Clifford theory for graded fusion categories

We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group $G$ as induced from module categories over fusion subcategories associated with the subgroups of $G$. We define invariant \C_e-module categories and extensions of \C_e-module categories. The construction of module categories over \C is reduced to determine invariant module categories for subgroups of $G$ and the indecomposable extensions of this modules categories. We associate a $G$-crossed product fusion category to each $G$-invariant \C_e-module category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extended.Comment: Corollary 5.5 has been corrected. Accepted by Israel Journal of Mathematic