Abstract: Cellular structure may be used to study the representation theory of non-semisimple algebras which are deformations of semisimple ones. I shall show how this technique may be applied to affine Temperley-Lieb algebras, and how this leads to the determination of certain decomposition numbers for standard modules of affine Hecke algebras. These numbers have alternative descriptions in terms of perverse sheaves on certain algebraic varieties.