  We construct the first known complex valued harmonic morphisms from the
non-compact Lie groups SL(n,R), SU*(2n) and 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(p,q), SU(p,q) and 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), SU(n) and Sp(n) equipped with
semi-Riemannian metrics.
