$\phi_2$ are cuspidal.

A bilinear form $\langle \; , \; \rangle$ on $V$ is defined by $$\langle (x_1,x_2) , (y_1,y_2) \rangle:=\Tr(x_1 y_1)+\Tr(x_2  y_2),$$ and the dual space of $V$ is identified with $V$ by $\langle \; , \; \rangle$. Let $\rho^\vee$ denote the contragredient representation of $\rho$ on $V$ with respect to $\langle \;\; , \;\; \rangle$, that is, $\langle x\cdot \rho(g) , y\cdot \rho^\vee(g) \rangle=\langle x , y \rangle$. Then, one has $Z_\rho=\mathrm{Ker}\rho^\vee$ and $$(y_1,y_2)\cdot \rho^\vee(g_1,g_2,g_3) =(g_2^{-1} y_1 g_1,g_2^{-1}y_2g_1)\, ^t{}\!g_3^{-1}.$$ For $s\in\C$ and $\Phi\in\cS(V(\A))$, we define the global zeta functions $Z(\Phi,\phi_1,\phi_2,s)$ and $Z^\vee(\Phi,\phi_1,\phi_2,s)$ as $$Z(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A)} |\omega(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho(h))\, \d h,$$ $$Z^\vee(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A)} |\omega^\vee(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho^\vee(h))\, \d h,$$ where $h=Z_\rho(g_1,g_2,g_3)\in H(\A)$, $\omega^\vee:=\omega^{-1}$ and $\d h$ is a Haar measure on $H(\A)$.

\[lem:abs\] There exists a sufficiently large constant $T>0$ such that $Z(\Phi,\phi_1,\phi_2,s)$ and $Z^\vee(\Phi,\phi_1,\phi_2,s)$ are absolutely and uniformly convergent on any compact set in $\{ s\in \C \mid \Re(s)>T \}$. Hence, they are holomorphic on the domain $\{ s\in \C \mid \Re(s)>T \}$.

See [@Saito2].

We set $$Z_+(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A) , |\omega(h)|\geq 1} |\omega(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho(h))\, \d h ,$$ $$Z_+^\vee(\Phi,\phi_1,\phi_2,s):=\int_{H(F)\bsl H(\A),|\omega^\vee(h)|\geq 1} |\omega^\vee(h)|^s \phi_1(g_1)\, \phi_2(g_2)\sum_{x\in V^0(F)} \Phi(x\cdot \rho^\vee(h))\, \d h .$$ 