Resolutions of locally analytic principal series representations of $$GL_2$$

Abstract

For a finite field extension $F/\\mathbb{Q}_p$ we associate a coefficient system attached on the Bruhat-Tits tree of $G:= {\\rm GL}_2(F)$ to a locally analytic representation $V$ of $G$. This is done in analogy to the work of Schneider and Stuhler for smooth representations. This coefficient system furnishes a chain-complex which is shown, in the case of locally analytic principal series representations $V$, to be a resolution of $V$.