• Corpus ID: 14396490

Hilbert schemes, Hecke algebras and the Calogero-Sutherland system

@article{Costello2003HilbertSH,
  title={Hilbert schemes, Hecke algebras and the Calogero-Sutherland system},
  author={Kevin J. Costello and Ian Grojnowski},
  journal={arXiv: Algebraic Geometry},
  year={2003}
}

• K. Costello, I. Grojnowski
• Published 13 October 2003
• Mathematics
• arXiv: Algebraic Geometry

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When X is C 2 , this was done in [13, 21] by realising this ring as a degeneration of the center of CSn. When the canonical class KX = 0, [14] extended this result by defining an algebra structure on H � ({(x, g) ∈ X n × Sn | gx = x}); the Sn-invariants of this algebra is the desired ring. But when KX 6 0 it seems no such algebra can exist. A completely different approach is needed. 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) [10, 19]. With this identification, Lehn computed the operator of intersection with the boundary. This 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. 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.