A reciprocity theorem for tensor products of group representations

Abstract:

Let G be a type I separable locally compact group. By studying a representation of $G \times G \times G$ we show that a measure class $\lambda$ on $G \times G \times G$ which describes the decompositions of tensor products is invariant under permutations, and that the multiplicity $n({\pi _1},{\pi _2},{\pi _3})$ of ${\bar \pi _3}$ in ${\pi _1} \otimes {\pi _2}$ is a symmetric function of its variables up to a $\lambda$ null set.