  This is a survey of results and conjectures on mirror symmetry phenomena in
the non-Abelian Hodge theory of a curve. We start with the conjecture of
Hausel-Thaddeus which claims that certain Hodge numbers of moduli spaces of
flat SL(n,C) and PGL(n,C)-connections on a smooth projective algebraic curve
agree. We then change our point of view in the non-Abelian Hodge theory of the
curve, and concentrate on the SL(n,C) and PGL(n,C) character varieties of the
curve. Here we discuss a recent conjecture of Hausel-Rodriguez-Villegas which
claims, analogously to the above conjecture, that certain Hodge numbers of
these character varieties also agree. We explain that for Hodge numbers of
character varieties one can use arithmetic methods, and thus we end up
explicitly calculating, in terms of Verlinde-type formulas, the number of
representations of the fundamental group into the finite groups SL(n,F_q) and
PGL(n,F_q), by using the character tables of these finite groups of Lie type.
Finally we explain a conjecture which enhances the previous result, and gives a
simple formula for the mixed Hodge polynomials, and in particular for the
Poincare polynomials of these character varieties, and detail the relationship
to results of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan. One consequence
of this conjecture is a curious Poincare duality type of symmetry, which leads
to a conjecture, similar to Faber's conjecture on the moduli space of curves,
about a strong Hard Lefschetz theorem for the character variety, which can be
considered as a generalization of both the Alvis-Curtis duality in the
representation theory of finite groups of Lie type and a recent result of the
author on the quaternionic geometry of matroids.
