Equivariant cohomology, symmetric functions and the Hilbert schemes of points on the total space of the invertible sheaf O(-2) over the projective line

Let X be the quasi-projective symplectic surface that is given by the total space of the invertible sheaf O(-2) over the projective line. Let Hilb X be the family of Hilbert schemes of points on X. We give and prove a closed formula expressing any multiplicative characteristic class evaluated on the Hilb X in terms of the standard Fock space description of the cohomology of the Hilb X. As a side result, we also deduce a formula for the Chern character of the tangent bundles of the Hilb X. The results found here are another step towards a complete description of the tangent bundle of the Hilbert scheme of a general quasi-projective surface as the formulas given here yield certain coefficients in that description. Finally, we also give a closed formula expressing any multiplicative characteristic class evaluated on some tautological bundles on the Hilb X.