Wreath Macdonald polynomials and the categorical McKay correspondence

Abstract

with an Appendix by Vadim Vologodsky To the memory of Andrei ZelevinskyMark 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 (Z/rZ) 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 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 [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.

Lectures on 𝐾-theoretic computations in enumerative geometry

“…The existence of the square root follows from Lemma 6.1.4. 8 The standard name for this term in algebraic geometry is the relative virtual tangent space, where the adjective relative refers to the map to the stack of domain deformations. Since the world relative is already infused with a very specific and different meaning for us, we avoid using it here.…”

On the Crossroads of Enumerative Geometry and Geometric Representation Theory

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