Harmonic morphisms from the classical non-compact semisimple Lie groups

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SLn(R)SLn(R), SU∗(2n)SU∗(2n) and Sp(n,R)Sp(n,R) equipped with their standard Riemannian metrics. We then introduce the notion of a bi-eigenfamily   and employ this to construct the first known solutions on the non-compact Riemannian SO∗(2n)SO∗(2n), SO(p,q)SO(p,q), SU(p,q)SU(p,q) and Sp(p,q)Sp(p,q). Applying a duality principle   we then show how to manufacture the first known complex-valued harmonic morphisms from the compact Lie groups SO(n)SO(n), SU(n)SU(n) and Sp(n)Sp(n) equipped with semi-Riemannian metrics.