Combinatorics and spherical functions on the Heisenberg group

Abstract:

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 $\mathbb {C}[V]$ is multiplicity free. One obtains a canonical basis $\{p_\alpha \}$ for the space $\mathbb {C}[V_{\mathbb {R}}]^K$ of $K$-invariant polynomials on $V_{\mathbb {R}}$ 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 $H=V\times \mathbb {R}$.