Appears in collection : Reductive groups and automorphic forms. Dedicated to the French school of automorphic forms and in memory of Roger Godement.

Let F be a non-archimedean local field and G be the group of F-rational points of a connected reductive group defined over F. The study of (complex smooth) representations of G imply various tools coming from different nature. These include in particular induction functors, Hecke algebras (seen as convolution algebras or as intertwinning algebras) and Bruhat-Tits buildings. When seeing Kac-Moody groups as a natural generalization of reductive groups, one can wonder how far the setting developed for reductive groups can be extended to the Kac-Moody case. Thanks to Rousseau, Gaussent-Rousseau and Bardy- Panse-Gaussent Rousseau, there is a suitable generalization of Bruhat-Tits buildings (called hovels, or masures) as well as handful definitions of spherical and Iwahori-Hecke algebras. Nevertheless, these algebras are not really fully satisfying as they do not, for instance, satisfy the analogue of Bernstein’s theorem in this setting. Another frustrating lack was that there was so far no natural construction attaching a Hecke algebra to a suitable analogue of open compact subgroups. In this talk, we discuss some results, obtained in collaboration with Auguste Hébert, addressing these questions for split Kac- Moody groups. In particular, we explain why the Iwahori-Hecke algebra as defined by Rousseau and his collaborators is not the right generalization of the usual Iwahori-Hecke algebra, as its center is « too small », then we define a suitable generalization (using a sort of completion process) that satisfies a Bernstein-like theorem. If enough time is left, we will also explain how to attach a suitable Hecke algebra to each type 0 spherical facet of the hovel that gives back the well-known Hecke algebras in the reductive case.