Let \(\mathbb{H}_n = \mathbb{C}^n \times \mathbb{R}\) be a \((2n+1)\)-dimensional Heisenberg group equipped with the group law \[ (z,t)(w,s)=(z+w, t+s-\frac{1}{2}\mathrm{Im }B(z,w)), \] where \[ B(z,w)=\sum_{j=1}^{p}z_{j}\bar{w}_{j}-\sum_{j=p+1}^{n}z_{j}\bar{w}_{j},\quad p+q=n \] and \[ U(p,q)=\{g\in Gl(n,\mathbb{C}): B(gz, gw)=B(z,w),\quad \forall (z,w)\in \mathbb{C}^{2n}\} \] which acts by automorphisms on \(\mathbb{H}_{n}\) via \[ g(z, t)=(gz,t)\forall (z, t) \in \mathbb{H}_{n}. \]
By using the identification of the spectrum of the (generalized) Gelfand pair \((U(p,q), \mathbb{H}_{n})\) with a subset \(\Sigma=\{(\lambda, (2k+p-q)|\lambda|):\lambda \neq 0, k \in \mathbb{Z} \}\cup \{(0, \sigma): \sigma \in \mathbb{R} \}\) of \(\mathbb{R}^2\), the spherical transform associated with the (generalized) Gelfand pair \(( U(p,q), \mathbb{H}_{n})\) is introduced.
The author proves a Paley-Wiener theorem for this spherical transform. Furthermore, the author also proves that the restrictions of the spherical trnsform of functions in \(C_{0}^{\infty}(\mathbb{H}_{n})\) to appropriated subsets of \(\Sigma\), can be extended to holomorphic functions on \(\mathbb{C}^2\).