Locally analytic vectors and local Shimura varieties

(Joint work with Gabriel Dospinescu). In this talk we report some progress in the study of the locally analytic vectors of proétale
period sheaves appearing in (rational) p-adic Hodge theory. The most important example for us will be local Shimura varieties,
in particular those arising from a local Shimura datum (G,b,μ) with b basic. We focus our attention in the Lubin-Tate and Drinfeld
tower, obtaining as a consequence that the Jacquet-Langlands functor of Scholze is compatible with the passage to locally analytic vectors.