On cartan invariants for graded Lie algebras of cartan type

Let L = X(m; n), X ∈ {W, S, H, K}, be a graded simple Lie algebra of Cartan type over an algebraically closed field of characteristic p > 3. Then L is a so-called generalized restricted Lie algebra. Let L be the primitive p-envelope of L, and G = X(m; 1), a subalgebra of L. In this paper, a close connection between Cartan invariants for U(L, χ) and U(G, χ) is established, where χ ∈ L* is extended to be a linear function on L trivially, and 1 ≤ ht(χ) < p-2+δ_XW. This reduces the study of projective representations of the generalized restricted Lie algebra L to the one of the corresponding restricted Lie algebra G. As a special case, we recover some results in [Shu and Jiang, On Cartan invariants and blocks of Zassenhaus algebras, Comm. Algebra 33(10) (2005) 3619-3630].