On the cohomology ring of the Hilbert scheme of \(\mathbb{C}^2\) (Q2708163)

Fix an integer \(n\). Let \(X_n\) be the Hilbert scheme parametrizing subschemes of \(\mathbb{C}^2\) of finite length \(n\). Let \(Z_n\) be the center of the algebra \(\mathbb{C} [S_n]\), where \(S_n\) denotes the symmetric group. The author proves that there is a natural filtration \(F\) on \(Z_n\), compatible with the product, such that the graded rings \(H^{2*} (X_n,\mathbb{C}^2)\) and \(Gr_*^F(Z_n)\) are isomorphic. Moreover the author conjectures that there exists an analogous isomorphism in the more general case of a crepant resolution \(M\to \mathbb{C}^{2n}/G\), where \(G\) is a finite subgroup of \(Sp_{2n} (\mathbb{C})\).