A Character variety $X_F G$ is a space of representations of a finitely generated group $F$ into a Lie group $G$. The most interesting cases are when $F$ is the fundamental group of a Kähler manifold $M$, and $G$ is a reductive group, since then $X_F G$ is homeomorphic to a space of so-called $G$-Higgs bundles over $M$.

Typically, $X_F G$ are singular algebraic varieties, defined over the integers, and many of its topological and arithmetic properties can be encoded in a polynomial generalization of the Euler-Poincaré characteristic: the $E$-polynomial.

In this seminar, concentrating in the case of the general linear group $G=GL(n,\mathbb{C})$, we present a remarkable relation between the $E$-polynomials of $X_F G$ and those of $X_F^{irr} G$, the locus of *irreducible representations* of $F$ into $G$. All concepts will be motivated with several examples, and we will give an overview of known explicit computations of $E$-polynomials, as well as some conjectures and open problems.

This is joint work with A. Nozad, J. Silva and A. Zamora.