Dedicated to Professor Zheng-fan Xu on the Occasion of His 70th Birthday

In this paper, we study the Cartan invariants for rank-one Cartan type Lie algebra W (1, n). This is done by reducing representations of generalized restricted Cartan type Lie algebra W (1, n) to representations of restricted Lie algebras W (1, 1) and of sl(2), and then extending Feldvoss-Nakano’s argument on W (1, 1) to the case W (1, n). Our discussion also shows some surprising difference between the representations of the restricted Lie algebra W (1, 1) and the non-restricted Lie algebra W (1, n) (n > 1), when the character heights of representations are zero. The method here can be applied to the generic case for Cartan type Lie algebras.