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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.

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  • $\begingroup$ One place you could start is Folland's A Course in Abstract Harmonic Analysis $\endgroup$
    – Yemon Choi
    Jul 26, 2014 at 1:28
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    $\begingroup$ As for your specific question: if $G$ is a finite group then your identity can be interpreted as a special case of: "columns of the character table are orthogonal" -- see e.g. the book of James and Liebeck for an introduction to some basic character theory on finite groups $\endgroup$
    – Yemon Choi
    Jul 26, 2014 at 1:30
  • $\begingroup$ Thanks! Do you know if there is also an inverse formula in the non-abelian case? $\endgroup$ Jul 26, 2014 at 2:53
  • $\begingroup$ I think the correct notion of "inverse formula" is that the character table, once suitably normalized is a unitary matrix, hence its inverse is its conjugate transpose. So this suggests if you want the inverse formula your Fourier transform must be between the space of conjugacy classes and the space of irreps. $\endgroup$
    – Will Sawin
    Jul 26, 2014 at 3:33
  • $\begingroup$ The book by Ruzhansky and Turunen studies the Fourier series on all compact Lie groups: www2.imperial.ac.uk/~ruzh/Book-Ruzhansky-Turunen-about.htm. The abelian case (tori) is quite classical. The book is available online. See section 7.6 for Fourier analysis. There are inversion formulas and Parseval identities and all that. (This is not helpful for finite groups, but might be a generalization in the direction you need.) $\endgroup$ Jul 26, 2014 at 5:35

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