  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 $\Lambda^k L^{[n]}$.
