On a finite group $G$, we have the second orthogonality relation $$\sum_{\chi\text{ irred.}} \chi(g)\overline{\chi(h)}=\begin{cases}|C_G(g)|&g=h\\0&\text{otherwise.}\end{cases}$$ for $g,h\in G$, where $|C_G(g)|$ is the size of the conjugacy class of $G$.
If $G$ is a compact group with Haar measure $\mu$, the first orthogonality relation generalizes: $$\int_G \chi_1(g)\overline{\chi_2(g)} d\mu(g)=\begin{cases}|G|&\chi_1\cong\chi_2\\0&\text{otherwise.}\end{cases}$$ for $\chi_1,\chi_2$ irreducible characters of $G$. On the other hand, it is not clear what happens about the second orthogonality relation. Does it generalize as well ? Can we compute the integral $$\int_{\chi\in\widehat G\text{ irred}} \chi(g)\overline{\chi(h)} dg$$ over the dual group $\widehat G$ (with respect to its Haar measure as a locally compact group) ? In particular, it is not clear what $|C_G(g)|$ would become.