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 $\text{bkrh}(L^{[n]})$ of the tautological bundle $L^{[n]}$ for the Bridgeland-King-Reid equivalence $\bkrh : \B{D}^b(X^{[n]}) \ra \B{D}^b_{\perm_n}(X^n)$intermsofacomplex$\comp{\mc{C}}L$of$\perm_n$-equivariantsheavesin$\B{D}^b{\perm_n}(X^n)$. We give, moreover, a characterization of the image $\bkrh(L^{[n]} \tens … \tens L^{[n]})$intermsofofthehyperderivedspectralsequence$E^{p,q}_1$associatedtothederived$k$-foldtensorpowerofthecomplex$\comp{\mc{C}}_L$\mathrm{.}Thestudyofthe$\perm_n$-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 $L^{[n]}\text{tens}{L}^{[n]}$ and ${\mathrm{\Lambda }}^k{L}^{[n]}$.