The Spherical Transform of a Schwartz Function on the Heisenberg Group

Suppose that K ⊂ U(n) is a compact Lie group acting on the (2n+1)dimensional Heisenberg group Hn. We say that (K,Hn) is a Gelfand pair if the convolution algebra LK(Hn) of integrable K-invariant functions on Hn is commutative. In this case, the Gelfand space ∆(K,Hn) is equipped with the GodementPlancherel measure, and the spherical transform ∧ : LK(Hn)→ L(∆(K,Hn)) is an isometry. The main result in this paper provides a complete characterization of the set SK(Hn) = {f̂ | f ∈ SK(Hn)} of spherical transforms of K-invariant Schwartz functions on Hn. We show that a function F on ∆(K,Hn) belongs to SK(Hn) if and only if the functions obtained from F via application of certain derivatives and difference operators satisfy decay conditions. We also consider spherical series expansions for K-invariant Schwartz functions on Hn modulo its center.