I found it quite hard to find a complete reference for the (adelic) reduction theory of algebraic groups used in automorphic representations. I want to show that for $G=GL(n)$ over a number field $F$, a standard Siegel set $\mathfrak{S}$ has finite volume modulo the action of the center $Z(\mathbb{A})$.
To be precise, the setting is as follows: $P$ is the standard minimal parabolic of upper triangular matrices in $GL(n)$, $N$ its unipotent radical, and $T$ the diagonal subgroup (a standard maximal torus as well as the Levi of $P$). Denote by $\alpha_i$ the simple root $\alpha_i(t)=t_{i,i}/t_{i+1,i+1}$ for $t=diag\{t_1,...,t_n\}\in T$. In the adelic setting, a Siegel set of $G$ is defined as follows.
First we let $K_\mathbb{A}$ be the standard maximal compact subgroup of $G_\mathbb{A}$. Choose a compact set $C\in N(\mathbb{A})$, and a positive real number $a$. Define a Siegel set to be: $$\begin{aligned} \mathfrak{S} &=\mathfrak{S}_{a, C}^{P} \\ &=\left\{g=n t k: n \in C, t \in T_{\mathbb{A}}, k \in K_{\mathbb{A}},\left|\alpha_{i}(t)\right| \geq a \text { for } 1 \leq i<n\right\}. \end{aligned}$$
O.K. Then one can prove that there exists a Siegel set (large enough), such that $G_F \mathfrak{S}=G_\mathbb{A}$ (the proof is quite nontrivial as I know, so I just accept it). Next, one can use the finiteness of the volume $Z(\mathbb{A})\backslash \frak{S}$ in $Z(\mathbb{A})\backslash G_\mathbb{A}$ to show the finiteness of the volume of the adelic automorphic quotient $Z(\mathbb{A})G_F\backslash G_\mathbb{A}$.
I'm stuck here. I have no idea how to work with this finiteness statement. Maybe to be more concrete, I have no idea how to play with the invariant measures on these adelic spaces. I want to ask for some help or hint related to this computation, or any reference would be very helpful. Thanks a lot in advance!