The index of a modular differential operator

The space of all weakly holomorphic modular forms and the space of all holomorphic period functions of a fixed weight for the modular group are realized as locally convex topological vector spaces that are topologically dual to each other. This framework is used to study the kernel and range of a linear differential operator that preserves modularity and to define and describe its adjoint. The main result is an index formula for such a differential operator that is holomorphic at infinity. The co-kernel of the operator is identified as a cohomology group of the modular group acting on the kernel.