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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 $L^{[n]}$ thetautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on$X$. We explicitely compute the image $\bkrh(L^{[n]})$ of the tautologicalbundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh :\B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$ in terms of a complex$\comp{\mc{C}}_L$ of $\perm_n$-equivariant sheaves in $\B{D}^b_{\perm_n}(X^n)$.We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens ...\tens L^{[n]})$ in terms of of the hyperderived spectral sequence $E^{p,q}_1$associated to the derived $k$-fold tensor power of the complex$\comp{\mc{C}}_L$. The study of the $\perm_n$-invariants of this spectralsequence allows to get the derived direct images of the double tensor power andof the general $k$-fold exterior power of the tautological bundle for theHilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases.This yields easily the computation of the cohomology of $X^{[n]}$ with valuesin $L^{[n]} \tens L^{[n]}$ and $\Lambda^k L^{[n]}$.

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