# In what way are Josephs completion functors analogous to algebraic principal series modules of Dixmier?

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 Josephs 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 Josephs 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 Josephs 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.

• I am a bit lost in your notation (for example, what is $\Delta$?). Have you looked at Jantzen's book "Einhüllende Algebren halbeinfacher Lie-Algebren" (in German)? It discusses the relation between Harish-Chandra modules such as $\mathcal{L}(N,M)$ and category $\mathcal{O}$ due to Joseph and Bernstein-Gelfand. Quantum groups are not particularly relevant here. – Victor Protsak Dec 18 '17 at 16:45
• Im sorry, Im gonna change notation. By $\Delta(\lambda)$ I mean a verma module with highest weight $\lambda$ and by $\nabla(.)$ the Dual Verma module. – Henrique Tyrrell Dec 18 '17 at 16:47