Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles

Let X a smooth quasi-projective algebraic surface, L a line bundle on X. Let X [n] the Hilbert scheme of n points on X and L the tautological bundle on X [n] naturally associated to the line bundle L on X. We explicitely compute the image Φ(L) of the tautological bundle L for the Bridgeland-King-Reid equivalence Φ : D(X ) → DbSn(X ), in terms of a complex C L of Snequivariant sheaves in DbSn(X ). By using some recent results by Haiman on polygraphs, we give a characterization of the image Φ(L ⊗ · · · ⊗ L) of the k-fold tensor power of the tautological bundle L in terms of of the hyperderived spectral sequence E 1 associated to the derived k-th tensor product C L ⊗ L . . . ⊗ C L. The study of the Sn-invariants of this spectral sequence allows to get the derived direct images Rμ∗(L [n] ⊗ L) and Rμ∗(Λ L) of the double tensor product and of the general k-fold exterior power of the tautological bundle for the Hilbert-Chow morphism μ, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of X [n] with values in L ⊗ L and ΛL.