Heisenberg Groups

In this chapter we introduce the Heisenberg group Hn and review some important aspects of its representation theory. Using Hermite and special Hermite functions we make a detailed study of the group Fourier transform on Hn. We define the Weyl correspondence of polynomials and establish a Hecke-Bochner type formula for the Weyl transform. In order to do that we develop the theory of bigraded spherical harmonics in terms of the representations of U(n). Finally, we establish several versions of Hardy’s theorem for the Fourier transform on Hn which are analogues of the results proved in Chapter 1 for the Euclidean Fourier transform.