Completed Iwahori-Hecke algebras and parahorical Hecke algebras for Kac-Moody groups over local fields

Ramla Abdellatif (1, 2) , Auguste Hébert (3, 4, 5)

1 UPJV - Université de Picardie Jules Verne
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 UPJV
3 UJM - Université Jean Monnet - Saint-Étienne
4 ICJ - Institut Camille Jordan
5 Université de Lyon

Let G be a split Kac-Moody group over a non-archimedean local field. We define a completion of the Iwahori-Hecke algebra of G. We determine its center and prove that it is isomorphic to the spherical Hecke algebra of G using the Satake isomorphism. This is thus similar to the situation of reductive groups. Our main tool is the masure I associated to this setting, which is the analogue of the Bruhat-Tits building for reductive groups. Then, for each special and spherical facet F , we associate a Hecke algebra. In the Kac-Moody setting, this construction was known only for the spherical subgroup and for the Iwahori subgroup.

Keywords: Kac-Moody groups; Hovel; Masure; Hecke algebras; Local fields; Iwahori-Hecke algebras

Domains: Representation Theory [math.RT]

Ramla Abdellatif, Auguste Hébert. Completed Iwahori-Hecke algebras and parahorical Hecke algebras for Kac-Moody groups over local fields. 2017. ⟨hal-01536067v2⟩