Graded extensions of Verma modules

TL;DR

This paper studies extensions between graded Verma modules in category O, revealing how R-polynomials encode this information, providing bounds, and explicitly determining extensions for rak{sl}_4.

Contribution

It precisely characterizes extension groups between graded Verma modules and connects them to Kazhdan-Lusztig combinatorics, including explicit computations for rak{sl}_4.

Findings

Extensions are determined by R-polynomial coefficients.

Upper bounds for extension dimensions are established.

Explicit extension structures are computed for rak{sl}_4.

Abstract

In this paper, we investigate extensions between graded Verma modules in the BGG category $\mathcal{O}$. In particular, we determine exactly which information about extensions between graded Verma modules is given by the coefficients of the $R$-polynomials. We also give some upper bounds for the dimensions of graded extensions between Verma modules in terms of Kazhdan-Lusztig combinatorics. We completely determine all extensions between Verma module in the regular block of category $\mathcal{O}$ for $\mathfrak{sl}_4$ and construct various ``unexpected'' higher extensions between Verma modules.