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Schubert varieties and Demazure’s character formula. (English) Zbl 0591.14036
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.
Reviewer: S.Priščepionok

MSC:
14M17 Homogeneous spaces and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds
20G10 Cohomology theory for linear algebraic groups
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