Vector bundles on flag varieties

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose (Formula presented.) (Formula presented.) to be the Grassmannian parameterizing linear subspaces of dimension d in (Formula presented.), where k is an algebraically closed field of characteristic (Formula presented.). Let E be a uniform vector bundle over G of rank (Formula presented.). We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle (Formula presented.) or its dual (Formula presented.) by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties (Formula presented.) in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in (Formula presented.). Furthermore, we generalize the Grauert–M (Formula presented.) lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i-semistable (Formula presented.) bundle over the complete flag variety splits as a direct sum of special line bundles.