Lattice representations of Heisenberg groups

In this article, we prove that a lattice representation of the Heisenberg group $H_{\mathbb R}^{(g,h)}$ associated to a lattice $L$ and a positive definite symmetric half-integral matrix M of degree $h$ is unitarily equivalent to the direct sum of $(det 2M)^g$ copies of the Schroedinger representation of $H_{\mathbb R}^{(g,h)}$.