# 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 $$#S_P$$.