Serre polynomials of $S{L}_n$- and $PG{L}_n$-character varieties of free groups

Let $G$ be a complex reductive group and ${\mathcal{X}}_{r}G$ denote the $G$-character variety of the free group of rank $r$. Using geometric methods, we prove that $E({\mathcal{X}}_{r}S{L}_n)=E({\mathcal{X}}_{r}PG{L}_n)$, for any $n,r\in \mathbb{N}$, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety $X$, settling a conjecture of Lawton-Mu~noz in [LM]. The proof involves the stratification by polystable type introduced in [FNZ], and shows moreover that the equality of E-polynomials holds for every stratum and, in particular, for the irreducible stratum of ${\mathcal{X}}_{r}S{L}_n$ and ${\mathcal{X}}_{r}PG{L}_n$. We also present explicit computations of these polynomials, and of the corresponding Euler characteristics, based on our previous results and on formulas of Mozgovoy-Reineke for $G{L}_n$-character varieties over finite fields.