First extension groups of Verma modules and $R$-polynomials

Noriyuki Abe

First extension groups of Verma modules and $R$-polynomials

Representation Theory 2025-01-20 v2

Authors: Noriyuki Abe

Abstract

We study the first extension groups between Verma modules. There was a conjecture which claims that the dimensions of the higher extension groups between Verma modules are the coefficients of $R$ -polynomials defined by Kazhdan-Lusztig. This conjecture was known as the Gabber-Joseph conjecture (although Gebber and Joseph did not state.) However, Boe gives a counterexample to this conjecture. In this paper, we study how far are the dimensions of extension groups from the coefficients of $R$ -polynomials.

Cite

@article{arxiv.1003.0169,
  title  = {First extension groups of Verma modules and $R$-polynomials},
  author = {Noriyuki Abe},
  journal= {arXiv preprint arXiv:1003.0169},
  year   = {2025}
}

Comments

16 pages, one of the main theorems was false, erratum is added

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