Abstract: Let S be a smooth complex surface, and let S^{[n]} = Hilb_n S, the Hilbert scheme of ideal sheaves on S with dim_C (H^0(O/I)) = n. These varieties posess many natural correspondences which define interesting actions of Lie algebras (and vertex algebras) on their cohomology groups. Surprisingly, many of these Lie algebras also appear in 2d conformal field theory. I'll construct the vertex operators on the equivariant cohomology of (C^2)^{[n]} with a torus action, by connecting them with a the cohomology groups of certain finite-dimensional approximations of the Sato Grassmannian. Then I'll use this to give an elementary proof of their
commutation relations, for this particular surface and group action.