Representations of the weak Weyl commutation relation

Abstract

Let $G$ be a locally compact abelian group with Pontraygin dual $\hat{G}$. Suppose $P$ is a closed subsemigroup of $G$ containing the identity element $0$. We assume that $P$ has dense interior and $P$ generates $G$. Let $U:=\{U_{\chi }{\}}_{\chi \in \hat{G}}$ be a strongly continuous group of unitaries and let $V:=\{V_a{\}}_{a\in P}$ be a strongly continuous semigroup of isometries. We call $(U,V)$ a weak Weyl pair if \[ U_χV_{a}=χ(a)V_{a}U_χ\] for every $\chi \in \hat{G}$ and for every $a\in P$. We work out the representation theory (the factorial and the irreducible representations) of the above commutation relation under the assumption that $\{V_a{V}_a^{\ast }:a\in P\}$ is a commuting family of projections. Not only does this generalise the results of [4] and [5], our proof brings out the Morita equivalence that lies behind the results. For $P=[0,\infty )\times [0,\infty )$, we demonstrate that if we drop the commutativity assumption on the range projections, then the representation theory of the weak Weyl commutation relation becomes very complicated.