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)