Wreath Macdonald polynomials and categorical McKay correspondence

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.

Appendix by Vadim Vologodsky. This is a slight modification (correction) of the version to appear in Cambridge Math Journal. The difference is in replacing $λ$ by $^t λ$ and such