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 ${\mathcal{S}}_n\mathrm{.}$ The proof is based on the study of the ${\mathbb{Z}}_k\wr {\mathcal{S}}_{n-1}$-conjugacy classes of ${\mathbb{Z}}_k\wr {\mathcal{S}}_n\mathrm{.}$ We define the generalized characters of ${\mathbb{Z}}_k\wr {\mathcal{S}}_n$ using the zonal spherical functions of $({\mathbb{Z}}_k\wr {\mathcal{S}}_n\times {\mathbb{Z}}_k\wr {\mathcal{S}}_{n-1},\text{diag}({\mathbb{Z}}_k\wr {\mathcal{S}}_{n-1}))\mathrm{.}$ 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].