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

Abstract

For a finite extension \(F/{{{\mathbb {Q}}}_p}\) we associate a coefficient system on the Bruhat-Tits tree of \(G:= \mathrm{GL}_2(F)\) 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.