If I have a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with a Lie group G as a symmetry,
$f(Ax)=f(x),\quad A\in G$
how might I go about obtaining a reduced function $\tilde{f}$ on $\mathbb{R}^n/\sim_G$, the original domain modulo the equivalence classes of G? Is the new domain also a lie group? How do I construct a projection/coordinate system from $x\rightarrow [x]_G$?
Thanks