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.