We study the heat kernel transform on a nilmanifold $M$ of the Heisenberg group. We show that the image of $L^2(M)$ under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces.
"Heat kernel transform for nilmanifolds associated to the Heisenberg group." Rev. Mat. Iberoamericana 24 (1) 243 - 266, April, 2008.