Grading of affinized Weyl semigroups of Kac–Moody type

Paul Philippe

Vol. 332 (2024), No. 2, 273–320

DOI: 10.2140/pjm.2024.332.273

Abstract

For any Kac–Moody root datum $\mathcal{𝒟}$, D. Muthiah and D. Orr have defined a partial order on the semidirect product $W_{+}^a$ of the integral Tits cone with the vectorial Weyl group of $\mathcal{𝒟}$, and a compatible length function. We classify covers for this order and show that this length function defines a $\mathbb{Z}$-grading of $W_{+}^a$, generalizing the case of affine ADE root systems and giving a positive answer to a conjecture of Muthiah and Orr.

Keywords

Kac–Moody groups, root systems, Coxeter groups, affine Weyl groups, affinized Bruhat order

Mathematical Subject Classification

Primary: 20F55, 20G44

Secondary: 20C08, 22E67

Milestones

Received: 12 June 2023

Revised: 4 October 2024

Accepted: 2 November 2024

Published: 6 December 2024