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

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