Structure of the module of vector-valued modular forms

Let V be a representation of the modular group Γ 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 Γ-modules to graded ℳ-lattices and give some applications, including the calculation of the Hilbert–Poincaré series of ℋ(V) in some cases.