Arithmetic Cohen-Macaulayness and Arithmetic Normality for Schubert Varieties

Let G be a semisimple, simply connected Chevalley group over a field k. Let T be a maximal torus, B, a Borel subgroup D T and W, the Weyl group of G (relative to T). Let P be a maximal parabolic subgroup of classical type (cf. 7, 8), i.e. a maximal parabolic subgroup such that if w is the associated fundamental weight, then I (w, av) I ( 1 2 (w, )/(c, a) ) is < 2, for all roots ae, (,) being a W-invariant scalar product on X(T) ( Q, where X(T) = character group of T. For T E W/Wp, let X(T) denote the Schubert variety in G/P associated to T, namely X(T) is the Zariski closure BTP (mod P) in G/P (of the Bruhat cell) endowed with the canonical reduced structure. For the canonical projective embedding G/P in P(H?(G/P, L)), let RT denote the homogeneous co-ordinate ring of X(T). Then the main result of this paper is given by