Let be a split Kac-Moody group over a non-Archimedean local field. We define a completion of the Iwahori-Hecke algebra of , then we compute its center and prove that it is isomorphic (via the Satake isomorphism) to the spherical Hecke algebra of . This is thus similar to the situation for reductive groups. Our main tool is the masure associated to this setting, which plays here the same role as Bruhat-Tits buildings do for reductive groups. In a second part, we associate a Hecke algebra to each spherical face of type , extending a construction that was only known, in the Kac-Moody setting, for the spherical subgroup and for the Iwahori subgroup.
Soit un groupe de Kac-Moody déployé sur un corps local non archimédien. Nous définissons une complétion de l’algèbre d’Iwahori-Hecke de , puis nous prouvons que son centre est isomorphe (via l’isomorphisme de Satake) à l’algèbre de Hecke sphérique de , ce qui est analogue au cas des groupes réductifs. Notre outil principal est la masure associée à , qui joue ici un rôle similaire à celui de l’immeuble de Bruhat-Tits dans le cas réductif. Dans une seconde partie, nous associons une algèbre de Hecke à chaque face sphérique de type . Jusqu’à présent cette construction n’était connue que pour le sous-groupe sphérique et le sous-groupe d’Iwahori.
Accepted:
Published online:
DOI: 10.5802/jep.88
Keywords: Kac-Moody groups, Hecke algebras, masure, local fields, Iwahori-Hecke algebras
Mot clés : Groupes de Kac-Moody, algèbres de Hecke, masure, corps locaux, algèbres d’Iwahori-Hecke
Ramla Abdellatif 1; Auguste Hébert 2
@article{JEP_2019__6__79_0, author = {Ramla Abdellatif and Auguste H\'ebert}, title = {Completed {Iwahori-Hecke} algebras and parahoric {Hecke} algebras for {Kac-Moody~groups} over local fields}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {79--118}, publisher = {\'Ecole polytechnique}, volume = {6}, year = {2019}, doi = {10.5802/jep.88}, mrnumber = {3915193}, zbl = {07033366}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.88/} }
TY - JOUR AU - Ramla Abdellatif AU - Auguste Hébert TI - Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields JO - Journal de l’École polytechnique — Mathématiques PY - 2019 SP - 79 EP - 118 VL - 6 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.88/ DO - 10.5802/jep.88 LA - en ID - JEP_2019__6__79_0 ER -
%0 Journal Article %A Ramla Abdellatif %A Auguste Hébert %T Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields %J Journal de l’École polytechnique — Mathématiques %D 2019 %P 79-118 %V 6 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.88/ %R 10.5802/jep.88 %G en %F JEP_2019__6__79_0
Ramla Abdellatif; Auguste Hébert. Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields. Journal de l’École polytechnique — Mathématiques, Volume 6 (2019), pp. 79-118. doi : 10.5802/jep.88. https://jep.centre-mersenne.org/articles/10.5802/jep.88/
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