Fundamental solutions of invariant differential operators on a semisimple Lie group

Let G be a connected semisimple Lie group of real rank one. We denote by U(g)(K) the algebra of left invariant differential operators on G right invariant by K, and let Z(U(g)(K)) be its center. In this paper we give a sufficient condition for a differential operator P epsilon Z(U(g)(K)) to have a fundamental solution on G. We verify that this condition implies PCinfinity(G) = C-infinity(G). If G has a compact Cartan subgroup, we also give a sufficient condition for a differential operator P epsilon Z(U(g)(K)) to have a parametrix on G. Finally we prove a necessary condition for the existence of parametrix of P epsilon Z(U(g)(K)) for a connected semisimple Lie group.