Fourier analysis of subgroup conjugacy invariant functions on finite groups

Given a finite group $G$ and a subgroup $H\le G$ , we develop a Fourier analysis for $H$ -conjugacy invariant functions on $G$ , without the assumption that $H$ is a multiplicity-free subgroup of $G$ . We also study the Fourier transform for functions in the center of the algebra of $H$ -conjugacy invariant functions on $G$ . We show that a recent calculation of Cesi is indeed a Fourier transform of a function in the center of the algebra of functions on the symmetric group that are conjugacy invariant with respect to a Young subgroup.