{em}\times E_{1}$ act on
${\tilde{{\mathbf{G}}}}^{F_{0}^{2em}}$ such that the first summand
$\operatorname{C}_{2em}$ of $E$ acts by $\langle F_{0}\rangle$ and the second
by the group generated by graph automorphisms. Note that this action is
faithful. Let $\widehat{F}_{0},\widehat{\gamma},\widehat{F}\in E$ be the
elements that act on ${\tilde{{\mathbf{G}}}}^{F^{2em}_{0}}$ by $F_{0}$,
$\gamma$ and $F$, respectively.

######  Lemma 2.6.

The torus ${\mathbf{T}}$ contains a Sylow $d$-torus ${\mathbf{S}}$ of
$({{\mathbf{G}}},vF)$. Moreover,
${\mathbf{T}}=\operatorname{C}_{{{\mathbf{G}}}}({\mathbf{S}})$ and $N=TV$,
where $N:=\operatorname{N}_{{\mathbf{G}}}({\mathbf{S}})^{vF}$,
$T:={\mathbf{T}}^{vF}$ and $V:={{\mathbf{V}}}^{vF}$.

###### Proof.

See [19, Lemma 3.2] and [6, Section 5.1]. ∎

Note that $E$ stabilises $N$, $T$, $V$ and hence $H:={{\mathbf{H}}}^{vF}$. In
what follows both the groups ${{\mathbf{G}}}^{F}$ and ${{\mathbf{G}}}^{vF}$
will be considered. Therefore, in addition to the notation in Malle–Späth [19]
the objects from $G_{0}:={{\mathbf{G}}}^{F}$ will be denoted with a subscript
$0$, e.g. $T_{0}:={\mathbf{T}}^{F}$,
$N_{0}:=\operatorname{N}_{G_{0}}({\mathbf{S}})$ and
$W_{0}:={{\mathbf{W}}}^{F}$. The following lemma provides a tool to pass
between the groups $G_{0}={{\mathbf{G}}}^{F}$ and $G:={{\mathbf{G}}}^{vF}$ and
compare them:

######  Lemma 2.7.

Let $g\in{{\mathbf{G}}}$ such that $gF(g)^{-1}=v$. Then the map

$\iota:\tilde{{\mathbf{G}}}^{F_{q}^{2e}}\rtimes
E\to\tilde{{\mathbf{G}}}^{F_{q}^{2e}}\rtimes E,\,x\mapsto x^{g^{-1}}$

is an isomorphism which maps ${{\mathbf{G}}}^{F}\rtimes E$ onto
${{\mathbf{G}}}^{vF}\rtimes E$.

###### Proof.

See the proof of [6, Proposition 5.3]. ∎

Since the image of $\hat{F}$ under $\iota$ is $v\hat{F}$ we obtain an
isomorphism $({{\mathbf{G}}}^{F}\rtimes
E)/\langle\hat{F}\rangle\cong({{\mathbf{G}}}^{vF}\rtimes E)/\langle
v\hat{F}\rangle$. From Theorem 2.3 we are now able to explicitly construct a
Sylow $2$-subgroup of $G:={{\mathbf{G}}}^{vF}$. First, we let $T_{2}$ and
$V_{2}$ be a Sylow $2$-subgroup of $T$ and $V$ respectively. We define
$P:=T_{2}V_{2}$ which forms a Sylow $2$-subgroup of $G$ and conclude that
$Q:=\iota^{-1}(P)$ is a Sylow $2$-subgroup of $G_{0}$. In the next section, we
show that $P$ can be chosen to be $E$-stable.

###  2.D. Automorphisms

######  Lemma 2.8.

Let $W$ be a Weyl group of irreducible type. If $W$ is of type $A_{n}$ ($n\geq
2$), $D_{n}$ ($n$ odd) or $E_{6}$, then the longest element $w_{0}\in W$ acts
as the (unique) non-trivial graph automorphism of order $2$ on $W$. In the
remaining cases, $w_{0}\in\operatorname{Z}(W)$.

###### Proof.

Follows from remarks following [20, Corollary B.23]. ∎

For ${{\mathbf{W}}}$ and ${{\mathbf{V}}}$ as in Section 2.C, it is an obvious
question whether the action of the representative $\tilde{w}_{0}$ of $w_{0}$
in ${{\mathbf{V}}}$ can be described in a similar way. The next lemma gives a
positive answer to this.

######  Lemma 2.9.

Whenever $w_{0}\in\operatorname{Z}({{\mathbf{W}}})$ then we have
$\tilde{w}_{0}\in Z({{\mathbf{V}}})$. In the remaining cases we have
$\mathrm{C}_{{{\mathbf{V}}}}(\tilde{w}_{0})=\mathrm{C}_{{{\mathbf{V}}}}(\gamma_{0})$,
where $\gamma_{0}$ is the graph automorphism which acts as $w_{0}$ on
${{\mathbf{W}}}$.

###### Proof.

This follows from the citations given in the proof of [19, Lemma 3.2]. ∎

######  Lemma 2.10.

There exists an $E$-stable Sylow $2$-subgroup $W_{2}$ of
${{\mathbf{W}}}^{w_{0}F}$ with $w_{0}\in Z(W_{2})$. Moreover, $W_{2}$ is a
Sylow $2$-subgroup of ${{\mathbf{W}}}^{F}$.

###### Proof.

Let us first assume that ${{\mathbf{W}}}$ is not of type $D_{2n}$. Using the
formulas given on the bottom of [4, page 121] together with the well-known
order formulas for Weyl groups, we deduce that
$|{{\mathbf{W}}}:{{\mathbf{W}}}^{\sigma}|$ is odd for any graph automorphism
$\sigma$. Moreover, $w_{0}$ is $\sigma$-stable so we can choose $W_{2}$ to be
a Sylow $2$-subgroup of ${{\mathbf{W}}}^{\sigma}$ with
$w_{0}\in\operatorname{Z}(W_{2})$ by Lemma 2.8.

In type $D_{2n}$ with $2n>4$, the element $w_{0}$ corresponds to a central
element of ${{\mathbf{W}}}$ and so
${{\mathbf{W}}}^{w_{0}F}={{\mathbf{W}}}^{F}$. If $2n>4$, it can be assumed
that $F$ is a field automorphism, otherwise, $E$ acts trivially on
${{\mathbf{W}}}^{F}$. It therefore suffices to find a $\sigma$-stable Sylow
$2$-subgroup of ${{\mathbf{W}}}$ for $\sigma$ the graph automorphism. However
$\sigma$ has order $2$, ${{\mathbf{W}}}$ has an odd number of Sylow
$2$-subgroups and so by the orbit-stabiliser theorem one must be fixed by
$\sigma$.

This leaves the case when ${{\mathbf{W}}}$ is of type $D_{4}$. As before, it
can be assumed $F$ is a field automorphism, otherwise the group $E$ acts
trivially on ${{\mathbf{W}}}^{F}$. In this case
${{\mathbf{W}}}^{F}={{\mathbf{W}}}$ and it was shown in the proof of Lemma 2.1
that $W(D_{4})$ has a Sylow $2$-subgroup which is $E$-stable. ∎

Let $V_{2}$ be the preimage of the Sylow $2$-subgroup $W_{2}$ from Lemma 2.10
under the natural projection map $V\to W$.

######  Corollary 2.11.

The Sylow $2$-subgroup $P:=T_{2}V_{2}$ of $G$ is $E$-stable.

###### Proof.

The group ${{\mathbf{H}}}$ is a characteristic subgroup of ${{\mathbf{V}}}$
and so $H\subseteq V_{2}$. Since $V/H\cong W$ and the image of $V_{2}$ in $W$
is $E$-stable it follows that $V_{2}$ is $E$-stable. ∎

As a consequence of this the Sylow $2$-subgroup $Q=\iota^{-1}(P)$ of $G_{0}$
is $D$-stable, where $D:=\iota^{-1}(E)/\langle\hat{F}\rangle$.

##  3\. Parametrisations of characters

###  3.A. Duality and character bijections of tori

We show how duality can be used to provide bijections between certain
characters of tori. For $({{\mathbf{G}}},{\mathbf{T}},F)$ from Section 2.C
take $({{\mathbf{G}}}^{\ast},{\mathbf{T}}^{\ast},F^{\ast})$ to be a triple in
duality as in [7, Definition 13.10]. Denote by $W_{2}^{\ast}$ and
$w_{0}^{\ast}$ the image of $W_{2}$ respectively $w_{0}$ under the isomorphism
${{\mathbf{W}}}\to{{\mathbf{W}}}^{\ast}$ induced by duality. In the following
we let $v^{\ast}$ be a fixed preimage in
$\operatorname{N}_{{{\mathbf{G}}}^{\ast}}({\mathbf{T}}^{\ast})$ of
$w_{0}^{\ast}$ whenever $d=2$, otherwise $v^{\ast}:=1$. Moreover, we will
denote the images of $v$ and $v^{\ast}$ in ${{\mathbf{W}}}$ respectively
${{\mathbf{W}}}^{\ast}$ by the same symbol.

######  Proposition 3.1.

Let $W_{2}$ be as in Lemma 2.10. Then there exists a bijection

$\alpha:\operatorname{Irr}({\mathbf{T}}^{F})^{W_{2}}\to\operatorname{Irr}({\mathbf{T}}^{vF})^{W_{2}}.$

Moreover, if $\sigma:{{\mathbf{G}}}\to{{\mathbf{G}}}$ is a bijective morphism
with $\sigma({\mathbf{T}})={\mathbf{T}}$ commuting with $F$ such that
$\sigma(v)=v$, then this bijection is equivariant with respect to $\sigma$.

###### Proof.

By duality we obtain a bijection
$\operatorname{Irr}({\mathbf{T}}^{F})\to({\mathbf{T}}^{\ast})^{F^{\ast}}$. Let
$\sigma$ be a bijective morphism of ${{\mathbf{G}}}$ which stabilises
${\mathbf{T}}$. Then there exists a unique bijective morphism (up to
$({\mathbf{T}}^{\ast})^{F^{\ast}}$-conjugation)
$\sigma^{\ast}:{{\mathbf{G}}}^{\ast}\to{{\mathbf{G}}}^{\ast}$ commuting with
$F^{\ast}$ and in duality with $\sigma$ such that this bijection is
$(\sigma,\sigma^{\ast})$-equivariant. Then we obtain a bijection
$\beta_{0}:\operatorname{Irr}({\mathbf{T}}^{F})^{W_{2}}\to(({\mathbf{T}}^{\ast})^{F^{\ast}})^{W_{2}^{\ast}}$.

The triple $({{\mathbf{G}}},{\mathbf{T}},vF)$ is in duality with
$({{\mathbf{G}}}^{\ast},{\mathbf{T}}^{\ast},F^{\ast}v^{\ast})$. Thus we
similarly obtain a $(\sigma,\sigma^{\ast})$-equivariant bijection
$\operatorname{Irr}({\mathbf{T}}^{vF})\to({\mathbf{T}}^{\ast})^{F^{\ast}v^{\ast}}$.
Furthermore, since $W_{2}\subset\operatorname{C}_{W}(w_{0})$, this induces a
bijection
$\beta:\operatorname{Irr}({\mathbf{T}}^{vF})^{W_{2}}\to(({\mathbf{T}}^{\ast})^{F^{\ast}v^{\ast}})^{W_{2}^{\ast}}$.
However $v^{\ast}\in W_{2}^{\ast}$ and so
$(({\mathbf{T}}^{\ast})^{F^{\ast}})^{W_{2}^{\ast}}=(({\mathbf{T}}^{\ast})^{F^{\ast}v^{\ast}})^{W_{2}^{\ast}}$.
In particular, we obtain a bijection

$\alpha:=\beta^{-1}\circ\beta_{0}:\operatorname{Irr}({\mathbf{T}}^{F})^{W_{2}}\to\operatorname{Irr}({\mathbf{T}}^{vF})^{W_{2}}$

which is $\sigma$-equivariant as both $\beta$ and $\beta_{0}$ are
$(\sigma,\sigma^{\ast})$-equivariant. ∎

######  Remark 3.2.

By [4, Equation (15.2)] duality induces bijections
$\mathrm{Z}(\tilde{{\mathbf{G}}}^{\ast})^{F^{\ast}}\to\operatorname{Irr}(\tilde{\mathbf{T}}^{F}/{\mathbf{T}}^{F})$
and
$\mathrm{Z}(\tilde{{\mathbf{G}}}^{\ast})^{F^{\ast}v^{\ast}}\to\operatorname{Irr}(\tilde{\mathbf{T}}^{vF}/{\mathbf{T}}^{vF})$.
In particular, if
$\theta_{0}\in\operatorname{Irr}(\tilde{\mathbf{T}}^{F}/{\mathbf{T}}^{F})$ is
the character corresponding to
$z\in\mathrm{Z}(\tilde{{\mathbf{G}}}^{\ast})^{F^{\ast}}$ then
$\theta:=\theta_{0}\circ\iota\in\operatorname{Irr}(\tilde{\mathbf{T}}^{vF}/{\mathbf{T}}^{vF})$
is the character corresponding to the same central element
$z\in\mathrm{Z}(\tilde{{\mathbf{G}}}^{\ast})^{F^{\ast}}$. Thus, we will denote
the character $\theta$ and $\theta_{0}$ by the same symbol $\hat{z}$.

In the following we will employ the notation introduced in Section 2.C with
respect to the dual group ${{\mathbf{G}}}^{\ast}$. Moreover, for
$s\in{\mathbf{T}}^{\ast}$ we denote by ${{\mathbf{W}}}^{\circ}(s)$ the Weyl
group of $\operatorname{C}_{{{\mathbf{G}}}^{\ast}}^{\circ}(s)$ with respect to
the maximal torus ${\mathbf{T}}^{\ast}$ and
${{\mathbf{W}}}(s):=\operatorname{C}_{{{\mathbf{W}}}^{\ast}}(s)$.

######  Proposition 3.3.

For $s\in(T_{2}^{\ast})^{W_{2}^{\ast}}$ we have $v^{\ast}\in W^{\circ}(s)$.

###### Proof.

It can be assumed that $q\equiv 3\,\mathrm{mod}\,4$, otherwise $v^{\ast}$ is
trivial. In particular, $v^{\ast}$ is our fixed preimage of $w_{0}^{\ast}$ in
$\operatorname{N}_{{{\mathbf{G}}}^{\ast}}({\mathbf{T}}^{\ast})$ and $s$
centralises a Sylow $2$-subgroup of $G^{\ast}$.

We have $w_{0}^{\ast}\in W(s)$, so $w_{0}^{\ast}\in W^{\circ}(s)$ whenever
${{\mathbf{C}}}:=\operatorname{C}_{{{\mathbf{G}}}^{\ast}}(s)$ is connected. We
can therefore assume that ${{\mathbf{C}}}$ is disconnected. Let us first
suppose that ${{\mathbf{G}}}$ is not of type $A_{n}$. By the proof of [19,
Theorem 8.7], using that $s$ centralises a Sylow $2$-subgroup of $G^{\ast}$,
the centraliser ${{\mathbf{C}}}^{\circ}$ contains a maximally split torus
$\mathbf{S}$ of $({{\mathbf{G}}},F^{\ast}v^{\ast})$. As
${\mathbf{T}}^{\ast}\subset{{\mathbf{C}}}^{\circ}$ there exists
$x\in{{\mathbf{C}}}^{\circ}$ such that
${\mathbf{S}}={}^{x}{\mathbf{T}}^{\ast}$. Let $h\in{{\mathbf{G}}}^{\ast}$ such
that $F^{\ast}(v^{\ast})=hF^{\ast}(h^{-1})$. In particular,
${}^{h^{-1}}{\mathbf{S}}$ is a maximally split torus of
$({{\mathbf{G}}}^{\ast},F^{\ast})$. Assume first that $F$ is untwisted, i.e.
$F^{\ast}$ induces the identity on ${{\mathbf{W}}}^{\ast}$. Since
${\mathbf{T}}^{\ast}$ is also a maximal $1$-split torus of
$({{\mathbf{G}}}^{\ast},F^{\ast})$ we have
$(h^{-1}x)^{-1}F^{\ast}(h^{-1}x)=x^{-1}h^{-1}F^{\ast}(h)F^{\ast}(x)\in{\mathbf{T}}^{\ast}$,
see [7, Application 3.23]. Since $x\in{{\mathbf{C}}}^{\circ}$ and the image of
$hF^{\ast}(h^{-1})$ in ${{\mathbf{W}}}^{\ast}$ is $w_{0}^{\ast}$, we find that
$w_{0}^{\ast}\in W^{\circ}(s)$.

Assume now that $F$ is twisted, i.e. $\phi:=F^{\ast}v^{\ast}$ induces the
identity on ${{\mathbf{W}}}^{\ast}$. Here, we use that both ${\mathbf{S}}$ and
${}^{h}{\mathbf{T}}^{\ast}$ are maximally split tori of
$({{\mathbf{G}}}^{\ast},\phi)$. In particular, $\phi$ acts trivially on the
Weyl group ${{\mathbf{W}}}({}^{h}{\mathbf{T}}^{\ast})$ and again by [7,
Application 3.23] we have
$(xh^{-1})^{-1}\phi(xh^{-1})\in{}^{h}{\mathbf{T}}^{\ast}$. This yields
$x^{-1}\phi(x)\phi(h^{-1})h\in{\mathbf{T}}^{\ast}$. As
$\phi(h^{-1})h=F^{\ast}(v_{0}^{\ast})$ we again deduce that $w_{0}^{\ast}\in
W^{\circ}(s)$.

Finally, if ${{\mathbf{G}}}^{F}$ is of type $A_{n}(\varepsilon q)$, $n>1$, we
use the proof of [18, Theorem 3.3]. As $s$ centralises a Sylow $2$-subgroup of
$G^{\ast}$ it follows by the arguments given there (together with the
information in [9, Table 4.5.1]) that $n+1$ is necessarily a power of $2$ and
${{\mathbf{C}}}$ is of rational type $A^{2}_{\frac{n-1}{2}}(\varepsilon q).2$
or $A_{\frac{n-1}{2}}(q^{2}).2$. A calculation shows that ${{\mathbf{C}}}$ can
only contain a Sylow $2$-subgroup of ${{\mathbf{G}}}^{F^{\ast}v^{\ast}}$ when
${{\mathbf{C}}}$ has rational type $A^{2}_{\frac{n-1}{2}}(q).2$. In this case
${{\mathbf{C}}}$ contains a maximal $1$-split torus of
$({{\mathbf{G}}}^{\ast},F^{\ast}v^{\ast})$ and the arguments from before apply
also here. ∎

The previous proposition provides a way to compare the characters of
$\tilde{{\mathbf{T}}}^{F}$ lying over a $W_{2}$-stable character of
${\mathbf{T}}^{F}$ with the analogous situation arising from
$\tilde{{\mathbf{T}}}^{vF}$. The following result will be used in Section 5.

######  Proposition 3.4.

Let $\alpha$ be the bijection as in Proposition 3.1. Then there exists a
bijection

$\tilde{\alpha}:\operatorname{Irr}(\tilde{\mathbf{T}}^{F}\mid\operatorname{Irr}({\mathbf{T}}^{F})_{2}^{W_{2}})\to\operatorname{Irr}(\tilde{\mathbf{T}}^{vF}\mid\operatorname{Irr}({\mathbf{T}}^{vF})_{2}^{W_{2}})$

such that
$\alpha\circ\mathrm{Res}_{{\mathbf{T}}^{F}}^{\tilde{\mathbf{T}}^{F}}=\mathrm{Res}_{{\mathbf{T}}^{vF}}^{\tilde{\mathbf{T}}^{vF}}\circ\tilde{\alpha}$
and if $\hat{z}\in\operatorname{Irr}(\tilde{\mathbf{T}}^{F}/{\mathbf{T}}^{F})$
then $\tilde{\alpha}(\hat{z})=\hat{z}$.

Additionally let $\sigma:\tilde{{{\mathbf{G}}}}\to\tilde{{{\mathbf{G}}}}$ be a
bijective morphism commuting with $F$ such that $\sigma|_{{{\mathbf{G}}}}$ is
as in Lemma 3.1. If
$\tilde{\lambda}\in\operatorname{Irr}(\tilde{\mathbf{T}}^{F}\mid\operatorname{Irr}({\mathbf{T}}^{F})_{2}^{W_{2}})$
is such that ${}^{\sigma}\tilde{\lambda}=\tilde{\lambda}\hat{z}$ for some
$z\in\mathrm{Z}(\tilde{{{\mathbf{G}}}}^{\ast})^{F^{\ast}}$, then we have
${}^{\sigma}\tilde{\alpha}(\tilde{\lambda})=\tilde{\alpha}(\tilde{\lambda})\hat{z}$.

###### Proof.

Duality yields again a bijection
$\operatorname{Irr}(\tilde{{\mathbf{T}}}^{F})\to(\tilde{{\mathbf{T}}}^{\ast})^{F^{\ast}}$.
Let $\tilde{s}\in(\tilde{{\mathbf{T}}}^{\ast})^{F^{\ast}}$ be a semisimple
element corresponding to a character
$\tilde{\lambda}\in\operatorname{Irr}(\tilde{\mathbf{T}}^{F}\mid\operatorname{Irr}({\mathbf{T}}^{F})^{W_{2}})$
under this bijection. The map $i:{{\mathbf{G}}}\to\tilde{{{\mathbf{G}}}}$
induces by duality a surjective map
$i^{\ast}:\tilde{{{\mathbf{G}}}}^{\ast}\to{{\mathbf{G}}}^{\ast}$ and the image
$s:=i^{\ast}(\tilde{s})$ of $\tilde{s}$ lies in
$(({\mathbf{T}}^{\ast})^{F^{\ast}})^{W_{2}^{\ast}}=(({\mathbf{T}}^{\ast})^{F^{\ast}v^{\ast}})^{W_{2}^{\ast}}$.
In particular, by Proposition 3.3 we have $v^{\ast}\in W^{\circ}(s)$. The map
$\iota^{\ast}$ yields an isomorphism $W^{\circ}(\tilde{s})\cong W^{\circ}(s)$
and so we deduce that ${}^{v^{\ast}}\tilde{s}=\tilde{s}$. In particular,
$\tilde{s}$ is $F^{\ast}v^{\ast}$-stable. Let
$\tilde{\alpha}(\tilde{\lambda})\in\operatorname{Irr}(\tilde{{\mathbf{T}}}^{vF})$
denote the character corresponding to $\tilde{s}$ under the bijection
$\operatorname{Irr}(\tilde{{\mathbf{T}}}^{vF})\to(\tilde{{\mathbf{T}}}^{\ast})^{F^{\ast}v^{\ast}}$.
One then checks easily that the so-obtained map

$\tilde{\alpha}:\operatorname{Irr}(\tilde{\mathbf{T}}^{F}\mid\operatorname{Irr}({\mathbf{T}}^{F})^{W_{2}})\to\operatorname{Irr}(\tilde{\mathbf{T}}^{vF}\mid\operatorname{Irr}({\mathbf{T}}^{vF})^{W_{2}})$

is a well-defined bijection which has all the required properties. ∎

###  3.B. Local characters

Recall that $P$ denotes the Sylow $2$-subgroup of $G={{\mathbf{G}}}^{vF}$
constructed in Corollary 2.11 and $Q=\iota^{-1}(P)$ its preimage under
$\iota$, which is a Sylow $2$-subgroup of $G_{0}={{\mathbf{G}}}^{F}$. In this
section we make use of the explicit description of $P$ to provide a
description of the odd degree characters in the principal $2$-block of
$\operatorname{N}_{G_{0}}(Q)$. For a finite group $H$ we denote by
$\operatorname{Irr}_{2^{\prime}}(H)$ its set of irreducible characters of odd
degree.

######  Proposition 3.5.

For $P=T_{2}V_{2}$ as in Corollary 2.11, there is a bijection

$\operatorname{Irr}_{2^{\prime}}(P)\to\operatorname{Irr}(T_{2})^{W_{2}}\times\operatorname{Irr}_{2^{\prime}}(W_{2}).$

###### Proof.

Any character of $\operatorname{Irr}_{2^{\prime}}(P)$ (that is any linear
character of $P$) covers a $P$-invariant character of the normal subgroup
$T_{2}$ of $P$. Since $P/T_{2}\cong V_{2}/H\cong W_{2}$ the statement follows
from [19, Corollary 3.13] and Gallagher’s theorem. ∎

We conjecture that the result in part (b) holds in general. Recall as in
Section 3.A that $W_{2}^{\ast}$ denotes the image of $W_{2}$ and
$T^{\ast}:=({\mathbf{T}}^{\ast})^{F^{\ast}v^{\ast}}$ corresponding to
$T:=({\mathbf{T}})^{vF}$ under duality.

######  Proposition 3.6.

Let $B$ be the principal $2$-block of $\operatorname{N}_{G_{0}}(Q)$. Then for
$Z:=(T_{2}^{\ast})^{W_{2}^{\ast}}$, there is a bijection
$\operatorname{Irr}_{0}(B)\to Z\times\operatorname{Irr}_{2^{\prime}}(W_{2})$.

###### Proof.

By Corollary 2.4 we have
$\operatorname{N}_{G_{0}}(Q)=\operatorname{C}_{G_{0}}(Q)Q$ and thus by [21,
Theorem 9.12] restriction defines a bijection
$\operatorname{Irr}_{0}(B)\to\operatorname{Irr}_{2^{\prime}}(Q)$. As in the
proof of Proposition 3.1, duality provides a bijection
$\operatorname{Irr}(T)^{W_{2}}\to(T^{\ast})^{W_{2}^{\ast}}$, which yields a
bijection

$\operatorname{Irr}(T_{2})^{W_{2}}\to(T_{2}^{\ast})^{W_{2}^{\ast}}.$

The result thus follows from Proposition 3.5 using that $P\cong Q$. ∎

######  Remark 3.7.

Let $P^{\ast}$ be a Sylow $2$-subgroup of $G^{\ast}$. As for $G$, it can be
obtained as an extension of $T_{2}^{\ast}$ by $W_{2}^{\ast}$. Therefore,
$Z:=(T_{2}^{\ast})^{W_{2}^{\ast}}$ is a central subgroup of $P^{\ast}$. We
believe that $Z$ should coincide in most cases with
$\operatorname{Z}(P^{\ast})$. For instance if $G^{\ast}$ is of type $A$ then
this is the case by [4, Lemma 13.17(ii)].

###  3.C. Global characters

This section focuses on the height zero characters of the principal block for
$G_{0}={{\mathbf{G}}}^{F}$ as in Section 2.C. First we count these characters
by counting those in ${{\mathbf{G}}}^{vF}$ using Malle’s parametrisation of
$2^{\prime}$-degree characters.

######  Lemma 3.8.

The principal $2$-block of $G={{\mathbf{G}}}^{vF}$ contains
$|Z|\times|\operatorname{Irr}_{2^{\prime}}(W_{2})|$ height zero characters,
where $W_{2}$ and $Z$ are taken from Lemma 2.10 and Proposition 3.6
respectively.

###### Proof.

The odd degree characters of $G$ have been parametrised by Malle [16,
Proposition 7.3]. However, by the proof of [8, Theorem A], the principal
$2$-block is the unique unipotent block of maximal defect. Therefore using
Malle’s explicit parametrisation, it follows that the height zero characters
of the principal block of $G$ are in bijection with pairs $(s,\phi)$, where
$s\in Z$ and $\phi\in\operatorname{Irr}_{2^{\prime}}(W(s))$, where
$W(s):=\operatorname{C}_{W}(s)$. As $s\in Z$, then
$W_{2}\leq\operatorname{C}_{W}(s)$ and thus by the main result of [19] we have
a McKay-bijection
$\operatorname{Irr}_{2^{\prime}}(W(s))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$.
∎

######  Corollary 3.9.

Recall that ${{\mathbf{G}}}$ is simple of simply connected and $F$ is a
Frobenius map with
${{\mathbf{G}}}^{F}\not\cong\\{\operatorname{Sp}_{2n}(q),{}^{3}D_{4}(q)\\}$
whenever $q\not\equiv 1\,\mathrm{mod}\,8$. Then the Alperin–McKay conjecture
holds for the principal $2$-block of ${{\mathbf{G}}}^{F}$.

###### Proof.

This follows from Proposition 3.6 and Theorem 3.8. ∎

Define

$\mathcal{P}_{0}:=\\{(\lambda_{0},\eta_{0})\mid\lambda_{0}\in\operatorname{Irr}(T_{0})\text{
and }\eta_{0}\in\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))\\},$

where $W_{0}(\lambda_{0}):=(N_{0})_{\lambda_{0}}/T_{0}$. From the proof of
[19, Theorem 6.3] there is a surjective map onto the principal Harish-Chandra
series

$\begin{array}[]{clcl}\Pi_{0}:&\mathcal{P}_{0}&\to&\displaystyle\bigcup_{\lambda_{0}\in\operatorname{Irr}(T_{0})}\mathcal{E}(G_{0},(T_{0},\lambda_{0}))\\\
&(\lambda_{0},\eta_{0})&\mapsto&R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}}\\\
\end{array}$

which becomes injective on $W_{0}$-orbits.

The main aim is to find a suitable subset of $\mathcal{P}_{0}$ to parametrise
the height zero characters of the principal block $b$ of
$G_{0}={{\mathbf{G}}}^{F}$. If $R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}}$ has
$2^{\prime}$-degree, then by [19, Lemma 8.9] it follows that
$2\nmid|W_{0}:W_{0}(\lambda_{0})|$. In other words, $W_{0}(\lambda_{0})$
contains a Sylow $2$-subgroup of $W_{0}$. Furthermore, the principal block of
$G_{0}$ is a subset of $\mathcal{E}_{2^{\prime}}(G_{0},1)$, see [4, Theorem
9.12(a)]. However for $s_{0}\in T_{0}^{\ast}$ in duality with
$\lambda_{0}\in\operatorname{Irr}(T_{0})$, it follows that $s_{0}$ has
$2$-power order if and only if $\lambda_{0}$ has $2$-power order. Therefore if
$\chi\in\operatorname{Irr}(b)$ lies in
$\mathcal{E}(G_{0},(T_{0},\lambda_{0}))$, then $\lambda_{0}$ must have
$2$-power order. Via the decomposition
$T_{0}=(T_{0})_{2}\times(T_{0})_{2^{\prime}}$, the $2$-power order characters
coincide with the set $\operatorname{Irr}((T_{0})_{2})$, which can be viewed
as the characters of $T_{0}$ with $(T_{0})_{2^{\prime}}$ in their kernel. Thus
for $W_{2}$ the fixed Sylow $2$-subgroup of $W$ from Lemma 2.10 define

$(\mathcal{P}_{0})_{2}:=\\{(\lambda_{0},\eta_{0})\in\mathcal{P}_{0}\mid\lambda_{0}\in\operatorname{Irr}((T_{0})_{2})^{W_{2}}\\}$

and set $\Pi_{\mathrm{glo}}$ to be the restriction of $\Pi_{0}$ to
$(\mathcal{P}_{0})_{2}$.

######  Theorem 3.10.

Let $b$ be the principal $2$-block of $G_{0}$. Then the map
$\Pi_{\mathrm{glo}}$ yields a bijection

$\Pi_{\mathrm{glo}}:(\mathcal{P}_{0})_{2}\to\operatorname{Irr}_{0}(b).$

###### Proof.

Every character of $\operatorname{Irr}_{0}(b)$ lies in the principal Harish-
Chandra series by [18, Theorem 3.3]. That is
$\operatorname{Irr}_{0}(b)\subset\Pi_{0}(\mathcal{P}_{0})$. If
$\chi=R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}}\in\operatorname{Irr}_{0}(b)$,
then as in the paragraph above, it follows that there is some
$W_{0}$-conjugate $(\lambda_{0}^{\prime},\eta_{0}^{\prime})$ of
$(\lambda_{0},\eta_{0})$ with
$\chi=R_{T_{0}}^{G_{0}}(\lambda_{0}^{\prime})_{\eta_{0}^{\prime}}$ and
$\lambda_{0}^{\prime}\in\operatorname{Irr}((T_{0})_{2})^{W_{2}}$; in other
words $(\lambda_{0}^{\prime},\eta_{0}^{\prime})\in(\mathcal{P}_{0})_{2}$.
Moreover, as $W_{2}$ is self-normalising in $W_{0}$ (Lemma 2.1), it follows
that $\lambda_{0}^{\prime}$ must be the unique character in its $W_{0}$-orbit
with $W_{2}\subseteq W_{0}(\lambda_{0}^{\prime})$. Hence
$\operatorname{Irr}_{0}(b)\subseteq\Pi_{\mathrm{glo}}((\mathcal{P}_{0})_{2})$
and each $\chi\in\operatorname{Irr}_{0}(b)$ has a unique preimage in
$(\mathcal{P}_{0})_{2}$ under $\Pi_{\mathrm{glo}}$.

It remains to show that $\Pi_{\mathrm{glo}}$ is indeed a bijection as stated
in the theorem. By Lemma 3.8, it suffices to show that
$|(\mathcal{P}_{0})_{2}|=|Z|\times|\operatorname{Irr}_{2^{\prime}}(W_{2})|$.
From the proof of Proposition 3.1, there is a bijection

$\operatorname{Irr}((T_{0})_{2})^{W_{2}}\to(T_{0}^{\ast})_{2}^{W_{2}^{\ast}}=(T_{2}^{\ast})^{W_{2}^{\ast}}=:Z.$

Furthermore, for each $\lambda_{0}\in(\mathcal{P}_{0})_{2}$, there is a McKay-
bijection
$\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$
by the main result of [19]. Thus
$|(\mathcal{P}_{0})_{2}|=|Z||\operatorname{Irr}_{2^{\prime}}(W_{2})|$. ∎

######  Example 3.11.

Consider $G=\operatorname{SL}_{2}(q)$ and assume that $q\equiv
3\,\mathrm{mod}\,4$. Recall that this case was excluded in Section 2.C. The
principal $2$-block $b$ of $G$ has four height zero characters. There are four
characters in the principal $1$-Harish-Chandra series corresponding to
characters in $(\mathcal{P}_{0})_{2}$, but only two of them are of
$2^{\prime}$-degree. On the other hand, all four $2^{\prime}$-degree
characters of $b$ lie in the principal $2$-Harish-Chandra series.

######  Remark 3.12.

Assume that ${{\mathbf{G}}}$ is not of type $A_{n}$, $D_{2n+1}$, $n>1$, or
$E_{6}$ so that the longest element $w_{0}\in{{\mathbf{W}}}$ acts by inversion
on the torus ${\mathbf{T}}$. It follows from the remarks after [19, Lemma 8.5]
that all $2^{\prime}$-characters lie in the union of Lusztig series
$\mathcal{E}(G_{0},s)$ with $s$ of $2$-power order. By the proof of [8,
Theorem A], the principal $2$-block is the unique unipotent block of maximal
defect. Hence, in these cases the Alperin–McKay conjecture for the principal
$2$-block is tantamount to the McKay conjecture for the prime $2$.

##  4\. Action of automorphisms

One of the key steps in the proof of Theorem 3.10 was the existence of a
McKay-bijection
$\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$.
We will now construct such a bijection with suitable equivariance properties.
For this we need the following lemma, whose proof follows [17, Lemma 2.1].

######  Lemma 4.1.

Let $H$ be a finite group and $A\subset\mathrm{Aut}(H)$ a cyclic group of
automorphisms stabilizing the normaliser $M$ of a Sylow $2$-subgroup of $H$.
Then there exists an $A$-equivariant McKay bijection
$\operatorname{Irr}_{2^{\prime}}(H)\to\operatorname{Irr}_{2^{\prime}}(M)$.

###### Proof.

According to the main result of [19] there exists such a McKay bijection. We
only need to show that it can be chosen to be $A$-equivariant. For $i\mid
r:=|A|$ let $a_{i}$ (resp. $b_{i}$) be the number of
$\theta\in\operatorname{Irr}_{2^{\prime}}(H)$ (resp.
$\theta\in\operatorname{Irr}_{2^{\prime}}(M)$) with $|A_{\theta}|=i$. As $A$
is cyclic, it suffices to show that $a_{i}=b_{i}$ for all $i\mid r$. Let $q$
be a prime dividing $r$ and set $s:=r/q$. By induction on $r$ we can assume
that $a_{i}=b_{i}$ for all $i\notin\\{r,s\\}$ and

$a_{s}+a_{r}=b_{s}+b_{r}.$

Let us first assume that $2\nmid r$. By Clifford theory we have
$|\operatorname{Irr}_{2^{\prime}}(HA)|=\sum_{i\mid r}a_{i}i^{2}/r$ and
similarly $|\operatorname{Irr}_{2^{\prime}}(MA)|=\sum_{i\mid r}b_{i}i^{2}/r$.
Since the McKay-conjecture holds for $HA$ we have
$|\operatorname{Irr}_{2^{\prime}}(HA)|=|\operatorname{Irr}_{2^{\prime}}(MA)|$
and so

$a_{s}s^{2}/r+a_{r}r=b_{s}s^{2}/r+b_{r}r.$

We therefore have two homogeneous linear equations in the variables
$a_{s}-b_{s}$ and $a_{r}-b_{r}$. As the associated coefficient matrix is
invertible we deduce that $a_{s}=b_{s}$ and $a_{r}=b_{r}$. Let’s now suppose
that $r$ is a power of $2$. In that case, we obtain
$|\operatorname{Irr}_{2^{\prime}}(HA)|=a_{r}r=|\operatorname{Irr}_{2^{\prime}}(MA)|=b_{r}r$.
We again deduce that $a_{s}=b_{s}$ and $a_{r}=b_{r}$. The general case follows
now by using the decomposition $A=A_{2}\times A_{2^{\prime}}$ and coprime
arguments. ∎

######  Remark 4.2.

We note that the existence of an automorphism-equivariant McKay-bijection
should also follow from a similar statement as [22, Theorem B]. As we only
need the result in the case of a cyclic automorphism group we have decided not
to pursue this.

Take $T_{0}={\mathbf{T}}^{F}$ and
$N_{0}=\operatorname{N}_{G_{0}}({\mathbf{S}})$ as in Section 2.C. For
$\tilde{\lambda}_{0}\in\operatorname{Irr}(\tilde{T}_{0})$ denote
$W_{0}(\tilde{\lambda}_{0}):=(N_{0})_{\tilde{\lambda}_{0}}/T_{0}$. Note that
if $\tilde{\lambda}_{0}\in\operatorname{Irr}(\tilde{T}_{0}\mid\lambda_{0})$
for some $\lambda_{0}\in\operatorname{Irr}(T_{0})$, then the factor group
$W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0})$ is an abelian group by the
proof of [19, Proposition 3.16].

######  Lemma 4.3.

Let $\lambda_{0}\in\operatorname{Irr}((T_{0})_{2})^{W_{2}}$ and let
$\tilde{\lambda}_{0}\in\operatorname{Irr}((\tilde{T}_{0})_{2}\mid\lambda_{0})$.
Then there exist an $E_{\lambda_{0}}$-equivariant bijection

$f_{\lambda_{0}}:\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$

such that
$f_{\lambda_{0}}(\eta_{0}\mu_{0})=f_{\lambda_{0}}(\eta_{0})\operatorname{Res}^{W_{0}(\lambda_{0})}_{W_{2}}(\mu_{0})$
for every character
$\eta_{0}\in\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))$ and
$\mu_{0}\in\operatorname{Irr}(W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0}))$.

###### Proof.

The group

$W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0})=\\{w\in
W_{0}\mid{}^{w}\tilde{\lambda}_{0}=\tilde{\lambda}_{0}\otimes\nu_{0}\text{
some
}\nu_{0}\in\operatorname{Irr}(\tilde{T}_{0}/T_{0})\\}/W_{0}(\tilde{\lambda}_{0})$

is always a $2$-group since $\tilde{\lambda}_{0}$ has $2$-power order and
$W_{0}$ acts trivially on $\tilde{T}_{0}/T_{0}$. As $\eta_{0}$ is a
$2^{\prime}$-character and the quotient is a $2$-group it follows that
$\eta_{0}$ restricts irreducibly to $W_{0}(\tilde{\lambda}_{0})$. By
Gallagher’s theorem, the group
$\operatorname{Irr}(W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0}))$ acts fixed
point freely on the orbit of
$\eta_{0}\in\operatorname{Irr}(W_{0}(\lambda_{0}))$. On the other hand, every
character of $\operatorname{Irr}_{2^{\prime}}(W_{2})$ is linear and thus
restricts irreducibly to $W_{2}(\tilde{\lambda}_{0}):=W_{2}\cap
W(\tilde{\lambda}_{0})$.

Let us first assume that $G_{0}$ is not of type $D_{2n}(q)$. Denote by $E_{0}$
the stabiliser of $\lambda_{0}$ in $E$. Observe that $E$ acts by inner
automorphisms on $W_{0}$ and centralises $W_{2}$ by Lemma 2.8. In particular,
every character of $W_{0}(\lambda_{0})$ and $W_{2}$ is $E_{0}$-stable in this
case. Since the Sylow $2$-subgroup $W_{2}$ is self-normalising in $W_{0}$
there exist a McKay bijection
$\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$.
By the previous discussion it’s now easy to construct a bijection
$f_{\lambda_{0}}:\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$
with the required properties.

Let us now assume that $G_{0}$ is of type $D_{2n}(q)$. We use the notation of
the proof of [19, Theorem 3.17]. Let $\Phi(\tilde{\lambda}_{0})$ be the root
system associated to the Weyl group $W_{0}(\tilde{\lambda}_{0})$. There exists
an $E_{0}$-stable base $\Delta_{0}$ of $\Phi(\tilde{\lambda}_{0})$. Denote
$A_{0}:=\mathrm{Stab}_{W_{0}}(\Delta_{0})$ which is $E_{0}$-stable as
$\Delta_{0}$ is. By the proof of [19, Theorem 3.17],
$W_{0}(\lambda_{0})=W_{0}(\tilde{\lambda}_{0})\rtimes A_{0}$. Moreover,
$A_{0}$ is a $2$-group as already observed above. Let
$\eta_{0}\in\operatorname{Irr}(W_{0}(\tilde{\lambda}_{0}))$ which extends to a
$2^{\prime}$-character of $W_{0}(\lambda_{0})$ and set
$\delta_{0}:=\mathrm{det}(\eta_{0})$. By [11, Lemma 6.24] there exists a
unique extension $\hat{\eta}_{0}\in\operatorname{Irr}(W_{0}(\lambda_{0}))$,
such that $\mathrm{det}(\hat{\eta}_{0})=\hat{\delta}_{0}$, where
$\hat{\delta}_{0}$ is the unique extension of $\delta_{0}$ with $A_{0}$ in its
kernel.

Similarly, we have $W_{2}=W_{2}(\tilde{\lambda}_{0})\rtimes A_{0}$. Thus, any
character $\eta_{0}\in\operatorname{Irr}(W_{2}(\tilde{\lambda}_{0}))$ covered
by a linear character of $W_{2}$ has a unique extension
$\hat{\eta}_{0}\in\operatorname{Irr}(W_{2})$ with $A_{0}$ in its kernel.

Let us now first assume that $G_{0}$ is not of type $D_{4}(q)$. Note that
$E/\operatorname{C}_{E}(W_{0})$ is cyclic and thus, by Lemma 4.1 there exists
an $E_{0}$-equivariant McKay-bijection $g_{\lambda_{0}}$ from
$\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))$ to
$\operatorname{Irr}_{2^{\prime}}(W_{2})$. This induces an $E_{0}$-equivariant
bijection

$\begin{array}[]{cccc}f_{0}:&\operatorname{Irr}(W_{0}(\tilde{\lambda}_{0}))\mid\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))&\to&\operatorname{Irr}(W_{2}(\tilde{\lambda}_{0})\mid\operatorname{Irr}_{2^{\prime}}(W_{2}))\\\
&\eta_{0}&\mapsto&\operatorname{Res}^{W_{2}}_{W_{2}(\tilde{\lambda}_{0})}(g_{\lambda_{0}}(\hat{\eta}_{0})).\\\
\end{array}$

We then define
$f_{\lambda_{0}}:\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$
by mapping the character $\hat{\eta}_{0}$ to $\widehat{f_{0}(\eta_{0})}$ and
extending this map
$\operatorname{Irr}(W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0}))$-equivariantly.
As $f_{0}$ is $E_{0}$-equivariant and $A_{0}$ is $E_{0}$-stable, so is
$f_{\lambda_{0}}$.

Finally, if $G_{0}$ is of type $D_{4}(q)$ then $W_{2}$ has index $3$ in
$W_{0}$. Hence, $W_{0}(\lambda_{0})=W_{2}$ or $W_{0}(\lambda_{0})=W_{0}$. In
the former case, we set $f_{\lambda_{0}}$ to be the identity map and in the
latter case it is easy to explicitly construct a bijection $f_{\lambda_{0}}$
with the required properties. ∎

We are also interested in the action of automorphisms on local characters. To
compute this action we use the following explicit parametrisation of
characters. Recall that
$T:=\operatorname{C}_{{{\mathbf{G}}}}({\mathbf{S}})^{vF}={\mathbf{T}}^{vF}$
and $N:=\operatorname{N}_{{{\mathbf{G}}}}({\mathbf{S}})^{vF}$.

######  Proposition 4.4.

Let $\Lambda$ be the extension map from [19, Corollary 3.13] with respect to
$T\lhd N$. Then the map

$\Pi:\mathcal{P}=\\{(\lambda,\eta)\mid\lambda\in\operatorname{Irr}(T),\eta\in\operatorname{Irr}(W(\lambda))\\}\to\operatorname{Irr}(N),\,(\lambda,\eta)\mapsto\operatorname{Ind}_{N_{\lambda}}^{N}(\Lambda(\lambda)\eta),$

is surjective and satisfies

  1. (1)

$\Pi(\lambda,\eta)=\Pi(\lambda^{\prime},\eta^{\prime})$ if and only if there
exists some $n\in N$ such that ${}^{n}\lambda=\lambda^{\prime}$ and
${}^{n}\eta=\eta^{\prime}$.

  2. (2)

${}^{\sigma}\Pi(\lambda,\eta)=\Pi({}^{\sigma}\lambda,{}^{\sigma}\eta)$ for all
$\sigma\in E$.

  3. (3)

Let $t\in\tilde{T}$,
$\tilde{\lambda}\in\operatorname{Irr}(\tilde{T}\mid\lambda)$ and
$\nu_{t}\in\operatorname{Irr}(N_{\lambda}/N_{\tilde{\lambda}})$ be the
faithful linear character given by
${}^{t}\Lambda(\lambda)=\Lambda(\lambda)\nu_{t}$. Then we have
${}^{t}\Pi(\lambda,\eta)=\Pi(\lambda,\eta\nu_{t})$.

###### Proof.

See [19, Proposition 3.15]. ∎

Recall that $P$ is a Sylow $2$-subgroup of $N$ whose image in $N/T$ is
$W_{2}$. We denote

$\mathcal{P}_{2}=\\{(\lambda,\eta)\mid\lambda\in\operatorname{Irr}(T_{2})^{W_{2}},\eta\in\operatorname{Irr}_{2^{\prime}}(W_{2})\\}.$

As in the proof of Proposition 3.5 we obtain that the map

$\begin{array}[]{clcl}\Pi_{\mathrm{loc}}:&\mathcal{P}_{2}&\to&\operatorname{Irr}_{2^{\prime}}(P)\\\
&(\lambda,\eta)&\mapsto&\operatorname{Res}^{N_{\lambda}}_{P}(\Lambda(\lambda))\eta,\end{array}$

is a bijection.

In the following we will compare the parametrisations arising in the two
groups $G_{0}:={{\mathbf{G}}}^{F}$ and $G:={{\mathbf{G}}}^{vF}$. For this,
denote by $\Lambda_{0}$ the extension map from [19, Corollary 3.13] with
respect to $T_{0}\lhd N_{0}$. To understand the action of automorphisms,
recall that $D:=\iota^{-1}(E)/\langle\hat{F}\rangle$, see the remarks after
Corollary 2.11 and $\tilde{{{\mathbf{G}}}}=\tilde{{\mathbf{T}}}{{\mathbf{G}}}$
is the regular embedding as in Section 2.C. Thus for
$\tilde{t}\in\tilde{{\mathbf{T}}}^{F}$, write $\tilde{t}=tz$ with
$t\in{\mathbf{T}}$ and $z\in\mathrm{Z}(\tilde{{{\mathbf{G}}}})$. Then for $g$
from Lemma 2.7 the element ${}^{g}\tilde{t}\in\tilde{{\mathbf{T}}}^{vF}$ and
we have a decomposition ${}^{g}\tilde{t}={}^{g}t{}^{g}z={}^{g}tz$.

######  Proposition 4.5.

Let $\lambda_{0}\in\operatorname{Irr}(T_{0})^{W_{2}}$ and set
$\lambda=\alpha(\lambda_{0})\in\operatorname{Irr}(T)^{W_{2}}$, for $\alpha$
from Lemma 3.1. Suppose that $\nu_{0}\in\operatorname{Irr}(W_{0}(\lambda))$
and $t_{0}\in\tilde{T}_{0}$ satisfy
${}^{t_{0}}\Lambda_{0}(\lambda_{0})=\Lambda(\lambda_{0})\nu_{0}$. Then we have
$\operatorname{Res}^{N_{\lambda}}_{P}({}^{t}\Lambda(\lambda))=\operatorname{Res}^{N_{\lambda}}_{P}(\Lambda(\lambda))\operatorname{Res}_{W_{2}}^{W_{0}(\lambda)}(\nu_{0})$,
where $t=\iota(t_{0})$.

###### Proof.

We first recall the general construction of the extension map $\Lambda$ with
respect to $T\lhd N$ (see in particular the proof of [19, Corollary 3.13]).
First one constructs an extension map ${{\mathbf{H}}}\lhd{{\mathbf{V}}}$. In a
second step one uses this to construct an extension map $\Gamma$ with respect
to $H={{\mathbf{H}}}^{vF}\lhd V={{\mathbf{V}}}^{vF}$. The extension map
$\Lambda$ is then obtained by sending $\lambda\in\operatorname{Irr}(T)$ to the
unique common extension $\Lambda(\lambda)$ in $N_{\lambda}=TV_{\lambda}$ of
$\lambda$ and the restriction of $\Gamma(\operatorname{Res}^{T}_{H}(\lambda))$
to $V_{\lambda}$.

Now, let $\nu\in\operatorname{Irr}(W(\lambda))$ such that
${}^{t}\Lambda(\lambda)=\Lambda(\lambda)\nu$. For $n\in V_{\lambda}$ we have

${}^{t}\Lambda(\lambda)(n)=\lambda([t,n])\Lambda(\lambda)(n).$

If $\tilde{\lambda}\in\operatorname{Irr}(\tilde{T})$ is an extension of
$\lambda$ then we can write
$\lambda([t,n])=\tilde{\lambda}(t){}^{n}\tilde{\lambda}(t^{-1})$. We have
${}^{n}\tilde{\lambda}=\tilde{\lambda}\hat{z}$ for some linear character
$\hat{z}\in\operatorname{Irr}(\tilde{T}/T)$. We conclude that
$\nu(w)=\hat{z}(t)$, where $w$ is the image of $n$ in $W(\lambda)$. Since
$\nu$ is a character of $N_{\lambda}/T\cong V_{\lambda}/H$ this uniquely
determines $\nu$. Now for $w\in W_{2}$ the equality
${}^{w}\tilde{\lambda}=\tilde{\lambda}\nu$ implies by Lemma 3.4 that
${}^{w}\tilde{\lambda}_{0}=\tilde{\lambda}_{0}\hat{z}$. The same reasoning as
above now equally applies to the extension map $\Lambda_{0}$ with respect to
$T_{0}\lhd N_{0}$. Therefore, for $w\in W_{2}$ we find that
$\nu_{0}(w)=\hat{z}(t_{0})=\hat{z}(t)=\nu(w)$. We thus obtain

$\operatorname{Res}^{N_{\lambda}}_{P}({}^{t}\Lambda(\lambda))=\operatorname{Res}^{N_{\lambda}}_{P}(\Lambda(\lambda))\operatorname{Res}_{W_{2}}^{W(\lambda)}(\nu)=\operatorname{Res}^{N_{\lambda}}_{P}(\Lambda(\lambda))\operatorname{Res}_{W_{2}}^{W(\lambda)}(\nu_{0}),$

which finishes the proof. ∎

We now turn to the action of automorphisms on the global characters.

######  Theorem 4.6.

Let $x\in\tilde{T}_{0}D$ and
$\delta_{\lambda_{0},x}\in\operatorname{Irr}(W_{0}({}^{x}\lambda_{0}))$ such
that
$\delta_{\lambda_{0},x}\Lambda_{0}({}^{x}\lambda_{0})={}^{x}\Lambda_{0}(\lambda_{0})$.
Then

${}^{x}(R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}})=R_{T_{0}}^{G_{0}}({}^{x}\lambda_{0})_{{}^{x}\eta_{0}\delta_{\lambda_{0},x}^{-1}}.$

###### Proof.

Follows from the results of [19, Theorem 5.7] as explained in the proof of
[19, Proposition 6.3]. ∎

######  Remark 4.7.

In the following theorem, to compensate for the inversion of
$\delta_{\lambda_{0},x}$ occurring in Theorem 4.6, a slightly altered version
of $f_{\lambda_{0}}$ from Lemma 4.3 is required. Fix $\mathcal{T}$ a
$\operatorname{Irr}(W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0}))\rtimes
E_{\lambda_{0}}$-transversal on
$\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))$. Then for
$\eta_{0}\in\mathcal{T}$, $\sigma\in E_{\lambda_{0}}$ and
$\mu_{0}\in\operatorname{Irr}(W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0}))$
define

$f_{\lambda_{0}}^{\prime}:\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))\to\operatorname{Irr}_{2^{\prime}}(W_{2})$

by setting
$f_{\lambda_{0}}^{\prime}({}^{\sigma}\eta_{0}\mu_{0}):=f_{\lambda_{0}}({}^{\sigma}\eta_{0}\mu_{0}^{-1})$.

It follows from construction for every character
$\eta_{0}\in\operatorname{Irr}_{2^{\prime}}(W_{0}(\lambda_{0}))$ and
$\mu_{0}\in\operatorname{Irr}(W_{0}(\lambda_{0})/W_{0}(\tilde{\lambda}_{0}))$
then
$f^{\prime}_{\lambda_{0}}(\eta_{0}\mu_{0})=f^{\prime}_{\lambda_{0}}(\eta_{0})\operatorname{Res}^{W_{0}(\lambda_{0})}_{W_{2}}(\mu_{0}^{-1})$.
Moreover, the definition implies that $f^{\prime}_{\lambda_{0}}$ is also an
$E_{\lambda_{0}}$-equivariant bijection.

######  Theorem 4.8.

Assume the setting of Section 2.C. For $b$ and $B$ the principal $2$-block of
$G_{0}$ respectively $\operatorname{N}_{G_{0}}(Q)$, there exists an
$\operatorname{N}_{\tilde{G}_{0}D}(Q)$-equivariant bijection
$\kappa:\operatorname{Irr}_{0}(b)\to\operatorname{Irr}_{0}(B)$.

###### Proof.

Restriction defines an $\operatorname{N}_{\tilde{G}_{0}D}(Q)$-equivariant
bijection $\operatorname{Irr}_{0}(B)\to\operatorname{Irr}_{2^{\prime}}(Q)$.
Additionally $\iota$ induces an equivariant bijection $\iota^{\prime}$ between
$\operatorname{Irr}_{2^{\prime}}(Q)$ and $\operatorname{Irr}_{2^{\prime}}(P)$,
that is
$\iota(\operatorname{N}_{\tilde{G}_{0}E}(Q))=\operatorname{N}_{\tilde{G}E}(P)$
and for $x\in\operatorname{N}_{\tilde{G}_{0}E}(Q)$, then
$\iota^{\prime}({}^{x}\chi)={}^{\iota(x)}\iota^{\prime}(\chi)$. Thus it
suffices to produce an equivariant bijection

$\kappa^{\prime}:\operatorname{Irr}_{0}(b)\to\operatorname{Irr}_{2^{\prime}}(P),$

that is $\kappa^{\prime}({}^{x}\chi)={}^{\iota(x)}\kappa^{\prime}(\chi)$, for
$x\in\operatorname{N}_{\tilde{G}_{0}D}(Q)$ and
$\chi\in\operatorname{Irr}_{0}(b)$. Note that as $P$ is $E$-stable, Corollary
2.4 implies
$\operatorname{N}_{\tilde{G}E}(P)=\operatorname{C}_{\tilde{G}}(\tilde{P})\tilde{P}E$
for $\tilde{P}=\tilde{T}_{2}P$. Thus the action on
$\operatorname{Irr}_{2^{\prime}}(P)$ arises from $\tilde{T}_{2}E$.

By the proof of Theorem 3.10 we have a bijection
$\Pi_{\mathrm{glo}}:(\mathcal{P}_{0})_{2}\to\operatorname{Irr}_{0}(b)$. On the
other hand
$\Pi_{\mathrm{loc}}:\mathcal{P}_{2}\to\operatorname{Irr}_{2^{\prime}}(P)$ is a
bijection. Finally by combining Proposition 3.1 and Remark 4.7 there is a
bijection $(\mathcal{P}_{0})_{2}\to\mathcal{P}_{2}$ which sends a pair
$(\lambda_{0},\eta_{0})$ to
$(\alpha(\lambda_{0}),f^{\prime}_{\lambda_{0}}(\eta_{0}))$ between parameter
sets. More explicitly, combining these yields a bijection

$\begin{array}[]{clcl}\kappa^{\prime}:&\operatorname{Irr}_{0}(b)&\to&\operatorname{Irr}_{2^{\prime}}(P)\\\
&R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}}&\mapsto&\operatorname{Res}_{P}^{N_{\alpha(\lambda_{0})}}(\Lambda(\alpha(\lambda_{0})))f^{\prime}_{\lambda_{0}}(\eta_{0}).\\\
\end{array}$

The equivariance of this bijection can be derived by combining the properties
of Harish-Chandra induction established in Theorem 4.6, the properties of the
parametrisation from Proposition 4.4 and Proposition 4.5:

Take $x\in\tilde{T}_{0}E$ and $\delta_{\lambda_{0},x}$ such that
${}^{\tilde{t}_{0}}\Lambda({}^{\sigma}\lambda_{0})=\Lambda({}^{\tilde{t}_{0}\sigma}\lambda_{0})\delta_{\lambda_{0},x}$
By Remark 4.7

$f^{\prime}_{{}^{x}\lambda_{0}}({}^{x}\eta_{0}\delta_{\lambda_{0},x}^{-1})=f^{\prime}_{{}^{x}\lambda_{0}}({}^{x}\eta_{0})\operatorname{Res}^{W_{0}({}^{x}\lambda_{0})}_{W_{2}}(\delta_{\lambda_{0},x})={}^{\iota(x)}f^{\prime}_{\lambda_{0}}(\eta_{0})\operatorname{Res}^{W_{0}({}^{x}\lambda_{0})}_{W_{2}}(\delta_{\lambda_{0},x}).$

While by Proposition 4.4 and Proposition 4.5

${}^{\iota(x)}\left(\operatorname{Res}_{P}^{N_{\alpha(\lambda_{0})}}(\Lambda(\alpha(\lambda_{0})))\right)=\operatorname{Res}_{P}^{N_{\alpha({}^{x}\lambda_{0})}}(\Lambda(\alpha({}^{x}\lambda_{0})))\operatorname{Res}^{W_{0}({}^{x}\lambda_{0})}_{W_{2}}(\delta_{\lambda_{0},x})$

Thus for $x\in\tilde{T_{0}}E$, the equivariance follows as

$\begin{array}[]{ccl}\kappa^{\prime}({}^{x}R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}})&=&\operatorname{Res}_{P}^{N_{\alpha({}^{x}\lambda_{0})}}(\Lambda(\alpha({}^{x}\lambda_{0})))f_{{}^{x}\lambda_{0}}({}^{x}\eta_{0}\delta_{\lambda_{0},x}^{-1})\\\
&=&{}^{\iota(x)}\left(\operatorname{Res}_{P}^{N_{\alpha(\lambda_{0})}}(\Lambda(\alpha(\lambda_{0})))f_{\lambda_{0}}(\eta_{0})\right)\\\
&=&{}^{\iota(x)}\kappa^{\prime}(R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}}).\end{array}$

∎

##  5\. Characters of $\tilde{G}_{0}$

In order to check the inductive conditions for $G_{0}:={{\mathbf{G}}}^{F}$ we
also need information on characters of
$\tilde{G}_{0}=\tilde{{{\mathbf{G}}}}^{F}$ covering characters of
$2^{\prime}$-degree of $G_{0}$. Recall that $T:={\mathbf{T}}^{vF}$,
$N:=\operatorname{N}_{{{\mathbf{G}}}}({\mathbf{S}})^{vF}$ and $\Lambda$ is an
extension map with respect to $T\lhd N$. Additionally
$\tilde{T}:=\tilde{{\mathbf{T}}}^{vF}$ and
$\tilde{N}:=\operatorname{N}_{\tilde{{{\mathbf{G}}}}}({\mathbf{S}})^{vF}$.

######  Proposition 5.1.

There exists an $NE$-equivariant extension map $\tilde{\Lambda}$ with respect
to $\tilde{T}\lhd\tilde{N}$ given by sending
$\tilde{\lambda}\in\operatorname{Irr}(\tilde{T})$ to the unique common
extension of $\tilde{\lambda}$ and
$\operatorname{Res}_{N_{\tilde{\lambda}}}^{N_{\lambda}}(\Lambda(\lambda))$,
where $\lambda=\operatorname{Res}_{T}^{\tilde{T}}(\tilde{\lambda})$.

###### Proof.

This was shown in the proof of [19, Proposition 3.20]. ∎

######  Definition 5.2.

We say that $(\lambda_{0},\eta_{0})\in(\mathcal{P}_{0})_{2}$ (as defined in
Section 3.C) is covered by the pair $(\tilde{\lambda}_{0},\tilde{\eta}_{0})$
if $\tilde{\lambda}_{0}\in\operatorname{Irr}(\tilde{T}_{0}\mid\lambda_{0})$
and
$\tilde{\eta}_{0}\in\operatorname{Irr}(W(\tilde{\lambda}_{0})\mid\eta_{0})$.
Note that
$\tilde{\eta}_{0}=\operatorname{Res}_{W(\tilde{\lambda}_{0})}^{W(\lambda_{0})}(\eta_{0})$
since $W(\lambda_{0})/W(\tilde{\lambda}_{0})$ is a $2$-group and $\eta_{0}$
has $2^{\prime}$-degree.

In the proof of Theorem 4.8, the set $(\mathcal{P}_{0})_{2}$ was used to
provide a bijection between the height zero characters of the principal blocks
of $G_{0}$ and $P$ by mapping $R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}}$ to
$\operatorname{Res}^{N_{\lambda}}_{P}(\Lambda(\lambda))f_{\lambda_{0}}(\eta_{0})$,
where $f_{\lambda_{0}}$ is from Lemma 4.3 and $\lambda:=\alpha(\lambda_{0})$
for $\alpha$ as defined in Section 3.A. The notion of covering defined for
$(\mathcal{P}_{0})_{2}$ can help understand those characters which cover the
height zero characters in the principal blocks of $G_{0}$ and $P$ under the
action of $\tilde{G}_{0}$ respectively $\tilde{P}:=\tilde{T}_{2}P$. Recall
that $\tilde{\alpha}$ from Lemma 3.4 is a bijection between
$\operatorname{Irr}(\tilde{T}_{0}\mid\operatorname{Irr}(T_{0})_{2}^{W_{2}})$
and $\operatorname{Irr}(\tilde{T}\mid\operatorname{Irr}(T)_{2}^{W_{2}})$.

######  Lemma 5.3.

Suppose that $(\tilde{\lambda}_{0},\tilde{\eta}_{0})$ covers
$(\lambda_{0},\eta_{0})\in(\mathcal{P}_{0})_{2}$ as in Definition 5.2.

  1. (a)

Then the character
$\tilde{\chi}:=R_{\tilde{T}_{0}}^{\tilde{G}_{0}}(\tilde{\lambda}_{0})_{\tilde{\eta}_{0}}$
covers $\chi:=R_{T_{0}}^{G_{0}}(\lambda_{0})_{\eta_{0}}$.

  2. (b)

Then the character
$\tilde{\psi}:=\operatorname{Ind}_{\tilde{P}_{\tilde{\lambda}}}^{\tilde{P}}\left(\operatorname{Res}^{\tilde{N}_{\tilde{\lambda}}}_{\tilde{P}_{\tilde{\lambda}}}(\tilde{\Lambda}(\tilde{\lambda}))\operatorname{Res}^{W_{2}}_{W_{2}(\tilde{\lambda})}(f_{\lambda_{0}}(\eta_{0}))\right)$
covers
$\psi:=\operatorname{Res}^{N_{\lambda}}_{P}(\Lambda(\lambda))f_{\lambda_{0}}(\eta_{0})$,
for $\lambda:=\alpha(\lambda_{0})$ and
$\tilde{\lambda}:=\tilde{\alpha}(\tilde{\lambda}_{0})$.

In particular, the characters $\tilde{\psi}$ and $\tilde{\chi}$ lie above the
same central character of $\operatorname{Z}(\tilde{G})$.

###### Proof.

Part (a) follows from [2, Theorem 13.9(b)], while part (b) is a consequence of
Proposition 5.1 and can be obtained as in [19, Corollary 3.21]. For the final
statement about central characters observe that $\tilde{\psi}$ lies above the
character $\tilde{\lambda}\in\operatorname{Irr}(\tilde{T})$ and $\tilde{\chi}$
lies above $\tilde{\lambda}_{0}\in\operatorname{Irr}(\tilde{T}_{0})$. By the
properties of the bijection in Lemma 3.4 they both lie above the same
character of $\mathrm{Z}(\tilde{G}_{0})=\mathrm{Z}(\tilde{G})$. ∎

The following lemma is crucial in verifying the inductive conditions.

######  Lemma 5.4.

Let $\tilde{\chi}\in\operatorname{Irr}(\tilde{G}_{0})$ and
$\tilde{\psi}\in\operatorname{Irr}(\tilde{P})$ as in Lemma 5.3. Let $\sigma\in
E$ and suppose that $\tilde{\chi}^{\sigma}=\tilde{\chi}\hat{z}$ for some
$\hat{z}\in\operatorname{Irr}(\tilde{G}_{0}/G_{0})$. Then we have
$\tilde{\psi}^{\sigma}=\tilde{\psi}\hat{z}$.

###### Proof.

By [19, Corollary 6.4] there exists a character
$\chi\in\operatorname{Irr}(G_{0}\mid\tilde{\chi})$ which satisfies
$(\tilde{G}_{0}E)_{\chi}=(\tilde{G}_{0})_{\chi}E_{\chi}$. Therefore, we have
$\chi^{\sigma}=\chi$ and consequently if $(\lambda_{0},\eta_{0})$ is the label
in $(\mathcal{P}_{0})_{2}$ of $\chi$ we have
$(\lambda_{0}^{\sigma},\eta_{0}^{\sigma})=(\lambda_{0},\eta_{0})$. We have
$\tilde{\lambda}_{0}^{\sigma}=\tilde{\lambda}_{0}\hat{z}$ for some
$\hat{z}\in\operatorname{Irr}(\tilde{T}/T)$ and so we obtain that
$W_{0}(\tilde{\lambda}_{0})$ is $\sigma$-stable. Moreover,
$\tilde{\eta}_{0}^{\sigma}=\tilde{\eta}_{0}$. We obtain
$\tilde{\Lambda}_{0}(\tilde{\lambda}_{0})^{\sigma}=\tilde{\Lambda}_{0}(\tilde{\lambda}_{0})\hat{z}$,
see Proposition 5.1. Thus,

$\tilde{\chi}^{\sigma}=R_{\tilde{T}_{0}}^{\tilde{G}_{0}}(\tilde{\lambda}_{0}^{\sigma})_{\tilde{\eta}_{0}^{\sigma}}=\hat{z}R_{\tilde{T}_{0}}^{\tilde{G}_{0}}(\tilde{\lambda}_{0})_{\tilde{\eta}_{0}}=\hat{z}\tilde{\chi},$

where the second to last equality is derived from [2, Proposition 13.15].
Moreover, $\tilde{\lambda}^{\sigma}=\tilde{\lambda}\hat{z}$ by Lemma 3.4. On
the other hand,
$\tilde{\Lambda}(\tilde{\lambda})^{\sigma}=\tilde{\Lambda}(\tilde{\lambda})\hat{z}$
by [19, Proposition 3.20] and so $\tilde{\psi}^{\sigma}=\tilde{\psi}\hat{z},$
which finishes the proof. ∎

##  6\. The inductive conditions

In this section, we show that the principal $2$-block of
$G_{0}:={{\mathbf{G}}}^{F}$ for $({{\mathbf{G}}},F)$ as in Section 2.C
satisfies the AM-condition. Recall that $Q:=\iota^{-1}(P)$ from Section 2.C is
a Sylow $2$-subgroup of $G_{0}$. In the following $b$ and $B$ denote the
principal $2$-block of $G_{0}$ respectively $\operatorname{N}_{G_{0}}(Q)$. We
need the following lemma.

######  Lemma 6.1.

  1. (a)

Let $\chi\in\operatorname{Irr}_{2^{\prime}}(G_{0})$. Then $\chi$ extends to
$G_{0}D_{\chi}$.

  2. (b)

Let $\chi^{\prime}\in\operatorname{Irr}_{0}(\operatorname{N}_{G_{0}}(Q),B)$.
Then $\chi^{\prime}$ extends to $\operatorname{N}_{G_{0}D}(Q)_{\chi^{\prime}}$
and $\operatorname{N}_{\tilde{G}_{0}}(Q)_{\chi^{\prime}}$.

###### Proof.

The first part was proved in [19, Proposition 8.10].

For the first statement of (b) we pass to the Sylow $2$-subgroup $P=\iota(Q)$
of ${{\mathbf{G}}}^{vF}$. We have
$\operatorname{N}_{G}(P)=\operatorname{C}_{G}(P)P$ by Theorem 2.3. In
particular, any height zero character of the principal block of
$\operatorname{N}_{G}(P)$ is a trivial extension of a linear character of $P$.
In other words, it is enough to show that every linear character
$\lambda\in\operatorname{Irr}(P)$ extends to a character
$\hat{\lambda}\in\operatorname{Irr}(PE_{\lambda})$ with $v\hat{F}$ in its
kernel. For this choose a linear character
$\nu\in\operatorname{Irr}(E_{\lambda})$ with $\nu(\hat{F})=\lambda(v)^{-1}$
and define $\hat{\lambda}(pe):=\lambda(p)\nu(e)$ for $p\in P$ and $e\in
E_{\lambda}$. The second part follows from Corollary 2.4 and Lemma 5.3(b). ∎

The following lemma also helps the checking of the inductive conditions (even
though we won’t use it in the upcoming arguments).

######  Lemma 6.2.

Any character in $\operatorname{Irr}_{0}(b)$ or $\operatorname{Irr}_{0}(B)$
has $Z(G_{0})$ in its kernel.

###### Proof.

Let $\chi\in\operatorname{Irr}_{2^{\prime}}(G_{0})$. Since $\chi$ is of
$2^{\prime}$-degree and $G_{0}$ is quasi-simple, the character $\chi$ has
$\mathrm{Z}(G_{0})_{2}$ in its kernel. If $\chi$ lies moreover in the
principal block, then $\chi\in\mathcal{E}(G_{0},s)$ for some $2$-element $s$.
Thus, $\chi$ is also trivial on $Z(G_{0})_{2^{\prime}}$ by [16, Lemma 2.2].
The local height zero characters were parametrised after Proposition 4.4.
Thus, for them the result follows from Lemma 5.3. ∎

We will use the following theorem to check the inductive condition for the
blocks in question. For the language of character triples and the definition
of the relation $\geq_{b}$ we refer the reader to [23, Section 1.1]. Moreover,
for $\chi\in\operatorname{Irr}(H)$, an irreducible character of a finite group
$H$, we denote by $\mathrm{bl}(\chi)$ the $2$-block of $H$ to which $\chi$
belongs.

######  Theorem 6.3.

Let $\chi\in\operatorname{Irr}(G_{0},b)$ and
$\chi^{\prime}\in\operatorname{Irr}(\operatorname{N}_{G_{0}}(Q),B)$ such that
the following holds:

  1. (i)

We have $(\tilde{G}_{0}{D})_{\chi}=(\tilde{G}_{0})_{\chi}{D}_{\chi}$ and
$\chi$ extends to $(G_{0}{D})_{\chi}$.

  2. (ii)

We have
$(\operatorname{N}_{\tilde{G}_{0}}(Q)\mathrm{N}_{G_{0}{D}}(Q))_{\chi^{\prime}}=\operatorname{N}_{\tilde{G}_{0}}(Q)_{\chi^{\prime}}\mathrm{N}_{G_{0}{D}}(Q)_{\chi^{\prime}}$
and $\chi^{\prime}$ extends to $\mathrm{N}_{G_{0}{D}}(Q)_{\chi^{\prime}}$ and
$\mathrm{N}_{\tilde{G}_{0}}(Q)_{\chi^{\prime}}$.

  3. (iii)

$(\tilde{G}_{0}{D})_{\chi}=G_{0}(\operatorname{N}_{\tilde{G}_{0}}(Q)\mathrm{N}_{G_{0}{D}}(Q))_{\chi^{\prime}}$.

  4. (iv)

There exists $\tilde{\chi}\in\operatorname{Irr}(\tilde{G}_{0}\mid\chi)$ and
$\tilde{\chi}^{\prime}\in\operatorname{Irr}(\operatorname{N}_{\tilde{G}_{0}}(Q)\mid\chi^{\prime})$
such that the following holds:

     * •

For all $m\in{\mathrm{N}_{G_{0}{D}}(Q)}_{\chi^{\prime}}$ there exists
$\nu\in\operatorname{Irr}(\tilde{G}_{0}/G_{0})$ with
$\tilde{\chi}^{m}=\nu\tilde{\chi}$ and $\tilde{\chi}^{\prime
m}=\mathrm{Res}^{\tilde{G}_{0}}_{\operatorname{N}_{\tilde{G}_{0}}(Q)}(\nu)\tilde{\chi}^{\prime}$.

     * •

The characters $\tilde{\chi}$ and $\tilde{\chi}^{\prime}$ cover the same
underlying central character of $\mathrm{Z}(\tilde{G}_{0})$.

  5. (v)

The Clifford correspondents
$\tilde{\chi}_{0}\in\operatorname{Irr}((\tilde{G}_{0})_{\chi}\mid\chi)$ and
$\tilde{\chi}^{\prime}_{0}\in\operatorname{Irr}(\operatorname{N}_{\tilde{G}_{0}}(Q)_{\chi^{\prime}}\mid\chi^{\prime})$
of $\tilde{\chi}$ and $\tilde{\chi}^{\prime}$ respectively satisfy
$\mathrm{bl}(\tilde{\chi}_{0})=\mathrm{bl}(\tilde{\chi}_{0}^{\prime})^{(\tilde{G}_{0})_{\chi}}$.

Let $Z_{0}:=\mathrm{Ker}(\chi)\cap\mathrm{Z}(G_{0})$. Then

$((\tilde{G}_{0}{D})_{\chi}/Z_{0},G_{0}/Z_{0},\overline{\chi})\geq_{b}((\operatorname{N}_{\tilde{G}_{0}}(Q)\mathrm{N}_{G_{0}{D}}(Q))_{\chi^{\prime}}/Z_{0},\operatorname{N}_{G_{0}}(Q)/Z_{0},\overline{\chi^{\prime}}),$

where $\overline{\chi}\in\operatorname{Irr}(G_{0}/Z_{0})$ and
$\overline{\chi^{\prime}}\in\operatorname{Irr}(\operatorname{N}_{G_{0}}(Q)/Z_{0})$
are the characters which inflate to $\chi$, respectively $\chi^{\prime}$.

###### Proof.

This is a consequence of [23, Theorem 2.1] and [23, Lemma 2.2]. ∎

Note that all conditions in Theorem 6.3 except condition (v) only depend on
the character theory of $G_{0}$ and $\tilde{G}_{0}$ (together with its
associated groups).

######  Theorem 6.4.

Let $({{\mathbf{G}}},F)$ be as in Section 2.C. Then the principal $2$-block
$b$ of $G_{0}$ satisfies the AM-condition.

###### Proof.

We show that the bijection
$\kappa:\operatorname{Irr}_{0}(b)\to\operatorname{Irr}_{0}(B)$ from Theorem
4.8 is a strong AM-bijection in the sense of [23, Definition 1.9]. Let
$\chi\in\operatorname{Irr}_{0}(b)$ and $\chi^{\prime}:=\kappa(\chi)$. By
possibly conjugating $\chi$ by an element of $\tilde{G}$ we can assume by [23,
Theorem 2.11] that the character $\chi$ satisfies condition (i) of Theorem
6.3. Using the Butterfly Theorem [23, Theorem 1.10] we see that it’s enough to
show that $\chi$ and $\chi^{\prime}$ satisfy the remaining conditions in
Theorem 6.3. Since $\kappa$ is equivariant we deduce that conditions (ii) and
(iii) hold (the extendibility of the local character follows from Lemma
6.1(b)). Let $\tilde{\chi}$ and $\tilde{\psi}$ be the characters constructed
in Lemma 5.3. By Lemma 5.4 these characters satisfy condition (iv) of Theorem
6.3. Finally for condition (v) let
$\tilde{\chi}_{0}\in\operatorname{Irr}((\tilde{G}_{0})_{\chi}\mid\chi)$ and
$\tilde{\chi}^{\prime}_{0}\in\operatorname{Irr}(\operatorname{N}_{\tilde{G}_{0}}(Q)_{\chi^{\prime}}\mid\chi^{\prime})$
be the Clifford correspondents of $\tilde{\chi}$ and $\tilde{\chi}^{\prime}$.
For $\tilde{b}$ the principal $2$-block of $\tilde{G}_{0}$, we obtain a
bijection

$Z(\tilde{G}_{0})_{2^{\prime}}\to\mathrm{Bl}(\tilde{G}_{0}\mid
b),\,z\mapsto\tilde{b}\otimes\hat{z},$

between the elements of odd order in $Z(\tilde{G}_{0})$ and the set of blocks
of $\tilde{G}_{0}$ covering the principal block of $G$. In particular, the
block of a character of $\tilde{G}$ covering a character in the principal
block of $G_{0}$ is uniquely determined by its underlying character of
$Z(\tilde{G}_{0})_{2^{\prime}}$. Let
$\hat{z}\in\operatorname{Irr}(Z(\tilde{G}_{0})\mid\tilde{\chi})$. Then we
deduce from this bijection that the character $\tilde{\chi}$ lies in the block
$\tilde{b}\otimes\hat{z}$. By the Harris–Knörr correspondence we deduce that
the character $\tilde{\psi}$ lies in the Harris–Knörr correspondent of
$\tilde{b}\otimes\hat{z}$. Observe that $\mathrm{bl}(\tilde{\chi}_{0})$ is
$\tilde{G}_{0}$-stable and hence the unique block of $(\tilde{G}_{0})_{\chi}$
below $\mathrm{bl}(\tilde{\chi})$. Similarly,
$\mathrm{bl}(\tilde{\chi}_{0}^{\prime})$ is the unique block of
$\mathrm{N}_{(\tilde{G}_{0})_{\chi}}(Q)$ below
$\mathrm{bl}(\tilde{\chi}^{\prime})$. From this it follows that
$\mathrm{bl}(\tilde{\chi}_{0})=\mathrm{bl}(\tilde{\chi}_{0}^{\prime})^{(\tilde{G}_{0})_{\chi}}$,
so condition (v) holds. ∎

##  7\. The remaining finite simple groups

For the remaining blocks of finite simple groups the following criterion will
be helpful:

######  Lemma 7.1.

Let $S$ be a finite simple non-abelian group and $\ell$ a prime. Let $b$ be an
$\ell$-block of the universal covering group $\hat{S}$ of $S$ with defect
group $Q$ such that $\mathrm{Out}(\hat{S})_{b}$ is cyclic. Assume that there
exists an $\operatorname{Aut}(\hat{S})_{b}$-equivariant Alperin–McKay
bijection
$f:\operatorname{Irr}_{0}(\hat{S},b)\to\operatorname{Irr}_{0}(\operatorname{N}_{\hat{S}}(Q),B)$
preserving central characters of $\operatorname{Z}(\hat{S})$ and that one of
the following holds:

  1. (i)

all characters of $\operatorname{Irr}_{0}(\hat{S},b)$, or
$\operatorname{Irr}_{0}(\operatorname{N}_{\hat{S}}(Q),B)$ have
$\operatorname{Z}(\hat{S})$ in their kernel.

  2. (ii)

$\mathrm{Out}(\hat{S})_{b}$ is an $\ell$-group.

Then the block $b$ satisfies the AM-condition.

###### Proof.

We check that the conditions in [27, Definition 4.4] are satisfied. Let
$X:=\hat{S}/(\mathrm{Ker(\chi)}\cap\operatorname{Z}(\hat{S}))$. There exists
an overgroup $Y$ of $X$ such that
$Y/\operatorname{C}_{Y}(X)X\cong\mathrm{Out}(X)_{\chi}$ and $Y/X$ is cyclic.
Let $\chi\in\operatorname{Irr}_{0}(\hat{S},b)$ and
$\chi^{\prime}:=f(\chi)\in\operatorname{Irr}_{0}(\operatorname{N}_{\hat{S}}(Q),B)$
considered as characters of $X$ respectively $\operatorname{N}_{X}(Q)$. Assume
that we are in case (i). There exist extensions
$\tilde{\chi}\in\operatorname{Irr}(Y\mid\chi)$ and
$\tilde{\chi}^{\prime}\in\operatorname{Irr}(\operatorname{N}_{Y}(Q)\mid
f(\chi))$ such that
$\operatorname{bl}(\tilde{\chi}^{\prime})^{Y}=\operatorname{bl}(\tilde{\chi})$.
As $\operatorname{C}_{X}(Y)=\operatorname{Z}(\hat{S})$, $\tilde{\chi}$ and
$\tilde{\chi}^{\prime}$ lie over the same central character of
$\operatorname{Z}(\hat{S})$. In particular, we have

$(Y,X,\chi)\geq_{b}(\operatorname{N}_{Y}(Q),\operatorname{N}_{X}(Q),f(\chi))$

by [27, Proposition 4.4].

In case (ii) we observe that $Y/\operatorname{C}_{Y}(X)X$ is an $\ell$-group.
In particular, every block of $\operatorname{C}_{Y}(X)X$ is covered by a
unique block of $Y$. By [27, Lemma 2.16] we find
$\tilde{\chi}\in\operatorname{Irr}(Y\mid\chi)$ and
$\tilde{\chi}^{\prime}\in\operatorname{Irr}(\operatorname{N}_{Y}(Q)\mid
f(\chi))$ which lie above the same character of $\operatorname{C}_{Y}(X)$. In
particular, we have
$\operatorname{bl}(\operatorname{Res}^{\operatorname{N}_{Y}(Q)}_{\operatorname{N}_{X}(Q)\operatorname{C}_{Y}(X)}(\tilde{\chi}^{\prime}))^{Y}=\operatorname{bl}(\tilde{\chi})$.
By [27, Proposition 4.4] this implies that

$(Y,X,\chi)\geq_{b}(\operatorname{N}_{Y}(Q),\operatorname{N}_{X}(Q),f(\chi)).$

In both cases, the Butterfly Theorem [27, Theorem 4.6] implies that the block
$b$ satisfies the AM-condition. ∎

We consider now the case excluded in Section 2.C. Together with Theorem 6.4
this completes the proof of Theorem 1.3 from the introduction.

######  Lemma 7.2.

The principal $2$-block of $G\in\\{\mathrm{Sp}_{2n}(q),{}^{3}D_{4}(q)\\}$
satisfies the AM-condition whenever $q$ is an odd power of an odd prime.

###### Proof.

In our case $\mathrm{Out}(G)$ is cyclic and every $2^{\prime}$-character lies
over the trivial character of $\operatorname{Z}(G)$. Moreover,
$\operatorname{Irr}_{2^{\prime}}(G)=\operatorname{Irr}_{0}(B_{0}(G))$ by
Remark 3.12. Let $Q$ be a Sylow $2$-subgroup of $G$. By Lemma 4.1 there exists
an $\mathrm{Aut}(G)_{Q}$-equivariant bijection
$\operatorname{Irr}_{2^{\prime}}(G)\to\operatorname{Irr}_{2^{\prime}}(\operatorname{N}_{G}(Q))$
preserving the underlying central characters of $\operatorname{Z}(G)$. In
particular, the principal block of $G$ satisfies the AM-condition by Lemma
7.1. ∎

We say that a simple group $S$ is AM-good for the prime $2$ if all $2$-blocks
of its universal covering group satisfy the inductive AM-condition.

######  Lemma 7.3.

Let $S$ be a simple group of Lie type defined over a field of characteristic
$p\neq 2$ with exceptional Schur multiplier. Then $S$ is AM-good for the prime
$2$.

###### Proof.

As argued in [24, Proposition 14.8] it suffices to consider as $S$ the simple
groups ${}^{2}A_{3}(3)$ and $B_{3}(3)$. Let $\hat{G}$ be the universal
covering group of $S$. By [17, Theorem 4.1] there exists a McKay-good
bijection
$f:\operatorname{Irr}_{2^{\prime}}(\hat{G})\to\operatorname{Irr}_{2^{\prime}}(\hat{M})$,
where $\hat{M}$ is the normaliser of a Sylow $2$-subgroup of $\hat{G}$. The
distribution of $2$-blocks of $\hat{G}$ is known by [3]. We observe that for
every character $\nu\in\operatorname{Irr}(\operatorname{Z}(\hat{G}))$ of
$2^{\prime}$-order there exists a unique $2$-block $b_{\nu}$ of $\hat{G}$ of
maximal defect associated to it. Moreover, as argued in the proof of [17,
Theorem 4.1] we have that $\mathrm{Out}(\hat{G})_{\nu}$ is a cyclic $2$-group
for every $1\neq\nu$ of $2^{\prime}$-order. The principal block satisfies the
AM-condition by Theorem 6.4. As a McKay-good bijection preserves central
characters we see that $f$ preserves the block decomposition. We deduce that
$b_{\nu}$, $\nu\neq 1$, also satisfies the AM-condition by Lemma 7.1. In
particular, by [3] the group ${}^{2}A_{3}(3)$ satisfies the AM-condition, as
all blocks with non-maximal defect are of central defect. We are left to
consider the three blocks $b$ of the universal covering group $\hat{G}$ of
$B_{3}(3)$ of defect $4$. Let $b$ be one of these blocks. We can use a proof
similar to [24, Proposition 14.6]. An inspection of [3] shows that
$|\operatorname{Irr}_{0}(b)|=4$. Moreover, these characters have
$\operatorname{Z}(\hat{G})_{2}$ in their kernel. By [15, Theorem 4.1] we
deduce that the Brauer correspondent $B$ of $b$ has also exactly four height
zero characters which all have $\operatorname{Z}(\hat{G})_{2}$ in their
kernel. Let $\bar{b}$ and $\bar{B}$ be the images of the blocks $b$ and $B$ in
the maximal 3-cover $\hat{G}^{\prime}$ of $S$. As $\bar{b}$ has defect $2^{3}$
and precisely $5$ ordinary characters (see [3]) its defect group is isomorphic
to the dihedral group $\mathrm{Dih}_{8}$, see [25, Theorem 8.1]. Using [24,
Proposition 14.4, Proposition 14.5] we deduce that there exists an
$\mathrm{Aut}(\hat{G}^{\prime})$-equivariant bijection
$\operatorname{Irr}(\bar{b})\to\operatorname{Irr}(\bar{B})$. Thus, $b$
satisfies the AM-condition by Lemma 7.1. ∎

######  Lemma 7.4.

Let $S$ be a simple group of Lie type defined over a field of characteristic
$p$. Then $S$ is AM-good for the prime $p$.

###### Proof.

Let $\hat{G}$ be the universal covering group of $S$ and $G$ the
$p^{\prime}$-cover of $S$. By [21, Theorem 9.10] there exists a bijection
between the set of $p$-blocks of $\hat{G}$ and the set of $p$-blocks of $G$.
With this observation the statement follows as in the proof of [26, Theorem
8.4]. ∎

##  8\. Consequences

In this section we derive some consequences of Theorem 6.4. We keep the
notation and setup of Section 2.C but we make no restriction on the type of
$G$.

######  Corollary 8.1.

Assume that the root system of ${{\mathbf{G}}}$ is of classical type. Then
every $2$-block of $G$ satisfies the AM-condition.

###### Proof.

By Lemma 7.3 we can assume that $S:=G/\operatorname{Z}(G)$ has non-exceptional
Schur multiplier. Observe that every subgraph of a Dynkin diagram of classical
type is again of classical type. According to [23, Theorem 3.12] it suffices
to prove that all strictly quasi-isolated $2$-blocks $b$ of $G$ are AM-good.
Suppose first that ${{\mathbf{G}}}$ is not of type $A$. Using the
classification of quasi-isolated elements in [1] together with [4, Theorem
21.14] we deduce that $b$ is the principal block of $G$. The claim follows
therefore from Theorem 6.4. Suppose therefore now that ${{\mathbf{G}}}$ is of
type $A$. Using the proof of the reduction theorem in [24, Theorem 13.4]
together with the results of [23, Section 3.3] we see that it is again
sufficient to prove the claim whenever $b$ is the principal block of $G$. This
again follows from Theorem 6.4. ∎

######  Corollary 8.2.

Suppose that the root system of ${{\mathbf{G}}}$ is of exceptional type and
let $b$ be a quasi-isolated $2$-block of $G$ of maximal defect. Then $b$ is
the principal block of $G$.

###### Proof.

Suppose first that $b$ is a unipotent block of $G$. Then the claim of the
corollary follows from the description of defect groups given in [8]. Suppose
now that $G$ is not of type $E_{6}$. Any block of maximal defect contains a
character of $2^{\prime}$-degree. According to Remark 3.12 such characters lie
in a unipotent block.

Finally for $G=E_{6}(\pm q)$ the non-unipotent quasi-isolated $2$-blocks are
given in [12, Table 3]. The order of the defect group is bounded by
$|C_{G^{\ast}}(s)|_{2}$, see [12, Lemma 2.6(a)], where $1\neq s\in G^{\ast}$
is the semisimple quasi-isolated element of $2^{\prime}$-order associated to
the block $b$. Going through the list given in [12] one checks that
$|C_{G^{\ast}}(s)|_{2}$ is always smaller than $|G|_{2}$. ∎

We can now complete the proof of Theorem 1.2 from the introduction.

######  Theorem 8.3.

The Alperin–McKay conjecture holds for $2$-blocks of maximal defect.

###### Proof.

By [5, Proposition 2.5] it suffices to establish that every block $b$ of
maximal defect of the universal central extension of a finite simple non-
abelian group $S$ satisfies the AM-condition. As explained in the proof of
[24, Proposition 14.8] alternating groups, Suzuki and Ree groups and sporadic
groups are AM-good. By Lemma 7.3 and Lemma 7.4 we can therefore assume that
$S=G/Z(G)$, such that $G$ is a group of Lie type defined over a field of odd
characteristic and $G$ is the universal covering group of $S$. By Corollary
8.1 we can assume that $G$ is an exceptional group of Lie type. By the main
result of [23] we can assume that $b$ is a quasi-isolated block. In this case
the result follows from Corollary 8.2 and Theorem 6.4. ∎

## References

  * [1] Cédric Bonnafé.  Quasi-isolated elements in reductive groups.  Comm. Algebra, 33(7):2315–2337, 2005.
  * [2] Cédric Bonnafé.  Sur les caractères des groupes réductifs finis à centre non connexe: applications aux groupes spéciaux linéaires et unitaires.  Astérisque, (306), 2006.
  * [3] Thomas Breuer.  Decomposition matrices.  http://www.math.rwth-aachen.de/ moc/decomposition/.
  * [4] Marc Cabanes and Michel Enguehard.  Representation theory of finite reductive groups, volume 1 of New Mathematical Monographs.  Cambridge University Press, Cambridge, 2004.
  * [5] Marc Cabanes and Britta Späth.  On the inductive Alperin-McKay condition for simple groups of type A.  J. Algebra, 442:104–123, 2015.
  * [6] Marc Cabanes and Britta Späth.  Equivariant character correspondences and inductive McKay condition for type $A$.  J. Reine Angew. Math., 728:153–194, 2017.
  * [7] François Digne and Jean Michel.  Representations of finite groups of Lie type, volume 21 of London Mathematical Society Student Texts.  Cambridge University Press, Cambridge, 1991.
  * [8] Michel Enguehard.  Sur les $l$-blocs unipotents des groupes réductifs finis quand $l$ est mauvais.  J. Algebra, 230(2):334–377, 2000.
  * [9] Daniel Gorenstein, Richard Lyons, and Ronald Solomon.  The classification of the finite simple groups. Number 3. Part I. Chapter A, volume 40 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 1998.
  * [10] James E. Humphreys.  Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 1990.
  * [11] I. Martin Isaacs.  Character theory of finite groups.  AMS Chelsea Publishing, Providence, RI, 2006.
  * [12] Radha Kessar and Gunter Malle.  Quasi-isolated blocks and Brauer’s height zero conjecture.  Ann. of Math. (2), 178(1):321–384, 2013.
  * [13] Radha Kessar and Gunter Malle.  Lusztig induction and $\ell$-blocks of finite reductive groups.  Pacific J. Math., 279(1-2):269–298, 2015.
  * [14] Anatolii Semenovich Kondrat’ev.  Normalizers of Sylow 2-subgroups in finite simple groups.  Mat. Zametki, 78(3):368–376, 2005.
  * [15] Burkhard Külshammer and Benjamin Sambale.  The 2-blocks of defect 4.  Represent. Theory, 17:226–236, 2013.
  * [16] Gunter Malle.  Height 0 characters of finite groups of Lie type.  Represent. Theory, 11:192–220, 2007.
  * [17] Gunter Malle.  The inductive McKay condition for simple groups not of Lie type.  Comm. Algebra, 36(2):455–463, 2008.
  * [18] Gunter Malle.  The Navarro-Tiep Galois conjecture for $p=2$.  Arch. Math. (Basel), 112(5):449–457, 2019.
  * [19] Gunter Malle and Britta Späth.  Characters of odd degree.  Ann. of Math. (2), 184(3):869–908, 2016.
  * [20] Gunter Malle and Donna Testerman.  Linear algebraic groups and finite groups of Lie type, volume 133 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2011.
  * [21] Gabriel Navarro.  Characters and blocks of finite groups, volume 250 of London Mathematical Society Lecture Note Series.  Cambridge University Press, Cambridge, 1998.
  * [22] Gabriel Navarro and Britta Späth.  On Brauer’s height zero conjecture.  J. Eur. Math. Soc. (JEMS), 16(4):695–747, 2014.
  * [23] Lucas Ruhstorfer.  Jordan decomposition for the Alperin-McKay conjecture.  arXiv e-prints, page arXiv:2010.04499, October 2020.
  * [24] Lucas Ruhstorfer.  Quasi-isolated blocks and the Alperin-McKay conjecture.  arXiv e-prints, page arXiv:2103.06394, March 2021.
  * [25] Benjamin Sambale.  Blocks of finite groups and their invariants, volume 2127 of Lecture Notes in Mathematics.  Springer, Cham, 2014.
  * [26] Britta Späth.  A reduction theorem for the Alperin-McKay conjecture.  J. Reine Angew. Math., 680:153–189, 2013.
  * [27] Britta Späth.  Reduction theorems for some global-local conjectures.  In Local representation theory and simple groups, EMS Ser. Lect. Math., pages 23–61. Eur. Math. Soc., Zürich, 2018.