Completed Iwahori-Hecke algebra for Kac-Moody groups over local fields

TL;DR

This paper constructs a completed Iwahori-Hecke algebra for split Kac-Moody groups over local fields, revealing a large center linked to Looijenga's invariant ring and contrasting with previous algebraic completions.

Contribution

It introduces a new topological completion of the Iwahori-Hecke algebra using support conditions, connecting it to geometric invariants and contrasting with algebraic approaches.

Findings

The completed algebra contains a large center isomorphic to Looijenga's invariant ring.

The construction uses Iwahori biinvariant functions with Weyl almost finite support.

Contrasts with earlier algebraic completions by Abdellatif and Hébert.

Abstract

Let $G$ be a split Kac-Moody group over a non-Archimedean local field, and let $\mathcal{H}$ be the Iwahori-Hecke algebra of $G$. In this paper, we construct a completed Iwahori-Hecke algebra $\widehat{\mathcal{H}}$ and prove that it contains a large center isomorphic to Looijenga's invariant ring. By the Kac-Moody Satake isomorphism, Looijenga's invariant ring is isomorphic to the spherical Hecke algebra. Our completion is constructed by considering Iwahori biinvariant functions on $G$ satisfying a support condition that we call Weyl almost finite support. We contrast our construction with another completion $\widetilde{\mathcal{H}}$, defined early by Abdellatif and H\'ebert, which is defined algebraically via the Bernstein-Lusztig presentation and not in terms of functions on $G$.