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 conjugation: I'd like to find out what is known in the literature for the $G$-equivariant cohomology of $X,Y$ (an the mixed Hodge structure on it).Moreover, is the cohomology of their GIT quotients $X//G$, $Y//G$ known too? Is there a relation between them?



Variety of commuting matrices
There's been a good deal of work since the papers Mark Grant cited.


The rational cohomology of $\mathrm{Hom}(\mathbb{Z}^n,K)//K$ for $K$ a compact connected Lie group was computed by Stafa (<https://arxiv.org/abs/1705.01443>). It's a theorem of Florentino and Lawton that if $G$ is a linearly reductive Lie group with maximal compact subgroup $K$, then $\mathrm{Hom}(\mathbb{Z}^n,G)//G$ deformation retracts to $\mathrm{Hom}(\mathbb{Z}^n,K)/K$, so for the general linear group, we can switch to working with the unitary groups instead. Stafa gives a general formula for the Poincare series, in terms of the order of the Weyl group and its action on the (dual of the) Lie algbra of a maximal torus. The formula reduces to $((1+t)^{2n}+(1-t^2)^n)/2$ for $G = U(n)$ (or $GL\_n (\mathbb{C})$). Florentino and Silva (<https://arxiv.org/abs/1711.07909>) computed algebro-geometric refinements of these Poincare series, and their work recovers Stafa's formula.


There's a similar story for the ordinary rational cohomology of $\mathrm{Hom}(\mathbb{Z}^n,G)$, discussed in a paper I wrote with Stafa, <https://arxiv.org/abs/1704.05793>. Since then D. Kishimoto and M. Takeda have made a good deal of progress, including information about the ring structure and torsion (also they gave a much shorter derivation of the Poincare series).


Regarding equivariant cohomology, Baird has some work in the compact case; see Section 4 of his paper <https://arxiv.org/abs/math/0610761>. Note that the inclusion of $\mathrm{Hom}(\mathbb{Z}^n,K)$ into $\mathrm{Hom}(\mathbb{Z}^n,G)$ is $K$-equivariant and a homotopy equivalence by a result of Pettet and Souto (Geom. and Topol. 17, 2013), and the inclusion of $K$ into $G$ is a homotopy equivalence, so $H\_K^\* (\mathrm{Hom}(\mathbb{Z}^n,K) \cong H\_K^\* (\mathrm{Hom}(\mathbb{Z}^n,G)) \cong H\_G^\* \mathrm{Hom}(\mathbb{Z}^n,G)$. It would be quite interesting to know more about the equivariant cohomology.


