I am reading Arthur's notes on the trace formula, and I would like to understand why sometimes the integral appearing there diverge. The example he gives is the following: $G=GL(2)$, $P_0$ the standard parabolic of upper triangular matrices, and $\gamma$ the matrix $\pmatrix{1&1\\1&1}$. He then considers the integral $$\int_{G_\gamma(\mathbb A) \backslash P_0(\mathbb A)} \int_{P_0(\mathbb A) \backslash G(\mathbb A)} f(k^{-1}p^{-1}\gamma p k) dp dk$$ Here $dp = |a|^{-1} da db du$ is a left Haar measure if $p = \pmatrix{a & u \\ & b}$ (I also do not understand how the left Haar measure takes this form). Now he claims that this integral reduces to a constant times $$\prod_{p} (1-p^{-1})^{-1} \int_{P_0(\mathbb A) \backslash G(\mathbb A)} \int_{\mathbb A} f(k^{-1}\pmatrix{1& u \\ & 1} k) du dk$$
I do not understands these computations. I tried to write explicitly the variable in $f$, making the $u$ disappear (yet it is still in Arthur's expression). Also, I tried to cut the adelic integral in product of local integrals to see the factor $(1-p^{-1})^{-1}$ (I think it should come from $|a|^{-1}$), but I only end up with $$\int_{G_\gamma(\mathbb A) \backslash P_0(\mathbb A)} \int_{P_0(\mathbb A) \backslash G(\mathbb A)} f(k^{-1}\pmatrix{1 & a/b \\ & 1} k) |a|^{-1} da db du dk$$ I would like to understand how to obtain his statement, e.g. if there are standard tricks to compute such integrals.
