Cohomology in singular blocks of parabolic category $\mathcal{O}$

TL;DR

This paper calculates the dimensions of Ext-groups between simple modules and dual generalized Verma modules within singular blocks of parabolic category O for complex semisimple Lie algebras and affine Kac-Moody algebras, advancing understanding of their homological structure.

Contribution

It provides explicit dimension formulas for Ext-groups in singular blocks of parabolic category O, extending previous results to more general algebraic settings.

Findings

Explicit Ext-group dimension formulas derived

Results apply to both complex semisimple Lie algebras and affine Kac-Moody algebras

Enhances understanding of homological properties in category O

Abstract

We determine the dimensions of $\mathrm{Ext}$-groups between simple modules and dual generalized Verma modules in singular blocks of parabolic versions of category $\mathcal{O}$ for complex semisimple Lie algebras and affine Kac-Moody algebras.