A Paley-Wiener Theorem for the Spherical Transform Associated with the Generalized Gelfand Pair $(U(p,q),H_{n})$, $p+q=n$

In this work we prove a Paley-Wiener theorem for the spherical transform associated to the generalized Gelfand pair $(H_n\ltimes U(p,q),H_n)$, where $H_n$ is the $2n+1$-dimensional Heisenberg group. In particular, by using the identification of the spectrum of $(U(p,q),H_n)$ with a subset $\Sigma$ of $\mathbb{R}^2$, we prove that the restrictions of the spherical transforms of functions in $C_{0}^{\infty}(H_n)$ to appropriated subsets of $\Sigma$, can be extended to holomorphic functions on $\mathbb{C}^2$. Also, we obtain a real variable characterizations of such transforms.