In this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work  for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.
Then we give two applications. In , we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in . In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.