Lattice representations of Heisenberg groups (Q1574449)

Let \(\mathbb{F}^{(k,l)}\) be the set of all \(k\times l\) matrices with entries in \(\mathbb{F}\). For positive integers \(g\) and \(h\), the Heisenberg group \(H^{(g,h)}_{\mathbb{R}}\) is defined as follows: \[ H^{(s,h)}_{\mathbb{R}}= \{(\lambda, \mu,\kappa)\mid \lambda, \mu\in \mathbb{R}^{(h,g)},\quad \kappa\in \mathbb{R}^{(h,h)},\quad \kappa+\mu^t\lambda\text{ symmetric}\} \] with multiplication \[ (\lambda, \mu,\kappa)\circ (\lambda', \mu',\kappa')= (\lambda+ \lambda', \mu+\mu', \kappa+\kappa'+ \lambda^t\mu'- \mu^t\lambda'). \] \textit{P. Cartier} [Proc. Symp. Pure Math. 9, 361-383 (1966; Zbl 0178.28401)] stated without proof that for \(h=1\) the lattice representation of \(H^{(g,1)}_{\mathbb{R}}\) associated to the lattice \(L\) is unitarily equivalent to the direct sum of \([L^*: L]^{1/2}\) copies of the Schrödinger representation of \(H^{(g,1)}_{\mathbb{R}}\), where \(L^*\) is the dual lattice of \(L\) with respect to a certain alternating bilinear form. \textit{R. Berndt} proved this fact for the case \(h=1\) in his lecture notes [Darstellungen der Heisenberggruppe und Thetafunktionen, Vorlesungsausarbeitung (Hamburg 1988)]. The main result of this paper is a complete proof of Cartier's theorem for \(H^{(g,h)}_{\mathbb{R}}\). Namely, let \({\mathcal M}\) be a positive definite, symmetric half-integral matrix of degree \(h\) and \(L\) be a self-dual lattice in \(\mathbb{C}^{(h,g)}\). Then the lattice representation \(\pi_{\mathcal M}\) of \(H^{(g,h)}_{\mathbb{R}}\) associated with \(L\) and \({\mathcal M}\) is unitarily equivalent to the direct sum of \((\text{det }2{\mathcal M})^g\) copies of the Schrödinger representation of \(H^{(g,h)}_{\mathbb{R}}\). A relation between lattice representations and theta functions is also explained.