Let $G$ be a $p$-adic Lie group with reductive Lie algebra $\mathfrak{ g}$. In analogy to the translation functors introduced by Bernstein and Gelfand on categories of $U(\mathfrak{ g})$-modules we consider similarly defined functors on the category of coadmissible modules over the locally analytic distribution algebra $D(G)$ on which the center of $U(\mathfrak{ g})$ acts locally finite. These functors induce equivalences between certain subcategories of the latter category. Furthermore, these translation functors are naturally related to those on category $\mathcal{ O}$ via the functors from category $\mathcal{ O}$ to the category of coadmissible modules. We also investigate the effect of the translation functors on locally analytic representations $\Pi(V)^{ \rm la}$ associated by the $p$-adic Langlands correspondence for ${ \rm GL}_2(\mathbb{ Q}_p)$ to 2-dimensional Galois representations $V$.
Categories: math.RT math.NT