Irreducible representations of infinite-dimensional Lie algebras of Cartan type

In this paper the irreducible representations of infinite-dimensional filtered Lie algebras are studied. The concept of the height of a representation is introduced, and it is proved that the representations of height greater than one of the Lie algebras $\mathbf W_n$, $\mathbf S_n$, $\mathbf H_n$ and $\mathbf K_n$ are induced. The representations of height one of the algebras $\mathbf W_n$ are also described.