We know that a line bundle $L$ on the complex flag variety $G/P$ is trivial iff $c\_1(E) = 0$. But if we have a homogeneous vector bundle $E$ of higher rank, then is it true that $c\_i(E) = 0$ $ \forall i$ implies that $E$ is trivial? If not, is there anything analogous we can say about such vector bundles? I apologize if this question is trivial as my Representation theory background isn't that great. 



Homogeneous vector bundles with zero chern classes
Take for instance $G/P = Gr(k,n)$, and let 
$$
0 \to U \to V \otimes \mathcal{O} \to Q \to 0
$$
be the tautological exact sequence. Then
$$
c\_\bullet(U \oplus Q) = c\_\bullet(U) \cdot c\_\bullet(Q) = c\_\bullet(V \otimes \mathcal{O}) = 1,
$$
but $U \oplus Q$ is a nontrivial vector bundle.


A similar example can be constructed on any flag variety.


