Invariant differential equations on reductive Lie groups and Lie algebras

Let G be a reductive Lie group with Lie algebra g, D be a non zero G-invari ant differential operator with constant coefficients on g and v be a G-inva riant distribution on f. We prove that the differential equation D . u = v has solutions in the space of G-invariant distributions on g; moreover, if v is tempered or of finite order, we can find solutions with the same prope rties. If D is a non zero bi-invariant differential operator on G, Benabdal lah and Rouviere gave a sufficient condition for D to have a central fundam ental solution on G. We prove that their condition is also sufficient for t he differential equation D . u. = v to have solutions in the space of finit e order central distributions on G.