A Necessary and Sufficient Condition for the Complete Reducibility of a g (A)-module

Suppose that A is an n×n complex matrix.The contragredient Lie algebraassociated to matrix A is denoted by g(A). A g(A)-module V is called h-diagonalizable if it admits a weight space decom-position V= V_λ,and dimV_λ<∞. Let P(V)={λ∈h外\*|V_λ≠0} be the set of weightsof V.For λ∈h*,set D(λ)={μ∈h*|μ≤λ}.