[Racines quantiques pour les systèmes de racines de Kac-Moody et propriétés de finitude de l’ordre de Bruhat affine de Kac-Moody]
The notion of quantum root comes up in the study of the quantum cohomology of generalized flag varieties, quantum Bruhat graphs or in the study of the affinized Bruhat order on an affinized Weyl semi-group. In this paper, we provide a complete classification of quantum roots associated to general Kac–Moody root systems. In particular we prove that the set of quantum roots of a given root system is always finite. We then deduce several finiteness properties of the affinized Bruhat order, generalizing results of A. Welch. In particular, we prove that its intervals are finite. This affinized Bruhat order was introduced by A. Braverman, D. Kazhdan and M. Patnaik in their study of Kac-Moody groups over discretely valued fields, and of the associated Iwahori-Hecke algebras.
Au sein d’un système de racines, les racines quantiques jouent un rôle particulier. Dans le cadre réductif elles aparaissent notamment dans l’étude de la cohomologie quantique des variétés de drapeaux, et la structure combinatoire des groupes de Weyl affines. Nous classifions les racines quantiques des systèmes de racines de Kac-Moody de type général, nous montrons en particulier que chaque système de racines n’en contient qu’un nombre fini. Nous en déduisons plusieurs propriétés de finitude pour l’ordre de Bruhat affinisé sur le semi-groupe de Weyl affinisé, généralisant des résultats de A. Welch. Nous montrons notamment que les intervalles sont finis. Cet ordre de Bruhat affinisé avait été introduit par A. Braverman, D. Kazhdan et M. Patnaik pour étudier les groupes de Kac-Moody sur les corps discrètement valués, et les algèbres de Iwahori-Hecke associées.
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Keywords: Kac–Moody groups, root systems, Bruhat order, Hecke algebras
Mots-clés : Groupes de Kac-Moody, systèmes de racines, ordre de Bruhat, algèbres de Hecke
Auguste Hébert 1 ; Paul Philippe 2
CC-BY 4.0
Auguste Hébert; Paul Philippe. Quantum roots for Kac–Moody root systems and finiteness properties of the Kac–Moody affinized Bruhat order. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 1637-1682. doi: 10.5802/jep.357
@article{JEP_2026__13__1637_0,
author = {Auguste H\'ebert and Paul Philippe},
title = {Quantum roots for {Kac{\textendash}Moody} root systems and finiteness properties of {the~Kac{\textendash}Moody~affinized} {Bruhat} order},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1637--1682},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.357},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.357/}
}
TY - JOUR AU - Auguste Hébert AU - Paul Philippe TI - Quantum roots for Kac–Moody root systems and finiteness properties of the Kac–Moody affinized Bruhat order JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 1637 EP - 1682 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.357/ DO - 10.5802/jep.357 LA - en ID - JEP_2026__13__1637_0 ER -
%0 Journal Article %A Auguste Hébert %A Paul Philippe %T Quantum roots for Kac–Moody root systems and finiteness properties of the Kac–Moody affinized Bruhat order %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 1637-1682 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.357/ %R 10.5802/jep.357 %G en %F JEP_2026__13__1637_0
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