*(English)*Zbl 1075.22005

An extension of the special linear mapping, called Segal-Bargmann integral transform (1993, 1997, Hall) or heat kernel transform, in the case of the non-compact spaces (Krötz, Stanton; in press) and non-compact groups is presented. In the paper the heat kernel transform ${H}_{t}$ for the $(2n+1)$-dimensional Heisenberg group $H$ and its universal complexification ${H}_{t}$ is studied in detail. As a main result, it is shown that the image of ${H}_{t}$ is a direct sum of two weighted Bergman spaces on ${H}_{C}$, in contrast to the classical case of the compact Euclidean space ${\mathbb{R}}^{n}$ (1961, Bargmann) and compact symmetric spaces (1999, Stenzel).

In this context, the corresponding partial weight functions are found, which turned out to be not nonnegative, and their oscillatory behavior is established.