Gutzmer's formula and poisson integrals on the Heisenberg group

Abstract

In 1978 M. Lassalle obtained an analogue of the Laurent series for holomorphic functions on the complexification of a compact symmetric space and proved a Plancherel type formula for such functions. In 2002 J. Faraut established such a formula, which he calls Gutzmer's formula, for all noncompact Riemannian symmetric spaces. This was immediately put into use by B. Krotz, G. Olafsson and R. Stanton to characterise the image of the heat kernel transform. In this article we prove an analogue of Gutzmer's formula for the Heisenberg motion group and use it to characterise Poisson integrals associated to the sublaplacian. We also use the Gutzmer's formula to study twisted Bergman spaces.