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

TL;DR

This paper computes the $E$-polynomials of certain parabolic $ ext{Sp}_{2n}$-character varieties of Riemann surfaces, revealing their structure and connectedness through advanced algebraic and combinatorial techniques.

Contribution

It introduces a novel stratification approach and applies character theory and M"obius inversion to explicitly calculate the $E$-polynomials of these varieties.

Findings

Explicit $E$-polynomials for the varieties.

Connectedness of the character varieties.

Euler characteristic calculations.

Abstract

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