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# Cambridge Journal of Mathematics

## Volume 2 (2014)

### Number 2

### Wreath Macdonald polynomials and the categorical McKay correspondence

Pages: 163 – 190

DOI: http://dx.doi.org/10.4310/CJM.2014.v2.n2.a1

#### Authors

#### 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 ${\mathfrak S}_n \ltimes (\mathbb{Z} / r \mathbb{Z})^n$. 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 ${\mathfrak S}_n$.

A short proof of a similar derived equivalence for any symplectic quotient singularity has been obtained by the first author and Kaledin [2] via quantization in positive characteristic. In the present note we prove various properties of these derived equivalences and then deduce generalized Macdonald positivity for wreath products.