Let $G$ be a finite abelian group. If $f,g$ are two functions on $G$, then the Fourier transform of $fg$ is the convolution of the Fourier transforms of $f$ and $g$.
When $G$ is non-abelian, such a formula no longer makes sense (but see this question on mathoverflow). However, we can still ask the following question:
Suppose that $f,g$ are supported on the isotypical components corresponding to the irreps $\rho_f,\rho_g$ (respectively). On what isotypical components is $fg$ supported?
Of course, the answer might depend on $f,g$. I suspect that the following holds:
The isotypical decomposition of $fg$ is supported on the support of the decomposition of the tensor product $\rho_f \otimes \rho_g$ into irreps of $G$.
This certainly holds in the abelian case. Does it hold in general?