Invariant distributions on p-adic analytic groups

Abstract

Let p be a prime number, let L be a finite extension of the field Qp 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