Representations of the weak Weyl commutation relation

TL;DR

This paper studies the representation theory of a generalized weak Weyl commutation relation involving groups and semigroups, extending previous results and highlighting the role of Morita equivalence, with complexity arising when certain projections do not commute.

Contribution

It generalizes existing results on weak Weyl relations by analyzing factorial and irreducible representations under a commutativity assumption, revealing Morita equivalence structures.

Findings

Representation theory is characterized under the assumption of commuting projections.

Dropping the commutativity assumption leads to complex representation structures.

The work extends prior results and connects to Morita equivalence concepts.

Abstract

Let $G$ be a locally compact abelian group with Pontraygin dual $\widehat{G}$. 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 $U:=\{U_{\chi}\}_{\chi \in \widehat{G}}$ 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 \[ U_{\chi}V_{a}=\chi(a)V_{a}U_{\chi}\] for every $\chi \in \widehat{G}$ 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 $\{V_{a}V_{a}^{*}:a \in P\}$ 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.