On the symmetric Gelfand pair $(\mathcal{H}_n\times \mathcal{H}_{n-1},diag (\mathcal{H}_{n-1}))$

Abstract: We show that the $\mathcal{H}{n-1}$-conjugacy classes of $\mathcal{H}_n,$ where $\mathcal{H}_n$ is the hyperoctahedral group on $2n$ elements, are indexed by marked bipartitions of $n.$ This will lead us to prove that $(\mathcal{H}_n\times \mathcal{H}{n-1},diag (\mathcal{H}{n-1}))$ is a symmetric Gelfand pair and that the induced representation $1{diag (\mathcal{H}{n-1})}^{\mathcal{H}_n\times \mathcal{H}{n-1}}$ is multiplicity free.