ON THE EQUIVARIANT COHOMOLOGY OF HILBERT SCHEMES OF POINTS IN THE PLANE

Let S be the affine plane regarded as a toric variety with an action of the 2-dimensional torus T. We study the equivariant Chow ring A(K)*(S-[n]) of the punctual Hilbert scheme S-[n] with equivariant coefficients inverted. We compute base change formulas in A(K)*(S-[n]) between the natural bases introduced by Nakajima, Ellingsrud and Str mme, and the classical basis associated to the fixed points. We compute the equivariant commutation relations between creation/annihilation operators. We express the class of the small diagonal in S-[n] in terms of the equivariant Chern classes of the tautological bundle. We prove that the nested Hilbert scheme S-0([n,n+1]) parametrizing nested punctual subschemes of degree n and n + 1 is irreducible.