Vanishing theorems for products of exterior and symmetric powers

For ample vector bundles $E$ over compact complex varieties $X$ and a Schur functor $S_I$ corresponding to an arbitrary partition $I$ of the integer $|I|$, one would like to know the optimal vanishing theorem for the cohomology groups $H^{p,q}(X, S_I(E))$, depending on the rank of $E$ and the dimension $n$ of $X$. Three years ago (Nov. 1995), in an unpublished paper one of us (W.N.) proved a vanishing theorem for the situation where the partition $I$ is a hook. Here we give a simpler proof of this theorem. We also treat the same problem under weaker positivity assumptions, in particular under the hypothesis of ample $\Lambda ^m E$ with $m\in \N^*$. In this case we also need some bound on the weight $|I|$ of the partition. Moreover, we prove that the same vanishing condition applies for $H^{q,p}(X, S_I(E))$, with $p,q$ interchanged.