A Jacquet-Langlands functor for $p$-adic locally analytic representations

TL;DR

This paper develops a theory showing the independence of locally analytic vectors from group actions in dual local Shimura varieties, extending previous results and applying it to Jacquet-Langlands functors and cohomology isomorphisms.

Contribution

It proves the independence of locally analytic vectors in dual local Shimura varieties and demonstrates compatibility of Scholze's p-adic Jacquet-Langlands functor with these vectors.

Findings

Locally analytic vectors are independent of actions of G and Gb in dual local Shimura varieties.

Scholze's p-adic Jacquet-Langlands functor commutes with passage to locally analytic vectors.

De Rham cohomology of the towers are isomorphic as smooth G×Gb representations.

Abstract

We study the locally analytic theory of infinite level local Shimura varieties. As a main result, we prove that in the case of a duality of local Shimura varieties, the locally analytic vectors of different period sheaves at infinite level are independent of the actions of the $p$-adic Lie groups $G$ and $G_b$ of the two towers; this generalizes a result of Pan for the Lubin-Tate and Drinfeld spaces for $\mathrm{GL}_2$. We apply this theory to show that Scholze's $p$-adic Jacquet-Langlands functor commutes with the passage to locally analytic vectors, and is compatible with central characters of Lie algebras. We also prove that the compactly supported de Rham cohomology of the two towers are isomorphic as smooth representations of $G\times G_b$.