Let be a complex semisimple Lie algebra. Let be a finite-dimensional g-module with weights , and an arbitrary g-module with infinitesimal character . Kostant proved that an infinitesimal character which occurs in the tensor product of is necessarily of the form . However, it is a difficult question whether a nonzero submodule with infinitesimal character indeed occurs in . Assume is a Harish-Chandra module with infinitesimal character . We prove a criterion when a nonzero submodule with infinitesimal character occurs in by using Dirac cohomology.