Let $G$ be a reductive algebraic group. Let $X$ be a $G$-variety and consider any closed subvariety $Z$ of $X$. Since any $g\in G$ acts as an automorphism, we know that $g.Z$ is again a closed subvariety of $X$. This yields an action of $G$ on the free module of cycles of $X$ which should induce an action of $G$ on the Chow ring of $X$. The invariants of this ring should be precisely the classes that correspond to linear combinations of $G$-orbits.
Has this action been studied before? Any kind of reference would be very welcome. Thanks!
Edit: It looks like my above idea is rather futile, so let me ask more broadly: Are there any techniques or results in intersection theory specifically on $G$-varieties? Could you name some references?