On the E-polynomial of a familiy of parabolic Sp2n-character varieties

In this thesis, we find the E-polynomials of a family of parabolic symplectic character varieties of Riemann surfaces by constructing a stratification, proving that each stratum has polynomial count, applying a result of Katz regarding the counting functions, and finally adding up the resulting E-polynomials of the strata. To count the number of rational points of the strata, we invoke a formula due to Frobenius. Our calculation make use of a formula for the evaluation of characters on semisimple elements coming from Deligne-Lusztig theory, applied to the character theory of the finite symplectic group, and Möbius inversion on the poset of set-partitions. We compute the Euler characteristic of the our character varieties with these polynomials, and show they are connected.