Left translation of a group function ($(L_{a}f)(x)=f(a^{-1}x)$)

For a scalar function on a group, left translation by $a$ is $(L_{a}f)(x)=f(a^{-1}x)$. With the positive-representation convention for Fourier analysis on a finite group, $\widehat{L_{a}f}(\rho )=\rho (a)\hat{f}(\rho )$.