A Unifying Framework for Complex-Valued Eigenfunctions via The Cartan Embedding
Sigmundur Gudmundsson, Adam Lindström
TL;DR
The paper presents a unifying Cartan-embedding framework to construct complex-valued $(\lambda,\mu)$-eigenfunctions on classical compact irreducible symmetric spaces, using the Cartan map $\hat{\Phi}$ to lift eigenfunctions from totally geodesic images and descend to quotients $G/K$. It provides explicit constructions on the quaternionic and complex Grassmannians, recasting many known eigenfunctions and producing new ones via invariant functions composed with $\hat{\Phi}$; it also shows how remaining symmetric spaces fit the framework and introduces a general product-compatibility principle that yields new harmonic morphisms on product manifolds. The work yields a coherent methodology for generating large eigenfamilies across many symmetric spaces and offers practical tools for harmonic morphisms, with potential applications in geometric analysis on homogeneous spaces. Overall, the Cartan-embedding approach unifies and extends explicit complex-valued eigenfunction constructions across the classical compact symmetric spaces.
Abstract
In this work we find a unifying scheme for the known explicit complex-valued eigenfunctions on the classical compact Riemannian symmetric spaces. For this we employ the well-known Cartan embedding for those spaces. This also leads to the construction of new eigenfunctions on the quaternionic Grassmannians.