TITLE: Variety of commuting matrices
QUESTION [18 upvotes]: 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?

REPLY [5 votes]: My new paper with Carlos Florentino and Jaime Silva answers this question:
Mixed Hodge structures on character varieties of nilpotent groups.

In particular, see Section 4.4.
For an implementation of the MHS in some special cases, please see the Mathematica NB here:
https://github.com/seanlawton/Mixed-Hodge-structures-on-character-varieties-of-nilpotent-groups