Variety of commuting matrices

Question

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?

Answer 1

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.

Answer 2

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

Answer 3

There has been some work done by Alejandro Adem and collaborators on the space of commuting tuples of elements in Lie groups, of which your $X=\mathrm{Hom}(\mathbb{Z}^2,G)$ is a special case. The paper

Adem, Alejandro; Cohen, Frederick R., Commuting elements and spaces of homomorphisms, Math. Ann. 338, No. 3, 587-626 (2007); erratum ibid. 347, No. 1, 245-248 (2010) ZBL1131.57003

contains some basic topological information about these spaces (note there is an erratum from 2010). The later paper

Adem, Alejandro; Gómez, José Manuel, On the structure of spaces of commuting elements in compact Lie groups, Björner, A. et al., Configuration spaces. Geometry, combinatorics and topology. Pisa: Edizioni della Normale (ISBN 978-88-7642-430-4/pbk; 978-88-7642-431-1/ebook). Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series 14, 1-26 (2012). ZBL1277.43011

has some information about equivariant K-theory of $\mathrm{Hom}(\mathbb{Z}^2,G)$ when $G$ is compact Lie.

Browsing through these references, I wouldn't be surprised if the equivariant cohomology of $\mathrm{Hom}(\mathbb{Z}^2,G)$ is unknown for $G=\mathrm{GL}(n,\mathbb{C})$.