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 ≥ max_{1 ≤ i ≤ 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+ δX, K∈m,[o]}. The descriptions of the blocks and a Brauer-type reciprocity are obtained.