Structure of the module of vector-valued modular forms

Let V be a representation of the modular group Gamma of dimension p. We show that the Z-graded space H(V) of holomorphic vector-valued modular forms associated to V is a free module of rank p over the algebra M of classical holomorphic modular forms. We study the nature of H considered as a functor from Gamma-modules to graded M-lattices and give some applications, including the calculation of the Hilbert-Poincare series of H(V) in some cases.