Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the Fourier transform by
$$ \hat{f}(\pi) = \int_{\mathrm{G}} f(x) \pi(x^{-1}) \, \mathrm{d} \mu(x),$$
where $\mu$ is a normalized Haar measure on $\mathrm{G}$, the Peter--Weyl theorem says that for $f \in L^2(\mathrm{G})$ one has the Fourier series expansion
$$ f(x) = \sum_{\pi \in \hat{\mathrm{G}}} d_\pi \operatorname{Tr}(\hat{f}(\pi) \pi(x)), $$
where $d_\pi$ is the dimension of $\pi$. Furthermore, if we define the convolution product $*$ by
$$ (f* g)(x) = \int_{\mathrm{G}} f(y) g(x y^{-1}) \, \mathrm{d} \mu(y), $$
then a version of the convolution theorem holds, namely that
$$ (\widehat{f * g})(\pi) = \hat{f}(\pi) \hat{g}(\pi). $$
**Question.** I am interested in whether there is a sense in which an ``inverse" convolution theorem might hold, something along the lines of
$$ (\hat{f} * \hat{g})(\pi) = (\widehat{fg})(\pi). $$
**Edit.** On reflection this might be asking for a somewhat unnatural identity. Perhaps a more natural question to ask would be whether there exists a convolution product on $\hat{\mathrm{G}}$ that makes a similar identity hold?

If it helps, I am more specifically interested in the case when $\mathrm{G} = \mathrm{SU}(2)$. I am fairly new to this so would appreciate any pointers to standard references if this is quite well-known!

meanon the dual side? (Maybe this is part of your question.) There's a measure on that side (the Plancherel measure), but the lack of algebraic structure means that I have a hard time guessing how convolution should behave. $\endgroup$ – LSpice Aug 6 '18 at 17:50mustbe if it is to satisfy your desired identity. $\endgroup$ – LSpice Aug 6 '18 at 17:57hypergroupstructure (so that the "product" of two points is a measure). In your setup such a structure is provided by the decomposition of tensor products of irreducible representations. $\endgroup$ – R W Aug 6 '18 at 20:05