Wreath Product Generalizations of the Triple $(S_{2n},H_n,\varphi )$ and Their Spherical Functions

The symmetric group $S_{2n}$ and the hyperoctaheadral group $H_n$ is a Gelfand triple for an arbitrary linear representation $\varphi$ of $H_n$. Their $\varphi$-spherical functions can be caught as 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 are always to be a Gelfand triple. Furthermore we study the relation between their spherical functions and multi-partition version of the ring of symmetric functions.