Counting points on generic character varieties
Masoud Kamgarpour, GyeongHyeon Nam, Bailey Whitbread, Stefano Giannini
TL;DR
This work develops a framework to count points on both multiplicative and additive character varieties associated with punctured surfaces for arbitrary connected reductive groups. It proves that multiplicative counts are polynomial in $q$, with the counting polynomials $f_X$ being palindromic after a finite base change, and it establishes a reductive analogue of the $P=W$ phenomenon, supported by topological implications like equal Euler characteristics and pure cohomology expectations. For the additive side, the paper derives polynomial counts after a quadratic base change, conjectures nonnegative coefficients, and links these counts to generalizations of Kac polynomials via endoscopy and $ rak g$-types; in the GL$_n$ case these recover known quiver-variety phenomena. The methodology combines the Frobenius mass formula with Deligne–Lusztig theory, endoscopy techniques, and intricate combinatorics of Levi and pseudo-Levi subgroups, yielding explicit counting polynomials and concrete examples for low-rank groups. Together, these results bridge geometric representation theory, number theory, and the topology of character varieties, and point toward broad conjectures about mixed Hodge structures, dualities, and reductive analogues of quiver polynomials.
Abstract
We count points on character varieties associated with punctured surfaces and regular semisimple generic conjugacy classes in reductive groups. We find that the number of points are palindromic polynomials. This suggests a $P=W$ conjecture for these varieties. We also count points on the corresponding additive character varieties and find that the number of points are also polynomials, which we conjecture have non-negative coefficients. These polynomials can be considered as the reductive analogues of the Kac polynomials of comet-shaped quivers.