Harmonic morphisms from the classical non-compact semisimple Lie groups

Abstract

We construct the first known complex-valued harmonic morphisms from the non-compact Lie groups SL n ( R ) , SU ∗ ( 2 n ) 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 ∗ ( 2 n ) , 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.