Abstract. We define the Iwahori-Hecke algebra I H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The fixer KI of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define I H as the algebra of some KI −bi-invariant functions on G with support consisting of a finite union of double classes. As two chambers in the hovel are not always in a same apartment, this support has to be in some large subsemigroup G + of G. In the split case, we prove that the structure constants of I H are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We give a presentation of this algebra I H, similar to the Bernstein-Lusztig presentation in the reductive case, and embed it in a greater algebra

We give a more building-oriented and somewhat simpler proof of the « saturation » theorem of Kapovich and Millson for any complex semisimple group. The main difference with their approach lies in the combinatorial part of the proof. We state a theorem of folding/unfolding triangles in the affine building, only in combinatorial terms. For the analytical part, we gather materials that appear in distinct papers of Kapovich, Leeb and Millson to complete the proof. * ) Le troisième auteur remercie le projet ANR-09-JCJC-0102-01 pour son soutien financier.

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