Locally analytic representations and sheaves on the Bruhat–Tits building

Abstract

Let $L$ be a finite field extension of ${\mathbb{Q}}_p$ and let $G$ be the group of $L$-rational points of a split connected reductive group over $L$. We view $G$ as a locally $L$-analytic group with Lie algebra $\mathfrak{g}$. The purpose of this work is to propose a construction which extends the localization of smooth $G$-representations of P. Schneider and U. Stuhler to the case of locally analytic $G$-representations. We define a functor from admissible locally analytic $G$-representations with prescribed infinitesimal character to a category of equivariant sheaves on the Bruhat–Tits building of $G$. For smooth representations, the corresponding sheaves are closely related to the sheaves of Schneider and Stuhler. The functor is also compatible, in a certain sense, with the localization of $\mathfrak{g}$-modules on the flag variety by A. Beilinson and J. Bernstein.