Extensions between Verma modules for dihedral groups

TL;DR

This paper computes the extensions between Verma modules for dihedral groups using Soergel bimodules, providing new insights into the structure of Category O for these groups.

Contribution

It introduces a method to explicitly calculate extensions between Verma modules for dihedral groups via Soergel bimodules, advancing understanding in this area.

Findings

Explicit extension formulas for dihedral groups

Application of Soergel bimodules to Category O

Enhanced understanding of module interactions in dihedral cases

Abstract

Computing the extensions between Verma modules is in general a very difficult problem. Using Soergel bimodules, one can construct a graded version of the principal block of Category $\mathcal{O}$ for any finite coxeter group. In this setting, we compute the extensions between Verma modules for dihedral groups.