Hecke-Bochner identity and eigenfunctions associated to Gelfand pairs on the Heisenberg group

Let Hn be the (2n+1)-dimensional Heisenberg group, and let K be a compact subgroup of U(n), such that (K,Hn) is a Gelfand pair. Also assume that the K-action on Cn is polar. We prove a Hecke-Bochner identity associated to the Gelfand pair (K,Hn). For the special case K=U(n), this was proved by Geller [6], giving a formula for the Weyl transform of a function f of the type f=Pg, where g is a radial function, and P a bigraded solid U(n)-harmonic polynomial. Using our general Hecke-Bochner identity we also characterize (under some conditions) joint eigenfunctions of all differential operators on Hn that are invariant under the action of K and the left action of Hn.