On k-ampleness equivalence

For a partition $a$ and a vector bundle $E$ on a projective variety $X$ let $\mathcal{F}{l}_s(E)$ be the corresponding flag manifold. There is a line bundle $Q_a^s$ on $\mathcal{F}{l}_s(E)$ with $p:\mathcal{F}{l}_s(E)\to X$ and $p_{\ast }{Q}_a^s={\mathcal{S}}_{a}E$. We prove, if ${\mathcal{S}}_{a}E$ is $k$-ample (in the sense of Sommese) then $Q_a^s$ is $k$-ample. For the inverse if $Q_a^s$ is $k$-ample, we prove that one of two the conditions of k-ampleness namely the cohomological vanishing is proved here but not yet the condition of semiamplenes of ${\mathcal{S}}_{a}E$ .