Some simple modules for the restricted Cartan-type Lie algebras

Every simple module having character height at most one for a restricted Cartan-type Lie algebra image can be realized as a quotient of a module obtained by starting with a simple module S for the homogeneous component of degree zero in the natural grading of image, extending the action trivially to positive components and inducing up to image. It is shown that if S is not restricted, or if it is restricted and its maximal vector does not have exceptional weight, then the induced module is already simple.