Let $G$ be a $p$-adic reductive group with $\mathfrak{g} = \mathrm{Lie}(G)$. I will summarize work with Matthias Strauch in which we construct an exact functor from category $\mathcal{O}^\infty$, the extension closure of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ inside the category of $U(\mathfrak{g})$-modules, into the category of admissible locally analytic representations of $G$. This expands on an earlier construction by Sascha Orlik and Matthias Strauch. A key role in our new construction is played by $p$-adic logarithms on tori, and representations in the image of this functor are related to some that are known to arise in the context of the $p$-adic Langlands program.

[pre-talk at 1:20PM]