Invent. math. 79, 611-618 (I985) Inve~ltione$ mathematicae 9 Springer-Verlag 1985 Schubert varieties and Demazure's character formula H.H. Andersen Matematisk Institut, Aarhus Universitet, DK-8000 Aarhus C, Denmark Introduction In [3] M. Demazure constructed the so-called Bott-Samelson scheme (see also [6]) which gives desingularizations of the Schubert varieties in the flag ma- nifold G/B. Here G denotes a semi-simple algebraic group over a field k and B is a Borel subgroup. He proved that this construction makes it easy to calculate the Chow ring of G/B and when k has characteristic zero he also used it to prove the following Theorem. Let X be a Schubert variety in G/B and let P, denote a line bundle on G/B with H~ Then Hi(X,P~)=O for i>0 and the restriction map H~ ~)---* H ~ Yd) is surjective. As a consequence of his proof he obtained Corollary. In characteristic zero all Schubert varieties in G/B have rational singularities. In particular, they are normal and Cohen-Macaulay. Furthermore he derived a formula for the character of H~ 9,) when X and !~ are as above. This formula has become known as Demazure's character formula. Recently, when attempts were made to carry over some of these arguments to Kac-Moody algebras, it was discovered that there is a gap in Demazure's proof. In fact, Proposition 11 (due to D.-N. Verma) is false unless the weight occurring is extremal - in which case A. Joseph has recently proved that the statement is equivalent to Demazure's character formula [7]. Joseph also proved (by a quite different argument) that the character formula holds for large dominant weights. In this paper we prove the above theorem in all characteristics. We are also able to deduce from our proof that Schubert varieties are normal and that Demazure's character formula holds in arbitrary characteristic. This paper is a revised version of a preprint with the same title appearing in Aarhus Universitet Preprint Series i983/84 (No. 44, June 1984). There I