Quantum roots for Kac-Moody root systems and finiteness properties of the Kac-Moody affine Bruhat order

TL;DR

This paper explores the structure of Kac-Moody groups over local fields, establishing finiteness properties of the affine Bruhat order and classifying quantum roots, thereby connecting quantum root systems with order finiteness.

Contribution

It proves the finiteness of intervals in the affine Bruhat order for Kac-Moody groups and classifies quantum roots, extending previous results to a broader setting.

Findings

Intervals in the affine Bruhat order are finite.

Finiteness of quantum roots for arbitrary Kac-Moody root systems.

Classification of quantum roots.

Abstract

Let $G$ be a split Kac-Moody group over a local field. In their study of the Iwahori-Hecke algebra of $G$, A.Braverman, D. Kazhdan and M. Patnaik defined a partial order - called the affine Bruhat order - on the extended affine Weyl semi-group $W^+$ of $G$. In this paper, we study finiteness questions for covers and co-covers of $W^+$, generalizing results of A. Welch. In particular we prove that the intervals for this order are finite. Our results rely on the finiteness of the set of quantum roots of arbitrary Kac-Moody root systems, which we prove. We also obtain a classification of quantum roots.