First extension groups of Verma modules and $R$-polynomials
Noriyuki Abe
TL;DR
This paper investigates the relationship between first extension groups of Verma modules in $\mathcal{O}$ and Kazhdan–Lusztig $R$-polynomials, clarifying the scope of the Gabber–Joseph conjecture. It introduces $V_\lambda(x,y)$ as the image of $\mathrm{Ext}^1(M(x\lambda),M(y\lambda))$ in $\mathrm{Ext}^1(M(w_0\lambda),M(\lambda))$ and constructs a Weyl-group action on these spaces using wall-crossing and translation functors. In the regular case, it proves that $\dim V_\lambda(x,y)$ equals the coefficient of $q$ in $(-1)^{\ell(y)-\ell(x)-1}R_{y,x}(q)$, extending Mazorchuk’s formula via graded category $\mathcal{O}$ methods; in the singular case, translation functors relate the regular and singular blocks with a described kernel. However, the Erratum shows that the original Theorem asserting exact equality with the $R$-polynomials is false in general, providing only an inequality and certain equalities in special cases. Together, these results refine our understanding of how extension dimensions approximate, but do not generally equal, $R$-polynomial coefficients, and they highlight the power and limits of graded and translation-theoretic techniques in category $\mathcal{O}$.
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.