The generalized restricted representations of graded Lie algebras of Cartan type

Let F be an algebraically closed field of characteristic p > 2. In this paper, the concepts of generalized restricted Lie algebras and their generalized restricted representations over F are introduced. Any graded Lie algebra of Cartan type L = X(m:n)((2)), X = W, S, H, or K, is a generalized restricted Lie algebra associated with a standard basis E of L and s a positive integer greater than or equal to max(1 less than or equal to i less than or equal to m){n(i)}. The irreducible generalized restricted representations of L are determined, which are in a one-to-one correspondence with p(sd) weight functions; here d is the dimension of the standard torus in L-1+delta X,L-K is an element of m,L-[0]. The descriptions of the blocks and a Brauer-type reciprocity are obtained. (C) 1997 Academic Press.