  We study character varieties arising as moduli of representations of an
orientable surface group into a reductive group $G$. We first show that if
$G/Z$ acts freely on the representation variety, then both the representation
variety and the character variety are smooth and equidimensional. Next, we
count points on a family of smooth character varieties; namely, those involving
both regular semisimple and regular unipotent monodromy. In particular, we show
that these varieties are polynomial count and obtain an explicit expression for
their $E$-polynomials. Finally, by analysing the $E$-polynomial, we determine
certain topological invariants of these varieties such as the Euler
characteristic and the number of connected components. As an application, we
give an example of a cohomologically rigid representation which is not
physically rigid.
