Serre polynomials of SLn- and PGLn-character varieties of free groups

Let G be a complex reductive group and chi(r)G denote the G-character variety of the free group of rank r. Using geometric methods, we prove that E(XrSLn) = E(chi(r)PGL(n)), for any n, r is an element of N, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety X, settling a conjecture of Lawton-Munoz in Lawton and Munoz (2016). The proof involves the stratification by polystable type introduced in Florentino et al. (2019), and shows moreover that the equality of E-polynomials holds for every stratum and, in particular, for the irreducible stratum of chi rSLn and chi(r)PGL(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 GL(n)-character varieties over finite fields. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This study uses geometric methods to prove the equality of E-polynomials for specific complex reductive groups, settling a conjecture by Lawton and Munoz. The proof involves stratification by polystable type and demonstrates the equality of E-polynomials across different strata, particularly in the irreducible strata.