. A G ] 1 3 D ec 1 99 9 Hilbert schemes , wreath products , and the McKay correspondence

Various algebraic structures have recently appeared in a parallel way in the framework of Hilbert schemes of points on a surface and respectively in the framework of equivariant K-theory [N1, Gr, S2, W], but direct connections are yet to be clarified to explain such a coincidence. We provide several non-trivial steps toward establishing our main conjecture on the isomorphism between the Hilbert quotient of the affine space C2n by the wreath product Γn = Γ ∼ Sn and Hilbert schemes of points on the minimal resolution of a simple singularity C2/Γ. We discuss further various implications of our main conjecture. We obtain a key ingredient toward a direct isomorphism between two forms of McKay correspondence in terms of Hilbert schemes [N1, Gr, N2] and respectively of wreath products [FJW]. We in addition establish a direct identification of various algebraic structures appearing in two different setups of equivariant K-theory [S2, W].