The Generalized Restricted Representations of Graded Lie Algebras of Cartan Type Original Research Article

LetFbe an algebraically closed field of characteristicp > 2. In this paper, the concepts of generalized restricted Lie algebras and their generalized restricted representations overFare introduced. Any graded Lie algebra of Cartan typeL = X(m : n)(2),X = W,S,H, orK, is a generalized restricted Lie algebra associated with a standard basisEofLandsa positive integer ≥ max1 ≤ i ≤ m{ni}. The irreducible generalized restricted representations ofLare determined, which are in a one-to-one correspondence withpsdweight functions; heredis the dimension of the standard torus inL1 + δX, Kvar epsilonm, [0]. The descriptions of the blocks and a Brauer-type reciprocity are obtained.