New biharmonic functions on the compact Lie groups SO(n), SU(n), Sp(n)

We develop a new scheme for the construction of explicit complex-valued proper biharmonic functions on Riemannian Lie groups. We exploit this and manufacture many infinite series of uncountable families of new solutions on the special unitary group SU(n). We then show that the special orthogonal group SO(n) and the quaternionic unitary group Sp(n) fall into the scheme. As a by-product we obtain new harmonic morphisms on these groups. All the constructed maps are defined on open and dense subsets of the corresponding spaces.