So I'm interested in Fourier transforms over G where G is $SU(2)$. That is to say things that look like this $$\widehat{f}\left(\rho_i\right)=\int_{u \in G} f(u) \rho_i(u) d\mu, \quad i=0,1,2, \ldots$$ where $\rho_i$ are the irreducible representations of SU(2).
Now in the case where I have $G\times G$ I'm aware that the irreps for this are $\rho_i\times\rho_j$ and my Fourier transform is basically the same thing with more indices about. However I'm interested in the situation where I have two copies of the group but the group action I'm interested in is $g\vartriangleright (G\times G) = (g\vartriangleright G) \times (g\vartriangleright G)$ for $g\in G$. Given this is essentially a restriction on the product groups action I'm unclear on how this affects the irreps and thus the resultant form of the Fourier transform.
Any help identifying form of the Fourier transform in this situation would be appreciated in the case where you have multiple copies of the group but the same action acting on each of them.