Completed Iwahori-Hecke algebra for Kac-Moody groups over local fields
Auguste Hébert, Dinakar Muthiah
TL;DR
The paper addresses how to realize the spherical and central structures of a Kac-Moody group over a non-Archimedean local field by constructing a completed Iwahori-Hecke algebra. It introduces Weyl almost finite (WAF) and Looijenga frameworks to define a robust completion $\widehat{\mathcal{H}}$ whose center is the Looijenga invariant ring $\mathbb{C}[\![Y^+]\!]^{W_0}$, aligning with the spherical Hecke algebra via the Satake isomorphism. The main contributions include a BL-style presentation in the completed setting, precise control of Z- and T-basis changes through finiteness lemmas, and a detailed comparison with Abdellatif–Hébert’s tilde construction, clarifying when and why left-right formulations can fail. The results provide a solid algebraic foundation for analyzing infinite-sum phenomena in Kac-Moody contexts and set the stage for further connections to Macdonald-type formulas and spherical harmonic analysis on Kac-Moody groups. Overall, the work unifies geometry, representation theory, and automorphic perspectives by embedding infinite-center phenomena into a controlled completed algebra, with explicit basis, center, and comparison results and clear implications for the spherical side via Satake.
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 Hébert, which is defined algebraically via the Bernstein-Lusztig presentation and not in terms of functions on $G$.