Let G be a locally compact, second countable, Hausdorff abelian group with Pontryagin dual 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_χ: χ ∈ G^*} be a strongly continuous group of unitaries and let V:={V_a: a ∈ P} be a strongly continuous semigroup of isometries. We call (U,V) a weak Weyl pair if U_χV_a=χ(a)V_aU_χ for every χ ∈ G^* and for every a ∈ P.

We work out the representation theory (the factorial and the irreducible representations) of the above commutation relation under the assumption that {V_aV_a^*: a ∈ 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=R₊², 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.

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