Hilbert schemes, Hecke algebras and the Calogero-Sutherland system

Abstract

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not generalize. Instead we recast this problem as a question of finding integrals of motion for a Hamiltonian which describes ``intersection with the boundary''. To do so, we use the identification of the cohomology of the Hilbert schemes with a Fock space modelled on the lattice H(X). With this identification, Lehn computed the operator of intersection with the boundary. It is essentially the Calogero-Sutherland Hamiltonian. We then solve the problem of finding integrals of motion by using the Dunkl-Cherednik operators to find an explicit commuting family of differential operators; these operators represent cup product on the Hilbert scheme. We provide two characterizations of the Hilbert scheme multiplication operators; the first as an algebra of operators that can be inductively built from functions and the CS Hamiltonian, and the second in terms of the centralizer of the CS Hamiltonian inside an appropriate ring of differential operators.