Generalized Verma modules induced from infinite dimensional simple Uq(sl2)-modules

Abstract

Let $g$ be a simple complex finite dimensional Lie algebra. For generic $q\in C^{⁎}$, let $U_q\left(g\right)$ be the quantum group with standard generators $E_i,F_i,q^{\pm {\lambda }_i}$. Fixing an index $i_0$, let $P$ be the subalgebra generated by all $E_i,q^{\pm {\lambda }_i}$ and $F_{i_0}$, which has sub-quotient algebras isomorphic to $U_{q_{i_0}}\left({\mathrm{sl}}_2\right)$. With each generalized Verma $U_q\left(g\right)$-module $M_P\left(V\right)$ induced from a simple $U_{q_{i_0}}\left({\mathrm{sl}}_2\right)$-module V, we associate a Verma module $M\left(V\right)$. In this paper, for infinite dimensional V we prove that $M_P\left(V\right)$ is simple if and only if $M\left(V\right)$ is simple.