Paley–Wiener theorems for the U(n)-spherical transform on the Heisenberg group

We prove several Paley–Wiener-type theorems related to the spherical transform on the Gelfand pair $$\big ({H_n}\rtimes {\text {U}(n)},{\text {U}(n)}\big )$$(Hn⋊U(n),U(n)), where $${H_n}$$Hn is the $$2n+1$$2n+1-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in $$\mathbb {R}^2$$R2, we prove that spherical transforms of $${\text {U}(n)}$$U(n)-invariant functions and distributions with compact support in $${H_n}$$Hn admit unique entire extensions to $$\mathbb {C}^2$$C2, and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations.