Indexing the Line Bundles of a Flag Manifold

Question

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?

Answer 1

Attempting to answer your last question: given a (topological, Lie, algebraic, etc.) group G and a closed subgroup P in G, the category of G-equivariant vector bundles on X=G/P is equivalent to the category of representations of P. So to a representation of P one can assign a vector bundle over G/P, and even a G-equivariant vector bundle. In addition, any line bundle over a generalized flag manifold of a simply connected semisimple group G has a unique, up to an isomorphism, G-equivariant structure. So line bundles over a generalized flag manifold are classified by one-dimensional representations of the parabolic subgroup.

Answer 2

The partial flag manifold $X$ which you mentioned is the set of flags $0\subset V_1\subset..\subset V_m=\Bbb C^n$, such that ${\rm dim}(V_j/V_{j-1})=k_j$. 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.