Abstract

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 suficient condition for a differential operator P in Z(U(g)^K) to have a fundamental solution on G. We verify that this condition implies P C^∞(G) = C^∞(G). If G has a compact Cartan subgroup, we also give a suficient condition for a differential operator P in Z(U(g)^K) to have a parametrix on G. Finally we prove a necessary condition for the existence of parametrix of P in Z(U(g)^K) for a connected semisimple Lie group.

Authors