The Laplacian on homogeneous spaces

TL;DR

This paper provides a solution to the eigenvalue problem of the Laplacian on homogeneous spaces G/H, revealing connections to Landau levels and Dirac operator indices, with implications for geometric analysis on Lie groups.

Contribution

It explicitly solves the Laplacian eigenvalue problem on G/H and links the eigenvalue multiplicity to Landau level degeneracy and Dirac operator indices.

Findings

Lowest eigenvalue multiplicity equals Landau level degeneracy.

Eigenspace corresponds to the G-equivariant index of Kostant's Dirac operator.

Provides explicit eigenfunctions for the Laplacian on G/H.

Abstract

The solution of the eigenvalue problem of the Laplacian on a general homogeneous space G/H is given. Here, G is a compact, semisimple Lie group, H is a closed subgroup of G, and the rank of H is equal to the rank of G. It is shown that the multiplicity of the lowest eigenvalue of the Laplacian on G/H is just the degeneracy of the lowest Landau level for a particle moving on G/H in the presence of the background gauge field. Moreover, the eigenspace of the lowest eigenvalue of the Laplacian on G/H is, up to a sign, equal to the G-equivariant index of the Dirac operator of Kostant on G/H.