New Biharmonic Functions on the Compact Lie Groups mathbf{SO}(n), mathbf{SU}(n), mathbf{Sp}(n)

Abstract

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 mathbf{SU}(n). We then show that the special orthogonal group mathbf{SO}(n) and the quaternionic unitary group mathbf{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.