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.



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> (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. 
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> (2). Is it possible to obtain the coinduced module from Verma module through, e.g., certain dualities? 
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Coinduced modules in the BGG category $\mathcal O$ over complex semisimple Lie algebras
This line of questioning has been pursued in greater generality. starting in prime characteristic by Ron Irving (and myself) and then in the analogous setting of category $\mathcal{O}$ for a semisimple Lie algebra over $\mathbb{C}$ [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=1231714) . Refinements involving related twisting functors were made by Henning Haahr Andersen and his collaborators: [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=1985191) and [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=2032059). 


The underlying idea is to study simultaneously the analogues of Verma modules (or Weyl modules in prime characteristic) which correspond to the various derived functors of induction. Though this is still incomplete, a survey is included in Chapter 12 of my AMS graduate text from 2008 on category $\mathcal{O}$. 


