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

  • Auguste Hebert
  • Paul Philippe
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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.

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Cite

BibTeXRIS
BibTeX
@misc{hebert2026quantum,
  title = {{Quantum roots for Kac-Moody root systems and finiteness properties of the Kac-Moody affine Bruhat order}},
  author = {Auguste Hebert and Paul Philippe},
  howpublished = {HAL},
  year = {2026},
  month = oct,
  eprint = {2405.12559},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-04579365v2},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quantum roots for Kac-Moody root systems and finiteness properties of the Kac-Moody affine Bruhat order
AU  - Auguste Hebert
AU  - Paul Philippe
PY  - 2026
DA  - 2026/10/08
PB  - HAL
UR  - https://hal.science/hal-04579365v2
AB  - 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.
ER  -