Wreath Macdonald polynomials and the categorical McKay correspondence
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Author(s)
Bezrukavnikov, Roman
Finkelberg, Michael
Vologodsky, Vadim
Date Issued
2014
Journal
Cambridge Journal of Mathematics
Publisher
International Press of Boston
Citation
Bezrukavnikov, Roman et al. “Wreath Macdonald Polynomials and the Categorical McKay Correspondence.” Cambridge Journal of Mathematics 2, 2 (2014): 163–190 © 2018 International Press of Boston
Version
Author's final manuscript
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 Sn[subscript ⋉](Z/rZ)[superscript 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 A[superscript 2n] by the symmetric group Sn.
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.4310/CJM.2014.V2.N2.A1
