Invariant distributions on p-adic analytic groups

Abstract

Let $p$ be a prime number, let $L$ be a finite extension of the field ${\mathbb{Q}}_p$ of $p$-adic numbers, let $K$ be a spherically complete extension field of $L$, and let $G$ be the group of $L$-rational points of a split reductive group over $L$. We derive several explicit descriptions of the center of the algebra $D(G,K)$ of locally analytic distributions on $G$ with values in $K$. The main result is a generalization of an isomorphism of Harish-Chandra which connects the center of $D(G,K)$ with the algebra of Weyl-invariant, centrally supported distributions on a maximal torus of G. This isomorphism is supposed to play a role in the theory of locally analytic representations of $G$ as studied by P. Schneider and J. Teitelbaum