Question: If $G$ is a compact Lie group, does the set of equivalence classes of unitary complex irreps $\widehat{G}$ carry an algebraic structure such as a natural binary operation or perhaps even a monoidal structure?
Motivation: If $G$ was the abelian compact Lie group $S^1$, we would have the first convolution theorem: $\widehat{f \ast g}(n) = \hat{f}(n) \hat{g}(n)$ and also the second convolution theorem: $\widehat{f g}(n) = (\hat{f} \ast \hat{g})(n)$.
In the case of $G$ non-abelian, the first theorem makes sense to generalize since I can convolve over any group and multiply functions pointwise over any set. However, the second convolution theorem does not immediately seem to be generalizable since $\widehat{G}$ is not a group in general--unless $\widehat{G}$ still carries some other kind of monoidal structure or perhaps some even weaker algebraic structure so that convolution still makes sense. (And I really WANT there to be a generalization because I am working with a fourier transform of a product of functions over $SO(3)$!)
Thanks! :)