TL;DR: In this article, the problem of whether there always exist meromorphic functions commuting with all substitutions of a function group is solved in the affirmative, and it is shown that there always exists a meromorphic function commuting with

TL;DR: In this article, a decomposition for the tensor product of two irreducible representations of the group of conformal mappings of the upper half-plane when one representation is taken in a Hilbert space of analytic functions and the other in the conjugate of such a space is obtained.