Proof. First we note the following isomorphisms

(2.22) $$ \begin{align} W( N_{G^\flat}(L^\flat), T^\flat) / W(L^\flat, T^\flat) \cong N_{G^\flat}(L^\flat, T^\flat) / N_{L^\flat} (T^\flat) \cong W(G^\flat,L^\flat)_{T^\flat}, \end{align} $$

where the subscript $T^\flat $ means that the $L^\flat $ -conjugacy class of $T^\flat $ is stabilized. Let $\mathcal S'$ be a maximal F-torus in ${\mathcal {L}}$ which contains a maximal F-split torus $\mathcal S$ of ${\mathcal {L}}$ . Since $G^\flat $ is quasi-split, the $\mathbf W_F$ -action on $\mathcal G^\flat $ comes from a diagam automorphism of $R(\mathcal G^\flat , \mathcal S^{\prime \flat })$ , and there are natural isomorphisms

$$ \begin{align*} W(G^\flat,L^\flat) & \cong N_{W(G^\flat, S^\flat)} \big( W(L^\flat,S^\flat) \big) / W(L^\flat,S^\flat) \\ & \cong N_{W(\mathcal G^\flat, \mathcal S^{\prime\flat})^{\mathbf W_F}} \big( W({\mathcal{L}}^\flat,\mathcal S^{\prime\flat})^{\mathbf W_F} \big) \big/ W({\mathcal{L}}^\flat,\mathcal S^{\prime\flat})^{\mathbf W_F} \cong W(\mathcal G^\flat, {\mathcal{L}}^\flat )^{\mathbf W_F}. \end{align*} $$

It follows that the natural maps

(2.23) $$ \begin{align} N_{G^\flat}(L^\flat) \to W(G^\flat,L^\flat) \to (N_{\mathcal G^\flat}({\mathcal{L}}^\flat)/ {\mathcal{L}}^\flat)(F) = W(\mathcal G^\flat,{\mathcal{L}}^\flat)^{\mathbf W_F} \end{align} $$

are surjective. Hence (2.22) is equal to $W(\mathcal G^\flat , {\mathcal {L}}^\flat )(F)_{T^\flat }$ . Clearly, any element of this group also stabilizes the ${\mathcal {L}}^\flat $ -conjugacy class of ${\mathcal {T}}^\flat $ . On the other hand, if an element of $W(\mathcal G^\flat , {\mathcal {L}}^\flat )$ stabilizes the ${\mathcal {L}}^\flat $ -conjugacy class of ${\mathcal {T}}^\flat $ , then a suitable representative of that element normalizes $T^\flat $ . Therefore (2.22) is naturally isomorphic to

(2.24) $$ \begin{align} W(\mathcal G^\flat, {\mathcal{L}}^\flat)_{{\mathcal{T}}^\flat}(F) \cong \big( W(N_{\mathcal G^\flat}({\mathcal{L}}^\flat), {\mathcal{T}}^\flat) / W({\mathcal{L}}^\flat, {\mathcal{T}}^\flat) \big) (F), \end{align} $$

which has a subgroup $W(N_{\mathcal G^\flat }({\mathcal {L}}^\flat ), {\mathcal {T}}^\flat )(F) / W({\mathcal {L}}^\flat , {\mathcal {T}}^\flat ) (F)$ . The natural isomorphism from the left hand side of (2.22) to the right hand side of (2.24) factors through (2.24), thus this subgroup is equal to (2.24). Combined with (2.21), it implies that the natural map $W(N_{G^\flat }(L^\flat ), T^\flat ) \to W(N_{\mathcal G^\flat }({\mathcal {L}}^\flat ), {\mathcal {T}}^\flat )(F)$ is an isomorphism.

Proof. (a) In the proof of [Reference KalethaKal4, Proposition 8.2], setwise splittings of (2.33) and (2.35) are chosen. A setwise splitting of (2.27) can then be obtained essentially as the product of these two splittings. It follows that the 2-cocycle classifying (2.27) is the sum of the 2-cocycles classifying (2.33) and (2.35), which means that (2.27) is isomorphic to the Baer sum of the other two extensions.

(b) As the difference between the extensions in part (a) and those in part (b) is given by pushout along $\theta $ in all three cases, the isomorphism here is a direct consequence of part (a). The construction of this Baer sum takes place in the category of groups with an action of $W(N_{\mathcal G^\flat }({\mathcal {L}}^\flat ), {\mathcal {T}}^\flat )(F)_{[x],\theta }$ , with the actions given above of this lemma.

Proof. First we reduce to the case of finite reductive groups. Let $P_y^\flat \subset G^\flat $ be the parahoric subgroup associated to the special vertex y from Lemma 2.6 (b). Similar to the extension (2.31), we consider the extension

(2.37) $$ \begin{align} 1 \to P_y^\flat \cap T^\flat \to P_y^\flat \cap N_{G^\flat}(L^\flat,T^\flat)_{\theta_{\mathfrak{f}}} \to W(N_{G^\flat}(L^\flat), T^\flat)_{\theta_{\mathfrak{f}}} \to 1. \end{align} $$

By Lemma 2.6 (c), pullback of (2.37) along $W(L^\flat , T^\flat )_{\theta } \to W(N_{G^\flat }(L^\flat ), T^\flat )_{\theta _{\mathfrak {f}}}$ , followed by pushout along $\theta _{\mathfrak {f}} : P_y \cap T^\flat \to {\mathbb {C}}^\times $ , recovers the extension (2.32). Since $\theta _{\mathfrak {f}}$ has depth zero, the pushout of (2.37) along $\theta _{\mathfrak {f}}$ can also be obtained from

(2.38) $$ \begin{align} 1 \to {\mathcal{T}}_{\mathfrak{f}} (k_F) \to N_{\mathcal G_y^\circ}({\mathcal{L}}_y^\circ,{\mathcal{T}}_{\mathfrak{f}})(k_F)_{\theta_{\mathfrak{f}}} \to W(N_{\mathcal G_y^\circ}({\mathcal{L}}_y^\circ), {\mathcal{T}}_{\mathfrak{f}})(k_F)_{\theta_{\mathfrak{f}}} \to 1. \end{align} $$

The image of $W(N_{G^\flat }(L^\flat ), T^\flat )_{\theta _{\mathfrak {f}}}$ in $W(\mathcal G_y^\circ , {\mathcal {T}}_{\mathfrak {f}})(k_F)$ is contained in $W(N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ ), {\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}}$ . If we can establish a $W(\mathcal G_y^\circ , {\mathcal {T}}_{\mathfrak {f}})(k_F)$ -equivariant splitting of (2.38), then we can extend it $W(N_{G^\flat }(L^\flat ), T^\flat )_{\mathfrak {X}^0 (L) \theta _{\mathfrak {f}}}$ -equivariantly to the versions of (2.38) for other $\theta _{\mathfrak {f}}$ .

Thus it suffices to construct an $N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}}$ -equivariant setwise splitting of

(2.39) $$ \begin{align} 1 \to {\mathcal{T}}_{\mathfrak{f}} (k_F) \to N_{{\mathcal{L}}_y^\circ}({\mathcal{T}}_{\mathfrak{f}})(k_F)_{\theta_{\mathfrak{f}}} \to W({\mathcal{L}}_y^\circ, {\mathcal{T}}_{\mathfrak{f}})(k_F)_{\theta_{\mathfrak{f}}} \to 1, \end{align} $$

which becomes a group homomorphism in the following pushout along $\theta _{\mathfrak {f}}$ :

(2.40) $$ \begin{align} 1 \to {\mathbb{C}}^\times \to \mathcal E_{\theta_{\mathfrak{f}}}^0 \to W({\mathcal{L}}_y^\circ, {\mathcal{T}}_{\mathfrak{f}})(k_F)_{\theta_{\mathfrak{f}}} \to 1. \end{align} $$

The existence of such a splitting was shown in [Reference KalethaKal3, Lemma 4.5.6 and Corollary 4.5.7]; it remains to prove its $N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}}$ -equivariance.

We denote the derived group of $\mathcal G$ by $\mathcal G_{\mathrm {der}}$ , and its simply connected cover by $\mathcal G_{\mathrm {sc}}$ . For disconnected algebraic groups, we define the derived and simply connected analogues by first passing to the neutral component of the group. Let ${\mathcal {T}}_{\mathfrak {f},c}$ be the preimage of ${\mathcal {T}}_{\mathfrak {f}}$ in $\mathcal G_{y,\mathrm {sc}}$ . Then $\theta _{\mathfrak {f}}$ can be pulled back to a character $\theta _{\mathfrak {f},c}$ of ${\mathcal {T}}_{\mathfrak {f},c}(k_F)$ . The preimage ${\mathcal {L}}_{y,c}$ of ${\mathcal {L}}_y^\circ $ in $\mathcal G_{y,\mathrm {sc}}$ is a Levi subgroup of the latter, so the derived subgroup of ${\mathcal {L}}_{y,c}$ is simply connected and we denote it by ${\mathcal {L}}_{y,\mathrm {sc}}$ . We write ${\mathcal {T}}_{\mathfrak {f},\mathrm {sc}} = {\mathcal {T}}_{\mathfrak {f},c} \cap {\mathcal {L}}_{\mathfrak {f},\mathrm {sc}}$ . By pushout along $\theta _{\mathfrak {f},c} : {\mathcal {T}}_{\mathfrak {f},c} (k_F) \to {\mathbb {C}}^\times $ and pullback along $W({\mathcal {L}}_y^\circ , {\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}} \to W({\mathcal {L}}_{y,c}, {\mathcal {T}}_{\mathfrak {f},c})(k_F)_{\theta _{\mathfrak {f},c}}$ of the extension

(2.41) $$ \begin{align} 1 \to {\mathcal{T}}_{\mathfrak{f},c}(k_F) \to N_{{\mathcal{L}}_{y,c}}({\mathcal{T}}_{\mathfrak{f},c})(k_F)_{\theta_{\mathfrak{f},c}} \to W({\mathcal{L}}_{y,c}, {\mathcal{T}}_{\mathfrak{f},c})_{\theta_{\mathfrak{f},c}} \to 1, \end{align} $$

one can obtain (2.40). The group $N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}}$ still acts by conjugation on (2.41), and we need to keep track of equivariance for that action. This may be replaced by the conjugation action of $N_{\mathcal G_{y,\mathrm {sc}}}({\mathcal {L}}_{\mathfrak {f},c},{\mathcal {T}}_{\mathfrak {f},c})(k_F)_{\theta _{\mathfrak {f},c}}$ , because the latter group has a larger image in $\mathcal G_{y,\mathrm {der}}(k_F)$ . Therefore we may assume without loss of generality that $\mathcal G_y^\circ $ is simply connected. Then (2.38) decomposes as a direct product of the analogous extensions for the $k_F$ -simple factors of $\mathcal G_y^\circ $ , thus we may assume without loss of generality that $\mathcal G_y^\circ $ is in addition $k_F$ -simple. By passing to a finite field extension of $k_F$ , we can make $\mathcal G_y^\circ $ absolutely simple.

From now on, we assume that $\mathcal G_y^\circ $ is absolutely simple and simply connected.

By Lemma 2.5, $W({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}}$ commutes with $N_{\mathcal G_y^\circ } ({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}} / {\mathcal {T}}_{\mathfrak {f}} (k_F)$ in the group $W (\mathcal G_y^\circ , {\mathcal {T}}_{\mathfrak {f}})(k_F)$ . Choose a pinning of $\mathcal G_y^\circ $ , associated to a maximal torus of ${\mathcal {L}}_y^\circ $ and stable under the action of $\mathrm {Frob}$ used to define $\mathcal G_y^\circ $ as $k_F$ -group. Then the Levi subgroup ${\mathcal {L}}_y^\circ $ is $\mathrm {Frob}$ -stable, and $\mathrm {Frob}$ stabilizes the pinning of ${\mathcal {L}}_y^\circ $ obtained by restriction from $\mathcal G_y^\circ $ . We can write $N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ ) = {\mathcal {L}}_y^\circ \rtimes \Gamma $ , where $\Gamma $ is a finite group of pinning-preserving automorphisms of ${\mathcal {L}}_y^\circ $ . For $\gamma \in \Gamma \subset N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ )$ , the element $\mathrm {Frob} \gamma \mathrm {Frob}^{-1}$ preserves the pinning and thus again lies in $\Gamma $ . In other words, $\mathrm {Frob}$ acts on the subgroup $\Gamma $ of $N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ )$ , and that action coincides with the $\mathrm {Frob}$ -action on $N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ ) / {\mathcal {L}}_y^\circ \cong \Gamma $ . This implies

$$\begin{align*}N_{\mathcal G_y^\circ}({\mathcal{L}}_y^\circ)(k_F) = N_{\mathcal G_y^\circ}({\mathcal{L}}_y^\circ)^{\mathrm{Frob}} = ({\mathcal{L}}_y^\circ \rtimes \Gamma)^{\mathrm{Frob}} = {\mathcal{L}}_y^{\circ,\mathrm{Frob}} \rtimes \Gamma^{\mathrm{Frob}} = {\mathcal{L}}_y^\circ (k_F) \rtimes \Gamma^{\mathrm{Frob}}. \end{align*}$$

Let ${\mathcal {L}}_{y,i}$ be an almost direct factor of ${\mathcal {L}}_y^\circ $ , coming from one $N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ )(k_F) \times \langle \mathrm {Frob} \rangle $ -orbit of simple factors of ${\mathcal {L}}_y^\circ $ . The group ${\mathcal {L}}_{y,\mathrm {der}}^\circ = {\mathcal {L}}_{y,\mathrm {sc}}$ is simply connected, thus is equal to the direct product of these ${\mathcal {L}}_{y,i}$ ’s. The group ${\mathcal {L}}_y^\circ $ is an almost direct product of ${\mathcal {L}}_{y,\mathrm {der}}^\circ $ and $Z({\mathcal {L}}_y^\circ )^\circ $ . For each i, let $\Gamma _i$ be the image of $\Gamma $ in the automorphism group of ${\mathcal {L}}_{y,i}$ . We write ${\mathcal {L}}_{y,i}^+ := {\mathcal {L}}_{y,i} \rtimes \Gamma _i$ . Similarly, we can define $\Gamma _Z \subset \mathrm {Aut}(Z_{\mathcal G_y^\circ }({\mathcal {L}}_{y,\mathrm {der}}^\circ )^\circ )$ and write

$$\begin{align*}Z_{\mathcal G_y^\circ}({\mathcal{L}}_{y,\mathrm{der}}^\circ)^+ := Z_{\mathcal G_y^\circ}({\mathcal{L}}_{y,\mathrm{der}}^\circ)^\circ \rtimes \Gamma_Z. \end{align*}$$

Then there is a natural embedding

(2.42) $$ \begin{align} W(N_{\mathcal G_y^\circ}({\mathcal{L}}_y^\circ), {\mathcal{T}}_{\mathfrak{f}}) \hookrightarrow W \big( Z_{\mathcal G_y^\circ}({\mathcal{L}}_{y,\mathrm{der}}^\circ)^+, Z({\mathcal{L}}_y^\circ)^\circ \big) \times \prod\nolimits_i \, W ({\mathcal{L}}_{y,i}^+, {\mathcal{T}}_{\mathfrak{f},i}) , \end{align} $$

where ${\mathcal {T}}_{\mathfrak {f},i} := {\mathcal {T}}_{\mathfrak {f}} \cap {\mathcal {L}}_{y,i}$ . We define $W ({\mathcal {L}}_{y,i}^+, {\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f}}}$ as the image of $W(N_{\mathcal G_y^\circ }({\mathcal {L}}_y^\circ ), {\mathcal {T}}_{\mathfrak {f}})_{\theta _{\mathfrak {f}}}$ under (2.42) followed by projection onto the i-th coordinate. This group satisfies

(2.43) $$ \begin{align} W({\mathcal{T}}_{\mathfrak{f}} {\mathcal{L}}_{y,i}, {\mathcal{T}}_{\mathfrak{f}})_{\theta_{\mathfrak{f}}} \; \subset \; W ({\mathcal{L}}_{y,i}^+, {\mathcal{T}}_{\mathfrak{f},i})_{\theta_{\mathfrak{f}}} \; \subset \; W ({\mathcal{L}}_{y,i}^+, {\mathcal{T}}_{\mathfrak{f},i})_{\theta_{\mathfrak{f},i}} , \end{align} $$

where $\theta _{\mathfrak {f},i} := \theta _{\mathfrak {f}} |_{{\mathcal {T}}_{\mathfrak {f},i} (k_F)}$ . In general, both of the above inclusions can be proper. The upshot of the construction is that Lemma 2.7 still applies. For every i, we can construct an extension similar to (2.38) as follows:

(2.44) $$ \begin{align} 1 \to {\mathcal{T}}_{\mathfrak{f},i}(k_F) \to N_{{\mathcal{L}}_{y,i}^+}({\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}} \to W ({\mathcal{L}}_{y,i}^+, {\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}} \to 1, \end{align} $$

where the subscript $\theta _{\mathfrak {f}}$ is defined as above and it guarantees the exactness of the above sequence (2.44). The extension (2.40) can be recovered from the extensions (2.44) for all i. Since the action of $N_{\mathcal G_y^\circ } ({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}}$ still shows up in (2.40), it suffices to construct splittings of

(2.45) $$ \begin{align} 1 \to {\mathcal{T}}_{\mathfrak{f},i}(k_F) \to N_{{\mathcal{L}}_{y,i}}({\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}} \to W ({\mathcal{L}}_{y,i}, {\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}} \to 1, \end{align} $$

which are invariant for the conjugation action of $W ({\mathcal {L}}_{y,i}^+, {\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ and become group homomorphisms in the following pushout along $\theta _{\mathfrak {f},i}$ :

(2.46) $$ \begin{align} 1 \to {\mathbb{C}}^\times \to \mathcal E_{\theta_{\mathfrak{f},i}}^0 \to W ({\mathcal{L}}_{y,i}, {\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}} \to 1. \end{align} $$

The existence of such a splitting was shown in [Reference KalethaKal3, Lemma 4.5.6 and Corollary 4.5.7]; it remains to show the invariance. The $N_{{\mathcal {L}}_{y,i}^+}({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ -invariants in the pushout of (2.45) are canonically isomorphic to the invariants in the version of (2.45) for one $k_F$ -simple factor of ${\mathcal {L}}_{y,i}$ except with invariants under a subgroup of $N_{{\mathcal {L}}_{y,i}^+}({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ . Therefore we may assume without loss of generality that in (2.45), ${\mathcal {L}}_{y,i}$ is $k_F$ -simple and simply connected.

Furthermore, ${\mathcal {L}}_{y,i}$ is the scalar restriction of a simple group ${\mathcal {L}}^{\prime }_{y,i}$ over a field extension $k'$ of $k_F$ . We may replace without loss of generality $k_F$ by $k'$ and $L_{y,i}$ by ${\mathcal {L}}^{\prime }_{y,i}$ . Thus we can reduce to the case where ${\mathcal {L}}_{y,i}$ is absolutely simple and simply connected. Lemma 2.7 still applies, and shows that

(2.47) $$ \begin{align} W ({\mathcal{L}}_{y,i}, {\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}} \text{ is central in } W ({\mathcal{L}}_{y,i}^+, {\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}}. \end{align} $$

Since $W(\mathcal G_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})$ is a Weyl group containing $W({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})$ , and $N_{\mathcal G_y^\circ } ({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})(k_F)_{\theta _{\mathfrak {f}}} / {\mathcal {T}}_{\mathfrak {f}} (k_F)$ normalizes $W({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})$ , the actions of $N_{{\mathcal {L}}_{y,i}^+}({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ on (2.45) and (2.46) are constructed from:

• • conjugation by elements of $N_{{\mathcal {L}}_{y,i}}({\mathcal {T}}_{\mathfrak {f},i})(k_F)$ ;

• • for each Dynkin diagram automorphism $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ , at most one coset of $N_{{\mathcal {L}}_{y,i}}({\mathcal {T}}_{\mathfrak {f},i})(k_F)$ .

We now check case-by-case.

Case I. $N_{{\mathcal {L}}_{y,i}^+}({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ acts on ${\mathcal {T}}_{\mathfrak {f},i}$ as conjugation by elements of $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ .

This holds whenever $W({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})$ has type $A_1, B_n, C_n, E_7, E_8, F_4$ or $G_2$ , because then the only Dynkin diagram automorphism is the identity. Then the action of $N_{{\mathcal {L}}_{y,i}^+}({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ can be viewed as coming from elements of $N_{{\mathcal {L}}_{y,i}}({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ . Hence any splitting of (2.46), as in [Reference KalethaKal3, §4.5], is $N_{{\mathcal {L}}_{y,i}^+}({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ -invariant.

Case II. $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ has type $E_6$ .

If $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}}$ is trivial, there is nothing to prove. Otherwise, we have $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f}}} \cong {\mathbb {Z}} / 3 {\mathbb {Z}}$ . However, its image in the automorphism group of the affine Dynkin diagram of $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ does not commute with the nontrivial diagram automorphism $\tau $ of $E_6$ , thus by (2.47), $\tau $ does not play a role in the picture. We conclude as in Case I.

Case III. $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ has type $A_{n-1}$ with $n> 2$ .

If ${\mathcal {L}}_{y,i}$ is an outer form of a split group, then $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})^{\mathrm {Frob}}$ becomes a Weyl group of type $B_{n-1}$ or $C_{n-1}$ . The associated root system does not admit nontrivial diagram automorphisms, thus we reduce back to case I. Therefore we may assume without loss of generality that ${\mathcal {L}}_{y,i}$ is split, and thus it is isomorphic to ${\mathrm {SL}}_n$ . By a change of coordinates for ${\mathrm {SL}}_n$ , we may assume without loss of generality that ${\mathcal {T}}_{\mathfrak {f},i}$ is the diagonal torus in ${\mathcal {L}}_{y,i}$ , with $\mathrm {Frob}$ -action given by the $q_F$ -th power map composed with conjugation by an elliptic element $F_A \in W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ . With respect to these coordinates, this is also how $\mathrm {Frob}$ acts on ${\mathrm {SL}}_n$ . Since elliptic elements in $S_n$ are n-cycles, we may assume that $F_A = (1 \, 2 \ldots n)$ . Note that ${\mathcal {T}}_{\mathfrak {f},i}$ splits over the degree n extension $k'$ of $k_F$

By (2.18), one can classify the possibilities for $\theta _{\mathfrak {f},i}$ in terms of points of a fundamental alcove. The non-singularity of $\theta _{\mathfrak {f}}$ and (2.18) force that $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f}}}$ is either trivial or comes from the barycenter of an alcove, in which case we have

$$\begin{align*}W ({\mathcal{L}}_{y,i},{\mathcal{T}}_{\mathfrak{f},i})_{\theta_{\mathfrak{f},i}} = \langle (1\,2 \ldots n) \rangle. \end{align*}$$

More precisely, ${\mathrm {GL}}_1 (k')$ admits an order n character represented by $\zeta _n \in k'$ , and $\theta _{\mathfrak {f},i}$ can be represented by $\mathrm {diag}(1,\zeta _n,\ldots , \zeta _n^{-1}) \in {\mathrm {PGL}}_n (k')$ . However, $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}}$ does not commute with the nontrivial diagram automorphism $\tau $ of $S_n$ ; from their actions on the affine Dynkin diagram of type $A_{n-1}$ , we see that $\tau $ acts by inversion. By (2.47), $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f}}}$ can only contain elements of $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}}$ of order $\leq 2$ . If $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f}}}$ is trivial, then there is nothing to show; thus we may assume that n is even and that $W ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f}}} = \langle (1 \, 2 \ldots n)^{n/2} \rangle $ . The group ${\mathcal {L}}_{y,i}^+ := {\mathcal {L}}_{y,i} \rtimes \Gamma $ is equal to ${\mathrm {SL}}_n \rtimes \langle -\top \rangle $ , where $-\top $ denotes the inverse transpose automorphism, because the nontrivial element of $\Gamma $ acts by $-\top $ composed with conjugation by an element of ${\mathrm {SL}}_n$ (because n is even). One can check that $W ({\mathcal {L}}_{y,i}^+,{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f}}} = \langle w_1, w_2 \rangle \cong ({\mathbb {Z}} / 2 {\mathbb {Z}})^2$ , where $w_1 = (1 \, 2 \ldots n)^{n/2}$ and $w_2 = -\top \circ (2 \, n) (3 \, n\text { - }1) \cdots (n/2 \, n/2 \!+\! 2)$ . However, the element $w_2$ does not commute with $\mathrm {Frob}$ in $W({\mathcal {L}}_{y,i}^+,{\mathcal {T}}_{\mathfrak {f},i})$ . Hence we have

$$\begin{align*}N_{{\mathcal{L}}_{y,i}^+}({\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}} = N_{{\mathcal{L}}_{y,i}}({\mathcal{T}}_{\mathfrak{f},i})(k_F)_{\theta_{\mathfrak{f}}}. \end{align*}$$

Case IV. $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ has type $D_n$ with $n>4$ .

Now $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}}$ embeds in ${\mathbb {Z}} / 4{\mathbb {Z}}$ (for n odd) or in ${\mathbb {Z}} / 2 {\mathbb {Z}} \times {\mathbb {Z}} / 2{\mathbb {Z}}$ (for n even). If we are not in Case I, the action of $N_{{\mathcal {L}}_{y,i}^+}({\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}}$ on the Dynkin diagram $D_n$ uses the nontrivial automorphism $\epsilon _n$ . We realize ${\mathcal {L}}_{y,i}$ as a spin group on a vector space of dimension $2n$ , and ${\mathcal {T}}_{\mathfrak {f},i}$ as the diagonal torus. Then $\epsilon _n$ becomes the reflection in the n-th coordinate of ${\mathcal {T}}_{\mathfrak {f},i}$ . It only fixes one nontrivial element of $X^* ({\mathcal {T}}_{\mathfrak {f},i}) / {\mathbb {Z}} R ({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ , thus $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}}$ has order two.

Again by (2.18), we have a classification of the possible $\theta _{\mathfrak {f},i}$ via points in a fundamental alcove. Via conjugation, we can reduce to the following situation: the character $\theta _{\mathfrak {f},i}$ of ${\mathcal {T}}_{\mathfrak {f},i} (k_F)$ has trivial restriction to the first coordinate and quadratic restriction to the n-th coordinate, while the restrictions to the other coordinates (as well as their inverses) differ and have higher order. Then $W({\mathcal {L}}_{y,i}^+,{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}} = \langle \epsilon _1 , \epsilon _n \rangle $ and $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}} = \langle \epsilon _1 \epsilon _n \rangle $ . Recall that (2.45) has a splitting, say it sends $\epsilon _1 \epsilon _n$ to $s_1 s_n$ with $s_1, s_n \in {\mathcal {L}}_{y,i}(k_F)$ representing reflections in these coordinates and $s_1^2 = s_n^{-2} \in Z({\mathcal {L}}_{y,i})(k_F)$ . Then $\epsilon _n$ acts as conjugation by $s_n$ on ${\mathcal {L}}_y^\circ $ , and fixes $s_1 s_n$ .

Case V. $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})$ has type $D_4$ .

In the automorphism group of the affine Dynkin diagram of $D_4$ , the order-three automorphisms of $D_4$ do not commute with (the image of) any nontrivial element of $W({\mathcal {L}}_{y,i},{\mathcal {T}}_{\mathfrak {f},i})_{\theta _{\mathfrak {f},i}} \subset {\mathbb {Z}} / 2 {\mathbb {Z}} \times {\mathbb {Z}} / 2{\mathbb {Z}}$ . If the action of $N_{{\mathcal {L}}_{y,i}^+} ({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f},i}}$ on the Dynkin diagram $D_4$ includes such exceptional automorphism, then $W({\mathcal {L}}_y^\circ ,{\mathcal {T}}_{\mathfrak {f}})_{\theta _{\mathfrak {f}}}$ is trivial and there is nothing to show. Otherwise either we are in Case I, or the action of $N_{{\mathcal {L}}_{y,i}^+} ({\mathcal {T}}_{\mathfrak {f},i})(k_F)_{\theta _{\mathfrak {f},i}}$ uses exactly one nontrivial diagram automorphism $\psi $ , of order two. But $\psi $ is conjugate to the standard Dynkin diagram automorphism $\epsilon _n$ by another diagram automorphism $\tau $ of $D_4$ . Conjugating everything by this $\tau $ brings us back to Case IV, which works just as well for $n = 4$ with these simplifications.

Proof. Consider a depth-zero supercuspidal L-parameter $\varphi $ for G. By [Reference KalethaKal3, Lemma 4.1.3.2], $Z_{G^\vee }(\varphi (\mathbf I_F))^\circ $ is a torus. (The torally wild assumption in [Reference KalethaKal3] is not needed in the proof.) Note that $Z_{G^\vee }(\varphi (\mathbf I_F)) \subset M^\vee $ because $\varphi (\mathbf P_F) = \mathbf P_F$ . By the proof of [Reference KalethaKal2, Lemma 5.2.2.2], we have that $T_M^\vee := Z_{M^{\vee ,\circ }}\big ( Z_{G^\vee } (\varphi (\mathbf I_F))^\circ \big )$ is a maximal torus of $M^\vee $ , normalized by $\varphi (\mathbf W_F)$ and contained in a Borel subgroup of $M^\vee $ normalized by $\varphi (\mathbf I_F)$ . Upon conjugating $\varphi $ by a suitable element of $M^\vee $ , we may assume without loss of generality that $T_M^\vee $ is contained in a $\mathbf W_F$ -stable Borel subgroup of $M^\vee $ , s.t.

(3.1) $$ \begin{align} \varphi (\mathbf W_F) \subset N_{M^\vee}(T_M^\vee) \rtimes \mathbf W_F. \end{align} $$

Now, Ad $(\varphi )$ gives an action of $\mathbf W_F / \mathbf P_F$ on $T_M^\vee $ , and the F-torus ${\mathcal {T}}_M$ dual to $T_M^\vee $ (with this $\mathbf W_F$ -action) is tamely ramified. Since $\varphi $ is discrete, $Z_{T_M^\vee }(\varphi (\mathbf W_F)) / Z(G^\vee )^{\mathbf W_F}$ is finite and ${\mathcal {T}}_M / Z(\mathcal G)$ is elliptic. By [Reference KalethaKal3, Remark 4.1.5], there exist canonical tamely ramified $\chi $ -data for $R(M^\vee ,T_M^\vee )$ . As in [Reference Langlands and ShelstadLaSh] and [Reference KalethaKal4, §6.1], these $\chi $ -data yield an embedding ${}^L T_M / \mathbf P_F \hookrightarrow M^\vee \rtimes \mathbf W_F / \mathbf P_F$ , whose $M^\vee $ -conjugacy class is canonical. It inflates to an L-embedding

(3.2) $$ \begin{align} {}^L j_M : {}^L T_M \hookrightarrow M^\vee \rtimes \mathbf W_F \: \subset \: {}^L G, \end{align} $$

such that ${}^L j_M (\mathbf W_F)$ stabilizes a pinning of $M^\vee $ . By [Reference KalethaKal3, Lemma 4.1.11], we may assume that

(3.3) $$ \begin{align} {}^L j_M (1,x) = (1,x) \text{ for all } x \in \mathbf I_F. \end{align} $$

For any embedding $j_M : {\mathcal {T}}_M \to \mathcal G$ whose dual L-homomorphism is ( $G^\vee $ -conjugate to) ${}^L j_M$ , we have

(3.4) $$ \begin{align} j_M (T_M) \text{ is a maximal tamely ramified torus of } G. \end{align} $$

In particular, $Z_{\mathcal G}(j_M ({\mathcal {T}}_M))$ is a maximal F-torus of $\mathcal G$ . Hence $T^\vee := Z_{G^\vee }(T_M^\vee )$ is a maximal torus of $G^\vee $ , and it is stable under Ad $({}^L j_M ({}^L T_M))$ -action because $T_M^\vee $ is. We denote ${}^L j_M$ with target ${}^L G$ by ${}^L j$ . Then

$$\begin{align*}{}^L T := T^\vee \rtimes {}^L j (\mathbf W_F) \end{align*}$$

is the L-group of an F-torus ${\mathcal {T}}$ . Since $\varphi $ is discrete, $Z_{T^\vee }(\varphi (\mathbf W_F)) / Z(G^\vee )^{\mathbf W_F}$ is finite and ${\mathcal {T}} / Z(\mathcal G)$ is elliptic.

Note that ${}^L j (\mathbf W_F)$ normalizes a Borel subgroup of $G^\vee $ , that is, the group generated by $T^\vee = Z_{G^\vee }(T_M^\vee )$ and the root subgroups $U_{\alpha ^\vee }$ for $\alpha ^\vee \in R (G^\vee ,T^\vee )$ such that $\alpha ^\vee |_{T_M^\vee }$ is positive with respect to the ${}^L j_M (\mathbf W_F)$ -stable Borel subgroup of $M^\vee $ from (3.1). The same arguments as for ${}^L j_M$ ensure that ${}^L j (\mathbf W_F)$ stabilizes a pinning of $G^\vee $ .

By construction, $\varphi $ factors as ${}^L j_M \circ \varphi _{T_M}$ , where $\varphi _{T_M} : \mathbf W_F \to {}^L T_M$ . Via the LLC for tori [Reference LanglandsLan2, Reference YuYu], $\varphi _{T_M}$ yields a character $\theta _{T_M}$ of $T_M$ . The LLC for tori preserves depth zero [Reference Solleveld and XuSoXu, Proposition 1.3], so $\theta _{T_M}$ has depth zero. We can also write that $\varphi $ factors as ${}^L j \circ \varphi _T$ , where $\varphi _T \in \Phi ^0 (T)$ . Then $\varphi _T$ determines a depth-zero character $\theta $ of T that extends $\theta _{T_M}$ . Since $Z_{G^\vee }(\varphi (\mathbf I_F))^\circ $ is a torus, by [Reference KalethaKal3, Lemma 4.1.10], $\theta _{T_M}$ is F-nonsingular. By (3.4), this implies that $\theta $ is F-nonsingular as well. More precisely, for any embedding $j : T \to G$ associated to ${}^L j$ , we see that $\theta $ determines a non-singular depth-zero character $j_* \theta $ of $j(T)$ .

Proof. The argument for bijectivity is given in [Reference KalethaKal1, Proposition 5.2.7] and [Reference KalethaKal3, Proposition 4.1.8]. With Lemma 3.1 at hand can apply these proofs in the special case of depth-zero L-parameters, then the tame ramification assumption on T and G in [Reference KalethaKal1, Reference KalethaKal3] is not needed.

Proof. Via (3.10) and the similar isomorphism

(3.11) $$ \begin{align} W (N_{G^\vee}(L^\vee),T^\vee )^{\mathbf W_F}_{\varphi_T} \cong W(N_{\mathcal G} ({\mathcal{L}}),{\mathcal{T}}) (F)_\theta , \end{align} $$

the desired statement is equivalent to Lemma 2.5.

Proof. (a) The following is shown in the proof of [Reference KalethaKal4, Proposition 8.2]. One has setwise splittings of (3.21) and (3.25), from which one constructs 2-cocycles in $Z^2 (W(L^\vee ,T^\vee )^{\mathbf W_F}_{\eta ,\phi _T})$ that classify these two extensions. Then the product of these 2-cocycles classifies the extension (3.17). Translating back from 2-cocycles to extensions establishes the desired isomorphism. To these arguments we only have to add that they all take place in the category of $\tilde N_{\varphi ,\eta }$ -groups.

(b) This is a direct consequence of part (a) and the earlier observations that, upon pushout along $\eta $ , the $\tilde N_{\varphi ,\eta }$ -actions on the said extensions factor through (3.19).

Proof. It suffices to show that (3.26) admits a $W (N_{G^\vee }(L^\vee ),T^\vee )^{\mathbf W_F}_{\eta ,\varphi _T}$ -equivariant splitting, for then the other required splittings are obtained by conjugating with elements of $W (N_{G^\vee }(L^\vee ),T^\vee )^{\mathbf W_F}_{\eta , \mathfrak {X}^0 (L^\vee ) \varphi _T}$ . By Lemma 2.6 (c), we have

(3.27) $$ \begin{align} W(G^\vee,T^\vee)^{\mathbf W_F} \cong W(\mathcal G,{\mathcal{T}})(F) \cong W(\mathcal G_y^\circ,{\mathcal{T}}_{\mathfrak{f}})(k_F). \end{align} $$

Since the group $\mathcal G_y^\circ $ and its maximal torus ${\mathcal {T}}_{\mathfrak {f}}$ split over an unramified extension of F, (3.27) shows that we can realize $W(G^\vee ,T^\vee )^{\mathbf W_F}$ in $W(G^{\vee ,\mathbf I_F,\circ }, T^{\vee ,\mathbf I_F,\circ })^{\mathbf W_F}$ . (Recall from (3.3) that $\mathbf I_F$ acts in the same way on $T^\vee $ and on $G^\vee $ .) Furthermore, as in the proof of Proposition 2.10, we may replace $\theta $ by its restriction $\theta _{\mathfrak {f}}$ to ${\mathcal {T}}_{\mathfrak {f}} (\mathfrak o_F)$ , which factors via ${\mathcal {T}}_{\mathfrak {f}} (k_F)$ . This makes the stabilizer of $\theta $ bigger but preserves the non-singularity. On the dual side, it means that we may replace $\varphi $ by $\varphi |_{\mathbf I_F}$ , which has image in $T^{\vee ,\mathbf I_F,\circ } \times \mathbf I_F$ . If $\imath _F \in \mathbf I_F$ is a topological generator of $\mathbf I_F / \mathbf P_F$ , then $\varphi (\imath _F)$ , that is, the semisimple parameter of $\theta _{\mathfrak {f}}$ , and $\varphi (\mathbf I_F)$ have the same centralizer in $G^{\vee ,\mathbf I_F,\circ }$ . These replacements for $G^\vee , T^\vee $ and $\varphi $ tell us that it suffices to prove the proposition assuming $\mathbf I_F$ acts trivially on $G^\vee $ , and with the centralizer of $\varphi (\imath _F) \in T^\vee $ instead of the centralizer of $\varphi $ .