Rigidity of formal characters of Lie algebras (II)

For a complex simple Lie algebra of type A_l,B_l,C_l or D_l, given a family of elements f_\lambda\ in commutative ring Z[\Lambda], we show that f_\lambda\ is just the formal character of the Weyl module V(\lambda) if f_\lambda\ satisfy several natural conditions. Hence we give a necessary and sufficient condition for constructing a family of g_l-modules from a family of g_{l-1}-modules.