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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?

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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 . Refinements involving related twisting functors were made by Henning Haahr Andersen and his collaborators: here and here.

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 $\mathcal{O}$.

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  • $\begingroup$ Thank you very much for this clarification. However I fail to identify twisted Verma modules with coinduced modules, that is, the former is a series of dual-Verma-module like modules while the later has no similar feature. So, I was wondering if you meant the coinduced module ($\text{Hom}_{U(\mathfrak b^-)}(U(\mathfrak g), \mathbb C_{\lambda})$) is one of them? $\endgroup$
    – GuNa
    Commented Sep 9, 2018 at 0:57
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    $\begingroup$ @GuNa: To answer your question at the end, I did mean this. But on revisiting Chapter 12 (for the first time), I realize that I didn't anywhere state explicitly that the coinduced module coincides with the categorical dual. The lilterature on induced and coinduced modules is extensive but rather formal. $\endgroup$
    – Jim Humphreys
    Commented Sep 9, 2018 at 21:45

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