Proof. Let us write $x = 1 + \varpi _K^{i} u$, with $u \in \mathcal {O}_K$. The binomial expansion gives us $x^p=1 + p \varpi _K^{i} u + \varpi _K^{2i} u'$, with $u' \in \mathcal {O}_K$. But $\varpi _K$ divides $p$ in $\mathcal {O}_K$ so $x^p \in U_K^{i+1}$.

Proof. Let $e$ be the ramification index of $K$ over $F$. Then $\varpi _K^{e} = \varpi _F u$, with $u \in \mathcal {O}_K^{\times }$. Now $\mathcal {O}_K^{\times } / U_K^1$ is isomorphic to $k_K^{\times }$ the residue field of $K$ of cardinal $q_K-1$. Therefore, $x:=u^{q_K-1} \in U_K^1$ and from Lemma 3.1, $x^{p^{N}} \in U_K^N$. We get the result with $m=e(q_K-1)p^N$ and $s=(q_K-1)p^N$.

Proof. (i) We apply a similar strategy as in [Reference Drevon and SécherreDS23, § 4]. Let $\eta$ be a character of $J$, trivial on $J^0$ such that $\eta (\varpi _F)=\bar {\alpha }$. Let $\Sigma :=\Lambda \eta ^{-1}$ such that $\varpi _F$ acts trivially on $\Sigma$.

Since $\Sigma$ is smooth and $J^0$ is compact, there exists an open subgroup $H^2$ of $J^1$ such that $H^2 \subset \ker (\Sigma _{|J^0})$ and $J^0/H^2$ is finite. Since $H^2$ is open in $J^0$, there exists an integer $N\geq 1$ such that $U_K^N \subset H^2$. By Lemma 3.2, there exist $m,n\geq 1$ such that $\varpi _K^m \in \varpi _F^s U_K^N$. We define a group $Q$ by

\[ Q=J/\langle H^2, \varpi_F \rangle. \]

The group $Q$ is finite. Moreover, since $\varpi _K^m \in \varpi _F^s H^2$, we have that $\langle H^2, \varpi _K^m \rangle =\langle H^2, \varpi _F^s \rangle$. Therefore, $\Sigma$ is trivial on $\langle H^2, \varpi _F \rangle$ and we can consider $\Sigma$ as a representation of $Q$.

Since $Q$ is a finite group, by [Reference SerreSer77, Proposition 41] we can consider the projective envelope of $\Sigma$ in the category of $\overline {\mathbf {F}}_{\ell }[Q]$-modules, denoted by $\overline {\mathcal {P}}$. We can regard $\overline {\mathcal {P}}$ as a representation of $J$ by inflation. Let

\[ Q_1= J/ \langle \varpi_F\rangle, \]

which is compact and $Q$ is a quotient of $Q_1$. Since $H^2$ is pro-$p$, a projective object of $Q$ remains projective as a representation of $Q_1$. Now we show that $\overline {\mathcal {P}}$ is projective in $\operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{1}(J)$. It is due to the fact that $\operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{1}(J)$ is equivalent to the category of smooth $\overline {\mathbf {F}}_{\ell }$-representations of $Q_1$. For arbitrary $\bar {\alpha }$, it follows from the fact that $\operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{1}(J)$ and $\operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{\bar {\alpha }}(J)$ are equivalent via twisting $\eta$ with each object in $\operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{1}(J)$.

(ii) By [Reference SerreSer77, Proposition 42], there is a unique projective $\overline {\mathbf {Z}}_{\ell }[Q]$-module $\mathcal {P}$ such that $\mathcal {P} \otimes \overline {\mathbf {F}}_{\ell } = \overline {\mathcal {P}}$. Like in the first part, $\mathcal {P}$ can be lifted to a projective object in $\operatorname {Rep}_{\overline {\mathbf {Z}}_{\ell }}^{\alpha }(J)$. If $\mathcal {P}'$ is another projective in $\operatorname {Rep}_{\overline {\mathbf {Z}}_{\ell }}^{\alpha }(J)$ with the same properties, then, by projectivity, any isomorphism from $\overline {\mathcal {P}}$ to $\mathcal {P}' \otimes \overline {\mathbf {F}}_{\ell }$ can be lifted to an isomorphism from $\mathcal {P}$ to $\mathcal {P}'$.

Proof. We write $\pi = \operatorname {c-Ind}_{J}^{G}(\Lambda )$, for an extended maximal simple type $(J,\Lambda )$. We have $\Lambda \in \operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{\bar {\alpha }}(J)$ for $\bar {\alpha }\in \overline {\mathbf {F}}_{\ell }^{\times }$. Let $\bar {\chi }$ be an $\overline {\mathbf {F}}_{\ell }$-character of $F^{\times }$ such that $\bar {\chi }$ is trivial on $\mathcal {O}_F^{\times }$ and $\bar {\chi }(\varpi _F)=\bar {\alpha }$. Frobenius reciprocity and Mackey formula give us

\[ 0\neq \operatorname{Hom}_{H}(\pi,\mathbf{1})\simeq \prod_{g\in J\backslash G/H} \operatorname{Hom}_{J^{g}\cap H}(\Lambda^{g},\mathbf{1}). \]

Thus, there exists a $g\in G$ such that $\operatorname {Hom}_{J^{g}\cap H}(\Lambda ^{g},\mathbf {1}) \neq 0$. Let $H':=H^{g^{-1}}$ such that $J^g \cap H = (J\cap H')^g$. Therefore,

\begin{align*} 0\neq \operatorname{Hom}_{J^{g}\cap H}(\Lambda^{g},\mathbf{1}) \simeq \operatorname{Hom}_{J\cap H'}(\Lambda,\mathbf{1}). \end{align*}

Define $I=\langle J\cap H',\varpi _F\rangle$ and so $\langle \varpi _F \rangle \subset I\subset J.$ Applying Frobenius reciprocity, we have

\[ \operatorname{Hom}_{J\cap H'}(\Lambda,\mathbf{1})\cong\operatorname{Hom}_{I}(\Lambda,\operatorname{Ind}_{J\cap H'}^{I}\mathbf{1}). \]

Let $\phi$ be a non-trivial element in $\operatorname {Hom}_{I}(\Lambda,\operatorname {Ind}_{J\cap H'}^{I}\mathbf {1})$. Let $V$ be the representation space of $\Lambda$. Taking the restriction to $\langle \varpi _F\rangle$, we have

\[ 0\neq \phi\in \operatorname{Hom}_{\langle \varpi_F\rangle}(\Lambda,\operatorname{Ind}_{J\cap H'}^{I}\mathbf{1}). \]

Since $\langle \varpi _F\rangle$ acts as the character $\bar {\chi }$ on $\Lambda$, $\phi (V)$ is contained in the $\bar {\chi }$-socle of $\mathrm {res}_{\langle \varpi _F\rangle }^{I}\operatorname {Ind}_{J\cap H'}^{I}\mathbf {1}$, which is the maximal subrepresentation where $\langle \varpi _F\rangle$ acts as a multiple of $\bar {\chi }$. Meanwhile,

\[ \operatorname{Ind}_{J\cap H'}^{I}\mathbf{1}\subset\operatorname{Ind}_{\{1\}}^{\langle \varpi_F\rangle}\mathbf{1}. \]

By Frobenius reciprocity, $\dim \operatorname {Hom}_{\langle \varpi _F\rangle }(\bar {\chi },\operatorname {Ind}_{\{1\}}^{\langle \varpi _F\rangle }\mathbf {1})=1$. On the other hand, $J\cap H'$ acts trivially on $\operatorname {Ind}_{J\cap H'}^{I}\mathbf {1}$. Since $\Lambda$ is $J\cap H'$-distinguished, one can inflate $\bar {\chi }$ to a character of $I$, still denoted by $\bar {\chi }$. Then the $\bar {\chi }$-socle of $\operatorname {Ind}_{J\cap H'}^{I}\mathbf {1}$ is equivalent to $\bar {\chi }$. Hence,

\[ \phi\in \operatorname{Hom}_{I}(\Lambda,\bar{\chi})\neq 0. \]

By Frobenius reciprocity and the fact that $J/ I$ is compact, we have

(3.1)\begin{equation} 0 \neq \operatorname{Hom}_{I}(\Lambda,\bar{\chi})\cong \operatorname{Hom}_{J}(\Lambda,\operatorname{c-Ind}_{I}^{J}\bar{\chi}), \end{equation}

where $\operatorname {c-Ind}_{I}^{J}\bar {\chi }\in \operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{\bar {\alpha }}(J)$ is admissible.

Let $\alpha$ be a root of unity in $\overline {\mathbf {Z}}_{\ell }$ whose image (after modulo $\ell$) in $\overline {\mathbf {F}}_{\ell }$ is equal to $\bar {\alpha }$, and denote by $\chi$ a $\overline {\mathbf {Z}}_{\ell }$-lift of $\bar {\chi }$ such that $\chi$ maps $\varpi _F$ to $\alpha$. Set $\tilde {\chi }=\chi \otimes _{\overline {\mathbf {Z}}_{\ell }}\overline {\mathbf {Q}}_{\ell }$. From Proposition 3.3, let $\overline {\mathcal {P}}$ be the projective envelope of $\Lambda$ in $\operatorname {Rep}_{\overline {\mathbf {F}}_{\ell }}^{\bar {\alpha }}(J)$ and $\mathcal {P}$ a projective object in $\operatorname {Rep}_{\overline {\mathbf {Z}}_{\ell }}^{\alpha }(J)$ such that $\mathcal {P} \otimes \overline {\mathbf {F}}_{\ell } = \overline {\mathcal {P}}$. Set $\Lambda |_{J^0}=\kappa \otimes \rho$. Since $\rho$ corresponds to a supercuspidal representation of $J^0/J^1$, a similar argument to [Reference VignérasVig96, Chapitre III, 2.9] gives us the structure of the projective envelope. Let us give some more details. From the structure of the projective envelope for finite groups, we see that if $\delta$ is an irreducible integral $\overline {\mathbf {Q}}_{\ell }$-representation of $J$, then the multiplicity of $\delta$ in $\mathcal {P} \otimes \overline {\mathbf {Q}}_{\ell }$ is equal to the multiplicity of $\Lambda$ in $r_\ell (\delta )$ (see [Reference Drevon and SécherreDS23, Proposition 4.9]). Moreover, if $\Lambda '$ is a subquotient of $\mathcal {P} \otimes \overline {\mathbf {F}}_{\ell }$, then [Reference Drevon and SécherreDS23, Lemma 4.13] says that there exists $\delta$ an irreducible integral $\overline {\mathbf {Q}}_{\ell }$-representation of $J$ such that its modulo $\ell$ reduction contains $\Lambda$ and $\Lambda '$. We show that $\Lambda \cong \Lambda '$ by proving that $\delta$ is an extended maximal simple type and that $\Lambda =r_\ell (\delta )$. In fact, there is an irreducible component $\delta _0$ of $\delta \vert _{J^0}$, such that $\kappa \otimes \rho$ is a subquotient of $r_{\ell }(\delta _0)$. Let $\tilde {\kappa }$ be a $\overline {\mathbf {Q}}_{\ell }$-lift of $\kappa$ as in [Reference VignérasVig96, Chapitre III, 4.20]. Denote by $\tilde {\eta }$ the restriction $\tilde {\kappa }\vert _{J^1}$. Since the $\tilde {\eta }$-isotypic part of $\delta _0$ is non-trivial, by taking $\tilde {\kappa }$-restriction as defined in [Reference VignérasVig01b, Definition 8.1 (c)], there is an irreducible $\overline {\mathbf {Q}}_{\ell }$-representation $\tilde {\rho }$ of $J^0/ J^1$ such that $\delta _0\cong \tilde {\kappa }\otimes \tilde {\rho }$. Then $\rho$ is a subquotient of $r_{\ell }(\tilde {\rho })$. Since $\rho$ is supercuspidal, by [Reference HelmHel16, Proposition 5.7] we know $\tilde {\rho }$ is supercuspidal and $\rho =r_{\ell }(\tilde {\rho })$, which implies that $\delta _0$ is a maximal simple type and $r_{\ell }(\delta _0)=\kappa \otimes \rho$. Hence, $\delta$ is an extended maximal simple type. We conclude that $r_{\ell }(\delta )=\Lambda$. Therefore, we have the following structure of $\mathcal {P}$:

1. (i) $\overline {\mathcal {P}}=\mathcal {P} \otimes _{\overline {\mathbf {Z}}_{\ell }} \overline {\mathbf {F}}_{\ell }$ is indecomposable of finite length, with each irreducible component isomorphic to $\Lambda$;

2. (ii) let $\widetilde {\mathcal {P}} := \mathcal {P} \otimes _{\overline {\mathbf {Z}}_{\ell }} \overline {\mathbf {Q}}_{\ell }$ be a $\overline {\mathbf {Q}}_{\ell }$-lift of $\mathcal {P}$, then $\widetilde {\mathcal {P}}$ is isomorphic to the direct sum of extended maximal simple types, which are all the $\overline {\mathbf {Q}}_{\ell }$-lifts of $\Lambda$.

By part (i) and (3.1), we get that

(3.2)\begin{equation} \operatorname{Hom}_{\overline{\mathbf{F}}_{\ell}[J]}(\overline{\mathcal{P}},\operatorname{c-Ind}_{I}^{J}\bar{\chi})\neq 0. \end{equation}

By § I,9.3,(viii) of [Reference VignérasVig96], $\operatorname {c-Ind}_{I}^{J}\chi$ is a $\overline {\mathbf {Z}}_{\ell }[J]$-lattice of $\operatorname {c-Ind}_{I}^{J}\tilde {\chi }$, and the reduction modulo-$\ell$ gives a surjective morphism from $\operatorname {c-Ind}_{I}^{J}\chi$ to $\operatorname {c-Ind}_{I}^{J}\bar {\chi }$. By the projectivity of $\mathcal {P}$, (3.2) implies that

\begin{align*} \operatorname{Hom}_{J}(\mathcal{P},\operatorname{c-Ind}_{I}^{J}\chi)\neq 0. \end{align*}

Hence, by tensoring with $\overline {\mathbf {Q}}_{\ell }$ on both sides,

(3.3)\begin{equation} 0\neq \operatorname{Hom}_{J}(\mathcal{P}\otimes\overline{\mathbf{Q}}_{\ell},\operatorname{c-Ind}_{I}^{J}\tilde{\chi})\subset \operatorname{Hom}_{J}(\mathcal{P}\otimes\overline{\mathbf{Q}}_{\ell},\operatorname{Ind}_{J\cap H'}^{J}\mathbf{1}). \end{equation}

From part (ii) and (3.3) there exists a $\overline {\mathbf {Q}}_{\ell }$-lifts $\widetilde {\Lambda }$ of $\Lambda$ such that $\operatorname {Hom}_{\overline {\mathbf {Q}}_{\ell }[J]}(\widetilde {\Lambda },\operatorname {Ind}_{J\cap H'}^{J}\mathbf {1}) \neq 0$. Hence, $\operatorname {Hom}_{J\cap H'}(\widetilde {\Lambda },\mathbf {1}) \neq 0$.

Let $\widetilde {\pi }:=\operatorname {c-Ind}_{J}^{G}(\widetilde {\Lambda })$, where $\tilde {\Lambda }$ is a $\overline {\mathbf {Q}}_{\ell }$-extended maximal simple type (the discussion before about $\delta$ shows that $\tilde {\Lambda }$ is an extended maximal simple type). Hence, $\widetilde {\pi }$ is irreducible and supercuspidal. Since the reduction modulo $\ell$ commutes with $\operatorname {c-Ind}$, we deduce that the representation $\widetilde {\pi }$ is a $\overline {\mathbf {Q}}_{\ell }$-lift of $\pi$. We are left to prove that it is $H$-distinguished. As before

\[ \operatorname{Hom}_{J^{g}\cap H}(\widetilde{\Lambda}^{g},\mathbf{1}) \simeq \operatorname{Hom}_{J\cap H'} (\widetilde{\Lambda},\mathbf{1}) \neq 0 \]

and Mackey formula and Frobenius reciprocity give us that $\widetilde {\pi }$ is $H$-distinguished.

Proof. There is no complex supercuspidal representation of $\operatorname {GL}_{2n}(F)$ distinguished by $\operatorname {Sp}_{2n}(F)$. When the field $F$ has characteristic zero, this follows from [Reference Heumos and RallisHR90, Theorem 3.2.2]; and when the field $F$ is of characteristic $p$ with $p\neq 2$, it follows from [Reference PrasadPra19, Theorem 1] (see [Reference PrasadPra19, § 8]). Therefore, the result follows from Theorem 3.4.

Proof. Let us denote by $\varphi$ (respectively, $\tilde {\varphi }_i$) the Langlands parameter of $\pi$ (respectively, $\tilde {\pi }_i$). For $i\in \{1,2\}$, since $\tilde {\pi }_2$ is $\sigma$-selfdual, $\tilde {\varphi }_i$ is either conjugate-symplectic or conjugate-orthogonal, and let us denote by $B_i$ the associated bilinear form. We also denote by $b_i$ the sign of $B_i$. Let us denote by $V_i$ the space of $\tilde {\varphi }_i$ and by $V$ the space of $\varphi$. The non-degenerate bilinear form $B_i:V_i \times V_i \to \overline {\mathbf {Q}}_{\ell }$ induces an isomorphism of $W_E$ representation $f_i : V_i^s \to V_i^{\vee }$, where $s \in W_F \setminus W_E$.

Since $\tilde {\varphi }_i$ is a $\overline {\mathbf {Q}}_{\ell }$-lift of $\varphi$, there exists an $\overline {\mathbf {Z}}_{\ell }$-lattice $L_i$ of $V_i$ such that $L_i \otimes \overline {\mathbf {Q}}_{\ell } \simeq V_i$ and $L_i \otimes \overline {\mathbf {F}}_{\ell } \simeq V$ (as $W_E$ representations). Let $M_i$ be the $\overline {\mathbf {Z}}_{\ell }$-lattice of $V_i$ such that $f_i$ induces an isomorphism $L_i^s \to M_i^{\vee }$. The modulo $\ell$ reduction does not depend on the lattice, so we also have $M_i \otimes \overline {\mathbf {Q}}_{\ell } \simeq V_i$ and $M_i \otimes \overline {\mathbf {F}}_{\ell } \simeq V$. Let $B_{i,\overline {\mathbf {Z}}_{\ell }}$ be the restriction of $B_i$ to $L_i\times M_i$. Its image is a $\overline {\mathbf {Z}}_{\ell }$-lattice of $\overline {\mathbf {Q}}_{\ell }$. In addition, by the definition of $L_i$ and $M_i$ it is non-degenerate as a $\overline {\mathbf {Z}}_{\ell }$-bilinear form. Therefore, its modulo $\ell$ reduction gives a non-degenerate bilinear form $\bar {B}_i$ on $V$. As $\ell \neq 2$ the sign $b_i$ is preserved by reduction modulo $\ell$, and is the sign of $\bar {B}_i$. Moreover, $\pi$ is supercuspidal, so $\varphi$ is irreducible. Schur's lemma implies that the sign of a bilinear form on $V$ is unique. Therefore, $b_1=b_2$ and this concludes the proof.

Proof. The implication $\rm (i) \Rightarrow (ii)$ is given by Theorem 3.4 applied with $\operatorname {GL}_n(E)$ and $H=\operatorname {GL}_n(F)$. The other direction $\rm (ii) \Rightarrow (i)$ follows from an argument of reduction of invariant linear forms as in [Reference Kurinczuk and MatringeKM20, Theorem 3.4].

To conclude, when $F$ has characteristic zero, if $\pi$ is distinguished, then all the $\sigma$-selfdual $\overline {\mathbf {Q}}_{\ell }$-lifts of $\pi$ are distinguished by Lemma 3.10. For $F$ of positive characteristic, this follows from [Reference SécherreSéc19, Theorem 10.11].

Proof. We prove the result by contradiction. Let us assume that $\pi$ is $\omega _{E/F}$-distinguished. Let $\chi$ be a character of $E^{\times }$ extending $\omega _{E/F}$. Let $\pi ':=\pi (\chi ^{-1} \circ \det )$. Then $\pi '$ is a supercuspidal representation and is distinguished since $\pi$ is $\omega _{E/F}$-distinguished. Applying Theorem 3.4 to $\pi '$ we find a lift $\tilde {\pi }$ of $\pi$ which is $\widetilde {\omega }_{E/F}$-distinguished, where $\widetilde {\omega }_{E/F}$ is the canonical $\ell$-adic lift of $\omega _{E/F}$, that is the $\overline {\mathbf {Q}}_{\ell }$-character of $F^{\times }$ of kernel $\operatorname {Nm}_{E/F}(F^{\times })$.

With Theorem 3.4 we also have a distinguished supercuspidal $\overline {\mathbf {Q}}_{\ell }$-lift of $\pi$. Thanks to Proposition 3.11, all $\sigma$-selfdual $\overline {\mathbf {Q}}_{\ell }$-lifts are distinguished by $\operatorname {GL}_{n}(F)$. In particular, $\tilde {\pi }$ is $\operatorname {GL}_{n}(F)$-distinguished, which contradicts the fact that a $\sigma$-selfdual supercuspidal $\overline {\mathbf {Q}}_{\ell }$-representation cannot be both distinguished and $\omega _{E/F}$-distinguished.

Proof. Let us denote by $\delta$ the action of $\sigma$-duality on $R$-representations, that is $V^\delta = (V^\sigma )^{\vee }$. A representation is $\sigma$-selfdual if and only if it is $\delta$ invariant.

Let $[\pi ]$ be the inertial class of $\pi$. Let $\mathcal {I}$ be the set of inertial classes of supercuspidal $\overline {\mathbf {Q}}_{\ell }$-lifts of $\pi$. This is a finite set of cardinal a power of $\ell$ (see [Reference Mínguez and SécherreMS17, Proposition 1.3]). In particular, since $\ell \neq 2$, this cardinal is odd. The action of $\delta$ induces an action on inertial classes of representations and since $\pi$ is $\delta$-invariant, $\delta$ induces an action on $\mathcal {I}$. But the cardinal of $\mathcal {I}$ is odd and $\delta$ is an involution, therefore there exists an inertial class which is $\delta$-invariant, that is there exists $\tilde {\pi }$ a supercuspidal $\overline {\mathbf {Q}}_{\ell }$-lift of $\pi$ such that $[\tilde {\pi }^\delta ]=[\tilde {\pi }]$. Let $\tilde {\chi }$ be an unramified character such that $\tilde {\pi }^\sigma =\tilde {\pi }^{\vee } \otimes \tilde {\chi }$.

Since $\tilde {\chi }$ is unramified, it is trivial on the kernel of the norm map, that is $\tilde {\chi } = \tilde {\chi }^{\sigma }$. Therefore, there exists an unramified character $\tilde {\eta }$ such that $\tilde {\chi } = \tilde {\eta } \tilde {\eta }^{\sigma } = \tilde {\eta }^2$. Since $\tilde {\pi }$ is $\ell$-integral, $\tilde {\chi }$ is also $\ell$-integral and so is $\tilde {\eta }$. Let $\tilde {\pi }':=\tilde {\pi } \otimes \tilde {\eta }^{-1}$ such that $\tilde {\pi }'$ is $\sigma$-selfdual. Let us denote by $\pi '=\pi \otimes \eta ^{-1}$ its reduction modulo $\ell$ where $\eta$ is the reduction modulo $\ell$ of $\tilde {\eta }$. Since $\tilde {\pi }'$ is $\sigma$-selfdual so is $\pi '$. The representations $\pi$ and $\pi '=\pi \otimes \eta ^{-1}$ are both $\sigma$-selfdual. Thus $\pi = \pi \otimes \eta ^{-2}$.

Let $X_u(\pi )$ be the set of unramified characters $\xi$ such that $\pi = \pi \otimes \xi$. It is contained in the cyclic group of order $n$ composed of the character $\xi$ such that $\xi ^n=1$ and is characterized by its order $f(\pi )$. Let $o(\eta )$ be the order of $\eta$. We have $o(\eta ) \mid 2f(\pi )$. Let $\tilde {\rho }$ be the canonical $\overline {\mathbf {Q}}_{\ell }$-lift of $\eta$. In particular $o(\tilde {\rho })=o(\eta ) \mid 2f(\pi )$. Let $\tilde {\pi }'':=\tilde {\pi }' \otimes \tilde {\rho }$. It is a supercuspidal $\overline {\mathbf {Q}}_{\ell }$-lift of $\pi$. To finish the proof, we are left to prove that it is $\sigma$-selfdual. The representation $\tilde {\pi }'$ being $\sigma$-selfdual, we just need to prove that $\tilde {\pi }'' = \tilde {\pi }'' \otimes \tilde {\rho }^2$, that is that $o(\tilde {\rho }) \mid 2f(\tilde {\pi }')$. The explicit formulas for $f(\tilde {\pi }')$ and $f(\pi )$ show that $f(\tilde {\pi }')=f(\pi )\ell ^k$ for some integer $k$ (see, for instance, [Reference Mínguez and SécherreMS14, Remark 3.21]). Thus $2f(\pi ) \mid 2f(\tilde {\pi }')$ and so $o(\tilde {\rho }) \mid 2f(\tilde {\pi }')$. Therefore, $\tilde {\pi }''$ is $\sigma$-selfdual.

Proof. Assume that $\pi$ is $\sigma$-selfdual. Then there exists a $\sigma$-selfdual lift $\tilde {\pi }$ by Proposition 3.13. From the dichotomy theorem in the complex case (see [Reference KableKab04] in characteristic zero and [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and StevensAKMSS21, Theorem A.2] in positive characteristic, or [Reference JoJo23]), $\tilde {\pi }$ is either distinguished or $\widetilde {\omega }_{E/F}$-distinguished. Since the reduction modulo $\ell$ preserves distinction (see [Reference Kurinczuk and MatringeKM20, Theorem 3.4]), it is also true for $\pi$. The disjunction is Proposition 3.12 and the reciprocity follows from Theorem 4.1.

Proof. A $\sigma$-selfdual supercuspidal $\overline {\mathbf {F}}_{\ell }$-representation is either distinguished or $\omega _{E/F}$-distinguished by Theorem 3.14. By Proposition 3.11 we can lift a distinguished representation to a distinguished representation. Hence, the result follows from the complex case in [Reference KableKab04] (the results of [Reference KableKab04] are still valid in positive characteristic by [Reference Anandavardhanan, Kurinczuk, Matringe, Sécherre and StevensAKMSS21, Appendix A] or [Reference JoJo23]).

Proof. We start with the same argument as in the proof of Proposition 3.13. The number of inertial classes of supercuspidal $\overline {\mathbf {Q}}_{\ell }$-lifts of $\pi$ is a power of $\ell$, hence is odd. The involution $\sigma$ permutes its elements, so there exists an inertial class $[\tilde {\pi }]$ which is $\sigma$-invariant. Let $\tilde {\chi }$ be an unramified character such that $\tilde {\pi }^\sigma =\tilde {\pi } \otimes \tilde {\chi }$.

Let us prove that $\tilde {\pi }$ is, in fact, $\sigma$-invariant. As in the proof of Proposition 3.13, let $X_u(\tilde {\pi })$ (respectively, $X_u(\pi )$) be the set of unramified $\overline {\mathbf {Q}}_{\ell }$-characters $\tilde {\xi }$ (respectively, unramified $\overline {\mathbf {F}}_{\ell }$-characters $\xi$) such that $\tilde {\pi } = \tilde {\pi } \otimes \tilde {\xi }$ (respectively, $\pi = \pi \otimes \xi$). It is characterized by its order $f(\tilde {\pi })$ (respectively, $f(\pi )$). Let $o(\tilde {\chi })$ be the order of $\tilde {\chi }$. To finish the proof, we need to prove that $o(\tilde {\chi })$ divides $f(\tilde {\pi })$. Indeed, this implies that $\tilde {\chi } \in X_u(\tilde {\pi })$ and $\tilde {\pi }^\sigma \simeq \tilde {\pi } \otimes \tilde {\chi } \simeq \tilde {\pi }$.

Let $\chi$ be the modulo $\ell$ reduction of $\tilde {\chi }$ of order $o(\chi )$. By taking the modulo $\ell$ reduction we have that $\pi \simeq \pi \otimes \chi$, thus $\chi \in X_u(\pi )$ and $o(\chi ) \mid f(\pi )$. In addition, there exists an integer $a\in \mathbb {N}$ such that $o(\tilde {\chi }) = o(\chi ) \ell ^{a}$. The explicit formulas for $f(\tilde {\pi })$ and $f(\pi )$ show that there exists an integer $b \in \mathbb {N}$ such that $f(\tilde {\pi }) = f(\pi ) \ell ^{b}$ and $f(\pi )$ is prime to $\ell$.

Since $\tilde {\chi }$ is unramified, it is trivial on the kernel of the norm map and $\tilde {\chi }=\tilde {\chi }^{\sigma }$. In particular, $\tilde {\pi }\simeq (\tilde {\pi } \otimes \tilde {\chi })^{\sigma } \simeq \tilde {\pi } \otimes \tilde {\chi }^2$ and $o(\tilde {\chi }) \mid 2f(\tilde {\pi })$. Since $\ell$ is odd, we get that $a \leq b$. At the end, $o(\chi ) \mid f(\pi )$ and $\ell ^a \mid \ell ^b$ thus $o(\tilde {\chi }) \mid f(\tilde {\pi })$ and this finishes the proof.

Proof. When $E$ has characteristic zero, since the modulo $\ell$ reduction preserves $\sigma$-invariance and the property of being distinguished for supercuspidal representations [Reference Kurinczuk and MatringeKM20, Theorem 3.4], the result follows from the complex case (see [Reference Hakim and MurnaghanHM02, § 4, Corollary] and [Reference Feigon, Lapid and OffenFLO12, Theorem 0.2]) by Theorem 3.4 and Proposition 3.16. For a field $E$ of positive characteristic this follows from [Reference ZouZou22, Theorem 1.1].

Proof. The condition $\chi \chi ^{\sigma }=\mathbf {1}$ is equivalent to $\chi _{|F^\times }$ being trivial on the norm group $\operatorname {Nm}_{E/F}(E^{\times })$. By the local class field theory the only two characters of $F^\times$ trivial on $\operatorname {Nm}_{E/F}(E^{\times })$ are $\mathbf {1}$ and $\omega _{E/F}$.

Proof. Recall that $B_2(E)$ is the standard Borel subgroup of $\operatorname {GL}_2(E)$. Note that $\operatorname {GL}_2(E)=B_2(E)\operatorname {GL}_2(F)\sqcup B_2(E)\eta \operatorname {GL}_2(F)$ with $\eta =\big (\begin{smallmatrix} 1 & \delta \\ 1 & -\delta \end{smallmatrix}\big )$ and $E=F[\delta ]$. Denote by $S(X)$ the Schwartz space of $X$ consisting of locally constant functions defined on $X$. There is a short exact sequence

\[ 0\to S(B_2(E)\eta\operatorname{GL}_2(F))\to S(\operatorname{GL}_2(E))\to S(B_2(E)\operatorname{GL}_2(F))\to0 \]

of $\operatorname {GL}_2(F)$-modules and $\pi$ can be embedded inside $S(\operatorname {GL}_2(E))$. Due to the geometric lemma, one has a short exact sequence

\[ 0\to \operatorname{c-Ind}_{E^\times}^{\operatorname{GL}_2(F)}\chi_1\chi_2^\sigma\to\pi\to \operatorname{Ind}_{B_2(F)}^{\operatorname{GL}_2(F)}\delta_{B_2(E)}^{1/2}\delta_{B_2(F)}^{-1/2}(\chi_1\boxtimes\chi_2)\to0 \]

with $\delta _{B_2(E)}^{1/2}=\delta _{B_2(F)}$. Applying the functor $\operatorname {Hom}_{\operatorname {GL}_2(F)}(-,\mathbf {1}_2)$,

\[ 0\to \operatorname{Hom}_{F^\times\times F^\times}(\chi_1\boxtimes\chi_2,\mathbf{1})\to \operatorname{Hom}_{\operatorname{GL}_2(F)}(\pi,\mathbf{1}_2)\to \operatorname{Hom}_{E^\times}(\chi_1\chi_2^\sigma,\mathbf{1})\to \operatorname{Ext}^1_{F^\times\times F^\times}(\chi_1\boxtimes\chi_2,\mathbf{1}), \]

where $\operatorname {Ext}^1_{F^\times \times F^\times }$ denotes the Ext functor in the category of the finite-dimensional representations of $F^\times \times F^\times$. Thanks to [Reference Drevon and SécherreDS23, Proposition 8.4] that $\operatorname {Ext}_{F^\times }^1(\chi,\mathbf {1})\neq 0$ if and only if $\chi$ is trivial, one has that $\operatorname {Hom}_{\operatorname {GL}_2(F)}(\pi,\mathbf {1})\neq 0$ if and only if $\chi _1\chi _2^\sigma =\mathbf {1}$ or $\chi _1|_{F^\times }=\chi _2|_{F^\times }=\mathbf {1}$. If $\chi _1\chi _2^\sigma =\mathbf {1}$ and $\chi _1|_{F^\times }=\chi _2|_{F^\times }=\mathbf {1}$, then $\chi _1=\chi _2$. Thus, $\pi$ is distinguished by $\operatorname {GL}_2(F)$ if and only if either $\chi _1\chi _2^\sigma =\mathbf {1}$ or $\chi _1|_{F^\times }=\chi _2|_{F^\times }=\mathbf {1}$ with $\chi _1\neq \chi _2$.

Now, let us assume that $\pi$ is distinguished by $\operatorname {GL}_2(F)$, that is $\chi _1\chi _2^\sigma =\mathbf {1}$ or $\chi _1|_{F^\times }= \chi _2|_{F^\times }=\mathbf {1}$.

If $\chi _1\neq \chi _2$, only one of the conditions $\chi _1\chi _2^\sigma =\mathbf {1}$ or $\chi _1|_{F^\times }=\chi _2|_{F^\times }=\mathbf {1}$ can be true. We then see from the previous exact sequence that $\operatorname {Hom}_{\operatorname {GL}_2(F)}(\pi,\mathbf {1}_2) = \operatorname {Hom}_{F^\times \times F^\times }(\chi _1\boxtimes \chi _2,\mathbf {1})$ or $\operatorname {Hom}_{\operatorname {GL}_2(F)}(\pi,\mathbf {1}_2) = \operatorname {Hom}_{E^\times }(\chi _1\chi _2^\sigma,\mathbf {1})$. Therefore,

\[ \dim\operatorname{Hom}_{\operatorname{GL}_2(F)}(\pi,\mathbf{1}_2)=1. \]

We are left to study the case $\chi _1=\chi _2$ with $\chi _1|_{F^\times }=\mathbf {1}$. Let $\chi :=\chi _1=\chi _2$. Note that if $q_E \not \equiv 1 \pmod {\ell }$ then $\pi (\chi,\chi )$ is irreducible. Thus, by Theorem 4.1, $\dim \operatorname {Hom}_{\operatorname {GL}_2(F)}(\pi (\chi,\chi ),\mathbf {1}_2)=1$. Thus, we assume that $q_E \equiv 1 \pmod {\ell }$. We have two cases.

1. (i) Suppose $q_F \equiv -1 \pmod {\ell }$. In this case $(\nu ^{-1/2})_{|F^{\times }}\neq \mathbf {1}$. The representation $\pi (\chi,\chi )$ is a direct sum of two irreducible representations: the character $(\chi \nu ^{-1/2}) \circ \det$ and a twisted Steinberg $\operatorname {St}_{\chi \nu ^{-1/2}}$ (see for instance [Reference VignérasVig89, Theorem 3]). As $\dim \operatorname {Hom}_{\operatorname {GL}_2(F)}((\chi \nu ^{-1/2}) \circ \det,\mathbf {1}_2)=0$ and

   \[ \dim\operatorname{Hom}_{\operatorname{GL}_2(F)}(\operatorname{St}_{\chi \nu^{-1/2}},\mathbf{1}_2) \leq 1 \]
   by Theorem 4.1, we get that $\dim \operatorname {Hom}_{\operatorname {GL}_2(F)}(\pi (\chi,\chi ),\mathbf {1}_2)=1$.

2. (ii) Suppose $q_F \equiv 1 \pmod {\ell }$. This time ${(\nu ^{-1/2})}_{|F^{\times }}= \mathbf {1}$. Therefore,

   \[ \dim\operatorname{Hom}_{\operatorname{GL}_2(F)}(\pi(\chi,\chi),\mathbf{1}_2)=\dim\operatorname{Hom}_{\operatorname{GL}_2(F)}(V(\mathbf{1},\mathbf{1}),\mathbf{1}_2) \]
   where $V(\mathbf {1},\mathbf {1})$ denotes the non-normalized induction.

Recall that

\[ \operatorname{GL}_2(E)=B_2(E)\operatorname{GL}_2(F)\sqcup B_2(E)\eta\operatorname{GL}_2(F) \]

where $\eta$ represents the open orbit in double coset $B_2(E)\backslash \operatorname {GL}_2(E)/\operatorname {GL}_2(F)$. There is a short exact sequence

\[ 0\to \operatorname{c-Ind}_{E^\times}^{\operatorname{GL}_2(F)}\mathbf{1}\to V(\mathbf{1},\mathbf{1})\to \operatorname{Ind}_{B_2(F)}^{\operatorname{GL}_2(F)}\delta_{B_2(E)}^{1/2}\delta_{B_2(F)}^{-1/2}\to0. \]

Let $f_1$ be a linear functional defined on a subset consisting of functions in $V(\mathbf {1},\mathbf {1})$ supported on the closed orbit $B_2(E)\operatorname {GL}_2(F)$ given by

\begin{align*} f_1(\varphi)=\int_{B_2(F)\backslash\operatorname{GL}_2(F) }\varphi(x)\,{dx} \end{align*}

for $\varphi \in V(\mathbf {1},\mathbf {1})$ supported on $B_2(E)\operatorname {GL}_2(F)$. There is a natural extension of $f_1$ to the whole space $V(\mathbf {1},\mathbf {1})$, still denoted by $f_1$, due to the embedding

\[ 0\to \operatorname{Hom}_{\operatorname{GL}_2(F)}(\operatorname{Ind}_{B_2(F)}^{\operatorname{GL}_2(F)}(\delta_{B_2(F)}^{1/2}),\mathbf{1}_2)\to \operatorname{Hom}_{\operatorname{GL}_2(F)}(V(\mathbf{1},\mathbf{1}),\mathbf{1}_2). \]

Then $f_1$ gives a $\operatorname {GL}_2(F)$-invariant linear functional on $\pi$ with support $B_2(E)\operatorname {GL}_2(F)$. Note that there is a $\operatorname {GL}_2(E)$-invariant distribution on $P^1(E)=B_2(E)\backslash \operatorname {GL}_2(E)$ if and only if $\delta _{B(E)}=\mathbf {1}$, i.e. $q_E \equiv 1 \pmod {\ell }$ (see [Reference BourbakiBou04, Chapter VII.6, Theorem 3]). Denote by $d\mu$ the $\operatorname {GL}_2(E)$-invariant measure on $P^1(E)$ and so it is $\operatorname {GL}_2(F)$-invariant. Then the restriction of $d\mu$ on $P^1(F)=B_2(F)\backslash \operatorname {GL}_2(F)$ is zero since $P^1(F)$ has measure zero with respect to $d\mu$. Therefore $d\mu$ and $f_1$ generate two different $\operatorname {GL}_2(F)$-invariant linear functionals on $V(\mathbf {1},\mathbf {1})$. Thus, $\dim \operatorname {Hom}_{\operatorname {GL}_2(F)}(V(\mathbf {1},\mathbf {1}),\mathbf {1}_2)=2$ (the dimension is bounded by 2 by the exact sequence at the beginning of the proof).

Proof. Let $M$ be an extension of $\mathbf {1}_2$ by $\chi \circ \det$ in the category of the smooth representations of $\operatorname {GL}_2(F)$ with trivial central character. The representation $M$ has the form

\[ g\mapsto\begin{pmatrix} \chi(\det(g)) & \alpha(g) \\0 & 1 \end{pmatrix} \]

for $g\in \operatorname {GL}_2(F)$. If $g_1,g_2 \in \operatorname {GL}_2(F)$, then $\alpha (g_1g_2)=\alpha (g_1)+\chi (\det (g_1))\alpha (g_2)$. Since $M$ has trivial central character, $\alpha (z)=0$ for every element $z$ in the center of $\operatorname {GL}_2(F)$. It is obvious that $\alpha (g)=0$ for all $g$ in $\operatorname {SL}_2(F)$ since $\operatorname {SL}_2(F)$ is a perfect group. Let $g \in \operatorname {GL}_2(F)$ and $t = \det (g) \in F^\times$. By writing $g=g_1g_2$ with $g_1=\big (\begin{smallmatrix} t & 0 \\0 & 1 \end{smallmatrix}\big )$ and $g_2=g_1^{-1}g \in \operatorname {SL}_2(F)$ we see that $\alpha (g)=\alpha (g_1)$. Let $\bar {\alpha }:F^\times \to \overline {\mathbf {F}}_{\ell }$ be defined by $\bar {\alpha }(t)=\alpha \big (\big (\begin{smallmatrix} t & 0 \\0 & 1 \end{smallmatrix}\big )\big )$. Then for all $g \in \operatorname {GL}_2(F)$, $\alpha (g)=\bar {\alpha }(\det (g))$. Moreover, $\bar {\alpha }$ is a cocycle satisfying $\bar {\alpha }(t_1t_2)=\bar {\alpha }(t_1)+\chi (t_1)\bar {\alpha }(t_2)$, for $t_1,t_2 \in F^\times$.

Let $t_1,t_2 \in F^\times$. The equality $\bar {\alpha }(t_1t_2)=\bar {\alpha }(t_2t_1)$ gives us that $\bar {\alpha }(t_1)+\chi (t_1)\bar {\alpha }(t_2)=\bar {\alpha }(t_2)+\chi (t_2)\bar {\alpha }(t_1)$. If $\chi \neq \mathbf {1}$, then there exists $t_1$ such that $\chi (t_1) \neq 1$. Therefore, for all $t_2 \in F^\times$, we have $\bar {\alpha }(t_2)=c(\chi (t_2)-1)$ where $c=\bar {\alpha }(t_1)/(\chi (t_1)-1)$ is a constant. Thus, for all $g \in \operatorname {GL}_2(F)$, $\alpha (g)=c\cdot (\chi (\det (g))-1)$ is a coboundary and the extension $M$ splits.