nt.number theory – Stabilizer in $G(mathbb{Z})$ of point in fundamental domain $G(mathbb{Z}) backslash G(mathbb{R}) / K$

Let $G$ be a semisimple group (the cases of primary interest to me are where $G$ is a special linear group or a special orthogonal group), let $K$ be a maximal compact subgroup of $G(mathbb{R})$, and let $Omega$ be a fundamental domain for the left-action of $G(mathbb{Z})$ on $G(mathbb{R})/K$.

Question: For $P in Omega$, let $S_P subset G(mathbb{Z})$ be the stabilizer of $P$. If $P$ is restricted to lie in the interior of $Omega$, is $#S_P$ a constant? If so, is it possible to compute this constant for, say, $G = operatorname{SL}_n$ or $G = operatorname{SO}(p,q)$?

What I know:

  • The answer is yes when $G = operatorname{SL}_2$. In this case, it is shown in Serre’s Cours d’arithmétique that the stabilizer of any $P$ in the interior of $Omega$ is given by ${pm 1}$.
  • The stabilizer in $G(mathbb{R})$ of $P$ is a conjugate of the subgroup $K$. Thus, $#S_P$ is the number of integral points in this conjugate subgroup, but I’m not sure how that varies with $P$.
  • In his paper Ensembles fondamenteaux pour les groupes arithmétiques, Borel constructs a finite union $U$ of Siegel sets such that the set of elements $gamma in G(mathbb{Z})$ for which $U cap gamma cdot U neq varnothing$ is finite. I guess this means that $#S_P$ is bounded independent of $P$, but I’m not sure if the argument in Borel’s paper (or in his previous paper together with Harish-Chandra entitled Arithmetic Subgroups of Algebraic Groups) allows one to effectively compute $#S_P$.