An algebro-geometric realization of the cohomology ring of Hilbert scheme of points in the affine plane

We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of ${\mathbb{G}}_m$-fixed point scheme of the $n$-th symmetric product of ${\mathbb{C}}^2$ for a natural ${\mathbb{G}}_m$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.