Geodesic
Bi-spherical functions on the symmetric group associated to hyperoctahedral subgroup
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 96-114

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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 $S_m$. 
@article{ZNSL_1997_240_a7,
     author = {V. N. Ivanov},
     title = {Bi-spherical functions on the symmetric group associated to hyperoctahedral subgroup},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {96--114},
     publisher = {mathdoc},
     volume = {240},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a7/}
}
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V. N. Ivanov. Bi-spherical functions on the symmetric group associated to hyperoctahedral subgroup. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part II, Tome 240 (1997), pp. 96-114. http://geodesic.mathdoc.fr/item/ZNSL_1997_240_a7/