Gelfand pairs on the Heisenberg group and Schwartz functions

Abstract

Let H n be the ( 2 n + 1 ) -dimensional Heisenberg group and K a compact group of automorphisms of H n such that ( K ⋉ 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 .