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The Saito determinant for extended affine Weyl discriminant strata

  • Andrea Brini
  • Karoline van Gemst
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Abstract

We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.

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Cite

BibTeXRIS
BibTeX
@misc{brini2026saito,
  title = {{The Saito determinant for extended affine Weyl discriminant strata}},
  author = {Andrea Brini and Karoline van Gemst},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.26817},
  eprint = {2608.26817},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26817},
}
RIS
TY  - UNPB
TI  - The Saito determinant for extended affine Weyl discriminant strata
AU  - Andrea Brini
AU  - Karoline van Gemst
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2608.26817
UR  - https://arxiv.org/abs/2608.26817
AB  - We investigate the form of the Saito metric on quotients of the reflection representation of an extended affine Weyl group, restricted to an arbitrary stratum of its discriminant. We compute the Saito determinant for all strata and Dynkin types and show that it is divisible by a product of q-analogues of restricted roots for the associated hyperplane arrangement. Our results simultaneously generalise the factorisation in the Weyl denominator formula to higher codimension strata, and provide q-analogue refinements of results of Antoniou-Feigin-Strachan for Coxeter groups.
ER  -