Let $G=\operatorname{GL}(n,\mathbb{C})$ and $\mathfrak{g}=\operatorname{Mat}(n,\mathbb{C})$ and let us consider the two varieties $X,Y$ defined as $$X=\{(x,y) \in G \times G \ | \ xy=yx\} $$ and $$Y=\{(x,y) \in \mathfrak{g} \times \mathfrak{g} \ | \ xy=yx\} .$$

The group $G$ acts on both of them by coniugation: I'd like to find out what is known in the literature for the $G$-equivariant cohomology of $X,Y$. Moreover, is the cohomology of their GIT quotients $X//G$, $Y//G$ known too? Is there a relation between them?