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In this paper we use Block's classification of simple modules over the firstWeyl algebra to obtain a complete classification of simple weight modules, inparticular, of Harish-Chandra modules, over the 1-spatial ageing algebraage(1). Most of these modules have infinite dimensional weight spaces and sofar the algebra age(1) is the only Lie algebra having simple weight moduleswith 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 Schrodinger algebra S that have a simpleage(1)-submodule thus constructing many new simple weight S-modules.

Classification of simple weight modules over the 1-spatial ageing algebra