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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 C2 for a natural Gm-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. [ABSTRACT FROM AUTHOR]