A complex valued function f:(M,g) \to \C$ is said to be a $(\lambda,mu)$-eigenfunction if it is eigen with respect to both the Laplace-Beltrami operator and the conformality operator kappa(f,f) = g(\nabla f,\nabla f). Recent developments have shown that $(\lambda,\mu)$-eigenfunctions can be used in the construction of harmonic morphisms, proper r-harmonic maps, and minimal submanifold of codimension two. There has also been interest in classifying eigenfunctions. In this talk we consider the case when (M,g) is a compact Lie group equipped with a bi-invariant metric, as the Laplace-Beltrami operator's spectral theory is described by the Peter-Weyl theorem. We use the Peter-Weyl theorem, and other representation theoretic techniques in order to classify all the pairs (\lambda, \mu) as well as describe conditions which ensure the existence of eigenfunctions. Finally, we produce new eigenfunctions on both compact Lie groups G and Riemannian symmetric spaces G/K. This is a joint project with Oskar Riedler. If you are interested please contact Prof. Dr. Anna Siffert (asiffert@uni-muenster.de)