Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal compact subgroup of $G_{\infty}$. We fix an embedding $G\hookrightarrow GL_N$ for some $N$. Let $\mathfrak{P}$ denote a prime of $\mathcal{O}_F$ lying over $p$. Let $G(\mathfrak{P})$ be the intersection of $G_{\infty}$ with the congruence subgroup of $GL_N(\mathcal{O}_F)$ at level $\mathfrak{P}$. Let $\Gamma$ be an arithmetic lattice of $G_{\infty}$.
Let us denote
$\Gamma (\mathfrak{P}):=\Gamma \cap G(\mathfrak{P})$.
Let us denote by $e$ and $f$ the ramification and inertia degrees of $\mathfrak{P}$ in $F$. Hence we get that $[F_{\mathfrak{P}}:\mathbb{Q}_p]=ef$ where $F_{\mathfrak{P}}$ is the completion of $F$ at $\mathfrak{P}$. We define $G_k:=G \cap (1+p^kM_{efN}(\mathbb{Z}_p))$ where $G:=\varprojlim_k\Gamma /\Gamma({\mathfrak{P}}^k)$
For $k\geqslant 0$, we define $Y_k:=\Gamma (\mathfrak{P}^{ek})\backslash G_{\infty} /K_{\infty}$.
Then let $vol(\mathfrak{P}^{ek})$ be the volume of $\Gamma(\mathfrak{P}^{ek})\backslash G_{\infty}$.
Let $\Gamma(\mathfrak{P}^{ek})\backslash G_{\infty}$ be compact.
Then I want to show that $vol(\mathfrak{P}^{ek})\sim [G:G_k]$. Here $\sim$ denotes that both of them are similar order as $k \rightarrow \infty$. It will be helpful if I can get some hints or some reference on this fact. Thank you in advance.