Cohomology of Graded Lie Algebras of Cartan Type of Characteristic 0

In D. Fux determined the structures of cohomology of formal vector fields Lie algebras with coefficients in trivial modules or dual adjoint modules. We determine the structure of H¹(L, V), where L = W(n), S(n) or H (n) is an infinite dimensional Lie algebra of characteristic 0 and V is a graded L-module V. (mixed product) or a graded irreducible L-module.