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

## TL;DR

This paper extends the theory of Hecke algebras to split Kac-Moody groups over local fields by defining a completion, analyzing its center, and establishing an isomorphism with the spherical Hecke algebra, using the masure structure.

## Contribution

It introduces a completion of the Iwahori-Hecke algebra for Kac-Moody groups and proves its center is isomorphic to the spherical Hecke algebra, generalizing known results for reductive groups.

## Key findings

- The completed Iwahori-Hecke algebra's center is isomorphic to the spherical Hecke algebra.
- The masure I serves as an analogue of the Bruhat-Tits building for Kac-Moody groups.
- Construction of Hecke algebras for special and spherical facets in the Kac-Moody setting.

## Abstract

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.