Bi-spherical functions on the symmetric group associated to hyperoctahedral subgroup

We consider an arbitrary irreducible representation of a symmetric group $S_{4m}$ that has both a $B_{2m}$-invariant vector and a $B_{2m}$-antiinvariant vector, where $B_{2m}$ is a hyperoctahedral subgroup of $S_{4m}$. The main result is an expression for a matrix element corresponding to these two vectors in terms of an irreducible character of the symmetric group $S_m$.