Gelfand pairs on the Heisenberg group and Schwartz functions

Let Hn be the (2n + 1)-dimensional Heisenberg group and K a compact group of automorphisms of Hn such that (K \ltimes Hn , K ) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K -invariant Schwartz functions on Hn 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 Hn.