Journal of the European Mathematical Society

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Volume 21, Issue 10, 2019, pp. 2995–3052
DOI: 10.4171/JEMS/896

Published online: 2019-06-06

Arithmetic and representation theory of wild character varieties

Tamás Hausel[1], Martin Mereb[2] and Michael Lennox Wong[3]

(1) Institute of Science and Technology Austria, Klosterneuburg, Austria
(2) Universidad de Buenos Aires, Argentina
(3) Universität Duisburg-Essen, Germany

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 the 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.

Keywords: Irregular connection, character variety, Hitchin system, Yokonuma–Hecke algebra, Macdonald polynomials

Hausel Tamás, Mereb Martin, Wong Michael Lennox: Arithmetic and representation theory of wild character varieties. J. Eur. Math. Soc. 21 (2019), 2995-3052. doi: 10.4171/JEMS/896