Heat kernel transform for nilmanifolds associated to the Heisenberg   group
  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. 
B. Kroetz, S. Thangavelu and Y. Xu