A note on Frobenius splitting of Schubert varieties and linear syzygies.

Let G be a connected, simply connected group and B be a Borel subgroup of G. Let X be a Schubert variety in G/B. Let [L.sub.1], . . ., [L.sub.r] be invertible sheaves on G/B. For [Alpha] [is an element of] [N.sup.r], let [Mathematical Expression Omitted]. The multi-graded coordinate rings R = [[symmetry].sub.[Alpha]][H.sup.0](G/B, [L.sup.[Alpha]]) and A = [[symmetry].sub.[Alpha]][H.sup.0](X, [L.sup.[Alpha]]), were shown to be wonderful in [1] assuming each of [L.sub.i] to be ample. In this note we extend the result obtained in [1] to effective invertible sheaves on G/B. An effective invertible sheaf on G/B is a pull-back of an ample invertible sheaf on G/P, for some parabolic subgroup P [contains] B of G. In this note, we use this fact to obtain the necessary vanishing conditions for effective invertible sheaves on G/B. This will prove that R and A are multi-wonderful.

An effective invertible sheaf L on G/B is associated to a dominant weight [Lambda]. Let [P.sub.[Lambda]] be the largest parabolic subgroup of G such that [Lambda] defines a character on [P.sub.[Lambda]]. We denote by [L.sub.[Lambda]] the associated invertible sheaf on G/[P.sub.[Lambda]]. Let [p.sub.[Lambda]]: G/B [right …