Laplace Eigenfunctions on Riemannian Symmetric Spaces and Borel–Weil Theorem

Abstract We identify a geometric relation between the Laplace–Beltrami spectra and eigenfunctions on compact Riemannian symmetric spaces Borel–Weil theory using ideas from symplectic geometry quantization. This is done by associating with each space, via Marsden–Weinstein reduction, generalized flag manifold that covers space parametrizing all of its maximal totally geodesic tori. In process, we notice direct Satake diagram painted Dynkin associated manifold. consider in detail examples classical simply connected rank one $SU(3)/SO(3)$. 2nd part paper, aid harmonic polynomials, induce holomorphic sections line bundle consider, show our construction provides eigenfunctions.