I have a question regarding highest-weight modules:
Let be $\mathfrak{g}$ a Lie algebra, $\mathfrak{b}$ a Borel subalgebra, $\mathfrak{h}$ a Cartan subalgebra and $U(\mathfrak{g})$ its universal enveloping algebra. Let be $M$ a $\mathfrak{b}$-module and $\text{Ind}^{\mathfrak{g}}_{\mathfrak{b}} M = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} M$ the induced module on $\mathfrak{g}$.
My question is: If $M=M(\lambda)$ is a highest-weight module to the weight $\lambda \in \mathfrak{h}^*$, is then $\text{Ind}^{\mathfrak{g}}_{\mathfrak{b}} M(\lambda)$ a highest-weight module with weight $\lambda$, too?
An example would be Verma modules, but is this correct in general? If yes, how would one proof it?
Thank you! Andreas