Schubert varieties and Demazure's character formula (Q1074682)

Let G be a semi-simple algebraic group over a field k (of any characteristic), B its Borel subgroup. The author studies Schubert varieties in the flag variety G/B and proves the following theorem: Let X be a Schubert variety in G/B and \({\mathcal L}^ a \)line bundle of G/B with \(H^ 0(G/B,{\mathcal L})\neq 0\). Then \(H^ i(X,{\mathcal L})=0\), for all \(i>0\) and the restriction map \(H^ 0(G/B,{\mathcal L})\to H^ 0(X,{\mathcal L})\) is surjective. From this follows, that, if char k\(=0\), all the Schubert varieties have rational singularities, are normal and Cohen-Macaulay. By using the theorem above, the author gives the proof of the Demazure character formula in arbitrary characteristic.