Spherical functions on Sp2 as a spherical homogeneous Sp2×(Sp1)2-space

We investigate spherical functions on Sp2 as a spherical homogeneous G=Sp2×(Sp1)2-space over a p-adic field k, which form a 4-dimensional vector space for each eigenvalue given by Satake parameter. Explicit expressions of spherical functions and Cartan decomposition of Sp2 are given. Using spherical transform, we determine Hecke module structure of the Schwartz–Bruhat space , which is free of rank 4.