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Wreath Macdonald polynomials and the categorical McKay correspondence. With an appendix by Vadim Vologodsky. (English) Zbl 1326.14037

Summary: M. Haiman [J. Am. Math. Soc. 14, No. 4, 941–1006 (2001; Zbl 1009.14001)] has reduced the 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 D. B. Kaledin [in: Algebraic geometry. Methods, relations, and applications. Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin. Moscow: Maik Nauka/Interperiodica. 13–33 (2004; Zbl 1137.14301)] 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.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14E16 McKay correspondence
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
16S32 Rings of differential operators (associative algebraic aspects)
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
14C05 Parametrization (Chow and Hilbert schemes)
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
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