On the $E$-polynomial of parabolic ${\mathrm{S}\mathrm{p}}_{2n}$-character varieties

We find the $E$-polynomials of a family of parabolic ${\mathrm{S}\mathrm{p}}_{2n}$-character varieties ${\mathcal{M}}_n^{\xi }$ 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 ${\mathbb{F}}_q$-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 $\mathrm{S}\mathrm{p}(2n,{\mathbb{F}}_q)$, and M"obius inversion on the poset of set-partitions. We compute the Euler characteristic of the ${\mathcal{M}}_n^{\xi }$ with these polynomials, and show they are connected.