Fundamental solutions of invariant differential operators on a semisimple Lie group II

Let G be a linear connected semisimple Lie group. We denote by U (g)(K) the algebra of left invariant differential operators on G that are also right invariant by K, and Z U (g)(K)) denotes center of U (g)(K). In this paper we give a sufficient condition for a differential operator P is an element of Z (U (g)(K)) to have a fundamental solution on G. This result extends the same one obtained previously for real rank one Lie groups and groups with only one conjugacy class of Cartan subgroups.