Let l∈N, l≥m∈N. In this paper, we introduce the notions of l-special skew shapes (resp. (l,m)-special skew shapes) and study their properties. Applications of these objects in the representation theory of type A affine Hecke algebras Hanaff with parameter q∈K are given, where q=l√ 1; or q=1 and charK=l. Using Young's seminormal construction, we show　that, for each l-special skew shape λ/μ with Stdl(λ,μ)≠θ (see Section 1 for definition of Stdl(λ,μ)), there is an indecomposable representation of Hanaff while for each (l,m)-special skew shape, there is an irreducible representation of Hanaffn. Our notion of (l,m)-special skew shape generalizes the notion of (l,m)-special partition in earlier work of O. Mathieu (see [M]) and of H. Wenzl (see [W]).