We introduce a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds. We then apply this for the important semisimple Lie groups SO(n), SU(n), Sp(n), SL n (R), Sp(n, R), SU(p, q), SO(p, q), Sp(p, q), SO * (2n) and SU * (2n). Keywords Biharmonic functions • p-harmonic functions • Semisimple Lie groups Mathematics Subject Classification 31B30 • 53C43 • 58E20 2 Preliminaries Let (M, g) be a smooth manifold equipped with a Riemannian metric g. We complexify the tangent bundle T M of M to T C M and extend the metric g to a complex-bilinear form on B Sigmundur Gudmundsson