Let $G=PGL_n(\mathbb{R})$, $K=PO_n(\mathbb{R})$ and $X=G/K$. Also suppose $\Gamma=SL_n(\mathbb{Z})$ acts on the left of $X$. We define a typical Hecke operator on $L^2(\Gamma\backslash X)$ by the double coset $T_a(p)=\Gamma g_a\Gamma$ for $a=(a_1,a_2,\dots,a_n)\in \mathbb{N}_0^n$ and $g_a=diag(p^{a_1},p^{a_2},\dots,p^{a_n})$ i.e.$$(T_a(p)f)(x)=\sum_{g\in\Gamma\backslash\Gamma g_a\Gamma}f(gx);$$with $T_a(p)=T_{\sigma(a)}(p)$ for all $\sigma\in S_n$ and $T_a(p)=Id$ if $a_1=a_2=\dots=a_n$. We also define a classical Hecke operator by $$(T(d)f)(x)=\sum_{g\in \Gamma\backslash\Gamma_d}f(gx),$$ where $\Gamma_d=\{\gamma\in M_n(\mathbb{Z}):\det(\gamma)=d\}.$ And relation between these two type of Hecke operators can be given by elementary divisor thoerem (or invariant factor theorem);

$$T(p^m)=\sum_{\substack{\sum_ia_i=m\\a_1\ge a_2\ge\dots\ge a_n}}T_a(p).$$

$\textbf{Question:}$

Let $B_r(x)$ be a ball of radius $r$ around $x$ in $X$ ($X$ is a Riemannian symmetric space). Let $a^0=(a_1^0,a_2^0,\dots,a_n^0)$ be a partition of $m$ in non-increasing order. Is it true that, $$\#\{\gamma\in \Gamma g_{a^0}\Gamma:\gamma x\in B_r(x)\} \sim\frac{\#\{\gamma\in\Gamma_{p^m}:\gamma x\in B_r(x)\}}{P(m)},$$ where $P(m)$ is number of partition of $m$ in non-increasing order?

Thanks for any help and reference.