I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks.


Let $G$ be a finite non-abelian group, $\hat{?}:L\_2(G)\to \hat L\_2(\hat G)$ be its Fourier transformation and $\check{?}:\hat L\_2(\hat G)\to L\_2(G)$ be the inverse Fourier transformation. The space $L\_2(G)$ carries the standard norm induced by the inner product $\langle x,y\rangle=\sum\_{g\in G}x(g)\cdot\overline{y(g)}$. The norm on $\hat L\_2(\hat G)$ is normalized so that the Fourier transformations $\hat?$ is an isometry.


Let $P:\hat L\_2(\hat G)\to \hat L\_2(\hat G)$ be the function assigning to a sequence of matrices $(M\_\alpha)\_{\alpha\in\hat G}$ in $\hat L\_2(\hat G)$ the sequence $(\frac 1{\dim(\alpha)}Tr(M\_\alpha)\cdot\mathrm{Id}\_{\alpha})\_{\alpha\in\hat G}$ of normalized traces of those matrices multiplied by the identity matrices. So, $P$ is a linear projection of $\hat L\_2(\hat G)$ onto its $|\hat G|$-dimensional subspace.



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> **Question 1.** What is the norm of a sequence $(M\_\alpha)\_{\alpha\in \hat G}\in P(\hat L\_2(\hat G))$ in the Hilbert space $\hat L\_2(\hat G)$? Is it equal to $(\frac1{|\hat G|}\sum\_{\alpha\in\hat G}|\frac1{\dim(\alpha)}Tr(M\_\alpha)|^2)^{1/2}$?
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I am interested in describing the projector $A=\check{?}\circ P\circ \hat{?}:L\_2(G)\to L\_2(G)$. It should assign to each function $f\in L\_2(G)$ some class function on $G$.



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> **Question 2.** Is $A$ the averaging of $f$ over conjugacy classes? If yes, where can I find a reference to this fact?
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What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?
Exercise 6.2 in Serre's book on representation theory of finite groups can be interpreted as saying that for functions $f,h$ on $G$ one has


$$\sum\_{g\in G}\overline{f(g)}{h(g)} = \dfrac{1}{|G|}\sum\_{\rho \in \hat G}\dim\rho\cdot \mathrm{Tr}(\hat f(\rho)^\*\hat h(\rho))$$ and so the correct formula for the norm should be $$\|(M\_{\rho})\_{\rho\in \hat G}\|^2 = \dfrac{1}{|G|}\sum\_{\rho \in \hat G}\dim\rho\cdot \mathrm{Tr}(M^\*\_{\rho}M\_\rho))$$ where $\*$ denotes the Hermitian adjoint. See Teras and CECCHERINI-SILBERSTEIN et al for this version. Serre uses a slightly different bilnear form that agrees with this one on characters.


For the second question you are correct I will use the second orthogonality relations which says that $$\sum\_{\rho\in \hat G}\overline{\mathrm{Tr}(\rho(g))}\mathrm{Tr}(\rho(h))$$ is $0$ unless $g,h$ are conjugate, in which case it is $\dfrac{|G|}{|\mathrm{Cl}(g)|}$ where $\mathrm{Cl}(g)$ is the conjugacy class of $g$.


Now $P\hat f$ is the sequence whose $\rho$-component is $$\sum\_{h\in G}\frac{f(h)}{\dim \rho}\mathrm{Tr}(\rho(h))I$$.


The Fourier inversion theorem says that if $k$ is the function with $\hat k=P\hat f$, then $$k(g) =\frac{1}{|G|}\sum\_{\rho\in \hat G}\dim \rho\cdot \mathrm{Tr}(\rho(g)^\*\hat k(\rho))$$
$$=\frac{1}{|G|}\sum\_{h\in G}f(h)\sum\_{\rho\in \hat G}\overline{\mathrm{Tr}(\rho(g))}\mathrm{Tr}(\rho(h))=\sum\_{h\sim g}\dfrac{f(h)}{|\mathrm{Cl}(g)|}$$ where $\sim$ is conjugacy by the second orthogonality relations, which is the average value of $f$ on the conjugacy class of $g$.


