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 a Gm-fixed point scheme of the n-th symmetric product of C-2 for a natural 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. We also prove similar results for type A S3-varieties and smooth hypertoric varieties. These results can be formulated in terms of the symplectic duality of Braden-Licata-Proudfoot-Webster.