Recently, I read a notes about Sakellaridis and Venkatesh conjecture. It mentions a technique called "unfolding" and gives an example:

Let X=A\G, X'=N\G, where G=PGL(2), A={ $\left[\begin{array}{cc} * & 0 \\ 0 & 1 \end{array}\right]$ }, N={ $\left[\begin{array}{cc} 1 & * \\ 0 & 1 \end{array}\right]$ }. Assume $\xi$ is a character of $N$, and $\xi(\left[\begin{array}{cc} 1 & x \\ 0 & 1 \end{array}\right])= \psi(x)$, where $\psi$ is a nontrivial character of a p-adic local field. Then we will have $L^2(X) \cong L^2(X',\xi)$ given by $\phi \mapsto \int_N\phi(ux')\xi(u)^{-1} $ and $\phi' \mapsto \int_A\phi'(ax)$.

It seems a direct computation, but I failed doing so. Can anyone give me a detailed calculation or tell me why the name is "unfolding"? Thank you all the time.