Generalized Verma modules over $\fr{sl}_{n+2}$ induced from $\ca{U}(\fr{h}_n)$-free $\fr{sl}_{n+1}$-modules

TL;DR

This paper constructs a new class of simple generalized Verma modules over r{sl}_{n+2} from r{sl}_{n+1}-modules that are free over a certain subalgebra, expanding the understanding of r{sl}_{n+2} representations.

Contribution

It introduces a novel construction of simple r{sl}_{n+2}-modules from r{sl}_{n+1}-modules that are free over r{h}_n, providing new examples and criteria for simplicity.

Findings

Derived necessary and sufficient conditions for simplicity.

Constructed a new class of simple r{sl}_{n+2}-modules.

Expanded the classification of r{sl}_{n+2} representations.

Abstract

A class of generalized Verma modules over $\fr{sl}_{n+2}$ is constructed from $\fr{sl}_{n+1}$-modules which are $\uhn$-free modules of rank $1$. The necessary and sufficient conditions for these $\fr{sl}_{n+2}$-modules to be simple are determined. This leads to a class of new simple $\fr{sl}_{n+2}$-modules.