Let $O(N)$ be the group of $N\times N$ real orthogonal matrices, and let $\chi_R(g)=\sum_i M^{(R)}_{ii}(g)$ denote the character of $g\in O(N)$ in the irreducible representation $R$, i.e. the trace of the matrix that represents it.
Orthogonality of matrix elements $\int M^{(R)}_{ik}(g)M^{(S)}_{lj}(g^{-1}) dg=\frac{1}{D_R}\delta_{RS}\delta_{ij}\delta_{kl},$ where $D_R$ is the dimension of $R$, implies the convolution equation $\int \chi_R(Ag)\chi_S(Bg^{-1}) dg=\delta_{RS}\frac{\chi_R(AB)}{D_R}$.
The average value of $\chi_R(Ag)$ can be written as $\int \chi_R(Ag)\chi_0(Bg^{-1}) dg$, where $\chi_0(g)=1$ is the character of the trivial representation.
I believe all of the above to be true. Yet, it is leading me to a confusion, because it means that the average value of $\chi_R(Ag)$ equals $\delta_{R0}$.
The problem is that this quantity does not depend on $A$, and this cannot be true. Characters are a basis for class functions, so this would mean that the average value of any class function of $Ag$ would be independent of $A$, which seems too strong. What about $({\rm Tr}(Ag))^2$, for instance?
What is the mistake?
In the end I am interested in class functions. Suppose a class function $F(g)$ for which I know the character expansion, $F(g)=\sum_R c_R\chi_R(g)$. Does this equation still hold if $g \notin O(N)$? I guess this would work for $GL(N,\mathbb{C})$, by some denseness argument, but it might not work for $O(N)$...