Jack Polynomials and Hilbert Schemes of Points on Surfaces

The Jack (symmetric) polynomials P (α) λ (x) form a class of symmetric polynomials which are indexed by a partition λ and depend rationally on a parameter α. They reduced to the Schur polynomials when α = 1, and to other classical families of symmetric polynomials for several specific parameters. Recently they attracts attention from various points of view, for example the integrable systems and combinatorics. They are simultaneous eigenfunctions of certain commuting families of differential operators, appearing in the integrable system called the Calogero-Sutherland system (see e.g., [2] and the reference therein). On the other hand, Macdonald studied their combinatorial properties. In fact, he introduced an even more general class of symmetric functions (a two parameter family), which have many common combinatorial features as Jack polynomials (see [17, Chapter 6]). It is well-known that Schur polynomials can be realized as certain elements of homology groups of Grassmann manifolds (see e.g., [9, Chapter 14]). The purpose of this paper is to give a similar geometric realization for Jack polynomials. However, spaces which we use are totally different. Our spaces are Hilbert schemes of points on a surface X which is the total space of a line bundle L over the projective line CP. The parameter α in Jack polynomials relates to our surface X by