Let k be an algebraically closed field. It is assumed that char k =p > 0. unless otherwise indicated. G is a simply connected semisimple algebraic group over k, T is a fixed maximal torus, and B a Bore1 subgroup of G containing T. Let Qi = @(G, T) be the root system of G, d the set of all simple roots, and @+ and @-the sets of all positive and negative roots relative to d, respectively. We assume that the set A is chosen such that B corresponds to 0‘.. Let X be the weight lattice of Sp, which may be considered as the character group of T or B. Therefore, if 1 E X, by abuse of notation, we also denote by L the one-dimensional Tor B-module determined by A. Let X’ be the set of all dominant weights, and w, the fundamental weight corresponding to a simple root a E A. We denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights. Let W be the Weyl group of 0. The reflection with respect to a root a is denoted by s,, the longest element of W by w”. There are two ways of W acting on X. (i) The ordinary action: if ;1 E X, M? E W, the image of i under w is denoted by ~2. (ii) Dot action: w . A= 5~(2 + 6) S. In particular, let A * = -w,,A. The dual root system of @ is denoted by @’ and the dual root of a E @ by a’. i.e., U” = 2a/(a, u), where ( , ) is the inner product (invariant under the ordinary action of w) of the Euclidean space spanned by @. For 1 E X, a E @, let (A, a) = (1, a’). All modules of an algebraic group we consider in this paper are assumed to be rational. In particular, for an infinite-dimensional module, the term “rational module” means this module is a direct limit of finite-dimensional rational modules. Let H be a closed subgroup of G containing T (such as G, B, or T), M an H-module; we define, for ,l E X,