Bispherical functions on the symmetric group associated with the hyperoctahedral subgroup
Given the partition \(\mu\vdash 2n\), let \(\pi^\mu\) be the irreducible character of \(S_{2n}\) associated with \(\mu\), and let \(V^\mu\) be the space of the representation. Let \(B_n\) be the hyperoctahedral subgroup of \(S_{2n}\). We choose a \(B_n\)-invariant (\(\pi^\mu(x)\eta=\eta\), \(\forall x\in B_n\)) element \(\eta\in V^\mu\) and a \(B_n\)-anti-invariant (\(\pi^\mu(x)\xi=\text{sgn}(x)\xi\), \(\forall x\in B_n\)) element \(\xi\in V^\mu\), and define the bispherical function \(x\mapsto(\pi^\mu(x)\xi,\eta)\). The partition \(\mu\) must have rows and columns of even length, and thus \(n\) is even. We write \(\mu=(2\lambda_1,2\lambda_1,2\lambda_2,2\lambda_2,\ldots)\) for some \(\lambda\vdash n/2\). The author proves that the bispherical function can be expressed in terms of the irreducible character of \(S_{n/2}\) that corresponds to \(\lambda\).
scientific article; zbMATH DE number 1408556