I don't really know why but I imagined the following product of representations, that only work for abelian groups. Let $G$ be a a finite abelian group and let $V, W$ be two representations (meaning $V$ and $W$ are vector spaces and for each $g∈G$, I write $g_V$ for the corresponding automorphism of $V$).
Let $V∗W$ be the (almost) representation whose underlying vector space is $V⊗W$ and such that $$g_{{}_{V∗W}} = \frac{1}{|G|} ∑_{hw=g} h_V ⊗ w_W \text{.}$$ This almost defines a representation: we can check that $g_{{}_{V∗W}} h_{{}_{V∗W}} = (gh)_{{}_{V∗W}}$ (we need $G$ to be abelian), but we don't have $e_{{}_{V∗W}} = \operatorname{id}_{V⊗W}$. So we can restrict the space to the image of $e_{{}_{V∗W}}$ (which is a projector) and this now defines a representation.
We see that this product is distributive on the direct sum, that for any irreducible representation $V$, we have $V∗V=V$ and that if $V,W$ are non isomorphic irreducible representations, then $V∗W=0$.
So if $\newcommand{\dG}{\widehat{G}}\dG$ denote the set of irreducible representations of $G$, we see that
$$ \left( ⨁_{V∈\dG} V^{⊕n_V} \right) ∗ \left( ⨁_{V∈\dG} V^{⊕m_V} \right) = \left( ⨁_{V∈\dG} V^{⊕(n_V m_V)} \right) \text{.} $$
Writing a representation as a sum of irreductibles is some sort of Fourier transform diagonalizing this product.
Is there some interpretation of this? Some link with the actual Fourier Transform? And is there a generalization to the non abelian case?