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.Suppose that there exists a separated scheme X over a finitely generated Z-algebra R, such that for some embedding R ֒→ C we havein such a case we say that X is a spreading out of X. If, further, there exists a polynomial P X (w) ∈ Z[w] such that for any homomorphism R → F q (where F q is the finite field of q elements), one hasthen we say that X has polynomial count and P X is the counting polynomial of X. The motivating result is then the following.Theorem 2.1.1. (N. Katz,[HV, Theorem 6.1.2]) Suppose that the complex algebraic variety X is of polynomial count with counting polynomial P X . Then E(X; x, y) = P X (xy).Remark 2.1.2. Thus, in the polynomial count case we find that the count polynomial P X (q) = E(X; q) agrees with the weight polynomial. We also expect our varieties to be Hodge-Tate, i.e., h p,q;j c (X) = 0 unless p = q, in which case H c (X; x, y, t) = W H(X; xy, t). Thus, in these cases we are not losing information by considering W H(X; xy, t) (resp. E(X; q)) instead of the usual H c (X; x, y, t) (resp. E(X; x, y)).
Wild character varietiesThe wild character varieties we study in this paper were first mentioned in [B2, §3 Remark 5], as a then new example in quasi-Hamiltonian geometry-a "multiplicative" variant of the theory of Hamiltonian group actions on symplectic manifolds-with a more thorough (and more general) construction given in [B3, §8]. We give a direct definition here for which knowledge of quasi-Hamiltonian geometry is not required; however, as we appeal to results of [B3, §9] on smoothness and the dimension of the varieties in question, we use some of the notation of [B3, §9] to justify the applicability of those results.
2.2.
