Laplace eigenfunctions on Riemannian symmetric spaces and Borel-Weil Theorem

TL;DR

This paper explores the connection between Laplace-Beltrami spectra on symmetric spaces and Borel-Weil theory, using symplectic geometry and geometric quantization to relate eigenfunctions to holomorphic sections on flag manifolds.

Contribution

It introduces a geometric framework linking Laplace eigenfunctions on symmetric spaces to Borel-Weil theory via Marsden-Weinstein reduction and flag manifolds, providing explicit constructions for classical cases.

Findings

Established a relation between Satake diagrams and Dynkin diagrams.

Constructed all eigenfunctions from holomorphic sections in studied examples.

Linked spectral theory with geometric quantization on symmetric spaces.

Abstract

We indicate a geometric relation between Laplace-Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces and the Borel-Weil theory using ideas from symplectic geometry and geometric quantization. This is done by associating to each compact Riemannian symmetric space, via Marsden-Weinstein reduction, a generalized flag manifold which covers the space parametrizing all of its maximal totally geodesic tori. In the process we notice a direct relation between the Satake diagram of the symmetric space and the painted Dynkin diagram of its associated flag manifold. We consider in detail the examples of the classical simply-connected spaces of rank one and the space SU(3)/SO(3). In the second part of the paper we provide a construction of harmonic polynomials inducing Laplace-Beltrami eigenfunctions on the symmetric space from holomorphic sections of the associated line bundle on the generalized flag manifold. We show that in the examples we consider the construction provides all of the eigenfunctions.