Fusion rule algebras related to a pair of compact groups

The purpose of the present paper is to investigate a fusion rule algebra arising from irreducible characters of a compact group $G$ and a closed subgroup $G_0$ of $G$ with finite index. The convolution of this fusion rule algebra is introduced by inducing irreducible representations of $G_0$ to $G$ and by restricting irreducible representations of $G$ to $G_0$.