I was recently looking back at an old question of mine, where I asked about the classification of the line bundles over a general complex flag manifold. Pavel Etingof gave the following excellent answer:

The partial flag manifold X which you mentioned is the set of flags $0 ⊂ V_1 ⊂ \ldots ⊂ V_m=ℂ^n$, such that $\dim(V_j/V_{j−1})=k_j$. So we have vector bundles $V_j$ on $X$ and line bundles $L_j$ on $X$ which are the top exterior powers of the bundles $V_j$, $j = 1,...,m−1$. Any line bundle on $X$ is a tensor product of powers of these line bundles, and the powers are uniquely determined. This is why line bundles on $M$ are labeled by a set of $m−1$ integers.

I then began to wonder how this relates to a more recent question of mine regarding the definition of a globally generated holomorphic vector bundle. So my question is: With respect to the above line bundle classification, and the obvious choice of holomorphic structure for these bundles (I am assuming that there is just one obvious choice here), which of these bundles are globally generated?

For the simplest case of complex projective space, we have of course that the line bundles are classified by the integers, and the positive ones are globally generated (or the negative, depending on convention). Now for the Grassmannians the line bundles are again classified by the integers, and I would again guess that the globally generated ones are exactly those of positive charge. So how does this extend?