Proof. Define the groups $\mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{{\mathbf {L}},{\mathbf {K}}}:=\mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{\mathbf {L}}\cap \mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{\mathbf {K}}$ and $\mathrm { Aut}_{\mathbb {F}}({\mathbf {G}}^{*F^*})_{{\mathbf {L}}^*,{\mathbf {K}}^*}:=\mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^{*F^*})_{{\mathbf {L}}^*}\cap \mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^{*F^*})_{{\mathbf {K}}^*}$ . If ${\mathbf {L}}={\mathbf {K}}$ the result follows from [Reference Cabanes and SpäthCS13, Proposition 2.2] while a similar argument applies in the general case.

Notation 2.3. Let ${\mathbf {G}}$ be a connected reductive linear algebraic group defined over an algebraic closure of a finite field of characteristic p and $F:{\mathbf {G}}\to {\mathbf {G}}$ a Frobenius endomorphism defining an $\mathbb {F}_q$ -structure on ${\mathbf {G}}$ , for a power q of p. Consider a prime $\ell $ different from p and denote by e the multiplicative order of q modulo $\ell $ (modulo $4$ if $\ell =2$ ). All blocks are considered with respect to the prime $\ell $ .

Hypothesis 2.4. Let ${\mathbf {G}}$ , $F:{\mathbf {G}}\to {\mathbf {G}}$ , q, $\ell $ and e be as in Notation 2.3. Assume that:

1. (i) $\ell \in \Gamma ({\mathbf {G}},F)$ with $\ell \geq 5$ and the Mackey formula holds for $({\mathbf {G}},F)$ ;

2. (ii) If ${\mathbf {K}}$ is an F-stable Levi subgroup of ${\mathbf {G}}$ , then

   $$\begin{align*}\left\lbrace\kappa\in\mathrm{Irr}\left({\mathbf{K}}^F\right)\enspace\middle|\enspace({\mathbf{L}},\lambda)\ll_e({\mathbf{K}},\kappa)\right\rbrace=\mathrm{Irr}\left({\mathbf{R}}_{\mathbf{L}}^{\mathbf{K}}(\lambda)\right)\end{align*}$$
   for every $(e,\ell ')$ -cuspidal pair $({\mathbf {L}},\lambda )$ of ${\mathbf {K}}$ .

Parametrisation 2.7. Let ${\mathbf {G}}$ , $F:{\mathbf {G}}\to {\mathbf {G}}$ , q, $\ell $ and e be as in Notation 2.3 and consider an e-cuspidal pair $({\mathbf {L}},\lambda )$ of ${\mathbf {G}}$ . Then there exists an $\ell $ -defect preserving $\mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -equivariant bijection

$$\begin{align*}\Omega^{\mathbf{G}}_{({\mathbf{L}},\lambda)}:{\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right)\to\mathrm{Irr}\left({\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\enspace\middle|\enspace \lambda\right) \end{align*}$$

such that

for every $\vartheta \in {\mathcal {E}}\left ({\mathbf {G}}^F,({\mathbf {L}},\lambda )\right )$ and where $X:={\mathbf {G}}^F\rtimes \mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ .

Hypothesis 3.1. Let ${\mathbf {G}}$ , $F:{\mathbf {G}}\to {\mathbf {G}}$ , q, $\ell $ and e be as in Notation 2.3 and suppose that $[{\mathbf {G}},{\mathbf {G}}]$ is simple and not of type $\mathbf {E}_6$ , $\mathbf {E}_7$ or $\mathbf {E}_8$ .

Proof. The required bijections are constructed in [Reference Digne and MichelDM90, Theorem 7.1] and satisfy (iv) by [Reference Digne and MichelDM90, Theorem 7.1 (iii)]. The properties (i) and (ii) follow from [Reference Cabanes and SpäthCS13, Theorem 3.1] and [Reference Geck and MalleGM20, Theorem 4.7.2 and Theorem 4.7.5] respectively. For (iii) see, for instance, the description given in [Reference MalleMal07, (2.1)].

Proof. The first statement follows from the proof of [Reference Cabanes and SpäthCS13, Corollary 3.3]. The second statement follows from Theorem 3.2 (iv).

Proof. This follows from the proof of [Reference Cabanes and SpäthCS13, Theorem 3.4] applied to arbitrary e-split Levi subgroups (see the comment in the proof of [Reference Brough and SpäthBS20, Proposition 5.5]). Regarding the statement on character degrees, see [Reference MalleMal07, Theorem 4.2] and the argument used to prove [Reference Brough and SpäthBS20, Lemma 5.3].

Proof. First, we observe that the map $I^{{\mathbf {K}}}_{({\mathbf {L}},\lambda )}$ is a bijection because of Theorem 3.2 (ii), in fact

$$ \begin{align*} I^{{\mathbf{K}}}_{({\mathbf{L}},\lambda)}\left(\mathrm{Irr}\left(W_{{\mathbf{K}}}\left({\mathbf{L}},\lambda\right)^F\right)\right)&=J_{{\mathbf{K}},s}^{-1}\left(\mathrm{ Irr}\left({\mathbf{R}}^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}_{{\mathbf{C}}_{{\mathbf{L}}^*}(s)}\left(J_{{\mathbf{L}},s}\left(\lambda\right)\right)\right)\right) \\ &=\mathrm{Irr}\left(J_{{\mathbf{K}},s}^{-1}\circ {\mathbf{R}}^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}_{{\mathbf{C}}_{{\mathbf{L}}^*}(s)}\circ J_{{\mathbf{L}},s}\left(\lambda\right)\right) \\ &=\mathrm{Irr}\left({\mathbf{R}}^{{\mathbf{K}}}_{{\mathbf{L}}}\left(\lambda\right)\right). \end{align*} $$

To show that the bijection is equivariant, let $\sigma \in \mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{{\mathbf {K}},{\mathbf {L}}}$ and consider $\sigma ^*\in \mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^{*F^*})_{{\mathbf {K}}^*,{\mathbf {L}}^*}$ with $\sigma \sim \sigma ^*$ (see Lemma 2.2). If $\sigma \in \mathrm { Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{{\mathbf {K}},({\mathbf {L}},\lambda )}$ , then $\sigma ^*$ stabilises the ${\mathbf {L}}^{*F^*}$ -orbit of s. Without loss of generality, we may assume that $\sigma ^*(s)=s$ . Then Theorem 3.2 (i) implies that $\sigma ^*$ stabilises $J_{{\mathbf {L}},s}(\lambda )$ . Applying Theorem 3.2 (i) and the equivariance properties of Corollary 3.3 and Theorem 3.4, we conclude that

$$ \begin{align*} I^{{\mathbf{K}}}_{({\mathbf{L}},\lambda)}\left(\eta\right)^\sigma &=J^{-1}_{{\mathbf{K}},s}\left(I^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))}\left(\eta\hspace{2pt}^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda}}\right)\right)^\sigma \\ &=J^{-1}_{{\mathbf{K}},s}\left(I^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))}\left(\left(\eta\hspace{2pt}^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda}}\right)^{\sigma^*}\right)\right) \\ &=J^{-1}_{{\mathbf{K}},s}\left(I^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))}\left(\left(\eta\hspace{2pt}^\sigma\right)^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda}}\right)\right) \\ &=I^{{\mathbf{K}}}_{({\mathbf{L}},\lambda)}\left(\eta\hspace{2pt}^\sigma\right) \end{align*} $$

for every $\eta \in \mathrm {Irr}(W_{{\mathbf {K}}}({\mathbf {L}},\lambda )^F)$ .

Proof. By the condition on character degrees given in Theorem 3.4 together with Theorem 3.2 (iii), we deduce that

$$ \begin{align*} \eta(1)_\ell &=\left(\eta\right)^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda}}(1)_\ell=\dfrac{I^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))}\left(\left(\eta\right)^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda}}\right)(1)_\ell}{J_{{\mathbf{L}},s}\left(\lambda\right)(1)_\ell\cdot \left|{\mathbf{C}}_{{\mathbf{K}}^*}(s)^{F^*}:{\mathbf{N}}_{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))^{F^*}\right|{}_\ell} \\ &=\dfrac{I^{{\mathbf{K}}}_{({\mathbf{L}},\lambda)}\left(\eta\right)(1)_\ell\cdot \left|{\mathbf{C}}_{{\mathbf{K}}^*}(s)^{F^*}\right|{}_\ell\cdot \left|{\mathbf{L}}^F\right|{}_\ell}{\lambda(1)_\ell\cdot |{\mathbf{C}}_{{\mathbf{L}}^*}(s)^{F^*}|_\ell\cdot |{\mathbf{K}}^F|_\ell\cdot \left|{\mathbf{C}}_{{\mathbf{K}}^*}(s)^{F^*}:{\mathbf{N}}_{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))^{F^*}\right|{}_\ell} \\ &=\dfrac{I^{{\mathbf{K}}}_{({\mathbf{L}},\lambda)}\left(\eta\right)(1)_\ell}{\lambda(1)_\ell\cdot \left|{\mathbf{K}}^F:{\mathbf{N}}_{{\mathbf{K}}}({\mathbf{L}},\lambda)^F\right|{}_\ell} \end{align*} $$

for every $\eta \in \mathrm {Irr}(W_{{\mathbf {K}}}({\mathbf {L}},\lambda )^F)$ . The result follows immediately from the above equality.

Proof. According to [Reference BonnaféBon06, Proposition 12.1] the character $\lambda \cdot \widehat {z}_{{\mathbf {L}}}$ is e-cuspidal, while Corollary 3.3 shows that $W_{{\mathbf {K}}}({\mathbf {L}},\lambda )^F=W_{{\mathbf {K}}}({\mathbf {L}},\lambda \cdot \widehat {z}_{{\mathbf {L}}})^F$ and that $i^{{\mathbf {K}}}_{{\mathbf {L}},\lambda }=i^{{\mathbf {K}}}_{{\mathbf {L}},\lambda \cdot \widehat {z}_{{\mathbf {L}}}}$ . Using Theorem 3.2 (iv) we obtain

$$\begin{align*}I_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))}^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}=I_{({\mathbf{C}}_{{\mathbf{L}}^*}(sz),J_{{\mathbf{L}},sz}\left(\lambda\cdot \widehat{z}_{{\mathbf{L}}}\right))}^{{\mathbf{C}}_{{\mathbf{K}}^*}(sz)}\end{align*}$$

and

$$ \begin{align*} I^{{\mathbf{K}}}_{({\mathbf{L}},\lambda)}\left(\eta\right)\cdot \widehat{z}_{{\mathbf{K}}}&=J_{{\mathbf{K}},s}^{-1}\left(I_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}\left(\lambda\right))}^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}\left(\left(\eta\right)^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda}}\right)\right)\cdot \widehat{z}_{{\mathbf{K}}} \\ &=J_{{\mathbf{K}},s}^{-1}\left(I_{({\mathbf{C}}_{{\mathbf{L}}^*}(sz),J_{{\mathbf{L}},sz}\left(\lambda\cdot \widehat{z}_{{\mathbf{L}}}\right))}^{{\mathbf{C}}_{{\mathbf{K}}^*}(sz)}\left(\left(\eta\right)^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda\cdot \widehat{z}_{{\mathbf{L}}}}}\right)\right)\cdot \widehat{z}_{{\mathbf{K}}} \\ &=J_{{\mathbf{K}},sz}^{-1}\left(I_{({\mathbf{C}}_{{\mathbf{L}}^*}(sz),J_{{\mathbf{L}},sz}\left(\lambda\cdot \widehat{z}_{{\mathbf{L}}}\right))}^{{\mathbf{C}}_{{\mathbf{K}}^*}(sz)}\left(\left(\eta\right)^{i^{{\mathbf{K}}}_{{\mathbf{L}},\lambda\cdot \widehat{z}_{{\mathbf{L}}}}}\right)\right) \\ &=I^{{\mathbf{K}}}_{({\mathbf{L}},\lambda\cdot\widehat{z}_{{\mathbf{L}}})}\left(\eta\right) \end{align*} $$

for every $\eta \in \mathrm {Irr}(W_{{\mathbf {K}}}({\mathbf {L}},\lambda )^F)$ .

Proof. As explained above, the maps are constructed as in (3.1) by replacing the maps $I_{({\mathbf {C}}_{{\mathbf {L}}^*}(s),J_{{\mathbf {L}},s}\left (\lambda \right ))}^{{\mathbf {C}}_{{\mathbf {K}}^*}(s)}$ given by Theorem 3.4 with those given by [Reference Broué, Malle and MichelBMM93, Theorem 3.2 (2)]. Consider ${\mathbf {K}}$ and $({\mathbf {L}},\lambda )$ as above and fix $\eta \in \mathrm {Irr}(W_{\mathbf {K}}({\mathbf {L}},\lambda )^F)$ . Since $W_{\mathbf {K}}({\mathbf {L}},\lambda )^F\leq W_{\mathbf {G}}({\mathbf {L}},\lambda )^F$ , the construction given in the proof of [Reference Cabanes and SpäthCS13, Corollary 3.3] shows that the map $i^{\mathbf {K}}_{{\mathbf {L}},\lambda }$ given by Corollary 3.3 coincides with the restriction of $i^{\mathbf {G}}_{{\mathbf {L}},\lambda }$ to $W_{\mathbf {K}}({\mathbf {L}},\lambda )^F$ . In particular, if we write $\rho ^{i^{\mathbf {K}}_{{\mathbf {L}},\lambda }}$ to denote the element of $\mathbb {Z}\mathrm { Irr}(W_{{\mathbf {C}}_{{\mathbf {K}}^*}(s)}({\mathbf {C}}_{{\mathbf {L}}^*}(s),J_{{\mathbf {L}},s}(\lambda ))^{F^*})$ corresponding to $\rho \in \mathbb {Z}\mathrm {Irr}(W_{\mathbf {K}}({\mathbf {L}},\lambda )^F)$ via the isomorphism $i^{{\mathbf {K}}}_{{\mathbf {L}},\lambda }$ , it follows by elementary character theory that

(3.3) $$ \begin{align} \left(\mathrm{Ind}^{W_{\mathbf{G}}({\mathbf{L}},\lambda)^F}_{W_{\mathbf{K}}({\mathbf{L}},\lambda)^F}(\eta)\right)^{i^{\mathbf{G}}_{{\mathbf{L}},\lambda}}=\mathrm{ Ind}^{W_{{\mathbf{C}}_{{\mathbf{G}}^*}(s)}({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))^{F^*}}_{W_{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))^{F^*}}\left(\eta^{i^{\mathbf{K}}_{{\mathbf{L}},\lambda}}\right). \end{align} $$

By the definition given in (3.1) and applying Theorem 3.2 (ii) and [Reference Broué, Malle and MichelBMM93, Theorem 3.2 (2.a)] we conclude that

$$ \begin{align*} {\mathbf{R}}^{\mathbf{G}}_{\mathbf{K}}\circ\mathcal{I}^{\mathbf{K}}_{({\mathbf{L}},\lambda)}(\eta)&={\mathbf{R}}_{\mathbf{K}}^{\mathbf{G}}\left(J^{-1}_{{\mathbf{K}},s}\left(\mathcal{I}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))}^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}\left(\eta^{i^{\mathbf{K}}_{{\mathbf{L}},\lambda}}\right)\right)\right) \\ &=J^{-1}_{{\mathbf{G}},s}\circ{\mathbf{R}}_{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}^{{\mathbf{C}}_{{\mathbf{G}}^*}(s)}\left(\mathcal{I}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))}^{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}\left(\eta^{i^{\mathbf{K}}_{{\mathbf{L}},\lambda}}\right)\right) \\ &=J^{-1}_{{\mathbf{G}},s}\circ\mathcal{I}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))}^{{\mathbf{C}}_{{\mathbf{G}}^*}(s)}\left(\mathrm{ Ind}^{W_{{\mathbf{C}}_{{\mathbf{G}}^*}(s)}({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))^{F^*}}_{W_{{\mathbf{C}}_{{\mathbf{K}}^*}(s)}({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))^{F^*}}\left(\eta^{i^{\mathbf{K}}_{{\mathbf{L}},\lambda}}\right)\right) \\ &=J^{-1}_{{\mathbf{G}},s}\circ\mathcal{I}_{({\mathbf{C}}_{{\mathbf{L}}^*}(s),J_{{\mathbf{L}},s}(\lambda))}^{{\mathbf{C}}_{{\mathbf{G}}^*}(s)}\left(\left(\mathrm{ Ind}^{W_{\mathbf{G}}({\mathbf{L}},\lambda)^F}_{W_{\mathbf{K}}({\mathbf{L}},\lambda)^F}(\eta)\right)^{i^{\mathbf{G}}_{{\mathbf{L}},\lambda}}\right) \\ &=\mathcal{I}^{\mathbf{G}}_{({\mathbf{L}},\lambda)}\circ\mathrm{Ind}^{W_{\mathbf{G}}({\mathbf{L}},\lambda)^F}_{W_{\mathbf{K}}({\mathbf{L}},\lambda)^F}(\eta) \end{align*} $$

where the penultimate equality holds because of (3.3). This proves (i).

The other properties follow by a similar argument. First, (ii) follows from [Reference Broué, Malle and MichelBMM93, Theorem 3.2 (2.b)] together with Theorem 3.2 (i) and recalling the compatibility with automorphisms obtained in Corollary 3.3. Secondly, to prove (iii) we observe that the trivial character of $W_{\mathbf {L}}({\mathbf {L}},\lambda )^F$ maps to the trivial character of $W_{{\mathbf {C}}_{{\mathbf {L}}^*}(s)}({\mathbf {C}}_{{\mathbf {L}}^*}(s),J_{{\mathbf {L}},s}(\lambda ))^F$ via the isomorphism $i^{\mathbf {L}}_{{\mathbf {L}},\lambda }$ , while the character $J_{{\mathbf {L}},s}(\lambda )$ is mapped to $\lambda $ via $J_{{\mathbf {L}},s}^{-1}$ . Then (iii) follows from [Reference Broué, Malle and MichelBMM93, Theorem 3.2 (2.c)].

Proof. This follows immediately from Lemma 3.6 applied to $\widetilde {{\mathbf {G}}}$ after noticing that induction of characters preserves the defect (this follows from the degree formula for induced characters).

Proof. First, by Clifford theory we deduce that

(3.5) $$ \begin{align} \mathrm{Irr}\left(\left(\left(\widetilde{\Lambda}\left(\widetilde{\lambda}\right)\cdot \widetilde{\eta}\right)^{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F}\right)_{{\mathbf{Z}}\left(\widetilde{{\mathbf{G}}}^F\right)}\right)=\mathrm{ Irr}\left(\widetilde{\lambda}_{{\mathbf{Z}}(\widetilde{{\mathbf{G}}}^F)}\right). \end{align} $$

On the other hand, by using the character formula [Reference Digne and MichelDM91, Proposition 12.2 (i)], we obtain

$$\begin{align*}{\mathbf{R}}^{\widetilde{{\mathbf{K}}}}_{\widetilde{{\mathbf{L}}}}(\widetilde{\lambda})_{{\mathbf{Z}}\left(\widetilde{{\mathbf{K}}}^F\right)}={\mathbf{R}}^{\widetilde{{\mathbf{K}}}}_{\widetilde{{\mathbf{L}}}}(\widetilde{\lambda})(1)\cdot \widetilde{\lambda}_{{\mathbf{Z}}\left(\widetilde{{\mathbf{K}}}^F\right)}\end{align*}$$

and hence

(3.6) $$ \begin{align} \mathrm{Irr}\left(I^{\widetilde{{\mathbf{K}}}}_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda})}\left(\widetilde{\eta}\right)_{{\mathbf{Z}}(\widetilde{{\mathbf{K}}}^F)}\right)=\mathrm{ Irr}\left(\widetilde{\lambda}_{{\mathbf{Z}}(\widetilde{{\mathbf{K}}}^F)}\right). \end{align} $$

Now the result follows by combining (3.5) with (3.6).

Proof. Notice that Hypothesis 2.4 holds for $\widetilde {{\mathbf {G}}}$ under our assumptions and so $\mathrm {bl}(\widetilde {\lambda })^{\widetilde {{\mathbf {K}}}^F}=\mathrm {bl}(\widetilde {\chi })$ by [Reference RossiRos24a, Proposition 4.8] and $\mathrm {bl}(\Upsilon _{(\widetilde {{\mathbf {L}}},\widetilde {\lambda })}^{\widetilde {{\mathbf {K}}}}(\widetilde {\chi }))=\mathrm { bl}(\widetilde {\lambda })^{{\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F}$ by [Reference RossiRos24a, Lemma 5.5]. Then the result follows from the transitivity of block induction.

Proof. Let $(x,z)\in ((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},{\mathbf {L}}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ . Since $\widetilde {\lambda }=\widetilde {\lambda }^x\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ , we have

$$\begin{align*}{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}},\widetilde{\lambda})^F={\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}},\widetilde{\lambda}^x\cdot \widehat{z}_{\widetilde{{\mathbf{L}}}})^F={\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}},\widetilde{\lambda}^x)^F.\end{align*}$$

By using the equivariance properties of $\widetilde {\Lambda }$ , we obtain

(3.7) $$ \begin{align} \left(\left(\widetilde{\Lambda}\left(\widetilde{\lambda}\right)\cdot \widetilde{\eta}\right)^{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F}\right)^{(x,z)}&=\left(\left(\widetilde{\Lambda}\left(\widetilde{\lambda}\right)\cdot \widetilde{\eta}\right)^x\right)^{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F}\cdot \widehat{z}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})} \nonumber \\ &=\left(\left(\widetilde{\Lambda}\left(\widetilde{\lambda}\right)\cdot \widetilde{\eta}\right)^x\cdot \widehat{z}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}},\widetilde{\lambda})^F}\right)^{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F} \\ &=\left(\widetilde{\Lambda}\left(\widetilde{\lambda}^x\cdot \widehat{z}_{\widetilde{{\mathbf{L}}}} \right)\cdot \widetilde{\eta}^x\right)^{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F} \nonumber \\ &=\left(\widetilde{\Lambda}\left(\widetilde{\lambda}\right)\cdot \widetilde{\eta}^x\right)^{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F}. \nonumber \end{align} $$

On the other hand, considering Lemma 3.5 and Lemma 3.7 with respect to $\widetilde {{\mathbf {G}}}$ it follows that

(3.8) $$ \begin{align} I^{\widetilde{{\mathbf{K}}}}_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda})}\left(\widetilde{\eta}\right)^{(x,z)}&=I^{\widetilde{{\mathbf{K}}}}_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda})}\left(\widetilde{\eta}\right)^x\cdot \widehat{z}_{\widetilde{{\mathbf{K}}}}\nonumber \\ &=I^{\widetilde{{\mathbf{K}}}}_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda}^x\cdot \widehat{z}_{\widetilde{{\mathbf{L}}}})}\left(\widetilde{\eta}^x\right) \\ &=I^{\widetilde{{\mathbf{K}}}}_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda})}\left(\widetilde{\eta}^x\right). \nonumber \end{align} $$

Now, the result follows immediately from (3.7) and (3.8).

Proof. First, assume ${\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }_1))={\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }_2))^z$ . By [Reference RossiRos24a, Proposition 4.10], there exists $u\in \widetilde {{\mathbf {K}}}^F$ such that $(\widetilde {{\mathbf {L}}},\widetilde {\lambda }_1)=(\widetilde {{\mathbf {L}}},\widetilde {\lambda }_2\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}})^u$ . This implies that $u\in {\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F$ and that $\widetilde {\lambda }_1=\widetilde {\lambda }_2^u\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ . Moreover, since $\widetilde {\lambda }_1$ lies over both $\lambda $ and $\lambda ^u$ , it follows from Clifford’s theorem that $\lambda =\lambda ^{uv}$ , for some $v\in \widetilde {{\mathbf {L}}}^F$ . Then $x:=uv\in {\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}},\lambda )^F$ and $\widetilde {\lambda }_1=\widetilde {\lambda }_2^x\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ . Conversely, if $\widetilde {\lambda }_1=\widetilde {\lambda }_2^x\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ for some $x\in {\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}},\lambda )^F$ , then [Reference BonnaféBon06, Proposition 10.11] yields the desired equality.

Proof. Let $z\in \mathcal {K}$ stabilise ${\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))$ . By Lemma 3.16 there exists $x\in {\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}},\lambda )^F$ such that $\widetilde {\lambda }=\widetilde {\lambda }^x\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ and hence $z=x^{-1}xz\in {\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}},\lambda )^F({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}},\lambda )^F\ltimes \mathcal {K})_{\widetilde {\lambda }}$ .

Proof. Set $\widetilde {\mathcal {G}}:=\mathrm {Irr}(\widetilde {{\mathbf {K}}}^F\mid {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda )))$ and $\widetilde {\mathcal {N}}:=\mathrm { Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \lambda )$ . First, we claim that $\widetilde {\mathcal {G}}$ is the union of the e-Harish-Chandra series in the set $\mathcal {HC}(\widetilde {{\mathbf {K}}}^F,({\mathbf {L}},\lambda ))$ . In fact, if $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ , then there exists $\chi \in {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda ))$ lying below $\widetilde {\chi }$ . By [Reference Geck and MalleGM20, Corollary 3.3.25], it follows that $\widetilde {\chi }$ is an irreducible constituent of ${\mathbf {R}}_{\widetilde {{\mathbf {L}}}}^{\widetilde {{\mathbf {K}}}}(\lambda ^{\widetilde {{\mathbf {L}}}^F})$ and therefore there exists $\widetilde {\nu }\in \mathrm {Irr}(\widetilde {{\mathbf {L}}}^F\mid \lambda )$ such that $\widetilde {\chi }\in {\mathcal {E}}(\widetilde {{\mathbf {K}}},(\widetilde {{\mathbf {L}}},\widetilde {\nu }))$ . On the other hand, if $\widetilde {\nu }\in \mathrm {Irr}(\widetilde {{\mathbf {L}}}^F\mid \lambda )$ and $\widetilde {\chi }\in {\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\nu }))$ , then [Reference Geck and MalleGM20, Corollary 3.3.25] implies that $\widetilde {\chi }$ lies over some character $\chi \in {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda ))$ . Since the action of $\mathcal {K}$ on $\mathcal {HC}(\widetilde {{\mathbf {K}}}^F,({\mathbf {L}},\lambda ))$ is transitive and recalling the definition of $\mathcal {T}$ , we hence obtain (3.9).

Now we prove (3.10). By Clifford theory, we know that every element of $\mathcal {G}$ lies above some character $\widetilde {\nu }\in \mathrm {Irr}(\widetilde {{\mathbf {L}}}\mid \lambda )$ . Since $\mathcal {K}$ is transitive on $\mathrm {Irr}(\widetilde {{\mathbf {L}}}^F\mid \lambda )$ , we deduce that $\widetilde {\mathcal {N}}$ is contained in the union

$$\begin{align*}\bigcup\limits_{z\in\mathcal{K}}\mathrm{Irr}\left({\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F\enspace\middle|\enspace \widetilde{\lambda}\right)\cdot \widehat{z}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})}.\end{align*}$$

Moreover, we claim that the above union coincides with

(3.11) $$ \begin{align} \bigcup\limits_{z\in\mathcal{T}}\mathrm{Irr}\left({\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F\enspace\middle|\enspace \widetilde{\lambda}\right)\cdot \widehat{z}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})}. \end{align} $$

To see this, let $z\in \mathcal {K}$ and write $z=z_0t$ , for some $z_0\in \mathcal {K}_{{\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))}$ and $t\in \mathcal {T}$ . By Corollary 3.17 we obtain $z_0\in {\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}},\lambda )^F({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}},\lambda )^F\ltimes \mathcal {K})_{\widetilde {\lambda }}$ and therefore

$$\begin{align*}\mathrm{Irr}\left({\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F\enspace\middle|\enspace \widetilde{\lambda}\right)\cdot \widehat{z}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})}=\mathrm{ Irr}\left({\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F\enspace\middle|\enspace \widetilde{\lambda}\right)\cdot \widehat{t}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})}.\end{align*}$$

This proves our claim and it remains to show that the union in (3.11) is disjoint. Assume that, for some $z\in \mathcal {T}$ , there exists a character $\widetilde {\psi }$ inside both $\mathrm {Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \widetilde {\lambda })$ and $\mathrm {Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \widetilde {\lambda })\cdot \widehat {z}_{{\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})}$ . By [Reference IsaacsIsa76, Problem 5.3] we deduce that $\mathrm { Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \widetilde {\lambda })\cdot \widehat {z}_{{\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})}=\mathrm { Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \widetilde {\lambda }\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}})$ and hence $\widetilde {\psi }$ lies above $\widetilde {\lambda }$ and $\widetilde {\lambda }\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ . By Clifford’s theorem $\widetilde {\lambda }=(\widetilde {\lambda }\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}})^u=\widetilde {\lambda }^u\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ , for some $u\in {\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F$ and now Lemma 3.16 implies ${\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))={\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))^z$ . By the definition of $\mathcal {T}$ it follows that the union in (3.11) is disjoint.

Proof. First observe that under our assumptions Hypothesis 2.4 holds for ${\mathbf {G}}$ and for $\widetilde {{\mathbf {G}}}$ . Set $\widetilde {\mathcal {G}}:=\mathrm {Irr}(\widetilde {{\mathbf {K}}}^F\mid {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda )))$ and $\widetilde {\mathcal {N}}:=\mathrm {Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \lambda )$ and fix $\widetilde {\lambda }\in \mathrm { Irr}(\widetilde {{\mathbf {L}}}^F\mid \lambda )$ . Let

$$\begin{align*}\Upsilon_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda})}^{\widetilde{{\mathbf{K}}}}:{\mathcal{E}}\left(\widetilde{{\mathbf{K}}}^F,\left(\widetilde{{\mathbf{L}}},\widetilde{\lambda}\right)\right)\to\mathrm{ Irr}\left({\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})^F\enspace\middle|\enspace\widetilde{\lambda}\right)\end{align*}$$

be the bijection constructed in (3.4). Let $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ be a $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},{\mathbf {L}}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ -transversal in ${\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))$ and observe that, by Lemma 3.14, the set $\widetilde {\mathbb {T}}_{\mathrm {loc}}:=\{\Upsilon _{(\widetilde {{\mathbf {L}}},\widetilde {\lambda })}^{\widetilde {{\mathbf {K}}}}(\widetilde {\chi })\mid \widetilde {\chi }\in \widetilde {\mathbb {T}}_{\mathrm {glo}}\}$ is a $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},{\mathbf {L}}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ -transversal in $\mathrm {Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \lambda )$ .

Next, we fix a transversal $\mathcal {T}$ for $\mathcal {K}_{{\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))}$ in $\mathcal {K}$ and we claim that

(3.12) $$ \begin{align} \widetilde{{\mathbf{K}}}^F_{\mathbf{L}}\left(\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{K}},{\mathbf{L}}}\ltimes \mathcal{K}\right)_{\widetilde{\lambda}}\cdot\mathcal{T}=\widetilde{{\mathbf{K}}}^F_{\mathbf{L}}\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{K}},({\mathbf{L}},\lambda)}\ltimes \mathcal{K}. \end{align} $$

To prove this equality, consider $xz\in (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}\ltimes \mathcal {K}$ . Then both $\widetilde {\lambda }$ and $\widetilde {\lambda }^x\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ lie over $\lambda $ and by [Reference IsaacsIsa76, Problem 6.2] there exists $u\in \mathcal {K}$ such that $\widetilde {\lambda }=\widetilde {\lambda }^x\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}\cdot \widehat {u}_{\widetilde {{\mathbf {L}}}}$ . Therefore $xz\in ((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},{\mathbf {L}}}\ltimes \mathcal {K})_{\widetilde {\lambda }}\cdot \mathcal {K}$ . On the other hand, applying Corollary 3.17, we obtain $\mathcal {K}_{{\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))}\leq \widetilde {{\mathbf {K}}}^F_{\mathbf {L}}(\widetilde {{\mathbf {K}}}^F_{\mathbf {L}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ and by the definition of $\mathcal {T}$ , we conclude that

$$\begin{align*}\widetilde{{\mathbf{K}}}^F_{\mathbf{L}}\left(\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{K}},{\mathbf{L}}}\ltimes \mathcal{K}\right)_{\widetilde{\lambda}}\cdot \mathcal{T}\geq \widetilde{{\mathbf{K}}}^F_{\mathbf{L}}\left(\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{K}},({\mathbf{L}},\lambda)}\ltimes \mathcal{K}\right).\end{align*}$$

To prove the remaining inclusion it’s enough to show that

$$\begin{align*}\widetilde{{\mathbf{L}}}^F\left(\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{K}},({\mathbf{L}},\lambda)}\ltimes \mathcal{K}\right)_{\widetilde{\lambda}}=\left(\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{K}},{\mathbf{L}}}\ltimes \mathcal{K}\right)_{\widetilde{\lambda}}.\end{align*}$$

Since $\widetilde {\lambda }$ is $\widetilde {{\mathbf {L}}}^F$ -invariant, one inclusion is trivial. So let $xz\in ((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},{\mathbf {L}}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ and observe that $\widetilde {\lambda }=\widetilde {\lambda }^x\cdot \widehat {z}_{\widetilde {{\mathbf {L}}}}$ lies both over $\lambda $ and over $\lambda ^x$ . By Clifford’s theorem there exists $y\in \widetilde {{\mathbf {L}}}^F$ such that $\lambda =\lambda ^{xy}$ and hence $xz\in \widetilde {{\mathbf {L}}}^F((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ . This proves the claim.

Now, using (3.12), we show that $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ is a $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}\ltimes \mathcal {K})$ -transversal in $\widetilde {\mathcal {G}}$ . Consider $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ . By Proposition 3.18 there exist unique $z\in \mathcal {T}$ and $\widetilde {\chi }_0'\in {\mathcal {E}}(\widetilde {{\mathbf {K}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))$ such that $\widetilde {\chi }=\widetilde {\chi }_0'\cdot \widehat {z}_{\widetilde {{\mathbf {K}}}}$ . Let $\widetilde {\chi }_0$ be the unique element in $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ such that $\widetilde {\chi }_0'=\widetilde {\chi }_0^x\cdot \widehat {u}_{\widetilde {{\mathbf {K}}}}$ , for some $xu\in ((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},{\mathbf {L}}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ . Then $\widetilde {\chi }=\widetilde {\chi }_0^x\cdot \widehat {u}_{\widehat {K}}\cdot \widehat {z}_{\widetilde {K}}$ , for $xuz\in ((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},{\mathbf {L}}}\ltimes \mathcal {K})_{\widetilde {\lambda }}\cdot \mathcal {T}$ . But using (3.12) and since $\widetilde {\chi }$ and $\widetilde {\chi }_0$ are ${\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F$ -invariant, we conclude that $\widetilde {\chi }=\widetilde {\chi }_0^y\cdot \widehat {v}_{\widetilde {{\mathbf {K}}}}$ , for some $y\in (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}$ and $v\in \mathcal {K}$ . This argument also shows that $\widetilde {\chi }_0$ is the unique element of $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ with this property.

Similarly, using (3.12), we deduce that the set $\widetilde {\mathbb {T}}_{\mathrm {loc}}$ is a $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}\ltimes \mathcal {K})$ -transversal in $\widetilde {\mathcal {N}}$ . Now, the map

$$\begin{align*}\widetilde{\Omega}_{({\mathbf{L}},\lambda)}^{\mathbf{K}}:\widetilde{\mathcal{G}}\to\widetilde{\mathcal{N}}\end{align*}$$

defined by

$$\begin{align*}\widetilde{\Omega}_{({\mathbf{L}},\lambda)}^{\mathbf{K}}\left(\widetilde{\chi}^x\cdot \widehat{z}_{\widetilde{{\mathbf{K}}}}\right):=\Upsilon_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda})}^{\widetilde{{\mathbf{K}}}}(\widetilde{\chi})^x\cdot \widehat{z}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})},\end{align*}$$

for every $\widetilde {\chi }\in \widetilde {\mathbb {T}}_{\mathrm {glo}}$ , $x\in (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}$ and $z\in \mathcal {K}$ , is a $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}\ltimes \mathcal {K})$ -equivariant bijection. The remaining properties follow from Lemma 3.11, Lemma 3.12 and Lemma 3.13 after noticing that $\widehat {z}_{\widetilde {{\mathbf {K}}}}$ and $\widehat {z}_{{\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})}$ are linear characters and that

$$\begin{align*}\mathrm{bl}\left(\widetilde{\psi}\cdot\widehat{z}_{{\mathbf{N}}_{\widetilde{{\mathbf{K}}}}({\mathbf{L}})}\right)^{\widetilde{{\mathbf{K}}}^F}=\mathrm{bl}\left(\widetilde{\psi}\right)^{\widetilde{{\mathbf{K}}}^F}\cdot \widehat{z}_{\widetilde{{\mathbf{K}}}}\end{align*}$$

for every $\widetilde {\psi }\in \mathrm {Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F)$ and $z\in \mathcal {K}$ .

Proof. Fix a $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}\ltimes \mathcal {K})$ -transversal $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ in $\mathrm {Irr}(\widetilde {{\mathbf {K}}}^F\mid {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda )))$ . By Theorem 3.19 the set $\widetilde {\mathbb {T}}_{\mathrm { loc}}:=\{\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {K}}(\widetilde {\chi })\mid \widetilde {\chi }\in \widetilde {\mathbb {T}}_{\mathrm {glo}}\}$ is a $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}\ltimes \mathcal {K})$ -transversal in $\mathrm {Irr}({\mathbf {N}}_{\widetilde {{\mathbf {K}}}}({\mathbf {L}})^F\mid \lambda )$ . For every $\widetilde {\chi }\in \widetilde {\mathbb {T}}_{\mathrm {glo}}$ fix an irreducible constituent $\chi \in {\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ of $\widetilde {\chi }_{{\mathbf {G}}^F}$ and define the set $\mathbb {T}_{\mathrm { glo}}$ consisting of such characters $\chi $ , while $\widetilde {\chi }$ runs over the elements of $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ . Similarly, for every $\widetilde {\psi }\in \widetilde {\mathbb {T}}_{\mathrm { loc}}$ , fix an irreducible constituent $\psi \in \mathrm {Irr}({\mathbf {N}}_{\mathbf {K}}({\mathbf {L}})^F\mid \lambda )$ of $\widetilde {\psi }_{{\mathbf {N}}_{\mathbf {K}}({\mathbf {L}})^F}$ and define the set $\mathbb {T}_{\mathrm {loc}}$ consisting of such characters $\psi $ , while $\widetilde {\psi }$ runs over the elements of $\widetilde {\mathbb {T}}_{\mathrm { loc}}$ . Then $\mathbb {T}_{\mathrm {glo}}$ and $\mathbb {T}_{\mathrm {loc}}$ are $(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}$ -transversals in ${\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ and $\mathrm {Irr}({\mathbf {N}}_{\mathbf {K}}({\mathbf {L}})^F\mid \lambda )$ respectively. Fix $\chi \in \mathbb {T}_{\mathrm {glo}}$ and let $\widetilde {\chi }$ be the unique element of $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ lying above $\chi $ . Let $\widetilde {\psi }:=\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {K}}(\widetilde {\chi })\in \widetilde {\mathbb {T}}_{\mathrm {loc}}$ and consider the unique element $\psi $ of $\mathbb {T}_{\mathrm {loc}}$ lying below $\widetilde {\psi }$ . This defines a bijection

(3.13) $$ \begin{align} \mathbb{T}_{\mathrm{glo}}\to\mathbb{T}_{\mathrm{loc}}. \end{align} $$

Then, defining

$$\begin{align*}\Omega_{({\mathbf{L}},\lambda)}^{\mathbf{K}}(\chi^x):=\psi^x\end{align*}$$

for every $x\in (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {K}},({\mathbf {L}},\lambda )}$ and every $\chi \in \mathbb {T}_{\mathrm {glo}}$ corresponding to $\psi \in \mathbb {T}_{\mathrm {loc}}$ via (3.13) we obtain the wanted bijection.

Assumption 4.1. Let $({\mathbf {L}},\lambda )$ be an e-cuspidal pair of ${\mathbf {G}}$ , set

$$\begin{align*}\mathcal{G}:={\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right) \hspace{10pt}\text{and}\hspace{10pt}\mathcal{N}:=\mathrm{Irr}\left({\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\enspace\middle|\enspace \lambda\right)\end{align*}$$

and consider

$$\begin{align*}\widetilde{\mathcal{G}}:=\mathrm{Irr}\left(\widetilde{{\mathbf{G}}}^F\enspace\middle|\enspace \mathcal{G}\right)\hspace{10pt}\text{and}\hspace{10pt}\widetilde{\mathcal{N}}:=\mathrm{ Irr}\left({\mathbf{N}}_{\widetilde{{\mathbf{G}}}}({\mathbf{L}})^F\enspace\middle|\enspace \mathcal{N}\right).\end{align*}$$

Assume that:

1. (i)

   1. (a) There is a semidirect decomposition $\widetilde {{\mathbf {G}}}^F\rtimes \mathcal {A}$ , with $\mathcal {A}$ a finite abelian group, such that

      $$\begin{align*}{\mathbf{C}}_{\widetilde{{\mathbf{G}}}^F\mathcal{A}}({\mathbf{G}}^F)={\mathbf{Z}}(\widetilde{{\mathbf{G}}}^F)\quad\text{and}\quad\widetilde{{\mathbf{G}}}^F\mathcal{A}/{\mathbf{Z}}(\widetilde{{\mathbf{G}}}^F)\simeq \mathrm{Aut}_{\mathbb{F}}({\mathbf{G}}^F);\end{align*}$$

   2. (b) Maximal extendibility holds with respect to ${\mathbf {G}}^F\unlhd \widetilde {{\mathbf {G}}}^F$ ;

   3. (c) Maximal extendibility holds with respect to ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F\unlhd {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ .

2. (ii) For $A:=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{({\mathbf {L}},\lambda )}$ there exists a defect preserving $(A\ltimes \mathcal {K})$ -equivariant bijection

   $$\begin{align*}\widetilde{\Omega}_{({\mathbf{L}},\lambda)}^{\mathbf{G}}:\widetilde{\mathcal{G}}\to \widetilde{\mathcal{N}}\end{align*}$$
   such that $\mathrm {Irr}\left (\widetilde {\chi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}\right )=\mathrm { Irr}\left (\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\widetilde {\chi })_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}\right )$ for every $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ .

3. (iii) For every $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ there exists $\chi \in \mathcal {G}\cap \mathrm {Irr}\left (\widetilde {\chi }_{{\mathbf {G}}^F}\right )$ such that:

   1. (a) $\left (\widetilde {{\mathbf {G}}}^F\mathcal {A}\right )_{\chi }=\widetilde {{\mathbf {G}}}^F_\chi \mathcal {A}_{\chi }$ ;

   2. (b) $\chi $ extends to $\chi '\in \mathrm {Irr}\left ({\mathbf {G}}^F\mathcal {A}_{\chi }\right )$ .

4. (iv) For every $\widetilde {\psi }\in \widetilde {\mathcal {N}}$ there exists $\psi \in \mathcal {N}\cap \mathrm {Irr}\left (\widetilde {\psi }_{{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F}\right )$ such that:

   1. (a) $\left (\widetilde {{\mathbf {G}}}^F \mathcal {A}\right )_{{\mathbf {L}},\psi }={\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi \left ({\mathbf {G}}^F\mathcal {A}\right )_{{\mathbf {L}},\psi }$ ;

   2. (b) $\psi $ extends to $\psi '\in \mathrm {Irr}\left (\left ({\mathbf {G}}^F \mathcal {A}\right )_{{\mathbf {L}},\psi }\right )$ .

Proof. As the two implications can be shown by similar arguments, we only show that (4.1) implies (4.2). To start, consider the subgroups

$$\begin{align*}T:={\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\left(\widetilde{{\mathbf{G}}}^F_{({\mathbf{L}},\lambda),\chi}\cdot ({\mathbf{G}}^F\mathcal{A})_{({\mathbf{L}},\lambda),\chi}\right)={\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\left(\widetilde{{\mathbf{G}}}^F_{({\mathbf{L}},\lambda),\psi}\cdot ({\mathbf{G}}^F\mathcal{A})_{({\mathbf{L}},\lambda),\psi}\right)\end{align*}$$

and

$$\begin{align*}V:={\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F(\widetilde{{\mathbf{G}}}^F \mathcal{A})_{({\mathbf{L}},\lambda),\chi}={\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F(\widetilde{{\mathbf{G}}}^F \mathcal{A})_{({\mathbf{L}},\lambda),\psi},\end{align*}$$

where the equalities follow since $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ is equivariant by assumption.

Define $U(\chi ):=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ and $U(\psi ):=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\psi }$ . We claim that $U(\chi )=U(\psi )$ . To prove this fact, notice that it is enough to show that $U(\chi )$ and $U(\psi )$ are contained in V, in fact this would imply ${U(\chi )=U(\chi )\cap V=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}}}\cap V=U(\psi )\cap V=U(\psi )}$ . If $x\in U(\chi )$ , then $\chi \in {\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))\cap {\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda )^x)$ and, by [Reference RossiRos24a, Proposition 4.10], there exists $y\in {\mathbf {G}}^F$ such that $({\mathbf {L}},\lambda )=({\mathbf {L}},\lambda )^{xy}$ . Notice that $y\in {\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ and hence $x\in V$ . On the other hand, if $x\in U(\psi )$ , then $\psi $ lies over $\lambda ^x$ and by Clifford’s theorem $\lambda ^{xy}=\lambda $ , for some $y\in {\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ . Also in this case $x\in V$ . Now $U(\chi )=U(\psi )$ and we denote this group by U.

Next, we claim that $T=U$ . If this is true, then we deduce that $T\leq {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi ({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\psi }\leq U=T$ and therefore (4.2) holds. First, observe that $T\leq U$ . As $T\cap {\mathbf {G}}^F={\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F=U\cap {\mathbf {G}}^F$ and $T\leq U\leq (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\chi }$ , it is enough to show that $T{\mathbf {G}}^F=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\chi }$ . First, repeating the same argument as before, a Frattini argument shows that

(4.3) $$ \begin{align} \left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{\chi}={\mathbf{G}}^F\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{({\mathbf{L}},\lambda),\chi} \end{align} $$

and

(4.4) $$ \begin{align} \widetilde{{\mathbf{G}}}^F_\chi={\mathbf{G}}^F\widetilde{{\mathbf{G}}}^F_{({\mathbf{L}},\lambda),\chi}. \end{align} $$

Then using the hypothesis we finally obtain

$$ \begin{align*} \left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{\chi}&\stackrel{(4.3)}{=}{\mathbf{G}}^F\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{({\mathbf{L}},\lambda),\chi}\\ &\stackrel{(4.1)}{=}{\mathbf{G}}^F\left(\widetilde{{\mathbf{G}}}_\chi^F\left({\mathbf{G}}^F\mathcal{A}\right)_{\chi}\right)_{({\mathbf{L}},\lambda)}\\ &\stackrel{(4.4)}{=}{\mathbf{G}}^F\left({\mathbf{G}}^F\widetilde{{\mathbf{G}}}^F_{({\mathbf{L}},\lambda),\chi}\left({\mathbf{G}}^F\mathcal{A}\right)_{\chi}\right)_{({\mathbf{L}},\lambda)}\\ &={\mathbf{G}}^F\widetilde{{\mathbf{G}}}^F_{({\mathbf{L}},\lambda),\chi}\left({\mathbf{G}}^F\mathcal{A}\right)_{({\mathbf{L}},\lambda),\chi}\\ &={\mathbf{G}}^FT. \end{align*} $$

This concludes the proof.

Proof. We start by fixing an $\left (A\ltimes \mathcal {K}\right )$ -transversal $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ in $\widetilde {\mathcal {G}}$ . As $\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ is $(A\ltimes \mathcal {K})$ -equivariant, we deduce that the set $\widetilde {\mathbb {T}}_{\mathrm {loc}}:=\{\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\widetilde {\chi })\mid \widetilde {\chi }\in \widetilde {\mathbb {T}}_{\mathrm { glo}}\}$ is an $(A\ltimes \mathcal {K})$ -transversal in $\widetilde {\mathcal {N}}$ . For every $\widetilde {\chi }\in \widetilde {\mathbb {T}}_{\mathrm {glo}}$ , we choose a character $\chi \in \mathcal {G}\cap \mathrm {Irr}(\widetilde {\chi }_{{\mathbf {G}}^F})$ satisfying Assumption 4.1 (iii). Denote by $\mathbb {T}_{\mathrm {glo}}$ the set of such characters $\chi $ , where $\widetilde {\chi }$ runs over $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ . Similarly, for every $\widetilde {\psi }\in \widetilde {\mathbb {T}}_{\mathrm {loc}}$ , fix a character $\psi \in \mathcal {N}\cap \mathrm { Irr}(\widetilde {\psi }_{{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F})$ satisfying Assumption 4.1 (iv) and denote by $\mathbb {T}_{\mathrm {loc}}$ the set of such characters $\psi $ . Now, arguing as in the proof of Theorem 3.20, we obtain an A-equivariant bijection between $\mathcal {G}$ and $\mathcal {N}$ by setting

$$\begin{align*}\Omega_{({\mathbf{L}},\lambda)}^{\mathbf{G}}\left(\chi^x\right):=\psi^x\end{align*}$$

for every $x\in A$ and $\chi \in \mathbb {T}_{\mathrm {glo}}$ , where $\psi $ is the unique character in $\mathbb {T}_{\mathrm {loc}}$ lying below $\widetilde {\psi }:=\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\widetilde {\chi })$ and $\widetilde {\chi }$ is the unique character in $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ lying over $\chi $ . By Assumption 4.1 (i.a) this means that $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ is $\mathrm {Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -equivariant.

To show that $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ preserves the defect, we use Assumption 4.1 (i.b) and (i.c). Clearly it’s enough to show that $d(\chi )=d(\psi )$ , for $\chi \in \mathbb {T}_{\mathrm {glo}}$ and $\psi :=\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\chi )\in \mathbb {T}_{\mathrm {loc}}$ . Let $\widetilde {\chi }$ (resp. $\widetilde {\psi }$ ) be the unique element of $\widetilde {\mathbb {T}}_{\mathrm {glo}}$ (resp. $\widetilde {\mathbb {T}}_{\mathrm {loc}}$ ) lying over $\chi $ (resp. $\psi $ ). Then $\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\widetilde {\chi })=\widetilde {\psi }$ and $d(\widetilde {\chi })=d(\widetilde {\psi })$ by Assumption 4.1 (ii). Moreover, since $\widetilde {{\mathbf {G}}}^F/{\mathbf {G}}^F\simeq {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F/{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ is abelian and using Assumption 4.1 (i.b) and (i.c), we deduce that the Clifford correspondent $\widehat {\chi }\in \mathrm {Irr}(\widetilde {{\mathbf {G}}}^F_\chi )$ of $\widetilde {\chi }$ over $\chi $ is an extension of $\chi $ and, similarly, that the Clifford correspondent $\widehat {\psi }\in \mathrm { Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi )$ of $\widetilde {\psi }$ over $\psi $ is an extension of $\psi $ . As a consequence

$$\begin{align*}\ell^{d(\chi)}=\ell^{d(\widehat{\chi})}\cdot \left|\widetilde{{\mathbf{G}}}^F_\chi:{\mathbf{G}}^F\right|{}_\ell\end{align*}$$

and

$$\begin{align*}\ell^{d(\psi)}=\ell^{d(\widehat{\psi})}\cdot \left|{\mathbf{N}}_{\widetilde{{\mathbf{G}}}}({\mathbf{L}})^F_\psi:{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\right|{}_\ell.\end{align*}$$

Therefore, as the defect is preserved by induction of characters, we obtain $d(\widehat {\chi })=d(\widetilde {\chi })=d(\widetilde {\psi })=d(\widehat {\psi })$ and it remains to show that $|\widetilde {{\mathbf {G}}}^F_\chi :{\mathbf {G}}^F|_\ell =|{\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi :{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F|_\ell $ . This follows from the proof of Lemma 4.2: in fact there it is shown that ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi ={\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\chi $ and therefore $\widetilde {{\mathbf {G}}}^F_\chi /{\mathbf {G}}^F\simeq {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\chi /{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F={\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi /{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ .

Next, we prove the condition on character triples. Applying a simplified version of [Reference SpäthSpä17, Theorem 5.3] adapted to ${\mathbf {G}}^F$ -central isomorphic character triples (this immediately follows by the argument used in the proof of [Reference SpäthSpä17, Theorem 5.3]), it is enough to show that

(4.5)

Moreover, as the equivalence relation is compatible with conjugation, it’s enough to prove this condition for a fixed $\chi \in \mathbb {T}_{\mathrm {glo}}$ and $\psi :=\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\chi )\in \mathbb {T}_{\mathrm {loc}}$ .

First of all, notice that the required group theoretical properties are satisfied by the proof of Lemma 4.2. In fact, there we have shown that $(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\chi }=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\psi }$ and that $(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\chi }={\mathbf {G}}^F(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ , while

$$ \begin{align*} {\mathbf{C}}_{\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{\chi}}\left({\mathbf{G}}^F\right)\leq{\mathbf{C}}_{\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{\chi}}\left({\mathbf{L}}^F\right)\leq \left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{L}},\chi}= \left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{L}},\psi}. \end{align*} $$

To construct the relevant projective representations, we make use of [Reference SpäthSpä12, Lemma 2.11]. As before, consider the corresponding $\widetilde {\chi }\in \widetilde {\mathbb {T}}_{\mathrm {glo}}$ and $\widetilde {\psi }\in \widetilde {\mathbb {T}}_{\mathrm {loc}}$ with $\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\widetilde {\chi })=\widetilde {\psi }$ , $\widetilde {\chi }$ lying over $\chi $ and $\widetilde {\psi }$ lying over $\psi $ . Furthermore, consider the Clifford correspondent $\widehat {\chi }\in \mathrm {Irr}(\widetilde {{\mathbf {G}}}^F_\chi \mid \chi )$ of $\widetilde {\chi }$ and the Clifford correspondent $\widehat {\psi }\in \mathrm { Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi \mid \psi )$ of $\widetilde {\psi }$ . Let $\widehat {\mathcal {D}}_{\mathrm {glo}}$ be a representation affording $\widehat {\chi }$ and notice that, by the choice of $\chi $ and using Assumption 4.1 (iii.b), there exists a representation $\mathcal {D}^{\prime }_{\mathrm {glo}}$ affording an extension $\chi '\in \mathrm {Irr}({\mathbf {G}}^F\mathcal {A}_{\chi })$ of $\chi $ . Additionally, we may chose $\mathcal {D}^{\prime }_{\mathrm {glo}}$ such that $\mathcal {D}^{\prime }_{\mathrm {glo}}(x)=\widehat {\mathcal {D}}_{\mathrm {glo}}(x)$ for all $x\in {\mathbf {G}}^F$ . Similarly, let $\widehat {\mathcal {D}}_{\mathrm {loc}}$ be a representation affording $\widehat {\psi }$ and observe that, by the choice of $\psi $ , there is a representation $\mathcal {D}^{\prime }_{\mathrm {loc}}$ affording an extension $\psi '\in \mathrm { Irr}(({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\psi })$ of $\psi $ . Also here, we may assume that $\widehat {\mathcal {D}}_{\mathrm {loc}}(x)=\mathcal {D}^{\prime }_{\mathrm {loc}}(x)$ for all $x\in {\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ . Applying [Reference SpäthSpä12, Lemma 2.11] with $L:={\mathbf {G}}^F$ , $\widetilde {L}:=\widetilde {{\mathbf {G}}}^F_\chi $ , $C:={\mathbf {G}}^F\mathcal {A}_{\chi }$ , $X:=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\chi }$ and recalling that $X=\widetilde {L}C$ because Assumption 4.1 (iii.a) holds for $\chi $ , we deduce that the map

$$\begin{align*}\mathcal{P}_{\mathrm{glo}}:\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{\chi}\to\mathrm{GL}_{\chi(1)}(\mathbb{C})\end{align*}$$

given by $\mathcal {P}_{\mathrm {glo}}(x_1x_2):=\widehat {\mathcal {D}}_{\mathrm {glo}}(x_1)\mathcal {D}^{\prime }_{\mathrm {glo}}(x_2)$ , for every $x_1\in \widetilde {{\mathbf {G}}}^F_\chi $ and $x_2\in {\mathbf {G}}^F\mathcal {A}_{\chi }$ , is a projective representation associated with $\chi $ whose factor set $\alpha _{\mathrm {glo}}$ satisfies

(4.6) $$ \begin{align} \alpha_{\mathrm{glo}}(x_1x_2,y_1y_2)=\mu_{x_2}^{\mathrm{glo}}(y_1) \end{align} $$

for every $x_1,y_1\in \widetilde {{\mathbf {G}}}^F_\chi $ and $x_2,y_2\in {\mathbf {G}}^F\mathcal {A}_{\chi }$ , where $\mu _{x_2}^{\mathrm {glo}}\in \mathrm {Irr}(\widetilde {{\mathbf {G}}}^F_\chi /{\mathbf {G}}^F)$ is determined by the equality $\widehat {\chi }=\mu _{x_2}^{\mathrm {glo}}\widehat {\chi }^{x_2}$ via Gallagher’s theorem. In a similar way, considering $L:={\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ , $\widetilde {L}:={\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\chi $ , $C:=({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ , $X:=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ and noticing that $X=\widetilde {L}C$ because Assumption 4.1 (iv.a) holds for $\psi $ , we deduce that the map

$$\begin{align*}\mathcal{P}_{\mathrm{loc}}:\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{L}},\chi}\to\mathrm{GL}_{\psi(1)}(\mathbb{C})\end{align*}$$

given by $\mathcal {P}_{\mathrm {loc}}(x_1x_2):=\widehat {\mathcal {D}}_{\mathrm {loc}}(x_1)\mathcal {D}^{\prime }_{\mathrm {loc}}(x_2)$ , for every $x_1\in {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\chi $ and $x_2\in ({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ , is a projective representation associated with $\psi $ whose factor set $\alpha _{\mathrm {loc}}$ satisfies

(4.7) $$ \begin{align} \alpha_{\mathrm{loc}}(x_1x_2,y_1y_2)=\mu_{x_2}^{\mathrm{loc}}(y_1) \end{align} $$

for every $x_1,y_1\in {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\chi $ and $x_2,y_2\in ({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ , where $\mu _{x_2}^{\mathrm {loc}}\in \mathrm { Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\chi /{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F)$ is determined by $\widehat {\psi }=\mu _{x_2}^{\mathrm {loc}}\widehat {\psi }^{x_2}$ . In order to obtain the condition on factor sets required to prove (4.5) we have to show that the restriction of $\alpha _{\mathrm {glo}}$ to $(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\chi }\times (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ coincides with $\alpha _{\mathrm {loc}}$ . Using (4.6) and (4.7), it is enough to show that

$$\begin{align*}\left(\mu_x^{\mathrm{glo}}\right)_{{\mathbf{N}}_{\widetilde{{\mathbf{G}}}}({\mathbf{L}})^F_\chi}=\mu_x^{\mathrm{loc}}\end{align*}$$

for every $x\in ({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ and where $\widehat {\chi }=\mu _x^{\mathrm {glo}}\widehat {\chi }^x$ and $\widehat {\psi }=\mu _x^{\mathrm {loc}}\widehat {\psi }^x$ . To prove this equality, since $({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\chi }={\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F A_\chi $ (see the proof of Lemma 4.2), we may assume $x\in A_\chi $ . Then, we conclude since $\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ is $(A\ltimes \mathcal {K})$ -equivariant.

To conclude we need to check one of the equivalent conditions of [Reference SpäthSpä17, Lemma 3.4]. Recalling that ${\mathbf {C}}_{(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\chi }}({\mathbf {G}}^F)={\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)$ by Assumption 4.1 (i.a), if $\zeta _{\mathrm {glo}}$ and $\zeta _{\mathrm {loc}}$ are the scalar functions of $\mathcal {P}_{\mathrm {glo}}$ and $\mathcal {P}_{\mathrm {loc}}$ respectively, we have to show that $\zeta _{\mathrm {glo}}$ and $\zeta _{\mathrm {loc}}$ coincide as characters of ${\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)$ . By the definition of $\mathcal {P}_{\mathrm {glo}}$ , it follows that $\zeta _{\mathrm {glo}}$ coincides with the unique irreducible constituent $\nu $ of $\widehat {\chi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}$ . Moreover, by Clifford theory we know that $\nu $ is also the unique irreducible constituent of $\widetilde {\chi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}$ . Therefore, we conclude that $\{\zeta _{\mathrm {glo}}\}=\mathrm {Irr}(\widetilde {\chi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)})$ and a similar argument shows that $\{\zeta _{\mathrm {loc}}\}=\mathrm { Irr}(\widetilde {\psi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)})$ . Then, Assumption 4.1 (ii) implies that $\zeta _{\mathrm {glo}}=\zeta _{\mathrm {loc}}$ . This completes the proof.

Assumption 4.4. Let $({\mathbf {L}},\lambda )$ be an e-cuspidal pair of ${\mathbf {G}}$ and suppose that $B:=\mathrm {bl}(\lambda )^{{\mathbf {G}}^F}$ is defined. Set

$$\begin{align*}\mathcal{G}:={\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right) \hspace{10pt}\text{and}\hspace{10pt}\mathcal{N}:=\mathrm{Irr}\left({\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\enspace\middle|\enspace \lambda\right)\end{align*}$$

and consider

$$\begin{align*}\widetilde{\mathcal{G}}:=\mathrm{Irr}\left(\widetilde{{\mathbf{G}}}^F\enspace\middle|\enspace \mathcal{G}\right)\hspace{10pt}\text{and}\hspace{10pt}\widetilde{\mathcal{N}}:=\mathrm{ Irr}\left({\mathbf{N}}_{\widetilde{{\mathbf{G}}}}({\mathbf{L}})^F\enspace\middle|\enspace \mathcal{N}\right).\end{align*}$$

Assume that:

1. (i)

   1. (a) There is a semidirect decomposition $\widetilde {{\mathbf {G}}}^F\rtimes \mathcal {A}$ , with $\mathcal {A}$ a finite abelian group, such that

      $$\begin{align*}{\mathbf{C}}_{(\widetilde{{\mathbf{G}}}^F\mathcal{A})}({\mathbf{G}}^F)={\mathbf{Z}}(\widetilde{{\mathbf{G}}}^F)\quad \text{and} \quad(\widetilde{{\mathbf{G}}}^F\mathcal{A})/{\mathbf{Z}}(\widetilde{{\mathbf{G}}}^F)\simeq \mathrm{Aut}_{\mathbb{F}}({\mathbf{G}}^F);\end{align*}$$

   2. (b) Maximal extendibility holds with respect to ${\mathbf {G}}^F\unlhd \widetilde {{\mathbf {G}}}^F$ ;

   3. (c) Maximal extendibility holds with respect to ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F\unlhd {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ .

2. (ii) For $A:=(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{({\mathbf {L}},\lambda )}$ there exists a defect preserving $(A\ltimes \mathcal {K})$ -equivariant bijection

   $$\begin{align*}\widetilde{\Omega}_{({\mathbf{L}},\lambda)}^{\mathbf{G}}:\widetilde{\mathcal{G}}\to \widetilde{\mathcal{N}}\end{align*}$$
   such that, for every $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ , the following conditions hold:

   1. (a) $\mathrm {Irr}\left (\widetilde {\chi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}\right )=\mathrm { Irr}\left (\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\widetilde {\chi })_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}\right )$ ;

   2. (b) $\mathrm {bl}\left (\widetilde {\chi }\right )=\mathrm {bl}\left (\widetilde {\Omega }_{({\mathbf {L}},\lambda )}^{\mathbf {G}}\left (\widetilde {\chi }\right )\right )^{\widetilde {{\mathbf {G}}}^F}$ .

3. (iii) For every $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ there exists $\chi \in \mathrm {Irr}\left (\widetilde {\chi }_{{\mathbf {G}}^F}\right )$ such that:

   1. (a) $\left (\widetilde {{\mathbf {G}}}^F\mathcal {A}\right )_\chi =\widetilde {{\mathbf {G}}}^F_\chi \mathcal {A}_\chi $ ;

   2. (b) $\chi $ extends to $\chi '\in \mathrm {Irr}\left ({\mathbf {G}}^F\mathcal {A}_\chi \right )$ .

4. (iv) For every $\widetilde {\psi }\in \widetilde {\mathcal {N}}$ there exists $\psi \in \mathcal {N}\cap \mathrm {Irr}\left (\widetilde {\psi }_{{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F}\right )$ such that:

   1. (a) $\left (\widetilde {{\mathbf {G}}}^F \mathcal {A}\right )_{{\mathbf {L}},\psi }={\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F_\psi \left ({\mathbf {G}}^F\mathcal {A}\right )_{{\mathbf {L}},\psi }$ ;

   2. (b) $\psi $ extends to $\psi '\in \mathrm {Irr}\left (\left ({\mathbf {G}}^F \mathcal {A}\right )_{{\mathbf {L}},\psi }\right )$ .

5. (v) Assume one of the following conditions:

   1. (a) $\mathrm {Out}({\mathbf {G}}^F)_{\mathcal {B}}$ is abelian, where $\mathcal {B}$ is the $\widetilde {{\mathbf {G}}}^F$ -orbit of B. In particular (iii) holds for every $\widetilde {{\mathbf {G}}}^F$ -conjugate of $\chi $ (see the proof of [Reference Brough and SpäthBS22, Lemma 4.7]); or

   2. (b) for every subgroup ${\mathbf {G}}^F\leq J\leq \widetilde {{\mathbf {G}}}^F$ we have that every block $C\in \mathrm {Bl}(J\mid B)$ is $\widetilde {{\mathbf {G}}}^F$ -invariant.

6. (vi) The pair $({\mathbf {L}},\lambda )$ is e-Brauer–Lusztig-cuspidal in the sense of Definition 2.5.