Is there any structure on the subset of $SL_n(\mathbb{R})$ commuting with a finite group upto integral matrices?

Question

Consider the group $$G = \left\langle \left[ \begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right] \right\rangle \subseteq SL_2(\mathbb{R}).$$

I want to consider the set of $ SL_2(\mathbb{R})$ defined by $$\{ g \in SL_2(\mathbb{R}) \ | \ g^{-1} h g \in SL_2(\mathbb{Z}) \text{ for each } h \in G \} \subseteq SL_2(\mathbb{R}) .$$

The condition says that $$\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in SL_2(\mathbb{Z}) $$ $$\Rightarrow \begin{bmatrix} dc + ab & d^2 + b^2 \\ -a^{2} - c^{2} & -cd-ab \end{bmatrix} \in SL_2(\mathbb{Z}).$$ It is not clear what are all the solutions for this in $SL_2(\mathbb{R})$.

More generally, I want to consider such structures with an arbitrary finite group $G \subseteq SL_d(\mathbb{Q})$ and analogous constructions in $SL_d(\mathbb{R})$ but I don't know what structures such objects have.

Answer 1

Some number theory preliminaries:

If $u,v,w$ are integers such that $w<0<v$ and $-wv=u^2+1,$ then there exists integers $a,b,c,d$ such that $-w=a^2+c^2, v=b^2+d^2,$ $u=dc+ab$ and $1=ad-bc.$

This is true due to unique factorization in the Gaussian integers, and is, really, beyond the scope of this question.

That means we get at least one matrix $M_0=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ in $ SL_2(\mathbb Z)$ which satisfies your condition to get $U=\begin{bmatrix}u&v\\w&-u\end{bmatrix}\in\mathbb SL_2(\mathbb Z).$

Let $J=\begin{bmatrix}0&1\\-1&0\end{bmatrix}.$ So $M_0^{-1}JM_0=U.$

If $g^{-1}Jg=U=M_0^{-1}JM_{0}$ then $M_{0}g^{-1}$ commutes with $J,$ and thus must share eigenvectors with $J.$

The eigenvectors of $J$ are $\begin{bmatrix}1\\\pm i\end{bmatrix}.$ If $M_{0}g^{-1}=(g_{ij})$ this means:

$$g_{11}+g_{12}i=\lambda\\g_{21}+g_{22}i=i\lambda$$

This yields $g_{11}=\lambda_1,g_{12}=\lambda_2,$ $g_{21}=-\lambda_2,$ $g_{22}=\lambda_1.$ This $M_0g^{-1}\in SL_2$ iff $|\lambda| =1.$

So the most general solution is any matrix $M$ of the form:

$$\begin{bmatrix}\lambda_1&\lambda_2\\-\lambda_2&\lambda_1\end{bmatrix}\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ where $\lambda_1^2+\lambda_2^2=1,$ and $a,b,c,d$ integers such that $ad-bc=1.$

That is, any rotation matrix times any element of $SL_2(\mathbb Z).$

You can restrict to $\lambda_1,\lambda_2\geq 0$ and still get all such matrices.

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The key of the number theory is not to find $a,b,c,d.$ It is to ensure that any such conjugate comes from some integers. Solving for $a,b,c,d$ would be messy in general, but we don't need to solve them.