Generalized characters of the generalized symmetric group

We prove that $(\mathbb{Z}_k \wr \mathcal{S}_n \times \mathbb{Z}_k \wr \mathcal{S}_{n-1}, \text{diag} (\mathbb{Z}_k \wr \mathcal{S}_{n-1}) )$ is a symmetric Gelfand pair, where $\mathbb{Z}_k \wr \mathcal{S}_n$ is the wreath product of the cyclic group $\mathbb{Z}_k$ with the symmetric group $\mathcal{S}_n.$ The proof is based on the study of the $\mathbb{Z}_k \wr \mathcal{S}_{n-1}$-conjugacy classes of $\mathbb{Z}_k \wr \mathcal{S}_n.$ We define the generalized characters of $\mathbb{Z}_k \wr \mathcal{S}_n$ using the zonal spherical functions of $(\mathbb{Z}_k \wr \mathcal{S}_n \times \mathbb{Z}_k \wr \mathcal{S}_{n-1}, \text{diag} (\mathbb{Z}_k \wr \mathcal{S}_{n-1}) ).$ We show that these generalized characters have properties similar to usual characters. A Murnaghan-Nakayama rule for the generalized characters of the hyperoctahedral group is presented. The generalized characters of the symmetric group were first studied by Strahov in [7].