Notation 2.1. Let $X\lhd Y$ be finite groups, and let $\mathbb {T}\subseteq \mathrm {Irr}(X)$ . An extension map with respect to $X\lhd Y$ for $\mathbb {T}$ is a map $\Lambda :\mathbb {T}\longrightarrow \coprod _{X\leq I \leq Y } \mathrm {Irr}(I)$ such that every $\lambda \in \mathbb {T}$ is mapped to an extension of $\lambda $ to ${Y_{\lambda }}$ , the inertia subgroup of $\lambda $ in Y. We say that maximal extendibility holds with respect to $X\lhd Y$ for $\mathbb {T}$ if such an extension map exists (see also [Reference Cabanes and SpäthCS2, Def. 5.7]). In such a case, the map can be chosen Y-equivariant, provided $\mathbb {T}$ is Y-stable (see [Reference Cabanes and SpäthCS2, Th. 4.1]). Whenever $\mathbb {T}=\mathrm {Irr}(X)$ , we omit to mention $\mathbb {T}$ . For $\lambda \in \mathrm {Irr}(X)$ and $\psi \in \mathrm {Irr}(Y)$ , we write ${\lambda ^Y}$ for the character induced to Y and ${\psi \rceil _X}$ for the restricted character. For any generalized character $\kappa $ , we denote by ${\mathrm {Irr}(\kappa )}$ the set of (irreducible) constituents of $\kappa $ . If $\sigma \in \mathrm {Aut}(X)$ and $\lambda \in \mathrm {Irr}(X)$ , we write ${\lambda ^{\sigma }}= \,^{\sigma ^{-1} }\lambda $ for the character with $\,^{\sigma ^{-1}} \lambda (x)= \lambda ^{\sigma }(x)=\lambda (\sigma (x))$ for $x\in X$ .

If two subgroups $H_1,H_2\leq Y$ satisfy $[H_1,H_2]=1$ , and $\lambda _i\in \mathrm {Irr}(H_i)$ for $i=1, 2$ with $\mathrm {Irr}({\left. \lambda _1\right\rceil _{{H_1\cap H_2}}})=\mathrm {Irr}({\left. \lambda _2\right\rceil _{{H_1\cap H_2}}})$ , then there exists a unique character $\phi \in \mathrm {Irr}({\left\langle H_1,H_2 \right\rangle })$ with $\mathrm {Irr}({\left. \phi \right\rceil _{{H_i}}})=\{\lambda _i\}$ according to [Reference Isaacs, Malle and NavarroIMN, §5] and we write ${\lambda _1\cdot \lambda _2}$ for this character. Let $\mathbb I$ be a finite set, and let Z, H, and $H_i$ ( $i\in \mathbb I$ ) be finite groups with $Z\leq H_i\leq H$ . If $[H_i,H_{i'}]=1$ , for every $i,i'\in \mathbb I$ with $i\neq i'$ and $H_i\cap {\left\langle H_{j}\mid j\in \mathbb I\setminus \{i\}\right\rangle } =Z$ , we consider $ {\left\langle H_{i}\mid i\in \mathbb I\right\rangle } \leq H$ the central product of the groups $H_i$ . Given $\nu \in \mathrm {Irr}(Z)$ and $\lambda _i\in \mathrm {Irr}(H_i\mid \nu )$ , we denote by ${\odot _{i \in \mathbb I}\lambda _i}\in \mathrm {Irr}({\left\langle H_{i}\mid i\in \mathbb I\right\rangle } )$ the character $\phi \in \mathrm {Irr}({\left\langle H_{i}\mid i\in \mathbb I\right\rangle })$ with $\mathrm {Irr}({\left. \phi \right\rceil _{{H_i}}})=\{\lambda _i\}$ for every $i\in \mathbb I$ (see also [Reference Isaacs, Malle and NavarroIMN, §5]).

Notation 2.2 (Simple groups of Lie type)

Let ${{\mathbf G}}$ be a simple linear algebraic group of simply connected type over an algebraic closure ${{\mathbb {F}}}$ of ${\mathbb {F}}_p$ for p a prime. Additionally, let $ F:{{\mathbf G}}\rightarrow {{\mathbf G}}$ be a Frobenius endomorphism defining an ${\mathbb {F}}_q$ -structure on ${{\mathbf G}}$ for q, a power of p. The automorphisms of ${{{{\mathbf G}}^F}}$ are restrictions to ${{{{\mathbf G}}^F}}$ of bijective endomorphisms of ${{\mathbf G}}$ commuting to F (see [Reference Gorenstein, Lyons and SolomonGLS, §1.15]), so it makes sense to consider stabilizers $\mathrm {Aut}({{{{\mathbf G}}^F}})_{{\mathbf H}}$ for F-stable subgroups ${{\mathbf H}}\leq {{\mathbf G}}$ . Let ${{\mathbf T}_0}$ be an F-stable maximally split torus, and let ${{\mathbf B}}$ be an F-stable Borel subgroup of ${{\mathbf G}}$ with ${\mathbf T}_0\subseteq {\mathbf B}$ and ${{\mathbf N}_0}:={\mathrm {N}}_{{\mathbf G}}({\mathbf T}_0)$ . According to [Reference MaslowskiMT, Th. 24.11], the group $ G:={{{{\mathbf G}}^F}}$ has a split $BN$ -pair with respect to $ B:={\mathbf B}^F$ , $ {T_0}:={\mathbf T}_0^F$ , and ${N_0}:={\mathbf N}_0^F$ . Let ${E({{{{\mathbf G}}^F}})}$ , often just $ E$ , be the subgroup of $\mathrm {Aut}({{{{\mathbf G}}^F}})_{({\mathbf B},{\mathbf T}_0)}$ generated by the restrictions to ${{{{\mathbf G}}^F}}$ of graph automorphisms and some Frobenius endomorphism ${F_0}$ stabilizing ${\mathbf T}_0$ and ${\mathbf B}$ as in [Reference Gorenstein, Lyons and SolomonGLS, Th. 2.5.1] and [Reference Cabanes and SpäthCS4, §2.A].

Let ${{{\mathbf G}}\leq \widetilde {{\mathbf G}}}$ be a regular embedding, that is, a closed inclusion of algebraic groups with ${\widetilde {{\mathbf G}}}=\operatorname Z( \widetilde {\mathbf G}){{\mathbf G}}$ and connected $\operatorname Z( \widetilde {{\mathbf G}})$ . Then ${\widetilde {\mathbf T}_0}:=\operatorname Z(\widetilde {\mathbf G}){\mathbf T}_0$ is a maximal torus of $\widetilde {{\mathbf G}}$ . Let ${\widetilde {T}_0}:=\widetilde {\mathbf T}_0^F$ . Assume that $F:\widetilde {\mathbf G}\rightarrow \widetilde {\mathbf G}$ is a Frobenius endomorphism extending the one of ${{\mathbf G}}$ (see also [Reference Malle and TestermanMS, §2]). Then $\widetilde {\mathbf G}^F$ has again a split $BN$ -pair with respect to the groups ${\widetilde B}:= \widetilde {T}_0 B$ and ${{\widetilde N}^{\prime }_0}:=\widetilde {T}_0 N_0$ (see [Reference MaslowskiMT, Th. 24.11]). Often the action of $\widetilde N^{\prime }_0$ on ${{\mathbf G}}$ will be studied via the group ${\widetilde N_0}:=\{x\in {\mathrm {N}}_{{\mathbf G}}({\mathbf T}_0)\mid x^{-1} F(x) \in \operatorname Z({{\mathbf G}})\}$ , which will be shown to induce the same automorphisms on ${{\mathbf G}}$ (see Remark 2.16).

Via the convention given in [Reference Malle and TestermanMS, §2], $E({{{{\mathbf G}}^F}})$ also acts on $\widetilde {\mathbf G}^F$ and the semi-direct product $\widetilde {\mathbf G}^F\rtimes E({{{{\mathbf G}}^F}})$ induces on ${{{{\mathbf G}}^F}}$ the whole automorphism group $\mathrm {Aut}({{{{\mathbf G}}^F}})$ .

Condition 2.3 (On stabilizers of irreducible characters of ${{{{\mathbf G}}^F}}$ )

1. A(∞): There exists some E-stable ${{{\widetilde {\mathbf G}}^F}}$ -transversal $\mathbb {T}$ in $\mathrm {Irr}({{{{\mathbf G}}^F}})$ , such that every $\chi \in \mathbb {T}$ extends to ${{{{\mathbf G}}^F}} E_{\chi }$ .

2. A ^′(∞): There exists some E-stable ${{{\widetilde {\mathbf G}}^F}}$ -transversal $\mathbb {T}$ in $\mathrm {Irr}({{{{\mathbf G}}^F}})$ .

Proof. This follows from [Reference CarterCSS, Rem. 3.3].

Notation 2.5. Let $ L$ be a standard Levi subgroup of G with respect to B and $T_0$ , that is, $L={{\mathbf L}}^F$ for some standard Levi subgroup ${{{\mathbf L}}}$ of ${{\mathbf G}}$ such that ${\mathbf T}_0\leq {{\mathbf L}}$ and ${{\mathbf L}} {\mathbf B}$ is an F-stable parabolic subgroup. We set $ N:={\mathrm {N}}_{{\mathbf G}}({{\mathbf L}})^F$ , $ W:=N/L$ , and we abbreviate

$$ \begin{align*}{E_L}={E({{{{\mathbf G}}^F}})}_{{{\mathbf L}}}.\end{align*} $$

We write ${\mathrm {Irr}_{cusp}(L)}$ for the set of cuspidal characters of L as defined in [Reference Cabanes and EnguehardC, 9.1] and ${\mathrm {Irr}_{cusp}(N)}:=\bigcup _{\lambda \in \mathrm {Irr}_{cusp}(L)}\mathrm {Irr}(\lambda ^N)$ . Let us denote by ${\operatorname {R}_L^G}$ the Harish-Chandra induction from L to G. For $\lambda \in \mathrm {Irr}_{cusp}(L)$ , let

$$ \begin{align*}{\mathrm{Irr}(G\mid (L,\lambda))}:=\mathrm{Irr}(\operatorname{R}_L^G(\lambda))\end{align*} $$

(sometimes denoted as ${\mathcal E(G,(L,\lambda ))}$ in the literature). Let also ${\mathrm {Irr}(G\mid (L,\mathbb {T}))}:=\bigcup _{\lambda \in \mathbb {T}}\mathrm {Irr}(G\mid (L,\lambda ))$ for $\mathbb {T}\subseteq \mathrm {Irr}_{cusp}(L)$ .

2.6. Let ${\mathrm {Aut}({{{{\mathbf G}}^F}})_{L,{\mathrm {HC}}}}$ be the subgroup of $\mathrm {Aut}({{{{\mathbf G}}^F}})$ generated by the automorphisms of ${{{{\mathbf G}}^F}}$ induced by N and $\mathrm {Aut}({{{{\mathbf G}}^F}})_{({\mathbf B}{{\mathbf L}},{{\mathbf L}})}$ . Note $E_L\leq \mathrm {Aut}({{{{\mathbf G}}^F}})_{L,{\mathrm {HC}}}$ . According to Howlett–Lehrer theory (see [Reference Cabanes and EnguehardC, §10]), fixing an extension $\widetilde {\lambda }\in \mathrm {Irr}(N_{\lambda })$ of $\lambda \in \mathrm {Irr}_{cusp}(L)$ defines a unique labeling of $\mathrm {Irr}(G\mid (L,\lambda ))$ by $\mathrm {Irr}(W(\lambda ))$ where ${W(\lambda )}:=N_{\lambda }/L$ . We write ${\operatorname {R}_L^G(\lambda )_{\eta }}$ for the character of $\mathrm {Irr}(G\mid (L,\lambda ))$ associated with $\eta \in \mathrm {Irr}(W(\lambda ))$ via the extension $\widetilde {\lambda }$ .

Accordingly, the parametrization of $\mathrm {Irr}(G\mid (L,\mathrm {Irr}_{cusp}(L)))$ depends on an extension map ${\Lambda _L}$ with respect to $L\lhd N$ for $\mathrm {Irr}_{cusp}(L)$ . For $\lambda \in \mathrm {Irr}_{cusp}(L)$ , let ${R(\lambda )}\lhd W(\lambda )$ be defined as in [Reference Cabanes and EnguehardC, Prop. 10.6.3]. If $\lambda \in \mathrm {Irr}_{cusp}(L)$ and $\sigma \in \mathrm {Aut}({{{{\mathbf G}}^F}})_{L,{\mathrm {HC}}}$ , let ${\delta _{\lambda ,\sigma }}$ be the unique linear character of $W(^{\sigma }\lambda )$ satisfying

(2.1) $$ \begin{align} ^{\sigma}\Lambda_L(\lambda)&=\Lambda_L(^{\sigma}\lambda)\delta_{\lambda,\sigma}. \end{align} $$

Notation 2.9. Assume that $\mathbb {T}$ is an $\widehat N$ -stable ${\widetilde {L}'}$ -transversal in $\mathrm {Irr}_{cusp}(L)$ . For each $\lambda \in \mathbb {T}$ , we denote by ${\mathcal O}_{\lambda }$ its N-orbit in $\mathrm {Irr}_{cusp}(L)$ . Note ${\mathcal O}_{\lambda }\subseteq \mathrm {Irr}_{cusp}(L)$ . Let $M(\lambda )\subseteq {\widetilde {L}'}$ be a set of representatives of the ${\widetilde {L}'}_{\lambda }$ -cosets in $\widetilde {L}'$ . We define an extension map $\Lambda _{L}$ on ${\mathcal O}_{\lambda }$ by

$$ \begin{align*} \Lambda_L(\lambda^{\prime m})= \Lambda_{L,\mathbb{T}}(\lambda')^m \quad\quad {\text{ for every }} \lambda'\in {\mathcal O}_{\lambda} {\text{ and }} m\in M(\lambda). \end{align*} $$

Hence, $\Lambda _L$ is defined, but depends on the choice of $M(\lambda )$ . The map $\Lambda _L': \mathrm {Irr}_{cusp}(L)\longrightarrow \coprod _{L\leq I \leq N} \mathrm {Irr}(I)$ with $\Lambda _L'(\mu ):={\left. \Lambda _L(\mu )\right\rceil _{{N_{\widetilde {\mu }}}}}$ for every $\mu \in \mathrm {Irr}_{cusp}(L)$ is well defined, where $\widetilde {\mu }\in \mathrm {Irr}({\widetilde {L}'}_{\mu })$ is an extension of $\mu $ . In contrast to $\Lambda _L$ , we see that $\Lambda _L'$ is independent of the choice of $M(\lambda )$ . Observe that $[N/L, {\widetilde {L}'}/L]=1$ . The map $\Lambda _L'$ is even $\widehat N {\widetilde {L}'}$ -equivariant since $\Lambda _L$ is N-equivariant and $\Lambda _{L,\mathbb {T}}$ is $\widehat N$ -equivariant.

We write ${\mathcal P '(L)}$ for the set of pairs $(\lambda ,\eta )$ with $\lambda \in \mathrm {Irr}_{cusp}(L)$ and $\eta \in \mathrm {Irr}(W(\lambda ))$ . The groups N and W act naturally via conjugation on ${\mathcal P ' (L)}$ . We denote by ${\mathcal P(L)}$ the set of N-orbits in $\mathcal P '(L)$ and by ${\overline {(\lambda ,\eta )}}$ the N-orbit containing $(\lambda ,\eta )$ . Since L is mostly clear from the context, we omit it, writing ${\mathcal P '}$ and ${\mathcal P}$ .

Proof. Clifford theory together with Gallagher’s lemma [Reference IsaacsI, 6.17] proves part (a). The definition of $\delta _{\lambda ,\sigma }$ in Equation (2.1) from 2.6 leads to part (b).

Proof of Theorem 2.8

For the application of Theorem 2.7, we have to ensure that under our assumptions, Equation (2.2) holds for characters $\lambda \in \mathbb {T}$ and $\sigma \in \mathrm {Aut}({{{{\mathbf G}}^F}})_{L,{\mathrm {HC}}}$ . For every $\lambda \in \mathrm {Irr}_{cusp}(L)$ , the character $\Lambda _L(\lambda )$ is an extension of $\Lambda _L'(\lambda )$ . Accordingly, $\delta _{\lambda ,\sigma }$ defined as the unique linear character of $W(^{\sigma } \lambda )$ such that $^{\sigma }\Lambda _L(\lambda )=\Lambda _L(^{\sigma }\lambda )\delta _{\lambda ,\sigma }$ satisfies as well ${\left. {}^{\sigma }\Lambda _L(\lambda )\right\rceil _{{N_{\widetilde {\lambda }^{\sigma } }}}} ={\left. \Lambda _L(^{\sigma }\lambda )\, \, \delta _{\lambda ,\sigma }\right\rceil _{{N_{\widetilde {\lambda } ^{\sigma } }}}}$ . Since $\Lambda _L'(\lambda )$ is ${\widetilde N'} E_L$ -equivariant, we see that ${\left. \delta _{\lambda ,\sigma }\right\rceil _{{N_{\widetilde {\lambda } ^{\sigma } }}}}$ is trivial. Accordingly, $\ker (\delta _{\lambda ,\sigma })\geq N_{^{\sigma } \widetilde {\lambda }}/L$ for every $\lambda \in \mathrm {Irr}_{cusp}(L)$ and $\sigma \in \mathrm {Aut}({{{{\mathbf G}}^F}})_{L,{\mathrm {HC}}}$ where $\widetilde {\lambda }$ denotes an extension of $\lambda $ to ${\widetilde {L}'}_{\lambda }$ . Recall $W(^{\sigma } \widetilde {\lambda })=N_{^{\sigma } \widetilde {\lambda }}/L$ . In combination with the inclusion $R(^{\sigma } \lambda ) \leq W(^{\sigma } \widetilde {\lambda }) $ from [Reference CarterCSS, Lem. 4.14], we obtain the required containment (2.2).

Via Harish-Chandra induction, the map

$$\begin{align*}{\Upsilon'}: \mathcal P \longrightarrow \mathrm{Irr}( G\mid (L,\mathrm{Irr}_{cusp}(L))){\text{ with }} \overline{(\lambda,\eta)}\longmapsto \operatorname{R}_L^G(\lambda)_{\eta}\end{align*}$$

is well defined according to [Reference Malle and TestermanMS, Th. 4.7] and bijective. Hence, $\Upsilon '\circ \Upsilon ^{-1}$ is a bijection between $\mathrm {Irr}_{cusp}(N)$ and $\mathrm {Irr}( G\mid (L,\mathrm {Irr}_{cusp}(L)))$ . Via $\Upsilon $ and $\Upsilon '$ , the group $\mathrm {Aut}({{{{\mathbf G}}^F}})_{L,{\mathrm {HC}}}$ and hence ${\widetilde N'} E_L$ act on $\mathcal P$ . By the description of this action given in Theorem 2.7 and Proposition 2.10, these actions coincide. Hence, $\Upsilon '\circ \Upsilon ^{-1}$ is ${\widetilde N'} E_L$ -equivariant. By Assumption 2.8 (ii), every $\psi _0\in \mathrm {Irr}_{cusp}(N)$ has an ${\widetilde {L}'}$ -conjugate $\psi $ such that $({\widetilde N'}E_L )_{\psi }= {\widetilde N'}_{\psi } (E_L)_{\psi } $ . Hence, every $\chi _0\in \mathrm {Irr}(G\mid (L,\mathrm {Irr}_{cusp}(L)))$ has an $\widetilde N'$ -conjugate $\chi $ with $(\widetilde {\mathbf G}^F E_L)_{\chi }=G(\widetilde N' E_L )_{\chi }=G(\widetilde N^{\prime }_{\chi }) (E_L)_{\chi }=\widetilde {\mathbf G}^F_{\chi } (E_L)_{\chi } $ . This implies the statement (see Lemma 2.4).

Proof. Recall $\psi =\Upsilon (\overline {(\lambda ,\eta )})=(\Lambda _{L,\mathbb {T}}(\lambda )\eta )^N$ . By the assumptions on $\mathbb {T}$ , $(\widehat N {\widetilde {L}'})_{\lambda }=\widehat N_{\lambda } {\widetilde {L}'}_{\lambda }$ for every $\lambda \in \mathbb {T}$ .

Let $\widetilde {\lambda }\in \mathrm {Irr}({\widetilde {L}'}_{\lambda }\mid \lambda )$ and $\eta _0\in \mathrm {Irr}({\left. \eta \right\rceil _{{W(\widetilde {\lambda })}}})$ . According to [Reference Cabanes, Schaeffer Fry and SpäthCE, 15.11], $\widetilde {\lambda }$ is an extension of $\lambda $ . The group ${\widetilde {L}'}_{\lambda }/(L\operatorname Z({{\widetilde G}}))$ acts by multiplication with linear characters of $W(\lambda )/W(\widetilde {\lambda })$ on $\mathrm {Irr}(W(\lambda )\mid \eta _0)$ . Computing the action of $W(\lambda )/W(\widetilde {\lambda })$ on $\mathrm {Irr}(\widetilde {L}^{\prime }_{\lambda }\mid \lambda )$ shows that the action of $\widetilde {L}^{\prime }_{\lambda }/L$ on $\mathrm {Irr}(W(\lambda )\mid \eta _0)$ is transitive. Hence, the characters $\{ (\Lambda _L(\lambda )\eta ')^N \mid \eta '\in \mathrm {Irr}(W(\lambda )\mid \eta _0)\}$ are the ${\widetilde {L}'}_{\lambda }$ -conjugates of $\psi $ .

Let $l\in {\widetilde {L}'}$ and $\widehat n\in \widehat N$ with $\psi ^{l}=(\psi )^{\widehat n}$ . Note that $\psi ^{\widehat n}\in \mathrm {Irr}(N\mid \mathbb {T})$ since $\mathbb {T}$ is $\widehat N$ -stable. Then $\mathrm {Irr}({\left. \psi ^{l}\right\rceil _{{L}}})$ is the N-orbit of $\lambda ^{l}$ . Recall $\mathbb {T}$ is an $\widetilde {L}'$ -transversal. If $l\notin {\widetilde {L}'}_{\lambda }$ , then $\lambda ^{l}\neq \lambda $ and $\lambda ^{l}\notin \mathbb {T}$ , in particular $\psi ^{l}\notin \mathrm {Irr}(N\mid \mathbb {T})$ . This implies $l\in {\widetilde {L}'}_{\lambda }$ and $\psi ^{l}=(\Lambda _L(\lambda )\eta \nu )^N$ for some linear character $\nu $ of $\mathrm {Irr}(W(\lambda )/W(\widetilde {\lambda }))$ . Accordingly, $(\psi )^{\widehat n}\in \mathrm {Irr}(N\mid \lambda )$ and hence $(\psi )^{\widehat n}=(\psi )^{\widehat n'}$ for some $\widehat n'\in \widehat N_{\lambda }$ . Note that

$$ \begin{align*}(\psi)^{\widehat n'}=((\Lambda_L(\lambda)\eta)^{\widehat n'})^N= (\Lambda_L(\lambda) \eta^{\widehat n'})^N.\end{align*} $$

The equality $\psi ^{l}=\psi ^{\widehat n'}$ implies $\eta ^{\widehat n'}=\eta \nu $ and hence $\widehat n' L\in W(\lambda ) \widehat K(\lambda )_{\eta _0}$ . As $\eta $ is $\widehat K(\lambda )_{\eta _0}$ -stable, $\eta ^{\widehat n'}=\eta $ and hence $\psi ^{\widehat n'}=\psi $ . This shows $(\widehat N {\widetilde {L}'} )_{\psi }=\widehat N_{\psi } {\widetilde {L}'}_{\psi }$ .

Proof. By the assumptions, there exists $\mathcal P_1 \subseteq \mathcal P$ such that:

• • if $\overline {(\lambda ,\eta )} \in \mathcal P_1$ , then $\lambda \in \mathbb {T}$ and $\eta $ is $\widehat K(\lambda )_{\eta _0}$ -stable for some $\widetilde {\lambda }\in \mathrm {Irr}(\widetilde {L}^{\prime }_{\lambda }\mid \lambda )$ and $\eta _0\in \mathrm {Irr}({\left. \eta \right\rceil _{{W(\widetilde {\lambda })}}})$ ; and

• • for each $\lambda \in \mathbb {T}$ , $\widetilde {\lambda }\in \mathrm {Irr}(\widetilde {L}^{\prime }_{\lambda }\mid \lambda )$ and $\eta _0\in \mathrm {Irr}(W(\widetilde {\lambda }))$ , there exists some $\eta \in \mathrm {Irr}(W(\lambda )\mid \eta _0)$ with $\overline { (\lambda ,\eta )}\in \mathcal P_1$ .

Proposition 2.11 tells us that the characters $\Upsilon (\mathcal P_1)$ can form part of an $E_L$ -stable ${\widetilde {L}'}$ -transversal.

According to Proposition 2.10, for every $\lambda \in \mathbb {T}$ and $\eta _0\in \mathrm {Irr}(W(\widetilde {\lambda }))$ , the group ${\widetilde {L}'}_{\lambda }$ acts transitively on the set $\mathrm {Irr}(W(\lambda )\mid \eta _0)$ . Since for each $\lambda \in \mathbb {T}$ and $\eta _0\in \mathrm {Irr}(W(\widetilde {\lambda }))$ there exists some $\eta \in \mathrm {Irr}(W(\lambda )\mid \eta _0)$ such that $\overline { (\lambda ,\eta )}\in \mathcal P_1$ , each ${\widetilde {L}'}$ -orbit has a nonempty intersection with $\Upsilon (\mathcal P_1)$ . This implies that every character in $\Upsilon (\mathcal P_1)$ has the property required (see Lemma 2.4).

Proof. We abbreviate $E({{{{\mathbf H}}}^F})$ as E. Let $\chi \in \mathrm {Irr}_{cusp}({{\mathbf H}}^F)$ . According to Lemma 2.4, it is sufficient to prove that $\chi $ has some $\widetilde {{\mathbf H}}^F$ -conjugate $\chi _0$ with $(\widetilde {{\mathbf H}}^F E)_{\chi _0}= \widetilde {{\mathbf H}}^F_{\chi _0} E_{\chi _0}$ . By [Reference Malle and SpäthMal2, Th. 1], there exists a semisimple character $\rho $ of ${{{{\mathbf H}}}^F}$ , such that $\rho $ and $\chi $ have the same stabilizer. By [Reference SpäthS4, Proof of 3.4(c)], the semisimple character $\rho $ has some $(\widetilde {{\mathbf H}})^F$ -conjugate $\rho _0$ with $(\widetilde {{\mathbf H}}^F E)_{\rho _0}= \widetilde {{\mathbf H}}^F_{\rho _0} E_{\rho _0}$ .

Hypothesis 2.14 (Extension of cuspidal characters of $\mathrm {D}_{l',sc}(q)$ )

Let ${{\mathbf H}}$ be a simply connected simple group of type $\mathrm {D}_{l'}$ ( $l'\geq 4$ ), and let $F:{{\mathbf H}}\rightarrow {{\mathbf H}}$ a standard Frobenius endomorphism. Then there exists some $E({{{{\mathbf H}}}^F})$ -stable $\widetilde {{\mathbf H}}^F$ -transversal $\mathbb {T}$ in $\mathrm {Irr}_{cusp}({{\mathbf H}}^F)$ such that every $\chi \in \mathbb {T}$ extends to ${{\mathbf H}}^FE({{{{\mathbf H}}}^F})_{\chi }$ .

Proof. For a finite group H with a split BN-pair of characteristic p, a given $\chi \in \mathrm {Irr} (H)$ is cuspidal if and only if the corresponding representation space has no nonzero fixed point under any O $_p(P)$ for any proper parabolic subgroup P of H. It is then clear that for any $H'\lhd H$ with $p'$ -index, one has $\chi \in \mathrm {Irr}_{cusp} (H)$ if and only if ${\left. \chi \right\rceil _{{H'}}}$ has a cuspidal irreducible component (and then all are). This gives (a) and (c). For (b), note that ${{\mathbf H}}_1\cap {{\mathbf H}}_2$ is a group of semi-simple elements, so that the O $_p(P)$ ’s as above for $H:={{\mathbf L}}_0^F$ are direct products of corresponding subgroups of ${{\mathbf H}}_1^F$ and ${{\mathbf H}}_2^F$ .

Proof. Let $\widetilde {\mathbf G}$ be a group with connected centre, such that there exists a regular embedding ${{\mathbf G}}\rightarrow \widetilde {\mathbf G}$ that is also an ${\mathbb {F}}_q$ -morphism as in 2.2. Then, according to a theorem of Lusztig (see [Reference Cabanes, Schaeffer Fry and SpäthCE, 15.11]), maximal extendibility holds with respect to ${{{{\mathbf G}}^F}}\lhd {{{\widetilde {\mathbf G}}^F}}$ , and $\chi $ has an extension $\widetilde {\chi }$ to ${{{\widetilde {\mathbf G}}^F}}_{\chi }$ . According to the above, ${{{\widetilde {\mathbf G}}^F}}\leq {\widetilde G}. \widetilde Z$ . Clearly, $\widetilde {\chi }$ extends to ${{{\widetilde {\mathbf G}}^F}}_{\chi }\widetilde Z$ since $\widetilde Z$ is abelian and $[\widetilde Z, {{{\widetilde {\mathbf G}}^F}}]=1$ . Now, we see that ${{{\widetilde {\mathbf G}}^F}}_{\chi }\widetilde Z={\widetilde G}_{\chi } \widetilde Z$ and hence $\chi $ extends to ${\widetilde G}_{\chi }$ as well.

Proof. We prove the statement only in the case where ${{\mathbf K}}={{\mathbf G}}$ . The results transfer to a general ${{\mathbf K}}$ as only the quotient groups are relevant to our considerations. Let $\widetilde {\chi }$ be a $\gamma $ -stable extension to ${\widetilde G}_{\chi }$ , then there exists an extension $\widetilde {\nu }\in \mathrm {Irr}(\widetilde Z')$ of $\nu $ such that $\widetilde {\chi }:=({\left. \widetilde {\chi }.\widetilde {\nu }\right\rceil _{{{{{\widetilde {\mathbf G}}^F}}_{\chi }}}})^{{{{\widetilde {\mathbf G}}^F}}}$ . We observe $(\widetilde {\chi }.\widetilde {\nu })^{\gamma }=\widetilde {\chi }.\widetilde {\nu }^{\gamma }$ . This leads to the given formula for $\mu $ in (ii).

For the other direction, let $\chi _0$ be the extension of $\chi $ to $\widetilde {{\mathbf G}}^F_{\chi }$ such that $\widetilde {\chi }=\chi _0^{\widetilde {\mathbf G}^F}$ . Then $\chi _0^{\gamma }=\chi _0{\left. \mu \right\rceil _{{{{{\widetilde {\mathbf G}}^F}}_{\chi }}}}$ and $ \chi _0.\widetilde {\nu }$ is an extension of $\chi $ to $\widetilde {{\mathbf G}}_{\chi }^F\widetilde Z'={\widetilde G}_{\chi } \widetilde Z'$ . The character $\widehat {\chi }:={\left. (\chi _0.\widetilde {\nu })\right\rceil _{{{\widetilde G}_{\chi }}}}$ satisfies

$$ \begin{align*} \widetilde{\chi}^{\gamma}.\widetilde{\nu}^{\gamma}=(\widetilde{\gamma}. \widetilde{\nu})^{\gamma}= (\chi_0.\widetilde{\nu})^{\gamma}= \chi_0^{\gamma}.\widetilde{\nu}^{\gamma}= \chi_0 {\left. \mu\right\rceil_{{\widetilde{{\mathbf G}}^F_{\chi}}}}.\widetilde{\nu}^{\gamma}. \end{align*} $$

There is some $\kappa \in \mathrm {Irr}({\widetilde G}_{\chi }/{{\mathbf G}}^F)$ with $\widetilde {\chi } ^{\gamma }=\widetilde {\chi } \kappa $ . According to [Reference IsaacsI, (6.17)], the above equality of characters implies $\kappa (t)\widetilde {\nu }^{\gamma } (z)(\widetilde {\nu } (z))^{-1}= \mu (tz)$ , whenever $t\in {\widetilde G}_{\chi }$ and $z\in \widetilde Z'$ with $tz\in \widetilde {{\mathbf G}}^F_{\chi }$ . By the assumption on $\mu $ and $\widetilde {\nu }$ , this leads to $\kappa =1$ . Then $\chi $ has a $\gamma $ -stable extension to ${\widetilde G}_{\chi }$ .

Proof. Part (a) follows from [Reference Cabanes and SpäthCS2, Th. 4.1] using Lemma 2.4.

Let $\gamma \in E(\operatorname {SL}_n(q))$ be the graph automorphism. Following the considerations in [Reference Cabanes and SpäthCS2, 3.2], we see that $\gamma '$ and $v_0\gamma $ induce the same automorphism of $\operatorname {SL}_n(q)$ , where $v_0\in \operatorname {SL}_n(p)$ is defined as in [Reference Cabanes and SpäthCS2, 3.2] and p is the prime dividing q. This proves that $\mathbb {T}$ is also $E'(\operatorname {SL}_n(q))$ -stable. For part (b), we have to prove that every $\chi \in \mathbb {T}$ extends to its inertia group in $\operatorname {SL}_n(q)E'(\operatorname {SL}_n(q))$ . This statement is clear whenever $E'(\operatorname {SL}_n(q))_{\chi }$ is cyclic (see [Reference IsaacsI, (9.12)]). If for $\chi \in \mathbb {T}$ the group $E'(\operatorname {SL}_n(q))_{\chi }$ is noncyclic, we see $\gamma '\in E'(\operatorname {SL}_n(q))_{\chi }$ . Let $F_{q'}\in E'(\operatorname {SL}_n(q))$ be a field automorphism such that $E'(\operatorname {SL}_n(q))_{\chi } ={\left\langle F_{q'},\gamma '\right\rangle }$ . By (a), there exists some $\gamma $ -stable extension of $\chi $ to $G{\left\langle F_{q'}\right\rangle }$ . This extension is then also $\gamma '$ and hence $\gamma v_0$ -stable as $[v_0,F_{q'}]=1$ . From this, we deduce that $\chi $ extends to $\operatorname {SL}_n(q)E'(\operatorname {SL}_n(q))_{\chi }$ .

Notation 3.3 (The groups ${{\mathbf G}}$ and ${\overline {\mathbf G}}$ , roots, and generators)

In this and the following section, we assume that the simply connected simple group ${{\mathbf G}}$ from 2.2 is of type $\mathrm {D}_l$ ( $l\geq 4$ ) over ${\mathbb {F}}$ the algebraic closure of ${\mathbb {F}}_p$ for $ p$ some odd prime. Hence, ${{\mathbf G}}\cong \mathrm {D}_{l,\mathrm {sc}}({\mathbb {F}})$ . Denote ${\underline {l}}:=\{1,\dots ,l\}$ . Let $ \Phi :=\{\pm e_i\pm e_j\mid i,j\in {\underline {l}},\, i\neq j \}$ be the root system of ${{\mathbf G}}$ with simple roots ${{\alpha }_2}:=e_2+e_1$ , ${{\alpha }_1}=e_2-e_1$ and ${{\alpha }_i}:=e_i-e_{i-1}$ ( $i\geq 3$ ),

$$ \begin{align*} \Delta:=\{ {\alpha}_{i}\mid i\in \underline {l} \} \end{align*} $$

(see [Reference Gorenstein, Lyons and SolomonGLS, Rem. 1.8.8]), where the set $\{{e_i}\}_{i\in {\underline {l}}}$ is an orthonormal basis of $\mathbb R^l$ whose scalar product is denoted by ${(x,y)}$ . The Chevalley generators ${\mathbf x _{\alpha }(t)}$ , ${\mathbf {n}_{\alpha }(t')}$ and ${{\mathbf h} _{\alpha }(t')}$ ( ${\alpha }\in \Phi $ , $t,t'\in {\mathbb {F}}$ with $t'\neq 0$ ) together with the Chevalley relations describe the group structure of ${{\mathbf G}}$ (see [Reference Gorenstein, Lyons and SolomonGLS, Th. 1.12.1]).

Let ${\overline \Phi }:=\{\pm e_i,\,\pm e_i\pm e_j\mid i,j\in {\underline {l}},\, i\neq j \}$ , ${{\overline {\mathbf G}}}:=\mathrm B_{l,sc}({\mathbb {F}})$ with Chevalley generators $\overline {\mathbf x }_{\alpha }(t)$ , $\overline {\mathbf {n}} _{\alpha }(t')$ and $\overline {\mathbf {h}} _{\alpha }(t')$ ( ${\alpha }\in \overline \Phi $ , $t,t'\in {\mathbb {F}}$ with $t'\neq 0$ ). Assume that the structure constants of ${{\mathbf G}}$ and $\overline {{\mathbf G}}$ are chosen such that $\mathbf x _{\alpha }(t)\mapsto \overline {\mathbf {x}} _{\alpha }(t)$ ( ${\alpha }\in \Phi $ , $t\in {\mathbb {F}}$ ) defines an embedding $\iota _{\mathrm {D}}:{{\mathbf G}}\rightarrow \overline {{\mathbf G}}$ . For simplicity of notation, we write $ \mathbf x _{\alpha }(t)$ , $ \mathbf {n}_{\alpha }(t')=\mathbf x _{\alpha }(t')\mathbf x _{-{\alpha }}(-t'{}^{-1})\mathbf x _{\alpha }(t')$ , and $ {\mathbf h} _{\alpha }(t')=\mathbf {n}_{\alpha }(t')\mathbf {n}_{\alpha }(1)^{-1}$ for the generators of ${\overline {\mathbf G}}$ and thus identify ${{\mathbf G}}$ with the corresponding subgroup of ${\overline {\mathbf G}}$ . This is possible according to [Reference SpäthS2, 10.1] (see also [Reference Malle and TestermanMS, 2.Reference Cabanes and EnguehardC]). Among the relations between Chevalley generators, the following will be the most useful to us. For $a,b\in {\mathbb R}^l\setminus \{0\}$ , recall ${\langle a,b\rangle }= 2(a,b)/(b,b)$ . Let ${\alpha },\beta \in \overline {\Phi }$ , $t\in {\mathbb {F}}$ , $t'\in {\mathbb {F}^{\times }}$ . Then

$$ \begin{align*} {\mathbf h} _{\alpha}(t'){\mathbf h}_{\beta}(t')&={\mathbf h} _{{\alpha}+\beta}(t')\ \ \text{whenever } {\alpha}+\beta\in\overline{\Phi},\\ \mathbf{n}_{\alpha}(t)^{{\mathbf h} _{\beta}(t')}&=\mathbf{n}_{\alpha}(t'{}^{{\left\langle {\alpha},\beta\right\rangle}}t), \\ {\mathbf h} _{\alpha}(t)^{\mathbf{n}_{\beta}(1)}&={\mathbf h} _{{\alpha}-{\left\langle {\alpha},\beta\right\rangle}\beta}(c_{{\alpha},\beta}t), \end{align*} $$

where the first line is from [Reference Gorenstein, Lyons and SolomonGLS, 1.12.1(e)], the second is easy from [Reference Gorenstein, Lyons and SolomonGLS, 1.12.1(g)], and the third, along with the definition of $c_{{\alpha },\beta }\in \{\pm 1\}$ , is from [Reference Gorenstein, Lyons and SolomonGLS, 1.12.1(i)].

Notation 3.5. Let ${{\mathbf L}}$ be a Levi subgroup of ${{\mathbf G}}$ such that ${\mathbf B} {{\mathbf L}}$ is a parabolic subgroup of ${{\mathbf G}}$ and ${\mathbf T}\subseteq {{\mathbf L}}$ . Let $ L:={{\mathbf L}}^F$ , and let ${\Phi '}$ be the root system of ${{\mathbf L}}$ , that is, ${{\mathbf L}}={\mathbf T}{\left\langle \left.{\mathbf X}_ {\alpha }\vphantom {{\alpha }\in \Phi '}\hskip .1em\right.\mid {\alpha }\in \Phi ' \right\rangle }$ . As $\Phi '$ is a parabolic root subsystem of $\Phi $ , it has as basis ${\Delta '}= \Delta \cap \Phi '$ . We assume that one of the following holds:

1. (i) $\Delta '\subseteq \{ e_2-e_1, e_3-e_2, \ldots , e_l-e_{l-1}\}$ , or

2. (ii) $\{ e_2-e_1, e_2+e_1\}\subseteq \Delta '$ .

3.7 (Decomposing  $\Phi '$ )

In the following, we decompose $\Phi '$ into smaller root systems, which are the disjoint union of irreducible root systems of the same type. By ${{\mathsf {type}}(\Psi )}$ , we denote the type of the root system $\Psi $ . Whenever $\Psi $ is a subset of $\overline {\Phi }$ , we also denote by ${W_{\Psi }}$ the subgroup of N $_{\overline {\mathbf G}} ({\mathbf T})/{\mathbf T}$ generated by reflections defined by elements of $\Psi $ .

Since $\Phi '$ is a parabolic root subsystem of $\Phi $ , $\Phi '$ decomposes as a disjoint union of indecomposable root systems of types $\mathrm {D}$ and $\mathrm A$ , that are called components of $\Phi '$ .

If $\Delta '$ satisfies Assumption 3.5 (i), let ${\Phi _{d}}$ be the union of the components of $\Phi '$ of type $\mathrm A_{d-1}$ ( $d\geq 2$ ). If $\Delta '$ satisfies Assumption 3.5 (ii), let ${\Phi _{-1}}$ be the union of components of $\Phi '$ that have a nontrivial intersection with $\{ e_2-e_1, e_1+e_2\}$ and let $\Phi _{d}$ be the union of components of $\Phi ' \setminus \Phi _{-1}$ of type $\mathrm A_{d-1}$ ( $d\geq 2$ ). If $\Delta '$ satisfies Assumption 3.5 (ii), ${\mathsf {type}}(\Phi _{{-1}})\in \{\mathrm A_3,\mathrm A_1\times \mathrm A_1, \mathrm {D}_m \mid m\geq 4\}$ .

Let ${\mathbb D '}$ be the set of integers d such that $\Phi _d$ is defined and nonempty, that is, $\operatorname {SL}_d({\mathbb {F}})$ is a summand of $[{{\mathbf L}},{{\mathbf L}}]$ . Then $\Phi '=\bigsqcup _{d\in \mathbb D'} \Phi _d$ , a disjoint union.

3.8 (Orbits of  $W_{\Phi '}$  on  ${\underline {l}}$ )

Let ${\mathcal O}$ be the set of orbits of $W_{\Phi '}$ on ${\underline {l}}$ , let ${{\mathcal O}_1}\subseteq {\mathcal O}$ be the set of singletons in ${\mathcal O}$ , and let ${{\mathcal O}_d}$ be the set of orbits of $W_{\Phi _d}$ on ${\underline {l}}$ contained in ${\mathcal O}\setminus {\mathcal O}_1$ , whenever $d\in \mathbb D'$ . We define

$$ \begin{align*} {\mathbb D(L)}=\mathbb D&=\begin{cases} \mathbb D'\cup \{1\},& \text{ if }{\mathcal O}_1\neq \emptyset,\\ \mathbb D',&\, \text{otherwise}. \end{cases} \end{align*} $$

For $d\in \mathbb D \setminus \{-1\}$ , let ${a_d}:=|{\mathcal O}_d|$ and note that $|I|=d$ for any $I\in {\mathcal O}_d$ .

For $I\subseteq {\underline {l}}$ , let ${\Phi _I}:=\Phi '\cap {\left\langle \left.e_k\vphantom {k\in I}\hskip .1em\right.\mid k\in I \right\rangle }$ and ${\overline {\Phi }_I}:=\overline \Phi \cap {\left\langle \left.e_k\vphantom {k\in I}\hskip .1em\right.\mid k\in I \right\rangle }$ . For $d\in \mathbb D$ , let ${J_d}:=\bigcup _{o\in {\mathcal O}_d}o$ , and ${\overline {\Phi }_d}:= \overline {\Phi }_{J_d}$ .

Notation 3.9 (Subgroups of ${{\mathbf L}}$ and L)

Let $\varpi \in {\mathbb {F}}^{\times }$ and $h_0$ as in Definition 3.4. Define ${{\mathbf h} _I(t)}:=\prod _{i\in I}{\mathbf h} _{e_i}(t)$ for $I\subseteq {\underline {l}}$ and $t\in {\mathbb {F}}^{\times }$ . For $I\in {\mathcal O}$ , let ${{{\mathbf G}_{I}}}={\left\langle \left.{\mathbf X}_{\alpha }\vphantom {{\alpha } \in \Phi _I}\hskip .1em\right.\mid {\alpha } \in \Phi _I \right\rangle }$ and $ {G_I}:={{\mathbf G}}_I^F$ . Note that for $I\in {\mathcal O}_1$ , the group ${{\mathbf G}_{I}}$ is trivial. Let ${H_0}:={\left\langle h_0,{\mathbf h} _{e_i}(\varpi ){\mathbf h} _{e_{i'}}(-\varpi )\mid i,i'\in {\underline {l}}\right\rangle } ={\left\langle h_{\alpha }(-1)\mid \alpha \in \Phi \right\rangle }$ . For $d\in \mathbb D$ , let ${{\widetilde H}_d}:={\left\langle h_0, {\mathbf h} _I(\varpi ) \mid I\in {\mathcal O}_d\right\rangle }$ , ${H_d}:={\left\langle h_0, {\mathbf h} _I(\varpi ) {\mathbf h} _{I'}(-\varpi )\mid I,I'\in {\mathcal O}_d\right\rangle }$ and

$$ \begin{align*} H:={\left\langle \widetilde H_d\mid d\in \mathbb D\right\rangle}\cap H_0. \end{align*} $$

Proof. An element $t\in {\mathbf T}$ can be written as $\prod _{i=1}^l{\mathbf h} _{e_i}(t_i)$ ( $t_i\in {\mathbb {F}}^{\times }$ ). We have $t\in H_0$ if $t_i\in {\left\langle \varpi \right\rangle }$ and $\prod _{i=1}^l t_i^2=1$ . In particular, ${\mathbf h} _I(\varpi )\in H_0$ if and only if $|I|$ even. This implies $H_d\leq H_0$ whenever $2\mid d$ . On the other hand, $\widetilde H_d\not \leq H_0$ for every $d\in \mathbb D_{\mathrm {odd}}$ .

Proof. We see that $[{\mathbf h} _I(\varpi ), {{\mathbf G}}_I]=1$ by the Chevalley relations and this implies the statement by the definition of H.

3.12 (Structure of  ${{\mathbf L}}$ )

We note that the Levi subgroup ${{\mathbf L}}$ satisfies ${{\mathbf L}}={\mathbf T} {\left\langle {{\mathbf G}}_I\mid I\in {\mathcal O}\right\rangle }$ . Let ${{\mathbf T}_I}:={\left\langle \left.{\mathbf h} _{e_i}(t)\vphantom {i\in I, t \in {\mathbb {F}}^{\times }}\hskip .1em\right.\mid i\in I, t \in {\mathbb {F}}^{\times } \right\rangle }$ for $I\in {\mathcal O}$ . For $I,I'\in {\mathcal O}$ with $I\neq I'$ , we see that no nontrivial linear combination of a root in $\Phi _I$ and one in $\Phi _{I'}$ is a root in $\Phi $ as well. Therefore, $[{{\mathbf G}}_I,{{\mathbf G}}_{I'}]=1$ according to Chevalley’s commutator formula. By the Steinberg relations, we see $[{{\mathbf G}}_I,{\mathbf T}_{I'}]=1$ . The group ${{\mathbf G}}_I$ is either trivial or a simply connected simple group unless $I= {\mathcal O}_{-1}$ and ${\mathsf {type}}(\Phi _{-1})= \mathrm A_1\times \mathrm A_1$ . Accordingly, $[{{\mathbf L}},{{\mathbf L}}]={\left\langle {{\mathbf G}}_I\mid I\in {\mathcal O}\right\rangle }$ .

We observe that ${{\mathbf G}}_I\cap {\mathbf T}\leq {\mathbf T}_I$ and computations with the coroot lattices prove that ${\mathbf T}$ is the central product of the groups ${\mathbf T}_I$ ( $I\in {\mathcal O}$ ) over ${\left\langle h_0\right\rangle }$ . This implies that ${{\mathbf L}}$ is the central product of the groups ${{\mathbf L}}_I$ ( $I\in {\mathcal O}$ ) where ${{{\mathbf L}}_I}:={\mathbf T}_I {{\mathbf G}}_I$ .

Analogously, we see that ${{\mathbf L}}$ is the central product of the groups ${{\mathbf L}}_d$ ( $d\in \mathbb D$ ) over the group ${\left\langle h_0\right\rangle }$ , where ${{{\mathbf L}}_d}:={\left\langle {{\mathbf L}}_I\mid I\in {\mathcal O}_d\right\rangle }$ .

Proof. Recall that ${{\mathbf L}}$ is the central product of the groups ${{\mathbf L}}_I$ , where each ${{\mathbf L}}_I$ is F-stable. Every $x\in {{\mathbf L}}$ can be written as $\prod _ {I \in {\mathcal O}} x_I$ with $x_I \in {{\mathbf L}}_I$ . Clearly, $x\in L$ if and only if $\mathcal L(x)=1$ . We see that $\mathcal L(x)=\prod _{I \in {\mathcal O}}\mathcal L(x_I)$ by the structure of $\mathcal L$ and hence $x\in L$ implies $\mathcal L(x_I)\in {\left\langle h_0\right\rangle }$ . The group $L_0$ is the group of elements $\prod _ {I \in {\mathcal O}} x_I$ with $x_I\in L_I:={{\mathbf L}}_I^F$ . The group $\widehat L:= \mathcal L({\left\langle h_0\right\rangle } ) \cap {{\mathbf L}}$ is the group of elements $\prod x_I$ with $x_I\in {{\mathbf L}}_I$ and $\mathcal L(x_I )\in {\left\langle h_0\right\rangle }$ . Hence, $\widehat L $ is the central product of $\widehat L_I$ ( $I\in {\mathcal O}$ ) over ${\left\langle h_0\right\rangle }$ . Clearly, $ L_0 \lhd \widehat L$ , $\widehat L_I={\left\langle L_I,t_I\right\rangle }$ and $L=L_0 {\left\langle t_I t_{I'}\mid I, I' \in {\mathcal O}\right\rangle }$ . This ensures the parts (a)–(c).

Part (d) follows from the fact that ${{\mathbf L}}/{\left\langle {{\mathbf G}}_I\mid I \in {\mathcal O}\right\rangle }$ is isomorphic to a quotient of ${\mathbf T}$ and hence abelian. For part (e), we observe $\mathcal L({\mathbf h} _Q(\zeta ))= {\mathbf h} _Q(\varpi )$ for every $Q\subseteq {\underline {l}}$ and recall that $\operatorname Z({{\mathbf G}})={\left\langle h_0,{\mathbf h} _{{\underline {l}}}(\varpi )\right\rangle }$ .

Notation 3.14 (Young-like subgroups, ${\mathcal {S}}_M$ and ${\mathcal {Y}}_J$ )

Let M be a set. Given a map $\|. \|: M\rightarrow \mathbb {Z}$ with $m\mapsto \| m\|$ , we define ${\mathcal {S}}_M$ to be the group of bijections $\pi : M\rightarrow M $ with $\|\pi (m)\|=\|m\|$ for every $m\in M$ and we write ${{\mathcal {S}}_{\pm M}}$ for the bijections $\pi : \{ \pm 1\}\times M \rightarrow \{ \pm 1\}\times M$ satisfying $\pi (-1,m)= (-\epsilon , m')$ and $\|m\|=\|m'\|$ , whenever $m, m'\in M$ with $\pi (1,m)=({\epsilon },m')$ . When no map $\|.\|$ is specified, we assume it is a constant map.

In order to denote the elements of ${\mathcal {S}}_M$ and ${\mathcal {S}}_{\pm M}$ , we fix a bijection $f:M \rightarrow \{1,\ldots , |M|\}$ . This induces a canonical embedding $\iota : {\mathcal {S}}_M \rightarrow {\mathcal {S}}_{\underline {{|M|}}}$ and an embedding $\iota _{\pm }: {\mathcal {S}}_{\pm M}\rightarrow {\mathcal {S}}_{\pm \underline {|M|}}$ . For r pairwise distinct elements $m_1, m_2,\ldots , m_r\in M$ , we write $(m_1,m_2,\ldots , m_r)\in {\mathcal {S}}_M$ for the element $\iota ^{-1}(f(m_1), f(m_2),\ldots , f(m_r))$ . Via $\iota _{\pm }$ , we obtain also a cycle notation for elements of ${\mathcal {S}}_{\pm M}$ .

If J is a partition of M, we write $J\vdash M$ for short. For $J\vdash M$ , we set

$$ \begin{align*} {{\mathcal{Y}}_J}&:=\{ \pi \in{\mathcal{S}}_M\mid \pi(J')=J' \text{ for every }J'\in J \}, {\text{ and }} \\ {{\mathcal{Y}}_{\pm J}}&:=\{ \pi \in{\mathcal{S}}_{\pm M}\mid \pi(\{\pm 1\}\times J')= \{\pm 1\}\times J' \text{ for every }J'\in J \}. \end{align*} $$

Let $M_{odd}:=\{ m\in M \mid \| m\| \text { odd }\}$ and

$$ \begin{align*} {{\mathcal{S}}_{\pm M}^{\mathrm{D}}} = \left \{ \pi\in {\mathcal{S}}_{\pm M} \mid |(\{1\}\times M_{{\mathrm {odd}}})\cap \pi^{-1}(\{-1\}\times M_{\mathrm {odd}})| \text{ is even} \right \}. \end{align*} $$

Proof. According to the considerations in [Reference Cabanes and EnguehardC, 9.2], $\rho _{{\mathbf T}}(N\cap N_0)/\rho _{{\mathbf T}}(L\cap N_0)\cong \operatorname {N}_{\overline W_0} (W_{\Phi '})/W_{\Phi '}$ , where ${\overline W_0}:=\overline N_0/\overline T_0$ . We then make routine considerations inside $\overline W_0$ (see, e.g., [Reference HowlettH]). Note that $\operatorname {N}_{\overline W_0} (W_{\Phi '})=\operatorname {Stab}_{\overline W_0}(\Phi ')=W_{\Phi '}\operatorname {Stab}_{\overline W_0}(\Delta ')$ .

From the definition of $\Phi _{-1}$ , one can check that $\operatorname {Stab}_{\overline W_0}(\Delta ')$ stabilizes $\Phi _{-1}\cap \Delta '$ . This implies that $\operatorname {Stab}_{\overline W_0}(\Delta ')$ stabilizes $\Phi _d\cap \Delta '$ for every $d\in \mathbb D$ , and

$$\begin{align*}\operatorname{Stab}_{W_{\overline{\Phi}_d}}(\Phi_d\cap \Delta')= {\mathcal{S}}_{\pm {\mathcal O}_d}.\end{align*}$$

We have $\operatorname {N}_{\overline W_0} (W_{\Phi '})=W_{\overline {\Phi }_1}\times \prod _{d\in \mathbb D}\operatorname {Stab}_{W_{\overline \Phi _{d}}}(\Phi _{d})= \operatorname {Stab}_{W_{\overline \Phi _{-1}}}(\Phi _{-1})\times W_{\overline \Phi _1}\times \prod _{\stackrel {d\in \mathbb D}{d>1}} \operatorname {Stab}_{W_{\overline \Phi _d}} (\Phi _d)$ with

$$ \begin{align*} \operatorname{Stab}_{W_{\overline \Phi_{-1}}}(\Phi_{-1})= W_{\Phi_{-1}} {\left\langle (1,-1)\right\rangle} =W_{\overline\Phi_{{-1}}}, \end{align*} $$

and

$$ \begin{align*}\operatorname{Stab}_{W_{\overline{\Phi}_d}}(\Phi_d)= W_{ \Phi_d} \rtimes {\mathcal{S}}_{\pm {\mathcal O}_d}\end{align*} $$

for $d\in \mathbb D$ with $d>1$ . Hence, $\rho _{{\mathbf T}}( N \cap N_0)/\rho _{{\mathbf T}}(L\cap N_0)\cong {\mathcal {S}}_{\pm {\mathcal O}}^{\mathrm {D}}$ .

Notation 3.17. For $d\in \mathbb D\setminus \{-1\}$ , we fix orderings on ${\mathcal O}_d$ and the sets $I\in {\mathcal O}_d$ : we write ${I_{d,j}}$ ( $ j \in \underline { a_d}$ ) for the sets in ${\mathcal O}_d $ and ${I_{d,j}(k)}\in I_{d,j}$ ( $ j \in \underline { a_d}$ , $k \in \underline d$ ) for the elements of $I_{d,j}$ .

For each $ k\in \underline d$ , let $f_k^{(d)}: {\underline {l}}\longrightarrow {\underline {l}}$ be a bijection such that $f_k^{(d)}(j)=I_{d,j}(k)$ for every $j\in \underline {a_d}$ and $f_k^{(d)}$ has the maximal number of fixed points. Then ${f_k^{(d)}}$ defines an element of ${\mathcal {S}}_{\pm {\underline {l}}}$ without sign changes, that we also denote by $f_k^{(d)}$ by abuse of notation.

Proof. For (a), we observe that the sets $\bigcup _{j\in {\underline {a_d}} }I_{d,j}(k)$ ( $k\in \underline d)$ form a partition of $J_d$ . This implies that the groups ${\mathcal {S}}_{\pm \underline {a_d}}^{f_k^{(d)}}$ and ${\mathcal {S}}_{\pm \underline {a_d}}^{f_{k'}^{(d)}}$ commute and are disjoint. We see that $\overline {\kappa }_d({\mathcal {S}}_{\pm \underline {a_d}})$ stabilizes ${\mathcal O}_d$ . This proves (a). Part (b) is clear from the definitions.

Notation 3.19 (The groups $\overline V_0$ , $\overline V_I$ , and $V_I$ )

The group ${V^{\prime }_{Weyl}}:={\left\langle \left.\overline {\mathbf {n}}_i'\vphantom { i\in {\underline {l}}}\hskip .1em\right.\mid i\in {\underline {l}} \right\rangle }$ with $\overline {\mathbf {n}}_1':=\mathbf {n}_{e_1}(1)$ and $\overline {\mathbf {n}}_i':=\mathbf {n}_{{\alpha }_i}(-1)$ , where ${\alpha }_i= e_i-e_{i-1}$ ( $2\leq i\leq l$ ) is known as the extended Weyl group of type $\mathrm B_l$ .