We associate with any affine Kac-Moody algebra g its stable moment graph. Such a graph turns out to be the main tool in order to get a categorical version of a result by Lusztig, stating certain stability property for affine Kazhdan- Lusztig polynomials. This stabilisation phenomenon bridges the Hecke algebra to its periodic module, which -according to the Feigin-Frenkel conjecture- governs the representation theory of g at the critical level. The stable moment graph is expected to enable us to apply moment graph techniques to the study of critical representations (joint with P. Fiebig).