A Nekrasov-Okounkov formula for Macdonald polynomials

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.

24 pages; Revised version includes an alternative proof (suggested by Jim Bryan) of an elliptic Nekrasov-Okounkov formula based on the equivariant DMVV formula. In v3 some final corrections have been carried out