1.1 Some results of Roche [Reference RocheRoc98]

Let $\mathfrak{o}$ be the ring of integers of $F$, $\mathfrak{p}$ the maximal ideal of $\mathfrak{o}$, and $\mathfrak{K}=\mathfrak{o}/\mathfrak{p}$ the residue field. We denote by $|\,\cdot \,|_{F}$ the absolute value on $F$ normalized by $|\unicode[STIX]{x1D71B}|_{F}=q^{-1}$ for a uniformizer element $\unicode[STIX]{x1D71B}$ of $F$, where $q$ is the number of elements of $\mathfrak{K}$. Let $(B,T)$ be a Borel pair of $G$, defined and split over $F$, and $U$ the radical unipotent of $B$. Let $X$ (respectively, $\check{X}$) be the group of algebraic characters (respectively, co-characters) of $T$. Let $T=T(\mathfrak{o})$ be the maximal compact subgroup of $T$, and $T_{+}\subset T$ the kernel of the natural morphism (reduction modulo $\mathfrak{p}$) $T(\mathfrak{o})\rightarrow T(\mathfrak{K})$. We fix a character $\unicode[STIX]{x1D712}_{0}$ of $T$. We assume that $\unicode[STIX]{x1D712}_{0}$ is depth zero, that is, trivial on $T_{+}$.

We fix a hyperspecial maximal compact subgroup $K$ of $G(F)$ corresponding to a hyperspecial vertex $x_{0}$ in the apartment of the Bruhat–Tits building of $G(F)$ associated to $T$. The base-point $x_{0}$ allows us to identify this apartment with the vector space $V=\check{X}\otimes _{\mathbb{Z}}\mathbb{R}$. Thus we have

$$\begin{eqnarray}K\,\cap \,B=(K\,\cap \,T)(K\,\cap \,U),\quad K\,\cap \,T=T.\end{eqnarray}$$

Let $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}(G,T)$ be the set of roots of $T$ in $G$, and $\check{\unicode[STIX]{x1D6F7}}$ the set of co-roots. Let $\unicode[STIX]{x1D6F7}_{+}=\unicode[STIX]{x1D6F7}(B,T)$ be the set of roots of $T$ in $B$, and $\unicode[STIX]{x1D6E5}\subset \unicode[STIX]{x1D6F7}_{+}$ the corresponding basis of $\unicode[STIX]{x1D6F7}$. Let $\unicode[STIX]{x2110}\subset K$ be the Iwahori subgroup of $G(F)$ corresponding to the alcove

$$\begin{eqnarray}A=\{v\in V:0<\unicode[STIX]{x1D6FC}(v)<1,\,\forall \unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}_{+}\},\quad V=\check{X}\otimes _{\mathbb{Z}}\mathbb{R}.\end{eqnarray}$$

Hence $\unicode[STIX]{x2110}$ is in good position with respect to the Borel pair $(B,T)$: we have the triangular decomposition

$$\begin{eqnarray}\unicode[STIX]{x2110}=(\unicode[STIX]{x2110}\,\cap \,\overline{U})(\unicode[STIX]{x2110}\,\cap \,T)(\unicode[STIX]{x2110}\,\cap \,U)\end{eqnarray}$$

where $\overline{U}$ is the unipotent radical of the Borel subgroup $\overline{B}$ opposite to $B$ with respect to $T$. Moreover, we have $T=\unicode[STIX]{x2110}\,\cap \,T$ and $T_{+}=\unicode[STIX]{x2110}_{+}\,\cap \,T$, where $\unicode[STIX]{x2110}_{+}$ is the pro-unipotent radical of $\unicode[STIX]{x2110}$, and the inclusion $T\subset \unicode[STIX]{x2110}$ induces an isomorphism $T/T_{+}\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x2110}/\unicode[STIX]{x2110}_{+}$. So the character $\unicode[STIX]{x1D712}_{0}$ of $T$ can be viewed as a one-dimensional representation of $\unicode[STIX]{x2110}$, denoted by $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D712}_{0}}$. It is given by

$$\begin{eqnarray}\unicode[STIX]{x1D70C}(\bar{u}tu)=\unicode[STIX]{x1D712}_{0}(t),\quad \bar{u}\in \unicode[STIX]{x2110}\,\cap \,\overline{U},~t\in T,~u\in \unicode[STIX]{x2110}\,\cap \,U.\end{eqnarray}$$

Let $\unicode[STIX]{x1D712}$ be a characterFootnote ⁵ of $T(F)$ extending $\unicode[STIX]{x1D712}_{0}$. Let $\mathfrak{s}=\mathfrak{s}_{\unicode[STIX]{x1D712}_{0}}$ be the $G(F)$-inertial class $[T,\unicode[STIX]{x1D712}]_{G}$ of the cuspidal pair $(T,\unicode[STIX]{x1D712})$ of $G(F)$. It does not depend on the choice of $\unicode[STIX]{x1D712}$, and we know by Roche [Reference RocheRoc98] that the pair $(\unicode[STIX]{x2110},\unicode[STIX]{x1D70C})$ is an $\mathfrak{s}$-type in the Bushnell–Kutzko sense: for an irreducible (smooth, complex) representation $\unicode[STIX]{x1D70B}$ of $G(F)$, we have $\operatorname{Hom}_{\unicode[STIX]{x2110}}(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D70B}|_{\unicode[STIX]{x2110}})\neq 0$ if and only if $\unicode[STIX]{x1D70B}$ is isomorphic to a subquotient of a principal series representation $i_{B}^{G}(\unicode[STIX]{x1D712}\unicode[STIX]{x1D709})$ of $G(F)$ for some character $\unicode[STIX]{x1D709}$ of $T(F)$ trivial on $T$, where $i_{B}^{G}$ denotes the normalized parabolic induction functor.

Let $N=N_{G}(T)$ be the normalizer of $T$ in $G$, and $W=N/T\;(=N(F)/T(F))$ the Weyl group of $G$. We denote by $N(F)_{\unicode[STIX]{x1D712}_{0}}$ (respectively, $W_{\unicode[STIX]{x1D712}_{0}}$) the subgroup of $N(F)$ (respectively, $W$) formed by the elements which fix $\unicode[STIX]{x1D712}_{0}$. The group $N(F)_{\unicode[STIX]{x1D712}_{0}}$ contains $T(F)$, and we have$W_{\unicode[STIX]{x1D712}_{0}}=N(F)_{\unicode[STIX]{x1D712}_{0}}/T(F)$.

The character $\unicode[STIX]{x1D712}_{0}$ of $T$ extends to a ($W_{\unicode[STIX]{x1D712}_{0}}$-invariant) character $\tilde{\unicode[STIX]{x1D712}}$ of $N(F)_{\unicode[STIX]{x1D712}_{0}}$. In fact, $\unicode[STIX]{x1D712}_{0}$ extends to a character $\tilde{\unicode[STIX]{x1D712}}_{0}$ of the subgroup $N_{\unicode[STIX]{x1D712}_{0}}$ of $N=N(\mathfrak{o})$ formed by the elements which fix $\unicode[STIX]{x1D712}_{0}$ [Reference Howlett and LehrerHL80, 6.11]. The choice a uniformizer element $\unicode[STIX]{x1D71B}$ of $F$ gives an identification $N(F)=\check{X}\,\rtimes \,N$, which induces an identification $N(F)_{\unicode[STIX]{x1D712}_{0}}=\check{X}\,\rtimes \,N_{\unicode[STIX]{x1D712}_{0}}$. Let $\tilde{\unicode[STIX]{x1D712}}=\tilde{\unicode[STIX]{x1D712}}_{0}^{\unicode[STIX]{x1D71B}}$ be the character of $N(F)_{\unicode[STIX]{x1D712}_{0}}$ extending $\tilde{\unicode[STIX]{x1D712}}_{0}$ trivially on $\check{X}$. It is $W_{\unicode[STIX]{x1D712}_{0}}$-invariant by construction, and the restriction $\tilde{\unicode[STIX]{x1D712}}|_{T(F)}$ is the character $\unicode[STIX]{x1D712}_{0}^{\unicode[STIX]{x1D71B}}$ of $T(F)$ extending $\unicode[STIX]{x1D712}_{0}$ trivially on $\check{X}$ for the identification $T(F)=\check{X}\times T$ given by $\unicode[STIX]{x1D71B}$. From now on, we suppose $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}_{0}^{\unicode[STIX]{x1D71B}}$ and $\tilde{\unicode[STIX]{x1D712}}=\tilde{\unicode[STIX]{x1D712}}_{0}^{\unicode[STIX]{x1D71B}}$ for a character $\tilde{\unicode[STIX]{x1D712}}_{0}$ of $N_{\unicode[STIX]{x1D712}_{0}}$ extending $\unicode[STIX]{x1D712}_{0}$ and a uniformizer element $\unicode[STIX]{x1D71B}$ of $F$.

Let

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F7}^{\prime }=\{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}:\unicode[STIX]{x1D712}_{0}\circ \check{\unicode[STIX]{x1D6FC}}|_{\mathfrak{o}^{\times }}=1\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \check{\unicode[STIX]{x1D6F7}}^{\prime }=\{\check{\unicode[STIX]{x1D6FC}}\in \check{\unicode[STIX]{x1D6F7}}:\unicode[STIX]{x1D712}_{0}\circ \check{\unicode[STIX]{x1D6FC}}|_{\mathfrak{o}^{\times }}=1\}. & \displaystyle \nonumber\end{eqnarray}$$

Hence we have $\check{\unicode[STIX]{x1D6F7}}^{\prime }=\{\check{\unicode[STIX]{x1D6FC}}\in \unicode[STIX]{x1D6F7}:\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}^{\prime }\}$. We also denote by $W^{\prime }$ the subgroup of $W_{\unicode[STIX]{x1D712}_{0}}$ generated by the reflections $s_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}^{\prime }$. By definition, $\check{\unicode[STIX]{x1D6F7}}^{\prime }$ is a closed sub-root system of $\check{\unicode[STIX]{x1D6F7}}$. Hence $\unicode[STIX]{x1D6F7}^{\prime }$ is a sub-root system of $\unicode[STIX]{x1D6F7}$, with Weyl group $W^{\prime }$. Let $G^{\prime }$ be a connected reductive group with root datum $(X,\unicode[STIX]{x1D6F7}^{\prime },\check{X},\check{\unicode[STIX]{x1D6F7}}^{\prime })$, defined and split over $F$. We identify $T$ with a maximal split torus $T^{\prime }$ in $G^{\prime }$. Then $\unicode[STIX]{x1D6F7}_{+}^{\prime }=\unicode[STIX]{x1D6F7}^{\prime }\,\cap \,\unicode[STIX]{x1D6F7}_{+}$ is a subset of positive roots of $\unicode[STIX]{x1D6F7}^{\prime }$. It defines a basis $\unicode[STIX]{x1D6E5}^{\prime }$ of $\unicode[STIX]{x1D6F7}^{\prime }$, and a Borel subgroup $B^{\prime }$ of $G^{\prime }$ containing $T^{\prime }$. We fix a hyperspecial maximal compact subgroup $K^{\prime }=K_{G^{\prime }}$ of $G^{\prime }(F)$ corresponding to a hyperspecial vertex $x_{0}^{\prime }$ in the apartment of the Bruhat–Tits building of $G^{\prime }(F)$ associated to $T^{\prime }$. The base-point $x_{0}^{\prime }$ allows us to identify this apartment with $V\;(=\check{X}\otimes _{\mathbb{Z}}\mathbb{R})$. Let $\unicode[STIX]{x2110}^{\prime }=\unicode[STIX]{x2110}_{G^{\prime }}\subset K^{\prime }$ be the Iwahori subgroup of $G^{\prime }(F)$ corresponding to the alcove

$$\begin{eqnarray}A^{\prime }=\{v\in V:0<\unicode[STIX]{x1D6FC}(v)<1,\,\forall \unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}_{+}^{\prime }\}.\end{eqnarray}$$

Put $C_{\unicode[STIX]{x1D712}_{0}}=\{w\in W_{\unicode[STIX]{x1D712}_{0}}:w(\unicode[STIX]{x1D6F7}_{+}^{\prime })=\unicode[STIX]{x1D6F7}_{+}^{\prime }\}$. Then we have [Reference RocheRoc98, Lemma 8.1]

$$\begin{eqnarray}W_{\unicode[STIX]{x1D712}_{0}}=W^{\prime }\,\rtimes \,C_{\unicode[STIX]{x1D712}_{0}},\end{eqnarray}$$

and the choice of an $F$-épinglage of the based root datum $(X,\unicode[STIX]{x1D6F7}^{\prime },\unicode[STIX]{x1D6E5}^{\prime },\check{X},\check{\unicode[STIX]{x1D6F7}}^{\prime },\check{\unicode[STIX]{x1D6E5}}^{\prime })$ gives an embedding of $C_{\unicode[STIX]{x1D712}_{0}}$ in $\operatorname{Aut}_{F}(G^{\prime })$. We put

$$\begin{eqnarray}\widetilde{G}^{\prime }=G^{\prime }\,\rtimes \,C_{\unicode[STIX]{x1D712}_{0}}.\end{eqnarray}$$

Then we have [Reference RocheRoc98]:

1. (1) $G^{\prime }$ is an endoscopic group of $G$;

2. (2) $\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\simeq \unicode[STIX]{x210B}(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\,\widetilde{\otimes }\,\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]$;

3. (3) there is a support-preserving isomorphism of $\mathbb{C}$-algebras

   $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}:\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }});\end{eqnarray}$$

4. (4) if the scheme-theoretic center of $G$ is a torus, then we have $C_{\unicode[STIX]{x1D712}_{0}}=\{1\}$.

In (1), $G^{\prime }$ is the underlying group of an endoscopic datum $\text{}\underline{G}^{\prime }=(G^{\prime },G^{\prime },s)$ of $G$, where $s$ is an element of ${\check{T}}=\operatorname{Hom}(\check{X},\mathbb{C}^{\times })$ whose connected centralizer ${\check{G}}_{s}$ in ${\check{G}}$ (the complex dual group of $G$) is a connected reductive group over $\mathbb{C}$ with root system $(\check{X},\check{\unicode[STIX]{x1D6F7}}^{\prime },X,\unicode[STIX]{x1D6F7}^{\prime })$, and $G^{\prime }={\check{G}}_{s}\times W_{F}(\subset \text{}^{L}G={\check{G}}\times W_{F})$. In (2), the twisted tensor product $\widetilde{\otimes }$ is given by the conjugation action of $C_{\unicode[STIX]{x1D712}_{0}}$ on $\check{X}\,\rtimes \,W^{\prime }$ (cf. [Reference RocheRoc98, § 8]). In (3), $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$ is the convolution $\mathbb{C}$-algebra of compactly supported functions on $G(F)$ which are $\unicode[STIX]{x1D70C}^{-1}$-spherical, and $\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ is the convolution $\mathbb{C}$-algebra of compactly supported $\unicode[STIX]{x2110}^{\prime }$-biinvariant functions on $\widetilde{G}^{\prime }(F)$. The isomorphism $\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}$ of (3) is described explicitly in § 1.2 (Proposition 2). In (4), the scheme-theoretic center of $G$ is the diagonalizable $F$-group scheme whose group of characters is $X/\mathbb{Z}\unicode[STIX]{x1D6F7}$. We denote it by $\mathfrak{Z}_{G}$. Hence $\mathfrak{Z}_{G}$ is a torus if and only if $X/\mathbb{Z}\unicode[STIX]{x1D6F7}$ is torsion-free, which is equivalent to ${\check{G}}$ having a simply connected derived group. Note that the reduced subscheme $(\mathfrak{Z}_{G})_{\text{red}}$ is a smooth group scheme that coincides with the ‘center’ $Z(G)$ of $G$ in the sense of Borel (whose group of characters is the quotient of $X/\mathbb{Z}\unicode[STIX]{x1D6F7}$ by its $p$-torsion). Hence $\mathfrak{Z}_{G}$ is a torus if and only if it is reduced and connected.

Remark 2. Let us make (1) precise. The group ${\check{T}}$ is the complex torus dual (in the sense of Langlands) to $T$: we have $X({\check{T}})=\check{X}$ and $\check{X}({\check{T}})=X$. Let $W_{F}\subset \unicode[STIX]{x1D6E4}_{F}$ be the absolute Weil group of $F$, and $\unicode[STIX]{x1D70F}_{F}:W_{F}\rightarrow W_{F}^{\text{ab}}\stackrel{\simeq }{\longrightarrow }F^{\times }$ be the morphism of local class field theory. The Langlands parameter of $\unicode[STIX]{x1D712}:T(F)\rightarrow \mathbb{C}^{\times }$ is the morphism $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}}:W_{F}\rightarrow {\check{T}}$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D708}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}}(\unicode[STIX]{x1D70E}))=\unicode[STIX]{x1D712}(\unicode[STIX]{x1D708}(\unicode[STIX]{x1D70F}_{F}(\unicode[STIX]{x1D70E}))),\quad \unicode[STIX]{x1D708}\in \check{X},~\unicode[STIX]{x1D70E}\in W_{F}.\end{eqnarray}$$

The restriction $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}}|_{I_{F}}$ to the inertia group $I_{F}=\unicode[STIX]{x1D70F}_{F}^{-1}(\mathfrak{o}^{\times })$ of $F$ depends only on $\unicode[STIX]{x1D712}_{0}$. We denote it by $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}_{0}}$. Since the group $\mathfrak{K}^{\times }$ is cyclic, the image $\operatorname{Im}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}_{0}})$ is a cyclic subgroup of ${\check{T}}$. Let us choose a generator $s$ of $\operatorname{Im}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}_{0}})$. Then we have

$$\begin{eqnarray}{\check{G}}^{\prime }=Z_{{\check{G}}}(\operatorname{Im}(\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}_{0}}))^{\circ }={\check{G}}_{s}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D719}\in W_{F}$ be a Frobenius element such that $\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D719})=\unicode[STIX]{x1D71B}$. Then $W_{F}=I_{F}\,\rtimes \,\langle \unicode[STIX]{x1D719}\rangle$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}}:W_{F}\rightarrow {\check{T}}$ is the morphism extending $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}_{0}}$ trivially on $\langle \unicode[STIX]{x1D719}\rangle$.

1.2 Hecke algebra isomorphism

Let $W=N(F)/T$ be the Iwahori–Weyl group of $G$. Recall that we have a (non-canonical) identification $N(F)=\check{X}\,\rtimes \,N$ given by the choice of the uniformizer element $\unicode[STIX]{x1D71B}$ of $F$. Since $N/T=N/T=W$, this identification also gives an identification $W=\check{X}\,\rtimes \,W$. Let $W_{\unicode[STIX]{x1D712}_{0}}=\check{X}\,\rtimes \,W_{\unicode[STIX]{x1D712}_{0}}$ be the subgroup of $W$ formed by the elements which fix $\unicode[STIX]{x1D712}_{0}$. As we have defined the subgroup $W^{\prime }\subset W_{\unicode[STIX]{x1D712}_{0}}$, we define a subgroup $W^{\prime }\subset W_{\unicode[STIX]{x1D712}_{0}}$ as follows. Let $\unicode[STIX]{x1D6F7}_{\text{aff}}^{\prime }$ be the set of affine roots $a=\unicode[STIX]{x1D6FC}+n$ where $\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}^{\prime }$ and $n\in \mathbb{Z}$. Then $W^{\prime }$ is the subgroup of $W_{\unicode[STIX]{x1D712}_{0}}$ generated by the affine reflections $s_{a}$ for $a\in \unicode[STIX]{x1D6F7}_{\text{aff}}^{\prime }$. Hence we have

$$\begin{eqnarray}W^{\prime }=\check{X}\,\rtimes \,W^{\prime }.\end{eqnarray}$$

Recall that we have defined in § 1.1 an alcove $A^{\prime }$ in $V$. This alcove is a chamber for the decomposition of $V$ induced by $\unicode[STIX]{x1D6F7}_{\text{aff}}^{\prime }$: $A^{\prime }$ is a connected component of $V\smallsetminus \bigcup _{a\in \unicode[STIX]{x1D6F7}_{\text{aff}}^{\prime }}H_{a}$ where $H_{a}=\{v\in V:a(v)=0\}$. It also defines a partial order on $\unicode[STIX]{x1D6F7}_{\text{aff}}^{\prime }$: for $a\in \unicode[STIX]{x1D6F7}_{\text{aff}}^{\prime }$, write $a>0$ if $a(v)>0$ for all $v\in A^{\prime }$. Let $\unicode[STIX]{x1D6E5}_{\text{aff}}^{\prime }$ be the subset of $\unicode[STIX]{x1D6F7}_{\text{aff}}^{\prime }$ formed of those affine roots $a$ which are minimal positive for this partial order. Put

$$\begin{eqnarray}\displaystyle & \displaystyle S^{\prime }=\{s_{a}:a\in \unicode[STIX]{x1D6E5}_{\text{aff}}^{\prime }\}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle C_{\unicode[STIX]{x1D712}_{0}}=\{w\in W_{\unicode[STIX]{x1D712}_{0}}:w(A^{\prime })=A^{\prime }\}. & \displaystyle \nonumber\end{eqnarray}$$

From [Reference RocheRoc98, Lemma 6.2], we have the following result.

Let $\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })$ be the Hecke algebra of the Coxeter system $(W^{\prime }\!,S^{\prime })$, that is, the associative $\mathbb{C}$-algebra with $\mathbb{C}$-basis $\{T_{w}:w\in W^{\prime }\}$ and multiplication given by

$$\begin{eqnarray}\displaystyle & \displaystyle T_{w_{1}w_{2}}=T_{w_{1}}T_{w_{2}}\quad \text{if }l^{\prime }(w_{1}w_{1})=l^{\prime }(w_{1})+l^{\prime }(w_{2}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle T_{s}T_{s}=(q-1)T_{s}+qT_{1}\quad ifs\in S^{\prime }. & \displaystyle \nonumber\end{eqnarray}$$

The group $C_{\unicode[STIX]{x1D712}_{0}}$ acts by $\mathbb{C}$-algebra automorphisms on $\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })$ via

$$\begin{eqnarray}\unicode[STIX]{x1D702}\cdot T_{w}=T_{\unicode[STIX]{x1D702}w\unicode[STIX]{x1D702}^{-1}},\quad \unicode[STIX]{x1D702}\in C_{\unicode[STIX]{x1D712}_{0}},w\in W^{\prime }.\end{eqnarray}$$

This action naturally extends to an injective morphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]\rightarrow \operatorname{End}_{\mathbb{C}\text{-}\text{alg}}(\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })).\end{eqnarray}$$

Via this morphism, we get a $\mathbb{C}$-algebra

$$\begin{eqnarray}\unicode[STIX]{x210B}(W_{\unicode[STIX]{x1D712}_{0}},S^{\prime })=\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })\,\widetilde{\otimes }\,\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]\end{eqnarray}$$

where the multiplication in the twisted tensor product is given by

$$\begin{eqnarray}(T_{w_{1}}\otimes e_{\unicode[STIX]{x1D702}_{1}})(T_{w_{2}}\otimes e_{\unicode[STIX]{x1D702}_{2}})=T_{w_{1}}T_{\unicode[STIX]{x1D702}_{1}w_{2}\unicode[STIX]{x1D702}_{1}^{-1}}\otimes e_{\unicode[STIX]{x1D702}_{1}\unicode[STIX]{x1D702}_{2}}\end{eqnarray}$$

for $w_{1},w_{2}\in W^{\prime }$ and $\unicode[STIX]{x1D702}_{1},\unicode[STIX]{x1D702}_{2}\in C_{\unicode[STIX]{x1D712}_{0}}$, where we have denoted by $e_{\unicode[STIX]{x1D702}_{1}}$ the element of $\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]$ given by $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D702}_{1}}(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D702}_{1},\unicode[STIX]{x1D702}}$ (Kronecker symbol). For $w\in W^{\prime }$ and $\unicode[STIX]{x1D702}\in C_{\unicode[STIX]{x1D712}_{0}}$, we write the basis element $T_{w}\otimes e_{\unicode[STIX]{x1D702}}\in \unicode[STIX]{x210B}(W_{\unicode[STIX]{x1D712}_{0}},S^{\prime })$ as $T_{w\unicode[STIX]{x1D702}}$. Thus we have $T_{w}T_{\unicode[STIX]{x1D702}}=T_{w\unicode[STIX]{x1D702}}$ and $T_{\unicode[STIX]{x1D702}}T_{w}T_{\unicode[STIX]{x1D702}^{-1}}=T_{\unicode[STIX]{x1D702}w\unicode[STIX]{x1D702}^{-1}}$.

For an element $w\in W_{\unicode[STIX]{x1D712}_{0}}=N(F)_{\unicode[STIX]{x1D712}_{0}}/T$, let $f_{\tilde{\unicode[STIX]{x1D712}},w}$ be the unique element of the Hecke algebra $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$ supported on $\unicode[STIX]{x2110}w\unicode[STIX]{x2110}=\unicode[STIX]{x2110}n\unicode[STIX]{x2110}$ and having value $q^{-l(w)/2}\tilde{\unicode[STIX]{x1D712}}(n)^{-1}$ at $n$ for any representative element $n$ of $w$ in $N(F)_{\unicode[STIX]{x1D712}_{0}}$, where $l$ is the length function on the affine Weyl group $W=N(F)/T$ of $G$. The functions $f_{\tilde{\unicode[STIX]{x1D712}},w}$ for $n\in W_{\unicode[STIX]{x1D712}_{0}}$ generate the $\mathbb{C}$-vector space $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$. On the other hand, let $\unicode[STIX]{x1D719}_{w}^{\prime }$ be the element $q^{-l^{\prime }(w)/2}T_{w}\in \unicode[STIX]{x210B}(W_{\unicode[STIX]{x1D712}_{0}},S^{\prime })$, where $l^{\prime }$ is the length function on $W_{\unicode[STIX]{x1D712}_{0}}$ defined by the lemma. From [Reference RocheRoc98, Theorem 6.3],Footnote ⁶ we have the following proposition.

Proposition 1. There is an isomorphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D717}_{\tilde{\unicode[STIX]{x1D712}}}:\unicode[STIX]{x210B}(W_{\unicode[STIX]{x1D712}_{0}},S^{\prime })\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\end{eqnarray}$$

which sends $\unicode[STIX]{x1D719}_{w}^{\prime }$ to $f_{\tilde{\unicode[STIX]{x1D712}},w}$ for all $w\in W_{\unicode[STIX]{x1D712}_{0}}$. Moreover, the restriction $\unicode[STIX]{x1D717}_{\tilde{\unicode[STIX]{x1D712}}}|_{\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })}$ is the unique injective morphism of $\mathbb{C}$-algebras $\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })\rightarrow \unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$ which is support-preserving in the sense that the image of $T_{w}$ has support $\unicode[STIX]{x2110}w\unicode[STIX]{x2110}$ for all $w\in W^{\prime }$.

Now let us recall the dual interpretation of Proposition 1 [Reference RocheRoc98]. The group $W^{\prime }=\check{X}\,\rtimes \,W^{\prime }$ is the Iwahori–Weyl group of $G^{\prime }$. Put

$$\begin{eqnarray}C^{\prime }=\{w\in W^{\prime }:w(A^{\prime })=A^{\prime }\}.\end{eqnarray}$$

Recall that we have the decomposition [Reference RocheRoc98, Lemma 8.1] $W_{\unicode[STIX]{x1D712}_{0}}=W^{\prime }\,\rtimes \,C_{\unicode[STIX]{x1D712}_{0}}$. From [Reference RocheRoc98, Lemma 8.1], we also have the decomposition

$$\begin{eqnarray}C_{\unicode[STIX]{x1D712}_{0}}=C^{\prime }\,\rtimes \,C_{\unicode[STIX]{x1D712}_{0}}.\end{eqnarray}$$

This gives a decomposition

$$\begin{eqnarray}\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]=\mathbb{C}[C^{\prime }]\,\widetilde{\otimes }\,\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]\end{eqnarray}$$

where the tensor product is defined by the conjugation action of $C_{\unicode[STIX]{x1D712}_{0}}$ on $C^{\prime }$. Hence we have

(1)$$\begin{eqnarray}\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })\,\widetilde{\otimes }\,\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]\,\simeq \,(\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })\,\widetilde{\otimes }\,\mathbb{C}[C^{\prime }])\,\widetilde{\otimes }\,\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}].\end{eqnarray}$$

By Iwahori and Matsumoto, we know there is a support-preserving isomorphism of $\mathbb{C}$-algebras

(2)$$\begin{eqnarray}\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })\,\widetilde{\otimes }\,\mathbb{C}[C^{\prime }]\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x210B}(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }}).\end{eqnarray}$$

For an element $w\in W_{\unicode[STIX]{x1D712}_{0}}=N(F)_{\unicode[STIX]{x1D712}_{0}}/T$, let $f_{w}^{\prime }$ be the unique element of the Hecke algebra $\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ supported on $\unicode[STIX]{x2110}^{\prime }w\unicode[STIX]{x2110}^{\prime }=\unicode[STIX]{x2110}^{\prime }n\unicode[STIX]{x2110}^{\prime }$ and having value $q^{-l^{\prime }(w)/2}$ at $n$, for any representative element $n$ of $w$ in $N(F)_{\unicode[STIX]{x1D712}_{0}}$. From (1) and (2), there is an isomorphism of $\mathbb{C}$-algebras

(3)$$\begin{eqnarray}\unicode[STIX]{x210B}(W_{\unicode[STIX]{x1D712}_{0}},S^{\prime })=\unicode[STIX]{x210B}(W^{\prime }\!,S^{\prime })\,\widetilde{\otimes }\,\mathbb{C}[C_{\unicode[STIX]{x1D712}_{0}}]\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }}).\end{eqnarray}$$

which sends $\unicode[STIX]{x1D719}_{w}^{\prime }$ to $f_{w}^{\prime }$ for all $w\in W_{\unicode[STIX]{x1D712}_{0}}$. Composing (3) with the isomorphism $\unicode[STIX]{x1D717}_{\tilde{\unicode[STIX]{x1D712}}}^{-1}$ of the Proposition 1, we obtain the following result.

Proposition 2. There is an isomorphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}:\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\end{eqnarray}$$

which sends $f_{\tilde{\unicode[STIX]{x1D712}},w}$ to $f_{w}^{\prime }$ for all $w\in W_{\unicode[STIX]{x1D712}_{0}}$.

1.3 Bernstein center

Most of the material in this subsection comes from [Reference HainesHai13, 4].

We denote by $\mathfrak{X}(\mathfrak{s})$ the complex algebraic variety (in fact a complex torus) associated to $\mathfrak{s}$, and by $\mathfrak{R}^{\mathfrak{s}}(G(F))$ the corresponding category of (complex, smooth) representations of $G(F)$. The elements of $\mathfrak{s}$ are the $G(F)$-conjugacy classes $(T,\unicode[STIX]{x1D713})_{G}$ of cuspidal pairs $(T,\unicode[STIX]{x1D713})$ of $G(F)$ such that $\unicode[STIX]{x1D713}$ is a character of $T(F)$ which extends a $W$-conjugate of $\unicode[STIX]{x1D712}_{0}$. Recall that we chose a base-point $(T,\unicode[STIX]{x1D712})_{G}\in \mathfrak{s}$ such that $\unicode[STIX]{x1D712}$ is $W_{\unicode[STIX]{x1D712}_{0}}$-invariant. Then, writing $\unicode[STIX]{x1D6F9}(T)=\operatorname{Hom}(T(F)/T,\mathbb{C}^{\times })$, the group of unramified characters of $G(F)$, we have an isomorphism of algebraic varieties

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(T)/W_{\unicode[STIX]{x1D712}_{0}}\stackrel{\simeq }{\longrightarrow }\mathfrak{X}(\mathfrak{s}),\quad \unicode[STIX]{x1D709}\mapsto (T,\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})_{G}.\end{eqnarray}$$

We also have an isomorphism of algebraic varieties

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(T)/W^{\prime }\stackrel{\simeq }{\longrightarrow }\mathfrak{X}(\mathfrak{s}^{\prime }),\quad \unicode[STIX]{x1D709}\mapsto (T,\unicode[STIX]{x1D709})_{G^{\prime }},\end{eqnarray}$$

where $\mathfrak{s}^{\prime }=[T,1]_{G^{\prime }}$ is the $G^{\prime }(F)$-inertial class of the cuspidal pair $(T,1)$ of $G^{\prime }(F)$. We deduce from these two isomorphisms a surjective morphism of algebraic varieties

$$\begin{eqnarray}\mathfrak{X}(\mathfrak{s}^{\prime })\rightarrow \mathfrak{X}(\mathfrak{s}),\quad (T,\unicode[STIX]{x1D709})_{G^{\prime }}\mapsto (T,\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})_{G},\end{eqnarray}$$

which gives dually an injective morphism of $\mathbb{C}$-algebras

(1)$$\begin{eqnarray}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D712}}:\mathbb{C}[\mathfrak{X}(\mathfrak{s})]\rightarrow \mathbb{C}[\mathfrak{X}(\mathfrak{s}^{\prime })].\end{eqnarray}$$

On the other hand, there is a natural equivalence between $\mathfrak{R}^{\mathfrak{s}}(G(F))$ and the category of left $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$-modules (given by $\unicode[STIX]{x1D70B}\mapsto \unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D70C}}$, where $\unicode[STIX]{x1D70B}^{\unicode[STIX]{x1D70C}}$ denotes the $\unicode[STIX]{x1D70C}$-isotypic component of $\unicode[STIX]{x1D70B}$) which gives an isomorphism between the centers of these categories (cf. [Reference HainesHai13, 3.1]), say

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}:\mathbb{C}[\mathfrak{X}(\mathfrak{s})]\stackrel{\simeq }{\longrightarrow }Z(G(F),\unicode[STIX]{x1D70C}).\end{eqnarray}$$

Here $Z(G(F),\unicode[STIX]{x1D70C})$ denotes the center of the (unital) algebra $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$. We also have an isomorphism

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}^{\prime }:\mathbb{C}[\mathfrak{X}(\mathfrak{s}^{\prime })]\stackrel{\simeq }{\longrightarrow }Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }}).\end{eqnarray}$$

The isomorphism $\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}$ of § 1.1.(3) gives by restriction an isomorphism of $\mathbb{C}$-algebras

(2)$$\begin{eqnarray}Z(G(F),\unicode[STIX]{x1D70C})\stackrel{\simeq }{\longrightarrow }Z(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }}).\end{eqnarray}$$

On the other hand, let

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}^{\unicode[STIX]{x2110}}:Z(G(F),\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\end{eqnarray}$$

be the unique morphism of $\mathbb{C}$-algebras which makes the diagram

(3)

commutative. It is injective by construction.

Now let $\unicode[STIX]{x1D713}$ be a character of $T(F)$ which extends a $W$-conjugate of $\unicode[STIX]{x1D712}_{0}$. Put $\unicode[STIX]{x1D713}^{\ast }=\unicode[STIX]{x1D6FF}_{B}^{1/2}\unicode[STIX]{x1D713}$. Let $i_{B}^{G}(\unicode[STIX]{x1D713})^{\unicode[STIX]{x1D70C}}$ be the space of functions $\unicode[STIX]{x1D719}:G(F)\rightarrow \mathbb{C}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D719}(tugk)=\unicode[STIX]{x1D713}^{\ast }(t)\unicode[STIX]{x1D719}(g)\unicode[STIX]{x1D70C}(k),\quad t\in T(F),~u\in U(F),~g\in G(F),~k\in \unicode[STIX]{x2110}.\end{eqnarray}$$

Then every function $f\in Z(G(F),\unicode[STIX]{x1D70C})$ acts on the left on $i_{B}^{G}(\unicode[STIX]{x1D713})^{\unicode[STIX]{x1D70C}}$ by multiplication by the scalar

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D713}}(f)=\unicode[STIX]{x1D6FD}^{-1}(f)((T,\unicode[STIX]{x1D713})_{G}),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}^{-1}(f)$ is viewed as a regular function on the complex algebraic variety $\mathfrak{X}(\mathfrak{s})$. In particular, we have the equality $\unicode[STIX]{x1D706}_{\text{}^{w}\unicode[STIX]{x1D713}}=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D713}}$ as functions on $Z(G(F),\unicode[STIX]{x1D70C})$. In the same manner, for an unramified character $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)=T(F)$, every function $f^{\prime }\in Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ acts on the left on the space $i_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x2110}^{\prime }}=i_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{1_{\unicode[STIX]{x2110}^{\prime }}}$ by multiplication by the scalar

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(f^{\prime })=\unicode[STIX]{x1D6FD}^{\prime -1}(f^{\prime })((T^{\prime }\!,\unicode[STIX]{x1D709})_{G^{\prime }}).\end{eqnarray}$$

Now take $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}$ for an unramified character $\unicode[STIX]{x1D709}$ of $T(F)$. From (3) we have

(4)$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}}(f)=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)),\quad f\in Z(G(F),\unicode[STIX]{x1D70C}).\end{eqnarray}$$

In other words, the morphism of $\mathbb{C}$-algebras $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}:Z(G(F),\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ is characterized by actions on principal series: for $f\in Z(G(F),\unicode[STIX]{x1D70C})$, $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)$ is the unique element of $Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ which acts on the left on the space $i_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x2110}^{\prime }}$ by multiplication by the scalar $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D712}\unicode[STIX]{x1D709}}(f)$, for all unramified characters $\unicode[STIX]{x1D709}$ of $T(F)$.

Proof. Let

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D712}}^{T}:\unicode[STIX]{x210B}(T(F),\unicode[STIX]{x1D712})\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x210B}(T^{\prime }(F),1_{T^{\prime }})\end{eqnarray}$$

be the isomorphism of $\mathbb{C}$-algebras obtained by replacing $G$ by $T$ in the definition of $\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}$. From [Reference RocheRoc98, Lemma 9.3] (see also [Reference Haines and RapoportHR12, equation 9.3.4]), there are morphisms of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D703}_{B,\unicode[STIX]{x1D712}}:\unicode[STIX]{x210B}(T(F),\unicode[STIX]{x1D712})\rightarrow \unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\end{eqnarray}$$

and

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D703}}_{B^{\prime }\!,1}:\unicode[STIX]{x210B}(T^{\prime }(F),1_{T^{\prime }})\rightarrow \unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\end{eqnarray}$$

which make the diagram

(5)

commutative. It induces a commutative diagram of functors

(6)

where ?-Mod denotes the category of left $?$-modules. From [Reference RocheRoc98, Theorem 9.4], $(\unicode[STIX]{x1D703}_{B,\unicode[STIX]{x1D712}})_{\ast }$ coincides with the induction functor $i_{B}^{G}$ (via the natural equivalences of categories), and $(\tilde{\unicode[STIX]{x1D703}}_{B^{\prime }\!,1})_{\ast }$ coincides with $i_{B^{\prime }}^{\tilde{G}}=i_{G^{\prime }}^{\widetilde{G}^{\prime }}\circ i_{B^{\prime }}^{G^{\prime }}$ where

$$\begin{eqnarray}i_{G^{\prime }}^{\widetilde{G}^{\prime }}:\unicode[STIX]{x210B}(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\text{-Mod}\rightarrow \unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\text{-Mod}\end{eqnarray}$$

is the ordinary induction functor. Here we used the equivalence of categories (cf. [Reference RocheRoc98, § 9, p. 400])

$$\begin{eqnarray}\mathfrak{R}^{1_{\unicode[STIX]{x2110}^{\prime }}}(\widetilde{G}^{\prime }(F))\rightarrow \unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\text{-Mod},\quad W\mapsto W^{\unicode[STIX]{x2110}^{\prime }}\end{eqnarray}$$

where $\mathfrak{R}^{1_{\unicode[STIX]{x2110}^{\prime }}}(\widetilde{G}^{\prime }(F))$ denotes the category of representations of $\widetilde{G}^{\prime }(F)$ generated by their $\unicode[STIX]{x2110}^{\prime }$-fixed vectors. In other words, for all unramified characters $\unicode[STIX]{x1D709}$ of $T(F)$, we have

$$\begin{eqnarray}(\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}})_{\ast }(i_{B}^{G}(\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})^{\unicode[STIX]{x2110}})=i_{B^{\prime }}^{\widetilde{G}^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x2110}^{\prime }}.\end{eqnarray}$$

Hence for $f\in Z(G(F),\unicode[STIX]{x1D70C})$, $\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}(f)$ acts on the space

$$\begin{eqnarray}i_{B^{\prime }}^{\widetilde{G}^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x2110}^{\prime }}=\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\otimes _{\unicode[STIX]{x210B}(G(F),1_{\unicode[STIX]{x2110}^{\prime }})}i_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x2110}^{\prime }}\end{eqnarray}$$

by multiplication by the scalar $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D712}\unicode[STIX]{x1D709}}(f)$, for all unramified characters $\unicode[STIX]{x1D709}$ of $T(F)$. But $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)\,\widetilde{\otimes }\,1\in \unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ acts on the space $i_{B^{\prime }}^{\widetilde{G}^{\prime }}(\unicode[STIX]{x1D709})$ by multiplication by the scalar $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D712}\unicode[STIX]{x1D709}}(f)$, for all unramified characters $\unicode[STIX]{x1D709}$ of $T(F)$. Hence

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}(f)-\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)\end{eqnarray}$$

acts by $0$ on each irreducible subquotient of the $\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$-module $i_{B^{\prime }}^{\widetilde{G}^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x2110}^{\prime }}$, for all unramified characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)$. This implies $\unicode[STIX]{x1D6F9}_{\tilde{\unicode[STIX]{x1D712}}}(f)=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)$ as elements of $\unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$. The lemma is proved.◻

1.4 Conjugation by $w\in W$

This subsection is a simple variant of [Reference HainesHai13, 4.3]. Let $w$ be an element of $W$. It can be lifted to an element $n\in N\,\cap \,K$. The character $\text{}^{w}\!\unicode[STIX]{x1D712}_{0}=\unicode[STIX]{x1D712}_{0}\circ \operatorname{Int}_{w^{-1}}$ of $T$ can be identified as in § 1.1 with a character of the Iwahori subgroup $\text{}^{w}\unicode[STIX]{x2110}=\operatorname{Int}_{n}(\unicode[STIX]{x2110})$ of $G(F)$, the inflation of $\text{}^{w}\!\unicode[STIX]{x1D712}_{0}$ to $\text{}^{w}\unicode[STIX]{x2110}$. This character is $\text{}^{w}\!\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}\circ \operatorname{Int}_{n^{-1}}$. The group $\text{}^{w}\unicode[STIX]{x2110}$ and the character $\text{}^{w}\!\unicode[STIX]{x1D70C}$ of $\text{}^{w}\unicode[STIX]{x2110}$, and more generally the function $\text{}^{w}\!f=f\circ \operatorname{Int}_{n^{-1}}\in \unicode[STIX]{x210B}(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})$ for any $f\in \unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$, do not depend on the choice of the representative element $n\in K\,\cap \,N$ of $w$: since $K\,\cap \,T=T$, another lift of $w$ in $N\,\cap \,K$ belongs to $nT=Tn$. The map

$$\begin{eqnarray}\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\rightarrow \unicode[STIX]{x210B}(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C}),\quad f\mapsto \text{}^{w}\!f\end{eqnarray}$$

is an isomorphism of $\mathbb{C}$-algebras which restricts to an isomorphism

$$\begin{eqnarray}[w]:Z(G(F),\unicode[STIX]{x1D70C})\stackrel{\simeq }{\longrightarrow }Z(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C}).\end{eqnarray}$$

From [Reference HainesHai13, 4.3.1], the diagram

(1)

is commutative. Here, the lower $\unicode[STIX]{x1D6FD}$ is the isomorphism of $\mathbb{C}$-algebras given by the natural equivalence between $\mathfrak{R}^{\mathfrak{s}}(G(F))$ and the category of left $\unicode[STIX]{x210B}(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})$-modules. As for the morphism $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$ in § 1.3.(

), the isomorphism of $\mathbb{C}$-algebras $[w]:Z(G(F),\unicode[STIX]{x1D70C})\rightarrow Z(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})$ can be characterized by actions on principal series (cf. § 1.3): for $f\in Z(G(F),\unicode[STIX]{x1D70C})$, $[w](f)=\text{}^{w}\!f$ is the unique element of $Z(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})$ which acts on the left on $i_{\text{}^{w}\!B}^{G}(\text{}^{w}\unicode[STIX]{x1D713})^{\text{}^{w}\!\unicode[STIX]{x1D70C}}\simeq i_{B}^{G}(\unicode[STIX]{x1D713})^{\unicode[STIX]{x1D70C}}$ by multiplication by the scalar

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\text{}^{w}\unicode[STIX]{x1D713}}(\text{}^{w}\!f)(=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D713}}(\text{}^{w}\!f)=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D713}}(f)),\end{eqnarray}$$

for all characters $\unicode[STIX]{x1D713}$ of $T(F)$ extending a $W$-conjugate of $\unicode[STIX]{x1D712}_{0}$.

If $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}$ for an unramified character $\unicode[STIX]{x1D709}$ of $T(F)$, then we have $\text{}^{w}\unicode[STIX]{x1D713}=(\text{}^{w}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w}\unicode[STIX]{x1D712}$ with

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{w}=(\text{}^{w}\unicode[STIX]{x1D712})\unicode[STIX]{x1D712}^{-1}.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D712}_{w}$ is a character of $T(F)$ extending $\unicode[STIX]{x1D712}_{w,0}=(\text{}^{w}\unicode[STIX]{x1D712}_{0})\unicode[STIX]{x1D712}_{0}^{-1}$. If $w\in W_{\unicode[STIX]{x1D712}_{0}}$, then we have $\unicode[STIX]{x1D712}_{w,0}=1$ and $\unicode[STIX]{x1D712}_{w}=1$. Moreover, if $w_{0}\in W_{\unicode[STIX]{x1D712}_{0}}$, since $\text{}^{w_{0}}\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D712}_{ww_{0}}=\unicode[STIX]{x1D712}_{w},\quad \unicode[STIX]{x1D712}_{w_{0}w}=\unicode[STIX]{x1D712}_{w_{0}ww_{0}^{-1}}.\end{eqnarray}$$

Let $\mathfrak{s}_{w}^{\prime }=\mathfrak{s}_{\unicode[STIX]{x1D712}_{w,0}}^{\prime }$ be the $G^{\prime }(F)$-inertial class $[T(F),\unicode[STIX]{x1D712}_{w}]_{G^{\prime }}$ of the cuspidal pair $(T(F),\unicode[STIX]{x1D712}_{w})$ of $G^{\prime }(F)$. Recall that we have an isomorphism of algebraic varieties

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(T)/W^{\prime }\stackrel{\simeq }{\longrightarrow }\mathfrak{X}(\mathfrak{s}^{\prime }),\quad \unicode[STIX]{x1D709}\mapsto (T,\unicode[STIX]{x1D709})_{G^{\prime }}.\end{eqnarray}$$

We also have an isomorphism of algebraic varieties

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(T)/W_{\text{}^{w}\!\unicode[STIX]{x1D712}_{0}}^{\prime }\stackrel{\simeq }{\longrightarrow }\mathfrak{X}(\mathfrak{s}_{w}^{\prime }),\quad \unicode[STIX]{x1D709}\mapsto (T,(\text{}^{w}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w})_{G^{\prime }},\end{eqnarray}$$

where $W_{\text{}^{w\!\unicode[STIX]{x1D712}_{0}}}^{\prime }=W^{\prime }\,\cap \,wW_{\unicode[STIX]{x1D712}_{0}}w^{-1}$ is the subgroup of $W^{\prime }$ formed by the elements which fix $\text{}^{w}\!\unicode[STIX]{x1D712}_{0}$. The surjective morphism of algebraic varieties

$$\begin{eqnarray}\mathfrak{X}(\mathfrak{s}_{w}^{\prime })\rightarrow \mathfrak{X}(\mathfrak{s}^{\prime }),\quad (T,(\text{}^{w}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w})_{G^{\prime }}\mapsto (T,\unicode[STIX]{x1D709})_{G^{\prime }}\end{eqnarray}$$

gives by duality an injective morphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{w}^{\prime }=\unicode[STIX]{x1D707}_{w,\unicode[STIX]{x1D712}}^{\prime }:\mathbb{C}[\mathfrak{X}(\mathfrak{s}^{\prime })]\rightarrow \mathbb{C}[\mathfrak{X}(\mathfrak{s}_{w}^{\prime })],\end{eqnarray}$$

which is the identity if $w\in W^{\prime }$. The natural isomorphism $T/T_{+}\stackrel{\simeq }{\longrightarrow }\unicode[STIX]{x2110}^{\prime }/\unicode[STIX]{x2110}_{+}^{\prime }$ allows us to view $\unicode[STIX]{x1D712}_{w,0}$ as a character $\unicode[STIX]{x1D70C}_{w}^{\prime }=\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D712}_{w,0}}^{\prime }$ of $\unicode[STIX]{x2110}^{\prime }$ trivial on $\unicode[STIX]{x2110}_{+}^{\prime }$ (cf. § 1.1), the inflation of $\unicode[STIX]{x1D712}_{w,0}$ to $\unicode[STIX]{x2110}^{\prime }$. Let

$$\begin{eqnarray}\{w\}^{\prime }=\{w\}_{\unicode[STIX]{x1D712}}^{\prime }:Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })\end{eqnarray}$$

be the injective morphism of $\mathbb{C}$-algebras which makes the diagram

commutative. Here the lower $\unicode[STIX]{x1D6FD}^{\prime }$ is the isomorphism of $\mathbb{C}$-algebras given by the natural equivalence between $\mathfrak{R}^{\mathfrak{s}_{w}^{\prime }}(G^{\prime }(F))$ and the category of left $\unicode[STIX]{x210B}(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$-modules. Note that if $w\in W_{\unicode[STIX]{x1D712}_{0}}$, then $\{w\}^{\prime }$ is an automorphism of $Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$, which is the identity if $w\in W^{\prime }$.

The morphism $\{w\}^{\prime }$ can be characterized by actions on principal series: for $f^{\prime }\in Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$, $\{w\}^{\prime }(f^{\prime })$ is the unique element of $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ which acts on the left on $i_{B^{\prime }}^{G^{\prime }}(\{w\}^{\prime }(\unicode[STIX]{x1D709}))^{\unicode[STIX]{x1D70C}_{w}^{\prime }}$ by multiplication by the scalar

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\{w\}^{\prime }(\unicode[STIX]{x1D709})}^{\prime }(\{w\}^{\prime }(f^{\prime }))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(f^{\prime }),\end{eqnarray}$$

for all unramified characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)$.

For $w_{0}\in W_{\unicode[STIX]{x1D712}_{0}}$, since $\text{}^{w_{0}}\!\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}$, we have $\unicode[STIX]{x1D712}_{w_{0}w}=\text{}^{w_{0}}(\unicode[STIX]{x1D712}_{w})$ and

$$\begin{eqnarray}\{w_{0}w\}^{\prime }(\unicode[STIX]{x1D709})=(\text{}^{w_{0}w}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w_{0}w}=\text{}^{w_{0}}(\{w\}^{\prime }(\unicode[STIX]{x1D709})).\end{eqnarray}$$

In particular, for $w^{\prime }\in W^{\prime }$, we have $\mathfrak{s}_{w^{\prime }w}^{\prime }=\mathfrak{s}_{w}^{\prime }$ and $\unicode[STIX]{x1D707}_{w^{\prime }w}^{\prime }=\unicode[STIX]{x1D707}_{w}^{\prime }$, and the morphism $\{w^{\prime }w\}^{\prime }:Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w^{\prime }w}^{\prime })$ is given by

$$\begin{eqnarray}\{w^{\prime }w\}^{\prime }=[w^{\prime }]_{w}^{\prime }\circ \{w\}^{\prime },\end{eqnarray}$$

where $[w^{\prime }]_{w}^{\prime }:Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w^{\prime }w}^{\prime })$ is the isomorphism of $\mathbb{C}$-algebras which makes the diagram

commutative. As for the morphism $\{w\}^{\prime }$ introduced above, the isomorphism $[w^{\prime }]_{w}^{\prime }$ can be characterized by actions on principal series: for $f^{\prime }\in Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$, $[w^{\prime }]_{w}(f^{\prime })$ is the unique element of $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w^{\prime }w}^{\prime })$ which acts on the left on $i_{B^{\prime }}^{G^{\prime }}(\{w^{\prime }w\}^{\prime }(\unicode[STIX]{x1D709}))^{\unicode[STIX]{x1D70C}_{w^{\prime }w}^{\prime }}$ by multiplication by the scalar

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\{w^{\prime }w\}^{\prime }(\unicode[STIX]{x1D709})}^{\prime }([w^{\prime }]_{w}^{\prime }(f^{\prime }))=\unicode[STIX]{x1D706}_{\{w\}(\unicode[STIX]{x1D709})}^{\prime }(f^{\prime })\end{eqnarray}$$

for all unramified characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)$.

On the other hand, for $w_{0}\in W_{\unicode[STIX]{x1D712}_{0}}$, since $\unicode[STIX]{x1D712}_{ww_{0},0}=\unicode[STIX]{x1D712}_{w,0}$, we have $\mathfrak{s}_{ww_{0}}^{\prime }=\mathfrak{s}_{w}^{\prime }$ and $\unicode[STIX]{x1D707}_{ww_{0}}^{\prime }=\unicode[STIX]{x1D707}_{w}^{\prime }$. We also have $\unicode[STIX]{x1D70C}_{ww_{0}}^{\prime }=\unicode[STIX]{x1D70C}_{w}^{\prime }$ and $\unicode[STIX]{x1D712}_{ww_{0}}=\unicode[STIX]{x1D712}_{w}$, and

$$\begin{eqnarray}\{ww_{0}\}^{\prime }(\unicode[STIX]{x1D709})=(\text{}^{ww_{0}}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w}=\text{}^{w}(\unicode[STIX]{x1D709}_{w_{0}})\cdot \{w\}^{\prime }(\unicode[STIX]{x1D709}),\quad \unicode[STIX]{x1D709}_{w_{0}}=(\text{}^{w_{0}}\unicode[STIX]{x1D709})\unicode[STIX]{x1D709}^{-1},\end{eqnarray}$$

for all unramified characters $\unicode[STIX]{x1D709}$ of $T(F)$. The automorphism of algebraic varieties

$$\begin{eqnarray}\mathbb{C}[\mathfrak{X}(\mathfrak{s}_{w}^{\prime })]\rightarrow \mathbb{C}[\mathfrak{X}(\mathfrak{s}_{w}^{\prime })],\quad (T^{\prime }\!,(\text{}^{ww_{0}}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w})_{G^{\prime }}\mapsto (T^{\prime }\!,(\text{}^{w}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w})_{G^{\prime }}\end{eqnarray}$$

gives by duality, and via the isomorphism $\unicode[STIX]{x1D6FD}^{\prime }:\mathbb{C}[\mathfrak{X}(\mathfrak{s}_{w}^{\prime })]\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$, an automorphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\{w_{0}\}_{w}^{\prime }:Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime }).\end{eqnarray}$$

By construction we have

$$\begin{eqnarray}\{ww_{0}\}^{\prime }=\{w_{0}\}_{w}^{\prime }\circ \{w\}^{\prime }.\end{eqnarray}$$

As before, it can be characterized via actions on principal series: for a function $f^{\prime }\in Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$, $\{w_{0}\}_{w}^{\prime }(f^{\prime })$ is the unique element of $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ which acts on the left on $i_{B^{\prime }}^{G^{\prime }}(\text{}^{w}(\unicode[STIX]{x1D709}_{w_{0}})\unicode[STIX]{x1D713})^{\unicode[STIX]{x1D70C}_{w}^{\prime }}$ with $\unicode[STIX]{x1D713}=\{w\}^{\prime }(\unicode[STIX]{x1D709})$ by multiplication by the scalar

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\text{}^{w}\!(\unicode[STIX]{x1D709}_{w_{0}})\unicode[STIX]{x1D713}}^{\prime }(\{w_{0}\}_{w}^{\prime }(f^{\prime }))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D713}}^{\prime }(f^{\prime })\end{eqnarray}$$

for all unramified characters $\unicode[STIX]{x1D709}$ of $T(F)$. Note that in general $\{w_{0}\}_{w}^{\prime }\neq \operatorname{id}$, even if $w_{0}\in W^{\prime }$.

The surjective morphism of algebraic varieties

$$\begin{eqnarray}\mathfrak{X}(\mathfrak{s}_{w}^{\prime })\rightarrow \mathfrak{X}(\mathfrak{s}),\quad (T,(\text{}^{w}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w})_{G^{\prime }}\mapsto (T,\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})_{G}\end{eqnarray}$$

gives by duality an injective morphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{w,\unicode[STIX]{x1D712}}:\mathbb{C}[\mathfrak{X}(\mathfrak{s})]\rightarrow \mathbb{C}[\mathfrak{X}(\mathfrak{s}_{w}^{\prime })].\end{eqnarray}$$

Let

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}:Z(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })\end{eqnarray}$$

be the injective morphism of $\mathbb{C}$-algebras which makes the diagram

(2)

commutative. By construction, the diagram

(3)

is commutative. As for the morphism $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$ of § 1.3.(

), the morphism $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}$ above can be characterized by actions on principal series: for $f\in Z(G(F),\unicode[STIX]{x1D70C})$, $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}f)$ is the unique element of $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ which acts on the left on $\unicode[STIX]{x1D704}_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x1D70C}_{w}^{\prime }}$ by multiplication by the scalar

(4)$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}}(f)\end{eqnarray}$$

for all characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)=T(F)$ extending a $W^{\prime }$-conjugate of $\unicode[STIX]{x1D712}_{w,0}$ (note that since $W^{\prime }\subset W_{\unicode[STIX]{x1D712}_{0}}$, $\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}$ extends a $W^{\prime }$-conjugate of $\text{}^{w}\!(\unicode[STIX]{x1D712}_{0})$, and in particular it extends a $W$-conjugate of $\unicode[STIX]{x1D712}_{0}$). From the characterization of the morphisms $\{w\}$ and $[w]^{\prime }$ by actions on principal series, we have also: for $f\in Z(G(F),\unicode[STIX]{x1D70C})$, $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f)\in Z(G(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ acts on the left on $i_{B^{\prime }}^{G^{\prime }}(\{w\}^{\prime }(\unicode[STIX]{x1D709}))^{\unicode[STIX]{x1D70C}_{w}^{\prime }}$ by multiplication by the scalar

(5)$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\{w\}^{\prime }(\unicode[STIX]{x1D709})}^{\prime }(\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f))=\unicode[STIX]{x1D706}_{\{w\}^{\prime }(\unicode[STIX]{x1D709})}^{\prime }(\{w\}^{\prime }(\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}}(f)\end{eqnarray}$$

for all unramified characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)$. Note that since $\{w\}^{\prime }(\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}=\text{}^{w}\!(\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}}(f)=\unicode[STIX]{x1D706}_{\text{}^{w}\!(\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})}(f)=\unicode[STIX]{x1D706}_{\{w\}^{\prime }(\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}}(f).\end{eqnarray}$$

Remark 1. For $w^{\prime }\in W^{\prime }$, since $\{w^{\prime }\}^{\prime }$ is the identity of $Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$, we have $\unicode[STIX]{x1D701}_{w^{\prime }\!,\unicode[STIX]{x1D712}}\circ [w^{\prime }]=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$. More generally, we have

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{w^{\prime }w,\unicode[STIX]{x1D712}}\circ [w^{\prime }w]=\{w^{\prime }w\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}=[w^{\prime }]_{w}^{\prime }\circ \{w\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}=[w^{\prime }]_{w}^{\prime }\circ \unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}\circ [w].\end{eqnarray}$$

On the other hand, for $w_{0}\in W_{\unicode[STIX]{x1D712}_{0}}$, we have

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{ww_{0},\unicode[STIX]{x1D712}}\circ [ww_{0}]=\{ww_{0}\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}=\{w_{0}\}_{w}^{\prime }\circ \{w\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}=\{w_{0}\}_{w}^{\prime }\circ \unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}\circ [w].\end{eqnarray}$$

Remark 2. As in § 1.1, the character $\text{}^{w}\unicode[STIX]{x1D712}_{0}$ of $T$ defines an $F$-split connected reductive group $G_{w}^{\prime }$ with maximal $F$-split torus $T_{w}^{\prime }=T$, an Iwahori subgroup $\unicode[STIX]{x2110}_{w}^{\prime }$ of $G_{w}^{\prime }(F)$, and an $F$-group$\widetilde{G}_{w}^{\prime }=G_{w}^{\prime }\,\rtimes \,C_{\text{}^{w}\!\unicode[STIX]{x1D712}_{0}}$. More precisely, $G_{w}^{\prime }$ is given by the root datum $(X,w(\unicode[STIX]{x1D6F7}^{\prime }),\check{X},w(\check{\unicode[STIX]{x1D6F7}}^{\prime }))$, $\unicode[STIX]{x2110}_{w}^{\prime }$ corresponds to the alcove $w(A^{\prime })\subset V$, and $C_{\text{}^{w}\!\unicode[STIX]{x1D712}_{0}}=wC_{\unicode[STIX]{x1D712}_{0}}w^{-1}$. In other words, $w$ induces by transport of structure an $F$-isomorphism $\widetilde{G}^{\prime }\stackrel{\simeq }{\longrightarrow }\widetilde{G}_{w}^{\prime }$ which allows us to identify $G_{w}^{\prime }(F)$ with $G(F)$ and $\unicode[STIX]{x2110}_{w}^{\prime }$ with $\unicode[STIX]{x2110}^{\prime }$. With these identifications, we have an injective morphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{\text{}^{w}\unicode[STIX]{x1D712}}^{\text{}^{w}\unicode[STIX]{x2110}}:Z(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})\rightarrow Z(G_{w}^{\prime }(F),1_{\unicode[STIX]{x2110}_{w}^{\prime }})=Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\end{eqnarray}$$

which is given by $\unicode[STIX]{x1D701}_{\text{}^{w}\unicode[STIX]{x1D712}}^{\text{}^{w}\unicode[STIX]{x2110}}=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}\circ [w]^{-1}$. Hence we have

$$\begin{eqnarray}\{w\}^{\prime }\circ \unicode[STIX]{x1D701}_{\text{}^{w}\unicode[STIX]{x1D712}}^{\text{}^{w}\unicode[STIX]{x2110}}=\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}.\end{eqnarray}$$

In particular, if $w\in W_{\unicode[STIX]{x1D712}_{0}}$, we have $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}=\{w\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}^{\text{}^{w}\unicode[STIX]{x2110}}$, and if $w\in W^{\prime }$, we have $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}^{\text{}^{w}\unicode[STIX]{x2110}}$.

1.5 Conjugation by $w\in W^{\prime }$

Let $w$ be an element of $W^{\prime }$. We choose a representative element $n_{w}^{\prime }$ of $w$ in $N^{\prime }\,\cap \,K^{\prime }$, and we put $\text{}^{w}\unicode[STIX]{x2110}^{\prime }=\text{}^{w}\!(\unicode[STIX]{x2110}^{\prime })=\operatorname{Int}_{n_{w}^{\prime }}(\unicode[STIX]{x2110}^{\prime })$. For a function $f^{\prime }\in Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$, we put $[w]^{\prime }(f^{\prime })=\text{}^{w}\!f^{\prime }=f^{\prime }\circ \operatorname{Int}_{n_{w}^{\prime -1}}$. Since $\text{}^{w}\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}$, by replacing $\unicode[STIX]{x2110}$ by $\text{}^{w}\unicode[STIX]{x2110}$ and $\unicode[STIX]{x2110}^{\prime }$ by $\text{}^{w}\unicode[STIX]{x2110}^{\prime }$ in the definition of $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$, we obtain an injective morphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\text{}^{w}\!\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}:Z(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),1_{\text{}^{w}\unicode[STIX]{x2110}^{\prime }})\end{eqnarray}$$

which makes the diagram

(1)

commutative. Here $\unicode[STIX]{x1D6FD}^{\prime }$ is the isomorphism of $\mathbb{C}$-algebras given by the natural equivalence between $\mathfrak{R}^{\mathfrak{s}^{\prime }}(G^{\prime }(F))$ and the category of left $\unicode[STIX]{x210B}(G^{\prime }(F),1_{\text{}^{w}\unicode[STIX]{x2110}^{\prime }})$-modules. By construction, the diagram

(2)

is commutative.

1.6 The theorem

The theory of orbital integrals and their $\unicode[STIX]{x1D705}$-variants has been developed in characteristic zero mainly by Harish-Chandra, Langlands, Kottwitz and Shelstad. More recently, Labesse simplified the study of stable conjugation of semi-simple elements thanks to a systematic use of non-abelian cohomological constructions. Ultimately, all these constructions are done in the derived category of bounded complexes of diagonalizable groups. Most of these constructions (in particular, all those used here) can be realized using complexes of tori, making them valid in characteristic $p>0$ (see [Reference Labesse and LemaireLL15]).

For a semi-simple element $\unicode[STIX]{x1D6FE}\in G(F)$, let $G_{\unicode[STIX]{x1D6FE}}=G^{\unicode[STIX]{x1D6FE},\circ }$ be its connected (i.e. stable) centralizer. For a function $f\in C_{\text{c}}^{\infty }(G(F))$, we define the orbital integral

$$\begin{eqnarray}\mathbf{O}_{\unicode[STIX]{x1D6FE}}(f)=\int _{G_{\unicode[STIX]{x1D6FE}}(F)\backslash G(F)}f(g^{-1}\unicode[STIX]{x1D6FE}g)\,\frac{dg}{dg_{\unicode[STIX]{x1D6FE}}}.\end{eqnarray}$$

This depends on the choice of Haar measures $dg$ on $G(F)$ and $dg_{\unicode[STIX]{x1D6FE}}$ on $G_{\unicode[STIX]{x1D6FE}}(F)$. Since the orbit $\{g^{-1}\unicode[STIX]{x1D6FE}g:g\in G(F)\}\subset G(F)$ is closed for the $\mathfrak{p}$-adic topology, the integral is absolutely convergent. Two semi-simple elements $\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D6FE}^{\prime }\in G(F^{\text{sep}})$ are stably conjugate (in $G$) if there exists an element $g\in G(F^{\text{sep}})$ such that $g^{-1}\unicode[STIX]{x1D6FE}g=\unicode[STIX]{x1D6FE}^{\prime }$ and $g\unicode[STIX]{x1D70E}(g)^{-1}\in G_{\unicode[STIX]{x1D6FE}}$ for all $\unicode[STIX]{x1D70E}\in \unicode[STIX]{x1D6E4}_{F}$. In that case, the connected centralizers $G_{\unicode[STIX]{x1D6FE}}$ and $G_{\unicode[STIX]{x1D6FE}^{\prime }}$ are inner forms of each other and we require the Haar measures $dg_{\unicode[STIX]{x1D6FE}}$ and $dg_{\unicode[STIX]{x1D6FE}^{\prime }}$ on $G_{\unicode[STIX]{x1D6FE}}(F)$ and on $G_{\unicode[STIX]{x1D6FE}^{\prime }}(F)$ to be compatible with each other, the Haar measure $dg$ on $G(F)$ being fixed. If, moreover, $\unicode[STIX]{x1D6FE}$ is regular (i.e. if the connected centralizer $G_{\unicode[STIX]{x1D6FE}}$ is a torus) then we also define the stable orbital integral

$$\begin{eqnarray}\mathbf{S}\mathbf{O}_{\unicode[STIX]{x1D6FE}}(f)=\mathop{\sum }_{\unicode[STIX]{x1D6FE}^{\prime }}\mathbf{O}_{\unicode[STIX]{x1D6FE}^{\prime }}(f)\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}^{\prime }$ runs over the elements of $G(F)$ which are stably conjugate to $\unicode[STIX]{x1D6FE}$ modulo conjugation by $G(F)$.

Two semi-simple elements $\unicode[STIX]{x1D6FF}\in G^{\prime }(F^{\text{sep}})$ and $\unicode[STIX]{x1D6FE}\in G(F^{\text{sep}})$correspond if their conjugacy classes (over $F^{\text{sep}}$) correspond via the natural surjective morphism $T^{\prime }/W^{\prime }\rightarrow T/W$. If $(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE})\in G^{\prime }(F)\times G(F)$ is a pair of corresponding elements, the connected centralizer $G_{\unicode[STIX]{x1D6FF}}^{\prime }$ of $\unicode[STIX]{x1D6FF}$ in $G^{\prime }$ is an inner form of $G_{\unicode[STIX]{x1D6FE}}$. If, moreover, $\unicode[STIX]{x1D6FE}$ is strongly regular (i.e. if the centralizer $G^{\unicode[STIX]{x1D6FE}}$ is a torus) then $\unicode[STIX]{x1D6FF}$ is said to be strongly $G$-regular.

A function $f^{\prime }\in C_{\text{c}}^{\infty }(G^{\prime }(F))$ is said to be an endoscopic transfer, or simply a transfer, of a function $f\in C_{\text{c}}^{\infty }(G(F))$ if $f$ and $f^{\prime }$ have matching strongly regular orbital integrals: for all (semi-simple) strongly $G$-regular elements $\unicode[STIX]{x1D6FF}\in G^{\prime }(F)$, we have

(1)$$\begin{eqnarray}\mathbf{S}\mathbf{O}_{\unicode[STIX]{x1D6FF}}(f^{\prime })=\mathop{\sum }_{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE})\mathbf{O}_{\unicode[STIX]{x1D6FE}}(f)\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ runs over the elements of $G(F)$ which correspond (in $G$) to $\unicode[STIX]{x1D6FF}$ modulo conjugation by $G(F)$. In particular, this means that $\mathbf{S}\mathbf{O}_{\unicode[STIX]{x1D6FF}}(f^{\prime })=0$ if $\unicode[STIX]{x1D6FF}$ does not correspond to any element of $G(F)$. In (1), $G_{\unicode[STIX]{x1D6FF}}^{\prime }$ is an inner form of $G_{\unicode[STIX]{x1D6FE}}$, and we choose compatible Haar measures on $G_{\unicode[STIX]{x1D6FF}}^{\prime }(F)$ and on $G_{\unicode[STIX]{x1D6FE}}(F)$. We use them, and the Haar measures $dg$ on $G(F)$ and $dg^{\prime }$ on $G^{\prime }(F)$ normalized by

$$\begin{eqnarray}\operatorname{vol}(\unicode[STIX]{x2110},dg)=1=\operatorname{vol}(\unicode[STIX]{x2110}^{\prime },dg^{\prime }),\end{eqnarray}$$

to define the stable orbital integral $\mathbf{S}\mathbf{O}_{\unicode[STIX]{x1D6FF}}$ on $G^{\prime }(F)$ and the orbital integral $\mathbf{O}_{\unicode[STIX]{x1D6FE}}$ on $G(F)$. The transfer factor $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE})$ is the canonical transfer factor defined by the choice of $K$ as in [Reference Mœglin and WaldspurgerMW16, 6.3].Footnote ⁷ In other words, $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE})$ is the transfer factor which makes $\operatorname{vol}(K^{\prime },dg^{\prime })^{-1}1_{K^{\prime }}$ a transfer of $\operatorname{vol}(K,dg)^{-1}1_{K}$ (fundamental lemma).

Recall that for all elements $w\in W$, we have defined an injective morphism of $\mathbb{C}$-algebras $\{w\}^{\prime }:Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ (cf. § 1.4). We are interested in the composition morphism

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}\circ [w]=\{w\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}:Z(G(F),\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime }).\end{eqnarray}$$

For $w_{0}\in W_{\unicode[STIX]{x1D712}_{0}}$, we have $\unicode[STIX]{x1D70C}_{ww_{0}}^{\prime }=\unicode[STIX]{x1D70C}_{w}^{\prime }$, but the composition morphism depends really on $w$, and not only on the projection of $w$ on $W/W^{\prime }$ (Remark 1 of § 1.4): we have $\{ww_{0}\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}=\{w_{0}\}_{w}^{\prime }\circ \{w\}^{\prime }\circ \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$ where $\{w_{0}\}_{w}^{\prime }$ is an automorphism of the $\mathbb{C}$-algebra $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ which is not trivial in general, even if $w_{0}\in W^{\prime }$.

For $f\in Z(G(F),\unicode[STIX]{x1D70C})$, we put

(2)$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f)=|W^{\prime }|^{-1}\mathop{\sum }_{w\in W}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f).\end{eqnarray}$$

By definition, the function $\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f)$ belongs to the subspace

$$\begin{eqnarray}\mathbf{Z}(G^{\prime }(F))_{\unicode[STIX]{x1D712}_{0}}=\mathop{\sum }_{w\in W/W_{\unicode[STIX]{x1D712}_{0}}}Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })\end{eqnarray}$$

of $\unicode[STIX]{x210B}(G^{\prime }(F),1_{\unicode[STIX]{x2110}_{+}^{\prime }})$. For $w_{1},w_{2}\in W$, we have $\mathfrak{s}_{w_{1}}^{\prime }=\mathfrak{s}_{w_{2}}^{\prime }$ if and only if there exists an element $w^{\prime }\in W^{\prime }$ such that $\text{}^{w^{\prime }}\!(\unicode[STIX]{x1D712}_{w_{1},0})=\unicode[STIX]{x1D712}_{w_{2},0}$, that is, $\unicode[STIX]{x1D712}_{w^{\prime }w_{1},0}=\unicode[STIX]{x1D712}_{w_{2},0}$. But the last equality is equivalent to $w_{2}^{-1}w^{\prime }w_{1}\in W_{\unicode[STIX]{x1D712}_{0}}$. Hence we have

(3)$$\begin{eqnarray}\mathbf{Z}(G^{\prime }(F))_{\unicode[STIX]{x1D712}_{0}}=\bigoplus _{w}Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })\end{eqnarray}$$

where $w$ runs over a set of representative elements of $W^{\prime }\backslash W/W_{\unicode[STIX]{x1D712}_{0}}$ in $W$.

By definition, the image of the Langlands parameter $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D712}}:W_{F}\rightarrow {\check{T}}$ of $\unicode[STIX]{x1D712}$ is contained in the center $Z({\check{G}}^{\prime })$ of the dual group ${\check{G}}^{\prime }$ (cf. Remark 2 of § 1.1), and from Langlands, it defines a character $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime }$ of $G^{\prime }(F)$. Of course $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime }$ extends the character $\unicode[STIX]{x1D712}_{0}$ of the maximal compact subgroup $T$ of $T^{\prime }(F)=T(F)$.

Proof. Let $\unicode[STIX]{x1D712}^{\natural }$ be another $W_{\unicode[STIX]{x1D712}_{0}}$-invariant character of $T(F)$ extending $\unicode[STIX]{x1D712}_{0}$. Put $\unicode[STIX]{x1D703}=(\unicode[STIX]{x1D712}^{\natural })\unicode[STIX]{x1D712}^{-1}$. It is an unramified character of $T(F)$ which coincides with the restriction of the character $w_{\unicode[STIX]{x1D712}^{\natural }}^{\prime }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime })^{-1}$ of $G^{\prime }(F)$ to $T(F)$. For $f\in Z(G(F),\unicode[STIX]{x1D70C})$ and $w\in W$, from § 1.4.(4), $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}^{\natural }}(\text{}^{w}\!f)$ is the unique element of $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ which acts on the left on $\unicode[STIX]{x1D704}_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x1D70C}_{w}^{\prime }}$ by multiplication by the scalar

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}^{\natural }}(\text{}^{w}\!f))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}^{\natural }}(f)=\unicode[STIX]{x1D706}_{(\unicode[STIX]{x1D709}\unicode[STIX]{x1D703})\unicode[STIX]{x1D712}}(f)=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D703}}^{\prime }(\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f)),\end{eqnarray}$$

for all characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)=T(F)$ extending a $W^{\prime }$-conjugate of $\unicode[STIX]{x1D712}_{w,0}$. Since $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}^{\natural }}^{\prime }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime })^{-1}|_{T(F)}$, we obtain

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}^{\natural }}(\text{}^{w}\!f)=\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}^{\natural }}^{\prime }(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime })^{-1}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f).\end{eqnarray}$$

Thus we have

$$\begin{eqnarray}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}^{\natural }}^{\prime })^{-1}\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}^{\natural }}(f)=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime })^{-1}\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f).\end{eqnarray}$$

This proves the lemma. ◻

For $f\in Z(G(F),\unicode[STIX]{x1D70C})$, we put

$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D709}}_{\unicode[STIX]{x1D712}_{0}}(f)=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime })^{-1}\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f).\end{eqnarray}$$

The function $\boldsymbol{\unicode[STIX]{x1D709}}_{\unicode[STIX]{x1D712}_{0}}(f)$ belongs to the subspace $\bigoplus _{w\in W^{\prime }\backslash W/W_{\unicode[STIX]{x1D712}_{0}}}Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{\text{}^{w}\!(\unicode[STIX]{x1D712}_{0})}^{\prime })$ of $\unicode[STIX]{x210B}(G^{\prime }(F),1_{\unicode[STIX]{x2110}_{+}^{\prime }})$.

Remark 1. The character $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime }$ of $G^{\prime }(F)$ is constant on each stable conjugacy class of $G^{\prime }(F)$ [Reference HalesHal95, Lemma 3.1]. Hence the theorem says we have

$$\begin{eqnarray}\mathbf{S}\mathbf{O}_{\unicode[STIX]{x1D6FF}}(\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f))=\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime }(\unicode[STIX]{x1D6FF})\widetilde{\mathbf{O}}_{\unicode[STIX]{x1D6FF}}(f)\end{eqnarray}$$

for all strongly $G$-regular elements $\unicode[STIX]{x1D6FF}\in G^{\prime }(F)$.

The proof is modeled on the method of Haines [Reference HainesHai13]. The final part of the proof uses local data, which are obtained via a local–global argument based on a simple trace formula, whose geometric side is the elliptic (strongly regular) part of the trace formula. Beforehand it is necessary to reduce the proof of the theorem to the proof of the following ‘elliptic version’.Footnote ⁸

For technical reasons, other reductions are needed. In fact, we will prove that we can suppose (in the theorem and also in Theorem*) that the derived group $G_{\text{der}}$ of $G$ is simply connected, and the scheme-theoretic center of $G$ is a (split) torus.

1.7 Variant with two depth-zero characters of $T$

As we will see in § 1.11, the reduction of the theorem to the Theorem* of § 1.6 is based on a descent formula (proposition of § 1.11) which leads us to consider a generalization of the Theorem* of § 1.6.

So let us fix another depth-zero character $\unicode[STIX]{x1D713}_{0}$ of $T$. Let $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}}^{\prime }$ be the inflation of $\unicode[STIX]{x1D713}_{0}$ to the Iwahori subgroup $\unicode[STIX]{x2110}^{\prime }$ of $G^{\prime }$. Here, $G^{\prime }$ is always the endoscopic group of $G$ defined by $\unicode[STIX]{x1D712}_{0}$ (cf. § 1.1). We define an injective morphism of $\mathbb{C}$-algebras (again denoted by $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$)

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}^{\unicode[STIX]{x2110}}:Z(G(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}}^{\prime }).\end{eqnarray}$$

Let us choose a character $\unicode[STIX]{x1D713}$ of $T(F)$ extending $\unicode[STIX]{x1D713}_{0}$ (it is not necessary to take it $W_{\unicode[STIX]{x1D713}_{0}}$-invariant). Let $\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}}=[T,\unicode[STIX]{x1D713}\unicode[STIX]{x1D712}]_{G}$ be the $G(F)$-inertial class of the cuspidal pair $(T,\unicode[STIX]{x1D713}\unicode[STIX]{x1D712})$ of $G(F)$, and $\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}}^{\prime }=[T,\unicode[STIX]{x1D713}]_{G^{\prime }}$ be the $G^{\prime }(F)$-inertial class of the cuspidal pair $(T,\unicode[STIX]{x1D713})$ of $G^{\prime }(F)$. Since

$$\begin{eqnarray}W_{\unicode[STIX]{x1D713}_{0}}^{\prime }\subset W_{\unicode[STIX]{x1D713}_{0}}\,\cap \,W_{\unicode[STIX]{x1D712}_{0}}\subset W_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}},\end{eqnarray}$$

the surjective morphism of algebraic varieties

$$\begin{eqnarray}\mathfrak{X}(\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}}^{\prime })\rightarrow \mathfrak{X}(\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}}),\quad (T,\unicode[STIX]{x1D709})\mapsto (T,\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})\end{eqnarray}$$

defines by duality an injective morphism of $\mathbb{C}$-algebras (again denoted by $\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D712}}$)

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D712}}:\mathbb{C}[\mathfrak{X}(\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})]\rightarrow \mathbb{C}[\mathfrak{X}(\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}}^{\prime })],\end{eqnarray}$$

and $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$ is the morphism of $\mathbb{C}$-algebras which makes the diagram

(1)

commutative. As for the morphism $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$ of § 1.3.(3), the morphism $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}$ above can be characterized by actions on principal series: for $f\in Z(G(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})$, $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f)$ is the unique element of $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}}^{\prime })$ which acts on the space $\unicode[STIX]{x1D704}_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}}^{\prime }}$ by multiplication by the scalar

(2)$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}(f))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}}(f)\end{eqnarray}$$

for all characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)=T(F)$ extending a $W^{\prime }$-conjugate of $\unicode[STIX]{x1D713}_{0}$ (note that since $W^{\prime }\subset W_{\unicode[STIX]{x1D712}_{0}}$, $\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}$ extends a $W^{\prime }$-conjugate of $\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}$).

Let $w$ be an element of $W$. As in § 1.4, we choose a lift $n$ of $w$ in $N\,\cap \,K$ (recall that $K$ is a hyperspecial maximal compact subgroup of $G(F)$ containing $\unicode[STIX]{x2110}$). We consider the character $\text{}^{w}\!(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})=\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}}\circ \operatorname{Int}_{n^{-1}}$ of $\text{}^{w}\unicode[STIX]{x2110}$, and the character $\unicode[STIX]{x1D70C}_{\text{}^{w}\!\unicode[STIX]{x1D713}_{0}}^{\prime }\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D712}_{w,0}}^{\prime }=\unicode[STIX]{x1D70C}_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }$ of $\unicode[STIX]{x2110}^{\prime }$. Recall that (by definition) $\unicode[STIX]{x1D712}_{w,0}=(\text{}^{w}\!\unicode[STIX]{x1D712}_{0})\unicode[STIX]{x1D712}_{0}^{-1}$, thus $\unicode[STIX]{x1D70C}_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }$ is the inflation of the character $(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}=\text{}^{w}\!(\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0})\unicode[STIX]{x1D712}_{0}^{-1}$ of $T$ to $\unicode[STIX]{x2110}^{\prime }$. We define an injective morphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}:Z(G(F),\text{}^{w}\!(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}}))\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }).\end{eqnarray}$$

Let $\mathfrak{s}_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }=[T,(\text{}^{w}\!\unicode[STIX]{x1D713})\unicode[STIX]{x1D712}_{w}]_{G^{\prime }}$ be the $G^{\prime }(F)$-inertial class of the cuspidal pair $(T,(\text{}^{w}\!\unicode[STIX]{x1D713})\unicode[STIX]{x1D712}_{w})$ of $G^{\prime }(F)$. Since

$$\begin{eqnarray}W_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }=W_{\text{}^{w}\!(\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0})}^{\prime }\subset W_{\unicode[STIX]{x1D712}_{0}},\end{eqnarray}$$

for a character $\unicode[STIX]{x1D709}$ of $T(F)$ extending $\unicode[STIX]{x1D713}_{0}$ and for $\unicode[STIX]{x1D702}\in W_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }$, we have

$$\begin{eqnarray}\text{}^{\unicode[STIX]{x1D702}}((\text{}^{w}\!\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w})=\text{}^{\unicode[STIX]{x1D702}w}(\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})\unicode[STIX]{x1D712}^{-1}=\text{}^{w}(\text{}^{w^{-1}\unicode[STIX]{x1D702}w}(\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}))\unicode[STIX]{x1D712}^{-1}=\text{}^{w}(\text{}^{w^{-1}\unicode[STIX]{x1D702}w}(\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})\unicode[STIX]{x1D712}^{-1})\unicode[STIX]{x1D712}_{w}\end{eqnarray}$$

(in this calculation we used $\text{}^{\unicode[STIX]{x1D702}}\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}$). Hence the surjective morphism of algebraic varieties

$$\begin{eqnarray}\mathfrak{X}(\mathfrak{s}_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime })\rightarrow \mathfrak{X}(\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}}),\quad (T,(\text{}^{w}\unicode[STIX]{x1D709})\unicode[STIX]{x1D712}_{w})_{G^{\prime }}\mapsto (T,\unicode[STIX]{x1D709}\unicode[STIX]{x1D712})_{G}\end{eqnarray}$$

is well defined. It defines by duality an injective morphism of $\mathbb{C}$-algebras

$$\begin{eqnarray}\unicode[STIX]{x1D702}_{w,\unicode[STIX]{x1D712}}:\mathbb{C}[\mathfrak{X}(\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})]\rightarrow \mathbb{C}[\mathfrak{X}(\mathfrak{s}_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime })],\end{eqnarray}$$

and $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}$ is the morphism of $\mathbb{C}$-algebras which makes the diagram

(3)

commutative. As for the morphism $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}$ of § 1.4.(

), the morphism $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}$ above can be characterized by actions on principal series: for $f\in Z(G(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})$, $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(f)$ is the unique element of $Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{(\text{}^{w}\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime })$ which acts on the left on $\unicode[STIX]{x1D704}_{B^{\prime }}^{G^{\prime }}(\unicode[STIX]{x1D709})^{\unicode[STIX]{x1D70C}_{(\text{}^{w}\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }}$ by multiplication by the scalar

(4)$$\begin{eqnarray}\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}}^{\prime }(\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f))=\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}}(f)\end{eqnarray}$$

for all characters $\unicode[STIX]{x1D709}$ of $T^{\prime }(F)=T(F)$ extending a $W^{\prime }$-conjugate of $(\text{}^{w}\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}$ (note that since $W^{\prime }\subset W_{\unicode[STIX]{x1D712}_{0}}$, $\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}$ extends a $W^{\prime }$-conjugate of $\text{}^{w}\!(\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0})$, and, in particular, it extends a $W$-conjugate of $\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}$).

Remark 2. If $\unicode[STIX]{x1D713}_{0}=1$, then (in the notation of §§ 1.1 and 1.4) $\mathfrak{s}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}}=\mathfrak{s}$ and $\mathfrak{s}_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }=\mathfrak{s}_{w}^{\prime }$, and $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}$ coincides with the injective morphism of $\mathbb{C}$-algebras $\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}:Z(G(F),\text{}^{w}\!\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{w}^{\prime })$ defined in § 1.4.(3). On the other hand, If $\unicode[STIX]{x1D713}_{0}\neq 1$, in general the composition morphism

$$\begin{eqnarray}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}\circ [w]:Z(G(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{(\text{}^{w}\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime })\end{eqnarray}$$

does not factorize through the morphism $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D712}}:Z(G(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})\rightarrow Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}}^{\prime })$ defined above; it does so if and only if the inclusion

$$\begin{eqnarray}W_{(\text{}^{w}\!\unicode[STIX]{x1D713}_{0})\unicode[STIX]{x1D712}_{w,0}}^{\prime }=W_{\text{}^{w}\!(\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0})}^{\prime }\subset W_{\unicode[STIX]{x1D713}_{0}}^{\prime }\end{eqnarray}$$

is satisfied. Nevertheless, equality (4) is true even if $\unicode[STIX]{x1D713}_{0}\neq 1$.

For $f\in Z(G(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})$, we put

(5)$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f)=|W^{\prime }|^{-1}\mathop{\sum }_{w\in W}\unicode[STIX]{x1D701}_{w,\unicode[STIX]{x1D712}}(\text{}^{w}\!f).\end{eqnarray}$$

The notation is coherent: if $\unicode[STIX]{x1D713}_{0}=1$, from Remark 2, the function $\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f)$ defined by (5) coincides with the one defined by § 1.4.(2).

For $f\in Z(G(F),\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0}})$, we put

$$\begin{eqnarray}\boldsymbol{\unicode[STIX]{x1D709}}_{\unicode[STIX]{x1D712}_{0}}(f)=(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}^{\prime })^{-1}\boldsymbol{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D712}}(f).\end{eqnarray}$$

Note that the function $\boldsymbol{\unicode[STIX]{x1D709}}_{\unicode[STIX]{x1D712}_{0}}(f)$ belongs to the subspace $\sum _{w\in W}Z(G^{\prime }(F),\unicode[STIX]{x1D70C}_{\text{}^{w}\!(\unicode[STIX]{x1D713}_{0}\unicode[STIX]{x1D712}_{0})}^{\prime })$ of $\unicode[STIX]{x210B}(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$. The following theorem (respectively, Theorem*) is a generalization of the theorem (respectively, Theorem*) of § 1.6: we recover the theorems of § 1.6 by taking $\unicode[STIX]{x1D713}_{0}=1$.

1.8 Parabolic descent

Let $P$ be a semi-standard parabolic subgroup of $G$ defined over $F$, where semi-standard means containing $T$. Let $U$ be the unipotent radical of $P$, and $M$ the semi-standard Levi component of $P$. The decomposition $P=M\,\rtimes \,U$ is defined over $F$. We denote by $W_{M}$ the Weyl group $N_{M}(T)/T$ of $M$. We put $B_{M}=B\,\cap \,M$ and $\unicode[STIX]{x2110}_{M}=\unicode[STIX]{x2110}\,\cap \,M$. The group $\unicode[STIX]{x2110}_{M}$ is an Iwahori subgroup of $M(F)$ in good position with respect to the Borel pair $(B_{M},T)$ of $M$.

Let $\mathfrak{s}_{M}=[T,\unicode[STIX]{x1D712}]_{M}$ be the $M(F)$-inertial class of the cuspidal pair $(T,\unicode[STIX]{x1D712})$ of $M(F)$. Let $\mathfrak{X}(\mathfrak{s}_{M})$ be the complex algebraic variety associated to $\mathfrak{s}_{M}$. There is a canonical surjective morphism of varieties