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$\DeclareMathOperator\GL{GL}$Can you describe the maps from $\GL(n, \mathbb{R})$ to $\GL(n, \mathbb{R})$ that are equivariant w.r.t. right multiplication by $\GL(n, \mathbb{Z})$? I'm interested even in classes of examples, not necessarily a full description.

Of course, there are maps that are equivariant to the whole $\GL(n, \mathbb{R})$. What else is there?

Thank you for your help.

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    $\begingroup$ The ones equivariant by all of GL(n,R) are the left multiplications. For what you like, writing $f(g)=s(g)g$, we see that $s(g)=s(gh)$ for all $h\in$GL(n,Z). Hence, the answer is the maps $f(g)=s(g)g$ for such maps $s$ (i.e., factoring through maps GL(n,R)/GL(n,Z)$\to$ GL(n,R). If you require the map to be continuous, this means that $s$ is continuous. $\endgroup$
    – YCor
    Commented Jun 21, 2023 at 12:25
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    $\begingroup$ What kind of description would satisfy you? If $n=1$, you are talking about all odd functions $\mathbb{R}_{\neq 0} \to \mathbb{R}_{\neq 0}$. $\endgroup$
    – David E Speyer
    Commented Jun 21, 2023 at 13:37

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