In developing a simple trace formula for $G$=GL$_2$ over a number field $F$ one encounters the following identity of local integrals (for example, in Gelbart-Jacquet, 1979):
$$\int_{Z_v N_v\backslash G_v}f_v(g^{-1}\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}g)=\lim_{a\to1}|1-a^{-1}|\int_{M_v\backslash G_v}f_v(g^{-1}\begin{pmatrix}a & 0\\ 0 & 1\end{pmatrix}g).$$
The above should follow from Iwasawa decompostion $G=NAK$ and a change of variables for $N$,
$$\begin{pmatrix}1 & x\\ 0 & 1\end{pmatrix}\mapsto\begin{pmatrix}1 & (1-a^{-1})x\\ 0 & 1\end{pmatrix}$$
or something of the sort. This would scale the Haar measure on $N$, which is taken as the measure on $F_v$ by $|1-a^{-1}|.$ (Note the variable $a$ and matrix $a$ are different.) Working backwards, I find:
\begin{align}
&k^{-1}a^{-1}\begin{pmatrix}1 & -x\\ 0 & 1\end{pmatrix}\begin{pmatrix}a & 0\\ 0 & 1\end{pmatrix}\begin{pmatrix}1 & x\\ 0 & 1\end{pmatrix}ak\\
=&\ k^{-1}a^{-1}\begin{pmatrix}a & 0\\ 0 & 1\end{pmatrix}\begin{pmatrix}1 & (1-a^{-1})x\\ 0 & 1\end{pmatrix}ak
\end{align}

And then apply the change of variables, leaving
$$k^{-1}a^{-1}\begin{pmatrix}a & ax\\ 0 & 1\end{pmatrix}ak\to k^{-1}a^{-1}\begin{pmatrix}1 & x\\ 0 & 1\end{pmatrix}ak\text{ (as }a\to1).$$
From here I have not been able to complete the computation.