A Nekrasov-Okounkov formula for Macdonald polynomials

TL;DR

This paper extends the Nekrasov-Okounkov hook-length formula to Macdonald polynomials and applies it to prove a conjecture related to the mixed Hodge polynomials of Higgs bundle moduli spaces.

Contribution

It introduces a Macdonald polynomial analogue of the Nekrasov-Okounkov formula and uses it to prove a significant conjecture in algebraic geometry.

Findings

Established a Macdonald polynomial version of the Nekrasov-Okounkov formula

Proved a conjecture on mixed Hodge polynomials of Higgs bundle moduli spaces

Connected combinatorial formulas with geometric invariants

Abstract

We prove a Macdonald polynomial analogue of the celebrated Nekrasov-Okounkov hook-length formula from the theory of random partitions. As an application we obtain a proof of one of the main conjectures of Hausel and Rodriguez-Villegas from their work on mixed Hodge polynomials of the moduli space of stable Higgs bundles on Riemann surfaces.