that $H_\rho\cap P$ is the Borel subgroup $B_\rho$ of $H_\rho$. Mackey’s theory gives us an exact sequence of $H_\rho$-modules: $$\label{eq2.2}0{\rightarrow}{{\mathrm{ind}}}_{N_{2,\rho}\cdot T_0}^{H_\rho}(\chi'_s){\rightarrow}I(s,\chi){\rightarrow}\textrm{n-}{{\mathrm{Ind}}}_{B_\rho}^{H_\rho}(\chi_{s}){\rightarrow}0,$$ where ${{\mathrm{ind}}}$ means compact induction, $\textrm{n-}{{\mathrm{Ind}}}$ means non-normalized induction, $\chi'_s$ is the character on $N_{2,\rho}\cdot T_0$ defined by $$\chi_s'(nh(a,1))=\chi_s(h(1,a))=\chi(a)|a|^{3s},\qquad (n \in N_{2, \rho}, \ a \in {{\mathrm{GL}}}_1)$$ and $\chi_s$ is viewed as character on $B_\rho$ by restriction, since $B_\rho \subset P.$ Here the embedding $${{\mathrm{ind}}}_{N_{2,\rho} \cdot T_0}^{H_\rho}(\chi'_s) {\hookrightarrow}I(s,\chi)$$ is defined as follows. For $f_s\in  {{\mathrm{ind}}}_{N_{2,\rho} \cdot T_0}^{H_\rho}(\chi'_s)$, then the corresponding $\tilde f_s\in  I(s,\chi)$ is defined by \[def of tilde f s\] $$\tilde f_s( mu w_\beta g)=\chi_s(m) f_s(g),$$ where $m\in M, u\in U^\alpha, g\in H_\rho$, and $$\tilde f_s(h)=0, \textrm{ if } h\notin Pw_\beta  H_\rho.$$ By , $\tilde f_s$ is well-defined.

The local zeta integrals
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For $\rho$ in $F^\times$ fix $\tau$ with $\tau^2=\rho,$ and let ${\widetilde}H_\rho$ be ${{\mathrm{GL}}}_3$ when $\rho$ is a square and the quasi-split unitary group ${\rm U}_{2,1}$ otherwise. Let $\pi$ be an irreducible generic representation of ${\widetilde}H_\rho(F)$ and $\chi$ be a quasi-character of $F^\times$. Let $\psi$ be a nontrivial additive character of $F$.

The maximal unipotent subgroup $U_\rho$ is given by $$\left\{\begin{pmatrix} 1&x+y\tau&\frac{(x^2-y^2\rho)}2+w\tau\\&1&-x+y\tau\\ &&1\end{pmatrix}x,y,z\in F\right\}.$$ If $\tau\in F$, this is simply a parametrization of the standard maximal 