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

Roman Bezrukavnikov (Dept. of Mathematics, M.I.T., Cambridge, Massachusetts, U.S.A.; National Research University Higher School of Economics, and International Laboratory of Representation Theory and Mathematical Physics, Moscow, Russia)

Michael Finkelberg (National Research University Higher School of Economics, Department of Mathematics, and IITP, Moscow, Russia)

Vadim Vologodsky (Department of Mathematics, University of Oregon, Eugene, Or., U.S.A.)

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.

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