$M_{Q}^{\prime}:=\operatorname{Ind}_{\mathrm{C}_{G}(Q)\times\mathrm{C}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\tilde{N}^{\prime}}(Q)}^{\mathrm{N}_{G}(Q)\times\mathrm{N}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\tilde{N}^{\prime}}(Q)}H^{d_{Q}}(\mathcal{C}_{Q}^{\prime})\mathrm{br}_{Q}(e_{s}^{L^{\prime}}).$

######  Lemma 8.3.

Suppose that $\ell\nmid|H^{1}(F,\mathrm{Z}(\mathbf{G}))|$. Then the bimodule
$M_{Q}^{\prime}$ almost extends to
$\mathrm{N}_{G}(Q)\times\mathrm{N}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\mathcal{N}}(Q)$.

###### Proof.

We first assume that $\mathbf{L}^{\prime}$ is $1$-split. Then
$M\cong\Lambda[\mathbf{G}^{F}/\mathbf{U}^{\prime F}]e_{s}^{L^{\prime}}$ is an
$\ell$-permutation module and $\mathcal{C}\cong M$. It follows that
$\mathcal{C}^{\prime}\cong M^{\prime}$ and thus $H^{0}(\mathrm{Br}_{\Delta
Q}(\mathcal{C}^{\prime}))\cong\mathrm{Br}_{\Delta Q}(M^{\prime})$. The
bimodule $M^{\prime}$ extends to a
$G\times(L^{\prime})^{\operatorname{opp}}\Delta\mathcal{N}_{\ell^{\prime}}$-module
$\hat{M}$, which is $\mathcal{N}$-stable. Thus, $\mathrm{Br}_{\Delta
Q}(\hat{M})$ is a
$k(\operatorname{C}_{G}(Q)\times(\operatorname{C}_{L^{\prime}}(Q))^{\operatorname{opp}}\Delta\operatorname{N}_{\mathcal{N}_{\ell^{\prime}}}(Q))$-module
extending $\mathrm{Br}_{\Delta Q}(M^{\prime})$. It follows that
$\mathrm{Br}_{\Delta Q}(\hat{M})$ is an $\ell$-permutation module as well.
Thus, there exists a unique $\ell$-permutation module $\hat{M}_{Q}$ which
lifts $\mathrm{Br}_{\Delta Q}(\hat{M})$ to $\Lambda$, see [1, Corollary
3.11.4]. Since this lift is unique and $\mathrm{Br}_{\Delta Q}(\hat{M})$ is
$\mathrm{N}_{\mathcal{N}}(Q)$-stable it follows that $\hat{M}_{Q}$ is
$\mathrm{N}_{\mathcal{N}}(Q)$-stable as well. In other words,
$H^{0}(\mathcal{C}_{Q}^{\prime})\mathrm{br}_{Q}(e_{s}^{L^{\prime}})$ almost
extends to
$\mathrm{C}_{G}(Q)\times\mathrm{C}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\mathcal{N}}(Q)$.
The claim follows by applying the induction functor.

Now let us assume that $\mathbf{L}^{\prime}$ is not $1$-split. By Lemma 8.2,
the bimodule
$H^{\mathrm{dim}}_{c}(\mathbf{Y}_{\mathrm{C}_{\mathbf{U}^{\prime}}(Q)}^{\mathrm{N}_{\mathbf{G}}(Q)},\Lambda)\mathrm{br}_{Q}(e_{s}^{L^{\prime}})$
almost extends to
$\mathrm{N}_{G}(Q)\times\mathrm{N}_{L^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\mathcal{N}}(Q)$.
On the other hand, the bimodule $M_{Q}^{\prime}$ is an extension of
$H^{\mathrm{dim}}_{c}(\mathbf{Y}_{\mathrm{C}_{\mathbf{U}^{\prime}}(Q)}^{\mathrm{N}_{\mathbf{G}}(Q)},\Lambda)\mathrm{br}_{Q}(e_{s}^{L^{\prime}})$.
The quotient group $\mathcal{N}/\tilde{L}^{\prime}$ is abelian by Lemma 6.11
and Lemma 6.13. Therefore, Lemma 7.3 shows that
$H^{d_{Q}}(\mathcal{C}_{Q}^{\prime})\mathrm{br}_{Q}(e_{s}^{L^{\prime}})$
almost extends as well to a
$\Lambda(\mathrm{N}_{G}(Q)\times\mathrm{N}_{L^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\mathcal{N}}(Q))$-module.
∎

According to Lemma 8.3 there exists an extension $\hat{M}_{Q}$ of
$M_{Q}^{\prime}$ to
$\Lambda(\mathrm{N}_{G}(Q)\times\mathrm{N}_{N}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\mathcal{N}_{\ell^{\prime}}}(Q))$.
With this notation the following is quite immediate.

######  Corollary 8.4.

Suppose the assumptions and notation as above. Then the bimodule
$M_{Q}^{\prime}C_{Q}$ induces a Morita equivalence between
$\Lambda\mathrm{N}_{N^{\prime}}(Q)C_{Q}$ and $\Lambda\mathrm{N}_{G}(Q)B_{Q}$
which lifts to the Morita equivalence $\hat{M}_{Q}$ between
$\Lambda\mathrm{N}_{\mathcal{N}_{\ell^{\prime}}}(Q,C_{Q})C_{Q}$ and
$\Lambda\mathrm{N}_{G\mathcal{A}_{\ell^{\prime}}}(Q,B_{Q})B_{Q}$.

###### Proof.

Note that
$M_{Q}^{\prime}\cong\operatorname{Ind}_{\mathrm{C}_{G}(Q)\times\mathrm{C}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{N^{\prime}}(Q)}^{\mathrm{N}_{G}(Q)\times\mathrm{N}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\tilde{N}^{\prime}}(Q)}(H^{d_{Q}}(\mathcal{C}_{Q}^{\prime}))\mathrm{br}_{Q}(e_{s}^{L^{\prime}})$.
From this we conclude that the bimodule $M_{Q}^{\prime}C_{Q}$ induces a Morita
equivalence between $\Lambda\mathrm{N}_{N^{\prime}}(Q)C_{Q}$ and
$\Lambda\mathrm{N}_{G}(Q)B_{Q}$. The second claim now follows from this and
[27, Theorem 3.4]. ∎

##  9\. The first reduction

Let $b$ be a block of $\Lambda Ge_{s}^{G}$, where
$s\in(\mathbf{G}^{\ast})^{F^{\ast}}$ is the fixed strictly-quasi isolated
element from before. By [35, Theorem 1.3] there exists a defect group $D$
contained in $N$ and since $\ell\nmid|N/L|$ we have $D\subset L$. In what
follows, we let $Q$ be a fixed characteristic subgroup of $D$. For a given
character $\chi$ we use $\mathrm{bl}(\chi)$ to denote the $\ell$-block $\chi$
belongs to. For the language of character triples and the definition of the
order relation $\geq_{b}$ on character triples we refer the reader to [36,
Section 1.1].

######  Theorem 9.1.

Let $\chi\in\operatorname{Irr}(G,b)$ and
$\chi^{\prime}\in\operatorname{Irr}(\operatorname{N}_{G}(Q),B_{Q})$ 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
$(\operatorname{N}_{\tilde{G}}(Q)\mathrm{N}_{G\mathcal{B}}(Q))_{\chi^{\prime}}=\operatorname{N}_{\tilde{G}}(Q)_{\chi^{\prime}}\mathrm{N}_{G\mathcal{B}}(Q)_{\chi^{\prime}}$
and $\chi^{\prime}$ extends to $\mathrm{N}_{G\mathcal{B}}(Q)_{\chi^{\prime}}$.

  3. (iii)

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

  4. (iv)

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

     * •

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

     * •

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

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

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

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

###### Proof.

This is a consequence of [36, Theorem 2.1] and [36, Lemma 2.2]. ∎

Note that all conditions in Theorem 9.1 except condition (v) only depend on
the character theory of $G$ and $\tilde{G}$ (together with its associated
groups).

Let $M^{\prime}$ be the bimodule constructed in Corollary 7.8 which induces a
Morita equivalence between $\Lambda Ge_{s}^{G}$ and $\Lambda
N^{\prime}e_{s}^{L^{\prime}}$. We let $c$ be the block of $\Lambda
N^{\prime}e_{s}^{L^{\prime}}$ corresponding to $b$ under this equivalence.
Recall the group $\mathcal{A}$ from Definition 7.6 and that
$\mathcal{N}=\mathrm{N}_{\tilde{G}\mathcal{A}}(\mathbf{L}^{\prime},e_{s}^{L^{\prime}})$.
We denote by $\mathcal{N}_{b}$ the stabilizer of the block $b$ in
$\mathcal{N}$.

With this notation we are now ready to state the following theorem which can
be seen as an analog of [36, Theorem 2.12].

######  Theorem 9.2.

Assume that $\ell\nmid 2|H^{1}(F,\mathrm{Z}(\mathbf{G}))|$ and suppose that
the following hold.

  1. (i)

There exists an
$\operatorname{Irr}(\tilde{N}^{\prime}/N^{\prime})\rtimes\mathrm{N}_{\mathcal{N}}(Q)$-equivariant
bijection
$\tilde{\varphi}:\operatorname{Irr}(\tilde{N^{\prime}}\mid\operatorname{Irr}_{0}(c))\to\operatorname{Irr}(\operatorname{N}_{\tilde{N}^{\prime}}(Q)\mid\operatorname{Irr}_{0}(C_{Q}))$
such that it maps characters covering the character
$\nu\in\operatorname{Irr}(\mathrm{Z}(\tilde{G}))$ to a character covering
$\nu$.

  2. (ii)

There exists an $\mathrm{N}_{\mathcal{N}}(Q,C_{Q})$-equivariant bijection
$\varphi:\operatorname{Irr}_{0}(N^{\prime},c)\to\operatorname{Irr}_{0}(\operatorname{N}_{N^{\prime}}(Q),C_{Q})$
which satisfies the following two conditions:

     * •

If $\chi\in\operatorname{Irr}_{0}(N^{\prime},c)$ extends to a subgroup $H$ of
$\mathcal{N}_{b}$ then $\varphi(\chi)$ extends to $\mathrm{N}_{H}(Q)$.

     * •

$\tilde{\varphi}(\operatorname{Irr}(\tilde{N}^{\prime}\mid\chi))=\operatorname{Irr}(\operatorname{N}_{\tilde{N}^{\prime}}(Q)\mid\varphi(\chi))$
for all $\chi\in\operatorname{Irr}_{0}(c)$.

  3. (iii)

For every $\theta\in\operatorname{Irr}_{0}(c)$ and
$\tilde{\theta}\in\operatorname{Irr}(\tilde{N}^{\prime}\mid\theta)$ the
following holds: If
$\theta_{0}\in\operatorname{Irr}(\tilde{N}^{\prime}_{\theta}\mid\theta)$ is
the Clifford correspondent of
$\tilde{\theta}\in\operatorname{Irr}(\tilde{N^{\prime}})$ then
$\mathrm{bl}(\theta_{0})=\mathrm{bl}(\theta_{0}^{\prime})^{\tilde{N}^{\prime}_{\theta}}$,
where
$\theta^{\prime}_{0}\in\operatorname{Irr}(\operatorname{N}_{\tilde{N}^{\prime}}(Q)_{\varphi(\theta)}\mid\varphi(\theta))$
is the Clifford correspondent of $\tilde{\varphi}(\tilde{\theta})$.

Then the block $b$ is AM-good.

###### Proof.

According to Corollary 7.8 the bimodule $M^{\prime}$ induces an
$\mathcal{N}$-equivariant bijection

$R:\operatorname{Irr}_{0}(N^{\prime},e_{s}^{L^{\prime}})\to\operatorname{Irr}_{0}(G,e_{s}^{G}).$

By Corollary 8.4 the bimodule $M_{Q}^{\prime}$ induces an
$\mathrm{N}_{\mathcal{N}}(Q,C_{Q})$-equivariant bijection

$R_{Q}:\operatorname{Irr}_{0}(\mathrm{N}_{N^{\prime}}(Q),C_{Q})\to\operatorname{Irr}_{0}(\mathrm{N}_{G}(Q),B_{Q}).$

We define $\Psi:=R_{Q}\circ\varphi\circ
R^{-1}:\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\mathrm{N}_{G}(Q),B_{Q})$,
which is by construction
$\mathrm{N}_{\tilde{G}\mathcal{A}}(Q,B_{Q})$-equivariant.

As in the proof of [36, Theorem 2.12] we can assume that the character $\chi$
satisfies condition (i) of Theorem 9.1. We denote $\chi^{\prime}:=\Psi(\chi)$
and show that the characters $\chi$ and $\chi^{\prime}$ satisfy the conditions
of Theorem 9.1.

Since the bijection
$\Psi:\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(Q),B_{Q})$
is $\mathrm{N}_{\tilde{G}\mathcal{A}}(Q,B_{Q})$-equivariant we deduce that
condition (iii) in Theorem 9.1 is satisfied and we have

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

The following lemma finishes the verification of condition (ii) in Theorem 9.1

######  Lemma 9.3.

The character $\chi^{\prime}$ extends to its inertia group in
$\operatorname{N}_{G\mathcal{B}}(Q)$.

###### Proof.

We have
$\operatorname{N}_{G\mathcal{B}}(Q)_{\chi^{\prime}}/\operatorname{N}_{G}(Q)\cong\mathcal{B}_{\chi}$.
Since the Sylow $r$-subgroups of $\mathcal{B}_{\chi}$ for $r\neq 2$ are cyclic
it suffices to show that $\chi^{\prime}$ extends to
$\operatorname{N}_{G\mathcal{B}_{2}}(Q)$, where $\mathcal{B}_{2}$ is the Sylow
$2$-subgroup of $\mathcal{B}_{\chi}$.

We may assume that $\mathcal{B}_{2}=\langle\phi^{i},\gamma\rangle$. There
exists some $\tilde{g}\in\tilde{L}^{\prime}$ such that
$F_{0}^{i}=\mathrm{ad}(\tilde{g})\phi^{i}$. Recall that the character $\chi$
extends to $G\mathcal{B}_{2}$. It follows that $\chi$ has a $\phi^{i}$-stable
extension to $G\langle\gamma\rangle$. Consequently, this extension is
$\tilde{g}^{-1}F_{0}^{i}$-stable. From this it follows that $\chi$ extends to
$G\langle\tilde{g}^{-1}F_{0}^{i}\rangle$. Thus, by Corollary 7.8 the character
$R^{-1}(\chi)$ extends to
$N^{\prime}\langle\tilde{g}^{-1}F_{0}^{i},\gamma\rangle$. By assumption (ii),
the character $\varphi(R^{-1}(\chi))$ therefore extends to its inertia group
in $\operatorname{N}_{G\langle\tilde{g}^{-1}F_{0}^{i},\gamma\rangle}(Q)$.
Hence, by Corollary 8.4 the character $\chi^{\prime}$ has an extension to
$\operatorname{N}_{G\langle\gamma\rangle}(Q)_{\chi^{\prime}}$ which is
$\operatorname{N}_{G\langle\phi,\gamma\rangle}(Q)_{\chi^{\prime}}$-stable. It
follows that $\chi^{\prime}$ extends to
$\operatorname{N}_{G\mathcal{B}_{2}}(Q)=\langle\phi^{i},\gamma\rangle$. ∎

Fix a character $\tilde{\chi}\in\operatorname{Irr}(\tilde{G}\mid\chi)$. We
define
$\tilde{\chi}^{\prime}:=\tilde{R}_{Q}\circ\tilde{\varphi}\circ\tilde{R}^{-1}(\tilde{\chi})$,
where $\tilde{R}$ and $\tilde{R}_{Q}$ are defined as in 1.8. From [36, Lemma
2.9] and assumption (i) it follows that the first part of condition (iv) in
Theorem 9.1 is satisfied. Moreover, [35, Lemma 2.10] and assumption (i) imply
that the second part of condition (iv) in Theorem 9.1 is satisfied.

We now verify condition (v) in Theorem 9.1. Let
$\tilde{\chi}_{0}\in\operatorname{Irr}(\tilde{G}_{\chi}\mid\chi)$ be the
Clifford correspondent of $\tilde{\chi}$. Moreover, let
$\tilde{\chi}_{0}^{\prime}\in\operatorname{Irr}(\operatorname{N}_{\tilde{G}}(Q)_{\chi^{\prime}}\mid\chi^{\prime})$
be the Clifford correspondent of $\tilde{\chi}^{\prime}$. Recall that there
exists a complex $\mathcal{C}^{\prime}$ of
$\Lambda(G\times(N^{\prime})^{\mathrm{opp}}\Delta\tilde{N})$-modules such that
$H^{d}(\mathcal{C}^{\prime})\cong M^{\prime}$ and $\mathcal{C}^{\prime}$
induces a splendid Rickard equivalence between $\Lambda Ge_{s}^{G}$ and
$\Lambda N^{\prime}e_{s}^{L^{\prime}}$, see Lemma 1.3.

######  Lemma 9.4.

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

###### Proof.

Define $J:=\tilde{G}_{\chi}$ and let $J_{0}$ be the subgroup of
$\tilde{N}^{\prime}/N^{\prime}$ corresponding to $J$ under the natural
isomorphism $\tilde{N}^{\prime}/N^{\prime}\cong\tilde{G}/G$.

Consider
$\mathcal{C}:=G\Gamma_{c}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\mathbf{G}},\Lambda)e_{s}^{L^{\prime}}$
as complex of
$\Lambda(G\times(L^{\prime})^{\operatorname{opp}}\Delta(\tilde{L}^{\prime}))$-modules
and define
$\tilde{\mathcal{C}}:=\operatorname{Ind}_{G\times(L^{\prime})^{\operatorname{opp}}\Delta(\tilde{L}^{\prime})}^{\tilde{G}\times(\tilde{L}^{\prime})^{\operatorname{opp}}}(\mathcal{C})$.
We have $\tilde{\mathcal{C}}\cong
G\Gamma_{c}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\tilde{\mathbf{G}}},\Lambda)e_{s}^{L^{\prime}}$
by [5, Proposition 1.1].

As in 1.8 we let $e\in\mathrm{Z}(\Lambda\tilde{L}^{\prime})$ be the central
idempotent such that $\sum_{n\in
N^{\prime}/L^{\prime}}{}^{n}e=e_{s}^{L^{\prime}}$. It follows similar to
arguments given in the proof of [4, Theorem 7.5] that the complex
$\tilde{\mathcal{C}}^{\prime}:=\tilde{\mathcal{C}}e\otimes_{\Lambda\tilde{L}^{\prime}}\Lambda\tilde{N}^{\prime}$
induces a splendid Rickard equivalence between
$\Lambda\tilde{N}^{\prime}e_{s}^{L^{\prime}}$ and $\Lambda\tilde{G}e_{s}^{G}$.
Moreover, we have
$\operatorname{Ind}_{G\times(L^{\prime})^{\operatorname{opp}}\Delta(\tilde{L}^{\prime})}^{\tilde{G}\times(\tilde{N}^{\prime})^{\operatorname{opp}}}(\mathcal{C}^{\prime})\cong\tilde{\mathcal{C}}^{\prime}$.
The cohomology of $\mathcal{C}^{\prime}$ is concentrated in degree
$d:=\mathrm{dim}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}})$ and
$H^{d}(\tilde{\mathcal{C}}_{0}^{\prime})\cong\mathrm{Ind}_{(G\times(L^{\prime})^{\mathrm{opp}})\Delta
J_{0}}^{J\times J_{0}^{\mathrm{opp}}}H^{d}(\mathcal{C}^{\prime})$. By [27,
Theorem 3.4] the bimodule $H^{d}(\tilde{\mathcal{C}}_{0}^{\prime})$ induces a
Morita equivalence between $\mathcal{O}J_{0}c$ and $\mathcal{O}Jb$. We denote
by

$R_{0}:\operatorname{Irr}(J_{0},c)\to\operatorname{Irr}(J,b)$

the associated bijection between irreducible characters. Using [35, Lemma 1.9]
we see that the complex
$\tilde{\mathcal{C}}_{0}^{\prime}:=\mathrm{Ind}_{(G\times
N^{\mathrm{opp}})\Delta(J_{0})}^{J\times
J_{0}^{\mathrm{opp}}}(\mathcal{C}^{\prime})c$ induces a splendid Rickard
equivalence between $\mathcal{O}Jb$ and $\mathcal{O}J_{0}c$. The bimodule

$(M^{\prime}_{Q})_{0}:=\operatorname{Ind}^{\operatorname{N}_{J}(Q)\times(\operatorname{N}_{J_{0}}(Q))^{\operatorname{opp}}}_{\operatorname{N}_{G}(Q)\times(\operatorname{N}_{L}(Q))^{\operatorname{opp}}\Delta\operatorname{N}_{J_{0}}(Q)}(\operatorname{Res}_{\operatorname{N}_{G}(Q)\times(\operatorname{N}_{L^{\prime}}(Q))^{\operatorname{opp}}\Delta\operatorname{N}_{J_{0}}(Q)}^{\operatorname{N}_{G}(Q)\times(\operatorname{N}_{L}(Q))^{\operatorname{opp}}\Delta\operatorname{N}_{\tilde{N}^{\prime}}(Q)}(M_{Q}^{\prime}))$

induces a Morita equivalence between $\mathcal{O}\mathrm{N}_{J_{0}}(Q)C_{Q}$
and $\mathcal{O}\mathrm{N}_{J}(Q)B_{Q}$. We denote the associated character
bijection by

$(R_{0})_{Q}:\operatorname{Irr}(\mathrm{N}_{J_{0}}(Q),C_{Q})\to\operatorname{Irr}(\mathrm{N}_{J}(Q),B_{Q}).$

By construction,
$\tilde{\chi}^{\prime}\in\operatorname{Irr}(\mathrm{N}_{\tilde{G}}(Q)\mid\chi^{\prime})$.
Let $\theta:=R^{-1}(\chi)$ and $\tilde{\theta}:=\tilde{R}^{-1}(\tilde{\chi})$.
We obtain that
$(R_{0})^{-1}_{Q}(\tilde{\chi}_{0}^{\prime})\in\operatorname{Irr}(\mathrm{N}_{J_{0}}(Q)\mid\varphi(\theta))$
is the Clifford correspondent of $\tilde{\varphi}(\tilde{\theta})$.
Consequently, we have

$\mathrm{bl}((R_{0})_{Q}^{-1}(\tilde{\chi^{\prime}_{0}}))^{J_{0}}=\mathrm{bl}(R^{-1}_{0}(\tilde{\chi}_{0}))$

by assumption (iii). By the definition of $M_{Q}^{\prime}$ after Lemma 8.3 we
have

$M_{Q}^{\prime}\otimes_{\Lambda}k\cong\operatorname{Ind}_{\mathrm{C}_{G}(Q)\times\mathrm{C}_{N}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{N^{\prime}}(Q)}^{\mathrm{N}_{G}(Q)\times\mathrm{N}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\tilde{N}^{\prime}}(Q)}H^{d_{Q}}(\mathrm{Br}_{\Delta
Q}(C^{\prime}))\mathrm{br}_{Q}(e_{s}^{L^{\prime}})$

and therefore,

$(M_{Q}^{\prime})_{0}\otimes_{\Lambda}k\cong\operatorname{Ind}_{\mathrm{C}_{G}(Q)\times\mathrm{C}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{J_{0}}(Q)}^{\mathrm{N}_{G}(Q)\times\mathrm{N}_{N^{\prime}}(Q)^{\mathrm{opp}}\Delta\mathrm{N}_{\tilde{J}_{0}}(Q)}H^{d_{Q}}(\mathrm{Br}_{\Delta
Q}(\mathcal{C}_{0}^{\prime}))\mathrm{br}_{Q}(e_{s}^{L^{\prime}}).$

Therefore, by [35, Remark 1.21] the equality
$\mathrm{bl}((R_{0})_{Q}^{-1}(\tilde{\chi^{\prime}_{0}}))^{J_{0}}=\mathrm{bl}(R^{-1}_{0}(\tilde{\chi}_{0}))$
implies that
$\mathrm{bl}(\tilde{\chi}^{\prime}_{0})^{J}=\mathrm{bl}(\tilde{\chi}_{0})$. ∎

Theorem 9.1 applies and we obtain

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

where $\mathrm{Z}:=\mathrm{Z}(G)\cap\mathrm{Ker}(\chi)$. Using [35, Theorem
1.10] we deduce that the bijection
$\Psi:\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(Q),B_{Q})$
is a strong iAM-bijection in the sense of [36, Definition 1.9], which shows
that $b$ is AM-good, see [36, Definition 1.11]. ∎

######  Remark 9.5.

The statement of the previous theorem remains true for the groups occurring in
Theorem 4.2 if we take $\mathbf{L}^{\prime\ast}$ to be the minimal Levi
subgroup containing $\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ and
replace $\mathcal{A}$ by the group $\langle F_{0}\rangle$. To see this, just
replace Corollary 7.8 and Corollary 8.4 by Theorem 4.2 (and its local version
which can be proved as in Corollary 8.4).

######  Remark 9.6.

In the proof of Theorem 9.2 we have shown that there exist a strong iAM-
bijection
$\Psi:\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(Q),B_{Q})$
for the characteristic subgroup $Q$ of a defect group $D$ of the block $b$. In
Section 12 it will be important to keep track of the subgroup $Q$. If such a
strong iAM-bijection exists then we will say that the block $b$ is AM-good
relative to the subgroup $Q$.

##  10\. Results on $\ell$-blocks when $\ell\mid(q-\varepsilon)$

###  10.1. Action of diagonal automorphisms on $2$-blocks

Let $Y$ be a finite group and $X$ be a normal subgroup of $Y$. For an
$\ell$-block $e$ of $X$ we denote by $Y[e]$ the Dade ramification group of $e$
in $Y$, see for instance [29].

######  Lemma 10.1.

Let $\mathbf{G}$ be a simple, simply connected group of type $A$ and let $b$
be a (non-necessarily quasi-isolated) $\ell$-block of $G$. If
$\ell\mid(q-\varepsilon)$ then block induction yields a bijective map
$\mathrm{Bl}(\tilde{G}[b]\mid b)\to\mathrm{Bl}(\tilde{G}\mid b)$.

###### Proof.

Suppose that $b$ is a block of $\Lambda Ge_{s}^{G}$ for a semisimple element
$s\in(\mathbf{G}^{\ast})^{F^{\ast}}$ of $\ell^{\prime}$-order. Consider a
regular embedding $\iota:\mathbf{G}\hookrightarrow\mathbf{\tilde{G}}$. Let
$\tilde{\chi}\in\mathcal{E}(\tilde{G},\tilde{s})$ be the unique semisimple
character in its Lusztig series, see [3, Theorem 15.10]. We fix a constituent
$\chi\in\operatorname{Irr}(G\mid\tilde{\chi})$ which lies in the block $b$. We
have $\operatorname{Irr}(\tilde{G}\mid\chi)=\\{\hat{z}\otimes\tilde{\chi}\mid
z\in\operatorname{Z}(\tilde{G})\\}$ and such a character
$\hat{z}\otimes\tilde{\chi}$ lies in the Lusztig series
$\mathcal{E}(\tilde{G},\tilde{s}z)$. Note that
$\mathcal{E}_{\ell}(\tilde{G},\tilde{s})=\operatorname{Irr}(\tilde{G},\tilde{b})$,
where $\tilde{b}$ is the block of $\tilde{\chi}$. Hence,
$\hat{z}\otimes\tilde{\chi}$ lies in $\tilde{b}$ if and only if the
$\ell^{\prime}$-part $(\tilde{s}z)_{\ell^{\prime}}$ of $\tilde{s}z$ is
$\tilde{\mathbf{G}}^{\ast}$-conjugate to $\tilde{s}$. This is equivalent to
$\tilde{s}z_{\ell^{\prime}}$ being $\tilde{\mathbf{G}}^{\ast}$-conjugate to
$\tilde{s}$. In this case we obtain that
$\tilde{\chi}=\hat{z}_{\ell^{\prime}}\otimes\tilde{\chi}$ since $\tilde{\chi}$
is the unique semisimple character in $\mathcal{E}(\tilde{G},\tilde{s})$.
Thus, $\tilde{b}\otimes\hat{z}=\tilde{b}$ if and only if
$\chi=\hat{z}_{\ell^{\prime}}\otimes\chi$. From this we deduce by Clifford
theory that $|\mathrm{Bl}(\tilde{G}\mid
b)|=|\tilde{G}_{\chi}:G|_{\ell^{\prime}}$.

On the other hand, by [29, Proposition 3.9] we deduce that
$|\mathrm{Bl}(\tilde{G}[b]\mid b)|=|\tilde{G}[b]:G|_{\ell^{\prime}}$.
According to [29, Theorem 3.5] and the Fong–Reynolds reduction block induction
yields a surjective map $\mathrm{Bl}(\tilde{G}[b]\mid
b)\to\mathrm{Bl}(\tilde{G}\mid b)$. This shows that
$|\tilde{G}[b]:G|_{\ell^{\prime}}\geq|\tilde{G}_{\chi}:G|_{\ell^{\prime}}$. On
the other hand, by [29, Theorem 4.1] we deduce that
$|\tilde{G}[b]:G|_{\ell^{\prime}}\leq|\tilde{G}_{\chi}:G|_{\ell^{\prime}}$.
This shows the statement. ∎

As in Section 9 we let $b$ be an $\ell$-block of $\Lambda Ge_{s}^{G}$, where
$s\in(\mathbf{G}^{\ast})^{F^{\ast}}$ is a strictly quasi-isolated element of
$\ell^{\prime}$-order. Recall the Levi subgroup $\mathbf{L}^{\prime}$ of
$\mathbf{G}$ defined in Lemma 6.2.

######  Lemma 10.2.

There exists a block $c_{1}$ of $\Lambda L^{\prime}e_{s}^{L^{\prime}}$, unique
up to $N^{\prime}$-conjugation, such that
$bH_{c}^{\mathrm{dim}}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\mathbf{G}},\Lambda)c_{1}\neq
0$.

###### Proof.

By Theorem 1.1 there exists a bimodule $M_{0}$ inducing a Morita equivalence
between $\Lambda Ge_{s}^{G}$ and $\Lambda N^{\prime}e_{s}^{L^{\prime}}$. We
let $c$ be the block of $\Lambda N^{\prime}e_{s}^{L^{\prime}}$ corresponding
to $b$ under the Morita equivalence induced by $M_{0}$. The $\Lambda(G\times
N^{\prime\operatorname{opp}})$-bimodule $bM_{0}c\cong bM_{0}$ is thus
indecomposable. We have $\mathrm{Res}^{G\times
N^{\prime\operatorname{opp}}}_{G\times(L^{\prime})^{\operatorname{opp}}}(M_{0})\cong
bH_{c}^{\mathrm{dim}}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\mathbf{G}},\Lambda)$.
Observe that $bM_{0}$ is a right $\Lambda N^{\prime}c$-module. Thus, if
$c_{1}$ is a block of $L^{\prime}$ such that
$bH_{c}^{\mathrm{dim}}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\mathbf{G}},\Lambda)c_{1}\neq
0$ then $c_{1}$ has to lie below $c$. This determines $c_{1}$ up to
$N^{\prime}$-conjugation. ∎

######  Definition 10.3.

If $\ell\nmid(q-\varepsilon)$ denote $T:=N^{\prime}$ and $\hat{c}:=c$.
Otherwise, we let $T$ be the largest subgroup of $N^{\prime}$ containing
$L^{\prime}$ such that there exists a unique block $\hat{c}$ covering $c_{1}$.
Additionally, we set $\tilde{T}:=T\tilde{L}^{\prime}$.

Let us now assume that $\ell\mid(q-\varepsilon)$. The block $b$ of $\Lambda
Ge_{s}^{G}$ is Morita equivalent to a block of $\Lambda Ne_{1}^{L^{F}}$. This
block of $N$ is covered by the principal block $\Lambda Le_{1}^{L}$ of $L$.
Thus, any Sylow $\ell$-subgroup $Q$ of $L$ is a defect group of $b$. A
Frattini argument shows that
$\operatorname{N}_{\tilde{N}^{\prime}}(Q)\tilde{L}^{\prime}=\tilde{N}^{\prime}$.
Thus, the subgroup $Q$ of $L$ satisfies the group-theoretic assumptions of
Lemma 8.2. We denote by $\hat{C}_{Q}$ the Brauer correspondent of $\hat{c}$ in
$\operatorname{N}_{T}(Q)$.

######  Theorem 10.4.

Assume that $\ell\mid(q-\varepsilon)$ and suppose that the following holds.

  1. (i)

There exists an
$\operatorname{Irr}(\tilde{T}/T)\rtimes\mathrm{N}_{\mathcal{N}}(Q)$-equivariant
bijection
$\tilde{\varphi}:\operatorname{Irr}(\tilde{T}\mid\operatorname{Irr}_{0}(\hat{c}))\to\operatorname{Irr}(\operatorname{N}_{\tilde{T}}(Q)\mid\operatorname{Irr}_{0}(\hat{C}_{Q}))$
such that it maps characters covering the character
$\nu\in\operatorname{Irr}(\mathrm{Z}(\tilde{G}))$ to a character covering
$\nu$.

  2. (ii)

There exists an $\mathrm{N}_{\mathcal{N}}(Q,\hat{C}_{Q})$-equivariant
bijection
$\varphi:\operatorname{Irr}_{0}(T,\hat{c})\to\operatorname{Irr}_{0}(\operatorname{N}_{T}(Q),\hat{C}_{Q})$
which satisfies the following two conditions:

     * •

If $\chi\in\operatorname{Irr}_{0}(T,\hat{c})$ extends to a subgroup $H$ of
$\mathcal{N}_{b}$ then $\varphi(\chi)$ extends to $\mathrm{N}_{H}(Q)$.

     * •

$\tilde{\varphi}(\operatorname{Irr}(\tilde{T}\mid\chi))=\operatorname{Irr}(\operatorname{N}_{\tilde{T}}(Q)\mid\varphi(\chi))$
for all $\chi\in\operatorname{Irr}_{0}(\hat{c})$.

  3. (iii)

For every $\theta\in\operatorname{Irr}_{0}(\hat{c})$ and
$\tilde{\theta}\in\operatorname{Irr}(\tilde{T}\mid\theta)$ the following
holds: If
$\theta_{0}\in\operatorname{Irr}(\tilde{T}^{\prime}_{\theta}\mid\theta)$ is
the Clifford correspondent of $\tilde{\theta}\in\operatorname{Irr}(\tilde{T})$
then
$\mathrm{bl}(\theta_{0})=\mathrm{bl}(\theta_{0}^{\prime})^{\tilde{T}_{\theta}}$,
where
$\theta^{\prime}_{0}\in\operatorname{Irr}(\operatorname{N}_{\tilde{T}}(Q)_{\varphi(\theta)}\mid\varphi(\theta))$
is the Clifford correspondent of $\tilde{\varphi}(\tilde{\theta})$.

Then the block $b$ is AM-good.

###### Proof.

According to the proof of Theorem 7.7 there exists a
$K[G\times(L^{\prime})^{\mathrm{opp}}\Delta\mathcal{N}]$-module extending
$H_{c}^{\mathrm{dim}}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\mathbf{G}},K)e_{s}^{L^{\prime}}$.
Denote by $M^{\prime}$ the restriction of $\hat{M}$ to
$G\times(N^{\prime})^{\mathrm{opp}}\Delta\tilde{N}^{\prime}$.

We claim that the bimodule $b\operatorname{Res}_{G\times
T^{\mathrm{opp}}}^{G\times(N^{\prime})^{\mathrm{opp}}\Delta\tilde{N}^{\prime}}(M^{\prime})\hat{c}$
induces an $\mathcal{N}_{b}$-equivariant bijection
$R:\operatorname{Irr}(T,\hat{c})\to\operatorname{Irr}(G,b)$. For
$T=N^{\prime}$ we observe that $\hat{c}$ is the unique block of $N^{\prime}$
covering the block $c_{1}$ of $L^{\prime}$. By Lemma 10.2 we deduce that
$bM^{\prime}=bM^{\prime}\hat{c}$. Since $N^{\prime}/L^{\prime}$ is cyclic,
Lemma 1.2 implies that $bM^{\prime}\hat{c}$ is multiplicity free as
$K(G\times(N^{\prime})^{\mathrm{opp}})$-module. Now Lemma 10.2 together with
Theorem 1.1 imply that
$|\operatorname{Irr}(G,b)|=|\operatorname{Irr}(N^{\prime},\hat{c})|$ and thus
the bimodule $bM^{\prime}\hat{c}$ necessarily induces a bijection between
these two sets.

Assume therefore now that $T=L^{\prime}$. Then $c_{1}$ is necessarily
$N^{\prime}$-stable and there are exactly $|N^{\prime}/L^{\prime}|$ different
blocks covering $c_{1}$. Consequently, by [29, Proposition 3.9] and [29,
Theorem 4.1] in this case the block $\Lambda N^{\prime}c$ is isomorphic to
$\Lambda L^{\prime}c_{0}$ via restriction. We have
$b\operatorname{Res}_{G\times(L^{\prime})^{\mathrm{opp}}}^{G\times(N^{\prime})^{\mathrm{opp}}\Delta\tilde{N}^{\prime}}(M^{\prime})\hat{c}\cong
H^{\mathrm{dim}}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\mathbf{G}},K)\hat{c}$.
Hence, Theorem 1.1 show that
$H^{\mathrm{dim}}(\mathbf{Y}_{\mathbf{U}^{\prime}}^{\mathbf{G}},\Lambda)\hat{c}$
induces a Morita equivalence between $\Lambda L^{\prime}\hat{c}$ and $\Lambda
Gb$. The claim thus also follows in this case.

As explained before the statement of this theorem, the subgroup $Q$ satisfies
the group theoretic requirement in Lemma 8.2. The proof of said lemma
therefore shows that the bimodule
$H^{\mathrm{dim}}_{c}(\mathbf{Y}_{\mathrm{N}_{\mathbf{U}^{\prime}}(Q)}^{\mathrm{N}_{\mathbf{G}}(Q)},K)\operatorname{br}_{Q}(e_{s}^{L^{\prime}})$
extends to an
$\operatorname{N}_{G}(Q)\times\operatorname{N}_{N^{\prime}}(Q)^{\operatorname{opp}}\Delta\operatorname{N}_{\mathcal{N}}(Q)$-module.
Let $\hat{M}_{Q}$ be such an extension and denote by $M_{Q}^{\prime}$ its
restriction to
$\operatorname{N}_{G}(Q)\times\operatorname{N}_{N^{\prime}}(Q)^{\operatorname{opp}}\Delta\operatorname{N}_{\tilde{N}^{\prime}}(Q)$.

Arguing as in the global case we obtain that the bimodule
$B_{Q}\operatorname{Res}_{\operatorname{N}_{G}(Q)\times\operatorname{N}_{T}(Q)^{\operatorname{opp}}}^{\operatorname{N}_{G}(Q)\times\operatorname{N}_{N^{\prime}}(Q)^{\operatorname{opp}}\Delta\operatorname{N}_{\tilde{N}^{\prime}}(Q)}(M_{Q}^{\prime})\hat{C}_{Q}$
induces a bijection
$R_{Q}:\operatorname{Irr}_{0}(\operatorname{N}_{T}(Q),\hat{C}_{Q})\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(Q),B_{Q})$.
As in Theorem 9.2 we define $\Psi:=R_{Q}\circ\varphi\circ
R^{-1}:\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\mathrm{N}_{G}(Q),B_{Q})$,
which is by construction
$\mathrm{N}_{\tilde{G}\mathcal{A}}(Q,B_{Q})$-equivariant. If $T=L^{\prime}$
then the arguments in the proof of [36, Theorem 3.12] show the result. We can
therefore from now on assume that $T=N^{\prime}$.

We fix a character $\chi\in\operatorname{Irr}_{0}(G,b)$ and denote
$\chi^{\prime}:=\Psi(\chi)$. As in Theorem 9.2 we want to show that the
characters $\chi$ and $\chi^{\prime}$ satisfy the conditions in Theorem 9.1.
Moreover, as in the proof of Theorem 9.2 we fix a character
$\tilde{\chi}\in\operatorname{Irr}(\tilde{G}\mid\chi)$ and define
$\tilde{\chi}^{\prime}:=\tilde{R}_{Q}\circ\tilde{\varphi}\circ\tilde{R}^{-1}(\tilde{\chi}).$
One easily sees that conditions (i)-(iv) of the theorem can now be verified
exactly in the same way as in Theorem 9.2. However, to show condition (v) we
crucially used Lemma 8.3. Nevertheless according to Lemma 1.5 we see that
$\mathrm{bl}(\tilde{\chi}^{\prime})^{\tilde{G}}=\mathrm{bl}(\tilde{\chi})$.
According to Lemma 10.1 the Clifford correspondents
$\tilde{\chi}_{0}\in\operatorname{Irr}(\tilde{G}_{\chi}\mid\chi)$ and
$\tilde{\chi}^{\prime}_{0}\in\operatorname{Irr}(\operatorname{N}_{\tilde{G}}(Q)_{\chi^{\prime}}\mid\chi^{\prime})$
of $\tilde{\chi}$ and $\tilde{\chi}^{\prime}$ respectively therefore satisfy
$\mathrm{bl}(\tilde{\chi}_{0})=\mathrm{bl}(\tilde{\chi}_{0}^{\prime})^{\tilde{G}_{\chi}}$.
In other words also condition (v) is satisfied. ∎

##  11\. Results on defect groups of blocks of groups of Lie type

###  11.1. Centralizers of defect groups

Following the terminology in [20, Section 3.4] we say that an $\ell$-group $D$
is Cabanes if it has a unique maximal abelian normal subgroup $Q$. Moreover,
in this case we say that $Q$ is the Cabanes subgroup of $D$. We recall [10,
Lemma 4.16]:

######  Theorem 11.1.

Let $\mathbf{H}$ be a connected reductive group defined over
$\overline{\mathbb{F}}_{p}$ such that $\ell\geq 5$ and $\ell\geq 7$ if
$\mathbf{H}$ has a component of type $E_{8}$. Then the defect group of any
$\ell$-block of $\mathbf{H}^{F}$ is Cabanes.

We keep the notation from the previous section. In particular, $\mathbf{G}$ is
simple, simply connected of type $A$ and $b$ denotes a strictly quasi-isolated
block of $\Lambda Ge_{s}^{G}$ and we let $\mathbf{L}^{\prime}$ be the proper
Levi subgroup of $\mathbf{G}$ constructed before. Moreover, $c$ denotes the
block of $\Lambda N^{\prime}e_{s}^{L^{\prime}}$ which corresponds to $b$ under
the Morita equivalence given by the bimodule $M^{\prime}$ defined after the
proof of Theorem 7.7.

We fix a defect group $D$ of the block $c$. Moreover if $\ell\geq 5$ we define
$Q$ to be the Cabanes subgroup of $D$ and if $\ell<5$ we let $Q:=D$. Let
$\tilde{b}$ be a block of $\tilde{G}$ covering $b$ with defect group
$\tilde{D}$ satisfying $\tilde{D}\cap G=D$. If $\ell\geq 5$ let $\tilde{Q}$ be
the Cabanes subgroup of $\tilde{D}$ otherwise we define
$\tilde{Q}:=\tilde{D}$.

######  Lemma 11.2.

Suppose that $\ell\neq 2$ and that $D$ is abelian if $\ell=2$. With the
notation as above we have
$\operatorname{C}_{G}(Q)=\operatorname{C}_{G}(\tilde{Q})$ and
$\operatorname{N}_{G}(Q)=\operatorname{N}_{G}(\tilde{Q})$.

###### Proof.

We may assume that
$\ell\mid(n+1,q-\varepsilon)=|\mathbf{\tilde{G}}^{F}:\operatorname{Z}(\tilde{\mathbf{G}}^{F})\mathbf{G}^{F}|$
since otherwise we have $\tilde{Q}=Q\mathrm{Z}(\mathbf{\tilde{G}}^{F})_{\ell}$
and the statement follows.

If $s$ has order $n+1$ then we have $\ell\nmid(n+1)$. As explained at the
beginning of the proof this implies the statement in this case. Hence, by [16,
Lemma 5.2] (see also Lemma 14.1 below) the statement of the lemma holds for
$\ell=2$. We can therefore assume that $\ell\geq 5$.

Suppose first that $s=1$. We consider again the Frobenius endomorphism
$F^{\prime}$ introduced in 5.2 instead of $F$. Let $\mathbf{S}$ be the
diagonal torus in $\mathbf{G}$. The Cabanes subgroup of a Sylow
$\ell$-subgroup of $\mathbf{G}^{F}$ is given by
$\mathbf{S}_{\ell}^{F^{\prime}}$. Since $\ell\neq 2$ we have
$\operatorname{C}_{\mathbf{G}}(\mathbf{S}_{\ell}^{F^{\prime}})=\mathbf{S}$,
see [11, Proposition 22.6]. From this the claim of the lemma follows easily in
this case.

Let $\mathbf{L}$ be a Levi subgroup of $\mathbf{G}$ dual to the Levi subgroup
$\mathrm{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ of $\mathbf{G}^{\ast}$. Note that
$\Lambda\mathbf{N}^{F}e_{1}^{\mathbf{L}^{F^{\prime}}}$ is Morita equivalent to
$\Lambda\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F^{\prime}}}$, see [4, Example 7.10].
Any block of $\Lambda\mathbf{N}^{F^{\prime}}e_{1}^{\mathbf{L}^{F^{\prime}}}$
has the same defect group as a block of $\mathbf{L}^{F^{\prime}}$ which is
covered by it. It is well known, see [9, Theorem 13], that since
$\ell\mid(q-\varepsilon)$ the algebra
$\Lambda\mathbf{L}^{F^{\prime}}e_{1}^{\mathbf{L}^{F^{\prime}}}$ consists of
one block only, the principal block of $\mathbf{L}^{F^{\prime}}$.

Therefore, $Q$ is the Cabanes subgroup of a Sylow $\ell$-subgroup of
$\mathbf{L}^{F^{\prime}}$. By Corollary 5.4 we have
$\tilde{\mathbf{L}}\cong(\mathrm{GL}_{e})^{d}\times\dots\times(\mathrm{GL}_{e})^{d}$
with the Frobenius endomorphism $F^{\prime}$ transitively permuting the $d$
copies of each $(\mathrm{GL}_{e})^{d}$ such that
$\tilde{\mathbf{L}}^{F^{\prime}}\cong\mathrm{GL}_{e}((\varepsilon
q)^{d})\times\dots\times\mathrm{GL}_{e}((\varepsilon q)^{d})$ with $e>1$.
Denote $Q_{0}:=Q\cap[\mathbf{L},\mathbf{L}]^{F^{\prime}}$. We conclude
$[\mathbf{L},\mathbf{L}]^{F^{\prime}}\cong\mathrm{SL}_{e}((\varepsilon
q)^{d})\times\dots\times\mathrm{SL}_{e}((\varepsilon q)^{d})$. Here, $d$
divides $|A(s)|$ which is of order prime to $\ell$, see Corollary 5.4. Let
$\mathbf{S}$ be the diagonal torus in
$\mathrm{SL}_{e}(\overline{\mathbb{F}}_{q})$. The Cabanes subgroup of the
Sylow $\ell$-subgroup of $\mathrm{SL}_{e}((\varepsilon q)^{d})$ is given by
$\mathbf{S}_{\ell}^{F^{\prime d}}$. Since $\ell\nmid d$ and $\ell\mid
q-\varepsilon$ it follows that $((\varepsilon q)^{d}-1)_{\ell}=(\varepsilon
q-1)_{\ell}$. We deduce that
$|\mathbf{S}_{\ell}^{F^{\prime}}|=|\mathbf{S}_{\ell}^{F^{\prime d}}|$. Denote
by $\Delta_{d}:\operatorname{GL}_{e}\to(\operatorname{GL}_{e})^{d}$ the
$d$-fold diagonal embedding. By the proof of [10, Lemma 4.16] we can therefore
assume that

$Q_{0}=\Delta_{d}(\mathbf{S}_{\ell}^{F^{\prime}})\times\dots\times\Delta_{d}(\mathbf{S}_{\ell}^{F^{\prime}}).$

Hence
$\operatorname{C}_{\tilde{\mathbf{G}}}(Q_{0})\subseteq\operatorname{GL}_{ed}\times\dots\times\operatorname{GL}_{ed}$.
Since $\ell\neq 2$ we have
$\operatorname{C}_{\operatorname{GL}_{e}}(\mathbf{S}_{\ell}^{F^{\prime}})=\mathbf{S}$,
see [11, Proposition 22.6]. Therefore,
$\operatorname{C}_{\mathbf{\tilde{G}}}(Q_{0})\cong(\mathrm{GL}_{d})^{e}\times\dots\times(\mathrm{GL}_{d})^{e}$.
A similar calculation shows that this coincides with
$\operatorname{C}_{\mathbf{\tilde{G}}}(\tilde{Q})$. This implies that
$\operatorname{C}_{\mathbf{G}}(\tilde{Q})=\operatorname{C}_{\mathbf{G}}(Q)=\operatorname{C}_{\mathbf{G}}(Q_{0})$.

Observe that $\mathbf{M}:=\mathrm{C}_{\mathbf{\tilde{G}}}(Q)$ is a Levi
subgroup of $\mathbf{\tilde{G}}$. It follows that
$\mathrm{N}_{\mathbf{\tilde{G}}}(Q)$ and
$\mathrm{N}_{\mathbf{\tilde{G}}}(\tilde{Q})$ are both contained in
$\mathrm{N}_{\mathbf{\tilde{G}}}(\mathbf{M})$. On the other hand,
$\mathrm{N}_{\mathbf{\tilde{G}}}(\mathbf{M})/\mathbf{M}\cong S_{(n+1/d)}$
given by permuting the components of
$\mathbf{M}=(\mathrm{GL}_{d})^{e}\times\dots\times(\mathrm{GL}_{d})^{e}$. By
the description of $Q_{0}$ is is clear that these automorphisms stabilize
$Q_{0}$. This shows that
$\mathrm{N}_{\mathbf{\tilde{G}}}(Q)=\mathrm{N}_{\mathbf{\tilde{G}}}(\tilde{Q})=\mathrm{N}_{\mathbf{\tilde{G}}}(\mathbf{M})$.
∎

######  Remark 11.3.

Suppose that $D$ is the Sylow $2$-subgroup of $\mathbf{G}$. Then the
conclusion of the previous lemma holds unless $\mathbf{G}$ is of type $A_{1}$
and $q\equiv\pm 3\,\mathrm{mod}\,8$, see [22, Theorem 1].

We denote $L_{0}:=[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}$ and we fix a
block $c_{0}$ of $L_{0}$ below $c$ such that $D_{0}:=D\cap L_{0}$ is a defect
group of $c_{0}$. Additionally, we set $Q_{0}:=Q\cap L_{0}$.

The induction step in the proof of Theorem 12.2 below requires the following
property of Cabanes subgroups.

######  Lemma 11.4.

Suppose that $\ell\neq 2$ and that $D/\mathrm{Z}(G)_{\ell}$ is abelian if
$\ell=2$. With the notation as introduced above we have
$\operatorname{C}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(Q_{0})=\operatorname{C}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(Q)$
and
$\operatorname{N}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(Q_{0})=\operatorname{N}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(Q)$.

###### Proof.

Let us first consider the special case, where $\ell=2$ and $D$ is non-abelian
but $D/\mathrm{Z}(G)_{\ell}$ is abelian. According to the proof of [16, Lemma
5.2] and [16, Proposition 3.4(a)] we observe that
$D\subset[\mathbf{L},\mathbf{L}]^{F}$. Since
$\mathbf{L}\subset\mathbf{L}^{\prime}$ by construction we have
$D\subset[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}$ and so $D_{0}=D$.
Hence, the statement holds trivially in this case. We can therefore assume
that the assumptions of Lemma 11.2 are satisfied.

We only prove the first part of the statement since the same arguments apply
when we replace the centralizer subgroups everywhere by their corresponding
normalizer subgroups. Consider the composition
$[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]\hookrightarrow\mathbf{L}^{\prime}\hookrightarrow\tilde{\mathbf{L}}^{\prime}$.
We let $\tilde{c}$ be a block of $\tilde{N}^{\prime}$ covering $c$. By [30,
Theorem 9.26] exists a defect group $\tilde{D}$ of $\tilde{c}$ such that
$\tilde{D}\cap N^{\prime}=\tilde{D}\cap L^{\prime}=D$. It follows that the
Cabanes subgroup $\tilde{Q}$ of $\tilde{D}$ satisfies $\tilde{Q}\cap
L^{\prime}=Q$. We have
$\operatorname{C}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(\tilde{Q})\subseteq\operatorname{C}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(Q)\subseteq\operatorname{C}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(Q_{0})$
and so it’s enough to show that
$\operatorname{C}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(\tilde{Q})=\operatorname{C}_{[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}}(Q_{0})$.

As in the proof of [36, Proposition 3.7] it follows that
$[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]=\mathbf{H}_{1}\times\dots\times\mathbf{H}_{r},$
where the $\mathbf{H}_{i}$ are simple algebraic groups of simply connected
type. The action of the Frobenius endomorphism $F$ induces a permutation $\pi$
on the set of simple components of
$[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]$. We let $\pi=\pi_{1}\cdots\pi_{t}$
be the decomposition of this permutation into disjoint cycles. For
$i=1,\dots,t$ choose $x_{i}\in\Pi_{i}$ in the support $\Pi_{i}$ of the
permutation $\pi_{i}$ and let $n_{i}=|\Pi_{i}|$ be the length of the cycle
$\pi_{i}$. We then have

$L_{0}:=[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]^{F}\cong\mathbf{H}^{F^{n_{1}}}_{x_{1}}\times\dots\times\mathbf{H}^{F^{n_{t}}}_{x_{t}}.$

Similarly, we can decompose $\tilde{\mathbf{L}}^{\prime}$ as
$\tilde{\mathbf{L}}^{\prime}=\tilde{\mathbf{H}}_{1}\times\dots\times\tilde{\mathbf{H}}_{r}$,
where
$\tilde{\mathbf{H}}_{i}\cap[\mathbf{L}^{\prime},\mathbf{L}^{\prime}]=\mathbf{H}_{i}$.
Therefore, $\tilde{\mathbf{L}}^{\prime
F}\cong\tilde{\mathbf{H}}^{F^{n_{1}}}_{x_{1}}\times\dots\times\tilde{\mathbf{H}}^{F^{n_{t}}}_{x_{t}}$.

Denote $\tilde{H}_{i}:=\tilde{\mathbf{H}}^{F^{n_{i}}}_{x_{i}}$ and
$H_{i}:=\mathbf{H}^{F^{n_{i}}}_{x_{i}}$. The block $c_{0}$ is strictly quasi-
isolated and decomposes as a direct product $c_{0}=c_{1}\otimes\dots\otimes
c_{t}$ of blocks which are strictly quasi-isolated in $H_{i}$, see the proof
of [36, Proposition 3.7]. It follows that $Q_{0}=Q_{1}\times\dots\times
Q_{t}$. Similarly, let $\tilde{c}^{\prime}$ be a block of $\tilde{L}^{\prime}$
below $\tilde{c}$ which covers $c_{0}$. We have a decomposition
$\tilde{c}^{\prime}=\tilde{c}_{1}\otimes\dots\otimes\tilde{c}_{t}$, where
$\tilde{c}_{i}$ is a block of $\tilde{H}_{i}$ covering $c_{i}$ and we obtain a
decomposition $\tilde{Q}=\tilde{Q}_{1}\times\dots\times\tilde{Q}_{t}$ with
$\tilde{Q}_{i}\cap H_{i}=Q_{i}$. It is therefore enough to show that
$\operatorname{C}_{H_{i}}(Q_{i})=\operatorname{C}_{H_{i}}(\tilde{Q}_{i})$.
This follows now from Lemma 11.2. ∎

######  Corollary 11.5.

Keep the assumptions of Lemma 11.2. If $D_{0}\leq\operatorname{Z}(L_{0})$ then
we have $D\leq\operatorname{Z}(L^{\prime})$.

###### Proof.

By Lemma 11.4 our assumption implies that
$Q\leq\operatorname{C}_{L^{\prime}}(L_{0})$. The proof of [3, Lemma 6.1] shows
that
$\operatorname{C}_{\mathbf{L}^{\prime}}(L_{0})=\operatorname{Z}(\mathbf{L}^{\prime})$
and so $Q\leq\operatorname{Z}(L^{\prime})$. This shows the claim for $\ell=2$.
For $\ell\geq 5$ observe that $Q$ is the maximal normal abelian subgroup of
$D$ and it follows from this that $Q=D$. ∎

##  12\. Reduction to isolated blocks

We first consider certain blocks of simple groups with non-exceptional Schur
multiplier.

######  Lemma 12.1.

Let $\ell\geq 5$ and $S$ be a simple group of Lie type $A$, $B$ or $C$ defined
over a field of characteristic $\neq\ell$ with exceptional Schur multiplier.
Then the Sylow $\ell$-subgroups of the universal covering group of $S$ are
cyclic. In particular, the inductive Alperin–McKay condition holds for all
$\ell$-blocks of the universal covering group of $S$.

###### Proof.

An examination of the groups $S$ with exceptional Schur multiplier shows that
$|S|_{\ell}\in\\{1,\ell\\}$ and $\ell\nmid|M(S)|$, where $M(S)$ is the Schur
multiplier of $S$. Therefore the Sylow $\ell$-subgroups of the universal
covering group of $S$ are cyclic. Therefore, the inductive Alperin–McKay
condition holds for all $\ell$-blocks of $S$ by the work of Koshitani–Späth,
see [23]. ∎

We can now prove our main theorem. The proof is essentially the same as the
one of [36, Theorem 3.12] using all the new ingredients proved up to here.
Recall the notation introduced in Remark 9.6.

######  Theorem 12.2.

Let $\ell\geq 5$ and assume that all isolated $\ell$-blocks of quasi-simple
groups of type $A$ defined over a field of characteristic $\neq\ell$ are AM-
good relative to the Cabanes subgroup of their defect group. Then all
$\ell$-blocks of quasi-simple groups of type $A$ are AM-good.

###### Proof.

Note that for $\ell\geq 5$ all blocks of simple groups of type $A$ with
exceptional Schur multiplier are AM-good (with respect to the prime $\ell$) by
Lemma 12.1. By [36, Theorem 3.12] it suffices to show that the strictly quasi-
isolated blocks of $\mathbf{G}^{F}$, where $\mathbf{G}$ is of type $A$, are
AM-good relative to the Cabanes subgroup of their defect group. We show this
statement by induction on the rank of $\mathbf{G}$.

Assume that $b$ is a strictly quasi-isolated block of $\mathbf{G}^{F}$ which
is not isolated. Recall that we have a decomposition
$L_{0}:=H_{1}\times\dots\times H_{t}$ where the finite groups $H_{i}$ are
either quasi-simple or solvable. In the former case our induction hypothesis
implies that the blocks $c_{i}$ are AM-good with respect to the Cabanes
subgroup of their defect group. By the proof of [36, Proposition 3.7] we
therefore obtain an iAM-bijection
$\varphi_{0}:\operatorname{Irr}_{0}(L_{0},c_{0})\to\operatorname{Irr}_{0}(\mathrm{N}_{L_{0}}(Q_{0}),(C_{0})_{Q_{0}})$.
Here, $Q_{0}$ is the Cabanes subgroup of the defect group $D_{0}$ of $c_{0}$
and $(C_{0})_{Q_{0}}=\operatorname{br}_{Q_{0}}(c_{0})$. We again use the
following notation as introduced in [36, Notation 3.9]:

######  Notation 12.3.

  * •

Assume that $D_{0}$ is central in $L_{0}$. Then we let $D$ be a defect group
of $c$ satisfying $D\cap L_{0}=D_{0}$. We define $L_{1}:=L_{0}D$ and we let
$e$ be the unique block of $L_{1}$ covering $c_{0}$. In addition, we set
$Q:=D$.

  * •

If $D_{0}$ is not central in $L_{0}$ then we set $L_{1}:=L_{0}$, $e:=c_{0}$
and we let $Q:=Q_{0}$.

Arguing as in the proof of [36, Lemma 3.10] we obtain a bijection

$\varphi_{0}:\operatorname{Irr}_{0}(L_{1},e)\to\operatorname{Irr}_{0}(\mathrm{N}_{L_{1}}(Q),E_{Q})$

which satisfies

$(\mathcal{N}_{\chi},L_{1},\chi)\geq_{b}(\mathrm{N}_{\mathcal{N}}(Q)_{\varphi_{0}(\chi)},\mathrm{N}_{L_{1}}(Q),\varphi_{0}(\chi)).$

for every character $\chi\in\operatorname{Irr}_{0}(L_{1},e)$. We show that
$\mathrm{N}_{L_{1}}(Q)=\mathrm{N}_{L_{1}}(\hat{Q})$, where $\hat{Q}$ is the
Cabanes subgroup of the defect group $D$. Assume first that we are in the case
that $D_{0}$ is central in $L_{0}$. It follows by Corollary 11.5 that $D$ is
abelian and thus $\hat{Q}=D$ is the Cabanes subgroup of $D$. If $D_{0}$ is not
central in $L_{0}$ then Lemma 11.4 shows that
$\operatorname{N}_{L_{1}}(\hat{Q})=\operatorname{N}_{L_{1}}(Q)$.

Recall the subgroup $T$ from Definition 10.3. We apply [36, Proposition 1.12]
to obtain an $\mathrm{N}_{\mathcal{N}}(\hat{Q},\hat{C}_{\hat{Q}})$-equivariant
bijection
$\varphi:\operatorname{Irr}_{0}(T,\hat{c})\to\operatorname{Irr}_{0}(\mathrm{N}_{T}(\hat{Q}),\hat{C}_{\hat{Q}})$
such that

$(\mathcal{N}_{\chi},T,\chi)\geq_{b}(\mathrm{N}_{\mathcal{N}}(\hat{Q})_{\varphi(\chi)},\mathrm{N}_{T}(\hat{Q}),\varphi(\chi))$

holds for every character $\chi\in\operatorname{Irr}_{0}(T,\hat{c})$.

The proof of [36, Lemma 3.11] now shows the existence of a bijection
$\tilde{\varphi}:\operatorname{Irr}(\tilde{T}\mid\operatorname{Irr}_{0}(\hat{c}))\to\operatorname{Irr}(\mathrm{N}_{T}(\hat{Q})\mid\operatorname{Irr}_{0}(\hat{C}_{\hat{Q}}))$
such that $\tilde{\varphi}$ together with the bijection
$\varphi:\operatorname{Irr}_{0}(T,\hat{c})\to\operatorname{Irr}_{0}(\mathrm{N}_{T}(\hat{Q}),\hat{C}_{\hat{Q}})$
satisfies assumptions (i)–(iii) of Theorem 9.2 and Theorem 10.4 respectively.

We can therefore apply Theorem 9.2 and obtain that the block $b$ is AM-good
with respect to the Cabanes subgroup $\hat{Q}$. ∎

######  Corollary 12.4.

The inductive Alperin–McKay condition holds for all $\ell$-blocks of quasi-
simple groups of type $A$, whenever $\ell\geq 5$ is a non defining prime.

###### Proof.

By Theorem 12.2 it is enough to show that the isolated (that means unipotent)
$\ell$-blocks of type $A$ are AM-good relative to the Cabanes subgroup of
their defect group. Let $d$ denote the order of $q$ modulo $\ell$.

Suppose first that $\ell\nmid(q-\varepsilon)$. Consider a unipotent block $b$
of $\mathbf{G}^{F}$. We fix a block $\tilde{b}$ of $\tilde{\mathbf{G}}^{F}$
covering $b$. We observe that $\tilde{b}$ has the same defect group as $b$.
There exists a $d$-cuspidal pair $(\mathbf{K},\zeta)$ of
$(\tilde{\mathbf{G}},F)$ associated to $\tilde{b}$. By the proof of [11,
Theorem 22.9] it follows that $Q:=\operatorname{Z}(\mathbf{K})^{F}_{\ell}$ is
the Cabanes subgroup of a defect group $D$ of $b$ and we have
$\mathbf{K}=\operatorname{C}_{\mathbf{\tilde{G}}}(Q)$. Since $Q$ is
characteristic in $\mathbf{K}$ it follows that
$\operatorname{N}_{\mathbf{\tilde{G}}}(\mathbf{K})=\operatorname{N}_{\mathbf{\tilde{G}}}(Q)$.
Note that unipotent blocks satisfy the requirements of [8, Corollary 6.1].
Therefore, [8, Corollary 6.1] shows that the iAM-condition holds for the
unipotent block $b$ relative to
$\operatorname{N}_{G}(\mathbf{K})=\operatorname{N}_{G}(Q)$.

Assume now that $\ell\mid(q-\varepsilon)$. It is well known, see [11, Example
22.10] and [11, Remark 22.11] that in this case $\mathbf{\tilde{G}}^{F}$ and
therefore also $\mathbf{G}^{F}$ has only one unipotent $\ell$-block. This is
the principal block of $\mathbf{G}^{F}$ and therefore has maximal defect
group. In particular, this block is AM-good relative to
$\operatorname{N}_{G}(\mathbf{S})$, where $\mathbf{S}$ is a Sylow
$\Phi_{d}$-torus of $\mathbf{G}$, by the main theorem of [12]. It is thus
sufficient to show that
$\operatorname{N}_{\mathbf{G}}(Q)=\operatorname{N}_{\mathbf{G}}(\mathbf{S})$.
Since $\ell\mid(q-\varepsilon)$ we know that $\mathbf{S}$ is a maximally split
torus of $\mathbf{G}$. Again by the proof of [11, Theorem 22.9] it follows
that $Q=\mathbf{S}_{\ell}^{F}$ is the Cabanes subgroup of a defect group of
$b$ and
$\operatorname{C}_{\mathbf{G}}(Q)=\mathbf{S}=\operatorname{C}_{\mathbf{G}}(\mathbf{S})$.
This implies
$\operatorname{N}_{\mathbf{G}}(Q)=\operatorname{N}_{\mathbf{G}}(\mathbf{S})$,
which proves the claim. ∎

######  Theorem 12.5.

Let $X$ be one of the symbols $B$ or $C$ and $\ell\geq 5$. Assume that all
isolated $\ell$-blocks of quasi-simple group of type $X$ and are AM-good
relative to the Cabanes subgroup of the defect group; Then all $\ell$-blocks
of quasi-simple groups of type $X$ are AM-good.

###### Proof.

Assume first that $n=2$ and $q$ is even. Then $\operatorname{Z}(\mathbf{G})$
is trival and so the isolated $\ell$-blocks of $\mathbf{G}^{F}$ are precisely
the quasi-isolated $\ell$-blocks. The statement is then a consequence of the
main theorem of [36]. Also we observe that for $\ell\geq 5$ all blocks of
simple groups of type $X$ with non-exceptional Schur multiplier are AM-good by
Lemma 12.1. We can therefore assume that we are in none of these exceptional
cases.

Fix an $\ell$-block $b$ of $\Lambda\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$ (not
necessarily quasi-isolated), where $\mathbf{G}$ is of type $X$ and
$s\in(\mathbf{G}^{\ast})^{F^{\ast}}$ is semisimple of $\ell^{\prime}$-order.

In contrast to the proof of Theorem 12.2 we don’t need to argue by induction.
We let $\mathbf{L}^{\ast}$ be the minimal Levi subgroup of $\mathbf{G}^{\ast}$
containing $\mathrm{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ and let $\mathbf{L}$ be
the Levi subgroup dual to $\mathbf{L}^{\ast}$. We let $\mathcal{A}:=\langle
F_{0}\rangle$ as in Theorem 4.2. Let $c$ be the block of
$\Lambda\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ corresponding to $b$ under the
Bonnafé–Dat–Rouquier equivalence from Theorem 4.2 and let $c_{0}$ be a block
of $L_{0}:=[\mathbf{L},\mathbf{L}]^{F}$ below $c$. We conclude that the block
$c_{0}$ is then an isolated block of $L_{0}$. As in the proof of Theorem 12.2
we obtain a decomposition $L_{0}=H_{1}\times\dots\times H_{t}$ into groups
which are either quasi-simple of type $A$ or $X$ or solvable and a
corresponding decomposition $c_{0}=c_{1}\otimes\dots\otimes c_{t}$ of $c_{0}$
into block $c_{i}$ which are isolated in $H_{i}$. Our assumption together with
the proof of Corollary 12.4 implies that these blocks are AM-good relative to
the Cabanes subgroup of their defect group. Following the proof of Theorem
12.2 and using Remark 9.5 instead of Theorem 9.2 we deducee that the block $b$
is AM-good relative to the $\ell$-subgroup $Q$, where $Q$ is defined as in
Notation 12.3. ∎

##  13\. A variant of Späth’s reduction theorem

In her paper [39] Späth shows that the Alperin–McKay conjecture holds for
every finite group if and only if the inductive Alperin–McKay condition holds
for every simple group. Our aim here is to modify her proof in order to get a
similar statement involving preferably only blocks with abelian defect.
Unfortunately in her proof, it is necessary to consider central extensions of
groups and a block might have abelian defect group but a block of a central
extension dominating it might not. Therefore, we need to consider blocks whose
defect group lies in a slightly larger class of groups.

For the following definition recall that if $G$ is a finite group its upper
central series is defined recursively as $Z_{0}(G):=1$ and
$\operatorname{Z}_{i}(G)$ is the unique subgroup of $G$ containing
$\operatorname{Z}_{i-1}(G)$ such that
$\operatorname{Z}_{i}(G)/\operatorname{Z}_{i-1}(G)=\operatorname{Z}(G/\operatorname{Z}_{i-1}(G))$.

######  Definition 13.1.

We say that a subgroup $D$ of a finite group $G$ is almost abelian in $G$ if
there exists an $i$ such that
$D\operatorname{Z}_{i}(G)/\operatorname{Z}_{i}(G)$ is abelian.

Observe that the property of $D$ being almost abelian can depend on the
ambient group $G$. Moreover, if one considers the hypercenter
$\operatorname{Z}_{\infty}(G)$ of $G$ (i.e. the union of all
$\operatorname{Z}_{i}(G)$ for $i\geq 0$) then $D$ is almost abelian in $G$ if
and only $D\operatorname{Z}_{\infty}(G)/\operatorname{Z}_{\infty}(G)$ is
abelian.

######  Lemma 13.2.

Suppose that $D$ is almost abelian in $G$.

  1. (a)

Let $H$ be a subgroup of $G$ and $E\leq H\cap D$. Then $E$ is almost abelian
in $H$.

  2. (b)

For $j=1,2$ the groups $D_{j}$ are almost abelian in $G_{j}$ if and only if
$D_{1}\times D_{2}$ is almost abelian in $G_{1}\times G_{2}$.

  3. (c)

A subgroup $E$ of $G$ is almost abelian in $G$ if and only if
$E\operatorname{Z}(G)/\operatorname{Z}(G)$ is almost abelian in
$G/\operatorname{Z}(G)$.

###### Proof.

Let us first prove part (a). By assumption, there is an $i$ such that
$D\operatorname{Z}_{i}(G)/\operatorname{Z}_{i}(G)$ is abelian. By induction
one easily shows that $\operatorname{Z}_{i}(G)\cap
H\lhd\operatorname{Z}_{i}(H)$. From this we deduce that $\frac{(D\cap
H)\operatorname{Z}_{i}(H)}{\operatorname{Z}_{i}(H)}$ is abelian. Therefore,
$D\cap H$ is almost abelian in $H$. Consequently, $E\leq D\cap H$ is almost
abelian as well.

For part (b) we observe that $\operatorname{Z}_{i}(G_{1}\times
G_{2})=\operatorname{Z}_{i}(G_{1})\times\operatorname{Z}_{i}(G_{2})$. Thus,

$\frac{(D_{1}\times D_{2})\operatorname{Z}_{i}(G_{1}\times
G_{2})}{\operatorname{Z}_{i}(G_{1}\times
G_{2})}\cong\frac{D_{1}\operatorname{Z}_{i}(G)}{\operatorname{Z}_{i}(G)}\times\frac{D_{2}\operatorname{Z}_{i}(G)}{\operatorname{Z}_{i}(G)}$

and the claim follows from this.

For part (c) one first shows by induction that
$\operatorname{Z}_{i-1}(G/\operatorname{Z}(G))=\operatorname{Z}_{i}(G)/Z(G)$
for all $i$. Hence, $D\operatorname{Z}(G)/\operatorname{Z}(G)$ is almost
abelian in $G/\operatorname{Z}(G)$ if and only if there exists an $i$ such
that

$\frac{\operatorname{Z}_{i}(G/\operatorname{Z}(G))D\operatorname{Z}(G)/\operatorname{Z}(G)}{\operatorname{Z}_{i}(G/\operatorname{Z}(G))}\cong\frac{\operatorname{Z}_{i+1}(G)D}{\operatorname{Z}_{i+1}(G)}$

is abelian. From this is follows that
$D\operatorname{Z}(G)/\operatorname{Z}(G)$ is almost abelian in
$G/\operatorname{Z}(G)$ if and only if $D$ is almost abelian in $G$. ∎

The aim of this section is to prove the following variant of [39, Theorem C].
We closely follow the proof of [12, Proposition 2.5].

######  Proposition 13.3.

Let $X$ be a finite group and $\ell$ a prime. Assume that for every non-
abelian simple subquotient $S$ of $X$ with $\ell\mid|S|$ the following holds:
Every $\ell$-block of the universal covering group $H$ of $S$ with almost
abelian defect group satisfies the iAM-condition. Then the Alperin–McKay
conjecture holds for any $\ell$-block of $X$ with almost abelian defect.

For a finite group $X$ let $F^{\ast}(X)$ be its generalized Fitting subgroup.

######  Proposition 13.4.

Let $X$ be a finite group and $b$ an $\ell$-block of $X$ with almost abelian
defect. Suppose that the Alperin–McKay conjecture is true for any $\ell$-block
with almost abelian defect group of any group $H$ with
$|H:\operatorname{Z}(H)|<|X:\operatorname{Z}(X)|$ and such that $H$ is
isomorphic to a subquotient of $X/\operatorname{Z}(X)$. Then one of the
following holds:

  1. (i)

The Alperin–McKay conjecture holds for $b$.

  2. (ii)

For any non-central normal subgroup $K$ of $X$ we have
$X=K\operatorname{N}_{X}(D)F^{\ast}(X)$.

###### Proof.

Let $K\lhd X$ be a non-central normal subgroup of $X$. Replacing $K$ by
$K\operatorname{Z}(X)$ if necessary we can assume that
$\operatorname{Z}(X)\leq K$. By assumption the Alperin–McKay conjecture is
true for every block with almost abelian defect of any central extension of
$X/K$. An analysis of the proof of [28, Theorem 6] then shows that the proof
of said theorem can be adapted and we obtain
$|\operatorname{Irr}_{0}(B)|=|\operatorname{Irr}_{0}(b)|$ where
$B\in\mathrm{Bl}(K\operatorname{N}_{X}(D))$ is the unique block with
$B^{X}=b$. If $K\operatorname{N}_{X}(D)$ is a proper subgroup of $X$ then our
assumption implies that the Alperin–McKay conjecture holds for the block $B$
of $K\operatorname{N}_{X}(D)$. Therefore,
$|\operatorname{Irr}_{0}(B)|=|\operatorname{Irr}_{0}(B_{D})|$, where $B_{D}$
is the Brauer correspondent of $b$. This would then imply that the
Alperin–McKay conjecture holds for $b$. Since the generalized Fitting subgroup
$F^{\ast}(X)$ is such a non-central normal subgroup of $X$ this argument shows
in particular that $X=F^{\ast}(X)\operatorname{N}_{X}(D)$. ∎

Proof of Proposition 13.3: The proof of the statement is by induction on
$|X:\operatorname{Z}(X)|$. We let $b$ be an $\ell$-block of $X$ with almost
abelian defect group $D$. We let $B_{D}$ be its Brauer correspondent in
$\operatorname{N}_{X}(D)$.

According to Proposition 13.4 we may assume that $X$ and $b$ satisfy the
statement in Proposition 13.4(ii). Consequently, any normal $\ell$-subgroup of
$X$ is central. As in [39, Section 6 and 7] we distinguish two cases.

Assume that there exists a normal non-central subgroup $K\lhd X$ with $K\leq
F^{\ast}(X)$. If $K\cap D\leq\operatorname{Z}(X)$ then
$|\operatorname{Irr}_{0}(b)|=|\operatorname{Irr}_{0}(B_{D})|$, according to
[39, Proposition 6.6].

Assume otherwise that the Fitting subgroup is central in $X$, and $E(X)$, the
group of components of $X$, is non-central. Inside $E(X)$ we can take a normal
subgroup $K\lhd X$ such that $K=[K,K]$ and $K/\operatorname{Z}(K)\cong S^{r}$
for a non-abelian simple group $S$ and an integer $r\geq 1$. The proof of [28,
Proposition 9(ii)] shows that the block $b$ covers a unique block $b_{0}$ of
$K$. Note that $D_{0}:=D\cap K$ is a defect group of $b_{0}$ which is almost
abelian by Lemma 13.2(a). Let $\tilde{G}=G^{r}$ be the univeral covering group
of $K/\operatorname{Z}(K)\cong S^{r}$, where $G$ is the universal covering
group of $S$. Let $\pi:\tilde{G}\to S^{r}$ be the associated quotient map. Let
$\tilde{b}_{0}$ be a block of $\tilde{G}$ dominating $b_{0}$. Again Lemma 13.2
ensures that $\tilde{b}_{0}$ has an almost abelian defect group $\tilde{D}$.
Following the proof of [39, Theorem 7.9] we obtain a bijection
$\tilde{\Omega}:\operatorname{Irr}(\tilde{G},\tilde{b}_{0})\to\operatorname{Irr}(\tilde{M},\tilde{B}_{0})$.
Here, $\tilde{M}$ is a suitably defined subgroup of $\tilde{G}$ containing
$\operatorname{N}_{\tilde{G}}(\tilde{D})$ and
$\tilde{B}_{0}^{\tilde{G}}=\tilde{b}_{0}$. From this the proof of [39, Theorem
7.9] then yields a bijection
$\hat{\Omega}:\operatorname{Irr}(K,b_{0})\to\operatorname{Irr}(M\operatorname{N}_{K}(D_{0}),B_{0})$
having the properties of the bijection in the statement of [39, Theorem 7.9].
Here, $M:=\pi(\tilde{M})$ and $B_{0}$ is the unique block of
$M\operatorname{N}_{K}(D_{0})$ with $B_{0}^{K}=b_{0}$. Using the counting
argument in the proof of [39, Theorem C] we are then able to deduce that
$|\operatorname{Irr}_{0}(X,b)|=|\operatorname{Irr}_{0}(M\operatorname{N}_{X}(D_{0}),B)|$.

Since
$|M\operatorname{N}_{X}(D_{0})/\operatorname{Z}(X)|<|X/\operatorname{Z}(X)|$
we can apply the induction hypothesis and we conclude that
$|\operatorname{Irr}_{0}(M\operatorname{N}_{X}(D_{0}),B)|=|\operatorname{Irr}_{0}(\operatorname{N}_{X}(D),B_{D})|$,
where $B_{D}$ is the Brauer correspondent of $b$. ∎

##  14\. On the Alperin–McKay conjecture for $2$-blocks with abelian defect
group

###  14.1. Classification of $2$-blocks with abelian defect group

We start by recalling the following classification of quasi-isolated
$2$-blocks of finite groups of Lie type, see [16, Lemma 5.2]:

######  Lemma 14.1.

Assume that $p$ is odd and that $\mathbf{G}$ is simple and simply-connected.
Let $b$ be a quasi-isolated $2$-block of $G$ with semi-simple label
$s\in\mathbf{G}^{\ast}$.

  1. (a)

Suppose that $b$ has abelian defect groups. Then one of the following holds.

    1. (i)

$\mathbf{G}$ is of type $A_{n}$, $n$ is even, and
$C^{\circ}_{\mathbf{G}^{\ast}}(s)$ is a torus.

    2. (ii)

$\mathbf{G}$ is of type $G_{2}$, $F_{4}$, $E_{6}$ or $E_{8}$, $s=1$ and $b$ is
of defect $0$.

  2. (b)

Suppose that $b$ has non-abelian defect groups, but for some central
$2$-subgroup $Z$ of $\mathbf{G}^{F}$, the image $\overline{b}$ in $G/Z$ has
abelian defect group. Then $Z$ is cyclic of order $2$ and one of the following
holds.

    1. (i)

$\mathbf{G}$ is of type $A_{n}$, $n\equiv 1\,\mathrm{mod}\,4$ and the defect
groups of $\overline{b}$ are $C_{2}\times C_{2}$.

    2. (ii)

$\mathbf{G}$ is of type $E_{7}$ and the defect groups of $\overline{b}$ are
$C_{2}\times C_{2}$.

In the following remark we collect some additional information from the proof
of [16, Lemma 5.2].

######  Remark 14.2.

  1. (a)

If we assume additionally that the blocks in part (b)(i) are isolated (i.e.
unipotent) then the proof of [16, Lemma 5.2] shows that $\mathbf{G}^{F}\cong
A_{1}(q)$ and $q\equiv\pm 3\,\mathrm{mod}\,8$. Moreover, their defect groups
are isomorphic to the quaternion group $Q_{8}$.

  2. (b)

The blocks in (b)(ii) occur only if $4\mid\mid(q-1)$. These blocks are
unipotent and their defect groups are isomorphic to the dihedral group
$D_{8}$. They correspond to lines 3 and 7 of the table on page 354 of [17].

Our aim is now to show the iAM-condition for all isolated blocks of positive
defect occuring in the classification of Lemma 14.1.

###  14.2. On a certain $2$-block of $E_{7}(q)$

In this subsection we consider the block $b$ of $E_{7}(q)$ occuring in part
(b)(ii) of the classification of Lemma 14.1. The author is very grateful to
Gunter Malle for pointing out the proof of the following proposition to him.

######  Proposition 14.3.

Let $b$ be one of the blocks of $G=E_{7}(q)$ occuring in Lemma 14.1(b)(ii).
Then we have $|\operatorname{Irr}_{0}(b)|=4$ and $|\operatorname{Irr}(b)|=5$.
Furthermore, the height zero characters of $b$ have $\mathrm{Z}(G)$ in their
kernel and are as follows:

  1. (i)

Two unipotent characters $\chi_{1},\chi_{2}$.

  2. (ii)

Two non-unipotent characters $\chi_{3},\chi_{4}$ which are conjugate under the
diagonal automorphism of $G$.

###### Proof.

Let $q$ be a prime power such that $4||(q-1)$. Set $G=E_{7}(q)_{sc}$ and
$S=G/Z(G)$, the simple group of type $E_{7}(q)$. We consider the unipotent
$2$-block $b$ of $G$ parametrized by the 1-cuspidal pair
$(E_{6},E_{6}[\theta])$ (and its Galois conjugate block
$(E_{6},E_{6}[\theta^{2}])$ for which all arguments apply similarly).

Firstly, we note that according to the description in [17, 3.2] the defect
group of $b$ is isomorphic to the dihedral group $D_{8}$. By [37, Theorem 8.1]
we can therefore deduce that $|\operatorname{Irr}_{0}(b)|=4$ and
$|\operatorname{Irr}(b)|=5$. Moreover, the unique block $\overline{b}$ of $S$
dominated by $b$ has defect group $C_{2}\times C_{2}$. Hence, by [37, Theorem
8.1] we know that
$4=|\operatorname{Irr}(\overline{b})|=|\operatorname{Irr}_{0}(\overline{b})|$
and thus the unique character in $\operatorname{Irr}(b)$ with positive height
is non-trivial on $\mathrm{Z}(G)$ and it is unique with this property among
the irreducible characters of $b$. We will now describe the character of
$\operatorname{Irr}(b)$ in more detail.

According to [17, Theorem B], the ordinary characters in $b$ are described as
follows: let $t\in G^{*}=E_{7}(q)_{\operatorname{ad}}$ be a (semisimple)
2-element such that $C^{\circ}_{\mathbf{G}^{*}}(t)$ has a Levi subgroup of
type $E_{6}$. Then $b$ contains those elements of $\mathcal{E}(G,t)$ which
under Jordan decomposition correspond to characters in the 1-Harish-Chandra
series of $\operatorname{C}^{\circ}_{\mathbf{G}^{*}}(t)$ above
$(E_{6},E_{6}[\theta])$, and moreover, these are the only characters in $b$.

In our particular case, the only centralizer in $\mathbf{G}^{*}$ that can
possibly contain a Levi subgroup of type $E_{6}$ is, apart from
$\mathbf{G}^{*}$ itself, also of type $E_{6}$. Now the centralizer of
$E_{6}(q)$ in $E_{7}(q)$ is $K:=E_{6}(q)(q-1)$, and so its center has order
$(q-1)$. Given that $4||(q-1)$, there are hence only four 2-elements $t$ in
$Z(K)$, the identity, an involution and two elements of order 4. Clearly the
centralizer of the identity is all of $G^{*}$. Now $N(K)=K.2$ (by a
calculation in the Weyl group), so for the involution
$s\in\operatorname{Z}(K)$ we have $\operatorname{C}_{G^{*}}(s)=K.2$ (that is,
the centralizer of $s$ in $\mathbf{G}^{\ast}$ is disconnected). Thus there are
two characters, say $\chi_{3}$ and $\chi_{4}$, of $b$ in that geometric
Lusztig series $\tilde{\mathcal{E}}(G,s)$ fused by the diagonal automorphism
of $G$. On the other hand, $N(K)$ acts non-trivially on $Z(K)$ (again by a
calculation in the Weyl group) so the two elements $t$ of order 4 in $Z(K)$
are $G^{*}$-conjugate and moreover $C_{G^{*}}(t)=K$. So here
$\mathcal{E}(G,t)$ contains one character $\chi_{5}$ in $b$, which is left
invariant under the diagonal automorphism.

From these calculations it necessarily follows from
$|\operatorname{Irr}(b)|=5$ that $\mathcal{E}(G,1)$ contains two characters
$\chi_{1}$ and $\chi_{2}$ of $b$ (in fact these are the irreducible characters
lying in the Harish-Chandra series of $E_{6}(\theta)$). These characters have
$\mathrm{Z}(G)$ in their kernel. The characters $\chi_{3}$ and $\chi_{4}$ are
$\tilde{G}$-conjugate and thus have the same underlying character of the
center. From this we deduce that $\chi_{5}$ has to be the unique character of
$b$ which is nontrival on $\mathrm{Z}(G)$. From this our arguments above show
that $\operatorname{Irr}_{0}(G,b)=\\{\chi_{1},\chi_{2},\chi_{3},\chi_{4}\\}$.

An entirely similar argument applies when $4||(q+1)$; here all groups
$E_{6}(q)$ have to be replaced by ${}^{2}E_{6}(q)$ and all $(q-1)$ by $(q+1)$.
∎

To prove the iAM-condition for the block $b$ we also need to compute the
action of group automorphisms on the height zero characters of its Brauer
correspondent. The information given in [17] does not seem sufficient for
this. Instead we will try to obtain all the necessary local information from
the invariants of the block $b$.

The following proposition is a consequence of [37, Proposition 10.26].

######  Proposition 14.4.

Let $H$ be a group isomorphic $C_{2}\times C_{2}$ or $D_{8}$ and for $n\geq 1$
consider a group extension

$1\xrightarrow{}H\xrightarrow{}D\xrightarrow{}C_{2^{n}}\xrightarrow{}1$

Then the invariants for every block of a finite group with defect group $D$
are known.

###### Proof.

We must show that the exceptions listed in [37, Proposition 10.26] do not
occur if $H$ is as in the statement of our proposition. The first assumption
on the coupling $\omega$ in [37, Proposition 10.26] can be easily verified by
examining the automorphism group structure of $C_{2}\times C_{2}$ and $D_{8}$
respectively. We are therefore left to show that $D$ is not isomorphic to
$C_{2^{m}}\wr C_{2}$ for all $m\geq 3$. For $H\cong C_{2}\times C_{2}$ this
follows from the proof of [37, Proposition 10.26].

Assume therefore that $H\cong D_{8}$ and $H\lhd(C_{2^{m}}\times
C_{2^{m}})\rtimes C_{2}$. There exists an element $a\in H$ which is not
contained in the base group $C_{2^{m}}\times C_{2^{m}}$. This element must act
on the base group by interchanging both components. Let $g\in C_{2^{m}}$ be a
generator. Then it follows that $a^{(g,1)}=(g,g^{-1})a$ and so $(g,g^{-1})\in
D_{8}$. This implies that $m\leq 2$. ∎

######  Lemma 14.5.

Let $b$ be a block of a finite group $G$ with defect group $D$. Suppose that
$B$ is a block of a subgroup $M$ of $G$ containing $\operatorname{N}_{G}(D)$
with $B^{G}=b$. Let $A\subseteq\mathrm{Aut}(G)$ be a finite cyclic subgroup
stabilizing $M$. Assume that the Alperin–McKay conjecture holds for every
block of $G\rtimes A$ and $M\rtimes A$ covering $b$ and $B$ respectively.

  1. (a)

If $A$ is a simple cyclic group or an $\ell$-group then the number of
$A$-invariant characters in $\operatorname{Irr}_{0}(b)$ is equal to the number
of $A$-invariant characters of $\operatorname{Irr}_{0}(B)$.

  2. (b)

We have $\operatorname{Irr}_{0}(b)=\operatorname{Irr}_{0}(b)^{A}$ if and only
if $\operatorname{Irr}_{0}(B)=\operatorname{Irr}_{0}(B)^{A}$.

###### Proof.

This follows from the proof of [39, Lemma 8.1]. ∎

######  Proposition 14.6.

Let $b$ be one of the blocks of $G=E_{7}(q)$ occuring in Lemma 14.1(b)(ii).
Then $b$ is AM-good relative to its defect group.

###### Proof.

Let $\mathbf{G}_{\mathrm{ad}}$ be the adjoint quotient of $\mathbf{G}$. There
exist a Frobenius endomorphism $F$ on $\mathbf{G}_{\mathrm{ad}}$ which
commutes with the quotient map $\pi:\mathbf{G}\to\mathbf{G}_{\mathrm{ad}}$.
Then $\pi$ induces an injective map $\pi:S\to G_{\mathrm{ad}}$, where
$G_{\mathrm{ad}}:=\mathbf{G}_{\mathrm{ad}}^{F}$ and $G_{\mathrm{ad}}$ induces
all diagonal automorphisms on $S$. More precisely, we have
$\tilde{G}/\mathrm{Z}(\tilde{G})\cong G_{\mathrm{ad}}$.

Let $D$ be a defect group of $b$ and $B$ be its Brauer correspondent in
$\operatorname{N}_{G}(D)$. We first construct an
$\mathrm{Aut}(G)_{b,D}$-equivariant bijection
$\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(D),B)$.
Let $\delta:G\to G$ be a diagonal automorphism induced by the action of an
element of $G_{\operatorname{ad}}$ and let $F_{0}:G\to G$ be a generator of
the group of field automorphisms of $G$ which together generate
$\mathrm{Out}(G)$. We first observe that the characters $\chi_{1},\chi_{2}$
are $\langle\delta,F_{0}\rangle$-stable by [24, Theorem 2.5]. Moreover,
Proposition 14.3 together with [36, Theorem 2.11] implies that
$\langle\delta,F_{0}\rangle_{\chi_{i}}=\langle F_{0}\rangle$ for $i=3,4$. In
particular, the block $b$ is $\langle\delta,F_{0}\rangle$-stable and thus we
can assume (by possibly replacing these automorphisms by a $G$-conjugate) that
$D$ is $\langle\delta,F_{0}\rangle$-stable.

We denote by $c$ either the block $b$ or $B$. Moreover, we let $\overline{c}$
be the unique block of $S:=G/Z(G)$ respectively
$\operatorname{N}_{G}(D)/\operatorname{Z}(G)$ which is dominated by $c$. As in
the proof of Proposition 14.3 we find that $|\operatorname{Irr}_{0}(c)|=4$ and
$|\operatorname{Irr}(\overline{c})|=|\operatorname{Irr}_{0}(\overline{c})|=4$.
Therefore, the quotient map induces a bijection
$\operatorname{Irr}_{0}(c)\to\operatorname{Irr}(\overline{c})$. Note that the
defect group $\overline{D}:=D/\operatorname{Z}(G)$ of the block $c$ has a
defect group isomorphic to $C_{2}\times C_{2}$. According to Proposition 14.4
we can therefore apply Lemma 14.5 to the automorphisms $\delta$ and $F_{0}$.
Using Proposition 14.3 we find that every character of
$\operatorname{Irr}_{0}(B)$ is $F_{0}$-stable. Moreover there exist two
characters $\psi_{1},\psi_{2}\in\operatorname{Irr}_{0}(B)$ which are
$\delta$-stable and the other two characters
$\psi_{3},\psi_{4}\in\operatorname{Irr}_{0}(B)$ are $\delta$-conjugate. We
therefore obtain that the bijection
$\operatorname{Irr}_{0}(b)\to\operatorname{Irr}(B),\chi_{i}\mapsto\psi_{i},$
is $\langle\delta,F_{0}\rangle$-equivariant.

We now show that the characters of $\operatorname{Irr}_{0}(\overline{b})$ and
$\operatorname{Irr}_{0}(\overline{B})$ extend to their inertia groups in
$G_{\operatorname{ad}}\langle F_{0}\rangle$ and
$\operatorname{N}_{G_{\operatorname{ad}}\langle F_{0}\rangle}(\overline{D})$
respectively. Since $\chi_{1},\chi_{2}$ are unipotent characters it follows
that they extend to $G_{\operatorname{ad}}\langle F_{0}\rangle$, see for
example [24, Theorem 2.4]. For $i=3,4$ the stabilizer quotient
$(G_{\operatorname{ad}}\langle F_{0}\rangle)_{\chi_{i}}/S$ of $\chi_{i}$ is
cyclic and so $\chi_{i}$ extends to its inertia group in
$G_{\operatorname{ad}}\langle F_{0}\rangle$. Similarly, for $i=3,4$ the local
character $\psi_{i}$ extends to its inertia group in
$\operatorname{N}_{G_{\operatorname{ad}}\langle F_{0}\rangle}(\overline{D})$
as well. It is therefore left to show that $\psi_{1}$ and $\psi_{2}$ extend to
$G_{\operatorname{ad}}\langle F_{0}\rangle$ as well. We let $b_{2}$ be a block
of $G_{\operatorname{ad}}\langle F_{0}\rangle$ covering $\overline{b}$. Let
$D_{2}$ be a defect group of $b_{2}$ such that $D_{2}\cap S=\overline{D}$. We
observe that $D_{1}:=D_{2}\cap G_{\operatorname{ad}}$ is a defect group of the
unique block $b_{1}$ of $G_{\operatorname{ad}}$ covering $\overline{b}$. Since
Brauer’s height zero conjecture holds for blocks with defect group of order
$8$ (see e.g. [37, Theorem 13.1] and [37, Theorem 8.1])it follows that $D_{1}$
is nonabelian. Since $|\operatorname{Irr}(b_{1})|=5$ we must have $D_{1}\cong
D_{8}$, see for instance [37, Theorem 8.1]. From this it follows that $D_{2}$
is a cyclic extension of $D_{8}$. Let $B_{1}$ be the Harris–Knörr
correspondent of $b_{1}$ in
$\operatorname{N}_{G_{\operatorname{ad}}}(\overline{D})$. Again, Proposition
14.4 ensures that Lemma 14.5 is applicable to the automorphism $F_{0}$. Since
every character of $\operatorname{Irr}_{0}(b_{1})$ is $F_{0}$-stable the same
is true for every character of $\operatorname{Irr}_{0}(B_{1})$. Since
$\operatorname{Irr}_{0}(B_{1})$ consists of the four extensions of $\psi_{1}$
and $\psi_{2}$ to $G_{\operatorname{ad}}$ it follows that $\psi_{1}$ and
$\psi_{2}$ must necessarily extend to $G_{\operatorname{ad}}\langle
F_{0}\rangle$.

It is now easy to show that the above information is enough to verifiy the
inductive conditions. For this we observe that we can choose extensions
$\hat{\chi}_{i}\in\operatorname{Irr}((G_{\operatorname{ad}}\langle
F_{0}\rangle)_{\chi_{i}})$ of $\chi_{i}$ and extensions
$\hat{\psi}_{i}\in\operatorname{Irr}((\operatorname{N}_{G_{\operatorname{ad}}\langle
F_{0}\rangle}(\overline{D}))_{\psi_{i}}))$ of $\psi_{i}$ lying in Harris–Knörr
corresponding blocks. It follows from [40, Lemma 2.15] and [40, Proposition
4.4] that the inductive Alperin–McKay condition (in the version of [40,
Definition 4.2]) is satisfied for the block $b$ of $G$. ∎

###  14.3. The inductive Alperin–McKay condition for
$\mathrm{SL}_{2}(\mathbb{F}_{q})$

######  Proposition 14.7.

The iAM-condition holds for the principal $2$-block of
$\mathbf{G}^{F}=\mathrm{SL}_{2}(\mathbb{F}_{q})$ relative to its defect group.

###### Proof.

By [6] we can assume that $\mathbf{G}^{F}$ has non-exceptional Schur
multiplier. We follow the proof (and notation) of Proposition 14.6. Using [11,
Theorem 21.14] it’s easy to see that the principal block $b$ is the unique
$2$-block of maximal defect. Hence,
$\operatorname{Irr}_{0}(b)=\operatorname{Irr}_{p^{\prime}}(G)$ and
$\operatorname{Irr}_{0}(B)=\operatorname{Irr}_{p^{\prime}}(\operatorname{N}_{G}(D))$,
where $B$ is the Brauer correspondent of $b$.

We place ourselves in the situation of [18, Section 15], where it was shown
that $S=G/\mathrm{Z}(G)$ is McKay-good. More precisely, it was shown that
there exists an intermediate subgroup $\operatorname{N}_{G}(D)\subset H$ and
an $\mathrm{Aut}(G)_{H}$-equivariant bijection
$\operatorname{Irr}_{p^{\prime}}(G)\to\operatorname{Irr}_{p^{\prime}}(H)$.
Modifying their bijection we obtain an $\mathrm{Aut}(G)_{D}$-equivariant
bijection
$\operatorname{Irr}_{0}(G,b)\to\operatorname{Irr}_{0}(\operatorname{N}_{G}(D),B)$.
This bijection preserves central characters since every $2^{\prime}$-character
of $G$ respectively $\operatorname{N}_{G}(D)$ has the $2$-group
$\operatorname{Z}(G)$ in its kernel. One then checks that all characters of
$\operatorname{Irr}_{0}(S,\overline{b})$ extend to their inertia group in
$G_{\mathrm{ad}}\langle F_{0}\rangle$ (the stabilizer is either cyclic or the
characters are unipotent). Similarly, one also checks that the characters of
$\operatorname{Irr}_{0}(\operatorname{N}_{S}(\overline{D}),\overline{B})$
extend to their inertia group: If $D$ is self-normalizing then every character
in $\operatorname{Irr}_{0}(\operatorname{N}_{S}(\overline{D}),\overline{B})$
is linear and one can check the claim by explicit computations. Otherwise,
both characters of $\operatorname{Irr}_{0}(\mathrm{N}_{G}(D),B)$ with non-
cyclic inertia group in $\mathrm{Out}(G)$ are unipotent characters of
$\operatorname{N}_{G}(D)\cong\mathrm{SL}_{2}(3)$ and their extension to the
inertia group follows again from explicit computations. Now using similar
arguments as in Proposition 14.6 shows that the block $b$ is AM-good. ∎

###  14.4. Groups with exceptional Schur multiplier

After having checked the iAM-condition for all quasi-isolated $2$-blocks of
groups of Lie type with almost abelian defect group in non-defining
characteristic we are left to show the iAM-condition for the remaining simple
groups and blocks under consideration.

######  Proposition 14.8.

Suppose that $S$ is a simple group not of Lie type in odd characteristic with
exceptional Schur multiplier. Then the inductive Alperin–McKay condition holds
for all $2$-blocks with almost abelian defect group of the universal covering
group $X$ of $S$.

###### Proof.

Observe first that the iAM-condition holds for all simple sporadic groups by
[6]. Assume that $S$ is a simple group of Lie type in characteristic different
from $2$ with exceptional Schur multiplier and let $X$ be its universal
covering group. Then $S$ is one of the following groups: $A_{1}(9)$,
${}^{2}A_{3}(3)$, $B_{3}(3)$ or $G_{2}(3)$. According to [6] both
$A_{1}(9)\cong A_{6}$ and $G_{2}(3)$ are AM-good. Observe that the blocks of
the universal covering groups of ${}^{2}A_{3}(3)$ and $B_{3}(3)$ with maximal
defect have a defect group which is not almost abelian. Recall that every
defect group of $X$ contains $\mathrm{Z}(X)_{\ell}$. An inspection of the
blocks of the universal covering group of ${}^{2}A_{3}(3)$ using [7] now shows
that all $2$-blocks with almost abelian defect groups have central defect.
Therefore, there is nothing to check in this case. For the universal covering
group of $B_{3}(3)$ the same arguments easily rule out all blocks except the
blocks denoted by $2$, $3$, $8$ and $9$ in [7].

We claim that the blocks $2$, $8$ and $9$ of $X$ have a defect group which is
not almost abelian. For this let $b$ be any of these blocks. Consider a
central extension $X\to X^{\prime}$ whose kernel is of order $2$. According to
[31, Theorem 9.10] we obtain a bijection $b\to\overline{b}$ between blocks of
$X$ and $X^{\prime}$. The block of $X^{\prime}$ corresponding to $b$ under
this bijection has according to [31, Theorem 9.10] a defect group of order $8$
and still two Brauer characters. The groups $C_{8}$ and $C_{2}\times C_{4}$
admit no automorphisms of odd order. By the remarks following [37, Theorem
1.30] we therefore deduce that these groups arise as defect groups of
nilpotent blocks only. Moreover, any block with defect group $C_{2}\times
C_{2}\times C_{2}$ can’t have exactly two Brauer characters, see [37, Theorem
13.1]. Since the block $\overline{b}$ is not nilpotent (it has $2$ Brauer
characters) its defect group is therefore not abelian. Since $X^{\prime}$ is a
$3$-cover of $S$, we deduce that the block $b$ cannot have an almost abelian
defect group.

It therefore remains to consider the block labeled $3$ in [7]. This block can
however be consider as a block of the $2$-cover of $S$, i.e. as a block of
$\mathbf{G}^{F}$, and can therefore be treated as a block with non-exceptional
covering group. Using the reduction theorem in [36] we can thus conclude that
this block is AM-good.

For alternating groups the inductive AM condition is known to hold by the main
result of [13] and [39, Corollary 8.3]. By [25] the iAM-condition holds for
Suzuki and Ree groups.

Let us finally assume that $S$ is a simple group of Lie type defined over a
field of characteristic $2$. Let $\mathbf{G}^{F}$ be such that
$\mathbf{G}^{F}/\mathrm{Z}(\mathbf{G}^{F})\cong S$. As before, let $X$ be the
universal covering group of $S$. Then there exists a surjective homorphism
$X\to\mathbf{G}^{F}$ whose kernel is the Sylow $2$-group $Z$ of
$\mathrm{Z}(\hat{G})$. According to [31, Theorem 9.10] we obtain a bijection
$B\to\overline{B}$ between blocks of $\hat{G}$ and $\mathbf{G}^{F}$ which maps
blocks with almost abelian defect to each other.

It is known (see [11, Theorem 6.18]) that the only $2$-blocks of
$\mathbf{G}^{F}$ are blocks of maximal defect and blocks of height zero. We
must therefore consider the cases where a Sylow $2$-subgroup of $S$ is
abelian. This is however precisely the case when $S\cong A_{1}(\pm 2^{f})$.
Again by the work of Breuer [6] we know that $S$ is AM-good whenver $S$ has an
exceptional Schur multiplier. Using the properties of the bijection
constructed in [38] in conjunction with [12, Theorem 4.1] shows that the
principal block of $S$ is AM-good. ∎

######  Theorem 14.9.

Suppose that $S$ is a simple group of Lie type in characteristic unequal to
$2$ with non-exceptional Schur multiplier. Then the inductive Alperin–McKay
condition holds for all $2$-blocks with almost abelian defect group of the
universal covering group $X$ of $S$.

###### Proof.

Let $\mathbf{G}$ be a simple, simply connected algebraic group and
$F:\mathbf{G}\to\mathbf{G}$ a Frobenius endomorphism such that
$S=\mathbf{G}^{F}/\mathrm{Z}(\mathbf{G}^{F})$. We fix a $2$-block $b$ of
$\mathbf{G}^{F}$ with almost abelian defect group. We want to show that $b$ is
AM-good relative to its defect group.

Using the proof of [36, Theorem 3.12] we see that the statement of [36,
Theorem 3.12] can be adapted to our situation as follows: If [36, Hypothesis
3.3] holds for all blocks with almost abelian defect groups then the block $b$
is AM-good. In other words, we can assume that the $2$-block $b$ of
$\mathbf{G}^{F}$ is strictly quasi-isolated. According to Lemma 14.1 all these
blocks are unipotent unless $\mathbf{G}$ is of type $A$. Suppose therefore now
that $\mathbf{G}$ is of type $A$ and $s\neq 1$. The assumptions of Lemma 11.2
applies. We can therefore use the proof of Theorem 12.2 shows that we can also
assume in this case that the block is unipotent. It therefore suffices to
check that the iAM-condition holds for the unipotent blocks occuring in Lemma
14.1. Using Remark 14.2 we see that this was checked in Proposition 14.6 and
Proposition 14.7. ∎

######  Theorem 14.10.

The Alperin–McKay conjecture holds for all $2$-blocks of almost abelian
defect.

###### Proof.

According to Proposition 13.3 to show the theorem it is sufficent to prove the
iAM-condition for all $\ell$-blocks with almost abelian defect of the
universal covering group of a simple finite group. The iAM-condition in these
cases has been verified in Theorem 14.9 and Proposition 14.8. ∎

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