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.

${\bf ~My ~question}$:

(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?

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?