The Laplacian on homogeneous spaces
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 Kostant's Dirac operator on G∕H.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- May 2008
- DOI:
- 10.1063/1.2924268
- arXiv:
- arXiv:0805.2531
- Bibcode:
- 2008JMP....49e3513H
- Keywords:
-
- Mathematical Physics;
- Condensed Matter - Mesoscopic Systems and Quantum Hall Effect;
- High Energy Physics - Theory;
- Mathematics - Representation Theory
- E-Print:
- J.Math.Phys.49:053513,2008