For a given finite-dimensional complex semisimple Lie algebera $\mathfrak g$, we fix Cartan $\mathfrak h$ and Borel subalgebras $\mathfrak b$, then we have the BGG category $\mathcal O$. As usual, we can define the Verma module $\text{Ind}_{\mathfrak b}^{\mathfrak g} \mathbb C_{\lambda}$, which is the induced module from one-dimensional $\mathfrak b$-module with $\lambda \in \mathfrak h^*$. We define the negative Borel $\mathfrak b^-$ as usual.

> (1). What is the role of the coinduced module $\text{Coind}_{\mathfrak b^-}^{\mathfrak g}\mathbb{C}_{\lambda}:= \text{Hom}_{U(\mathfrak b^-)}(U(\mathfrak g), \mathbb C_{\lambda})$ in $\mathcal O$? That is, we have fruitful results about the Verma modules, it is reasonable to guess that there should be some similar results for the coinduced module. 
 
> (2). Is it possible to obtain the coinduced module from Verma module through, e.g., certain dualities?