The Spherical Transform of a Schwartz Function on the Heisenberg Group

Suppose thatK⊂U(n) is a compact Lie group acting on the (2n+1)-dimensional Heisenberg groupHn. We say that (K, Hn) is a Gelfand pair if the convolution algebraL1K(Hn) of integrableK-invariant functions onHnis commutative. In this case, the Gelfand spaceΔ:(K, Hn) is equipped with the Godement–Plancherel measure, and the spherical transform∧:L2K(Hn)→L2(Δ(K, Hn)) is an isometry. The main result in this paper provides a complete characterization of the set SK(Hn)∧={f | f∈SK(Hn)} of spherical transforms ofK-invariant Schwartz functions onHn. We show that a functionFonΔ(K, Hn) belongs to SK(Hn)∧if and only if the functions obtained fromFvia application of certain derivatives and difference operators satisfy decay conditions. We also consider spherical series expansions forK-invariant Schwartz functions onHnmodulo its center.