Harmonic Analysis On Heisenberg Nilmanifolds

In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold Gamma<span class='textbac'>k</span>slash}H-n. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of L-2 (Gamma<span class='textbac'>k</span>slash}H-n) into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and characterise the image of L-2 (Gamma<span class='textbac'>k</span>slashH-n) in terms of twisted Bergman and Hermite Bergman spaces.