The Laplacian on homogeneous spaces

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 Kostant’s Dirac operator on $G∕H$⁠.