Filtrations and completions of certain positive level modules of affine algebras

We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the module is irreducible and in the second case we prove that the intersection of all the modules is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration. These are certain highest weight integrable modules and the multiplicity and the highest weight are the same as that obtained by decomposing the tensor product of the highest weight crystal bases with the crystal bases of a loop module.