Let $G$ be an affine algebraic group over $\mathbb{C}$ and $V=\textrm{Spec}(A)$ an affine $G$-variety. Assume that $G$ is reductive. My understanding is that a central object in studying the action of $G$ on $V$ is the quotient $V\to V/G$ that is defined by the inclusion of the invariant ring $A^G$ in $A$. However, in my opinion there is a drawback: The quotient $V/G$ keeps track only of the closed orbits. I would like to have an object that represents the orbit closures rather than only the closed orbits. Furthermore, it should be able to tell me when one orbit closure is contained in another.
To be more precise, I want to associate to every affine $G$-variety $V$ a scheme(?) $V_G$ (optimally in a functorial way) such that every two different orbit closures $X,Y$ in $V$ correspond to two different points $x,y$ in $V_G$. Moreover $X\subset Y$ if and only if $x\in\overline{\{y\}}$.
In return I would give up on the existence of a natural map $V\to V_G$ and on $V_G$ being of finite type over $\mathbb{C}$.
Maybe the following example illustrates what I have in mind.
If we let $G=\mathbb{C}^*$ the torus act on the affine line $V=\mathbb{A}^1$, then $V/G$ is just $\textrm{Spec}(\mathbb{C})$ - there is only one closed orbit, namely $\{0\}$. But we have two orbit closures, namely $\{0\}$ and $\mathbb{A}^1$. So, as a guess, my desired object could be the spectrum of a local integral domain $B$ of dimension one. Then $\textrm{Spec}(B)$ consists of two points: A closed point $s$ that should correspond to the closed orbit $\{0\}$ and the generic point $t$ that corresponds to the orbit closure $\mathbb{A}^1$. The fact that $s\in\overline{\{t\}}$ would correspond to the fact that $\{0\}\subset\mathbb{A}^1$.
Is there such an object? Is there such a theory?
