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PreprintarXiv licencearXiv

Number of left cells in the second lowest two-sided cell

  • Xun Xie
  • Qifan Zhou
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Abstract

We prove Shi's conjecture that the number of left cells in the second lowest two-sided cell of the affine Weyl group attached to a simply connected simple complex group G is exactly |W0|/2, where W0 is the corresponding finite Weyl group. The same formula holds for the corresponding extended affine Weyl group. The key step is to show that the reductive centralizer in G of a minimal nilpotent element has identity component with simply connected derived subgroup and admits no nontrivial algebraic central extensions by 𝔾m. This gives an isomorphism of based rings between the asymptotic ring of the extended cell and a matrix ring over the representation ring of the centralizer, with matrix size equal to the Euler characteristic of the Springer fiber, namely |W0|/2.

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Cite

BibTeXRIS
BibTeX
@misc{xie2026number,
  title = {{Number of left cells in the second lowest two-sided cell}},
  author = {Xun Xie and Qifan Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35795},
  eprint = {2610.35795},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35795},
}
RIS
TY  - UNPB
TI  - Number of left cells in the second lowest two-sided cell
AU  - Xun Xie
AU  - Qifan Zhou
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35795
UR  - https://arxiv.org/abs/2610.35795
AB  - We prove Shi's conjecture that the number of left cells in the second lowest two-sided cell of the affine Weyl group attached to a simply connected simple complex group G is exactly |W_0|/2, where W_0 is the corresponding finite Weyl group. The same formula holds for the corresponding extended affine Weyl group. The key step is to show that the reductive centralizer in G of a minimal nilpotent element has identity component with simply connected derived subgroup and admits no nontrivial algebraic central extensions by 𝔾_m. This gives an isomorphism of based rings between the asymptotic ring of the extended cell and a matrix ring over the representation ring of the centralizer, with matrix size equal to the Euler characteristic of the Springer fiber, namely |W_0|/2.
ER  -