Proof. In the proof of [Reference Bonnafé, Dat and Rouquier4, Theorem 7.5], use the fact that $M'$ extends $H_c^{\dim }(\mathbf {Y}_{\mathbf {U}}^{\mathbf {G}},\Lambda ) e_s^{\mathbf {L}^F}$ instead of [Reference Bonnafé, Dat and Rouquier4, Proposition 7.3]. The rest of the proof of the theorem is as in [Reference Bonnafé, Dat and Rouquier4, Section 7].

Proof. According to the proof of [Reference Rickard34, Theorem 5.2], there exists a unique complex $\mathcal C_Q'$ of $\ell $ -permutation $\Lambda (\operatorname {C}_{\mathbf {G}^F}(Q) \times \operatorname {C}_{\mathbf {N}^F}(Q)^{\operatorname {opp}})$ -modules lifting the complex $\operatorname {Br}_{\Delta Q}(\mathcal C')$ from k to $\Lambda $ . Note that

$$ \begin{align*} \operatorname{Res}_{\operatorname{C}_{\mathbf{G}^F}(Q) \times \operatorname{C}_{\mathbf{L}^F}(Q)^{\operatorname{opp}} }^{\operatorname{C}_{\mathbf{G}^F}(Q) \times \operatorname{C}_{\mathbf{N}^F}(Q)^{\operatorname{opp}}} (\mathrm{Br}_{\Delta Q}(\mathcal C')) \cong G \Gamma_c(\mathbf{Y}_{\operatorname{C}_{\mathbf{U}}(Q)}^{\mathrm{C}_{\mathbf{G}}(Q)},k) \mathrm{br}_Q(e_s^{\mathbf{L}^F}) \end{align*} $$

in $\mathrm {Ho}^b(k(\mathrm {C}_{\mathbf {G}^F}(Q) \times \mathrm {C}_{\mathbf {L}^F}(Q)^{\mathrm {opp}}))$ ; see, for example, [Reference Ruhstorfer37, Lemma 2.9]. Let $M^{\prime }_Q:=H^{d_Q}(\mathcal {C}_Q')$ , where by [Reference Bonnafé, Dat and Rouquier4, Theorem 4.14], the integer $d_Q:=\mathrm {dim}( \mathbf {Y}_{\operatorname {C}_{\mathbf {U}}(Q)}^{\operatorname {C}_{\mathbf {G}}(Q)})$ is the unique degree in which $\mathrm {Br}_{\Delta Q}(\mathcal {C})$ has nonzero cohomology. Thus, $M^{\prime }_Q c_Q$ induces a Morita equivalence between $\Lambda \mathrm {C}_{\mathbf {G}^F}(Q) b_Q$ and $\Lambda \mathrm {C}_{\mathbf {N}^F}(Q) c_Q$ . By the proof of [Reference Ruhstorfer37, Proposition 1.12], the bimodule $\operatorname {Ind}_{\operatorname {C}_{\mathbf {G}^F}(Q) \times \operatorname {C}_{\mathbf {N}^F}(Q)^{\mathrm {opp}} \Delta \operatorname {N}_{\mathbf {N}^F}(Q) }^{\operatorname {N}_{\mathbf {G}^F}(Q) \times \operatorname {N}_{\mathbf {N}^F}(Q)^{\mathrm {opp}}}(M^{\prime }_Q c_Q)$ induces a Morita equivalence between $\Lambda \mathrm {N}_{\mathbf {G}^F}(Q) B_Q$ and $\Lambda \mathrm {N}_{\mathbf {N}^F}(Q) C_Q$ , where $B_Q$ (respectively $C_Q$ ) is the block covering $b_Q$ (respectively $c_Q$ ).

In particular, $\operatorname {Ind}_{\operatorname {C}_{\mathbf {G}^F}(Q) \times \operatorname {C}_{\mathbf {N}^F}(Q)^{\mathrm {opp}} \Delta \operatorname {N}_{\mathbf {N}^F}(Q) }^{\operatorname {N}_{\mathbf {G}^F}(Q) \times \operatorname {N}_{\mathbf {N}^F}(Q)^{\mathrm {opp}}}(M^{\prime }_Q) C_Q$ is indecomposable. Since this is true for all Brauer pairs of all blocks c of $\Lambda \mathbf {N}^F e_s^{\mathbf {L}^F}$ , it follows that $\operatorname {Ind}_{\operatorname {C}_{\mathbf {G}^F}(Q) \times \operatorname {C}_{\mathbf {N}^F}(Q)^{\mathrm {opp}} \Delta \operatorname {N}_{\mathbf {N}^F}(Q) }^{\operatorname {N}_{\mathbf {G}^F}(Q) \times \operatorname {N}_{\mathbf {N}^F}(Q)^{\mathrm {opp}}}(M^{\prime }_Q)$ is multiplicity free as well. By Mackey’s formula, this module is an extension of

$$ \begin{align*} \operatorname{Ind}_{\operatorname{C}_{\mathbf{G}^F}(Q) \times \operatorname{C}_{\mathbf{L}^F}(Q)^{\mathrm{opp}} \Delta \operatorname{N}_{\mathbf{L}^F}(Q) }^{\operatorname{N}_{\mathbf{G}^F}(Q) \times \operatorname{N}_{\mathbf{L}^F}(Q)^{\mathrm{opp}}}(H^{\mathrm{dim}}_c(\mathbf{Y}_{\operatorname{C}_{\mathbf{U}}(Q)}^{\mathrm{C}_{\mathbf{G}}(Q)},\Lambda) \mathrm{br}_Q(c_Q)) \cong (H^{\mathrm{dim}}_c(\mathbf{Y}_{\operatorname{C}_{\mathbf{U}}(Q)}^{\mathrm{N}_{\mathbf{G}}(Q)},\Lambda) \mathrm{br}_Q(c_Q)). \end{align*} $$

Since $\mathbf {N}^F/\mathbf {L}^F$ is cyclic of $\ell '$ -order and the Morita bimodule is indecomposable, the result therefore follows from Clifford theory.

Proof. Let us first assume that $\Lambda =k$ . We consider the complex of $\ell $ -permutation $k(\mathbf {G}^F \times (\mathbf {L}^F)^{\operatorname {opp}} \Delta (\tilde {\mathbf {L}}^F))$ -modules $ \mathcal {C}:=G\Gamma _c(\mathbf {Y}_{\mathbf {U}},k)^{\mathrm {red}}e_s^{\mathbf {L}^F}$ . Consider an $\ell $ -subgroup $\tilde R$ of $\mathbf {G}^F \times (\mathbf {L}^F)^{\operatorname {opp}} \Delta (\tilde {\mathbf {L}}^F)$ . Let $Z:=\mathrm {Z}(\tilde {\mathbf {G}}^F)_{\ell }$ , and denote $R:=\tilde {R} \cap (\mathbf {G}^F \times (\mathbf {L}^F)^{\operatorname {opp}})$ . By assumption, $\ell \nmid |H^1(F,\mathrm {Z}(\mathbf {G}))|=|\mathbf {\tilde {G}}^F/\mathbf {G}^F\mathrm {Z}(\mathbf {\tilde {G}}^F)|$ , and therefore $\tilde {R} \subset R \Delta (Z)$ . The subgroup $\Delta (Z)$ centralises the variety $\mathbf {Y}_{\mathbf {U}}^{\mathbf {G}}$ . Hence, $\mathcal {C}$ can be regarded as complex of $(\mathbf {G}^F \times (\mathbf {L}^F)^{\operatorname {opp}} \Delta (\tilde {\mathbf {L}}^F))/ \Delta (Z)$ -modules. Since $\tilde {R} \subset R \Delta (Z)$ , we deduce that $\operatorname {Br}_{R}(\mathcal {C}) \cong \operatorname {Br}_{\tilde R}(\mathcal {C})$ . Using [Reference Bonnafé, Dat and Rouquier4, Corollary 3.8], we deduce that $\operatorname {Br}_{R}(\mathcal {C})$ is acyclic unless R is $\mathbf {G}^F \times (\mathbf {L}^F)^{\operatorname {opp}}$ -conjugate to a subgroup of $\Delta \mathbf {L}^F$ . Thus, $\tilde R$ is $\mathbf {G}^F \times (\mathbf {L}^F)^{\operatorname {opp}}$ -conjugate to a subgroup of $\Delta \tilde {\mathbf {L}}^F$ . By [Reference Bonnafé, Dat and Rouquier4, Lemma A.2], we deduce that the vertices of the indecomposable summands of the components $\mathcal {C}$ are all contained in $\Delta \tilde {\mathbf {L}}^F$ .

Consider now the complex $\mathrm {End}^{\bullet }_{k\mathbf {G}^F}(\mathcal {C})$ of $k(\mathbf {L}^F \times (\mathbf {L}^F)^{\operatorname {opp}} \Delta (\tilde {\mathbf {L}}^F))$ -bimodules. The same argument as above (for $\operatorname {End}^{\bullet }_{k\mathbf {G}^F}(\mathcal {C})$ instead of $\mathcal {C}$ ) shows that $\operatorname {Br}_{\tilde R}(\mathrm {End}^{\bullet }_{k\mathbf {G}^F}(\mathcal {C})) \cong \operatorname {Br}_{R}(\mathrm {End}^{\bullet }_{k\mathbf {G}^F}(\mathcal {C}))$ . The cohomology of the latter complex is concentrated in degree $0$ only by the proof of Step 1 of [Reference Bonnafé, Dat and Rouquier4, Theorem 7.5]. Hence, by [Reference Bonnafé, Dat and Rouquier4, Lemma A.3], we have $\operatorname {End}_{k \mathbf {G}^F}^{\bullet }(\mathcal {C}) \cong \operatorname {End}_{D^b(k \mathbf {G}^F)}(\mathcal {C})$ in $\operatorname {Ho}^b(k(\mathbf {L}^F \times (\mathbf {L}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F)).$

The rest of the proof is now almost identical to the proof of [Reference Bonnafé, Dat and Rouquier4, Theorem 7.6]. For completeness, we provide most of the details here. Denote $\mathcal {C}'=\operatorname {Ind}_{\mathbf {G}^F\times (\mathbf {L}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F }^{\mathbf {G}^F\times (\mathbf {N}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F}(\mathcal {C})$ . Let $\mathcal P$ be a projective resolution of $k\mathbf {N}^F$ – that is, a complex of projective $k(\mathbf {N}^F\times (\mathbf {N}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F)$ -modules such that its terms $\mathcal P^i=0$ for $i>0$ – together with a quasi-isomorphism $\mathcal P\to k\mathbf {N}^F$ of $k(\mathbf {N}^F\times (\mathbf {N}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F)$ -modules.

Let $\mathcal X$ be a complex of $k(\mathbf {G}^F \times (\mathbf {N}^F)^{\mathrm {opp}} \Delta \tilde {\mathbf {L}}^F)$ -modules. Observe that we can consider the complex $\operatorname {End}_{k \mathbf {G}^F}^{\bullet }(\mathcal X)$ as a complex of $k(\mathbf {N}^F \times (\mathbf {N}^F)^{\mathrm {opp}} \Delta \tilde {\mathbf {L}}^F)$ -modules. We have a natural isomorphism $\operatorname {End}_{k(\mathbf {G}^F \times (\mathbf {N}^F)^{\mathrm {opp}} \Delta \tilde {\mathbf {L}}^F)}^{\bullet }(\mathcal X) \cong \operatorname {Hom}^{\bullet }_{k (\mathbf {N}^F \times (\mathbf {N}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F)}(k \mathbf {N}^F, \operatorname {End}_{k \mathbf {G}^F}^{\bullet }(\mathcal {X}))$ . The terms of $\mathcal C'$ are projective $k\mathbf {G}^F$ -modules. Therefore, as in the proof of [Reference Bonnafé, Dat and Rouquier4, Theorem 7.6], we can consider the following commutative diagram:

Using the isomorphisms of complexes in $\operatorname {Ho}^b(k(\mathbf {N}^F\times (\mathbf {N}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F))$

$$ \begin{align*}\operatorname{End}_{k\mathbf{G}^F}^{\bullet}(\mathcal{C}') \cong \operatorname{Ind}_{\mathbf{L}^F\times (\mathbf{L}^F)^{\operatorname{opp}} \Delta \tilde{\mathbf{L}}^F}^{\mathbf{N}^F\times (\mathbf{N}^F)^{\operatorname{opp}} \Delta \tilde{\mathbf{L}}^F}( \operatorname{End}_{k\mathbf{G}^F}^{\bullet}(C))\end{align*} $$

and

$$ \begin{align*}\operatorname{End}_{D^b(k\mathbf{G}^F)}(\mathcal{C}') \cong \operatorname{Ind}_{\mathbf{L}^F\times (\mathbf{L}^F)^{\operatorname{opp}} \Delta \tilde{\mathbf{L}}^F}^{\mathbf{N}^F\times (\mathbf{N}^F)^{\operatorname{opp}} \Delta \tilde{\mathbf{L}}^F}( \operatorname{End}_{D^b(k\mathbf{G}^F)}(C)),\end{align*} $$

we deduce that

$$ \begin{align*}\operatorname{End}_{k\mathbf{G}^F}^{\bullet}(\mathcal{C}') \cong \operatorname{End}_{D^b(k\mathbf{G}^F)}(\mathcal{C}')\text{ in }\operatorname{Ho}^b(k(\mathbf{N}^F\times (\mathbf{N}^F)^{\operatorname{opp}} \Delta \tilde{\mathbf{L}}^F)).\end{align*} $$

Now, the canonical map

$$ \begin{align*}\operatorname{Hom}_{\operatorname{Ho}^b(k(\mathbf{N}^F\times (\mathbf{N}^F)^{\operatorname{opp}}))}(k\mathbf{N}^F,\operatorname{End}_{D^b(k\mathbf{G}^F)}(\mathcal{C}')) \to \operatorname{Hom}_{\operatorname{Ho}^b(k(\mathbf{N}^F\times (\mathbf{N}^F)^{\operatorname{opp}}))}(\mathcal P,\operatorname{End}_{D^b(k\mathbf{G}^F)}(\mathcal{C}'))\end{align*} $$

is an isomorphism, since $\operatorname {End}_{D^b(k\mathbf {G}^F)}(\mathcal {C}')$ is a complex concentrated in degree $0$ . It follows that the top horizontal map in the commutative diagram above is an isomorphism, hence we have canonical isomorphisms

$$ \begin{align*}\operatorname{End}_{\operatorname{Ho}^b(k(\mathbf{G}^F\times(\mathbf{N}^F)^{\operatorname{opp}} \Delta(\tilde{\mathbf{L}}^F)))}(\mathcal{C}')\xrightarrow{\cong} \operatorname{End}_{D^b(k(\mathbf{G}^F\times(\mathbf{N}^F)^{\operatorname{opp}} \Delta(\tilde{\mathbf{L}}^F)))}(\mathcal{C}')\xrightarrow{\cong} \end{align*} $$

$$ \begin{align*}\operatorname{End}_{k(\mathbf{G}^F\times(\mathbf{N}^F)^{\operatorname{opp}} \Delta \tilde{\mathbf{L}}^F)}(\operatorname{Ind}_{\mathbf{G}^F\times(\mathbf{L}^F)^{\operatorname{opp}} \Delta(\tilde{\mathbf{L}}^F)}^{\mathbf{G}^F\times(\mathbf{N}^F)^{\operatorname{opp}} \Delta (\tilde{\mathbf{L}}^F)} H_c^d(\mathbf{Y}_{\mathbf{U}},k) e_s^{\mathbf{L}^F}) .\end{align*} $$

Using the proof of Step 3 of [Reference Bonnafé, Dat and Rouquier4, Theorem 7.6], we deduce that there exists a summand $\tilde {\mathcal C}$ of $\mathcal {C}'$ that is quasi-isomorphic to $M'$ . As in Step 4 and 5 of the proof of [Reference Bonnafé, Dat and Rouquier4, Theorem 7.6], we see that $\mathrm {Res}_{\mathbf {G}^F\times (\mathbf {L}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F}^{\mathbf {G}^F\times (\mathbf {N}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F} (\tilde {\mathcal {C}}) \cong \mathcal {C}$ in $\operatorname {Ho}^b(k({\mathbf {G}^F\times (\mathbf {L}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F}))$ . Furthermore, Step 5 of the proof of [Reference Bonnafé, Dat and Rouquier4, Theorem 7.6] shows that $\tilde {\mathcal C}$ induces a splendid Rickard equivalence between $k \mathbf {G}^F e_s^{\mathbf {G}^F}$ and $k \mathbf {N}^F e_s^{\mathbf {L}^F}$ .

Finally, let us consider the case $\Lambda =\mathcal {O}$ . Using [Reference Rickard34, Lemma 5.1] together with the arguments in the first paragraph of the proof of [Reference Rickard34, Theorem 5.2], we observe that there exists a unique complex of $\mathcal {O} (\mathbf {G}^F \times (\mathbf {L}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F)$ -modules lifting $\tilde {\mathcal C}$ . Moreover, by the proof of [Reference Rickard34, Theorem 5.2], this complex induces a splendid Rickard equivalence between $\mathcal {O} \mathbf {G}^F e_s^{\mathbf {G}^F}$ and $\mathcal {O} \mathbf {N}^F e_s^{\mathbf {L}^F}$ .

Proof. The pair $(\underline {\mathbf {L}},\underline {\mathbf {P}}')$ is $\langle \tau , \gamma \rangle $ -stable. Therefore, we can consider $\mathbf {Y}_{\underline {\mathbf {U}}'}^{\underline {\mathbf {G}}}$ a $\underline {\mathbf {G}}^{\tau F_0} \times (\underline {\mathbf {L}}^{\tau F_0})^{\operatorname {opp}} \Delta ( \tilde {\underline {\mathbf {L}}}^{\tau F_0} \langle \tau ,\gamma \rangle )$ -variety. By [Reference Bonnafé, Dat and Rouquier4, Theorem 7.2], we have an isomorphism

$$ \begin{align*}H^{\mathrm{dim}}_c(\mathbf{Y}_{\underline{\mathbf{U}}}^{\underline{\mathbf{G}}, \tau F_0},\Lambda) e_{\underline{s}}^{\underline{\mathbf{L}}^{\tau F_0}} \cong H^{\mathrm{dim}}_c(\mathbf{Y}_{\underline{\mathbf{U}'}}^{\underline{\mathbf{G}}, \tau F_0},\Lambda) e_{\underline{s}}^{\underline{\mathbf{L}}^{\tau F_0}}\end{align*} $$

of $\Lambda ((\underline {\mathbf {G}}^{\tau F_0} \times (\underline {\mathbf {L}}^{\tau F_0})^{\mathrm {opp}}) \Delta ( \underline {\tilde {\mathbf {L}}}^F ))$ -modules. The projection $\mathrm {pr}:\underline {\mathbf {G}} \to \mathbf {G}$ onto the first coordinate defines an isomorphism

$$ \begin{align*}H_c^{\mathrm{dim}}(\mathbf{Y}_{\underline{\mathbf{U}}}^{\underline{\mathbf{G}},\tau F_0},\Lambda) e_{\underline{s}}^{\mathbf{L}^{\tau F_0}} \cong H_c^{\mathrm{dim}}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G},F},\Lambda) e_s^{\mathbf{L}^F}\end{align*} $$

of $\mathbf {G}^F \times (\mathbf {L}^F)^{\mathrm {opp}} \Delta \tilde {\mathbf {L}}^F$ -modules. By transport of structure, we can endow $H_c^{\mathrm {dim}}(\mathbf {Y}_{\mathbf {U}}^{\mathbf {G},F},\Lambda ) e_s^{\mathbf {L}^F}$ with a $\mathbf {G}^F \times (\mathbf {L}^F)^{\mathrm {opp}} \Delta \tilde {\mathbf {L}}^F \langle F_0,\gamma \rangle $ -structure.

Assume now that $\mathbf {N}/\mathbf {L}$ is centralised by $\mathcal {A}$ , and let $n \in \mathbf {N}^{F}$ be a generator of the quotient group $\mathbf {N}^F/ \mathbf {L}^F$ . Then we have $\gamma (n)n^{-1},F_0(n)n^{-1} \in \mathbf {L}^F$ . Thus, conjugation by $\underline {n}$ defines an automorphism of $\underline {\mathbf {G}} \langle \tau , \gamma \rangle $ , which normalises $\underline {\mathbf {L}} \langle \tau , \gamma \rangle $ . Hence, by [Reference Ruhstorfer37, Lemma 3.1], the $\Lambda ((\underline {\mathbf {G}}^{\tau F_0} \times (\underline {\mathbf {L}}^{\tau F_0})^{\mathrm {opp}}) \Delta ( \underline {\tilde {\mathbf {L}}}^{\tau F_0} \langle \tau , \gamma \rangle ))$ -module $H^{\mathrm {dim}}_c(\mathbf {Y}_{\underline {\mathbf {U}'}}^{\underline {\mathbf {G}}, \tau F_0}) e_{\underline {s}}^{\underline {\mathbf {L}}^{\tau F_0}}$ is $\underline {n}$ -stable. By transport of structure, we deduce that $H_c^{\mathrm {dim}}(\mathbf {Y}_{\mathbf {U}}^{\mathbf {G},F},\Lambda ) e_s^{\mathbf {L}^F}$ is $\mathbf {N}^F$ -stable as a $\mathbf {G}^F \times (\mathbf {L}^F)^{\mathrm {opp}} \Delta (\tilde {\mathbf {L}}^F \mathcal {A})$ -module. Thus, it extends to a $\mathbf {G}^F \times (\mathbf {N}^F)^{\operatorname {opp}} \Delta \tilde {\mathbf {L}}^F \langle F_0,\gamma \rangle $ -module by [Reference Rouquier35, Lemma 10.2.13].

Proof. Recall that the projection map $\mathrm {pr}:\underline {\mathbf {G}}^{\tau F_0} \to \mathbf {G}^F$ is an isomorphism of finite groups. If H is a subgroup of $\mathbf {G}^F$ , we let $\underline {H}:=\mathrm {pr}^{-1}(H)$ ; and if $x \in \Lambda H$ , we let $\underline {x}:= \mathrm {pr}^{-1}(x) \in \Lambda \underline {H}$ .

The quotient group $\mathrm {N}_{\mathcal {N}}(Q)/ \mathrm {N}_{\tilde {\mathbf {L}}^F \mathcal {A} }(Q)$ is cyclic as it embeds into $\mathcal {N}/\tilde {\mathbf {L}}^F \mathcal {A} \cong \mathbf {N}^F/\mathbf {L}^F$ . Let $x \in \mathrm {N}_{\mathbf {N}^F}(Q)$ be a generator of said quotient. Since $\mathbf {N}^F/\mathbf {L}^F$ is centralised by $F_0$ , we have $x F_0(x)^{-1} \in \mathrm {N}_{\mathbf {L}^F}(Q)$ . Let $\underline {x}:=(x,F^{r-1}_0(x),\dots ,F_0(x)) \in \underline {\mathbf {G}}^{\tau F_0}$ such that $\mathrm {pr}(\underline {x})=x$ . Consider the bijective morphism $\underline {\phi }:\underline {\tilde {\mathbf {G}}} \langle \tau \rangle \to \underline {\tilde {\mathbf {G}}} \langle \tau \rangle $ given by conjugation with $\underline {x}$ . Note that $\underline {\phi }$ stabilises $\underline {\tilde {\mathbf {G}}}$ and commutes with the Frobenius endomorphism $\tau F_0$ of $\underline {\tilde {\mathbf {G}}} \rtimes \langle \tau \rangle $ . Moreover, $\underline {\phi }$ also stabilises the Levi subgroup $\tilde {\underline {\mathbf {L}}} \langle \tau \rangle $ of $\underline {\tilde {\mathbf {G}}} \rtimes \langle \tau \rangle $ . We denote $e:=\operatorname {br}_Q(e_s^{\mathbf {L}^F})$ . By [Reference Ruhstorfer37, Lemma 2.23], we obtain an isomorphism

$$ \begin{align*}{}^{\underline{\phi}} (H_c^{\mathrm{dim}}(\mathbf{Y}_{\operatorname{C}_{\underline{\mathbf{U}'}}(\underline{Q})}^{\mathrm{N}_{\underline{\mathbf{G}}}(\underline{Q}),\tau F_0},\Lambda ) \underline{e} )^{\underline{\phi}} \cong H_c^{\mathrm{dim}}(\mathbf{Y}_{\operatorname{C}_{\underline{\phi}(\underline{\mathbf{U}'})}(\underline{Q})}^{\mathrm{N}_{\underline{\mathbf{G}}}(\underline{Q}),\tau F_0},\Lambda ) \underline{e}\end{align*} $$

of $\Lambda ((\mathrm {N}_{\underline {\mathbf {G}}^{\tau F_0}}(\underline {Q}) \times \mathrm {N}_{\underline {\mathbf {L}}^{\tau F_0}}(\underline {Q})^{\mathrm {opp}}) \Delta (\mathrm {N}_{\underline {\tilde {\mathbf {L}}}^{\tau F_0} \langle \tau \rangle }(\underline {Q}))$ -modules. We have two Levi decompositions

$$ \begin{align*}\underline{\tilde{\mathbf{P}}} \langle \tau \rangle= \underline{\tilde{\mathbf{L}}} \langle \tau \rangle \ltimes \underline{\mathbf{U}'} \text{ and } \underline{\phi}(\underline{\tilde{\mathbf{P}}} \langle \tau \rangle)= \underline{\tilde{\mathbf{L}}} \langle \tau \rangle \ltimes \underline{\phi}(\underline{\mathbf{U}}') \end{align*} $$

with the same Levi subgroup $\underline {\tilde {\mathbf {L}}} \langle \tau \rangle $ of $\tilde {\mathbf {G}} \langle \tau \rangle $ . Therefore, [Reference Ruhstorfer37, Theorem 5.2] yields

$$ \begin{align*} H_c^{\mathrm{dim}}(\mathbf{Y}_{\operatorname{C}_{\underline{\phi}(\underline{\mathbf{U}'})}(\underline{Q})}^{\mathrm{N}_{\underline{\mathbf{G}}}(\underline{Q}),\tau F_0},\Lambda ) \underline{e} \cong H_c^{\mathrm{dim}}(\mathbf{Y}_{\operatorname{C}_{\underline{\mathbf{U}'}}(\underline{Q})}^{\mathrm{N}_{\underline{\mathbf{G}}}(\underline{Q}),\tau F_0},\Lambda ) \underline{e}. \end{align*} $$

It follows from this that $H_c^{\mathrm {dim}}(\mathbf {Y}_{\operatorname {C}_{\underline {\mathbf {U}'}}(\underline {Q})}^{\mathrm {N}_{\underline {\mathbf {G}}}(\underline {Q}),\tau F_0},\Lambda ) \underline {e}$ is $( \underline {\phi },\underline {\phi }^{-1})$ -invariant. Hence, the bimodule $H^{\mathrm {dim}}_c(\mathbf {Y}^{\mathrm {N}_{\mathbf {G}}(Q)}_{\operatorname {C}_{\mathbf {U}}(Q)},\Lambda )e$ is by transport of structure x-invariant as $\Lambda ((\mathrm {N}_{\mathbf {G}^F}(Q) \times \mathrm {N}_{\mathbf {L}^F}(Q)^{\operatorname {opp}}) \Delta (\mathrm {N}_{\tilde {\mathbf {L}}^F \langle F_0 \rangle }(Q)))$ -module. The claim now follows from [Reference Rouquier35, Lemma 10.2.13].

Proof. Let $\iota ^{\ast }: \mathbf {\tilde {G}}^{\ast } \to \mathbf {G}^{\ast }$ be the map dual to the map $\iota : \mathbf {G} \hookrightarrow \mathbf {\tilde {G}}$ , and let $ \tilde s\in (\mathbf {\tilde {G}}^{\ast })^{F^{\ast }}$ such that $\iota ^{\ast }(\tilde {s})=s$ . Consider the injective morphism $\omega _s: A(s) \to \mathrm {Z}(\mathbf {\tilde {G}}^{\ast })$ as in [Reference Bonnafé2, Corollary 2.8]. Let e denote the exponent of the subgroup $A(s)^{F^{\ast }}$ of $A(s)$ . Let $g \in \mathrm {C}^{\circ }_{\mathbf {G}^{\ast }}(s) \mathrm {C}_{(\mathbf {G}^{\ast })^{F^{\ast }}}(s)$ so that $g^e \in \mathrm {C}_{\mathbf {G}^{\ast }}^{\circ }(s)$ . Then $1=\omega _s(g)^e=\omega _{s^e}(g)$ , and therefore $g \in \mathrm {Ker}(\omega _s^e)=\iota ^{\ast }(\mathrm {C}_{\mathbf {\tilde {G}}^{\ast }}(\tilde {s}^e))$ . Consequently, $\iota ^{\ast }(\mathrm {C}_{\mathbf {\tilde {G}}^{\ast }}(\tilde {s}^e))=\mathrm {C}^{\circ }_{\mathbf {G}^{\ast }}(s^e)$ is a Levi subgroup of $\mathbf {G}^{\ast }$ containing $\mathrm {C}^{\circ }_{\mathbf {G}^{\ast }}(s) \mathrm {C}_{(\mathbf {G}^{\ast })^{F^{\ast }}}(s)$ . This is a proper Levi subgroup of $\mathbf {G}^{\ast }$ unless $s^e=1$ . By the classification of quasi-isolated elements of $\mathbf {G}^{\ast }=\mathrm {PGL}_{n+1}(\overline {\mathbb {F}_p})$ in [Reference Bonnafé2, Proposition 5.2], we have $\mathrm {o}(s)=|A(s)|$ , which implies that $e=|A(s)|$ . Since e is the exponent of $A(s)^{F^{\ast }}$ , we must necessarily have $A(s)=A(s)^{F^{\ast }}$ .

Proof. We keep the notations of Remark 5.3. Assume first that $\varepsilon =1$ . Let $w \in \langle v_m \rangle $ as in Remark 5.3 such that $\mathbf {L}^{\ast }$ is of type $W^{\circ }(t) w$ . Then w has order d with d dividing $m=|A(s)|$ , and we can decompose $w=w_1 \cdots w_r$ into disjoint cycles, each $w_i$ of order d with $d \mid m$ and $n+1=r d$ . Note that $\mathbf {L}^{\ast }$ is d-split if and only if $\tilde {\mathbf {L}}^{\ast }$ is d-split; see [Reference Cabanes and Enguehard11, Proposition 13.2].

Furthermore, $\tilde {\mathbf {L}}^{F} \cong \tilde {\mathbf {L}}_m^{w F} \cong \mathrm {GL}_r({\mathbb {F}_{q^d}}) \times \dots \mathrm {GL}_r({\mathbb {F}_{q^d}})$ . Hence $\tilde {\mathbf {L}}$ and therefore also $\mathbf {L}$ is d-split; see [Reference Cabanes and Enguehard11, Example 13.4]. By Ennola-duality, we obtain that if $ \varepsilon =-1$ , then $\mathbf {L}$ is $d'$ -split where

$$ \begin{align*} d'= \begin{cases} 2d & \text{if } d \text{ is odd},\\ d/2 & \text{if } d \equiv 2 \, \mathrm{mod} \, 4,\\ d &\text{if } d \equiv 0 \, \mathrm{mod} \, 4. \end{cases} \end{align*} $$

This also shows the result in the twisted case.

Proof. Observe that $\gamma $ is a quasi-central morphism in the sense of [Reference Digne and Michel16, Definition-Theorem 1.15]. According to [Reference Digne and Michel16, Remark 1.30], the group $\mathbf {G}^{\gamma }$ is connected. The Levi subgroup $\mathbf {L}_m$ is a standard Levi subgroup of $\mathbf {G}$ relative to the $\gamma $ -stable pair $(\mathbf {T}_0,\mathbf {B}_0)$ whose associated set of simple roots is $\gamma $ -stable. Thus, $\mathbf {L}_m$ is a $\gamma $ -stable Levi subgroup contained in a $\gamma $ -stable parabolic subgroup $\mathbf {P}_m$ of $\mathbf {G}$ . From [Reference Digne and Michel16, Proposition 1.11], it therefore follows that $\mathbf {L}_m^{\gamma }$ is a Levi subgroup in the connected reductive group $\mathbf {G}^{\gamma }$ with parabolic subgroup $\mathbf {P}_m^{\gamma }$ .

Proof. Define $V:=\langle n_{(i,i+1)} \mid i=1,\dots ,n \rangle $ , where $n_{(i,i+1)}$ is the matrix obtained by taking the permutation matrix corresponding to $(i,i+1)$ and multiplying its ith row with $-1$ . We have a group epimorphism $V \to W,\, n_{(i,i+1)} \mapsto (i,i+1)$ . By construction, we have $\gamma ( n_{(i,i+1)})=n_{\gamma ((i,i+1))}$ and every $n_{(i,i+1)}$ is $F_p$ -stable. We can write $\sigma \in W$ as a reduced expression $\sigma =s_1 \dots s_r$ with $s_i \in \{(1,2), \dots , (n-1,n)\}$ . We define a map $r:W \to V, \, \sigma \mapsto n_{s_1} \dots n_{s_r}$ . By Matsumoto’s Lemma, the map r is well-defined and injective; see also the proof of [Reference Späth40, Lemma 2.23]. Since the map r is bijective onto its image and $\gamma $ -equivariant, it follows that $\sigma $ is $\gamma $ -stable if and only if $r(\sigma )$ is $\gamma $ -stable.

Proof. Let $W_{\mathbf {L}_m}$ be the Weyl group of $\mathbf {L}_m$ with respect to the maximal torus $\mathbf {T}_0$ . By duality, $W_{\mathbf {L}_m}$ is (anti-)isomorphic to the Weyl group $W^{\circ }(t_m)$ of $\mathbf {L}_m^{\ast }$ . Since $W^{\circ }(t_m) \cong (S_e)^m$ , by [Reference Bonnafé2, Proposition 5.2], a computation in $W \cong S_{n+1}$ shows that $\mathrm {N}_{W}(W_{\mathbf {L}_m}) \cong S_e \wr S_m$ . Using [Reference Malle and Testerman27, Corollary 12.11], we thus obtain $\mathrm {N}_{\mathbf {G}}(\mathbf {L}_m)/\mathbf {L}_m\cong \mathrm {N}_{W}(W_{\mathbf {L}_m})/W_{\mathbf {L}_m} \cong S_m$ . The last isomorphism is given by mapping $\sigma \in \mathrm {N}_W(W_{\mathbf {L}_m})$ to the associated permutation $\pi \in S_m$ on the set of the m simple components of the Levi subgroup $\mathbf {L}_m$ . For any such permutation $\pi \in S_m$ , we define an element $\sigma _{\pi } \in W \cong S_{n+1}$ by defining

$$ \begin{align*}\sigma_{\pi}(j):= \pi(j-km)+km \text{ for } j \in \{km+1,\dots,(k+1) m\}.\end{align*} $$

By direct computation we obtain

$$ \begin{align*}\sigma_{{}^{\gamma} \pi}(j)=(m+1)-\pi((m+1)-(j-km))+km\end{align*} $$

and

$$ \begin{align*} {}^{\gamma} \sigma_{\pi}(j)=n+2-\pi((n+2-j)-(e-k-1) m )-m (e-k-1). \end{align*} $$

Comparing these two expressions, we obtain $\sigma _{{}^{\gamma } \pi }={}^{\gamma } \sigma _{\pi }$ . In particular, $\sigma _{\pi }$ is $\gamma $ -invariant whenever $\pi $ is. The claim now follows from Lemma 6.4.

Proof. We abbreviate $v:=v_m$ .According to Lemma 6.5, it is enough to show that if $w \in \operatorname {N}_{W}(W_{\mathbf {L}_m})$ with $\gamma (w)w^{-1} \in \langle v \rangle W_{\mathbf {L}_m}$ , then there exists some $y \in \langle v \rangle $ such that $\gamma (yw) (yw)^{-1} \in W_{\mathbf {L}_m}$ .

Denote by $\bar {\,}:\operatorname {N}_{W}(W_{L_m}) \to \operatorname {N}_{W}(W_{L_m})/W_{L_m}$ the projection map. In what follows, we identify $\operatorname {N}_{W}(W_{L_m})/W_{L_m}$ with $S_m$ as in the proof of Lemma 6.5. Note that the element $\overline v$ corresponds to an m-cycle of $S_m$ . Assume first that $\overline {v}$ has odd order. Then the map $\langle \overline {v} \rangle \to \langle \overline {v} \rangle , \overline {v} \mapsto \gamma (\overline {v})\overline {v}^{-1}=\overline {v}^{-2},$ is surjective. Thus, there exists $\overline {x} \in \langle \overline {v} \rangle $ such that $\gamma (\overline {w}) \overline {w}^{-1}=\gamma (\overline {x}^{-1}) \overline {x}$ . In other words, $\overline {xw}$ is $\gamma $ -stable.

Assume now that $\overline {v}$ has even order. We denote by $\mathrm {sgn}$ the sign map on the symmetric group $\operatorname {N}_{W}(W_{L_m})/W_{L_m} \cong S_m$ and by $A_m$ its kernel. Since $\overline {v}$ has even order, we have $\mathrm {sgn}(\overline {v})=-1$ . On the other hand, $\mathrm {sgn}(\gamma (\overline {w}) \overline {w}^{-1})=1$ , and thus $\gamma (\overline {w}) \overline {w}^{-1} \in \langle \overline {v}^{-2} \rangle = \langle \overline {v} \rangle \cap A_m$ . However, $\gamma (\overline {v}) \overline {v}^{-1}=\overline {v}^2$ . Thus, there exists $\overline {x} \in \langle \overline {v} \rangle $ such that $\gamma (\overline {w})\overline {w}^{-1}=\gamma (\overline {x}^{-1}) \overline {x}$ . In other words, $\overline {xw}$ is $\gamma $ -stable.

Proof. Recall that $s={}^g t \in (\mathbf {G}^{\ast })^{F^{\ast }}$ is a semisimple element with connected centraliser $\mathbf {L}^{\ast }=\operatorname {C}^{\circ }_{\mathbf {G}^{\ast }}(s)$ . We assume that the element $g \in \mathbf {G}$ has the property that $\mathbf {T}={}^g \mathbf {T}_0$ is a maximally split torus of $\mathbf {L}={}^g \mathbf {L}_I$ . Denote $w:=g^{-1} F(g) \mathbf {T}_0 \in W$ .

Let $\phi \in \operatorname {N}_{\mathbf {G}^F \mathcal {B}}(\mathbf {L},e_s^{\mathbf {L}^F})$ . Since $\phi (\mathbf {T})$ is a second maximally split torus of $\mathbf {L}$ , we can by possibly multiplying $\phi $ with an element of $\mathbf {L}^F$ assume that $\phi $ stabilises $\mathbf {T}$ . Denote $\phi _0:={}^{g^{-1}} \phi $ , and observe that $\phi _0=n_v f$ for some $f \in \mathcal {B}'$ and $n_v \in \operatorname {N}_{\mathbf {G}}(\mathbf {T}_0)$ with image $v=n_v \mathbf {T}_0$ in W. Since $[{}^g \phi ,{}^g F]={}^g [F,\phi ]=1$ and f, F centralise W, we conclude that $v \in \operatorname {C}_W(w)$ . We observe that the morphism $\phi _0$ of $\mathbf {L}_I$ is dual to the morphism $\phi _0^{\ast }:=f^{\ast } n_{v^{\ast }}$ of $\mathbf {L}_I^{\ast }$ . Let $\phi ^{\ast }:={}^{\pi (g)} \phi _0^{\ast }$ . Since $v^{\ast } \in \operatorname {C}_W(w^{\ast })$ , we find that $[F^{\ast },\phi ^{\ast }] \in \mathbf {T}^{\ast }$ . Hence, by applying Lang’s theorem, we find $t' \in \mathbf {T}^{\ast }$ such that $t' \phi ^{\ast }$ and $F^{\ast }$ commute. Observe that such an element $t'$ is unique up to multiplication with $(\mathbf {T}^{\ast })^{F^{\ast }}$ . Therefore, replacing $\phi ^{\ast }$ by $t' \phi ^{\ast }$ , we can assume that $\phi ^{\ast }$ and $F^{\ast }$ commute. The bijective morphisms $\phi :\mathbf {L} \to \mathbf {L}$ and $\phi ^{\ast }:\mathbf {L}^{\ast } \to \mathbf {L}^{\ast }$ are therefore in duality. Since $\phi (e_s^{\mathbf {L}^F})=e_s^{\mathbf {L}^F}$ , it follows that $\phi ^{\ast }$ stabilises the $(\mathbf {L}^{\ast })^{F^{\ast }}$ -conjugacy class of s: that is, $\phi ^{\ast }(s)=s$ . Hence, $\phi ^{\ast } \in \mathrm {C}_{(\mathbf {G}^{\ast })^{F^{\ast }} \mathcal {B}^{\prime \ast }}(s)$ . This defines a map between the two quotient groups by sending the $\mathbf {L}^F$ -coset of $\phi $ to the $(\mathbf {L}^{\ast })^{F^{\ast }}$ -coset of $\phi ^{\ast }$ .

Assume conversely that $\phi ^{\ast } \in \mathrm {C}_{(\mathbf {G}^{\ast })^{F^{\ast }} \mathcal {B}^{\prime \ast }}(s) (\mathbf {L}^{\ast })^{F^{\ast }}$ , and we assume again that $\phi ^{\ast }(\mathbf {T}^{\ast })=\mathbf {T}^{\ast }$ . We set $\phi _0^{\ast }={}^{\pi (g)^{-1}} \phi ^{\ast }$ and write $\phi _0^{\ast }=f^{\ast } n_v^{\ast }$ such that $n_v^{\ast } \mathbf {T}_0=v^{\ast }$ . Since $[\phi ^{\ast },F^{\ast }]=1$ , we obtain again $v \in \operatorname {C}_W(w)$ . Thus, denoting $\phi :={}^g (n_v f)$ , we find again some $t' \in \mathbf {T}$ such that $t'\phi $ commutes with F.

Proof. Let $\gamma ^{\ast }: \mathbf {G}^{\ast } \to \mathbf {G}^{\ast }$ be the dual of the graph automorphism $\gamma $ . There exists some $g \in \mathbf {G}^{\ast }$ such that ${}^g s =t$ . Consider the element

$$ \begin{align*} \tilde{t}=\tilde{t}_m:=(1,\zeta,\dots,\zeta^{m-1}) \otimes I_e\in \mathbf{\tilde{G}}^{\ast}=\mathrm{GL}_{n+1}(\overline{\mathbb{F}_p}), \end{align*} $$

a preimage of t under the natural map $\iota ^{\ast }:\mathrm {GL}_{n+1}(\overline {\mathbb {F}_p}) \to \mathrm {PGL}_{n+1}(\overline {\mathbb {F}_p})$ . This element satisfies $\iota ^{\ast }(\tilde {t})=t$ , and we observe that $\gamma ^{\ast }(\tilde {t})=tz$ , where $z:=\zeta I_{n+1} \in \mathrm {Z}(\mathbf {\tilde {G}})$ . So, t is $\gamma ^{\ast }$ -stable, and we conclude that ${}^{g^{-1} \gamma ^{\ast }(g) \gamma ^{\ast }} s={}^{g^{-1} \gamma ^{\ast } g} s=s$ . Consider the element $x:=g^{-1} \gamma ^{\ast }(g)$ . By assumption, there exists some $y \in (\mathbf {G}^{\ast })^{F^{\ast }}$ such that s is $y \gamma ^{\ast }$ -stable. Therefore $x y^{-1} \in \mathrm {C}_{\mathbf {G}^{\ast }}(s)$ . Since $A(s)^{F^{\ast }}=A(s)$ by Lemma 5.2, we deduce that

$$ \begin{align*}F^{\ast}(xy^{-1}) (xy^{-1})^{-1}=F^{\ast}(x)x^{-1} \in \mathrm{C}^{\circ}_{\mathbf{G}^{\ast}}(s)=\mathbf{L}^{\ast}.\end{align*} $$

Thus, by Lang’s theorem, there exists some $l \in \mathrm {C}_{\mathbf {G}^{\ast }}^{\circ }(s)$ such that $lx$ is $F^{\ast }$ -stable. We have $(g^{-1} \gamma ^{\ast } g)^2=1$ , which implies that $(lx \gamma ^{\ast })^2 \in \mathrm {C}^{\circ }_{\mathbf {G}^{\ast }}(s)$ . Recall that $\mathbf {L}^{\ast }={}^{g^{-1}} \mathbf {L}_m^{\ast }$ and that $W^{\circ }(t_m)$ denotes the Weyl group of $\mathrm {C}^{\circ }_{\mathbf {G}^{\ast }}(t_m)$ relative to the maximal torus $\mathbf {T}_0^{\ast }$ . Let $\mathbf {P}_m^{\ast }$ be the standard parabolic subgroup associated with the parabolic subsystem $W^{\circ }(t_m)$ of $W^{\ast }$ with Levi complement $\mathbf {L}_m^{\ast }$ . We define $\mathbf {P}^{\ast }:={}^{g^{-1}} \mathbf {P}_m^{\ast }$ , and we observe that the pair $(\mathbf {L}_m^{\ast },\mathbf {P}_m^{\ast })$ is $\gamma ^{\ast }$ -stable. Consequently, $x \gamma ^{\ast }$ stabilises the pair $(\mathbf {L}^{\ast },\mathbf {P}^{\ast })$ ; and since $l \in \mathbf {L}^{\ast }$ , we observe that $lx \gamma ^{\ast }$ also stabilises this pair. The statement follows now by using duality; see Lemma 6.7.

Proof. By construction, we have $((\mathbf {L}')^{\ast },(\mathbf {P}')^{\ast })={}^{g^{-1}} (\mathbf {L}_{m'},\mathbf {P}_{m'})$ , where g is as in the proof of Lemma 6.8. The graph automorphism $\gamma $ stabilises $((\mathbf {L}_{m'})^{\ast },(\mathbf {P}_{m'})^{\ast })$ , and we have $\mathbf {L}^{\ast } \subseteq (\mathbf {L}')^{\ast }$ . It follows that the element $l g^{-1} \gamma ^{\ast }(g) \gamma ^{\ast }$ , where $l \in (\mathbf {L}^{\ast })^{F^{\ast }}$ is as in the proof of Lemma 6.8, stabilises $((\mathbf {L}')^{\ast },(\mathbf {P}')^{\ast })$ and the $((\mathbf {L}')^{\ast })^{F^{\ast }}$ -conjugacy class of s. This yields the claim.

Proof. We have $t={}^g s$ for some $g\in \mathbf {G}^{\ast }$ . By the construction in Lemma 6.2, the quotient group $\mathrm {C}_{\mathbf {G}^{\ast }}(s)(\mathbf {L}')^{\ast }/(\mathbf {L}')^{\ast }$ is cyclic of prime order. In particular, $n':=g^{-1} F^{\ast }(g) \in \mathrm {C}_{\mathbf {G}^{\ast }}(s)$ , so either $n'$ generates $\mathrm {C}_{\mathbf {G}^{\ast }}(s)(\mathbf {L}')^{\ast }/(\mathbf {L}')^{\ast }$ or $n' \in (\mathbf {L}')^{\ast }$ .

Recall that $\phi $ is a power of $F_p$ or $F_p \gamma $ and $\gamma $ fixes t; see the proof of Lemma 6.8. Consequently, $\phi $ acts as a permutation on the different eigenvalues of $\tilde {t} \in (\mathbf {\tilde {G}}^{\ast })^{F^{\ast }}$ . Therefore, there exists some $z \in W^{\ast }$ of order dividing r such that ${}^{z \phi } t = t$ . We let $z_0 \in \mathrm {N}_{\mathbf {G}^{\ast }}(\mathbf {T}_0^{\ast })$ be the permutation matrix corresponding to z.

We deduce that s is $x \phi ^{\ast }$ -stable, where $x:=g^{-1}z_0 \phi ^{\ast }(g)$ . By assumption, there exists some $y \in (\mathbf {G}^{\ast })^{F^{\ast }}$ such that s is $y \phi ^{\ast }$ -stable. Therefore $x y^{-1} \in \mathrm {C}_{\mathbf {G}^{\ast }}(s)$ . Since $A(s)^{F^{\ast }}=A(s)$ by Lemma 5.2, we deduce that $F^{\ast }(xy^{-1}) (xy^{-1})^{-1}=F^{\ast }(x)x^{-1} \in \mathrm {C}^{\circ }_{\mathbf {G}^{\ast }}(s)$ . Thus, by Lang’s theorem, there exists some $l \in \mathrm {C}_{\mathbf {G}^{\ast }}^{\circ }(s)$ such that $lx$ is $F^{\ast }$ -stable. We have

$$ \begin{align*} (x\phi^{\ast})^r=(g^{-1} z_0 \phi^{\ast} g)^r=g^{-1} (z_0 \phi^{\ast})^r g=g^{-1} F^{\ast}(g) F^{\ast}=n' F^{\ast}. \end{align*} $$

If $n'$ generates $(\mathbf {N}')^{\ast }/(\mathbf {L}')^{\ast }= \mathrm {C}_{(\mathbf {G}^{\ast })^{F^{\ast }}}(s)(\mathbf {L}')^{\ast }/ (\mathbf {L}')^{\ast }$ , then we can conclude by duality that $\mathrm {N}_{G \langle \phi \rangle }(\mathbf {L}',e_s^{L'})/L'$ is cyclic. On the other hand, if $g^{-1} F^{\ast }(g) \in (\mathbf {L}')^{\ast }$ , then there exists $l' \in (\mathbf {L}')^{\ast }$ such that $gl'$ is $F^{\ast }$ -stable. We conclude that $(\mathbf {L}')^{\ast }={}^{gl}(\mathbf {L}_{m'})^{\ast }$ is $(\mathbf {G}^{\ast })^{F^{\ast }}$ -conjugate to $(\mathbf {L}_{m'})^{\ast }$ and therefore maximally split.

Proof. We keep the notation of Lemma 6.10. As in Lemma 6.10, there exists some permutation $z_0 \in \operatorname {N}_{\mathbf {G}^{\ast }}(\mathbf {T}_0^{\ast })$ of order dividing r such that ${}^{z_0 \phi ^{\ast }} t = t$ . The element $\gamma ^{\ast }(z_0)z_0^{-1}$ stabilises t, so $\gamma ^{\ast }(z_0)z_0^{-1} \in \mathbf {N}_{m}^{\ast }$ . By Lemma 6.6, we first see that there exists some $y \in \mathbf {N}^{\ast }_{m}$ such that $yz_0$ is $\langle \phi ^{\ast }, \gamma ^{\ast } \rangle $ -stable. Observe that we still have ${}^{yz_0 \phi ^{\ast }} t=t$ , and therefore $N_{(\phi ^{\ast })^r/\phi ^{\ast }}(yz_0)=(yz_0)^r \in \mathbf {N}^{\ast }_{m} \subset \mathbf {N}_{m'}^{\ast }$ . Furthermore, since $t_{m'}$ is a power of $t_m$ , the element $t_{m'}$ is $yz_0 \phi ^{\ast }$ -stable as well so that $yz_0 \in \mathrm {N}_{\mathbf {G}^{\ast }}(\mathbf {L}^{\ast }_{m'})$ . Note that $ v_{m'}$ corresponds to an $m'$ -cycle under the isomorphism $\mathrm {N}_{\mathbf {G}^{\ast }}(\mathbf {L}^{\ast }_{m'})/\mathbf {L}^{\ast }_{m'} \cong S_{m'}$ . Since $m'$ is a prime number, it follows that if some power of $\sigma \in S_{m'}$ is an $m'$ -cycle, then $\sigma $ itself must be an $m'$ -cycle. Since $(yz_0)^r \in \mathbf {N}^{\ast }_{m'}$ this argument shows that we must have either $yz_0 \in \mathbf {N}^{\ast }_{m'}$ or $(yz_0)^r \in \mathbf {L}^{\ast }_{m'}$ . In the first case, we deduce that $z_0 \in \mathbf {N}^{\ast }_{m'}=\mathbf {L}_{m'}^{\ast } \mathbf {N}_m^{\ast }$ . In particular, there exists some $z_0' \in \mathbf {L}_{m'}$ such that ${}^{z_0' \phi ^{\ast }} t=t$ . In the second case, we define $z_0':=yz_0$ . Consider $x_{\phi }:=g^{-1} z_0' \phi ^{\ast }(g)$ and $x_{\gamma _0}:=g^{-1} \gamma _0^{\ast }(g)$ so that $x_{\phi } \phi ^{\ast }$ and $x_{\gamma _0} \gamma _0^{\ast }$ stabilise the semisimple element s. By the construction of $z_0'$ , we have $N_{(\phi ^{\ast })^r/\phi ^{\ast }}(z_0') \in \mathbf {L}_{m'}^{\ast }$ . Therefore, as in the proof of Lemma 6.10, we deduce that $(x_{\phi } \phi ^{\ast })^r=n' F^{\ast }$ for some element $n'$ generating the quotient group $\mathbf {N}^{\prime \ast }/\mathbf {L}^{\prime \ast }$ . Since $\gamma _0^{\ast }(z_0') z_0^{\prime -1} \in \mathbf {L}_{m'}^{\ast }$ , it follows moreover that

$$ \begin{align*}[x_{\phi} \phi^{\ast},x_{\gamma_0} \gamma_0^{\ast}]={}^g [z_0' \phi^{\ast}, \gamma_0^{\ast}] \in {}^g \mathbf{L}_{m'}=\mathbf{L}'.\end{align*} $$

As in the proof of Lemma 6.10, we find $l_{\phi },l_{\gamma } \in \mathbf {L}^{\ast }$ such that $x^{\prime }_{\phi }:=l_{\phi } x_{\phi }$ and $x_{\gamma _0}':= l_{\gamma _0} x_{\gamma _0}$ are $F^{\ast }$ -stable. We deduce that the morphisms dual to $x^{\prime }_{\phi } \phi ^{\ast }$ and $x^{\prime }_{\gamma _0} \gamma _0^{\ast }$ generate the quotient group $\mathrm {N}_{G \mathcal {B}}(\mathbf {L}',e_s^{L'})/L'$ . The statement therefore follows by duality.