Vector bundles on flag varieties

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose $$G=G_k(d,n)$$ $$(2\le d\le n-d)$$ to be the Grassmannian parameterizing linear subspaces of dimension d in $$k^n$$, where k is an algebraically closed field of characteristic $$p>0$$. Let E be a uniform vector bundle over G of rank $$r\le d$$. We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle $$H_d$$ or its dual $$H_d^{\vee }$$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $$F(d_1,\ldots ,d_s)$$ 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 $$F(d_1,\ldots ,d_s)$$. Furthermore, we generalize the Grauert–M$$\ddot{\text{u}}$$lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i-semistable $$(1\le i\le n-1)$$ bundle over the complete flag variety splits as a direct sum of special line bundles.