Abstract
In this paper we use Block’s classification of simple modules over the first Weyl algebra to obtain a complete classification of simple weight modules, in particular, of Harish-Chandra modules, over the 1-spatial ageing algebra $\mathfrak {age(1)}$ . Most of these modules have infinite dimensional weight spaces and so far the algebra $\mathfrak {age(1)}$ is the only Lie algebra having simple weight modules with infinite dimensional weight spaces for which such a classification exists. As an application we classify all simple weight modules over the (1+1)-dimensional space-time Schrödinger algebra $\mathcal {S}$ that have a simple $\mathfrak {age(1)}$ -submodule thus constructing many new simple weight $\mathcal {S}$ -modules.
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