On first extensions in $\mathcal{S}$-subcategories of $\mathcal{O}$

TL;DR

This paper calculates the initial extension groups between simple and standard objects within certain subcategories of the BGG category , providing insights into their homological structure in representation theory.

Contribution

It explicitly computes the first extension groups in -subcategories of category , advancing understanding of their homological properties in Lie algebra representations.

Findings

Computed first extension groups from simple to proper standard objects.

Determined cases for extensions from simple to standard objects in principal blocks.

Enhanced understanding of homological relationships in -subcategories.

Abstract

We compute the first extension group from a simple object to a proper standard object and, in some cases, the first extension group from a simple object to a standard object in the principal block of an $\mathcal{S}$-subcategory of the BGG category $\mathcal{O}$ associated to a triangular decomposition of a semi-simple finite dimensional complex Lie algebra.