Following on from this question link text, how are the line bundles of a complex flag variety indexed? Are they still labeled by the integers? If so, why? A representation theory explanition in terms of the homogenous space description of the variety $U(n)/U(k_1) \times \cdots \times U(k_m)$ would be the most useful.

Also, is there an 'obvious' reason why the line bundles should be associated to a representation of $U(k_1) \times \cdots \times U(k_m)$, as opposed to a representation of another quotient description of the variety. For example, $\mathbb{CP}^{n-1}$ can be described as a $SU(n)/U(n)$ or as $S^{2n-1}/U(1)$, but the tangent bundle can only be described as associated to a representation of $U(n)$. In general, for a principal $G$-bundle $P$, what vector bundles over $P$ can be described as associated to a representation of $G$? All of them?