Proof In $\mathrm {char}(F)=0$ , it is a part of [Reference Gan and Ichino18, Proposition C.4]. The key ingredients of the proof therein are the Howe duality and the Kudla’s filtrartion on the normalized Jacquet module of the Weil representation. However, Howe duality does hold for $\mathrm {char}(F)\ne 2$ case and the Kudla’s computation on the Jacquet module of the Weil representation also holds for $\mathrm {char}(F) \ne 2$ (see [Reference Kudla37, Section III.8]. Except for these, other arguments in [Reference Gan and Ichino18, Theorem C.4] equally apply to $\mathrm {char}(F) =p$ case too.

Proof Write $(\omega _{\psi ,V_n,W_n})_{\mathrm {U}^{\prime }(F),(\mu _{-c'}^{\prime })^{-1}}$ for the twisted Jacquet module of $\omega _{\psi ,V_n,W_n}$ with respect to $\mathrm {U}'(F)$ and $(\mu _{-c'}^{\prime })^{-1}$ (i.e., the quotient space $\omega _{\psi ,V_n,W_n} / \mathcal {V}$ , where $\mathcal {V}$ is a subspace spanned by $\{\omega _{\psi ,V_n,W_n}(u)\cdot \phi -(\mu _{-c'}^{\prime })^{-1}(u)\cdot \phi \}_{u\in \mathrm {U}', \phi \in \omega _{\psi ,V_n,W_n}}$ .) By Theorem A.2, we have

$$ \begin{align*} &\operatorname{Hom}_{\mathrm{G}_n(F)\times \mathrm{U}'(F)}\big(\omega_{\psi,V_n,W_n}, \ \widetilde{\pi} \otimes (\mu_{-c'}^{\prime})^{-1}\big) \cong \operatorname{Hom}_{\mathrm{G}_n(F)}\big((\omega_{\psi,V_n,W_n})_{\mathrm{U}'(F),(\mu_{-c'}^{\prime})^{-1}}, \ \widetilde{\pi}\big)\\ &\qquad\cong \operatorname{Hom}_{\mathrm{G}_n(F)}\big(\operatorname{ind}_{\widetilde{\mathrm{U}}(F)}^{\mathrm{G}_n(F)} (\mu_{c'}^{+}), \ \widetilde{\pi} \big) \cong \operatorname{Hom}_{\widetilde{\mathrm{U}}(F)}\big(\mu_{c'}^{+}, \ (\widetilde{\pi}^{\vee}\mid_{\widetilde{\mathrm{U}}(F)})^{\vee} \big)\\ &\qquad\cong \operatorname{Hom}_{\widetilde{\mathrm{U}}(F)}\big(\widetilde{\pi}^{\vee}, \ (\mu_{c'}^{+})^{\vee} \big)\cong \operatorname{Hom}_{\widetilde{\mathrm{U}}(F)}\big(\widetilde{\pi}, \ \mu_{c'}^{+} \big), \end{align*} $$

where the last equality follows from the facts that $\widetilde {\pi }$ and $\mu _{c'}^{+}$ are unitary.

On the other hand,

$$\begin{align*}\operatorname{Hom}_{\mathrm{G}_n(F) \times \mathrm{U}'(F)}\big(\omega_{\psi,V_n,W_n}, \ \widetilde{\pi} \otimes (\mu_{c-'}^{\prime})^{-1}\big)\cong \operatorname{Hom}_{\mathrm{U}'(F)}\big(\Theta_{\psi,V_n,W_n}(\widetilde{\pi}), \ (\mu_{-c'}^{\prime})^{-1}\big). \end{align*}$$

If $\widetilde {\pi }$ is $\mu _{c'}^{+}$ -generic, then $\operatorname {Hom}_{\widetilde {\mathrm {U}}(F)}\big (\widetilde {\pi }, \ \mu _{c'}^{+} \big ) \ne 0$ , and henceforth, $\operatorname {Hom}_{\mathrm {U}'(F)}\big (\Theta _{\psi ,V_n,W_n}(\widetilde {\pi }), \ (\mu _{-c'}^{\prime })^{-1}\big ) \ne 0$ . Therefore, we have $\Theta _{\psi ,V_n,W_n}(\widetilde {\pi })\ne 0$ , and by Proposition 3.2, $\operatorname {Hom}_{\mathrm {U}'(F)}\big (\theta _{\psi ,V_n,W_n}(\widetilde {\pi }), \ (\mu _{-c'}^{\prime })^{-1}\big ) \ne 0$ . This proves (i).

If $\widetilde {\pi }$ is not $\mu _{c'}^{+}$ -generic, then $\operatorname {Hom}_{\widetilde {\mathrm {U}}(F)}\big (\widetilde {\pi }, \ \mu _{c'}^{+} \big )=0$ , and from the above argument we have

$$\begin{align*}\operatorname{Hom}_{\mathrm{U}'(F)}\big(\Theta_{\psi,V_n,W_n}(\widetilde{\pi}), (\mu_{-c'}^{\prime})^{-1}\big)=0\end{align*}$$

which proves (ii).

Proof We keep the notation in Proposition 4.1, i.e., $\widetilde {\pi }$ and $\tau $ have supercuspidal supports $(\rho _1,\ldots ,\rho _r;\widetilde {\pi _0})$ with $\widetilde {\pi _0} \in \operatorname {Irr}(\mathrm {G}_{n_0}(F))$ and $(\rho _1^{\prime },\ldots ,\rho _s^{\prime };\tau _0)$ with $\tau _0 \in \operatorname {Irr}(\mathrm {J}_{m_0}(F))$ , respectively. For convenience in notation, set $\chi =\chi _{V_n}$ . Note that $\chi =\chi ^{-1}$ . Since $V_{n_0}$ (resp. $W_{m_0}$ ) is the anisotropic kernel of $V_n$ (resp. $W_m$ ), $n_0=0 \text { or }1$ (resp. $m_0=0$ ). Furthermore, $\rho _i,\rho _j^{\prime }$ are 1-dimensional character of $\mathrm {GL}_1(F)$ and $\widetilde {\pi _0}=\mathbb {I}_{V_{n_0}}$ (resp. $\tau _0=\mathbb {I}_{W_{m_0}}$ ) the trivial representation of $\mathrm {G}_{n_0}(F)$ (resp. $\mathrm {J}_{m_0}(F)$ ).

We first prove (i). By Proposition 4.1 (ii),

$$ \begin{align*}\gamma(s,\widetilde{\pi} \times \sigma\chi,\psi) & =\gamma(s,\mathbb{I}_{V_{n_0}} \times \sigma\chi,\psi) \cdot\! \prod_{j=1}^{s} \big(\gamma(s,\rho_i^{\prime} \chi^{-1} \times \sigma\chi,\psi) \cdot \gamma(s,(\rho_i^{\prime} \chi^{-1})^{-1} \times \sigma\chi,\psi)\!\big)\\ & \quad \times \prod_{i=1}^{\frac{l-l_0}{2}} \gamma\left(s+\frac{l+1}{2}-i,\sigma\chi,\psi\right) \cdot \gamma\left(s+i-\frac{l+1}{2},\sigma\chi,\psi\right) \\ & =\frac{\gamma(s,\mathbb{I}_{V_{n_0}} \times \sigma\chi,\psi)}{\gamma(s,\mathbb{I}_{W_{m_0}}\times \sigma,\psi)} \cdot \gamma(s,\mathbb{I}_{W_{m_0}}\times \sigma ,\psi) \prod_{j=1}^{s}\gamma(s,\rho_i^{\prime} \times \sigma,\psi)\cdot \gamma(s,(\rho_i^{\prime} )^{-1} \times \sigma,\psi)\\ & \quad \times \frac{\prod_{i=1}^{l} \gamma(s+\frac{l+1}{2}-i,\sigma\chi,\psi)}{\prod_{i=1}^{l_0} \gamma(s+\frac{l_0+1}{2}-i,\sigma\chi,\psi) }\\ & = \gamma(s,\tau \times \sigma,\psi) \cdot \frac{\gamma(s,\mathbb{I}_{V_{n_0}} \times \sigma\chi,\psi)}{\gamma(s,\mathbb{I}_{W_{m_0}}\times \sigma,\psi)} \cdot \frac{\prod_{i=1}^{l} \gamma(s+\frac{l+1}{2}-i,\sigma\chi,\psi) }{\prod_{i=1}^{l_0} \gamma(s+\frac{l_0+1}{2}-i,\sigma\chi,\psi)}. \end{align*} $$

Here, $\prod _{i=1}^{l_0} \gamma (s+\frac {l_0+1}{2}-i,\sigma \chi ,\psi )$ is defined as $\gamma (s,\sigma \chi ,\psi )^{-1}$ if $l_0=-1$ .

If $n_0=0$ , then $l_0=-1$ . By Remark 2.6, we have

$$\begin{align*}\gamma(s,\widetilde{\pi} \times \sigma\chi,\psi) = \gamma(s,\tau \times \sigma,\psi) \cdot \frac{\gamma(s,\sigma\chi,\psi)}{\gamma(s,\sigma,\psi)} \cdot \prod_{i=1}^{l} \gamma\left(s+\frac{l+1}{2}-i,\sigma\chi,\psi\right). \end{align*}$$

If $n_0=1$ , then $l_0=1$ . By Property 2.5 (viii) and Remark 2.6, we have

$$ \begin{align*} \gamma(s,\widetilde{\pi} \times \sigma\chi,\psi) &= \gamma(s,\tau \times \sigma,\psi) \cdot \frac{\gamma(s,\sigma\chi,\psi)\cdot \gamma(s,\sigma,\psi)}{\gamma(s,\sigma,\psi)\cdot \gamma(s, \sigma\chi,\psi)} \cdot \prod_{i=1}^{l} \gamma\left(s+\frac{l+1}{2}-i,\sigma\chi,\psi\right) \\ & =\gamma(s,\tau \times \sigma,\psi) \cdot \prod_{i=1}^{l} \gamma\left(s+\frac{l+1}{2}-i,\sigma\chi,\psi\right). \end{align*} $$

Next, we prove (ii). By Proposition 4.1 (iii),

$$ \begin{align*} \gamma(s,\tau \times \sigma,\psi) & =\gamma(s,\mathbb{I}_{W_{m_0}} \times \sigma,\psi)\cdot \prod_{j=1}^{r} \gamma(s,\rho_j \chi \times \sigma,\psi)\cdot \gamma(s,(\rho_j \chi)^{-1} \times \sigma,\psi)\\ & \quad \times \prod_{i=1}^{\frac{l_0-l}{2}} \gamma\left(s-\frac{(l-1)}{2}-i,\sigma\chi,\psi\right) \cdot \gamma\left(s+i+\frac{l-1}{2},\sigma\chi^{-1},\psi\right)\\& =\frac{\gamma(s,\mathbb{I}_{W_{m_0}} \times \sigma,\psi)}{\gamma(s,\mathbb{I}_{V_{n_0}}\times \sigma\chi ,\psi)} \cdot \gamma(s,\mathbb{I}_{V_{n_0}}\times \sigma \chi,\psi) \cdot \\&\quad\prod_{j=1}^{r} \gamma(s,\rho_j \times \sigma\chi ,\psi)\cdot \gamma(s,(\rho_j )^{-1} \times \sigma\chi ,\psi)\\& \quad \times \frac{ \prod_{i=1}^{-l} \gamma(s-\frac{(l-1)}{2}-i,\sigma\chi,\psi)}{\prod_{i=1}^{-l_0} \gamma(s-\frac{(l_0-1)}{2}-i,\sigma\chi,\psi)}\\& =\gamma(s,\widetilde{\pi} \times \sigma\chi,\psi) \cdot \frac{\gamma(s,\mathbb{I}_{W_{m_0}} \times \sigma,\psi)}{\gamma(s,\mathbb{I}_{V_{n_0}}\times \sigma\chi,\psi)} \cdot \frac{\prod_{i=1}^{-l} \gamma(s-\frac{(l-1)}{2}-i,\sigma\chi,\psi) }{\prod_{i=1}^{-l_0} \gamma(s-\frac{(l_0-1)}{2}-i,\sigma\chi,\psi)}. \end{align*} $$

Here, $\prod _{i=1}^{-l_0} \gamma (s-\frac {(l_0-1)}{2}-i,\sigma \chi ,\psi )$ is defined as $\gamma (s,\sigma \chi ,\psi )^{-1}$ if $l_0=1$ .

If $n_0=0$ , then $l_0=-1$ . By Remark 2.6,

$$\begin{align*}\gamma(s,\tau \times \sigma,\psi)=\gamma(s,\widetilde{\pi} \times \sigma\chi,\psi)\cdot\frac{\gamma(s,\sigma,\psi)}{\gamma(s, \sigma\chi,\psi)} \cdot \prod_{i=1}^{-l} \gamma\left(s-\frac{(l-1)}{2}-i,\sigma\chi,\psi\right).\end{align*}$$

If $n_0=1$ , then $l_0=1$ . By Property 2.5 (viii) and Remark 2.6,

$$ \begin{align*} \gamma(s,\tau \times \sigma,\psi) &=\gamma(s,\widetilde{\pi} \times \sigma\chi,\psi)\cdot \frac{\gamma(s,\sigma,\psi)}{\gamma(s,\sigma,\psi) \cdot \gamma(s,\sigma\chi,\psi) \cdot \gamma(s,\sigma\chi,\psi)^{-1}} \cdot\\&\quad \prod_{i=1}^{-l} \gamma\left(s-\frac{(l-1)}{2}-i,\sigma\chi,\psi\right)\\ & =\gamma(s,\widetilde{\pi} \times \sigma\chi,\psi) \cdot \prod_{i=1}^{-l} \gamma\left(s-\frac{(l-1)}{2}-i,\sigma\chi,\psi\right). \end{align*} $$

This completes the proof.

Proof

The proof follows a similar line of reasoning in [Reference Gan and Ichino18, Theorem 11.5]. Write $V=V_n$ .

By the multiplicative property of $\gamma $ -factors, it is enough to consider the case when $\widetilde {\pi }$ and $\sigma $ are supercuspidal. This can be proved using a global-to-local argument. Let $\pi $ be an irreducible supercuspidal constituent of $\widetilde {\pi }\vert _{\mathrm {SO}(V)}$ . (See [Reference Ban and Jantzen6, Section 2] for supecuspidality of $\pi $ .) Suppose we have the following data:

• • L, a totally imaginary number field (or a global function field) with $\mathbb {A}$ its adele ring such that $L_{v_0}=F$ for some finite place $v_0$ of L;

• • $\Psi $ , a nontrivial additive character of $L \backslash \mathbb {A}$ such that $\Psi _{v_0}=\psi $ ;

• • $\mathfrak {U}$ , a generic automorphic character of $U(L) \backslash U(\mathbb {A})$ associated with $\Psi $ such that $\mathfrak {U}_{v_0}=\mu _{c'}$ ;

• • $\mathbb {\mathbb {W}}$ , a symplectic space over L of dimension $2m$ , with associated isometry group $\mathrm {Sp}(\mathbb {W})$ ;

• • $\mathbb {V}$ , a symmetric space over L of dimension $2n$ such that $\mathbb {V}_{v_0}=V$ , with associated isometry group $\mathrm {O}(\mathbb {V})$ and Hecke character $\chi $ of $\mathbb {A}^{\times }$ such that $\chi _{v_0}=\chi _{V}$ ;

• • $\{\mathbb {W}_k\}$ , the tower of symplectic spaces over L containing $\mathbb {W}$ ;

• • S, a finite set consisting all archimedean places of L and $v_0$ (if L is a global function field, $S=\{v_0\}$ );

• • $\Pi $ , a globally $\mathfrak {U}$ -generic cuspidal representation of $\mathrm {SO}(\mathbb {V})(\mathbb {A})$ such that $\Pi _{v_0}=\pi $ and for all places $v \notin S$ of L, $\Pi _v$ is unramified;

• • $\Sigma $ , a globally generic cuspidal representation of $\mathrm {GL}_r(\mathbb {A})$ such that $\Sigma _{v_0}=\sigma $ and for all places $v \notin S$ of L, $\Sigma _{v}$ is unramified;

• • $k_0$ , the first index of the global theta lift tower $\{\Theta _{\mathbb {V},\mathbb {W}_k}(\Pi )\}$ with respect to $\Psi $ and $\chi $ such that $\Theta _{\mathbb {V},\mathbb {W}_{k_0}}(\Pi )$ is nonzero;

• • $\Xi $ , an irreducible constituent of $\Theta _{\mathbb {V},\mathbb {W}_{k_0}}(\Pi ).$

The existence of such L follows from a similar argument in [Reference Morimoto41, Lemma 5.2]. We can always construct such $\Pi $ and $\Sigma $ . (For the $\mathrm {char}(F)=0$ case, see [Reference Shahidi44, Proposition 5.1] and for the $\mathrm {char}(F)=p$ case, see [Reference Gan and Lomelí19, Theorem 1.1].) Furthermore, Theorem B.1 tells that such $k_0$ exists and $\Xi $ is cuspidal. By Proposition 2.4, there is an automorphic generic character $\widetilde {\mathfrak {U}}$ of $\widetilde {\mathrm {U}}(\mathbb {A})$ such that $\widetilde {\mathfrak {U}}\vert _{\mathrm {U}(\mathbb {A})}=\mathfrak {U}$ and a globally $\widetilde {\mathfrak {U}}$ -generic cuspidal representation $\widetilde {\Pi }$ of $\mathrm {O}(\mathbb {V})(\mathbb {A})$ such that $\operatorname {Res}(\widetilde {\Pi })=\Pi $ . By Proposition 2.1, we have $\widetilde {\Pi }_{v_0}=\widetilde {\pi } \text { or } \widetilde {\pi }\otimes \det $ .

Write $l_0=2n-2k_0-1$ . For simplicity, we assume $l_0\ge 1$ . (The case $l_0 \le -1$ can be proved similarly.) For unramified places $v \notin S$ , an unramified representation $\Sigma _{v}$ is considered the unique unramified quotient of a principal series $\operatorname {Ind}_{\mathbb {B}(L_v)}^{\mathrm {GL}_r(L_v)} (\chi _1\otimes \cdots \otimes \chi _r)$ , where $\chi _i$ are unramified character of $F^{\times }$ and $\mathbb {B}$ is a Borel subgroup of $\mathrm {GL}_r$ . Therefore, multiplicativity property of $\gamma $ -factors together with Lemma 4.2 imply the following:

(4.2) $$ \begin{align} \gamma(s,\Xi_{v} \times \Sigma_{v},\Psi_v)=\gamma(\Sigma_v,\mathbb{V}_{v})\cdot\gamma(s,\widetilde{\Pi}_v \times \Sigma_{v} \chi_{v},\Psi_v)\cdot \prod_{i=1}^{l_0}\gamma(s+\frac{l_0+1}{2}-i,\Sigma_{v} \chi_{v},\Psi_v)^{-1}. \end{align} $$

Note that for all places $v \notin S$ , $\Xi _v, \Pi _v$ and $\Sigma _v$ are unramified. Therefore, Property 2.5 (iv) of $\gamma $ -factors implies that by taking products of (4.2) over all places $v \notin S$ we have

$$\begin{align*}\frac{L^S(1-s,\Xi^{\vee} \times \Sigma^{\vee})}{L^S(s,\Xi \times \Sigma)}&=\gamma^S(\Sigma_v,\mathbb{V}_{v}) \cdot \frac{L^S(1-s,\widetilde{\Pi}^{\vee} \times \Sigma^{\vee} \chi^{\vee})}{L^S(s,\widetilde{\Pi} \times \Sigma \chi)}\\&\quad \times \prod_{i=1}^{l_0} \frac{L^S(1-s-\frac{l_0+1}{2}+i,\Sigma^{\vee} \chi^{-1})^{-1}}{L^S(s+\frac{l_0+1}{2}-i,\Sigma \chi)^{-1}}.\end{align*}$$

For any archimedean place v of L, $L_v = \mathbb {C}$ . The theta correspondence for complex groups is well-understood and can be described in terms of the local Langlands correspondence (see [Reference Adams and Barbasch1]). Applying the archimedean property of $\gamma $ -factors, we have the identity (4.2) at the archimedean places v of L as well.

By the global functional equation of $\gamma $ -factors,

$$\begin{align*}\frac{L^S(1-s,\Xi^{\vee} \times \Sigma^{\vee})}{L^S(s,\Xi \times \Sigma)} \times \prod_{v \in S}\gamma(s,\Xi_{v} \times \Sigma_{v},\Psi_{v}) =1 \end{align*}$$

$$\begin{align*}=\gamma^S(\Sigma_v,\mathbb{V}_{v}) \cdot \frac{L^S(1-s,\widetilde{\Pi}^{\vee} \times \Sigma^{\vee} \chi^{\vee})}{L^S(s,\widetilde{\Pi} \times \Sigma \chi)} \times \prod_{i=1}^{l_0} \frac{L^S(1-s-\frac{l_0+1}{2}+i,\Sigma^{\vee} \chi^{-1})^{-1}}{L^S(s+\frac{l_0+1}{2}-i,\Sigma \chi)^{-1}} \end{align*}$$

$$\begin{align*}\times \prod_{v \in S} \left( \gamma(\Sigma_v,\mathbb{V}_{v}) \cdot \gamma(s,\widetilde{\Pi}_{v}\times \Sigma_{v} \chi_{v},\Psi_{v}) \times \prod_{i=1}^{l_0}\gamma(s+\frac{l_0+1}{2}-i,\Sigma_{v} \chi_{v},\Psi_{v})^{-1} \right) .\end{align*}$$

By canceling the identities at places outside $v_0$ , we have

$$\begin{align*}\gamma(s,\Xi_{v_0} \times \Sigma_{v_0},\Psi_{v_0}) &=\gamma(\Sigma_{v_0},\mathbb{V}_{v_0}) \cdot \gamma(s,\widetilde{\Pi}_{v_0}\times \Sigma_{v_0} \chi_{v_0},\Psi_{v_0})\cdot\\ &\quad\prod_{i=1}^{l_0}\gamma\left(s+\frac{l_0+1}{2}-i,\Sigma_{v_0} \chi_{v_0},\Psi_{v_0}\right)^{-1}. \end{align*}$$

Since $\widetilde {\Pi }_{v_0}=\widetilde {\pi } \text { or } \widetilde {\pi } \otimes \det $ , $\Sigma _{v_0}=\sigma $ , $\mathbb {V}_{v_0}=V$ and $\Xi _{v_0}=\theta _{\psi ,\mathbb {V}_{v_0},(\mathbb {W}_{k})_{v_0}}(\widetilde {\pi })$ , (2.7) implies the desired identity for this specific theta lift $\Xi _{v_0}$ . Moreover, Lemma 4.5 establishes the desired identity for any arbitrary non-zero theta lift.

Proof Note that $\tau $ is nonzero and tempered by Propositions 3.2 and 3.3. Then this is a special case of Theorem 4.3 when $n=m$ , $l=-1$ , and $\sigma $ is a supercuspidal representation.

Proof Let $W_n$ be a $2n$ -dimesional symplectic space. Put

$$\begin{align*}(\widetilde{\pi_1}',\widetilde{\pi_2}')=\begin{cases} (\widetilde{\pi_1},\widetilde{\pi_2}) \quad \quad \quad \quad \quad \quad \ \text{if } \widetilde{\pi_1},\widetilde{\pi_2} \text{ are } \mu_{c'}^{+}\text{-generic }\\ (\widetilde{\pi_1}\otimes \det,\widetilde{\pi_2}\otimes \det) \quad \text{if } \widetilde{\pi_1},\widetilde{\pi_2} \text{ are } \mu_{c'}^{-}\text{-generic but not } \mu_{c'}^{+}\text{-generic}.\end{cases} \end{align*}$$

Then by Proposition 3.3, $\Theta _{\psi ,V_n,W_n}(\widetilde {\pi _1}')$ and $\Theta _{\psi ,V_n,W_n}(\widetilde {\pi _2}')$ are nonzero and $(\mu _{-c'}^{\prime })^{-1}$ -generic. Put $\tau _1:=\theta _{\psi ,V_n,W_n}(\widetilde {\pi _1}')$ and $\tau _2:=\theta _{\psi ,V_n,W_n}(\widetilde {\pi _2}')$ . Since $\widetilde {\pi _1}$ and $\widetilde {\pi _2}$ have the same central characters, $\tau _1,\tau _2$ also have the same central characters by Proposition A.1. Furthermore, by Corollary 4.6,

$$ \begin{align*} \gamma(s,\tau_1 \times \rho,\psi)=\gamma(s,\widetilde{\pi_1}' \times \rho \chi_{V_n},\psi)\cdot\gamma(s,\rho \chi_{V_n},\psi), \\ \gamma(s,\tau_2 \times \rho,\psi)=\gamma(s,\widetilde{\pi_2}' \times \rho \chi_{V_n},\psi)\cdot \gamma(s,\rho \chi_{V_n},\psi) \end{align*} $$

for any irreducible supercuspidal representation $\rho $ of $\mathrm {GL}_i(F)$ where $1 \le i \le n$ .

Therefore, we obtain

$$\begin{align*}\gamma(s,\tau_1 \times \rho,\psi)=\gamma(s,\tau_2 \times \rho,\psi)\end{align*}$$

and due to property of the dependence on $\psi $ of $\gamma $ -factors, i.e., Property 2.5(iii), we have

$$\begin{align*}\gamma(s,\tau_1 \times \rho,\psi^{-1})=\gamma(s,\tau_2 \times \rho,\psi^{-1}).\end{align*}$$

Then by Theorem 2.12, we have $\tau _1 \simeq \tau _2$ and by Howe duality, $\widetilde {\pi _1}'\simeq \widetilde {\pi _2}'$ . Henceforth, we have $\widetilde {\pi _1} \simeq \widetilde {\pi _2}$ .

Proof $(\mathrm i)$ and $(\mathrm {ii})$ follows a similar line reasoning in [Reference Jiang and Soudry33, Section 3.2]. Note that both $\tau $ and $\rho $ are tempered and generic. From the definition of local L-function (2.4), the zero of $\gamma (s,\tau \times \rho ,\psi )$ comes from the poles of $L(s, \tau \times \rho )$ . However, Property 2.5 (vii) implies that $L(s,\tau \times \rho )$ has no pole for $\operatorname {Re}(s)>0$ . Therefore $(\mathrm {iii})$ of the theorem follows.

When $\mathrm {G}=\mathrm {J}_n$ , (iv) follows from [Reference Jo34, Corollary 5.9] and in the proof of [Reference Jo34, Corollary 5.9], the restriction that $\tau $ is supercuspidal is given to apply “Casselman–Shahidi Lemma” ([Reference Jo34, Lemma 5.8]). However, “Casselman–Shahidi Lemma” also holds for tempered representations by [Reference Kim36, Proposition 12.3]. Except that, other arguments of [Reference Jo34, Corollary 5.9] equally work for tempered representations. This completes the proof of $(\mathrm {iv})$ when $\mathrm {G}=\mathrm {J}_n$ .