Wreath product generalizations of the triple (S2n,Hn,φ) and their spherical functions

The symmetric group S2n and the hyperoctahedral group Hn is a Gelfand triple for an arbitrary linear representation φ of Hn. Their φ-spherical functions can be caught as a transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet is always a Gelfand triple. Furthermore we study the relation between their spherical functions and a multi-partition version of the ring of symmetric functions.