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The heat kernel transform for the Heisenberg group. (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 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.

22E30Analysis on real and complex Lie groups
22F05General theory of group and pseudogroup actions