Fourier analysis of subgroup conjugacy invariant functions on finite groups

Given a finite group and a subgroup , we develop a Fourier analysis for -conjugacy invariant functions on , without the assumption that is a multiplicity-free subgroup of . We also study the Fourier transform for functions in the center of the algebra of -conjugacy invariant functions on . 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.