Decomposition of tensor products of irreducible unitary representations

Abstract:

It is shown that the tensor product of an irreducible unitary representation of a (discrete) group $G$ and an $n$-dimensional $(n < \infty )$ unitary representation of $G$ decomposes into at most ${n^2}$ irreducible subrepresentations; the multiplicity of each irreducible constituent is not greater than $n$. As an application it is shown that the restriction of an irreducible unitary representation to a subgroup of finite index is a finite sum of irreducible subrepresentations.