Let $G$ be a compact (Hausdorff) topological group, and let $\sigma, \tau$ fixed irreducible unitary representation of $G$. Let us identify two representations if they are (unitarily) equivalent, and let us denote the (equivalence classes of) irreducible representations by $\widehat{{G}}$.
In a paper I am reading, (I think) it is implicitly used that the set $$\{\pi\in \widehat{G}: \sigma\subseteq \pi\otimes \tau\}$$ is finite. Why is this the case? Here, $\pi\otimes \tau$ denotes the tensor product of the two representations. Note that $\pi\otimes \tau$, being a finite-dimensional representation, decomposes (uniquely) as a finite direct sum of irreducible representations. The statement $\sigma\subseteq \pi\otimes \tau$ then means that $\sigma$ occurs as an irreducible component in this decomposition.
I tried to play around a bit with Frobenius reciprocity but did not get very far.
Thanks in advance for your help!
Any nonzero map $\sigma \to \pi \otimes \tau$ dualizes to a nonzero map $\sigma \otimes \tau^{\ast} \to \pi$ and vice versa, so the set in question is exactly the set of irreducibles occuring in $\sigma \otimes \tau^{\ast}$, which is clearly finite.