Wreath Macdonald polynomials and categorical McKay correspondence

TL;DR

This paper explores the geometric and categorical aspects of Macdonald polynomials related to wreath products, extending Haiman's work on Hilbert schemes and derived equivalences to prove generalized positivity conjectures.

Contribution

It demonstrates properties of derived equivalences that imply the generalized Macdonald positivity for wreath products, extending previous geometric and categorical results.

Findings

Derived equivalence properties imply generalized Macdonald positivity.

Extension of Haiman's geometric approach to wreath products.

Proof of positivity conjecture for wreath product cases.

Abstract

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert scheme, as a byproduct he obtained a derived equivalence between coherent sheaves on the Hilbert scheme and coherent sheaves on the orbifold quotient of ${\mathbb A}^{2n}$ by the symmetric group $S_n$. A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin via quantization in positive characteristic. In the present note we show the properties of the derived equivalence which imply the generalized Macdonald positivity for wreath products.