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.

References in corpus (5)

• Collective fields, Calogero-Sutherland model and generalized matrix models
• Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation
• The cup product of the Hilbert scheme for K3 surfaces
• Lectures on Dunkl operators
• Matrix Theory, Hilbert Scheme and Integrable System

Cited by in corpus (8)

• Affine wreath product algebras
• Quantum cohomology of the Hilbert scheme of points in the plane
• K-Theory of Hilbert Schemes as a Formal Quantum Field Theory
• The classes of the quasihomogeneous Hilbert schemes of points on the plane
• The Chow ring of punctual Hilbert schemes of toric surfaces
• Integral generators for the cohomology ring of moduli spaces of sheaves over Poisson surfaces
• Two point extremal Gromov-Witten invariants of Hilbert schemes of points on surfaces
• Generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes