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Lusztig's conjectures for BnB_n, B~n\widetilde B_n, and C~n\widetilde C_n

Primarily AI-generated textHuman understanding: some partsmath.RT — Representation Theorymath.GR — Group Theory

Contributed by Qiyue Tang

Version 2 / Nov 10, 2026 / CC BY 4.0

Abstract

We prove Lusztig's conjectures P1\mathrm{P1}–P15\mathrm{P15} for every positive integer-valued weight function on the finite Coxeter group of type BnB_n and the nonextended affine Weyl groups of types B~n\widetilde B_n and C~n\widetilde C_n.

Author comments: v2:（2026/11/12）：Version 1 underwent a Rethlas-OSV review, including a supplementary round that completed coverage of all 36 review scopes. The final report recommended revision and classified all four findings as minor. It gave a counterexample to the assertion that the largest surviving cuspidal odd part always equals \(4u-2\). The report recommended using \(4u-2\) as an upper bound and establishing equality at the later step where it is needed. The other three findings concerned the attribution of the general parameter range in the introduction, the citation supporting the sharp asymptotic threshold, and the explanation of the parameter normalization used to obtain the residual centers. The report proposed a specific correction for each finding and identified no issue requiring a change to the main conclusions. The revised manuscript underwent a second Rethlas-OSV review, which identified only two issues: an incorrect citation in the introduction and an inaccurate article title in the bibliography. I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.

Provenance statement

The research reported in this paper was developed during an eight-hour research run using GPT-6-Pro. GPT-6-Pro and codex was subsequently used to audit the mathematical arguments and to revise and polish the manuscript.

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Version history

  1. v1 · 78 pagesSubmitted by Qiyue TangInitial depositSupersededNov 08, 2026
  2. v2 · 78 pagesSubmitted by Qiyue Tangv2:（2026/11/12）：Version 1 underwent a Rethlas-OSV review, including a supplementary round that completed coverage of all 36 review scopes. The final report recommended revision and classified all four findings as minor. It gave a counterexample to the assertion that the largest surviving cuspidal odd part always equals \(4u-2\). The report recommended using \(4u-2\) as an upper bound and establishing equality at the later step where it is needed. The other three findings concerned the attribution of the general parameter range in the introduction, the citation supporting the sharp asymptotic threshold, and the explanation of the parameter normalization used to obtain the residual centers. The report proposed a specific correction for each finding and identified no issue requiring a change to the main conclusions. The revised manuscript underwent a second Rethlas-OSV review, which identified only two issues: an incorrect citation in the introduction and an inaccurate article title in the bibliography. I am posting this manuscript to share the mathematical results themselves. In doing so, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in checking the arguments or preparing a finished account, and I believe the eventual scholarly credit should go to those who undertake that work.CurrentNov 11, 2026