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?



Intersection theory for $G$-varieties - an action on the chow ring?
If you are interested in intersection theory of varieties with $G$-actions, then you want to study equivariant intersection theory. This theory exploits the $G$-action in a way that leads to deeper invariants than ordinary intersection theory. The three references I would recommend if you are first learning the subject are:


[Fulton's lectures notes on equivariant cohomology](http://www.math.washington.edu/~dandersn/eilenberg/) (compiled by Dave Anderson)


[Equivariant Cohomology and Equivariant Intersection Theory](http://arxiv.org/abs/math/9802063) by Michel Brion


[Equivariant Chow Groups for Torus Actions](http://www-fourier.ujf-grenoble.fr/PUBLIS/publications/REF_365.pdf) by Michel Brion


