  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 $\C^{2n}$ by the wreath product $\G ~ S_n$
and Hilbert schemes of points on the minimal resolution of a simple singularity
$\C^2 /\G$. 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].
