Fourier analysis of subgroup conjugacy invariant functions on finite groups (Q1955631)

The article deals with a variant of the Fourier transform on the convolution algebra \(\mathcal{C}(G,H)\) of complex-valued functions on a finite group \(G\), invariant under the conjugation action of its subgroup \(H\). An orthogonal basis of \(\mathcal{C}(G,H)\) is found and used to establish an algebra isomorphism -- a ``Fourier transform'' -- between \(\mathcal{C}(G,H)\) and a certain direct sum of matrix algebras. It is accompanied by a corresponding inversion formula and Plancherel's formula. The theory is then specialized to the center of \(\mathcal{C}(G,H)\), yielding a spherical Fourier transform. The authors also provide an example involving the symmetric group, and propose, under some additional assumptions, a canonical choice of the basis of \(\mathcal{C}(G,H)\).