and $\mathcal{J}(\chi)$ have the same stabilizer in
$\tilde{G}_{b}\langle\phi\rangle$. By construction of the map $\mathcal{J}$,
for $\chi\in\operatorname{Irr}_{0}(b)$ and $\tau\in\langle\phi\rangle_{\chi}$
it follows that $\tau(\mathcal{J}(\chi))$ and $\mathcal{J}(\tau(\chi))$ are in
the same $\tilde{G}(\tilde{s})_{c}$-orbit. However since the order of $\phi$
is coprime to the order of
$\tilde{G}(\tilde{s})_{c}/G(s)\operatorname{Z}(\tilde{G})$ it follows that
necessarily $\tau(\mathcal{J}(\chi))=\mathcal{J}(\tau(\chi))$. ∎

This deals with all cases where the block stabilizer is cyclic except for case
(2) and (8) in type $E_{6}$ and their Ennola duals.

######  Lemma 8.4.

Suppose that we are either in case (2) or (8) of type $E_{6}$. Let $c_{0}$ be
the unique block associated to the union $\mathcal{E}_{2}(G^{\circ}(s),s)$ of
Lusztig series. Then there exists an
$\operatorname{N}_{\tilde{G}\mathcal{B}}(G(s))_{b}$-equivariant bijection
$\operatorname{Irr}_{0}(G^{\circ}(s),c_{0})\to\operatorname{Irr}_{0}(G,b)$.

###### Proof.

Using Theorem 1.2 and the proof of Lemma 2.3, we find that the Jordan
decomposition
$\tilde{\mathcal{J}}:\mathcal{E}_{2}(\tilde{G},\tilde{s})\to\mathcal{E}_{2}(\tilde{G}(\tilde{s}),\tilde{s})$
is
$\operatorname{N}_{\tilde{G}\mathcal{B}}(G(s),s)\times\operatorname{Irr}(\tilde{G}/G)_{2}$-equivariant.

Our assumption implies by Lemma 2.5 that every $\tilde{G}$-orbit of a
character in $\mathcal{E}_{2}(G,s)$ has size three and there exists a unique
character in this orbit which lies in the block $b$. It follows that the map
$b\operatorname{Res}_{G}^{\tilde{G}}:\mathcal{E}_{2}(\tilde{G},\tilde{s})/\operatorname{Irr}(\tilde{G}/G)_{2}\to\operatorname{Irr}(G,b)$
is a bijection. It follows from the proof of Lemma 2.5 that the map
$\operatorname{Res}_{G^{\circ}(s)}^{\tilde{G}(\tilde{s})}:\mathcal{E}_{2}(\tilde{G}(\tilde{s}),\tilde{s})/\operatorname{Irr}(\tilde{G}(\tilde{s})/G^{\circ}(s))_{2}\to\operatorname{Irr}(G^{\circ}(s),c_{0})$
is a bijection as well. The composition of these bijections therefore yields
an $\mathrm{N}_{\tilde{G}\mathcal{B}}(G(s))_{b}$-equivariant bijection as
claimed. ∎

###  8.3. Central characters

Let $\mathbf{H}$ be a semisimple algebraic group with Frobenius endomorphism
$F:\mathbf{H}\to\mathbf{H}$ definining an $\mathbb{F}_{q}$-structure. Recall
that by [MT11, Proposition 9.15] there exist natural isogenies
$\tau:\mathbf{H}_{\operatorname{sc}}\to\mathbf{H}$ and
$\pi:\mathbf{H}\to\mathbf{H}_{\operatorname{ad}}$.

######  Lemma 8.5.

Assume that $\mathbf{H}$ is semisimple and
$\mathbf{H}_{\operatorname{sc}}^{F}$ is perfect and suppose that
$\tau(\operatorname{Z}(\mathbf{H}_{\operatorname{sc}})^{F})=\operatorname{Z}(\mathbf{H})^{F}$.
Then $\operatorname{Z}(\mathbf{H})^{F}\subset[\mathbf{H}^{F},\mathbf{H}^{F}]$.

###### Proof.

By [MT11, Proposition 24.21] (whose assumption can be weakened to $\mathbf{H}$
semisimple)
$\tau(\mathbf{H}_{\operatorname{sc}}^{F})=[\mathbf{H}^{F},\mathbf{H}^{F}]$.
Therefore,
$\operatorname{Z}(\mathbf{H})^{F}=\tau(\operatorname{Z}(\mathbf{H}_{\operatorname{sc}})^{F})\subseteq\tau(\mathbf{H}_{\operatorname{sc}}^{F})=[\mathbf{H}^{F},\mathbf{H}^{F}]$.
∎

The assumption
$\tau(\operatorname{Z}(\mathbf{H}_{\operatorname{sc}})^{F})=\operatorname{Z}(\mathbf{H})^{F}$
on the center in the previous lemma is for instance satisfied if
$\mathbf{H}=\mathbf{H}_{\operatorname{sc}},\mathbf{H}_{\operatorname{ad}}$ or
$\operatorname{Z}(\mathbf{H}_{\operatorname{sc}})^{F}=\operatorname{Z}(\mathbf{H}_{\operatorname{sc}})$.

The following lemma can be compared to [Mal08, Proposition 4.8]:

######  Lemma 8.6.

Let $H$ be as in the previous lemma and $\ell\neq p$ be arbitrary. Then the
characters of $\operatorname{Irr}_{0}(B_{0}(H))$ have $\operatorname{Z}(H)$ in
their kernel.

Moreover, there exists a bijection between the set of
$H^{1}(F,\mathrm{Z}(\mathbf{H}))_{\ell}$-stable characters of
$\operatorname{Irr}_{0}(B_{0}(H))$ and orbits
$\operatorname{Irr}_{0}(B_{0}(H_{\operatorname{ad}}))/\operatorname{Irr}(H_{\operatorname{ad}}/\pi(H))_{\ell}$.

###### Proof.

Let $\chi\in\operatorname{Irr}_{0}(B_{0}(H))$ and $\lambda=\mathrm{det}(\chi)$
its associated determinantal character. Since $\lambda$ is linear it follows
from Lemma 8.5 that $\lambda$ has $\operatorname{Z}(H)$ in its kernel. For
$z\in\operatorname{Z}(H)_{\ell}$ there exists an $\ell$-power root of unity
$\zeta$ such that $\chi(z)=\chi(1)\zeta$ and $1=\lambda(z)=\zeta^{\chi(1)}$.
Since $\chi$ is of $\ell^{\prime}$-degree, it follows that $\zeta=1$ and we
conclude that $\operatorname{Z}(H)_{\ell}$ is in the kernel of $\chi$.
Moreover, $\chi$ has $\operatorname{Z}(H)_{\ell^{\prime}}$ in its kernel since
it lies in the principal block. Hence, any $\ell^{\prime}$-character of the
principal block of $H$ is trivial on the center of $H$.

Now we proceed as in the proof of [CE04, Section 17]. Take
$\iota:\mathbf{H}\to\tilde{\mathbf{H}}$ a regular embedding and
$\pi:\tilde{\mathbf{H}}\to\mathbf{H}_{\operatorname{ad}}$ the quotient map. We
define a map from the set of
$H^{1}(F,\mathrm{Z}(\mathbf{H}))_{\ell}$-invariant characters of
$\operatorname{Irr}_{0}(B_{0}(H))$ to
$\operatorname{Irr}_{0}(B_{0}(H_{\mathrm{ad}}))$ as follows. Any
$H^{1}(F,\mathrm{Z}(\mathbf{H}))_{\ell}$-stable character
$\chi\in\operatorname{Irr}_{0}(B_{0}(H))$ extends to an
$\ell^{\prime}$-character $\tilde{\chi}$ of $\tilde{H}$. This character can be
assumed to have $\operatorname{Z}(\tilde{H})$ in its kernel and to lie in the
principal block of $\tilde{H}$ by [CE04, Theorem 15.11]. The character
$\tilde{\chi}$ considered as a character of
$\tilde{H}/\operatorname{Z}(\tilde{H})$ lies again in the principal block of
$H_{\operatorname{ad}}$. Note that if we were to choose a different character
$\tilde{\chi}^{\prime}$ with the same properties as $\tilde{\chi}$, then
$\tilde{\chi}^{\prime}=\lambda\tilde{\chi}$ for some
$\lambda\in\operatorname{Irr}(\tilde{H}/H\mathrm{Z}(\tilde{H}))$ of
$\ell$-power order. As
$H_{\operatorname{ad}}/\pi(H)\cong\tilde{H}/H\mathrm{Z}(\tilde{H})$ by [CE04,
Equation 15.4] it follows that the images of $\tilde{\chi}^{\prime}$ and
$\tilde{\chi}$ in $H_{\operatorname{ad}}$ are in the same
$\operatorname{Irr}(H_{\operatorname{ad}}/\pi(H))_{\ell}$-orbit. The
bijectivity of the map follows easily from this. ∎

In particular, if $\ell\nmid|H^{1}(F,\operatorname{Z}(\mathbf{H}))|$ then the
sets $\operatorname{Irr}_{0}(B_{0}(H))$ and
$\operatorname{Irr}_{0}(B_{0}(H_{\operatorname{ad}}))$ are in bijection. We
obtain the following consequence for our situation:

######  Lemma 8.7.

As before, let $G$ be an exceptional group of Lie type and $b$ be a quasi-
isolated $2$-block of $G$. Then
$\operatorname{Z}(G)_{2}\leq\mathrm{Ker}(\chi)$ for every height zero
character $\chi\in\operatorname{Irr}_{0}(G,b)$.

###### Proof.

We can assume that $G=E_{7}(q)$, since $\operatorname{Z}(G)_{2}=1$ otherwise.
Let $s\in G^{\ast}$ be the quasi-isolated element associated to the block $b$.
By Lemma 2.3 it suffices to show that every height zero character in the
principal block $B_{0}(G(s))$ has $\operatorname{Z}(G)_{2}$ in its kernel. Let
$\tau:\mathbf{G}(s)_{\operatorname{sc}}\to\mathbf{G}(s)$ be the simply
connected covering of the connected reductive group $\mathbf{G}(s)$. An
inspection of [KM13, Table 4] shows that
$\operatorname{Z}(\mathbf{G}(s)_{\operatorname{sc}})=\operatorname{Z}(\mathbf{G}(s)_{\operatorname{sc}})^{F}$.
Therefore, by Lemma 8.6 every $2^{\prime}$-character of $G(s)$ has
$\operatorname{Z}(G(s))$ in its kernel. ∎

##  9\. The inductive AM-condition

###  9.1. A criterion for the iAM-condition

We recall the definition of AM-good, see also [Ruh22, Definition 1.9]. For the
precise definition of the order relation ”$\geq_{b}$” (which we will not need
in the following) see also [Ruh22, Section 1].

######  Definition 9.1.

Let $H$ be a finite group and $b$ a block of $H$ with non-central defect group
$D$. Assume that for $\Gamma:=\mathrm{N}_{\mathrm{Aut}(H)}(D,b)$ there exist

  1. (i)

a $\Gamma$-stable subgroup $M$ with $\mathrm{N}_{H}(D)\leq M\lneq H$;

  2. (ii)

a $\Gamma$-equivariant bijection
$\Psi:\operatorname{Irr}_{0}(H,b)\to\operatorname{Irr}_{0}(M,B)$ where
$B\in\mathrm{Bl}(M\mid D)$ is the unique block with $B^{H}=b$;

  3. (iii)

$(H/Z\rtimes\Gamma_{\chi},H/Z,\overline{\chi})\geq_{b}(M/Z\rtimes\Gamma_{\chi},M/Z,\overline{\Psi(\chi)}),$

for every $\chi\in\operatorname{Irr}_{0}(H,b)$ and
$Z=\mathrm{Ker}(\chi)\cap\mathrm{Z}(H)$, where $\overline{\chi}$ and
$\overline{\Psi(\chi)}$ lift to $\chi$ and $\Psi(\chi)$, respectively.

Then we say that $b$ is AM-good or satisfies the iAM-condition (with respect
to the subgroup $M$). Moreover, we say that
$\Psi:\operatorname{Irr}_{0}(H,b)\to\operatorname{Irr}_{0}(M,B)$ is a strong
iAM-bijection for the block $b$ with respect to the subgroup $M$.

######  Remark 9.2.

If $\Psi$ is a strong iAM-bijection as in the definition above, then we always
have (see [NS14, Lemma 3.12])

$(H\rtimes\Gamma_{\chi},H,\chi)\geq_{b}(M\rtimes\Gamma_{\chi},M,\Psi(\chi)).$

If $\Psi$ satisfies only this weaker condition, we say that $\Psi$ is an iAM-
bijection.

###  9.2. Groups of Lie type

We now assume again that $G$ is an exceptional group of Lie type as in Section
2. We recall the following theorem:

######  Theorem 9.3.

Let $b$ be a block of $G$ with defect group $P$ and $M$ a proper
$\mathrm{Aut}(G)_{b}$-stable subgroup of $G$ containing
$\operatorname{N}_{G}(P)$. In addition, denote by $B$ the unique block of $M$
with $B^{G}=b$ and we set $\tilde{M}:=M\operatorname{N}_{\tilde{G}}(P)$. Let
$\chi\in\operatorname{Irr}(G,b)$ and $\chi^{\prime}\in\operatorname{Irr}(M,B)$
such that the following holds:

  1. (i)

We have $(\tilde{G}\mathcal{B})_{\chi}=\tilde{G}_{\chi}\mathcal{B}_{\chi}$ and
$\chi$ extends to $G\mathcal{B}_{\chi}$.

  2. (ii)

We have
$(\operatorname{N}_{\tilde{G}}(M)\mathrm{N}_{G\mathcal{B}}(M))_{\chi^{\prime}}=\operatorname{N}_{\tilde{G}}(M)_{\chi^{\prime}}\mathrm{N}_{G\mathcal{B}}(M)_{\chi^{\prime}}$
and $\chi^{\prime}$ extends to $\mathrm{N}_{G\mathcal{B}}(M)_{\chi^{\prime}}$.

  3. (iii)

$(\tilde{G}\mathcal{B})_{\chi}=G(\operatorname{N}_{\tilde{G}}(M)\mathrm{N}_{G\mathcal{B}}(M))_{\chi^{\prime}}$.

  4. (iv)

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

     * •

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

     * •

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

  5. (v)

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

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

$((\tilde{G}\mathcal{B})_{\chi}/Z,G/Z,\overline{\chi})\geq_{b}((\operatorname{N}_{\tilde{G}}(M)\mathrm{N}_{G\mathcal{B}}(M))_{\chi^{\prime}}/Z,M/Z,\overline{\chi^{\prime}}),$

where $\overline{\chi}$ and $\overline{\chi^{\prime}}$ are the characters
which inflate to $\chi$, respectively $\chi^{\prime}$.

###### Proof.

This is a consequence of [Ruh22, Theorem 2.1] and [Ruh22, Lemma 2.2]. ∎

Assumption (i) can essentially always be satisfied:

######  Theorem 9.4.

In every $\tilde{G}$-orbit of $\operatorname{Irr}(G)$ there exists a character
$\chi$ satisfying assumption (i) of Theorem 9.3.

###### Proof.

See [CS19, Theorem B]. ∎

######  Lemma 9.5.

Let $\chi\in\operatorname{Irr}(G,b)$. Then block induction yields a bijective
map $\mathrm{Bl}(\tilde{G}_{\chi}\mid b)\to\mathrm{Bl}(\tilde{G}\mid b)$.

###### Proof.

By [Nav98, Corollary 6.2] we deduce that block induction yields a surjective
map $\mathrm{Bl}(\tilde{G}_{\chi}\mid b)\to\mathrm{Bl}(\tilde{G}\mid b)$. We
therefore only need to show injectivity.

If $\tilde{G}_{b}=\tilde{G}_{\chi}$, then this follows from [Nav98, Theorem
9.14]. Otherwise, since $\tilde{G}/G\operatorname{Z}(\tilde{G})$ has prime
order (or order $1$) in our situation we have $\tilde{G}_{b}=\tilde{G}$ and
$\tilde{G}_{\chi}=G\operatorname{Z}(\tilde{G})$. Since
$\tilde{G}_{\chi}=G\operatorname{Z}(\tilde{G})$ is a central product, we
deduce that $\mathrm{Bl}(\tilde{G}_{\chi}\mid b)$ is in bijection with
$(\operatorname{Z}(\tilde{G})/\operatorname{Z}(G))_{2^{\prime}}$. On the other
hand, by [Bon05, Corollary 2.8, Corollary 2.9] we have
$|\mathrm{Bl}(\tilde{G}\mid
b)|\geq|(\operatorname{Z}(\tilde{G})/\operatorname{Z}(G))|_{2^{\prime}}$.
Hence, the map is necessarily bijective. ∎

As a consequence of the previous lemma, we can in Theorem 9.3 replace
condition (v) by the requirement that
$\mathrm{bl}(\tilde{\chi}^{\prime})^{\tilde{G}}=\mathrm{bl}(\tilde{\chi})$.

###  9.3. A criterion for blocks with cyclic stabilizer

In cases where we have additional information about the block the criterion of
Theorem 9.3 simplifies.

######  Lemma 9.6.

Let $b$ be a strictly quasi-isolated $2$-block of $G$ such that
$\mathrm{Out}(G)_{b}$ is cyclic and let $D$ be a characteristic subgroup of a
defect group $P$ of $b$. Suppose that
$f:\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(D),B)$
is an $\operatorname{N}_{\tilde{G}\mathcal{B}}(D,B)$-equivariant bijection
which preserves central characters. Assume that one of the following holds:

  1. (a)

$\tilde{G}_{b}=G\operatorname{Z}(\tilde{G})$.

  2. (b)

$\operatorname{Z}(G)$ is a $2$-group and every character of
$\operatorname{Irr}_{0}(G,b)$ has $\operatorname{Z}(G)$ in its kernel.

Then the block $b$ is AM-good with respect to the subgroup
$\operatorname{N}_{G}(D)$.

###### Proof.

We show that $\chi\in\operatorname{Irr}_{0}(G,b)$ and $\chi^{\prime}:=f(\chi)$
satisfy the assumptions of Theorem 9.3. Firstly, by Theorem 9.4 we can assume
by possibly replacing $\chi$ by a $\tilde{G}$-conjugate of it that $\chi$ and
$\chi^{\prime}$ satisfy assumptions (i)-(iii) of Theorem 9.3.

Let’s suppose first that (a) holds. By assumption $\chi$ and $\chi^{\prime}$
lie over the same character of $\operatorname{Z}(G)$. Observe that
$G\operatorname{Z}(\tilde{G})$ is a central product of $G$ and
$\operatorname{Z}(\tilde{G})$. Hence, there exist extensions
$\tilde{\chi}_{0}$ and $\tilde{\chi}_{0}^{\prime}$ which lie over the same
character of $\operatorname{Z}(\tilde{G})$. Therefore, we have
$\mathrm{bl}(\tilde{\chi}_{0})=\mathrm{bl}(\tilde{\chi}_{0}^{\prime})^{\operatorname{Z}(\tilde{G})G}$.
Hence, condition (v). Since the extension of $\chi$ to
$G\operatorname{Z}(\tilde{G})$ is uniquely determined by its values on
$\operatorname{Z}(\tilde{G})$ it follows that the characters $\tilde{\chi}$
and $\tilde{\chi}^{\prime}$ satisfy assumption (iv).

Assume now that (b) holds. We can consider $\chi$ as a character of the simple
group $S=G/\operatorname{Z}(G)$. As $\mathrm{Out}(G)_{b}$ is cyclic, the
character $\chi$ extends to a character $\hat{\chi}$ of
$(\tilde{G}\mathcal{B})_{\chi}$ with $\operatorname{Z}(\tilde{G})$ in its
kernel. Denote by $\tilde{\chi}_{0}$ the restriction of $\hat{\chi}$ to
$\tilde{G}_{\chi}$. Let $\overline{b}$ be the unique block of
$G/\operatorname{Z}(G)$ which is dominated by $b$, see [Nav98, Theorem 7.6].
By assumption $G/\operatorname{Z}(G)$ is a normal subgroup of $2$-power index
in $\tilde{G}/\operatorname{Z}(\tilde{G})$. Hence, there exists a unique block
$\overline{\tilde{b}}$ of $\tilde{G}/\operatorname{Z}(\tilde{G})$ which covers
$\overline{b}$, see [Nav98, Corollary 9.6]. Thus, the character $\tilde{\chi}$
necessarily lies in the block $\tilde{b}$ dominating $\overline{\tilde{b}}$.
We observe that the same arguments apply to the local character
$\chi^{\prime}$.

Now assumption (iv) in Theorem 9.3 follows from observing that $\tilde{\chi}$
and $\tilde{\chi}^{\prime}$ are both
$\operatorname{N}_{G\mathcal{B}}(D)_{\chi^{\prime}}$-stable as $\tilde{\chi}$
extends to the character
$\hat{\chi}\in\operatorname{Irr}((\tilde{G}\mathcal{B})_{\chi})$ and similarly
$\chi^{\prime}$ extends to
$\operatorname{N}_{(\tilde{G}\mathcal{B})_{\chi}}(D)$. For assumption (v)
observe that $\tilde{\chi}$ and $\tilde{\chi}^{\prime}$ lie in the unique
block covering $b$ resp. $B_{D}$ whose characters have
$\operatorname{Z}(\tilde{G})_{2^{\prime}}$ in their kernel. ∎

##  10\. Blocks with non cyclic-stablizer

###  10.1. Quasi-isolated blocks with non-cyclic stabilizer

As before, let $b$ be a $2$-block associated to a quasi-isolated semisimple
element $s$. In this section, we assume that $\mathrm{Out}(G)_{b}$ is non-
cyclic. In particular, this means by Lemma 8.1 and Lemma 8.2 that we are in
one of the following cases, where $\varepsilon\in\\{\pm 1\\}$:

  * •

$G=E_{6}(\varepsilon q)$ and $s$ has centralizer $A_{2}(\varepsilon q)^{3}.3$

  * •

$G=E_{6}(\varepsilon q)$ and $s$ has centralizer $D_{4}(\varepsilon q).3$

  * •

$G=E_{7}$ with $e=1$ and $s$ with centralizer $A_{5}(q)A_{2}(q)$.

We make the following useful observation:

######  Corollary 10.1.

Let $b$ be a quasi-isolated block of $G$ associated to the semisimple element
$s$ such that $\mathrm{Out}(G)_{b}$ is non-cyclic. Then the following hold:

  1. (a)

We have $b=e_{s}^{\mathbf{G}^{F}}$ and $\lambda\in\mathcal{E}(L,s)$ is a
character of central $2$-defect with $b_{L}(\lambda)=e_{s}^{\mathbf{L}^{F}}$.

  2. (b)

Assume that $s$ is not a semisimple element in $E_{6}$ with centralizer of
type $A_{2}^{3}$ and let $\mathbf{L}_{0}$ be the $1$-split Levi subgroup
associated to $s$. Then we have
$\mathbf{L}^{\ast}=\operatorname{C}_{\mathbf{L}^{\ast}}(s)$ and
$\mathbf{L}_{0}^{\ast}=\operatorname{C}_{\mathbf{L}_{0}^{\ast}}(s)$. In
particular, $\mathbf{L}^{\ast}$ (resp. $\mathbf{L}_{0}^{\ast}$) is the
centralizer of a Sylow $e$-torus (resp. Sylow $1$-torus) of
$(\mathbf{G}^{\ast},F)$.

###### Proof.

That $b=e_{s}^{\mathbf{G}^{F}}$ follows from Lemma 2.5. Moreover, $\lambda$ is
of central defect by using [KM13, Proposition 2.7(g)] and consulting the
relevant tables in [KM13]. Moreover, we see from these tables that
$\operatorname{C}_{\mathbf{L}^{\ast}}(s)=\operatorname{C}^{\circ}_{\mathbf{L}^{\ast}}(s)$
is a torus and hence $e_{s}^{\mathbf{L}^{F}}$ is a $2$-block which yields
$b_{L}(\lambda)=e_{s}^{\mathbf{L}^{F}}$. Part (b) follows again by an
inspection of the relevant tables in [KM13]. ∎

We let $B$ be the unique block of $\operatorname{N}_{G}(\mathbf{L})$ covering
$e_{s}^{\mathbf{L}^{F}}$ such that $B^{G}=b$, see Lemma 6.6. Our aim in this
and the next section is to show that there exists an iAM-bijection
$\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)$.
Because of part (b) of Corollary 10.1 we will need to focus particulary on the
semisimple element in $E_{6}(\varepsilon q)$ with centralizer
$A_{2}(\varepsilon q)^{3}.3$.

###  10.2. Chevalley presentation and the extended Weyl group

Let $\mathbf{G}$ be a simply connected simple algebraic group with root system
$\Phi$ and base $\Delta$. We let
$x_{\alpha}:\overline{\mathbb{F}}_{q}\to\mathbf{G}$, $\alpha\in\Phi$, be the
Chevalley generators of $\mathbf{G}$ so that $\mathbf{G}=\langle
x_{\alpha}(t)\mid\alpha\in\Phi,t\in\overline{\mathbb{F}}_{q}\rangle$. We let
$\mathbf{X}_{\alpha}$ denote the image of $x_{\alpha}$ and set
$\mathbf{T}:=\langle h_{\alpha}(t)\mid\alpha\in\Delta\rangle$. As $\mathbf{G}$
is simply connected the torus $\mathbf{T}$ is the direct product of the
subgroups $h_{\alpha}(\overline{\mathbb{F}}_{q}^{\times})$, $\alpha\in\Delta$.
Let $W:=\operatorname{N}_{\mathbf{G}}(\mathbf{T})/\mathbf{T}$ the Weyl group
of $\mathbf{G}$ with respect to the maximal torus $\mathbf{T}$ and $w_{0}\in
W$ the longest element in $W$. Denote by
$S:=\\{s_{\alpha}\mid\alpha\in\Delta\\}$ the set of simple reflections such
that $W=\langle s_{\alpha}\mid\alpha\in W\rangle$.

Set $V:=\langle
n_{\alpha}(1)\mid\alpha\in\Phi\rangle\subset\operatorname{N}_{\mathbf{G}}(\mathbf{T})$.
We have a surjective map $V\to W$ with kernel $H:=V\cap\mathbf{T}$ which is an
elementary abelian $2$-group of rank $|\Delta|$. By Matsumotos’s lemma we have
a canonical set-theoretic splitting $r:W\to V,w\,\mapsto
n_{\alpha_{1}}(1)\cdots n_{\alpha_{r}}(1)$, where $w=s_{\alpha_{1}}\cdots
s_{\alpha_{r}}$ is a reduced expression of $w$.

We will slightly deviate from the definition of the Frobenius endomorphism
used in Section 4.2, which will make it easier to exploit Ennola duality. For
this let $F_{q}$ be the Frobenius endomorphism acting as $x_{\alpha}(t)\mapsto
x_{\alpha}(t^{q})$ for $t\in\overline{\mathbb{F}}_{q}$, $\alpha\in\Delta$.
Moreover, if $\mathbf{G}$ is of type $E_{6}$, let
$\gamma:\mathbf{G}\to\mathbf{G}$ be the graph automorphism given by
$\gamma(x_{\alpha}(t))=x_{\tau(\alpha)}(t)$ for
$t\in\overline{\mathbb{F}}_{q}$, $\alpha\in\Delta$. Here, $\tau$ denotes the
unique non-trivial symmetry of the associated Dynkin diagram. For
$\varepsilon\in\\{1,-1\\}$ we then define the Frobenius endomorphism $F$ to be

$F:=F_{q}\text{ if }\varepsilon=1\text{ and }F:=F_{q}\gamma_{0}\text{ if
}\varepsilon=-1,$

where $\gamma_{0}:=\gamma\tilde{w}_{0}$ with $\tilde{w}_{0}:=r(w_{0})\in V$
the canonical representative of the longest element in the Weyl group. By
[MS16, Lemma 3.12] we know that $\operatorname{C}_{V}(\tilde{w}_{0}\gamma)=V$.

Let $s\in(\mathbf{T}^{\ast})^{F}$ be the representative constructed in [Bon05,
Theorem 5.1] of the rational conjugacy class of one of the elements relevant
in Corollary 10.1. Set $W(s):=\\{w\in W^{\ast}\mid{}^{w}s=s\\}$ and let
$W^{\circ}(s)$ the Weyl group of
$\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ relative to the maximal
torus $\mathbf{T}^{\ast}$. We let $v\in V$ be the canonical representative (of
a $W^{\circ}(s)$-conjugate) of the longest element of the Weyl group
$W^{\circ}(s)$. For $s$ with centralizer of type $A_{2}(\varepsilon q)^{3}.3$
the above constructions will be made more explicit later. As the image of $v$
in $W$ lies in $W^{\circ}(s)$ there exists by Lang’s theorem an element
$g\in\mathbf{G}^{\circ}(s)$ such that $gF(g)^{-1}=v$.

######  Notation 10.2.

For an integer $d\in\\{1,2\\}$ and a sign $\varepsilon\in\\{\pm\\}$ we define
a new integer $d_{\varepsilon}$ as $d_{\varepsilon}:=d$ if $\varepsilon=1$ and
$d_{\varepsilon}:=d+(-1)^{d+1}$ if $\varepsilon=-1$. We define
$v_{(d_{\varepsilon})}:=1$ if $d=1$ and $v_{(d_{\varepsilon})}:=v$ if $d=2$.

###  10.3. Construction of the semisimple element

In this subsection, let $\mathbf{G}$ be of type $E_{6}$. We will now focus on
the element $s$ whose centralizer has type $A_{2}(\varepsilon q)^{3}.3$. We
will first construct a concrete representative of the conjugacy class of $s$.
For this, let $\tilde{\alpha}$ be the highest root of $\Phi$ and set
$\alpha_{0}=-\tilde{\alpha}$. We label the simple roots in the affine Dynkin
diagram of $E_{6}$ as follows:

[extended,label,label macro/.code=α_#1, edge length=.75cm]E6

Recall that the highest root satisfies
$\tilde{\alpha}=\alpha_{1}+\alpha_{6}+2\alpha_{3}+2\alpha_{5}+2\alpha_{2}+3\alpha_{4}$
and that

$\operatorname{Z}(\mathbf{G})=\\{h_{\alpha_{1}}(\omega)h_{\alpha_{3}}(\omega^{2})h_{\alpha_{5}}(\omega)h_{\alpha_{6}}(\omega^{2})\mid\omega^{3}=1\\}$

by [GLS98, Table 1.12.6].

For $\omega\in\overline{\mathbb{F}}_{p}^{\times}$ of order $3$, we consider
the semisimple element

$s_{0}:=h_{\alpha_{1}}(\omega^{2})h_{\alpha_{3}}(\omega)h_{\alpha_{5}}(\omega)h_{\alpha_{6}}(\omega^{2})\in\mathbf{T}^{F}$

with image $s:=\pi(s_{0})\in\mathbf{T}^{\ast}$.

It follows that $s_{0}$ is $\gamma$-stable, where $\gamma$ is the graph
automorphism of order $2$ of $E_{6}$, and
$F_{p}(s_{0})s_{0}^{-1}\in\operatorname{Z}(\mathbf{G})$ if and only if
$p\equiv 1\mod 3$. In particular, $s$ is $F$-stable if and only if $q\equiv
1_{\varepsilon}\mod 3$. In this case, the element $s$ is isolated with
connected centralizer of type $(A_{2}(\varepsilon q))^{3}$. On the other hand,
if $q\equiv 2_{\varepsilon}\mod 3$, then $s$ is $w_{0}F$-stable.

A base of the root system of $\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$
is given by $\Delta(s)=(\Delta\setminus\\{\alpha_{4}\\})\cup\\{\alpha_{0}\\}.$
Let $A_{1}:=\\{\alpha_{1},\alpha_{3}\\}$, $A_{2}:=\\{\alpha_{5},\alpha_{6}\\}$
and $A_{3}:=\\{\alpha_{0},\alpha_{2}\\}$. Denote by $\mathbf{G}_{i}:=\langle
x_{\alpha}(t)\mid t\in\overline{\mathbb{F}}_{p},\alpha\in A_{i}\rangle$ the
reductive group with base $A_{i}$. Recall that
$\mathrm{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ is a semisimple algebraic group as
the element $s$ is isolated. We obtain
$\mathbf{G}^{\circ}(s)=\mathbf{G}_{1}\mathbf{G}_{2}\mathbf{G}_{3}$, where
$\mathbf{G}_{i}\cong\operatorname{SL}_{3}(\overline{\mathbb{F}}_{p})$. There
exists an $F$-stable central subgroup $\mathbf{Z}$ of order $3$ of
$\mathbf{G}(s)$ such that $\mathbf{G}_{i}\cap\mathbf{G}_{j}=\mathbf{Z}$ for
all $i\neq j$. We let $\mathbf{\mathbf{T}}_{i}=\mathbf{T}\cap\mathbf{G}_{i}$
such that $\mathbf{T}$ is a central product
$\mathbf{T}=\mathbf{T}_{1}\mathbf{T}_{2}\mathbf{T}_{3}$. We define
$v_{1}:=n_{\alpha_{1}}(1)$, $v_{2}:=n_{\alpha_{0}}(1)$,
$v_{3}:=n_{\alpha_{6}}(1)$, and denote

$v:=v_{1}v_{2}v_{3}.$

As before, we let $g\in\mathbf{G}^{\circ}(s)$ such that $gF(g)^{-1}=v$.

###  10.4. Construction of Levi subgroups

Let $\mathbf{G}$ be again of exceptional type. For a subset $I$ of the simple
roots $S$ we denote by
$\mathbf{L}_{I}=\mathbf{T}\langle\mathbf{X}_{\alpha}\mid\alpha\in\Phi_{I}\rangle$
the standard Levi subgroup of $\mathbf{G}$ associated to it.

######  Lemma 10.3.

Assume the notation from above and let $s\in(\mathbf{T}^{\ast})^{F}$ be one of
the quasi-isolated elements appearing in Corollary 10.1. Then the following
hold:

  1. (a)

The torus $\mathbf{T}$ is the $1_{\varepsilon}$-split Levi subgroup associated
to the semisimple element $s\in(\mathbf{G}^{\ast})^{F}$.

  2. (b)

Assume that $\mathbf{G}$ is of type $E_{6}$ and let $I=\\{\alpha_{4}\\}$ if
$s$ has rational type $A_{2}(\varepsilon q)^{3}.3$ and $I=\emptyset$
otherwise. Then the Levi subgroup ${}^{g}\mathbf{L}_{I}$ is the
$2_{\varepsilon}$-split Levi subgroup associated to the semisimple element
$s\in(\mathbf{G}^{\ast})^{F^{\ast}}$ with $F$-fixed points isomorphic to
$({}^{g}\mathbf{L}_{I})^{F}\cong\mathbf{L}_{I}^{vF}$.

###### Proof.

Recall that by construction $s\in(\mathbf{T}^{\ast})^{F}$ and
$(\mathbf{T}^{\ast})^{F}$ has polynomial order $\Phi_{1_{\varepsilon}}^{6}$.
Since $\mathbf{T}$ is a maximal torus of $\mathbf{G}$ it follows that
$\operatorname{C}_{\mathbf{G}}(\mathbf{T})=\mathbf{T}$, whence part (a).

For part (b) recall that the Lang image under $F$ of the element
$g\in\mathbf{G}^{\circ}(s)$ is $v$ and so
$\mathbf{L}^{F}\cong\mathbf{L}_{I}^{vF}$. If $\operatorname{C}_{G^{\ast}}(s)$
has type $D_{4}(\varepsilon q)^{3}.3$ the result in part (b) follows from
Corollary 10.1(b) and the fact that $v\in W^{\circ}(s)$ is conjugate to the
longest element in $W$. If $s$ has centralizer of type $A_{2}(\varepsilon
q)^{3}.3$, then the torus
$(\mathbf{T}^{g})^{F}\leq\operatorname{C}_{\mathbf{G}^{\ast}}(s)$ has order

$|\mathbf{T}^{vF}|=|\mathbf{T}_{1}^{vF}||\mathbf{T}_{2}^{vF}||\mathbf{T}_{3}^{vF}|=\Phi_{1}(q)^{3}\Phi_{2}(q)^{3}.$

In particular, $\mathbf{T}^{g}$ contains a Sylow $2_{\varepsilon}$-torus of
$(\mathbf{G}^{\circ}(s),F)$. Hence, by Lemma 1.8 the centralizer
$\operatorname{C}_{\mathbf{G}}(({}^{g}\mathbf{T})_{\Phi_{2_{\varepsilon}}})$
in $\mathbf{G}$ of the Sylow $\Phi_{2_{\varepsilon}}$-torus of
$({}^{g}\mathbf{T},F)$ is the $2_{\varepsilon}$-split Levi subgroup associated
to $s$. We compute that
$\mathbf{L}_{I}\cong\operatorname{C}_{\mathbf{G}}(\mathbf{T}_{\Phi_{2_{\varepsilon}}})$,
where

$\mathbf{T}_{\Phi_{2_{\varepsilon}}}=\\{h_{\alpha_{0}}(t_{0})h_{\alpha_{1}}(t_{1})h_{\alpha_{6}}(t_{6})\mid
t_{0},t_{1},t_{6}\in\overline{\mathbb{F}}_{q}^{\times}\\}$

is the Sylow $\Phi_{2_{\varepsilon}}$-torus of $(\mathbf{T},vF)$. It follows
from this that $\mathbf{L}={}^{g}\mathbf{L}_{I}$ is the
$2_{\varepsilon}$-split Levi subgroup associated to $s$. ∎

In view of the previous lemma, we define
$L_{(1_{\varepsilon})}:=\mathbf{T}^{F}$ and
$N_{(1_{\varepsilon})}:=\operatorname{N}_{\mathbf{G}^{F}}(\mathbf{T},e_{s}^{\mathbf{T}^{F}})$.
Whereas, we define $L_{(2_{\varepsilon})}:=\mathbf{L}_{I}^{vF}$ and
$N_{(2_{\varepsilon})}:=\operatorname{N}_{\mathbf{G}^{vF}}(\mathbf{L}_{I},e_{s}^{\mathbf{L}_{I}^{vF}})$.
For $d\in\\{1,2\\}$ we let $W_{(d)}:=N_{(d)}/L_{(d)}$ and observe that
$W_{(d)}\cong\operatorname{C}_{W(s)}(v_{(d)})$.

###  10.5. Construction of the normalizer

Our aim in this section is to construct a subgroup
$V_{(2_{\varepsilon})}\leq\operatorname{C}_{V}(v)$ such that
$N_{(2_{\varepsilon})}=L_{(2_{\varepsilon})}V_{(2_{\varepsilon})}$.

For this, we first need the following two general observations about Chevalley
relations.

######  Lemma 10.4.

Let $n\in V$ representing $w\in W$. For $\beta\in\Phi$ and
$u\in\overline{\mathbb{F}}_{q}^{\times}$ we have
${}^{n}n_{\beta}(u)=n_{w(\beta)}(\pm u)$ and similarly
${}^{n}x_{\beta}(u)=x_{w(\beta)}(\pm u)$. In particular, if $n$ has odd order
and $w(\beta)=\beta$, then ${}^{n}x_{\beta}(u)=x_{\beta}(u)$.

###### Proof.

The group $V$ is generated by $\\{n_{\alpha}(1)\mid\alpha\in\Phi\\}$, so that
we can write $n=n_{\alpha_{1}}(1)\cdots n_{\alpha_{l}}(1)$ for some $l$. If
$n=n_{\alpha}(1)$ for some $\alpha$ we have
${}^{n}x_{\beta}(u)=x_{w_{\alpha}(\beta)}(\pm u)$ (and
${}^{n}n_{\beta}(u)=n_{w_{\alpha}(\beta)}(\pm u)$) by [Spä06, Satz 2.1.6]. The
statement follows now by induction on $l$. ∎

Recall that $r:W\to V$ denotes the canonical set theoretic splitting of the
quotient map $V\to W$. We have the following lemma, see [AH17, Lemma 3.2].

######  Lemma 10.5.

Suppose $I\subset S$ is a set of simple roots with corresponding Levi factor
$\mathbf{L}_{I}$. Let $\rho^{\vee}(I)$ be one-half the sum of the positive
coroots of $\mathbf{L}_{I}$ and let $z_{I}:=(2\rho^{\vee}(I))(-1)$ be the
principal involution in $\mathbf{L}_{I}$. Then

$r(w_{0}(I))^{2}=z_{I}.$

######  Proposition 10.6.

Let $s$ be the quasi-siolated semisimple element with centralizer
$A_{2}(\varepsilon q)^{3}.3$ constructed above. There exists a section
$V_{(2_{\varepsilon})}$ of $\operatorname{C}_{W(s)}(v)$ under the natural map
$V\to W$ which has the following properties:

  1. (a)

$V_{(2_{\varepsilon})}$ is $\gamma$-stable.

  2. (b)

$V_{(2_{\varepsilon})}\cong C_{4}\wr C_{3}$.

  3. (c)

$V_{(2_{\varepsilon})}\cap H=\langle
h_{\alpha_{1}}(-1),h_{\alpha_{4}}(-1),h_{\alpha_{6}}(-1)\rangle\cong
C_{2}^{3}$.

  4. (d)

$V_{(2_{\varepsilon})}$ centralizes the subgroup
$\langle\mathbf{X}_{\alpha_{4}},\mathbf{X}_{-\alpha_{4}}\rangle$.

###### Proof.

Let $V^{\circ}_{(2_{\varepsilon})}=\langle v_{1},v_{2},v_{3}\rangle$. For
$i\neq j$ the sets $A_{i}$ and $A_{j}$ are orthogonal by definition and all
the roots of the root system $E_{6}$ have the same length. Hence, [Spä06,
Bemerkung 2.1.7(b)] ensures that $[v_{i},v_{j}]=1$. Hence,
$V^{\circ}_{(2_{\varepsilon})}$ is abelian.

Let $\tilde{w}_{0}=r(w_{0})$ be the canonical representative of the longest
element of the Weyl group $W$. Take $\tilde{w}_{0}^{\prime}=r(w_{0}^{\prime})$
to be the canonical representative of the longest element $w_{0}^{\prime}$ of
the Weyl group of type $D_{5}$ associated to the base
$\Delta^{\prime}=\\{\alpha_{2},\alpha_{3},\alpha_{4},\alpha_{5},\alpha_{6}\\}$.
We define $x:=\tilde{w}_{0}\tilde{w}_{0}^{\prime}$. By construction, the
element $x$ induces a triality graph automorphism $\sigma$ on the extended
Dynkin diagram of $E_{6}$. It follows from Lemma 10.5 that $\tilde{w}_{0}$ and
$\tilde{w}_{0}^{\prime}$ both have order $2$. Consequently,

$\gamma(x)=\tilde{w}_{0}{}^{\tilde{w}_{0}}\tilde{w}_{0}^{\prime}=\tilde{w}_{0}^{\prime}\tilde{w}_{0}=x^{-1}.$

The image of $x$ in $W$ has order $3$. Observe that the kernel of the natural
map $V\to W$ is the elementary $2$-group $H$ and thus $x$ has order $3$ or
$6$. We define $v^{\prime}:=x^{4}$ if $x$ has order $6$ and $v^{\prime}:=x$ if
$x$ has order $3$. Thus, the element $v^{\prime}$ has order $3$ and still
induces the graph automorphism $\sigma$ on the extended Dynkin diagram. We
denote $V_{(2_{\varepsilon})}:=\langle
V^{\circ}_{(2_{\varepsilon})},v^{\prime}\rangle$. By construction, the group
$V_{(2_{\varepsilon})}$ is a $\gamma$-stable section of
$\operatorname{C}_{W(s)}(v)$. From the Chevalley relations [Spä06, Satz 2.1.6]
(see also Lemma 10.4) it follows that
${}^{v^{\prime}}n_{\delta}(1)=n_{\sigma(\delta)}(\varepsilon_{\delta})$ for
$\delta\in\\{\alpha,\beta\\}$ and signs $\varepsilon_{\delta}\in\\{\pm 1\\}$
(possibly depending on the choice of structure constants in the Chevalley
groups). This ensures that $v^{\prime}$ normalizes
$V^{\circ}_{(2_{\varepsilon})}$. Since $V^{\circ}_{(2_{\varepsilon})}\cong
C_{4}^{3}$ it follows from this that $V_{(2_{\varepsilon})}\cong C_{4}\wr
C_{3}$ (see Lemma 6.4). From this, we also see that $V_{(2_{\varepsilon})}\cap
H=\langle h_{\alpha_{1}}(-1),h_{\alpha_{0}}(-1),h_{\alpha_{6}}(-1)\rangle\cong
C_{2}^{3}$ with
$h_{\alpha_{0}}(-1)=h_{\alpha_{6}}(-1)h_{\alpha_{4}}(-1)h_{\alpha_{1}}(-1)$.

We are left to show that $V_{(2_{\varepsilon})}$ centralizes the root subgroup
$\mathbf{X}_{\pm\alpha_{4}}$. Note that the roots
$\alpha_{0},\alpha_{1},\alpha_{6}$ are all orthogonal to $\alpha_{4}$. Hence,
by the Chevalley relations (see [Spä06, Bemerkung 2.1.7(b)]) the elements
$n_{\delta}(1)$ with $\delta\in\\{\alpha_{0},\alpha_{1},\alpha_{6}\\}$
centralize the root subgroup $\mathbf{X}_{\pm\alpha_{4}}$. Moreover, the
element $v^{\prime}$ has odd order and thus it centralizes
$\mathbf{X}_{\pm\alpha_{4}}$ by Lemma 10.4. ∎

###  10.6. Automorphisms and Lusztig series of twisted groups

The following notation which follows [MS16, Notation 3.3] will be used for
working with twisted groups.

######  Notation 10.7.

As before let $F$ be the fixed Frobenius endomorphism from 10.2 defining an
$\mathbb{F}_{q}$-structure on $\mathbf{G}$ with $q=p^{m}$. Set
$h:=\mathrm{o}(\gamma)\mathrm{o}(v)$. Let
$E:=\operatorname{C}_{hm}\times\langle\gamma\rangle$ if $\varepsilon=1$ and
$E:=\operatorname{C}_{hm}$ if $\varepsilon=-1$ such that the first factor of
$E$ acts on ${\tilde{\mathbf{G}}}^{F_{p}^{2hm}}$ by $\langle F_{p}\rangle$ and
the second by the group $\langle\gamma\rangle$ generated by graph
automorphisms. Note that this action is faithful. Let
$\widehat{F}_{p},\widehat{\gamma},\widehat{F}\in E$ be the elements that act
on ${\tilde{\mathbf{G}}}^{F_{p}^{2hm}}$ by $F_{p}$, $\gamma$ and $F$,
respectively.

By construction, the element $\hat{F}$ centralizes $\mathbf{\tilde{G}}^{F}$.
We can therefore identify the subgroup
$\mathcal{B}\subset\mathrm{Aut}(\mathbf{\tilde{G}}^{F})$ from before with the
quotient group $E/\langle\hat{F}\rangle$. The following lemma is a consequence
of the proof of [MS16, Proposition 3.4].

######  Lemma 10.8.

Let $v\in V$ be as before and $g\in\mathbf{G}$ with $v=gF(g)^{-1}$. Then the
map

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

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

It is a natural question to ask whether the isomorphism from Lemma 10.8
preserves Lusztig series.

######  Lemma 10.9.

Let $s$ be one of the the semisimple elements in Corollary 10.1 and $v\in
G^{\circ}(s)$ as before. Then the isomorphism
$\iota:\mathbf{G}^{F}\to\mathbf{G}^{vF},x\mapsto{}^{g}x,$ induces a bijection
$\mathcal{E}(\mathbf{G}^{vF},s)\to\mathcal{E}(\mathbf{G}^{F},s)$.

###### Proof.

Recall that $s\in\mathbf{T}^{\ast}$ and $W^{\circ}(s)$ is defined as the Weyl
group of $\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ with respect to its
maximal torus $\mathbf{T}^{\ast}$. Assume that for $z\in\\{1,v\\}$ the duality
between $(\mathbf{G},zF)$ and $(\mathbf{G}^{\ast},Fz^{\ast})$ is defined
around the maximal tori $\mathbf{T}$ and $\mathbf{T}^{\ast}$ with the same
choices, see [DM91, Definition 13.10] and the remarks following it. The
characters in $\mathcal{E}(\mathbf{G}^{vF},s)$ are the constituents of
$R_{\mathbf{S}}^{\mathbf{G}}(\theta^{\prime})$, where
$(\mathbf{S},\theta^{\prime})$ is in duality with
$(\mathbf{S}^{\ast},s^{\prime})$ and $s^{\prime}$ is rationally conjugate to
$s$. In other words, by [DM91, 3.25] there exists $x\in\mathbf{G}$ such that
$\mathbf{T}={}^{x^{-1}}\mathbf{S}$ such that the image $w$ of $x^{-1}(vF)(x)$
in $W^{\ast}$ lies in $W^{\circ}(s)$ and the character
$\theta:={}^{x^{-1}}\theta^{\prime}\in\operatorname{Irr}(\mathbf{T}^{wvF})$
corresponds to $s\in(\mathbf{T}^{\ast})^{w^{\ast}v^{\ast}F^{\ast}}$ under the
isomorphism
$\operatorname{Irr}(\mathbf{T}^{wvF})\to\mathbf{T}^{F^{\ast}v^{\ast}w^{\ast}}$
induced by duality, see [DM91, Proposition 13.11]. It therefore suffices to
show that the irreducible constituents of
${}^{g}R_{\mathbf{S}}^{\mathbf{G}}(\theta^{\prime})$ lie in
$\mathcal{E}(\mathbf{G}^{F},s)$. We have
${}^{g}R_{\mathbf{S}}^{\mathbf{G}}(\theta^{\prime})=R_{{}^{g}\mathbf{S}}^{\mathbf{G}}({}^{g}\theta^{\prime})$
by the proof of [CE04, Proposition 7.13]. It therefore suffices to show that
the pair $({}^{g}\mathbf{S},{}^{g}\theta^{\prime})$ corresponds to
$(\mathbf{S}^{\ast},s)$ under duality. We have
${}^{(gx)^{-1}}({}^{g}\mathbf{S},{}^{g}\theta^{\prime})=(\mathbf{T},\theta)$
with $wv=(gx)^{-1}F(gx)\in W^{\circ}(s)$. By the consistent choice of duality,
the character $\theta$ corresponds to the element
$s\in(\mathbf{T}^{\ast})^{w^{\ast}v^{\ast}F^{\ast}}$ under the duality
isomorphism
$\operatorname{Irr}(\mathbf{T}^{wvF})\to(\mathbf{T}^{\ast})^{Fv^{\ast}w^{\ast}}$.
Thus, $({}^{g}\mathbf{S},{}^{g}\theta^{\prime})$ corresponds to
$(\mathbf{S}^{\ast},s)$ under duality. ∎

The proof of Lemma 10.9 also shows the following:

######  Corollary 10.10.

Let ${}^{g}\mathbf{L}_{I}$ be the $2_{\varepsilon}$-split Levi subgroup
associated to $s$ as in Lemma 10.3. The isomorphism
$\iota:\mathbf{L}_{I}^{F}\to\mathbf{L}_{I}^{vF},x\mapsto{}^{g}x,$ induces a
bijection
$\mathcal{E}(\mathbf{L}_{I}^{vF},s)\to\mathcal{E}(\mathbf{L}_{I}^{F},s)$.

Recall that $W_{(e)}=N_{(e)}/L_{(e)}$. We have $W_{(1_{\varepsilon})}=W(s)$
and $W_{(2_{\varepsilon})}=\operatorname{C}_{W(s)}(v)$ by Lemma 1.10. The
groups $W_{(1)}$ and $W_{(e)}$ have a common Sylow $2$-subgroup $W_{2}$ with
$v_{(1)},v_{(e)}\in\operatorname{Z}(W_{2})$.

######  Lemma 10.11.

There exists a natural bijection
$\Xi:\mathcal{E}_{2}(L_{(1_{\varepsilon})},s)^{W_{2}}\to\mathcal{E}_{2}(L_{(2_{\varepsilon})},s)^{W_{2}}$.

###### Proof.

Let $z\in\\{v_{(1)},v_{(e)}\\}$. By duality we have a bijection
$\operatorname{Irr}(\mathbf{T}^{zF})\to(\mathbf{T}^{\ast})^{F^{\ast}z^{\ast}}$,
which restricts to a $\operatorname{C}_{W(s)}(z)$-equivariant bijection

$\mathcal{E}_{2}(\mathbf{T}^{zF},s)\to
A_{z}:=\\{t\in(\mathbf{T}^{\ast})^{F^{\ast}z^{\ast}}\mid t_{2^{\prime}}=s\\}.$

As $z\in W_{2}$ it follows that $A_{v_{(1)}}^{W_{2}}=A_{v_{(e)}}^{W_{2}}$.
Since $W_{2}\leq\operatorname{C}_{W(s)}(z)$ we therefore obtain a bijection

$\mathcal{E}_{2}(\mathbf{T}^{F},s)^{W_{2}}\to\mathcal{E}_{2}(\mathbf{T}^{vF},s)^{W_{2}}.$

Moreover, by definition
$\mathbf{T}^{\ast}=\operatorname{C}_{\mathbf{L}_{I}^{\ast}}(s)$ and so
Deligne–Lusztig induction yields a bijection
$R_{\mathbf{T}^{vF}}^{\mathbf{L}_{I}^{vF}}:\mathcal{E}_{2}(\mathbf{T}^{vF},s)\to\mathcal{E}_{2}(\mathbf{L}_{I}^{vF},s)$,
which is equivariant for all automorphisms of $\mathbf{L}_{I}$ which commute
with $vF$ and stabilize $\mathbf{T}$. The composition of these bijections
yields a bijection
$\mathcal{E}_{2}(\mathbf{T}^{F},s)^{W_{2}}\to\mathcal{E}_{2}(\mathbf{L}_{I}^{vF},s)^{W_{2}}$
as desired. ∎

###  10.7. Extension maps

Whenever $\mathbf{L}_{(d)}$ is a maximal torus for $d\in\\{1,2\\}$ an
extension map $L_{(d)}\lhd N_{(d)}$ an automorphism-equivariant extension map
is known by the work of Malle–Späth.

By Corollary 10.1 we therefore only need to consider the case when the
centralizer of $s$ is of type $A_{2}(\varepsilon q)^{3}.3$ and
$d=2_{\varepsilon}$. The aim of this section is therefore to give an
automorphism-equivariant extension map for the extension
$L_{(2_{\varepsilon})}\lhd N_{(2_{\varepsilon})}$. For this, first observe
that the semisimple element $s$ is $E^{\prime}:=E$-stable if $p\equiv 1\mod 3$
and $E^{\prime}:=\langle\gamma,w_{0}F_{p}\rangle$-stable if $p\equiv 2\mod 3$.
Hence, for $d\in\\{1,2\\}$ and $G_{(d)}:=\mathbf{G}^{v_{(d)}F}$ we have

$\operatorname{N}_{G_{(d)}E}(L_{(d)},e_{s}^{L_{(d)}})=N_{(d)}E^{\prime}.$

To prove the existence of such an extension map, we need the following lemma.

######  Lemma 10.12.

Let $s$ be the semisimple element with centralizer $A_{2}(\varepsilon
q)^{3}.3$. Then there exists an $E^{\prime}$-equivariant extension map
$\Gamma$ for $Z:=V_{(2_{\varepsilon})}\cap H\lhd V_{(2_{\varepsilon})}$.

###### Proof.

Observe that $w_{0}F_{p}$ acts in the same way as $\gamma$ on $V$. We identify
$V^{\circ}_{(2_{\varepsilon})}=\langle v_{1},v_{2},v_{3}\rangle$ with
$C_{4}^{3}$. For $\nu=\nu_{1}\times\nu_{2}\times\nu_{3}$ we let
$\theta=\theta_{1}\times\theta_{2}\times\theta_{3}\in\operatorname{Irr}(V^{\circ}_{(2_{\varepsilon})}\mid\nu)$
such that $\theta_{i}=\theta_{j}$ whenever $\nu_{i}=\nu_{j}$. Hence,
$(V_{(2_{\varepsilon})}E^{\prime})_{\theta}=(V_{(2_{\varepsilon})}E^{\prime})_{\nu}$.
If $v^{\prime}\notin V_{(2_{\varepsilon})}$ we let $\Gamma(\nu):=\theta$.
Otherwise, since $V_{(2_{\varepsilon})}\cong C_{4}\wr C_{3}$, there exists a
character $\Gamma(\nu)\in\operatorname{Irr}(V_{(2_{\varepsilon})}\mid\theta)$
with $v^{\prime}$ in its kernel. We show that there exists a choice of
$\theta$ such that the so-obtained character $\Gamma(\nu)$ is
$E^{\prime}_{\nu}$-stable. For this we can assume that $\nu$ is
$\langle\gamma,v^{\prime}\rangle$ since the claim is clear otherwise. The
character $\theta:=\theta_{1}\times\theta_{2}\times\theta_{3}$ in
$\operatorname{Irr}(V^{\circ}_{(2_{\varepsilon})}\mid\nu)$ is $\gamma$-stable
if and only if $\theta_{1}=\theta_{3}$. Moreover, $\theta$ can only be
$v^{\prime}$-stable if all $\theta_{i}$ have order $4$. Now, we know that for
$i=1,2,3$ there are signs $\varepsilon_{i}\in\\{\pm 1\\}$ such that
${}^{v^{\prime}}v_{i}=v_{i+1}^{\varepsilon_{i}}$. Since $v^{\prime}$ has order
$3$ this forces $\varepsilon_{1}\varepsilon_{2}\varepsilon_{3}=1$. Moreover,
$\gamma(v^{\prime})=v^{\prime-1}$. In particular, ${}^{\gamma
v^{\prime}\gamma}v_{2}={}^{v^{\prime-1}}v_{2}$. Since $\gamma(v_{2})=v_{2}$
and $\gamma(v_{1})=v_{3}$, we obtain
$v_{1}^{\varepsilon_{2}}=v_{1}^{\varepsilon_{1}}$. We deduce
$\varepsilon_{1}\varepsilon_{2}=1$ which forces $\varepsilon_{3}=1$. In
particular, for any character $\theta_{1}\in\operatorname{Irr}(C_{4})$ of
order $4$, the character
$\theta=\theta_{1}\times\theta_{1}^{\varepsilon_{1}}\times\theta_{1}$ is
$\langle\gamma,v^{\prime}\rangle$-stable and covers the character $\nu$. ∎

For constructing extensions we use the following lemma, which is a consequence
of [Spä12, Lemma 2.11].

######  Lemma 10.13.

Let $\tilde{X}$ be a normal subgroup of a finite group $\tilde{Y}$ and $Y$ a
subgroup of $\tilde{Y}$ with $\tilde{Y}=\tilde{X}Y$. Assume that
$\vartheta\in\operatorname{Irr}(\tilde{X})$ is a $\tilde{Y}$-stable character
which restricts irreducibly to the intersection $X:=\tilde{X}\cap Y$. If the
character $\mathrm{Res}_{X}^{\tilde{X}}(\vartheta)$ extends to a character
$\mu\in\operatorname{Irr}(Y)$, then $\vartheta$ extends to a character of
$\operatorname{Irr}(\tilde{Y}\mid\mu)$.

######  Lemma 10.14.

Let $s$ be the semisimple element with centralizer $A_{2}(\varepsilon
q)^{3}.3$. There exists an extension map $\Lambda_{(2_{\varepsilon})}$ for
$\mathcal{E}_{2}(L_{(2_{\varepsilon})},s)$ with respect to
$L_{(2_{\varepsilon})}\lhd N_{(2_{\varepsilon})}$ such that
$\Lambda_{(2_{\varepsilon})}$ is
$\operatorname{N}_{G_{(2_{\varepsilon})}E}(L_{(2_{\varepsilon})},e_{s}^{L_{(2_{\varepsilon})}})$-equivariant.

###### Proof.

Let $V_{(2_{\varepsilon})}$ be the section of $\operatorname{C}_{W(s)}(v)$
constructed in Proposition 10.6. We can therefore write
$N_{(2_{\varepsilon})}$ as
$N_{(2_{\varepsilon})}=L_{(2_{\varepsilon})}V_{(2_{\varepsilon})}$. The
intersection

$Z=L_{(2_{\varepsilon})}\cap V_{(2_{\varepsilon})}=T_{(2_{\varepsilon})}\cap
V_{(2_{\varepsilon})}=H\cap V_{(2_{\varepsilon})}=\langle
h_{\alpha_{0}}(-1),h_{\alpha_{1}}(-1),h_{\alpha_{6}}(-1)\rangle$

is a central subgroup of $\mathbf{L}_{I}$ by Proposition 10.6. Note that
$\mathbf{L}_{I}=\operatorname{Z}^{\circ}(\mathbf{L}_{I})[\mathbf{L}_{I},\mathbf{L}_{I}]$
and
$[\mathbf{L}_{I},\mathbf{L}_{I}]\cong\mathrm{SL}_{2}(\overline{\mathbb{F}}_{p})$.
By Bonnafé–Rouquier, see the main theorem of [BDR17], the block
$e_{s}^{L_{(2_{\varepsilon})}}$ is Morita equivalent to the principal block of
the torus $T_{(2_{\varepsilon})}$. In particular, this block has
$|(T_{(2_{\varepsilon})})|_{2}$ many irreducible characters.

Let $s_{0}$ be the image of $s$ under the map
$\mathbf{L}_{I}^{\ast}\to[\mathbf{L}_{I},\mathbf{L}_{I}]^{\ast}$ with central
kernel obtained by duality from the inclusion map
$[\mathbf{L}_{I},\mathbf{L}_{I}]\subset\mathbf{L}_{I}$, see [CE04, Section
15.1]. Since $1\neq s_{0}$ is a $2^{\prime}$-element it follows that its
centralizer in
$[\mathbf{L}_{I},\mathbf{L}_{I}]^{\ast}\cong\mathrm{PGL}_{2}(\overline{\mathbb{F}}_{p})$
is again a torus. In particular,
$e_{s_{0}}^{[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]}$ is again a block
with
$|(T_{(2_{\varepsilon})})\cap[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]|_{2}$
many irreducible characters. Since
$e_{s_{0}}^{[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]}$ is the unique
block below $e_{s}^{L_{(2_{\varepsilon})}}$ we deduce by Clifford theory that
every character in
$\operatorname{Irr}([L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}],e_{s_{0}}^{[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]})$
is $L_{(2_{\varepsilon})}$-stable.

Let us now first construct an extension map $\Lambda_{0}$ for
$[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]Z\lhd[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]V_{(2_{\varepsilon})}$.
The group $V_{(2_{\varepsilon})}$ centralizes
$[\mathbf{L}_{I},\mathbf{L}_{I}]=\langle\mathbf{X}_{\alpha_{4}},\mathbf{X}_{-\alpha_{4}}\rangle$
by Proposition 10.6. Hence,
$[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]V_{(2_{\varepsilon})}$ is a
central product. Recall the extension map $\Gamma$ for $Z\lhd
V_{(2_{\varepsilon})}$ from Lemma 10.12. We can therefore define an extension
map $\Lambda_{0}$ by sending a character
$\psi_{0}\in\operatorname{Irr}([L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]Z)$
with $\nu\in\operatorname{Irr}(Z\mid\psi_{0})$ below $\psi_{0}$ to the
character
$\psi_{0}.\Gamma(\nu)\in\operatorname{Irr}([L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]V_{(2_{\varepsilon})})$.
By the equivariance properties of $\Gamma$, it follows that $\Lambda_{0}$ is
$\operatorname{N}_{G_{(2_{\varepsilon})}E}(L_{(2_{\varepsilon})},e_{s}^{L_{(2_{\varepsilon})}})$-equivariant
as well. We now define an extension map $\Lambda_{(2_{\varepsilon})}$ for
$\mathcal{E}_{2}(L_{(2_{\varepsilon})}s)$ with respect to
$L_{(2_{\varepsilon})}\lhd
N_{\Lambda_{(2_{\varepsilon})}}=L_{(2_{\varepsilon})}V_{(2_{\varepsilon})}$ as
follows. For $\psi\in\mathcal{E}_{2}(L_{(2_{\varepsilon})},s)$ its restriction
$\psi_{0}:=\mathrm{Res}_{[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]Z}^{L_{(2_{\varepsilon})}}(\psi)$
to $[L_{(2_{\varepsilon})},L_{(2_{\varepsilon})}]$ is irreducible. By Lemma
10.13 we can therefore define
$\Lambda_{(2_{\varepsilon})}(\psi)\in\operatorname{Irr}(L_{(2_{\varepsilon})}(V_{(2_{\varepsilon})})_{\psi})$
as the unique character which restricts to both
$\psi\in\operatorname{Irr}(L_{(2_{\varepsilon})})$ and
$\Lambda_{0}(\psi_{0})$. By construction the map $\Lambda_{(2_{\varepsilon})}$
is
$\operatorname{N}_{GE}(L_{(2_{\varepsilon})},e_{s}^{L_{(2_{\varepsilon})}})$-equivariant.
∎

###  10.8. Construction of the AM-bijection

To show the automorphism equivariance of the map constructed in Theorem 10.17
we need the following statements which are analogues of [BR21, Proposition
4.4, Proposition 4.5]. For $d\in\\{1,2\\}$ we define
$\tilde{T}_{(d)}:=\mathbf{\tilde{T}}^{v_{(d)}F}$ and let
$\tilde{L}_{(d)}:=L_{(d)}\tilde{T}_{(d)}$ and
$\tilde{N}_{(d)}:=N_{(d)}\tilde{L}_{(d)}$.

######  Proposition 10.15.

Let $d\in\\{1,2\\}$ and $\Lambda_{(d)}$ be the extension map for
$\mathcal{E}_{2}(L_{(d)},s)$ from [MS16, Corollary 3.13] or Corollary 10.14
with respect to $L_{(d)}\lhd N_{(d)}$. Then the map

$\Pi:\mathcal{P}:=\\{(\mu,\eta)\mid\mu\in\operatorname{Irr}(L_{(d)},e_{s}^{L_{(d)}}),\eta\in\operatorname{Irr}(W_{(d)}(\mu))\\}\to\operatorname{Irr}(N_{(d)},e_{s}^{L_{(d)}})$

with
$\Pi(\mu,\eta)=\operatorname{Ind}_{N_{(d)}(\mu)}^{N_{(d)}}(\Lambda_{(d)}(\mu)\eta)$
is surjective and satisfies

  1. (a)

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

  2. (b)

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

  3. (c)

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

###### Proof.

The proof of part (a) and (b) works as in [MS16, Proposition 3.15]. For part
(c) observe that our assumption on the semisimple element $s$ implies that
$\operatorname{C}_{\mathbf{L}^{\ast}}(s)$ is a torus. Hence, for all elements
$t\in\operatorname{C}_{\mathbf{G}^{\ast}}(s)_{2}$ the centralizer
$\operatorname{C}_{\mathbf{L}^{\ast}}(st)$ is connected. Thus, every character
of $\mathcal{E}_{2}(L_{(d)},s)$ is $\tilde{T}_{(d)}$. Hence, also part (c)
follows as in [MS16, Proposition 3.15]. ∎

For the following proposition recall that $W_{(2_{\varepsilon})}$ can be
viewed as a subgroup of $W_{(1_{\varepsilon})}$. The formulation of the
following proposition is slightly different from [BR21, Proposition 4.5] but
the proof is identical.

######  Proposition 10.16.

Let $\mu_{0}\in\mathcal{E}_{2}(L_{(1_{\varepsilon})},s)^{W_{2}}$ with image
$\mu=\Xi(\mu_{0})\in\mathcal{E}_{2}(L_{(2_{\varepsilon})},s)^{W_{2}}$. Suppose
that $\nu_{0}\in\operatorname{Irr}(W_{(1_{\varepsilon})}(\mu_{0}))$ and
$t_{0}\in\tilde{T}_{(1_{\varepsilon})}$ satisfy
${}^{t_{0}}\Lambda_{(1_{\varepsilon})}(\mu_{0})=\Lambda_{(1_{\varepsilon})}(\mu_{0})\nu_{0}$.
Then we have

${}^{t}\Lambda_{(2_{\varepsilon})}(\mu)=\Lambda_{(2_{\varepsilon})}(\mu)\nu,$

where $t:=\iota(t_{0})$ and
$\nu:=\operatorname{Res}^{W_{(1_{\varepsilon})}(\mu_{0})}_{W_{(2_{\varepsilon})}(\mu_{0})}(\nu_{0})$.

######  Theorem 10.17.

Let $b=e_{s}^{G}$ be one of the blocks in Corollary 10.1 and $B$ the unique
block of $\operatorname{N}_{G}(\mathbf{L})$ covering $e_{s}^{L}$. Then there
exists an
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{L})_{B}$-equivariant
bijection
$f:\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)\to\operatorname{Irr}_{0}(G,e_{s}^{G})$.

###### Proof.

We first parametrize the local characters. For this we use the $e$-split Levi
subgroup $\mathbf{L}_{(e)}$. Let $\Lambda^{(e)}$ be the extension map
associated to $L_{(e)}\lhd N_{(e)}$ from Lemma 10.14. Recall that we denote
$W_{(e)}=N_{(e)}/L_{(e)}$. By Proposition 10.15 we obtain a surjective map

$\Pi:\mathcal{P}=\\{(\mu,\eta)\mid\mu\in\operatorname{Irr}(L_{(e)},e_{s}^{L_{(e)}}),\eta\in\operatorname{Irr}(W_{(e)}(\mu))\\}\to\operatorname{Irr}(N_{(e)},e_{s}^{L_{(e)}}),(\mu,\eta)\to\operatorname{Ind}_{N_{(e)}(\mu)}^{N_{(e)}}(\Lambda_{(e)}(\mu)\eta).$

The characters of height zero in the unique block $N_{(e)}$ covering the block
$e_{s}^{L_{(e)}}$ are precisely those which correspond to tuples
$(\mu,\eta)\in\mathcal{P}$ with $2\nmid|W_{(e)}:W_{(e)}(\mu)|$ and
$\eta\in\operatorname{Irr}_{2^{\prime}}(W_{(e)}(\mu))$. By Proposition
10.15(c) we can assume that the pair $(\mu,\eta)$ satisfies additionally
$W_{2}\leq W_{(e)}(\mu)$.

Let us now consider the global characters. We abbreviate
$G_{(1)}:=\mathbf{G}^{v_{(1)}F}$. Every height zero character $\chi$ of
$\mathcal{E}_{2}(G_{(1)},s)$ lies in the principal series, i.e. it is of the
form $\chi=R_{L_{(1)}}^{G_{(1)}}(\mu_{0})_{\eta_{0}}$ for some
$\mu_{0}\in\mathcal{E}_{2}(L_{(1)},s)$ and
$\eta_{0}\in\operatorname{Irr}(W_{(1)}(\mu_{0}))$, see Corollary 3.2. For such
a character to have height zero we must by Corollary 3.3 necessarily have that
$\mu_{0}$ is of height zero with $2\nmid|W_{(1)}:W_{(1)}(\mu_{0})|$ and
$\eta_{0}\in\operatorname{Irr}_{2^{\prime}}(W_{(1)}(\mu_{0}))$.

We now construct our bijection. We can choose the Sylow $2$-subgroup $W_{2}$
of $W_{(1)}$ and $W_{(e)}$ to be $E^{\prime}$-stable and we observe that
$\operatorname{N}_{W_{(e)}}(W_{2})E^{\prime}=\operatorname{N}_{W_{(1)}}(W_{2})E^{\prime}$.
Moreover, recall that by Lemma 10.11 and its proof we have an
$\operatorname{N}_{W_{(e)}}(W_{2})E^{\prime}$-equivariant bijection

$\Xi:\mathcal{E}_{2}(L_{(e)},s)^{W_{2}}\to\mathcal{E}_{2}(L_{(1)},s)^{W_{2}}.$

Fix an $\operatorname{N}_{W_{(e)}}(W_{2})E^{\prime}$-transversal $\mathcal{T}$
of $\mathcal{E}_{2}(L_{(e)},s)^{W_{2}}$. For $\mu\in\mathcal{T}$, the group
$W_{2}$ is a common Sylow $2$-subgroup of $W_{(e)}(\mu)$ and
$W_{(1)}(\mu_{0})$ where $\mu_{0}:=\Xi(\mu)$. As in [BR21, Lemma 4.3] we
construct for every $\mu\in\mathcal{T}$ an
$(\operatorname{N}_{W_{(e)}}(W_{2})E^{\prime})_{\mu}$-equivariant McKay-
bijection

$f_{\mu}:\operatorname{Irr}_{2^{\prime}}(W_{(e)}(\mu))\to\operatorname{Irr}_{2^{\prime}}(W_{(1)}(\mu_{0})),$

compatible with the multiplication of linear characters. For
$\mu\in\mathcal{E}_{2}(L_{(e)},s)^{W_{2}}$ with ${}^{x}\mu\in\mathcal{T}$ for
$x\in\operatorname{N}_{W_{(e)}}(W_{2})E^{\prime}$ we define
${}^{x}f_{\mu}:=f_{{}^{x}\mu}$. Denote by $B_{(e)}$ the unique block of
$N_{(e)}$ covering $e_{s}^{L_{(e)}}$ and set $b_{(1)}=e_{s}^{G_{(1)}}$. By
Proposition 10.15 we obtain a bijection

$\operatorname{Irr}_{0}(N_{(e)},B_{(e)})\to\operatorname{Irr}_{0}(G_{(1)},b_{(1)}),\quad\operatorname{Ind}_{N_{(e)}(\mu)}^{N_{(e)}}(\Lambda^{(e)}(\mu)\eta)\mapsto
R_{L_{(1)}}^{G_{(1)}}(\Xi(\mu))_{f_{\mu}(\eta)}.$

Note that it is actually not clear that the so-obtained map is well-defined as
the character $R_{L_{(1)}}^{G_{(1)}}(\Xi(\mu))_{f_{\mu}(\eta)}$ is not
necessarily of height zero. However, by Lemma 10.9 and Corollary 10.10, using
the map $\iota$, we obtain bijections
$\operatorname{Irr}_{0}(N_{(e)},B_{(e)})\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)$
and $\operatorname{Irr}_{0}(G_{(1)},b_{(1)})\to\operatorname{Irr}_{0}(G,b)$
and by Remark 6.3 we have
$|\operatorname{Irr}_{0}(B)|=|\operatorname{Irr}_{0}(b)|$. In addition, the
map is injective by the properties of Harish-Chandra series and its image
contains all height zero characters by the arguments from above. Therefore,
the map is necessarily a well-defined bijection and yields by applying $\iota$
a bijection
$f:\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)\to\operatorname{Irr}_{0}(G,b)$.

It is therefore left to show that $f$ is
$\operatorname{N}_{\tilde{G}E}(\mathbf{L})_{B}$-equivariant. This can however
by checked exactly as in the proof of [BR21, Theorem 4.8], where we replace
[BR21, Proposition 4,4, Proposition 4.5] by Propositions 10.15 and 10.16. ∎

##  11\. Verifying the inductive condition

In this section we show the bijection $f$ from Theorem 10.17 is a strong iAM-
bijection. For this we need the following auxiliary result:

######  Proposition 11.1.

In the situation of Theorem 10.17, a character
$\chi\in\operatorname{Irr}_{0}(G,b)$ satisfies assumption (i) of Theorem 9.3
if and only if its AM-correspondent $\psi:=f(\chi)$ satisfies assumption (ii).

###### Proof.

By the equivariance of $f$ it is clear that
$(\tilde{G}\mathcal{B})_{\chi}=\tilde{G}_{\chi}\mathcal{B}_{\chi}$ if and only
if
$(\operatorname{N}_{\tilde{G}}(L)\mathrm{N}_{G\mathcal{B}}(L))_{\chi^{\prime}}=\operatorname{N}_{\tilde{G}}(L)_{\chi^{\prime}}\mathrm{N}_{G\mathcal{B}}(L)_{\chi^{\prime}}$.
To finish the proof of the proposition we show that every character in
$\operatorname{Irr}_{0}(G,b)$ (resp.
$\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)$) extends to its
inertia group in $G\mathcal{B}_{\chi}$ (resp.
$\operatorname{N}_{G\mathcal{B}}(\mathbf{L})$).

Let
$\chi\in\operatorname{Irr}_{0}(G,b)\cup\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)$
and observe that $\mathcal{B}_{\chi}$ is only non-cyclic if $\mathbf{G}$ is of
type $E_{6}$ and $\gamma\in\mathcal{B}_{\chi}$. However if
$\gamma\in\mathcal{B}_{\chi}$ then for $g\in\tilde{G}$ we have
$\gamma\in\mathcal{B}_{{}^{g}\chi}$ if and only if
$\gamma(g)g^{-1}\in\tilde{G}_{\chi}$. Since $H^{1}(F,\mathrm{Z}(G))$ has odd
order and $\gamma$ acts by inversion on it this is equivalent to
$g\in\tilde{G}_{\chi}$. Hence it suffices to show that there exists a
character in the $\tilde{L}$-orbit of $\chi$ which satisfies condition (i)
resp. (ii) of Theorem 9.3.

In particular for $\chi\in\operatorname{Irr}_{0}(G,b)$ this follows from
Theorem 9.4. For the characters of $\operatorname{N}_{G}(\mathbf{L})$, the
claim is a consequence of [MS16, Proposition 3.18] whenever the associated
$e$-split Levi subgroup $\mathbf{L}$ is a maximal torus. On the other hand, if
$\varepsilon=-1$ then $\mathcal{B}$ is cyclic. If $\varepsilon=1$, then we
only need to consider the case $e=2$, i.e. when $q\equiv 3\mod 4$, since
$\mathbf{L}$ is a maximal torus otherwise. In that case $q$ is not a square
and thus the group of field automorphisms has odd order. It thus follows again
that $\mathcal{B}$ is a cyclic group. ∎

We now discuss the Clifford theory for the normal inclusion $G\lhd\tilde{G}$.
For this, we fix a character
$\psi=\operatorname{Ind}_{N_{(e)}(\mu_{(e)})}^{N_{(e)}}(\Lambda^{(e)}(\mu_{(e)})\eta_{(e)})$
and we let $\chi:=R_{L_{(1)}}^{G_{(1)}}(\mu_{(1)})_{\eta_{(1)}}$ be its image
under the map $f$. Moreover, we fix a semisimple element
$\tilde{s}\in(\mathbf{\tilde{G}}^{\ast})^{F}$ of $2^{\prime}$-order with
$\iota^{\ast}(\tilde{s})=s$. By the proof of Proposition 10.15 we find for
every character $\mu_{(d)}\in\mathcal{E}_{2}(L_{(d)},s)$ an extension
$\tilde{\mu}_{(d)}\in\mathcal{E}_{2}(\tilde{L}_{(d)},\tilde{s})$. We also fix
characters
$\tilde{\eta}_{(d)}\in\operatorname{Irr}(W_{(d)}(\tilde{\mu}_{(d)})\mid\eta_{(d)})$
below $\eta_{(d)}$.

######  Corollary 11.2.

For $d\in\\{1,2\\}$, there exists an $N_{(d)}E$-equivariant extension map
$\tilde{\Lambda}_{(d)}$ for the set
$\mathcal{E}_{2}(\tilde{L}_{(d)},\tilde{s})$ with respect to
$\tilde{L}_{(d)}\lhd\tilde{N}_{(d)}$ by sending
$\tilde{\mu}\in\mathcal{E}_{2}(\tilde{L}_{(d)},\tilde{s})$ to the unique
common extension of $\tilde{\mu}$ and
$\operatorname{Res}_{N_{\tilde{\mu}}}^{N_{\mu}}(\Lambda(\mu))$ where
$\mu\in\mathcal{E}_{2}(L_{(d)},s)$ is the restriction of $\tilde{\mu}$ to
$L_{(d)}$.

###### Proof.

This follows exactly as in the proof [MS16, Proposition 3.20]. ∎

We can now complete the proof of the inductive conditions:

######  Theorem 11.3.

Let $b$ be one of the blocks from Theorem 10.17. Then $b$ is AM-good relative
to the subgroup $\operatorname{N}_{G}(\mathbf{L})$.

###### Proof.

Let $B$ be the unique block of $\operatorname{N}_{G}(\mathbf{L})$ covering
$e_{s}^{L}$. We show that the bijection
$f:\operatorname{Irr}_{0}(b)\to\operatorname{Irr}_{0}(B)$ from Theorem 10.17
is a strong AM-bijection, see Definition 9.1. To show this we want to use
Theorem 9.3. First observe that the subgroup
$M:=\operatorname{N}_{G}(\mathbf{L})$ satisfies the assumptions of Theorem 9.3
by Lemma 6.6. Let $\chi\in\operatorname{Irr}_{0}(b)$ and
$\chi^{\prime}:=f(\chi)$. By possibly conjugating $\chi$ by an element of
$\tilde{G}$ we can assume by Theorem 9.4 that the character $\chi$ satisfies
condition (i) of Theorem 9.3. Using the Butterfly Theorem [Ruh22, Theorem
1.10] we see that it’s enough to show that $\chi$ and $\chi^{\prime}$ satisfy
the remaining conditions in Theorem 9.3. Since $f$ is equivariant we deduce
that conditions (ii) and (iii) hold, see Proposition 11.1.

As in [BR21, Lemma 5.3] one sees that
$\tilde{\chi}:=R_{\tilde{L}_{(1)}}^{\tilde{G}_{(1)}}(\tilde{\mu}_{(1)})_{\tilde{\eta}_{(1)}}\in\operatorname{Irr}(\tilde{G},e_{\tilde{s}}^{\tilde{G}})$
and
$\tilde{\psi}:=\tilde{\Lambda}(\tilde{\mu})\tilde{\eta}\in\operatorname{Irr}(\tilde{N}_{(e)},\tilde{B}_{e})$.
These characters satisfy condition (iv) of Theorem 9.3 by Proposition 10.15.
Finally for condition (v) by the remarks after Lemma 9.5 it suffices to show
that $\mathrm{bl}(\tilde{\psi})^{\tilde{G}}=\mathrm{bl}(\tilde{\chi})$. This
is however guaranteed by Proposition 1.11. ∎

##  12\. Centralizers of type $A_{2}(\varepsilon q^{3}).3$

###  12.1. Setup and notation

After having introduced the notation for dealing with the semisimple element
$s$ with centralizer of type $A_{2}(\varepsilon q)^{3}.3$ in
$E_{6}(\varepsilon q)$, we use this subsection to complete the proof of the
iAM-condition in cases (2) and (8) of Table 1, i.e. of the blocks associated
to the semisimple element of $E_{6}(\varepsilon q)$ with centralizer of type
$A_{2}(\varepsilon q^{3}).3$. As before, we let $\mathbf{G}$ be a simple,
simply connected algebraic group of type $E_{6}$ with Frobenius endomorphism
$F$ as in Section 10. In addition, we let $s\in\mathbf{T}^{\ast}$ be the
semisimple element with centralizer $A_{2}^{3}.3$ as in Section 10, but we
will consider $\mathbf{G}$ with a different Frobenius endomorphism in order to
simplify the calculations for these cases. Throughout this section, we will
consider the Frobenius endomorphism

$F:=F_{q}v^{\prime}\text{ if }\varepsilon=1\text{ and
}F:=F_{q}\gamma_{0}v^{\prime}\text{ if }\varepsilon=-1,$

where $v^{\prime}\in V$ is the Weyl group element constructed in the proof of
Proposition 10.6. We observe that $\mathbf{G}(s)^{F}$ is of type
$A_{2}(\varepsilon q^{3}).3$ and $(\mathbf{T},F)$ has polynomial order
$\Phi_{1}^{3}\Phi_{2}^{3}$.

We define

$m:=n_{\alpha_{0}}(1)\,{}^{v^{\prime}}n_{\alpha_{0}}(1)\,{}^{v^{\prime
2}}n_{\alpha_{0}}(1)$

and

$n:=n_{\alpha_{2}}(1)\,{}^{v^{\prime}}n_{\alpha_{2}}(1)\,{}^{v^{\prime
2}}n_{\alpha_{2}}(1).$

Our construction together with [Spä06, Bemerkung 2.1.7(b)] implies that
$m,n\in V^{F}=\operatorname{C}_{V}(v^{\prime})$. We fix an element
$g\in\mathbf{G}$ such that $m=g^{-1}F(g)$. Moreover,
$m_{(d_{\varepsilon})}:=1$ if $d=1$ and $m_{(d_{\varepsilon})}:=m$ if $d=2$.

######  Notation 12.1.

Set $h:=\mathrm{o}(\gamma_{0})\mathrm{o}(m)\mathrm{ord}(v^{\prime})$. Let
$E:=\operatorname{C}_{hm}\times\langle\gamma_{0}\rangle$ if $\varepsilon=1$
and $E:=\operatorname{C}_{hm}$ if $\varepsilon=-1$ such that the first factor
of $E$ acts on ${\tilde{\mathbf{G}}}^{F_{p}^{2hm}}$ by $\langle F_{p}\rangle$
and the second by the group $\langle\gamma_{0}\rangle$ generated by graph
automorphisms.

Note that we changed the graph automorphism $\gamma$ to $\gamma_{0}$ in order
to ensure that $E$ stabilizes $\mathbf{G}^{F}$, where $F$ is defined as above.
Again we denote by

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

the isomorphism induces by conjugation with the element $g$. Observe, that
this is an isomorphism which maps $\mathbf{G}^{F}\rtimes E$ onto
$\mathbf{G}^{mF}\rtimes E$.

###  12.2. Construction of Levi subgroups

We now construct the Levi subgroups associated to the semisimple element
$s\in(\mathbf{G}^{\ast})^{F}$. The Levi subgroup
$\mathbf{L}_{(1_{\varepsilon})}$ of $\mathbf{G}$ associated to
$s\in(\mathbf{T}^{\ast})^{F}$ is the centralizer of the
$\Phi_{1_{\varepsilon}}$-torus

$\mathbf{S}_{(1_{\varepsilon})}:=\\{h_{\alpha_{2}+\alpha_{3}+\alpha_{5}}(t_{1})h_{\alpha_{0}+\alpha_{1}+\alpha_{6}}(t_{2})\mid
t_{1},t_{2}\in\overline{\mathbb{F}}_{q}^{\times}\\}.$

Calculations in the root system of $E_{6}$ show that

$\mathbf{L}_{(1_{\varepsilon})}=\mathbf{G}_{1}\mathbf{G}_{2}\mathbf{S}_{(1_{\varepsilon})},$

where $\mathbf{G}_{1}$ has base

$A_{2}^{(1)}=\\{\alpha_{2}+\alpha_{4}+\alpha_{5}+\alpha_{6},\alpha_{1}+\alpha_{3}+\alpha_{4}+\alpha_{5}\\}$

and $\mathbf{G}_{2}$ has base

$A_{2}^{(2)}=\\{\alpha_{3}+\alpha_{4}+\alpha_{5}+\alpha_{6},\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}\\}.$

Observe that the roots in the sets $A_{1}^{(1)},A_{2}^{(2)}$ are orthogonal to
$\alpha_{2}+\alpha_{3}+\alpha_{5}$ and $\alpha_{1}+\alpha_{0}+\alpha_{6}$.
Moreover, the second root in each of theses sets is the image under
$v^{\prime}$ of the first. In particular, $\mathbf{L}_{(1_{\varepsilon})}^{F}$
has rational type $A_{2}(\varepsilon q)^{2}\Phi_{1_{\varepsilon}}^{2}$, as
already predicted by Table 1.

The $2_{\varepsilon}$-split Levi subgroup $\mathbf{L}_{(2_{\varepsilon})}$
associated to $s\in(\mathbf{T}^{\ast})^{mF}$ is the $2_{\varepsilon}$-split
Levi subgroup associated to the $2_{\varepsilon}$-split torus
$\mathbf{S}_{(2_{\varepsilon})}:=\\{h_{\alpha_{0}+\alpha_{1}+\alpha_{6}}(t)\mid
t\in\overline{\mathbb{F}}_{q}^{\times}\\}$ of $(\mathbf{G},mF)$. In
particular, we have

$\mathbf{L}_{(2_{\varepsilon})}=\mathbf{G}_{1}\mathbf{G}_{2}\mathbf{G}_{3}\mathbf{S}_{(2_{\varepsilon})},$

where $\mathbf{G}_{1},\mathbf{G}_{2}$ are defined as above and
$\mathbf{G}_{3}$ is of type $A_{1}$ with base $\\{\alpha_{4}\\}$. Here, the
roots of the root system of
$[\mathbf{L}_{(2_{\varepsilon})},\mathbf{L}_{(2_{\varepsilon})}]$ are
orthogonal to the root $\alpha_{1}+\alpha_{0}+\alpha_{6}$. Observe that the
element $m$ interchanges $\mathbf{G}_{1}$ and $\mathbf{G}_{2}$ so that the
Levi subgroup $\mathbf{L}_{(2_{\varepsilon})}^{mF}$ has rational type
$A_{2}(q^{2})A_{1}(\varepsilon q)\Phi_{(2_{\varepsilon})}$ in accordance with
Table 1.

Summarizing our computations we have thus proved:

######  Lemma 12.2.

Assume the notation from above. Then the following hold:

  1. (a)

The Levi subgroup $\mathbf{L}_{(1_{\varepsilon})}$ is the
$1_{\varepsilon}$-split Levi subgroup associated to the semisimple element
$s\in(\mathbf{G}^{\ast})^{F^{\prime}}$.

  2. (b)

The Levi subgroup $({}^{g}\mathbf{L}_{(2_{\varepsilon})})$ is the
$2_{\varepsilon}$-split Levi subgroup associated to the semisimple element
$s\in(\mathbf{G}^{\ast})^{F}$.

As before for $d\in\\{1,2\\}$ we define $L_{(d)}:=\mathbf{L}_{(d)}^{m_{(d)}F}$
and
$N_{(d)}:=\operatorname{N}_{\mathbf{G}^{v_{(d)}F}}(\mathbf{L}_{(d)},e_{s}^{L_{(d)}})$
and $W_{(d)}:=N_{(d)}/L_{(d)}$.

######  Proposition 12.3.

The subgroup $V_{(d_{\varepsilon})}=\langle m,n^{d^{2}}\rangle$ is a section
of $W_{(d_{\varepsilon})}$. In particular, $V_{(d_{\varepsilon})}$ has central
intersection

$Z_{(d_{\varepsilon})}:=\mathbf{L}_{(d_{\varepsilon})}\cap
V_{d_{\varepsilon}}=\langle
h_{\alpha_{0}}(-1)h_{\alpha_{1}}(-1)h_{\alpha_{6}}(-1),(h_{\alpha_{2}}(-1)h_{\alpha_{3}}(-1)h_{\alpha_{5}}(-1))^{d}\rangle\subset\mathbf{S}_{(d_{\varepsilon})}$

with $\mathbf{L}_{(d_{\varepsilon})}$.

###### Proof.

From Lemma 1.10 it follows that $V_{(d_{\varepsilon})}$ is a section of
$W_{(d_{\varepsilon})}$. Moreover, the Chevalley relations [Spä06, Bemerkung
2.1.7(b)] imply that the unique map $\langle
n_{\alpha_{0}}(1),n_{\alpha_{2}}(1)\rangle\to\langle m,n\rangle$ with
$n_{\alpha_{0}}(1)\mapsto m$ and $n_{\alpha_{2}}(1)\mapsto n$ is an
isomorphism. Observe that $H\cap\langle
n_{\alpha_{0}}(1),n_{\alpha_{2}}(1)\rangle=\langle
h_{\alpha_{0}}(-1),h_{\alpha_{2}}(-1)\rangle$. The claim follows from this. ∎

As before, for $d\in\\{1,2\\}$ we define $G_{(d)}:=\mathbf{G}^{v_{(d)}F}$.

######  Remark 12.4.

By the tables in [How80], we have

$\operatorname{N}_{G_{(1_{\varepsilon})}}(\mathbf{L}_{(1_{\varepsilon})})/L_{(1_{\varepsilon})}\cong
S_{3}\times C_{2},$

where the $S_{3}$ component centralizes
$[\mathbf{L}_{(1_{\varepsilon})},\mathbf{L}_{(1_{\varepsilon})}]$ and the
central $C_{2}$ copy permutes the two simple components of
$\mathbf{L}_{(1_{\varepsilon})}$. As $W_{(1_{\varepsilon})}\cong S_{3}$ does
not centralize
$[\mathbf{L}_{(1_{\varepsilon})},\mathbf{L}_{(1_{\varepsilon})}]$, we observe
that $W_{(1_{\varepsilon})}$ embedds diagonally in that decomposition of the
normalizer.

We now consider the action of the automorphisms in $E$ on the semisimple
element $s$. Recall that ${}^{w_{0}F_{p}}s=s$ if $p\equiv 2\mod 3$ while
${}^{F_{p}}s=s$ if $p\equiv 1\mod 3$. Moreover, $w_{0}\gamma_{0}$ stabilizes
$s$. However, $w_{0}\notin\operatorname{C}_{W}(v^{\prime})$ and so either
$\operatorname{C}_{WE}(v^{\prime},s)=\operatorname{C}_{W(s)}(v^{\prime})\langle
F_{p}\gamma_{0}\rangle$ or
$\operatorname{C}_{WE}(v^{\prime},s)=\operatorname{C}_{W(s)}(v^{\prime})\langle
F_{p}\rangle$. This agrees with the computation in the proof of Lemma 8.1. We
set $E^{\prime}:=\operatorname{C}_{WE}(v^{\prime},s)$ and deduce that
$\operatorname{N}_{G_{(d)}E}(L_{(d)},e_{s}^{L_{(d)}})=N_{(d)}E^{\prime}$.

######  Lemma 12.5.

There exists an $E^{\prime}$-equivariant extension map $\Gamma$ for
$Z_{(d)}\lhd V_{(d)}$.

###### Proof.

Observe that $E^{\prime}$ centralizes $V$. In particular, every extension map
for $Z_{(d)}\lhd V_{(d)}$ is $E^{\prime}$-equivariant and therefore an
equivariant extension map $\Gamma$ always exists by [Isa06, Corollary 11.13]
since $V_{(d)}/Z_{(d)}\cong W_{(d)}$ has only cyclic Sylow subgroups. ∎

The groups $W_{(1)}$ and $W_{(e)}$ have a common ($E^{\prime}$-stable) Sylow
$2$-subgroup $W_{2}$ with $v_{(1)},v_{(e)}\in\operatorname{Z}(W_{2})$. Note
that in contrast to the situation of Section 10 the group $W_{2}$ is
selfnormalizing in $W_{(1)}$ and $W_{(e)}$.

######  Lemma 12.6.

For $d\in\\{1,e\\}$ there exists an extension map $\Lambda_{(d)}$ for
$\mathcal{E}_{2}(L_{(d)},s)^{W_{2}}$ with respect to $L_{(d)}\lhd N_{(d)}$
such that $\Lambda_{(d)}$ is
$\operatorname{N}_{G_{(d)}E}(L_{(d)},e_{s}^{L_{(d)}})$-equivariant.

###### Proof.

Observe that it suffices to construct $\Lambda_{(d)}$ on an
$E^{\prime}$-transversal $\mathcal{T}$ of $\mathcal{E}_{2}(L_{(d)},s)^{W_{2}}$
and extend it $E^{\prime}$-equivariantly. Hence, for $\psi\in\mathcal{T}$ it
is enough to construct an extension
$\Lambda(\psi)\in\operatorname{Irr}((N_{(d)})_{\psi}\mid\psi)$ with the same
stabilizer in $E^{\prime}$. Observe furthermore that the Levi subgroup
$\mathbf{L}_{(2_{\varepsilon})}$ is only relevant, when $q\equiv 3\mod 4$ or
$\varepsilon=-1$ and therefore $q\equiv 2\mod 3$. In both cases $q$ is not a
square. Hence, the group $E^{\prime}/\langle
m_{(2_{\varepsilon})}\hat{F}\rangle$ has always odd order. Since
$N_{(2_{\varepsilon})}/L_{(2_{\varepsilon})}\cong C_{2}$, an extension map
with the desired properties always exists. We therefore only need to consider
the case $d=1_{\varepsilon}$. Let $V_{(d)}$ be the section of $W_{(d)}$
constructed in Proposition 12.3. We can therefore write $N_{(d)}$ as
$N_{(d)}=L_{(d)}V_{(d)}$. Since $W_{(d)}\cong S_{3}$, any character of
$\mathcal{T}$ has $2$ extensions to its inertia group in $N_{(d)}$. Hence, we
may replace $E^{\prime}$ without loss of generality by its Sylow $2$-subgroup.
The intersection $Z_{(d)}=L_{(d)}\cap V_{(d)}$ is a central subgroup of
$L_{(d)}$ by Proposition 12.3. Denote
$L_{0}:=[L_{(d)},L_{(d)}]\cong\operatorname{SL}_{3}(q)\times\operatorname{SL}_{3}(q)$.
We let $\psi\in\mathcal{E}_{2}(L_{(d)},s)$ and fix
$\psi_{0}\in\operatorname{Irr}_{0}(L_{0}\mid\psi)$ and denote by
$\psi^{\prime}\in\operatorname{Irr}((L_{0})_{\psi_{0}}\mid\psi_{0})$ the
Clifford correspondent of $\psi$. Note that the involutions in $W_{(d)}$ act
by permuting the two components of $L_{0}$, while the Sylow $3$-subgroup of
$W_{(d)}$ centralizes $L_{0}$, see also Remark 12.4. Since
$L_{(d)}/\operatorname{Z}(L_{0})L_{0}$ is a $2^{\prime}$-group and
$E^{\prime}$ is a $2$-group it follows that we can choose $\psi_{0}$ with the
additional property that

$(W_{(d)}E^{\prime})_{\psi}\leq(W_{(d)}E^{\prime})_{\psi_{0}}.$

Let $\nu\in\operatorname{Irr}(Z_{(d)}\mid\psi)$ be the unique character below
$\psi$. We consider the semidirect product $L_{0}\rtimes W_{(d)}$. We have a
surjective group homomorphism $L_{0}\rtimes W_{(d)}\to L_{0}\rtimes
W_{(d)}/\operatorname{C}_{W_{(d)}}(L_{0})$. Since
$W_{(d)}/\operatorname{C}_{W_{(d)}}(L_{0})$ acts by permuting the two
components of $L_{0}$, we can extend $\psi_{0}$ to a representation of
$L_{0}\rtimes W_{(d)}$ with $W_{(d)}$ in its kernel. By [BS19, Proposition
4.2], the character
$\psi_{0}\Gamma(\nu)\in\operatorname{Irr}(L_{0}V_{(d)}\mid\psi_{0})$ is
therefore a well-defined character which extends $\psi_{0}$. By Lemma 10.13
there exists a unique character
$\hat{\psi}^{\prime}\in\operatorname{Irr}((L_{0}V_{(d)})_{\psi^{\prime}})$
extending both $\psi^{\prime}$ and $\psi_{0}\Gamma(\nu)$.

We then define
$\Lambda(\psi):=\operatorname{Ind}_{(L_{0}V_{(d)})_{\psi^{\prime}}}^{L(V_{(d)})_{\psi}}(\hat{\psi}^{\prime})$.
Note that $(V_{(d)})_{\psi}=(V_{(d)})_{\psi^{\prime}}$ since $\psi_{0}$ is
$V_{(d)}$-stable. Hence, the so-constructed character is indeed irreducible.
By Mackey’s formula $\Lambda(\psi)$ is indeed an extension of $\psi$. We must
check that $\Lambda(\psi)$ is $E^{\prime}_{\psi}$-invariant. This follows
however from the fact that $\psi^{\prime}$,$\nu$ are both
$E^{\prime}_{\psi}$-invariant and $\Gamma$ is an
$E^{\prime}_{\psi}$-equivariant extension map. ∎

###  12.3. Construction of the AM-bijection

Let $b$ be one of the $2$-blocks of $G=\mathbf{G}^{F}$ associated to the
semisimple element $s\in G^{\ast}$. We let $b_{(d)}$ be the block of $G_{(d)}$
corresponding to the block $b$ of $\mathbf{G}^{F}$ under the isomorphism
induced by $\iota$. Furthermore, we let $(\mathbf{L}_{(e)},\lambda_{(e)})$ be
the $e$-cuspidal pair of $(\mathbf{G},m_{(e)}F)$ associated to the block
$b_{(e)}$ of $\mathbf{G}^{m_{(e)}F}$. By the proof of [KM13, Proposition 4.3],
there exists a unique character $\lambda_{(1)}\in\mathcal{E}(L_{(1)},s)$ such
that all constituents of $R_{L_{(1)}}^{G_{(1)}}(\lambda_{(1)})$ lie in the
$2$-block $b_{(1)}$.

######  Lemma 12.7.

There exists a natural bijection

$\Xi:\operatorname{Irr}(L_{(1)},b_{L_{(1)}}(\lambda_{(1)}))^{W_{2}}\to\operatorname{Irr}(L_{(e)},b_{L_{(e)}}(\lambda_{(e)}))^{W_{2}}.$

###### Proof.

We follow the proof of Lemma 10.11. For $z\in\\{1,m\\}$, we obtain again
$\operatorname{C}_{W(s)}(z)$-equivariant bijections

$\mathcal{E}_{2}(\mathbf{T}^{zF},s)\to
A_{z}:=\\{t\in(\mathbf{T}^{\ast})^{F^{\ast}z^{\ast}}\mid t_{2^{\prime}}=s\\}.$

and from this we get again a bijection

$\mathcal{E}_{2}(\mathbf{T}^{F},s)^{W_{2}}\to\mathcal{E}_{2}(\mathbf{T}^{zF},s)^{W_{2}}.$

Moreover, Deligne–Lusztig induction yields for $d\in\\{1,2\\}$ a bijection
$b_{L}(\lambda_{(d)})R_{T_{(d)}(s)}^{L_{(d)}}:\mathcal{E}_{2}(T_{(d)},s)\to\operatorname{Irr}(L_{(d)},b_{L_{(d)}}(\lambda_{(d)}))$,
see the proof of Lemma 6.5. The composition is again the sought bijection. ∎

######  Theorem 12.8.

Let $b$ be one of the blocks of Table 1 associated to the $e$-cuspidal pair
$(\mathbf{L},\lambda)$ and $B$ the unique block of
$\operatorname{N}_{G}(\mathbf{L})$ covering $b_{L}(\lambda)$. Then there
exists an
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{L})_{B}$-equivariant
bijection
$f:\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)\to\operatorname{Irr}_{0}(G,b)$.
In particular, the block $b$ is AM-good.

###### Proof.

The proof is similar to the proof of Theorem 10.17. We will therefore only
sketch the proof with emphasis on the differences to the proof of Theorem
10.17. We first parametrize the local characters. Let $\Lambda^{(e)}$ be the
extension map associated to $L_{(e)}\lhd N_{(e)}$ from Lemma 12.6. Again we
have a parametrization

$\Pi:\mathcal{P}=\\{(\mu,\eta)\mid\mu\in\operatorname{Irr}(L_{(e)},b_{L_{(e)}}(\lambda{(e)})),\eta\in\operatorname{Irr}(W_{(e)}(\mu))\\}\to\operatorname{Irr}(N_{(e)},b_{L_{(e)}}(\lambda_{e}))$

with
$\Pi(\mu,\eta)=\operatorname{Ind}_{N_{(e)}(\mu)}^{N_{(e)}}(\Lambda_{(e)}(\mu)\eta)$.
Observe that $W_{2}$ is self-normalizing in $W_{(e)}$ and
$\mathcal{E}_{2}(T_{(e)},s)^{W_{2}}$ is in bijection with
$\operatorname{Irr}(L_{(e)},b_{L_{(e)}}(\lambda_{(e)}))^{W_{2}}$. A count of
the height zero characters therefore gives
$|\operatorname{Irr}_{0}(N_{(e)},b_{L_{(e)}}(\lambda))|=(q-\varepsilon)_{2}2$.

Hence, every height zero character is of the form
$\chi=R_{L_{(1)}}^{G_{(1)}}(\mu_{0})_{\eta_{0}}$ for some
$\mu_{0}\in\mathcal{E}_{2}(L_{(1)},b_{L_{(1)}}(\lambda))$ and
$\eta_{0}\in\operatorname{Irr}(W_{(1)}(\mu_{0}))$, see Corollary 3.2. For such
a character to have height zero we must by Corollary 3.3 necessarily have that
$\mu_{0}$ is of height zero with $2\nmid|W_{(1)}:W_{(1)}(\mu_{0})|$ and
$\eta_{0}\in\operatorname{Irr}_{2^{\prime}}(W_{(1)}(\mu_{0}))$.

By Lemma 12.7 we have a bijection

$\Xi:\mathcal{E}_{2}(L_{(e)},s)^{W_{2}}\to\mathcal{E}_{2}(L_{(1)},s)^{W_{2}}.$

For $(\mu,\eta)\in\mathcal{P}$ a pair with $W_{2}\leq W_{(e)}(\mu)$ and
$\mu_{0}:=\Xi(\mu)$, we have again an $E^{\prime}_{\mu}$-equivariant McKay-
bijection

$f_{\mu}:\operatorname{Irr}_{2^{\prime}}(W_{(e)}(\mu))\to\operatorname{Irr}_{2^{\prime}}(W_{(1)}(\mu_{0})).$

Denote by $B_{(e)}$ the unique block of $N_{(e)}$ covering
$b_{L}(\lambda_{(e)})$. By Proposition 10.15 we obtain a bijection

$\operatorname{Irr}_{0}(N_{(e)},B_{(e)})\to\operatorname{Irr}_{0}(G_{(1)},b_{(1)}),\quad\operatorname{Ind}_{N_{(e)}(\mu)}^{N_{(e)}}(\Lambda^{(e)}(\mu)\eta)\mapsto
R_{L_{(1)}}^{G_{(1)}}(\Xi(\mu))_{f_{\mu}(\eta)}.$

Again, we need to check that the so-obtained map is well-defined. For this,
observe that the number of height zero characters in the block $b$ is equal to
the number of height zero characters in the principal block of
$A_{2}(\varepsilon q^{3})$, see Lemma 8.4. By [BR21, Lemma 3.8], the latter
number is equal to $(q-\varepsilon)_{2}2$. Comparing this with the number of
height zero characters of the block $B$ obtained above, we deduce that
$|\operatorname{Irr}_{0}(B)|=|\operatorname{Irr}_{0}(b)|$. In addition, the
map is injective by the properties of Harish-Chandra series and its image
contains all height zero characters by the arguments from above. Therefore,
the map is necessarily a well-defined bijection and yields by applying $\iota$
a bijection
$f:\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)\to\operatorname{Irr}_{0}(G,b)$.
It is left to check that $f$ is
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{L},B)=\operatorname{N}_{G\mathcal{B}}(\mathbf{L},B)$-equivariant.
But this is again checked as in Theorem 10.17. Hence, the block $b$ is AM-good
by Proposition 9.6. ∎

###  12.4. Some more information on defect groups

We now turn to the description of defect groups in case (2) or (8) of Table 1.
We keep the notation from the beginning of this section. We denote
$F^{\prime}:=Fm_{(e)}$, where the Frobenius $F$ and $m_{(e)}\in V^{F}$ are
defined as before. Let $s\in(\mathbf{T}^{\ast})^{F^{\prime}}$ the semisimple
element with centralizer $A_{2}(\varepsilon q^{3}).3$ from before and $b$ one
of the blocks of $\mathbf{G}^{F^{\prime}}$ associated to $s$. By Theorem 4.5,
the group $P^{\prime}=\mathbf{T}_{2}^{F^{\prime}}\langle m\rangle$ is
therefore a Sylow $2$-subgroup of $\mathbf{G}(s)^{F^{\prime}}$. Let
$D^{\prime}:=\mathbf{T}_{2}^{F^{\prime}}$ and
$\mathbf{M}:=\operatorname{C}_{\mathbf{L}}(D^{\prime})$. Using [CE04, Lemma
13.17] we obtain $\mathbf{M}=\mathbf{G}_{1}\mathbf{G}_{2}\mathbf{T}$ in both
cases. We use the description of Lemma 1.11 to see that one of the blocks
$b_{D^{\prime}}$ of $e_{s}^{\mathbf{M}^{F^{\prime}}}$ associated to $s$ is a
$b$-Brauer pair. Finally, observe that the unique character
$\lambda^{\prime}\in\mathcal{E}({\mathbf{M}}^{F^{\prime}},s)\cap\operatorname{Irr}(\mathbf{M}^{F^{\prime}},b_{D^{\prime}})$
is the canonical character associated to the self-centralizing Brauer pair
$(D^{\prime},b_{D^{\prime}})$ and that
$D^{\prime}=\operatorname{Z}(\mathbf{M})_{2}^{F^{\prime}}$.

We need the following representation theoretic lemma:

######  Lemma 12.9.

Let $H_{0}$ be a normal subgroup of a finite group $H$ and $U\leq H_{0}$ with
$\operatorname{Z}(U)\cong C_{3}$ and
$\operatorname{Z}(U)\leq\operatorname{Z}(H)$. Let
$Z\subset\operatorname{Z}(H)$ and $\chi\in\operatorname{Irr}(H)$ with
$\operatorname{Res}_{U}^{H}(\chi)=\sum_{\phi\in\operatorname{Irr}(U)}a_{\phi}\phi$.
Then

  1. (a)

$\operatorname{Res}^{U}_{\operatorname{Z}(U)}(\phi)\phi(1)=\operatorname{Res}^{H}_{\operatorname{Z}(U)}(\chi)\chi(1)$
for all $\phi\in\operatorname{Irr}(U)$ with $a_{\phi}\neq 0$.

  2. (b)

$\operatorname{Res}_{UZ}^{H}(\chi)=\sum_{\phi\in\operatorname{Irr}(U)}a_{\phi}\hat{\phi}$,
where $\hat{\phi}$ is the unique extension of $\phi$ with
$\operatorname{Res}^{U}_{Z\operatorname{Z}(U)}(\phi)\hat{\phi}(1)=\operatorname{Res}^{H}_{Z\operatorname{Z}(U)}(\chi)\hat{\chi}(1)$

###### Proof.

Let $1\neq z\in\operatorname{Z}(U)$. Assume first that $\chi(z)=n\omega$ for
$n=\chi(1)$ and a primitive third root of unity $\omega$. For
$\xi\in\\{1,\omega^{\pm 1}\\}$ denote by $a_{\xi}$ the sum of degrees of
constituents $\phi$ of the restriction $\operatorname{Res}^{H}_{U}(\chi)$ with
$\phi(z)=\phi(1)\xi$. We obtain $n=\chi(1)=a_{1}+a_{\omega}+a_{\omega^{2}}$
and $\omega n=a_{1}+a_{\omega}\omega+a_{\omega^{2}}\omega^{2}$. Since
$-1-\omega=\omega^{2}$ the second equation rewrites as
$0=(a_{1}-a_{\omega^{2}})+(a_{\omega}-a_{\omega^{2}}-n)\omega$. This forces
$a_{1}-a_{\omega^{2}}=0$ and $a_{\omega}-a_{\omega^{2}}-n=0$ which together
with the first equation implies $a_{1}=a_{\omega^{2}}=0$. Hence, in the
restriction of $\chi$ only characters with the same character on the center
appear.

If on the other hand $\operatorname{Z}(U)$ is in the kernel of $\chi$, then we
obtain the two equations $n=a_{1}+a_{\omega}+a_{\omega^{2}}$ and
$n=a_{1}+a_{\omega}\omega+a_{\omega^{2}}\omega^{2}$. This forces again
$a_{\omega}=a_{\omega^{2}}=0$. This shows part (a) also in this case.

Part (b) is now obtained by comparing the character values on $Z$ on both
sides. ∎

######  Proposition 12.10.

There exists a character
$\zeta^{\prime}\in\operatorname{Irr}(\operatorname{C}_{G}(P^{\prime})\mid\zeta)$
which appears with odd multiplicity in $\zeta$ and such that
$\zeta^{\prime}(1)_{2}=|\operatorname{C}_{G}(P^{\prime}):\operatorname{Z}(D^{\prime})\cap\operatorname{C}_{G}(P^{\prime})|_{2}$.

###### Proof.

We start by examining the right hand side of the equality in the proposition.
We note that
$\operatorname{Z}(D^{\prime})\cap\operatorname{C}_{G}(P^{\prime})=\operatorname{Z}(P^{\prime})$,
as $\operatorname{Z}(P^{\prime})\leq D^{\prime}$. Denote
$M:=\mathbf{M}^{F^{\prime}}$. We observe that
$\operatorname{C}_{[M,M]}(P^{\prime})\operatorname{Z}(P^{\prime})$ has odd
index in $\operatorname{C}_{M}(P^{\prime})$. This is because
$M/[M,M]\operatorname{Z}(M)$, $[M,M]\cap\operatorname{Z}(M)$ are both of odd
order and $\operatorname{Z}(M)_{2}=D^{\prime}$.

Moreover, we have
$\operatorname{C}_{[M,M]}(P^{\prime})\operatorname{Z}(P^{\prime})/\operatorname{Z}(P^{\prime})\cong\operatorname{C}_{[M,M]}(P^{\prime})/\operatorname{C}_{[M,M]}(P^{\prime})\cap\operatorname{Z}(P^{\prime})$
and
$\operatorname{C}_{[M,M]}(P^{\prime})\cap\operatorname{Z}(P^{\prime})\leq\operatorname{Z}(\operatorname{C}_{[M,M]}(P^{\prime}))$
which is a $2^{\prime}$-group. Hence, we need to find a constituent with
degree $\zeta^{\prime}(1)_{2}=|\operatorname{C}_{[M,M]}(P^{\prime})|_{2}$.

We first consider case (2). Let
$\zeta_{0}=\chi\times\chi\in\operatorname{Irr}([M,M]\mid\zeta)$. Our aim is to
construct a constituent
$\zeta_{0}^{\prime}\in\operatorname{Irr}(\operatorname{C}_{[M,M]}(P^{\prime})\mid\zeta_{0})$
with odd multiplicity. The subgroup
$\operatorname{C}_{[M,M]}(P^{\prime})\cong\operatorname{SL}_{3}(q)$ embedds
diagonally into
$[M,M]\cong\operatorname{SL}_{3}(q)\times\operatorname{SL}_{3}(q)$. Hence, the
constituents of the restriction of a character
$\zeta=\chi\times\chi\in\operatorname{Irr}([M,M])$ are precisely the
constituents $\zeta^{\prime}\in\operatorname{Irr}(\mathrm{SL}_{3}(q))$ of the
character $\chi^{2}\in\operatorname{Irr}(\mathrm{SL}_{3}(q))$. Here, the
character $\chi$ is a character of degree
$\frac{1}{3}\Phi_{1}(q)^{2}\Phi_{2}(q)$ of $\operatorname{SL}_{3}(q)$, lying
in a Lusztig series with centralizer of type $\Phi_{3}(q).3$. Also observe
that the diagonal automorphisms induced by $M$ on
$[M,M]\cong\operatorname{SL}_{3}(q)\times\operatorname{SL}_{3}(q)$ are the
ones induced by the diagonal action of $\Delta\operatorname{GL}_{3}(q)$. It
can now be checked with CHEVIE [GHL+96], that there exists a unique semisimple
character $\zeta_{0}^{\prime}$ of degree
$\frac{1}{3}\Phi_{1}(q)^{2}\Phi_{2}(q)$ which appears as a constituent (of odd
multiplicity) in $\zeta_{0}$.

Denote $C:=\operatorname{C}_{M}(P^{\prime})$ and observe that $C$ induces all
diagonal automorphisms of $M$ on $[M,M]$. We let
$\hat{\zeta}\in\operatorname{Irr}(M_{\zeta_{0}}\mid\zeta_{0})$ be the Clifford
correspondent of $\zeta$ and we denote by
$\hat{\zeta}_{0}^{\prime}\in\operatorname{Irr}(C_{\zeta_{0}}\mid\zeta_{0}^{\prime})$
the character with the same values as $\zeta_{0}$ on $\operatorname{Z}(M)\cap
C$ as in Lemma 12.9. Denote
$\zeta^{\prime}:=\operatorname{Ind}_{C_{\zeta_{0}}}^{C}(\hat{\zeta}_{0}^{\prime})$.
Observe that $M=M_{\zeta_{0}}C$ and $C_{\zeta_{0}}=C_{\zeta_{0}^{\prime}}$.
Hence, for the scalar product of $\zeta$ and $\zeta^{\prime}$, we have by
Mackey’s formula

$\langle\zeta,\zeta^{\prime}\rangle=\langle\operatorname{Res}_{C}^{M}\operatorname{Ind}_{M_{\zeta_{0}}}^{M}(\hat{\zeta}_{0}),\operatorname{Ind}_{C_{\zeta_{0}^{\prime}}}^{C}(\hat{\zeta}_{0}^{\prime})\rangle=\langle\operatorname{Ind}_{C_{\zeta_{0}^{\prime}}}^{C}(\operatorname{Res}_{C_{\zeta_{0}}}^{M_{\zeta_{0}}}(\hat{\zeta}_{0})),\operatorname{Ind}_{C_{\zeta_{0}^{\prime}}}^{C}(\hat{\zeta}_{0}^{\prime})\rangle.$

By construction, the character $\hat{\zeta}_{0}^{\prime}$ is the unique
character in its $C$-orbit that appears with non-zero multiplicity in
$\operatorname{Res}_{C_{\zeta_{0}}}^{M_{\zeta_{0}}}(\hat{\zeta}_{0})$. Hence,
by Clifford correspondence, we have
$\langle\zeta,\zeta^{\prime}\rangle=\langle\operatorname{Res}_{C_{\zeta_{0}}}^{M_{\zeta_{0}}}(\hat{\zeta}_{0}),\zeta^{\prime}_{0}\rangle$,
which is odd.

Let us now consider case (8). Note that here $M=[M,M]\operatorname{Z}(M)$. Let
$\zeta_{0}=\operatorname{Res}_{[M,M]}^{M}(\zeta)$. This character of
$[M,M]\cong\operatorname{SL}_{3}(q^{2})$ lies in a Lusztig series with
centralizer of type $\Phi_{3}(q^{2}).3$. Moreover, it is non-trivial on the
center and has degree $\frac{1}{3}\Phi_{1}(q^{2})^{2}\Phi_{2}(q^{2})$. We now
consider its restriction to the centralizer
$\operatorname{C}_{[M,M]}(P^{\prime})\cong\operatorname{SL}_{3}(q)$.

We let $x\in\operatorname{SL}_{3}(q)$ such that $x$ is conjugate to the
diagonal matrix $\mathrm{diag}(\tau,\tau^{q},\tau^{q^{2}})$ with
$\tau\in\mathbb{F}_{q^{3}}^{\times}$ of order $q^{2}+q+1$. By [SF73],
$\zeta_{0}(x)$ is a third root of unity. Moreover in the restriction of
$\zeta_{0}$ to $\operatorname{SL}_{3}(q)$ by [SF73] and Lemma 12.9 only
characters $\psi$ of degree $\psi(1)_{2}=(\Phi_{1}^{2}(q)\Phi_{2}(q))_{2}$
contribute to $\zeta(x)$ (as all the other characters either vanish on $x$ or
are trivial on the center). If all these characters were to appear with even
multiplicity in the restriction of $\zeta_{0}$, then $\zeta_{0}(x)/2$ would be
an algebraic integer. This is however not the cases as $\zeta_{0}(x)$ is a
third root of unity. Hence, there exists a constituent with odd multiplicity
as required. We conclude with Lemma 12.9. ∎

######  Corollary 12.11.

Let $b$ be one of the blocks of $G$ associated to case (2) or (8) in 1 and
$P^{\prime}$ a Sylow $2$-subgroup of $G(s)$. Then there exists a maximal
$b$-Brauer pair $(P^{\prime},b_{P^{\prime}})$ with
$(D^{\prime},b_{D^{\prime}})\leq(P^{\prime},b_{P^{\prime}})$. Moreover, the
pair $(D^{\prime},P^{\prime})$ is conjugate to the pair $(D,P)$ from [KM13,
Theorem 1.2].

###### Proof.

Let $b$ be associated to the quasi-central $e$-cuspidal pair
$(\mathbf{L},\lambda)$ with $\lambda\in\mathcal{E}(\mathbf{L}^{F^{\prime}},s)$
and $b_{P}^{\prime}$ be the block associated to the character $\zeta^{\prime}$
constructed in Proposition 12.10. Hence, [CE04, Proposition 22.14] is
applicable and we obtain that $(P^{\prime},b_{P^{\prime}})$ is a $b$-Brauer
pair. Since $|P^{\prime}|=|\mathbf{G}(s)^{F^{\prime}}|_{2}$ it follows that
$(P^{\prime},b_{P^{\prime}})$ is a maximal $b$-Brauer pair.

Comparing $(D^{\prime},P^{\prime})$ with the description of the tuple $(D,P)$
from [KM13, Proposition 2.7] shows again that these are
$\mathbf{G}^{F^{\prime}}$-conjugate. ∎

##  13\. Proof of the inductive condition for non-maximal unipotent blocks

In this section, we consider the remaining non-maximal unipotent blocks. In
particular, $b=b_{G}(\mathbf{L},\lambda)$ is a unipotent block of $G$
associated to a unipotent $e$-cuspidal pair $(\mathbf{L},\lambda)$ of central
defect.

######  Lemma 13.1.

Let $b=b_{G}(\mathbf{L},\lambda)$ be a unipotent non-maximal block of $G$
associated to a unipotent $e$-cuspidal pair $(\mathbf{L},\lambda)$ of central
defect. Moreover, let $\mathbf{H}\in\\{\mathbf{L},\mathbf{G}\\}$ and
$\mu\in\mathcal{E}(\mathbf{H},1)$ a unipotent character. Then $\mu$ extends to
$\mathrm{N}_{\tilde{G}\mathcal{B}}(\mathbf{H})$.

###### Proof.

By [CE04, Proposition 15.9], the character $\mu\in\mathcal{E}(H,1)$ restricts
irreducibly to a character $\mu_{0}$ of $H_{0}$, where
$\mathbf{H}_{0}=[\mathbf{H},\mathbf{H}]$. Note that by the classification of
Lemma 7.1, the group $\mathbf{H}_{0}$ is always simple of simply connected
type and we have an injective map
$H_{0}/\operatorname{Z}(H_{0})\hookrightarrow\mathbf{H}_{\operatorname{ad}}^{F}$,
which factors through $H/\operatorname{Z}(H)$. Since $\mu_{0}$ is unipotent,
we can consider it as a character of $S:=H_{0}/\operatorname{Z}(H_{0})$, which
is a non-abelian simple group. Now, the group
$N:=\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{H})$ acts by automorphisms
on $S$. Hence, we obtain $N/\operatorname{C}_{N}(S)\cong A$, where $S\leq
A\leq\mathrm{Aut}(S)$. By [Mal08, Theorem 2.4, Theorem 2.5] and Lemma 7.1 the
unipotent character $\mu_{0}\in\operatorname{Irr}(S)$ extends to
$\mathrm{Aut}(S)$. Therefore, the character $\mu_{0}$ extends to a character
$\hat{\mu}\in\operatorname{Irr}(N)$, which is trivial on
$\operatorname{C}_{N}(S)$. Moreover, by the proof of [Mal08, Theorem 2.4,
Theorem 2.5], the so-obtained extension $\hat{\mu}$ is an extension of $\mu$.
∎

Let $\mathbf{T}$ be an $F$-stable maximal torus of $\mathbf{L}$. As
$\mathbf{L}^{F}=\mathbf{T}^{F}[\mathbf{L},\mathbf{L}]^{F}$ we can consider
$\operatorname{Irr}(\mathbf{L}^{F}/[\mathbf{L},\mathbf{L}]^{F})$ as a subgroup
of $\operatorname{Irr}(\mathbf{T}^{F})$. By duality we obtain a bijection
$(\mathbf{T}^{\ast})^{F^{\ast}}\to\operatorname{Irr}(\mathbf{T}^{F})$ which by
[CE04, Equation 8.19] restricts to a bijection

$\operatorname{Irr}(\operatorname{Z}(\mathbf{L}^{\ast})^{F^{\ast}})\to\operatorname{Irr}(\mathbf{L}^{F}/[\mathbf{L},\mathbf{L}]^{F}),\quad
z\mapsto\hat{z}.$

With this in mind, we can formulate the next lemma.

######  Lemma 13.2.

Let $\mu=\hat{z}\in\operatorname{Irr}_{(}L)$ be a linear character.

  1. (a)

The character $\mu$ extends to its stabilizer in
$\operatorname{N}_{G\mathcal{B}}(\mathbf{L})$.

  2. (b)

Any extension of $\mu$ to $\operatorname{N}_{G}(\mathbf{L})_{\mu}$ is
$\tilde{L}$-stable if and only if
$\mathrm{N}_{\operatorname{C}_{G^{\ast}}(z)}(\mathbf{L}^{\ast})=\operatorname{C}^{\circ}_{G^{\ast}}(z)$.

###### Proof.

As $\mathbf{L}$ is an $e$-split Levi subgroup it follows that $\mathbf{L}$ is
the centralizer of an $e$-split torus $\mathbf{S}_{0}$ of $\mathbf{G}$.
Consequently, $\mathbf{S}_{0}$ is contained in a Sylow $e$-torus $\mathbf{S}$
of $\mathbf{G}$. Its centralizer
$\mathbf{T}:=\operatorname{C}_{\mathbf{G}}(\mathbf{S})$ is by [KM15, Lemma
3.17] a maximal torus of $\mathbf{G}$ and by construction it centralizes
$\mathbf{S}_{0}$. In particular, $\mathbf{T}$ is a maximal torus of
$\mathbf{L}$. Consider the restriction
$\mu_{0}:=\operatorname{Res}_{\mathbf{T}^{F}}^{\mathbf{L}^{F}}(\mu)\in\operatorname{Irr}(\mathbf{T}^{F})$
of $\mu$. By [MS16, Theorem 3.1], the character $\mu_{0}$ extends to a
character
$\mu_{1}\in\operatorname{Irr}(\operatorname{N}_{G\mathcal{B}}(\mathbf{T})_{\mu_{0}})$.
Since all Sylow $e$-tori of $\mathbf{L}$ are $L$-conjugate by [MT11, Theorem
25.11], it follows that
$\operatorname{N}_{G\mathcal{B}}(\mathbf{L})=\mathbf{L}^{F}\operatorname{N}_{G\mathcal{B}}(\mathbf{L},\mathbf{T})$.
From this we conclude that $\mu$ extends to
$\operatorname{N}_{G\mathcal{B}}(\mathbf{L})_{\mu}$ by Lemma 10.13.

For part (b), let $\hat{z}^{\prime}$ be an extension of $\hat{z}$ to its
inertia group in $\operatorname{N}_{G}(\mathbf{L})$ and fix $t\in\tilde{L}$
with $\tilde{L}=\langle L,t\rangle$. Observe that by [CS13, Lemma 5.8 (b)]
there exists a character
$\nu\in\operatorname{Irr}(\operatorname{N}_{G}(\mathbf{L})_{\hat{z}})$ with
${}^{t}\hat{z}^{\prime}=\hat{z}^{\prime}\nu$, whose kernel is equal to
$\mathrm{ker}(\nu)=\operatorname{N}_{G}(\mathbf{L})_{\widehat{\tilde{z}}}$,
where $\tilde{z}\in\operatorname{Z}(\tilde{L}^{\ast})$ satisfies
$\iota^{\ast}(\tilde{z})=z$. By Clifford theory, the extension
$\hat{z}^{\prime}$ is $\tilde{L}$-stable if and only if
$\operatorname{N}_{G}(\mathbf{L})_{\hat{z}}=\operatorname{N}_{G}(\mathbf{L})_{\widehat{\tilde{z}}}$.
By Proposition 1.6, the subgroup $W_{G}(\mathbf{L})_{\hat{z}}$ corresponds via
duality to the subgroup
$\mathrm{N}_{\operatorname{C}_{G^{\ast}}(z)}(\mathbf{L}^{\ast})/L^{\ast}$ and
for $w\in W_{G}(\mathbf{L})_{\hat{z}}$ we have $w\in
W_{G}(\mathbf{L})_{\widehat{\tilde{z}}}$ if and only if
$w^{\ast}\in\operatorname{C}^{\circ}_{G^{\ast}}(z)=\iota^{\ast}(\operatorname{C}^{\circ}_{\tilde{G}^{\ast}}(\tilde{z}))$.
∎

Using the previous two statements we can now parametrize the height zero
characters of a local block associated to $b$. The character $\lambda$ is of
central defect and so the block $b_{L}(\lambda)$ is nilpotent with defect
group $\operatorname{Z}(L)_{2}$. We deduce that

$\operatorname{Irr}_{0}(L,b_{L}(\lambda))=\\{\lambda\hat{z}\mid
z\in\operatorname{Z}(\mathbf{L}^{\ast})_{2}^{F}\\}.$

For each $\mu=\lambda\hat{z}$ choose a character $\Lambda(\mu)$ of
$\operatorname{N}_{G}(L,{\mu})$ extending it (such a character always exists
whenever the Sylow subgroups of $W_{G}(L,\lambda)$ are cyclic; for the
remaining cases this follows from Lemma 13.1 and Lemma 13.2) According to (the
proof of) Lemma 6.6(ii) there exists a unique block $B$ of
$\operatorname{N}_{G}(\mathbf{L})$ covering $b_{L}(\lambda)$. By Clifford
theory, we obtain

$\operatorname{Irr}(\operatorname{N}_{G}(\mathbf{L}),B)=\\{\operatorname{Ind}_{{\operatorname{N}_{G}(\mathbf{L},\lambda)_{\eta}}}^{\operatorname{N}_{G}(\mathbf{L})}(\Lambda(\mu)\eta)\mid\eta\in\operatorname{Irr}_{0}(W_{G}(\mathbf{L},\lambda)_{\mu})\text{
and }\mu\in\operatorname{Irr}(L,b_{L}(\lambda))\\}.$

We deduce that a character
$\operatorname{Ind}_{{\operatorname{N}_{G}(\mathbf{L},\lambda)_{\eta}}}^{\operatorname{N}_{G}(\mathbf{L})}(\Lambda(\mu)\eta)$
has height zero if and only if the inertia group of $\mu$ in
$W_{G}(\mathbf{L},\lambda)$ contains a Sylow $2$-subgroup of
$W_{G}(\mathbf{L},\lambda)$. Fix a Sylow $2$-subgroup $W_{2}$ of
$W_{G}(\mathbf{L},\lambda)$ and observe that $W_{2}$ is self-normalizing in
$W_{G}(\mathbf{L},\lambda)$. Therefore, we obtain

$\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)=\\{\operatorname{Ind}_{{\operatorname{N}_{G}(\mathbf{L},\lambda)_{\eta}}}^{\operatorname{N}_{G}(\mathbf{L})}(\Lambda(\mu)\eta)\mid\eta\in\operatorname{Irr}_{0}(W_{G}(\mathbf{L},\lambda)_{\mu})\text{
and }\mu\in\operatorname{Irr}(L,b_{L}(\lambda))^{W_{2}}\\},$

where every character in the set on the right-hand side appears exactly once.

######  Lemma 13.3.

Let $B$ be the unique block of $\operatorname{N}_{G}(\mathbf{L})$ covering
$b_{L}(\lambda)$. Then the number of height zero characters in $B$ for the
unipotent blocks in Lemma 7.1 is as follows:

  1. (i)

$|\operatorname{Irr}_{0}(B)|=|P/P^{\prime}|=4$;

  2. (ii)

$|\operatorname{Irr}_{0}(B)|=|P/P^{\prime}|=8$.

###### Proof.

As already observed in Lemma 7.4 the defect groups in case (i) are dihedral
and the equality follows again from [Sam14, Theorem 8.1] or by a similar
argument as in case (ii). Suppose therefore that we are in case (ii): The
action of $W_{G^{\ast}}(\mathbf{L}^{\ast})\cong S_{3}\times C_{2}$ on
$\operatorname{Z}(L^{\ast})=\Phi_{1}^{2}$ is described in the proof of [KM13,
Proposition 6.4]. We may therefore choose a Sylow $2$-subgroup $W_{2}\cong
C_{2}\times C_{2}$ of $W_{G^{\ast}}(\mathbf{L}^{\ast})$ such that $C_{2}\leq
S_{3}$ permutes the two copies of $\Phi_{1}$ and the central subgroup $C_{2}$
of $W_{G}(\mathbf{L})$ acts on $\operatorname{Z}(\mathbf{L})_{2}^{F}$ by
inversion. The only characters
$\mu=\lambda\hat{z}\in\operatorname{Irr}(L,b_{L}(\lambda))$ which are
therefore fixed under $W_{2}$ (isomorphic to $C_{2}\times C_{2}$) are the ones
with
$z=\pm(1,1)\in\operatorname{Z}(L^{\ast})\cong\mathbb{F}_{q}^{\times}\times\mathbb{F}_{q}^{\times}$.
Both of these characters contribute four height zero characters to the block
$B$. Hence, there are $8$ height zero characters in total. ∎

######  Proposition 13.4.

Assume that we are either in case (i) or (ii) of Lemma 7.1. Then there exists
an $\operatorname{N}_{G\mathcal{B}}(\mathbf{L})_{B}$-equivariant bijection
$\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)\to\operatorname{Irr}_{0}(G,b)$.
Moreover, the block $b$ is AM-good.

###### Proof.

Let us first describe the action of automorphisms on the height zero
characters of the block $b$. In case (i) the same reasoning applies as in
[Ruh21]. In particular, the two unipotent characters are invariant under
$\mathrm{Out}(G)$ and the two non-unipotent characters are permuted by the
diagonal automorphisms, see also Lemma 13.2(b). In case (ii) we note that by
Example 1.3 all 8 characters are necessarily stable under all automorphisms of
$G$.

Let us now consider the action of automorphisms on the height zero characters
of the block $B$. We claim that every height zero character
$\psi\in\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)$ extends to
its inertia group in $\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{L})$. We
first observe that $\lambda$ has an extension $\hat{\lambda}$ to
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{L})$ by Lemma 13.1. It
therefore remains to describe extensions of the characters
$\psi\in\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)$ which
cover the character $\mu=\lambda\hat{z}$, with
$z\in\operatorname{Z}(\mathbf{L})_{2}^{F}$ of order $2$ as described in the
proof of Lemma 13.3. First note that by Lemma 13.2, the character $\hat{z}$
extend to a character $\hat{z}^{\prime}$ in its inertia group in
$\operatorname{N}_{G\mathcal{B}}(\mathbf{L})$. In case (i) note that we have
$\operatorname{C}_{\mathbf{G}^{\ast}}(z)=(E_{6}(q)\Phi_{1}).2$. Therefore, by
Lemma 13.2 any such extension
$\hat{z}^{\prime}\in\operatorname{Irr}(\operatorname{N}_{G}(\mathbf{L}))$ of
the character $\hat{z}$ to $\operatorname{N}_{G}(\mathbf{L})$ is not invariant
under diagonal automorphisms. In particular, we have
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{L})_{\psi}=\operatorname{N}_{G\mathcal{B}}(\mathbf{L})_{\psi}$.
Since
$\operatorname{N}_{G\mathcal{B}}(\mathbf{L})_{\psi}/\operatorname{N}_{G}(\mathbf{L})$
is cyclic, the character $\psi$ thus extends to its inertia group,

We now discuss the underlying central characters. Note that in case (ii),
$\operatorname{Z}(G)=1$ so there is nothing to do. In case (i) the two
unipotent characters in $\operatorname{Irr}_{0}(G,b)$ (resp. in
$\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)$) have
$\operatorname{Z}(G)$ in their kernel. Moreover, for $1\neq z\in G^{\ast}$
with $\mathcal{E}(G,z)\cap\operatorname{Irr}_{0}(G,b)\neq\emptyset$ we have
$z\in[G^{\ast},G^{\ast}]$ by [Lüb] and thus, the characters in
$\mathcal{E}(G,z)$ and $\mathcal{E}(L,z)$ have $\operatorname{Z}(G)$ in their
kernel as well.

By the description of automorphisms it is now clear how to construct an
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{L})_{B}$-equivariant
bijection
$\operatorname{Irr}_{0}(\operatorname{N}_{G}(\mathbf{L}),B)\to\operatorname{Irr}_{0}(G,b)$.
The block $b$ is therefore AM-good by Lemma 9.6. ∎

##  14\. Proof of the main theorem

###  14.1. Navarro–Späth’s Theorem

Recall that a group $X$ is said to be involved in a finite group $Y$, if there
exist $H\lhd K\leq Y$ such that $X\cong K/H$.

We recall [NS14, Theorem 7.1]. In the statement of the theorem,
$\operatorname{Irr}_{0}(Y\mid P)$ denotes the set of height zero characters in
a block of $Y$ with defect group $P$.

######  Theorem 14.1.

Let $\mathcal{S}$ be a collection of finite non-abelian simple groups of order
divisible by $\ell$ that satisfy the inductive AM-condition for the prime
$\ell$. Assume that $N$ is a finite group such that all the non-abelian simple
groups involved in $N$ of order divisible by $\ell$ are in $\mathcal{S}$.
Suppose that $N\lhd H$ for some finite group $H$, and let $P$ be any
$\ell$-subgroup of $N$. Then there exists an
$\operatorname{N}_{H}(P)$-equivariant bijection
$\Pi:\operatorname{Irr}_{0}(N\mid
P)\to\operatorname{Irr}_{0}(\operatorname{N}_{N}(P)\mid P)$ with

$(H_{\tau},N,\tau)\geq_{b}(\operatorname{N}_{H}(P)_{\Pi(\tau)},\operatorname{N}_{N}(P),\Pi(\tau))$

for every $\tau\in\operatorname{Irr}_{0}(N\mid P)$.

######  Definition 14.2.

Let $S$ be a finite non-abelian simple group. We say that $S$ is a minimal
counterexample if $S$ is not AM-good (with respect to the prime $\ell$) but
all finite simple groups $S^{\prime}$ with $|S^{\prime}|<|S|$ are AM-good
(with respect to the prime $\ell$).

We use the following consequence of Proposition 14.1. This will become
important in the proof of Theorem 14.6.

######  Corollary 14.3.

Let $b$ be a block of the universal covering group $G$ of a finite simple
group $S$ of Lie type and assume that all finite simple groups $S^{\prime}$
with $|S^{\prime}|<|S|$ are AM-good. Then the following are equivalent:

  1. (i)

The block $b$ is AM-good relative to the normalizer of its defect group.

  2. (ii)

The block $b$ is AM-good.

###### Proof.

Clearly (i) implies (ii). So we need to show that if $b$ is AM-good, then it
is also AM-good relative to the normalizer of its defect group $P$. By the
assumption of (ii), there exists a strong iAM-bijection
$\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(M,B)$ for some suitable
$\mathrm{Aut}(G)_{b}$-stable subgroup $M$ with $\operatorname{N}_{G}(P)\leq
M$. Now by assumption all simple non-abelian groups involved in $M$ are AM-
good. Therefore, by Theorem 14.1 there exist an iAM-bijection
$\operatorname{Irr}_{0}(M,B)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(P),B_{P})$.
In particular, the composition
$\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(P),B_{P})$
is again an iAM-bijection, see [NS14, Lemma 3.8(a)]. By [Ruh22, Remark 3.15],
the block $b$ is therefore AM-good relative to the normalizer of its defect
group. ∎

###  14.2. Proof of the inductive conditions

######  Proposition 14.4.

Let $b$ be a strictly quasi-isolated $2$-block of an exceptional group of Lie
type $G$ as in Section 2 which is not associated to a semisimple element with
centralizer $A_{2}(\pm q^{3}).3$ or ${}^{3}D_{4}(\pm q)$ in $E_{6}(\pm q)$.
Assume that $\mathrm{Out}(G)_{b}$ is cyclic and assume that
$S:=G/\mathrm{Z}(G)$ is a minimal counterexample. Then the block $b$ is AM-
good.

###### Proof.

We use the criterion of Lemma 9.6. By Proposition 8.3 there exists a block $c$
of $G^{\circ}(s)$ and an
$\operatorname{N}_{\tilde{G}\mathcal{B}}(G^{\circ}(s),c)$-equivariant
bijection
$\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(G^{\circ}(s),c)$, which
preserves central characters.

Let $(D,b_{D})$ be the $b$-Brauer pair constructed in Lemma 6.1 resp. Lemma
7.2 such that $\operatorname{N}_{G}(D,b_{D})=\operatorname{N}_{G(s)}(D)$. We
obtain that $C_{D}=b_{D}$.

By assumption, $S$ is a minimal counterexample. Hence, by Theorem 14.1 there
exists an $\mathrm{Aut}(G^{\circ}(s))_{c,P}$-equivariant bijection
$\operatorname{Irr}_{0}(G^{\circ}(s),c)\to\operatorname{Irr}_{0}(\operatorname{N}_{G^{\circ}(s)}(D),C_{P})$,
which preserves central character. In addition, Theorem 14.1 also yields an
$\mathrm{Aut}(G^{\circ}(s))_{c,D}$-equivariant bijection
$\operatorname{Irr}_{0}(\operatorname{N}_{G^{\circ}(s)}(D),C_{D})\to\operatorname{Irr}_{0}(\operatorname{N}_{G^{\circ}(s)}(D),C_{P})$
preserving central characters. Combining these bijections yields an
$\mathrm{Aut}(G^{\circ}(s))_{c,D}$-equivariant bijection
$\operatorname{Irr}_{0}(G^{\circ}(s),c)\to\operatorname{Irr}_{0}(\operatorname{N}_{G^{\circ}(s)}(D),C_{D})$,
which preserves central character.

Now induction of characters yields a natural bijection
$\operatorname{Ind}_{\operatorname{N}_{G}(D,b_{D})}^{\operatorname{N}_{G}(D)}:\operatorname{Irr}_{0}(\operatorname{N}_{G}(D,b_{D}),b_{D})\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(D),B_{D})$.
We conclude that the composition of these bijections yields an
$\operatorname{N}_{\tilde{G}\mathcal{B}}(D,B_{D})$-equivariant bijection
$\Psi:\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(D),B_{D})$,
which preserves central characters. Let $\chi\in\operatorname{Irr}_{0}(G,b)$
and $\chi^{\prime}=\Psi(\chi)$. If $G$ is not of type $E_{7}(q)$, then $\chi$
and $\chi^{\prime}$ satisfy the conditions of Lemma 9.6 by Lemma 2.5. On the
other hand, if $G$ is of type $E_{7}(q)$ then the condition of Lemma 9.6 are
satisfied by Lemma 8.7. Hence, by Lemma 9.6 the block $b$ is AM-good. ∎

######  Lemma 14.5.

Let $b$ be one of the blocks of $G=E_{6}(\pm q)$ associated to a semisimple
element with centralizer ${}^{3}D_{4}(q)$. Then $b$ is AM-good relative to the
normalizer of its defect group.

###### Proof.

By the main result of [BDR17] there exists a Morita equivalence between
$\mathcal{O}G(s)e_{s}^{G^{\circ}(s)}$ and $\mathcal{O}Ge_{s}^{G}$ which is
obtained by lifting the Deligne–Lusztig induction functor
$R_{G^{\circ}(s)}^{G}$. In particular, there exists a maximal $b$-Brauer pair
$(P,b_{P})$ with $P\leq G(s)$. By Lemma 2.4 and Lemma 2.5, the blocks of
$\mathcal{O}G(s)e_{s}^{G^{\circ}(s)}$ are Morita equivalent by restriction to
the unique block $c$ of $\mathcal{O}G^{\circ}(s)e_{s}^{G^{\circ}(s)}$. We
deduce that Deligne–Lusztig induction followed by truncation with the
idempotent $b$ induces a Morita equivalence between $\mathcal{O}G^{\circ}(s)c$
and $\mathcal{O}Gb$. This induces a character bijection
$bR_{G^{\circ}(s)}^{G}:\operatorname{Irr}_{0}(G^{\circ}(s),c)\to\operatorname{Irr}_{0}(G,b)$
which is
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{G}^{\circ}(s),b)$-equivariant.
Hence, as in the proof of [Ruh21, Theorem 10.6], there exists a maximal
$c$-Brauer pair $(P,c_{P})$ together with an
$\operatorname{N}_{\tilde{G}\mathcal{B}}(\mathbf{G}^{\circ}(s),P,C_{P})$-equivariant
bijection

$R_{\operatorname{N}_{G^{\circ}(s)}(P)}^{\operatorname{N}_{G}(P)}:\operatorname{Irr}_{0}(\operatorname{N}_{G^{\circ}(s)}(P),C_{P})\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(P),B_{P}).$

We are thus exactly in the situation of Lemma 14.4 and the proof of said lemma
shows the statement. ∎

The blocks in Lemma 14.5 could have been treated alternatively with the same
methods as the ones developed in Section 9.3.

We are now ready to prove our main theorem.

######  Theorem 14.6.

The Alperin–McKay conjecture holds for the prime $2$.

###### Proof.

According to Späth’s reduction theorem [Spä13, Theorem C], it suffices to show
that all finite simple non-abelian groups $S$ are AM-good for the prime $2$.

We assume that $S$ is a minimal counterexample. In particular, by [BR21,
Section 7,Section 8] we can assume that its universal covering group $G$ is a
group of Lie type defined over a field of odd characteristic with exceptional
root system. Assume first that the block $b$ of $G$ is not quasi-isolated and
let $\mathbf{M}^{\ast}$ be the minimal Levi subgroup of $\mathbf{G}^{\ast}$
containing $\operatorname{C}_{\mathbf{G}^{\ast}}(s)$. In particular, by
assumption on the minimality of $S$, the $2$-blocks of the simple components
of $[\mathbf{M},\mathbf{M}]^{F}$ are AM-good relative to the normalizer of
their defect group, see Corollary 14.3. By the main theorem of [Ruh22], the
block $b$ is therefore AM-good relative to the normalizer of its defect group.
We may therefore assume that $b$ is a quasi-isolated block. Suppose first that
$b$ is a unipotent block. If $b$ is the principal block, then the result
follows from [BR21, Theorem 8.3] and if $b$ is not the principal block, the
result follows from Proposition 13.4. Therefore, we can assume that $b$ is not
unipotent. If $\mathrm{Out}(G)_{b}$ is cyclic, the result follows from
Proposition 14.4, while if $\mathrm{Out}(G)_{b}$ is non-cyclic, the result
follows from Corollary 14.3 and Proposition 11.3. ∎

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