Some remarks on tautological sheaves on Hilbert schemes of points on a surface

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland-King-Reid transform. We obtain Brion-Danila's Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert-Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.