A Character variety X_FG 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_FG is homeomorphic to a space of so-called G-Higgs bundles over M. Typically, X_FG 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, C), we present a remarkable relation between the E-polynomials of XFG and those of X_F^{irr}G the locus of *irreducible representations* of F into G. All concepts will be motivated with F 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.

# Topology and Arithmetic of GL(n,C)-Character varieties

Carlos Florentino

Universidade de Lisboa, CMAFcIO