The Cohomology Rings of Hilbert Schemes via Jack Polynomials

Fundamental and deep connections have been developed in recent years between the geometry of Hilbert schemes X [n] of points on a (quasi-)projective surface X and combinatorics of symmetric functions. Among distinguished classes of symmetric functions, let us mention the monomial symmetric functions, Schur polynomials, Jack polynomials (which depend on a Jack parameter), and Macdonald polynomials, etc (cf. [Mac]). The monomial symmetric functions can be realized as certain ordinary cohomology classes of the Hilbert schemes associated to an embedded curve in a surface (cf. [Na1]). Nakajima [Na2] further showed that the Jack polynomials whose Jack parameter is a positive integer γ can be realized as certain T-equivariant cohomology classes of the Hilbert schemes of points on the surface X(γ) which is the total space of the line bundle OP1(−γ) over the complex projective line P. Here and below T stands for the one-dimensional complex torus. In other words, the Jack parameter is interpreted as minus the self-intersection number of the zero-section in X(γ). With very different motivations, Haiman (cf. [Hai] and the references therein) developed connections between the Macdonald polynomials and the geometry of Hilbert schemes, and in particular realized the Macdonald polynomials as certain T-equivariant K-homology classes of the Hilbert schemes of points on the affine plane C (A similar result has been conjectured in [Na2]). In this note, we shall establish a link somewhat different from [Na2] between equivariant cohomology of Hilbert schemes and Jack polynomials, and then use this to describe the equivariant and ordinary cohomology rings of the Hilbert schemes of points on the surface X(γ). We first show that the Jack polynomials can be realized in terms of certain T-equivariant cohomology classes of the Hilbert schemes of points on the affine plane, and the Jack parameter comes from the ratio of the Tweights on the two affine lines preserved by the T-action. In our view, the present construction is conceptually simpler than the original one in [Na2]. This result is probably not very surprising however and could be well anticipated by experts (as it is done by elaborating the ideas of [Na2] with new inputs from [Vas]). But we