to the ${{\mathrm {U}}}_{2,1}$ case in [@H12]. Let $F$ be a local field in this section. If $F$ is a non-archimedean local field, let ${{\mathfrak{o}}}$ be the ring of integers of $F$ and let $q$ be the number of the residue field of $F$. We also fix a uniformizer $\varpi$ of $F$ when $F$ is non-archimedean.

The local zeta integrals defined by Ginzburg involve the unique split exceptional group of type $G_2$. We denote this group $G_2$ and realize it as a set of $8 \times 8$ matrices as in [@H12]. Let $B=TU$ be the upper triangular Borel subgroup of $G_2$ under the realization in [@H12] with torus $T$ and maximal unipotent subgroup $U$.

The group $G_2$ has two simple roots $\alpha,\beta$, where $\alpha$ is the short root and $\beta$ is the long root. Then the set of positive roots of $G_2$ is ${\left\{{\alpha,\beta,\alpha+\beta,2\alpha+\beta,3\alpha+\beta,3\alpha+2\beta}\right\}}$. We denote the set of all roots by $\Phi$ and the set of positive roots by $\Phi^+.$ For a root $\delta$, we denote $U_\delta$ the corresponding root space of $\delta$ and ${{\bf {x}}}_\delta: {{\mathbb {G}}}_a{\rightarrow}U_\delta$ a choice of isomorphism. We let $X_\delta = d{{\bf {x}}}_\delta(1).$ (Here, $d{{\bf {x}}}_\delta$ is the differential.) We assume that $X_\alpha = 