I'm currently thinking to generalize a known result on abelian groups to non-abelian groups. This is the problem. Fix an abelian group $G$. We know that:
$\mathbb{E}_{\chi\in \hat{G}}\chi(g)=\left\{\begin{array}{cc}1 & g=0 \\ 0 & g\neq 0\end{array} \right.$
It's not possible to easily generalize this identity in the non-abelian case, because if we replace characters with representations, they are matrices of different dimension so the product doesn't make sense. My question is whether there exist a plausible generalization of this identity in the case where $G$ is a non-abelian group.
We also have a Fourier inversion formula in the Abelian case. Do we have a suitable inverse formula in the non-abelian case?
I would also appreciate if one could introduce me some reference about non-abelian Fourier Analysis.