Ramla Abdellatif - Iwahori - Hecke algebras and hovels for split Kac - Moody groups
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2018-05-20

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.