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.

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, then we compute its center and prove that it is isomorphic (via the Satake isomorphism) to the spherical Hecke algebra of G. 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 F of type 0, extending a construction that was only known, in the Kac-Moody setting, for the spherical subgroup and for the Iwahori subgroup.

Soit G un groupe de Kac-Moody déployé sur un corps local non archimédien. Nous définissons une complétion de l’algèbre d’Iwahori-Hecke de G, puis nous prouvons que son centre est isomorphe (via l’isomorphisme de Satake) à l’algèbre de Hecke sphérique de G, ce qui est analogue au cas des groupes réductifs. Notre outil principal est la masure associée à G, 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 F de type 0. Jusqu’à présent cette construction n’était connue que pour le sous-groupe sphérique et le sous-groupe d’Iwahori.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.88
Classification: 20G44, 20C08, 20E42
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

1 LAMFA – UPJV, UMR CNRS 7352 80 039 Amiens Cedex 1, France
2 Université de Lyon, UJM-Saint-Étienne CNRS, UMR CNRS 5208 F-42023, Saint-Étienne, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     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},
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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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