Laplace Eigenfunctions on Riemannian Symmetric Spaces and Borel–Weil Theorem

Abstract

AbstractWe identify a geometric relation between the 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 with each compact Riemannian symmetric space, via Marsden–Weinstein reduction, a generalized flag manifold that 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 2nd part of the paper, with the aid of harmonic polynomials, we induce Laplace–Beltrami eigenfunctions on the symmetric space from holomorphic sections of the associated line bundle on the generalized flag manifold. In the examples we consider, we show that our construction provides all of the eigenfunctions.