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

Let $\mathbb{H}^{n}$ be the $(2n+1)$-dimensional Heisenberg group, and let $K$ be a compact subgroup of U(n), such that $(K,\mathbb{H}^{n})$ is a Gelfand pair. Also assume that the $K$-action on $\mathbb{C}^n$ is polar. We prove a Hecke-Bochner identity associated to the Gelfand pair $(K,\mathbb{H}^{n})$. For the special case $K=U(n)$, this was proved by Geller, 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 $\mathbb{H}^{n}$ that are invariant under the action of $K$ and the left action of $\mathbb{H}^{n}$.