Let $R_r(G)$ be the (connected component of the identity of the) variety of commuting $r$-tuples of elements of a complex reductive group $G$. We determine the mixed Hodge structure on the cohomology of the representation variety $R_r(G)$ and of the character variety $R_r(G)/G$, for general $r$ and $G$. We also obtain explicit formulae (both closed and recursive) for the mixed Hodge polynomials, Poincaré polynomials and Euler characteristics of these representation and character varieties. In the character variety case, this gives the counting polynomial over finite fields, and some results also apply to character varieties of nilpotent groups.

This is joint work with S. Lawton and J. Silva (arXiv:2110.07060).