Zonal polynomials for wreath products

Abstract

The pair of groups, symmetric group S 2n and hyperoctohedral group H n , form a Gelfand pair. The characteristic map is a mapping from the graded algebra generated by the zonal spherical functions of (S 2n ,H n ) into the ring of symmetric functions. The images of the zonal spherical functions under this map are called the zonal polynomials. A wreath product generalization of the Gelfand pair (S 2n ,H n ) is discussed in this paper. Then a multi-partition versions of the theory is constructed. The multi-partition version of zonal polynomials are products of zonal polynomials and Schur functions and are obtained from a characteristic map from the graded Hecke algebra into a multipartition version of the ring of symmetric functions.