An important invariant of a finitely generated group \(\Gamma\) is its moduli space of Lie group-valued representations. From the case when the Lie group is \(\mathrm{GL}_n(\mathbb{C})\), these moduli spaces are called character varieties. The study of character varieties relates to the theory of bundles of various types, locally homogeneous geometry, knot theory, and mathematical physics.
Here is how to define a character variety (over \(\mathbb{C}\)). Start with a complex reductive affine algebraic group \(G\) and a finitely generated (discrete) group \(\Gamma\). The group \(G\) acts by conjugation on the affine variety of homomorphisms \(\mathrm{Hom}(\Gamma, G)\). The \(G\)-character variety of \(\Gamma\) is the affine (geometric invariant theoretic) quotient \(\mathfrak{X}_\Gamma(G):=\mathrm{Hom}(\Gamma, G)/\!\!/G\) by this action.
As \(\mathfrak{X}_\Gamma(G)\) is a complex affine variety, its cohomology admits a mixed Hodge structure (MHS). The MHS is encoded by the mixed Hodge polynomial \(\mu_{\Gamma, G}\), which generalizes the Poincaré polynomial.
The paper under review addresses the case when \(\Gamma\cong \mathbb{Z}^r\). Although for many cases of \(r\) and \(G\), the variety \(\mathfrak{X}_{\mathbb{Z}^r}(G)\) is irreducible and even normal, it is known to not always be irreducible, and is not known to always be normal. So the authors consider a normalization of a special component, denoted by \(\mathfrak{X}^*_{\mathbb{Z}^r}(G)\), that contains the identity representation. In general, \(\mathfrak{X}^*_{\mathbb{Z}^r}(G)\cong T^r/W\) where \(T\) is a maximal torus in \(G\) and \(W\) is the Weyl group of \(G\). The first main theorem in this well-written and interesting paper is an explicit computation of the mixed Hodge polynomial of \(\mathfrak{X}^*_{\mathbb{Z}^r}(G)\).
Specializing the mixed Hodge polynomial in a certain way one obtains a polynomial, called the \(E\)-polynomial. The \(E\)-polynomial, with respect to compactly supported cohomology, is related to arithmetic geometry. Let \(\mathbb{F}_q\) be a field of order \(q\). If the variety in question admits a model over \(\mathbb{Z}\) and the function that counts the number of \(\mathbb{F}_q\)-points is a polynomial in \(q\), then the \(E\)-polynomial and the counting function coincide. The second main theorem in this paper computes the (compactly supported) \(E\)-polynomial of \(\mathfrak{X}_{\mathbb{Z}^r}(\mathrm{SL}(n,\mathbb{C}))\) explicitly, in terms of partitions.
Some of the main results in this paper have recently been generalized by the reviewer and the authors of this paper [“Mixed Hodge structures on character varieties of nilpotent groups”, Preprint, arXiv:2110.07060]). In particular, we compute the MHS of the identity component of \(G\)-character varieties when \(\Gamma\) is nilpotent (which includes the abelian case). We also show there is only one irreducible component containing the identity representation (in the abelian case) and moreover that the normalization step used in the paper under review is unnecessary.
Finally, the reviewer notes that the paper under review, in addition to being mathematically interesting, can serve as a very good introduction to MHSs on character varieties. Overall, the paper is a fine piece of scholarship!