Gelfand pairs on the Heisenberg group and Schwartz functions

Let H_n be the (2n + 1)-dimensional Heisenberg group and K a compact group of automorphisms of H_n such that (K \ltimes H_n,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on H_n and the space of Schwartz function on a closed subset of R^s homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on H_n.