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 $L^{[n]}$ the tautological bundle on $X^{[n]}$ naturally associated to the line bundle $L$ on $X$. We explicitely compute the image \bkrh(L^{[n]}) of the tautological bundle $L^{[n]}$ 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 spectral sequence allows to get the derived direct images of the double tensor power and of the general $k$-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This yields easily the computation of the cohomology of $X^{[n]}$ with values in L^{[n]} \tens L^{[n]} and $\Lambda^k L^{[n]}$.Comment: 41 pages; revised version, exposition improve