Abstract: A sufficient and necessary condition was obtained for an irreducible generalized $\chi$-reduced module of a Cartan type Lie algebra over $k=\bar{\mathbb{F}}_q$ being split over ${\mathbb{F}}_q$, where the height of the character $\chi$ is no more than 0. For the Witt algebra, the corresponding result for general $\chi$ was given.

Key words: Cartan type Lie algebra, generalized restricted Lie algebra, modular representation, Frobenius morphism, ${\mathbb{F}}_q$-form