Quantum roots for Kac–Moody root systems and finiteness properties of the Kac–Moody affinized Bruhat order

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.