Combinatorics and spherical functions on the Heisenberg group

Let $V$ be a finite dimensional Hermitian vector space and $K$ be a compact Lie subgroup of $U(V)$ for which the representation of $K$ on $\hbox{${\mathbb C}$}[V]$ is multiplicity free. One obtains a canonical basis $\{p_\alpha\}$ for the space $\hbox{${\mathbb C}$}[\hbox{$V_{\mathbb R}$}]^K$ of $K$-invariant polynomials on $\hbox{$V_{\mathbb R}$}$ and also a basis $\{q_\alpha\}$ via orthogonalization of the $p_\alpha$'s. The polynomial $p_\alpha$ yields the homogeneous component of highest degree in $q_\alpha$. The coefficients that express the $q_\alpha$'s in terms of the $p_\beta$'s are the generalized binomial coefficients of Z. Yan. We present some new combinatorial identities that involve these coefficients. These have applications to analysis on Heisenberg groups. Indeed, the polynomials $q_\alpha$ completely determine the generic bounded spherical functions for a Gelfand pair obtained from the action of $K$ on a Heisenberg group $H=V\times \hbox{${\mathbb R}$}$.