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

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

2010-02-28

arxiv.org/abs/1003.0169v1

Mathematics

Representation Theory

15 pages

Scientific paper

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.

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