A complex reductive algebraic group G has a special kind of mixed Hodge structure which is called Hodge-Tate type. These structures can then be described by their E-polynomial (or Hodge-Euler polynomial), which in turn is related to counting polynomials for the number of points of G in finite fields.

In this seminar, we consider analogous structures on character varieties: spaces of representations of a finitely generated group F into G. In the case that F is a free group, we can show that character varieties are Hodge-Tate type. Moreover, we will present a simple proof that the Hodge-Euler polynomials for G = SLn(C) and for G = PSLn(C) coincide. This can be viewed as a special instance of geometric Langlands duality.