Let $f:X_1\to X_2$ be a closed submanifold. Let $\rho:G_1\to G_2$ be a closed Lie subgroup. Let $G_1$ acts on $X_1$ and $G_2$ on $X_2$ and suppose $f$ is $\rho$-equivariant. I would like to get a morphism $\overline{f}:X_1/G_1\to X_2/G_2$ and conditions for $\overline{f}$ to be also a closed embedding. Do you know a reference where I can find this situation treated, in the category of analytic manifolds(or algebraic varieties)?
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