Arithmetic and representation theory of wild character varieties

TL;DR

This paper counts points over finite fields on wild character varieties of Riemann surfaces, revealing new algebraic structures and proposing a connection between their mixed Hodge polynomials and Higgs moduli spaces, suggesting a wild P=W conjecture.

Contribution

It introduces counting formulas involving Yokonuma-Hecke algebra characters for wild character varieties and conjectures a link to twisted parabolic Higgs moduli spaces.

Findings

Counting formulas involve Yokonuma-Hecke characters

Conjecture of equality between mixed Hodge and perverse Hodge polynomials

Indication of a wild P=W conjecture for Hitchin systems

Abstract

We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma-Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P=W conjecture for a suitable wild Hitchin system.