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

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?