Having been introduced by S. Gudmundsson and A. Sakovich as a machinery for constructing complex valued harmonic morphisms, (λ, μ)-eigenfunctions have found several more applications in the recent years. They have been used to construct complex-valued p-harmonic maps (Gudmundsson & Sobak) and as a tool for finding new minimal submanifolds of codimension two (Gudmundsson & Munn). They have also increasingly been studied as interesting objects in their own right, for example by A. Siffert and O. Riedler.

In this talk I will begin by giving an introduction to (λ, μ)-eigenfunctions and their applications. I will then present a recently discovered characterisation of the known examples on the compact classical Riemannian symmetric spaces in terms of the well known Cartan embedding. Special attention will be paid to the case of the Quaternionic Grassmannians, where this characterisation has led to the finding of new eigenfunctions.