Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $x \in W$ be an element of the Weyl group (which acts on weights by $x \cdot \lambda = x(\lambda + \rho) - \rho$, $\rho$ being the half sum of positive roots and if $M, N \in \mathcal{O}$, define $ \mathcal{L}(N, M) = \text{Hom}_\mathcal{O}(N, M)^{ad-fin}$, and define Joseph`s completion functor

$\textbf{C}_x: \mathcal{O}_0 \rightarrow \mathcal{O}_0, M \mapsto \mathcal{L}(\Delta(x^{-1} \cdot 0), M) \otimes_{U(\mathfrak{g})}\Delta(0)$

Where $\Delta(\lambda)$ is the Verma module with highest weight $\lambda$ and $\nabla(\lambda)$ its dual Verma module.

In *Homomorphisms and Extensions of Principal Series Representations*, by Catharina Stroppel, the author says in the introduction with regards to Joseph`s completion functor on the trivial block of the category $\mathcal{O}$:

The principal series representations from the title are just the modules $\textbf{C}_x(M)$, where $M$ is a dual Verma module. They can also be described as the (co-)induced representations from some minimal parabolic subalgebra ([7, 9.3 and 9.6] = J. Dixmier, Enveloping Algebras).

But I have no idea how they are equal, since Dixmier's coinduced modules are $\text{Hom}$ spaces over parabolic subalgebras while this functor is a tensor product over the "Borel" subalgebra. Does anyone know this is possible? I have been trying to read Joseph`s book on quantum groups but the language and content are ... intense. I would be happy with and consider a correct answer to this question any readable references with information about this or an overall explanation of why these modules are supposed to correspond to each other.

`m sorry, I`

m gonna change notation. By $\Delta(\lambda)$ I mean a verma module with highest weight $\lambda$ and by $\nabla(.)$ the Dual Verma module. $\endgroup$ – Henrique Tyrrell Dec 18 '17 at 16:47